}[rr] [ld][rrd] & & {\\rm Chow}^n(T^*_{U_i}/{U_i})[d][ld]\\\\\\overline{{\\rm Chow}^n_{\\mu }(T^*_X/X)}@{^(->}[rr][rrd]& & {\\rm Chow}^n(T^*_X/X)[d] & U_i[ld], \\\\& & X &}$ where $\\overline{{\\rm Chow}_{\\mu }^n(T^*_{U_i} /U_i)}$ is the closure of ${\\rm Chow}_{\\mu }^n(T^*_{U_i} /U_i)$ in ${\\rm Chow}^n(T^*_{U_i}/{U_i})$ .","We have the isomorphisms $\\overline{{\\rm Chow}_{\\mu }^n(T^*_{U_i} /U_i)} & \\cong \\overline{{\\rm Chow}_{\\mu }^n(\\mathbb {A}^2)} \\times U_i,\\\\\\prod ^s_{j=1, U_i}{\\rm Chow}^{\\alpha _j}(T^*_{U_i}/U_i) & \\cong \\big (\\prod ^s_{j=1}{\\rm Chow}^{\\alpha _j}(\\mathbb {A}^2)\\big )\\times U_i.$ Thus by Lemma REF , $\\tau ^{\\prime }_{\\mu }$ is the normalization of $\\overline{{\\rm Chow}^n_{\\mu }(T^*_{U_i}/{U_i})}$ .","Applying Lemma REF , we get $\\tau _{\\mu }$ is the normalization of $\\overline{{\\rm Chow}^n_{\\mu }(T^*_X/X)}$ .","Lemma 3.4 Let $X$ be an integral scheme and $\\tilde{X}$ a normal integral scheme.","Given a morphism $\\nu : \\tilde{X}\\rightarrow X$ , suppose there exists an open covering ${U_i}$ of $X$ such that $\\nu _i: \\tilde{U_i}=\\nu ^{-1}(U_i)\\rightarrow U_i$ is the normalization for each $i$ .","Then $\\nu $ is the normalization.","This follows from the universal property of normalization [5].","With respect to Proposition REF , we obtain the decomposition of spectral data.","It will be used to prove the main result (Theorem REF ) in §4.","Theorem 3.5 Let $X$ be a smooth projective surface.","Given a spectral datum $a: X \\rightarrow {\\rm Chow}^{n}(T^*_X /X)$ , suppose the generic point of $X$ is mapped to the stratum ${\\rm Chow}^{n}_{\\mu }(T^*_X /X)$ with $\\mu =(1^{\\alpha _1}2^{\\alpha _2}\\dots s^{\\alpha _s})$ .","Then there exist spectral data $a_i: X \\rightarrow \\overline{{\\rm Chow}^{\\alpha _i}_{(1^{\\alpha _i})}(T^*_X /X)}$ , $i=1, \\cdots , s$ , such that $a=\\sum ^s_{i=1} ia_i.$ By the assumption, $a$ induces a morphism $X \\rightarrow \\overline{{\\rm Chow}^n_{\\mu }(T^*_X /X)}$ over $X$ .","By Proposition REF , it holds that $\\overline{{\\rm Chow}^n_{\\mu }(T^*_X /X)}^{\\rm nor} \\cong {\\rm Chow}^{\\alpha _1}(T^*_X /X) \\times _X \\dots \\times _X {\\rm Chow}^{\\alpha _s}(T^*_X /X).$ Since $X$ is smooth, there exists a unique $a^{\\prime }$ fitting into the diagram Chown(T*X /X)nor [d] [rd, \"\"] X [r][ru,\"a'\", dotted] Chown(T*X /X) [r, hook] Chown(T*X /X) Here $a^{\\prime }$ is equivalent to data $(a_i)_{1 \\le i \\le s}$ , where $a_i: X \\rightarrow \\overline{{\\rm Chow}^{\\alpha _i}_{(1^{\\alpha _i})}(T^*_X /X)}.$ Finally by the commutativity of the diagram, we conclude that $a=\\sum \\limits _{i=1}^s i a_i$ ."],["The image of the Hitchin morphism for surfaces","Given a spectral datum $a: X \\rightarrow {\\rm Chow}^{n}(T^*_X/X)$ , we first briefly review the construction of the spectral cover $\\widetilde{X}_a \\rightarrow X$ , and the correspondence between Cohen-Macaulay sheaves of generic rank one over $\\widetilde{X}_a$ and Higgs bundles over $X$ [3].","Next, we show that given two Higgs bundles, their direct sum corresponds to the sum of the corresponding spectral data (Lemma REF ).","Finally, we prove the main theorem that $sd_X: {M}_X \\rightarrow {A}_X$ is surjective (Theorem REF ) with the help of Theorem REF .","Define $X_a$ as the pullback in the following diagram Xa [d,\"a\"] [rr] Cayleyn(T*X/X) [d,\"\"] X [rr,\"a\"] Chown(T*X/X) where ${\\rm Cayley}^n(T^*_X/X)$ is a subscheme of ${\\rm Chow}^n(T^*_X/X) \\times _X T^*_X$ ([3]).","Theorem 4.1 (Theorems 7.1 and 7.3 in [3]) Given a spectral datum $a: X \\rightarrow {\\rm Chow}^{n}(T^*_X/X)$ such that $a$ maps the generic point of $X$ into ${\\rm Chow}^n_{(1^n)}(T^*_X/X)$ , there exists a unique finite flat covering $\\widetilde{\\pi }_a: \\widetilde{X}_a \\rightarrow X$ such that there exists an open subset $U \\subseteq X$ of codimension at least 2 such that for every point $x \\in U$ , the fiber $(\\widetilde{\\pi }_a)^{-1}$ is a point of ${\\rm Hilb}^n(T^*_X/X)$ lying over the point $a(x) \\in {\\rm Chow}^n(T^*_X/X)$ ; there is a natural morphism $\\widetilde{\\iota }_a: \\widetilde{X}_a \\rightarrow T^*_X$ factoring through $\\iota _a: X_a \\hookrightarrow T^*_X$ such that the following diagram is commutative; Xa [rd,\"a\"] [r, \"\"] Xa [r,\"a\", hook] [d, \"a\"] T*X [ld, \"\"] X $\\widetilde{X}_a$ is the Cohen-Macaulayfication of $X_a$ ; the fiber $sd_X^{-1}(a)$ is isomorphic to the stack of Cohen-Macaulay sheaves of generic rank one over $\\widetilde{X}_a$ .","The correspondence demonstrated in the theorem can be understood in the following way.","Given a Cohen-Macaulay sheaf $L$ of generic rank one over $\\widetilde{X}_a$ , its pushfoward $\\iota _* L$ is a sheaf over $X_a$ .","Via $\\iota _a$ , the sheaf $(\\widetilde{\\iota }_a)_* L$ has a natural structure as a $S(T_X)$ -module, where $S(T_X)$ is the symmetric product of the tangent bundle $T_X$ .","Furthermore, it is a finite $S(T_X)$ -module.","Therefore, it corresponds to a spectral data in ${A}_X$ , which is exactly $a$ , and also corresponds to a Higgs bundle $(E,\\phi )$ over $X$ .","The correspondence between a finite $S(T_X)$ -module $L$ and a spectral data $a$ can be understood in the following way.","For each $x \\in X$ , we can equip $L|_{\\pi ^{-1}(x)}$ with a cycle ${\\rm cyc}(L|_{\\pi ^{-1}(x)}):=\\sum _{y \\in \\pi ^{-1}(x)} {\\rm len}_{y}(L_{\\pi ^{-1}(x)}) \\cdot y.$ Then, the $S(T_X)$ -module $L$ corresponds to $a$ if for each $x \\in X$ , we have $a(x)={\\rm cyc}(L|_{\\pi ^{-1}(x)}).$ Theorem REF only works for spectral data which map the generic point of $X$ into ${\\rm Chow}^n_{(1^n)}(T^*_X/X)$ .","Given an arbitrary spectral datum, we can follow the approach in the theorem to construct $X_a$ , but the correspondence is not clear.","We will discuss this issue in Remark REF .","Lemma 4.2 Given a pair of spectral data $a_i: X\\rightarrow {\\rm Chow}^{n_i}(T^*_X/X)$ , $i=1, 2$ , let $(E_i, \\phi _i)$ be Higgs bundles on $X$ whose spectral data are $a_i$ .","Then $(E_1\\oplus E_2, \\phi _1\\oplus \\phi _2)$ is a Higgs bundle with the spectral datum $a_1+a_2: X\\rightarrow {\\rm Chow}^{n_1+n_2}(T^*_X/X)$ .","The pair $(E_1\\oplus E_2, \\phi _1\\oplus \\phi _2)$ is obviously a Higgs bundle on $X$ .","By the correspondence [11], there exist coherent sheaves $F_i$ on $T^*_X$ such that $\\pi _* F_i\\simeq E_i$ .","Thus, $\\pi _*(F_1\\oplus F_2)\\simeq E_1\\oplus E_2$ .","For any $x\\in X$ , $a_1(x)+a_2(x)={\\rm cyc}( L_1 |_{\\pi ^{-1}(x)})+{\\rm cyc}(L_2|_{\\pi ^{-1}(x)})={\\rm cyc}((L_1+L_2)|_{\\pi ^{-1}(x)})=(a_1+a_2)(x).$ This finishes the proof.","Theorem 4.3 Let $X$ be a smooth projective surface.","The image of the Hitchin map $h_X: {M}_X \\rightarrow {B}_X$ is ${A}_X$ , i.e.","$sd_X : {M}_X \\rightarrow {A}_X$ is surjective.","We will show that given any spectral datum $a \\in {A}_X$ , we can construct a Higgs bundle $(E,\\phi ) \\in sd_X^{-1}(a) \\subseteq {M}_X$ .","By Theorem REF , this is equivalent to finding a $S(T_X)$ -module $L$ corresponding to the spectral datum $a$ .","Now given a spectral datum $a: X \\rightarrow {\\rm Chow}^n(T^*_X/X)$ , suppose that the generic point of $X$ is mapped into some stratum ${\\rm Chow}^n_{\\mu }(T^*_X/X)$ , where $\\mu =(1^{\\alpha _1}, 2^{\\alpha _2}, \\dots , s^{\\alpha _s})$ .","By Theorem REF , we have $a=a_1+2a_2+\\dots +sa_s,$ where the spectral datum $a_i$ maps the generic point of $X$ into ${\\rm Chow}^{\\alpha _i}_{(1^{\\alpha _i})}(T^*_X /X)$ .","Therefore, by Theorem REF , for each $i$ , one can construct a spectral cover $\\widetilde{X}_{a_i}$ of $X$ Xai [rd,\"i\"] [r] Xai [r,\"i\", hook] [d, \"i\"] T*X [ld, \"\"] X such that $\\widetilde{\\pi }_{a_i}: \\widetilde{X}_{a_i} \\rightarrow X$ is a finite flat morphism and the morphism $\\widetilde{\\iota }_{a_i}: \\widetilde{X}_{a_i} \\rightarrow T^*_X$ factors through $\\iota _{i}: X_{a_i} \\hookrightarrow T^*_X$ .","Then, the $S(T_X)$ -module $(\\widetilde{\\iota }_i)_* \\mathcal {O}_{\\widetilde{X}_{a_i}}$ corresponds to the spectral datum $a_i$ .","By Lemma REF , the sheaf $L:=\\bigoplus \\limits _{i=1}^s (\\widetilde{\\iota }_i)_* \\mathcal {O}_{\\widetilde{X}_{a_i}}^{\\oplus i}$ corresponds to the spectral data $\\sum \\limits _{i=1}^s i a_i$ , which is exactly $a$ .","Therefore, $sd_X^{-1}(a)$ is nonempty.","Remark 4.4 Although Theorem REF gives the surjectivity of the morphism $sd_X: {M}_X \\rightarrow {A}_X$ , we do not find a cover $\\widetilde{X}_a \\rightarrow X$ such that Higgs bundles over $X$ with spectral datum $a$ will correspond to some special sheaves over $\\widetilde{X}_a$ .","On the other hand, for any $a \\in {A}_X(k)$ , we can construct a finite, flat cover $\\widetilde{X}_a \\rightarrow X$ following Chen and Ngô's idea in [3].","However, the generic rank one sheaves over $\\widetilde{X}_a$ do not correspond to Higgs bundles of rank $n$ over $X$ .","We give the construction and explain the reason in the following.","Let $a: X \\rightarrow \\overline{{\\rm Chow}^n_{\\mu }(T^*_X/X)}$ be a spectral datum, where $\\mu =(1^{\\alpha _1}, \\dots , s^{\\alpha _s})$ .","As discussed in §3, it can be regarded as a morphism $a=(a_1,\\dots ,a_s): X \\rightarrow {\\rm Chow}^{\\alpha _1}(T^*_X /X) \\times _X \\dots \\times _X {\\rm Chow}^{\\alpha _s}(T^*_X /X),$ where $a_i$ maps the generic point of $X$ to ${\\rm Chow}^{\\alpha _i}_{(1^{\\alpha _i})}(T^*_X /X)$ .","Let $U^{\\prime }$ be an open subset of $X$ such that $a_i(U^{\\prime })\\subseteq {\\rm Chow}^{\\alpha _i}_{(1^{\\alpha _i})}(T^*_X /X)$ for all $i$ .","By the properness of the Hilbert-Chow morphism, one can find an open subset $U$ of $X$ with ${\\rm codim}(X\\backslash U, X)\\ge 2$ and there exists a unique morphism $\\tilde{a}: U \\rightarrow {\\rm Hilb}^{\\alpha _1}(T^*_X/X) \\times _X \\dots \\times _X {\\rm Hilb}^{\\alpha _s}(T^*_X/X)$ fitting into the commutative diagram U' [rr] [d] Hilb1(T*X/X) X ...X Hilbs(T*X/X) [d,\"Hilbert-Chow\"] U [urr, dotted] [rr,\"a|U\"] Chow1(T*X /X) X ...X Chows(T*X /X) and this morphism also induces an inclusion $U \\hookrightarrow T^*_X$ .","Now pulling back via $\\tilde{a}$ the product of universal families ${Z}_{\\alpha _1}(T^*_X/X)\\times _X \\cdots \\times _X {Z}_{\\alpha _s}(T^*_X/X),$ we get a finite, flat cover of $U$ , which is a closed subscheme of $T^*_U$ .","Then applying Serre's extension theorem [9], we get a finite, flat cover $\\widetilde{X}_a \\rightarrow X$ factoring through $T^*_X$ .","This cover has degree $\\sum ^s_{i=1}\\alpha _i}[d]^{i_X}& {A}_{X^{\\prime }}@{^(->}[d]^{i_{X^{\\prime }}} \\\\{B}_X[r]^{\\chi ^*}& {B}_{X^{\\prime }},}$ where $\\mathfrak {i}_X$ and $\\mathfrak {i}_{X^{\\prime }}$ are closed immersions by [2] and $\\chi ^*: \\bigoplus ^n_{i=1} H^0(X, S^i T^*_X)\\rightarrow \\bigoplus ^n_{i=1} H^0(X^{\\prime }, S^i T^*_{X^{\\prime }})$ is an injection.","From the diagram, we conclude that the natural morphism is a closed immersion.","Lemma 5.2 Let $U$ be an open subset of a smooth variety $X$ with $\\text{codim}(X-U, X)\\ge 2$ .","Then ${A}_X\\rightarrow {A}_U$ is an isomorphism.","Recall for any $k$ -scheme $T$ , the $T$ -points of ${A}_X$ are ${A}_X(T)=\\big \\lbrace {X_T[r] [rd]_{id_{X_T}} & \\text{Chow}^n(T^*_{X_T/T}/{X_T})[d] \\\\& X_T,}\\big \\rbrace $ By Lemma REF and the Yoneda Lemma, we shall show that the natural map ${A}_X(T)\\rightarrow {A}_U(T)$ induced by $j$ is surjective for all such $T$ .","First we consider the case $T^*_X=T^*_{X/k}$ is trivial, i.e.","$T^*_X\\simeq X\\times V$ , where $\\dim V=r$ .","Then it holds that $\\text{Chow}^n(T^*_{X_T/T}/{X_T})\\simeq \\text{Chow}^n(V)\\times X_T$ and an analogous isomorphism for $U_T$ .","Given a spectral datum $a\\in {A}_U(T)$ , it is equivalent to a morphism $a: U_T\\rightarrow \\text{Chow}^n(V)$ , which for ease of notation will also be denoted by $a$ .","By the key lemma [2], it gives rise to a morphism $U_T\\rightarrow V\\times S^2V\\times \\cdots \\times S^nV,$ which in turn amounts to $U_T\\rightarrow T^*_{X_T/T}\\times S^2T^*_{X_T/T}\\times \\cdots \\times S^nT^*_{X_T/T}$ by the triviality of the relative cotangent bundle.","This is exactly an element in $\\bigoplus ^n_{i=1} H^0(U_T, S^i T^*_{U_T/T})$ Claim: for each $i>0$ , we have $H^0(X_T, S^i T^*_{X_T/T})\\simeq H^0(T, \\mathcal {O}_{T})\\otimes _k H^0(X, S^i T^*_{X/k}).$ Given this Claim and from Remark REF , we obtain that the natural map $H^0(X_T, S^i T^*_{X_T/T})\\rightarrow H^0(U_T, S^i T^*_{U_T/T})$ is an isomorphism for each $i>0$ .","Thus we get $\\bar{a}\\in \\bigoplus ^n_{i=1} H^0(X_T, S^i T^*_{X_T/T}).$ We can view $\\bar{a}$ as a morphism $ \\bar{a}: X_T\\rightarrow V\\times S^2V\\times \\cdots \\times S^nV,$ which is equivalent to a section $\\bar{a}\\in {A}_X(T)$ .","To prove the Claim above, consider the fibred product ${X_T [r]^{g}[d]_{\\varphi _T} & X[d]^{\\varphi } \\\\T[r]^{h}& {\\rm Spec}(k).","}$ It holds that $ T^*_{X_T/T}\\simeq g^* T^*_{X/k}.$ Since $X$ is smooth, $T^*_{X/k}$ is locally free, hence $S^i T^*_{X_T/T}\\simeq g^*S^i( T^*_{X/k})$ for all $i>0$ .","By flat base change, it therefore follows that $&&H^0(X_T, S^iT^*_{X_T/T})\\\\&\\simeq & H^0(X_T, g^*S^i( T^*_{X/k}))\\\\&\\simeq & H^0({\\rm Spec}{k}, h_*(\\varphi _T)_*g^*S^i( T^*_{X/k}))\\\\&\\simeq &H^0({\\rm Spec}{k}, h_*h^*\\varphi _*S^i(T^*_{X/k}))\\\\&\\simeq &H^0({\\rm Spec}{k}, h_*\\mathcal {O}_{T}\\otimes _k \\varphi _*S^i(T^*_{X/k}))\\\\&\\simeq & H^0(T, \\mathcal {O}_{T})\\otimes _k H^0(X, S^i(T^*_{X/k})).","$ This proves the Claim.","In general, we take an affine open covering $\\lbrace X_i\\rbrace $ of $X$ such that $T^*_{X/k}|_{X_i}$ is trivial for every $i$ .","Put $U_i=U\\cap X_i$ .","For all $i$ , it holds that $\\text{codim}(X_i-U_i, X_i)\\ge 2$ .","By the preceding argument, ${A}_{X_i}(T)\\rightarrow {A}_{U_i}(T)$ is surjective for all $i$ .","Since $j^*$ is actually injective, we can lift section locally and then glue the resulting sections together.","This finishes the proof of this lemma.","Theorem 5.3 Let $\\chi : X^{\\prime }\\rightarrow X$ be a birational proper morphism of smooth varieties.","Then the natural morphism ${A}_X\\rightarrow {A}_{X^{\\prime }}$ induced by $\\chi $ is an isomorphism as $k$ -schemes.","By the valuative criterion for properness, one can find an open subset $j: U\\rightarrow X$ with $\\text{codim}(X-U, X)\\ge 2$ and a morphism $\\nu : U\\rightarrow X^{\\prime }$ such that $\\chi \\circ \\nu =j$ .","As a result, we have the commutative diagram of schemes ${{A}_{X^{\\prime }} @{^(->}[dr] & \\\\{A}_X[r]^{\\simeq } @{^(->}[u] & {A}_{U},}$ where the bottom map is an isomorphism by Lemma REF .","Therefore ${A}_X\\rightarrow {A}_{X^{\\prime }}$ is an isomorphism.","Example 5.4 Let $Y \\hookrightarrow X$ be a smooth subvariety.","Denote by $\\pi : \\widetilde{X}={\\rm BL}_Y(X)\\rightarrow X$ the blow up of $X$ along $Y$ .","Then one has the injection of vector bundles $0 \\rightarrow \\pi ^* T^*_X \\rightarrow T^*_{\\widetilde{X}}.$ Note that ${\\rm Chow}^n(T^*_X/X) \\times _X \\widetilde{X} \\cong {\\rm Chow}^n(\\pi ^* T^*_X/ \\widetilde{X})$ , and the injection above induces a morphism ${\\rm Chow}^n(\\pi ^* T^*_X/ \\widetilde{X}) \\rightarrow {\\rm Chow}^n(T^*_{\\widetilde{X}} / \\widetilde{X})$ .","Now given a section $a_X: X \\rightarrow {\\rm Chow}_n(T^*_X /X)$ , we define the section $a_{\\widetilde{X}}: \\widetilde{X} \\rightarrow {\\rm Chow}_n(T^*_{\\widetilde{X}}/ \\widetilde{X})$ as the compositions of the following morphisms $a_{\\widetilde{X}}: \\widetilde{X} \\cong X \\times _X \\widetilde{X} \\xrightarrow{} {\\rm Chow}^n(T^*_X/X) \\times _X \\widetilde{X} \\cong {\\rm Chow}^n(\\pi ^* T^*_X/ \\widetilde{X}) \\rightarrow {\\rm Chow}^n(T^*_{\\widetilde{X}}/ \\widetilde{X}),$ which gives a map ${A}_X(k) \\rightarrow {A}_{\\widetilde{X}}(k)$ between the $k$ -points.","By Theorem REF , the map is bijective.","Corollary 5.5 For any projective ruled surface $X$ , ${A}_X={B}_X$ .","In particular it is an affine space.","By Theorem REF , we can assume $X$ is minimal, i.e.","it does not contain a smooth rational curve $\\Gamma $ with $\\Gamma ^2=-1$ .","If $X=\\mathbb {P}^{2}$ , then ${A}_X={B}_X$ is a point; otherwise $X\\simeq \\mathbb {P}_C(E)$ for some smooth curve $C$ and a vector bundle $E$ of rank two, so ${A}_X\\simeq {A}_C$ , and hence ${A}_X={B}_X$ , see [3] and the following remark.","School of Mathematics, Sun Yat-Sen University W. 135 Xingang Rd., Guangzhou, Guangdong 510275, P.R.","China E-mail address: songlei3@mail.sysu.edu.cn Department of Mathematics, South China University of Technology 381 Wushan Rd., Guangzhou, Guangdong 510641, P.R.","China E-mail address: hsun71275@scut.edu.cn"]],"string":"[\n [\n \"On the image of Hitchin morphism for algebraic surfaces: The case ${\\\\rm\\n GL}_n$\"\n ],\n [\n \"Abstract The Hitchin morphism is a map from the moduli space of Higgs bundles $\\\\mathscr{M}_X$ to the Hitchin base $\\\\mathscr{B}_X$, where $X$ is a smooth projective variety.\",\n \"When $X$ has dimension at least two, this morphism is not surjective in general.\",\n \"Recently, Chen-Ng\\\\^o introduced a closed subscheme $\\\\mathscr{A}_X$ of $\\\\mathscr{B}_X$, which is called the space of spectral data.\",\n \"They proved that the Hitchin morphism factors through $\\\\mathscr{A}_X$ and conjectured that $\\\\mathscr{A}_X$ is the image of the Hitchin morphism.\",\n \"We prove that when $X$ is a smooth projective surface, this conjecture is true for vector bundles.\",\n \"Moreover, we show that $\\\\mathscr{A}_X$, for any dimension, is invariant under proper birational morphisms, and apply the result to study $\\\\mathscr{A}_X$ for ruled surfaces.\"\n ],\n [\n \"Introduction\",\n \"Throughout the paper, we work over an algebraically closed field $k$ of characteristic zero.\",\n \"Let $\\\\mathcal {M}_X$ be the moduli space of semistable Higgs bundles of rank $n$ over a smooth projective variety $X$ over $k$ .\",\n \"Let ${B}_X= \\\\bigoplus \\\\limits _{i=1}^n H^0(X, S^n T^*X)$ be the Hitchin base, where $T^*X$ is the cotangent bundle of $X$ .\",\n \"The Hitchin morphism $h_X: \\\\mathcal {M}_X \\\\rightarrow {B}_X$ was introduced in Hitchin's seminal work [6] for algebraic curves.\",\n \"The morphism was proved to be dominant by Beauville-Narasimhan-Ramanan [1], and was later on proved to be proper by Nisture [7] and Simpson [11], and hence surjective.\",\n \"In a more general setting and for higher dimensional varieties, Simpson showed that the Hitchin morphism is still proper [10], [11], but it is not surjective in general.\",\n \"Recently, Chen and Ngô took an attempt to understand the image of the Hitchin morphism and obtained a higher dimensional analogue of the BNR correspondence [1].\",\n \"In [2], [3], they considered the moduli stack ${M}_X$ of Higgs bundles on $X$ , and introduced a closed subscheme $\\\\mathfrak {i}_X: {A}_X \\\\hookrightarrow {B}_X$ , where ${A}_X={\\\\rm Sect}(X,{\\\\rm Chow}^n(T^*_X/X))$ is called the space of spectral data (see §2).\",\n \"They showed the Hitchin morphism factors through the space of spectral data MX [d,\\\"hX\\\"] [ld, dotted, \\\"sdX\\\" description] AX [r, hook, \\\"iX\\\"] BX, and conjectured that the morphism $sd_X$ is surjective.\",\n \"Conjecture 1.1 (Conjecture 5.2 in [3]) For every point $a \\\\in {A}_X(k)$ , the fiber $sd^{-1}_X(a)$ is nonempty.\",\n \"In fact, the conjecture is stated for $G$ -Higgs bundles, where $G$ is a split reductive group.\",\n \"In case the dimension $d=2$ and $G={\\\\rm GL}_n$ , the conjecture is proved by Chen and Ngô but only for spectral data in an open subset ${A}^{\\\\heartsuit }_X(k) \\\\subseteq {A}_X(k)$ , and it is verified for some minimal surfaces, such as ruled surfaces and nonisotrivial elliptic surfaces for all $a\\\\in {A}_X(k)$ , see [3] for more precise statements.\",\n \"In this article, we shall confirm the conjecture in case $d=2$ and $G={\\\\rm GL}_n$ : Theorem 1.2 (Theorem REF ) Let $X$ be a smooth projective surface.\",\n \"Let ${M}_X$ be the moduli stack of Higgs bundles of rank $n$ over $X$ .\",\n \"Then the image of the Hitchin morphism $h_X: {M}_X \\\\rightarrow {B}_X$ is ${A}_X$ , i.e., $sd_X : {M}_X \\\\rightarrow {A}_X$ is surjective.\",\n \"Here is an overview of the proof of the main result and the structure of the paper.\",\n \"Let ${\\\\rm Chow}^n(T^*_X/X)$ be the relative Chow variety of $n$ -points on $T^*_X$ over $X$ .\",\n \"Chen and Ngô defined the space of spectral data ${A}_X$ to be the space of sections $X \\\\rightarrow {\\\\rm Chow}^n(T^*_X/X)$ , and a section $a$ is called a spectral datum.\",\n \"Note that ${A}_X$ has a natural scheme structure, cf.\",\n \"§2 for an elaboration.\",\n \"For the relative Chow variety, there is a natural stratification, as in the absolute case, ${\\\\rm Chow}^{n}(T^*_X/X) = \\\\coprod _{\\\\mu } {\\\\rm Chow}^n_{\\\\mu }(T^*_X/X),$ where the union is taken over all partitions of $n$ .\",\n \"Chen and Ngô showed that given a spectral datum $a: X \\\\rightarrow {\\\\rm Chow}^n(T^*_X/X)$ , if the generic point of $X$ under $a$ lies in ${\\\\rm Chow}^n_{(1^n)}(T^*_X/X)$ , then there is a finite flat cover $\\\\widetilde{X}_a \\\\rightarrow X$ such that the Higgs bundles over $X$ with spectral datum $a$ correspond to the Cohen-Macaulay sheaves of rank one on $\\\\widetilde{X}_a$ (see [3]).\",\n \"In particular, for such general $a$ , then $sd_X^{-1}(a)$ is nonempty.\",\n \"Now given an arbitrary $a \\\\in {A}_X(k)$ , we assume that $a$ maps the generic point of $X$ to the stratum ${\\\\rm Chow}^n_{\\\\mu }(T^*_X/X)$ , where $\\\\mu =(1^{\\\\alpha _1}\\\\dots s^{\\\\alpha _s})$ is a partition of $n$ .\",\n \"Therefore, $a$ induces a morphism $X \\\\rightarrow \\\\overline{{\\\\rm Chow}^n_{\\\\mu }(T^*_X /X)}$ .\",\n \"Since $X$ is smooth, the spectral datum $a$ factors through the normalization $\\\\overline{{\\\\rm Chow}^n_{\\\\mu }(T^*_X /X)}^{\\\\rm nor}$ .\",\n \"Abusing the notation, we use the same notation $a: X \\\\rightarrow \\\\overline{{\\\\rm Chow}^n_{\\\\mu }(T^*_X /X)}^{\\\\rm nor}$ .\",\n \"We observe the isomorphism, which is analogous to [4], $\\\\overline{{\\\\rm Chow}^n_{\\\\mu }(T^*_X/X)}^{\\\\rm nor} \\\\cong {\\\\rm Chow}^{\\\\alpha _1}(T^*_X/X)\\\\times _X \\\\cdots \\\\times _X {\\\\rm Chow}^{\\\\alpha _s}(T^*_X/X)$ still holds (Proposition REF ).\",\n \"It follows that $a$ yields a collection of spectral data $(a_i)_{1\\\\le i \\\\le s}$ , $a_i: X \\\\rightarrow \\\\overline{{\\\\rm Chow}^{\\\\alpha _i}_{(1^{\\\\alpha _i})}(T^*_X/X)}={\\\\rm Chow}^{\\\\alpha _i}(T^*_X/X).$ This gives a decomposition of the given spectral datum $a$ , which is the key point to prove the main result.\",\n \"Theorem 1.3 (Theorem REF ) Let $X$ a smooth projective surface.\",\n \"Given a spectral datum $a: X \\\\rightarrow {\\\\rm Chow}^{n}(T^*_X /X)$ , suppose the generic point of $X$ is mapped to the stratum ${\\\\rm Chow}^{n}_{\\\\mu }(T^*_X /X)$ with $\\\\mu =(1^{\\\\alpha _1}2^{\\\\alpha _2}\\\\dots s^{\\\\alpha _s})$ .\",\n \"Then there exist spectral data $a_i: X \\\\rightarrow \\\\overline{{\\\\rm Chow}^{\\\\alpha _i}_{(1^{\\\\alpha _i})}(T^*_X /X)}$ , $i=1, \\\\cdots , s$ , such that $a=\\\\sum ^s_{i=1} ia_i.$ Since $a_i$ maps the generic point of $X$ to ${\\\\rm Chow}^{\\\\alpha _i}_{(1^{\\\\alpha _i})}(T^*_X/X)$ , there exists a Higgs bundle $(E_i,\\\\phi _i)$ corresponds to $a_i$ according to [3].\",\n \"Then the direct sum $(E,\\\\phi ):=\\\\bigoplus _{i=1}^s (E_i,\\\\phi _i)^{\\\\oplus i},$ is a Higgs bundle of $n$ corresponding to the spectral datum $a$ .\",\n \"This gives the surjectivity of $sd_X: {M}_X(k) \\\\rightarrow {A}_X(k)$ .\",\n \"In a different direction, in §5, we prove the birational invariance of ${A}_X$ for any dimensional variety $X$ , and apply the result to study the spaces of spectral data of ruled surfaces.\",\n \"Theorem 1.4 (Theorem REF ) Let $\\\\chi : X^{\\\\prime }\\\\rightarrow X$ be a birational proper morphism of smooth varieties.\",\n \"Then the natural morphism ${A}_X\\\\rightarrow {A}_{X^{\\\\prime }}$ induced by $\\\\chi $ is an isomorphism as $k$ -schemes.\",\n \"There are a number of interesting questions to be addressed in the future.\",\n \"As we stated above, in the surface case, we can construct a Higgs bundle for any given spectral data; however, we do not find a “cover\\\" $\\\\widetilde{X}_a \\\\rightarrow X$ such that we can get a correspondence between Higgs bundles with spectral data $a$ over $X$ and some special rank 1 sheaves over $\\\\widetilde{X}_a$ .\",\n \"In Remark REF , we discuss a finite, flat cover $\\\\widetilde{X}_a \\\\rightarrow X$ based on an arbitrary spectral datum $a$ , which may help to give the correspondence.\",\n \"In this article, we only consider Higgs bundles (${\\\\rm GL}_n$ -case).\",\n \"It would be very interesting to generalize the result to $G$ -Higgs bundles over higher dimensional varieties.\",\n \"This is of course the full statement of Chen and Ngô's conjecture.\",\n \"The classical Hitchin morphism is given for the moduli space of semistable Higgs bundles, and the conjectural image of the Hitchin morphism is for the moduli stack of Higgs bundles.\",\n \"It is natural to ask if ${A}_X$ is the image of the Hitchin morphism is for the moduli space $\\\\mathcal {M}^{ss}_X(P)$ of Higgs bundles with respect to some Hilbert polynomial $P$ ?\",\n \"Acknowledgments.\",\n \"The authors would like to thank X. Hu, G. Kydonakis, M. Li, J. Xu and L. Zhao for helpful discussions and conversations.\",\n \"The authors are very grateful to G. Kydonakis for invaluable suggestions on an early draft.\",\n \"During the preparation of the paper, L. S. and H. S. were partially supported respectively by the Guangdong Basic and Applied Basic Research Foundation 2020A1515010876 and 2019A1515110961.\"\n ],\n [\n \"Moduli Stacks of Higgs Bundles, Hitchin Base and Spectral data\",\n \"In this section, fix the rank $n$ , we discuss three moduli stacks on smooth quasi-projective variety $X$ over $k$ : the moduli stack of Higgs bundles ${M}_X$ ; the moduli stack of the Hitchin base ${B}_X$ ; the moduli stack of spectral data ${A}_X$ , and they are the main objects we work on.\",\n \"In this paper, we only focus on the Higgs bundles, i.e.\",\n \"${\\\\rm GL}_n$ -Higgs bundles, but some of the constructions can be extended to $G$ -Higgs bundles (see [2], [3] for more details).\",\n \"Let $\\\\mathfrak {C}^d_{{\\\\rm GL}_n} \\\\subseteq \\\\mathfrak {gl}^d_n$ be the closed subscheme consisting of $d$ -tuples $(x_1,\\\\dots ,x_d) \\\\in \\\\mathfrak {gl}^d_n$ such that $x_i$ and $x_j$ commutes, i.e.\",\n \"$[x_i,x_j]=0$ for all indices $i,j$ .\",\n \"The scheme $\\\\mathfrak {C}^d_{{\\\\rm GL}_n}$ is also known as the commuting scheme (see [8]).\",\n \"There are two group actions on $\\\\mathfrak {C}^d_{{\\\\rm GL}_n}$ : Given $g \\\\in {\\\\rm GL}_n$ , the ${\\\\rm GL}_n$ -action on $\\\\mathfrak {C}^d_{{\\\\rm GL}_n}$ is defined as $(x_1,\\\\dots ,x_d) \\\\rightarrow ({\\\\rm ad}(g)(x_1),\\\\dots ,{\\\\rm ad}(g)(x_d)).$ Given $h \\\\in {\\\\rm GL}_d$ , the ${\\\\rm GL}_d$ -action on $\\\\mathfrak {C}^d_{{\\\\rm GL}_n}$ is defined as $(x_1,\\\\dots ,x_d) \\\\rightarrow (x_1,\\\\dots ,x_d)h.$ We first consider the ${\\\\rm GL}_n$ -action on $\\\\mathfrak {C}^d_{{\\\\rm GL}_n}$ .\",\n \"The Chevalley restriction theorem gives us a ${\\\\rm GL}_d$ -equivariant morphism $\\\\mathbb {C}[\\\\mathfrak {gl}_n^d]^{{\\\\rm GL}_n} \\\\rightarrow \\\\mathbb {C}[\\\\mathfrak {t}^d]^{\\\\mathfrak {S}_n},$ where $\\\\mathfrak {t} \\\\subseteq \\\\mathfrak {gl}_n$ is a Cartan subalgebra and $\\\\mathfrak {S}_n$ is the permutation group.\",\n \"Restricting the morphism to $\\\\mathfrak {C}^d_{{\\\\rm GL}_n}$ , we have a ${\\\\rm GL}_d$ -equivariant lifting $[\\\\mathfrak {C}^{d}_{{\\\\rm GL}_n}/{\\\\rm GL}_n] \\\\rightarrow {\\\\rm Spec}(\\\\mathbb {C}[\\\\mathfrak {t}^d]^{\\\\mathfrak {S}_n})$ of $[\\\\mathfrak {C}^{d}_{{\\\\rm GL}_n}/{\\\\rm GL}_n] \\\\rightarrow \\\\mathfrak {C}^{d}_{{\\\\rm GL}_n}/{\\\\rm GL}_n$ along with the Chevalley restriction morphism.\",\n \"Let $V$ be a vector space of dimension $d$ .\",\n \"Therefore, we have ${\\\\rm Spec}(\\\\mathbb {C}[\\\\mathfrak {t}^d]^{\\\\mathfrak {S}_n}) \\\\cong {\\\\rm Chow}^n(V),$ where ${\\\\rm Chow}^n(V)$ is the Chow variety of $n$ points on $V$ .\",\n \"Then we have the morphism $[\\\\mathfrak {C}^{d}_{{\\\\rm GL}_n}/{\\\\rm GL}_n] \\\\rightarrow {\\\\rm Chow}^n(V).$ By [2], we know that there is a closed embedding ${\\\\rm Chow}^n(V) \\\\hookrightarrow V \\\\times S^2 V \\\\times \\\\dots \\\\times S^n V,$ and this induces the following morphism $[\\\\mathfrak {C}^{d}_{{\\\\rm GL}_n}/{\\\\rm GL}_n] \\\\rightarrow {\\\\rm Chow}^n(V)\\\\hookrightarrow V \\\\times S^2 V \\\\times \\\\dots \\\\times S^n V.$ As a $d$ -dimensional vector space, there is a canonical ${\\\\rm GL}_d$ -action on $V$ .\",\n \"We define the following quotient stacks ${A}:=[{\\\\rm Chow}^n(V)/{\\\\rm GL}_d], \\\\quad {B}:= [V \\\\times S^2V \\\\times \\\\dots \\\\times S^n V/ {\\\\rm GL}_d].$ By the definition of quotient stacks, for each $k$ -scheme $S$ , ${B}(S)$ is the set of pairs $(E,\\\\phi )$ , where $E$ is a rank $d$ vector bundle (as a ${\\\\rm GL}_d$ -principal bundle) over $S$ and $\\\\phi : E \\\\rightarrow V \\\\times S^2V \\\\times \\\\dots \\\\times S^n V$ is a ${\\\\rm GL}_d$ -equivariant map.\",\n \"Note that the ${\\\\rm GL}_d$ -equivariant map $\\\\phi $ corresponds to a unique element in $\\\\bigoplus \\\\limits _{i=1}^{n}H^0(S, S^i E)$ .\",\n \"Therefore, we have the following isomorphism as sets (or groupoids) by restricting to a fixed vector bundle $E$ over $S$ , ${B}(S)|_E \\\\cong \\\\bigoplus _{i=1}^{n}H^0(S, S^i E).$ This isomorphism also implies that ${B}(S)|_E$ has a scheme structure.\",\n \"For the quotient stack ${A}$ , given a $k$ -scheme $S$ , ${A}(S)$ is the set of pairs $(E,\\\\varphi )$ , where $E$ is a rank $d$ vector bundle over $S$ and $\\\\varphi : E \\\\rightarrow {\\\\rm Chow}^n(V)$ is a ${\\\\rm GL}_d$ -equivariant map.\",\n \"This map is equivalent to an element in ${\\\\rm Chow}^n(E/S)$ , where ${\\\\rm Chow}^n(E/S)$ is the relative Chow variety.\",\n \"In other words, we have ${A}(S)|_E \\\\cong {\\\\rm Sect}(S,{\\\\rm Chow}^n(E/S)),$ where ${\\\\rm Sect}(S,{\\\\rm Chow}^n(E/S))$ is the set of sections.\"\n ],\n [\n \"Moduli Stack of Higgs Bundles\",\n \"Recall that there is a natural ${\\\\rm GL}_d$ -morphism on $[\\\\mathfrak {C}^{d}_{{\\\\rm GL}_n}/{\\\\rm GL}_n]$ .\",\n \"Denote by ${M}:=[\\\\mathfrak {C}^{d}_{{\\\\rm GL}_n}/{\\\\rm GL}_n \\\\times {\\\\rm GL}_d]$ the quotient stack.\",\n \"Let $S$ be a scheme, and then the $S$ -points of ${M}$ are triples $(V,E,\\\\theta )$ , where $V$ is a vector bundle of rank $d$ (${\\\\rm GL}_d$ -principal bundle) over $S$ ; $E$ is a vector bundle of rank $n$ (${\\\\rm GL}_n$ -principal bundle) over $S$ ; $\\\\theta \\\\in H^0(S, End(E) \\\\otimes V)$ is a $\\\\mathcal {O}_S$ -linear morphism such that $\\\\theta \\\\wedge \\\\theta =0$ .\",\n \"Now we fix a smooth projective (or quasi-projective) variety $X$ of dimension $d$ .\",\n \"Let $T^*_X$ be the cotangent bundle over $X$ .\",\n \"We define a contravariant functor ${M}_X: ({\\\\rm Sch}/k)^{\\\\rm op} \\\\rightarrow {\\\\rm Sets}$ as follows.\",\n \"Let $S$ be a scheme, and define $X_S:=X \\\\times _{k} S$ .\",\n \"Denote by $\\\\pi : X_S \\\\rightarrow X$ the natural projection.\",\n \"There is a natural morphism $[\\\\mathfrak {C}^{d}_{{\\\\rm GL}_n}/{\\\\rm GL}_n \\\\times {\\\\rm GL}_d] \\\\rightarrow [\\\\ast / {\\\\rm GL}_d]=\\\\mathbb {B}{\\\\rm GL}_d,$ which induces the morphism $[\\\\mathfrak {C}^{d}_{{\\\\rm GL}_n}/{\\\\rm GL}_n \\\\times {\\\\rm GL}_d](X_S) \\\\rightarrow \\\\mathbb {B}{\\\\rm GL}_d(X_S).$ Note that $\\\\pi ^* T^*_X$ is a dimension $d$ vector bundle over $X_S$ , which corresponds to a point $[\\\\pi ^* T^*_X] \\\\in \\\\mathbb {B}{\\\\rm GL}_d(X_S)$ .\",\n \"We define ${M}_X(S)$ to be the preimage of $[\\\\pi ^* T^*_X]$ in $[\\\\mathfrak {C}^{d}_{{\\\\rm GL}_n}/{\\\\rm GL}_n \\\\times {\\\\rm GL}_d](X_S)$ , which parametrizes pairs $(E,\\\\theta )$ such that $E$ is a rank $n$ vector bundle over $X_S$ ; $\\\\theta \\\\in H^0(X_S, \\\\mathcal {E}nd(E) \\\\otimes \\\\pi ^* T^*_X)$ is a section such that $\\\\theta \\\\wedge \\\\theta =0$ .\",\n \"It is easy to check that ${M}_X$ is a stack, which is actually the stack of Higgs bundles on $X$.\"\n ],\n [\n \"Hitchin Base ${B}_X$\",\n \"We have a natural morphism ${B} \\\\rightarrow \\\\mathbb {B}{\\\\rm GL}_d$ of stacks.\",\n \"Fixing a dimension $d$ smooth projective variety $X$ , we define the following contravariant functor as follows ${B}_X: ({\\\\rm Sch}/k)^{\\\\rm op} \\\\rightarrow {\\\\rm Sets}$ For each scheme $S$ , we define the space ${B}_X(S):={B}(X_S)|_{[\\\\pi ^* T^*_X]} \\\\subseteq {B}(X_S)$ to be the preimage of the point $[\\\\pi ^*T^*_X] \\\\in \\\\mathbb {B}{\\\\rm GL}_d(X_S)$ under the map ${B}(X_S) \\\\rightarrow \\\\mathbb {B}{\\\\rm GL}_d(X_S)$ .\",\n \"Thus, ${B}_X(S) \\\\cong \\\\bigoplus _{i=1}^{n}H^0(X_S, S^i \\\\pi ^* T^*_X),$ where $S^i \\\\pi ^* T^*_X$ is the $i$ -th symmetric product of $\\\\pi ^* T^*_X$ .\",\n \"Based on the above definition, it is easy to check that ${B}_X$ has a natural stack structure.\",\n \"Furthermore, we can rewrite $H^0(X_S, S^i \\\\pi ^* T^*_X)$ in the following way $& H^0(X_S, S^i \\\\pi ^* T^*_X) \\\\cong {\\\\rm Hom}(\\\\pi ^* T^*_X, S^i V)^{{\\\\rm GL}_d} \\\\cong {\\\\rm Hom}(X_S \\\\times _X T^*_X, S^i V)^{{\\\\rm GL}_d}\\\\\\\\\\\\cong & {\\\\rm Hom}(S \\\\times T^*_X, S^i V)^{{\\\\rm GL}_d} \\\\cong {\\\\rm Hom}(S, {\\\\rm Hom}(T^*_X, S^i V)^{{\\\\rm GL}_d}) \\\\cong {\\\\rm Hom}(S, H^0(X, S^i T^*_X)).$ Therefore, ${B}_X(S) \\\\cong \\\\bigoplus _{i=1}^{n} {\\\\rm Hom}(S, H^0(X, S^i T^*_X)).$ Then, the stack ${B}_X$ is representable by the scheme $\\\\bigoplus _{i=1}^{n} H^0(X, S^i T^*_X)$ , i.e.\",\n \"${B}_X(\\\\ast ) \\\\cong {\\\\rm Hom}(\\\\ast , \\\\bigoplus _{i=1}^{n} H^0(X, S^i T^*_X)).$ This gives the following lemma.\",\n \"Lemma 2.1 The moduli stack ${B}_X$ is represented by $\\\\bigoplus \\\\limits _{i=1}^{n} H^0(X, S^i T^*_X)$ .\",\n \"By abusing notation, we also use ${B}_X$ for the scheme $\\\\bigoplus \\\\limits _{i=1}^{n} H^0(X, S^i T^*_X)$ , which is known as the Hitchin base.\"\n ],\n [\n \"Space of Spectral Data ${A}_X$\",\n \"Similar to ${B}_X$ , we can define the the functor ${A}_X: ({\\\\rm Sch}/k)^{\\\\rm op} \\\\rightarrow {\\\\rm Sets}$ such that for each scheme $S$ , define ${A}_X(S) \\\\subseteq {A}(X_S)$ to be the preimage of the point $[\\\\pi ^*T^*_X]$ under the map ${A}(X_S) \\\\rightarrow \\\\mathbb {B}{\\\\rm GL}_d(X_S)$ .\",\n \"We know that ${A}_X(S) \\\\cong {\\\\rm Sect}(X_S,{\\\\rm Chow}^n(\\\\pi ^* T^*_X/X_S)).$ With the same discussion as for ${B}_X$ , we have ${A}_X(S) \\\\cong {\\\\rm Sect}(X_S, {\\\\rm Chow}^n( \\\\pi ^*T^*_X/X_S ) ) \\\\cong {\\\\rm Hom}(S, {\\\\rm Sect}(X, {\\\\rm Chow}^n( T^*_X/X ) )).$ This gives us the following lemma.\",\n \"Lemma 2.2 The stack ${A}_X$ is representable by the scheme ${\\\\rm Sect}(X, {\\\\rm Chow}^n( T^*_X/X ) )$ , i.e.\",\n \"${A}_X(\\\\ast ) \\\\cong {\\\\rm Hom}(\\\\ast , {\\\\rm Sect}(X, {\\\\rm Chow}^n( T^*_X/X ) )).$ We use the same notation ${A}_X$ for the corresponding scheme ${\\\\rm Sect}(X, {\\\\rm Chow}^n( T^*_X/X ) )$ .\",\n \"An element $a \\\\in {A}_X$ is called a spectral datum and ${A}_X$ is called the space of spectral data.\",\n \"Sometimes, we use the notation ${A}_X^n$ to highlight the integer $n$ .\",\n \"Remark 2.3 ${B}_X$ and ${A}_X$ are claimed to be schemes [2], [3], and here we give the constructions from the viewpoint of moduli stack.\",\n \"Remark 2.4 Let $U \\\\rightarrow X$ be a morphism of schemes.\",\n \"Since ${M}$ is a stack, we have a natural map ${M}(X) \\\\rightarrow {M}(U)$ .\",\n \"Furthermore, this map induces a morphism ${M}_X \\\\rightarrow {M}_U$ as stacks.\",\n \"The same argument also works for morphisms ${A}_X \\\\rightarrow {A}_U$ and ${B}_X \\\\rightarrow {B}_U$ .\",\n \"As a special case, if $U$ is an open subscheme of $X$ , then we have a natural morphism ${A}_X \\\\rightarrow {A}_U$ .\",\n \"Remark 2.5 Recall that the closed embedding ${\\\\rm Chow}^n(V) \\\\hookrightarrow V \\\\times S^2 V \\\\times \\\\dots \\\\times S^n V,$ induces the closed embedding $\\\\mathfrak {i}_X: {A}_X \\\\hookrightarrow {B}_X$ ([3]).\",\n \"Chen and Ngô showed that the Hitchin map $h_X: {M}_X \\\\rightarrow {B}_X$ factors through ${A}_X$ .\",\n \"More precisely, there exists a map $sd_X: {M}_X \\\\rightarrow {A}_X$ such that $h_X=\\\\mathfrak {i}_X \\\\circ sd_X$ .\",\n \"MX [d,\\\"hX\\\"] [ld, dotted, \\\"sdX\\\" description] AX [r, hook, \\\"iX\\\"] BX When $X$ is a smooth curve, we have ${A}_X \\\\cong {B}_X$ , and hence $sd_X$ is surjective.\",\n \"But for higher dimensions, ${A}_X$ is smaller than ${B}_X$ in general.\"\n ],\n [\n \"Decomposition of spectral data\",\n \"In this section, we prove a decomposition theorem (Theorem REF ) for spectral datum $a \\\\in {A}_X$ , where $X$ is an algebraic surface.\",\n \"This decomposition theorem is an important tool to prove the main theorem (Theorem REF ).\",\n \"In §3.1, we briefly review some relevant facts about Chow varieties, and we refer the reader to [4] for more details.\",\n \"In §3.2, we generalize some results to the relative case, and prove the decomposition theorem.\"\n ],\n [\n \"Classical Case\",\n \"Let $X$ be a smooth projective or quasi-projective surface.\",\n \"Let ${\\\\rm Chow}^n(X)$ be the Chow variety of $n$ points on $X$ .\",\n \"Let $\\\\mu $ be a partition of $n$ .\",\n \"We will write $\\\\mu =(n_1,\\\\dots ,n_k)$ or $\\\\mu =(1^{\\\\alpha _1}2^{\\\\alpha _2}\\\\dots s^{\\\\alpha _s})$ interchangeably.\",\n \"Define a locally closed subset ${\\\\rm Chow}^n_{\\\\mu }(X)$ of ${\\\\rm Chow}^n(X)$ as ${\\\\rm Chow}^n_{\\\\mu }(X):=\\\\lbrace \\\\sum _i^k n_i z_i \\\\text{ } | \\\\text{ } z_i \\\\in X \\\\text{ distinct points } \\\\rbrace .$ Then, there is a stratification of ${\\\\rm Chow}^{n}(X)$ : ${\\\\rm Chow}^{n}(X) = \\\\coprod _{\\\\mu } {\\\\rm Chow}^n_{\\\\mu }(X),$ where $\\\\mu $ ranges over all partitions of $n$ .\",\n \"Given a partition $\\\\mu =(1^{\\\\alpha _1}2^{\\\\alpha _2}\\\\dots s^{\\\\alpha _s})$ , we have a natural morphism ${\\\\rm Chow}^{\\\\alpha _1}(X) \\\\times {\\\\rm Chow}^{\\\\alpha _2}(X) \\\\times \\\\dots \\\\times {\\\\rm Chow}^{\\\\alpha _s}(X) \\\\rightarrow {\\\\rm Chow}^n(X)$ defined as $(z_1,z_2,\\\\dots , z_s) \\\\mapsto z_1+ 2z_2 + \\\\dots + s z_s.$ Indeed this morphism gives the normalization of $\\\\overline{{\\\\rm Chow}^n_{\\\\mu }(X)}$ .\",\n \"Lemma 3.1 (Exercise 7.4.2 in [4]) Given a partition $\\\\mu =(1^{\\\\alpha _1}, 2^{\\\\alpha _2}, \\\\dots , s^{\\\\alpha _s})$ , we have $\\\\overline{{\\\\rm Chow}^n_{\\\\mu }(X)}^{\\\\rm nor} \\\\cong {\\\\rm Chow}^{\\\\alpha _1}(X) \\\\times \\\\dots \\\\times {\\\\rm Chow}^{\\\\alpha _s}(X),$ where $\\\\overline{{\\\\rm Chow}^n_{\\\\mu }(X)}^{\\\\rm nor}$ is the normalization of the closure of ${\\\\rm Chow}^n_{\\\\mu }(X)$ .\"\n ],\n [\n \"Relative Case\",\n \"Let $X$ be a smooth projective surface.\",\n \"Denote by $T^*_X$ the cotangent bundle over $X$ .\",\n \"The Chow variety ${\\\\rm Chow}^n(T^*_X/X)$ is defined as follows ${\\\\rm Chow}^n(T^*_X/X):= \\\\underbrace{T^*_X \\\\times _X \\\\dots \\\\times _X T^*_X}_n/ \\\\mathfrak {S}_n,$ where $\\\\mathfrak {S}_n$ is the symmetric group.\",\n \"Similar to the classical case, we define a locally closed subset ${\\\\rm Chow}^n_{\\\\mu }(T^*_X/X)$ of ${\\\\rm Chow}^n(T^*_X/X)$ with respect to the partition $\\\\mu $ , and then we have a stratification of ${\\\\rm Chow}^{n}(T^*_X/X)$ , ${\\\\rm Chow}^{n}(T^*_X/X) = \\\\coprod _{\\\\mu } {\\\\rm Chow}^n_{\\\\mu }(T^*_X/X),$ where $\\\\mu $ ranges over all partitions of $n$ .\",\n \"For ease of notation, set $\\\\prod ^s_{j=1, X}{\\\\rm Chow}^{\\\\alpha _j}(T^*_X/X):={\\\\rm Chow}^{\\\\alpha _1}(T^*_X/X) \\\\times _X {\\\\rm Chow}^{\\\\alpha _2}(T^*_X/X) \\\\times _X \\\\dots \\\\times _X {\\\\rm Chow}^{\\\\alpha _s}(T^*_X/X),$ where the subscript $X$ under the symbol $\\\\prod ^s_{j=1, X}$ indicates that the product is taken over $X$ .\",\n \"Similarly set $\\\\prod ^n_{j=1, X}T^*_X:=\\\\underbrace{T^*_X \\\\times _X \\\\dots \\\\times _X T^*_X}_n.$ We summarize some facts: Lemma 3.2 For any $n\\\\ge 1$ and a partition $\\\\mu =(1^{\\\\alpha _1} \\\\dots s^{\\\\alpha _s})$ , the following hold: (i) $\\\\prod \\\\limits ^n_{j=1, X}T^*_X$ is smooth over $X$ .\",\n \"(ii) ${\\\\rm Chow}^n_{\\\\mu }(T^*_X /X)$ is smooth over $X$ .\",\n \"(iii) ${\\\\rm Chow}^n(T^*_X /X)$ has rational singularities, in particular it is normal and Cohen-Macaulay.\",\n \"(iv) $\\\\prod \\\\limits ^s_{j=1, X}{\\\\rm Chow}^{\\\\alpha _j}(T^*_X/X)\\\\cong \\\\prod \\\\limits ^{\\\\alpha _1+\\\\dots +\\\\alpha _s}_{j=1, X}T^*_X/{\\\\mathfrak {S}_{\\\\alpha _1}\\\\times \\\\cdots \\\\times \\\\mathfrak {S}_{\\\\alpha _s}}$ has rational singularities, in particular, it is normal and Cohen-Macaulay.$\\\\Box $ Given a partition $\\\\mu =(1^{\\\\alpha _1}2^{\\\\alpha _2}\\\\dots s^{\\\\alpha _s})$ of $n$ , we define a morphism $\\\\tau _{\\\\mu }: \\\\prod ^s_{j=1, X}{\\\\rm Chow}^{\\\\alpha _j}(T^*_X/X) \\\\rightarrow {\\\\rm Chow}^n(T^*_X/X)\\\\qquad \\\\mathrm {(\\\\ast )}$ as $(z_1,z_2,\\\\dots , z_s) \\\\mapsto z_1+ 2z_2 + \\\\dots + s z_s.$ The following is a key observation, which is a relative version of Lemma REF .\",\n \"Proposition 3.3 Given a partition $\\\\mu =(1^{\\\\alpha _1} \\\\dots s^{\\\\alpha _s})$ , the morphism $\\\\tau _{\\\\mu }$ gives the normalization $\\\\overline{{\\\\rm Chow}^n_{\\\\mu }(T^*_X/X)}$ .\",\n \"The generic point of the product is mapped to the generic point of ${\\\\rm Chow}^n_{\\\\mu }(T^*_X/X)$ by $\\\\tau _{\\\\mu }$ .\",\n \"Thus $\\\\tau _{\\\\mu }$ induces a proper, surjective morphism from $\\\\prod ^s_{j=1, X}{\\\\rm Chow}^{\\\\alpha _j}(T^*_X/X)$ to $\\\\overline{{\\\\rm Chow}^n_{\\\\mu }(T^*_X/X)}$ .\",\n \"Also, note that by Lemma REF (iv), the product $\\\\prod ^s_{j=1, X}{\\\\rm Chow}^{\\\\alpha _j}(T^*_X/X)$ is a normal variety.\",\n \"To see it is indeed the normalization, we shall utilize Lemma REF , which says one can check normalization locally.\",\n \"Take an open covering $\\\\lbrace U_i\\\\rbrace $ of $X$ such that $T^*_X |_{U_i} \\\\cong U_i \\\\times \\\\mathbb {A}^2$ .\",\n \"We have the commutative diagram with Cartesian squares ${&\\\\prod ^s_{j=1, U_i}{\\\\rm Chow}^{\\\\alpha _j}(T^*_{U_i}/{U_i})[d]^{\\\\tau ^{\\\\prime }_{\\\\mu }}[ld][rrd]&&\\\\\\\\\\\\prod ^s_{j=1, X}{\\\\rm Chow}^{\\\\alpha _j}(T^*_X/X)[rrd][d]^{\\\\tau _{\\\\mu }}& \\\\overline{{\\\\rm Chow}^n_{\\\\mu }(T^*_{U_i}/{U_i})}@{^(->}[rr] [ld][rrd] & & {\\\\rm Chow}^n(T^*_{U_i}/{U_i})[d][ld]\\\\\\\\\\\\overline{{\\\\rm Chow}^n_{\\\\mu }(T^*_X/X)}@{^(->}[rr][rrd]& & {\\\\rm Chow}^n(T^*_X/X)[d] & U_i[ld], \\\\\\\\& & X &}$ where $\\\\overline{{\\\\rm Chow}_{\\\\mu }^n(T^*_{U_i} /U_i)}$ is the closure of ${\\\\rm Chow}_{\\\\mu }^n(T^*_{U_i} /U_i)$ in ${\\\\rm Chow}^n(T^*_{U_i}/{U_i})$ .\",\n \"We have the isomorphisms $\\\\overline{{\\\\rm Chow}_{\\\\mu }^n(T^*_{U_i} /U_i)} & \\\\cong \\\\overline{{\\\\rm Chow}_{\\\\mu }^n(\\\\mathbb {A}^2)} \\\\times U_i,\\\\\\\\\\\\prod ^s_{j=1, U_i}{\\\\rm Chow}^{\\\\alpha _j}(T^*_{U_i}/U_i) & \\\\cong \\\\big (\\\\prod ^s_{j=1}{\\\\rm Chow}^{\\\\alpha _j}(\\\\mathbb {A}^2)\\\\big )\\\\times U_i.$ Thus by Lemma REF , $\\\\tau ^{\\\\prime }_{\\\\mu }$ is the normalization of $\\\\overline{{\\\\rm Chow}^n_{\\\\mu }(T^*_{U_i}/{U_i})}$ .\",\n \"Applying Lemma REF , we get $\\\\tau _{\\\\mu }$ is the normalization of $\\\\overline{{\\\\rm Chow}^n_{\\\\mu }(T^*_X/X)}$ .\",\n \"Lemma 3.4 Let $X$ be an integral scheme and $\\\\tilde{X}$ a normal integral scheme.\",\n \"Given a morphism $\\\\nu : \\\\tilde{X}\\\\rightarrow X$ , suppose there exists an open covering ${U_i}$ of $X$ such that $\\\\nu _i: \\\\tilde{U_i}=\\\\nu ^{-1}(U_i)\\\\rightarrow U_i$ is the normalization for each $i$ .\",\n \"Then $\\\\nu $ is the normalization.\",\n \"This follows from the universal property of normalization [5].\",\n \"With respect to Proposition REF , we obtain the decomposition of spectral data.\",\n \"It will be used to prove the main result (Theorem REF ) in §4.\",\n \"Theorem 3.5 Let $X$ be a smooth projective surface.\",\n \"Given a spectral datum $a: X \\\\rightarrow {\\\\rm Chow}^{n}(T^*_X /X)$ , suppose the generic point of $X$ is mapped to the stratum ${\\\\rm Chow}^{n}_{\\\\mu }(T^*_X /X)$ with $\\\\mu =(1^{\\\\alpha _1}2^{\\\\alpha _2}\\\\dots s^{\\\\alpha _s})$ .\",\n \"Then there exist spectral data $a_i: X \\\\rightarrow \\\\overline{{\\\\rm Chow}^{\\\\alpha _i}_{(1^{\\\\alpha _i})}(T^*_X /X)}$ , $i=1, \\\\cdots , s$ , such that $a=\\\\sum ^s_{i=1} ia_i.$ By the assumption, $a$ induces a morphism $X \\\\rightarrow \\\\overline{{\\\\rm Chow}^n_{\\\\mu }(T^*_X /X)}$ over $X$ .\",\n \"By Proposition REF , it holds that $\\\\overline{{\\\\rm Chow}^n_{\\\\mu }(T^*_X /X)}^{\\\\rm nor} \\\\cong {\\\\rm Chow}^{\\\\alpha _1}(T^*_X /X) \\\\times _X \\\\dots \\\\times _X {\\\\rm Chow}^{\\\\alpha _s}(T^*_X /X).$ Since $X$ is smooth, there exists a unique $a^{\\\\prime }$ fitting into the diagram Chown(T*X /X)nor [d] [rd, \\\"\\\"] X [r][ru,\\\"a'\\\", dotted] Chown(T*X /X) [r, hook] Chown(T*X /X) Here $a^{\\\\prime }$ is equivalent to data $(a_i)_{1 \\\\le i \\\\le s}$ , where $a_i: X \\\\rightarrow \\\\overline{{\\\\rm Chow}^{\\\\alpha _i}_{(1^{\\\\alpha _i})}(T^*_X /X)}.$ Finally by the commutativity of the diagram, we conclude that $a=\\\\sum \\\\limits _{i=1}^s i a_i$ .\"\n ],\n [\n \"The image of the Hitchin morphism for surfaces\",\n \"Given a spectral datum $a: X \\\\rightarrow {\\\\rm Chow}^{n}(T^*_X/X)$ , we first briefly review the construction of the spectral cover $\\\\widetilde{X}_a \\\\rightarrow X$ , and the correspondence between Cohen-Macaulay sheaves of generic rank one over $\\\\widetilde{X}_a$ and Higgs bundles over $X$ [3].\",\n \"Next, we show that given two Higgs bundles, their direct sum corresponds to the sum of the corresponding spectral data (Lemma REF ).\",\n \"Finally, we prove the main theorem that $sd_X: {M}_X \\\\rightarrow {A}_X$ is surjective (Theorem REF ) with the help of Theorem REF .\",\n \"Define $X_a$ as the pullback in the following diagram Xa [d,\\\"a\\\"] [rr] Cayleyn(T*X/X) [d,\\\"\\\"] X [rr,\\\"a\\\"] Chown(T*X/X) where ${\\\\rm Cayley}^n(T^*_X/X)$ is a subscheme of ${\\\\rm Chow}^n(T^*_X/X) \\\\times _X T^*_X$ ([3]).\",\n \"Theorem 4.1 (Theorems 7.1 and 7.3 in [3]) Given a spectral datum $a: X \\\\rightarrow {\\\\rm Chow}^{n}(T^*_X/X)$ such that $a$ maps the generic point of $X$ into ${\\\\rm Chow}^n_{(1^n)}(T^*_X/X)$ , there exists a unique finite flat covering $\\\\widetilde{\\\\pi }_a: \\\\widetilde{X}_a \\\\rightarrow X$ such that there exists an open subset $U \\\\subseteq X$ of codimension at least 2 such that for every point $x \\\\in U$ , the fiber $(\\\\widetilde{\\\\pi }_a)^{-1}$ is a point of ${\\\\rm Hilb}^n(T^*_X/X)$ lying over the point $a(x) \\\\in {\\\\rm Chow}^n(T^*_X/X)$ ; there is a natural morphism $\\\\widetilde{\\\\iota }_a: \\\\widetilde{X}_a \\\\rightarrow T^*_X$ factoring through $\\\\iota _a: X_a \\\\hookrightarrow T^*_X$ such that the following diagram is commutative; Xa [rd,\\\"a\\\"] [r, \\\"\\\"] Xa [r,\\\"a\\\", hook] [d, \\\"a\\\"] T*X [ld, \\\"\\\"] X $\\\\widetilde{X}_a$ is the Cohen-Macaulayfication of $X_a$ ; the fiber $sd_X^{-1}(a)$ is isomorphic to the stack of Cohen-Macaulay sheaves of generic rank one over $\\\\widetilde{X}_a$ .\",\n \"The correspondence demonstrated in the theorem can be understood in the following way.\",\n \"Given a Cohen-Macaulay sheaf $L$ of generic rank one over $\\\\widetilde{X}_a$ , its pushfoward $\\\\iota _* L$ is a sheaf over $X_a$ .\",\n \"Via $\\\\iota _a$ , the sheaf $(\\\\widetilde{\\\\iota }_a)_* L$ has a natural structure as a $S(T_X)$ -module, where $S(T_X)$ is the symmetric product of the tangent bundle $T_X$ .\",\n \"Furthermore, it is a finite $S(T_X)$ -module.\",\n \"Therefore, it corresponds to a spectral data in ${A}_X$ , which is exactly $a$ , and also corresponds to a Higgs bundle $(E,\\\\phi )$ over $X$ .\",\n \"The correspondence between a finite $S(T_X)$ -module $L$ and a spectral data $a$ can be understood in the following way.\",\n \"For each $x \\\\in X$ , we can equip $L|_{\\\\pi ^{-1}(x)}$ with a cycle ${\\\\rm cyc}(L|_{\\\\pi ^{-1}(x)}):=\\\\sum _{y \\\\in \\\\pi ^{-1}(x)} {\\\\rm len}_{y}(L_{\\\\pi ^{-1}(x)}) \\\\cdot y.$ Then, the $S(T_X)$ -module $L$ corresponds to $a$ if for each $x \\\\in X$ , we have $a(x)={\\\\rm cyc}(L|_{\\\\pi ^{-1}(x)}).$ Theorem REF only works for spectral data which map the generic point of $X$ into ${\\\\rm Chow}^n_{(1^n)}(T^*_X/X)$ .\",\n \"Given an arbitrary spectral datum, we can follow the approach in the theorem to construct $X_a$ , but the correspondence is not clear.\",\n \"We will discuss this issue in Remark REF .\",\n \"Lemma 4.2 Given a pair of spectral data $a_i: X\\\\rightarrow {\\\\rm Chow}^{n_i}(T^*_X/X)$ , $i=1, 2$ , let $(E_i, \\\\phi _i)$ be Higgs bundles on $X$ whose spectral data are $a_i$ .\",\n \"Then $(E_1\\\\oplus E_2, \\\\phi _1\\\\oplus \\\\phi _2)$ is a Higgs bundle with the spectral datum $a_1+a_2: X\\\\rightarrow {\\\\rm Chow}^{n_1+n_2}(T^*_X/X)$ .\",\n \"The pair $(E_1\\\\oplus E_2, \\\\phi _1\\\\oplus \\\\phi _2)$ is obviously a Higgs bundle on $X$ .\",\n \"By the correspondence [11], there exist coherent sheaves $F_i$ on $T^*_X$ such that $\\\\pi _* F_i\\\\simeq E_i$ .\",\n \"Thus, $\\\\pi _*(F_1\\\\oplus F_2)\\\\simeq E_1\\\\oplus E_2$ .\",\n \"For any $x\\\\in X$ , $a_1(x)+a_2(x)={\\\\rm cyc}( L_1 |_{\\\\pi ^{-1}(x)})+{\\\\rm cyc}(L_2|_{\\\\pi ^{-1}(x)})={\\\\rm cyc}((L_1+L_2)|_{\\\\pi ^{-1}(x)})=(a_1+a_2)(x).$ This finishes the proof.\",\n \"Theorem 4.3 Let $X$ be a smooth projective surface.\",\n \"The image of the Hitchin map $h_X: {M}_X \\\\rightarrow {B}_X$ is ${A}_X$ , i.e.\",\n \"$sd_X : {M}_X \\\\rightarrow {A}_X$ is surjective.\",\n \"We will show that given any spectral datum $a \\\\in {A}_X$ , we can construct a Higgs bundle $(E,\\\\phi ) \\\\in sd_X^{-1}(a) \\\\subseteq {M}_X$ .\",\n \"By Theorem REF , this is equivalent to finding a $S(T_X)$ -module $L$ corresponding to the spectral datum $a$ .\",\n \"Now given a spectral datum $a: X \\\\rightarrow {\\\\rm Chow}^n(T^*_X/X)$ , suppose that the generic point of $X$ is mapped into some stratum ${\\\\rm Chow}^n_{\\\\mu }(T^*_X/X)$ , where $\\\\mu =(1^{\\\\alpha _1}, 2^{\\\\alpha _2}, \\\\dots , s^{\\\\alpha _s})$ .\",\n \"By Theorem REF , we have $a=a_1+2a_2+\\\\dots +sa_s,$ where the spectral datum $a_i$ maps the generic point of $X$ into ${\\\\rm Chow}^{\\\\alpha _i}_{(1^{\\\\alpha _i})}(T^*_X /X)$ .\",\n \"Therefore, by Theorem REF , for each $i$ , one can construct a spectral cover $\\\\widetilde{X}_{a_i}$ of $X$ Xai [rd,\\\"i\\\"] [r] Xai [r,\\\"i\\\", hook] [d, \\\"i\\\"] T*X [ld, \\\"\\\"] X such that $\\\\widetilde{\\\\pi }_{a_i}: \\\\widetilde{X}_{a_i} \\\\rightarrow X$ is a finite flat morphism and the morphism $\\\\widetilde{\\\\iota }_{a_i}: \\\\widetilde{X}_{a_i} \\\\rightarrow T^*_X$ factors through $\\\\iota _{i}: X_{a_i} \\\\hookrightarrow T^*_X$ .\",\n \"Then, the $S(T_X)$ -module $(\\\\widetilde{\\\\iota }_i)_* \\\\mathcal {O}_{\\\\widetilde{X}_{a_i}}$ corresponds to the spectral datum $a_i$ .\",\n \"By Lemma REF , the sheaf $L:=\\\\bigoplus \\\\limits _{i=1}^s (\\\\widetilde{\\\\iota }_i)_* \\\\mathcal {O}_{\\\\widetilde{X}_{a_i}}^{\\\\oplus i}$ corresponds to the spectral data $\\\\sum \\\\limits _{i=1}^s i a_i$ , which is exactly $a$ .\",\n \"Therefore, $sd_X^{-1}(a)$ is nonempty.\",\n \"Remark 4.4 Although Theorem REF gives the surjectivity of the morphism $sd_X: {M}_X \\\\rightarrow {A}_X$ , we do not find a cover $\\\\widetilde{X}_a \\\\rightarrow X$ such that Higgs bundles over $X$ with spectral datum $a$ will correspond to some special sheaves over $\\\\widetilde{X}_a$ .\",\n \"On the other hand, for any $a \\\\in {A}_X(k)$ , we can construct a finite, flat cover $\\\\widetilde{X}_a \\\\rightarrow X$ following Chen and Ngô's idea in [3].\",\n \"However, the generic rank one sheaves over $\\\\widetilde{X}_a$ do not correspond to Higgs bundles of rank $n$ over $X$ .\",\n \"We give the construction and explain the reason in the following.\",\n \"Let $a: X \\\\rightarrow \\\\overline{{\\\\rm Chow}^n_{\\\\mu }(T^*_X/X)}$ be a spectral datum, where $\\\\mu =(1^{\\\\alpha _1}, \\\\dots , s^{\\\\alpha _s})$ .\",\n \"As discussed in §3, it can be regarded as a morphism $a=(a_1,\\\\dots ,a_s): X \\\\rightarrow {\\\\rm Chow}^{\\\\alpha _1}(T^*_X /X) \\\\times _X \\\\dots \\\\times _X {\\\\rm Chow}^{\\\\alpha _s}(T^*_X /X),$ where $a_i$ maps the generic point of $X$ to ${\\\\rm Chow}^{\\\\alpha _i}_{(1^{\\\\alpha _i})}(T^*_X /X)$ .\",\n \"Let $U^{\\\\prime }$ be an open subset of $X$ such that $a_i(U^{\\\\prime })\\\\subseteq {\\\\rm Chow}^{\\\\alpha _i}_{(1^{\\\\alpha _i})}(T^*_X /X)$ for all $i$ .\",\n \"By the properness of the Hilbert-Chow morphism, one can find an open subset $U$ of $X$ with ${\\\\rm codim}(X\\\\backslash U, X)\\\\ge 2$ and there exists a unique morphism $\\\\tilde{a}: U \\\\rightarrow {\\\\rm Hilb}^{\\\\alpha _1}(T^*_X/X) \\\\times _X \\\\dots \\\\times _X {\\\\rm Hilb}^{\\\\alpha _s}(T^*_X/X)$ fitting into the commutative diagram U' [rr] [d] Hilb1(T*X/X) X ...X Hilbs(T*X/X) [d,\\\"Hilbert-Chow\\\"] U [urr, dotted] [rr,\\\"a|U\\\"] Chow1(T*X /X) X ...X Chows(T*X /X) and this morphism also induces an inclusion $U \\\\hookrightarrow T^*_X$ .\",\n \"Now pulling back via $\\\\tilde{a}$ the product of universal families ${Z}_{\\\\alpha _1}(T^*_X/X)\\\\times _X \\\\cdots \\\\times _X {Z}_{\\\\alpha _s}(T^*_X/X),$ we get a finite, flat cover of $U$ , which is a closed subscheme of $T^*_U$ .\",\n \"Then applying Serre's extension theorem [9], we get a finite, flat cover $\\\\widetilde{X}_a \\\\rightarrow X$ factoring through $T^*_X$ .\",\n \"This cover has degree $\\\\sum ^s_{i=1}\\\\alpha _i}[d]^{i_X}& {A}_{X^{\\\\prime }}@{^(->}[d]^{i_{X^{\\\\prime }}} \\\\\\\\{B}_X[r]^{\\\\chi ^*}& {B}_{X^{\\\\prime }},}$ where $\\\\mathfrak {i}_X$ and $\\\\mathfrak {i}_{X^{\\\\prime }}$ are closed immersions by [2] and $\\\\chi ^*: \\\\bigoplus ^n_{i=1} H^0(X, S^i T^*_X)\\\\rightarrow \\\\bigoplus ^n_{i=1} H^0(X^{\\\\prime }, S^i T^*_{X^{\\\\prime }})$ is an injection.\",\n \"From the diagram, we conclude that the natural morphism is a closed immersion.\",\n \"Lemma 5.2 Let $U$ be an open subset of a smooth variety $X$ with $\\\\text{codim}(X-U, X)\\\\ge 2$ .\",\n \"Then ${A}_X\\\\rightarrow {A}_U$ is an isomorphism.\",\n \"Recall for any $k$ -scheme $T$ , the $T$ -points of ${A}_X$ are ${A}_X(T)=\\\\big \\\\lbrace {X_T[r] [rd]_{id_{X_T}} & \\\\text{Chow}^n(T^*_{X_T/T}/{X_T})[d] \\\\\\\\& X_T,}\\\\big \\\\rbrace $ By Lemma REF and the Yoneda Lemma, we shall show that the natural map ${A}_X(T)\\\\rightarrow {A}_U(T)$ induced by $j$ is surjective for all such $T$ .\",\n \"First we consider the case $T^*_X=T^*_{X/k}$ is trivial, i.e.\",\n \"$T^*_X\\\\simeq X\\\\times V$ , where $\\\\dim V=r$ .\",\n \"Then it holds that $\\\\text{Chow}^n(T^*_{X_T/T}/{X_T})\\\\simeq \\\\text{Chow}^n(V)\\\\times X_T$ and an analogous isomorphism for $U_T$ .\",\n \"Given a spectral datum $a\\\\in {A}_U(T)$ , it is equivalent to a morphism $a: U_T\\\\rightarrow \\\\text{Chow}^n(V)$ , which for ease of notation will also be denoted by $a$ .\",\n \"By the key lemma [2], it gives rise to a morphism $U_T\\\\rightarrow V\\\\times S^2V\\\\times \\\\cdots \\\\times S^nV,$ which in turn amounts to $U_T\\\\rightarrow T^*_{X_T/T}\\\\times S^2T^*_{X_T/T}\\\\times \\\\cdots \\\\times S^nT^*_{X_T/T}$ by the triviality of the relative cotangent bundle.\",\n \"This is exactly an element in $\\\\bigoplus ^n_{i=1} H^0(U_T, S^i T^*_{U_T/T})$ Claim: for each $i>0$ , we have $H^0(X_T, S^i T^*_{X_T/T})\\\\simeq H^0(T, \\\\mathcal {O}_{T})\\\\otimes _k H^0(X, S^i T^*_{X/k}).$ Given this Claim and from Remark REF , we obtain that the natural map $H^0(X_T, S^i T^*_{X_T/T})\\\\rightarrow H^0(U_T, S^i T^*_{U_T/T})$ is an isomorphism for each $i>0$ .\",\n \"Thus we get $\\\\bar{a}\\\\in \\\\bigoplus ^n_{i=1} H^0(X_T, S^i T^*_{X_T/T}).$ We can view $\\\\bar{a}$ as a morphism $ \\\\bar{a}: X_T\\\\rightarrow V\\\\times S^2V\\\\times \\\\cdots \\\\times S^nV,$ which is equivalent to a section $\\\\bar{a}\\\\in {A}_X(T)$ .\",\n \"To prove the Claim above, consider the fibred product ${X_T [r]^{g}[d]_{\\\\varphi _T} & X[d]^{\\\\varphi } \\\\\\\\T[r]^{h}& {\\\\rm Spec}(k).\",\n \"}$ It holds that $ T^*_{X_T/T}\\\\simeq g^* T^*_{X/k}.$ Since $X$ is smooth, $T^*_{X/k}$ is locally free, hence $S^i T^*_{X_T/T}\\\\simeq g^*S^i( T^*_{X/k})$ for all $i>0$ .\",\n \"By flat base change, it therefore follows that $&&H^0(X_T, S^iT^*_{X_T/T})\\\\\\\\&\\\\simeq & H^0(X_T, g^*S^i( T^*_{X/k}))\\\\\\\\&\\\\simeq & H^0({\\\\rm Spec}{k}, h_*(\\\\varphi _T)_*g^*S^i( T^*_{X/k}))\\\\\\\\&\\\\simeq &H^0({\\\\rm Spec}{k}, h_*h^*\\\\varphi _*S^i(T^*_{X/k}))\\\\\\\\&\\\\simeq &H^0({\\\\rm Spec}{k}, h_*\\\\mathcal {O}_{T}\\\\otimes _k \\\\varphi _*S^i(T^*_{X/k}))\\\\\\\\&\\\\simeq & H^0(T, \\\\mathcal {O}_{T})\\\\otimes _k H^0(X, S^i(T^*_{X/k})).\",\n \"$ This proves the Claim.\",\n \"In general, we take an affine open covering $\\\\lbrace X_i\\\\rbrace $ of $X$ such that $T^*_{X/k}|_{X_i}$ is trivial for every $i$ .\",\n \"Put $U_i=U\\\\cap X_i$ .\",\n \"For all $i$ , it holds that $\\\\text{codim}(X_i-U_i, X_i)\\\\ge 2$ .\",\n \"By the preceding argument, ${A}_{X_i}(T)\\\\rightarrow {A}_{U_i}(T)$ is surjective for all $i$ .\",\n \"Since $j^*$ is actually injective, we can lift section locally and then glue the resulting sections together.\",\n \"This finishes the proof of this lemma.\",\n \"Theorem 5.3 Let $\\\\chi : X^{\\\\prime }\\\\rightarrow X$ be a birational proper morphism of smooth varieties.\",\n \"Then the natural morphism ${A}_X\\\\rightarrow {A}_{X^{\\\\prime }}$ induced by $\\\\chi $ is an isomorphism as $k$ -schemes.\",\n \"By the valuative criterion for properness, one can find an open subset $j: U\\\\rightarrow X$ with $\\\\text{codim}(X-U, X)\\\\ge 2$ and a morphism $\\\\nu : U\\\\rightarrow X^{\\\\prime }$ such that $\\\\chi \\\\circ \\\\nu =j$ .\",\n \"As a result, we have the commutative diagram of schemes ${{A}_{X^{\\\\prime }} @{^(->}[dr] & \\\\\\\\{A}_X[r]^{\\\\simeq } @{^(->}[u] & {A}_{U},}$ where the bottom map is an isomorphism by Lemma REF .\",\n \"Therefore ${A}_X\\\\rightarrow {A}_{X^{\\\\prime }}$ is an isomorphism.\",\n \"Example 5.4 Let $Y \\\\hookrightarrow X$ be a smooth subvariety.\",\n \"Denote by $\\\\pi : \\\\widetilde{X}={\\\\rm BL}_Y(X)\\\\rightarrow X$ the blow up of $X$ along $Y$ .\",\n \"Then one has the injection of vector bundles $0 \\\\rightarrow \\\\pi ^* T^*_X \\\\rightarrow T^*_{\\\\widetilde{X}}.$ Note that ${\\\\rm Chow}^n(T^*_X/X) \\\\times _X \\\\widetilde{X} \\\\cong {\\\\rm Chow}^n(\\\\pi ^* T^*_X/ \\\\widetilde{X})$ , and the injection above induces a morphism ${\\\\rm Chow}^n(\\\\pi ^* T^*_X/ \\\\widetilde{X}) \\\\rightarrow {\\\\rm Chow}^n(T^*_{\\\\widetilde{X}} / \\\\widetilde{X})$ .\",\n \"Now given a section $a_X: X \\\\rightarrow {\\\\rm Chow}_n(T^*_X /X)$ , we define the section $a_{\\\\widetilde{X}}: \\\\widetilde{X} \\\\rightarrow {\\\\rm Chow}_n(T^*_{\\\\widetilde{X}}/ \\\\widetilde{X})$ as the compositions of the following morphisms $a_{\\\\widetilde{X}}: \\\\widetilde{X} \\\\cong X \\\\times _X \\\\widetilde{X} \\\\xrightarrow{} {\\\\rm Chow}^n(T^*_X/X) \\\\times _X \\\\widetilde{X} \\\\cong {\\\\rm Chow}^n(\\\\pi ^* T^*_X/ \\\\widetilde{X}) \\\\rightarrow {\\\\rm Chow}^n(T^*_{\\\\widetilde{X}}/ \\\\widetilde{X}),$ which gives a map ${A}_X(k) \\\\rightarrow {A}_{\\\\widetilde{X}}(k)$ between the $k$ -points.\",\n \"By Theorem REF , the map is bijective.\",\n \"Corollary 5.5 For any projective ruled surface $X$ , ${A}_X={B}_X$ .\",\n \"In particular it is an affine space.\",\n \"By Theorem REF , we can assume $X$ is minimal, i.e.\",\n \"it does not contain a smooth rational curve $\\\\Gamma $ with $\\\\Gamma ^2=-1$ .\",\n \"If $X=\\\\mathbb {P}^{2}$ , then ${A}_X={B}_X$ is a point; otherwise $X\\\\simeq \\\\mathbb {P}_C(E)$ for some smooth curve $C$ and a vector bundle $E$ of rank two, so ${A}_X\\\\simeq {A}_C$ , and hence ${A}_X={B}_X$ , see [3] and the following remark.\",\n \"School of Mathematics, Sun Yat-Sen University W. 135 Xingang Rd., Guangzhou, Guangdong 510275, P.R.\",\n \"China E-mail address: songlei3@mail.sysu.edu.cn Department of Mathematics, South China University of Technology 381 Wushan Rd., Guangzhou, Guangdong 510641, P.R.\",\n \"China E-mail address: hsun71275@scut.edu.cn\"\n ]\n]"},"id":{"kind":"string","value":"2107.01679"}}},{"rowIdx":1576,"cells":{"text":{"kind":"list like","value":[["The $\\sigma_-$ Cohomology Analysis for Symmetric Higher-Spin Fields"],["Abstract In this paper, we present a complete proof of the so-called First On-Shell Theorem that determines dynamical content of the unfolded equations for free symmetric massless fields of arbitrary integer spin in any dimension and arbitrary integer or half-integer spin in four dimensions.","This is achieved by calculation of the respective $\\sigma_-$ cohomology both in the tensor language in Minkowski space of any dimension and in terms of spinors in $AdS_4$.","In the $d$-dimensional case $H^p(\\sigma_-)$ is computed for any $p$ and interpretation of $H^p(\\sigma_-)$ is given both for the original Fronsdal system and for the associated systems of higher form fields."],["Introduction","Higher-spin (HS) gauge theory is based on works of Fronsdal [1] and Fang and Fronsdal [2], where the action and equations of motion for massless gauge fields of any spin were originally obtained in flat four-dimensional Minkowski space.","Even earlier, important restrictions on low-energy HS vertices were obtained by Weinberg in [3], [4] and so-called no-go theorems restricting $S$ -matrix possessing too high symmetries in flat space-time were proven in [5], [6].","(For a review see [7].)","The no-go theorems implied the existence of the $s=2$ barrier suggesting that the construction of an interacting local HS theory in Minkowski space-time is impossible.","The proof of these theorems essentially uses the specific form of the algebra of isometries of Minkowski space.","The $s=2$ barrier in flat space can be overcome in the space-time with non-zero sectional curvature, for example, in the anti-de Sitter space [8].","In these spaces it becomes possible to formulate a consistent nonlinear theory of fields of all spins [9], [10].","The construction of a nonlinear HS theory is essentially based on the so-called unfolded approach [11], [12], which is a far-going generalization of the Cartan formulation of gravity ($s=2$ ) in terms of differential forms to fields of any spin $s>2$ .","Via introducing appropriate auxiliary variables, the unfolding procedure allows one to replace the system of partial differential equations of any order on a smooth manifold by a larger system of first-order equations on vector-valued differential forms.","One of the essential features of this approach, which is very useful for analysing symmetries of a given system, is that the variables in the equations are valued in one or another representation of the underlying symmetry algebra.","The dynamical content of the HS theory can be reconstructed from its unfolded formulation using the $\\sigma _-$ cohomology technique [13].","As is recalled below, the dynamical data of the theory are in one-to-one correspondence with the cohomology of certain linear nilpotent operator $\\sigma _-$ that can be read of the unfolded equations in question.","The statement that unfolded equations of free HS fields are equivalent to the Fronsdal equations was made in the original papers in the spinor [14] and tensor [15] formalisms.","In the tensor formulation of HS theory the idea of the proof was illustrated in [16], where however the analysis of the trace part of the Fronsdal equations was not completed, while general arguments for mixed-symmetry HS fields were given in [17].","In [18] the unfolded equations for massless fields were derived from the Fronsdal theory by the BRST methods.","To the best of our knowledge, no detailed analysis of the problem in the spinor formalism was available in the literature.","In this paper we present a complete proof of the so called First On-Shell Theorem by computation of the cohomology rings of $\\sigma _-$ for the physically important cases of the integer-spin symmetric fields both in flat space-time of any dimension and $AdS_d$ as well as for the fields of any integer and half-integer spin in ${AdS}_4$ .","The computation technique analogous to the Hodge theory for differential forms is performed in terms of so-called $\\sigma _-$ cohomology and provides a complete analysis of the dynamical content of the free unfolded equations for symmetric massless fields of any spin.","Giving a direct proof of the equivalence between the Fronsdal formulation of the HS gauge theory and its unfolded formulation this paper fills in some gaps in the literature also illustrating a general approach applicable to a broad class of unfolded systems.","In addition, in the tensor case we compute higher $\\sigma _-$ cohomology groups and interpret them in terms of higher Bianchi identities and more general dynamical systems.","In particular, we discuss the matching between the Bianchi identities in terms of one-form gauge fields and zero-form field strengths.","The rest of the paper is organized as follows.","In Section 2 we briefly recall different approaches to the description of HS massless fields.","Main idea of the $\\sigma _-$ cohomology approach is explained in Section 3.","Cohomology calculation method used in this paper is discussed in Section 4.","Section 5 contains derivation of $H^p(\\sigma _-)$ in Minkowski space of any dimension.","In particular, the cases of $\\mathrm {GL}(d)$ and $\\mathrm {O}(d)$ representations are analysed here.","In Section 6 calculation of the low-order cohomology groups in $AdS_4$ is performed.","Obtained results are discussed in Section 7.","Index conventions and normalisations of the tensor Young diagrams are presented in the Appendix A."],["Metric formulation","According to Fronsdal [1], a spin-$s$ massless symmetric field can be described in terms of two symmetric traceless tensors (for index conventions see Appendix A) $\\phi ^{a(s)} \\equiv \\phi ^{a_1..a_s}, \\quad \\phi ^{a(s-2)} \\equiv \\phi ^{a_1..a_{s-2}}, \\quad \\eta _{b_1 b_2}\\phi ^{b_1 b_2 a_3 .. a_s} \\equiv \\phi ^{a(s-2)b}{}_b = 0, \\quad \\phi ^{a(s-4)b}{}_b = 0\\,.$ These two fields can be combined into a single rank-$s$ totally symmetric tensor $\\varphi ^{a(s)} = \\phi ^{a(s)} + \\eta ^{aa}\\phi ^{a(s-2)}$ obeying the double-tracelessness condition $\\varphi ^{a(s-4)bc}{}_{bc} = 0\\,.$ The field equations in Fronsdal theory have the form $R^{a(s)}(\\varphi ) = \\square \\varphi ^{a(s)} - s \\partial ^a \\partial _k \\varphi ^{k a(s-1)} + \\frac{s(s-1)}{2} \\partial ^a \\partial ^a \\varphi ^{a(s-2)k}{}_k = 0\\,,$ where $\\partial _a = \\frac{\\partial }{\\partial x^a}$ .","The tensor $R^{a(s)}(\\varphi )$ is invariant under the gauge transformations with a rank-$(s-1)$ traceless gauge parameter $\\varepsilon (x)$ $\\delta \\varphi ^{a(s)} = \\partial ^a \\varepsilon ^{a(s-1)}, \\qquad \\varepsilon ^{a(s-3)k}{}_k = 0\\,.$ Fronsdal equation (REF ) is a generalization of the well-known equations of fields with spins $s = 0,1,2$ .","For the case of $s = 1$ the last term in the Fronsdal tensor disappears and Eq.","(REF ) reproduces Maxwell equations for the field $A^a$ .","Without the last two terms at $s = 0$ it gives Klein-Gordon equation for a massless scalar field.","The case of $s = 2$ reproduces the equations of linearized gravity [19].","Gauge transformation (REF ) gives the known gauge transformations of low-spin fields and its absence for a scalar field."],["Tensor formalism","The unified description of massless fields of arbitrary spin can be given in the so-called frame-like formalism that generalizes Cartan formulation of gravity, operating in terms of differential forms [14], [15], [20].","Frame-like formulation of the HS gauge theory in any dimension is given in terms of the one-form fields $ \\omega ^{a(s-1),b(t)} =dx^\\nu \\omega _\\nu ^{a(s-1),b(t)}$ valued in two-row Young diagrams corresponding to irreducible $\\mathfrak {o}(d-1,1)$ (i.e., traceless) modules [15], obeying conditions $& \\omega ^{a(s-1),a b(t-1)} = 0, \\\\& \\omega ^{a(s-3)k}{}_{k}{}^{,b(t)} = 0\\,.$ (For index conventions see Appendix A.)","By introducing auxiliary fields it is possible to put a system of partial differential equations into the first-order unfolded form [11], [12].","Generally, unfolded equations read as $d W^A = \\sum _{n = 1}^{\\infty } G^A_{B_1,..,B_n}W^{B_1}\\wedge ...\\wedge W^{B_n}\\,,\\qquad d:=dx^\\nu \\partial _\\nu \\,.$ Here $W^A$ is a set of differential forms over some manifold.","(Indices are treated formally and can take an infinite number of values.)","The coefficients $G^A_{B_1,..,B_n}$ satisfy the (anti)symmetry condition $G^A_{B_1,..,B_i,..,B_j,..,B_n} = (-1)^{|B_i||B_j|} G^A_{B_1,..,B_j,..,B_i,..,B_n}\\,,$ where $|B_i|$ denotes the form-degree of $W^{B_i}$ .","Also $G^A_{B_1,..,B_n}$ are restricted by the integrability conditions expressing that $d^2=0$ .","In the tensor language the unfolded HS equations in Minkowski space proposed in [15] read as $D_L\\omega ^{a(s-1),b(t)} + h_m \\wedge \\omega ^{a(s-1),b(t)m} = 0, \\quad t \\in \\lbrace 0,..,s-2\\rbrace ,$ $D_L\\omega ^{a(s-1),b(s-1)} = h_n \\wedge h_m \\wedge C^{a(s-1)n,b(s-1)m},$ where $h_n$ is a soldering form (vielbein, frame field, tetrad) and $D_L = d + \\varpi $ is the background Lorentz covariant derivative that satisfies relations $D_L h^a = 0, \\qquad D_L^2 = 0\\,.$ In the Cartesian coordinate system with $\\varpi =0$ the equations simplify to $d\\omega ^{a(s-1),b(t)} + h_m \\wedge \\omega ^{a(s-1),b(t)m} = 0, \\quad t \\in \\lbrace 0,..,s-2\\rbrace ,$ $d\\omega ^{a(s-1),b(s-1)} = h_n \\wedge h_m \\wedge C^{a(s-1)n,b(s-1)m}\\,,$ where $C$ satisfies the Lorentz irreducibility conditions $C^{a(n),a b(m-1)} = 0\\,, \\qquad C^{a(n-2)k}{}_k{}^{,b(m)} = 0\\,.$ The traceless tensor $C$ on the r.h.s.","of (REF ) is a generalized Weyl tensor.","There are also unfolded equations on $C$ and on additional auxiliary fields [10] (for reviews see [16], [21]).","This system constitutes an infinite chain of zero-form equations.","Zero-form sector, that contains equations on spin-zero and spin-one fields, will not be considered in this paper.","Equations (REF ) are invariant under the gauge transformations $\\delta \\omega ^{a(s-1),b(t)} = d\\varepsilon ^{a(s-1),b(t)} + h_m \\varepsilon ^{a(s-1),b(t)m}, \\quad t \\in \\lbrace 0,..,s-2\\rbrace $ and eq.","(REF ) is invariant under $&\\delta \\omega ^{a(s-1),b(s-1)} = d \\varepsilon ^{a(s-1),b(s-1)},\\\\&\\delta C^{a(s),b(s)} = 0,$ where $\\varepsilon $ are zero-forms valued in the appropriate two-row irreducible $\\mathfrak {o}(d-1,1)$ -modules obeying conditions analogous to (REF ).","The Fronsdal field is embedded into the frame-like one-form $e^{a(s-1)}\\equiv \\omega ^{a(s-1)}$ in the following manner.","Converting the form index into the fiber one using vielbein $h$ , $e^{a(s-1)|b} = e^{a(s-1)}_\\mu h^{\\mu b}\\,,$ the resulting tensor (REF ) can be decomposed into irreducible $\\mathfrak {o}(d-1,1)$ -modules.","In terms of traceless Young diagrams this decomposition is $\\begin{picture}(15,10)(0,7){\\put (05,10){\\line (1,0){10}}\\put (05,20){\\line (1,0){10}}\\put (05,10){\\line (0,1){10}}\\put (15,10){\\line (0,1){10}}}\\end{picture} \\,\\,\\,\\otimes _{so}\\,\\,\\,\\begin{picture}(65,18)(10,7){\\put (25,13){\\scriptsize {s-1}}\\put (05,20){\\line (1,0){60}}\\put (05,10){\\line (1,0){60}}\\put (05,10){\\line (0,1){10}}\\put (65,10){\\line (0,1){10}}}\\end{picture}\\,\\ \\cong \\,\\,\\,\\ \\begin{picture}(65,18)(10,7){\\put (30,13){\\scriptsize {s}}\\put (05,20){\\line (1,0){60}}\\put (05,10){\\line (1,0){60}}\\put (05,10){\\line (0,1){10}}\\put (65,10){\\line (0,1){10}}}\\end{picture}\\,\\ \\oplus \\,\\,\\,\\ \\begin{picture}(65,18)(10,7){\\put (25,13){\\scriptsize {s-2}}\\put (05,20){\\line (1,0){60}}\\put (05,10){\\line (1,0){60}}\\put (05,10){\\line (0,1){10}}\\put (65,10){\\line (0,1){10}}}\\end{picture}\\,\\,\\,\\ \\oplus \\,\\,\\,\\ \\begin{picture}(10,18)(0,7){\\put (05,00){\\line (1,0){10}}\\put (05,10){\\line (1,0){10}}\\put (05,0){\\line (0,1){10}}\\put (15,0){\\line (0,1){10}}}\\end{picture}$ ${s-1}$ .","The first two components give the Fronsdal field, while the third one is an excess of the components of the frame-like field in comparison with the Fronsdal field.","At the tensor level, this decomposition is represented as: $e^{a(s-1)|b} = \\psi _1^{a(s-1)b} + \\beta _1\\eta ^{aa}\\psi _2^{a(s-3)b} + \\beta _2\\eta ^{ab}\\psi _2^{a(s-2)} + \\psi _3^{a(s-1),b},$ where $\\psi _i$ are traceless and correspond to the $i$ -th diagram.","The relative coefficient $\\frac{\\beta _2}{\\beta _1}$ is fixed by the tracelessness condition with respect to indices $a$ .","This decomposition shows that the Fronsdal field identifies with the symmetric part of the frame-like field, since the contribution of the third diagram disappears upon symmetrization.","The resulting field $\\varphi ^{a(s)} := e^{a(s-1)|a}\\,$ is symmetric and double-traceless.","The extra term $\\psi _3^{a(s-1),b}$ is pure gauge.","Its contribution can be canceled by the gauge transformation $\\delta e^{a(s-1)|b} = \\varepsilon ^{a(s-1)|b} $ with suitable gauge parameter.","For detailed discussion of Fronsdal field embedding see [16], [20], [21].","It is not difficult to check [15], [20] (for reviews see [16], [21]) that the Fronsdal equations and gauge transformations follow from the unfolded system (REF ), (REF ).","A more complicated question is whether the Fronsdal fields and equations are the only ones that result from (REF ), (REF ).","The answer can be obtained via the $\\sigma _-$ cohomology technique [13]."],["Spinor language in $AdS_4$","The physically important case of the unfolded system for HS connection (REF ), (REF ) is that of $AdS_4$ space-time in which case the language of two-component spinors is most appropriate.","In this language instead of using Lorentz indices $a,b,...= 0,1,2,3$ , one uses two pairs of dotted and undotted spinor indices $\\alpha ,\\beta ,...$ and $\\dot{\\alpha },\\dot{\\beta },...$ taking values $\\lbrace 1,2\\rbrace $ .","The two languages are related via Pauli matrices.","The $AdS_4$ background geometry is described in terms of the Lorentz connection $\\varpi $ and frame field $h$ , that satisfy equations $\\begin{array}{l}d h^{\\alpha \\dot{\\beta }}+\\varpi ^{\\alpha }{ }_{\\gamma } \\wedge h^{\\gamma \\dot{\\beta }}+\\overline{\\varpi }^{\\dot{\\beta }}{}_{\\dot{\\gamma }} \\wedge h^{\\alpha \\dot{\\gamma }}=0\\,, \\\\d \\varpi ^{\\alpha \\beta }+\\varpi ^{\\alpha }{ }_{\\gamma } \\wedge \\varpi ^{\\gamma \\beta }=-\\lambda ^{2} h^{\\alpha }{}_{\\dot{\\gamma }} \\wedge h^{\\beta \\dot{\\gamma }}\\,, \\\\d \\overline{\\varpi }^{\\dot{\\alpha } \\dot{\\beta }}+\\overline{\\varpi }^{\\dot{\\alpha }}{}_{\\dot{\\gamma }} \\wedge \\overline{\\varpi }^{\\dot{\\gamma } \\dot{\\beta }}=-\\lambda ^{2} h_{\\gamma }{ }^{\\dot{\\alpha }} \\wedge h^{\\gamma \\dot{\\beta }}\\,,\\end{array}$ where $\\lambda ^2$ is proportional to the curvature of $AdS_4$ and we adopt the following rules $A_{\\alpha }=A^{\\beta } \\epsilon _{\\beta \\alpha }\\,, \\quad A^{\\alpha }=\\epsilon ^{\\alpha \\beta } A_{\\beta }\\,, \\quad \\epsilon _{\\alpha \\beta } \\epsilon ^{\\gamma \\beta }=\\epsilon _{\\alpha }{}^{\\gamma }=\\delta _{\\alpha }^{\\gamma }=-\\epsilon ^{\\gamma }{}_{\\alpha }\\,,$ where $\\epsilon _{\\alpha \\beta } = -\\epsilon _{\\beta \\alpha }\\,, \\quad \\epsilon _{12} = 1\\,.$ The spinor version of the unfolded system for one-form $\\omega $ reads as follows.","First, the HS curvatures in the spinor language are [14] $R^{\\alpha (n),\\dot{\\alpha }(m)}=D_L\\omega ^{\\alpha (n),\\dot{\\alpha }(m)} + \\lambda ^2(n h^\\alpha _{\\ \\dot{\\gamma }}\\wedge \\omega ^{\\alpha (n-1),\\dot{\\gamma }\\dot{\\alpha }(m)} + m h_\\gamma ^{\\ \\dot{\\alpha }}\\wedge \\omega ^{\\gamma \\alpha (n),\\dot{\\alpha }(m-1)})$ and $D_L=d+\\varpi +\\overline{\\varpi }$ is a Lorentz-covariant derivative with Cartan's spin-connection $(\\varpi \\oplus \\overline{\\varpi })$ $D_L \\omega ^{\\alpha (n),\\dot{\\alpha }(m)}=d\\omega ^{\\alpha (n),\\dot{\\alpha }(m)}+n\\varpi _\\alpha ^{\\ \\beta }\\wedge \\omega _{\\beta \\alpha (n-1),\\dot{\\alpha }(m)} + m\\overline{\\varpi }_{\\dot{\\alpha }}^{\\ \\dot{\\beta }}\\wedge \\omega _{\\alpha (n),\\dot{\\beta }\\dot{\\alpha }(m-1)}\\,.$ The $AdS_4$ deformation of the unfolded equations (REF ), (REF ) then takes the form [12] $R^{\\alpha (n),\\dot{\\alpha }(m)}=\\delta _{0,n}\\,h_{\\beta \\dot{\\alpha }}\\wedge h^{\\beta }_{\\ \\dot{\\alpha }}\\overline{C}^{\\dot{\\alpha }(m+2)} + \\delta _{0, m}\\, h_{\\alpha \\dot{\\beta }}\\wedge h_{\\alpha }^{\\ \\dot{\\beta }} C^{\\alpha (n+2)}.$ The main advantage of the two-component spinor notation is that it makes the representation theory of the Lorentz group very simple.","Namely, every Lorentz irreducible multispinor representing a traceless tensor is totally symmetric in its spinor indices.","Since the only Lorentz invariant objects are antisymmetric bispinors $\\epsilon _{\\alpha \\beta }$ and $\\overline{\\epsilon }_{\\dot{\\alpha }\\dot{\\beta }}$ irreducible multispinors $X^{\\alpha (n),\\dot{\\alpha }(m)}$ are necessarily symmetric with respect to the indices in the groups $\\alpha (n)$ and $\\dot{\\alpha }(m)$ separately.","Thus, working with the two-component spinor notation one can happily forget about painful calculations with the traces of Lorentz-tensors."],["The idea of $\\sigma _-$ cohomology analysis: example of integer spin massless fields","The l.h.s.","'s of unfolded HS equations and gauge transformations in $d$ -dimensional Minkowski space are [15], [16] $&R^{a(s-1),b(k)} = D_L \\omega ^{a(s-1),b(k)} + \\sigma _-(\\omega )^{{a(s-1),b(k)}},\\\\&\\delta \\omega ^{a(s-1),b(k)} = D_L \\varepsilon ^{a(s-1),b(k)} + \\sigma _-(\\varepsilon )^{{a(s-1),b(k)}},$ where $(\\sigma _-\\omega )^{{a(s-1),b(k)}} := h_c \\wedge \\omega ^{a(s-1),b(k)c}\\,,\\qquad (\\sigma _-\\varepsilon )^{{a(s-1),b(k)}} := h_c \\wedge \\varepsilon ^{a(s-1),b(k)c} .$ $R^{a(s-1),b(k)}$ is referred to as (linearized) HS curvature.","For simplicity we study the Minkowski case.","Since $\\sigma _-$ in $AdS_d$ is defined analogously, our analysis applies to that case as well.","Due to their definition, HS curvatures obey the Bianchi identities $D_L R^{a(s-1),b(k)} + \\sigma _-(R)^{a(s-1),b(k)} = 0\\,.$ The appearance of $\\sigma _-$ allows one to clarify the role of the fields $\\omega ^{a(s-1),b(k)}$ and gauge parameters $\\varepsilon ^{a(s-1),b(k)}$ .","Working with the zero-forms $\\varepsilon ^{a(s-1),b(k)}$ and one-forms $\\omega ^{a(s-1),b(k)}$ valued in two-row Young diagrams, we consider the space $V^p$ of $p$ -forms valued in two-row Young diagrams with any $p$ .","Defining $\\sigma _-$ to annihilate the forms with an empty second row, we find that $\\sigma _- V^p\\subset V^{p+1}$ and $\\sigma _-\\,\\sigma _-=0$ .","As originally proposed in [13], the $\\sigma _-$ cohomology $H(\\sigma _-)=\\ker ({\\sigma _-})/{\\mathrm {im\\,}(\\sigma _-)}$ classifies fields, their equations and gauge symmetries.","Indeed, those components of the fields $\\omega ^{a(s-1),b(t)}$ , that are not annihilated by $\\sigma _-$ , can be expressed via derivatives of the fields with lower $t$ by setting suitable components of the HS curvatures to zero.","Such fields are called auxiliary.","Conversely, those components of the fields $\\omega ^{a(s-1),b(t)}$ , that cannot be expressed in terms of derivatives of lower fields via zero-curvature conditions, are in $\\ker (\\sigma _-)$ .","By Stueckelberg fields we mean $\\sigma _-$ -exact fields (i.e.","fields of the form $\\sigma _-\\chi $ ) as they can be eliminated by an appropriate $\\sigma _-$ -exact term in the gauge transformation (REF ).","Fields that are not expressed via derivatives of other fields and are not Stueckelberg are called dynamical.","These describe the physical degrees of freedom of the theory.","Thus, the dynamical HS fields are associated with $H^1(\\sigma _-)$ .","The classification for the gauge parameters is analogous.","The parameters, that are not annihilated by $\\sigma _-$ , describe algebraic Stueckelberg shifts.","The leftover symmetries are described by the parameters in $\\ker (\\sigma _-)$ .","$\\sigma _-$ -exact parameters correspond to the so called gauge for gauge transformations.","Parameters, which are $\\sigma _-$ -closed and not $\\sigma _-$ -exact, are referred to as genuine differential gauge parameters.","Note that since in the HS example in question the gauge parameters are zero-forms there is no room for gauge for gauge symmetries in that case.","Let $V$ be a graded vector space, $\\mathcal {C}$ be an element of $\\Lambda ^p(\\mathcal {M}^d)\\otimes V$ over some smooth $d$ -dimensional manifold $\\mathcal {M}^d$ .","We demand the grading of $V$ to be bounded from below, that is $V$ is $\\mathbb {N}$ -graded.","Let $\\sigma _ {\\pm }$ be operators that act ”vertically”, i.e.","do not affect the space-time coordinates, and shift grading by $\\pm 1$ , $D_L$ be the Grassmann-odd operator that does not affect the grading and is allowed to act non-trivially on the space-time coordinates.","Consider the covariant constancy condition of a general form along with the zero-curvature condition $\\mathcal {D} \\mathcal {C} = (D_L + \\sigma _- + \\sigma _+) \\mathcal {C} = 0, \\quad \\mathcal {D}^2 = 0.$ Notice that eq.","(REF ) remains invariant under the gauge transformations $\\delta \\mathcal {C} = \\mathcal {D} \\varepsilon ,$ where $\\varepsilon \\in \\Lambda ^{p-1}(\\mathcal {M}^d) \\otimes V$ .","One can prove the following proposition [13] (see also [16], [22], [23], [24]): Theorem 3.1 The following is true: 1) Differential gauge symmetry parameters $\\varepsilon $ span $H^{p-1}(\\sigma _-)$ 2) Nontrivial dynamical fields $\\mathcal {C}$ span $H^p(\\sigma _-)$ 3) Physically distinguishable differential field equations on the nontrivial dynamical fields, contained in $\\mathcal {D} \\mathcal {C} = 0$ , span $H^{p+1}(\\sigma _-)$ Thus, taking into account that HS gauge fields are described by the one-forms $\\omega $ , to prove that the Fronsdal metric formulation is equivalent to the unfolded one, we have to calculate $H^{0}(\\sigma _-), H^{1}(\\sigma _-)$ and $H^{2}(\\sigma _-)$ .","More generally, higher cohomology $H^{k}(\\sigma _-)$ with $k>p+1$ describes Bianchi identities for dynamical equations at $k=p+2$ and Bianchi for Bianchi identites at $k>p+2$ [24].","Similarly, the lower cohomology $H^{k}(\\sigma _-)$ with $k4$ .","are now realized by the first-order differential operators $\\left(t_{\\mathfrak {gl}(d)}{}\\right)^a_b = Y^a \\partial _{Yb} + Z^a \\partial _{Zb} + \\theta ^a \\partial _{\\theta ^b}, \\quad \\left(t^{\\mathfrak {so}(d)}{}\\right)_{ab} = \\frac{1}{2}\\bigg (\\eta _{ac}t_{\\mathfrak {gl}}{}^c_b - \\eta _{bc}t_{\\mathfrak {gl}}{}^c_a\\bigg )\\,,$ where $\\theta ^c$ is a Grassmann-odd element of the exterior algebra associated with the frame one-form $e^a$ .","In these terms $\\sigma _-$ acts as $\\sigma _- \\omega = \\theta ^a {\\omega }{Z^a} = m \\theta ^c\\omega _{a(n),c b(m-1)}(x,\\theta )Y^{a(n)}Z^{b(m-1)}\\,.$ It differs from the definition of Section 3 by an additional numerical factor introduced for future convenience.","In the sequel we sometimes do not write variables $Y,Z,\\theta $ explicitly, that are always assumed to be present implicitly.","We adopt the convention that index $a$ is contracted with $Y$ , $b$ with $Z$ and $c_i$ with $\\theta ^{c_i}$ with $\\theta $ s ordered as $c_1,...,c_p$ .","The space $\\Lambda (\\mathcal {M}^d) \\otimes \\mathbb {R}[Y,Z]$ can be equipped with the scalar product $\\langle f,g\\rangle = \\frac{1}{\\pi ^{2n}}\\int _{\\mathbb {C}^d \\times \\mathbb {C}^d} d^{2d}Z\\, d^{2d}Y\\, d^d \\theta \\, d^d \\overline{\\theta }\\, f(Z,Y,\\theta )\\,\\overline{g(Z,Y,\\theta )}\\, e^{-|Z|^2-|Y|^2-\\overline{\\theta }\\theta },$ where $f,g \\in \\Lambda (\\mathcal {M}^d) \\otimes \\mathbb {R}[Y,Z]$ with complex $Y,Z,\\theta $ and Berezin integral over anticommuting variables.","(We work with the polynomials of complex variables with real coefficients).","The space $\\Lambda (\\mathcal {M}^d) \\otimes \\mathbb {R}[Y,Z]$ with the scalar product (REF ) is a Hilbert space in the Euclidean metric signature case used in this section.","This scalar product yields the following conjugation rules: $(Z^a)^* = \\partial _{Z}{}_{a}\\,,\\qquad (Y^a)^* = \\partial _{Y}{}_a\\,,\\qquad (\\theta ^a)^* = \\partial _{\\theta }{}_a\\,.$"],["$\\mathrm {GL}(d)$ example","To illustrate the idea of our construction let us first consider a simpler case where fields and gauge parameters take values in the irreps of $\\mathfrak {gl}(d)$ described by two-row Young diagrams (no tracelessness conditions are imposed).","Define the following operators, that form $\\mathfrak {gl}(2)$ $t_{1} = Y^{a}\\frac{\\partial }{\\partial Z^{a}}, \\quad t_{2} = Z^{a}\\frac{\\partial }{\\partial Y^{a}}, \\quad t_{0} = Y^{a}\\frac{\\partial }{\\partial Y^{a}} - Z^{a}\\frac{\\partial }{\\partial Z^{a}},$ ${t_{1}}{t_{2}} = t_{0}, \\quad {t_{0}}{t_{1}} = 2 t_{1}, \\quad {t_{0}}{t_{2}} = -2 t_{2},$ $h_{1} = Y^{a}\\frac{\\partial }{\\partial Y^{a}}, \\quad h_2= Z^{a}\\frac{\\partial }{\\partial Z^{a}}\\,.$ Namely, $t_i$ form $\\mathfrak {sl}(2)$ while $h_1+h_2$ is central.","In terms of these operators the space of $p$ -forms valued in two-row Young diagrams is identified as $\\ker (t_1)$ $V^p = \\lbrace F \\in \\Lambda ^p(\\mathcal {M}^d) \\otimes \\mathbb {R}[Y,Z]| t_1 F = 0 \\rbrace \\,.$ Here $\\Lambda ^p(\\mathcal {M}^d)$ is generated by the Grassmann variables $\\theta ^a$ .","Let us introduce auxiliary operators $Z_{\\theta } = Z^{a}\\frac{\\partial }{\\partial \\theta ^{a} }, \\quad Y_{\\theta } = Y^{a}\\frac{\\partial }{\\partial \\theta ^{a} }, \\quad D = \\theta ^a \\frac{\\partial }{\\partial \\theta ^{a} }, \\quad \\theta _Y = \\theta ^a \\frac{\\partial }{\\partial Y^{a} }, \\quad \\theta _Z = \\theta ^a \\frac{\\partial }{\\partial Z^{a} }\\,.$ Among the auxiliary operators $D$ plays the most important role as it gives differential form degree.","Now we should construct $\\sigma _+: \\sigma _+^2 = 0$ on $V^p$ and $\\Im (\\sigma _+) \\subset V^p$ .","Consider the following operator: $\\sigma _+ = f(t_{0})Z_{\\theta } + g(t_{0})Y_{\\theta }t_2\\,,$ where $f(t_{0}) = \\sum _{n=0}^{\\infty }f_{n}t_{0}^{n}$ and $g(t_{0}) = \\sum _{n=0}^{\\infty }g_{n}t_{0}^{n}$ .","Functions $f$ and $g$ have to be found from the conditions $\\sigma _+^2F=0\\,,\\qquad t_{1}\\sigma _+ F =0\\,, \\qquad \\forall F \\in V^p\\,.$ After some re-ordering of operators this yields two equations $& 0 = Y_{\\theta } \\bigg (f(t_{0} - 1)F + g(t_{0}-1)t_{0}F \\bigg ),\\\\& 0 = Z_{\\theta }Y_{\\theta } \\bigg (f(t_{0})g(t_{0}+1)t_{2}F - g(t_{0})f(t_{0}+1)t_{2}F - g(t_{0})g(t_{0}+1)t_{2}F \\bigg )\\,$ verified by $f(t_0) = -(t_0+1)g(t_0)$ giving $\\sigma _+ = -(t_0+1)g(t_0)Z_{\\theta }+g(t_0)Y_{\\theta }t_2\\,.$ The free coefficient $g(t_0)$ is determined from the conjugacy requirement: $(f,\\sigma _+ g) = -(\\theta _Z \\bigg (g(t_0)(t_0+1)+g(t_0)\\bigg )f, g) = (\\sigma _- f,g)\\,$ giving $g(t_0) = - \\frac{1}{t_0+2}$ and hence $\\sigma _+:=(\\sigma _-)^*= \\frac{t_0+1}{t_0+2}Z_{\\theta }-\\frac{1}{t_0+2}Y_{\\theta }t_2\\,.$ One can notice, that $\\sigma _+$ in (REF ) differs from what one would expect from the conjugation rules (REF ).","The reason is that in (REF ) we work with $C := \\Lambda ^p(\\mathcal {M}^d) \\otimes \\mathbb {R}[Y,Z]$ complex.","In the $\\mathrm {GL}(d)$ -case we deal with $\\big (\\Lambda ^p(\\mathcal {M}^d) \\otimes \\mathbb {R}[Y,Z]\\big ) \\cap \\ker (t_1)$ complex, therefore one should project on the highest weight vectors of the underlying $\\mathfrak {sl}(2)$ in the complex C. The same procedure applies to the $\\mathrm {O}(d)$ -case.","Though general formulae for extremal projectors are known for any simple Lie algebraWe are grateful to the referee for bringing this fact to our attention.","[26], [27], [28], [29](for reviews see [30], [31]), to keep the paper self-contained we derive the relevant projectors straightforwardly.","Knowing $\\sigma _+$ , it remains to construct the Laplace operator $\\Delta = \\lbrace \\sigma _-,\\sigma _+\\rbrace $ and find its zeros.","Elementary computation gives $\\Delta = \\frac{t_0}{t_0+1}(D+h_2-1) + \\frac{1}{t_0+1}Y_\\theta \\theta _Y - \\frac{1}{(t_0+1)(t_0+2)}t_2 \\theta _Z Y_\\theta \\,.$ Being built from the manifestly $\\mathfrak {gl}(d)$ -invariant operators, $\\Delta $ commutes with $\\mathfrak {gl}(d)$ hence being diagonal on its irreducible submodules.","Thus, it suffices to analyze zeros of $\\Delta $ on $\\mathfrak {gl}(d)$ irreducible components of the forms."],["$H^0(\\sigma _-)$","Any element of $V^0$ has the form $F = F_{a(n),b(m)}Y^{a(n)}Z^{b(m)}$ .","It is easy to see that $\\Delta F = h_2 F\\,.$ Therefore $H^0(\\sigma _-) = \\lbrace F = F_{a(n)}Y^{a(n)}| \\forall F_{a(n)}\\in \\mathbb {R}\\rbrace \\,.$"],["$H^p(\\sigma _-)$ , {{formula:0ba4be03-0dbe-402f-9169-c5a2a3811fb6}}","For $p>0$ , a general element of $V^p$ is $F = F_{a(n),b(m)|c_1,..,c_p}Y^{a(n)}Z^{b(m)}\\theta ^{c_1}..\\theta ^{c_p}$ .","Generally it forms a reducible $\\mathfrak {gl}(d)$ -module associated with the tensor product of two diagrams.","In terms of Young diagrams it decomposes into the following irreducible components: $\\begin{picture}(15,10)(0,7){\\put (05,0){\\line (1,0){10}}\\put (05,20){\\line (1,0){10}}\\put (05,0){\\line (0,1){20}}\\put (15,0){\\line (0,1){20}}\\put (8,10){\\scriptsize {p}}}\\end{picture} \\,\\,\\,\\otimes _{gl}\\,\\,\\,\\begin{picture}(10,18)(0,7){\\put (05,00){\\line (1,0){35}}\\put (05,10){\\line (1,0){35}}\\put (05,0){\\line (0,1){10}}\\put (40,0){\\line (0,1){10}}\\put (17,3){\\scriptsize {m}}}\\end{picture}\\begin{picture}(65,18)(10,7){\\put (35,13){\\scriptsize {n}}\\put (05,20){\\line (1,0){60}}\\put (05,10){\\line (1,0){60}}\\put (05,10){\\line (0,1){10}}\\put (65,10){\\line (0,1){10}}}\\end{picture}\\cong \\begin{picture}(10,18)(0,7){\\put (05,00){\\line (1,0){35}}\\put (05,10){\\line (1,0){35}}\\put (05,0){\\line (0,1){10}}\\put (40,0){\\line (0,1){10}}\\put (17,3){\\scriptsize {m}}\\put (05,-23){\\line (1,0){10}}\\put (05,-23){\\line (0,1){23}}\\put (15,-23){\\line (0,1){23}}\\put (7,-21){\\begin{turn}{90}\\scriptsize p-1\\end{turn}}}\\end{picture}\\begin{picture}(65,18)(10,7){\\put (26,13){\\scriptsize {n+1}}\\put (05,20){\\line (1,0){60}}\\put (05,10){\\line (1,0){60}}\\put (05,10){\\line (0,1){10}}\\put (65,10){\\line (0,1){10}}}\\end{picture}\\oplus \\begin{picture}(10,18)(0,7){\\put (05,00){\\line (1,0){35}}\\put (05,10){\\line (1,0){35}}\\put (05,0){\\line (0,1){10}}\\put (40,0){\\line (0,1){10}}\\put (12,3){\\scriptsize {m+1}}\\put (05,-23){\\line (1,0){10}}\\put (05,-23){\\line (0,1){23}}\\put (15,-23){\\line (0,1){23}}\\put (7,-21){\\begin{turn}{90}\\scriptsize p-1\\end{turn}}}\\end{picture}\\begin{picture}(65,18)(10,7){\\put (35,13){\\scriptsize {n}}\\put (05,20){\\line (1,0){60}}\\put (05,10){\\line (1,0){60}}\\put (05,10){\\line (0,1){10}}\\put (65,10){\\line (0,1){10}}}\\end{picture}\\oplus \\begin{picture}(10,18)(0,7){\\put (05,00){\\line (1,0){35}}\\put (05,10){\\line (1,0){35}}\\put (05,0){\\line (0,1){10}}\\put (40,0){\\line (0,1){10}}\\put (17,3){\\scriptsize {m}}\\put (05,-23){\\line (1,0){10}}\\put (05,-23){\\line (0,1){23}}\\put (15,-23){\\line (0,1){23}}\\put (7,-14){\\begin{turn}{90}\\scriptsize p\\end{turn}}}\\end{picture}\\begin{picture}(65,18)(10,7){\\put (35,13){\\scriptsize {n}}\\put (05,20){\\line (1,0){60}}\\put (05,10){\\line (1,0){60}}\\put (05,10){\\line (0,1){10}}\\put (65,10){\\line (0,1){10}}}\\end{picture}\\oplus \\begin{picture}(10,18)(0,7){\\put (05,00){\\line (1,0){35}}\\put (05,10){\\line (1,0){35}}\\put (05,0){\\line (0,1){10}}\\put (40,0){\\line (0,1){10}}\\put (12,3){\\scriptsize {m+1}}\\put (05,-23){\\line (1,0){10}}\\put (05,-23){\\line (0,1){23}}\\put (15,-23){\\line (0,1){23}}\\put (7,-21){\\begin{turn}{90}\\scriptsize p-2\\end{turn}}}\\end{picture}\\begin{picture}(65,18)(10,7){\\put (26,13){\\scriptsize {n+1}}\\put (05,20){\\line (1,0){60}}\\put (05,10){\\line (1,0){60}}\\put (05,10){\\line (0,1){10}}\\put (65,10){\\line (0,1){10}}}\\end{picture}\\,.\\\\$ At $p = 1$ the last diagram is absent.","The manifest decomposition of $F_{a(n),b(m)|c_1,c_2,..,c_p}$ into irreducible components is $F_{a(n),b(m)|c_1,c_2,..,c_p} = F_1{}_{a(n)c_1,b(m),c_2,..,c_p} + \\frac{m}{n-m+2} F_1{}_{a(n)b,b(m-1)c_1,c_2,..,c_p} + F_2{}_{a(n),b(m)c_1,c_2,..,c_p} + \\\\ + F_3{}_{a(n),b(m),c_1,..,c_p} + F_4{}_{a(n)c_1,b(m)c_2,c_3,..,c_p},$ where $F_i$ corresponds to the $i$ -th diagram.","There are no restrictions on $n,m,p$ in (REF ) except for $n \\ge m$ .","If for some $n,m$ tensor expression has a wrong Young shape, it is zero.","To simplify calculations we derive restrictions on $n,m,p$ for each diagram from the condition of being $\\sigma _-$ -closed.","For the second and fourth diagrams we find no restrictions, but for others we have $\\sigma _- \\bigg (\\text{1st diagram}\\bigg ) = -m\\bigg (1-\\frac{1}{n-m+2}\\bigg )F_1{}_{a(n)c_0,b(m-1)c_1,c_2,..,c_p} \\Rightarrow m = 0,$ $\\sigma _- F_3{}_{a(n),b(m),c_1,..,c_p} = m F_3{}_{a(n),b(m-1)c_0,c_1,..,c_p} \\Rightarrow m = 0\\,.$ Using this we obtain the action of $\\Delta $ on the rest diagrams: $\\Delta F_1{}_{a(n)c_1,c_2,..,c_p} = \\frac{1}{n+1}\\Delta \\theta _Y F_1(Y,Z,\\theta ) = \\frac{n(p-1)}{(n+1)^2}\\theta _Y F_1(Y,Z,\\theta ),$ $\\Delta F_2{}_{a(n),b(m)c_1,c_2,..,c_p} = \\frac{1}{m+1}\\Delta \\theta _Z F_2(Y,Z,\\theta ) = \\frac{m+p}{m+1}\\theta _Z F_2(Y,Z,\\theta ),$ $\\Delta F_4{}_{a(n)c_1,b(m)c_2,c_3,..,c_p} = \\frac{1}{(m+1)(n+1)}\\Delta \\theta _Y\\theta _Z F_4(Y,Z,\\theta )\\\\ = \\frac{(n-m)(p+m-1)}{(n-m+1)(m+1)(n+1)} \\theta _Y\\theta _Z F_4(Y,Z,\\theta )\\,.$ As a result, $H^1(\\sigma _-) = \\lbrace \\phi = F_{a(n)c}\\theta ^c Y^{a(n)}| h_2 F = 0, F\\in V^1\\rbrace \\,,$ $H^p(\\sigma _-) = \\lbrace W = \\theta _Y\\theta _Z C(Y,Z,\\theta )| t_0 C = 0, C \\in V^{p-2} \\,,\\quad p>1\\rbrace ,$ $C =C{}_{a(n),b(n),c_1,..,c_{p-2}}Y^{a(n)}Z^{b(n)}\\theta ^{c_1}..\\theta ^{c_{p-2}} \\in V^{p-2}\\,.$ The dynamical interpretation of the obtained results is as follows.","The system has one symmetric gauge field with gauge transformation described by a symmetric parameter.","The second cohomology group $H^2(\\sigma _-)$ is spanned by a single tensor corresponding to the generalized (traceful) Weyl tensor.","If the latter is set to zero, the system becomes topological with the zero-curvature field equations.","Otherwise the unfolded equations encode a set of constraints expressing all fields and Weyl tensor via derivatives of the physical fields.","Proceeding further with the equations on the Weyl tensor and its descendants results in an infinite set of constraints with no differential equations on the physical field.","Such off-shell unfolded equations were considered in [32].","The off-shell systems are for interest in many contexts such as, e.g., construction of actions and quantization [33], [34].","The lower cohomology groups (REF ), (REF ) and (REF ) match with those obtained, e.g., in [16]."],["Irreducibility conditions","The $\\mathrm {O}(d)$ case is in many respects analogous to that of $\\mathrm {GL}(d)$ .","The difference is due to the tracelessness condition (REF ).","The algebra of the operators encoding irreducibility conditions is extended since the metric allows the new types of contractions between $\\theta , Y, Z$ and their derivatives.","From the representation theory perspective new terms associated with traces appear in the diagram decomposition of the form coefficients, affecting the cohomology analysis.","The following operators form the algebra $\\mathfrak {sp}(4)$ : $&t_{1} = Y^{a}\\frac{\\partial }{\\partial Z^{a}}, \\quad t_{2} = Z^{a}\\frac{\\partial }{\\partial Y^{a}}, \\quad h_{1} = Y^{a}\\frac{\\partial }{\\partial Y^{a}}, \\quad h_{2}= Z^{a}\\frac{\\partial }{\\partial Z^{a}},\\\\&t_{0} = Y^{a}\\frac{\\partial }{\\partial Y^{a}} - Z^{a}\\frac{\\partial }{\\partial Z^{a}},\\\\&f_{1} = \\partial _{Y}^a\\partial _{Y a}, \\quad f_{2} = \\partial _{Z}^a\\partial _{Z a}, \\quad f_{3} = \\partial _{Y}^a\\partial _{Z a}, \\\\&e_{1} = Y^a Y_a, \\quad e_{2} = Z^a Z_a, \\quad e_{3} = Y^a Z_a\\,.$ Evidently, these operators commute with the $\\mathfrak {so}(d)$ generators (REF ).","$\\mathfrak {sp}(4)$ and $\\mathfrak {so}(d)$ form a Howe-dual pair [35].","Young condition (REF ) and tracelessness condition (REF ) impose highest weight conditions on a $\\mathfrak {sp}(4)$ -module.","In addition, we introduce the following $\\mathrm {O}(d)$ invariant operators: $&Z_{\\theta } = Z^a \\partial _{\\theta a}, \\quad Y_{\\theta } = Y^a \\partial _{\\theta a},\\\\&\\partial _{\\theta Z} = \\partial _{\\theta a}\\partial _{Z}^a, \\quad \\partial _{\\theta Y} = \\partial _{\\theta a}\\partial _{Y}^a,\\\\&\\theta _Z = \\theta ^a \\partial _{Z a}, \\quad \\theta _Y = \\theta ^a \\partial _{Y a}\\,,$ which, along with $D$ (REF ) counting differential form degree, extend $\\mathfrak {sp}(4)$ to $\\mathfrak {osp}(2|4)$ .","The simplest way to see this is to let index $a$ take a single value, treating the operators $Z,Y,\\partial _Z, \\partial _Y, \\theta , \\partial _\\theta $ as creation and annihilation operators, and apply the oscillator realization of $\\mathfrak {osp}(2|4)$ .","In the problem in question, the form space is $V^p = \\lbrace F \\in \\Lambda ^p(\\mathcal {M}^d) \\otimes \\mathbb {R}[Y,Z]| t_1 F = 0, f_1 F = 0 \\rbrace \\,.$ Note that these restrictions imply the tracelessness over indices $(Y,Z)$ and $(Z,Z)$ as a consequence of the form of commutators of $t_1$ with $f_{1,2}$ ."],["$\\sigma _+$","Let us look for $\\sigma _+ = \\sigma _-^* $ in the form $\\sigma _+ = g_1(h_1,h_2)Z_{\\theta } + g_2(h_1,h_2)t_2Y_{\\theta } + g_3(h_1,h_2)e_3 \\partial _{\\theta Y} + g_4(h_1,h_2)e_1t_2 \\partial _{\\theta Y} + g_5(h_1,h_2)e_2 \\partial _{\\theta Z}+\\\\+g_6(h_1,h_2)e_3 t_2\\partial _{\\theta Z} + g_7(h_1,h_2)e_1 t_2^2 \\partial _{\\theta Z}\\,.$ The condition $\\Im (\\sigma _+) \\subset V^p$ gives $0 = f_1 \\sigma _+ F = \\bigg (2 g_2(h_1+2,h_2)+2g_3(h_1+2,h_2)+2g_4(h_1+2,h_2)(d + 2h_1)\\bigg )t_2 \\partial _{\\theta Y} F + \\\\+ \\bigg (2g_6(h_1+2,h_2)+2g_7(h_1+2,h_2)(d+2h_1)\\bigg )t_2^2\\partial _{\\theta Z}F,$ $0 = t_1 \\sigma _+ F = \\bigg (g_1(h_1-1,h_2+1)+g_2(h_1-1,h_2+1)t_0\\bigg )Y_\\theta F + \\bigg (-g_3(-1,1)+2g_5(-1,1)+ \\\\ + g_6(-1,1)t_0 \\bigg )e_3 \\partial _{\\theta Z} + \\bigg (g_3(-1,1)+g_4(-1,1)(t_0-2) \\bigg )e_1 \\partial _{\\theta Y} + \\bigg (-g_4(-1,1)+g_6(-1,1)+ \\\\ + 2g_7(-1,1)(t_0-1)\\bigg )e_1 t_2 \\partial _{\\theta Z}\\,.$ This imposes the following six equations on seven coefficients $g_1(h_1,h_2)+g_2(h_1,h_2)(t_0+2) = 0, \\\\-g_3(h_1,h_2)+2g_5(h_1,h_2)+g_6(h_1,h_2)(t_0+2) = 0, \\\\g_3(h_1,h_2)+g_4(h_1,h_2) t_0 = 0, \\\\-g_4(h_1,h_2)+g_6(h_1,h_2)+2g_7(h_1,h_2)(t_0+1) = 0, \\\\g_2(h_1,h_2)+g_3(h_1,h_2)+g_4(h_1,h_2)(d+2h_1-4) = 0, \\\\g_6(h_1,h_2)+g_7(h_1,h_2)(d+2h_1-4) = 0\\,.$ Choosing $g_7(h_1,h_2)$ as a free parameter, we obtain $g_1(h_1,h_2) = -(t_0+2)(d-4+h_1+h_2)(d-6+2h_2)g_7(h_1,h_2),\\\\g_2(h_1,h_2) = (d-4+h_1+h_2)(d-6+2h_2)g_7(h_1,h_2),\\\\g_3(h_1,h_2) = t_0(d-6+2h_2)g_7(h_1,h_2),\\\\g_4(h_1,h_2) = -(d-6+2h_2)g_7(h_1,h_2),\\\\g_5(h_1,h_2) = (t_0+1)(d-4+h_1+h_2)g_7(h_1,h_2),\\\\g_6(h_1,h_2) = -(d-4+2h_1)g_7(h_1,h_2)\\,.$ Now using the conjugation rules (REF ) and highest weight conditions (REF ) we get $(F_1,\\sigma _+ F_2) = \\Big ( -g_7(h_1,h_2)(t_0+2)(d-4+h_1+h_2)(d-6+2h_2) \\Big ) (\\sigma _{-}F_1,F_2)\\,.$ The condition $\\sigma _+ = \\sigma ^*_-$ demands $g_7(h_1,h_2) = - \\frac{1}{(t_0+2)(d-4+h_1+h_2)(d-6+2h_2)}$ giving $\\sigma _+ = Z_{\\theta } - \\frac{1}{t_0+2} t_2 Y_{\\theta } - \\frac{t_0}{(t_0+2)(d-4+h_1+h_2)} e_3 \\partial _{\\theta Y} + \\frac{1}{(t_0+2)(d-4+h_1+h_2)}e_1t_2 \\partial _{\\theta Y} - \\\\-\\frac{t_0+1}{(t_0+2)(d-6+2h_2)} e_2 \\partial _{\\theta Z} +\\frac{d-4+2h_1}{(t_0+2)(d-4+h_1+h_2)(d-6+2h_2)} e_3 t_2\\partial _{\\theta Z} - \\\\-\\frac{1}{(t_0+2)(d-4+h_1+h_2)(d-6+2h_2)} e_1 t_2^2 \\partial _{\\theta Z}\\,.$ This yields operator $\\sigma _+$ such that $\\sigma _+^2 = 0$ on $V$ , $\\Im (\\sigma _+) \\subset V$ and $\\sigma _-^* = \\sigma _+$ .","To calculate the Laplace operator $\\Delta = \\sigma _- \\sigma _+ + \\sigma _+ \\sigma _-$ on $V^p$ we obtain straightforwardly that $\\sigma _- \\sigma _+ = \\theta _Z Z_\\theta - \\frac{1}{t_0+1}t_2 \\theta _Z Y_\\theta - \\frac{1}{t_0+1}\\theta _Y Y_\\theta - \\frac{t_0-1}{(t_0+1)(d-3+h_1+h_2)}\\bigg (e_3 \\theta _Z \\partial _{\\theta Y} + \\theta ^a Y_a \\partial _{\\theta Y}\\bigg ) \\\\+ \\frac{1}{(t_0+1)(d-3+h_1+h_2)}\\bigg (e_1 t_2 \\theta _Z \\partial _{\\theta Y} + e_1 \\theta _Y \\partial _{\\theta Y} \\bigg ) - \\frac{t_0}{(t_0+1)(d-4+2h_2)}\\bigg (e_2 \\theta _Z \\partial _{\\theta Z} + 2 \\theta ^a Z_a \\partial _{\\theta Z} \\bigg )\\\\+ \\frac{d-4+2h_1}{(t_0+1)(d-3+h_1+h_2)(d-4+2h_2)}\\bigg (e_3 t_2 \\theta _Z \\partial _{\\theta Z} + e_3 \\theta _Y \\partial _{\\theta Z} + t_2 \\theta ^a Y_a \\partial _{\\theta Z} - \\theta ^a Z_a \\partial _{\\theta Z}\\bigg )\\\\- \\frac{1}{(t_0+1)(d-3+h_1+h_2)(d-4+2h_2)}\\bigg (e_1 t_2^2\\theta _Z \\partial _{\\theta Z} + 2e_1 t_2 \\theta _Y \\partial _{\\theta Z} \\bigg )\\,.$ $\\sigma _+ \\sigma _-= Z_\\theta \\theta _Z + \\frac{1}{t_0+2}t_2 \\theta _Z Y_\\theta + \\frac{t_0}{(t_0+2)(d-4+h_1+h_2)}e_3 \\theta _Z \\partial _{\\theta Y}- \\frac{1}{(t_0+2)(d-4+h_1+h_2)}\\times \\\\ \\times e_1 t_2 \\theta _Z \\partial _{\\theta Y}+ \\frac{t_0+1}{(t_0+2)(d-6+2h_2)}e_2 \\theta _Z \\partial _{\\theta Z}- \\frac{d-4+2h_1}{(t_0+2)(d-4+h_1+h_2)(d-6+2h_2)}e_3 t_2 \\theta _Z \\partial _{\\theta Z}\\\\+ \\frac{1}{(t_0+2)(d-4+h_1+h_2)(d-6+2h_2)}e_1 t_2^2\\theta _Z \\partial _{\\theta Z}\\,.$ Since, by construction, both $\\sigma _- \\sigma _+$ and $\\sigma _+ \\sigma _-$ and hence $\\Delta $ are $\\mathrm {O}(d)$ invariant, $\\Delta $ is diagonal on irreducible $\\mathrm {O}(d)$ -modules and, to compute $H(\\sigma _-)$ , it suffices to find its zeros on the irreducible components."],["$H^0(\\sigma _-)$","In the sector of zero-forms, all terms that contain $\\frac{\\partial }{\\partial \\theta ^c}$ trivialize.","Hence, $\\Delta F = h_2 F = m F \\,$ and $H^0(\\sigma _-) = \\lbrace F = F_{a(n)}Y^{a(n)}| \\forall F_{a(n)}\\in \\mathbb {R}\\rbrace \\,.$ Comparing the resulting differential gauge parameters with (REF ), we find that, as anticipated, differential gauge symmetries in the unfolded formulation coincide with those of the Fronsdal theory."],["$H^p(\\sigma _-)$ , {{formula:b937a896-0a47-49ab-98e2-b9277cda9f42}}","The main difference between $\\mathfrak {o}(d)$ - and $\\mathfrak {gl}(d)$ - cases is due to the traceful terms in the decomposition of the $p$ -forms into the irreducible parts depicted as $\\begin{picture}(15,10)(0,7){\\put (05,0){\\line (1,0){10}}\\put (05,20){\\line (1,0){10}}\\put (05,0){\\line (0,1){20}}\\put (15,0){\\line (0,1){20}}\\put (8,10){\\scriptsize {p}}}\\end{picture} \\,\\,\\,\\otimes _{so}\\,\\,\\,\\begin{picture}(10,18)(0,7){\\put (05,00){\\line (1,0){35}}\\put (05,10){\\line (1,0){35}}\\put (05,0){\\line (0,1){10}}\\put (40,0){\\line (0,1){10}}\\put (17,3){\\scriptsize {m}}}\\end{picture}\\begin{picture}(65,18)(10,7){\\put (35,13){\\scriptsize {n}}\\put (05,20){\\line (1,0){60}}\\put (05,10){\\line (1,0){60}}\\put (05,10){\\line (0,1){10}}\\put (65,10){\\line (0,1){10}}}\\end{picture}\\cong \\begin{picture}(10,18)(0,7){\\put (05,00){\\line (1,0){35}}\\put (05,10){\\line (1,0){35}}\\put (05,0){\\line (0,1){10}}\\put (40,0){\\line (0,1){10}}\\put (17,3){\\scriptsize {m}}\\put (05,-25){\\line (1,0){10}}\\put (05,-25){\\line (0,1){25}}\\put (15,-25){\\line (0,1){25}}\\put (7,-15){\\begin{turn}{90}\\scriptsize p\\end{turn}}}\\end{picture}\\begin{picture}(65,18)(10,7){\\put (35,13){\\scriptsize {n}}\\put (05,20){\\line (1,0){60}}\\put (05,10){\\line (1,0){60}}\\put (05,10){\\line (0,1){10}}\\put (65,10){\\line (0,1){10}}}\\end{picture}\\oplus \\begin{picture}(10,18)(0,7){\\put (05,00){\\line (1,0){35}}\\put (05,10){\\line (1,0){35}}\\put (05,0){\\line (0,1){10}}\\put (40,0){\\line (0,1){10}}\\put (12,3){\\scriptsize {m+1}}\\put (05,-25){\\line (1,0){10}}\\put (05,-25){\\line (0,1){25}}\\put (15,-25){\\line (0,1){25}}\\put (7,-21){\\begin{turn}{90}\\scriptsize p-1\\end{turn}}}\\end{picture}\\begin{picture}(65,18)(10,7){\\put (35,13){\\scriptsize {n}}\\put (05,20){\\line (1,0){60}}\\put (05,10){\\line (1,0){60}}\\put (05,10){\\line (0,1){10}}\\put (65,10){\\line (0,1){10}}}\\end{picture}\\oplus \\begin{picture}(10,18)(0,7){\\put (05,00){\\line (1,0){35}}\\put (05,10){\\line (1,0){35}}\\put (05,0){\\line (0,1){10}}\\put (40,0){\\line (0,1){10}}\\put (17,3){\\scriptsize {m}}\\put (05,-25){\\line (1,0){10}}\\put (05,-25){\\line (0,1){25}}\\put (15,-25){\\line (0,1){25}}\\put (7,-21){\\begin{turn}{90}\\scriptsize p-1\\end{turn}}}\\end{picture}$ ${n+1}$ $\\oplus \\begin{picture}(10,18)(0,7){\\put (05,00){\\line (1,0){35}}\\put (05,10){\\line (1,0){35}}\\put (05,0){\\line (0,1){10}}\\put (40,0){\\line (0,1){10}}\\put (12,3){\\scriptsize {m+1}}\\put (05,-25){\\line (1,0){10}}\\put (05,-25){\\line (0,1){25}}\\put (15,-25){\\line (0,1){25}}\\put (7,-21){\\begin{turn}{90}\\scriptsize p-2\\end{turn}}}\\end{picture}\\begin{picture}(65,18)(10,7){\\put (26,13){\\scriptsize {n+1}}\\put (05,20){\\line (1,0){60}}\\put (05,10){\\line (1,0){60}}\\put (05,10){\\line (0,1){10}}\\put (65,10){\\line (0,1){10}}}\\end{picture}\\oplus \\begin{picture}(10,18)(0,7){\\put (05,00){\\line (1,0){35}}\\put (05,10){\\line (1,0){35}}\\put (05,0){\\line (0,1){10}}\\put (40,0){\\line (0,1){10}}\\put (17,3){\\scriptsize {m}}\\put (05,-25){\\line (1,0){10}}\\put (05,-25){\\line (0,1){25}}\\put (15,-25){\\line (0,1){25}}\\put (7,-21){\\begin{turn}{90}\\scriptsize p-1\\end{turn}}}\\end{picture}\\begin{picture}(65,18)(10,7){\\put (26,13){\\scriptsize {n-1}}\\put (05,20){\\line (1,0){60}}\\put (05,10){\\line (1,0){60}}\\put (05,10){\\line (0,1){10}}\\put (65,10){\\line (0,1){10}}}\\end{picture}\\oplus \\begin{picture}(10,18)(0,7){\\put (05,00){\\line (1,0){35}}\\put (05,10){\\line (1,0){35}}\\put (05,0){\\line (0,1){10}}\\put (40,0){\\line (0,1){10}}\\put (12,3){\\scriptsize {m-1}}\\put (05,-25){\\line (1,0){10}}\\put (05,-25){\\line (0,1){25}}\\put (15,-25){\\line (0,1){25}}\\put (7,-21){\\begin{turn}{90}\\scriptsize p-1\\end{turn}}}\\end{picture}\\begin{picture}(65,18)(10,7){\\put (35,13){\\scriptsize {n}}\\put (05,20){\\line (1,0){60}}\\put (05,10){\\line (1,0){60}}\\put (05,10){\\line (0,1){10}}\\put (65,10){\\line (0,1){10}}}\\end{picture}\\oplus \\begin{picture}(10,18)(0,7){\\put (05,00){\\line (1,0){35}}\\put (05,10){\\line (1,0){35}}\\put (05,0){\\line (0,1){10}}\\put (40,0){\\line (0,1){10}}\\put (12,3){\\scriptsize {m-1}}\\put (05,-25){\\line (1,0){10}}}\\put (05,-25){\\line (0,1){25}}\\put (15,-25){\\line (0,1){25}}\\put (7,-21){\\begin{turn}{90}\\scriptsize p-2\\end{turn}}\\end{picture}$ ${n+1}$ ${m+1}$90$p-2$ ${n-1}$ $\\oplus \\begin{picture}(10,18)(0,7){\\put (05,00){\\line (1,0){35}}\\put (05,10){\\line (1,0){35}}\\put (05,0){\\line (0,1){10}}\\put (40,0){\\line (0,1){10}}\\put (12,3){\\scriptsize {m-1}}\\put (05,-25){\\line (1,0){10}}\\put (05,-25){\\line (0,1){25}}\\put (15,-25){\\line (0,1){25}}\\put (7,-21){\\begin{turn}{90}\\scriptsize p-2\\end{turn}}}\\end{picture}\\begin{picture}(65,18)(10,7){\\put (26,13){\\scriptsize {n-1}}\\put (05,20){\\line (1,0){60}}\\put (05,10){\\line (1,0){60}}\\put (05,10){\\line (0,1){10}}\\put (65,10){\\line (0,1){10}}}\\end{picture}\\oplus \\begin{picture}(10,18)(0,7){\\put (05,00){\\line (1,0){35}}\\put (05,10){\\line (1,0){35}}\\put (05,0){\\line (0,1){10}}\\put (40,0){\\line (0,1){10}}\\put (17,3){\\scriptsize {m}}\\put (05,-25){\\line (1,0){10}}\\put (05,-25){\\line (0,1){25}}\\put (15,-25){\\line (0,1){25}}\\put (7,-21){\\begin{turn}{90}\\scriptsize p-2\\end{turn}}}\\end{picture}\\begin{picture}(65,18)(10,7){\\put (35,13){\\scriptsize {n}}\\put (05,20){\\line (1,0){60}}\\put (05,10){\\line (1,0){60}}\\put (05,10){\\line (0,1){10}}\\put (65,10){\\line (0,1){10}}}\\end{picture}\\oplus \\begin{picture}(10,18)(0,7){\\put (05,00){\\line (1,0){35}}\\put (05,10){\\line (1,0){35}}\\put (05,0){\\line (0,1){10}}\\put (40,0){\\line (0,1){10}}\\put (17,3){\\scriptsize {m}}\\put (05,-25){\\line (1,0){10}}\\put (05,-25){\\line (0,1){25}}\\put (15,-25){\\line (0,1){25}}\\put (7,-21){\\begin{turn}{90}\\scriptsize p-2\\end{turn}}}\\end{picture}\\begin{picture}(65,18)(10,7){\\put (35,13){\\scriptsize {n}}\\put (05,20){\\line (1,0){60}}\\put (05,10){\\line (1,0){60}}\\put (05,10){\\line (0,1){10}}\\put (65,10){\\line (0,1){10}}}\\end{picture}\\oplus \\begin{picture}(10,18)(0,7){\\put (05,00){\\line (1,0){35}}\\put (05,10){\\line (1,0){35}}\\put (05,0){\\line (0,1){10}}\\put (40,0){\\line (0,1){10}}\\put (17,3){\\scriptsize {m}}\\put (05,-25){\\line (1,0){10}}\\put (05,-25){\\line (0,1){25}}\\put (15,-25){\\line (0,1){25}}\\put (7,-21){\\begin{turn}{90}\\scriptsize p-3\\end{turn}}}\\end{picture}\\begin{picture}(65,18)(10,7){\\put (26,13){\\scriptsize {n+1}}\\put (05,20){\\line (1,0){60}}\\put (05,10){\\line (1,0){60}}\\put (05,10){\\line (0,1){10}}\\put (65,10){\\line (0,1){10}}}\\end{picture}\\oplus $ $\\oplus \\begin{picture}(10,18)(0,7){\\put (05,00){\\line (1,0){35}}\\put (05,10){\\line (1,0){35}}\\put (05,0){\\line (0,1){10}}\\put (40,0){\\line (0,1){10}}\\put (12,3){\\scriptsize {m+1}}\\put (05,-25){\\line (1,0){10}}\\put (05,-25){\\line (0,1){25}}\\put (15,-25){\\line (0,1){25}}\\put (7,-21){\\begin{turn}{90}\\scriptsize p-3\\end{turn}}}\\end{picture}\\begin{picture}(65,18)(10,7){\\put (35,13){\\scriptsize {n}}\\put (05,20){\\line (1,0){60}}\\put (05,10){\\line (1,0){60}}\\put (05,10){\\line (0,1){10}}\\put (65,10){\\line (0,1){10}}}\\end{picture}\\oplus \\begin{picture}(10,18)(0,7){\\put (05,00){\\line (1,0){35}}\\put (05,10){\\line (1,0){35}}\\put (05,0){\\line (0,1){10}}\\put (40,0){\\line (0,1){10}}\\put (17,3){\\scriptsize {m}}\\put (05,-25){\\line (1,0){10}}\\put (05,-25){\\line (0,1){25}}\\put (15,-25){\\line (0,1){25}}\\put (7,-21){\\begin{turn}{90}\\scriptsize p-3\\end{turn}}}\\end{picture}\\begin{picture}(65,18)(10,7){\\put (26,13){\\scriptsize {n-1}}\\put (05,20){\\line (1,0){60}}\\put (05,10){\\line (1,0){60}}\\put (05,10){\\line (0,1){10}}\\put (65,10){\\line (0,1){10}}}\\end{picture}\\oplus \\begin{picture}(10,18)(0,7){\\put (05,00){\\line (1,0){35}}\\put (05,10){\\line (1,0){35}}\\put (05,0){\\line (0,1){10}}\\put (40,0){\\line (0,1){10}}\\put (12,3){\\scriptsize {m-1}}\\put (05,-25){\\line (1,0){10}}\\put (05,-25){\\line (0,1){25}}\\put (15,-25){\\line (0,1){25}}\\put (7,-21){\\begin{turn}{90}\\scriptsize p-3\\end{turn}}}\\end{picture}\\begin{picture}(65,18)(10,7){\\put (35,13){\\scriptsize {n}}\\put (05,20){\\line (1,0){60}}\\put (05,10){\\line (1,0){60}}\\put (05,10){\\line (0,1){10}}\\put (65,10){\\line (0,1){10}}}\\end{picture}\\oplus \\begin{picture}(10,18)(0,7){\\put (05,00){\\line (1,0){35}}\\put (05,10){\\line (1,0){35}}\\put (05,0){\\line (0,1){10}}\\put (40,0){\\line (0,1){10}}\\put (17,3){\\scriptsize {m}}\\put (05,-25){\\line (1,0){10}}\\put (05,-25){\\line (0,1){25}}\\put (15,-25){\\line (0,1){25}}\\put (7,-21){\\begin{turn}{90}\\scriptsize p-4\\end{turn}}}\\end{picture}\\begin{picture}(65,18)(10,7){\\put (35,13){\\scriptsize {n}}\\put (05,20){\\line (1,0){60}}\\put (05,10){\\line (1,0){60}}\\put (05,10){\\line (0,1){10}}\\put (65,10){\\line (0,1){10}}}\\end{picture}$ For $1 \\le p \\le 3$ , the diagrams carrying negative $p$ -dependent labels are absent.","It is important to note that the diagram $(n,m,p-2)$ is present twice: one copy results from the contraction of one of the form indices with the first row followed by the symmetrization of another form index with the same row.","Another one results from the application of the same procedure to the second row.","This fact leads to two different tensor implementations.","Also note that some of the diagrams vanish for special dimensions by virtue of the Two-Column Theorem: Theorem 5.1 $\\mathfrak {so}(d)$ traceless tensors with the symmetry properties of such Young diagrams that the sum of the heights of the first two columns exceeds $d$ , are identically zero.","[36] The cohomology $H^p(\\sigma _-)$ is empty for $p>d$ .","Analogously, some potential elements of $H^p(\\sigma _-)$ are zero by the Two-Column Theorem for large $p \\le d$ .","Now we are in a position to consider the action of the Laplace operator on each of the diagrams (REF ) separately.","In the following restrictions on $n,m,p$ will be imposed: if for some $n,m,p$ a tensor has a wrong Young shape, it is zero.","In most cases we will simplify calculation by demanding $p$ -forms be $\\sigma _-$ -closed.","Another simplification is due to the fact that $\\Delta F = 0$ is equivalent to the two equations $\\sigma _- F = 0$ and $\\sigma _+ F = 0$ .","Diagram (n,m;p), $n\\ge 0, m\\ge 0, p\\ge 0$ has the tensor form $T_{a(n),b(m),c_1,..,c_p}$ .","It is $\\sigma _-$ -closed, if $m = 0$ .","Then $\\Delta T_{a(n),c_1,..,c_p} = p T_{a(n),c_1,..,c_p}\\,.$ This diagram is in $\\ker (\\Delta )$ at $p = 0$ that reproduces the already obtained result for $H^0(\\sigma _-)$ .","Diagram (n+1,m+1;p-2), $n\\ge 0, m\\ge 0, p\\ge 2$ has the tensor form $T_{a(n)c_1,b(m)c_2,..,c_p}$ .","This diagram is $\\sigma _-$ -closed.","The action of $\\Delta $ is $\\Delta T_{a(n)c_1,b(m)c_2,..,c_p} = \\frac{(n-m)(p+m-1)}{(n-m+1)}T_{a(n)c_1,b(m)c_2,..,c_p}\\,.$ This diagram is in $\\ker (\\Delta )$ , if $n=m$ , thus belonging to $H^p(\\sigma _-)$ with $p\\ge 2$ .","Diagram (n,m+1;p-1), $n\\ge 1, m\\ge 0, p\\ge 1$ has the tensor form $T_{a(n),b(m)c_0,c_1,..,c_{p-1}}$ .","Then $\\Delta T_{a(n),b(m)c_0,c_1,..,c_{p-1}} = (m+p) T_{a(n),b(m)c_0,c_1,..,c_{p-1}}\\,.$ This equation admits no solutions since $p\\ge 1$ in the case in question.","Diagram (n+1,m;p-1), $n\\ge 0, m\\ge 0, p\\ge 1$ has the tensor form $T_{a(n)c_0,b(m),c_1,..,c_{p-1}} + \\frac{m}{n-m+2}T_{a(n)b,b(m-1)c_0,c_1,..,c_{p-1}}\\,.$ It is $\\sigma _-$ -closed iff $m = 0$ .","Then $\\Delta T_{a(n)c_0,c_1,..,c_{p-1}} = \\frac{n(p-1)}{n+1} T_{a(n)c_0,c_1,..,c_{p-1}}\\,.$ This expression vanishes at $p = 1$ .","Hence, $T_{a(n)c_0} \\in H^1(\\sigma _-)\\,.$ As one can see, $n=0$ in (REF ) also leads to zero Laplace action.","However, this is not a new result, since it has been already accounted in the diagrams $(n,m;p)$ for $n=m=p=0$ (REF ), $(n+1,m+1;p-2)$ for $n=m=0,p\\ge 2$ (REF ) and the $n=m=0,p=1$ case of $(n+1,m;p-1)$ (REF ).","This fact is a simple consequence of the tensor multiplication of a column by a scalar.","Diagram (n-1,m;p-1), $n\\ge 1, m\\ge 0, p\\ge 1$ has the tensor form $T = \\eta _{ac_1}\\rho _{a(n-1),b(m),c_2,..,c_p} - \\frac{n-1}{d-4+2n}\\eta _{aa}\\rho _{a(n-2)c_1,b(m),c_2,..,c_p} + \\frac{m(n-1)}{(d-4+m+n)(d-4+2n)}\\times \\\\\\times \\eta _{aa}\\rho _{a(n-2)b,b(m-1)c_1,c_2,..,c_p} - \\frac{m}{d-4+m+n}\\eta _{ab}\\rho _{a(n-1),b(m-1)c_1,c_2,..,c_p}\\,,$ where $\\rho $ is an arbitrary $(n-1,m;p-1)$ tensor.","One can check that $T \\in \\ker (\\sigma _-)$ demands $m = 0$ with $T = \\eta _{ac_1}\\rho _{a(n-1),c_2,..,c_p} - \\frac{n-1}{d-4+2n}\\eta _{aa}\\rho _{a(n-2)c_1,c_2,..,c_p}.$ After some calculation we obtain $\\Delta T = \\frac{(p-1)(d-2+n)}{d-3+n} T\\,.$ This implies that $\\Delta $ has zero at $p = 1$ contributing to $H^1(\\sigma _-)$ .","The case of $d=2,n=1$ must be considered separately because of the divergent denominator.","The seeming divergence emerges due to the second term in (REF ), which is absent at $d=2,n=1$ , $T = \\eta _{ac_1}\\rho _{c_2,..,c_p} \\Rightarrow \\sigma _+ T =\\frac{p-1}{2}\\Big (\\eta _{bc_1}\\rho _{a,c_2,..,c_{p-1}} - \\eta _{ac_1}\\rho _{b,c_2,..,c_{p-1}}\\Big )\\,,$ leading to the same answer with $p=1$ .","Diagram (n,m-1;p-1) $n\\ge 1, m\\ge 1, p\\ge 1$ has the tensor form $T = (n-1)\\eta _{aa} \\rho _{a(n-2)bc_1,b(m-1),c_2,..,c_p} - \\frac{(n-1)(m-1)}{d-6+2m} \\eta _{aa} \\rho _{a(n-2)bb,b(m-2)c_1,c_2,..,c_p} - (d-4+m+n) \\times \\\\\\times \\eta _{ac_1} \\rho _{a(n-1)b,b(m-1),c_2,..,c_p} - (n-m)\\eta _{ab} \\rho _{a(n-1)c_1,b(m-1),c_2,..,c_p} + \\frac{(m-1)(d-4+2n)}{d-6+2m}\\eta _{ab} \\times \\\\\\times \\rho _{a(n-1)b,b(m-2)c_1,c_2,..,c_p} + \\frac{(n-m+1)(d-4+m+n)}{n} \\eta _{bc_1} \\rho _{a(n),b(m-1),c_2,..,c_p} - \\\\-\\frac{(m-1)(n-m+1)(d-4+m+n)}{(d-6+2m)n}\\eta _{bb} \\rho _{a(n),b(m-2)c_1,c_2,..,c_p}\\,,$ where $\\rho $ is an arbitrary $(n,m-1;p-1)$ tensor.","Let us show that this diagram can never be annihilated by $\\sigma _-$ .","Indeed, $\\sigma _- T = \\bigg (d-4+m+n-(n-m)\\bigg )\\eta _{ac_0} \\rho _{a(n-1)c_1,b(m-1),c_2,..,c_p} + (m-1)(\\emph {lit}),$ where (lit) denotes some terms that are linearly independent from the first one.","If $m=1$ the above expression reduces to $\\sigma _- T = \\big (d-2\\big )\\eta _{ac_0} \\rho _{a(n-1)c_1,c_2,..,c_p}.$ The expression in brackets vanishes at $d = 2$ .","However, such diagram is zero by virtue of the Two-column theorem 5.1, since the heights of the first two columns sum up to $3>d$ .","Thus, the nontrivial $T$ is never in $\\ker (\\sigma _-)$ .","Diagram (n+1,m-1;p-2) $n\\ge 1, m\\ge 1, p\\ge 2$ has the tensor form $T = \\eta _{ac_1}\\rho _{a(n-1)bc_2,b(m-1),c_3,..c_p}- \\frac{n-m+1}{n}\\eta _{bc_1}\\rho _{a(n)c_2,b(m-1),c_3,..c_p} + \\\\ + \\frac{m-1}{n-m+3}\\eta _{ac_1}\\rho _{a(n-1)bb,b(m-2)c_2,c_3,..c_p}-\\frac{(m-1)(n-1)}{(d-6+2m)(n-m+3)}\\eta _{aa}\\rho _{a(n-2)bbc_1,b(m-2)c_2,c_3,..c_p} + \\\\ + \\frac{2(m-1)(n-m+1)}{(d-6+2m)(n-m+3)}\\eta _{ab}\\rho _{a(n-1)bc_1,b(m-2)c_2,c_3,..c_p}- \\frac{(m-1)(n-m+1)}{n(n-m+3)}\\times \\\\ \\times \\eta _{bc_1}\\rho _{a(n)b,b(m-2)c_2,c_3,..c_p} - \\frac{(m-1)(n-m+2)(n-m+1)}{(d-6+2m)(n-m+3)n} \\eta _{bb}\\rho _{a(n)c_1,b(m-2)c_2,c_3,..c_p}\\,,$ where $\\rho $ is an arbitrary $(n+1,m-1;p-2)$ tensor.","Though the tensor realization (REF ) may look complicated, the problem is simplified by the observation that all terms except for the first and second ones carry a factor of $(m-1)$ .","The action of $\\sigma _-$ on the first and second terms produces a factor of $(m-1)$ in front of each $\\eta \\rho $ combination.","It can be checked that $\\sigma _- T$ has an overall factor of $(m-1)$ so that the only possible solution for $T \\in \\ker (\\sigma _-)$ is at $m=1$ in which case the tensor decomposition acquires the form $T = \\eta _{ac_1}\\rho _{a(n-1)bc_2,..,c_p} - \\eta _{bc_1}\\rho _{a(n)c_2,..,c_p}.$ At $p = 2$ , after some calculations one can check that $\\sigma _+ T = 0\\,.$ For $p>2$ it is not difficult to see that $\\sigma _+ T \\ne 0$ .","Indeed, $\\sigma _+ T = (p-2)\\bigg (1 + \\frac{1}{n}\\bigg )\\eta _{ac_1}\\rho _{a(n-1)bc_2,b,c_3,..,c_{p-1}} + (\\emph {lit})\\,,$ where (lit) denotes other linearly independent terms.","The first term is never zero.","Thus, $T = \\eta _{ac_1}\\rho _{a(n-1)bc_2} - \\eta _{bc_1}\\rho _{a(n)c_2} \\in H^2(\\sigma _-)\\,.$ Diagram (n-1,m+1;p-2), $n\\ge 1, m\\ge 0, p\\ge 2$ has the tensor form $T = \\eta _{ac_1} \\rho _{a(n-1),b(m)c_2,..,c_p} - \\frac{n-1}{d-4+2n}\\eta _{aa} \\rho _{a(n-2)c_1,b(m)c_2,..,c_p}\\,,$ where $\\rho $ is an arbitrary $(n-1,m+1;p-2)$ tensor.","Though it is obviously in $\\ker ( \\sigma _-)$ for any $m$ , it is not hard to see that it is never in $\\ker (\\sigma _+)$ .","$\\sigma _+ T = -\\bigg (1 + \\frac{p-2}{m+1} \\bigg ) \\eta _{ac_1} \\rho _{a(n-1),b(m+1),..,c_{p-1}} + (\\emph {lit} )\\,.$ Hence this diagram does not contribute to $H^p(\\sigma _-)$ .","Diagram (n-1,m-1;p-2), $n\\ge 1, m\\ge 1, p\\ge 2$ has an involved tensor form.","Since the coefficients in the expression below are complicated, we extract the factor of $(m-1)$ once present denoting the leftover coefficients as $\\alpha _i$ , 0.75em $T = \\eta _{ac_1}\\eta _{bc_2}\\rho _{a(n-1),b(m-1),..,c_p} + (m-1)\\alpha _1 \\eta _{aa}\\eta _{aa}\\rho _{a(n-4)bbc_1,b(m-2)c_2,..,c_p} + \\\\ + \\alpha _2 \\eta _{aa}\\eta _{ac_1}\\rho _{a(n-3)bc_2,b(m-1),..,c_p}+ (m-1)\\alpha _3\\eta _{aa}\\eta _{ac_1}\\rho _{a(n-3)bb,b(m-2)c_2,..,c_p} + \\\\ + (m-1)\\alpha _4 \\eta _{aa}\\eta _{ab}\\rho _{a(n-3)bc_1,b(m-2)c_2,..,c_p}+\\alpha _5 \\eta _{aa}\\eta _{bc_1}\\rho _{a(n-2)c_2,b(m-1),..,c_p} + \\\\ + (m-1)\\alpha _6\\eta _{aa}\\eta _{bc_1}\\rho _{a(n-2)b,b(m-2)c_2,..,c_p}+ (m-1)\\alpha _7 \\eta _{aa}\\eta _{bb}\\rho _{a(n-2)c_1,b(m-2)c_2,..,c_p} + \\\\ + \\alpha _8 \\eta _{ac_1}\\eta _{ab}\\rho _{a(n-2)c_2,b(m-1),..,c_p}+ (m-1)\\alpha _9 \\eta _{ac_1}\\eta _{ab}\\rho _{a(n-2)b,b(m-2)c_2,..,c_p} + \\\\ + (m-1)\\alpha _{10} \\eta _{ac_1}\\eta _{bb}\\rho _{a(n-1),b(m-2)c_2,..,c_p}+ (m-1) \\alpha _{11} \\eta _{ab}\\eta _{ab}\\rho _{a(n-2)c_1,b(m-2)c_2,..,c_p} + \\\\ + (m-1) \\alpha _{12} \\eta _{ab}\\eta _{bc_1}\\rho _{a(n-1),b(m-2)c_2,..,c_p}\\,,$ where $\\rho $ is an arbitrary $(n-1,m-1;p-2)$ tensor.","The explicit form of $\\alpha _i$ is given in the Appendix A.","Now we observe that the action of $\\sigma _-$ on the terms free of the factor of $(m-1)$ produces such factor.","Hence, $\\sigma _-(T)$ has the form of the sum of linearly independent terms with the common factor of $(m-1)$ .","Consequently, $\\sigma _- T = 0, \\text{ iff\\quad } m = 1.$ At $m=1$ , the only terms that remain are $T = \\eta _{ac_1}\\eta _{bc_2}\\rho _{a(n-1),..,c_p} + \\frac{(n-2)(n-1)}{(d-3+n)(d-4+2n)} \\eta _{aa}\\eta _{ac_1}\\rho _{a(n-3)bc_2,..,c_p} + \\\\ + \\frac{(n-1)(d-2+n)}{(d-3+n)(d-4+2n)} \\eta _{aa}\\eta _{bc_1}\\rho _{a(n-2)c_2,..,c_p} - \\frac{n-1}{d-3+n} \\eta _{ac_1}\\eta _{ab}\\rho _{a(n-2)c_2,..,c_p}.$ It can be checked that for $p = 2$ $\\sigma _+ T = 0\\,.$ For $p > 2$ it is not difficult to see that $\\sigma _+ T = \\frac{(p-2)d}{(d-2)(d-3+n)(d-2+n)} e_3^2 \\theta _Y \\rho + (\\emph {lit} ),$ where $\\rho = \\rho _{a(n-1),c_1,..,c_{p-2}}Y^{a(n-1)}\\theta ^{c_1}..\\theta ^{c_{p-2}}$ .","Therefore, $H^2(\\sigma _-) \\ni T = \\eta _{ac_1}\\eta _{bc_2}\\rho {}_{a(n-1)} + \\frac{(n-2)(n-1)}{(d-3+n)(d-4+2n)} \\eta _{aa}\\eta _{ac_1}\\rho {}_{a(n-3)bc_2} + \\\\+ \\frac{(n-1)(d-2+n)}{(d-3+n)(d-4+2n)}\\eta _{aa}\\eta _{bc_1}\\rho {}_{a(n-2)c_2} -\\frac{n-1}{d-3+n} \\eta _{ac_1}\\eta _{ab}\\rho {}_{a(n-2)c_2} \\,.$ Diagram (n+1,m;p-3), $n\\ge 1, m\\ge 1, p\\ge 3$ has the tensor form $T = \\eta _{bc_1}\\rho _{a(n)c_2,b(m-1)c_3,..,c_p} - \\frac{n}{n-m+1}\\eta _{ac_1}\\rho _{a(n-1)bc_2,b(m-1)c_3,..,c_p}\\,,$ where $\\rho $ is an arbitrary $(n+1,m;p-3)$ tensor.","Explicit computation gives $\\Delta T = \\frac{t_0}{t_0+1}\\bigg (p + h_2 - 1\\bigg )T - \\frac{2 t_0}{(t_0+1)(d-4+2h_2)}\\theta ^a Z_a \\partial _{\\theta Z} T + \\\\ + \\frac{d-4+2h_1}{(t_0+1)(d-3+h_1+h_2)(d-4+2h_2)} \\bigg (t_2 \\theta ^a Y_a \\partial _{\\theta Z} - \\theta ^a Z_a \\partial _{\\theta Z}\\bigg )T - \\\\ - \\frac{t_0-1}{(t_0+1)(d-3+h_1+h_2)}\\theta ^a Y_a \\partial _{\\theta Y}T\\,.$ This expression vanishes at $n = m$ .","Indeed, in this case $t_0 T = 0, h_1 T= h_2 T = n T, t_2 T = 0$ so that $\\Delta T = \\frac{1}{d-3+2n} \\bigg (t_2 \\theta ^a Y_a \\partial _{\\theta Z} - \\theta ^a Z_a \\partial _{\\theta Z}\\bigg )T + \\frac{1}{d-3+2n}\\theta ^a Y_a \\partial _{\\theta Y}T\\,.$ Since $t_2 \\theta ^a Y_a \\partial _{\\theta Z}T = \\theta ^a Y_a \\partial _{\\theta Z}t_2 T - \\theta ^a Y_a \\partial _{\\theta Y}T + \\theta ^a Z_a \\partial _{\\theta Z}T = - \\theta ^a Y_a \\partial _{\\theta Y}T + \\theta ^a Z_a \\partial _{\\theta Z}T\\,,$ it follows that $\\Delta T=0$ at $n=m$ .","To check that $\\Delta T\\ne 0$ at $n\\ne m$ one should substitute the expression for $T$ noticing that different linearly independent terms have no common factor to vanish, that implies nontriviality of $\\Delta T$ .","Diagram (n,m+1;p-3), $n\\ge 1, m\\ge 0, p\\ge 3$ has the tensor form $T = \\eta _{ac_1}\\rho _{a(n-1)c_2,b(m)c_3,..,c_p}$ .","It belongs to $\\ker (\\sigma _-)$ , but not to $\\ker (\\sigma _+)$ .","$\\sigma _+ T = \\bigg (1 + \\frac{p-3}{m+1}\\bigg )\\eta _{ac_1}\\rho _{a(n-1)c_2,b(m+1),..,c_p} + (\\emph {lit})\\,.$ Diagram (n,m;p-2) $n\\ge 1, m\\ge 0, p\\ge 2$ admits two tensor realizations $T_i$ due to the double presence of this diagram in the result of tensor product.","The tensors $T_1 = \\eta _{ac_1} \\rho _1{}_{a(n-1)c_2,b(m),c_3,..,c_p} + \\frac{m}{n}\\eta _{bc_1} \\rho _1{}_{a(n),b(m-1)c_2,c_3,..,c_p},$ $T_2 = \\eta _{aa} \\rho _2{}_{a(n-2)bc_1,b(m-1)c_2,c_3,..,c_p} - \\frac{n-m}{n-1}\\eta _{ab} \\rho _2{}_{a(n-1)c_1,b(m-1)c_2,c_3,..,c_p} - \\frac{d-4+n+m}{n-1}\\eta _{ac_1} \\times \\\\\\times \\rho _2{}_{a(n-1)b,b(m-1)c_2,c_3,..,c_p} + \\frac{(n-m+1)(d-4+m+n)}{n(n-1)}\\eta _{bc_1} \\rho _2{}_{a(n),b(m-1)c_2,c_3,..,c_p}$ are linearly independent.","That Laplace operator acts diagonally on $T_i$ , $\\Delta T_i = \\lambda _i(n,m,p) T_i$ , allows us to separately consider each of these diagrams.","Firstly, we check if these are in $\\ker (\\sigma _-)$ computing $&\\sigma _- T_1 = m \\eta _{ac_1} \\rho _1{}_{a(n-1)c_2,b(m-1)c_3,..,c_{p+1}},\\\\&\\sigma _- T_2 = \\frac{d-4+2m}{n-1} \\eta _{ac_1} \\rho _2{}_{a(n-1)c_2,b(m-1)c_3,..,c_{p+1}}.$ $T_1 \\in \\ker (\\sigma _-)$ at $m = 0$ .","Formally, $T_2$ is annihilated by $\\sigma _-$ at $d=2, m = 1$ , but this is not allowed by the Two-column theorem.","So, the only candidate for cohomology is $T_1$ .","However, $\\sigma _+ T_1 = \\frac{1}{n+1}\\Big (1 + \\frac{p-2}{n} \\Big )\\eta _{bc_1}\\rho _1{}_{a(n),c_2,..,c_{p-1}} + (\\emph {lit})\\,,$ which is never zero.","Diagram (n-1,m;p-3), $n\\ge 2, m\\ge 1, p\\ge 3$ has the tensor form $T = \\eta _{ac_1}\\eta _{bc_2} \\rho _{a(n-1),b(m-1)c_3,..,c_p} - \\frac{(n-1)(d-3+m+n)}{(d-4+n+m)(d-4+2n)}\\eta _{aa}\\eta _{bc_1} \\rho _{a(n-2)c_2,b(m-1)c_3,..,c_p} -\\\\- \\frac{(n-1)(n-2)}{(d-4+m+n)(d-4+2n)} \\eta _{aa}\\eta _{ac_1} \\rho _{a(n-3)bc_2,b(m-1)c_3,..,c_p} + \\frac{n-1}{d-4+m+n} \\times \\\\\\times \\eta _{ab}\\eta _{ac_1} \\rho _{a(n-2)c_2,b(m-1)c_3,..,c_p}\\,,$ where $\\rho $ is an arbitrary $(n-1,m;p-3)$ tensor.","Obviously, $T \\in \\ker (\\sigma _-)$ .","However, $T \\notin \\ker (\\sigma _+)$ .","$\\sigma _+ T = \\bigg (1 + \\frac{p-3}{m}\\bigg ) \\eta _{ac_1}\\eta _{bc_2} \\rho _{a(n-1),b(m),c_3,..,c_{p-1}} + (\\emph {lit})\\,,$ hence not contributing to cohomology.","Diagram (n,m-1;p-3), $n\\ge 1, m\\ge 1, p\\ge 3$ has the tensor form $T = \\eta _{ac_1}\\eta _{bc_2}\\rho _{a(n-1)c_3,b(m-1),..,c_p} + (m-1)(\\emph {lit})\\,$ with all terms except for the first one carrying a factor of $(m-1)$ .","The action of $\\sigma _-$ on the first term brings a factor of $(m-1)$ in front of $\\eta \\rho $ .","Since all $\\eta \\rho $ terms in the decomposition are linearly independent we conclude that $\\sigma _- T = 0, \\text{ if } m = 1.$ At $p = 3$ one can check that $\\sigma _+ T = 0$ .","However, for $p>3$ $T$ does not belong to $\\ker (\\sigma _+)$ , $\\sigma _+ T = - (p-3) \\eta _{ac_1}\\eta _{bc_2} \\rho _{a(n-1)c_3,b,..,c_{p-1}} + (\\emph {lit})\\,.$ Consequently, the only contribution to $H^3(\\sigma _-)$ is $T = \\eta _{ac_1}\\eta _{bc_2}\\rho _{a(n-1)c_3} \\in H^3(\\sigma _-)\\,.$ Diagram (n,m;p-4), $n\\ge 1, m\\ge 1, p\\ge 4$ has the tensor form $T = \\eta _{ac_1}\\eta _{bc_2}\\rho _{a(n-1)c_3,b(m-1)c_4,..,c_p}\\,.$ This is obviously annihilated by $\\sigma _-$ , but not by $\\sigma _+$ .","$\\sigma _+ T = - \\bigg (1 + \\frac{p-4}{m} \\bigg ) \\eta _{ac_1}\\eta _{bc_2}\\rho _{a(n-1)c_3,b(m),..,c_{p-1}} + (\\emph {lit})\\,.$ Hence it does not contribute to $H^p(\\sigma _-)$ ."],["Summary","Summarizing the results of Sections REF and REF we found the following cohomology groups: $H^0(\\sigma _-) = \\lbrace F = F_{a(n)}Y^{a(n)}| F \\in V^0\\rbrace \\,,$ $H^1(\\sigma _-) = \\lbrace \\phi = F_1{}_{a(n)c}Y^{a(n)}\\theta ^c, F_1 \\in V^1;\\\\ \\phi ^{tr} = \\Big [(n-1)\\eta _{aa}F_2{}_{a(n-2)c} - (d-4+2n)\\eta _{ac}F_2{}_{a(n-1)}\\Big ]Y^{a(n)}\\theta ^c \\in V^1 \\rbrace \\,,$ $H^2(\\sigma _-) = \\lbrace W = \\theta _Y\\theta _Z C(Y,Z): t_0 C = 0, C \\in V^{0}; \\\\ \\mathcal {E}_A = \\Big [\\eta _{ac_1}\\rho _1{}_{a(n-1)bc_2} - \\eta _{bc_1}\\rho _1{}_{a(n)c_2}\\Big ]Y^{a(n)}Z^b\\theta ^{c_1}\\theta ^{c_2} \\in V^2; \\\\ \\mathcal {E}_B = \\Big [\\eta _{ac_1}\\eta _{bc_2}\\rho _2{}_{a(n-1)} + \\frac{(n-2)(n-1)}{(d-3+n)(d-4+2n)} \\eta _{aa}\\eta _{ac_1}\\rho _2{}_{a(n-3)bc_2} +\\\\+ \\frac{(n-1)(d-2+n)}{(d-3+n)(d-4+2n)}\\eta _{aa}\\eta _{bc_1}\\rho _2{}_{a(n-2)c_2} - \\frac{n-1}{d-3+n} \\eta _{ac_1}\\eta _{ab}\\rho _2{}_{a(n-2)c_2}\\Big ]Y^{a(n)}Z^b\\theta ^{c_1}\\theta ^{c_2} \\in V^2 \\rbrace \\,,$ $H^3(\\sigma _-) = \\lbrace B^{fr} = \\eta _{ac_1}\\eta _{bc_2}\\rho _{a(n-1)c_3} Y^{a(n)}Z^{b}\\theta ^{c_1}\\theta ^{c_2}\\theta ^{c_3} \\in V^3; \\\\B_1 = \\theta _Y\\theta _Z C(Y,Z,\\theta ): t_0 C = 0, C = C_{a(n),b(n),c}Y^{a(n)}Z^{b(n)}\\theta ^{c} \\in V^1; \\\\B_2 = \\Big [\\eta _{bc_1}\\rho _{a(n)c_2,b(n-1)c_3} - n\\eta _{ac_1}\\rho _{a(n-1)bc_2,b(n-1)c_3}\\Big ]Y^{a(n)}Z^{b(n)}\\theta ^{c_1}\\theta ^{c_2}\\theta ^{c_3} \\in V^3\\rbrace \\,.$ At $p>3$ $H^p(\\sigma _-) = \\Big \\lbrace B_1 = \\theta _Y\\theta _Z C(Y,Z,\\theta ): t_0 C = 0, C = C_{a(n),b(n),c_1,..,c_{p-2}}Y^{a(n)}Z^{b(n)}\\theta ^{c_1}..\\theta ^{c_{p-2}} \\in V^{p-2}; \\\\B_2 = \\Big [\\eta _{bc_1}\\rho _{a(n)c_2,b(n-1)c_3,..,c_p} - n\\eta _{ac_1}\\rho _{a(n-1)bc_2,b(n-1)c_3,..,c_p}\\Big ]Y^{a(n)}Z^{b(n)}\\theta ^{c_1}..\\theta ^{c_{p}} \\in V^{p} \\Big \\rbrace .$ According to Theorem REF , the differential gauge transformation parameters are described by $H^0(\\sigma _-)$ (REF ).","The gauge parameter in the Fronsdal theory is known to be a symmetric traceless tensor.","Since tensors $(n,0,0)$ constitute the cohomology group $H^0(\\sigma _-)$ , the unfolded differential gauge transformation is shown to coincide with the Fronsdal one.","As recalled in Section REF , the Fronsdal field consists of two symmetric traceless fields (REF ).","These fields are represented by the cohomology groups $H^1(\\sigma _-)$ (REF ).","Cohomology group $H^1(\\sigma _-)$ consists of two elements $(n+1,0,0)$ and $(n-1,0,0)$ matching the components of the Fronsdal field.","Thus, the physical fields in the unfolded approach indeed coincide with the Fronsdal field.","The cohomology group $H^2(\\sigma _-)$ (REF ) describes gauge invariant combinations of derivatives of the physical fields that can be used to impose differential equations on the latter.","The Fronsdal cohomology classes $\\mathcal {E}_A$ and $\\mathcal {E}_B$ match with the Fronsdal equations: $\\mathcal {E}_A$ is associated with the traceless part of the Fronsdal equations, while $\\mathcal {E}_B$ with the trace part, that is the equations $\\mathcal {E}_A = 0, \\mathcal {E}_B = 0$ just reproduce the Fronsdal equations.","Note that the number of resulting equations is the same as the number of fields, as it should be in a Lagrangian system.","$W$ in (REF ) represents \"Weyl\" cohomology.","Imposing $W = 0$ in the case of gravity one gets conformally flat metrics and in the case of higher spins \"conformally flat\" fields.","In Einstein gravity and HS theory, the equation $W=0$ is not imposed.","Instead, elements of $W$ are interpreted as new fields $C$ that describe generalized Weyl tensors by virtue of the unfolded equations (REF ).","Thus, calculation of the cohomology group $H^2(\\sigma _ -)$ shows that unfolded equations (REF ), (REF ) contain Fronsdal equations along with constraints on auxiliary fields.","In accordance with the general discussion of Section elements of $H^3(\\sigma _-)$ (REF ) correspond to Bianchi identities.","Class $B^{fr}$ describes the Bianchi identities for the Fronsdal equations.","Note that their number coincides with the number of differential gauge parameters.","The remaining classes $B_1$ and $B_2$ correspond to the Bianchi identities on the Weyl tensor.","It is noteworthy that the latter can be checked to coincide with the first $\\widetilde{\\sigma }_-$ cohomology in the Weyl sector of zero-forms of [10] for $s>1$ .","This fact exhibits the connection between the gauge and Weyl sectors.","For $p>3$ cohomology groups $H^p(\\sigma _-)$ describe the higher Bianchi identities for Bianchi identities on the Weyl tensor also interpreted as syzygies [24].","Obtained lower cohomology groups match with the results of [16], [17], [18].","In HS theory the fields are realized by one-forms.","Formally, one can consider field equations (REF ), (REF ) for $p$ -forms $\\omega _{a(n),b(m)}$ valued in a two-row irreducible $\\mathfrak {o}(d)$ -module.","From our results and physical interpretation of the $\\sigma _-$ cohomology groups it follows that for $p>1$ the unfolded system in the gauge sector is off-shell.","To answer the question whether the full unfolded system including both the gauge $p$ -form sector and the Weyl $(p-1)$ -form sector is off-shell, the analysis of $H(\\tilde{\\sigma }_{-})$ has to be performed in the Weyl sector.","The case of $p>1$ may be somewhat similar to the $s=1$ case, where the equation on $A_\\mu $ lies in the Weyl sector.","Finally, let us stress that the results of this section for HS fields in Minkowski space admit a straightforward deformation to $AdS_d$ with the same operator $\\sigma _-$ .","This is because in that case dynamical fields are described by rectangular diagrams of the $AdS_d$ algebra $\\mathfrak {o}(d-1,2)$ [37].","In general, in the flat limit, irreducible massless (gauge) fields in $AdS_d$ decompose into nontrivial sets of irreducible flat space massless fields [38], [39], [40] and there is no one-to-one correspondence between massless fields in Minkowski space and $AdS_d$ .","Namely, a generic irreducible field in Minkowski space may admit no deformation to $AdS_d$ (see also [41])."],["$\\sigma _-$ cohomology in {{formula:6304f7a9-7e1d-44fd-b47c-3df99bd81cdd}} in the spinor language","$4d$ HS theories admit a description in terms of two-component spinors instead of tensors.","That is, instead of using the generating functions in the tensor form $\\omega (x,dx\\,|Y,Z)$ , where $Y^a$ and $Z^a$ carried vector Lorentz indices $ a=\\lbrace 0,1,2,3\\rbrace $ , we will use $\\omega (y,\\overline{y}\\,|x,dx) = \\sum \\limits _{k,m}\\omega ^{\\alpha _1\\dots \\alpha _k,\\dot{\\alpha }_1\\dots \\dot{\\alpha }_m}(x,dx)\\, y_{\\alpha _1}\\dots y_{\\alpha _k}\\,\\overline{y}_{\\dot{\\alpha }_1}\\dots \\overline{y}_{\\dot{\\alpha }_m}\\,,$ where the indices $\\alpha $ and $\\dot{\\alpha }$ of the two-component commuting spinors $y^\\alpha $ and $\\overline{y}^{\\dot{\\alpha }}$ take two values $\\lbrace 1,2\\rbrace $ .","Analogously to Sections 3 and 4, we have to introduce the grading on the space of $\\Lambda ^\\bullet (M)\\otimes \\mathbb {C}[[y,\\overline{y}]]$ .","Consider the homogeneous element of $\\Lambda ^\\bullet (M)\\otimes \\mathbb {C}[[y,\\overline{y}]]$ of degree $N$ and $\\overline{N}$ in $y$ and $\\overline{y}$ , respectively $\\omega (\\mu y, \\overline{\\mu }\\overline{y}\\,|x,dx) = \\mu ^N\\,\\overline{\\mu }{}^{\\overline{N}}\\,\\omega (y,\\overline{y}\\,|x,dx).$ Define the grading operator $G$ on $\\Lambda ^\\bullet (M)\\otimes \\mathbb {C}[[y,\\overline{y}]]$ as follows: $G\\omega (y,\\overline{y}\\,|x,dx) = |N-\\overline{N}|\\,\\omega (y,\\overline{y}\\,|x,dx) \\equiv |\\deg _y(\\omega (y,\\overline{y}\\,|x,dx))-\\deg _{\\overline{y}}(\\omega (y,\\overline{y}\\,|x,dx))|.$ Note that in the bosonic sector the frame-like fields $e^{m_1\\dots m_{s-1}} \\ \\leftrightarrow \\ e^{\\alpha _1\\dots \\alpha _{s-1},\\dot{\\alpha }_1\\dots \\dot{\\alpha }_{s-1}}$ have the lowest possible grading $G=0$ .","For our later computations to match with the Fronsdal theory, we define the action of $\\sigma _-$ on $\\omega (y,\\overline{y})$ to decrease the $G$ -grading: $\\sigma _-\\omega (y,\\overline{y}) &:= i\\, \\overline{y}^{\\dot{\\alpha }}h^\\alpha _{\\ \\dot{\\alpha }}\\partial _\\alpha \\ \\omega (y,\\overline{y}), \\quad \\quad \\text{at } \\deg _y(\\omega )>\\deg _{\\overline{y}}(\\omega ),\\\\\\sigma _-\\omega (y,\\overline{y}) &:= i\\, y^{\\alpha }h_\\alpha ^{\\ \\dot{\\alpha }}\\overline{\\partial }_{\\dot{\\alpha }}\\ \\omega (y,\\overline{y}), \\quad \\quad \\text{at } \\deg _y(\\omega )<\\deg _{\\overline{y}}(\\omega ),\\\\\\sigma _-\\omega (y,\\overline{y}) &:= 0\\qquad \\qquad \\qquad \\qquad \\quad \\,\\,\\text{at } \\deg _y(\\omega )=\\deg _{\\overline{y}}(\\omega )\\,,$ where $_\\alpha :=\\frac{}{y^\\alpha }\\,,\\qquad \\overline{}_{\\dot{\\alpha }}:=\\frac{}{\\overline{y}^{\\dot{\\alpha }}}\\,,$ and the dependence on $x$ and $dx$ in $\\omega (y,\\overline{y}) = \\omega (y,\\overline{y}\\,|x,dx)$ is always implicit.","It is easy to check that so defined $\\sigma _-$ is nilpotent, $(\\sigma _-)^2=0$ .","Note that $\\sigma _\\pm $ change the grading $G$ by 2.","This agrees in particular with the fact that the bosonic and fermionic sectors, where the grading is even and odd respectively, are independent.","We consider in detail the more complicated bosonic case, observing in the end that the computation for fermionic fields is quite similar.","Note that the analysis of $\\sigma _-$ cohomology in the $4d$ conformal HS theory was also performed in terms of two-component spinors in [42], [43].","It is more complicated since the generating functions of conformal HS theory depend on twice as many independent spinors, but simpler since it is free of the module factors like $|N- N|$ in the grading definition (REF ).","Next, we define a scalar productBeing $\\mathrm {SL}(2,\\mathbb {C})$ -invariant this scalar product is not positive-definite.","Analogously to the tensorial case, without affecting the $\\sigma _-$ cohomology analysis it can be made positive-definite by going to the $\\mathfrak {su}(2)\\oplus \\mathfrak {su}(2)$ algebra, which is the compact real form of $\\mathfrak {sl}(2,\\mathbb {C})\\oplus \\mathfrak {sl}(2,\\mathbb {C})$ with altered conjugation rules $\\overline{y}^\\alpha =y_\\alpha $ , $\\overline{y}^{\\dot{\\alpha }} =y_{\\dot{\\alpha }}$ .","on generating elements of $\\Lambda ^\\bullet (M)\\otimes \\mathbb {C}[[y,\\overline{y}]]$ by $\\langle q^\\alpha |q^\\beta \\rangle =\\langle y^\\alpha |y^\\beta \\rangle =i\\epsilon ^{\\alpha \\beta }\\,, &&\\langle \\overline{p}_{\\dot{\\alpha }}|\\overline{p}_{\\dot{\\beta }}\\rangle =-\\langle \\overline{\\partial }_{\\dot{\\alpha }}|\\overline{\\partial }_{\\dot{\\beta }}\\rangle =i\\overline{\\epsilon }_{\\dot{\\alpha }\\dot{\\beta }} \\,, && \\langle h_{\\alpha \\dot{\\alpha }}|h_{\\beta \\dot{\\beta }}\\rangle =\\epsilon _{\\alpha \\beta }\\epsilon _{\\dot{\\alpha }\\dot{\\beta }}\\,,$ where $q^\\alpha =y^\\alpha $ , $p_\\alpha =i\\partial _\\alpha $ and $h^{\\alpha \\dot{\\beta }}$ is a vierbein one-form.","In some local coordinates $x^\\mu $ on the base manifold (which in our case is $AdS_4$ ) the vierbein one-forms $h^\\alpha _{\\ \\dot{\\alpha }}$ can be expressed as $h^\\alpha _{\\ \\dot{\\alpha }} = \\left(h_{\\mu }\\right)^\\alpha _{\\ \\dot{\\alpha }}\\,dx^\\mu .$ The $AdS_4$ vierbein $\\left(h_{\\mu }\\right)^\\alpha _{\\ \\dot{\\alpha }}$ is demanded to be non-degenerate at any point of $AdS_4$ .","The next step is to obtain $\\sigma _+:=\\sigma _-^\\dagger $ with respect to the scalar product $\\langle \\,,\\,\\rangle $ , i.e., $\\langle \\phi |\\sigma _-\\psi \\rangle = \\langle \\sigma _+\\phi |\\psi \\rangle $ .","It is not hard to check that $\\sigma _+\\omega (y,\\overline{y}) &:= -iy^\\alpha D_\\alpha ^{\\ \\dot{\\alpha }}\\overline{\\partial }_{\\dot{\\alpha }}\\ \\omega (y,\\overline{y}) \\quad \\quad \\text{at } \\deg _y(\\omega )>\\deg _{\\overline{y}}(\\omega )\\,,\\\\\\sigma _+\\omega (y,\\overline{y}) &:= -i \\overline{y}^{\\dot{\\alpha }}D^\\alpha _{\\ \\dot{\\alpha }}\\partial _\\alpha \\ \\omega (y,\\overline{y}) \\quad \\quad \\text{at } \\deg _y(\\omega )<\\deg _{\\overline{y}}(\\omega )\\,,\\\\\\sigma _+ \\omega (y,\\overline{y})&:= -i\\left(y^\\alpha D_\\alpha ^{\\ \\dot{\\alpha }}\\overline{\\partial }_{\\dot{\\alpha }}+\\overline{y}^{\\dot{\\alpha }}D^\\alpha _{\\ \\dot{\\alpha }}\\partial _\\alpha \\right)\\omega (y,\\overline{y})\\quad \\quad \\text{at } \\deg _y(\\omega )=\\deg _{\\overline{y}}(\\omega )\\,,$ where $D_\\alpha ^{\\ \\dot{\\alpha }} := \\frac{}{h^\\alpha _{\\ \\dot{\\alpha }}} \\,,\\qquad D^\\alpha _{\\ \\dot{\\alpha }} := \\frac{}{h_\\alpha ^{\\ \\dot{\\alpha }}}\\,.$ By $\\omega = \\omega (y,\\overline{y}\\,|x,h)$ we mean a general $p$ -form polynomial in $y$ and $\\overline{y}$ with the coordinate one-forms $dx$ replaced by $h$ via (REF ), that is $\\omega (y,\\overline{y}\\,|x,h) = \\sum _{n,m} \\omega _{\\alpha _1\\dots \\alpha _p|\\mu (n)|\\dot{\\alpha }_1\\dots \\dot{\\alpha }_p|\\dot{\\mu }(m)}(x)\\,h^{\\alpha _1\\dot{\\alpha }_1}\\wedge \\dots \\wedge h^{\\alpha _p\\dot{\\alpha }_p}\\,y^{\\mu (n)}\\,\\overline{y}^{\\dot{\\mu }(m)}\\,.$ So defined $\\sigma _+$ increases the grading.","The Laplace operator $\\Delta :=\\sigma _-\\sigma _++\\sigma _+\\sigma _-$ is by construction self-adjoint with respect to $\\langle \\,|\\,\\rangle $ and non-negative definite for the compact version of the space-time symmetry algebra."],["Bosonic case in $AdS_4$","To calculate cohomology of $\\sigma _-$ we have to compute the action of $\\Delta $ .","Since $\\sigma _-$ and $\\sigma _+$ are defined differently in the different regions of the $(N, \\overline{N})$ plane, we compute the Laplacian action in the these regions separately.","Direct computation yields: $\\Delta _{N>\\overline{N}+2}&= N(\\overline{N}+2) + y^\\beta _\\alpha h^\\alpha _{\\ \\dot{\\gamma }} D_\\beta ^{\\ \\dot{\\gamma }} + \\overline{y}^{\\dot{\\alpha }}\\overline{}_{\\dot{\\beta }}h_{\\gamma \\dot{\\alpha }}D^{\\gamma \\dot{\\beta }}\\,,\\\\\\Delta _{N<\\overline{N}-2}&=\\overline{N}(N+2)+y^\\alpha _\\beta h_{\\alpha \\dot{\\gamma }}D^{\\beta \\dot{\\gamma }}+\\overline{y}^{\\dot{\\alpha }}\\overline{}_{\\dot{\\beta }} h_\\gamma ^{\\ \\dot{\\beta }}D^{\\gamma }_{\\ \\dot{\\alpha }}\\,,\\\\\\Delta _{N=\\overline{N}+2} &= \\Delta _{N>\\overline{N}+2} + \\overline{y}^{\\dot{\\alpha }}\\overline{y}^{\\dot{\\beta }}_\\alpha _\\beta h^\\beta _{\\ \\dot{\\beta }}D^\\alpha _{\\ \\dot{\\alpha }}\\,,\\\\\\Delta _{N=\\overline{N}-2} &= \\Delta _{N<\\overline{N}-2} + y^\\alpha y^\\beta \\overline{}_{\\dot{\\alpha }}\\overline{}_{\\dot{\\beta }}h_\\beta ^{\\ \\dot{\\beta }}D_\\alpha ^{\\ \\dot{\\alpha }}\\,,\\\\\\Delta _{N=\\overline{N}}&=\\overline{y}^{\\dot{\\alpha }}\\overline{}_{\\dot{\\beta }} h_{\\gamma \\dot{\\alpha }} D^{\\gamma \\dot{\\beta }}+y^\\alpha _\\beta h_{\\alpha \\dot{\\gamma }}D^{\\beta \\dot{\\gamma }}-\\overline{y}^{\\dot{\\alpha }} y^\\beta _\\alpha \\overline{}_{\\dot{\\beta }}h^\\alpha _{\\ \\dot{\\alpha }} D_\\beta ^{\\ \\dot{\\beta }}- y^\\alpha \\overline{y}^{\\dot{\\beta }}_\\beta \\overline{}_{\\dot{\\alpha }}h_\\alpha ^{\\ \\dot{\\alpha }}D^\\beta _{\\ \\dot{\\beta }}\\,.$ The computation of the cohomology $H^p(\\sigma _-)$ will be performed as follows.","Taking a general $p$ -form $\\omega (y,\\overline{y}\\,|x)$ , we decompose it into Lorentz irreducible components.","As we will observe, the projectors onto irreducible parts will commute with the action of the Laplacian.","Thus, instead of involved calculation of the action of $\\Delta $ on all of the irreducible components of $\\omega (y,\\overline{y}\\,|x)$ we can first calculate its action on the general $\\omega (y,\\overline{y}\\,|x)$ and then project."],[" $H^0(\\sigma _-)$","Evidently, $\\Delta _{N=\\overline{N}}\\Big |_\\text{0-forms}= 0$ since all the terms in (REF ) contain derivatives in $h$ 's.","At the same time, $\\Delta _{N\\ne \\overline{N}}\\Big |_\\text{0-forms} > 0$ .","Thus, we conclude $H^0(\\sigma _-)=\\ker \\left(\\Delta \\Big |_\\text{0-forms}\\right)= \\left\\lbrace F(y,\\overline{y})=F_{\\alpha (n),\\dot{\\alpha }(n)}y^{\\alpha (n)}\\overline{y}^{\\dot{\\alpha }(n)},\\ n\\in \\mathbb {N}_0\\right\\rbrace .$ By Theorem REF , elements of this cohomology space correspond to the parameters of differential (non-Stueckelberg) linearized HS gauge transformations.","This result fits the pattern of the spin-$s$ Fronsdal gauge symmetry parameters with $n=s-1$ ."],[" $H^1(\\sigma _-)$","The decomposition of a one-form $\\Theta (y,\\overline{y}\\,|x)$ into Lorentz irreps reads as $\\Theta (y,\\overline{y}\\,|x)=\\underbrace{\\Theta _{\\mu (n+1)|\\dot{\\mu }(m+1)}\\ h^{\\mu \\dot{\\mu }}y^{\\mu (n)}y^{\\dot{\\mu }(m)}}_{\\Theta _\\mathrm {A}(y,\\overline{y})}-\\frac{1}{2}\\underbrace{\\Theta _{\\mu (n-1)|\\dot{\\mu }(m+1)}\\ h_\\nu ^{\\ \\dot{\\mu }}y^\\nu y^{\\mu (n-1)}\\overline{y}^{\\dot{\\mu }(m)}}_{\\Theta _\\mathrm {B}(y,\\overline{y})} -\\\\-\\frac{1}{2}\\underbrace{\\Theta _{\\mu (n+1)|\\dot{\\mu }(m-1)}\\ h^{\\mu }_{\\ \\dot{\\nu }}y^{\\mu (n)}\\overline{y}^{\\dot{\\nu }}\\overline{y}^{\\dot{\\mu }(m-1)}}_{\\Theta _\\mathrm {C}(y,\\overline{y})} +\\frac{1}{4}\\underbrace{\\Theta _{\\mu (n-1)|\\dot{\\mu }(m-1)}\\ h_{\\nu \\dot{\\nu }} y^\\nu y^{\\mu (n-1)}\\overline{y}^{\\dot{\\nu }}\\overline{y}^{\\dot{\\mu }(m-1)}}_{\\Theta _\\mathrm {D}(y,\\overline{y})}.$ Thus, for fixed $n$ and $m$ , there are four Lorentz-irreducible one-forms: $\\Theta _\\text{A}$ , $\\Theta _\\mathrm {B}$ , $\\Theta _\\mathrm {C}$ , $\\Theta _\\mathrm {D}$ .","For direct computations it will be convenient to separate two of the indices of the $y$ group: $\\Theta (y,\\overline{y}) = \\Theta _{\\lambda ,\\nu ,\\mu (n-1)|\\dot{\\lambda },\\dot{\\nu },\\dot{\\mu }(m-1)}\\ h^{\\lambda \\dot{\\lambda }}y^{\\nu }y^{\\mu (n-1)}\\overline{y}^{\\dot{\\nu }}\\overline{y}^{\\dot{\\mu }(m-1)}.$ In terms of the basis one-forms $h^{\\lambda \\dot{\\lambda }}y^{\\nu }y^{\\mu (n-1)}\\overline{y}^{\\dot{\\nu }}\\overline{y}^{\\dot{\\mu }(m-1)}$ the projectors onto irreducible components are $\\mathcal {P}_\\mathrm {A} &= {S}_{(\\lambda ,\\mu ,\\nu )}{S}_{(\\dot{\\lambda },\\dot{\\mu },\\dot{\\nu })}, && \\mathcal {P}_\\mathrm {C} = {S}_{(\\lambda ,\\mu ,\\nu )}\\epsilon _{\\dot{\\lambda }\\dot{\\nu }},\\\\\\mathcal {P}_\\mathrm {B} &= \\epsilon _{\\lambda \\nu }{S}_{(\\dot{\\lambda },\\dot{\\mu },\\dot{\\nu })}, &&\\mathcal {P}_\\mathrm {D} = \\epsilon _{\\lambda \\nu }\\epsilon _{\\dot{\\lambda }\\dot{\\nu }},$ where ${S}_{(\\lambda ,\\mu ,\\nu )}$ implies symmetrization over indices $\\lambda ,\\mu ,\\nu $ and similarly for the dotted indices."],["$H^1(\\sigma _-)$ in the diagonal sector {{formula:a426b6e9-6846-47b5-abc4-9ff268867159}}","In the diagonal sector with $n=m$ the Laplacian is a sum of the following four terms: $\\Delta _{N=\\overline{N}}\\Theta (y,\\overline{y}) =\\underbrace{\\overline{y}^{\\dot{\\alpha }}\\overline{}_{\\dot{\\beta }} h_{\\gamma \\dot{\\alpha }} D^{\\gamma \\dot{\\beta }} \\Theta (y,\\overline{y}) }_{T_1(y,\\overline{y})}+\\underbrace{y^\\alpha _\\beta h_{\\alpha \\dot{\\gamma }}D^{\\beta \\dot{\\gamma }}\\Theta (y,\\overline{y}) }_{T_2(y,\\overline{y})}-\\\\\\underbrace{-\\overline{y}^{\\dot{\\alpha }} y^\\beta _\\alpha \\overline{}_{\\dot{\\beta }}h^\\alpha _{\\ \\dot{\\alpha }} D_\\beta ^{\\ \\dot{\\beta }}\\Theta (y,\\overline{y}) }_{T_3(y,\\overline{y})}\\underbrace{-y^\\alpha \\overline{y}^{\\dot{\\beta }}_\\beta \\overline{}_{\\dot{\\alpha }}h_\\alpha ^{\\ \\dot{\\alpha }}D^\\beta _{\\ \\dot{\\beta }}\\Theta (y,\\overline{y}) }_{T_4(y,\\overline{y})}.$ Consider the first term in (REF ).","$\\overline{y}^{\\dot{\\alpha }}\\overline{}_{\\dot{\\beta }} h_{\\gamma \\dot{\\alpha }} D^{\\gamma \\dot{\\beta }} \\Theta (y,\\overline{y}) = \\Theta _{\\lambda ,\\nu ,\\mu (n-1)|\\dot{\\lambda },\\dot{\\nu },\\dot{\\mu }(m-1)}\\ \\underbrace{\\Big [\\overline{y}^{\\dot{\\alpha }}\\overline{}_{\\dot{\\beta }} h_{\\gamma \\dot{\\alpha }} D^{\\gamma \\dot{\\beta }} \\big ( h^{\\lambda \\dot{\\lambda }}y^{\\nu }y^{\\mu (n-1)}\\overline{y}^{\\dot{\\nu }}\\overline{y}^{\\dot{\\mu }(m-1)}\\big )\\Big ]}_{T_1^{\\lambda ,\\nu ,\\mu (n-1)|\\dot{\\lambda },\\dot{\\nu },\\dot{\\mu }(n-1)}}\\,.$ The expression in square brackets is denoted by $T_1^{\\lambda ,\\nu ,\\mu (n-1)|\\dot{\\lambda },\\dot{\\nu },\\dot{\\mu }(n-1)}$ .","The notation for other irreducible components $T_2$ , $T_3$ and $T_4$ is analogous.","Straightforward computation yields $T_1^{\\lambda ,\\nu ,\\mu (n-1)|\\dot{\\lambda },\\dot{\\nu },\\dot{\\mu }(n-1)} &=& -h^\\lambda _{\\ \\dot{\\alpha }}\\epsilon ^{\\dot{\\nu }\\dot{\\lambda }} y^\\nu y^{\\mu (n-1)}\\overline{y}^{\\dot{\\alpha }}\\overline{y}^{\\dot{\\mu }(n-1)} -{}\\nonumber \\\\&&- (n-1) h^\\lambda _{\\ \\dot{\\alpha }}\\epsilon ^{\\dot{\\mu }\\dot{\\lambda }}y^\\nu y^{\\mu (n-1)}\\overline{y}^{\\dot{\\alpha }}\\overline{y}^{\\dot{\\nu }}\\overline{y}^{\\dot{\\mu }(n-2)}\\,,\\\\T_2^{\\lambda ,\\nu ,\\mu (n-1)|\\dot{\\lambda },\\dot{\\nu },\\dot{\\mu }(n-1)} &=& -h_\\alpha ^{\\ \\dot{\\lambda }}\\epsilon ^{\\nu \\lambda }y^{\\alpha }y^{\\mu (n-1)}\\overline{y}^{\\dot{\\nu }}\\overline{y}^{\\dot{\\mu }(n-1)} -{}\\nonumber \\\\&&- (n-1)h_\\alpha ^{\\ \\dot{\\lambda }}\\epsilon ^{\\mu \\lambda }y^\\alpha y^\\nu y^{\\mu (n-2)}\\overline{y}^{\\dot{\\nu }}\\overline{y}^{\\dot{\\mu }(n-1)}\\,,\\\\T_3^{\\lambda ,\\nu ,\\mu (n-1)|\\dot{\\lambda },\\dot{\\nu },\\dot{\\mu }(n-1)} &=& -\\left(h^\\nu _{\\ \\dot{\\alpha }}y^\\lambda y^{\\mu (n-1)}+(n-1)h^\\mu _{\\ \\dot{\\alpha }}y^\\lambda y^\\nu y^{\\mu (n-2)}\\right){}\\nonumber \\times \\\\&&\\times \\left(\\epsilon ^{\\dot{\\nu }\\dot{\\lambda }}\\overline{y}^{\\dot{\\alpha }}\\overline{y}^{\\dot{\\mu }(n-1)} + (n-1)\\epsilon ^{\\dot{\\mu }\\dot{\\lambda }}\\overline{y}^{\\dot{\\alpha }}\\overline{y}^{\\dot{\\nu }}\\overline{y}^{\\dot{\\mu }(n-2)}\\right)\\,,\\\\T_4^{\\lambda ,\\nu ,\\mu (n-1)|\\dot{\\lambda },\\dot{\\nu },\\dot{\\mu }(n-1)} &=& -\\left(h_\\alpha ^{\\ \\dot{\\nu }}\\overline{y}^{\\dot{\\lambda }} \\overline{y}^{\\dot{\\mu }(n-1)}+(n-1)h_{\\alpha }^{\\ \\dot{\\nu }} \\overline{y}^{\\dot{\\lambda }} \\overline{y}^{\\dot{\\nu }} \\overline{y}^{\\dot{\\mu }(n-2)}\\right){}\\nonumber \\times \\\\&&\\times \\left(\\epsilon ^{\\nu \\lambda } y^{\\alpha } y^{\\mu (n-1)} + (n-1)\\epsilon ^{\\mu \\lambda } y^{\\alpha } y^{\\nu } y^{\\mu (n-2)}\\right).$ Projecting onto the irreducible parts of $\\Theta (y,\\overline{y})$ we find $\\Delta _{N=\\overline{N}}\\left(\\Theta _{\\mathrm {A}}\\right)&=0\\,,\\\\\\Delta _{N=\\overline{N}}\\left(\\Theta _{\\mathrm {B}}\\right)&=(n+1)^2\\Theta _{\\mathrm {B}}\\ne 0\\,,\\\\\\Delta _{N=\\overline{N}}\\left(\\Theta _{\\text{C}}\\right)&=(n+1)^2\\Theta _{\\text{C}}\\ne 0\\,,\\\\\\Delta _{N=\\overline{N}}\\left(\\Theta _{\\mathrm {D}}\\right)&=0.$ Thus, the only elements of the kernel of $\\Delta ^\\text{1-forms}\\Big |_{N=\\overline{N}}$ are $\\Theta _{\\mathrm {A}}(y,\\overline{y})$ and $\\Theta _{\\mathrm {D}}(y,\\overline{y})$ .","By the Hodge theorem of Section 4 this yields that $H^1(\\sigma _-) = \\ker \\left(\\Delta ^\\text{1-forms}\\Big |_{N=\\overline{N}}\\right)$ is $H^1(\\sigma _-) &= \\bigoplus _{n\\ge 0} H^1_{(n)}(\\sigma _-),\\\\H^1_{(n)}(\\sigma _-) &= \\Big \\lbrace \\phi _{(n)}(y,\\overline{y}\\,|x) + \\phi _{(n)}^\\text{tr}(y,\\overline{y}\\,|x),\\\\\\phi _{(n)}(y,\\overline{y}\\,|x) &:= \\phi _{\\mu (n+1),\\dot{\\mu }(n+1)}(x)\\, h^{\\mu \\dot{\\mu }} \\, y^{\\mu (n)}\\overline{y}^{\\dot{\\mu }(n)},\\\\\\phi _{(n)}^\\text{tr}(y,\\overline{y}\\,|x) &:= \\phi ^\\mathrm {tr}_{\\mu (n-1),\\dot{\\mu }(n-1)}(x)\\, h_{\\nu \\dot{\\nu }} \\, y^\\nu y^{\\mu (n-1)}\\overline{y}^{\\dot{\\nu }}\\overline{y}^{\\dot{\\mu }(n-1)}\\,, \\text{ if } n>0 \\Big \\rbrace ,$ where $n$ is the number of indices of the corresponding cocycles.","Equivalently, $H^1(\\sigma _-) = \\Big \\lbrace h^{\\mu \\dot{\\mu }}\\,\\partial _\\mu \\overline{\\partial }_{\\dot{\\mu }}\\,F_1(y,\\overline{y}\\,|x) + h_{\\mu \\dot{\\mu }}\\,y^\\mu \\overline{y}^{\\dot{\\mu }} F_2(y,\\overline{y}\\,|x)\\Big \\rbrace ,$ where $F_{1,2}(y,\\overline{y}\\,|x)$ belongs to the diagonal $N=\\overline{N}$ , that is, $\\left(y^\\alpha \\frac{\\partial }{\\partial y^\\alpha } - \\overline{y}^{\\dot{\\alpha }}\\frac{\\partial }{\\partial \\overline{y}^{\\dot{\\alpha }}}\\right)F_{1,2}(y,\\overline{y}\\,|x) = 0.$ The fields $\\phi (y,\\overline{y})$ and $\\phi ^\\text{tr}(y,\\overline{y})$ exactly correspond to the irreducible components of the double-traceless Fronsdal field.","It remains to prove that there are no other nontrivial cocycles in $H^1(\\sigma _- )$ at $N\\ne \\overline{N}$ ."],["$H^1(\\sigma _-)$ in the far-from-diagonal sector {{formula:f3550507-9a9d-493c-9968-364e086c8c2f}}","Consider the action of the Laplace operator $\\Delta _{N>\\overline{N}+2}$ on general one-forms at $N>\\overline{N}+2$ $\\Delta _{N>\\overline{N}+2}\\Theta (y, \\overline{y}) =\\left(n(m+2) + y^\\beta _\\alpha h^\\alpha _{\\ \\dot{\\gamma }} D_\\beta ^{\\ \\dot{\\gamma }} + \\overline{y}^{\\dot{\\alpha }}\\overline{}_{\\dot{\\beta }}h_{\\gamma \\dot{\\alpha }}D^{\\gamma \\dot{\\beta }}\\right)\\Theta (y, \\overline{y}) =\\\\=\\underbrace{n(m+2)\\Theta (y, \\overline{y})}_{T_1(y,\\overline{y})} +\\underbrace{y^\\beta _\\alpha h^\\alpha _{\\ \\dot{\\gamma }} D_\\beta ^{\\ \\dot{\\gamma }}\\Theta (y, \\overline{y})}_{T_2(y,\\overline{y})} +\\underbrace{\\overline{y}^{\\dot{\\alpha }}\\overline{}_{\\dot{\\beta }}h_{\\gamma \\dot{\\alpha }}D^{\\gamma \\dot{\\beta }}\\Theta (y,\\overline{y})}_{T_3(y,\\overline{y})}.$ Analogously to (REF ), we denote $T_2^{\\lambda ,\\nu ,\\mu (n-1)|\\dot{\\lambda },\\dot{\\nu },\\dot{\\mu }(m-1)} = y^\\beta _\\alpha h^\\alpha _{\\ \\dot{\\gamma }} D_\\beta ^{\\ \\dot{\\gamma }} \\big ( h^{\\lambda \\dot{\\lambda }}y^{\\nu }y^{\\mu (n-1)}\\overline{y}^{\\dot{\\nu }}\\overline{y}^{\\dot{\\mu }(m-1)}\\big )$ and similarly for $T_1$ and $T_3$ .","In this sector we obtain $T_1^{\\lambda ,\\nu ,\\mu (n-1)|\\dot{\\lambda },\\dot{\\nu },\\dot{\\mu }(m-1)} &=n(m+2)h^{\\lambda \\dot{\\lambda }}y^{\\nu }y^{\\mu (n-1)}\\overline{y}^{\\dot{\\nu }}\\overline{y}^{\\dot{\\mu }(m-1)}\\,,\\\\T_2^{\\lambda ,\\nu ,\\mu (n-1)|\\dot{\\lambda },\\dot{\\nu },\\dot{\\mu }(m-1)} &=-h^{\\nu \\dot{\\lambda }}y^{\\lambda }y^{\\mu (n-1)}\\overline{y}^{\\dot{\\nu }}\\overline{y}^{\\dot{\\mu }(m-1)} - (n-1)h^{\\mu \\dot{\\lambda }}y^{\\lambda }y^{\\nu }y^{\\mu (n-2)}\\overline{y}^{\\dot{\\nu }}\\overline{y}^{\\dot{\\mu }(m-1)}\\,,\\\\T_3^{\\lambda ,\\nu ,\\mu (n-1)|\\dot{\\lambda },\\dot{\\nu },\\dot{\\mu }(m-1)} &= - h^{\\lambda }_{\\ \\dot{\\alpha }}\\epsilon ^{\\dot{\\nu }\\dot{\\lambda }}y^{\\nu }y^{\\mu (n-1)}\\overline{y}^{\\dot{\\alpha }}\\overline{y}^{\\dot{\\mu }(m-1)} - (m-1) h^{\\lambda }_{\\ \\dot{\\alpha }}\\epsilon ^{\\dot{\\lambda }\\dot{\\mu }}y^{\\nu }y^{\\mu (n-1)}\\overline{y}^{\\dot{\\alpha }}\\overline{y}^{\\dot{\\nu }}\\overline{y}^{\\dot{\\mu }(m-2)}\\,.$ Projection onto the irreducible components according to (REF ) yields $\\Delta _{N>\\overline{N}+2}\\left(\\Theta _\\mathrm {A}\\right) &= n(m+1)\\Theta _\\mathrm {A}\\ne 0\\,,\\\\\\Delta _{N>\\overline{N}+2}\\left(\\Theta _\\mathrm {B}\\right) &= (nm+2n+1)\\Theta _\\mathrm {B}\\ne 0\\,,\\\\\\Delta _{N>\\overline{N}+2}\\left(\\Theta _\\mathrm {C}\\right) &= (nm-n+2m)\\Theta _\\mathrm {C}\\ne 0\\,,\\\\\\Delta _{N>\\overline{N}+2}\\left(\\Theta _\\mathrm {D}\\right) &= (nm+2n-m+4)\\Theta _\\mathrm {D}\\ne 0\\,.$ Thus, there are no nontrivial cocycles in this sector.","For $N<\\overline{N}-2$ the computation is analogous.","Thus, $H^1(\\sigma _-) =0$ in the far-from-diagonal sector."],["Subtlety in the near-diagonal sector $|N-\\overline{N}|=2$","In this case we face certain peculiarity.","Denote the space of $p$ -forms ($p=1$ for $H^{1}$ ) with $N$ chiral and $\\overline{N}$ anti-chiral indices by $\\mathcal {V}_{(N,\\overline{N})}$ .","Recall that the grading operator is $G=|N-\\overline{N}|$ .","Consider the case with $N-\\overline{N}=2$ .","Namely, let $N=n+1$ and $\\overline{N}=n-1$ .","At $G=2$ the operator $\\sigma _-$ maps a state $X\\in \\mathcal {V}_{(n+1,n-1)}$ onto the diagonal, $\\sigma _-(X)\\in \\mathcal {V}_{(n,n)}$ , where in accordance with (REF ), $\\sigma _+$ acts 'both up and down': $\\mathcal {V}_{(n+1,n-1)} \\xrightarrow{} \\mathcal {V}_{(n,n)} \\xrightarrow{} \\mathcal {V}_{(n-1,n+1)} \\oplus \\mathcal {V}_{(n+1,n-1)}\\,.$ Thus, $\\mathcal {V}_{(n+1,n-1)} \\xrightarrow{}\\mathcal {V}_{(n-1,n+1)} \\oplus \\mathcal {V}_{(n+1,n-1)}\\,.$ As a result, $\\Delta _{(n+1,n-1)} : \\mathcal {V}_{(n+1,n-1)} \\longrightarrow \\mathcal {V}_{(n-1,n+1)} \\oplus \\mathcal {V}_{(n+1,n-1)}\\,,\\\\\\Delta _{(n-1,n+1)} : \\mathcal {V}_{(n-1,n+1)} \\longrightarrow \\mathcal {V}_{(n-1,n+1)} \\oplus \\mathcal {V}_{(n+1,n-1)}\\,.$ Consequently, $\\ker (\\Delta )$ should be searched in the form of a linear combination of the vectors both from $\\mathcal {V}_{(n+1,n-1)}$ and from $\\mathcal {V}_{(n-1,n+1)}$ .","Indeed, let $X$ be a vector in $\\mathcal {V}_{(n+1,n-1)}$ .","Consider the complex conjugated vector $\\overline{X}\\in \\mathcal {V}_{(n-1,n+1)}$ and compute the action of the Laplacian on them.","Let $\\Delta X = \\Delta _{(n+1,n-1)}X=\\alpha (n) X + \\beta (n) \\overline{X}\\,,\\\\\\Delta \\overline{X} = \\Delta _{(n-1,n+1)}\\overline{X}= \\gamma (n) X + \\delta (n) \\overline{X}$ with some coefficients $\\alpha $ , $\\beta $ , $\\gamma $ , and $\\delta $ .","That $X$ and $\\overline{X}$ are conjugated and operator $\\Delta $ is self-adjoint implies the relations $\\alpha = \\overline{\\delta }$ and $\\beta =\\overline{\\gamma }$ .","Looking for $\\ker (\\Delta )$ in the form $Y=F(n)X+G(n)\\overline{X}\\in \\ker (\\Delta )$ and acting on $Y$ by the Laplace operator we find that the condition $\\Delta Y =0$ yields $\\Delta Y = F(n)\\Delta _{(n+1,n-1)}X + G(n)\\Delta _{(n-1,n+1)}\\overline{X} =\\\\=\\big (\\alpha (n)F(n)+\\overline{\\beta (n)}G(n)\\big )X + \\big (\\beta (n)F(n)+\\overline{\\alpha (n)}G(n)\\big )\\overline{X} = 0\\,.$ Since $X$ and $\\overline{X}$ are linearly independent, the problem of finding such $Y=\\alpha X+\\beta \\overline{X}$ that $\\Delta Y=0$ , amounts to the linear system $\\begin{bmatrix}\\alpha (n) & \\overline{\\beta (n)} \\\\\\beta (n) & \\overline{\\alpha (n)}\\end{bmatrix}\\begin{bmatrix}F(n) \\\\G(n)\\end{bmatrix}=\\begin{bmatrix}0 \\\\0\\end{bmatrix}\\,,$ which admits non-trivial solutions iff $\\det \\begin{bmatrix}\\alpha (n) & \\overline{\\beta (n)} \\\\\\beta (n) & \\overline{\\alpha (n)}\\end{bmatrix}=|\\alpha (n)|^2 - |\\beta (n)|^2 = 0.$ Hence, we conclude that $\\alpha (n)=\\beta (n)\\cdot e^{i\\chi }, \\quad \\quad \\chi \\in [0,2\\pi )\\,.$ In the next section coefficients $\\alpha (n)$ and $\\beta (n)$ will be shown to be real, i.e., $e^{i\\chi }=\\pm 1$ .","Summarizing, if we find that the coefficients $\\alpha (n)$ and $\\beta (n)$ coincide up to a sign, $\\alpha (n)=\\pm \\beta (n)$ , this would imply the existence of a non-trivial $\\sigma _-$ -cocycle $Y = X \\mp \\overline{X}\\,.$ Otherwise the cohomology is trivial."],["$H^1(\\sigma _-)$ in the near-diagonal sector {{formula:1655b858-2c9a-47bf-8194-18e799b3a71f}}","To compute $H^1(\\sigma _-)$ in the leftover sector of $N=\\overline{N}+2$ (analysis at $N=\\overline{N}-2$ is analogous) consider a general one-form $\\Theta (y,\\overline{y})$ (REF ) with $N=\\overline{N}+2$ .","In this sector, the Laplacian differs form that at $N>\\overline{N}+2$ by the $T_4(y,\\overline{y})$ term in $\\Delta _{N=\\overline{N}+2}\\Theta (y, \\overline{y}) = \\Big (\\underbrace{N(\\overline{N}+2) + y^\\beta _\\alpha h^\\alpha _{\\ \\dot{\\gamma }} D_\\beta ^{\\ \\dot{\\gamma }} + \\overline{y}^{\\dot{\\alpha }}\\overline{}_{\\dot{\\beta }}h_{\\gamma \\dot{\\alpha }}D^{\\gamma \\dot{\\beta }}}_{\\Delta _{N>\\overline{N}+2}}\\Big )\\Theta (y, \\overline{y}) +\\underbrace{\\overline{y}^{\\dot{\\alpha }}\\overline{y}^{\\dot{\\beta }}_\\alpha _\\beta h^\\beta _{\\ \\dot{\\beta }}D^\\alpha _{\\ \\dot{\\alpha }}\\Theta (y, \\overline{y})}_{T_4(y,\\overline{y})}.$ Consequently, it is essential to compute the action of this additional term.","As before (cf.","(REF )), denote $T_4^{\\lambda ,\\nu ,\\mu (n-1)|\\dot{\\lambda },\\dot{\\nu },\\dot{\\mu }(m-1)} = \\overline{y}^{\\dot{\\alpha }}\\overline{y}^{\\dot{\\beta }}_\\alpha _\\beta h^\\beta _{\\ \\dot{\\beta }}D^\\alpha _{\\ \\dot{\\alpha }} \\big (h^{\\lambda \\dot{\\lambda }}y^{\\nu }y^{\\mu (n-1)}\\overline{y}^{\\dot{\\nu }}\\overline{y}^{\\dot{\\mu }(m-1)}\\big )\\,.$ This yields $T_4^{\\lambda ,\\nu ,\\mu (n-1)|\\dot{\\lambda },\\dot{\\nu },\\dot{\\mu }(m-1)}=(n-1)\\epsilon ^{\\mu \\lambda }h^{\\nu }_{\\ \\dot{\\beta }}y^{\\mu (n-2)}\\overline{y}^{\\dot{\\lambda }}\\overline{y}^{\\dot{\\beta }}\\overline{y}^{\\dot{\\nu }}\\overline{y}^{\\dot{\\mu }(m-1)} + (n-1)\\epsilon ^{\\nu \\lambda }h^{\\mu }_{\\ \\dot{\\beta }}y^{\\mu (n-2)}\\overline{y}^{\\dot{\\lambda }}\\overline{y}^{\\dot{\\beta }}\\overline{y}^{\\dot{\\nu }}\\overline{y}^{\\dot{\\mu }(m-1)}+\\\\+(n-1)(n-2)\\epsilon ^{\\mu \\lambda }h^{\\mu }_{\\ \\dot{\\beta }}y^{\\nu }y^{\\mu (n-3)}\\overline{y}^{\\dot{\\lambda }}\\overline{y}^{\\dot{\\beta }}\\overline{y}^{\\dot{\\nu }}\\overline{y}^{\\dot{\\mu }(m-1)}.$ Projecting $T_4$ onto the irreducible parts, we find: $\\text{(A)}:&& {S}_{(\\lambda ,\\nu ,\\mu )}{S}_{(\\dot{\\lambda },\\dot{\\nu },\\dot{\\mu })}T_4^{\\lambda ,\\nu ,\\mu (n-1)|\\dot{\\lambda },\\dot{\\nu },\\dot{\\mu }(m-1)}&=0\\,,\\\\\\text{(B)}:&& \\ \\ {S}_{(\\dot{\\lambda },\\dot{\\nu },\\dot{\\mu })}\\epsilon _{\\lambda \\nu }T_4^{\\lambda ,\\nu ,\\mu (n-1)|\\dot{\\lambda },\\dot{\\nu },\\dot{\\mu }(m-1)} &= -(n-1)(2n-1)h^{\\mu }_{\\ \\dot{\\beta }} y^{\\mu (n-2)}\\overline{y}^{\\dot{\\beta }}\\overline{y}^{\\dot{\\mu }(m+1)}\\,,\\\\\\text{(C)}:&& \\ \\ \\epsilon _{\\dot{\\lambda }\\dot{\\nu }}{S}_{(\\lambda ,\\nu ,\\mu )}T_4^{\\lambda ,\\nu ,\\mu (n-1)|\\dot{\\lambda },\\dot{\\nu },\\dot{\\mu }(m-1)}&=0\\,,\\\\\\text{(D)}:&& \\ \\ \\epsilon _{\\dot{\\lambda }\\dot{\\nu }}\\epsilon _{\\lambda \\nu }T_4^{\\lambda ,\\nu ,\\mu (n-1)|\\dot{\\lambda },\\dot{\\nu },\\dot{\\mu }(m-1)}&=0\\,.$ We observe that the action of the Laplacian in this sector differs from the previously computed one only in the type-(B) sector, namely, $\\Delta _{N=\\overline{N}+2}\\left(\\Theta _\\mathrm {B}\\right) = \\left(n(m+3)\\Theta _\\mathrm {B}+(2n^2-3n+1)\\Theta _\\mathrm {C}\\right)\\Big |_{m=n-2}=\\underbrace{(n^2+n)}_{\\alpha (n)}\\Theta _\\mathrm {B}+\\underbrace{(2n^2-3n+1)}_{\\beta (n)}\\Theta _\\mathrm {C}\\,.$ That $|\\alpha (n)|\\ne |\\beta (n)|$ at integer $n$ implies triviality of $H^1(\\sigma _-)$ in the near-diagonal sector."],["$H^2(\\sigma _-)$","The calculation of $H^2(\\sigma _-)$ is in main features analogous to that of $H^1(\\sigma _-)$ .","To decompose a general two-form $\\Omega (y,\\overline{y}\\,|x)$ into irreducible parts we use the following useful identity $h^{\\nu \\dot{\\nu }}\\wedge h^{\\lambda \\dot{\\lambda }} = \\frac{1}{2}H^{\\nu \\lambda }\\epsilon ^{\\dot{\\nu }\\dot{\\lambda }}+\\frac{1}{2}\\overline{H}^{\\dot{\\nu }\\dot{\\lambda }}\\epsilon ^{\\nu \\lambda },$ where $H^{\\nu \\lambda }=H^{(\\nu \\lambda )}:=h^{\\nu }_{\\ \\dot{\\gamma }}\\wedge h^{\\lambda \\dot{\\gamma }}\\,,\\qquad \\overline{H}^{\\dot{\\nu }\\dot{\\lambda }}=H^{(\\dot{\\nu }\\dot{\\lambda })}:=h^{\\ \\dot{\\nu }}_{\\gamma }\\wedge h^{\\gamma \\dot{\\lambda }}\\,.$ The decomposition reads $\\Omega (y,\\overline{y}\\, |x) = \\underbrace{\\Omega ^\\mathrm {A}_{\\mu (n+2)|\\dot{\\mu }(m)}\\ H^{\\mu \\mu }y^{\\mu (n)}\\overline{y}^{\\dot{\\mu }(m)}}_{\\Phi _{\\mathrm {A}(n,m)}}+\\underbrace{\\overline{\\Omega }^\\mathrm {A}_{\\mu (n)|\\dot{\\mu }(m+2)}\\ \\overline{H}^{\\dot{\\mu }\\dot{\\mu }}y^{\\mu (n)}\\overline{y}^{\\dot{\\mu }(m)}}_{\\overline{\\Phi }_{\\mathrm {A}(n,m)}}+\\\\+\\underbrace{\\Omega ^\\mathrm {B}_{\\mu (n-2)|\\dot{\\mu }(m)}\\ H_{\\nu \\nu }y^\\nu y^\\nu y^{\\mu (n-2)}\\overline{y}^{\\dot{\\mu }(m)}}_{\\Phi _{\\mathrm {B}(n,m)}}+\\underbrace{\\overline{\\Omega }^\\mathrm {B}_{\\mu (n)|\\dot{\\mu }(m-2)}\\overline{H}_{\\dot{\\nu }\\dot{\\nu }} y^{\\mu (n)}\\overline{y}^{\\dot{\\nu }}\\overline{y}^{\\dot{\\nu }}\\overline{y}^{\\dot{\\mu }(m-2)}}_{\\overline{\\Phi }_{\\mathrm {B}(n,m)}}+\\\\+\\underbrace{\\Omega ^\\mathrm {C}_{\\mu (n)|\\dot{\\mu }(m)}\\ H_{\\nu }^{\\ \\mu }y^{\\nu }y^{\\mu (n-1)}\\overline{y}^{\\dot{\\mu }(m)}}_{\\Phi _{\\mathrm {C}(n,m)}}+\\underbrace{\\overline{\\Omega }^\\mathrm {C}_{\\mu (n)|\\dot{\\mu }(m)}\\ \\overline{H}_{\\dot{\\nu }}^{\\ \\dot{\\mu }}y^{\\mu (n)}\\overline{y}^{\\dot{\\nu }}\\overline{y}^{\\dot{\\mu }(m-1)}}_{{\\overline{\\Phi }_{\\mathrm {C}(n,m)}}}.$ Consider now the reducible two-forms $\\Phi ^{\\lambda (2),\\nu (2),\\mu (n-2)|\\dot{\\mu }(m)} = H^{\\lambda \\lambda }y^{\\nu }y^{\\nu }y^{\\mu (n-2)}\\overline{y}^{\\dot{\\mu }(m)},\\\\\\overline{\\Phi }^{\\mu (n)|\\dot{\\lambda }(2),\\dot{\\nu }(2),\\dot{\\mu }(m-2)} = \\overline{H}^{\\dot{\\lambda }\\dot{\\lambda }}y^{\\mu (n)}\\overline{y}^{\\dot{\\nu }}\\overline{y}^{\\dot{\\nu }}\\overline{y}^{\\dot{\\mu }(m-2)}.","$ In these terms, the projectors onto irreducible components are $\\mathcal {P}_\\mathrm {A}&={S}_{(\\lambda ,\\nu ,\\mu )}\\,, && \\mathcal {P}_\\mathrm {B} = \\epsilon _{\\lambda \\nu }\\epsilon _{\\lambda \\nu }\\,, && \\mathcal {P}_\\mathrm {C} = {S}_{(\\lambda ,\\nu ,\\mu )}\\circ \\epsilon _{\\lambda \\nu }\\,,\\\\\\overline{\\mathcal {P}}_\\mathrm {A}&={S}_{(\\dot{\\lambda },\\dot{\\nu },\\dot{\\mu })}\\,, && \\overline{\\mathcal {P}}_\\mathrm {B}=\\epsilon _{\\dot{\\lambda }\\dot{\\nu }}\\epsilon _{\\dot{\\lambda }\\dot{\\nu }}\\,, && \\overline{\\mathcal {P}}_\\mathrm {C}={S}_{(\\dot{\\lambda },\\dot{\\nu },\\dot{\\mu })}\\circ \\epsilon _{\\dot{\\lambda }\\dot{\\nu }}\\,$ and the decomposition (REF ) reads as $\\Phi _\\mathrm {A}&=\\mathcal {P}_\\mathrm {A}\\Phi \\,, && \\Phi _\\mathrm {B}=\\mathcal {P}_\\mathrm {B}\\Phi \\,, && \\Phi _\\mathrm {C}=\\mathcal {P}_\\mathrm {C}\\Phi \\,, \\\\ \\overline{\\Phi }_\\mathrm {A}&=\\overline{\\mathcal {P}}_\\mathrm {A}\\overline{\\Phi }\\,, && \\overline{\\Phi }_\\mathrm {B}=\\overline{\\mathcal {P}}_\\mathrm {B}\\overline{\\Phi }\\,, && \\overline{\\Phi }_\\mathrm {C}=\\overline{\\mathcal {P}}_\\mathrm {C}\\overline{\\Phi }\\,.$ For practical calculations we have to find the result of the action of the operator $D=\\frac{\\partial }{\\partial h}$ on the two-form $H$ .","The result is $D_{\\alpha \\dot{\\beta }}H^{\\nu \\lambda } = D_{\\alpha \\dot{\\beta }}\\left(h^\\nu _{\\ \\dot{\\gamma }}\\wedge h^{\\lambda \\dot{\\gamma }}\\right) = \\epsilon _\\alpha ^{\\ \\nu }\\epsilon _{\\dot{\\beta }\\dot{\\gamma }} h^{\\lambda \\dot{\\gamma }} - h^{\\nu }_{\\ \\dot{\\gamma }}\\,\\epsilon _\\alpha ^{\\ \\lambda }\\epsilon _{\\dot{\\beta }}^{\\ \\dot{\\gamma }} = - \\epsilon _\\alpha ^{\\ \\nu } h^\\lambda _{\\ \\dot{\\beta }} - \\epsilon _\\alpha ^{\\ \\lambda }h^\\nu _{\\ \\dot{\\beta }} = -2\\epsilon _\\alpha ^{\\ (\\nu } h^{\\lambda )}_{\\ \\dot{\\beta }}\\,,$ or, in the condensed notation for symmetrized indices, $D_{\\alpha \\dot{\\beta }}H^{\\nu \\nu }=-2\\epsilon _\\alpha ^{\\ \\nu }h^{\\nu }_{\\ \\dot{\\beta }}.$"],["$H^2(\\sigma _-)$ in the far-from-diagonal sector {{formula:76b17993-ba2a-4ad4-89f9-a21723194d83}}","Compute $\\Delta _{N>\\overline{N}+2}$ on the general two-forms $\\Phi $ and $\\overline{\\Phi }$ , $\\Delta _{N>\\overline{N}+2}(\\Phi )=\\underbrace{n(m+2)\\Phi (y,\\overline{y})}_{T_1(y,\\overline{y})} + \\underbrace{y^\\beta _\\alpha h^\\alpha _{\\ \\dot{\\gamma }} D_\\beta ^{\\ \\dot{\\gamma }}\\Phi (y,\\overline{y})}_{T_2(y,\\overline{y})} + \\underbrace{\\overline{y}^{\\dot{\\alpha }}\\overline{}_{\\dot{\\beta }}h_{\\gamma \\dot{\\alpha }}D^{\\gamma \\dot{\\beta }}\\Phi (y,\\overline{y})}_{T_3(y,\\overline{y})}\\,.$ As in (REF ), denote $T_2^{\\lambda (2),\\nu (2),\\mu (n-2)|\\dot{\\mu }(m)} = y^\\beta _\\alpha h^\\alpha _{\\ \\dot{\\gamma }} D_\\beta ^{\\ \\dot{\\gamma }} H^{\\lambda \\lambda }y^{\\nu }y^{\\nu }y^{\\mu (n-2)}\\overline{y}^{\\dot{\\mu }(m)}$ and similarly for $T_1$ and $T_3$ .","Straightforward computation yields $T_1^{\\lambda (2),\\nu (2),\\mu (n-2)|\\dot{\\mu }(m)} &= n(m+2) \\, H^{\\lambda \\lambda }y^{\\nu }y^{\\nu }y^{\\mu (n-2)}\\overline{y}^{\\dot{\\mu }(m)}\\,,\\\\T_2^{\\lambda (2),\\nu (2),\\mu (n-2)|\\dot{\\mu }(m)} &=-4 H^{\\nu \\lambda } y^{\\lambda }y^{\\nu }y^{\\mu (n-2)}\\overline{y}^{\\dot{\\mu }(m)} - 2(n-2)\\,H^{\\mu \\lambda }y^{\\lambda }y^{\\nu (2)}y^{\\mu (n-3)}\\overline{y}^{\\dot{\\mu }(m)}\\,,\\\\T_3^{\\lambda (2),\\nu (2),\\mu (n-2)|\\dot{\\mu }(m)} &=m\\, H^{\\lambda \\lambda }y^{\\nu (2)} y^{\\mu (n-2)}\\overline{y}^{\\dot{\\mu }(m)}.$ Projecting onto the irreducible part $\\Phi _\\mathrm {A}$ we obtain $\\Delta _{N>\\overline{N}+2}\\Phi _\\mathrm {A}=\\mathcal {P}_\\mathrm {A}\\left(T_1+T_2+T_3\\right)=\\left[n(m+2)-2n+m\\right]\\Phi _\\mathrm {A}=m(n+1)\\Phi _\\mathrm {A}.$ We see that $\\Phi _\\mathrm {A}\\in \\ker (\\Delta )$ whenever $m=0$ .","This gives a 2-cocycle of the form $H^{\\mu \\mu }y^{\\mu (n)}$ .","It can be represented in terms of the generating function as follows.","Contract all the indices in $H^{\\mu \\mu }y^{\\mu (n)}$ with some symmetric coefficients $\\Omega ^\\mathrm {A}_{\\mu \\mu \\mu (n)}$ to obtain $\\Omega ^\\mathrm {A}_{\\mu (n+2)}H^{\\mu \\mu }y^{\\mu (n)}\\equiv h^{(\\lambda }_{\\ \\dot{\\gamma }}\\wedge h^{\\nu \\dot{\\gamma })}\\Omega _{(\\lambda \\nu \\mu (n))}y^{\\mu (n)}\\Rightarrow h^{\\lambda }_{\\ \\dot{\\gamma }}\\wedge h^{\\nu \\dot{\\gamma }}\\,_\\lambda _\\nu C(y,0\\, |x) \\in \\ker \\left(\\Delta _{N>\\overline{N}+2}\\Big |_\\text{2-forms}\\right),$ where $C(y,0\\, |x)=\\Omega _{\\mu \\mu \\mu (n)} y^\\mu y^{\\mu } y^{\\mu (n)}$ .","Summarizing, we found a part of the kernel of $\\Delta $ represented by the two-forms $W(y,0\\,|x)=H^{\\mu \\nu }_\\mu _\\nu C(y,0\\,|x)$ with $C(y,0|x)$ being a general polynomial of $y$ 's of degree $\\ge 4$ .","Let us now project (REF ) onto the second irreducible part $\\Phi _\\mathrm {B}$ , $\\Delta _{N>\\overline{N}+2}\\Phi _\\mathrm {B}=\\mathcal {P}_\\mathrm {B}\\left(T_1+T_2+T_3\\right)=\\left[n(m+2)+0+m\\right]\\Phi _\\mathrm {B}\\ne 0\\quad \\forall n,m\\in \\mathbb {N}_0.$ Since $\\Phi _{B(n,m)}$ is proportional to $H_{\\nu \\nu }y^\\nu y^\\nu y^{\\mu (n-2)}\\overline{y}^{\\dot{\\mu }(m)}$ , the case $n=m=0$ is beyond the allowed region.","Thus, $\\Phi _\\mathrm {B}$ does not contribute to $H^2(\\sigma _-)$ .","Projecting (REF ) onto $\\Phi _\\mathrm {C}$ , we find $\\Delta _{N>\\overline{N}+2}\\Phi _\\mathrm {C}=\\mathcal {P}_\\mathrm {C}\\left(T_1+T_2+T_3\\right) = [n(m+2)-2-(n-2)+m]\\Phi _\\mathrm {C} = (nm+n+m)\\Phi _\\mathrm {C}\\,.$ Again, $\\Phi _\\mathrm {C}$ does not contribute to $H^2(\\sigma _-)$ since $m>0$ , $n\\ge 0$ .","Next, we consider the anti-holomorphic two-form $\\overline{\\Phi }$ .","The action of the Laplacian yields $\\Delta _{N>\\overline{N}+2}(\\overline{\\Phi })=\\underbrace{n(m+2)\\overline{\\Phi }(y,\\overline{y})}_{T_1(y,\\overline{y})} + \\underbrace{y^\\beta _\\alpha h^\\alpha _{\\ \\dot{\\gamma }} D_\\beta ^{\\ \\dot{\\gamma }}\\overline{\\Phi }(y,\\overline{y})}_{T_2(y,\\overline{y})} + \\underbrace{\\overline{y}^{\\dot{\\alpha }}\\overline{}_{\\dot{\\beta }}h_{\\gamma \\dot{\\alpha }}D^{\\gamma \\dot{\\beta }}\\overline{\\Phi }(y,\\overline{y})}_{T_3(y,\\overline{y})}\\,.$ As in (REF ) we set $T_2^{\\mu (n)|\\dot{\\lambda }(2),\\dot{\\nu }(2),\\dot{\\mu }(m-2)} = y^\\beta _\\alpha h^\\alpha _{\\ \\dot{\\gamma }} D_\\beta ^{\\ \\dot{\\gamma }}\\,\\overline{H}^{\\dot{\\lambda }\\dot{\\lambda }}y^{\\mu (n)}\\overline{y}^{\\dot{\\nu }}\\overline{y}^{\\dot{\\nu }}\\overline{y}^{\\dot{\\mu }(m-2)}$ and analogously for $T_1$ and $T_3$ .","The computation in components yields $T_1^{\\mu (n)|\\dot{\\lambda }(2),\\dot{\\nu }(2),\\dot{\\mu }(m-2)} &= n(m+2) \\overline{H}^{\\dot{\\lambda }\\dot{\\lambda }}y^{\\mu (n)}\\overline{y}^{\\dot{\\nu }}\\overline{y}^{\\dot{\\nu }}\\overline{y}^{\\dot{\\mu }(m-2)},\\\\T_2^{\\mu (n)|\\dot{\\lambda }(2),\\dot{\\nu }(2),\\dot{\\mu }(m-2)} &=-n\\,\\overline{H}^{\\dot{\\lambda }\\dot{\\lambda }} y^{\\mu (n)}\\overline{y}^{\\dot{\\nu }(2)}\\overline{y}^{\\dot{\\mu }(m-2)},\\\\T_3^{\\mu (n)|\\dot{\\lambda }(2),\\dot{\\nu }(2),\\dot{\\mu }(m-2)} &=-4\\,\\epsilon ^{\\dot{\\nu }\\dot{\\lambda }}\\overline{H}_{\\dot{\\alpha }}^{\\ \\dot{\\lambda }} y^{\\mu (n)}\\overline{y}^{\\dot{\\alpha }}\\overline{y}^{\\dot{\\nu }}\\overline{y}^{\\dot{\\mu }(m-2)} - 2(m-2)\\,\\epsilon ^{\\dot{\\mu }\\dot{\\lambda }}\\overline{H}_{\\dot{\\alpha }}^{\\ \\dot{\\lambda }}y^{\\mu (n)}\\overline{y}^{\\dot{\\alpha }}\\overline{y}^{\\dot{\\nu }(2)}\\overline{y}^{\\dot{\\mu }(m-3)}.$ Projecting onto the irreducible components we obtain $\\Delta _{N>\\overline{N}+2}\\overline{\\Phi }_\\mathrm {A} &= \\overline{\\mathcal {P}}_\\mathrm {A}\\left(T_1+T_2+T_3\\right)=[n(m+1)]\\overline{\\Phi }_\\mathrm {A}\\,,\\\\\\Delta _{N>\\overline{N}+2}\\overline{\\Phi }_\\mathrm {B} &= \\overline{\\mathcal {P}}_\\mathrm {B}\\left(T_1+T_2+T_3\\right)=(nm+n+4m)\\overline{\\Phi }_\\mathrm {B}\\,,\\\\\\Delta _{N>\\overline{N}+2}\\overline{\\Phi }_\\mathrm {C} &= \\overline{\\mathcal {P}}_\\mathrm {C}\\left(T_1+T_2+T_3\\right) =(nm+n-m)\\overline{\\Phi }_\\mathrm {C}\\,.$ The condition $n>m+2$ valid in the far-from-diagonal sector does not allow $\\overline{\\Phi }_\\mathrm {A,B,C}$ to be in the kernel of $ \\Delta $ .","The analysis of the opposite sector $N<\\overline{N}-2$ is analogous via swapping dotted and undotted indices.","As a result, the final answer for the under-diagonal sector is $W(0,\\overline{y}\\, |x)=\\overline{H}^{\\dot{\\mu }\\dot{\\nu }}\\overline{}_{\\dot{\\mu }}\\overline{}_{\\dot{\\nu }}C(0,\\overline{y}\\, |x).$ This completes the analysis of $H^2(\\sigma _-)$ in the sector $|N-\\overline{N}|>2$ .","The cohomology is represented by the two-forms $W(y,\\overline{y}\\, |x) = h^{\\mu }_{\\ \\dot{\\gamma }}\\wedge h^{\\nu \\dot{\\gamma }}\\,_\\mu _\\nu C(y,0\\, |x) + h_\\gamma ^{\\ \\dot{\\mu }}\\wedge h^{\\gamma \\dot{\\nu }}\\,\\overline{}_{\\dot{\\mu }}\\overline{}_{\\dot{\\nu }} C(0,\\overline{y}\\, |x).$ These two-forms are known to represent the so-called Weyl cocycle in the HS theory.","It is thus shown that there are no other non-trivial 2-cocycles in this sector."],["$H^2(\\sigma _-)$ on the diagonal {{formula:bf16bd86-cdaa-48f9-9718-beef40a13be5}}","Now we prove that there are no non-trivial cocycles at $N=\\overline{N}$ except for the Weyl cohomology (REF ).","As before, act by the operator $\\Delta _{N=\\overline{N}}$ on the two-form $\\Phi ^{\\lambda (2)|\\nu (2)|\\mu (n-2)|\\dot{\\mu }(n)}(y,\\overline{y})=H^{\\lambda \\lambda }y^{\\nu }y^{\\nu }y^{\\mu (n-2)}\\overline{y}^{\\dot{\\mu }(n)}$ $\\Delta _{N=\\overline{N}}\\Phi (y,\\overline{y})=\\underbrace{\\overline{y}^{\\dot{\\alpha }}\\overline{}_{\\dot{\\beta }} h_{\\gamma \\dot{\\alpha }} D^{\\gamma \\dot{\\beta }} \\Phi (y,\\overline{y})}_{T_1(y,\\overline{y})}+\\underbrace{y^\\alpha _\\beta h_{\\alpha \\dot{\\gamma }}D^{\\beta \\dot{\\gamma }}\\Phi (y,\\overline{y})}_{T_2(y,\\overline{y})}-\\\\\\underbrace{-\\overline{y}^{\\dot{\\alpha }} y^\\beta _\\alpha \\overline{}_{\\dot{\\beta }}h^\\alpha _{\\ \\dot{\\alpha }} D_\\beta ^{\\ \\dot{\\beta }}\\Phi (y,\\overline{y})}_{T_3(y,\\overline{y})}\\underbrace{-y^\\alpha \\overline{y}^{\\dot{\\beta }}_\\beta \\overline{}_{\\dot{\\alpha }}h_\\alpha ^{\\ \\dot{\\alpha }}D^\\beta _{\\ \\dot{\\beta }}\\Phi (y,\\overline{y})}_{T_4(y,\\overline{y})}\\,.$ Denoting $T_1^{\\lambda (2),\\nu (2),\\mu (n-2)|\\dot{\\mu }(n)} = \\overline{y}^{\\dot{\\alpha }}\\overline{}_{\\dot{\\beta }} h_{\\gamma \\dot{\\alpha }} D^{\\gamma \\dot{\\beta }}\\, H^{\\lambda \\lambda }y^{\\nu (2)}y^{\\mu (n-2)}\\overline{y}^{\\dot{\\mu }(n)}$ and analogously for $T_2$ , $T_3$ and $T_4$ , straightforward computation yields $T_1^{\\lambda (2),\\nu (2),\\mu (n-2)|\\dot{\\mu }(n)} & = & n\\, H^{\\lambda \\lambda }y^{\\nu (2)}y^{\\mu (n-2)}\\overline{y}^{\\dot{\\mu }(n)}\\,,\\\\T_2^{\\lambda (2),\\nu (2),\\mu (n-2)|\\dot{\\mu }(n)} & = & -4 \\epsilon ^{\\nu \\lambda } H_{\\alpha }^{\\ \\lambda } y^{\\alpha }y^{\\nu }y^{\\mu (n-2)}\\overline{y}^{\\dot{\\mu }(n)} -{}\\nonumber \\\\&& - 2(n-2)\\epsilon ^{\\mu \\lambda } H_{\\alpha }^{\\ \\lambda }y^{\\alpha }y^{\\nu (2)}y^{\\mu (n-3)}\\overline{y}^{\\dot{\\mu }(n)}\\,,\\\\T_3^{\\lambda (2),\\nu (2),\\mu (n-2)|\\dot{\\mu }(n)} & = & 4n\\, h^{\\nu }_{\\ \\dot{\\alpha }}\\wedge h^{\\lambda \\dot{\\mu }}\\, y^{\\lambda } y^{\\nu } y^{\\mu (n-2)}\\overline{y}^{\\dot{\\alpha }}\\overline{y}^{\\dot{\\mu }(n-2)} +{}\\nonumber \\\\& &+ 2n(n-2)\\, h^{\\mu }_{\\ \\dot{\\alpha }}\\wedge h^{\\lambda \\dot{\\mu }} y^{\\lambda }y^{\\nu (2)}y^{\\mu (n-3)}\\overline{y}^{\\dot{\\alpha }}\\overline{y}^{\\dot{\\mu }(n-1)}\\,,\\\\T_4^{\\lambda (2),\\nu (2),\\mu (n-2)|\\dot{\\mu }(n)} & = & 4n\\, \\epsilon ^{\\nu \\lambda }\\, h_{\\alpha }^{\\ \\dot{\\mu }}\\wedge h^{\\lambda }_{\\ \\dot{\\beta }}\\,y^{\\alpha }y^{\\nu }y^{\\mu (n-2)}\\overline{y}^{\\dot{\\beta }}\\overline{y}^{\\dot{\\mu }(n-1)} +{}\\nonumber \\\\& & + 2n(n-2)\\, \\epsilon ^{\\mu \\lambda }\\,h_{\\alpha }^{\\ \\dot{\\mu }}\\wedge h^{\\lambda }_{\\ \\dot{\\beta }}\\,y^{\\alpha }y^{\\nu (2)}y^{\\mu (n-3)}\\overline{y}^{\\dot{\\beta }}\\overline{y}^{\\dot{\\mu }(n-1)}.$ and $\\Delta _{N=\\overline{N}}\\Phi _\\mathrm {A}&= \\mathcal {P}_\\mathrm {A}(T_1+T_2+T_3+T_4) = n^2\\Phi _\\mathrm {A},\\\\\\Delta _{N=\\overline{N}}\\Phi _\\mathrm {B}&= \\mathcal {P}_\\mathrm {B}(T_1+T_2+T_3+T_4) =(2n^2+5n+4)\\Phi _\\mathrm {B}\\ne 0,\\\\\\Delta _{N=\\overline{N}}\\Phi _\\mathrm {C}&= \\mathcal {P}_\\mathrm {C}(T_1+T_2+T_3+T_4) = 2(n^2+n+1)\\Phi _\\mathrm {C} -2n(n+1)\\overline{\\Phi }_\\mathrm {C}.$ We observe that the only way for some of $\\Phi _\\text{A,B,C}$ to be in $H^2(\\sigma _-)$ is at $n=0$ .","But in the diagonal sector with $N=\\overline{N} = n$ this implies $N=\\overline{N}=0$ .","This case extends formula (REF ) to the spin-one $y,\\overline{y}$ -independent sector.","The analysis of the anti-holomorphic part $\\overline{\\Phi }$ is analogous.","The resulting cohomology parameterizes the spin-one field strength, i.e., Faraday field strength."],["$H^2(\\sigma _-)$ in the near-diagonal sector {{formula:f4e6b1a7-8a76-4307-9345-3c00844fb11a}}","In the near-diagonal sector a subtlety considered in Section 7.2.3 takes place.","We should search for a kernel of $\\Delta $ in the form of a linear combination of the two-forms lying under the diagonal and above the diagonal.","Our strategy is to act separately on the general holomorphic (REF ) and anti-holomorphic () two-forms placed below the diagonal $N=\\overline{N}-2$ and then determine which two-forms are in $\\ker (\\Delta )$ .","(The computation with $N>\\overline{N}$ only differs by the complex conjugation.)","We start with the general holomorphic two-form below the diagonal $\\Phi _{(n-1,n+1)}^{\\lambda (2)|\\nu (2)|\\mu (n-3)|\\dot{\\mu }(n+1)}(y,\\overline{y})=H^{\\lambda \\lambda }y^{\\nu (2)}y^{\\mu (n-3)}\\overline{y}^{\\dot{\\mu }(n+1)}.$ Firstly, we set $n\\ge 4$ considering the cases of $n\\le 3$ , that are special in our computation scheme, because $n-3$ is the number of indices $\\mu $ , later.","This yields $\\Delta _{N=\\overline{N}-2}\\Phi _{(n-1,n+1)}(y,\\overline{y}) = \\underbrace{\\Delta _{N<\\overline{N}- 2}\\Phi _{(n-1,n+1)}(y,\\overline{y})}_{T_1(y,\\overline{y})} +\\underbrace{y^\\alpha y^\\beta \\overline{}_{\\dot{\\alpha }}\\overline{}_{\\dot{\\beta }}h_\\beta ^{\\ \\dot{\\beta }}D_\\alpha ^{\\ \\dot{\\alpha }}\\Phi _{(n-1,n+1)}(y,\\overline{y})}_{T_2(y,\\overline{y})}\\,.$ The first term is computed the same way as in (REF ) giving $T_1^{\\lambda (2),\\nu (2),\\mu (n-3)|\\dot{\\mu }(n+1)}= (n+1)^2 \\, H^{\\lambda \\lambda }y^{\\nu }y^{\\nu }y^{\\mu (n-3)}\\overline{y}^{\\dot{\\mu }(n+1)} -4 H^{\\nu \\lambda } y^{\\lambda }y^{\\nu }y^{\\mu (n-3)}\\overline{y}^{\\dot{\\mu }(n+1)} -\\\\-2(n-3)\\,H^{\\mu \\lambda }y^{\\lambda }y^{\\nu (2)}y^{\\mu (n-4)}\\overline{y}^{\\dot{\\mu }(n+1)} +(n-1)\\, H^{\\lambda \\lambda }y^{\\nu (2)} y^{\\mu (n-3)}\\overline{y}^{\\dot{\\mu }(n+1)}.$ The computation of the additional term $T_2(y,\\overline{y})$ yields $T_2^{\\lambda (2),\\nu (2),\\mu (n-3)|\\dot{\\mu }(n+1)} = y^\\alpha y^\\beta \\overline{}_{\\dot{\\alpha }}\\overline{}_{\\dot{\\beta }}h_\\beta ^{\\ \\dot{\\beta }}D_\\alpha ^{\\ \\dot{\\alpha }}H^{\\lambda \\lambda }y^{\\nu (2)}y^{\\mu (n-3)}\\overline{y}^{\\dot{\\mu }(n+1)} =\\\\=-2n(n+1)\\, h_{\\beta }^{\\ \\dot{\\mu }}\\wedge h^{\\lambda \\dot{\\mu }} y^{\\lambda }y^{\\beta }y^{\\nu (2)}y^{\\mu (n-3)}\\overline{y}^{\\dot{\\mu }(n-1)}=\\\\=-n(n+1)\\, \\overline{H}^{\\dot{\\mu }\\dot{\\mu }}y^{\\lambda (2)}y^{\\nu (2)}y^{\\mu (n-3)}\\overline{y}^{\\dot{\\mu }(n-1)}.$ Projection onto the irreducible parts $A$ , $B$ and $C$ yields $\\Delta \\Phi _{\\mathrm {A}(n-1,n+1)} &= \\underbrace{n(n+1)}_{\\alpha (n)}\\Phi _{\\mathrm {A}(n-1,n+1)} \\underbrace{ - n(n+1)}_{\\beta (n)}\\overline{\\Phi }_{\\mathrm {A}(n+1,n-1)}\\,,\\\\\\Delta \\Phi _{\\mathrm {B}(n-1,n+1)} &= (n^2+5n-4)\\Phi _{\\mathrm {B}(n-1,n+1)}\\,,\\\\\\Delta \\Phi _{\\mathrm {C}(n-1,n+1)} &= (n^2+2n-1)\\Phi _{\\mathrm {C}(n-1,n+1)}\\,.$ Let us stress that the complex conjugation denoted by $\\dagger $ swaps dotted and undotted indices $(y^\\alpha )^\\dagger = \\overline{y}^{\\dot{\\alpha }}, \\quad \\quad (H^{\\alpha \\alpha })^\\dagger = \\overline{H}^{\\dot{\\alpha }\\dot{\\alpha }}$ and relates $\\Phi $ and $\\overline{\\Phi }$ in the following way: $(\\Phi _{\\mathrm {A,B,C}(n-1,n+1)})^\\dagger = \\overline{\\Phi }_{\\mathrm {A,B,C}(n+1,n-1)}.$ The computation for the complex-conjugated objects $\\overline{\\Phi }_{\\mathrm {A,B,C}(n+1,n-1)}$ is analogous giving $\\Delta \\overline{\\Phi }_{\\mathrm {A}(n+1,n-1)} &= n(n+1)\\,\\overline{\\Phi }_{\\mathrm {A}(n+1,n-1)} - n(n+1)\\,\\Phi _{\\mathrm {A}(n-1,n+1)}\\,,\\\\\\Delta \\overline{\\Phi }_{\\mathrm {B}(n+1,n-1)} &= (n^2+5n-4)\\overline{\\Phi }_{\\mathrm {B}(n+1,n-1)}\\,,\\\\\\Delta \\overline{\\Phi }_{\\mathrm {C}(n+1,n-1)} &= (n^2+2n-1)\\overline{\\Phi }_{\\mathrm {C}(n+1,n-1)}\\,.$ From (REF ) and (REF ) we observe that there is a non-trivial 2-cocycle $\\mathcal {E}_\\mathrm {A}=\\Phi _{\\mathrm {A}(n-1,n+1)}+\\overline{\\Phi }_{\\mathrm {A}(n+1,n-1)}= \\mathcal {E}^\\mathrm {A}_{\\mu (n+1),\\dot{\\mu }(n+1)}\\,\\Big (H^{\\mu \\mu }y^{\\mu (n-1)}\\overline{y}^{\\dot{\\mu }(n+1)} + \\overline{H}^{\\dot{\\mu }\\dot{\\mu }}y^{\\mu (n+1)}\\overline{y}^{\\dot{\\mu }(n-1)}\\Big )$ with arbitrary coefficients $\\mathcal {E}^\\mathrm {A}_{\\mu (n+1),\\dot{\\mu }(n+1)}(x)$ .","This answer agrees with the analysis of Section 7.2.3.","Indeed, the coefficients on the r.h.s.","of (REF ) coincide up to a sign $\\alpha (n) = -\\beta (n)$ , and by (REF ) of Section 7.2.3 this implies a non-trivial 2-cocycle (REF ).","This cocycle represents the traceless part of the free Fronsdal HS equations.","The irreducible representations of types $(B)$ and $(C)$ do not contribute to cohomology since they are not in $\\ker (\\Delta )$ (recall that we are assuming $n\\ge 4$ ).","Now consider the cases of $n=1,2,3$ .","Computing the action of the Laplace operator on the following objects: $\\Phi _{(0,2)}^{\\lambda \\lambda |\\dot{\\mu }\\dot{\\mu }}(y,\\overline{y}) &= H^{\\lambda \\lambda }\\overline{y}^{\\dot{\\mu }}\\overline{y}^{\\dot{\\mu }},\\\\\\Phi _{(1,3)}^{\\lambda \\lambda |\\mu |\\dot{\\mu }(3)}(y,\\overline{y}) &= H^{\\lambda \\lambda }y^\\mu \\overline{y}^{\\dot{\\mu }(3)},\\\\\\Phi _{(2,4)}^{\\lambda \\lambda |\\nu \\nu |\\dot{\\mu }(4)}(y,\\overline{y}) &= H^{\\lambda \\lambda }y^\\nu y^\\nu \\overline{y}^{\\dot{\\mu }(4)}\\,,$ it is not difficult to obtain $(\\Delta \\Phi _{(0,2)})^{\\lambda \\lambda |\\dot{\\mu }\\dot{\\mu }}(y,\\overline{y}) &= 2 H^{\\lambda \\lambda }\\overline{y}^{\\dot{\\mu }(2)} - 2\\overline{H}^{\\dot{\\mu }\\dot{\\mu }}y^{\\lambda (2)},\\\\(\\Delta \\Phi _{(1,3)})^{\\lambda \\lambda |\\mu |\\dot{\\mu }(3)}(y,\\overline{y}) &= 8H^{\\lambda \\lambda }y^\\mu \\overline{y}^{\\dot{\\mu }(3)} - 2H^{\\mu \\lambda }y^\\lambda \\overline{y}^{\\dot{\\mu }(3)} - 6\\overline{H}^{\\dot{\\mu }\\dot{\\mu }}y^{\\lambda (2)}y^\\mu \\overline{y}^{\\dot{\\mu }},\\\\(\\Delta \\Phi _{(2,4)})^{\\lambda \\lambda |\\nu \\nu |\\dot{\\mu }(4)}(y,\\overline{y}) &= 16H^{\\lambda \\lambda }y^{\\nu \\nu }\\overline{y}^{\\dot{\\mu }(4)} - 4H^{\\nu \\lambda }y^\\lambda y^\\nu \\overline{y}^{\\dot{\\mu }(4)} - 12\\overline{H}^{\\dot{\\mu }\\dot{\\mu }}y^{\\lambda (2)}y^{\\nu (2)}\\overline{y}^{\\dot{\\mu }(2)}.$ We see that these results for $n=1,2,3$ extend the traceless part of the Fronsdal cohomology (REF ) to spins $s=2,3,4$ .","It remains to analyze the case of anti-holomorphic two-form below the diagonal $N=\\overline{N}-2$ $\\overline{\\Phi }_{(n-1,n+1)}^{\\dot{\\nu }(2)|\\mu (n-1)|\\dot{\\lambda }(2)|\\dot{\\mu }(n-1)}(y,\\overline{y}) = \\overline{H}^{\\dot{\\lambda }\\dot{\\lambda }}y^{\\mu (n-1)}\\overline{y}^{\\dot{\\nu }(2)}\\overline{y}^{\\dot{\\mu }(n-1)}.$ Unlike Eq.","(REF ), the number of indices $\\mu $ and $\\dot{\\mu }$ in (REF ) is $n-1$ , not $n-3$ .","Hence, there is no need to consider separately the cases of $n\\ge 4$ and $n\\le 3$ .","Instead, we set $n\\ge 2$ and then analyze the $n=1$ case separately.","Let $n\\ge 2$ .","The action of the corresponding Laplace operator on (REF ) yields $\\Delta _{N=\\overline{N}-2}\\overline{\\Phi }_{(n-1,n+1)}(y,\\overline{y}) = \\underbrace{\\Delta _{N<\\overline{N}+2}\\overline{\\Phi }_{(n-1,n+1)}(y,\\overline{y})}_{T_3(y,\\overline{y})} +\\underbrace{y^\\alpha y^\\beta \\overline{}_{\\dot{\\alpha }}\\overline{}_{\\dot{\\beta }}h_\\beta ^{\\ \\dot{\\beta }}D_\\alpha ^{\\ \\dot{\\alpha }}\\overline{\\Phi }_{(n-1,n+1)}(y,\\overline{y})}_{T_4(y,\\overline{y})}.$ The computation is completely analogous to that for the holomorphic two-form.","After projecting onto the irreducible components it gives $\\Delta \\overline{\\Phi }_{\\mathrm {A}(n-1,n+1)} &= (n^2+n-4)\\overline{\\Phi }_{\\mathrm {A}(n-1,n+1)}\\,,\\\\\\Delta \\overline{\\Phi }_{\\mathrm {B}(n-1,n+1)} &= \\underbrace{(n^2+4n-1)}_{\\alpha (n)}\\overline{\\Phi }_{\\mathrm {B}(n-1,n+1)} \\underbrace{ - (n^2+4n-1)}_{\\beta (n)}\\Phi _{\\mathrm {B}(n+1,n-1)}\\,,\\\\\\Delta \\overline{\\Phi }_{\\mathrm {C}(n-1,n+1)} &= (n^2+2n+1)\\overline{\\Phi }_{\\mathrm {C}(n-1,n+1)}\\,.$ Applying once again the result (REF ) of Section 7.2.3 to (REF ), on the r.h.s.","of which the coefficients coincide up to a sign, $\\alpha (n) = -\\beta (n)$ , we obtain the 2-cocycle of the form $\\mathcal {E}_\\mathrm {B}=\\Phi _{\\mathrm {B}(n+1,n-1)}+\\overline{\\Phi }_{\\mathrm {B}(n-1,n+1)} =\\\\= \\mathcal {E}^\\mathrm {B}_{\\mu (n-1)\\dot{\\mu }(n-1)}\\,\\Big (H_{\\nu \\nu }y^{\\nu (2)} y^{\\mu (n-1)}\\overline{y}^{\\dot{\\mu }(n-1)} + \\overline{H}_{\\dot{\\nu }\\dot{\\nu }}y^{\\mu (n-1)}\\overline{y}^{\\dot{\\nu }(2)}\\overline{y}^{\\dot{\\mu }(n-1)}\\big )\\,,$ that represents the trace part of the Fronsdal equations.","Having considered $n\\ge 2$ , now consider the case of $n=1$ .","Computation of the action of the Laplace operator on the following two-form: $\\overline{\\Phi }_{(0,2)}^{\\dot{\\lambda }\\dot{\\lambda }|\\dot{\\mu }\\dot{\\mu }}(y,\\overline{y}) = \\overline{H}^{\\dot{\\lambda }\\dot{\\lambda }}\\overline{y}^{\\dot{\\mu }}\\overline{y}^{\\dot{\\mu }}$ yields $(\\Delta \\overline{\\Phi }_{(0,2)})^{\\dot{\\lambda }\\dot{\\lambda }|\\dot{\\mu }\\dot{\\mu }}(y,\\overline{y}) = 4 \\overline{H}^{\\dot{\\lambda }\\dot{\\lambda }}\\overline{y}^{\\dot{\\mu }(2)}-4\\overline{H}^{\\dot{\\mu }\\dot{\\lambda }}\\overline{y}^{\\dot{\\lambda }}\\overline{y}^{\\dot{\\mu }} - 2 H_{\\alpha \\alpha }y^{\\alpha }y^{\\alpha }\\epsilon ^{\\dot{\\mu }\\dot{\\lambda }}\\epsilon ^{\\dot{\\mu }\\dot{\\lambda }}.$ After projecting onto the irreducible components, we find that the case of $n=1$ extends the trace part of the Fronsdal cocycle $\\mathcal {E}_\\mathrm {B}$ (REF ) to spin $s=2$ .","In addition, (REF ) also contributes to the antiholomorphic part of the Weyl cocycle represented by the second term on the r.h.s.","of (REF ).","The holomorphic part of the latter lies in the opposite (complex-conjugated) region, in which the analysis is completely analogous.","This 2-cocycle represents the Weyl tensor for the linearized gravity ($s=2$ ) in $AdS_4$ .","This completes the analysis of $H^2(\\sigma _-)$ in the near-diagonal sector $N=\\overline{N} \\pm 2$ ."],["Summary for bosonic $H^{0,1,2}(\\sigma _-)$","Here we collect the final results for the cocycles associated with the bosonic HS gauge parameters, fields and field equations in $AdS_4$ .","Recall that $H^0(\\sigma _-)$ represents parameters of the differential HS gauge symmetries.","It is spanned by the zero-forms $F(y,\\overline{y}|\\,x) = F_{\\alpha (n)\\,\\dot{\\alpha }(n)}(x)\\,y^{\\alpha (n)}\\overline{y}^{\\dot{\\alpha }(n)}\\,,\\qquad n\\in \\mathbb {N}_0\\,.$ $H^1(\\sigma _-)$ represents the dynamical HS fields.","For the bosonic HS fields in $AdS_4$ it is spanned by the two 1-cocycles $\\phi (y,\\overline{y}\\,|x)$ and $\\phi ^\\text{tr}(y,\\overline{y}\\,|x)$ corresponding, respectively, to the traceless and trace components of the original Fronsdal field in the metric formalism: $\\phi (y,\\overline{y}\\,|x) = h^{\\mu \\dot{\\mu }}\\,\\partial _\\mu \\overline{\\partial }_{\\dot{\\mu }}\\, F_1(y,\\overline{y}\\,|x),\\\\\\phi ^\\text{tr}(y,\\overline{y}\\,|x) = h_{\\mu \\dot{\\mu }}\\,y^\\mu \\overline{y}^{\\dot{\\mu }}\\, F_2(y,\\overline{y}\\,|x),$ where $F_{1,2}(y,\\overline{y}\\,|x)$ are $(N,\\overline{N})$ -diagonal, that is $\\left(y^\\alpha \\frac{\\partial }{\\partial y^\\alpha } - \\overline{y}^{\\dot{\\alpha }}\\frac{\\partial }{\\partial \\overline{y}^{\\dot{\\alpha }}}\\right)F_{1,2}(y,\\overline{y}\\,|x) = 0.$ Finally, $H^2(\\sigma _-)$ , which represents gauge invariant differential operators on the bosonic HS fields, are spanned by three different 2-cocycles: the so-called Weyl cocycle $W(y,\\overline{y}\\,|x)$ and two irreducible components of the Fronsdal cocycle $\\mathcal {E}_\\mathrm {A}(y,\\overline{y}\\,|x)$ (REF ) and $\\mathcal {E}_\\mathrm {B}(y,\\overline{y}\\,|x)$ (REF ).","The latter correspond to the ${\\it l.h.s.}","$ 's of the dynamical equations for the fields of spin $s>1$ (spin $s\\le 1$ field equations are in the zero-form sector of unfolded equations [12]).","Note that these cocycles are real since they contain equal numbers of dotted and undotted indices.","$W(y,\\overline{y}\\, |x) &= H^{\\mu \\nu } _\\mu _\\nu C(y,0\\, |x) + \\overline{H}^{\\dot{\\mu }\\dot{\\nu }} \\overline{}_{\\dot{\\mu }}\\overline{}_{\\dot{\\nu }} C(0,\\overline{y}\\, |x)\\,,\\\\\\mathcal {E}_\\mathrm {A}(y,\\overline{y}\\,|x) &= \\Big (H^{\\mu \\nu }\\partial _\\mu \\partial _\\nu + \\overline{H}^{\\dot{\\mu }\\dot{\\nu }}\\overline{\\partial }_{\\dot{\\mu }}\\overline{\\partial }_{\\dot{\\nu }}\\Big ) C^\\text{diag}(y,\\overline{y}\\,|x)\\,,\\\\\\mathcal {E}_\\mathrm {B}(y,\\overline{y}\\,|x) &= \\Big (H^{\\mu \\nu } y_\\mu y_\\nu + \\overline{H}^{\\dot{\\mu }\\dot{\\nu }}\\overline{y}_{\\dot{\\mu }}\\overline{y}_{\\dot{\\nu }}\\Big ) C^\\text{diag}(y,\\overline{y}\\,|x),$ where $C^\\text{diag}(y,\\overline{y})$ obey (REF )."],["Fermionic HS fields in $AdS_4$","So far we considered the bosonic case with even grading $G=|N-\\overline{N}|$ .","By (REF ) odd $G$ corresponds to fields of half-integer spins, i.e., oddness of $G$ determines the field statistics.","To extend the results for $H^p(\\sigma _-)$ to fermionic fields we should first define the operator $\\sigma _-$ on multispinors of odd ranks.","In the fermionic case the lowest possible odd grading is $G=|N-\\overline{N}|=1$ .","In this sector we define the action of $\\sigma _-$ to vanish.","In all other gradings $\\sigma _\\pm $ is defined analogously to the bosonic case.","Namely, $\\sigma _-\\omega (y,\\overline{y}) &:= i\\, \\overline{y}^{\\dot{\\alpha }}h^\\alpha _{\\ \\dot{\\alpha }}\\partial _\\alpha \\ \\omega (y,\\overline{y}), \\quad \\quad \\text{at } N\\ge \\overline{N}+3\\,,\\\\\\sigma _-\\omega (y,\\overline{y}) &:= i\\, y^{\\alpha }h_\\alpha ^{\\ \\dot{\\alpha }}\\overline{\\partial }_{\\dot{\\alpha }}\\ \\omega (y,\\overline{y}), \\quad \\quad \\text{at } N\\le \\overline{N}-3\\,.$ Analogously, the operator $\\sigma _+$ is defined as $\\sigma _+\\omega (y,\\overline{y}) &:= -i\\, y^\\alpha D_\\alpha ^{\\ \\dot{\\alpha }}\\overline{\\partial }_{\\dot{\\alpha }}\\ \\omega (y,\\overline{y}), \\quad \\quad \\text{at } N\\ge \\overline{N}+3\\,,\\\\\\sigma _+\\omega (y,\\overline{y}) &:= -i\\, \\overline{y}^{\\dot{\\alpha }}D^\\alpha _{\\ \\dot{\\alpha }}\\partial _\\alpha \\ \\omega (y,\\overline{y}), \\quad \\quad \\text{at } N\\le \\overline{N}-3\\,.$ For the lowest grading $|N-\\overline{N}|=1$ , $\\sigma _+$ is defined by $\\sigma _+ = -i\\left(y^\\alpha D_\\alpha ^{\\ \\dot{\\alpha }}\\overline{\\partial }_{\\dot{\\alpha }}+\\overline{y}^{\\dot{\\alpha }}D^\\alpha _{\\ \\dot{\\alpha }}\\partial _\\alpha \\right), \\quad \\quad \\text{at } |N-\\overline{N}|=1.$ Notice that the action of the fermionic Laplace operator is analogous to that of the bosonic one () with the grading shifted by one, $\\Delta ^\\text{fermionic}_G=\\Delta ^\\text{bosonic}_{G-1}$ .","The final result is $\\bullet \\quad \\Delta ^\\text{fermionic}_{N>\\overline{N}+3}&= \\Delta ^\\text{bosonic}_{N>\\overline{N}+2} = N(\\overline{N}+2) + y^\\beta _\\alpha h^\\alpha _{\\ \\dot{\\gamma }} D_\\beta ^{\\ \\dot{\\gamma }} + \\overline{y}^{\\dot{\\alpha }}\\overline{}_{\\dot{\\beta }}h_{\\gamma \\dot{\\alpha }}D^{\\gamma \\dot{\\beta }}\\,,\\\\\\bullet \\quad \\Delta ^\\text{fermionic}_{N=\\overline{N}+3}&= \\Delta ^\\text{bosonic}_{N=\\overline{N}+2} = \\Delta _{N>\\overline{N}+2} + \\overline{y}^{\\dot{\\alpha }}\\overline{y}^{\\dot{\\beta }}_\\alpha _\\beta h^\\beta _{\\ \\dot{\\beta }}D^\\alpha _{\\ \\dot{\\alpha }}\\,,\\\\\\bullet \\quad \\Delta ^\\text{fermionic}_{N=\\overline{N}+1}&= \\Delta ^\\text{fermionic}_{N=\\overline{N}-1} = \\Delta ^\\text{bosonic}_{N=\\overline{N}} =\\\\ \\qquad \\qquad \\qquad \\qquad =y^\\alpha _\\beta h_{\\alpha \\dot{\\gamma }}D^{\\beta \\dot{\\gamma }}+ \\overline{y}^{\\dot{\\alpha }}\\overline{}_{\\dot{\\beta }} &h_{\\gamma \\dot{\\alpha }} D^{\\gamma \\dot{\\beta }}-\\overline{y}^{\\dot{\\alpha }} y^\\beta _\\alpha \\overline{}_{\\dot{\\beta }}h^\\alpha _{\\ \\dot{\\alpha }} D_\\beta ^{\\ \\dot{\\beta }}- y^\\alpha \\overline{y}^{\\dot{\\beta }}_\\beta \\overline{}_{\\dot{\\alpha }}h_\\alpha ^{\\ \\dot{\\alpha }}D^\\beta _{\\ \\dot{\\beta }}\\,.\\nonumber $ This allows us do deduce the fermionic cohomology from the bosonic one arriving at the following final results."],["Fermionic $H^0(\\sigma _-)$","The space $H^0(\\sigma _-)$ for fermionic HS fields is spanned by two independent zero-forms with $N-\\overline{N} = \\pm 1:$ $H^0(\\sigma _-) = \\Big \\lbrace F(y,\\overline{y}\\,|x) + \\overline{F}(y,\\overline{y}\\,|x) = F_{\\alpha (n+1),\\dot{\\alpha }(n)}(x)\\,y^{\\alpha (n+1)}\\overline{y}^{\\dot{\\alpha }(n)}+\\overline{F}_{\\alpha (n),\\dot{\\alpha }(n+1)}(x)\\,y^{\\alpha (n)}\\overline{y}^{\\dot{\\alpha }(n+1)}\\Big \\rbrace \\,.$ Recall that, by Theorem 3.1, $H^0(\\sigma _-)$ represents parameters of differential HS gauge transformations."],["Fermionic $H^1(\\sigma _-)$","In the bosonic case, we had two physically different cocycles in $H^1$ (REF ) corresponding to traceless $\\phi (y,\\overline{y}\\,|x)$ and trace $\\phi ^\\text{tr}(y,\\overline{y}\\,|x)$ parts of the Fronsdal field.","These belong to the diagonal $N=\\overline{N}$ .","For the fermionic case the situation is analogous.","The lowest grading is now $G = |N-\\overline{N}| = 1$ .","So, in this sector there are four (not two) different 1-cocycles: $\\psi $ , $\\psi ^\\text{tr}$ , $\\overline{\\psi }$ and $\\overline{\\psi }^\\text{tr}$ given by $\\psi (y,\\overline{y}\\,|x) &= \\psi _{\\mu (n+2),\\dot{\\mu }(n+1)}(x)\\,h^{\\mu \\dot{\\mu }}\\,y^{\\mu (n+1)}\\overline{y}^{\\dot{\\mu }(n)},\\\\\\overline{\\psi }(y,\\overline{y}\\,|x) &= \\overline{\\psi }_{\\mu (n+1),\\dot{\\mu }(n+2)}(x)\\,h^{\\mu \\dot{\\mu }}\\,y^{\\mu (n)}\\overline{y}^{\\dot{\\mu }(n+1)},\\\\\\psi ^\\text{tr}(y,\\overline{y}\\,|x) &= \\psi ^\\text{tr}_{\\mu (n),\\dot{\\mu }(n-1)}(x)\\,h_{\\nu \\dot{\\nu }}\\,y^\\nu y^{\\mu (n)}\\overline{y}^{\\dot{\\nu }}\\overline{y}^{\\dot{\\mu }(n-1)},\\\\\\overline{\\psi }^\\text{tr}(y,\\overline{y}\\,|x) &= \\overline{\\psi }^\\text{tr}_{\\mu (n-1),\\dot{\\mu }(n)}(x)\\,h_{\\nu \\dot{\\nu }}\\,y^\\nu y^{\\mu (n-1)}\\overline{y}^{\\dot{\\nu }}\\overline{y}^{\\dot{\\mu }(n)}$ with a non-negative integer $n$ (positive for $\\psi ^\\text{tr}$ and $\\overline{\\psi }^\\text{tr}$ ).","Cocycles $\\psi $ and $\\psi ^\\text{tr}$ belong to the upper near-diagonal line $N=\\overline{N}+1$ , whereas $\\overline{\\psi }$ and $\\overline{\\psi }^\\text{tr}$ belong to the lower near-diagonal line $N=\\overline{N} -1$ .","All four of them have grading $G=1$ .","$\\psi $ and $\\overline{\\psi }$ are mutually conjugated.","These results can be put into the following concise form $\\psi (y,\\overline{y}\\,|x) &= h^{\\mu \\dot{\\mu }}\\,\\partial _\\mu \\overline{\\partial }_{\\dot{\\mu }}\\,F_1(y,\\overline{y}\\,|x),\\\\\\psi ^\\text{tr}(y,\\overline{y}\\,|x) &= h_{\\mu \\dot{\\mu }}\\,y^\\mu \\overline{y}^{\\dot{\\mu }}\\,F_2(y,\\overline{y}\\,|x),$ where $F_{1,2}(y,\\overline{y}\\,|x)$ are of the homogeneity degree $N-\\overline{N}=1$ , i.e., $\\left(y^\\alpha \\frac{\\partial }{\\partial y^\\alpha } - \\overline{y}^{\\dot{\\alpha }}\\frac{\\partial }{\\partial \\overline{y}^{\\dot{\\alpha }}}\\right) F_{1,2}(y,\\overline{y}\\,|x) = F_{1,2}(y,\\overline{y}\\,|x)\\,.$ For $\\overline{\\psi }$ and $\\overline{\\psi }^\\text{tr}$ the results are analogous except that $\\overline{F}_{1,2}(y,\\overline{y}\\,|x)$ have degree $N -\\overline{N} = -1$ ."],["Fermionic $H^2(\\sigma _-)$","By the same arguments the fermionic $H^2$ is analogous to the bosonic one.","Recall that the bosonic 2-cocycles are represented by three different two-forms: Weyl tensor, traceless and traceful parts of the generalized Einstein tensors (near diagonal, $G=3$ ).","The fermionic Weyl cohomology is given by the same formula as the bosonic one: $W^\\text{ferm}(y,\\overline{y}\\,|x) = H^{\\mu \\nu }\\,\\partial _\\mu \\partial _\\nu C(y,0\\,|x) + \\overline{H}^{\\dot{\\mu }\\dot{\\nu }}\\,\\overline{\\partial }_{\\dot{\\mu }}\\overline{\\partial }_{\\dot{\\nu }}C(0,\\overline{y}\\,|x),$ where $C(y,0\\,|x)$ and $C(0,\\overline{y}\\,|x)$ are polynomials of $y$ and $\\overline{y}$ , respectively.","The two bosonic Fronsdal cocycles (REF ) were represented by the two zero-forms $C^\\text{diag}(y,\\overline{y})$ with the support on the diagonal $N=\\overline{N}$ .","In the fermionic case the two Fronsdal cocycles split into four.","The bosonic diagonal polynomial $C^\\text{diag}(y,\\overline{y})$ is replaced by a pair of near-diagonal $C^\\text{near-diag}(y,\\overline{y})$ and $\\overline{C}^\\text{near-diag}(y,\\overline{y})$ satisfying the relations $\\left(y^\\alpha \\frac{\\partial }{\\partial y^\\alpha } - \\overline{y}^{\\dot{\\alpha }}\\frac{\\partial }{\\partial \\overline{y}^{\\dot{\\alpha }}}\\right) C^\\text{near-diag}(y,\\overline{y}\\,|x) &= C^\\text{near-diag}(y,\\overline{y}\\,|x),\\\\\\left(y^\\alpha \\frac{\\partial }{\\partial y^\\alpha } - \\overline{y}^{\\dot{\\alpha }}\\frac{\\partial }{\\partial \\overline{y}^{\\dot{\\alpha }}}\\right) \\overline{C}^\\text{near-diag}(y,\\overline{y}\\,|x) &= -\\overline{C}^\\text{near-diag}(y,\\overline{y}\\,|x)\\,$ These support the fermionic 2-cocycles associated with the ${\\it l.h.s.}","$ 's of the fermionic field equations for spin $s\\ge 3/2$ massless fields as follows $\\mathcal {E}^\\text{ferm}_\\mathrm {A}(y,\\overline{y}\\,|x) &= \\Big (H^{\\mu \\nu }\\partial _\\mu \\partial _\\nu + \\overline{H}^{\\dot{\\mu }\\dot{\\nu }}\\overline{\\partial }_{\\dot{\\mu }}\\overline{\\partial }_{\\dot{\\nu }}\\Big ) C^\\text{near-diag}(y,\\overline{y}\\,|x),\\\\\\overline{\\mathcal {E}}^\\text{ferm}_\\mathrm {A}(y,\\overline{y}\\,|x) &= \\Big (H^{\\mu \\nu }\\partial _\\mu \\partial _\\nu + \\overline{H}^{\\dot{\\mu }\\dot{\\nu }}\\overline{\\partial }_{\\dot{\\mu }}\\overline{\\partial }_{\\dot{\\nu }}\\Big ) \\overline{C}^\\text{near-diag}(y,\\overline{y}\\,|x),\\\\\\mathcal {E}^\\text{ferm}_\\mathrm {B}(y,\\overline{y}\\,|x) &= \\Big (H^{\\mu \\nu } y_\\mu y_\\nu + \\overline{H}^{\\dot{\\mu }\\dot{\\nu }}\\overline{y}_{\\dot{\\mu }}\\overline{y}_{\\dot{\\nu }}\\Big ) C^\\text{near-diag}(y,\\overline{y}\\,|x),\\\\\\overline{\\mathcal {E}}^\\text{ferm}_\\mathrm {B}(y,\\overline{y}\\,|x) &= \\Big (H^{\\mu \\nu } y_\\mu y_\\nu + \\overline{H}^{\\dot{\\mu }\\dot{\\nu }}\\overline{y}_{\\dot{\\mu }}\\overline{y}_{\\dot{\\nu }}\\Big ) \\overline{C}^\\text{near-diag}(y,\\overline{y}\\,|x)\\,.$"],["Conclusion","In this paper, free unfolded equations for massless HS fields are analyzed in detail in terms of $\\sigma _-$ cohomology.","This is done both in flat space of arbitrary dimension in the tensor formalism for bosonic fields and in $AdS_4$ in the spinor formalism for both bosonic and fermionic fields.","Not surprisingly, the final results agree with those stated long ago in the original papers [12], [15].","Our aim is to present the detailed analysis of the $\\sigma _-$ cohomology providing an exhaustive proof of the so-called First On-Shell Theorem of the form of free unfolded HS equations, allowing the interested reader to check every step.","In the tensor case the full set of cohomology groups $H^p(\\sigma _-)$ was found both for the groups $\\mathrm {GL}(d)$ and $\\mathrm {O}(d)$ .","Our results for $\\mathrm {GL}(d)$ and $p<3$ coincide with those found in [16].","For the $\\mathrm {O}(d)$ case of traceless fields lower cohomology groups matched against those in [16], [17], [18].","In $AdS_4$ we used spinor formalism to analyze $H^{0,1,2}(\\sigma _-)$ for both bosonic and fermionic HS fields.","To the best of our knowledge such analysis was not available in the literature.","Practically, to compute $H^k(\\sigma _-)$ in both $\\mathrm {Mink}^d$ and $AdS_4$ cases we used the analogue of the Hodge theorem.","Namely, the problem of finding the cohomology $H^k(\\sigma _-) = \\mathrm {ker}(\\sigma ^{k}_-)/\\mathrm {im}(\\sigma ^{k-1}_-)$ was reduced to the calculation of the kernel of an appropriate positive-definite Laplace-Hodge operator $\\Delta $ invariant under the action of compact version of the space-time symmetry algebra.","This technique was shown to be lucid and efficient.","Having found the cohomology groups $H^k(\\sigma _-)$ for $k=0,1,2$ , in accordance with [13] we obtained the exhaustive information about the differential HS gauge parameters, dynamical HS gauge fields and their field equations.","Thus, we have explicitly proven the so-called First On-Shell Theorem for bosonic HS fields in $\\mathrm {Mink}^d$ (which case is straightforwardly extendable to $AdS_d$ ) and all massless fields in $AdS_4$ .","The technique used in this paper can be further applied to the calculation of $H^p(\\sigma _-)$ in the zero-form sector of HS fields studied in [13], [22] that describes dynamics of a scalar field and $s=1$ particle as well as to more general systems considered in [44], [45], [46].","One of the byproduct results of this paper is the interpretation of the matching between $\\sigma _-$ cohomology of the one-form sector against zero-form sector expressing the matching between Bianchi identities in the two sectors."],["Acknowledgement","We are grateful to Vyacheslav Didenko, Anatoly Korybut and Alexander Tarusov for helpful and stimulating discussions and Maxim Grigoriev for a useful comment.","We are particularly grateful to the referee for useful suggestions and the correspondence.","We acknowledge a partial support from the Russian Basic Research Foundation Grant No 20-02-00208.","Since most of the tensors encountered in the course of this paper are Young tensors in the symmetric basis, it is convenient to accept the following notation.","A tensor without a certain type of index symmetry will be denoted as $T^{a|b|c|..}$ , where the vertical line $|$ separates groups of indices not related by any symmetries to each other.","A tensor that has a symmetric set of $n$ indices, say, $(a_1,a_2,\\dots ,a_n)$ will be denoted $T^{a(n)|...}\\equiv T^{(a_1a_2\\dots a_n)|...}$ .","A tensor corresponding to a certain Young diagram in the symmetric basis then has the form: $T^{a(n), b(m), c(k),...}$ .","Symmetrization over $n$ indices is performed by the formula $\\mathrm {Sym} = \\frac{1}{n!","}\\sum _{\\text{all permutations}}$ .","We will denote all symmetrized tensor indices by the same letter.","For example, $T^{a(n)|a} \\equiv \\frac{1}{(n+1)!}","\\sum _{\\sigma \\in S_{n+1}} T^{\\sigma (a_1..a_n|a_{n+1})}\\,.$ In Section we omit $Y,Z,\\theta $ and assume that all indices are contracted with the corresponding variables.","The rules for raising and lowering $\\mathfrak {sl}(2)$ -indices are $A_\\alpha = A^\\beta \\epsilon _{\\beta \\alpha }, \\quad \\quad A^\\alpha = \\epsilon ^{\\alpha \\beta }A_\\beta , \\quad \\quad \\epsilon _{\\alpha \\beta }\\epsilon ^{\\gamma \\beta } = \\epsilon _\\alpha ^{\\ \\gamma } = \\delta _\\alpha ^{\\ \\gamma } = -\\epsilon ^\\gamma _{\\ \\alpha }$ with $\\epsilon _{\\alpha \\beta } = -\\epsilon _{\\beta \\alpha },\\quad \\quad \\quad \\epsilon _{12} = 1.$ 0.75em $\\alpha _1 = \\tfrac{(-3 + n) (-2 + n) (-1 + n) (-3 + d + 2 n)}{(-5 + d +m + n) (-4 + d + m + n) (-4 + d + 2 n) (d^2 + d (-8 + m + 3 n) -2 (-8 + m + m^2 + 7 n - 3 m n))},$ 0.75em $\\alpha _2 = \\tfrac{(-2 + n) (-1 + n)}{(-4 + d + m + n) (-4 + d + 2 n)},$ 0.75em $\\alpha _3 = -\\tfrac{2 (-2 + n) (-1 + n) (-3 + d + 2 n)}{(-4 + d + m +n) (-4 + d + 2 n) (d^2 + d (-8 + m + 3 n) -2 (-8 + m + m^2 + 7 n - 3 m n))},$ 0.75em $\\alpha _4 = \\tfrac{2(1 + m - n) (-2 + n) (-1 + n) (-3 + d + 2 n)}{(-5 +d + m + n) (-4 + d + m + n) (-4 + d + 2 n) (d^2 +d (-8 + m + 3 n) - 2 (-8 + m + m^2 + 7 n - 3 m n))},$ 0.75em $\\alpha _5 = \\tfrac{ (-1 + n) (-3 + d + m + n)}{(-4 + d + m + n) (-4 + d + 2 n)},$ 0.75em $\\alpha _6 = -\\tfrac{(-2 + d + 2 m) (-1 + n)}{(-4 + d + 2 n) (d^2 +d (-8 + m + 3 n) - 2 (-8 + m + m^2 + 7 n - 3 m n))},$ 0.75em $\\alpha _7 = \\tfrac{(-1 + n) (-3 + d + 2 n) (16 + d^2 - 7 m - 9 n +2 d (-4 + m + n) + (m + n)^2)}{(-5 + d + m + n) (-4 + d + m +n) (-4 + d + 2 n) (d^2 + d (-8 + m + 3 n) -2 (-8 + m + m^2 + 7 n - 3 m n))},$ 0.75em $\\alpha _8 = -\\tfrac{(-1 + n)}{(-4 + d + m + n)},$ 0.75em $\\alpha _9 = \\tfrac{2 (-1 + n) (-3 + d + 2 n)}{(-4 + d + m + n) (d^2 +d (-8 + m + 3 n) - 2 (-8 + m + m^2 + 7 n - 3 m n))},$ 0.75em $\\alpha _{10} = -\\tfrac{(-3 + d + 2 n)}{d^2 + d (-8 + m + 3 n) - 2 (-8 + m + m^2 + 7 n - 3 m n)},$ 0.75em $\\alpha _{11} = -\\tfrac{(-4 + d + 2 m) (-1 + n) (-3 + d + 2 n)}{(-5 + d + m + n) (-4 + d + m + n) (d^2 + d (-8 + m + 3 n) - 2 (-8 + m + m^2 + 7 n - 3 m n))},$ 0.75em $\\alpha _{12} = \\tfrac{(14 + d^2 - 6 n + 2 m (-5 + m + n) +d (-8 + 3 m + n))}{(-4 + d + m + n) (d^2 + d (-8 + m + 3 n) -2 (-8 + m + m^2 + 7 n - 3 m n))}\\,.$"]],"string":"[\n [\n \"The $\\\\sigma_-$ Cohomology Analysis for Symmetric Higher-Spin Fields\"\n ],\n [\n \"Abstract In this paper, we present a complete proof of the so-called First On-Shell Theorem that determines dynamical content of the unfolded equations for free symmetric massless fields of arbitrary integer spin in any dimension and arbitrary integer or half-integer spin in four dimensions.\",\n \"This is achieved by calculation of the respective $\\\\sigma_-$ cohomology both in the tensor language in Minkowski space of any dimension and in terms of spinors in $AdS_4$.\",\n \"In the $d$-dimensional case $H^p(\\\\sigma_-)$ is computed for any $p$ and interpretation of $H^p(\\\\sigma_-)$ is given both for the original Fronsdal system and for the associated systems of higher form fields.\"\n ],\n [\n \"Introduction\",\n \"Higher-spin (HS) gauge theory is based on works of Fronsdal [1] and Fang and Fronsdal [2], where the action and equations of motion for massless gauge fields of any spin were originally obtained in flat four-dimensional Minkowski space.\",\n \"Even earlier, important restrictions on low-energy HS vertices were obtained by Weinberg in [3], [4] and so-called no-go theorems restricting $S$ -matrix possessing too high symmetries in flat space-time were proven in [5], [6].\",\n \"(For a review see [7].)\",\n \"The no-go theorems implied the existence of the $s=2$ barrier suggesting that the construction of an interacting local HS theory in Minkowski space-time is impossible.\",\n \"The proof of these theorems essentially uses the specific form of the algebra of isometries of Minkowski space.\",\n \"The $s=2$ barrier in flat space can be overcome in the space-time with non-zero sectional curvature, for example, in the anti-de Sitter space [8].\",\n \"In these spaces it becomes possible to formulate a consistent nonlinear theory of fields of all spins [9], [10].\",\n \"The construction of a nonlinear HS theory is essentially based on the so-called unfolded approach [11], [12], which is a far-going generalization of the Cartan formulation of gravity ($s=2$ ) in terms of differential forms to fields of any spin $s>2$ .\",\n \"Via introducing appropriate auxiliary variables, the unfolding procedure allows one to replace the system of partial differential equations of any order on a smooth manifold by a larger system of first-order equations on vector-valued differential forms.\",\n \"One of the essential features of this approach, which is very useful for analysing symmetries of a given system, is that the variables in the equations are valued in one or another representation of the underlying symmetry algebra.\",\n \"The dynamical content of the HS theory can be reconstructed from its unfolded formulation using the $\\\\sigma _-$ cohomology technique [13].\",\n \"As is recalled below, the dynamical data of the theory are in one-to-one correspondence with the cohomology of certain linear nilpotent operator $\\\\sigma _-$ that can be read of the unfolded equations in question.\",\n \"The statement that unfolded equations of free HS fields are equivalent to the Fronsdal equations was made in the original papers in the spinor [14] and tensor [15] formalisms.\",\n \"In the tensor formulation of HS theory the idea of the proof was illustrated in [16], where however the analysis of the trace part of the Fronsdal equations was not completed, while general arguments for mixed-symmetry HS fields were given in [17].\",\n \"In [18] the unfolded equations for massless fields were derived from the Fronsdal theory by the BRST methods.\",\n \"To the best of our knowledge, no detailed analysis of the problem in the spinor formalism was available in the literature.\",\n \"In this paper we present a complete proof of the so called First On-Shell Theorem by computation of the cohomology rings of $\\\\sigma _-$ for the physically important cases of the integer-spin symmetric fields both in flat space-time of any dimension and $AdS_d$ as well as for the fields of any integer and half-integer spin in ${AdS}_4$ .\",\n \"The computation technique analogous to the Hodge theory for differential forms is performed in terms of so-called $\\\\sigma _-$ cohomology and provides a complete analysis of the dynamical content of the free unfolded equations for symmetric massless fields of any spin.\",\n \"Giving a direct proof of the equivalence between the Fronsdal formulation of the HS gauge theory and its unfolded formulation this paper fills in some gaps in the literature also illustrating a general approach applicable to a broad class of unfolded systems.\",\n \"In addition, in the tensor case we compute higher $\\\\sigma _-$ cohomology groups and interpret them in terms of higher Bianchi identities and more general dynamical systems.\",\n \"In particular, we discuss the matching between the Bianchi identities in terms of one-form gauge fields and zero-form field strengths.\",\n \"The rest of the paper is organized as follows.\",\n \"In Section 2 we briefly recall different approaches to the description of HS massless fields.\",\n \"Main idea of the $\\\\sigma _-$ cohomology approach is explained in Section 3.\",\n \"Cohomology calculation method used in this paper is discussed in Section 4.\",\n \"Section 5 contains derivation of $H^p(\\\\sigma _-)$ in Minkowski space of any dimension.\",\n \"In particular, the cases of $\\\\mathrm {GL}(d)$ and $\\\\mathrm {O}(d)$ representations are analysed here.\",\n \"In Section 6 calculation of the low-order cohomology groups in $AdS_4$ is performed.\",\n \"Obtained results are discussed in Section 7.\",\n \"Index conventions and normalisations of the tensor Young diagrams are presented in the Appendix A.\"\n ],\n [\n \"Metric formulation\",\n \"According to Fronsdal [1], a spin-$s$ massless symmetric field can be described in terms of two symmetric traceless tensors (for index conventions see Appendix A) $\\\\phi ^{a(s)} \\\\equiv \\\\phi ^{a_1..a_s}, \\\\quad \\\\phi ^{a(s-2)} \\\\equiv \\\\phi ^{a_1..a_{s-2}}, \\\\quad \\\\eta _{b_1 b_2}\\\\phi ^{b_1 b_2 a_3 .. a_s} \\\\equiv \\\\phi ^{a(s-2)b}{}_b = 0, \\\\quad \\\\phi ^{a(s-4)b}{}_b = 0\\\\,.$ These two fields can be combined into a single rank-$s$ totally symmetric tensor $\\\\varphi ^{a(s)} = \\\\phi ^{a(s)} + \\\\eta ^{aa}\\\\phi ^{a(s-2)}$ obeying the double-tracelessness condition $\\\\varphi ^{a(s-4)bc}{}_{bc} = 0\\\\,.$ The field equations in Fronsdal theory have the form $R^{a(s)}(\\\\varphi ) = \\\\square \\\\varphi ^{a(s)} - s \\\\partial ^a \\\\partial _k \\\\varphi ^{k a(s-1)} + \\\\frac{s(s-1)}{2} \\\\partial ^a \\\\partial ^a \\\\varphi ^{a(s-2)k}{}_k = 0\\\\,,$ where $\\\\partial _a = \\\\frac{\\\\partial }{\\\\partial x^a}$ .\",\n \"The tensor $R^{a(s)}(\\\\varphi )$ is invariant under the gauge transformations with a rank-$(s-1)$ traceless gauge parameter $\\\\varepsilon (x)$ $\\\\delta \\\\varphi ^{a(s)} = \\\\partial ^a \\\\varepsilon ^{a(s-1)}, \\\\qquad \\\\varepsilon ^{a(s-3)k}{}_k = 0\\\\,.$ Fronsdal equation (REF ) is a generalization of the well-known equations of fields with spins $s = 0,1,2$ .\",\n \"For the case of $s = 1$ the last term in the Fronsdal tensor disappears and Eq.\",\n \"(REF ) reproduces Maxwell equations for the field $A^a$ .\",\n \"Without the last two terms at $s = 0$ it gives Klein-Gordon equation for a massless scalar field.\",\n \"The case of $s = 2$ reproduces the equations of linearized gravity [19].\",\n \"Gauge transformation (REF ) gives the known gauge transformations of low-spin fields and its absence for a scalar field.\"\n ],\n [\n \"Tensor formalism\",\n \"The unified description of massless fields of arbitrary spin can be given in the so-called frame-like formalism that generalizes Cartan formulation of gravity, operating in terms of differential forms [14], [15], [20].\",\n \"Frame-like formulation of the HS gauge theory in any dimension is given in terms of the one-form fields $ \\\\omega ^{a(s-1),b(t)} =dx^\\\\nu \\\\omega _\\\\nu ^{a(s-1),b(t)}$ valued in two-row Young diagrams corresponding to irreducible $\\\\mathfrak {o}(d-1,1)$ (i.e., traceless) modules [15], obeying conditions $& \\\\omega ^{a(s-1),a b(t-1)} = 0, \\\\\\\\& \\\\omega ^{a(s-3)k}{}_{k}{}^{,b(t)} = 0\\\\,.$ (For index conventions see Appendix A.)\",\n \"By introducing auxiliary fields it is possible to put a system of partial differential equations into the first-order unfolded form [11], [12].\",\n \"Generally, unfolded equations read as $d W^A = \\\\sum _{n = 1}^{\\\\infty } G^A_{B_1,..,B_n}W^{B_1}\\\\wedge ...\\\\wedge W^{B_n}\\\\,,\\\\qquad d:=dx^\\\\nu \\\\partial _\\\\nu \\\\,.$ Here $W^A$ is a set of differential forms over some manifold.\",\n \"(Indices are treated formally and can take an infinite number of values.)\",\n \"The coefficients $G^A_{B_1,..,B_n}$ satisfy the (anti)symmetry condition $G^A_{B_1,..,B_i,..,B_j,..,B_n} = (-1)^{|B_i||B_j|} G^A_{B_1,..,B_j,..,B_i,..,B_n}\\\\,,$ where $|B_i|$ denotes the form-degree of $W^{B_i}$ .\",\n \"Also $G^A_{B_1,..,B_n}$ are restricted by the integrability conditions expressing that $d^2=0$ .\",\n \"In the tensor language the unfolded HS equations in Minkowski space proposed in [15] read as $D_L\\\\omega ^{a(s-1),b(t)} + h_m \\\\wedge \\\\omega ^{a(s-1),b(t)m} = 0, \\\\quad t \\\\in \\\\lbrace 0,..,s-2\\\\rbrace ,$ $D_L\\\\omega ^{a(s-1),b(s-1)} = h_n \\\\wedge h_m \\\\wedge C^{a(s-1)n,b(s-1)m},$ where $h_n$ is a soldering form (vielbein, frame field, tetrad) and $D_L = d + \\\\varpi $ is the background Lorentz covariant derivative that satisfies relations $D_L h^a = 0, \\\\qquad D_L^2 = 0\\\\,.$ In the Cartesian coordinate system with $\\\\varpi =0$ the equations simplify to $d\\\\omega ^{a(s-1),b(t)} + h_m \\\\wedge \\\\omega ^{a(s-1),b(t)m} = 0, \\\\quad t \\\\in \\\\lbrace 0,..,s-2\\\\rbrace ,$ $d\\\\omega ^{a(s-1),b(s-1)} = h_n \\\\wedge h_m \\\\wedge C^{a(s-1)n,b(s-1)m}\\\\,,$ where $C$ satisfies the Lorentz irreducibility conditions $C^{a(n),a b(m-1)} = 0\\\\,, \\\\qquad C^{a(n-2)k}{}_k{}^{,b(m)} = 0\\\\,.$ The traceless tensor $C$ on the r.h.s.\",\n \"of (REF ) is a generalized Weyl tensor.\",\n \"There are also unfolded equations on $C$ and on additional auxiliary fields [10] (for reviews see [16], [21]).\",\n \"This system constitutes an infinite chain of zero-form equations.\",\n \"Zero-form sector, that contains equations on spin-zero and spin-one fields, will not be considered in this paper.\",\n \"Equations (REF ) are invariant under the gauge transformations $\\\\delta \\\\omega ^{a(s-1),b(t)} = d\\\\varepsilon ^{a(s-1),b(t)} + h_m \\\\varepsilon ^{a(s-1),b(t)m}, \\\\quad t \\\\in \\\\lbrace 0,..,s-2\\\\rbrace $ and eq.\",\n \"(REF ) is invariant under $&\\\\delta \\\\omega ^{a(s-1),b(s-1)} = d \\\\varepsilon ^{a(s-1),b(s-1)},\\\\\\\\&\\\\delta C^{a(s),b(s)} = 0,$ where $\\\\varepsilon $ are zero-forms valued in the appropriate two-row irreducible $\\\\mathfrak {o}(d-1,1)$ -modules obeying conditions analogous to (REF ).\",\n \"The Fronsdal field is embedded into the frame-like one-form $e^{a(s-1)}\\\\equiv \\\\omega ^{a(s-1)}$ in the following manner.\",\n \"Converting the form index into the fiber one using vielbein $h$ , $e^{a(s-1)|b} = e^{a(s-1)}_\\\\mu h^{\\\\mu b}\\\\,,$ the resulting tensor (REF ) can be decomposed into irreducible $\\\\mathfrak {o}(d-1,1)$ -modules.\",\n \"In terms of traceless Young diagrams this decomposition is $\\\\begin{picture}(15,10)(0,7){\\\\put (05,10){\\\\line (1,0){10}}\\\\put (05,20){\\\\line (1,0){10}}\\\\put (05,10){\\\\line (0,1){10}}\\\\put (15,10){\\\\line (0,1){10}}}\\\\end{picture} \\\\,\\\\,\\\\,\\\\otimes _{so}\\\\,\\\\,\\\\,\\\\begin{picture}(65,18)(10,7){\\\\put (25,13){\\\\scriptsize {s-1}}\\\\put (05,20){\\\\line (1,0){60}}\\\\put (05,10){\\\\line (1,0){60}}\\\\put (05,10){\\\\line (0,1){10}}\\\\put (65,10){\\\\line (0,1){10}}}\\\\end{picture}\\\\,\\\\ \\\\cong \\\\,\\\\,\\\\,\\\\ \\\\begin{picture}(65,18)(10,7){\\\\put (30,13){\\\\scriptsize {s}}\\\\put (05,20){\\\\line (1,0){60}}\\\\put (05,10){\\\\line (1,0){60}}\\\\put (05,10){\\\\line (0,1){10}}\\\\put (65,10){\\\\line (0,1){10}}}\\\\end{picture}\\\\,\\\\ \\\\oplus \\\\,\\\\,\\\\,\\\\ \\\\begin{picture}(65,18)(10,7){\\\\put (25,13){\\\\scriptsize {s-2}}\\\\put (05,20){\\\\line (1,0){60}}\\\\put (05,10){\\\\line (1,0){60}}\\\\put (05,10){\\\\line (0,1){10}}\\\\put (65,10){\\\\line (0,1){10}}}\\\\end{picture}\\\\,\\\\,\\\\,\\\\ \\\\oplus \\\\,\\\\,\\\\,\\\\ \\\\begin{picture}(10,18)(0,7){\\\\put (05,00){\\\\line (1,0){10}}\\\\put (05,10){\\\\line (1,0){10}}\\\\put (05,0){\\\\line (0,1){10}}\\\\put (15,0){\\\\line (0,1){10}}}\\\\end{picture}$ ${s-1}$ .\",\n \"The first two components give the Fronsdal field, while the third one is an excess of the components of the frame-like field in comparison with the Fronsdal field.\",\n \"At the tensor level, this decomposition is represented as: $e^{a(s-1)|b} = \\\\psi _1^{a(s-1)b} + \\\\beta _1\\\\eta ^{aa}\\\\psi _2^{a(s-3)b} + \\\\beta _2\\\\eta ^{ab}\\\\psi _2^{a(s-2)} + \\\\psi _3^{a(s-1),b},$ where $\\\\psi _i$ are traceless and correspond to the $i$ -th diagram.\",\n \"The relative coefficient $\\\\frac{\\\\beta _2}{\\\\beta _1}$ is fixed by the tracelessness condition with respect to indices $a$ .\",\n \"This decomposition shows that the Fronsdal field identifies with the symmetric part of the frame-like field, since the contribution of the third diagram disappears upon symmetrization.\",\n \"The resulting field $\\\\varphi ^{a(s)} := e^{a(s-1)|a}\\\\,$ is symmetric and double-traceless.\",\n \"The extra term $\\\\psi _3^{a(s-1),b}$ is pure gauge.\",\n \"Its contribution can be canceled by the gauge transformation $\\\\delta e^{a(s-1)|b} = \\\\varepsilon ^{a(s-1)|b} $ with suitable gauge parameter.\",\n \"For detailed discussion of Fronsdal field embedding see [16], [20], [21].\",\n \"It is not difficult to check [15], [20] (for reviews see [16], [21]) that the Fronsdal equations and gauge transformations follow from the unfolded system (REF ), (REF ).\",\n \"A more complicated question is whether the Fronsdal fields and equations are the only ones that result from (REF ), (REF ).\",\n \"The answer can be obtained via the $\\\\sigma _-$ cohomology technique [13].\"\n ],\n [\n \"Spinor language in $AdS_4$\",\n \"The physically important case of the unfolded system for HS connection (REF ), (REF ) is that of $AdS_4$ space-time in which case the language of two-component spinors is most appropriate.\",\n \"In this language instead of using Lorentz indices $a,b,...= 0,1,2,3$ , one uses two pairs of dotted and undotted spinor indices $\\\\alpha ,\\\\beta ,...$ and $\\\\dot{\\\\alpha },\\\\dot{\\\\beta },...$ taking values $\\\\lbrace 1,2\\\\rbrace $ .\",\n \"The two languages are related via Pauli matrices.\",\n \"The $AdS_4$ background geometry is described in terms of the Lorentz connection $\\\\varpi $ and frame field $h$ , that satisfy equations $\\\\begin{array}{l}d h^{\\\\alpha \\\\dot{\\\\beta }}+\\\\varpi ^{\\\\alpha }{ }_{\\\\gamma } \\\\wedge h^{\\\\gamma \\\\dot{\\\\beta }}+\\\\overline{\\\\varpi }^{\\\\dot{\\\\beta }}{}_{\\\\dot{\\\\gamma }} \\\\wedge h^{\\\\alpha \\\\dot{\\\\gamma }}=0\\\\,, \\\\\\\\d \\\\varpi ^{\\\\alpha \\\\beta }+\\\\varpi ^{\\\\alpha }{ }_{\\\\gamma } \\\\wedge \\\\varpi ^{\\\\gamma \\\\beta }=-\\\\lambda ^{2} h^{\\\\alpha }{}_{\\\\dot{\\\\gamma }} \\\\wedge h^{\\\\beta \\\\dot{\\\\gamma }}\\\\,, \\\\\\\\d \\\\overline{\\\\varpi }^{\\\\dot{\\\\alpha } \\\\dot{\\\\beta }}+\\\\overline{\\\\varpi }^{\\\\dot{\\\\alpha }}{}_{\\\\dot{\\\\gamma }} \\\\wedge \\\\overline{\\\\varpi }^{\\\\dot{\\\\gamma } \\\\dot{\\\\beta }}=-\\\\lambda ^{2} h_{\\\\gamma }{ }^{\\\\dot{\\\\alpha }} \\\\wedge h^{\\\\gamma \\\\dot{\\\\beta }}\\\\,,\\\\end{array}$ where $\\\\lambda ^2$ is proportional to the curvature of $AdS_4$ and we adopt the following rules $A_{\\\\alpha }=A^{\\\\beta } \\\\epsilon _{\\\\beta \\\\alpha }\\\\,, \\\\quad A^{\\\\alpha }=\\\\epsilon ^{\\\\alpha \\\\beta } A_{\\\\beta }\\\\,, \\\\quad \\\\epsilon _{\\\\alpha \\\\beta } \\\\epsilon ^{\\\\gamma \\\\beta }=\\\\epsilon _{\\\\alpha }{}^{\\\\gamma }=\\\\delta _{\\\\alpha }^{\\\\gamma }=-\\\\epsilon ^{\\\\gamma }{}_{\\\\alpha }\\\\,,$ where $\\\\epsilon _{\\\\alpha \\\\beta } = -\\\\epsilon _{\\\\beta \\\\alpha }\\\\,, \\\\quad \\\\epsilon _{12} = 1\\\\,.$ The spinor version of the unfolded system for one-form $\\\\omega $ reads as follows.\",\n \"First, the HS curvatures in the spinor language are [14] $R^{\\\\alpha (n),\\\\dot{\\\\alpha }(m)}=D_L\\\\omega ^{\\\\alpha (n),\\\\dot{\\\\alpha }(m)} + \\\\lambda ^2(n h^\\\\alpha _{\\\\ \\\\dot{\\\\gamma }}\\\\wedge \\\\omega ^{\\\\alpha (n-1),\\\\dot{\\\\gamma }\\\\dot{\\\\alpha }(m)} + m h_\\\\gamma ^{\\\\ \\\\dot{\\\\alpha }}\\\\wedge \\\\omega ^{\\\\gamma \\\\alpha (n),\\\\dot{\\\\alpha }(m-1)})$ and $D_L=d+\\\\varpi +\\\\overline{\\\\varpi }$ is a Lorentz-covariant derivative with Cartan's spin-connection $(\\\\varpi \\\\oplus \\\\overline{\\\\varpi })$ $D_L \\\\omega ^{\\\\alpha (n),\\\\dot{\\\\alpha }(m)}=d\\\\omega ^{\\\\alpha (n),\\\\dot{\\\\alpha }(m)}+n\\\\varpi _\\\\alpha ^{\\\\ \\\\beta }\\\\wedge \\\\omega _{\\\\beta \\\\alpha (n-1),\\\\dot{\\\\alpha }(m)} + m\\\\overline{\\\\varpi }_{\\\\dot{\\\\alpha }}^{\\\\ \\\\dot{\\\\beta }}\\\\wedge \\\\omega _{\\\\alpha (n),\\\\dot{\\\\beta }\\\\dot{\\\\alpha }(m-1)}\\\\,.$ The $AdS_4$ deformation of the unfolded equations (REF ), (REF ) then takes the form [12] $R^{\\\\alpha (n),\\\\dot{\\\\alpha }(m)}=\\\\delta _{0,n}\\\\,h_{\\\\beta \\\\dot{\\\\alpha }}\\\\wedge h^{\\\\beta }_{\\\\ \\\\dot{\\\\alpha }}\\\\overline{C}^{\\\\dot{\\\\alpha }(m+2)} + \\\\delta _{0, m}\\\\, h_{\\\\alpha \\\\dot{\\\\beta }}\\\\wedge h_{\\\\alpha }^{\\\\ \\\\dot{\\\\beta }} C^{\\\\alpha (n+2)}.$ The main advantage of the two-component spinor notation is that it makes the representation theory of the Lorentz group very simple.\",\n \"Namely, every Lorentz irreducible multispinor representing a traceless tensor is totally symmetric in its spinor indices.\",\n \"Since the only Lorentz invariant objects are antisymmetric bispinors $\\\\epsilon _{\\\\alpha \\\\beta }$ and $\\\\overline{\\\\epsilon }_{\\\\dot{\\\\alpha }\\\\dot{\\\\beta }}$ irreducible multispinors $X^{\\\\alpha (n),\\\\dot{\\\\alpha }(m)}$ are necessarily symmetric with respect to the indices in the groups $\\\\alpha (n)$ and $\\\\dot{\\\\alpha }(m)$ separately.\",\n \"Thus, working with the two-component spinor notation one can happily forget about painful calculations with the traces of Lorentz-tensors.\"\n ],\n [\n \"The idea of $\\\\sigma _-$ cohomology analysis: example of integer spin massless fields\",\n \"The l.h.s.\",\n \"'s of unfolded HS equations and gauge transformations in $d$ -dimensional Minkowski space are [15], [16] $&R^{a(s-1),b(k)} = D_L \\\\omega ^{a(s-1),b(k)} + \\\\sigma _-(\\\\omega )^{{a(s-1),b(k)}},\\\\\\\\&\\\\delta \\\\omega ^{a(s-1),b(k)} = D_L \\\\varepsilon ^{a(s-1),b(k)} + \\\\sigma _-(\\\\varepsilon )^{{a(s-1),b(k)}},$ where $(\\\\sigma _-\\\\omega )^{{a(s-1),b(k)}} := h_c \\\\wedge \\\\omega ^{a(s-1),b(k)c}\\\\,,\\\\qquad (\\\\sigma _-\\\\varepsilon )^{{a(s-1),b(k)}} := h_c \\\\wedge \\\\varepsilon ^{a(s-1),b(k)c} .$ $R^{a(s-1),b(k)}$ is referred to as (linearized) HS curvature.\",\n \"For simplicity we study the Minkowski case.\",\n \"Since $\\\\sigma _-$ in $AdS_d$ is defined analogously, our analysis applies to that case as well.\",\n \"Due to their definition, HS curvatures obey the Bianchi identities $D_L R^{a(s-1),b(k)} + \\\\sigma _-(R)^{a(s-1),b(k)} = 0\\\\,.$ The appearance of $\\\\sigma _-$ allows one to clarify the role of the fields $\\\\omega ^{a(s-1),b(k)}$ and gauge parameters $\\\\varepsilon ^{a(s-1),b(k)}$ .\",\n \"Working with the zero-forms $\\\\varepsilon ^{a(s-1),b(k)}$ and one-forms $\\\\omega ^{a(s-1),b(k)}$ valued in two-row Young diagrams, we consider the space $V^p$ of $p$ -forms valued in two-row Young diagrams with any $p$ .\",\n \"Defining $\\\\sigma _-$ to annihilate the forms with an empty second row, we find that $\\\\sigma _- V^p\\\\subset V^{p+1}$ and $\\\\sigma _-\\\\,\\\\sigma _-=0$ .\",\n \"As originally proposed in [13], the $\\\\sigma _-$ cohomology $H(\\\\sigma _-)=\\\\ker ({\\\\sigma _-})/{\\\\mathrm {im\\\\,}(\\\\sigma _-)}$ classifies fields, their equations and gauge symmetries.\",\n \"Indeed, those components of the fields $\\\\omega ^{a(s-1),b(t)}$ , that are not annihilated by $\\\\sigma _-$ , can be expressed via derivatives of the fields with lower $t$ by setting suitable components of the HS curvatures to zero.\",\n \"Such fields are called auxiliary.\",\n \"Conversely, those components of the fields $\\\\omega ^{a(s-1),b(t)}$ , that cannot be expressed in terms of derivatives of lower fields via zero-curvature conditions, are in $\\\\ker (\\\\sigma _-)$ .\",\n \"By Stueckelberg fields we mean $\\\\sigma _-$ -exact fields (i.e.\",\n \"fields of the form $\\\\sigma _-\\\\chi $ ) as they can be eliminated by an appropriate $\\\\sigma _-$ -exact term in the gauge transformation (REF ).\",\n \"Fields that are not expressed via derivatives of other fields and are not Stueckelberg are called dynamical.\",\n \"These describe the physical degrees of freedom of the theory.\",\n \"Thus, the dynamical HS fields are associated with $H^1(\\\\sigma _-)$ .\",\n \"The classification for the gauge parameters is analogous.\",\n \"The parameters, that are not annihilated by $\\\\sigma _-$ , describe algebraic Stueckelberg shifts.\",\n \"The leftover symmetries are described by the parameters in $\\\\ker (\\\\sigma _-)$ .\",\n \"$\\\\sigma _-$ -exact parameters correspond to the so called gauge for gauge transformations.\",\n \"Parameters, which are $\\\\sigma _-$ -closed and not $\\\\sigma _-$ -exact, are referred to as genuine differential gauge parameters.\",\n \"Note that since in the HS example in question the gauge parameters are zero-forms there is no room for gauge for gauge symmetries in that case.\",\n \"Let $V$ be a graded vector space, $\\\\mathcal {C}$ be an element of $\\\\Lambda ^p(\\\\mathcal {M}^d)\\\\otimes V$ over some smooth $d$ -dimensional manifold $\\\\mathcal {M}^d$ .\",\n \"We demand the grading of $V$ to be bounded from below, that is $V$ is $\\\\mathbb {N}$ -graded.\",\n \"Let $\\\\sigma _ {\\\\pm }$ be operators that act ”vertically”, i.e.\",\n \"do not affect the space-time coordinates, and shift grading by $\\\\pm 1$ , $D_L$ be the Grassmann-odd operator that does not affect the grading and is allowed to act non-trivially on the space-time coordinates.\",\n \"Consider the covariant constancy condition of a general form along with the zero-curvature condition $\\\\mathcal {D} \\\\mathcal {C} = (D_L + \\\\sigma _- + \\\\sigma _+) \\\\mathcal {C} = 0, \\\\quad \\\\mathcal {D}^2 = 0.$ Notice that eq.\",\n \"(REF ) remains invariant under the gauge transformations $\\\\delta \\\\mathcal {C} = \\\\mathcal {D} \\\\varepsilon ,$ where $\\\\varepsilon \\\\in \\\\Lambda ^{p-1}(\\\\mathcal {M}^d) \\\\otimes V$ .\",\n \"One can prove the following proposition [13] (see also [16], [22], [23], [24]): Theorem 3.1 The following is true: 1) Differential gauge symmetry parameters $\\\\varepsilon $ span $H^{p-1}(\\\\sigma _-)$ 2) Nontrivial dynamical fields $\\\\mathcal {C}$ span $H^p(\\\\sigma _-)$ 3) Physically distinguishable differential field equations on the nontrivial dynamical fields, contained in $\\\\mathcal {D} \\\\mathcal {C} = 0$ , span $H^{p+1}(\\\\sigma _-)$ Thus, taking into account that HS gauge fields are described by the one-forms $\\\\omega $ , to prove that the Fronsdal metric formulation is equivalent to the unfolded one, we have to calculate $H^{0}(\\\\sigma _-), H^{1}(\\\\sigma _-)$ and $H^{2}(\\\\sigma _-)$ .\",\n \"More generally, higher cohomology $H^{k}(\\\\sigma _-)$ with $k>p+1$ describes Bianchi identities for dynamical equations at $k=p+2$ and Bianchi for Bianchi identites at $k>p+2$ [24].\",\n \"Similarly, the lower cohomology $H^{k}(\\\\sigma _-)$ with $k4$ .\",\n \"are now realized by the first-order differential operators $\\\\left(t_{\\\\mathfrak {gl}(d)}{}\\\\right)^a_b = Y^a \\\\partial _{Yb} + Z^a \\\\partial _{Zb} + \\\\theta ^a \\\\partial _{\\\\theta ^b}, \\\\quad \\\\left(t^{\\\\mathfrak {so}(d)}{}\\\\right)_{ab} = \\\\frac{1}{2}\\\\bigg (\\\\eta _{ac}t_{\\\\mathfrak {gl}}{}^c_b - \\\\eta _{bc}t_{\\\\mathfrak {gl}}{}^c_a\\\\bigg )\\\\,,$ where $\\\\theta ^c$ is a Grassmann-odd element of the exterior algebra associated with the frame one-form $e^a$ .\",\n \"In these terms $\\\\sigma _-$ acts as $\\\\sigma _- \\\\omega = \\\\theta ^a {\\\\omega }{Z^a} = m \\\\theta ^c\\\\omega _{a(n),c b(m-1)}(x,\\\\theta )Y^{a(n)}Z^{b(m-1)}\\\\,.$ It differs from the definition of Section 3 by an additional numerical factor introduced for future convenience.\",\n \"In the sequel we sometimes do not write variables $Y,Z,\\\\theta $ explicitly, that are always assumed to be present implicitly.\",\n \"We adopt the convention that index $a$ is contracted with $Y$ , $b$ with $Z$ and $c_i$ with $\\\\theta ^{c_i}$ with $\\\\theta $ s ordered as $c_1,...,c_p$ .\",\n \"The space $\\\\Lambda (\\\\mathcal {M}^d) \\\\otimes \\\\mathbb {R}[Y,Z]$ can be equipped with the scalar product $\\\\langle f,g\\\\rangle = \\\\frac{1}{\\\\pi ^{2n}}\\\\int _{\\\\mathbb {C}^d \\\\times \\\\mathbb {C}^d} d^{2d}Z\\\\, d^{2d}Y\\\\, d^d \\\\theta \\\\, d^d \\\\overline{\\\\theta }\\\\, f(Z,Y,\\\\theta )\\\\,\\\\overline{g(Z,Y,\\\\theta )}\\\\, e^{-|Z|^2-|Y|^2-\\\\overline{\\\\theta }\\\\theta },$ where $f,g \\\\in \\\\Lambda (\\\\mathcal {M}^d) \\\\otimes \\\\mathbb {R}[Y,Z]$ with complex $Y,Z,\\\\theta $ and Berezin integral over anticommuting variables.\",\n \"(We work with the polynomials of complex variables with real coefficients).\",\n \"The space $\\\\Lambda (\\\\mathcal {M}^d) \\\\otimes \\\\mathbb {R}[Y,Z]$ with the scalar product (REF ) is a Hilbert space in the Euclidean metric signature case used in this section.\",\n \"This scalar product yields the following conjugation rules: $(Z^a)^* = \\\\partial _{Z}{}_{a}\\\\,,\\\\qquad (Y^a)^* = \\\\partial _{Y}{}_a\\\\,,\\\\qquad (\\\\theta ^a)^* = \\\\partial _{\\\\theta }{}_a\\\\,.$\"\n ],\n [\n \"$\\\\mathrm {GL}(d)$ example\",\n \"To illustrate the idea of our construction let us first consider a simpler case where fields and gauge parameters take values in the irreps of $\\\\mathfrak {gl}(d)$ described by two-row Young diagrams (no tracelessness conditions are imposed).\",\n \"Define the following operators, that form $\\\\mathfrak {gl}(2)$ $t_{1} = Y^{a}\\\\frac{\\\\partial }{\\\\partial Z^{a}}, \\\\quad t_{2} = Z^{a}\\\\frac{\\\\partial }{\\\\partial Y^{a}}, \\\\quad t_{0} = Y^{a}\\\\frac{\\\\partial }{\\\\partial Y^{a}} - Z^{a}\\\\frac{\\\\partial }{\\\\partial Z^{a}},$ ${t_{1}}{t_{2}} = t_{0}, \\\\quad {t_{0}}{t_{1}} = 2 t_{1}, \\\\quad {t_{0}}{t_{2}} = -2 t_{2},$ $h_{1} = Y^{a}\\\\frac{\\\\partial }{\\\\partial Y^{a}}, \\\\quad h_2= Z^{a}\\\\frac{\\\\partial }{\\\\partial Z^{a}}\\\\,.$ Namely, $t_i$ form $\\\\mathfrak {sl}(2)$ while $h_1+h_2$ is central.\",\n \"In terms of these operators the space of $p$ -forms valued in two-row Young diagrams is identified as $\\\\ker (t_1)$ $V^p = \\\\lbrace F \\\\in \\\\Lambda ^p(\\\\mathcal {M}^d) \\\\otimes \\\\mathbb {R}[Y,Z]| t_1 F = 0 \\\\rbrace \\\\,.$ Here $\\\\Lambda ^p(\\\\mathcal {M}^d)$ is generated by the Grassmann variables $\\\\theta ^a$ .\",\n \"Let us introduce auxiliary operators $Z_{\\\\theta } = Z^{a}\\\\frac{\\\\partial }{\\\\partial \\\\theta ^{a} }, \\\\quad Y_{\\\\theta } = Y^{a}\\\\frac{\\\\partial }{\\\\partial \\\\theta ^{a} }, \\\\quad D = \\\\theta ^a \\\\frac{\\\\partial }{\\\\partial \\\\theta ^{a} }, \\\\quad \\\\theta _Y = \\\\theta ^a \\\\frac{\\\\partial }{\\\\partial Y^{a} }, \\\\quad \\\\theta _Z = \\\\theta ^a \\\\frac{\\\\partial }{\\\\partial Z^{a} }\\\\,.$ Among the auxiliary operators $D$ plays the most important role as it gives differential form degree.\",\n \"Now we should construct $\\\\sigma _+: \\\\sigma _+^2 = 0$ on $V^p$ and $\\\\Im (\\\\sigma _+) \\\\subset V^p$ .\",\n \"Consider the following operator: $\\\\sigma _+ = f(t_{0})Z_{\\\\theta } + g(t_{0})Y_{\\\\theta }t_2\\\\,,$ where $f(t_{0}) = \\\\sum _{n=0}^{\\\\infty }f_{n}t_{0}^{n}$ and $g(t_{0}) = \\\\sum _{n=0}^{\\\\infty }g_{n}t_{0}^{n}$ .\",\n \"Functions $f$ and $g$ have to be found from the conditions $\\\\sigma _+^2F=0\\\\,,\\\\qquad t_{1}\\\\sigma _+ F =0\\\\,, \\\\qquad \\\\forall F \\\\in V^p\\\\,.$ After some re-ordering of operators this yields two equations $& 0 = Y_{\\\\theta } \\\\bigg (f(t_{0} - 1)F + g(t_{0}-1)t_{0}F \\\\bigg ),\\\\\\\\& 0 = Z_{\\\\theta }Y_{\\\\theta } \\\\bigg (f(t_{0})g(t_{0}+1)t_{2}F - g(t_{0})f(t_{0}+1)t_{2}F - g(t_{0})g(t_{0}+1)t_{2}F \\\\bigg )\\\\,$ verified by $f(t_0) = -(t_0+1)g(t_0)$ giving $\\\\sigma _+ = -(t_0+1)g(t_0)Z_{\\\\theta }+g(t_0)Y_{\\\\theta }t_2\\\\,.$ The free coefficient $g(t_0)$ is determined from the conjugacy requirement: $(f,\\\\sigma _+ g) = -(\\\\theta _Z \\\\bigg (g(t_0)(t_0+1)+g(t_0)\\\\bigg )f, g) = (\\\\sigma _- f,g)\\\\,$ giving $g(t_0) = - \\\\frac{1}{t_0+2}$ and hence $\\\\sigma _+:=(\\\\sigma _-)^*= \\\\frac{t_0+1}{t_0+2}Z_{\\\\theta }-\\\\frac{1}{t_0+2}Y_{\\\\theta }t_2\\\\,.$ One can notice, that $\\\\sigma _+$ in (REF ) differs from what one would expect from the conjugation rules (REF ).\",\n \"The reason is that in (REF ) we work with $C := \\\\Lambda ^p(\\\\mathcal {M}^d) \\\\otimes \\\\mathbb {R}[Y,Z]$ complex.\",\n \"In the $\\\\mathrm {GL}(d)$ -case we deal with $\\\\big (\\\\Lambda ^p(\\\\mathcal {M}^d) \\\\otimes \\\\mathbb {R}[Y,Z]\\\\big ) \\\\cap \\\\ker (t_1)$ complex, therefore one should project on the highest weight vectors of the underlying $\\\\mathfrak {sl}(2)$ in the complex C. The same procedure applies to the $\\\\mathrm {O}(d)$ -case.\",\n \"Though general formulae for extremal projectors are known for any simple Lie algebraWe are grateful to the referee for bringing this fact to our attention.\",\n \"[26], [27], [28], [29](for reviews see [30], [31]), to keep the paper self-contained we derive the relevant projectors straightforwardly.\",\n \"Knowing $\\\\sigma _+$ , it remains to construct the Laplace operator $\\\\Delta = \\\\lbrace \\\\sigma _-,\\\\sigma _+\\\\rbrace $ and find its zeros.\",\n \"Elementary computation gives $\\\\Delta = \\\\frac{t_0}{t_0+1}(D+h_2-1) + \\\\frac{1}{t_0+1}Y_\\\\theta \\\\theta _Y - \\\\frac{1}{(t_0+1)(t_0+2)}t_2 \\\\theta _Z Y_\\\\theta \\\\,.$ Being built from the manifestly $\\\\mathfrak {gl}(d)$ -invariant operators, $\\\\Delta $ commutes with $\\\\mathfrak {gl}(d)$ hence being diagonal on its irreducible submodules.\",\n \"Thus, it suffices to analyze zeros of $\\\\Delta $ on $\\\\mathfrak {gl}(d)$ irreducible components of the forms.\"\n ],\n [\n \"$H^0(\\\\sigma _-)$\",\n \"Any element of $V^0$ has the form $F = F_{a(n),b(m)}Y^{a(n)}Z^{b(m)}$ .\",\n \"It is easy to see that $\\\\Delta F = h_2 F\\\\,.$ Therefore $H^0(\\\\sigma _-) = \\\\lbrace F = F_{a(n)}Y^{a(n)}| \\\\forall F_{a(n)}\\\\in \\\\mathbb {R}\\\\rbrace \\\\,.$\"\n ],\n [\n \"$H^p(\\\\sigma _-)$ , {{formula:0ba4be03-0dbe-402f-9169-c5a2a3811fb6}}\",\n \"For $p>0$ , a general element of $V^p$ is $F = F_{a(n),b(m)|c_1,..,c_p}Y^{a(n)}Z^{b(m)}\\\\theta ^{c_1}..\\\\theta ^{c_p}$ .\",\n \"Generally it forms a reducible $\\\\mathfrak {gl}(d)$ -module associated with the tensor product of two diagrams.\",\n \"In terms of Young diagrams it decomposes into the following irreducible components: $\\\\begin{picture}(15,10)(0,7){\\\\put (05,0){\\\\line (1,0){10}}\\\\put (05,20){\\\\line (1,0){10}}\\\\put (05,0){\\\\line (0,1){20}}\\\\put (15,0){\\\\line (0,1){20}}\\\\put (8,10){\\\\scriptsize {p}}}\\\\end{picture} \\\\,\\\\,\\\\,\\\\otimes _{gl}\\\\,\\\\,\\\\,\\\\begin{picture}(10,18)(0,7){\\\\put (05,00){\\\\line (1,0){35}}\\\\put (05,10){\\\\line (1,0){35}}\\\\put (05,0){\\\\line (0,1){10}}\\\\put (40,0){\\\\line (0,1){10}}\\\\put (17,3){\\\\scriptsize {m}}}\\\\end{picture}\\\\begin{picture}(65,18)(10,7){\\\\put (35,13){\\\\scriptsize {n}}\\\\put (05,20){\\\\line (1,0){60}}\\\\put (05,10){\\\\line (1,0){60}}\\\\put (05,10){\\\\line (0,1){10}}\\\\put (65,10){\\\\line (0,1){10}}}\\\\end{picture}\\\\cong \\\\begin{picture}(10,18)(0,7){\\\\put (05,00){\\\\line (1,0){35}}\\\\put (05,10){\\\\line (1,0){35}}\\\\put (05,0){\\\\line (0,1){10}}\\\\put (40,0){\\\\line (0,1){10}}\\\\put (17,3){\\\\scriptsize {m}}\\\\put (05,-23){\\\\line (1,0){10}}\\\\put (05,-23){\\\\line (0,1){23}}\\\\put (15,-23){\\\\line (0,1){23}}\\\\put (7,-21){\\\\begin{turn}{90}\\\\scriptsize p-1\\\\end{turn}}}\\\\end{picture}\\\\begin{picture}(65,18)(10,7){\\\\put (26,13){\\\\scriptsize {n+1}}\\\\put (05,20){\\\\line (1,0){60}}\\\\put (05,10){\\\\line (1,0){60}}\\\\put (05,10){\\\\line (0,1){10}}\\\\put (65,10){\\\\line (0,1){10}}}\\\\end{picture}\\\\oplus \\\\begin{picture}(10,18)(0,7){\\\\put (05,00){\\\\line (1,0){35}}\\\\put (05,10){\\\\line (1,0){35}}\\\\put (05,0){\\\\line (0,1){10}}\\\\put (40,0){\\\\line (0,1){10}}\\\\put (12,3){\\\\scriptsize {m+1}}\\\\put (05,-23){\\\\line (1,0){10}}\\\\put (05,-23){\\\\line (0,1){23}}\\\\put (15,-23){\\\\line (0,1){23}}\\\\put (7,-21){\\\\begin{turn}{90}\\\\scriptsize p-1\\\\end{turn}}}\\\\end{picture}\\\\begin{picture}(65,18)(10,7){\\\\put (35,13){\\\\scriptsize {n}}\\\\put (05,20){\\\\line (1,0){60}}\\\\put (05,10){\\\\line (1,0){60}}\\\\put (05,10){\\\\line (0,1){10}}\\\\put (65,10){\\\\line (0,1){10}}}\\\\end{picture}\\\\oplus \\\\begin{picture}(10,18)(0,7){\\\\put (05,00){\\\\line (1,0){35}}\\\\put (05,10){\\\\line (1,0){35}}\\\\put (05,0){\\\\line (0,1){10}}\\\\put (40,0){\\\\line (0,1){10}}\\\\put (17,3){\\\\scriptsize {m}}\\\\put (05,-23){\\\\line (1,0){10}}\\\\put (05,-23){\\\\line (0,1){23}}\\\\put (15,-23){\\\\line (0,1){23}}\\\\put (7,-14){\\\\begin{turn}{90}\\\\scriptsize p\\\\end{turn}}}\\\\end{picture}\\\\begin{picture}(65,18)(10,7){\\\\put (35,13){\\\\scriptsize {n}}\\\\put (05,20){\\\\line (1,0){60}}\\\\put (05,10){\\\\line (1,0){60}}\\\\put (05,10){\\\\line (0,1){10}}\\\\put (65,10){\\\\line (0,1){10}}}\\\\end{picture}\\\\oplus \\\\begin{picture}(10,18)(0,7){\\\\put (05,00){\\\\line (1,0){35}}\\\\put (05,10){\\\\line (1,0){35}}\\\\put (05,0){\\\\line (0,1){10}}\\\\put (40,0){\\\\line (0,1){10}}\\\\put (12,3){\\\\scriptsize {m+1}}\\\\put (05,-23){\\\\line (1,0){10}}\\\\put (05,-23){\\\\line (0,1){23}}\\\\put (15,-23){\\\\line (0,1){23}}\\\\put (7,-21){\\\\begin{turn}{90}\\\\scriptsize p-2\\\\end{turn}}}\\\\end{picture}\\\\begin{picture}(65,18)(10,7){\\\\put (26,13){\\\\scriptsize {n+1}}\\\\put (05,20){\\\\line (1,0){60}}\\\\put (05,10){\\\\line (1,0){60}}\\\\put (05,10){\\\\line (0,1){10}}\\\\put (65,10){\\\\line (0,1){10}}}\\\\end{picture}\\\\,.\\\\\\\\$ At $p = 1$ the last diagram is absent.\",\n \"The manifest decomposition of $F_{a(n),b(m)|c_1,c_2,..,c_p}$ into irreducible components is $F_{a(n),b(m)|c_1,c_2,..,c_p} = F_1{}_{a(n)c_1,b(m),c_2,..,c_p} + \\\\frac{m}{n-m+2} F_1{}_{a(n)b,b(m-1)c_1,c_2,..,c_p} + F_2{}_{a(n),b(m)c_1,c_2,..,c_p} + \\\\\\\\ + F_3{}_{a(n),b(m),c_1,..,c_p} + F_4{}_{a(n)c_1,b(m)c_2,c_3,..,c_p},$ where $F_i$ corresponds to the $i$ -th diagram.\",\n \"There are no restrictions on $n,m,p$ in (REF ) except for $n \\\\ge m$ .\",\n \"If for some $n,m$ tensor expression has a wrong Young shape, it is zero.\",\n \"To simplify calculations we derive restrictions on $n,m,p$ for each diagram from the condition of being $\\\\sigma _-$ -closed.\",\n \"For the second and fourth diagrams we find no restrictions, but for others we have $\\\\sigma _- \\\\bigg (\\\\text{1st diagram}\\\\bigg ) = -m\\\\bigg (1-\\\\frac{1}{n-m+2}\\\\bigg )F_1{}_{a(n)c_0,b(m-1)c_1,c_2,..,c_p} \\\\Rightarrow m = 0,$ $\\\\sigma _- F_3{}_{a(n),b(m),c_1,..,c_p} = m F_3{}_{a(n),b(m-1)c_0,c_1,..,c_p} \\\\Rightarrow m = 0\\\\,.$ Using this we obtain the action of $\\\\Delta $ on the rest diagrams: $\\\\Delta F_1{}_{a(n)c_1,c_2,..,c_p} = \\\\frac{1}{n+1}\\\\Delta \\\\theta _Y F_1(Y,Z,\\\\theta ) = \\\\frac{n(p-1)}{(n+1)^2}\\\\theta _Y F_1(Y,Z,\\\\theta ),$ $\\\\Delta F_2{}_{a(n),b(m)c_1,c_2,..,c_p} = \\\\frac{1}{m+1}\\\\Delta \\\\theta _Z F_2(Y,Z,\\\\theta ) = \\\\frac{m+p}{m+1}\\\\theta _Z F_2(Y,Z,\\\\theta ),$ $\\\\Delta F_4{}_{a(n)c_1,b(m)c_2,c_3,..,c_p} = \\\\frac{1}{(m+1)(n+1)}\\\\Delta \\\\theta _Y\\\\theta _Z F_4(Y,Z,\\\\theta )\\\\\\\\ = \\\\frac{(n-m)(p+m-1)}{(n-m+1)(m+1)(n+1)} \\\\theta _Y\\\\theta _Z F_4(Y,Z,\\\\theta )\\\\,.$ As a result, $H^1(\\\\sigma _-) = \\\\lbrace \\\\phi = F_{a(n)c}\\\\theta ^c Y^{a(n)}| h_2 F = 0, F\\\\in V^1\\\\rbrace \\\\,,$ $H^p(\\\\sigma _-) = \\\\lbrace W = \\\\theta _Y\\\\theta _Z C(Y,Z,\\\\theta )| t_0 C = 0, C \\\\in V^{p-2} \\\\,,\\\\quad p>1\\\\rbrace ,$ $C =C{}_{a(n),b(n),c_1,..,c_{p-2}}Y^{a(n)}Z^{b(n)}\\\\theta ^{c_1}..\\\\theta ^{c_{p-2}} \\\\in V^{p-2}\\\\,.$ The dynamical interpretation of the obtained results is as follows.\",\n \"The system has one symmetric gauge field with gauge transformation described by a symmetric parameter.\",\n \"The second cohomology group $H^2(\\\\sigma _-)$ is spanned by a single tensor corresponding to the generalized (traceful) Weyl tensor.\",\n \"If the latter is set to zero, the system becomes topological with the zero-curvature field equations.\",\n \"Otherwise the unfolded equations encode a set of constraints expressing all fields and Weyl tensor via derivatives of the physical fields.\",\n \"Proceeding further with the equations on the Weyl tensor and its descendants results in an infinite set of constraints with no differential equations on the physical field.\",\n \"Such off-shell unfolded equations were considered in [32].\",\n \"The off-shell systems are for interest in many contexts such as, e.g., construction of actions and quantization [33], [34].\",\n \"The lower cohomology groups (REF ), (REF ) and (REF ) match with those obtained, e.g., in [16].\"\n ],\n [\n \"Irreducibility conditions\",\n \"The $\\\\mathrm {O}(d)$ case is in many respects analogous to that of $\\\\mathrm {GL}(d)$ .\",\n \"The difference is due to the tracelessness condition (REF ).\",\n \"The algebra of the operators encoding irreducibility conditions is extended since the metric allows the new types of contractions between $\\\\theta , Y, Z$ and their derivatives.\",\n \"From the representation theory perspective new terms associated with traces appear in the diagram decomposition of the form coefficients, affecting the cohomology analysis.\",\n \"The following operators form the algebra $\\\\mathfrak {sp}(4)$ : $&t_{1} = Y^{a}\\\\frac{\\\\partial }{\\\\partial Z^{a}}, \\\\quad t_{2} = Z^{a}\\\\frac{\\\\partial }{\\\\partial Y^{a}}, \\\\quad h_{1} = Y^{a}\\\\frac{\\\\partial }{\\\\partial Y^{a}}, \\\\quad h_{2}= Z^{a}\\\\frac{\\\\partial }{\\\\partial Z^{a}},\\\\\\\\&t_{0} = Y^{a}\\\\frac{\\\\partial }{\\\\partial Y^{a}} - Z^{a}\\\\frac{\\\\partial }{\\\\partial Z^{a}},\\\\\\\\&f_{1} = \\\\partial _{Y}^a\\\\partial _{Y a}, \\\\quad f_{2} = \\\\partial _{Z}^a\\\\partial _{Z a}, \\\\quad f_{3} = \\\\partial _{Y}^a\\\\partial _{Z a}, \\\\\\\\&e_{1} = Y^a Y_a, \\\\quad e_{2} = Z^a Z_a, \\\\quad e_{3} = Y^a Z_a\\\\,.$ Evidently, these operators commute with the $\\\\mathfrak {so}(d)$ generators (REF ).\",\n \"$\\\\mathfrak {sp}(4)$ and $\\\\mathfrak {so}(d)$ form a Howe-dual pair [35].\",\n \"Young condition (REF ) and tracelessness condition (REF ) impose highest weight conditions on a $\\\\mathfrak {sp}(4)$ -module.\",\n \"In addition, we introduce the following $\\\\mathrm {O}(d)$ invariant operators: $&Z_{\\\\theta } = Z^a \\\\partial _{\\\\theta a}, \\\\quad Y_{\\\\theta } = Y^a \\\\partial _{\\\\theta a},\\\\\\\\&\\\\partial _{\\\\theta Z} = \\\\partial _{\\\\theta a}\\\\partial _{Z}^a, \\\\quad \\\\partial _{\\\\theta Y} = \\\\partial _{\\\\theta a}\\\\partial _{Y}^a,\\\\\\\\&\\\\theta _Z = \\\\theta ^a \\\\partial _{Z a}, \\\\quad \\\\theta _Y = \\\\theta ^a \\\\partial _{Y a}\\\\,,$ which, along with $D$ (REF ) counting differential form degree, extend $\\\\mathfrak {sp}(4)$ to $\\\\mathfrak {osp}(2|4)$ .\",\n \"The simplest way to see this is to let index $a$ take a single value, treating the operators $Z,Y,\\\\partial _Z, \\\\partial _Y, \\\\theta , \\\\partial _\\\\theta $ as creation and annihilation operators, and apply the oscillator realization of $\\\\mathfrak {osp}(2|4)$ .\",\n \"In the problem in question, the form space is $V^p = \\\\lbrace F \\\\in \\\\Lambda ^p(\\\\mathcal {M}^d) \\\\otimes \\\\mathbb {R}[Y,Z]| t_1 F = 0, f_1 F = 0 \\\\rbrace \\\\,.$ Note that these restrictions imply the tracelessness over indices $(Y,Z)$ and $(Z,Z)$ as a consequence of the form of commutators of $t_1$ with $f_{1,2}$ .\"\n ],\n [\n \"$\\\\sigma _+$\",\n \"Let us look for $\\\\sigma _+ = \\\\sigma _-^* $ in the form $\\\\sigma _+ = g_1(h_1,h_2)Z_{\\\\theta } + g_2(h_1,h_2)t_2Y_{\\\\theta } + g_3(h_1,h_2)e_3 \\\\partial _{\\\\theta Y} + g_4(h_1,h_2)e_1t_2 \\\\partial _{\\\\theta Y} + g_5(h_1,h_2)e_2 \\\\partial _{\\\\theta Z}+\\\\\\\\+g_6(h_1,h_2)e_3 t_2\\\\partial _{\\\\theta Z} + g_7(h_1,h_2)e_1 t_2^2 \\\\partial _{\\\\theta Z}\\\\,.$ The condition $\\\\Im (\\\\sigma _+) \\\\subset V^p$ gives $0 = f_1 \\\\sigma _+ F = \\\\bigg (2 g_2(h_1+2,h_2)+2g_3(h_1+2,h_2)+2g_4(h_1+2,h_2)(d + 2h_1)\\\\bigg )t_2 \\\\partial _{\\\\theta Y} F + \\\\\\\\+ \\\\bigg (2g_6(h_1+2,h_2)+2g_7(h_1+2,h_2)(d+2h_1)\\\\bigg )t_2^2\\\\partial _{\\\\theta Z}F,$ $0 = t_1 \\\\sigma _+ F = \\\\bigg (g_1(h_1-1,h_2+1)+g_2(h_1-1,h_2+1)t_0\\\\bigg )Y_\\\\theta F + \\\\bigg (-g_3(-1,1)+2g_5(-1,1)+ \\\\\\\\ + g_6(-1,1)t_0 \\\\bigg )e_3 \\\\partial _{\\\\theta Z} + \\\\bigg (g_3(-1,1)+g_4(-1,1)(t_0-2) \\\\bigg )e_1 \\\\partial _{\\\\theta Y} + \\\\bigg (-g_4(-1,1)+g_6(-1,1)+ \\\\\\\\ + 2g_7(-1,1)(t_0-1)\\\\bigg )e_1 t_2 \\\\partial _{\\\\theta Z}\\\\,.$ This imposes the following six equations on seven coefficients $g_1(h_1,h_2)+g_2(h_1,h_2)(t_0+2) = 0, \\\\\\\\-g_3(h_1,h_2)+2g_5(h_1,h_2)+g_6(h_1,h_2)(t_0+2) = 0, \\\\\\\\g_3(h_1,h_2)+g_4(h_1,h_2) t_0 = 0, \\\\\\\\-g_4(h_1,h_2)+g_6(h_1,h_2)+2g_7(h_1,h_2)(t_0+1) = 0, \\\\\\\\g_2(h_1,h_2)+g_3(h_1,h_2)+g_4(h_1,h_2)(d+2h_1-4) = 0, \\\\\\\\g_6(h_1,h_2)+g_7(h_1,h_2)(d+2h_1-4) = 0\\\\,.$ Choosing $g_7(h_1,h_2)$ as a free parameter, we obtain $g_1(h_1,h_2) = -(t_0+2)(d-4+h_1+h_2)(d-6+2h_2)g_7(h_1,h_2),\\\\\\\\g_2(h_1,h_2) = (d-4+h_1+h_2)(d-6+2h_2)g_7(h_1,h_2),\\\\\\\\g_3(h_1,h_2) = t_0(d-6+2h_2)g_7(h_1,h_2),\\\\\\\\g_4(h_1,h_2) = -(d-6+2h_2)g_7(h_1,h_2),\\\\\\\\g_5(h_1,h_2) = (t_0+1)(d-4+h_1+h_2)g_7(h_1,h_2),\\\\\\\\g_6(h_1,h_2) = -(d-4+2h_1)g_7(h_1,h_2)\\\\,.$ Now using the conjugation rules (REF ) and highest weight conditions (REF ) we get $(F_1,\\\\sigma _+ F_2) = \\\\Big ( -g_7(h_1,h_2)(t_0+2)(d-4+h_1+h_2)(d-6+2h_2) \\\\Big ) (\\\\sigma _{-}F_1,F_2)\\\\,.$ The condition $\\\\sigma _+ = \\\\sigma ^*_-$ demands $g_7(h_1,h_2) = - \\\\frac{1}{(t_0+2)(d-4+h_1+h_2)(d-6+2h_2)}$ giving $\\\\sigma _+ = Z_{\\\\theta } - \\\\frac{1}{t_0+2} t_2 Y_{\\\\theta } - \\\\frac{t_0}{(t_0+2)(d-4+h_1+h_2)} e_3 \\\\partial _{\\\\theta Y} + \\\\frac{1}{(t_0+2)(d-4+h_1+h_2)}e_1t_2 \\\\partial _{\\\\theta Y} - \\\\\\\\-\\\\frac{t_0+1}{(t_0+2)(d-6+2h_2)} e_2 \\\\partial _{\\\\theta Z} +\\\\frac{d-4+2h_1}{(t_0+2)(d-4+h_1+h_2)(d-6+2h_2)} e_3 t_2\\\\partial _{\\\\theta Z} - \\\\\\\\-\\\\frac{1}{(t_0+2)(d-4+h_1+h_2)(d-6+2h_2)} e_1 t_2^2 \\\\partial _{\\\\theta Z}\\\\,.$ This yields operator $\\\\sigma _+$ such that $\\\\sigma _+^2 = 0$ on $V$ , $\\\\Im (\\\\sigma _+) \\\\subset V$ and $\\\\sigma _-^* = \\\\sigma _+$ .\",\n \"To calculate the Laplace operator $\\\\Delta = \\\\sigma _- \\\\sigma _+ + \\\\sigma _+ \\\\sigma _-$ on $V^p$ we obtain straightforwardly that $\\\\sigma _- \\\\sigma _+ = \\\\theta _Z Z_\\\\theta - \\\\frac{1}{t_0+1}t_2 \\\\theta _Z Y_\\\\theta - \\\\frac{1}{t_0+1}\\\\theta _Y Y_\\\\theta - \\\\frac{t_0-1}{(t_0+1)(d-3+h_1+h_2)}\\\\bigg (e_3 \\\\theta _Z \\\\partial _{\\\\theta Y} + \\\\theta ^a Y_a \\\\partial _{\\\\theta Y}\\\\bigg ) \\\\\\\\+ \\\\frac{1}{(t_0+1)(d-3+h_1+h_2)}\\\\bigg (e_1 t_2 \\\\theta _Z \\\\partial _{\\\\theta Y} + e_1 \\\\theta _Y \\\\partial _{\\\\theta Y} \\\\bigg ) - \\\\frac{t_0}{(t_0+1)(d-4+2h_2)}\\\\bigg (e_2 \\\\theta _Z \\\\partial _{\\\\theta Z} + 2 \\\\theta ^a Z_a \\\\partial _{\\\\theta Z} \\\\bigg )\\\\\\\\+ \\\\frac{d-4+2h_1}{(t_0+1)(d-3+h_1+h_2)(d-4+2h_2)}\\\\bigg (e_3 t_2 \\\\theta _Z \\\\partial _{\\\\theta Z} + e_3 \\\\theta _Y \\\\partial _{\\\\theta Z} + t_2 \\\\theta ^a Y_a \\\\partial _{\\\\theta Z} - \\\\theta ^a Z_a \\\\partial _{\\\\theta Z}\\\\bigg )\\\\\\\\- \\\\frac{1}{(t_0+1)(d-3+h_1+h_2)(d-4+2h_2)}\\\\bigg (e_1 t_2^2\\\\theta _Z \\\\partial _{\\\\theta Z} + 2e_1 t_2 \\\\theta _Y \\\\partial _{\\\\theta Z} \\\\bigg )\\\\,.$ $\\\\sigma _+ \\\\sigma _-= Z_\\\\theta \\\\theta _Z + \\\\frac{1}{t_0+2}t_2 \\\\theta _Z Y_\\\\theta + \\\\frac{t_0}{(t_0+2)(d-4+h_1+h_2)}e_3 \\\\theta _Z \\\\partial _{\\\\theta Y}- \\\\frac{1}{(t_0+2)(d-4+h_1+h_2)}\\\\times \\\\\\\\ \\\\times e_1 t_2 \\\\theta _Z \\\\partial _{\\\\theta Y}+ \\\\frac{t_0+1}{(t_0+2)(d-6+2h_2)}e_2 \\\\theta _Z \\\\partial _{\\\\theta Z}- \\\\frac{d-4+2h_1}{(t_0+2)(d-4+h_1+h_2)(d-6+2h_2)}e_3 t_2 \\\\theta _Z \\\\partial _{\\\\theta Z}\\\\\\\\+ \\\\frac{1}{(t_0+2)(d-4+h_1+h_2)(d-6+2h_2)}e_1 t_2^2\\\\theta _Z \\\\partial _{\\\\theta Z}\\\\,.$ Since, by construction, both $\\\\sigma _- \\\\sigma _+$ and $\\\\sigma _+ \\\\sigma _-$ and hence $\\\\Delta $ are $\\\\mathrm {O}(d)$ invariant, $\\\\Delta $ is diagonal on irreducible $\\\\mathrm {O}(d)$ -modules and, to compute $H(\\\\sigma _-)$ , it suffices to find its zeros on the irreducible components.\"\n ],\n [\n \"$H^0(\\\\sigma _-)$\",\n \"In the sector of zero-forms, all terms that contain $\\\\frac{\\\\partial }{\\\\partial \\\\theta ^c}$ trivialize.\",\n \"Hence, $\\\\Delta F = h_2 F = m F \\\\,$ and $H^0(\\\\sigma _-) = \\\\lbrace F = F_{a(n)}Y^{a(n)}| \\\\forall F_{a(n)}\\\\in \\\\mathbb {R}\\\\rbrace \\\\,.$ Comparing the resulting differential gauge parameters with (REF ), we find that, as anticipated, differential gauge symmetries in the unfolded formulation coincide with those of the Fronsdal theory.\"\n ],\n [\n \"$H^p(\\\\sigma _-)$ , {{formula:b937a896-0a47-49ab-98e2-b9277cda9f42}}\",\n \"The main difference between $\\\\mathfrak {o}(d)$ - and $\\\\mathfrak {gl}(d)$ - cases is due to the traceful terms in the decomposition of the $p$ -forms into the irreducible parts depicted as $\\\\begin{picture}(15,10)(0,7){\\\\put (05,0){\\\\line (1,0){10}}\\\\put (05,20){\\\\line (1,0){10}}\\\\put (05,0){\\\\line (0,1){20}}\\\\put (15,0){\\\\line (0,1){20}}\\\\put (8,10){\\\\scriptsize {p}}}\\\\end{picture} \\\\,\\\\,\\\\,\\\\otimes _{so}\\\\,\\\\,\\\\,\\\\begin{picture}(10,18)(0,7){\\\\put (05,00){\\\\line (1,0){35}}\\\\put (05,10){\\\\line (1,0){35}}\\\\put (05,0){\\\\line (0,1){10}}\\\\put (40,0){\\\\line (0,1){10}}\\\\put (17,3){\\\\scriptsize {m}}}\\\\end{picture}\\\\begin{picture}(65,18)(10,7){\\\\put (35,13){\\\\scriptsize {n}}\\\\put (05,20){\\\\line (1,0){60}}\\\\put (05,10){\\\\line (1,0){60}}\\\\put (05,10){\\\\line (0,1){10}}\\\\put (65,10){\\\\line (0,1){10}}}\\\\end{picture}\\\\cong \\\\begin{picture}(10,18)(0,7){\\\\put (05,00){\\\\line (1,0){35}}\\\\put (05,10){\\\\line (1,0){35}}\\\\put (05,0){\\\\line (0,1){10}}\\\\put (40,0){\\\\line (0,1){10}}\\\\put (17,3){\\\\scriptsize {m}}\\\\put (05,-25){\\\\line (1,0){10}}\\\\put (05,-25){\\\\line (0,1){25}}\\\\put (15,-25){\\\\line (0,1){25}}\\\\put (7,-15){\\\\begin{turn}{90}\\\\scriptsize p\\\\end{turn}}}\\\\end{picture}\\\\begin{picture}(65,18)(10,7){\\\\put (35,13){\\\\scriptsize {n}}\\\\put (05,20){\\\\line (1,0){60}}\\\\put (05,10){\\\\line (1,0){60}}\\\\put (05,10){\\\\line (0,1){10}}\\\\put (65,10){\\\\line (0,1){10}}}\\\\end{picture}\\\\oplus \\\\begin{picture}(10,18)(0,7){\\\\put (05,00){\\\\line (1,0){35}}\\\\put (05,10){\\\\line (1,0){35}}\\\\put (05,0){\\\\line (0,1){10}}\\\\put (40,0){\\\\line (0,1){10}}\\\\put (12,3){\\\\scriptsize {m+1}}\\\\put (05,-25){\\\\line (1,0){10}}\\\\put (05,-25){\\\\line (0,1){25}}\\\\put (15,-25){\\\\line (0,1){25}}\\\\put (7,-21){\\\\begin{turn}{90}\\\\scriptsize p-1\\\\end{turn}}}\\\\end{picture}\\\\begin{picture}(65,18)(10,7){\\\\put (35,13){\\\\scriptsize {n}}\\\\put (05,20){\\\\line (1,0){60}}\\\\put (05,10){\\\\line (1,0){60}}\\\\put (05,10){\\\\line (0,1){10}}\\\\put (65,10){\\\\line (0,1){10}}}\\\\end{picture}\\\\oplus \\\\begin{picture}(10,18)(0,7){\\\\put (05,00){\\\\line (1,0){35}}\\\\put (05,10){\\\\line (1,0){35}}\\\\put (05,0){\\\\line (0,1){10}}\\\\put (40,0){\\\\line (0,1){10}}\\\\put (17,3){\\\\scriptsize {m}}\\\\put (05,-25){\\\\line (1,0){10}}\\\\put (05,-25){\\\\line (0,1){25}}\\\\put (15,-25){\\\\line (0,1){25}}\\\\put (7,-21){\\\\begin{turn}{90}\\\\scriptsize p-1\\\\end{turn}}}\\\\end{picture}$ ${n+1}$ $\\\\oplus \\\\begin{picture}(10,18)(0,7){\\\\put (05,00){\\\\line (1,0){35}}\\\\put (05,10){\\\\line (1,0){35}}\\\\put (05,0){\\\\line (0,1){10}}\\\\put (40,0){\\\\line (0,1){10}}\\\\put (12,3){\\\\scriptsize {m+1}}\\\\put (05,-25){\\\\line (1,0){10}}\\\\put (05,-25){\\\\line (0,1){25}}\\\\put (15,-25){\\\\line (0,1){25}}\\\\put (7,-21){\\\\begin{turn}{90}\\\\scriptsize p-2\\\\end{turn}}}\\\\end{picture}\\\\begin{picture}(65,18)(10,7){\\\\put (26,13){\\\\scriptsize {n+1}}\\\\put (05,20){\\\\line (1,0){60}}\\\\put (05,10){\\\\line (1,0){60}}\\\\put (05,10){\\\\line (0,1){10}}\\\\put (65,10){\\\\line (0,1){10}}}\\\\end{picture}\\\\oplus \\\\begin{picture}(10,18)(0,7){\\\\put (05,00){\\\\line (1,0){35}}\\\\put (05,10){\\\\line (1,0){35}}\\\\put (05,0){\\\\line (0,1){10}}\\\\put (40,0){\\\\line (0,1){10}}\\\\put (17,3){\\\\scriptsize {m}}\\\\put (05,-25){\\\\line (1,0){10}}\\\\put (05,-25){\\\\line (0,1){25}}\\\\put (15,-25){\\\\line (0,1){25}}\\\\put (7,-21){\\\\begin{turn}{90}\\\\scriptsize p-1\\\\end{turn}}}\\\\end{picture}\\\\begin{picture}(65,18)(10,7){\\\\put (26,13){\\\\scriptsize {n-1}}\\\\put (05,20){\\\\line (1,0){60}}\\\\put (05,10){\\\\line (1,0){60}}\\\\put (05,10){\\\\line (0,1){10}}\\\\put (65,10){\\\\line (0,1){10}}}\\\\end{picture}\\\\oplus \\\\begin{picture}(10,18)(0,7){\\\\put (05,00){\\\\line (1,0){35}}\\\\put (05,10){\\\\line (1,0){35}}\\\\put (05,0){\\\\line (0,1){10}}\\\\put (40,0){\\\\line (0,1){10}}\\\\put (12,3){\\\\scriptsize {m-1}}\\\\put (05,-25){\\\\line (1,0){10}}\\\\put (05,-25){\\\\line (0,1){25}}\\\\put (15,-25){\\\\line (0,1){25}}\\\\put (7,-21){\\\\begin{turn}{90}\\\\scriptsize p-1\\\\end{turn}}}\\\\end{picture}\\\\begin{picture}(65,18)(10,7){\\\\put (35,13){\\\\scriptsize {n}}\\\\put (05,20){\\\\line (1,0){60}}\\\\put (05,10){\\\\line (1,0){60}}\\\\put (05,10){\\\\line (0,1){10}}\\\\put (65,10){\\\\line (0,1){10}}}\\\\end{picture}\\\\oplus \\\\begin{picture}(10,18)(0,7){\\\\put (05,00){\\\\line (1,0){35}}\\\\put (05,10){\\\\line (1,0){35}}\\\\put (05,0){\\\\line (0,1){10}}\\\\put (40,0){\\\\line (0,1){10}}\\\\put (12,3){\\\\scriptsize {m-1}}\\\\put (05,-25){\\\\line (1,0){10}}}\\\\put (05,-25){\\\\line (0,1){25}}\\\\put (15,-25){\\\\line (0,1){25}}\\\\put (7,-21){\\\\begin{turn}{90}\\\\scriptsize p-2\\\\end{turn}}\\\\end{picture}$ ${n+1}$ ${m+1}$90$p-2$ ${n-1}$ $\\\\oplus \\\\begin{picture}(10,18)(0,7){\\\\put (05,00){\\\\line (1,0){35}}\\\\put (05,10){\\\\line (1,0){35}}\\\\put (05,0){\\\\line (0,1){10}}\\\\put (40,0){\\\\line (0,1){10}}\\\\put (12,3){\\\\scriptsize {m-1}}\\\\put (05,-25){\\\\line (1,0){10}}\\\\put (05,-25){\\\\line (0,1){25}}\\\\put (15,-25){\\\\line (0,1){25}}\\\\put (7,-21){\\\\begin{turn}{90}\\\\scriptsize p-2\\\\end{turn}}}\\\\end{picture}\\\\begin{picture}(65,18)(10,7){\\\\put (26,13){\\\\scriptsize {n-1}}\\\\put (05,20){\\\\line (1,0){60}}\\\\put (05,10){\\\\line (1,0){60}}\\\\put (05,10){\\\\line (0,1){10}}\\\\put (65,10){\\\\line (0,1){10}}}\\\\end{picture}\\\\oplus \\\\begin{picture}(10,18)(0,7){\\\\put (05,00){\\\\line (1,0){35}}\\\\put (05,10){\\\\line (1,0){35}}\\\\put (05,0){\\\\line (0,1){10}}\\\\put (40,0){\\\\line (0,1){10}}\\\\put (17,3){\\\\scriptsize {m}}\\\\put (05,-25){\\\\line (1,0){10}}\\\\put (05,-25){\\\\line (0,1){25}}\\\\put (15,-25){\\\\line (0,1){25}}\\\\put (7,-21){\\\\begin{turn}{90}\\\\scriptsize p-2\\\\end{turn}}}\\\\end{picture}\\\\begin{picture}(65,18)(10,7){\\\\put (35,13){\\\\scriptsize {n}}\\\\put (05,20){\\\\line (1,0){60}}\\\\put (05,10){\\\\line (1,0){60}}\\\\put (05,10){\\\\line (0,1){10}}\\\\put (65,10){\\\\line (0,1){10}}}\\\\end{picture}\\\\oplus \\\\begin{picture}(10,18)(0,7){\\\\put (05,00){\\\\line (1,0){35}}\\\\put (05,10){\\\\line (1,0){35}}\\\\put (05,0){\\\\line (0,1){10}}\\\\put (40,0){\\\\line (0,1){10}}\\\\put (17,3){\\\\scriptsize {m}}\\\\put (05,-25){\\\\line (1,0){10}}\\\\put (05,-25){\\\\line (0,1){25}}\\\\put (15,-25){\\\\line (0,1){25}}\\\\put (7,-21){\\\\begin{turn}{90}\\\\scriptsize p-2\\\\end{turn}}}\\\\end{picture}\\\\begin{picture}(65,18)(10,7){\\\\put (35,13){\\\\scriptsize {n}}\\\\put (05,20){\\\\line (1,0){60}}\\\\put (05,10){\\\\line (1,0){60}}\\\\put (05,10){\\\\line (0,1){10}}\\\\put (65,10){\\\\line (0,1){10}}}\\\\end{picture}\\\\oplus \\\\begin{picture}(10,18)(0,7){\\\\put (05,00){\\\\line (1,0){35}}\\\\put (05,10){\\\\line (1,0){35}}\\\\put (05,0){\\\\line (0,1){10}}\\\\put (40,0){\\\\line (0,1){10}}\\\\put (17,3){\\\\scriptsize {m}}\\\\put (05,-25){\\\\line (1,0){10}}\\\\put (05,-25){\\\\line (0,1){25}}\\\\put (15,-25){\\\\line (0,1){25}}\\\\put (7,-21){\\\\begin{turn}{90}\\\\scriptsize p-3\\\\end{turn}}}\\\\end{picture}\\\\begin{picture}(65,18)(10,7){\\\\put (26,13){\\\\scriptsize {n+1}}\\\\put (05,20){\\\\line (1,0){60}}\\\\put (05,10){\\\\line (1,0){60}}\\\\put (05,10){\\\\line (0,1){10}}\\\\put (65,10){\\\\line (0,1){10}}}\\\\end{picture}\\\\oplus $ $\\\\oplus \\\\begin{picture}(10,18)(0,7){\\\\put (05,00){\\\\line (1,0){35}}\\\\put (05,10){\\\\line (1,0){35}}\\\\put (05,0){\\\\line (0,1){10}}\\\\put (40,0){\\\\line (0,1){10}}\\\\put (12,3){\\\\scriptsize {m+1}}\\\\put (05,-25){\\\\line (1,0){10}}\\\\put (05,-25){\\\\line (0,1){25}}\\\\put (15,-25){\\\\line (0,1){25}}\\\\put (7,-21){\\\\begin{turn}{90}\\\\scriptsize p-3\\\\end{turn}}}\\\\end{picture}\\\\begin{picture}(65,18)(10,7){\\\\put (35,13){\\\\scriptsize {n}}\\\\put (05,20){\\\\line (1,0){60}}\\\\put (05,10){\\\\line (1,0){60}}\\\\put (05,10){\\\\line (0,1){10}}\\\\put (65,10){\\\\line (0,1){10}}}\\\\end{picture}\\\\oplus \\\\begin{picture}(10,18)(0,7){\\\\put (05,00){\\\\line (1,0){35}}\\\\put (05,10){\\\\line (1,0){35}}\\\\put (05,0){\\\\line (0,1){10}}\\\\put (40,0){\\\\line (0,1){10}}\\\\put (17,3){\\\\scriptsize {m}}\\\\put (05,-25){\\\\line (1,0){10}}\\\\put (05,-25){\\\\line (0,1){25}}\\\\put (15,-25){\\\\line (0,1){25}}\\\\put (7,-21){\\\\begin{turn}{90}\\\\scriptsize p-3\\\\end{turn}}}\\\\end{picture}\\\\begin{picture}(65,18)(10,7){\\\\put (26,13){\\\\scriptsize {n-1}}\\\\put (05,20){\\\\line (1,0){60}}\\\\put (05,10){\\\\line (1,0){60}}\\\\put (05,10){\\\\line (0,1){10}}\\\\put (65,10){\\\\line (0,1){10}}}\\\\end{picture}\\\\oplus \\\\begin{picture}(10,18)(0,7){\\\\put (05,00){\\\\line (1,0){35}}\\\\put (05,10){\\\\line (1,0){35}}\\\\put (05,0){\\\\line (0,1){10}}\\\\put (40,0){\\\\line (0,1){10}}\\\\put (12,3){\\\\scriptsize {m-1}}\\\\put (05,-25){\\\\line (1,0){10}}\\\\put (05,-25){\\\\line (0,1){25}}\\\\put (15,-25){\\\\line (0,1){25}}\\\\put (7,-21){\\\\begin{turn}{90}\\\\scriptsize p-3\\\\end{turn}}}\\\\end{picture}\\\\begin{picture}(65,18)(10,7){\\\\put (35,13){\\\\scriptsize {n}}\\\\put (05,20){\\\\line (1,0){60}}\\\\put (05,10){\\\\line (1,0){60}}\\\\put (05,10){\\\\line (0,1){10}}\\\\put (65,10){\\\\line (0,1){10}}}\\\\end{picture}\\\\oplus \\\\begin{picture}(10,18)(0,7){\\\\put (05,00){\\\\line (1,0){35}}\\\\put (05,10){\\\\line (1,0){35}}\\\\put (05,0){\\\\line (0,1){10}}\\\\put (40,0){\\\\line (0,1){10}}\\\\put (17,3){\\\\scriptsize {m}}\\\\put (05,-25){\\\\line (1,0){10}}\\\\put (05,-25){\\\\line (0,1){25}}\\\\put (15,-25){\\\\line (0,1){25}}\\\\put (7,-21){\\\\begin{turn}{90}\\\\scriptsize p-4\\\\end{turn}}}\\\\end{picture}\\\\begin{picture}(65,18)(10,7){\\\\put (35,13){\\\\scriptsize {n}}\\\\put (05,20){\\\\line (1,0){60}}\\\\put (05,10){\\\\line (1,0){60}}\\\\put (05,10){\\\\line (0,1){10}}\\\\put (65,10){\\\\line (0,1){10}}}\\\\end{picture}$ For $1 \\\\le p \\\\le 3$ , the diagrams carrying negative $p$ -dependent labels are absent.\",\n \"It is important to note that the diagram $(n,m,p-2)$ is present twice: one copy results from the contraction of one of the form indices with the first row followed by the symmetrization of another form index with the same row.\",\n \"Another one results from the application of the same procedure to the second row.\",\n \"This fact leads to two different tensor implementations.\",\n \"Also note that some of the diagrams vanish for special dimensions by virtue of the Two-Column Theorem: Theorem 5.1 $\\\\mathfrak {so}(d)$ traceless tensors with the symmetry properties of such Young diagrams that the sum of the heights of the first two columns exceeds $d$ , are identically zero.\",\n \"[36] The cohomology $H^p(\\\\sigma _-)$ is empty for $p>d$ .\",\n \"Analogously, some potential elements of $H^p(\\\\sigma _-)$ are zero by the Two-Column Theorem for large $p \\\\le d$ .\",\n \"Now we are in a position to consider the action of the Laplace operator on each of the diagrams (REF ) separately.\",\n \"In the following restrictions on $n,m,p$ will be imposed: if for some $n,m,p$ a tensor has a wrong Young shape, it is zero.\",\n \"In most cases we will simplify calculation by demanding $p$ -forms be $\\\\sigma _-$ -closed.\",\n \"Another simplification is due to the fact that $\\\\Delta F = 0$ is equivalent to the two equations $\\\\sigma _- F = 0$ and $\\\\sigma _+ F = 0$ .\",\n \"Diagram (n,m;p), $n\\\\ge 0, m\\\\ge 0, p\\\\ge 0$ has the tensor form $T_{a(n),b(m),c_1,..,c_p}$ .\",\n \"It is $\\\\sigma _-$ -closed, if $m = 0$ .\",\n \"Then $\\\\Delta T_{a(n),c_1,..,c_p} = p T_{a(n),c_1,..,c_p}\\\\,.$ This diagram is in $\\\\ker (\\\\Delta )$ at $p = 0$ that reproduces the already obtained result for $H^0(\\\\sigma _-)$ .\",\n \"Diagram (n+1,m+1;p-2), $n\\\\ge 0, m\\\\ge 0, p\\\\ge 2$ has the tensor form $T_{a(n)c_1,b(m)c_2,..,c_p}$ .\",\n \"This diagram is $\\\\sigma _-$ -closed.\",\n \"The action of $\\\\Delta $ is $\\\\Delta T_{a(n)c_1,b(m)c_2,..,c_p} = \\\\frac{(n-m)(p+m-1)}{(n-m+1)}T_{a(n)c_1,b(m)c_2,..,c_p}\\\\,.$ This diagram is in $\\\\ker (\\\\Delta )$ , if $n=m$ , thus belonging to $H^p(\\\\sigma _-)$ with $p\\\\ge 2$ .\",\n \"Diagram (n,m+1;p-1), $n\\\\ge 1, m\\\\ge 0, p\\\\ge 1$ has the tensor form $T_{a(n),b(m)c_0,c_1,..,c_{p-1}}$ .\",\n \"Then $\\\\Delta T_{a(n),b(m)c_0,c_1,..,c_{p-1}} = (m+p) T_{a(n),b(m)c_0,c_1,..,c_{p-1}}\\\\,.$ This equation admits no solutions since $p\\\\ge 1$ in the case in question.\",\n \"Diagram (n+1,m;p-1), $n\\\\ge 0, m\\\\ge 0, p\\\\ge 1$ has the tensor form $T_{a(n)c_0,b(m),c_1,..,c_{p-1}} + \\\\frac{m}{n-m+2}T_{a(n)b,b(m-1)c_0,c_1,..,c_{p-1}}\\\\,.$ It is $\\\\sigma _-$ -closed iff $m = 0$ .\",\n \"Then $\\\\Delta T_{a(n)c_0,c_1,..,c_{p-1}} = \\\\frac{n(p-1)}{n+1} T_{a(n)c_0,c_1,..,c_{p-1}}\\\\,.$ This expression vanishes at $p = 1$ .\",\n \"Hence, $T_{a(n)c_0} \\\\in H^1(\\\\sigma _-)\\\\,.$ As one can see, $n=0$ in (REF ) also leads to zero Laplace action.\",\n \"However, this is not a new result, since it has been already accounted in the diagrams $(n,m;p)$ for $n=m=p=0$ (REF ), $(n+1,m+1;p-2)$ for $n=m=0,p\\\\ge 2$ (REF ) and the $n=m=0,p=1$ case of $(n+1,m;p-1)$ (REF ).\",\n \"This fact is a simple consequence of the tensor multiplication of a column by a scalar.\",\n \"Diagram (n-1,m;p-1), $n\\\\ge 1, m\\\\ge 0, p\\\\ge 1$ has the tensor form $T = \\\\eta _{ac_1}\\\\rho _{a(n-1),b(m),c_2,..,c_p} - \\\\frac{n-1}{d-4+2n}\\\\eta _{aa}\\\\rho _{a(n-2)c_1,b(m),c_2,..,c_p} + \\\\frac{m(n-1)}{(d-4+m+n)(d-4+2n)}\\\\times \\\\\\\\\\\\times \\\\eta _{aa}\\\\rho _{a(n-2)b,b(m-1)c_1,c_2,..,c_p} - \\\\frac{m}{d-4+m+n}\\\\eta _{ab}\\\\rho _{a(n-1),b(m-1)c_1,c_2,..,c_p}\\\\,,$ where $\\\\rho $ is an arbitrary $(n-1,m;p-1)$ tensor.\",\n \"One can check that $T \\\\in \\\\ker (\\\\sigma _-)$ demands $m = 0$ with $T = \\\\eta _{ac_1}\\\\rho _{a(n-1),c_2,..,c_p} - \\\\frac{n-1}{d-4+2n}\\\\eta _{aa}\\\\rho _{a(n-2)c_1,c_2,..,c_p}.$ After some calculation we obtain $\\\\Delta T = \\\\frac{(p-1)(d-2+n)}{d-3+n} T\\\\,.$ This implies that $\\\\Delta $ has zero at $p = 1$ contributing to $H^1(\\\\sigma _-)$ .\",\n \"The case of $d=2,n=1$ must be considered separately because of the divergent denominator.\",\n \"The seeming divergence emerges due to the second term in (REF ), which is absent at $d=2,n=1$ , $T = \\\\eta _{ac_1}\\\\rho _{c_2,..,c_p} \\\\Rightarrow \\\\sigma _+ T =\\\\frac{p-1}{2}\\\\Big (\\\\eta _{bc_1}\\\\rho _{a,c_2,..,c_{p-1}} - \\\\eta _{ac_1}\\\\rho _{b,c_2,..,c_{p-1}}\\\\Big )\\\\,,$ leading to the same answer with $p=1$ .\",\n \"Diagram (n,m-1;p-1) $n\\\\ge 1, m\\\\ge 1, p\\\\ge 1$ has the tensor form $T = (n-1)\\\\eta _{aa} \\\\rho _{a(n-2)bc_1,b(m-1),c_2,..,c_p} - \\\\frac{(n-1)(m-1)}{d-6+2m} \\\\eta _{aa} \\\\rho _{a(n-2)bb,b(m-2)c_1,c_2,..,c_p} - (d-4+m+n) \\\\times \\\\\\\\\\\\times \\\\eta _{ac_1} \\\\rho _{a(n-1)b,b(m-1),c_2,..,c_p} - (n-m)\\\\eta _{ab} \\\\rho _{a(n-1)c_1,b(m-1),c_2,..,c_p} + \\\\frac{(m-1)(d-4+2n)}{d-6+2m}\\\\eta _{ab} \\\\times \\\\\\\\\\\\times \\\\rho _{a(n-1)b,b(m-2)c_1,c_2,..,c_p} + \\\\frac{(n-m+1)(d-4+m+n)}{n} \\\\eta _{bc_1} \\\\rho _{a(n),b(m-1),c_2,..,c_p} - \\\\\\\\-\\\\frac{(m-1)(n-m+1)(d-4+m+n)}{(d-6+2m)n}\\\\eta _{bb} \\\\rho _{a(n),b(m-2)c_1,c_2,..,c_p}\\\\,,$ where $\\\\rho $ is an arbitrary $(n,m-1;p-1)$ tensor.\",\n \"Let us show that this diagram can never be annihilated by $\\\\sigma _-$ .\",\n \"Indeed, $\\\\sigma _- T = \\\\bigg (d-4+m+n-(n-m)\\\\bigg )\\\\eta _{ac_0} \\\\rho _{a(n-1)c_1,b(m-1),c_2,..,c_p} + (m-1)(\\\\emph {lit}),$ where (lit) denotes some terms that are linearly independent from the first one.\",\n \"If $m=1$ the above expression reduces to $\\\\sigma _- T = \\\\big (d-2\\\\big )\\\\eta _{ac_0} \\\\rho _{a(n-1)c_1,c_2,..,c_p}.$ The expression in brackets vanishes at $d = 2$ .\",\n \"However, such diagram is zero by virtue of the Two-column theorem 5.1, since the heights of the first two columns sum up to $3>d$ .\",\n \"Thus, the nontrivial $T$ is never in $\\\\ker (\\\\sigma _-)$ .\",\n \"Diagram (n+1,m-1;p-2) $n\\\\ge 1, m\\\\ge 1, p\\\\ge 2$ has the tensor form $T = \\\\eta _{ac_1}\\\\rho _{a(n-1)bc_2,b(m-1),c_3,..c_p}- \\\\frac{n-m+1}{n}\\\\eta _{bc_1}\\\\rho _{a(n)c_2,b(m-1),c_3,..c_p} + \\\\\\\\ + \\\\frac{m-1}{n-m+3}\\\\eta _{ac_1}\\\\rho _{a(n-1)bb,b(m-2)c_2,c_3,..c_p}-\\\\frac{(m-1)(n-1)}{(d-6+2m)(n-m+3)}\\\\eta _{aa}\\\\rho _{a(n-2)bbc_1,b(m-2)c_2,c_3,..c_p} + \\\\\\\\ + \\\\frac{2(m-1)(n-m+1)}{(d-6+2m)(n-m+3)}\\\\eta _{ab}\\\\rho _{a(n-1)bc_1,b(m-2)c_2,c_3,..c_p}- \\\\frac{(m-1)(n-m+1)}{n(n-m+3)}\\\\times \\\\\\\\ \\\\times \\\\eta _{bc_1}\\\\rho _{a(n)b,b(m-2)c_2,c_3,..c_p} - \\\\frac{(m-1)(n-m+2)(n-m+1)}{(d-6+2m)(n-m+3)n} \\\\eta _{bb}\\\\rho _{a(n)c_1,b(m-2)c_2,c_3,..c_p}\\\\,,$ where $\\\\rho $ is an arbitrary $(n+1,m-1;p-2)$ tensor.\",\n \"Though the tensor realization (REF ) may look complicated, the problem is simplified by the observation that all terms except for the first and second ones carry a factor of $(m-1)$ .\",\n \"The action of $\\\\sigma _-$ on the first and second terms produces a factor of $(m-1)$ in front of each $\\\\eta \\\\rho $ combination.\",\n \"It can be checked that $\\\\sigma _- T$ has an overall factor of $(m-1)$ so that the only possible solution for $T \\\\in \\\\ker (\\\\sigma _-)$ is at $m=1$ in which case the tensor decomposition acquires the form $T = \\\\eta _{ac_1}\\\\rho _{a(n-1)bc_2,..,c_p} - \\\\eta _{bc_1}\\\\rho _{a(n)c_2,..,c_p}.$ At $p = 2$ , after some calculations one can check that $\\\\sigma _+ T = 0\\\\,.$ For $p>2$ it is not difficult to see that $\\\\sigma _+ T \\\\ne 0$ .\",\n \"Indeed, $\\\\sigma _+ T = (p-2)\\\\bigg (1 + \\\\frac{1}{n}\\\\bigg )\\\\eta _{ac_1}\\\\rho _{a(n-1)bc_2,b,c_3,..,c_{p-1}} + (\\\\emph {lit})\\\\,,$ where (lit) denotes other linearly independent terms.\",\n \"The first term is never zero.\",\n \"Thus, $T = \\\\eta _{ac_1}\\\\rho _{a(n-1)bc_2} - \\\\eta _{bc_1}\\\\rho _{a(n)c_2} \\\\in H^2(\\\\sigma _-)\\\\,.$ Diagram (n-1,m+1;p-2), $n\\\\ge 1, m\\\\ge 0, p\\\\ge 2$ has the tensor form $T = \\\\eta _{ac_1} \\\\rho _{a(n-1),b(m)c_2,..,c_p} - \\\\frac{n-1}{d-4+2n}\\\\eta _{aa} \\\\rho _{a(n-2)c_1,b(m)c_2,..,c_p}\\\\,,$ where $\\\\rho $ is an arbitrary $(n-1,m+1;p-2)$ tensor.\",\n \"Though it is obviously in $\\\\ker ( \\\\sigma _-)$ for any $m$ , it is not hard to see that it is never in $\\\\ker (\\\\sigma _+)$ .\",\n \"$\\\\sigma _+ T = -\\\\bigg (1 + \\\\frac{p-2}{m+1} \\\\bigg ) \\\\eta _{ac_1} \\\\rho _{a(n-1),b(m+1),..,c_{p-1}} + (\\\\emph {lit} )\\\\,.$ Hence this diagram does not contribute to $H^p(\\\\sigma _-)$ .\",\n \"Diagram (n-1,m-1;p-2), $n\\\\ge 1, m\\\\ge 1, p\\\\ge 2$ has an involved tensor form.\",\n \"Since the coefficients in the expression below are complicated, we extract the factor of $(m-1)$ once present denoting the leftover coefficients as $\\\\alpha _i$ , 0.75em $T = \\\\eta _{ac_1}\\\\eta _{bc_2}\\\\rho _{a(n-1),b(m-1),..,c_p} + (m-1)\\\\alpha _1 \\\\eta _{aa}\\\\eta _{aa}\\\\rho _{a(n-4)bbc_1,b(m-2)c_2,..,c_p} + \\\\\\\\ + \\\\alpha _2 \\\\eta _{aa}\\\\eta _{ac_1}\\\\rho _{a(n-3)bc_2,b(m-1),..,c_p}+ (m-1)\\\\alpha _3\\\\eta _{aa}\\\\eta _{ac_1}\\\\rho _{a(n-3)bb,b(m-2)c_2,..,c_p} + \\\\\\\\ + (m-1)\\\\alpha _4 \\\\eta _{aa}\\\\eta _{ab}\\\\rho _{a(n-3)bc_1,b(m-2)c_2,..,c_p}+\\\\alpha _5 \\\\eta _{aa}\\\\eta _{bc_1}\\\\rho _{a(n-2)c_2,b(m-1),..,c_p} + \\\\\\\\ + (m-1)\\\\alpha _6\\\\eta _{aa}\\\\eta _{bc_1}\\\\rho _{a(n-2)b,b(m-2)c_2,..,c_p}+ (m-1)\\\\alpha _7 \\\\eta _{aa}\\\\eta _{bb}\\\\rho _{a(n-2)c_1,b(m-2)c_2,..,c_p} + \\\\\\\\ + \\\\alpha _8 \\\\eta _{ac_1}\\\\eta _{ab}\\\\rho _{a(n-2)c_2,b(m-1),..,c_p}+ (m-1)\\\\alpha _9 \\\\eta _{ac_1}\\\\eta _{ab}\\\\rho _{a(n-2)b,b(m-2)c_2,..,c_p} + \\\\\\\\ + (m-1)\\\\alpha _{10} \\\\eta _{ac_1}\\\\eta _{bb}\\\\rho _{a(n-1),b(m-2)c_2,..,c_p}+ (m-1) \\\\alpha _{11} \\\\eta _{ab}\\\\eta _{ab}\\\\rho _{a(n-2)c_1,b(m-2)c_2,..,c_p} + \\\\\\\\ + (m-1) \\\\alpha _{12} \\\\eta _{ab}\\\\eta _{bc_1}\\\\rho _{a(n-1),b(m-2)c_2,..,c_p}\\\\,,$ where $\\\\rho $ is an arbitrary $(n-1,m-1;p-2)$ tensor.\",\n \"The explicit form of $\\\\alpha _i$ is given in the Appendix A.\",\n \"Now we observe that the action of $\\\\sigma _-$ on the terms free of the factor of $(m-1)$ produces such factor.\",\n \"Hence, $\\\\sigma _-(T)$ has the form of the sum of linearly independent terms with the common factor of $(m-1)$ .\",\n \"Consequently, $\\\\sigma _- T = 0, \\\\text{ iff\\\\quad } m = 1.$ At $m=1$ , the only terms that remain are $T = \\\\eta _{ac_1}\\\\eta _{bc_2}\\\\rho _{a(n-1),..,c_p} + \\\\frac{(n-2)(n-1)}{(d-3+n)(d-4+2n)} \\\\eta _{aa}\\\\eta _{ac_1}\\\\rho _{a(n-3)bc_2,..,c_p} + \\\\\\\\ + \\\\frac{(n-1)(d-2+n)}{(d-3+n)(d-4+2n)} \\\\eta _{aa}\\\\eta _{bc_1}\\\\rho _{a(n-2)c_2,..,c_p} - \\\\frac{n-1}{d-3+n} \\\\eta _{ac_1}\\\\eta _{ab}\\\\rho _{a(n-2)c_2,..,c_p}.$ It can be checked that for $p = 2$ $\\\\sigma _+ T = 0\\\\,.$ For $p > 2$ it is not difficult to see that $\\\\sigma _+ T = \\\\frac{(p-2)d}{(d-2)(d-3+n)(d-2+n)} e_3^2 \\\\theta _Y \\\\rho + (\\\\emph {lit} ),$ where $\\\\rho = \\\\rho _{a(n-1),c_1,..,c_{p-2}}Y^{a(n-1)}\\\\theta ^{c_1}..\\\\theta ^{c_{p-2}}$ .\",\n \"Therefore, $H^2(\\\\sigma _-) \\\\ni T = \\\\eta _{ac_1}\\\\eta _{bc_2}\\\\rho {}_{a(n-1)} + \\\\frac{(n-2)(n-1)}{(d-3+n)(d-4+2n)} \\\\eta _{aa}\\\\eta _{ac_1}\\\\rho {}_{a(n-3)bc_2} + \\\\\\\\+ \\\\frac{(n-1)(d-2+n)}{(d-3+n)(d-4+2n)}\\\\eta _{aa}\\\\eta _{bc_1}\\\\rho {}_{a(n-2)c_2} -\\\\frac{n-1}{d-3+n} \\\\eta _{ac_1}\\\\eta _{ab}\\\\rho {}_{a(n-2)c_2} \\\\,.$ Diagram (n+1,m;p-3), $n\\\\ge 1, m\\\\ge 1, p\\\\ge 3$ has the tensor form $T = \\\\eta _{bc_1}\\\\rho _{a(n)c_2,b(m-1)c_3,..,c_p} - \\\\frac{n}{n-m+1}\\\\eta _{ac_1}\\\\rho _{a(n-1)bc_2,b(m-1)c_3,..,c_p}\\\\,,$ where $\\\\rho $ is an arbitrary $(n+1,m;p-3)$ tensor.\",\n \"Explicit computation gives $\\\\Delta T = \\\\frac{t_0}{t_0+1}\\\\bigg (p + h_2 - 1\\\\bigg )T - \\\\frac{2 t_0}{(t_0+1)(d-4+2h_2)}\\\\theta ^a Z_a \\\\partial _{\\\\theta Z} T + \\\\\\\\ + \\\\frac{d-4+2h_1}{(t_0+1)(d-3+h_1+h_2)(d-4+2h_2)} \\\\bigg (t_2 \\\\theta ^a Y_a \\\\partial _{\\\\theta Z} - \\\\theta ^a Z_a \\\\partial _{\\\\theta Z}\\\\bigg )T - \\\\\\\\ - \\\\frac{t_0-1}{(t_0+1)(d-3+h_1+h_2)}\\\\theta ^a Y_a \\\\partial _{\\\\theta Y}T\\\\,.$ This expression vanishes at $n = m$ .\",\n \"Indeed, in this case $t_0 T = 0, h_1 T= h_2 T = n T, t_2 T = 0$ so that $\\\\Delta T = \\\\frac{1}{d-3+2n} \\\\bigg (t_2 \\\\theta ^a Y_a \\\\partial _{\\\\theta Z} - \\\\theta ^a Z_a \\\\partial _{\\\\theta Z}\\\\bigg )T + \\\\frac{1}{d-3+2n}\\\\theta ^a Y_a \\\\partial _{\\\\theta Y}T\\\\,.$ Since $t_2 \\\\theta ^a Y_a \\\\partial _{\\\\theta Z}T = \\\\theta ^a Y_a \\\\partial _{\\\\theta Z}t_2 T - \\\\theta ^a Y_a \\\\partial _{\\\\theta Y}T + \\\\theta ^a Z_a \\\\partial _{\\\\theta Z}T = - \\\\theta ^a Y_a \\\\partial _{\\\\theta Y}T + \\\\theta ^a Z_a \\\\partial _{\\\\theta Z}T\\\\,,$ it follows that $\\\\Delta T=0$ at $n=m$ .\",\n \"To check that $\\\\Delta T\\\\ne 0$ at $n\\\\ne m$ one should substitute the expression for $T$ noticing that different linearly independent terms have no common factor to vanish, that implies nontriviality of $\\\\Delta T$ .\",\n \"Diagram (n,m+1;p-3), $n\\\\ge 1, m\\\\ge 0, p\\\\ge 3$ has the tensor form $T = \\\\eta _{ac_1}\\\\rho _{a(n-1)c_2,b(m)c_3,..,c_p}$ .\",\n \"It belongs to $\\\\ker (\\\\sigma _-)$ , but not to $\\\\ker (\\\\sigma _+)$ .\",\n \"$\\\\sigma _+ T = \\\\bigg (1 + \\\\frac{p-3}{m+1}\\\\bigg )\\\\eta _{ac_1}\\\\rho _{a(n-1)c_2,b(m+1),..,c_p} + (\\\\emph {lit})\\\\,.$ Diagram (n,m;p-2) $n\\\\ge 1, m\\\\ge 0, p\\\\ge 2$ admits two tensor realizations $T_i$ due to the double presence of this diagram in the result of tensor product.\",\n \"The tensors $T_1 = \\\\eta _{ac_1} \\\\rho _1{}_{a(n-1)c_2,b(m),c_3,..,c_p} + \\\\frac{m}{n}\\\\eta _{bc_1} \\\\rho _1{}_{a(n),b(m-1)c_2,c_3,..,c_p},$ $T_2 = \\\\eta _{aa} \\\\rho _2{}_{a(n-2)bc_1,b(m-1)c_2,c_3,..,c_p} - \\\\frac{n-m}{n-1}\\\\eta _{ab} \\\\rho _2{}_{a(n-1)c_1,b(m-1)c_2,c_3,..,c_p} - \\\\frac{d-4+n+m}{n-1}\\\\eta _{ac_1} \\\\times \\\\\\\\\\\\times \\\\rho _2{}_{a(n-1)b,b(m-1)c_2,c_3,..,c_p} + \\\\frac{(n-m+1)(d-4+m+n)}{n(n-1)}\\\\eta _{bc_1} \\\\rho _2{}_{a(n),b(m-1)c_2,c_3,..,c_p}$ are linearly independent.\",\n \"That Laplace operator acts diagonally on $T_i$ , $\\\\Delta T_i = \\\\lambda _i(n,m,p) T_i$ , allows us to separately consider each of these diagrams.\",\n \"Firstly, we check if these are in $\\\\ker (\\\\sigma _-)$ computing $&\\\\sigma _- T_1 = m \\\\eta _{ac_1} \\\\rho _1{}_{a(n-1)c_2,b(m-1)c_3,..,c_{p+1}},\\\\\\\\&\\\\sigma _- T_2 = \\\\frac{d-4+2m}{n-1} \\\\eta _{ac_1} \\\\rho _2{}_{a(n-1)c_2,b(m-1)c_3,..,c_{p+1}}.$ $T_1 \\\\in \\\\ker (\\\\sigma _-)$ at $m = 0$ .\",\n \"Formally, $T_2$ is annihilated by $\\\\sigma _-$ at $d=2, m = 1$ , but this is not allowed by the Two-column theorem.\",\n \"So, the only candidate for cohomology is $T_1$ .\",\n \"However, $\\\\sigma _+ T_1 = \\\\frac{1}{n+1}\\\\Big (1 + \\\\frac{p-2}{n} \\\\Big )\\\\eta _{bc_1}\\\\rho _1{}_{a(n),c_2,..,c_{p-1}} + (\\\\emph {lit})\\\\,,$ which is never zero.\",\n \"Diagram (n-1,m;p-3), $n\\\\ge 2, m\\\\ge 1, p\\\\ge 3$ has the tensor form $T = \\\\eta _{ac_1}\\\\eta _{bc_2} \\\\rho _{a(n-1),b(m-1)c_3,..,c_p} - \\\\frac{(n-1)(d-3+m+n)}{(d-4+n+m)(d-4+2n)}\\\\eta _{aa}\\\\eta _{bc_1} \\\\rho _{a(n-2)c_2,b(m-1)c_3,..,c_p} -\\\\\\\\- \\\\frac{(n-1)(n-2)}{(d-4+m+n)(d-4+2n)} \\\\eta _{aa}\\\\eta _{ac_1} \\\\rho _{a(n-3)bc_2,b(m-1)c_3,..,c_p} + \\\\frac{n-1}{d-4+m+n} \\\\times \\\\\\\\\\\\times \\\\eta _{ab}\\\\eta _{ac_1} \\\\rho _{a(n-2)c_2,b(m-1)c_3,..,c_p}\\\\,,$ where $\\\\rho $ is an arbitrary $(n-1,m;p-3)$ tensor.\",\n \"Obviously, $T \\\\in \\\\ker (\\\\sigma _-)$ .\",\n \"However, $T \\\\notin \\\\ker (\\\\sigma _+)$ .\",\n \"$\\\\sigma _+ T = \\\\bigg (1 + \\\\frac{p-3}{m}\\\\bigg ) \\\\eta _{ac_1}\\\\eta _{bc_2} \\\\rho _{a(n-1),b(m),c_3,..,c_{p-1}} + (\\\\emph {lit})\\\\,,$ hence not contributing to cohomology.\",\n \"Diagram (n,m-1;p-3), $n\\\\ge 1, m\\\\ge 1, p\\\\ge 3$ has the tensor form $T = \\\\eta _{ac_1}\\\\eta _{bc_2}\\\\rho _{a(n-1)c_3,b(m-1),..,c_p} + (m-1)(\\\\emph {lit})\\\\,$ with all terms except for the first one carrying a factor of $(m-1)$ .\",\n \"The action of $\\\\sigma _-$ on the first term brings a factor of $(m-1)$ in front of $\\\\eta \\\\rho $ .\",\n \"Since all $\\\\eta \\\\rho $ terms in the decomposition are linearly independent we conclude that $\\\\sigma _- T = 0, \\\\text{ if } m = 1.$ At $p = 3$ one can check that $\\\\sigma _+ T = 0$ .\",\n \"However, for $p>3$ $T$ does not belong to $\\\\ker (\\\\sigma _+)$ , $\\\\sigma _+ T = - (p-3) \\\\eta _{ac_1}\\\\eta _{bc_2} \\\\rho _{a(n-1)c_3,b,..,c_{p-1}} + (\\\\emph {lit})\\\\,.$ Consequently, the only contribution to $H^3(\\\\sigma _-)$ is $T = \\\\eta _{ac_1}\\\\eta _{bc_2}\\\\rho _{a(n-1)c_3} \\\\in H^3(\\\\sigma _-)\\\\,.$ Diagram (n,m;p-4), $n\\\\ge 1, m\\\\ge 1, p\\\\ge 4$ has the tensor form $T = \\\\eta _{ac_1}\\\\eta _{bc_2}\\\\rho _{a(n-1)c_3,b(m-1)c_4,..,c_p}\\\\,.$ This is obviously annihilated by $\\\\sigma _-$ , but not by $\\\\sigma _+$ .\",\n \"$\\\\sigma _+ T = - \\\\bigg (1 + \\\\frac{p-4}{m} \\\\bigg ) \\\\eta _{ac_1}\\\\eta _{bc_2}\\\\rho _{a(n-1)c_3,b(m),..,c_{p-1}} + (\\\\emph {lit})\\\\,.$ Hence it does not contribute to $H^p(\\\\sigma _-)$ .\"\n ],\n [\n \"Summary\",\n \"Summarizing the results of Sections REF and REF we found the following cohomology groups: $H^0(\\\\sigma _-) = \\\\lbrace F = F_{a(n)}Y^{a(n)}| F \\\\in V^0\\\\rbrace \\\\,,$ $H^1(\\\\sigma _-) = \\\\lbrace \\\\phi = F_1{}_{a(n)c}Y^{a(n)}\\\\theta ^c, F_1 \\\\in V^1;\\\\\\\\ \\\\phi ^{tr} = \\\\Big [(n-1)\\\\eta _{aa}F_2{}_{a(n-2)c} - (d-4+2n)\\\\eta _{ac}F_2{}_{a(n-1)}\\\\Big ]Y^{a(n)}\\\\theta ^c \\\\in V^1 \\\\rbrace \\\\,,$ $H^2(\\\\sigma _-) = \\\\lbrace W = \\\\theta _Y\\\\theta _Z C(Y,Z): t_0 C = 0, C \\\\in V^{0}; \\\\\\\\ \\\\mathcal {E}_A = \\\\Big [\\\\eta _{ac_1}\\\\rho _1{}_{a(n-1)bc_2} - \\\\eta _{bc_1}\\\\rho _1{}_{a(n)c_2}\\\\Big ]Y^{a(n)}Z^b\\\\theta ^{c_1}\\\\theta ^{c_2} \\\\in V^2; \\\\\\\\ \\\\mathcal {E}_B = \\\\Big [\\\\eta _{ac_1}\\\\eta _{bc_2}\\\\rho _2{}_{a(n-1)} + \\\\frac{(n-2)(n-1)}{(d-3+n)(d-4+2n)} \\\\eta _{aa}\\\\eta _{ac_1}\\\\rho _2{}_{a(n-3)bc_2} +\\\\\\\\+ \\\\frac{(n-1)(d-2+n)}{(d-3+n)(d-4+2n)}\\\\eta _{aa}\\\\eta _{bc_1}\\\\rho _2{}_{a(n-2)c_2} - \\\\frac{n-1}{d-3+n} \\\\eta _{ac_1}\\\\eta _{ab}\\\\rho _2{}_{a(n-2)c_2}\\\\Big ]Y^{a(n)}Z^b\\\\theta ^{c_1}\\\\theta ^{c_2} \\\\in V^2 \\\\rbrace \\\\,,$ $H^3(\\\\sigma _-) = \\\\lbrace B^{fr} = \\\\eta _{ac_1}\\\\eta _{bc_2}\\\\rho _{a(n-1)c_3} Y^{a(n)}Z^{b}\\\\theta ^{c_1}\\\\theta ^{c_2}\\\\theta ^{c_3} \\\\in V^3; \\\\\\\\B_1 = \\\\theta _Y\\\\theta _Z C(Y,Z,\\\\theta ): t_0 C = 0, C = C_{a(n),b(n),c}Y^{a(n)}Z^{b(n)}\\\\theta ^{c} \\\\in V^1; \\\\\\\\B_2 = \\\\Big [\\\\eta _{bc_1}\\\\rho _{a(n)c_2,b(n-1)c_3} - n\\\\eta _{ac_1}\\\\rho _{a(n-1)bc_2,b(n-1)c_3}\\\\Big ]Y^{a(n)}Z^{b(n)}\\\\theta ^{c_1}\\\\theta ^{c_2}\\\\theta ^{c_3} \\\\in V^3\\\\rbrace \\\\,.$ At $p>3$ $H^p(\\\\sigma _-) = \\\\Big \\\\lbrace B_1 = \\\\theta _Y\\\\theta _Z C(Y,Z,\\\\theta ): t_0 C = 0, C = C_{a(n),b(n),c_1,..,c_{p-2}}Y^{a(n)}Z^{b(n)}\\\\theta ^{c_1}..\\\\theta ^{c_{p-2}} \\\\in V^{p-2}; \\\\\\\\B_2 = \\\\Big [\\\\eta _{bc_1}\\\\rho _{a(n)c_2,b(n-1)c_3,..,c_p} - n\\\\eta _{ac_1}\\\\rho _{a(n-1)bc_2,b(n-1)c_3,..,c_p}\\\\Big ]Y^{a(n)}Z^{b(n)}\\\\theta ^{c_1}..\\\\theta ^{c_{p}} \\\\in V^{p} \\\\Big \\\\rbrace .$ According to Theorem REF , the differential gauge transformation parameters are described by $H^0(\\\\sigma _-)$ (REF ).\",\n \"The gauge parameter in the Fronsdal theory is known to be a symmetric traceless tensor.\",\n \"Since tensors $(n,0,0)$ constitute the cohomology group $H^0(\\\\sigma _-)$ , the unfolded differential gauge transformation is shown to coincide with the Fronsdal one.\",\n \"As recalled in Section REF , the Fronsdal field consists of two symmetric traceless fields (REF ).\",\n \"These fields are represented by the cohomology groups $H^1(\\\\sigma _-)$ (REF ).\",\n \"Cohomology group $H^1(\\\\sigma _-)$ consists of two elements $(n+1,0,0)$ and $(n-1,0,0)$ matching the components of the Fronsdal field.\",\n \"Thus, the physical fields in the unfolded approach indeed coincide with the Fronsdal field.\",\n \"The cohomology group $H^2(\\\\sigma _-)$ (REF ) describes gauge invariant combinations of derivatives of the physical fields that can be used to impose differential equations on the latter.\",\n \"The Fronsdal cohomology classes $\\\\mathcal {E}_A$ and $\\\\mathcal {E}_B$ match with the Fronsdal equations: $\\\\mathcal {E}_A$ is associated with the traceless part of the Fronsdal equations, while $\\\\mathcal {E}_B$ with the trace part, that is the equations $\\\\mathcal {E}_A = 0, \\\\mathcal {E}_B = 0$ just reproduce the Fronsdal equations.\",\n \"Note that the number of resulting equations is the same as the number of fields, as it should be in a Lagrangian system.\",\n \"$W$ in (REF ) represents \\\"Weyl\\\" cohomology.\",\n \"Imposing $W = 0$ in the case of gravity one gets conformally flat metrics and in the case of higher spins \\\"conformally flat\\\" fields.\",\n \"In Einstein gravity and HS theory, the equation $W=0$ is not imposed.\",\n \"Instead, elements of $W$ are interpreted as new fields $C$ that describe generalized Weyl tensors by virtue of the unfolded equations (REF ).\",\n \"Thus, calculation of the cohomology group $H^2(\\\\sigma _ -)$ shows that unfolded equations (REF ), (REF ) contain Fronsdal equations along with constraints on auxiliary fields.\",\n \"In accordance with the general discussion of Section elements of $H^3(\\\\sigma _-)$ (REF ) correspond to Bianchi identities.\",\n \"Class $B^{fr}$ describes the Bianchi identities for the Fronsdal equations.\",\n \"Note that their number coincides with the number of differential gauge parameters.\",\n \"The remaining classes $B_1$ and $B_2$ correspond to the Bianchi identities on the Weyl tensor.\",\n \"It is noteworthy that the latter can be checked to coincide with the first $\\\\widetilde{\\\\sigma }_-$ cohomology in the Weyl sector of zero-forms of [10] for $s>1$ .\",\n \"This fact exhibits the connection between the gauge and Weyl sectors.\",\n \"For $p>3$ cohomology groups $H^p(\\\\sigma _-)$ describe the higher Bianchi identities for Bianchi identities on the Weyl tensor also interpreted as syzygies [24].\",\n \"Obtained lower cohomology groups match with the results of [16], [17], [18].\",\n \"In HS theory the fields are realized by one-forms.\",\n \"Formally, one can consider field equations (REF ), (REF ) for $p$ -forms $\\\\omega _{a(n),b(m)}$ valued in a two-row irreducible $\\\\mathfrak {o}(d)$ -module.\",\n \"From our results and physical interpretation of the $\\\\sigma _-$ cohomology groups it follows that for $p>1$ the unfolded system in the gauge sector is off-shell.\",\n \"To answer the question whether the full unfolded system including both the gauge $p$ -form sector and the Weyl $(p-1)$ -form sector is off-shell, the analysis of $H(\\\\tilde{\\\\sigma }_{-})$ has to be performed in the Weyl sector.\",\n \"The case of $p>1$ may be somewhat similar to the $s=1$ case, where the equation on $A_\\\\mu $ lies in the Weyl sector.\",\n \"Finally, let us stress that the results of this section for HS fields in Minkowski space admit a straightforward deformation to $AdS_d$ with the same operator $\\\\sigma _-$ .\",\n \"This is because in that case dynamical fields are described by rectangular diagrams of the $AdS_d$ algebra $\\\\mathfrak {o}(d-1,2)$ [37].\",\n \"In general, in the flat limit, irreducible massless (gauge) fields in $AdS_d$ decompose into nontrivial sets of irreducible flat space massless fields [38], [39], [40] and there is no one-to-one correspondence between massless fields in Minkowski space and $AdS_d$ .\",\n \"Namely, a generic irreducible field in Minkowski space may admit no deformation to $AdS_d$ (see also [41]).\"\n ],\n [\n \"$\\\\sigma _-$ cohomology in {{formula:6304f7a9-7e1d-44fd-b47c-3df99bd81cdd}} in the spinor language\",\n \"$4d$ HS theories admit a description in terms of two-component spinors instead of tensors.\",\n \"That is, instead of using the generating functions in the tensor form $\\\\omega (x,dx\\\\,|Y,Z)$ , where $Y^a$ and $Z^a$ carried vector Lorentz indices $ a=\\\\lbrace 0,1,2,3\\\\rbrace $ , we will use $\\\\omega (y,\\\\overline{y}\\\\,|x,dx) = \\\\sum \\\\limits _{k,m}\\\\omega ^{\\\\alpha _1\\\\dots \\\\alpha _k,\\\\dot{\\\\alpha }_1\\\\dots \\\\dot{\\\\alpha }_m}(x,dx)\\\\, y_{\\\\alpha _1}\\\\dots y_{\\\\alpha _k}\\\\,\\\\overline{y}_{\\\\dot{\\\\alpha }_1}\\\\dots \\\\overline{y}_{\\\\dot{\\\\alpha }_m}\\\\,,$ where the indices $\\\\alpha $ and $\\\\dot{\\\\alpha }$ of the two-component commuting spinors $y^\\\\alpha $ and $\\\\overline{y}^{\\\\dot{\\\\alpha }}$ take two values $\\\\lbrace 1,2\\\\rbrace $ .\",\n \"Analogously to Sections 3 and 4, we have to introduce the grading on the space of $\\\\Lambda ^\\\\bullet (M)\\\\otimes \\\\mathbb {C}[[y,\\\\overline{y}]]$ .\",\n \"Consider the homogeneous element of $\\\\Lambda ^\\\\bullet (M)\\\\otimes \\\\mathbb {C}[[y,\\\\overline{y}]]$ of degree $N$ and $\\\\overline{N}$ in $y$ and $\\\\overline{y}$ , respectively $\\\\omega (\\\\mu y, \\\\overline{\\\\mu }\\\\overline{y}\\\\,|x,dx) = \\\\mu ^N\\\\,\\\\overline{\\\\mu }{}^{\\\\overline{N}}\\\\,\\\\omega (y,\\\\overline{y}\\\\,|x,dx).$ Define the grading operator $G$ on $\\\\Lambda ^\\\\bullet (M)\\\\otimes \\\\mathbb {C}[[y,\\\\overline{y}]]$ as follows: $G\\\\omega (y,\\\\overline{y}\\\\,|x,dx) = |N-\\\\overline{N}|\\\\,\\\\omega (y,\\\\overline{y}\\\\,|x,dx) \\\\equiv |\\\\deg _y(\\\\omega (y,\\\\overline{y}\\\\,|x,dx))-\\\\deg _{\\\\overline{y}}(\\\\omega (y,\\\\overline{y}\\\\,|x,dx))|.$ Note that in the bosonic sector the frame-like fields $e^{m_1\\\\dots m_{s-1}} \\\\ \\\\leftrightarrow \\\\ e^{\\\\alpha _1\\\\dots \\\\alpha _{s-1},\\\\dot{\\\\alpha }_1\\\\dots \\\\dot{\\\\alpha }_{s-1}}$ have the lowest possible grading $G=0$ .\",\n \"For our later computations to match with the Fronsdal theory, we define the action of $\\\\sigma _-$ on $\\\\omega (y,\\\\overline{y})$ to decrease the $G$ -grading: $\\\\sigma _-\\\\omega (y,\\\\overline{y}) &:= i\\\\, \\\\overline{y}^{\\\\dot{\\\\alpha }}h^\\\\alpha _{\\\\ \\\\dot{\\\\alpha }}\\\\partial _\\\\alpha \\\\ \\\\omega (y,\\\\overline{y}), \\\\quad \\\\quad \\\\text{at } \\\\deg _y(\\\\omega )>\\\\deg _{\\\\overline{y}}(\\\\omega ),\\\\\\\\\\\\sigma _-\\\\omega (y,\\\\overline{y}) &:= i\\\\, y^{\\\\alpha }h_\\\\alpha ^{\\\\ \\\\dot{\\\\alpha }}\\\\overline{\\\\partial }_{\\\\dot{\\\\alpha }}\\\\ \\\\omega (y,\\\\overline{y}), \\\\quad \\\\quad \\\\text{at } \\\\deg _y(\\\\omega )<\\\\deg _{\\\\overline{y}}(\\\\omega ),\\\\\\\\\\\\sigma _-\\\\omega (y,\\\\overline{y}) &:= 0\\\\qquad \\\\qquad \\\\qquad \\\\qquad \\\\quad \\\\,\\\\,\\\\text{at } \\\\deg _y(\\\\omega )=\\\\deg _{\\\\overline{y}}(\\\\omega )\\\\,,$ where $_\\\\alpha :=\\\\frac{}{y^\\\\alpha }\\\\,,\\\\qquad \\\\overline{}_{\\\\dot{\\\\alpha }}:=\\\\frac{}{\\\\overline{y}^{\\\\dot{\\\\alpha }}}\\\\,,$ and the dependence on $x$ and $dx$ in $\\\\omega (y,\\\\overline{y}) = \\\\omega (y,\\\\overline{y}\\\\,|x,dx)$ is always implicit.\",\n \"It is easy to check that so defined $\\\\sigma _-$ is nilpotent, $(\\\\sigma _-)^2=0$ .\",\n \"Note that $\\\\sigma _\\\\pm $ change the grading $G$ by 2.\",\n \"This agrees in particular with the fact that the bosonic and fermionic sectors, where the grading is even and odd respectively, are independent.\",\n \"We consider in detail the more complicated bosonic case, observing in the end that the computation for fermionic fields is quite similar.\",\n \"Note that the analysis of $\\\\sigma _-$ cohomology in the $4d$ conformal HS theory was also performed in terms of two-component spinors in [42], [43].\",\n \"It is more complicated since the generating functions of conformal HS theory depend on twice as many independent spinors, but simpler since it is free of the module factors like $|N- N|$ in the grading definition (REF ).\",\n \"Next, we define a scalar productBeing $\\\\mathrm {SL}(2,\\\\mathbb {C})$ -invariant this scalar product is not positive-definite.\",\n \"Analogously to the tensorial case, without affecting the $\\\\sigma _-$ cohomology analysis it can be made positive-definite by going to the $\\\\mathfrak {su}(2)\\\\oplus \\\\mathfrak {su}(2)$ algebra, which is the compact real form of $\\\\mathfrak {sl}(2,\\\\mathbb {C})\\\\oplus \\\\mathfrak {sl}(2,\\\\mathbb {C})$ with altered conjugation rules $\\\\overline{y}^\\\\alpha =y_\\\\alpha $ , $\\\\overline{y}^{\\\\dot{\\\\alpha }} =y_{\\\\dot{\\\\alpha }}$ .\",\n \"on generating elements of $\\\\Lambda ^\\\\bullet (M)\\\\otimes \\\\mathbb {C}[[y,\\\\overline{y}]]$ by $\\\\langle q^\\\\alpha |q^\\\\beta \\\\rangle =\\\\langle y^\\\\alpha |y^\\\\beta \\\\rangle =i\\\\epsilon ^{\\\\alpha \\\\beta }\\\\,, &&\\\\langle \\\\overline{p}_{\\\\dot{\\\\alpha }}|\\\\overline{p}_{\\\\dot{\\\\beta }}\\\\rangle =-\\\\langle \\\\overline{\\\\partial }_{\\\\dot{\\\\alpha }}|\\\\overline{\\\\partial }_{\\\\dot{\\\\beta }}\\\\rangle =i\\\\overline{\\\\epsilon }_{\\\\dot{\\\\alpha }\\\\dot{\\\\beta }} \\\\,, && \\\\langle h_{\\\\alpha \\\\dot{\\\\alpha }}|h_{\\\\beta \\\\dot{\\\\beta }}\\\\rangle =\\\\epsilon _{\\\\alpha \\\\beta }\\\\epsilon _{\\\\dot{\\\\alpha }\\\\dot{\\\\beta }}\\\\,,$ where $q^\\\\alpha =y^\\\\alpha $ , $p_\\\\alpha =i\\\\partial _\\\\alpha $ and $h^{\\\\alpha \\\\dot{\\\\beta }}$ is a vierbein one-form.\",\n \"In some local coordinates $x^\\\\mu $ on the base manifold (which in our case is $AdS_4$ ) the vierbein one-forms $h^\\\\alpha _{\\\\ \\\\dot{\\\\alpha }}$ can be expressed as $h^\\\\alpha _{\\\\ \\\\dot{\\\\alpha }} = \\\\left(h_{\\\\mu }\\\\right)^\\\\alpha _{\\\\ \\\\dot{\\\\alpha }}\\\\,dx^\\\\mu .$ The $AdS_4$ vierbein $\\\\left(h_{\\\\mu }\\\\right)^\\\\alpha _{\\\\ \\\\dot{\\\\alpha }}$ is demanded to be non-degenerate at any point of $AdS_4$ .\",\n \"The next step is to obtain $\\\\sigma _+:=\\\\sigma _-^\\\\dagger $ with respect to the scalar product $\\\\langle \\\\,,\\\\,\\\\rangle $ , i.e., $\\\\langle \\\\phi |\\\\sigma _-\\\\psi \\\\rangle = \\\\langle \\\\sigma _+\\\\phi |\\\\psi \\\\rangle $ .\",\n \"It is not hard to check that $\\\\sigma _+\\\\omega (y,\\\\overline{y}) &:= -iy^\\\\alpha D_\\\\alpha ^{\\\\ \\\\dot{\\\\alpha }}\\\\overline{\\\\partial }_{\\\\dot{\\\\alpha }}\\\\ \\\\omega (y,\\\\overline{y}) \\\\quad \\\\quad \\\\text{at } \\\\deg _y(\\\\omega )>\\\\deg _{\\\\overline{y}}(\\\\omega )\\\\,,\\\\\\\\\\\\sigma _+\\\\omega (y,\\\\overline{y}) &:= -i \\\\overline{y}^{\\\\dot{\\\\alpha }}D^\\\\alpha _{\\\\ \\\\dot{\\\\alpha }}\\\\partial _\\\\alpha \\\\ \\\\omega (y,\\\\overline{y}) \\\\quad \\\\quad \\\\text{at } \\\\deg _y(\\\\omega )<\\\\deg _{\\\\overline{y}}(\\\\omega )\\\\,,\\\\\\\\\\\\sigma _+ \\\\omega (y,\\\\overline{y})&:= -i\\\\left(y^\\\\alpha D_\\\\alpha ^{\\\\ \\\\dot{\\\\alpha }}\\\\overline{\\\\partial }_{\\\\dot{\\\\alpha }}+\\\\overline{y}^{\\\\dot{\\\\alpha }}D^\\\\alpha _{\\\\ \\\\dot{\\\\alpha }}\\\\partial _\\\\alpha \\\\right)\\\\omega (y,\\\\overline{y})\\\\quad \\\\quad \\\\text{at } \\\\deg _y(\\\\omega )=\\\\deg _{\\\\overline{y}}(\\\\omega )\\\\,,$ where $D_\\\\alpha ^{\\\\ \\\\dot{\\\\alpha }} := \\\\frac{}{h^\\\\alpha _{\\\\ \\\\dot{\\\\alpha }}} \\\\,,\\\\qquad D^\\\\alpha _{\\\\ \\\\dot{\\\\alpha }} := \\\\frac{}{h_\\\\alpha ^{\\\\ \\\\dot{\\\\alpha }}}\\\\,.$ By $\\\\omega = \\\\omega (y,\\\\overline{y}\\\\,|x,h)$ we mean a general $p$ -form polynomial in $y$ and $\\\\overline{y}$ with the coordinate one-forms $dx$ replaced by $h$ via (REF ), that is $\\\\omega (y,\\\\overline{y}\\\\,|x,h) = \\\\sum _{n,m} \\\\omega _{\\\\alpha _1\\\\dots \\\\alpha _p|\\\\mu (n)|\\\\dot{\\\\alpha }_1\\\\dots \\\\dot{\\\\alpha }_p|\\\\dot{\\\\mu }(m)}(x)\\\\,h^{\\\\alpha _1\\\\dot{\\\\alpha }_1}\\\\wedge \\\\dots \\\\wedge h^{\\\\alpha _p\\\\dot{\\\\alpha }_p}\\\\,y^{\\\\mu (n)}\\\\,\\\\overline{y}^{\\\\dot{\\\\mu }(m)}\\\\,.$ So defined $\\\\sigma _+$ increases the grading.\",\n \"The Laplace operator $\\\\Delta :=\\\\sigma _-\\\\sigma _++\\\\sigma _+\\\\sigma _-$ is by construction self-adjoint with respect to $\\\\langle \\\\,|\\\\,\\\\rangle $ and non-negative definite for the compact version of the space-time symmetry algebra.\"\n ],\n [\n \"Bosonic case in $AdS_4$\",\n \"To calculate cohomology of $\\\\sigma _-$ we have to compute the action of $\\\\Delta $ .\",\n \"Since $\\\\sigma _-$ and $\\\\sigma _+$ are defined differently in the different regions of the $(N, \\\\overline{N})$ plane, we compute the Laplacian action in the these regions separately.\",\n \"Direct computation yields: $\\\\Delta _{N>\\\\overline{N}+2}&= N(\\\\overline{N}+2) + y^\\\\beta _\\\\alpha h^\\\\alpha _{\\\\ \\\\dot{\\\\gamma }} D_\\\\beta ^{\\\\ \\\\dot{\\\\gamma }} + \\\\overline{y}^{\\\\dot{\\\\alpha }}\\\\overline{}_{\\\\dot{\\\\beta }}h_{\\\\gamma \\\\dot{\\\\alpha }}D^{\\\\gamma \\\\dot{\\\\beta }}\\\\,,\\\\\\\\\\\\Delta _{N<\\\\overline{N}-2}&=\\\\overline{N}(N+2)+y^\\\\alpha _\\\\beta h_{\\\\alpha \\\\dot{\\\\gamma }}D^{\\\\beta \\\\dot{\\\\gamma }}+\\\\overline{y}^{\\\\dot{\\\\alpha }}\\\\overline{}_{\\\\dot{\\\\beta }} h_\\\\gamma ^{\\\\ \\\\dot{\\\\beta }}D^{\\\\gamma }_{\\\\ \\\\dot{\\\\alpha }}\\\\,,\\\\\\\\\\\\Delta _{N=\\\\overline{N}+2} &= \\\\Delta _{N>\\\\overline{N}+2} + \\\\overline{y}^{\\\\dot{\\\\alpha }}\\\\overline{y}^{\\\\dot{\\\\beta }}_\\\\alpha _\\\\beta h^\\\\beta _{\\\\ \\\\dot{\\\\beta }}D^\\\\alpha _{\\\\ \\\\dot{\\\\alpha }}\\\\,,\\\\\\\\\\\\Delta _{N=\\\\overline{N}-2} &= \\\\Delta _{N<\\\\overline{N}-2} + y^\\\\alpha y^\\\\beta \\\\overline{}_{\\\\dot{\\\\alpha }}\\\\overline{}_{\\\\dot{\\\\beta }}h_\\\\beta ^{\\\\ \\\\dot{\\\\beta }}D_\\\\alpha ^{\\\\ \\\\dot{\\\\alpha }}\\\\,,\\\\\\\\\\\\Delta _{N=\\\\overline{N}}&=\\\\overline{y}^{\\\\dot{\\\\alpha }}\\\\overline{}_{\\\\dot{\\\\beta }} h_{\\\\gamma \\\\dot{\\\\alpha }} D^{\\\\gamma \\\\dot{\\\\beta }}+y^\\\\alpha _\\\\beta h_{\\\\alpha \\\\dot{\\\\gamma }}D^{\\\\beta \\\\dot{\\\\gamma }}-\\\\overline{y}^{\\\\dot{\\\\alpha }} y^\\\\beta _\\\\alpha \\\\overline{}_{\\\\dot{\\\\beta }}h^\\\\alpha _{\\\\ \\\\dot{\\\\alpha }} D_\\\\beta ^{\\\\ \\\\dot{\\\\beta }}- y^\\\\alpha \\\\overline{y}^{\\\\dot{\\\\beta }}_\\\\beta \\\\overline{}_{\\\\dot{\\\\alpha }}h_\\\\alpha ^{\\\\ \\\\dot{\\\\alpha }}D^\\\\beta _{\\\\ \\\\dot{\\\\beta }}\\\\,.$ The computation of the cohomology $H^p(\\\\sigma _-)$ will be performed as follows.\",\n \"Taking a general $p$ -form $\\\\omega (y,\\\\overline{y}\\\\,|x)$ , we decompose it into Lorentz irreducible components.\",\n \"As we will observe, the projectors onto irreducible parts will commute with the action of the Laplacian.\",\n \"Thus, instead of involved calculation of the action of $\\\\Delta $ on all of the irreducible components of $\\\\omega (y,\\\\overline{y}\\\\,|x)$ we can first calculate its action on the general $\\\\omega (y,\\\\overline{y}\\\\,|x)$ and then project.\"\n ],\n [\n \" $H^0(\\\\sigma _-)$\",\n \"Evidently, $\\\\Delta _{N=\\\\overline{N}}\\\\Big |_\\\\text{0-forms}= 0$ since all the terms in (REF ) contain derivatives in $h$ 's.\",\n \"At the same time, $\\\\Delta _{N\\\\ne \\\\overline{N}}\\\\Big |_\\\\text{0-forms} > 0$ .\",\n \"Thus, we conclude $H^0(\\\\sigma _-)=\\\\ker \\\\left(\\\\Delta \\\\Big |_\\\\text{0-forms}\\\\right)= \\\\left\\\\lbrace F(y,\\\\overline{y})=F_{\\\\alpha (n),\\\\dot{\\\\alpha }(n)}y^{\\\\alpha (n)}\\\\overline{y}^{\\\\dot{\\\\alpha }(n)},\\\\ n\\\\in \\\\mathbb {N}_0\\\\right\\\\rbrace .$ By Theorem REF , elements of this cohomology space correspond to the parameters of differential (non-Stueckelberg) linearized HS gauge transformations.\",\n \"This result fits the pattern of the spin-$s$ Fronsdal gauge symmetry parameters with $n=s-1$ .\"\n ],\n [\n \" $H^1(\\\\sigma _-)$\",\n \"The decomposition of a one-form $\\\\Theta (y,\\\\overline{y}\\\\,|x)$ into Lorentz irreps reads as $\\\\Theta (y,\\\\overline{y}\\\\,|x)=\\\\underbrace{\\\\Theta _{\\\\mu (n+1)|\\\\dot{\\\\mu }(m+1)}\\\\ h^{\\\\mu \\\\dot{\\\\mu }}y^{\\\\mu (n)}y^{\\\\dot{\\\\mu }(m)}}_{\\\\Theta _\\\\mathrm {A}(y,\\\\overline{y})}-\\\\frac{1}{2}\\\\underbrace{\\\\Theta _{\\\\mu (n-1)|\\\\dot{\\\\mu }(m+1)}\\\\ h_\\\\nu ^{\\\\ \\\\dot{\\\\mu }}y^\\\\nu y^{\\\\mu (n-1)}\\\\overline{y}^{\\\\dot{\\\\mu }(m)}}_{\\\\Theta _\\\\mathrm {B}(y,\\\\overline{y})} -\\\\\\\\-\\\\frac{1}{2}\\\\underbrace{\\\\Theta _{\\\\mu (n+1)|\\\\dot{\\\\mu }(m-1)}\\\\ h^{\\\\mu }_{\\\\ \\\\dot{\\\\nu }}y^{\\\\mu (n)}\\\\overline{y}^{\\\\dot{\\\\nu }}\\\\overline{y}^{\\\\dot{\\\\mu }(m-1)}}_{\\\\Theta _\\\\mathrm {C}(y,\\\\overline{y})} +\\\\frac{1}{4}\\\\underbrace{\\\\Theta _{\\\\mu (n-1)|\\\\dot{\\\\mu }(m-1)}\\\\ h_{\\\\nu \\\\dot{\\\\nu }} y^\\\\nu y^{\\\\mu (n-1)}\\\\overline{y}^{\\\\dot{\\\\nu }}\\\\overline{y}^{\\\\dot{\\\\mu }(m-1)}}_{\\\\Theta _\\\\mathrm {D}(y,\\\\overline{y})}.$ Thus, for fixed $n$ and $m$ , there are four Lorentz-irreducible one-forms: $\\\\Theta _\\\\text{A}$ , $\\\\Theta _\\\\mathrm {B}$ , $\\\\Theta _\\\\mathrm {C}$ , $\\\\Theta _\\\\mathrm {D}$ .\",\n \"For direct computations it will be convenient to separate two of the indices of the $y$ group: $\\\\Theta (y,\\\\overline{y}) = \\\\Theta _{\\\\lambda ,\\\\nu ,\\\\mu (n-1)|\\\\dot{\\\\lambda },\\\\dot{\\\\nu },\\\\dot{\\\\mu }(m-1)}\\\\ h^{\\\\lambda \\\\dot{\\\\lambda }}y^{\\\\nu }y^{\\\\mu (n-1)}\\\\overline{y}^{\\\\dot{\\\\nu }}\\\\overline{y}^{\\\\dot{\\\\mu }(m-1)}.$ In terms of the basis one-forms $h^{\\\\lambda \\\\dot{\\\\lambda }}y^{\\\\nu }y^{\\\\mu (n-1)}\\\\overline{y}^{\\\\dot{\\\\nu }}\\\\overline{y}^{\\\\dot{\\\\mu }(m-1)}$ the projectors onto irreducible components are $\\\\mathcal {P}_\\\\mathrm {A} &= {S}_{(\\\\lambda ,\\\\mu ,\\\\nu )}{S}_{(\\\\dot{\\\\lambda },\\\\dot{\\\\mu },\\\\dot{\\\\nu })}, && \\\\mathcal {P}_\\\\mathrm {C} = {S}_{(\\\\lambda ,\\\\mu ,\\\\nu )}\\\\epsilon _{\\\\dot{\\\\lambda }\\\\dot{\\\\nu }},\\\\\\\\\\\\mathcal {P}_\\\\mathrm {B} &= \\\\epsilon _{\\\\lambda \\\\nu }{S}_{(\\\\dot{\\\\lambda },\\\\dot{\\\\mu },\\\\dot{\\\\nu })}, &&\\\\mathcal {P}_\\\\mathrm {D} = \\\\epsilon _{\\\\lambda \\\\nu }\\\\epsilon _{\\\\dot{\\\\lambda }\\\\dot{\\\\nu }},$ where ${S}_{(\\\\lambda ,\\\\mu ,\\\\nu )}$ implies symmetrization over indices $\\\\lambda ,\\\\mu ,\\\\nu $ and similarly for the dotted indices.\"\n ],\n [\n \"$H^1(\\\\sigma _-)$ in the diagonal sector {{formula:a426b6e9-6846-47b5-abc4-9ff268867159}}\",\n \"In the diagonal sector with $n=m$ the Laplacian is a sum of the following four terms: $\\\\Delta _{N=\\\\overline{N}}\\\\Theta (y,\\\\overline{y}) =\\\\underbrace{\\\\overline{y}^{\\\\dot{\\\\alpha }}\\\\overline{}_{\\\\dot{\\\\beta }} h_{\\\\gamma \\\\dot{\\\\alpha }} D^{\\\\gamma \\\\dot{\\\\beta }} \\\\Theta (y,\\\\overline{y}) }_{T_1(y,\\\\overline{y})}+\\\\underbrace{y^\\\\alpha _\\\\beta h_{\\\\alpha \\\\dot{\\\\gamma }}D^{\\\\beta \\\\dot{\\\\gamma }}\\\\Theta (y,\\\\overline{y}) }_{T_2(y,\\\\overline{y})}-\\\\\\\\\\\\underbrace{-\\\\overline{y}^{\\\\dot{\\\\alpha }} y^\\\\beta _\\\\alpha \\\\overline{}_{\\\\dot{\\\\beta }}h^\\\\alpha _{\\\\ \\\\dot{\\\\alpha }} D_\\\\beta ^{\\\\ \\\\dot{\\\\beta }}\\\\Theta (y,\\\\overline{y}) }_{T_3(y,\\\\overline{y})}\\\\underbrace{-y^\\\\alpha \\\\overline{y}^{\\\\dot{\\\\beta }}_\\\\beta \\\\overline{}_{\\\\dot{\\\\alpha }}h_\\\\alpha ^{\\\\ \\\\dot{\\\\alpha }}D^\\\\beta _{\\\\ \\\\dot{\\\\beta }}\\\\Theta (y,\\\\overline{y}) }_{T_4(y,\\\\overline{y})}.$ Consider the first term in (REF ).\",\n \"$\\\\overline{y}^{\\\\dot{\\\\alpha }}\\\\overline{}_{\\\\dot{\\\\beta }} h_{\\\\gamma \\\\dot{\\\\alpha }} D^{\\\\gamma \\\\dot{\\\\beta }} \\\\Theta (y,\\\\overline{y}) = \\\\Theta _{\\\\lambda ,\\\\nu ,\\\\mu (n-1)|\\\\dot{\\\\lambda },\\\\dot{\\\\nu },\\\\dot{\\\\mu }(m-1)}\\\\ \\\\underbrace{\\\\Big [\\\\overline{y}^{\\\\dot{\\\\alpha }}\\\\overline{}_{\\\\dot{\\\\beta }} h_{\\\\gamma \\\\dot{\\\\alpha }} D^{\\\\gamma \\\\dot{\\\\beta }} \\\\big ( h^{\\\\lambda \\\\dot{\\\\lambda }}y^{\\\\nu }y^{\\\\mu (n-1)}\\\\overline{y}^{\\\\dot{\\\\nu }}\\\\overline{y}^{\\\\dot{\\\\mu }(m-1)}\\\\big )\\\\Big ]}_{T_1^{\\\\lambda ,\\\\nu ,\\\\mu (n-1)|\\\\dot{\\\\lambda },\\\\dot{\\\\nu },\\\\dot{\\\\mu }(n-1)}}\\\\,.$ The expression in square brackets is denoted by $T_1^{\\\\lambda ,\\\\nu ,\\\\mu (n-1)|\\\\dot{\\\\lambda },\\\\dot{\\\\nu },\\\\dot{\\\\mu }(n-1)}$ .\",\n \"The notation for other irreducible components $T_2$ , $T_3$ and $T_4$ is analogous.\",\n \"Straightforward computation yields $T_1^{\\\\lambda ,\\\\nu ,\\\\mu (n-1)|\\\\dot{\\\\lambda },\\\\dot{\\\\nu },\\\\dot{\\\\mu }(n-1)} &=& -h^\\\\lambda _{\\\\ \\\\dot{\\\\alpha }}\\\\epsilon ^{\\\\dot{\\\\nu }\\\\dot{\\\\lambda }} y^\\\\nu y^{\\\\mu (n-1)}\\\\overline{y}^{\\\\dot{\\\\alpha }}\\\\overline{y}^{\\\\dot{\\\\mu }(n-1)} -{}\\\\nonumber \\\\\\\\&&- (n-1) h^\\\\lambda _{\\\\ \\\\dot{\\\\alpha }}\\\\epsilon ^{\\\\dot{\\\\mu }\\\\dot{\\\\lambda }}y^\\\\nu y^{\\\\mu (n-1)}\\\\overline{y}^{\\\\dot{\\\\alpha }}\\\\overline{y}^{\\\\dot{\\\\nu }}\\\\overline{y}^{\\\\dot{\\\\mu }(n-2)}\\\\,,\\\\\\\\T_2^{\\\\lambda ,\\\\nu ,\\\\mu (n-1)|\\\\dot{\\\\lambda },\\\\dot{\\\\nu },\\\\dot{\\\\mu }(n-1)} &=& -h_\\\\alpha ^{\\\\ \\\\dot{\\\\lambda }}\\\\epsilon ^{\\\\nu \\\\lambda }y^{\\\\alpha }y^{\\\\mu (n-1)}\\\\overline{y}^{\\\\dot{\\\\nu }}\\\\overline{y}^{\\\\dot{\\\\mu }(n-1)} -{}\\\\nonumber \\\\\\\\&&- (n-1)h_\\\\alpha ^{\\\\ \\\\dot{\\\\lambda }}\\\\epsilon ^{\\\\mu \\\\lambda }y^\\\\alpha y^\\\\nu y^{\\\\mu (n-2)}\\\\overline{y}^{\\\\dot{\\\\nu }}\\\\overline{y}^{\\\\dot{\\\\mu }(n-1)}\\\\,,\\\\\\\\T_3^{\\\\lambda ,\\\\nu ,\\\\mu (n-1)|\\\\dot{\\\\lambda },\\\\dot{\\\\nu },\\\\dot{\\\\mu }(n-1)} &=& -\\\\left(h^\\\\nu _{\\\\ \\\\dot{\\\\alpha }}y^\\\\lambda y^{\\\\mu (n-1)}+(n-1)h^\\\\mu _{\\\\ \\\\dot{\\\\alpha }}y^\\\\lambda y^\\\\nu y^{\\\\mu (n-2)}\\\\right){}\\\\nonumber \\\\times \\\\\\\\&&\\\\times \\\\left(\\\\epsilon ^{\\\\dot{\\\\nu }\\\\dot{\\\\lambda }}\\\\overline{y}^{\\\\dot{\\\\alpha }}\\\\overline{y}^{\\\\dot{\\\\mu }(n-1)} + (n-1)\\\\epsilon ^{\\\\dot{\\\\mu }\\\\dot{\\\\lambda }}\\\\overline{y}^{\\\\dot{\\\\alpha }}\\\\overline{y}^{\\\\dot{\\\\nu }}\\\\overline{y}^{\\\\dot{\\\\mu }(n-2)}\\\\right)\\\\,,\\\\\\\\T_4^{\\\\lambda ,\\\\nu ,\\\\mu (n-1)|\\\\dot{\\\\lambda },\\\\dot{\\\\nu },\\\\dot{\\\\mu }(n-1)} &=& -\\\\left(h_\\\\alpha ^{\\\\ \\\\dot{\\\\nu }}\\\\overline{y}^{\\\\dot{\\\\lambda }} \\\\overline{y}^{\\\\dot{\\\\mu }(n-1)}+(n-1)h_{\\\\alpha }^{\\\\ \\\\dot{\\\\nu }} \\\\overline{y}^{\\\\dot{\\\\lambda }} \\\\overline{y}^{\\\\dot{\\\\nu }} \\\\overline{y}^{\\\\dot{\\\\mu }(n-2)}\\\\right){}\\\\nonumber \\\\times \\\\\\\\&&\\\\times \\\\left(\\\\epsilon ^{\\\\nu \\\\lambda } y^{\\\\alpha } y^{\\\\mu (n-1)} + (n-1)\\\\epsilon ^{\\\\mu \\\\lambda } y^{\\\\alpha } y^{\\\\nu } y^{\\\\mu (n-2)}\\\\right).$ Projecting onto the irreducible parts of $\\\\Theta (y,\\\\overline{y})$ we find $\\\\Delta _{N=\\\\overline{N}}\\\\left(\\\\Theta _{\\\\mathrm {A}}\\\\right)&=0\\\\,,\\\\\\\\\\\\Delta _{N=\\\\overline{N}}\\\\left(\\\\Theta _{\\\\mathrm {B}}\\\\right)&=(n+1)^2\\\\Theta _{\\\\mathrm {B}}\\\\ne 0\\\\,,\\\\\\\\\\\\Delta _{N=\\\\overline{N}}\\\\left(\\\\Theta _{\\\\text{C}}\\\\right)&=(n+1)^2\\\\Theta _{\\\\text{C}}\\\\ne 0\\\\,,\\\\\\\\\\\\Delta _{N=\\\\overline{N}}\\\\left(\\\\Theta _{\\\\mathrm {D}}\\\\right)&=0.$ Thus, the only elements of the kernel of $\\\\Delta ^\\\\text{1-forms}\\\\Big |_{N=\\\\overline{N}}$ are $\\\\Theta _{\\\\mathrm {A}}(y,\\\\overline{y})$ and $\\\\Theta _{\\\\mathrm {D}}(y,\\\\overline{y})$ .\",\n \"By the Hodge theorem of Section 4 this yields that $H^1(\\\\sigma _-) = \\\\ker \\\\left(\\\\Delta ^\\\\text{1-forms}\\\\Big |_{N=\\\\overline{N}}\\\\right)$ is $H^1(\\\\sigma _-) &= \\\\bigoplus _{n\\\\ge 0} H^1_{(n)}(\\\\sigma _-),\\\\\\\\H^1_{(n)}(\\\\sigma _-) &= \\\\Big \\\\lbrace \\\\phi _{(n)}(y,\\\\overline{y}\\\\,|x) + \\\\phi _{(n)}^\\\\text{tr}(y,\\\\overline{y}\\\\,|x),\\\\\\\\\\\\phi _{(n)}(y,\\\\overline{y}\\\\,|x) &:= \\\\phi _{\\\\mu (n+1),\\\\dot{\\\\mu }(n+1)}(x)\\\\, h^{\\\\mu \\\\dot{\\\\mu }} \\\\, y^{\\\\mu (n)}\\\\overline{y}^{\\\\dot{\\\\mu }(n)},\\\\\\\\\\\\phi _{(n)}^\\\\text{tr}(y,\\\\overline{y}\\\\,|x) &:= \\\\phi ^\\\\mathrm {tr}_{\\\\mu (n-1),\\\\dot{\\\\mu }(n-1)}(x)\\\\, h_{\\\\nu \\\\dot{\\\\nu }} \\\\, y^\\\\nu y^{\\\\mu (n-1)}\\\\overline{y}^{\\\\dot{\\\\nu }}\\\\overline{y}^{\\\\dot{\\\\mu }(n-1)}\\\\,, \\\\text{ if } n>0 \\\\Big \\\\rbrace ,$ where $n$ is the number of indices of the corresponding cocycles.\",\n \"Equivalently, $H^1(\\\\sigma _-) = \\\\Big \\\\lbrace h^{\\\\mu \\\\dot{\\\\mu }}\\\\,\\\\partial _\\\\mu \\\\overline{\\\\partial }_{\\\\dot{\\\\mu }}\\\\,F_1(y,\\\\overline{y}\\\\,|x) + h_{\\\\mu \\\\dot{\\\\mu }}\\\\,y^\\\\mu \\\\overline{y}^{\\\\dot{\\\\mu }} F_2(y,\\\\overline{y}\\\\,|x)\\\\Big \\\\rbrace ,$ where $F_{1,2}(y,\\\\overline{y}\\\\,|x)$ belongs to the diagonal $N=\\\\overline{N}$ , that is, $\\\\left(y^\\\\alpha \\\\frac{\\\\partial }{\\\\partial y^\\\\alpha } - \\\\overline{y}^{\\\\dot{\\\\alpha }}\\\\frac{\\\\partial }{\\\\partial \\\\overline{y}^{\\\\dot{\\\\alpha }}}\\\\right)F_{1,2}(y,\\\\overline{y}\\\\,|x) = 0.$ The fields $\\\\phi (y,\\\\overline{y})$ and $\\\\phi ^\\\\text{tr}(y,\\\\overline{y})$ exactly correspond to the irreducible components of the double-traceless Fronsdal field.\",\n \"It remains to prove that there are no other nontrivial cocycles in $H^1(\\\\sigma _- )$ at $N\\\\ne \\\\overline{N}$ .\"\n ],\n [\n \"$H^1(\\\\sigma _-)$ in the far-from-diagonal sector {{formula:f3550507-9a9d-493c-9968-364e086c8c2f}}\",\n \"Consider the action of the Laplace operator $\\\\Delta _{N>\\\\overline{N}+2}$ on general one-forms at $N>\\\\overline{N}+2$ $\\\\Delta _{N>\\\\overline{N}+2}\\\\Theta (y, \\\\overline{y}) =\\\\left(n(m+2) + y^\\\\beta _\\\\alpha h^\\\\alpha _{\\\\ \\\\dot{\\\\gamma }} D_\\\\beta ^{\\\\ \\\\dot{\\\\gamma }} + \\\\overline{y}^{\\\\dot{\\\\alpha }}\\\\overline{}_{\\\\dot{\\\\beta }}h_{\\\\gamma \\\\dot{\\\\alpha }}D^{\\\\gamma \\\\dot{\\\\beta }}\\\\right)\\\\Theta (y, \\\\overline{y}) =\\\\\\\\=\\\\underbrace{n(m+2)\\\\Theta (y, \\\\overline{y})}_{T_1(y,\\\\overline{y})} +\\\\underbrace{y^\\\\beta _\\\\alpha h^\\\\alpha _{\\\\ \\\\dot{\\\\gamma }} D_\\\\beta ^{\\\\ \\\\dot{\\\\gamma }}\\\\Theta (y, \\\\overline{y})}_{T_2(y,\\\\overline{y})} +\\\\underbrace{\\\\overline{y}^{\\\\dot{\\\\alpha }}\\\\overline{}_{\\\\dot{\\\\beta }}h_{\\\\gamma \\\\dot{\\\\alpha }}D^{\\\\gamma \\\\dot{\\\\beta }}\\\\Theta (y,\\\\overline{y})}_{T_3(y,\\\\overline{y})}.$ Analogously to (REF ), we denote $T_2^{\\\\lambda ,\\\\nu ,\\\\mu (n-1)|\\\\dot{\\\\lambda },\\\\dot{\\\\nu },\\\\dot{\\\\mu }(m-1)} = y^\\\\beta _\\\\alpha h^\\\\alpha _{\\\\ \\\\dot{\\\\gamma }} D_\\\\beta ^{\\\\ \\\\dot{\\\\gamma }} \\\\big ( h^{\\\\lambda \\\\dot{\\\\lambda }}y^{\\\\nu }y^{\\\\mu (n-1)}\\\\overline{y}^{\\\\dot{\\\\nu }}\\\\overline{y}^{\\\\dot{\\\\mu }(m-1)}\\\\big )$ and similarly for $T_1$ and $T_3$ .\",\n \"In this sector we obtain $T_1^{\\\\lambda ,\\\\nu ,\\\\mu (n-1)|\\\\dot{\\\\lambda },\\\\dot{\\\\nu },\\\\dot{\\\\mu }(m-1)} &=n(m+2)h^{\\\\lambda \\\\dot{\\\\lambda }}y^{\\\\nu }y^{\\\\mu (n-1)}\\\\overline{y}^{\\\\dot{\\\\nu }}\\\\overline{y}^{\\\\dot{\\\\mu }(m-1)}\\\\,,\\\\\\\\T_2^{\\\\lambda ,\\\\nu ,\\\\mu (n-1)|\\\\dot{\\\\lambda },\\\\dot{\\\\nu },\\\\dot{\\\\mu }(m-1)} &=-h^{\\\\nu \\\\dot{\\\\lambda }}y^{\\\\lambda }y^{\\\\mu (n-1)}\\\\overline{y}^{\\\\dot{\\\\nu }}\\\\overline{y}^{\\\\dot{\\\\mu }(m-1)} - (n-1)h^{\\\\mu \\\\dot{\\\\lambda }}y^{\\\\lambda }y^{\\\\nu }y^{\\\\mu (n-2)}\\\\overline{y}^{\\\\dot{\\\\nu }}\\\\overline{y}^{\\\\dot{\\\\mu }(m-1)}\\\\,,\\\\\\\\T_3^{\\\\lambda ,\\\\nu ,\\\\mu (n-1)|\\\\dot{\\\\lambda },\\\\dot{\\\\nu },\\\\dot{\\\\mu }(m-1)} &= - h^{\\\\lambda }_{\\\\ \\\\dot{\\\\alpha }}\\\\epsilon ^{\\\\dot{\\\\nu }\\\\dot{\\\\lambda }}y^{\\\\nu }y^{\\\\mu (n-1)}\\\\overline{y}^{\\\\dot{\\\\alpha }}\\\\overline{y}^{\\\\dot{\\\\mu }(m-1)} - (m-1) h^{\\\\lambda }_{\\\\ \\\\dot{\\\\alpha }}\\\\epsilon ^{\\\\dot{\\\\lambda }\\\\dot{\\\\mu }}y^{\\\\nu }y^{\\\\mu (n-1)}\\\\overline{y}^{\\\\dot{\\\\alpha }}\\\\overline{y}^{\\\\dot{\\\\nu }}\\\\overline{y}^{\\\\dot{\\\\mu }(m-2)}\\\\,.$ Projection onto the irreducible components according to (REF ) yields $\\\\Delta _{N>\\\\overline{N}+2}\\\\left(\\\\Theta _\\\\mathrm {A}\\\\right) &= n(m+1)\\\\Theta _\\\\mathrm {A}\\\\ne 0\\\\,,\\\\\\\\\\\\Delta _{N>\\\\overline{N}+2}\\\\left(\\\\Theta _\\\\mathrm {B}\\\\right) &= (nm+2n+1)\\\\Theta _\\\\mathrm {B}\\\\ne 0\\\\,,\\\\\\\\\\\\Delta _{N>\\\\overline{N}+2}\\\\left(\\\\Theta _\\\\mathrm {C}\\\\right) &= (nm-n+2m)\\\\Theta _\\\\mathrm {C}\\\\ne 0\\\\,,\\\\\\\\\\\\Delta _{N>\\\\overline{N}+2}\\\\left(\\\\Theta _\\\\mathrm {D}\\\\right) &= (nm+2n-m+4)\\\\Theta _\\\\mathrm {D}\\\\ne 0\\\\,.$ Thus, there are no nontrivial cocycles in this sector.\",\n \"For $N<\\\\overline{N}-2$ the computation is analogous.\",\n \"Thus, $H^1(\\\\sigma _-) =0$ in the far-from-diagonal sector.\"\n ],\n [\n \"Subtlety in the near-diagonal sector $|N-\\\\overline{N}|=2$\",\n \"In this case we face certain peculiarity.\",\n \"Denote the space of $p$ -forms ($p=1$ for $H^{1}$ ) with $N$ chiral and $\\\\overline{N}$ anti-chiral indices by $\\\\mathcal {V}_{(N,\\\\overline{N})}$ .\",\n \"Recall that the grading operator is $G=|N-\\\\overline{N}|$ .\",\n \"Consider the case with $N-\\\\overline{N}=2$ .\",\n \"Namely, let $N=n+1$ and $\\\\overline{N}=n-1$ .\",\n \"At $G=2$ the operator $\\\\sigma _-$ maps a state $X\\\\in \\\\mathcal {V}_{(n+1,n-1)}$ onto the diagonal, $\\\\sigma _-(X)\\\\in \\\\mathcal {V}_{(n,n)}$ , where in accordance with (REF ), $\\\\sigma _+$ acts 'both up and down': $\\\\mathcal {V}_{(n+1,n-1)} \\\\xrightarrow{} \\\\mathcal {V}_{(n,n)} \\\\xrightarrow{} \\\\mathcal {V}_{(n-1,n+1)} \\\\oplus \\\\mathcal {V}_{(n+1,n-1)}\\\\,.$ Thus, $\\\\mathcal {V}_{(n+1,n-1)} \\\\xrightarrow{}\\\\mathcal {V}_{(n-1,n+1)} \\\\oplus \\\\mathcal {V}_{(n+1,n-1)}\\\\,.$ As a result, $\\\\Delta _{(n+1,n-1)} : \\\\mathcal {V}_{(n+1,n-1)} \\\\longrightarrow \\\\mathcal {V}_{(n-1,n+1)} \\\\oplus \\\\mathcal {V}_{(n+1,n-1)}\\\\,,\\\\\\\\\\\\Delta _{(n-1,n+1)} : \\\\mathcal {V}_{(n-1,n+1)} \\\\longrightarrow \\\\mathcal {V}_{(n-1,n+1)} \\\\oplus \\\\mathcal {V}_{(n+1,n-1)}\\\\,.$ Consequently, $\\\\ker (\\\\Delta )$ should be searched in the form of a linear combination of the vectors both from $\\\\mathcal {V}_{(n+1,n-1)}$ and from $\\\\mathcal {V}_{(n-1,n+1)}$ .\",\n \"Indeed, let $X$ be a vector in $\\\\mathcal {V}_{(n+1,n-1)}$ .\",\n \"Consider the complex conjugated vector $\\\\overline{X}\\\\in \\\\mathcal {V}_{(n-1,n+1)}$ and compute the action of the Laplacian on them.\",\n \"Let $\\\\Delta X = \\\\Delta _{(n+1,n-1)}X=\\\\alpha (n) X + \\\\beta (n) \\\\overline{X}\\\\,,\\\\\\\\\\\\Delta \\\\overline{X} = \\\\Delta _{(n-1,n+1)}\\\\overline{X}= \\\\gamma (n) X + \\\\delta (n) \\\\overline{X}$ with some coefficients $\\\\alpha $ , $\\\\beta $ , $\\\\gamma $ , and $\\\\delta $ .\",\n \"That $X$ and $\\\\overline{X}$ are conjugated and operator $\\\\Delta $ is self-adjoint implies the relations $\\\\alpha = \\\\overline{\\\\delta }$ and $\\\\beta =\\\\overline{\\\\gamma }$ .\",\n \"Looking for $\\\\ker (\\\\Delta )$ in the form $Y=F(n)X+G(n)\\\\overline{X}\\\\in \\\\ker (\\\\Delta )$ and acting on $Y$ by the Laplace operator we find that the condition $\\\\Delta Y =0$ yields $\\\\Delta Y = F(n)\\\\Delta _{(n+1,n-1)}X + G(n)\\\\Delta _{(n-1,n+1)}\\\\overline{X} =\\\\\\\\=\\\\big (\\\\alpha (n)F(n)+\\\\overline{\\\\beta (n)}G(n)\\\\big )X + \\\\big (\\\\beta (n)F(n)+\\\\overline{\\\\alpha (n)}G(n)\\\\big )\\\\overline{X} = 0\\\\,.$ Since $X$ and $\\\\overline{X}$ are linearly independent, the problem of finding such $Y=\\\\alpha X+\\\\beta \\\\overline{X}$ that $\\\\Delta Y=0$ , amounts to the linear system $\\\\begin{bmatrix}\\\\alpha (n) & \\\\overline{\\\\beta (n)} \\\\\\\\\\\\beta (n) & \\\\overline{\\\\alpha (n)}\\\\end{bmatrix}\\\\begin{bmatrix}F(n) \\\\\\\\G(n)\\\\end{bmatrix}=\\\\begin{bmatrix}0 \\\\\\\\0\\\\end{bmatrix}\\\\,,$ which admits non-trivial solutions iff $\\\\det \\\\begin{bmatrix}\\\\alpha (n) & \\\\overline{\\\\beta (n)} \\\\\\\\\\\\beta (n) & \\\\overline{\\\\alpha (n)}\\\\end{bmatrix}=|\\\\alpha (n)|^2 - |\\\\beta (n)|^2 = 0.$ Hence, we conclude that $\\\\alpha (n)=\\\\beta (n)\\\\cdot e^{i\\\\chi }, \\\\quad \\\\quad \\\\chi \\\\in [0,2\\\\pi )\\\\,.$ In the next section coefficients $\\\\alpha (n)$ and $\\\\beta (n)$ will be shown to be real, i.e., $e^{i\\\\chi }=\\\\pm 1$ .\",\n \"Summarizing, if we find that the coefficients $\\\\alpha (n)$ and $\\\\beta (n)$ coincide up to a sign, $\\\\alpha (n)=\\\\pm \\\\beta (n)$ , this would imply the existence of a non-trivial $\\\\sigma _-$ -cocycle $Y = X \\\\mp \\\\overline{X}\\\\,.$ Otherwise the cohomology is trivial.\"\n ],\n [\n \"$H^1(\\\\sigma _-)$ in the near-diagonal sector {{formula:1655b858-2c9a-47bf-8194-18e799b3a71f}}\",\n \"To compute $H^1(\\\\sigma _-)$ in the leftover sector of $N=\\\\overline{N}+2$ (analysis at $N=\\\\overline{N}-2$ is analogous) consider a general one-form $\\\\Theta (y,\\\\overline{y})$ (REF ) with $N=\\\\overline{N}+2$ .\",\n \"In this sector, the Laplacian differs form that at $N>\\\\overline{N}+2$ by the $T_4(y,\\\\overline{y})$ term in $\\\\Delta _{N=\\\\overline{N}+2}\\\\Theta (y, \\\\overline{y}) = \\\\Big (\\\\underbrace{N(\\\\overline{N}+2) + y^\\\\beta _\\\\alpha h^\\\\alpha _{\\\\ \\\\dot{\\\\gamma }} D_\\\\beta ^{\\\\ \\\\dot{\\\\gamma }} + \\\\overline{y}^{\\\\dot{\\\\alpha }}\\\\overline{}_{\\\\dot{\\\\beta }}h_{\\\\gamma \\\\dot{\\\\alpha }}D^{\\\\gamma \\\\dot{\\\\beta }}}_{\\\\Delta _{N>\\\\overline{N}+2}}\\\\Big )\\\\Theta (y, \\\\overline{y}) +\\\\underbrace{\\\\overline{y}^{\\\\dot{\\\\alpha }}\\\\overline{y}^{\\\\dot{\\\\beta }}_\\\\alpha _\\\\beta h^\\\\beta _{\\\\ \\\\dot{\\\\beta }}D^\\\\alpha _{\\\\ \\\\dot{\\\\alpha }}\\\\Theta (y, \\\\overline{y})}_{T_4(y,\\\\overline{y})}.$ Consequently, it is essential to compute the action of this additional term.\",\n \"As before (cf.\",\n \"(REF )), denote $T_4^{\\\\lambda ,\\\\nu ,\\\\mu (n-1)|\\\\dot{\\\\lambda },\\\\dot{\\\\nu },\\\\dot{\\\\mu }(m-1)} = \\\\overline{y}^{\\\\dot{\\\\alpha }}\\\\overline{y}^{\\\\dot{\\\\beta }}_\\\\alpha _\\\\beta h^\\\\beta _{\\\\ \\\\dot{\\\\beta }}D^\\\\alpha _{\\\\ \\\\dot{\\\\alpha }} \\\\big (h^{\\\\lambda \\\\dot{\\\\lambda }}y^{\\\\nu }y^{\\\\mu (n-1)}\\\\overline{y}^{\\\\dot{\\\\nu }}\\\\overline{y}^{\\\\dot{\\\\mu }(m-1)}\\\\big )\\\\,.$ This yields $T_4^{\\\\lambda ,\\\\nu ,\\\\mu (n-1)|\\\\dot{\\\\lambda },\\\\dot{\\\\nu },\\\\dot{\\\\mu }(m-1)}=(n-1)\\\\epsilon ^{\\\\mu \\\\lambda }h^{\\\\nu }_{\\\\ \\\\dot{\\\\beta }}y^{\\\\mu (n-2)}\\\\overline{y}^{\\\\dot{\\\\lambda }}\\\\overline{y}^{\\\\dot{\\\\beta }}\\\\overline{y}^{\\\\dot{\\\\nu }}\\\\overline{y}^{\\\\dot{\\\\mu }(m-1)} + (n-1)\\\\epsilon ^{\\\\nu \\\\lambda }h^{\\\\mu }_{\\\\ \\\\dot{\\\\beta }}y^{\\\\mu (n-2)}\\\\overline{y}^{\\\\dot{\\\\lambda }}\\\\overline{y}^{\\\\dot{\\\\beta }}\\\\overline{y}^{\\\\dot{\\\\nu }}\\\\overline{y}^{\\\\dot{\\\\mu }(m-1)}+\\\\\\\\+(n-1)(n-2)\\\\epsilon ^{\\\\mu \\\\lambda }h^{\\\\mu }_{\\\\ \\\\dot{\\\\beta }}y^{\\\\nu }y^{\\\\mu (n-3)}\\\\overline{y}^{\\\\dot{\\\\lambda }}\\\\overline{y}^{\\\\dot{\\\\beta }}\\\\overline{y}^{\\\\dot{\\\\nu }}\\\\overline{y}^{\\\\dot{\\\\mu }(m-1)}.$ Projecting $T_4$ onto the irreducible parts, we find: $\\\\text{(A)}:&& {S}_{(\\\\lambda ,\\\\nu ,\\\\mu )}{S}_{(\\\\dot{\\\\lambda },\\\\dot{\\\\nu },\\\\dot{\\\\mu })}T_4^{\\\\lambda ,\\\\nu ,\\\\mu (n-1)|\\\\dot{\\\\lambda },\\\\dot{\\\\nu },\\\\dot{\\\\mu }(m-1)}&=0\\\\,,\\\\\\\\\\\\text{(B)}:&& \\\\ \\\\ {S}_{(\\\\dot{\\\\lambda },\\\\dot{\\\\nu },\\\\dot{\\\\mu })}\\\\epsilon _{\\\\lambda \\\\nu }T_4^{\\\\lambda ,\\\\nu ,\\\\mu (n-1)|\\\\dot{\\\\lambda },\\\\dot{\\\\nu },\\\\dot{\\\\mu }(m-1)} &= -(n-1)(2n-1)h^{\\\\mu }_{\\\\ \\\\dot{\\\\beta }} y^{\\\\mu (n-2)}\\\\overline{y}^{\\\\dot{\\\\beta }}\\\\overline{y}^{\\\\dot{\\\\mu }(m+1)}\\\\,,\\\\\\\\\\\\text{(C)}:&& \\\\ \\\\ \\\\epsilon _{\\\\dot{\\\\lambda }\\\\dot{\\\\nu }}{S}_{(\\\\lambda ,\\\\nu ,\\\\mu )}T_4^{\\\\lambda ,\\\\nu ,\\\\mu (n-1)|\\\\dot{\\\\lambda },\\\\dot{\\\\nu },\\\\dot{\\\\mu }(m-1)}&=0\\\\,,\\\\\\\\\\\\text{(D)}:&& \\\\ \\\\ \\\\epsilon _{\\\\dot{\\\\lambda }\\\\dot{\\\\nu }}\\\\epsilon _{\\\\lambda \\\\nu }T_4^{\\\\lambda ,\\\\nu ,\\\\mu (n-1)|\\\\dot{\\\\lambda },\\\\dot{\\\\nu },\\\\dot{\\\\mu }(m-1)}&=0\\\\,.$ We observe that the action of the Laplacian in this sector differs from the previously computed one only in the type-(B) sector, namely, $\\\\Delta _{N=\\\\overline{N}+2}\\\\left(\\\\Theta _\\\\mathrm {B}\\\\right) = \\\\left(n(m+3)\\\\Theta _\\\\mathrm {B}+(2n^2-3n+1)\\\\Theta _\\\\mathrm {C}\\\\right)\\\\Big |_{m=n-2}=\\\\underbrace{(n^2+n)}_{\\\\alpha (n)}\\\\Theta _\\\\mathrm {B}+\\\\underbrace{(2n^2-3n+1)}_{\\\\beta (n)}\\\\Theta _\\\\mathrm {C}\\\\,.$ That $|\\\\alpha (n)|\\\\ne |\\\\beta (n)|$ at integer $n$ implies triviality of $H^1(\\\\sigma _-)$ in the near-diagonal sector.\"\n ],\n [\n \"$H^2(\\\\sigma _-)$\",\n \"The calculation of $H^2(\\\\sigma _-)$ is in main features analogous to that of $H^1(\\\\sigma _-)$ .\",\n \"To decompose a general two-form $\\\\Omega (y,\\\\overline{y}\\\\,|x)$ into irreducible parts we use the following useful identity $h^{\\\\nu \\\\dot{\\\\nu }}\\\\wedge h^{\\\\lambda \\\\dot{\\\\lambda }} = \\\\frac{1}{2}H^{\\\\nu \\\\lambda }\\\\epsilon ^{\\\\dot{\\\\nu }\\\\dot{\\\\lambda }}+\\\\frac{1}{2}\\\\overline{H}^{\\\\dot{\\\\nu }\\\\dot{\\\\lambda }}\\\\epsilon ^{\\\\nu \\\\lambda },$ where $H^{\\\\nu \\\\lambda }=H^{(\\\\nu \\\\lambda )}:=h^{\\\\nu }_{\\\\ \\\\dot{\\\\gamma }}\\\\wedge h^{\\\\lambda \\\\dot{\\\\gamma }}\\\\,,\\\\qquad \\\\overline{H}^{\\\\dot{\\\\nu }\\\\dot{\\\\lambda }}=H^{(\\\\dot{\\\\nu }\\\\dot{\\\\lambda })}:=h^{\\\\ \\\\dot{\\\\nu }}_{\\\\gamma }\\\\wedge h^{\\\\gamma \\\\dot{\\\\lambda }}\\\\,.$ The decomposition reads $\\\\Omega (y,\\\\overline{y}\\\\, |x) = \\\\underbrace{\\\\Omega ^\\\\mathrm {A}_{\\\\mu (n+2)|\\\\dot{\\\\mu }(m)}\\\\ H^{\\\\mu \\\\mu }y^{\\\\mu (n)}\\\\overline{y}^{\\\\dot{\\\\mu }(m)}}_{\\\\Phi _{\\\\mathrm {A}(n,m)}}+\\\\underbrace{\\\\overline{\\\\Omega }^\\\\mathrm {A}_{\\\\mu (n)|\\\\dot{\\\\mu }(m+2)}\\\\ \\\\overline{H}^{\\\\dot{\\\\mu }\\\\dot{\\\\mu }}y^{\\\\mu (n)}\\\\overline{y}^{\\\\dot{\\\\mu }(m)}}_{\\\\overline{\\\\Phi }_{\\\\mathrm {A}(n,m)}}+\\\\\\\\+\\\\underbrace{\\\\Omega ^\\\\mathrm {B}_{\\\\mu (n-2)|\\\\dot{\\\\mu }(m)}\\\\ H_{\\\\nu \\\\nu }y^\\\\nu y^\\\\nu y^{\\\\mu (n-2)}\\\\overline{y}^{\\\\dot{\\\\mu }(m)}}_{\\\\Phi _{\\\\mathrm {B}(n,m)}}+\\\\underbrace{\\\\overline{\\\\Omega }^\\\\mathrm {B}_{\\\\mu (n)|\\\\dot{\\\\mu }(m-2)}\\\\overline{H}_{\\\\dot{\\\\nu }\\\\dot{\\\\nu }} y^{\\\\mu (n)}\\\\overline{y}^{\\\\dot{\\\\nu }}\\\\overline{y}^{\\\\dot{\\\\nu }}\\\\overline{y}^{\\\\dot{\\\\mu }(m-2)}}_{\\\\overline{\\\\Phi }_{\\\\mathrm {B}(n,m)}}+\\\\\\\\+\\\\underbrace{\\\\Omega ^\\\\mathrm {C}_{\\\\mu (n)|\\\\dot{\\\\mu }(m)}\\\\ H_{\\\\nu }^{\\\\ \\\\mu }y^{\\\\nu }y^{\\\\mu (n-1)}\\\\overline{y}^{\\\\dot{\\\\mu }(m)}}_{\\\\Phi _{\\\\mathrm {C}(n,m)}}+\\\\underbrace{\\\\overline{\\\\Omega }^\\\\mathrm {C}_{\\\\mu (n)|\\\\dot{\\\\mu }(m)}\\\\ \\\\overline{H}_{\\\\dot{\\\\nu }}^{\\\\ \\\\dot{\\\\mu }}y^{\\\\mu (n)}\\\\overline{y}^{\\\\dot{\\\\nu }}\\\\overline{y}^{\\\\dot{\\\\mu }(m-1)}}_{{\\\\overline{\\\\Phi }_{\\\\mathrm {C}(n,m)}}}.$ Consider now the reducible two-forms $\\\\Phi ^{\\\\lambda (2),\\\\nu (2),\\\\mu (n-2)|\\\\dot{\\\\mu }(m)} = H^{\\\\lambda \\\\lambda }y^{\\\\nu }y^{\\\\nu }y^{\\\\mu (n-2)}\\\\overline{y}^{\\\\dot{\\\\mu }(m)},\\\\\\\\\\\\overline{\\\\Phi }^{\\\\mu (n)|\\\\dot{\\\\lambda }(2),\\\\dot{\\\\nu }(2),\\\\dot{\\\\mu }(m-2)} = \\\\overline{H}^{\\\\dot{\\\\lambda }\\\\dot{\\\\lambda }}y^{\\\\mu (n)}\\\\overline{y}^{\\\\dot{\\\\nu }}\\\\overline{y}^{\\\\dot{\\\\nu }}\\\\overline{y}^{\\\\dot{\\\\mu }(m-2)}.\",\n \"$ In these terms, the projectors onto irreducible components are $\\\\mathcal {P}_\\\\mathrm {A}&={S}_{(\\\\lambda ,\\\\nu ,\\\\mu )}\\\\,, && \\\\mathcal {P}_\\\\mathrm {B} = \\\\epsilon _{\\\\lambda \\\\nu }\\\\epsilon _{\\\\lambda \\\\nu }\\\\,, && \\\\mathcal {P}_\\\\mathrm {C} = {S}_{(\\\\lambda ,\\\\nu ,\\\\mu )}\\\\circ \\\\epsilon _{\\\\lambda \\\\nu }\\\\,,\\\\\\\\\\\\overline{\\\\mathcal {P}}_\\\\mathrm {A}&={S}_{(\\\\dot{\\\\lambda },\\\\dot{\\\\nu },\\\\dot{\\\\mu })}\\\\,, && \\\\overline{\\\\mathcal {P}}_\\\\mathrm {B}=\\\\epsilon _{\\\\dot{\\\\lambda }\\\\dot{\\\\nu }}\\\\epsilon _{\\\\dot{\\\\lambda }\\\\dot{\\\\nu }}\\\\,, && \\\\overline{\\\\mathcal {P}}_\\\\mathrm {C}={S}_{(\\\\dot{\\\\lambda },\\\\dot{\\\\nu },\\\\dot{\\\\mu })}\\\\circ \\\\epsilon _{\\\\dot{\\\\lambda }\\\\dot{\\\\nu }}\\\\,$ and the decomposition (REF ) reads as $\\\\Phi _\\\\mathrm {A}&=\\\\mathcal {P}_\\\\mathrm {A}\\\\Phi \\\\,, && \\\\Phi _\\\\mathrm {B}=\\\\mathcal {P}_\\\\mathrm {B}\\\\Phi \\\\,, && \\\\Phi _\\\\mathrm {C}=\\\\mathcal {P}_\\\\mathrm {C}\\\\Phi \\\\,, \\\\\\\\ \\\\overline{\\\\Phi }_\\\\mathrm {A}&=\\\\overline{\\\\mathcal {P}}_\\\\mathrm {A}\\\\overline{\\\\Phi }\\\\,, && \\\\overline{\\\\Phi }_\\\\mathrm {B}=\\\\overline{\\\\mathcal {P}}_\\\\mathrm {B}\\\\overline{\\\\Phi }\\\\,, && \\\\overline{\\\\Phi }_\\\\mathrm {C}=\\\\overline{\\\\mathcal {P}}_\\\\mathrm {C}\\\\overline{\\\\Phi }\\\\,.$ For practical calculations we have to find the result of the action of the operator $D=\\\\frac{\\\\partial }{\\\\partial h}$ on the two-form $H$ .\",\n \"The result is $D_{\\\\alpha \\\\dot{\\\\beta }}H^{\\\\nu \\\\lambda } = D_{\\\\alpha \\\\dot{\\\\beta }}\\\\left(h^\\\\nu _{\\\\ \\\\dot{\\\\gamma }}\\\\wedge h^{\\\\lambda \\\\dot{\\\\gamma }}\\\\right) = \\\\epsilon _\\\\alpha ^{\\\\ \\\\nu }\\\\epsilon _{\\\\dot{\\\\beta }\\\\dot{\\\\gamma }} h^{\\\\lambda \\\\dot{\\\\gamma }} - h^{\\\\nu }_{\\\\ \\\\dot{\\\\gamma }}\\\\,\\\\epsilon _\\\\alpha ^{\\\\ \\\\lambda }\\\\epsilon _{\\\\dot{\\\\beta }}^{\\\\ \\\\dot{\\\\gamma }} = - \\\\epsilon _\\\\alpha ^{\\\\ \\\\nu } h^\\\\lambda _{\\\\ \\\\dot{\\\\beta }} - \\\\epsilon _\\\\alpha ^{\\\\ \\\\lambda }h^\\\\nu _{\\\\ \\\\dot{\\\\beta }} = -2\\\\epsilon _\\\\alpha ^{\\\\ (\\\\nu } h^{\\\\lambda )}_{\\\\ \\\\dot{\\\\beta }}\\\\,,$ or, in the condensed notation for symmetrized indices, $D_{\\\\alpha \\\\dot{\\\\beta }}H^{\\\\nu \\\\nu }=-2\\\\epsilon _\\\\alpha ^{\\\\ \\\\nu }h^{\\\\nu }_{\\\\ \\\\dot{\\\\beta }}.$\"\n ],\n [\n \"$H^2(\\\\sigma _-)$ in the far-from-diagonal sector {{formula:76b17993-ba2a-4ad4-89f9-a21723194d83}}\",\n \"Compute $\\\\Delta _{N>\\\\overline{N}+2}$ on the general two-forms $\\\\Phi $ and $\\\\overline{\\\\Phi }$ , $\\\\Delta _{N>\\\\overline{N}+2}(\\\\Phi )=\\\\underbrace{n(m+2)\\\\Phi (y,\\\\overline{y})}_{T_1(y,\\\\overline{y})} + \\\\underbrace{y^\\\\beta _\\\\alpha h^\\\\alpha _{\\\\ \\\\dot{\\\\gamma }} D_\\\\beta ^{\\\\ \\\\dot{\\\\gamma }}\\\\Phi (y,\\\\overline{y})}_{T_2(y,\\\\overline{y})} + \\\\underbrace{\\\\overline{y}^{\\\\dot{\\\\alpha }}\\\\overline{}_{\\\\dot{\\\\beta }}h_{\\\\gamma \\\\dot{\\\\alpha }}D^{\\\\gamma \\\\dot{\\\\beta }}\\\\Phi (y,\\\\overline{y})}_{T_3(y,\\\\overline{y})}\\\\,.$ As in (REF ), denote $T_2^{\\\\lambda (2),\\\\nu (2),\\\\mu (n-2)|\\\\dot{\\\\mu }(m)} = y^\\\\beta _\\\\alpha h^\\\\alpha _{\\\\ \\\\dot{\\\\gamma }} D_\\\\beta ^{\\\\ \\\\dot{\\\\gamma }} H^{\\\\lambda \\\\lambda }y^{\\\\nu }y^{\\\\nu }y^{\\\\mu (n-2)}\\\\overline{y}^{\\\\dot{\\\\mu }(m)}$ and similarly for $T_1$ and $T_3$ .\",\n \"Straightforward computation yields $T_1^{\\\\lambda (2),\\\\nu (2),\\\\mu (n-2)|\\\\dot{\\\\mu }(m)} &= n(m+2) \\\\, H^{\\\\lambda \\\\lambda }y^{\\\\nu }y^{\\\\nu }y^{\\\\mu (n-2)}\\\\overline{y}^{\\\\dot{\\\\mu }(m)}\\\\,,\\\\\\\\T_2^{\\\\lambda (2),\\\\nu (2),\\\\mu (n-2)|\\\\dot{\\\\mu }(m)} &=-4 H^{\\\\nu \\\\lambda } y^{\\\\lambda }y^{\\\\nu }y^{\\\\mu (n-2)}\\\\overline{y}^{\\\\dot{\\\\mu }(m)} - 2(n-2)\\\\,H^{\\\\mu \\\\lambda }y^{\\\\lambda }y^{\\\\nu (2)}y^{\\\\mu (n-3)}\\\\overline{y}^{\\\\dot{\\\\mu }(m)}\\\\,,\\\\\\\\T_3^{\\\\lambda (2),\\\\nu (2),\\\\mu (n-2)|\\\\dot{\\\\mu }(m)} &=m\\\\, H^{\\\\lambda \\\\lambda }y^{\\\\nu (2)} y^{\\\\mu (n-2)}\\\\overline{y}^{\\\\dot{\\\\mu }(m)}.$ Projecting onto the irreducible part $\\\\Phi _\\\\mathrm {A}$ we obtain $\\\\Delta _{N>\\\\overline{N}+2}\\\\Phi _\\\\mathrm {A}=\\\\mathcal {P}_\\\\mathrm {A}\\\\left(T_1+T_2+T_3\\\\right)=\\\\left[n(m+2)-2n+m\\\\right]\\\\Phi _\\\\mathrm {A}=m(n+1)\\\\Phi _\\\\mathrm {A}.$ We see that $\\\\Phi _\\\\mathrm {A}\\\\in \\\\ker (\\\\Delta )$ whenever $m=0$ .\",\n \"This gives a 2-cocycle of the form $H^{\\\\mu \\\\mu }y^{\\\\mu (n)}$ .\",\n \"It can be represented in terms of the generating function as follows.\",\n \"Contract all the indices in $H^{\\\\mu \\\\mu }y^{\\\\mu (n)}$ with some symmetric coefficients $\\\\Omega ^\\\\mathrm {A}_{\\\\mu \\\\mu \\\\mu (n)}$ to obtain $\\\\Omega ^\\\\mathrm {A}_{\\\\mu (n+2)}H^{\\\\mu \\\\mu }y^{\\\\mu (n)}\\\\equiv h^{(\\\\lambda }_{\\\\ \\\\dot{\\\\gamma }}\\\\wedge h^{\\\\nu \\\\dot{\\\\gamma })}\\\\Omega _{(\\\\lambda \\\\nu \\\\mu (n))}y^{\\\\mu (n)}\\\\Rightarrow h^{\\\\lambda }_{\\\\ \\\\dot{\\\\gamma }}\\\\wedge h^{\\\\nu \\\\dot{\\\\gamma }}\\\\,_\\\\lambda _\\\\nu C(y,0\\\\, |x) \\\\in \\\\ker \\\\left(\\\\Delta _{N>\\\\overline{N}+2}\\\\Big |_\\\\text{2-forms}\\\\right),$ where $C(y,0\\\\, |x)=\\\\Omega _{\\\\mu \\\\mu \\\\mu (n)} y^\\\\mu y^{\\\\mu } y^{\\\\mu (n)}$ .\",\n \"Summarizing, we found a part of the kernel of $\\\\Delta $ represented by the two-forms $W(y,0\\\\,|x)=H^{\\\\mu \\\\nu }_\\\\mu _\\\\nu C(y,0\\\\,|x)$ with $C(y,0|x)$ being a general polynomial of $y$ 's of degree $\\\\ge 4$ .\",\n \"Let us now project (REF ) onto the second irreducible part $\\\\Phi _\\\\mathrm {B}$ , $\\\\Delta _{N>\\\\overline{N}+2}\\\\Phi _\\\\mathrm {B}=\\\\mathcal {P}_\\\\mathrm {B}\\\\left(T_1+T_2+T_3\\\\right)=\\\\left[n(m+2)+0+m\\\\right]\\\\Phi _\\\\mathrm {B}\\\\ne 0\\\\quad \\\\forall n,m\\\\in \\\\mathbb {N}_0.$ Since $\\\\Phi _{B(n,m)}$ is proportional to $H_{\\\\nu \\\\nu }y^\\\\nu y^\\\\nu y^{\\\\mu (n-2)}\\\\overline{y}^{\\\\dot{\\\\mu }(m)}$ , the case $n=m=0$ is beyond the allowed region.\",\n \"Thus, $\\\\Phi _\\\\mathrm {B}$ does not contribute to $H^2(\\\\sigma _-)$ .\",\n \"Projecting (REF ) onto $\\\\Phi _\\\\mathrm {C}$ , we find $\\\\Delta _{N>\\\\overline{N}+2}\\\\Phi _\\\\mathrm {C}=\\\\mathcal {P}_\\\\mathrm {C}\\\\left(T_1+T_2+T_3\\\\right) = [n(m+2)-2-(n-2)+m]\\\\Phi _\\\\mathrm {C} = (nm+n+m)\\\\Phi _\\\\mathrm {C}\\\\,.$ Again, $\\\\Phi _\\\\mathrm {C}$ does not contribute to $H^2(\\\\sigma _-)$ since $m>0$ , $n\\\\ge 0$ .\",\n \"Next, we consider the anti-holomorphic two-form $\\\\overline{\\\\Phi }$ .\",\n \"The action of the Laplacian yields $\\\\Delta _{N>\\\\overline{N}+2}(\\\\overline{\\\\Phi })=\\\\underbrace{n(m+2)\\\\overline{\\\\Phi }(y,\\\\overline{y})}_{T_1(y,\\\\overline{y})} + \\\\underbrace{y^\\\\beta _\\\\alpha h^\\\\alpha _{\\\\ \\\\dot{\\\\gamma }} D_\\\\beta ^{\\\\ \\\\dot{\\\\gamma }}\\\\overline{\\\\Phi }(y,\\\\overline{y})}_{T_2(y,\\\\overline{y})} + \\\\underbrace{\\\\overline{y}^{\\\\dot{\\\\alpha }}\\\\overline{}_{\\\\dot{\\\\beta }}h_{\\\\gamma \\\\dot{\\\\alpha }}D^{\\\\gamma \\\\dot{\\\\beta }}\\\\overline{\\\\Phi }(y,\\\\overline{y})}_{T_3(y,\\\\overline{y})}\\\\,.$ As in (REF ) we set $T_2^{\\\\mu (n)|\\\\dot{\\\\lambda }(2),\\\\dot{\\\\nu }(2),\\\\dot{\\\\mu }(m-2)} = y^\\\\beta _\\\\alpha h^\\\\alpha _{\\\\ \\\\dot{\\\\gamma }} D_\\\\beta ^{\\\\ \\\\dot{\\\\gamma }}\\\\,\\\\overline{H}^{\\\\dot{\\\\lambda }\\\\dot{\\\\lambda }}y^{\\\\mu (n)}\\\\overline{y}^{\\\\dot{\\\\nu }}\\\\overline{y}^{\\\\dot{\\\\nu }}\\\\overline{y}^{\\\\dot{\\\\mu }(m-2)}$ and analogously for $T_1$ and $T_3$ .\",\n \"The computation in components yields $T_1^{\\\\mu (n)|\\\\dot{\\\\lambda }(2),\\\\dot{\\\\nu }(2),\\\\dot{\\\\mu }(m-2)} &= n(m+2) \\\\overline{H}^{\\\\dot{\\\\lambda }\\\\dot{\\\\lambda }}y^{\\\\mu (n)}\\\\overline{y}^{\\\\dot{\\\\nu }}\\\\overline{y}^{\\\\dot{\\\\nu }}\\\\overline{y}^{\\\\dot{\\\\mu }(m-2)},\\\\\\\\T_2^{\\\\mu (n)|\\\\dot{\\\\lambda }(2),\\\\dot{\\\\nu }(2),\\\\dot{\\\\mu }(m-2)} &=-n\\\\,\\\\overline{H}^{\\\\dot{\\\\lambda }\\\\dot{\\\\lambda }} y^{\\\\mu (n)}\\\\overline{y}^{\\\\dot{\\\\nu }(2)}\\\\overline{y}^{\\\\dot{\\\\mu }(m-2)},\\\\\\\\T_3^{\\\\mu (n)|\\\\dot{\\\\lambda }(2),\\\\dot{\\\\nu }(2),\\\\dot{\\\\mu }(m-2)} &=-4\\\\,\\\\epsilon ^{\\\\dot{\\\\nu }\\\\dot{\\\\lambda }}\\\\overline{H}_{\\\\dot{\\\\alpha }}^{\\\\ \\\\dot{\\\\lambda }} y^{\\\\mu (n)}\\\\overline{y}^{\\\\dot{\\\\alpha }}\\\\overline{y}^{\\\\dot{\\\\nu }}\\\\overline{y}^{\\\\dot{\\\\mu }(m-2)} - 2(m-2)\\\\,\\\\epsilon ^{\\\\dot{\\\\mu }\\\\dot{\\\\lambda }}\\\\overline{H}_{\\\\dot{\\\\alpha }}^{\\\\ \\\\dot{\\\\lambda }}y^{\\\\mu (n)}\\\\overline{y}^{\\\\dot{\\\\alpha }}\\\\overline{y}^{\\\\dot{\\\\nu }(2)}\\\\overline{y}^{\\\\dot{\\\\mu }(m-3)}.$ Projecting onto the irreducible components we obtain $\\\\Delta _{N>\\\\overline{N}+2}\\\\overline{\\\\Phi }_\\\\mathrm {A} &= \\\\overline{\\\\mathcal {P}}_\\\\mathrm {A}\\\\left(T_1+T_2+T_3\\\\right)=[n(m+1)]\\\\overline{\\\\Phi }_\\\\mathrm {A}\\\\,,\\\\\\\\\\\\Delta _{N>\\\\overline{N}+2}\\\\overline{\\\\Phi }_\\\\mathrm {B} &= \\\\overline{\\\\mathcal {P}}_\\\\mathrm {B}\\\\left(T_1+T_2+T_3\\\\right)=(nm+n+4m)\\\\overline{\\\\Phi }_\\\\mathrm {B}\\\\,,\\\\\\\\\\\\Delta _{N>\\\\overline{N}+2}\\\\overline{\\\\Phi }_\\\\mathrm {C} &= \\\\overline{\\\\mathcal {P}}_\\\\mathrm {C}\\\\left(T_1+T_2+T_3\\\\right) =(nm+n-m)\\\\overline{\\\\Phi }_\\\\mathrm {C}\\\\,.$ The condition $n>m+2$ valid in the far-from-diagonal sector does not allow $\\\\overline{\\\\Phi }_\\\\mathrm {A,B,C}$ to be in the kernel of $ \\\\Delta $ .\",\n \"The analysis of the opposite sector $N<\\\\overline{N}-2$ is analogous via swapping dotted and undotted indices.\",\n \"As a result, the final answer for the under-diagonal sector is $W(0,\\\\overline{y}\\\\, |x)=\\\\overline{H}^{\\\\dot{\\\\mu }\\\\dot{\\\\nu }}\\\\overline{}_{\\\\dot{\\\\mu }}\\\\overline{}_{\\\\dot{\\\\nu }}C(0,\\\\overline{y}\\\\, |x).$ This completes the analysis of $H^2(\\\\sigma _-)$ in the sector $|N-\\\\overline{N}|>2$ .\",\n \"The cohomology is represented by the two-forms $W(y,\\\\overline{y}\\\\, |x) = h^{\\\\mu }_{\\\\ \\\\dot{\\\\gamma }}\\\\wedge h^{\\\\nu \\\\dot{\\\\gamma }}\\\\,_\\\\mu _\\\\nu C(y,0\\\\, |x) + h_\\\\gamma ^{\\\\ \\\\dot{\\\\mu }}\\\\wedge h^{\\\\gamma \\\\dot{\\\\nu }}\\\\,\\\\overline{}_{\\\\dot{\\\\mu }}\\\\overline{}_{\\\\dot{\\\\nu }} C(0,\\\\overline{y}\\\\, |x).$ These two-forms are known to represent the so-called Weyl cocycle in the HS theory.\",\n \"It is thus shown that there are no other non-trivial 2-cocycles in this sector.\"\n ],\n [\n \"$H^2(\\\\sigma _-)$ on the diagonal {{formula:bf16bd86-cdaa-48f9-9718-beef40a13be5}}\",\n \"Now we prove that there are no non-trivial cocycles at $N=\\\\overline{N}$ except for the Weyl cohomology (REF ).\",\n \"As before, act by the operator $\\\\Delta _{N=\\\\overline{N}}$ on the two-form $\\\\Phi ^{\\\\lambda (2)|\\\\nu (2)|\\\\mu (n-2)|\\\\dot{\\\\mu }(n)}(y,\\\\overline{y})=H^{\\\\lambda \\\\lambda }y^{\\\\nu }y^{\\\\nu }y^{\\\\mu (n-2)}\\\\overline{y}^{\\\\dot{\\\\mu }(n)}$ $\\\\Delta _{N=\\\\overline{N}}\\\\Phi (y,\\\\overline{y})=\\\\underbrace{\\\\overline{y}^{\\\\dot{\\\\alpha }}\\\\overline{}_{\\\\dot{\\\\beta }} h_{\\\\gamma \\\\dot{\\\\alpha }} D^{\\\\gamma \\\\dot{\\\\beta }} \\\\Phi (y,\\\\overline{y})}_{T_1(y,\\\\overline{y})}+\\\\underbrace{y^\\\\alpha _\\\\beta h_{\\\\alpha \\\\dot{\\\\gamma }}D^{\\\\beta \\\\dot{\\\\gamma }}\\\\Phi (y,\\\\overline{y})}_{T_2(y,\\\\overline{y})}-\\\\\\\\\\\\underbrace{-\\\\overline{y}^{\\\\dot{\\\\alpha }} y^\\\\beta _\\\\alpha \\\\overline{}_{\\\\dot{\\\\beta }}h^\\\\alpha _{\\\\ \\\\dot{\\\\alpha }} D_\\\\beta ^{\\\\ \\\\dot{\\\\beta }}\\\\Phi (y,\\\\overline{y})}_{T_3(y,\\\\overline{y})}\\\\underbrace{-y^\\\\alpha \\\\overline{y}^{\\\\dot{\\\\beta }}_\\\\beta \\\\overline{}_{\\\\dot{\\\\alpha }}h_\\\\alpha ^{\\\\ \\\\dot{\\\\alpha }}D^\\\\beta _{\\\\ \\\\dot{\\\\beta }}\\\\Phi (y,\\\\overline{y})}_{T_4(y,\\\\overline{y})}\\\\,.$ Denoting $T_1^{\\\\lambda (2),\\\\nu (2),\\\\mu (n-2)|\\\\dot{\\\\mu }(n)} = \\\\overline{y}^{\\\\dot{\\\\alpha }}\\\\overline{}_{\\\\dot{\\\\beta }} h_{\\\\gamma \\\\dot{\\\\alpha }} D^{\\\\gamma \\\\dot{\\\\beta }}\\\\, H^{\\\\lambda \\\\lambda }y^{\\\\nu (2)}y^{\\\\mu (n-2)}\\\\overline{y}^{\\\\dot{\\\\mu }(n)}$ and analogously for $T_2$ , $T_3$ and $T_4$ , straightforward computation yields $T_1^{\\\\lambda (2),\\\\nu (2),\\\\mu (n-2)|\\\\dot{\\\\mu }(n)} & = & n\\\\, H^{\\\\lambda \\\\lambda }y^{\\\\nu (2)}y^{\\\\mu (n-2)}\\\\overline{y}^{\\\\dot{\\\\mu }(n)}\\\\,,\\\\\\\\T_2^{\\\\lambda (2),\\\\nu (2),\\\\mu (n-2)|\\\\dot{\\\\mu }(n)} & = & -4 \\\\epsilon ^{\\\\nu \\\\lambda } H_{\\\\alpha }^{\\\\ \\\\lambda } y^{\\\\alpha }y^{\\\\nu }y^{\\\\mu (n-2)}\\\\overline{y}^{\\\\dot{\\\\mu }(n)} -{}\\\\nonumber \\\\\\\\&& - 2(n-2)\\\\epsilon ^{\\\\mu \\\\lambda } H_{\\\\alpha }^{\\\\ \\\\lambda }y^{\\\\alpha }y^{\\\\nu (2)}y^{\\\\mu (n-3)}\\\\overline{y}^{\\\\dot{\\\\mu }(n)}\\\\,,\\\\\\\\T_3^{\\\\lambda (2),\\\\nu (2),\\\\mu (n-2)|\\\\dot{\\\\mu }(n)} & = & 4n\\\\, h^{\\\\nu }_{\\\\ \\\\dot{\\\\alpha }}\\\\wedge h^{\\\\lambda \\\\dot{\\\\mu }}\\\\, y^{\\\\lambda } y^{\\\\nu } y^{\\\\mu (n-2)}\\\\overline{y}^{\\\\dot{\\\\alpha }}\\\\overline{y}^{\\\\dot{\\\\mu }(n-2)} +{}\\\\nonumber \\\\\\\\& &+ 2n(n-2)\\\\, h^{\\\\mu }_{\\\\ \\\\dot{\\\\alpha }}\\\\wedge h^{\\\\lambda \\\\dot{\\\\mu }} y^{\\\\lambda }y^{\\\\nu (2)}y^{\\\\mu (n-3)}\\\\overline{y}^{\\\\dot{\\\\alpha }}\\\\overline{y}^{\\\\dot{\\\\mu }(n-1)}\\\\,,\\\\\\\\T_4^{\\\\lambda (2),\\\\nu (2),\\\\mu (n-2)|\\\\dot{\\\\mu }(n)} & = & 4n\\\\, \\\\epsilon ^{\\\\nu \\\\lambda }\\\\, h_{\\\\alpha }^{\\\\ \\\\dot{\\\\mu }}\\\\wedge h^{\\\\lambda }_{\\\\ \\\\dot{\\\\beta }}\\\\,y^{\\\\alpha }y^{\\\\nu }y^{\\\\mu (n-2)}\\\\overline{y}^{\\\\dot{\\\\beta }}\\\\overline{y}^{\\\\dot{\\\\mu }(n-1)} +{}\\\\nonumber \\\\\\\\& & + 2n(n-2)\\\\, \\\\epsilon ^{\\\\mu \\\\lambda }\\\\,h_{\\\\alpha }^{\\\\ \\\\dot{\\\\mu }}\\\\wedge h^{\\\\lambda }_{\\\\ \\\\dot{\\\\beta }}\\\\,y^{\\\\alpha }y^{\\\\nu (2)}y^{\\\\mu (n-3)}\\\\overline{y}^{\\\\dot{\\\\beta }}\\\\overline{y}^{\\\\dot{\\\\mu }(n-1)}.$ and $\\\\Delta _{N=\\\\overline{N}}\\\\Phi _\\\\mathrm {A}&= \\\\mathcal {P}_\\\\mathrm {A}(T_1+T_2+T_3+T_4) = n^2\\\\Phi _\\\\mathrm {A},\\\\\\\\\\\\Delta _{N=\\\\overline{N}}\\\\Phi _\\\\mathrm {B}&= \\\\mathcal {P}_\\\\mathrm {B}(T_1+T_2+T_3+T_4) =(2n^2+5n+4)\\\\Phi _\\\\mathrm {B}\\\\ne 0,\\\\\\\\\\\\Delta _{N=\\\\overline{N}}\\\\Phi _\\\\mathrm {C}&= \\\\mathcal {P}_\\\\mathrm {C}(T_1+T_2+T_3+T_4) = 2(n^2+n+1)\\\\Phi _\\\\mathrm {C} -2n(n+1)\\\\overline{\\\\Phi }_\\\\mathrm {C}.$ We observe that the only way for some of $\\\\Phi _\\\\text{A,B,C}$ to be in $H^2(\\\\sigma _-)$ is at $n=0$ .\",\n \"But in the diagonal sector with $N=\\\\overline{N} = n$ this implies $N=\\\\overline{N}=0$ .\",\n \"This case extends formula (REF ) to the spin-one $y,\\\\overline{y}$ -independent sector.\",\n \"The analysis of the anti-holomorphic part $\\\\overline{\\\\Phi }$ is analogous.\",\n \"The resulting cohomology parameterizes the spin-one field strength, i.e., Faraday field strength.\"\n ],\n [\n \"$H^2(\\\\sigma _-)$ in the near-diagonal sector {{formula:f4e6b1a7-8a76-4307-9345-3c00844fb11a}}\",\n \"In the near-diagonal sector a subtlety considered in Section 7.2.3 takes place.\",\n \"We should search for a kernel of $\\\\Delta $ in the form of a linear combination of the two-forms lying under the diagonal and above the diagonal.\",\n \"Our strategy is to act separately on the general holomorphic (REF ) and anti-holomorphic () two-forms placed below the diagonal $N=\\\\overline{N}-2$ and then determine which two-forms are in $\\\\ker (\\\\Delta )$ .\",\n \"(The computation with $N>\\\\overline{N}$ only differs by the complex conjugation.)\",\n \"We start with the general holomorphic two-form below the diagonal $\\\\Phi _{(n-1,n+1)}^{\\\\lambda (2)|\\\\nu (2)|\\\\mu (n-3)|\\\\dot{\\\\mu }(n+1)}(y,\\\\overline{y})=H^{\\\\lambda \\\\lambda }y^{\\\\nu (2)}y^{\\\\mu (n-3)}\\\\overline{y}^{\\\\dot{\\\\mu }(n+1)}.$ Firstly, we set $n\\\\ge 4$ considering the cases of $n\\\\le 3$ , that are special in our computation scheme, because $n-3$ is the number of indices $\\\\mu $ , later.\",\n \"This yields $\\\\Delta _{N=\\\\overline{N}-2}\\\\Phi _{(n-1,n+1)}(y,\\\\overline{y}) = \\\\underbrace{\\\\Delta _{N<\\\\overline{N}- 2}\\\\Phi _{(n-1,n+1)}(y,\\\\overline{y})}_{T_1(y,\\\\overline{y})} +\\\\underbrace{y^\\\\alpha y^\\\\beta \\\\overline{}_{\\\\dot{\\\\alpha }}\\\\overline{}_{\\\\dot{\\\\beta }}h_\\\\beta ^{\\\\ \\\\dot{\\\\beta }}D_\\\\alpha ^{\\\\ \\\\dot{\\\\alpha }}\\\\Phi _{(n-1,n+1)}(y,\\\\overline{y})}_{T_2(y,\\\\overline{y})}\\\\,.$ The first term is computed the same way as in (REF ) giving $T_1^{\\\\lambda (2),\\\\nu (2),\\\\mu (n-3)|\\\\dot{\\\\mu }(n+1)}= (n+1)^2 \\\\, H^{\\\\lambda \\\\lambda }y^{\\\\nu }y^{\\\\nu }y^{\\\\mu (n-3)}\\\\overline{y}^{\\\\dot{\\\\mu }(n+1)} -4 H^{\\\\nu \\\\lambda } y^{\\\\lambda }y^{\\\\nu }y^{\\\\mu (n-3)}\\\\overline{y}^{\\\\dot{\\\\mu }(n+1)} -\\\\\\\\-2(n-3)\\\\,H^{\\\\mu \\\\lambda }y^{\\\\lambda }y^{\\\\nu (2)}y^{\\\\mu (n-4)}\\\\overline{y}^{\\\\dot{\\\\mu }(n+1)} +(n-1)\\\\, H^{\\\\lambda \\\\lambda }y^{\\\\nu (2)} y^{\\\\mu (n-3)}\\\\overline{y}^{\\\\dot{\\\\mu }(n+1)}.$ The computation of the additional term $T_2(y,\\\\overline{y})$ yields $T_2^{\\\\lambda (2),\\\\nu (2),\\\\mu (n-3)|\\\\dot{\\\\mu }(n+1)} = y^\\\\alpha y^\\\\beta \\\\overline{}_{\\\\dot{\\\\alpha }}\\\\overline{}_{\\\\dot{\\\\beta }}h_\\\\beta ^{\\\\ \\\\dot{\\\\beta }}D_\\\\alpha ^{\\\\ \\\\dot{\\\\alpha }}H^{\\\\lambda \\\\lambda }y^{\\\\nu (2)}y^{\\\\mu (n-3)}\\\\overline{y}^{\\\\dot{\\\\mu }(n+1)} =\\\\\\\\=-2n(n+1)\\\\, h_{\\\\beta }^{\\\\ \\\\dot{\\\\mu }}\\\\wedge h^{\\\\lambda \\\\dot{\\\\mu }} y^{\\\\lambda }y^{\\\\beta }y^{\\\\nu (2)}y^{\\\\mu (n-3)}\\\\overline{y}^{\\\\dot{\\\\mu }(n-1)}=\\\\\\\\=-n(n+1)\\\\, \\\\overline{H}^{\\\\dot{\\\\mu }\\\\dot{\\\\mu }}y^{\\\\lambda (2)}y^{\\\\nu (2)}y^{\\\\mu (n-3)}\\\\overline{y}^{\\\\dot{\\\\mu }(n-1)}.$ Projection onto the irreducible parts $A$ , $B$ and $C$ yields $\\\\Delta \\\\Phi _{\\\\mathrm {A}(n-1,n+1)} &= \\\\underbrace{n(n+1)}_{\\\\alpha (n)}\\\\Phi _{\\\\mathrm {A}(n-1,n+1)} \\\\underbrace{ - n(n+1)}_{\\\\beta (n)}\\\\overline{\\\\Phi }_{\\\\mathrm {A}(n+1,n-1)}\\\\,,\\\\\\\\\\\\Delta \\\\Phi _{\\\\mathrm {B}(n-1,n+1)} &= (n^2+5n-4)\\\\Phi _{\\\\mathrm {B}(n-1,n+1)}\\\\,,\\\\\\\\\\\\Delta \\\\Phi _{\\\\mathrm {C}(n-1,n+1)} &= (n^2+2n-1)\\\\Phi _{\\\\mathrm {C}(n-1,n+1)}\\\\,.$ Let us stress that the complex conjugation denoted by $\\\\dagger $ swaps dotted and undotted indices $(y^\\\\alpha )^\\\\dagger = \\\\overline{y}^{\\\\dot{\\\\alpha }}, \\\\quad \\\\quad (H^{\\\\alpha \\\\alpha })^\\\\dagger = \\\\overline{H}^{\\\\dot{\\\\alpha }\\\\dot{\\\\alpha }}$ and relates $\\\\Phi $ and $\\\\overline{\\\\Phi }$ in the following way: $(\\\\Phi _{\\\\mathrm {A,B,C}(n-1,n+1)})^\\\\dagger = \\\\overline{\\\\Phi }_{\\\\mathrm {A,B,C}(n+1,n-1)}.$ The computation for the complex-conjugated objects $\\\\overline{\\\\Phi }_{\\\\mathrm {A,B,C}(n+1,n-1)}$ is analogous giving $\\\\Delta \\\\overline{\\\\Phi }_{\\\\mathrm {A}(n+1,n-1)} &= n(n+1)\\\\,\\\\overline{\\\\Phi }_{\\\\mathrm {A}(n+1,n-1)} - n(n+1)\\\\,\\\\Phi _{\\\\mathrm {A}(n-1,n+1)}\\\\,,\\\\\\\\\\\\Delta \\\\overline{\\\\Phi }_{\\\\mathrm {B}(n+1,n-1)} &= (n^2+5n-4)\\\\overline{\\\\Phi }_{\\\\mathrm {B}(n+1,n-1)}\\\\,,\\\\\\\\\\\\Delta \\\\overline{\\\\Phi }_{\\\\mathrm {C}(n+1,n-1)} &= (n^2+2n-1)\\\\overline{\\\\Phi }_{\\\\mathrm {C}(n+1,n-1)}\\\\,.$ From (REF ) and (REF ) we observe that there is a non-trivial 2-cocycle $\\\\mathcal {E}_\\\\mathrm {A}=\\\\Phi _{\\\\mathrm {A}(n-1,n+1)}+\\\\overline{\\\\Phi }_{\\\\mathrm {A}(n+1,n-1)}= \\\\mathcal {E}^\\\\mathrm {A}_{\\\\mu (n+1),\\\\dot{\\\\mu }(n+1)}\\\\,\\\\Big (H^{\\\\mu \\\\mu }y^{\\\\mu (n-1)}\\\\overline{y}^{\\\\dot{\\\\mu }(n+1)} + \\\\overline{H}^{\\\\dot{\\\\mu }\\\\dot{\\\\mu }}y^{\\\\mu (n+1)}\\\\overline{y}^{\\\\dot{\\\\mu }(n-1)}\\\\Big )$ with arbitrary coefficients $\\\\mathcal {E}^\\\\mathrm {A}_{\\\\mu (n+1),\\\\dot{\\\\mu }(n+1)}(x)$ .\",\n \"This answer agrees with the analysis of Section 7.2.3.\",\n \"Indeed, the coefficients on the r.h.s.\",\n \"of (REF ) coincide up to a sign $\\\\alpha (n) = -\\\\beta (n)$ , and by (REF ) of Section 7.2.3 this implies a non-trivial 2-cocycle (REF ).\",\n \"This cocycle represents the traceless part of the free Fronsdal HS equations.\",\n \"The irreducible representations of types $(B)$ and $(C)$ do not contribute to cohomology since they are not in $\\\\ker (\\\\Delta )$ (recall that we are assuming $n\\\\ge 4$ ).\",\n \"Now consider the cases of $n=1,2,3$ .\",\n \"Computing the action of the Laplace operator on the following objects: $\\\\Phi _{(0,2)}^{\\\\lambda \\\\lambda |\\\\dot{\\\\mu }\\\\dot{\\\\mu }}(y,\\\\overline{y}) &= H^{\\\\lambda \\\\lambda }\\\\overline{y}^{\\\\dot{\\\\mu }}\\\\overline{y}^{\\\\dot{\\\\mu }},\\\\\\\\\\\\Phi _{(1,3)}^{\\\\lambda \\\\lambda |\\\\mu |\\\\dot{\\\\mu }(3)}(y,\\\\overline{y}) &= H^{\\\\lambda \\\\lambda }y^\\\\mu \\\\overline{y}^{\\\\dot{\\\\mu }(3)},\\\\\\\\\\\\Phi _{(2,4)}^{\\\\lambda \\\\lambda |\\\\nu \\\\nu |\\\\dot{\\\\mu }(4)}(y,\\\\overline{y}) &= H^{\\\\lambda \\\\lambda }y^\\\\nu y^\\\\nu \\\\overline{y}^{\\\\dot{\\\\mu }(4)}\\\\,,$ it is not difficult to obtain $(\\\\Delta \\\\Phi _{(0,2)})^{\\\\lambda \\\\lambda |\\\\dot{\\\\mu }\\\\dot{\\\\mu }}(y,\\\\overline{y}) &= 2 H^{\\\\lambda \\\\lambda }\\\\overline{y}^{\\\\dot{\\\\mu }(2)} - 2\\\\overline{H}^{\\\\dot{\\\\mu }\\\\dot{\\\\mu }}y^{\\\\lambda (2)},\\\\\\\\(\\\\Delta \\\\Phi _{(1,3)})^{\\\\lambda \\\\lambda |\\\\mu |\\\\dot{\\\\mu }(3)}(y,\\\\overline{y}) &= 8H^{\\\\lambda \\\\lambda }y^\\\\mu \\\\overline{y}^{\\\\dot{\\\\mu }(3)} - 2H^{\\\\mu \\\\lambda }y^\\\\lambda \\\\overline{y}^{\\\\dot{\\\\mu }(3)} - 6\\\\overline{H}^{\\\\dot{\\\\mu }\\\\dot{\\\\mu }}y^{\\\\lambda (2)}y^\\\\mu \\\\overline{y}^{\\\\dot{\\\\mu }},\\\\\\\\(\\\\Delta \\\\Phi _{(2,4)})^{\\\\lambda \\\\lambda |\\\\nu \\\\nu |\\\\dot{\\\\mu }(4)}(y,\\\\overline{y}) &= 16H^{\\\\lambda \\\\lambda }y^{\\\\nu \\\\nu }\\\\overline{y}^{\\\\dot{\\\\mu }(4)} - 4H^{\\\\nu \\\\lambda }y^\\\\lambda y^\\\\nu \\\\overline{y}^{\\\\dot{\\\\mu }(4)} - 12\\\\overline{H}^{\\\\dot{\\\\mu }\\\\dot{\\\\mu }}y^{\\\\lambda (2)}y^{\\\\nu (2)}\\\\overline{y}^{\\\\dot{\\\\mu }(2)}.$ We see that these results for $n=1,2,3$ extend the traceless part of the Fronsdal cohomology (REF ) to spins $s=2,3,4$ .\",\n \"It remains to analyze the case of anti-holomorphic two-form below the diagonal $N=\\\\overline{N}-2$ $\\\\overline{\\\\Phi }_{(n-1,n+1)}^{\\\\dot{\\\\nu }(2)|\\\\mu (n-1)|\\\\dot{\\\\lambda }(2)|\\\\dot{\\\\mu }(n-1)}(y,\\\\overline{y}) = \\\\overline{H}^{\\\\dot{\\\\lambda }\\\\dot{\\\\lambda }}y^{\\\\mu (n-1)}\\\\overline{y}^{\\\\dot{\\\\nu }(2)}\\\\overline{y}^{\\\\dot{\\\\mu }(n-1)}.$ Unlike Eq.\",\n \"(REF ), the number of indices $\\\\mu $ and $\\\\dot{\\\\mu }$ in (REF ) is $n-1$ , not $n-3$ .\",\n \"Hence, there is no need to consider separately the cases of $n\\\\ge 4$ and $n\\\\le 3$ .\",\n \"Instead, we set $n\\\\ge 2$ and then analyze the $n=1$ case separately.\",\n \"Let $n\\\\ge 2$ .\",\n \"The action of the corresponding Laplace operator on (REF ) yields $\\\\Delta _{N=\\\\overline{N}-2}\\\\overline{\\\\Phi }_{(n-1,n+1)}(y,\\\\overline{y}) = \\\\underbrace{\\\\Delta _{N<\\\\overline{N}+2}\\\\overline{\\\\Phi }_{(n-1,n+1)}(y,\\\\overline{y})}_{T_3(y,\\\\overline{y})} +\\\\underbrace{y^\\\\alpha y^\\\\beta \\\\overline{}_{\\\\dot{\\\\alpha }}\\\\overline{}_{\\\\dot{\\\\beta }}h_\\\\beta ^{\\\\ \\\\dot{\\\\beta }}D_\\\\alpha ^{\\\\ \\\\dot{\\\\alpha }}\\\\overline{\\\\Phi }_{(n-1,n+1)}(y,\\\\overline{y})}_{T_4(y,\\\\overline{y})}.$ The computation is completely analogous to that for the holomorphic two-form.\",\n \"After projecting onto the irreducible components it gives $\\\\Delta \\\\overline{\\\\Phi }_{\\\\mathrm {A}(n-1,n+1)} &= (n^2+n-4)\\\\overline{\\\\Phi }_{\\\\mathrm {A}(n-1,n+1)}\\\\,,\\\\\\\\\\\\Delta \\\\overline{\\\\Phi }_{\\\\mathrm {B}(n-1,n+1)} &= \\\\underbrace{(n^2+4n-1)}_{\\\\alpha (n)}\\\\overline{\\\\Phi }_{\\\\mathrm {B}(n-1,n+1)} \\\\underbrace{ - (n^2+4n-1)}_{\\\\beta (n)}\\\\Phi _{\\\\mathrm {B}(n+1,n-1)}\\\\,,\\\\\\\\\\\\Delta \\\\overline{\\\\Phi }_{\\\\mathrm {C}(n-1,n+1)} &= (n^2+2n+1)\\\\overline{\\\\Phi }_{\\\\mathrm {C}(n-1,n+1)}\\\\,.$ Applying once again the result (REF ) of Section 7.2.3 to (REF ), on the r.h.s.\",\n \"of which the coefficients coincide up to a sign, $\\\\alpha (n) = -\\\\beta (n)$ , we obtain the 2-cocycle of the form $\\\\mathcal {E}_\\\\mathrm {B}=\\\\Phi _{\\\\mathrm {B}(n+1,n-1)}+\\\\overline{\\\\Phi }_{\\\\mathrm {B}(n-1,n+1)} =\\\\\\\\= \\\\mathcal {E}^\\\\mathrm {B}_{\\\\mu (n-1)\\\\dot{\\\\mu }(n-1)}\\\\,\\\\Big (H_{\\\\nu \\\\nu }y^{\\\\nu (2)} y^{\\\\mu (n-1)}\\\\overline{y}^{\\\\dot{\\\\mu }(n-1)} + \\\\overline{H}_{\\\\dot{\\\\nu }\\\\dot{\\\\nu }}y^{\\\\mu (n-1)}\\\\overline{y}^{\\\\dot{\\\\nu }(2)}\\\\overline{y}^{\\\\dot{\\\\mu }(n-1)}\\\\big )\\\\,,$ that represents the trace part of the Fronsdal equations.\",\n \"Having considered $n\\\\ge 2$ , now consider the case of $n=1$ .\",\n \"Computation of the action of the Laplace operator on the following two-form: $\\\\overline{\\\\Phi }_{(0,2)}^{\\\\dot{\\\\lambda }\\\\dot{\\\\lambda }|\\\\dot{\\\\mu }\\\\dot{\\\\mu }}(y,\\\\overline{y}) = \\\\overline{H}^{\\\\dot{\\\\lambda }\\\\dot{\\\\lambda }}\\\\overline{y}^{\\\\dot{\\\\mu }}\\\\overline{y}^{\\\\dot{\\\\mu }}$ yields $(\\\\Delta \\\\overline{\\\\Phi }_{(0,2)})^{\\\\dot{\\\\lambda }\\\\dot{\\\\lambda }|\\\\dot{\\\\mu }\\\\dot{\\\\mu }}(y,\\\\overline{y}) = 4 \\\\overline{H}^{\\\\dot{\\\\lambda }\\\\dot{\\\\lambda }}\\\\overline{y}^{\\\\dot{\\\\mu }(2)}-4\\\\overline{H}^{\\\\dot{\\\\mu }\\\\dot{\\\\lambda }}\\\\overline{y}^{\\\\dot{\\\\lambda }}\\\\overline{y}^{\\\\dot{\\\\mu }} - 2 H_{\\\\alpha \\\\alpha }y^{\\\\alpha }y^{\\\\alpha }\\\\epsilon ^{\\\\dot{\\\\mu }\\\\dot{\\\\lambda }}\\\\epsilon ^{\\\\dot{\\\\mu }\\\\dot{\\\\lambda }}.$ After projecting onto the irreducible components, we find that the case of $n=1$ extends the trace part of the Fronsdal cocycle $\\\\mathcal {E}_\\\\mathrm {B}$ (REF ) to spin $s=2$ .\",\n \"In addition, (REF ) also contributes to the antiholomorphic part of the Weyl cocycle represented by the second term on the r.h.s.\",\n \"of (REF ).\",\n \"The holomorphic part of the latter lies in the opposite (complex-conjugated) region, in which the analysis is completely analogous.\",\n \"This 2-cocycle represents the Weyl tensor for the linearized gravity ($s=2$ ) in $AdS_4$ .\",\n \"This completes the analysis of $H^2(\\\\sigma _-)$ in the near-diagonal sector $N=\\\\overline{N} \\\\pm 2$ .\"\n ],\n [\n \"Summary for bosonic $H^{0,1,2}(\\\\sigma _-)$\",\n \"Here we collect the final results for the cocycles associated with the bosonic HS gauge parameters, fields and field equations in $AdS_4$ .\",\n \"Recall that $H^0(\\\\sigma _-)$ represents parameters of the differential HS gauge symmetries.\",\n \"It is spanned by the zero-forms $F(y,\\\\overline{y}|\\\\,x) = F_{\\\\alpha (n)\\\\,\\\\dot{\\\\alpha }(n)}(x)\\\\,y^{\\\\alpha (n)}\\\\overline{y}^{\\\\dot{\\\\alpha }(n)}\\\\,,\\\\qquad n\\\\in \\\\mathbb {N}_0\\\\,.$ $H^1(\\\\sigma _-)$ represents the dynamical HS fields.\",\n \"For the bosonic HS fields in $AdS_4$ it is spanned by the two 1-cocycles $\\\\phi (y,\\\\overline{y}\\\\,|x)$ and $\\\\phi ^\\\\text{tr}(y,\\\\overline{y}\\\\,|x)$ corresponding, respectively, to the traceless and trace components of the original Fronsdal field in the metric formalism: $\\\\phi (y,\\\\overline{y}\\\\,|x) = h^{\\\\mu \\\\dot{\\\\mu }}\\\\,\\\\partial _\\\\mu \\\\overline{\\\\partial }_{\\\\dot{\\\\mu }}\\\\, F_1(y,\\\\overline{y}\\\\,|x),\\\\\\\\\\\\phi ^\\\\text{tr}(y,\\\\overline{y}\\\\,|x) = h_{\\\\mu \\\\dot{\\\\mu }}\\\\,y^\\\\mu \\\\overline{y}^{\\\\dot{\\\\mu }}\\\\, F_2(y,\\\\overline{y}\\\\,|x),$ where $F_{1,2}(y,\\\\overline{y}\\\\,|x)$ are $(N,\\\\overline{N})$ -diagonal, that is $\\\\left(y^\\\\alpha \\\\frac{\\\\partial }{\\\\partial y^\\\\alpha } - \\\\overline{y}^{\\\\dot{\\\\alpha }}\\\\frac{\\\\partial }{\\\\partial \\\\overline{y}^{\\\\dot{\\\\alpha }}}\\\\right)F_{1,2}(y,\\\\overline{y}\\\\,|x) = 0.$ Finally, $H^2(\\\\sigma _-)$ , which represents gauge invariant differential operators on the bosonic HS fields, are spanned by three different 2-cocycles: the so-called Weyl cocycle $W(y,\\\\overline{y}\\\\,|x)$ and two irreducible components of the Fronsdal cocycle $\\\\mathcal {E}_\\\\mathrm {A}(y,\\\\overline{y}\\\\,|x)$ (REF ) and $\\\\mathcal {E}_\\\\mathrm {B}(y,\\\\overline{y}\\\\,|x)$ (REF ).\",\n \"The latter correspond to the ${\\\\it l.h.s.}\",\n \"$ 's of the dynamical equations for the fields of spin $s>1$ (spin $s\\\\le 1$ field equations are in the zero-form sector of unfolded equations [12]).\",\n \"Note that these cocycles are real since they contain equal numbers of dotted and undotted indices.\",\n \"$W(y,\\\\overline{y}\\\\, |x) &= H^{\\\\mu \\\\nu } _\\\\mu _\\\\nu C(y,0\\\\, |x) + \\\\overline{H}^{\\\\dot{\\\\mu }\\\\dot{\\\\nu }} \\\\overline{}_{\\\\dot{\\\\mu }}\\\\overline{}_{\\\\dot{\\\\nu }} C(0,\\\\overline{y}\\\\, |x)\\\\,,\\\\\\\\\\\\mathcal {E}_\\\\mathrm {A}(y,\\\\overline{y}\\\\,|x) &= \\\\Big (H^{\\\\mu \\\\nu }\\\\partial _\\\\mu \\\\partial _\\\\nu + \\\\overline{H}^{\\\\dot{\\\\mu }\\\\dot{\\\\nu }}\\\\overline{\\\\partial }_{\\\\dot{\\\\mu }}\\\\overline{\\\\partial }_{\\\\dot{\\\\nu }}\\\\Big ) C^\\\\text{diag}(y,\\\\overline{y}\\\\,|x)\\\\,,\\\\\\\\\\\\mathcal {E}_\\\\mathrm {B}(y,\\\\overline{y}\\\\,|x) &= \\\\Big (H^{\\\\mu \\\\nu } y_\\\\mu y_\\\\nu + \\\\overline{H}^{\\\\dot{\\\\mu }\\\\dot{\\\\nu }}\\\\overline{y}_{\\\\dot{\\\\mu }}\\\\overline{y}_{\\\\dot{\\\\nu }}\\\\Big ) C^\\\\text{diag}(y,\\\\overline{y}\\\\,|x),$ where $C^\\\\text{diag}(y,\\\\overline{y})$ obey (REF ).\"\n ],\n [\n \"Fermionic HS fields in $AdS_4$\",\n \"So far we considered the bosonic case with even grading $G=|N-\\\\overline{N}|$ .\",\n \"By (REF ) odd $G$ corresponds to fields of half-integer spins, i.e., oddness of $G$ determines the field statistics.\",\n \"To extend the results for $H^p(\\\\sigma _-)$ to fermionic fields we should first define the operator $\\\\sigma _-$ on multispinors of odd ranks.\",\n \"In the fermionic case the lowest possible odd grading is $G=|N-\\\\overline{N}|=1$ .\",\n \"In this sector we define the action of $\\\\sigma _-$ to vanish.\",\n \"In all other gradings $\\\\sigma _\\\\pm $ is defined analogously to the bosonic case.\",\n \"Namely, $\\\\sigma _-\\\\omega (y,\\\\overline{y}) &:= i\\\\, \\\\overline{y}^{\\\\dot{\\\\alpha }}h^\\\\alpha _{\\\\ \\\\dot{\\\\alpha }}\\\\partial _\\\\alpha \\\\ \\\\omega (y,\\\\overline{y}), \\\\quad \\\\quad \\\\text{at } N\\\\ge \\\\overline{N}+3\\\\,,\\\\\\\\\\\\sigma _-\\\\omega (y,\\\\overline{y}) &:= i\\\\, y^{\\\\alpha }h_\\\\alpha ^{\\\\ \\\\dot{\\\\alpha }}\\\\overline{\\\\partial }_{\\\\dot{\\\\alpha }}\\\\ \\\\omega (y,\\\\overline{y}), \\\\quad \\\\quad \\\\text{at } N\\\\le \\\\overline{N}-3\\\\,.$ Analogously, the operator $\\\\sigma _+$ is defined as $\\\\sigma _+\\\\omega (y,\\\\overline{y}) &:= -i\\\\, y^\\\\alpha D_\\\\alpha ^{\\\\ \\\\dot{\\\\alpha }}\\\\overline{\\\\partial }_{\\\\dot{\\\\alpha }}\\\\ \\\\omega (y,\\\\overline{y}), \\\\quad \\\\quad \\\\text{at } N\\\\ge \\\\overline{N}+3\\\\,,\\\\\\\\\\\\sigma _+\\\\omega (y,\\\\overline{y}) &:= -i\\\\, \\\\overline{y}^{\\\\dot{\\\\alpha }}D^\\\\alpha _{\\\\ \\\\dot{\\\\alpha }}\\\\partial _\\\\alpha \\\\ \\\\omega (y,\\\\overline{y}), \\\\quad \\\\quad \\\\text{at } N\\\\le \\\\overline{N}-3\\\\,.$ For the lowest grading $|N-\\\\overline{N}|=1$ , $\\\\sigma _+$ is defined by $\\\\sigma _+ = -i\\\\left(y^\\\\alpha D_\\\\alpha ^{\\\\ \\\\dot{\\\\alpha }}\\\\overline{\\\\partial }_{\\\\dot{\\\\alpha }}+\\\\overline{y}^{\\\\dot{\\\\alpha }}D^\\\\alpha _{\\\\ \\\\dot{\\\\alpha }}\\\\partial _\\\\alpha \\\\right), \\\\quad \\\\quad \\\\text{at } |N-\\\\overline{N}|=1.$ Notice that the action of the fermionic Laplace operator is analogous to that of the bosonic one () with the grading shifted by one, $\\\\Delta ^\\\\text{fermionic}_G=\\\\Delta ^\\\\text{bosonic}_{G-1}$ .\",\n \"The final result is $\\\\bullet \\\\quad \\\\Delta ^\\\\text{fermionic}_{N>\\\\overline{N}+3}&= \\\\Delta ^\\\\text{bosonic}_{N>\\\\overline{N}+2} = N(\\\\overline{N}+2) + y^\\\\beta _\\\\alpha h^\\\\alpha _{\\\\ \\\\dot{\\\\gamma }} D_\\\\beta ^{\\\\ \\\\dot{\\\\gamma }} + \\\\overline{y}^{\\\\dot{\\\\alpha }}\\\\overline{}_{\\\\dot{\\\\beta }}h_{\\\\gamma \\\\dot{\\\\alpha }}D^{\\\\gamma \\\\dot{\\\\beta }}\\\\,,\\\\\\\\\\\\bullet \\\\quad \\\\Delta ^\\\\text{fermionic}_{N=\\\\overline{N}+3}&= \\\\Delta ^\\\\text{bosonic}_{N=\\\\overline{N}+2} = \\\\Delta _{N>\\\\overline{N}+2} + \\\\overline{y}^{\\\\dot{\\\\alpha }}\\\\overline{y}^{\\\\dot{\\\\beta }}_\\\\alpha _\\\\beta h^\\\\beta _{\\\\ \\\\dot{\\\\beta }}D^\\\\alpha _{\\\\ \\\\dot{\\\\alpha }}\\\\,,\\\\\\\\\\\\bullet \\\\quad \\\\Delta ^\\\\text{fermionic}_{N=\\\\overline{N}+1}&= \\\\Delta ^\\\\text{fermionic}_{N=\\\\overline{N}-1} = \\\\Delta ^\\\\text{bosonic}_{N=\\\\overline{N}} =\\\\\\\\ \\\\qquad \\\\qquad \\\\qquad \\\\qquad =y^\\\\alpha _\\\\beta h_{\\\\alpha \\\\dot{\\\\gamma }}D^{\\\\beta \\\\dot{\\\\gamma }}+ \\\\overline{y}^{\\\\dot{\\\\alpha }}\\\\overline{}_{\\\\dot{\\\\beta }} &h_{\\\\gamma \\\\dot{\\\\alpha }} D^{\\\\gamma \\\\dot{\\\\beta }}-\\\\overline{y}^{\\\\dot{\\\\alpha }} y^\\\\beta _\\\\alpha \\\\overline{}_{\\\\dot{\\\\beta }}h^\\\\alpha _{\\\\ \\\\dot{\\\\alpha }} D_\\\\beta ^{\\\\ \\\\dot{\\\\beta }}- y^\\\\alpha \\\\overline{y}^{\\\\dot{\\\\beta }}_\\\\beta \\\\overline{}_{\\\\dot{\\\\alpha }}h_\\\\alpha ^{\\\\ \\\\dot{\\\\alpha }}D^\\\\beta _{\\\\ \\\\dot{\\\\beta }}\\\\,.\\\\nonumber $ This allows us do deduce the fermionic cohomology from the bosonic one arriving at the following final results.\"\n ],\n [\n \"Fermionic $H^0(\\\\sigma _-)$\",\n \"The space $H^0(\\\\sigma _-)$ for fermionic HS fields is spanned by two independent zero-forms with $N-\\\\overline{N} = \\\\pm 1:$ $H^0(\\\\sigma _-) = \\\\Big \\\\lbrace F(y,\\\\overline{y}\\\\,|x) + \\\\overline{F}(y,\\\\overline{y}\\\\,|x) = F_{\\\\alpha (n+1),\\\\dot{\\\\alpha }(n)}(x)\\\\,y^{\\\\alpha (n+1)}\\\\overline{y}^{\\\\dot{\\\\alpha }(n)}+\\\\overline{F}_{\\\\alpha (n),\\\\dot{\\\\alpha }(n+1)}(x)\\\\,y^{\\\\alpha (n)}\\\\overline{y}^{\\\\dot{\\\\alpha }(n+1)}\\\\Big \\\\rbrace \\\\,.$ Recall that, by Theorem 3.1, $H^0(\\\\sigma _-)$ represents parameters of differential HS gauge transformations.\"\n ],\n [\n \"Fermionic $H^1(\\\\sigma _-)$\",\n \"In the bosonic case, we had two physically different cocycles in $H^1$ (REF ) corresponding to traceless $\\\\phi (y,\\\\overline{y}\\\\,|x)$ and trace $\\\\phi ^\\\\text{tr}(y,\\\\overline{y}\\\\,|x)$ parts of the Fronsdal field.\",\n \"These belong to the diagonal $N=\\\\overline{N}$ .\",\n \"For the fermionic case the situation is analogous.\",\n \"The lowest grading is now $G = |N-\\\\overline{N}| = 1$ .\",\n \"So, in this sector there are four (not two) different 1-cocycles: $\\\\psi $ , $\\\\psi ^\\\\text{tr}$ , $\\\\overline{\\\\psi }$ and $\\\\overline{\\\\psi }^\\\\text{tr}$ given by $\\\\psi (y,\\\\overline{y}\\\\,|x) &= \\\\psi _{\\\\mu (n+2),\\\\dot{\\\\mu }(n+1)}(x)\\\\,h^{\\\\mu \\\\dot{\\\\mu }}\\\\,y^{\\\\mu (n+1)}\\\\overline{y}^{\\\\dot{\\\\mu }(n)},\\\\\\\\\\\\overline{\\\\psi }(y,\\\\overline{y}\\\\,|x) &= \\\\overline{\\\\psi }_{\\\\mu (n+1),\\\\dot{\\\\mu }(n+2)}(x)\\\\,h^{\\\\mu \\\\dot{\\\\mu }}\\\\,y^{\\\\mu (n)}\\\\overline{y}^{\\\\dot{\\\\mu }(n+1)},\\\\\\\\\\\\psi ^\\\\text{tr}(y,\\\\overline{y}\\\\,|x) &= \\\\psi ^\\\\text{tr}_{\\\\mu (n),\\\\dot{\\\\mu }(n-1)}(x)\\\\,h_{\\\\nu \\\\dot{\\\\nu }}\\\\,y^\\\\nu y^{\\\\mu (n)}\\\\overline{y}^{\\\\dot{\\\\nu }}\\\\overline{y}^{\\\\dot{\\\\mu }(n-1)},\\\\\\\\\\\\overline{\\\\psi }^\\\\text{tr}(y,\\\\overline{y}\\\\,|x) &= \\\\overline{\\\\psi }^\\\\text{tr}_{\\\\mu (n-1),\\\\dot{\\\\mu }(n)}(x)\\\\,h_{\\\\nu \\\\dot{\\\\nu }}\\\\,y^\\\\nu y^{\\\\mu (n-1)}\\\\overline{y}^{\\\\dot{\\\\nu }}\\\\overline{y}^{\\\\dot{\\\\mu }(n)}$ with a non-negative integer $n$ (positive for $\\\\psi ^\\\\text{tr}$ and $\\\\overline{\\\\psi }^\\\\text{tr}$ ).\",\n \"Cocycles $\\\\psi $ and $\\\\psi ^\\\\text{tr}$ belong to the upper near-diagonal line $N=\\\\overline{N}+1$ , whereas $\\\\overline{\\\\psi }$ and $\\\\overline{\\\\psi }^\\\\text{tr}$ belong to the lower near-diagonal line $N=\\\\overline{N} -1$ .\",\n \"All four of them have grading $G=1$ .\",\n \"$\\\\psi $ and $\\\\overline{\\\\psi }$ are mutually conjugated.\",\n \"These results can be put into the following concise form $\\\\psi (y,\\\\overline{y}\\\\,|x) &= h^{\\\\mu \\\\dot{\\\\mu }}\\\\,\\\\partial _\\\\mu \\\\overline{\\\\partial }_{\\\\dot{\\\\mu }}\\\\,F_1(y,\\\\overline{y}\\\\,|x),\\\\\\\\\\\\psi ^\\\\text{tr}(y,\\\\overline{y}\\\\,|x) &= h_{\\\\mu \\\\dot{\\\\mu }}\\\\,y^\\\\mu \\\\overline{y}^{\\\\dot{\\\\mu }}\\\\,F_2(y,\\\\overline{y}\\\\,|x),$ where $F_{1,2}(y,\\\\overline{y}\\\\,|x)$ are of the homogeneity degree $N-\\\\overline{N}=1$ , i.e., $\\\\left(y^\\\\alpha \\\\frac{\\\\partial }{\\\\partial y^\\\\alpha } - \\\\overline{y}^{\\\\dot{\\\\alpha }}\\\\frac{\\\\partial }{\\\\partial \\\\overline{y}^{\\\\dot{\\\\alpha }}}\\\\right) F_{1,2}(y,\\\\overline{y}\\\\,|x) = F_{1,2}(y,\\\\overline{y}\\\\,|x)\\\\,.$ For $\\\\overline{\\\\psi }$ and $\\\\overline{\\\\psi }^\\\\text{tr}$ the results are analogous except that $\\\\overline{F}_{1,2}(y,\\\\overline{y}\\\\,|x)$ have degree $N -\\\\overline{N} = -1$ .\"\n ],\n [\n \"Fermionic $H^2(\\\\sigma _-)$\",\n \"By the same arguments the fermionic $H^2$ is analogous to the bosonic one.\",\n \"Recall that the bosonic 2-cocycles are represented by three different two-forms: Weyl tensor, traceless and traceful parts of the generalized Einstein tensors (near diagonal, $G=3$ ).\",\n \"The fermionic Weyl cohomology is given by the same formula as the bosonic one: $W^\\\\text{ferm}(y,\\\\overline{y}\\\\,|x) = H^{\\\\mu \\\\nu }\\\\,\\\\partial _\\\\mu \\\\partial _\\\\nu C(y,0\\\\,|x) + \\\\overline{H}^{\\\\dot{\\\\mu }\\\\dot{\\\\nu }}\\\\,\\\\overline{\\\\partial }_{\\\\dot{\\\\mu }}\\\\overline{\\\\partial }_{\\\\dot{\\\\nu }}C(0,\\\\overline{y}\\\\,|x),$ where $C(y,0\\\\,|x)$ and $C(0,\\\\overline{y}\\\\,|x)$ are polynomials of $y$ and $\\\\overline{y}$ , respectively.\",\n \"The two bosonic Fronsdal cocycles (REF ) were represented by the two zero-forms $C^\\\\text{diag}(y,\\\\overline{y})$ with the support on the diagonal $N=\\\\overline{N}$ .\",\n \"In the fermionic case the two Fronsdal cocycles split into four.\",\n \"The bosonic diagonal polynomial $C^\\\\text{diag}(y,\\\\overline{y})$ is replaced by a pair of near-diagonal $C^\\\\text{near-diag}(y,\\\\overline{y})$ and $\\\\overline{C}^\\\\text{near-diag}(y,\\\\overline{y})$ satisfying the relations $\\\\left(y^\\\\alpha \\\\frac{\\\\partial }{\\\\partial y^\\\\alpha } - \\\\overline{y}^{\\\\dot{\\\\alpha }}\\\\frac{\\\\partial }{\\\\partial \\\\overline{y}^{\\\\dot{\\\\alpha }}}\\\\right) C^\\\\text{near-diag}(y,\\\\overline{y}\\\\,|x) &= C^\\\\text{near-diag}(y,\\\\overline{y}\\\\,|x),\\\\\\\\\\\\left(y^\\\\alpha \\\\frac{\\\\partial }{\\\\partial y^\\\\alpha } - \\\\overline{y}^{\\\\dot{\\\\alpha }}\\\\frac{\\\\partial }{\\\\partial \\\\overline{y}^{\\\\dot{\\\\alpha }}}\\\\right) \\\\overline{C}^\\\\text{near-diag}(y,\\\\overline{y}\\\\,|x) &= -\\\\overline{C}^\\\\text{near-diag}(y,\\\\overline{y}\\\\,|x)\\\\,$ These support the fermionic 2-cocycles associated with the ${\\\\it l.h.s.}\",\n \"$ 's of the fermionic field equations for spin $s\\\\ge 3/2$ massless fields as follows $\\\\mathcal {E}^\\\\text{ferm}_\\\\mathrm {A}(y,\\\\overline{y}\\\\,|x) &= \\\\Big (H^{\\\\mu \\\\nu }\\\\partial _\\\\mu \\\\partial _\\\\nu + \\\\overline{H}^{\\\\dot{\\\\mu }\\\\dot{\\\\nu }}\\\\overline{\\\\partial }_{\\\\dot{\\\\mu }}\\\\overline{\\\\partial }_{\\\\dot{\\\\nu }}\\\\Big ) C^\\\\text{near-diag}(y,\\\\overline{y}\\\\,|x),\\\\\\\\\\\\overline{\\\\mathcal {E}}^\\\\text{ferm}_\\\\mathrm {A}(y,\\\\overline{y}\\\\,|x) &= \\\\Big (H^{\\\\mu \\\\nu }\\\\partial _\\\\mu \\\\partial _\\\\nu + \\\\overline{H}^{\\\\dot{\\\\mu }\\\\dot{\\\\nu }}\\\\overline{\\\\partial }_{\\\\dot{\\\\mu }}\\\\overline{\\\\partial }_{\\\\dot{\\\\nu }}\\\\Big ) \\\\overline{C}^\\\\text{near-diag}(y,\\\\overline{y}\\\\,|x),\\\\\\\\\\\\mathcal {E}^\\\\text{ferm}_\\\\mathrm {B}(y,\\\\overline{y}\\\\,|x) &= \\\\Big (H^{\\\\mu \\\\nu } y_\\\\mu y_\\\\nu + \\\\overline{H}^{\\\\dot{\\\\mu }\\\\dot{\\\\nu }}\\\\overline{y}_{\\\\dot{\\\\mu }}\\\\overline{y}_{\\\\dot{\\\\nu }}\\\\Big ) C^\\\\text{near-diag}(y,\\\\overline{y}\\\\,|x),\\\\\\\\\\\\overline{\\\\mathcal {E}}^\\\\text{ferm}_\\\\mathrm {B}(y,\\\\overline{y}\\\\,|x) &= \\\\Big (H^{\\\\mu \\\\nu } y_\\\\mu y_\\\\nu + \\\\overline{H}^{\\\\dot{\\\\mu }\\\\dot{\\\\nu }}\\\\overline{y}_{\\\\dot{\\\\mu }}\\\\overline{y}_{\\\\dot{\\\\nu }}\\\\Big ) \\\\overline{C}^\\\\text{near-diag}(y,\\\\overline{y}\\\\,|x)\\\\,.$\"\n ],\n [\n \"Conclusion\",\n \"In this paper, free unfolded equations for massless HS fields are analyzed in detail in terms of $\\\\sigma _-$ cohomology.\",\n \"This is done both in flat space of arbitrary dimension in the tensor formalism for bosonic fields and in $AdS_4$ in the spinor formalism for both bosonic and fermionic fields.\",\n \"Not surprisingly, the final results agree with those stated long ago in the original papers [12], [15].\",\n \"Our aim is to present the detailed analysis of the $\\\\sigma _-$ cohomology providing an exhaustive proof of the so-called First On-Shell Theorem of the form of free unfolded HS equations, allowing the interested reader to check every step.\",\n \"In the tensor case the full set of cohomology groups $H^p(\\\\sigma _-)$ was found both for the groups $\\\\mathrm {GL}(d)$ and $\\\\mathrm {O}(d)$ .\",\n \"Our results for $\\\\mathrm {GL}(d)$ and $p<3$ coincide with those found in [16].\",\n \"For the $\\\\mathrm {O}(d)$ case of traceless fields lower cohomology groups matched against those in [16], [17], [18].\",\n \"In $AdS_4$ we used spinor formalism to analyze $H^{0,1,2}(\\\\sigma _-)$ for both bosonic and fermionic HS fields.\",\n \"To the best of our knowledge such analysis was not available in the literature.\",\n \"Practically, to compute $H^k(\\\\sigma _-)$ in both $\\\\mathrm {Mink}^d$ and $AdS_4$ cases we used the analogue of the Hodge theorem.\",\n \"Namely, the problem of finding the cohomology $H^k(\\\\sigma _-) = \\\\mathrm {ker}(\\\\sigma ^{k}_-)/\\\\mathrm {im}(\\\\sigma ^{k-1}_-)$ was reduced to the calculation of the kernel of an appropriate positive-definite Laplace-Hodge operator $\\\\Delta $ invariant under the action of compact version of the space-time symmetry algebra.\",\n \"This technique was shown to be lucid and efficient.\",\n \"Having found the cohomology groups $H^k(\\\\sigma _-)$ for $k=0,1,2$ , in accordance with [13] we obtained the exhaustive information about the differential HS gauge parameters, dynamical HS gauge fields and their field equations.\",\n \"Thus, we have explicitly proven the so-called First On-Shell Theorem for bosonic HS fields in $\\\\mathrm {Mink}^d$ (which case is straightforwardly extendable to $AdS_d$ ) and all massless fields in $AdS_4$ .\",\n \"The technique used in this paper can be further applied to the calculation of $H^p(\\\\sigma _-)$ in the zero-form sector of HS fields studied in [13], [22] that describes dynamics of a scalar field and $s=1$ particle as well as to more general systems considered in [44], [45], [46].\",\n \"One of the byproduct results of this paper is the interpretation of the matching between $\\\\sigma _-$ cohomology of the one-form sector against zero-form sector expressing the matching between Bianchi identities in the two sectors.\"\n ],\n [\n \"Acknowledgement\",\n \"We are grateful to Vyacheslav Didenko, Anatoly Korybut and Alexander Tarusov for helpful and stimulating discussions and Maxim Grigoriev for a useful comment.\",\n \"We are particularly grateful to the referee for useful suggestions and the correspondence.\",\n \"We acknowledge a partial support from the Russian Basic Research Foundation Grant No 20-02-00208.\",\n \"Since most of the tensors encountered in the course of this paper are Young tensors in the symmetric basis, it is convenient to accept the following notation.\",\n \"A tensor without a certain type of index symmetry will be denoted as $T^{a|b|c|..}$ , where the vertical line $|$ separates groups of indices not related by any symmetries to each other.\",\n \"A tensor that has a symmetric set of $n$ indices, say, $(a_1,a_2,\\\\dots ,a_n)$ will be denoted $T^{a(n)|...}\\\\equiv T^{(a_1a_2\\\\dots a_n)|...}$ .\",\n \"A tensor corresponding to a certain Young diagram in the symmetric basis then has the form: $T^{a(n), b(m), c(k),...}$ .\",\n \"Symmetrization over $n$ indices is performed by the formula $\\\\mathrm {Sym} = \\\\frac{1}{n!\",\n \"}\\\\sum _{\\\\text{all permutations}}$ .\",\n \"We will denote all symmetrized tensor indices by the same letter.\",\n \"For example, $T^{a(n)|a} \\\\equiv \\\\frac{1}{(n+1)!}\",\n \"\\\\sum _{\\\\sigma \\\\in S_{n+1}} T^{\\\\sigma (a_1..a_n|a_{n+1})}\\\\,.$ In Section we omit $Y,Z,\\\\theta $ and assume that all indices are contracted with the corresponding variables.\",\n \"The rules for raising and lowering $\\\\mathfrak {sl}(2)$ -indices are $A_\\\\alpha = A^\\\\beta \\\\epsilon _{\\\\beta \\\\alpha }, \\\\quad \\\\quad A^\\\\alpha = \\\\epsilon ^{\\\\alpha \\\\beta }A_\\\\beta , \\\\quad \\\\quad \\\\epsilon _{\\\\alpha \\\\beta }\\\\epsilon ^{\\\\gamma \\\\beta } = \\\\epsilon _\\\\alpha ^{\\\\ \\\\gamma } = \\\\delta _\\\\alpha ^{\\\\ \\\\gamma } = -\\\\epsilon ^\\\\gamma _{\\\\ \\\\alpha }$ with $\\\\epsilon _{\\\\alpha \\\\beta } = -\\\\epsilon _{\\\\beta \\\\alpha },\\\\quad \\\\quad \\\\quad \\\\epsilon _{12} = 1.$ 0.75em $\\\\alpha _1 = \\\\tfrac{(-3 + n) (-2 + n) (-1 + n) (-3 + d + 2 n)}{(-5 + d +m + n) (-4 + d + m + n) (-4 + d + 2 n) (d^2 + d (-8 + m + 3 n) -2 (-8 + m + m^2 + 7 n - 3 m n))},$ 0.75em $\\\\alpha _2 = \\\\tfrac{(-2 + n) (-1 + n)}{(-4 + d + m + n) (-4 + d + 2 n)},$ 0.75em $\\\\alpha _3 = -\\\\tfrac{2 (-2 + n) (-1 + n) (-3 + d + 2 n)}{(-4 + d + m +n) (-4 + d + 2 n) (d^2 + d (-8 + m + 3 n) -2 (-8 + m + m^2 + 7 n - 3 m n))},$ 0.75em $\\\\alpha _4 = \\\\tfrac{2(1 + m - n) (-2 + n) (-1 + n) (-3 + d + 2 n)}{(-5 +d + m + n) (-4 + d + m + n) (-4 + d + 2 n) (d^2 +d (-8 + m + 3 n) - 2 (-8 + m + m^2 + 7 n - 3 m n))},$ 0.75em $\\\\alpha _5 = \\\\tfrac{ (-1 + n) (-3 + d + m + n)}{(-4 + d + m + n) (-4 + d + 2 n)},$ 0.75em $\\\\alpha _6 = -\\\\tfrac{(-2 + d + 2 m) (-1 + n)}{(-4 + d + 2 n) (d^2 +d (-8 + m + 3 n) - 2 (-8 + m + m^2 + 7 n - 3 m n))},$ 0.75em $\\\\alpha _7 = \\\\tfrac{(-1 + n) (-3 + d + 2 n) (16 + d^2 - 7 m - 9 n +2 d (-4 + m + n) + (m + n)^2)}{(-5 + d + m + n) (-4 + d + m +n) (-4 + d + 2 n) (d^2 + d (-8 + m + 3 n) -2 (-8 + m + m^2 + 7 n - 3 m n))},$ 0.75em $\\\\alpha _8 = -\\\\tfrac{(-1 + n)}{(-4 + d + m + n)},$ 0.75em $\\\\alpha _9 = \\\\tfrac{2 (-1 + n) (-3 + d + 2 n)}{(-4 + d + m + n) (d^2 +d (-8 + m + 3 n) - 2 (-8 + m + m^2 + 7 n - 3 m n))},$ 0.75em $\\\\alpha _{10} = -\\\\tfrac{(-3 + d + 2 n)}{d^2 + d (-8 + m + 3 n) - 2 (-8 + m + m^2 + 7 n - 3 m n)},$ 0.75em $\\\\alpha _{11} = -\\\\tfrac{(-4 + d + 2 m) (-1 + n) (-3 + d + 2 n)}{(-5 + d + m + n) (-4 + d + m + n) (d^2 + d (-8 + m + 3 n) - 2 (-8 + m + m^2 + 7 n - 3 m n))},$ 0.75em $\\\\alpha _{12} = \\\\tfrac{(14 + d^2 - 6 n + 2 m (-5 + m + n) +d (-8 + 3 m + n))}{(-4 + d + m + n) (d^2 + d (-8 + m + 3 n) -2 (-8 + m + m^2 + 7 n - 3 m n))}\\\\,.$\"\n ]\n]"},"id":{"kind":"string","value":"2107.01736"}}},{"rowIdx":1577,"cells":{"text":{"kind":"list like","value":[["Linear-Time Model Checking Branching Processes"],["Abstract (Multi-type) branching processes are a natural and well-studied model for generating random infinite trees.","Branching processes feature both nondeterministic and probabilistic branching, generalizing both transition systems and Markov chains (but not generally Markov decision processes).","We study the complexity of model checking branching processes against linear-time omega-regular specifications: is it the case almost surely that every branch of a tree randomly generated by the branching process satisfies the omega-regular specification?","The main result is that for LTL specifications this problem is in PSPACE, subsuming classical results for transition systems and Markov chains, respectively.","The underlying general model-checking algorithm is based on the automata-theoretic approach, using unambiguous B\\\"uchi automata."],["Introduction","Checking whether a (labelled) transition system satisfies a linear-time specification is a staple in verification.","The specification is often given as a formula of linear temporal logic (LTL).","While early procedures for LTL model checking work directly with the formula [24], the automata-theoretic approach translates LTL formulas into finite automata on infinite words, such as Büchi automata, and analyzes a product of the system and the automaton [36].","This approach can lead to clean and modular model-checking algorithms.","Although LTL captures only a subset of $\\omega $ -regular languages, model-checking algorithms based on the automata-theoretic approach can be made optimal from the point of view of computational complexity.","In particular, model checking finite transition systems against LTL specifications is PSPACE-complete [31], and the algorithm [36] that, loosely speaking, translates (the negation of) the LTL formula into a Büchi automaton and checks the product with the transition system for emptiness can indeed be implemented in PSPACE.","The same approach does not directly work for probabilistic systems modelled as finite Markov chains: intuitively, the nondeterminism in a Büchi automaton causes issues in a stochastic setting where the specification should hold with probability 1, i.e., almost surely but not necessarily surely.","A possible remedy is to translate the nondeterministic Büchi automaton further into a deterministic automaton, e.g., a deterministic Rabin automaton (deterministic Büchi automata are less expressive), with which the Markov chain can be naturally instrumented and subsequently analyzed.","This determinization step causes a (second) exponential blowup and does not lead to algorithms that are optimal from a computational-complexity point of view.","However, for Markov decision processes (MDPs), which allow for nondeterminism in the probabilistic system, this approach is adequate and leads to an optimal, double-exponential time, model-checking algorithm.","Checking whether a Markov chain satisfies an LTL specification with probability 1 is PSPACE-complete, but membership in PSPACE was proved only in [10], [11], not using the automata-theoretic approach but by a recursive procedure on the formula.","This raised the question if there is also an optimal algorithm based on the automata-theoretic approach; see [35] for a survey of the state of the art at the end of the 90s.","The answer is yes and was first given in [12], using a single-exponential translation from LTL to separated Büchi automata.","Such automata are special unambiguous Büchi automata, which restrict nondeterministic Büchi automata by requiring that every word have at most one accepting run.","Another algorithm, using alternating Büchi automata, was proposed in [6], exploiting reverse determinism, a property also related to unambiguousness.","A polynomial-time (even NC) model-checking algorithm for Markov chains against general unambiguous Büchi automata was given in [2].","These works all imply optimal PSPACE algorithms for LTL model checking of Markov chains via the automata-theoretic approach.","In this paper we exhibit an LTL model checking algorithm that has the following features: (1) it applies to (multi-type) branching processes, a well established model for random trees, generalizing both nondeterministic transition systems and Markov chains; (2) it runs in PSPACE, which is the optimal complexity both for nondeterministic transition systems and Markov chains; and (3) it is based on the automata-theoretic approach (using unambiguous Büchi automata).","The fact that there exists an algorithm with the first two features might seem surprising, as one might think that any system model that encompasses both nondeterminism and probability will generalize MDPs, for which LTL model checking is 2EXPTIME-complete [11].","Branching processes (BPs) are a well-studied model in mathematics with applications in numerous fields including biology, physics and natural language processing; see, e.g., [23], [1], [22].","BPs randomly generate infinite trees, and, from a computer-science point of view, they might be the most natural model to do so: (multi-type) BPs can be thought of as a version of stochastic context-free grammars without terminal symbols, randomly generating infinite derivation trees.","For example, consider the following BP, taken from [8], with 3 types $I, B, D$ : $&I {0.9} I && B {0.2} D && D {1} D \\nonumber \\\\[-1mm]&I {0.1} I B && B {0.5} B \\\\[-1mm]& && B {0.3} B B \\nonumber $ This BP might generate a tree with the following prefix: [yscale=0.8] 0) at (0,0) $I$ ; 1) at (-2,-1) $I$ ; 2) at (2,-1) $B$ ; 11) at (-2,-2) $I$ ; 111) at (-3,-3) $I$ ; 112) at (-1,-3) $B$ ; 21) at (1,-2) $B$ ; 22) at (3,-2) $B$ ; 211) at (1,-3) $B$ ; 221) at (3,-3) $D$ ; (0)–(1); (0)–(2); (1)–(11); (11)–(111); (11)–(112); (2)–(21); (2)–(22); (21)–(211); (22)–(221); [fill] (-3,-3.4) circle (0.7pt); [fill] (-3,-3.6) circle (0.7pt); [fill] (-3,-3.8) circle (0.7pt); [fill] (-1,-3.4) circle (0.7pt); [fill] (-1,-3.6) circle (0.7pt); [fill] (-1,-3.8) circle (0.7pt); [fill] (1,-3.4) circle (0.7pt); [fill] (1,-3.6) circle (0.7pt); [fill] (1,-3.8) circle (0.7pt); [fill] (3,-3.4) circle (0.7pt); [fill] (3,-3.6) circle (0.7pt); [fill] (3,-3.8) circle (0.7pt); The probability that the BP generates a tree with the shown prefix is the product of the probabilities of the fired transition rules, i.e., (in breadth-first order) $0.1 \\cdot 0.9 \\cdot 0.3 \\cdot 0.1 \\cdot 0.5 \\cdot 0.2$ .","BPs generalize transition systems.","Consider the following transition system: MDPrand] (X) at (0,0) $X$ ; MDPrand] (Y) at (2,0) $Y$ ; [->] (-1,0) to (X); [->] (X) edge[bend left] (Y); [->] (Y) edge[bend left] (X); [->] (Y) edge[loop right,looseness=12] (Y); It is equivalent to the BP with $X {1} Y$ and $Y {1} X Y$ , which generates with probability 1 the following unique tree: [yscale=0.8] 0) at (0,0) $X$ ; 1) at (0,-1) $Y$ ; 2l) at (-2,-2) $X$ ; 2r) at ( 2,-2) $Y$ ; 3l) at (-2,-3) $Y$ ; 3rl) at (1,-3) $X$ ; 3rr) at (3,-3) $Y$ ; [fill] (-2,-3.4) circle (0.7pt); [fill] (-2,-3.6) circle (0.7pt); [fill] (-2,-3.8) circle (0.7pt); [fill] ( 1,-3.4) circle (0.7pt); [fill] ( 1,-3.6) circle (0.7pt); [fill] ( 1,-3.8) circle (0.7pt); [fill] ( 3,-3.4) circle (0.7pt); [fill] ( 3,-3.6) circle (0.7pt); [fill] ( 3,-3.8) circle (0.7pt); (0) – (1); (1) – (2l); (1) – (2r); (2l) – (3l); (2r) – (3rl); (2r) – (3rr); The branches of this unique tree are exactly the executions of the transition system.","As a consequence, any LTL formula holds on all executions of the transition system if and only if it holds (with probability 1) on all branches of the generated tree.","BPs also generalize Markov chains.","Consider the following Markov chain: MDPrand] (X) at (0,0) $X$ ; MDPrand] (Y) at (2,0) $Y$ ; [->] (-1,0) to (X); [->] (X) edge[bend left] node[above] 1 (Y); [->] (Y) edge[bend left] node[below] $0.3$ (X); [->] (Y) edge[loop right,looseness=12] node[right] $0.7$ (Y); It is equivalent to the BP with $X {1} Y$ and $Y {0.3} X$ and $Y {0.7} Y$ , which generates, with probabilities $0.3$ , $0.7 \\cdot 0.3$ , $0.7 \\cdot 0.7$ , respectively, the following prefixes of (degenerated) trees: [yscale=0.8] 00) at (0, 0) $X$ ; 10) at (2, 0) $X$ ; 20) at (4, 0) $X$ ; 01) at (0,-1) $Y$ ; 11) at (2,-1) $Y$ ; 21) at (4,-1) $Y$ ; 02) at (0,-2) $X$ ; 12) at (2,-2) $Y$ ; 22) at (4,-2) $Y$ ; 03) at (0,-3) $Y$ ; 13) at (2,-3) $X$ ; 23) at (4,-3) $Y$ ; [fill] (0,-3.4) circle (0.7pt); [fill] (0,-3.6) circle (0.7pt); [fill] (0,-3.8) circle (0.7pt); [fill] (2,-3.4) circle (0.7pt); [fill] (2,-3.6) circle (0.7pt); [fill] (2,-3.8) circle (0.7pt); [fill] (4,-3.4) circle (0.7pt); [fill] (4,-3.6) circle (0.7pt); [fill] (4,-3.8) circle(0.7pt); (00) – (01) – (02) – (03); (10) – (11) – (12) – (13); (20) – (21) – (22) – (23); Here, each possible “tree” has only a single branch, and the possible “trees” are distributed in the same way as the possible executions of the Markov chain.","As a consequence, any LTL formula holds with probability 1 on a random execution of the Markov chain if and only if it holds with probability 1 on the (single) branch of the generated tree.","Hence, both for the transition system and for the Markov chain, the respective model-checking question reduces to the BP model-checking problem which asks whether with probability 1 the property holds on all branches.","For LTL specifications, we refer to this BP model-checking problem as $\\mathbb {P}(\\textup {LTL})=1$ .","Our main result is that it is in PSPACE, generalizing the corresponding classical results on transition systems and Markov chains.","As mentioned, our model-checking algorithm is based on the automata-theoretic approach, in particular on unambiguous Büchi automata.","Another important technical ingredient is the algorithmic analysis of certain nonnegative matrices in terms of their spectral radius.","The latter points to the fact that the numbers in the system generally matter, even though we only consider the qualitative problem of comparing the satisfaction probability with 1.","For example, for the BP given in (REF ), one can show that the probability that all branches eventually hit a node of type $D$ is less than 1 (in fact, it is 0).","Intuitively, this is because the probability of “branching” via $B {0.3} B B$ is larger than the probability of “dying” via $B {0.2} D$ .","Were the probabilities $0.3$ and $0.2$ swapped, the probability that all branches eventually hit a node of type $D$ would be 1; cf. [8].","We also consider the problem $\\mathbb {P}(\\textup {LTL}= 0)$ , which asks whether the probability that all branches satisfy a given LTL formula is 0.","Even though it is trivial to negate an LTL formula, this problem is (unlike in Markov chains) not equivalent to the complement of $\\mathbb {P}(\\textup {LTL}= 1)$ , because even when the probability is less than 1 that the formula holds on all branches, the probability may still be 0 that the negated formula holds on all branches.","We will show that $\\mathbb {P}(\\textup {LTL}= 0)$ is much more computationally complex than $\\mathbb {P}(\\textup {LTL}=1)$ : it is 2EXPTIME-complete.","Besides LTL, we also consider automata-based specifications.","Büchi automata are relevant from a verification point of view, as a way of specifying desired or undesired executions of the system.","Unambiguous Büchi automata are useful from a technical point of view, in particular, to facilitate our main result on $\\mathbb {P}(\\textup {LTL}=1)$ .","See sec:prelims,tab:map for definitions of our problems and a map of our results.","Readers familiar with MDPs may wonder how the problem $\\mathbb {P}(\\textup {LTL})=1$ can have lower computational complexity than the problem whether all schedulers of an MDP satisfy an LTL specification almost surely.","Consider the BP $X &{1} Y_1 Y_2 & \\qquad Y_1 &{0.7} X & \\qquad Y_1 &{0.3} Z & \\qquad Y_2 &{0.5} X & \\qquad Y_2 &{0.5} Z& \\qquad Z &{1} Z\\,,$ which might be depicted graphically as follows: MDPrand] (X) at (0,0) $X$ ; MDPrand] (Y1) at (3,1) $Y_1$ ; MDPrand] (Y2) at (3,-1) $Y_2$ ; MDPrand] (Z) at (6,0) $Z$ ; [->] (-1,0) to (X); [->] (X) edge[bend left=20] (Y1); [->] (X) edge[bend right=20] (Y2); [->] (Y1) edge[bend left=0] node[pos=0.4,below] $0.7$ (X); [->] (Y2) edge[bend right=0] node[pos=0.4,above] $0.5$ (X); [->] (Y1) edge node[pos=0.4,below] $0.3$ (Z); [->] (Y2) edge node[pos=0.4,above] $0.5$ (Z); [->] (Z) edge[loop right,looseness=12] node[right] 1 (Z); One might view this BP as an MDP where in an $X$ -node the scheduler nondeterministically picks either the $Y_1$ - or the $Y_2$ -successor, and in an $Y_i$ -node, the $X$ - or the $Z$ -successor is chosen randomly.","In such an MDP, regardless of the scheduler, a random run reaches with probability 1 a $Z$ -node.","However, in the BP above, the probability is positive that some branch of a random tree never reaches a $Z$ -node.","Although each branch of a random tree could be thought of as being witnessed by at least one scheduler, this is not a contradiction, as there are uncountably many schedulers (over which one cannot take a sum).","Hence, if an MDP is interpreted as a BP in the way sketched above, then the requirement that the BP satisfy an LTL formula almost surely on all branches is stronger, and computationally less complex to check, than the requirement that the MDP satisfy, for each scheduler, the formula almost surely."],["Related work.","We have already discussed related work concerning model checking transition systems and Markov chains.","In addition to the mentioned applications of BPs in various fields, there has also been work on BPs in computer science, especially in the last 10 years.","This paper builds on [8], where specifications in terms of deterministic parity tree automata are considered.","The work [8] implies decidability of the problems considered in this paper and some basic upper complexity bounds.","For example, it is not hard to derive from [8] that $\\mathbb {P}(\\textup {LTL}=1)$ is in 2EXPTIME.","Lowering this to PSPACE is the main achievement of this paper.","A related strand of work considers regular tree languages; i.e., the specification is not in terms of a word automaton that is run on each branch but in terms of tree automata.","Even measurability is not easy to show in this case [20], and fundamental decidability questions around computing the measure have been answered positively only for subclasses of regular tree languages [25], [26].","Fundamental results on the complexity of algorithmically analyzing BPs have been obtained in [18].","Indeed, in sec:as-finite we build on and improve results from [18] on finiteness (more often called “extinction” in the literature) of BPs.","Another recent line of work considers extensions of BPs with nondeterminism, focusing on algorithmic questions about properties such as reachability.","Branching MDPs, which are BPs where a controller chooses actions to influence the evolution of the tree, have been investigated, e.g., in [16], [17].","Even branching games, featuring two adversarial controllers, have been studied recently [14].","The work [21] also considers BPs with “internal” nondeterminism (as opposed to the “external” nondeterminism manifested as branching in the generated tree), along with model-checking problems against the logic GPL.","This expressive, $\\mu $ -calculus based modal logic had been introduced in [9].","The system model therein, called reactive probabilistic labeled transition systems (RPLTSs), is essentially equivalent to BPs as considered in this paper.","BPs are related to models for probabilistic programs with recursion, such as Recursive Markov chains, for which model-checking problems have been studied in detail; see, in particular, [19].","Very loosely speaking, a run of a (“1-exit”) Recursive Markov chain can be viewed as a depth-first traversal of a tree generated by a BP.","Indeed, for a lower bound in the present paper (thm:NBA-0) we adapt a proof from [19].","However, most qualitative model-checking problems for Recursive Markov chains are EXPTIME-complete [19], and so many of the BP problems we study turn out to have different computational complexity.","As a key technical tool we use unambiguous Büchi automata, as recently proposed for Markov chains [2].","It is non-trivial to extend their use to random trees, as the branching behaviour of BPs interferes with the spectral-radius based analysis from [2].","One may view as the main technical insight of this paper that the limited nondeterminism in unambiguous automata can be combined with the tree branching of BPs, so that, in a sense, BP model checking reduces to comparing the spectral radius of a certain nonnegative matrix with 1 (prop:coUBA-1)."],["Preliminaries","Let $\\mathbb {N}$ and $\\mathbb {N}_0$ denote the set of positive and nonnegative integers, respectively.","For a finite set $\\Gamma $ , we write $\\Gamma ^*$ (resp., $\\Gamma ^+$ ) for the set of words (resp., nonempty words) over $\\Gamma $ ."],["Branching processes.","A (multi-type) branching process (BP) is a tuple $\\mathcal {B}= (\\Gamma , \\mathord {{}}, \\mathit {Prob}, X_0)$ , where $\\Gamma $ is a finite set of types, $\\mathord {{}} \\subseteq \\Gamma \\times \\Gamma ^+$ is a finite set of transition rules, $\\mathit {Prob}$ is a function assigning positive rational probabilities to transition rules so that for every $X\\in \\Gamma $ we have $\\sum _{X {} w} \\mathit {Prob}(X {} w)=1$ , and $X_0\\in \\Gamma $ is the start type.","We write $X {p} w$ to denote that $\\mathit {Prob}(X {} w) = p$ .","Given a BP $\\mathcal {B}$ and a type $X \\in \\Gamma $ we write $\\mathcal {B}[X]$ for the BP obtained from $\\mathcal {B}$ by making $X$ the start type.","For $X, Y \\in \\Gamma $ we call $Y$ a successor of $X$ if there is a rule $X {} u Y v$ for some $u, v \\in \\Gamma ^*$ .","A BP with $\\varepsilon $ -rules allowed relaxes the requirement $\\mathord {{}} \\subseteq \\Gamma \\times \\Gamma ^+$ to $\\mathord {{}} \\subseteq \\Gamma \\times \\Gamma ^*$ , i.e., there may be rules of the form $X {} \\varepsilon $ , where $\\varepsilon $ denotes the empty word.","In the following, we disallow $\\varepsilon $ -rules unless specified otherwise; but the definitions generalize in a natural way.","Fix a BP $\\mathcal {B}= (\\Gamma , \\mathord {{}}, \\mathit {Prob}, X_0)$ for the rest of the section.","Write $\\llbracket \\mathcal {B} \\rrbracket $ for the set of trees generated by $\\mathcal {B}$ ; i.e., $\\llbracket \\mathcal {B} \\rrbracket $ denotes the set of ordered $\\Gamma $ -labelled trees $t$ such that for each $X \\in \\Gamma $ and each $X$ -labelled node $v$ in $t$ , there is a rule $X {} X_1 \\cdots X_k$ , denoted by $\\mathit {rule}(v)$ , such that the $k$ ordered children of $v$ are labelled with $X_1, \\ldots , X_k$ , respectively.","We say a node has type $X \\in \\Gamma $ if the node is labelled with $X$ .","A finite prefix of a tree $t \\in \\llbracket \\mathcal {B} \\rrbracket $ is an ordered $\\Gamma $ -labelled finite tree obtained from $t$ by designating some nodes as leaves, and removing all their children, grandchildren, etc.","Write $\\llparenthesis \\mathcal {B} \\rrparenthesis $ for the set of finite prefixes of trees generated by $\\mathcal {B}$ .","For $t \\in \\llparenthesis \\mathcal {B} \\rrparenthesis $ write $t{\\downarrow } \\subseteq \\llbracket \\mathcal {B} \\rrbracket $ for the (“cylinder”) set of trees $t^{\\prime } \\in \\llbracket \\mathcal {B} \\rrbracket $ such that $t$ is a finite prefix of $t^{\\prime }$ .","For $X \\in \\Gamma $ write $\\llbracket \\mathcal {B} \\rrbracket _X \\subseteq \\llbracket \\mathcal {B} \\rrbracket $ and $\\llparenthesis \\mathcal {B} \\rrparenthesis _X \\subseteq \\llparenthesis \\mathcal {B} \\rrparenthesis $ for the subsets of trees whose root has type $X$ ; the trees in $\\llbracket \\mathcal {B} \\rrbracket _X$ are called $X$ -trees.","A branch of a tree $t$ is a sequence $v_0 v_1 \\cdots $ of nodes in $t$ , where $v_0$ is the root of $t$ and $v_{i+1}$ is a child of $v_i$ for all $i \\in \\mathbb {N}_0$ .","See [8] for equivalent, more formal tree-related definitions.","For each $X \\in \\Gamma $ we define the probability space $(\\llbracket \\mathcal {B} \\rrbracket _X,\\Sigma _X,\\mathbb {P}_X)$ , where $\\Sigma _X$ is the $\\sigma $ -algebra generated by $\\lbrace t{\\downarrow } \\mid t\\in \\llparenthesis \\mathcal {B} \\rrparenthesis _X\\rbrace $ , and $\\mathbb {P}_X$ is the probability measure generated by $\\mathbb {P}_X(t{\\downarrow }) := \\prod _{v} \\mathit {Prob}(\\mathit {rule}(v))$ for all $t \\in \\llparenthesis \\mathcal {B} \\rrparenthesis _X$ , where the product extends over all non-leaf nodes $v$ in $t$ .","This is analogous to the standard definition of the probability space of a Markov chain.","We may write $\\mathbb {P}_\\mathcal {B}$ for $\\mathbb {P}_{X_0}$ , omitting the subscript when $\\mathcal {B}$ is understood.","We often talk about events (i.e., measurable sets of trees) and their probability in text form.","For example, by saying “a $\\mathcal {B}$ -tree has with positive probability infinitely many nodes of type $X$ ” we mean that $\\mathbb {P}_\\mathcal {B}(E) > 0$ where $E \\subseteq \\llbracket \\mathcal {B} \\rrbracket _{X_0}$ is the set of $X_0$ -trees with infinitely many nodes of type $X$ .","We are particularly interested in sets of trees all whose branches (more precisely, their associated sequences of types) satisfy an $\\omega $ -regular linear-time property $L \\subseteq \\Gamma ^\\omega $ .","Given $L \\subseteq \\Gamma ^\\omega $ , we write $\\mathbb {P}_\\mathcal {B}(L)$ for the probability that all branches of a $\\mathcal {B}$ -tree satisfy $L$ .","Linear temporal logic (LTL) formulas specify linear-time properties; see, e.g., [33] for a definition of LTL.","An important example for us are formulas of the form $\\mathsf {F}T$ , where $T \\subseteq \\Gamma $ , which denotes the linear-time property $\\lbrace u X w \\mid u \\in \\Gamma ^*,\\ X \\in T,\\ w \\in \\Gamma ^\\omega \\rbrace $ .","Accordingly, $\\mathbb {P}_\\mathcal {B}(\\mathsf {F}T)$ denotes the probability that all branches of a $\\mathcal {B}$ -tree have a node whose type is in $T$ (equivalently, the probability that a $\\mathcal {B}$ -tree has a finite prefix all whose leaves have a type in $T$ ).","We use finite automata on infinite words over $\\Gamma $ , where $\\Gamma $ is the set of types of a BP.","We use deterministic parity automata (DPAs), deterministic Büchi automata (DBAs), nondeterministic Büchi automata (NBAs), and unambiguous Büchi automata (UBAs).","The definitions are standard; see, e.g., [33].","In the following we fix some terms and notation.","Let $\\mathcal {A}= (Q, \\Gamma , \\delta , Q_0, F)$ be an NBA, where $Q$ is a finite set of states, $\\Gamma $ is the alphabet, $\\delta \\subseteq Q \\times \\Gamma \\times Q$ is the transition relation, $Q_0 \\subseteq Q$ is the set of initial states, and $F \\subseteq Q$ is the set of accepting states.","We write $q \\xrightarrow{} r$ to denote that $(q,X,r) \\in \\delta $ .","A finite sequence $q_0 \\xrightarrow{} q_1 \\xrightarrow{} \\cdots \\xrightarrow{} q_n$ is called a path and can be summarized as $q_0 {\\;\\xrightarrow{}{}\\hspace{0.0pt}^{*}\\;} q_n$ .","An infinite sequence $q_0 \\xrightarrow{} q_1 \\xrightarrow{} \\cdots $ is called a run of $X_1 X_2 \\cdots $ .","We call the run accepting if $q_0 \\in Q_0$ and $q_i \\in F$ holds for infinitely many $q_i$ .","The NBA $\\mathcal {A}$ accepts (resp., rejects) an infinite word $w \\in \\Gamma ^\\omega $ if $w$ has (resp., does not have) an accepting run in $\\mathcal {A}$ .","The NBA $\\mathcal {A}$ is called an unambiguous Büchi automaton (UBA) if every $w \\in \\Gamma ^\\omega $ has at most one accepting run.","An automaton $\\mathcal {A}$ defines $\\omega $ -regular linear-time properties $\\lbrace w \\in \\Gamma ^\\omega \\mid \\mathcal {A}\\text{ accepts } w\\rbrace $ and $\\lbrace w \\in \\Gamma ^\\omega \\mid \\mathcal {A}\\text{ rejects } w\\rbrace $ .","In keeping with previous definitions, we write $\\mathbb {P}_\\mathcal {B}(\\mathcal {A}\\text{ accepts})$ (resp., $\\mathbb {P}_\\mathcal {B}(\\mathcal {A}\\text{ rejects})$ ) for the probability that all branches of a $\\mathcal {B}$ -tree (more precisely, their associated sequences of types) are accepted (resp., rejected) by $\\mathcal {A}$ .","We consider the following computational problems.","The problem $\\mathbb {P}(\\text{finite}) = 1$ asks, given a BP $\\mathcal {B}$ with $\\varepsilon $ -rules allowed, whether the probability that a $\\mathcal {B}$ -tree is finite is 1.","The problem $\\mathbb {P}(\\textup {LTL}) = 1$ asks, given a BP $\\mathcal {B}$ and an LTL formula $\\varphi $ , whether $\\mathbb {P}_\\mathcal {B}(\\varphi ) = 1$ .","The problems $\\mathbb {P}(\\textup {DPA}) = 1$ (resp., $\\mathbb {P}(\\textup {NBA}) = 1$ ) ask, given a BP $\\mathcal {B}$ and a DPA (resp., NBA) $\\mathcal {A}$ , whether $\\mathbb {P}_\\mathcal {B}(\\mathcal {A}\\text{ accepts}) = 1$ .","The problems $\\mathbb {P}(\\textup {coNBA}) = 1$ (resp., $\\mathbb {P}(\\textup {coUBA}) = 1$ )We do not explicitly define or use a notion of “co-Büchi automata” to avoid possible confusion about accepting/rejecting.","If one were to do so, one would define a “co-NBA” $\\mathcal {A}$ like an NBA $\\mathcal {A}$ , but the “co-NBA” $\\mathcal {A}$ would accept a word $w \\in \\Gamma ^\\omega $ if and only if $\\mathcal {A}$ viewed as an NBA rejects $w$ .","Similarly for “co-UBAs”.","ask, given a BP $\\mathcal {B}$ and an NBA (resp., UBA) $\\mathcal {A}$ , whether $\\mathbb {P}_\\mathcal {B}(\\mathcal {A}\\text{ rejects}) = 1$ .","The problems $\\mathbb {P}(\\textup {LTL}) = 0, \\mathbb {P}(\\textup {DPA}) = 0, \\ldots $ are defined similarly, where “${=}\\,1$ ” is replaced with “${=}\\,0$ ”.","Table: Results and organization of the paper.The complexity classes indicate completeness results, except “in NC”, which only means membership in NC.See tab:map for a map of our results in those terms, as well as for an overview of the rest of the paper.","As explained in the introduction, the problem $\\mathbb {P}(\\textup {LTL})=1$ is of particular interest from a model-checking point of view, and the technically most challenging one.","In addition to standard complexity classes between P and 2EXPTIME, we use the class NC, the subclass of P comprising those problems solvable in polylogarithmic time by a parallel random-access machine using polynomially many processors; see, e.g., [27].","To prove membership in PSPACE in a modular way, we will use the following pattern: Let $P_1, P_2$ be two problems, where $P_2$ is in NC.","Suppose there is a reduction from $P_1$ to $P_2$ implemented by a PSPACE transducer, i.e., a Turing machine whose work tape (but not necessarily its output tape) is PSPACE-bounded.","Then $P_1$ is in PSPACE.","Note that the output of the transducer is (at most) exponential.","Problems in NC can be decided in polylogarithmic space [4].","Using standard techniques for composing space-bounded transducers (see, e.g., [27]), it follows that $P_1$ is in PSPACE.","We use finite sets $S$ to index matrices $M \\in \\mathbb {R}^{S \\times S}$ and vectors $v \\in \\mathbb {R}^S$ .","The graph of a nonnegative matrix $M \\in [0, \\infty )^{S \\times S}$ is the directed graph $(S,E)$ with $E = \\lbrace (s,t) \\in S \\times S \\mid M_{s,t} > 0\\rbrace $ .","The spectral radius of a matrix is the largest absolute value of its eigenvalues.","The following lemma allows to efficiently compare the spectral radius of a nonnegative matrix with 1.","Given a nonnegative rational matrix $M$ , one can determine in NC whether $\\rho < 1$ or $\\rho = 1$ or $\\rho > 1$ , where $\\rho $ denotes the spectral radius of $M$ .","Use the algorithm from [13], but not with Gaussian elimination as suggested there, but by solving the systems of linear equations described in [13] in NC.","The latter is possible in NC [5]."],["Basic Results","In this section we develop the more basic results indicated in tab:map, on finiteness (sec:as-finite), deterministic parity automata (sec:DPA), and Büchi automata (sec:NBA), on the one hand rounding off the complexity map in tab:map, and on the other hand building the foundation for more challenging results in the following sections.","In particular, prop:as-finite-NC is indirectly used throughout the paper."],["Finiteness","In this section we consider BPs with $\\varepsilon $ -rules allowed, i.e., rules of the form $X {} \\varepsilon $ .","Such BPs may generate finite trees.","We are interested in the almost-sure finiteness problem, also denoted as $\\mathbb {P}(\\text{finite})=1$ , i.e., the problem whether the probability that a given BP with $\\varepsilon $ -rules allowed generates a finite tree is equal to 1.","In prop:as-finite-NC below we show that this problem is in NC.","All upper bounds on the complexity of $\\mathbb {P}(\\cdot ) = 1$ problems in this paper build directly or indirectly on this result.","While the almost-sure finiteness (or “extinction”) problem has often been studied and is known to be in (strongly) polynomial time [18], [13], its membership in NC is, to the best of the authors' knowledge, new.","For instance, since linear programming is P-complete, one cannot use linear programming (as in [18]) to show membership in NC.","Nor can one directly use the strongly polynomial-time algorithm of [13], as it computes, in a sub-procedure, the set of types $X$ for which there exists a finite $X$ -tree.","But the latter problem is P-complete.","For the rest of the section, fix a BP $\\mathcal {B}= (\\Gamma , \\mathord {{}}, \\mathit {Prob}, X_0)$ with $\\varepsilon $ -rules allowed.","Define a directed graph $G = (\\Gamma ,E)$ (i.e., the types of $\\mathcal {B}$ are the vertices of $G$ ) with an edge $(X,Y) \\in E$ if and only if $Y$ is a successor of $X$ (i.e., there is a rule $X {} u Y v$ for some $u,v \\in \\Gamma ^*$ ).","Given a strongly connected component (SCC) $S \\subseteq \\Gamma $ of $G$ and $X \\in S$ , define a BP $\\mathcal {B}[S,X] = (S,\\mathord {{}_S},\\mathit {Prob}_S,X)$ obtained from $\\mathcal {B}$ by restricting the types to $S$ and deleting on all right-hand sides of the rules those types not in $S$ .","The following lemma is straightforward: A $\\mathcal {B}$ -tree is infinite with positive probability if and only if there exist an SCC $S \\subseteq \\Gamma $ of $G$ and $X \\in S$ such that $X$ is reachable from $X_0$ in $G$ and a $\\mathcal {B}[S,X]$ -tree is infinite with positive probability.","Let $M \\in \\mathbb {Q}^{\\Gamma \\times \\Gamma }$ be the nonnegative $\\Gamma \\times \\Gamma $ -matrix with $M_{X,Y} = \\sum _{X {p} w} p |w|_Y$ , where $|w|_Y \\in \\mathbb {N}_0$ is the number of occurrences of $Y$ in $w$ .","That is, $M_{X,Y}$ is the expected number of direct $Y$ -successors of the root of a $\\mathcal {B}[X]$ -tree.","By induction, $M^i$ , the $i$ th power of $M$ , is such that $(M^i)_{X,Y}$ is the expected number of $Y$ -nodes that are exactly $i$ levels under the root of a $\\mathcal {B}[X]$ -tree.","The graph of $M$ is exactly the previously defined graph $G$ .","Let $S \\subseteq \\Gamma $ be an SCC of $G$ .","Denote by $M_S \\in \\mathbb {Q}^{S \\times S}$ the (square) principal submatrix obtained from $M$ by restricting it to the rows and columns indexed by elements of $S$ .","Let $\\rho _S$ denote the spectral radius of $M_S$ .","Call $S$ supercritical if $\\rho _S > 1$ .","Call $S$ linear if for all rules $X {} w$ with $X \\in S$ there is exactly one occurrence in $w$ of a type in $S$ .","Observe that if $S$ is linear then $M_S$ is stochastic, i.e., $M_S \\vec{1} = \\vec{1}$ where $\\vec{1}$ is the all-1 vector, i.e., the element of $\\lbrace 1\\rbrace ^S$ .","In that case, by the Perron-Frobenius theorem [3], we have $\\rho _S = 1$ and, thus, $S$ is not supercritical.","The following characterization can be proved using [13] (which builds on [18]): lemmalemasfinitenesschar A $\\mathcal {B}$ -tree is infinite with positive probability if and only if there exist an SCC $S \\subseteq \\Gamma $ of $G$ and $X \\in S$ such that $X$ is reachable from $X_0$ in $G$ and $S$ is supercritical or linear.","It follows: propositionpropasfiniteNC The problem $\\mathbb {P}(\\text{finite})=1$ is in NC."],["Deterministic Parity Automata","In this section we consider deterministic parity automata (DPAs) on words.","In [8] it was shown that the problem $\\mathbb {P}(\\textup {DPA}) = 1$ can be decided in polynomial time.","We improve this to membership in NC.","By the following lemma we can check in NC whether a $\\mathcal {B}$ -tree almost surely has a finite prefix all whose leaves have types in a given set $T$ .","The proof is by reduction to almost-sure finiteness.","lemmalemAFTone Given a BP $\\mathcal {B}= (\\Gamma , \\mathord {{}}, \\mathit {Prob}, X_0)$ and a set of types $T \\subseteq \\Gamma $ , the problem whether $\\mathbb {P}_{X_0}(\\mathsf {F}T) = 1$ is in NC.","By combining lem:AFT-1 with results from [8] we obtain: theoremthmDPAone The problem $\\mathbb {P}(\\textup {DPA}) = 1$ is in NC.","The hardness result in the following theorem highlights the different complexities of $\\mathbb {P}(\\cdot ) = 0$ and $\\mathbb {P}(\\cdot ) = 1$ problems in this paper.","theoremthmDPAzero The problem $\\mathbb {P}(\\textup {DPA}) = 0$ is P-complete.","It is P-hard even for deterministic Büchi automata with two states, the accepting state being a sink."],["Büchi Automata","The problem $\\mathbb {P}(\\textup {NBA}) = 1$ is PSPACE-complete.","PSPACE-hardness is immediate in two different ways.","It follows from the PSPACE-hardness of model checking Markov chains against NBAs [34].","It also follows from the PSPACE-hardness of model checking transition systems against NBAs.","(The latter follows easily from the PSPACE-hardness of NBA universality [31].)","Both model-checking problems are special cases of $\\mathbb {P}(\\textup {NBA}) = 1$ .","Towards membership in PSPACE, we use a translation from NBA to DPA [28].","This translation causes an exponential blow-up, but an inspection of the construction [28] reveals that it can be computed by a PSPACE transducer.","By thm:DPA-1 the problem $\\mathbb {P}(\\textup {DPA}) = 1$ is in NC.","By lem:PSPACE-transducer it follows that $\\mathbb {P}(\\textup {NBA}) = 1$ is in PSPACE.","theoremthmNBAzero The problem $\\mathbb {P}(\\textup {NBA}) = 0$ is EXPTIME-complete.","It is EXPTIME-hard even for NBAs whose only accepting state is a sink.","Towards membership in EXPTIME, an NBA can be translated, in exponential time, to a DPA of exponential size; see, e.g., [28].","Since $\\mathbb {P}(\\textup {DPA}) = 0$ is in P by thm:DPA-0, it follows that $\\mathbb {P}(\\textup {NBA}) = 0$ is in EXPTIME.","Concerning EXPTIME-hardness, we adapt the proof (in the online appendix) of [19] on model checking recursive Markov chains against NBAs.","The details are in app:NBA-0."],["Co-Büchi Automata","In this section we consider the problem $\\mathbb {P}(\\textup {coNBA}) = 1$ , which asks, given a BP $\\mathcal {B}$ and a Büchi automaton $\\mathcal {A}$ , whether $\\mathcal {B}$ almost surely generates a tree whose branches are all rejected by $\\mathcal {A}$ ; i.e., whether $\\mathbb {P}_{\\mathcal {B}}(\\mathcal {A}\\text{ rejects}) = 1$ .","Dually, one might ask whether the probability is positive that a $\\mathcal {B}$ -tree has a branch accepted by $\\mathcal {A}$ .","Intuitively, we view the Büchi automaton $\\mathcal {A}$ as specifying “bad” branches, and we would like the tree almost surely not to have any bad branches.","This problem is in PSPACE, which can be shown via a translation to DPAs, as in thm:NBA-1.","However, with a view on the following sections, in particular on LTL specifications, we pursue a different approach to the problem $\\mathbb {P}(\\textup {coNBA}) = 1$ .","In this section we lay the groundwork for arbitrary Büchi automata $\\mathcal {A}$ .","By building on these results, we will show in the next section that if $\\mathcal {A}$ is unambiguous then the problem is in NC, which will allow us to derive our headline result, namely that $\\mathbb {P}(\\textup {LTL})=1$ is in PSPACE.","Let $\\mathcal {B}= (\\Gamma , \\mathord {{}}, \\mathit {Prob}, X_0)$ be a BP and $\\mathcal {A}= (Q, \\Gamma , \\delta , Q_0, F)$ a (not necessarily unambiguous) Büchi automaton.","Define a Büchi automaton, $\\mathcal {A}\\times \\mathcal {B}$ , by $\\mathcal {A}\\times \\mathcal {B}:= (Q \\times \\Gamma , \\Gamma , \\delta _{\\mathcal {A}\\times \\mathcal {B}}, Q_0 \\times \\lbrace X_0\\rbrace , F \\times \\Gamma )$ , where $&\\delta _{\\mathcal {A}\\times \\mathcal {B}}((q_1, X_1), X_2)={\\left\\lbrace \\begin{array}{ll}\\delta (q_1, X_1) \\times \\lbrace X_2\\rbrace & \\text{if $X_2$ is a successor of~$X_1$}\\\\\\emptyset & \\text{otherwise.}\\end{array}\\right.","}$ The remainder of the section is organized as follows.","In sub:UBA-X1-f we show that the problem $\\mathbb {P}(\\textup {coNBA}) = 1$ reduces to the analysis of certain SCCs within $\\mathcal {A}\\times \\mathcal {B}$ .","In sub:Bdet we introduce a key lemma, lem:Bdet, which allows us to “forget” about the distinction between accepting and non-accepting states: the lemma reduces $\\mathbb {P}(\\textup {coNBA}) = 1$ to a pure reachability problem in an exponential-sized BP, $\\mathcal {B}_{\\mathit {det}}$ .","This leads us to prove PSPACE-completeness of $\\mathbb {P}(\\textup {coNBA}) = 1$ , but more importantly, lem:Bdet plays a key role in the rest of the paper.","We prove it in sub:Bdet-proof."],["The Automaton $\\mathcal {A}[f,X_f]$","For any $(f,X_f) \\in F \\times \\Gamma $ on a cycle of the transition graph of $\\mathcal {A}\\times \\mathcal {B}$ , define the Büchi automaton $\\mathcal {A}[f,X_f] \\ := \\ (\\lbrace \\bar{q}_0\\rbrace \\cup Q[f,X_f], \\Gamma , \\delta [f,X_f], \\lbrace \\bar{q}_0\\rbrace , \\lbrace (f,X_f)\\rbrace )$ as the Büchi automaton obtained from $\\mathcal {A}\\times \\mathcal {B}$ by making $(f,X_f)$ the only accepting state, restricting the set of states, $Q[f,X_f] \\subseteq Q \\times \\Gamma $ , to those $(q,X)$ that, in the transition graph of $\\mathcal {A}\\times \\mathcal {B}$ , are reachable from $(f,X_f)$ and can reach $(f,X_f)$ , i.e., those $(q,X)$ in the SCC containing $(f, X_f)$ , restricting the transition function $\\delta [f,X_f]$ accordingly, i.e., $\\delta [f,X_f]((q,X),Y) \\ := \\ \\delta _{\\mathcal {A}\\times \\mathcal {B}}((q,X),Y) \\cap Q[f,X_f]\\,,$ making $\\bar{q}_0$ the only initial state, and setting $\\delta [f,X_f](\\bar{q}_0,X_f) := \\lbrace (f,X_f)\\rbrace $ and $\\delta [f,X_f](\\bar{q}_0,X) := \\emptyset $ for all $X \\in \\Gamma \\setminus \\lbrace X_f\\rbrace $ .","The following lemma follows from the pigeonhole principle and basic probability arguments:lemmalemUBAXonef The probability that some branch of a $\\mathcal {B}$ -tree is accepted by $\\mathcal {A}$ is positive if and only if there are $q_0 \\in Q_0$ and $f \\in F$ and $X_f \\in \\Gamma $ such that $(f, X_f)$ is reachable from $(q_0, X_0)$ in the transition graph of $\\mathcal {A}\\times \\mathcal {B}$ and the probability that some branch of a $\\mathcal {B}[X_f]$ -tree is accepted by $\\mathcal {A}[f,X_f]$ is positive.","For the rest of the section let $(f,X_f) \\in F \\times \\Gamma $ be on a cycle of the transition graph of $\\mathcal {A}\\times \\mathcal {B}$ ."],["The Determinization $\\mathcal {A}_{\\mathit {det}}$ and the BP {{formula:82465720-6fff-453e-a5bf-44f8cad75b0e}}","Let $\\mathcal {A}_{\\mathit {det}}:= (2^{\\lbrace \\bar{q}_0\\rbrace \\cup Q[f,X_f]}, \\Gamma , \\delta _{\\mathit {det}}, \\lbrace \\bar{q}_0\\rbrace , 2^{\\lbrace \\bar{q}_0\\rbrace \\cup Q[f,X_f]} \\setminus \\lbrace \\emptyset \\rbrace )$ be the determinization of $\\mathcal {A}[f,X_f]$ obtained by the standard subset construction.","Which states are accepting will not actually be relevant.","Note that every state reachable via a nonempty path from $\\lbrace \\bar{q}_0\\rbrace $ is of the form $P \\times \\lbrace X\\rbrace $ with $P \\subseteq Q$ and $X \\in \\Gamma $ .","Define a BP $\\mathcal {B}_{\\mathit {det}}$ based on $\\mathcal {A}_{\\mathit {det}}$ as $\\mathcal {B}_{\\mathit {det}}\\ := \\ (\\Gamma ^{\\prime }, \\mathord {{}^{\\prime }}, \\mathit {Prob}^{\\prime }, \\lbrace (f,X_f)\\rbrace )\\,,$ where the set of types $\\Gamma ^{\\prime } \\subseteq 2^{Q[f,X_f]}$ is the set of those states in $\\mathcal {A}_{\\mathit {det}}$ that are reachable (in $\\mathcal {A}_{\\mathit {det}}$ ) from $\\lbrace \\bar{q_0}\\rbrace $ via a nonempty path (recall that they are of the form $P \\times \\lbrace X\\rbrace $ with $P \\subseteq Q$ and $X \\in \\Gamma $ ), and $X^{\\prime } &{\\;{p}{}\\hspace{0.0pt}{^{\\prime }}\\;} \\delta _{\\mathit {det}}(X^{\\prime },X_1) \\cdots \\delta _{\\mathit {det}}(X^{\\prime },X_k)$ for all $X^{\\prime } = P \\times \\lbrace X\\rbrace \\in \\Gamma ^{\\prime }$ with $P \\ne \\emptyset $ and all $X {p} X_1 \\cdots X_k$ , and $\\emptyset {\\;{1}{}\\hspace{0.0pt}{^{\\prime }}\\;} \\emptyset $ .","Here is the key lemma of this section: lemmalemBdet The following statements are equivalent: The probability that some branch of a $\\mathcal {B}[X_f]$ -tree is accepted by $\\mathcal {A}[f,X_f]$ is positive.","The probability that some branch of a $\\mathcal {B}_{\\mathit {det}}$ -tree does not have any nodes of type $\\emptyset $ is positive.","We prove lem:Bdet in sub:Bdet-proof.","It will be used in the proof of thm:coNBA-1 below; but more importantly, lem:Bdet is the foundation of sec:coUBA.","Given that lem:Bdet reflects the key insight of this section, let us comment further.","Considering that condition (ii) does not mention a notion of acceptance, one might have two concerns at this point: Condition (ii) does not obviously imply that with positive probability there is even a branch with infinitely many nodes of types containing $(f,X_f)$ .","Even if with positive probability there is such a branch, it is not obvious that such branches would necessarily correspond to branches of $\\mathcal {B}[X_f]$ that are accepted by $\\mathcal {A}[f,X_f]$ .","Even for the special case of Markov chains (i.e., every tree has only a single branch), lem:Bdet is not at all obvious, and both concerns (a) and (b) apply.","Indeed, for Markov chains, Courcoubetis and Yannakakis prove a statement related to lem:Bdet, namely [11], with a proof related to ours and dealing explicitly with concern (b) above.","For the special case of transition systems (i.e., the BP generates exactly one tree), lem:Bdet is simple though: consider the branch that follows a cycle around $(f,X_f)$ .","For the general case, we need a result on BPs from [8], dealing with concern (a) above.","The high-level principle behind the proof of lem:Bdet is often used in the analysis of Markov chains: if it is possible, infinitely often, to reach a state with a probability bounded away from 0, then this state is almost surely reached infinitely often.","See sub:Bdet-proof for a full proof of lem:Bdet.","We can now derive a PSPACE procedure for the problem $\\mathbb {P}(\\textup {coNBA}) = 1$ without resorting to DPAs: theoremthmcoNBAone The problem $\\mathbb {P}(\\textup {coNBA}) = 1$ is PSPACE-complete.","thm:NBA-0 (for NBAs) has a coNBA-analogue: theoremthmcoNBAzero The problem $\\mathbb {P}(\\textup {coNBA}) = 0$ is EXPTIME-complete.","It is EXPTIME-hard even for NBAs all whose states are accepting."],["Co-Unambiguous Büchi Automata","In this section we build on the previous section, in particular on lem:Bdet, to derive our main technical result: given a BP $\\mathcal {B}$ and an unambiguous Büchi automaton (UBA) $\\mathcal {A}$ , one can decide in NC whether $\\mathcal {B}$ almost surely generates a tree all whose branches are rejected by $\\mathcal {A}$ : The problem $\\mathbb {P}(\\textup {coUBA}) = 1$ is in NC.","The rest of the section is devoted to the proof of this theorem.","Fix a BP $\\mathcal {B}$ and a UBA $\\mathcal {A}$ .","Since NC is closed under complement, we can focus on the problem whether the probability is positive that a $\\mathcal {B}$ -tree has some branch accepted by $\\mathcal {A}$ .","We use lem:UBA-X1-f.","Since reachability in a graph is in NL and, hence, in NC, it suffices to decide in NC whether the probability that some branch of a $\\mathcal {B}[X_f]$ -tree is accepted by $\\mathcal {A}[f,X_f]$ is positive.","By lem:Bdet it suffices to decide in NC whether the probability that some branch of a $\\mathcal {B}_{\\mathit {det}}$ -tree does not have any nodes of type $\\emptyset $ is positive.","The challenge is that $\\mathcal {B}_{\\mathit {det}}$ may be exponentially larger than $\\mathcal {A}$ , so we need to exploit the unambiguousness of $\\mathcal {A}$ and the regular structure it gives to $\\mathcal {B}_{\\mathit {det}}$ .","Let $\\mathcal {B}_{\\mathit {det}}^{\\prime \\prime }$ be the BP (with $\\varepsilon $ -rules allowed) obtained from $\\mathcal {B}_{\\mathit {det}}$ by removing the type $\\emptyset $ and eliminating all occurrences of type $\\emptyset $ from all right-hand sides.","The probability that a $\\mathcal {B}_{\\mathit {det}}$ -tree has an infinite branch of non-$\\emptyset $ nodes is equal to the probability that a $\\mathcal {B}_{\\mathit {det}}^{\\prime \\prime }$ -tree is infinite.","Hence, it remains to show that one can decide in NC whether the probability that a $\\mathcal {B}_{\\mathit {det}}^{\\prime \\prime }$ -tree is infinite is positive.","Define a matrix $M \\in \\mathbb {Q}^{Q[f,X_f] \\times Q[f,X_f]}$ whose rows and columns are indexed with the non-$\\bar{q}_0$ states of $\\mathcal {A}[f,X_f]$ : $M_{(q,X),(r,Y)} \\ := \\ {\\left\\lbrace \\begin{array}{ll} \\displaystyle \\sum _{X {p} u} p |u|_Y & \\text{if } (q,X) \\xrightarrow{} (r,Y) \\ \\text{ in } \\mathcal {A}[f,X_f] \\\\[1mm] 0 & \\text{otherwise\\,,}\\end{array}\\right.","}$ where $|u|_Y \\in \\mathbb {N}_0$ is the number of occurrences of $Y$ in $u$ .","(Think of $M_{(q,X),(r,Y)}$ as the expected number of $(r,Y)$ -“successors” of $(q,X)$ .)","The graph of $M$ is equal to the transition graph of $\\mathcal {A}[f,X_f]$ (excluding $\\bar{q}_0$ ), which is strongly connected.","Say that $\\mathcal {A}[f,X_f]$ has proper branching if there exist $(q,Y) \\xrightarrow{} (r_1,Z_1)$ and $(q,Y) \\xrightarrow{} (r_2,Z_2)$ in $\\mathcal {A}[f,X_f]$ and a rule $Y {p} u_1 Z_1 u_2 Z_2 u_3$ in $\\mathcal {B}$ with $u_1, u_2, u_3 \\in \\Gamma ^*$ .","Now we can state the key lemma: lemmalemkey Let $\\rho $ be the spectral radius of $M$ .","The probability that a $\\mathcal {B}_{\\mathit {det}}^{\\prime \\prime }$ -tree is infinite is positive if and only if either $\\rho > 1$ or $\\rho = 1$ and $\\mathcal {A}[f,X_f]$ does not have proper branching.","Observe the similarity between lem:key,lem:as-finiteness-char.","In fact, the proof of lem:key, given below, is based on lem:as-finiteness-char.","lem:key shows that properties of $\\mathcal {A}[f,X_f]$ and $M$ (which are polynomial-sized objects) determine a property of the exponential-sized BP $\\mathcal {B}_{\\mathit {det}}^{\\prime \\prime }$ .","Unambiguousness of $\\mathcal {A}[f,X_f]$ is crucial for that connection.","Given that lem:key reflects the key insight of this section (if not of this paper), let us comment further.","Suppose $\\mathcal {A}[f,X_f]$ has two outgoing transitions in a state $(q,Y)$ , say $(q,Y) \\xrightarrow{} (r_1,Z_1)$ and $(q,Y) \\xrightarrow{} (r_2,Z_2)$ .","This branching could be “proper branching” as defined before lem:key, or the original UBA $\\mathcal {A}$ could be nondeterministic when reading $Y$ in $q$ and have transitions $q \\xrightarrow{} r_1$ and $q \\xrightarrow{} r_2$ .","Either type of branching causes non-0 entries in the matrix $M$ and, intuitively, increases its spectral radius $\\rho $ .","lem:key tells us that the probability that a $\\mathcal {B}_{\\mathit {det}}^{\\prime \\prime }$ -tree is infinite is governed by the combined effect on $\\rho $ of both types of branching: if $\\rho > 1$ then a $\\mathcal {B}_{\\mathit {det}}^{\\prime \\prime }$ -tree is infinite with positive probability; only in the borderline case, $\\rho =1$ , the type of branching matters.","Again, this characterization is only correct if the nondeterminism in $\\mathcal {A}$ does not cause ambiguousness.","Let us consider what lem:key states for the special case of Markov chains.","In that case, clearly there is no proper branching.","One can show, using unambiguousness, that for Markov chains the spectral radius $\\rho $ of the matrix $M$ is at most 1.","Hence, lem:key states for Markov chains that the probability that a $\\mathcal {B}_{\\mathit {det}}^{\\prime \\prime }$ -tree (consisting of a single branch) is infinite is positive if and only if $\\rho = 1$ .","Indeed, a related statement can be found in [2].","To finish the proof of prop:coUBA-1 it suffices to show that we can check the conditions of lem:key in NC.","Indeed, for comparing the spectral radius with 1, we employ lem:determine-spectral-radius.","One can check for proper branching in logarithmic space, hence in NC.","This completes the proof of prop:coUBA-1."],["LTL","With prop:coUBA-1 from the previous section, we can now show our headline result: The problem $\\mathbb {P}(\\textup {LTL})=1$ is PSPACE-complete.","PSPACE-hardness is immediate in two different ways.","It follows both from the PSPACE-hardness of model checking Markov chains against LTL and from the PSPACE-hardness of model checking transition systems against LTL [30].","Both model-checking problems are special cases of $\\mathbb {P}(\\textup {LTL}) = 1$ .","Towards membership in PSPACE, there is a classical PSPACE procedure that translates an LTL formula into an (exponential-sized) Büchi automaton [36].","As noted by several authors (e.g., [12], [7]), this procedure can easily be adapted to ensure that the Büchi automaton be a UBA.","By applying this translation to the negation $\\lnot \\varphi $ of the input formula $\\varphi $ , we obtain a UBA that rejects exactly those words that satisfy $\\varphi $ .","By prop:coUBA-1 the problem $\\mathbb {P}(\\textup {coUBA}) = 1$ is in NC.","By lem:PSPACE-transducer it follows that $\\mathbb {P}(\\textup {LTL}) = 1$ is in PSPACE.","Finally we show the following result, exhibiting a big complexity gap between the problems $\\mathbb {P}(\\textup {LTL})=1$ and $\\mathbb {P}(\\textup {LTL})=0$ .","theoremthmLTLzero The problem $\\mathbb {P}(\\textup {LTL})=0$ is 2EXPTIME-complete.","For membership in 2EXPTIME, we use again the classical procedure that translates an LTL formula into an exponential-sized Büchi automaton [36] and then invoke thm:NBA-0.","For 2EXPTIME-hardness we adapt the reduction from [11] for MDPs.","The details are in app:LTL-0."],["Conclusions","We have devised a PSPACE procedure for $\\mathbb {P}(\\textup {LTL}) = 1$ , i.e., qualitative LTL model checking of BPs.","The best previously known procedure ran in 2EXPTIME [8].","Since BPs naturally generalize both transition systems and Markov chains (for both of which LTL model checking is PSPACE-complete), one might view our model-checking algorithm as an optimal general procedure.","The same holds for NBA-specifications instead of LTL.","The main technical ingredients have been the automata-theoretic approach and the algorithmic analysis of UBAs, nonnegative matrices, and finiteness of BPs.","Our proofs were inspired by the observation that the spectral radii of certain nonnegative matrices are central to model checking Markov chains against UBAs, and also determine fundamental properties of BPs.","Very loosely speaking, when model checking Markov chains against UBAs, the spectral radius measures the amount of nondeterministic branching in the UBA, whereas when analyzing BPs, the spectral radius measures the amount of tree branching.","The “general case”, i.e., model checking BPs, features both kinds of branching.","Serendipitously, an analysis of spectral radii still leads, as we have seen, to optimal algorithms.","We have also established the complexities of related problems, partially as a tool for the mentioned LTL and NBA problems and partially to map out the landscape.","We have shown that the $\\mathbb {P}(\\cdot ) = 0$ variants are more complex than their $\\mathbb {P}(\\cdot ) = 1$ counterparts.","An intuitive explanation of this phenomenon is that for an instance of an $\\mathbb {P}(\\cdot )=1$ problems to be negative, tree branching and probabilistic branching “work together” to falsify the specification on some branch.","In contrast, for $\\mathbb {P}(\\cdot )=0$ problems, tree branching and probabilistic branching are “adversaries”, like in MDPs.","Indeed, for lower bounds on $\\mathbb {P}(\\cdot )=0$ problems we have encoded alternation in various forms.","One might ask about the complexity of $\\mathbb {P}(\\textup {UBA}) = 1$ .","Indeed, in trying to solve $\\mathbb {P}(\\textup {LTL}) = 1$ efficiently, the authors set out to solve $\\mathbb {P}(\\textup {UBA})=1$ efficiently (perhaps in P or even NC), with the PSPACE transduction from LTL to UBA in mind.","However, the complexity of UBA universality is an open problem [29]; only membership in PSPACE is known.","So even for the fixed transition system with $a {1} a b$ and $b {1} a b$ the problem $\\mathbb {P}(\\textup {UBA})=1$ cannot be placed in P without improving the complexity of UBA universality.","A PSPACE-hardness proof of $\\mathbb {P}(\\textup {UBA})=1$ might have to make use of both types of branching in BPs, as $\\mathbb {P}(\\textup {UBA})=1$ is in NC for Markov chains [2].","Model checking BPs quantitatively, i.e., computing the satisfaction probability, comparing it with a threshold, or approximating it, is left for future work.","Exact versions of these problems are computationally complex, as they are at least as hard as the corresponding $\\mathbb {P}(\\cdot )=0$ problem.","The paper [8] describes, for DPAs, nonlinear equation systems whose least nonnegative solution characterizes the satisfaction probabilities.","Newton's method is efficient for approximating the solution of such equation systems; see [32], [15].","sectionappendix"],["Proof of lem:as-finiteness-char","* By lem:as-finite-SCC it suffices to show that if $G$ is strongly connected then a $\\mathcal {B}$ -tree is infinite with positive probability if and only if $\\Gamma $ is supercritical or linear.","So, let $G$ be strongly connected.","Call a type $X \\in \\Gamma $ immortal if there is no finite $X$ -tree.","If a type $X$ is immortal, then the probability that a $\\mathcal {B}[X]$ -tree is finite is 0, and for all $Y \\in \\Gamma $ the probability that a $\\mathcal {B}[Y]$ -tree is finite is less than 1 (as $G$ is strongly connected).","If $\\Gamma $ is linear then all types are immortal.","So we can assume in the following that $\\Gamma $ is not linear.","Since $\\Gamma $ is not linear and $G$ is strongly connected, for all $X \\in \\Gamma $ there is a finite prefix of an $X$ -tree that has at least two leaves of type $X$ ; and an $X$ -tree has, with positive probability, that finite prefix.","Suppose some type, say $X$ , is immortal.","Then every $X$ -tree that has a finite prefix with $k$ ($k \\in \\mathbb {N}$ ) leaves of type $X$ has at least $k$ (infinite) branches that go through those $k$ nodes of type $X$ .","By strong connectedness, it follows that an $X$ -tree that has a finite prefix with $k$ leaves of type $X$ has, with probability 1 (i.e., $\\mathbb {P}_X$ -almost surely), a finite prefix with $k+1$ leaves of type $X$ .","Hence, with probability 1 we have $\\lim _{i \\rightarrow \\infty } Z_i = \\infty $ , where $Z_i$ is the number of nodes that are exactly $i$ levels under the root of an $X$ -tree.","Denoting by $\\mathbb {E}$ the expectation with respect to $\\mathbb {P}_X$ , it follows with Fatou's lemma that $\\lim _{i \\rightarrow \\infty } \\mathbb {E}Z_i = \\infty $ .","We can write $\\mathbb {E}Z_i = (M^i \\vec{1})_X$ , where $\\vec{1}$ denotes the vector in $\\lbrace 1\\rbrace ^\\Gamma $ .","Then $ \\lim _{i \\rightarrow \\infty } \\left\\Vert M^i \\vec{1}\\right\\Vert \\ = \\ \\infty \\,,$ where $\\left\\Vert \\cdot \\right\\Vert $ denotes the 1-norm.","Since $G$ is strongly connected, by the Perron-Frobenius theorem [3], $M$ has an eigenvector $v \\in \\mathbb {R}^\\Gamma $ with $v_Y > 0$ for all $Y \\in \\Gamma $ and $M v = \\rho v$ , where $\\rho $ denotes the spectral radius of $M$ .","Since $M^i v = \\rho ^i v$ , it follows that $\\lim _{i \\rightarrow \\infty } \\rho ^i \\left\\Vert v\\right\\Vert \\ = \\ \\lim _{i \\rightarrow \\infty } \\left\\Vert M^i v\\right\\Vert \\ = \\ \\infty \\,.$ Hence $\\rho >1$ .","We conclude that if some type is immortal, then we have $\\rho >1$ and the probability that a $\\mathcal {B}$ -tree is finite is less than 1.","On the other hand, if no type is immortal, then it follows from [13] (building on [18]) that a $\\mathcal {B}$ -tree is infinite with positive probability if and only if $\\rho > 1$ .","We conclude that, regardless of whether there exists an immortal type, a $\\mathcal {B}$ -tree is infinite with positive probability if and only if $\\Gamma $ is supercritical."],["Proof of prop:as-finite-NC","* We use the characterization from lem:as-finiteness-char.","One can compute in NL, and hence in NC, the directed acyclic graph of SCCs of $G$ and the set of types that are reachable from $X_0$ .","Therefore, we assume without loss of generality that $G$ is strongly connected.","By lem:determine-spectral-radius one can check in NC whether $\\Gamma $ is supercritical.","Whether $\\Gamma $ is linear can be checked in logarithmic space, hence in NC."],["Proof of lem:AFT-1","* The problem can be rephrased as almost-sure finiteness of BPs with $\\varepsilon $ -rules allowed.","Indeed, given $\\mathcal {B}$ and $T$ , one can eliminate all occurrences of types in $T$ from all right-hand sides; then, $\\mathsf {F}T$ in the original BP corresponds to finiteness in the new BP.","Hence, the result follows from prop:as-finite-NC."],["Proof of thm:DPA-1","* In [8] it was shown that the problem $\\mathbb {P}(\\textup {DPA}) = 1$ can be decided in polynomial time.","We show how to implement this approach in NC.","In [8] a product of the BP and the DPA is computed and analyzed; the product is a BP whose types are coloured with priorities (natural numbers).","The product, call it $\\mathcal {B}= (\\Gamma , \\mathord {{}}, \\mathit {Prob}, X_0)$ , can be computed in logarithmic space.","The question is then whether the probability is 1 that all branches of a $\\mathcal {B}$ -tree are such that the highest priority that appears infinitely often is even.","It is shown in [8] that this is the case if and only if all types $X$ that are reachable from $X_0$ and are associated with an odd priority satisfy $\\mathbb {P}_X(\\mathsf {F}N_X) = 1$ , where $N_X \\subseteq \\Gamma $ is a certain set of types that can be computed in NL by a simple reachability analysis in $\\mathcal {B}$ .","By lem:AFT-1 one can determine in NC whether $\\mathbb {P}_X(\\mathsf {F}N_X) = 1$ .","The theorem follows."],["Proof of thm:DPA-0","* Membership in P was shown in [8].","For P-hardness we reduce from the monotone circuit value problem.","Given a monotone circuit with output gate $g_{\\mathit {out}}$ , we construct a BP $\\mathcal {B}= (\\Gamma ,\\mathord {{}},\\mathit {Prob},X_{g_{\\mathit {out}}})$ and a DBA $\\mathcal {A}$ such that $\\mathbb {P}_{X_{g_{\\mathit {out}}}}(\\mathcal {A}\\text{ accepts}) > 0$ if and only if $g_{\\mathit {out}}$ evaluates to 1.","For each $\\wedge $ - and each $\\vee $ -gate $g$ include a type $X_g \\in \\Gamma $ .","Also include types $X_0, X_1 \\in \\Gamma $ for the inputs $0, 1$ , respectively.","For each $\\wedge $ -gate $g$ with children $g_1, \\ldots , g_k$ include a rule $X_g &{1} X_{g_1} \\cdots X_{g_k}\\,.\\multicolumn{2}{l}{\\text{For each $\\vee $-gate~$g$ with children $g_1, \\ldots , g_k$ include rules}}\\\\X_g &{1/k} X_{g_1},\\ \\ldots ,\\ X_g {1/k} X_{g_k}\\,.\\multicolumn{2}{l}{\\text{Include rules}}\\\\X_0 &{1} X_0 \\text{ and }X_1 {1} X_1\\,.$ Construct a two-state DBA $\\mathcal {A}$ that accepts exactly those $w \\in \\Gamma ^\\omega $ that contain $X_1$ .","It follows from a straightforward induction over the gate height (longest distance to an input) that, for any gate $g$ , it evaluates to 1 if and only if $\\mathbb {P}_{X_g}(\\mathcal {A}\\text{ accepts}) > 0$ ."],["Additional Definitions Concerning Alternating Turing Machines","An alternating Turing machine is a 6-tuple $(S_\\exists , S_\\forall , \\Sigma , T, s_0, s_{\\mathit {acc}})$ , where $S = S_\\exists \\cup S_\\forall \\cup \\lbrace s_{\\mathit {acc}}\\rbrace $ is a finite set of (control) states partitioned into existential states $S_\\exists $ and universal states $S_\\forall $ and the (only) accepting state $s_{\\mathit {acc}}$ , $\\Sigma $ is a finite alphabet, $T \\subseteq (S_\\exists \\cup S_\\forall ) \\times \\Sigma \\times \\Sigma \\times \\lbrace -1,+1\\rbrace \\times S$ is a transition relation, and $s_0$ is the initial state.","A transition $(s,a,a^{\\prime },D,s^{\\prime }) \\in T$ means that if $M$ is in state $s$ and its head reads letter $a$ , then it rewrites the content of the current cell with the letter $a^{\\prime }$ , it moves the head in direction $D$ (either left if $D=-1$ , or right if $D=+1$ ), and it changes its state to $s^{\\prime }$ .","We assume that for all $s \\in S_\\exists \\cup S_\\forall $ and $a \\in \\Sigma $ there is at least one outgoing transition.","A configuration of an alternating Turing machine is given by a 3-tuple $(i, s, w)$ where $i \\in \\mathbb {N}$ indicates the header position, $s \\in S$ is the current state of the Turing machine, and $w \\in \\Sigma ^*$ is the contents of the memory tape.","For a word $w \\in \\Sigma ^*$ we will write $w(i)$ to denote its $i$ 'th component, and $w[i=a]$ to denote the string $w(1)\\ldots w(i-1) a w(i+1)\\ldots w(n)$ .","The initial configuration of an alternating Turing machine on a word $w \\in \\Sigma ^*$ is $(1, s_0, w)$ , and given a transition $t = (s, a, a^{\\prime }, D, s^{\\prime })$ and a configuration $c = (i, s^{\\prime \\prime }, w)$ we will use $c \\triangleright t$ to denote the configuration $(i+D, s^{\\prime }, w[i=a^{\\prime }])$ if $s^{\\prime \\prime } = s$ and $w_i = a$ (and leave it undefined otherwise).","We extend this notation to sequences of transitions $t_1\\ldots t_n \\in T^*$ by writing $c \\triangleright t_1\\ldots t_n = (c \\triangleright t_1) \\triangleright t_2\\ldots t_n$ .","We will use $\\pi _\\mathbb {N}$ , $\\pi _S$ , and $\\pi _{\\Sigma ^*}$ to denote the projections of a configuration $(i, s, w)$ to $i$ , $s$ , and $w$ , respectively.","For any $(s, a) \\in (S_\\exists \\cup S_\\forall ) \\times \\Sigma $ , let $T_{s,a} := \\lbrace (s,a,a^{\\prime },D,s^{\\prime }) \\in T \\mid a^{\\prime } \\in \\Sigma ,\\ D \\in \\lbrace -1,+1\\rbrace ,\\ s^{\\prime } \\in S\\rbrace $ .","A string $r = r_1r_2\\ldots \\in T^*\\cup T^\\omega $ is called a run of $M$ on $w$ if for all $i$ , $r_{i+1} \\in T_{\\pi _S(c_{i}), \\pi _{\\Sigma ^*}(c_{i})(\\pi _\\mathbb {N}(c_{i}))}$ , where $c_i = c_0\\triangleright r_1\\ldots r_i$ whenever $r_{i+1}$ exists.","Any run $r$ is called an accepting run if $r \\in T^*$ and $\\pi _S(c_0\\triangleright r) = s_{\\mathit {acc}}$ .","A strategy is a function $\\sigma : \\lbrace (i, s, w, r) \\in \\mathbb {N}\\times S \\times \\Sigma ^* \\times T^* \\mid s \\in S_\\exists \\rbrace \\rightarrow T$ such that $\\sigma (i, s, w, r) \\in T_{s, w_i}$ for any $i, s, w, r$ .","For a configuration $c$ and a run $r$ , by abuse of notation, we will write $\\sigma (c, r)$ to mean $\\sigma (\\pi _\\mathbb {N}(c), \\pi _S(c), \\pi _{\\Sigma ^*}(c), r)$ .","A run is consistent with a strategy $\\sigma $ if for all $i$ , $r_{i+1} = \\sigma (c_0\\triangleright r_1\\ldots r_i, r_1\\ldots r_i)$ whenever the latter is defined.","The behaviour of $\\sigma $ is the set of runs consistent with $\\sigma $ .","A winning strategy is a strategy $\\sigma $ such that every run $r$ consistent with $\\sigma $ is an accepting run.","Note that whenever $\\sigma $ is winning, its behaviour is finite.","Let $\\sigma $ be a strategy, then we call $\\sigma $ $N$ -bounded if for any $r$ in the behavior of $\\sigma $ , and any prefix $r_1\\ldots r_k$ of $r$ , $|\\pi _{\\Sigma ^*}(c_0\\triangleright r_1\\ldots r_k)| < N$ .","A Turing machine is $N$ -bounded if it has an $N$ -bounded winning strategy."],["Proof of thm:NBA-0","* Given the main body, it remains to prove EXPTIME-hardness of $\\mathbb {P}(\\textup {NBA}) = 0$ .","Recall that alternating PSPACE equals EXPTIME.","We give a polynomial-time reduction from the problem of acceptance of a word by a PSPACE-bounded alternating Turing machine.","Without loss of generality, we can assume that the Turing machine is linear-bounded, i.e., uses only the space occupied by the input word.","Let $M = (S_\\exists , S_\\forall , \\Sigma , T, s_0, s_{\\mathit {acc}})$ be a linear-bounded alternating Turing machine.","Let $w = a_1 \\cdots a_n \\in \\Sigma ^*$ be the input word.","As mentioned before, we can assume that $M$ uses exactly $n$ tape cells.","We construct a BP $\\mathcal {B}$ and an NBA $\\mathcal {A}$ such that $\\mathbb {P}_\\mathcal {B}(\\mathcal {A}\\text{ accepts}) > 0$ if and only $M$ accepts $w$ .","The BP $\\mathcal {B}$ has the following set of types: $\\Gamma \\ = \\ (\\lbrace 1, \\ldots , n\\rbrace \\times S \\times \\Sigma ) \\ \\cup \\ (\\lbrace 1, \\ldots , n\\rbrace \\times T) \\ \\cup \\ (\\lbrace 1, \\ldots , n\\rbrace \\times \\lbrace \\mathit {chk}\\rbrace ) \\ \\cup \\ \\lbrace E\\rbrace $ Intuitively, a type $(i,s,a)$ means that the head is at position $i$ , the current state is $s$ , and the head is reading letter $a$ ; a type $(i,t)$ means that the head is at position $i$ , and transition $t$ is being executed; a type $(i,\\mathit {chk})$ means that the accepting state has been reached, and cell $i$ is being “checked” (in a sense to be explained later); type $E$ indicates an error.","The type $(1,s_0,a_1)$ is the start type of $\\mathcal {B}$ .","For all $(i,s,a) \\in \\Gamma $ with $s \\in S_\\exists $ , include rules $(i,s,a) {1/k} (i,t_j) \\quad (1 \\le j \\le k)\\,,$ where $\\lbrace t_1, \\ldots , t_k\\rbrace = T_{s,a}$ .","(Intuitively, a transition going out of an existential state is chosen as the only child, uniformly at random.)","For all $(i,s,a) \\in \\Gamma $ with $s \\in S_\\forall $ , include a rule $(i,s,a) {1} (i,t_1) \\cdots (i,t_k) \\,,$ where $\\lbrace t_1, \\ldots , t_k\\rbrace = T_{s,a}$ .","(Intuitively, all possible transitions going out of a universal state are children.)","For all $(i,(s,a,a^{\\prime },D,s^{\\prime })) \\in \\Gamma $ with $1 \\le i{+}D \\le n$ , include rules $(i,(s,a,a^{\\prime },D,s^{\\prime })) {1/|\\Sigma |} (i{+}D, s^{\\prime }, b_j) \\quad (1 \\le j \\le |\\Sigma |)\\,,$ where $\\lbrace b_1, \\ldots , b_{|\\Sigma |}\\rbrace = \\Sigma $ .","(Intuitively, the letter in the cell at the new head position $i{+}D$ is guessed uniformly at random.)","For all $(i,(s,a,a^{\\prime },D,s^{\\prime })) \\in \\Gamma $ with $i{+}D \\in \\lbrace 0,n{+}1\\rbrace $ , include a rule $(i,(s,a,a^{\\prime },D,s^{\\prime })) {1} E\\,.$ (Intuitively, when the space bound is exceeded, move to the error type $E$ .)","Include a rule $E {1} E$ (i.e., a self-loop).","For all $(i, s_{\\mathit {acc}}, a) \\in \\Gamma $ , include a rule $(i, s_{\\mathit {acc}}, a) {1} (1,\\mathit {chk}) \\cdots (n,\\mathit {chk})\\,.$ (Intuitively, after reaching $s_{\\mathit {acc}}$ all $n$ cells are “checked”.)","For all $(i,\\mathit {chk}) \\in \\Gamma $ , include a rule $(i,\\mathit {chk}) {1} (i,\\mathit {chk})$ (i.e., a self-loop).","The NBA $\\mathcal {A}= (Q, \\Gamma , \\delta , Q_0, \\lbrace f\\rbrace )$ has the following set of states: $Q \\ = \\ (\\lbrace 1, \\ldots , n\\rbrace \\times \\Sigma ) \\ \\cup \\ \\lbrace f\\rbrace $ The set of initial states is $Q_0 = \\lbrace (1,a_1), \\ldots , (n,a_n)\\rbrace $ (recall that $a_1 \\cdots a_n$ is the input word).","The idea is that if a prefix $X_1 \\cdots X_k \\in \\Gamma ^*$ of a tree branch corresponds to a prefix of a correct computation of $M$ , then the set of automaton states in $\\delta (Q_0,X_1 \\cdots X_k)$ corresponds to the tape after this computation prefix.","In fact, the transition relation $\\delta \\subseteq Q \\times \\Gamma \\times Q$ is deterministic, i.e., for all $q \\in Q$ and $X \\in \\Gamma $ there is at most one $q^{\\prime }$ with $(q,X,q^{\\prime }) \\in \\delta $ .","Moreover, for any transition $((i,a), X, (j,a^{\\prime })) \\in \\delta $ we will have $i=j$ .","For $(i,a) \\in Q$ and all $(j,s,b) \\in \\Gamma $ with $i \\ne j$ or $a=b$ , include a self-loop $((i,a),(j,s,b),(i,a)) \\in \\delta \\,.$ (Intuitively, letter $a$ in cell $i$ is compatible with the head being on cell $j$ and reading letter $b$ .)","For all $(i,a) \\in Q$ and all $(j,t) \\in \\Gamma $ with $i \\ne j$ , include a self-loop $((i,a),(j,t),(i,a)) \\in \\delta \\,.$ (Intuitively, the content of cell $i$ stays unchanged when the head is at position $j$ .)","For all $i,s,a,a^{\\prime },D,s^{\\prime }$ with $(i,a) \\in Q$ and $(i,(s,a,a^{\\prime },D,s^{\\prime })) \\in \\Gamma $ , include a transition $((i,a), (i,(s,a,a^{\\prime },D,s^{\\prime })), (i,a^{\\prime }))\\in \\delta \\,.$ (Intuitively, the transition $(s,a,a^{\\prime },D,s^{\\prime })$ changes the content of cell $i$ from $a$ to $a^{\\prime }$ .)","For all $(i,a) \\in Q$ , include a transition $((i,a),(i,\\mathit {chk}),f) \\in \\delta \\,.$ (Intuitively, the type $(i,\\mathit {chk})$ checks if the computation has been consistent in cell $i$ .)","For all $X \\in \\Gamma $ , include a self-loop $(f,X,f) \\in \\delta $ .","We will show that in this case, $\\mathbb {P}_\\mathcal {B}(\\mathcal {A}\\text{ accepts}) > 0$ if and only if $M$ accepts $w$ .","Firstly note that $f$ is a sink state in $\\mathcal {A}$ , and hence, if $\\mathcal {A}$ reaches $f$ after reading some prefix of a branch, it will accept any branch with this prefix.","This means that a tree $t$ is accepted by $\\mathcal {A}$ if and only if there exists a finite prefix of $t$ such that $\\mathcal {A}$ reaches $f$ on all of its branches.","Assume that $M$ accepts $w$ .","Then there exists a linear-bounded winning strategy $\\sigma $ for the existential player.","We will write $c_0$ for the initial configuration of $M$ on $w$ , $R$ for the behaviour of $\\sigma $ , and $N$ for the length of the longest run in $R$ .","We will show that there exists a finite prefix generated by $\\mathcal {B}$ with nonzero probability that precisely models this strategy, and that is such that $\\mathcal {A}$ reaches $f$ on all of its branches.","Since any tree with this prefix is accepted, this implies that $\\mathbb {P}_\\mathcal {B}(\\mathcal {A}\\text{ accepts}) > 0$ if $M$ accepts $w$ .","Let $t$ be the tree defined as follows: the root of $t$ is $(1, s_0, a_1)$ , for any node $(p_i, s_i, b_i)$ on level $2i$ with $s_i \\in S_\\exists $ , let $t_1\\ldots t_{i}$ be such that the branch from the root to $(p_i, s_i, b_i)$ has nodes $(p_j, t_{j+1})$ on level $2j+1$ for each $0 \\le j < i$ .","Then $(p_i, s_i, b_i)$ has a child $(p_i, \\sigma (c_0 \\triangleright t_1\\ldots t_{i}, t_1\\ldots t_i))$ , for any node $(p_i, s_i, b_i)$ on level $2i$ with $s_i \\in S_\\forall $ , let $t_1\\ldots t_{i}$ be such that the branch from the root to $(p_i, s_i, b_i)$ has nodes $(p_j, t_{j+1})$ on level $2j+1$ for each $0 \\le j < i$ .","Let $w_i = \\pi _{\\Sigma ^*}(c_0 \\triangleright t_1\\ldots t_{i})$ .","Then $(p_i, s_i, b_i)$ has children $(p_i, t_{i+1})$ for each $t_{i+1} \\in T_{s_i, w_i(p_i)}$ , for any node $(p_i, t_{i+1})$ with $t_{i+1} = (s, a, a^{\\prime }, D, s^{\\prime })$ on level $2i+1$ , let $t_1\\ldots t_{i}$ be such that the branch from the root to $(p_i, t_{i+1})$ has nodes $(p_j, t_{j+1})$ on level $2j+1$ for each $0 \\le j < i$ .","Let $c_{i+1} = c_0 \\triangleright t_1\\ldots t_{i+1}$ .","Then $(p_i, t_{i+1})$ has a child $(p_i+D, s^{\\prime }, \\pi _{\\Sigma ^*}(c_{i+1})(p_i+D))$ , any node $(p_i, s_{\\mathit {acc}}, b_i)$ has children $(j, \\mathit {chk})$ for each $1 \\le j \\le n$ , and any node $(j, \\mathit {chk})$ has a child $(j, \\mathit {chk})$ .","By construction, $t$ is now such that for any node $(p_i, t_{i+1})$ with $t_{i+1} = (s, a, a^{\\prime }, D, s^{\\prime })$ on level $2i+1$ , there exist runs in $R$ prefixed by $t_1 \\ldots t_{i+1}$ , where $t_1\\ldots t_{i}$ are such that the branch from the root to $(p_i, t_{i+1})$ has nodes $(p_j, t_{j+1})$ on level $2j+1$ for each $0 \\le j < i$ .","Hence, by the fact that $\\sigma $ is linear-bounded, $1 \\le p_i+D \\le n$ and thus all the transitions in $t$ are according to the rules of $\\mathcal {B}$ .","Moreover, since the length of runs in $R$ is bounded by $N$ , all the nodes at level $2N+1$ must be of the form $(i, \\mathit {chk})$ , and since all states $(i, \\mathit {chk})$ are sink states, $t$ is generated with a nonzero probability.","Finally pick any state $(i, \\mathit {chk})$ in $t$ .","The only types from which $(i, a) \\in Q$ does not have an outgoing edge are of the form $(i, s, b)$ .","However, for any state $(i, s, b)$ at level $2j$ on the branch to $(i, \\mathit {chk})$ , let $t_1\\ldots t_{j}$ be such that the branch from the root to $(i,s,b)$ has nodes $(p_k, t_{k+1})$ on level $2k+1$ for each $0 \\le k < j$ .","Then $(i, b) = (i, \\pi _{\\Sigma ^*}(c_0 \\triangleright t_1\\ldots t_{j})(i)) = \\delta ((i,w(i)), (p_0, t_1)\\ldots (p_{j-1}, t_{j}))$ and hence $(i, a)$ survives, and reaches $f$ upon reading $(i, \\mathit {chk})$ .","For the other direction, assume that $\\mathbb {P}_\\mathcal {B}(\\mathcal {A}\\text{ accepts}) > 0$ .","Then there exists a prefix $t$ generated by $\\mathcal {B}$ with branches of length $N$ for some $N$ such that $\\mathcal {A}$ reaches $f$ on all of its branches.","W.l.o.g.","we can assume that $\\pi _S(c_0) \\ne s_{\\mathit {acc}}$ , because otherwise any strategy is winning.","Note that for any accepted tree $t$ and any branch prefix $(1, s_0, a_1)(1, t_1)\\ldots (p_i, s_i, b_i)(p_i, t_{i+1})$ in $t$ , the set of reachable states in the automaton from $Q_0$ reflects the tape contents of $M$ , ie.","$\\delta (Q_0, (1, s_0, a_1)(1, t_1)\\ldots (p_i, s_i, b_i)(p_i, t_{i+1}))=\\lbrace (j, w(j))\\mid 1 \\le j \\le n, w = \\pi _{\\Sigma ^*}(c_0 \\triangleright t_1\\ldots t_{i+1})\\rbrace $ .","Also $s_i = \\pi _S(c_0 \\triangleright t_1\\ldots t_i)$ .","If this was not the case, then there would exist $j$ such that $\\delta ((j, a_j), (1, s_0, a_1)(1, t_1)\\ldots (p_i, s_i, b_i)(p_i, t_{i+1})) = \\emptyset $ and hence the branch reaching $(j, \\mathit {chk})$ prefixed by $(1, s_0, a_1)(1, t_1)\\ldots (p_i, s_i, b_i)(p_i, t_{i+1})$ does not reach $f$ .","Let $n_0n_1\\ldots $ be any branch in $t$ .","Pick $m$ such that $n_{2i+1} = (a_i, t_{i+1})$ for all $0 \\le i < m$ and $n_{2m+1} = (a_m, t_{m+1})$ .","Let $c = c_0 \\triangleright t_1\\ldots t_m$ .","Then we define $\\sigma $ to be any strategy such that for any such $c$ , if $\\pi _S(c) \\in S_\\exists $ , then $\\sigma (c, t_1\\ldots t_m) = t_{m+1}$ .","This is a valid strategy since for any branch in $t$ , the set of reachable states of the automaton after reading a branch prefix reflects the tape contents of $M$ and the alphabet characters contained in the nodes of the branch prefix have to reflect the automaton states.","Note that $\\sigma $ is $n$ -bounded.","We claim that $\\sigma $ is a winning strategy and hence $M$ accepts $w$ .","Let $r = t_1t_2\\ldots $ be a run consistent with $\\sigma $ .","We will show that there exists a branch $(1, s_0, a_1)(1, t_1)\\ldots (p_i, t_{i+1})(p_{i+1}, s_{\\mathit {acc}}, w_{i+1}(p_{i+1}))$ in $t$ with $p_i = \\pi _\\mathbb {N}(c_0\\triangleright t_1\\ldots t_i)$ , $s_i = \\pi _S(c_0\\triangleright t_1\\ldots t_i)$ , and $w_i = \\pi _{\\Sigma ^*}(c_0\\triangleright t_1\\ldots t_i)$ (and hence $|r| = i+1$ and $\\pi _S(c_0\\triangleright r) = s_{\\mathit {acc}}$ ).","For any branch prefix $(1, s_0, a_1)(1, t_1)\\ldots (p_j, t_{j+1})$ where $t_{j+1} = (s, a, a^{\\prime }, D, s^{\\prime })$ , according to the rules of $\\mathcal {B}$ , $(p_j, t_{j+1})$ has a single child $(p_i+D, s^{\\prime }, b)$ .","Let $c_{j+1} = c_0 \\triangleright t_1\\ldots t_{j+1}$ .","Since trees prefixed by $t$ are accepted, $p_i+D = \\pi _\\mathbb {N}(c_{j+1})$ , $s^{\\prime } = \\pi _S(c_{j+1})$ , and $b = \\pi _{\\Sigma ^*}(c_{j+1})(\\pi _\\mathbb {N}(c_{j+1}))$ .","For any branch prefix $(1, s_0, a_1)(1, t_1)\\ldots (p_{j-1}, t_{j})(p_j, s_j, b_j)$ where $s_j \\in S_\\forall $ , according to the rules of $\\mathcal {B}$ , $(p_j, s_j, b_j)$ has children $(p_j, t^{\\prime })$ for each $t^{\\prime } \\in T_{s_j, b_j}$ .","Let $c_j = c_0\\triangleright t_1\\ldots t_j$ .","Since trees prefixed by $t$ are accepted, $s_j = \\pi _S(c_j)$ and $b_j = \\pi _{\\Sigma ^*}(c_j)(\\pi _\\mathbb {N}(c_j))$ .","Hence, $t_{j+1} \\in T_{s_j, b_j}$ and $(p_j, s_j, b_j)$ has a child $(p_j, t_{j+1})$ .","For any branch prefix $(1, s_0, a_1)(1, t_1)\\ldots (p_{j-1}, t_{j})(p_j, s_j, b_j)$ where $s_j \\in S_\\exists $ , according to the rules of $\\mathcal {B}$ , $(p_j, s_j, b_j)$ has a single child $(p_j, t^{\\prime })$ .","By definition of $\\sigma $ , $\\sigma (c_0\\triangleright t_1\\ldots t_j, t_1\\ldots t_j) = t^{\\prime }$ .","Since $t$ is finite, every branch reaches states of the form $(p, s_{\\mathit {acc}}, b)$ in a finite number of steps.","Hence, $r$ is finite and reaches $s_{\\mathit {acc}}$ and since $r$ is any run consistent with $\\sigma $ , $\\sigma $ is an winning strategy.","Thus, $M$ accepts $w$ ."],["Proof of lem:UBA-X1-f","* Consider a branch $X_0 X_1 X_2 \\cdots $ of a $\\mathcal {B}$ -tree accepted by $\\mathcal {A}$ .","Then $\\mathcal {A}$ has an accepting run $q_0 \\xrightarrow{} q_1 \\xrightarrow{} q_2 \\xrightarrow{} \\cdots $ with $q_0 \\in Q_0$ .","By the pigeonhole principle, there are $f \\in F$ and $X_f \\in \\Gamma $ such that this run contains the segment $f \\xrightarrow{}$ infinitely often.","By its construction, the Büchi automaton $\\mathcal {A}\\times \\mathcal {B}$ has the accepting run $(q_0,X_0) \\xrightarrow{} (q_1,X_1) \\xrightarrow{} \\cdots \\,,$ which contains the state $(f,X_f)$ infinitely often.","Let $k \\ge 1$ be such that $(q_k,X_k) = (f,X_f)$ .","Then $\\mathcal {A}[f,X_f]$ accepts $X_k X_{k+1} \\cdots $ via the run $\\bar{q}_0 \\xrightarrow{} (q_k,X_k) \\xrightarrow{} (q_{k+1},X_{k+1}) \\cdots \\,.$ Therefore, denoting by $E(f,X_f,n)$ for $n \\in \\mathbb {N}$ the event that there exists a branch $X_k X_{k+1} \\cdots $ emanating from the $n$ th (in a breadth-first order) node in the tree (necessarily a node of type $X_k = X_f$ ) such that $\\mathcal {A}[f,X_f]$ has an accepting run $\\bar{q}_0 \\xrightarrow{} (q_k,X_k) \\xrightarrow{} (q_{k+1},X_{k+1}) \\cdots \\,,$ we have $\\mathbb {P}_{X_0}(\\mathcal {A}\\text{ accepts some branch})\\ \\le \\ &\\sum _{f \\in F} \\sum _{X_f \\in \\Gamma } \\sum _{n \\in \\mathbb {N}} \\mathbb {P}_{X_0}(E(f,X_f,n))\\,.$ Further, $&\\ \\mathbb {P}_{X_0}(E(f,X_f,n)) \\\\=&\\ \\mathbb {P}_{X_0}(\\text{the $n$th node has type~$X_f$}) \\cdot \\mathbb {P}_{X_f}(\\mathcal {A}[f,X_f] \\text{ accepts some branch}) \\\\\\le &\\ \\mathbb {P}_{X_f}(\\mathcal {A}[f,X_f] \\text{ accepts some branch})\\,.$ The “only if” direction follows.","Towards the “if” direction, suppose that $\\mathcal {A}\\times \\mathcal {B}$ has a path $(q_0,X_0) {\\;\\xrightarrow{}{}\\hspace{0.0pt}^{*}\\;} (f,X_f)$ with $q_0 \\in Q_0$ and $f \\in F$ such that the probability that some branch of a $\\mathcal {B}[X_f]$ -tree is accepted by $\\mathcal {A}[f,X_f]$ is positive.","Then there is a successor, say $X_f^{\\prime }$ , of $X_f$ such that the probability that some branch of a $\\mathcal {B}[X_f^{\\prime }]$ -tree is accepted by $\\mathcal {A}\\times \\mathcal {B}$ when started in $(f,X_f)$ is positive.","Thus, the probability that some branch of a $\\mathcal {B}$ -tree (starts with $X_0 w X_f^{\\prime }$ and) is accepted by $\\mathcal {A}$ is positive."],["Proof of lem:Bdet","In this subsection we prove lem:Bdet, which is instrumental for the main results of the paper.","* Fix a word $w \\in \\Gamma ^+$ such that $\\delta [f,X_f]((f,X_f),w)$ is maximal, i.e., there is no $w^{\\prime } \\in \\Gamma ^+$ such that $\\delta [f,X_f]((f,X_f),w^{\\prime }) \\supsetneq \\delta [f,X_f]((f,X_f),w)$ .","Let $v \\in \\Gamma ^*$ be a word such that $\\delta [f,X_f]((f,X_f),v) \\ni (f,X_f)$ .","Then, $\\delta [f,X_f]((f,X_f),v w) = \\delta [f,X_f]((f,X_f), w)$ ; i.e., for any path $(f,X_f) {\\;\\xrightarrow{}{}\\hspace{0.0pt}^{*}\\;} (q,X)$ there is a path $(f,X_f) {\\;\\xrightarrow{}{}\\hspace{0.0pt}^{*}\\;} (f,X_f) {\\;\\xrightarrow{}{}\\hspace{0.0pt}^{*}\\;} (q,X)$ .","Since $\\delta [f,X_f]((f,X_f),v) \\ni (f,X_f)$ , we have $\\delta [f,X_f]((f,X_f),v w) \\ \\supseteq \\ \\delta [f,X_f]((f,X_f), w)\\,.$ But $w$ is maximal.","We enrich $\\mathcal {A}_{\\mathit {det}}$ to obtain a DBA, $\\mathcal {A}_{\\mathit {det}}^{\\prime }$ , whose states have an additional component keeping track of whether the word $w \\in \\Gamma ^+$ from above is being seen.","The accepting runs of $\\mathcal {A}_{\\mathit {det}}^{\\prime }$ contain infinitely many $w$ -labelled segments that, loosely speaking, “start from $(f,X_f)$ ”.","Formally, let $W = \\lbrace 0\\rbrace \\cup \\lbrace v \\in \\Gamma ^* \\mid v \\text{ is a suffix of } w\\rbrace \\,,$ including $w$ and the empty word $\\varepsilon $ .","We assume $0 \\notin \\Gamma $ .","Define $\\mathcal {A}_{\\mathit {det}}^{\\prime } \\ :=\\ (2^{\\lbrace \\bar{q}_0\\rbrace \\cup Q[f,X_f]} \\times W, \\delta _{\\mathit {det}}^{\\prime }, (\\lbrace \\bar{q}_0\\rbrace ,0), (2^{Q[f,X_f]} \\setminus \\lbrace \\emptyset \\rbrace ) \\times \\lbrace \\varepsilon \\rbrace )\\,,$ where $\\delta _{\\mathit {det}}^{\\prime }((U,X v), X) \\ &= \\ (\\delta _{\\mathit {det}}(U,X), v) \\\\\\delta _{\\mathit {det}}^{\\prime }((U,X v), Y) \\ &= \\ (\\delta _{\\mathit {det}}(U,X), 0) \\quad &&\\text{for } X \\ne Y \\\\\\delta _{\\mathit {det}}^{\\prime }((U,v), X) \\ &= \\ (\\delta _{\\mathit {det}}(U,X), w) \\quad &&\\text{for } v \\in \\lbrace 0, \\varepsilon \\rbrace ,\\ (f,X_f) \\in \\delta _{\\mathit {det}}(U,X) \\\\\\delta _{\\mathit {det}}^{\\prime }((U,v), X) \\ &= \\ (\\delta _{\\mathit {det}}(U,X), 0) \\quad &&\\text{for } v \\in \\lbrace 0, \\varepsilon \\rbrace ,\\ (f,X_f) \\notin \\delta _{\\mathit {det}}(U,X)\\,.$ It follows from lem:UBA-w and the construction of $\\mathcal {A}_{\\mathit {det}}^{\\prime }$ that $\\mathcal {A}_{\\mathit {det}}^{\\prime }$ has a single accepting state reachable from $(\\lbrace \\bar{q}_0\\rbrace ,0)$ , namely $(\\delta _{\\mathit {det}}(\\lbrace (f,X_f)\\rbrace ,w), \\varepsilon )$ .","The following statements are equivalent: The probability that some branch of a $\\mathcal {B}[X_f]$ -tree is accepted by $\\mathcal {A}[f,X_f]$ is positive.","The probability that some branch of a $\\mathcal {B}[X_f]$ -tree has a run (accepting or not) in $\\mathcal {A}[f,X_f]$ is positive.","The probability that some branch of a $\\mathcal {B}[X_f]$ -tree is accepted by $\\mathcal {A}_{\\mathit {det}}^{\\prime }$ is positive.","lem:UBA-3-equivalences implies lem:Bdet, as it follows from the definition of $\\mathcal {B}_{\\mathit {det}}$ that the probability that some branch of a $\\mathcal {B}[X_f]$ -tree has a run in $\\mathcal {A}[f,X_f]$ (cf.","condition (ii) in lem:UBA-3-equivalences) equals the probability that some branch of a $\\mathcal {B}_{\\mathit {det}}$ -tree does not have any nodes of type $\\emptyset $ (cf.","condition (ii) of lem:Bdet).","So it remains to prove lem:UBA-3-equivalences.","[Proof of lem:UBA-3-equivalences] (i) $\\Longrightarrow $ (ii).","Trivial.","(iii) $\\Longrightarrow $ (i).","Let $X_f X_1 X_2 \\cdots $ be accepted by $\\mathcal {A}_{\\mathit {det}}^{\\prime }$ .","Then $X_1 X_2 \\cdots $ can be decomposed in $v_1 w v_2 w \\cdots $ with $v_1, v_2, \\ldots \\in \\Gamma ^*$ such that $\\mathcal {A}[f,X_f]$ has paths $(f,X_f) {\\;\\xrightarrow{}{}\\hspace{0.0pt}^{*}\\;} (f,X_f)$ for all $i \\ge 1$ .","Let $i \\ge 2$ .","There is $(q,X)$ with $(f,X_f) {\\;\\xrightarrow{}{}\\hspace{0.0pt}^{*}\\;} (q,X) {\\;\\xrightarrow{}{}\\hspace{0.0pt}^{*}\\;} (f,X_f)\\,.$ By lem:UBA-w there is a path $(f,X_f) {\\;\\xrightarrow{}{}\\hspace{0.0pt}^{*}\\;} (f,X_f) {\\;\\xrightarrow{}{}\\hspace{0.0pt}^{*}\\;} (q,X)\\,.$ Thus also $(f,X_f) {\\;\\xrightarrow{}{}\\hspace{0.0pt}^{*}\\;} (f,X_f)$ .","Since $i \\ge 2$ was arbitrary, it follows that $\\mathcal {A}[f,X_f]$ has an accepting run $\\bar{q}_0 \\xrightarrow{} (f,X_f) {\\;\\xrightarrow{}{}\\hspace{0.0pt}^{*}\\;} (f,X_f) {\\;\\xrightarrow{}{}\\hspace{0.0pt}^{*}\\;} (f,X_f) {\\;\\xrightarrow{}{}\\hspace{0.0pt}^{*}\\;} \\cdots \\,.$ (ii) $\\Longrightarrow $ (iii).","For this part we use results from [8], which considers the problem of model checking BPs against DPAs.","We need only a special case of such automata: (a) the same (word) automaton is run on every branch of the tree, and (b) our automaton $\\mathcal {A}_{\\mathit {det}}^{\\prime }$ is a DBA, which can be viewed as a DPA whose states are labelled with only 2 priorities: priority 0 for non-accepting states and priority 1 for accepting states.","The paper [8] constructs a product BP from the BP and the automaton, and subsequently considers BPs whose types are coloured with a priority.","In this way the model-checking problem reduces to computing the probability that there is a branch on which the highest priority that occurs infinitely often is odd.","Since our automaton $\\mathcal {A}_{\\mathit {det}}^{\\prime }$ already embeds a BP, instead of taking another product, we define a BP, $\\mathcal {B}_{\\mathit {det}}^{\\prime }$ , more directly based on $\\mathcal {A}_{\\mathit {det}}^{\\prime }$ , in the same way that $\\mathcal {B}_{\\mathit {det}}$ was defined based on $\\mathcal {A}_{\\mathit {det}}$ in sub:Bdet.","More explicitly, $\\mathcal {B}_{\\mathit {det}}^{\\prime } \\ := \\ (\\Gamma ^{\\prime }, \\mathord {{}^{\\prime }}, \\mathit {Prob}^{\\prime }, (\\lbrace (f,X_f)\\rbrace ,w))\\,,$ where the set of types $\\Gamma ^{\\prime } \\subseteq 2^{Q[f,X_f]} \\times W$ is the set of those states in $\\mathcal {A}_{\\mathit {det}}^{\\prime }$ that are reachable (in $\\mathcal {A}_{\\mathit {det}}^{\\prime }$ ) from $(\\lbrace \\bar{q}\\rbrace ,0)$ via a nonempty path (recall that they are of the form $(P \\times \\lbrace X\\rbrace ,v)$ with $P \\subseteq Q$ and $X \\in \\Gamma $ and $v \\in W$ ), and $X^{\\prime } &{\\;{p}{}\\hspace{0.0pt}{^{\\prime }}\\;} \\delta _{\\mathit {det}}^{\\prime }(X^{\\prime },X_1) \\cdots \\delta _{\\mathit {det}}^{\\prime }(X^{\\prime },X_k)\\multicolumn{2}{l}{\\text{for all $X^{\\prime } = (P \\times \\lbrace X\\rbrace ,v) \\in \\Gamma ^{\\prime }$ with $P \\ne \\emptyset $ and all $X {p} X_1 \\cdots X_k$, and}}\\\\(\\emptyset ,v) &{\\;{1}{}\\hspace{0.0pt}{^{\\prime }}\\;} (\\emptyset ,v)$ for all $v \\in W$ .","Recall that $(\\delta _{\\mathit {det}}(\\lbrace (f,X_f)\\rbrace ,w), \\varepsilon ) =: X_w^{\\prime }$ is the single accepting state in $\\mathcal {A}_{\\mathit {det}}^{\\prime }$ that is reachable from $(\\lbrace \\bar{q}_0\\rbrace ,0)$ .","We call a branch of $\\mathcal {B}_{\\mathit {det}}^{\\prime }$ accepting if it contains $X_w^{\\prime }$ infinitely often.","Suppose “(ii)”, i.e., the probability that some branch of a $\\mathcal {B}[X_f]$ -tree has a (non-accepting or accepting) run in $\\mathcal {A}[f,X_f]$ is positive.","Thus, the probability that some branch of a $\\mathcal {B}[X_f]$ -tree has a run in $\\mathcal {A}_{\\mathit {det}}$ that does not enter the state $\\emptyset $ is positive.","Let $X_1 w^{\\prime } \\in \\Gamma ^+$ be such that $(f,X_f) \\in \\delta _{\\mathit {det}}(\\lbrace (f,X_f)\\rbrace ,w X_1 w^{\\prime })\\,.$ Then, also the probability that some branch of a $\\mathcal {B}[X_f]$ -tree starts with $X_f w X_1 w^{\\prime }$ and has a run in $\\mathcal {A}_{\\mathit {det}}$ that does not enter $\\emptyset $ is positive.","Thus, the probability that some branch of a $\\mathcal {B}[X_1]$ -tree starts with $X_1 w^{\\prime }$ and has a run in $\\mathcal {A}_{\\mathit {det}}$ , started in $\\delta _{\\mathit {det}}(\\lbrace (f,X_f)\\rbrace ,w)$ , that does not enter $\\emptyset $ is positive.","Hence, the probability that some branch of a $\\mathcal {B}[X_1]$ -tree has a run in $\\mathcal {A}_{\\mathit {det}}^{\\prime }$ , started in $(\\delta _{\\mathit {det}}(\\lbrace (f,X_f)\\rbrace ,w), \\varepsilon )$ , that does not enter a state of the form $(\\emptyset ,v)$ is positive.","From the construction of $\\mathcal {B}_{\\mathit {det}}^{\\prime }$ it follows that the probability that some branch of a $\\mathcal {B}_{\\mathit {det}}^{\\prime }[X_w^{\\prime }]$ -tree (i.e., with $X_w^{\\prime } = (\\delta _{\\mathit {det}}(\\lbrace (f,X_f)\\rbrace ,w), \\varepsilon )$ as start type) does not have a node of a type of the form $(\\emptyset ,v)$ is positive.","Consider any type $(U_0,v_0)$ in $\\mathcal {B}_{\\mathit {det}}^{\\prime }$ with $U_0 \\ne \\emptyset $ and view it as a state in $\\mathcal {A}_{\\mathit {det}}^{\\prime }$ .","Then there is $u_1 \\in \\Gamma ^*$ with $(U_0,v_0) {\\;\\xrightarrow{}{}\\hspace{0.0pt}^{*}\\;} (U_1,v_1)$ where $U_1 \\ne \\emptyset $ and $v_1 \\in \\lbrace 0, \\varepsilon \\rbrace $ .","Let $u_2 \\in \\Gamma ^*$ be a shortest word such that there is $(q_1,X_1) \\in U_1$ with $(q_1,X_1) {\\;\\xrightarrow{}{}\\hspace{0.0pt}^{*}\\;} (f,X_f)$ in $\\mathcal {A}[f,X_f]$ .","It follows that in $\\mathcal {A}_{\\mathit {det}}^{\\prime }$ we have $(U_0,v_0) {\\;\\xrightarrow{}{}\\hspace{0.0pt}^{*}\\;} (U_2,w) {\\;\\xrightarrow{}{}\\hspace{0.0pt}^{*}\\;} (U_3,\\varepsilon )$ with $(f,X_f) \\in U_2$ .","By lem:UBA-w we have $U_3 = \\delta _{\\mathit {det}}(\\lbrace (f,X_f)\\rbrace ,w)\\,.$ Hence $(U_0,v_0) {\\;\\xrightarrow{}{}\\hspace{0.0pt}^{*}\\;} X_w^{\\prime }$ .","Combining this reachability fact with the previous argument, we infer that the probability that some branch of a $\\mathcal {B}_{\\mathit {det}}^{\\prime }[X_w^{\\prime }]$ -tree has only nodes of types from which $X_w^{\\prime }$ is reachable (in $\\mathcal {B}_{\\mathit {det}}^{\\prime }$ ) is positive.","It follows from [8] that the probability that some branch of a $\\mathcal {B}_{\\mathit {det}}^{\\prime }[X_w^{\\prime }]$ -tree is accepting is positive.In terms of the notation therein, we instantiate [8] with $X := X_w^{\\prime }$ .","By the reachability argument above, $N_X = N_{X_w^{\\prime }}$ does not include types of the form $(U,v)$ with $U \\ne \\emptyset $ .","Thus, we have argued that the probability of $\\mathsf {F}N_{X_w^{\\prime }}$ is $p < 1$ .","Hence, [8] asserts that the probability that a $\\mathcal {B}_{\\mathit {det}}^{\\prime }[X_w^{\\prime }]$ -tree has an $X_w^{\\prime }$ -branch equals $1-p > 0$ , where $X_w^{\\prime }$ -branch means accepting path in terms of our definition.","Hence, the probability that some branch of a $\\mathcal {B}_{\\mathit {det}}^{\\prime }$ -tree is accepting is positive.","From the construction of $\\mathcal {B}_{\\mathit {det}}^{\\prime }$ it follows that the probability that some branch of a $\\mathcal {B}[X_f]$ -tree is accepted by $\\mathcal {A}_{\\mathit {det}}^{\\prime }$ is positive, i.e., “(iii)”."],["Proof of thm:coNBA-1","* Towards membership in PSPACE, fix a BP $\\mathcal {B}$ and an NBA $\\mathcal {A}$ .","Since PSPACE is closed under complement, we can focus on the problem whether the probability is positive that a $\\mathcal {B}$ -tree has a branch accepted by $\\mathcal {A}$ .","We use lem:UBA-X1-f.","Since reachability in a graph is in NL and, hence, in PSPACE, it suffices to decide in PSPACE whether the probability that some branch of a $\\mathcal {B}[X_f]$ -tree is accepted by $\\mathcal {A}[f,X_f]$ is positive.","In order to check that, by lem:Bdet it suffices to construct the BP $\\mathcal {B}_{\\mathit {det}}$ and then invoke the NC procedure of lem:AFT-1 to check if $\\mathbb {P}_{\\mathcal {B}_{\\mathit {det}}}(\\mathsf {F}\\lbrace \\emptyset \\rbrace ) < 1$ .","The BP $\\mathcal {B}_{\\mathit {det}}$ has exponential size but can be computed with a PSPACE transducer.","(In particular, whether a state in $\\mathcal {A}_{\\mathit {det}}$ is reachable from $\\lbrace \\bar{q}\\rbrace $ via a nonempty path can be determined in NPSPACE $=$ PSPACE.)","By lem:PSPACE-transducer it follows that $\\mathbb {P}(\\textup {coNBA}) = 1$ is in PSPACE.","PSPACE-hardness follows from the PSPACE-hardness [34] of the probabilistic emptiness problem, which, given a Markov chain and an NBA, asks if the probability is 0 that the Markov chain generates a word accepted by the NBA.","(We remark that, in contrast, the problem whether a given transition system has a run accepted by a given NBA is in NL: search the product for an accepting cycle)."],["Proof of thm:coNBA-0","* Towards membership in EXPTIME, an NBA can be translated, in exponential time, to a DPA of exponential size; see, e.g., [28].","By shifting the priorities (colours) in the DPA by 1, we can make the DPA accept exactly those words that are rejected by the NBA.","Since $\\mathbb {P}(\\textup {DPA}) = 0$ is in P by thm:DPA-0, it follows that $\\mathbb {P}(\\textup {coNBA}) = 0$ is in EXPTIME.","Concerning EXPTIME-hardness, we adapt the construction of the proof of thm:NBA-0.","As in that proof, let $M = (S_\\exists , S_\\forall , \\Sigma , T, s_0, s_{\\mathit {acc}})$ be a linear-bounded alternating Turing machine.","Let $w = a_1 \\cdots a_n \\in \\Sigma ^*$ be the input word, and assume again that $M$ uses exactly $n$ tape cells.","We construct a BP $\\mathcal {B}$ and an NBA $\\mathcal {A}$ such that the probability that all branches of the random tree are rejected by $\\mathcal {A}$ is positive if and only $M$ accepts $w$ .","For $\\mathcal {B}$ we use almost the same construction as in thm:NBA-0, except that we do not need the types $(1,\\mathit {chk}), \\ldots , (n,\\mathit {chk})$ .","We replace them by a single type $\\mathit {end}$ .","Accordingly, for all $(i, s_{\\mathit {acc}}, a) \\in \\Gamma $ , we replace the rule $(i, s_{\\mathit {acc}}, a) &{1} (1,\\mathit {chk}) \\cdots (n,\\mathit {chk})\\multicolumn{2}{l}{\\text{by a rule}}\\\\(i, s_{\\mathit {acc}}, a) &{1} \\mathit {end}\\,,$ and include a rule $\\mathit {end}{1} \\mathit {end}$ (i.e., a self-loop).","We want to construct the NBA $\\mathcal {A}$ so that it accepts exactly those branches that correspond to infinite computations that do not arrive at $s_{\\mathit {acc}}$ , or to “non-computations”, i.e., where the “guessing” in a rule $(i,(s,a,a^{\\prime },D,s^{\\prime })) {1/|\\Sigma |} (i{+}D, s^{\\prime }, b_j) \\quad (1 \\le j \\le |\\Sigma |)\\,,$ has been wrong.","The NBA $\\mathcal {A}= (Q, \\Gamma , \\delta , Q_0, Q)$ has the same set of states as in thm:NBA-0: $Q \\ = \\ (\\lbrace 1, \\ldots , n\\rbrace \\times \\Sigma ) \\ \\cup \\ \\lbrace f\\rbrace $ As in thm:NBA-0, the set of initial states is $Q_0 = \\lbrace (1,a_1), \\ldots , (n,a_n)\\rbrace $ (recall that $a_1 \\cdots a_n$ is the input word).","As in thm:NBA-0, the idea is that if a prefix $X_1 \\cdots X_k \\in \\Gamma ^*$ of a tree branch corresponds to a prefix of a correct computation of $M$ , then the set of automaton states in $\\delta (Q_0,X_1 \\cdots X_k)$ corresponds to the tape after this computation prefix.","In fact, the transition relation $\\delta \\subseteq Q \\times \\Gamma \\times Q$ is deterministic, i.e., for all $q \\in Q$ and $X \\in \\Gamma $ there is at most one $q^{\\prime }$ with $(q,X,q^{\\prime }) \\in \\delta $ .","Moreover, for any transition $((i,a), X, (j,a^{\\prime })) \\in \\delta $ we will have $i=j$ .","Unlike in thm:NBA-0, all states are accepting.","For $(i,a) \\in Q$ and all $(j,s,b) \\in \\Gamma $ with $i \\ne j$ or $a=b$ , include a self-loop $((i,a),(j,s,b),(i,a)) \\in \\delta \\,.$ (Intuitively, letter $a$ in cell $i$ is compatible with the head being on cell $j$ and reading letter $b$ .)","For $(i,a) \\in Q$ and all $(i,s,b) \\in \\Gamma $ with $a \\ne b$ , include a transition $((i,a),(i,s,b),f) \\in \\delta \\,.$ (Intuitively, letter $a$ in cell $i$ is not compatible with the head being on cell $i$ and reading letter $b$ ; i.e., the prefix of the branch does not correspond to a correct computation.)","For all $(i,a) \\in Q$ and all $(j,t) \\in \\Gamma $ with $i \\ne j$ , include a self-loop $((i,a),(j,t),(i,a)) \\in \\delta \\,.$ (Intuitively, the content of cell $i$ stays unchanged when the head is at position $j$ .)","For all $i,s,a,a^{\\prime },D,s^{\\prime }$ with $(i,a) \\in Q$ and $(i,(s,a,a^{\\prime },D,s^{\\prime })) \\in \\Gamma $ , include a transition $((i,a), (i,(s,a,a^{\\prime },D,s^{\\prime })), (i,a^{\\prime }))\\in \\delta \\,.$ (Intuitively, the transition $(s,a,a^{\\prime },D,s^{\\prime })$ changes the content of cell $i$ from $a$ to $a^{\\prime }$ .)","For all $X \\in \\Gamma $ , include a self-loop $(f,X,f) \\in \\delta $ .","Note that $(f,\\mathit {end},f)$ is the only transition labeled with $\\mathit {end}$ .","In this way: If a tree branch does not correspond to a correct computation, the automaton $\\mathcal {A}$ enters the state $f$ , remains there forever, and, thus, accepts.","If a tree branch corresponds to an infinite computation not entering $s_{\\mathit {acc}}$ , the set of states that $\\mathcal {A}$ can be in always reflects the tape.","Thus, $\\mathcal {A}$ accepts.","If a tree branch corresponds to a computation entering $s_{\\mathit {acc}}$ , the branch also enters $\\mathit {end}$ , and $\\mathcal {A}$ does not enter $f$ .","Thus, $\\mathcal {A}$ rejects.","It follows that the probability that all branches of the random tree are rejected by $\\mathcal {A}$ is positive if and only $M$ accepts $w$ .","A more detailed argument would follow very similar lines as the proof of thm:NBA-0."],["Proof of lem:key","* The automaton $\\mathcal {A}[f,X_f]$ is unambiguous, as $\\mathcal {A}$ is unambiguous, and has $(f,X_f)$ as (the only) accepting state.","Recall also that $(f,X_f)$ is reachable from all states in $\\mathcal {A}[f,X_f]$ .","It follows that $\\mathcal {A}[f,X_f]$ does not have diamonds, i.e., $\\mathcal {A}[f,X_f]$ does not have states $(q,X), (q^{\\prime },X^{\\prime })$ and a word $u \\in \\Gamma ^*$ such that $\\mathcal {A}[f,X_f]$ has two different paths $(q,X) {\\;\\xrightarrow{}{}\\hspace{0.0pt}^{*}\\;} (q^{\\prime },X^{\\prime })$ .","By the Perron-Frobenius theorem [3], $M$ has an eigenvector $v \\in (0,\\infty )^{Q[f,X_f]}$ (all entries positive) with $M v = \\rho v$ .","(Think of the entries of $v$ as “weights” of the states in $\\mathcal {A}[f,X_f]$ .","Loosely speaking, the equality $(M v)_{(q,X)} = \\rho v_{(q,X)}$ expresses that the expected combined weight of the “successors” of $(q,X)$ is equal to the weight of $(q,X)$ multiplied by $\\rho $ .)","We “lift” $v$ to define a vector $\\bar{v} \\in [0,\\infty )^{\\Gamma ^{\\prime }}$ where $\\Gamma ^{\\prime }$ is the set of types in $\\mathcal {B}_{\\mathit {det}}$ (recall that they are of the form $P \\times \\lbrace X\\rbrace $ with $P \\subseteq Q$ and $X \\in \\Gamma $ ): $\\bar{v}_{P \\times \\lbrace X\\rbrace } \\ := \\ \\sum _{q \\in P} v_{(q,X)}$ Denote by $\\bar{M} \\in \\mathbb {Q}^{\\Gamma ^{\\prime } \\times \\Gamma ^{\\prime }}$ the matrix defined before lem:as-finiteness-char, but for $\\mathcal {B}_{\\mathit {det}}$ .","Then we have for all $P \\times \\lbrace X\\rbrace \\in \\Gamma ^{\\prime }$ : $(\\bar{M} \\bar{v})_{P \\times \\lbrace X\\rbrace }&\\ =\\ \\sum _{X {p} X_1 \\cdots X_k} p \\sum _{i=1}^k \\bar{v}_{\\delta _{\\mathit {det}}(P \\times \\lbrace X\\rbrace , X_i)} \\\\&\\ =\\ \\sum _{X {p} X_1 \\cdots X_k} p \\sum _{i=1}^k \\ \\sum _{(r,X_i) \\in \\delta _{\\mathit {det}}(P \\times \\lbrace X\\rbrace , X_i)} v_{(r,X_i)} \\\\&\\ \\mathop {=}^*\\ \\sum _{X {p} X_1 \\cdots X_k} p \\sum _{i=1}^k \\ \\sum _{q \\in P} \\ \\sum _{r : (q,X) \\xrightarrow{} (r,X_i)} v_{(r,X_i)} \\\\&\\ =\\ \\sum _{q \\in P} \\ \\sum _{X {p} X_1 \\cdots X_k} p \\sum _{i=1}^k \\ \\sum _{r : (q,X) \\xrightarrow{} (r,X_i)} v_{(r,X_i)} \\\\&\\ =\\ \\sum _{q \\in P} (M v)_{(q,X)}\\\\&\\ =\\ \\sum _{q \\in P} \\rho v_{(q,X)}\\\\&\\ =\\ \\rho \\bar{v}_{P \\times \\lbrace X\\rbrace }\\;,$ where the third equality (marked with $\\displaystyle \\mathop {=}^*$ ) holds as for any $(r,X_i) \\in \\delta _{\\mathit {det}}(P \\times \\lbrace X\\rbrace , X_i)$ there is exactly one $q \\in P$ with $(q,X) \\xrightarrow{} (r,X_i)$ in $\\mathcal {A}[f,X_f]$ .","Indeed, towards a contradiction, suppose there are $q_1, q_2 \\in Q$ with $q_1 \\ne q_2$ and $(q_j,X) \\xrightarrow{} (r,X_i)$ for both $j \\in \\lbrace 1,2\\rbrace $ .","Since $P \\times \\lbrace X\\rbrace $ is reachable in $\\mathcal {A}_{\\mathit {det}}$ from $\\lbrace (f,X_f)\\rbrace $ , there is $u \\in \\Gamma ^*$ with $(f,X_f) {\\;\\xrightarrow{}{}\\hspace{0.0pt}^{*}\\;} (q_j,X) \\xrightarrow{} (r,X_i)$ in $\\mathcal {A}[f,X_f]$ for both $j \\in \\lbrace 1,2\\rbrace $ , contradicting the absence of diamonds in $\\mathcal {A}[f,X_f]$ .","We conclude from the above computation that $\\bar{M} \\bar{v} = \\rho \\bar{v}$ ; i.e., $\\bar{v}$ is an eigenvector of $\\bar{M}$ with eigenvalue $\\rho $ .","In the following, for subsets $\\Delta \\subseteq \\Gamma ^{\\prime }$ , we write $\\bar{M}_{\\Delta } \\in \\mathbb {Q}^{\\Delta \\times \\Delta }$ for the (square) principal submatrix obtained from $\\bar{M}$ by restricting it to the rows and columns indexed by elements of $\\Delta $ .","Similarly, define $\\bar{v}_{\\Delta } \\in [0,\\infty )^{\\Delta }$ by restricting $\\bar{v}$ to the entries indexed by elements of $\\Delta $ .","Since $\\bar{M}$ and $\\bar{v}$ are nonnegative, we have $\\bar{M}_{\\Delta } \\bar{v}_{\\Delta } \\le \\rho \\bar{v}_{\\Delta }$ (the inequality is meant componentwise).","Note that $\\bar{v}_\\emptyset = 0$ .","Define $\\Gamma ^{\\prime \\prime } := \\Gamma ^{\\prime } \\setminus \\lbrace \\emptyset \\rbrace $ .","Thus we have $\\bar{M}_{\\Gamma ^{\\prime \\prime }} \\bar{v}_{\\Gamma ^{\\prime \\prime }} = \\rho \\bar{v}_{\\Gamma ^{\\prime \\prime }}$ .","All entries of $\\bar{v}_{\\Gamma ^{\\prime \\prime }}$ are positive, as all entries of $v$ are.","By Perron-Frobenius theory [3] it follows that $\\rho $ is the spectral radius of $\\bar{M}_{\\Gamma ^{\\prime \\prime }}$ .","The matrix $\\bar{M}_{\\Gamma ^{\\prime \\prime }}$ is equal to the matrix defined before lem:as-finiteness-char, but for $\\mathcal {B}_{\\mathit {det}}^{\\prime \\prime }$ .","We complete the proof with the following case distinction.","Suppose $\\rho > 1$ .","Let $\\Delta \\subseteq \\Gamma ^{\\prime \\prime }$ be a bottom SCC of the graph of $\\bar{M}_{\\Gamma ^{\\prime \\prime }}$ .","As $\\Delta $ is bottom, $\\bar{M}_\\Delta \\bar{v}_\\Delta = \\rho \\bar{v}_\\Delta $ .","So the spectral radius of $\\bar{M}_\\Delta $ is at least $\\rho > 1$ (in fact, it must be equal to $\\rho $ ).","It follows that $\\Delta $ is supercritical in $\\mathcal {B}_{\\mathit {det}}^{\\prime \\prime }$ .","Thus, by lem:as-finiteness-char, a $\\mathcal {B}_{\\mathit {det}}^{\\prime \\prime }$ -tree is infinite with positive probability.","Suppose that $\\rho = 1$ and that $\\mathcal {A}[f,X_f]$ does not have proper branching.","Let $\\Delta \\subseteq \\Gamma ^{\\prime \\prime }$ be a bottom SCC of the graph of $\\bar{M}_{\\Gamma ^{\\prime \\prime }}$ .","Then $\\bar{M}_\\Delta \\bar{v}_\\Delta = \\bar{v}_\\Delta $ , so by the Perron-Frobenius theorem [3] the spectral radius of $\\bar{M}_\\Delta $ is 1.","By the absence of proper branching we also have $\\bar{M}_\\Delta \\vec{1} \\le \\vec{1}$ , where $\\vec{1}$ denotes the all-1 vector, i.e., the element of $\\lbrace 1\\rbrace ^\\Delta $ .","By Perron-Frobenius theory [3] it follows that $\\bar{M}_\\Delta \\vec{1} = \\vec{1}$ .","Thus, $\\Delta $ is linear in $\\mathcal {B}_{\\mathit {det}}^{\\prime \\prime }$ .","Hence, by lem:as-finiteness-char, a $\\mathcal {B}_{\\mathit {det}}^{\\prime \\prime }$ -tree is infinite with positive probability.","Suppose that $\\rho =1$ and that $\\mathcal {A}[f,X_f]$ has proper branching, i.e., there exist $(q,Y) \\xrightarrow{} (r_1,Z_1) \\text{ and } (q,Y) \\xrightarrow{} (r_2,Z_2) \\text{ in } \\mathcal {A}[f,X_f]$ and a rule $Y {p} u_1 Z_1 u_2 Z_2 u_3$ with $u_1, u_2, u_3 \\in \\Gamma ^*$ .","Consider any SCC $\\Delta \\subseteq \\Gamma ^{\\prime \\prime }$ of the graph of $\\bar{M}_{\\Gamma ^{\\prime \\prime }}$ .","Denote by $\\rho _\\Delta $ the spectral radius of $\\bar{M}_\\Delta $ .","As $\\bar{M}_\\Delta $ is a principal submatrix of $\\bar{M}$ , we have $\\rho _\\Delta \\le \\rho = 1$ [3].","So $\\Delta $ is not supercritical.","$\\rho _\\Delta < 1$ .","We have argued before lem:as-finiteness-char that $\\Delta $ being linear would imply $\\rho _\\Delta =1$ .","Hence, $\\Delta $ is not linear.","$\\rho _\\Delta = 1$ .","Recall that $\\bar{M}_{\\Delta } \\bar{v}_{\\Delta } \\le \\bar{v}_{\\Delta }$ , so by Perron-Frobenius theory [3] we must have $\\bar{M}_{\\Delta } \\bar{v}_{\\Delta } = \\bar{v}_{\\Delta }$ .","Let $P \\times \\lbrace X\\rbrace \\in \\Delta $ and $p \\in P$ .","Let $u_0 \\in \\Gamma ^*$ with $(p,X) {\\;\\xrightarrow{}{}\\hspace{0.0pt}^{*}\\;} (q,Y)$ in $\\mathcal {A}[f,X_f]$ .","Hence $(p,X) {\\;\\xrightarrow{}{}\\hspace{0.0pt}^{*}\\;} (q,Y) \\xrightarrow{} (r_j,Z_j)$ in $\\mathcal {A}[f,X_f]$ for both $j \\in \\lbrace 1,2\\rbrace $ .","Towards a contradiction, suppose $\\Delta $ is linear.","If $\\delta _{\\mathit {det}}(P \\times \\lbrace X\\rbrace , u_0)$ is not in $\\Delta $ , then neither are $\\delta _{\\mathit {det}}(P \\times \\lbrace X\\rbrace , u_0 Z_1)$ or $\\delta _{\\mathit {det}}(P \\times \\lbrace X\\rbrace , u_0 Z_2)$ .","Otherwise (i.e., $\\delta _{\\mathit {det}}(P \\times \\lbrace X\\rbrace , u_0) \\in \\Delta $ ) there is $j \\in \\lbrace 1,2\\rbrace $ such that $\\delta _{\\mathit {det}}(P \\times \\lbrace X\\rbrace , u_0 Z_j) \\notin \\Delta $ , as $\\Delta $ is linear.","Either way there is $j \\in \\lbrace 1,2\\rbrace $ such that $\\delta _{\\mathit {det}}(P \\times \\lbrace X\\rbrace , u_0 Z_j) \\notin \\Delta \\,.$ Note that $(r_j,Z_j) \\in \\delta _{\\mathit {det}}(P \\times \\lbrace X\\rbrace , u_0 Z_j) \\ne \\emptyset $ .","Let $x Z$ (with $x \\in \\Gamma ^*$ and $Z \\in \\Gamma $ ) be the shortest prefix of $u_0 Z_j$ such that $U := \\delta _{\\mathit {det}}(P \\times \\lbrace X\\rbrace , x) \\in \\Delta $ but $\\delta _{\\mathit {det}}(U,Z) \\notin \\Delta $ .","Then we have: $\\bar{v}_U&\\ =\\ (\\bar{M}_\\Delta \\bar{v}_\\Delta )_U \\\\&\\ <\\ (\\bar{M}_\\Delta \\bar{v}_\\Delta )_U + \\bar{M}_{U,\\delta _{\\mathit {det}}(U,Z)} \\bar{v}_{\\delta _{\\mathit {det}}(U,Z)} \\\\&\\ \\le \\ (\\bar{M} \\bar{v})_U \\\\&\\ =\\ \\bar{v}_U\\,,$ a contradiction.","Thus, $\\Delta $ is not linear.","We conclude that in both cases $\\Delta $ is neither linear nor supercritical.","Since $\\Delta $ was an arbitrary SCC, we conclude from lem:as-finiteness-char that a $\\mathcal {B}_{\\mathit {det}}^{\\prime \\prime }$ -tree is almost surely finite.","Suppose $\\rho < 1$ .","Then, for any SCC $\\Delta $ the spectral radius of $\\bar{M}_{\\Delta }$ is also less than 1 [3], so $\\Delta $ is neither supercritical nor linear.","Thus, by lem:as-finiteness-char, a $\\mathcal {B}_{\\mathit {det}}^{\\prime \\prime }$ -tree is almost surely finite.","Hence, the probability that a $\\mathcal {B}_{\\mathit {det}}^{\\prime \\prime }$ -tree is infinite is positive if and only if either $\\rho > 1$ or $\\rho = 1$ and $\\mathcal {A}[f,X_f]$ does not have proper branching."],["Proof of thm:LTL-0","* Given the main body, it remains to show 2EXPTIME-hardness of $\\mathbb {P}(\\textup {LTL}) = 0$ .","Our construction is inspired by the proof of 2EXPTIME-hardness of model-checking a concurrent probabilistic program (another name for MDP) against an LTL formula, see Theorem 3.2.1 in [11].","We will use the fact that 2EXPTIME is equal to alternating EXPSPACE.","Let $M = (S_\\exists , S_\\forall , \\Sigma , T, s_0, s_{\\mathit {acc}})$ be an alternating Turing machine whose work tape usage is bounded by $2^n$ on any input of length $n$ .","Without loss of generality, we can assume that the machine has two possible next moves for each configuration and that it halts when it reaches the accepting state $s_{\\mathit {acc}}$ .","For a given alternating TM $M$ and an input $w$ of length $n$ , we will construct a BP $\\mathcal {B}$ and an LTL formula $\\varphi $ both of size $\\mathcal {O}(n)$ such that $\\mathbb {P}_\\mathcal {B}(\\varphi ) > 0$ if and only if $M$ accepts $w$ .","The branching process $\\mathcal {B}$ is defined by the diagram in Fig.","REF .","Every node in the diagram has a unique label that corresponds to a type of $\\mathcal {B}$ , although not all nodes are explicitly labelled.","The $$ -nodes represent randomising branching and $$ -nodes represent tree branching.","Namely, if a $$ -node $x$ has $k$ successors $y_1,\\ldots , y_k$ : $@1@R=10pt@C=10pt{ & *+[l]{x\\,} [dr] [dl] &\\\\y_1 & \\cdots &y_k}$$, then $ B$ has the rules $ x 1/k yi$ for $ i=1,...,k$.","On the other hand, if a $$-node $ x$ has $ k$ successors $ y1,..., yk$: $$, then $ B$ has the rule $ x 1 y1yk$.$ The start type of $\\mathcal {B}$ is $a$ .","The detailed diagram of the blocks $I$ , $O$ , $N$ , $D_1$ , $D_2$ and $D_3$ is shown in Fig.","REF on the left.","All nodes in these blocks are $$ -nodes.","The diagram of the blocks $C_1$ and $C_2$ is shown in Fig.","REF on the right.","These blocks have $$ -nodes in the first $n+1$ levels, and the rest are $$ -nodes.","The initial and final nodes are labelled by $u$ and $v$ , respectively.","The nodes below $u$ are labelled with $(\\ell _i,0)$ and $(\\ell _i,1)$ , $i=1,\\ldots ,n$ , as shown in the picture.","Every block has its own unique set of labels $u$ , $v$ , $(\\ell _i,0)$ and $(\\ell _i,1)$ , $i=1,\\ldots ,n$ ; however we do not distinguish them in the diagram for simplicity.","In every block, except for $I$ , the nodes above $v$ are labelled by $(\\ell _{n+1},\\delta )$ , where $\\delta \\in \\Sigma \\;\\cup \\; (S_\\exists \\cup S_\\forall )\\times \\Sigma $ ranges over the symbols of the extended work tape alphabet.","In block $I$ , there are $n+1$ nodes above $v$ which are labelled by $(\\ell _{n+1},\\gamma _i)$ for $i=1,\\ldots ,n+1$ such that for an input word $w=w_1\\ldots w_n$ we have $\\gamma _1=(s_0,w_1)$ , $\\gamma _i=w_i$ for $1 0$ .","Conversely, suppose there is a positive probability that $\\mathcal {B}$ generates a tree $T$ all whose branches satisfy $\\varphi $ .","Hence every branch of $T$ eventually reaches state $d$ (and then always repeats it).","Since $T$ is finitely branching, it follows by König's lemma that there exists a finite prefix of $T$ whose every leaf is labelled by $d$ .","This prefix encodes the moves of the existential player (after each appearance of state $m_1$ in the prefix) that allow him to reach the accepting configuration, no matter what the universal player does.","In other words, the existential player has a winning strategy, and hence $M$ accepts $w$ .","Formally, this can be shown along similar lines as in the proof of thm:NBA-0.","Therefore, we proved that $M$ accepts $w$ if and only if $\\mathbb {P}_\\mathcal {B}(\\varphi ) > 0$ .","Figure: Diagrams of the randomising (left) and tree (right) branching blocks.Figure: Diagram of the branching process ℬ\\mathcal {B}."]],"string":"[\n [\n \"Linear-Time Model Checking Branching Processes\"\n ],\n [\n \"Abstract (Multi-type) branching processes are a natural and well-studied model for generating random infinite trees.\",\n \"Branching processes feature both nondeterministic and probabilistic branching, generalizing both transition systems and Markov chains (but not generally Markov decision processes).\",\n \"We study the complexity of model checking branching processes against linear-time omega-regular specifications: is it the case almost surely that every branch of a tree randomly generated by the branching process satisfies the omega-regular specification?\",\n \"The main result is that for LTL specifications this problem is in PSPACE, subsuming classical results for transition systems and Markov chains, respectively.\",\n \"The underlying general model-checking algorithm is based on the automata-theoretic approach, using unambiguous B\\\\\\\"uchi automata.\"\n ],\n [\n \"Introduction\",\n \"Checking whether a (labelled) transition system satisfies a linear-time specification is a staple in verification.\",\n \"The specification is often given as a formula of linear temporal logic (LTL).\",\n \"While early procedures for LTL model checking work directly with the formula [24], the automata-theoretic approach translates LTL formulas into finite automata on infinite words, such as Büchi automata, and analyzes a product of the system and the automaton [36].\",\n \"This approach can lead to clean and modular model-checking algorithms.\",\n \"Although LTL captures only a subset of $\\\\omega $ -regular languages, model-checking algorithms based on the automata-theoretic approach can be made optimal from the point of view of computational complexity.\",\n \"In particular, model checking finite transition systems against LTL specifications is PSPACE-complete [31], and the algorithm [36] that, loosely speaking, translates (the negation of) the LTL formula into a Büchi automaton and checks the product with the transition system for emptiness can indeed be implemented in PSPACE.\",\n \"The same approach does not directly work for probabilistic systems modelled as finite Markov chains: intuitively, the nondeterminism in a Büchi automaton causes issues in a stochastic setting where the specification should hold with probability 1, i.e., almost surely but not necessarily surely.\",\n \"A possible remedy is to translate the nondeterministic Büchi automaton further into a deterministic automaton, e.g., a deterministic Rabin automaton (deterministic Büchi automata are less expressive), with which the Markov chain can be naturally instrumented and subsequently analyzed.\",\n \"This determinization step causes a (second) exponential blowup and does not lead to algorithms that are optimal from a computational-complexity point of view.\",\n \"However, for Markov decision processes (MDPs), which allow for nondeterminism in the probabilistic system, this approach is adequate and leads to an optimal, double-exponential time, model-checking algorithm.\",\n \"Checking whether a Markov chain satisfies an LTL specification with probability 1 is PSPACE-complete, but membership in PSPACE was proved only in [10], [11], not using the automata-theoretic approach but by a recursive procedure on the formula.\",\n \"This raised the question if there is also an optimal algorithm based on the automata-theoretic approach; see [35] for a survey of the state of the art at the end of the 90s.\",\n \"The answer is yes and was first given in [12], using a single-exponential translation from LTL to separated Büchi automata.\",\n \"Such automata are special unambiguous Büchi automata, which restrict nondeterministic Büchi automata by requiring that every word have at most one accepting run.\",\n \"Another algorithm, using alternating Büchi automata, was proposed in [6], exploiting reverse determinism, a property also related to unambiguousness.\",\n \"A polynomial-time (even NC) model-checking algorithm for Markov chains against general unambiguous Büchi automata was given in [2].\",\n \"These works all imply optimal PSPACE algorithms for LTL model checking of Markov chains via the automata-theoretic approach.\",\n \"In this paper we exhibit an LTL model checking algorithm that has the following features: (1) it applies to (multi-type) branching processes, a well established model for random trees, generalizing both nondeterministic transition systems and Markov chains; (2) it runs in PSPACE, which is the optimal complexity both for nondeterministic transition systems and Markov chains; and (3) it is based on the automata-theoretic approach (using unambiguous Büchi automata).\",\n \"The fact that there exists an algorithm with the first two features might seem surprising, as one might think that any system model that encompasses both nondeterminism and probability will generalize MDPs, for which LTL model checking is 2EXPTIME-complete [11].\",\n \"Branching processes (BPs) are a well-studied model in mathematics with applications in numerous fields including biology, physics and natural language processing; see, e.g., [23], [1], [22].\",\n \"BPs randomly generate infinite trees, and, from a computer-science point of view, they might be the most natural model to do so: (multi-type) BPs can be thought of as a version of stochastic context-free grammars without terminal symbols, randomly generating infinite derivation trees.\",\n \"For example, consider the following BP, taken from [8], with 3 types $I, B, D$ : $&I {0.9} I && B {0.2} D && D {1} D \\\\nonumber \\\\\\\\[-1mm]&I {0.1} I B && B {0.5} B \\\\\\\\[-1mm]& && B {0.3} B B \\\\nonumber $ This BP might generate a tree with the following prefix: [yscale=0.8] 0) at (0,0) $I$ ; 1) at (-2,-1) $I$ ; 2) at (2,-1) $B$ ; 11) at (-2,-2) $I$ ; 111) at (-3,-3) $I$ ; 112) at (-1,-3) $B$ ; 21) at (1,-2) $B$ ; 22) at (3,-2) $B$ ; 211) at (1,-3) $B$ ; 221) at (3,-3) $D$ ; (0)–(1); (0)–(2); (1)–(11); (11)–(111); (11)–(112); (2)–(21); (2)–(22); (21)–(211); (22)–(221); [fill] (-3,-3.4) circle (0.7pt); [fill] (-3,-3.6) circle (0.7pt); [fill] (-3,-3.8) circle (0.7pt); [fill] (-1,-3.4) circle (0.7pt); [fill] (-1,-3.6) circle (0.7pt); [fill] (-1,-3.8) circle (0.7pt); [fill] (1,-3.4) circle (0.7pt); [fill] (1,-3.6) circle (0.7pt); [fill] (1,-3.8) circle (0.7pt); [fill] (3,-3.4) circle (0.7pt); [fill] (3,-3.6) circle (0.7pt); [fill] (3,-3.8) circle (0.7pt); The probability that the BP generates a tree with the shown prefix is the product of the probabilities of the fired transition rules, i.e., (in breadth-first order) $0.1 \\\\cdot 0.9 \\\\cdot 0.3 \\\\cdot 0.1 \\\\cdot 0.5 \\\\cdot 0.2$ .\",\n \"BPs generalize transition systems.\",\n \"Consider the following transition system: MDPrand] (X) at (0,0) $X$ ; MDPrand] (Y) at (2,0) $Y$ ; [->] (-1,0) to (X); [->] (X) edge[bend left] (Y); [->] (Y) edge[bend left] (X); [->] (Y) edge[loop right,looseness=12] (Y); It is equivalent to the BP with $X {1} Y$ and $Y {1} X Y$ , which generates with probability 1 the following unique tree: [yscale=0.8] 0) at (0,0) $X$ ; 1) at (0,-1) $Y$ ; 2l) at (-2,-2) $X$ ; 2r) at ( 2,-2) $Y$ ; 3l) at (-2,-3) $Y$ ; 3rl) at (1,-3) $X$ ; 3rr) at (3,-3) $Y$ ; [fill] (-2,-3.4) circle (0.7pt); [fill] (-2,-3.6) circle (0.7pt); [fill] (-2,-3.8) circle (0.7pt); [fill] ( 1,-3.4) circle (0.7pt); [fill] ( 1,-3.6) circle (0.7pt); [fill] ( 1,-3.8) circle (0.7pt); [fill] ( 3,-3.4) circle (0.7pt); [fill] ( 3,-3.6) circle (0.7pt); [fill] ( 3,-3.8) circle (0.7pt); (0) – (1); (1) – (2l); (1) – (2r); (2l) – (3l); (2r) – (3rl); (2r) – (3rr); The branches of this unique tree are exactly the executions of the transition system.\",\n \"As a consequence, any LTL formula holds on all executions of the transition system if and only if it holds (with probability 1) on all branches of the generated tree.\",\n \"BPs also generalize Markov chains.\",\n \"Consider the following Markov chain: MDPrand] (X) at (0,0) $X$ ; MDPrand] (Y) at (2,0) $Y$ ; [->] (-1,0) to (X); [->] (X) edge[bend left] node[above] 1 (Y); [->] (Y) edge[bend left] node[below] $0.3$ (X); [->] (Y) edge[loop right,looseness=12] node[right] $0.7$ (Y); It is equivalent to the BP with $X {1} Y$ and $Y {0.3} X$ and $Y {0.7} Y$ , which generates, with probabilities $0.3$ , $0.7 \\\\cdot 0.3$ , $0.7 \\\\cdot 0.7$ , respectively, the following prefixes of (degenerated) trees: [yscale=0.8] 00) at (0, 0) $X$ ; 10) at (2, 0) $X$ ; 20) at (4, 0) $X$ ; 01) at (0,-1) $Y$ ; 11) at (2,-1) $Y$ ; 21) at (4,-1) $Y$ ; 02) at (0,-2) $X$ ; 12) at (2,-2) $Y$ ; 22) at (4,-2) $Y$ ; 03) at (0,-3) $Y$ ; 13) at (2,-3) $X$ ; 23) at (4,-3) $Y$ ; [fill] (0,-3.4) circle (0.7pt); [fill] (0,-3.6) circle (0.7pt); [fill] (0,-3.8) circle (0.7pt); [fill] (2,-3.4) circle (0.7pt); [fill] (2,-3.6) circle (0.7pt); [fill] (2,-3.8) circle (0.7pt); [fill] (4,-3.4) circle (0.7pt); [fill] (4,-3.6) circle (0.7pt); [fill] (4,-3.8) circle(0.7pt); (00) – (01) – (02) – (03); (10) – (11) – (12) – (13); (20) – (21) – (22) – (23); Here, each possible “tree” has only a single branch, and the possible “trees” are distributed in the same way as the possible executions of the Markov chain.\",\n \"As a consequence, any LTL formula holds with probability 1 on a random execution of the Markov chain if and only if it holds with probability 1 on the (single) branch of the generated tree.\",\n \"Hence, both for the transition system and for the Markov chain, the respective model-checking question reduces to the BP model-checking problem which asks whether with probability 1 the property holds on all branches.\",\n \"For LTL specifications, we refer to this BP model-checking problem as $\\\\mathbb {P}(\\\\textup {LTL})=1$ .\",\n \"Our main result is that it is in PSPACE, generalizing the corresponding classical results on transition systems and Markov chains.\",\n \"As mentioned, our model-checking algorithm is based on the automata-theoretic approach, in particular on unambiguous Büchi automata.\",\n \"Another important technical ingredient is the algorithmic analysis of certain nonnegative matrices in terms of their spectral radius.\",\n \"The latter points to the fact that the numbers in the system generally matter, even though we only consider the qualitative problem of comparing the satisfaction probability with 1.\",\n \"For example, for the BP given in (REF ), one can show that the probability that all branches eventually hit a node of type $D$ is less than 1 (in fact, it is 0).\",\n \"Intuitively, this is because the probability of “branching” via $B {0.3} B B$ is larger than the probability of “dying” via $B {0.2} D$ .\",\n \"Were the probabilities $0.3$ and $0.2$ swapped, the probability that all branches eventually hit a node of type $D$ would be 1; cf. [8].\",\n \"We also consider the problem $\\\\mathbb {P}(\\\\textup {LTL}= 0)$ , which asks whether the probability that all branches satisfy a given LTL formula is 0.\",\n \"Even though it is trivial to negate an LTL formula, this problem is (unlike in Markov chains) not equivalent to the complement of $\\\\mathbb {P}(\\\\textup {LTL}= 1)$ , because even when the probability is less than 1 that the formula holds on all branches, the probability may still be 0 that the negated formula holds on all branches.\",\n \"We will show that $\\\\mathbb {P}(\\\\textup {LTL}= 0)$ is much more computationally complex than $\\\\mathbb {P}(\\\\textup {LTL}=1)$ : it is 2EXPTIME-complete.\",\n \"Besides LTL, we also consider automata-based specifications.\",\n \"Büchi automata are relevant from a verification point of view, as a way of specifying desired or undesired executions of the system.\",\n \"Unambiguous Büchi automata are useful from a technical point of view, in particular, to facilitate our main result on $\\\\mathbb {P}(\\\\textup {LTL}=1)$ .\",\n \"See sec:prelims,tab:map for definitions of our problems and a map of our results.\",\n \"Readers familiar with MDPs may wonder how the problem $\\\\mathbb {P}(\\\\textup {LTL})=1$ can have lower computational complexity than the problem whether all schedulers of an MDP satisfy an LTL specification almost surely.\",\n \"Consider the BP $X &{1} Y_1 Y_2 & \\\\qquad Y_1 &{0.7} X & \\\\qquad Y_1 &{0.3} Z & \\\\qquad Y_2 &{0.5} X & \\\\qquad Y_2 &{0.5} Z& \\\\qquad Z &{1} Z\\\\,,$ which might be depicted graphically as follows: MDPrand] (X) at (0,0) $X$ ; MDPrand] (Y1) at (3,1) $Y_1$ ; MDPrand] (Y2) at (3,-1) $Y_2$ ; MDPrand] (Z) at (6,0) $Z$ ; [->] (-1,0) to (X); [->] (X) edge[bend left=20] (Y1); [->] (X) edge[bend right=20] (Y2); [->] (Y1) edge[bend left=0] node[pos=0.4,below] $0.7$ (X); [->] (Y2) edge[bend right=0] node[pos=0.4,above] $0.5$ (X); [->] (Y1) edge node[pos=0.4,below] $0.3$ (Z); [->] (Y2) edge node[pos=0.4,above] $0.5$ (Z); [->] (Z) edge[loop right,looseness=12] node[right] 1 (Z); One might view this BP as an MDP where in an $X$ -node the scheduler nondeterministically picks either the $Y_1$ - or the $Y_2$ -successor, and in an $Y_i$ -node, the $X$ - or the $Z$ -successor is chosen randomly.\",\n \"In such an MDP, regardless of the scheduler, a random run reaches with probability 1 a $Z$ -node.\",\n \"However, in the BP above, the probability is positive that some branch of a random tree never reaches a $Z$ -node.\",\n \"Although each branch of a random tree could be thought of as being witnessed by at least one scheduler, this is not a contradiction, as there are uncountably many schedulers (over which one cannot take a sum).\",\n \"Hence, if an MDP is interpreted as a BP in the way sketched above, then the requirement that the BP satisfy an LTL formula almost surely on all branches is stronger, and computationally less complex to check, than the requirement that the MDP satisfy, for each scheduler, the formula almost surely.\"\n ],\n [\n \"Related work.\",\n \"We have already discussed related work concerning model checking transition systems and Markov chains.\",\n \"In addition to the mentioned applications of BPs in various fields, there has also been work on BPs in computer science, especially in the last 10 years.\",\n \"This paper builds on [8], where specifications in terms of deterministic parity tree automata are considered.\",\n \"The work [8] implies decidability of the problems considered in this paper and some basic upper complexity bounds.\",\n \"For example, it is not hard to derive from [8] that $\\\\mathbb {P}(\\\\textup {LTL}=1)$ is in 2EXPTIME.\",\n \"Lowering this to PSPACE is the main achievement of this paper.\",\n \"A related strand of work considers regular tree languages; i.e., the specification is not in terms of a word automaton that is run on each branch but in terms of tree automata.\",\n \"Even measurability is not easy to show in this case [20], and fundamental decidability questions around computing the measure have been answered positively only for subclasses of regular tree languages [25], [26].\",\n \"Fundamental results on the complexity of algorithmically analyzing BPs have been obtained in [18].\",\n \"Indeed, in sec:as-finite we build on and improve results from [18] on finiteness (more often called “extinction” in the literature) of BPs.\",\n \"Another recent line of work considers extensions of BPs with nondeterminism, focusing on algorithmic questions about properties such as reachability.\",\n \"Branching MDPs, which are BPs where a controller chooses actions to influence the evolution of the tree, have been investigated, e.g., in [16], [17].\",\n \"Even branching games, featuring two adversarial controllers, have been studied recently [14].\",\n \"The work [21] also considers BPs with “internal” nondeterminism (as opposed to the “external” nondeterminism manifested as branching in the generated tree), along with model-checking problems against the logic GPL.\",\n \"This expressive, $\\\\mu $ -calculus based modal logic had been introduced in [9].\",\n \"The system model therein, called reactive probabilistic labeled transition systems (RPLTSs), is essentially equivalent to BPs as considered in this paper.\",\n \"BPs are related to models for probabilistic programs with recursion, such as Recursive Markov chains, for which model-checking problems have been studied in detail; see, in particular, [19].\",\n \"Very loosely speaking, a run of a (“1-exit”) Recursive Markov chain can be viewed as a depth-first traversal of a tree generated by a BP.\",\n \"Indeed, for a lower bound in the present paper (thm:NBA-0) we adapt a proof from [19].\",\n \"However, most qualitative model-checking problems for Recursive Markov chains are EXPTIME-complete [19], and so many of the BP problems we study turn out to have different computational complexity.\",\n \"As a key technical tool we use unambiguous Büchi automata, as recently proposed for Markov chains [2].\",\n \"It is non-trivial to extend their use to random trees, as the branching behaviour of BPs interferes with the spectral-radius based analysis from [2].\",\n \"One may view as the main technical insight of this paper that the limited nondeterminism in unambiguous automata can be combined with the tree branching of BPs, so that, in a sense, BP model checking reduces to comparing the spectral radius of a certain nonnegative matrix with 1 (prop:coUBA-1).\"\n ],\n [\n \"Preliminaries\",\n \"Let $\\\\mathbb {N}$ and $\\\\mathbb {N}_0$ denote the set of positive and nonnegative integers, respectively.\",\n \"For a finite set $\\\\Gamma $ , we write $\\\\Gamma ^*$ (resp., $\\\\Gamma ^+$ ) for the set of words (resp., nonempty words) over $\\\\Gamma $ .\"\n ],\n [\n \"Branching processes.\",\n \"A (multi-type) branching process (BP) is a tuple $\\\\mathcal {B}= (\\\\Gamma , \\\\mathord {{}}, \\\\mathit {Prob}, X_0)$ , where $\\\\Gamma $ is a finite set of types, $\\\\mathord {{}} \\\\subseteq \\\\Gamma \\\\times \\\\Gamma ^+$ is a finite set of transition rules, $\\\\mathit {Prob}$ is a function assigning positive rational probabilities to transition rules so that for every $X\\\\in \\\\Gamma $ we have $\\\\sum _{X {} w} \\\\mathit {Prob}(X {} w)=1$ , and $X_0\\\\in \\\\Gamma $ is the start type.\",\n \"We write $X {p} w$ to denote that $\\\\mathit {Prob}(X {} w) = p$ .\",\n \"Given a BP $\\\\mathcal {B}$ and a type $X \\\\in \\\\Gamma $ we write $\\\\mathcal {B}[X]$ for the BP obtained from $\\\\mathcal {B}$ by making $X$ the start type.\",\n \"For $X, Y \\\\in \\\\Gamma $ we call $Y$ a successor of $X$ if there is a rule $X {} u Y v$ for some $u, v \\\\in \\\\Gamma ^*$ .\",\n \"A BP with $\\\\varepsilon $ -rules allowed relaxes the requirement $\\\\mathord {{}} \\\\subseteq \\\\Gamma \\\\times \\\\Gamma ^+$ to $\\\\mathord {{}} \\\\subseteq \\\\Gamma \\\\times \\\\Gamma ^*$ , i.e., there may be rules of the form $X {} \\\\varepsilon $ , where $\\\\varepsilon $ denotes the empty word.\",\n \"In the following, we disallow $\\\\varepsilon $ -rules unless specified otherwise; but the definitions generalize in a natural way.\",\n \"Fix a BP $\\\\mathcal {B}= (\\\\Gamma , \\\\mathord {{}}, \\\\mathit {Prob}, X_0)$ for the rest of the section.\",\n \"Write $\\\\llbracket \\\\mathcal {B} \\\\rrbracket $ for the set of trees generated by $\\\\mathcal {B}$ ; i.e., $\\\\llbracket \\\\mathcal {B} \\\\rrbracket $ denotes the set of ordered $\\\\Gamma $ -labelled trees $t$ such that for each $X \\\\in \\\\Gamma $ and each $X$ -labelled node $v$ in $t$ , there is a rule $X {} X_1 \\\\cdots X_k$ , denoted by $\\\\mathit {rule}(v)$ , such that the $k$ ordered children of $v$ are labelled with $X_1, \\\\ldots , X_k$ , respectively.\",\n \"We say a node has type $X \\\\in \\\\Gamma $ if the node is labelled with $X$ .\",\n \"A finite prefix of a tree $t \\\\in \\\\llbracket \\\\mathcal {B} \\\\rrbracket $ is an ordered $\\\\Gamma $ -labelled finite tree obtained from $t$ by designating some nodes as leaves, and removing all their children, grandchildren, etc.\",\n \"Write $\\\\llparenthesis \\\\mathcal {B} \\\\rrparenthesis $ for the set of finite prefixes of trees generated by $\\\\mathcal {B}$ .\",\n \"For $t \\\\in \\\\llparenthesis \\\\mathcal {B} \\\\rrparenthesis $ write $t{\\\\downarrow } \\\\subseteq \\\\llbracket \\\\mathcal {B} \\\\rrbracket $ for the (“cylinder”) set of trees $t^{\\\\prime } \\\\in \\\\llbracket \\\\mathcal {B} \\\\rrbracket $ such that $t$ is a finite prefix of $t^{\\\\prime }$ .\",\n \"For $X \\\\in \\\\Gamma $ write $\\\\llbracket \\\\mathcal {B} \\\\rrbracket _X \\\\subseteq \\\\llbracket \\\\mathcal {B} \\\\rrbracket $ and $\\\\llparenthesis \\\\mathcal {B} \\\\rrparenthesis _X \\\\subseteq \\\\llparenthesis \\\\mathcal {B} \\\\rrparenthesis $ for the subsets of trees whose root has type $X$ ; the trees in $\\\\llbracket \\\\mathcal {B} \\\\rrbracket _X$ are called $X$ -trees.\",\n \"A branch of a tree $t$ is a sequence $v_0 v_1 \\\\cdots $ of nodes in $t$ , where $v_0$ is the root of $t$ and $v_{i+1}$ is a child of $v_i$ for all $i \\\\in \\\\mathbb {N}_0$ .\",\n \"See [8] for equivalent, more formal tree-related definitions.\",\n \"For each $X \\\\in \\\\Gamma $ we define the probability space $(\\\\llbracket \\\\mathcal {B} \\\\rrbracket _X,\\\\Sigma _X,\\\\mathbb {P}_X)$ , where $\\\\Sigma _X$ is the $\\\\sigma $ -algebra generated by $\\\\lbrace t{\\\\downarrow } \\\\mid t\\\\in \\\\llparenthesis \\\\mathcal {B} \\\\rrparenthesis _X\\\\rbrace $ , and $\\\\mathbb {P}_X$ is the probability measure generated by $\\\\mathbb {P}_X(t{\\\\downarrow }) := \\\\prod _{v} \\\\mathit {Prob}(\\\\mathit {rule}(v))$ for all $t \\\\in \\\\llparenthesis \\\\mathcal {B} \\\\rrparenthesis _X$ , where the product extends over all non-leaf nodes $v$ in $t$ .\",\n \"This is analogous to the standard definition of the probability space of a Markov chain.\",\n \"We may write $\\\\mathbb {P}_\\\\mathcal {B}$ for $\\\\mathbb {P}_{X_0}$ , omitting the subscript when $\\\\mathcal {B}$ is understood.\",\n \"We often talk about events (i.e., measurable sets of trees) and their probability in text form.\",\n \"For example, by saying “a $\\\\mathcal {B}$ -tree has with positive probability infinitely many nodes of type $X$ ” we mean that $\\\\mathbb {P}_\\\\mathcal {B}(E) > 0$ where $E \\\\subseteq \\\\llbracket \\\\mathcal {B} \\\\rrbracket _{X_0}$ is the set of $X_0$ -trees with infinitely many nodes of type $X$ .\",\n \"We are particularly interested in sets of trees all whose branches (more precisely, their associated sequences of types) satisfy an $\\\\omega $ -regular linear-time property $L \\\\subseteq \\\\Gamma ^\\\\omega $ .\",\n \"Given $L \\\\subseteq \\\\Gamma ^\\\\omega $ , we write $\\\\mathbb {P}_\\\\mathcal {B}(L)$ for the probability that all branches of a $\\\\mathcal {B}$ -tree satisfy $L$ .\",\n \"Linear temporal logic (LTL) formulas specify linear-time properties; see, e.g., [33] for a definition of LTL.\",\n \"An important example for us are formulas of the form $\\\\mathsf {F}T$ , where $T \\\\subseteq \\\\Gamma $ , which denotes the linear-time property $\\\\lbrace u X w \\\\mid u \\\\in \\\\Gamma ^*,\\\\ X \\\\in T,\\\\ w \\\\in \\\\Gamma ^\\\\omega \\\\rbrace $ .\",\n \"Accordingly, $\\\\mathbb {P}_\\\\mathcal {B}(\\\\mathsf {F}T)$ denotes the probability that all branches of a $\\\\mathcal {B}$ -tree have a node whose type is in $T$ (equivalently, the probability that a $\\\\mathcal {B}$ -tree has a finite prefix all whose leaves have a type in $T$ ).\",\n \"We use finite automata on infinite words over $\\\\Gamma $ , where $\\\\Gamma $ is the set of types of a BP.\",\n \"We use deterministic parity automata (DPAs), deterministic Büchi automata (DBAs), nondeterministic Büchi automata (NBAs), and unambiguous Büchi automata (UBAs).\",\n \"The definitions are standard; see, e.g., [33].\",\n \"In the following we fix some terms and notation.\",\n \"Let $\\\\mathcal {A}= (Q, \\\\Gamma , \\\\delta , Q_0, F)$ be an NBA, where $Q$ is a finite set of states, $\\\\Gamma $ is the alphabet, $\\\\delta \\\\subseteq Q \\\\times \\\\Gamma \\\\times Q$ is the transition relation, $Q_0 \\\\subseteq Q$ is the set of initial states, and $F \\\\subseteq Q$ is the set of accepting states.\",\n \"We write $q \\\\xrightarrow{} r$ to denote that $(q,X,r) \\\\in \\\\delta $ .\",\n \"A finite sequence $q_0 \\\\xrightarrow{} q_1 \\\\xrightarrow{} \\\\cdots \\\\xrightarrow{} q_n$ is called a path and can be summarized as $q_0 {\\\\;\\\\xrightarrow{}{}\\\\hspace{0.0pt}^{*}\\\\;} q_n$ .\",\n \"An infinite sequence $q_0 \\\\xrightarrow{} q_1 \\\\xrightarrow{} \\\\cdots $ is called a run of $X_1 X_2 \\\\cdots $ .\",\n \"We call the run accepting if $q_0 \\\\in Q_0$ and $q_i \\\\in F$ holds for infinitely many $q_i$ .\",\n \"The NBA $\\\\mathcal {A}$ accepts (resp., rejects) an infinite word $w \\\\in \\\\Gamma ^\\\\omega $ if $w$ has (resp., does not have) an accepting run in $\\\\mathcal {A}$ .\",\n \"The NBA $\\\\mathcal {A}$ is called an unambiguous Büchi automaton (UBA) if every $w \\\\in \\\\Gamma ^\\\\omega $ has at most one accepting run.\",\n \"An automaton $\\\\mathcal {A}$ defines $\\\\omega $ -regular linear-time properties $\\\\lbrace w \\\\in \\\\Gamma ^\\\\omega \\\\mid \\\\mathcal {A}\\\\text{ accepts } w\\\\rbrace $ and $\\\\lbrace w \\\\in \\\\Gamma ^\\\\omega \\\\mid \\\\mathcal {A}\\\\text{ rejects } w\\\\rbrace $ .\",\n \"In keeping with previous definitions, we write $\\\\mathbb {P}_\\\\mathcal {B}(\\\\mathcal {A}\\\\text{ accepts})$ (resp., $\\\\mathbb {P}_\\\\mathcal {B}(\\\\mathcal {A}\\\\text{ rejects})$ ) for the probability that all branches of a $\\\\mathcal {B}$ -tree (more precisely, their associated sequences of types) are accepted (resp., rejected) by $\\\\mathcal {A}$ .\",\n \"We consider the following computational problems.\",\n \"The problem $\\\\mathbb {P}(\\\\text{finite}) = 1$ asks, given a BP $\\\\mathcal {B}$ with $\\\\varepsilon $ -rules allowed, whether the probability that a $\\\\mathcal {B}$ -tree is finite is 1.\",\n \"The problem $\\\\mathbb {P}(\\\\textup {LTL}) = 1$ asks, given a BP $\\\\mathcal {B}$ and an LTL formula $\\\\varphi $ , whether $\\\\mathbb {P}_\\\\mathcal {B}(\\\\varphi ) = 1$ .\",\n \"The problems $\\\\mathbb {P}(\\\\textup {DPA}) = 1$ (resp., $\\\\mathbb {P}(\\\\textup {NBA}) = 1$ ) ask, given a BP $\\\\mathcal {B}$ and a DPA (resp., NBA) $\\\\mathcal {A}$ , whether $\\\\mathbb {P}_\\\\mathcal {B}(\\\\mathcal {A}\\\\text{ accepts}) = 1$ .\",\n \"The problems $\\\\mathbb {P}(\\\\textup {coNBA}) = 1$ (resp., $\\\\mathbb {P}(\\\\textup {coUBA}) = 1$ )We do not explicitly define or use a notion of “co-Büchi automata” to avoid possible confusion about accepting/rejecting.\",\n \"If one were to do so, one would define a “co-NBA” $\\\\mathcal {A}$ like an NBA $\\\\mathcal {A}$ , but the “co-NBA” $\\\\mathcal {A}$ would accept a word $w \\\\in \\\\Gamma ^\\\\omega $ if and only if $\\\\mathcal {A}$ viewed as an NBA rejects $w$ .\",\n \"Similarly for “co-UBAs”.\",\n \"ask, given a BP $\\\\mathcal {B}$ and an NBA (resp., UBA) $\\\\mathcal {A}$ , whether $\\\\mathbb {P}_\\\\mathcal {B}(\\\\mathcal {A}\\\\text{ rejects}) = 1$ .\",\n \"The problems $\\\\mathbb {P}(\\\\textup {LTL}) = 0, \\\\mathbb {P}(\\\\textup {DPA}) = 0, \\\\ldots $ are defined similarly, where “${=}\\\\,1$ ” is replaced with “${=}\\\\,0$ ”.\",\n \"Table: Results and organization of the paper.The complexity classes indicate completeness results, except “in NC”, which only means membership in NC.See tab:map for a map of our results in those terms, as well as for an overview of the rest of the paper.\",\n \"As explained in the introduction, the problem $\\\\mathbb {P}(\\\\textup {LTL})=1$ is of particular interest from a model-checking point of view, and the technically most challenging one.\",\n \"In addition to standard complexity classes between P and 2EXPTIME, we use the class NC, the subclass of P comprising those problems solvable in polylogarithmic time by a parallel random-access machine using polynomially many processors; see, e.g., [27].\",\n \"To prove membership in PSPACE in a modular way, we will use the following pattern: Let $P_1, P_2$ be two problems, where $P_2$ is in NC.\",\n \"Suppose there is a reduction from $P_1$ to $P_2$ implemented by a PSPACE transducer, i.e., a Turing machine whose work tape (but not necessarily its output tape) is PSPACE-bounded.\",\n \"Then $P_1$ is in PSPACE.\",\n \"Note that the output of the transducer is (at most) exponential.\",\n \"Problems in NC can be decided in polylogarithmic space [4].\",\n \"Using standard techniques for composing space-bounded transducers (see, e.g., [27]), it follows that $P_1$ is in PSPACE.\",\n \"We use finite sets $S$ to index matrices $M \\\\in \\\\mathbb {R}^{S \\\\times S}$ and vectors $v \\\\in \\\\mathbb {R}^S$ .\",\n \"The graph of a nonnegative matrix $M \\\\in [0, \\\\infty )^{S \\\\times S}$ is the directed graph $(S,E)$ with $E = \\\\lbrace (s,t) \\\\in S \\\\times S \\\\mid M_{s,t} > 0\\\\rbrace $ .\",\n \"The spectral radius of a matrix is the largest absolute value of its eigenvalues.\",\n \"The following lemma allows to efficiently compare the spectral radius of a nonnegative matrix with 1.\",\n \"Given a nonnegative rational matrix $M$ , one can determine in NC whether $\\\\rho < 1$ or $\\\\rho = 1$ or $\\\\rho > 1$ , where $\\\\rho $ denotes the spectral radius of $M$ .\",\n \"Use the algorithm from [13], but not with Gaussian elimination as suggested there, but by solving the systems of linear equations described in [13] in NC.\",\n \"The latter is possible in NC [5].\"\n ],\n [\n \"Basic Results\",\n \"In this section we develop the more basic results indicated in tab:map, on finiteness (sec:as-finite), deterministic parity automata (sec:DPA), and Büchi automata (sec:NBA), on the one hand rounding off the complexity map in tab:map, and on the other hand building the foundation for more challenging results in the following sections.\",\n \"In particular, prop:as-finite-NC is indirectly used throughout the paper.\"\n ],\n [\n \"Finiteness\",\n \"In this section we consider BPs with $\\\\varepsilon $ -rules allowed, i.e., rules of the form $X {} \\\\varepsilon $ .\",\n \"Such BPs may generate finite trees.\",\n \"We are interested in the almost-sure finiteness problem, also denoted as $\\\\mathbb {P}(\\\\text{finite})=1$ , i.e., the problem whether the probability that a given BP with $\\\\varepsilon $ -rules allowed generates a finite tree is equal to 1.\",\n \"In prop:as-finite-NC below we show that this problem is in NC.\",\n \"All upper bounds on the complexity of $\\\\mathbb {P}(\\\\cdot ) = 1$ problems in this paper build directly or indirectly on this result.\",\n \"While the almost-sure finiteness (or “extinction”) problem has often been studied and is known to be in (strongly) polynomial time [18], [13], its membership in NC is, to the best of the authors' knowledge, new.\",\n \"For instance, since linear programming is P-complete, one cannot use linear programming (as in [18]) to show membership in NC.\",\n \"Nor can one directly use the strongly polynomial-time algorithm of [13], as it computes, in a sub-procedure, the set of types $X$ for which there exists a finite $X$ -tree.\",\n \"But the latter problem is P-complete.\",\n \"For the rest of the section, fix a BP $\\\\mathcal {B}= (\\\\Gamma , \\\\mathord {{}}, \\\\mathit {Prob}, X_0)$ with $\\\\varepsilon $ -rules allowed.\",\n \"Define a directed graph $G = (\\\\Gamma ,E)$ (i.e., the types of $\\\\mathcal {B}$ are the vertices of $G$ ) with an edge $(X,Y) \\\\in E$ if and only if $Y$ is a successor of $X$ (i.e., there is a rule $X {} u Y v$ for some $u,v \\\\in \\\\Gamma ^*$ ).\",\n \"Given a strongly connected component (SCC) $S \\\\subseteq \\\\Gamma $ of $G$ and $X \\\\in S$ , define a BP $\\\\mathcal {B}[S,X] = (S,\\\\mathord {{}_S},\\\\mathit {Prob}_S,X)$ obtained from $\\\\mathcal {B}$ by restricting the types to $S$ and deleting on all right-hand sides of the rules those types not in $S$ .\",\n \"The following lemma is straightforward: A $\\\\mathcal {B}$ -tree is infinite with positive probability if and only if there exist an SCC $S \\\\subseteq \\\\Gamma $ of $G$ and $X \\\\in S$ such that $X$ is reachable from $X_0$ in $G$ and a $\\\\mathcal {B}[S,X]$ -tree is infinite with positive probability.\",\n \"Let $M \\\\in \\\\mathbb {Q}^{\\\\Gamma \\\\times \\\\Gamma }$ be the nonnegative $\\\\Gamma \\\\times \\\\Gamma $ -matrix with $M_{X,Y} = \\\\sum _{X {p} w} p |w|_Y$ , where $|w|_Y \\\\in \\\\mathbb {N}_0$ is the number of occurrences of $Y$ in $w$ .\",\n \"That is, $M_{X,Y}$ is the expected number of direct $Y$ -successors of the root of a $\\\\mathcal {B}[X]$ -tree.\",\n \"By induction, $M^i$ , the $i$ th power of $M$ , is such that $(M^i)_{X,Y}$ is the expected number of $Y$ -nodes that are exactly $i$ levels under the root of a $\\\\mathcal {B}[X]$ -tree.\",\n \"The graph of $M$ is exactly the previously defined graph $G$ .\",\n \"Let $S \\\\subseteq \\\\Gamma $ be an SCC of $G$ .\",\n \"Denote by $M_S \\\\in \\\\mathbb {Q}^{S \\\\times S}$ the (square) principal submatrix obtained from $M$ by restricting it to the rows and columns indexed by elements of $S$ .\",\n \"Let $\\\\rho _S$ denote the spectral radius of $M_S$ .\",\n \"Call $S$ supercritical if $\\\\rho _S > 1$ .\",\n \"Call $S$ linear if for all rules $X {} w$ with $X \\\\in S$ there is exactly one occurrence in $w$ of a type in $S$ .\",\n \"Observe that if $S$ is linear then $M_S$ is stochastic, i.e., $M_S \\\\vec{1} = \\\\vec{1}$ where $\\\\vec{1}$ is the all-1 vector, i.e., the element of $\\\\lbrace 1\\\\rbrace ^S$ .\",\n \"In that case, by the Perron-Frobenius theorem [3], we have $\\\\rho _S = 1$ and, thus, $S$ is not supercritical.\",\n \"The following characterization can be proved using [13] (which builds on [18]): lemmalemasfinitenesschar A $\\\\mathcal {B}$ -tree is infinite with positive probability if and only if there exist an SCC $S \\\\subseteq \\\\Gamma $ of $G$ and $X \\\\in S$ such that $X$ is reachable from $X_0$ in $G$ and $S$ is supercritical or linear.\",\n \"It follows: propositionpropasfiniteNC The problem $\\\\mathbb {P}(\\\\text{finite})=1$ is in NC.\"\n ],\n [\n \"Deterministic Parity Automata\",\n \"In this section we consider deterministic parity automata (DPAs) on words.\",\n \"In [8] it was shown that the problem $\\\\mathbb {P}(\\\\textup {DPA}) = 1$ can be decided in polynomial time.\",\n \"We improve this to membership in NC.\",\n \"By the following lemma we can check in NC whether a $\\\\mathcal {B}$ -tree almost surely has a finite prefix all whose leaves have types in a given set $T$ .\",\n \"The proof is by reduction to almost-sure finiteness.\",\n \"lemmalemAFTone Given a BP $\\\\mathcal {B}= (\\\\Gamma , \\\\mathord {{}}, \\\\mathit {Prob}, X_0)$ and a set of types $T \\\\subseteq \\\\Gamma $ , the problem whether $\\\\mathbb {P}_{X_0}(\\\\mathsf {F}T) = 1$ is in NC.\",\n \"By combining lem:AFT-1 with results from [8] we obtain: theoremthmDPAone The problem $\\\\mathbb {P}(\\\\textup {DPA}) = 1$ is in NC.\",\n \"The hardness result in the following theorem highlights the different complexities of $\\\\mathbb {P}(\\\\cdot ) = 0$ and $\\\\mathbb {P}(\\\\cdot ) = 1$ problems in this paper.\",\n \"theoremthmDPAzero The problem $\\\\mathbb {P}(\\\\textup {DPA}) = 0$ is P-complete.\",\n \"It is P-hard even for deterministic Büchi automata with two states, the accepting state being a sink.\"\n ],\n [\n \"Büchi Automata\",\n \"The problem $\\\\mathbb {P}(\\\\textup {NBA}) = 1$ is PSPACE-complete.\",\n \"PSPACE-hardness is immediate in two different ways.\",\n \"It follows from the PSPACE-hardness of model checking Markov chains against NBAs [34].\",\n \"It also follows from the PSPACE-hardness of model checking transition systems against NBAs.\",\n \"(The latter follows easily from the PSPACE-hardness of NBA universality [31].)\",\n \"Both model-checking problems are special cases of $\\\\mathbb {P}(\\\\textup {NBA}) = 1$ .\",\n \"Towards membership in PSPACE, we use a translation from NBA to DPA [28].\",\n \"This translation causes an exponential blow-up, but an inspection of the construction [28] reveals that it can be computed by a PSPACE transducer.\",\n \"By thm:DPA-1 the problem $\\\\mathbb {P}(\\\\textup {DPA}) = 1$ is in NC.\",\n \"By lem:PSPACE-transducer it follows that $\\\\mathbb {P}(\\\\textup {NBA}) = 1$ is in PSPACE.\",\n \"theoremthmNBAzero The problem $\\\\mathbb {P}(\\\\textup {NBA}) = 0$ is EXPTIME-complete.\",\n \"It is EXPTIME-hard even for NBAs whose only accepting state is a sink.\",\n \"Towards membership in EXPTIME, an NBA can be translated, in exponential time, to a DPA of exponential size; see, e.g., [28].\",\n \"Since $\\\\mathbb {P}(\\\\textup {DPA}) = 0$ is in P by thm:DPA-0, it follows that $\\\\mathbb {P}(\\\\textup {NBA}) = 0$ is in EXPTIME.\",\n \"Concerning EXPTIME-hardness, we adapt the proof (in the online appendix) of [19] on model checking recursive Markov chains against NBAs.\",\n \"The details are in app:NBA-0.\"\n ],\n [\n \"Co-Büchi Automata\",\n \"In this section we consider the problem $\\\\mathbb {P}(\\\\textup {coNBA}) = 1$ , which asks, given a BP $\\\\mathcal {B}$ and a Büchi automaton $\\\\mathcal {A}$ , whether $\\\\mathcal {B}$ almost surely generates a tree whose branches are all rejected by $\\\\mathcal {A}$ ; i.e., whether $\\\\mathbb {P}_{\\\\mathcal {B}}(\\\\mathcal {A}\\\\text{ rejects}) = 1$ .\",\n \"Dually, one might ask whether the probability is positive that a $\\\\mathcal {B}$ -tree has a branch accepted by $\\\\mathcal {A}$ .\",\n \"Intuitively, we view the Büchi automaton $\\\\mathcal {A}$ as specifying “bad” branches, and we would like the tree almost surely not to have any bad branches.\",\n \"This problem is in PSPACE, which can be shown via a translation to DPAs, as in thm:NBA-1.\",\n \"However, with a view on the following sections, in particular on LTL specifications, we pursue a different approach to the problem $\\\\mathbb {P}(\\\\textup {coNBA}) = 1$ .\",\n \"In this section we lay the groundwork for arbitrary Büchi automata $\\\\mathcal {A}$ .\",\n \"By building on these results, we will show in the next section that if $\\\\mathcal {A}$ is unambiguous then the problem is in NC, which will allow us to derive our headline result, namely that $\\\\mathbb {P}(\\\\textup {LTL})=1$ is in PSPACE.\",\n \"Let $\\\\mathcal {B}= (\\\\Gamma , \\\\mathord {{}}, \\\\mathit {Prob}, X_0)$ be a BP and $\\\\mathcal {A}= (Q, \\\\Gamma , \\\\delta , Q_0, F)$ a (not necessarily unambiguous) Büchi automaton.\",\n \"Define a Büchi automaton, $\\\\mathcal {A}\\\\times \\\\mathcal {B}$ , by $\\\\mathcal {A}\\\\times \\\\mathcal {B}:= (Q \\\\times \\\\Gamma , \\\\Gamma , \\\\delta _{\\\\mathcal {A}\\\\times \\\\mathcal {B}}, Q_0 \\\\times \\\\lbrace X_0\\\\rbrace , F \\\\times \\\\Gamma )$ , where $&\\\\delta _{\\\\mathcal {A}\\\\times \\\\mathcal {B}}((q_1, X_1), X_2)={\\\\left\\\\lbrace \\\\begin{array}{ll}\\\\delta (q_1, X_1) \\\\times \\\\lbrace X_2\\\\rbrace & \\\\text{if $X_2$ is a successor of~$X_1$}\\\\\\\\\\\\emptyset & \\\\text{otherwise.}\\\\end{array}\\\\right.\",\n \"}$ The remainder of the section is organized as follows.\",\n \"In sub:UBA-X1-f we show that the problem $\\\\mathbb {P}(\\\\textup {coNBA}) = 1$ reduces to the analysis of certain SCCs within $\\\\mathcal {A}\\\\times \\\\mathcal {B}$ .\",\n \"In sub:Bdet we introduce a key lemma, lem:Bdet, which allows us to “forget” about the distinction between accepting and non-accepting states: the lemma reduces $\\\\mathbb {P}(\\\\textup {coNBA}) = 1$ to a pure reachability problem in an exponential-sized BP, $\\\\mathcal {B}_{\\\\mathit {det}}$ .\",\n \"This leads us to prove PSPACE-completeness of $\\\\mathbb {P}(\\\\textup {coNBA}) = 1$ , but more importantly, lem:Bdet plays a key role in the rest of the paper.\",\n \"We prove it in sub:Bdet-proof.\"\n ],\n [\n \"The Automaton $\\\\mathcal {A}[f,X_f]$\",\n \"For any $(f,X_f) \\\\in F \\\\times \\\\Gamma $ on a cycle of the transition graph of $\\\\mathcal {A}\\\\times \\\\mathcal {B}$ , define the Büchi automaton $\\\\mathcal {A}[f,X_f] \\\\ := \\\\ (\\\\lbrace \\\\bar{q}_0\\\\rbrace \\\\cup Q[f,X_f], \\\\Gamma , \\\\delta [f,X_f], \\\\lbrace \\\\bar{q}_0\\\\rbrace , \\\\lbrace (f,X_f)\\\\rbrace )$ as the Büchi automaton obtained from $\\\\mathcal {A}\\\\times \\\\mathcal {B}$ by making $(f,X_f)$ the only accepting state, restricting the set of states, $Q[f,X_f] \\\\subseteq Q \\\\times \\\\Gamma $ , to those $(q,X)$ that, in the transition graph of $\\\\mathcal {A}\\\\times \\\\mathcal {B}$ , are reachable from $(f,X_f)$ and can reach $(f,X_f)$ , i.e., those $(q,X)$ in the SCC containing $(f, X_f)$ , restricting the transition function $\\\\delta [f,X_f]$ accordingly, i.e., $\\\\delta [f,X_f]((q,X),Y) \\\\ := \\\\ \\\\delta _{\\\\mathcal {A}\\\\times \\\\mathcal {B}}((q,X),Y) \\\\cap Q[f,X_f]\\\\,,$ making $\\\\bar{q}_0$ the only initial state, and setting $\\\\delta [f,X_f](\\\\bar{q}_0,X_f) := \\\\lbrace (f,X_f)\\\\rbrace $ and $\\\\delta [f,X_f](\\\\bar{q}_0,X) := \\\\emptyset $ for all $X \\\\in \\\\Gamma \\\\setminus \\\\lbrace X_f\\\\rbrace $ .\",\n \"The following lemma follows from the pigeonhole principle and basic probability arguments:lemmalemUBAXonef The probability that some branch of a $\\\\mathcal {B}$ -tree is accepted by $\\\\mathcal {A}$ is positive if and only if there are $q_0 \\\\in Q_0$ and $f \\\\in F$ and $X_f \\\\in \\\\Gamma $ such that $(f, X_f)$ is reachable from $(q_0, X_0)$ in the transition graph of $\\\\mathcal {A}\\\\times \\\\mathcal {B}$ and the probability that some branch of a $\\\\mathcal {B}[X_f]$ -tree is accepted by $\\\\mathcal {A}[f,X_f]$ is positive.\",\n \"For the rest of the section let $(f,X_f) \\\\in F \\\\times \\\\Gamma $ be on a cycle of the transition graph of $\\\\mathcal {A}\\\\times \\\\mathcal {B}$ .\"\n ],\n [\n \"The Determinization $\\\\mathcal {A}_{\\\\mathit {det}}$ and the BP {{formula:82465720-6fff-453e-a5bf-44f8cad75b0e}}\",\n \"Let $\\\\mathcal {A}_{\\\\mathit {det}}:= (2^{\\\\lbrace \\\\bar{q}_0\\\\rbrace \\\\cup Q[f,X_f]}, \\\\Gamma , \\\\delta _{\\\\mathit {det}}, \\\\lbrace \\\\bar{q}_0\\\\rbrace , 2^{\\\\lbrace \\\\bar{q}_0\\\\rbrace \\\\cup Q[f,X_f]} \\\\setminus \\\\lbrace \\\\emptyset \\\\rbrace )$ be the determinization of $\\\\mathcal {A}[f,X_f]$ obtained by the standard subset construction.\",\n \"Which states are accepting will not actually be relevant.\",\n \"Note that every state reachable via a nonempty path from $\\\\lbrace \\\\bar{q}_0\\\\rbrace $ is of the form $P \\\\times \\\\lbrace X\\\\rbrace $ with $P \\\\subseteq Q$ and $X \\\\in \\\\Gamma $ .\",\n \"Define a BP $\\\\mathcal {B}_{\\\\mathit {det}}$ based on $\\\\mathcal {A}_{\\\\mathit {det}}$ as $\\\\mathcal {B}_{\\\\mathit {det}}\\\\ := \\\\ (\\\\Gamma ^{\\\\prime }, \\\\mathord {{}^{\\\\prime }}, \\\\mathit {Prob}^{\\\\prime }, \\\\lbrace (f,X_f)\\\\rbrace )\\\\,,$ where the set of types $\\\\Gamma ^{\\\\prime } \\\\subseteq 2^{Q[f,X_f]}$ is the set of those states in $\\\\mathcal {A}_{\\\\mathit {det}}$ that are reachable (in $\\\\mathcal {A}_{\\\\mathit {det}}$ ) from $\\\\lbrace \\\\bar{q_0}\\\\rbrace $ via a nonempty path (recall that they are of the form $P \\\\times \\\\lbrace X\\\\rbrace $ with $P \\\\subseteq Q$ and $X \\\\in \\\\Gamma $ ), and $X^{\\\\prime } &{\\\\;{p}{}\\\\hspace{0.0pt}{^{\\\\prime }}\\\\;} \\\\delta _{\\\\mathit {det}}(X^{\\\\prime },X_1) \\\\cdots \\\\delta _{\\\\mathit {det}}(X^{\\\\prime },X_k)$ for all $X^{\\\\prime } = P \\\\times \\\\lbrace X\\\\rbrace \\\\in \\\\Gamma ^{\\\\prime }$ with $P \\\\ne \\\\emptyset $ and all $X {p} X_1 \\\\cdots X_k$ , and $\\\\emptyset {\\\\;{1}{}\\\\hspace{0.0pt}{^{\\\\prime }}\\\\;} \\\\emptyset $ .\",\n \"Here is the key lemma of this section: lemmalemBdet The following statements are equivalent: The probability that some branch of a $\\\\mathcal {B}[X_f]$ -tree is accepted by $\\\\mathcal {A}[f,X_f]$ is positive.\",\n \"The probability that some branch of a $\\\\mathcal {B}_{\\\\mathit {det}}$ -tree does not have any nodes of type $\\\\emptyset $ is positive.\",\n \"We prove lem:Bdet in sub:Bdet-proof.\",\n \"It will be used in the proof of thm:coNBA-1 below; but more importantly, lem:Bdet is the foundation of sec:coUBA.\",\n \"Given that lem:Bdet reflects the key insight of this section, let us comment further.\",\n \"Considering that condition (ii) does not mention a notion of acceptance, one might have two concerns at this point: Condition (ii) does not obviously imply that with positive probability there is even a branch with infinitely many nodes of types containing $(f,X_f)$ .\",\n \"Even if with positive probability there is such a branch, it is not obvious that such branches would necessarily correspond to branches of $\\\\mathcal {B}[X_f]$ that are accepted by $\\\\mathcal {A}[f,X_f]$ .\",\n \"Even for the special case of Markov chains (i.e., every tree has only a single branch), lem:Bdet is not at all obvious, and both concerns (a) and (b) apply.\",\n \"Indeed, for Markov chains, Courcoubetis and Yannakakis prove a statement related to lem:Bdet, namely [11], with a proof related to ours and dealing explicitly with concern (b) above.\",\n \"For the special case of transition systems (i.e., the BP generates exactly one tree), lem:Bdet is simple though: consider the branch that follows a cycle around $(f,X_f)$ .\",\n \"For the general case, we need a result on BPs from [8], dealing with concern (a) above.\",\n \"The high-level principle behind the proof of lem:Bdet is often used in the analysis of Markov chains: if it is possible, infinitely often, to reach a state with a probability bounded away from 0, then this state is almost surely reached infinitely often.\",\n \"See sub:Bdet-proof for a full proof of lem:Bdet.\",\n \"We can now derive a PSPACE procedure for the problem $\\\\mathbb {P}(\\\\textup {coNBA}) = 1$ without resorting to DPAs: theoremthmcoNBAone The problem $\\\\mathbb {P}(\\\\textup {coNBA}) = 1$ is PSPACE-complete.\",\n \"thm:NBA-0 (for NBAs) has a coNBA-analogue: theoremthmcoNBAzero The problem $\\\\mathbb {P}(\\\\textup {coNBA}) = 0$ is EXPTIME-complete.\",\n \"It is EXPTIME-hard even for NBAs all whose states are accepting.\"\n ],\n [\n \"Co-Unambiguous Büchi Automata\",\n \"In this section we build on the previous section, in particular on lem:Bdet, to derive our main technical result: given a BP $\\\\mathcal {B}$ and an unambiguous Büchi automaton (UBA) $\\\\mathcal {A}$ , one can decide in NC whether $\\\\mathcal {B}$ almost surely generates a tree all whose branches are rejected by $\\\\mathcal {A}$ : The problem $\\\\mathbb {P}(\\\\textup {coUBA}) = 1$ is in NC.\",\n \"The rest of the section is devoted to the proof of this theorem.\",\n \"Fix a BP $\\\\mathcal {B}$ and a UBA $\\\\mathcal {A}$ .\",\n \"Since NC is closed under complement, we can focus on the problem whether the probability is positive that a $\\\\mathcal {B}$ -tree has some branch accepted by $\\\\mathcal {A}$ .\",\n \"We use lem:UBA-X1-f.\",\n \"Since reachability in a graph is in NL and, hence, in NC, it suffices to decide in NC whether the probability that some branch of a $\\\\mathcal {B}[X_f]$ -tree is accepted by $\\\\mathcal {A}[f,X_f]$ is positive.\",\n \"By lem:Bdet it suffices to decide in NC whether the probability that some branch of a $\\\\mathcal {B}_{\\\\mathit {det}}$ -tree does not have any nodes of type $\\\\emptyset $ is positive.\",\n \"The challenge is that $\\\\mathcal {B}_{\\\\mathit {det}}$ may be exponentially larger than $\\\\mathcal {A}$ , so we need to exploit the unambiguousness of $\\\\mathcal {A}$ and the regular structure it gives to $\\\\mathcal {B}_{\\\\mathit {det}}$ .\",\n \"Let $\\\\mathcal {B}_{\\\\mathit {det}}^{\\\\prime \\\\prime }$ be the BP (with $\\\\varepsilon $ -rules allowed) obtained from $\\\\mathcal {B}_{\\\\mathit {det}}$ by removing the type $\\\\emptyset $ and eliminating all occurrences of type $\\\\emptyset $ from all right-hand sides.\",\n \"The probability that a $\\\\mathcal {B}_{\\\\mathit {det}}$ -tree has an infinite branch of non-$\\\\emptyset $ nodes is equal to the probability that a $\\\\mathcal {B}_{\\\\mathit {det}}^{\\\\prime \\\\prime }$ -tree is infinite.\",\n \"Hence, it remains to show that one can decide in NC whether the probability that a $\\\\mathcal {B}_{\\\\mathit {det}}^{\\\\prime \\\\prime }$ -tree is infinite is positive.\",\n \"Define a matrix $M \\\\in \\\\mathbb {Q}^{Q[f,X_f] \\\\times Q[f,X_f]}$ whose rows and columns are indexed with the non-$\\\\bar{q}_0$ states of $\\\\mathcal {A}[f,X_f]$ : $M_{(q,X),(r,Y)} \\\\ := \\\\ {\\\\left\\\\lbrace \\\\begin{array}{ll} \\\\displaystyle \\\\sum _{X {p} u} p |u|_Y & \\\\text{if } (q,X) \\\\xrightarrow{} (r,Y) \\\\ \\\\text{ in } \\\\mathcal {A}[f,X_f] \\\\\\\\[1mm] 0 & \\\\text{otherwise\\\\,,}\\\\end{array}\\\\right.\",\n \"}$ where $|u|_Y \\\\in \\\\mathbb {N}_0$ is the number of occurrences of $Y$ in $u$ .\",\n \"(Think of $M_{(q,X),(r,Y)}$ as the expected number of $(r,Y)$ -“successors” of $(q,X)$ .)\",\n \"The graph of $M$ is equal to the transition graph of $\\\\mathcal {A}[f,X_f]$ (excluding $\\\\bar{q}_0$ ), which is strongly connected.\",\n \"Say that $\\\\mathcal {A}[f,X_f]$ has proper branching if there exist $(q,Y) \\\\xrightarrow{} (r_1,Z_1)$ and $(q,Y) \\\\xrightarrow{} (r_2,Z_2)$ in $\\\\mathcal {A}[f,X_f]$ and a rule $Y {p} u_1 Z_1 u_2 Z_2 u_3$ in $\\\\mathcal {B}$ with $u_1, u_2, u_3 \\\\in \\\\Gamma ^*$ .\",\n \"Now we can state the key lemma: lemmalemkey Let $\\\\rho $ be the spectral radius of $M$ .\",\n \"The probability that a $\\\\mathcal {B}_{\\\\mathit {det}}^{\\\\prime \\\\prime }$ -tree is infinite is positive if and only if either $\\\\rho > 1$ or $\\\\rho = 1$ and $\\\\mathcal {A}[f,X_f]$ does not have proper branching.\",\n \"Observe the similarity between lem:key,lem:as-finiteness-char.\",\n \"In fact, the proof of lem:key, given below, is based on lem:as-finiteness-char.\",\n \"lem:key shows that properties of $\\\\mathcal {A}[f,X_f]$ and $M$ (which are polynomial-sized objects) determine a property of the exponential-sized BP $\\\\mathcal {B}_{\\\\mathit {det}}^{\\\\prime \\\\prime }$ .\",\n \"Unambiguousness of $\\\\mathcal {A}[f,X_f]$ is crucial for that connection.\",\n \"Given that lem:key reflects the key insight of this section (if not of this paper), let us comment further.\",\n \"Suppose $\\\\mathcal {A}[f,X_f]$ has two outgoing transitions in a state $(q,Y)$ , say $(q,Y) \\\\xrightarrow{} (r_1,Z_1)$ and $(q,Y) \\\\xrightarrow{} (r_2,Z_2)$ .\",\n \"This branching could be “proper branching” as defined before lem:key, or the original UBA $\\\\mathcal {A}$ could be nondeterministic when reading $Y$ in $q$ and have transitions $q \\\\xrightarrow{} r_1$ and $q \\\\xrightarrow{} r_2$ .\",\n \"Either type of branching causes non-0 entries in the matrix $M$ and, intuitively, increases its spectral radius $\\\\rho $ .\",\n \"lem:key tells us that the probability that a $\\\\mathcal {B}_{\\\\mathit {det}}^{\\\\prime \\\\prime }$ -tree is infinite is governed by the combined effect on $\\\\rho $ of both types of branching: if $\\\\rho > 1$ then a $\\\\mathcal {B}_{\\\\mathit {det}}^{\\\\prime \\\\prime }$ -tree is infinite with positive probability; only in the borderline case, $\\\\rho =1$ , the type of branching matters.\",\n \"Again, this characterization is only correct if the nondeterminism in $\\\\mathcal {A}$ does not cause ambiguousness.\",\n \"Let us consider what lem:key states for the special case of Markov chains.\",\n \"In that case, clearly there is no proper branching.\",\n \"One can show, using unambiguousness, that for Markov chains the spectral radius $\\\\rho $ of the matrix $M$ is at most 1.\",\n \"Hence, lem:key states for Markov chains that the probability that a $\\\\mathcal {B}_{\\\\mathit {det}}^{\\\\prime \\\\prime }$ -tree (consisting of a single branch) is infinite is positive if and only if $\\\\rho = 1$ .\",\n \"Indeed, a related statement can be found in [2].\",\n \"To finish the proof of prop:coUBA-1 it suffices to show that we can check the conditions of lem:key in NC.\",\n \"Indeed, for comparing the spectral radius with 1, we employ lem:determine-spectral-radius.\",\n \"One can check for proper branching in logarithmic space, hence in NC.\",\n \"This completes the proof of prop:coUBA-1.\"\n ],\n [\n \"LTL\",\n \"With prop:coUBA-1 from the previous section, we can now show our headline result: The problem $\\\\mathbb {P}(\\\\textup {LTL})=1$ is PSPACE-complete.\",\n \"PSPACE-hardness is immediate in two different ways.\",\n \"It follows both from the PSPACE-hardness of model checking Markov chains against LTL and from the PSPACE-hardness of model checking transition systems against LTL [30].\",\n \"Both model-checking problems are special cases of $\\\\mathbb {P}(\\\\textup {LTL}) = 1$ .\",\n \"Towards membership in PSPACE, there is a classical PSPACE procedure that translates an LTL formula into an (exponential-sized) Büchi automaton [36].\",\n \"As noted by several authors (e.g., [12], [7]), this procedure can easily be adapted to ensure that the Büchi automaton be a UBA.\",\n \"By applying this translation to the negation $\\\\lnot \\\\varphi $ of the input formula $\\\\varphi $ , we obtain a UBA that rejects exactly those words that satisfy $\\\\varphi $ .\",\n \"By prop:coUBA-1 the problem $\\\\mathbb {P}(\\\\textup {coUBA}) = 1$ is in NC.\",\n \"By lem:PSPACE-transducer it follows that $\\\\mathbb {P}(\\\\textup {LTL}) = 1$ is in PSPACE.\",\n \"Finally we show the following result, exhibiting a big complexity gap between the problems $\\\\mathbb {P}(\\\\textup {LTL})=1$ and $\\\\mathbb {P}(\\\\textup {LTL})=0$ .\",\n \"theoremthmLTLzero The problem $\\\\mathbb {P}(\\\\textup {LTL})=0$ is 2EXPTIME-complete.\",\n \"For membership in 2EXPTIME, we use again the classical procedure that translates an LTL formula into an exponential-sized Büchi automaton [36] and then invoke thm:NBA-0.\",\n \"For 2EXPTIME-hardness we adapt the reduction from [11] for MDPs.\",\n \"The details are in app:LTL-0.\"\n ],\n [\n \"Conclusions\",\n \"We have devised a PSPACE procedure for $\\\\mathbb {P}(\\\\textup {LTL}) = 1$ , i.e., qualitative LTL model checking of BPs.\",\n \"The best previously known procedure ran in 2EXPTIME [8].\",\n \"Since BPs naturally generalize both transition systems and Markov chains (for both of which LTL model checking is PSPACE-complete), one might view our model-checking algorithm as an optimal general procedure.\",\n \"The same holds for NBA-specifications instead of LTL.\",\n \"The main technical ingredients have been the automata-theoretic approach and the algorithmic analysis of UBAs, nonnegative matrices, and finiteness of BPs.\",\n \"Our proofs were inspired by the observation that the spectral radii of certain nonnegative matrices are central to model checking Markov chains against UBAs, and also determine fundamental properties of BPs.\",\n \"Very loosely speaking, when model checking Markov chains against UBAs, the spectral radius measures the amount of nondeterministic branching in the UBA, whereas when analyzing BPs, the spectral radius measures the amount of tree branching.\",\n \"The “general case”, i.e., model checking BPs, features both kinds of branching.\",\n \"Serendipitously, an analysis of spectral radii still leads, as we have seen, to optimal algorithms.\",\n \"We have also established the complexities of related problems, partially as a tool for the mentioned LTL and NBA problems and partially to map out the landscape.\",\n \"We have shown that the $\\\\mathbb {P}(\\\\cdot ) = 0$ variants are more complex than their $\\\\mathbb {P}(\\\\cdot ) = 1$ counterparts.\",\n \"An intuitive explanation of this phenomenon is that for an instance of an $\\\\mathbb {P}(\\\\cdot )=1$ problems to be negative, tree branching and probabilistic branching “work together” to falsify the specification on some branch.\",\n \"In contrast, for $\\\\mathbb {P}(\\\\cdot )=0$ problems, tree branching and probabilistic branching are “adversaries”, like in MDPs.\",\n \"Indeed, for lower bounds on $\\\\mathbb {P}(\\\\cdot )=0$ problems we have encoded alternation in various forms.\",\n \"One might ask about the complexity of $\\\\mathbb {P}(\\\\textup {UBA}) = 1$ .\",\n \"Indeed, in trying to solve $\\\\mathbb {P}(\\\\textup {LTL}) = 1$ efficiently, the authors set out to solve $\\\\mathbb {P}(\\\\textup {UBA})=1$ efficiently (perhaps in P or even NC), with the PSPACE transduction from LTL to UBA in mind.\",\n \"However, the complexity of UBA universality is an open problem [29]; only membership in PSPACE is known.\",\n \"So even for the fixed transition system with $a {1} a b$ and $b {1} a b$ the problem $\\\\mathbb {P}(\\\\textup {UBA})=1$ cannot be placed in P without improving the complexity of UBA universality.\",\n \"A PSPACE-hardness proof of $\\\\mathbb {P}(\\\\textup {UBA})=1$ might have to make use of both types of branching in BPs, as $\\\\mathbb {P}(\\\\textup {UBA})=1$ is in NC for Markov chains [2].\",\n \"Model checking BPs quantitatively, i.e., computing the satisfaction probability, comparing it with a threshold, or approximating it, is left for future work.\",\n \"Exact versions of these problems are computationally complex, as they are at least as hard as the corresponding $\\\\mathbb {P}(\\\\cdot )=0$ problem.\",\n \"The paper [8] describes, for DPAs, nonlinear equation systems whose least nonnegative solution characterizes the satisfaction probabilities.\",\n \"Newton's method is efficient for approximating the solution of such equation systems; see [32], [15].\",\n \"sectionappendix\"\n ],\n [\n \"Proof of lem:as-finiteness-char\",\n \"* By lem:as-finite-SCC it suffices to show that if $G$ is strongly connected then a $\\\\mathcal {B}$ -tree is infinite with positive probability if and only if $\\\\Gamma $ is supercritical or linear.\",\n \"So, let $G$ be strongly connected.\",\n \"Call a type $X \\\\in \\\\Gamma $ immortal if there is no finite $X$ -tree.\",\n \"If a type $X$ is immortal, then the probability that a $\\\\mathcal {B}[X]$ -tree is finite is 0, and for all $Y \\\\in \\\\Gamma $ the probability that a $\\\\mathcal {B}[Y]$ -tree is finite is less than 1 (as $G$ is strongly connected).\",\n \"If $\\\\Gamma $ is linear then all types are immortal.\",\n \"So we can assume in the following that $\\\\Gamma $ is not linear.\",\n \"Since $\\\\Gamma $ is not linear and $G$ is strongly connected, for all $X \\\\in \\\\Gamma $ there is a finite prefix of an $X$ -tree that has at least two leaves of type $X$ ; and an $X$ -tree has, with positive probability, that finite prefix.\",\n \"Suppose some type, say $X$ , is immortal.\",\n \"Then every $X$ -tree that has a finite prefix with $k$ ($k \\\\in \\\\mathbb {N}$ ) leaves of type $X$ has at least $k$ (infinite) branches that go through those $k$ nodes of type $X$ .\",\n \"By strong connectedness, it follows that an $X$ -tree that has a finite prefix with $k$ leaves of type $X$ has, with probability 1 (i.e., $\\\\mathbb {P}_X$ -almost surely), a finite prefix with $k+1$ leaves of type $X$ .\",\n \"Hence, with probability 1 we have $\\\\lim _{i \\\\rightarrow \\\\infty } Z_i = \\\\infty $ , where $Z_i$ is the number of nodes that are exactly $i$ levels under the root of an $X$ -tree.\",\n \"Denoting by $\\\\mathbb {E}$ the expectation with respect to $\\\\mathbb {P}_X$ , it follows with Fatou's lemma that $\\\\lim _{i \\\\rightarrow \\\\infty } \\\\mathbb {E}Z_i = \\\\infty $ .\",\n \"We can write $\\\\mathbb {E}Z_i = (M^i \\\\vec{1})_X$ , where $\\\\vec{1}$ denotes the vector in $\\\\lbrace 1\\\\rbrace ^\\\\Gamma $ .\",\n \"Then $ \\\\lim _{i \\\\rightarrow \\\\infty } \\\\left\\\\Vert M^i \\\\vec{1}\\\\right\\\\Vert \\\\ = \\\\ \\\\infty \\\\,,$ where $\\\\left\\\\Vert \\\\cdot \\\\right\\\\Vert $ denotes the 1-norm.\",\n \"Since $G$ is strongly connected, by the Perron-Frobenius theorem [3], $M$ has an eigenvector $v \\\\in \\\\mathbb {R}^\\\\Gamma $ with $v_Y > 0$ for all $Y \\\\in \\\\Gamma $ and $M v = \\\\rho v$ , where $\\\\rho $ denotes the spectral radius of $M$ .\",\n \"Since $M^i v = \\\\rho ^i v$ , it follows that $\\\\lim _{i \\\\rightarrow \\\\infty } \\\\rho ^i \\\\left\\\\Vert v\\\\right\\\\Vert \\\\ = \\\\ \\\\lim _{i \\\\rightarrow \\\\infty } \\\\left\\\\Vert M^i v\\\\right\\\\Vert \\\\ = \\\\ \\\\infty \\\\,.$ Hence $\\\\rho >1$ .\",\n \"We conclude that if some type is immortal, then we have $\\\\rho >1$ and the probability that a $\\\\mathcal {B}$ -tree is finite is less than 1.\",\n \"On the other hand, if no type is immortal, then it follows from [13] (building on [18]) that a $\\\\mathcal {B}$ -tree is infinite with positive probability if and only if $\\\\rho > 1$ .\",\n \"We conclude that, regardless of whether there exists an immortal type, a $\\\\mathcal {B}$ -tree is infinite with positive probability if and only if $\\\\Gamma $ is supercritical.\"\n ],\n [\n \"Proof of prop:as-finite-NC\",\n \"* We use the characterization from lem:as-finiteness-char.\",\n \"One can compute in NL, and hence in NC, the directed acyclic graph of SCCs of $G$ and the set of types that are reachable from $X_0$ .\",\n \"Therefore, we assume without loss of generality that $G$ is strongly connected.\",\n \"By lem:determine-spectral-radius one can check in NC whether $\\\\Gamma $ is supercritical.\",\n \"Whether $\\\\Gamma $ is linear can be checked in logarithmic space, hence in NC.\"\n ],\n [\n \"Proof of lem:AFT-1\",\n \"* The problem can be rephrased as almost-sure finiteness of BPs with $\\\\varepsilon $ -rules allowed.\",\n \"Indeed, given $\\\\mathcal {B}$ and $T$ , one can eliminate all occurrences of types in $T$ from all right-hand sides; then, $\\\\mathsf {F}T$ in the original BP corresponds to finiteness in the new BP.\",\n \"Hence, the result follows from prop:as-finite-NC.\"\n ],\n [\n \"Proof of thm:DPA-1\",\n \"* In [8] it was shown that the problem $\\\\mathbb {P}(\\\\textup {DPA}) = 1$ can be decided in polynomial time.\",\n \"We show how to implement this approach in NC.\",\n \"In [8] a product of the BP and the DPA is computed and analyzed; the product is a BP whose types are coloured with priorities (natural numbers).\",\n \"The product, call it $\\\\mathcal {B}= (\\\\Gamma , \\\\mathord {{}}, \\\\mathit {Prob}, X_0)$ , can be computed in logarithmic space.\",\n \"The question is then whether the probability is 1 that all branches of a $\\\\mathcal {B}$ -tree are such that the highest priority that appears infinitely often is even.\",\n \"It is shown in [8] that this is the case if and only if all types $X$ that are reachable from $X_0$ and are associated with an odd priority satisfy $\\\\mathbb {P}_X(\\\\mathsf {F}N_X) = 1$ , where $N_X \\\\subseteq \\\\Gamma $ is a certain set of types that can be computed in NL by a simple reachability analysis in $\\\\mathcal {B}$ .\",\n \"By lem:AFT-1 one can determine in NC whether $\\\\mathbb {P}_X(\\\\mathsf {F}N_X) = 1$ .\",\n \"The theorem follows.\"\n ],\n [\n \"Proof of thm:DPA-0\",\n \"* Membership in P was shown in [8].\",\n \"For P-hardness we reduce from the monotone circuit value problem.\",\n \"Given a monotone circuit with output gate $g_{\\\\mathit {out}}$ , we construct a BP $\\\\mathcal {B}= (\\\\Gamma ,\\\\mathord {{}},\\\\mathit {Prob},X_{g_{\\\\mathit {out}}})$ and a DBA $\\\\mathcal {A}$ such that $\\\\mathbb {P}_{X_{g_{\\\\mathit {out}}}}(\\\\mathcal {A}\\\\text{ accepts}) > 0$ if and only if $g_{\\\\mathit {out}}$ evaluates to 1.\",\n \"For each $\\\\wedge $ - and each $\\\\vee $ -gate $g$ include a type $X_g \\\\in \\\\Gamma $ .\",\n \"Also include types $X_0, X_1 \\\\in \\\\Gamma $ for the inputs $0, 1$ , respectively.\",\n \"For each $\\\\wedge $ -gate $g$ with children $g_1, \\\\ldots , g_k$ include a rule $X_g &{1} X_{g_1} \\\\cdots X_{g_k}\\\\,.\\\\multicolumn{2}{l}{\\\\text{For each $\\\\vee $-gate~$g$ with children $g_1, \\\\ldots , g_k$ include rules}}\\\\\\\\X_g &{1/k} X_{g_1},\\\\ \\\\ldots ,\\\\ X_g {1/k} X_{g_k}\\\\,.\\\\multicolumn{2}{l}{\\\\text{Include rules}}\\\\\\\\X_0 &{1} X_0 \\\\text{ and }X_1 {1} X_1\\\\,.$ Construct a two-state DBA $\\\\mathcal {A}$ that accepts exactly those $w \\\\in \\\\Gamma ^\\\\omega $ that contain $X_1$ .\",\n \"It follows from a straightforward induction over the gate height (longest distance to an input) that, for any gate $g$ , it evaluates to 1 if and only if $\\\\mathbb {P}_{X_g}(\\\\mathcal {A}\\\\text{ accepts}) > 0$ .\"\n ],\n [\n \"Additional Definitions Concerning Alternating Turing Machines\",\n \"An alternating Turing machine is a 6-tuple $(S_\\\\exists , S_\\\\forall , \\\\Sigma , T, s_0, s_{\\\\mathit {acc}})$ , where $S = S_\\\\exists \\\\cup S_\\\\forall \\\\cup \\\\lbrace s_{\\\\mathit {acc}}\\\\rbrace $ is a finite set of (control) states partitioned into existential states $S_\\\\exists $ and universal states $S_\\\\forall $ and the (only) accepting state $s_{\\\\mathit {acc}}$ , $\\\\Sigma $ is a finite alphabet, $T \\\\subseteq (S_\\\\exists \\\\cup S_\\\\forall ) \\\\times \\\\Sigma \\\\times \\\\Sigma \\\\times \\\\lbrace -1,+1\\\\rbrace \\\\times S$ is a transition relation, and $s_0$ is the initial state.\",\n \"A transition $(s,a,a^{\\\\prime },D,s^{\\\\prime }) \\\\in T$ means that if $M$ is in state $s$ and its head reads letter $a$ , then it rewrites the content of the current cell with the letter $a^{\\\\prime }$ , it moves the head in direction $D$ (either left if $D=-1$ , or right if $D=+1$ ), and it changes its state to $s^{\\\\prime }$ .\",\n \"We assume that for all $s \\\\in S_\\\\exists \\\\cup S_\\\\forall $ and $a \\\\in \\\\Sigma $ there is at least one outgoing transition.\",\n \"A configuration of an alternating Turing machine is given by a 3-tuple $(i, s, w)$ where $i \\\\in \\\\mathbb {N}$ indicates the header position, $s \\\\in S$ is the current state of the Turing machine, and $w \\\\in \\\\Sigma ^*$ is the contents of the memory tape.\",\n \"For a word $w \\\\in \\\\Sigma ^*$ we will write $w(i)$ to denote its $i$ 'th component, and $w[i=a]$ to denote the string $w(1)\\\\ldots w(i-1) a w(i+1)\\\\ldots w(n)$ .\",\n \"The initial configuration of an alternating Turing machine on a word $w \\\\in \\\\Sigma ^*$ is $(1, s_0, w)$ , and given a transition $t = (s, a, a^{\\\\prime }, D, s^{\\\\prime })$ and a configuration $c = (i, s^{\\\\prime \\\\prime }, w)$ we will use $c \\\\triangleright t$ to denote the configuration $(i+D, s^{\\\\prime }, w[i=a^{\\\\prime }])$ if $s^{\\\\prime \\\\prime } = s$ and $w_i = a$ (and leave it undefined otherwise).\",\n \"We extend this notation to sequences of transitions $t_1\\\\ldots t_n \\\\in T^*$ by writing $c \\\\triangleright t_1\\\\ldots t_n = (c \\\\triangleright t_1) \\\\triangleright t_2\\\\ldots t_n$ .\",\n \"We will use $\\\\pi _\\\\mathbb {N}$ , $\\\\pi _S$ , and $\\\\pi _{\\\\Sigma ^*}$ to denote the projections of a configuration $(i, s, w)$ to $i$ , $s$ , and $w$ , respectively.\",\n \"For any $(s, a) \\\\in (S_\\\\exists \\\\cup S_\\\\forall ) \\\\times \\\\Sigma $ , let $T_{s,a} := \\\\lbrace (s,a,a^{\\\\prime },D,s^{\\\\prime }) \\\\in T \\\\mid a^{\\\\prime } \\\\in \\\\Sigma ,\\\\ D \\\\in \\\\lbrace -1,+1\\\\rbrace ,\\\\ s^{\\\\prime } \\\\in S\\\\rbrace $ .\",\n \"A string $r = r_1r_2\\\\ldots \\\\in T^*\\\\cup T^\\\\omega $ is called a run of $M$ on $w$ if for all $i$ , $r_{i+1} \\\\in T_{\\\\pi _S(c_{i}), \\\\pi _{\\\\Sigma ^*}(c_{i})(\\\\pi _\\\\mathbb {N}(c_{i}))}$ , where $c_i = c_0\\\\triangleright r_1\\\\ldots r_i$ whenever $r_{i+1}$ exists.\",\n \"Any run $r$ is called an accepting run if $r \\\\in T^*$ and $\\\\pi _S(c_0\\\\triangleright r) = s_{\\\\mathit {acc}}$ .\",\n \"A strategy is a function $\\\\sigma : \\\\lbrace (i, s, w, r) \\\\in \\\\mathbb {N}\\\\times S \\\\times \\\\Sigma ^* \\\\times T^* \\\\mid s \\\\in S_\\\\exists \\\\rbrace \\\\rightarrow T$ such that $\\\\sigma (i, s, w, r) \\\\in T_{s, w_i}$ for any $i, s, w, r$ .\",\n \"For a configuration $c$ and a run $r$ , by abuse of notation, we will write $\\\\sigma (c, r)$ to mean $\\\\sigma (\\\\pi _\\\\mathbb {N}(c), \\\\pi _S(c), \\\\pi _{\\\\Sigma ^*}(c), r)$ .\",\n \"A run is consistent with a strategy $\\\\sigma $ if for all $i$ , $r_{i+1} = \\\\sigma (c_0\\\\triangleright r_1\\\\ldots r_i, r_1\\\\ldots r_i)$ whenever the latter is defined.\",\n \"The behaviour of $\\\\sigma $ is the set of runs consistent with $\\\\sigma $ .\",\n \"A winning strategy is a strategy $\\\\sigma $ such that every run $r$ consistent with $\\\\sigma $ is an accepting run.\",\n \"Note that whenever $\\\\sigma $ is winning, its behaviour is finite.\",\n \"Let $\\\\sigma $ be a strategy, then we call $\\\\sigma $ $N$ -bounded if for any $r$ in the behavior of $\\\\sigma $ , and any prefix $r_1\\\\ldots r_k$ of $r$ , $|\\\\pi _{\\\\Sigma ^*}(c_0\\\\triangleright r_1\\\\ldots r_k)| < N$ .\",\n \"A Turing machine is $N$ -bounded if it has an $N$ -bounded winning strategy.\"\n ],\n [\n \"Proof of thm:NBA-0\",\n \"* Given the main body, it remains to prove EXPTIME-hardness of $\\\\mathbb {P}(\\\\textup {NBA}) = 0$ .\",\n \"Recall that alternating PSPACE equals EXPTIME.\",\n \"We give a polynomial-time reduction from the problem of acceptance of a word by a PSPACE-bounded alternating Turing machine.\",\n \"Without loss of generality, we can assume that the Turing machine is linear-bounded, i.e., uses only the space occupied by the input word.\",\n \"Let $M = (S_\\\\exists , S_\\\\forall , \\\\Sigma , T, s_0, s_{\\\\mathit {acc}})$ be a linear-bounded alternating Turing machine.\",\n \"Let $w = a_1 \\\\cdots a_n \\\\in \\\\Sigma ^*$ be the input word.\",\n \"As mentioned before, we can assume that $M$ uses exactly $n$ tape cells.\",\n \"We construct a BP $\\\\mathcal {B}$ and an NBA $\\\\mathcal {A}$ such that $\\\\mathbb {P}_\\\\mathcal {B}(\\\\mathcal {A}\\\\text{ accepts}) > 0$ if and only $M$ accepts $w$ .\",\n \"The BP $\\\\mathcal {B}$ has the following set of types: $\\\\Gamma \\\\ = \\\\ (\\\\lbrace 1, \\\\ldots , n\\\\rbrace \\\\times S \\\\times \\\\Sigma ) \\\\ \\\\cup \\\\ (\\\\lbrace 1, \\\\ldots , n\\\\rbrace \\\\times T) \\\\ \\\\cup \\\\ (\\\\lbrace 1, \\\\ldots , n\\\\rbrace \\\\times \\\\lbrace \\\\mathit {chk}\\\\rbrace ) \\\\ \\\\cup \\\\ \\\\lbrace E\\\\rbrace $ Intuitively, a type $(i,s,a)$ means that the head is at position $i$ , the current state is $s$ , and the head is reading letter $a$ ; a type $(i,t)$ means that the head is at position $i$ , and transition $t$ is being executed; a type $(i,\\\\mathit {chk})$ means that the accepting state has been reached, and cell $i$ is being “checked” (in a sense to be explained later); type $E$ indicates an error.\",\n \"The type $(1,s_0,a_1)$ is the start type of $\\\\mathcal {B}$ .\",\n \"For all $(i,s,a) \\\\in \\\\Gamma $ with $s \\\\in S_\\\\exists $ , include rules $(i,s,a) {1/k} (i,t_j) \\\\quad (1 \\\\le j \\\\le k)\\\\,,$ where $\\\\lbrace t_1, \\\\ldots , t_k\\\\rbrace = T_{s,a}$ .\",\n \"(Intuitively, a transition going out of an existential state is chosen as the only child, uniformly at random.)\",\n \"For all $(i,s,a) \\\\in \\\\Gamma $ with $s \\\\in S_\\\\forall $ , include a rule $(i,s,a) {1} (i,t_1) \\\\cdots (i,t_k) \\\\,,$ where $\\\\lbrace t_1, \\\\ldots , t_k\\\\rbrace = T_{s,a}$ .\",\n \"(Intuitively, all possible transitions going out of a universal state are children.)\",\n \"For all $(i,(s,a,a^{\\\\prime },D,s^{\\\\prime })) \\\\in \\\\Gamma $ with $1 \\\\le i{+}D \\\\le n$ , include rules $(i,(s,a,a^{\\\\prime },D,s^{\\\\prime })) {1/|\\\\Sigma |} (i{+}D, s^{\\\\prime }, b_j) \\\\quad (1 \\\\le j \\\\le |\\\\Sigma |)\\\\,,$ where $\\\\lbrace b_1, \\\\ldots , b_{|\\\\Sigma |}\\\\rbrace = \\\\Sigma $ .\",\n \"(Intuitively, the letter in the cell at the new head position $i{+}D$ is guessed uniformly at random.)\",\n \"For all $(i,(s,a,a^{\\\\prime },D,s^{\\\\prime })) \\\\in \\\\Gamma $ with $i{+}D \\\\in \\\\lbrace 0,n{+}1\\\\rbrace $ , include a rule $(i,(s,a,a^{\\\\prime },D,s^{\\\\prime })) {1} E\\\\,.$ (Intuitively, when the space bound is exceeded, move to the error type $E$ .)\",\n \"Include a rule $E {1} E$ (i.e., a self-loop).\",\n \"For all $(i, s_{\\\\mathit {acc}}, a) \\\\in \\\\Gamma $ , include a rule $(i, s_{\\\\mathit {acc}}, a) {1} (1,\\\\mathit {chk}) \\\\cdots (n,\\\\mathit {chk})\\\\,.$ (Intuitively, after reaching $s_{\\\\mathit {acc}}$ all $n$ cells are “checked”.)\",\n \"For all $(i,\\\\mathit {chk}) \\\\in \\\\Gamma $ , include a rule $(i,\\\\mathit {chk}) {1} (i,\\\\mathit {chk})$ (i.e., a self-loop).\",\n \"The NBA $\\\\mathcal {A}= (Q, \\\\Gamma , \\\\delta , Q_0, \\\\lbrace f\\\\rbrace )$ has the following set of states: $Q \\\\ = \\\\ (\\\\lbrace 1, \\\\ldots , n\\\\rbrace \\\\times \\\\Sigma ) \\\\ \\\\cup \\\\ \\\\lbrace f\\\\rbrace $ The set of initial states is $Q_0 = \\\\lbrace (1,a_1), \\\\ldots , (n,a_n)\\\\rbrace $ (recall that $a_1 \\\\cdots a_n$ is the input word).\",\n \"The idea is that if a prefix $X_1 \\\\cdots X_k \\\\in \\\\Gamma ^*$ of a tree branch corresponds to a prefix of a correct computation of $M$ , then the set of automaton states in $\\\\delta (Q_0,X_1 \\\\cdots X_k)$ corresponds to the tape after this computation prefix.\",\n \"In fact, the transition relation $\\\\delta \\\\subseteq Q \\\\times \\\\Gamma \\\\times Q$ is deterministic, i.e., for all $q \\\\in Q$ and $X \\\\in \\\\Gamma $ there is at most one $q^{\\\\prime }$ with $(q,X,q^{\\\\prime }) \\\\in \\\\delta $ .\",\n \"Moreover, for any transition $((i,a), X, (j,a^{\\\\prime })) \\\\in \\\\delta $ we will have $i=j$ .\",\n \"For $(i,a) \\\\in Q$ and all $(j,s,b) \\\\in \\\\Gamma $ with $i \\\\ne j$ or $a=b$ , include a self-loop $((i,a),(j,s,b),(i,a)) \\\\in \\\\delta \\\\,.$ (Intuitively, letter $a$ in cell $i$ is compatible with the head being on cell $j$ and reading letter $b$ .)\",\n \"For all $(i,a) \\\\in Q$ and all $(j,t) \\\\in \\\\Gamma $ with $i \\\\ne j$ , include a self-loop $((i,a),(j,t),(i,a)) \\\\in \\\\delta \\\\,.$ (Intuitively, the content of cell $i$ stays unchanged when the head is at position $j$ .)\",\n \"For all $i,s,a,a^{\\\\prime },D,s^{\\\\prime }$ with $(i,a) \\\\in Q$ and $(i,(s,a,a^{\\\\prime },D,s^{\\\\prime })) \\\\in \\\\Gamma $ , include a transition $((i,a), (i,(s,a,a^{\\\\prime },D,s^{\\\\prime })), (i,a^{\\\\prime }))\\\\in \\\\delta \\\\,.$ (Intuitively, the transition $(s,a,a^{\\\\prime },D,s^{\\\\prime })$ changes the content of cell $i$ from $a$ to $a^{\\\\prime }$ .)\",\n \"For all $(i,a) \\\\in Q$ , include a transition $((i,a),(i,\\\\mathit {chk}),f) \\\\in \\\\delta \\\\,.$ (Intuitively, the type $(i,\\\\mathit {chk})$ checks if the computation has been consistent in cell $i$ .)\",\n \"For all $X \\\\in \\\\Gamma $ , include a self-loop $(f,X,f) \\\\in \\\\delta $ .\",\n \"We will show that in this case, $\\\\mathbb {P}_\\\\mathcal {B}(\\\\mathcal {A}\\\\text{ accepts}) > 0$ if and only if $M$ accepts $w$ .\",\n \"Firstly note that $f$ is a sink state in $\\\\mathcal {A}$ , and hence, if $\\\\mathcal {A}$ reaches $f$ after reading some prefix of a branch, it will accept any branch with this prefix.\",\n \"This means that a tree $t$ is accepted by $\\\\mathcal {A}$ if and only if there exists a finite prefix of $t$ such that $\\\\mathcal {A}$ reaches $f$ on all of its branches.\",\n \"Assume that $M$ accepts $w$ .\",\n \"Then there exists a linear-bounded winning strategy $\\\\sigma $ for the existential player.\",\n \"We will write $c_0$ for the initial configuration of $M$ on $w$ , $R$ for the behaviour of $\\\\sigma $ , and $N$ for the length of the longest run in $R$ .\",\n \"We will show that there exists a finite prefix generated by $\\\\mathcal {B}$ with nonzero probability that precisely models this strategy, and that is such that $\\\\mathcal {A}$ reaches $f$ on all of its branches.\",\n \"Since any tree with this prefix is accepted, this implies that $\\\\mathbb {P}_\\\\mathcal {B}(\\\\mathcal {A}\\\\text{ accepts}) > 0$ if $M$ accepts $w$ .\",\n \"Let $t$ be the tree defined as follows: the root of $t$ is $(1, s_0, a_1)$ , for any node $(p_i, s_i, b_i)$ on level $2i$ with $s_i \\\\in S_\\\\exists $ , let $t_1\\\\ldots t_{i}$ be such that the branch from the root to $(p_i, s_i, b_i)$ has nodes $(p_j, t_{j+1})$ on level $2j+1$ for each $0 \\\\le j < i$ .\",\n \"Then $(p_i, s_i, b_i)$ has a child $(p_i, \\\\sigma (c_0 \\\\triangleright t_1\\\\ldots t_{i}, t_1\\\\ldots t_i))$ , for any node $(p_i, s_i, b_i)$ on level $2i$ with $s_i \\\\in S_\\\\forall $ , let $t_1\\\\ldots t_{i}$ be such that the branch from the root to $(p_i, s_i, b_i)$ has nodes $(p_j, t_{j+1})$ on level $2j+1$ for each $0 \\\\le j < i$ .\",\n \"Let $w_i = \\\\pi _{\\\\Sigma ^*}(c_0 \\\\triangleright t_1\\\\ldots t_{i})$ .\",\n \"Then $(p_i, s_i, b_i)$ has children $(p_i, t_{i+1})$ for each $t_{i+1} \\\\in T_{s_i, w_i(p_i)}$ , for any node $(p_i, t_{i+1})$ with $t_{i+1} = (s, a, a^{\\\\prime }, D, s^{\\\\prime })$ on level $2i+1$ , let $t_1\\\\ldots t_{i}$ be such that the branch from the root to $(p_i, t_{i+1})$ has nodes $(p_j, t_{j+1})$ on level $2j+1$ for each $0 \\\\le j < i$ .\",\n \"Let $c_{i+1} = c_0 \\\\triangleright t_1\\\\ldots t_{i+1}$ .\",\n \"Then $(p_i, t_{i+1})$ has a child $(p_i+D, s^{\\\\prime }, \\\\pi _{\\\\Sigma ^*}(c_{i+1})(p_i+D))$ , any node $(p_i, s_{\\\\mathit {acc}}, b_i)$ has children $(j, \\\\mathit {chk})$ for each $1 \\\\le j \\\\le n$ , and any node $(j, \\\\mathit {chk})$ has a child $(j, \\\\mathit {chk})$ .\",\n \"By construction, $t$ is now such that for any node $(p_i, t_{i+1})$ with $t_{i+1} = (s, a, a^{\\\\prime }, D, s^{\\\\prime })$ on level $2i+1$ , there exist runs in $R$ prefixed by $t_1 \\\\ldots t_{i+1}$ , where $t_1\\\\ldots t_{i}$ are such that the branch from the root to $(p_i, t_{i+1})$ has nodes $(p_j, t_{j+1})$ on level $2j+1$ for each $0 \\\\le j < i$ .\",\n \"Hence, by the fact that $\\\\sigma $ is linear-bounded, $1 \\\\le p_i+D \\\\le n$ and thus all the transitions in $t$ are according to the rules of $\\\\mathcal {B}$ .\",\n \"Moreover, since the length of runs in $R$ is bounded by $N$ , all the nodes at level $2N+1$ must be of the form $(i, \\\\mathit {chk})$ , and since all states $(i, \\\\mathit {chk})$ are sink states, $t$ is generated with a nonzero probability.\",\n \"Finally pick any state $(i, \\\\mathit {chk})$ in $t$ .\",\n \"The only types from which $(i, a) \\\\in Q$ does not have an outgoing edge are of the form $(i, s, b)$ .\",\n \"However, for any state $(i, s, b)$ at level $2j$ on the branch to $(i, \\\\mathit {chk})$ , let $t_1\\\\ldots t_{j}$ be such that the branch from the root to $(i,s,b)$ has nodes $(p_k, t_{k+1})$ on level $2k+1$ for each $0 \\\\le k < j$ .\",\n \"Then $(i, b) = (i, \\\\pi _{\\\\Sigma ^*}(c_0 \\\\triangleright t_1\\\\ldots t_{j})(i)) = \\\\delta ((i,w(i)), (p_0, t_1)\\\\ldots (p_{j-1}, t_{j}))$ and hence $(i, a)$ survives, and reaches $f$ upon reading $(i, \\\\mathit {chk})$ .\",\n \"For the other direction, assume that $\\\\mathbb {P}_\\\\mathcal {B}(\\\\mathcal {A}\\\\text{ accepts}) > 0$ .\",\n \"Then there exists a prefix $t$ generated by $\\\\mathcal {B}$ with branches of length $N$ for some $N$ such that $\\\\mathcal {A}$ reaches $f$ on all of its branches.\",\n \"W.l.o.g.\",\n \"we can assume that $\\\\pi _S(c_0) \\\\ne s_{\\\\mathit {acc}}$ , because otherwise any strategy is winning.\",\n \"Note that for any accepted tree $t$ and any branch prefix $(1, s_0, a_1)(1, t_1)\\\\ldots (p_i, s_i, b_i)(p_i, t_{i+1})$ in $t$ , the set of reachable states in the automaton from $Q_0$ reflects the tape contents of $M$ , ie.\",\n \"$\\\\delta (Q_0, (1, s_0, a_1)(1, t_1)\\\\ldots (p_i, s_i, b_i)(p_i, t_{i+1}))=\\\\lbrace (j, w(j))\\\\mid 1 \\\\le j \\\\le n, w = \\\\pi _{\\\\Sigma ^*}(c_0 \\\\triangleright t_1\\\\ldots t_{i+1})\\\\rbrace $ .\",\n \"Also $s_i = \\\\pi _S(c_0 \\\\triangleright t_1\\\\ldots t_i)$ .\",\n \"If this was not the case, then there would exist $j$ such that $\\\\delta ((j, a_j), (1, s_0, a_1)(1, t_1)\\\\ldots (p_i, s_i, b_i)(p_i, t_{i+1})) = \\\\emptyset $ and hence the branch reaching $(j, \\\\mathit {chk})$ prefixed by $(1, s_0, a_1)(1, t_1)\\\\ldots (p_i, s_i, b_i)(p_i, t_{i+1})$ does not reach $f$ .\",\n \"Let $n_0n_1\\\\ldots $ be any branch in $t$ .\",\n \"Pick $m$ such that $n_{2i+1} = (a_i, t_{i+1})$ for all $0 \\\\le i < m$ and $n_{2m+1} = (a_m, t_{m+1})$ .\",\n \"Let $c = c_0 \\\\triangleright t_1\\\\ldots t_m$ .\",\n \"Then we define $\\\\sigma $ to be any strategy such that for any such $c$ , if $\\\\pi _S(c) \\\\in S_\\\\exists $ , then $\\\\sigma (c, t_1\\\\ldots t_m) = t_{m+1}$ .\",\n \"This is a valid strategy since for any branch in $t$ , the set of reachable states of the automaton after reading a branch prefix reflects the tape contents of $M$ and the alphabet characters contained in the nodes of the branch prefix have to reflect the automaton states.\",\n \"Note that $\\\\sigma $ is $n$ -bounded.\",\n \"We claim that $\\\\sigma $ is a winning strategy and hence $M$ accepts $w$ .\",\n \"Let $r = t_1t_2\\\\ldots $ be a run consistent with $\\\\sigma $ .\",\n \"We will show that there exists a branch $(1, s_0, a_1)(1, t_1)\\\\ldots (p_i, t_{i+1})(p_{i+1}, s_{\\\\mathit {acc}}, w_{i+1}(p_{i+1}))$ in $t$ with $p_i = \\\\pi _\\\\mathbb {N}(c_0\\\\triangleright t_1\\\\ldots t_i)$ , $s_i = \\\\pi _S(c_0\\\\triangleright t_1\\\\ldots t_i)$ , and $w_i = \\\\pi _{\\\\Sigma ^*}(c_0\\\\triangleright t_1\\\\ldots t_i)$ (and hence $|r| = i+1$ and $\\\\pi _S(c_0\\\\triangleright r) = s_{\\\\mathit {acc}}$ ).\",\n \"For any branch prefix $(1, s_0, a_1)(1, t_1)\\\\ldots (p_j, t_{j+1})$ where $t_{j+1} = (s, a, a^{\\\\prime }, D, s^{\\\\prime })$ , according to the rules of $\\\\mathcal {B}$ , $(p_j, t_{j+1})$ has a single child $(p_i+D, s^{\\\\prime }, b)$ .\",\n \"Let $c_{j+1} = c_0 \\\\triangleright t_1\\\\ldots t_{j+1}$ .\",\n \"Since trees prefixed by $t$ are accepted, $p_i+D = \\\\pi _\\\\mathbb {N}(c_{j+1})$ , $s^{\\\\prime } = \\\\pi _S(c_{j+1})$ , and $b = \\\\pi _{\\\\Sigma ^*}(c_{j+1})(\\\\pi _\\\\mathbb {N}(c_{j+1}))$ .\",\n \"For any branch prefix $(1, s_0, a_1)(1, t_1)\\\\ldots (p_{j-1}, t_{j})(p_j, s_j, b_j)$ where $s_j \\\\in S_\\\\forall $ , according to the rules of $\\\\mathcal {B}$ , $(p_j, s_j, b_j)$ has children $(p_j, t^{\\\\prime })$ for each $t^{\\\\prime } \\\\in T_{s_j, b_j}$ .\",\n \"Let $c_j = c_0\\\\triangleright t_1\\\\ldots t_j$ .\",\n \"Since trees prefixed by $t$ are accepted, $s_j = \\\\pi _S(c_j)$ and $b_j = \\\\pi _{\\\\Sigma ^*}(c_j)(\\\\pi _\\\\mathbb {N}(c_j))$ .\",\n \"Hence, $t_{j+1} \\\\in T_{s_j, b_j}$ and $(p_j, s_j, b_j)$ has a child $(p_j, t_{j+1})$ .\",\n \"For any branch prefix $(1, s_0, a_1)(1, t_1)\\\\ldots (p_{j-1}, t_{j})(p_j, s_j, b_j)$ where $s_j \\\\in S_\\\\exists $ , according to the rules of $\\\\mathcal {B}$ , $(p_j, s_j, b_j)$ has a single child $(p_j, t^{\\\\prime })$ .\",\n \"By definition of $\\\\sigma $ , $\\\\sigma (c_0\\\\triangleright t_1\\\\ldots t_j, t_1\\\\ldots t_j) = t^{\\\\prime }$ .\",\n \"Since $t$ is finite, every branch reaches states of the form $(p, s_{\\\\mathit {acc}}, b)$ in a finite number of steps.\",\n \"Hence, $r$ is finite and reaches $s_{\\\\mathit {acc}}$ and since $r$ is any run consistent with $\\\\sigma $ , $\\\\sigma $ is an winning strategy.\",\n \"Thus, $M$ accepts $w$ .\"\n ],\n [\n \"Proof of lem:UBA-X1-f\",\n \"* Consider a branch $X_0 X_1 X_2 \\\\cdots $ of a $\\\\mathcal {B}$ -tree accepted by $\\\\mathcal {A}$ .\",\n \"Then $\\\\mathcal {A}$ has an accepting run $q_0 \\\\xrightarrow{} q_1 \\\\xrightarrow{} q_2 \\\\xrightarrow{} \\\\cdots $ with $q_0 \\\\in Q_0$ .\",\n \"By the pigeonhole principle, there are $f \\\\in F$ and $X_f \\\\in \\\\Gamma $ such that this run contains the segment $f \\\\xrightarrow{}$ infinitely often.\",\n \"By its construction, the Büchi automaton $\\\\mathcal {A}\\\\times \\\\mathcal {B}$ has the accepting run $(q_0,X_0) \\\\xrightarrow{} (q_1,X_1) \\\\xrightarrow{} \\\\cdots \\\\,,$ which contains the state $(f,X_f)$ infinitely often.\",\n \"Let $k \\\\ge 1$ be such that $(q_k,X_k) = (f,X_f)$ .\",\n \"Then $\\\\mathcal {A}[f,X_f]$ accepts $X_k X_{k+1} \\\\cdots $ via the run $\\\\bar{q}_0 \\\\xrightarrow{} (q_k,X_k) \\\\xrightarrow{} (q_{k+1},X_{k+1}) \\\\cdots \\\\,.$ Therefore, denoting by $E(f,X_f,n)$ for $n \\\\in \\\\mathbb {N}$ the event that there exists a branch $X_k X_{k+1} \\\\cdots $ emanating from the $n$ th (in a breadth-first order) node in the tree (necessarily a node of type $X_k = X_f$ ) such that $\\\\mathcal {A}[f,X_f]$ has an accepting run $\\\\bar{q}_0 \\\\xrightarrow{} (q_k,X_k) \\\\xrightarrow{} (q_{k+1},X_{k+1}) \\\\cdots \\\\,,$ we have $\\\\mathbb {P}_{X_0}(\\\\mathcal {A}\\\\text{ accepts some branch})\\\\ \\\\le \\\\ &\\\\sum _{f \\\\in F} \\\\sum _{X_f \\\\in \\\\Gamma } \\\\sum _{n \\\\in \\\\mathbb {N}} \\\\mathbb {P}_{X_0}(E(f,X_f,n))\\\\,.$ Further, $&\\\\ \\\\mathbb {P}_{X_0}(E(f,X_f,n)) \\\\\\\\=&\\\\ \\\\mathbb {P}_{X_0}(\\\\text{the $n$th node has type~$X_f$}) \\\\cdot \\\\mathbb {P}_{X_f}(\\\\mathcal {A}[f,X_f] \\\\text{ accepts some branch}) \\\\\\\\\\\\le &\\\\ \\\\mathbb {P}_{X_f}(\\\\mathcal {A}[f,X_f] \\\\text{ accepts some branch})\\\\,.$ The “only if” direction follows.\",\n \"Towards the “if” direction, suppose that $\\\\mathcal {A}\\\\times \\\\mathcal {B}$ has a path $(q_0,X_0) {\\\\;\\\\xrightarrow{}{}\\\\hspace{0.0pt}^{*}\\\\;} (f,X_f)$ with $q_0 \\\\in Q_0$ and $f \\\\in F$ such that the probability that some branch of a $\\\\mathcal {B}[X_f]$ -tree is accepted by $\\\\mathcal {A}[f,X_f]$ is positive.\",\n \"Then there is a successor, say $X_f^{\\\\prime }$ , of $X_f$ such that the probability that some branch of a $\\\\mathcal {B}[X_f^{\\\\prime }]$ -tree is accepted by $\\\\mathcal {A}\\\\times \\\\mathcal {B}$ when started in $(f,X_f)$ is positive.\",\n \"Thus, the probability that some branch of a $\\\\mathcal {B}$ -tree (starts with $X_0 w X_f^{\\\\prime }$ and) is accepted by $\\\\mathcal {A}$ is positive.\"\n ],\n [\n \"Proof of lem:Bdet\",\n \"In this subsection we prove lem:Bdet, which is instrumental for the main results of the paper.\",\n \"* Fix a word $w \\\\in \\\\Gamma ^+$ such that $\\\\delta [f,X_f]((f,X_f),w)$ is maximal, i.e., there is no $w^{\\\\prime } \\\\in \\\\Gamma ^+$ such that $\\\\delta [f,X_f]((f,X_f),w^{\\\\prime }) \\\\supsetneq \\\\delta [f,X_f]((f,X_f),w)$ .\",\n \"Let $v \\\\in \\\\Gamma ^*$ be a word such that $\\\\delta [f,X_f]((f,X_f),v) \\\\ni (f,X_f)$ .\",\n \"Then, $\\\\delta [f,X_f]((f,X_f),v w) = \\\\delta [f,X_f]((f,X_f), w)$ ; i.e., for any path $(f,X_f) {\\\\;\\\\xrightarrow{}{}\\\\hspace{0.0pt}^{*}\\\\;} (q,X)$ there is a path $(f,X_f) {\\\\;\\\\xrightarrow{}{}\\\\hspace{0.0pt}^{*}\\\\;} (f,X_f) {\\\\;\\\\xrightarrow{}{}\\\\hspace{0.0pt}^{*}\\\\;} (q,X)$ .\",\n \"Since $\\\\delta [f,X_f]((f,X_f),v) \\\\ni (f,X_f)$ , we have $\\\\delta [f,X_f]((f,X_f),v w) \\\\ \\\\supseteq \\\\ \\\\delta [f,X_f]((f,X_f), w)\\\\,.$ But $w$ is maximal.\",\n \"We enrich $\\\\mathcal {A}_{\\\\mathit {det}}$ to obtain a DBA, $\\\\mathcal {A}_{\\\\mathit {det}}^{\\\\prime }$ , whose states have an additional component keeping track of whether the word $w \\\\in \\\\Gamma ^+$ from above is being seen.\",\n \"The accepting runs of $\\\\mathcal {A}_{\\\\mathit {det}}^{\\\\prime }$ contain infinitely many $w$ -labelled segments that, loosely speaking, “start from $(f,X_f)$ ”.\",\n \"Formally, let $W = \\\\lbrace 0\\\\rbrace \\\\cup \\\\lbrace v \\\\in \\\\Gamma ^* \\\\mid v \\\\text{ is a suffix of } w\\\\rbrace \\\\,,$ including $w$ and the empty word $\\\\varepsilon $ .\",\n \"We assume $0 \\\\notin \\\\Gamma $ .\",\n \"Define $\\\\mathcal {A}_{\\\\mathit {det}}^{\\\\prime } \\\\ :=\\\\ (2^{\\\\lbrace \\\\bar{q}_0\\\\rbrace \\\\cup Q[f,X_f]} \\\\times W, \\\\delta _{\\\\mathit {det}}^{\\\\prime }, (\\\\lbrace \\\\bar{q}_0\\\\rbrace ,0), (2^{Q[f,X_f]} \\\\setminus \\\\lbrace \\\\emptyset \\\\rbrace ) \\\\times \\\\lbrace \\\\varepsilon \\\\rbrace )\\\\,,$ where $\\\\delta _{\\\\mathit {det}}^{\\\\prime }((U,X v), X) \\\\ &= \\\\ (\\\\delta _{\\\\mathit {det}}(U,X), v) \\\\\\\\\\\\delta _{\\\\mathit {det}}^{\\\\prime }((U,X v), Y) \\\\ &= \\\\ (\\\\delta _{\\\\mathit {det}}(U,X), 0) \\\\quad &&\\\\text{for } X \\\\ne Y \\\\\\\\\\\\delta _{\\\\mathit {det}}^{\\\\prime }((U,v), X) \\\\ &= \\\\ (\\\\delta _{\\\\mathit {det}}(U,X), w) \\\\quad &&\\\\text{for } v \\\\in \\\\lbrace 0, \\\\varepsilon \\\\rbrace ,\\\\ (f,X_f) \\\\in \\\\delta _{\\\\mathit {det}}(U,X) \\\\\\\\\\\\delta _{\\\\mathit {det}}^{\\\\prime }((U,v), X) \\\\ &= \\\\ (\\\\delta _{\\\\mathit {det}}(U,X), 0) \\\\quad &&\\\\text{for } v \\\\in \\\\lbrace 0, \\\\varepsilon \\\\rbrace ,\\\\ (f,X_f) \\\\notin \\\\delta _{\\\\mathit {det}}(U,X)\\\\,.$ It follows from lem:UBA-w and the construction of $\\\\mathcal {A}_{\\\\mathit {det}}^{\\\\prime }$ that $\\\\mathcal {A}_{\\\\mathit {det}}^{\\\\prime }$ has a single accepting state reachable from $(\\\\lbrace \\\\bar{q}_0\\\\rbrace ,0)$ , namely $(\\\\delta _{\\\\mathit {det}}(\\\\lbrace (f,X_f)\\\\rbrace ,w), \\\\varepsilon )$ .\",\n \"The following statements are equivalent: The probability that some branch of a $\\\\mathcal {B}[X_f]$ -tree is accepted by $\\\\mathcal {A}[f,X_f]$ is positive.\",\n \"The probability that some branch of a $\\\\mathcal {B}[X_f]$ -tree has a run (accepting or not) in $\\\\mathcal {A}[f,X_f]$ is positive.\",\n \"The probability that some branch of a $\\\\mathcal {B}[X_f]$ -tree is accepted by $\\\\mathcal {A}_{\\\\mathit {det}}^{\\\\prime }$ is positive.\",\n \"lem:UBA-3-equivalences implies lem:Bdet, as it follows from the definition of $\\\\mathcal {B}_{\\\\mathit {det}}$ that the probability that some branch of a $\\\\mathcal {B}[X_f]$ -tree has a run in $\\\\mathcal {A}[f,X_f]$ (cf.\",\n \"condition (ii) in lem:UBA-3-equivalences) equals the probability that some branch of a $\\\\mathcal {B}_{\\\\mathit {det}}$ -tree does not have any nodes of type $\\\\emptyset $ (cf.\",\n \"condition (ii) of lem:Bdet).\",\n \"So it remains to prove lem:UBA-3-equivalences.\",\n \"[Proof of lem:UBA-3-equivalences] (i) $\\\\Longrightarrow $ (ii).\",\n \"Trivial.\",\n \"(iii) $\\\\Longrightarrow $ (i).\",\n \"Let $X_f X_1 X_2 \\\\cdots $ be accepted by $\\\\mathcal {A}_{\\\\mathit {det}}^{\\\\prime }$ .\",\n \"Then $X_1 X_2 \\\\cdots $ can be decomposed in $v_1 w v_2 w \\\\cdots $ with $v_1, v_2, \\\\ldots \\\\in \\\\Gamma ^*$ such that $\\\\mathcal {A}[f,X_f]$ has paths $(f,X_f) {\\\\;\\\\xrightarrow{}{}\\\\hspace{0.0pt}^{*}\\\\;} (f,X_f)$ for all $i \\\\ge 1$ .\",\n \"Let $i \\\\ge 2$ .\",\n \"There is $(q,X)$ with $(f,X_f) {\\\\;\\\\xrightarrow{}{}\\\\hspace{0.0pt}^{*}\\\\;} (q,X) {\\\\;\\\\xrightarrow{}{}\\\\hspace{0.0pt}^{*}\\\\;} (f,X_f)\\\\,.$ By lem:UBA-w there is a path $(f,X_f) {\\\\;\\\\xrightarrow{}{}\\\\hspace{0.0pt}^{*}\\\\;} (f,X_f) {\\\\;\\\\xrightarrow{}{}\\\\hspace{0.0pt}^{*}\\\\;} (q,X)\\\\,.$ Thus also $(f,X_f) {\\\\;\\\\xrightarrow{}{}\\\\hspace{0.0pt}^{*}\\\\;} (f,X_f)$ .\",\n \"Since $i \\\\ge 2$ was arbitrary, it follows that $\\\\mathcal {A}[f,X_f]$ has an accepting run $\\\\bar{q}_0 \\\\xrightarrow{} (f,X_f) {\\\\;\\\\xrightarrow{}{}\\\\hspace{0.0pt}^{*}\\\\;} (f,X_f) {\\\\;\\\\xrightarrow{}{}\\\\hspace{0.0pt}^{*}\\\\;} (f,X_f) {\\\\;\\\\xrightarrow{}{}\\\\hspace{0.0pt}^{*}\\\\;} \\\\cdots \\\\,.$ (ii) $\\\\Longrightarrow $ (iii).\",\n \"For this part we use results from [8], which considers the problem of model checking BPs against DPAs.\",\n \"We need only a special case of such automata: (a) the same (word) automaton is run on every branch of the tree, and (b) our automaton $\\\\mathcal {A}_{\\\\mathit {det}}^{\\\\prime }$ is a DBA, which can be viewed as a DPA whose states are labelled with only 2 priorities: priority 0 for non-accepting states and priority 1 for accepting states.\",\n \"The paper [8] constructs a product BP from the BP and the automaton, and subsequently considers BPs whose types are coloured with a priority.\",\n \"In this way the model-checking problem reduces to computing the probability that there is a branch on which the highest priority that occurs infinitely often is odd.\",\n \"Since our automaton $\\\\mathcal {A}_{\\\\mathit {det}}^{\\\\prime }$ already embeds a BP, instead of taking another product, we define a BP, $\\\\mathcal {B}_{\\\\mathit {det}}^{\\\\prime }$ , more directly based on $\\\\mathcal {A}_{\\\\mathit {det}}^{\\\\prime }$ , in the same way that $\\\\mathcal {B}_{\\\\mathit {det}}$ was defined based on $\\\\mathcal {A}_{\\\\mathit {det}}$ in sub:Bdet.\",\n \"More explicitly, $\\\\mathcal {B}_{\\\\mathit {det}}^{\\\\prime } \\\\ := \\\\ (\\\\Gamma ^{\\\\prime }, \\\\mathord {{}^{\\\\prime }}, \\\\mathit {Prob}^{\\\\prime }, (\\\\lbrace (f,X_f)\\\\rbrace ,w))\\\\,,$ where the set of types $\\\\Gamma ^{\\\\prime } \\\\subseteq 2^{Q[f,X_f]} \\\\times W$ is the set of those states in $\\\\mathcal {A}_{\\\\mathit {det}}^{\\\\prime }$ that are reachable (in $\\\\mathcal {A}_{\\\\mathit {det}}^{\\\\prime }$ ) from $(\\\\lbrace \\\\bar{q}\\\\rbrace ,0)$ via a nonempty path (recall that they are of the form $(P \\\\times \\\\lbrace X\\\\rbrace ,v)$ with $P \\\\subseteq Q$ and $X \\\\in \\\\Gamma $ and $v \\\\in W$ ), and $X^{\\\\prime } &{\\\\;{p}{}\\\\hspace{0.0pt}{^{\\\\prime }}\\\\;} \\\\delta _{\\\\mathit {det}}^{\\\\prime }(X^{\\\\prime },X_1) \\\\cdots \\\\delta _{\\\\mathit {det}}^{\\\\prime }(X^{\\\\prime },X_k)\\\\multicolumn{2}{l}{\\\\text{for all $X^{\\\\prime } = (P \\\\times \\\\lbrace X\\\\rbrace ,v) \\\\in \\\\Gamma ^{\\\\prime }$ with $P \\\\ne \\\\emptyset $ and all $X {p} X_1 \\\\cdots X_k$, and}}\\\\\\\\(\\\\emptyset ,v) &{\\\\;{1}{}\\\\hspace{0.0pt}{^{\\\\prime }}\\\\;} (\\\\emptyset ,v)$ for all $v \\\\in W$ .\",\n \"Recall that $(\\\\delta _{\\\\mathit {det}}(\\\\lbrace (f,X_f)\\\\rbrace ,w), \\\\varepsilon ) =: X_w^{\\\\prime }$ is the single accepting state in $\\\\mathcal {A}_{\\\\mathit {det}}^{\\\\prime }$ that is reachable from $(\\\\lbrace \\\\bar{q}_0\\\\rbrace ,0)$ .\",\n \"We call a branch of $\\\\mathcal {B}_{\\\\mathit {det}}^{\\\\prime }$ accepting if it contains $X_w^{\\\\prime }$ infinitely often.\",\n \"Suppose “(ii)”, i.e., the probability that some branch of a $\\\\mathcal {B}[X_f]$ -tree has a (non-accepting or accepting) run in $\\\\mathcal {A}[f,X_f]$ is positive.\",\n \"Thus, the probability that some branch of a $\\\\mathcal {B}[X_f]$ -tree has a run in $\\\\mathcal {A}_{\\\\mathit {det}}$ that does not enter the state $\\\\emptyset $ is positive.\",\n \"Let $X_1 w^{\\\\prime } \\\\in \\\\Gamma ^+$ be such that $(f,X_f) \\\\in \\\\delta _{\\\\mathit {det}}(\\\\lbrace (f,X_f)\\\\rbrace ,w X_1 w^{\\\\prime })\\\\,.$ Then, also the probability that some branch of a $\\\\mathcal {B}[X_f]$ -tree starts with $X_f w X_1 w^{\\\\prime }$ and has a run in $\\\\mathcal {A}_{\\\\mathit {det}}$ that does not enter $\\\\emptyset $ is positive.\",\n \"Thus, the probability that some branch of a $\\\\mathcal {B}[X_1]$ -tree starts with $X_1 w^{\\\\prime }$ and has a run in $\\\\mathcal {A}_{\\\\mathit {det}}$ , started in $\\\\delta _{\\\\mathit {det}}(\\\\lbrace (f,X_f)\\\\rbrace ,w)$ , that does not enter $\\\\emptyset $ is positive.\",\n \"Hence, the probability that some branch of a $\\\\mathcal {B}[X_1]$ -tree has a run in $\\\\mathcal {A}_{\\\\mathit {det}}^{\\\\prime }$ , started in $(\\\\delta _{\\\\mathit {det}}(\\\\lbrace (f,X_f)\\\\rbrace ,w), \\\\varepsilon )$ , that does not enter a state of the form $(\\\\emptyset ,v)$ is positive.\",\n \"From the construction of $\\\\mathcal {B}_{\\\\mathit {det}}^{\\\\prime }$ it follows that the probability that some branch of a $\\\\mathcal {B}_{\\\\mathit {det}}^{\\\\prime }[X_w^{\\\\prime }]$ -tree (i.e., with $X_w^{\\\\prime } = (\\\\delta _{\\\\mathit {det}}(\\\\lbrace (f,X_f)\\\\rbrace ,w), \\\\varepsilon )$ as start type) does not have a node of a type of the form $(\\\\emptyset ,v)$ is positive.\",\n \"Consider any type $(U_0,v_0)$ in $\\\\mathcal {B}_{\\\\mathit {det}}^{\\\\prime }$ with $U_0 \\\\ne \\\\emptyset $ and view it as a state in $\\\\mathcal {A}_{\\\\mathit {det}}^{\\\\prime }$ .\",\n \"Then there is $u_1 \\\\in \\\\Gamma ^*$ with $(U_0,v_0) {\\\\;\\\\xrightarrow{}{}\\\\hspace{0.0pt}^{*}\\\\;} (U_1,v_1)$ where $U_1 \\\\ne \\\\emptyset $ and $v_1 \\\\in \\\\lbrace 0, \\\\varepsilon \\\\rbrace $ .\",\n \"Let $u_2 \\\\in \\\\Gamma ^*$ be a shortest word such that there is $(q_1,X_1) \\\\in U_1$ with $(q_1,X_1) {\\\\;\\\\xrightarrow{}{}\\\\hspace{0.0pt}^{*}\\\\;} (f,X_f)$ in $\\\\mathcal {A}[f,X_f]$ .\",\n \"It follows that in $\\\\mathcal {A}_{\\\\mathit {det}}^{\\\\prime }$ we have $(U_0,v_0) {\\\\;\\\\xrightarrow{}{}\\\\hspace{0.0pt}^{*}\\\\;} (U_2,w) {\\\\;\\\\xrightarrow{}{}\\\\hspace{0.0pt}^{*}\\\\;} (U_3,\\\\varepsilon )$ with $(f,X_f) \\\\in U_2$ .\",\n \"By lem:UBA-w we have $U_3 = \\\\delta _{\\\\mathit {det}}(\\\\lbrace (f,X_f)\\\\rbrace ,w)\\\\,.$ Hence $(U_0,v_0) {\\\\;\\\\xrightarrow{}{}\\\\hspace{0.0pt}^{*}\\\\;} X_w^{\\\\prime }$ .\",\n \"Combining this reachability fact with the previous argument, we infer that the probability that some branch of a $\\\\mathcal {B}_{\\\\mathit {det}}^{\\\\prime }[X_w^{\\\\prime }]$ -tree has only nodes of types from which $X_w^{\\\\prime }$ is reachable (in $\\\\mathcal {B}_{\\\\mathit {det}}^{\\\\prime }$ ) is positive.\",\n \"It follows from [8] that the probability that some branch of a $\\\\mathcal {B}_{\\\\mathit {det}}^{\\\\prime }[X_w^{\\\\prime }]$ -tree is accepting is positive.In terms of the notation therein, we instantiate [8] with $X := X_w^{\\\\prime }$ .\",\n \"By the reachability argument above, $N_X = N_{X_w^{\\\\prime }}$ does not include types of the form $(U,v)$ with $U \\\\ne \\\\emptyset $ .\",\n \"Thus, we have argued that the probability of $\\\\mathsf {F}N_{X_w^{\\\\prime }}$ is $p < 1$ .\",\n \"Hence, [8] asserts that the probability that a $\\\\mathcal {B}_{\\\\mathit {det}}^{\\\\prime }[X_w^{\\\\prime }]$ -tree has an $X_w^{\\\\prime }$ -branch equals $1-p > 0$ , where $X_w^{\\\\prime }$ -branch means accepting path in terms of our definition.\",\n \"Hence, the probability that some branch of a $\\\\mathcal {B}_{\\\\mathit {det}}^{\\\\prime }$ -tree is accepting is positive.\",\n \"From the construction of $\\\\mathcal {B}_{\\\\mathit {det}}^{\\\\prime }$ it follows that the probability that some branch of a $\\\\mathcal {B}[X_f]$ -tree is accepted by $\\\\mathcal {A}_{\\\\mathit {det}}^{\\\\prime }$ is positive, i.e., “(iii)”.\"\n ],\n [\n \"Proof of thm:coNBA-1\",\n \"* Towards membership in PSPACE, fix a BP $\\\\mathcal {B}$ and an NBA $\\\\mathcal {A}$ .\",\n \"Since PSPACE is closed under complement, we can focus on the problem whether the probability is positive that a $\\\\mathcal {B}$ -tree has a branch accepted by $\\\\mathcal {A}$ .\",\n \"We use lem:UBA-X1-f.\",\n \"Since reachability in a graph is in NL and, hence, in PSPACE, it suffices to decide in PSPACE whether the probability that some branch of a $\\\\mathcal {B}[X_f]$ -tree is accepted by $\\\\mathcal {A}[f,X_f]$ is positive.\",\n \"In order to check that, by lem:Bdet it suffices to construct the BP $\\\\mathcal {B}_{\\\\mathit {det}}$ and then invoke the NC procedure of lem:AFT-1 to check if $\\\\mathbb {P}_{\\\\mathcal {B}_{\\\\mathit {det}}}(\\\\mathsf {F}\\\\lbrace \\\\emptyset \\\\rbrace ) < 1$ .\",\n \"The BP $\\\\mathcal {B}_{\\\\mathit {det}}$ has exponential size but can be computed with a PSPACE transducer.\",\n \"(In particular, whether a state in $\\\\mathcal {A}_{\\\\mathit {det}}$ is reachable from $\\\\lbrace \\\\bar{q}\\\\rbrace $ via a nonempty path can be determined in NPSPACE $=$ PSPACE.)\",\n \"By lem:PSPACE-transducer it follows that $\\\\mathbb {P}(\\\\textup {coNBA}) = 1$ is in PSPACE.\",\n \"PSPACE-hardness follows from the PSPACE-hardness [34] of the probabilistic emptiness problem, which, given a Markov chain and an NBA, asks if the probability is 0 that the Markov chain generates a word accepted by the NBA.\",\n \"(We remark that, in contrast, the problem whether a given transition system has a run accepted by a given NBA is in NL: search the product for an accepting cycle).\"\n ],\n [\n \"Proof of thm:coNBA-0\",\n \"* Towards membership in EXPTIME, an NBA can be translated, in exponential time, to a DPA of exponential size; see, e.g., [28].\",\n \"By shifting the priorities (colours) in the DPA by 1, we can make the DPA accept exactly those words that are rejected by the NBA.\",\n \"Since $\\\\mathbb {P}(\\\\textup {DPA}) = 0$ is in P by thm:DPA-0, it follows that $\\\\mathbb {P}(\\\\textup {coNBA}) = 0$ is in EXPTIME.\",\n \"Concerning EXPTIME-hardness, we adapt the construction of the proof of thm:NBA-0.\",\n \"As in that proof, let $M = (S_\\\\exists , S_\\\\forall , \\\\Sigma , T, s_0, s_{\\\\mathit {acc}})$ be a linear-bounded alternating Turing machine.\",\n \"Let $w = a_1 \\\\cdots a_n \\\\in \\\\Sigma ^*$ be the input word, and assume again that $M$ uses exactly $n$ tape cells.\",\n \"We construct a BP $\\\\mathcal {B}$ and an NBA $\\\\mathcal {A}$ such that the probability that all branches of the random tree are rejected by $\\\\mathcal {A}$ is positive if and only $M$ accepts $w$ .\",\n \"For $\\\\mathcal {B}$ we use almost the same construction as in thm:NBA-0, except that we do not need the types $(1,\\\\mathit {chk}), \\\\ldots , (n,\\\\mathit {chk})$ .\",\n \"We replace them by a single type $\\\\mathit {end}$ .\",\n \"Accordingly, for all $(i, s_{\\\\mathit {acc}}, a) \\\\in \\\\Gamma $ , we replace the rule $(i, s_{\\\\mathit {acc}}, a) &{1} (1,\\\\mathit {chk}) \\\\cdots (n,\\\\mathit {chk})\\\\multicolumn{2}{l}{\\\\text{by a rule}}\\\\\\\\(i, s_{\\\\mathit {acc}}, a) &{1} \\\\mathit {end}\\\\,,$ and include a rule $\\\\mathit {end}{1} \\\\mathit {end}$ (i.e., a self-loop).\",\n \"We want to construct the NBA $\\\\mathcal {A}$ so that it accepts exactly those branches that correspond to infinite computations that do not arrive at $s_{\\\\mathit {acc}}$ , or to “non-computations”, i.e., where the “guessing” in a rule $(i,(s,a,a^{\\\\prime },D,s^{\\\\prime })) {1/|\\\\Sigma |} (i{+}D, s^{\\\\prime }, b_j) \\\\quad (1 \\\\le j \\\\le |\\\\Sigma |)\\\\,,$ has been wrong.\",\n \"The NBA $\\\\mathcal {A}= (Q, \\\\Gamma , \\\\delta , Q_0, Q)$ has the same set of states as in thm:NBA-0: $Q \\\\ = \\\\ (\\\\lbrace 1, \\\\ldots , n\\\\rbrace \\\\times \\\\Sigma ) \\\\ \\\\cup \\\\ \\\\lbrace f\\\\rbrace $ As in thm:NBA-0, the set of initial states is $Q_0 = \\\\lbrace (1,a_1), \\\\ldots , (n,a_n)\\\\rbrace $ (recall that $a_1 \\\\cdots a_n$ is the input word).\",\n \"As in thm:NBA-0, the idea is that if a prefix $X_1 \\\\cdots X_k \\\\in \\\\Gamma ^*$ of a tree branch corresponds to a prefix of a correct computation of $M$ , then the set of automaton states in $\\\\delta (Q_0,X_1 \\\\cdots X_k)$ corresponds to the tape after this computation prefix.\",\n \"In fact, the transition relation $\\\\delta \\\\subseteq Q \\\\times \\\\Gamma \\\\times Q$ is deterministic, i.e., for all $q \\\\in Q$ and $X \\\\in \\\\Gamma $ there is at most one $q^{\\\\prime }$ with $(q,X,q^{\\\\prime }) \\\\in \\\\delta $ .\",\n \"Moreover, for any transition $((i,a), X, (j,a^{\\\\prime })) \\\\in \\\\delta $ we will have $i=j$ .\",\n \"Unlike in thm:NBA-0, all states are accepting.\",\n \"For $(i,a) \\\\in Q$ and all $(j,s,b) \\\\in \\\\Gamma $ with $i \\\\ne j$ or $a=b$ , include a self-loop $((i,a),(j,s,b),(i,a)) \\\\in \\\\delta \\\\,.$ (Intuitively, letter $a$ in cell $i$ is compatible with the head being on cell $j$ and reading letter $b$ .)\",\n \"For $(i,a) \\\\in Q$ and all $(i,s,b) \\\\in \\\\Gamma $ with $a \\\\ne b$ , include a transition $((i,a),(i,s,b),f) \\\\in \\\\delta \\\\,.$ (Intuitively, letter $a$ in cell $i$ is not compatible with the head being on cell $i$ and reading letter $b$ ; i.e., the prefix of the branch does not correspond to a correct computation.)\",\n \"For all $(i,a) \\\\in Q$ and all $(j,t) \\\\in \\\\Gamma $ with $i \\\\ne j$ , include a self-loop $((i,a),(j,t),(i,a)) \\\\in \\\\delta \\\\,.$ (Intuitively, the content of cell $i$ stays unchanged when the head is at position $j$ .)\",\n \"For all $i,s,a,a^{\\\\prime },D,s^{\\\\prime }$ with $(i,a) \\\\in Q$ and $(i,(s,a,a^{\\\\prime },D,s^{\\\\prime })) \\\\in \\\\Gamma $ , include a transition $((i,a), (i,(s,a,a^{\\\\prime },D,s^{\\\\prime })), (i,a^{\\\\prime }))\\\\in \\\\delta \\\\,.$ (Intuitively, the transition $(s,a,a^{\\\\prime },D,s^{\\\\prime })$ changes the content of cell $i$ from $a$ to $a^{\\\\prime }$ .)\",\n \"For all $X \\\\in \\\\Gamma $ , include a self-loop $(f,X,f) \\\\in \\\\delta $ .\",\n \"Note that $(f,\\\\mathit {end},f)$ is the only transition labeled with $\\\\mathit {end}$ .\",\n \"In this way: If a tree branch does not correspond to a correct computation, the automaton $\\\\mathcal {A}$ enters the state $f$ , remains there forever, and, thus, accepts.\",\n \"If a tree branch corresponds to an infinite computation not entering $s_{\\\\mathit {acc}}$ , the set of states that $\\\\mathcal {A}$ can be in always reflects the tape.\",\n \"Thus, $\\\\mathcal {A}$ accepts.\",\n \"If a tree branch corresponds to a computation entering $s_{\\\\mathit {acc}}$ , the branch also enters $\\\\mathit {end}$ , and $\\\\mathcal {A}$ does not enter $f$ .\",\n \"Thus, $\\\\mathcal {A}$ rejects.\",\n \"It follows that the probability that all branches of the random tree are rejected by $\\\\mathcal {A}$ is positive if and only $M$ accepts $w$ .\",\n \"A more detailed argument would follow very similar lines as the proof of thm:NBA-0.\"\n ],\n [\n \"Proof of lem:key\",\n \"* The automaton $\\\\mathcal {A}[f,X_f]$ is unambiguous, as $\\\\mathcal {A}$ is unambiguous, and has $(f,X_f)$ as (the only) accepting state.\",\n \"Recall also that $(f,X_f)$ is reachable from all states in $\\\\mathcal {A}[f,X_f]$ .\",\n \"It follows that $\\\\mathcal {A}[f,X_f]$ does not have diamonds, i.e., $\\\\mathcal {A}[f,X_f]$ does not have states $(q,X), (q^{\\\\prime },X^{\\\\prime })$ and a word $u \\\\in \\\\Gamma ^*$ such that $\\\\mathcal {A}[f,X_f]$ has two different paths $(q,X) {\\\\;\\\\xrightarrow{}{}\\\\hspace{0.0pt}^{*}\\\\;} (q^{\\\\prime },X^{\\\\prime })$ .\",\n \"By the Perron-Frobenius theorem [3], $M$ has an eigenvector $v \\\\in (0,\\\\infty )^{Q[f,X_f]}$ (all entries positive) with $M v = \\\\rho v$ .\",\n \"(Think of the entries of $v$ as “weights” of the states in $\\\\mathcal {A}[f,X_f]$ .\",\n \"Loosely speaking, the equality $(M v)_{(q,X)} = \\\\rho v_{(q,X)}$ expresses that the expected combined weight of the “successors” of $(q,X)$ is equal to the weight of $(q,X)$ multiplied by $\\\\rho $ .)\",\n \"We “lift” $v$ to define a vector $\\\\bar{v} \\\\in [0,\\\\infty )^{\\\\Gamma ^{\\\\prime }}$ where $\\\\Gamma ^{\\\\prime }$ is the set of types in $\\\\mathcal {B}_{\\\\mathit {det}}$ (recall that they are of the form $P \\\\times \\\\lbrace X\\\\rbrace $ with $P \\\\subseteq Q$ and $X \\\\in \\\\Gamma $ ): $\\\\bar{v}_{P \\\\times \\\\lbrace X\\\\rbrace } \\\\ := \\\\ \\\\sum _{q \\\\in P} v_{(q,X)}$ Denote by $\\\\bar{M} \\\\in \\\\mathbb {Q}^{\\\\Gamma ^{\\\\prime } \\\\times \\\\Gamma ^{\\\\prime }}$ the matrix defined before lem:as-finiteness-char, but for $\\\\mathcal {B}_{\\\\mathit {det}}$ .\",\n \"Then we have for all $P \\\\times \\\\lbrace X\\\\rbrace \\\\in \\\\Gamma ^{\\\\prime }$ : $(\\\\bar{M} \\\\bar{v})_{P \\\\times \\\\lbrace X\\\\rbrace }&\\\\ =\\\\ \\\\sum _{X {p} X_1 \\\\cdots X_k} p \\\\sum _{i=1}^k \\\\bar{v}_{\\\\delta _{\\\\mathit {det}}(P \\\\times \\\\lbrace X\\\\rbrace , X_i)} \\\\\\\\&\\\\ =\\\\ \\\\sum _{X {p} X_1 \\\\cdots X_k} p \\\\sum _{i=1}^k \\\\ \\\\sum _{(r,X_i) \\\\in \\\\delta _{\\\\mathit {det}}(P \\\\times \\\\lbrace X\\\\rbrace , X_i)} v_{(r,X_i)} \\\\\\\\&\\\\ \\\\mathop {=}^*\\\\ \\\\sum _{X {p} X_1 \\\\cdots X_k} p \\\\sum _{i=1}^k \\\\ \\\\sum _{q \\\\in P} \\\\ \\\\sum _{r : (q,X) \\\\xrightarrow{} (r,X_i)} v_{(r,X_i)} \\\\\\\\&\\\\ =\\\\ \\\\sum _{q \\\\in P} \\\\ \\\\sum _{X {p} X_1 \\\\cdots X_k} p \\\\sum _{i=1}^k \\\\ \\\\sum _{r : (q,X) \\\\xrightarrow{} (r,X_i)} v_{(r,X_i)} \\\\\\\\&\\\\ =\\\\ \\\\sum _{q \\\\in P} (M v)_{(q,X)}\\\\\\\\&\\\\ =\\\\ \\\\sum _{q \\\\in P} \\\\rho v_{(q,X)}\\\\\\\\&\\\\ =\\\\ \\\\rho \\\\bar{v}_{P \\\\times \\\\lbrace X\\\\rbrace }\\\\;,$ where the third equality (marked with $\\\\displaystyle \\\\mathop {=}^*$ ) holds as for any $(r,X_i) \\\\in \\\\delta _{\\\\mathit {det}}(P \\\\times \\\\lbrace X\\\\rbrace , X_i)$ there is exactly one $q \\\\in P$ with $(q,X) \\\\xrightarrow{} (r,X_i)$ in $\\\\mathcal {A}[f,X_f]$ .\",\n \"Indeed, towards a contradiction, suppose there are $q_1, q_2 \\\\in Q$ with $q_1 \\\\ne q_2$ and $(q_j,X) \\\\xrightarrow{} (r,X_i)$ for both $j \\\\in \\\\lbrace 1,2\\\\rbrace $ .\",\n \"Since $P \\\\times \\\\lbrace X\\\\rbrace $ is reachable in $\\\\mathcal {A}_{\\\\mathit {det}}$ from $\\\\lbrace (f,X_f)\\\\rbrace $ , there is $u \\\\in \\\\Gamma ^*$ with $(f,X_f) {\\\\;\\\\xrightarrow{}{}\\\\hspace{0.0pt}^{*}\\\\;} (q_j,X) \\\\xrightarrow{} (r,X_i)$ in $\\\\mathcal {A}[f,X_f]$ for both $j \\\\in \\\\lbrace 1,2\\\\rbrace $ , contradicting the absence of diamonds in $\\\\mathcal {A}[f,X_f]$ .\",\n \"We conclude from the above computation that $\\\\bar{M} \\\\bar{v} = \\\\rho \\\\bar{v}$ ; i.e., $\\\\bar{v}$ is an eigenvector of $\\\\bar{M}$ with eigenvalue $\\\\rho $ .\",\n \"In the following, for subsets $\\\\Delta \\\\subseteq \\\\Gamma ^{\\\\prime }$ , we write $\\\\bar{M}_{\\\\Delta } \\\\in \\\\mathbb {Q}^{\\\\Delta \\\\times \\\\Delta }$ for the (square) principal submatrix obtained from $\\\\bar{M}$ by restricting it to the rows and columns indexed by elements of $\\\\Delta $ .\",\n \"Similarly, define $\\\\bar{v}_{\\\\Delta } \\\\in [0,\\\\infty )^{\\\\Delta }$ by restricting $\\\\bar{v}$ to the entries indexed by elements of $\\\\Delta $ .\",\n \"Since $\\\\bar{M}$ and $\\\\bar{v}$ are nonnegative, we have $\\\\bar{M}_{\\\\Delta } \\\\bar{v}_{\\\\Delta } \\\\le \\\\rho \\\\bar{v}_{\\\\Delta }$ (the inequality is meant componentwise).\",\n \"Note that $\\\\bar{v}_\\\\emptyset = 0$ .\",\n \"Define $\\\\Gamma ^{\\\\prime \\\\prime } := \\\\Gamma ^{\\\\prime } \\\\setminus \\\\lbrace \\\\emptyset \\\\rbrace $ .\",\n \"Thus we have $\\\\bar{M}_{\\\\Gamma ^{\\\\prime \\\\prime }} \\\\bar{v}_{\\\\Gamma ^{\\\\prime \\\\prime }} = \\\\rho \\\\bar{v}_{\\\\Gamma ^{\\\\prime \\\\prime }}$ .\",\n \"All entries of $\\\\bar{v}_{\\\\Gamma ^{\\\\prime \\\\prime }}$ are positive, as all entries of $v$ are.\",\n \"By Perron-Frobenius theory [3] it follows that $\\\\rho $ is the spectral radius of $\\\\bar{M}_{\\\\Gamma ^{\\\\prime \\\\prime }}$ .\",\n \"The matrix $\\\\bar{M}_{\\\\Gamma ^{\\\\prime \\\\prime }}$ is equal to the matrix defined before lem:as-finiteness-char, but for $\\\\mathcal {B}_{\\\\mathit {det}}^{\\\\prime \\\\prime }$ .\",\n \"We complete the proof with the following case distinction.\",\n \"Suppose $\\\\rho > 1$ .\",\n \"Let $\\\\Delta \\\\subseteq \\\\Gamma ^{\\\\prime \\\\prime }$ be a bottom SCC of the graph of $\\\\bar{M}_{\\\\Gamma ^{\\\\prime \\\\prime }}$ .\",\n \"As $\\\\Delta $ is bottom, $\\\\bar{M}_\\\\Delta \\\\bar{v}_\\\\Delta = \\\\rho \\\\bar{v}_\\\\Delta $ .\",\n \"So the spectral radius of $\\\\bar{M}_\\\\Delta $ is at least $\\\\rho > 1$ (in fact, it must be equal to $\\\\rho $ ).\",\n \"It follows that $\\\\Delta $ is supercritical in $\\\\mathcal {B}_{\\\\mathit {det}}^{\\\\prime \\\\prime }$ .\",\n \"Thus, by lem:as-finiteness-char, a $\\\\mathcal {B}_{\\\\mathit {det}}^{\\\\prime \\\\prime }$ -tree is infinite with positive probability.\",\n \"Suppose that $\\\\rho = 1$ and that $\\\\mathcal {A}[f,X_f]$ does not have proper branching.\",\n \"Let $\\\\Delta \\\\subseteq \\\\Gamma ^{\\\\prime \\\\prime }$ be a bottom SCC of the graph of $\\\\bar{M}_{\\\\Gamma ^{\\\\prime \\\\prime }}$ .\",\n \"Then $\\\\bar{M}_\\\\Delta \\\\bar{v}_\\\\Delta = \\\\bar{v}_\\\\Delta $ , so by the Perron-Frobenius theorem [3] the spectral radius of $\\\\bar{M}_\\\\Delta $ is 1.\",\n \"By the absence of proper branching we also have $\\\\bar{M}_\\\\Delta \\\\vec{1} \\\\le \\\\vec{1}$ , where $\\\\vec{1}$ denotes the all-1 vector, i.e., the element of $\\\\lbrace 1\\\\rbrace ^\\\\Delta $ .\",\n \"By Perron-Frobenius theory [3] it follows that $\\\\bar{M}_\\\\Delta \\\\vec{1} = \\\\vec{1}$ .\",\n \"Thus, $\\\\Delta $ is linear in $\\\\mathcal {B}_{\\\\mathit {det}}^{\\\\prime \\\\prime }$ .\",\n \"Hence, by lem:as-finiteness-char, a $\\\\mathcal {B}_{\\\\mathit {det}}^{\\\\prime \\\\prime }$ -tree is infinite with positive probability.\",\n \"Suppose that $\\\\rho =1$ and that $\\\\mathcal {A}[f,X_f]$ has proper branching, i.e., there exist $(q,Y) \\\\xrightarrow{} (r_1,Z_1) \\\\text{ and } (q,Y) \\\\xrightarrow{} (r_2,Z_2) \\\\text{ in } \\\\mathcal {A}[f,X_f]$ and a rule $Y {p} u_1 Z_1 u_2 Z_2 u_3$ with $u_1, u_2, u_3 \\\\in \\\\Gamma ^*$ .\",\n \"Consider any SCC $\\\\Delta \\\\subseteq \\\\Gamma ^{\\\\prime \\\\prime }$ of the graph of $\\\\bar{M}_{\\\\Gamma ^{\\\\prime \\\\prime }}$ .\",\n \"Denote by $\\\\rho _\\\\Delta $ the spectral radius of $\\\\bar{M}_\\\\Delta $ .\",\n \"As $\\\\bar{M}_\\\\Delta $ is a principal submatrix of $\\\\bar{M}$ , we have $\\\\rho _\\\\Delta \\\\le \\\\rho = 1$ [3].\",\n \"So $\\\\Delta $ is not supercritical.\",\n \"$\\\\rho _\\\\Delta < 1$ .\",\n \"We have argued before lem:as-finiteness-char that $\\\\Delta $ being linear would imply $\\\\rho _\\\\Delta =1$ .\",\n \"Hence, $\\\\Delta $ is not linear.\",\n \"$\\\\rho _\\\\Delta = 1$ .\",\n \"Recall that $\\\\bar{M}_{\\\\Delta } \\\\bar{v}_{\\\\Delta } \\\\le \\\\bar{v}_{\\\\Delta }$ , so by Perron-Frobenius theory [3] we must have $\\\\bar{M}_{\\\\Delta } \\\\bar{v}_{\\\\Delta } = \\\\bar{v}_{\\\\Delta }$ .\",\n \"Let $P \\\\times \\\\lbrace X\\\\rbrace \\\\in \\\\Delta $ and $p \\\\in P$ .\",\n \"Let $u_0 \\\\in \\\\Gamma ^*$ with $(p,X) {\\\\;\\\\xrightarrow{}{}\\\\hspace{0.0pt}^{*}\\\\;} (q,Y)$ in $\\\\mathcal {A}[f,X_f]$ .\",\n \"Hence $(p,X) {\\\\;\\\\xrightarrow{}{}\\\\hspace{0.0pt}^{*}\\\\;} (q,Y) \\\\xrightarrow{} (r_j,Z_j)$ in $\\\\mathcal {A}[f,X_f]$ for both $j \\\\in \\\\lbrace 1,2\\\\rbrace $ .\",\n \"Towards a contradiction, suppose $\\\\Delta $ is linear.\",\n \"If $\\\\delta _{\\\\mathit {det}}(P \\\\times \\\\lbrace X\\\\rbrace , u_0)$ is not in $\\\\Delta $ , then neither are $\\\\delta _{\\\\mathit {det}}(P \\\\times \\\\lbrace X\\\\rbrace , u_0 Z_1)$ or $\\\\delta _{\\\\mathit {det}}(P \\\\times \\\\lbrace X\\\\rbrace , u_0 Z_2)$ .\",\n \"Otherwise (i.e., $\\\\delta _{\\\\mathit {det}}(P \\\\times \\\\lbrace X\\\\rbrace , u_0) \\\\in \\\\Delta $ ) there is $j \\\\in \\\\lbrace 1,2\\\\rbrace $ such that $\\\\delta _{\\\\mathit {det}}(P \\\\times \\\\lbrace X\\\\rbrace , u_0 Z_j) \\\\notin \\\\Delta $ , as $\\\\Delta $ is linear.\",\n \"Either way there is $j \\\\in \\\\lbrace 1,2\\\\rbrace $ such that $\\\\delta _{\\\\mathit {det}}(P \\\\times \\\\lbrace X\\\\rbrace , u_0 Z_j) \\\\notin \\\\Delta \\\\,.$ Note that $(r_j,Z_j) \\\\in \\\\delta _{\\\\mathit {det}}(P \\\\times \\\\lbrace X\\\\rbrace , u_0 Z_j) \\\\ne \\\\emptyset $ .\",\n \"Let $x Z$ (with $x \\\\in \\\\Gamma ^*$ and $Z \\\\in \\\\Gamma $ ) be the shortest prefix of $u_0 Z_j$ such that $U := \\\\delta _{\\\\mathit {det}}(P \\\\times \\\\lbrace X\\\\rbrace , x) \\\\in \\\\Delta $ but $\\\\delta _{\\\\mathit {det}}(U,Z) \\\\notin \\\\Delta $ .\",\n \"Then we have: $\\\\bar{v}_U&\\\\ =\\\\ (\\\\bar{M}_\\\\Delta \\\\bar{v}_\\\\Delta )_U \\\\\\\\&\\\\ <\\\\ (\\\\bar{M}_\\\\Delta \\\\bar{v}_\\\\Delta )_U + \\\\bar{M}_{U,\\\\delta _{\\\\mathit {det}}(U,Z)} \\\\bar{v}_{\\\\delta _{\\\\mathit {det}}(U,Z)} \\\\\\\\&\\\\ \\\\le \\\\ (\\\\bar{M} \\\\bar{v})_U \\\\\\\\&\\\\ =\\\\ \\\\bar{v}_U\\\\,,$ a contradiction.\",\n \"Thus, $\\\\Delta $ is not linear.\",\n \"We conclude that in both cases $\\\\Delta $ is neither linear nor supercritical.\",\n \"Since $\\\\Delta $ was an arbitrary SCC, we conclude from lem:as-finiteness-char that a $\\\\mathcal {B}_{\\\\mathit {det}}^{\\\\prime \\\\prime }$ -tree is almost surely finite.\",\n \"Suppose $\\\\rho < 1$ .\",\n \"Then, for any SCC $\\\\Delta $ the spectral radius of $\\\\bar{M}_{\\\\Delta }$ is also less than 1 [3], so $\\\\Delta $ is neither supercritical nor linear.\",\n \"Thus, by lem:as-finiteness-char, a $\\\\mathcal {B}_{\\\\mathit {det}}^{\\\\prime \\\\prime }$ -tree is almost surely finite.\",\n \"Hence, the probability that a $\\\\mathcal {B}_{\\\\mathit {det}}^{\\\\prime \\\\prime }$ -tree is infinite is positive if and only if either $\\\\rho > 1$ or $\\\\rho = 1$ and $\\\\mathcal {A}[f,X_f]$ does not have proper branching.\"\n ],\n [\n \"Proof of thm:LTL-0\",\n \"* Given the main body, it remains to show 2EXPTIME-hardness of $\\\\mathbb {P}(\\\\textup {LTL}) = 0$ .\",\n \"Our construction is inspired by the proof of 2EXPTIME-hardness of model-checking a concurrent probabilistic program (another name for MDP) against an LTL formula, see Theorem 3.2.1 in [11].\",\n \"We will use the fact that 2EXPTIME is equal to alternating EXPSPACE.\",\n \"Let $M = (S_\\\\exists , S_\\\\forall , \\\\Sigma , T, s_0, s_{\\\\mathit {acc}})$ be an alternating Turing machine whose work tape usage is bounded by $2^n$ on any input of length $n$ .\",\n \"Without loss of generality, we can assume that the machine has two possible next moves for each configuration and that it halts when it reaches the accepting state $s_{\\\\mathit {acc}}$ .\",\n \"For a given alternating TM $M$ and an input $w$ of length $n$ , we will construct a BP $\\\\mathcal {B}$ and an LTL formula $\\\\varphi $ both of size $\\\\mathcal {O}(n)$ such that $\\\\mathbb {P}_\\\\mathcal {B}(\\\\varphi ) > 0$ if and only if $M$ accepts $w$ .\",\n \"The branching process $\\\\mathcal {B}$ is defined by the diagram in Fig.\",\n \"REF .\",\n \"Every node in the diagram has a unique label that corresponds to a type of $\\\\mathcal {B}$ , although not all nodes are explicitly labelled.\",\n \"The $$ -nodes represent randomising branching and $$ -nodes represent tree branching.\",\n \"Namely, if a $$ -node $x$ has $k$ successors $y_1,\\\\ldots , y_k$ : $@1@R=10pt@C=10pt{ & *+[l]{x\\\\,} [dr] [dl] &\\\\\\\\y_1 & \\\\cdots &y_k}$$, then $ B$ has the rules $ x 1/k yi$ for $ i=1,...,k$.\",\n \"On the other hand, if a $$-node $ x$ has $ k$ successors $ y1,..., yk$: $$, then $ B$ has the rule $ x 1 y1yk$.$ The start type of $\\\\mathcal {B}$ is $a$ .\",\n \"The detailed diagram of the blocks $I$ , $O$ , $N$ , $D_1$ , $D_2$ and $D_3$ is shown in Fig.\",\n \"REF on the left.\",\n \"All nodes in these blocks are $$ -nodes.\",\n \"The diagram of the blocks $C_1$ and $C_2$ is shown in Fig.\",\n \"REF on the right.\",\n \"These blocks have $$ -nodes in the first $n+1$ levels, and the rest are $$ -nodes.\",\n \"The initial and final nodes are labelled by $u$ and $v$ , respectively.\",\n \"The nodes below $u$ are labelled with $(\\\\ell _i,0)$ and $(\\\\ell _i,1)$ , $i=1,\\\\ldots ,n$ , as shown in the picture.\",\n \"Every block has its own unique set of labels $u$ , $v$ , $(\\\\ell _i,0)$ and $(\\\\ell _i,1)$ , $i=1,\\\\ldots ,n$ ; however we do not distinguish them in the diagram for simplicity.\",\n \"In every block, except for $I$ , the nodes above $v$ are labelled by $(\\\\ell _{n+1},\\\\delta )$ , where $\\\\delta \\\\in \\\\Sigma \\\\;\\\\cup \\\\; (S_\\\\exists \\\\cup S_\\\\forall )\\\\times \\\\Sigma $ ranges over the symbols of the extended work tape alphabet.\",\n \"In block $I$ , there are $n+1$ nodes above $v$ which are labelled by $(\\\\ell _{n+1},\\\\gamma _i)$ for $i=1,\\\\ldots ,n+1$ such that for an input word $w=w_1\\\\ldots w_n$ we have $\\\\gamma _1=(s_0,w_1)$ , $\\\\gamma _i=w_i$ for $1 0$ .\",\n \"Conversely, suppose there is a positive probability that $\\\\mathcal {B}$ generates a tree $T$ all whose branches satisfy $\\\\varphi $ .\",\n \"Hence every branch of $T$ eventually reaches state $d$ (and then always repeats it).\",\n \"Since $T$ is finitely branching, it follows by König's lemma that there exists a finite prefix of $T$ whose every leaf is labelled by $d$ .\",\n \"This prefix encodes the moves of the existential player (after each appearance of state $m_1$ in the prefix) that allow him to reach the accepting configuration, no matter what the universal player does.\",\n \"In other words, the existential player has a winning strategy, and hence $M$ accepts $w$ .\",\n \"Formally, this can be shown along similar lines as in the proof of thm:NBA-0.\",\n \"Therefore, we proved that $M$ accepts $w$ if and only if $\\\\mathbb {P}_\\\\mathcal {B}(\\\\varphi ) > 0$ .\",\n \"Figure: Diagrams of the randomising (left) and tree (right) branching blocks.Figure: Diagram of the branching process ℬ\\\\mathcal {B}.\"\n ]\n]"},"id":{"kind":"string","value":"2107.01687"}}},{"rowIdx":1578,"cells":{"text":{"kind":"list like","value":[["The Discovery of a Remnant Radio Galaxy in A2065 Using GMRT"],["Abstract The upgraded Giant Metrewave Radio Telescope (GMRT) has been used to map the cluster A2065 at z = 0.0726.","We report the discovery of a remnant radio galaxy at the peripheral cluster region.","The spatially resolved radio emission from the remnant radio galaxy shows an elongated, bar-shaped structure, whose size is $\\approx$ 52$^{\\prime\\prime}$ $\\times$ 110$^{\\prime\\prime}$ ($\\simeq$ 72 $\\times$ 152 kpc$^2$).","Our study with the multiwavelength GMRT data and \\textit{Chandra} data shows that across the remnant radio galaxy there is a hint of a surface-brightness edge in the hot X-ray gas.","We detect tentative flattening of the radio spectral index as the old plasma at the near end of the surface-brightness edge is reinvigorated by the passage of possible shock front and shows the expected change in radio emission characteristics.","We suggest that the remnant radio galaxy has been seeded by the lobes of the active galactic nucleus (AGN), hosted by the WISEA J152228.01$+$274141.3 source, demonstrating the connection between AGNs and remnant radio sources.","Although the number of known remnant radio sources is beginning to increase, we emphasize the need for better data to understand the physics and nature of poorly understood remnant radio sources."],["Introduction","The hierarchical structure formation scenario suggests that cluster mergers are the mechanisms by which galaxy clusters form and these mergers are the most energetic phenomena in the universe since the Big Bang.","A large number of clusters of galaxies are known to contain large-scale, low-surface-brightness diffuse radio emission whose origin is not often related to the active galactic nuclei (AGNs) but to the intracluster medium (ICM).","According to the location with respect to the cluster center, these sources are of two key types, radio halos and relics.","Radio halos show a regular, amorphous emission around the cluster center of a size from a few hundred kiloparsecs (minihalo) to megaparsecs (halo), whereas radio relics show an irregular emission located in peripheral cluster regions of a size from several hundreds of kiloparsecs to megaparsecs.","Another third broad type of sources comprises of revived fossil plasma sources and phoenices.","These do not have any preferred location in the ICM are of size from several tens of kpc to a few hundred kpc and trace AGN radio plasma that has somehow been reenergised through the processes in the ICM [27], [16], [9].","The radio jets emanating from an AGN arise from the accretion onto a supermassive black hole.","These form synchrotron-emitting radio lobes in a radio-loud AGN in the environments of their host galaxies.","The active phase of an AGN can last several tens of megayears, following this the radio jets stop and their features, namely an unresolved radio core, radio jets, hotspots, and radio lobes, all start to fade away due to synchrotron cooling.","This phase once the jets have switched off is called as the remnant phase of a radio galaxy, or merely the remnant radio galaxy.","The remnant radio galaxy falls into the third broad type of sources in the ICM, which is poorly understood and presently very few sources are known [25], [2], [23].","The high spatial resolution of the Chandra and upgraded Giant Metrewave Radio Telescope (GMRT) allow us to study the interplay between X-ray-emitting hot gas and nonthermal radio emission.","We have obtained new upgraded GMRT data, and have used archive GMRT and Chandra data of the nearby $z$ = 0.0726 [26] richness class 2 cluster, A2065, which is one of the 10 galaxy clusters that make up the Corona Borealis supercluster.","[24] noted that two cD galaxies dominate the cluster center in the optical and their radial velocities differ by $\\sim $ 600 km s$^{-1}$ .","Table: Summary of the radio observations and image properties.Previous radio study using 100 m Green Bank Telescope at 1.4 GHz [7] detected a smooth diffuse structure $\\simeq $ 1 Mpc in extent, a giant radio halo, whereas a recent study showed a slightly smaller extent for the radio halo using Low-Frequency Array (LOFAR) at 153 MHz [6].","Similarly, previous X-ray studies led to detections of two surface-brightness peaks coincident with the two cD galaxies observed in the cluster's center.","The data indicated that the cluster is an unequal-mass merger and only one of the two cooling cores has survived the merger.","The northern cluster seems to have fallen into the more massive southern cluster from the southeast, and having lost its gas content, it has now moved to its present location to the northwest.","Evidence of shock with $M$ $\\approx $ 1.7 at $\\sim $ 140 kpc from the southern cD galaxy too is noted and the cooling flow is displaced to the southeast of the southern cD galaxy [3], [20].","In this study, we report the discovery of a remnant radio galaxy in cluster A2065 using the GMRT.","We also present an X-ray analysis of archive Chandra data, which enhances the interpretation of the current discovery.","We define spectral index, $\\alpha $ as, $S_\\nu $ $\\propto $ $\\nu ^{\\alpha }$ , where S$_\\nu $ is the flux density at frequency, $\\nu $ .","To estimate the intrinsic parameters, we use a $\\Lambda $ CDM cosmology with $\\Omega _{\\rm m}$ = 0.27, $\\Omega _{\\Lambda }$ = 0.73, and H$_0$ = 70 km s$^{-1}$ Mpc$^{-1}$ .","At the redshift of the cluster, 1 arcsec corresponds to 1.384 kpc.","All uncertainties are at 1$\\sigma $ error bars unless otherwise stated.","The position angles (P.A.s) are measured from north to east.","Throughout, positions are given in J2000 coordinates.","The organization of the paper is as follows.","We present the X-ray and radio observations and their data analyses in Sec. .","Our results, i.e., radio morphology, spectral structure, and X-ray gas properties are presented in Sec. .","In Sec.",", we discuss our observational results, source diagnostics, and dynamical interpretation of the remnant radio galaxy.","Sec.","summarizes our conclusions.","A2065 was observed by Chandra with the acis-i detector on 2002 August 18 and November 24 for 28 and 22 ks (Obs_ID 3182) respectively, and on 2007 September 13 for 5 ks (Obs_ID 7689).","The data reduction of these three archival data sets has been performed independently.","We used the calibration databases, caldb version 4.7.6 and ciao version 4.9.","After the initial calibration, i.e., standard filtering and removal of bad pixels, the event files of the three observations were projected to a common reference point and then merged to create Gaussian-smoothed, background-subtracted, exposure-corrected image in the broadband 0.5–7.0 keV energy range.","We excised the central AGN and background point sources.","Using the ciao tool specextract, we extracted the source, background spectra, and the response files of selected regions in the Chandra images separately for the three data sets.","The spectra for each data set were grouped so that each bin contained at least 20 counts.","We restricted our spectral extraction in the energy range 0.5–7.5 keV, and fitted for the thermal gas emission with xspec [1] using the isothermal apec model.","The abundance was fixed to 0.3 $\\times $ solar [3] and all spectral fits include absorption by the Galactic column [5].","Figure: Exposure-corrected, background-subtracted, broadband (0.5–7.0 keV) Chandra image of A2065.","The image was binned (bin = 16) and smoothed using a Gaussian kernel of 10 '' ^{\\prime \\prime }.","The color bar shows the surface-brightness, in 10 -6 ^{-6} counts s -1 ^{-1} per 8 ×\\times 8 pixel binning.","The X-ray peak is located at R.A. 15:22:29.18, decl.","27:42:21.05.","The peak surface-brightness and the faintest regions transitioning between blue and green have a surface-brightness of 6.0 ×\\times 10 -5 ^{-5} counts s -1 ^{-1} arcsec -2 ^{-2} and 1.6 ×\\times 10 -6 ^{-6} counts s -1 ^{-1} arcsec -2 ^{-2} respectively.The surface-brightness is displayed in logarithmic scales to emphasize low-surface-brightness emission."],["GMRT data","A summary of the radio observations is presented in Table REF .","We inspected the data for bad antennas, bad time stamps and data impacted by radio frequency interference.","The data were analyzed using the standard reduction methodologies [17].","After the initial steps of flux density and bandpass calibration, all corrupted data were removed using standard routines.","The bandpass calibrated data for wide-bandwith GMRT data, i.e., the 250–500 MHz band and 1050–1450 MHz band data were split into five 40 MHz (from 300–500 MHz) and eight 50 MHz (from 1050–1450 MHz) subbands respectively.","Subsequently, each subband data was analyzed separately, similar to narrowband “classic\" GMRT data.","Several cycles of phase-only self-calibration and imaging were performed and the calibration solutions were applied to the target data to correct for residual phase errors.","To take care of the wide-field imaging at low GMRT frequencies, we performed faceting imaging.","Problematic bright sources in the field of view whose side lobes affected the area of the target were subtracted, a.k.a.","peeled; this (direction-dependent) calibration step improved the dynamic range of images.","The calibrated data for five 40 MHz 250–500 MHz band data and eight 50 MHz 1050–1450 MHz band data were joined to form full 200 MHz and 400 MHz, respectively, wide-bandwidth-calibrated visibility data sets.","The combined wide-bandwidth-calibrated data (and similarly the narrowband-calibrated “classic\" GMRT data) were imaged using the casa task tclean.","We used 3D imaging (gridder = `widefield'), two Taylor coefficients (nterms = 2, tt0 and tt1) and Briggs weighting (robust = 0.5) in task tclean.","A final amplitude-and-phase self-calibration with solution interval equal to the length of observation was then carried out using the gaincal and applycal tasks in casa.","The two Stokes polarizations, RR and LL, were combined to obtain the final total intensity Stokes I image, and corrected for the GMRT primary beam response.","This provides images at center frequencies (tt0) and the corresponding in-band spectral index maps (tt1/tt0).","The amplitude errors are estimated to be within 5% or less for archival narrow-bandwidth and wide-bandwidth GMRT data.","Table REF also provides image properties, including the restoring beams and root mean square (rms) noise levels at half-power points of the final images.","The uncertainty in the flux density measurements are estimated as $\\left[(S_\\nu \\times f)^2 + (\\textsc {rms})^2 \\times N_{\\rm beams}\\right]^{0.5},$ where $S_\\nu $ is the flux density, $f$ is an absolute flux density calibration uncertainty, rms is the noise, and $N_{\\rm beams}$ is the number of synthesized beams.","We use these error estimates for the flux density measurements when computing the error estimates for spectral index measurements.","Figure: The central 3 ' ^{\\prime } ×\\times 3 ' ^{\\prime } cluster region with 250–500 MHz band radio contours overlaid on the Digitized Sky Survey image of A2065.","Note the northern cD galaxy registers well with the northern radio peak, but the southern cD galaxy is displaced by ∼\\sim 24 north of the radio peak.","The X-ray peak at the cluster center is coincident with the southern radio peak (see also Figure ).The radio emission from the two cD galaxies (the brighter southern galaxy 2MASS J152229.17++274227.5 and the fainter northern galaxy 2MASX J152228.92++274244.1) of the cluster is shown on the east of the remnant radio galaxy emission.The radio contour levels are --3, 3, 6, 9, 15, 42 and 84 ×\\times 80 μ\\mu Jy beam -1 ^{-1}.The cluster-member galaxies are marked and a majority (≈\\approx 60%) show radio emission.","The two bright cluster galaxies, SDSS ++230.6++27.7++0.08 (R.A.: 15:22:24.0, decl.","++27:42:51; zz = 0.0723) and 2MASS J152229.17++274222.75 (zz = 0.074875) are marked in blue colour ."],["Results","The exposure-corrected background-subtracted X-ray binned (bin = 16) and Gaussian-smoothed (10$^{\\prime \\prime }$ kernel) image in the 0.5–7.0 keV band is shown in Figure REF .","Figure REF shows Chandra X-ray contours from Figure REF and 250–500 MHz band radio contours overlaid on the exposure-corrected, background-subtracted broadband (0.5–7.0 keV) Chandra (bin = 8) image (left panel) and Digitized Sky Survey image (right panel) of A2065.","The central 3$^{\\prime }$ $\\times $ 3$^{\\prime }$ cluster region with 250–500 MHz band radio contours overlaid on the Digitized Sky Survey image of A2065 is shown in Figure REF .","Figure REF also marks the cluster member galaxies, including the two bright cluster galaxies in marked in blue colour, SDSS $+$ 230.6$+$ 27.7$+$ 0.08 (NED; R.A.: 15:22:24.0, decl.","$+$ 27:42:51; $z$ = 0.0723, $m$ = 16.1) and 2MASS J152229.17$+$ 274222.75 [24], and a majority of them, $\\approx $ 60%, show radio emission.","Figure REF shows clear detections for the southern cD galaxy and remnant radio galaxy at the GMRT bands, namely the 240 MHz band (top-left panel), 250–500 MHz band (top-right panel), 610 MHz band (bottom-left panel), and 1050–1450 MHz band (bottom-right panel).","The northern cD galaxy is only detected in our 250–500 MHz band data.","The images have rms noise values in the range 0.56–0.06 mJy beam$^{-1}$ at the half-power points.","The 150 MHz band and 1050–1450 MHz band data sets had very short on-source integration times, $\\lesssim $ 0.9 hr, and therefore these images have relatively high rms noise values (see Table REF ).","Note that we also made images at several angular resolutions, including imaging data sets using the same range of baseline lengths, and do not find the presence or absence of additional morphological features.","The integrated flux densities computed thus are also consistent, within error bars with those reported in Table REF ."],["Southern and northern cD galaxies","Figure REF shows a diffuse X-ray tail extending to the north at the center and that the cluster emission is elongated along northwest-southeast direction.","The hot gas emission (i) shows a compact, bright region and a diffuse tail extending to the north at the center, and (ii) is extended along the northwest–southeast direction showing pieces of evidence for surface-brightness edges.","Though the X-ray peak emission and peak of the radio emission register well (see Figure REF , left), the position of the optical host of the southern cD galaxy (R.A.: 15:22:29.17, decl.",": $+$ 27:42:27.5) is displaced by $\\sim $ 24 from the compact, bright, X-ray peak emission at the cluster center (see Figure REF , right, and also Figure REF ).","A low-surface-brightness diffuse tail, both in X-ray as well as in radio images, extending to the north at the center of the image, reaches out to the second cD galaxy (R.A.: 15:22:28.92, decl.",": $+$ 27:42:44.1; Figures REF and REF , top-right panels).","The X-ray peak corresponding to the second northern cD galaxy and the peak of the second weaker component of radio emission at this location also register well.","The southern cD galaxy seems to have retained a cool core residing at the center of its subcluster and there is no evidence for a cool core corresponding to the northern cD galaxy [3], [20].","The closely associated radio and X-ray morphology at the center and its connection with the southern cD galaxy suggest that both X-ray and radio emissions are due to ram pressure stripping as the southern cD galaxy moves relative to the ICM.","The two blue plus signs in the 250–500 MHz band image marked in Figure REF (top-right panel) correspond to the hosts of the brighter southern cD galaxy, 2MASS J152229.17$+$ 274222.75 at $z$ = 0.074875 and fainter northern cD galaxy, 2MASX J152228.92$+$ 274244.1 at $z$ = 0.072854 (see also Figure REF ).","The northern cD galaxy is only detected in our 250–500 MHz band image.","However, we not only detected the southern cD galaxy in our long observations, we also detected the peak radio emissions at the location of the southern cD galaxy at R.A.: 15:22:29.17, decl.",": $+$ 27:42:27.5 in our short, 150 MHz band, and 1050–1450 MHz band data sets.","The two cD galaxies are barely resolved using LOFAR at 153 MHz [6].","The southern cD galaxy also had a 2.8$\\sigma $ (upper limit) detection at 74 MHz [4].","It seems that around the southern cD radio galaxy there is some extended radio emission, which is more clearly seen in our 250–500 MHz band image (Figure REF , top-right panel).","Since several cluster-member galaxies also show radio emission, it is unclear if this extended radio emission is indeed associated with the southern cD radio galaxy or with the two adjacent cluster-member galaxies southeast of it (see Figure REF ).","Table: Radio flux densities for the remnant radio galaxy and the southern cD galaxy"],["Remnant radio galaxy","Figures REF and REF also suggest that the remnant radio galaxy is hosted by the 19.19 magnitude WISEA J152228.01$+$ 274141.3 source that is 78 away from the optical galaxy V1CG 136 (R.A.: 15:22:28.5, decl.",": $+$ 27:41:46) at redshift $z$ = 0.072 [12].","The cluster contains gas not only with an extremely wide range of temperatures and complex structure but that also shows a cold front at 30$^{\\prime \\prime }$ ($\\simeq $ 41.5 kpc) and a shock-like bow-shaped feature beyond $\\sim $ 100$^{\\prime \\prime }$ ($\\simeq $ 140 kpc) on the southeast of the southern cD [3].","Figure REF shows that the northeastern boundary of the remnant radio galaxy coincides with the 1.3 $\\times $ 10$^{-7}$ counts s$^{-1}$ arcsec$^{-2}$ surface-brightness contour level; gas in false color transitioning from the green region to the blue region at a distance of $\\sim $ 33$^{\\prime \\prime }$ ($\\simeq $ 45.7 kpc) from the southern cD galaxy.","All four panels of Figure REF suggest that the radio emission from the remnant radio galaxy has an elongated, bar-shaped structure.","The remnant radio galaxy, buried in the radio halo emission that is barely detected using LOFAR at 153 MHz seems to possess an elongated, bar-shaped structure [6], consistent with our results.","This suggests that the remnant radio galaxy has a steep spectrum, and its consistency in radio morphology gives us additional confidence in our image fidelity, imaging, calibration and data reduction processes.","The best sensitivity image for the low surface-brightness was obtained from the 250–500 MHz band data and we now discuss below the morphology of the remnant radio galaxy.","The remnant radio galaxy broadly consists of a high-surface-brightness southeastern part and a fainter low-surface-brightness extended northwestern part.","The two oppositely directed radio jets emanating from the apex of the WISEA J152228.01$+$ 274141.3 source initially traverse toward the southeast and northwest directions.","As the galaxy plows through the dense intracluster gas, these jets traversing in opposite directions form a trail, after sharp bends in the jets, behind the WISEA J152228.01$+$ 274141.3 source due to interaction with the ICM.","The width of the remnant radio galaxy increases from $\\sim $ 17$^{\\prime \\prime }$ ($\\simeq $ 23.5 kpc) at the southeastern end to $\\sim $ 27$^{\\prime \\prime }$ ($\\simeq $ 37.4 kpc) and reduces again before flaring to reach a maximum width of $\\sim $ 35$^{\\prime \\prime }$ ($\\simeq $ 48.4 kpc) and finally fading completely.","The average width of the narrower southeastern part is a factor of $\\sim $ 2.3 smaller than the broader northwestern part.","It seems that a pinch is present at $\\sim $ 35$^{\\prime \\prime }$ ($\\simeq $ 48.4 kpc) of the $\\sim $ 162 ($\\simeq $ 22.4 kpc) width from the southeastern end, i.e., between the two narrower and the broader parts.","The northeastern boundary toward the two cD galaxies of the remnant radio galaxy is sharp, while the emission fades more slowly at the southwestern boundary.","The overall source size, assuming elongation, of the bar-shaped structure is $\\approx $ 52$^{\\prime \\prime }$ $\\times $ 110$^{\\prime \\prime }$ , corresponding to 72 $\\times $ 152 kpc$^2$ at the assumed distance.","Figure: Integrated radio spectrum of the remnant radio galaxy using the data in Table .Peak flux density is reported from our low-resolution, 15 '' ^{\\prime \\prime } image at the 1050–1450 MHz band for the remnant radio galaxy (see also Sec.","for a discussion).We also show the spectra of the southern cD galaxy for comparison; note that the (nearly) straight spectra of it provides a good check of our calibration.","A 2.8σ\\sigma limit is reported at the 74 MHz , peak flux densities at the 150 MHz band, and 1050–1450ṀHz band are our measurements; the 1400 MHz data is from the VLA FIRST survey for the southern cD galaxy."],["Radio spectra","The total flux densities were determined from integration in polygons with measured in areas encompassing the remnant radio galaxy, i.e., we determined the surface-brightness and divided it by the number of pixels in the polygon at each frequency.","These results are presented in Table REF along with the total intensity spectra of the remnant radio galaxy, and the southern cD galaxy is shown in Figure REF .","A 2.8$\\sigma $ upper limit at the 74 MHz [4] and the peak flux density from our 150 MHz band data for the southern cD galaxy are also reported.","We see that all these radio measurements, within the error bars, fall nearly along with a `straight' power law, $\\alpha $ = $-$ 1.2 $\\pm $ 0.2 with some hints of energy losses by synchrotron cooling for the dominant southern cD galaxy.","This provides confidence in our data calibration and its reduction methodologies.","The remnant radio galaxy is not detected in our 150 MHz band short observation, and is barely detected using LOFAR at 153 MHz (see also Sec.","REF ).","However in the 1050–1450 MHz band short observation, we see hints of radio emission from the remnant radio galaxy.","We made a low-resolution, 15$^{\\prime \\prime }$ image at the 1050–1450 MHz band and report peak flux density for the radio emission from the remnant radio galaxy in Figure REF .","Since this 1050–1450 MHz band measurement comes from a short $\\lesssim $ 0.9 hr duration observation, has relatively high rms noise (see also Sec.","), and the observation does not sample similar ($u,v$ )-spacings as the rest of the low-frequency observations, we henceforth no longer include it in our spectral index analysis.","It appears that the integrated spectrum for the remnant radio galaxy cannot be represented by a single power law.","The segment of the spectrum with the best data is $<$ 1250 MHz, and the low-frequency spectral index $\\alpha $ (240–610 MHz) = $-$ 1.4 $\\pm $ 0.2.","Our data cannot rule out the possibility of spectral shape to be due to two populations of the relativistic electrons: a low-frequency component characterized by $\\alpha $ (240–610 MHz) $\\approx $ $-$ 1.4 and a second component having $\\alpha $ ($>$ 610 MHz) $<$ $-$ 1.4 with a cutoff or a spectral break between 400 and 1250 MHz.","Figure: The plot shows the radio spectra between the 240 MHz and 250–500 MHz band data sets (red upper triangles) and between the 250–500 MHz band and 610 MHz band data sets (blue lower triangles) as a function of distance from the head.","The location of the pinch is shown as a vertical line.Figure: The averaged in-band spectral index image showing spectra between 300 MHz and 500 MHz using 250–500 MHz band data.","The total intensity surface-brightness contours are at 0.23 and 0.46 mJy beam -1 ^{-1} from the 300–500 MHz radio image (Figure , top-right panel).","We used three irregular polygon regions, marked in the image, and determined the mean (and rms) spectral indices that are denoted at the upper-right corner of the image.","The averaged integrated in-band spectral index for the remnant radio galaxy is denoted at the lower-left corner of the image.","The cross marks correspond to the positions of two cD galaxies and the position of the optical host galaxy, discussed in Sec.","and .","The dashed line indicates the best-fitting position of the X-ray surface-brightness edge discussed in Sec.",".Figure: Background-subtracted and exposure-corrected Chandra image at ∼\\sim 4 '' ^{\\prime \\prime } angular resolution in the broad (0.5–7.0 keV) energy band.","The color bar shows the surface-brightness, in photon cm -2 ^{-2} s -1 ^{-1} pixels -1 ^{-1}.","The black surface-brightness contours are displayed using the 250–500 MHz band image.","The inset shows the enlargement of the inner region.","Also shown are two sets of sectors (in cyan and blue colors) that were used for the surface-brightness and temperature measurements.The spectral index measurements between the 240 MHz band and 250–500 MHz band data sets, and between the 250–500 MHz band and 610 band MHz band data sets show that the radio spectrum steepens with distance from the head, from $-$ 1.2 $\\pm $ 0.1 to $-$ 1.4 $\\pm $ 0.2, and from $-$ 1.3 $\\pm $ 0.1 to $-$ 1.6 $\\pm $ 0.2, respectively, due to synchrotron and inverse Compton losses (Figure REF ).","The location after the pinch, where the radio plasma from the AGN connects to the remnant radio galaxy, the spectrum, between the 240 MHz band and 250–500 MHz band data, remains flattened, $\\alpha $ = $-$ 1.3 $\\pm $ 0.1.","The wide-bandwith data sets were imaged to make multiscale multifrequency synthesis images, which provides images at center frequencies and the corresponding in-band spectral index maps (see also Sec.","REF ).","The in-band spectral index map averaged over a region of interest, to increase signal-to-noise ratio, for the 250–500 MHz band data is shown in Figure REF .","We divided the remnant radio galaxy into three distinct regions: (1) the high-surface-brightness narrower southeastern part ahead of the pinch (in cyan).","The latter low-surface-brightness broader northwestern part is divided into two; (2) the northeastern region (in yellow), and (3) the southwestern far end region (in magenta).","The size of the regions is chosen such that the signal in the total intensity image is (approximately) more than five times the rms noise.","The southern cD galaxy has an in-band spectral index of $-$ 0.9 $\\pm $ 0.2.","Similarly, the spectra of three regions of the remnant radio galaxy are $-$ 1.2 $\\pm $ 0.2, $-$ 1.3 $\\pm $ 0.2, and $-$ 1.4 $\\pm $ 0.2 for regions (1), (2), and (3) respectively.","This is consistent with the averaged in-band spectral index of $-$ 1.3 $\\pm $ 0.2 within the error bars for the remnant radio galaxy.","It is also consistent with the crude estimates of two 50 MHz subband images, 300–350 MHz and 450–500 MHz, at two extreme ends of the 250–500 MHz band data.","The region ahead of the pinch (in cyan) and the northeastern region close to two cD galaxies (in yellow) have comparable spectral indices.","It appears that the high-surface-brightness narrower region (in cyan) ahead of the pinch, is flatter than the two regions, together forming low-surface-brightness broader northwestern region (in yellow and magenta) after the pinch, consistent with our result discussed above.","Furthermore, the area between the two regions of the broader northwestern part of the source, i.e., the southwestern far-end region (in magenta) have a significantly steeper spectrum than the northeastern region close to two cD galaxies (in yellow)."],["ICM gas properties","[3] reported that the cluster contains gas with an extremely wide range of temperatures and a wealth of structure.","They also reported a discontinuity at $\\sim $ 30$^{\\prime \\prime }$ ($\\simeq $ 41.5 kpc) in the southeast region from the southern cD galaxy.","It is at nearly this same distance where we report an enhancement of radio surface-brightness, toward the two cD galaxies, showing a sharp feature.","We aim to search an X-ray surface-brightness edge, i.e., the shock front that is coincident with the northeastern sharp edge toward two cD galaxies of the remnant radio galaxy.","In order to understand the gas physics at this region of the cluster, we have extracted the surface-brightness profile from two box-shaped regions, one encompassing the remnant radio galaxy (called “outside\" here) and the other adjacent to it toward cD galaxies (called “inside\" here) as shown in Figure REF (in magenta).","We extracted the surface-brightnesses and the spectra from these regions, which are displayed in Figure REF .","A single temperature model with Galactic absorption and the abundance fixed provided a good fit to the spectrum.","We noticed the X-ray surface-brightness edge, where the surface-brightness drops (see Figures REF and REF ) and therefore implies a discontinuity, or jump, in the gas density.","The surface-brightnesses are $S_{\\rm x}$ (inside) and $S_{\\rm x}$ (outside) $\\approx $ 8.2 $\\pm $ 0.3 $\\times $ 10$^{-6}$ counts s$^{-1}$ arcsec$^{-2}$ and 3.7 $\\pm $ 0.3 $\\times $ 10$^{-6}$ counts s$^{-1}$ arcsec$^{-2}$ , respectively.","Consequently, our modeled densities in the two regions, $\\rho $ (inside) and $\\rho $ (outside) are approximately 2.9$^{+0.2}_{-0.1}$ $\\times $ 10$^{-3}$ cm$^{-3}$ and 1.9$^{+0.2}_{-0.2}$ $\\times $ 10$^{-3}$ cm$^{-3}$ , respectively.","Our corresponding derived temperatures for the two regions, $kT_{\\rm (inside)}$ and $kT_{\\rm (outside)}$ are approximately 6.1$^{+0.9}_{-0.8}$ keV ($\\chi ^2$ = 0.93 for 221 degrees of freedom) and 5.5$^{+0.5}_{-0.4}$ keV ($\\chi ^2$ = 1.08 for 462 degrees of freedom), respectively.","We also repeated this exercise for a pie-circular aperture (shown in blue in Figure REF ) centered on the southern cD galaxy and lying toward the radio emission from the remnant radio galaxy with two regions containing the radio emission from the remnant radio galaxy and the smaller adjacent region toward the cD galaxy.","The quantitative estimates from this are nearly identical and the fit statistics are at 90% confidence intervals.","Although with large error bars, these density and temperature measurements are hinting that the surface-brightness edge toward the two cD galaxies is possibly a shock front.","Consequently, the observed density jump and the temperature jump across the edge, $\\rho _{\\rm (inside)}$ /$\\rho _{\\rm (outside)}$ = 1.5$^{+0.2}_{-0.2}$ and $kT_{\\rm (inside)}$ /$kT_{\\rm (outside)}$ = 1.1$^{+0.7}_{-0.5}$ suggest the Mach numbers, $M$ = 1.3$^{+0.2}_{-0.2}$ and 1.1$^{+0.4}_{-0.3}$ , respectively under the Rankine-Hugoniot shock conditions for the intracluster gas, $\\gamma $ = 5/3, which are all obviously in agreement, are within the error bars of [3].","We also believe that a smaller measure for the temperature jump is due to the contribution of cool thermal emission from the core to the adjacent region toward two cD galaxies, $kT_{\\rm (inside)}$ .","Note, it is equally possible that the surface-brightness edge is a cold front.","Here, we assume it to be a shock front because of consistencies between our measurements and those of [3] and [20].","Figure: Plot of the radial surface-brightness profile and the temperature profile for the two red sectors (shown in Figure ) as a function of distance from the southern cD galaxy.","The error bars represent 90% confidence intervals."],["Discussion","In passing, we note that we do not detect the radio halo of A2065 in our current $\\approx 8^{\\prime \\prime }$ resolution 250–500 MHz band image.","This is likely to be due to the fairly short duration of our observing run (just $\\approx 1.3$ hr), which results in poor ($u,v$ )-coverage at the short spacings that are critical to detect $\\approx 10^\\prime $ -sized structures like the radio halo [7].","However, the large fractional bandwidth at 250–500 MHz band implies that it has excellent ($u,v$ )-coverage at the intermediate and long baselines that are needed to accurately map the arcminute-sized structures like our target, the remnant radio galaxy (angular size $\\approx 1^\\prime \\times 2^\\prime $ ).","In addition, we have performed deep clean, where we have used the multiscale multifrequency synthesis process of combining data from multiple spectral channels onto the same spatial-frequency grid during imaging to take advantage of the increased ($u,v$ )-coverage and imaging sensitivity, and hence we do not foresee any loss of flux density that could be linked to the remnant radio galaxy in the data presented above.","Below we discuss the nature of this newly discovered remnant radio galaxy in A2065.","As the remnant radio galaxy ages, its electrons will lose so much energy that their spectra will eventually steepen.","This steepening would reach to a point where they do not emit significant radiation at high frequencies, instead, the radio emission may only be observable at low frequencies.","However, [2] argues that in addition to the aged population of remnant radio galaxies, there should be younger population as well in which lobes have not had time to steepen over the observed frequency range [8], [25].","It is possible that the remnant phase of a radio galaxy is governed by a Sedov-like expansion, which will cause the dimming of radio emission due to a decrease in the magnetic field strength and a decrease in particle energies due to adiabatic expansion losses.","Unfortunately, the models of the spectra of remnant radio galaxy are highly degenerate, and the modeling of an individual remnant radio galaxy cannot constrain the remnant phase of radio galaxy [15].","Therefore, a possible way to constrain the remnant phase is via a statistical approach, and sadly, due to their small number, the physics of remnant radio galaxies remains poorly understood.","Of course, alongside their must exist a significant population of remnant radio galaxies, which show the absence of detectable radio emission in many clusters associated with merger activity [11].","Thus, searching for low-surface-brightness profiles, say $\\lesssim $ 50 mJy arcmin$^{-2}$ , with an absence of radio cores and hotspots, and with no well-defined radio morphologies, is a possible way to identify remnant radio sources in large survey images.","Radio morphology of the remnant radio galaxy in A2065 shows that it is an elongated, bar-shaped structure (52$^{\\prime \\prime }$ $\\times $ 110$^{\\prime \\prime }$ $\\simeq $ 72 $\\times $ 152 kpc$^2$ ), which is buried within a radio halo emission of size $\\lesssim $ 1 Mpc in size [6] and [7].","The remnant radio galaxy is located in the near vicinity of the southern cD galaxy; the northeastern boundary of the remnant radio galaxy is merely at a distance of $\\sim $ 33$^{\\prime \\prime }$ ($\\simeq $ 45.7 kpc) from the southern cD galaxy.","The possible radio core of the remnant radio galaxy is coincident with the WISEA J152228.01$+$ 274141.3 source that is 78 away from the optical galaxy V1CG 136.","Our spectral index measurements show that the radio spectrum steepens with distance from the possible radio core from $\\alpha $ = $-$ 1.2 $\\pm $ 0.1 to $\\alpha $ = $-$ 1.4 $\\pm $ 0.2 in our low-frequency (between the 240 MHz band and the 250–500 MHz band) spectra, and from $\\alpha $ = $-$ 1.3 $\\pm $ 0.1 to $\\alpha $ = $-$ 1.6 $\\pm $ 0.2 in our high-frequency (between the 250–500 MHz band and the 610 MHz band) spectra (Figure REF ), as is expected for synchrotron and inverse Compton losses.","We also find evidence for spectral steepening across the remnant radio galaxy in our in-band spectral index measurements, from $\\alpha $ = $-$ 1.3 $\\pm $ 0.2 to $\\alpha $ = $-$ 1.4 $\\pm $ 0.2 (Figure REF ), in the direction away from the two cD galaxies.","Our X-ray measurements show a surface-brightness edge that is coincident with the northeastern boundary of the remnant radio galaxy and its nature (shock or cold front) is not clear from the data.","If it is a shock front, it is rather weak, with $M$ being between 1.1 and 1.3; however it is unclear if an (inward?)","traveling shock is responsible for spectral steepening across the remnant radio galaxy.","Below we discuss the possible progenitor of the remnant radio galaxy in A2065."],["Remnant source diagnostic","Remnant radio sources are old, and are bound by model-dependent physical parameters because of the required assumptions of volume-filling factors, etc.","We use the parameter $N$ , which is the number of relativistic electrons found by integrating the energy spectrum of radiating electrons between Lorentz factors of 1000 and 10,000 for a typical equipartition magnetic field of 5 $\\mu $ G in the 2–200 MHz band for the remnant radio galaxy.","The number distribution of relativistic electrons drops rapidly with energy for a steep spectrum remnant radio galaxy.","Hence the number of low-energy electrons is a good measure of the total acceleration of relativistic electrons during the lifetime of $\\sim $ 10$^8$ yr for the small redshift of the source, and indicates the entire history of the source.","We thus determine $N$ $\\approx $ 2.1 $\\times $ 10$^{60}$ and the total energy stored in the magnetic field $\\approx $ 1.1 $\\times $ 10$^{57}$ erg assuming cylindrical geometry for the size of the remnant radio galaxy [21], [22].","Here, we have assumed the volume filling factor is unity, and hence the volume and the number of relativistic electrons $N$ are thus the upper limits.","The emitting plasma will most likely occupy a smaller volume, leading to larger energy densities and therefore larger values of the magnetic field strength and will require a smaller number of electrons to produce the observed luminosity.","These basic energetics are similar to the Coma radio relic source 1253$+$ 275 [10].","Moreover, the number of relativistic electrons, $N$ , is similar to that of the typical FR I radio galaxy 3C 31 but two orders of magnitude smaller than the FR II radio galaxy Cygnus A [13].","This suggests that the FR I radio galaxy is possibly the progenitor of the remnant radio galaxy in A2065.","Thus, our high-resolution, high-sensitivity radio image at the 250–500 MHz band along with the presence of the WISEA J152228.01$+$ 274141.3 source suggest that the source is a wide-angle-tailed radio source (see also Sec.","REF ).","Furthermore, the position of the WISEA J152228.01$+$ 274141.3 source exactly coincides with the likely radio core position of the wide-angle-tailed radio source.","The source diagnostic based on the number of low-energy relativistic electrons also suggests that the remnant radio galaxy is fed by the small radio tail of the wide-angle-tailed FR I radio galaxy.","If this proposition is correct, the two oppositely directed radio jets emanating from the apex of the WISEA J152228.01$+$ 274141.3 source initially traverse toward the northwest and southeast.","As the galaxy plows through the dense ICM, these jets traversing in opposite directions form a trail after a sharp bend in the southeastern jet due to interaction with the ICM.","The jets probably overlap and twist forming a pinch and then expand, and the jet becomes weaker [18].","The sharpness in surface-brightness toward the northeast, i.e., the side facing two cD galaxies and diffuse extensions toward the far southwest side, along with corresponding spectral gradients, suggests the passage of an (inward?)","traveling shock front."],["Summary and conclusions","The remnant radio source seems to trace AGN radio plasma that has somehow been reenergised through processes in the ICM.","It is possible that, similar to radio relic sources, remnant radio sources are also believed to be tracers of merger shocks, unfortunately, only a few clusters have shown to possess corresponding features in hot X-ray gas.","The remnant radio galaxy in A2065 may belong to this (rare) class possessing an X-ray surface-brightness edge across the remnant radio galaxy, indicating a possible connection with the shock front.","Using our deep high-resolution low-frequency observations, we discovered a bar-shaped remnant radio galaxy, whose size is $\\approx $ 52$^{\\prime \\prime }$ $\\times $ 110$^{\\prime \\prime }$ (= 72 $\\times $ 152 kpc$^2$ ), remnant radio galaxy in the massive unequal-mass-merger galaxy cluster A2065 at $z$ = 0.072.","It has a steep spectral index $\\alpha $ (240–610 MHz) = $-$ 1.4 $\\pm $ 0.2 and $\\alpha $ ($>$ 610 MHz) $<$ $-$ 1.4 $\\pm $ 0.2, corresponding to the line representing the best-fitting regression to the radio data.","The Chandra data, and the upgraded and “classic\" GMRT data, show (i) the presence of an X-ray surface-brightness edge attributed to a shock front, though a possibility of cold front cannot be ruled out, at the remnant radio galaxy's edge toward the two cD galaxies, (ii) the tentative flattening of the radio spectral index, in our 250–500 MHz in-band data, of the remnant radio galaxy at the near end of the X-ray surface-brightness edge, and (iii) possibly a wide-angle-tailed, FR I radio galaxy is the progenitor of the remnant radio galaxy.","Thus, the remnant radio galaxy has possibly been reinvigorated by the passage of the shock front and adiabatic compression, and shows the expected change in radio emission.","We also suggest that the remnant radio galaxy was seeded by the AGN during its past active phase, and has been hosted by the WISEA J152228.01$+$ 274141.3 source, demonstrating the connection between AGNs and remnant radio galaxy, similar to the connection between AGNs and radio relic sources [28].","Although low-frequency observations are starting to reveal more and more of these type of sources, presently there are a very few known remnant radio systems; clearly an outstanding difficulty is the small sample size.","Although the number of known remnant radio sources is increasing, better data are necessary for several of them to allow a detailed study.","Fortunately, forthcoming radio surveys using the LOFAR, MeerKAT, Square Kilometre Array, etc., will greatly increase the sample size, along with deep multifrequency radio data, e.g., using the upgraded GMRT, which will constrain the physics and nature of poorly understood remnant radio sources.","We are undertaking an upgraded GMRT study at the 125–250 MHz, 250–500 MHz, and 550–850 MHz bands of several clusters in order to detect such ultra-steep-spectrum sources, map radio halo emission, and provide exact flux density estimates, models, and statistics."],["Acknowledgments","I would like to thank the anonymous referee for helpful remarks.","D.V.L.","acknowledges the support of the Department of Atomic Energy, Government of India, under project No.","12-R&D-TFR-5.02-0700, and thanks the staff of the GMRT who made these observations possible.","The GMRT is run by the National Centre for Radio Astrophysics of the Tata Institute of Fundamental Research.","This research made use of the radio astronomical database GalaxyClusters.com, maintained by the Observatory of Hamburg.","This research has made use of the NED, which is operated by the Jet Propulsion Laboratory, Caltech, under contract with the NASA, and NASA's Astrophysics Data System.","Facilities: Chandra, GMRT Data Availability: The GMRT and Chandra data underlying this article are available via the GMRT Online Archive, naps.ncra.tifr.res.in/goa and the Chandra Data Archive, cxc.harvard.edu/cda.","All data analyses packages used in this work are publicly available."]],"string":"[\n [\n \"The Discovery of a Remnant Radio Galaxy in A2065 Using GMRT\"\n ],\n [\n \"Abstract The upgraded Giant Metrewave Radio Telescope (GMRT) has been used to map the cluster A2065 at z = 0.0726.\",\n \"We report the discovery of a remnant radio galaxy at the peripheral cluster region.\",\n \"The spatially resolved radio emission from the remnant radio galaxy shows an elongated, bar-shaped structure, whose size is $\\\\approx$ 52$^{\\\\prime\\\\prime}$ $\\\\times$ 110$^{\\\\prime\\\\prime}$ ($\\\\simeq$ 72 $\\\\times$ 152 kpc$^2$).\",\n \"Our study with the multiwavelength GMRT data and \\\\textit{Chandra} data shows that across the remnant radio galaxy there is a hint of a surface-brightness edge in the hot X-ray gas.\",\n \"We detect tentative flattening of the radio spectral index as the old plasma at the near end of the surface-brightness edge is reinvigorated by the passage of possible shock front and shows the expected change in radio emission characteristics.\",\n \"We suggest that the remnant radio galaxy has been seeded by the lobes of the active galactic nucleus (AGN), hosted by the WISEA J152228.01$+$274141.3 source, demonstrating the connection between AGNs and remnant radio sources.\",\n \"Although the number of known remnant radio sources is beginning to increase, we emphasize the need for better data to understand the physics and nature of poorly understood remnant radio sources.\"\n ],\n [\n \"Introduction\",\n \"The hierarchical structure formation scenario suggests that cluster mergers are the mechanisms by which galaxy clusters form and these mergers are the most energetic phenomena in the universe since the Big Bang.\",\n \"A large number of clusters of galaxies are known to contain large-scale, low-surface-brightness diffuse radio emission whose origin is not often related to the active galactic nuclei (AGNs) but to the intracluster medium (ICM).\",\n \"According to the location with respect to the cluster center, these sources are of two key types, radio halos and relics.\",\n \"Radio halos show a regular, amorphous emission around the cluster center of a size from a few hundred kiloparsecs (minihalo) to megaparsecs (halo), whereas radio relics show an irregular emission located in peripheral cluster regions of a size from several hundreds of kiloparsecs to megaparsecs.\",\n \"Another third broad type of sources comprises of revived fossil plasma sources and phoenices.\",\n \"These do not have any preferred location in the ICM are of size from several tens of kpc to a few hundred kpc and trace AGN radio plasma that has somehow been reenergised through the processes in the ICM [27], [16], [9].\",\n \"The radio jets emanating from an AGN arise from the accretion onto a supermassive black hole.\",\n \"These form synchrotron-emitting radio lobes in a radio-loud AGN in the environments of their host galaxies.\",\n \"The active phase of an AGN can last several tens of megayears, following this the radio jets stop and their features, namely an unresolved radio core, radio jets, hotspots, and radio lobes, all start to fade away due to synchrotron cooling.\",\n \"This phase once the jets have switched off is called as the remnant phase of a radio galaxy, or merely the remnant radio galaxy.\",\n \"The remnant radio galaxy falls into the third broad type of sources in the ICM, which is poorly understood and presently very few sources are known [25], [2], [23].\",\n \"The high spatial resolution of the Chandra and upgraded Giant Metrewave Radio Telescope (GMRT) allow us to study the interplay between X-ray-emitting hot gas and nonthermal radio emission.\",\n \"We have obtained new upgraded GMRT data, and have used archive GMRT and Chandra data of the nearby $z$ = 0.0726 [26] richness class 2 cluster, A2065, which is one of the 10 galaxy clusters that make up the Corona Borealis supercluster.\",\n \"[24] noted that two cD galaxies dominate the cluster center in the optical and their radial velocities differ by $\\\\sim $ 600 km s$^{-1}$ .\",\n \"Table: Summary of the radio observations and image properties.Previous radio study using 100 m Green Bank Telescope at 1.4 GHz [7] detected a smooth diffuse structure $\\\\simeq $ 1 Mpc in extent, a giant radio halo, whereas a recent study showed a slightly smaller extent for the radio halo using Low-Frequency Array (LOFAR) at 153 MHz [6].\",\n \"Similarly, previous X-ray studies led to detections of two surface-brightness peaks coincident with the two cD galaxies observed in the cluster's center.\",\n \"The data indicated that the cluster is an unequal-mass merger and only one of the two cooling cores has survived the merger.\",\n \"The northern cluster seems to have fallen into the more massive southern cluster from the southeast, and having lost its gas content, it has now moved to its present location to the northwest.\",\n \"Evidence of shock with $M$ $\\\\approx $ 1.7 at $\\\\sim $ 140 kpc from the southern cD galaxy too is noted and the cooling flow is displaced to the southeast of the southern cD galaxy [3], [20].\",\n \"In this study, we report the discovery of a remnant radio galaxy in cluster A2065 using the GMRT.\",\n \"We also present an X-ray analysis of archive Chandra data, which enhances the interpretation of the current discovery.\",\n \"We define spectral index, $\\\\alpha $ as, $S_\\\\nu $ $\\\\propto $ $\\\\nu ^{\\\\alpha }$ , where S$_\\\\nu $ is the flux density at frequency, $\\\\nu $ .\",\n \"To estimate the intrinsic parameters, we use a $\\\\Lambda $ CDM cosmology with $\\\\Omega _{\\\\rm m}$ = 0.27, $\\\\Omega _{\\\\Lambda }$ = 0.73, and H$_0$ = 70 km s$^{-1}$ Mpc$^{-1}$ .\",\n \"At the redshift of the cluster, 1 arcsec corresponds to 1.384 kpc.\",\n \"All uncertainties are at 1$\\\\sigma $ error bars unless otherwise stated.\",\n \"The position angles (P.A.s) are measured from north to east.\",\n \"Throughout, positions are given in J2000 coordinates.\",\n \"The organization of the paper is as follows.\",\n \"We present the X-ray and radio observations and their data analyses in Sec. .\",\n \"Our results, i.e., radio morphology, spectral structure, and X-ray gas properties are presented in Sec. .\",\n \"In Sec.\",\n \", we discuss our observational results, source diagnostics, and dynamical interpretation of the remnant radio galaxy.\",\n \"Sec.\",\n \"summarizes our conclusions.\",\n \"A2065 was observed by Chandra with the acis-i detector on 2002 August 18 and November 24 for 28 and 22 ks (Obs_ID 3182) respectively, and on 2007 September 13 for 5 ks (Obs_ID 7689).\",\n \"The data reduction of these three archival data sets has been performed independently.\",\n \"We used the calibration databases, caldb version 4.7.6 and ciao version 4.9.\",\n \"After the initial calibration, i.e., standard filtering and removal of bad pixels, the event files of the three observations were projected to a common reference point and then merged to create Gaussian-smoothed, background-subtracted, exposure-corrected image in the broadband 0.5–7.0 keV energy range.\",\n \"We excised the central AGN and background point sources.\",\n \"Using the ciao tool specextract, we extracted the source, background spectra, and the response files of selected regions in the Chandra images separately for the three data sets.\",\n \"The spectra for each data set were grouped so that each bin contained at least 20 counts.\",\n \"We restricted our spectral extraction in the energy range 0.5–7.5 keV, and fitted for the thermal gas emission with xspec [1] using the isothermal apec model.\",\n \"The abundance was fixed to 0.3 $\\\\times $ solar [3] and all spectral fits include absorption by the Galactic column [5].\",\n \"Figure: Exposure-corrected, background-subtracted, broadband (0.5–7.0 keV) Chandra image of A2065.\",\n \"The image was binned (bin = 16) and smoothed using a Gaussian kernel of 10 '' ^{\\\\prime \\\\prime }.\",\n \"The color bar shows the surface-brightness, in 10 -6 ^{-6} counts s -1 ^{-1} per 8 ×\\\\times 8 pixel binning.\",\n \"The X-ray peak is located at R.A. 15:22:29.18, decl.\",\n \"27:42:21.05.\",\n \"The peak surface-brightness and the faintest regions transitioning between blue and green have a surface-brightness of 6.0 ×\\\\times 10 -5 ^{-5} counts s -1 ^{-1} arcsec -2 ^{-2} and 1.6 ×\\\\times 10 -6 ^{-6} counts s -1 ^{-1} arcsec -2 ^{-2} respectively.The surface-brightness is displayed in logarithmic scales to emphasize low-surface-brightness emission.\"\n ],\n [\n \"GMRT data\",\n \"A summary of the radio observations is presented in Table REF .\",\n \"We inspected the data for bad antennas, bad time stamps and data impacted by radio frequency interference.\",\n \"The data were analyzed using the standard reduction methodologies [17].\",\n \"After the initial steps of flux density and bandpass calibration, all corrupted data were removed using standard routines.\",\n \"The bandpass calibrated data for wide-bandwith GMRT data, i.e., the 250–500 MHz band and 1050–1450 MHz band data were split into five 40 MHz (from 300–500 MHz) and eight 50 MHz (from 1050–1450 MHz) subbands respectively.\",\n \"Subsequently, each subband data was analyzed separately, similar to narrowband “classic\\\" GMRT data.\",\n \"Several cycles of phase-only self-calibration and imaging were performed and the calibration solutions were applied to the target data to correct for residual phase errors.\",\n \"To take care of the wide-field imaging at low GMRT frequencies, we performed faceting imaging.\",\n \"Problematic bright sources in the field of view whose side lobes affected the area of the target were subtracted, a.k.a.\",\n \"peeled; this (direction-dependent) calibration step improved the dynamic range of images.\",\n \"The calibrated data for five 40 MHz 250–500 MHz band data and eight 50 MHz 1050–1450 MHz band data were joined to form full 200 MHz and 400 MHz, respectively, wide-bandwidth-calibrated visibility data sets.\",\n \"The combined wide-bandwidth-calibrated data (and similarly the narrowband-calibrated “classic\\\" GMRT data) were imaged using the casa task tclean.\",\n \"We used 3D imaging (gridder = `widefield'), two Taylor coefficients (nterms = 2, tt0 and tt1) and Briggs weighting (robust = 0.5) in task tclean.\",\n \"A final amplitude-and-phase self-calibration with solution interval equal to the length of observation was then carried out using the gaincal and applycal tasks in casa.\",\n \"The two Stokes polarizations, RR and LL, were combined to obtain the final total intensity Stokes I image, and corrected for the GMRT primary beam response.\",\n \"This provides images at center frequencies (tt0) and the corresponding in-band spectral index maps (tt1/tt0).\",\n \"The amplitude errors are estimated to be within 5% or less for archival narrow-bandwidth and wide-bandwidth GMRT data.\",\n \"Table REF also provides image properties, including the restoring beams and root mean square (rms) noise levels at half-power points of the final images.\",\n \"The uncertainty in the flux density measurements are estimated as $\\\\left[(S_\\\\nu \\\\times f)^2 + (\\\\textsc {rms})^2 \\\\times N_{\\\\rm beams}\\\\right]^{0.5},$ where $S_\\\\nu $ is the flux density, $f$ is an absolute flux density calibration uncertainty, rms is the noise, and $N_{\\\\rm beams}$ is the number of synthesized beams.\",\n \"We use these error estimates for the flux density measurements when computing the error estimates for spectral index measurements.\",\n \"Figure: The central 3 ' ^{\\\\prime } ×\\\\times 3 ' ^{\\\\prime } cluster region with 250–500 MHz band radio contours overlaid on the Digitized Sky Survey image of A2065.\",\n \"Note the northern cD galaxy registers well with the northern radio peak, but the southern cD galaxy is displaced by ∼\\\\sim 24 north of the radio peak.\",\n \"The X-ray peak at the cluster center is coincident with the southern radio peak (see also Figure ).The radio emission from the two cD galaxies (the brighter southern galaxy 2MASS J152229.17++274227.5 and the fainter northern galaxy 2MASX J152228.92++274244.1) of the cluster is shown on the east of the remnant radio galaxy emission.The radio contour levels are --3, 3, 6, 9, 15, 42 and 84 ×\\\\times 80 μ\\\\mu Jy beam -1 ^{-1}.The cluster-member galaxies are marked and a majority (≈\\\\approx 60%) show radio emission.\",\n \"The two bright cluster galaxies, SDSS ++230.6++27.7++0.08 (R.A.: 15:22:24.0, decl.\",\n \"++27:42:51; zz = 0.0723) and 2MASS J152229.17++274222.75 (zz = 0.074875) are marked in blue colour .\"\n ],\n [\n \"Results\",\n \"The exposure-corrected background-subtracted X-ray binned (bin = 16) and Gaussian-smoothed (10$^{\\\\prime \\\\prime }$ kernel) image in the 0.5–7.0 keV band is shown in Figure REF .\",\n \"Figure REF shows Chandra X-ray contours from Figure REF and 250–500 MHz band radio contours overlaid on the exposure-corrected, background-subtracted broadband (0.5–7.0 keV) Chandra (bin = 8) image (left panel) and Digitized Sky Survey image (right panel) of A2065.\",\n \"The central 3$^{\\\\prime }$ $\\\\times $ 3$^{\\\\prime }$ cluster region with 250–500 MHz band radio contours overlaid on the Digitized Sky Survey image of A2065 is shown in Figure REF .\",\n \"Figure REF also marks the cluster member galaxies, including the two bright cluster galaxies in marked in blue colour, SDSS $+$ 230.6$+$ 27.7$+$ 0.08 (NED; R.A.: 15:22:24.0, decl.\",\n \"$+$ 27:42:51; $z$ = 0.0723, $m$ = 16.1) and 2MASS J152229.17$+$ 274222.75 [24], and a majority of them, $\\\\approx $ 60%, show radio emission.\",\n \"Figure REF shows clear detections for the southern cD galaxy and remnant radio galaxy at the GMRT bands, namely the 240 MHz band (top-left panel), 250–500 MHz band (top-right panel), 610 MHz band (bottom-left panel), and 1050–1450 MHz band (bottom-right panel).\",\n \"The northern cD galaxy is only detected in our 250–500 MHz band data.\",\n \"The images have rms noise values in the range 0.56–0.06 mJy beam$^{-1}$ at the half-power points.\",\n \"The 150 MHz band and 1050–1450 MHz band data sets had very short on-source integration times, $\\\\lesssim $ 0.9 hr, and therefore these images have relatively high rms noise values (see Table REF ).\",\n \"Note that we also made images at several angular resolutions, including imaging data sets using the same range of baseline lengths, and do not find the presence or absence of additional morphological features.\",\n \"The integrated flux densities computed thus are also consistent, within error bars with those reported in Table REF .\"\n ],\n [\n \"Southern and northern cD galaxies\",\n \"Figure REF shows a diffuse X-ray tail extending to the north at the center and that the cluster emission is elongated along northwest-southeast direction.\",\n \"The hot gas emission (i) shows a compact, bright region and a diffuse tail extending to the north at the center, and (ii) is extended along the northwest–southeast direction showing pieces of evidence for surface-brightness edges.\",\n \"Though the X-ray peak emission and peak of the radio emission register well (see Figure REF , left), the position of the optical host of the southern cD galaxy (R.A.: 15:22:29.17, decl.\",\n \": $+$ 27:42:27.5) is displaced by $\\\\sim $ 24 from the compact, bright, X-ray peak emission at the cluster center (see Figure REF , right, and also Figure REF ).\",\n \"A low-surface-brightness diffuse tail, both in X-ray as well as in radio images, extending to the north at the center of the image, reaches out to the second cD galaxy (R.A.: 15:22:28.92, decl.\",\n \": $+$ 27:42:44.1; Figures REF and REF , top-right panels).\",\n \"The X-ray peak corresponding to the second northern cD galaxy and the peak of the second weaker component of radio emission at this location also register well.\",\n \"The southern cD galaxy seems to have retained a cool core residing at the center of its subcluster and there is no evidence for a cool core corresponding to the northern cD galaxy [3], [20].\",\n \"The closely associated radio and X-ray morphology at the center and its connection with the southern cD galaxy suggest that both X-ray and radio emissions are due to ram pressure stripping as the southern cD galaxy moves relative to the ICM.\",\n \"The two blue plus signs in the 250–500 MHz band image marked in Figure REF (top-right panel) correspond to the hosts of the brighter southern cD galaxy, 2MASS J152229.17$+$ 274222.75 at $z$ = 0.074875 and fainter northern cD galaxy, 2MASX J152228.92$+$ 274244.1 at $z$ = 0.072854 (see also Figure REF ).\",\n \"The northern cD galaxy is only detected in our 250–500 MHz band image.\",\n \"However, we not only detected the southern cD galaxy in our long observations, we also detected the peak radio emissions at the location of the southern cD galaxy at R.A.: 15:22:29.17, decl.\",\n \": $+$ 27:42:27.5 in our short, 150 MHz band, and 1050–1450 MHz band data sets.\",\n \"The two cD galaxies are barely resolved using LOFAR at 153 MHz [6].\",\n \"The southern cD galaxy also had a 2.8$\\\\sigma $ (upper limit) detection at 74 MHz [4].\",\n \"It seems that around the southern cD radio galaxy there is some extended radio emission, which is more clearly seen in our 250–500 MHz band image (Figure REF , top-right panel).\",\n \"Since several cluster-member galaxies also show radio emission, it is unclear if this extended radio emission is indeed associated with the southern cD radio galaxy or with the two adjacent cluster-member galaxies southeast of it (see Figure REF ).\",\n \"Table: Radio flux densities for the remnant radio galaxy and the southern cD galaxy\"\n ],\n [\n \"Remnant radio galaxy\",\n \"Figures REF and REF also suggest that the remnant radio galaxy is hosted by the 19.19 magnitude WISEA J152228.01$+$ 274141.3 source that is 78 away from the optical galaxy V1CG 136 (R.A.: 15:22:28.5, decl.\",\n \": $+$ 27:41:46) at redshift $z$ = 0.072 [12].\",\n \"The cluster contains gas not only with an extremely wide range of temperatures and complex structure but that also shows a cold front at 30$^{\\\\prime \\\\prime }$ ($\\\\simeq $ 41.5 kpc) and a shock-like bow-shaped feature beyond $\\\\sim $ 100$^{\\\\prime \\\\prime }$ ($\\\\simeq $ 140 kpc) on the southeast of the southern cD [3].\",\n \"Figure REF shows that the northeastern boundary of the remnant radio galaxy coincides with the 1.3 $\\\\times $ 10$^{-7}$ counts s$^{-1}$ arcsec$^{-2}$ surface-brightness contour level; gas in false color transitioning from the green region to the blue region at a distance of $\\\\sim $ 33$^{\\\\prime \\\\prime }$ ($\\\\simeq $ 45.7 kpc) from the southern cD galaxy.\",\n \"All four panels of Figure REF suggest that the radio emission from the remnant radio galaxy has an elongated, bar-shaped structure.\",\n \"The remnant radio galaxy, buried in the radio halo emission that is barely detected using LOFAR at 153 MHz seems to possess an elongated, bar-shaped structure [6], consistent with our results.\",\n \"This suggests that the remnant radio galaxy has a steep spectrum, and its consistency in radio morphology gives us additional confidence in our image fidelity, imaging, calibration and data reduction processes.\",\n \"The best sensitivity image for the low surface-brightness was obtained from the 250–500 MHz band data and we now discuss below the morphology of the remnant radio galaxy.\",\n \"The remnant radio galaxy broadly consists of a high-surface-brightness southeastern part and a fainter low-surface-brightness extended northwestern part.\",\n \"The two oppositely directed radio jets emanating from the apex of the WISEA J152228.01$+$ 274141.3 source initially traverse toward the southeast and northwest directions.\",\n \"As the galaxy plows through the dense intracluster gas, these jets traversing in opposite directions form a trail, after sharp bends in the jets, behind the WISEA J152228.01$+$ 274141.3 source due to interaction with the ICM.\",\n \"The width of the remnant radio galaxy increases from $\\\\sim $ 17$^{\\\\prime \\\\prime }$ ($\\\\simeq $ 23.5 kpc) at the southeastern end to $\\\\sim $ 27$^{\\\\prime \\\\prime }$ ($\\\\simeq $ 37.4 kpc) and reduces again before flaring to reach a maximum width of $\\\\sim $ 35$^{\\\\prime \\\\prime }$ ($\\\\simeq $ 48.4 kpc) and finally fading completely.\",\n \"The average width of the narrower southeastern part is a factor of $\\\\sim $ 2.3 smaller than the broader northwestern part.\",\n \"It seems that a pinch is present at $\\\\sim $ 35$^{\\\\prime \\\\prime }$ ($\\\\simeq $ 48.4 kpc) of the $\\\\sim $ 162 ($\\\\simeq $ 22.4 kpc) width from the southeastern end, i.e., between the two narrower and the broader parts.\",\n \"The northeastern boundary toward the two cD galaxies of the remnant radio galaxy is sharp, while the emission fades more slowly at the southwestern boundary.\",\n \"The overall source size, assuming elongation, of the bar-shaped structure is $\\\\approx $ 52$^{\\\\prime \\\\prime }$ $\\\\times $ 110$^{\\\\prime \\\\prime }$ , corresponding to 72 $\\\\times $ 152 kpc$^2$ at the assumed distance.\",\n \"Figure: Integrated radio spectrum of the remnant radio galaxy using the data in Table .Peak flux density is reported from our low-resolution, 15 '' ^{\\\\prime \\\\prime } image at the 1050–1450 MHz band for the remnant radio galaxy (see also Sec.\",\n \"for a discussion).We also show the spectra of the southern cD galaxy for comparison; note that the (nearly) straight spectra of it provides a good check of our calibration.\",\n \"A 2.8σ\\\\sigma limit is reported at the 74 MHz , peak flux densities at the 150 MHz band, and 1050–1450ṀHz band are our measurements; the 1400 MHz data is from the VLA FIRST survey for the southern cD galaxy.\"\n ],\n [\n \"Radio spectra\",\n \"The total flux densities were determined from integration in polygons with measured in areas encompassing the remnant radio galaxy, i.e., we determined the surface-brightness and divided it by the number of pixels in the polygon at each frequency.\",\n \"These results are presented in Table REF along with the total intensity spectra of the remnant radio galaxy, and the southern cD galaxy is shown in Figure REF .\",\n \"A 2.8$\\\\sigma $ upper limit at the 74 MHz [4] and the peak flux density from our 150 MHz band data for the southern cD galaxy are also reported.\",\n \"We see that all these radio measurements, within the error bars, fall nearly along with a `straight' power law, $\\\\alpha $ = $-$ 1.2 $\\\\pm $ 0.2 with some hints of energy losses by synchrotron cooling for the dominant southern cD galaxy.\",\n \"This provides confidence in our data calibration and its reduction methodologies.\",\n \"The remnant radio galaxy is not detected in our 150 MHz band short observation, and is barely detected using LOFAR at 153 MHz (see also Sec.\",\n \"REF ).\",\n \"However in the 1050–1450 MHz band short observation, we see hints of radio emission from the remnant radio galaxy.\",\n \"We made a low-resolution, 15$^{\\\\prime \\\\prime }$ image at the 1050–1450 MHz band and report peak flux density for the radio emission from the remnant radio galaxy in Figure REF .\",\n \"Since this 1050–1450 MHz band measurement comes from a short $\\\\lesssim $ 0.9 hr duration observation, has relatively high rms noise (see also Sec.\",\n \"), and the observation does not sample similar ($u,v$ )-spacings as the rest of the low-frequency observations, we henceforth no longer include it in our spectral index analysis.\",\n \"It appears that the integrated spectrum for the remnant radio galaxy cannot be represented by a single power law.\",\n \"The segment of the spectrum with the best data is $<$ 1250 MHz, and the low-frequency spectral index $\\\\alpha $ (240–610 MHz) = $-$ 1.4 $\\\\pm $ 0.2.\",\n \"Our data cannot rule out the possibility of spectral shape to be due to two populations of the relativistic electrons: a low-frequency component characterized by $\\\\alpha $ (240–610 MHz) $\\\\approx $ $-$ 1.4 and a second component having $\\\\alpha $ ($>$ 610 MHz) $<$ $-$ 1.4 with a cutoff or a spectral break between 400 and 1250 MHz.\",\n \"Figure: The plot shows the radio spectra between the 240 MHz and 250–500 MHz band data sets (red upper triangles) and between the 250–500 MHz band and 610 MHz band data sets (blue lower triangles) as a function of distance from the head.\",\n \"The location of the pinch is shown as a vertical line.Figure: The averaged in-band spectral index image showing spectra between 300 MHz and 500 MHz using 250–500 MHz band data.\",\n \"The total intensity surface-brightness contours are at 0.23 and 0.46 mJy beam -1 ^{-1} from the 300–500 MHz radio image (Figure , top-right panel).\",\n \"We used three irregular polygon regions, marked in the image, and determined the mean (and rms) spectral indices that are denoted at the upper-right corner of the image.\",\n \"The averaged integrated in-band spectral index for the remnant radio galaxy is denoted at the lower-left corner of the image.\",\n \"The cross marks correspond to the positions of two cD galaxies and the position of the optical host galaxy, discussed in Sec.\",\n \"and .\",\n \"The dashed line indicates the best-fitting position of the X-ray surface-brightness edge discussed in Sec.\",\n \".Figure: Background-subtracted and exposure-corrected Chandra image at ∼\\\\sim 4 '' ^{\\\\prime \\\\prime } angular resolution in the broad (0.5–7.0 keV) energy band.\",\n \"The color bar shows the surface-brightness, in photon cm -2 ^{-2} s -1 ^{-1} pixels -1 ^{-1}.\",\n \"The black surface-brightness contours are displayed using the 250–500 MHz band image.\",\n \"The inset shows the enlargement of the inner region.\",\n \"Also shown are two sets of sectors (in cyan and blue colors) that were used for the surface-brightness and temperature measurements.The spectral index measurements between the 240 MHz band and 250–500 MHz band data sets, and between the 250–500 MHz band and 610 band MHz band data sets show that the radio spectrum steepens with distance from the head, from $-$ 1.2 $\\\\pm $ 0.1 to $-$ 1.4 $\\\\pm $ 0.2, and from $-$ 1.3 $\\\\pm $ 0.1 to $-$ 1.6 $\\\\pm $ 0.2, respectively, due to synchrotron and inverse Compton losses (Figure REF ).\",\n \"The location after the pinch, where the radio plasma from the AGN connects to the remnant radio galaxy, the spectrum, between the 240 MHz band and 250–500 MHz band data, remains flattened, $\\\\alpha $ = $-$ 1.3 $\\\\pm $ 0.1.\",\n \"The wide-bandwith data sets were imaged to make multiscale multifrequency synthesis images, which provides images at center frequencies and the corresponding in-band spectral index maps (see also Sec.\",\n \"REF ).\",\n \"The in-band spectral index map averaged over a region of interest, to increase signal-to-noise ratio, for the 250–500 MHz band data is shown in Figure REF .\",\n \"We divided the remnant radio galaxy into three distinct regions: (1) the high-surface-brightness narrower southeastern part ahead of the pinch (in cyan).\",\n \"The latter low-surface-brightness broader northwestern part is divided into two; (2) the northeastern region (in yellow), and (3) the southwestern far end region (in magenta).\",\n \"The size of the regions is chosen such that the signal in the total intensity image is (approximately) more than five times the rms noise.\",\n \"The southern cD galaxy has an in-band spectral index of $-$ 0.9 $\\\\pm $ 0.2.\",\n \"Similarly, the spectra of three regions of the remnant radio galaxy are $-$ 1.2 $\\\\pm $ 0.2, $-$ 1.3 $\\\\pm $ 0.2, and $-$ 1.4 $\\\\pm $ 0.2 for regions (1), (2), and (3) respectively.\",\n \"This is consistent with the averaged in-band spectral index of $-$ 1.3 $\\\\pm $ 0.2 within the error bars for the remnant radio galaxy.\",\n \"It is also consistent with the crude estimates of two 50 MHz subband images, 300–350 MHz and 450–500 MHz, at two extreme ends of the 250–500 MHz band data.\",\n \"The region ahead of the pinch (in cyan) and the northeastern region close to two cD galaxies (in yellow) have comparable spectral indices.\",\n \"It appears that the high-surface-brightness narrower region (in cyan) ahead of the pinch, is flatter than the two regions, together forming low-surface-brightness broader northwestern region (in yellow and magenta) after the pinch, consistent with our result discussed above.\",\n \"Furthermore, the area between the two regions of the broader northwestern part of the source, i.e., the southwestern far-end region (in magenta) have a significantly steeper spectrum than the northeastern region close to two cD galaxies (in yellow).\"\n ],\n [\n \"ICM gas properties\",\n \"[3] reported that the cluster contains gas with an extremely wide range of temperatures and a wealth of structure.\",\n \"They also reported a discontinuity at $\\\\sim $ 30$^{\\\\prime \\\\prime }$ ($\\\\simeq $ 41.5 kpc) in the southeast region from the southern cD galaxy.\",\n \"It is at nearly this same distance where we report an enhancement of radio surface-brightness, toward the two cD galaxies, showing a sharp feature.\",\n \"We aim to search an X-ray surface-brightness edge, i.e., the shock front that is coincident with the northeastern sharp edge toward two cD galaxies of the remnant radio galaxy.\",\n \"In order to understand the gas physics at this region of the cluster, we have extracted the surface-brightness profile from two box-shaped regions, one encompassing the remnant radio galaxy (called “outside\\\" here) and the other adjacent to it toward cD galaxies (called “inside\\\" here) as shown in Figure REF (in magenta).\",\n \"We extracted the surface-brightnesses and the spectra from these regions, which are displayed in Figure REF .\",\n \"A single temperature model with Galactic absorption and the abundance fixed provided a good fit to the spectrum.\",\n \"We noticed the X-ray surface-brightness edge, where the surface-brightness drops (see Figures REF and REF ) and therefore implies a discontinuity, or jump, in the gas density.\",\n \"The surface-brightnesses are $S_{\\\\rm x}$ (inside) and $S_{\\\\rm x}$ (outside) $\\\\approx $ 8.2 $\\\\pm $ 0.3 $\\\\times $ 10$^{-6}$ counts s$^{-1}$ arcsec$^{-2}$ and 3.7 $\\\\pm $ 0.3 $\\\\times $ 10$^{-6}$ counts s$^{-1}$ arcsec$^{-2}$ , respectively.\",\n \"Consequently, our modeled densities in the two regions, $\\\\rho $ (inside) and $\\\\rho $ (outside) are approximately 2.9$^{+0.2}_{-0.1}$ $\\\\times $ 10$^{-3}$ cm$^{-3}$ and 1.9$^{+0.2}_{-0.2}$ $\\\\times $ 10$^{-3}$ cm$^{-3}$ , respectively.\",\n \"Our corresponding derived temperatures for the two regions, $kT_{\\\\rm (inside)}$ and $kT_{\\\\rm (outside)}$ are approximately 6.1$^{+0.9}_{-0.8}$ keV ($\\\\chi ^2$ = 0.93 for 221 degrees of freedom) and 5.5$^{+0.5}_{-0.4}$ keV ($\\\\chi ^2$ = 1.08 for 462 degrees of freedom), respectively.\",\n \"We also repeated this exercise for a pie-circular aperture (shown in blue in Figure REF ) centered on the southern cD galaxy and lying toward the radio emission from the remnant radio galaxy with two regions containing the radio emission from the remnant radio galaxy and the smaller adjacent region toward the cD galaxy.\",\n \"The quantitative estimates from this are nearly identical and the fit statistics are at 90% confidence intervals.\",\n \"Although with large error bars, these density and temperature measurements are hinting that the surface-brightness edge toward the two cD galaxies is possibly a shock front.\",\n \"Consequently, the observed density jump and the temperature jump across the edge, $\\\\rho _{\\\\rm (inside)}$ /$\\\\rho _{\\\\rm (outside)}$ = 1.5$^{+0.2}_{-0.2}$ and $kT_{\\\\rm (inside)}$ /$kT_{\\\\rm (outside)}$ = 1.1$^{+0.7}_{-0.5}$ suggest the Mach numbers, $M$ = 1.3$^{+0.2}_{-0.2}$ and 1.1$^{+0.4}_{-0.3}$ , respectively under the Rankine-Hugoniot shock conditions for the intracluster gas, $\\\\gamma $ = 5/3, which are all obviously in agreement, are within the error bars of [3].\",\n \"We also believe that a smaller measure for the temperature jump is due to the contribution of cool thermal emission from the core to the adjacent region toward two cD galaxies, $kT_{\\\\rm (inside)}$ .\",\n \"Note, it is equally possible that the surface-brightness edge is a cold front.\",\n \"Here, we assume it to be a shock front because of consistencies between our measurements and those of [3] and [20].\",\n \"Figure: Plot of the radial surface-brightness profile and the temperature profile for the two red sectors (shown in Figure ) as a function of distance from the southern cD galaxy.\",\n \"The error bars represent 90% confidence intervals.\"\n ],\n [\n \"Discussion\",\n \"In passing, we note that we do not detect the radio halo of A2065 in our current $\\\\approx 8^{\\\\prime \\\\prime }$ resolution 250–500 MHz band image.\",\n \"This is likely to be due to the fairly short duration of our observing run (just $\\\\approx 1.3$ hr), which results in poor ($u,v$ )-coverage at the short spacings that are critical to detect $\\\\approx 10^\\\\prime $ -sized structures like the radio halo [7].\",\n \"However, the large fractional bandwidth at 250–500 MHz band implies that it has excellent ($u,v$ )-coverage at the intermediate and long baselines that are needed to accurately map the arcminute-sized structures like our target, the remnant radio galaxy (angular size $\\\\approx 1^\\\\prime \\\\times 2^\\\\prime $ ).\",\n \"In addition, we have performed deep clean, where we have used the multiscale multifrequency synthesis process of combining data from multiple spectral channels onto the same spatial-frequency grid during imaging to take advantage of the increased ($u,v$ )-coverage and imaging sensitivity, and hence we do not foresee any loss of flux density that could be linked to the remnant radio galaxy in the data presented above.\",\n \"Below we discuss the nature of this newly discovered remnant radio galaxy in A2065.\",\n \"As the remnant radio galaxy ages, its electrons will lose so much energy that their spectra will eventually steepen.\",\n \"This steepening would reach to a point where they do not emit significant radiation at high frequencies, instead, the radio emission may only be observable at low frequencies.\",\n \"However, [2] argues that in addition to the aged population of remnant radio galaxies, there should be younger population as well in which lobes have not had time to steepen over the observed frequency range [8], [25].\",\n \"It is possible that the remnant phase of a radio galaxy is governed by a Sedov-like expansion, which will cause the dimming of radio emission due to a decrease in the magnetic field strength and a decrease in particle energies due to adiabatic expansion losses.\",\n \"Unfortunately, the models of the spectra of remnant radio galaxy are highly degenerate, and the modeling of an individual remnant radio galaxy cannot constrain the remnant phase of radio galaxy [15].\",\n \"Therefore, a possible way to constrain the remnant phase is via a statistical approach, and sadly, due to their small number, the physics of remnant radio galaxies remains poorly understood.\",\n \"Of course, alongside their must exist a significant population of remnant radio galaxies, which show the absence of detectable radio emission in many clusters associated with merger activity [11].\",\n \"Thus, searching for low-surface-brightness profiles, say $\\\\lesssim $ 50 mJy arcmin$^{-2}$ , with an absence of radio cores and hotspots, and with no well-defined radio morphologies, is a possible way to identify remnant radio sources in large survey images.\",\n \"Radio morphology of the remnant radio galaxy in A2065 shows that it is an elongated, bar-shaped structure (52$^{\\\\prime \\\\prime }$ $\\\\times $ 110$^{\\\\prime \\\\prime }$ $\\\\simeq $ 72 $\\\\times $ 152 kpc$^2$ ), which is buried within a radio halo emission of size $\\\\lesssim $ 1 Mpc in size [6] and [7].\",\n \"The remnant radio galaxy is located in the near vicinity of the southern cD galaxy; the northeastern boundary of the remnant radio galaxy is merely at a distance of $\\\\sim $ 33$^{\\\\prime \\\\prime }$ ($\\\\simeq $ 45.7 kpc) from the southern cD galaxy.\",\n \"The possible radio core of the remnant radio galaxy is coincident with the WISEA J152228.01$+$ 274141.3 source that is 78 away from the optical galaxy V1CG 136.\",\n \"Our spectral index measurements show that the radio spectrum steepens with distance from the possible radio core from $\\\\alpha $ = $-$ 1.2 $\\\\pm $ 0.1 to $\\\\alpha $ = $-$ 1.4 $\\\\pm $ 0.2 in our low-frequency (between the 240 MHz band and the 250–500 MHz band) spectra, and from $\\\\alpha $ = $-$ 1.3 $\\\\pm $ 0.1 to $\\\\alpha $ = $-$ 1.6 $\\\\pm $ 0.2 in our high-frequency (between the 250–500 MHz band and the 610 MHz band) spectra (Figure REF ), as is expected for synchrotron and inverse Compton losses.\",\n \"We also find evidence for spectral steepening across the remnant radio galaxy in our in-band spectral index measurements, from $\\\\alpha $ = $-$ 1.3 $\\\\pm $ 0.2 to $\\\\alpha $ = $-$ 1.4 $\\\\pm $ 0.2 (Figure REF ), in the direction away from the two cD galaxies.\",\n \"Our X-ray measurements show a surface-brightness edge that is coincident with the northeastern boundary of the remnant radio galaxy and its nature (shock or cold front) is not clear from the data.\",\n \"If it is a shock front, it is rather weak, with $M$ being between 1.1 and 1.3; however it is unclear if an (inward?)\",\n \"traveling shock is responsible for spectral steepening across the remnant radio galaxy.\",\n \"Below we discuss the possible progenitor of the remnant radio galaxy in A2065.\"\n ],\n [\n \"Remnant source diagnostic\",\n \"Remnant radio sources are old, and are bound by model-dependent physical parameters because of the required assumptions of volume-filling factors, etc.\",\n \"We use the parameter $N$ , which is the number of relativistic electrons found by integrating the energy spectrum of radiating electrons between Lorentz factors of 1000 and 10,000 for a typical equipartition magnetic field of 5 $\\\\mu $ G in the 2–200 MHz band for the remnant radio galaxy.\",\n \"The number distribution of relativistic electrons drops rapidly with energy for a steep spectrum remnant radio galaxy.\",\n \"Hence the number of low-energy electrons is a good measure of the total acceleration of relativistic electrons during the lifetime of $\\\\sim $ 10$^8$ yr for the small redshift of the source, and indicates the entire history of the source.\",\n \"We thus determine $N$ $\\\\approx $ 2.1 $\\\\times $ 10$^{60}$ and the total energy stored in the magnetic field $\\\\approx $ 1.1 $\\\\times $ 10$^{57}$ erg assuming cylindrical geometry for the size of the remnant radio galaxy [21], [22].\",\n \"Here, we have assumed the volume filling factor is unity, and hence the volume and the number of relativistic electrons $N$ are thus the upper limits.\",\n \"The emitting plasma will most likely occupy a smaller volume, leading to larger energy densities and therefore larger values of the magnetic field strength and will require a smaller number of electrons to produce the observed luminosity.\",\n \"These basic energetics are similar to the Coma radio relic source 1253$+$ 275 [10].\",\n \"Moreover, the number of relativistic electrons, $N$ , is similar to that of the typical FR I radio galaxy 3C 31 but two orders of magnitude smaller than the FR II radio galaxy Cygnus A [13].\",\n \"This suggests that the FR I radio galaxy is possibly the progenitor of the remnant radio galaxy in A2065.\",\n \"Thus, our high-resolution, high-sensitivity radio image at the 250–500 MHz band along with the presence of the WISEA J152228.01$+$ 274141.3 source suggest that the source is a wide-angle-tailed radio source (see also Sec.\",\n \"REF ).\",\n \"Furthermore, the position of the WISEA J152228.01$+$ 274141.3 source exactly coincides with the likely radio core position of the wide-angle-tailed radio source.\",\n \"The source diagnostic based on the number of low-energy relativistic electrons also suggests that the remnant radio galaxy is fed by the small radio tail of the wide-angle-tailed FR I radio galaxy.\",\n \"If this proposition is correct, the two oppositely directed radio jets emanating from the apex of the WISEA J152228.01$+$ 274141.3 source initially traverse toward the northwest and southeast.\",\n \"As the galaxy plows through the dense ICM, these jets traversing in opposite directions form a trail after a sharp bend in the southeastern jet due to interaction with the ICM.\",\n \"The jets probably overlap and twist forming a pinch and then expand, and the jet becomes weaker [18].\",\n \"The sharpness in surface-brightness toward the northeast, i.e., the side facing two cD galaxies and diffuse extensions toward the far southwest side, along with corresponding spectral gradients, suggests the passage of an (inward?)\",\n \"traveling shock front.\"\n ],\n [\n \"Summary and conclusions\",\n \"The remnant radio source seems to trace AGN radio plasma that has somehow been reenergised through processes in the ICM.\",\n \"It is possible that, similar to radio relic sources, remnant radio sources are also believed to be tracers of merger shocks, unfortunately, only a few clusters have shown to possess corresponding features in hot X-ray gas.\",\n \"The remnant radio galaxy in A2065 may belong to this (rare) class possessing an X-ray surface-brightness edge across the remnant radio galaxy, indicating a possible connection with the shock front.\",\n \"Using our deep high-resolution low-frequency observations, we discovered a bar-shaped remnant radio galaxy, whose size is $\\\\approx $ 52$^{\\\\prime \\\\prime }$ $\\\\times $ 110$^{\\\\prime \\\\prime }$ (= 72 $\\\\times $ 152 kpc$^2$ ), remnant radio galaxy in the massive unequal-mass-merger galaxy cluster A2065 at $z$ = 0.072.\",\n \"It has a steep spectral index $\\\\alpha $ (240–610 MHz) = $-$ 1.4 $\\\\pm $ 0.2 and $\\\\alpha $ ($>$ 610 MHz) $<$ $-$ 1.4 $\\\\pm $ 0.2, corresponding to the line representing the best-fitting regression to the radio data.\",\n \"The Chandra data, and the upgraded and “classic\\\" GMRT data, show (i) the presence of an X-ray surface-brightness edge attributed to a shock front, though a possibility of cold front cannot be ruled out, at the remnant radio galaxy's edge toward the two cD galaxies, (ii) the tentative flattening of the radio spectral index, in our 250–500 MHz in-band data, of the remnant radio galaxy at the near end of the X-ray surface-brightness edge, and (iii) possibly a wide-angle-tailed, FR I radio galaxy is the progenitor of the remnant radio galaxy.\",\n \"Thus, the remnant radio galaxy has possibly been reinvigorated by the passage of the shock front and adiabatic compression, and shows the expected change in radio emission.\",\n \"We also suggest that the remnant radio galaxy was seeded by the AGN during its past active phase, and has been hosted by the WISEA J152228.01$+$ 274141.3 source, demonstrating the connection between AGNs and remnant radio galaxy, similar to the connection between AGNs and radio relic sources [28].\",\n \"Although low-frequency observations are starting to reveal more and more of these type of sources, presently there are a very few known remnant radio systems; clearly an outstanding difficulty is the small sample size.\",\n \"Although the number of known remnant radio sources is increasing, better data are necessary for several of them to allow a detailed study.\",\n \"Fortunately, forthcoming radio surveys using the LOFAR, MeerKAT, Square Kilometre Array, etc., will greatly increase the sample size, along with deep multifrequency radio data, e.g., using the upgraded GMRT, which will constrain the physics and nature of poorly understood remnant radio sources.\",\n \"We are undertaking an upgraded GMRT study at the 125–250 MHz, 250–500 MHz, and 550–850 MHz bands of several clusters in order to detect such ultra-steep-spectrum sources, map radio halo emission, and provide exact flux density estimates, models, and statistics.\"\n ],\n [\n \"Acknowledgments\",\n \"I would like to thank the anonymous referee for helpful remarks.\",\n \"D.V.L.\",\n \"acknowledges the support of the Department of Atomic Energy, Government of India, under project No.\",\n \"12-R&D-TFR-5.02-0700, and thanks the staff of the GMRT who made these observations possible.\",\n \"The GMRT is run by the National Centre for Radio Astrophysics of the Tata Institute of Fundamental Research.\",\n \"This research made use of the radio astronomical database GalaxyClusters.com, maintained by the Observatory of Hamburg.\",\n \"This research has made use of the NED, which is operated by the Jet Propulsion Laboratory, Caltech, under contract with the NASA, and NASA's Astrophysics Data System.\",\n \"Facilities: Chandra, GMRT Data Availability: The GMRT and Chandra data underlying this article are available via the GMRT Online Archive, naps.ncra.tifr.res.in/goa and the Chandra Data Archive, cxc.harvard.edu/cda.\",\n \"All data analyses packages used in this work are publicly available.\"\n ]\n]"},"id":{"kind":"string","value":"2107.01795"}}},{"rowIdx":1579,"cells":{"text":{"kind":"list like","value":[["Restless and Uncertain: Robust Policies for Restless Bandits via Deep\n Multi-Agent Reinforcement Learning"],["Abstract We introduce robustness in \\textit{restless multi-armed bandits} (RMABs), a popular model for constrained resource allocation among independent stochastic processes (arms).","Nearly all RMAB techniques assume stochastic dynamics are precisely known.","However, in many real-world settings, dynamics are estimated with significant \\emph{uncertainty}, e.g., via historical data, which can lead to bad outcomes if ignored.","To address this, we develop an algorithm to compute minimax regret -- robust policies for RMABs.","Our approach uses a double oracle framework (oracles for \\textit{agent} and \\textit{nature}), which is often used for single-process robust planning but requires significant new techniques to accommodate the combinatorial nature of RMABs.","Specifically, we design a deep reinforcement learning (RL) algorithm, DDLPO, which tackles the combinatorial challenge by learning an auxiliary \"$\\lambda$-network\" in tandem with policy networks per arm, greatly reducing sample complexity, with guarantees on convergence.","DDLPO, of general interest, implements our reward-maximizing agent oracle.","We then tackle the challenging regret-maximizing nature oracle, a non-stationary RL challenge, by formulating it as a multi-agent RL problem between a policy optimizer and adversarial nature.","This formulation is of general interest -- we solve it for RMABs by creating a multi-agent extension of DDLPO with a shared critic.","We show our approaches work well in three experimental domains."],["Introduction","The restless multi-armed bandit (RMAB) problem models sequential decision making tasks under a limited resource constraint.","An RMAB instance consists of a budget $B$ and a set of $N$ arms, where an arm corresponds to an action-dependent Markov state-transition process.","At every timestep, a planner selects a subset of $N$ arms and decides what actions to take on each selected arm such that the total cost of actions per timestep is less than a budget $B$ .","The arms transition to new states and generate rewards, depending on the action taken on them.","The goal is to plan a policy that maximizes the total expected reward.","This problem naturally captures a wide range of real-world problems.","For example, in a healthcare intervention problem [26], a health worker (planner) is in charge of the well-being of a set of patients (arms) who have enrolled in a health program, e.g., by ensuring adherence to daily tuberculosis medication.","The healthcare worker provides timely interventions to the patients via phone call or in-person visit to ensure that the patients remain in the adhering state.","However, phone calls and in-person visits require dedicated time from the health worker, thereby imposing a budget $B$ on the number of patients she can call or visit in-person each day.","The problem is challenging because of three key reasons: (1) the state of each patient representing the extent of adherence evolves over time; (2) the intervention decisions at each timestep influence the state transitions of all patients and, consequently, affect future rewards; and (3) different actions incur different costs, and the total cost of allocation must be at most $B$ at every timestep.","The combinatorial nature of the problem makes it computationally expensive to solve optimally [31].","Prior works have considered specific variants of the RMAB problem that are applicable to real-world scenarios, including healthcare intervention planning [23], [26], anti-poaching patrol planning [34], sensor monitoring tasks [11], [17], machine replacement [35], and many more.","A common assumption across existing literature on RMABs is that the state transition dynamics of all the arms are known beforehand.","Unfortunately, in real-world scenarios, even in the presence of prior data, there is still often significant uncertainty in transition dynamics.","A natural way to encode this uncertainty is interval uncertainty—rather than assuming that the probability of transitioning from one state to another is fixed to one value, we assume that it can take any value from a given closed continuous interval.","This additional uncertainty makes the RMAB problem even more challenging since any solution must be robust to variations in transition probabilities within the given interval.","In the presence of interval uncertainty, we formulate a generalization of RMAB that we call Robust Restless Bandits.","Our goal is to achieve minimax regret—minimize the worst-case regret against the optimal policy over all possible values of transition probabilities within the given uncertainty intervals.","Our contributions are as follows: We introduce the Robust RMAB problem for interval uncertainty settings and optimize for minimax regret.","We are the first to provide a minimax-regret solution for the restless bandits problem, while also generalizing to multi-action RMABs.","We develop a double oracle algorithm for solving Robust RMABs and demonstrate its effectiveness on experimental domains, including two inspired by real-world problems.","To enable our double oracle approach, we propose RMABPPO, a deep reinforcement learning (RL) algorithm for solving multi-action RMABs.","RMABPPO hinges on learning an auxiliary “$\\lambda $ -network” allowing us to decouple each arm's learning, greatly reducing sample complexity.","Under minimax regret, the adversary in the double oracle approach is notoriously difficult to implement due to non-stationarity.","To address this, we formulate the adversary oracle as a multi-agent reinforcement learning problem and solve it with a multi-agent extension of RMABPPO."],["Restless bandits","[40] introduced RMABs, following which, there is a vast literature on binary-action RMABs.","In the classic setting, a planner selects $k$ out of $N$ state-transitioning arms, resulting in a two-action decision problem with the goal of maximizing reward.","Although the problem is PSPACE-Hard [31], heuristics based on Lagrangian relaxation of the optimization have been proposed [40] which turns out to be optimal under the indexability criterion.","However, the Whittle index solution does not extend to budgeted multi-action settings, where there are more than two actions, each with an associated cost.","[12] and [16] have extended index-based solutions to multi-action RMABs for instances with special monotonic structure.","Along similar lines, [20] proposed a method that leverages the convexity of an approximate Lagrangian version of the multi-action RMAB problem.","Another relevant category of research is taking decisions on weakly coupled Markov decision processes (WCMDP).","[15] proposed a Lagrangian relaxation of the general WCMDP problem and proposed an LP for minimizing the Lagrange bound.","[3] and [13] provided less scalable solutions that provide better approximations to WCMDPs.","All these papers have assumed perfect knowledge of the state transition dynamics to compute policies.","A few recent works have studied RMABs in online settings with unknown transition probabilities but make the limiting assumptions that the planner takes only one non-combinatorial action [8] or that states periodically reset [18].","None of the above papers consider robust planning under environment uncertainty, which we address.","Despite the recent successes of reinforcement learning (RL) for solving large-scale games [28], [37], RL has so far seen little application to RMABs, except for a few recent works that learn Whittle indices for indexable binary-action RMABs using (i) deep RL [29] and (ii) Q-learning when states are observable [5], [7] or when arms are homogeneous [4].","In contrast, our deep RL approach provides a more general solution to binary and multi-action RMAB domains that performs well regardless of indexability.","One recent work does also address learning in the multi-action setting [19], but their approach is based on tabular Q-learning which can be sample inefficient and scale poorly, unlike function-approximation methods like ours."],["Robust planning","The RL literature has a large body of work on robust planning, mainly focused on maximizing the minimum (maximin) reward through robust adversarial RL [33] or multi-agent RL settings [22], [24].","The minimax regret criterion [6] has been offered as a compelling alternative to maximin reward, which often leads to overly conservative policies that buffer against worst-possible realizations.","A challenge to computing minimax regret–optimal strategies is that such games often involve very large and sometimes continuous strategy spaces, rendering it impossible to explore the entire strategy space.","The double oracle approach [27] has been applied as an effective means by which to optimally explore only a small subset of these large spaces while still guaranteeing optimal performance [30], [10].","Double oracle has been extended to optimize multi-agent RL problems with multiple selfish agents [22].","More recently, [41] offered an algorithm for minimax regret planning for a non-RMAB problem using RL to instantiate the two oracles.","However, their approach cannot handle budget-constrained discrete actions, which we have in the RMAB setting.","Additionally, we formulate the nature oracle as a multi-agent RL problem, allowing us to handle the joint discrete and continuous action spaces that arise in the minimax regret nature oracle.","Robustness objectives have been considered for bandit applications in the two-action stochastic setting, where each pull of an arm draws a reward sampled from an unknown Bernoulli distribution.","[39] address minimax regret of time-varying reward shifts using heuristics to trade off remembering vs. forgetting, and [9] consider a maximin objective to guide exploration in Monte Carlo Tree Search.","However, RMABs are significantly harder than the stochastic settings since, in RMABs, the rewards are dependent on the current state which in turn depends on the actions taken on the arms."],["Problem Statement","We consider the multi-action RMAB setting with $N$ arms.","Each arm $n\\in [N]$ follows a Markov decision process (MDP) $(\\mathcal {S}_n, \\mathcal {A}_n, \\mathcal {C}_n, T_n, R_n, \\beta )$ , where $\\mathcal {S}_n$ is a set of finite, discrete states, $\\mathcal {A}_n$ is a set of finite, discrete actions, $\\mathcal {C}_n : \\mathcal {A}_n \\xrightarrow{} \\mathbb {R}$ corresponds to action costs, $T_n: \\mathcal {S}_n\\times \\mathcal {A}_n \\times \\mathcal {S}_n \\xrightarrow{} [0,1]$ gives the probability of transitioning from one state to another given an action, $R_n:\\mathcal {S}_n\\times \\mathcal {A}_n \\times \\mathcal {S}_n \\xrightarrow{} \\mathbb {R}$ is a reward function, and $\\beta \\in [0, 1)$ is the discount parameter.","The agent must select actions for each arm, each round, such that the sum cost of actions does not exceed a per-round budget $B$ .","The aim of multi-action RMABs is to maximize total reward over a fixed number of $H$ rounds, subject to this budget constraint, generalizing the well-studied binary-action RMAB.","In this work, we extend multi-action RMABs to the robust setting in which the exact transition probabilities are unknown.","Instead, the transitions of each arm are determined by a set of parameters $\\mathcal {P}_n$ .","For a given arm $n \\in [N]$ and parameter $p_n \\in \\mathcal {P}_n$ , let $\\langle {\\omega _{n,p_n}}\\rangle :=[\\underline{\\omega }_{n,p_n}, \\overline{\\omega }_{n,p_n}]$ represent the range, i.e., uncertainty over transition probabilities.","We ask the following question: how to find a solution to the multi-action RMAB problem that is robust to such uncertainties?","We consider an objective that incorporates our uncertainty over $\\mathcal {P}_n$ .","Let $\\omega $ be a given realization of the transition probabilities such that $\\omega _{n,p_n} \\in \\langle {\\omega _{n,p_n}}\\rangle $ for all $n \\in [N]$ and $p_n \\in \\mathcal {P}_n$ .","Let $G(\\pi ,\\omega )$ be the planner's expected reward under policy $\\pi $ and a realization $\\omega $ of the uncertainty.","Regret is defined: $L(\\pi ,\\omega ) = G(\\pi ^*_{\\omega },\\omega ) - G(\\pi ,\\omega ) \\ ,$ where $\\pi ^*_{\\omega }$ is the optimal policy under $\\omega $ .","In our robust planning formulation, our objective is to compute a policy $\\pi $ that minimizes the maximum regret $L$ possible for any realization of $\\omega $ , leading to the following minimax objective: $\\min _{\\pi }\\max _{{\\omega }}{L(\\pi ,{\\omega })} \\ .$ This problem is computationally expensive to solve since simply computing a policy $\\pi $ that maximizes the expected reward $G(\\pi ,\\omega )$ is PSPACE-Hard [31] even for the special case when the transition rules are known i.e., $\\omega $ is given.","This challenge is likely a key reason why the robust formulation has not yet been addressed in the literature.","To overcome the complexity of the minimax optimization, we take a double oracle approach [27], which requires key innovations to work in the RMAB setting."],["Preliminaries","The double oracle approach achieves the minimax regret objective in Eq.","REF ) by casting the optimization problem as a zero-sum game between two players, the agent and nature [41].","The agent selects a policy that minimizes regret for some realization of the transition probabilities.","Nature then adversarially selects the values of $\\mathcal {P}_n$ that maximize regret for a given policy of the planner.","This framework is desirable since it converges to an $\\varepsilon $ –optimal solution [2], [41], assuming there are oracles that return the best response for both players.","The key technical contributions of this paper arise from the design of the agent and nature oracles for best response computations.","For the agent, minimizing regret with respect to a fixed nature strategy is equivalent to maximizing reward w.r.t.","that strategy, so the agent objective is the same as solving a multi-action RMAB to find the best policy.","A policy $\\pi $ maps states to decision matrices $\\mathbf {A} \\in \\lbrace 0,1\\rbrace ^{N\\times |\\mathcal {A}|}$ where the total number of active actions is constrained by a budget for each round.","Let $\\mathbf {s} = (s^1, \\ldots , s^{N})$ represent the initial state of each arm.","Then, for a given parameter $\\omega $ , the optimal policy $\\pi ^*_{\\omega }$ maximizes the expected discounted sum of rewards of all arms as given by the constrained Bellman equation: $\\begin{aligned}J(\\mathbf {s}) &= \\max _{\\mathbf {A}}\\left\\lbrace \\sum _{n=1}^{N} R_n(s_n) + \\beta \\mathbb {E}_{\\omega }[J(\\mathbf {s}^\\prime ) \\mid \\mathbf {s}, \\mathbf {A}]\\right\\rbrace \\\\\\text{s.t. }","& \\sum _{j=1}^{|\\mathcal {A}|} a_{nj} = 1 \\hspace{8.53581pt} \\forall n \\in [N]\\qquad \\sum _{n=1}^{N}\\sum _{j=1}^{|\\mathcal {A}|} a_{nj}c_{nj} \\le B\\end{aligned}$ where $a_{nj} \\in \\mathbf {A}_n$ and $c_{nj}$ are the corresponding action costs in $\\mathcal {C}_n$ .","We then relax the problem by taking the Lagrangian relaxation of the budget constraint [15], giving: $\\begin{aligned}&J(\\mathbf {s}, \\lambda ^*) = \\min _{\\lambda } \\left( \\frac{\\lambda B}{1-\\beta } + \\sum _{n=1}^{N}\\max _{a_{nj}\\in \\mathcal {A}_n}Q_n(s_n, a_{nj}, \\lambda ) \\right)\\end{aligned}$ $\\begin{aligned}\\text{where }\\hspace{2.84526pt} Q_n(s_n, a_{nj}, \\lambda ) =R_n(s_n) - \\lambda c_{nj} + \\beta \\mathbb {E}_{\\omega } \\left[ Q_n(s^{\\prime }, a_{nj}, \\lambda ) \\mid \\pi ^{La}_{\\omega }(\\lambda ) \\right] \\ .\\end{aligned}$ Here, $Q$ is the value function and $\\pi ^{La}_{\\omega }(\\lambda )$ is the optimal policy for a given $\\lambda $ .","See [3] for a detailed derivation.","Note that for a given value of $\\lambda $ , Eq.","REF could be solved using $N$ individual value iterations.","However, setting $\\lambda :=$ $\\lambda ^*$ is critical to finding good policies for a multi-action RMAB and is known to be asymptotically optimal in the binary-action case [38], i.e., $\\pi ^{La}_{\\omega }(\\lambda ^*) \\xrightarrow{} \\pi ^{*}_{\\omega }$ .","Given this relationship, in the remainder of the paper, we focus of computing $\\pi ^{La}_{\\omega }(\\lambda ^*)$ and denote it as $\\pi ^{*}_{\\omega }$ for convenience."],["Solving Robust Restless Bandits","We now build up our approach for finding robust RMAB policies.","The underlying pure strategy space for the agent is the set of all feasible policies $\\pi : \\mathcal {S}_1\\times \\cdots \\times \\mathcal {S}_N \\mapsto \\mathcal {A}_1\\times \\cdots \\times \\mathcal {A}_N$ .","The pure strategy space for nature is a closed set of parameters $\\omega $ within the given uncertainty intervals.","The agent oracle's goal is to find a policy $\\pi $ , or pure strategy, to minimize regret (Eq.","REF ) given a mixed strategy $\\tilde{\\omega }$ , where a mixed strategy is a probability distribution over a set of pure strategies.","That is, the agent minimizes $L(\\pi ,\\tilde{\\omega })$ w.r.t.","$\\pi $ , while $\\tilde{\\omega }$ is constant.","Recall that $L(\\pi ,\\tilde{\\omega }) = G(\\pi ^{*}_{\\tilde{\\omega }},\\tilde{\\omega }) - G(\\pi ,\\tilde{\\omega })$ .","Since the first term $G(\\pi ^{*}_{\\tilde{\\omega }},\\tilde{\\omega })$ is constant, minimizing $L(\\pi ,\\tilde{\\omega })$ is equivalent to maximizing the second term $G(\\pi ,\\tilde{\\omega })$ , which is maximal at $\\pi =\\pi ^{*}_{\\tilde{\\omega }}$ .","In other words, the agent oracle must compute an optimal reward-maximizing policy w.r.t.","$\\tilde{\\omega }$ .","The objective of maximizing reward is standard in reinforcement learning, enabling us to build off existing RL techniques as we show in Section REF .","On the other hand, the nature oracle's goal is to find a parameter setting $\\omega $ , or pure strategy, that maximizes the agent's regret given a mixed strategy $\\tilde{\\pi }$ , i.e., maximize $L(\\tilde{\\pi },\\omega )$ with respect to $\\omega $ , while $\\tilde{\\pi }$ is fixed.","This objective is far more challenging because both $G(\\tilde{\\pi }^*_{\\omega },\\omega )$ and $G(\\tilde{\\pi },\\omega )$ are functions of $\\omega $ .","Moreover, computing $G(\\tilde{\\pi }^*_{\\omega },\\omega )$ requires obtaining an optimal policy $\\tilde{\\pi }^*_{\\omega }$ as $\\omega $ changes in the optimization.","Unfortunately, $\\omega $ comes from an infinite continuous strategy space, making this problem difficult.","However, as one of our main contributions, we propose a novel method for implementing the regret-maximizing nature oracle by casting it as a multi-agent reinforcement learning problem, simultaneously solving for policies $\\pi ^*_{\\omega }$ while computing worst-case parameters $\\omega $ to maximize $L(\\tilde{\\pi },\\omega )$ .","Solving a multi-agent reinforcement learning problem first requires an RL algorithm to optimize the underlying policy; hence we first introduce a novel RL approach, RMABPPO, to solve RMABs (Sec.","REF ) as a part of our agent oracle and then use the algorithm as the backbone of our nature oracle (Sec.","REF )."],["Agent Oracle: Deep RL for RMABs via RMABPPO","Existing deep RL approaches can be applied to the objective in Eq.","REF , but they fail to scale past trivially sized RMAB problems since the action and state spaces grow exponentially in $N$ .","For example, for a binary-action RMAB with $N=50$ and $B=20$ , the action space would be of size $\\binom{50}{20}\\approx 10^{12}$ , which is not feasible to learn, even with a neural network.","To overcome this, we develop a novel deep RL algorithm that instead solves the decoupled problem (Eq.","REF ).","The key benefit of decoupling is to render policies and $Q$ values of each arm independent, allowing us to learn $N$ independent networks with linearly sized state and action spaces, relieving the combinatorial burden of the learning problem — the above example would now only have $N \\times 2 = 100$ actions.","However, this approach introduces a new technical challenge in solving the dual objective which maximizes over policies but minimizes over $\\lambda $ , as discussed in Sec. .","We derive a dual gradient update procedure that iteratively optimizes each objective as follows: (1) holding $\\lambda $ constant, learn $N$ independent policies via a policy gradient procedure, augmenting the state space to include $\\lambda $ as input, as in Eq.","REF ; (2) use sampled trajectories from those learned policies as an estimate to update $\\lambda $ towards its minimizing value via a novel gradient update rule.","Another challenge is that $\\lambda ^*$ of Eq.","REF depends on the current state of each arm — therefore, a key element of our approach is to learn this function concurrently with our iterative optimization, using a neural network we call the $\\lambda $ -network that is parameterized by $\\Lambda $ .","To train the $\\lambda $ -network, we use the following gradient update rule.","[]propositionlambdaUpdate A gradient rule for updating the $\\lambda $ -network, parameterized by $\\Lambda $ , such that for a state $\\mathbf {s}$ , the $\\lambda $ -network predicts the value $\\lambda $ that minimizes Eq.","REF is as follows: $\\begin{aligned}\\Lambda _t = \\Lambda _{t-1} - \\alpha \\left( \\frac{B}{1-\\beta } + \\sum _{n=1}^{N}D_n(s_n, \\lambda _{t-1}(\\mathbf {s})) \\right)\\end{aligned}$ where $\\alpha $ is the learning rate and $D_n(s_n, \\lambda )$ is the negative of the expected $\\beta $ -discounted sum of action costs for arm $n$ starting at state $s_n$ under the optimal policy for arm $n$ for a given value of $\\lambda $ .","Although $D_n$ cannot be computed exactly as we do not know the optimal policy, it can be estimated from samples of multiple rollouts of the policy during training.","As long as arm policies are trained for adequate time on the given value of $\\lambda $ , the gradient estimate will be accurate, i.e., $D_n(s_n, \\lambda _{t-1}(\\mathbf {s})) \\approx -\\sum _{k=0}^{K-1} \\beta ^k c_{nk}$ where $K$ is the number of samples collected in an epoch and $c_{nk}$ is the action cost of arm $n$ in round $k$ .","Moreover, this procedure will converge to the optimal parameters $\\Lambda $ if the arm policies are optimal.","[]propositionlambdaConvergence Given arm policies corresponding to optimal $Q$ -functions, the gradient update rule of Prop.","REF will lead $\\Lambda $ to converge to the optimal as the number of training epochs and $K\\xrightarrow{}\\infty $ .","Note that to collect samples that reflect the proper gradient, the RMAB budget must not be imposed at training time — rather the policy networks and $\\lambda $ -network must learn to play the Lagrange policy of Eq.","REF which spends the correct budget in expectation.","At training time, actions are sampled randomly according to the actor network distributions.","At test time, actions are taken deterministically by greedily selecting the highest probability actions until the budget is spent.","In theory, the policy networks could be trained via any deep RL procedure, as long as the above characteristics for training the $\\lambda $ -network are ensured.","In practice, we train with proximal policy optimization (PPO) [36], which has been demonstrated to have state-of-the-art performance while being relatively simple to implement, and will allow flexibility to handle both discrete and continuous actions—the latter will be important for the nature oracle.","We also navigate the important trade-off between exploring new policy actions after an update to $\\Lambda $ and allowing the policy networks to converge before updating $\\Lambda $ , by using an entropy regularization term on the loss function of the policy networks for each arm that is controlled by a cyclical temperature parameter.","We call our algorithm RMABPPO, give pseudocode in Algorithm REF , and give more implementation details in the appendix."],["Nature Oracle: Multi-Agent RL for RMABs via MA-RMABPPO","Armed with a deep RL procedure for learning RMAB policies, we are now ready to create the multi-agent extension needed to implement the nature oracle.","Recall that the challenge of the nature oracle is to simultaneously optimize a policy as well as the environment parameters $\\omega $ .","We propose to treat this optimization as a multi-agent RL problem designed to handle this form of non-stationarity [25] via a centralized critic.","To implement the nature oracle, we introduce two agents $A$ and $B$ , where $A$ 's goal is to optimize the RMAB policy $\\pi ^*_{\\tilde{\\omega }}$ and $B$ 's goal is to find parameters $\\omega $ that maximize regret of the current agent mixed strategy $\\tilde{\\pi }$ .","The shared environment transition function is $T: \\mathcal {S} \\times \\mathcal {A}_A \\times \\mathcal {A}_B \\xrightarrow{} \\mathcal {S}$ .","The action space $\\mathcal {A}_A$ will be the same as the action space of multi-action RMAB.","At a given state $\\mathbf {s}$ , the action space $\\mathcal {A}_B$ will allow agent $B$ to select $\\omega $ which, in general, depends on $\\mathbf {s}$ .","That is, at each step, agent $B$ will select environment parameters $\\omega $ , and thus state/action transition probabilities that will determine the outcome of agent $A$ 's actions.","We adopt the centralized critic idea from multi-agent PPO [43] to our RMAB setting to create MA-RMABPPO.","Since the policy space of agent $A$ is discrete while that of agent $B$ is continuous, a notable benefit to PPO is that it offers a convenient way to train both policies.","The final step is to define the rewards for each agent $A$ and $B$ to match their respective objectives.","Since agent $A$ 's objective is to maximize $\\pi ^*_{\\tilde{\\omega }}$ , it simply adopts the reward defined by the RMAB, i.e., ${R}^{(A)} = \\sum _{n=1}^N R_n$ .","However, agent $B$ 's objective is to learn the regret-maximizing parameters $\\omega $ .","This objective is challenging because it requires computing and optimizing over the returns of the fixed input policy $\\tilde{\\pi }$ with respect to all possible $\\omega $ values, which in general is non-convex.","In practice, to estimate the returns of $\\tilde{\\pi }$ , we execute a series of single-step roll-outs for computational efficiency.","That is, given $\\mathbf {s}$ and $\\mathbf {a}$ at a given round, we sample the next state profile $\\mathbf {s^\\prime }$ , and define the reward of agent $B$ (i.e., the regret of input policy $\\tilde{\\pi }$ ) as ${R}^{(B)} = \\sum _{n=1}^N R_n(s_n,a_n,s^\\prime _n) - \\frac{1}{Y}\\sum _{y=1}^{Y}r_y^{\\tilde{\\pi }}$ , where $r_y^{\\tilde{\\pi }}$ is the reward obtained from each of $Y$ 1-step Monte Carlo simulations of the mixed strategy $\\tilde{\\pi }$ .","Since agent $A$ has the same policy network architecture as RMABPPO, i.e., N discrete policy networks and one $\\lambda $ -network.","The agent $B$ actor network is a single continuous-action policy network.","Since agent $A$ and $B$ have separate objectives, they each have their own centralized critic networks which take the actions of the other as inputs to their state space.","For agent $A$ , this is implemented as one critic network per arm, where each network is augmented with agent $B$ 's actions.","Finally, to ensure good gradient estimates for the $\\lambda $ -network in MA-RMABPPO, we keep agent $B$ 's network — and thus the environment — constant between $\\lambda $ updates, updating $B$ 's network at the same frequency as the $\\lambda $ -network updates.","Pseudocode for MA-RMABPPO and further details of its implementation are given in the appendix."],["The Minimax Regret–Robust RMAB Double Oracle","We now have all the pieces we need to present our robust algorithm RMAB Double Oracle (RMABDO), with pseudocode presented in Algorithm REF , adapted from the MIRROR framework [41].","We use RMABPPO to instantiate the agent oracle and MA-RMABPPO for the nature oracle.","The double oracle approach proceeds as follows.","The agent maintains strategy set $\\Pi $ , initially empty, and nature maintains strategy set $\\Omega $ , initialized with an arbitrary parameter setting.","In each iteration, we solve for a mixed Nash equilibrium in the regret game between the agent and nature to learn a mixed strategy ($\\tilde{\\pi }, \\tilde{\\omega }$ ) for each player.","We then call the agent and nature oracles to each compute a best response $\\pi $ and $\\omega $ to their opponent's strategy, which get added to their respective strategy sets $\\Pi $ and $\\Omega $ .","We repeat this process until the improvement in value for each player is within the tolerance $\\varepsilon $ , which is guaranteed to converge within finite steps to within $\\epsilon $ value of the minimax regret–optimal policy (Theorem 2 of [41]), or until a set number of iterations.","Additionally, we show that a policy that maximizes reward assuming a fixed parameter set can incur arbitrarily large regret when the parameters are changed (proof in appendix).","Formally, [tb] RMABPPO Input: Initial state $\\mathbf {s}_0$ , nature mixed strategy $\\tilde{\\omega }$ Parameters: num epochs n_epochs, num subepochs n_subepochs [1] Randomly initialize a policy network $\\pi _{\\theta _n}$ for each arm $n \\in [N]$ Randomly initialize $\\lambda $ -network Lambda Initialize an empty buffer $\\textit {epoch} = 1, 2, \\ldots , \\texttt {n\\_epochs}$ Sample $\\lambda = \\textsc {Lambda}(\\mathbf {s})$ $\\textit {subepoch} = 1, \\ldots , \\texttt {n\\_subepochs}$ timestep $t = 1, \\ldots , \\texttt {n\\_simulation\\_steps}$ Sample action $a_n = \\pi _{\\theta _n}(s_n, \\lambda )$ for all $n \\in [N]$ Add trajectories $(\\mathbf {s}, \\mathbf {a}, r, \\mathbf {s}^\\prime , \\lambda )$ to buffer Update policy network $\\pi _{\\theta _n}$ of all arms via PPO, using trajectories in buffer Update Lambda using discounted sum of trajectories of final subepoch return $\\pi _{\\theta _1}, \\ldots , \\pi _{\\theta _N}$ and Lambda [tb] RMABDO Input: Environment simulator and parameter uncertainty interval $\\langle \\omega _{n, p_n} \\rangle $ for all $n \\in [N]$ Parameters: Convergence threshold $\\varepsilon $ Output: Best agent mixed strategy $\\tilde{\\pi }$ [1] $\\Omega _0 = \\lbrace \\omega _0\\rbrace $ , with $\\omega _0$ selected at random $\\Pi _0 = \\lbrace \\pi _{B_1}, \\pi _{B_2}, \\ldots \\rbrace $ , where $\\pi _{B_i}$ are baseline and heuristic strategies epoch $e = 1, 2, \\ldots $ Solve for $(\\tilde{\\pi }_e, \\tilde{\\omega }_e)$ , mixed Nash equilibrium of regret game with strategy sets $\\Omega _{e-1}$ and $\\Pi _{e-1}$ $\\pi _e = \\textsc {RMABPPO}(\\tilde{\\omega }_e)$ $\\omega _e = \\textsc {MA-RMABPPO}(\\tilde{\\pi }_e)$ $\\Omega _e = \\Omega _{e-1} \\cup \\lbrace \\omega _e\\rbrace , \\Pi _e = \\Pi _{e-1} \\cup \\lbrace \\pi _e\\rbrace $ $L(\\tilde{\\pi }_e, \\omega _e) - L(\\tilde{\\pi }_{e-1}, \\tilde{\\omega }_{e-1}) \\le \\varepsilon $ and $L(\\pi _e, \\tilde{\\omega }_e) - L(\\tilde{\\pi }_{e-1}, \\tilde{\\omega }_{e-1}) \\le \\varepsilon $ return $\\tilde{\\pi }_e$ []propositionregretProposition In the RMAB problem with interval uncertainty, the max regret of a reward-maximizing policy can be arbitrarily large compared to a minimax regret–optimal policy."],["Experimental Evaluation","We compare our algorithm against five baselines.","Namely, we compare against three variations of the approach from [15], which computes a reward-maximizing Lagrange policy for each step of a multi-action RMAB problem for fixed transition probabilities.","The three variations are, pessimistic (HP), mean (HM), and optimistic (HO), which assume the transition probabilities have been set to lower bound, mean, and upper bound of the intervals, respectively.","We also compare against RLvMid, which learns a policy via RMABPPO assuming mean parameter values, and Rand, which acts randomly each round.","All results are averaged over 50 random seeds and were executed on a cluster running CentOS with Intel(R) Xeon(R) CPU E5-2683 v4 @ 2.1 GHz with 8GB of RAM using Python 3.7.10.","Our RMABPPO implementation builds on OpenAI Spinning Up [1] and RMABDO builds on the MIRROR implementation [41], computing Nash equilibria using Nashpy 0.0.21 [21].","Code is available on github.Code: https://github.com/killian-34/RobustRMAB Hyperparameter settings are included in the appendix.","We evaluate all the approaches on three experimental domains, which are as follows.","Synthetic domain: We create this setup to show that the reward-maximizing Lagrange policies (HP, HM, and HO) may incur large regret.","There are three arm types $\\lbrace U,V,W\\rbrace $ , each with two actions, $\\mathcal {C} = [0, 1]$ , two states $\\mathcal {S}=\\lbrace 0,1\\rbrace $ , $R(s)=s$ , and the following transition probabilities, with rows and columns corresponding to actions and next states, respectively: $T^n_{s=0}=\\begin{bmatrix}0.5 & 0.5 \\\\0.5 & 0.5\\end{bmatrix}, \\hspace{5.69054pt}T^n_{s=1}=\\begin{bmatrix}1.0 & 0.0 \\\\1-p_n & p_n\\end{bmatrix}\\hspace{5.69054pt}\\text{where}\\hspace{5.69054pt}\\begin{matrix}p_U \\in [0.00, 1.00] \\\\p_V \\in [0.05, 0.90] \\\\p_W \\in [0.10, 0.95]\\end{matrix} \\ .$ Note that, when an arm is at $s=0$ , both the actions have equal impact on the state transition.","When the arms are at $s=1$ , selecting the arm with higher $p_n$ is optimal.","Accordingly, $\\pi _\\textit {HP} = [W,V,U]$ , $\\pi _\\textit {HM} = [W,U,V]$ , and $\\pi _\\textit {HO} = [U,W,V]$ .","Observe that there exist values of $p_n$ that can make each of the policies incur large regret, e.g., $p=[0.0, 0.9, 0.1]$ for $\\pi _\\textit {HM}$ , which would induce an optimal policy $[1,2,0]$ that is the reverse of $\\pi _\\textit {HM}$ .","ARMMAN: The maternal healthcare intervention problem [5] is modeled as a binary-action RMAB that selects a subset of beneficiaries each week to intervene on with tailored maternal health messaging to encourage engagement.","The behavior of each enrolled woman is modeled by an MDP with three states: Self-motivated, Persuadable, and Lost Cause.","We use the summary statistics mentioned in their paper and assume uncertainty intervals of $0.5$ centered around the transition parameters, resulting in 6 uncertain parameters per arm (details in appendix).","Similar to the setup in [5], we assume 1:1:3 split of arms with high, medium, and low probability of increasing their engagement upon intervention.","In our experiments, we scale the value of $N$ in multiples of 5 to keep the same split of arm categories of 1:1:3.","SIS Epidemic Model: A discrete-state model in which arms represent distinct geographic regions and each member of an arm's population of size $N_{\\textit {p}}$ is tracked to measure whether they are already infected by a disease or are susceptible.","For a particular region, the fraction of susceptible members represents the state of the region.","The model is defined by parameters $\\lambda _{\\textit {c}}$ , the average number of contacts per timestep, and $r_{\\textit {infect}}$ , the probability of infection given contact with an infectious person.","Details on computing discrete state transition probabilities are derived from [42] and given in the appendix.","We augment the model to include three actions $\\lbrace a_0, a_1, a_2\\rbrace $ with costs $c=\\lbrace 0, 1, 2\\rbrace $ .","Action $a_0$ represents no action, $a_1$ divides the contacts per day $\\lambda _{\\textit {c}}$ by $a^{\\textit {eff}}_1$ , e.g., by communicating messaging about physical distancing, and $a_2$ divides the probability of infection given contact $r_{\\textit {infect}}$ by $a^{\\textit {eff}}_2$ , e.g., by distributing face masks.","We impose the following uncertainty intervals: $\\lambda _{\\textit {c}} \\in [1, 10]$ , $r_{\\textit {infect}} \\in [0.5, 0.99]$ , $a^{\\textit {eff}}_1 \\in [1, 10]$ , and $a^{\\textit {eff}}_2 \\in [1, 10]$ ."],["Robust Double Oracle","To evaluate the algorithms for robustness, we compute the regret of an agent's strategy against a nature pure strategy $\\omega $ as the difference between the average reward obtained by the agent's strategy on $\\omega $ and the average reward of the optimal strategy against $\\omega $ .","The average reward is computed as the discounted sum of rewards over all arms for a horizon of length 10, averaged over 25 simulations.","In each setting, double oracle runs for 6 iterations, using 100 rollout steps and 100 training epochs for each oracle.","After completion, each baseline strategy is evaluated, querying the nature oracle for the best response against that strategy.","We report the max regret against all nature's pure strategies.","Figure REF shows that our double oracle method has the lowest regret-per-arm and beats all the baselines across all experimental settings.","The top row shows the results on the synthetic domain, demonstrating that our approach can reduce regret by about $50\\%$ against existing benchmarks, across various values of $N$ and $B$ .","This is expected because the domain is designed to ensure large regret for HP, HM, and HO baselines.","Here, a benefit of $50\\%$ in the regret corresponds to, in the worst case, keeping $1/3$ of the arms in the good state for an extra round compared to the baseline.","Similarly, for the ARMMAN domain, our algorithm performs consistently better than the baselines, achieving regret that is around $50\\%$ lower than the best baselines.","The third domain (SIS) is interesting since the state space is large (number of states was restricted to 2 and 3 in the previous domains).","Similar to the earlier results, in the SIS model with populations size of 50, our algorithm outperforms in terms of regret.","We discuss the runtime of Figure 1 in Section REF .","Additional results included in the appendix."],["Agent Oracle","We evaluate the performance of RMABPPO, our novel RL approach to find a reward-maximizing policy for multi-action RMABs, against an upper bound, namely, the solution by Hawkins, given exact transition probabilities.","This will create an upper bound because, given access to the exact transition dynamics, Hawkins will compute the exact Lagrange policy, whereas RMABPPO must learn to approximate the Lagrange policy from samples.","Each setting instantiates the environment with a random sample of valid parameter settings for each seed.","Figure REF shows that the reward accumulated by our agent oracle is comparable to the Hawkins algorithm.","In the synthetic domain (top row), the policy learns to act on the $33\\%$ of arms who belong to category $W$ .","The mean reward of RMABPPO almost matches that of Hawkins algorithm as $N$ scales with a commensurate budget (Fig.","REF left).","As we fix $N$ and vary the budget (Fig.","REF right), the optimal policy accumulates more reward, and RMABPPO is almost equal to the optimal.","We observe similar results on the ARMMAN domain (middle row), where it is optimal to act on $20\\%$ of arms (that forms category $A$ ; details in appendix).","Additionally, on the third large-scale domain (SIS, bottom left), we show the strong performance of RMABPPO holds in a multi-action setting even as we increase the number of states from 50 to 500 to test the scalability.","Moreover, computationally, RMABPPO beats Hawkins: in the bottom-right of Fig.","REF , a single rollout (10 time steps) of the Hawkins policy takes around 100 seconds when there are 500 states (population size), and steeply increases with more states.","This demonstrates that it would be prohibitive to run Hawkins within the loop of the double oracle, since the agent policy needs to be evaluated thousands of times to compute the regret matrices—for merely 25 simulations, computation would take around 42 minutes to evaluate a single cell in the regret matrix over all pure strategy combinations, where the matrix has size $|\\Pi | \\times |\\Omega |$ ."],["Limitations","We believe these advancements have the potential to improve resource allocation in low-resource settings, but acknowledge they are not without tradeoffs.","For example, the baseline methods we compare against, while less robust, can provide interpretable `index' policies that capture the value for acting on an arm, whereas our solution's output can be difficult for a domain expert to interpret.","Further, such optimization tools have the risk of amplifying underlying biases in the data and translating that to unfair resource allocation.","However, by addressing the robust version of the problem, we directly address this concern by providing a flexible tool for mitigating biases, by allowing users to tune their uncertainties and thus, providing natural ways to develop good policies even when data availability is skewed."],["Conclusion","In this paper, we address a blocking limitation that inhibits restless bandits to be used for many real-world planning problems: we often lack perfect knowledge of the environment dynamics.","To plan effective and risk-averse policies, it is therefore essential to take a robust approach to account for this uncertainty.","Our approach enables us to learn minimax regret–optimal policies by providing RMABPPO, a novel RL algorithm to decouple the budget-constrained problem by training an auxiliary $\\lambda $ -network, and MA-RMABPPO, a new approach to instantiating the nature oracle of an adversarial double oracle game setup by using multi-agent RL to handle nonstationary in nature's challenging optimization problem.","Notably, RMABPPO can also extend to solving RMAB problems with continuous states and/or actions, a setting which has not previously been addressed in the literature.","We hope these contributions bring us closer to deploying restless bandits in the real world.","This work was supported in part by the Army Research Office by Multidisciplinary University Research Initiative (MURI) grant number W911NF1810208.","J.A.K.","was supported by an NSF Graduate Research Fellowship under grant DGE1745303.","A.B.","was supported by the Harvard Center for Research on Computation and Society."],["Proof of Proposition ","* This follows from taking the gradient of Eq.","REF with respect to $\\lambda $ .","To compute the gradient of $Q_n$ , we simply roll out $Q_n$ over time then take the derivative, yielding $\\frac{dQ_n}{d\\lambda } = \\sum _{t=0}^{1} \\beta ^t c_{n,t}$ , i.e., the negative of the expected discounted sum of action costs under the optimal policy."],["Proof of Proposition ","* This follows from the convexity of Eq.","REF with respect to $\\lambda $ .","The convexity of Eq.","REF can be seen from the definition of $Q_n$ , i.e., the max over piece-wise linear functions of $\\lambda $ is convex."],["Proof of Proposition ","* Consider a binary-action RMAB problem with two arms A and B.","Let the reward from each arm be $R$ when the arm is in a good state and 0 in a bad state.","Our problem is to plan the best action with a budget of 1 and horizon of 1.","Supposing the initial state is bad for each arm, the transition probabilities for the transition matrix for each arm $n$ is $\\begin{bmatrix} 1 & 0 \\\\ 1 - p_n & p_n \\end{bmatrix}$ where the uncertain variable $p_n$ is constrained to be within $p_A, p_B \\in [0, 1]$ .","Each value in the matrix corresponds to the probability of an arm at state bad transitioning to bad (column 1) or good (column 2) if we take the passive (row 1) or active action (row 2).","To compute a reward-maximizing policy that does not consider robustness to uncertainty, we must optimize for one instantiation of the uncertainty set, which requires making one of three assumptions.","Case 1: If we assume $p_A = p_B$ , then an optimal policy is to act with probability $a_A$ on arm A and $a_B$ on arm B as long as $a_A + a_B = 1$ .","W.l.o.g., suppose $a_A \\ge a_B$ ; then nature would set $p_A = 0$ and $p_B = 1$ , imposing regret at least $R/2$ .","Case 2: If $p_A > p_B$ , then the optimal policy would be to always act on arm A with probability $a_A = 1$ and never act on B ($a_B = 0$ ).","Nature would then set $p_A = 0$ and $p_B = 1$ to impose regret $R$ .","Case 3: If $p_A < p_B$ , the case is symmetric to Case 2 and result in regret $R$ .","Clearly, max regret is minimized when our action is such that $a_A + a_B = 1$ ; in this setting, we learn this optimal policy only under Case 1.","Following Case 2 or 3, the difference between our regret and the minimax regret is $R/2$ , which grows arbitrarily higher as $R \\rightarrow \\infty $ .","A slight modification to this problem renders Case 1 non-optimal.","Let the reward be $R$ when arm A is in a good state and $R-1$ for arm B, so the optimal policy learned under the assumption from Case 1 leads to $a_A = 1$ and $a_B = 0$ .","Then nature could respond with $p_A = 0$ and $p_B = 1$ , yielding reward 0 and regret $R-1$ , while the minimax regret–optimal policy achieves a minimum reward of $(R-1)/2$ (by playing $a_A = 0.5$ and $a_B = 0.5$ where nature responds with $p_A = 0$ and $p_B = 1$ ).","Thus, the gap again can grow arbitrarily high as $R \\rightarrow \\infty $ provided that $R > 1$ .","We therefore have that in all cases, any reward-maximizing policy can achieve arbitrarily bad performance in terms of regret."],["MA-RMABPPO: Nature Oracle Algorithm","We provide the pseudocode for MA-RMABPPO in Alg.",", which is used to implement the nature oracle.","[tb] MA-RMABPPO Input: Agent mixed strategy $\\tilde{\\pi }$ Parameters: n_epochs, n_subepochs, n_simsteps, n_sims [1] Initialize agent A: arm policies $\\pi _{n}^{(A)}~ \\forall n \\in [N]$ , $\\lambda $ -network $\\textsc {Lambda}$ , arm critic networks $\\phi _{n}^{(A)}~\\forall n \\in [N]$ Initialize agent B: nature parameter policy $\\pi ^{(B)}$ , critic network $\\phi ^{(B)}$ Initialize empty buffer $\\textit {epoch} = 1, 2, \\ldots , \\texttt {n\\_epochs}$ Sample $\\mathbf {s}$ at random Sample $\\lambda = \\textsc {Lambda}(\\mathbf {s})$ $\\textit {subepoch} = 1, \\ldots , \\texttt {n\\_subepochs}$ $t = 1, \\ldots , \\texttt {n\\_simsteps}$ Sample agent A action $a_n^{(A)} \\sim \\pi _{n}^{(A)}(s_n, \\lambda )$ for each $n \\in [N]$ Sample agent B action $\\omega ^{(B)} \\sim \\pi ^{(B)}$ ($\\pi ^{(B)}$ may take $\\mathbf {s}$ as input) $r^{(A)}, \\mathbf {s}^\\prime = \\textsc {simulate}(\\mathbf {s}, \\mathbf {a}^{(A)}, \\omega ^{(B)})$ $\\tilde{r} = \\textsc {simulate}(\\mathbf {s}, \\tilde{\\pi }(\\mathbf {s}), \\omega ^{(B)}, \\texttt {n\\_sims})$ (mean of 1-step rollouts of $\\tilde{\\pi }$ ) $r^{(B)} = r^{(B)} - \\tilde{r}$ (agent regret) Store $(\\mathbf {s}, \\mathbf {a}^{(A)}, \\omega ^{(B)}, r^{(A)}, r^{(B)}, \\mathbf {s}^\\prime )$ in buffer $\\mathbf {s} = \\mathbf {s}^\\prime $ Update $\\pi _n^{(A)}$ , $\\phi _n^{(A)}$ for all $n$ using trajectories in buffer.","$\\pi _n^{(A)}$ gets $\\omega ^{(B)}$ as part of state Update Lambda via discounted trajectories in buffer Update $\\pi ^{(B)}, \\phi ^{(B)}$ from trajectories in buffer.","$\\phi ^{(B)}$ gets $a^{(A)}$ as part of state return $\\pi ^{(B)}$"],["ARMMAN","The MDPs in the ARMMAN domain [5] have three ordered states representing the level of engagement of the beneficiaries in the previous week.","Rewards are better for lower states, i.e., $R(0)=1, R(1)=0.5, R(2)=0$ .","At each step, the beneficiary may only change by one level, e.g., low-to-medium or high-to-medium but not low-to-high.","They also assume that beneficiaries follow one of three typical patterns, A, B, and C, resulting in three MDPs with different transition probabilities.","There are two patterns of effects present that differentiate the beneficiary types.","(1) For each of the above types, the planner can only make a difference when the patient is in state 1.","Type A responds very positively to interventions, but regresses to low reward states in absence.","Type B has a similar but less amplified effect, and type C is likely to stay in state 1, but can be prevented from regressing to state 2 when an action is taken.","(2) Further, types A and C have only a 10% chance of staying in the high reward state, while type B has a 90% chance of staying there.","We converted these patient types to robust versions where the transition probabilities are uncertain as follows: $T^i_{s=0}=\\begin{bmatrix}p^i_{000} & 1 - p^i_{000} & 0.0 \\\\p^i_{010} & 1 - p^i_{010} & 0.0\\end{bmatrix}, \\hspace{5.69054pt}T^i_{s=1}=\\begin{bmatrix}0.0 & 1 - p^i_{102} & p^i_{102} \\\\p^i_{110} & 1 - p^i_{110} & 0.0\\end{bmatrix}, \\ $ $T^i_{s=2}=\\begin{bmatrix}0.0 & 1 - p^i_{202} & p^i_{202} \\\\0.0 & 1 - p^i_{212} & p^i_{212}\\end{bmatrix},$ where $i$ indexes the type (i.e., A, B or C).","We then set each $p^i_{sas^\\prime }$ to be in a range of width 0.5 centered on the entries from each of the A, B, C beneficiary types for $s\\in \\lbrace 1,2\\rbrace $ .","To add additional heterogeneity to the experiments, for $s=0$ , we set the range to 1.0 so that any beneficiary type can be made to have some non-negligible chance of staying in the good state, rather than only type B beneficiaries.","The full set of parameter ranges are given in the Table REF below.","Table: Upper and lower parameter ranges for the robust ARMMAN environment.In all experiments, 20% of arms were sampled from type A, 20% from type B and 60% for type C. To add additional heterogeneity, for each of the 50 random seeds we uniformly sample a sub-range contained within the ranges given in Table REF .","In the agent oracle experiments, for each of the 50 random seeds, since these require fully instantiated transition matrices, we uniformly sample each parameter value for each arm according to its type such that the values are contained in the ranges given in Table REF ."],["SIS Epidemic Model","In this domain, each arm follows its own compartmental SIS epidemic model.","Each arm's SIS model tracks whether each of $N_p$ members of a population is in a susceptible (S) or infectious (I) state.","This can be tracked with $N_p$ states, since it can be computed how many people are in state I if only the number of people in state S and the population size $N_p$ is known.","To define a discrete SIS model, we instantiate the model given in [42] section 4.1 with a $\\Delta t$ of 1.","We also augment the model to include action effects and rewards.","Specifically, $R(N_S) = N_S/N_p$ , where $N_S$ is the number of susceptible (non-infected) people.","Further, there are three actions $\\lbrace a_0, a_1, a_2\\rbrace $ with costs $c=\\lbrace 0, 1, 2\\rbrace $ .","Action $a_0$ represents no action, $a_1$ divides the contacts per day $\\lambda _{\\textit {c}}$ ($\\lambda $ in [42]) by $a^{\\textit {eff}}_1$ , and $a_2$ divides the infectiousness $r_{\\textit {infect}}$ ($r(t)$ in [42]) by $a^{\\textit {eff}}_2$ .","That is, taking action $a_1$ will reduce the average number of contacts per day in a given arm, and taking action $a_2$ will reduce the probability of infection given contact in a given arm, thus reducing the expected number of people that will become infected in the next round.","However, to make this a robust problem, the relative effect sizes of each action for each arm will not be known to the planner, nor will the $\\lambda _\\textit {c}$ or $r_{\\textit {infect}}$ .","We impose the following uncertainty intervals for all arms: $\\lambda _{\\textit {c}} \\in [1, 10]$ , $r_{\\textit {infect}} \\in [0.5, 0.99]$ , $a^{\\textit {eff}}_1 \\in [1, 10]$ , and $a^{\\textit {eff}}_2 \\in [1, 10]$ .","In the robust double oracle experiments, to add additional heterogeneity, for each of the 50 random seeds we uniformly sample a sub-range contained within the ranges given above for each arm.","In the agent oracle experiments, for each of the 50 random seeds, since these require fully instantiated transition matrices, we uniformly sample each parameter value for each arm such that the values are contained in the ranges given above."],["Hyperparameter Settings and Implementation Details","Neural networks: All neural networks in experiments are implemented using PyTorch 1.3.1 [32] with 2 fully connected layers each with 16 units and tanh activation functions, and a final layer of appropriate size for the relevant output dimension with an identity activation function.","The output of discrete actor networks (i.e., the policy network from the agent oracle, and the policy network of agent A in the nature oracle) pass through a categorical distribution from which actions are randomly sampled at training time, without a budget imposed.","It is critical not to impose the budget at training time, so that the budget spent by the optimal policy under a given $\\lambda $ will result in a meaningful gradient for updating the $\\lambda $ -network.","The output of continuous actor networks (i.e., agent B in the nature oracle which selects environment parameter settings) instead are passed as the means of Gaussian distributions – with the log standard deviations learned as individual parameters separate from the network – from which continuous actions are sampled at training time.","At test time, actions are sampled from both types of networks deterministically.","For categorical distributions, we greedily select the highest probability actions.","For Gaussian distributions, we act according to the means.","All discount factors were set to 0.9.","The remaining hyperparameters that were constant for all experiments for the agent and nature oracles are indicated in Table REF .","For Robust Double Oracle experiments, all agent and nature oracles were run for 100 training epochs.","For Agent Oracle experiments, RMABPPO was run for 100 training epochs for the synthetic and ARMMAN domains and 200 epochs for the SIS domain.","$\\lambda $ -network: Critical to training the $\\lambda $ -network is cyclical control of the temperature parameter that weights the entropy term in the actor loss functions.","Recall that the $\\lambda $ -network is only updated every n_subepochs.","In general, after each update to the $\\lambda $ -network, we want to encourage exploration so that actor networks explore the new part of the state space defined by updated predictions of $\\lambda $ .","However, after n_subepochs rounds, we will use the cost of the sampled actor policies as a gradient for updating the $\\lambda $ -network, and that gradient will only be accurate if the actor policy has converged to the optimal policies for the given $\\lambda $ predictions.","Therefore, we also want to have little or no exploration in the round before we update the $\\lambda $ -network.","In general, we would also like the entropy of the policy network to reduce over time so that the actor networks and $\\lambda $ -networks eventually both converge.","To accomplish both of these tasks, the weight (temperature) of the entropy regularization term in the loss function of the actor network will decay/reset according to two processes.","The first process will linearly decay the temperature from some positive, but time-decaying starting value (see next process) $\\tau _t$ immediately after each $\\lambda $ -network update, down to 0 after $\\texttt {n\\_subepochs}$ .","The second process will linearly decay the temperature from a maximum $\\tau _0$ (start entropy coeff in Table REF ) down to $\\tau _{\\min }$ (end entropy coeff in Table REF ) by the end of training.","We found that it also helps to train the actor network with no entropy and with the $\\lambda $ -network frozen for some number of rounds before training is stopped (lambda freeze epochs in Table REF ).","Double Oracle: In all experiments in the main text, we initialize the agent strategy list with HO, HM, and HP, and the nature strategy list with pessimistic, mean, and optimistic nature strategies, then run RMABDO for 5 iterations.","This produces a set of 8 agent strategies, 8 nature strategies, a table where each entry represents the regret of each agent pure strategy (row) against each nature pure strategy (column), and an optimal mixed strategy over each set that represents a Nash equilibrium of the minimax regret game given in the table.","The regret table is computed by first computing the returns of each agent/nature pure strategy combination, then subtracting the max value of each column from all entries in that column (i.e., the best agent strategy for a given nature strategy gets 0 regret).","The regret of RMABDO is reported as the expected utility corresponding to the Nash equilibrium of the regret game given by the table, once that regret table is normalized to account for the returns of baselines (see next paragraph).","After this main loop completes, we then compute the regret of the baselines by evaluating each baseline policy against each pure strategy in the nature strategy list.","Then, we also run the nature oracle against each baseline policy to find a nature strategy that should maximize the regret of that baseline.","The regret for each baseline is reported as the max regret against this new nature strategy, as well as all pure nature strategies from the main RMABDO loop.","Hawkins Baselines: The Hawkins policies are implemented with gurobipy 9.1.2, a Python wrapper for Gurobi (9.0.3) [14] following the LP given in [15] equation 2.5 to compute $\\lambda $ and $Q(s,a,\\lambda )$ for each arm and the integer program in equation 2.12 to select actions.","RLvMid Baseline: We found that RLvMid found effective policies for the nature strategy it was trained against (as evidenced in Figure REF ), but that that learned policy could be brittle against other nature strategies.","This is likely because different nature strategies produce different distributions of states, meaning RLvMid would fit policies well to states seen when planning against the mean nature strategy, but underfit its policies for states seen more often in different distributions.","However, the lone RLvMid baseline policy can somewhat correct for this effect by training an ensemble of policies against slight perturbations of the mean nature strategy that adjust the parameter values output by nature by a small $\\epsilon $ .","In all experiments we train 3 RLvMid policies against 3 random perturbations of the mean nature strategy, then report the regret of RLvMid as the minimum of the max regrets returned by any of the 3.","Table: Hyperparameter settings for agent and nature oracles for all experiments."],["Additional Experimental Results","Fig.","REF shows regret results from additional robust experiments which scale up the number of arms for Synthetic (top left), ARMMAN (top right) and SIS (bottom left; $N_p=50$ ), as well as and the size of the state space of SIS (bottom right; $N=5, B=4$ ).","RMABDO continues to outperform all baselines by a large margin.","When the size of the state space is scaled up for SIS, it becomes infeasible to run the Hawkins baselines due to its long query time which grows quadratically with the size of the state space.","Because of this, we exclude the Hawkins baselines from the main double oracle loop in these experiments.","Further, Hawkins cannot even be evaluated as a baseline as the state space increases since to compute its maximum regret, we must get one best response from the nature oracle against the baseline, which requires querying the Hawkins baseline policies tens of thousands of times, which is prohibitive when the query time takes even ~$1s$ to run.","Figure: Additional experiments showing maximum policy regret in robust setting for Synthetic (top left), ARMMAN (top right) and SIS (bottom) domains, respectively.","Synthetic is scaled by 3 and ARMMAN by 5 to maintain the distributions of arm types specified in Section .","RMABDO beats all baselines by a large margin across various parameter settings.","When the state space is scaled up (bottom right) Hawkins baselines become infeasible to run due to its long query time (see Section for a discussion), even for a small number of arms (N=5,B=4N = 5, B = 4)."]],"string":"[\n [\n \"Restless and Uncertain: Robust Policies for Restless Bandits via Deep\\n Multi-Agent Reinforcement Learning\"\n ],\n [\n \"Abstract We introduce robustness in \\\\textit{restless multi-armed bandits} (RMABs), a popular model for constrained resource allocation among independent stochastic processes (arms).\",\n \"Nearly all RMAB techniques assume stochastic dynamics are precisely known.\",\n \"However, in many real-world settings, dynamics are estimated with significant \\\\emph{uncertainty}, e.g., via historical data, which can lead to bad outcomes if ignored.\",\n \"To address this, we develop an algorithm to compute minimax regret -- robust policies for RMABs.\",\n \"Our approach uses a double oracle framework (oracles for \\\\textit{agent} and \\\\textit{nature}), which is often used for single-process robust planning but requires significant new techniques to accommodate the combinatorial nature of RMABs.\",\n \"Specifically, we design a deep reinforcement learning (RL) algorithm, DDLPO, which tackles the combinatorial challenge by learning an auxiliary \\\"$\\\\lambda$-network\\\" in tandem with policy networks per arm, greatly reducing sample complexity, with guarantees on convergence.\",\n \"DDLPO, of general interest, implements our reward-maximizing agent oracle.\",\n \"We then tackle the challenging regret-maximizing nature oracle, a non-stationary RL challenge, by formulating it as a multi-agent RL problem between a policy optimizer and adversarial nature.\",\n \"This formulation is of general interest -- we solve it for RMABs by creating a multi-agent extension of DDLPO with a shared critic.\",\n \"We show our approaches work well in three experimental domains.\"\n ],\n [\n \"Introduction\",\n \"The restless multi-armed bandit (RMAB) problem models sequential decision making tasks under a limited resource constraint.\",\n \"An RMAB instance consists of a budget $B$ and a set of $N$ arms, where an arm corresponds to an action-dependent Markov state-transition process.\",\n \"At every timestep, a planner selects a subset of $N$ arms and decides what actions to take on each selected arm such that the total cost of actions per timestep is less than a budget $B$ .\",\n \"The arms transition to new states and generate rewards, depending on the action taken on them.\",\n \"The goal is to plan a policy that maximizes the total expected reward.\",\n \"This problem naturally captures a wide range of real-world problems.\",\n \"For example, in a healthcare intervention problem [26], a health worker (planner) is in charge of the well-being of a set of patients (arms) who have enrolled in a health program, e.g., by ensuring adherence to daily tuberculosis medication.\",\n \"The healthcare worker provides timely interventions to the patients via phone call or in-person visit to ensure that the patients remain in the adhering state.\",\n \"However, phone calls and in-person visits require dedicated time from the health worker, thereby imposing a budget $B$ on the number of patients she can call or visit in-person each day.\",\n \"The problem is challenging because of three key reasons: (1) the state of each patient representing the extent of adherence evolves over time; (2) the intervention decisions at each timestep influence the state transitions of all patients and, consequently, affect future rewards; and (3) different actions incur different costs, and the total cost of allocation must be at most $B$ at every timestep.\",\n \"The combinatorial nature of the problem makes it computationally expensive to solve optimally [31].\",\n \"Prior works have considered specific variants of the RMAB problem that are applicable to real-world scenarios, including healthcare intervention planning [23], [26], anti-poaching patrol planning [34], sensor monitoring tasks [11], [17], machine replacement [35], and many more.\",\n \"A common assumption across existing literature on RMABs is that the state transition dynamics of all the arms are known beforehand.\",\n \"Unfortunately, in real-world scenarios, even in the presence of prior data, there is still often significant uncertainty in transition dynamics.\",\n \"A natural way to encode this uncertainty is interval uncertainty—rather than assuming that the probability of transitioning from one state to another is fixed to one value, we assume that it can take any value from a given closed continuous interval.\",\n \"This additional uncertainty makes the RMAB problem even more challenging since any solution must be robust to variations in transition probabilities within the given interval.\",\n \"In the presence of interval uncertainty, we formulate a generalization of RMAB that we call Robust Restless Bandits.\",\n \"Our goal is to achieve minimax regret—minimize the worst-case regret against the optimal policy over all possible values of transition probabilities within the given uncertainty intervals.\",\n \"Our contributions are as follows: We introduce the Robust RMAB problem for interval uncertainty settings and optimize for minimax regret.\",\n \"We are the first to provide a minimax-regret solution for the restless bandits problem, while also generalizing to multi-action RMABs.\",\n \"We develop a double oracle algorithm for solving Robust RMABs and demonstrate its effectiveness on experimental domains, including two inspired by real-world problems.\",\n \"To enable our double oracle approach, we propose RMABPPO, a deep reinforcement learning (RL) algorithm for solving multi-action RMABs.\",\n \"RMABPPO hinges on learning an auxiliary “$\\\\lambda $ -network” allowing us to decouple each arm's learning, greatly reducing sample complexity.\",\n \"Under minimax regret, the adversary in the double oracle approach is notoriously difficult to implement due to non-stationarity.\",\n \"To address this, we formulate the adversary oracle as a multi-agent reinforcement learning problem and solve it with a multi-agent extension of RMABPPO.\"\n ],\n [\n \"Restless bandits\",\n \"[40] introduced RMABs, following which, there is a vast literature on binary-action RMABs.\",\n \"In the classic setting, a planner selects $k$ out of $N$ state-transitioning arms, resulting in a two-action decision problem with the goal of maximizing reward.\",\n \"Although the problem is PSPACE-Hard [31], heuristics based on Lagrangian relaxation of the optimization have been proposed [40] which turns out to be optimal under the indexability criterion.\",\n \"However, the Whittle index solution does not extend to budgeted multi-action settings, where there are more than two actions, each with an associated cost.\",\n \"[12] and [16] have extended index-based solutions to multi-action RMABs for instances with special monotonic structure.\",\n \"Along similar lines, [20] proposed a method that leverages the convexity of an approximate Lagrangian version of the multi-action RMAB problem.\",\n \"Another relevant category of research is taking decisions on weakly coupled Markov decision processes (WCMDP).\",\n \"[15] proposed a Lagrangian relaxation of the general WCMDP problem and proposed an LP for minimizing the Lagrange bound.\",\n \"[3] and [13] provided less scalable solutions that provide better approximations to WCMDPs.\",\n \"All these papers have assumed perfect knowledge of the state transition dynamics to compute policies.\",\n \"A few recent works have studied RMABs in online settings with unknown transition probabilities but make the limiting assumptions that the planner takes only one non-combinatorial action [8] or that states periodically reset [18].\",\n \"None of the above papers consider robust planning under environment uncertainty, which we address.\",\n \"Despite the recent successes of reinforcement learning (RL) for solving large-scale games [28], [37], RL has so far seen little application to RMABs, except for a few recent works that learn Whittle indices for indexable binary-action RMABs using (i) deep RL [29] and (ii) Q-learning when states are observable [5], [7] or when arms are homogeneous [4].\",\n \"In contrast, our deep RL approach provides a more general solution to binary and multi-action RMAB domains that performs well regardless of indexability.\",\n \"One recent work does also address learning in the multi-action setting [19], but their approach is based on tabular Q-learning which can be sample inefficient and scale poorly, unlike function-approximation methods like ours.\"\n ],\n [\n \"Robust planning\",\n \"The RL literature has a large body of work on robust planning, mainly focused on maximizing the minimum (maximin) reward through robust adversarial RL [33] or multi-agent RL settings [22], [24].\",\n \"The minimax regret criterion [6] has been offered as a compelling alternative to maximin reward, which often leads to overly conservative policies that buffer against worst-possible realizations.\",\n \"A challenge to computing minimax regret–optimal strategies is that such games often involve very large and sometimes continuous strategy spaces, rendering it impossible to explore the entire strategy space.\",\n \"The double oracle approach [27] has been applied as an effective means by which to optimally explore only a small subset of these large spaces while still guaranteeing optimal performance [30], [10].\",\n \"Double oracle has been extended to optimize multi-agent RL problems with multiple selfish agents [22].\",\n \"More recently, [41] offered an algorithm for minimax regret planning for a non-RMAB problem using RL to instantiate the two oracles.\",\n \"However, their approach cannot handle budget-constrained discrete actions, which we have in the RMAB setting.\",\n \"Additionally, we formulate the nature oracle as a multi-agent RL problem, allowing us to handle the joint discrete and continuous action spaces that arise in the minimax regret nature oracle.\",\n \"Robustness objectives have been considered for bandit applications in the two-action stochastic setting, where each pull of an arm draws a reward sampled from an unknown Bernoulli distribution.\",\n \"[39] address minimax regret of time-varying reward shifts using heuristics to trade off remembering vs. forgetting, and [9] consider a maximin objective to guide exploration in Monte Carlo Tree Search.\",\n \"However, RMABs are significantly harder than the stochastic settings since, in RMABs, the rewards are dependent on the current state which in turn depends on the actions taken on the arms.\"\n ],\n [\n \"Problem Statement\",\n \"We consider the multi-action RMAB setting with $N$ arms.\",\n \"Each arm $n\\\\in [N]$ follows a Markov decision process (MDP) $(\\\\mathcal {S}_n, \\\\mathcal {A}_n, \\\\mathcal {C}_n, T_n, R_n, \\\\beta )$ , where $\\\\mathcal {S}_n$ is a set of finite, discrete states, $\\\\mathcal {A}_n$ is a set of finite, discrete actions, $\\\\mathcal {C}_n : \\\\mathcal {A}_n \\\\xrightarrow{} \\\\mathbb {R}$ corresponds to action costs, $T_n: \\\\mathcal {S}_n\\\\times \\\\mathcal {A}_n \\\\times \\\\mathcal {S}_n \\\\xrightarrow{} [0,1]$ gives the probability of transitioning from one state to another given an action, $R_n:\\\\mathcal {S}_n\\\\times \\\\mathcal {A}_n \\\\times \\\\mathcal {S}_n \\\\xrightarrow{} \\\\mathbb {R}$ is a reward function, and $\\\\beta \\\\in [0, 1)$ is the discount parameter.\",\n \"The agent must select actions for each arm, each round, such that the sum cost of actions does not exceed a per-round budget $B$ .\",\n \"The aim of multi-action RMABs is to maximize total reward over a fixed number of $H$ rounds, subject to this budget constraint, generalizing the well-studied binary-action RMAB.\",\n \"In this work, we extend multi-action RMABs to the robust setting in which the exact transition probabilities are unknown.\",\n \"Instead, the transitions of each arm are determined by a set of parameters $\\\\mathcal {P}_n$ .\",\n \"For a given arm $n \\\\in [N]$ and parameter $p_n \\\\in \\\\mathcal {P}_n$ , let $\\\\langle {\\\\omega _{n,p_n}}\\\\rangle :=[\\\\underline{\\\\omega }_{n,p_n}, \\\\overline{\\\\omega }_{n,p_n}]$ represent the range, i.e., uncertainty over transition probabilities.\",\n \"We ask the following question: how to find a solution to the multi-action RMAB problem that is robust to such uncertainties?\",\n \"We consider an objective that incorporates our uncertainty over $\\\\mathcal {P}_n$ .\",\n \"Let $\\\\omega $ be a given realization of the transition probabilities such that $\\\\omega _{n,p_n} \\\\in \\\\langle {\\\\omega _{n,p_n}}\\\\rangle $ for all $n \\\\in [N]$ and $p_n \\\\in \\\\mathcal {P}_n$ .\",\n \"Let $G(\\\\pi ,\\\\omega )$ be the planner's expected reward under policy $\\\\pi $ and a realization $\\\\omega $ of the uncertainty.\",\n \"Regret is defined: $L(\\\\pi ,\\\\omega ) = G(\\\\pi ^*_{\\\\omega },\\\\omega ) - G(\\\\pi ,\\\\omega ) \\\\ ,$ where $\\\\pi ^*_{\\\\omega }$ is the optimal policy under $\\\\omega $ .\",\n \"In our robust planning formulation, our objective is to compute a policy $\\\\pi $ that minimizes the maximum regret $L$ possible for any realization of $\\\\omega $ , leading to the following minimax objective: $\\\\min _{\\\\pi }\\\\max _{{\\\\omega }}{L(\\\\pi ,{\\\\omega })} \\\\ .$ This problem is computationally expensive to solve since simply computing a policy $\\\\pi $ that maximizes the expected reward $G(\\\\pi ,\\\\omega )$ is PSPACE-Hard [31] even for the special case when the transition rules are known i.e., $\\\\omega $ is given.\",\n \"This challenge is likely a key reason why the robust formulation has not yet been addressed in the literature.\",\n \"To overcome the complexity of the minimax optimization, we take a double oracle approach [27], which requires key innovations to work in the RMAB setting.\"\n ],\n [\n \"Preliminaries\",\n \"The double oracle approach achieves the minimax regret objective in Eq.\",\n \"REF ) by casting the optimization problem as a zero-sum game between two players, the agent and nature [41].\",\n \"The agent selects a policy that minimizes regret for some realization of the transition probabilities.\",\n \"Nature then adversarially selects the values of $\\\\mathcal {P}_n$ that maximize regret for a given policy of the planner.\",\n \"This framework is desirable since it converges to an $\\\\varepsilon $ –optimal solution [2], [41], assuming there are oracles that return the best response for both players.\",\n \"The key technical contributions of this paper arise from the design of the agent and nature oracles for best response computations.\",\n \"For the agent, minimizing regret with respect to a fixed nature strategy is equivalent to maximizing reward w.r.t.\",\n \"that strategy, so the agent objective is the same as solving a multi-action RMAB to find the best policy.\",\n \"A policy $\\\\pi $ maps states to decision matrices $\\\\mathbf {A} \\\\in \\\\lbrace 0,1\\\\rbrace ^{N\\\\times |\\\\mathcal {A}|}$ where the total number of active actions is constrained by a budget for each round.\",\n \"Let $\\\\mathbf {s} = (s^1, \\\\ldots , s^{N})$ represent the initial state of each arm.\",\n \"Then, for a given parameter $\\\\omega $ , the optimal policy $\\\\pi ^*_{\\\\omega }$ maximizes the expected discounted sum of rewards of all arms as given by the constrained Bellman equation: $\\\\begin{aligned}J(\\\\mathbf {s}) &= \\\\max _{\\\\mathbf {A}}\\\\left\\\\lbrace \\\\sum _{n=1}^{N} R_n(s_n) + \\\\beta \\\\mathbb {E}_{\\\\omega }[J(\\\\mathbf {s}^\\\\prime ) \\\\mid \\\\mathbf {s}, \\\\mathbf {A}]\\\\right\\\\rbrace \\\\\\\\\\\\text{s.t. }\",\n \"& \\\\sum _{j=1}^{|\\\\mathcal {A}|} a_{nj} = 1 \\\\hspace{8.53581pt} \\\\forall n \\\\in [N]\\\\qquad \\\\sum _{n=1}^{N}\\\\sum _{j=1}^{|\\\\mathcal {A}|} a_{nj}c_{nj} \\\\le B\\\\end{aligned}$ where $a_{nj} \\\\in \\\\mathbf {A}_n$ and $c_{nj}$ are the corresponding action costs in $\\\\mathcal {C}_n$ .\",\n \"We then relax the problem by taking the Lagrangian relaxation of the budget constraint [15], giving: $\\\\begin{aligned}&J(\\\\mathbf {s}, \\\\lambda ^*) = \\\\min _{\\\\lambda } \\\\left( \\\\frac{\\\\lambda B}{1-\\\\beta } + \\\\sum _{n=1}^{N}\\\\max _{a_{nj}\\\\in \\\\mathcal {A}_n}Q_n(s_n, a_{nj}, \\\\lambda ) \\\\right)\\\\end{aligned}$ $\\\\begin{aligned}\\\\text{where }\\\\hspace{2.84526pt} Q_n(s_n, a_{nj}, \\\\lambda ) =R_n(s_n) - \\\\lambda c_{nj} + \\\\beta \\\\mathbb {E}_{\\\\omega } \\\\left[ Q_n(s^{\\\\prime }, a_{nj}, \\\\lambda ) \\\\mid \\\\pi ^{La}_{\\\\omega }(\\\\lambda ) \\\\right] \\\\ .\\\\end{aligned}$ Here, $Q$ is the value function and $\\\\pi ^{La}_{\\\\omega }(\\\\lambda )$ is the optimal policy for a given $\\\\lambda $ .\",\n \"See [3] for a detailed derivation.\",\n \"Note that for a given value of $\\\\lambda $ , Eq.\",\n \"REF could be solved using $N$ individual value iterations.\",\n \"However, setting $\\\\lambda :=$ $\\\\lambda ^*$ is critical to finding good policies for a multi-action RMAB and is known to be asymptotically optimal in the binary-action case [38], i.e., $\\\\pi ^{La}_{\\\\omega }(\\\\lambda ^*) \\\\xrightarrow{} \\\\pi ^{*}_{\\\\omega }$ .\",\n \"Given this relationship, in the remainder of the paper, we focus of computing $\\\\pi ^{La}_{\\\\omega }(\\\\lambda ^*)$ and denote it as $\\\\pi ^{*}_{\\\\omega }$ for convenience.\"\n ],\n [\n \"Solving Robust Restless Bandits\",\n \"We now build up our approach for finding robust RMAB policies.\",\n \"The underlying pure strategy space for the agent is the set of all feasible policies $\\\\pi : \\\\mathcal {S}_1\\\\times \\\\cdots \\\\times \\\\mathcal {S}_N \\\\mapsto \\\\mathcal {A}_1\\\\times \\\\cdots \\\\times \\\\mathcal {A}_N$ .\",\n \"The pure strategy space for nature is a closed set of parameters $\\\\omega $ within the given uncertainty intervals.\",\n \"The agent oracle's goal is to find a policy $\\\\pi $ , or pure strategy, to minimize regret (Eq.\",\n \"REF ) given a mixed strategy $\\\\tilde{\\\\omega }$ , where a mixed strategy is a probability distribution over a set of pure strategies.\",\n \"That is, the agent minimizes $L(\\\\pi ,\\\\tilde{\\\\omega })$ w.r.t.\",\n \"$\\\\pi $ , while $\\\\tilde{\\\\omega }$ is constant.\",\n \"Recall that $L(\\\\pi ,\\\\tilde{\\\\omega }) = G(\\\\pi ^{*}_{\\\\tilde{\\\\omega }},\\\\tilde{\\\\omega }) - G(\\\\pi ,\\\\tilde{\\\\omega })$ .\",\n \"Since the first term $G(\\\\pi ^{*}_{\\\\tilde{\\\\omega }},\\\\tilde{\\\\omega })$ is constant, minimizing $L(\\\\pi ,\\\\tilde{\\\\omega })$ is equivalent to maximizing the second term $G(\\\\pi ,\\\\tilde{\\\\omega })$ , which is maximal at $\\\\pi =\\\\pi ^{*}_{\\\\tilde{\\\\omega }}$ .\",\n \"In other words, the agent oracle must compute an optimal reward-maximizing policy w.r.t.\",\n \"$\\\\tilde{\\\\omega }$ .\",\n \"The objective of maximizing reward is standard in reinforcement learning, enabling us to build off existing RL techniques as we show in Section REF .\",\n \"On the other hand, the nature oracle's goal is to find a parameter setting $\\\\omega $ , or pure strategy, that maximizes the agent's regret given a mixed strategy $\\\\tilde{\\\\pi }$ , i.e., maximize $L(\\\\tilde{\\\\pi },\\\\omega )$ with respect to $\\\\omega $ , while $\\\\tilde{\\\\pi }$ is fixed.\",\n \"This objective is far more challenging because both $G(\\\\tilde{\\\\pi }^*_{\\\\omega },\\\\omega )$ and $G(\\\\tilde{\\\\pi },\\\\omega )$ are functions of $\\\\omega $ .\",\n \"Moreover, computing $G(\\\\tilde{\\\\pi }^*_{\\\\omega },\\\\omega )$ requires obtaining an optimal policy $\\\\tilde{\\\\pi }^*_{\\\\omega }$ as $\\\\omega $ changes in the optimization.\",\n \"Unfortunately, $\\\\omega $ comes from an infinite continuous strategy space, making this problem difficult.\",\n \"However, as one of our main contributions, we propose a novel method for implementing the regret-maximizing nature oracle by casting it as a multi-agent reinforcement learning problem, simultaneously solving for policies $\\\\pi ^*_{\\\\omega }$ while computing worst-case parameters $\\\\omega $ to maximize $L(\\\\tilde{\\\\pi },\\\\omega )$ .\",\n \"Solving a multi-agent reinforcement learning problem first requires an RL algorithm to optimize the underlying policy; hence we first introduce a novel RL approach, RMABPPO, to solve RMABs (Sec.\",\n \"REF ) as a part of our agent oracle and then use the algorithm as the backbone of our nature oracle (Sec.\",\n \"REF ).\"\n ],\n [\n \"Agent Oracle: Deep RL for RMABs via RMABPPO\",\n \"Existing deep RL approaches can be applied to the objective in Eq.\",\n \"REF , but they fail to scale past trivially sized RMAB problems since the action and state spaces grow exponentially in $N$ .\",\n \"For example, for a binary-action RMAB with $N=50$ and $B=20$ , the action space would be of size $\\\\binom{50}{20}\\\\approx 10^{12}$ , which is not feasible to learn, even with a neural network.\",\n \"To overcome this, we develop a novel deep RL algorithm that instead solves the decoupled problem (Eq.\",\n \"REF ).\",\n \"The key benefit of decoupling is to render policies and $Q$ values of each arm independent, allowing us to learn $N$ independent networks with linearly sized state and action spaces, relieving the combinatorial burden of the learning problem — the above example would now only have $N \\\\times 2 = 100$ actions.\",\n \"However, this approach introduces a new technical challenge in solving the dual objective which maximizes over policies but minimizes over $\\\\lambda $ , as discussed in Sec. .\",\n \"We derive a dual gradient update procedure that iteratively optimizes each objective as follows: (1) holding $\\\\lambda $ constant, learn $N$ independent policies via a policy gradient procedure, augmenting the state space to include $\\\\lambda $ as input, as in Eq.\",\n \"REF ; (2) use sampled trajectories from those learned policies as an estimate to update $\\\\lambda $ towards its minimizing value via a novel gradient update rule.\",\n \"Another challenge is that $\\\\lambda ^*$ of Eq.\",\n \"REF depends on the current state of each arm — therefore, a key element of our approach is to learn this function concurrently with our iterative optimization, using a neural network we call the $\\\\lambda $ -network that is parameterized by $\\\\Lambda $ .\",\n \"To train the $\\\\lambda $ -network, we use the following gradient update rule.\",\n \"[]propositionlambdaUpdate A gradient rule for updating the $\\\\lambda $ -network, parameterized by $\\\\Lambda $ , such that for a state $\\\\mathbf {s}$ , the $\\\\lambda $ -network predicts the value $\\\\lambda $ that minimizes Eq.\",\n \"REF is as follows: $\\\\begin{aligned}\\\\Lambda _t = \\\\Lambda _{t-1} - \\\\alpha \\\\left( \\\\frac{B}{1-\\\\beta } + \\\\sum _{n=1}^{N}D_n(s_n, \\\\lambda _{t-1}(\\\\mathbf {s})) \\\\right)\\\\end{aligned}$ where $\\\\alpha $ is the learning rate and $D_n(s_n, \\\\lambda )$ is the negative of the expected $\\\\beta $ -discounted sum of action costs for arm $n$ starting at state $s_n$ under the optimal policy for arm $n$ for a given value of $\\\\lambda $ .\",\n \"Although $D_n$ cannot be computed exactly as we do not know the optimal policy, it can be estimated from samples of multiple rollouts of the policy during training.\",\n \"As long as arm policies are trained for adequate time on the given value of $\\\\lambda $ , the gradient estimate will be accurate, i.e., $D_n(s_n, \\\\lambda _{t-1}(\\\\mathbf {s})) \\\\approx -\\\\sum _{k=0}^{K-1} \\\\beta ^k c_{nk}$ where $K$ is the number of samples collected in an epoch and $c_{nk}$ is the action cost of arm $n$ in round $k$ .\",\n \"Moreover, this procedure will converge to the optimal parameters $\\\\Lambda $ if the arm policies are optimal.\",\n \"[]propositionlambdaConvergence Given arm policies corresponding to optimal $Q$ -functions, the gradient update rule of Prop.\",\n \"REF will lead $\\\\Lambda $ to converge to the optimal as the number of training epochs and $K\\\\xrightarrow{}\\\\infty $ .\",\n \"Note that to collect samples that reflect the proper gradient, the RMAB budget must not be imposed at training time — rather the policy networks and $\\\\lambda $ -network must learn to play the Lagrange policy of Eq.\",\n \"REF which spends the correct budget in expectation.\",\n \"At training time, actions are sampled randomly according to the actor network distributions.\",\n \"At test time, actions are taken deterministically by greedily selecting the highest probability actions until the budget is spent.\",\n \"In theory, the policy networks could be trained via any deep RL procedure, as long as the above characteristics for training the $\\\\lambda $ -network are ensured.\",\n \"In practice, we train with proximal policy optimization (PPO) [36], which has been demonstrated to have state-of-the-art performance while being relatively simple to implement, and will allow flexibility to handle both discrete and continuous actions—the latter will be important for the nature oracle.\",\n \"We also navigate the important trade-off between exploring new policy actions after an update to $\\\\Lambda $ and allowing the policy networks to converge before updating $\\\\Lambda $ , by using an entropy regularization term on the loss function of the policy networks for each arm that is controlled by a cyclical temperature parameter.\",\n \"We call our algorithm RMABPPO, give pseudocode in Algorithm REF , and give more implementation details in the appendix.\"\n ],\n [\n \"Nature Oracle: Multi-Agent RL for RMABs via MA-RMABPPO\",\n \"Armed with a deep RL procedure for learning RMAB policies, we are now ready to create the multi-agent extension needed to implement the nature oracle.\",\n \"Recall that the challenge of the nature oracle is to simultaneously optimize a policy as well as the environment parameters $\\\\omega $ .\",\n \"We propose to treat this optimization as a multi-agent RL problem designed to handle this form of non-stationarity [25] via a centralized critic.\",\n \"To implement the nature oracle, we introduce two agents $A$ and $B$ , where $A$ 's goal is to optimize the RMAB policy $\\\\pi ^*_{\\\\tilde{\\\\omega }}$ and $B$ 's goal is to find parameters $\\\\omega $ that maximize regret of the current agent mixed strategy $\\\\tilde{\\\\pi }$ .\",\n \"The shared environment transition function is $T: \\\\mathcal {S} \\\\times \\\\mathcal {A}_A \\\\times \\\\mathcal {A}_B \\\\xrightarrow{} \\\\mathcal {S}$ .\",\n \"The action space $\\\\mathcal {A}_A$ will be the same as the action space of multi-action RMAB.\",\n \"At a given state $\\\\mathbf {s}$ , the action space $\\\\mathcal {A}_B$ will allow agent $B$ to select $\\\\omega $ which, in general, depends on $\\\\mathbf {s}$ .\",\n \"That is, at each step, agent $B$ will select environment parameters $\\\\omega $ , and thus state/action transition probabilities that will determine the outcome of agent $A$ 's actions.\",\n \"We adopt the centralized critic idea from multi-agent PPO [43] to our RMAB setting to create MA-RMABPPO.\",\n \"Since the policy space of agent $A$ is discrete while that of agent $B$ is continuous, a notable benefit to PPO is that it offers a convenient way to train both policies.\",\n \"The final step is to define the rewards for each agent $A$ and $B$ to match their respective objectives.\",\n \"Since agent $A$ 's objective is to maximize $\\\\pi ^*_{\\\\tilde{\\\\omega }}$ , it simply adopts the reward defined by the RMAB, i.e., ${R}^{(A)} = \\\\sum _{n=1}^N R_n$ .\",\n \"However, agent $B$ 's objective is to learn the regret-maximizing parameters $\\\\omega $ .\",\n \"This objective is challenging because it requires computing and optimizing over the returns of the fixed input policy $\\\\tilde{\\\\pi }$ with respect to all possible $\\\\omega $ values, which in general is non-convex.\",\n \"In practice, to estimate the returns of $\\\\tilde{\\\\pi }$ , we execute a series of single-step roll-outs for computational efficiency.\",\n \"That is, given $\\\\mathbf {s}$ and $\\\\mathbf {a}$ at a given round, we sample the next state profile $\\\\mathbf {s^\\\\prime }$ , and define the reward of agent $B$ (i.e., the regret of input policy $\\\\tilde{\\\\pi }$ ) as ${R}^{(B)} = \\\\sum _{n=1}^N R_n(s_n,a_n,s^\\\\prime _n) - \\\\frac{1}{Y}\\\\sum _{y=1}^{Y}r_y^{\\\\tilde{\\\\pi }}$ , where $r_y^{\\\\tilde{\\\\pi }}$ is the reward obtained from each of $Y$ 1-step Monte Carlo simulations of the mixed strategy $\\\\tilde{\\\\pi }$ .\",\n \"Since agent $A$ has the same policy network architecture as RMABPPO, i.e., N discrete policy networks and one $\\\\lambda $ -network.\",\n \"The agent $B$ actor network is a single continuous-action policy network.\",\n \"Since agent $A$ and $B$ have separate objectives, they each have their own centralized critic networks which take the actions of the other as inputs to their state space.\",\n \"For agent $A$ , this is implemented as one critic network per arm, where each network is augmented with agent $B$ 's actions.\",\n \"Finally, to ensure good gradient estimates for the $\\\\lambda $ -network in MA-RMABPPO, we keep agent $B$ 's network — and thus the environment — constant between $\\\\lambda $ updates, updating $B$ 's network at the same frequency as the $\\\\lambda $ -network updates.\",\n \"Pseudocode for MA-RMABPPO and further details of its implementation are given in the appendix.\"\n ],\n [\n \"The Minimax Regret–Robust RMAB Double Oracle\",\n \"We now have all the pieces we need to present our robust algorithm RMAB Double Oracle (RMABDO), with pseudocode presented in Algorithm REF , adapted from the MIRROR framework [41].\",\n \"We use RMABPPO to instantiate the agent oracle and MA-RMABPPO for the nature oracle.\",\n \"The double oracle approach proceeds as follows.\",\n \"The agent maintains strategy set $\\\\Pi $ , initially empty, and nature maintains strategy set $\\\\Omega $ , initialized with an arbitrary parameter setting.\",\n \"In each iteration, we solve for a mixed Nash equilibrium in the regret game between the agent and nature to learn a mixed strategy ($\\\\tilde{\\\\pi }, \\\\tilde{\\\\omega }$ ) for each player.\",\n \"We then call the agent and nature oracles to each compute a best response $\\\\pi $ and $\\\\omega $ to their opponent's strategy, which get added to their respective strategy sets $\\\\Pi $ and $\\\\Omega $ .\",\n \"We repeat this process until the improvement in value for each player is within the tolerance $\\\\varepsilon $ , which is guaranteed to converge within finite steps to within $\\\\epsilon $ value of the minimax regret–optimal policy (Theorem 2 of [41]), or until a set number of iterations.\",\n \"Additionally, we show that a policy that maximizes reward assuming a fixed parameter set can incur arbitrarily large regret when the parameters are changed (proof in appendix).\",\n \"Formally, [tb] RMABPPO Input: Initial state $\\\\mathbf {s}_0$ , nature mixed strategy $\\\\tilde{\\\\omega }$ Parameters: num epochs n_epochs, num subepochs n_subepochs [1] Randomly initialize a policy network $\\\\pi _{\\\\theta _n}$ for each arm $n \\\\in [N]$ Randomly initialize $\\\\lambda $ -network Lambda Initialize an empty buffer $\\\\textit {epoch} = 1, 2, \\\\ldots , \\\\texttt {n\\\\_epochs}$ Sample $\\\\lambda = \\\\textsc {Lambda}(\\\\mathbf {s})$ $\\\\textit {subepoch} = 1, \\\\ldots , \\\\texttt {n\\\\_subepochs}$ timestep $t = 1, \\\\ldots , \\\\texttt {n\\\\_simulation\\\\_steps}$ Sample action $a_n = \\\\pi _{\\\\theta _n}(s_n, \\\\lambda )$ for all $n \\\\in [N]$ Add trajectories $(\\\\mathbf {s}, \\\\mathbf {a}, r, \\\\mathbf {s}^\\\\prime , \\\\lambda )$ to buffer Update policy network $\\\\pi _{\\\\theta _n}$ of all arms via PPO, using trajectories in buffer Update Lambda using discounted sum of trajectories of final subepoch return $\\\\pi _{\\\\theta _1}, \\\\ldots , \\\\pi _{\\\\theta _N}$ and Lambda [tb] RMABDO Input: Environment simulator and parameter uncertainty interval $\\\\langle \\\\omega _{n, p_n} \\\\rangle $ for all $n \\\\in [N]$ Parameters: Convergence threshold $\\\\varepsilon $ Output: Best agent mixed strategy $\\\\tilde{\\\\pi }$ [1] $\\\\Omega _0 = \\\\lbrace \\\\omega _0\\\\rbrace $ , with $\\\\omega _0$ selected at random $\\\\Pi _0 = \\\\lbrace \\\\pi _{B_1}, \\\\pi _{B_2}, \\\\ldots \\\\rbrace $ , where $\\\\pi _{B_i}$ are baseline and heuristic strategies epoch $e = 1, 2, \\\\ldots $ Solve for $(\\\\tilde{\\\\pi }_e, \\\\tilde{\\\\omega }_e)$ , mixed Nash equilibrium of regret game with strategy sets $\\\\Omega _{e-1}$ and $\\\\Pi _{e-1}$ $\\\\pi _e = \\\\textsc {RMABPPO}(\\\\tilde{\\\\omega }_e)$ $\\\\omega _e = \\\\textsc {MA-RMABPPO}(\\\\tilde{\\\\pi }_e)$ $\\\\Omega _e = \\\\Omega _{e-1} \\\\cup \\\\lbrace \\\\omega _e\\\\rbrace , \\\\Pi _e = \\\\Pi _{e-1} \\\\cup \\\\lbrace \\\\pi _e\\\\rbrace $ $L(\\\\tilde{\\\\pi }_e, \\\\omega _e) - L(\\\\tilde{\\\\pi }_{e-1}, \\\\tilde{\\\\omega }_{e-1}) \\\\le \\\\varepsilon $ and $L(\\\\pi _e, \\\\tilde{\\\\omega }_e) - L(\\\\tilde{\\\\pi }_{e-1}, \\\\tilde{\\\\omega }_{e-1}) \\\\le \\\\varepsilon $ return $\\\\tilde{\\\\pi }_e$ []propositionregretProposition In the RMAB problem with interval uncertainty, the max regret of a reward-maximizing policy can be arbitrarily large compared to a minimax regret–optimal policy.\"\n ],\n [\n \"Experimental Evaluation\",\n \"We compare our algorithm against five baselines.\",\n \"Namely, we compare against three variations of the approach from [15], which computes a reward-maximizing Lagrange policy for each step of a multi-action RMAB problem for fixed transition probabilities.\",\n \"The three variations are, pessimistic (HP), mean (HM), and optimistic (HO), which assume the transition probabilities have been set to lower bound, mean, and upper bound of the intervals, respectively.\",\n \"We also compare against RLvMid, which learns a policy via RMABPPO assuming mean parameter values, and Rand, which acts randomly each round.\",\n \"All results are averaged over 50 random seeds and were executed on a cluster running CentOS with Intel(R) Xeon(R) CPU E5-2683 v4 @ 2.1 GHz with 8GB of RAM using Python 3.7.10.\",\n \"Our RMABPPO implementation builds on OpenAI Spinning Up [1] and RMABDO builds on the MIRROR implementation [41], computing Nash equilibria using Nashpy 0.0.21 [21].\",\n \"Code is available on github.Code: https://github.com/killian-34/RobustRMAB Hyperparameter settings are included in the appendix.\",\n \"We evaluate all the approaches on three experimental domains, which are as follows.\",\n \"Synthetic domain: We create this setup to show that the reward-maximizing Lagrange policies (HP, HM, and HO) may incur large regret.\",\n \"There are three arm types $\\\\lbrace U,V,W\\\\rbrace $ , each with two actions, $\\\\mathcal {C} = [0, 1]$ , two states $\\\\mathcal {S}=\\\\lbrace 0,1\\\\rbrace $ , $R(s)=s$ , and the following transition probabilities, with rows and columns corresponding to actions and next states, respectively: $T^n_{s=0}=\\\\begin{bmatrix}0.5 & 0.5 \\\\\\\\0.5 & 0.5\\\\end{bmatrix}, \\\\hspace{5.69054pt}T^n_{s=1}=\\\\begin{bmatrix}1.0 & 0.0 \\\\\\\\1-p_n & p_n\\\\end{bmatrix}\\\\hspace{5.69054pt}\\\\text{where}\\\\hspace{5.69054pt}\\\\begin{matrix}p_U \\\\in [0.00, 1.00] \\\\\\\\p_V \\\\in [0.05, 0.90] \\\\\\\\p_W \\\\in [0.10, 0.95]\\\\end{matrix} \\\\ .$ Note that, when an arm is at $s=0$ , both the actions have equal impact on the state transition.\",\n \"When the arms are at $s=1$ , selecting the arm with higher $p_n$ is optimal.\",\n \"Accordingly, $\\\\pi _\\\\textit {HP} = [W,V,U]$ , $\\\\pi _\\\\textit {HM} = [W,U,V]$ , and $\\\\pi _\\\\textit {HO} = [U,W,V]$ .\",\n \"Observe that there exist values of $p_n$ that can make each of the policies incur large regret, e.g., $p=[0.0, 0.9, 0.1]$ for $\\\\pi _\\\\textit {HM}$ , which would induce an optimal policy $[1,2,0]$ that is the reverse of $\\\\pi _\\\\textit {HM}$ .\",\n \"ARMMAN: The maternal healthcare intervention problem [5] is modeled as a binary-action RMAB that selects a subset of beneficiaries each week to intervene on with tailored maternal health messaging to encourage engagement.\",\n \"The behavior of each enrolled woman is modeled by an MDP with three states: Self-motivated, Persuadable, and Lost Cause.\",\n \"We use the summary statistics mentioned in their paper and assume uncertainty intervals of $0.5$ centered around the transition parameters, resulting in 6 uncertain parameters per arm (details in appendix).\",\n \"Similar to the setup in [5], we assume 1:1:3 split of arms with high, medium, and low probability of increasing their engagement upon intervention.\",\n \"In our experiments, we scale the value of $N$ in multiples of 5 to keep the same split of arm categories of 1:1:3.\",\n \"SIS Epidemic Model: A discrete-state model in which arms represent distinct geographic regions and each member of an arm's population of size $N_{\\\\textit {p}}$ is tracked to measure whether they are already infected by a disease or are susceptible.\",\n \"For a particular region, the fraction of susceptible members represents the state of the region.\",\n \"The model is defined by parameters $\\\\lambda _{\\\\textit {c}}$ , the average number of contacts per timestep, and $r_{\\\\textit {infect}}$ , the probability of infection given contact with an infectious person.\",\n \"Details on computing discrete state transition probabilities are derived from [42] and given in the appendix.\",\n \"We augment the model to include three actions $\\\\lbrace a_0, a_1, a_2\\\\rbrace $ with costs $c=\\\\lbrace 0, 1, 2\\\\rbrace $ .\",\n \"Action $a_0$ represents no action, $a_1$ divides the contacts per day $\\\\lambda _{\\\\textit {c}}$ by $a^{\\\\textit {eff}}_1$ , e.g., by communicating messaging about physical distancing, and $a_2$ divides the probability of infection given contact $r_{\\\\textit {infect}}$ by $a^{\\\\textit {eff}}_2$ , e.g., by distributing face masks.\",\n \"We impose the following uncertainty intervals: $\\\\lambda _{\\\\textit {c}} \\\\in [1, 10]$ , $r_{\\\\textit {infect}} \\\\in [0.5, 0.99]$ , $a^{\\\\textit {eff}}_1 \\\\in [1, 10]$ , and $a^{\\\\textit {eff}}_2 \\\\in [1, 10]$ .\"\n ],\n [\n \"Robust Double Oracle\",\n \"To evaluate the algorithms for robustness, we compute the regret of an agent's strategy against a nature pure strategy $\\\\omega $ as the difference between the average reward obtained by the agent's strategy on $\\\\omega $ and the average reward of the optimal strategy against $\\\\omega $ .\",\n \"The average reward is computed as the discounted sum of rewards over all arms for a horizon of length 10, averaged over 25 simulations.\",\n \"In each setting, double oracle runs for 6 iterations, using 100 rollout steps and 100 training epochs for each oracle.\",\n \"After completion, each baseline strategy is evaluated, querying the nature oracle for the best response against that strategy.\",\n \"We report the max regret against all nature's pure strategies.\",\n \"Figure REF shows that our double oracle method has the lowest regret-per-arm and beats all the baselines across all experimental settings.\",\n \"The top row shows the results on the synthetic domain, demonstrating that our approach can reduce regret by about $50\\\\%$ against existing benchmarks, across various values of $N$ and $B$ .\",\n \"This is expected because the domain is designed to ensure large regret for HP, HM, and HO baselines.\",\n \"Here, a benefit of $50\\\\%$ in the regret corresponds to, in the worst case, keeping $1/3$ of the arms in the good state for an extra round compared to the baseline.\",\n \"Similarly, for the ARMMAN domain, our algorithm performs consistently better than the baselines, achieving regret that is around $50\\\\%$ lower than the best baselines.\",\n \"The third domain (SIS) is interesting since the state space is large (number of states was restricted to 2 and 3 in the previous domains).\",\n \"Similar to the earlier results, in the SIS model with populations size of 50, our algorithm outperforms in terms of regret.\",\n \"We discuss the runtime of Figure 1 in Section REF .\",\n \"Additional results included in the appendix.\"\n ],\n [\n \"Agent Oracle\",\n \"We evaluate the performance of RMABPPO, our novel RL approach to find a reward-maximizing policy for multi-action RMABs, against an upper bound, namely, the solution by Hawkins, given exact transition probabilities.\",\n \"This will create an upper bound because, given access to the exact transition dynamics, Hawkins will compute the exact Lagrange policy, whereas RMABPPO must learn to approximate the Lagrange policy from samples.\",\n \"Each setting instantiates the environment with a random sample of valid parameter settings for each seed.\",\n \"Figure REF shows that the reward accumulated by our agent oracle is comparable to the Hawkins algorithm.\",\n \"In the synthetic domain (top row), the policy learns to act on the $33\\\\%$ of arms who belong to category $W$ .\",\n \"The mean reward of RMABPPO almost matches that of Hawkins algorithm as $N$ scales with a commensurate budget (Fig.\",\n \"REF left).\",\n \"As we fix $N$ and vary the budget (Fig.\",\n \"REF right), the optimal policy accumulates more reward, and RMABPPO is almost equal to the optimal.\",\n \"We observe similar results on the ARMMAN domain (middle row), where it is optimal to act on $20\\\\%$ of arms (that forms category $A$ ; details in appendix).\",\n \"Additionally, on the third large-scale domain (SIS, bottom left), we show the strong performance of RMABPPO holds in a multi-action setting even as we increase the number of states from 50 to 500 to test the scalability.\",\n \"Moreover, computationally, RMABPPO beats Hawkins: in the bottom-right of Fig.\",\n \"REF , a single rollout (10 time steps) of the Hawkins policy takes around 100 seconds when there are 500 states (population size), and steeply increases with more states.\",\n \"This demonstrates that it would be prohibitive to run Hawkins within the loop of the double oracle, since the agent policy needs to be evaluated thousands of times to compute the regret matrices—for merely 25 simulations, computation would take around 42 minutes to evaluate a single cell in the regret matrix over all pure strategy combinations, where the matrix has size $|\\\\Pi | \\\\times |\\\\Omega |$ .\"\n ],\n [\n \"Limitations\",\n \"We believe these advancements have the potential to improve resource allocation in low-resource settings, but acknowledge they are not without tradeoffs.\",\n \"For example, the baseline methods we compare against, while less robust, can provide interpretable `index' policies that capture the value for acting on an arm, whereas our solution's output can be difficult for a domain expert to interpret.\",\n \"Further, such optimization tools have the risk of amplifying underlying biases in the data and translating that to unfair resource allocation.\",\n \"However, by addressing the robust version of the problem, we directly address this concern by providing a flexible tool for mitigating biases, by allowing users to tune their uncertainties and thus, providing natural ways to develop good policies even when data availability is skewed.\"\n ],\n [\n \"Conclusion\",\n \"In this paper, we address a blocking limitation that inhibits restless bandits to be used for many real-world planning problems: we often lack perfect knowledge of the environment dynamics.\",\n \"To plan effective and risk-averse policies, it is therefore essential to take a robust approach to account for this uncertainty.\",\n \"Our approach enables us to learn minimax regret–optimal policies by providing RMABPPO, a novel RL algorithm to decouple the budget-constrained problem by training an auxiliary $\\\\lambda $ -network, and MA-RMABPPO, a new approach to instantiating the nature oracle of an adversarial double oracle game setup by using multi-agent RL to handle nonstationary in nature's challenging optimization problem.\",\n \"Notably, RMABPPO can also extend to solving RMAB problems with continuous states and/or actions, a setting which has not previously been addressed in the literature.\",\n \"We hope these contributions bring us closer to deploying restless bandits in the real world.\",\n \"This work was supported in part by the Army Research Office by Multidisciplinary University Research Initiative (MURI) grant number W911NF1810208.\",\n \"J.A.K.\",\n \"was supported by an NSF Graduate Research Fellowship under grant DGE1745303.\",\n \"A.B.\",\n \"was supported by the Harvard Center for Research on Computation and Society.\"\n ],\n [\n \"Proof of Proposition \",\n \"* This follows from taking the gradient of Eq.\",\n \"REF with respect to $\\\\lambda $ .\",\n \"To compute the gradient of $Q_n$ , we simply roll out $Q_n$ over time then take the derivative, yielding $\\\\frac{dQ_n}{d\\\\lambda } = \\\\sum _{t=0}^{1} \\\\beta ^t c_{n,t}$ , i.e., the negative of the expected discounted sum of action costs under the optimal policy.\"\n ],\n [\n \"Proof of Proposition \",\n \"* This follows from the convexity of Eq.\",\n \"REF with respect to $\\\\lambda $ .\",\n \"The convexity of Eq.\",\n \"REF can be seen from the definition of $Q_n$ , i.e., the max over piece-wise linear functions of $\\\\lambda $ is convex.\"\n ],\n [\n \"Proof of Proposition \",\n \"* Consider a binary-action RMAB problem with two arms A and B.\",\n \"Let the reward from each arm be $R$ when the arm is in a good state and 0 in a bad state.\",\n \"Our problem is to plan the best action with a budget of 1 and horizon of 1.\",\n \"Supposing the initial state is bad for each arm, the transition probabilities for the transition matrix for each arm $n$ is $\\\\begin{bmatrix} 1 & 0 \\\\\\\\ 1 - p_n & p_n \\\\end{bmatrix}$ where the uncertain variable $p_n$ is constrained to be within $p_A, p_B \\\\in [0, 1]$ .\",\n \"Each value in the matrix corresponds to the probability of an arm at state bad transitioning to bad (column 1) or good (column 2) if we take the passive (row 1) or active action (row 2).\",\n \"To compute a reward-maximizing policy that does not consider robustness to uncertainty, we must optimize for one instantiation of the uncertainty set, which requires making one of three assumptions.\",\n \"Case 1: If we assume $p_A = p_B$ , then an optimal policy is to act with probability $a_A$ on arm A and $a_B$ on arm B as long as $a_A + a_B = 1$ .\",\n \"W.l.o.g., suppose $a_A \\\\ge a_B$ ; then nature would set $p_A = 0$ and $p_B = 1$ , imposing regret at least $R/2$ .\",\n \"Case 2: If $p_A > p_B$ , then the optimal policy would be to always act on arm A with probability $a_A = 1$ and never act on B ($a_B = 0$ ).\",\n \"Nature would then set $p_A = 0$ and $p_B = 1$ to impose regret $R$ .\",\n \"Case 3: If $p_A < p_B$ , the case is symmetric to Case 2 and result in regret $R$ .\",\n \"Clearly, max regret is minimized when our action is such that $a_A + a_B = 1$ ; in this setting, we learn this optimal policy only under Case 1.\",\n \"Following Case 2 or 3, the difference between our regret and the minimax regret is $R/2$ , which grows arbitrarily higher as $R \\\\rightarrow \\\\infty $ .\",\n \"A slight modification to this problem renders Case 1 non-optimal.\",\n \"Let the reward be $R$ when arm A is in a good state and $R-1$ for arm B, so the optimal policy learned under the assumption from Case 1 leads to $a_A = 1$ and $a_B = 0$ .\",\n \"Then nature could respond with $p_A = 0$ and $p_B = 1$ , yielding reward 0 and regret $R-1$ , while the minimax regret–optimal policy achieves a minimum reward of $(R-1)/2$ (by playing $a_A = 0.5$ and $a_B = 0.5$ where nature responds with $p_A = 0$ and $p_B = 1$ ).\",\n \"Thus, the gap again can grow arbitrarily high as $R \\\\rightarrow \\\\infty $ provided that $R > 1$ .\",\n \"We therefore have that in all cases, any reward-maximizing policy can achieve arbitrarily bad performance in terms of regret.\"\n ],\n [\n \"MA-RMABPPO: Nature Oracle Algorithm\",\n \"We provide the pseudocode for MA-RMABPPO in Alg.\",\n \", which is used to implement the nature oracle.\",\n \"[tb] MA-RMABPPO Input: Agent mixed strategy $\\\\tilde{\\\\pi }$ Parameters: n_epochs, n_subepochs, n_simsteps, n_sims [1] Initialize agent A: arm policies $\\\\pi _{n}^{(A)}~ \\\\forall n \\\\in [N]$ , $\\\\lambda $ -network $\\\\textsc {Lambda}$ , arm critic networks $\\\\phi _{n}^{(A)}~\\\\forall n \\\\in [N]$ Initialize agent B: nature parameter policy $\\\\pi ^{(B)}$ , critic network $\\\\phi ^{(B)}$ Initialize empty buffer $\\\\textit {epoch} = 1, 2, \\\\ldots , \\\\texttt {n\\\\_epochs}$ Sample $\\\\mathbf {s}$ at random Sample $\\\\lambda = \\\\textsc {Lambda}(\\\\mathbf {s})$ $\\\\textit {subepoch} = 1, \\\\ldots , \\\\texttt {n\\\\_subepochs}$ $t = 1, \\\\ldots , \\\\texttt {n\\\\_simsteps}$ Sample agent A action $a_n^{(A)} \\\\sim \\\\pi _{n}^{(A)}(s_n, \\\\lambda )$ for each $n \\\\in [N]$ Sample agent B action $\\\\omega ^{(B)} \\\\sim \\\\pi ^{(B)}$ ($\\\\pi ^{(B)}$ may take $\\\\mathbf {s}$ as input) $r^{(A)}, \\\\mathbf {s}^\\\\prime = \\\\textsc {simulate}(\\\\mathbf {s}, \\\\mathbf {a}^{(A)}, \\\\omega ^{(B)})$ $\\\\tilde{r} = \\\\textsc {simulate}(\\\\mathbf {s}, \\\\tilde{\\\\pi }(\\\\mathbf {s}), \\\\omega ^{(B)}, \\\\texttt {n\\\\_sims})$ (mean of 1-step rollouts of $\\\\tilde{\\\\pi }$ ) $r^{(B)} = r^{(B)} - \\\\tilde{r}$ (agent regret) Store $(\\\\mathbf {s}, \\\\mathbf {a}^{(A)}, \\\\omega ^{(B)}, r^{(A)}, r^{(B)}, \\\\mathbf {s}^\\\\prime )$ in buffer $\\\\mathbf {s} = \\\\mathbf {s}^\\\\prime $ Update $\\\\pi _n^{(A)}$ , $\\\\phi _n^{(A)}$ for all $n$ using trajectories in buffer.\",\n \"$\\\\pi _n^{(A)}$ gets $\\\\omega ^{(B)}$ as part of state Update Lambda via discounted trajectories in buffer Update $\\\\pi ^{(B)}, \\\\phi ^{(B)}$ from trajectories in buffer.\",\n \"$\\\\phi ^{(B)}$ gets $a^{(A)}$ as part of state return $\\\\pi ^{(B)}$\"\n ],\n [\n \"ARMMAN\",\n \"The MDPs in the ARMMAN domain [5] have three ordered states representing the level of engagement of the beneficiaries in the previous week.\",\n \"Rewards are better for lower states, i.e., $R(0)=1, R(1)=0.5, R(2)=0$ .\",\n \"At each step, the beneficiary may only change by one level, e.g., low-to-medium or high-to-medium but not low-to-high.\",\n \"They also assume that beneficiaries follow one of three typical patterns, A, B, and C, resulting in three MDPs with different transition probabilities.\",\n \"There are two patterns of effects present that differentiate the beneficiary types.\",\n \"(1) For each of the above types, the planner can only make a difference when the patient is in state 1.\",\n \"Type A responds very positively to interventions, but regresses to low reward states in absence.\",\n \"Type B has a similar but less amplified effect, and type C is likely to stay in state 1, but can be prevented from regressing to state 2 when an action is taken.\",\n \"(2) Further, types A and C have only a 10% chance of staying in the high reward state, while type B has a 90% chance of staying there.\",\n \"We converted these patient types to robust versions where the transition probabilities are uncertain as follows: $T^i_{s=0}=\\\\begin{bmatrix}p^i_{000} & 1 - p^i_{000} & 0.0 \\\\\\\\p^i_{010} & 1 - p^i_{010} & 0.0\\\\end{bmatrix}, \\\\hspace{5.69054pt}T^i_{s=1}=\\\\begin{bmatrix}0.0 & 1 - p^i_{102} & p^i_{102} \\\\\\\\p^i_{110} & 1 - p^i_{110} & 0.0\\\\end{bmatrix}, \\\\ $ $T^i_{s=2}=\\\\begin{bmatrix}0.0 & 1 - p^i_{202} & p^i_{202} \\\\\\\\0.0 & 1 - p^i_{212} & p^i_{212}\\\\end{bmatrix},$ where $i$ indexes the type (i.e., A, B or C).\",\n \"We then set each $p^i_{sas^\\\\prime }$ to be in a range of width 0.5 centered on the entries from each of the A, B, C beneficiary types for $s\\\\in \\\\lbrace 1,2\\\\rbrace $ .\",\n \"To add additional heterogeneity to the experiments, for $s=0$ , we set the range to 1.0 so that any beneficiary type can be made to have some non-negligible chance of staying in the good state, rather than only type B beneficiaries.\",\n \"The full set of parameter ranges are given in the Table REF below.\",\n \"Table: Upper and lower parameter ranges for the robust ARMMAN environment.In all experiments, 20% of arms were sampled from type A, 20% from type B and 60% for type C. To add additional heterogeneity, for each of the 50 random seeds we uniformly sample a sub-range contained within the ranges given in Table REF .\",\n \"In the agent oracle experiments, for each of the 50 random seeds, since these require fully instantiated transition matrices, we uniformly sample each parameter value for each arm according to its type such that the values are contained in the ranges given in Table REF .\"\n ],\n [\n \"SIS Epidemic Model\",\n \"In this domain, each arm follows its own compartmental SIS epidemic model.\",\n \"Each arm's SIS model tracks whether each of $N_p$ members of a population is in a susceptible (S) or infectious (I) state.\",\n \"This can be tracked with $N_p$ states, since it can be computed how many people are in state I if only the number of people in state S and the population size $N_p$ is known.\",\n \"To define a discrete SIS model, we instantiate the model given in [42] section 4.1 with a $\\\\Delta t$ of 1.\",\n \"We also augment the model to include action effects and rewards.\",\n \"Specifically, $R(N_S) = N_S/N_p$ , where $N_S$ is the number of susceptible (non-infected) people.\",\n \"Further, there are three actions $\\\\lbrace a_0, a_1, a_2\\\\rbrace $ with costs $c=\\\\lbrace 0, 1, 2\\\\rbrace $ .\",\n \"Action $a_0$ represents no action, $a_1$ divides the contacts per day $\\\\lambda _{\\\\textit {c}}$ ($\\\\lambda $ in [42]) by $a^{\\\\textit {eff}}_1$ , and $a_2$ divides the infectiousness $r_{\\\\textit {infect}}$ ($r(t)$ in [42]) by $a^{\\\\textit {eff}}_2$ .\",\n \"That is, taking action $a_1$ will reduce the average number of contacts per day in a given arm, and taking action $a_2$ will reduce the probability of infection given contact in a given arm, thus reducing the expected number of people that will become infected in the next round.\",\n \"However, to make this a robust problem, the relative effect sizes of each action for each arm will not be known to the planner, nor will the $\\\\lambda _\\\\textit {c}$ or $r_{\\\\textit {infect}}$ .\",\n \"We impose the following uncertainty intervals for all arms: $\\\\lambda _{\\\\textit {c}} \\\\in [1, 10]$ , $r_{\\\\textit {infect}} \\\\in [0.5, 0.99]$ , $a^{\\\\textit {eff}}_1 \\\\in [1, 10]$ , and $a^{\\\\textit {eff}}_2 \\\\in [1, 10]$ .\",\n \"In the robust double oracle experiments, to add additional heterogeneity, for each of the 50 random seeds we uniformly sample a sub-range contained within the ranges given above for each arm.\",\n \"In the agent oracle experiments, for each of the 50 random seeds, since these require fully instantiated transition matrices, we uniformly sample each parameter value for each arm such that the values are contained in the ranges given above.\"\n ],\n [\n \"Hyperparameter Settings and Implementation Details\",\n \"Neural networks: All neural networks in experiments are implemented using PyTorch 1.3.1 [32] with 2 fully connected layers each with 16 units and tanh activation functions, and a final layer of appropriate size for the relevant output dimension with an identity activation function.\",\n \"The output of discrete actor networks (i.e., the policy network from the agent oracle, and the policy network of agent A in the nature oracle) pass through a categorical distribution from which actions are randomly sampled at training time, without a budget imposed.\",\n \"It is critical not to impose the budget at training time, so that the budget spent by the optimal policy under a given $\\\\lambda $ will result in a meaningful gradient for updating the $\\\\lambda $ -network.\",\n \"The output of continuous actor networks (i.e., agent B in the nature oracle which selects environment parameter settings) instead are passed as the means of Gaussian distributions – with the log standard deviations learned as individual parameters separate from the network – from which continuous actions are sampled at training time.\",\n \"At test time, actions are sampled from both types of networks deterministically.\",\n \"For categorical distributions, we greedily select the highest probability actions.\",\n \"For Gaussian distributions, we act according to the means.\",\n \"All discount factors were set to 0.9.\",\n \"The remaining hyperparameters that were constant for all experiments for the agent and nature oracles are indicated in Table REF .\",\n \"For Robust Double Oracle experiments, all agent and nature oracles were run for 100 training epochs.\",\n \"For Agent Oracle experiments, RMABPPO was run for 100 training epochs for the synthetic and ARMMAN domains and 200 epochs for the SIS domain.\",\n \"$\\\\lambda $ -network: Critical to training the $\\\\lambda $ -network is cyclical control of the temperature parameter that weights the entropy term in the actor loss functions.\",\n \"Recall that the $\\\\lambda $ -network is only updated every n_subepochs.\",\n \"In general, after each update to the $\\\\lambda $ -network, we want to encourage exploration so that actor networks explore the new part of the state space defined by updated predictions of $\\\\lambda $ .\",\n \"However, after n_subepochs rounds, we will use the cost of the sampled actor policies as a gradient for updating the $\\\\lambda $ -network, and that gradient will only be accurate if the actor policy has converged to the optimal policies for the given $\\\\lambda $ predictions.\",\n \"Therefore, we also want to have little or no exploration in the round before we update the $\\\\lambda $ -network.\",\n \"In general, we would also like the entropy of the policy network to reduce over time so that the actor networks and $\\\\lambda $ -networks eventually both converge.\",\n \"To accomplish both of these tasks, the weight (temperature) of the entropy regularization term in the loss function of the actor network will decay/reset according to two processes.\",\n \"The first process will linearly decay the temperature from some positive, but time-decaying starting value (see next process) $\\\\tau _t$ immediately after each $\\\\lambda $ -network update, down to 0 after $\\\\texttt {n\\\\_subepochs}$ .\",\n \"The second process will linearly decay the temperature from a maximum $\\\\tau _0$ (start entropy coeff in Table REF ) down to $\\\\tau _{\\\\min }$ (end entropy coeff in Table REF ) by the end of training.\",\n \"We found that it also helps to train the actor network with no entropy and with the $\\\\lambda $ -network frozen for some number of rounds before training is stopped (lambda freeze epochs in Table REF ).\",\n \"Double Oracle: In all experiments in the main text, we initialize the agent strategy list with HO, HM, and HP, and the nature strategy list with pessimistic, mean, and optimistic nature strategies, then run RMABDO for 5 iterations.\",\n \"This produces a set of 8 agent strategies, 8 nature strategies, a table where each entry represents the regret of each agent pure strategy (row) against each nature pure strategy (column), and an optimal mixed strategy over each set that represents a Nash equilibrium of the minimax regret game given in the table.\",\n \"The regret table is computed by first computing the returns of each agent/nature pure strategy combination, then subtracting the max value of each column from all entries in that column (i.e., the best agent strategy for a given nature strategy gets 0 regret).\",\n \"The regret of RMABDO is reported as the expected utility corresponding to the Nash equilibrium of the regret game given by the table, once that regret table is normalized to account for the returns of baselines (see next paragraph).\",\n \"After this main loop completes, we then compute the regret of the baselines by evaluating each baseline policy against each pure strategy in the nature strategy list.\",\n \"Then, we also run the nature oracle against each baseline policy to find a nature strategy that should maximize the regret of that baseline.\",\n \"The regret for each baseline is reported as the max regret against this new nature strategy, as well as all pure nature strategies from the main RMABDO loop.\",\n \"Hawkins Baselines: The Hawkins policies are implemented with gurobipy 9.1.2, a Python wrapper for Gurobi (9.0.3) [14] following the LP given in [15] equation 2.5 to compute $\\\\lambda $ and $Q(s,a,\\\\lambda )$ for each arm and the integer program in equation 2.12 to select actions.\",\n \"RLvMid Baseline: We found that RLvMid found effective policies for the nature strategy it was trained against (as evidenced in Figure REF ), but that that learned policy could be brittle against other nature strategies.\",\n \"This is likely because different nature strategies produce different distributions of states, meaning RLvMid would fit policies well to states seen when planning against the mean nature strategy, but underfit its policies for states seen more often in different distributions.\",\n \"However, the lone RLvMid baseline policy can somewhat correct for this effect by training an ensemble of policies against slight perturbations of the mean nature strategy that adjust the parameter values output by nature by a small $\\\\epsilon $ .\",\n \"In all experiments we train 3 RLvMid policies against 3 random perturbations of the mean nature strategy, then report the regret of RLvMid as the minimum of the max regrets returned by any of the 3.\",\n \"Table: Hyperparameter settings for agent and nature oracles for all experiments.\"\n ],\n [\n \"Additional Experimental Results\",\n \"Fig.\",\n \"REF shows regret results from additional robust experiments which scale up the number of arms for Synthetic (top left), ARMMAN (top right) and SIS (bottom left; $N_p=50$ ), as well as and the size of the state space of SIS (bottom right; $N=5, B=4$ ).\",\n \"RMABDO continues to outperform all baselines by a large margin.\",\n \"When the size of the state space is scaled up for SIS, it becomes infeasible to run the Hawkins baselines due to its long query time which grows quadratically with the size of the state space.\",\n \"Because of this, we exclude the Hawkins baselines from the main double oracle loop in these experiments.\",\n \"Further, Hawkins cannot even be evaluated as a baseline as the state space increases since to compute its maximum regret, we must get one best response from the nature oracle against the baseline, which requires querying the Hawkins baseline policies tens of thousands of times, which is prohibitive when the query time takes even ~$1s$ to run.\",\n \"Figure: Additional experiments showing maximum policy regret in robust setting for Synthetic (top left), ARMMAN (top right) and SIS (bottom) domains, respectively.\",\n \"Synthetic is scaled by 3 and ARMMAN by 5 to maintain the distributions of arm types specified in Section .\",\n \"RMABDO beats all baselines by a large margin across various parameter settings.\",\n \"When the state space is scaled up (bottom right) Hawkins baselines become infeasible to run due to its long query time (see Section for a discussion), even for a small number of arms (N=5,B=4N = 5, B = 4).\"\n ]\n]"},"id":{"kind":"string","value":"2107.01689"}}},{"rowIdx":1580,"cells":{"text":{"kind":"list like","value":[["Percolation transition for random forests in $d\\geq 3$"],["Abstract The arboreal gas is the probability measure on (unrooted spanning) forests of a graph in which each forest is weighted by a factor $\\beta>0$ per edge.","It arises as the $q\\to 0$ limit with $p=\\beta q$ of the $q$-state random cluster model.","We prove that in dimensions $d\\geq 3$ the arboreal gas undergoes a percolation phase transition.","This contrasts with the case of $d=2$ where all trees are finite for all $\\beta>0$.","The starting point for our analysis is an exact relationship between the arboreal gas and a non-linear sigma model with target space the fermionic hyperbolic plane $\\mathbb{H}^{0|2}$.","This latter model can be thought of as the $0$-state Potts model, with the arboreal gas being its random cluster representation.","Unlike the $q>0$ Potts models, the $\\mathbb{H}^{0|2}$ model has continuous symmetries.","By combining a renormalisation group analysis with Ward identities we prove that this symmetry is spontaneously broken at low temperatures.","In terms of the arboreal gas, this symmetry breaking translates into the existence of infinite trees in the thermodynamic limit.","Our analysis also establishes massless free field correlations at low temperatures and the existence of a macroscopic tree on finite tori."],["Introduction","This paper has two distinct motivations.","The first is to study the percolative properties of the arboreal gas, and the second is to understand spontaneously broken continuous symmetries.","We first present our results from the percolation perspective, and then turn to continuous symmetries."],["Main results for the arboreal gas","The arboreal gas is the uniform measure on (unrooted spanning) forests of a weighted graph.","More precisely, given an undirected graph $G=(\\Lambda ,E)$ , a forest $F=(\\Lambda ,E(F))$ is an acyclic subgraph of $G$ having the same vertex set as $G$ .","Given an edge weight $\\beta >0$ (inverse temperature) and a vertex weight $h\\ge 0$ (external field), the probability of a forest $F$ under the arboreal gas measure is $ ¶_{\\beta ,h}^{G}[F] \\frac{1}{Z_{\\beta ,h}^{G}} \\beta ^{|E(F)|} \\prod _{T\\in F} (1+h|V(T)|)$ where $T\\in F$ denotes that $T$ is a tree in the forest, i.e., a connected component of $F$ , $|E(F)|$ is the number of edges in $F$ , and $|V(T)|$ is the number of vertices in $T$ .","The arboreal gas is also known as the (weighted) uniform forest model, as Bernoulli bond percolation conditioned to be acyclic, and as the $q\\rightarrow 0$ limit of the $q$ -state random cluster model with $p/q$ converging to $\\beta $ , see [52].","We study the arboreal gas on a sequence of tori $\\Lambda _N = ^d/L^N^d$ with $L$ fixed and $N\\rightarrow \\infty $ .","To simplify notation, we will use $\\Lambda _N$ to denote both the graph and its vertex set.","From the percolation point of view, the most fundamental question concerns whether a typical forest $F$ under the law (REF ) contains a giant tree.","In all dimensions, elementary arguments show that giant trees can exist only if $h=0$ and if $\\beta $ is large enough, in the sense that connection probabilities decay exponentially whenever $h>0$ or $\\beta $ is small; see Appendix REF .","The existence of a percolative phase for $h=0$ and $\\beta $ large does not, however, follow from standard techniques.","The subtlety of the existence of a percolative phase is perhaps best evidenced by considering the case $d=2$ : in this case giant trees do not exist for any $\\beta >0$ [18].","Our main result is that for $d\\ge 3$ giant trees do exist for $\\beta $ large and $h=0$ , and that truncated correlations have massless free field decay.","To state our result precisely, let $\\lbrace 0 \\leftrightarrow x\\rbrace $ denote the event that 0 and $x$ are connected, i.e., in the same tree.","Let $d\\ge 3$ and $L \\ge L_0(d)$ .","Then there is $\\beta _0 \\in (0,\\infty )$ such that for $\\beta \\ge \\beta _0$ there exist $\\zeta _d(\\beta ) = 1-O(1/\\beta )$ , ${\\sf c}(\\beta ) = {\\sf c}+ O(1/\\beta )$ with ${\\sf c}>0$ , and $\\kappa >0$ such that $¶_{\\beta ,0}^{\\Lambda _N}[0\\leftrightarrow x]= \\zeta _d(\\beta ) + \\frac{{\\sf c}(\\beta )}{\\beta |x|^{d-2}} + O(\\frac{1}{\\beta |x|^{d-2+\\kappa }}) + O(\\frac{1}{\\beta L^{\\kappa N}}),$ where $|x|$ denotes the Euclidean norm.","Numerical evidence for this phase transition of the arboreal gas was given in [37].","More broadly our work was inspired by [34], [35], [19], [37], [54], [55], and we discuss further motivation later.","Although both the arboreal gas and Bernoulli bond percolation have phase transitions for $d\\ge 3$ , the supercritical phases of these models behave very differently: (REF ) shows that the arboreal gas behaves like a critical model even in the supercritical phase, in the sense that it has massless free field truncated correlation decay.","While this behaviour looks unusual when viewed through the lens of supercritical percolation, it is natural from the viewpoint of broken continuous symmetries.","We will return to this point in Section REF .","Theorem REF concerns the arboreal gas on large finite tori.","Another limit to consider the arboreal gas in is the weak infinite volume limit.","To this end, we consider the limit obtained by first taking $N\\rightarrow \\infty $ with $h>0$ and then taking $h\\downarrow 0$ .","In manner similar to that for Bernoulli bond percolation in [43] and [2], the external field is equivalent to considering the arboreal gas on an extended graph $G^{\\mathfrak {g}} = (\\Lambda \\cup \\lbrace \\mathfrak {g}\\rbrace , E\\cup E^{\\mathfrak {g}})$ where $E^{\\mathfrak {g}} = \\Lambda \\times \\lbrace \\mathfrak {g}\\rbrace $ and each edge in $E^{\\mathfrak {g}}$ has weight $h$ .","The additional vertex $\\mathfrak {g}$ is called the ghost vertex.","The measure (REF ) is then obtained by forgetting the connections to the ghost.","This rephrases that the product in (REF ) is equivalent to connecting a uniformly chosen vertex in each tree $T$ to $\\mathfrak {g}$ with probability $h|V(T)|/(1+h|V(T)|)$ .","For vertices $x,y \\in \\Lambda $ , we continue to denote by $\\lbrace x\\leftrightarrow y\\rbrace $ the event that $x$ and $y$ are connected in the random forest subgraph of $G$ with law (REF ), i.e., without using the edges in $E^\\mathfrak {g}$ .","We write $\\lbrace x\\leftrightarrow \\mathfrak {g}\\rbrace $ to denote the event that $x$ is connected to $\\mathfrak {g}$ .","The event $\\lbrace 0\\leftrightarrow \\mathfrak {g}\\rbrace $ is a finite volume proxy for the event that the tree $T_0$ containing 0 becomes infinite in the infinite volume limit when $h\\downarrow 0$ .","Indeed, let us define $\\theta _d(\\beta )\\lim _{h\\downarrow 0}\\lim _{N\\rightarrow \\infty } ¶^{\\Lambda _N}_{\\beta ,h}[0\\leftrightarrow \\mathfrak {g}],$ and let $¶_\\beta ^{^d}$ be any (possibly subsequential) weak infinite volume limit $\\lim _{h\\downarrow 0}\\lim _{N\\rightarrow \\infty }¶^{\\Lambda _N}_{\\beta ,h}$ .","Then $\\theta _d(\\beta )=¶_\\beta ^{^d}[|T_0|=\\infty ],$ see Proposition REF .","By a stochastic domination argument it is straightforward to show that $\\theta _d(\\beta ) =0 \\qquad \\text{for $0\\le \\beta 0$ by [18].","The next theorem shows that for $d\\ge 3$ the arboreal gas also has a phase transition in this infinite volume limit.","Let $d\\ge 3$ and $L \\ge L_0(d)$ .","Then there is $\\beta _0 \\in (0,\\infty )$ such that for $\\beta \\ge \\beta _0$ the limit (REF ) exists and $\\theta _{d}(\\beta )^2 = \\zeta _d(\\beta )= 1-O(1/\\beta ),$ where $\\zeta _d(\\beta )$ is the finite volume density of the tree containing 0 from Theorem REF .","In fact, our proof shows that $\\theta _d(\\beta )\\sim 1-c/\\beta $ with $c=(-\\Delta ^{^d})^{-1}(0,0)>0$ the expected time a simple random walk spends at the origin.","This behaviour is different from that of Bernoulli bond percolation and more generally that of the random cluster model with $q>0$ .","For these models the percolation probability is governed by Peierls' contours and is $1-O((1-p)^{2d})$ by [63].","That the arboreal gas behaves critically within its supercritical phase can be further quantified in terms of the following truncated two-point functions: $\\tau _{\\beta }(x) = \\lim _{h\\downarrow 0} \\tau _{\\beta ,h}(x),& \\qquad \\tau _{\\beta ,h}(x)=\\lim _{N\\rightarrow \\infty }¶_{\\beta ,h}^{\\Lambda _N}[0\\leftrightarrow x, 0 \\lnot \\leftrightarrow \\mathfrak {g}],\\\\\\sigma _{\\beta }(x) = \\lim _{h\\downarrow 0} \\sigma _{\\beta ,h}(x),& \\qquad \\sigma _{\\beta ,h}(x) = \\lim _{N\\rightarrow \\infty }{¶_{\\beta ,h}^{\\Lambda _N}[0\\lnot \\leftrightarrow \\mathfrak {g}]^2 - ¶_{\\beta ,h}^{\\Lambda _N}[0\\lnot \\leftrightarrow x, 0\\lnot \\leftrightarrow \\mathfrak {g},x\\lnot \\leftrightarrow \\mathfrak {g}]}.$ Under the assumptions of Theorem REF , for $\\beta \\ge \\beta _0$ , the limits (REF )–() exist and there exist constants ${\\sf c}_i(\\beta ) = {\\sf c}_i+O(1/\\beta )$ and $\\kappa >0$ such that $\\tau _{\\beta }(x) &= \\frac{{\\sf c}_1(\\beta )}{\\beta |x|^{d-2}} +O(\\frac{1}{\\beta |x|^{d-2+\\kappa }}),\\\\\\sigma _{\\beta }(x) &= \\frac{{\\sf c}_2(\\beta )}{\\beta ^2|x|^{2d-4}} + O(\\frac{1}{\\beta ^2 |x|^{2d-4+\\kappa }}).$ The constants satisfy $({\\sf c}_2(\\beta )/{\\sf c}_1(\\beta )^2)\\theta _d(\\beta )^2=1$ and ${\\sf c}(\\beta ) = 2{\\sf c}_1(\\beta )$ , ${\\sf c}(\\beta )$ from Theorem REF .","Further results could be deduced from our analysis, but to maintain focus we have not carried these out in detail.","We mention some of them below in Section REF when discussing our results and open problems."],["The $^{0|2}$ model and continuous symmetries","In [34], [35], the arboreal gas was related to a fermionic field theory and a supersymmetric non-linear sigma model with target space one half of the degenerate super-sphere $\\mathbb {S}^{0|2}$ .","In [18] this was reinterpreted as a non-linear sigma model with hyperbolic target space $^{0|2}$ (which we refer to as the $^{0|2}$ model for short).","The reinterpretation was essential in [18]; it is less essential for the present work, but nevertheless, we continue to use the $^{0|2}$ formulation of the model.","Briefly, the $^{0|2}$ model is defined as follows; see [18] for further details.","For every vertex $x\\in \\Lambda $ , there are two (anticommuting) Grassmann variables $\\xi _x$ and $\\eta _x$ and we then set $z_x \\sqrt{1-2\\xi _x\\eta _x} 1-\\xi _x\\eta _x.$ Thus the $z_x$ commute with each other and with the odd elements $\\xi _x$ and $\\eta _x$ .","The formal triples $u_x(\\xi _x,\\eta _x,z_x)$ are supervectors with two odd components $\\xi _x,\\eta _x$ and an even component $z_x$ .","These supervectors satisfy the sigma model constraint $u_x\\cdot u_x=-1$ for the super inner product $u_x\\cdot u_y -\\xi _x\\eta _y-\\xi _y\\eta _x - z_xz_y.$ In analogy with the tetrahedral representation of the Potts model, the sigma model constraint can be thought of as $u_x\\cdot u_x = q-1$ with $q=0$ .","The constraint is also reminiscent of the embedding of the hyperbolic space $^2$ in $^3$ equipped with the standard quadratic form with Lorentzian signature $(1, 1, -1)$ .","Indeed, $-\\xi _x\\eta _y-\\xi _y\\eta _x$ is the fermionic analogue of the Euclidean inner product on $^2$ .","The expectation of the $^{0|2}$ model is ${F}_{\\beta ,h} \\frac{1}{Z_{\\beta ,h}}\\int (\\prod _{x\\in \\Lambda } \\partial _{\\eta _x}\\partial _{\\xi _x} \\frac{1}{z_x}) e^{\\frac{\\beta }{2} (u,\\Delta u) - h(1,z-1)} F.$ In this expression, $\\int \\prod _{x\\in \\Lambda } \\partial _{\\eta _x}\\partial _{\\xi _x}$ denotes the Grassmann integral (i.e., the coefficient of the top degree monomial of the integrand), $Z_{\\beta ,h}$ is a normalising constant, and $(u,\\Delta u) = -\\frac{1}{2} \\sum _{xy\\in E(\\Lambda ) }(u_x-u_y)\\cdot (u_x-u_y)= \\sum _{xy\\in E(\\Lambda ) }(u_x\\cdot u_y+1),\\qquad (1,z) = \\sum _{x\\in \\Lambda } z_x,$ where $xy\\in E(\\Lambda )$ denotes that $x$ and $y$ are nearest neighbours (counting every pair once), and the inner products are given by (REF ).","The factors $1/z_x$ in (REF ) are the canonical fermionic volume form invariant under the symmetries associated with (REF ) as discussed further below.","As explained in [18] (see also [34] where such relations were first observed) connection and edge probabilities of the arboreal gas are equivalent to correlation functions of the $^{0|2}$ model.","The following proposition summarises the ones we need, see Appendix .","For any finite graph $G$ , any $\\beta \\ge 0$ and $h \\ge 0$ , $ ¶_{\\beta ,h}[0\\leftrightarrow \\mathfrak {g}] &= {z_0}_{\\beta ,h},\\\\¶_{\\beta ,h}[0\\leftrightarrow x, 0 \\lnot \\leftrightarrow \\mathfrak {g}] &= {\\xi _0\\eta _x}_{\\beta ,h},\\\\¶_{\\beta ,h}[0\\leftrightarrow x]+¶_{\\beta ,h}[0\\lnot \\leftrightarrow x, 0\\leftrightarrow \\mathfrak {g}, x\\leftrightarrow \\mathfrak {g}] &= -{u_0\\cdot u_x}_{\\beta ,h},$ and the normalising constants in (REF ) and (REF ) are equal.","In particular, $ ¶_{\\beta ,0}[0\\leftrightarrow x] = -{u_0\\cdot u_x}_{\\beta ,0} = -{z_0z_x}_{\\beta ,0} = {\\xi _0\\eta _x}_{\\beta ,0} = 1-{\\xi _0\\eta _0\\xi _x\\eta _x}_{\\beta ,0}.$ These relations resemble those between the Potts model and the random cluster model, giving further credence to our heuristic that the $^{0|2}$ model may be interpreted as the 0-state Potts model, with the arboreal gas playing the role of the 0-state random cluster model.","Nevertheless, there are important differences from the $q$ -state Potts model with $q \\ge 1$ .","Chief amongst them is that the $^{0|2}$ model has continuous symmetries.","To make this precise, let $ T=\\sum _{x\\in \\Lambda } z_x \\partial _{\\xi _x}, \\qquad \\bar{T}=\\sum _{x\\in \\Lambda } z_x \\partial _{\\eta _x}.$ One way to understand the significance of $T, \\bar{T}$ is via the identities ${TF}_{\\beta ,0}={\\bar{T} F}_{\\beta ,0}=0$ for any (noncommutative) polynomial $F$ in the variables $\\xi $ and $\\eta $ .","For example, ${T\\xi _{0}}_{\\beta ,0}={z_{0}}_{\\beta ,0}=0$ .","Identities derived in this way are conventionally called Ward identities.","The maps $T$ and $\\bar{T}$ are infinitesimal generators of two global internal supersymmetries of the $^{0|2}$ model.","These supersymmetries are explicitly broken if $h \\ne 0$ .","They are analogues of infinitesimal Lorentz boosts or infinitesimal rotations.","Together with a further internal symmetry corresponding to rotations in the $\\xi , \\eta $ plane, these operators generate the symmetry algebra $\\mathfrak {osp}(1|2)$ of the $^{0|2}$ model.","For details and further explanations, see [18].","As generators of continuous symmetries, $T$ and $\\bar{T}$ imply Ward identities that are not available for the Potts model with $q\\ge 1$ .","These identities are crucial for our analysis and will be discussed below.","The phase transition of the arboreal gas corresponds to a spontaneous breaking of the above supersymmetries in the infinite volume limit.","This is shown in our next theorem for the $^{0|2}$ model from which Theorems REF and REF follow immediately by (REF )–() (except for the same statements relating the constants, which we omitted here); a similar reformulation applies to Theorem REF .","Let $d\\ge 3$ and $L\\ge L_0(d)$ .","There exists $\\beta _0\\in (0,\\infty )$ and constants $\\theta _d(\\beta )=1+O(1/\\beta )$ and ${\\sf c}_i(\\beta ) = {\\sf c}_i+O(1/\\beta )$ and $\\kappa >0$ (all dependent on $d$ ) such that for $\\beta \\ge \\beta _0$ , $\\lim _{h\\downarrow 0}\\lim _{N\\rightarrow \\infty }{z_0}_{\\beta ,h}&= \\theta _d(\\beta )\\\\\\lim _{h\\downarrow 0}\\lim _{N\\rightarrow \\infty }{\\xi _0\\eta _x}_{\\beta ,h}&= \\frac{{\\sf c}_1(\\beta )}{\\beta |x|^{d-2}} + O(\\frac{1}{\\beta |x|^{d-2+\\kappa }})\\\\\\lim _{h\\downarrow 0}\\lim _{N\\rightarrow \\infty }{{z_0z_x}_{\\beta ,h}-{z_0}_{\\beta ,h} {z_x}_{\\beta ,h}}&= -\\frac{{\\sf c}_2(\\beta )}{\\beta ^2 |x|^{2d-4}} + O(\\frac{1}{\\beta ^2|x|^{2d-4+\\kappa }}).$ In particular, $\\lim _{h\\downarrow 0}\\lim _{N\\rightarrow \\infty }{u_0\\cdot u_x}_{\\beta ,h} = -\\theta _d(\\beta )^2 -\\frac{2{\\sf c}_1(\\beta )}{\\beta |x|^{d-2}} + O(\\frac{1}{\\beta |x|^{d-2+\\kappa }}).$ In fact, the constants ${\\sf c_{i}}(\\beta )$ both satisfy ${\\sf c_{i}}(\\beta ) = c_{d}^{i} + O(1/\\beta )$ , where $c_{d}$ is the leading constant in the asymptotics of the Green function of the Laplacian $-\\Delta ^{^{d}}$ on $^{d}$ : $(-\\Delta ^{^d})^{-1}(0,x)= \\frac{c_{d}}{|x|^{d-2}} + O(|x|^{-(d-2)-1}).$ Our proof of Theorem REF is by a rigorous renormalisation group analysis aided by Ward identities.","The starting point is setting $\\psi =\\sqrt{\\beta }\\eta $ and $\\bar{\\psi }= \\sqrt{\\beta }\\xi $ ; the fermionic density in (REF ) is then equivalent to $\\exp {- (\\psi ,-\\Delta \\bar{\\psi })-\\frac{1}{\\beta }(1+h) \\sum _{x\\in \\Lambda }\\psi _x\\bar{\\psi }_x - \\frac{1}{2\\beta } \\sum _{x\\in \\Lambda } \\psi _x\\bar{\\psi }_x \\sum _{e\\in \\mathcal {E}_d} \\psi _{x+e} \\bar{\\psi }_{x+e}},$ where the 1 in the quadratic term arises from putting the volume form $1/z = e^{+\\xi \\eta }=e^{-\\eta \\xi }$ into the exponential, and $\\mathcal {E}_{d}=\\lbrace e_{1},\\dots , e_{2d}\\rbrace $ are the standard unit vectors (where $e_{d+j}=-e_{j}$ ).","The reformulation (REF ) looks very much like a fermionic version of the $\\varphi ^4$ spin model.","However, the following differences are important: The coupling constants of the quadratic and quartic terms are related.","This relation is due to the geometric origin of the model as a non-linear sigma model and analogous relations are present in intrinsic coordinates for other sigma models like the vector $O(n)$ model.","We will use the following Ward identity for the $^{0|2}$ model to address this point: $ {z_0}_{\\beta ,h} = {T\\xi _0}_{\\beta ,h}= -\\sum _{x\\in \\Lambda } h {\\xi _0 Tz_x}_{\\beta ,h}= h \\sum _{x\\in \\Lambda } {\\xi _0 \\eta _x}_{\\beta ,h},$ where $T$ is the symmetry generator (REF ).","Due to the fermionic nature of the field, the quartic term actually has gradients in it: denoting the discrete gradient in direction $e$ by $(\\nabla _e\\psi )_x = \\psi _{x+e}-\\psi _x$ , it can be written as $\\frac{1}{2}\\psi _x\\bar{\\psi }_x \\sum _{e\\in \\mathcal {E}_d} \\psi _{x+e}\\bar{\\psi }_{x+e}=\\frac{1}{2}\\psi _x\\bar{\\psi }_x \\sum _{e\\in \\mathcal {E}_d} (\\nabla _e \\psi )_x(\\nabla _e \\bar{\\psi })_x\\psi _x\\bar{\\psi }_x (\\nabla \\psi )_x(\\nabla \\bar{\\psi })_x,$ where we introduced the shorthand notation $(\\nabla \\psi )_x(\\nabla \\bar{\\psi })_x= \\frac{1}{2} \\sum _{e\\in \\mathcal {E}_d}(\\nabla _e \\psi )_x(\\nabla _e \\bar{\\psi })_x$ .","After taking into account the points above, power counting heuristics (which we expect can be generalised to all non-linear sigma models with continuous symmetry) predict that the lower critical dimension for spontaneous symmetry breaking with free field low temperature fluctuations is two for the $^{0|2}$ model.","In conjunction with [18], our results rigorously establish that the lower critical dimension is two for the $^{0|2}$ model."],["Background on non-linear sigma models and renormalisation","The low temperature renormalisation group analysis of non-linear sigma models with non-abelian continuous symmetry is a notorious problem that was famously considered by Balaban for the case of $O(n)$ symmetry, see [9], [10] and references therein.","Our comparatively simple analysis of the $^{0|2}$ model, which is a non-linear sigma model with non-abelian continuous $OSp(1|2)$ symmetry, is made possible mainly by the fact that it does not suffer a large field problem because it has a fermionic representation.","In addition to this, our approach to the $^{0|2}$ model differs from Balaban's approach to the $O(n)$ model on a conceptual level, in that it is based on intrinsic coordinates as opposed to extrinsic ones.","It is unclear to us how to implement an extrinsic approach in our situation of $OSp(1|2)$ symmetry.","Somewhat remarkably, despite its simplicity, the $^{0|2}$ model has all of the main features present in the non-abelian $O(n)$ models, including: absence of spontaneous symmetry breaking in 2d (proven in [18]); mass generation in 2d (conjectured in [35]); and a spontaneous symmetry breaking phase transition with massless low temperature fluctuations in $d\\ge 3$ (the main result of this work).","The $^{0|2}$ model is a member of the family of hyperbolic sigma models with target spaces $^{n|2m}$ , see [36] for a discussion of some aspects of this.","By supersymmetric localisation the observables of the $^{0|2}$ model considered in Theorem REF are equivalent to the analogous ones of the non-linear sigma model with target $^{2|4}$ .","While this relation does not play a role in this paper, it leads to a more direct representation of the continuous symmetry breaking observed here.","In brief, in the $^{2|4}$ model each vertex comes equipped with two real and four Grassmann fields.","By expressing these fields in horospherical coordinates one of the real fields and the four Grassmann fields can be integrated out.","The marginal distribution of the remaining real field, which is called the $t$ -field, may be viewed as a `$\\nabla \\phi $ ' random surface model, albeit with a nonconvex and nonlocal Hamiltonian.","By this we mean that the potential is invariant under the global translation $t_x\\mapsto t_x +r$ for $r\\in $ .","See [18] for more details, where this perspective was used to prove the absence of symmetry breaking in $d=2$ .","The full $^{n|2m}$ family has been important for advancing our understanding of other aspects of these models [18], [36].","Of particular note, we mention that the $^{2|2}$ model has received substantial prior attention due to its exact connection to linearly reinforced random walks and its motivation from random matrix theory, see [64], [71], [72], [70], [40].","For hyperbolic sigma models with target $^n$ , $n \\ge 1$ , spontaneous symmetry breaking for all $\\beta >0$ was shown in [70], and with target $^{2|2}$ for $\\beta $ large in [40] (see also [39]).","For motivation from random matrix theory and the Anderson transition see [68], [69].","These proofs make essential use of the horospherical coordinates mentioned above.","Moreover, the proof of symmetry breaking for the $^{2|2}$ model in [40] relies on an infinite number of Ward identifies resulting from supersymmetric localisation.","These identities are absent in the $^{0|2}$ model, limiting the applicability of the methods of [40] to our setting.","At the same time, the $^{2|2}$ model has no purely fermionic representation, and so our methods do not apply there, at least without significant further developments.","Introductions to fermionic renormalisation include [20], [59], [65], see also [47].","Recent probabilistic applications of these approaches to fermionic renormalisation include the study of interacting dimers [48], [49] and two-dimensional finite range Ising models [46], [45], [7].","Our organisation of the renormalisation group is instead based on a finite range decomposition, and follows [27] and its further developments in [30], [31], [15], [32], [33], [16], [28], [11].","This approach has its origins in [26].","For an introduction to this approach in a hierarchical bosonic context see [17].","Previous applications of this approach include the study of 4d weakly self-avoiding walks [14], [13]; the nearest-neighbour critical 4d $|\\varphi |^4$ model [12], [67] and long-range versions thereof [66], [56]; the ultraviolet $\\varphi ^4_3$ problem [25], [29]; analysis of the Kosterlitz–Thousless transition of the 2d Coulomb gas [38], [42]; the Cauchy–Born problem [1]; and others.","While the construction of the bulk renormalisation group flow is simpler for the intrinsic representation of the $^{0|2}$ model than in many of the previous references, a crucial novelty of our present work is the combination of the finite range renormalisation group approach with Ward identities, together with a precise analysis of a nontrivial zero mode.","This has enabled us to apply these methods to a non-linear sigma model in the phase of broken symmetry.","It would be extremely interesting to understand this approach for bosonic non-linear sigma models where, while `large fields' cause serious complications, the formal perturbative analysis is very much in parallel to the fermionic version we study in this paper.","Ward identities of a different type have previously been used in the renormalisation group analyses in [8] and [21] and many follow-up works including [48], [49].","Finally, we mention that Theorem REF yields quantitative finite volume statements.","The proof implements a rigorous finite size analysis along the lines of that proposed in [24].","It would be very interesting to extend this to even higher precision as discussed in Section REF below."],["Future directions for the arboreal gas","In this section we discuss several interesting open directions, including the geometric structure of the weak infinite volume limits of the arboreal gas and its relation to the uniform spanning tree, and a conjectural finite size universality similar to Wigner–Dyson universality from random matrix theory."],["Finite volume behaviour","The detailed finite volume behaviour of the arboreal gas would be very interesting to understand beyond the precision of Theorem REF .","On the complete graph at supercritical temperatures it is known that there is a unique macroscopic cluster, and that there are an unbounded number of clusters whose sizes are of order $|\\Lambda |^{2/3}$ [57].","The fluctuations of the macroscopic cluster are non-Gaussian of scale $|\\Lambda |^{2/3}$ and the distribution of the ordered cluster sizes of the mesoscopic clusters has been determined [57].","The joint law of the mesoscopic clusters can be characterised [58].","Intriguingly, $|\\Lambda |^{2/3}$ is the size of the largest tree at criticality on the complete graph.","The order statistics of the supercritical mesoscopic clusters follow the critical point order statistics [58].","Going beyond the complete graph, is this distribution of ordered cluster sizes universal, at least in sufficiently high dimensions?","This would be similar to the conjectured universality of Wigner–Dyson statistics from random matrix theory [60] or the conjectured universality of the distribution of macroscopic loops in loop representations of $O(n)$ (and other) spin systems [61], [51].","More generally it would be an instance of the universality of low temperature fluctuations in finite volume in models with continuous symmetries.","Finally, we mention that on expander graphs the existence of a phase transition for the arboreal gas is not difficult to show by using a natural split–merge dynamics [50].","It would be interesting if this dynamical approach could also be used to obtain information about the cluster size distribution.","As mentioned previously, the arboreal gas is also known as the uniform forest model [52].","We emphasise that the arboreal gas is not what is typically known as the uniform spanning forest (USF), which is in fact the weak limit as $\\Lambda _N \\uparrow ^d$ of a uniform spanning tree (UST) [62].","On a finite graph, the UST is the $\\beta \\rightarrow \\infty $ limit of the arboreal gas.","The correct scaling of the external field for this limit is $h=\\beta \\kappa $ and we thus write $¶_{\\text{UST},\\kappa } = \\lim _{\\beta \\rightarrow \\infty }¶_{\\beta ,\\beta \\kappa }$ for the UST on a finite graph (plus ghost vertex if $\\kappa >0$ ).","For $\\kappa >0$ , this measure is also known as the rooted spanning forest, because disregarding the connections to the ghost vertex disconnects the tree of the UST, with vertices previously connected to the ghost becoming roots.","The distributions of rooted and unrooted forests are not the same.","To help prevent confusion we will refer to the rooted spanning forests as (a special case of) the UST.","It is trivial that $¶_{\\text{UST},0}^{\\Lambda _N}[0\\leftrightarrow x]=1$ .","Nevertheless, the behaviour of the UST in the weak infinite volume limit depends on the dimension $d$ .","This limit can be defined as $¶_{\\rm UST}^{^d} =\\lim _{\\kappa \\downarrow 0}\\lim _{N\\rightarrow \\infty }¶_{\\text{UST},\\kappa }^{\\Lambda _N}$ and is independent of the finite volume boundary conditions (e.g.","free, wired, or periodic as above) imposed on $\\Lambda _N$ , see [62].","Even though the function $1_{0\\leftrightarrow x}$ is not continuous with respect to the topology of weak convergence, it is still true that $¶_{{\\rm UST}}^{^d}[0\\leftrightarrow x] = \\lim _{\\kappa \\downarrow 0}\\lim _{N\\rightarrow \\infty }¶_{{\\rm UST},\\kappa }^{\\Lambda _N}[0 \\leftrightarrow x].$ The order of limits here is essential.","In this infinite volume limit the UST disconnects into infinitely many infinite trees if $d>4$ , but remains a single connected tree if $d\\le 4$ , see [62].","Moreover, $¶_{\\rm UST}^{^d}[0 \\leftrightarrow x] + ¶_{\\rm UST}^{^d}[0\\lnot \\leftrightarrow x,|T_0|=\\infty , |T_x|=\\infty ] = 1.$ On the left-hand side, the second term vanishes if $d\\le 4$ whereas the first term tends to 0 as $|x|\\rightarrow \\infty $ if $d>4$ .","Furthermore, the geometric structure of the trees under $¶_{\\rm UST}^{^d}$ is well understood.","In particular, all trees are one-ended, meaning that removing one edge from a tree results in two trees, of which one is finite [62], [23].","For the arboreal gas, the existence and uniqueness of infinite volume limits is an open question.","Nonetheless, subsequential limits exist, and in such an infinite volume limit all trees are finite almost surely when $\\beta $ is small, while Theorem REF implies the existence of an infinite tree for $\\beta $ large.","Moreover, by Theorem REF , $ \\lim _{h\\downarrow 0}\\lim _{N\\rightarrow \\infty }{ ¶_{\\beta ,h}^{\\Lambda _N}[0 \\leftrightarrow x] + ¶_{\\beta ,h}^{\\Lambda _N}[0\\lnot \\leftrightarrow x, 0\\leftrightarrow \\mathfrak {g}, x\\leftrightarrow \\mathfrak {g}]}= \\theta _d(\\beta )^2 + \\frac{2{\\sf c}_1(\\beta )}{\\beta |x|^{d-2}} + O(\\frac{1}{\\beta |x|^{d-2+\\kappa }}).$ By analogy with the UST, we expect that only the first term on the left-hand side contributes for $d\\le 4$ and that only the second term contributes asymptotically as $|x|\\rightarrow \\infty $ for $d>4$ .","The tempting conjecture that the UST stochastically dominates the arboreal gas on the torus is consistent with these expectations.","The analogue of the left-hand side of (REF ) plays an important role in the proof of uniqueness of the infinite cluster in Bernoulli percolation in [5]; this is related to the vanishing of the second term.","As already mentioned, for the arboreal gas we only expect this to be true in $d\\le 4$ .","Beyond the questions above, it would be interesting to analyse more detailed geometric aspects of the arboreal gas.","For example, can one construct scaling limits as has been done for some spanning tree models [3], [4], [44], [6]?","Finally, we mention that a detailed analysis of the infinite volume behaviour of the arboreal gas on regular trees with wired boundary conditions has been carried out [41].","This infinite volume behaviour is consistent with the finite volume behaviour of the complete graph, e.g., at all supercritical temperatures the sizes of finite clusters have the same distribution as those of critical percolation.","Our analysis could be extended to a detailed study of the approach $h\\downarrow 0$ .","To keep the length of this paper within bounds, we do not carry this out, but here briefly comment on what we expect can be shown by extensions of our analysis.","As discussed above, a natural object is the magnetisation $M(\\beta ,h) = \\lim _{N\\rightarrow \\infty } M_{N}(\\beta ,h), \\qquad M_{N}(\\beta ,h) = ¶^{\\Lambda _{N}}_{\\beta ,h}[0\\leftrightarrow \\mathfrak {g}],$ and the corresponding susceptibility (neglecting questions concerning the order of limits) $\\chi (\\beta ,h)= {}{h}M(\\beta ,h)= \\sum _{x} \\sigma _{\\beta ,h}(x).$ Thus for the arboreal gas, the susceptibility is not the sum over $\\tau _{\\beta ,h}(x)$ as is the case for Bernoulli bond percolation, but the sum over $\\sigma _{\\beta ,h}(x)$ .","In terms of the sigma model, $\\chi $ maybe viewed as the longitudinal susceptibility, often denoted $\\chi _{||}$ .","In this interpretation, the sum over $\\tau _{\\beta ,h}(x)$ is the transversal susceptibility $\\chi _{\\perp }$ and satisfies the Ward identity $\\chi _{\\perp }(\\beta ,h)=\\sum _{x} \\tau _{\\beta ,h}(x) = h^{-1}M(\\beta ,h)$ .","This identity is crucial in our analysis.","For the longitudinal susceptibility, we expect that it would be possible to extend our analysis to show $\\chi (\\beta ,h) \\sim {\\left\\lbrace \\begin{array}{ll}C(\\beta ) h^{-1/2} & (d=3)\\\\C(\\beta ) |\\log h|& (d=4)\\\\C(\\beta ) & (d>4).\\end{array}\\right.","}$ Defining the free energy $f(\\beta ,h)=\\lim _{N\\rightarrow \\infty } |\\Lambda _N|^{-1}\\log Z_{\\beta ,h}^{\\Lambda _{N}}$ , for $\\beta \\ge \\beta _0$ the previous asymptotics suggest that $h\\mapsto f(\\beta ,h)$ is $C^2$ in $d>4$ but only $C^1$ for $d=3,4$ .","In fact, extrapolating from our renormalisation group analysis we believe that for $\\beta \\ge \\beta _0$ the free energy is $C^n$ but not $C^{n+1}$ as a function of $h\\ge 0$ for $n={\\frac{d-1}{2}}$ .","It is unclear how this is connected to the geometry of the component graph of the UST, which also changes as the dimension is varied [22], [53]."],["Organisation and notation","This paper is organised as follows.","In Section , we show how Theorem REF is reduced to renormalisation group results with the help of the Ward identity (REF ).","In Sections – we then prove these renormalisation group results.","Section is concerned with the construction of the bulk renormalisation group flow and Section uses this analysis to compute the susceptibility.","Section then constructs the renormalisation group flow for observables, which is used in Section to compute pointwise correlation functions.","The short Section then collects the results.","Finally, in Appendix we collect relations between the arboreal gas and the $^{0|2}$ model as well as basic percolation and high temperature properties of the arboreal gas, and in Appendix we include some background material about the renormalisation group method.","Throughout we use $a_{n}\\sim b_{n}$ to denote $\\lim _{n\\rightarrow \\infty }a_{n}/b_{n}=1$ , $a_{n}\\asymp b_{n}$ to denote the existence of $c,C>0$ such that $c a_{n}\\le b_{n}\\le Ca_{n}$ , $a_n \\lesssim b_n$ if $a_n \\le Cb_n$ , and $a_{n}=O(b_{n})$ if $|a_{n}|\\lesssim |b_{n}|$ .","We consider the dimension $d \\ge 3$ to be fixed, and hence allow implicit constants to depend on $d$ .","In Sections 1 and 2 we allow implicit constants to depend on $L$ as well, as this dependence does not play a role.","In subsequent sections $L$ -dependence is made explicit, though uniformity in $L$ is only crucial in the contractive estimate of Theorem REF .","Our main theorems hypothesise $L=L(d)$ is large, and for geometric convenience we will assume throughout that $L$ is at least $2^{d+2}$ ."],["Consequences of combining renormalisation and Ward\nidentities","In our renormalisation group analysis, which provides the foundation for the proofs of the theorems stated in Section , we will first drop the constraint between the coupling constants of the quadratic and quartic terms in (REF ).","The constraint will be restored in the end with the help of the Ward identity (REF ), i.e., $ {z_0}_{\\beta ,h}= h \\sum _{x\\in \\Lambda } {\\xi _0 \\eta _x}_{\\beta ,h},\\quad \\text{ and in particular }{z_0}_{\\beta ,0}= 0.$ The application of this Ward identity is the subject of this section.","In our analysis we distinguish between two orders of limits.","We first analyse the `infinite volume' limit $\\lim _{h\\downarrow 0} \\lim _{N\\rightarrow \\infty }$ , and prove Theorem REF (and thus Theorems REF –REF ).","Using results of this analysis (and with several applications of the Ward identity), we then also analyse the much more delicate `finite volume' limit $\\lim _{N\\rightarrow \\infty }\\lim _{h\\downarrow 0}$ in order to prove Theorem REF ."],["Infinite volume correlation functions","For $m^2 >0$ arbitrary and coupling constants $s_0,a_0,b_0$ , which eventually will be taken small, we consider the model with fermionic Gaussian reference measure with covariance $ C= (-\\Delta +m^2)^{-1}$ on $\\Lambda _N$ and interaction $V_0= V_0(\\Lambda _N)=\\sum _{x\\in \\Lambda _N}{s_0 (\\nabla \\psi )_x(\\nabla \\bar{\\psi })_x + a_0 \\psi _x\\bar{\\psi }_x + b_0 \\psi _x\\bar{\\psi }_x(\\nabla \\psi )_x(\\nabla \\bar{\\psi })_x},$ where we recall the squared gradient notation from (REF ).","Thus the corresponding expectation is ${F}_{m^2,s_0,a_0,b_0} =\\frac{1}{Z_{m^2,s_0,a_0,b_0}}\\frac{1}{\\det (-\\Delta +m^2)} \\int \\partial _\\psi \\partial _{\\bar{\\psi }}\\, e^{-(\\psi ,(-\\Delta +m^2)\\bar{\\psi }) - V_0} F,$ where $\\int \\partial _\\psi \\partial _{\\bar{\\psi }}$ denotes the Grassmann integral, and $Z_{m^2,s_0,a_0,b_0}$ is defined such that ${1}_{m^2,s_0,a_0,b_0}=1$ .","The following result resembles those in [14], [13], [67] for weakly self-avoiding walks in dimension 4.","Compared to the latter results, our analysis is substantially simplified since the $^{0|2}$ model can be studied in terms of only fermionic variables with a quartic interaction that is irrelevant in dimensions $d > 2$ .","However, in Section REF , we state an improvement of the following result that sees the full zero mode of the low temperature phase.","This analysis, which relies crucially on the interplay with Ward identities, goes beyond the analysis of [14], [13], [67].","Let $d \\ge 3$ and $L \\ge L_0(d)$ .","For $b_0$ sufficiently small and $m^2 \\ge 0$ , there are $s_0 = s_0^c(b_0,m^2)$ and $a_0= a_0^c(b_0,m^2)$ independent of $N$ so that the following hold: The functions $s_0^c$ and $a_0^c$ are continuous in both variables, differentiable in $b_0$ with uniformly bounded $b_0$ -derivatives, and satisfy the estimates $ s_0^c(b_0,m^2) = O(b_0), \\qquad a_0^c(b_0,m^2) = O(b_0)$ uniformly in $m^2\\ge 0$ .","There exists $\\kappa >0$ such that if the torus sidelength satisfies $L^{-N} \\le m$ , $\\sum _{x\\in \\Lambda _N} {\\bar{\\psi }_0\\psi _x}_{m^2,s_0,a_0,b_0} = \\frac{1}{m^2}+ \\frac{O(b_0L^{-(2+\\kappa )N})}{m^4}.$ Moreover, there are functions $\\lambda = \\lambda (b_0,m^2) = 1+O(b_0), \\qquad \\gamma = \\gamma (b_0,m^2) = (-\\Delta ^{^d}+m^2)^{-1}(0,0) + O(b_0),$ having the same continuity properties as $s_0^c$ and $a_0^c$ such that ${\\bar{\\psi }_0\\psi _0}_{m^2,s_0,a_0,b_0}&= \\gamma + O(b_0L^{-\\kappa N}),\\\\{\\bar{\\psi }_0\\psi _x}_{m^2,s_0,a_0,b_0}&= (-\\Delta +m^2)^{-1}(0,x) + O(b_0|x|^{-(d-2)-\\kappa })+ O(b_0L^{-\\kappa N}),\\\\{\\bar{\\psi }_0\\psi _0;\\bar{\\psi }_x\\psi _x}_{m^2,s_0,a_0,b_0}&= - \\lambda ^{2}(-\\Delta +m^2)^{-1}(0,x)^2 + O(b_0|x|^{-2(d-2)-\\kappa })+ O(b_0L^{-\\kappa N}).$ Here ${A;B}={AB}-{A}{B}$ .","The proof of this theorem is given in Sections – and occupies most of this paper.","We now show how to derive Theorem REF for the $^{0|2}$ model from it together with the Ward identity (REF ).","To this end, assuming $s_0>-1$ we further rescale $\\psi $ by $1/\\sqrt{1+s_0}$ (and likewise for $\\bar{\\psi }$ ) in (REF ), setting $ \\xi = \\sqrt{\\frac{1+s_0}{\\beta }} \\bar{\\psi }, \\qquad \\eta =\\sqrt{\\frac{1+s_0}{\\beta }} \\psi ,$ Up to a normalisation constant, the fermionic density with respect to the fermionic Gaussian measure with covariance $C=(-\\Delta +m^2)^{-1}$ becomes, see (REF ), $\\exp {-\\sum _{x\\in \\Lambda _N}{(1+s_0) (\\nabla \\psi )_x(\\nabla \\bar{\\psi })_x+\\frac{1+s_0}{\\beta }(1+h) \\psi _x\\bar{\\psi }_x + \\frac{(1+s_0)^2}{\\beta } \\psi _x\\bar{\\psi }_x (\\nabla \\psi )_x(\\nabla \\bar{\\psi })_x}}.$ For any $m^2 \\ge 0$ and $s_0>-1$ , (REF ) is of the form (REF ) with $a_0 = \\frac{1+s_0}{\\beta }(1+h) -m^2, \\qquad b_0 = \\frac{(1+s_0)^2}{\\beta }.$ To use Theorem REF we need to invert this implicit relation between $(\\beta ,h)$ and $(m^2,s_0,a_0,b_0)$ .","This is achieved by the following corollary.","A key observation is that the Ward identity (REF ) allows us to identify the critical point with $h=0$ .","To make this precise, with $s_0^c$ and $a_0^c$ as in Theorem REF , define the functions $\\beta (b_0,m^2) &= \\frac{(1+s_0^c(b_0,m^2))^2}{b_0},\\\\h(b_0,m^2) &= -1+ \\frac{a_0^c(b_0,m^2)+m^2}{b_0} (1+s_0^c(b_0,m^2)).$ By Theorem REF , both functions are continuous in $b_0>0$ small enough and $m^2 \\ge 0$ .","(i) Assume $b_0>0$ is small enough.","Then $h(b_0,m^2) = m^2 \\beta (b_0,m^2)(1+O(b_0)).$ In particular, $h(b_0,0) = 0$ and $h(b_0,m^2)>0$ if $m^2>0$ .","(ii) For $\\beta $ large enough and $h\\ge 0$ , there are functions $\\tilde{b}_0(\\beta ,h)>0$ and $\\tilde{m}^2(\\beta ,h)\\ge 0$ such that $h(\\tilde{b}_0,\\tilde{m}^2)=h$ and $\\beta (\\tilde{b}_0,\\tilde{m}^2)=\\beta $ .","Both functions are right-continuous as $h\\downarrow 0$ when $\\beta $ is fixed.","To prove (i), we use the Ward identity (REF ) with $(\\beta ,h)$ given by (REF )–().","The left- and right-hand sides of (REF ) are, respectively, ${z_0}_{\\beta ,h} &= 1-\\frac{1+s_0^c(b_0,m^2)}{\\beta } {\\bar{\\psi }_0\\psi _0}_{m^2,s_0,a_0,b_0} ,\\\\h \\sum _{x\\in \\Lambda _N} {\\xi _0 \\eta _x}_{\\beta ,h}&= \\frac{(1+s_0^c(b_0,m^2))h(b_0,m^2)}{\\beta (b_0,m^2)} \\sum _{x\\in \\Lambda _N} {\\bar{\\psi }_0\\psi _x}_{m^2,s_0,a_0,b_0} .$ By Theorem REF , in the limit $N\\rightarrow \\infty $ , we obtain from (REF ) that if $m^{2}>0$ , the identity $1-\\frac{1+s_0^c(b_0,m^2)}{\\beta (b_0,m^2)}\\gamma (b_0,m^2) = \\frac{(1+s_0^c(b_0,m^2))h(b_0,m^2)}{\\beta (b_0,m^2) m^2}$ holds.","Solving for $h$ , we have $h(b_0,m^2)= m^2 { \\frac{\\beta (b_0,m^2)}{1+s_0^c(b_0,m^2)} - \\gamma (b_0,m^2) }.$ Since $s_0^c(b_0,m^2)=O(b_0)$ , $\\beta (b_{0},m^{2}) \\asymp 1/b_0$ , and $\\gamma (b_0,m^2)=O(1)$ , all uniformly in $m^2\\ge 0$ , we obtain $h(b_0,m^2) = m^2 \\beta (b_0,m^2)(1+O(b_0))$ .","In particular, $h(b_0,0)=0$ .","Claim (ii) follows from an implicit function theorem argument that uses that $s_0^c$ and $a_0^c$ are continuous in $m^2\\ge 0$ and differentiable in $b_0$ if $m^2>0$ with $b_0$ -derivatives uniformly bounded in $m^2>0$ .","This argument is the same as the proof of [14] (with our notation $s_0$ instead of $z_0$ , $a_0$ instead of $\\nu _0$ , $b_0$ instead of $g_0$ , and with $1/\\beta $ instead of $g$ and $h$ instead of $\\nu $ ) and is omitted here.","Assuming Theorem REF , the proof of Theorem REF is immediate from the last corollary.","The main statements of Theorems REF and REF then follow immediately, except for the identifications $\\theta _{d}(\\beta )^{2}=\\zeta _{d}(\\beta )$ , $({\\sf c}_2(\\beta )/{\\sf c}_1(\\beta )^{2})\\theta _{d}(\\beta )^{2}=1$ , and ${\\sf c}(\\beta )=2{\\sf c}_2(\\beta )$ which we will obtain in Section REF .","Given $\\beta \\ge \\beta _0$ and $h>0$ we choose $b_0>0$ and $m^2>0$ as in Corollary REF (ii).","Since $z_x = 1-\\xi _x\\eta _x$ and using (REF ) we then have ${z_0}_{\\beta ,h}&= 1-{\\xi _0\\eta _0}_{\\beta ,h}=1- \\frac{1+s_0}{\\beta } {\\bar{\\psi }_0\\psi _0}_{m^2,s_0,a_0,b_0},\\\\{\\xi _0\\eta _x}_{\\beta ,h}&= \\frac{1+s_0}{\\beta } {\\bar{\\psi }_0\\psi _x}_{m^2,s_0,a_0,b_0},\\\\{z_0z_x}_{\\beta ,h}-{z_0}_{\\beta ,h}^2&= {\\xi _0\\eta _0\\xi _x\\eta _x}_{\\beta ,h} - {\\xi _0\\eta _0}_{\\beta ,h}^2= \\frac{(1+s_0)^2}{\\beta ^2} {\\bar{\\psi }_0\\psi _0;\\bar{\\psi }_x\\psi _x}_{m^2,s_0,a_0,b_0}.$ Taking $N\\rightarrow \\infty $ and then $h\\downarrow 0$ , the results follow from Corollary REF (i) and Theorem REF with $ \\theta _d(\\beta ) = 1- \\frac{b_0\\gamma }{1+s_0^c},\\qquad {\\sf c}_1(\\beta ) = (1+s_0^c)c_d,\\qquad {\\sf c}_2(\\beta ) = \\lambda ^{2} (1+s_0^c)^2 c_d^2,$ where the functions $\\lambda $ and $\\gamma $ are evaluated at $m^2=0$ and $b_0$ given as above, $c_d$ is the constant in the asymptotics of the free Green function on $^{d}$ , see (REF ), and we have used the simplification of the error terms $O(|x|^{-(d-2)-1})+O(b_0 |x|^{-(d-2+\\kappa )}) = O(|x|^{-(d-2+\\kappa )})$ and $O(|x|^{-2(d-2)-1})+O(b_0 |x|^{-2(d-2)-\\kappa )}) = O(|x|^{-2(d-2)-\\kappa )})$ ."],["Finite volume limit","The next theorem extends Theorem REF by more precise estimates valid in the limit $m^2\\downarrow 0$ with $\\Lambda _N$ fixed.","In these estimates $t_N \\in (0,1/m^{2})$ is a continuous function of $m^2>0$ that satisfies $t_N = \\frac{1}{m^2} - O(L^{2N}),\\qquad \\text{and}\\\\\\lim _{m\\downarrow 0}{ (-\\Delta +m^2)^{-1}(0,x) - \\frac{t_N}{|\\Lambda _N|}}= (-\\Delta ^{^d})^{-1}(0,x) + O(L^{-(d-2)N}),$ where on the right-hand side $\\Delta ^{^d}$ is the Laplacian on $^d$ , on the left-hand side $\\Delta $ is the Laplacian on $\\Lambda _N$ , and $|\\Lambda _N| = L^{dN}$ denotes the volume of the torus $\\Lambda _N$ .","We define $W_N(x) = W_{N,m^2}(x) = (-\\Delta +m^2)^{-1}(0,x) - \\frac{t_N}{|\\Lambda _N|}.$ In the following, $\\Lambda _{N}$ is fixed and the parameters $(\\beta ,h)$ are related to $(m^2,s_0,a_0,b_0)$ as in Corollary REF .","We will write ${\\cdot }_{m^2,b_0} ={\\cdot }_{m^2,s_0^c(b_0,m^2),a_0^c(b_0,m^2),b_0}$ for the corresponding expectation and similarly for the partition function $Z_{m^2,b_0}$ .","Under the conditions of Theorem REF except that we no longer restrict $L^{-N}\\le m$ , in addition to the functions $a^{c}_{0}$ , $s^{c}_{0}$ , $\\lambda $ , and $\\gamma $ , there are functions $\\tilde{a}_{N,N}^c = \\tilde{a}_{N,N}^c(b_0,m^2)$ and $u_N^c = u_N^c(b_0,m^2)$ , both continuous in $b_0$ small and $m^2\\ge 0$ , as well as $\\tilde{u}_{N,N}^c = \\tilde{u}_{N,N}^c(b_0,m^2)= t_N\\tilde{a}_{N,N}^c(b_0,m^2) + O(b_0L^{-\\kappa N}),$ continuous in $b_0$ small and $m^2> 0$ , such that, for $x\\in \\Lambda _N$ , $ \\sum _{x\\in \\Lambda _N} {\\bar{\\psi }_0\\psi _x}_{m^2, b_0}&= \\frac{1}{m^2}- \\frac{1}{m^4}\\frac{\\tilde{a}_{N,N}^c}{1+\\tilde{u}_{N,N}^c}, \\\\{\\bar{\\psi }_0\\psi _0}_{m^2,b_0}&=\\gamma + \\frac{\\lambda t_N |\\Lambda _N|^{-1}}{1+\\tilde{u}_{N,N}^c} +E_{00},\\\\{\\bar{\\psi }_0\\psi _0\\bar{\\psi }_x\\psi _x}_{m^2,b_0}&=-\\lambda ^2W_N(x)^2+\\gamma ^2+\\frac{-2\\lambda ^2W_N(x)+ 2\\lambda \\gamma }{1+\\tilde{u}_{N,N}^c} t_N|\\Lambda _N|^{-1}+E_{00xx},$ and $Z_{m^{2},b_{0}} = e^{-u_{N}^{c}|\\Lambda _N|}(1+\\tilde{u}_{N,N}^{c}).$ The remainder terms satisfy $E_{00}&=\\frac{O(b_0L^{-(d-2+\\kappa )N}+b_0L^{-\\kappa N}(m^2|\\Lambda _N|)^{-1})}{1+\\tilde{u}_{N,N}^c},\\\\E_{00xx}&= O(b_0|x|^{-2(d-2)-\\kappa }) + O(b_0L^{-(d-2+\\kappa )N})&\\qquad +(O(b_0|x|^{-(d-2+\\kappa )})+ O(b_0L^{-\\kappa N})) \\frac{(m^2|\\Lambda _N|)^{-1}}{1+\\tilde{u}_{N,N}^c}.$ The proof of this theorem is again given in Sections –.","The proof of Theorem REF is based on Theorem REF and several upcoming lemmas.","These lemmas exploit Ward identities to relate the functions given by the theorem to one another.","To orient the reader, the first two lemmas below can be viewed as preparatory for the key computation in Lemma REF .","Under the conditions of Theorem REF , $^{\\Lambda _{N}}_{\\beta ,0}|T_0|=\\frac{b_0}{1+s_0^c(b_0,0)} \\frac{1+O(b_0L^{-\\kappa N})}{\\tilde{a}_{N,N}^{c}(b_{0},0)} + O(b_0L^{2N}).$ In particular, if $b_0>0$ this implies $1/\\tilde{a}_{N,N}^{c}(b_{0},0) = O(|\\Lambda _{N}|/b_{0})$ and $\\tilde{a}_{N,N}^{c}(b_{0},0)>0$ .","From (REF ), we have that $^{\\Lambda _{N}}_{\\beta ,0}|T_0|=\\sum _{x\\in \\Lambda _N} ¶_{\\beta ,0}[0\\leftrightarrow x] = \\sum _{x\\in \\Lambda _N} {\\xi _0\\eta _x}_{\\beta ,0}=\\lim _{h\\rightarrow 0} \\sum _{x\\in \\Lambda _N} {\\xi _0\\eta _x}_{\\beta ,h}.$ Changing variables, $\\sum _{x\\in \\Lambda _N} {\\xi _0\\eta _x}_{\\beta ,h}= \\frac{b_0}{1+s_0^c(b_0,m^2)} \\sum _{x\\in \\Lambda _N} {\\bar{\\psi }_0\\psi _x}_{m^2,b_0},$ where $(\\beta , h)$ and $(b_0, m^2)$ are related as in (REF ) and ().","To evaluate the right-hand side we use (REF ).","Note that $\\frac{1}{m^2}- \\frac{1}{m^4}\\frac{\\tilde{a}_{N,N}^c}{1+\\tilde{u}_{N,N}^c}&= \\frac{1}{m^2}\\frac{1+\\tilde{u}_{N,N}^c - \\tilde{a}_{N,N}^cm^{-2}}{1+\\tilde{u}_{N,N}^c}&= \\frac{1+\\tilde{a}_{N,N}^c (t_N-m^{-2})+O(b_0L^{-\\kappa N})}{m^2+\\tilde{a}_{N,N}^ct_Nm^2 + O(b_0m^2L^{-\\kappa N})}&= \\frac{1+\\tilde{a}_{N,N}^c O(L^{2N})+O(b_0L^{-\\kappa N})}{m^2+\\tilde{a}_{N,N}^c(1+O(m^2L^{2N})) + O(b_0m^2L^{-\\kappa N})},$ where the second equality is due to (REF ) and the third follows from (REF ).","As $m^2\\downarrow 0$ , the right-hand side of the third equality behaves asymptotically as $\\frac{1+\\tilde{a}_{N,N}^c O(L^{2N})+O(b_0L^{-\\kappa N})}{\\tilde{a}_{N,N}^c}=\\frac{1+O(b_0L^{-\\kappa N})}{\\tilde{a}_{N,N}^c}+O(L^{2N}).$ Since $s_{0}^{c}(b_{0},0)=O(b_{0})$ by Theorem REF we therefore obtain the first claim: $_{\\beta ,0}^{\\Lambda _N}|T_0|=\\frac{b_0}{1+s_0^c(b_0,0)}\\lim _{m^2\\downarrow 0} \\sum _{x\\in \\Lambda _N} {\\bar{\\psi }_0\\psi _x}_{m^2,b_0}=\\frac{b_0}{1+s_0^c(b_0,0)} \\frac{1+O(b_0L^{-\\kappa N})}{\\tilde{a}_{N,N}^{c}(b_{0},0)} + O(b_0L^{2N})$ For the second claim, let us observe that on the one hand $ Z_{\\beta ,h}= {\\frac{\\beta }{1+s_0^c}}^{|\\Lambda _N|}(\\det (-\\Delta +m^2)) Z_{m^2, b_0}= {\\frac{\\beta e^{-u_{N}^{c}}}{1+s_0^c}}^{|\\Lambda _N|}(\\det (-\\Delta +m^2))(1+\\tilde{u}_{N,N}^{c}),$ where the first equality is by Proposition REF and (REF ), (REF ), and (REF ), and the second equality is (REF ).","On the other hand, by (REF ), $\\lim _{h\\rightarrow 0} Z_{\\beta ,h}= Z_{\\beta ,0}>0.$ Since, by Theorem REF , $u_{N}^{c}$ and $s_0^c$ remain bounded as $m^2\\downarrow 0$ with $\\Lambda _N$ fixed (and thus also $\\beta $ which is given by (REF )), from $\\det (-\\Delta +m^2) \\downarrow 0$ , we conclude that $1+\\tilde{u}_{N,N}^c$ diverges as $m^2\\downarrow 0$ .","By (REF ), this implies $\\tilde{a}_{N,N}^{c}(b_{0},0)>0$ .","The upper bound on $1/\\tilde{a}_{N,N}^c(b_0,0)$ follows by re-arranging (REF ) and using the trivial bound $|T_0| \\le |\\Lambda _N|$ .","Under the conditions of Theorem REF and if $b_{0}>0$ , $1=\\frac{b_0}{1+s_0^c(b_0,0)}{\\gamma (b_0,0) + \\frac{\\lambda (b_0,0)}{|\\Lambda _N|\\tilde{a}_{N,N}^c(b_0,0)}(1+O(b_0L^{-\\kappa N}))}.$ The Ward identity ${z_0}_{\\beta ,0}=0$ implies $0={z_0}_{\\beta ,0}= 1-{\\xi _0\\eta _0}_{\\beta ,0}&=1-\\lim _{m^2\\downarrow 0} \\frac{1+s_0^c(b_0,m^2)}{\\beta (b_0,m^2)}{\\bar{\\psi }_0\\psi _0}_{m^2,b_0}& =1-\\lim _{m^2\\downarrow 0} \\frac{b_0}{1+s_0^c(b_0,m^2)}{\\bar{\\psi }_0\\psi _0}_{m^2,b_0},$ where we used (REF ) and that $\\beta =\\beta (b_0,m^2)$ is as in (REF ).","To compute ${\\bar{\\psi }_0\\psi _0}_{m^2,b_0}$ , we apply ().","Since $\\tilde{u}_{N,N}^c = \\tilde{a}_{N,N}^ct_N +O(b_0L^{-\\kappa N})$ and $t_N = m^{-2}+O(L^{2N})$ , $\\lim _{m^2\\downarrow 0}{\\bar{\\psi }_0\\psi _0}_{m^2,b_0}&= \\gamma (b_{0},0) + \\lim _{m^{2}\\downarrow 0} \\frac{\\lambda (b_{0},m^{2})t_{N}|\\Lambda _{N}|^{-1}}{1+\\tilde{a}_{N,N}^{c}(b_{0},m^{2})t_{N}+O(b_{0}L^{-\\kappa N})} +\\lim _{m^{2}\\downarrow 0}E_{00}&= \\gamma (b_{0},0) +\\frac{\\lambda (b_{0},0)|\\Lambda _{N}|^{-1}}{\\tilde{a}_{N,N}^c(b_{0},0)} + \\lim _{m^{2}\\downarrow 0}E_{00}.$ The limits in the second line exist by Theorem REF and Lemma REF , which in particular implies $\\tilde{a}_{N,N}^c(b_0,0) > 0$ since $b_0>0$ .","As $m^2\\downarrow 0$ , the error term $E_{00}$ is bounded by $O(b_0L^{-\\kappa N}/(|\\Lambda _N|\\tilde{a}_{N,N}^c))= (\\lambda (b_{0},0) |\\Lambda _N|^{-1}/\\tilde{a}_{N,N}^c)O(b_0L^{-\\kappa N})$ since $\\lambda (b_{0},0) =1-O(b_0)\\ge 1/2$ , finishing the proof.","Given Theorem REF , the following lemma is the main step in the proof of Theorem REF .","Under the conditions of Theorem REF and if $b_{0}>0$ , $¶_{\\beta ,0}^{\\Lambda _N}[0\\leftrightarrow x]= \\theta _d(\\beta )^2+2 \\frac{b_0}{1+s_0^c}\\lambda \\theta _d(\\beta )(-\\Delta ^{^d})^{-1}(0,x)\\\\+ O(b_0^2|x|^{-(d-2)-\\kappa })+ O(b_0L^{-(d-2)N})+ O(b_0^2L^{-\\kappa N}),$ where $\\theta _d(\\beta )$ is defined in (REF ).","By the last expression for $¶_{\\beta ,0}[0\\leftrightarrow x]$ in (REF ) and (REF ), (REF ): $ ¶_{\\beta ,0}^{\\Lambda _N}[0\\leftrightarrow x]= 1-\\lim _{h\\downarrow 0}{\\xi _0\\eta _0\\xi _x\\eta _x}_{\\beta ,h}= 1- \\lim _{m^2\\downarrow 0}{ \\frac{b_0^2}{(1+s_0^c)^2}{\\bar{\\psi }_0\\psi _0\\bar{\\psi }_x\\psi _x}_{m^{2},b_{0}}}.$ To compute $\\lim _{m^2\\downarrow 0}{\\bar{\\psi }_0\\psi _0\\bar{\\psi }_x\\psi _x}_{m^{2},b_{0}}$ we start from ().","By Lemma REF , as $m^2\\downarrow 0$ with $\\Lambda _N$ fixed, $\\frac{1}{1+\\tilde{u}_{N,N}^c} \\sim \\frac{1}{m^{-2}\\tilde{a}_{N,N}^c(b_0,0)}=O(\\frac{m^2|\\Lambda _N|}{b_0}).$ This implies the error term in () is, as $m^2\\downarrow 0$ with $\\Lambda _N$ fixed, $|E_{00xx}|\\le O(|x|^{-(d-2)-\\kappa }) + O(L^{-\\kappa N}).$ For the main term we have $\\lim _{m^{2}\\downarrow 0}{\\bar{\\psi }_0\\psi _0\\bar{\\psi }_x\\psi _x}_{m^2,b_0}&=-\\lambda ^2 W_{N,0}(x)^2 +\\gamma ^2+\\lim _{m^{2}\\downarrow 0}\\frac{-2\\lambda ^2 W_{N}(x)+ 2\\lambda \\gamma }{1+\\tilde{u}_{N,N}^c} {t_N|\\Lambda _N|^{-1}}&= -\\lambda ^2 W_{N,0}(x)^2 +\\gamma ^2+ 2(-\\lambda W_{N,0}(x)+\\gamma )\\frac{\\lambda }{\\tilde{a}_{N,N}^c|\\Lambda _{N}|},$ where on the right-hand side the functions $\\lambda $ , $\\gamma $ , and $\\tilde{a}_{N,N}^c$ are evaluated at $m^2=0$ .","By Lemma REF , $\\frac{b_{0}}{1+s_{0}^{c}}\\frac{\\lambda }{\\tilde{a}_{N,N}^{c}|\\Lambda _{N}| } ={1-\\frac{b_{0}\\gamma }{1+s_{0}^{c}}}(1+O(b_0L^{-\\kappa N}))$ so that $-{\\frac{b_{0}}{1+s_{0}^{c}}}^{2}\\frac{2\\lambda ^2 W_{N,0}(x)}{\\tilde{a}_{N,N}^c|\\Lambda _{N}|}{ (1+O(b_0L^{-\\kappa N}))}&= -\\frac{2b_{0}}{1+s_{0}^{c}}(1-\\frac{b_{0}\\gamma }{1+s_{0}^{c}})\\lambda W_{N,0}(x)\\\\{\\frac{b_{0}}{1+s_{0}^{c}}}^{2}\\frac{2\\lambda \\gamma }{\\tilde{a}_{N,N}^c|\\Lambda _{N}|}{ (1+O(b_0L^{-\\kappa N}))}&= 2\\gamma \\frac{b_{0}}{1+s_{0}} - 2\\gamma ^{2}{\\frac{b_{0}}{1+s_{0}^{c}}}^{2}.$ Substituting these bounds into (REF ) and then (REF ) we obtain $¶_{\\beta ,0}^{\\Lambda _N}[0\\leftrightarrow x]&=1-{\\frac{b_{0}}{1+s_{0}^{c}}}^{2}\\lim _{m^{2}\\downarrow 0}{\\bar{\\psi }_0\\psi _0\\bar{\\psi }_x\\psi _x}_{m^2,b_0}&= (1-\\frac{\\gamma b_{0}}{1+s_{0}})^{2}+\\frac{2b_{0}\\lambda }{1+s_{0}^{c}} (1-\\frac{b_{0}\\gamma }{1+s_{0}^{c}}) W_{N,0}(x)+ {\\frac{b_{0}\\lambda W_{N,0}(x)}{1+s_{0}^{c}}}^{2}&\\qquad +O(b_0^2 L^{-\\kappa N} W_{N,0}(x)) + O(b_0^2 L^{-\\kappa N})+ O(b_0^2|E_{00xx}|).$ Using the definition (REF ) of $\\theta _d(\\beta )$ , that $W_{N,0}(x) = (-\\Delta ^{^d})^{-1}(0,x)+O(L^{-(d-2)N})$ by (), and in particular $W_{N,0}(x) = O(|x|^{-(d-2)})$ , the claim follows.","The next (and final) lemma is inessential for the main conclusions, but will allow us to identify the constants from the infinite volume and the finite volume analyses.","Under the conditions of Theorem REF and if $b_{0}>0$ , then $\\lambda \\theta _d(\\beta ) = 1$ .","Let $ w_{N} \\frac{b_{0}}{1+s^{c}_{0}} \\frac{1}{\\tilde{a}_{N,N}^{c}|\\Lambda _{N}|}= ^{\\Lambda _N}_{\\beta ,0} \\frac{|T_0|}{|\\Lambda _N|} + O(b_0L^{-\\kappa N}+b_0L^{-(d-2)N}),$ where the second equality is due to Lemma REF .","The density $^{\\Lambda _N}_{\\beta ,0} |T_0|/|\\Lambda _N|$ can also be computed by summing the estimate in Lemma REF and dividing by $|\\Lambda _N|$ .","Subtracting this result from (REF ) gives $w_N-\\theta _d(\\beta )^2=O(b_0L^{-\\kappa N}).$ On the other hand, (REF ) shows that $\\lambda w_N-\\theta _d(\\beta )= O(b_0L^{-\\kappa N}).$ The limit $w= \\lim _{N\\rightarrow \\infty } w_N$ thus exists and satisfies $\\lambda w = \\theta _d(\\beta )$ and $w=\\theta _d(\\beta )^2$ .","Since $\\theta _d(\\beta )=1- O(1/\\beta )\\ne 0$ this implies $\\lambda \\theta _d(\\beta ) = 1$ .","The proof follows by rewriting Lemma REF .","Let $c_d$ be the constant in the Green function asymptotics of (REF ), and recall the constants $\\theta _d(\\beta )$ and ${\\sf c}_i(\\beta )$ from (REF ).","Theorem REF then follows from Lemma REF by setting $\\zeta _d(\\beta )= \\theta _d(\\beta )^2, \\qquad {\\sf c}(\\beta )=(1+s_{0}^{c})2\\lambda \\theta _{d}(\\beta ) c_d,$ and simplifying the error terms using $O(b_0|x|^{-(d-2)-1})+O(b_0^2 |x|^{-(d-2+\\kappa )}) = O(\\beta ^{-1}|x|^{-(d-2+\\kappa )})$ and $O(b_0L^{-(d-2)N})+O(b_0^2 L^{-\\kappa N}) = O(\\beta ^{-1}L^{-\\kappa N})$ .","For Theorem REF , $\\zeta _d(\\beta ) =\\theta _d(\\beta )^2$ was established in the previous proof.","For Theorem REF , the identity $({\\sf c}_2(\\beta )/{\\sf c}_1(\\beta )^{2})\\theta _{d}(\\beta )^{2}=1$ is equivalent to $\\theta _{d}(\\beta )\\lambda =1$ , i.e., Lemma REF .","Similarly, ${\\sf c}(\\beta )=2\\lambda \\theta _{d}(\\beta ) {\\sf c}_1(\\beta ) = 2{\\sf c}_1(\\beta )$ .","To compute $¶_{\\beta ,0}[0\\leftrightarrow x]$ we started from the expression $1-{\\xi _0\\eta _0\\xi _x\\eta _x}_{\\beta ,0}$ in (REF ).","An alternative route would have been to start from ${\\xi _0\\eta _x}_{\\beta ,0}$ .","For technical reasons arising in Section it is, however, easier to obtain sufficient precision when working with ${\\xi _0\\eta _0\\xi _x\\eta _x}_{\\beta ,0}$ ."],["The bulk renormalisation group flow","We will prove Theorems REF and REF by a renormalisation group analysis that is set up following [27], [33] and [13], [14]; see also [17] for a conceptual introduction.","Our proof is largely self-contained.","The exceptions to self-containment concern general properties about finite range decomposition, norms, and approximation by local polynomials that were developed systematically in [11], [30], [31].","The properties we need are all reviewed in this section.","The first six subsections set up the framework of the analysis, and the remaining three define and analyse the renormalisation group flow.","Throughout $\\Lambda \\Lambda _N$ is the discrete torus of side length $L^N$ .","We leave $L$ implicit; it will eventually be chosen large.","We sometimes omit the $N$ when it does not play a role."],["Finite range decomposition","Let $\\Delta $ denote the lattice Laplacian on $\\Lambda _N$ , and let $m^2>0$ .","Our starting point for the analysis is the decomposition $ C = (-\\Delta +m^2)^{-1} = C_1 + \\cdots +C_{N-1}+C_{N,N}$ where the $C_j$ and $C_{N,N}$ are positive semidefinite $m^2$ -dependent matrices indexed by $\\Lambda _N$ .","These covariances can be chosen with the following properties, see [17] and Appendix REF ."],["Finite range property","For $jL^j$ then, by (REF ), $_{C_j}\\theta (F_1F_2) = (_{C_j}\\theta F_1)(_{C_j}\\theta F_2).$"],["Symmetries","We briefly discuss symmetries, which are important in extracting the relevant and marginal contributions in each renormalisation group step (see Section REF below).","To use the terminology of [30], [31], we call an element $F \\in (\\Lambda )$ (globally) gauge invariant if every monomial in its representation has the same number of factors of $\\bar{\\psi }$ and $\\psi $ .","Some readers may be more familiar with the terminology symplectically invariant or $U(1)$ invariant.","Similarly, $F \\in (\\Lambda \\sqcup \\Lambda )$ is gauge invariant if the combined number of factors of $\\bar{\\psi }$ and $\\bar{\\zeta }$ is the same as the combined number of factors of $\\psi $ and $\\zeta $ .","We denote by $_{\\rm sym}(X)$ the subalgebra of $(X)$ of gauge invariant elements and likewise for $_{\\rm sym}(\\Lambda \\sqcup \\Lambda )$ .","The maps $\\theta $ and $_{C}$ preserve gauge symmetry.","A bijection $E\\colon \\Lambda \\rightarrow \\Lambda $ is an automorphism of the torus $\\Lambda $ if it maps nearest neighbours to nearest neighbours.","Bijections act as homomorphisms on the algebra $(\\Lambda )$ by $E\\psi _x = \\psi _{Ex}$ and $E\\bar{\\psi }_x = \\bar{\\psi }_{Ex}$ and similarly for $(\\Lambda \\sqcup \\Lambda )$ .","If $C$ is invariant under lattice symmetries, i.e., $C(Ex,Ey)=C(x,y)$ for all automorphisms $E$ , then the convolution $_C\\theta $ commutes with automorphisms of $\\Lambda $ , i.e., $E _{C}\\theta F = _{C}\\theta E F$ .","In particular $E_{C_{j}}\\theta F = _{C_{j}}\\theta E F$ for the covariances of the finite range decomposition (REF ).","An important consequence of this discussion is that if $F\\in _{\\rm sym}(\\Lambda )$ and $F$ is invariant under lattice symmetries, then $_{C_{j}}\\theta F\\in _{\\rm sym}(\\Lambda )$ is also invariant under lattice symmetries."],["Polymer coordinates","We will use (REF ) and the decomposition (REF ) to study the progressive integration $ Z_{j+1} = _{C_{j+1}} \\theta Z_j,$ for a given $Z_{0}\\in (\\Lambda )$ .","To be concrete here, the reader may keep $Z_0=e^{-V_0(\\Lambda )}$ with $V_{0}(\\Lambda )$ from (REF ) in mind, but to compute correlation functions we will consider generalisations of this choice of $Z_{0}$ in Section .","The analysis is performed by defining suitable coordinates and norms that enable the progressive integration to be treated as a dynamical system: this is the renormalisation group.","Towards this end, this section defines local polymer coordinates as in [27], [33].","Figure: Illustration of jj-blocks when L=2L=2."],["Blocks and Polymers","Recall $\\Lambda \\Lambda _{N}$ denotes a torus of side length $L^{N}$ .","Partition $\\Lambda _N$ into nested scale-$j$ blocks $_j$ of side lengths $L^j$ where $j=0,\\dots , N$ .","Thus scale-0 blocks are simply the points in $\\Lambda $ , while the only scale-$N$ block is $\\Lambda $ itself, see Figure REF .","The set of $j$ -polymers $_j=_j(\\Lambda )$ consists of finite unions of blocks in $_j$ .","To define a notion of connectedness, say $X,Y \\in _j$ do not touch if $\\inf _{x\\in X,y\\in Y} |x-y|_\\infty > 1$ .","A polymer is connected if it is not empty and there is a path of touching blocks between any two blocks of the polymer.","The subset of connected $j$ -polymers is denoted $_j$ .","We will drop $j$ - prefixes when the scale is clear.","For a fixed $j$ -polymer $X$ , let $_j(X)$ denote the $j$ -blocks contained in $X$ and let $|_j(X)|$ be the number of such blocks.","Connected polymers $X$ with $|_j(X)| \\le 2^d$ are called small sets and the collection of all small sets is denoted $_j$ .","Polymers which are not small will be called large.","Finally, for $X \\in _j$ we define its small set neighbourhood $X^\\square \\in _j$ as the union over all small sets containing a block in $_j(X)$ , and its closure $\\bar{X}$ as the smallest $Y\\in _{j+1}$ such that $X\\subset Y$ ."],["Coordinates","For coupling constants $V_j=(z_j,y_j,a_j,b_j) \\in 4$ and a set $X \\subset \\Lambda _{N}$ , let $ V_j(X)=\\\\\\sum _{x\\in X} { y_j (\\nabla \\psi )_x (\\nabla \\bar{\\psi })_x + \\frac{z_j}{2}((-\\Delta \\psi )_x\\bar{\\psi }_x + \\psi _x(-\\Delta \\bar{\\psi })_x) + a_j \\psi _x\\bar{\\psi }_x + b_j \\psi _x\\bar{\\psi }_x (\\nabla \\psi )_x(\\nabla \\bar{\\psi })_x}.$ For the scale $j=0$ , if we set $ Z_0 = e^{-V_0(\\Lambda _{N})}$ , then the polymer coordinates take the simple form $ Z_0 = e^{-V_0(\\Lambda _{N})} = e^{-u_{0}|\\Lambda _{N}|}\\sum _{X \\subset \\Lambda _{N}} e^{-V_0(\\Lambda _{N} \\setminus X)} K_0(X), \\quad K_0(X) =1_{X=\\varnothing }, \\quad u_{0}=0.$ To study the recursion $ Z_{j+1}=_{C_{j+1}}\\theta Z_{j}$ at a general scale $j=1,\\dots ,N$ , we will make a choice of coupling constants $V_{j}$ and of polymer coordinates $(K_j(X))_{X\\in _j(X)}$ such that $ Z_{j}= e^{-u_{j}|\\Lambda _{N}|} \\sum _{X\\in _{j}} e^{-V_{j}(\\Lambda _{N}\\setminus X)} K_{j}(X).$ The $K_j(X)$ 's will be defined in such a way that they satisfy the locality and symmetry property $K_j(X) \\in _{\\rm sym}(X^\\square )$ and the following important component factorisation property: for $X,Y\\in _{j}$ that do not touch, $ K_j(X \\cup Y) = K_j(X)K_j(Y).$ Note that since they are gauge symmetric, the $K_j(X)$ are even elements of $$ , so they commute and the product on the right-hand side is unambiguous.","Using the previous identity, $K_j(X) = \\prod _{Y \\in \\operatorname{Comp}(X)} K_j(Y),$ where $\\operatorname{Comp}(X)$ denotes the set of connected components of the polymer $X$ .","In particular, each $K_j= (K_{j}(X))_{X\\in _{j}(\\Lambda _{N})}$ satisfying (REF ) can be identified with $K_j= (K_{j}(X))_{X\\in _{j}(\\Lambda _{N})}$ .","We say that $K_{j}$ is automorphism invariant if $EK_{j}(X) = K_{j}(E(X))$ and all $X\\in _{j}(\\Lambda _{N})$ for all torus automorphisms $E \\in {\\rm Aut}(\\Lambda _N)$ that map blocks in $_j$ to blocks in $_j$ .","Let $^\\varnothing _j(\\Lambda _N)$ be the linear space of automorphism invariant $K_j = (K_j(X))_{X\\in _j(\\Lambda _N)}$ with $K_j(X) \\in _{\\rm sym}(X^\\square )$ for every $X \\in _j$ .","Polymer coordinates at scale $j$ are thus a choice of $V_j$ together with an element of the space $^\\varnothing _{j}$ .","The renormalisation group map is a particular choice of a map $(V_j,K_j)\\mapsto (V_{j+1},K_{j+1})$ .","There is freedom in the choice because of the flexibility in the definition of coordinates.","The goal is to choose $V_{j}$ such that the size of the $K_{j}$ decrease rapidly as $j$ increases.","To achieve this goal we need norms to quantify the sizes of these expressions."],["Norms","We now define the $T_j(\\ell )$ norms we will use on $(\\Lambda )$ .","General properties of these norms were systematically developed in [30], to which we will refer for some proofs.","To help the reader, in places where we specialise the definitions of [30] we indicate the more general notation that is used in [30].","We start with some notation.","For any set $S$ , we write $S^*$ for the set of finite sequences in $S$ .","We write $\\Lambda _f = \\Lambda \\times \\lbrace \\pm 1\\rbrace $ and for $(x,\\sigma ) \\in \\Lambda _f$ we write $\\psi _{x,\\sigma } = \\psi _x$ if $\\sigma =+1$ and $\\psi _{x,\\sigma }=\\bar{\\psi }_x$ if $\\sigma =-1$ .","Then every element $F \\in (\\Lambda )$ can be written in the form $ F = \\sum _{z\\in \\Lambda _f^*} \\frac{1}{z!","}F_z \\psi ^z$ where $\\psi ^z = \\psi _{z_1}\\cdots \\psi _{z_n}$ if $z=(z_1,\\dots , z_n)$ .","We are using the notation that $z!=n!$ if the sequence $z$ has length $n$ .","The representation in (REF ) is in general not unique.","To obtain a unique representation we require that the $F_{z}$ are antisymmetric with respect to permutations of the components of $z$ (this is possible due to the antisymmetry of the Grassmann variables).","Antisymmetry implies that $F_{z}=0$ if $z$ has length exceeding $2|\\Lambda |$ or if $z$ has any repeated entries.","Let $p_{\\Phi }=2d$ .","The space of test functions $\\Phi _j(\\ell )$ is defined as the set of functions $g\\colon \\Lambda _f^* \\rightarrow $ , $z\\mapsto g_{z}$ together with norm $ \\Vert g\\Vert _{\\Phi _j(\\ell )} =\\sup _{n\\ge 0}\\sup _{z\\in \\Lambda _f^n} \\sup _{|\\alpha _i| \\le p_\\Phi }\\ell ^{-n}L^{j(|\\alpha _1|+\\cdots |\\alpha _n|)}|\\nabla ^{\\alpha _1}_{z_1}\\cdots \\nabla ^{\\alpha _n}_{z_n} g_z|.$ In this definition, $\\nabla _{z_i}^{\\alpha _i}$ denotes the discrete derivative $\\nabla ^{\\alpha _i}$ with multiindex $\\alpha _i$ acting on the spatial part of the $i$ th component of the finite sequence $z$ .","The $\\Phi _j(\\ell )$ norm measures spatial smoothness of test functions, which act as substitutes for fields.","Restricted to sequences of fixed length, it is a lattice $C^{p_\\Phi }$ norm at spatial scale $L^j$ and field scale $\\ell $ .","We will mainly use the following choice of $\\ell $ for $\\Phi _j(\\ell )$ : $\\ell _j = \\ell _0 L^{-\\frac{1}{2}(d-2)j}$ for a large constant $\\ell _0$ .","This choice captures the size of the covariances in the decomposition (REF ).","Indeed, since the covariances $C_{j}$ are functions of sequences of length 2, the bounds (REF ) imply $ \\Vert C_{j}\\Vert _{\\Phi _j(\\ell _j)} \\le 1,$ when $\\ell _0$ is chosen as a large ($L$ -dependent, due to the index $j+1$ on the left-hand side of (REF )) constant relative to the constants in (REF ) with $|\\alpha |\\le p_\\Phi $ ; we will always assume that $\\ell _0$ is fixed in this way.","In (REF ) we have made a slight abuse of notation to identify $C_{j}$ with the coefficient in (REF ) of $F= \\sum _{x,y} \\bar{\\psi }_{x}\\psi _{y}C_{j}(x,y)$ .","We define $T_j(\\ell )$ to be the algebra $ (\\Lambda )$ together with the dual norm $ \\Vert F\\Vert _{T_j(\\ell )} = \\sup _{\\Vert g\\Vert _{\\Phi _j(\\ell )} \\le 1}| {F,g}|,\\qquad \\text{where }{F,g} = \\sum _{z \\in \\Lambda _f^*} \\frac{1}{z!}","F_z g_z$ when $F\\in (\\Lambda )$ is expressed as in (REF ).","An analogous definition applies to $(\\Lambda \\sqcup \\Lambda )$ and we then write $T_j(\\ell \\sqcup \\ell ) T_j(\\ell )$ for this norm (with the first notation to emphasise the doubled algebra).","The $T_j(\\ell )$ norm measures smoothness of field functionals $F\\in (\\Lambda )$ with respect to fields whose size is measured by $\\Phi _j(\\ell )$ .","They therefore implement the power counting on which renormalisation relies.","Important, but relatively straightforwardly verified, properties of these norms are systematically developed in [30]; we summarise the ones we need now.","First, $T_j(\\ell )$ is a Banach algebra, i.e., the following product property holds (see [30]): $ \\Vert F_1F_2\\Vert _{T_j(\\ell )} \\le \\Vert F_1\\Vert _{T_j(\\ell )}\\Vert F_2\\Vert _{T_j(\\ell )}.$ Using the product property, we may gain some intuition regarding these norms by considering the following simple examples: $\\Vert \\psi _x\\bar{\\psi }_x\\Vert _{T_j(\\ell )} \\le \\Vert \\psi _x\\Vert _{T_j(\\ell )} \\Vert \\bar{\\psi }_x\\Vert _{T_j(\\ell )}= \\ell ^2, \\\\\\Vert (\\nabla _e\\psi )_x\\bar{\\psi }_x\\Vert _{T_j(\\ell )}\\le \\Vert \\nabla _e\\psi _x\\Vert _{T_j(\\ell )} \\Vert \\bar{\\psi }_x\\Vert _{T_j(\\ell )} = \\ell ^2 L^{-j}.$ The following more subtle example relies crucially on the antisymmetry of the coefficients $F_z$ and plays an important role for our model: $\\Vert \\psi _x\\bar{\\psi }_x\\psi _{x+e}\\bar{\\psi }_{x+e}\\Vert _{T_j(\\ell )}= \\Vert \\psi _x\\bar{\\psi }_x(\\nabla _e\\psi )_{x}(\\nabla _e\\bar{\\psi })_{x}\\Vert _{T_j(\\ell )}\\asymp \\ell ^4L^{-2j}.$ In general, and bearing in mind the antisymmetry, each factor of the fields contributes a factor $\\ell $ and each derivative a factor $L^{-j}$ .","Second, from the definition, the following monotonicity properties hold: for $\\ell \\le \\ell ^{\\prime }$ , $ \\Vert F\\Vert _{T_j(\\ell )} \\le \\Vert F\\Vert _{T_j(\\ell ^{\\prime })}, \\qquad \\Vert F\\Vert _{T_{j+1}(\\ell )} \\le \\Vert F\\Vert _{T_j(\\ell ^{\\prime })}.$ Third, the doubling map satisfies (see [30]): for $F \\in (\\Lambda )$ , $ \\Vert \\theta F\\Vert _{T_j(\\ell )} \\le \\Vert F\\Vert _{T_j(2\\ell )}$ where the norm on the left-hand side is the $T_j(\\ell ) T_j(\\ell \\sqcup \\ell )$ norm on $(\\Lambda \\sqcup \\Lambda )$ .","Finally, the following contraction bound for the fermionic Gaussian expectation is an application of the Gram inequality whose importance is well-known in fermionic renormalisation.","It is proved in [30].","Assume $C$ is a covariance matrix with $\\Vert C\\Vert _{T_j(\\ell )} \\le 1$ .","For $F \\in (\\Lambda \\sqcup \\Lambda )$ , then $ \\Vert _{C}F\\Vert _{T_j(\\ell )} \\le \\Vert F\\Vert _{T_j(\\ell )}.$ In particular, by (REF ) the fermionic Gaussian convolution satisfies $ \\Vert _{C} \\theta F\\Vert _{T_{j}(\\ell )}\\le \\Vert F\\Vert _{T_{j}(2\\ell )}.$ For our choices of $\\ell _j$ and of the finite range covariance matrices $C_j$ , the inequalities (REF ) and (REF ) in particular imply $ \\Vert F\\Vert _{T_{j+1}(\\ell _{j+1})} \\le \\Vert F\\Vert _{T_{j+1}(2\\ell _{j+1})} \\le \\Vert F\\Vert _{T_j(\\ell _j)},\\qquad \\Vert _{C_{j+1}}\\theta F\\Vert _{T_{j+1}(\\ell _{j+1})} \\le \\Vert F\\Vert _{T_j(\\ell _j)}.$ We remark that the existence of this contraction estimate for the expectation combined with (REF ) below is what makes renormalisation of fermionic fields much simpler than that of bosonic ones."],["Localisation","To define the renormalisation group map we need one more important ingredient: the localisation operators $\\text{Loc}_{X,Y}$ that will be used to extract the relevant and marginal terms from the $K_j$ coordinate to incorporate them in the renormalisation from $V_j$ into $V_{j+1}$ .","These operators are generalised Taylor approximations which take as inputs $F\\in (X)$ and produce best approximations of $F$ in a finite dimensional space of local field polynomials."],["Local field polynomials","Formal local field polynomials are formal polynomials in the symbols $\\psi ,\\bar{\\psi },\\nabla \\psi ,\\nabla \\bar{\\psi },\\nabla ^2\\psi ,...$ (without spatial index).","The dimension of a formal local field monomial is given by $(d-2)/2$ times the number of factors of $\\psi $ or $\\bar{\\psi }$ plus the number of discrete derivatives $\\nabla $ in its representation.","Concretely, we consider the following space of formal local field polynomials.","Let $^\\varnothing \\cong 4$ be the linear space of formal local field monomials of the form $ V= y (\\nabla \\psi ) (\\nabla \\bar{\\psi }) + \\frac{z}{2}((-\\Delta \\psi )\\bar{\\psi }+ \\psi (-\\Delta \\bar{\\psi }))+ a \\psi \\bar{\\psi }+ b \\psi \\bar{\\psi }(\\nabla \\psi )(\\nabla \\bar{\\psi }).$ We will identify elements $V\\in ^\\varnothing $ with their coupling constants $(z,y,a,b) \\in 4$ .","Sometimes we include a constant term and write $u+V \\in ^\\varnothing $ with $u+V\\cong (u,z,y,a,b)\\in 5$ .","Given a set $X\\subset \\Lambda $ , a formal local field polynomial $P$ can be specialised to an element of $(\\Lambda )$ by replacing formal monomials by evaluations.","For example, if $P=\\bar{\\psi }\\psi $ , $P(X) = \\sum _{x\\in X}\\bar{\\psi }_{x}\\psi _{x}$ .","We call polynomials arising in this way local polynomials.","The most important case is $V \\mapsto V(X)$ , with $ V(X) =\\sum _{x\\in X} { y (\\nabla \\psi )_x (\\nabla \\bar{\\psi })_x + \\frac{z}{2}((-\\Delta \\psi )_x\\bar{\\psi }_x + \\psi _x(-\\Delta \\bar{\\psi })_x) + a \\psi _x\\bar{\\psi }_x + b \\psi _x\\bar{\\psi }_x (\\nabla \\psi )_x(\\nabla \\bar{\\psi })_x},$ where $\\Delta -\\frac{1}{2} \\sum _{e\\in _d} \\nabla _{-e}\\nabla _{e}$ and $(\\nabla \\psi )_{x}(\\nabla \\bar{\\psi })_{x} \\frac{1}{2}\\sum _{e\\in _d}\\nabla _{e}\\psi _{x}\\nabla _{e}\\bar{\\psi }_{x}$ are the lattice Laplacian and the square of the lattice gradient; recall that $_d = \\lbrace e_1, \\dots , e_{2d}\\rbrace $ .","For a constant $u\\in we write$ u(X) = u|X|$, where $ |X|$ is the number of points in $ X $.","Thus $ (u+V)(X)= u(X)+V(X)=u|X|+V(X)$.$ For $X\\subset \\Lambda $ , define $^\\varnothing (X) = \\lbrace V(X): V\\in ^\\varnothing \\rbrace \\subset (\\Lambda )$ and analogously $(^\\varnothing )(X) = \\lbrace u|X|+V(X): u\\in \\,V \\in ^\\varnothing \\rbrace \\subset (\\Lambda )$ .","The space $^\\varnothing $ contains all formal local field polynomials of dimension at most $d$ that are (i) gauge invariant, (ii) respect lattice symmetries (if $EX=X$ for an automorphism $E$ , then $EV(X)=V(X)$ ) and (iii) have no constant terms.","Respecting lattice symmetries means that $^\\varnothing $ does not contain terms such as $(\\nabla _{e_1}\\psi )(\\nabla _{e_2}\\bar{\\psi })$ , which cannot occur if $X$ and $V(X)$ are invariant under lattice rotations.","We also emphasise that there is no $(\\bar{\\psi }\\psi )^2$ term, which would be consistent with power counting and symmetries, because it vanishes upon specialisation by anticommutativity of the fermionic variables.","Two further remarks are in order.","First, the monomial $\\psi \\bar{\\psi }(\\nabla \\psi )(\\nabla \\bar{\\psi })$ has dimension $2d-2>d$ for $d\\ge 3$ ; we include it in $^\\varnothing $ since it occurs in the initial potential.","Second, the monomials multiplying $z$ and $y$ are equivalent upon specialisation when $X=\\Lambda $ by summation by parts, and differ only by boundary terms for general $X \\subset \\Lambda $ .","This would allow us to keep only one of them, but it will be simpler to keep both.","The localisation operators $\\operatorname{Loc}_{X,Y}$ associate local field monomials to elements of $(X)$ .","In renormalisation group terminology, the image of $\\operatorname{Loc}$ projects onto the space of all relevant and marginal local polynomials.","The precise definitions of the localisation operators do not play a direct role in this paper.","Rather, only their abstract properties, summarised in the following Proposition REF , will be required.","We use the general framework developed in [31] to define these operators.","In short, the definitions of $\\operatorname{Loc}_X$ and $\\operatorname{Loc}_{X,Y}$ are those of [31].","These definitions require a choice of field dimensions, which we choose as $[\\psi ]=[\\bar{\\psi }] = (d-2)/2$ , a choice of maximal field dimension $d_+$ , which we choose as $d_{+}=d$ , and a choice of a space $\\hat{P}$ of test polynomials, which we define exactly as in [31] with the substitution $\\nabla _e\\nabla _e\\rightarrow -\\nabla _e\\nabla _{-e}$ explained in [31].","The following properties are then almost immediate from [31].","For $L=L(d)$ sufficiently large there is a universal $\\bar{C}>0$ such that: for $j1$ is a parameter that will be chosen sufficiently large.","Note that $^\\varnothing \\oplus ^\\varnothing _j(\\Lambda _N)$ is a finite-dimensional complex normed vector space with the above norms since $N<\\infty $ .","As a consequence it is a Banach space.","Let $d \\ge 3$ , $L \\ge L_0(d)$ , and $A \\ge A_0(L,d)$ .","Assume that $u_j=0$ .","There exists $\\epsilon = \\epsilon (L,A)>0$ such that for $j+10$ depending only on $d$ such that $ |_{j}(X)| \\ge (1+2\\eta ) |_{j+1}(\\bar{X})|.$ By (REF ), if $A=A(L)$ is large enough, $A^{|_{j+1}(U)|} \\sum _{X \\in _j\\setminus _{j}: \\bar{X}=U} (A/2)^{-|_j(X)|}\\le (2^{L^d} 2^{1+2\\eta }A^{-2\\eta })^{|_{j+1}(U)|} \\le A^{-\\eta |_{j+1}(U)|},$ as the set of $X\\in _{j}\\setminus _{j}$ with $\\bar{X}=U$ has size at most $2^{L^{d}|_{j+1}(U)|}$ .","Now, the contribution to (REF ) from large sets $X \\in _j \\setminus _j$ is $\\sum _{X\\in _j \\setminus _j: \\bar{X}=U} e^{-V_{j+1}(U\\setminus X)+u_{j+1}|X|} { _{C_{j+1}}\\theta K_j(X)+ _{C_{j+1}} (\\delta I)^X }.$ If $U\\in \\mathcal {C}_{j+1}\\backslash \\mathcal {S}_{j+1}$ , we proceed as follows: For $\\Vert K_j\\Vert _j + \\Vert V_j\\Vert _j \\le \\epsilon $ with $\\epsilon $ sufficiently small, by Lemma REF the $T_{j+1}(\\ell _{j+1})$ norm of the $K$ contribution to (REF ) is bounded by $ A^{|_{j+1}(U)|-2^d}\\sum _{X\\in _j \\setminus _j: \\bar{X}=U} 2^{|_j(X)|} \\Vert _{C_{j+1}}\\theta K_j(X)\\Vert _{T_{j+1}(\\ell _{j+1})}.$ By the definition of $\\Vert K_{j}\\Vert _{j}$ and noting that $(|_{j}(X)|-2^{d})_{+}=|_{j}(X)|-2^{d}$ since $X\\notin _{j}$ , $\\Vert _{C_{j+1}}\\theta K_j(X)\\Vert _{T_{j+1}(\\ell _{j+1})}\\le A^{-(|_{j}(X)|-2^{d})}\\Vert K_j\\Vert _j,$ where we have also used the contraction estimates (REF ), (REF ).","Inserting this bound into (REF ) and using (REF ) gives that the $K$ contribution to (REF ) is bounded by $ A^{|_{j+1}(U)|}\\sum _{X\\in _j \\setminus _j: \\bar{X}=U} (A/2)^{-|_j(X)|}\\Vert K_j\\Vert _{j} \\le A^{-\\eta } \\Vert K_j\\Vert _{j}.$ This is the desired bound.","Examining the $\\delta I$ contribution to (REF ), Lemmas REF and REF and the product property yield $& A^{|_{j+1}(U)|-2^d} \\Vert \\sum _{X\\in _j \\setminus _j: \\bar{X}=U} e^{-V_{j+1}(U\\setminus X)+u_{j+1}|X|} { _{C_{j+1}} (\\delta I)^X }\\Vert _{T_{j+1}(\\ell _{j+1})}&\\le A^{|_{j+1}(U)|-2^d}\\sum _{X\\in _j \\setminus _j: \\bar{X}=U} [2O(\\Vert V_j\\Vert _j + \\Vert K_j\\Vert _j)]^{|_j(X)|} .$ If $\\Vert V_j\\Vert _j + \\Vert K_j\\Vert _j<\\epsilon $ and $\\epsilon $ is sufficiently small (depending on $A$ ), then the quantity in brackets is less than $1/(4 A^{2})$ .","By the elementary inequality $(c^{2})^{n-2}\\le c^{n}$ for $c\\in (0,1)$ , $n>4$ and using that $|_{j}(X)|\\ge 2^{d}+1>4$ for each summand, we obtain the upper bound $[O(\\Vert V_j\\Vert _j + \\Vert K_j\\Vert _j)]^{2} A^{|_{j+1}(U)|} \\sum _{X\\in _j\\setminus _j: \\bar{X}=U} (A/2)^{-|_{j}(X)|}.$ Using again (REF ), it follows that the $\\delta I$ contribution to (REF ) is bounded by $O(A^{-\\eta }[\\Vert V_j\\Vert _j + \\Vert K_j\\Vert _j]^{2})$ for $A$ sufficiently large.","We have now completed the bound on (REF ) provided $U\\in \\mathcal {C}_{j+1}\\backslash \\mathcal {S}_{j+1}$ .","The argument is similar if $U \\in _{j+1}$ .","In this case the prefactor $A^{|_{j+1}(U)|-2^d}$ gets replaced by 1 in (REF ) and (REF ).","In place of (REF ) we obtain, since $1+ 2^d\\le |_j(X)|\\le L^d |_{j+1}(U)|\\le [2L]^d$ and the number of summands in this case is at most $2^{[2L]^d}$ , $\\Vert \\sum _{X\\in _j \\setminus _j: \\bar{X}=U} e^{-V_{j+1}(U\\setminus X)+u_{j+1}|X|} _{C_{j+1}}\\theta K_j(X)\\Vert _{T_{j+1}(\\ell _{j+1})}\\le A^{-1}2^{2[2L]^d}\\Vert K_j\\Vert _j=O(A^{-\\eta } \\Vert K_j\\Vert _j)$ for $A$ large enough depending on $L$ and $d$ .","On the other hand in place of (REF ) we have $\\Vert \\sum _{X\\in _j \\setminus _j: \\bar{X}=U} e^{-V_{j+1}(U\\setminus X)+u_{j+1}|X|} _{C_{j+1}}\\delta I^X\\Vert _{T_{j+1}(\\ell _{j+1})}\\le O(A^{-\\eta }[\\Vert V_j\\Vert _j + \\Vert K_j\\Vert _j]^{2})$ provided $\\epsilon $ is chosen sufficiently small depending on $L$ , since each summand on the left-hand side has $|_j(X))|\\ge 3$ .","Thus for $A=A(L,d)$ sufficiently large and $\\epsilon =\\epsilon (A,L)$ sufficiently small, the expression (REF ) is bounded in the $T_{j+1}(\\ell _{j+1})$ norm by $O(A^{-\\eta }(\\Vert K_j\\Vert _j+[\\Vert V_j\\Vert _j + \\Vert K_j\\Vert _j]^{2}))$ in all cases."],["Non-linear part","To conclude the proof of the theorem, we show $\\Vert K_{j+1}(U)-\\mathcal {L}_{j+1}(U)\\Vert _{j+1}\\le O(\\Vert K_j\\Vert _j^2+\\Vert V_j\\Vert _j^2)$ .","Recall the definition of $K_{j+1}$ from (REF ), see also Appendix REF .","We have $K_{j+1}(U)-\\mathcal {L}_{j+1}(U)=e^{u_{j+1}|U|} \\sum _{_2(U)}e^{-(u+V)_{j+1}(X_{I})} _{C_{j+1}} \\check{K}_j(\\check{X})\\prod _{(B,X)\\in }\\theta J(B,X) ,$ where $\\sum _{_{2}(U)}$ indicates that the system of polymers $(X_, \\check{X})$ contains at least two components.","In particular, if $=\\varnothing $ , $\\check{X}$ has least two components and if $\\check{X}=\\varnothing $ then $||\\ge 2$ , where $||$ is the number of pairs in $$ , see Appendix REF .","If $\\Vert V_j\\Vert _j + \\Vert K_j\\Vert _j$ is small enough, arguing as in (REF ) implies $\\Vert e^{u_{j+1}(U)}e^{-(u+V)_{j+1}( X_{I})}\\Vert _{T_{j+1}(\\ell _{j+1})}\\le 2^{|_{j}(U)|}$ .","By (REF ), $\\Vert J(B, X)\\Vert _{T_{j}(\\ell _{j})} =O(\\Vert K_j\\Vert _{j})$ .","Thus using that $_{C_{j+1}}\\theta $ contracts from $T_{j+1}(\\ell _{j+1})$ into $T_{j}(\\ell _{j})$ we obtain $\\Vert K_{j+1}(U)-\\mathcal {L}_{j+1}(U)\\Vert _{T_{j+1}(\\ell _{j+1})}\\le \\sum _{_2(U)}[O(\\Vert K_j\\Vert _j)]^{||}2^{|_j(U)|} \\Vert \\check{K}_j(\\check{X})\\Vert _{T_{j+1}(\\ell _{j+1})}.$ We first estimate the norm of $\\check{K}_j(\\check{X})$ , and then the resulting sum.","If $\\Vert V_{j}\\Vert _{j}+\\Vert K_{j}\\Vert _{j}\\le \\epsilon $ and $\\epsilon =\\epsilon (A, L)$ is sufficiently small, then $\\Vert \\check{K}(\\check{X})\\Vert _{T_{j+1}(\\ell _{j+1})} \\le [O( \\Vert V_j\\Vert _j +\\Vert K_j\\Vert _j)]^{|\\operatorname{Comp}(\\check{X})|}(\\frac{A}{2})^{-\\sum _{Y\\in \\operatorname{Comp}(\\check{X})}(|_j(Y)|-2^{d})_{+}}.$ For notational convenience, for $Y\\in \\mathcal {C}_j$ let $\\tilde{K}(Y)= \\sum _{W \\in _j(Y)} \\theta K_j(Y\\setminus W) (\\delta I)^W.$ The claimed bound follows from the definition of $\\check{K}(X)$ in (REF ) if we show $\\Vert \\tilde{K}(Y)-\\sum _{B \\in _j(Y)} {\\theta } J(B,Y)\\Vert _{T_{j+1}(\\ell _{j+1})} \\le 2^{d+1}O (\\Vert V_j\\Vert _j + \\Vert K_j\\Vert _j) (A/2)^{-(|_j(Y)|-2^{d})_{+}}$ for $Y$ a connected component of $\\check{X}$ .","We apply the triangle inequality.","Since $J(B,Y)=0$ if $Y\\notin _{j}$ , $\\Vert \\sum _{B \\in _j(Y)}_{C_{j+1}}\\theta J(B,Y)\\Vert _{T_{j+1}(\\ell _{j+1})} \\le O(\\Vert K_{j}\\Vert _{j})$ where we have used $\\Vert J(B, X)\\Vert _{T_{j}(\\ell _{j})}=O(\\Vert K_j\\Vert _{j})$ .","By (REF ), component factorisation of $K_j$ , and the contraction property of the norms and $\\theta $ , for $B\\in _j$ and $Z\\in _j$ , $\\Vert \\delta I(B)\\Vert _{T_{j+1}(\\ell _{j+1})}&\\le O(\\Vert V_j\\Vert _j+\\Vert K_j\\Vert _j), \\\\\\Vert \\theta K_j(Z)\\Vert _{T_{j+1}(\\ell _{j+1})} &\\le A^{-\\sum _{W\\in \\operatorname{Comp}(Z)} (|_j(W)|-2^d)_+}\\Vert K_j\\Vert _j^{|\\operatorname{Comp}(Z)|}.$ We now impose the condition that $\\epsilon \\le A^{-2^d}$ .","Then plugging the previous bounds into the expression for $\\tilde{K}(Y)$ we have $\\Vert \\tilde{K}(Y)\\Vert _{T_{j+1}(\\ell _{j+1})}& \\le \\sum _{Z \\in _j(Y)}\\Vert (\\delta I)^Z \\theta K_j(Y \\setminus Z)\\Vert _{T_{j+1}(\\ell _{j+1})}&\\le \\sum _{Z \\in _j(Y)}(O(\\Vert V_j\\Vert _{j}+ \\Vert K_j\\Vert _{j}))^{|_j(Z)|+|\\operatorname{Comp}(Y\\backslash Z)|} A^{-\\sum _{W\\in \\operatorname{Comp}(Y\\backslash Z)}(|_j(W)|-2^d)_+}&\\le \\sum _{Z \\in _j(Y)}\\left(A^{2^{d}}O(\\Vert V_j\\Vert _{j}+\\Vert K_j\\Vert _{j})\\right)^{|\\operatorname{Comp}(Y\\backslash Z)|}&\\qquad \\qquad \\qquad \\qquad \\times (O(\\Vert V_j\\Vert _{j}+\\Vert K_j\\Vert _{j}))^{|_j(Z)|}A^{-\\sum _{W\\in \\operatorname{Comp}(Y\\backslash Z)}|_{j}(W)|}&\\le (A^{2^{d}}O(\\Vert V_j\\Vert _{j}+ \\Vert K_j\\Vert _{j}))(O(\\Vert V_j\\Vert _{j}+ \\Vert K_j\\Vert _{j}) + A^{-1})^{|_{j}(Y)|}&\\le A^{2^{d}}\\left(\\frac{A}{2}\\right)^{-|_{j}(Y)|} O(\\Vert V_j\\Vert _{j}+\\Vert K_j\\Vert _{j}).$ (Note that if $Y$ is a small set, the factors of $A$ could have been omitted.)","Together with (REF ) this proves the lemma.","We are now ready to complete the bound on the non-linear part.","For brevity let us write $b$ for the factors $O(\\Vert V_j\\Vert _{j}+\\Vert K_j\\Vert _{j})$ above.","We have reduced verification of () to showing $A^{|_{j+1}(U)|}2^{|_{j}(U)|}\\sum _{_{2}(U)} (b2^{d+1}A^{2^{d}})^{||+|\\operatorname{Comp}(\\check{X})|}(\\frac{A}{2})^{-|_{j}(\\check{X})|} \\le O(b^2)$ for all $U\\in _{j+1}$ .","Since $|_{j}(U)|\\le L^{d}|_{j+1}(U)|$ , for any $c>0$ the prefactor can be bounded by $A^{|_{j+1}(U)|}2^{|_{j}(U)|} \\le (\\frac{A}{2})^{(1-c)|_{j+1}(U)|}2^{(L^{d}+1)|_{j}(U)|} (\\frac{A}{2})^{c|_{j+1}(U)|}.$ Taking $c>1$ , the product of the first two terms on the last right-hand side is less than 1 for $A$ sufficiently large.","It suffices to prove that for some $c>1$ , one has $(\\frac{A}{2})^{c|_{j+1}(U)|}\\sum _{_{2}(U)} (b2^{d+1}A^{2^{d}})^{||+|\\operatorname{Comp}(\\check{X})|}(\\frac{A}{2})^{-|_{j}(\\check{X})|} \\le O(b^2).$ At this point we appeal to [27]; this result estimates the same sum but over $(U)$ instead of $_{2}(U)$ .","As we are estimating $\\sum _{_{2}(U)}$ the supremum over $n\\ge 1$ in [27] becomes a supremum over $n\\ge 2$ since $|| + \\operatorname{Comp}(\\check{X})|\\ge 2$ .","This yields that if $A=A(L,d)$ is large enough, then there is a $m$ such that for all $U\\in _{j+1}$ $(\\frac{A}{2})^{c|_{j+1}(U)|}\\sum _{_{2}(U)} (b2^{d+1}A^{2^{d}})^{||+|\\operatorname{Comp}(\\check{X})|}(\\frac{A}{2})^{-|_{j}(\\check{X})|} = O( (bA^{m})^{2}),$ which is $A^{\\nu }O(b^{2})$ as needed."],["Flow of the renormalisation group","Recall the infinite volume limit of the renormalisation group maps $\\Phi _{j+1,\\infty }$ discussed below Proposition REF .","We equip $^\\varnothing _{j}(^{d})$ with the norm $\\Vert K\\Vert _{j}$ defined by ().","Next we study the iteration of the renormalisation group maps as a dynamical system.","In what follows $K_0=0$ means $K_0(X)=1_{X=\\varnothing }$ for $X\\in _j$ .","Let $d\\ge 3$ , $L\\ge L_0$ , and $A \\ge A_0(L)$ .","For $m^2 \\ge 0$ arbitrary and $b_0$ small, there exist $V_0^c(b_0,m^2)$ and $\\kappa >0$ such that if $(V_0,K_0)=(V_0^c(m^2,b_0),0)$ and $(V_{j+1},K_{j+1})= \\Phi _{j+1,\\infty }(V_j,K_j,m^2)$ is the flow of the infinite volume renormalisation group map then $ \\Vert V_j\\Vert _j = O(b_0L^{-\\kappa j}), \\qquad \\Vert K_j\\Vert _j = O(b_0^2 L^{-\\kappa j}).$ The components of $V_0^c(m^2,b_0)$ are continuous and uniformly bounded in $m^2 \\ge 0$ and differentiable in $b_0$ with uniformly bounded derivative.","The proof is by applying the well-known stable manifold theorem for smooth dynamical systems in the version given in [27].","This theorem applies to dynamical systems in Banach spaces, and hence two preparatory observations are needed.","First, for each $j$ , the linear space $^\\varnothing _{j}(^{d})$ equipped with $\\Vert \\cdot \\Vert _{j}$ is complete (when restricted to elements of finite norm), i.e., a Banach space.","Second, by the consistency of the finite volume renormalisation group maps (Proposition REF ), the estimates given in Theorem REF also hold for the infinite volume limit.","To prove the theorem we first write down the dynamical system corresponding to the renormalisation group map.","The definition of $V_{j+1}$ is (REF ).","We start with the contribution to $V_{j+1}$ arising from $_{C_{j+1}}\\theta V_{j}(B) (\\tilde{u}_{j+1}|B|+\\tilde{V}_{j+1}(B))$ .","This can be computed by the Wick formula (REF ): $\\tilde{z}_{j+1} = z_{j},\\qquad \\tilde{y}_{j+1} = y_{j} + \\kappa _{j}^{yb}b_{j},\\qquad \\tilde{a}_{j+1} = a_{j} + \\kappa _{j}^{ab}b_{j},\\qquad \\tilde{b}_{j+1} = b_{j},$ with $\\kappa _{j}^{yb} = -C_{j+1}(0)$ and $\\kappa _{j}^{ab}=\\Delta C_{j+1}(0)$ .","Since $\\Vert V_j(B)\\Vert _{T_j(\\ell _j)}$ is comparable with $|z_j| + |y_j| + L^{2j}|a_j| + L^{-(d-2)j} |b_j|$ , it is natural to rescale $\\hat{z}_{j}=z_{j}$ , $\\hat{y}_{j}=y_{j}$ , $\\hat{a}_{j}=L^{2j}a_{j}$ , $\\hat{b}_{j}=L^{-(d-2)j}b_{j}$ , $\\hat{\\kappa }_{j}^{ab}=L^{dj}\\kappa _{j}^{ab}$ , and $\\hat{\\kappa }_{j}^{yb}=L^{(d-2)j}\\kappa _{j}^{yb}$ .","The definition (REF ) of $V_{j+1}$ then becomes $\\hat{z}_{j+1} &= \\hat{z}_{j}+ \\hat{r}^{z}_{j},&\\qquad \\hat{y}_{j+1}&= \\hat{y}_{j} + \\hat{\\kappa }_{j}^{yb}\\hat{b}_{j} + \\hat{r}^{y}_{j}, \\\\\\hat{a}_{j+1}&= L^{2}\\hat{a}_{j} + \\hat{\\kappa }_{j}^{ab}\\hat{b}_{j} + \\hat{r}^{a}_{j}, &\\qquad \\hat{b}_{j+1}&= L^{-(d-2)}\\hat{b}_{j} + \\hat{r}^{b}_{j},$ where the $\\hat{r}_{j}$ are linear maps in $K_j$ and have size $O({K_{j}}_{j})$ by (REF ) of Theorem REF , and the $\\hat{\\kappa }_{j}$ are uniformly bounded in $j$ by the covariance estimates (REF ).","We set $v_j=(\\hat{y}_j,\\hat{z}_j,\\hat{a}_j)$ and $w_j=(\\hat{b}_j,K_j)$ and use $\\Vert \\cdot \\Vert $ for the norm given by maximum of the respective components.","By the computation above and Theorem REF the infinite volume renormalisation group map can be written in block diagonal form $\\begin{pmatrix}v_{j+1} \\\\ w_{j+1}\\end{pmatrix}= \\begin{pmatrix} E & B_j \\\\ 0 & D_j \\end{pmatrix}\\begin{pmatrix}v_{j} \\\\ w_{j}\\end{pmatrix}+ \\begin{pmatrix} 0 \\\\ g_{j+1}(v_j,w_j) \\end{pmatrix}$ with $\\Vert E^{-1}\\Vert = 1$ , $\\Vert B_j\\Vert $ bounded, and $\\Vert D_j\\Vert \\le \\max \\lbrace L^{-(d-2)}, O(L^{-3}+A^{-\\eta })\\rbrace \\le L^{-\\kappa }$ provided $A$ is large enough, and with $g_j(0,0)=0$ and $Dg_j(0,0)=0$ .","For every $m^2\\ge 0$ , the existence of $V_{0}^{c}(m^{2},b_{0})$ and its differentiability in $b_{0}$ now follow by [27].","To see that $V_0^c(m^2,b_0)$ is also continuous in $m^2$ , regard $v_j$ and $w_j$ as bounded continuous functions of $m^2$ , i.e., consider $v_j \\in C_b([0,\\infty ),^3)$ and $w_j\\in C_b([0,\\infty ), \\times ^\\varnothing _j(^d))$ .","Since all the estimates above are uniform in $m^2 \\ge 0$ , the previous argument then gives a solution $V_0^c \\in C_b((-\\epsilon ,\\epsilon ),C_b([0,\\infty ),^3))$ .","The bounds (REF ) are not part of the statement of [27], but are immediate from the proof.","By consistency, the finite volume renormalisation group flow for $V_j$ agrees with the infinite volume renormalisation group flow up to scale $j0$ , the $V^\\star _j$ have terms not present in $V^\\star _{0}$ , for example the terms involving $q$ in the next definition.","The nilpotency of the observable fields $\\sigma _{a}, \\sigma _{b}$ limits the possibilities.","Let $^\\star $ be the space of formal field polynomials $u^\\star +V^\\star $ of the form: $&\\left.\\begin{aligned}V^\\star &=- \\lambda _a \\sigma _a \\bar{\\psi }_a - \\lambda _b \\psi _b\\bar{\\sigma }_b+ \\sigma _a \\bar{\\sigma }_b\\frac{r}{2}(\\bar{\\psi }_a\\psi _a+\\bar{\\psi }_b\\psi _b),\\;\\,\\quad \\\\\\nonumber u^\\star &= - { \\sigma _a\\bar{\\sigma }_b } q,\\end{aligned}\\right\\rbrace \\qquad \\text{in Case (1),}\\\\& \\left.\\begin{aligned}V^\\star &={ -}\\sigma _a \\lambda _a\\bar{\\psi }_a\\psi _a - \\sigma _b\\lambda _b\\bar{\\psi }_b\\psi _b{ -}\\sigma _a\\sigma _b\\frac{\\eta }{2} (\\bar{\\psi }_a\\psi _b+\\bar{\\psi }_b\\psi _a)\\\\& \\nonumber \\quad \\quad \\quad \\quad \\quad + \\sigma _a \\sigma _b \\frac{r}{2}(\\bar{\\psi }_a\\psi _a+\\bar{\\psi }_b\\psi _b)),\\\\\\nonumber u^\\star &=- \\sigma _a \\gamma _a- \\sigma _b \\gamma _b- { \\sigma _a\\sigma _b } q ,\\end{aligned}\\right\\rbrace \\qquad \\text{in Case (2),}$ for observable coupling constants $(\\lambda _{a},\\lambda _{b},q, r)\\in {4}$ respectively $(\\lambda _{a},\\lambda _{b},\\gamma _{a},\\gamma _b,q,\\eta , r)\\in {7}$ .","For $X \\subset \\Lambda $ , we define $(u^{\\star }+V^\\star )(X) \\in ^\\star (X \\cap \\lbrace a,b\\rbrace )$ by $ (u^\\star +V^\\star )(X) =-\\lambda _a \\sigma _a \\bar{\\psi }_a 1_{a\\in X} -\\lambda _b\\psi _b \\bar{\\sigma }_b 1_{b \\in X} + \\sigma _a \\bar{\\sigma }_b q 1_{a\\in X,b \\in X}+\\sigma _a \\bar{\\sigma }_b \\frac{r}{2}(\\bar{\\psi }_a\\psi _a+\\bar{\\psi }_b\\psi _b)1_{a\\in X,b\\in X}$ in Case (1), and analogously in Case (2).","The terms corresponding to $r$ do not appear at any step of the noninteracting iteration (REF ) if we start with them equal to 0.","We include them here in preparation for the interacting model.","The evolutions of $u^\\star _j+V^\\star _j \\rightarrow u^{\\star }_{j+1}$ and $V^\\star _j \\rightarrow V^\\star _{j+1}$ are equivalent to the evolution of the coupling constants $(\\lambda _a,\\lambda _b,q, r)$ respectively $(\\lambda _a,\\lambda _b,\\gamma _a,\\gamma _b,q,\\eta ,r)$ .","By computation of the fermionic Gaussian moments in (REF ), the flow of the observable coupling constants according to (REF ) is then given as follows.","Note that the evolution of couplings constants in $V^\\star $ is independent of the coupling constants in $u^\\star $ .","Let $V^\\varnothing _0=0$ , and let $u^\\star _{j}$ and $V^\\star _j$ be of the form as in Definition REF .","The map (REF ) is then given as follows.","In Case (1), for $x\\in \\lbrace a,b\\rbrace $ , $\\lambda _{x,j+1} &=\\lambda _{x,j}\\\\q_{j+1} &= q_j + \\lambda _{a,j}\\lambda _{b,j} C_{j+1}(a,b) + r_jC_{j+1}(0,0)\\\\r_{j+1} &= r_j,\\multicolumn{2}{l}{\\text{whereas in Case (2), for $x\\in \\lbrace a,b\\rbrace $,}}\\\\\\lambda _{x,j+1} &= \\lambda _{x,j}\\\\\\gamma _{x,j+1} &= \\gamma _{x,j} + \\lambda _{x,j}C_{j+1}(0,0)\\\\q_{j+1} &= q_j + \\eta _j C_{j+1}(a,b)+ r_{j} C_{j+1}(0,0)- \\lambda _{a,j}\\lambda _{b,j} C_{j+1}(a,b)^2\\\\\\eta _{j+1} &= \\eta _j - 2\\lambda _{a,j}\\lambda _{b,j} C_{j+1}(a,b)\\\\r_{j+1} & =r_{j}.$ This follows from straightforward evaluation of (REF ) using (REF ).","To continue the warm-up for the interacting case, we illustrate how these equations reproduce the direct computations of correlation functions.","When $r_{0}=0$ by a computation using Definition REF , the formulas (REF ) and (REF )–() imply the correlation functions in Cases (1) and (2) are given by ${\\bar{\\psi }_a\\psi _b} = \\frac{ q_{N+1}}{\\lambda _{a,0}\\lambda _{b,0}},\\qquad {\\bar{\\psi }_a\\psi _a} = \\frac{\\gamma _{a,N+1}}{\\lambda _{a,0}},\\qquad {\\bar{\\psi }_a\\psi _a;\\bar{\\psi }_b\\psi _b} = \\frac{q_{N+1}}{\\lambda _{a,0}\\lambda _{b,0}},$ with $qq^{(1)}$ and $\\lambda \\lambda ^{(1)}$ for the first equation $qq^{(2)}$ , $\\gamma \\gamma ^{(2)}$ , and $\\lambda \\lambda ^{(2)}$ for the last two.","Recalling the convention $C_{N+1}=t_{N}Q_{N}$ , for Case (2) with $r_{0}=0$ we have $q_{N+1}= - \\lambda _{0,a}\\lambda _{0,b}{ \\sum _{k \\le N} C_k(a,b) + t_NQ_N(a,b)}^2 = -\\lambda _{0,a}\\lambda _{0,b}(-\\Delta +m^2)^{-1}(a,b)^2,$ by (REF ).","When combined with (REF ) this gives the expected result.","In the preceding computation we kept the potential in the exponential for the entire computation, whereas in Sections and the zero mode is integrated out directly without rewriting the integrand in this form (see, e.g., (REF )).","We distinguish these two approaches by using $N+1$ subscripts for the former and $(N,N)$ for the latter, and by putting tildes on quantities associated with the $(N,N)$ th step as was done in Section .","Before moving to the interacting model, we introduce the coalescence scale $j_{ab}$ as the largest integer $j$ such that $C_{k+1}(a,b)=0$ for all $k < j$ , i.e., $j_{ab} = {\\log _L (2|a-b|_{{\\infty }})}.$ In the degenerate cases $\\lambda _a=0$ or $\\lambda _b=0$ when only one of the observable fields is present we use the convention $j_{ab}=+\\infty $ .","Note that the finite range property (REF ) implies that $q_j=\\eta _j=r_{j}=0$ for $j0$ such that: for $j L^{j+1}$ then $P^\\star (B,B^{\\prime })=0$ , by the finite range of the covariance $C_{j+1}$ .","As a result, $P^{\\star }(B)\\in ^{\\star }(B^\\square )$ .","With these definitions in place, $ u^\\star _{j+1}+V^\\star _{j+1}$ is defined in the same way as $ u_{j+1}+ V_{j+1}$ with the addition of the second order term $P^\\star $ , and $K^\\star _{j+1}$ is then defined in the same way as $K_{j+1}$ : The map $(V_j,K_j) \\mapsto ({ u^\\star _{j+1}},V^\\star _{j+1})$ is defined, for $B\\in _j$ , by $u^\\star _{j+1}(B)+V^\\star _{j+1}(B) =_{C_{j+1}}\\theta (V^\\star _j(B) - Q^\\star (B)) - P^\\star (B)$ where $u^\\star _{j+1}$ consists of all monomials that do not contain factors of $\\psi $ or $\\bar{\\psi }$ .","Explicitly, $u^\\star _{j+1}={\\left\\lbrace \\begin{array}{ll}-\\sigma _a\\bar{\\sigma }_b q_{j+1}, &\\text{Case (1)}, \\\\-\\sigma _a\\sigma _b q_{j+1} - \\sigma _a \\gamma _{a,j+1}-\\sigma _b\\gamma _{b,j+1}, &\\text{Case (2)},\\end{array}\\right.","}$ The map $(V_j,K_j)\\mapsto K^\\star _{j+1}$ is defined as in Definition REF except that $V^\\varnothing $ and $u^\\varnothing $ are replaced by $V=V^\\varnothing +V^\\star $ and $u=u^\\varnothing +u^\\star $ .","Propositions REF and REF also hold for this extended definition of the renormalisation group map.","The proofs are the same; for Proposition REF we provide a proof in Appendix REF ."],["Estimates for the renormalisation group map with observables","In this section, the $O$ -notation refers to scale $j+1$ norms, i.e., for $F,G \\in (\\Lambda )$ , we write $F=G+O(t)$ to denote that $\\Vert F-G\\Vert _{T_{j+1}(\\ell _{j+1})} \\le O(t)$ .","Under the assumptions of Theorem REF , if also $\\Vert V^\\star _j\\Vert _j+\\Vert K^\\star _j\\Vert _j \\le \\epsilon $ and $u^\\star _j=0$ , then for $j+10$ , $\\lambda _{j} &= \\lambda _0 + O_L(\\lambda _0b_0)\\\\r_j &= O_L(\\lambda _0b_0|a-b|^{-\\kappa })1_{j\\ge j_{ab}}\\multicolumn{2}{l}{\\text{and, in Case (2), if $q_0=r_0=\\gamma _{x,0}=\\eta _0=0$ and $\\lambda _0>0$,}}\\\\\\lambda _{j} &= \\lambda _0 + O_L(\\lambda _0b_0)\\\\\\eta _j &= O_L(\\lambda _0^2|a-b|^{-(d-2)})1_{j\\ge j_{ab}}\\\\r_{j}&= O_L(\\lambda _0b_0|a-b|^{-(d-2)-\\kappa })1_{j\\ge j_{ab}},$ where $\\lambda _{j}=\\lambda _{x,j}$ for either $x=a$ or $x=b$ .","In both Cases (1) and (2), $\\Vert V^\\star _{j}\\Vert _{j} \\le \\lambda _0 + O_L(\\lambda _0^2) + O_L(\\lambda _0b_0).$ The bounds on the coupling constants follow from Lemma REF ; the hypothesis regarding $K_{j}(X)=0$ for $jj_{ab}$ since we have started the flow with $r_{0}=0$ in Case (1), and $q_{0}=\\eta _{0}=r_{0}=0$ in Case (2).","What remains is to analyse the recurrences.","For $\\lambda _{x,j}$ , since $\\ell _{x,j}^{-1}\\ell _j^{-p} = 1$ in Case ($p$ ), using (REF ), respectively (), $\\lambda _{x,j} = \\lambda _0 + \\sum _{k=0}^{j-1} O_L(\\Vert K^\\star _k\\Vert _k)= \\lambda _0 + \\sum _{k=0}^{j-1} O_L(\\lambda _0b_0L^{-\\kappa k})=\\lambda _{0} + O_L(\\lambda _{0}b_{0}).$ The bounds on $r_{j}$ follow from the fact that all contributions are 0 for scales $j0$ such that $\\Vert V^\\varnothing _j\\Vert _j = O(b_0L^{-\\alpha j}), \\qquad \\Vert K^\\varnothing _j\\Vert _j = O(b_0L^{-\\alpha j}).$ Using this as input, we iterate the observable flow (REF )–(), with initial condition $\\lambda _{a,0}=\\lambda _{b,0}=\\lambda _0$ small enough and all other observable coupling constants equal to 0.","Assume that the bulk renormalisation group flow $(V^\\varnothing _j,K^\\varnothing _j)$ obeys $\\Vert V^\\varnothing _j\\Vert _j+\\Vert K^\\varnothing _j\\Vert _j = O(b_0L^{-\\alpha j})$ for some $\\alpha >0$ .","Then there is $\\kappa >0$ such that for $\\lambda _{0,a}=\\lambda _{0,b}=\\lambda _0 >0$ sufficiently small and all other observable coupling constants initially 0, $ \\Vert V^\\star _j\\Vert _j \\le O_L(\\lambda _0), \\qquad \\Vert K^\\star _j\\Vert _j \\le O_L(\\lambda _0b_0L^{-\\kappa j}).$ The proof is by induction from a scale $j_0$ which will be determined below.","When $j=0$ we have $\\Vert V^\\star _{0}\\Vert _{0}\\asymp \\lambda _{0,a}+\\lambda _{0,b}$ and $\\Vert K^\\star _{0}\\Vert _{0}=0$ .","Now let us choose $C_j>0$ so that $\\Vert K^\\star _k\\Vert _{k} \\le C_j b_0\\lambda _0L^{-\\kappa k}$ for all $k \\le j$ .","We start the proof by studying the behaviour of $C_j$ as $j$ increases.","We may assume $\\kappa <\\alpha /2$ and also that $\\kappa $ is less than the exponents of $L$ and $A$ in ().","Then Lemma REF shows that $\\Vert V^\\star _{k}\\Vert _k \\le C_j\\lambda _0$ for $k\\le j+1$ provided $\\lambda _0<1$ and $b_0$ is sufficiently small (depending on $\\kappa $ ).","We now use Theorem REF to control $K^\\star _{j+1}$ .","Since $A \\gg L$ and taking $\\lambda _0$ sufficiently small we obtain $\\Vert K^\\star _{j+1}\\Vert _{j+1}&\\le \\frac{1}{2} L^{-\\kappa }\\Vert K^\\star _j\\Vert _j +O_L(1) C_j\\lambda _0 b_0 L^{-\\alpha j}& \\le C_jb_{0}\\lambda _{0} (\\frac{1}{2} L^{-\\kappa (j+1)} +\\frac{1}{2} O_L(1)L^{-(\\alpha /2)(j-1)}L^{-(\\alpha /2)(j+1)})&\\le C_jb_{0}\\lambda _{0} (\\frac{1}{2} +\\frac{1}{2} O_L(1)L^{-(\\alpha /2)(j-1)}) L^{-\\kappa (j+1)}$ since $\\alpha >2\\kappa $ .","Now there is $j_0\\in \\mathbb {N}$ so that for all $j\\ge j_0$ , the term $O_L(1)L^{-(\\alpha /2)(j-1)}$ on the right-hand side is bounded by 1.","Thus choosing $C=C_{j_0}$ , by induction for all $j\\ge j_0$ , $\\Vert K^\\star _j\\Vert _{j} \\le C b_0\\lambda _0L^{-\\kappa j}$ and $\\Vert V^\\star _{j}\\Vert _j \\le C\\lambda _0$ and the claim follows."],["Computation of pointwise correlation functions","In this section we use the results of Section to prove the following estimates for the pointwise correlation functions ${\\bar{\\psi }_a\\psi _b}$ , ${\\bar{\\psi }_a\\psi _a}$ , and ${\\bar{\\psi }_a\\psi _a\\bar{\\psi }_b\\psi _b}$ .","Recall the definition (REF ) of $W_{N}(x)=W_{N,m^{2}}(x)$ .","For $b_0$ sufficiently small and $m^2 \\ge 0$ , there exists continuous functions $\\lambda =\\lambda (b_0,m^2)= 1+O_L(b_0), \\qquad \\gamma = \\gamma (b_0,m^2) = (-\\Delta ^{^d}+m^2)^{-1}(0,0) + O_L(b_0),$ such that if $V^\\varnothing _{0}=V_{0}^{c}(m^{2},b_{0})$ is as in Theorem REF , $V^\\star _{0}$ is as in Proposition REF , and $\\tilde{u}_{N,N}^c=\\tilde{u}_{N,N}^c(b_0,m^2)$ is as in Proposition with initial condition $V^\\varnothing _0=V_0^c$ , $ {\\bar{\\psi }_a\\psi _a} =\\gamma + \\frac{\\gamma t_N |\\Lambda _N|^{-1}+ O_L(b_0L^{-(d-2+\\kappa )N}) + O_L(b_0L^{-\\kappa N}(m^{2}|\\Lambda _N|)^{-1})}{1+\\tilde{u}_{N,N}}.$ Under the same assumptions as in Proposition , ${\\bar{\\psi }_a\\psi _b}&=W_N(a-b)+ \\frac{t_N|\\Lambda _N|^{-1}}{1+\\tilde{u}_{N,N}}&\\qquad + O_L(b_0|a-b|^{-(d-2+\\kappa )})+\\frac{O_L(b_0|a-b|^{-\\kappa } (m^2|\\Lambda _N|)^{-1})}{1+\\tilde{u}_{N,N}}\\\\{\\bar{\\psi }_a\\psi _a\\bar{\\psi }_b\\psi _b}&= -\\lambda ^2 W_N(a-b)^2 + \\gamma ^2+\\frac{-2\\lambda ^2 W_N(a-b)+ 2\\lambda \\gamma }{1+\\tilde{u}_{N,N}}t_N|\\Lambda _N|^{-1}&\\qquad + O_L(b_0 |a-b|^{-2(d-2)-\\kappa })+ O_L(b_0L^{-(d-2+\\kappa )N})&\\qquad + (O_L(b_0L^{-\\kappa N}) + O_L(b_0 |a-b|^{-(d-2+\\kappa )}))\\frac{(m^{2}|\\Lambda _N|)^{-1}}{1+\\tilde{u}_{N,N}}.$ Throughout this section, we assume that the renormalisation group flow $(V_j,K_j)_{j \\le N}$ is given as in Corollary REF (bulk) and Proposition REF (observables)."],["Integration of the zero mode","As in the analysis of the susceptibility in Section , we treat the final integration over the zero mode explicitly.","Again we will only require the restriction to constant $\\psi ,\\bar{\\psi }$ (as discussed below (REF )) of $_{C} \\theta Z_0= _{t_N Q_N} \\theta Z_N = e^{-u^\\varnothing _N|\\Lambda _N|} \\tilde{Z}_{N,N},$ where the last equation defines $\\tilde{Z}_{N,N}$ .","We write $\\tilde{Z}_{N,N}=\\tilde{Z}_{N,N}^{\\varnothing }+\\tilde{Z}_{N,N}^{\\star }$ for its decomposition into bulk and observable parts.","The bulk term was already computed in Proposition .","The observable term $\\tilde{Z}_{N,N}^\\star $ is computed by the next lemma; in the lemma we only give explicit formulas for the terms that will be used in the proofs of Propositions and .","Restricted to constant $\\psi ,\\bar{\\psi }$ , in Case (1), $\\tilde{Z}_{N,N}^\\star &=\\sigma _a\\bar{\\psi }\\tilde{Z}_{N,N}^{\\sigma _a\\bar{\\psi }} + \\bar{\\sigma }_b\\psi \\tilde{Z}_{N,N}^{\\sigma _b\\psi }+\\sigma _a\\bar{\\sigma }_b \\tilde{Z}_{N,N}^{\\bar{\\sigma }_b\\sigma _a}+\\sigma _a\\bar{\\sigma }_b \\psi \\bar{\\psi }\\tilde{Z}_{N,N}^{\\bar{\\sigma }_b\\sigma _a\\psi \\bar{\\psi }}\\multicolumn{2}{l}{\\text{where}}\\\\\\tilde{Z}_{N,N}^{\\sigma _a\\bar{\\psi }}&= \\lambda _{a,N} +O_L(\\ell _{x,N}^{-1}\\ell _N^{-1}\\Vert K^\\star _N\\Vert _N)\\\\\\tilde{Z}_{N,N}^{\\bar{\\sigma }_b\\psi }&= \\lambda _{b,N} +O_L(\\ell _{x,N}^{-1}\\ell _N^{-1}\\Vert K^\\star _N\\Vert _N)\\\\\\tilde{Z}_{N,N}^{\\bar{\\sigma }_b\\sigma _a}&= q_N (1+\\tilde{u}_{N,N})+ \\lambda _{a,N}\\lambda _{b,N} t_N|\\Lambda _N|^{-1} { + r_N t_N |\\Lambda _N|^{-1}}&\\qquad \\qquad + O_L(m^{-2}|\\Lambda _N|^{-1}\\ell _N^{-2}\\ell _{ab,N}^{-1})\\Vert K^\\star _N\\Vert _N.$ In Case (2), $\\tilde{Z}_{N,N}^\\star &=\\sigma _a\\tilde{Z}_{N,N}^{\\sigma _a}+ \\sigma _a\\bar{\\psi }\\psi \\tilde{Z}_{N,N}^{\\sigma _a\\bar{\\psi }\\psi }+\\sigma _b \\tilde{Z}_{N,N}^{\\sigma _b} + \\sigma _b\\bar{\\psi }\\psi \\tilde{Z}_{N,N}^{\\sigma _b\\bar{\\psi }\\psi }+ \\sigma _a\\sigma _b \\tilde{Z}_{N,N}^{\\sigma _a\\sigma _b}+ \\sigma _a\\sigma _b \\psi \\bar{\\psi }\\tilde{Z}_{N,N}^{\\sigma _a\\sigma _b\\psi \\bar{\\psi }}\\multicolumn{2}{l}{\\text{where, setting $\\tilde{\\lambda }_{x,N,N} = \\lambda _{x,N}+O_L(\\ell _{x,N}^{-1} \\ell _N^{-2} \\Vert K^\\star _N\\Vert _N)$,}}\\\\\\tilde{Z}_{N,N}^{\\sigma _x}&=\\gamma _{x,N} (1+\\tilde{u}_{N,N}) +\\tilde{\\lambda }_{x,N,N}t_N|\\Lambda _N|^{-1}+ O_L(\\ell _{x,N}^{-1})\\Vert K^\\star _N\\Vert _N.\\\\\\tilde{Z}_{N,N}^{\\sigma _a\\sigma _b}&= (q_N+\\gamma _{a,N}\\gamma _{b,N}) (1+\\tilde{u}_{N,N})+ (\\eta _N+r_N +\\tilde{\\lambda }_{a,N,N}\\gamma _{b,N}+\\tilde{\\lambda }_{b,N,N}\\gamma _{a,N})t_N|\\Lambda _N|^{-1}&\\qquad \\qquad + O_L((|\\gamma _{a,N}|+|\\gamma _{b,N}|)\\ell _{x,N}^{-1}+ \\ell _{ab,N}^{-1}+ m^{-2}|\\Lambda _N|^{-1}\\ell _N^{-2}\\ell _{ab,N}^{-1})\\Vert K^\\star _N\\Vert _N.$ The error bounds above reveal the tension in the explicit choices of $\\ell _{x,j}^{-1}$ and $\\ell _{ab,j}^{-1}$ .","To obtain effective error estimates, we want $\\ell _{x,N}^{-1}$ and $\\ell _{ab,N}^{-1}$ to be as small as possible.","On the other hand, to control the iterative estimates of Theorem REF over the entire trajectory, i.e., to prove Proposition REF , we cannot have $\\ell _{x,j}$ and $\\ell _{ab,j}$ too large.","In particular, either of the more naive choices $\\ell _{ab,j}=\\ell _{x, j}^{2}$ and $\\ell _{ab,j} = \\ell _{x,j_{ab}}^{2}$ in Case (2) would lead to difficulties, both in terms of forcing us to track additional terms in the flow and in terms of controlling norms inductively, or to error bounds that are not strong enough to capture the zero mode sufficiently accurately.","Throughout the proof, we restrict to constant $\\psi ,\\bar{\\psi }$ .","Since ${e^{+u_N^\\varnothing (\\Lambda )}Z_N}^\\star &= {e^{-u_N^\\star (\\Lambda )}(e^{-V_N(\\Lambda )}+K_N(\\Lambda ))}^\\star &=(e^{-u_N^\\star (\\Lambda )}-1)(e^{-V^\\varnothing _N(\\Lambda )}+K^\\varnothing _N(\\Lambda ))+ e^{-u_N^\\star (\\Lambda )}(e^{-V_N(\\Lambda )}+K_N(\\Lambda ))^\\star ,$ by applying $_{t_NQ_N}\\theta $ we obtain $\\tilde{Z}_{N,N}^\\star = \\underbrace{(e^{-u^\\star _N(\\Lambda )}-1)(1+\\tilde{u}_{N,N}-|\\Lambda _N|\\tilde{a}_{N,N}\\psi \\bar{\\psi })}_{A}+ \\underbrace{e^{-u_N^\\star (\\Lambda )}_{t_NQ_N}\\theta (e^{-V_N(\\Lambda )}+K_N(\\Lambda ))^\\star }_{B}.$ In obtaining $A$ we used (REF ) which gives $_{t_NQ_N}\\theta (e^{-V_N^\\varnothing (\\Lambda )}+K_N^\\varnothing (\\Lambda ))= 1+\\tilde{u}_{N,N}-|\\Lambda _N|\\tilde{a}_{N,N}\\psi \\bar{\\psi }$ .","Since each term in $V_N^\\varnothing (\\Lambda )$ contains a factor $\\psi \\bar{\\psi }$ and each term in $V_N^\\star (\\Lambda )$ either $\\psi $ or $\\bar{\\psi }$ , we have $V_N^\\varnothing (\\Lambda )V_N^\\star (\\Lambda )=0$ .","Thus $B = e^{-u_N^\\star (\\Lambda )}_{t_NQ_N}\\theta (-V_N^\\star (\\Lambda )+\\frac{1}{2} V_N^\\star (\\Lambda )^2+K_N^\\star (\\Lambda )).$ Case (1).","Since $\\sigma _a^2=\\bar{\\sigma }_b^2=0$ , $e^{-u_N^\\star (\\Lambda )}-1=-u^\\star _N(\\Lambda ) = \\sigma _a\\bar{\\sigma }_b q_N,$ we get $A&=\\sigma _a\\bar{\\sigma }_b q_N (1+\\tilde{u}_{N,N}-|\\Lambda _N|\\tilde{a}_{N,N}\\psi \\bar{\\psi })\\\\B &=\\sigma _a\\bar{\\psi }(\\lambda _{a,N}+k^{\\sigma _a\\bar{\\psi }}_{N})+{\\psi \\bar{\\sigma }_{b}}(\\lambda _{b,N}+k^{\\bar{\\sigma }_b\\psi }_{N})+\\sigma _a\\bar{\\sigma }_b _{t_NQ_N}\\theta \\bar{\\psi }\\psi (\\lambda _{a,N}\\lambda _{b,N} - r_N + k^{\\sigma _a\\bar{\\sigma }_b \\bar{\\psi }\\psi }_{N}).$ The constants $k_N^\\#$ are given in terms of derivatives of $K_N(\\Lambda )$ and bounded analogously as in (REF ).","For example, $k_N^{\\sigma _a\\bar{\\psi }} = O_L(\\ell _{a,N}^{-1}\\ell _N^{-1}\\Vert K^\\star _N\\Vert _N)$ , and similarly for the other $k_N^\\#$ terms, the rule being that we have a factor $\\ell _{x,N}^{-1}$ if there is a superscript $\\sigma _a$ or $\\bar{\\sigma }_b$ but not both, a factor $\\ell _{ab,N}^{-1}$ for $\\sigma _a\\bar{\\sigma }_b$ and a factor $\\ell _N^{-1}$ for each superscript $\\psi $ or $\\bar{\\psi }$ .","These bounds follow from the definition of the $T_j(\\ell _j)$ norm.","Since $_{t_NQ_N}\\theta \\psi \\bar{\\psi }= -t_N|\\Lambda _N|^{-1}+\\psi \\bar{\\psi }$ the claim follows by collecting terms and using (REF ).","Case (2).","Using again that $\\sigma _a^2=\\sigma _b^2=0$ , but now taking in account that $u^\\star (\\Lambda )$ has additional terms compared to Case (1), $e^{-u_N^\\star (\\Lambda )}-1=-u^\\star _N(\\Lambda )+\\frac{1}{2} u^\\star _N(\\Lambda )^2 = \\sigma _a\\sigma _b (q_N+\\gamma _{a,N}\\gamma _{b,N}) + \\sigma _a \\gamma _{a,N} +\\sigma _b\\gamma _{b,N},$ and therefore $A =(\\sigma _a\\sigma _b(q_N+\\gamma _{a,N}\\gamma _{b,N})+\\sigma _a \\gamma _{a,N}+\\sigma _b\\gamma _{b,N})(1 + \\tilde{u}_{N,N} - |\\Lambda _N|\\tilde{a}_{N,N} \\psi \\bar{\\psi }).$ Since in Case (2) each term in $V^\\star _N(\\Lambda )$ contains a factor of $\\psi \\bar{\\psi }$ , we have $V^\\star _N(\\Lambda )^2=0$ and thus $B = (1-u_N^\\star (\\Lambda ))_{t_NQ_N}\\theta (-V^\\star _N(\\Lambda )+K^\\star _N(\\Lambda ))$ Therefore $B&= \\sigma _a k_N^{\\sigma _a}+\\sigma _b k_N^{\\sigma _b}+\\sigma _a\\sigma _b (\\gamma _{a,N} k_N^{\\sigma _b} + \\gamma _{b,N} k_N^{\\sigma _a} + k_N^{\\sigma _a\\sigma _b})&\\qquad +_{t_NQ_N}[\\sigma _a\\bar{\\psi }\\psi \\tilde{\\lambda }_{a,N}+ \\sigma _a \\sigma _b\\bar{\\psi }\\psi (\\eta _N+r_N +\\gamma _{a,N}\\tilde{\\lambda }_{b,N}+\\gamma _{b,N}\\tilde{\\lambda }_{a,N} + k^{\\sigma _a\\sigma _b\\bar{\\psi }\\psi })]$ where we have set $\\tilde{\\lambda }_{x,N} = \\lambda _{x,N} + k_N^{\\sigma _x\\bar{\\psi }\\psi }$ .","Taking the expectation and collecting all terms gives $\\tilde{Z}_{N,N}^{\\sigma _a\\sigma _b}&= (q_N+\\gamma _{a,N}\\gamma _{b,N}) (1+\\tilde{u}_{N,N})&\\qquad \\qquad + (\\eta _N+r_N +\\tilde{\\lambda }_{a,N}\\gamma _{b,N}+\\tilde{\\lambda }_{b,N}\\gamma _{a,N} + k^{\\sigma _a\\sigma _b\\bar{\\psi }\\psi })t_N|\\Lambda _N|^{-1}&\\qquad \\qquad +\\gamma _{a,N} k_N^{\\sigma _b} + \\gamma _{b,N} k_N^{\\sigma _a} + k_N^{\\sigma _a\\sigma _b}\\\\\\tilde{Z}_{N,N}^{\\sigma _a}&=\\gamma _{a,N} (1+\\tilde{u}_{N,N}) + \\tilde{\\lambda }_{a,N}t_N|\\Lambda _N|^{-1}+ k_N^{\\sigma _a}.$ The bounds on the constants $k_N^\\#$ are analogous to those in Case (1)."],["Analysis of one-point functions","We now analyse the observable flow given by Lemma REF to derive the asymptotics of the correlation functions.","Note that the coupling constants $\\lambda _{x,j}$ and $\\gamma _{x,j}$ can possibly depend on $x=a,b$ as the contributions from $K$ can depend on the relative position of the points in the division of $\\Lambda _N$ into blocks.","The following lemma shows that in the limit $j\\rightarrow \\infty $ they become independent of $x$ ; an analogous argument was used in [13].","Under the hypotheses of Proposition there are $\\lambda _\\infty ^{(p)} = \\lambda _0^{(p)} +O_L(\\lambda _0^{(p)}b_0)$ and $\\gamma _\\infty = O_L(\\lambda _0^{(2)})$ , all continuous in $m^{2}\\ge 0$ and $b_{0}$ small, such that for $x\\in \\lbrace a,b\\rbrace $ , $ \\lambda _{x,j}^{(p)} = \\lambda _\\infty ^{(p)} + O_L(\\lambda _0^{(p)} b_0 L^{-\\kappa j}),\\qquad \\gamma _{x,j}=\\gamma _\\infty + O_L(\\lambda _0^{(2)} b_0 L^{-(d-2+\\kappa )j}).$ In Case (1), $\\lambda _\\infty ^{(1)} = \\lambda _0^{(1)}$ .","In Case (2), $\\lambda _\\infty ^{(2)} = \\lambda _0^{(2)} + O_L(\\lambda _0^{(2)}b_0)$ and $\\gamma _\\infty ^{(2)} =\\lambda _\\infty ^{(2)}(-\\Delta ^{^d}+m^2)^{-1}(0,0)+O_L(\\lambda _0^{(2)}b_0)$ , and with the abbreviations $\\lambda \\lambda ^{(2)}$ and $\\gamma \\gamma ^{(2)}$ , $ {\\bar{\\psi }_a\\psi _a} =\\frac{\\gamma _\\infty }{\\lambda _0}+ \\frac{\\frac{\\lambda _{\\infty }}{\\lambda _0} t_N |\\Lambda _N|^{-1}+ O_L(b_0L^{-(d-2+\\kappa )N}) + O_L(b_0L^{-\\kappa N}(m^{2}|\\Lambda _N|)^{-1})}{1+\\tilde{u}_{N,N}}.$ We will typically drop the superscript $(p)$ .","In both cases, we have already seen that $\\lambda _{x,j} = \\lambda _0 + \\sum _{k=0}^{j-1} O_L(\\Vert K^\\star _k\\Vert _k) = \\lambda _0 + \\sum _{k=0}^{j-1} O_L(\\lambda _0b_0L^{-\\kappa k}) .$ Since the $K^\\star _k$ are independent of $N$ for $k0$ fixed, $\\partial _{\\bar{\\psi }} \\partial _{\\sigma _a} \\tilde{Z}_{N,N}|_{0}= \\lambda _{a,N} + O_L(\\ell _N^{-1}\\ell _{x,N}^{-1} \\Vert K^\\star _N\\Vert _N)= \\lambda _{a,N} + O_L(\\Vert K^\\star _N\\Vert _N)\\xrightarrow[N\\rightarrow \\infty ]{}\\lambda _{a,\\infty },$ where $|_0$ denotes projection onto the degree 0 part, i.e., $\\psi =\\bar{\\psi }=\\sigma =\\bar{\\sigma }=0$ .","On the other hand, we claim $\\partial _{\\bar{\\psi }} \\partial _{\\sigma _a} \\tilde{Z}_{N,N}|_{0}= \\lambda _{0,a}m^2(1+\\tilde{u}_{N,N})\\sum _{x\\in \\Lambda _N}{\\bar{\\psi }_0\\psi _x}= \\lambda _{0,a}{1+\\tilde{u}_{N,N}-\\frac{\\tilde{a}_{N,N}}{m^2}}.$ Indeed, the first equality in (REF ) follows analogously to [13]: let $\\Gamma (\\rho ,\\bar{\\rho })$ be as in (REF ), except that $Z_0$ now includes the observable terms $\\sigma _a$ and $\\bar{\\sigma }_b$ and we write $\\rho $ and $\\bar{\\rho }$ for the constant external field to distinguish them from $\\sigma _a$ and $\\bar{\\sigma }_b$ .","Then as in (REF ) $-\\sum _{x\\in \\Lambda _N} {\\psi _x}_{\\sigma _a,\\bar{\\sigma }_b} =\\partial _{\\bar{\\rho }} \\Gamma (\\rho ,\\bar{\\rho })|_{\\rho =\\bar{\\rho }=0} = m^{-2} \\frac{\\partial _{\\bar{\\psi }}\\tilde{Z}_{N,N}|_{\\psi =\\bar{\\psi }=0}}{\\tilde{Z}_{N,N}|_{\\psi =\\bar{\\psi }=0}}$ and ${\\cdot }_{\\sigma _a,\\bar{\\sigma }_b}$ denotes the expectation that still depends on the observable fields $\\sigma _a$ and $\\bar{\\sigma }_b$ .","Differentiating with respect to $\\sigma _a$ and setting $\\bar{\\sigma }_b=0$ gives $\\lambda _{0,a}\\sum _{x\\in \\Lambda _N} {\\bar{\\psi }_a\\psi _x}= -m^{-2} \\frac{\\partial _{\\sigma _a}\\partial _{\\bar{\\psi }}\\tilde{Z}_{N,N}|_{0}}{\\tilde{Z}_{N,N}|_0}= m^{-2} \\frac{\\partial _{\\bar{\\psi }}\\partial _{\\sigma _a}\\tilde{Z}_{N,N}|_{0}}{1+\\tilde{u}_{N,N}}$ which is the first equality of (REF ) upon rearranging.","The second equality in (REF ) follows from Proposition .","The right-hand side of (REF ) converges to $\\lambda _0$ in the limit $N\\rightarrow \\infty $ with $m^2>0$ fixed since $\\tilde{a}_{N,N}= a_N - k_N^2/|\\Lambda _N| =O_L(L^{-2N}\\Vert V_N\\Vert _N)+O_L(L^{-2N}\\Vert K_N\\Vert _N) \\rightarrow 0$ and $\\tilde{u}_{N,N}= k_N^0 + \\tilde{a}_{N,N}t_N = O(\\Vert K_N\\Vert _N) + \\tilde{a}_{N,N}t_N \\rightarrow 0$ when $m^2>0$ is fixed.","Since the left-hand sides of (REF )–(REF ) are equal, we conclude that $\\lambda _{a,\\infty }=\\lambda _0$ when $m^2>0$ .","By continuity this identity then extends to $m^2=0$ .","In Case (2), to show (REF ), we use (REF ), that Proposition REF implies $\\Vert K^\\star _N\\Vert _N = O_L(\\lambda _0b_0L^{-\\kappa N})$ , and $\\ell _{x,N}^{-1}\\ell _N^{-2} = 1$ and $\\ell _{x,N}^{-1}= \\ell _N^{2} = O_L(L^{-(d-2)N})$ to obtain $\\frac{\\tilde{Z}_{N,N}^{\\sigma _a}}{1+\\tilde{u}_{N,N}}= \\gamma _{a,N} +\\frac{\\lambda _{a,\\infty } t_N |\\Lambda _N|^{-1} + O_L(\\lambda _0 b_0L^{-(d-2+\\kappa )N}) + O_L( \\lambda _0b_0 L^{-\\kappa N} (m^{2}|\\Lambda _N|)^{-1})}{1+\\tilde{u}_{N,N}}.$ Since ${\\bar{\\psi }_a\\psi _a} =\\tilde{Z}_{N,N}^{\\sigma _a}/(\\lambda _0 (1+\\tilde{u}_{N,N}))$ this gives (REF ).","In particular, by the translation invariance of ${\\bar{\\psi }_a\\psi _a}$ , taking $N\\rightarrow \\infty $ implies both $\\gamma _{a,\\infty }$ and $\\lambda _{a,\\infty }$ are in fact independent of $a$ .","Taking $\\lambda _0>0$ small enough, the proposition follows immediately from Lemma REF with $\\lambda = \\lambda _\\infty ^{(2)}/\\lambda _0^{(2)}$ and $\\gamma = \\gamma _\\infty ^{(2)}/\\lambda _0^{(2)}$ ,"],["Analysis of two-point functions","Next we derive estimates for the two-point functions.","Under the hypotheses of Proposition , and in terms of the same $\\lambda _\\infty $ and $\\gamma _\\infty $ as in Lemma REF , ${\\bar{\\psi }_a\\psi _b}&=W_N(a-b)+ \\frac{t_N|\\Lambda _N|^{-1}}{1+\\tilde{u}_{N,N}}&\\qquad + O_L(\\frac{b_0}{\\lambda _0}|a-b|^{-(d-2+\\kappa )})+O_L(\\frac{b_0}{\\lambda _0} |a-b|^{-\\kappa })\\frac{(m^2|\\Lambda _N|)^{-1}}{1+\\tilde{u}_{N,N}}\\\\{\\bar{\\psi }_a\\psi _a\\bar{\\psi }_b\\psi _b}&= -\\frac{\\lambda _\\infty ^2}{\\lambda _0^2} W_N(a-b)^2 +\\frac{\\gamma _\\infty ^2}{\\lambda _0^2}+\\frac{-2\\lambda _\\infty ^2 W_N(a-b)+ 2\\lambda _{\\infty }\\gamma _{\\infty }}{\\lambda _0^2(1+\\tilde{u}_{N,N})}t_N|\\Lambda _N|^{-1}&\\qquad + O_L(\\frac{b_0}{\\lambda _0} |a-b|^{-2(d-2)-\\kappa })+ O_L(\\frac{b_0}{\\lambda _0}L^{-(d-2+\\kappa )N})&\\qquad +(O_L(\\frac{b_0}{\\lambda _0}|a-b|^{-(d-2+\\kappa )})+O_L(\\frac{b_0}{\\lambda _0}L^{-\\kappa N}))\\frac{(m^{2}|\\Lambda _N|)^{-1}}{1+\\tilde{u}_{N,N}}.$ The proofs of (REF ) and () corresponding to Cases (1) and (2) are again analogous.","Case (1).","By Lemma REF and Lemmas REF and REF , $ \\lambda _{x,j}= \\lambda _\\infty + O_L( \\lambda _0b_0 L^{-\\kappa })= \\lambda _0 + O_L( \\lambda _0b_0 L^{-\\kappa }),\\qquad r_j=O_L(\\lambda _0b_0 |a-b|^{-\\kappa })1_{j\\ge j_{ab}}.$ Using that $\\ell _{ab,j}^{-1}\\Vert K^{ab}_j\\Vert _j \\le O_L(\\lambda _0b_0L^{-(d-2+\\kappa )j})1_{j\\ge j_{ab}}$ and $C_{j+1}(a,b) \\le C_{j+1}(0,0) \\le O_L(L^{-(d-2)j})$ it then follows from Lemma REF that $q_{N}&= \\sum _{j=j_{ab}-1}^N { \\lambda _{a,j}\\lambda _{b,j} C_{j+1}(a,b) + r_j C_{j+1}(0,0) + O_L(\\lambda _0 b_0 L^{-(d-2+\\kappa )j}) }&= \\lambda _0^2 \\sum _{j=1}^N C_j(a,b)+ O_L(\\lambda _0 b_0 |a-b|^{-(d-2)-\\kappa })&= \\lambda _0^2 W_N(a-b) + O_L(\\lambda _0 b_0 |a-b|^{-(d-2)-\\kappa }),$ where we have used (REF ), $|a-b|\\le L$ , that $C_j(a,b)=0$ for $j0$ and suppose that for all $x$ , $h_x= h$ .","Then there are $c,C>0$ depending on $d,\\beta ,h$ such that $¶^{\\Lambda _{N}}_{\\beta ,h}[0\\leftrightarrow x,0\\mathfrak {g}] \\le Ce^{-c|x|},\\qquad ¶^{\\Lambda _{N}}_{\\beta ,h}[0\\leftrightarrow x,0\\lnot \\leftrightarrow \\mathfrak {g}] \\le Ce^{-c|x|}$ We begin with the inequality on the left of (REF ).","Define $(0\\leftrightarrow x)$ to be the set of forests in which both 0 is connected to $x$ and $T_{0}$ is rooted at 0, and $$ the set of all forests.","In this argument we treat $$ as being a set of (possibly) rooted forests, i.e., we identify edges to $\\mathfrak {g}$ with roots.","Without loss of generality we may assume $x\\cdot e_{1} \\ge \\alpha |x|$ for a fixed $\\alpha >0$ .","Note that if $F\\in (0\\leftrightarrow x)$ there is a unique path $\\gamma _{F}$ from 0 to $x$ in $F$ , and there are at least $\\alpha |x|$ edges of the form $\\lbrace u,u+e_{1}\\rbrace $ in $\\gamma _{F}$ .","We define a map $S\\colon (0\\leftrightarrow x) \\rightarrow 2^{}$ by, for $F\\in (0\\leftrightarrow x)$ , choosing a subset $\\lbrace u_{i},v_{i}\\rbrace $ of the edges $\\lbrace \\lbrace u,v\\rbrace \\in \\gamma _{F} \\mid v=u+e_{1}\\rbrace $ , and removing each $\\lbrace u_{i},v_{i}\\rbrace $ and rooting the tree containing $v_{i}$ at $v_{i}$ .","Thus $S(F)$ is the set of forests that results from all possible choices in the first step.","The second step does yield an element of $2^{}$ since $T_{0}$ is rooted at 0, so it cannot be the case that the tree containing $v_{i}$ is already rooted (connected to $\\mathfrak {g}$ ).","The map $S$ is injective, meaning that given $\\bar{F}\\in \\bigcup _{F\\in (0\\leftrightarrow x)}S(F)$ there is a unique $F$ such that $\\bar{F} \\in S(F)$ .","Indeed, given $\\bar{F}\\in S(F)$ , $F$ can be reconstructed as follows.","In $\\bar{F}$ , either the tree containing $x$ contains 0, or else it is rooted at a unique vertex $v^{\\prime }$ and it is not connected to $u^{\\prime }=v^{\\prime }-e_{1}$ .","Set $\\bar{F}^{\\prime }=\\bar{F}\\cup \\lbrace u^{\\prime },v^{\\prime }\\rbrace $ .","The previous sentence applies to $\\bar{F}^{\\prime }$ as well, and continuing until a connection to 0 is formed we recover $F$ .","This reconstruction was independent of $F$ , and hence if $\\bar{F}_{1}=\\bar{F}_{2}$ , $\\bar{F}_{i}\\in S(F_{i})$ , we have $F_{1}=F_{2}$ .","Let $w(F)= h \\beta ^{F}\\prod _{T\\ne T_0}(1+h|V(T)|)$ .","Then for $\\bar{F} \\in S(F)$ , $w(\\bar{F}) = w(F) (\\frac{h}{\\beta })^{k}$ if $\\bar{F}$ had $k$ edges removed.","Hence if the connection from 0 to $x$ in $F$ has $k$ edges of the form $\\lbrace u,v\\rbrace $ , $v=u+e_{1}$ , $\\sum _{\\bar{F}\\in S(F)}w(\\bar{F}) = (1+\\frac{h}{\\beta })^{k}w(F).$ Let $_{k}(x) \\subset (0\\leftrightarrow x)$ be the set of forests where the connection from 0 to $x$ contains $k$ edges of the form $\\lbrace u,v\\rbrace $ , $v=u+e_{1}$ .","We have the lower bound $Z_{\\beta ,h}^{\\Lambda _{N}} = \\sum _{F\\in } \\beta ^{F}\\prod _{T\\in F}(1+h|V(T)|) \\ge \\sum _{k\\ge 0} \\sum _{F\\in _{k}(x)} \\sum _{\\bar{F}\\in S(F)} w(\\bar{F})$ since $S$ is injective and all of the summands are non-negative.","Hence we obtain, using (REF ), $¶^{\\Lambda _{N}}_{\\beta ,h}[0\\leftrightarrow x,0\\mathfrak {g}]&\\le \\frac{\\sum _{k\\ge \\alpha |x|} \\sum _{F\\in _{k}(x)}w(F)}{\\sum _{k\\ge 0} \\sum _{F\\in _{k}(x)} \\sum _{\\bar{F}\\in S(F)} w(\\bar{F})} &= \\frac{\\sum _{k\\ge \\alpha |x|} \\sum _{F\\in _{k}(x)}(1+\\frac{h}{\\beta })^{-k}\\sum _{\\bar{F}\\in S(F)}w(\\bar{F})}{\\sum _{k\\ge 0} \\sum _{F\\in _{k}(x)} \\sum _{\\bar{F}\\in S(F)} w(\\bar{F})}\\le (1+\\frac{h}{\\beta })^{-\\alpha |x|}.$ A similar argument applies when $0\\lnot \\leftrightarrow \\mathfrak {g}$ ; this condition is used in the second step defining $S$ to ensure the trees containing the vertices $v_{i}$ are not already connected to $\\mathfrak {g}$ .","In this case the weight $w(F)$ does not have the factor $h$ , but the remainder of the argument is identical.","If $h>0$ , then there are $c,C>0$ depending on $d,\\beta ,h$ such that Let $h>0$ and suppose that for all $x$ , $h_x= h$ .","Then there are $c,C>0$ depending on $d,\\beta ,h$ such that $¶^{\\Lambda _{N}}_{\\beta ,h}[0\\leftrightarrow x] \\le Ce^{-c |x |}.$ Since $¶^{\\Lambda _{N}}_{\\beta ,h}[0\\leftrightarrow x] =¶^{\\Lambda _{N}}_{\\beta ,h}[0\\leftrightarrow x, 0\\leftrightarrow \\mathfrak {g}] +¶^{\\Lambda _{N}}_{\\beta ,h}[0\\leftrightarrow x, 0\\lnot \\leftrightarrow \\mathfrak {g}],$ it is enough to estimate the first term, as the second is covered by Lemma REF .","Note $¶^{\\Lambda _{N}}_{\\beta ,h}[0\\leftrightarrow x, 0\\leftrightarrow \\mathfrak {g}]=\\sum _{y} ¶^{\\Lambda _{N}}_{\\beta ,h} [1_{0\\leftrightarrow x}1_{0\\leftrightarrow y}1_{y\\mathfrak {g}}] = \\sum _{y} ¶^{\\Lambda _{N}}_{\\beta ,h} [1_{0\\leftrightarrow x}1_{0\\leftrightarrow y}1_{0\\mathfrak {g}}].$ where the first equality follows from the fact that the only one vertex per component may connect to $\\mathfrak {g}$ , and the second follows from exchangeability of the choice of root.","Examining the rightmost expression, there are most $c_{d} |x |^{d}$ summands in which $|y |\\le |x|$ ; for these terms we drop the condition $0\\leftrightarrow y$ .","For the rest we drop $0\\leftrightarrow x$ .","This gives, by Lemma REF , $¶^{\\Lambda _{N}}_{\\beta ,h}[0\\leftrightarrow x, 0\\leftrightarrow \\mathfrak {g}]\\le C |x|^{d}e^{-c |x |} + \\sum _{ |y |> |x |} Ce^{-c |y |} \\le Ce^{-c |x |},$ where $c,C$ are changing from location to location but depend on $d,\\beta ,h$ only."],["Infinite volume limit","We now discuss weak limits $¶^{^d}_{\\beta }$ obtained by (i) first taking a (possibly subsequential) infinite-volume weak limit $¶^{^d}_{\\beta ,h} = \\lim _{N}¶^{\\Lambda _N}_{\\beta ,h}$ and (ii) subsequently taking a (possibly subsequential) limit $¶^{^d}_{\\beta }=\\lim _{h\\downarrow 0 }¶^{^d}_{\\beta ,h}$ .","We do not explicitly indicate the convergent subsequence chosen as what follows applies to any fixed choice.","Define $\\theta _{d,N}(\\beta ,h) ¶^{\\Lambda _N}_{\\beta ,h}[0\\leftrightarrow \\mathfrak {g}]=1-h^{-1}¶^{\\Lambda _N}_{\\beta ,h}[0\\mathfrak {g}]$ where the second equality is due to (REF ).","Since this last display only involves cylinder events, $\\lim _{N\\rightarrow \\infty }\\theta _{d,N}(\\beta ,h) = 1-h^{-1}¶^{^d}_{\\beta ,h}[0\\mathfrak {g}] \\theta _{d}(\\beta ,h),$ where the last equality defines $\\theta _{d}(\\beta ,h)$ .","Assume $\\lim _{h\\downarrow 0}\\theta _{d}(\\beta ,h)=\\theta _{d}(\\beta )$ exists.","Then $¶^{^d}_{\\beta }[|T_{0}|=\\infty ] = \\theta _{d}(\\beta ).$ Write $¶_{\\beta ,h}=¶^{^d}_{\\beta ,h}$ .","We claim that $¶_{\\beta ,h}[0\\mathfrak {g}] = \\sum _{n\\ge 1} ¶_{\\beta ,h}[|T_{0}|=n] \\frac{h}{1+nh},$ and hence, since $\\theta _{d}(\\beta ,h) = 1-h^{-1}¶_{\\beta ,h}[0\\mathfrak {g}]$ , $\\theta _{d}(\\beta ,h) = 1-\\sum _{n\\ge 1} ¶_{\\beta ,h}[|T_{0}|=n] \\frac{1}{1+nh}.$ Granting the claim, by dominated convergence we obtain $¶_{\\beta ,0}[|T_{0}|<\\infty ] = \\sum _{n\\ge 1}¶_{\\beta ,0}[|T_{0}|=n]=1-\\theta _{d}(\\beta ),$ as desired.","To prove the claim, rewrite it as $¶_{\\beta ,h}[|T_{0}|=\\infty , 0\\mathfrak {g}]=\\lim _{r\\rightarrow \\infty }¶_{\\beta ,h}[|T_{0}|\\ge r, 0\\mathfrak {g}]=0.$ The probabilities inside the limit are probabilities of cylinder events, and hence are limits of finite volume probabilities.","For a fixed $r$ the probability is at most $h/(1+rh)$ in finite volume, which vanishes as $r\\rightarrow \\infty $ ."],["Finite range decomposition","In this appendix, we give the precise references for the construction of the finite range decomposition (REF ).","The general method we use was introduced in [11], and presented in the special case we use in [17] and we will use this reference.","For $t>0$ , first recall the polynomials $P_t$ from [17] (these polynomials are called $W_t^*$ in [11]).","These are polynomials of degree bounded by $t$ satisfying $ \\frac{1}{\\lambda } = \\int _{0}^\\infty t^2 P_t(\\lambda ) \\, \\frac{dt}{t},\\qquad 0 \\le P_t(u) \\le O_s(1+t^2u)^{-s}$ for any $s>0$ and $u \\in [0,2]$ .","Our decomposition (REF ) is defined by $C_1(x,y) &= \\frac{1}{(2d+m^2)|\\Lambda _N|}\\sum _{k \\in \\Lambda _N^*} e^{ik\\cdot (x-y)} \\int _{0}^{\\frac{1}{2} L} t^2 P_t(\\frac{\\lambda (k)+m^2}{2d+m^2}) \\, \\frac{dt}{t}\\\\C_j(x,y) &= \\frac{1}{(2d+m^2)|\\Lambda _N|}\\sum _{k \\in \\Lambda _N^*} e^{ik\\cdot (x-y)} \\int _{\\frac{1}{2} L^{j-1}}^{\\frac{1}{2} L^j} t^2 P_t(\\frac{\\lambda (k)+m^2}{2d+m^2}) \\, \\frac{dt}{t}\\\\C_{N,N}(x,y) &= \\frac{1}{(2d+m^2)|\\Lambda _N|}\\sum _{k \\in \\Lambda _N^*} e^{ik\\cdot (x-y)} \\int _{\\frac{1}{2} L^N}^\\infty t^2 P_t(\\frac{\\lambda (k)+m^2}{2d+m^2}) \\, \\frac{dt}{t},$ where $\\lambda (k) = 4\\sum _{j=1}^{d}\\sin ^{2}(k_{j}/2)$ and $\\Lambda _N^* \\subset [-\\pi ,\\pi )^d$ is the dual torus.","The estimates for $C_1, \\dots , C_{N-1}$ are straightforward from these Fourier representations and can be found in [17].","We remark that in [17], the torus covariances are defined by periodisation of the finite range covariances on $^d$ ; by Poisson summation this is equivalent to the above definition.","The decomposition of $C_{N,N}$ in (REF ) is defined by removing the zero mode from $C_{N,N}$ : $C_{N}(x,y) &= \\frac{1}{(2d+m^2)|\\Lambda _N|}\\sum _{k \\in \\Lambda _N^*:k \\ne 0} e^{ik\\cdot (x-y)} \\int _{\\frac{1}{2} L^N}^\\infty t^2 P_t(\\frac{\\lambda (k)+m^2}{2d+m^2}) \\, \\frac{dt}{t}\\\\t_N &= \\frac{1}{2d+m^2} \\int _{\\frac{1}{2} L^N}^\\infty t^2 P_t(\\frac{m^2}{2d+m^2}) \\, \\frac{dt}{t},$ from which (REF ) is immediate.","For $C_{N}$ estimates follows as in [17]: $|C_N(x,y)| &\\le \\frac{1}{|\\Lambda _N|} \\sum _{k \\in \\Lambda _N^*: k \\ne 0} {\\int _{\\frac{1}{2} L^N}^\\infty t^2 P_t(\\frac{\\lambda (k)+m^2}{2d+m^2}) \\, \\frac{dt}{t}}&\\lesssim \\frac{1}{|\\Lambda _N|} \\sum _{k \\in \\Lambda _N^*: k \\ne 0} {\\int _{\\frac{1}{2} L^N}^\\infty t^{2}t^{-2s}|k|^{-2s} \\, \\frac{dt}{t}}&\\lesssim \\frac{L^{2N}}{|\\Lambda _N|} \\sum _{k \\in \\Lambda _N^*: k \\ne 0}L^{-2sN}|k|^{-2s}\\lesssim L^{-(d-2)N} \\int _1^\\infty r^{-2s+d-1} dr \\lesssim L^{-(d-2)N}$ and analogously for the discrete gradients.","Finally, by (REF ), $t_N&= \\frac{1}{(2d+m^2)} \\int _{\\frac{1}{2} L^N}^\\infty t^2P_t(\\frac{m^2}{2d+m^2}) \\, \\frac{dt}{t}&= \\frac{1}{m^2}-\\frac{1}{(2d+m^2)} \\int _0^{\\frac{1}{2} L^N} t^2P_t(\\frac{m^2}{2d+m^2}) \\, \\frac{dt}{t}=\\frac{1}{m^2} - O(L^{2N}).$"],["Proof of Proposition ","Proposition REF is essentially [27], with the minor changes of the separation of the coupling constant $u_j$ and the explicit inclusion of $\\theta $ .","We include a proof here for convenience.","We begin by algebraically manipulating $Z_{j}$ .","These manipulations only rely on factorisation properties and not on the precise definitions of $I$ and $K$ .","We will clearly state below when we restrict to the context of Proposition REF .","Consider $Z_j = \\sum _{X\\in _j} I^{\\Lambda \\setminus X}K(X)$ where $I(B) = e^{-V_j(B)}$ and $K(X)= K_j(X)$ .","We will use that $I^Y = \\prod _{B \\in _j(Y)} I(B)$ factors over blocks and $K_{j}(X)$ factors over connected components of $X$ .","Given any $\\tilde{I}(B) \\in (B)$ for $B \\in _j$ , $ \\theta Z_j = \\sum _{X\\in _j} \\tilde{I}^{\\Lambda \\setminus X} \\tilde{K}(X),\\qquad \\tilde{K}(X)= \\sum _{Y \\in _j(X)} (\\delta I)^Y \\theta K(X\\setminus Y),$ where $\\delta I(B) = \\theta I(B)-\\tilde{I}(B)$ and $\\tilde{I}^{Y}\\prod _{B\\in _{j}(Y)}\\tilde{I}(B)$ .","To see this, insert the identity $\\theta I^{\\Lambda \\setminus X} = \\sum _{Y\\subset \\Lambda \\setminus X} \\tilde{I}^{Y}(\\delta I)^{\\Lambda \\setminus (X\\cup Y)}$ into (REF ); (REF ) then follows by changing the summation index.","Using that $\\tilde{K}$ factors over components since $K$ factors over components, we can then define $\\check{K}(Y)$ to be $\\tilde{K}(Y) - \\sum _{B\\in _{j}(Y)}\\theta J(B,Y)$ for any connected polymer $Y$ (and zero otherwise), where $J(B,Y)\\in $ are given.","This yields the formula $\\tilde{K}(X) = \\prod _{Y \\in \\operatorname{Comp}(X)} {\\check{K}(Y) + \\sum _{B \\in _j(Y)} \\theta J(B,Y)}.$ Expanding the product, this can be written as $\\tilde{K}(X) = \\sum _{\\check{X} \\subset \\operatorname{Comp}(X)} \\check{K}(\\check{X}) \\prod _{Y \\in \\operatorname{Comp}(X \\setminus \\check{X})} \\sum _{B \\in _j(Y)} \\theta J(B,Y).$ Given the polymer $X \\setminus \\check{X}$ , now define $=(X,\\check{X}) \\subset \\lbrace (B,Y): B\\in _{j}, Y\\in _{j}\\rbrace $ so that $\\prod _{Y \\in \\operatorname{Comp}(X \\setminus \\check{X})} \\sum _{B \\in _j(Y)}J(B,Y) = \\sum _{} \\prod _{(B,Y)\\in } J(B,Y).$ Then $\\tilde{K}(X) = \\sum _{(\\check{X}, )} \\check{K}(\\check{X}) \\prod _{(B,Y) \\in } \\theta J(B,Y).$ By (REF ), since $\\tilde{I}\\in (\\Lambda )$ is a constant with respect to $_{C_{j+1}}$ , $_{C_{j+1}} \\theta Z_j&= \\sum _{X \\in _{j}} \\tilde{I}^{\\Lambda \\setminus X} _{C_{j+1}}\\tilde{K}(X)&= \\sum _{U \\in _{j+1}} \\tilde{I}^{\\Lambda \\setminus U} \\sum _{X \\in _j: \\bar{X} = U} \\tilde{I}^{U\\setminus X} _{C_{j+1}} \\tilde{K}(X)= \\sum _{U \\in _{j+1}} \\tilde{I}^{\\Lambda \\setminus U} K^{\\prime }(U),$ where by (REF ) $K^{\\prime }(U) = \\sum _{X \\in _j: \\bar{X} = U} \\tilde{I}^{U\\setminus X}\\sum _{(\\check{X}, )} _{C_{j+1}} \\check{K}(\\check{X})\\prod _{(B,Y) \\in } \\theta J(B,Y).$ We now specialise to the setting of Proposition REF .","Thus we set $\\tilde{I}(B)= e^{-(u+V)_{j+1}(B)}$ and $K_{j+1}$ as in (REF ).","Then $_{C_{j+1}} \\theta Z_j$ is $e^{-u_{j+1}|\\Lambda |}\\sum _{U \\in _{j+1}} e^{-V_{j+1}(\\Lambda \\setminus U)} K_{j+1}(U).$ To confirm this we explain the notation used in (REF ).","In the definition of $K_{j+1}$ , $X_{}$ is by definition $\\bigcup _{(B,Y)\\in }Y$ .","Given $X$ and $\\check{X}$ a subset of components of $X$ , $(X,\\check{X})$ is the set of sets of pairs $(B,Y)$ where (i) $B$ is a block in $Y$ , (ii) $Y$ is a component of $X\\setminus \\check{X}$ , (iii) each component $Y$ occurs in exactly one pair, and (iv) the closure of $\\check{X}\\cup \\cup _{(B,Y)}Y$ is $U$ .","To conclude the proof we make the definition that $\\sum _{(U)}$ is a shorthand for the double sum $\\sum _{X\\in _{j}:\\bar{X}=U}\\sum _{(\\check{X},)}$ , as an explicit description of the set $(U)$ will not be needed.","What remains is to prove the claims regarding factorisation and automorphism invariance.","For factorisation, note that $\\sum _{(U_{1}\\cup U_{2})} = \\sum _{(U_{1})}\\sum _{(U_{2})}$ for $U_{1},U_{2}\\in _{j+1}$ that do not touch.","Since $U_{1}$ and $U_{2}$ are distance $L^{j+1}>\\frac{1}{2}L^{j+1}+2^{d+1}L^{j}$ apart in this case the Grassmann integrals in the definition of $K_{j+1}(U_{1}\\cup U_{2})$ factor.","Here we have used our standing assumption that $L>2^{d+2}$ , that $J(B,Y)=0$ if $Y\\notin _{j}$ , and that the range of $C_{j+1}$ is $\\frac{1}{2}L^{j+1}$ .","Automorphism invariance follows directly from the formula for $K_{j+1}$ ."],["Acknowledgements","We thank David Brydges and Gordon Slade.","This article would not have been possible in this form without their previous contributions to the renormalisation group method.","We also thank them for the permission to include Figure REF from [17].","R.B.","was supported by the European Research Council under the European Union's Horizon 2020 research and innovation programme (grant agreement No.","851682 SPINRG).","N.C. was supported by Israel Science Foundation grant number 1692/17."]],"string":"[\n [\n \"Percolation transition for random forests in $d\\\\geq 3$\"\n ],\n [\n \"Abstract The arboreal gas is the probability measure on (unrooted spanning) forests of a graph in which each forest is weighted by a factor $\\\\beta>0$ per edge.\",\n \"It arises as the $q\\\\to 0$ limit with $p=\\\\beta q$ of the $q$-state random cluster model.\",\n \"We prove that in dimensions $d\\\\geq 3$ the arboreal gas undergoes a percolation phase transition.\",\n \"This contrasts with the case of $d=2$ where all trees are finite for all $\\\\beta>0$.\",\n \"The starting point for our analysis is an exact relationship between the arboreal gas and a non-linear sigma model with target space the fermionic hyperbolic plane $\\\\mathbb{H}^{0|2}$.\",\n \"This latter model can be thought of as the $0$-state Potts model, with the arboreal gas being its random cluster representation.\",\n \"Unlike the $q>0$ Potts models, the $\\\\mathbb{H}^{0|2}$ model has continuous symmetries.\",\n \"By combining a renormalisation group analysis with Ward identities we prove that this symmetry is spontaneously broken at low temperatures.\",\n \"In terms of the arboreal gas, this symmetry breaking translates into the existence of infinite trees in the thermodynamic limit.\",\n \"Our analysis also establishes massless free field correlations at low temperatures and the existence of a macroscopic tree on finite tori.\"\n ],\n [\n \"Introduction\",\n \"This paper has two distinct motivations.\",\n \"The first is to study the percolative properties of the arboreal gas, and the second is to understand spontaneously broken continuous symmetries.\",\n \"We first present our results from the percolation perspective, and then turn to continuous symmetries.\"\n ],\n [\n \"Main results for the arboreal gas\",\n \"The arboreal gas is the uniform measure on (unrooted spanning) forests of a weighted graph.\",\n \"More precisely, given an undirected graph $G=(\\\\Lambda ,E)$ , a forest $F=(\\\\Lambda ,E(F))$ is an acyclic subgraph of $G$ having the same vertex set as $G$ .\",\n \"Given an edge weight $\\\\beta >0$ (inverse temperature) and a vertex weight $h\\\\ge 0$ (external field), the probability of a forest $F$ under the arboreal gas measure is $ ¶_{\\\\beta ,h}^{G}[F] \\\\frac{1}{Z_{\\\\beta ,h}^{G}} \\\\beta ^{|E(F)|} \\\\prod _{T\\\\in F} (1+h|V(T)|)$ where $T\\\\in F$ denotes that $T$ is a tree in the forest, i.e., a connected component of $F$ , $|E(F)|$ is the number of edges in $F$ , and $|V(T)|$ is the number of vertices in $T$ .\",\n \"The arboreal gas is also known as the (weighted) uniform forest model, as Bernoulli bond percolation conditioned to be acyclic, and as the $q\\\\rightarrow 0$ limit of the $q$ -state random cluster model with $p/q$ converging to $\\\\beta $ , see [52].\",\n \"We study the arboreal gas on a sequence of tori $\\\\Lambda _N = ^d/L^N^d$ with $L$ fixed and $N\\\\rightarrow \\\\infty $ .\",\n \"To simplify notation, we will use $\\\\Lambda _N$ to denote both the graph and its vertex set.\",\n \"From the percolation point of view, the most fundamental question concerns whether a typical forest $F$ under the law (REF ) contains a giant tree.\",\n \"In all dimensions, elementary arguments show that giant trees can exist only if $h=0$ and if $\\\\beta $ is large enough, in the sense that connection probabilities decay exponentially whenever $h>0$ or $\\\\beta $ is small; see Appendix REF .\",\n \"The existence of a percolative phase for $h=0$ and $\\\\beta $ large does not, however, follow from standard techniques.\",\n \"The subtlety of the existence of a percolative phase is perhaps best evidenced by considering the case $d=2$ : in this case giant trees do not exist for any $\\\\beta >0$ [18].\",\n \"Our main result is that for $d\\\\ge 3$ giant trees do exist for $\\\\beta $ large and $h=0$ , and that truncated correlations have massless free field decay.\",\n \"To state our result precisely, let $\\\\lbrace 0 \\\\leftrightarrow x\\\\rbrace $ denote the event that 0 and $x$ are connected, i.e., in the same tree.\",\n \"Let $d\\\\ge 3$ and $L \\\\ge L_0(d)$ .\",\n \"Then there is $\\\\beta _0 \\\\in (0,\\\\infty )$ such that for $\\\\beta \\\\ge \\\\beta _0$ there exist $\\\\zeta _d(\\\\beta ) = 1-O(1/\\\\beta )$ , ${\\\\sf c}(\\\\beta ) = {\\\\sf c}+ O(1/\\\\beta )$ with ${\\\\sf c}>0$ , and $\\\\kappa >0$ such that $¶_{\\\\beta ,0}^{\\\\Lambda _N}[0\\\\leftrightarrow x]= \\\\zeta _d(\\\\beta ) + \\\\frac{{\\\\sf c}(\\\\beta )}{\\\\beta |x|^{d-2}} + O(\\\\frac{1}{\\\\beta |x|^{d-2+\\\\kappa }}) + O(\\\\frac{1}{\\\\beta L^{\\\\kappa N}}),$ where $|x|$ denotes the Euclidean norm.\",\n \"Numerical evidence for this phase transition of the arboreal gas was given in [37].\",\n \"More broadly our work was inspired by [34], [35], [19], [37], [54], [55], and we discuss further motivation later.\",\n \"Although both the arboreal gas and Bernoulli bond percolation have phase transitions for $d\\\\ge 3$ , the supercritical phases of these models behave very differently: (REF ) shows that the arboreal gas behaves like a critical model even in the supercritical phase, in the sense that it has massless free field truncated correlation decay.\",\n \"While this behaviour looks unusual when viewed through the lens of supercritical percolation, it is natural from the viewpoint of broken continuous symmetries.\",\n \"We will return to this point in Section REF .\",\n \"Theorem REF concerns the arboreal gas on large finite tori.\",\n \"Another limit to consider the arboreal gas in is the weak infinite volume limit.\",\n \"To this end, we consider the limit obtained by first taking $N\\\\rightarrow \\\\infty $ with $h>0$ and then taking $h\\\\downarrow 0$ .\",\n \"In manner similar to that for Bernoulli bond percolation in [43] and [2], the external field is equivalent to considering the arboreal gas on an extended graph $G^{\\\\mathfrak {g}} = (\\\\Lambda \\\\cup \\\\lbrace \\\\mathfrak {g}\\\\rbrace , E\\\\cup E^{\\\\mathfrak {g}})$ where $E^{\\\\mathfrak {g}} = \\\\Lambda \\\\times \\\\lbrace \\\\mathfrak {g}\\\\rbrace $ and each edge in $E^{\\\\mathfrak {g}}$ has weight $h$ .\",\n \"The additional vertex $\\\\mathfrak {g}$ is called the ghost vertex.\",\n \"The measure (REF ) is then obtained by forgetting the connections to the ghost.\",\n \"This rephrases that the product in (REF ) is equivalent to connecting a uniformly chosen vertex in each tree $T$ to $\\\\mathfrak {g}$ with probability $h|V(T)|/(1+h|V(T)|)$ .\",\n \"For vertices $x,y \\\\in \\\\Lambda $ , we continue to denote by $\\\\lbrace x\\\\leftrightarrow y\\\\rbrace $ the event that $x$ and $y$ are connected in the random forest subgraph of $G$ with law (REF ), i.e., without using the edges in $E^\\\\mathfrak {g}$ .\",\n \"We write $\\\\lbrace x\\\\leftrightarrow \\\\mathfrak {g}\\\\rbrace $ to denote the event that $x$ is connected to $\\\\mathfrak {g}$ .\",\n \"The event $\\\\lbrace 0\\\\leftrightarrow \\\\mathfrak {g}\\\\rbrace $ is a finite volume proxy for the event that the tree $T_0$ containing 0 becomes infinite in the infinite volume limit when $h\\\\downarrow 0$ .\",\n \"Indeed, let us define $\\\\theta _d(\\\\beta )\\\\lim _{h\\\\downarrow 0}\\\\lim _{N\\\\rightarrow \\\\infty } ¶^{\\\\Lambda _N}_{\\\\beta ,h}[0\\\\leftrightarrow \\\\mathfrak {g}],$ and let $¶_\\\\beta ^{^d}$ be any (possibly subsequential) weak infinite volume limit $\\\\lim _{h\\\\downarrow 0}\\\\lim _{N\\\\rightarrow \\\\infty }¶^{\\\\Lambda _N}_{\\\\beta ,h}$ .\",\n \"Then $\\\\theta _d(\\\\beta )=¶_\\\\beta ^{^d}[|T_0|=\\\\infty ],$ see Proposition REF .\",\n \"By a stochastic domination argument it is straightforward to show that $\\\\theta _d(\\\\beta ) =0 \\\\qquad \\\\text{for $0\\\\le \\\\beta 0$ by [18].\",\n \"The next theorem shows that for $d\\\\ge 3$ the arboreal gas also has a phase transition in this infinite volume limit.\",\n \"Let $d\\\\ge 3$ and $L \\\\ge L_0(d)$ .\",\n \"Then there is $\\\\beta _0 \\\\in (0,\\\\infty )$ such that for $\\\\beta \\\\ge \\\\beta _0$ the limit (REF ) exists and $\\\\theta _{d}(\\\\beta )^2 = \\\\zeta _d(\\\\beta )= 1-O(1/\\\\beta ),$ where $\\\\zeta _d(\\\\beta )$ is the finite volume density of the tree containing 0 from Theorem REF .\",\n \"In fact, our proof shows that $\\\\theta _d(\\\\beta )\\\\sim 1-c/\\\\beta $ with $c=(-\\\\Delta ^{^d})^{-1}(0,0)>0$ the expected time a simple random walk spends at the origin.\",\n \"This behaviour is different from that of Bernoulli bond percolation and more generally that of the random cluster model with $q>0$ .\",\n \"For these models the percolation probability is governed by Peierls' contours and is $1-O((1-p)^{2d})$ by [63].\",\n \"That the arboreal gas behaves critically within its supercritical phase can be further quantified in terms of the following truncated two-point functions: $\\\\tau _{\\\\beta }(x) = \\\\lim _{h\\\\downarrow 0} \\\\tau _{\\\\beta ,h}(x),& \\\\qquad \\\\tau _{\\\\beta ,h}(x)=\\\\lim _{N\\\\rightarrow \\\\infty }¶_{\\\\beta ,h}^{\\\\Lambda _N}[0\\\\leftrightarrow x, 0 \\\\lnot \\\\leftrightarrow \\\\mathfrak {g}],\\\\\\\\\\\\sigma _{\\\\beta }(x) = \\\\lim _{h\\\\downarrow 0} \\\\sigma _{\\\\beta ,h}(x),& \\\\qquad \\\\sigma _{\\\\beta ,h}(x) = \\\\lim _{N\\\\rightarrow \\\\infty }{¶_{\\\\beta ,h}^{\\\\Lambda _N}[0\\\\lnot \\\\leftrightarrow \\\\mathfrak {g}]^2 - ¶_{\\\\beta ,h}^{\\\\Lambda _N}[0\\\\lnot \\\\leftrightarrow x, 0\\\\lnot \\\\leftrightarrow \\\\mathfrak {g},x\\\\lnot \\\\leftrightarrow \\\\mathfrak {g}]}.$ Under the assumptions of Theorem REF , for $\\\\beta \\\\ge \\\\beta _0$ , the limits (REF )–() exist and there exist constants ${\\\\sf c}_i(\\\\beta ) = {\\\\sf c}_i+O(1/\\\\beta )$ and $\\\\kappa >0$ such that $\\\\tau _{\\\\beta }(x) &= \\\\frac{{\\\\sf c}_1(\\\\beta )}{\\\\beta |x|^{d-2}} +O(\\\\frac{1}{\\\\beta |x|^{d-2+\\\\kappa }}),\\\\\\\\\\\\sigma _{\\\\beta }(x) &= \\\\frac{{\\\\sf c}_2(\\\\beta )}{\\\\beta ^2|x|^{2d-4}} + O(\\\\frac{1}{\\\\beta ^2 |x|^{2d-4+\\\\kappa }}).$ The constants satisfy $({\\\\sf c}_2(\\\\beta )/{\\\\sf c}_1(\\\\beta )^2)\\\\theta _d(\\\\beta )^2=1$ and ${\\\\sf c}(\\\\beta ) = 2{\\\\sf c}_1(\\\\beta )$ , ${\\\\sf c}(\\\\beta )$ from Theorem REF .\",\n \"Further results could be deduced from our analysis, but to maintain focus we have not carried these out in detail.\",\n \"We mention some of them below in Section REF when discussing our results and open problems.\"\n ],\n [\n \"The $^{0|2}$ model and continuous symmetries\",\n \"In [34], [35], the arboreal gas was related to a fermionic field theory and a supersymmetric non-linear sigma model with target space one half of the degenerate super-sphere $\\\\mathbb {S}^{0|2}$ .\",\n \"In [18] this was reinterpreted as a non-linear sigma model with hyperbolic target space $^{0|2}$ (which we refer to as the $^{0|2}$ model for short).\",\n \"The reinterpretation was essential in [18]; it is less essential for the present work, but nevertheless, we continue to use the $^{0|2}$ formulation of the model.\",\n \"Briefly, the $^{0|2}$ model is defined as follows; see [18] for further details.\",\n \"For every vertex $x\\\\in \\\\Lambda $ , there are two (anticommuting) Grassmann variables $\\\\xi _x$ and $\\\\eta _x$ and we then set $z_x \\\\sqrt{1-2\\\\xi _x\\\\eta _x} 1-\\\\xi _x\\\\eta _x.$ Thus the $z_x$ commute with each other and with the odd elements $\\\\xi _x$ and $\\\\eta _x$ .\",\n \"The formal triples $u_x(\\\\xi _x,\\\\eta _x,z_x)$ are supervectors with two odd components $\\\\xi _x,\\\\eta _x$ and an even component $z_x$ .\",\n \"These supervectors satisfy the sigma model constraint $u_x\\\\cdot u_x=-1$ for the super inner product $u_x\\\\cdot u_y -\\\\xi _x\\\\eta _y-\\\\xi _y\\\\eta _x - z_xz_y.$ In analogy with the tetrahedral representation of the Potts model, the sigma model constraint can be thought of as $u_x\\\\cdot u_x = q-1$ with $q=0$ .\",\n \"The constraint is also reminiscent of the embedding of the hyperbolic space $^2$ in $^3$ equipped with the standard quadratic form with Lorentzian signature $(1, 1, -1)$ .\",\n \"Indeed, $-\\\\xi _x\\\\eta _y-\\\\xi _y\\\\eta _x$ is the fermionic analogue of the Euclidean inner product on $^2$ .\",\n \"The expectation of the $^{0|2}$ model is ${F}_{\\\\beta ,h} \\\\frac{1}{Z_{\\\\beta ,h}}\\\\int (\\\\prod _{x\\\\in \\\\Lambda } \\\\partial _{\\\\eta _x}\\\\partial _{\\\\xi _x} \\\\frac{1}{z_x}) e^{\\\\frac{\\\\beta }{2} (u,\\\\Delta u) - h(1,z-1)} F.$ In this expression, $\\\\int \\\\prod _{x\\\\in \\\\Lambda } \\\\partial _{\\\\eta _x}\\\\partial _{\\\\xi _x}$ denotes the Grassmann integral (i.e., the coefficient of the top degree monomial of the integrand), $Z_{\\\\beta ,h}$ is a normalising constant, and $(u,\\\\Delta u) = -\\\\frac{1}{2} \\\\sum _{xy\\\\in E(\\\\Lambda ) }(u_x-u_y)\\\\cdot (u_x-u_y)= \\\\sum _{xy\\\\in E(\\\\Lambda ) }(u_x\\\\cdot u_y+1),\\\\qquad (1,z) = \\\\sum _{x\\\\in \\\\Lambda } z_x,$ where $xy\\\\in E(\\\\Lambda )$ denotes that $x$ and $y$ are nearest neighbours (counting every pair once), and the inner products are given by (REF ).\",\n \"The factors $1/z_x$ in (REF ) are the canonical fermionic volume form invariant under the symmetries associated with (REF ) as discussed further below.\",\n \"As explained in [18] (see also [34] where such relations were first observed) connection and edge probabilities of the arboreal gas are equivalent to correlation functions of the $^{0|2}$ model.\",\n \"The following proposition summarises the ones we need, see Appendix .\",\n \"For any finite graph $G$ , any $\\\\beta \\\\ge 0$ and $h \\\\ge 0$ , $ ¶_{\\\\beta ,h}[0\\\\leftrightarrow \\\\mathfrak {g}] &= {z_0}_{\\\\beta ,h},\\\\\\\\¶_{\\\\beta ,h}[0\\\\leftrightarrow x, 0 \\\\lnot \\\\leftrightarrow \\\\mathfrak {g}] &= {\\\\xi _0\\\\eta _x}_{\\\\beta ,h},\\\\\\\\¶_{\\\\beta ,h}[0\\\\leftrightarrow x]+¶_{\\\\beta ,h}[0\\\\lnot \\\\leftrightarrow x, 0\\\\leftrightarrow \\\\mathfrak {g}, x\\\\leftrightarrow \\\\mathfrak {g}] &= -{u_0\\\\cdot u_x}_{\\\\beta ,h},$ and the normalising constants in (REF ) and (REF ) are equal.\",\n \"In particular, $ ¶_{\\\\beta ,0}[0\\\\leftrightarrow x] = -{u_0\\\\cdot u_x}_{\\\\beta ,0} = -{z_0z_x}_{\\\\beta ,0} = {\\\\xi _0\\\\eta _x}_{\\\\beta ,0} = 1-{\\\\xi _0\\\\eta _0\\\\xi _x\\\\eta _x}_{\\\\beta ,0}.$ These relations resemble those between the Potts model and the random cluster model, giving further credence to our heuristic that the $^{0|2}$ model may be interpreted as the 0-state Potts model, with the arboreal gas playing the role of the 0-state random cluster model.\",\n \"Nevertheless, there are important differences from the $q$ -state Potts model with $q \\\\ge 1$ .\",\n \"Chief amongst them is that the $^{0|2}$ model has continuous symmetries.\",\n \"To make this precise, let $ T=\\\\sum _{x\\\\in \\\\Lambda } z_x \\\\partial _{\\\\xi _x}, \\\\qquad \\\\bar{T}=\\\\sum _{x\\\\in \\\\Lambda } z_x \\\\partial _{\\\\eta _x}.$ One way to understand the significance of $T, \\\\bar{T}$ is via the identities ${TF}_{\\\\beta ,0}={\\\\bar{T} F}_{\\\\beta ,0}=0$ for any (noncommutative) polynomial $F$ in the variables $\\\\xi $ and $\\\\eta $ .\",\n \"For example, ${T\\\\xi _{0}}_{\\\\beta ,0}={z_{0}}_{\\\\beta ,0}=0$ .\",\n \"Identities derived in this way are conventionally called Ward identities.\",\n \"The maps $T$ and $\\\\bar{T}$ are infinitesimal generators of two global internal supersymmetries of the $^{0|2}$ model.\",\n \"These supersymmetries are explicitly broken if $h \\\\ne 0$ .\",\n \"They are analogues of infinitesimal Lorentz boosts or infinitesimal rotations.\",\n \"Together with a further internal symmetry corresponding to rotations in the $\\\\xi , \\\\eta $ plane, these operators generate the symmetry algebra $\\\\mathfrak {osp}(1|2)$ of the $^{0|2}$ model.\",\n \"For details and further explanations, see [18].\",\n \"As generators of continuous symmetries, $T$ and $\\\\bar{T}$ imply Ward identities that are not available for the Potts model with $q\\\\ge 1$ .\",\n \"These identities are crucial for our analysis and will be discussed below.\",\n \"The phase transition of the arboreal gas corresponds to a spontaneous breaking of the above supersymmetries in the infinite volume limit.\",\n \"This is shown in our next theorem for the $^{0|2}$ model from which Theorems REF and REF follow immediately by (REF )–() (except for the same statements relating the constants, which we omitted here); a similar reformulation applies to Theorem REF .\",\n \"Let $d\\\\ge 3$ and $L\\\\ge L_0(d)$ .\",\n \"There exists $\\\\beta _0\\\\in (0,\\\\infty )$ and constants $\\\\theta _d(\\\\beta )=1+O(1/\\\\beta )$ and ${\\\\sf c}_i(\\\\beta ) = {\\\\sf c}_i+O(1/\\\\beta )$ and $\\\\kappa >0$ (all dependent on $d$ ) such that for $\\\\beta \\\\ge \\\\beta _0$ , $\\\\lim _{h\\\\downarrow 0}\\\\lim _{N\\\\rightarrow \\\\infty }{z_0}_{\\\\beta ,h}&= \\\\theta _d(\\\\beta )\\\\\\\\\\\\lim _{h\\\\downarrow 0}\\\\lim _{N\\\\rightarrow \\\\infty }{\\\\xi _0\\\\eta _x}_{\\\\beta ,h}&= \\\\frac{{\\\\sf c}_1(\\\\beta )}{\\\\beta |x|^{d-2}} + O(\\\\frac{1}{\\\\beta |x|^{d-2+\\\\kappa }})\\\\\\\\\\\\lim _{h\\\\downarrow 0}\\\\lim _{N\\\\rightarrow \\\\infty }{{z_0z_x}_{\\\\beta ,h}-{z_0}_{\\\\beta ,h} {z_x}_{\\\\beta ,h}}&= -\\\\frac{{\\\\sf c}_2(\\\\beta )}{\\\\beta ^2 |x|^{2d-4}} + O(\\\\frac{1}{\\\\beta ^2|x|^{2d-4+\\\\kappa }}).$ In particular, $\\\\lim _{h\\\\downarrow 0}\\\\lim _{N\\\\rightarrow \\\\infty }{u_0\\\\cdot u_x}_{\\\\beta ,h} = -\\\\theta _d(\\\\beta )^2 -\\\\frac{2{\\\\sf c}_1(\\\\beta )}{\\\\beta |x|^{d-2}} + O(\\\\frac{1}{\\\\beta |x|^{d-2+\\\\kappa }}).$ In fact, the constants ${\\\\sf c_{i}}(\\\\beta )$ both satisfy ${\\\\sf c_{i}}(\\\\beta ) = c_{d}^{i} + O(1/\\\\beta )$ , where $c_{d}$ is the leading constant in the asymptotics of the Green function of the Laplacian $-\\\\Delta ^{^{d}}$ on $^{d}$ : $(-\\\\Delta ^{^d})^{-1}(0,x)= \\\\frac{c_{d}}{|x|^{d-2}} + O(|x|^{-(d-2)-1}).$ Our proof of Theorem REF is by a rigorous renormalisation group analysis aided by Ward identities.\",\n \"The starting point is setting $\\\\psi =\\\\sqrt{\\\\beta }\\\\eta $ and $\\\\bar{\\\\psi }= \\\\sqrt{\\\\beta }\\\\xi $ ; the fermionic density in (REF ) is then equivalent to $\\\\exp {- (\\\\psi ,-\\\\Delta \\\\bar{\\\\psi })-\\\\frac{1}{\\\\beta }(1+h) \\\\sum _{x\\\\in \\\\Lambda }\\\\psi _x\\\\bar{\\\\psi }_x - \\\\frac{1}{2\\\\beta } \\\\sum _{x\\\\in \\\\Lambda } \\\\psi _x\\\\bar{\\\\psi }_x \\\\sum _{e\\\\in \\\\mathcal {E}_d} \\\\psi _{x+e} \\\\bar{\\\\psi }_{x+e}},$ where the 1 in the quadratic term arises from putting the volume form $1/z = e^{+\\\\xi \\\\eta }=e^{-\\\\eta \\\\xi }$ into the exponential, and $\\\\mathcal {E}_{d}=\\\\lbrace e_{1},\\\\dots , e_{2d}\\\\rbrace $ are the standard unit vectors (where $e_{d+j}=-e_{j}$ ).\",\n \"The reformulation (REF ) looks very much like a fermionic version of the $\\\\varphi ^4$ spin model.\",\n \"However, the following differences are important: The coupling constants of the quadratic and quartic terms are related.\",\n \"This relation is due to the geometric origin of the model as a non-linear sigma model and analogous relations are present in intrinsic coordinates for other sigma models like the vector $O(n)$ model.\",\n \"We will use the following Ward identity for the $^{0|2}$ model to address this point: $ {z_0}_{\\\\beta ,h} = {T\\\\xi _0}_{\\\\beta ,h}= -\\\\sum _{x\\\\in \\\\Lambda } h {\\\\xi _0 Tz_x}_{\\\\beta ,h}= h \\\\sum _{x\\\\in \\\\Lambda } {\\\\xi _0 \\\\eta _x}_{\\\\beta ,h},$ where $T$ is the symmetry generator (REF ).\",\n \"Due to the fermionic nature of the field, the quartic term actually has gradients in it: denoting the discrete gradient in direction $e$ by $(\\\\nabla _e\\\\psi )_x = \\\\psi _{x+e}-\\\\psi _x$ , it can be written as $\\\\frac{1}{2}\\\\psi _x\\\\bar{\\\\psi }_x \\\\sum _{e\\\\in \\\\mathcal {E}_d} \\\\psi _{x+e}\\\\bar{\\\\psi }_{x+e}=\\\\frac{1}{2}\\\\psi _x\\\\bar{\\\\psi }_x \\\\sum _{e\\\\in \\\\mathcal {E}_d} (\\\\nabla _e \\\\psi )_x(\\\\nabla _e \\\\bar{\\\\psi })_x\\\\psi _x\\\\bar{\\\\psi }_x (\\\\nabla \\\\psi )_x(\\\\nabla \\\\bar{\\\\psi })_x,$ where we introduced the shorthand notation $(\\\\nabla \\\\psi )_x(\\\\nabla \\\\bar{\\\\psi })_x= \\\\frac{1}{2} \\\\sum _{e\\\\in \\\\mathcal {E}_d}(\\\\nabla _e \\\\psi )_x(\\\\nabla _e \\\\bar{\\\\psi })_x$ .\",\n \"After taking into account the points above, power counting heuristics (which we expect can be generalised to all non-linear sigma models with continuous symmetry) predict that the lower critical dimension for spontaneous symmetry breaking with free field low temperature fluctuations is two for the $^{0|2}$ model.\",\n \"In conjunction with [18], our results rigorously establish that the lower critical dimension is two for the $^{0|2}$ model.\"\n ],\n [\n \"Background on non-linear sigma models and renormalisation\",\n \"The low temperature renormalisation group analysis of non-linear sigma models with non-abelian continuous symmetry is a notorious problem that was famously considered by Balaban for the case of $O(n)$ symmetry, see [9], [10] and references therein.\",\n \"Our comparatively simple analysis of the $^{0|2}$ model, which is a non-linear sigma model with non-abelian continuous $OSp(1|2)$ symmetry, is made possible mainly by the fact that it does not suffer a large field problem because it has a fermionic representation.\",\n \"In addition to this, our approach to the $^{0|2}$ model differs from Balaban's approach to the $O(n)$ model on a conceptual level, in that it is based on intrinsic coordinates as opposed to extrinsic ones.\",\n \"It is unclear to us how to implement an extrinsic approach in our situation of $OSp(1|2)$ symmetry.\",\n \"Somewhat remarkably, despite its simplicity, the $^{0|2}$ model has all of the main features present in the non-abelian $O(n)$ models, including: absence of spontaneous symmetry breaking in 2d (proven in [18]); mass generation in 2d (conjectured in [35]); and a spontaneous symmetry breaking phase transition with massless low temperature fluctuations in $d\\\\ge 3$ (the main result of this work).\",\n \"The $^{0|2}$ model is a member of the family of hyperbolic sigma models with target spaces $^{n|2m}$ , see [36] for a discussion of some aspects of this.\",\n \"By supersymmetric localisation the observables of the $^{0|2}$ model considered in Theorem REF are equivalent to the analogous ones of the non-linear sigma model with target $^{2|4}$ .\",\n \"While this relation does not play a role in this paper, it leads to a more direct representation of the continuous symmetry breaking observed here.\",\n \"In brief, in the $^{2|4}$ model each vertex comes equipped with two real and four Grassmann fields.\",\n \"By expressing these fields in horospherical coordinates one of the real fields and the four Grassmann fields can be integrated out.\",\n \"The marginal distribution of the remaining real field, which is called the $t$ -field, may be viewed as a `$\\\\nabla \\\\phi $ ' random surface model, albeit with a nonconvex and nonlocal Hamiltonian.\",\n \"By this we mean that the potential is invariant under the global translation $t_x\\\\mapsto t_x +r$ for $r\\\\in $ .\",\n \"See [18] for more details, where this perspective was used to prove the absence of symmetry breaking in $d=2$ .\",\n \"The full $^{n|2m}$ family has been important for advancing our understanding of other aspects of these models [18], [36].\",\n \"Of particular note, we mention that the $^{2|2}$ model has received substantial prior attention due to its exact connection to linearly reinforced random walks and its motivation from random matrix theory, see [64], [71], [72], [70], [40].\",\n \"For hyperbolic sigma models with target $^n$ , $n \\\\ge 1$ , spontaneous symmetry breaking for all $\\\\beta >0$ was shown in [70], and with target $^{2|2}$ for $\\\\beta $ large in [40] (see also [39]).\",\n \"For motivation from random matrix theory and the Anderson transition see [68], [69].\",\n \"These proofs make essential use of the horospherical coordinates mentioned above.\",\n \"Moreover, the proof of symmetry breaking for the $^{2|2}$ model in [40] relies on an infinite number of Ward identifies resulting from supersymmetric localisation.\",\n \"These identities are absent in the $^{0|2}$ model, limiting the applicability of the methods of [40] to our setting.\",\n \"At the same time, the $^{2|2}$ model has no purely fermionic representation, and so our methods do not apply there, at least without significant further developments.\",\n \"Introductions to fermionic renormalisation include [20], [59], [65], see also [47].\",\n \"Recent probabilistic applications of these approaches to fermionic renormalisation include the study of interacting dimers [48], [49] and two-dimensional finite range Ising models [46], [45], [7].\",\n \"Our organisation of the renormalisation group is instead based on a finite range decomposition, and follows [27] and its further developments in [30], [31], [15], [32], [33], [16], [28], [11].\",\n \"This approach has its origins in [26].\",\n \"For an introduction to this approach in a hierarchical bosonic context see [17].\",\n \"Previous applications of this approach include the study of 4d weakly self-avoiding walks [14], [13]; the nearest-neighbour critical 4d $|\\\\varphi |^4$ model [12], [67] and long-range versions thereof [66], [56]; the ultraviolet $\\\\varphi ^4_3$ problem [25], [29]; analysis of the Kosterlitz–Thousless transition of the 2d Coulomb gas [38], [42]; the Cauchy–Born problem [1]; and others.\",\n \"While the construction of the bulk renormalisation group flow is simpler for the intrinsic representation of the $^{0|2}$ model than in many of the previous references, a crucial novelty of our present work is the combination of the finite range renormalisation group approach with Ward identities, together with a precise analysis of a nontrivial zero mode.\",\n \"This has enabled us to apply these methods to a non-linear sigma model in the phase of broken symmetry.\",\n \"It would be extremely interesting to understand this approach for bosonic non-linear sigma models where, while `large fields' cause serious complications, the formal perturbative analysis is very much in parallel to the fermionic version we study in this paper.\",\n \"Ward identities of a different type have previously been used in the renormalisation group analyses in [8] and [21] and many follow-up works including [48], [49].\",\n \"Finally, we mention that Theorem REF yields quantitative finite volume statements.\",\n \"The proof implements a rigorous finite size analysis along the lines of that proposed in [24].\",\n \"It would be very interesting to extend this to even higher precision as discussed in Section REF below.\"\n ],\n [\n \"Future directions for the arboreal gas\",\n \"In this section we discuss several interesting open directions, including the geometric structure of the weak infinite volume limits of the arboreal gas and its relation to the uniform spanning tree, and a conjectural finite size universality similar to Wigner–Dyson universality from random matrix theory.\"\n ],\n [\n \"Finite volume behaviour\",\n \"The detailed finite volume behaviour of the arboreal gas would be very interesting to understand beyond the precision of Theorem REF .\",\n \"On the complete graph at supercritical temperatures it is known that there is a unique macroscopic cluster, and that there are an unbounded number of clusters whose sizes are of order $|\\\\Lambda |^{2/3}$ [57].\",\n \"The fluctuations of the macroscopic cluster are non-Gaussian of scale $|\\\\Lambda |^{2/3}$ and the distribution of the ordered cluster sizes of the mesoscopic clusters has been determined [57].\",\n \"The joint law of the mesoscopic clusters can be characterised [58].\",\n \"Intriguingly, $|\\\\Lambda |^{2/3}$ is the size of the largest tree at criticality on the complete graph.\",\n \"The order statistics of the supercritical mesoscopic clusters follow the critical point order statistics [58].\",\n \"Going beyond the complete graph, is this distribution of ordered cluster sizes universal, at least in sufficiently high dimensions?\",\n \"This would be similar to the conjectured universality of Wigner–Dyson statistics from random matrix theory [60] or the conjectured universality of the distribution of macroscopic loops in loop representations of $O(n)$ (and other) spin systems [61], [51].\",\n \"More generally it would be an instance of the universality of low temperature fluctuations in finite volume in models with continuous symmetries.\",\n \"Finally, we mention that on expander graphs the existence of a phase transition for the arboreal gas is not difficult to show by using a natural split–merge dynamics [50].\",\n \"It would be interesting if this dynamical approach could also be used to obtain information about the cluster size distribution.\",\n \"As mentioned previously, the arboreal gas is also known as the uniform forest model [52].\",\n \"We emphasise that the arboreal gas is not what is typically known as the uniform spanning forest (USF), which is in fact the weak limit as $\\\\Lambda _N \\\\uparrow ^d$ of a uniform spanning tree (UST) [62].\",\n \"On a finite graph, the UST is the $\\\\beta \\\\rightarrow \\\\infty $ limit of the arboreal gas.\",\n \"The correct scaling of the external field for this limit is $h=\\\\beta \\\\kappa $ and we thus write $¶_{\\\\text{UST},\\\\kappa } = \\\\lim _{\\\\beta \\\\rightarrow \\\\infty }¶_{\\\\beta ,\\\\beta \\\\kappa }$ for the UST on a finite graph (plus ghost vertex if $\\\\kappa >0$ ).\",\n \"For $\\\\kappa >0$ , this measure is also known as the rooted spanning forest, because disregarding the connections to the ghost vertex disconnects the tree of the UST, with vertices previously connected to the ghost becoming roots.\",\n \"The distributions of rooted and unrooted forests are not the same.\",\n \"To help prevent confusion we will refer to the rooted spanning forests as (a special case of) the UST.\",\n \"It is trivial that $¶_{\\\\text{UST},0}^{\\\\Lambda _N}[0\\\\leftrightarrow x]=1$ .\",\n \"Nevertheless, the behaviour of the UST in the weak infinite volume limit depends on the dimension $d$ .\",\n \"This limit can be defined as $¶_{\\\\rm UST}^{^d} =\\\\lim _{\\\\kappa \\\\downarrow 0}\\\\lim _{N\\\\rightarrow \\\\infty }¶_{\\\\text{UST},\\\\kappa }^{\\\\Lambda _N}$ and is independent of the finite volume boundary conditions (e.g.\",\n \"free, wired, or periodic as above) imposed on $\\\\Lambda _N$ , see [62].\",\n \"Even though the function $1_{0\\\\leftrightarrow x}$ is not continuous with respect to the topology of weak convergence, it is still true that $¶_{{\\\\rm UST}}^{^d}[0\\\\leftrightarrow x] = \\\\lim _{\\\\kappa \\\\downarrow 0}\\\\lim _{N\\\\rightarrow \\\\infty }¶_{{\\\\rm UST},\\\\kappa }^{\\\\Lambda _N}[0 \\\\leftrightarrow x].$ The order of limits here is essential.\",\n \"In this infinite volume limit the UST disconnects into infinitely many infinite trees if $d>4$ , but remains a single connected tree if $d\\\\le 4$ , see [62].\",\n \"Moreover, $¶_{\\\\rm UST}^{^d}[0 \\\\leftrightarrow x] + ¶_{\\\\rm UST}^{^d}[0\\\\lnot \\\\leftrightarrow x,|T_0|=\\\\infty , |T_x|=\\\\infty ] = 1.$ On the left-hand side, the second term vanishes if $d\\\\le 4$ whereas the first term tends to 0 as $|x|\\\\rightarrow \\\\infty $ if $d>4$ .\",\n \"Furthermore, the geometric structure of the trees under $¶_{\\\\rm UST}^{^d}$ is well understood.\",\n \"In particular, all trees are one-ended, meaning that removing one edge from a tree results in two trees, of which one is finite [62], [23].\",\n \"For the arboreal gas, the existence and uniqueness of infinite volume limits is an open question.\",\n \"Nonetheless, subsequential limits exist, and in such an infinite volume limit all trees are finite almost surely when $\\\\beta $ is small, while Theorem REF implies the existence of an infinite tree for $\\\\beta $ large.\",\n \"Moreover, by Theorem REF , $ \\\\lim _{h\\\\downarrow 0}\\\\lim _{N\\\\rightarrow \\\\infty }{ ¶_{\\\\beta ,h}^{\\\\Lambda _N}[0 \\\\leftrightarrow x] + ¶_{\\\\beta ,h}^{\\\\Lambda _N}[0\\\\lnot \\\\leftrightarrow x, 0\\\\leftrightarrow \\\\mathfrak {g}, x\\\\leftrightarrow \\\\mathfrak {g}]}= \\\\theta _d(\\\\beta )^2 + \\\\frac{2{\\\\sf c}_1(\\\\beta )}{\\\\beta |x|^{d-2}} + O(\\\\frac{1}{\\\\beta |x|^{d-2+\\\\kappa }}).$ By analogy with the UST, we expect that only the first term on the left-hand side contributes for $d\\\\le 4$ and that only the second term contributes asymptotically as $|x|\\\\rightarrow \\\\infty $ for $d>4$ .\",\n \"The tempting conjecture that the UST stochastically dominates the arboreal gas on the torus is consistent with these expectations.\",\n \"The analogue of the left-hand side of (REF ) plays an important role in the proof of uniqueness of the infinite cluster in Bernoulli percolation in [5]; this is related to the vanishing of the second term.\",\n \"As already mentioned, for the arboreal gas we only expect this to be true in $d\\\\le 4$ .\",\n \"Beyond the questions above, it would be interesting to analyse more detailed geometric aspects of the arboreal gas.\",\n \"For example, can one construct scaling limits as has been done for some spanning tree models [3], [4], [44], [6]?\",\n \"Finally, we mention that a detailed analysis of the infinite volume behaviour of the arboreal gas on regular trees with wired boundary conditions has been carried out [41].\",\n \"This infinite volume behaviour is consistent with the finite volume behaviour of the complete graph, e.g., at all supercritical temperatures the sizes of finite clusters have the same distribution as those of critical percolation.\",\n \"Our analysis could be extended to a detailed study of the approach $h\\\\downarrow 0$ .\",\n \"To keep the length of this paper within bounds, we do not carry this out, but here briefly comment on what we expect can be shown by extensions of our analysis.\",\n \"As discussed above, a natural object is the magnetisation $M(\\\\beta ,h) = \\\\lim _{N\\\\rightarrow \\\\infty } M_{N}(\\\\beta ,h), \\\\qquad M_{N}(\\\\beta ,h) = ¶^{\\\\Lambda _{N}}_{\\\\beta ,h}[0\\\\leftrightarrow \\\\mathfrak {g}],$ and the corresponding susceptibility (neglecting questions concerning the order of limits) $\\\\chi (\\\\beta ,h)= {}{h}M(\\\\beta ,h)= \\\\sum _{x} \\\\sigma _{\\\\beta ,h}(x).$ Thus for the arboreal gas, the susceptibility is not the sum over $\\\\tau _{\\\\beta ,h}(x)$ as is the case for Bernoulli bond percolation, but the sum over $\\\\sigma _{\\\\beta ,h}(x)$ .\",\n \"In terms of the sigma model, $\\\\chi $ maybe viewed as the longitudinal susceptibility, often denoted $\\\\chi _{||}$ .\",\n \"In this interpretation, the sum over $\\\\tau _{\\\\beta ,h}(x)$ is the transversal susceptibility $\\\\chi _{\\\\perp }$ and satisfies the Ward identity $\\\\chi _{\\\\perp }(\\\\beta ,h)=\\\\sum _{x} \\\\tau _{\\\\beta ,h}(x) = h^{-1}M(\\\\beta ,h)$ .\",\n \"This identity is crucial in our analysis.\",\n \"For the longitudinal susceptibility, we expect that it would be possible to extend our analysis to show $\\\\chi (\\\\beta ,h) \\\\sim {\\\\left\\\\lbrace \\\\begin{array}{ll}C(\\\\beta ) h^{-1/2} & (d=3)\\\\\\\\C(\\\\beta ) |\\\\log h|& (d=4)\\\\\\\\C(\\\\beta ) & (d>4).\\\\end{array}\\\\right.\",\n \"}$ Defining the free energy $f(\\\\beta ,h)=\\\\lim _{N\\\\rightarrow \\\\infty } |\\\\Lambda _N|^{-1}\\\\log Z_{\\\\beta ,h}^{\\\\Lambda _{N}}$ , for $\\\\beta \\\\ge \\\\beta _0$ the previous asymptotics suggest that $h\\\\mapsto f(\\\\beta ,h)$ is $C^2$ in $d>4$ but only $C^1$ for $d=3,4$ .\",\n \"In fact, extrapolating from our renormalisation group analysis we believe that for $\\\\beta \\\\ge \\\\beta _0$ the free energy is $C^n$ but not $C^{n+1}$ as a function of $h\\\\ge 0$ for $n={\\\\frac{d-1}{2}}$ .\",\n \"It is unclear how this is connected to the geometry of the component graph of the UST, which also changes as the dimension is varied [22], [53].\"\n ],\n [\n \"Organisation and notation\",\n \"This paper is organised as follows.\",\n \"In Section , we show how Theorem REF is reduced to renormalisation group results with the help of the Ward identity (REF ).\",\n \"In Sections – we then prove these renormalisation group results.\",\n \"Section is concerned with the construction of the bulk renormalisation group flow and Section uses this analysis to compute the susceptibility.\",\n \"Section then constructs the renormalisation group flow for observables, which is used in Section to compute pointwise correlation functions.\",\n \"The short Section then collects the results.\",\n \"Finally, in Appendix we collect relations between the arboreal gas and the $^{0|2}$ model as well as basic percolation and high temperature properties of the arboreal gas, and in Appendix we include some background material about the renormalisation group method.\",\n \"Throughout we use $a_{n}\\\\sim b_{n}$ to denote $\\\\lim _{n\\\\rightarrow \\\\infty }a_{n}/b_{n}=1$ , $a_{n}\\\\asymp b_{n}$ to denote the existence of $c,C>0$ such that $c a_{n}\\\\le b_{n}\\\\le Ca_{n}$ , $a_n \\\\lesssim b_n$ if $a_n \\\\le Cb_n$ , and $a_{n}=O(b_{n})$ if $|a_{n}|\\\\lesssim |b_{n}|$ .\",\n \"We consider the dimension $d \\\\ge 3$ to be fixed, and hence allow implicit constants to depend on $d$ .\",\n \"In Sections 1 and 2 we allow implicit constants to depend on $L$ as well, as this dependence does not play a role.\",\n \"In subsequent sections $L$ -dependence is made explicit, though uniformity in $L$ is only crucial in the contractive estimate of Theorem REF .\",\n \"Our main theorems hypothesise $L=L(d)$ is large, and for geometric convenience we will assume throughout that $L$ is at least $2^{d+2}$ .\"\n ],\n [\n \"Consequences of combining renormalisation and Ward\\nidentities\",\n \"In our renormalisation group analysis, which provides the foundation for the proofs of the theorems stated in Section , we will first drop the constraint between the coupling constants of the quadratic and quartic terms in (REF ).\",\n \"The constraint will be restored in the end with the help of the Ward identity (REF ), i.e., $ {z_0}_{\\\\beta ,h}= h \\\\sum _{x\\\\in \\\\Lambda } {\\\\xi _0 \\\\eta _x}_{\\\\beta ,h},\\\\quad \\\\text{ and in particular }{z_0}_{\\\\beta ,0}= 0.$ The application of this Ward identity is the subject of this section.\",\n \"In our analysis we distinguish between two orders of limits.\",\n \"We first analyse the `infinite volume' limit $\\\\lim _{h\\\\downarrow 0} \\\\lim _{N\\\\rightarrow \\\\infty }$ , and prove Theorem REF (and thus Theorems REF –REF ).\",\n \"Using results of this analysis (and with several applications of the Ward identity), we then also analyse the much more delicate `finite volume' limit $\\\\lim _{N\\\\rightarrow \\\\infty }\\\\lim _{h\\\\downarrow 0}$ in order to prove Theorem REF .\"\n ],\n [\n \"Infinite volume correlation functions\",\n \"For $m^2 >0$ arbitrary and coupling constants $s_0,a_0,b_0$ , which eventually will be taken small, we consider the model with fermionic Gaussian reference measure with covariance $ C= (-\\\\Delta +m^2)^{-1}$ on $\\\\Lambda _N$ and interaction $V_0= V_0(\\\\Lambda _N)=\\\\sum _{x\\\\in \\\\Lambda _N}{s_0 (\\\\nabla \\\\psi )_x(\\\\nabla \\\\bar{\\\\psi })_x + a_0 \\\\psi _x\\\\bar{\\\\psi }_x + b_0 \\\\psi _x\\\\bar{\\\\psi }_x(\\\\nabla \\\\psi )_x(\\\\nabla \\\\bar{\\\\psi })_x},$ where we recall the squared gradient notation from (REF ).\",\n \"Thus the corresponding expectation is ${F}_{m^2,s_0,a_0,b_0} =\\\\frac{1}{Z_{m^2,s_0,a_0,b_0}}\\\\frac{1}{\\\\det (-\\\\Delta +m^2)} \\\\int \\\\partial _\\\\psi \\\\partial _{\\\\bar{\\\\psi }}\\\\, e^{-(\\\\psi ,(-\\\\Delta +m^2)\\\\bar{\\\\psi }) - V_0} F,$ where $\\\\int \\\\partial _\\\\psi \\\\partial _{\\\\bar{\\\\psi }}$ denotes the Grassmann integral, and $Z_{m^2,s_0,a_0,b_0}$ is defined such that ${1}_{m^2,s_0,a_0,b_0}=1$ .\",\n \"The following result resembles those in [14], [13], [67] for weakly self-avoiding walks in dimension 4.\",\n \"Compared to the latter results, our analysis is substantially simplified since the $^{0|2}$ model can be studied in terms of only fermionic variables with a quartic interaction that is irrelevant in dimensions $d > 2$ .\",\n \"However, in Section REF , we state an improvement of the following result that sees the full zero mode of the low temperature phase.\",\n \"This analysis, which relies crucially on the interplay with Ward identities, goes beyond the analysis of [14], [13], [67].\",\n \"Let $d \\\\ge 3$ and $L \\\\ge L_0(d)$ .\",\n \"For $b_0$ sufficiently small and $m^2 \\\\ge 0$ , there are $s_0 = s_0^c(b_0,m^2)$ and $a_0= a_0^c(b_0,m^2)$ independent of $N$ so that the following hold: The functions $s_0^c$ and $a_0^c$ are continuous in both variables, differentiable in $b_0$ with uniformly bounded $b_0$ -derivatives, and satisfy the estimates $ s_0^c(b_0,m^2) = O(b_0), \\\\qquad a_0^c(b_0,m^2) = O(b_0)$ uniformly in $m^2\\\\ge 0$ .\",\n \"There exists $\\\\kappa >0$ such that if the torus sidelength satisfies $L^{-N} \\\\le m$ , $\\\\sum _{x\\\\in \\\\Lambda _N} {\\\\bar{\\\\psi }_0\\\\psi _x}_{m^2,s_0,a_0,b_0} = \\\\frac{1}{m^2}+ \\\\frac{O(b_0L^{-(2+\\\\kappa )N})}{m^4}.$ Moreover, there are functions $\\\\lambda = \\\\lambda (b_0,m^2) = 1+O(b_0), \\\\qquad \\\\gamma = \\\\gamma (b_0,m^2) = (-\\\\Delta ^{^d}+m^2)^{-1}(0,0) + O(b_0),$ having the same continuity properties as $s_0^c$ and $a_0^c$ such that ${\\\\bar{\\\\psi }_0\\\\psi _0}_{m^2,s_0,a_0,b_0}&= \\\\gamma + O(b_0L^{-\\\\kappa N}),\\\\\\\\{\\\\bar{\\\\psi }_0\\\\psi _x}_{m^2,s_0,a_0,b_0}&= (-\\\\Delta +m^2)^{-1}(0,x) + O(b_0|x|^{-(d-2)-\\\\kappa })+ O(b_0L^{-\\\\kappa N}),\\\\\\\\{\\\\bar{\\\\psi }_0\\\\psi _0;\\\\bar{\\\\psi }_x\\\\psi _x}_{m^2,s_0,a_0,b_0}&= - \\\\lambda ^{2}(-\\\\Delta +m^2)^{-1}(0,x)^2 + O(b_0|x|^{-2(d-2)-\\\\kappa })+ O(b_0L^{-\\\\kappa N}).$ Here ${A;B}={AB}-{A}{B}$ .\",\n \"The proof of this theorem is given in Sections – and occupies most of this paper.\",\n \"We now show how to derive Theorem REF for the $^{0|2}$ model from it together with the Ward identity (REF ).\",\n \"To this end, assuming $s_0>-1$ we further rescale $\\\\psi $ by $1/\\\\sqrt{1+s_0}$ (and likewise for $\\\\bar{\\\\psi }$ ) in (REF ), setting $ \\\\xi = \\\\sqrt{\\\\frac{1+s_0}{\\\\beta }} \\\\bar{\\\\psi }, \\\\qquad \\\\eta =\\\\sqrt{\\\\frac{1+s_0}{\\\\beta }} \\\\psi ,$ Up to a normalisation constant, the fermionic density with respect to the fermionic Gaussian measure with covariance $C=(-\\\\Delta +m^2)^{-1}$ becomes, see (REF ), $\\\\exp {-\\\\sum _{x\\\\in \\\\Lambda _N}{(1+s_0) (\\\\nabla \\\\psi )_x(\\\\nabla \\\\bar{\\\\psi })_x+\\\\frac{1+s_0}{\\\\beta }(1+h) \\\\psi _x\\\\bar{\\\\psi }_x + \\\\frac{(1+s_0)^2}{\\\\beta } \\\\psi _x\\\\bar{\\\\psi }_x (\\\\nabla \\\\psi )_x(\\\\nabla \\\\bar{\\\\psi })_x}}.$ For any $m^2 \\\\ge 0$ and $s_0>-1$ , (REF ) is of the form (REF ) with $a_0 = \\\\frac{1+s_0}{\\\\beta }(1+h) -m^2, \\\\qquad b_0 = \\\\frac{(1+s_0)^2}{\\\\beta }.$ To use Theorem REF we need to invert this implicit relation between $(\\\\beta ,h)$ and $(m^2,s_0,a_0,b_0)$ .\",\n \"This is achieved by the following corollary.\",\n \"A key observation is that the Ward identity (REF ) allows us to identify the critical point with $h=0$ .\",\n \"To make this precise, with $s_0^c$ and $a_0^c$ as in Theorem REF , define the functions $\\\\beta (b_0,m^2) &= \\\\frac{(1+s_0^c(b_0,m^2))^2}{b_0},\\\\\\\\h(b_0,m^2) &= -1+ \\\\frac{a_0^c(b_0,m^2)+m^2}{b_0} (1+s_0^c(b_0,m^2)).$ By Theorem REF , both functions are continuous in $b_0>0$ small enough and $m^2 \\\\ge 0$ .\",\n \"(i) Assume $b_0>0$ is small enough.\",\n \"Then $h(b_0,m^2) = m^2 \\\\beta (b_0,m^2)(1+O(b_0)).$ In particular, $h(b_0,0) = 0$ and $h(b_0,m^2)>0$ if $m^2>0$ .\",\n \"(ii) For $\\\\beta $ large enough and $h\\\\ge 0$ , there are functions $\\\\tilde{b}_0(\\\\beta ,h)>0$ and $\\\\tilde{m}^2(\\\\beta ,h)\\\\ge 0$ such that $h(\\\\tilde{b}_0,\\\\tilde{m}^2)=h$ and $\\\\beta (\\\\tilde{b}_0,\\\\tilde{m}^2)=\\\\beta $ .\",\n \"Both functions are right-continuous as $h\\\\downarrow 0$ when $\\\\beta $ is fixed.\",\n \"To prove (i), we use the Ward identity (REF ) with $(\\\\beta ,h)$ given by (REF )–().\",\n \"The left- and right-hand sides of (REF ) are, respectively, ${z_0}_{\\\\beta ,h} &= 1-\\\\frac{1+s_0^c(b_0,m^2)}{\\\\beta } {\\\\bar{\\\\psi }_0\\\\psi _0}_{m^2,s_0,a_0,b_0} ,\\\\\\\\h \\\\sum _{x\\\\in \\\\Lambda _N} {\\\\xi _0 \\\\eta _x}_{\\\\beta ,h}&= \\\\frac{(1+s_0^c(b_0,m^2))h(b_0,m^2)}{\\\\beta (b_0,m^2)} \\\\sum _{x\\\\in \\\\Lambda _N} {\\\\bar{\\\\psi }_0\\\\psi _x}_{m^2,s_0,a_0,b_0} .$ By Theorem REF , in the limit $N\\\\rightarrow \\\\infty $ , we obtain from (REF ) that if $m^{2}>0$ , the identity $1-\\\\frac{1+s_0^c(b_0,m^2)}{\\\\beta (b_0,m^2)}\\\\gamma (b_0,m^2) = \\\\frac{(1+s_0^c(b_0,m^2))h(b_0,m^2)}{\\\\beta (b_0,m^2) m^2}$ holds.\",\n \"Solving for $h$ , we have $h(b_0,m^2)= m^2 { \\\\frac{\\\\beta (b_0,m^2)}{1+s_0^c(b_0,m^2)} - \\\\gamma (b_0,m^2) }.$ Since $s_0^c(b_0,m^2)=O(b_0)$ , $\\\\beta (b_{0},m^{2}) \\\\asymp 1/b_0$ , and $\\\\gamma (b_0,m^2)=O(1)$ , all uniformly in $m^2\\\\ge 0$ , we obtain $h(b_0,m^2) = m^2 \\\\beta (b_0,m^2)(1+O(b_0))$ .\",\n \"In particular, $h(b_0,0)=0$ .\",\n \"Claim (ii) follows from an implicit function theorem argument that uses that $s_0^c$ and $a_0^c$ are continuous in $m^2\\\\ge 0$ and differentiable in $b_0$ if $m^2>0$ with $b_0$ -derivatives uniformly bounded in $m^2>0$ .\",\n \"This argument is the same as the proof of [14] (with our notation $s_0$ instead of $z_0$ , $a_0$ instead of $\\\\nu _0$ , $b_0$ instead of $g_0$ , and with $1/\\\\beta $ instead of $g$ and $h$ instead of $\\\\nu $ ) and is omitted here.\",\n \"Assuming Theorem REF , the proof of Theorem REF is immediate from the last corollary.\",\n \"The main statements of Theorems REF and REF then follow immediately, except for the identifications $\\\\theta _{d}(\\\\beta )^{2}=\\\\zeta _{d}(\\\\beta )$ , $({\\\\sf c}_2(\\\\beta )/{\\\\sf c}_1(\\\\beta )^{2})\\\\theta _{d}(\\\\beta )^{2}=1$ , and ${\\\\sf c}(\\\\beta )=2{\\\\sf c}_2(\\\\beta )$ which we will obtain in Section REF .\",\n \"Given $\\\\beta \\\\ge \\\\beta _0$ and $h>0$ we choose $b_0>0$ and $m^2>0$ as in Corollary REF (ii).\",\n \"Since $z_x = 1-\\\\xi _x\\\\eta _x$ and using (REF ) we then have ${z_0}_{\\\\beta ,h}&= 1-{\\\\xi _0\\\\eta _0}_{\\\\beta ,h}=1- \\\\frac{1+s_0}{\\\\beta } {\\\\bar{\\\\psi }_0\\\\psi _0}_{m^2,s_0,a_0,b_0},\\\\\\\\{\\\\xi _0\\\\eta _x}_{\\\\beta ,h}&= \\\\frac{1+s_0}{\\\\beta } {\\\\bar{\\\\psi }_0\\\\psi _x}_{m^2,s_0,a_0,b_0},\\\\\\\\{z_0z_x}_{\\\\beta ,h}-{z_0}_{\\\\beta ,h}^2&= {\\\\xi _0\\\\eta _0\\\\xi _x\\\\eta _x}_{\\\\beta ,h} - {\\\\xi _0\\\\eta _0}_{\\\\beta ,h}^2= \\\\frac{(1+s_0)^2}{\\\\beta ^2} {\\\\bar{\\\\psi }_0\\\\psi _0;\\\\bar{\\\\psi }_x\\\\psi _x}_{m^2,s_0,a_0,b_0}.$ Taking $N\\\\rightarrow \\\\infty $ and then $h\\\\downarrow 0$ , the results follow from Corollary REF (i) and Theorem REF with $ \\\\theta _d(\\\\beta ) = 1- \\\\frac{b_0\\\\gamma }{1+s_0^c},\\\\qquad {\\\\sf c}_1(\\\\beta ) = (1+s_0^c)c_d,\\\\qquad {\\\\sf c}_2(\\\\beta ) = \\\\lambda ^{2} (1+s_0^c)^2 c_d^2,$ where the functions $\\\\lambda $ and $\\\\gamma $ are evaluated at $m^2=0$ and $b_0$ given as above, $c_d$ is the constant in the asymptotics of the free Green function on $^{d}$ , see (REF ), and we have used the simplification of the error terms $O(|x|^{-(d-2)-1})+O(b_0 |x|^{-(d-2+\\\\kappa )}) = O(|x|^{-(d-2+\\\\kappa )})$ and $O(|x|^{-2(d-2)-1})+O(b_0 |x|^{-2(d-2)-\\\\kappa )}) = O(|x|^{-2(d-2)-\\\\kappa )})$ .\"\n ],\n [\n \"Finite volume limit\",\n \"The next theorem extends Theorem REF by more precise estimates valid in the limit $m^2\\\\downarrow 0$ with $\\\\Lambda _N$ fixed.\",\n \"In these estimates $t_N \\\\in (0,1/m^{2})$ is a continuous function of $m^2>0$ that satisfies $t_N = \\\\frac{1}{m^2} - O(L^{2N}),\\\\qquad \\\\text{and}\\\\\\\\\\\\lim _{m\\\\downarrow 0}{ (-\\\\Delta +m^2)^{-1}(0,x) - \\\\frac{t_N}{|\\\\Lambda _N|}}= (-\\\\Delta ^{^d})^{-1}(0,x) + O(L^{-(d-2)N}),$ where on the right-hand side $\\\\Delta ^{^d}$ is the Laplacian on $^d$ , on the left-hand side $\\\\Delta $ is the Laplacian on $\\\\Lambda _N$ , and $|\\\\Lambda _N| = L^{dN}$ denotes the volume of the torus $\\\\Lambda _N$ .\",\n \"We define $W_N(x) = W_{N,m^2}(x) = (-\\\\Delta +m^2)^{-1}(0,x) - \\\\frac{t_N}{|\\\\Lambda _N|}.$ In the following, $\\\\Lambda _{N}$ is fixed and the parameters $(\\\\beta ,h)$ are related to $(m^2,s_0,a_0,b_0)$ as in Corollary REF .\",\n \"We will write ${\\\\cdot }_{m^2,b_0} ={\\\\cdot }_{m^2,s_0^c(b_0,m^2),a_0^c(b_0,m^2),b_0}$ for the corresponding expectation and similarly for the partition function $Z_{m^2,b_0}$ .\",\n \"Under the conditions of Theorem REF except that we no longer restrict $L^{-N}\\\\le m$ , in addition to the functions $a^{c}_{0}$ , $s^{c}_{0}$ , $\\\\lambda $ , and $\\\\gamma $ , there are functions $\\\\tilde{a}_{N,N}^c = \\\\tilde{a}_{N,N}^c(b_0,m^2)$ and $u_N^c = u_N^c(b_0,m^2)$ , both continuous in $b_0$ small and $m^2\\\\ge 0$ , as well as $\\\\tilde{u}_{N,N}^c = \\\\tilde{u}_{N,N}^c(b_0,m^2)= t_N\\\\tilde{a}_{N,N}^c(b_0,m^2) + O(b_0L^{-\\\\kappa N}),$ continuous in $b_0$ small and $m^2> 0$ , such that, for $x\\\\in \\\\Lambda _N$ , $ \\\\sum _{x\\\\in \\\\Lambda _N} {\\\\bar{\\\\psi }_0\\\\psi _x}_{m^2, b_0}&= \\\\frac{1}{m^2}- \\\\frac{1}{m^4}\\\\frac{\\\\tilde{a}_{N,N}^c}{1+\\\\tilde{u}_{N,N}^c}, \\\\\\\\{\\\\bar{\\\\psi }_0\\\\psi _0}_{m^2,b_0}&=\\\\gamma + \\\\frac{\\\\lambda t_N |\\\\Lambda _N|^{-1}}{1+\\\\tilde{u}_{N,N}^c} +E_{00},\\\\\\\\{\\\\bar{\\\\psi }_0\\\\psi _0\\\\bar{\\\\psi }_x\\\\psi _x}_{m^2,b_0}&=-\\\\lambda ^2W_N(x)^2+\\\\gamma ^2+\\\\frac{-2\\\\lambda ^2W_N(x)+ 2\\\\lambda \\\\gamma }{1+\\\\tilde{u}_{N,N}^c} t_N|\\\\Lambda _N|^{-1}+E_{00xx},$ and $Z_{m^{2},b_{0}} = e^{-u_{N}^{c}|\\\\Lambda _N|}(1+\\\\tilde{u}_{N,N}^{c}).$ The remainder terms satisfy $E_{00}&=\\\\frac{O(b_0L^{-(d-2+\\\\kappa )N}+b_0L^{-\\\\kappa N}(m^2|\\\\Lambda _N|)^{-1})}{1+\\\\tilde{u}_{N,N}^c},\\\\\\\\E_{00xx}&= O(b_0|x|^{-2(d-2)-\\\\kappa }) + O(b_0L^{-(d-2+\\\\kappa )N})&\\\\qquad +(O(b_0|x|^{-(d-2+\\\\kappa )})+ O(b_0L^{-\\\\kappa N})) \\\\frac{(m^2|\\\\Lambda _N|)^{-1}}{1+\\\\tilde{u}_{N,N}^c}.$ The proof of this theorem is again given in Sections –.\",\n \"The proof of Theorem REF is based on Theorem REF and several upcoming lemmas.\",\n \"These lemmas exploit Ward identities to relate the functions given by the theorem to one another.\",\n \"To orient the reader, the first two lemmas below can be viewed as preparatory for the key computation in Lemma REF .\",\n \"Under the conditions of Theorem REF , $^{\\\\Lambda _{N}}_{\\\\beta ,0}|T_0|=\\\\frac{b_0}{1+s_0^c(b_0,0)} \\\\frac{1+O(b_0L^{-\\\\kappa N})}{\\\\tilde{a}_{N,N}^{c}(b_{0},0)} + O(b_0L^{2N}).$ In particular, if $b_0>0$ this implies $1/\\\\tilde{a}_{N,N}^{c}(b_{0},0) = O(|\\\\Lambda _{N}|/b_{0})$ and $\\\\tilde{a}_{N,N}^{c}(b_{0},0)>0$ .\",\n \"From (REF ), we have that $^{\\\\Lambda _{N}}_{\\\\beta ,0}|T_0|=\\\\sum _{x\\\\in \\\\Lambda _N} ¶_{\\\\beta ,0}[0\\\\leftrightarrow x] = \\\\sum _{x\\\\in \\\\Lambda _N} {\\\\xi _0\\\\eta _x}_{\\\\beta ,0}=\\\\lim _{h\\\\rightarrow 0} \\\\sum _{x\\\\in \\\\Lambda _N} {\\\\xi _0\\\\eta _x}_{\\\\beta ,h}.$ Changing variables, $\\\\sum _{x\\\\in \\\\Lambda _N} {\\\\xi _0\\\\eta _x}_{\\\\beta ,h}= \\\\frac{b_0}{1+s_0^c(b_0,m^2)} \\\\sum _{x\\\\in \\\\Lambda _N} {\\\\bar{\\\\psi }_0\\\\psi _x}_{m^2,b_0},$ where $(\\\\beta , h)$ and $(b_0, m^2)$ are related as in (REF ) and ().\",\n \"To evaluate the right-hand side we use (REF ).\",\n \"Note that $\\\\frac{1}{m^2}- \\\\frac{1}{m^4}\\\\frac{\\\\tilde{a}_{N,N}^c}{1+\\\\tilde{u}_{N,N}^c}&= \\\\frac{1}{m^2}\\\\frac{1+\\\\tilde{u}_{N,N}^c - \\\\tilde{a}_{N,N}^cm^{-2}}{1+\\\\tilde{u}_{N,N}^c}&= \\\\frac{1+\\\\tilde{a}_{N,N}^c (t_N-m^{-2})+O(b_0L^{-\\\\kappa N})}{m^2+\\\\tilde{a}_{N,N}^ct_Nm^2 + O(b_0m^2L^{-\\\\kappa N})}&= \\\\frac{1+\\\\tilde{a}_{N,N}^c O(L^{2N})+O(b_0L^{-\\\\kappa N})}{m^2+\\\\tilde{a}_{N,N}^c(1+O(m^2L^{2N})) + O(b_0m^2L^{-\\\\kappa N})},$ where the second equality is due to (REF ) and the third follows from (REF ).\",\n \"As $m^2\\\\downarrow 0$ , the right-hand side of the third equality behaves asymptotically as $\\\\frac{1+\\\\tilde{a}_{N,N}^c O(L^{2N})+O(b_0L^{-\\\\kappa N})}{\\\\tilde{a}_{N,N}^c}=\\\\frac{1+O(b_0L^{-\\\\kappa N})}{\\\\tilde{a}_{N,N}^c}+O(L^{2N}).$ Since $s_{0}^{c}(b_{0},0)=O(b_{0})$ by Theorem REF we therefore obtain the first claim: $_{\\\\beta ,0}^{\\\\Lambda _N}|T_0|=\\\\frac{b_0}{1+s_0^c(b_0,0)}\\\\lim _{m^2\\\\downarrow 0} \\\\sum _{x\\\\in \\\\Lambda _N} {\\\\bar{\\\\psi }_0\\\\psi _x}_{m^2,b_0}=\\\\frac{b_0}{1+s_0^c(b_0,0)} \\\\frac{1+O(b_0L^{-\\\\kappa N})}{\\\\tilde{a}_{N,N}^{c}(b_{0},0)} + O(b_0L^{2N})$ For the second claim, let us observe that on the one hand $ Z_{\\\\beta ,h}= {\\\\frac{\\\\beta }{1+s_0^c}}^{|\\\\Lambda _N|}(\\\\det (-\\\\Delta +m^2)) Z_{m^2, b_0}= {\\\\frac{\\\\beta e^{-u_{N}^{c}}}{1+s_0^c}}^{|\\\\Lambda _N|}(\\\\det (-\\\\Delta +m^2))(1+\\\\tilde{u}_{N,N}^{c}),$ where the first equality is by Proposition REF and (REF ), (REF ), and (REF ), and the second equality is (REF ).\",\n \"On the other hand, by (REF ), $\\\\lim _{h\\\\rightarrow 0} Z_{\\\\beta ,h}= Z_{\\\\beta ,0}>0.$ Since, by Theorem REF , $u_{N}^{c}$ and $s_0^c$ remain bounded as $m^2\\\\downarrow 0$ with $\\\\Lambda _N$ fixed (and thus also $\\\\beta $ which is given by (REF )), from $\\\\det (-\\\\Delta +m^2) \\\\downarrow 0$ , we conclude that $1+\\\\tilde{u}_{N,N}^c$ diverges as $m^2\\\\downarrow 0$ .\",\n \"By (REF ), this implies $\\\\tilde{a}_{N,N}^{c}(b_{0},0)>0$ .\",\n \"The upper bound on $1/\\\\tilde{a}_{N,N}^c(b_0,0)$ follows by re-arranging (REF ) and using the trivial bound $|T_0| \\\\le |\\\\Lambda _N|$ .\",\n \"Under the conditions of Theorem REF and if $b_{0}>0$ , $1=\\\\frac{b_0}{1+s_0^c(b_0,0)}{\\\\gamma (b_0,0) + \\\\frac{\\\\lambda (b_0,0)}{|\\\\Lambda _N|\\\\tilde{a}_{N,N}^c(b_0,0)}(1+O(b_0L^{-\\\\kappa N}))}.$ The Ward identity ${z_0}_{\\\\beta ,0}=0$ implies $0={z_0}_{\\\\beta ,0}= 1-{\\\\xi _0\\\\eta _0}_{\\\\beta ,0}&=1-\\\\lim _{m^2\\\\downarrow 0} \\\\frac{1+s_0^c(b_0,m^2)}{\\\\beta (b_0,m^2)}{\\\\bar{\\\\psi }_0\\\\psi _0}_{m^2,b_0}& =1-\\\\lim _{m^2\\\\downarrow 0} \\\\frac{b_0}{1+s_0^c(b_0,m^2)}{\\\\bar{\\\\psi }_0\\\\psi _0}_{m^2,b_0},$ where we used (REF ) and that $\\\\beta =\\\\beta (b_0,m^2)$ is as in (REF ).\",\n \"To compute ${\\\\bar{\\\\psi }_0\\\\psi _0}_{m^2,b_0}$ , we apply ().\",\n \"Since $\\\\tilde{u}_{N,N}^c = \\\\tilde{a}_{N,N}^ct_N +O(b_0L^{-\\\\kappa N})$ and $t_N = m^{-2}+O(L^{2N})$ , $\\\\lim _{m^2\\\\downarrow 0}{\\\\bar{\\\\psi }_0\\\\psi _0}_{m^2,b_0}&= \\\\gamma (b_{0},0) + \\\\lim _{m^{2}\\\\downarrow 0} \\\\frac{\\\\lambda (b_{0},m^{2})t_{N}|\\\\Lambda _{N}|^{-1}}{1+\\\\tilde{a}_{N,N}^{c}(b_{0},m^{2})t_{N}+O(b_{0}L^{-\\\\kappa N})} +\\\\lim _{m^{2}\\\\downarrow 0}E_{00}&= \\\\gamma (b_{0},0) +\\\\frac{\\\\lambda (b_{0},0)|\\\\Lambda _{N}|^{-1}}{\\\\tilde{a}_{N,N}^c(b_{0},0)} + \\\\lim _{m^{2}\\\\downarrow 0}E_{00}.$ The limits in the second line exist by Theorem REF and Lemma REF , which in particular implies $\\\\tilde{a}_{N,N}^c(b_0,0) > 0$ since $b_0>0$ .\",\n \"As $m^2\\\\downarrow 0$ , the error term $E_{00}$ is bounded by $O(b_0L^{-\\\\kappa N}/(|\\\\Lambda _N|\\\\tilde{a}_{N,N}^c))= (\\\\lambda (b_{0},0) |\\\\Lambda _N|^{-1}/\\\\tilde{a}_{N,N}^c)O(b_0L^{-\\\\kappa N})$ since $\\\\lambda (b_{0},0) =1-O(b_0)\\\\ge 1/2$ , finishing the proof.\",\n \"Given Theorem REF , the following lemma is the main step in the proof of Theorem REF .\",\n \"Under the conditions of Theorem REF and if $b_{0}>0$ , $¶_{\\\\beta ,0}^{\\\\Lambda _N}[0\\\\leftrightarrow x]= \\\\theta _d(\\\\beta )^2+2 \\\\frac{b_0}{1+s_0^c}\\\\lambda \\\\theta _d(\\\\beta )(-\\\\Delta ^{^d})^{-1}(0,x)\\\\\\\\+ O(b_0^2|x|^{-(d-2)-\\\\kappa })+ O(b_0L^{-(d-2)N})+ O(b_0^2L^{-\\\\kappa N}),$ where $\\\\theta _d(\\\\beta )$ is defined in (REF ).\",\n \"By the last expression for $¶_{\\\\beta ,0}[0\\\\leftrightarrow x]$ in (REF ) and (REF ), (REF ): $ ¶_{\\\\beta ,0}^{\\\\Lambda _N}[0\\\\leftrightarrow x]= 1-\\\\lim _{h\\\\downarrow 0}{\\\\xi _0\\\\eta _0\\\\xi _x\\\\eta _x}_{\\\\beta ,h}= 1- \\\\lim _{m^2\\\\downarrow 0}{ \\\\frac{b_0^2}{(1+s_0^c)^2}{\\\\bar{\\\\psi }_0\\\\psi _0\\\\bar{\\\\psi }_x\\\\psi _x}_{m^{2},b_{0}}}.$ To compute $\\\\lim _{m^2\\\\downarrow 0}{\\\\bar{\\\\psi }_0\\\\psi _0\\\\bar{\\\\psi }_x\\\\psi _x}_{m^{2},b_{0}}$ we start from ().\",\n \"By Lemma REF , as $m^2\\\\downarrow 0$ with $\\\\Lambda _N$ fixed, $\\\\frac{1}{1+\\\\tilde{u}_{N,N}^c} \\\\sim \\\\frac{1}{m^{-2}\\\\tilde{a}_{N,N}^c(b_0,0)}=O(\\\\frac{m^2|\\\\Lambda _N|}{b_0}).$ This implies the error term in () is, as $m^2\\\\downarrow 0$ with $\\\\Lambda _N$ fixed, $|E_{00xx}|\\\\le O(|x|^{-(d-2)-\\\\kappa }) + O(L^{-\\\\kappa N}).$ For the main term we have $\\\\lim _{m^{2}\\\\downarrow 0}{\\\\bar{\\\\psi }_0\\\\psi _0\\\\bar{\\\\psi }_x\\\\psi _x}_{m^2,b_0}&=-\\\\lambda ^2 W_{N,0}(x)^2 +\\\\gamma ^2+\\\\lim _{m^{2}\\\\downarrow 0}\\\\frac{-2\\\\lambda ^2 W_{N}(x)+ 2\\\\lambda \\\\gamma }{1+\\\\tilde{u}_{N,N}^c} {t_N|\\\\Lambda _N|^{-1}}&= -\\\\lambda ^2 W_{N,0}(x)^2 +\\\\gamma ^2+ 2(-\\\\lambda W_{N,0}(x)+\\\\gamma )\\\\frac{\\\\lambda }{\\\\tilde{a}_{N,N}^c|\\\\Lambda _{N}|},$ where on the right-hand side the functions $\\\\lambda $ , $\\\\gamma $ , and $\\\\tilde{a}_{N,N}^c$ are evaluated at $m^2=0$ .\",\n \"By Lemma REF , $\\\\frac{b_{0}}{1+s_{0}^{c}}\\\\frac{\\\\lambda }{\\\\tilde{a}_{N,N}^{c}|\\\\Lambda _{N}| } ={1-\\\\frac{b_{0}\\\\gamma }{1+s_{0}^{c}}}(1+O(b_0L^{-\\\\kappa N}))$ so that $-{\\\\frac{b_{0}}{1+s_{0}^{c}}}^{2}\\\\frac{2\\\\lambda ^2 W_{N,0}(x)}{\\\\tilde{a}_{N,N}^c|\\\\Lambda _{N}|}{ (1+O(b_0L^{-\\\\kappa N}))}&= -\\\\frac{2b_{0}}{1+s_{0}^{c}}(1-\\\\frac{b_{0}\\\\gamma }{1+s_{0}^{c}})\\\\lambda W_{N,0}(x)\\\\\\\\{\\\\frac{b_{0}}{1+s_{0}^{c}}}^{2}\\\\frac{2\\\\lambda \\\\gamma }{\\\\tilde{a}_{N,N}^c|\\\\Lambda _{N}|}{ (1+O(b_0L^{-\\\\kappa N}))}&= 2\\\\gamma \\\\frac{b_{0}}{1+s_{0}} - 2\\\\gamma ^{2}{\\\\frac{b_{0}}{1+s_{0}^{c}}}^{2}.$ Substituting these bounds into (REF ) and then (REF ) we obtain $¶_{\\\\beta ,0}^{\\\\Lambda _N}[0\\\\leftrightarrow x]&=1-{\\\\frac{b_{0}}{1+s_{0}^{c}}}^{2}\\\\lim _{m^{2}\\\\downarrow 0}{\\\\bar{\\\\psi }_0\\\\psi _0\\\\bar{\\\\psi }_x\\\\psi _x}_{m^2,b_0}&= (1-\\\\frac{\\\\gamma b_{0}}{1+s_{0}})^{2}+\\\\frac{2b_{0}\\\\lambda }{1+s_{0}^{c}} (1-\\\\frac{b_{0}\\\\gamma }{1+s_{0}^{c}}) W_{N,0}(x)+ {\\\\frac{b_{0}\\\\lambda W_{N,0}(x)}{1+s_{0}^{c}}}^{2}&\\\\qquad +O(b_0^2 L^{-\\\\kappa N} W_{N,0}(x)) + O(b_0^2 L^{-\\\\kappa N})+ O(b_0^2|E_{00xx}|).$ Using the definition (REF ) of $\\\\theta _d(\\\\beta )$ , that $W_{N,0}(x) = (-\\\\Delta ^{^d})^{-1}(0,x)+O(L^{-(d-2)N})$ by (), and in particular $W_{N,0}(x) = O(|x|^{-(d-2)})$ , the claim follows.\",\n \"The next (and final) lemma is inessential for the main conclusions, but will allow us to identify the constants from the infinite volume and the finite volume analyses.\",\n \"Under the conditions of Theorem REF and if $b_{0}>0$ , then $\\\\lambda \\\\theta _d(\\\\beta ) = 1$ .\",\n \"Let $ w_{N} \\\\frac{b_{0}}{1+s^{c}_{0}} \\\\frac{1}{\\\\tilde{a}_{N,N}^{c}|\\\\Lambda _{N}|}= ^{\\\\Lambda _N}_{\\\\beta ,0} \\\\frac{|T_0|}{|\\\\Lambda _N|} + O(b_0L^{-\\\\kappa N}+b_0L^{-(d-2)N}),$ where the second equality is due to Lemma REF .\",\n \"The density $^{\\\\Lambda _N}_{\\\\beta ,0} |T_0|/|\\\\Lambda _N|$ can also be computed by summing the estimate in Lemma REF and dividing by $|\\\\Lambda _N|$ .\",\n \"Subtracting this result from (REF ) gives $w_N-\\\\theta _d(\\\\beta )^2=O(b_0L^{-\\\\kappa N}).$ On the other hand, (REF ) shows that $\\\\lambda w_N-\\\\theta _d(\\\\beta )= O(b_0L^{-\\\\kappa N}).$ The limit $w= \\\\lim _{N\\\\rightarrow \\\\infty } w_N$ thus exists and satisfies $\\\\lambda w = \\\\theta _d(\\\\beta )$ and $w=\\\\theta _d(\\\\beta )^2$ .\",\n \"Since $\\\\theta _d(\\\\beta )=1- O(1/\\\\beta )\\\\ne 0$ this implies $\\\\lambda \\\\theta _d(\\\\beta ) = 1$ .\",\n \"The proof follows by rewriting Lemma REF .\",\n \"Let $c_d$ be the constant in the Green function asymptotics of (REF ), and recall the constants $\\\\theta _d(\\\\beta )$ and ${\\\\sf c}_i(\\\\beta )$ from (REF ).\",\n \"Theorem REF then follows from Lemma REF by setting $\\\\zeta _d(\\\\beta )= \\\\theta _d(\\\\beta )^2, \\\\qquad {\\\\sf c}(\\\\beta )=(1+s_{0}^{c})2\\\\lambda \\\\theta _{d}(\\\\beta ) c_d,$ and simplifying the error terms using $O(b_0|x|^{-(d-2)-1})+O(b_0^2 |x|^{-(d-2+\\\\kappa )}) = O(\\\\beta ^{-1}|x|^{-(d-2+\\\\kappa )})$ and $O(b_0L^{-(d-2)N})+O(b_0^2 L^{-\\\\kappa N}) = O(\\\\beta ^{-1}L^{-\\\\kappa N})$ .\",\n \"For Theorem REF , $\\\\zeta _d(\\\\beta ) =\\\\theta _d(\\\\beta )^2$ was established in the previous proof.\",\n \"For Theorem REF , the identity $({\\\\sf c}_2(\\\\beta )/{\\\\sf c}_1(\\\\beta )^{2})\\\\theta _{d}(\\\\beta )^{2}=1$ is equivalent to $\\\\theta _{d}(\\\\beta )\\\\lambda =1$ , i.e., Lemma REF .\",\n \"Similarly, ${\\\\sf c}(\\\\beta )=2\\\\lambda \\\\theta _{d}(\\\\beta ) {\\\\sf c}_1(\\\\beta ) = 2{\\\\sf c}_1(\\\\beta )$ .\",\n \"To compute $¶_{\\\\beta ,0}[0\\\\leftrightarrow x]$ we started from the expression $1-{\\\\xi _0\\\\eta _0\\\\xi _x\\\\eta _x}_{\\\\beta ,0}$ in (REF ).\",\n \"An alternative route would have been to start from ${\\\\xi _0\\\\eta _x}_{\\\\beta ,0}$ .\",\n \"For technical reasons arising in Section it is, however, easier to obtain sufficient precision when working with ${\\\\xi _0\\\\eta _0\\\\xi _x\\\\eta _x}_{\\\\beta ,0}$ .\"\n ],\n [\n \"The bulk renormalisation group flow\",\n \"We will prove Theorems REF and REF by a renormalisation group analysis that is set up following [27], [33] and [13], [14]; see also [17] for a conceptual introduction.\",\n \"Our proof is largely self-contained.\",\n \"The exceptions to self-containment concern general properties about finite range decomposition, norms, and approximation by local polynomials that were developed systematically in [11], [30], [31].\",\n \"The properties we need are all reviewed in this section.\",\n \"The first six subsections set up the framework of the analysis, and the remaining three define and analyse the renormalisation group flow.\",\n \"Throughout $\\\\Lambda \\\\Lambda _N$ is the discrete torus of side length $L^N$ .\",\n \"We leave $L$ implicit; it will eventually be chosen large.\",\n \"We sometimes omit the $N$ when it does not play a role.\"\n ],\n [\n \"Finite range decomposition\",\n \"Let $\\\\Delta $ denote the lattice Laplacian on $\\\\Lambda _N$ , and let $m^2>0$ .\",\n \"Our starting point for the analysis is the decomposition $ C = (-\\\\Delta +m^2)^{-1} = C_1 + \\\\cdots +C_{N-1}+C_{N,N}$ where the $C_j$ and $C_{N,N}$ are positive semidefinite $m^2$ -dependent matrices indexed by $\\\\Lambda _N$ .\",\n \"These covariances can be chosen with the following properties, see [17] and Appendix REF .\"\n ],\n [\n \"Finite range property\",\n \"For $jL^j$ then, by (REF ), $_{C_j}\\\\theta (F_1F_2) = (_{C_j}\\\\theta F_1)(_{C_j}\\\\theta F_2).$\"\n ],\n [\n \"Symmetries\",\n \"We briefly discuss symmetries, which are important in extracting the relevant and marginal contributions in each renormalisation group step (see Section REF below).\",\n \"To use the terminology of [30], [31], we call an element $F \\\\in (\\\\Lambda )$ (globally) gauge invariant if every monomial in its representation has the same number of factors of $\\\\bar{\\\\psi }$ and $\\\\psi $ .\",\n \"Some readers may be more familiar with the terminology symplectically invariant or $U(1)$ invariant.\",\n \"Similarly, $F \\\\in (\\\\Lambda \\\\sqcup \\\\Lambda )$ is gauge invariant if the combined number of factors of $\\\\bar{\\\\psi }$ and $\\\\bar{\\\\zeta }$ is the same as the combined number of factors of $\\\\psi $ and $\\\\zeta $ .\",\n \"We denote by $_{\\\\rm sym}(X)$ the subalgebra of $(X)$ of gauge invariant elements and likewise for $_{\\\\rm sym}(\\\\Lambda \\\\sqcup \\\\Lambda )$ .\",\n \"The maps $\\\\theta $ and $_{C}$ preserve gauge symmetry.\",\n \"A bijection $E\\\\colon \\\\Lambda \\\\rightarrow \\\\Lambda $ is an automorphism of the torus $\\\\Lambda $ if it maps nearest neighbours to nearest neighbours.\",\n \"Bijections act as homomorphisms on the algebra $(\\\\Lambda )$ by $E\\\\psi _x = \\\\psi _{Ex}$ and $E\\\\bar{\\\\psi }_x = \\\\bar{\\\\psi }_{Ex}$ and similarly for $(\\\\Lambda \\\\sqcup \\\\Lambda )$ .\",\n \"If $C$ is invariant under lattice symmetries, i.e., $C(Ex,Ey)=C(x,y)$ for all automorphisms $E$ , then the convolution $_C\\\\theta $ commutes with automorphisms of $\\\\Lambda $ , i.e., $E _{C}\\\\theta F = _{C}\\\\theta E F$ .\",\n \"In particular $E_{C_{j}}\\\\theta F = _{C_{j}}\\\\theta E F$ for the covariances of the finite range decomposition (REF ).\",\n \"An important consequence of this discussion is that if $F\\\\in _{\\\\rm sym}(\\\\Lambda )$ and $F$ is invariant under lattice symmetries, then $_{C_{j}}\\\\theta F\\\\in _{\\\\rm sym}(\\\\Lambda )$ is also invariant under lattice symmetries.\"\n ],\n [\n \"Polymer coordinates\",\n \"We will use (REF ) and the decomposition (REF ) to study the progressive integration $ Z_{j+1} = _{C_{j+1}} \\\\theta Z_j,$ for a given $Z_{0}\\\\in (\\\\Lambda )$ .\",\n \"To be concrete here, the reader may keep $Z_0=e^{-V_0(\\\\Lambda )}$ with $V_{0}(\\\\Lambda )$ from (REF ) in mind, but to compute correlation functions we will consider generalisations of this choice of $Z_{0}$ in Section .\",\n \"The analysis is performed by defining suitable coordinates and norms that enable the progressive integration to be treated as a dynamical system: this is the renormalisation group.\",\n \"Towards this end, this section defines local polymer coordinates as in [27], [33].\",\n \"Figure: Illustration of jj-blocks when L=2L=2.\"\n ],\n [\n \"Blocks and Polymers\",\n \"Recall $\\\\Lambda \\\\Lambda _{N}$ denotes a torus of side length $L^{N}$ .\",\n \"Partition $\\\\Lambda _N$ into nested scale-$j$ blocks $_j$ of side lengths $L^j$ where $j=0,\\\\dots , N$ .\",\n \"Thus scale-0 blocks are simply the points in $\\\\Lambda $ , while the only scale-$N$ block is $\\\\Lambda $ itself, see Figure REF .\",\n \"The set of $j$ -polymers $_j=_j(\\\\Lambda )$ consists of finite unions of blocks in $_j$ .\",\n \"To define a notion of connectedness, say $X,Y \\\\in _j$ do not touch if $\\\\inf _{x\\\\in X,y\\\\in Y} |x-y|_\\\\infty > 1$ .\",\n \"A polymer is connected if it is not empty and there is a path of touching blocks between any two blocks of the polymer.\",\n \"The subset of connected $j$ -polymers is denoted $_j$ .\",\n \"We will drop $j$ - prefixes when the scale is clear.\",\n \"For a fixed $j$ -polymer $X$ , let $_j(X)$ denote the $j$ -blocks contained in $X$ and let $|_j(X)|$ be the number of such blocks.\",\n \"Connected polymers $X$ with $|_j(X)| \\\\le 2^d$ are called small sets and the collection of all small sets is denoted $_j$ .\",\n \"Polymers which are not small will be called large.\",\n \"Finally, for $X \\\\in _j$ we define its small set neighbourhood $X^\\\\square \\\\in _j$ as the union over all small sets containing a block in $_j(X)$ , and its closure $\\\\bar{X}$ as the smallest $Y\\\\in _{j+1}$ such that $X\\\\subset Y$ .\"\n ],\n [\n \"Coordinates\",\n \"For coupling constants $V_j=(z_j,y_j,a_j,b_j) \\\\in 4$ and a set $X \\\\subset \\\\Lambda _{N}$ , let $ V_j(X)=\\\\\\\\\\\\sum _{x\\\\in X} { y_j (\\\\nabla \\\\psi )_x (\\\\nabla \\\\bar{\\\\psi })_x + \\\\frac{z_j}{2}((-\\\\Delta \\\\psi )_x\\\\bar{\\\\psi }_x + \\\\psi _x(-\\\\Delta \\\\bar{\\\\psi })_x) + a_j \\\\psi _x\\\\bar{\\\\psi }_x + b_j \\\\psi _x\\\\bar{\\\\psi }_x (\\\\nabla \\\\psi )_x(\\\\nabla \\\\bar{\\\\psi })_x}.$ For the scale $j=0$ , if we set $ Z_0 = e^{-V_0(\\\\Lambda _{N})}$ , then the polymer coordinates take the simple form $ Z_0 = e^{-V_0(\\\\Lambda _{N})} = e^{-u_{0}|\\\\Lambda _{N}|}\\\\sum _{X \\\\subset \\\\Lambda _{N}} e^{-V_0(\\\\Lambda _{N} \\\\setminus X)} K_0(X), \\\\quad K_0(X) =1_{X=\\\\varnothing }, \\\\quad u_{0}=0.$ To study the recursion $ Z_{j+1}=_{C_{j+1}}\\\\theta Z_{j}$ at a general scale $j=1,\\\\dots ,N$ , we will make a choice of coupling constants $V_{j}$ and of polymer coordinates $(K_j(X))_{X\\\\in _j(X)}$ such that $ Z_{j}= e^{-u_{j}|\\\\Lambda _{N}|} \\\\sum _{X\\\\in _{j}} e^{-V_{j}(\\\\Lambda _{N}\\\\setminus X)} K_{j}(X).$ The $K_j(X)$ 's will be defined in such a way that they satisfy the locality and symmetry property $K_j(X) \\\\in _{\\\\rm sym}(X^\\\\square )$ and the following important component factorisation property: for $X,Y\\\\in _{j}$ that do not touch, $ K_j(X \\\\cup Y) = K_j(X)K_j(Y).$ Note that since they are gauge symmetric, the $K_j(X)$ are even elements of $$ , so they commute and the product on the right-hand side is unambiguous.\",\n \"Using the previous identity, $K_j(X) = \\\\prod _{Y \\\\in \\\\operatorname{Comp}(X)} K_j(Y),$ where $\\\\operatorname{Comp}(X)$ denotes the set of connected components of the polymer $X$ .\",\n \"In particular, each $K_j= (K_{j}(X))_{X\\\\in _{j}(\\\\Lambda _{N})}$ satisfying (REF ) can be identified with $K_j= (K_{j}(X))_{X\\\\in _{j}(\\\\Lambda _{N})}$ .\",\n \"We say that $K_{j}$ is automorphism invariant if $EK_{j}(X) = K_{j}(E(X))$ and all $X\\\\in _{j}(\\\\Lambda _{N})$ for all torus automorphisms $E \\\\in {\\\\rm Aut}(\\\\Lambda _N)$ that map blocks in $_j$ to blocks in $_j$ .\",\n \"Let $^\\\\varnothing _j(\\\\Lambda _N)$ be the linear space of automorphism invariant $K_j = (K_j(X))_{X\\\\in _j(\\\\Lambda _N)}$ with $K_j(X) \\\\in _{\\\\rm sym}(X^\\\\square )$ for every $X \\\\in _j$ .\",\n \"Polymer coordinates at scale $j$ are thus a choice of $V_j$ together with an element of the space $^\\\\varnothing _{j}$ .\",\n \"The renormalisation group map is a particular choice of a map $(V_j,K_j)\\\\mapsto (V_{j+1},K_{j+1})$ .\",\n \"There is freedom in the choice because of the flexibility in the definition of coordinates.\",\n \"The goal is to choose $V_{j}$ such that the size of the $K_{j}$ decrease rapidly as $j$ increases.\",\n \"To achieve this goal we need norms to quantify the sizes of these expressions.\"\n ],\n [\n \"Norms\",\n \"We now define the $T_j(\\\\ell )$ norms we will use on $(\\\\Lambda )$ .\",\n \"General properties of these norms were systematically developed in [30], to which we will refer for some proofs.\",\n \"To help the reader, in places where we specialise the definitions of [30] we indicate the more general notation that is used in [30].\",\n \"We start with some notation.\",\n \"For any set $S$ , we write $S^*$ for the set of finite sequences in $S$ .\",\n \"We write $\\\\Lambda _f = \\\\Lambda \\\\times \\\\lbrace \\\\pm 1\\\\rbrace $ and for $(x,\\\\sigma ) \\\\in \\\\Lambda _f$ we write $\\\\psi _{x,\\\\sigma } = \\\\psi _x$ if $\\\\sigma =+1$ and $\\\\psi _{x,\\\\sigma }=\\\\bar{\\\\psi }_x$ if $\\\\sigma =-1$ .\",\n \"Then every element $F \\\\in (\\\\Lambda )$ can be written in the form $ F = \\\\sum _{z\\\\in \\\\Lambda _f^*} \\\\frac{1}{z!\",\n \"}F_z \\\\psi ^z$ where $\\\\psi ^z = \\\\psi _{z_1}\\\\cdots \\\\psi _{z_n}$ if $z=(z_1,\\\\dots , z_n)$ .\",\n \"We are using the notation that $z!=n!$ if the sequence $z$ has length $n$ .\",\n \"The representation in (REF ) is in general not unique.\",\n \"To obtain a unique representation we require that the $F_{z}$ are antisymmetric with respect to permutations of the components of $z$ (this is possible due to the antisymmetry of the Grassmann variables).\",\n \"Antisymmetry implies that $F_{z}=0$ if $z$ has length exceeding $2|\\\\Lambda |$ or if $z$ has any repeated entries.\",\n \"Let $p_{\\\\Phi }=2d$ .\",\n \"The space of test functions $\\\\Phi _j(\\\\ell )$ is defined as the set of functions $g\\\\colon \\\\Lambda _f^* \\\\rightarrow $ , $z\\\\mapsto g_{z}$ together with norm $ \\\\Vert g\\\\Vert _{\\\\Phi _j(\\\\ell )} =\\\\sup _{n\\\\ge 0}\\\\sup _{z\\\\in \\\\Lambda _f^n} \\\\sup _{|\\\\alpha _i| \\\\le p_\\\\Phi }\\\\ell ^{-n}L^{j(|\\\\alpha _1|+\\\\cdots |\\\\alpha _n|)}|\\\\nabla ^{\\\\alpha _1}_{z_1}\\\\cdots \\\\nabla ^{\\\\alpha _n}_{z_n} g_z|.$ In this definition, $\\\\nabla _{z_i}^{\\\\alpha _i}$ denotes the discrete derivative $\\\\nabla ^{\\\\alpha _i}$ with multiindex $\\\\alpha _i$ acting on the spatial part of the $i$ th component of the finite sequence $z$ .\",\n \"The $\\\\Phi _j(\\\\ell )$ norm measures spatial smoothness of test functions, which act as substitutes for fields.\",\n \"Restricted to sequences of fixed length, it is a lattice $C^{p_\\\\Phi }$ norm at spatial scale $L^j$ and field scale $\\\\ell $ .\",\n \"We will mainly use the following choice of $\\\\ell $ for $\\\\Phi _j(\\\\ell )$ : $\\\\ell _j = \\\\ell _0 L^{-\\\\frac{1}{2}(d-2)j}$ for a large constant $\\\\ell _0$ .\",\n \"This choice captures the size of the covariances in the decomposition (REF ).\",\n \"Indeed, since the covariances $C_{j}$ are functions of sequences of length 2, the bounds (REF ) imply $ \\\\Vert C_{j}\\\\Vert _{\\\\Phi _j(\\\\ell _j)} \\\\le 1,$ when $\\\\ell _0$ is chosen as a large ($L$ -dependent, due to the index $j+1$ on the left-hand side of (REF )) constant relative to the constants in (REF ) with $|\\\\alpha |\\\\le p_\\\\Phi $ ; we will always assume that $\\\\ell _0$ is fixed in this way.\",\n \"In (REF ) we have made a slight abuse of notation to identify $C_{j}$ with the coefficient in (REF ) of $F= \\\\sum _{x,y} \\\\bar{\\\\psi }_{x}\\\\psi _{y}C_{j}(x,y)$ .\",\n \"We define $T_j(\\\\ell )$ to be the algebra $ (\\\\Lambda )$ together with the dual norm $ \\\\Vert F\\\\Vert _{T_j(\\\\ell )} = \\\\sup _{\\\\Vert g\\\\Vert _{\\\\Phi _j(\\\\ell )} \\\\le 1}| {F,g}|,\\\\qquad \\\\text{where }{F,g} = \\\\sum _{z \\\\in \\\\Lambda _f^*} \\\\frac{1}{z!}\",\n \"F_z g_z$ when $F\\\\in (\\\\Lambda )$ is expressed as in (REF ).\",\n \"An analogous definition applies to $(\\\\Lambda \\\\sqcup \\\\Lambda )$ and we then write $T_j(\\\\ell \\\\sqcup \\\\ell ) T_j(\\\\ell )$ for this norm (with the first notation to emphasise the doubled algebra).\",\n \"The $T_j(\\\\ell )$ norm measures smoothness of field functionals $F\\\\in (\\\\Lambda )$ with respect to fields whose size is measured by $\\\\Phi _j(\\\\ell )$ .\",\n \"They therefore implement the power counting on which renormalisation relies.\",\n \"Important, but relatively straightforwardly verified, properties of these norms are systematically developed in [30]; we summarise the ones we need now.\",\n \"First, $T_j(\\\\ell )$ is a Banach algebra, i.e., the following product property holds (see [30]): $ \\\\Vert F_1F_2\\\\Vert _{T_j(\\\\ell )} \\\\le \\\\Vert F_1\\\\Vert _{T_j(\\\\ell )}\\\\Vert F_2\\\\Vert _{T_j(\\\\ell )}.$ Using the product property, we may gain some intuition regarding these norms by considering the following simple examples: $\\\\Vert \\\\psi _x\\\\bar{\\\\psi }_x\\\\Vert _{T_j(\\\\ell )} \\\\le \\\\Vert \\\\psi _x\\\\Vert _{T_j(\\\\ell )} \\\\Vert \\\\bar{\\\\psi }_x\\\\Vert _{T_j(\\\\ell )}= \\\\ell ^2, \\\\\\\\\\\\Vert (\\\\nabla _e\\\\psi )_x\\\\bar{\\\\psi }_x\\\\Vert _{T_j(\\\\ell )}\\\\le \\\\Vert \\\\nabla _e\\\\psi _x\\\\Vert _{T_j(\\\\ell )} \\\\Vert \\\\bar{\\\\psi }_x\\\\Vert _{T_j(\\\\ell )} = \\\\ell ^2 L^{-j}.$ The following more subtle example relies crucially on the antisymmetry of the coefficients $F_z$ and plays an important role for our model: $\\\\Vert \\\\psi _x\\\\bar{\\\\psi }_x\\\\psi _{x+e}\\\\bar{\\\\psi }_{x+e}\\\\Vert _{T_j(\\\\ell )}= \\\\Vert \\\\psi _x\\\\bar{\\\\psi }_x(\\\\nabla _e\\\\psi )_{x}(\\\\nabla _e\\\\bar{\\\\psi })_{x}\\\\Vert _{T_j(\\\\ell )}\\\\asymp \\\\ell ^4L^{-2j}.$ In general, and bearing in mind the antisymmetry, each factor of the fields contributes a factor $\\\\ell $ and each derivative a factor $L^{-j}$ .\",\n \"Second, from the definition, the following monotonicity properties hold: for $\\\\ell \\\\le \\\\ell ^{\\\\prime }$ , $ \\\\Vert F\\\\Vert _{T_j(\\\\ell )} \\\\le \\\\Vert F\\\\Vert _{T_j(\\\\ell ^{\\\\prime })}, \\\\qquad \\\\Vert F\\\\Vert _{T_{j+1}(\\\\ell )} \\\\le \\\\Vert F\\\\Vert _{T_j(\\\\ell ^{\\\\prime })}.$ Third, the doubling map satisfies (see [30]): for $F \\\\in (\\\\Lambda )$ , $ \\\\Vert \\\\theta F\\\\Vert _{T_j(\\\\ell )} \\\\le \\\\Vert F\\\\Vert _{T_j(2\\\\ell )}$ where the norm on the left-hand side is the $T_j(\\\\ell ) T_j(\\\\ell \\\\sqcup \\\\ell )$ norm on $(\\\\Lambda \\\\sqcup \\\\Lambda )$ .\",\n \"Finally, the following contraction bound for the fermionic Gaussian expectation is an application of the Gram inequality whose importance is well-known in fermionic renormalisation.\",\n \"It is proved in [30].\",\n \"Assume $C$ is a covariance matrix with $\\\\Vert C\\\\Vert _{T_j(\\\\ell )} \\\\le 1$ .\",\n \"For $F \\\\in (\\\\Lambda \\\\sqcup \\\\Lambda )$ , then $ \\\\Vert _{C}F\\\\Vert _{T_j(\\\\ell )} \\\\le \\\\Vert F\\\\Vert _{T_j(\\\\ell )}.$ In particular, by (REF ) the fermionic Gaussian convolution satisfies $ \\\\Vert _{C} \\\\theta F\\\\Vert _{T_{j}(\\\\ell )}\\\\le \\\\Vert F\\\\Vert _{T_{j}(2\\\\ell )}.$ For our choices of $\\\\ell _j$ and of the finite range covariance matrices $C_j$ , the inequalities (REF ) and (REF ) in particular imply $ \\\\Vert F\\\\Vert _{T_{j+1}(\\\\ell _{j+1})} \\\\le \\\\Vert F\\\\Vert _{T_{j+1}(2\\\\ell _{j+1})} \\\\le \\\\Vert F\\\\Vert _{T_j(\\\\ell _j)},\\\\qquad \\\\Vert _{C_{j+1}}\\\\theta F\\\\Vert _{T_{j+1}(\\\\ell _{j+1})} \\\\le \\\\Vert F\\\\Vert _{T_j(\\\\ell _j)}.$ We remark that the existence of this contraction estimate for the expectation combined with (REF ) below is what makes renormalisation of fermionic fields much simpler than that of bosonic ones.\"\n ],\n [\n \"Localisation\",\n \"To define the renormalisation group map we need one more important ingredient: the localisation operators $\\\\text{Loc}_{X,Y}$ that will be used to extract the relevant and marginal terms from the $K_j$ coordinate to incorporate them in the renormalisation from $V_j$ into $V_{j+1}$ .\",\n \"These operators are generalised Taylor approximations which take as inputs $F\\\\in (X)$ and produce best approximations of $F$ in a finite dimensional space of local field polynomials.\"\n ],\n [\n \"Local field polynomials\",\n \"Formal local field polynomials are formal polynomials in the symbols $\\\\psi ,\\\\bar{\\\\psi },\\\\nabla \\\\psi ,\\\\nabla \\\\bar{\\\\psi },\\\\nabla ^2\\\\psi ,...$ (without spatial index).\",\n \"The dimension of a formal local field monomial is given by $(d-2)/2$ times the number of factors of $\\\\psi $ or $\\\\bar{\\\\psi }$ plus the number of discrete derivatives $\\\\nabla $ in its representation.\",\n \"Concretely, we consider the following space of formal local field polynomials.\",\n \"Let $^\\\\varnothing \\\\cong 4$ be the linear space of formal local field monomials of the form $ V= y (\\\\nabla \\\\psi ) (\\\\nabla \\\\bar{\\\\psi }) + \\\\frac{z}{2}((-\\\\Delta \\\\psi )\\\\bar{\\\\psi }+ \\\\psi (-\\\\Delta \\\\bar{\\\\psi }))+ a \\\\psi \\\\bar{\\\\psi }+ b \\\\psi \\\\bar{\\\\psi }(\\\\nabla \\\\psi )(\\\\nabla \\\\bar{\\\\psi }).$ We will identify elements $V\\\\in ^\\\\varnothing $ with their coupling constants $(z,y,a,b) \\\\in 4$ .\",\n \"Sometimes we include a constant term and write $u+V \\\\in ^\\\\varnothing $ with $u+V\\\\cong (u,z,y,a,b)\\\\in 5$ .\",\n \"Given a set $X\\\\subset \\\\Lambda $ , a formal local field polynomial $P$ can be specialised to an element of $(\\\\Lambda )$ by replacing formal monomials by evaluations.\",\n \"For example, if $P=\\\\bar{\\\\psi }\\\\psi $ , $P(X) = \\\\sum _{x\\\\in X}\\\\bar{\\\\psi }_{x}\\\\psi _{x}$ .\",\n \"We call polynomials arising in this way local polynomials.\",\n \"The most important case is $V \\\\mapsto V(X)$ , with $ V(X) =\\\\sum _{x\\\\in X} { y (\\\\nabla \\\\psi )_x (\\\\nabla \\\\bar{\\\\psi })_x + \\\\frac{z}{2}((-\\\\Delta \\\\psi )_x\\\\bar{\\\\psi }_x + \\\\psi _x(-\\\\Delta \\\\bar{\\\\psi })_x) + a \\\\psi _x\\\\bar{\\\\psi }_x + b \\\\psi _x\\\\bar{\\\\psi }_x (\\\\nabla \\\\psi )_x(\\\\nabla \\\\bar{\\\\psi })_x},$ where $\\\\Delta -\\\\frac{1}{2} \\\\sum _{e\\\\in _d} \\\\nabla _{-e}\\\\nabla _{e}$ and $(\\\\nabla \\\\psi )_{x}(\\\\nabla \\\\bar{\\\\psi })_{x} \\\\frac{1}{2}\\\\sum _{e\\\\in _d}\\\\nabla _{e}\\\\psi _{x}\\\\nabla _{e}\\\\bar{\\\\psi }_{x}$ are the lattice Laplacian and the square of the lattice gradient; recall that $_d = \\\\lbrace e_1, \\\\dots , e_{2d}\\\\rbrace $ .\",\n \"For a constant $u\\\\in we write$ u(X) = u|X|$, where $ |X|$ is the number of points in $ X $.\",\n \"Thus $ (u+V)(X)= u(X)+V(X)=u|X|+V(X)$.$ For $X\\\\subset \\\\Lambda $ , define $^\\\\varnothing (X) = \\\\lbrace V(X): V\\\\in ^\\\\varnothing \\\\rbrace \\\\subset (\\\\Lambda )$ and analogously $(^\\\\varnothing )(X) = \\\\lbrace u|X|+V(X): u\\\\in \\\\,V \\\\in ^\\\\varnothing \\\\rbrace \\\\subset (\\\\Lambda )$ .\",\n \"The space $^\\\\varnothing $ contains all formal local field polynomials of dimension at most $d$ that are (i) gauge invariant, (ii) respect lattice symmetries (if $EX=X$ for an automorphism $E$ , then $EV(X)=V(X)$ ) and (iii) have no constant terms.\",\n \"Respecting lattice symmetries means that $^\\\\varnothing $ does not contain terms such as $(\\\\nabla _{e_1}\\\\psi )(\\\\nabla _{e_2}\\\\bar{\\\\psi })$ , which cannot occur if $X$ and $V(X)$ are invariant under lattice rotations.\",\n \"We also emphasise that there is no $(\\\\bar{\\\\psi }\\\\psi )^2$ term, which would be consistent with power counting and symmetries, because it vanishes upon specialisation by anticommutativity of the fermionic variables.\",\n \"Two further remarks are in order.\",\n \"First, the monomial $\\\\psi \\\\bar{\\\\psi }(\\\\nabla \\\\psi )(\\\\nabla \\\\bar{\\\\psi })$ has dimension $2d-2>d$ for $d\\\\ge 3$ ; we include it in $^\\\\varnothing $ since it occurs in the initial potential.\",\n \"Second, the monomials multiplying $z$ and $y$ are equivalent upon specialisation when $X=\\\\Lambda $ by summation by parts, and differ only by boundary terms for general $X \\\\subset \\\\Lambda $ .\",\n \"This would allow us to keep only one of them, but it will be simpler to keep both.\",\n \"The localisation operators $\\\\operatorname{Loc}_{X,Y}$ associate local field monomials to elements of $(X)$ .\",\n \"In renormalisation group terminology, the image of $\\\\operatorname{Loc}$ projects onto the space of all relevant and marginal local polynomials.\",\n \"The precise definitions of the localisation operators do not play a direct role in this paper.\",\n \"Rather, only their abstract properties, summarised in the following Proposition REF , will be required.\",\n \"We use the general framework developed in [31] to define these operators.\",\n \"In short, the definitions of $\\\\operatorname{Loc}_X$ and $\\\\operatorname{Loc}_{X,Y}$ are those of [31].\",\n \"These definitions require a choice of field dimensions, which we choose as $[\\\\psi ]=[\\\\bar{\\\\psi }] = (d-2)/2$ , a choice of maximal field dimension $d_+$ , which we choose as $d_{+}=d$ , and a choice of a space $\\\\hat{P}$ of test polynomials, which we define exactly as in [31] with the substitution $\\\\nabla _e\\\\nabla _e\\\\rightarrow -\\\\nabla _e\\\\nabla _{-e}$ explained in [31].\",\n \"The following properties are then almost immediate from [31].\",\n \"For $L=L(d)$ sufficiently large there is a universal $\\\\bar{C}>0$ such that: for $j1$ is a parameter that will be chosen sufficiently large.\",\n \"Note that $^\\\\varnothing \\\\oplus ^\\\\varnothing _j(\\\\Lambda _N)$ is a finite-dimensional complex normed vector space with the above norms since $N<\\\\infty $ .\",\n \"As a consequence it is a Banach space.\",\n \"Let $d \\\\ge 3$ , $L \\\\ge L_0(d)$ , and $A \\\\ge A_0(L,d)$ .\",\n \"Assume that $u_j=0$ .\",\n \"There exists $\\\\epsilon = \\\\epsilon (L,A)>0$ such that for $j+10$ depending only on $d$ such that $ |_{j}(X)| \\\\ge (1+2\\\\eta ) |_{j+1}(\\\\bar{X})|.$ By (REF ), if $A=A(L)$ is large enough, $A^{|_{j+1}(U)|} \\\\sum _{X \\\\in _j\\\\setminus _{j}: \\\\bar{X}=U} (A/2)^{-|_j(X)|}\\\\le (2^{L^d} 2^{1+2\\\\eta }A^{-2\\\\eta })^{|_{j+1}(U)|} \\\\le A^{-\\\\eta |_{j+1}(U)|},$ as the set of $X\\\\in _{j}\\\\setminus _{j}$ with $\\\\bar{X}=U$ has size at most $2^{L^{d}|_{j+1}(U)|}$ .\",\n \"Now, the contribution to (REF ) from large sets $X \\\\in _j \\\\setminus _j$ is $\\\\sum _{X\\\\in _j \\\\setminus _j: \\\\bar{X}=U} e^{-V_{j+1}(U\\\\setminus X)+u_{j+1}|X|} { _{C_{j+1}}\\\\theta K_j(X)+ _{C_{j+1}} (\\\\delta I)^X }.$ If $U\\\\in \\\\mathcal {C}_{j+1}\\\\backslash \\\\mathcal {S}_{j+1}$ , we proceed as follows: For $\\\\Vert K_j\\\\Vert _j + \\\\Vert V_j\\\\Vert _j \\\\le \\\\epsilon $ with $\\\\epsilon $ sufficiently small, by Lemma REF the $T_{j+1}(\\\\ell _{j+1})$ norm of the $K$ contribution to (REF ) is bounded by $ A^{|_{j+1}(U)|-2^d}\\\\sum _{X\\\\in _j \\\\setminus _j: \\\\bar{X}=U} 2^{|_j(X)|} \\\\Vert _{C_{j+1}}\\\\theta K_j(X)\\\\Vert _{T_{j+1}(\\\\ell _{j+1})}.$ By the definition of $\\\\Vert K_{j}\\\\Vert _{j}$ and noting that $(|_{j}(X)|-2^{d})_{+}=|_{j}(X)|-2^{d}$ since $X\\\\notin _{j}$ , $\\\\Vert _{C_{j+1}}\\\\theta K_j(X)\\\\Vert _{T_{j+1}(\\\\ell _{j+1})}\\\\le A^{-(|_{j}(X)|-2^{d})}\\\\Vert K_j\\\\Vert _j,$ where we have also used the contraction estimates (REF ), (REF ).\",\n \"Inserting this bound into (REF ) and using (REF ) gives that the $K$ contribution to (REF ) is bounded by $ A^{|_{j+1}(U)|}\\\\sum _{X\\\\in _j \\\\setminus _j: \\\\bar{X}=U} (A/2)^{-|_j(X)|}\\\\Vert K_j\\\\Vert _{j} \\\\le A^{-\\\\eta } \\\\Vert K_j\\\\Vert _{j}.$ This is the desired bound.\",\n \"Examining the $\\\\delta I$ contribution to (REF ), Lemmas REF and REF and the product property yield $& A^{|_{j+1}(U)|-2^d} \\\\Vert \\\\sum _{X\\\\in _j \\\\setminus _j: \\\\bar{X}=U} e^{-V_{j+1}(U\\\\setminus X)+u_{j+1}|X|} { _{C_{j+1}} (\\\\delta I)^X }\\\\Vert _{T_{j+1}(\\\\ell _{j+1})}&\\\\le A^{|_{j+1}(U)|-2^d}\\\\sum _{X\\\\in _j \\\\setminus _j: \\\\bar{X}=U} [2O(\\\\Vert V_j\\\\Vert _j + \\\\Vert K_j\\\\Vert _j)]^{|_j(X)|} .$ If $\\\\Vert V_j\\\\Vert _j + \\\\Vert K_j\\\\Vert _j<\\\\epsilon $ and $\\\\epsilon $ is sufficiently small (depending on $A$ ), then the quantity in brackets is less than $1/(4 A^{2})$ .\",\n \"By the elementary inequality $(c^{2})^{n-2}\\\\le c^{n}$ for $c\\\\in (0,1)$ , $n>4$ and using that $|_{j}(X)|\\\\ge 2^{d}+1>4$ for each summand, we obtain the upper bound $[O(\\\\Vert V_j\\\\Vert _j + \\\\Vert K_j\\\\Vert _j)]^{2} A^{|_{j+1}(U)|} \\\\sum _{X\\\\in _j\\\\setminus _j: \\\\bar{X}=U} (A/2)^{-|_{j}(X)|}.$ Using again (REF ), it follows that the $\\\\delta I$ contribution to (REF ) is bounded by $O(A^{-\\\\eta }[\\\\Vert V_j\\\\Vert _j + \\\\Vert K_j\\\\Vert _j]^{2})$ for $A$ sufficiently large.\",\n \"We have now completed the bound on (REF ) provided $U\\\\in \\\\mathcal {C}_{j+1}\\\\backslash \\\\mathcal {S}_{j+1}$ .\",\n \"The argument is similar if $U \\\\in _{j+1}$ .\",\n \"In this case the prefactor $A^{|_{j+1}(U)|-2^d}$ gets replaced by 1 in (REF ) and (REF ).\",\n \"In place of (REF ) we obtain, since $1+ 2^d\\\\le |_j(X)|\\\\le L^d |_{j+1}(U)|\\\\le [2L]^d$ and the number of summands in this case is at most $2^{[2L]^d}$ , $\\\\Vert \\\\sum _{X\\\\in _j \\\\setminus _j: \\\\bar{X}=U} e^{-V_{j+1}(U\\\\setminus X)+u_{j+1}|X|} _{C_{j+1}}\\\\theta K_j(X)\\\\Vert _{T_{j+1}(\\\\ell _{j+1})}\\\\le A^{-1}2^{2[2L]^d}\\\\Vert K_j\\\\Vert _j=O(A^{-\\\\eta } \\\\Vert K_j\\\\Vert _j)$ for $A$ large enough depending on $L$ and $d$ .\",\n \"On the other hand in place of (REF ) we have $\\\\Vert \\\\sum _{X\\\\in _j \\\\setminus _j: \\\\bar{X}=U} e^{-V_{j+1}(U\\\\setminus X)+u_{j+1}|X|} _{C_{j+1}}\\\\delta I^X\\\\Vert _{T_{j+1}(\\\\ell _{j+1})}\\\\le O(A^{-\\\\eta }[\\\\Vert V_j\\\\Vert _j + \\\\Vert K_j\\\\Vert _j]^{2})$ provided $\\\\epsilon $ is chosen sufficiently small depending on $L$ , since each summand on the left-hand side has $|_j(X))|\\\\ge 3$ .\",\n \"Thus for $A=A(L,d)$ sufficiently large and $\\\\epsilon =\\\\epsilon (A,L)$ sufficiently small, the expression (REF ) is bounded in the $T_{j+1}(\\\\ell _{j+1})$ norm by $O(A^{-\\\\eta }(\\\\Vert K_j\\\\Vert _j+[\\\\Vert V_j\\\\Vert _j + \\\\Vert K_j\\\\Vert _j]^{2}))$ in all cases.\"\n ],\n [\n \"Non-linear part\",\n \"To conclude the proof of the theorem, we show $\\\\Vert K_{j+1}(U)-\\\\mathcal {L}_{j+1}(U)\\\\Vert _{j+1}\\\\le O(\\\\Vert K_j\\\\Vert _j^2+\\\\Vert V_j\\\\Vert _j^2)$ .\",\n \"Recall the definition of $K_{j+1}$ from (REF ), see also Appendix REF .\",\n \"We have $K_{j+1}(U)-\\\\mathcal {L}_{j+1}(U)=e^{u_{j+1}|U|} \\\\sum _{_2(U)}e^{-(u+V)_{j+1}(X_{I})} _{C_{j+1}} \\\\check{K}_j(\\\\check{X})\\\\prod _{(B,X)\\\\in }\\\\theta J(B,X) ,$ where $\\\\sum _{_{2}(U)}$ indicates that the system of polymers $(X_, \\\\check{X})$ contains at least two components.\",\n \"In particular, if $=\\\\varnothing $ , $\\\\check{X}$ has least two components and if $\\\\check{X}=\\\\varnothing $ then $||\\\\ge 2$ , where $||$ is the number of pairs in $$ , see Appendix REF .\",\n \"If $\\\\Vert V_j\\\\Vert _j + \\\\Vert K_j\\\\Vert _j$ is small enough, arguing as in (REF ) implies $\\\\Vert e^{u_{j+1}(U)}e^{-(u+V)_{j+1}( X_{I})}\\\\Vert _{T_{j+1}(\\\\ell _{j+1})}\\\\le 2^{|_{j}(U)|}$ .\",\n \"By (REF ), $\\\\Vert J(B, X)\\\\Vert _{T_{j}(\\\\ell _{j})} =O(\\\\Vert K_j\\\\Vert _{j})$ .\",\n \"Thus using that $_{C_{j+1}}\\\\theta $ contracts from $T_{j+1}(\\\\ell _{j+1})$ into $T_{j}(\\\\ell _{j})$ we obtain $\\\\Vert K_{j+1}(U)-\\\\mathcal {L}_{j+1}(U)\\\\Vert _{T_{j+1}(\\\\ell _{j+1})}\\\\le \\\\sum _{_2(U)}[O(\\\\Vert K_j\\\\Vert _j)]^{||}2^{|_j(U)|} \\\\Vert \\\\check{K}_j(\\\\check{X})\\\\Vert _{T_{j+1}(\\\\ell _{j+1})}.$ We first estimate the norm of $\\\\check{K}_j(\\\\check{X})$ , and then the resulting sum.\",\n \"If $\\\\Vert V_{j}\\\\Vert _{j}+\\\\Vert K_{j}\\\\Vert _{j}\\\\le \\\\epsilon $ and $\\\\epsilon =\\\\epsilon (A, L)$ is sufficiently small, then $\\\\Vert \\\\check{K}(\\\\check{X})\\\\Vert _{T_{j+1}(\\\\ell _{j+1})} \\\\le [O( \\\\Vert V_j\\\\Vert _j +\\\\Vert K_j\\\\Vert _j)]^{|\\\\operatorname{Comp}(\\\\check{X})|}(\\\\frac{A}{2})^{-\\\\sum _{Y\\\\in \\\\operatorname{Comp}(\\\\check{X})}(|_j(Y)|-2^{d})_{+}}.$ For notational convenience, for $Y\\\\in \\\\mathcal {C}_j$ let $\\\\tilde{K}(Y)= \\\\sum _{W \\\\in _j(Y)} \\\\theta K_j(Y\\\\setminus W) (\\\\delta I)^W.$ The claimed bound follows from the definition of $\\\\check{K}(X)$ in (REF ) if we show $\\\\Vert \\\\tilde{K}(Y)-\\\\sum _{B \\\\in _j(Y)} {\\\\theta } J(B,Y)\\\\Vert _{T_{j+1}(\\\\ell _{j+1})} \\\\le 2^{d+1}O (\\\\Vert V_j\\\\Vert _j + \\\\Vert K_j\\\\Vert _j) (A/2)^{-(|_j(Y)|-2^{d})_{+}}$ for $Y$ a connected component of $\\\\check{X}$ .\",\n \"We apply the triangle inequality.\",\n \"Since $J(B,Y)=0$ if $Y\\\\notin _{j}$ , $\\\\Vert \\\\sum _{B \\\\in _j(Y)}_{C_{j+1}}\\\\theta J(B,Y)\\\\Vert _{T_{j+1}(\\\\ell _{j+1})} \\\\le O(\\\\Vert K_{j}\\\\Vert _{j})$ where we have used $\\\\Vert J(B, X)\\\\Vert _{T_{j}(\\\\ell _{j})}=O(\\\\Vert K_j\\\\Vert _{j})$ .\",\n \"By (REF ), component factorisation of $K_j$ , and the contraction property of the norms and $\\\\theta $ , for $B\\\\in _j$ and $Z\\\\in _j$ , $\\\\Vert \\\\delta I(B)\\\\Vert _{T_{j+1}(\\\\ell _{j+1})}&\\\\le O(\\\\Vert V_j\\\\Vert _j+\\\\Vert K_j\\\\Vert _j), \\\\\\\\\\\\Vert \\\\theta K_j(Z)\\\\Vert _{T_{j+1}(\\\\ell _{j+1})} &\\\\le A^{-\\\\sum _{W\\\\in \\\\operatorname{Comp}(Z)} (|_j(W)|-2^d)_+}\\\\Vert K_j\\\\Vert _j^{|\\\\operatorname{Comp}(Z)|}.$ We now impose the condition that $\\\\epsilon \\\\le A^{-2^d}$ .\",\n \"Then plugging the previous bounds into the expression for $\\\\tilde{K}(Y)$ we have $\\\\Vert \\\\tilde{K}(Y)\\\\Vert _{T_{j+1}(\\\\ell _{j+1})}& \\\\le \\\\sum _{Z \\\\in _j(Y)}\\\\Vert (\\\\delta I)^Z \\\\theta K_j(Y \\\\setminus Z)\\\\Vert _{T_{j+1}(\\\\ell _{j+1})}&\\\\le \\\\sum _{Z \\\\in _j(Y)}(O(\\\\Vert V_j\\\\Vert _{j}+ \\\\Vert K_j\\\\Vert _{j}))^{|_j(Z)|+|\\\\operatorname{Comp}(Y\\\\backslash Z)|} A^{-\\\\sum _{W\\\\in \\\\operatorname{Comp}(Y\\\\backslash Z)}(|_j(W)|-2^d)_+}&\\\\le \\\\sum _{Z \\\\in _j(Y)}\\\\left(A^{2^{d}}O(\\\\Vert V_j\\\\Vert _{j}+\\\\Vert K_j\\\\Vert _{j})\\\\right)^{|\\\\operatorname{Comp}(Y\\\\backslash Z)|}&\\\\qquad \\\\qquad \\\\qquad \\\\qquad \\\\times (O(\\\\Vert V_j\\\\Vert _{j}+\\\\Vert K_j\\\\Vert _{j}))^{|_j(Z)|}A^{-\\\\sum _{W\\\\in \\\\operatorname{Comp}(Y\\\\backslash Z)}|_{j}(W)|}&\\\\le (A^{2^{d}}O(\\\\Vert V_j\\\\Vert _{j}+ \\\\Vert K_j\\\\Vert _{j}))(O(\\\\Vert V_j\\\\Vert _{j}+ \\\\Vert K_j\\\\Vert _{j}) + A^{-1})^{|_{j}(Y)|}&\\\\le A^{2^{d}}\\\\left(\\\\frac{A}{2}\\\\right)^{-|_{j}(Y)|} O(\\\\Vert V_j\\\\Vert _{j}+\\\\Vert K_j\\\\Vert _{j}).$ (Note that if $Y$ is a small set, the factors of $A$ could have been omitted.)\",\n \"Together with (REF ) this proves the lemma.\",\n \"We are now ready to complete the bound on the non-linear part.\",\n \"For brevity let us write $b$ for the factors $O(\\\\Vert V_j\\\\Vert _{j}+\\\\Vert K_j\\\\Vert _{j})$ above.\",\n \"We have reduced verification of () to showing $A^{|_{j+1}(U)|}2^{|_{j}(U)|}\\\\sum _{_{2}(U)} (b2^{d+1}A^{2^{d}})^{||+|\\\\operatorname{Comp}(\\\\check{X})|}(\\\\frac{A}{2})^{-|_{j}(\\\\check{X})|} \\\\le O(b^2)$ for all $U\\\\in _{j+1}$ .\",\n \"Since $|_{j}(U)|\\\\le L^{d}|_{j+1}(U)|$ , for any $c>0$ the prefactor can be bounded by $A^{|_{j+1}(U)|}2^{|_{j}(U)|} \\\\le (\\\\frac{A}{2})^{(1-c)|_{j+1}(U)|}2^{(L^{d}+1)|_{j}(U)|} (\\\\frac{A}{2})^{c|_{j+1}(U)|}.$ Taking $c>1$ , the product of the first two terms on the last right-hand side is less than 1 for $A$ sufficiently large.\",\n \"It suffices to prove that for some $c>1$ , one has $(\\\\frac{A}{2})^{c|_{j+1}(U)|}\\\\sum _{_{2}(U)} (b2^{d+1}A^{2^{d}})^{||+|\\\\operatorname{Comp}(\\\\check{X})|}(\\\\frac{A}{2})^{-|_{j}(\\\\check{X})|} \\\\le O(b^2).$ At this point we appeal to [27]; this result estimates the same sum but over $(U)$ instead of $_{2}(U)$ .\",\n \"As we are estimating $\\\\sum _{_{2}(U)}$ the supremum over $n\\\\ge 1$ in [27] becomes a supremum over $n\\\\ge 2$ since $|| + \\\\operatorname{Comp}(\\\\check{X})|\\\\ge 2$ .\",\n \"This yields that if $A=A(L,d)$ is large enough, then there is a $m$ such that for all $U\\\\in _{j+1}$ $(\\\\frac{A}{2})^{c|_{j+1}(U)|}\\\\sum _{_{2}(U)} (b2^{d+1}A^{2^{d}})^{||+|\\\\operatorname{Comp}(\\\\check{X})|}(\\\\frac{A}{2})^{-|_{j}(\\\\check{X})|} = O( (bA^{m})^{2}),$ which is $A^{\\\\nu }O(b^{2})$ as needed.\"\n ],\n [\n \"Flow of the renormalisation group\",\n \"Recall the infinite volume limit of the renormalisation group maps $\\\\Phi _{j+1,\\\\infty }$ discussed below Proposition REF .\",\n \"We equip $^\\\\varnothing _{j}(^{d})$ with the norm $\\\\Vert K\\\\Vert _{j}$ defined by ().\",\n \"Next we study the iteration of the renormalisation group maps as a dynamical system.\",\n \"In what follows $K_0=0$ means $K_0(X)=1_{X=\\\\varnothing }$ for $X\\\\in _j$ .\",\n \"Let $d\\\\ge 3$ , $L\\\\ge L_0$ , and $A \\\\ge A_0(L)$ .\",\n \"For $m^2 \\\\ge 0$ arbitrary and $b_0$ small, there exist $V_0^c(b_0,m^2)$ and $\\\\kappa >0$ such that if $(V_0,K_0)=(V_0^c(m^2,b_0),0)$ and $(V_{j+1},K_{j+1})= \\\\Phi _{j+1,\\\\infty }(V_j,K_j,m^2)$ is the flow of the infinite volume renormalisation group map then $ \\\\Vert V_j\\\\Vert _j = O(b_0L^{-\\\\kappa j}), \\\\qquad \\\\Vert K_j\\\\Vert _j = O(b_0^2 L^{-\\\\kappa j}).$ The components of $V_0^c(m^2,b_0)$ are continuous and uniformly bounded in $m^2 \\\\ge 0$ and differentiable in $b_0$ with uniformly bounded derivative.\",\n \"The proof is by applying the well-known stable manifold theorem for smooth dynamical systems in the version given in [27].\",\n \"This theorem applies to dynamical systems in Banach spaces, and hence two preparatory observations are needed.\",\n \"First, for each $j$ , the linear space $^\\\\varnothing _{j}(^{d})$ equipped with $\\\\Vert \\\\cdot \\\\Vert _{j}$ is complete (when restricted to elements of finite norm), i.e., a Banach space.\",\n \"Second, by the consistency of the finite volume renormalisation group maps (Proposition REF ), the estimates given in Theorem REF also hold for the infinite volume limit.\",\n \"To prove the theorem we first write down the dynamical system corresponding to the renormalisation group map.\",\n \"The definition of $V_{j+1}$ is (REF ).\",\n \"We start with the contribution to $V_{j+1}$ arising from $_{C_{j+1}}\\\\theta V_{j}(B) (\\\\tilde{u}_{j+1}|B|+\\\\tilde{V}_{j+1}(B))$ .\",\n \"This can be computed by the Wick formula (REF ): $\\\\tilde{z}_{j+1} = z_{j},\\\\qquad \\\\tilde{y}_{j+1} = y_{j} + \\\\kappa _{j}^{yb}b_{j},\\\\qquad \\\\tilde{a}_{j+1} = a_{j} + \\\\kappa _{j}^{ab}b_{j},\\\\qquad \\\\tilde{b}_{j+1} = b_{j},$ with $\\\\kappa _{j}^{yb} = -C_{j+1}(0)$ and $\\\\kappa _{j}^{ab}=\\\\Delta C_{j+1}(0)$ .\",\n \"Since $\\\\Vert V_j(B)\\\\Vert _{T_j(\\\\ell _j)}$ is comparable with $|z_j| + |y_j| + L^{2j}|a_j| + L^{-(d-2)j} |b_j|$ , it is natural to rescale $\\\\hat{z}_{j}=z_{j}$ , $\\\\hat{y}_{j}=y_{j}$ , $\\\\hat{a}_{j}=L^{2j}a_{j}$ , $\\\\hat{b}_{j}=L^{-(d-2)j}b_{j}$ , $\\\\hat{\\\\kappa }_{j}^{ab}=L^{dj}\\\\kappa _{j}^{ab}$ , and $\\\\hat{\\\\kappa }_{j}^{yb}=L^{(d-2)j}\\\\kappa _{j}^{yb}$ .\",\n \"The definition (REF ) of $V_{j+1}$ then becomes $\\\\hat{z}_{j+1} &= \\\\hat{z}_{j}+ \\\\hat{r}^{z}_{j},&\\\\qquad \\\\hat{y}_{j+1}&= \\\\hat{y}_{j} + \\\\hat{\\\\kappa }_{j}^{yb}\\\\hat{b}_{j} + \\\\hat{r}^{y}_{j}, \\\\\\\\\\\\hat{a}_{j+1}&= L^{2}\\\\hat{a}_{j} + \\\\hat{\\\\kappa }_{j}^{ab}\\\\hat{b}_{j} + \\\\hat{r}^{a}_{j}, &\\\\qquad \\\\hat{b}_{j+1}&= L^{-(d-2)}\\\\hat{b}_{j} + \\\\hat{r}^{b}_{j},$ where the $\\\\hat{r}_{j}$ are linear maps in $K_j$ and have size $O({K_{j}}_{j})$ by (REF ) of Theorem REF , and the $\\\\hat{\\\\kappa }_{j}$ are uniformly bounded in $j$ by the covariance estimates (REF ).\",\n \"We set $v_j=(\\\\hat{y}_j,\\\\hat{z}_j,\\\\hat{a}_j)$ and $w_j=(\\\\hat{b}_j,K_j)$ and use $\\\\Vert \\\\cdot \\\\Vert $ for the norm given by maximum of the respective components.\",\n \"By the computation above and Theorem REF the infinite volume renormalisation group map can be written in block diagonal form $\\\\begin{pmatrix}v_{j+1} \\\\\\\\ w_{j+1}\\\\end{pmatrix}= \\\\begin{pmatrix} E & B_j \\\\\\\\ 0 & D_j \\\\end{pmatrix}\\\\begin{pmatrix}v_{j} \\\\\\\\ w_{j}\\\\end{pmatrix}+ \\\\begin{pmatrix} 0 \\\\\\\\ g_{j+1}(v_j,w_j) \\\\end{pmatrix}$ with $\\\\Vert E^{-1}\\\\Vert = 1$ , $\\\\Vert B_j\\\\Vert $ bounded, and $\\\\Vert D_j\\\\Vert \\\\le \\\\max \\\\lbrace L^{-(d-2)}, O(L^{-3}+A^{-\\\\eta })\\\\rbrace \\\\le L^{-\\\\kappa }$ provided $A$ is large enough, and with $g_j(0,0)=0$ and $Dg_j(0,0)=0$ .\",\n \"For every $m^2\\\\ge 0$ , the existence of $V_{0}^{c}(m^{2},b_{0})$ and its differentiability in $b_{0}$ now follow by [27].\",\n \"To see that $V_0^c(m^2,b_0)$ is also continuous in $m^2$ , regard $v_j$ and $w_j$ as bounded continuous functions of $m^2$ , i.e., consider $v_j \\\\in C_b([0,\\\\infty ),^3)$ and $w_j\\\\in C_b([0,\\\\infty ), \\\\times ^\\\\varnothing _j(^d))$ .\",\n \"Since all the estimates above are uniform in $m^2 \\\\ge 0$ , the previous argument then gives a solution $V_0^c \\\\in C_b((-\\\\epsilon ,\\\\epsilon ),C_b([0,\\\\infty ),^3))$ .\",\n \"The bounds (REF ) are not part of the statement of [27], but are immediate from the proof.\",\n \"By consistency, the finite volume renormalisation group flow for $V_j$ agrees with the infinite volume renormalisation group flow up to scale $j0$ , the $V^\\\\star _j$ have terms not present in $V^\\\\star _{0}$ , for example the terms involving $q$ in the next definition.\",\n \"The nilpotency of the observable fields $\\\\sigma _{a}, \\\\sigma _{b}$ limits the possibilities.\",\n \"Let $^\\\\star $ be the space of formal field polynomials $u^\\\\star +V^\\\\star $ of the form: $&\\\\left.\\\\begin{aligned}V^\\\\star &=- \\\\lambda _a \\\\sigma _a \\\\bar{\\\\psi }_a - \\\\lambda _b \\\\psi _b\\\\bar{\\\\sigma }_b+ \\\\sigma _a \\\\bar{\\\\sigma }_b\\\\frac{r}{2}(\\\\bar{\\\\psi }_a\\\\psi _a+\\\\bar{\\\\psi }_b\\\\psi _b),\\\\;\\\\,\\\\quad \\\\\\\\\\\\nonumber u^\\\\star &= - { \\\\sigma _a\\\\bar{\\\\sigma }_b } q,\\\\end{aligned}\\\\right\\\\rbrace \\\\qquad \\\\text{in Case (1),}\\\\\\\\& \\\\left.\\\\begin{aligned}V^\\\\star &={ -}\\\\sigma _a \\\\lambda _a\\\\bar{\\\\psi }_a\\\\psi _a - \\\\sigma _b\\\\lambda _b\\\\bar{\\\\psi }_b\\\\psi _b{ -}\\\\sigma _a\\\\sigma _b\\\\frac{\\\\eta }{2} (\\\\bar{\\\\psi }_a\\\\psi _b+\\\\bar{\\\\psi }_b\\\\psi _a)\\\\\\\\& \\\\nonumber \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad + \\\\sigma _a \\\\sigma _b \\\\frac{r}{2}(\\\\bar{\\\\psi }_a\\\\psi _a+\\\\bar{\\\\psi }_b\\\\psi _b)),\\\\\\\\\\\\nonumber u^\\\\star &=- \\\\sigma _a \\\\gamma _a- \\\\sigma _b \\\\gamma _b- { \\\\sigma _a\\\\sigma _b } q ,\\\\end{aligned}\\\\right\\\\rbrace \\\\qquad \\\\text{in Case (2),}$ for observable coupling constants $(\\\\lambda _{a},\\\\lambda _{b},q, r)\\\\in {4}$ respectively $(\\\\lambda _{a},\\\\lambda _{b},\\\\gamma _{a},\\\\gamma _b,q,\\\\eta , r)\\\\in {7}$ .\",\n \"For $X \\\\subset \\\\Lambda $ , we define $(u^{\\\\star }+V^\\\\star )(X) \\\\in ^\\\\star (X \\\\cap \\\\lbrace a,b\\\\rbrace )$ by $ (u^\\\\star +V^\\\\star )(X) =-\\\\lambda _a \\\\sigma _a \\\\bar{\\\\psi }_a 1_{a\\\\in X} -\\\\lambda _b\\\\psi _b \\\\bar{\\\\sigma }_b 1_{b \\\\in X} + \\\\sigma _a \\\\bar{\\\\sigma }_b q 1_{a\\\\in X,b \\\\in X}+\\\\sigma _a \\\\bar{\\\\sigma }_b \\\\frac{r}{2}(\\\\bar{\\\\psi }_a\\\\psi _a+\\\\bar{\\\\psi }_b\\\\psi _b)1_{a\\\\in X,b\\\\in X}$ in Case (1), and analogously in Case (2).\",\n \"The terms corresponding to $r$ do not appear at any step of the noninteracting iteration (REF ) if we start with them equal to 0.\",\n \"We include them here in preparation for the interacting model.\",\n \"The evolutions of $u^\\\\star _j+V^\\\\star _j \\\\rightarrow u^{\\\\star }_{j+1}$ and $V^\\\\star _j \\\\rightarrow V^\\\\star _{j+1}$ are equivalent to the evolution of the coupling constants $(\\\\lambda _a,\\\\lambda _b,q, r)$ respectively $(\\\\lambda _a,\\\\lambda _b,\\\\gamma _a,\\\\gamma _b,q,\\\\eta ,r)$ .\",\n \"By computation of the fermionic Gaussian moments in (REF ), the flow of the observable coupling constants according to (REF ) is then given as follows.\",\n \"Note that the evolution of couplings constants in $V^\\\\star $ is independent of the coupling constants in $u^\\\\star $ .\",\n \"Let $V^\\\\varnothing _0=0$ , and let $u^\\\\star _{j}$ and $V^\\\\star _j$ be of the form as in Definition REF .\",\n \"The map (REF ) is then given as follows.\",\n \"In Case (1), for $x\\\\in \\\\lbrace a,b\\\\rbrace $ , $\\\\lambda _{x,j+1} &=\\\\lambda _{x,j}\\\\\\\\q_{j+1} &= q_j + \\\\lambda _{a,j}\\\\lambda _{b,j} C_{j+1}(a,b) + r_jC_{j+1}(0,0)\\\\\\\\r_{j+1} &= r_j,\\\\multicolumn{2}{l}{\\\\text{whereas in Case (2), for $x\\\\in \\\\lbrace a,b\\\\rbrace $,}}\\\\\\\\\\\\lambda _{x,j+1} &= \\\\lambda _{x,j}\\\\\\\\\\\\gamma _{x,j+1} &= \\\\gamma _{x,j} + \\\\lambda _{x,j}C_{j+1}(0,0)\\\\\\\\q_{j+1} &= q_j + \\\\eta _j C_{j+1}(a,b)+ r_{j} C_{j+1}(0,0)- \\\\lambda _{a,j}\\\\lambda _{b,j} C_{j+1}(a,b)^2\\\\\\\\\\\\eta _{j+1} &= \\\\eta _j - 2\\\\lambda _{a,j}\\\\lambda _{b,j} C_{j+1}(a,b)\\\\\\\\r_{j+1} & =r_{j}.$ This follows from straightforward evaluation of (REF ) using (REF ).\",\n \"To continue the warm-up for the interacting case, we illustrate how these equations reproduce the direct computations of correlation functions.\",\n \"When $r_{0}=0$ by a computation using Definition REF , the formulas (REF ) and (REF )–() imply the correlation functions in Cases (1) and (2) are given by ${\\\\bar{\\\\psi }_a\\\\psi _b} = \\\\frac{ q_{N+1}}{\\\\lambda _{a,0}\\\\lambda _{b,0}},\\\\qquad {\\\\bar{\\\\psi }_a\\\\psi _a} = \\\\frac{\\\\gamma _{a,N+1}}{\\\\lambda _{a,0}},\\\\qquad {\\\\bar{\\\\psi }_a\\\\psi _a;\\\\bar{\\\\psi }_b\\\\psi _b} = \\\\frac{q_{N+1}}{\\\\lambda _{a,0}\\\\lambda _{b,0}},$ with $qq^{(1)}$ and $\\\\lambda \\\\lambda ^{(1)}$ for the first equation $qq^{(2)}$ , $\\\\gamma \\\\gamma ^{(2)}$ , and $\\\\lambda \\\\lambda ^{(2)}$ for the last two.\",\n \"Recalling the convention $C_{N+1}=t_{N}Q_{N}$ , for Case (2) with $r_{0}=0$ we have $q_{N+1}= - \\\\lambda _{0,a}\\\\lambda _{0,b}{ \\\\sum _{k \\\\le N} C_k(a,b) + t_NQ_N(a,b)}^2 = -\\\\lambda _{0,a}\\\\lambda _{0,b}(-\\\\Delta +m^2)^{-1}(a,b)^2,$ by (REF ).\",\n \"When combined with (REF ) this gives the expected result.\",\n \"In the preceding computation we kept the potential in the exponential for the entire computation, whereas in Sections and the zero mode is integrated out directly without rewriting the integrand in this form (see, e.g., (REF )).\",\n \"We distinguish these two approaches by using $N+1$ subscripts for the former and $(N,N)$ for the latter, and by putting tildes on quantities associated with the $(N,N)$ th step as was done in Section .\",\n \"Before moving to the interacting model, we introduce the coalescence scale $j_{ab}$ as the largest integer $j$ such that $C_{k+1}(a,b)=0$ for all $k < j$ , i.e., $j_{ab} = {\\\\log _L (2|a-b|_{{\\\\infty }})}.$ In the degenerate cases $\\\\lambda _a=0$ or $\\\\lambda _b=0$ when only one of the observable fields is present we use the convention $j_{ab}=+\\\\infty $ .\",\n \"Note that the finite range property (REF ) implies that $q_j=\\\\eta _j=r_{j}=0$ for $j0$ such that: for $j L^{j+1}$ then $P^\\\\star (B,B^{\\\\prime })=0$ , by the finite range of the covariance $C_{j+1}$ .\",\n \"As a result, $P^{\\\\star }(B)\\\\in ^{\\\\star }(B^\\\\square )$ .\",\n \"With these definitions in place, $ u^\\\\star _{j+1}+V^\\\\star _{j+1}$ is defined in the same way as $ u_{j+1}+ V_{j+1}$ with the addition of the second order term $P^\\\\star $ , and $K^\\\\star _{j+1}$ is then defined in the same way as $K_{j+1}$ : The map $(V_j,K_j) \\\\mapsto ({ u^\\\\star _{j+1}},V^\\\\star _{j+1})$ is defined, for $B\\\\in _j$ , by $u^\\\\star _{j+1}(B)+V^\\\\star _{j+1}(B) =_{C_{j+1}}\\\\theta (V^\\\\star _j(B) - Q^\\\\star (B)) - P^\\\\star (B)$ where $u^\\\\star _{j+1}$ consists of all monomials that do not contain factors of $\\\\psi $ or $\\\\bar{\\\\psi }$ .\",\n \"Explicitly, $u^\\\\star _{j+1}={\\\\left\\\\lbrace \\\\begin{array}{ll}-\\\\sigma _a\\\\bar{\\\\sigma }_b q_{j+1}, &\\\\text{Case (1)}, \\\\\\\\-\\\\sigma _a\\\\sigma _b q_{j+1} - \\\\sigma _a \\\\gamma _{a,j+1}-\\\\sigma _b\\\\gamma _{b,j+1}, &\\\\text{Case (2)},\\\\end{array}\\\\right.\",\n \"}$ The map $(V_j,K_j)\\\\mapsto K^\\\\star _{j+1}$ is defined as in Definition REF except that $V^\\\\varnothing $ and $u^\\\\varnothing $ are replaced by $V=V^\\\\varnothing +V^\\\\star $ and $u=u^\\\\varnothing +u^\\\\star $ .\",\n \"Propositions REF and REF also hold for this extended definition of the renormalisation group map.\",\n \"The proofs are the same; for Proposition REF we provide a proof in Appendix REF .\"\n ],\n [\n \"Estimates for the renormalisation group map with observables\",\n \"In this section, the $O$ -notation refers to scale $j+1$ norms, i.e., for $F,G \\\\in (\\\\Lambda )$ , we write $F=G+O(t)$ to denote that $\\\\Vert F-G\\\\Vert _{T_{j+1}(\\\\ell _{j+1})} \\\\le O(t)$ .\",\n \"Under the assumptions of Theorem REF , if also $\\\\Vert V^\\\\star _j\\\\Vert _j+\\\\Vert K^\\\\star _j\\\\Vert _j \\\\le \\\\epsilon $ and $u^\\\\star _j=0$ , then for $j+10$ , $\\\\lambda _{j} &= \\\\lambda _0 + O_L(\\\\lambda _0b_0)\\\\\\\\r_j &= O_L(\\\\lambda _0b_0|a-b|^{-\\\\kappa })1_{j\\\\ge j_{ab}}\\\\multicolumn{2}{l}{\\\\text{and, in Case (2), if $q_0=r_0=\\\\gamma _{x,0}=\\\\eta _0=0$ and $\\\\lambda _0>0$,}}\\\\\\\\\\\\lambda _{j} &= \\\\lambda _0 + O_L(\\\\lambda _0b_0)\\\\\\\\\\\\eta _j &= O_L(\\\\lambda _0^2|a-b|^{-(d-2)})1_{j\\\\ge j_{ab}}\\\\\\\\r_{j}&= O_L(\\\\lambda _0b_0|a-b|^{-(d-2)-\\\\kappa })1_{j\\\\ge j_{ab}},$ where $\\\\lambda _{j}=\\\\lambda _{x,j}$ for either $x=a$ or $x=b$ .\",\n \"In both Cases (1) and (2), $\\\\Vert V^\\\\star _{j}\\\\Vert _{j} \\\\le \\\\lambda _0 + O_L(\\\\lambda _0^2) + O_L(\\\\lambda _0b_0).$ The bounds on the coupling constants follow from Lemma REF ; the hypothesis regarding $K_{j}(X)=0$ for $jj_{ab}$ since we have started the flow with $r_{0}=0$ in Case (1), and $q_{0}=\\\\eta _{0}=r_{0}=0$ in Case (2).\",\n \"What remains is to analyse the recurrences.\",\n \"For $\\\\lambda _{x,j}$ , since $\\\\ell _{x,j}^{-1}\\\\ell _j^{-p} = 1$ in Case ($p$ ), using (REF ), respectively (), $\\\\lambda _{x,j} = \\\\lambda _0 + \\\\sum _{k=0}^{j-1} O_L(\\\\Vert K^\\\\star _k\\\\Vert _k)= \\\\lambda _0 + \\\\sum _{k=0}^{j-1} O_L(\\\\lambda _0b_0L^{-\\\\kappa k})=\\\\lambda _{0} + O_L(\\\\lambda _{0}b_{0}).$ The bounds on $r_{j}$ follow from the fact that all contributions are 0 for scales $j0$ such that $\\\\Vert V^\\\\varnothing _j\\\\Vert _j = O(b_0L^{-\\\\alpha j}), \\\\qquad \\\\Vert K^\\\\varnothing _j\\\\Vert _j = O(b_0L^{-\\\\alpha j}).$ Using this as input, we iterate the observable flow (REF )–(), with initial condition $\\\\lambda _{a,0}=\\\\lambda _{b,0}=\\\\lambda _0$ small enough and all other observable coupling constants equal to 0.\",\n \"Assume that the bulk renormalisation group flow $(V^\\\\varnothing _j,K^\\\\varnothing _j)$ obeys $\\\\Vert V^\\\\varnothing _j\\\\Vert _j+\\\\Vert K^\\\\varnothing _j\\\\Vert _j = O(b_0L^{-\\\\alpha j})$ for some $\\\\alpha >0$ .\",\n \"Then there is $\\\\kappa >0$ such that for $\\\\lambda _{0,a}=\\\\lambda _{0,b}=\\\\lambda _0 >0$ sufficiently small and all other observable coupling constants initially 0, $ \\\\Vert V^\\\\star _j\\\\Vert _j \\\\le O_L(\\\\lambda _0), \\\\qquad \\\\Vert K^\\\\star _j\\\\Vert _j \\\\le O_L(\\\\lambda _0b_0L^{-\\\\kappa j}).$ The proof is by induction from a scale $j_0$ which will be determined below.\",\n \"When $j=0$ we have $\\\\Vert V^\\\\star _{0}\\\\Vert _{0}\\\\asymp \\\\lambda _{0,a}+\\\\lambda _{0,b}$ and $\\\\Vert K^\\\\star _{0}\\\\Vert _{0}=0$ .\",\n \"Now let us choose $C_j>0$ so that $\\\\Vert K^\\\\star _k\\\\Vert _{k} \\\\le C_j b_0\\\\lambda _0L^{-\\\\kappa k}$ for all $k \\\\le j$ .\",\n \"We start the proof by studying the behaviour of $C_j$ as $j$ increases.\",\n \"We may assume $\\\\kappa <\\\\alpha /2$ and also that $\\\\kappa $ is less than the exponents of $L$ and $A$ in ().\",\n \"Then Lemma REF shows that $\\\\Vert V^\\\\star _{k}\\\\Vert _k \\\\le C_j\\\\lambda _0$ for $k\\\\le j+1$ provided $\\\\lambda _0<1$ and $b_0$ is sufficiently small (depending on $\\\\kappa $ ).\",\n \"We now use Theorem REF to control $K^\\\\star _{j+1}$ .\",\n \"Since $A \\\\gg L$ and taking $\\\\lambda _0$ sufficiently small we obtain $\\\\Vert K^\\\\star _{j+1}\\\\Vert _{j+1}&\\\\le \\\\frac{1}{2} L^{-\\\\kappa }\\\\Vert K^\\\\star _j\\\\Vert _j +O_L(1) C_j\\\\lambda _0 b_0 L^{-\\\\alpha j}& \\\\le C_jb_{0}\\\\lambda _{0} (\\\\frac{1}{2} L^{-\\\\kappa (j+1)} +\\\\frac{1}{2} O_L(1)L^{-(\\\\alpha /2)(j-1)}L^{-(\\\\alpha /2)(j+1)})&\\\\le C_jb_{0}\\\\lambda _{0} (\\\\frac{1}{2} +\\\\frac{1}{2} O_L(1)L^{-(\\\\alpha /2)(j-1)}) L^{-\\\\kappa (j+1)}$ since $\\\\alpha >2\\\\kappa $ .\",\n \"Now there is $j_0\\\\in \\\\mathbb {N}$ so that for all $j\\\\ge j_0$ , the term $O_L(1)L^{-(\\\\alpha /2)(j-1)}$ on the right-hand side is bounded by 1.\",\n \"Thus choosing $C=C_{j_0}$ , by induction for all $j\\\\ge j_0$ , $\\\\Vert K^\\\\star _j\\\\Vert _{j} \\\\le C b_0\\\\lambda _0L^{-\\\\kappa j}$ and $\\\\Vert V^\\\\star _{j}\\\\Vert _j \\\\le C\\\\lambda _0$ and the claim follows.\"\n ],\n [\n \"Computation of pointwise correlation functions\",\n \"In this section we use the results of Section to prove the following estimates for the pointwise correlation functions ${\\\\bar{\\\\psi }_a\\\\psi _b}$ , ${\\\\bar{\\\\psi }_a\\\\psi _a}$ , and ${\\\\bar{\\\\psi }_a\\\\psi _a\\\\bar{\\\\psi }_b\\\\psi _b}$ .\",\n \"Recall the definition (REF ) of $W_{N}(x)=W_{N,m^{2}}(x)$ .\",\n \"For $b_0$ sufficiently small and $m^2 \\\\ge 0$ , there exists continuous functions $\\\\lambda =\\\\lambda (b_0,m^2)= 1+O_L(b_0), \\\\qquad \\\\gamma = \\\\gamma (b_0,m^2) = (-\\\\Delta ^{^d}+m^2)^{-1}(0,0) + O_L(b_0),$ such that if $V^\\\\varnothing _{0}=V_{0}^{c}(m^{2},b_{0})$ is as in Theorem REF , $V^\\\\star _{0}$ is as in Proposition REF , and $\\\\tilde{u}_{N,N}^c=\\\\tilde{u}_{N,N}^c(b_0,m^2)$ is as in Proposition with initial condition $V^\\\\varnothing _0=V_0^c$ , $ {\\\\bar{\\\\psi }_a\\\\psi _a} =\\\\gamma + \\\\frac{\\\\gamma t_N |\\\\Lambda _N|^{-1}+ O_L(b_0L^{-(d-2+\\\\kappa )N}) + O_L(b_0L^{-\\\\kappa N}(m^{2}|\\\\Lambda _N|)^{-1})}{1+\\\\tilde{u}_{N,N}}.$ Under the same assumptions as in Proposition , ${\\\\bar{\\\\psi }_a\\\\psi _b}&=W_N(a-b)+ \\\\frac{t_N|\\\\Lambda _N|^{-1}}{1+\\\\tilde{u}_{N,N}}&\\\\qquad + O_L(b_0|a-b|^{-(d-2+\\\\kappa )})+\\\\frac{O_L(b_0|a-b|^{-\\\\kappa } (m^2|\\\\Lambda _N|)^{-1})}{1+\\\\tilde{u}_{N,N}}\\\\\\\\{\\\\bar{\\\\psi }_a\\\\psi _a\\\\bar{\\\\psi }_b\\\\psi _b}&= -\\\\lambda ^2 W_N(a-b)^2 + \\\\gamma ^2+\\\\frac{-2\\\\lambda ^2 W_N(a-b)+ 2\\\\lambda \\\\gamma }{1+\\\\tilde{u}_{N,N}}t_N|\\\\Lambda _N|^{-1}&\\\\qquad + O_L(b_0 |a-b|^{-2(d-2)-\\\\kappa })+ O_L(b_0L^{-(d-2+\\\\kappa )N})&\\\\qquad + (O_L(b_0L^{-\\\\kappa N}) + O_L(b_0 |a-b|^{-(d-2+\\\\kappa )}))\\\\frac{(m^{2}|\\\\Lambda _N|)^{-1}}{1+\\\\tilde{u}_{N,N}}.$ Throughout this section, we assume that the renormalisation group flow $(V_j,K_j)_{j \\\\le N}$ is given as in Corollary REF (bulk) and Proposition REF (observables).\"\n ],\n [\n \"Integration of the zero mode\",\n \"As in the analysis of the susceptibility in Section , we treat the final integration over the zero mode explicitly.\",\n \"Again we will only require the restriction to constant $\\\\psi ,\\\\bar{\\\\psi }$ (as discussed below (REF )) of $_{C} \\\\theta Z_0= _{t_N Q_N} \\\\theta Z_N = e^{-u^\\\\varnothing _N|\\\\Lambda _N|} \\\\tilde{Z}_{N,N},$ where the last equation defines $\\\\tilde{Z}_{N,N}$ .\",\n \"We write $\\\\tilde{Z}_{N,N}=\\\\tilde{Z}_{N,N}^{\\\\varnothing }+\\\\tilde{Z}_{N,N}^{\\\\star }$ for its decomposition into bulk and observable parts.\",\n \"The bulk term was already computed in Proposition .\",\n \"The observable term $\\\\tilde{Z}_{N,N}^\\\\star $ is computed by the next lemma; in the lemma we only give explicit formulas for the terms that will be used in the proofs of Propositions and .\",\n \"Restricted to constant $\\\\psi ,\\\\bar{\\\\psi }$ , in Case (1), $\\\\tilde{Z}_{N,N}^\\\\star &=\\\\sigma _a\\\\bar{\\\\psi }\\\\tilde{Z}_{N,N}^{\\\\sigma _a\\\\bar{\\\\psi }} + \\\\bar{\\\\sigma }_b\\\\psi \\\\tilde{Z}_{N,N}^{\\\\sigma _b\\\\psi }+\\\\sigma _a\\\\bar{\\\\sigma }_b \\\\tilde{Z}_{N,N}^{\\\\bar{\\\\sigma }_b\\\\sigma _a}+\\\\sigma _a\\\\bar{\\\\sigma }_b \\\\psi \\\\bar{\\\\psi }\\\\tilde{Z}_{N,N}^{\\\\bar{\\\\sigma }_b\\\\sigma _a\\\\psi \\\\bar{\\\\psi }}\\\\multicolumn{2}{l}{\\\\text{where}}\\\\\\\\\\\\tilde{Z}_{N,N}^{\\\\sigma _a\\\\bar{\\\\psi }}&= \\\\lambda _{a,N} +O_L(\\\\ell _{x,N}^{-1}\\\\ell _N^{-1}\\\\Vert K^\\\\star _N\\\\Vert _N)\\\\\\\\\\\\tilde{Z}_{N,N}^{\\\\bar{\\\\sigma }_b\\\\psi }&= \\\\lambda _{b,N} +O_L(\\\\ell _{x,N}^{-1}\\\\ell _N^{-1}\\\\Vert K^\\\\star _N\\\\Vert _N)\\\\\\\\\\\\tilde{Z}_{N,N}^{\\\\bar{\\\\sigma }_b\\\\sigma _a}&= q_N (1+\\\\tilde{u}_{N,N})+ \\\\lambda _{a,N}\\\\lambda _{b,N} t_N|\\\\Lambda _N|^{-1} { + r_N t_N |\\\\Lambda _N|^{-1}}&\\\\qquad \\\\qquad + O_L(m^{-2}|\\\\Lambda _N|^{-1}\\\\ell _N^{-2}\\\\ell _{ab,N}^{-1})\\\\Vert K^\\\\star _N\\\\Vert _N.$ In Case (2), $\\\\tilde{Z}_{N,N}^\\\\star &=\\\\sigma _a\\\\tilde{Z}_{N,N}^{\\\\sigma _a}+ \\\\sigma _a\\\\bar{\\\\psi }\\\\psi \\\\tilde{Z}_{N,N}^{\\\\sigma _a\\\\bar{\\\\psi }\\\\psi }+\\\\sigma _b \\\\tilde{Z}_{N,N}^{\\\\sigma _b} + \\\\sigma _b\\\\bar{\\\\psi }\\\\psi \\\\tilde{Z}_{N,N}^{\\\\sigma _b\\\\bar{\\\\psi }\\\\psi }+ \\\\sigma _a\\\\sigma _b \\\\tilde{Z}_{N,N}^{\\\\sigma _a\\\\sigma _b}+ \\\\sigma _a\\\\sigma _b \\\\psi \\\\bar{\\\\psi }\\\\tilde{Z}_{N,N}^{\\\\sigma _a\\\\sigma _b\\\\psi \\\\bar{\\\\psi }}\\\\multicolumn{2}{l}{\\\\text{where, setting $\\\\tilde{\\\\lambda }_{x,N,N} = \\\\lambda _{x,N}+O_L(\\\\ell _{x,N}^{-1} \\\\ell _N^{-2} \\\\Vert K^\\\\star _N\\\\Vert _N)$,}}\\\\\\\\\\\\tilde{Z}_{N,N}^{\\\\sigma _x}&=\\\\gamma _{x,N} (1+\\\\tilde{u}_{N,N}) +\\\\tilde{\\\\lambda }_{x,N,N}t_N|\\\\Lambda _N|^{-1}+ O_L(\\\\ell _{x,N}^{-1})\\\\Vert K^\\\\star _N\\\\Vert _N.\\\\\\\\\\\\tilde{Z}_{N,N}^{\\\\sigma _a\\\\sigma _b}&= (q_N+\\\\gamma _{a,N}\\\\gamma _{b,N}) (1+\\\\tilde{u}_{N,N})+ (\\\\eta _N+r_N +\\\\tilde{\\\\lambda }_{a,N,N}\\\\gamma _{b,N}+\\\\tilde{\\\\lambda }_{b,N,N}\\\\gamma _{a,N})t_N|\\\\Lambda _N|^{-1}&\\\\qquad \\\\qquad + O_L((|\\\\gamma _{a,N}|+|\\\\gamma _{b,N}|)\\\\ell _{x,N}^{-1}+ \\\\ell _{ab,N}^{-1}+ m^{-2}|\\\\Lambda _N|^{-1}\\\\ell _N^{-2}\\\\ell _{ab,N}^{-1})\\\\Vert K^\\\\star _N\\\\Vert _N.$ The error bounds above reveal the tension in the explicit choices of $\\\\ell _{x,j}^{-1}$ and $\\\\ell _{ab,j}^{-1}$ .\",\n \"To obtain effective error estimates, we want $\\\\ell _{x,N}^{-1}$ and $\\\\ell _{ab,N}^{-1}$ to be as small as possible.\",\n \"On the other hand, to control the iterative estimates of Theorem REF over the entire trajectory, i.e., to prove Proposition REF , we cannot have $\\\\ell _{x,j}$ and $\\\\ell _{ab,j}$ too large.\",\n \"In particular, either of the more naive choices $\\\\ell _{ab,j}=\\\\ell _{x, j}^{2}$ and $\\\\ell _{ab,j} = \\\\ell _{x,j_{ab}}^{2}$ in Case (2) would lead to difficulties, both in terms of forcing us to track additional terms in the flow and in terms of controlling norms inductively, or to error bounds that are not strong enough to capture the zero mode sufficiently accurately.\",\n \"Throughout the proof, we restrict to constant $\\\\psi ,\\\\bar{\\\\psi }$ .\",\n \"Since ${e^{+u_N^\\\\varnothing (\\\\Lambda )}Z_N}^\\\\star &= {e^{-u_N^\\\\star (\\\\Lambda )}(e^{-V_N(\\\\Lambda )}+K_N(\\\\Lambda ))}^\\\\star &=(e^{-u_N^\\\\star (\\\\Lambda )}-1)(e^{-V^\\\\varnothing _N(\\\\Lambda )}+K^\\\\varnothing _N(\\\\Lambda ))+ e^{-u_N^\\\\star (\\\\Lambda )}(e^{-V_N(\\\\Lambda )}+K_N(\\\\Lambda ))^\\\\star ,$ by applying $_{t_NQ_N}\\\\theta $ we obtain $\\\\tilde{Z}_{N,N}^\\\\star = \\\\underbrace{(e^{-u^\\\\star _N(\\\\Lambda )}-1)(1+\\\\tilde{u}_{N,N}-|\\\\Lambda _N|\\\\tilde{a}_{N,N}\\\\psi \\\\bar{\\\\psi })}_{A}+ \\\\underbrace{e^{-u_N^\\\\star (\\\\Lambda )}_{t_NQ_N}\\\\theta (e^{-V_N(\\\\Lambda )}+K_N(\\\\Lambda ))^\\\\star }_{B}.$ In obtaining $A$ we used (REF ) which gives $_{t_NQ_N}\\\\theta (e^{-V_N^\\\\varnothing (\\\\Lambda )}+K_N^\\\\varnothing (\\\\Lambda ))= 1+\\\\tilde{u}_{N,N}-|\\\\Lambda _N|\\\\tilde{a}_{N,N}\\\\psi \\\\bar{\\\\psi }$ .\",\n \"Since each term in $V_N^\\\\varnothing (\\\\Lambda )$ contains a factor $\\\\psi \\\\bar{\\\\psi }$ and each term in $V_N^\\\\star (\\\\Lambda )$ either $\\\\psi $ or $\\\\bar{\\\\psi }$ , we have $V_N^\\\\varnothing (\\\\Lambda )V_N^\\\\star (\\\\Lambda )=0$ .\",\n \"Thus $B = e^{-u_N^\\\\star (\\\\Lambda )}_{t_NQ_N}\\\\theta (-V_N^\\\\star (\\\\Lambda )+\\\\frac{1}{2} V_N^\\\\star (\\\\Lambda )^2+K_N^\\\\star (\\\\Lambda )).$ Case (1).\",\n \"Since $\\\\sigma _a^2=\\\\bar{\\\\sigma }_b^2=0$ , $e^{-u_N^\\\\star (\\\\Lambda )}-1=-u^\\\\star _N(\\\\Lambda ) = \\\\sigma _a\\\\bar{\\\\sigma }_b q_N,$ we get $A&=\\\\sigma _a\\\\bar{\\\\sigma }_b q_N (1+\\\\tilde{u}_{N,N}-|\\\\Lambda _N|\\\\tilde{a}_{N,N}\\\\psi \\\\bar{\\\\psi })\\\\\\\\B &=\\\\sigma _a\\\\bar{\\\\psi }(\\\\lambda _{a,N}+k^{\\\\sigma _a\\\\bar{\\\\psi }}_{N})+{\\\\psi \\\\bar{\\\\sigma }_{b}}(\\\\lambda _{b,N}+k^{\\\\bar{\\\\sigma }_b\\\\psi }_{N})+\\\\sigma _a\\\\bar{\\\\sigma }_b _{t_NQ_N}\\\\theta \\\\bar{\\\\psi }\\\\psi (\\\\lambda _{a,N}\\\\lambda _{b,N} - r_N + k^{\\\\sigma _a\\\\bar{\\\\sigma }_b \\\\bar{\\\\psi }\\\\psi }_{N}).$ The constants $k_N^\\\\#$ are given in terms of derivatives of $K_N(\\\\Lambda )$ and bounded analogously as in (REF ).\",\n \"For example, $k_N^{\\\\sigma _a\\\\bar{\\\\psi }} = O_L(\\\\ell _{a,N}^{-1}\\\\ell _N^{-1}\\\\Vert K^\\\\star _N\\\\Vert _N)$ , and similarly for the other $k_N^\\\\#$ terms, the rule being that we have a factor $\\\\ell _{x,N}^{-1}$ if there is a superscript $\\\\sigma _a$ or $\\\\bar{\\\\sigma }_b$ but not both, a factor $\\\\ell _{ab,N}^{-1}$ for $\\\\sigma _a\\\\bar{\\\\sigma }_b$ and a factor $\\\\ell _N^{-1}$ for each superscript $\\\\psi $ or $\\\\bar{\\\\psi }$ .\",\n \"These bounds follow from the definition of the $T_j(\\\\ell _j)$ norm.\",\n \"Since $_{t_NQ_N}\\\\theta \\\\psi \\\\bar{\\\\psi }= -t_N|\\\\Lambda _N|^{-1}+\\\\psi \\\\bar{\\\\psi }$ the claim follows by collecting terms and using (REF ).\",\n \"Case (2).\",\n \"Using again that $\\\\sigma _a^2=\\\\sigma _b^2=0$ , but now taking in account that $u^\\\\star (\\\\Lambda )$ has additional terms compared to Case (1), $e^{-u_N^\\\\star (\\\\Lambda )}-1=-u^\\\\star _N(\\\\Lambda )+\\\\frac{1}{2} u^\\\\star _N(\\\\Lambda )^2 = \\\\sigma _a\\\\sigma _b (q_N+\\\\gamma _{a,N}\\\\gamma _{b,N}) + \\\\sigma _a \\\\gamma _{a,N} +\\\\sigma _b\\\\gamma _{b,N},$ and therefore $A =(\\\\sigma _a\\\\sigma _b(q_N+\\\\gamma _{a,N}\\\\gamma _{b,N})+\\\\sigma _a \\\\gamma _{a,N}+\\\\sigma _b\\\\gamma _{b,N})(1 + \\\\tilde{u}_{N,N} - |\\\\Lambda _N|\\\\tilde{a}_{N,N} \\\\psi \\\\bar{\\\\psi }).$ Since in Case (2) each term in $V^\\\\star _N(\\\\Lambda )$ contains a factor of $\\\\psi \\\\bar{\\\\psi }$ , we have $V^\\\\star _N(\\\\Lambda )^2=0$ and thus $B = (1-u_N^\\\\star (\\\\Lambda ))_{t_NQ_N}\\\\theta (-V^\\\\star _N(\\\\Lambda )+K^\\\\star _N(\\\\Lambda ))$ Therefore $B&= \\\\sigma _a k_N^{\\\\sigma _a}+\\\\sigma _b k_N^{\\\\sigma _b}+\\\\sigma _a\\\\sigma _b (\\\\gamma _{a,N} k_N^{\\\\sigma _b} + \\\\gamma _{b,N} k_N^{\\\\sigma _a} + k_N^{\\\\sigma _a\\\\sigma _b})&\\\\qquad +_{t_NQ_N}[\\\\sigma _a\\\\bar{\\\\psi }\\\\psi \\\\tilde{\\\\lambda }_{a,N}+ \\\\sigma _a \\\\sigma _b\\\\bar{\\\\psi }\\\\psi (\\\\eta _N+r_N +\\\\gamma _{a,N}\\\\tilde{\\\\lambda }_{b,N}+\\\\gamma _{b,N}\\\\tilde{\\\\lambda }_{a,N} + k^{\\\\sigma _a\\\\sigma _b\\\\bar{\\\\psi }\\\\psi })]$ where we have set $\\\\tilde{\\\\lambda }_{x,N} = \\\\lambda _{x,N} + k_N^{\\\\sigma _x\\\\bar{\\\\psi }\\\\psi }$ .\",\n \"Taking the expectation and collecting all terms gives $\\\\tilde{Z}_{N,N}^{\\\\sigma _a\\\\sigma _b}&= (q_N+\\\\gamma _{a,N}\\\\gamma _{b,N}) (1+\\\\tilde{u}_{N,N})&\\\\qquad \\\\qquad + (\\\\eta _N+r_N +\\\\tilde{\\\\lambda }_{a,N}\\\\gamma _{b,N}+\\\\tilde{\\\\lambda }_{b,N}\\\\gamma _{a,N} + k^{\\\\sigma _a\\\\sigma _b\\\\bar{\\\\psi }\\\\psi })t_N|\\\\Lambda _N|^{-1}&\\\\qquad \\\\qquad +\\\\gamma _{a,N} k_N^{\\\\sigma _b} + \\\\gamma _{b,N} k_N^{\\\\sigma _a} + k_N^{\\\\sigma _a\\\\sigma _b}\\\\\\\\\\\\tilde{Z}_{N,N}^{\\\\sigma _a}&=\\\\gamma _{a,N} (1+\\\\tilde{u}_{N,N}) + \\\\tilde{\\\\lambda }_{a,N}t_N|\\\\Lambda _N|^{-1}+ k_N^{\\\\sigma _a}.$ The bounds on the constants $k_N^\\\\#$ are analogous to those in Case (1).\"\n ],\n [\n \"Analysis of one-point functions\",\n \"We now analyse the observable flow given by Lemma REF to derive the asymptotics of the correlation functions.\",\n \"Note that the coupling constants $\\\\lambda _{x,j}$ and $\\\\gamma _{x,j}$ can possibly depend on $x=a,b$ as the contributions from $K$ can depend on the relative position of the points in the division of $\\\\Lambda _N$ into blocks.\",\n \"The following lemma shows that in the limit $j\\\\rightarrow \\\\infty $ they become independent of $x$ ; an analogous argument was used in [13].\",\n \"Under the hypotheses of Proposition there are $\\\\lambda _\\\\infty ^{(p)} = \\\\lambda _0^{(p)} +O_L(\\\\lambda _0^{(p)}b_0)$ and $\\\\gamma _\\\\infty = O_L(\\\\lambda _0^{(2)})$ , all continuous in $m^{2}\\\\ge 0$ and $b_{0}$ small, such that for $x\\\\in \\\\lbrace a,b\\\\rbrace $ , $ \\\\lambda _{x,j}^{(p)} = \\\\lambda _\\\\infty ^{(p)} + O_L(\\\\lambda _0^{(p)} b_0 L^{-\\\\kappa j}),\\\\qquad \\\\gamma _{x,j}=\\\\gamma _\\\\infty + O_L(\\\\lambda _0^{(2)} b_0 L^{-(d-2+\\\\kappa )j}).$ In Case (1), $\\\\lambda _\\\\infty ^{(1)} = \\\\lambda _0^{(1)}$ .\",\n \"In Case (2), $\\\\lambda _\\\\infty ^{(2)} = \\\\lambda _0^{(2)} + O_L(\\\\lambda _0^{(2)}b_0)$ and $\\\\gamma _\\\\infty ^{(2)} =\\\\lambda _\\\\infty ^{(2)}(-\\\\Delta ^{^d}+m^2)^{-1}(0,0)+O_L(\\\\lambda _0^{(2)}b_0)$ , and with the abbreviations $\\\\lambda \\\\lambda ^{(2)}$ and $\\\\gamma \\\\gamma ^{(2)}$ , $ {\\\\bar{\\\\psi }_a\\\\psi _a} =\\\\frac{\\\\gamma _\\\\infty }{\\\\lambda _0}+ \\\\frac{\\\\frac{\\\\lambda _{\\\\infty }}{\\\\lambda _0} t_N |\\\\Lambda _N|^{-1}+ O_L(b_0L^{-(d-2+\\\\kappa )N}) + O_L(b_0L^{-\\\\kappa N}(m^{2}|\\\\Lambda _N|)^{-1})}{1+\\\\tilde{u}_{N,N}}.$ We will typically drop the superscript $(p)$ .\",\n \"In both cases, we have already seen that $\\\\lambda _{x,j} = \\\\lambda _0 + \\\\sum _{k=0}^{j-1} O_L(\\\\Vert K^\\\\star _k\\\\Vert _k) = \\\\lambda _0 + \\\\sum _{k=0}^{j-1} O_L(\\\\lambda _0b_0L^{-\\\\kappa k}) .$ Since the $K^\\\\star _k$ are independent of $N$ for $k0$ fixed, $\\\\partial _{\\\\bar{\\\\psi }} \\\\partial _{\\\\sigma _a} \\\\tilde{Z}_{N,N}|_{0}= \\\\lambda _{a,N} + O_L(\\\\ell _N^{-1}\\\\ell _{x,N}^{-1} \\\\Vert K^\\\\star _N\\\\Vert _N)= \\\\lambda _{a,N} + O_L(\\\\Vert K^\\\\star _N\\\\Vert _N)\\\\xrightarrow[N\\\\rightarrow \\\\infty ]{}\\\\lambda _{a,\\\\infty },$ where $|_0$ denotes projection onto the degree 0 part, i.e., $\\\\psi =\\\\bar{\\\\psi }=\\\\sigma =\\\\bar{\\\\sigma }=0$ .\",\n \"On the other hand, we claim $\\\\partial _{\\\\bar{\\\\psi }} \\\\partial _{\\\\sigma _a} \\\\tilde{Z}_{N,N}|_{0}= \\\\lambda _{0,a}m^2(1+\\\\tilde{u}_{N,N})\\\\sum _{x\\\\in \\\\Lambda _N}{\\\\bar{\\\\psi }_0\\\\psi _x}= \\\\lambda _{0,a}{1+\\\\tilde{u}_{N,N}-\\\\frac{\\\\tilde{a}_{N,N}}{m^2}}.$ Indeed, the first equality in (REF ) follows analogously to [13]: let $\\\\Gamma (\\\\rho ,\\\\bar{\\\\rho })$ be as in (REF ), except that $Z_0$ now includes the observable terms $\\\\sigma _a$ and $\\\\bar{\\\\sigma }_b$ and we write $\\\\rho $ and $\\\\bar{\\\\rho }$ for the constant external field to distinguish them from $\\\\sigma _a$ and $\\\\bar{\\\\sigma }_b$ .\",\n \"Then as in (REF ) $-\\\\sum _{x\\\\in \\\\Lambda _N} {\\\\psi _x}_{\\\\sigma _a,\\\\bar{\\\\sigma }_b} =\\\\partial _{\\\\bar{\\\\rho }} \\\\Gamma (\\\\rho ,\\\\bar{\\\\rho })|_{\\\\rho =\\\\bar{\\\\rho }=0} = m^{-2} \\\\frac{\\\\partial _{\\\\bar{\\\\psi }}\\\\tilde{Z}_{N,N}|_{\\\\psi =\\\\bar{\\\\psi }=0}}{\\\\tilde{Z}_{N,N}|_{\\\\psi =\\\\bar{\\\\psi }=0}}$ and ${\\\\cdot }_{\\\\sigma _a,\\\\bar{\\\\sigma }_b}$ denotes the expectation that still depends on the observable fields $\\\\sigma _a$ and $\\\\bar{\\\\sigma }_b$ .\",\n \"Differentiating with respect to $\\\\sigma _a$ and setting $\\\\bar{\\\\sigma }_b=0$ gives $\\\\lambda _{0,a}\\\\sum _{x\\\\in \\\\Lambda _N} {\\\\bar{\\\\psi }_a\\\\psi _x}= -m^{-2} \\\\frac{\\\\partial _{\\\\sigma _a}\\\\partial _{\\\\bar{\\\\psi }}\\\\tilde{Z}_{N,N}|_{0}}{\\\\tilde{Z}_{N,N}|_0}= m^{-2} \\\\frac{\\\\partial _{\\\\bar{\\\\psi }}\\\\partial _{\\\\sigma _a}\\\\tilde{Z}_{N,N}|_{0}}{1+\\\\tilde{u}_{N,N}}$ which is the first equality of (REF ) upon rearranging.\",\n \"The second equality in (REF ) follows from Proposition .\",\n \"The right-hand side of (REF ) converges to $\\\\lambda _0$ in the limit $N\\\\rightarrow \\\\infty $ with $m^2>0$ fixed since $\\\\tilde{a}_{N,N}= a_N - k_N^2/|\\\\Lambda _N| =O_L(L^{-2N}\\\\Vert V_N\\\\Vert _N)+O_L(L^{-2N}\\\\Vert K_N\\\\Vert _N) \\\\rightarrow 0$ and $\\\\tilde{u}_{N,N}= k_N^0 + \\\\tilde{a}_{N,N}t_N = O(\\\\Vert K_N\\\\Vert _N) + \\\\tilde{a}_{N,N}t_N \\\\rightarrow 0$ when $m^2>0$ is fixed.\",\n \"Since the left-hand sides of (REF )–(REF ) are equal, we conclude that $\\\\lambda _{a,\\\\infty }=\\\\lambda _0$ when $m^2>0$ .\",\n \"By continuity this identity then extends to $m^2=0$ .\",\n \"In Case (2), to show (REF ), we use (REF ), that Proposition REF implies $\\\\Vert K^\\\\star _N\\\\Vert _N = O_L(\\\\lambda _0b_0L^{-\\\\kappa N})$ , and $\\\\ell _{x,N}^{-1}\\\\ell _N^{-2} = 1$ and $\\\\ell _{x,N}^{-1}= \\\\ell _N^{2} = O_L(L^{-(d-2)N})$ to obtain $\\\\frac{\\\\tilde{Z}_{N,N}^{\\\\sigma _a}}{1+\\\\tilde{u}_{N,N}}= \\\\gamma _{a,N} +\\\\frac{\\\\lambda _{a,\\\\infty } t_N |\\\\Lambda _N|^{-1} + O_L(\\\\lambda _0 b_0L^{-(d-2+\\\\kappa )N}) + O_L( \\\\lambda _0b_0 L^{-\\\\kappa N} (m^{2}|\\\\Lambda _N|)^{-1})}{1+\\\\tilde{u}_{N,N}}.$ Since ${\\\\bar{\\\\psi }_a\\\\psi _a} =\\\\tilde{Z}_{N,N}^{\\\\sigma _a}/(\\\\lambda _0 (1+\\\\tilde{u}_{N,N}))$ this gives (REF ).\",\n \"In particular, by the translation invariance of ${\\\\bar{\\\\psi }_a\\\\psi _a}$ , taking $N\\\\rightarrow \\\\infty $ implies both $\\\\gamma _{a,\\\\infty }$ and $\\\\lambda _{a,\\\\infty }$ are in fact independent of $a$ .\",\n \"Taking $\\\\lambda _0>0$ small enough, the proposition follows immediately from Lemma REF with $\\\\lambda = \\\\lambda _\\\\infty ^{(2)}/\\\\lambda _0^{(2)}$ and $\\\\gamma = \\\\gamma _\\\\infty ^{(2)}/\\\\lambda _0^{(2)}$ ,\"\n ],\n [\n \"Analysis of two-point functions\",\n \"Next we derive estimates for the two-point functions.\",\n \"Under the hypotheses of Proposition , and in terms of the same $\\\\lambda _\\\\infty $ and $\\\\gamma _\\\\infty $ as in Lemma REF , ${\\\\bar{\\\\psi }_a\\\\psi _b}&=W_N(a-b)+ \\\\frac{t_N|\\\\Lambda _N|^{-1}}{1+\\\\tilde{u}_{N,N}}&\\\\qquad + O_L(\\\\frac{b_0}{\\\\lambda _0}|a-b|^{-(d-2+\\\\kappa )})+O_L(\\\\frac{b_0}{\\\\lambda _0} |a-b|^{-\\\\kappa })\\\\frac{(m^2|\\\\Lambda _N|)^{-1}}{1+\\\\tilde{u}_{N,N}}\\\\\\\\{\\\\bar{\\\\psi }_a\\\\psi _a\\\\bar{\\\\psi }_b\\\\psi _b}&= -\\\\frac{\\\\lambda _\\\\infty ^2}{\\\\lambda _0^2} W_N(a-b)^2 +\\\\frac{\\\\gamma _\\\\infty ^2}{\\\\lambda _0^2}+\\\\frac{-2\\\\lambda _\\\\infty ^2 W_N(a-b)+ 2\\\\lambda _{\\\\infty }\\\\gamma _{\\\\infty }}{\\\\lambda _0^2(1+\\\\tilde{u}_{N,N})}t_N|\\\\Lambda _N|^{-1}&\\\\qquad + O_L(\\\\frac{b_0}{\\\\lambda _0} |a-b|^{-2(d-2)-\\\\kappa })+ O_L(\\\\frac{b_0}{\\\\lambda _0}L^{-(d-2+\\\\kappa )N})&\\\\qquad +(O_L(\\\\frac{b_0}{\\\\lambda _0}|a-b|^{-(d-2+\\\\kappa )})+O_L(\\\\frac{b_0}{\\\\lambda _0}L^{-\\\\kappa N}))\\\\frac{(m^{2}|\\\\Lambda _N|)^{-1}}{1+\\\\tilde{u}_{N,N}}.$ The proofs of (REF ) and () corresponding to Cases (1) and (2) are again analogous.\",\n \"Case (1).\",\n \"By Lemma REF and Lemmas REF and REF , $ \\\\lambda _{x,j}= \\\\lambda _\\\\infty + O_L( \\\\lambda _0b_0 L^{-\\\\kappa })= \\\\lambda _0 + O_L( \\\\lambda _0b_0 L^{-\\\\kappa }),\\\\qquad r_j=O_L(\\\\lambda _0b_0 |a-b|^{-\\\\kappa })1_{j\\\\ge j_{ab}}.$ Using that $\\\\ell _{ab,j}^{-1}\\\\Vert K^{ab}_j\\\\Vert _j \\\\le O_L(\\\\lambda _0b_0L^{-(d-2+\\\\kappa )j})1_{j\\\\ge j_{ab}}$ and $C_{j+1}(a,b) \\\\le C_{j+1}(0,0) \\\\le O_L(L^{-(d-2)j})$ it then follows from Lemma REF that $q_{N}&= \\\\sum _{j=j_{ab}-1}^N { \\\\lambda _{a,j}\\\\lambda _{b,j} C_{j+1}(a,b) + r_j C_{j+1}(0,0) + O_L(\\\\lambda _0 b_0 L^{-(d-2+\\\\kappa )j}) }&= \\\\lambda _0^2 \\\\sum _{j=1}^N C_j(a,b)+ O_L(\\\\lambda _0 b_0 |a-b|^{-(d-2)-\\\\kappa })&= \\\\lambda _0^2 W_N(a-b) + O_L(\\\\lambda _0 b_0 |a-b|^{-(d-2)-\\\\kappa }),$ where we have used (REF ), $|a-b|\\\\le L$ , that $C_j(a,b)=0$ for $j0$ and suppose that for all $x$ , $h_x= h$ .\",\n \"Then there are $c,C>0$ depending on $d,\\\\beta ,h$ such that $¶^{\\\\Lambda _{N}}_{\\\\beta ,h}[0\\\\leftrightarrow x,0\\\\mathfrak {g}] \\\\le Ce^{-c|x|},\\\\qquad ¶^{\\\\Lambda _{N}}_{\\\\beta ,h}[0\\\\leftrightarrow x,0\\\\lnot \\\\leftrightarrow \\\\mathfrak {g}] \\\\le Ce^{-c|x|}$ We begin with the inequality on the left of (REF ).\",\n \"Define $(0\\\\leftrightarrow x)$ to be the set of forests in which both 0 is connected to $x$ and $T_{0}$ is rooted at 0, and $$ the set of all forests.\",\n \"In this argument we treat $$ as being a set of (possibly) rooted forests, i.e., we identify edges to $\\\\mathfrak {g}$ with roots.\",\n \"Without loss of generality we may assume $x\\\\cdot e_{1} \\\\ge \\\\alpha |x|$ for a fixed $\\\\alpha >0$ .\",\n \"Note that if $F\\\\in (0\\\\leftrightarrow x)$ there is a unique path $\\\\gamma _{F}$ from 0 to $x$ in $F$ , and there are at least $\\\\alpha |x|$ edges of the form $\\\\lbrace u,u+e_{1}\\\\rbrace $ in $\\\\gamma _{F}$ .\",\n \"We define a map $S\\\\colon (0\\\\leftrightarrow x) \\\\rightarrow 2^{}$ by, for $F\\\\in (0\\\\leftrightarrow x)$ , choosing a subset $\\\\lbrace u_{i},v_{i}\\\\rbrace $ of the edges $\\\\lbrace \\\\lbrace u,v\\\\rbrace \\\\in \\\\gamma _{F} \\\\mid v=u+e_{1}\\\\rbrace $ , and removing each $\\\\lbrace u_{i},v_{i}\\\\rbrace $ and rooting the tree containing $v_{i}$ at $v_{i}$ .\",\n \"Thus $S(F)$ is the set of forests that results from all possible choices in the first step.\",\n \"The second step does yield an element of $2^{}$ since $T_{0}$ is rooted at 0, so it cannot be the case that the tree containing $v_{i}$ is already rooted (connected to $\\\\mathfrak {g}$ ).\",\n \"The map $S$ is injective, meaning that given $\\\\bar{F}\\\\in \\\\bigcup _{F\\\\in (0\\\\leftrightarrow x)}S(F)$ there is a unique $F$ such that $\\\\bar{F} \\\\in S(F)$ .\",\n \"Indeed, given $\\\\bar{F}\\\\in S(F)$ , $F$ can be reconstructed as follows.\",\n \"In $\\\\bar{F}$ , either the tree containing $x$ contains 0, or else it is rooted at a unique vertex $v^{\\\\prime }$ and it is not connected to $u^{\\\\prime }=v^{\\\\prime }-e_{1}$ .\",\n \"Set $\\\\bar{F}^{\\\\prime }=\\\\bar{F}\\\\cup \\\\lbrace u^{\\\\prime },v^{\\\\prime }\\\\rbrace $ .\",\n \"The previous sentence applies to $\\\\bar{F}^{\\\\prime }$ as well, and continuing until a connection to 0 is formed we recover $F$ .\",\n \"This reconstruction was independent of $F$ , and hence if $\\\\bar{F}_{1}=\\\\bar{F}_{2}$ , $\\\\bar{F}_{i}\\\\in S(F_{i})$ , we have $F_{1}=F_{2}$ .\",\n \"Let $w(F)= h \\\\beta ^{F}\\\\prod _{T\\\\ne T_0}(1+h|V(T)|)$ .\",\n \"Then for $\\\\bar{F} \\\\in S(F)$ , $w(\\\\bar{F}) = w(F) (\\\\frac{h}{\\\\beta })^{k}$ if $\\\\bar{F}$ had $k$ edges removed.\",\n \"Hence if the connection from 0 to $x$ in $F$ has $k$ edges of the form $\\\\lbrace u,v\\\\rbrace $ , $v=u+e_{1}$ , $\\\\sum _{\\\\bar{F}\\\\in S(F)}w(\\\\bar{F}) = (1+\\\\frac{h}{\\\\beta })^{k}w(F).$ Let $_{k}(x) \\\\subset (0\\\\leftrightarrow x)$ be the set of forests where the connection from 0 to $x$ contains $k$ edges of the form $\\\\lbrace u,v\\\\rbrace $ , $v=u+e_{1}$ .\",\n \"We have the lower bound $Z_{\\\\beta ,h}^{\\\\Lambda _{N}} = \\\\sum _{F\\\\in } \\\\beta ^{F}\\\\prod _{T\\\\in F}(1+h|V(T)|) \\\\ge \\\\sum _{k\\\\ge 0} \\\\sum _{F\\\\in _{k}(x)} \\\\sum _{\\\\bar{F}\\\\in S(F)} w(\\\\bar{F})$ since $S$ is injective and all of the summands are non-negative.\",\n \"Hence we obtain, using (REF ), $¶^{\\\\Lambda _{N}}_{\\\\beta ,h}[0\\\\leftrightarrow x,0\\\\mathfrak {g}]&\\\\le \\\\frac{\\\\sum _{k\\\\ge \\\\alpha |x|} \\\\sum _{F\\\\in _{k}(x)}w(F)}{\\\\sum _{k\\\\ge 0} \\\\sum _{F\\\\in _{k}(x)} \\\\sum _{\\\\bar{F}\\\\in S(F)} w(\\\\bar{F})} &= \\\\frac{\\\\sum _{k\\\\ge \\\\alpha |x|} \\\\sum _{F\\\\in _{k}(x)}(1+\\\\frac{h}{\\\\beta })^{-k}\\\\sum _{\\\\bar{F}\\\\in S(F)}w(\\\\bar{F})}{\\\\sum _{k\\\\ge 0} \\\\sum _{F\\\\in _{k}(x)} \\\\sum _{\\\\bar{F}\\\\in S(F)} w(\\\\bar{F})}\\\\le (1+\\\\frac{h}{\\\\beta })^{-\\\\alpha |x|}.$ A similar argument applies when $0\\\\lnot \\\\leftrightarrow \\\\mathfrak {g}$ ; this condition is used in the second step defining $S$ to ensure the trees containing the vertices $v_{i}$ are not already connected to $\\\\mathfrak {g}$ .\",\n \"In this case the weight $w(F)$ does not have the factor $h$ , but the remainder of the argument is identical.\",\n \"If $h>0$ , then there are $c,C>0$ depending on $d,\\\\beta ,h$ such that Let $h>0$ and suppose that for all $x$ , $h_x= h$ .\",\n \"Then there are $c,C>0$ depending on $d,\\\\beta ,h$ such that $¶^{\\\\Lambda _{N}}_{\\\\beta ,h}[0\\\\leftrightarrow x] \\\\le Ce^{-c |x |}.$ Since $¶^{\\\\Lambda _{N}}_{\\\\beta ,h}[0\\\\leftrightarrow x] =¶^{\\\\Lambda _{N}}_{\\\\beta ,h}[0\\\\leftrightarrow x, 0\\\\leftrightarrow \\\\mathfrak {g}] +¶^{\\\\Lambda _{N}}_{\\\\beta ,h}[0\\\\leftrightarrow x, 0\\\\lnot \\\\leftrightarrow \\\\mathfrak {g}],$ it is enough to estimate the first term, as the second is covered by Lemma REF .\",\n \"Note $¶^{\\\\Lambda _{N}}_{\\\\beta ,h}[0\\\\leftrightarrow x, 0\\\\leftrightarrow \\\\mathfrak {g}]=\\\\sum _{y} ¶^{\\\\Lambda _{N}}_{\\\\beta ,h} [1_{0\\\\leftrightarrow x}1_{0\\\\leftrightarrow y}1_{y\\\\mathfrak {g}}] = \\\\sum _{y} ¶^{\\\\Lambda _{N}}_{\\\\beta ,h} [1_{0\\\\leftrightarrow x}1_{0\\\\leftrightarrow y}1_{0\\\\mathfrak {g}}].$ where the first equality follows from the fact that the only one vertex per component may connect to $\\\\mathfrak {g}$ , and the second follows from exchangeability of the choice of root.\",\n \"Examining the rightmost expression, there are most $c_{d} |x |^{d}$ summands in which $|y |\\\\le |x|$ ; for these terms we drop the condition $0\\\\leftrightarrow y$ .\",\n \"For the rest we drop $0\\\\leftrightarrow x$ .\",\n \"This gives, by Lemma REF , $¶^{\\\\Lambda _{N}}_{\\\\beta ,h}[0\\\\leftrightarrow x, 0\\\\leftrightarrow \\\\mathfrak {g}]\\\\le C |x|^{d}e^{-c |x |} + \\\\sum _{ |y |> |x |} Ce^{-c |y |} \\\\le Ce^{-c |x |},$ where $c,C$ are changing from location to location but depend on $d,\\\\beta ,h$ only.\"\n ],\n [\n \"Infinite volume limit\",\n \"We now discuss weak limits $¶^{^d}_{\\\\beta }$ obtained by (i) first taking a (possibly subsequential) infinite-volume weak limit $¶^{^d}_{\\\\beta ,h} = \\\\lim _{N}¶^{\\\\Lambda _N}_{\\\\beta ,h}$ and (ii) subsequently taking a (possibly subsequential) limit $¶^{^d}_{\\\\beta }=\\\\lim _{h\\\\downarrow 0 }¶^{^d}_{\\\\beta ,h}$ .\",\n \"We do not explicitly indicate the convergent subsequence chosen as what follows applies to any fixed choice.\",\n \"Define $\\\\theta _{d,N}(\\\\beta ,h) ¶^{\\\\Lambda _N}_{\\\\beta ,h}[0\\\\leftrightarrow \\\\mathfrak {g}]=1-h^{-1}¶^{\\\\Lambda _N}_{\\\\beta ,h}[0\\\\mathfrak {g}]$ where the second equality is due to (REF ).\",\n \"Since this last display only involves cylinder events, $\\\\lim _{N\\\\rightarrow \\\\infty }\\\\theta _{d,N}(\\\\beta ,h) = 1-h^{-1}¶^{^d}_{\\\\beta ,h}[0\\\\mathfrak {g}] \\\\theta _{d}(\\\\beta ,h),$ where the last equality defines $\\\\theta _{d}(\\\\beta ,h)$ .\",\n \"Assume $\\\\lim _{h\\\\downarrow 0}\\\\theta _{d}(\\\\beta ,h)=\\\\theta _{d}(\\\\beta )$ exists.\",\n \"Then $¶^{^d}_{\\\\beta }[|T_{0}|=\\\\infty ] = \\\\theta _{d}(\\\\beta ).$ Write $¶_{\\\\beta ,h}=¶^{^d}_{\\\\beta ,h}$ .\",\n \"We claim that $¶_{\\\\beta ,h}[0\\\\mathfrak {g}] = \\\\sum _{n\\\\ge 1} ¶_{\\\\beta ,h}[|T_{0}|=n] \\\\frac{h}{1+nh},$ and hence, since $\\\\theta _{d}(\\\\beta ,h) = 1-h^{-1}¶_{\\\\beta ,h}[0\\\\mathfrak {g}]$ , $\\\\theta _{d}(\\\\beta ,h) = 1-\\\\sum _{n\\\\ge 1} ¶_{\\\\beta ,h}[|T_{0}|=n] \\\\frac{1}{1+nh}.$ Granting the claim, by dominated convergence we obtain $¶_{\\\\beta ,0}[|T_{0}|<\\\\infty ] = \\\\sum _{n\\\\ge 1}¶_{\\\\beta ,0}[|T_{0}|=n]=1-\\\\theta _{d}(\\\\beta ),$ as desired.\",\n \"To prove the claim, rewrite it as $¶_{\\\\beta ,h}[|T_{0}|=\\\\infty , 0\\\\mathfrak {g}]=\\\\lim _{r\\\\rightarrow \\\\infty }¶_{\\\\beta ,h}[|T_{0}|\\\\ge r, 0\\\\mathfrak {g}]=0.$ The probabilities inside the limit are probabilities of cylinder events, and hence are limits of finite volume probabilities.\",\n \"For a fixed $r$ the probability is at most $h/(1+rh)$ in finite volume, which vanishes as $r\\\\rightarrow \\\\infty $ .\"\n ],\n [\n \"Finite range decomposition\",\n \"In this appendix, we give the precise references for the construction of the finite range decomposition (REF ).\",\n \"The general method we use was introduced in [11], and presented in the special case we use in [17] and we will use this reference.\",\n \"For $t>0$ , first recall the polynomials $P_t$ from [17] (these polynomials are called $W_t^*$ in [11]).\",\n \"These are polynomials of degree bounded by $t$ satisfying $ \\\\frac{1}{\\\\lambda } = \\\\int _{0}^\\\\infty t^2 P_t(\\\\lambda ) \\\\, \\\\frac{dt}{t},\\\\qquad 0 \\\\le P_t(u) \\\\le O_s(1+t^2u)^{-s}$ for any $s>0$ and $u \\\\in [0,2]$ .\",\n \"Our decomposition (REF ) is defined by $C_1(x,y) &= \\\\frac{1}{(2d+m^2)|\\\\Lambda _N|}\\\\sum _{k \\\\in \\\\Lambda _N^*} e^{ik\\\\cdot (x-y)} \\\\int _{0}^{\\\\frac{1}{2} L} t^2 P_t(\\\\frac{\\\\lambda (k)+m^2}{2d+m^2}) \\\\, \\\\frac{dt}{t}\\\\\\\\C_j(x,y) &= \\\\frac{1}{(2d+m^2)|\\\\Lambda _N|}\\\\sum _{k \\\\in \\\\Lambda _N^*} e^{ik\\\\cdot (x-y)} \\\\int _{\\\\frac{1}{2} L^{j-1}}^{\\\\frac{1}{2} L^j} t^2 P_t(\\\\frac{\\\\lambda (k)+m^2}{2d+m^2}) \\\\, \\\\frac{dt}{t}\\\\\\\\C_{N,N}(x,y) &= \\\\frac{1}{(2d+m^2)|\\\\Lambda _N|}\\\\sum _{k \\\\in \\\\Lambda _N^*} e^{ik\\\\cdot (x-y)} \\\\int _{\\\\frac{1}{2} L^N}^\\\\infty t^2 P_t(\\\\frac{\\\\lambda (k)+m^2}{2d+m^2}) \\\\, \\\\frac{dt}{t},$ where $\\\\lambda (k) = 4\\\\sum _{j=1}^{d}\\\\sin ^{2}(k_{j}/2)$ and $\\\\Lambda _N^* \\\\subset [-\\\\pi ,\\\\pi )^d$ is the dual torus.\",\n \"The estimates for $C_1, \\\\dots , C_{N-1}$ are straightforward from these Fourier representations and can be found in [17].\",\n \"We remark that in [17], the torus covariances are defined by periodisation of the finite range covariances on $^d$ ; by Poisson summation this is equivalent to the above definition.\",\n \"The decomposition of $C_{N,N}$ in (REF ) is defined by removing the zero mode from $C_{N,N}$ : $C_{N}(x,y) &= \\\\frac{1}{(2d+m^2)|\\\\Lambda _N|}\\\\sum _{k \\\\in \\\\Lambda _N^*:k \\\\ne 0} e^{ik\\\\cdot (x-y)} \\\\int _{\\\\frac{1}{2} L^N}^\\\\infty t^2 P_t(\\\\frac{\\\\lambda (k)+m^2}{2d+m^2}) \\\\, \\\\frac{dt}{t}\\\\\\\\t_N &= \\\\frac{1}{2d+m^2} \\\\int _{\\\\frac{1}{2} L^N}^\\\\infty t^2 P_t(\\\\frac{m^2}{2d+m^2}) \\\\, \\\\frac{dt}{t},$ from which (REF ) is immediate.\",\n \"For $C_{N}$ estimates follows as in [17]: $|C_N(x,y)| &\\\\le \\\\frac{1}{|\\\\Lambda _N|} \\\\sum _{k \\\\in \\\\Lambda _N^*: k \\\\ne 0} {\\\\int _{\\\\frac{1}{2} L^N}^\\\\infty t^2 P_t(\\\\frac{\\\\lambda (k)+m^2}{2d+m^2}) \\\\, \\\\frac{dt}{t}}&\\\\lesssim \\\\frac{1}{|\\\\Lambda _N|} \\\\sum _{k \\\\in \\\\Lambda _N^*: k \\\\ne 0} {\\\\int _{\\\\frac{1}{2} L^N}^\\\\infty t^{2}t^{-2s}|k|^{-2s} \\\\, \\\\frac{dt}{t}}&\\\\lesssim \\\\frac{L^{2N}}{|\\\\Lambda _N|} \\\\sum _{k \\\\in \\\\Lambda _N^*: k \\\\ne 0}L^{-2sN}|k|^{-2s}\\\\lesssim L^{-(d-2)N} \\\\int _1^\\\\infty r^{-2s+d-1} dr \\\\lesssim L^{-(d-2)N}$ and analogously for the discrete gradients.\",\n \"Finally, by (REF ), $t_N&= \\\\frac{1}{(2d+m^2)} \\\\int _{\\\\frac{1}{2} L^N}^\\\\infty t^2P_t(\\\\frac{m^2}{2d+m^2}) \\\\, \\\\frac{dt}{t}&= \\\\frac{1}{m^2}-\\\\frac{1}{(2d+m^2)} \\\\int _0^{\\\\frac{1}{2} L^N} t^2P_t(\\\\frac{m^2}{2d+m^2}) \\\\, \\\\frac{dt}{t}=\\\\frac{1}{m^2} - O(L^{2N}).$\"\n ],\n [\n \"Proof of Proposition \",\n \"Proposition REF is essentially [27], with the minor changes of the separation of the coupling constant $u_j$ and the explicit inclusion of $\\\\theta $ .\",\n \"We include a proof here for convenience.\",\n \"We begin by algebraically manipulating $Z_{j}$ .\",\n \"These manipulations only rely on factorisation properties and not on the precise definitions of $I$ and $K$ .\",\n \"We will clearly state below when we restrict to the context of Proposition REF .\",\n \"Consider $Z_j = \\\\sum _{X\\\\in _j} I^{\\\\Lambda \\\\setminus X}K(X)$ where $I(B) = e^{-V_j(B)}$ and $K(X)= K_j(X)$ .\",\n \"We will use that $I^Y = \\\\prod _{B \\\\in _j(Y)} I(B)$ factors over blocks and $K_{j}(X)$ factors over connected components of $X$ .\",\n \"Given any $\\\\tilde{I}(B) \\\\in (B)$ for $B \\\\in _j$ , $ \\\\theta Z_j = \\\\sum _{X\\\\in _j} \\\\tilde{I}^{\\\\Lambda \\\\setminus X} \\\\tilde{K}(X),\\\\qquad \\\\tilde{K}(X)= \\\\sum _{Y \\\\in _j(X)} (\\\\delta I)^Y \\\\theta K(X\\\\setminus Y),$ where $\\\\delta I(B) = \\\\theta I(B)-\\\\tilde{I}(B)$ and $\\\\tilde{I}^{Y}\\\\prod _{B\\\\in _{j}(Y)}\\\\tilde{I}(B)$ .\",\n \"To see this, insert the identity $\\\\theta I^{\\\\Lambda \\\\setminus X} = \\\\sum _{Y\\\\subset \\\\Lambda \\\\setminus X} \\\\tilde{I}^{Y}(\\\\delta I)^{\\\\Lambda \\\\setminus (X\\\\cup Y)}$ into (REF ); (REF ) then follows by changing the summation index.\",\n \"Using that $\\\\tilde{K}$ factors over components since $K$ factors over components, we can then define $\\\\check{K}(Y)$ to be $\\\\tilde{K}(Y) - \\\\sum _{B\\\\in _{j}(Y)}\\\\theta J(B,Y)$ for any connected polymer $Y$ (and zero otherwise), where $J(B,Y)\\\\in $ are given.\",\n \"This yields the formula $\\\\tilde{K}(X) = \\\\prod _{Y \\\\in \\\\operatorname{Comp}(X)} {\\\\check{K}(Y) + \\\\sum _{B \\\\in _j(Y)} \\\\theta J(B,Y)}.$ Expanding the product, this can be written as $\\\\tilde{K}(X) = \\\\sum _{\\\\check{X} \\\\subset \\\\operatorname{Comp}(X)} \\\\check{K}(\\\\check{X}) \\\\prod _{Y \\\\in \\\\operatorname{Comp}(X \\\\setminus \\\\check{X})} \\\\sum _{B \\\\in _j(Y)} \\\\theta J(B,Y).$ Given the polymer $X \\\\setminus \\\\check{X}$ , now define $=(X,\\\\check{X}) \\\\subset \\\\lbrace (B,Y): B\\\\in _{j}, Y\\\\in _{j}\\\\rbrace $ so that $\\\\prod _{Y \\\\in \\\\operatorname{Comp}(X \\\\setminus \\\\check{X})} \\\\sum _{B \\\\in _j(Y)}J(B,Y) = \\\\sum _{} \\\\prod _{(B,Y)\\\\in } J(B,Y).$ Then $\\\\tilde{K}(X) = \\\\sum _{(\\\\check{X}, )} \\\\check{K}(\\\\check{X}) \\\\prod _{(B,Y) \\\\in } \\\\theta J(B,Y).$ By (REF ), since $\\\\tilde{I}\\\\in (\\\\Lambda )$ is a constant with respect to $_{C_{j+1}}$ , $_{C_{j+1}} \\\\theta Z_j&= \\\\sum _{X \\\\in _{j}} \\\\tilde{I}^{\\\\Lambda \\\\setminus X} _{C_{j+1}}\\\\tilde{K}(X)&= \\\\sum _{U \\\\in _{j+1}} \\\\tilde{I}^{\\\\Lambda \\\\setminus U} \\\\sum _{X \\\\in _j: \\\\bar{X} = U} \\\\tilde{I}^{U\\\\setminus X} _{C_{j+1}} \\\\tilde{K}(X)= \\\\sum _{U \\\\in _{j+1}} \\\\tilde{I}^{\\\\Lambda \\\\setminus U} K^{\\\\prime }(U),$ where by (REF ) $K^{\\\\prime }(U) = \\\\sum _{X \\\\in _j: \\\\bar{X} = U} \\\\tilde{I}^{U\\\\setminus X}\\\\sum _{(\\\\check{X}, )} _{C_{j+1}} \\\\check{K}(\\\\check{X})\\\\prod _{(B,Y) \\\\in } \\\\theta J(B,Y).$ We now specialise to the setting of Proposition REF .\",\n \"Thus we set $\\\\tilde{I}(B)= e^{-(u+V)_{j+1}(B)}$ and $K_{j+1}$ as in (REF ).\",\n \"Then $_{C_{j+1}} \\\\theta Z_j$ is $e^{-u_{j+1}|\\\\Lambda |}\\\\sum _{U \\\\in _{j+1}} e^{-V_{j+1}(\\\\Lambda \\\\setminus U)} K_{j+1}(U).$ To confirm this we explain the notation used in (REF ).\",\n \"In the definition of $K_{j+1}$ , $X_{}$ is by definition $\\\\bigcup _{(B,Y)\\\\in }Y$ .\",\n \"Given $X$ and $\\\\check{X}$ a subset of components of $X$ , $(X,\\\\check{X})$ is the set of sets of pairs $(B,Y)$ where (i) $B$ is a block in $Y$ , (ii) $Y$ is a component of $X\\\\setminus \\\\check{X}$ , (iii) each component $Y$ occurs in exactly one pair, and (iv) the closure of $\\\\check{X}\\\\cup \\\\cup _{(B,Y)}Y$ is $U$ .\",\n \"To conclude the proof we make the definition that $\\\\sum _{(U)}$ is a shorthand for the double sum $\\\\sum _{X\\\\in _{j}:\\\\bar{X}=U}\\\\sum _{(\\\\check{X},)}$ , as an explicit description of the set $(U)$ will not be needed.\",\n \"What remains is to prove the claims regarding factorisation and automorphism invariance.\",\n \"For factorisation, note that $\\\\sum _{(U_{1}\\\\cup U_{2})} = \\\\sum _{(U_{1})}\\\\sum _{(U_{2})}$ for $U_{1},U_{2}\\\\in _{j+1}$ that do not touch.\",\n \"Since $U_{1}$ and $U_{2}$ are distance $L^{j+1}>\\\\frac{1}{2}L^{j+1}+2^{d+1}L^{j}$ apart in this case the Grassmann integrals in the definition of $K_{j+1}(U_{1}\\\\cup U_{2})$ factor.\",\n \"Here we have used our standing assumption that $L>2^{d+2}$ , that $J(B,Y)=0$ if $Y\\\\notin _{j}$ , and that the range of $C_{j+1}$ is $\\\\frac{1}{2}L^{j+1}$ .\",\n \"Automorphism invariance follows directly from the formula for $K_{j+1}$ .\"\n ],\n [\n \"Acknowledgements\",\n \"We thank David Brydges and Gordon Slade.\",\n \"This article would not have been possible in this form without their previous contributions to the renormalisation group method.\",\n \"We also thank them for the permission to include Figure REF from [17].\",\n \"R.B.\",\n \"was supported by the European Research Council under the European Union's Horizon 2020 research and innovation programme (grant agreement No.\",\n \"851682 SPINRG).\",\n \"N.C. was supported by Israel Science Foundation grant number 1692/17.\"\n ]\n]"},"id":{"kind":"string","value":"2107.01878"}}},{"rowIdx":1581,"cells":{"text":{"kind":"list like","value":[["Sub-Feller Semigroups Generated by Pseudodifferential Operators on\n Symmetric Spaces of Noncompact Type"],["Abstract We consider global pseudodifferential operators on symmetric spaces of noncompact type, defined using spherical functions.","The associated symbols have a natural probabilistic form that extend the notion of the characteristic exponent appearing in Gangolli's L\\'evy-Khinchine formula to a function of two variables.","The Hille-Yosida-Ray theorem is used to obtain conditions on such a symbol so that the corresponding pseudodifferential operator has an extension that generates a sub-Feller semigroup, generalising existing results for Euclidean space."],["Introduction","Pseudodifferential operator theory is a powerful tool in the study of Feller–Markov processes on Euclidean space (see for example [31], [37], [38], [39], [40], or [11] §5 for a summary).","Primarily developed by Niels Jacob and collaborators (see e.g.","[26], [27], [24]), this framework characterises sub-Feller semigroups and their generators as pseudodifferential operators ($\\Psi $ DOs) acting on $C_0(\\mathbb {R}^d)$ , the Banach space of continuous, real-valued functions on $\\mathbb {R}^d$ that vanish at infinity.","The associated symbols capture many properties of the sub-Feller processes and semigroups they represent, generalising the well-established relationship between Lévy processes and their characteristic exponents given by the Lévy–Khinchine formula.","The key difference is that the Feller–Markov symbols typically depend on two variables instead of one — in the case of Feller processes, this is sometimes described as the Lévy characteristics having gained spatial dependence.","Manifold-valued Feller–Markov processes have also attracted interest in recent years (see [13], [25] and [32] §7 for excellent summaries), though the absence of a global harmonic analysis on general manifolds has so far prevented a $\\Psi $ DO approach.","Lie groups and symmetric spaces come with their own harmonic analysis, however, in the form of the spherical transform (see Harish-Chandra [18], [19] and Helgason [22], [21]).","A natural question to ask then is to what extent a $\\Psi $ DO-based approach can be applied to the study of sub-Feller processes on Lie groups and symmetric spaces.","Much work has already been done in this area, especially in the “constant coefficient” case of Lévy processes, in which the symbol depends only on its second argument.","Here, the symbols are given by Gangolli's Lévy–Khinchine formula ([16] Theorem 6.2), a direct analogue to the classical result.","The first paper to use probabilistic $\\Psi $ DO methods on a Lie group was [6], where $\\Psi $ DOs are used to study Lévy processes on the Heisenberg group.","Pseudodifferential operator representations of semigroups and generators have also been found for Lévy processes on a general Lie group — see [2], [3] for the compact case and [4] Section 5 for the noncompact case.","For Feller processes, $\\Psi $ DO representations have been found when the Lie group or symmetric space is compact — see for example the final sections of [7] and [8].","This paper seeks to develop a more general theory of $\\Psi $ DOs for symmetric spaces of noncompact type, and apply it to seek conditions on a symbol so that the corresponding $\\Psi $ DO extends to the generator of some sub-Feller process.","For the $\\mathbb {R}^d$ case, this question has been studied thoroughly by both Jacob and Hoh — see [26], [27], as well as [24] Chapter 4.","Our approach is similar to Jacob [27] and Hoh [24], but where they have used Fourier transform, we use the spherical transform.","The spherical transform enjoys many of the same properties as the Fourier transform on $\\mathbb {R}^d$ , and we find that several of the arguments in [27] and [24] generalise directly to this setting.","However, there are significant differences between the two settings, and a direct transcription of Jacob and Hoh's arguments is certainly not possible.","One notable difference, for example, is the condition (2.2) in Jacob [27], which features a derivative that on a manifold makes little sense.","Even on a symmetric space, elements of a tangent space cannot be easily translated from point to point, and a different approach is needed.","We take a more operator-theoretic approach, and replace each of the partial derivatives $\\frac{\\partial }{\\partial x_i}$ ($i=1,\\ldots ,d$ ) with the fractional Laplacian $\\sqrt{-\\Delta }$ .","This is an exciting object to work with, and has the added advantage that spherical functions form a system of eigenfunctions for this operator.","The structure of this paper will be the following.","Section presents a summary of necessary concepts and results from harmonic analysis on symmetric spaces, and introduces the system of symbols and $\\Psi $ DOs that will be used later on.","We also introduce here the spherical anisotropic Sobolev spaces, a generalisation of the anisotropic Sobolev spaces first considered by Niels Jacob [26], [27].","In Section , we consider (a version of) the Hille–Yosida–Ray theorem (see Theorem REF ), and note that, thanks to the work of Applebaum and Ngan [7], [8], one of the conditionsNamely, that an operator must satisfy the positive maximum principle.","of the Hille–Yosida–Ray theorem has already been fully addressed.","This work motivates our introduction of a class of operators we will call Gangolli operators, which will automatically each be densely defined linear operators on $C_0(K|G|K)$ that satisfy the positive maximum principle.","We then prove that Gangolli operators are $\\Psi $ DOs in the sense of Section REF , and derive a formula for their symbols (Theorem REF ).","Section is concerned with seeking sufficient conditions for a Gangolli operator $q(\\sigma ,D)$ to extend to the generator a sub-Feller semigroup.","Informed by the work of the previous section, this amounts to finding conditions that ensure $\\overline{\\operatorname{Ran}(\\alpha +q(\\sigma ,D))}=C_0(K|G|K)$ for some $\\alpha >0$ (see Theorem REF (REF )).","This section perhaps most closely follows the approach of Jacob [27] and Hoh [24] Chapter 4, and where proofs are similar to these sources, we omit detail, and instead aim to emphasise what is different about the symmetric space setting.","Full proofs may also be found in my PhD thesis [42].","Finally, in Section we present a large class of examples of symbols that satisfy the conditions found in Sections and .","Notation.","For a topological space $X$ , $\\mathcal {B}(X)$ will denote the Borel $\\sigma $ -algebra associated with $X$ , and $B_b(X)$ the space of bounded, Borel functions from $X\\rightarrow \\mathbb {R}$ , a Banach space with respect to the supremum norm.","If $X$ is a locally compact Hausdorff space, then we write $C_0(X)$ for the closed subspace of $B_b(X)$ consisting of continuous functions vanishing at infinity, and $C_c(X)$ for the dense subspace of compactly supported continuous functions.","If $X$ is a smooth manifold and $M\\in \\mathbb {N}\\cup \\lbrace \\infty \\rbrace $ , then we write $C^M_c(X)$ for the space of compactly supported $M$ -times continuously differentiable functions on $X$ ."],["Preliminaries","Let $(G,K)$ is a Riemannian symmetric pair, so that $G$ is a connected, semisimple Lie group with finite centre, $K$ is a maximal compact subgroup of $G$ , and, for some nontrivial involution $\\Theta $ on $G$ , $G^\\Theta _0\\subseteq K\\subseteq G^\\Theta ,$ where $G^\\Theta $ is the fixed point set of $\\Theta $ , and $G^\\Theta _0$ is the identity component of $G^\\Theta $ .","Let $\\mathfrak {g}$ and $\\mathfrak {k}$ denote the Lie algebras of $G$ and $K$ , respectively.","Note that $\\mathfrak {k}$ is the $+1$ eigenspace of the differential $\\theta :=d\\Theta $ ; let $\\mathfrak {p}$ denote the $-1$ eigenspace.","In fact, $\\theta $ is a Cartan involution on $\\mathfrak {g}$ , and the corresponding Cartan decomposition is $\\mathfrak {g} = \\mathfrak {p} \\oplus \\mathfrak {k}.$ Let $B$ denote the Killing form of $G$ , defined for each for all $X,Y\\in \\mathfrak {g}$ by $B(X,Y) = \\operatorname{tr}(\\operatorname{ad}X\\operatorname{ad}Y)$ .","Since $(G,K)$ is of noncompact type, $B$ is nondegenerate, negative definite on $\\mathfrak {k}$ , and positive definite on $\\mathfrak {p}$ .","Fix an $\\operatorname{Ad}(K)$ -invariant inner product $\\langle \\cdot ,\\cdot \\rangle $ on $\\mathfrak {g}$ .","Associated with $(G,K)$ is the symmetric space $G/K$ , a smooth Riemannian manifold, with Riemannian structure induced by the restriction of $\\langle \\cdot ,\\cdot \\rangle $ to $\\mathfrak {p}$ .","There is a one to one correspondence between functions on $G/K$ and $K$ -right-invariant functions on $G$ , we denote both by $\\mathcal {F}(G/K)$ .","Similarly, we identify $K$ -invariant functions on $G/K$ with $K$ -bi-invariant functions on $G$ , and denote both by $\\mathcal {F}(K|G|K)$ .","Similar conventions will be used to denote standard subspaces of $\\mathcal {F}(G/K)$ and $\\mathcal {F}(K|G|K)$ ; for example $C(K|G|K)$ will denote both the space of continuous, $K$ -invariant functions on $G/K$ , and the space of continuous, $K$ -bi-invariant functions on $G$ ."],["Harmonic Analysis on Symmetric Spaces of Noncompact Type","For a thorough treatment of this topic, see Helgason [22], [21].","Let ${\\bf D}(G)$ denote the set of all left invariant differential operators on $G$ , and let ${\\bf D}_K(G)$ denote the subspace of those operators that are also $K$ -right-invariant.","A mapping $\\phi :G\\rightarrow \\mathbb {C}$ is called a spherical if it is $K$ -bi-invariant, satisfies $\\phi (e) =1$ , and is a simultaneous eigenfunction of every element of ${\\bf D}_K(G)$ .","Fix an Iwasawa decomposition $G=NAK$ , where $N$ is a nilpotent Lie subgroup of $G$ , and $A$ is Abelian.","Let $\\mathfrak {n}$ and $\\mathfrak {a}$ denote respectively the Lie algebras of $N$ and $A$ .","For each $\\sigma \\in G$ , let $A(\\sigma )$ denote the unique element of $\\operatorname{\\mathfrak {a}}$ such that $\\sigma \\in Ne^{A(\\sigma )}K$ .","Harish-Chandra's integral formula states that every spherical function on $G$ takes the form $\\phi _\\lambda (\\sigma ) = \\int _Ke^{(\\rho +i\\lambda )(A(k\\sigma ))}dk, \\hspace{20.0pt} \\forall \\sigma \\in G,$ for some $\\lambda \\in *$ .","Moreover, $\\phi _\\lambda =\\phi _{\\lambda ^{\\prime }}$ if an only if $s(\\lambda )=\\lambda ^{\\prime }$ for some element $s$ of the Weyl group $W$ .","A spherical function $\\phi _\\lambda $ is positive definite if and only if $\\lambda \\in $ .","The spherical transform of a function $f\\in L^1(K|G|K)$ is the function $\\hat{f}:\\rightarrow \\mathbb {C}$ given by $\\hat{f}(\\lambda ) = \\int _G\\phi _{-\\lambda }(\\sigma )f(\\sigma )d\\sigma , \\hspace{20.0pt} \\forall \\lambda \\in .$ Similarly, given a finite Borel measure $\\mu $ on $G$ , the spherical transform of $\\mu $ is the mapping $\\hat{\\mu }:\\rightarrow \\mathbb {C}$ given by $\\hat{\\mu }(\\lambda ) = \\int _G\\phi _{-\\lambda }(\\sigma )\\mu (d\\sigma ).$ The spherical transform enjoys many useful properties, the most powerful being that it defines an isomorphism of the Banach convolution algebra $L^1(K|G|K)$ with the space $L^1(,\\omega )^W$ of Weyl group invariant elements of $L^1(,\\omega )$ .","The Borel measure $\\omega $ is called Plancherel measure, and is given by $\\omega (d\\lambda ) = |\\operatorname{\\bf c}(\\lambda )|^{-2}d\\lambda ,$ where $\\operatorname{\\bf c}$ denotes Harish-Chandra's $\\operatorname{\\bf c}$ -function.","According to the spherical inversion theorem, for all $f\\in C_c^\\infty (K|G|K)$ and all $\\sigma \\in G$ , $f(\\sigma ) = \\int _{}\\phi _\\lambda (\\sigma )\\hat{f}(\\lambda )\\omega (d\\lambda ).$ There is also a version of Plancherel's identity for the spherical transform, namely $\\Vert f\\Vert _{L^2(K|G|K)} = \\Vert \\hat{f}\\Vert _{L^2(,\\omega )}, \\hspace{20.0pt} \\forall f\\in C_c^\\infty (K|G|K).$ Let $L^2(,\\omega )^W$ denote the subspace of $L^2(,\\omega )$ consisting of $W$ -invariants.","Then the image of $C_c^\\infty (K|G|K)$ under spherical transformation is a dense subspace of $L^2(,\\omega )^W$ , and as such the spherical transform extends to an isometric isomorphism between the Hilbert spaces $L^2(K|G|K)$ and $L^2(,\\omega )^W$ .","For more details, see for example Helgason [21] Chapter IV § 7.3, pp. 454.","Similarly to classical Fourier theory, the most natural setting for the spherical transform is Schwarz space.","A function $f\\in C^\\infty (G)$ is called rapidly decreasing if $\\sup _{\\sigma \\in G}(1+|\\sigma |)^q\\phi _0(\\sigma )^{-1}(Df)(\\sigma ) < \\infty , \\hspace{20.0pt} \\forall D\\in {\\bf D}(G), \\;q\\in \\mathbb {N}\\cup \\lbrace 0\\rbrace ,$ where $|\\sigma |$ denotes the geodesic distance between the double coset spaces $K\\sigma K$ and $KeK$ .","Equivalently, if $\\sigma \\in G$ is decomposed according to the Cartan decomposition $G=K\\overline{\\exp \\operatorname{\\mathfrak {a}}^+}K$ , so that $\\sigma =ke^Hk^{\\prime }$ for some unique $H\\in \\overline{\\operatorname{\\mathfrak {a}}^+}$ , then $|\\sigma |=|H|$ — see Gangolli and Varadarajan [17] pp.167 for more details on this object.","The ($K$ -bi-invariant) Schwarz space $\\mathcal {S}(K|G|K)$ is the Fréchet space comprising of all rapidly decreasing, $K$ -bi-invariant functions $f\\in C^\\infty (G)$ , together with the family of seminorms given by the left-hand side of (REF ).","By viewing the spaces $\\operatorname{\\mathfrak {a}}$ and $$ as finite dimensional vector spaces, we also consider the classical Schwartz spaces $\\mathcal {S}()$ and $\\mathcal {S}(\\operatorname{\\mathfrak {a}})$ , as well as $W$ -invariant subspaces $\\mathcal {S}(\\operatorname{\\mathfrak {a}})^W$ and $\\mathcal {S}()^W$ .","The Euclidean Fourier transform ${F}(f)(\\lambda ) = \\int _{\\operatorname{\\mathfrak {a}}}e^{-i\\lambda (H)}f(H)dH, \\hspace{20.0pt} \\forall f\\in \\mathcal {S}(\\operatorname{\\mathfrak {a}}),\\lambda \\in $ defines a topological isomorphism between the spaces $\\mathcal {S}(\\operatorname{\\mathfrak {a}})^W$ and $\\mathcal {S}()^W$ in the usual way.","Given $f\\in \\mathcal {S}(K|G|K)$ and $H\\in \\operatorname{\\mathfrak {a}}$ , the Abel transform is defined by ${A}f (H) = e^{\\rho (H)}\\int _Nf((\\exp H)n)dn.$ The Abel transform is fascinating in its own right, and we refer to Sawyer [36] for more information.","However, for our purposes we are mainly interested in its role in the following: Theorem 2.1 Writing ${H}$ for the spherical transform, the diagram [scale=2] 1) at (0,1)$\\mathcal {S}()^W$ ; 2) at (1,0)$\\mathcal {S}(\\operatorname{\\mathfrak {a}})^W$ ; 3) at (-1,0)$\\mathcal {S}(K|G|K)$ ; [->] (2) – (1); [->] (3) – (1); [->] (3) – (2); t (.63,.63) ${F}$ ; t (-.63,.63) ${H}$ ; t (0,-.2) ${A}$ ; commutes, up to normalizing constants.","Each arrow describes an isomorphism of Fréchet algebras.","This result will be extremely useful in later sections, especially when proving Theorem REF .","For more details, see Proposition 3 in Anker [1], Gangolli and Varadarajan [17] page 265, and Helgason [21] pp.","450."],["Probability on Lie Groups and Symmetric spaces","In this subsection, we summarise a few key notions from probability theory on Lie groups and symmetric spaces.","Books by Liao [33], [34] are excellent sources for this material, as is the paper [35] by Liao and Wang.","Fix a probability space $(\\Omega ,\\mathcal {F},P)$ .","Just as with functions on $G$ and $G/K$ , we may view stochastic processes on $G/K$ as projections of processes on $G$ whose laws are $K$ -right invariant.","Equipped with its natural filtration $\\lbrace \\mathcal {F}^X_t,t\\ge 0\\rbrace $ , $X$ is said to have independent increments if for all $t>s\\ge 0$ , $X(s)^{-1}X(t)$ is independent of $\\mathcal {F}^X_s$ , and stationary increments if $X(s)^{-1}X(t) \\sim X(0)^{-1}X(t-s) \\hspace{20.0pt} \\forall t>s\\ge 0.$ A process $X=(X(t),t\\ge 0)$ on $G$ is stochastically continuous if, for all $s\\ge 0$ and all $B\\in \\mathcal {B}(G)$ with $e\\notin B$ , $\\lim _{t\\rightarrow s}P(X(s)^{-1}X(t)\\in B)=0.$ A stochastically continuous process $X$ on $G$ with stationary and independent increments is called a Lévy process on $G$ .","A process on $G/K$ is called a Lévy process if it is the projection of a Lévy process on $G$ , under the canonical surjection $\\pi :G\\mapsto G/K$ .","The convolution product of two Borel measures $\\mu _1,\\mu _2$ on $G$ is defined for each $B\\in \\mathcal {B}(G)$ by $(\\mu _1\\ast \\mu _2)(B) = \\int _G\\int _G\\operatorname{\\bf 1}_B(\\sigma \\tau )\\mu _1(d\\sigma )\\mu _2(d\\tau ).$ Note that since $G$ is semisimple, it is unimodular, and hence this operation is commutative.","It is also clear from the definition that $\\mu _1\\ast \\mu _2$ is $K$ -bi-invariant whenever $\\mu _1$ and $\\mu _2$ are.","Definition 2.2 A family $(\\mu _t,t\\ge 0)$ of finite Borel measures on $G$ will be called a convolution semigroup (of probability measures) if $\\mu _t(G)=1$ for all $t\\ge 0$ , $\\mu _{s+t} = \\mu _s\\ast \\mu _t$ for all $s,t\\ge 0$ , and $\\mu _t\\rightarrow \\mu _0$ weakly as $t\\rightarrow 0$ .","Note that $\\mu _0$ must be an idempotent measure, in the sense that $\\mu _0\\ast \\mu _0=\\mu _0$ .","By Theorem 1.2.10 on page 34 of Heyer [23], $\\mu _0$ must coincide with Haar measure on a compact subgroup of $G$ .","We we will frequently take $\\mu _0$ to be normalised Haar measure on $K$ , so that the image of $\\mu _0$ after projecting onto $G/K$ is $\\delta _o$ , the delta mass at $o:=eK$ .","One may also define convolution of measures on $G/K$ , and convolution semigroups on $G/K$ are defined analogously — see Liao [34] Section 1.3 for more details.","In fact, the projection map $\\pi :G\\rightarrow G/K$ induces a bijection between the set of all convolution semigroups on $G/K$ and the set of all $K$ -bi-invariant convolution semigroups on $G$ — see Liao [34] Propositions 1.9 and 1.12, pp. 11–13.","We henceforth identify these two sets, but generally opt to perform calculations using objects defined on $G$ , for simplicity.","If $X=(X(t),t\\ge 0)$ is a Lévy process on $G$ (resp.","$G/K$ ), and if $\\mu _t$ denotes the law of $X$ at time $t$ , then $(\\mu _t,t\\ge 0)$ is a convolution semigroup on $G$ (resp.","$G/K$ ).","A well-known construction shows that all convolution semigroups on $G$ and $G/K$ arise this way — see for example Theorem 1.7 on page 8 of [34].","Let $(\\mu _t, t\\ge 0)$ be a $K$ -bi-invariant convolution semigroup of probability measures on $G$ associated with a Lévy process $X$ on $G/K$ .","The Feller semigroup $(T_t,t\\ge 0)$ associated with $X$ (also called the Hunt semigroup of $(\\mu _t,t\\ge 0)$ ) is given by $T_tf(\\sigma ) = \\int _Gf(\\sigma \\tau )\\mu _t(d\\tau ) \\hspace{20.0pt} \\forall f\\in C_0(K|G|K), \\;\\sigma \\in G.$ Definition 2.3 Let $X_1,\\ldots ,X_l$ be a basis of $\\mathfrak {g}$ , ordered so that $X_1,\\ldots ,X_d$ is a basis of $\\mathfrak {p}$ .","A collection $\\lbrace x_1,\\ldots ,x_l\\rbrace $ of smooth functions of compact support is called a system of exponential coordinate functions if there is a neighbourhood $U$ of $e$ for which $\\sigma = \\exp \\left(\\sum _{i=1}^lx_i(\\sigma )X_i\\right) \\hspace{20.0pt} \\forall \\sigma \\in U.$ The $x_i$ may be chosen so as to be $K$ -right-invariant for $i=1,\\ldots , m$ , and such that $\\sum _{i=1}^dx_i(k\\sigma )X_i = \\sum _{i=1}^dx_i(\\sigma )\\operatorname{Ad}(k)X_i \\hspace{20.0pt} \\forall k\\in K.$ For more details, see Liao [34] pp.36–37, 83.","The choice of basis $X_1,\\ldots ,X_d$ of $\\mathfrak {p}$ enables us to view $\\operatorname{Ad}k$ as a $m\\times m$ matrix, for each $k\\in K$ .","A vector $b\\in \\mathbb {R}^m$ is said to be $\\operatorname{Ad}(K)$ -invariant if $b=\\operatorname{Ad}(k)^Tb, \\hspace{20.0pt} \\forall k\\in K.$ Similarly, an $m\\times m$ real-valued matrix $a=(a_{ij})$ is $\\operatorname{Ad}(K)$ -invariant if $a=\\operatorname{Ad}(k)^Ta\\operatorname{Ad}(k) \\hspace{20.0pt} \\forall k\\in K.$ A Borel measure $\\nu $ on $G$ is called a Lévy measure if $\\nu (\\lbrace e\\rbrace ) = 0$ , $\\int _G\\sum _{i=1}^lx_i(\\sigma )^2\\nu (d\\sigma )$ , and $\\nu (B^c)<\\infty $ for any neighbourhood $B$ of $e$ .","We state a useful corollary of the famous Hunt formula.","For more details, including a proof, see Section 3.2 Liao [34], pp. 78.","Theorem 2.4 Let $\\mathcal {L}$ be the infinitesimal generator associated with a $K$ -bi-invariant Lévy process on $G$ .","Then $C_c^\\infty (G)\\subseteq \\operatorname{Dom}\\mathcal {L}$ , and there is an $\\operatorname{Ad}(K)$ -invariant vector $b\\in \\mathbb {R}^m$ , an $\\operatorname{Ad}(K)$ -invariant covariance matrix $a:=(a_{ij})$ , and a $K$ -bi-invariant Lévy measure $\\nu $ such that $\\begin{aligned}\\mathcal {L}f(\\sigma ) = \\sum _{i=1}^db_iX_if(\\sigma ) &+ \\frac{1}{2}\\sum _{j,k=1}^da_{jk}X_jX_kf(\\sigma ) \\\\&+ \\int _G\\left(f(\\sigma \\tau )-f(\\sigma )-\\sum _{i=1}^dx_i(\\tau )X_if(\\sigma )\\right)\\nu (d\\sigma ),\\end{aligned}$ for all $f\\in C_c^\\infty (G)$ and $\\sigma \\in G$ .","Moreover, the triple $(b,a,\\nu )$ is completely determined by $\\mathcal {L}$ , and independent of the choice of exponential coordinate functions $x_i,\\;i=1,\\ldots ,d$ .","Conversely, given a triple $(b,a,\\nu )$ of this kind, there is a unique $K$ -bi-invariant convolution semigroup of probability measures on $G$ with infinitesimal generator given by $\\mathcal {L}$ .","Let $\\mathcal {L}$ be the Lévy generator described in Theorem REF above, and let $\\mathcal {L}_D$ be the diffusion part of $\\mathcal {L}$ , so that $\\mathcal {L}_D = \\sum _{i=1}^db_iX_i + \\frac{1}{2}\\sum _{i,j=1}^da_{jk}X_jX_k.$ Now by the discussion surrounding equation (3.3) of Liao [34], pp.","75, $\\mathcal {L}_D\\in {\\bf D}_K(G)$ , and so for each $\\lambda \\in $ there is a constant $\\beta (\\mathcal {L}_D,\\lambda )\\in \\mathbb {C}$ such that $\\mathcal {L}_D\\phi _\\lambda = \\beta (\\mathcal {L}_D,\\lambda )\\phi _\\lambda .$ The mapping $\\lambda \\mapsto \\beta (\\mathcal {L}_D,\\lambda )$ is a $W$ -invariant quadratic polynomial function on $$ .","Theorem 2.5 (Gangolli's Lévy–Khinchine formula) Let $(\\mu _t,t\\ge 0)$ be a $K$ -bi-invariant convolution semigroup of probability measures on $G$ with infinitesimal generator $\\mathcal {L}$ , and let $\\mathcal {L}_D$ denote the diffusion part of $\\mathcal {L}$ .","Then $\\hat{\\mu }_t=e^{-t\\psi }$ , where $\\psi (\\lambda ) = -\\beta (\\mathcal {L}_D,\\lambda ) + \\int _G(1-\\phi _\\lambda (\\sigma ))\\nu (d\\sigma ) \\hspace{20.0pt} \\forall \\lambda \\in ,$ and $\\beta (\\mathcal {L}_D,\\lambda )$ is given by (REF ).","This result was first proven by Ramesh Gangolli in [16], see also Liao and Wang [35] .","For a proof of the specific statement above, see page 139 of Liao [34].","The function $\\psi $ given by (REF ) will be called the Gangolli exponent of the process $X$ .","Remark 2.6 If Definition REF (REF ) is relaxed so that each $\\mu _t$ need only satisfy $\\mu _t(G)\\le 1$ , all of the results described in this subsection continue to hold, except “sub-” must be added to some to the terms: convolution semigroups of sub-probability measures, sub-Lévy generators, sub-diffusion operators, and so on."],["Positive and Negative Definite Functions","By viewing $$ as a finite-dimensional real vector space, we may consider positive and negative definite functions on $$ , defined in the usual way.","Proposition 2.7 For all $\\sigma \\in G$ , $\\lambda \\mapsto \\phi _\\lambda (\\sigma )$ is positive definite.","Let $\\mu $ be a finite $K$ -bi-invariant Borel measure.","Then $\\hat{\\mu }$ is positive definite.","Let $\\sigma \\in G$ , $n\\in \\mathbb {N}$ , $\\lambda _1,\\ldots ,\\lambda _n\\in $ , and $c_1,\\ldots ,c_n\\in \\mathbb {C}$ , and note that $\\sum _{\\alpha ,\\beta =1}^nc_\\alpha \\overline{c_\\beta }e^{(i(\\lambda _\\alpha -\\lambda _\\beta )+\\rho )A(k\\sigma )} = \\left|\\sum _{\\alpha =1}^nc_\\alpha e^{(i\\lambda _\\alpha +\\frac{\\rho }{2})A(k\\sigma )}\\right|^2 \\ge 0.$ Therefore, by the Harish-Chandra integral formula, $\\sum _{\\alpha ,\\beta =1}^nc_\\alpha \\overline{c_\\beta }\\phi _{\\lambda _\\alpha -\\lambda _\\beta }(\\sigma ) = \\int _K\\sum _{\\alpha ,\\beta =1}^nc_\\alpha \\overline{c_\\beta }e^{(i(\\lambda _\\alpha -\\lambda _\\beta )+\\rho )A(k\\sigma )}dk \\ge 0.$ Part 1 follows.","For part 2, observe that since (REF ) holds for all $c_1,\\ldots ,c_n$ , we can replace each $c_j$ by its complex conjugate.","Therefore, $\\sum _{\\alpha ,\\beta =1}^n\\overline{c_\\alpha }c_\\beta \\phi _{\\lambda _\\alpha -\\lambda _\\beta }(\\sigma )\\ge 0$ for all $\\sigma \\in G$ , $n\\in \\mathbb {N}$ , $\\lambda _1,\\ldots ,\\lambda _n\\in $ , and $c_1,\\ldots ,c_n\\in \\mathbb {C}$ .","Taking complex conjugates and using the fact that $\\phi _{-\\lambda }=\\overline{\\phi _\\lambda }$ , it follows that for all $\\sigma \\in G$ , $n\\in \\mathbb {N}$ , $\\lambda _1,\\ldots ,\\lambda _n\\in $ , and $c_1,\\ldots ,c_n\\in \\mathbb {C}$ , $\\sum _{\\alpha ,\\beta =1}^nc_\\alpha \\overline{c_\\beta }\\phi _{-(\\lambda _\\alpha -\\lambda _\\beta )}(\\sigma ) = \\overline{\\sum _{\\alpha ,\\beta =1}^n\\overline{c_\\alpha }c_\\beta \\phi _{\\lambda _\\alpha -\\lambda _\\beta }}\\ge 0,$ and hence $\\sum _{\\alpha ,\\beta =1}^nc_\\alpha \\overline{c_\\beta }\\hat{\\mu }(\\lambda _\\alpha -\\lambda _\\beta ) = \\int _{}\\sum _{\\alpha ,\\beta =1}^nc_\\alpha \\overline{c_\\beta }\\phi _{-(\\lambda _\\alpha -\\lambda _\\beta )}(\\sigma )\\mu (d\\sigma )\\ge 0.$ By choosing a basis of $$ , we may identify it with $\\mathbb {R}^m$ , and apply classical results about positive (resp.","negative) definite functions on Euclidean space to functions on $$ , to obtain results about positive (resp.","negative) definite functions in this new setting.","One useful application of this is the Schoenberg correspondence, which states that a map $\\psi :\\rightarrow \\mathbb {C}$ is negative definite if and only if $\\psi (0)\\ge 0$ and $e^{-t\\psi }$ is positive definite for all $t>0$ .","This is immediate by the Schoenberg correspondence on $\\mathbb {R}^m$ — see Berg and Forst [10] page 41 for a proof.","Proposition 2.8 Let $\\psi :\\rightarrow \\mathbb {C}$ be the Gangolli exponent of a Lévy process on $G/K$ .","Then $\\psi $ is negative definite.","Let $X=(X(t),t\\ge 0)$ is a Lévy process on $G/K$ , and let $\\nu _t$ be the law of $X(t)$ , for all $t\\ge 0$ .","Then $(\\nu _t,t\\ge 0)$ forms a convolution semigroup on $G/K$ .","By Proposition 1.12 of Liao [34] (pp.","13), $(\\nu _t,t\\ge 0)$ arises as the projection onto $G/K$ of a $K$ -bi-invariant convolution semigroup $(\\mu _t,t\\ge 0)$ on $G$ .","By Proposition REF , the spherical transform of each $\\mu _t$ is positive definite, and by the Schoenberg correspondence, for each $t\\ge 0$ , there is a negative definite function $\\psi _t$ on $$ such that $\\psi _t(0)\\ge 0$ and $\\hat{\\mu }_t=e^{-\\psi _t}$ .","In fact, since $(\\mu _t,t\\ge 0)$ is a convolution semigroup, it must be the case that $\\hat{\\mu }_t = e^{-t\\psi _1}, \\hspace{20.0pt} \\forall t\\ge 0.$ By uniqueness of Gangolli exponents, $\\psi =\\psi _1$ , a negative definite function.","We finish this subsection with a collection of results about negative definite functions, which will be particularly useful for calculation in later sections.","Proposition 2.9 Let $\\psi :\\rightarrow \\mathbb {C}$ be a continuous negative definite function.","Then For all $\\lambda ,\\eta \\in $ , $\\left|\\sqrt{|\\psi (\\lambda )|}-\\sqrt{|\\psi (\\eta )|}\\right|\\le \\sqrt{|\\psi (\\lambda -\\eta )|}$ (Generalised Peetre inequality) For all $s\\in \\mathbb {R}$ and $\\lambda ,\\eta \\in $ , $\\left(\\frac{1+|\\psi (\\lambda )|}{1+|\\psi (\\eta )|}\\right)^s \\le 2^{|s|}(1+|\\psi (\\lambda -\\eta )|^2)^{|s|}.$ There is a constant $c_\\psi >0$ such that $|\\psi (\\lambda )|\\le c_\\psi (1+|\\lambda |^2) \\hspace{20.0pt} \\forall \\lambda \\in .$ These results follow from their analogues on $\\mathbb {R}^m$ — see Hoh [24] page 16."],["Spherical Anisotropic Sobolev Spaces","Suppose $\\psi $ is a real-valued continuous negative definite function, and let $s\\in \\mathbb {R}$ .","We define the (spherical) anisotropic Sobolev space associated with $\\psi $ and $s$ to be $H^{\\psi ,s} := \\left\\lbrace u\\in \\mathcal {S}^{\\prime }(K|G|K):\\int _G(1+\\psi (\\lambda ))^s|\\hat{u}(\\lambda )|^2\\omega (d\\lambda )<\\infty \\right\\rbrace ,$ where $\\mathcal {S}^{\\prime }(K|G|K)$ denotes the space of $K$ -bi-invariant tempered distributions.","One can check that each $H^{\\psi ,s}$ is a Hilbert space with respect to the inner product $\\langle u,v\\rangle _{\\psi ,s} := \\int _{}(1+\\psi (\\lambda ))^s\\hat{u}(\\lambda )\\overline{\\hat{v}(\\lambda )}\\omega (d\\lambda ), \\hspace{20.0pt} \\forall u,v\\in H^{\\psi ,s}.$ These spaces are a generalisation of the anisotropic Sobolev spaces first introduced by Niels Jacob, see [26], and developed further by Hoh, see [24].","For the special case $\\psi (\\lambda )=|\\rho |^2+|\\lambda |^2$ , we will write $H^{\\psi ,s}=H^s$ .","Note also that $H^{\\psi ,0}=L^2(K|G|K)$ , by the Plancherel theorem.","In this case, we will omit subscripts and just write $\\langle \\cdot ,\\cdot \\rangle $ for the inner product on $L^2(K|G|K)$ .","Note that $\\psi $ is a non-negative function, since it is negative definite and real-valued.","We impose an additional assumption, namely that there exist constants $r,c>0$ such that $\\psi (\\lambda )\\ge c\\left(|\\rho |^2+|\\lambda |^2\\right)^r \\hspace{20.0pt} \\forall \\lambda \\in .$ Analogous assumptions can be found in Jacob [27] (1.5) and Hoh [24] (4.4), and the role of (REF ) will be very similar.","Theorem 2.10 Let $\\psi $ be a real-valued, continuous negative definite symbol, satisfying (REF ).","Then $C_c^\\infty (K|G|K)$ and $\\mathcal {S}(K|G|K)$ are dense in each $H^{\\psi ,s}$ , and we have continuous embeddings $\\mathcal {S}(K|G|K)\\hookrightarrow H^{\\psi ,s}\\hookrightarrow \\mathcal {S}^{\\prime }(K|G|K)$ We have continuous embeddings $H^{\\psi ,s_2}\\hookrightarrow H^{\\psi ,s_1} $ whenever $s_1,s_2\\in \\mathbb {R}$ with $s_2\\ge s_1$ .","In particular, $H^{\\psi ,s}\\hookrightarrow L^2(K|G|K)$ for all $s\\ge 0$ .","Under the standard identification of $L^2(K|G|K)$ with its dual, the dual space of each $H^{\\psi ,s}$ is isomorphic to $H^{\\psi ,-s}$ , with $\\Vert u\\Vert _{\\psi ,-s} = \\sup \\left\\lbrace \\frac{|\\langle u,v\\rangle |}{\\Vert v\\Vert _{\\psi ,s}}:v\\in C_c^\\infty (K|G|K),\\;v\\ne 0\\right\\rbrace ,$ for all $s\\in \\mathbb {R}$ .","For $r>0$ as in equation (REF ), we have continuous embeddings $H^s \\hookrightarrow H^{\\psi ,s} \\hookrightarrow H^{rs},$ for all $s\\ge 0$ .","Let $s_3>s_2>s_1$ .","Then for all $\\epsilon >0$ , there is $c(\\epsilon )\\ge 0$ such that $\\Vert u\\Vert _{\\psi ,s_2} \\le \\epsilon \\Vert u\\Vert _{\\psi ,s_3} + c(\\epsilon )\\Vert u\\Vert _{\\psi ,s_1}$ for all $u\\in H^{\\psi ,s_3}$ .","There exist continuous embeddings $H^{\\psi ,s}\\hookrightarrow C_0(K|G|K)$ for all $s>\\frac{d}{r}$ , where $d=\\dim (G/K)$ .","For brevity, let $\\langle \\lambda \\rangle := \\sqrt{1+|\\lambda |^2}, \\hspace{20.0pt} \\forall \\lambda \\in ,$ and $\\Psi (\\lambda ) := \\sqrt{1+\\psi (\\lambda )}, \\hspace{20.0pt} \\forall \\lambda \\in .$ The proof of Theorem REF will be given after the next lemma.","Lemma 2.11 Let $M>d=\\dim (G/K)$ .","Then $\\langle \\cdot \\rangle ^{-M} \\in L^1(,\\omega )$ .","By standard arguments, one may check that $\\int _{\\mathbb {R}^d}\\langle \\xi \\rangle ^{-M}d\\xi <\\infty $ , for all $M>d$ .","Writing $p=\\frac{\\dim N}{2}$ , we have $d= \\dim + 2p$ , and hence $\\int _{}\\langle \\lambda \\rangle ^{-M+2p}d\\lambda < \\infty $ whenever $M>d$ .","By Proposition 7.2 on page 450 of Helgason [21], there are $C_1,C_2>0$ such that $|\\operatorname{\\bf c}(\\lambda )|^{-1} \\le C_1 + C_2|\\lambda |^p \\hspace{20.0pt} \\forall \\lambda \\in .$ Let $C>0$ be such that $(C_1 + C_2|\\lambda |^p)^2 < C(1+|\\lambda |^2)^p$ for all $\\lambda \\in $ .","Then $\\int _{}\\langle \\lambda \\rangle ^{-M}\\omega (d\\lambda ) = \\int _{}\\langle \\lambda \\rangle ^{-M}|\\operatorname{\\bf c}(\\lambda )|^{-2}d\\lambda \\le C\\int _{}\\langle \\lambda \\rangle ^{-M+2p}d\\lambda < \\infty ,$ whenever $M>d$ .","[Proof of Theorem REF ] Much of this theorem may be proved by adapting proofs from the $\\mathbb {R}^d$ case.","For example, to prove Theorem REF (REF ), let $\\mathcal {L}^{\\psi ,s}$ denote the space of all measurable functions $v$ on $$ for which $\\Psi ^sv\\in L^2(,\\omega )^W$ , a Hilbert space with respect to the inner product $\\langle u,v\\rangle = \\int _{\\operatorname{\\mathfrak {a}}^\\ast }\\Psi (\\lambda )^{2s}u(\\lambda )\\overline{v(\\lambda )}\\omega (d\\lambda ), \\hspace{20.0pt} \\forall u,v \\in \\mathcal {L}^{\\psi ,s}.$ By viewing $$ as a real vector space and using inequality (REF ) to relate $\\omega $ to Lebesgue measure, the proof of Theorem 3.10.3 on page 208 of Jacob [28] may be easily adapted to show that $\\mathcal {S}()^W\\hookrightarrow \\mathcal {L}^{\\psi ,s}\\hookrightarrow \\mathcal {S}^{\\prime }()^W$ is continuous.","Noting Theorem REF , Theorem REF (REF ) follows.","Proofs of Theorem REF (REF )–(REF ) are almost identical to their $\\mathbb {R}^d$ -based counterparts, see Jacob [27] §1, or Hoh [24] pp. 46–48.","By Theorem REF (REF ), if we can prove the existence of a continuous embedding $H^s \\hookrightarrow C_0(K|G|K)$ for all $s>\\frac{d}{2}$ , then Theorem REF (REF ) will follow.","Let $s>\\frac{d}{2}$ and $u\\in \\mathcal {S}(K|G|K)$ .","By Lemma REF , $\\langle \\cdot \\rangle ^{-s}\\in L^2(\\operatorname{\\mathfrak {a}}^\\ast ,\\omega )$ , and by spherical inversion, $|u(\\sigma )| = \\left|\\int _{}\\phi _\\lambda (\\sigma )\\hat{u}(\\lambda )\\omega (d\\lambda )\\right| \\le \\int _{}|\\hat{u}(\\lambda )|\\omega (d\\lambda ) = \\int _{}\\langle \\lambda \\rangle ^{-s}\\langle \\lambda \\rangle ^s|\\hat{u}(\\lambda )|\\omega (d\\lambda ),$ for all $\\sigma \\in G$ .","By the Cauchy–Schwarz inequality, $|u(\\sigma )| \\le \\Vert \\langle \\cdot \\rangle ^{-s}\\Vert _{L^2(,\\omega )}\\Vert \\langle \\cdot \\rangle ^s\\hat{u}\\Vert _{L^2(,\\omega )} = C\\Vert u\\Vert _s$ for all $\\sigma \\in G$ , where $C= \\Vert \\langle \\cdot \\rangle ^{-s}\\Vert _{L^2(,\\omega )}$ .","It follows that $\\Vert u \\Vert _{C_0(K|G|K)} := \\sup _{\\sigma \\in G}|u(\\sigma )| \\le C\\Vert u\\Vert _s.$ The embedding (REF ) may then be obtained using a density argument."],["Pseudodifferential Operators and Their Symbols","A measurable mapping $q:G\\times \\rightarrow \\mathbb {C}$ will be called a negative definite symbol if it is locally bounded, and if for each $\\sigma \\in G$ , $q(\\sigma ,\\cdot )$ is negative definite and continuous.","If in addition $q$ is continuous in its first argument, we will call $q$ a continuous negative definite symbol.","Let $\\mathcal {M}(G)$ denote the set of all measurable functions on $G$ .","Theorem 2.12 Let $q$ be a negative definite symbol, and for each $f\\in C^\\infty _c(K|G|K)$ and $\\sigma \\in G$ , define $q(\\sigma , D)f(\\sigma ) = \\int _{}\\hat{f}(\\lambda )\\phi _\\lambda (\\sigma )q(\\sigma ,\\lambda )\\omega (d\\lambda ).$ Then Equation (REF ) defines a linear operator $q(\\sigma ,D):C_c^\\infty (K|G|K)\\rightarrow \\mathcal {M}(G)$ .","If $q$ is a continuous negative definite symbol, then $q(\\sigma ,D):C_c^\\infty (K|G|K)\\rightarrow C(G)$ .","If $q$ is $K$ -bi-invariant in its first argument, then $q(\\sigma ,D)f$ is $K$ -bi-invariant for all $f\\in C_c^\\infty (K|G|K)$ .","Theorem REF (REF ) and (REF ) are proved in a similar manner to Theorem 4.5.7 of Jacob [28], while (REF ) is immediate from the $K$ -bi-invariance of each spherical function $\\phi _\\lambda $ .","Definition 2.13 Operators of the form (REF ), where $q$ is a negative definite symbol, will be called (spherical) pseudodifferential operators on $G$ .","Example 2.14 The following examples of pseudodifferential operators arise as the generators of Lévy processes on symmetric spaces.","Their symbols are constant in $\\sigma $ , a property that is well-known in the case of $G=\\mathbb {R}^d$ , $K=\\lbrace 0\\rbrace $ .","Section will introduce a large class of examples pseudodifferential operators with spatial dependence.","Diffusion operators with constant coefficients.","Consider the pure diffusion operator $\\mathcal {L} := \\sum _{i=1}^db_iX_i + \\sum _{i,j=1}^da_{ij}X_iX_j$ where $b$ and $a$ are as in Theorem REF .","Recall that such operators belong to ${\\bf D}_K(G)$ ; for each $\\lambda \\in $ , let $\\beta (\\mathcal {L},\\lambda )$ denote the $\\phi _\\lambda $ -eigenvalue of $\\mathcal {L}$ .","Then $(\\sigma ,\\lambda )\\mapsto -\\beta (T,\\lambda )$ is a continuous negative definite symbol, and the associated pseudodifferential operator is $-T$ .","To see this, let $(\\mu _t,t\\ge 0)$ denote the convolution semigroup generated by $\\mathcal {L}$ , and let $(T_t,t\\ge 0)$ be the associated Hunt semigroup, as defined in (REF ).","Then $\\mathcal {L}f(\\sigma ) = \\left.\\frac{d}{dt}T_tf(\\sigma )\\right|_{t=0},$ for all $f\\in C_c^\\infty (K|G|K)$ and $\\sigma \\in G$ .","By the spherical inversion formula (REF ), $T_tf(\\sigma ) = \\int _G\\int _{}\\hat{f}(\\lambda )\\phi _\\lambda (\\sigma \\tau )\\omega (d\\lambda )p_t(d\\tau ), \\hspace{20.0pt} \\forall f\\in C_c^\\infty (K|G|K),\\;\\sigma \\in G.$ Recalling that $\\hat{f}\\in \\mathcal {S}()$ whenever $f\\in C_c^\\infty (K|G|K)$ , a Fubini argument may be applied to conclude that $T_tf(\\sigma ) = \\int _{}\\hat{f}(\\lambda )T_t\\phi _\\lambda (\\sigma )\\omega (d\\lambda ) = \\int _{}\\hat{f}(\\lambda )e^{-t\\beta (\\mathcal {L},\\lambda )}\\phi _\\lambda (\\sigma )\\omega (d\\lambda ),$ for all $f\\in C_c^\\infty (K|G|K)$ and $\\sigma \\in G$ .","By (REF ), given $f\\in C_c^\\infty (K|G|K)$ and $\\sigma \\in G$ , $\\begin{aligned}\\mathcal {L}f(\\sigma ) &= \\left.\\frac{d}{dt}\\int _{}\\hat{f}(\\lambda )e^{-t\\beta (\\mathcal {L},\\lambda )}\\phi _\\lambda (\\sigma )\\omega (d\\lambda )\\right|_{t=0} \\\\&= \\lim _{t\\rightarrow 0}\\int _{}\\hat{f}(\\lambda )\\left(\\frac{e^{-t\\beta (\\mathcal {L},\\lambda )}-1}{t}\\right)\\phi _\\lambda (\\sigma )\\omega (d\\lambda ).\\end{aligned}$ Now, for all $t>0$ and $\\lambda \\in $ , $\\left|\\hat{f}(\\lambda )\\left(\\frac{e^{-t\\beta (\\mathcal {L},\\lambda )}-1}{t}\\right) \\phi _\\lambda (\\sigma )\\right| \\le \\left|\\hat{f}(\\lambda )\\right|\\left|\\frac{e^{-t\\beta (\\mathcal {L},\\lambda )}-1}{t}\\right| \\le \\left|\\hat{f}(\\lambda )\\right||\\beta (\\mathcal {L},\\lambda )|.$ Moreover, $|\\hat{f}||\\beta (\\mathcal {L},\\cdot )|\\in L^1(,\\omega )^W$ , since $\\hat{f}\\in \\mathcal {S}()^W$ , and $\\beta (\\mathcal {L},\\cdot )$ is a $W$ -invariant polynomial function.","By the dominated convergence theorem, we may bring the limit through the integral sign in (REF ) to conclude that $\\begin{aligned}\\mathcal {L}f(\\sigma ) &= \\int _{}\\hat{f}(\\lambda )\\lim _{t\\rightarrow 0}\\left(\\frac{e^{-t\\beta (\\mathcal {L},\\lambda )}-1}{t}\\right)\\phi _\\lambda (\\sigma )\\omega (d\\lambda ) \\\\&= -\\int _{}\\hat{f}(\\lambda )\\phi _\\lambda (\\sigma )\\beta (\\mathcal {L} ,\\lambda )\\omega (d\\lambda )\\end{aligned}$ for all $f\\in C_c^\\infty (K|G|K)$ and $\\sigma \\in G$ .","Brownian motion.","As a special case of the above, $-\\Delta $ is a pseudodifferential operator with symbol $|\\rho |^2+|\\lambda |^2$ .","Lévy generators.","More generally, if $\\mathcal {A}$ is the infinitesimal generator of a $K$ -bi-invariant Lévy process on $G$ , and if $\\psi $ is the corresponding Gangolli exponent, then $(\\sigma ,\\lambda )\\mapsto \\psi (\\lambda )$ is a continuous negative definite symbol, and $-\\mathcal {A}$ is the corresponding pseudodifferential operator — see Applebaum [4] Theorem 5.1."],["Gangolli Operators and the Hille–Yosida–Ray Theorem","We will soon define the class of pseudodifferential operators that will be of primary interest.","In this section, we motivate this definition with a short discussion of the Hille–Yosida–Ray theorem, and prove that our class of operators are pseudodifferential operators in the sense of Definition REF .","We finish the section with some examples.","Let $E$ be a locally compact, Hausdorff space, let $\\mathcal {C}$ be a closed subspace of $C_0(E)$ , and let $\\mathcal {F}(E)$ denote the space of all real-valued functions on $E$ .","A $C_0$ -semigroup $(T_t,t\\ge 0)$ defined on $C_0(K|G|K)$ is called sub-Feller if for all $f\\in C_0(E)$ , and all $t\\ge 0$ , $0\\le f\\le 1 ~\\Rightarrow ~ 0\\le T_tf\\le 1.$ A linear operator $A:\\mathcal {D}(A)\\rightarrow \\mathcal {F}(E)$ is said to satisfy the positive maximum principle, if for all $f\\in \\mathcal {D}(A)$ and $x_0\\in E$ such that $f(x_0)=\\sup _{x\\in E}f(x)\\ge 0$ , we have $Af(x_0)\\le 0$ .","The following theorem is an extended version of the Hille–Yosida–Ray theorem, which fully characterises the operators that extend to generators of sub-Feller semigroups on $C_0(E)$ .","Similar versions in which $E=\\mathbb {R}^d$ may found in [24], pp.","53, and [28], pp. 333.","For a proof, see Ethier and Kurz [14], pp. 165.","Theorem 3.1 (Hille–Yosida–Ray) A linear operator $(\\mathcal {A},\\operatorname{Dom}\\mathcal {A})$ on $C_0(E)$ is closable and its closure generates a strongly continuous, sub-Feller semigroup on $C_0(E)$ if and only if the following is satisfied: $\\operatorname{Dom}\\mathcal {A}$ is dense in $C_0(E)$ , $\\mathcal {A}$ satisfies the positive maximum principle, and There exists $\\alpha >0$ such that $\\operatorname{Ran}(\\alpha I-\\mathcal {A})$ is dense in $C_0(E)$ .","In their papers [7] and [8], Applebaum and Ngan found necessary and sufficient conditions for an operator defined on $C_c^\\infty (K|G|K)$ to satisfy Theorem REF (REF ), for the cases $E=G$ , $G/K$ and $K|G|K$ .","We will focus primarily on the case $E=K|G|K$ , since this is the realm in which the spherical transform is available.","A mapping $\\nu :G\\times \\mathcal {B}\\rightarrow [0,\\infty ]$ will be called a $K$ -bi-invariant Lévy kernel if it is $K$ -bi-invariant in its first argument, and if for all $\\sigma \\in G$ , $\\nu (\\sigma ,\\cdot )$ is a $K$ -bi-invariant Lévy measure.","Fix a system of exponential coordinate functions, as defined in Definition REF , and adopt all of the notation conventions from this definition.","Definition 3.2 An operator $\\mathcal {A}:C_c^\\infty (K|G|K)\\rightarrow \\mathcal {F}(G)$ will be called a Gangolli operator if there exist mappings $c, b_i, a_{jk}\\in C(K|G|K)$ ($1\\le i,j,k\\le d$ ), as well as a $K$ -bi-invariant Lévy kernel $\\nu $ , such that for all $f\\in C_c^\\infty (K|G|K)$ , $\\begin{aligned}\\mathcal {A}f(\\sigma ) = -c(\\sigma )f(\\sigma ) + \\sum _{i=1}^db_i&(\\sigma )X_if(\\sigma ) + \\sum _{j,k=1}^da_{jk}(\\sigma )X_jX_kf(\\sigma ) \\\\&+ \\int _G\\left(f(\\sigma \\tau )-f(\\sigma )-\\sum _{i=1}^dx_i(\\tau )X_if(\\sigma )\\right)\\nu (\\sigma ,d\\tau ),\\end{aligned}$ and if, in addition, $c$ is a non-negative mapping.","For all $\\sigma \\in G$ , $b(\\sigma ):=(b_1(\\sigma ),\\ldots ,b_d(\\sigma ))$ is an $\\operatorname{Ad}(K)$ -invariant vector, For all $\\sigma \\in G$ , $a(\\sigma ):=(a_{jk}(\\sigma ))$ is an $\\operatorname{Ad}(K)$ -invariant, non-negative definite, symmetric matrix.","For each $f\\in C_b(K|G|K)$ , the mappings $\\sigma \\mapsto \\int _Uf(\\tau )\\sum _{i=1}^dx(\\tau )^2\\nu (\\sigma ,d\\tau )$ and $\\sigma \\mapsto \\int _{U^c}f(\\tau )\\nu (\\sigma ,d\\tau )$ are continuous from $G$ to $[0,\\infty )$ .","$\\lim _{\\sigma \\rightarrow \\infty }\\nu (\\sigma ,U^c)=0$ .","The form (REF ) is necessary for $\\mathcal {A}$ to satisfy the positive maximum principle.","The various continuity assumptions, together with condition (REF ), ensure that $\\mathcal {A}$ maps into $C_0(K|G|K)$ .","By Theorem 3.2 (3) of [8] as well as Theorems 3.7 and 3.8 of Applebaum and Ngan [7], Gangolli operators are densely defined linear operators on $C_0(K|G|K)$ that satisfy the positive maximum principle.","Remarks 3.3 Gangolli operators were first introduced in [8] in compact symmetric spaces and for a more restrictive form of (REF ).","Equation (REF ) may be viewed as a spatially dependent generalisation of (REF ), with an additional killing term $c$ .","Indeed, if the spatial dependence of the coefficients $c$ , $b$ , $a$ and $\\nu $ is removed from (REF ), then the resulting operator extends to the generator of a killed Lévy process, with state space $G\\cup \\lbrace \\infty \\rbrace $ , the one-point compactification of $G$ .","For a more detailed discussion, see [7], pp. 149.","For a Gangolli operator $\\mathcal {A}$ given by (REF ), the local part of $\\mathcal {A}$ is defined $\\mathcal {L}^\\sigma = -c(\\sigma ) + \\sum _{i=1}^db_i(\\sigma )X_i + \\sum _{j,k=1}^da_{jk}(\\sigma )X_jX_k \\hspace{20.0pt} \\forall \\sigma \\in G.$ Holding $\\sigma \\in G$ fixed, $\\mathcal {L}^\\sigma +c(\\sigma )$ can be viewed as a diffusion operator with constant coefficients.","In particular, $\\mathcal {L}^\\sigma \\in {\\bf D}_K(G)$ for all $\\sigma \\in G$ .","Let $\\beta (\\mathcal {L}^\\sigma ,\\lambda )$ denote the $\\phi _\\lambda $ -eigenvalue of $\\mathcal {L}^\\sigma $ , so that $\\mathcal {L}^\\sigma \\phi _\\lambda = \\beta (\\mathcal {L}^\\sigma ,\\lambda )\\phi _\\lambda \\hspace{20.0pt} \\forall \\sigma \\in G,\\;\\lambda \\in .$ Lemma 3.4 Let $\\mathcal {A}$ be a Gangolli operator given by (REF ), and, using the notation introduced above, define $q:G\\times \\rightarrow \\mathbb {C}$ by $q(\\sigma ,\\lambda ) = -\\beta (\\mathcal {L}^\\sigma ,\\lambda ) + \\int _G(1-\\phi _\\lambda (\\tau ))\\nu (\\sigma ,d\\tau ), \\hspace{20.0pt} \\forall \\sigma \\in G,\\lambda \\in .$ Then $q$ is a continuous negative definite symbol.","That $q$ is continuous in its first argument is immediate from Definition REF .","Fix $\\sigma \\in G$ and consider $q(\\sigma ,\\cdot )-c(\\sigma )$ .","By Theorem REF , there is a convolution semigroup $(\\mu ^\\sigma _t,t\\ge 0)$ generated by $\\mathcal {A}+c(\\sigma )$ , and by Theorem REF , the corresponding Gangolli exponent is a continuous negative definite mapping on $$ , given by $\\psi ^\\sigma (\\lambda ) = q(\\sigma ,\\lambda )-c(\\sigma ) \\hspace{20.0pt} \\forall \\lambda \\in .$ Therefore $q(\\sigma ,\\cdot )$ is continuous, and negative definite since for fixed $\\sigma $ , $c(\\sigma )$ is a non-negative constant.","Definition 3.5 The symbols described by Lemma REF will be referred to as Gangolli symbols, due to their connection with Gangolli's Lévy–Khinchine formula.","Remarks 3.6 Gangolli exponents are precisely those Gangolli symbols constant in their first argument.","The set of all Gangolli symbols forms a convex cone.","Theorem 3.7 Let $\\mathcal {A}$ be a Gangolli operator of the form (REF ), and let $q$ be a Gangolli symbol given by (REF ).","Then $\\mathcal {A}$ is a pseudodifferential operator with symbol $q$ .","By Theorem REF and Lemma REF , $f\\mapsto -\\int _{}\\hat{f}(\\lambda )\\phi _\\lambda (\\sigma )q(\\sigma ,\\lambda )\\omega (d\\lambda )$ is a well-defined mapping from $C_c^\\infty (K|G|K)\\rightarrow C(G)$ .","We show that it is equal to $\\mathcal {A}$ .","Continue to denote the local part of $\\mathcal {A}$ by $\\mathcal {L}^\\sigma $ , as in (REF ), and let $\\mathcal {P}$ denote the non-local part, so that $\\mathcal {P}f(\\sigma ) = \\int _G\\left(f(\\sigma \\tau )-f(\\sigma )-\\sum _{i=1}^dx_i(\\tau )X_if(\\sigma )\\right)\\nu (\\sigma ,d\\tau )$ for all $f\\in C_c^\\infty (K|G|K)$ and $\\sigma \\in G$ .","By design, $\\mathcal {A}f(\\sigma ) = \\mathcal {L}^\\sigma f(\\sigma ) + \\mathcal {P}f(\\sigma ), \\hspace{20.0pt} \\forall f\\in C_c^\\infty (K|G|K),\\;\\sigma \\in G.$ Fix $\\sigma \\in G$ , and view $\\mathcal {L}^\\sigma + c(\\sigma )$ as a $K$ -bi-invariant diffusion operator with constant coefficients.","By (REF ) in Example REF (REF ), $\\left(\\mathcal {L}^\\sigma + c(\\sigma )\\right)f(\\tau ) = -\\int _{}\\hat{f}(\\lambda )\\phi _\\lambda (\\tau )\\beta (\\mathcal {L}^\\sigma +c(\\sigma ),\\lambda )\\omega (d\\lambda )$ for all $f\\in C_c^\\infty (K|G|K)$ and $\\tau \\in G$ , where $\\beta (\\mathcal {L}^\\sigma +c(\\sigma ),\\lambda )$ denotes the $\\phi _\\lambda $ -eigenvalue of $\\mathcal {L}^\\sigma +c(\\sigma )\\in {\\bf D}_K(G)$ .","Clearly $\\beta (\\mathcal {L}^\\sigma +c(\\sigma ),\\lambda ) = \\beta (\\mathcal {L}^\\sigma ,\\lambda )+c(\\sigma ), \\hspace{20.0pt} \\forall \\sigma \\in G,\\;\\lambda \\in ,$ and hence, expanding the right-hand side of (REF ) and applying the spherical inversion formula (REF ), $\\mathcal {L}^\\sigma f(\\tau ) = -\\int _{}\\hat{f}(\\lambda )\\beta (\\mathcal {L}^\\sigma ,\\lambda )\\phi _\\lambda (\\tau )\\omega (d\\lambda )$ for all $f\\in C_c^\\infty (K|G|K)$ and $\\sigma ,\\tau \\in G$ .","In particular, $\\mathcal {L}^\\sigma f(\\sigma ) = -\\int _{}\\hat{f}(\\lambda )\\beta (\\mathcal {L}^\\sigma ,\\lambda )\\phi _\\lambda (\\sigma )\\omega (d\\lambda ) \\hspace{20.0pt}\\forall f\\in C_c^\\infty (K|G|K),\\;\\sigma \\in G.$ Consider next the action of the non-local part $\\mathcal {P}$ .","By Lemma 2.3 on page 39 of Liao [34], for each fixed $\\sigma \\in G$ , and for all $f\\in C_b^2(K|G|K)$ , the integrand on the right-hand side of (REF ) is absolutely integrable with respect to $\\nu (\\sigma ,\\cdot )$ .","Therefore, (REF ) may be used to extend the domain of $\\mathcal {P}$ so as to include $C_b^2(K|G|K)$ .","We do so now, and (without any loss of precision) denote the extension by $\\mathcal {P}$ .","Let us proceed similarly to [8] Section 5, and define for each $\\sigma \\in G$ a linear functional $\\mathcal {P}^\\sigma :C_b^2(K|G|K)\\rightarrow \\mathbb {C}$ by $\\mathcal {P}^\\sigma f:= \\mathcal {P}\\left(L_\\sigma ^{-1}f\\right)(\\sigma ),$ for each $\\sigma \\in G$ and $f\\in C_b^2(K|G|K)$ .","Then $\\mathcal {P}f(\\sigma ) = \\mathcal {P}^\\sigma (L_\\sigma f)$ , for all $\\sigma \\in G$ and $f\\in C_b^2(K|G|K)$ , and hence $\\mathcal {P}^\\sigma \\phi _\\lambda = \\int _G\\left(L_\\sigma ^{-1}\\phi _\\lambda (\\sigma \\tau ) - L_\\sigma ^{-1}\\phi _\\lambda (\\sigma ) - \\sum _{i=1}^dx_i(\\tau )X_iL_\\sigma ^{-1}\\phi _\\lambda (\\sigma )\\right)\\nu (\\sigma ,d\\tau ),$ for all $\\lambda \\in $ and $\\sigma \\in G$ .","Moreover, by the above discussion, the integrand on the right-hand side of (REF ) is absolutely $\\nu (\\sigma ,\\cdot )$ -integrable, for all $\\lambda \\in $ and $\\sigma \\in G$ .","Now, $L_\\sigma ^{-1}\\phi _\\lambda (\\sigma \\tau ) - L_\\sigma ^{-1}\\phi _\\lambda (\\sigma ) - \\sum _{i=1}^dx_i(\\tau )X_iL_\\sigma ^{-1}\\phi _\\lambda (\\sigma ) &= \\phi _\\lambda (\\tau ) - \\phi _\\lambda (e) - \\sum _{i=1}^dx_i(\\tau )X_i\\phi _\\lambda (e) \\\\&= \\phi _\\lambda (\\tau ) - 1,$ since $\\phi _\\lambda (e)=1$ , and, by Theorem 5.3 (b) on page 139 of Liao [34], $X\\phi _\\lambda (e)=0$ for all $X\\in \\mathfrak {p}$ .","Thus, for all $\\lambda \\in $ and $\\sigma \\in G$ , $\\phi _\\lambda -1$ is absolutely $\\nu (\\sigma ,\\cdot )$ -integrable, and $\\mathcal {P}^\\sigma \\phi _\\lambda = \\int _G\\left(\\phi _\\lambda (\\tau ) - 1\\right)\\nu (\\sigma ,d\\tau ).$ A standard argument involving the functional equation $\\phi _\\lambda (\\sigma )\\phi _\\lambda (\\tau ) = \\int _K\\phi _\\lambda (\\sigma k\\tau )dk, \\hspace{20.0pt} \\forall \\lambda \\in ,\\sigma ,\\tau \\in G,$ for spherical functions (see [21] Proposition 2.2, pp.","400) may now be applied in precisely the same way as in [8] (5.3)–(5.7), to infer that $\\mathcal {P}\\phi _\\lambda (\\sigma ) = \\int _G(\\phi _\\lambda (\\sigma \\tau ) - \\phi _\\lambda (\\sigma ))\\nu (\\sigma ,d\\tau ) = \\int _G(\\phi _\\lambda (\\tau ) - 1)\\phi _\\lambda (\\sigma )\\nu (\\sigma ,d\\tau )$ for all $\\sigma \\in G$ and $\\lambda \\in $ .","Finally, let $f\\in C_c^\\infty (K|G|K)$ , and observe that by the spherical inversion formula $\\begin{aligned}\\mathcal {P}f(\\sigma ) &= \\int _G\\Bigg (\\int _{}\\phi _\\lambda (\\sigma \\tau )\\hat{f}(\\lambda )\\omega (d\\lambda ) - \\int _{}\\phi _\\lambda (\\sigma )\\hat{f}(\\lambda )\\omega (d\\lambda ) \\\\&\\hspace{100.0pt}-\\sum _{i=1}^dx_i(\\tau )X_i\\left[\\int _{}\\phi _\\lambda \\hat{f}(\\lambda )\\omega (d\\lambda )\\right](\\sigma )\\Bigg )\\nu (\\sigma ,d\\tau )\\end{aligned}$ Claim For all $X\\in \\mathfrak {p}$ and $f\\in C_c^\\infty (K|G|K)$ , $X\\left[\\int _{}\\phi _\\lambda \\hat{f}(\\lambda )\\omega (d\\lambda )\\right](\\sigma )=\\int _{}X\\phi _\\lambda (\\sigma )\\hat{f}(\\lambda )\\omega (d\\lambda ).$ Proof of Claim This is a fairly standard differentiation-through-integration-sign argument.","First note that by translation invariance of $X$ , it suffices to prove the claim for $\\sigma =e$ .","Now, $X\\left[\\int _{}\\phi _\\lambda \\hat{f}(\\lambda )\\omega (d\\lambda )\\right](e) &= \\left.\\frac{d}{dt}\\int _{}\\phi _\\lambda (\\exp tX)\\hat{f}(\\lambda )\\omega (d\\lambda )\\right|_{t=0} \\\\&= \\lim _{t\\rightarrow 0}\\int _{}\\frac{\\phi _\\lambda (\\exp tX)-1}{t}\\hat{f}(\\lambda )\\omega (d\\lambda ).$ By the mean value theorem, for each $t>0$ and $\\lambda \\in $ , $\\frac{\\phi _\\lambda (\\exp tX)-1}{t}=X\\phi _\\lambda (\\exp t^{\\prime }X),$ for some $00$ .","By Helgason [20] Theorem 1.1 (iii), $\\Vert X\\phi _\\lambda \\Vert _\\infty \\le C(1+|\\lambda |)$ , for some some constant $C>0$ .","Thus, for $f\\in C_c^\\infty (K|G|K)$ , $\\lambda \\in $ and $t>0$ , $\\left|\\frac{\\phi _\\lambda (\\exp tX)-1}{t}\\hat{f}(\\lambda )\\right| \\le C(1+|\\lambda |)|\\hat{f}(\\lambda )|,$ and clearly $C(1+|\\cdot |)\\hat{f} \\in L^1()^W$ , since $\\hat{f}\\in \\mathcal {S}()$ .","Hence by dominated convergence, $X\\left[\\int _{}\\phi _\\lambda \\hat{f}(\\lambda )\\omega (d\\lambda )\\right](e) = \\int _{}\\lim _{t\\rightarrow 0}\\frac{\\phi _\\lambda (\\exp tX)-1}{t}\\hat{f}(\\lambda )\\omega (d\\lambda ) = \\int _{}X\\phi _\\lambda (e)\\hat{f}(\\lambda )\\omega (d\\lambda ),$ which completes the proof of the claim.","Applying the claim to (REF ), for $f\\in C_c^\\infty (K|G|K)$ and $\\sigma \\in G$ , $\\mathcal {P}f(\\sigma ) &= \\int _G\\Bigg (\\int _{}\\phi _\\lambda (\\sigma \\tau )\\hat{f}(\\lambda )\\omega (d\\lambda ) - \\int _{}\\phi _\\lambda (\\sigma )\\hat{f}(\\lambda )\\omega (d\\lambda ) \\\\&\\hspace{150.0pt}-\\sum _{i=1}^dx_i(\\tau )\\int _{}X_i\\phi _\\lambda (\\sigma )\\hat{f}(\\lambda )\\omega (d\\lambda )\\Bigg )\\nu (\\sigma ,d\\tau ) \\\\&=\\int _G\\int _{}\\hat{f}(\\lambda )\\left(\\phi _\\lambda (\\sigma \\tau )-\\phi _\\lambda (\\sigma )-\\sum _{i=1}^dx_i(\\tau )X_i\\phi _\\lambda (\\sigma )\\right)\\omega (d\\lambda )\\nu (\\sigma ,d\\tau ).$ By the Fubini theorem, $\\mathcal {P}f(\\sigma ) &= \\int _{}\\hat{f}(\\lambda )\\int _G\\left(\\phi _\\lambda (\\sigma \\tau )-\\phi _\\lambda (\\sigma )-\\sum _{i=1}^dx_i(\\tau )X_i\\phi _\\lambda (\\sigma )\\right)\\nu (\\sigma ,d\\tau )\\omega (d\\lambda ) \\\\&= \\int _{}\\hat{f}(\\lambda )\\mathcal {P}\\phi _\\lambda (\\sigma )\\omega (d\\lambda )$ for all $f\\in C_c^\\infty (K|G|K)$ and $\\sigma \\in G$ .","It follows by (REF ) that $\\mathcal {P}f(\\sigma ) = \\int _{}\\hat{f}(\\lambda )\\phi _\\lambda (\\sigma )\\int _G(\\phi _\\lambda (\\tau ) - 1)\\nu (\\sigma ,d\\tau )\\omega (d\\lambda )$ for all $f\\in C_c^\\infty (K|G|K)$ and $\\sigma \\in G$ .","The result now follows by substituting (REF ) and (REF ) into (REF ).","Example 3.8 As already noted, all Gangolli exponents are Gangolli symbols.","Let $u\\in C(K|G|K)$ be non-negative, and let $v:\\rightarrow \\mathbb {C}$ be a Gangolli exponent.","Then $q:G\\times \\rightarrow \\mathbb {C}$ given by $q(\\sigma ,\\lambda ) = u(\\sigma )v(\\lambda ) \\hspace{20.0pt} \\forall \\sigma \\in G,\\;\\lambda \\in $ is a Gangolli symbol.","Indeed, by Theorem REF , there exists a sub-diffusion operator $T\\in {\\bf D}_K(G)$ and a $K$ -bi-invariant Lévy measure $\\nu $ such that, $v(\\lambda ) = -\\beta (T,\\lambda ) + \\int _G(1-\\phi _\\lambda (\\sigma ))\\nu (d\\tau )$ for all $\\lambda \\in $ .","But then, for each $\\sigma \\in G$ and $\\lambda \\in $ $q(\\sigma ,\\lambda ) = -\\beta (u(\\sigma )T,\\lambda ) + \\int _G(1-\\phi _\\lambda (\\sigma ))\\nu (d\\tau ).$ Writing $\\mathcal {L}^\\sigma =u(\\sigma )T$ and $\\nu (\\sigma ,d\\tau ) = u(\\sigma )\\nu (d\\tau )$ for each $\\sigma \\in G$ , $q$ resembles (REF ).","Since $T$ is a sub-diffusion operator, there is a non-negative constant $c$ , a vector $b\\in \\mathbb {R}^d$ and a non-negative definite symmetric matrix $(a_{jk})$ such that $T = -c + \\sum _{i=1}^db_iX_i+\\sum _{j,k=1}^da_{jk}X_jX_k.$ For each $\\sigma \\in G$ , and for $i,j,k=1,\\ldots ,d$ , let $c(\\sigma ) = cu(\\sigma ), \\hspace{20.0pt} b_i(\\sigma ) = b_iu(\\sigma ), \\hspace{20.0pt} a_{jk}(\\sigma ) = a_{jk}u(\\sigma )\\hspace{10.0pt} \\text{and} \\hspace{10.0pt} \\nu (\\sigma ,\\cdot )=\\nu .$ Since $u$ is non-negative, continuous and $K$ -bi-invariant, the conditions of Definition REF are easily verified for these characteristics."],["Construction of Sub-Feller Semigroups","In the previous section, we used results from Applebaum and Ngan [7] and [8] to introduce a class of pseudodifferential operators that satisfy all but the last condition of the Hille–Yosida–Ray theorem, Theorem REF (REF ), in the case where $E=K|G|K$ .","In this section we tackle this the third condition, and seek conditions on a symbol $q$ so that, for some $\\alpha >0$ , $\\overline{\\operatorname{Ran}(\\alpha +q(\\sigma ,D))}=C_0(K|G|K).$ We use an approach based primarily on Jacob [27] and Hoh [24] Section 4.","Now that we are on the level of operators, there are more arguments that closely resemble these sources.","In these cases, proofs are not expanded in great detail, and may be omitted entirely to save space.","Instead, we aim to emphasise what does not carry over from the Euclidean space setting.","As in previous work, let $\\psi :\\rightarrow \\mathbb {R}$ be a fixed real-valued, continuous negative definite function satisfying (REF ) for some fixed $r>0$ .","The next lemma will be needed later on for proving regularity properties of pseudodifferential operators — see [24] Lemma 4.2 on page 48 for comparison.","Before stating the result, we introduce some notation.","For a mapping $q:G\\times \\rightarrow \\mathbb {R}$ and for each $\\lambda ,\\eta \\in $ , $\\sigma \\in G$ , define $F_{\\lambda ,\\eta }(\\sigma ) = \\phi _{-\\lambda }(\\sigma )q(\\sigma ,\\eta ).$ Observe that if $q(\\cdot ,\\eta )\\in L^2(K|G|K)$ for all $\\eta \\in $ , then $F_{\\lambda ,\\eta }\\in L^2(K|G|K)$ , and we may consider the spherical transform $\\hat{F}_{\\lambda ,\\eta }\\in L^2(,\\omega )$ , given by $\\hat{F}_{\\lambda ,\\eta }(\\mu ) = \\int _G\\phi _{-\\mu }(\\sigma )\\phi _{-\\lambda }(\\sigma )q(\\sigma ,\\eta )d\\sigma , \\hspace{20.0pt} \\forall \\mu \\in .$ To give some intuition to the introduction of $F_{\\lambda ,\\eta }$ , consider the case $G=\\mathbb {R}^d$ , $K=\\lbrace 0\\rbrace $ .","In this case, the so-called frequency shift property for the Fourier transform says that $\\begin{aligned}\\hat{F}_{\\lambda ,\\eta }(\\mu ) &= \\frac{1}{(2\\pi )^{d/2}}\\int _{\\mathbb {R}^d}e^{-i\\mu \\cdot x}e^{-\\lambda \\cdot x}q(x,\\eta )dx \\\\&= \\frac{1}{(2\\pi )^{d/2}}\\int _{\\mathbb {R}^d}e^{-i(\\mu +\\lambda )\\cdot x}q(x,\\eta )dx = \\hat{q}(\\lambda +\\mu ,\\eta ),\\end{aligned}$ where $^{\\wedge }$ denotes the Fourier transform taken in the first argument of $q$ .","Hoh [24] and Jacob [29] make use of bounds on $\\hat{q}(\\lambda -\\mu ,\\eta )$ , and $\\hat{F}_{\\lambda ,\\eta }(-\\mu )$ will assume an analogous role in work to come.","The next lemma is an analogue of Lemma 2.1 of Jacob [27].","See also Hoh [24] Lemma 4.2, pp. 48.","It's proof is similar to these other lemmas, but some additional steps are required to work around some new difficulties that arise in this setting.","Lemma 4.1 Let $M\\in \\mathbb {N}$ , $q:G\\times \\rightarrow \\mathbb {R}$ and suppose $q(\\cdot ,\\lambda )\\in C^M_c(K|G|K)$ for all $\\lambda \\in $ .","Suppose further that for each $\\beta \\in \\lbrace 0,1,\\ldots ,M\\rbrace $ , there is a non-negative function $\\Phi _\\beta \\in L^1(K|G|K)$ such that, in the notation above, $\\left|(-\\Delta )^{\\beta /2}F_{\\lambda ,\\eta }(\\sigma )\\right| \\le \\Phi _\\beta (\\sigma )\\langle \\lambda \\rangle ^M(1+\\psi (\\eta )),$ for all $\\lambda ,\\eta \\in $ , $\\sigma \\in G$ .","Then there is a constant $C_M>0$ such that $\\left|\\hat{F}_{\\lambda ,\\eta }(\\mu )\\right| \\le C_M\\sum _{\\beta =0}^M\\Vert \\Phi _\\beta \\Vert _1\\langle \\lambda +\\mu \\rangle ^{-M}(1+\\psi (\\eta )),$ for all $\\lambda ,\\mu ,\\eta \\in $ , where $\\Vert \\cdot \\Vert _1$ denotes the usual norm on the Banach space $L^1(K|G|K)$ (taken with respect to Haar measure).","Remarks 4.2 Here, we are again using the notation $\\langle \\lambda \\rangle =\\sqrt{1+|\\lambda |^2}$ , introduced in (REF ).","The condition (REF ) may seem quite obscure.","The role of $\\langle \\lambda +\\mu \\rangle $ will hopefully become apparent in the proof of Theorem REF .","In Section , we will develop a class of examples where it is satisfied.","Under the conditions of the lemma, and using the Fubini theorem, we have the following: for all $u\\in C_c^\\infty (K|G|K)$ and $\\lambda \\in $ , $\\begin{aligned}(q(\\sigma ,D)u)^{\\wedge }(\\lambda ) &= \\int _C\\int _{}\\phi _{-\\lambda }(\\sigma )\\phi _\\eta (\\sigma )q(\\sigma ,\\eta )\\hat{u}(\\eta )\\omega (d\\eta )d\\sigma \\\\&= \\int _{}\\left(\\int _C\\phi _\\eta (\\sigma )F_{\\lambda ,\\eta }(\\sigma )d\\sigma \\right)\\hat{u}(\\eta )\\omega (d\\eta ) \\\\&= \\int _{}\\hat{F}_{\\lambda ,\\eta }(-\\eta )\\hat{u}(\\eta )\\omega (d\\eta ).\\end{aligned}$ Fubini's theorem does indeed apply here — a suitable bound for the integrand on the first line of (REF ) may be found by noting that for all $\\lambda ,\\eta \\in $ and $\\sigma \\in G$ , $|\\phi _{-\\lambda }(\\sigma )\\phi _\\eta (\\sigma )q(\\sigma ,\\eta )\\hat{u}(\\eta )|\\le |q(\\sigma ,\\eta )||\\hat{u}(\\eta )|\\le \\Phi _0(\\sigma )\\big (1+\\psi (\\eta )\\big )|\\hat{u}(\\eta )|,$ for some $\\Phi _0\\in L^1(K|G|K)$ , by (REF ).","By Theorem REF , $\\hat{u}\\in \\mathcal {S}()$ , and the usual bound (REF ) on the density of Plancherel measure may be applied, similarly to (REF ), to conclude that the right-hand side of (REF ) is $\\omega (d\\eta )\\times d\\sigma $ -integrable.","[Proof of Lemma REF ] Let $\\beta \\in \\lbrace 0,1,\\ldots ,M\\rbrace $ and $\\lambda ,\\eta \\in $ be fixed.","The fractional Laplacian $(-\\Delta )^{\\beta /2}$ satisfies a well-known eigenrelation $(-\\Delta )^{\\beta /2}\\phi _\\mu = \\left(|\\rho |^2+|\\mu |^2\\right)^{\\beta /2}\\phi _\\mu , \\hspace{20.0pt} \\forall \\mu \\in ,$ which may be proven using subordination methods and properties of the Laplace-Beltrami operator on a symmetric space, using similar techniques to Section 5.7 of [5], pp. 154–7.","One can also show using standard methods that $\\left((-\\Delta )^{\\beta /2}f\\right)^{\\wedge }(\\mu ) = \\left(|\\rho |^2+|\\mu |^2\\right)^{\\beta /2}\\hat{f}(\\mu ),$ for all $f\\in C^M_c(K|G|K)$ and $\\mu \\in $ .","Then, using the definition of the spherical transform, $(|\\rho |^2+|\\mu |^2)^{\\beta /2}\\hat{f}(\\mu ) = \\int _G\\phi _{-\\mu }(\\sigma )(-\\Delta )^{\\beta /2}f(\\sigma )d\\sigma , $ for all $f\\in C_c^M(K|G|K)$ and all $\\mu \\in $ .","Applying this to $f=F_{\\lambda ,\\eta }$ , we have for all $\\mu \\in \\operatorname{\\mathfrak {a}}^\\ast $ , $\\left|\\left(|\\rho |^2+|\\mu |^2\\right)^{\\beta /2}\\hat{F}_{\\lambda ,\\eta }(\\mu )\\right| &\\le \\int _G|\\phi _{-\\mu }(\\sigma )|\\left|(-\\Delta )^{\\beta /2}F_{\\lambda ,\\eta }(\\sigma )\\right|d\\sigma \\\\&\\le \\int _G\\Phi _\\beta (\\sigma )\\langle \\lambda \\rangle ^M(1+\\psi (\\eta ))d\\sigma = \\Vert \\Phi _\\beta \\Vert _1\\langle \\lambda \\rangle ^M(1+\\psi (\\eta )),$ and summing over $\\beta $ , $\\sum _{\\beta =0}^M\\left(|\\rho |^2+|\\mu |^2\\right)^{\\beta /2}\\left|\\hat{F}_{\\lambda ,\\eta }(\\mu )\\right| \\le \\sum _{\\beta =0}^M\\Vert \\Phi _\\beta \\Vert _1\\langle \\lambda \\rangle ^M(1+\\psi (\\eta )),$ for all $\\lambda ,\\mu ,\\eta \\in $ .","Let $C^{\\prime }_M>0$ be the smallest positive number such that $\\langle \\mu \\rangle ^M \\le C^{\\prime }_M\\sum _{\\beta =0}^M\\left(|\\rho |^2+|\\mu |^2\\right)^{\\beta /2} \\hspace{20.0pt} \\forall \\mu \\in .$ Then, rearranging (REF ), $\\left|\\hat{F}_{\\lambda ,\\eta }(\\mu )\\right| \\le C^{\\prime }_M\\sum _{\\beta =0}^M\\Vert \\Phi _\\beta \\Vert _1\\langle \\mu \\rangle ^{-M}\\langle \\lambda \\rangle ^M(1+\\psi (\\eta )),$ for all $\\lambda ,\\mu ,\\eta \\in $ .","Finally, observe that by Peetre's inequality (see Proposition REF (REF )), $\\langle \\lambda \\rangle ^M\\langle \\lambda +\\mu \\rangle ^{-M} = \\left(\\frac{1+|\\lambda |^2}{1+|\\lambda +\\mu |^2}\\right)^{M/2} \\le 2^{M/2}(1+|\\mu |^2)^{M/2} = 2^{M/2}\\langle \\mu \\rangle ^M$ for all $\\lambda ,\\mu \\in $ .","Therefore, for all $\\lambda ,\\mu \\in $ , $\\langle \\mu \\rangle ^{-M}\\langle \\lambda \\rangle ^M\\le 2^{M/2}\\langle \\lambda +\\mu \\rangle ^{-M}$ and by (REF ), $\\left|\\hat{F}_{\\lambda ,\\eta }(\\mu )\\right| \\le 2^{M/2}C^{\\prime }_M\\sum _{\\beta =0}^M\\Vert \\Phi _\\beta \\Vert _1\\langle \\lambda +\\mu \\rangle ^{-M}(1+\\psi (\\eta ))$ The result now follows by taking $C_M=2^{M/2}C^{\\prime }_M$ .","Remark 4.3 The constant $C_M:=2^{M/2}\\sup _{\\lambda \\in }\\frac{\\langle \\lambda \\rangle ^M}{\\sum _{\\beta =0}^M\\big (|\\rho |^2+|\\lambda |^2\\big )^{\\beta /2}}$ appearing in the proof of Lemma REF will remain relevant throughout this chapter.","Let now $q:G\\times \\rightarrow \\mathbb {R}$ be a continuous negative definite symbol, $K$ -bi-invariant in its first argument, and $W$ -invariant in its second (for example, $q$ could be taken to be a Gangolli symbol, as in (REF )).","Similarly to Jacob [27] §4 and Hoh [24] (4.26), we write $q(\\sigma ,\\lambda ) = q_1(\\lambda )+q_2(\\sigma ,\\lambda ), \\hspace{20.0pt} \\forall \\sigma \\in G,\\lambda \\in ,$ where $q_1(\\lambda )=q(\\sigma _0,\\lambda )$ and $q_2(\\sigma ,\\lambda )=q(\\sigma ,\\lambda )-q(\\sigma _0,\\lambda )$ , for some fixed $\\sigma _0\\in G$ .","Observe that $q_1$ is necessarily a negative definite symbol.","Though $q_2$ may not be, we may still define the operator $q_2(\\sigma ,D)$ in a meaningful way, by $q_2(\\sigma ,D) := q(\\sigma ,D) - q_1(D) = \\int _{}\\phi _\\lambda (\\sigma )q_2(\\sigma ,\\lambda )\\hat{f}(\\lambda )\\omega (d\\lambda ), \\hspace{20.0pt} \\forall \\sigma \\in G.$ By decomposing $q$ in this way, we view it as a perturbation of a negative definite function $q_1$ by $q_2$ .","The assumptions we place on $q$ will control the size of this perturbation, as well ensuring certain regularity properties of $q(\\sigma ,D)$ acting on the anisotropic Sobolev spaces introduced in Section REF .","Assumptions 4.4 In the notation above, we impose the following: There exist constants $c_0,c_1>0$ such that for all $\\lambda \\in $ with $|\\lambda |\\ge 1$ , $c_0(1+\\psi (\\lambda ))\\le q_1(\\lambda ) \\le c_1(1+\\psi (\\lambda )).$ Let $M\\in \\mathbb {N}$ , $M>\\dim (G/K)$ , and suppose that $q_2(\\cdot ,\\lambda )\\in C^M_c(K|G|K)$ for all $\\lambda \\in $ .","Suppose further that for $\\beta =0,1,\\ldots ,M$ , there exists $\\Phi _\\beta \\in L^1(K|G|K)$ such that $\\left|(-\\Delta )^{\\beta /2}F_{\\lambda ,\\eta }(\\sigma )\\right| \\le \\Phi _\\beta (\\sigma )\\langle \\lambda \\rangle ^M\\big (1+\\psi (\\eta )\\big ),$ for all $\\lambda ,\\eta \\in $ , $\\sigma \\in G$ , where $F_{\\lambda ,\\eta }(\\sigma )=\\phi _{-\\lambda }(\\sigma )q_2(\\sigma ,\\eta )$ (c.f.","(REF )).","Remarks 4.5 These assumptions are analogues to P.1, P.2.q of Jacob [27], pp.","156, or (A.1), (A.2.M) of Hoh [24], pp.54.","As noted in Remark REF (REF ), the conditions in Assumption REF (REF ) imply that $(q_2(\\sigma ,D)u)^{\\wedge }(\\lambda ) = \\int _{}\\hat{F}_{\\lambda ,\\eta }(-\\eta )\\hat{u}(\\eta )\\omega (d\\eta ),$ for all $\\lambda \\in $ and $u\\in C_c^\\infty (K|G|K)$ , a fact that will be useful several times more.","Theorem 4.6 Subject to Assumptions REF , for all $s\\in \\mathbb {R}$ , $q_1(D)$ extends to a continuous operator from $H^{\\psi ,s+2}$ to $H^{\\psi ,s}$ , and $q(\\sigma ,D)$ extends to a continuous operator from $H^{\\psi ,2}$ to $L^2(K|G|K)$ .","The proof of the first part is omitted, since it is an easy adaptation of the proof of Theorem 4.8 on page 55 of Hoh [24] — first proved as Corollary 3.1 in Jacob [27].","The second part is also proved similarly to Theorem 4.8 of [24], the main difference being that $\\hat{F}_{\\lambda ,\\eta }(-\\eta )$ takes the place of the transformed symbol, as discussed previously (see (REF )).","By (REF ) and the Plancherel theorem, $|\\langle q_2(\\sigma ,D)u,v\\rangle | &= \\left|\\int _{}(q_2(\\sigma ,D)u)^{\\wedge }(\\lambda )\\overline{\\hat{v}(\\lambda )}\\omega (d\\lambda )\\right| \\\\&= \\left|\\int _{}\\int _{}\\hat{F}_{\\lambda ,\\eta }(-\\eta )\\hat{u}(\\eta )\\overline{\\hat{v}(\\lambda )}\\omega (d\\eta )\\omega (d\\lambda )\\right| \\\\&\\le \\int _{}\\int _{}\\left|\\hat{F}_{\\lambda ,\\eta }(-\\eta )\\right||\\hat{u}(\\eta )||\\hat{v}(\\lambda )|\\omega (d\\eta )\\omega (d\\lambda ).$ Then, using (REF ), Lemma REF and Young's convolution inequalitySee Simon [43] Theorem 6.6.3, page 550.","Here, we are again identifying $$ with a Euclidean space., $|\\langle q_2(\\sigma ,D)u,v\\rangle | &\\le C_M\\sum _{\\beta =0}^M\\Vert \\Phi _\\beta \\Vert _1\\int _{}\\int _{}\\langle \\lambda -\\eta \\rangle ^{-M}\\Psi (\\eta )^2|\\hat{u}(\\eta )||\\hat{v}(\\lambda )|\\omega (d\\eta )\\omega (d\\lambda ) \\\\&= C_M\\sum _{\\beta =0}^M\\Vert \\Phi _\\beta \\Vert _1\\int _{}\\left[\\langle \\cdot \\rangle ^{-M}\\ast \\big (\\Psi ^2|\\hat{u}|\\big )\\right](\\lambda )|\\hat{v}(\\lambda )|\\omega (d\\lambda ) \\\\&\\le C_M\\sum _{\\beta =0}^M\\Vert \\Phi _\\beta \\Vert _1\\left\\Vert \\langle \\cdot \\rangle ^{-M}\\ast \\big (\\Psi ^2|\\hat{u}|\\big )\\right\\Vert _{L^2(,\\omega )}\\Vert \\hat{v}\\Vert _{L^2(,\\omega )} \\\\&\\le C_M\\sum _{\\beta =0}^M\\Vert \\Phi _\\beta \\Vert _1\\left\\Vert \\langle \\cdot \\rangle ^{-M}\\right\\Vert _{L^1(,\\omega )}\\Vert u\\Vert _{\\psi ,2}\\Vert v\\Vert ,$ for all $u,v\\in C_c^\\infty (K|G|K)$ .","Hence, for all $u\\in C_c^\\infty (K|G|K)$ , $\\Vert q_2(\\sigma ,D)u\\Vert &= \\sup _{\\begin{array}{c}v\\in C_c^\\infty (K|G|K) \\\\ \\Vert v\\Vert =1\\end{array}}|\\langle q_2(\\sigma ,D)u,v\\rangle | \\le C_M\\sum _{\\beta =0}^M\\Vert \\Phi _\\beta \\Vert _1\\left\\Vert \\langle \\cdot \\rangle ^{-M}\\right\\Vert _{L^1(,\\omega )}\\Vert u\\Vert _{\\psi ,2},$ and $q_2(\\sigma ,D)$ extends to a bounded linear operator $H^{\\psi ,2}\\rightarrow L^2(K|G|K)$ .","Under an additional assumption, we are able to obtain a more powerful result.","Theorem 4.7 Suppose Assumptions REF hold, and suppose further that $s\\in \\mathbb {R}$ satisfies $|s-1|+1+\\dim (G/K) < M$ .","Then $q(\\sigma ,D)$ extends to a continuous linear operator from $H^{\\psi ,s+2}\\rightarrow H^{\\psi ,s}$ .","We first need a technical lemma.","Lemma 4.8 Let $s\\in \\mathbb {R}$ and $M\\in \\mathbb {N}$ be such that $|s-1|+1+\\dim (G/K)0$ .","To do this, we seek solutions $u$ to the equation $(q(\\sigma ,D)+\\alpha )u=f,$ for a given function $f$ and $\\alpha >0$ .","For $\\alpha >0$ , consider the bilinear form $B_\\alpha $ defined for all $u,v\\in C_c^\\infty (K|G|K)$ by $B_\\alpha (u,v) = \\langle (q(\\sigma ,D)+\\alpha )u,v\\rangle .$ The next result is an analogue of Jacob [27] Lemma 3.2, pp.","160, and Hoh [24] Theorem 4.9, pp. 56.","Theorem 4.9 Subject to Assumptions REF , $B_\\alpha $ extends continuously to $H^{\\psi ,1}\\times H^{\\psi ,1}$ .","This proof is very similar to those of Jacob [27] Lemma 3.2, pp.","160, and Hoh [24] Theorem 4.9, pp.","56, and so we give only a sketch.","Let $u,v\\in H^{\\psi ,1}$ .","Using Assumption REF (REF ) and the fact that $q_1$ is continuous, there is $\\kappa _1>0$ such that $|q_1|\\le \\kappa _1\\Psi ^2$ .","Plancherel's identity may then be used to show that $\\left|\\langle q_1(D)u,v\\rangle \\right| \\le \\int _{}|q_1(\\lambda )||\\hat{u}(\\lambda )||\\hat{v}(\\lambda )|\\omega (d\\lambda ) \\le \\kappa _1\\Vert u\\Vert _{\\psi ,1}\\Vert v\\Vert _{\\psi ,1}.$ Furthermore, methods similar to the proof of Theorem REF are used to show that $\\left|\\langle q_2(\\sigma ,D)u,v\\rangle \\right| \\le \\kappa _2\\left\\Vert \\langle \\cdot \\rangle ^{-M+1}\\right\\Vert _{L^1(,\\omega )}\\Vert u\\Vert _{\\psi ,1}\\Vert v\\Vert _{\\psi ,1},$ where $\\kappa _2=C_M\\sqrt{2(1+c_\\psi )}\\sum _{\\beta =0}^M\\Vert \\Phi _\\beta \\Vert _{L^1(,\\omega )}.$ By Theorem REF (REF ), there is $\\kappa _3>0$ such that $\\Vert u\\Vert \\le \\kappa _3\\Vert u\\Vert _{\\psi ,1}$ , and thus for all $u,v\\in H^{\\psi ,1}$ , $\\left|B_\\alpha (u,v)\\right| \\le \\left|\\langle q_1(D)u,v\\rangle \\right| + \\left|\\langle q_2(\\sigma , D)u,v\\rangle \\right| + \\alpha \\left|\\langle u,v\\rangle \\right| \\le \\left(\\kappa _1 + \\kappa _2 + \\alpha \\kappa _3^2\\right)\\Vert u\\Vert _{\\psi ,1}\\Vert v\\Vert _{\\psi ,2},$ which proves the theorem.","Recall that a bilinear form $B$ defined on a Hilbert space $(H,\\langle \\cdot ,\\cdot \\rangle )$ is coercive if there is $c>0$ such that $B(u,u)\\ge c\\langle u,u\\rangle \\hspace{20.0pt} \\forall u\\in H.$ Theorem 4.10 (Lax–Milgram) Let $B$ be a bounded bilinear form, defined on a Hilbert space $(H,\\langle \\cdot ,\\cdot ,\\rangle )$ , and suppose $B$ is coercive with constant $c$ .","Then given $f\\in H^{\\prime }$ , there is a unique $u\\in H$ such that $B(u,v) = f(v) \\hspace{20.0pt} \\forall v\\in H.$ See for example Theorem 1 of Evans [15], pp. 297–9.","We would like to apply the Lax–Milgram theorem to $B_\\alpha $ .","For this we impose an additional assumption, which will ensure $B_\\alpha $ is coercive on $H^{\\psi ,1}$ .","Assumption 4.11 Let $M\\in \\mathbb {N}$ , $M>\\dim (G/K)+1$ , and write $\\gamma _M = \\left(8C_M(2(1+c_\\psi ))^{1/2}\\Vert \\langle \\cdot \\rangle ^{-M+1}\\Vert _{L^1(,\\omega )}\\right)^{-1},$ where $c_{\\psi }$ and $C_M$ are constants given by (REF ) and (REF ), respectively.","For $c_0$ is as in Assumption REF (REF ), assume that $\\sum _{\\beta =0}^M\\Vert \\Phi _\\beta \\Vert _1 \\le \\gamma _Mc_0.$ Remark 4.12 See Jacob [27] P.3 and P.4, pp.","161, or Hoh [24] (A.3.M), pp.","54, for comparison.","Examples where Assumption REF is satisfied are considered in Section .","The next theorem is an analogue of Theorem 3.1 of Jacob [27].","Theorem 4.13 Suppose Assumptions REF and REF hold, with $M>\\dim (G/K)+1$ .","Then there is $\\alpha _0>0$ such that $B_\\alpha (u,u)\\ge \\frac{c_0}{2}\\Vert u\\Vert _{1,\\lambda }^2,$ for all $\\alpha \\ge \\alpha _0$ and $u\\in H^{\\psi ,1}$ .","Proceed exactly as in Hoh [24] page 57, lines 8–17.","By Assumption REF (REF ), there is $\\alpha _0>0$ such that $q_1(\\lambda ) \\ge c_0\\Psi (\\lambda )^2-\\alpha _0 \\hspace{20.0pt} \\forall \\lambda \\in .$ This may be used to prove that for all $u\\in H^{\\psi ,1}$ , $\\langle q_1(D)u,u\\rangle \\ge c_0\\Vert u\\Vert _{\\psi ,1}^2 - \\alpha _0\\Vert u\\Vert ^2,$ at which point we can apply (REF ) and (REF ), as well as Assumption REF , to conclude $\\left|\\langle q_2(\\sigma ,D)u,u\\rangle \\right| &\\le C_M\\sqrt{2(1+c_\\psi )}\\sum _{\\beta =0}^M\\Vert \\Phi _\\beta \\Vert _{L^1(,\\omega )}\\left\\Vert \\langle \\cdot \\rangle ^{-M+1}\\right\\Vert _{L^1(,\\omega )}\\Vert u\\Vert _{\\psi ,1}^2 \\\\&= \\frac{1}{8\\gamma _M}\\sum _{\\beta =0}^M\\Vert \\Phi _\\beta \\Vert _{L^1(,\\omega )}\\Vert u\\Vert _{\\psi ,1}^2 \\le \\frac{c_0}{8}\\Vert u\\Vert _{\\psi ,1},$ for all $u\\in H^{\\psi ,1}$ .","Thus, for all $u\\in H^{\\psi ,1}$ $\\langle q(\\sigma ,D)u,u\\rangle &\\ge \\langle q_1(D)u,u\\rangle - \\left|\\langle q_2(\\sigma ,D)u,u\\rangle \\right| \\\\&\\ge (c_0-\\frac{c_0}{8})\\Vert u\\Vert _{\\psi ,1}^2 - \\alpha _0\\Vert u\\Vert _{\\psi ,1}^2 \\ge \\frac{c_0}{2}\\Vert u\\Vert _{\\psi ,1}^2 - \\alpha _0\\Vert u\\Vert _{\\psi ,1}^2.$ Therefore, for all $\\alpha \\ge \\alpha _0$ and $u\\in H^{\\psi ,1}$ $B_\\alpha (u,u) &= \\langle q(\\sigma ,D)u,u\\rangle + \\alpha \\Vert u\\Vert \\ge \\langle q(\\sigma ,D)u,u\\rangle + \\alpha _0\\Vert u\\Vert \\ge \\frac{c_0}{2}\\Vert u\\Vert _{\\psi ,1}^2,$ We may now apply the Lax–Milgram theorem (Theorem REF ) to $B_\\alpha $ , together with the linear functional $v\\mapsto \\langle f,\\cdot \\rangle $ , and conclude the following: Theorem 4.14 Let $\\alpha \\ge \\alpha _0$ .","Then (REF ) has a weak solution in the following sense: for all $f\\in L^2(K|G|K)$ there is a unique $u\\in H^{\\psi ,1}$ such that for all $v\\in H^{\\psi ,1}$ , $B_\\alpha (u,v) = \\langle f,v\\rangle .$ Theorem REF provides a weak solution to (REF ).","The next task is to prove that this solution is in fact a strong solution that belongs to $C_0(K|G|K)$ .","This will be achieved using the Sobolev embedding of Theorem REF (REF ).","Just as in Jacob [27] Theorem 3.1 and Hoh [24] Theorem 4.11, we have a useful lower bound for the pseudodifferential operator $q(\\sigma ,D)$ acting on $H^{\\psi ,s}$ , when $s\\ge 0$ .","Theorem 4.15 Let $s\\ge 0$ , and suppose the symbol $q$ satisfies Assumptions REF and REF , for some $M>|s-1|+1+\\dim (G/K)$ .","Then there is $\\kappa >0$ such that for all $u\\in H^{\\psi ,s+2}$ , $\\Vert q(\\sigma ,D)u\\Vert _{\\psi ,s} \\ge \\frac{c_0}{4}\\Vert u\\Vert _{\\psi ,s+2}-\\kappa \\Vert u \\Vert .$ The proof is formally no different to the sources mentioned: let $u\\in H^{\\psi ,s+2}$ , and use (REF ) and Theorem REF (REF ) to prove that $\\Vert q_1(D)u\\Vert _{\\psi ,s} \\ge \\frac{c_0}{2}\\Vert u\\Vert _{\\psi ,s+2} - \\kappa _1\\Vert u\\Vert ,$ for some $\\kappa _1>0$ .","Recall the estimate (REF ) of $\\Vert q_2(\\sigma ,D)u\\Vert _{\\psi ,s}$ from the proof of Theorem REF .","In light of Assumption REF and the particular form chosen for $\\gamma _M$ , one can use (REF ) to show that $\\Vert q_2(\\sigma ,D)u\\Vert _{\\psi ,s} &\\le C_M\\sum _{\\beta =0}^M\\Vert \\Phi _\\beta \\Vert _{L^1(,\\omega )}\\Big (\\left\\Vert \\langle \\cdot \\rangle ^{-M}\\right\\Vert _{L^1(,\\omega )}\\Vert u\\Vert _{\\psi ,s+2} + C_{s,\\psi }\\Vert u\\Vert _{\\psi ,s+1}\\Big ) \\\\&\\le C_Mc_0\\gamma _M\\Big (\\left\\Vert \\langle \\cdot \\rangle ^{-M}\\right\\Vert _{L^1(,\\omega )}\\Vert u\\Vert _{\\psi ,s+2} + C_{s,\\psi }\\Vert u\\Vert _{\\psi ,s+1}\\Big ) \\\\&\\le \\frac{c_0}{8}\\Vert u\\Vert _{\\psi ,s+2} + c\\Vert u\\Vert _{\\psi ,s+1},$ where $c>0$ is a constant.","Using Theorem REF (REF ) once again, let $\\kappa _2>0$ such that $c\\Vert u\\Vert _{\\psi ,s+1} \\le \\frac{c_0}{8}\\Vert u\\Vert _{\\psi ,s+2} + \\kappa _2\\Vert u\\Vert .$ Then, by the above, $\\Vert q_2(\\sigma ,D)u\\Vert _{\\psi ,s} \\le \\frac{c_0}{4}\\Vert u\\Vert _{\\psi ,s+2} + \\kappa _2\\Vert u\\Vert .$ Combining (REF ) and (REF ), we get $\\Vert q(\\sigma ,D)u\\Vert _{\\psi ,s} \\ge \\Vert q_1(D)u\\Vert _{\\psi ,s} - \\Vert q_2(\\sigma ,D)u\\Vert _{\\psi ,s} \\ge \\frac{c_0}{4}\\Vert u\\Vert _{\\psi ,s+2} - (\\kappa _1+\\kappa _2)\\Vert u\\Vert .$ The proof of the next theorem makes use of a particular family $(J_\\epsilon ,0<\\epsilon \\le 1)$ of bounded linear operators $L^2(G)$ , which will play the role of a Friedrich mollifier, but in the noncompact symmetric space setting.","First note that by identifying $\\operatorname{\\mathfrak {a}}$ with $\\mathbb {R}^m$ via our chosen basis, it makes sense to consider Friedrich mollifiers on $\\operatorname{\\mathfrak {a}}$ .","For $0<\\epsilon \\le 1$ and $H\\in \\operatorname{\\mathfrak {a}}$ , let $l(H) := C_0e^{\\frac{1}{|H|^2-1}}\\operatorname{\\bf 1}_{B_1(0)}(H), ~~\\text{ and }~~ l_\\epsilon (H) := \\epsilon ^{-m}l(H/\\epsilon ),$ where $C_0>0$ is a constant chosen so that $\\int _{\\operatorname{\\mathfrak {a}}}l(H)dH=1$ .","This mollifier (acting on Euclidean space) is used frequently in Evans [15], where it is called the standard mollifier — see Appendix C.4, pp. 629.","Jacob [27] and Hoh [24] both use this mollifier to show that the weak solution $u$ in Theorem REF is also a strong solution, in the sense that it satisfies (REF ), and that $u\\in H^{\\psi ,s}$ for suitably large $s$ .","Observe that $l,l_\\epsilon \\in \\mathcal {S}(\\operatorname{\\mathfrak {a}})^W$ for all $0<\\epsilon \\le 1$ .","Let $j,j_\\epsilon \\in \\mathcal {S}(K|G|K)$ be the corresponding mappings obtained via the commutative diagram of Theorem REF , so that $\\hat{j_\\epsilon } = {F}(l_\\epsilon ), \\hspace{20.0pt} \\forall 0<\\epsilon \\le 1,$ where ${F}$ denotes the Euclidean Fourier transform (see equation (REF )).","For $0<\\epsilon \\le 1$ , let $J_\\epsilon $ be the convolution operator defined on $L^2(K|G|K)$ by $J_\\epsilon u = j_\\epsilon \\ast u.$ The most important properties of $(J_\\epsilon ,0<\\epsilon \\le 1)$ needed for the proof of Theorem REF are stated below, and proven in the appendix.","Proposition 4.16 $\\hat{j_\\epsilon }(\\lambda ) = \\hat{j}(\\epsilon \\lambda )$ for all $0<\\epsilon \\le 1$ and $\\lambda \\in $ .","For all $0<\\epsilon \\le 1$ , $J_\\epsilon $ is a self-adjoint contraction of $L^2(K|G|K)$ .","$J_\\epsilon u\\in H^{\\psi ,s}$ for all $s\\ge 0$ , $u\\in L^2(K|G|K)$ and $0<\\epsilon \\le 1$ , and if $u\\in H^{\\psi ,s}$ , then $\\Vert J_\\epsilon u\\Vert _{\\psi ,s} \\le \\Vert u\\Vert _{\\psi ,s}.$ $\\Vert J_\\epsilon u - u\\Vert _{\\psi ,s}\\rightarrow 0$ as $\\epsilon \\rightarrow 0$ .","The following commutator estimate will also be useful in the proof of Theorem REF .","Lemma 4.17 Let $s\\ge 0$ , and suppose $q$ is a continuous negative definite symbol satisfying Assumption REF (REF ) for $M>|s-1|+1+\\dim (G/K)$ .","Then there is $c>0$ such that for all $ 0<\\epsilon \\le 1$ and all $u\\in C_c^\\infty (K|G|K)$ , $\\Vert [J_\\epsilon ,q(\\sigma ,D)]u\\Vert _{\\psi ,s} \\le c\\Vert u\\Vert _{\\psi ,s+1}.$ Let $0<\\epsilon \\le 1$ and $u\\in C_c^\\infty (K|G|K)$ , and observe that by Proposition REF (REF ), $[J_\\epsilon ,q_1(D)]u)^\\wedge (\\lambda ) = \\hat{j}(\\epsilon \\lambda )q_1(\\lambda )\\hat{u}(\\lambda ) - q_1(\\lambda )\\hat{j}(\\epsilon \\lambda )\\hat{u}(\\lambda ) = 0,$ for all $\\lambda \\in $ , so $[J_\\epsilon ,q_1(D)]u = 0$ .","For $\\lambda ,\\eta \\in $ , let $F_{\\lambda ,\\eta }=\\phi _{-\\lambda }q_2(\\cdot ,\\eta )$ , as previously (c.f.","(REF )).","Then by (REF ), for all $\\lambda \\in $ , $\\begin{aligned}\\left([J_\\epsilon ,q(\\sigma ,D)]u\\right)^\\wedge (\\lambda ) &= (J_\\epsilon q_2(\\sigma ,D)u)^\\wedge (\\lambda ) - (q_2(\\sigma ,D)J_\\epsilon u)^\\wedge (\\lambda ) \\\\&=\\hat{j}(\\epsilon \\lambda )(q_2(\\sigma ,D)u)^\\wedge (\\lambda ) - \\int _{}\\hat{F}_{\\lambda ,\\eta }(-\\eta )\\hat{j}(\\epsilon \\eta )\\hat{u}(\\eta )\\omega (d\\eta ).\\end{aligned}$ Now, $(q_2(\\sigma ,D)u)^\\wedge (\\lambda ) &= \\int _G\\phi _{-\\lambda }(\\sigma )(q_2(\\sigma ,D)u)(\\sigma )d\\sigma \\\\&= \\int _G\\int _{}\\phi _{-\\lambda }(\\sigma )\\phi _{-\\eta }(\\sigma )q_2(\\sigma ,\\eta )\\hat{u}(\\eta )\\omega (d\\eta )d\\sigma .$ Also, since $q_2(\\sigma ,\\cdot )$ is continuous and $\\hat{u}\\in \\mathcal {S}()$ , we have $q_2(\\sigma ,\\cdot )\\hat{u}\\in L^1(,\\omega )$ .","Therefore, by the Fubini theorem, $(q_2(\\sigma ,D)u)^\\wedge (\\lambda ) &= \\int _{}\\int _G\\phi _{-\\eta }(\\sigma )\\phi _{-\\lambda }(\\sigma )q_2(\\sigma ,\\eta )\\hat{u}(\\eta )d\\sigma \\omega (d\\eta ) \\\\&=\\int _{}\\int _G\\phi _{-\\eta }(\\sigma )F_{\\lambda ,\\eta }(\\sigma )d\\sigma \\hat{u}(\\eta )\\omega (d\\eta ) = \\int _{}\\hat{F}_{\\lambda ,\\eta }(-\\eta )\\hat{u}(\\eta )\\omega (d\\eta ),$ and so, substituting back into (REF ), $\\left([J_\\epsilon ,q(\\sigma ,D)]u\\right)^\\wedge (\\lambda ) = \\int _{}\\hat{F}_{\\lambda ,\\eta }(-\\eta )\\left(\\hat{j}(\\epsilon \\lambda )-\\hat{j}(\\epsilon \\eta )\\right)\\hat{u}(\\eta )\\omega (d\\eta ),$ for all $\\lambda \\in $ .","From here, a straightforward adaptation to the proof of Hoh [24] Theorem 4.4, pp.","51–52, with (REF ) replacing Hoh [24] (4.23), completes the proof of the lemma.","We are now ready to state and prove that, subject to our conditions, a strong solution to (REF ) exists, and belongs to an anisotropic Sobolev space of suitably high order.","Theorem 4.18 Let $\\alpha _0$ be as in Theorem REF , let $\\alpha \\ge \\alpha _0$ , and let $s\\ge 0$ .","Suppose that the negative definite symbol $q$ satisfies Assumptions REF and REF , where $M>|s-1|+1+\\dim (G/K)$ .","Then for all $f\\in H^{\\psi ,s}$ , there is a unique $u\\in H^{\\psi ,s+2}$ such that $(q(\\sigma ,D)+\\alpha )u = f.$ Let $f\\in H^{\\psi ,s}$ .","By Theorem REF we also have $f\\in L^2(K|G|K)$ , and so by Theorem REF there is a unique $u\\in H^{\\psi ,1}$ such that $B_\\alpha (u,v) =\\langle f,v\\rangle \\hspace{20.0pt} \\forall v\\in C_c^\\infty (K|G|K).$ The proof follows that of [27] Theorem 4.3, pp.","163 and [24] Theorem 4.12, pp.","59, using induction to show that that $u\\in H^{\\psi ,t}$ for $1\\le t\\le s+2$ , and in particular, that $u\\in H^{\\psi ,s+2}$ .","The family of operators $(J_\\epsilon ,0<\\epsilon \\le 1)$ take over role of the Friedrich mollifiers of [27] and [24].","By Proposition REF these operators satisfy the properties needed for the proof to carry over with little alteration.","Lemma REF and Theorem REF replace [24] Theorem 4.4 and 4.11, respectively.","Theorem 4.19 Let $q$ be a continuous negative definite function on $$ , satisfying Assumptions REF and REF with $M>\\max \\left\\lbrace 1,\\frac{d}{r}\\right\\rbrace +d$ , where $d=\\dim (G/K)$ .","Then for all $\\alpha \\ge \\alpha _0$ , $\\overline{\\operatorname{Ran}(\\alpha +q(\\sigma ,D))}=C_0(K|G|K).$ Fix $s\\in \\mathbb {R}$ with $\\max \\left\\lbrace \\frac{d}{r},1\\right\\rbrace < s\\min \\lbrace 1,d/r\\rbrace +d$ .","Then $-q(\\sigma ,D)$ extends to the infinitesimal generator of a strongly continuous sub-Feller semigroup on $C_0(K|G|K)$ .","By construction, $-q(\\sigma ,D)$ is a Gangolli operator, and therefore is a densely defined linear operator on $C_0(K|G|K)$ that satisfies the positive maximum principle.","By Theorems REF and REF , $-q(\\sigma ,D)$ is closable, and its closure generates a strongly continuous sub-Feller semigroup."],["A Class of Examples","We now present a class of Gangolli symbols for which the conditions of Corollary REF are satisfied.","Let $M\\in \\mathbb {N}$ such that $M>\\min \\lbrace 1,d/r\\rbrace +d+1$ .","We consider symbols $q:G\\times \\rightarrow \\mathbb {R}$ of the form $q(\\sigma ,\\lambda ) = \\kappa \\psi (\\lambda ) + u(\\sigma )v(\\lambda ), \\hspace{20.0pt} \\forall \\sigma \\in G,\\lambda \\in ,$ where $\\kappa $ is a positive constant, $u\\in C^M_c(K|G|K)$ is non-negative, and $v:\\rightarrow \\mathbb {R}$ is a negative definite function satisfying $|v(\\lambda )|\\le c_v(1+\\psi (\\lambda )) \\hspace{20.0pt} \\forall \\lambda \\in ,$ for some constant $c_v>0$ .","As in previous sections, $\\psi :\\rightarrow \\mathbb {R}$ is a fixed continuous negative definite function, satisfying (REF ).","In addition to this, we assume $\\psi $ is a Gangolli exponent.","By Example REF , the mappings $(\\sigma ,\\lambda )\\mapsto c_0\\psi (\\lambda )$ and $(\\sigma ,\\lambda )\\mapsto u(\\sigma )v(\\lambda )$ are both Gangolli symbols, and hence so is $q$ .","For each $\\lambda \\in $ and $\\sigma \\in G$ , $q_1(\\lambda ) = \\kappa \\psi (\\lambda ), \\hspace{20.0pt} \\forall \\lambda \\in ,$ and $q_2(\\sigma ,\\lambda ) = u(\\sigma )v(\\lambda ), \\hspace{20.0pt} \\forall \\sigma \\in G,\\;\\lambda \\in .$ Observe that since $v$ has compact support, $\\operatorname{Supp}(v)\\ne G$ , and for any $\\sigma _0\\in G\\setminus \\operatorname{Supp}(v)$ , $q_1(\\lambda ) = q(\\sigma _0,\\lambda ) \\hspace{20.0pt} \\forall \\lambda \\in ,$ and so $q$ is of the form (REF ).","Proposition 5.1 $q_1$ satisfies Assumption REF (REF ).","Let $c_0=\\frac{\\kappa }{2}\\min \\lbrace 1,c\\rbrace $ and $c_1=\\kappa $ .","Certainly we have $q_1(\\lambda ) \\le c_1(1+\\psi (\\lambda )) \\hspace{20.0pt} \\forall \\lambda \\in ;$ in particular this inequality holds when $|\\lambda |\\ge 1$ .","Also, by (REF ), if $|\\lambda |\\ge 1$ , then $\\psi (\\lambda )\\ge c|\\lambda |^r\\ge c$ and so, if $|\\lambda |\\ge 1$ , $q_1(\\lambda ) = \\frac{\\kappa }{2}(\\psi (\\lambda )+\\psi (\\lambda )) \\ge \\frac{\\kappa }{2}(c+\\psi (\\lambda )) \\ge c_0(1+\\psi (\\lambda )).$ That is, Assumption REF (REF ) holds.","Proving that Assumption REF (REF ) is satisfied will take more work.","However, once we have done so, Assumption REF will follow by taking $\\kappa $ sufficiently large.","Note that for the case we are considering, $F_{\\lambda ,\\eta }(\\sigma ) = \\phi _{-\\lambda }(\\sigma )u(\\sigma )v(\\eta ), \\hspace{20.0pt} \\forall \\sigma \\in G,\\;\\lambda ,\\eta \\in ,$ and so, for $\\beta =0,1,\\ldots ,M$ , $(-\\Delta )^{\\beta /2}F_{\\lambda ,\\eta }(\\sigma ) = v(\\eta )(-\\Delta )^{\\beta /2}(\\phi _{-\\lambda }u)(\\sigma ),$ for all $\\lambda ,\\eta \\in $ and $\\sigma \\in G$ .","By (REF ), $\\left|(-\\Delta )^{\\beta /2}F_{\\lambda ,\\eta }(\\sigma )\\right| = |v(\\eta )|\\left|(-\\Delta )^{\\beta /2}(\\phi _{-\\lambda }u)(\\sigma )\\right| \\le c_v\\left|(-\\Delta )^{\\beta /2}(\\phi _{-\\lambda }u)(\\sigma )\\right|\\big (1+\\psi (\\eta )\\big ).$ To verify Assumption REF (REF ), we seek for each $\\beta \\in \\lbrace 0,1,\\ldots ,M\\rbrace $ a mapping $\\Phi _\\beta \\in L^1(K|G|K)$ such that $\\left|(-\\Delta )^{\\beta /2}(\\phi _{-\\lambda }u)(\\sigma )\\right| \\le \\Phi _\\beta \\langle \\lambda \\rangle ^M, \\hspace{20.0pt} \\forall \\sigma \\in G,\\;\\lambda \\in .$ Let us first introduce some notation.","For each $n\\in \\mathbb {N}$ , a noncommutative version of the multinomial theorem tells us that $(-\\Delta )^n(\\phi _{-\\lambda }u) = (-1)^n\\left(\\sum _{j=1}^dX_j^2\\right)^n(\\phi _{-\\lambda }u) = \\sum _{\\begin{array}{c}\\alpha \\in \\mathbb {N}_0^d,\\\\|\\alpha |\\le r\\end{array}}c_\\alpha X^{\\alpha }(\\phi _{-\\lambda }u)$ for some coefficients $c_\\alpha $ , where $|\\alpha |=\\alpha _1+\\ldots +\\alpha _d$ and $X^{\\alpha }:=X_1^{\\alpha _1}\\ldots X_d^{\\alpha _d}$ .","This may also be seen by noting that $(-\\Delta )^n$ belongs to the universal enveloping algebra $\\mathcal {U}(\\mathfrak {p})$ , and applying the Poincaré-Birkoff–Witt theorem (see Knapp [30] Theorem 3.8, pp.","217) to write $(-\\Delta )^n$ in the basis $\\lbrace X^\\alpha :\\alpha \\in \\mathbb {N}_0^d\\rbrace $ of $\\mathcal {U}(\\mathfrak {p})$ .","Expanding the right-hand side of (REF ) using the fact that each $X_j$ is a derivation will give a large sum of terms of the form $\\kappa _{X,Y}X\\phi _{-\\lambda }Yu,$ where the $\\kappa _{X,Y}$ are constants, and $X,Y\\in {\\bf D}(G)$ are products of powers of $X_1,\\ldots , X_d$ , each with degree at most $2l$ .","Let ${U}_n$ be the set of all the $X$ 's and ${V}_n$ the set of all the $Y$ 's, so that $(-\\Delta )^n(\\phi _{-\\lambda }u) = \\sum _{\\begin{array}{c}X\\in {U}_n,\\\\Y\\in {V}_n\\end{array}}\\kappa _{X,Y}X\\phi _{-\\lambda }Yu.$ The following bound will be useful.","Lemma 5.2 For all $X\\in {\\bf D}(G)$ , there is a constant $C_X>0$ such that $|X\\phi _\\lambda (\\sigma )| \\le C_X\\langle \\lambda \\rangle ^{\\deg X}\\phi _0(\\sigma ),$ for all $\\lambda \\in $ and $\\sigma \\in G$ .","This is a straightforward corollary of Theorem 1.1 (iii) of [20] — see also Lemma 46 of Harish-Chandra [18], pp. 294.","Proposition 5.3 The mapping $q_2:G\\times \\rightarrow \\mathbb {R}$ given by (REF ) satisfies Assumption REF (REF ).","It is clear by construction that $q_2(\\cdot ,\\lambda )\\in C_c^M(K|G|K)$ for all $\\lambda \\in $ .","To verify the rest of Assumption REF (REF ), it will be useful to assume that $M$ is even.","Note that this is an acceptable assumption, since if $M$ is odd, we may replace it with $M-1$ .","We have chosen $M>\\min \\lbrace 1,d/r\\rbrace +d+1$ , and hence both $M$ and $M-1$ are strictly greater than $\\min \\lbrace 1,d/r\\rbrace +d$ , as required by Corollary REF .","Let $\\beta \\in \\lbrace 0,1,\\ldots ,M\\rbrace $ .","We seek $\\Phi _\\beta \\in L^1(K|G|K)$ for which (REF ) holds.","Let $n=\\lfloor \\beta \\rfloor $ .","Assume first that $\\beta $ is even, so that $n=\\beta /2$ .","By (REF ) and Lemma REF , $\\left|(-\\Delta )^{\\beta /2}(\\phi _{-\\lambda }u)\\right| \\le \\sum _{\\begin{array}{c}X\\in {U}_n,\\\\Y\\in {V}_n\\end{array}}|\\kappa _{X,Y}||X\\phi _{-\\lambda }||Yu| &\\le \\sum _{\\begin{array}{c}X\\in {U}_n,\\\\Y\\in {V}_n\\end{array}}C_X|\\kappa _{X,Y}||Yu|\\langle \\lambda \\rangle ^{\\deg Y}|\\phi _0| \\\\&\\le \\sum _{\\begin{array}{c}X\\in {U}_n,\\\\Y\\in {V}_n\\end{array}}C_X|\\kappa _{X,Y}||Yu|\\langle \\lambda \\rangle ^{\\deg X},$ since $|\\phi _0|\\le 1$ .","Now, $\\deg X\\le 2l=\\beta \\le M$ for all $X\\in {U}_n$ , and therefore, $\\left|(-\\Delta )^{\\beta /2}(\\phi _{-\\lambda }u)\\right| \\le \\kappa _\\beta \\sum _{Y\\in {V}_{\\beta /2}}|Yu|\\langle \\lambda \\rangle ^M$ where $\\kappa _\\beta = \\sup \\left\\lbrace C_X|\\kappa _{X,Y}|:X\\in {U}_{\\beta /2},Y\\in {V}_{\\beta /2}\\right\\rbrace .$ Let $\\Phi _\\beta :=\\kappa _\\beta \\sum _{Y\\in {V}_{\\beta /2}}|Yu|.$ Then $\\Phi _\\beta \\in L^1(K|G|K)$ , since each $Yu$ is a continuous function of compact support.","Moreover, $\\Vert \\Phi _\\beta \\Vert _1 \\le \\kappa _\\beta \\sum _{Y\\in {V}_{\\beta /2}}\\Vert Yu\\Vert _1$ In particular, we have verified (REF ) when $\\beta $ is even.","For the remainder of the proof of Proposition REF , assume $\\beta $ is an odd integer, so that $\\beta =2n+1$ .","Since we're assuming $M$ is even, note that $1\\le \\beta \\le M-1$ .","By (REF ), $\\left|(-\\Delta )^{\\beta /2}(\\phi _\\lambda u)\\right| = \\left|\\sqrt{-\\Delta }(-\\Delta )^n(\\phi _\\lambda u)\\right| \\le \\sum _{\\begin{array}{c}X\\in {U}_n,\\\\Y\\in {V}_n\\end{array}}|\\kappa _{X,Y}|\\left|\\sqrt{-\\Delta }\\big (X\\phi _{-\\lambda }Yu\\big )\\right|.$ The families ${U}_n$ and ${V}_n$ each consist of differential operators of degree at most $2n=\\beta -1$ .","Now, $-\\sqrt{-\\Delta }$ is the infinitesimal generator of the process obtained by subordinating Brownian motion on $G/K$ by the standard $\\frac{1}{2}$ -stable subordinator on $\\mathbb {R}$ .","By standard subordination theory (see [5] Section 5.7, pp.","154, or [41] §13 for more general theory) $\\sqrt{-\\Delta }$ may be expressed as a Bochner integral $\\sqrt{-\\Delta } = \\frac{1}{2\\sqrt{\\pi }}\\int _{0+}^\\infty t^{-3/2}(1-T_t)dt,$ where $(T_t,t\\ge 0)$ denotes the heat semigroup generated by $\\Delta $ .","Alternatively, equation (REF ) may also be obtained using the spectral theorem, see for example [9] §3.1.","Given $X\\in {U}_n$ and $Y\\in {V}_n$ and $\\sigma \\in G$ , $\\begin{aligned}\\left|\\sqrt{-\\Delta }(X\\phi _{-\\lambda }Yu)(\\sigma )\\right| &= \\frac{1}{2\\sqrt{\\pi }}\\left|\\int _{0+}^\\infty t^{-3/2}(1-T_t)\\big (X\\phi _{-\\lambda }Yu\\big )dt\\right| \\\\&\\le \\frac{1}{2\\sqrt{\\pi }}\\Bigg [\\left|\\int _{0+}^1 t^{-3/2}(1-T_t)\\big (X\\phi _{-\\lambda }Yu\\big )(\\sigma )dt\\right| \\\\&\\hspace{90.0pt}+ \\left|\\int _1^\\infty t^{-3/2}(1-T_t)\\big (X\\phi _{-\\lambda }Yu\\big )(\\sigma )dt\\right|\\Bigg ].\\end{aligned}$ Let $(h_t,t\\ge 0)$ denote the heat kernel associated with $(T_t,t\\ge 0)$ .","For the $\\int _1^\\infty $ term of (REF ), note that $\\int _1^\\infty t^{-3/2}dt=2$ , and so $&\\left|\\int _1^\\infty t^{-3/2}(1-T_t)\\big (X\\phi _{-\\lambda }Yu\\big )(\\sigma )dt\\right| \\\\&\\hspace{100.0pt}= \\left|\\int _1^\\infty t^{-3/2}X\\phi _{-\\lambda }(\\sigma )Yu(\\sigma )dt - \\int _1^\\infty t^{-3/2}T_t\\big (X\\phi _{-\\lambda }Yu\\big )(\\sigma )dt\\right| \\\\&\\hspace{100.0pt}\\le 2|X\\phi _{-\\lambda }(\\sigma )||Yu(\\sigma )| + \\left|\\int _1^\\infty t^{-3/2}\\int _GX\\phi _{-\\lambda }(\\sigma \\tau )Yu(\\sigma \\tau )h_t(\\tau )d\\tau dt\\right|.$ By Lemma REF and the fact that $\\deg X\\le \\beta -1$ , $|X\\phi _{-\\lambda }| \\le C_X\\langle \\lambda \\rangle ^{\\deg X} \\le C\\langle \\lambda \\rangle ^{\\beta -1},$ for all $\\lambda \\in $ , where $C_X$ is as in (REF ), and $C=\\max \\lbrace C_X:X\\in {U}_n\\rbrace $ .","Thus $&\\left|\\int _1^\\infty t^{-3/2}(1-T_t)\\big (X\\phi _{-\\lambda }Yu\\big )(\\sigma )dt\\right| \\\\&\\hspace{80.0pt}\\le 2|X\\phi _{-\\lambda }(\\sigma )||Yu(\\sigma )| + \\int _1^\\infty t^{-3/2}\\int _G|X\\phi _{-\\lambda }(\\sigma \\tau )||Yu(\\sigma \\tau )|h_t(\\tau )d\\tau dt \\\\&\\hspace{80.0pt}\\le C\\left(2|Yu(\\sigma )| + \\int _1^\\infty t^{-3/2}\\int _G|Yu(\\sigma \\tau )|h_t(\\tau )d\\tau dt\\right)\\langle \\lambda \\rangle ^{\\beta -1} \\\\&\\hspace{80.0pt}= \\Phi _{\\beta ,Y}^{(1)}(\\sigma )\\langle \\lambda \\rangle ^{\\beta -1},$ where $\\Phi _{\\beta ,Y}^{(1)} := C\\left(2|Yu| + \\int _1^\\infty t^{-3/2}T_t\\big (|Yu|\\big )dt\\right).$ Since $\\beta -1\\le M$ and $\\langle \\lambda \\rangle \\ge 1$ for all $\\lambda \\in $ , it follows that $\\left|\\int _1^\\infty t^{-3/2}(1-T_t)\\big (X\\phi _{-\\lambda }Yu\\big )dt\\right| \\le \\Phi _{\\beta ,Y}^{(1)}\\langle \\lambda \\rangle ^M,$ for all $\\lambda \\in $ .","We claim that $\\Phi _{\\beta ,Y}^{(1)}\\in L^1(K|G|K)$ .","Clearly $|Yu|\\in L^1(K|G|K)$ , since it is a continuous function of compact support.","Each of the operators $T_t$ is a positivity preserving contraction of $L^1(K|G|K)$ , and so $\\int _1^\\infty t^{-3/2}\\int _GT_t\\big (|Yu|\\big )(\\sigma )d\\sigma dt = \\int _1^\\infty t^{-3/2}\\left\\Vert T_t\\big (|Yu|\\big )\\right\\Vert _1 dt \\le \\int _1^\\infty t^{-3/2}\\Vert Yu\\Vert _1dt = 2\\Vert Yu\\Vert _1.$ By Fubini's theorem, $\\int _1^\\infty t^{-3/2}T_t\\big (|Yu|\\big )dt\\in L^1(K|G|K)$ , with $\\left\\Vert \\int _1^\\infty t^{-3/2}T_t\\big (|Yu|\\big )dt\\right\\Vert _{L_1(K|G|K)} \\le 2\\Vert Yu\\Vert _1.$ It follows by (REF ) that $\\Phi _{\\beta ,Y}^{(1)}\\in L^1(K|G|K)$ , and that $\\Vert \\Phi _{\\beta ,Y}^{(1)}\\Vert _1 \\le 4C\\Vert Yu\\Vert _1.$ For the $\\int _{0+}^1$ term of (REF ), observe that by Lemma 6.1.12 of Davies [12], pp.","169, as well as the Fubini theorem, $\\int _{0+}^1t^{-3/2}(1-T_t)\\big (X\\phi _{-\\lambda }Yu\\big )dt &= -\\int _{0+}^1t^{-3/2}\\int _0^t T_s\\Delta \\big (X\\phi _{-\\lambda }Yu\\big )dsdt \\\\&= -\\int _0^1\\int _s^1t^{-3/2}T_s\\Delta \\big (X\\phi _{-\\lambda }Yu\\big )dtds \\\\&= -\\int _0^12(s^{-1/2}-1)T_s\\Delta \\big (X\\phi _{-\\lambda }Yu\\big )ds.$ Hence, using the product formula for $\\Delta $ , $\\begin{aligned}\\int _{0+}^1t^{-3/2}(1-T_t)\\big (X\\phi _{-\\lambda }Yu\\big )dt &= -2\\int _0^1(s^{-1/2}-1)\\Big \\lbrace T_s\\big (X\\phi _{-\\lambda }\\Delta Yu\\big ) \\\\&\\hspace{6.0pt}+ 2\\sum _{j=1}^dT_s\\big (X_jX\\phi _{-\\lambda }X_jYu\\big ) + T_s\\big (\\Delta X\\phi _{-\\lambda }Yu\\big )\\Big \\rbrace ds.\\end{aligned}$ Let $C$ and $C_X$ be as in (REF ).","Then for all $\\sigma \\in G$ , $\\left|T_s\\big (X\\phi _{-\\lambda }\\Delta Yu\\big )(\\sigma )\\right| &= \\left|\\int _GX\\phi _{-\\lambda }(\\sigma \\tau )\\Delta Yu(\\sigma \\tau )h_s(\\tau )d\\tau \\right| \\\\&\\le \\int _G|X\\phi _{-\\lambda }(\\sigma \\tau )||\\Delta Yu(\\sigma \\tau )|h_s(\\tau )d\\tau \\\\&\\le C_X\\langle \\lambda \\rangle ^{\\deg X}\\int _G|\\Delta Yu(\\sigma \\tau )|h_s(\\tau )d\\tau \\le C\\langle \\lambda \\rangle ^{\\beta -1}T_s|\\Delta Yu|(\\sigma ).$ In exactly the same way, for $j=1,\\ldots ,d$ , $\\left|T_s\\big (X_jX\\phi _{-\\lambda }X_jYu\\big )\\right| \\le C_X^{(j)}\\langle \\lambda \\rangle ^{\\deg X+1}T_s|X_jYu| \\le C^{\\prime }\\langle \\lambda \\rangle ^{\\beta }T_s|X_jYu|,$ and also $\\left|T_s\\big (\\Delta X\\phi _{-\\lambda }Yu\\big )\\right| \\le C_X^{(0)}\\langle \\lambda \\rangle ^{\\deg X+2}T_s|Yu| \\le C^{\\prime }\\langle \\lambda \\rangle ^{\\beta +1}T_s|Yu|,$ where the constants $C_X^{(j)},C^{(j)}$ are chosen so that for all $\\lambda \\in $ and $j=1,\\ldots ,d$ , $|X\\phi _{-\\lambda }| \\le C_X^{(0)}\\langle \\lambda \\rangle ^{\\deg X +2}, \\hspace{40.0pt} |X_jX\\phi _{-\\lambda }| \\le C_X^{(j)}\\langle \\lambda \\rangle ^{\\deg X +1},$ and $C^{\\prime }:=\\max \\lbrace C_X^{(j)}:X\\in {U}_n,j=0,1,\\ldots ,d\\rbrace $ .","Such constants exist by Lemma REF .","Now, $\\langle \\lambda \\rangle ^{\\beta -1} \\le \\langle \\lambda \\rangle ^\\beta \\le \\langle \\lambda \\rangle ^{\\beta +1}$ for all $\\lambda \\in $ , and hence by (REF ), $&\\left|\\int _{0+}^1t^{-3/2}(1-T_t)\\big (X\\phi _{-\\lambda }Yu\\big )dt\\right| \\\\&\\hspace{20.0pt}\\le 2\\int _0^1(s^{-1/2}-1)\\Big \\lbrace \\left|T_s\\big (X\\phi _{-\\lambda }\\Delta Yu\\big )\\right| + 2\\sum _{j=1}^d\\left|T_s\\big (X_jX\\phi _{-\\lambda }X_jYu\\big )\\right| \\\\&\\hspace{240.0pt}+ \\left|T_s\\big (\\Delta X\\phi _{-\\lambda }Yu\\big )\\right|\\Big \\rbrace ds \\\\&\\hspace{20.0pt}\\le 2C^{\\prime }\\langle \\lambda \\rangle ^{\\beta +1}\\int _0^1(s^{-1/2}-1)T_s\\left(|\\Delta Yu|+2\\sum _{j=1}^d|X_jYu|+|Yu|\\right)ds.$ Since $\\beta \\le M-1$ , it follows that for all $X\\in {U}_n$ and $Y\\in {V}_n$ , $\\left|\\int _{0+}^1t^{-3/2}(1-T_t)\\big (X\\phi _{-\\lambda }Yu\\big )dt\\right| \\le \\Phi _{\\beta ,Y}^{(2)}\\langle \\lambda \\rangle ^M,$ where $\\Phi _{\\beta ,Y}^{(2)} = C^{\\prime }\\int _0^1(s^{-1/2}-1)T_s\\left(|\\Delta Yu|+2\\sum _{j=1}^d|X_jYu|+|Yu|\\right)ds.$ Observe that $\\Phi _{\\beta ,Y}^{(2)}\\in L^1(K|G|K)$ for all $Y\\in {V}_n$ .","Indeed, $u\\in C_c^M(K|G|K)$ , and $\\deg Y\\le \\beta -1\\le M-2$ , hence $|\\Delta Yu|$ , $\\sum _{j=1}^d|X_jYu|$ and $|Yu|$ are all continuous functions of compact support.","Thus $T_s\\big (|\\Delta Yu|+2\\sum _{j=1}^d|X_jYu|+|Yu|\\big )\\in L^1(K|G|K)$ , and, since $T_s$ is an $L^1(K|G|K)$ -contraction, $\\left\\Vert T_s\\left(|\\Delta Yu|+2\\sum _{j=1}^d|X_jYu|+|Yu|\\right)\\right\\Vert _1 \\le \\Vert \\Delta Yu\\Vert _1 +2\\sum _{j=1}^d\\Vert X_jYu\\Vert _1 +\\Vert Yu\\Vert _1.$ Noting that $\\int _0^1(s^{-1/2}-1)ds = 1$ , it follows by Fubini's theorem that $\\Phi _{\\beta ,Y}^{(2)}\\in L^1(K|G|K)$ , with $\\left\\Vert \\Phi _{\\beta ,Y}^{(2)}\\right\\Vert _1 \\le C_X^{\\prime }\\left(\\Vert \\Delta Yu\\Vert _1 + 2\\sum _{j=1}^d\\Vert X_jYu\\Vert _1 +\\Vert Yu\\Vert _1\\right).$ Substituting (REF ) and (REF ) into (REF ), we obtain the pointwise estimate $\\left|\\sqrt{-\\Delta }(X\\phi _{-\\lambda }Yu)\\right| \\le \\frac{1}{2\\sqrt{\\pi }}\\left(\\Phi _{\\beta ,Y}^{(1)}+\\Phi _{\\beta ,Y}^{(2)}\\right)\\langle \\lambda \\rangle ^M,$ for all $X\\in {U}_n$ , $Y\\in {V}_n$ and $\\lambda \\in $ , where the $\\Phi _{\\beta ,Y}^{(j)}$ ($j=1,2$ ) are given by (REF ) and (REF ).","Hence by (REF ), for all $\\lambda \\in $ , $\\left|(-\\Delta )^{\\beta /2}(\\phi _\\lambda u)\\right| \\le \\sum _{\\begin{array}{c}X\\in {U}_n,\\\\Y\\in {V}_n\\end{array}}|\\kappa _{X,Y}|\\left|\\sqrt{-\\Delta }\\big (X\\phi _{-\\lambda }Yu\\big )\\right| \\le \\Phi _\\beta \\langle \\lambda \\rangle ^M,$ where $\\Phi _\\beta :=\\frac{1}{2\\sqrt{\\pi }}\\sum _{\\begin{array}{c}X\\in {U}_n,\\\\Y\\in {V}_n\\end{array}}|\\kappa _{X,Y}|\\left(\\Phi _{\\beta ,Y}^{(1)}+\\Phi _{\\beta ,Y}^{(2)}\\right),$ and $\\beta $ is still assumed to be odd.","It was shown previously that $\\Phi _{\\beta ,Y}^{(1)},\\Phi _{\\beta ,Y}^{(2)}\\in L^1(K|G|K)$ , for all $Y\\in {V}_n$ , and hence $\\Phi _\\beta \\in L^1(K|G|K)$ .","Moreover, by (REF ) and (REF ), $\\begin{aligned}\\left\\Vert \\Phi _\\beta \\right\\Vert _1 &\\le \\kappa _\\beta ^{\\prime }\\sum _{Y\\in {V}_n}\\Bigg (\\Vert \\Delta Yu\\Vert _1+\\sum _{j=1}^d\\Vert X_jYu\\Vert _1 +\\Vert Yu\\Vert _1\\Bigg ),\\end{aligned}$ for some positive constant $\\kappa _\\beta ^{\\prime }$ .","In particular, we have verified (REF ) when $\\beta $ is odd.","Corollary 5.4 Let $q:G\\times \\rightarrow \\mathbb {R}$ be of the form (REF ).","Then for $\\kappa $ sufficiently large, the conditions of Corollary REF are satisfied.","In particular, $-q(\\sigma ,D)$ extends to the infinitesimal generator of a strongly continuous sub-Feller semigroup on $C_0(K|G|K)$ ."],["Appendix: The Friedrich Mollifier $J_\\epsilon $","Recall the Friedrich mollifier on $\\operatorname{\\mathfrak {a}}$ from Section , defined for each $0<\\epsilon \\le 1$ and $H\\in \\operatorname{\\mathfrak {a}}$ by $l_\\epsilon (H) := \\epsilon ^{-m}l(H/\\epsilon ), \\hspace{20.0pt} \\text{ where } \\hspace{20.0pt} l(H) := C_0e^{\\frac{1}{|H|^2-1}}\\operatorname{\\bf 1}_{B_1(0)}(H),$ where $C_0>0$ is a normalising constant, and the associated mappings $j,j_\\epsilon \\in \\mathcal {S}(K|G|K)$ are again given by $\\hat{j_\\epsilon } = {F}(l_\\epsilon ), \\hspace{20.0pt} \\forall 0<\\epsilon \\le 1.$ For $0<\\epsilon \\le 1$ , the operators $J_\\epsilon $ were defined $J_\\epsilon u = j_\\epsilon \\ast u \\hspace{20.0pt} \\forall f\\in L^2(K|G|K).$ This appendix will be devoted to proving Proposition REF , which we re-state below.","Proposition 6.1 $\\hat{j_\\epsilon }(\\lambda ) = \\hat{j}(\\epsilon \\lambda )$ for all $0<\\epsilon \\le 1$ and $\\lambda \\in $ .","For all $0<\\epsilon \\le 1$ , $J_\\epsilon $ is a self-adjoint contraction of $L^2(K|G|K)$ .","$J_\\epsilon u\\in H^{\\psi ,s}$ for all $s\\ge 0$ , $u\\in L^2(K|G|K)$ and $0<\\epsilon \\le 1$ , and if $u\\in H^{\\psi ,s}$ , then $\\Vert J_\\epsilon u\\Vert _{\\psi ,s} \\le \\Vert u\\Vert _{\\psi ,s}.$ $\\Vert J_\\epsilon u - u\\Vert _{\\psi ,s}\\rightarrow 0$ as $\\epsilon \\rightarrow 0$ .","Let $0<\\epsilon \\le 1$ and $\\lambda \\in $ .","Using a change of variable $H\\mapsto \\epsilon ^{-1}H$ , $\\hat{j_\\epsilon }(\\lambda ) = {F}(l_\\epsilon )(\\lambda ) &= \\int _{\\operatorname{\\mathfrak {a}}}e^{i\\lambda (H)}\\epsilon ^{-m}l(\\epsilon ^{-1}H)dH \\\\&= \\int _{\\operatorname{\\mathfrak {a}}}e^{i\\epsilon \\lambda (H)}l(H)dH = {F}(l)(\\epsilon \\lambda ) = \\hat{j}(\\epsilon \\lambda ).$ The map $l$ is symmetric under $H\\mapsto -H$ , and hence ${F}(l)=\\hat{j}$ is real-valued.","Therefore, given $u,v\\in L^2(K|G|K)$ and $0<\\epsilon \\le 1$ , $\\langle J_\\epsilon u,v\\rangle = \\int _{}\\hat{j}(\\epsilon \\lambda )\\hat{u}(\\lambda )\\overline{\\hat{v}(\\lambda )}\\omega (d\\lambda ) = \\int _{}\\hat{u}(\\lambda )\\overline{\\hat{j}(\\epsilon \\lambda )\\hat{v}(\\lambda )}\\omega (d\\lambda ) = \\langle u, J_\\epsilon v\\rangle .$ To see that $J_\\epsilon $ is a contraction, note that $|\\hat{j_\\epsilon }(\\lambda )|=|\\hat{j}(\\epsilon \\lambda )|\\le \\hat{j}(0)= 1$ for all $\\lambda \\in $ , and so by Plancherel's identity $\\Vert J_\\epsilon u\\Vert = \\Vert \\hat{j}_\\epsilon \\hat{u}\\Vert _{L^2(,\\omega )} \\le \\Vert \\hat{u}\\Vert _{L^2(K|G|K)} = \\Vert u \\Vert ,$ for all $u\\in L^2(K|G|K)$ and all $0<\\epsilon \\le 1$ .","Let $s\\ge 0$ , $0<\\epsilon \\le 1$ .","By Theorem REF , $\\hat{j}\\in \\mathcal {S}()^W$ , and hence there is $\\kappa >0$ such that $\\langle \\lambda \\rangle ^s\\left|\\hat{j}(\\epsilon \\lambda )\\right| \\le \\kappa , \\hspace{20.0pt} \\forall \\lambda \\in .$ Then, using Proposition REF (REF ), $\\Psi (\\lambda )^s\\left|\\hat{j}(\\epsilon \\lambda )\\right| \\le c_\\psi ^{s/2}\\langle \\lambda \\rangle ^s\\left|\\hat{j}(\\epsilon \\lambda )\\right|\\le c_\\psi ^{s/2}\\kappa ,$ for all $\\lambda \\in $ .","Let $u\\in L^2(K|G|K)$ .","By Plancherel's identity, $\\int _{}\\Psi (\\lambda )^{2s}\\left|\\hat{j}(\\epsilon \\lambda )\\right|^2|\\hat{u}(\\lambda )|^2\\omega (d\\lambda ) \\le c_\\psi ^s\\kappa ^2\\Vert u \\Vert ^2 < \\infty .$ By Proposition REF (REF ), $(J_\\epsilon u)^\\wedge (\\lambda ) = \\hat{j}(\\epsilon \\lambda )\\hat{u}(\\lambda )$ , for all $\\lambda \\in $ , and hence $\\int _{}\\Psi (\\lambda )^{2s}|(J_\\epsilon u)^\\wedge (\\lambda )|^2\\omega (d\\lambda )<\\infty .$ That is, $J_\\epsilon u\\in H^{\\psi ,s}$ .","Next, suppose $u\\in H^{\\psi ,s}$ .","Then, since $|\\hat{j_\\epsilon }|\\le 1$ , $\\Vert J_\\epsilon u\\Vert _{\\psi ,s} = \\Vert \\Psi ^s\\hat{j_\\epsilon }\\hat{u}\\Vert _{L^2(,\\omega )} \\le \\Vert \\Psi ^s\\hat{u}\\Vert _{L^2(,\\omega )} = \\Vert u\\Vert _{\\psi ,s},$ as desired.","By Theorem 1 on page 250 of Evans [15], for all $v\\in \\mathcal {S}(\\operatorname{\\mathfrak {a}})^W$ , $l_\\epsilon \\ast v\\rightarrow v$ as $\\epsilon \\rightarrow 0$ , in the classical Sobolev space $W^s()$ , for all $s\\ge 0$ .","Therefore, for all $s\\ge 0$ and $v\\in \\mathcal {S}(\\operatorname{\\mathfrak {a}})^W$ , $\\lim _{\\epsilon \\rightarrow 0}\\int _{}(1+|\\lambda |^2)^s|{F}(l_\\epsilon \\ast v-v)(\\lambda )|^2d\\lambda = 0,$ Let $u\\in C_c^\\infty (K|G|K)$ .","Then by Theorem REF , ${F}^{-1}(\\hat{u})\\in \\mathcal {S}(\\operatorname{\\mathfrak {a}})^W$ .","Observe that if $v={F}^{-1}(\\hat{u})$ , then ${F}(l_\\epsilon \\ast v - v) = (\\hat{j}_\\epsilon -1)\\hat{u} = (J_\\epsilon u - u)^\\wedge ,$ and so by (REF ), $\\lim _{\\epsilon \\rightarrow 0}\\int _{}(1+|\\lambda |^2)^s|(J_\\epsilon u - u)^\\wedge (\\lambda )|^2d\\lambda = 0,$ for all $s\\ge 0$ .","By (REF ), $\\int _{}(1+|\\lambda |^2)^s|(J_\\epsilon u - u)^\\wedge (\\lambda )|^2\\omega (d\\lambda )&\\le \\int _{}(1+|\\lambda |^2)^s|(J_\\epsilon u - u)^\\wedge (\\lambda )|^2(C_1+C_2|\\lambda |^p)^2d\\lambda \\\\&\\le \\kappa \\int _{}(1+|\\lambda |^2)^{s+p}|(J_\\epsilon u - u)^\\wedge (\\lambda )|^2d\\lambda ,$ for some constant $\\kappa >0$ , and where $p=\\frac{\\dim N}{2}$ .","Hence $\\lim _{\\epsilon \\rightarrow 0}\\int _{}(1+|\\lambda |^2)^s|(J_\\epsilon u - u)^\\wedge (\\lambda )|^2\\omega (d\\lambda ) = 0$ By Proposition REF (REF ), $\\Vert J_\\epsilon u-u\\Vert _{\\psi ,s}^2 &= \\int _{}(1+\\psi (\\lambda ))^s|(J_\\epsilon u - u)^\\wedge (\\lambda )|^2\\omega (d\\lambda ) \\\\&\\le c_\\psi \\int _{}(1+|\\lambda |^2)^s|(J_\\epsilon u - u)^\\wedge (\\lambda )|^2\\omega (d\\lambda ) \\rightarrow 0$ as $\\epsilon \\rightarrow 0$ .","Since $C_c^\\infty (K|G|K)$ is dense in $H^{\\psi ,s}$ , Proposition REF (REF ) follows.","Acknowledgement Many thanks to David Applebaum for his advice and support with writing this paper.","Thanks also to the University of Sheffield's School of Mathematics and Statistics, and to the EPSRC for providing PhD funding while this research was carried out."]],"string":"[\n [\n \"Sub-Feller Semigroups Generated by Pseudodifferential Operators on\\n Symmetric Spaces of Noncompact Type\"\n ],\n [\n \"Abstract We consider global pseudodifferential operators on symmetric spaces of noncompact type, defined using spherical functions.\",\n \"The associated symbols have a natural probabilistic form that extend the notion of the characteristic exponent appearing in Gangolli's L\\\\'evy-Khinchine formula to a function of two variables.\",\n \"The Hille-Yosida-Ray theorem is used to obtain conditions on such a symbol so that the corresponding pseudodifferential operator has an extension that generates a sub-Feller semigroup, generalising existing results for Euclidean space.\"\n ],\n [\n \"Introduction\",\n \"Pseudodifferential operator theory is a powerful tool in the study of Feller–Markov processes on Euclidean space (see for example [31], [37], [38], [39], [40], or [11] §5 for a summary).\",\n \"Primarily developed by Niels Jacob and collaborators (see e.g.\",\n \"[26], [27], [24]), this framework characterises sub-Feller semigroups and their generators as pseudodifferential operators ($\\\\Psi $ DOs) acting on $C_0(\\\\mathbb {R}^d)$ , the Banach space of continuous, real-valued functions on $\\\\mathbb {R}^d$ that vanish at infinity.\",\n \"The associated symbols capture many properties of the sub-Feller processes and semigroups they represent, generalising the well-established relationship between Lévy processes and their characteristic exponents given by the Lévy–Khinchine formula.\",\n \"The key difference is that the Feller–Markov symbols typically depend on two variables instead of one — in the case of Feller processes, this is sometimes described as the Lévy characteristics having gained spatial dependence.\",\n \"Manifold-valued Feller–Markov processes have also attracted interest in recent years (see [13], [25] and [32] §7 for excellent summaries), though the absence of a global harmonic analysis on general manifolds has so far prevented a $\\\\Psi $ DO approach.\",\n \"Lie groups and symmetric spaces come with their own harmonic analysis, however, in the form of the spherical transform (see Harish-Chandra [18], [19] and Helgason [22], [21]).\",\n \"A natural question to ask then is to what extent a $\\\\Psi $ DO-based approach can be applied to the study of sub-Feller processes on Lie groups and symmetric spaces.\",\n \"Much work has already been done in this area, especially in the “constant coefficient” case of Lévy processes, in which the symbol depends only on its second argument.\",\n \"Here, the symbols are given by Gangolli's Lévy–Khinchine formula ([16] Theorem 6.2), a direct analogue to the classical result.\",\n \"The first paper to use probabilistic $\\\\Psi $ DO methods on a Lie group was [6], where $\\\\Psi $ DOs are used to study Lévy processes on the Heisenberg group.\",\n \"Pseudodifferential operator representations of semigroups and generators have also been found for Lévy processes on a general Lie group — see [2], [3] for the compact case and [4] Section 5 for the noncompact case.\",\n \"For Feller processes, $\\\\Psi $ DO representations have been found when the Lie group or symmetric space is compact — see for example the final sections of [7] and [8].\",\n \"This paper seeks to develop a more general theory of $\\\\Psi $ DOs for symmetric spaces of noncompact type, and apply it to seek conditions on a symbol so that the corresponding $\\\\Psi $ DO extends to the generator of some sub-Feller process.\",\n \"For the $\\\\mathbb {R}^d$ case, this question has been studied thoroughly by both Jacob and Hoh — see [26], [27], as well as [24] Chapter 4.\",\n \"Our approach is similar to Jacob [27] and Hoh [24], but where they have used Fourier transform, we use the spherical transform.\",\n \"The spherical transform enjoys many of the same properties as the Fourier transform on $\\\\mathbb {R}^d$ , and we find that several of the arguments in [27] and [24] generalise directly to this setting.\",\n \"However, there are significant differences between the two settings, and a direct transcription of Jacob and Hoh's arguments is certainly not possible.\",\n \"One notable difference, for example, is the condition (2.2) in Jacob [27], which features a derivative that on a manifold makes little sense.\",\n \"Even on a symmetric space, elements of a tangent space cannot be easily translated from point to point, and a different approach is needed.\",\n \"We take a more operator-theoretic approach, and replace each of the partial derivatives $\\\\frac{\\\\partial }{\\\\partial x_i}$ ($i=1,\\\\ldots ,d$ ) with the fractional Laplacian $\\\\sqrt{-\\\\Delta }$ .\",\n \"This is an exciting object to work with, and has the added advantage that spherical functions form a system of eigenfunctions for this operator.\",\n \"The structure of this paper will be the following.\",\n \"Section presents a summary of necessary concepts and results from harmonic analysis on symmetric spaces, and introduces the system of symbols and $\\\\Psi $ DOs that will be used later on.\",\n \"We also introduce here the spherical anisotropic Sobolev spaces, a generalisation of the anisotropic Sobolev spaces first considered by Niels Jacob [26], [27].\",\n \"In Section , we consider (a version of) the Hille–Yosida–Ray theorem (see Theorem REF ), and note that, thanks to the work of Applebaum and Ngan [7], [8], one of the conditionsNamely, that an operator must satisfy the positive maximum principle.\",\n \"of the Hille–Yosida–Ray theorem has already been fully addressed.\",\n \"This work motivates our introduction of a class of operators we will call Gangolli operators, which will automatically each be densely defined linear operators on $C_0(K|G|K)$ that satisfy the positive maximum principle.\",\n \"We then prove that Gangolli operators are $\\\\Psi $ DOs in the sense of Section REF , and derive a formula for their symbols (Theorem REF ).\",\n \"Section is concerned with seeking sufficient conditions for a Gangolli operator $q(\\\\sigma ,D)$ to extend to the generator a sub-Feller semigroup.\",\n \"Informed by the work of the previous section, this amounts to finding conditions that ensure $\\\\overline{\\\\operatorname{Ran}(\\\\alpha +q(\\\\sigma ,D))}=C_0(K|G|K)$ for some $\\\\alpha >0$ (see Theorem REF (REF )).\",\n \"This section perhaps most closely follows the approach of Jacob [27] and Hoh [24] Chapter 4, and where proofs are similar to these sources, we omit detail, and instead aim to emphasise what is different about the symmetric space setting.\",\n \"Full proofs may also be found in my PhD thesis [42].\",\n \"Finally, in Section we present a large class of examples of symbols that satisfy the conditions found in Sections and .\",\n \"Notation.\",\n \"For a topological space $X$ , $\\\\mathcal {B}(X)$ will denote the Borel $\\\\sigma $ -algebra associated with $X$ , and $B_b(X)$ the space of bounded, Borel functions from $X\\\\rightarrow \\\\mathbb {R}$ , a Banach space with respect to the supremum norm.\",\n \"If $X$ is a locally compact Hausdorff space, then we write $C_0(X)$ for the closed subspace of $B_b(X)$ consisting of continuous functions vanishing at infinity, and $C_c(X)$ for the dense subspace of compactly supported continuous functions.\",\n \"If $X$ is a smooth manifold and $M\\\\in \\\\mathbb {N}\\\\cup \\\\lbrace \\\\infty \\\\rbrace $ , then we write $C^M_c(X)$ for the space of compactly supported $M$ -times continuously differentiable functions on $X$ .\"\n ],\n [\n \"Preliminaries\",\n \"Let $(G,K)$ is a Riemannian symmetric pair, so that $G$ is a connected, semisimple Lie group with finite centre, $K$ is a maximal compact subgroup of $G$ , and, for some nontrivial involution $\\\\Theta $ on $G$ , $G^\\\\Theta _0\\\\subseteq K\\\\subseteq G^\\\\Theta ,$ where $G^\\\\Theta $ is the fixed point set of $\\\\Theta $ , and $G^\\\\Theta _0$ is the identity component of $G^\\\\Theta $ .\",\n \"Let $\\\\mathfrak {g}$ and $\\\\mathfrak {k}$ denote the Lie algebras of $G$ and $K$ , respectively.\",\n \"Note that $\\\\mathfrak {k}$ is the $+1$ eigenspace of the differential $\\\\theta :=d\\\\Theta $ ; let $\\\\mathfrak {p}$ denote the $-1$ eigenspace.\",\n \"In fact, $\\\\theta $ is a Cartan involution on $\\\\mathfrak {g}$ , and the corresponding Cartan decomposition is $\\\\mathfrak {g} = \\\\mathfrak {p} \\\\oplus \\\\mathfrak {k}.$ Let $B$ denote the Killing form of $G$ , defined for each for all $X,Y\\\\in \\\\mathfrak {g}$ by $B(X,Y) = \\\\operatorname{tr}(\\\\operatorname{ad}X\\\\operatorname{ad}Y)$ .\",\n \"Since $(G,K)$ is of noncompact type, $B$ is nondegenerate, negative definite on $\\\\mathfrak {k}$ , and positive definite on $\\\\mathfrak {p}$ .\",\n \"Fix an $\\\\operatorname{Ad}(K)$ -invariant inner product $\\\\langle \\\\cdot ,\\\\cdot \\\\rangle $ on $\\\\mathfrak {g}$ .\",\n \"Associated with $(G,K)$ is the symmetric space $G/K$ , a smooth Riemannian manifold, with Riemannian structure induced by the restriction of $\\\\langle \\\\cdot ,\\\\cdot \\\\rangle $ to $\\\\mathfrak {p}$ .\",\n \"There is a one to one correspondence between functions on $G/K$ and $K$ -right-invariant functions on $G$ , we denote both by $\\\\mathcal {F}(G/K)$ .\",\n \"Similarly, we identify $K$ -invariant functions on $G/K$ with $K$ -bi-invariant functions on $G$ , and denote both by $\\\\mathcal {F}(K|G|K)$ .\",\n \"Similar conventions will be used to denote standard subspaces of $\\\\mathcal {F}(G/K)$ and $\\\\mathcal {F}(K|G|K)$ ; for example $C(K|G|K)$ will denote both the space of continuous, $K$ -invariant functions on $G/K$ , and the space of continuous, $K$ -bi-invariant functions on $G$ .\"\n ],\n [\n \"Harmonic Analysis on Symmetric Spaces of Noncompact Type\",\n \"For a thorough treatment of this topic, see Helgason [22], [21].\",\n \"Let ${\\\\bf D}(G)$ denote the set of all left invariant differential operators on $G$ , and let ${\\\\bf D}_K(G)$ denote the subspace of those operators that are also $K$ -right-invariant.\",\n \"A mapping $\\\\phi :G\\\\rightarrow \\\\mathbb {C}$ is called a spherical if it is $K$ -bi-invariant, satisfies $\\\\phi (e) =1$ , and is a simultaneous eigenfunction of every element of ${\\\\bf D}_K(G)$ .\",\n \"Fix an Iwasawa decomposition $G=NAK$ , where $N$ is a nilpotent Lie subgroup of $G$ , and $A$ is Abelian.\",\n \"Let $\\\\mathfrak {n}$ and $\\\\mathfrak {a}$ denote respectively the Lie algebras of $N$ and $A$ .\",\n \"For each $\\\\sigma \\\\in G$ , let $A(\\\\sigma )$ denote the unique element of $\\\\operatorname{\\\\mathfrak {a}}$ such that $\\\\sigma \\\\in Ne^{A(\\\\sigma )}K$ .\",\n \"Harish-Chandra's integral formula states that every spherical function on $G$ takes the form $\\\\phi _\\\\lambda (\\\\sigma ) = \\\\int _Ke^{(\\\\rho +i\\\\lambda )(A(k\\\\sigma ))}dk, \\\\hspace{20.0pt} \\\\forall \\\\sigma \\\\in G,$ for some $\\\\lambda \\\\in *$ .\",\n \"Moreover, $\\\\phi _\\\\lambda =\\\\phi _{\\\\lambda ^{\\\\prime }}$ if an only if $s(\\\\lambda )=\\\\lambda ^{\\\\prime }$ for some element $s$ of the Weyl group $W$ .\",\n \"A spherical function $\\\\phi _\\\\lambda $ is positive definite if and only if $\\\\lambda \\\\in $ .\",\n \"The spherical transform of a function $f\\\\in L^1(K|G|K)$ is the function $\\\\hat{f}:\\\\rightarrow \\\\mathbb {C}$ given by $\\\\hat{f}(\\\\lambda ) = \\\\int _G\\\\phi _{-\\\\lambda }(\\\\sigma )f(\\\\sigma )d\\\\sigma , \\\\hspace{20.0pt} \\\\forall \\\\lambda \\\\in .$ Similarly, given a finite Borel measure $\\\\mu $ on $G$ , the spherical transform of $\\\\mu $ is the mapping $\\\\hat{\\\\mu }:\\\\rightarrow \\\\mathbb {C}$ given by $\\\\hat{\\\\mu }(\\\\lambda ) = \\\\int _G\\\\phi _{-\\\\lambda }(\\\\sigma )\\\\mu (d\\\\sigma ).$ The spherical transform enjoys many useful properties, the most powerful being that it defines an isomorphism of the Banach convolution algebra $L^1(K|G|K)$ with the space $L^1(,\\\\omega )^W$ of Weyl group invariant elements of $L^1(,\\\\omega )$ .\",\n \"The Borel measure $\\\\omega $ is called Plancherel measure, and is given by $\\\\omega (d\\\\lambda ) = |\\\\operatorname{\\\\bf c}(\\\\lambda )|^{-2}d\\\\lambda ,$ where $\\\\operatorname{\\\\bf c}$ denotes Harish-Chandra's $\\\\operatorname{\\\\bf c}$ -function.\",\n \"According to the spherical inversion theorem, for all $f\\\\in C_c^\\\\infty (K|G|K)$ and all $\\\\sigma \\\\in G$ , $f(\\\\sigma ) = \\\\int _{}\\\\phi _\\\\lambda (\\\\sigma )\\\\hat{f}(\\\\lambda )\\\\omega (d\\\\lambda ).$ There is also a version of Plancherel's identity for the spherical transform, namely $\\\\Vert f\\\\Vert _{L^2(K|G|K)} = \\\\Vert \\\\hat{f}\\\\Vert _{L^2(,\\\\omega )}, \\\\hspace{20.0pt} \\\\forall f\\\\in C_c^\\\\infty (K|G|K).$ Let $L^2(,\\\\omega )^W$ denote the subspace of $L^2(,\\\\omega )$ consisting of $W$ -invariants.\",\n \"Then the image of $C_c^\\\\infty (K|G|K)$ under spherical transformation is a dense subspace of $L^2(,\\\\omega )^W$ , and as such the spherical transform extends to an isometric isomorphism between the Hilbert spaces $L^2(K|G|K)$ and $L^2(,\\\\omega )^W$ .\",\n \"For more details, see for example Helgason [21] Chapter IV § 7.3, pp. 454.\",\n \"Similarly to classical Fourier theory, the most natural setting for the spherical transform is Schwarz space.\",\n \"A function $f\\\\in C^\\\\infty (G)$ is called rapidly decreasing if $\\\\sup _{\\\\sigma \\\\in G}(1+|\\\\sigma |)^q\\\\phi _0(\\\\sigma )^{-1}(Df)(\\\\sigma ) < \\\\infty , \\\\hspace{20.0pt} \\\\forall D\\\\in {\\\\bf D}(G), \\\\;q\\\\in \\\\mathbb {N}\\\\cup \\\\lbrace 0\\\\rbrace ,$ where $|\\\\sigma |$ denotes the geodesic distance between the double coset spaces $K\\\\sigma K$ and $KeK$ .\",\n \"Equivalently, if $\\\\sigma \\\\in G$ is decomposed according to the Cartan decomposition $G=K\\\\overline{\\\\exp \\\\operatorname{\\\\mathfrak {a}}^+}K$ , so that $\\\\sigma =ke^Hk^{\\\\prime }$ for some unique $H\\\\in \\\\overline{\\\\operatorname{\\\\mathfrak {a}}^+}$ , then $|\\\\sigma |=|H|$ — see Gangolli and Varadarajan [17] pp.167 for more details on this object.\",\n \"The ($K$ -bi-invariant) Schwarz space $\\\\mathcal {S}(K|G|K)$ is the Fréchet space comprising of all rapidly decreasing, $K$ -bi-invariant functions $f\\\\in C^\\\\infty (G)$ , together with the family of seminorms given by the left-hand side of (REF ).\",\n \"By viewing the spaces $\\\\operatorname{\\\\mathfrak {a}}$ and $$ as finite dimensional vector spaces, we also consider the classical Schwartz spaces $\\\\mathcal {S}()$ and $\\\\mathcal {S}(\\\\operatorname{\\\\mathfrak {a}})$ , as well as $W$ -invariant subspaces $\\\\mathcal {S}(\\\\operatorname{\\\\mathfrak {a}})^W$ and $\\\\mathcal {S}()^W$ .\",\n \"The Euclidean Fourier transform ${F}(f)(\\\\lambda ) = \\\\int _{\\\\operatorname{\\\\mathfrak {a}}}e^{-i\\\\lambda (H)}f(H)dH, \\\\hspace{20.0pt} \\\\forall f\\\\in \\\\mathcal {S}(\\\\operatorname{\\\\mathfrak {a}}),\\\\lambda \\\\in $ defines a topological isomorphism between the spaces $\\\\mathcal {S}(\\\\operatorname{\\\\mathfrak {a}})^W$ and $\\\\mathcal {S}()^W$ in the usual way.\",\n \"Given $f\\\\in \\\\mathcal {S}(K|G|K)$ and $H\\\\in \\\\operatorname{\\\\mathfrak {a}}$ , the Abel transform is defined by ${A}f (H) = e^{\\\\rho (H)}\\\\int _Nf((\\\\exp H)n)dn.$ The Abel transform is fascinating in its own right, and we refer to Sawyer [36] for more information.\",\n \"However, for our purposes we are mainly interested in its role in the following: Theorem 2.1 Writing ${H}$ for the spherical transform, the diagram [scale=2] 1) at (0,1)$\\\\mathcal {S}()^W$ ; 2) at (1,0)$\\\\mathcal {S}(\\\\operatorname{\\\\mathfrak {a}})^W$ ; 3) at (-1,0)$\\\\mathcal {S}(K|G|K)$ ; [->] (2) – (1); [->] (3) – (1); [->] (3) – (2); t (.63,.63) ${F}$ ; t (-.63,.63) ${H}$ ; t (0,-.2) ${A}$ ; commutes, up to normalizing constants.\",\n \"Each arrow describes an isomorphism of Fréchet algebras.\",\n \"This result will be extremely useful in later sections, especially when proving Theorem REF .\",\n \"For more details, see Proposition 3 in Anker [1], Gangolli and Varadarajan [17] page 265, and Helgason [21] pp.\",\n \"450.\"\n ],\n [\n \"Probability on Lie Groups and Symmetric spaces\",\n \"In this subsection, we summarise a few key notions from probability theory on Lie groups and symmetric spaces.\",\n \"Books by Liao [33], [34] are excellent sources for this material, as is the paper [35] by Liao and Wang.\",\n \"Fix a probability space $(\\\\Omega ,\\\\mathcal {F},P)$ .\",\n \"Just as with functions on $G$ and $G/K$ , we may view stochastic processes on $G/K$ as projections of processes on $G$ whose laws are $K$ -right invariant.\",\n \"Equipped with its natural filtration $\\\\lbrace \\\\mathcal {F}^X_t,t\\\\ge 0\\\\rbrace $ , $X$ is said to have independent increments if for all $t>s\\\\ge 0$ , $X(s)^{-1}X(t)$ is independent of $\\\\mathcal {F}^X_s$ , and stationary increments if $X(s)^{-1}X(t) \\\\sim X(0)^{-1}X(t-s) \\\\hspace{20.0pt} \\\\forall t>s\\\\ge 0.$ A process $X=(X(t),t\\\\ge 0)$ on $G$ is stochastically continuous if, for all $s\\\\ge 0$ and all $B\\\\in \\\\mathcal {B}(G)$ with $e\\\\notin B$ , $\\\\lim _{t\\\\rightarrow s}P(X(s)^{-1}X(t)\\\\in B)=0.$ A stochastically continuous process $X$ on $G$ with stationary and independent increments is called a Lévy process on $G$ .\",\n \"A process on $G/K$ is called a Lévy process if it is the projection of a Lévy process on $G$ , under the canonical surjection $\\\\pi :G\\\\mapsto G/K$ .\",\n \"The convolution product of two Borel measures $\\\\mu _1,\\\\mu _2$ on $G$ is defined for each $B\\\\in \\\\mathcal {B}(G)$ by $(\\\\mu _1\\\\ast \\\\mu _2)(B) = \\\\int _G\\\\int _G\\\\operatorname{\\\\bf 1}_B(\\\\sigma \\\\tau )\\\\mu _1(d\\\\sigma )\\\\mu _2(d\\\\tau ).$ Note that since $G$ is semisimple, it is unimodular, and hence this operation is commutative.\",\n \"It is also clear from the definition that $\\\\mu _1\\\\ast \\\\mu _2$ is $K$ -bi-invariant whenever $\\\\mu _1$ and $\\\\mu _2$ are.\",\n \"Definition 2.2 A family $(\\\\mu _t,t\\\\ge 0)$ of finite Borel measures on $G$ will be called a convolution semigroup (of probability measures) if $\\\\mu _t(G)=1$ for all $t\\\\ge 0$ , $\\\\mu _{s+t} = \\\\mu _s\\\\ast \\\\mu _t$ for all $s,t\\\\ge 0$ , and $\\\\mu _t\\\\rightarrow \\\\mu _0$ weakly as $t\\\\rightarrow 0$ .\",\n \"Note that $\\\\mu _0$ must be an idempotent measure, in the sense that $\\\\mu _0\\\\ast \\\\mu _0=\\\\mu _0$ .\",\n \"By Theorem 1.2.10 on page 34 of Heyer [23], $\\\\mu _0$ must coincide with Haar measure on a compact subgroup of $G$ .\",\n \"We we will frequently take $\\\\mu _0$ to be normalised Haar measure on $K$ , so that the image of $\\\\mu _0$ after projecting onto $G/K$ is $\\\\delta _o$ , the delta mass at $o:=eK$ .\",\n \"One may also define convolution of measures on $G/K$ , and convolution semigroups on $G/K$ are defined analogously — see Liao [34] Section 1.3 for more details.\",\n \"In fact, the projection map $\\\\pi :G\\\\rightarrow G/K$ induces a bijection between the set of all convolution semigroups on $G/K$ and the set of all $K$ -bi-invariant convolution semigroups on $G$ — see Liao [34] Propositions 1.9 and 1.12, pp. 11–13.\",\n \"We henceforth identify these two sets, but generally opt to perform calculations using objects defined on $G$ , for simplicity.\",\n \"If $X=(X(t),t\\\\ge 0)$ is a Lévy process on $G$ (resp.\",\n \"$G/K$ ), and if $\\\\mu _t$ denotes the law of $X$ at time $t$ , then $(\\\\mu _t,t\\\\ge 0)$ is a convolution semigroup on $G$ (resp.\",\n \"$G/K$ ).\",\n \"A well-known construction shows that all convolution semigroups on $G$ and $G/K$ arise this way — see for example Theorem 1.7 on page 8 of [34].\",\n \"Let $(\\\\mu _t, t\\\\ge 0)$ be a $K$ -bi-invariant convolution semigroup of probability measures on $G$ associated with a Lévy process $X$ on $G/K$ .\",\n \"The Feller semigroup $(T_t,t\\\\ge 0)$ associated with $X$ (also called the Hunt semigroup of $(\\\\mu _t,t\\\\ge 0)$ ) is given by $T_tf(\\\\sigma ) = \\\\int _Gf(\\\\sigma \\\\tau )\\\\mu _t(d\\\\tau ) \\\\hspace{20.0pt} \\\\forall f\\\\in C_0(K|G|K), \\\\;\\\\sigma \\\\in G.$ Definition 2.3 Let $X_1,\\\\ldots ,X_l$ be a basis of $\\\\mathfrak {g}$ , ordered so that $X_1,\\\\ldots ,X_d$ is a basis of $\\\\mathfrak {p}$ .\",\n \"A collection $\\\\lbrace x_1,\\\\ldots ,x_l\\\\rbrace $ of smooth functions of compact support is called a system of exponential coordinate functions if there is a neighbourhood $U$ of $e$ for which $\\\\sigma = \\\\exp \\\\left(\\\\sum _{i=1}^lx_i(\\\\sigma )X_i\\\\right) \\\\hspace{20.0pt} \\\\forall \\\\sigma \\\\in U.$ The $x_i$ may be chosen so as to be $K$ -right-invariant for $i=1,\\\\ldots , m$ , and such that $\\\\sum _{i=1}^dx_i(k\\\\sigma )X_i = \\\\sum _{i=1}^dx_i(\\\\sigma )\\\\operatorname{Ad}(k)X_i \\\\hspace{20.0pt} \\\\forall k\\\\in K.$ For more details, see Liao [34] pp.36–37, 83.\",\n \"The choice of basis $X_1,\\\\ldots ,X_d$ of $\\\\mathfrak {p}$ enables us to view $\\\\operatorname{Ad}k$ as a $m\\\\times m$ matrix, for each $k\\\\in K$ .\",\n \"A vector $b\\\\in \\\\mathbb {R}^m$ is said to be $\\\\operatorname{Ad}(K)$ -invariant if $b=\\\\operatorname{Ad}(k)^Tb, \\\\hspace{20.0pt} \\\\forall k\\\\in K.$ Similarly, an $m\\\\times m$ real-valued matrix $a=(a_{ij})$ is $\\\\operatorname{Ad}(K)$ -invariant if $a=\\\\operatorname{Ad}(k)^Ta\\\\operatorname{Ad}(k) \\\\hspace{20.0pt} \\\\forall k\\\\in K.$ A Borel measure $\\\\nu $ on $G$ is called a Lévy measure if $\\\\nu (\\\\lbrace e\\\\rbrace ) = 0$ , $\\\\int _G\\\\sum _{i=1}^lx_i(\\\\sigma )^2\\\\nu (d\\\\sigma )$ , and $\\\\nu (B^c)<\\\\infty $ for any neighbourhood $B$ of $e$ .\",\n \"We state a useful corollary of the famous Hunt formula.\",\n \"For more details, including a proof, see Section 3.2 Liao [34], pp. 78.\",\n \"Theorem 2.4 Let $\\\\mathcal {L}$ be the infinitesimal generator associated with a $K$ -bi-invariant Lévy process on $G$ .\",\n \"Then $C_c^\\\\infty (G)\\\\subseteq \\\\operatorname{Dom}\\\\mathcal {L}$ , and there is an $\\\\operatorname{Ad}(K)$ -invariant vector $b\\\\in \\\\mathbb {R}^m$ , an $\\\\operatorname{Ad}(K)$ -invariant covariance matrix $a:=(a_{ij})$ , and a $K$ -bi-invariant Lévy measure $\\\\nu $ such that $\\\\begin{aligned}\\\\mathcal {L}f(\\\\sigma ) = \\\\sum _{i=1}^db_iX_if(\\\\sigma ) &+ \\\\frac{1}{2}\\\\sum _{j,k=1}^da_{jk}X_jX_kf(\\\\sigma ) \\\\\\\\&+ \\\\int _G\\\\left(f(\\\\sigma \\\\tau )-f(\\\\sigma )-\\\\sum _{i=1}^dx_i(\\\\tau )X_if(\\\\sigma )\\\\right)\\\\nu (d\\\\sigma ),\\\\end{aligned}$ for all $f\\\\in C_c^\\\\infty (G)$ and $\\\\sigma \\\\in G$ .\",\n \"Moreover, the triple $(b,a,\\\\nu )$ is completely determined by $\\\\mathcal {L}$ , and independent of the choice of exponential coordinate functions $x_i,\\\\;i=1,\\\\ldots ,d$ .\",\n \"Conversely, given a triple $(b,a,\\\\nu )$ of this kind, there is a unique $K$ -bi-invariant convolution semigroup of probability measures on $G$ with infinitesimal generator given by $\\\\mathcal {L}$ .\",\n \"Let $\\\\mathcal {L}$ be the Lévy generator described in Theorem REF above, and let $\\\\mathcal {L}_D$ be the diffusion part of $\\\\mathcal {L}$ , so that $\\\\mathcal {L}_D = \\\\sum _{i=1}^db_iX_i + \\\\frac{1}{2}\\\\sum _{i,j=1}^da_{jk}X_jX_k.$ Now by the discussion surrounding equation (3.3) of Liao [34], pp.\",\n \"75, $\\\\mathcal {L}_D\\\\in {\\\\bf D}_K(G)$ , and so for each $\\\\lambda \\\\in $ there is a constant $\\\\beta (\\\\mathcal {L}_D,\\\\lambda )\\\\in \\\\mathbb {C}$ such that $\\\\mathcal {L}_D\\\\phi _\\\\lambda = \\\\beta (\\\\mathcal {L}_D,\\\\lambda )\\\\phi _\\\\lambda .$ The mapping $\\\\lambda \\\\mapsto \\\\beta (\\\\mathcal {L}_D,\\\\lambda )$ is a $W$ -invariant quadratic polynomial function on $$ .\",\n \"Theorem 2.5 (Gangolli's Lévy–Khinchine formula) Let $(\\\\mu _t,t\\\\ge 0)$ be a $K$ -bi-invariant convolution semigroup of probability measures on $G$ with infinitesimal generator $\\\\mathcal {L}$ , and let $\\\\mathcal {L}_D$ denote the diffusion part of $\\\\mathcal {L}$ .\",\n \"Then $\\\\hat{\\\\mu }_t=e^{-t\\\\psi }$ , where $\\\\psi (\\\\lambda ) = -\\\\beta (\\\\mathcal {L}_D,\\\\lambda ) + \\\\int _G(1-\\\\phi _\\\\lambda (\\\\sigma ))\\\\nu (d\\\\sigma ) \\\\hspace{20.0pt} \\\\forall \\\\lambda \\\\in ,$ and $\\\\beta (\\\\mathcal {L}_D,\\\\lambda )$ is given by (REF ).\",\n \"This result was first proven by Ramesh Gangolli in [16], see also Liao and Wang [35] .\",\n \"For a proof of the specific statement above, see page 139 of Liao [34].\",\n \"The function $\\\\psi $ given by (REF ) will be called the Gangolli exponent of the process $X$ .\",\n \"Remark 2.6 If Definition REF (REF ) is relaxed so that each $\\\\mu _t$ need only satisfy $\\\\mu _t(G)\\\\le 1$ , all of the results described in this subsection continue to hold, except “sub-” must be added to some to the terms: convolution semigroups of sub-probability measures, sub-Lévy generators, sub-diffusion operators, and so on.\"\n ],\n [\n \"Positive and Negative Definite Functions\",\n \"By viewing $$ as a finite-dimensional real vector space, we may consider positive and negative definite functions on $$ , defined in the usual way.\",\n \"Proposition 2.7 For all $\\\\sigma \\\\in G$ , $\\\\lambda \\\\mapsto \\\\phi _\\\\lambda (\\\\sigma )$ is positive definite.\",\n \"Let $\\\\mu $ be a finite $K$ -bi-invariant Borel measure.\",\n \"Then $\\\\hat{\\\\mu }$ is positive definite.\",\n \"Let $\\\\sigma \\\\in G$ , $n\\\\in \\\\mathbb {N}$ , $\\\\lambda _1,\\\\ldots ,\\\\lambda _n\\\\in $ , and $c_1,\\\\ldots ,c_n\\\\in \\\\mathbb {C}$ , and note that $\\\\sum _{\\\\alpha ,\\\\beta =1}^nc_\\\\alpha \\\\overline{c_\\\\beta }e^{(i(\\\\lambda _\\\\alpha -\\\\lambda _\\\\beta )+\\\\rho )A(k\\\\sigma )} = \\\\left|\\\\sum _{\\\\alpha =1}^nc_\\\\alpha e^{(i\\\\lambda _\\\\alpha +\\\\frac{\\\\rho }{2})A(k\\\\sigma )}\\\\right|^2 \\\\ge 0.$ Therefore, by the Harish-Chandra integral formula, $\\\\sum _{\\\\alpha ,\\\\beta =1}^nc_\\\\alpha \\\\overline{c_\\\\beta }\\\\phi _{\\\\lambda _\\\\alpha -\\\\lambda _\\\\beta }(\\\\sigma ) = \\\\int _K\\\\sum _{\\\\alpha ,\\\\beta =1}^nc_\\\\alpha \\\\overline{c_\\\\beta }e^{(i(\\\\lambda _\\\\alpha -\\\\lambda _\\\\beta )+\\\\rho )A(k\\\\sigma )}dk \\\\ge 0.$ Part 1 follows.\",\n \"For part 2, observe that since (REF ) holds for all $c_1,\\\\ldots ,c_n$ , we can replace each $c_j$ by its complex conjugate.\",\n \"Therefore, $\\\\sum _{\\\\alpha ,\\\\beta =1}^n\\\\overline{c_\\\\alpha }c_\\\\beta \\\\phi _{\\\\lambda _\\\\alpha -\\\\lambda _\\\\beta }(\\\\sigma )\\\\ge 0$ for all $\\\\sigma \\\\in G$ , $n\\\\in \\\\mathbb {N}$ , $\\\\lambda _1,\\\\ldots ,\\\\lambda _n\\\\in $ , and $c_1,\\\\ldots ,c_n\\\\in \\\\mathbb {C}$ .\",\n \"Taking complex conjugates and using the fact that $\\\\phi _{-\\\\lambda }=\\\\overline{\\\\phi _\\\\lambda }$ , it follows that for all $\\\\sigma \\\\in G$ , $n\\\\in \\\\mathbb {N}$ , $\\\\lambda _1,\\\\ldots ,\\\\lambda _n\\\\in $ , and $c_1,\\\\ldots ,c_n\\\\in \\\\mathbb {C}$ , $\\\\sum _{\\\\alpha ,\\\\beta =1}^nc_\\\\alpha \\\\overline{c_\\\\beta }\\\\phi _{-(\\\\lambda _\\\\alpha -\\\\lambda _\\\\beta )}(\\\\sigma ) = \\\\overline{\\\\sum _{\\\\alpha ,\\\\beta =1}^n\\\\overline{c_\\\\alpha }c_\\\\beta \\\\phi _{\\\\lambda _\\\\alpha -\\\\lambda _\\\\beta }}\\\\ge 0,$ and hence $\\\\sum _{\\\\alpha ,\\\\beta =1}^nc_\\\\alpha \\\\overline{c_\\\\beta }\\\\hat{\\\\mu }(\\\\lambda _\\\\alpha -\\\\lambda _\\\\beta ) = \\\\int _{}\\\\sum _{\\\\alpha ,\\\\beta =1}^nc_\\\\alpha \\\\overline{c_\\\\beta }\\\\phi _{-(\\\\lambda _\\\\alpha -\\\\lambda _\\\\beta )}(\\\\sigma )\\\\mu (d\\\\sigma )\\\\ge 0.$ By choosing a basis of $$ , we may identify it with $\\\\mathbb {R}^m$ , and apply classical results about positive (resp.\",\n \"negative) definite functions on Euclidean space to functions on $$ , to obtain results about positive (resp.\",\n \"negative) definite functions in this new setting.\",\n \"One useful application of this is the Schoenberg correspondence, which states that a map $\\\\psi :\\\\rightarrow \\\\mathbb {C}$ is negative definite if and only if $\\\\psi (0)\\\\ge 0$ and $e^{-t\\\\psi }$ is positive definite for all $t>0$ .\",\n \"This is immediate by the Schoenberg correspondence on $\\\\mathbb {R}^m$ — see Berg and Forst [10] page 41 for a proof.\",\n \"Proposition 2.8 Let $\\\\psi :\\\\rightarrow \\\\mathbb {C}$ be the Gangolli exponent of a Lévy process on $G/K$ .\",\n \"Then $\\\\psi $ is negative definite.\",\n \"Let $X=(X(t),t\\\\ge 0)$ is a Lévy process on $G/K$ , and let $\\\\nu _t$ be the law of $X(t)$ , for all $t\\\\ge 0$ .\",\n \"Then $(\\\\nu _t,t\\\\ge 0)$ forms a convolution semigroup on $G/K$ .\",\n \"By Proposition 1.12 of Liao [34] (pp.\",\n \"13), $(\\\\nu _t,t\\\\ge 0)$ arises as the projection onto $G/K$ of a $K$ -bi-invariant convolution semigroup $(\\\\mu _t,t\\\\ge 0)$ on $G$ .\",\n \"By Proposition REF , the spherical transform of each $\\\\mu _t$ is positive definite, and by the Schoenberg correspondence, for each $t\\\\ge 0$ , there is a negative definite function $\\\\psi _t$ on $$ such that $\\\\psi _t(0)\\\\ge 0$ and $\\\\hat{\\\\mu }_t=e^{-\\\\psi _t}$ .\",\n \"In fact, since $(\\\\mu _t,t\\\\ge 0)$ is a convolution semigroup, it must be the case that $\\\\hat{\\\\mu }_t = e^{-t\\\\psi _1}, \\\\hspace{20.0pt} \\\\forall t\\\\ge 0.$ By uniqueness of Gangolli exponents, $\\\\psi =\\\\psi _1$ , a negative definite function.\",\n \"We finish this subsection with a collection of results about negative definite functions, which will be particularly useful for calculation in later sections.\",\n \"Proposition 2.9 Let $\\\\psi :\\\\rightarrow \\\\mathbb {C}$ be a continuous negative definite function.\",\n \"Then For all $\\\\lambda ,\\\\eta \\\\in $ , $\\\\left|\\\\sqrt{|\\\\psi (\\\\lambda )|}-\\\\sqrt{|\\\\psi (\\\\eta )|}\\\\right|\\\\le \\\\sqrt{|\\\\psi (\\\\lambda -\\\\eta )|}$ (Generalised Peetre inequality) For all $s\\\\in \\\\mathbb {R}$ and $\\\\lambda ,\\\\eta \\\\in $ , $\\\\left(\\\\frac{1+|\\\\psi (\\\\lambda )|}{1+|\\\\psi (\\\\eta )|}\\\\right)^s \\\\le 2^{|s|}(1+|\\\\psi (\\\\lambda -\\\\eta )|^2)^{|s|}.$ There is a constant $c_\\\\psi >0$ such that $|\\\\psi (\\\\lambda )|\\\\le c_\\\\psi (1+|\\\\lambda |^2) \\\\hspace{20.0pt} \\\\forall \\\\lambda \\\\in .$ These results follow from their analogues on $\\\\mathbb {R}^m$ — see Hoh [24] page 16.\"\n ],\n [\n \"Spherical Anisotropic Sobolev Spaces\",\n \"Suppose $\\\\psi $ is a real-valued continuous negative definite function, and let $s\\\\in \\\\mathbb {R}$ .\",\n \"We define the (spherical) anisotropic Sobolev space associated with $\\\\psi $ and $s$ to be $H^{\\\\psi ,s} := \\\\left\\\\lbrace u\\\\in \\\\mathcal {S}^{\\\\prime }(K|G|K):\\\\int _G(1+\\\\psi (\\\\lambda ))^s|\\\\hat{u}(\\\\lambda )|^2\\\\omega (d\\\\lambda )<\\\\infty \\\\right\\\\rbrace ,$ where $\\\\mathcal {S}^{\\\\prime }(K|G|K)$ denotes the space of $K$ -bi-invariant tempered distributions.\",\n \"One can check that each $H^{\\\\psi ,s}$ is a Hilbert space with respect to the inner product $\\\\langle u,v\\\\rangle _{\\\\psi ,s} := \\\\int _{}(1+\\\\psi (\\\\lambda ))^s\\\\hat{u}(\\\\lambda )\\\\overline{\\\\hat{v}(\\\\lambda )}\\\\omega (d\\\\lambda ), \\\\hspace{20.0pt} \\\\forall u,v\\\\in H^{\\\\psi ,s}.$ These spaces are a generalisation of the anisotropic Sobolev spaces first introduced by Niels Jacob, see [26], and developed further by Hoh, see [24].\",\n \"For the special case $\\\\psi (\\\\lambda )=|\\\\rho |^2+|\\\\lambda |^2$ , we will write $H^{\\\\psi ,s}=H^s$ .\",\n \"Note also that $H^{\\\\psi ,0}=L^2(K|G|K)$ , by the Plancherel theorem.\",\n \"In this case, we will omit subscripts and just write $\\\\langle \\\\cdot ,\\\\cdot \\\\rangle $ for the inner product on $L^2(K|G|K)$ .\",\n \"Note that $\\\\psi $ is a non-negative function, since it is negative definite and real-valued.\",\n \"We impose an additional assumption, namely that there exist constants $r,c>0$ such that $\\\\psi (\\\\lambda )\\\\ge c\\\\left(|\\\\rho |^2+|\\\\lambda |^2\\\\right)^r \\\\hspace{20.0pt} \\\\forall \\\\lambda \\\\in .$ Analogous assumptions can be found in Jacob [27] (1.5) and Hoh [24] (4.4), and the role of (REF ) will be very similar.\",\n \"Theorem 2.10 Let $\\\\psi $ be a real-valued, continuous negative definite symbol, satisfying (REF ).\",\n \"Then $C_c^\\\\infty (K|G|K)$ and $\\\\mathcal {S}(K|G|K)$ are dense in each $H^{\\\\psi ,s}$ , and we have continuous embeddings $\\\\mathcal {S}(K|G|K)\\\\hookrightarrow H^{\\\\psi ,s}\\\\hookrightarrow \\\\mathcal {S}^{\\\\prime }(K|G|K)$ We have continuous embeddings $H^{\\\\psi ,s_2}\\\\hookrightarrow H^{\\\\psi ,s_1} $ whenever $s_1,s_2\\\\in \\\\mathbb {R}$ with $s_2\\\\ge s_1$ .\",\n \"In particular, $H^{\\\\psi ,s}\\\\hookrightarrow L^2(K|G|K)$ for all $s\\\\ge 0$ .\",\n \"Under the standard identification of $L^2(K|G|K)$ with its dual, the dual space of each $H^{\\\\psi ,s}$ is isomorphic to $H^{\\\\psi ,-s}$ , with $\\\\Vert u\\\\Vert _{\\\\psi ,-s} = \\\\sup \\\\left\\\\lbrace \\\\frac{|\\\\langle u,v\\\\rangle |}{\\\\Vert v\\\\Vert _{\\\\psi ,s}}:v\\\\in C_c^\\\\infty (K|G|K),\\\\;v\\\\ne 0\\\\right\\\\rbrace ,$ for all $s\\\\in \\\\mathbb {R}$ .\",\n \"For $r>0$ as in equation (REF ), we have continuous embeddings $H^s \\\\hookrightarrow H^{\\\\psi ,s} \\\\hookrightarrow H^{rs},$ for all $s\\\\ge 0$ .\",\n \"Let $s_3>s_2>s_1$ .\",\n \"Then for all $\\\\epsilon >0$ , there is $c(\\\\epsilon )\\\\ge 0$ such that $\\\\Vert u\\\\Vert _{\\\\psi ,s_2} \\\\le \\\\epsilon \\\\Vert u\\\\Vert _{\\\\psi ,s_3} + c(\\\\epsilon )\\\\Vert u\\\\Vert _{\\\\psi ,s_1}$ for all $u\\\\in H^{\\\\psi ,s_3}$ .\",\n \"There exist continuous embeddings $H^{\\\\psi ,s}\\\\hookrightarrow C_0(K|G|K)$ for all $s>\\\\frac{d}{r}$ , where $d=\\\\dim (G/K)$ .\",\n \"For brevity, let $\\\\langle \\\\lambda \\\\rangle := \\\\sqrt{1+|\\\\lambda |^2}, \\\\hspace{20.0pt} \\\\forall \\\\lambda \\\\in ,$ and $\\\\Psi (\\\\lambda ) := \\\\sqrt{1+\\\\psi (\\\\lambda )}, \\\\hspace{20.0pt} \\\\forall \\\\lambda \\\\in .$ The proof of Theorem REF will be given after the next lemma.\",\n \"Lemma 2.11 Let $M>d=\\\\dim (G/K)$ .\",\n \"Then $\\\\langle \\\\cdot \\\\rangle ^{-M} \\\\in L^1(,\\\\omega )$ .\",\n \"By standard arguments, one may check that $\\\\int _{\\\\mathbb {R}^d}\\\\langle \\\\xi \\\\rangle ^{-M}d\\\\xi <\\\\infty $ , for all $M>d$ .\",\n \"Writing $p=\\\\frac{\\\\dim N}{2}$ , we have $d= \\\\dim + 2p$ , and hence $\\\\int _{}\\\\langle \\\\lambda \\\\rangle ^{-M+2p}d\\\\lambda < \\\\infty $ whenever $M>d$ .\",\n \"By Proposition 7.2 on page 450 of Helgason [21], there are $C_1,C_2>0$ such that $|\\\\operatorname{\\\\bf c}(\\\\lambda )|^{-1} \\\\le C_1 + C_2|\\\\lambda |^p \\\\hspace{20.0pt} \\\\forall \\\\lambda \\\\in .$ Let $C>0$ be such that $(C_1 + C_2|\\\\lambda |^p)^2 < C(1+|\\\\lambda |^2)^p$ for all $\\\\lambda \\\\in $ .\",\n \"Then $\\\\int _{}\\\\langle \\\\lambda \\\\rangle ^{-M}\\\\omega (d\\\\lambda ) = \\\\int _{}\\\\langle \\\\lambda \\\\rangle ^{-M}|\\\\operatorname{\\\\bf c}(\\\\lambda )|^{-2}d\\\\lambda \\\\le C\\\\int _{}\\\\langle \\\\lambda \\\\rangle ^{-M+2p}d\\\\lambda < \\\\infty ,$ whenever $M>d$ .\",\n \"[Proof of Theorem REF ] Much of this theorem may be proved by adapting proofs from the $\\\\mathbb {R}^d$ case.\",\n \"For example, to prove Theorem REF (REF ), let $\\\\mathcal {L}^{\\\\psi ,s}$ denote the space of all measurable functions $v$ on $$ for which $\\\\Psi ^sv\\\\in L^2(,\\\\omega )^W$ , a Hilbert space with respect to the inner product $\\\\langle u,v\\\\rangle = \\\\int _{\\\\operatorname{\\\\mathfrak {a}}^\\\\ast }\\\\Psi (\\\\lambda )^{2s}u(\\\\lambda )\\\\overline{v(\\\\lambda )}\\\\omega (d\\\\lambda ), \\\\hspace{20.0pt} \\\\forall u,v \\\\in \\\\mathcal {L}^{\\\\psi ,s}.$ By viewing $$ as a real vector space and using inequality (REF ) to relate $\\\\omega $ to Lebesgue measure, the proof of Theorem 3.10.3 on page 208 of Jacob [28] may be easily adapted to show that $\\\\mathcal {S}()^W\\\\hookrightarrow \\\\mathcal {L}^{\\\\psi ,s}\\\\hookrightarrow \\\\mathcal {S}^{\\\\prime }()^W$ is continuous.\",\n \"Noting Theorem REF , Theorem REF (REF ) follows.\",\n \"Proofs of Theorem REF (REF )–(REF ) are almost identical to their $\\\\mathbb {R}^d$ -based counterparts, see Jacob [27] §1, or Hoh [24] pp. 46–48.\",\n \"By Theorem REF (REF ), if we can prove the existence of a continuous embedding $H^s \\\\hookrightarrow C_0(K|G|K)$ for all $s>\\\\frac{d}{2}$ , then Theorem REF (REF ) will follow.\",\n \"Let $s>\\\\frac{d}{2}$ and $u\\\\in \\\\mathcal {S}(K|G|K)$ .\",\n \"By Lemma REF , $\\\\langle \\\\cdot \\\\rangle ^{-s}\\\\in L^2(\\\\operatorname{\\\\mathfrak {a}}^\\\\ast ,\\\\omega )$ , and by spherical inversion, $|u(\\\\sigma )| = \\\\left|\\\\int _{}\\\\phi _\\\\lambda (\\\\sigma )\\\\hat{u}(\\\\lambda )\\\\omega (d\\\\lambda )\\\\right| \\\\le \\\\int _{}|\\\\hat{u}(\\\\lambda )|\\\\omega (d\\\\lambda ) = \\\\int _{}\\\\langle \\\\lambda \\\\rangle ^{-s}\\\\langle \\\\lambda \\\\rangle ^s|\\\\hat{u}(\\\\lambda )|\\\\omega (d\\\\lambda ),$ for all $\\\\sigma \\\\in G$ .\",\n \"By the Cauchy–Schwarz inequality, $|u(\\\\sigma )| \\\\le \\\\Vert \\\\langle \\\\cdot \\\\rangle ^{-s}\\\\Vert _{L^2(,\\\\omega )}\\\\Vert \\\\langle \\\\cdot \\\\rangle ^s\\\\hat{u}\\\\Vert _{L^2(,\\\\omega )} = C\\\\Vert u\\\\Vert _s$ for all $\\\\sigma \\\\in G$ , where $C= \\\\Vert \\\\langle \\\\cdot \\\\rangle ^{-s}\\\\Vert _{L^2(,\\\\omega )}$ .\",\n \"It follows that $\\\\Vert u \\\\Vert _{C_0(K|G|K)} := \\\\sup _{\\\\sigma \\\\in G}|u(\\\\sigma )| \\\\le C\\\\Vert u\\\\Vert _s.$ The embedding (REF ) may then be obtained using a density argument.\"\n ],\n [\n \"Pseudodifferential Operators and Their Symbols\",\n \"A measurable mapping $q:G\\\\times \\\\rightarrow \\\\mathbb {C}$ will be called a negative definite symbol if it is locally bounded, and if for each $\\\\sigma \\\\in G$ , $q(\\\\sigma ,\\\\cdot )$ is negative definite and continuous.\",\n \"If in addition $q$ is continuous in its first argument, we will call $q$ a continuous negative definite symbol.\",\n \"Let $\\\\mathcal {M}(G)$ denote the set of all measurable functions on $G$ .\",\n \"Theorem 2.12 Let $q$ be a negative definite symbol, and for each $f\\\\in C^\\\\infty _c(K|G|K)$ and $\\\\sigma \\\\in G$ , define $q(\\\\sigma , D)f(\\\\sigma ) = \\\\int _{}\\\\hat{f}(\\\\lambda )\\\\phi _\\\\lambda (\\\\sigma )q(\\\\sigma ,\\\\lambda )\\\\omega (d\\\\lambda ).$ Then Equation (REF ) defines a linear operator $q(\\\\sigma ,D):C_c^\\\\infty (K|G|K)\\\\rightarrow \\\\mathcal {M}(G)$ .\",\n \"If $q$ is a continuous negative definite symbol, then $q(\\\\sigma ,D):C_c^\\\\infty (K|G|K)\\\\rightarrow C(G)$ .\",\n \"If $q$ is $K$ -bi-invariant in its first argument, then $q(\\\\sigma ,D)f$ is $K$ -bi-invariant for all $f\\\\in C_c^\\\\infty (K|G|K)$ .\",\n \"Theorem REF (REF ) and (REF ) are proved in a similar manner to Theorem 4.5.7 of Jacob [28], while (REF ) is immediate from the $K$ -bi-invariance of each spherical function $\\\\phi _\\\\lambda $ .\",\n \"Definition 2.13 Operators of the form (REF ), where $q$ is a negative definite symbol, will be called (spherical) pseudodifferential operators on $G$ .\",\n \"Example 2.14 The following examples of pseudodifferential operators arise as the generators of Lévy processes on symmetric spaces.\",\n \"Their symbols are constant in $\\\\sigma $ , a property that is well-known in the case of $G=\\\\mathbb {R}^d$ , $K=\\\\lbrace 0\\\\rbrace $ .\",\n \"Section will introduce a large class of examples pseudodifferential operators with spatial dependence.\",\n \"Diffusion operators with constant coefficients.\",\n \"Consider the pure diffusion operator $\\\\mathcal {L} := \\\\sum _{i=1}^db_iX_i + \\\\sum _{i,j=1}^da_{ij}X_iX_j$ where $b$ and $a$ are as in Theorem REF .\",\n \"Recall that such operators belong to ${\\\\bf D}_K(G)$ ; for each $\\\\lambda \\\\in $ , let $\\\\beta (\\\\mathcal {L},\\\\lambda )$ denote the $\\\\phi _\\\\lambda $ -eigenvalue of $\\\\mathcal {L}$ .\",\n \"Then $(\\\\sigma ,\\\\lambda )\\\\mapsto -\\\\beta (T,\\\\lambda )$ is a continuous negative definite symbol, and the associated pseudodifferential operator is $-T$ .\",\n \"To see this, let $(\\\\mu _t,t\\\\ge 0)$ denote the convolution semigroup generated by $\\\\mathcal {L}$ , and let $(T_t,t\\\\ge 0)$ be the associated Hunt semigroup, as defined in (REF ).\",\n \"Then $\\\\mathcal {L}f(\\\\sigma ) = \\\\left.\\\\frac{d}{dt}T_tf(\\\\sigma )\\\\right|_{t=0},$ for all $f\\\\in C_c^\\\\infty (K|G|K)$ and $\\\\sigma \\\\in G$ .\",\n \"By the spherical inversion formula (REF ), $T_tf(\\\\sigma ) = \\\\int _G\\\\int _{}\\\\hat{f}(\\\\lambda )\\\\phi _\\\\lambda (\\\\sigma \\\\tau )\\\\omega (d\\\\lambda )p_t(d\\\\tau ), \\\\hspace{20.0pt} \\\\forall f\\\\in C_c^\\\\infty (K|G|K),\\\\;\\\\sigma \\\\in G.$ Recalling that $\\\\hat{f}\\\\in \\\\mathcal {S}()$ whenever $f\\\\in C_c^\\\\infty (K|G|K)$ , a Fubini argument may be applied to conclude that $T_tf(\\\\sigma ) = \\\\int _{}\\\\hat{f}(\\\\lambda )T_t\\\\phi _\\\\lambda (\\\\sigma )\\\\omega (d\\\\lambda ) = \\\\int _{}\\\\hat{f}(\\\\lambda )e^{-t\\\\beta (\\\\mathcal {L},\\\\lambda )}\\\\phi _\\\\lambda (\\\\sigma )\\\\omega (d\\\\lambda ),$ for all $f\\\\in C_c^\\\\infty (K|G|K)$ and $\\\\sigma \\\\in G$ .\",\n \"By (REF ), given $f\\\\in C_c^\\\\infty (K|G|K)$ and $\\\\sigma \\\\in G$ , $\\\\begin{aligned}\\\\mathcal {L}f(\\\\sigma ) &= \\\\left.\\\\frac{d}{dt}\\\\int _{}\\\\hat{f}(\\\\lambda )e^{-t\\\\beta (\\\\mathcal {L},\\\\lambda )}\\\\phi _\\\\lambda (\\\\sigma )\\\\omega (d\\\\lambda )\\\\right|_{t=0} \\\\\\\\&= \\\\lim _{t\\\\rightarrow 0}\\\\int _{}\\\\hat{f}(\\\\lambda )\\\\left(\\\\frac{e^{-t\\\\beta (\\\\mathcal {L},\\\\lambda )}-1}{t}\\\\right)\\\\phi _\\\\lambda (\\\\sigma )\\\\omega (d\\\\lambda ).\\\\end{aligned}$ Now, for all $t>0$ and $\\\\lambda \\\\in $ , $\\\\left|\\\\hat{f}(\\\\lambda )\\\\left(\\\\frac{e^{-t\\\\beta (\\\\mathcal {L},\\\\lambda )}-1}{t}\\\\right) \\\\phi _\\\\lambda (\\\\sigma )\\\\right| \\\\le \\\\left|\\\\hat{f}(\\\\lambda )\\\\right|\\\\left|\\\\frac{e^{-t\\\\beta (\\\\mathcal {L},\\\\lambda )}-1}{t}\\\\right| \\\\le \\\\left|\\\\hat{f}(\\\\lambda )\\\\right||\\\\beta (\\\\mathcal {L},\\\\lambda )|.$ Moreover, $|\\\\hat{f}||\\\\beta (\\\\mathcal {L},\\\\cdot )|\\\\in L^1(,\\\\omega )^W$ , since $\\\\hat{f}\\\\in \\\\mathcal {S}()^W$ , and $\\\\beta (\\\\mathcal {L},\\\\cdot )$ is a $W$ -invariant polynomial function.\",\n \"By the dominated convergence theorem, we may bring the limit through the integral sign in (REF ) to conclude that $\\\\begin{aligned}\\\\mathcal {L}f(\\\\sigma ) &= \\\\int _{}\\\\hat{f}(\\\\lambda )\\\\lim _{t\\\\rightarrow 0}\\\\left(\\\\frac{e^{-t\\\\beta (\\\\mathcal {L},\\\\lambda )}-1}{t}\\\\right)\\\\phi _\\\\lambda (\\\\sigma )\\\\omega (d\\\\lambda ) \\\\\\\\&= -\\\\int _{}\\\\hat{f}(\\\\lambda )\\\\phi _\\\\lambda (\\\\sigma )\\\\beta (\\\\mathcal {L} ,\\\\lambda )\\\\omega (d\\\\lambda )\\\\end{aligned}$ for all $f\\\\in C_c^\\\\infty (K|G|K)$ and $\\\\sigma \\\\in G$ .\",\n \"Brownian motion.\",\n \"As a special case of the above, $-\\\\Delta $ is a pseudodifferential operator with symbol $|\\\\rho |^2+|\\\\lambda |^2$ .\",\n \"Lévy generators.\",\n \"More generally, if $\\\\mathcal {A}$ is the infinitesimal generator of a $K$ -bi-invariant Lévy process on $G$ , and if $\\\\psi $ is the corresponding Gangolli exponent, then $(\\\\sigma ,\\\\lambda )\\\\mapsto \\\\psi (\\\\lambda )$ is a continuous negative definite symbol, and $-\\\\mathcal {A}$ is the corresponding pseudodifferential operator — see Applebaum [4] Theorem 5.1.\"\n ],\n [\n \"Gangolli Operators and the Hille–Yosida–Ray Theorem\",\n \"We will soon define the class of pseudodifferential operators that will be of primary interest.\",\n \"In this section, we motivate this definition with a short discussion of the Hille–Yosida–Ray theorem, and prove that our class of operators are pseudodifferential operators in the sense of Definition REF .\",\n \"We finish the section with some examples.\",\n \"Let $E$ be a locally compact, Hausdorff space, let $\\\\mathcal {C}$ be a closed subspace of $C_0(E)$ , and let $\\\\mathcal {F}(E)$ denote the space of all real-valued functions on $E$ .\",\n \"A $C_0$ -semigroup $(T_t,t\\\\ge 0)$ defined on $C_0(K|G|K)$ is called sub-Feller if for all $f\\\\in C_0(E)$ , and all $t\\\\ge 0$ , $0\\\\le f\\\\le 1 ~\\\\Rightarrow ~ 0\\\\le T_tf\\\\le 1.$ A linear operator $A:\\\\mathcal {D}(A)\\\\rightarrow \\\\mathcal {F}(E)$ is said to satisfy the positive maximum principle, if for all $f\\\\in \\\\mathcal {D}(A)$ and $x_0\\\\in E$ such that $f(x_0)=\\\\sup _{x\\\\in E}f(x)\\\\ge 0$ , we have $Af(x_0)\\\\le 0$ .\",\n \"The following theorem is an extended version of the Hille–Yosida–Ray theorem, which fully characterises the operators that extend to generators of sub-Feller semigroups on $C_0(E)$ .\",\n \"Similar versions in which $E=\\\\mathbb {R}^d$ may found in [24], pp.\",\n \"53, and [28], pp. 333.\",\n \"For a proof, see Ethier and Kurz [14], pp. 165.\",\n \"Theorem 3.1 (Hille–Yosida–Ray) A linear operator $(\\\\mathcal {A},\\\\operatorname{Dom}\\\\mathcal {A})$ on $C_0(E)$ is closable and its closure generates a strongly continuous, sub-Feller semigroup on $C_0(E)$ if and only if the following is satisfied: $\\\\operatorname{Dom}\\\\mathcal {A}$ is dense in $C_0(E)$ , $\\\\mathcal {A}$ satisfies the positive maximum principle, and There exists $\\\\alpha >0$ such that $\\\\operatorname{Ran}(\\\\alpha I-\\\\mathcal {A})$ is dense in $C_0(E)$ .\",\n \"In their papers [7] and [8], Applebaum and Ngan found necessary and sufficient conditions for an operator defined on $C_c^\\\\infty (K|G|K)$ to satisfy Theorem REF (REF ), for the cases $E=G$ , $G/K$ and $K|G|K$ .\",\n \"We will focus primarily on the case $E=K|G|K$ , since this is the realm in which the spherical transform is available.\",\n \"A mapping $\\\\nu :G\\\\times \\\\mathcal {B}\\\\rightarrow [0,\\\\infty ]$ will be called a $K$ -bi-invariant Lévy kernel if it is $K$ -bi-invariant in its first argument, and if for all $\\\\sigma \\\\in G$ , $\\\\nu (\\\\sigma ,\\\\cdot )$ is a $K$ -bi-invariant Lévy measure.\",\n \"Fix a system of exponential coordinate functions, as defined in Definition REF , and adopt all of the notation conventions from this definition.\",\n \"Definition 3.2 An operator $\\\\mathcal {A}:C_c^\\\\infty (K|G|K)\\\\rightarrow \\\\mathcal {F}(G)$ will be called a Gangolli operator if there exist mappings $c, b_i, a_{jk}\\\\in C(K|G|K)$ ($1\\\\le i,j,k\\\\le d$ ), as well as a $K$ -bi-invariant Lévy kernel $\\\\nu $ , such that for all $f\\\\in C_c^\\\\infty (K|G|K)$ , $\\\\begin{aligned}\\\\mathcal {A}f(\\\\sigma ) = -c(\\\\sigma )f(\\\\sigma ) + \\\\sum _{i=1}^db_i&(\\\\sigma )X_if(\\\\sigma ) + \\\\sum _{j,k=1}^da_{jk}(\\\\sigma )X_jX_kf(\\\\sigma ) \\\\\\\\&+ \\\\int _G\\\\left(f(\\\\sigma \\\\tau )-f(\\\\sigma )-\\\\sum _{i=1}^dx_i(\\\\tau )X_if(\\\\sigma )\\\\right)\\\\nu (\\\\sigma ,d\\\\tau ),\\\\end{aligned}$ and if, in addition, $c$ is a non-negative mapping.\",\n \"For all $\\\\sigma \\\\in G$ , $b(\\\\sigma ):=(b_1(\\\\sigma ),\\\\ldots ,b_d(\\\\sigma ))$ is an $\\\\operatorname{Ad}(K)$ -invariant vector, For all $\\\\sigma \\\\in G$ , $a(\\\\sigma ):=(a_{jk}(\\\\sigma ))$ is an $\\\\operatorname{Ad}(K)$ -invariant, non-negative definite, symmetric matrix.\",\n \"For each $f\\\\in C_b(K|G|K)$ , the mappings $\\\\sigma \\\\mapsto \\\\int _Uf(\\\\tau )\\\\sum _{i=1}^dx(\\\\tau )^2\\\\nu (\\\\sigma ,d\\\\tau )$ and $\\\\sigma \\\\mapsto \\\\int _{U^c}f(\\\\tau )\\\\nu (\\\\sigma ,d\\\\tau )$ are continuous from $G$ to $[0,\\\\infty )$ .\",\n \"$\\\\lim _{\\\\sigma \\\\rightarrow \\\\infty }\\\\nu (\\\\sigma ,U^c)=0$ .\",\n \"The form (REF ) is necessary for $\\\\mathcal {A}$ to satisfy the positive maximum principle.\",\n \"The various continuity assumptions, together with condition (REF ), ensure that $\\\\mathcal {A}$ maps into $C_0(K|G|K)$ .\",\n \"By Theorem 3.2 (3) of [8] as well as Theorems 3.7 and 3.8 of Applebaum and Ngan [7], Gangolli operators are densely defined linear operators on $C_0(K|G|K)$ that satisfy the positive maximum principle.\",\n \"Remarks 3.3 Gangolli operators were first introduced in [8] in compact symmetric spaces and for a more restrictive form of (REF ).\",\n \"Equation (REF ) may be viewed as a spatially dependent generalisation of (REF ), with an additional killing term $c$ .\",\n \"Indeed, if the spatial dependence of the coefficients $c$ , $b$ , $a$ and $\\\\nu $ is removed from (REF ), then the resulting operator extends to the generator of a killed Lévy process, with state space $G\\\\cup \\\\lbrace \\\\infty \\\\rbrace $ , the one-point compactification of $G$ .\",\n \"For a more detailed discussion, see [7], pp. 149.\",\n \"For a Gangolli operator $\\\\mathcal {A}$ given by (REF ), the local part of $\\\\mathcal {A}$ is defined $\\\\mathcal {L}^\\\\sigma = -c(\\\\sigma ) + \\\\sum _{i=1}^db_i(\\\\sigma )X_i + \\\\sum _{j,k=1}^da_{jk}(\\\\sigma )X_jX_k \\\\hspace{20.0pt} \\\\forall \\\\sigma \\\\in G.$ Holding $\\\\sigma \\\\in G$ fixed, $\\\\mathcal {L}^\\\\sigma +c(\\\\sigma )$ can be viewed as a diffusion operator with constant coefficients.\",\n \"In particular, $\\\\mathcal {L}^\\\\sigma \\\\in {\\\\bf D}_K(G)$ for all $\\\\sigma \\\\in G$ .\",\n \"Let $\\\\beta (\\\\mathcal {L}^\\\\sigma ,\\\\lambda )$ denote the $\\\\phi _\\\\lambda $ -eigenvalue of $\\\\mathcal {L}^\\\\sigma $ , so that $\\\\mathcal {L}^\\\\sigma \\\\phi _\\\\lambda = \\\\beta (\\\\mathcal {L}^\\\\sigma ,\\\\lambda )\\\\phi _\\\\lambda \\\\hspace{20.0pt} \\\\forall \\\\sigma \\\\in G,\\\\;\\\\lambda \\\\in .$ Lemma 3.4 Let $\\\\mathcal {A}$ be a Gangolli operator given by (REF ), and, using the notation introduced above, define $q:G\\\\times \\\\rightarrow \\\\mathbb {C}$ by $q(\\\\sigma ,\\\\lambda ) = -\\\\beta (\\\\mathcal {L}^\\\\sigma ,\\\\lambda ) + \\\\int _G(1-\\\\phi _\\\\lambda (\\\\tau ))\\\\nu (\\\\sigma ,d\\\\tau ), \\\\hspace{20.0pt} \\\\forall \\\\sigma \\\\in G,\\\\lambda \\\\in .$ Then $q$ is a continuous negative definite symbol.\",\n \"That $q$ is continuous in its first argument is immediate from Definition REF .\",\n \"Fix $\\\\sigma \\\\in G$ and consider $q(\\\\sigma ,\\\\cdot )-c(\\\\sigma )$ .\",\n \"By Theorem REF , there is a convolution semigroup $(\\\\mu ^\\\\sigma _t,t\\\\ge 0)$ generated by $\\\\mathcal {A}+c(\\\\sigma )$ , and by Theorem REF , the corresponding Gangolli exponent is a continuous negative definite mapping on $$ , given by $\\\\psi ^\\\\sigma (\\\\lambda ) = q(\\\\sigma ,\\\\lambda )-c(\\\\sigma ) \\\\hspace{20.0pt} \\\\forall \\\\lambda \\\\in .$ Therefore $q(\\\\sigma ,\\\\cdot )$ is continuous, and negative definite since for fixed $\\\\sigma $ , $c(\\\\sigma )$ is a non-negative constant.\",\n \"Definition 3.5 The symbols described by Lemma REF will be referred to as Gangolli symbols, due to their connection with Gangolli's Lévy–Khinchine formula.\",\n \"Remarks 3.6 Gangolli exponents are precisely those Gangolli symbols constant in their first argument.\",\n \"The set of all Gangolli symbols forms a convex cone.\",\n \"Theorem 3.7 Let $\\\\mathcal {A}$ be a Gangolli operator of the form (REF ), and let $q$ be a Gangolli symbol given by (REF ).\",\n \"Then $\\\\mathcal {A}$ is a pseudodifferential operator with symbol $q$ .\",\n \"By Theorem REF and Lemma REF , $f\\\\mapsto -\\\\int _{}\\\\hat{f}(\\\\lambda )\\\\phi _\\\\lambda (\\\\sigma )q(\\\\sigma ,\\\\lambda )\\\\omega (d\\\\lambda )$ is a well-defined mapping from $C_c^\\\\infty (K|G|K)\\\\rightarrow C(G)$ .\",\n \"We show that it is equal to $\\\\mathcal {A}$ .\",\n \"Continue to denote the local part of $\\\\mathcal {A}$ by $\\\\mathcal {L}^\\\\sigma $ , as in (REF ), and let $\\\\mathcal {P}$ denote the non-local part, so that $\\\\mathcal {P}f(\\\\sigma ) = \\\\int _G\\\\left(f(\\\\sigma \\\\tau )-f(\\\\sigma )-\\\\sum _{i=1}^dx_i(\\\\tau )X_if(\\\\sigma )\\\\right)\\\\nu (\\\\sigma ,d\\\\tau )$ for all $f\\\\in C_c^\\\\infty (K|G|K)$ and $\\\\sigma \\\\in G$ .\",\n \"By design, $\\\\mathcal {A}f(\\\\sigma ) = \\\\mathcal {L}^\\\\sigma f(\\\\sigma ) + \\\\mathcal {P}f(\\\\sigma ), \\\\hspace{20.0pt} \\\\forall f\\\\in C_c^\\\\infty (K|G|K),\\\\;\\\\sigma \\\\in G.$ Fix $\\\\sigma \\\\in G$ , and view $\\\\mathcal {L}^\\\\sigma + c(\\\\sigma )$ as a $K$ -bi-invariant diffusion operator with constant coefficients.\",\n \"By (REF ) in Example REF (REF ), $\\\\left(\\\\mathcal {L}^\\\\sigma + c(\\\\sigma )\\\\right)f(\\\\tau ) = -\\\\int _{}\\\\hat{f}(\\\\lambda )\\\\phi _\\\\lambda (\\\\tau )\\\\beta (\\\\mathcal {L}^\\\\sigma +c(\\\\sigma ),\\\\lambda )\\\\omega (d\\\\lambda )$ for all $f\\\\in C_c^\\\\infty (K|G|K)$ and $\\\\tau \\\\in G$ , where $\\\\beta (\\\\mathcal {L}^\\\\sigma +c(\\\\sigma ),\\\\lambda )$ denotes the $\\\\phi _\\\\lambda $ -eigenvalue of $\\\\mathcal {L}^\\\\sigma +c(\\\\sigma )\\\\in {\\\\bf D}_K(G)$ .\",\n \"Clearly $\\\\beta (\\\\mathcal {L}^\\\\sigma +c(\\\\sigma ),\\\\lambda ) = \\\\beta (\\\\mathcal {L}^\\\\sigma ,\\\\lambda )+c(\\\\sigma ), \\\\hspace{20.0pt} \\\\forall \\\\sigma \\\\in G,\\\\;\\\\lambda \\\\in ,$ and hence, expanding the right-hand side of (REF ) and applying the spherical inversion formula (REF ), $\\\\mathcal {L}^\\\\sigma f(\\\\tau ) = -\\\\int _{}\\\\hat{f}(\\\\lambda )\\\\beta (\\\\mathcal {L}^\\\\sigma ,\\\\lambda )\\\\phi _\\\\lambda (\\\\tau )\\\\omega (d\\\\lambda )$ for all $f\\\\in C_c^\\\\infty (K|G|K)$ and $\\\\sigma ,\\\\tau \\\\in G$ .\",\n \"In particular, $\\\\mathcal {L}^\\\\sigma f(\\\\sigma ) = -\\\\int _{}\\\\hat{f}(\\\\lambda )\\\\beta (\\\\mathcal {L}^\\\\sigma ,\\\\lambda )\\\\phi _\\\\lambda (\\\\sigma )\\\\omega (d\\\\lambda ) \\\\hspace{20.0pt}\\\\forall f\\\\in C_c^\\\\infty (K|G|K),\\\\;\\\\sigma \\\\in G.$ Consider next the action of the non-local part $\\\\mathcal {P}$ .\",\n \"By Lemma 2.3 on page 39 of Liao [34], for each fixed $\\\\sigma \\\\in G$ , and for all $f\\\\in C_b^2(K|G|K)$ , the integrand on the right-hand side of (REF ) is absolutely integrable with respect to $\\\\nu (\\\\sigma ,\\\\cdot )$ .\",\n \"Therefore, (REF ) may be used to extend the domain of $\\\\mathcal {P}$ so as to include $C_b^2(K|G|K)$ .\",\n \"We do so now, and (without any loss of precision) denote the extension by $\\\\mathcal {P}$ .\",\n \"Let us proceed similarly to [8] Section 5, and define for each $\\\\sigma \\\\in G$ a linear functional $\\\\mathcal {P}^\\\\sigma :C_b^2(K|G|K)\\\\rightarrow \\\\mathbb {C}$ by $\\\\mathcal {P}^\\\\sigma f:= \\\\mathcal {P}\\\\left(L_\\\\sigma ^{-1}f\\\\right)(\\\\sigma ),$ for each $\\\\sigma \\\\in G$ and $f\\\\in C_b^2(K|G|K)$ .\",\n \"Then $\\\\mathcal {P}f(\\\\sigma ) = \\\\mathcal {P}^\\\\sigma (L_\\\\sigma f)$ , for all $\\\\sigma \\\\in G$ and $f\\\\in C_b^2(K|G|K)$ , and hence $\\\\mathcal {P}^\\\\sigma \\\\phi _\\\\lambda = \\\\int _G\\\\left(L_\\\\sigma ^{-1}\\\\phi _\\\\lambda (\\\\sigma \\\\tau ) - L_\\\\sigma ^{-1}\\\\phi _\\\\lambda (\\\\sigma ) - \\\\sum _{i=1}^dx_i(\\\\tau )X_iL_\\\\sigma ^{-1}\\\\phi _\\\\lambda (\\\\sigma )\\\\right)\\\\nu (\\\\sigma ,d\\\\tau ),$ for all $\\\\lambda \\\\in $ and $\\\\sigma \\\\in G$ .\",\n \"Moreover, by the above discussion, the integrand on the right-hand side of (REF ) is absolutely $\\\\nu (\\\\sigma ,\\\\cdot )$ -integrable, for all $\\\\lambda \\\\in $ and $\\\\sigma \\\\in G$ .\",\n \"Now, $L_\\\\sigma ^{-1}\\\\phi _\\\\lambda (\\\\sigma \\\\tau ) - L_\\\\sigma ^{-1}\\\\phi _\\\\lambda (\\\\sigma ) - \\\\sum _{i=1}^dx_i(\\\\tau )X_iL_\\\\sigma ^{-1}\\\\phi _\\\\lambda (\\\\sigma ) &= \\\\phi _\\\\lambda (\\\\tau ) - \\\\phi _\\\\lambda (e) - \\\\sum _{i=1}^dx_i(\\\\tau )X_i\\\\phi _\\\\lambda (e) \\\\\\\\&= \\\\phi _\\\\lambda (\\\\tau ) - 1,$ since $\\\\phi _\\\\lambda (e)=1$ , and, by Theorem 5.3 (b) on page 139 of Liao [34], $X\\\\phi _\\\\lambda (e)=0$ for all $X\\\\in \\\\mathfrak {p}$ .\",\n \"Thus, for all $\\\\lambda \\\\in $ and $\\\\sigma \\\\in G$ , $\\\\phi _\\\\lambda -1$ is absolutely $\\\\nu (\\\\sigma ,\\\\cdot )$ -integrable, and $\\\\mathcal {P}^\\\\sigma \\\\phi _\\\\lambda = \\\\int _G\\\\left(\\\\phi _\\\\lambda (\\\\tau ) - 1\\\\right)\\\\nu (\\\\sigma ,d\\\\tau ).$ A standard argument involving the functional equation $\\\\phi _\\\\lambda (\\\\sigma )\\\\phi _\\\\lambda (\\\\tau ) = \\\\int _K\\\\phi _\\\\lambda (\\\\sigma k\\\\tau )dk, \\\\hspace{20.0pt} \\\\forall \\\\lambda \\\\in ,\\\\sigma ,\\\\tau \\\\in G,$ for spherical functions (see [21] Proposition 2.2, pp.\",\n \"400) may now be applied in precisely the same way as in [8] (5.3)–(5.7), to infer that $\\\\mathcal {P}\\\\phi _\\\\lambda (\\\\sigma ) = \\\\int _G(\\\\phi _\\\\lambda (\\\\sigma \\\\tau ) - \\\\phi _\\\\lambda (\\\\sigma ))\\\\nu (\\\\sigma ,d\\\\tau ) = \\\\int _G(\\\\phi _\\\\lambda (\\\\tau ) - 1)\\\\phi _\\\\lambda (\\\\sigma )\\\\nu (\\\\sigma ,d\\\\tau )$ for all $\\\\sigma \\\\in G$ and $\\\\lambda \\\\in $ .\",\n \"Finally, let $f\\\\in C_c^\\\\infty (K|G|K)$ , and observe that by the spherical inversion formula $\\\\begin{aligned}\\\\mathcal {P}f(\\\\sigma ) &= \\\\int _G\\\\Bigg (\\\\int _{}\\\\phi _\\\\lambda (\\\\sigma \\\\tau )\\\\hat{f}(\\\\lambda )\\\\omega (d\\\\lambda ) - \\\\int _{}\\\\phi _\\\\lambda (\\\\sigma )\\\\hat{f}(\\\\lambda )\\\\omega (d\\\\lambda ) \\\\\\\\&\\\\hspace{100.0pt}-\\\\sum _{i=1}^dx_i(\\\\tau )X_i\\\\left[\\\\int _{}\\\\phi _\\\\lambda \\\\hat{f}(\\\\lambda )\\\\omega (d\\\\lambda )\\\\right](\\\\sigma )\\\\Bigg )\\\\nu (\\\\sigma ,d\\\\tau )\\\\end{aligned}$ Claim For all $X\\\\in \\\\mathfrak {p}$ and $f\\\\in C_c^\\\\infty (K|G|K)$ , $X\\\\left[\\\\int _{}\\\\phi _\\\\lambda \\\\hat{f}(\\\\lambda )\\\\omega (d\\\\lambda )\\\\right](\\\\sigma )=\\\\int _{}X\\\\phi _\\\\lambda (\\\\sigma )\\\\hat{f}(\\\\lambda )\\\\omega (d\\\\lambda ).$ Proof of Claim This is a fairly standard differentiation-through-integration-sign argument.\",\n \"First note that by translation invariance of $X$ , it suffices to prove the claim for $\\\\sigma =e$ .\",\n \"Now, $X\\\\left[\\\\int _{}\\\\phi _\\\\lambda \\\\hat{f}(\\\\lambda )\\\\omega (d\\\\lambda )\\\\right](e) &= \\\\left.\\\\frac{d}{dt}\\\\int _{}\\\\phi _\\\\lambda (\\\\exp tX)\\\\hat{f}(\\\\lambda )\\\\omega (d\\\\lambda )\\\\right|_{t=0} \\\\\\\\&= \\\\lim _{t\\\\rightarrow 0}\\\\int _{}\\\\frac{\\\\phi _\\\\lambda (\\\\exp tX)-1}{t}\\\\hat{f}(\\\\lambda )\\\\omega (d\\\\lambda ).$ By the mean value theorem, for each $t>0$ and $\\\\lambda \\\\in $ , $\\\\frac{\\\\phi _\\\\lambda (\\\\exp tX)-1}{t}=X\\\\phi _\\\\lambda (\\\\exp t^{\\\\prime }X),$ for some $00$ .\",\n \"By Helgason [20] Theorem 1.1 (iii), $\\\\Vert X\\\\phi _\\\\lambda \\\\Vert _\\\\infty \\\\le C(1+|\\\\lambda |)$ , for some some constant $C>0$ .\",\n \"Thus, for $f\\\\in C_c^\\\\infty (K|G|K)$ , $\\\\lambda \\\\in $ and $t>0$ , $\\\\left|\\\\frac{\\\\phi _\\\\lambda (\\\\exp tX)-1}{t}\\\\hat{f}(\\\\lambda )\\\\right| \\\\le C(1+|\\\\lambda |)|\\\\hat{f}(\\\\lambda )|,$ and clearly $C(1+|\\\\cdot |)\\\\hat{f} \\\\in L^1()^W$ , since $\\\\hat{f}\\\\in \\\\mathcal {S}()$ .\",\n \"Hence by dominated convergence, $X\\\\left[\\\\int _{}\\\\phi _\\\\lambda \\\\hat{f}(\\\\lambda )\\\\omega (d\\\\lambda )\\\\right](e) = \\\\int _{}\\\\lim _{t\\\\rightarrow 0}\\\\frac{\\\\phi _\\\\lambda (\\\\exp tX)-1}{t}\\\\hat{f}(\\\\lambda )\\\\omega (d\\\\lambda ) = \\\\int _{}X\\\\phi _\\\\lambda (e)\\\\hat{f}(\\\\lambda )\\\\omega (d\\\\lambda ),$ which completes the proof of the claim.\",\n \"Applying the claim to (REF ), for $f\\\\in C_c^\\\\infty (K|G|K)$ and $\\\\sigma \\\\in G$ , $\\\\mathcal {P}f(\\\\sigma ) &= \\\\int _G\\\\Bigg (\\\\int _{}\\\\phi _\\\\lambda (\\\\sigma \\\\tau )\\\\hat{f}(\\\\lambda )\\\\omega (d\\\\lambda ) - \\\\int _{}\\\\phi _\\\\lambda (\\\\sigma )\\\\hat{f}(\\\\lambda )\\\\omega (d\\\\lambda ) \\\\\\\\&\\\\hspace{150.0pt}-\\\\sum _{i=1}^dx_i(\\\\tau )\\\\int _{}X_i\\\\phi _\\\\lambda (\\\\sigma )\\\\hat{f}(\\\\lambda )\\\\omega (d\\\\lambda )\\\\Bigg )\\\\nu (\\\\sigma ,d\\\\tau ) \\\\\\\\&=\\\\int _G\\\\int _{}\\\\hat{f}(\\\\lambda )\\\\left(\\\\phi _\\\\lambda (\\\\sigma \\\\tau )-\\\\phi _\\\\lambda (\\\\sigma )-\\\\sum _{i=1}^dx_i(\\\\tau )X_i\\\\phi _\\\\lambda (\\\\sigma )\\\\right)\\\\omega (d\\\\lambda )\\\\nu (\\\\sigma ,d\\\\tau ).$ By the Fubini theorem, $\\\\mathcal {P}f(\\\\sigma ) &= \\\\int _{}\\\\hat{f}(\\\\lambda )\\\\int _G\\\\left(\\\\phi _\\\\lambda (\\\\sigma \\\\tau )-\\\\phi _\\\\lambda (\\\\sigma )-\\\\sum _{i=1}^dx_i(\\\\tau )X_i\\\\phi _\\\\lambda (\\\\sigma )\\\\right)\\\\nu (\\\\sigma ,d\\\\tau )\\\\omega (d\\\\lambda ) \\\\\\\\&= \\\\int _{}\\\\hat{f}(\\\\lambda )\\\\mathcal {P}\\\\phi _\\\\lambda (\\\\sigma )\\\\omega (d\\\\lambda )$ for all $f\\\\in C_c^\\\\infty (K|G|K)$ and $\\\\sigma \\\\in G$ .\",\n \"It follows by (REF ) that $\\\\mathcal {P}f(\\\\sigma ) = \\\\int _{}\\\\hat{f}(\\\\lambda )\\\\phi _\\\\lambda (\\\\sigma )\\\\int _G(\\\\phi _\\\\lambda (\\\\tau ) - 1)\\\\nu (\\\\sigma ,d\\\\tau )\\\\omega (d\\\\lambda )$ for all $f\\\\in C_c^\\\\infty (K|G|K)$ and $\\\\sigma \\\\in G$ .\",\n \"The result now follows by substituting (REF ) and (REF ) into (REF ).\",\n \"Example 3.8 As already noted, all Gangolli exponents are Gangolli symbols.\",\n \"Let $u\\\\in C(K|G|K)$ be non-negative, and let $v:\\\\rightarrow \\\\mathbb {C}$ be a Gangolli exponent.\",\n \"Then $q:G\\\\times \\\\rightarrow \\\\mathbb {C}$ given by $q(\\\\sigma ,\\\\lambda ) = u(\\\\sigma )v(\\\\lambda ) \\\\hspace{20.0pt} \\\\forall \\\\sigma \\\\in G,\\\\;\\\\lambda \\\\in $ is a Gangolli symbol.\",\n \"Indeed, by Theorem REF , there exists a sub-diffusion operator $T\\\\in {\\\\bf D}_K(G)$ and a $K$ -bi-invariant Lévy measure $\\\\nu $ such that, $v(\\\\lambda ) = -\\\\beta (T,\\\\lambda ) + \\\\int _G(1-\\\\phi _\\\\lambda (\\\\sigma ))\\\\nu (d\\\\tau )$ for all $\\\\lambda \\\\in $ .\",\n \"But then, for each $\\\\sigma \\\\in G$ and $\\\\lambda \\\\in $ $q(\\\\sigma ,\\\\lambda ) = -\\\\beta (u(\\\\sigma )T,\\\\lambda ) + \\\\int _G(1-\\\\phi _\\\\lambda (\\\\sigma ))\\\\nu (d\\\\tau ).$ Writing $\\\\mathcal {L}^\\\\sigma =u(\\\\sigma )T$ and $\\\\nu (\\\\sigma ,d\\\\tau ) = u(\\\\sigma )\\\\nu (d\\\\tau )$ for each $\\\\sigma \\\\in G$ , $q$ resembles (REF ).\",\n \"Since $T$ is a sub-diffusion operator, there is a non-negative constant $c$ , a vector $b\\\\in \\\\mathbb {R}^d$ and a non-negative definite symmetric matrix $(a_{jk})$ such that $T = -c + \\\\sum _{i=1}^db_iX_i+\\\\sum _{j,k=1}^da_{jk}X_jX_k.$ For each $\\\\sigma \\\\in G$ , and for $i,j,k=1,\\\\ldots ,d$ , let $c(\\\\sigma ) = cu(\\\\sigma ), \\\\hspace{20.0pt} b_i(\\\\sigma ) = b_iu(\\\\sigma ), \\\\hspace{20.0pt} a_{jk}(\\\\sigma ) = a_{jk}u(\\\\sigma )\\\\hspace{10.0pt} \\\\text{and} \\\\hspace{10.0pt} \\\\nu (\\\\sigma ,\\\\cdot )=\\\\nu .$ Since $u$ is non-negative, continuous and $K$ -bi-invariant, the conditions of Definition REF are easily verified for these characteristics.\"\n ],\n [\n \"Construction of Sub-Feller Semigroups\",\n \"In the previous section, we used results from Applebaum and Ngan [7] and [8] to introduce a class of pseudodifferential operators that satisfy all but the last condition of the Hille–Yosida–Ray theorem, Theorem REF (REF ), in the case where $E=K|G|K$ .\",\n \"In this section we tackle this the third condition, and seek conditions on a symbol $q$ so that, for some $\\\\alpha >0$ , $\\\\overline{\\\\operatorname{Ran}(\\\\alpha +q(\\\\sigma ,D))}=C_0(K|G|K).$ We use an approach based primarily on Jacob [27] and Hoh [24] Section 4.\",\n \"Now that we are on the level of operators, there are more arguments that closely resemble these sources.\",\n \"In these cases, proofs are not expanded in great detail, and may be omitted entirely to save space.\",\n \"Instead, we aim to emphasise what does not carry over from the Euclidean space setting.\",\n \"As in previous work, let $\\\\psi :\\\\rightarrow \\\\mathbb {R}$ be a fixed real-valued, continuous negative definite function satisfying (REF ) for some fixed $r>0$ .\",\n \"The next lemma will be needed later on for proving regularity properties of pseudodifferential operators — see [24] Lemma 4.2 on page 48 for comparison.\",\n \"Before stating the result, we introduce some notation.\",\n \"For a mapping $q:G\\\\times \\\\rightarrow \\\\mathbb {R}$ and for each $\\\\lambda ,\\\\eta \\\\in $ , $\\\\sigma \\\\in G$ , define $F_{\\\\lambda ,\\\\eta }(\\\\sigma ) = \\\\phi _{-\\\\lambda }(\\\\sigma )q(\\\\sigma ,\\\\eta ).$ Observe that if $q(\\\\cdot ,\\\\eta )\\\\in L^2(K|G|K)$ for all $\\\\eta \\\\in $ , then $F_{\\\\lambda ,\\\\eta }\\\\in L^2(K|G|K)$ , and we may consider the spherical transform $\\\\hat{F}_{\\\\lambda ,\\\\eta }\\\\in L^2(,\\\\omega )$ , given by $\\\\hat{F}_{\\\\lambda ,\\\\eta }(\\\\mu ) = \\\\int _G\\\\phi _{-\\\\mu }(\\\\sigma )\\\\phi _{-\\\\lambda }(\\\\sigma )q(\\\\sigma ,\\\\eta )d\\\\sigma , \\\\hspace{20.0pt} \\\\forall \\\\mu \\\\in .$ To give some intuition to the introduction of $F_{\\\\lambda ,\\\\eta }$ , consider the case $G=\\\\mathbb {R}^d$ , $K=\\\\lbrace 0\\\\rbrace $ .\",\n \"In this case, the so-called frequency shift property for the Fourier transform says that $\\\\begin{aligned}\\\\hat{F}_{\\\\lambda ,\\\\eta }(\\\\mu ) &= \\\\frac{1}{(2\\\\pi )^{d/2}}\\\\int _{\\\\mathbb {R}^d}e^{-i\\\\mu \\\\cdot x}e^{-\\\\lambda \\\\cdot x}q(x,\\\\eta )dx \\\\\\\\&= \\\\frac{1}{(2\\\\pi )^{d/2}}\\\\int _{\\\\mathbb {R}^d}e^{-i(\\\\mu +\\\\lambda )\\\\cdot x}q(x,\\\\eta )dx = \\\\hat{q}(\\\\lambda +\\\\mu ,\\\\eta ),\\\\end{aligned}$ where $^{\\\\wedge }$ denotes the Fourier transform taken in the first argument of $q$ .\",\n \"Hoh [24] and Jacob [29] make use of bounds on $\\\\hat{q}(\\\\lambda -\\\\mu ,\\\\eta )$ , and $\\\\hat{F}_{\\\\lambda ,\\\\eta }(-\\\\mu )$ will assume an analogous role in work to come.\",\n \"The next lemma is an analogue of Lemma 2.1 of Jacob [27].\",\n \"See also Hoh [24] Lemma 4.2, pp. 48.\",\n \"It's proof is similar to these other lemmas, but some additional steps are required to work around some new difficulties that arise in this setting.\",\n \"Lemma 4.1 Let $M\\\\in \\\\mathbb {N}$ , $q:G\\\\times \\\\rightarrow \\\\mathbb {R}$ and suppose $q(\\\\cdot ,\\\\lambda )\\\\in C^M_c(K|G|K)$ for all $\\\\lambda \\\\in $ .\",\n \"Suppose further that for each $\\\\beta \\\\in \\\\lbrace 0,1,\\\\ldots ,M\\\\rbrace $ , there is a non-negative function $\\\\Phi _\\\\beta \\\\in L^1(K|G|K)$ such that, in the notation above, $\\\\left|(-\\\\Delta )^{\\\\beta /2}F_{\\\\lambda ,\\\\eta }(\\\\sigma )\\\\right| \\\\le \\\\Phi _\\\\beta (\\\\sigma )\\\\langle \\\\lambda \\\\rangle ^M(1+\\\\psi (\\\\eta )),$ for all $\\\\lambda ,\\\\eta \\\\in $ , $\\\\sigma \\\\in G$ .\",\n \"Then there is a constant $C_M>0$ such that $\\\\left|\\\\hat{F}_{\\\\lambda ,\\\\eta }(\\\\mu )\\\\right| \\\\le C_M\\\\sum _{\\\\beta =0}^M\\\\Vert \\\\Phi _\\\\beta \\\\Vert _1\\\\langle \\\\lambda +\\\\mu \\\\rangle ^{-M}(1+\\\\psi (\\\\eta )),$ for all $\\\\lambda ,\\\\mu ,\\\\eta \\\\in $ , where $\\\\Vert \\\\cdot \\\\Vert _1$ denotes the usual norm on the Banach space $L^1(K|G|K)$ (taken with respect to Haar measure).\",\n \"Remarks 4.2 Here, we are again using the notation $\\\\langle \\\\lambda \\\\rangle =\\\\sqrt{1+|\\\\lambda |^2}$ , introduced in (REF ).\",\n \"The condition (REF ) may seem quite obscure.\",\n \"The role of $\\\\langle \\\\lambda +\\\\mu \\\\rangle $ will hopefully become apparent in the proof of Theorem REF .\",\n \"In Section , we will develop a class of examples where it is satisfied.\",\n \"Under the conditions of the lemma, and using the Fubini theorem, we have the following: for all $u\\\\in C_c^\\\\infty (K|G|K)$ and $\\\\lambda \\\\in $ , $\\\\begin{aligned}(q(\\\\sigma ,D)u)^{\\\\wedge }(\\\\lambda ) &= \\\\int _C\\\\int _{}\\\\phi _{-\\\\lambda }(\\\\sigma )\\\\phi _\\\\eta (\\\\sigma )q(\\\\sigma ,\\\\eta )\\\\hat{u}(\\\\eta )\\\\omega (d\\\\eta )d\\\\sigma \\\\\\\\&= \\\\int _{}\\\\left(\\\\int _C\\\\phi _\\\\eta (\\\\sigma )F_{\\\\lambda ,\\\\eta }(\\\\sigma )d\\\\sigma \\\\right)\\\\hat{u}(\\\\eta )\\\\omega (d\\\\eta ) \\\\\\\\&= \\\\int _{}\\\\hat{F}_{\\\\lambda ,\\\\eta }(-\\\\eta )\\\\hat{u}(\\\\eta )\\\\omega (d\\\\eta ).\\\\end{aligned}$ Fubini's theorem does indeed apply here — a suitable bound for the integrand on the first line of (REF ) may be found by noting that for all $\\\\lambda ,\\\\eta \\\\in $ and $\\\\sigma \\\\in G$ , $|\\\\phi _{-\\\\lambda }(\\\\sigma )\\\\phi _\\\\eta (\\\\sigma )q(\\\\sigma ,\\\\eta )\\\\hat{u}(\\\\eta )|\\\\le |q(\\\\sigma ,\\\\eta )||\\\\hat{u}(\\\\eta )|\\\\le \\\\Phi _0(\\\\sigma )\\\\big (1+\\\\psi (\\\\eta )\\\\big )|\\\\hat{u}(\\\\eta )|,$ for some $\\\\Phi _0\\\\in L^1(K|G|K)$ , by (REF ).\",\n \"By Theorem REF , $\\\\hat{u}\\\\in \\\\mathcal {S}()$ , and the usual bound (REF ) on the density of Plancherel measure may be applied, similarly to (REF ), to conclude that the right-hand side of (REF ) is $\\\\omega (d\\\\eta )\\\\times d\\\\sigma $ -integrable.\",\n \"[Proof of Lemma REF ] Let $\\\\beta \\\\in \\\\lbrace 0,1,\\\\ldots ,M\\\\rbrace $ and $\\\\lambda ,\\\\eta \\\\in $ be fixed.\",\n \"The fractional Laplacian $(-\\\\Delta )^{\\\\beta /2}$ satisfies a well-known eigenrelation $(-\\\\Delta )^{\\\\beta /2}\\\\phi _\\\\mu = \\\\left(|\\\\rho |^2+|\\\\mu |^2\\\\right)^{\\\\beta /2}\\\\phi _\\\\mu , \\\\hspace{20.0pt} \\\\forall \\\\mu \\\\in ,$ which may be proven using subordination methods and properties of the Laplace-Beltrami operator on a symmetric space, using similar techniques to Section 5.7 of [5], pp. 154–7.\",\n \"One can also show using standard methods that $\\\\left((-\\\\Delta )^{\\\\beta /2}f\\\\right)^{\\\\wedge }(\\\\mu ) = \\\\left(|\\\\rho |^2+|\\\\mu |^2\\\\right)^{\\\\beta /2}\\\\hat{f}(\\\\mu ),$ for all $f\\\\in C^M_c(K|G|K)$ and $\\\\mu \\\\in $ .\",\n \"Then, using the definition of the spherical transform, $(|\\\\rho |^2+|\\\\mu |^2)^{\\\\beta /2}\\\\hat{f}(\\\\mu ) = \\\\int _G\\\\phi _{-\\\\mu }(\\\\sigma )(-\\\\Delta )^{\\\\beta /2}f(\\\\sigma )d\\\\sigma , $ for all $f\\\\in C_c^M(K|G|K)$ and all $\\\\mu \\\\in $ .\",\n \"Applying this to $f=F_{\\\\lambda ,\\\\eta }$ , we have for all $\\\\mu \\\\in \\\\operatorname{\\\\mathfrak {a}}^\\\\ast $ , $\\\\left|\\\\left(|\\\\rho |^2+|\\\\mu |^2\\\\right)^{\\\\beta /2}\\\\hat{F}_{\\\\lambda ,\\\\eta }(\\\\mu )\\\\right| &\\\\le \\\\int _G|\\\\phi _{-\\\\mu }(\\\\sigma )|\\\\left|(-\\\\Delta )^{\\\\beta /2}F_{\\\\lambda ,\\\\eta }(\\\\sigma )\\\\right|d\\\\sigma \\\\\\\\&\\\\le \\\\int _G\\\\Phi _\\\\beta (\\\\sigma )\\\\langle \\\\lambda \\\\rangle ^M(1+\\\\psi (\\\\eta ))d\\\\sigma = \\\\Vert \\\\Phi _\\\\beta \\\\Vert _1\\\\langle \\\\lambda \\\\rangle ^M(1+\\\\psi (\\\\eta )),$ and summing over $\\\\beta $ , $\\\\sum _{\\\\beta =0}^M\\\\left(|\\\\rho |^2+|\\\\mu |^2\\\\right)^{\\\\beta /2}\\\\left|\\\\hat{F}_{\\\\lambda ,\\\\eta }(\\\\mu )\\\\right| \\\\le \\\\sum _{\\\\beta =0}^M\\\\Vert \\\\Phi _\\\\beta \\\\Vert _1\\\\langle \\\\lambda \\\\rangle ^M(1+\\\\psi (\\\\eta )),$ for all $\\\\lambda ,\\\\mu ,\\\\eta \\\\in $ .\",\n \"Let $C^{\\\\prime }_M>0$ be the smallest positive number such that $\\\\langle \\\\mu \\\\rangle ^M \\\\le C^{\\\\prime }_M\\\\sum _{\\\\beta =0}^M\\\\left(|\\\\rho |^2+|\\\\mu |^2\\\\right)^{\\\\beta /2} \\\\hspace{20.0pt} \\\\forall \\\\mu \\\\in .$ Then, rearranging (REF ), $\\\\left|\\\\hat{F}_{\\\\lambda ,\\\\eta }(\\\\mu )\\\\right| \\\\le C^{\\\\prime }_M\\\\sum _{\\\\beta =0}^M\\\\Vert \\\\Phi _\\\\beta \\\\Vert _1\\\\langle \\\\mu \\\\rangle ^{-M}\\\\langle \\\\lambda \\\\rangle ^M(1+\\\\psi (\\\\eta )),$ for all $\\\\lambda ,\\\\mu ,\\\\eta \\\\in $ .\",\n \"Finally, observe that by Peetre's inequality (see Proposition REF (REF )), $\\\\langle \\\\lambda \\\\rangle ^M\\\\langle \\\\lambda +\\\\mu \\\\rangle ^{-M} = \\\\left(\\\\frac{1+|\\\\lambda |^2}{1+|\\\\lambda +\\\\mu |^2}\\\\right)^{M/2} \\\\le 2^{M/2}(1+|\\\\mu |^2)^{M/2} = 2^{M/2}\\\\langle \\\\mu \\\\rangle ^M$ for all $\\\\lambda ,\\\\mu \\\\in $ .\",\n \"Therefore, for all $\\\\lambda ,\\\\mu \\\\in $ , $\\\\langle \\\\mu \\\\rangle ^{-M}\\\\langle \\\\lambda \\\\rangle ^M\\\\le 2^{M/2}\\\\langle \\\\lambda +\\\\mu \\\\rangle ^{-M}$ and by (REF ), $\\\\left|\\\\hat{F}_{\\\\lambda ,\\\\eta }(\\\\mu )\\\\right| \\\\le 2^{M/2}C^{\\\\prime }_M\\\\sum _{\\\\beta =0}^M\\\\Vert \\\\Phi _\\\\beta \\\\Vert _1\\\\langle \\\\lambda +\\\\mu \\\\rangle ^{-M}(1+\\\\psi (\\\\eta ))$ The result now follows by taking $C_M=2^{M/2}C^{\\\\prime }_M$ .\",\n \"Remark 4.3 The constant $C_M:=2^{M/2}\\\\sup _{\\\\lambda \\\\in }\\\\frac{\\\\langle \\\\lambda \\\\rangle ^M}{\\\\sum _{\\\\beta =0}^M\\\\big (|\\\\rho |^2+|\\\\lambda |^2\\\\big )^{\\\\beta /2}}$ appearing in the proof of Lemma REF will remain relevant throughout this chapter.\",\n \"Let now $q:G\\\\times \\\\rightarrow \\\\mathbb {R}$ be a continuous negative definite symbol, $K$ -bi-invariant in its first argument, and $W$ -invariant in its second (for example, $q$ could be taken to be a Gangolli symbol, as in (REF )).\",\n \"Similarly to Jacob [27] §4 and Hoh [24] (4.26), we write $q(\\\\sigma ,\\\\lambda ) = q_1(\\\\lambda )+q_2(\\\\sigma ,\\\\lambda ), \\\\hspace{20.0pt} \\\\forall \\\\sigma \\\\in G,\\\\lambda \\\\in ,$ where $q_1(\\\\lambda )=q(\\\\sigma _0,\\\\lambda )$ and $q_2(\\\\sigma ,\\\\lambda )=q(\\\\sigma ,\\\\lambda )-q(\\\\sigma _0,\\\\lambda )$ , for some fixed $\\\\sigma _0\\\\in G$ .\",\n \"Observe that $q_1$ is necessarily a negative definite symbol.\",\n \"Though $q_2$ may not be, we may still define the operator $q_2(\\\\sigma ,D)$ in a meaningful way, by $q_2(\\\\sigma ,D) := q(\\\\sigma ,D) - q_1(D) = \\\\int _{}\\\\phi _\\\\lambda (\\\\sigma )q_2(\\\\sigma ,\\\\lambda )\\\\hat{f}(\\\\lambda )\\\\omega (d\\\\lambda ), \\\\hspace{20.0pt} \\\\forall \\\\sigma \\\\in G.$ By decomposing $q$ in this way, we view it as a perturbation of a negative definite function $q_1$ by $q_2$ .\",\n \"The assumptions we place on $q$ will control the size of this perturbation, as well ensuring certain regularity properties of $q(\\\\sigma ,D)$ acting on the anisotropic Sobolev spaces introduced in Section REF .\",\n \"Assumptions 4.4 In the notation above, we impose the following: There exist constants $c_0,c_1>0$ such that for all $\\\\lambda \\\\in $ with $|\\\\lambda |\\\\ge 1$ , $c_0(1+\\\\psi (\\\\lambda ))\\\\le q_1(\\\\lambda ) \\\\le c_1(1+\\\\psi (\\\\lambda )).$ Let $M\\\\in \\\\mathbb {N}$ , $M>\\\\dim (G/K)$ , and suppose that $q_2(\\\\cdot ,\\\\lambda )\\\\in C^M_c(K|G|K)$ for all $\\\\lambda \\\\in $ .\",\n \"Suppose further that for $\\\\beta =0,1,\\\\ldots ,M$ , there exists $\\\\Phi _\\\\beta \\\\in L^1(K|G|K)$ such that $\\\\left|(-\\\\Delta )^{\\\\beta /2}F_{\\\\lambda ,\\\\eta }(\\\\sigma )\\\\right| \\\\le \\\\Phi _\\\\beta (\\\\sigma )\\\\langle \\\\lambda \\\\rangle ^M\\\\big (1+\\\\psi (\\\\eta )\\\\big ),$ for all $\\\\lambda ,\\\\eta \\\\in $ , $\\\\sigma \\\\in G$ , where $F_{\\\\lambda ,\\\\eta }(\\\\sigma )=\\\\phi _{-\\\\lambda }(\\\\sigma )q_2(\\\\sigma ,\\\\eta )$ (c.f.\",\n \"(REF )).\",\n \"Remarks 4.5 These assumptions are analogues to P.1, P.2.q of Jacob [27], pp.\",\n \"156, or (A.1), (A.2.M) of Hoh [24], pp.54.\",\n \"As noted in Remark REF (REF ), the conditions in Assumption REF (REF ) imply that $(q_2(\\\\sigma ,D)u)^{\\\\wedge }(\\\\lambda ) = \\\\int _{}\\\\hat{F}_{\\\\lambda ,\\\\eta }(-\\\\eta )\\\\hat{u}(\\\\eta )\\\\omega (d\\\\eta ),$ for all $\\\\lambda \\\\in $ and $u\\\\in C_c^\\\\infty (K|G|K)$ , a fact that will be useful several times more.\",\n \"Theorem 4.6 Subject to Assumptions REF , for all $s\\\\in \\\\mathbb {R}$ , $q_1(D)$ extends to a continuous operator from $H^{\\\\psi ,s+2}$ to $H^{\\\\psi ,s}$ , and $q(\\\\sigma ,D)$ extends to a continuous operator from $H^{\\\\psi ,2}$ to $L^2(K|G|K)$ .\",\n \"The proof of the first part is omitted, since it is an easy adaptation of the proof of Theorem 4.8 on page 55 of Hoh [24] — first proved as Corollary 3.1 in Jacob [27].\",\n \"The second part is also proved similarly to Theorem 4.8 of [24], the main difference being that $\\\\hat{F}_{\\\\lambda ,\\\\eta }(-\\\\eta )$ takes the place of the transformed symbol, as discussed previously (see (REF )).\",\n \"By (REF ) and the Plancherel theorem, $|\\\\langle q_2(\\\\sigma ,D)u,v\\\\rangle | &= \\\\left|\\\\int _{}(q_2(\\\\sigma ,D)u)^{\\\\wedge }(\\\\lambda )\\\\overline{\\\\hat{v}(\\\\lambda )}\\\\omega (d\\\\lambda )\\\\right| \\\\\\\\&= \\\\left|\\\\int _{}\\\\int _{}\\\\hat{F}_{\\\\lambda ,\\\\eta }(-\\\\eta )\\\\hat{u}(\\\\eta )\\\\overline{\\\\hat{v}(\\\\lambda )}\\\\omega (d\\\\eta )\\\\omega (d\\\\lambda )\\\\right| \\\\\\\\&\\\\le \\\\int _{}\\\\int _{}\\\\left|\\\\hat{F}_{\\\\lambda ,\\\\eta }(-\\\\eta )\\\\right||\\\\hat{u}(\\\\eta )||\\\\hat{v}(\\\\lambda )|\\\\omega (d\\\\eta )\\\\omega (d\\\\lambda ).$ Then, using (REF ), Lemma REF and Young's convolution inequalitySee Simon [43] Theorem 6.6.3, page 550.\",\n \"Here, we are again identifying $$ with a Euclidean space., $|\\\\langle q_2(\\\\sigma ,D)u,v\\\\rangle | &\\\\le C_M\\\\sum _{\\\\beta =0}^M\\\\Vert \\\\Phi _\\\\beta \\\\Vert _1\\\\int _{}\\\\int _{}\\\\langle \\\\lambda -\\\\eta \\\\rangle ^{-M}\\\\Psi (\\\\eta )^2|\\\\hat{u}(\\\\eta )||\\\\hat{v}(\\\\lambda )|\\\\omega (d\\\\eta )\\\\omega (d\\\\lambda ) \\\\\\\\&= C_M\\\\sum _{\\\\beta =0}^M\\\\Vert \\\\Phi _\\\\beta \\\\Vert _1\\\\int _{}\\\\left[\\\\langle \\\\cdot \\\\rangle ^{-M}\\\\ast \\\\big (\\\\Psi ^2|\\\\hat{u}|\\\\big )\\\\right](\\\\lambda )|\\\\hat{v}(\\\\lambda )|\\\\omega (d\\\\lambda ) \\\\\\\\&\\\\le C_M\\\\sum _{\\\\beta =0}^M\\\\Vert \\\\Phi _\\\\beta \\\\Vert _1\\\\left\\\\Vert \\\\langle \\\\cdot \\\\rangle ^{-M}\\\\ast \\\\big (\\\\Psi ^2|\\\\hat{u}|\\\\big )\\\\right\\\\Vert _{L^2(,\\\\omega )}\\\\Vert \\\\hat{v}\\\\Vert _{L^2(,\\\\omega )} \\\\\\\\&\\\\le C_M\\\\sum _{\\\\beta =0}^M\\\\Vert \\\\Phi _\\\\beta \\\\Vert _1\\\\left\\\\Vert \\\\langle \\\\cdot \\\\rangle ^{-M}\\\\right\\\\Vert _{L^1(,\\\\omega )}\\\\Vert u\\\\Vert _{\\\\psi ,2}\\\\Vert v\\\\Vert ,$ for all $u,v\\\\in C_c^\\\\infty (K|G|K)$ .\",\n \"Hence, for all $u\\\\in C_c^\\\\infty (K|G|K)$ , $\\\\Vert q_2(\\\\sigma ,D)u\\\\Vert &= \\\\sup _{\\\\begin{array}{c}v\\\\in C_c^\\\\infty (K|G|K) \\\\\\\\ \\\\Vert v\\\\Vert =1\\\\end{array}}|\\\\langle q_2(\\\\sigma ,D)u,v\\\\rangle | \\\\le C_M\\\\sum _{\\\\beta =0}^M\\\\Vert \\\\Phi _\\\\beta \\\\Vert _1\\\\left\\\\Vert \\\\langle \\\\cdot \\\\rangle ^{-M}\\\\right\\\\Vert _{L^1(,\\\\omega )}\\\\Vert u\\\\Vert _{\\\\psi ,2},$ and $q_2(\\\\sigma ,D)$ extends to a bounded linear operator $H^{\\\\psi ,2}\\\\rightarrow L^2(K|G|K)$ .\",\n \"Under an additional assumption, we are able to obtain a more powerful result.\",\n \"Theorem 4.7 Suppose Assumptions REF hold, and suppose further that $s\\\\in \\\\mathbb {R}$ satisfies $|s-1|+1+\\\\dim (G/K) < M$ .\",\n \"Then $q(\\\\sigma ,D)$ extends to a continuous linear operator from $H^{\\\\psi ,s+2}\\\\rightarrow H^{\\\\psi ,s}$ .\",\n \"We first need a technical lemma.\",\n \"Lemma 4.8 Let $s\\\\in \\\\mathbb {R}$ and $M\\\\in \\\\mathbb {N}$ be such that $|s-1|+1+\\\\dim (G/K)0$ .\",\n \"To do this, we seek solutions $u$ to the equation $(q(\\\\sigma ,D)+\\\\alpha )u=f,$ for a given function $f$ and $\\\\alpha >0$ .\",\n \"For $\\\\alpha >0$ , consider the bilinear form $B_\\\\alpha $ defined for all $u,v\\\\in C_c^\\\\infty (K|G|K)$ by $B_\\\\alpha (u,v) = \\\\langle (q(\\\\sigma ,D)+\\\\alpha )u,v\\\\rangle .$ The next result is an analogue of Jacob [27] Lemma 3.2, pp.\",\n \"160, and Hoh [24] Theorem 4.9, pp. 56.\",\n \"Theorem 4.9 Subject to Assumptions REF , $B_\\\\alpha $ extends continuously to $H^{\\\\psi ,1}\\\\times H^{\\\\psi ,1}$ .\",\n \"This proof is very similar to those of Jacob [27] Lemma 3.2, pp.\",\n \"160, and Hoh [24] Theorem 4.9, pp.\",\n \"56, and so we give only a sketch.\",\n \"Let $u,v\\\\in H^{\\\\psi ,1}$ .\",\n \"Using Assumption REF (REF ) and the fact that $q_1$ is continuous, there is $\\\\kappa _1>0$ such that $|q_1|\\\\le \\\\kappa _1\\\\Psi ^2$ .\",\n \"Plancherel's identity may then be used to show that $\\\\left|\\\\langle q_1(D)u,v\\\\rangle \\\\right| \\\\le \\\\int _{}|q_1(\\\\lambda )||\\\\hat{u}(\\\\lambda )||\\\\hat{v}(\\\\lambda )|\\\\omega (d\\\\lambda ) \\\\le \\\\kappa _1\\\\Vert u\\\\Vert _{\\\\psi ,1}\\\\Vert v\\\\Vert _{\\\\psi ,1}.$ Furthermore, methods similar to the proof of Theorem REF are used to show that $\\\\left|\\\\langle q_2(\\\\sigma ,D)u,v\\\\rangle \\\\right| \\\\le \\\\kappa _2\\\\left\\\\Vert \\\\langle \\\\cdot \\\\rangle ^{-M+1}\\\\right\\\\Vert _{L^1(,\\\\omega )}\\\\Vert u\\\\Vert _{\\\\psi ,1}\\\\Vert v\\\\Vert _{\\\\psi ,1},$ where $\\\\kappa _2=C_M\\\\sqrt{2(1+c_\\\\psi )}\\\\sum _{\\\\beta =0}^M\\\\Vert \\\\Phi _\\\\beta \\\\Vert _{L^1(,\\\\omega )}.$ By Theorem REF (REF ), there is $\\\\kappa _3>0$ such that $\\\\Vert u\\\\Vert \\\\le \\\\kappa _3\\\\Vert u\\\\Vert _{\\\\psi ,1}$ , and thus for all $u,v\\\\in H^{\\\\psi ,1}$ , $\\\\left|B_\\\\alpha (u,v)\\\\right| \\\\le \\\\left|\\\\langle q_1(D)u,v\\\\rangle \\\\right| + \\\\left|\\\\langle q_2(\\\\sigma , D)u,v\\\\rangle \\\\right| + \\\\alpha \\\\left|\\\\langle u,v\\\\rangle \\\\right| \\\\le \\\\left(\\\\kappa _1 + \\\\kappa _2 + \\\\alpha \\\\kappa _3^2\\\\right)\\\\Vert u\\\\Vert _{\\\\psi ,1}\\\\Vert v\\\\Vert _{\\\\psi ,2},$ which proves the theorem.\",\n \"Recall that a bilinear form $B$ defined on a Hilbert space $(H,\\\\langle \\\\cdot ,\\\\cdot \\\\rangle )$ is coercive if there is $c>0$ such that $B(u,u)\\\\ge c\\\\langle u,u\\\\rangle \\\\hspace{20.0pt} \\\\forall u\\\\in H.$ Theorem 4.10 (Lax–Milgram) Let $B$ be a bounded bilinear form, defined on a Hilbert space $(H,\\\\langle \\\\cdot ,\\\\cdot ,\\\\rangle )$ , and suppose $B$ is coercive with constant $c$ .\",\n \"Then given $f\\\\in H^{\\\\prime }$ , there is a unique $u\\\\in H$ such that $B(u,v) = f(v) \\\\hspace{20.0pt} \\\\forall v\\\\in H.$ See for example Theorem 1 of Evans [15], pp. 297–9.\",\n \"We would like to apply the Lax–Milgram theorem to $B_\\\\alpha $ .\",\n \"For this we impose an additional assumption, which will ensure $B_\\\\alpha $ is coercive on $H^{\\\\psi ,1}$ .\",\n \"Assumption 4.11 Let $M\\\\in \\\\mathbb {N}$ , $M>\\\\dim (G/K)+1$ , and write $\\\\gamma _M = \\\\left(8C_M(2(1+c_\\\\psi ))^{1/2}\\\\Vert \\\\langle \\\\cdot \\\\rangle ^{-M+1}\\\\Vert _{L^1(,\\\\omega )}\\\\right)^{-1},$ where $c_{\\\\psi }$ and $C_M$ are constants given by (REF ) and (REF ), respectively.\",\n \"For $c_0$ is as in Assumption REF (REF ), assume that $\\\\sum _{\\\\beta =0}^M\\\\Vert \\\\Phi _\\\\beta \\\\Vert _1 \\\\le \\\\gamma _Mc_0.$ Remark 4.12 See Jacob [27] P.3 and P.4, pp.\",\n \"161, or Hoh [24] (A.3.M), pp.\",\n \"54, for comparison.\",\n \"Examples where Assumption REF is satisfied are considered in Section .\",\n \"The next theorem is an analogue of Theorem 3.1 of Jacob [27].\",\n \"Theorem 4.13 Suppose Assumptions REF and REF hold, with $M>\\\\dim (G/K)+1$ .\",\n \"Then there is $\\\\alpha _0>0$ such that $B_\\\\alpha (u,u)\\\\ge \\\\frac{c_0}{2}\\\\Vert u\\\\Vert _{1,\\\\lambda }^2,$ for all $\\\\alpha \\\\ge \\\\alpha _0$ and $u\\\\in H^{\\\\psi ,1}$ .\",\n \"Proceed exactly as in Hoh [24] page 57, lines 8–17.\",\n \"By Assumption REF (REF ), there is $\\\\alpha _0>0$ such that $q_1(\\\\lambda ) \\\\ge c_0\\\\Psi (\\\\lambda )^2-\\\\alpha _0 \\\\hspace{20.0pt} \\\\forall \\\\lambda \\\\in .$ This may be used to prove that for all $u\\\\in H^{\\\\psi ,1}$ , $\\\\langle q_1(D)u,u\\\\rangle \\\\ge c_0\\\\Vert u\\\\Vert _{\\\\psi ,1}^2 - \\\\alpha _0\\\\Vert u\\\\Vert ^2,$ at which point we can apply (REF ) and (REF ), as well as Assumption REF , to conclude $\\\\left|\\\\langle q_2(\\\\sigma ,D)u,u\\\\rangle \\\\right| &\\\\le C_M\\\\sqrt{2(1+c_\\\\psi )}\\\\sum _{\\\\beta =0}^M\\\\Vert \\\\Phi _\\\\beta \\\\Vert _{L^1(,\\\\omega )}\\\\left\\\\Vert \\\\langle \\\\cdot \\\\rangle ^{-M+1}\\\\right\\\\Vert _{L^1(,\\\\omega )}\\\\Vert u\\\\Vert _{\\\\psi ,1}^2 \\\\\\\\&= \\\\frac{1}{8\\\\gamma _M}\\\\sum _{\\\\beta =0}^M\\\\Vert \\\\Phi _\\\\beta \\\\Vert _{L^1(,\\\\omega )}\\\\Vert u\\\\Vert _{\\\\psi ,1}^2 \\\\le \\\\frac{c_0}{8}\\\\Vert u\\\\Vert _{\\\\psi ,1},$ for all $u\\\\in H^{\\\\psi ,1}$ .\",\n \"Thus, for all $u\\\\in H^{\\\\psi ,1}$ $\\\\langle q(\\\\sigma ,D)u,u\\\\rangle &\\\\ge \\\\langle q_1(D)u,u\\\\rangle - \\\\left|\\\\langle q_2(\\\\sigma ,D)u,u\\\\rangle \\\\right| \\\\\\\\&\\\\ge (c_0-\\\\frac{c_0}{8})\\\\Vert u\\\\Vert _{\\\\psi ,1}^2 - \\\\alpha _0\\\\Vert u\\\\Vert _{\\\\psi ,1}^2 \\\\ge \\\\frac{c_0}{2}\\\\Vert u\\\\Vert _{\\\\psi ,1}^2 - \\\\alpha _0\\\\Vert u\\\\Vert _{\\\\psi ,1}^2.$ Therefore, for all $\\\\alpha \\\\ge \\\\alpha _0$ and $u\\\\in H^{\\\\psi ,1}$ $B_\\\\alpha (u,u) &= \\\\langle q(\\\\sigma ,D)u,u\\\\rangle + \\\\alpha \\\\Vert u\\\\Vert \\\\ge \\\\langle q(\\\\sigma ,D)u,u\\\\rangle + \\\\alpha _0\\\\Vert u\\\\Vert \\\\ge \\\\frac{c_0}{2}\\\\Vert u\\\\Vert _{\\\\psi ,1}^2,$ We may now apply the Lax–Milgram theorem (Theorem REF ) to $B_\\\\alpha $ , together with the linear functional $v\\\\mapsto \\\\langle f,\\\\cdot \\\\rangle $ , and conclude the following: Theorem 4.14 Let $\\\\alpha \\\\ge \\\\alpha _0$ .\",\n \"Then (REF ) has a weak solution in the following sense: for all $f\\\\in L^2(K|G|K)$ there is a unique $u\\\\in H^{\\\\psi ,1}$ such that for all $v\\\\in H^{\\\\psi ,1}$ , $B_\\\\alpha (u,v) = \\\\langle f,v\\\\rangle .$ Theorem REF provides a weak solution to (REF ).\",\n \"The next task is to prove that this solution is in fact a strong solution that belongs to $C_0(K|G|K)$ .\",\n \"This will be achieved using the Sobolev embedding of Theorem REF (REF ).\",\n \"Just as in Jacob [27] Theorem 3.1 and Hoh [24] Theorem 4.11, we have a useful lower bound for the pseudodifferential operator $q(\\\\sigma ,D)$ acting on $H^{\\\\psi ,s}$ , when $s\\\\ge 0$ .\",\n \"Theorem 4.15 Let $s\\\\ge 0$ , and suppose the symbol $q$ satisfies Assumptions REF and REF , for some $M>|s-1|+1+\\\\dim (G/K)$ .\",\n \"Then there is $\\\\kappa >0$ such that for all $u\\\\in H^{\\\\psi ,s+2}$ , $\\\\Vert q(\\\\sigma ,D)u\\\\Vert _{\\\\psi ,s} \\\\ge \\\\frac{c_0}{4}\\\\Vert u\\\\Vert _{\\\\psi ,s+2}-\\\\kappa \\\\Vert u \\\\Vert .$ The proof is formally no different to the sources mentioned: let $u\\\\in H^{\\\\psi ,s+2}$ , and use (REF ) and Theorem REF (REF ) to prove that $\\\\Vert q_1(D)u\\\\Vert _{\\\\psi ,s} \\\\ge \\\\frac{c_0}{2}\\\\Vert u\\\\Vert _{\\\\psi ,s+2} - \\\\kappa _1\\\\Vert u\\\\Vert ,$ for some $\\\\kappa _1>0$ .\",\n \"Recall the estimate (REF ) of $\\\\Vert q_2(\\\\sigma ,D)u\\\\Vert _{\\\\psi ,s}$ from the proof of Theorem REF .\",\n \"In light of Assumption REF and the particular form chosen for $\\\\gamma _M$ , one can use (REF ) to show that $\\\\Vert q_2(\\\\sigma ,D)u\\\\Vert _{\\\\psi ,s} &\\\\le C_M\\\\sum _{\\\\beta =0}^M\\\\Vert \\\\Phi _\\\\beta \\\\Vert _{L^1(,\\\\omega )}\\\\Big (\\\\left\\\\Vert \\\\langle \\\\cdot \\\\rangle ^{-M}\\\\right\\\\Vert _{L^1(,\\\\omega )}\\\\Vert u\\\\Vert _{\\\\psi ,s+2} + C_{s,\\\\psi }\\\\Vert u\\\\Vert _{\\\\psi ,s+1}\\\\Big ) \\\\\\\\&\\\\le C_Mc_0\\\\gamma _M\\\\Big (\\\\left\\\\Vert \\\\langle \\\\cdot \\\\rangle ^{-M}\\\\right\\\\Vert _{L^1(,\\\\omega )}\\\\Vert u\\\\Vert _{\\\\psi ,s+2} + C_{s,\\\\psi }\\\\Vert u\\\\Vert _{\\\\psi ,s+1}\\\\Big ) \\\\\\\\&\\\\le \\\\frac{c_0}{8}\\\\Vert u\\\\Vert _{\\\\psi ,s+2} + c\\\\Vert u\\\\Vert _{\\\\psi ,s+1},$ where $c>0$ is a constant.\",\n \"Using Theorem REF (REF ) once again, let $\\\\kappa _2>0$ such that $c\\\\Vert u\\\\Vert _{\\\\psi ,s+1} \\\\le \\\\frac{c_0}{8}\\\\Vert u\\\\Vert _{\\\\psi ,s+2} + \\\\kappa _2\\\\Vert u\\\\Vert .$ Then, by the above, $\\\\Vert q_2(\\\\sigma ,D)u\\\\Vert _{\\\\psi ,s} \\\\le \\\\frac{c_0}{4}\\\\Vert u\\\\Vert _{\\\\psi ,s+2} + \\\\kappa _2\\\\Vert u\\\\Vert .$ Combining (REF ) and (REF ), we get $\\\\Vert q(\\\\sigma ,D)u\\\\Vert _{\\\\psi ,s} \\\\ge \\\\Vert q_1(D)u\\\\Vert _{\\\\psi ,s} - \\\\Vert q_2(\\\\sigma ,D)u\\\\Vert _{\\\\psi ,s} \\\\ge \\\\frac{c_0}{4}\\\\Vert u\\\\Vert _{\\\\psi ,s+2} - (\\\\kappa _1+\\\\kappa _2)\\\\Vert u\\\\Vert .$ The proof of the next theorem makes use of a particular family $(J_\\\\epsilon ,0<\\\\epsilon \\\\le 1)$ of bounded linear operators $L^2(G)$ , which will play the role of a Friedrich mollifier, but in the noncompact symmetric space setting.\",\n \"First note that by identifying $\\\\operatorname{\\\\mathfrak {a}}$ with $\\\\mathbb {R}^m$ via our chosen basis, it makes sense to consider Friedrich mollifiers on $\\\\operatorname{\\\\mathfrak {a}}$ .\",\n \"For $0<\\\\epsilon \\\\le 1$ and $H\\\\in \\\\operatorname{\\\\mathfrak {a}}$ , let $l(H) := C_0e^{\\\\frac{1}{|H|^2-1}}\\\\operatorname{\\\\bf 1}_{B_1(0)}(H), ~~\\\\text{ and }~~ l_\\\\epsilon (H) := \\\\epsilon ^{-m}l(H/\\\\epsilon ),$ where $C_0>0$ is a constant chosen so that $\\\\int _{\\\\operatorname{\\\\mathfrak {a}}}l(H)dH=1$ .\",\n \"This mollifier (acting on Euclidean space) is used frequently in Evans [15], where it is called the standard mollifier — see Appendix C.4, pp. 629.\",\n \"Jacob [27] and Hoh [24] both use this mollifier to show that the weak solution $u$ in Theorem REF is also a strong solution, in the sense that it satisfies (REF ), and that $u\\\\in H^{\\\\psi ,s}$ for suitably large $s$ .\",\n \"Observe that $l,l_\\\\epsilon \\\\in \\\\mathcal {S}(\\\\operatorname{\\\\mathfrak {a}})^W$ for all $0<\\\\epsilon \\\\le 1$ .\",\n \"Let $j,j_\\\\epsilon \\\\in \\\\mathcal {S}(K|G|K)$ be the corresponding mappings obtained via the commutative diagram of Theorem REF , so that $\\\\hat{j_\\\\epsilon } = {F}(l_\\\\epsilon ), \\\\hspace{20.0pt} \\\\forall 0<\\\\epsilon \\\\le 1,$ where ${F}$ denotes the Euclidean Fourier transform (see equation (REF )).\",\n \"For $0<\\\\epsilon \\\\le 1$ , let $J_\\\\epsilon $ be the convolution operator defined on $L^2(K|G|K)$ by $J_\\\\epsilon u = j_\\\\epsilon \\\\ast u.$ The most important properties of $(J_\\\\epsilon ,0<\\\\epsilon \\\\le 1)$ needed for the proof of Theorem REF are stated below, and proven in the appendix.\",\n \"Proposition 4.16 $\\\\hat{j_\\\\epsilon }(\\\\lambda ) = \\\\hat{j}(\\\\epsilon \\\\lambda )$ for all $0<\\\\epsilon \\\\le 1$ and $\\\\lambda \\\\in $ .\",\n \"For all $0<\\\\epsilon \\\\le 1$ , $J_\\\\epsilon $ is a self-adjoint contraction of $L^2(K|G|K)$ .\",\n \"$J_\\\\epsilon u\\\\in H^{\\\\psi ,s}$ for all $s\\\\ge 0$ , $u\\\\in L^2(K|G|K)$ and $0<\\\\epsilon \\\\le 1$ , and if $u\\\\in H^{\\\\psi ,s}$ , then $\\\\Vert J_\\\\epsilon u\\\\Vert _{\\\\psi ,s} \\\\le \\\\Vert u\\\\Vert _{\\\\psi ,s}.$ $\\\\Vert J_\\\\epsilon u - u\\\\Vert _{\\\\psi ,s}\\\\rightarrow 0$ as $\\\\epsilon \\\\rightarrow 0$ .\",\n \"The following commutator estimate will also be useful in the proof of Theorem REF .\",\n \"Lemma 4.17 Let $s\\\\ge 0$ , and suppose $q$ is a continuous negative definite symbol satisfying Assumption REF (REF ) for $M>|s-1|+1+\\\\dim (G/K)$ .\",\n \"Then there is $c>0$ such that for all $ 0<\\\\epsilon \\\\le 1$ and all $u\\\\in C_c^\\\\infty (K|G|K)$ , $\\\\Vert [J_\\\\epsilon ,q(\\\\sigma ,D)]u\\\\Vert _{\\\\psi ,s} \\\\le c\\\\Vert u\\\\Vert _{\\\\psi ,s+1}.$ Let $0<\\\\epsilon \\\\le 1$ and $u\\\\in C_c^\\\\infty (K|G|K)$ , and observe that by Proposition REF (REF ), $[J_\\\\epsilon ,q_1(D)]u)^\\\\wedge (\\\\lambda ) = \\\\hat{j}(\\\\epsilon \\\\lambda )q_1(\\\\lambda )\\\\hat{u}(\\\\lambda ) - q_1(\\\\lambda )\\\\hat{j}(\\\\epsilon \\\\lambda )\\\\hat{u}(\\\\lambda ) = 0,$ for all $\\\\lambda \\\\in $ , so $[J_\\\\epsilon ,q_1(D)]u = 0$ .\",\n \"For $\\\\lambda ,\\\\eta \\\\in $ , let $F_{\\\\lambda ,\\\\eta }=\\\\phi _{-\\\\lambda }q_2(\\\\cdot ,\\\\eta )$ , as previously (c.f.\",\n \"(REF )).\",\n \"Then by (REF ), for all $\\\\lambda \\\\in $ , $\\\\begin{aligned}\\\\left([J_\\\\epsilon ,q(\\\\sigma ,D)]u\\\\right)^\\\\wedge (\\\\lambda ) &= (J_\\\\epsilon q_2(\\\\sigma ,D)u)^\\\\wedge (\\\\lambda ) - (q_2(\\\\sigma ,D)J_\\\\epsilon u)^\\\\wedge (\\\\lambda ) \\\\\\\\&=\\\\hat{j}(\\\\epsilon \\\\lambda )(q_2(\\\\sigma ,D)u)^\\\\wedge (\\\\lambda ) - \\\\int _{}\\\\hat{F}_{\\\\lambda ,\\\\eta }(-\\\\eta )\\\\hat{j}(\\\\epsilon \\\\eta )\\\\hat{u}(\\\\eta )\\\\omega (d\\\\eta ).\\\\end{aligned}$ Now, $(q_2(\\\\sigma ,D)u)^\\\\wedge (\\\\lambda ) &= \\\\int _G\\\\phi _{-\\\\lambda }(\\\\sigma )(q_2(\\\\sigma ,D)u)(\\\\sigma )d\\\\sigma \\\\\\\\&= \\\\int _G\\\\int _{}\\\\phi _{-\\\\lambda }(\\\\sigma )\\\\phi _{-\\\\eta }(\\\\sigma )q_2(\\\\sigma ,\\\\eta )\\\\hat{u}(\\\\eta )\\\\omega (d\\\\eta )d\\\\sigma .$ Also, since $q_2(\\\\sigma ,\\\\cdot )$ is continuous and $\\\\hat{u}\\\\in \\\\mathcal {S}()$ , we have $q_2(\\\\sigma ,\\\\cdot )\\\\hat{u}\\\\in L^1(,\\\\omega )$ .\",\n \"Therefore, by the Fubini theorem, $(q_2(\\\\sigma ,D)u)^\\\\wedge (\\\\lambda ) &= \\\\int _{}\\\\int _G\\\\phi _{-\\\\eta }(\\\\sigma )\\\\phi _{-\\\\lambda }(\\\\sigma )q_2(\\\\sigma ,\\\\eta )\\\\hat{u}(\\\\eta )d\\\\sigma \\\\omega (d\\\\eta ) \\\\\\\\&=\\\\int _{}\\\\int _G\\\\phi _{-\\\\eta }(\\\\sigma )F_{\\\\lambda ,\\\\eta }(\\\\sigma )d\\\\sigma \\\\hat{u}(\\\\eta )\\\\omega (d\\\\eta ) = \\\\int _{}\\\\hat{F}_{\\\\lambda ,\\\\eta }(-\\\\eta )\\\\hat{u}(\\\\eta )\\\\omega (d\\\\eta ),$ and so, substituting back into (REF ), $\\\\left([J_\\\\epsilon ,q(\\\\sigma ,D)]u\\\\right)^\\\\wedge (\\\\lambda ) = \\\\int _{}\\\\hat{F}_{\\\\lambda ,\\\\eta }(-\\\\eta )\\\\left(\\\\hat{j}(\\\\epsilon \\\\lambda )-\\\\hat{j}(\\\\epsilon \\\\eta )\\\\right)\\\\hat{u}(\\\\eta )\\\\omega (d\\\\eta ),$ for all $\\\\lambda \\\\in $ .\",\n \"From here, a straightforward adaptation to the proof of Hoh [24] Theorem 4.4, pp.\",\n \"51–52, with (REF ) replacing Hoh [24] (4.23), completes the proof of the lemma.\",\n \"We are now ready to state and prove that, subject to our conditions, a strong solution to (REF ) exists, and belongs to an anisotropic Sobolev space of suitably high order.\",\n \"Theorem 4.18 Let $\\\\alpha _0$ be as in Theorem REF , let $\\\\alpha \\\\ge \\\\alpha _0$ , and let $s\\\\ge 0$ .\",\n \"Suppose that the negative definite symbol $q$ satisfies Assumptions REF and REF , where $M>|s-1|+1+\\\\dim (G/K)$ .\",\n \"Then for all $f\\\\in H^{\\\\psi ,s}$ , there is a unique $u\\\\in H^{\\\\psi ,s+2}$ such that $(q(\\\\sigma ,D)+\\\\alpha )u = f.$ Let $f\\\\in H^{\\\\psi ,s}$ .\",\n \"By Theorem REF we also have $f\\\\in L^2(K|G|K)$ , and so by Theorem REF there is a unique $u\\\\in H^{\\\\psi ,1}$ such that $B_\\\\alpha (u,v) =\\\\langle f,v\\\\rangle \\\\hspace{20.0pt} \\\\forall v\\\\in C_c^\\\\infty (K|G|K).$ The proof follows that of [27] Theorem 4.3, pp.\",\n \"163 and [24] Theorem 4.12, pp.\",\n \"59, using induction to show that that $u\\\\in H^{\\\\psi ,t}$ for $1\\\\le t\\\\le s+2$ , and in particular, that $u\\\\in H^{\\\\psi ,s+2}$ .\",\n \"The family of operators $(J_\\\\epsilon ,0<\\\\epsilon \\\\le 1)$ take over role of the Friedrich mollifiers of [27] and [24].\",\n \"By Proposition REF these operators satisfy the properties needed for the proof to carry over with little alteration.\",\n \"Lemma REF and Theorem REF replace [24] Theorem 4.4 and 4.11, respectively.\",\n \"Theorem 4.19 Let $q$ be a continuous negative definite function on $$ , satisfying Assumptions REF and REF with $M>\\\\max \\\\left\\\\lbrace 1,\\\\frac{d}{r}\\\\right\\\\rbrace +d$ , where $d=\\\\dim (G/K)$ .\",\n \"Then for all $\\\\alpha \\\\ge \\\\alpha _0$ , $\\\\overline{\\\\operatorname{Ran}(\\\\alpha +q(\\\\sigma ,D))}=C_0(K|G|K).$ Fix $s\\\\in \\\\mathbb {R}$ with $\\\\max \\\\left\\\\lbrace \\\\frac{d}{r},1\\\\right\\\\rbrace < s\\\\min \\\\lbrace 1,d/r\\\\rbrace +d$ .\",\n \"Then $-q(\\\\sigma ,D)$ extends to the infinitesimal generator of a strongly continuous sub-Feller semigroup on $C_0(K|G|K)$ .\",\n \"By construction, $-q(\\\\sigma ,D)$ is a Gangolli operator, and therefore is a densely defined linear operator on $C_0(K|G|K)$ that satisfies the positive maximum principle.\",\n \"By Theorems REF and REF , $-q(\\\\sigma ,D)$ is closable, and its closure generates a strongly continuous sub-Feller semigroup.\"\n ],\n [\n \"A Class of Examples\",\n \"We now present a class of Gangolli symbols for which the conditions of Corollary REF are satisfied.\",\n \"Let $M\\\\in \\\\mathbb {N}$ such that $M>\\\\min \\\\lbrace 1,d/r\\\\rbrace +d+1$ .\",\n \"We consider symbols $q:G\\\\times \\\\rightarrow \\\\mathbb {R}$ of the form $q(\\\\sigma ,\\\\lambda ) = \\\\kappa \\\\psi (\\\\lambda ) + u(\\\\sigma )v(\\\\lambda ), \\\\hspace{20.0pt} \\\\forall \\\\sigma \\\\in G,\\\\lambda \\\\in ,$ where $\\\\kappa $ is a positive constant, $u\\\\in C^M_c(K|G|K)$ is non-negative, and $v:\\\\rightarrow \\\\mathbb {R}$ is a negative definite function satisfying $|v(\\\\lambda )|\\\\le c_v(1+\\\\psi (\\\\lambda )) \\\\hspace{20.0pt} \\\\forall \\\\lambda \\\\in ,$ for some constant $c_v>0$ .\",\n \"As in previous sections, $\\\\psi :\\\\rightarrow \\\\mathbb {R}$ is a fixed continuous negative definite function, satisfying (REF ).\",\n \"In addition to this, we assume $\\\\psi $ is a Gangolli exponent.\",\n \"By Example REF , the mappings $(\\\\sigma ,\\\\lambda )\\\\mapsto c_0\\\\psi (\\\\lambda )$ and $(\\\\sigma ,\\\\lambda )\\\\mapsto u(\\\\sigma )v(\\\\lambda )$ are both Gangolli symbols, and hence so is $q$ .\",\n \"For each $\\\\lambda \\\\in $ and $\\\\sigma \\\\in G$ , $q_1(\\\\lambda ) = \\\\kappa \\\\psi (\\\\lambda ), \\\\hspace{20.0pt} \\\\forall \\\\lambda \\\\in ,$ and $q_2(\\\\sigma ,\\\\lambda ) = u(\\\\sigma )v(\\\\lambda ), \\\\hspace{20.0pt} \\\\forall \\\\sigma \\\\in G,\\\\;\\\\lambda \\\\in .$ Observe that since $v$ has compact support, $\\\\operatorname{Supp}(v)\\\\ne G$ , and for any $\\\\sigma _0\\\\in G\\\\setminus \\\\operatorname{Supp}(v)$ , $q_1(\\\\lambda ) = q(\\\\sigma _0,\\\\lambda ) \\\\hspace{20.0pt} \\\\forall \\\\lambda \\\\in ,$ and so $q$ is of the form (REF ).\",\n \"Proposition 5.1 $q_1$ satisfies Assumption REF (REF ).\",\n \"Let $c_0=\\\\frac{\\\\kappa }{2}\\\\min \\\\lbrace 1,c\\\\rbrace $ and $c_1=\\\\kappa $ .\",\n \"Certainly we have $q_1(\\\\lambda ) \\\\le c_1(1+\\\\psi (\\\\lambda )) \\\\hspace{20.0pt} \\\\forall \\\\lambda \\\\in ;$ in particular this inequality holds when $|\\\\lambda |\\\\ge 1$ .\",\n \"Also, by (REF ), if $|\\\\lambda |\\\\ge 1$ , then $\\\\psi (\\\\lambda )\\\\ge c|\\\\lambda |^r\\\\ge c$ and so, if $|\\\\lambda |\\\\ge 1$ , $q_1(\\\\lambda ) = \\\\frac{\\\\kappa }{2}(\\\\psi (\\\\lambda )+\\\\psi (\\\\lambda )) \\\\ge \\\\frac{\\\\kappa }{2}(c+\\\\psi (\\\\lambda )) \\\\ge c_0(1+\\\\psi (\\\\lambda )).$ That is, Assumption REF (REF ) holds.\",\n \"Proving that Assumption REF (REF ) is satisfied will take more work.\",\n \"However, once we have done so, Assumption REF will follow by taking $\\\\kappa $ sufficiently large.\",\n \"Note that for the case we are considering, $F_{\\\\lambda ,\\\\eta }(\\\\sigma ) = \\\\phi _{-\\\\lambda }(\\\\sigma )u(\\\\sigma )v(\\\\eta ), \\\\hspace{20.0pt} \\\\forall \\\\sigma \\\\in G,\\\\;\\\\lambda ,\\\\eta \\\\in ,$ and so, for $\\\\beta =0,1,\\\\ldots ,M$ , $(-\\\\Delta )^{\\\\beta /2}F_{\\\\lambda ,\\\\eta }(\\\\sigma ) = v(\\\\eta )(-\\\\Delta )^{\\\\beta /2}(\\\\phi _{-\\\\lambda }u)(\\\\sigma ),$ for all $\\\\lambda ,\\\\eta \\\\in $ and $\\\\sigma \\\\in G$ .\",\n \"By (REF ), $\\\\left|(-\\\\Delta )^{\\\\beta /2}F_{\\\\lambda ,\\\\eta }(\\\\sigma )\\\\right| = |v(\\\\eta )|\\\\left|(-\\\\Delta )^{\\\\beta /2}(\\\\phi _{-\\\\lambda }u)(\\\\sigma )\\\\right| \\\\le c_v\\\\left|(-\\\\Delta )^{\\\\beta /2}(\\\\phi _{-\\\\lambda }u)(\\\\sigma )\\\\right|\\\\big (1+\\\\psi (\\\\eta )\\\\big ).$ To verify Assumption REF (REF ), we seek for each $\\\\beta \\\\in \\\\lbrace 0,1,\\\\ldots ,M\\\\rbrace $ a mapping $\\\\Phi _\\\\beta \\\\in L^1(K|G|K)$ such that $\\\\left|(-\\\\Delta )^{\\\\beta /2}(\\\\phi _{-\\\\lambda }u)(\\\\sigma )\\\\right| \\\\le \\\\Phi _\\\\beta \\\\langle \\\\lambda \\\\rangle ^M, \\\\hspace{20.0pt} \\\\forall \\\\sigma \\\\in G,\\\\;\\\\lambda \\\\in .$ Let us first introduce some notation.\",\n \"For each $n\\\\in \\\\mathbb {N}$ , a noncommutative version of the multinomial theorem tells us that $(-\\\\Delta )^n(\\\\phi _{-\\\\lambda }u) = (-1)^n\\\\left(\\\\sum _{j=1}^dX_j^2\\\\right)^n(\\\\phi _{-\\\\lambda }u) = \\\\sum _{\\\\begin{array}{c}\\\\alpha \\\\in \\\\mathbb {N}_0^d,\\\\\\\\|\\\\alpha |\\\\le r\\\\end{array}}c_\\\\alpha X^{\\\\alpha }(\\\\phi _{-\\\\lambda }u)$ for some coefficients $c_\\\\alpha $ , where $|\\\\alpha |=\\\\alpha _1+\\\\ldots +\\\\alpha _d$ and $X^{\\\\alpha }:=X_1^{\\\\alpha _1}\\\\ldots X_d^{\\\\alpha _d}$ .\",\n \"This may also be seen by noting that $(-\\\\Delta )^n$ belongs to the universal enveloping algebra $\\\\mathcal {U}(\\\\mathfrak {p})$ , and applying the Poincaré-Birkoff–Witt theorem (see Knapp [30] Theorem 3.8, pp.\",\n \"217) to write $(-\\\\Delta )^n$ in the basis $\\\\lbrace X^\\\\alpha :\\\\alpha \\\\in \\\\mathbb {N}_0^d\\\\rbrace $ of $\\\\mathcal {U}(\\\\mathfrak {p})$ .\",\n \"Expanding the right-hand side of (REF ) using the fact that each $X_j$ is a derivation will give a large sum of terms of the form $\\\\kappa _{X,Y}X\\\\phi _{-\\\\lambda }Yu,$ where the $\\\\kappa _{X,Y}$ are constants, and $X,Y\\\\in {\\\\bf D}(G)$ are products of powers of $X_1,\\\\ldots , X_d$ , each with degree at most $2l$ .\",\n \"Let ${U}_n$ be the set of all the $X$ 's and ${V}_n$ the set of all the $Y$ 's, so that $(-\\\\Delta )^n(\\\\phi _{-\\\\lambda }u) = \\\\sum _{\\\\begin{array}{c}X\\\\in {U}_n,\\\\\\\\Y\\\\in {V}_n\\\\end{array}}\\\\kappa _{X,Y}X\\\\phi _{-\\\\lambda }Yu.$ The following bound will be useful.\",\n \"Lemma 5.2 For all $X\\\\in {\\\\bf D}(G)$ , there is a constant $C_X>0$ such that $|X\\\\phi _\\\\lambda (\\\\sigma )| \\\\le C_X\\\\langle \\\\lambda \\\\rangle ^{\\\\deg X}\\\\phi _0(\\\\sigma ),$ for all $\\\\lambda \\\\in $ and $\\\\sigma \\\\in G$ .\",\n \"This is a straightforward corollary of Theorem 1.1 (iii) of [20] — see also Lemma 46 of Harish-Chandra [18], pp. 294.\",\n \"Proposition 5.3 The mapping $q_2:G\\\\times \\\\rightarrow \\\\mathbb {R}$ given by (REF ) satisfies Assumption REF (REF ).\",\n \"It is clear by construction that $q_2(\\\\cdot ,\\\\lambda )\\\\in C_c^M(K|G|K)$ for all $\\\\lambda \\\\in $ .\",\n \"To verify the rest of Assumption REF (REF ), it will be useful to assume that $M$ is even.\",\n \"Note that this is an acceptable assumption, since if $M$ is odd, we may replace it with $M-1$ .\",\n \"We have chosen $M>\\\\min \\\\lbrace 1,d/r\\\\rbrace +d+1$ , and hence both $M$ and $M-1$ are strictly greater than $\\\\min \\\\lbrace 1,d/r\\\\rbrace +d$ , as required by Corollary REF .\",\n \"Let $\\\\beta \\\\in \\\\lbrace 0,1,\\\\ldots ,M\\\\rbrace $ .\",\n \"We seek $\\\\Phi _\\\\beta \\\\in L^1(K|G|K)$ for which (REF ) holds.\",\n \"Let $n=\\\\lfloor \\\\beta \\\\rfloor $ .\",\n \"Assume first that $\\\\beta $ is even, so that $n=\\\\beta /2$ .\",\n \"By (REF ) and Lemma REF , $\\\\left|(-\\\\Delta )^{\\\\beta /2}(\\\\phi _{-\\\\lambda }u)\\\\right| \\\\le \\\\sum _{\\\\begin{array}{c}X\\\\in {U}_n,\\\\\\\\Y\\\\in {V}_n\\\\end{array}}|\\\\kappa _{X,Y}||X\\\\phi _{-\\\\lambda }||Yu| &\\\\le \\\\sum _{\\\\begin{array}{c}X\\\\in {U}_n,\\\\\\\\Y\\\\in {V}_n\\\\end{array}}C_X|\\\\kappa _{X,Y}||Yu|\\\\langle \\\\lambda \\\\rangle ^{\\\\deg Y}|\\\\phi _0| \\\\\\\\&\\\\le \\\\sum _{\\\\begin{array}{c}X\\\\in {U}_n,\\\\\\\\Y\\\\in {V}_n\\\\end{array}}C_X|\\\\kappa _{X,Y}||Yu|\\\\langle \\\\lambda \\\\rangle ^{\\\\deg X},$ since $|\\\\phi _0|\\\\le 1$ .\",\n \"Now, $\\\\deg X\\\\le 2l=\\\\beta \\\\le M$ for all $X\\\\in {U}_n$ , and therefore, $\\\\left|(-\\\\Delta )^{\\\\beta /2}(\\\\phi _{-\\\\lambda }u)\\\\right| \\\\le \\\\kappa _\\\\beta \\\\sum _{Y\\\\in {V}_{\\\\beta /2}}|Yu|\\\\langle \\\\lambda \\\\rangle ^M$ where $\\\\kappa _\\\\beta = \\\\sup \\\\left\\\\lbrace C_X|\\\\kappa _{X,Y}|:X\\\\in {U}_{\\\\beta /2},Y\\\\in {V}_{\\\\beta /2}\\\\right\\\\rbrace .$ Let $\\\\Phi _\\\\beta :=\\\\kappa _\\\\beta \\\\sum _{Y\\\\in {V}_{\\\\beta /2}}|Yu|.$ Then $\\\\Phi _\\\\beta \\\\in L^1(K|G|K)$ , since each $Yu$ is a continuous function of compact support.\",\n \"Moreover, $\\\\Vert \\\\Phi _\\\\beta \\\\Vert _1 \\\\le \\\\kappa _\\\\beta \\\\sum _{Y\\\\in {V}_{\\\\beta /2}}\\\\Vert Yu\\\\Vert _1$ In particular, we have verified (REF ) when $\\\\beta $ is even.\",\n \"For the remainder of the proof of Proposition REF , assume $\\\\beta $ is an odd integer, so that $\\\\beta =2n+1$ .\",\n \"Since we're assuming $M$ is even, note that $1\\\\le \\\\beta \\\\le M-1$ .\",\n \"By (REF ), $\\\\left|(-\\\\Delta )^{\\\\beta /2}(\\\\phi _\\\\lambda u)\\\\right| = \\\\left|\\\\sqrt{-\\\\Delta }(-\\\\Delta )^n(\\\\phi _\\\\lambda u)\\\\right| \\\\le \\\\sum _{\\\\begin{array}{c}X\\\\in {U}_n,\\\\\\\\Y\\\\in {V}_n\\\\end{array}}|\\\\kappa _{X,Y}|\\\\left|\\\\sqrt{-\\\\Delta }\\\\big (X\\\\phi _{-\\\\lambda }Yu\\\\big )\\\\right|.$ The families ${U}_n$ and ${V}_n$ each consist of differential operators of degree at most $2n=\\\\beta -1$ .\",\n \"Now, $-\\\\sqrt{-\\\\Delta }$ is the infinitesimal generator of the process obtained by subordinating Brownian motion on $G/K$ by the standard $\\\\frac{1}{2}$ -stable subordinator on $\\\\mathbb {R}$ .\",\n \"By standard subordination theory (see [5] Section 5.7, pp.\",\n \"154, or [41] §13 for more general theory) $\\\\sqrt{-\\\\Delta }$ may be expressed as a Bochner integral $\\\\sqrt{-\\\\Delta } = \\\\frac{1}{2\\\\sqrt{\\\\pi }}\\\\int _{0+}^\\\\infty t^{-3/2}(1-T_t)dt,$ where $(T_t,t\\\\ge 0)$ denotes the heat semigroup generated by $\\\\Delta $ .\",\n \"Alternatively, equation (REF ) may also be obtained using the spectral theorem, see for example [9] §3.1.\",\n \"Given $X\\\\in {U}_n$ and $Y\\\\in {V}_n$ and $\\\\sigma \\\\in G$ , $\\\\begin{aligned}\\\\left|\\\\sqrt{-\\\\Delta }(X\\\\phi _{-\\\\lambda }Yu)(\\\\sigma )\\\\right| &= \\\\frac{1}{2\\\\sqrt{\\\\pi }}\\\\left|\\\\int _{0+}^\\\\infty t^{-3/2}(1-T_t)\\\\big (X\\\\phi _{-\\\\lambda }Yu\\\\big )dt\\\\right| \\\\\\\\&\\\\le \\\\frac{1}{2\\\\sqrt{\\\\pi }}\\\\Bigg [\\\\left|\\\\int _{0+}^1 t^{-3/2}(1-T_t)\\\\big (X\\\\phi _{-\\\\lambda }Yu\\\\big )(\\\\sigma )dt\\\\right| \\\\\\\\&\\\\hspace{90.0pt}+ \\\\left|\\\\int _1^\\\\infty t^{-3/2}(1-T_t)\\\\big (X\\\\phi _{-\\\\lambda }Yu\\\\big )(\\\\sigma )dt\\\\right|\\\\Bigg ].\\\\end{aligned}$ Let $(h_t,t\\\\ge 0)$ denote the heat kernel associated with $(T_t,t\\\\ge 0)$ .\",\n \"For the $\\\\int _1^\\\\infty $ term of (REF ), note that $\\\\int _1^\\\\infty t^{-3/2}dt=2$ , and so $&\\\\left|\\\\int _1^\\\\infty t^{-3/2}(1-T_t)\\\\big (X\\\\phi _{-\\\\lambda }Yu\\\\big )(\\\\sigma )dt\\\\right| \\\\\\\\&\\\\hspace{100.0pt}= \\\\left|\\\\int _1^\\\\infty t^{-3/2}X\\\\phi _{-\\\\lambda }(\\\\sigma )Yu(\\\\sigma )dt - \\\\int _1^\\\\infty t^{-3/2}T_t\\\\big (X\\\\phi _{-\\\\lambda }Yu\\\\big )(\\\\sigma )dt\\\\right| \\\\\\\\&\\\\hspace{100.0pt}\\\\le 2|X\\\\phi _{-\\\\lambda }(\\\\sigma )||Yu(\\\\sigma )| + \\\\left|\\\\int _1^\\\\infty t^{-3/2}\\\\int _GX\\\\phi _{-\\\\lambda }(\\\\sigma \\\\tau )Yu(\\\\sigma \\\\tau )h_t(\\\\tau )d\\\\tau dt\\\\right|.$ By Lemma REF and the fact that $\\\\deg X\\\\le \\\\beta -1$ , $|X\\\\phi _{-\\\\lambda }| \\\\le C_X\\\\langle \\\\lambda \\\\rangle ^{\\\\deg X} \\\\le C\\\\langle \\\\lambda \\\\rangle ^{\\\\beta -1},$ for all $\\\\lambda \\\\in $ , where $C_X$ is as in (REF ), and $C=\\\\max \\\\lbrace C_X:X\\\\in {U}_n\\\\rbrace $ .\",\n \"Thus $&\\\\left|\\\\int _1^\\\\infty t^{-3/2}(1-T_t)\\\\big (X\\\\phi _{-\\\\lambda }Yu\\\\big )(\\\\sigma )dt\\\\right| \\\\\\\\&\\\\hspace{80.0pt}\\\\le 2|X\\\\phi _{-\\\\lambda }(\\\\sigma )||Yu(\\\\sigma )| + \\\\int _1^\\\\infty t^{-3/2}\\\\int _G|X\\\\phi _{-\\\\lambda }(\\\\sigma \\\\tau )||Yu(\\\\sigma \\\\tau )|h_t(\\\\tau )d\\\\tau dt \\\\\\\\&\\\\hspace{80.0pt}\\\\le C\\\\left(2|Yu(\\\\sigma )| + \\\\int _1^\\\\infty t^{-3/2}\\\\int _G|Yu(\\\\sigma \\\\tau )|h_t(\\\\tau )d\\\\tau dt\\\\right)\\\\langle \\\\lambda \\\\rangle ^{\\\\beta -1} \\\\\\\\&\\\\hspace{80.0pt}= \\\\Phi _{\\\\beta ,Y}^{(1)}(\\\\sigma )\\\\langle \\\\lambda \\\\rangle ^{\\\\beta -1},$ where $\\\\Phi _{\\\\beta ,Y}^{(1)} := C\\\\left(2|Yu| + \\\\int _1^\\\\infty t^{-3/2}T_t\\\\big (|Yu|\\\\big )dt\\\\right).$ Since $\\\\beta -1\\\\le M$ and $\\\\langle \\\\lambda \\\\rangle \\\\ge 1$ for all $\\\\lambda \\\\in $ , it follows that $\\\\left|\\\\int _1^\\\\infty t^{-3/2}(1-T_t)\\\\big (X\\\\phi _{-\\\\lambda }Yu\\\\big )dt\\\\right| \\\\le \\\\Phi _{\\\\beta ,Y}^{(1)}\\\\langle \\\\lambda \\\\rangle ^M,$ for all $\\\\lambda \\\\in $ .\",\n \"We claim that $\\\\Phi _{\\\\beta ,Y}^{(1)}\\\\in L^1(K|G|K)$ .\",\n \"Clearly $|Yu|\\\\in L^1(K|G|K)$ , since it is a continuous function of compact support.\",\n \"Each of the operators $T_t$ is a positivity preserving contraction of $L^1(K|G|K)$ , and so $\\\\int _1^\\\\infty t^{-3/2}\\\\int _GT_t\\\\big (|Yu|\\\\big )(\\\\sigma )d\\\\sigma dt = \\\\int _1^\\\\infty t^{-3/2}\\\\left\\\\Vert T_t\\\\big (|Yu|\\\\big )\\\\right\\\\Vert _1 dt \\\\le \\\\int _1^\\\\infty t^{-3/2}\\\\Vert Yu\\\\Vert _1dt = 2\\\\Vert Yu\\\\Vert _1.$ By Fubini's theorem, $\\\\int _1^\\\\infty t^{-3/2}T_t\\\\big (|Yu|\\\\big )dt\\\\in L^1(K|G|K)$ , with $\\\\left\\\\Vert \\\\int _1^\\\\infty t^{-3/2}T_t\\\\big (|Yu|\\\\big )dt\\\\right\\\\Vert _{L_1(K|G|K)} \\\\le 2\\\\Vert Yu\\\\Vert _1.$ It follows by (REF ) that $\\\\Phi _{\\\\beta ,Y}^{(1)}\\\\in L^1(K|G|K)$ , and that $\\\\Vert \\\\Phi _{\\\\beta ,Y}^{(1)}\\\\Vert _1 \\\\le 4C\\\\Vert Yu\\\\Vert _1.$ For the $\\\\int _{0+}^1$ term of (REF ), observe that by Lemma 6.1.12 of Davies [12], pp.\",\n \"169, as well as the Fubini theorem, $\\\\int _{0+}^1t^{-3/2}(1-T_t)\\\\big (X\\\\phi _{-\\\\lambda }Yu\\\\big )dt &= -\\\\int _{0+}^1t^{-3/2}\\\\int _0^t T_s\\\\Delta \\\\big (X\\\\phi _{-\\\\lambda }Yu\\\\big )dsdt \\\\\\\\&= -\\\\int _0^1\\\\int _s^1t^{-3/2}T_s\\\\Delta \\\\big (X\\\\phi _{-\\\\lambda }Yu\\\\big )dtds \\\\\\\\&= -\\\\int _0^12(s^{-1/2}-1)T_s\\\\Delta \\\\big (X\\\\phi _{-\\\\lambda }Yu\\\\big )ds.$ Hence, using the product formula for $\\\\Delta $ , $\\\\begin{aligned}\\\\int _{0+}^1t^{-3/2}(1-T_t)\\\\big (X\\\\phi _{-\\\\lambda }Yu\\\\big )dt &= -2\\\\int _0^1(s^{-1/2}-1)\\\\Big \\\\lbrace T_s\\\\big (X\\\\phi _{-\\\\lambda }\\\\Delta Yu\\\\big ) \\\\\\\\&\\\\hspace{6.0pt}+ 2\\\\sum _{j=1}^dT_s\\\\big (X_jX\\\\phi _{-\\\\lambda }X_jYu\\\\big ) + T_s\\\\big (\\\\Delta X\\\\phi _{-\\\\lambda }Yu\\\\big )\\\\Big \\\\rbrace ds.\\\\end{aligned}$ Let $C$ and $C_X$ be as in (REF ).\",\n \"Then for all $\\\\sigma \\\\in G$ , $\\\\left|T_s\\\\big (X\\\\phi _{-\\\\lambda }\\\\Delta Yu\\\\big )(\\\\sigma )\\\\right| &= \\\\left|\\\\int _GX\\\\phi _{-\\\\lambda }(\\\\sigma \\\\tau )\\\\Delta Yu(\\\\sigma \\\\tau )h_s(\\\\tau )d\\\\tau \\\\right| \\\\\\\\&\\\\le \\\\int _G|X\\\\phi _{-\\\\lambda }(\\\\sigma \\\\tau )||\\\\Delta Yu(\\\\sigma \\\\tau )|h_s(\\\\tau )d\\\\tau \\\\\\\\&\\\\le C_X\\\\langle \\\\lambda \\\\rangle ^{\\\\deg X}\\\\int _G|\\\\Delta Yu(\\\\sigma \\\\tau )|h_s(\\\\tau )d\\\\tau \\\\le C\\\\langle \\\\lambda \\\\rangle ^{\\\\beta -1}T_s|\\\\Delta Yu|(\\\\sigma ).$ In exactly the same way, for $j=1,\\\\ldots ,d$ , $\\\\left|T_s\\\\big (X_jX\\\\phi _{-\\\\lambda }X_jYu\\\\big )\\\\right| \\\\le C_X^{(j)}\\\\langle \\\\lambda \\\\rangle ^{\\\\deg X+1}T_s|X_jYu| \\\\le C^{\\\\prime }\\\\langle \\\\lambda \\\\rangle ^{\\\\beta }T_s|X_jYu|,$ and also $\\\\left|T_s\\\\big (\\\\Delta X\\\\phi _{-\\\\lambda }Yu\\\\big )\\\\right| \\\\le C_X^{(0)}\\\\langle \\\\lambda \\\\rangle ^{\\\\deg X+2}T_s|Yu| \\\\le C^{\\\\prime }\\\\langle \\\\lambda \\\\rangle ^{\\\\beta +1}T_s|Yu|,$ where the constants $C_X^{(j)},C^{(j)}$ are chosen so that for all $\\\\lambda \\\\in $ and $j=1,\\\\ldots ,d$ , $|X\\\\phi _{-\\\\lambda }| \\\\le C_X^{(0)}\\\\langle \\\\lambda \\\\rangle ^{\\\\deg X +2}, \\\\hspace{40.0pt} |X_jX\\\\phi _{-\\\\lambda }| \\\\le C_X^{(j)}\\\\langle \\\\lambda \\\\rangle ^{\\\\deg X +1},$ and $C^{\\\\prime }:=\\\\max \\\\lbrace C_X^{(j)}:X\\\\in {U}_n,j=0,1,\\\\ldots ,d\\\\rbrace $ .\",\n \"Such constants exist by Lemma REF .\",\n \"Now, $\\\\langle \\\\lambda \\\\rangle ^{\\\\beta -1} \\\\le \\\\langle \\\\lambda \\\\rangle ^\\\\beta \\\\le \\\\langle \\\\lambda \\\\rangle ^{\\\\beta +1}$ for all $\\\\lambda \\\\in $ , and hence by (REF ), $&\\\\left|\\\\int _{0+}^1t^{-3/2}(1-T_t)\\\\big (X\\\\phi _{-\\\\lambda }Yu\\\\big )dt\\\\right| \\\\\\\\&\\\\hspace{20.0pt}\\\\le 2\\\\int _0^1(s^{-1/2}-1)\\\\Big \\\\lbrace \\\\left|T_s\\\\big (X\\\\phi _{-\\\\lambda }\\\\Delta Yu\\\\big )\\\\right| + 2\\\\sum _{j=1}^d\\\\left|T_s\\\\big (X_jX\\\\phi _{-\\\\lambda }X_jYu\\\\big )\\\\right| \\\\\\\\&\\\\hspace{240.0pt}+ \\\\left|T_s\\\\big (\\\\Delta X\\\\phi _{-\\\\lambda }Yu\\\\big )\\\\right|\\\\Big \\\\rbrace ds \\\\\\\\&\\\\hspace{20.0pt}\\\\le 2C^{\\\\prime }\\\\langle \\\\lambda \\\\rangle ^{\\\\beta +1}\\\\int _0^1(s^{-1/2}-1)T_s\\\\left(|\\\\Delta Yu|+2\\\\sum _{j=1}^d|X_jYu|+|Yu|\\\\right)ds.$ Since $\\\\beta \\\\le M-1$ , it follows that for all $X\\\\in {U}_n$ and $Y\\\\in {V}_n$ , $\\\\left|\\\\int _{0+}^1t^{-3/2}(1-T_t)\\\\big (X\\\\phi _{-\\\\lambda }Yu\\\\big )dt\\\\right| \\\\le \\\\Phi _{\\\\beta ,Y}^{(2)}\\\\langle \\\\lambda \\\\rangle ^M,$ where $\\\\Phi _{\\\\beta ,Y}^{(2)} = C^{\\\\prime }\\\\int _0^1(s^{-1/2}-1)T_s\\\\left(|\\\\Delta Yu|+2\\\\sum _{j=1}^d|X_jYu|+|Yu|\\\\right)ds.$ Observe that $\\\\Phi _{\\\\beta ,Y}^{(2)}\\\\in L^1(K|G|K)$ for all $Y\\\\in {V}_n$ .\",\n \"Indeed, $u\\\\in C_c^M(K|G|K)$ , and $\\\\deg Y\\\\le \\\\beta -1\\\\le M-2$ , hence $|\\\\Delta Yu|$ , $\\\\sum _{j=1}^d|X_jYu|$ and $|Yu|$ are all continuous functions of compact support.\",\n \"Thus $T_s\\\\big (|\\\\Delta Yu|+2\\\\sum _{j=1}^d|X_jYu|+|Yu|\\\\big )\\\\in L^1(K|G|K)$ , and, since $T_s$ is an $L^1(K|G|K)$ -contraction, $\\\\left\\\\Vert T_s\\\\left(|\\\\Delta Yu|+2\\\\sum _{j=1}^d|X_jYu|+|Yu|\\\\right)\\\\right\\\\Vert _1 \\\\le \\\\Vert \\\\Delta Yu\\\\Vert _1 +2\\\\sum _{j=1}^d\\\\Vert X_jYu\\\\Vert _1 +\\\\Vert Yu\\\\Vert _1.$ Noting that $\\\\int _0^1(s^{-1/2}-1)ds = 1$ , it follows by Fubini's theorem that $\\\\Phi _{\\\\beta ,Y}^{(2)}\\\\in L^1(K|G|K)$ , with $\\\\left\\\\Vert \\\\Phi _{\\\\beta ,Y}^{(2)}\\\\right\\\\Vert _1 \\\\le C_X^{\\\\prime }\\\\left(\\\\Vert \\\\Delta Yu\\\\Vert _1 + 2\\\\sum _{j=1}^d\\\\Vert X_jYu\\\\Vert _1 +\\\\Vert Yu\\\\Vert _1\\\\right).$ Substituting (REF ) and (REF ) into (REF ), we obtain the pointwise estimate $\\\\left|\\\\sqrt{-\\\\Delta }(X\\\\phi _{-\\\\lambda }Yu)\\\\right| \\\\le \\\\frac{1}{2\\\\sqrt{\\\\pi }}\\\\left(\\\\Phi _{\\\\beta ,Y}^{(1)}+\\\\Phi _{\\\\beta ,Y}^{(2)}\\\\right)\\\\langle \\\\lambda \\\\rangle ^M,$ for all $X\\\\in {U}_n$ , $Y\\\\in {V}_n$ and $\\\\lambda \\\\in $ , where the $\\\\Phi _{\\\\beta ,Y}^{(j)}$ ($j=1,2$ ) are given by (REF ) and (REF ).\",\n \"Hence by (REF ), for all $\\\\lambda \\\\in $ , $\\\\left|(-\\\\Delta )^{\\\\beta /2}(\\\\phi _\\\\lambda u)\\\\right| \\\\le \\\\sum _{\\\\begin{array}{c}X\\\\in {U}_n,\\\\\\\\Y\\\\in {V}_n\\\\end{array}}|\\\\kappa _{X,Y}|\\\\left|\\\\sqrt{-\\\\Delta }\\\\big (X\\\\phi _{-\\\\lambda }Yu\\\\big )\\\\right| \\\\le \\\\Phi _\\\\beta \\\\langle \\\\lambda \\\\rangle ^M,$ where $\\\\Phi _\\\\beta :=\\\\frac{1}{2\\\\sqrt{\\\\pi }}\\\\sum _{\\\\begin{array}{c}X\\\\in {U}_n,\\\\\\\\Y\\\\in {V}_n\\\\end{array}}|\\\\kappa _{X,Y}|\\\\left(\\\\Phi _{\\\\beta ,Y}^{(1)}+\\\\Phi _{\\\\beta ,Y}^{(2)}\\\\right),$ and $\\\\beta $ is still assumed to be odd.\",\n \"It was shown previously that $\\\\Phi _{\\\\beta ,Y}^{(1)},\\\\Phi _{\\\\beta ,Y}^{(2)}\\\\in L^1(K|G|K)$ , for all $Y\\\\in {V}_n$ , and hence $\\\\Phi _\\\\beta \\\\in L^1(K|G|K)$ .\",\n \"Moreover, by (REF ) and (REF ), $\\\\begin{aligned}\\\\left\\\\Vert \\\\Phi _\\\\beta \\\\right\\\\Vert _1 &\\\\le \\\\kappa _\\\\beta ^{\\\\prime }\\\\sum _{Y\\\\in {V}_n}\\\\Bigg (\\\\Vert \\\\Delta Yu\\\\Vert _1+\\\\sum _{j=1}^d\\\\Vert X_jYu\\\\Vert _1 +\\\\Vert Yu\\\\Vert _1\\\\Bigg ),\\\\end{aligned}$ for some positive constant $\\\\kappa _\\\\beta ^{\\\\prime }$ .\",\n \"In particular, we have verified (REF ) when $\\\\beta $ is odd.\",\n \"Corollary 5.4 Let $q:G\\\\times \\\\rightarrow \\\\mathbb {R}$ be of the form (REF ).\",\n \"Then for $\\\\kappa $ sufficiently large, the conditions of Corollary REF are satisfied.\",\n \"In particular, $-q(\\\\sigma ,D)$ extends to the infinitesimal generator of a strongly continuous sub-Feller semigroup on $C_0(K|G|K)$ .\"\n ],\n [\n \"Appendix: The Friedrich Mollifier $J_\\\\epsilon $\",\n \"Recall the Friedrich mollifier on $\\\\operatorname{\\\\mathfrak {a}}$ from Section , defined for each $0<\\\\epsilon \\\\le 1$ and $H\\\\in \\\\operatorname{\\\\mathfrak {a}}$ by $l_\\\\epsilon (H) := \\\\epsilon ^{-m}l(H/\\\\epsilon ), \\\\hspace{20.0pt} \\\\text{ where } \\\\hspace{20.0pt} l(H) := C_0e^{\\\\frac{1}{|H|^2-1}}\\\\operatorname{\\\\bf 1}_{B_1(0)}(H),$ where $C_0>0$ is a normalising constant, and the associated mappings $j,j_\\\\epsilon \\\\in \\\\mathcal {S}(K|G|K)$ are again given by $\\\\hat{j_\\\\epsilon } = {F}(l_\\\\epsilon ), \\\\hspace{20.0pt} \\\\forall 0<\\\\epsilon \\\\le 1.$ For $0<\\\\epsilon \\\\le 1$ , the operators $J_\\\\epsilon $ were defined $J_\\\\epsilon u = j_\\\\epsilon \\\\ast u \\\\hspace{20.0pt} \\\\forall f\\\\in L^2(K|G|K).$ This appendix will be devoted to proving Proposition REF , which we re-state below.\",\n \"Proposition 6.1 $\\\\hat{j_\\\\epsilon }(\\\\lambda ) = \\\\hat{j}(\\\\epsilon \\\\lambda )$ for all $0<\\\\epsilon \\\\le 1$ and $\\\\lambda \\\\in $ .\",\n \"For all $0<\\\\epsilon \\\\le 1$ , $J_\\\\epsilon $ is a self-adjoint contraction of $L^2(K|G|K)$ .\",\n \"$J_\\\\epsilon u\\\\in H^{\\\\psi ,s}$ for all $s\\\\ge 0$ , $u\\\\in L^2(K|G|K)$ and $0<\\\\epsilon \\\\le 1$ , and if $u\\\\in H^{\\\\psi ,s}$ , then $\\\\Vert J_\\\\epsilon u\\\\Vert _{\\\\psi ,s} \\\\le \\\\Vert u\\\\Vert _{\\\\psi ,s}.$ $\\\\Vert J_\\\\epsilon u - u\\\\Vert _{\\\\psi ,s}\\\\rightarrow 0$ as $\\\\epsilon \\\\rightarrow 0$ .\",\n \"Let $0<\\\\epsilon \\\\le 1$ and $\\\\lambda \\\\in $ .\",\n \"Using a change of variable $H\\\\mapsto \\\\epsilon ^{-1}H$ , $\\\\hat{j_\\\\epsilon }(\\\\lambda ) = {F}(l_\\\\epsilon )(\\\\lambda ) &= \\\\int _{\\\\operatorname{\\\\mathfrak {a}}}e^{i\\\\lambda (H)}\\\\epsilon ^{-m}l(\\\\epsilon ^{-1}H)dH \\\\\\\\&= \\\\int _{\\\\operatorname{\\\\mathfrak {a}}}e^{i\\\\epsilon \\\\lambda (H)}l(H)dH = {F}(l)(\\\\epsilon \\\\lambda ) = \\\\hat{j}(\\\\epsilon \\\\lambda ).$ The map $l$ is symmetric under $H\\\\mapsto -H$ , and hence ${F}(l)=\\\\hat{j}$ is real-valued.\",\n \"Therefore, given $u,v\\\\in L^2(K|G|K)$ and $0<\\\\epsilon \\\\le 1$ , $\\\\langle J_\\\\epsilon u,v\\\\rangle = \\\\int _{}\\\\hat{j}(\\\\epsilon \\\\lambda )\\\\hat{u}(\\\\lambda )\\\\overline{\\\\hat{v}(\\\\lambda )}\\\\omega (d\\\\lambda ) = \\\\int _{}\\\\hat{u}(\\\\lambda )\\\\overline{\\\\hat{j}(\\\\epsilon \\\\lambda )\\\\hat{v}(\\\\lambda )}\\\\omega (d\\\\lambda ) = \\\\langle u, J_\\\\epsilon v\\\\rangle .$ To see that $J_\\\\epsilon $ is a contraction, note that $|\\\\hat{j_\\\\epsilon }(\\\\lambda )|=|\\\\hat{j}(\\\\epsilon \\\\lambda )|\\\\le \\\\hat{j}(0)= 1$ for all $\\\\lambda \\\\in $ , and so by Plancherel's identity $\\\\Vert J_\\\\epsilon u\\\\Vert = \\\\Vert \\\\hat{j}_\\\\epsilon \\\\hat{u}\\\\Vert _{L^2(,\\\\omega )} \\\\le \\\\Vert \\\\hat{u}\\\\Vert _{L^2(K|G|K)} = \\\\Vert u \\\\Vert ,$ for all $u\\\\in L^2(K|G|K)$ and all $0<\\\\epsilon \\\\le 1$ .\",\n \"Let $s\\\\ge 0$ , $0<\\\\epsilon \\\\le 1$ .\",\n \"By Theorem REF , $\\\\hat{j}\\\\in \\\\mathcal {S}()^W$ , and hence there is $\\\\kappa >0$ such that $\\\\langle \\\\lambda \\\\rangle ^s\\\\left|\\\\hat{j}(\\\\epsilon \\\\lambda )\\\\right| \\\\le \\\\kappa , \\\\hspace{20.0pt} \\\\forall \\\\lambda \\\\in .$ Then, using Proposition REF (REF ), $\\\\Psi (\\\\lambda )^s\\\\left|\\\\hat{j}(\\\\epsilon \\\\lambda )\\\\right| \\\\le c_\\\\psi ^{s/2}\\\\langle \\\\lambda \\\\rangle ^s\\\\left|\\\\hat{j}(\\\\epsilon \\\\lambda )\\\\right|\\\\le c_\\\\psi ^{s/2}\\\\kappa ,$ for all $\\\\lambda \\\\in $ .\",\n \"Let $u\\\\in L^2(K|G|K)$ .\",\n \"By Plancherel's identity, $\\\\int _{}\\\\Psi (\\\\lambda )^{2s}\\\\left|\\\\hat{j}(\\\\epsilon \\\\lambda )\\\\right|^2|\\\\hat{u}(\\\\lambda )|^2\\\\omega (d\\\\lambda ) \\\\le c_\\\\psi ^s\\\\kappa ^2\\\\Vert u \\\\Vert ^2 < \\\\infty .$ By Proposition REF (REF ), $(J_\\\\epsilon u)^\\\\wedge (\\\\lambda ) = \\\\hat{j}(\\\\epsilon \\\\lambda )\\\\hat{u}(\\\\lambda )$ , for all $\\\\lambda \\\\in $ , and hence $\\\\int _{}\\\\Psi (\\\\lambda )^{2s}|(J_\\\\epsilon u)^\\\\wedge (\\\\lambda )|^2\\\\omega (d\\\\lambda )<\\\\infty .$ That is, $J_\\\\epsilon u\\\\in H^{\\\\psi ,s}$ .\",\n \"Next, suppose $u\\\\in H^{\\\\psi ,s}$ .\",\n \"Then, since $|\\\\hat{j_\\\\epsilon }|\\\\le 1$ , $\\\\Vert J_\\\\epsilon u\\\\Vert _{\\\\psi ,s} = \\\\Vert \\\\Psi ^s\\\\hat{j_\\\\epsilon }\\\\hat{u}\\\\Vert _{L^2(,\\\\omega )} \\\\le \\\\Vert \\\\Psi ^s\\\\hat{u}\\\\Vert _{L^2(,\\\\omega )} = \\\\Vert u\\\\Vert _{\\\\psi ,s},$ as desired.\",\n \"By Theorem 1 on page 250 of Evans [15], for all $v\\\\in \\\\mathcal {S}(\\\\operatorname{\\\\mathfrak {a}})^W$ , $l_\\\\epsilon \\\\ast v\\\\rightarrow v$ as $\\\\epsilon \\\\rightarrow 0$ , in the classical Sobolev space $W^s()$ , for all $s\\\\ge 0$ .\",\n \"Therefore, for all $s\\\\ge 0$ and $v\\\\in \\\\mathcal {S}(\\\\operatorname{\\\\mathfrak {a}})^W$ , $\\\\lim _{\\\\epsilon \\\\rightarrow 0}\\\\int _{}(1+|\\\\lambda |^2)^s|{F}(l_\\\\epsilon \\\\ast v-v)(\\\\lambda )|^2d\\\\lambda = 0,$ Let $u\\\\in C_c^\\\\infty (K|G|K)$ .\",\n \"Then by Theorem REF , ${F}^{-1}(\\\\hat{u})\\\\in \\\\mathcal {S}(\\\\operatorname{\\\\mathfrak {a}})^W$ .\",\n \"Observe that if $v={F}^{-1}(\\\\hat{u})$ , then ${F}(l_\\\\epsilon \\\\ast v - v) = (\\\\hat{j}_\\\\epsilon -1)\\\\hat{u} = (J_\\\\epsilon u - u)^\\\\wedge ,$ and so by (REF ), $\\\\lim _{\\\\epsilon \\\\rightarrow 0}\\\\int _{}(1+|\\\\lambda |^2)^s|(J_\\\\epsilon u - u)^\\\\wedge (\\\\lambda )|^2d\\\\lambda = 0,$ for all $s\\\\ge 0$ .\",\n \"By (REF ), $\\\\int _{}(1+|\\\\lambda |^2)^s|(J_\\\\epsilon u - u)^\\\\wedge (\\\\lambda )|^2\\\\omega (d\\\\lambda )&\\\\le \\\\int _{}(1+|\\\\lambda |^2)^s|(J_\\\\epsilon u - u)^\\\\wedge (\\\\lambda )|^2(C_1+C_2|\\\\lambda |^p)^2d\\\\lambda \\\\\\\\&\\\\le \\\\kappa \\\\int _{}(1+|\\\\lambda |^2)^{s+p}|(J_\\\\epsilon u - u)^\\\\wedge (\\\\lambda )|^2d\\\\lambda ,$ for some constant $\\\\kappa >0$ , and where $p=\\\\frac{\\\\dim N}{2}$ .\",\n \"Hence $\\\\lim _{\\\\epsilon \\\\rightarrow 0}\\\\int _{}(1+|\\\\lambda |^2)^s|(J_\\\\epsilon u - u)^\\\\wedge (\\\\lambda )|^2\\\\omega (d\\\\lambda ) = 0$ By Proposition REF (REF ), $\\\\Vert J_\\\\epsilon u-u\\\\Vert _{\\\\psi ,s}^2 &= \\\\int _{}(1+\\\\psi (\\\\lambda ))^s|(J_\\\\epsilon u - u)^\\\\wedge (\\\\lambda )|^2\\\\omega (d\\\\lambda ) \\\\\\\\&\\\\le c_\\\\psi \\\\int _{}(1+|\\\\lambda |^2)^s|(J_\\\\epsilon u - u)^\\\\wedge (\\\\lambda )|^2\\\\omega (d\\\\lambda ) \\\\rightarrow 0$ as $\\\\epsilon \\\\rightarrow 0$ .\",\n \"Since $C_c^\\\\infty (K|G|K)$ is dense in $H^{\\\\psi ,s}$ , Proposition REF (REF ) follows.\",\n \"Acknowledgement Many thanks to David Applebaum for his advice and support with writing this paper.\",\n \"Thanks also to the University of Sheffield's School of Mathematics and Statistics, and to the EPSRC for providing PhD funding while this research was carried out.\"\n ]\n]"},"id":{"kind":"string","value":"2107.01817"}}},{"rowIdx":1582,"cells":{"text":{"kind":"list like","value":[["End-to-end Neural Coreference Resolution Revisited: A Simple yet\n Effective Baseline"],["Abstract Since the first end-to-end neural coreference resolution model was introduced, many extensions to the model have been proposed, ranging from using higher-order inference to directly optimizing evaluation metrics using reinforcement learning.","Despite improving the coreference resolution performance by a large margin, these extensions add substantial extra complexity to the original model.","Motivated by this observation and the recent advances in pre-trained Transformer language models, we propose a simple yet effective baseline for coreference resolution.","Even though our model is a simplified version of the original neural coreference resolution model, it achieves impressive performance, outperforming all recent extended works on the public English OntoNotes benchmark.","Our work provides evidence for the necessity of carefully justifying the complexity of existing or newly proposed models, as introducing a conceptual or practical simplification to an existing model can still yield competitive results."],["Introduction","Coreference resolution is the task of clustering mentions in text that refer to the same real-world entities [1] (Figure REF ).","As a fundamental task of natural language processing, coreference resolution can be an essential component for many downstream applications [2], [3].","Many traditional coreference resolution systems are pipelined systems, each consists of two separate components: (1) a mention detector for identifying entity mentions from text (2) a coreference resolver for clustering the extracted mentions [4], [5], [6], [7], [8].","These models typically rely heavily on syntatic parsers and use highly engineered mention proposal algorithms.","Figure: An example of coreference resolution.","There are two coreference chains in this example.Figure: An overview of coreference resolution research in the last decade.","Pipelined systems were heavily used before the introduction of e2e-coref.","Since 2017, various extensions to the model have been proposed.In 2017, the first end-to-end coreference resolution model named e2e-coref was proposed [9].","It outperforms previous pipelined systems without using any syntactic parser or complicated hand-engineered features.","Since then, many extensions to the e2e-coref model have been introduced, ranging from using higher-order inference to directly optimizing evaluation metrics using reinforcement learning [10], [11], [12], [13], [14], [15], [16], [17] (Figure REF ).","Despite improving the coreference resolution performance by a large margin, these extensions add a lot of extra complexity to the original model.","Motivated by this observation and the recent advances in pre-trained Transformer language models, we propose a simple yet effective baseline for coreference resolution.","We introduce simplifications to the original e2e-coref model, creating a conceptually simpler model for coreference resolution.","Despite its simplicity, our model achieves promising performance, outperforming all aforementioned methods on the public English OntoNotes benchmark.","Our work provides evidence for the necessity of carefully justifying the complexity of existing or newly proposed models, as introducing a conceptual or practical simplification to an existing model can still yield competitive results.","The findings of our work agree with the results of several recent work [18], [19].","For example, in [19], the authors also introduced a minimalist approach that performs on par with more complicated models.","In Section , we first describe our baseline model for coreference resolution.","After that, we describe the conducted experiments and their results in Section .","Finally, we conclude this work and discuss future work in Section"],["Method","At a high level, our coreference resolution model is similar to the e2e-coref model (Figure REF ).","Given a sequence of tokens from an input document, the model first forms a contextualized representation for each token using a Transformer-based encoder.","After that, all the spans (up to a certain length) in the document are enumerated.","The model then assigns a score to each candidate span indicating whether the span is an entity mention.","A portion of top-scoring spans is extracted and fed to the next stage where the model predicts distributions over possible antecedents for each extracted span.","The final coreference clusters can be naturally constructed from the antecedent predictions.","In the following subsections, we go into more specific details.","Figure: An example illustrating the strategy of concatenating the speaker’s name with the corresponding utterance (assuming the model utilizes WordPiece for tokenization)."],["Notations and Preliminaries","Given an input document $D = (t_1, t_2, ..., t_n)$ consisting of $n$ tokens, the total number of possible text spans is $N = n(n+1)/2$ .","For each span $i$ , we denote the start and end indices of the span by $\\texttt {START}(i)$ and $\\texttt {END}(i)$ respectively.","We also assume an ordering of the spans based on $\\texttt {START}(i)$ ; spans with the same start index are ordered by $\\texttt {END}(i)$ .","Furthermore, we only consider spans that are entirely within a sentence and limit spans to a max length of $L$ .","Since the speaker information is known to contain useful information for coreference resolution, it has been extensively used in previous works [5], [9], [11], [15], [17].","For example, the original e2e-coref model converts speaker information into binary features indicating whether two candidate mentions are from the same speaker.","In this work, we employ a more intuitive strategy that directly concatenates the speaker’s name with the corresponding utterance [20].","This straightforward strategy is simple to implement and has been shown to be more effective than the feature-based method [20].","Figure REF illustrates the concatenation strategy."],["Encoder Layer","Given the input document $D = (t_1, t_2, ..., t_n)$ , the model simply forms a contextualized representation for each input token, using a Transformer-based encoder such as BERT [21] or SpanBERT [17].","These pretrained language models typically can only run on sequences with at most 512 tokens.","Therefore, to encode a long document (i.e, $n > 512$ ), we split the document into overlapping segments by creating a $n$ -sized segment after every $n/2$ tokens.","These segments are then passed on to the Transformer-based encoder independently.","The final token representations are derived by taking the token representations with maximum context.","Let $\\textbf {X} = (\\textbf {x}_1, \\textbf {x}_2, ..., \\textbf {x}_n)$ be the output of the Transformer encoder.","Note that the e2e-coref model uses the GloVe and Turian embeddings [22], [23] and character embeddings produced by 1-dimensional convolution neural networks.","From an implementation point of view, it is easier to use a Transformer encoder than combining these traditional embeddings.","For example, the Transformers library https://github.com/huggingface/transformers allows users to experiment with various state-of-the-art Transformer-based models by simply writing few lines of code.","Now, for each span $i$ , its span representation $\\textbf {g}_i$ is defined as: $\\textbf {g}_i = \\big [\\textbf {x}_{\\texttt {START}(i)}, \\textbf {x}_{\\texttt {END}(i)}, \\hat{\\textbf {x}}_{i} \\big ]$ where $\\textbf {x}_{\\texttt {START}(i)}$ and $\\textbf {x}_{\\texttt {END}(i)}$ are the boundary representations, consisting of the first and the last token representations of the span $i$ .","And $\\hat{\\textbf {x}}_{i}$ is computed using an attention mechanism [24] as follows: $\\begin{split}\\alpha _t & = \\text{FFNN}_{\\alpha }(\\textbf {x}_t) \\\\\\beta _{i,t} & = \\frac{\\exp {(\\alpha _t)}}{\\sum \\limits _{j=\\texttt {START}(i)}^{\\texttt {END}(i)} \\exp {(\\alpha _j)}} \\\\\\hat{\\textbf {x}}_{i} &= \\sum \\limits _{j=\\texttt {START}(i)}^{\\texttt {END}(i)} \\beta _{i,j} \\,\\textbf {x}_j\\end{split}$ where $\\text{FFNN}_{\\alpha }$ is a multi-layer feedforward neural network that maps each token-level representation $\\textbf {x}_t$ into an unnormalized attention score.","$\\hat{\\textbf {x}}_{i}$ is a weighted sum of token vectors in the span $i$ .","Our span representation generation process closely follows that of e2e-coref.","However, a simplification we make is that we do not include any additional features such as the size of span $i$ in its representation $\\textbf {g}_i$ ."],["Mentions Extractor Layer","In this layer, we first enumerate all the spans (up to a certain length $L$ ) in the document.","For each span $i$ , we simply use a feedforward neural network $\\text{FFNN}_\\text{m}$ to compute its mention score: $s_m(i) = \\text{FFNN}_\\text{m}(\\textbf {g}_i)$ After this step, we only keep up to $\\lambda n$ spans with the highest mention scores.","In previous works, to maintain a high recall of gold mentions, $\\lambda $ is typically set to be $0.4$ [9], [11].","These works do not directly train the mention extractor.","The mention extractor and the mention linker are jointly trained to only maximize the marginal likelihood of gold antecedent spans.","In coreference resolution datasets such as the OntoNotes benchmark [25], singleton mentions are not explicitly labeled, because the annotations contain only mentions that belong to a coreference chain.","However, these annotations of non-singleton mentions can still provide useful signals for training an efficient mention extractor [10].","Thus, we also propose to pre-train our mention extractor using these annotations.","In Section , we will empirically demonstrate that this pre-training step greatly improves the performance of our mention extractor layer.","As a result, we only need set the parameter $\\lambda $ to be 0.25 in order to maintain a high recall of gold mentions.","To this end, the pretraining loss is calculated as follows: $\\begin{split}\\mathcal {L}_{\\text{detect}}(i) &= y_i \\log {\\hat{y}_i} + (1-y_i) \\log {(1-\\hat{y}_i)} \\\\\\mathcal {L}_{\\text{detect}} &= - \\sum _{i\\,\\in \\,\\text{S}} \\mathcal {L}_{\\text{detect}}(i)\\end{split}$ where $\\hat{y}_i = \\text{sigmoid}(s_m(i))$ , and $y_i = 1$ if and only if the span $i$ is a mention in one of the coreference chains.","$S$ is the set of the top scoring spans (and so $|S| \\le \\lambda n$ )."],["Mentions Linker Layer","For each span $i$ extracted by the mention extractor, the mention linker needs to assign an antecedent $a_i$ from all preceding spans or a dummy antecedent $\\epsilon $ : $a_i \\in Y(i) = \\lbrace \\epsilon , 1, \\dots , i-1\\rbrace $ (the ordering of spans was discussed in Subsection REF ).","The dummy antecedent $\\epsilon $ represents two possible cases.","One case is the span itself is not an entity mention.","The other case is the span is an entity mention but it is not coreferent with any previous span extracted by the mention extractor.","The coreference score $s(i,j)$ of two spans $i$ and $j$ is computed as follows: $\\begin{split}s_a(i, j) &= \\text{FFNN}_\\text{a}\\big (\\big [\\textbf {g}_i, \\textbf {g}_j, \\textbf {g}_i \\circ \\textbf {g}_j \\big ]\\big ) \\\\s(i, j) &= s_m(i) + s_m(j) + s_a(i,j)\\end{split}$ where $\\text{FFNN}_\\text{a}$ is a feedforward network.","$s_m(i)$ and $s_m(j)$ are calculated using Equation REF .","The score $s(i,j)$ is affected by three factors: (1) $s_m(i)$ , whether span $i$ is a mention, (2) $s_m(j)$ , whether span $j$ is a mention, and (3) $s_a(i,j)$ whether $j$ is an antecedent of $i$ .","In the special case of the dummy antecedent, $s(i, \\epsilon )$ is fixed to 0.","In the e2e-coref model, when computing $s_a(i,j)$ , a vector encoding additional features such as genre information and the distance between the two spans is also used.","We do not use such feature vector when computing $s_a(i,j)$ to simplify the implementation.","We want to maximize the marginal log-likelihood of all antecedents in the correct coreference chain for each mention: $\\log {\\prod _{i \\in S}{\\sum _{\\hat{y}\\in Y(i) \\cap \\text{GOLD}(i)} P(\\hat{y})}}$ where $S$ is the set of the top scoring spans extracted by the mention extractor (i.e., the set of unpruned spans).","$\\text{GOLD}(i)$ is the set of spans in the gold cluster containing span $i$ .","If span $i$ does not belong to any coreference chain or all gold antecedents have been pruned, then $\\text{GOLD}(i) = \\lbrace \\epsilon \\rbrace $ .","Table: Performance on the OntoNotes coreference resolution benchmark.Table: Proportion of gold mentions covered in the development data by different mention extractors.To summarize, we first pre-train the mention extractor to minimize the loss function defined in Equation REF .","After that, we jointly train the mention extractor and the mention linker to optimize the objective function defined in Equation REF in an end-to-end manner."],["Experiments and Results","Dataset and Experiments Setup To evaluate the effectiveness of the proposed approach, we use the CoNLL-2012 Shared Task English data [25] which is based on the OntoNotes corpus.","This dataset has 2802/343/348 documents for the train/dev/test split.","Similar to previous works, we report precision, recall, and F1 of the MUC, $\\text{B}^3$ , and $\\text{CEAF}_{\\phi _4}$ metrics, and also average the F1 score of all three metrics.","We used SpanBERT (spanbert-large-cased) [17] as the encoder.","Two different learning rates are used, one for the lower pretrained SpanBERT encoder (5e-05) and one for the upper layers (1e-4).","We also use learning rate decay.","The number of training epochs is set to be 100.","The batch size is set to be 32.","We did hyper-parameter tuning using the provided dev set.","To train our model, we use two 16GB V100 GPUs and use techniques such as gradient checkpointing and gradient accumulation to avoid running out of GPUs' memory.","Comparison with Previous Methods Table REF compares our model with several state-of-the-art coreference resolution systems.","Overall, our model outperforms the original e2e-coref model and also all recent extended works.","For example, compared to the variant [c2f-coref + SpanBERT-large], our model achieves higher F1-scores for the MUC and $\\text{B}^3$ metrics.","Even though our model achieves a slightly lower F1-score for the $\\text{CEAF}_{\\phi _4}$ metric, the overall averaged F1 score of our model is still better.","It is worth mentioning that the variant [c2f-coref + SpanBERT-large] is more complex than our method, because it has some other additional components such as coarse-to-fine antecedent pruning and higher-order inference [11], [17].","Recently, a model named CorefQA has been proposed [20], and it achieves an averaged F1 score of $83.1$ on the English OntoNotes benchmark.","The work takes a complete departure from the paradigm used by the e2e-coref model, and instead, proposes to formulate the coreference resolution problem as a span prediction task, like in question answering.","Despite its impressive performance, the CorefQA model is very computationally expensive.","In order to predict coreference clusters for a single document, CorefQA needs to run a Transformer-based model on the same document many times (each time a different query is appended to the document).","Analysis on the Performance of the Mention Extractor As mentioned in Subsection REF , in our work, the value of the parameter $\\lambda $ for pruning is set to be 0.25.","On the other hand, it is set to be 0.4 in the e2e-coref model.","Table REF shows the comparison in more details.","Our mention extractor is able to extract 95.7% of all the gold mentions in the dev set, while the mention extractor of the e2e-coref model is only able to extract 92.7% of them.","Furthermore, by proposing less candidate spans, the workload of our mention linker is also reduced."],["Conclusions","In this work, we propose a simple yet effective baseline for the task of coreference resolution.","Despite its simplicity, our model still can achieve impressive performance, outperforming all recent extended works on the popular English OntoNotes benchmark.","In future work, we are interested in reducing the computational complexity of our baseline model using compression techniques [26], [27], [28].","We also plan to extend our work to address the task of event coreference resolution [29], [30]."]],"string":"[\n [\n \"End-to-end Neural Coreference Resolution Revisited: A Simple yet\\n Effective Baseline\"\n ],\n [\n \"Abstract Since the first end-to-end neural coreference resolution model was introduced, many extensions to the model have been proposed, ranging from using higher-order inference to directly optimizing evaluation metrics using reinforcement learning.\",\n \"Despite improving the coreference resolution performance by a large margin, these extensions add substantial extra complexity to the original model.\",\n \"Motivated by this observation and the recent advances in pre-trained Transformer language models, we propose a simple yet effective baseline for coreference resolution.\",\n \"Even though our model is a simplified version of the original neural coreference resolution model, it achieves impressive performance, outperforming all recent extended works on the public English OntoNotes benchmark.\",\n \"Our work provides evidence for the necessity of carefully justifying the complexity of existing or newly proposed models, as introducing a conceptual or practical simplification to an existing model can still yield competitive results.\"\n ],\n [\n \"Introduction\",\n \"Coreference resolution is the task of clustering mentions in text that refer to the same real-world entities [1] (Figure REF ).\",\n \"As a fundamental task of natural language processing, coreference resolution can be an essential component for many downstream applications [2], [3].\",\n \"Many traditional coreference resolution systems are pipelined systems, each consists of two separate components: (1) a mention detector for identifying entity mentions from text (2) a coreference resolver for clustering the extracted mentions [4], [5], [6], [7], [8].\",\n \"These models typically rely heavily on syntatic parsers and use highly engineered mention proposal algorithms.\",\n \"Figure: An example of coreference resolution.\",\n \"There are two coreference chains in this example.Figure: An overview of coreference resolution research in the last decade.\",\n \"Pipelined systems were heavily used before the introduction of e2e-coref.\",\n \"Since 2017, various extensions to the model have been proposed.In 2017, the first end-to-end coreference resolution model named e2e-coref was proposed [9].\",\n \"It outperforms previous pipelined systems without using any syntactic parser or complicated hand-engineered features.\",\n \"Since then, many extensions to the e2e-coref model have been introduced, ranging from using higher-order inference to directly optimizing evaluation metrics using reinforcement learning [10], [11], [12], [13], [14], [15], [16], [17] (Figure REF ).\",\n \"Despite improving the coreference resolution performance by a large margin, these extensions add a lot of extra complexity to the original model.\",\n \"Motivated by this observation and the recent advances in pre-trained Transformer language models, we propose a simple yet effective baseline for coreference resolution.\",\n \"We introduce simplifications to the original e2e-coref model, creating a conceptually simpler model for coreference resolution.\",\n \"Despite its simplicity, our model achieves promising performance, outperforming all aforementioned methods on the public English OntoNotes benchmark.\",\n \"Our work provides evidence for the necessity of carefully justifying the complexity of existing or newly proposed models, as introducing a conceptual or practical simplification to an existing model can still yield competitive results.\",\n \"The findings of our work agree with the results of several recent work [18], [19].\",\n \"For example, in [19], the authors also introduced a minimalist approach that performs on par with more complicated models.\",\n \"In Section , we first describe our baseline model for coreference resolution.\",\n \"After that, we describe the conducted experiments and their results in Section .\",\n \"Finally, we conclude this work and discuss future work in Section\"\n ],\n [\n \"Method\",\n \"At a high level, our coreference resolution model is similar to the e2e-coref model (Figure REF ).\",\n \"Given a sequence of tokens from an input document, the model first forms a contextualized representation for each token using a Transformer-based encoder.\",\n \"After that, all the spans (up to a certain length) in the document are enumerated.\",\n \"The model then assigns a score to each candidate span indicating whether the span is an entity mention.\",\n \"A portion of top-scoring spans is extracted and fed to the next stage where the model predicts distributions over possible antecedents for each extracted span.\",\n \"The final coreference clusters can be naturally constructed from the antecedent predictions.\",\n \"In the following subsections, we go into more specific details.\",\n \"Figure: An example illustrating the strategy of concatenating the speaker’s name with the corresponding utterance (assuming the model utilizes WordPiece for tokenization).\"\n ],\n [\n \"Notations and Preliminaries\",\n \"Given an input document $D = (t_1, t_2, ..., t_n)$ consisting of $n$ tokens, the total number of possible text spans is $N = n(n+1)/2$ .\",\n \"For each span $i$ , we denote the start and end indices of the span by $\\\\texttt {START}(i)$ and $\\\\texttt {END}(i)$ respectively.\",\n \"We also assume an ordering of the spans based on $\\\\texttt {START}(i)$ ; spans with the same start index are ordered by $\\\\texttt {END}(i)$ .\",\n \"Furthermore, we only consider spans that are entirely within a sentence and limit spans to a max length of $L$ .\",\n \"Since the speaker information is known to contain useful information for coreference resolution, it has been extensively used in previous works [5], [9], [11], [15], [17].\",\n \"For example, the original e2e-coref model converts speaker information into binary features indicating whether two candidate mentions are from the same speaker.\",\n \"In this work, we employ a more intuitive strategy that directly concatenates the speaker’s name with the corresponding utterance [20].\",\n \"This straightforward strategy is simple to implement and has been shown to be more effective than the feature-based method [20].\",\n \"Figure REF illustrates the concatenation strategy.\"\n ],\n [\n \"Encoder Layer\",\n \"Given the input document $D = (t_1, t_2, ..., t_n)$ , the model simply forms a contextualized representation for each input token, using a Transformer-based encoder such as BERT [21] or SpanBERT [17].\",\n \"These pretrained language models typically can only run on sequences with at most 512 tokens.\",\n \"Therefore, to encode a long document (i.e, $n > 512$ ), we split the document into overlapping segments by creating a $n$ -sized segment after every $n/2$ tokens.\",\n \"These segments are then passed on to the Transformer-based encoder independently.\",\n \"The final token representations are derived by taking the token representations with maximum context.\",\n \"Let $\\\\textbf {X} = (\\\\textbf {x}_1, \\\\textbf {x}_2, ..., \\\\textbf {x}_n)$ be the output of the Transformer encoder.\",\n \"Note that the e2e-coref model uses the GloVe and Turian embeddings [22], [23] and character embeddings produced by 1-dimensional convolution neural networks.\",\n \"From an implementation point of view, it is easier to use a Transformer encoder than combining these traditional embeddings.\",\n \"For example, the Transformers library https://github.com/huggingface/transformers allows users to experiment with various state-of-the-art Transformer-based models by simply writing few lines of code.\",\n \"Now, for each span $i$ , its span representation $\\\\textbf {g}_i$ is defined as: $\\\\textbf {g}_i = \\\\big [\\\\textbf {x}_{\\\\texttt {START}(i)}, \\\\textbf {x}_{\\\\texttt {END}(i)}, \\\\hat{\\\\textbf {x}}_{i} \\\\big ]$ where $\\\\textbf {x}_{\\\\texttt {START}(i)}$ and $\\\\textbf {x}_{\\\\texttt {END}(i)}$ are the boundary representations, consisting of the first and the last token representations of the span $i$ .\",\n \"And $\\\\hat{\\\\textbf {x}}_{i}$ is computed using an attention mechanism [24] as follows: $\\\\begin{split}\\\\alpha _t & = \\\\text{FFNN}_{\\\\alpha }(\\\\textbf {x}_t) \\\\\\\\\\\\beta _{i,t} & = \\\\frac{\\\\exp {(\\\\alpha _t)}}{\\\\sum \\\\limits _{j=\\\\texttt {START}(i)}^{\\\\texttt {END}(i)} \\\\exp {(\\\\alpha _j)}} \\\\\\\\\\\\hat{\\\\textbf {x}}_{i} &= \\\\sum \\\\limits _{j=\\\\texttt {START}(i)}^{\\\\texttt {END}(i)} \\\\beta _{i,j} \\\\,\\\\textbf {x}_j\\\\end{split}$ where $\\\\text{FFNN}_{\\\\alpha }$ is a multi-layer feedforward neural network that maps each token-level representation $\\\\textbf {x}_t$ into an unnormalized attention score.\",\n \"$\\\\hat{\\\\textbf {x}}_{i}$ is a weighted sum of token vectors in the span $i$ .\",\n \"Our span representation generation process closely follows that of e2e-coref.\",\n \"However, a simplification we make is that we do not include any additional features such as the size of span $i$ in its representation $\\\\textbf {g}_i$ .\"\n ],\n [\n \"Mentions Extractor Layer\",\n \"In this layer, we first enumerate all the spans (up to a certain length $L$ ) in the document.\",\n \"For each span $i$ , we simply use a feedforward neural network $\\\\text{FFNN}_\\\\text{m}$ to compute its mention score: $s_m(i) = \\\\text{FFNN}_\\\\text{m}(\\\\textbf {g}_i)$ After this step, we only keep up to $\\\\lambda n$ spans with the highest mention scores.\",\n \"In previous works, to maintain a high recall of gold mentions, $\\\\lambda $ is typically set to be $0.4$ [9], [11].\",\n \"These works do not directly train the mention extractor.\",\n \"The mention extractor and the mention linker are jointly trained to only maximize the marginal likelihood of gold antecedent spans.\",\n \"In coreference resolution datasets such as the OntoNotes benchmark [25], singleton mentions are not explicitly labeled, because the annotations contain only mentions that belong to a coreference chain.\",\n \"However, these annotations of non-singleton mentions can still provide useful signals for training an efficient mention extractor [10].\",\n \"Thus, we also propose to pre-train our mention extractor using these annotations.\",\n \"In Section , we will empirically demonstrate that this pre-training step greatly improves the performance of our mention extractor layer.\",\n \"As a result, we only need set the parameter $\\\\lambda $ to be 0.25 in order to maintain a high recall of gold mentions.\",\n \"To this end, the pretraining loss is calculated as follows: $\\\\begin{split}\\\\mathcal {L}_{\\\\text{detect}}(i) &= y_i \\\\log {\\\\hat{y}_i} + (1-y_i) \\\\log {(1-\\\\hat{y}_i)} \\\\\\\\\\\\mathcal {L}_{\\\\text{detect}} &= - \\\\sum _{i\\\\,\\\\in \\\\,\\\\text{S}} \\\\mathcal {L}_{\\\\text{detect}}(i)\\\\end{split}$ where $\\\\hat{y}_i = \\\\text{sigmoid}(s_m(i))$ , and $y_i = 1$ if and only if the span $i$ is a mention in one of the coreference chains.\",\n \"$S$ is the set of the top scoring spans (and so $|S| \\\\le \\\\lambda n$ ).\"\n ],\n [\n \"Mentions Linker Layer\",\n \"For each span $i$ extracted by the mention extractor, the mention linker needs to assign an antecedent $a_i$ from all preceding spans or a dummy antecedent $\\\\epsilon $ : $a_i \\\\in Y(i) = \\\\lbrace \\\\epsilon , 1, \\\\dots , i-1\\\\rbrace $ (the ordering of spans was discussed in Subsection REF ).\",\n \"The dummy antecedent $\\\\epsilon $ represents two possible cases.\",\n \"One case is the span itself is not an entity mention.\",\n \"The other case is the span is an entity mention but it is not coreferent with any previous span extracted by the mention extractor.\",\n \"The coreference score $s(i,j)$ of two spans $i$ and $j$ is computed as follows: $\\\\begin{split}s_a(i, j) &= \\\\text{FFNN}_\\\\text{a}\\\\big (\\\\big [\\\\textbf {g}_i, \\\\textbf {g}_j, \\\\textbf {g}_i \\\\circ \\\\textbf {g}_j \\\\big ]\\\\big ) \\\\\\\\s(i, j) &= s_m(i) + s_m(j) + s_a(i,j)\\\\end{split}$ where $\\\\text{FFNN}_\\\\text{a}$ is a feedforward network.\",\n \"$s_m(i)$ and $s_m(j)$ are calculated using Equation REF .\",\n \"The score $s(i,j)$ is affected by three factors: (1) $s_m(i)$ , whether span $i$ is a mention, (2) $s_m(j)$ , whether span $j$ is a mention, and (3) $s_a(i,j)$ whether $j$ is an antecedent of $i$ .\",\n \"In the special case of the dummy antecedent, $s(i, \\\\epsilon )$ is fixed to 0.\",\n \"In the e2e-coref model, when computing $s_a(i,j)$ , a vector encoding additional features such as genre information and the distance between the two spans is also used.\",\n \"We do not use such feature vector when computing $s_a(i,j)$ to simplify the implementation.\",\n \"We want to maximize the marginal log-likelihood of all antecedents in the correct coreference chain for each mention: $\\\\log {\\\\prod _{i \\\\in S}{\\\\sum _{\\\\hat{y}\\\\in Y(i) \\\\cap \\\\text{GOLD}(i)} P(\\\\hat{y})}}$ where $S$ is the set of the top scoring spans extracted by the mention extractor (i.e., the set of unpruned spans).\",\n \"$\\\\text{GOLD}(i)$ is the set of spans in the gold cluster containing span $i$ .\",\n \"If span $i$ does not belong to any coreference chain or all gold antecedents have been pruned, then $\\\\text{GOLD}(i) = \\\\lbrace \\\\epsilon \\\\rbrace $ .\",\n \"Table: Performance on the OntoNotes coreference resolution benchmark.Table: Proportion of gold mentions covered in the development data by different mention extractors.To summarize, we first pre-train the mention extractor to minimize the loss function defined in Equation REF .\",\n \"After that, we jointly train the mention extractor and the mention linker to optimize the objective function defined in Equation REF in an end-to-end manner.\"\n ],\n [\n \"Experiments and Results\",\n \"Dataset and Experiments Setup To evaluate the effectiveness of the proposed approach, we use the CoNLL-2012 Shared Task English data [25] which is based on the OntoNotes corpus.\",\n \"This dataset has 2802/343/348 documents for the train/dev/test split.\",\n \"Similar to previous works, we report precision, recall, and F1 of the MUC, $\\\\text{B}^3$ , and $\\\\text{CEAF}_{\\\\phi _4}$ metrics, and also average the F1 score of all three metrics.\",\n \"We used SpanBERT (spanbert-large-cased) [17] as the encoder.\",\n \"Two different learning rates are used, one for the lower pretrained SpanBERT encoder (5e-05) and one for the upper layers (1e-4).\",\n \"We also use learning rate decay.\",\n \"The number of training epochs is set to be 100.\",\n \"The batch size is set to be 32.\",\n \"We did hyper-parameter tuning using the provided dev set.\",\n \"To train our model, we use two 16GB V100 GPUs and use techniques such as gradient checkpointing and gradient accumulation to avoid running out of GPUs' memory.\",\n \"Comparison with Previous Methods Table REF compares our model with several state-of-the-art coreference resolution systems.\",\n \"Overall, our model outperforms the original e2e-coref model and also all recent extended works.\",\n \"For example, compared to the variant [c2f-coref + SpanBERT-large], our model achieves higher F1-scores for the MUC and $\\\\text{B}^3$ metrics.\",\n \"Even though our model achieves a slightly lower F1-score for the $\\\\text{CEAF}_{\\\\phi _4}$ metric, the overall averaged F1 score of our model is still better.\",\n \"It is worth mentioning that the variant [c2f-coref + SpanBERT-large] is more complex than our method, because it has some other additional components such as coarse-to-fine antecedent pruning and higher-order inference [11], [17].\",\n \"Recently, a model named CorefQA has been proposed [20], and it achieves an averaged F1 score of $83.1$ on the English OntoNotes benchmark.\",\n \"The work takes a complete departure from the paradigm used by the e2e-coref model, and instead, proposes to formulate the coreference resolution problem as a span prediction task, like in question answering.\",\n \"Despite its impressive performance, the CorefQA model is very computationally expensive.\",\n \"In order to predict coreference clusters for a single document, CorefQA needs to run a Transformer-based model on the same document many times (each time a different query is appended to the document).\",\n \"Analysis on the Performance of the Mention Extractor As mentioned in Subsection REF , in our work, the value of the parameter $\\\\lambda $ for pruning is set to be 0.25.\",\n \"On the other hand, it is set to be 0.4 in the e2e-coref model.\",\n \"Table REF shows the comparison in more details.\",\n \"Our mention extractor is able to extract 95.7% of all the gold mentions in the dev set, while the mention extractor of the e2e-coref model is only able to extract 92.7% of them.\",\n \"Furthermore, by proposing less candidate spans, the workload of our mention linker is also reduced.\"\n ],\n [\n \"Conclusions\",\n \"In this work, we propose a simple yet effective baseline for the task of coreference resolution.\",\n \"Despite its simplicity, our model still can achieve impressive performance, outperforming all recent extended works on the popular English OntoNotes benchmark.\",\n \"In future work, we are interested in reducing the computational complexity of our baseline model using compression techniques [26], [27], [28].\",\n \"We also plan to extend our work to address the task of event coreference resolution [29], [30].\"\n ]\n]"},"id":{"kind":"string","value":"2107.01700"}}},{"rowIdx":1583,"cells":{"text":{"kind":"list like","value":[["KAISA: An Adaptive Second-Order Optimizer Framework for Deep Neural\n Networks"],["Abstract Kronecker-factored Approximate Curvature (K-FAC) has recently been shown to converge faster in deep neural network (DNN) training than stochastic gradient descent (SGD); however, K-FAC's larger memory footprint hinders its applicability to large models.","We present KAISA, a K-FAC-enabled, Adaptable, Improved, and ScAlable second-order optimizer framework that adapts the memory footprint, communication, and computation given specific models and hardware to improve performance and increase scalability.","We quantify the tradeoffs between memory and communication cost and evaluate KAISA on large models, including ResNet-50, Mask R-CNN, U-Net, and BERT, on up to 128 NVIDIA A100 GPUs.","Compared to the original optimizers, KAISA converges 18.1-36.3% faster across applications with the same global batch size.","Under a fixed memory budget, KAISA converges 32.5% and 41.6% faster in ResNet-50 and BERT-Large, respectively.","KAISA can balance memory and communication to achieve scaling efficiency equal to or better than the baseline optimizers.","KAISA is open source and available at https://github.com/gpauloski/kfac_pytorch."],["Introduction","Deep neural networks (DNNs) have driven breakthroughs in many research domains, including image classification [22], [24], object detection and segmentation [21], machine translation [53], and language modeling [17].","DNNs are typically trained with stochastic gradient descent (SGD), or variants thereof.","As models and training datasets become larger, training must increasingly be performed in parallel on many CPUs, GPUs, or TPUs [56], [30].","For example, the BERT [17] model with 330 million parameters would take weeks to months to train on a single GPU, and GPT-3 [11] with 175 billion parameters cannot fit in the memory of any commercially available GPU.","While distributed training on many processors can reduce training time, the need to communicate weight updates and other information among processors can limit scalability [39].","Figure: SGD vs. K-FAC training for ResNet-32 with the CIFAR-10 dataset.","K-FAC reduces iterations needed for convergence.Recent theoretical [36], [20], [35] and empirical [8], [42], [50] studies have shown that Kronecker-factored Approximate Curvature (K-FAC) can accelerate training by enabling convergence with fewer iterations than SGD—for example, achieving baseline validation accuracy in 40% fewer epochs than SGD on ResNet-32 with the CIFAR10 dataset (see Figure REF ).","Technically, researchers use K-FAC to approximate the inverse of the Fisher Information Matrix (FIM)—an approximation of the Hessian—and precondition the gradients before parameter updates [36].","The K-FAC approximation is 1) compute intensive, due to the required inverse or eigen decomposition calculations with $O(N^3)$ complexity ($N$ is the number of parameters); 2) memory intensive, as it stores per-layer activations and gradients, which may take $O(N^2)$ space compared to $O(N)$ for SGD; and 3) communication intensive for distributed training because the Kronecker factors and accompanying inverses or eigen decompositions must be communicated to all workers.","To overcome these overheads and achieve faster training times than SGD at scale, K-FAC updates are often decoupled from gradient preconditioning so the expensive computations are performed less frequently.","Existing K-FAC systems either cache all Kronecker factor eigen decompositions needed to precondition gradients locally [44] or distribute gradient preconditioning across processors [42].","Neither approach enables K-FAC DNN deployments that are both memory and communication efficient, thus inhibiting scalability.","The first approach avoids communication when local memory is sufficient.","The second approach reduces memory footprint by not caching eigen decompositions locally but increases communication.","These various considerations make it difficult to use K-FAC efficiently in practice given the specific requirements of the DNN model or hardware and require users to choose between optimizing memory or communication.","To address these concerns, we propose KAISA , a K-FAC-enabled, Adaptable, Improved, and ScAlable second-order optimizer framework that can adapt execution to a given model size and memory limit.","KAISA allows tuning the ratio between communication and memory footprint to optimally apply K-FAC to distributed training by controlling the fraction of processes with local access to data.","This adaptation is made possible by grouping the processes, then distributing and communicating data within each group, as detailed in §REF and §REF .","K-FAC distribution strategies described in other work are effectively special cases of KAISA in which either all processes [44] or one process [42] cache data.","We evaluate KAISA on a range of classification, segmentation, and language modeling applications, including ResNet-50 [22], Mask R-CNN [21], U-Net [46], and BERT [17], on clusters of 448 NVIDIA Tesla V100s and 192 Ampere A100 GPUs.","Our results show that: 1) with fixed global batch size, KAISA trains deep neural networks 18.1–36.3% faster than the original optimizers for these applications; 2) with a fixed memory budget, KAISA trains ResNet-50 and BERT-Large phase 2 in 32.5% and 41.6% less time compared to momemtum SGD and Fused LAMB, respectively; 3) by varying the number of gradient workers, the K-FAC memory overhead can be reduced by 1.5–$2.9\\times $ ; 4) for high communication applications such as ResNet-50, extra processor memory can be used to train 24.4% faster; and 5) using state-of-the-art NVIDIA A100 GPUs, KAISA converges in fewer iterations at all scales, and KAISA 's scaling performance is on-par with SGD even with KAISA 's additional communication overhead.","Our code is open source with the MIT license and available at https://github.com/gpauloski/kfac_pytorch.","Our contributions in this paper are: An adaptive second-order optimizer framework that trains faster than SGD and its variants, while preserving convergence.","A quantitative study of the tradeoff between memory footprint and communication and its impact on training time in K-FAC design.","The first large scale evaluation of K-FAC convergence and speedup relative to SGD for Mask R-CNN, U-Net, and BERT on up to 128 A100 GPUs.","The rest of the paper is as follows.","We present the mathematical background and distributed implementation of K-FAC in §.","We describe KAISA 's design in § and implementation in §.","We present our experiments and results in §.","In §, we summarize existing DNN optimization frameworks and memory management techniques.","Finally, we conclude in §."],["Background","We first introduce K-FAC and its distributed implementation.","K-FAC is an efficient approximation of the Fisher Information Matrix (FIM), which has been shown to be equivalent to the Generalized Gauss-Newton (GGN) matrix in specific cases and can be viewed as an approximation of the Hessian [36].","In a standard SGD update step, the weights are updated using the gradients of the loss, as illustrated in Equation REF .","The K-FAC update step in Equation uses the FIM $F$ to precondition the gradients prior to update [44].","$w^{(k)}$ is the weight at iteration $k$ , ${\\alpha }^{(k)}$ is the learning rate at iteration $k$ , $n$ is the mini-batch size, $\\nabla {L_i}(w^{(k)})$ is the gradient of the loss function $L_i$ for the $i^\\text{th}$ example with regard to $w^{(k)}$ , and $F^{-1}(w^{(k)})$ is the inverse of the FIM.","$\\text{SGD: } w^{(k+1)} &= w^{(k)} - \\frac{{\\alpha }^{(k)}}{n}\\sum _{i=1}^{n}\\nabla {L_i}(w^{(k)}) \\\\\\text{K-FAC: } w^{(k+1)} &= w^{(k)} - \\frac{{\\alpha }^{(k)}F^{-1}(w^{(k)})}{n}\\sum _{i=1}^{n} \\nabla {L_i}(w^{(k)}) $ It has been shown empirically that training DNNs with the K-FAC second-order method enables convergence with fewer iterations than with SGD alone.","Theoretical understandings of the convergence rates of natural gradient methods, such as K-FAC, are an area of active research.","Previous work has shown that K-FAC [57] has linear convergence to the global minimum given a sufficiently over-parameterized model.","In strongly-convex problems, natural gradient methods have a quadratic convergence rate compared to the linear convergence of SGD [10].","The strongly-convex case provides some understanding of how K-FAC can improve convergence in non-convex cases.","Further, it has been shown that natural gradient methods enable larger learning rates improving the rate of convergence [57].","K-FAC makes greater per-iteration progress in minimizing the objective function at the cost of more computationally expensive iterations."],["K-FAC Approximation","K-FAC is based on the Kronecker product, a block matrix factorization that can reduce a large matrix inverse into two smaller matrix inverses.","K-FAC exploits the properties of the Kronecker product and the geometry of the FIM for DNNs to greatly reduce the complexity of computing the approximate FIM inverse.","The Kronecker product is written as $A \\otimes B$ where $A$ has size $m\\times n$ and $B$ has size $p \\times q$ .","The resulting matrix has shape $mp\\times nq$ .","$A \\otimes B=\\begin{bmatrix}a_{11}B & \\dots & a_{1n}B \\\\\\vdots & \\ddots & \\vdots \\\\a_{m1}B & \\dots & a_{mn}B\\end{bmatrix}$ The Kronecker product has two convenient properties: $(A \\otimes B)^{-1} &= A^{-1} \\otimes B^{-1} \\\\(A \\otimes B)\\vec{c} &= B^\\top \\vec{c} A .$"],["K-FAC Approximation","The FIM for a deep neural network is a block matrix where each block maps to layers in the model.","The block corresponding to the $i$ and $j^{\\text{th}}$ layers is approximated as $F_{i,j}\\approx a_{i-1}a_{j-1}^\\top \\otimes g_{i}g_{j}^\\top .$ Here, $a_{i-1}$ and $g_i$ are the activation of the $i-1^\\text{th}$ layer and the gradients of the $i^\\text{th}$ layer in the model, respectively [36], [44].Formally, $F$ represents an expected value, and the expectation of a Kronecker product is not equivalent to the Kronecker product of the expected factors.","However, this approximation still reasonably represents the structure of the FIM [36].","A fundamental assumption of K-FAC is the independence between layers [36], [44].This assumption is sufficient to produce an effective approximation for $F$ and necessary to produce a tractable algorithm.","Using this assumption, K-FAC approximates the FIM as a diagonal block matrix $\\hat{F}$ .","$\\hat{F}=\\texttt {diag}(\\hat{F}_1,...,\\hat{F}_i,...,\\hat{F}_L)$ As shown in Figure REF , the inverse of $F$ is a diagonal block matrix composed of the inverses of each diagonal block $\\hat{F}_i$ : $\\hat{F}^{-1}=\\texttt {diag}(\\hat{F}_1^{-1},...,\\hat{F}_i^{-1},...,\\hat{F}_L^{-1})$ where $\\hat{F}_{i}= a_{i-1}a_{i-1}^\\top \\otimes g_{i}g_{i}^\\top =A_{i-1}\\otimes G_i.","$ We refer to $A_{i-1}$ and $G_{i}$ as the Kronecker factors.","In practice, $A_{i-1}$ and $G_{i}$ are estimated with a running average of the factors computed over training batches [36], [44].","Due to the layer-wise independence of K-FAC, the gradient preconditioning and weight update for a single layer $i$ at iteration $k$ can be written as: $w_i^{(k+1)}=w_i^{(k)}-\\alpha ^{(k)}\\hat{F}_i^{-1}\\nabla L_i(w_i^{(k)}).$ We can apply properties REF and to reduce the gradient preconditioning, $\\hat{F}_i^{-1}\\nabla L_i(w_i^{(k)})$ , to an efficient form where the smaller Kronecker factors, rather than the large FIM, are inverted.","$\\hat{F}_i^{-1}\\nabla L_i(w_i^{(k)})=G_i^{-1}\\nabla L_i(w_i^{(k)})A_{i-1}^{-1}$ Tikhonov regularization is used to avoid ill-conditioned matrix inverses with K-FAC by adding a damping parameter $\\gamma $ to the diagonal of $\\hat{F}_i$ [20], [42].","In most implementations, instead of computing $\\hat{F}^{-1}$ , we compute $(\\hat{F}_i+\\gamma I)^{-1}$ as: $(\\hat{F}_i+\\gamma I)^{-1}=(A_{i-1}+\\gamma I)^{-1}\\otimes (G_i+\\gamma I)^{-1}.","$ Thus, the final update step for the parameters of layer $i$ at iteration $k$ is: $w_i^{(k+1)}=&w_i^{(k)}-\\alpha ^{(k)}(\\hat{F}_i+\\gamma I)^{-1}\\nabla L_i(w_i^{(k)}) \\\\=&w_i^{(k)}-\\alpha ^{(k)}(G_i+\\gamma I)^{-1}\\nabla L_i(w_i^{(k)})(A_{i-1}+\\gamma I)^{-1}.$"],["Alternative Approximation","It has been shown that an empirically more stableIn this context, stable means produces more consistent validation results across batch sizes and hyperparameter settings.","approximation for $(\\hat{F}+\\gamma I)^{-1}\\nabla L_i(w_i^{(k)})$ can be computed using an eigen decomposition of the Kronecker factors, $A$ and $G$ , from Equation REF [20], [44].","Given $Q_A$ and $Q_G$ , the eigenvectors of the factors, and $\\upsilon _A$ and $\\upsilon _G$ , the eigenvalues of the factors, the preconditioned gradient can be computed as follows.","$V_1&=Q_G^\\top \\nabla L_i(w_i^{(k)}) Q_A \\\\V_2&=V_1/(\\upsilon _G\\upsilon _A^\\top +\\gamma ) \\\\(\\hat{F}_i+\\gamma I)^{-1}\\nabla L_i(w_i^{(k)})&=Q_GV_2Q_A^\\top $ The composition of the factors $A$ and $G$ in Equation REF guarantees the factors are symmetric and therefore the factor eigen decompostions have real eigenvalues and orthogonal eigenvectors.","In this work, we use the eigen decomposition method for gradient preconditioning."],["Infrequent K-FAC Updates","A common strategy in second-order optimization methods is to update the second-order information every few iterations [36], [42], [44].","Intuitively, second-order information does not change as rapidly from one iteration to the next like first-order information.","The K-FAC update interval parameter controls the number of iterations between second-order updates, i.e., iterations between eigen decomposition recomputations.","Larger K-FAC update intervals result in more stale information, so tuning this parameter is key to achieving fast training with K-FAC.","Practically, K-FAC can maintain convergence with K-FAC update intervals of 100–2000 iterations [44].","Figure: Hybrid-parallel distributed K-FAC implementation overview.Blue boxes are standard computations in data-parallel training.Red boxes are computations required by K-FAC.Workers maintain identical copies of the model.The output of each layer zz is computed during the forward pass for the local batch xx.Then, the loss between the true output yy and predicted output z L z_L is calculated and used to compute gradients in the backward pass.The gradients are then allreduced across workers.During the forward/backward pass, K-FAC caches intermediate data for computing factors.In the K-FAC stage, workers compute factors AA and GG in data-parallel and allreduce the results.The eigen decompositions and preconditioned gradients 𝒢\\mathcal {G} are computed in the K-FAC approximation stage.The existing K-FAC methods, MEM-OPT and COMM-OPT, implement the model-parallel stage differently as shown at the bottom of the figure.After the K-FAC stage, all workers have 𝒢\\mathcal {G} and can update weights locally using 𝒢\\mathcal {G} and a standard optimizer (e.g., SGD or ADAM)."],["Distributed Implementation","Existing distributed K-FAC implementations are hybrid-parallel, with first-order information (e.g., gradients) computed in data-parallel and second-order information (e.g., K-FAC approximation) in model-parallel.","Figure REF outlines the model-parallel K-FAC computation performed between standard data-parallel training steps.","We refer to the two existing distributed K-FAC implementation strategies as MEM-OPT (memory optimized K-FAC) and COMM-OPT (communication optimized K-FAC).","Both work by replicating the DNN across all processes and assigning a random local batch of training data to each process at each iteration.","The data-parallel forward pass, backward pass, and SGD weight update stages outlined in Figure REF are the same in both methods.","The key differences between the two approaches are the model-parallel computation implementations in the gradient preconditioning stage."],["MEM-OPT","MEM-OPT [42] assigns each layer in the DNN to a different process during the preconditioning stage.","Each process computes the eigen decompositions of $A_{i-1}$ and $G_i$ needed for preconditioning the gradients using Equations REF –.","Each process then broadcasts the preconditioned gradients to all processes such that subsequent SGD weight updates can be done in data-parallel.","MEM-OPT has a low memory footprint because no eigen decompositions are duplicated: each layer's eigen decompositions are stored on only one process.","MEM-OPT has communication in three places: a) gradient allreduce, b) factor allreduce, and c) preconditioned gradient broadcast, all shown in Figure REF .","In non-K-FAC update steps (e.g., steps where the eigen decompositions are not updated), MEM-OPT avoids the factor allreduce because eigen decompositions from a previous step are reused; however, the preconditioned gradient broadcast is still required because the gradient being preconditioned changes every iteration."],["COMM-OPT","COMM-OPT [44] decouples the eigen decomposition from gradient preconditioning.","Instead of assigning each layer to a process, individual factors are assigned to a process to be eigen decomposed in parallel.","The eigen decompositions are broadcast back to all processes such that every process holds a copy of all eigen decompositions.","Each process computes all preconditioned gradients locally prior to weight updates.","COMM-OPT has a larger memory-footprint because every process must maintain a copy of the eigen decompositions.","COMM-OPT has communication in three places: a) gradient allreduce, b) factor allreduce, and c) eigen decomposition broadcast, also shown in Figure REF .","Decoupling the eigen decompositions from the gradient preconditioning achieves two goals: 1) $A_{i-1}$ and $G_i$ can be computed in different processes, doubling the maximum worker utilization, and 2) in non-K-FAC update steps, the communication in (b) and (c) can be avoided because every worker can locally precondition gradients with the cached eigen decompositions.","Thus, in non-K-FAC update intervals, COMM-OPT has no additional communication overhead compared to SGD.","COMM-OPT has been shown to be 4–16% faster than MEM-OPT for ResNet-50 training on 16–256 V100 GPUs [44].","These two implementations convey the fundamental tradeoff between caching the decompositions locally to avoid communication and communicating the preconditioned gradients every iteration to avoid additional memory overheads.","This tradeoff impacts the communication, computation, and memory overhead of K-FAC and motivates our research in this paper."],["Design","We introduce features of KAISA 's design that enable its tunable memory footprint and explain how KAISA performs load balancing and half precision training to improve scalability."],["Tunable Memory Footprint","KAISA introduces HYBRID-OPT, a distributed K-FAC strategy that achieves variable memory overhead in K-FAC.","In this strategy, each layer has a subset of the processes assigned to be gradient preconditioners, referred to as the gradient workers.","grad_worker_frac defines the size of this subset, i.e., the gradient worker count is $\\texttt {max}(1, \\textit {grad\\_worker\\_frac}\\times \\textit {world\\_size})$ .","One of the gradient workers is also responsible for computing the eigen decompositions for the layer and broadcasting the results to the remaining gradient workers.","At the end of the eigen decomposition stage, each gradient worker has a copy of the eigen decompositions and can precondition the gradient.","Gradient receivers are the processes not assigned as gradient workers for the layer.","Each gradient worker is responsible for broadcasting the preconditioned gradient to a subset of the gradient receivers.","With multiple gradient workers, the broadcast groups are smaller and simultaneous broadcasts are possible, reducing overall communication time.","For example, given two gradient workers and two gradient receivers, each gradient worker sends the preconditioned gradients to just one receiver; both gradient workers perform this communication at the same time.","After the gradient broadcast, all processes have a copy of the preconditioned gradient with which local weights can be updated.","Observe that KAISA unifies existing distributed K-FAC strategies because COMM-OPT and MEM-OPT are special cases of HYBRID-OPT.","COMM-OPT is the case where grad_worker_frac = 1 (all processes precondition the layer's gradient), and MEM-OPT is the case where grad_worker_frac = 1/world_size (a single process preconditions a layer's gradient and broadcasts the result).","Figure REF compares MEM-OPT, COMM-OPT, and HYBRID-OPT in an eight process environment.","As grad_worker_frac increases, more processes cache the eigen decompositions and the memory footprint increases.","Visually, the number of processes that cache the eigen decompositions is represented by the processes in the dashed red box in Figure REF .","Continuing with Figure REF , HYBRID-OPT has a lower preconditioned gradient broadcast cost than MEM-OPT because the broadcast groups are smaller and broadcasts are overlapped.","HYBRID-OPT requires four separate broadcasts to groups of size two in comparison to MEM-OPT which requires one large broadcast to a group of eight.","Since these broadcasts involve non-overlapping processes, we can execute all four broadcast calls simultaneously in the HYBRID-OPT case.","Given the complexity of broadcasting using the minimum spanning tree algorithm is $O(\\log p)$ , where $p$ is the number of processes in the broadcast group, the complexity is reduced from $O(\\log 8)$ in MEM-OPT to $O(\\log 2)$ in HYBRID-OPT.","Note in steps where the eigen decompositions are updated, HYBRID-OPT incurs an additional eigen decomposition broadcast with $O(\\log 4)$ complexity; however, as mentioned in §REF , eigen decompositions are updated infrequently so the average complexity for HYBRID-OPT is still less than that of MEM-OPT.","The problem addressed by KAISA's HYBRID-OPT strategy is akin to that of 2.5D matrix multiplication [48].","Both algorithms can utilize extra processor memory to reduce communication costs by controlling the replication factor of data.","In 2.5D matrix multiplication, a parameter $c$ determines the number of data copies, and in KAISA, the grad_worker_frac determines the number of workers that store the eigen decompositions."],["Greedy Factor Distribution","The eigen decompositions required for K-FAC are expensive to compute.","Distributed K-FAC optimizes this stage by computing eigen decompositions in a model-parallel fashion.","To most efficiently use available resources, the eigen decomposition computations should be distributed in a manner that minimizes the makespan $T$ , the time it takes for all processes to complete their assigned computations.","We use the longest processing time greedy algorithm which produces an assignment with makespan $T\\le \\frac{3}{2}T^*$ where $T^*$ is the optimal makespan [28].","The longest processing time algorithm sorts jobs by decreasing length and then iteratively assigns each job to the worker with the lowest current work load.","For each factor to be eigen decomposed, the processing time is approximated as $O(N^3)$ where each factor is an $N\\times N$ matrix [16].","Alternatively, memory usage can be optimized for by using $O(N^2)$ as the approximation since $N^2$ is the size of the factor.","KAISA uses the longest processing time strategy at the start of training to assign each factor to a process."],["Half Precision Storage and Computation","Mixed precision training is a common strategy to reduce training times and memory usage on supported hardware [37].","Popular frameworks for automatic mixed precision (AMP) training, such as NVIDIA AMP and PyTorch AMP, cast forward and backward pass operations to half precision where possible.","Gradient calculations in the backward pass can often produce very small values that would be clipped when cast to half precision, so these frameworks scale gradients to a larger value during the backward pass and unscale the gradients appropriately before the optimization step.","KAISA adapts to the precision being used for training.","When training with AMP, factors are stored in half precision, reducing memory costs.","K-FAC computations are performed in half precision where possible.","Eigen decompositions are generally unstable in half precision so factors are cast to single precision before decomposition.","KAISA can store eigen decompositions in half precision to further reduce memory consumption if needed.","Half precision training has become a staple in achieving state-of-the-art results in deep learning, and KAISA can particularly benefit from half precision training due to the K-FAC computation and communication overhead."],["K-FAC Usage","PythonStyle model = DistributedDataParallel(model) optimizer = optim.SGD(model.parameters(), ...) preconditioner = KFAC(model, grad_worker_frac=0.5) for data, target in train_loader: optimizer.zero_grad() output = model(data) loss = criterion(output, target) loss.backward() preconditioner.step() optimizer.step() We design KAISA to implement K-FAC as a preconditioner to standard optimizers with support for Conv2d and Linear layers.","KAISA has an easy-to-use interface and can be incorporated into existing training scripts in two lines: one to initialize and one to call KFAC.step() prior to optimizer.step() (see Listing ).","KAISA automatically registers the model and determines the distributed communication backend (e.g., Torch, Horovod, single-process).","A call to KFAC.step(): 1) computes the factors using the forward/backward pass data, 2) computes the eigen decompositions in parallel and broadcasts the results, 3) computes the preconditioned gradients and broadcasts the results if necessary, and 4) scales the preconditioned gradients.","We implement KAISA using PyTorch [43], with communication, interface, large-batch training, and gradient preconditioning implemented to collectively enable efficient second-order optimization."],["AMP and Distributed Training","We use PyTorch AMP for mixed precision training.","With PyTorch AMP, the GradScaler object responsible for scaling and unscaling the gradient in the backward pass can be passed to KAISA .","KAISA uses the GradScaler to correctly unscale the $G$ factors, since the scale factor can change from iteration to iteration, causing problems when computing the running average of $G$ over the course of training.","All communication operations are performed in the precision of the data to reduce bandwidth requirements.","KAISA supports torch.distributed and Horovod [47] for distributed training.","In this work, we use torch.distributed for all experiments because the DistributedDataParallel model wrapper overlaps gradient communication with backpropogation, works seamlessly with PyTorch AMP, and provides a broadcast group abstraction needed for HYBRID-OPT."],["Factor Accumulation","Processor memory (e.g., GPU VRAM) limits the maximum per-processor batch size during training.","A common strategy to achieve larger effective batch sizes is gradient accumulation, a method where gradient values for multiple forward and backward passes are accumulated between optimization steps.","For example, if processor memory limits the batch size to 8, an effective batch size of 32 can be achieved by accumulating gradients over four forward/backward passes prior to the optimization step.","This strategy is common in applications such as BERT, where effective batch sizes are often $>2^{15}$ , but modern GPUs are limited to local batch sizes $<2^7$ .","The forward/backward pass data needed by KAISA to compute the factors is accumulated over each mini-batch between calls to KFAC.step().","As the number of gradient accumulation steps increases, the memory needed to accumulate the forward/backward pass data grows linearly.","KAISA can efficiently support gradient accumulation by computing the factors for the current mini-batch during the forward/backward pass instead of during KFAC.step().","The factor communication is still performed during KFAC.step()."],["Triangular Factor Communication","Previous K-FAC work has exploited the symmetric nature of the Kronecker factors to reduce communication volume by sending only the upper triangle for each factor [49], [50].","Our implementation supports extracting the upper triangle for the factor allreduce and reconstructing the full factor before the eigen decomposition stage.","For the models studied in this work, this optimization did not yield performance improvements for two reasons.","First, network latency impacted overall communication time more than bandwidth; second, this optimization has an additional overhead for extracting the upper triangle and reconstructing the factor.","For models with larger individual layers—and therefore factors—this optimization could yield greater benefits."],["Gradient Preconditioning","The largest overhead of K-FAC in non-K-FAC update steps is computing the preconditioned gradients.","This process, described by Equations REF –, involves a series of matrix additions, divisions, and multiplications.","Observe that the gradient $\\nabla L_i(w_i^{(k)})$ is the only variable that changes between non-K-FAC update iterations.","In particular, the computation involving the outer product of the eigenvalues, $1/(\\upsilon _G\\upsilon _A^\\top +\\gamma )$ , in Equation does not need to be recomputed every iteration—only after the eigen decompositions are updated.","We also observe that when $\\emph {grad\\_worker\\_frac}>1/world\\_size$ , multiple processes perform this computation redundantly.","To reduce the total number of operations during the preconditioning stage, we move the computation of the outer product into the eigen decomposition stage.","The process assigned to eigen decompose $G$ computes $1/(\\upsilon _G\\upsilon _A^\\top +\\gamma )$ and broadcasts the result to all gradient workers instead of broadcasting $\\upsilon _A$ and $\\upsilon _G$ .","This ensures that $1/(\\upsilon _G\\upsilon _A^\\top +\\gamma )$ is computed once (on a single worker) and then reused many times by other workers.","In practice, this reduced the time to precondition the gradients for a single layer by up to 53%."],["Experiments","We report on experiments that address four issues.","1) Convergence to baseline evaluation metrics and 2) time to convergence with and without KAISA to validate the design and implementation.","3) An exploration of the memory and communication tradeoff using KAISA to develop a quantitative understanding of the impact of the grad_worker_frac configuration.","4) An evaluation of KAISA's scaling performance."],["Hardware and Software Stack","We performed experiments on two computers.","The first is the GPU subsystem of the Frontera supercomputer at the Texas Advanced Computing Center, which has 112 nodes, each powered by IBM Power9 processors, with four 16 GB NVIDIA V100 GPUs (448 GPUs in total).","Nodes are connected by an InfiniBand EDR network.","We use PyTorch 1.6, CUDA 10.2, CUDNN 7.6.5, and NCCL 2.5.6, and MVAPICH2-GDR 2.3.4 to launch processes on multiple nodes for distributed training.","The second is the GPU subsystem of the Theta supercomputer at Argonne National Laboratory.","This system has 24 NVIDIA DGXA100 nodes with eight 40GB A100 GPUs each (192 A100 GPUs in total).","On DGXA100 nodes, we use PyTorch 1.7, CUDA 11.0, CUDNN 8.0.4, and NCCL 2.7.8.","We distinguish between these two systems in the following text by specifying either V100 or A100 GPUs."],["Applications","We evaluate KAISA for classification, segmentation, and language modeling applications.","Classification: We use ResNet-50 [22] with the ImageNet-1k dataset [29], which has 1000 categories with approximately 1.3M training images and 50K validation images.","We use K-FAC to precondition all convolutional and linear layers in ResNet-50 and SGD for the weight updates.","Segmentation: We explore two segmentation tasks.","First, we use the NVIDIA reference PyTorch implementation of Mask R-CNN [21], [5] with the Common Objects in Context (COCO) 2014 dataset [32].","We use K-FAC to precondition the convolutional and linear layers in the region of interest (ROI) heads of Mask R-CNN and SGD for the weight updates.","Second, we use a U-Net [46] architecture for segmenting brain tumor sub-regions.","We extend a Kaggle competition implementation [2] to enable multi-GPU training.","The test case was run on the LGG Segmentation Dataset [3], which contains Magnetic Resonance (MR) images of the brain from 110 patients across five hospitals.","Images from a random subset of 100 patients are used as the training dataset and the remaining 10 patients are used for validation.","We apply K-FAC to all convolutional layers in the model.","Language Modeling: We train the BERT-Large Uncased model using a modified version of the NVIDIA reference PyTorch implementation for BERT [17], [5] with the English Wikipedia [52] and Toronto BookCorpus datasets [59].","Each transformer in BERT is implemented using a series of Linear layers, and we apply K-FAC to all linear layers.","We do not use K-FAC to precondition the embedding layer and prediction head because both of these layers have a Kronecker factor with shape $vocab\\_size\\times vocab\\_size$ , and since the vocab size for BERT-Large is 30K, these factors cannot be efficiently eigen decomposed.","Fused LAMB is used as the optimizer [56].","The strategy outlined in §REF is used to reduce memory consumption since gradient accumulation is used for training."],["Convergence with Fixed Batch Size","We compare KAISA performance (converged accuracy, epochs to convergence, and time to convergence) on ResNet-50, Mask R-CNN, U-Net, and BERT-Large against the baseline implementations listed in Table REF .","For ResNet-50 and Mask R-CNN, we use MLPerf benchmark target results [4].","For U-Net, we use the baseline validation Dice similarity coefficient (DSC) from the model's GitHub repository [2].","For the BERT-Large baseline, researchers have reported F1 scores of 91.08% [5], 91.0% [1], and 90.4% [56] for fine-tuning on the SQuAD v1.1 dataset.","We use the best reported F1 score (i.e., the NVIDIA implementation [5]) with the LAMB optimizer.","Due to the partial unavailability of the Toronto BookCorpus training dataset, our measurement only converges to 90.8%.","(The Toronto BookCorpus dataset is no longer available online as a holistic package; we could recover only 14,155 of the 16,846 books.)","Table: Baseline performance and hardware summary for ResNet-50, Mask R-CNN, U-Net, and BERT-Large.","“val acc” is validation accuracy.","“mAP” is mean average precision.","“DSC” is Dice similarity coefficient.In all comparisons, we use the same global batch size for KAISA and the original optimizers to isolate the improvement from second-order information.","Table REF summarizes the hyperparameters for each application.","For ResNet and K-FAC specific hyperparameters, we use values from [44] to provide direct performance comparisons to previous K-FAC works.","For Mask R-CNN and BERT-Large, we use the NVIDIA reference hyperparameters [5].","Further performance improvements could be gained through more extensive hyperparameter tuning; however, this was not necessary to achieve results better than the original optimizers.","Table: Summary of hyperparameters used for each application.","BS = global batch size, LR = learning rate, WU = warm up iterations, K_freq = number of iterations between eigen decomposition re-computations, F_freq = iterations between factor updates.","grad_worker_frac = 1 and damping = 0.003 for all cases.ResNet-50: We train ResNet-50 for 55 and 90 epochs for KAISA and momentum SGD, respectively, using FP16 precision on eight NVIDIA A100 GPUs.","Figure REF shows the validation accuracy curve using the two optimization methods.","KAISA converges to the baseline validation accuracy at epoch 46 and momentum SGD at epoch 65.","The time-to-convergence is 268.1 minutes for KAISA : 24.3% less than the 354.0 minutes for momentum SGD.","Figure: Validation Metric Curve Comparison between KAISA and SGD/ADAM on ResNet-50, Mask R-CNN, and U-Net.","The dotted lines represent the target metric.= 0pt Mask R-CNN: Our baseline measurement of Mask R-CNN on 32 NVIDIA V100 GPUs takes 25,640 iterations to converge to 0.377 bbox mAP and 0.342 segm mAP in FP32.","With K-FAC enabled for the ROI heads, training converges to 0.379 bbox mAP and 0.350 segm mAP in 21,000 iterations.","The bbox mAP comparision for SGD and KAISA is shown in Figure REF .","KAISA reduces the training time from 136.1 minutes to 115.8 minutes, a 14.9% improvement.","With a batch size of 128 on 64 V100 GPUs, KAISA converges in 12,000 iterations compared to 15,000 with SGD and reduces training time by 18.1%.","U-Net: The reference U-Net implementation [2] with ADAM converges to 91.0% validation DSC within 50 epochs with four NVIDIA A100 GPUs using FP32 training.","Using KAISA , the training converges above 91.0% in 30 epochs as seen in Figure REF .","KAISA reduces the training time by 25.4% (10.9 minutes vs. 14.6 minutes).","BERT: BERT pretraining has two phases.","The first trains with maximum sequence length of 128 for 7038 iterations and the second trains with maximum sequence length of 512 for 1,563 iterations.","We fine-tune the pretrained BERT model for three epochs using the SQuAD dataset.","All phases and fine-tuning are done in FP16.","Tuning the hyperparameters of BERT is expensive as each run can take over 130 hours with 8 A100 GPUs, so we showcase the effectiveness of KAISA with the second phase of BERT pretraining.","For phase two pretraining, we start with the same model pretrained with LAMB during phase one.","We train with KAISA for {800, 1,000, 1,200} iterations then fine-tune SQuAD.","Table REF summarizes the validation SQuAD F1 scores after fine-tuning and the pretraining performance improvements.","Table: BERT performance comparison: KAISA vs. LAMBKAISA converges to the 90.8 F1 baseline in 800 iterations, 47.9% less iterations than required for LAMB, and KAISA takes 36.3% less time than LAMB to converge."],["Convergence with Fixed Memory Budget","To understand how a grad_worker_frac can enable second-order optimization in memory-constrained environments, we compare KAISA's performance against baselines with fixed memory budgets.","In particular, we train ResNet-50 on 64 V100 GPUs and BERT-Large phase two on eight A100 GPUs.","For each experiment, we use the maximum possible local batch size and measure the convergence, epochs to convergence, and time to convergence.","Table REF summarizes the hyperparameters and time to convergence.","Table: Time to convergence and hyperparameters for each optimizer.","BS = global batch size, LR = learning rate, g_frac = gradient worker fraction, K_f = iterations between eigen decomposition re-computations, F_f = iterations between factor updates, T_conv is time to convergence in minutes.ResNet-50: With momentum SGD, the maximum local batch size is 128 per GPU and the global batch size is 8,192.","We train ResNet-50 for 90 epochs using momentum SGD which achieves 75.0% validation accuracy—0.9% lower than the MLPerf baseline.","Next, we use KAISA with grad_worker_frac = 1 and a local batch size of 80; however, the training runs out of memory.","Lowering grad_worker_frac to 1/2, training takes 83 minutes to converge to 75.9% in the 48th epoch.","The complete 55 epoch training takes 95 minutes and the validation accuracy reaches 76.0%.","So, even if momentum SGD converges to 75.9% by the 90th epoch, KAISA still reduces the time to convergence by 32.5%.","Finally, we run the same training process with MEM_OPT by setting grad_worker_frac to 1/64.","It converges at the 47th epoch and the time to convergence is 96 minutes.","The complete 55 epoch training takes 111 minutes.","This experiment highlights the benefit of KAISA in cases where compute resource can not be efficiently utilized with the original optimizers.","With a grad_worker_frac value of 1/2, KAISA offers benefits over COMM_OPT and MEM_OPT by enabling second-order optimization under a tight memory budget that is 13.9% faster than COMM_OPT and not feasible with MEM_OPT.","BERT: With LAMB, the maximum possible local batch size per GPU is 12 for the second phase of this BERT implementation [5], and the global batch size is 24,576.","For KAISA, we use a local batch size of 8 and global batch size of 32,768.","This experiment uses the same hyperparameters as in §REF , so all cases with LAMB and BERT should converge to the baseline, thus we only project the training time with the first 100 steps.","As the global batch size with LAMB is 24,576, 2,084 training steps are required to finish three epochs, and the training time is 2,917.6 minutes.","KAISA, with grad_worker_frac = 1/2, takes 3,268.8 minutes to finish the three epochs.","However, KAISA converges to baseline after 800 steps, as shown in Table REF , so the time to converge is 1,702.5 minutes—41.6% faster than LAMB.","Setting grad_worker_frac = 1 takes 1,703.5 minutes to converge for KAISA.","The performance is comparable to the case with grad_worker_frac = 1/2.","We conduct a detailed study on this convergences phenomena for different grad_worker_frac values in §REF ."],["Memory vs. Communication","To understand the impact of grad_worker_frac on training times, we train ResNet-{18, 50, 101, 152}, Mask R-CNN, and BERT-Large on 64 V100 GPUs for grad_worker_frac $\\in $ {1/64, 1/32, 1/16, 1/8, 1/4, 1/2, 1}.","For each experiment, we record the average iteration time, i.e., time between weight updates, and the GPU memory usage.","We refer to the K-FAC overhead as the memory required to store the factors and eigen decompositions, and the K-FAC overhead is computed as the difference between the K-FAC and SGD memory usage.","For all ResNet models, we use the same hyperparameters used for ResNet-50 (Section REF ) except for ResNet-152 where the local batch size was lowered to 24.","The results are presented in Figure REF , and a summary of the memory usage is provided in Table REF .","The K-FAC memory overhead increases linearly as a function of grad_worker_frac for all models.","KAISA requires 1.5–45.8% more memory than SGD depending on the application and value of grad_worker_frac (Table REF ).","The maximum K-FAC overhead (i.e., when grad_worker_frac = 1) is 1.5–2.9$\\times $ that of the minimum K-FAC overhead (i.e., when grad_worker_frac = $1/64$ ).","Table: Summary of per-GPU memory usage for training, in MB.Abs.","is the absolute memory required for training.Δ\\Delta is the %-increase in memory required over SGD.The K-FAC overhead is the K-FAC abs.","memory minus the SGD abs.","memory.With respect to iteration times, the ResNet models scale well with the number of gradient workers with ResNet-50 scaling the best.","For ResNet-50, the speedup from a gradient worker count of 1 to 64 is 24.4% for FP32 with a 22% increase in total memory usage.","The average iteration times for Mask R-CNN and BERT-Large remain constant as the number of gradient workers is increased.","For comparison, the same ResNet-50 experiment with 64 V100s in [44] only shows a 7.6% speedup when increasing the gradient worker count from 1 to 64.","This improvement in KAISA over previous work is due to the unique contributions presented in §REF , §REF , §REF , and §REF .","These promising results for ResNet-50, a de facto standard benchmark for deep learning systems, are important as the performance characteristics of ResNet-50 represent a large set of commonly used models (e.g.","VGG16, U-Nets, etc.).","We can understand why ResNet model performance varies across grad_worker_frac values while the Mask R-CNN and BERT-Large performance remains constant by considering the bandwidth requirements of these applications.","The bandwidth required by KAISA is a function of the size of the factors, eigen decompositions, and frequency of K-FAC updates.","With 64 V100s, ResNet-50 calls KFAC.step() frequently (4–6 calls/second) and incurs a K-FAC memory overhead between 634 MB and 1.8 GB.","In comparison, Mask R-CNN calls KFAC.step() with a lower frequency (3 calls/second) and has a much smaller K-FAC overhead (100–200 MB).","Thus, the changes in how KAISA communicates data with respect to grad_worker_frac are less apparent in Mask R-CNN.","BERT-Large has the lowest bandwidth requirements of all applications even though it has the largest K-FAC overhead.","BERT-Large uses gradient accumulation to achieve very large batch sizes (32K for phase 2) and as a result only calls KFAC.step() every $\\sim $ 120 seconds.","While the iteration times for low-communication models such as BERT and Mask R-CNN are invariant to the grad_worker_frac value in KAISA, KAISA still produces faster-than-SGD training times with small increases in memory-overhead.","This is due to KAISA's unique features outlined in § and §.","Further, practitioners training these models at larger scales, e.g., 100s or 1000s of GPUs, where communication becomes a greater bottleneck will benefit more from the flexibility KAISA provides to adapt training to environments with increasing communication costs.","Tuning the grad_worker_frac hyperparameter, to determine an optimal balance between iteration time and memory usage, is simple as it only requires profiling the average iteration time for each grad_worker_frac value over a few iterations.","We find that with respect to training times, grad_worker_frac has the most impact in applications that spend a larger proportion of time doing communication.","To further understand how grad_worker_frac impacts training times, we analyze the execution time for each section within KFAC.step() with ResNet-50 on 64 V100s.","Figure REF provides the time spent in each section for all layers in the model during a call to KFAC.step().","Times are averaged over 10,000 iterations and across all workers.","Eigen decompositions are updated every 500 iterations.","As shown in Figure REF , factor computation and communication, eigen decomposition, and scaling and updating the gradients, are invariant to the grad_worker_frac.","Figure: Average function execution time during calls to KFAC.step() for ResNet-50 on 64 GPUs.The time required to broadcast the eigen decompositions increases substantially as the number of gradient workers is increased.","Referring back to Figure REF , the more gradient workers there are, the more processes that need to receive the eigen decompositions.","However, the eigen decompositions, in this case, are only recomputed every 500 iterations so the eigen decomposition broadcast has a negligible effect on the average iteration time.","On the other hand, the gradient preconditioning and broadcast occur every iteration regardless of if the factors or eigen decompositons are updated and have the greatest influence on average iteration time.","We see that the time to precondition the gradients increases with the gradient worker count because each process is assigned as a gradient worker for more layers.","This discovery highlights the importance of the gradient preconditioning optimizations made in §REF .","The time to broadcast the preconditioned gradients decreases to 0 as the gradient worker count approaches world_size, and notably, the time decreases at a faster rate than the increase in time required in the preconditioning stage.","This trend is a result of each gradient worker needing to send the results to fewer other processes as the gradient worker count increases."],["Scaling","To examine the scaling characteristic of KAISA, we measure the average time per epoch for ResNet-50 and average time per iteration for BERT-Large phase 2.","We choose ResNet-50 and BERT-Large because they represent a high-communication and low-communication model, respectively (see §REF ).","We study three KAISA variants: COMM-OPT, MEM-OPT, and HYBRID-OPT with a grad_worker_frac of 1/2, and we report the projected end-to-end training time speedup for the KAISA variants over the base optimizers (SGD and LAMB) in Figure REF .","For ResNet-50, we project the training time to 90 epochs for SGD and 55 for KAISA.","For BERT-Large, we project the phase 2 training time to 1,563 steps for LAMB and 800 for KAISA based on the study in §REF .","The hyperparameters in Table REF are used, and for ResNet-50, the K-FAC update frequency is scaled inversely with the global batch size to keep the number of K-FAC updates per training samples constant.","We store and communicate the factors and eigen decompositions in FP16 for BERT-Large.","In Figure REF and REF , MEM-OPT maintains a constant speedup over SGD and LAMB, respectively, across all scales.","In contrast, the speedup for COMM-OPT improves in both cases as the scale increases indicating the tradeoff of more memory for reduced communication does have scaling benefits.","HYBRID-OPT sees performance improvements on par with COMM-OPT with BERT-Large while using less GPU memory.","Overall, HYBRID-OPT has the best balance of scaling and memory usage for large-scale BERT pretraining.","As a whole, KAISA achieves better speedups with BERT-Large which is likely due to ResNet-50 being more communication bound than BERT-Large as noted in §REF .","Further scaling experiments on a machine with more GPUs is needed to understand the characteristics and limits of KAISA's speedup over SGD.","The emergence of DNNs has motivated revisiting many aspects of system design.","TensorFlow [6], PyTorch [43], and MXNet [14] are examples of deep learning frameworks that support end-to-end training.","TensorFlow Serving [41], DLHub [13], and Clipper [15] focus on low-latency inference serving.","Ray [38] provides a platform to integrate simulation, training, and serving for reinforcement learning applications.","Researchers have also designed new scheduling approaches in GPU clusters with informed hardware heterogeneity [40], sharing capability [55], and early user feedback [54] to optimize cost and hardware utilization.","Our work here focuses on a lower-level question: given model architecture, communication bandwidth, processor count, and memory size, can the hybrid-parallel aspect of distributed K-FAC be adaptable and reduce training time?","In the rest of this section, we briefly review other works in optimization frameworks.","Optimizer Frameworks: Distributed optimization frameworks take many forms.","In a synchronous optimizer such as Horovod [47], all variables are updated in every iteration.","Asynchronous optimizers relax variable update consistency, for example by passing values via parameter servers [31], to achieve higher performance than synchronous methods [45], [33].","The pipeline parallel paradigm, such as GPipe [25] and PipeDream [39] which hold multiple versions of a model partition in a processor and exploit asynchronous optimizers, has shown comparable training convergence.","BytePS [27] proposes a unified interface for synchronous and asynchronous SGD.","Generally, asynchronous SGD has a non-linear slowdown compared to synchronous SGD [7].","KAISA implements K-FAC in a synchronous manner as a preconditioner such that it can work with any SGD variant.","Memory Shortage and Remedies: Training DNNs is memory intensive.","A common technique for training large models is to swap between processor memory and host memory, such as in SwapAdvisor [23] and SuperNeurons [51].","An alternative is to discard some activation tensors and rematerialize them when needed for back-propagation.","Checkmate [26] formulates this tradeoff as an optimization problem and provides an optimal rematerialization schedule.","In KAISA , we apply techniques such as precision relaxing to reduce the memory consumption of K-FAC and controlling the distribution of eigen decomposition results across processors to maintain a minimal cost in training time.","K-FAC Convergence: Prior work on distributed K-FAC has reported training convergence primarily on ResNet-like convolutional neural networks.","One study [8] used asynchronous distributed K-FAC to train ResNet-50 with ImageNet 2$\\times $ faster than standard SGD, but only achieved 70% validation accuracy, a 5.9% loss compared to the 75.9% MLPerf [4] baseline.","Another distributed K-FAC implementation [42] trained ResNet-50 with ImageNet to 74.9% validation accuracy in just 978 iterations; however, comparisons to SGD are not provided.","Later work iterated on this implementation to achieve 75.0% validation accuracy on ImageNet with ResNet-50 on 2048 V100 GPUs in 2 minutes by carefully optimizing the baseline SGD training and introducing a 21-bit floating point (FP21) specification for the factors along with other optimizations from previous works such as triangular matrix communication and infrequent K-FAC updates.","FP21 was introduced due to worse convergence when using FP16 for factor communication; however, with KAISA , we found similar validation accuracy with ResNet-50 for FP32 and FP16 factor communication.","Differences in the numerical stability of the eigen decomposition in KAISA rather than matrix inversion could be a possible reason for the discrepancies.","A fourth study [44] trained ResNet-50 to the 75.9% MLPerf baseline in 18-25% less time than with SGD by replacing the factor inverse with eigen decomposition and optimizing for reduced communication in non-K-FAC update steps, at the cost of a high memory footprint.","KAISA generalizes previous distributed K-FAC strategies via a tunable memory footprint to balance the memory and communication costs.","This design enables efficient, distributed K-FAC research across a wider range of hardware.","Further, we showcase KAISA 's effectiveness on a variety of domains and models.","Alternative K-FAC Methods: Eigenvalue-corrected Kronecker-factorization (EK-FAC) [18], a more accurate approximation of the FIM, can perform cheap, partial updates.","Noisy K-FAC [58] and noisy EK-FAC [9] are functionally similar to standard K-FAC but introduce adaptive weight noise.","K-BFGS and K-BFGS(L) [19] apply BFGS [12] and L-BFGS [34] in an analogous method to K-FAC (e.g., block-diagonal, Kronecker-factored).","These works have shown better-than-K-FAC performance in many small scale (e.g., single GPU) and small dataset/model (e.g., MNIST or CIFAR-10 with VGG16) cases.","Kronecker-factor based FIM approximation variants are a growing area of research; however, large-scale studies have largely been limited to standard K-FAC.","KAISA introduces a valuable, unified design paradigm that can be applied to these K-FAC variants to efficiently deploy and evaluate their effectiveness on large models at scale."],["Conclusion","We have presented KAISA , a K-FAC-enabled, Adaptable, Improved, and ScAlable second-order optimizer framework.","To enable scalable DNN training, KAISA adapts memory and communication usage, and appropriately distributes the complex K-FAC computations, to best suit the model and hardware characteristics.","We design KAISA to be adaptable to hardware with limited memory (e.g., gaming GPUs) or environments with high communication costs (e.g., Ethernet or massively parallel).","We study the fundamental tradeoff between data access on local and remote memory, and evaluate KAISA's correctness and impact on training time by using four real-world applications.","With the same global batch size, our experiments show a 18.1–36.3% training time reduction for ResNet-50, Mask R-CNN, U-Net, and BERT-Large, while preserving convergence to the baseline.","Under the same memory budget, ResNet-50 and BERT phase 2 converge to the baseline in 32.5% and 41.6% less time compared to momentum SGD and Fused LAMB.","In high-communication applications, such as ResNet models, we show that extra processor memory can be used to improve iteration times by reducing communication.","In low-communication applications, such as Mask R-CNN and BERT-Large, we show the optimal KAISA usage and efficient scaling on par with SGD up to 128 GPUs."]],"string":"[\n [\n \"KAISA: An Adaptive Second-Order Optimizer Framework for Deep Neural\\n Networks\"\n ],\n [\n \"Abstract Kronecker-factored Approximate Curvature (K-FAC) has recently been shown to converge faster in deep neural network (DNN) training than stochastic gradient descent (SGD); however, K-FAC's larger memory footprint hinders its applicability to large models.\",\n \"We present KAISA, a K-FAC-enabled, Adaptable, Improved, and ScAlable second-order optimizer framework that adapts the memory footprint, communication, and computation given specific models and hardware to improve performance and increase scalability.\",\n \"We quantify the tradeoffs between memory and communication cost and evaluate KAISA on large models, including ResNet-50, Mask R-CNN, U-Net, and BERT, on up to 128 NVIDIA A100 GPUs.\",\n \"Compared to the original optimizers, KAISA converges 18.1-36.3% faster across applications with the same global batch size.\",\n \"Under a fixed memory budget, KAISA converges 32.5% and 41.6% faster in ResNet-50 and BERT-Large, respectively.\",\n \"KAISA can balance memory and communication to achieve scaling efficiency equal to or better than the baseline optimizers.\",\n \"KAISA is open source and available at https://github.com/gpauloski/kfac_pytorch.\"\n ],\n [\n \"Introduction\",\n \"Deep neural networks (DNNs) have driven breakthroughs in many research domains, including image classification [22], [24], object detection and segmentation [21], machine translation [53], and language modeling [17].\",\n \"DNNs are typically trained with stochastic gradient descent (SGD), or variants thereof.\",\n \"As models and training datasets become larger, training must increasingly be performed in parallel on many CPUs, GPUs, or TPUs [56], [30].\",\n \"For example, the BERT [17] model with 330 million parameters would take weeks to months to train on a single GPU, and GPT-3 [11] with 175 billion parameters cannot fit in the memory of any commercially available GPU.\",\n \"While distributed training on many processors can reduce training time, the need to communicate weight updates and other information among processors can limit scalability [39].\",\n \"Figure: SGD vs. K-FAC training for ResNet-32 with the CIFAR-10 dataset.\",\n \"K-FAC reduces iterations needed for convergence.Recent theoretical [36], [20], [35] and empirical [8], [42], [50] studies have shown that Kronecker-factored Approximate Curvature (K-FAC) can accelerate training by enabling convergence with fewer iterations than SGD—for example, achieving baseline validation accuracy in 40% fewer epochs than SGD on ResNet-32 with the CIFAR10 dataset (see Figure REF ).\",\n \"Technically, researchers use K-FAC to approximate the inverse of the Fisher Information Matrix (FIM)—an approximation of the Hessian—and precondition the gradients before parameter updates [36].\",\n \"The K-FAC approximation is 1) compute intensive, due to the required inverse or eigen decomposition calculations with $O(N^3)$ complexity ($N$ is the number of parameters); 2) memory intensive, as it stores per-layer activations and gradients, which may take $O(N^2)$ space compared to $O(N)$ for SGD; and 3) communication intensive for distributed training because the Kronecker factors and accompanying inverses or eigen decompositions must be communicated to all workers.\",\n \"To overcome these overheads and achieve faster training times than SGD at scale, K-FAC updates are often decoupled from gradient preconditioning so the expensive computations are performed less frequently.\",\n \"Existing K-FAC systems either cache all Kronecker factor eigen decompositions needed to precondition gradients locally [44] or distribute gradient preconditioning across processors [42].\",\n \"Neither approach enables K-FAC DNN deployments that are both memory and communication efficient, thus inhibiting scalability.\",\n \"The first approach avoids communication when local memory is sufficient.\",\n \"The second approach reduces memory footprint by not caching eigen decompositions locally but increases communication.\",\n \"These various considerations make it difficult to use K-FAC efficiently in practice given the specific requirements of the DNN model or hardware and require users to choose between optimizing memory or communication.\",\n \"To address these concerns, we propose KAISA , a K-FAC-enabled, Adaptable, Improved, and ScAlable second-order optimizer framework that can adapt execution to a given model size and memory limit.\",\n \"KAISA allows tuning the ratio between communication and memory footprint to optimally apply K-FAC to distributed training by controlling the fraction of processes with local access to data.\",\n \"This adaptation is made possible by grouping the processes, then distributing and communicating data within each group, as detailed in §REF and §REF .\",\n \"K-FAC distribution strategies described in other work are effectively special cases of KAISA in which either all processes [44] or one process [42] cache data.\",\n \"We evaluate KAISA on a range of classification, segmentation, and language modeling applications, including ResNet-50 [22], Mask R-CNN [21], U-Net [46], and BERT [17], on clusters of 448 NVIDIA Tesla V100s and 192 Ampere A100 GPUs.\",\n \"Our results show that: 1) with fixed global batch size, KAISA trains deep neural networks 18.1–36.3% faster than the original optimizers for these applications; 2) with a fixed memory budget, KAISA trains ResNet-50 and BERT-Large phase 2 in 32.5% and 41.6% less time compared to momemtum SGD and Fused LAMB, respectively; 3) by varying the number of gradient workers, the K-FAC memory overhead can be reduced by 1.5–$2.9\\\\times $ ; 4) for high communication applications such as ResNet-50, extra processor memory can be used to train 24.4% faster; and 5) using state-of-the-art NVIDIA A100 GPUs, KAISA converges in fewer iterations at all scales, and KAISA 's scaling performance is on-par with SGD even with KAISA 's additional communication overhead.\",\n \"Our code is open source with the MIT license and available at https://github.com/gpauloski/kfac_pytorch.\",\n \"Our contributions in this paper are: An adaptive second-order optimizer framework that trains faster than SGD and its variants, while preserving convergence.\",\n \"A quantitative study of the tradeoff between memory footprint and communication and its impact on training time in K-FAC design.\",\n \"The first large scale evaluation of K-FAC convergence and speedup relative to SGD for Mask R-CNN, U-Net, and BERT on up to 128 A100 GPUs.\",\n \"The rest of the paper is as follows.\",\n \"We present the mathematical background and distributed implementation of K-FAC in §.\",\n \"We describe KAISA 's design in § and implementation in §.\",\n \"We present our experiments and results in §.\",\n \"In §, we summarize existing DNN optimization frameworks and memory management techniques.\",\n \"Finally, we conclude in §.\"\n ],\n [\n \"Background\",\n \"We first introduce K-FAC and its distributed implementation.\",\n \"K-FAC is an efficient approximation of the Fisher Information Matrix (FIM), which has been shown to be equivalent to the Generalized Gauss-Newton (GGN) matrix in specific cases and can be viewed as an approximation of the Hessian [36].\",\n \"In a standard SGD update step, the weights are updated using the gradients of the loss, as illustrated in Equation REF .\",\n \"The K-FAC update step in Equation uses the FIM $F$ to precondition the gradients prior to update [44].\",\n \"$w^{(k)}$ is the weight at iteration $k$ , ${\\\\alpha }^{(k)}$ is the learning rate at iteration $k$ , $n$ is the mini-batch size, $\\\\nabla {L_i}(w^{(k)})$ is the gradient of the loss function $L_i$ for the $i^\\\\text{th}$ example with regard to $w^{(k)}$ , and $F^{-1}(w^{(k)})$ is the inverse of the FIM.\",\n \"$\\\\text{SGD: } w^{(k+1)} &= w^{(k)} - \\\\frac{{\\\\alpha }^{(k)}}{n}\\\\sum _{i=1}^{n}\\\\nabla {L_i}(w^{(k)}) \\\\\\\\\\\\text{K-FAC: } w^{(k+1)} &= w^{(k)} - \\\\frac{{\\\\alpha }^{(k)}F^{-1}(w^{(k)})}{n}\\\\sum _{i=1}^{n} \\\\nabla {L_i}(w^{(k)}) $ It has been shown empirically that training DNNs with the K-FAC second-order method enables convergence with fewer iterations than with SGD alone.\",\n \"Theoretical understandings of the convergence rates of natural gradient methods, such as K-FAC, are an area of active research.\",\n \"Previous work has shown that K-FAC [57] has linear convergence to the global minimum given a sufficiently over-parameterized model.\",\n \"In strongly-convex problems, natural gradient methods have a quadratic convergence rate compared to the linear convergence of SGD [10].\",\n \"The strongly-convex case provides some understanding of how K-FAC can improve convergence in non-convex cases.\",\n \"Further, it has been shown that natural gradient methods enable larger learning rates improving the rate of convergence [57].\",\n \"K-FAC makes greater per-iteration progress in minimizing the objective function at the cost of more computationally expensive iterations.\"\n ],\n [\n \"K-FAC Approximation\",\n \"K-FAC is based on the Kronecker product, a block matrix factorization that can reduce a large matrix inverse into two smaller matrix inverses.\",\n \"K-FAC exploits the properties of the Kronecker product and the geometry of the FIM for DNNs to greatly reduce the complexity of computing the approximate FIM inverse.\",\n \"The Kronecker product is written as $A \\\\otimes B$ where $A$ has size $m\\\\times n$ and $B$ has size $p \\\\times q$ .\",\n \"The resulting matrix has shape $mp\\\\times nq$ .\",\n \"$A \\\\otimes B=\\\\begin{bmatrix}a_{11}B & \\\\dots & a_{1n}B \\\\\\\\\\\\vdots & \\\\ddots & \\\\vdots \\\\\\\\a_{m1}B & \\\\dots & a_{mn}B\\\\end{bmatrix}$ The Kronecker product has two convenient properties: $(A \\\\otimes B)^{-1} &= A^{-1} \\\\otimes B^{-1} \\\\\\\\(A \\\\otimes B)\\\\vec{c} &= B^\\\\top \\\\vec{c} A .$\"\n ],\n [\n \"K-FAC Approximation\",\n \"The FIM for a deep neural network is a block matrix where each block maps to layers in the model.\",\n \"The block corresponding to the $i$ and $j^{\\\\text{th}}$ layers is approximated as $F_{i,j}\\\\approx a_{i-1}a_{j-1}^\\\\top \\\\otimes g_{i}g_{j}^\\\\top .$ Here, $a_{i-1}$ and $g_i$ are the activation of the $i-1^\\\\text{th}$ layer and the gradients of the $i^\\\\text{th}$ layer in the model, respectively [36], [44].Formally, $F$ represents an expected value, and the expectation of a Kronecker product is not equivalent to the Kronecker product of the expected factors.\",\n \"However, this approximation still reasonably represents the structure of the FIM [36].\",\n \"A fundamental assumption of K-FAC is the independence between layers [36], [44].This assumption is sufficient to produce an effective approximation for $F$ and necessary to produce a tractable algorithm.\",\n \"Using this assumption, K-FAC approximates the FIM as a diagonal block matrix $\\\\hat{F}$ .\",\n \"$\\\\hat{F}=\\\\texttt {diag}(\\\\hat{F}_1,...,\\\\hat{F}_i,...,\\\\hat{F}_L)$ As shown in Figure REF , the inverse of $F$ is a diagonal block matrix composed of the inverses of each diagonal block $\\\\hat{F}_i$ : $\\\\hat{F}^{-1}=\\\\texttt {diag}(\\\\hat{F}_1^{-1},...,\\\\hat{F}_i^{-1},...,\\\\hat{F}_L^{-1})$ where $\\\\hat{F}_{i}= a_{i-1}a_{i-1}^\\\\top \\\\otimes g_{i}g_{i}^\\\\top =A_{i-1}\\\\otimes G_i.\",\n \"$ We refer to $A_{i-1}$ and $G_{i}$ as the Kronecker factors.\",\n \"In practice, $A_{i-1}$ and $G_{i}$ are estimated with a running average of the factors computed over training batches [36], [44].\",\n \"Due to the layer-wise independence of K-FAC, the gradient preconditioning and weight update for a single layer $i$ at iteration $k$ can be written as: $w_i^{(k+1)}=w_i^{(k)}-\\\\alpha ^{(k)}\\\\hat{F}_i^{-1}\\\\nabla L_i(w_i^{(k)}).$ We can apply properties REF and to reduce the gradient preconditioning, $\\\\hat{F}_i^{-1}\\\\nabla L_i(w_i^{(k)})$ , to an efficient form where the smaller Kronecker factors, rather than the large FIM, are inverted.\",\n \"$\\\\hat{F}_i^{-1}\\\\nabla L_i(w_i^{(k)})=G_i^{-1}\\\\nabla L_i(w_i^{(k)})A_{i-1}^{-1}$ Tikhonov regularization is used to avoid ill-conditioned matrix inverses with K-FAC by adding a damping parameter $\\\\gamma $ to the diagonal of $\\\\hat{F}_i$ [20], [42].\",\n \"In most implementations, instead of computing $\\\\hat{F}^{-1}$ , we compute $(\\\\hat{F}_i+\\\\gamma I)^{-1}$ as: $(\\\\hat{F}_i+\\\\gamma I)^{-1}=(A_{i-1}+\\\\gamma I)^{-1}\\\\otimes (G_i+\\\\gamma I)^{-1}.\",\n \"$ Thus, the final update step for the parameters of layer $i$ at iteration $k$ is: $w_i^{(k+1)}=&w_i^{(k)}-\\\\alpha ^{(k)}(\\\\hat{F}_i+\\\\gamma I)^{-1}\\\\nabla L_i(w_i^{(k)}) \\\\\\\\=&w_i^{(k)}-\\\\alpha ^{(k)}(G_i+\\\\gamma I)^{-1}\\\\nabla L_i(w_i^{(k)})(A_{i-1}+\\\\gamma I)^{-1}.$\"\n ],\n [\n \"Alternative Approximation\",\n \"It has been shown that an empirically more stableIn this context, stable means produces more consistent validation results across batch sizes and hyperparameter settings.\",\n \"approximation for $(\\\\hat{F}+\\\\gamma I)^{-1}\\\\nabla L_i(w_i^{(k)})$ can be computed using an eigen decomposition of the Kronecker factors, $A$ and $G$ , from Equation REF [20], [44].\",\n \"Given $Q_A$ and $Q_G$ , the eigenvectors of the factors, and $\\\\upsilon _A$ and $\\\\upsilon _G$ , the eigenvalues of the factors, the preconditioned gradient can be computed as follows.\",\n \"$V_1&=Q_G^\\\\top \\\\nabla L_i(w_i^{(k)}) Q_A \\\\\\\\V_2&=V_1/(\\\\upsilon _G\\\\upsilon _A^\\\\top +\\\\gamma ) \\\\\\\\(\\\\hat{F}_i+\\\\gamma I)^{-1}\\\\nabla L_i(w_i^{(k)})&=Q_GV_2Q_A^\\\\top $ The composition of the factors $A$ and $G$ in Equation REF guarantees the factors are symmetric and therefore the factor eigen decompostions have real eigenvalues and orthogonal eigenvectors.\",\n \"In this work, we use the eigen decomposition method for gradient preconditioning.\"\n ],\n [\n \"Infrequent K-FAC Updates\",\n \"A common strategy in second-order optimization methods is to update the second-order information every few iterations [36], [42], [44].\",\n \"Intuitively, second-order information does not change as rapidly from one iteration to the next like first-order information.\",\n \"The K-FAC update interval parameter controls the number of iterations between second-order updates, i.e., iterations between eigen decomposition recomputations.\",\n \"Larger K-FAC update intervals result in more stale information, so tuning this parameter is key to achieving fast training with K-FAC.\",\n \"Practically, K-FAC can maintain convergence with K-FAC update intervals of 100–2000 iterations [44].\",\n \"Figure: Hybrid-parallel distributed K-FAC implementation overview.Blue boxes are standard computations in data-parallel training.Red boxes are computations required by K-FAC.Workers maintain identical copies of the model.The output of each layer zz is computed during the forward pass for the local batch xx.Then, the loss between the true output yy and predicted output z L z_L is calculated and used to compute gradients in the backward pass.The gradients are then allreduced across workers.During the forward/backward pass, K-FAC caches intermediate data for computing factors.In the K-FAC stage, workers compute factors AA and GG in data-parallel and allreduce the results.The eigen decompositions and preconditioned gradients 𝒢\\\\mathcal {G} are computed in the K-FAC approximation stage.The existing K-FAC methods, MEM-OPT and COMM-OPT, implement the model-parallel stage differently as shown at the bottom of the figure.After the K-FAC stage, all workers have 𝒢\\\\mathcal {G} and can update weights locally using 𝒢\\\\mathcal {G} and a standard optimizer (e.g., SGD or ADAM).\"\n ],\n [\n \"Distributed Implementation\",\n \"Existing distributed K-FAC implementations are hybrid-parallel, with first-order information (e.g., gradients) computed in data-parallel and second-order information (e.g., K-FAC approximation) in model-parallel.\",\n \"Figure REF outlines the model-parallel K-FAC computation performed between standard data-parallel training steps.\",\n \"We refer to the two existing distributed K-FAC implementation strategies as MEM-OPT (memory optimized K-FAC) and COMM-OPT (communication optimized K-FAC).\",\n \"Both work by replicating the DNN across all processes and assigning a random local batch of training data to each process at each iteration.\",\n \"The data-parallel forward pass, backward pass, and SGD weight update stages outlined in Figure REF are the same in both methods.\",\n \"The key differences between the two approaches are the model-parallel computation implementations in the gradient preconditioning stage.\"\n ],\n [\n \"MEM-OPT\",\n \"MEM-OPT [42] assigns each layer in the DNN to a different process during the preconditioning stage.\",\n \"Each process computes the eigen decompositions of $A_{i-1}$ and $G_i$ needed for preconditioning the gradients using Equations REF –.\",\n \"Each process then broadcasts the preconditioned gradients to all processes such that subsequent SGD weight updates can be done in data-parallel.\",\n \"MEM-OPT has a low memory footprint because no eigen decompositions are duplicated: each layer's eigen decompositions are stored on only one process.\",\n \"MEM-OPT has communication in three places: a) gradient allreduce, b) factor allreduce, and c) preconditioned gradient broadcast, all shown in Figure REF .\",\n \"In non-K-FAC update steps (e.g., steps where the eigen decompositions are not updated), MEM-OPT avoids the factor allreduce because eigen decompositions from a previous step are reused; however, the preconditioned gradient broadcast is still required because the gradient being preconditioned changes every iteration.\"\n ],\n [\n \"COMM-OPT\",\n \"COMM-OPT [44] decouples the eigen decomposition from gradient preconditioning.\",\n \"Instead of assigning each layer to a process, individual factors are assigned to a process to be eigen decomposed in parallel.\",\n \"The eigen decompositions are broadcast back to all processes such that every process holds a copy of all eigen decompositions.\",\n \"Each process computes all preconditioned gradients locally prior to weight updates.\",\n \"COMM-OPT has a larger memory-footprint because every process must maintain a copy of the eigen decompositions.\",\n \"COMM-OPT has communication in three places: a) gradient allreduce, b) factor allreduce, and c) eigen decomposition broadcast, also shown in Figure REF .\",\n \"Decoupling the eigen decompositions from the gradient preconditioning achieves two goals: 1) $A_{i-1}$ and $G_i$ can be computed in different processes, doubling the maximum worker utilization, and 2) in non-K-FAC update steps, the communication in (b) and (c) can be avoided because every worker can locally precondition gradients with the cached eigen decompositions.\",\n \"Thus, in non-K-FAC update intervals, COMM-OPT has no additional communication overhead compared to SGD.\",\n \"COMM-OPT has been shown to be 4–16% faster than MEM-OPT for ResNet-50 training on 16–256 V100 GPUs [44].\",\n \"These two implementations convey the fundamental tradeoff between caching the decompositions locally to avoid communication and communicating the preconditioned gradients every iteration to avoid additional memory overheads.\",\n \"This tradeoff impacts the communication, computation, and memory overhead of K-FAC and motivates our research in this paper.\"\n ],\n [\n \"Design\",\n \"We introduce features of KAISA 's design that enable its tunable memory footprint and explain how KAISA performs load balancing and half precision training to improve scalability.\"\n ],\n [\n \"Tunable Memory Footprint\",\n \"KAISA introduces HYBRID-OPT, a distributed K-FAC strategy that achieves variable memory overhead in K-FAC.\",\n \"In this strategy, each layer has a subset of the processes assigned to be gradient preconditioners, referred to as the gradient workers.\",\n \"grad_worker_frac defines the size of this subset, i.e., the gradient worker count is $\\\\texttt {max}(1, \\\\textit {grad\\\\_worker\\\\_frac}\\\\times \\\\textit {world\\\\_size})$ .\",\n \"One of the gradient workers is also responsible for computing the eigen decompositions for the layer and broadcasting the results to the remaining gradient workers.\",\n \"At the end of the eigen decomposition stage, each gradient worker has a copy of the eigen decompositions and can precondition the gradient.\",\n \"Gradient receivers are the processes not assigned as gradient workers for the layer.\",\n \"Each gradient worker is responsible for broadcasting the preconditioned gradient to a subset of the gradient receivers.\",\n \"With multiple gradient workers, the broadcast groups are smaller and simultaneous broadcasts are possible, reducing overall communication time.\",\n \"For example, given two gradient workers and two gradient receivers, each gradient worker sends the preconditioned gradients to just one receiver; both gradient workers perform this communication at the same time.\",\n \"After the gradient broadcast, all processes have a copy of the preconditioned gradient with which local weights can be updated.\",\n \"Observe that KAISA unifies existing distributed K-FAC strategies because COMM-OPT and MEM-OPT are special cases of HYBRID-OPT.\",\n \"COMM-OPT is the case where grad_worker_frac = 1 (all processes precondition the layer's gradient), and MEM-OPT is the case where grad_worker_frac = 1/world_size (a single process preconditions a layer's gradient and broadcasts the result).\",\n \"Figure REF compares MEM-OPT, COMM-OPT, and HYBRID-OPT in an eight process environment.\",\n \"As grad_worker_frac increases, more processes cache the eigen decompositions and the memory footprint increases.\",\n \"Visually, the number of processes that cache the eigen decompositions is represented by the processes in the dashed red box in Figure REF .\",\n \"Continuing with Figure REF , HYBRID-OPT has a lower preconditioned gradient broadcast cost than MEM-OPT because the broadcast groups are smaller and broadcasts are overlapped.\",\n \"HYBRID-OPT requires four separate broadcasts to groups of size two in comparison to MEM-OPT which requires one large broadcast to a group of eight.\",\n \"Since these broadcasts involve non-overlapping processes, we can execute all four broadcast calls simultaneously in the HYBRID-OPT case.\",\n \"Given the complexity of broadcasting using the minimum spanning tree algorithm is $O(\\\\log p)$ , where $p$ is the number of processes in the broadcast group, the complexity is reduced from $O(\\\\log 8)$ in MEM-OPT to $O(\\\\log 2)$ in HYBRID-OPT.\",\n \"Note in steps where the eigen decompositions are updated, HYBRID-OPT incurs an additional eigen decomposition broadcast with $O(\\\\log 4)$ complexity; however, as mentioned in §REF , eigen decompositions are updated infrequently so the average complexity for HYBRID-OPT is still less than that of MEM-OPT.\",\n \"The problem addressed by KAISA's HYBRID-OPT strategy is akin to that of 2.5D matrix multiplication [48].\",\n \"Both algorithms can utilize extra processor memory to reduce communication costs by controlling the replication factor of data.\",\n \"In 2.5D matrix multiplication, a parameter $c$ determines the number of data copies, and in KAISA, the grad_worker_frac determines the number of workers that store the eigen decompositions.\"\n ],\n [\n \"Greedy Factor Distribution\",\n \"The eigen decompositions required for K-FAC are expensive to compute.\",\n \"Distributed K-FAC optimizes this stage by computing eigen decompositions in a model-parallel fashion.\",\n \"To most efficiently use available resources, the eigen decomposition computations should be distributed in a manner that minimizes the makespan $T$ , the time it takes for all processes to complete their assigned computations.\",\n \"We use the longest processing time greedy algorithm which produces an assignment with makespan $T\\\\le \\\\frac{3}{2}T^*$ where $T^*$ is the optimal makespan [28].\",\n \"The longest processing time algorithm sorts jobs by decreasing length and then iteratively assigns each job to the worker with the lowest current work load.\",\n \"For each factor to be eigen decomposed, the processing time is approximated as $O(N^3)$ where each factor is an $N\\\\times N$ matrix [16].\",\n \"Alternatively, memory usage can be optimized for by using $O(N^2)$ as the approximation since $N^2$ is the size of the factor.\",\n \"KAISA uses the longest processing time strategy at the start of training to assign each factor to a process.\"\n ],\n [\n \"Half Precision Storage and Computation\",\n \"Mixed precision training is a common strategy to reduce training times and memory usage on supported hardware [37].\",\n \"Popular frameworks for automatic mixed precision (AMP) training, such as NVIDIA AMP and PyTorch AMP, cast forward and backward pass operations to half precision where possible.\",\n \"Gradient calculations in the backward pass can often produce very small values that would be clipped when cast to half precision, so these frameworks scale gradients to a larger value during the backward pass and unscale the gradients appropriately before the optimization step.\",\n \"KAISA adapts to the precision being used for training.\",\n \"When training with AMP, factors are stored in half precision, reducing memory costs.\",\n \"K-FAC computations are performed in half precision where possible.\",\n \"Eigen decompositions are generally unstable in half precision so factors are cast to single precision before decomposition.\",\n \"KAISA can store eigen decompositions in half precision to further reduce memory consumption if needed.\",\n \"Half precision training has become a staple in achieving state-of-the-art results in deep learning, and KAISA can particularly benefit from half precision training due to the K-FAC computation and communication overhead.\"\n ],\n [\n \"K-FAC Usage\",\n \"PythonStyle model = DistributedDataParallel(model) optimizer = optim.SGD(model.parameters(), ...) preconditioner = KFAC(model, grad_worker_frac=0.5) for data, target in train_loader: optimizer.zero_grad() output = model(data) loss = criterion(output, target) loss.backward() preconditioner.step() optimizer.step() We design KAISA to implement K-FAC as a preconditioner to standard optimizers with support for Conv2d and Linear layers.\",\n \"KAISA has an easy-to-use interface and can be incorporated into existing training scripts in two lines: one to initialize and one to call KFAC.step() prior to optimizer.step() (see Listing ).\",\n \"KAISA automatically registers the model and determines the distributed communication backend (e.g., Torch, Horovod, single-process).\",\n \"A call to KFAC.step(): 1) computes the factors using the forward/backward pass data, 2) computes the eigen decompositions in parallel and broadcasts the results, 3) computes the preconditioned gradients and broadcasts the results if necessary, and 4) scales the preconditioned gradients.\",\n \"We implement KAISA using PyTorch [43], with communication, interface, large-batch training, and gradient preconditioning implemented to collectively enable efficient second-order optimization.\"\n ],\n [\n \"AMP and Distributed Training\",\n \"We use PyTorch AMP for mixed precision training.\",\n \"With PyTorch AMP, the GradScaler object responsible for scaling and unscaling the gradient in the backward pass can be passed to KAISA .\",\n \"KAISA uses the GradScaler to correctly unscale the $G$ factors, since the scale factor can change from iteration to iteration, causing problems when computing the running average of $G$ over the course of training.\",\n \"All communication operations are performed in the precision of the data to reduce bandwidth requirements.\",\n \"KAISA supports torch.distributed and Horovod [47] for distributed training.\",\n \"In this work, we use torch.distributed for all experiments because the DistributedDataParallel model wrapper overlaps gradient communication with backpropogation, works seamlessly with PyTorch AMP, and provides a broadcast group abstraction needed for HYBRID-OPT.\"\n ],\n [\n \"Factor Accumulation\",\n \"Processor memory (e.g., GPU VRAM) limits the maximum per-processor batch size during training.\",\n \"A common strategy to achieve larger effective batch sizes is gradient accumulation, a method where gradient values for multiple forward and backward passes are accumulated between optimization steps.\",\n \"For example, if processor memory limits the batch size to 8, an effective batch size of 32 can be achieved by accumulating gradients over four forward/backward passes prior to the optimization step.\",\n \"This strategy is common in applications such as BERT, where effective batch sizes are often $>2^{15}$ , but modern GPUs are limited to local batch sizes $<2^7$ .\",\n \"The forward/backward pass data needed by KAISA to compute the factors is accumulated over each mini-batch between calls to KFAC.step().\",\n \"As the number of gradient accumulation steps increases, the memory needed to accumulate the forward/backward pass data grows linearly.\",\n \"KAISA can efficiently support gradient accumulation by computing the factors for the current mini-batch during the forward/backward pass instead of during KFAC.step().\",\n \"The factor communication is still performed during KFAC.step().\"\n ],\n [\n \"Triangular Factor Communication\",\n \"Previous K-FAC work has exploited the symmetric nature of the Kronecker factors to reduce communication volume by sending only the upper triangle for each factor [49], [50].\",\n \"Our implementation supports extracting the upper triangle for the factor allreduce and reconstructing the full factor before the eigen decomposition stage.\",\n \"For the models studied in this work, this optimization did not yield performance improvements for two reasons.\",\n \"First, network latency impacted overall communication time more than bandwidth; second, this optimization has an additional overhead for extracting the upper triangle and reconstructing the factor.\",\n \"For models with larger individual layers—and therefore factors—this optimization could yield greater benefits.\"\n ],\n [\n \"Gradient Preconditioning\",\n \"The largest overhead of K-FAC in non-K-FAC update steps is computing the preconditioned gradients.\",\n \"This process, described by Equations REF –, involves a series of matrix additions, divisions, and multiplications.\",\n \"Observe that the gradient $\\\\nabla L_i(w_i^{(k)})$ is the only variable that changes between non-K-FAC update iterations.\",\n \"In particular, the computation involving the outer product of the eigenvalues, $1/(\\\\upsilon _G\\\\upsilon _A^\\\\top +\\\\gamma )$ , in Equation does not need to be recomputed every iteration—only after the eigen decompositions are updated.\",\n \"We also observe that when $\\\\emph {grad\\\\_worker\\\\_frac}>1/world\\\\_size$ , multiple processes perform this computation redundantly.\",\n \"To reduce the total number of operations during the preconditioning stage, we move the computation of the outer product into the eigen decomposition stage.\",\n \"The process assigned to eigen decompose $G$ computes $1/(\\\\upsilon _G\\\\upsilon _A^\\\\top +\\\\gamma )$ and broadcasts the result to all gradient workers instead of broadcasting $\\\\upsilon _A$ and $\\\\upsilon _G$ .\",\n \"This ensures that $1/(\\\\upsilon _G\\\\upsilon _A^\\\\top +\\\\gamma )$ is computed once (on a single worker) and then reused many times by other workers.\",\n \"In practice, this reduced the time to precondition the gradients for a single layer by up to 53%.\"\n ],\n [\n \"Experiments\",\n \"We report on experiments that address four issues.\",\n \"1) Convergence to baseline evaluation metrics and 2) time to convergence with and without KAISA to validate the design and implementation.\",\n \"3) An exploration of the memory and communication tradeoff using KAISA to develop a quantitative understanding of the impact of the grad_worker_frac configuration.\",\n \"4) An evaluation of KAISA's scaling performance.\"\n ],\n [\n \"Hardware and Software Stack\",\n \"We performed experiments on two computers.\",\n \"The first is the GPU subsystem of the Frontera supercomputer at the Texas Advanced Computing Center, which has 112 nodes, each powered by IBM Power9 processors, with four 16 GB NVIDIA V100 GPUs (448 GPUs in total).\",\n \"Nodes are connected by an InfiniBand EDR network.\",\n \"We use PyTorch 1.6, CUDA 10.2, CUDNN 7.6.5, and NCCL 2.5.6, and MVAPICH2-GDR 2.3.4 to launch processes on multiple nodes for distributed training.\",\n \"The second is the GPU subsystem of the Theta supercomputer at Argonne National Laboratory.\",\n \"This system has 24 NVIDIA DGXA100 nodes with eight 40GB A100 GPUs each (192 A100 GPUs in total).\",\n \"On DGXA100 nodes, we use PyTorch 1.7, CUDA 11.0, CUDNN 8.0.4, and NCCL 2.7.8.\",\n \"We distinguish between these two systems in the following text by specifying either V100 or A100 GPUs.\"\n ],\n [\n \"Applications\",\n \"We evaluate KAISA for classification, segmentation, and language modeling applications.\",\n \"Classification: We use ResNet-50 [22] with the ImageNet-1k dataset [29], which has 1000 categories with approximately 1.3M training images and 50K validation images.\",\n \"We use K-FAC to precondition all convolutional and linear layers in ResNet-50 and SGD for the weight updates.\",\n \"Segmentation: We explore two segmentation tasks.\",\n \"First, we use the NVIDIA reference PyTorch implementation of Mask R-CNN [21], [5] with the Common Objects in Context (COCO) 2014 dataset [32].\",\n \"We use K-FAC to precondition the convolutional and linear layers in the region of interest (ROI) heads of Mask R-CNN and SGD for the weight updates.\",\n \"Second, we use a U-Net [46] architecture for segmenting brain tumor sub-regions.\",\n \"We extend a Kaggle competition implementation [2] to enable multi-GPU training.\",\n \"The test case was run on the LGG Segmentation Dataset [3], which contains Magnetic Resonance (MR) images of the brain from 110 patients across five hospitals.\",\n \"Images from a random subset of 100 patients are used as the training dataset and the remaining 10 patients are used for validation.\",\n \"We apply K-FAC to all convolutional layers in the model.\",\n \"Language Modeling: We train the BERT-Large Uncased model using a modified version of the NVIDIA reference PyTorch implementation for BERT [17], [5] with the English Wikipedia [52] and Toronto BookCorpus datasets [59].\",\n \"Each transformer in BERT is implemented using a series of Linear layers, and we apply K-FAC to all linear layers.\",\n \"We do not use K-FAC to precondition the embedding layer and prediction head because both of these layers have a Kronecker factor with shape $vocab\\\\_size\\\\times vocab\\\\_size$ , and since the vocab size for BERT-Large is 30K, these factors cannot be efficiently eigen decomposed.\",\n \"Fused LAMB is used as the optimizer [56].\",\n \"The strategy outlined in §REF is used to reduce memory consumption since gradient accumulation is used for training.\"\n ],\n [\n \"Convergence with Fixed Batch Size\",\n \"We compare KAISA performance (converged accuracy, epochs to convergence, and time to convergence) on ResNet-50, Mask R-CNN, U-Net, and BERT-Large against the baseline implementations listed in Table REF .\",\n \"For ResNet-50 and Mask R-CNN, we use MLPerf benchmark target results [4].\",\n \"For U-Net, we use the baseline validation Dice similarity coefficient (DSC) from the model's GitHub repository [2].\",\n \"For the BERT-Large baseline, researchers have reported F1 scores of 91.08% [5], 91.0% [1], and 90.4% [56] for fine-tuning on the SQuAD v1.1 dataset.\",\n \"We use the best reported F1 score (i.e., the NVIDIA implementation [5]) with the LAMB optimizer.\",\n \"Due to the partial unavailability of the Toronto BookCorpus training dataset, our measurement only converges to 90.8%.\",\n \"(The Toronto BookCorpus dataset is no longer available online as a holistic package; we could recover only 14,155 of the 16,846 books.)\",\n \"Table: Baseline performance and hardware summary for ResNet-50, Mask R-CNN, U-Net, and BERT-Large.\",\n \"“val acc” is validation accuracy.\",\n \"“mAP” is mean average precision.\",\n \"“DSC” is Dice similarity coefficient.In all comparisons, we use the same global batch size for KAISA and the original optimizers to isolate the improvement from second-order information.\",\n \"Table REF summarizes the hyperparameters for each application.\",\n \"For ResNet and K-FAC specific hyperparameters, we use values from [44] to provide direct performance comparisons to previous K-FAC works.\",\n \"For Mask R-CNN and BERT-Large, we use the NVIDIA reference hyperparameters [5].\",\n \"Further performance improvements could be gained through more extensive hyperparameter tuning; however, this was not necessary to achieve results better than the original optimizers.\",\n \"Table: Summary of hyperparameters used for each application.\",\n \"BS = global batch size, LR = learning rate, WU = warm up iterations, K_freq = number of iterations between eigen decomposition re-computations, F_freq = iterations between factor updates.\",\n \"grad_worker_frac = 1 and damping = 0.003 for all cases.ResNet-50: We train ResNet-50 for 55 and 90 epochs for KAISA and momentum SGD, respectively, using FP16 precision on eight NVIDIA A100 GPUs.\",\n \"Figure REF shows the validation accuracy curve using the two optimization methods.\",\n \"KAISA converges to the baseline validation accuracy at epoch 46 and momentum SGD at epoch 65.\",\n \"The time-to-convergence is 268.1 minutes for KAISA : 24.3% less than the 354.0 minutes for momentum SGD.\",\n \"Figure: Validation Metric Curve Comparison between KAISA and SGD/ADAM on ResNet-50, Mask R-CNN, and U-Net.\",\n \"The dotted lines represent the target metric.= 0pt Mask R-CNN: Our baseline measurement of Mask R-CNN on 32 NVIDIA V100 GPUs takes 25,640 iterations to converge to 0.377 bbox mAP and 0.342 segm mAP in FP32.\",\n \"With K-FAC enabled for the ROI heads, training converges to 0.379 bbox mAP and 0.350 segm mAP in 21,000 iterations.\",\n \"The bbox mAP comparision for SGD and KAISA is shown in Figure REF .\",\n \"KAISA reduces the training time from 136.1 minutes to 115.8 minutes, a 14.9% improvement.\",\n \"With a batch size of 128 on 64 V100 GPUs, KAISA converges in 12,000 iterations compared to 15,000 with SGD and reduces training time by 18.1%.\",\n \"U-Net: The reference U-Net implementation [2] with ADAM converges to 91.0% validation DSC within 50 epochs with four NVIDIA A100 GPUs using FP32 training.\",\n \"Using KAISA , the training converges above 91.0% in 30 epochs as seen in Figure REF .\",\n \"KAISA reduces the training time by 25.4% (10.9 minutes vs. 14.6 minutes).\",\n \"BERT: BERT pretraining has two phases.\",\n \"The first trains with maximum sequence length of 128 for 7038 iterations and the second trains with maximum sequence length of 512 for 1,563 iterations.\",\n \"We fine-tune the pretrained BERT model for three epochs using the SQuAD dataset.\",\n \"All phases and fine-tuning are done in FP16.\",\n \"Tuning the hyperparameters of BERT is expensive as each run can take over 130 hours with 8 A100 GPUs, so we showcase the effectiveness of KAISA with the second phase of BERT pretraining.\",\n \"For phase two pretraining, we start with the same model pretrained with LAMB during phase one.\",\n \"We train with KAISA for {800, 1,000, 1,200} iterations then fine-tune SQuAD.\",\n \"Table REF summarizes the validation SQuAD F1 scores after fine-tuning and the pretraining performance improvements.\",\n \"Table: BERT performance comparison: KAISA vs. LAMBKAISA converges to the 90.8 F1 baseline in 800 iterations, 47.9% less iterations than required for LAMB, and KAISA takes 36.3% less time than LAMB to converge.\"\n ],\n [\n \"Convergence with Fixed Memory Budget\",\n \"To understand how a grad_worker_frac can enable second-order optimization in memory-constrained environments, we compare KAISA's performance against baselines with fixed memory budgets.\",\n \"In particular, we train ResNet-50 on 64 V100 GPUs and BERT-Large phase two on eight A100 GPUs.\",\n \"For each experiment, we use the maximum possible local batch size and measure the convergence, epochs to convergence, and time to convergence.\",\n \"Table REF summarizes the hyperparameters and time to convergence.\",\n \"Table: Time to convergence and hyperparameters for each optimizer.\",\n \"BS = global batch size, LR = learning rate, g_frac = gradient worker fraction, K_f = iterations between eigen decomposition re-computations, F_f = iterations between factor updates, T_conv is time to convergence in minutes.ResNet-50: With momentum SGD, the maximum local batch size is 128 per GPU and the global batch size is 8,192.\",\n \"We train ResNet-50 for 90 epochs using momentum SGD which achieves 75.0% validation accuracy—0.9% lower than the MLPerf baseline.\",\n \"Next, we use KAISA with grad_worker_frac = 1 and a local batch size of 80; however, the training runs out of memory.\",\n \"Lowering grad_worker_frac to 1/2, training takes 83 minutes to converge to 75.9% in the 48th epoch.\",\n \"The complete 55 epoch training takes 95 minutes and the validation accuracy reaches 76.0%.\",\n \"So, even if momentum SGD converges to 75.9% by the 90th epoch, KAISA still reduces the time to convergence by 32.5%.\",\n \"Finally, we run the same training process with MEM_OPT by setting grad_worker_frac to 1/64.\",\n \"It converges at the 47th epoch and the time to convergence is 96 minutes.\",\n \"The complete 55 epoch training takes 111 minutes.\",\n \"This experiment highlights the benefit of KAISA in cases where compute resource can not be efficiently utilized with the original optimizers.\",\n \"With a grad_worker_frac value of 1/2, KAISA offers benefits over COMM_OPT and MEM_OPT by enabling second-order optimization under a tight memory budget that is 13.9% faster than COMM_OPT and not feasible with MEM_OPT.\",\n \"BERT: With LAMB, the maximum possible local batch size per GPU is 12 for the second phase of this BERT implementation [5], and the global batch size is 24,576.\",\n \"For KAISA, we use a local batch size of 8 and global batch size of 32,768.\",\n \"This experiment uses the same hyperparameters as in §REF , so all cases with LAMB and BERT should converge to the baseline, thus we only project the training time with the first 100 steps.\",\n \"As the global batch size with LAMB is 24,576, 2,084 training steps are required to finish three epochs, and the training time is 2,917.6 minutes.\",\n \"KAISA, with grad_worker_frac = 1/2, takes 3,268.8 minutes to finish the three epochs.\",\n \"However, KAISA converges to baseline after 800 steps, as shown in Table REF , so the time to converge is 1,702.5 minutes—41.6% faster than LAMB.\",\n \"Setting grad_worker_frac = 1 takes 1,703.5 minutes to converge for KAISA.\",\n \"The performance is comparable to the case with grad_worker_frac = 1/2.\",\n \"We conduct a detailed study on this convergences phenomena for different grad_worker_frac values in §REF .\"\n ],\n [\n \"Memory vs. Communication\",\n \"To understand the impact of grad_worker_frac on training times, we train ResNet-{18, 50, 101, 152}, Mask R-CNN, and BERT-Large on 64 V100 GPUs for grad_worker_frac $\\\\in $ {1/64, 1/32, 1/16, 1/8, 1/4, 1/2, 1}.\",\n \"For each experiment, we record the average iteration time, i.e., time between weight updates, and the GPU memory usage.\",\n \"We refer to the K-FAC overhead as the memory required to store the factors and eigen decompositions, and the K-FAC overhead is computed as the difference between the K-FAC and SGD memory usage.\",\n \"For all ResNet models, we use the same hyperparameters used for ResNet-50 (Section REF ) except for ResNet-152 where the local batch size was lowered to 24.\",\n \"The results are presented in Figure REF , and a summary of the memory usage is provided in Table REF .\",\n \"The K-FAC memory overhead increases linearly as a function of grad_worker_frac for all models.\",\n \"KAISA requires 1.5–45.8% more memory than SGD depending on the application and value of grad_worker_frac (Table REF ).\",\n \"The maximum K-FAC overhead (i.e., when grad_worker_frac = 1) is 1.5–2.9$\\\\times $ that of the minimum K-FAC overhead (i.e., when grad_worker_frac = $1/64$ ).\",\n \"Table: Summary of per-GPU memory usage for training, in MB.Abs.\",\n \"is the absolute memory required for training.Δ\\\\Delta is the %-increase in memory required over SGD.The K-FAC overhead is the K-FAC abs.\",\n \"memory minus the SGD abs.\",\n \"memory.With respect to iteration times, the ResNet models scale well with the number of gradient workers with ResNet-50 scaling the best.\",\n \"For ResNet-50, the speedup from a gradient worker count of 1 to 64 is 24.4% for FP32 with a 22% increase in total memory usage.\",\n \"The average iteration times for Mask R-CNN and BERT-Large remain constant as the number of gradient workers is increased.\",\n \"For comparison, the same ResNet-50 experiment with 64 V100s in [44] only shows a 7.6% speedup when increasing the gradient worker count from 1 to 64.\",\n \"This improvement in KAISA over previous work is due to the unique contributions presented in §REF , §REF , §REF , and §REF .\",\n \"These promising results for ResNet-50, a de facto standard benchmark for deep learning systems, are important as the performance characteristics of ResNet-50 represent a large set of commonly used models (e.g.\",\n \"VGG16, U-Nets, etc.).\",\n \"We can understand why ResNet model performance varies across grad_worker_frac values while the Mask R-CNN and BERT-Large performance remains constant by considering the bandwidth requirements of these applications.\",\n \"The bandwidth required by KAISA is a function of the size of the factors, eigen decompositions, and frequency of K-FAC updates.\",\n \"With 64 V100s, ResNet-50 calls KFAC.step() frequently (4–6 calls/second) and incurs a K-FAC memory overhead between 634 MB and 1.8 GB.\",\n \"In comparison, Mask R-CNN calls KFAC.step() with a lower frequency (3 calls/second) and has a much smaller K-FAC overhead (100–200 MB).\",\n \"Thus, the changes in how KAISA communicates data with respect to grad_worker_frac are less apparent in Mask R-CNN.\",\n \"BERT-Large has the lowest bandwidth requirements of all applications even though it has the largest K-FAC overhead.\",\n \"BERT-Large uses gradient accumulation to achieve very large batch sizes (32K for phase 2) and as a result only calls KFAC.step() every $\\\\sim $ 120 seconds.\",\n \"While the iteration times for low-communication models such as BERT and Mask R-CNN are invariant to the grad_worker_frac value in KAISA, KAISA still produces faster-than-SGD training times with small increases in memory-overhead.\",\n \"This is due to KAISA's unique features outlined in § and §.\",\n \"Further, practitioners training these models at larger scales, e.g., 100s or 1000s of GPUs, where communication becomes a greater bottleneck will benefit more from the flexibility KAISA provides to adapt training to environments with increasing communication costs.\",\n \"Tuning the grad_worker_frac hyperparameter, to determine an optimal balance between iteration time and memory usage, is simple as it only requires profiling the average iteration time for each grad_worker_frac value over a few iterations.\",\n \"We find that with respect to training times, grad_worker_frac has the most impact in applications that spend a larger proportion of time doing communication.\",\n \"To further understand how grad_worker_frac impacts training times, we analyze the execution time for each section within KFAC.step() with ResNet-50 on 64 V100s.\",\n \"Figure REF provides the time spent in each section for all layers in the model during a call to KFAC.step().\",\n \"Times are averaged over 10,000 iterations and across all workers.\",\n \"Eigen decompositions are updated every 500 iterations.\",\n \"As shown in Figure REF , factor computation and communication, eigen decomposition, and scaling and updating the gradients, are invariant to the grad_worker_frac.\",\n \"Figure: Average function execution time during calls to KFAC.step() for ResNet-50 on 64 GPUs.The time required to broadcast the eigen decompositions increases substantially as the number of gradient workers is increased.\",\n \"Referring back to Figure REF , the more gradient workers there are, the more processes that need to receive the eigen decompositions.\",\n \"However, the eigen decompositions, in this case, are only recomputed every 500 iterations so the eigen decomposition broadcast has a negligible effect on the average iteration time.\",\n \"On the other hand, the gradient preconditioning and broadcast occur every iteration regardless of if the factors or eigen decompositons are updated and have the greatest influence on average iteration time.\",\n \"We see that the time to precondition the gradients increases with the gradient worker count because each process is assigned as a gradient worker for more layers.\",\n \"This discovery highlights the importance of the gradient preconditioning optimizations made in §REF .\",\n \"The time to broadcast the preconditioned gradients decreases to 0 as the gradient worker count approaches world_size, and notably, the time decreases at a faster rate than the increase in time required in the preconditioning stage.\",\n \"This trend is a result of each gradient worker needing to send the results to fewer other processes as the gradient worker count increases.\"\n ],\n [\n \"Scaling\",\n \"To examine the scaling characteristic of KAISA, we measure the average time per epoch for ResNet-50 and average time per iteration for BERT-Large phase 2.\",\n \"We choose ResNet-50 and BERT-Large because they represent a high-communication and low-communication model, respectively (see §REF ).\",\n \"We study three KAISA variants: COMM-OPT, MEM-OPT, and HYBRID-OPT with a grad_worker_frac of 1/2, and we report the projected end-to-end training time speedup for the KAISA variants over the base optimizers (SGD and LAMB) in Figure REF .\",\n \"For ResNet-50, we project the training time to 90 epochs for SGD and 55 for KAISA.\",\n \"For BERT-Large, we project the phase 2 training time to 1,563 steps for LAMB and 800 for KAISA based on the study in §REF .\",\n \"The hyperparameters in Table REF are used, and for ResNet-50, the K-FAC update frequency is scaled inversely with the global batch size to keep the number of K-FAC updates per training samples constant.\",\n \"We store and communicate the factors and eigen decompositions in FP16 for BERT-Large.\",\n \"In Figure REF and REF , MEM-OPT maintains a constant speedup over SGD and LAMB, respectively, across all scales.\",\n \"In contrast, the speedup for COMM-OPT improves in both cases as the scale increases indicating the tradeoff of more memory for reduced communication does have scaling benefits.\",\n \"HYBRID-OPT sees performance improvements on par with COMM-OPT with BERT-Large while using less GPU memory.\",\n \"Overall, HYBRID-OPT has the best balance of scaling and memory usage for large-scale BERT pretraining.\",\n \"As a whole, KAISA achieves better speedups with BERT-Large which is likely due to ResNet-50 being more communication bound than BERT-Large as noted in §REF .\",\n \"Further scaling experiments on a machine with more GPUs is needed to understand the characteristics and limits of KAISA's speedup over SGD.\",\n \"The emergence of DNNs has motivated revisiting many aspects of system design.\",\n \"TensorFlow [6], PyTorch [43], and MXNet [14] are examples of deep learning frameworks that support end-to-end training.\",\n \"TensorFlow Serving [41], DLHub [13], and Clipper [15] focus on low-latency inference serving.\",\n \"Ray [38] provides a platform to integrate simulation, training, and serving for reinforcement learning applications.\",\n \"Researchers have also designed new scheduling approaches in GPU clusters with informed hardware heterogeneity [40], sharing capability [55], and early user feedback [54] to optimize cost and hardware utilization.\",\n \"Our work here focuses on a lower-level question: given model architecture, communication bandwidth, processor count, and memory size, can the hybrid-parallel aspect of distributed K-FAC be adaptable and reduce training time?\",\n \"In the rest of this section, we briefly review other works in optimization frameworks.\",\n \"Optimizer Frameworks: Distributed optimization frameworks take many forms.\",\n \"In a synchronous optimizer such as Horovod [47], all variables are updated in every iteration.\",\n \"Asynchronous optimizers relax variable update consistency, for example by passing values via parameter servers [31], to achieve higher performance than synchronous methods [45], [33].\",\n \"The pipeline parallel paradigm, such as GPipe [25] and PipeDream [39] which hold multiple versions of a model partition in a processor and exploit asynchronous optimizers, has shown comparable training convergence.\",\n \"BytePS [27] proposes a unified interface for synchronous and asynchronous SGD.\",\n \"Generally, asynchronous SGD has a non-linear slowdown compared to synchronous SGD [7].\",\n \"KAISA implements K-FAC in a synchronous manner as a preconditioner such that it can work with any SGD variant.\",\n \"Memory Shortage and Remedies: Training DNNs is memory intensive.\",\n \"A common technique for training large models is to swap between processor memory and host memory, such as in SwapAdvisor [23] and SuperNeurons [51].\",\n \"An alternative is to discard some activation tensors and rematerialize them when needed for back-propagation.\",\n \"Checkmate [26] formulates this tradeoff as an optimization problem and provides an optimal rematerialization schedule.\",\n \"In KAISA , we apply techniques such as precision relaxing to reduce the memory consumption of K-FAC and controlling the distribution of eigen decomposition results across processors to maintain a minimal cost in training time.\",\n \"K-FAC Convergence: Prior work on distributed K-FAC has reported training convergence primarily on ResNet-like convolutional neural networks.\",\n \"One study [8] used asynchronous distributed K-FAC to train ResNet-50 with ImageNet 2$\\\\times $ faster than standard SGD, but only achieved 70% validation accuracy, a 5.9% loss compared to the 75.9% MLPerf [4] baseline.\",\n \"Another distributed K-FAC implementation [42] trained ResNet-50 with ImageNet to 74.9% validation accuracy in just 978 iterations; however, comparisons to SGD are not provided.\",\n \"Later work iterated on this implementation to achieve 75.0% validation accuracy on ImageNet with ResNet-50 on 2048 V100 GPUs in 2 minutes by carefully optimizing the baseline SGD training and introducing a 21-bit floating point (FP21) specification for the factors along with other optimizations from previous works such as triangular matrix communication and infrequent K-FAC updates.\",\n \"FP21 was introduced due to worse convergence when using FP16 for factor communication; however, with KAISA , we found similar validation accuracy with ResNet-50 for FP32 and FP16 factor communication.\",\n \"Differences in the numerical stability of the eigen decomposition in KAISA rather than matrix inversion could be a possible reason for the discrepancies.\",\n \"A fourth study [44] trained ResNet-50 to the 75.9% MLPerf baseline in 18-25% less time than with SGD by replacing the factor inverse with eigen decomposition and optimizing for reduced communication in non-K-FAC update steps, at the cost of a high memory footprint.\",\n \"KAISA generalizes previous distributed K-FAC strategies via a tunable memory footprint to balance the memory and communication costs.\",\n \"This design enables efficient, distributed K-FAC research across a wider range of hardware.\",\n \"Further, we showcase KAISA 's effectiveness on a variety of domains and models.\",\n \"Alternative K-FAC Methods: Eigenvalue-corrected Kronecker-factorization (EK-FAC) [18], a more accurate approximation of the FIM, can perform cheap, partial updates.\",\n \"Noisy K-FAC [58] and noisy EK-FAC [9] are functionally similar to standard K-FAC but introduce adaptive weight noise.\",\n \"K-BFGS and K-BFGS(L) [19] apply BFGS [12] and L-BFGS [34] in an analogous method to K-FAC (e.g., block-diagonal, Kronecker-factored).\",\n \"These works have shown better-than-K-FAC performance in many small scale (e.g., single GPU) and small dataset/model (e.g., MNIST or CIFAR-10 with VGG16) cases.\",\n \"Kronecker-factor based FIM approximation variants are a growing area of research; however, large-scale studies have largely been limited to standard K-FAC.\",\n \"KAISA introduces a valuable, unified design paradigm that can be applied to these K-FAC variants to efficiently deploy and evaluate their effectiveness on large models at scale.\"\n ],\n [\n \"Conclusion\",\n \"We have presented KAISA , a K-FAC-enabled, Adaptable, Improved, and ScAlable second-order optimizer framework.\",\n \"To enable scalable DNN training, KAISA adapts memory and communication usage, and appropriately distributes the complex K-FAC computations, to best suit the model and hardware characteristics.\",\n \"We design KAISA to be adaptable to hardware with limited memory (e.g., gaming GPUs) or environments with high communication costs (e.g., Ethernet or massively parallel).\",\n \"We study the fundamental tradeoff between data access on local and remote memory, and evaluate KAISA's correctness and impact on training time by using four real-world applications.\",\n \"With the same global batch size, our experiments show a 18.1–36.3% training time reduction for ResNet-50, Mask R-CNN, U-Net, and BERT-Large, while preserving convergence to the baseline.\",\n \"Under the same memory budget, ResNet-50 and BERT phase 2 converge to the baseline in 32.5% and 41.6% less time compared to momentum SGD and Fused LAMB.\",\n \"In high-communication applications, such as ResNet models, we show that extra processor memory can be used to improve iteration times by reducing communication.\",\n \"In low-communication applications, such as Mask R-CNN and BERT-Large, we show the optimal KAISA usage and efficient scaling on par with SGD up to 128 GPUs.\"\n ]\n]"},"id":{"kind":"string","value":"2107.01739"}}},{"rowIdx":1584,"cells":{"text":{"kind":"list like","value":[["A precise bare simulation approach to the minimization of some\n distances. Foundations"],["Abstract In information theory -- as well as in the adjacent fields of statistics, machine learning, artificial intelligence, signal processing and pattern recognition -- many flexibilizations of the omnipresent Kullback-Leibler information distance (relative entropy) and of the closely related Shannon entropy have become frequently used tools.","To tackle corresponding constrained minimization (respectively maximization) problems by a newly developed dimension-free bare (pure) simulation method, is the main goal of this paper.","Almost no assumptions (like convexity) on the set of constraints are needed, within our discrete setup of arbitrary dimension, and our method is precise (i.e., converges in the limit).","As a side effect, we also derive an innovative way of constructing new useful distances/divergences.","To illustrate the core of our approach, we present numerous examples.","The potential for widespread applicability is indicated, too; in particular, we deliver many recent references for uses of the involved distances/divergences and entropies in various different research fields (which may also serve as an interdisciplinary interface)."],["Introduction","Directed (i.e., not necessarily symmetric) distances $D(\\mathbf {P},\\mathbf {Q})$ between two finite discretefor reasons of technicality, in this paper we only deal with such kind of distributions; for instance, these can be also achieved from more involved systems by quantizations of observations represented by finite partitions of the observation/data space, or by making use of the dual representation for CASM $\\varphi -$ divergences (cf.","Liese & Vajda [218], Broniatowski & Keziou [61]).","(probability) distributions $\\mathbf {P},\\mathbf {Q}$ or between two general Euclidean vectors $\\mathbf {P},\\mathbf {Q}$ are known as divergences; they serve as important (dis)similarity measures, proximity measures and discrepancy measures in various different research areas such as information theory, statistics, artificial intelligence, machine learning, signal processing, pattern recognition, physics, finance, etc.since there exists a vast literature on divergences and connected entropies in these fields, for the sake of brevity we will give in this introduction only some basic references; many corresponding concrete applications will be mentioned in the following sections, in the course of the method-illuminating examples..","Besides Bregman distances/divergences (with which we do not deal here), another major class are the $\\varphi -$ divergences $D_{\\varphi }( \\mathbf {P},\\mathbf {Q})$ of Csiszar-Ali-Silvey-Morimoto (in short CASM $\\varphi -$ divergences, cf.","[94], [11], [266]).","The latter covers — with corresponding choices of $\\varphi $ — e.g.","the omnipresent Kullback-Leibler information distance/divergence [200] (also known as relative entropy), the Jensen-Shannon distance/divergence, as well as the power divergences (also known as alpha-divergences, Cressie-Read measures, and Tsallis cross-entropies).","For some comprehensive overviews on CASM $\\varphi -$ divergences, the reader is referred to the insightful books of e.g.","Liese & Vajda [217], Read & Cressie [303], Vajda [371], Csiszar & Shields [99], Stummer [344], Pardo [282], Liese & Miescke [216], the survey articles of e.g.","Liese & Vajda [218], Vajda & van der Meulen [374], Reid & Williamson [304], Basseville [34], and the references therein; an imbedding of CASM $\\varphi -$ divergences to more general frameworks can be found e.g.","in Stummer & Vajda [350], Broniatowski & Vajda [65], Stummer & Kißlinger [346] and Broniatowski & Stummer [64].","0.5cm Frequently used special cases of the above-mentioned power divergences are e.g.","the (squared) Hellinger distance, the Pearson chi-square divergence, and the Neyman chi-square divergence.","Moreover, several deterministic transformations of power divergences are also prominently used in research, most notably the Bhattacharyya distance [48] and the more general Renyi divergences [309] (also known as Renyi cross-entropies); a comprehensive exposition of the latter is given e.g.","in van Erven & Harremoes [380].","Some other important deterministic transformations of power divergences include the Bhattacharyya coefficient (cf.","[48],[49],[50]) — which is also called affinity (cf.","Matusita [256]) and fidelity similarity (cf.","e.g.","Deza & Deza [113]) — as well as the Bhattacharyya arccos distance (cf.","[50]) and the Fisher distance (also known as Rao distance, geodesic distance, cf.","e.g.","Deza & Deza [113]).","As shown below, by further explicit transformations we can also recover Sundaresan’s divergence [352] [353].","The minimization $\\inf _{\\mathbf {Q}\\in \\mathbf {\\Omega }} D( \\mathbf {Q}, \\mathbf {P} )$ of divergences from one distribution (respectively, its equivalent vector of frequencies) $\\mathbf {P}$ to an appropriate set $\\mathbf {\\Omega }$ of distributions (frequency vectors) appears in a natural way in various different contexts, as indicated in the following.","For instance, let $\\mathbf {P} = \\mathbf {P}_{true}$ be the true distribution of a mechanism which generates non-deterministic data and $\\mathbf {\\Omega }$ is a pregiven model in the sense of a (parametric or non-parametric) family of distributions which serves as an “approximation” (in fact, a collection of approximations) of the “truth” $\\mathbf {P}_{true}$ .","If $\\mathbf {P}_{true} \\notin \\mathbf {\\Omega }$ — e.g.","since $\\mathbf {\\Omega }$ reflects some simplifications of $\\mathbf {P}_{true}$ which is in line with the general scientific procedure — then the positive quantity $\\Phi _{\\mathbf {P}_{true}}(\\mathbf {\\Omega }) := \\inf _{\\mathbf {Q}\\in \\mathbf {\\Omega }}D( \\mathbf {Q}, \\mathbf {P}_{true} )$ can be used as an index of model adequacy in the sense of a degree of departure between the model and the truth (cf.","Lindsay [222], see also e.g.","Lindsay et al.","[223], Markatou & Sofikitou [248], Markatou & Chen [249]); small index values should indicate high adequacy.","If $\\mathbf {P}_{true} \\in \\mathbf {\\Omega }$ , then $\\Phi _{\\mathbf {P}_{true}}(\\mathbf {\\Omega }) = 0$ which corresponds to full adequacy.","This index of model adequacy $\\Phi _{\\mathbf {P}_{true}}(\\mathbf {\\Omega })$ can also be seen as index of goodness/quality of approximation to the truth or as model misspecification error, and it can be used for model assessment as well as for model search (model selection, model hunting) by comparing the indices $\\Phi _{\\mathbf {P}_{true}}(\\mathbf {\\Omega _{1}}),\\Phi _{\\mathbf {P}_{true}}(\\mathbf {\\Omega _{2}}), \\ldots $ of competing models $\\mathbf {\\Omega _{1}}, \\mathbf {\\Omega _{2}}, \\ldots $ and choosing the one with the smallest index; this idea can be also used for classification (e.g.","analogously to Bilik & Khomchuk [54] who deal with continuous (rather than discrete) distributions) where the $\\mathbf {\\Omega _{i}}$ are interpreted as (possibly data-derived but fixed) classes which are disjoint and non-exhaustive.","Typically, in statistical analyses the true distribution $\\mathbf {P}_{true}$ is unknown and is either replaced by a hypothesis-distribution $\\mathbf {P}_{hyp}$ or by a distribution $\\mathbf {P}_{data}$ derived from data (generated by $\\mathbf {P}_{true}$ ) which converges to $\\mathbf {P}_{true}$ as the data/sample size tends to infinity (e.g.","$\\mathbf {P}_{data}$ may be the well-known empirical distribution or a conditional distribution).","Correspondingly, $\\Phi _{\\mathbf {P}_{data}}(\\mathbf {\\Omega })$ reflects a data-derived approximation (estimate) of the index of model adequacy (resp.","of the model misspecification error) from which one can cast corresponding model-adequacy tests and related goodness-of-fit tests.","Moreover, for the case of i.i.d.","data-generation and $\\mathbf {P}_{data}$ to be the corresponding empirical distribution, the (not necessarily existent or unique) best-model-member/element choice $\\arg \\min _{\\mathbf {Q}\\in \\mathbf {\\Omega }} D( \\mathbf {Q}, \\mathbf {P}_{data})$ amounts to the well-known minimum distance estimator which for the Kullback-Leibler information divergence $D$ is equal to the omnipresent maximum likelihood estimator; for comprehensive surveys on divergence-based statistical testing and estimation, the reader is referred to e.g.","the references in the second half of the first paragraph in the current introduction.","Most of the above-mentioned considerations also hold for deterministic (rather than non-deterministic) frameworks where $\\mathbf {P}$ is a general Euclidean vector (rather than a probability-distribution describing frequency vector in the probability simplex) and $\\mathbf {\\Omega }$ is a model in the sense of a family of general Euclidean vectors (which may be encodings of more complicated context descriptions).","Returning to the general context, let us mention that from CASM $\\varphi -$ divergences one can also derive the widely used $\\varphi -$ entropies $\\mathcal {E}_{\\varphi }(\\mathbf {Q})$ of a distribution $\\mathbf {Q}$ (and non-probability versions thereof) in the sense of Burbea & Rao [68] (see also Csiszar [95], Ben-Bassat [38], Ben-Tal & Teboulle [40], Kesavan & Kapur [187], Dacunha-Castelle & Gamboa [102], Teboulle & Vajda [357], Gamboa & Gassiat [132], Vajda & Zvarova [376]); these entropies can e.g.","be basically constructed from $D_{\\varphi }(\\mathbf {Q},\\mathbf {P}^{unif})$ where $\\mathbf {P}^{unif}$ denotes the uniform distribution.","Moreover, by use of certain deterministic transformations $h$ one can also deduce the more general $(h,\\varphi )-$ entropies (and non-probability versions thereof) in the sense of Salicru et al.","[314] (see also e.g.","Pardo [282]).","As will be worked out in detail in Subsection REF below, from this one can deduce as special cases a variety of prominently used quantities in research, such as for instance: the omnipresent Shannon entropy [328], the $\\gamma -$ order Renyi entropy [309], the $\\gamma -$ order entropy of Havrda-Charvat [157] (also called non-additive $\\gamma -$ order Tsallis entropy [363] in statistical physics), the $\\widetilde{\\gamma }-$ order entropy of Arimoto [16], Vajda's quadratic entropy [371], Sharma-Mittal entropies [329], the Euclidean $\\gamma -$ norms, as well as measures of diversity, heterogeneity and unevenness, like the Gini-Simpson diversity index, the diversity index of Hill [160], the Simpson-Herfindahl index (which is also known as index of coincidence, cf.","Harremoes & Topsoe [155] and its generalization in Harremoes & Vajda [156]), the diversity index of Patil & Taillie [286], the $\\gamma -$ mean heterogeneity index (see e.g.","van der Lubbe [379]); see also Nayak [271] and Jost [176] for some interrelations with the above-mentioned entropies.","Given that the constraint set $\\mathbf {\\Omega }$ reflects some incomplete/partial information about a system (e.g.","moment constraints), the maximization over $\\mathbf {Q}\\in \\mathbf {\\Omega }$ of the above-mentioned entropies, norms and diversity indices (and the more general $(h,\\varphi )-$ entropies) is important for many research topics, most notably manifested in Jaynes’s [165],[166] omnipresent, “universally applicable” maximum entropy principle (which employs the Shannon entropy), and its generalizations (see e.g.","the books of Kapur [184], Kapur & Kesavan [185], Arndt [17], and Gzyl et al.","[150] for comprehensive surveys).","Besides the above-mentioned principal overview, let us now briefly discuss some existing technical issues for the minimization of CASM $\\varphi -$ divergences $\\Phi _{\\mathbf {P}}(\\mathbf {\\Omega }) :=\\inf _{\\mathbf {Q}\\in \\mathbf {\\Omega }} D_{\\varphi }( \\mathbf {Q}, \\mathbf {P} )$ .","For (not necessarily discrete) probability distributions/measures $\\mathbf {P}$ and sets $\\mathbf {\\Omega }$ of probability distributions/measures satisfying a finite set of linear equality constraints, $\\Phi _{\\mathbf {P}}(\\mathbf {\\Omega })$ has been characterized in Csiszar [96] and more recently by Csiszar & Matus [98], Broniatowski & Keziou [60], Leonard [213], and Pelletier [288] among others, in various contexts; those results extend to inequality constraints.","Minimizations of $\\gamma -$ order Renyi divergences on $\\gamma -$ convex sets $\\mathbf {\\Omega }$ are studied e.g.","in Kumar & Sason [202], whereas Kumar & Sundaresan [203] [204] investigate minimizations of Sundaresan’s divergence on certain convex sets $\\mathbf {\\Omega }$ .","To our knowledge, no general representation for $\\Phi _{\\mathbf {P}}(\\mathbf {\\Omega })$ for a positive distribution/measure $\\mathbf {P}$ (respectively, for an Euclidean vector with positive components) and a general set $\\mathbf {\\Omega }$ of signed measures (respectively, of Euclidean vector with components of arbitrary sign) exists.","At the contrary, many algorithmic approaches for such minimization problems have been proposed; they mostly aim at finding minimizers more than at the evaluation of the minimum divergence itself, which is obtained as a by-product.","Moreover, it is well-known that such kind of CASM $\\varphi -$ divergence minimization problems may be hard to tackle or even intractable via usual methods such as the omnipresent gradient descent method and versions thereof, especially for non-parametric or semi-parametric $\\mathbf {\\Omega }$ in sufficiently high-dimensional situations.","For instance, $\\mathbf {\\Omega }$ may consist (only) of constraints on moments or on L-moments (see e.g.","Broniatowski & Decurninge [59]); alternatively, $\\mathbf {\\Omega }$ may be e.g.","a tubular neighborhood of a parametric model (see e.g.","Liu & Lindsay [225], Ghosh & Basu [135]).","The same intractability problem holds for the above-mentioned $(h,\\varphi )-$ entropy maximization problems.","In the light of this, the goals of this paper are: to solve constrained minimization problems of a large range of CASM $\\varphi -$ divergences and deterministic transformations thereof (respectively constrained maximization problems of $(h,\\varphi )-$ entropies including Euclidean norms and diversity indices), by means of a newly developed dimension-free bare (pure) simulation method which is precise (i.e., converges in the limit) and which needs almost no assumptions (like convexity) on the set $\\mathbf {\\Omega }$ of constraints; in doing so, for the sake of brevity we concentrate on finding/computing the minimum divergences themselves rather than the corresponding minimizers (to achieve the latter, e.g.","dichotomous search could be used in a subsequent step, however); to derive a method of constructing new useful distances/divergences; to present numerous examples in order to illuminate our method and its potential for wide-spread applicability; as we go along, we also deliver many recent references for uses of the outcoming distances/divergences and entropies (covering in particular all the above-mentioned ones).","This agenda is achieved in the following way.","In the next Section , we briefly introduce the principal idea of our new bare-simulation optimization paradigm.","After manifesting the fundamentally employed class of CASM $\\varphi -$ divergences in Section , we give in Section the main cornerstones, construction principles and theorems, for deterministic as well as for statistical divergence-minimization problems; the maximization of generalized entropies is addressed, too.","Section deals with the concrete determination of the involved simulation-weights, as well as with the interrelated issue of creating associated CASM $\\varphi -$ divergences.","Some sampling-concerning details for the principal implementation of our bare-simulation optimization approach are worked out in Section .","The main proofs are presented in the appendices.","A first simulation-based algorithm in vein with the present proposal has been developed by Broniatowski [58], in the restricted setup of risk estimation for power divergences.","The present paper extends this considerably by considering general CASM $\\varphi -$ divergences and related entropies, and by dealing with corresponding general optimization problems, of both deterministic respectively stochastic type."],["A new minimization paradigm ","We concern with minimization problems of the following type, where $\\mathcal {M}$ is a topological space and $\\mathcal {T}$ is the Borel $\\sigma -$ field over a given base on $\\mathcal {M}$ ; e.g.","take $\\mathcal {M} = \\mathbb {R}^{K}$ to be the $K-$ dimensional Euclidean space equipped with the Borel $\\sigma -$ field $\\mathcal {T}$ .","Definition 1 A measurable function $\\Phi : \\mathcal {M} \\mapsto [0,\\infty ]$ and measurable set $\\Omega \\subset \\mathcal {M}$ i.e.","$\\Omega \\in \\mathcal {T}$ are called “bare-simulation minimizable” (BS-minimizable) respectively “bare-simulation maximizable” (BS-maximizable) if for $\\Phi (\\Omega ):=\\inf _{Q\\in \\Omega }\\left\\lbrace \\Phi (Q) \\right\\rbrace < \\infty \\qquad \\textrm {respectively} \\qquad \\Phi (\\Omega ):=\\sup _{Q\\in \\Omega }\\left\\lbrace \\Phi (Q) \\right\\rbrace < \\infty $ there exists a measurable function $G: [0,\\infty [ \\mapsto [0,\\infty [$ as well as a sequence $\\left((\\mathfrak {X}_{n},\\mathcal {A}_{n},\\mathbb {\\Pi }_{n})\\right)_{n \\in \\mathbb {N}}$ of probability spaces and on them a sequence $(\\xi _{n})_{n\\in \\mathbb {N}}$ in order to emphasize the dependence on $\\Phi $ , one should use the notations $(\\xi _{\\Phi ,n})_{n\\in \\mathbb {N}}$ , $\\mathbb {\\Pi }_{\\Phi ,n}$ , etc.","; this is avoided for the sake of a better readability.","of $\\mathcal {M}-$ valued random variables such that $G\\left(-\\lim _{n\\rightarrow \\infty }\\frac{1}{n}\\log \\mathbb {\\Pi }_{n}\\left[\\xi _{n}\\in \\Omega \\right] \\right) =\\inf _{Q\\in \\Omega }\\Phi (Q)=\\Phi (\\Omega )$ respectively $G\\left(-\\lim _{n\\rightarrow \\infty }\\frac{1}{n}\\log \\mathbb {\\Pi }_{n}\\left[\\xi _{n}\\in \\Omega \\right] \\right) =\\sup _{Q\\in \\Omega }\\Phi (Q) =\\Phi (\\Omega );$ in situations where $\\Phi $ is fixed and different $\\Omega $ ’s are considered, we say that “$\\Phi $ is bare-simulation minimizable (BS-minimizable) on $\\Omega $ ” respectively “$\\Phi $ is bare-simulation maximizable (BS-maximizable) on $\\Omega $ ”.","Remark 2 (a) Even in situations where one can uniformly choose $(\\mathfrak {X}_{n},\\mathcal {A}_{n},\\mathbb {\\Pi }_{n}) \\equiv (\\widetilde{\\mathfrak {X}},\\widetilde{\\mathcal {A}},\\widetilde{\\mathbb {\\Pi }})$ , the sequence $(\\xi _{n})_{n\\in \\mathbb {N}}$ may be not “independent and identically distributed” .","(b) Throughout the paper, we shall mainly deal with $BS-$ minimizability.","The basic idea/incentive of this new approach is: if a minimization problem (REF ) has no explicit solution and is computationally intractable (or unfeasible) but can be shown to be BS-minimizable with concretely constructable $(\\xi _{n})_{n\\in \\mathbb {N}}$ and $(\\mathbb {\\Pi }_{n})_{n\\in \\mathbb {N}}$ , then one can basically simulate the log-probabilities $-\\frac{1}{n} \\mathbb {\\Pi }_{n}\\left[\\xi _{n}\\in \\Omega \\right]$ for large enough integer $n \\in \\mathbb {N}$ to obtain an approximation of (REF ) without having to evaluate the corresponding (not necessarily unique) minimizer, where the latter is typically time-costly.","Finding minimizers can be performed through dichotomic search, once an algorithm leading to the minimal value of the divergence on adequate families of sets $\\Omega $ is at hand; for the sake of brevity, this is omitted in the current paper.","For reasons of transparency, we start to demonstrate this approach for the following important/prominent class of minimization problems with the following components: $\\mathcal {M}$ is the $K-$ dimensional Euclidean space $\\mathbb {R}^{K}$ , i.e.","$\\Omega $ is a set of vectors $Q$ with a number of $K$ components (where $K$ may be huge, as it is e.g.","the case in big data contexts); $\\Phi (\\cdot ) := \\Phi _{P}(\\cdot )$ depends on some known vector $P$ in $\\mathbb {R}^{K}$ with $K$ nonnegative components; $\\Phi _{P}(\\cdot )$ is a “directed distance” (divergence) from $P$ into $\\Omega $ in the sense of $\\Omega \\ni Q \\mapsto \\Phi _{P}(Q):= D(Q, P)$ , where $D(\\cdot ,\\cdot )$ has the the two properties “$D(Q, P)\\ge 0$ ” and “$D(Q, P) = 0$ if and only if $Q=P$ ”.","In particular, $D(\\cdot ,\\cdot )$ needs neither satisfy the symmetry $D(Q,P)= D(P,Q)$ nor the triangular inequality.","In other words, (1) together with (i)-(iii) constitutes a distance/divergence-minimization problem; we design a “universal” method to solve such problems by constructing appropriate (cf.","(REF )) sequences $(\\xi _{n})_{n\\in \\mathbb {N}}$ of $\\mathbb {R}^{K}-$ valued random variables, for all directed distances $D(\\cdot ,\\cdot )$ from a large subclass of the important omnipresent Csiszar-Ali-Silvey-Morimoto CASM $\\varphi -$ divergences (also called $f-$ divergences).","As a second demonstration for the workability of our paradigm, we “extend” (i) to (iii) to the setup where $P$ is a random element of the simplex $\\mathbb {S}^{K}$ of $K-$ component probability (frequency) vectors (cf.","the exact definition below) and $\\Omega \\subset \\mathbb {S}^{K}$ ; for the sub-setup where $P$ corresponds to a data-observation-dependent probability distribution and $\\Omega $ corresponds to a pregiven model in the sense of a family of probability distributions, the formula (REF ) amounts to the corresponding (discrete) “minimization-distance estimation problem (MDEP)” of choosing the best model element/member under given data an alternative naming also used in literature is to call $\\Omega $ a model class (rather than model), and each $P \\in \\Omega $ a model (rather than model element).","This is important/prominent in statistics and in the adjacent research fields of artificial intelligence and machine learning; the concrete solving of the MDEP is especially “hard” for nonparametric respectively semiparametric problems, and our BS method is predestined for such kind of contexts."],["Directed distances ","In detail, concerning the above-mentioned point (i) we take the $K-$ dimensional Euclidean space $\\mathcal {M}=\\mathbb {R}^{K}$ , denote from now on — as usual — its elements (i.e.","vectors) in boldface letters, and also employ the subsets $&& \\mathbb {R}_{\\ne 0}^{K} := \\lbrace \\mathbf {Q}:= (q_{1},\\ldots ,q_{K}) \\in \\mathbb {R}^K: \\,q_{i} \\ne 0 \\ \\text{for all} \\ i=1,\\ldots ,K \\rbrace , \\\\&& \\mathbb {R}_{> 0}^{K} := \\lbrace \\mathbf {Q}:= (q_{1},\\ldots ,q_{K}) \\in \\mathbb {R}^K: \\,q_{i} > 0 \\ \\text{for all} \\ i=1,\\ldots ,K \\rbrace , \\\\&& \\mathbb {R}_{\\ge 0}^{K} := \\lbrace \\mathbf {Q}:= (q_{1},\\ldots ,q_{K}) \\in \\mathbb {R}^K: \\, q_{i} \\ge 0 \\ \\text{for all} \\ i=1,\\ldots ,K \\rbrace , \\\\&& \\mathbb {R}_{\\le 0}^{K} := \\lbrace \\mathbf {Q}:= (q_{1},\\ldots ,q_{K}) \\in \\mathbb {R}^K: \\, q_{i} \\le 0 \\ \\text{for all} \\ i=1,\\ldots ,K \\rbrace , \\\\&& \\mathbb {S}^{K} := \\lbrace \\mathbf {Q} := (q_{1},\\ldots ,q_{K}) \\in \\mathbb {R}_{\\ge 0}^{K}:\\, \\sum _{i=1}^{K} q_{i} =1 \\rbrace \\quad \\text{(simplex of probability vectors)},\\\\&& \\mathbb {S}_{> 0}^{K} := \\lbrace \\mathbf {Q}:= (q_{1},\\ldots ,q_{K}) \\in \\mathbb {R}_{>0}^{K}: \\, \\sum _{i=1}^{K} q_{i} =1 \\rbrace .","$ Concerning the directed distances $D(\\cdot ,\\cdot )$ in (ii) and (iii), we deal with the important omnipresent Csiszar-Ali-Silvey-Morimoto $\\varphi -$ divergences (CASM $\\varphi -$ divergences) — adapted to our context: Definition 3 (a) Let the “divergence-generator” be a lower semicontinuous convex function $\\varphi : \\, ]-\\infty ,\\infty [ \\rightarrow [0,\\infty ]$ satisfying $\\varphi (1)=0$ .","Furthermore, for the effective domain $dom(\\varphi ) := \\lbrace t \\in \\mathbb {R} : \\varphi (t) < \\infty \\rbrace $ we assume that its interior $int(dom(\\varphi ))$ is non-empty which implies that $int(dom(\\varphi )) = ]a,b[$ for some $-\\infty \\le a < 1 < b \\le \\infty $ .","Moreover, we suppose that $\\varphi $ is strictly convex in a non-empty neighborhood $]t_{-}^{sc},t_{+}^{sc}[ \\subseteq ]a,b[$ of one ($t_{-}^{sc} < 1 < t_{+}^{sc}$ ).","Also, we set $\\varphi (a) := \\lim _{t \\downarrow a} \\varphi (t)$ and $\\varphi (b) := \\lim _{t \\uparrow b} \\varphi (t)$ (these limits always exist).","The class of all such functions $\\varphi $ will be denoted by $\\widetilde{\\Upsilon }(]a,b[)$ .","A frequent choice is e.g.","$]a,b[=]0,\\infty [$ or $]a,b[=]-\\infty ,\\infty [$ .","(b) For $\\varphi \\in \\widetilde{\\Upsilon }(]a,b[)$ , $\\mathbf {P} := (p_{1},\\ldots ,p_{K}) \\in \\mathbb {R}_{\\ge 0}^{K}$ and $\\mathbf {Q} := (q_{1},\\ldots ,q_{K}) \\in \\mathbf {\\Omega } \\subset \\mathbb {R}^{K}$ , we define the Csiszar-Ali-Silvey-Morimoto $\\varphi -$ divergence $\\Phi _{\\mathbf {P}} \\left(\\mathbf {Q}\\right) := D_{\\varphi }( \\mathbf {Q}, \\mathbf {P} ) :=\\sum _{k=1}^{K} p_{k} \\cdot \\varphi \\left( \\frac{q_{k}}{p_{k}}\\right) \\, \\ge 0.$ As usual, in (REF ) we employ the three conventions that $p\\cdot \\varphi \\left( \\frac{0}{p}\\right) = p \\cdot \\varphi (0) >0$ for all $p > 0$ , and $0 \\cdot \\varphi \\left( \\frac{q}{0}\\right) = q \\cdot \\lim _{x\\rightarrow \\infty } \\frac{\\varphi (x \\cdot \\textrm {sgn}(q))}{x\\cdot \\textrm {sgn}(q)} >0$ for $q \\ne 0$ (employing the sign of $q$ ), and $0 \\cdot \\varphi \\left( \\frac{0}{0}\\right) :=0$ .","Throughout the paper, we only consider constellations $(\\varphi ,\\mathbf {P},\\mathbf {\\Omega })$ for which the very mild condition $\\Phi _{\\mathbf {P}}(\\Omega ) := \\inf _{\\mathbf {Q}\\in \\mathbf {\\Omega }}D_{\\varphi }( \\mathbf {Q}, \\mathbf {P} ) \\ne \\infty \\ \\ \\footnote {i.e.","dom(\\varphi ) covers (at least) a non-empty part of \\lbrace 1\\rbrace \\cup \\mathcal {R}\\big (\\frac{\\mathbf {\\Omega }}{\\mathbf {P}}\\big ), where \\mathcal {R}\\big (\\frac{\\mathbf {\\Omega }}{\\mathbf {P}}\\big ) := \\big \\lbrace \\frac{q_{k}}{p_{k}}: \\, k \\in \\lbrace 1,\\ldots ,K\\rbrace , \\mathbf {Q}:=(q_{1},\\ldots ,q_{K}) \\in \\Omega \\big \\rbrace is the range of all possibleentry-ratios.","}\\nonumber $ holds.","For probability vectors ${P}$ and ${Q}$ in $\\mathbb {S}^{K}$ , the $\\varphi -$ divergences $D_{\\varphi }( {Q}, {P} )$ were introduced by Csiszar [94], Ali & Silvey [11] and Morimoto [266] (where the first two references even deal with more general probability distributions); for some comprehensive overviews — including statistical applications to goodness-of-fit testing and minimum distance estimation — the reader is referred to the insightful books of e.g.","Liese & Vajda [217], Read & Cressie [303], Vajda [371], Csiszar & Shields [99], Stummer [344], Pardo [282], Liese & Miescke [216], the survey articles of e.g.","Liese & Vajda [218], Vajda & van der Meulen [374], Reid & Williamson [304], Basseville [34], and the references therein.","Some exemplary recent studies and applications of CASM $\\varphi -$ divergences appear e.g.","in Qiao & Minematsu [298] for invariances in speech recognition, Nguyen et al.","[273] in connection with empirical risk optimization, Feixas et al.","[124] for various different image processing tasks, Luo et al.","[232] for video clip segmentation and key frame generation, Kißlinger & Stummer [189] for model preselection (structure detection) in the context of nonlinear recursive models with additional exogenous inputs, Mahboubi & Kochenderfer [241] within a context of traffic-pattern learning from flight tracks, Guo et al.","[145] for local contrastive descriptors in image classification through e.g.","regional color distributions, Csiszar & Breuer [100] for modelling generalized-ball type constraints in expectation minimization problems, Kißlinger & Stummer [191] for the detection of distributional changes in random data (streams and clouds), Noh et al.","[275] within a context of generative local metric learning for nearest neighbor classification, Yu et al.","[416] for adversarial learning within oil spill segmentation, Arslan [19] for automated active reconfiguration in mobile sensor networks, Sason [320] in connection with with data-processing and majorization inequalities, Ciftci et al.","[89] for the optimization of multienergy microgrids in energy infrastructure systems, and Stummer [345] for solving some new optimal transport (OT) problems which flexibilize some Wasserstein-distance based OTs.","For the setup of $D_{\\varphi }( \\mathbf {Q}, \\mathbf {P} )$ for vectors $\\mathbf {P}$ , $\\mathbf {Q}$ with non-negative components the reader is referred to e.g.","Stummer & Vajda [349] (who deal with even more general nonnegative measures and giving some statistical as well as information-theoretic applications) and Gietl & Reffel [137] (including applications to iterative proportional fitting).","The case of $\\varphi -$ divergences for vectors with arbitrary components can be extracted from e.g.","Broniatowski & Keziou [60] who actually deal with finite signed measures.","For a comprehensive technical treatment, see also Broniatowski & Stummer [64].","Clearly, from (REF ) it is obvious that in general $D_{\\varphi }( \\mathbf {Q}, \\mathbf {P} ) \\ne D_{\\varphi }(\\mathbf {P}, \\mathbf {Q})$ (non-symmetry).","Moreover, it is straightforward to deduce that $D_{\\varphi }(\\mathbf {Q}, \\mathbf {P}) = 0$ if and only if $\\mathbf {Q}=\\mathbf {P}$ (reflexivity).","Very prominent and important examples of CASM $\\varphi -$ divergences are the power divergences in the scaling of e.g.","Liese & Vajda [217] (in other scalings also called Rathie & Kannapan’s non-additive directed divergences of order $\\gamma $ [302], Cressie-Read divergences [93] [303], relative Tsallis entropies or Tsallis cross-entropies [364] (see also Shiino [331]), Amari’s alpha-divergences [12]) where basically (up to technicalities) $\\varphi (t): = \\varphi _{\\gamma }(t) :=\\frac{t^\\gamma -\\gamma \\cdot t+ \\gamma - 1}{\\gamma \\cdot (\\gamma -1)}$ ($\\gamma \\in \\mathbb {R}\\backslash \\lbrace 0,1\\rbrace $ ), $\\varphi (t): = \\varphi _{0}(t) :=\\lim _{\\gamma \\rightarrow 0} \\varphi _{\\gamma }(t) = - \\log t + t - 1$ , $\\varphi (t): = \\varphi _{1}(t) :=\\lim _{\\gamma \\rightarrow 1} \\varphi _{\\gamma }(t) = t \\cdot \\log t + 1 - t$ .","Usually, in the literature one takes $t \\in \\, ]0,\\infty [$ (and the limit as $t \\rightarrow 0$ ), except for the case $\\gamma =2$ where one handles $t \\in \\, ]-\\infty ,\\infty [$ ; for our purposes, we have to essentially extend these divergence generators $\\varphi _{\\gamma }$ for $t<0$ , which will be carried out and discussed in detail below, namely in (REF ), (REF ) (see also Table 1), as well as at several other places in this paper.","Notice that $D_{\\varphi _{1}}(\\mathbf {Q},\\mathbf {P})$ basically corresponds to the (extended form of) the omnipresent Kullback-Leibler information resp.","relative entropy.","Below, we shall also consider the minimization/maximization of important transforms of power divergences such as Renyi divergences/entropies, Sundaresan’s divergence, etc., which are frequently used in information theory and its applications to e.g.","artificial intelligence, machine learning, and physics.","Remark 4 Since, in general, our methods work also for non-probability vectors $\\mathbf {Q},\\mathbf {P}$ , we can also deal with — plain versions and transformations of — weighted $\\varphi -$ divergences of the form $D_{\\varphi }^{wei}( \\mathbf {Q}, \\mathbf {P} ) :=\\sum _{k=1}^{K} c_{k} \\cdot p_{k} \\cdot \\varphi \\left( \\frac{q_{k}}{p_{k}}\\right) \\, \\ge 0$ where $c_{k} >0$ ($k=1,\\ldots ,K$ ) are weights which not necessarily add up to one.","Indeed, by means of (REF ) we formally end up with $\\inf _{\\mathbf {Q}\\in \\mathbf {\\Omega }}D_{\\varphi }^{wei}( \\mathbf {Q}, \\mathbf {P} ) =\\inf _{\\mathbf {Q}^{wei}\\in \\mathbf {\\Omega }^{wei}}D_{\\varphi }( \\mathbf {Q}^{wei}, \\mathbf {P}^{wei} )\\nonumber $ where $\\mathbf {P}^{wei} := (c_{1} \\cdot p_{1},\\ldots ,c_{K} \\cdot p_{K})$ , $\\mathbf {Q}^{wei} := (c_{1} \\cdot q_{1},\\ldots ,c_{K} \\cdot q_{K})$ and $\\mathbf {\\Omega }^{wei}$ is the corresponding rescaling of $\\mathbf {\\Omega }$ .","Of course, all the necessary technicalities for the $\\varphi -$ divergences (see below) have to be adapted to the weighted $\\varphi -$ divergences; for the sake of brevity, this will not be discussed in detail.","Notice that $\\mathbf {P}^{wei}$ , $\\mathbf {Q}^{wei}$ are generally not probability vectors anymore, even if $\\mathbf {Q},\\mathbf {P}$ are probability vectors.","In the latter case, and under the assumption $\\sum _{k=1}^{K} c_{k} = 1$ , the divergences (REF ) coincide with the discrete versions of the ($\\mathbf {c}-$ )local divergences of Avlogiaris et al.","[22], [23] who also give absolutely-continuous versions and beyond (see also Broniatowski & Stummer [64] for an imbedding in a general divergence framework)."],["The cornerstone ","In this Section , we show that a number of deterministic optimization problems and and problems in statistical minimum risk based approaches pertaining to non- or semi-parametric contexts are BS-minimizable/amenable in the sense of Definition REF .","The below-mentioned Sections and will draw conclusions, proposing effective solutions.","For the construction of the desired sequence $(\\xi _{n})_{n\\in \\mathbb {N}}$ of $\\mathbb {R}^{K}-$ valued random variables (viz.","random vectors) and a corresponding probability distribution $\\mathbb {\\Pi }$ (which will not depend on $n$ ), we will assume that the divergence generator $\\varphi \\in \\widetilde{\\Upsilon } (]a,b[)$ has the additional property that it can be represented as $\\varphi (t)=\\sup _{z\\in \\mathbb {R}}\\Big ( z\\cdot t-\\log \\int _{\\mathbb {R}}e^{z \\cdot y}d\\mathbb {} (y)\\Big ), \\qquad t\\in \\mathbb {R},\\ \\ $ for some probability distribution/measure $\\mathbb {} $ on the real line such that the function $z\\mapsto MGF_{\\mathbb {} }(z):=\\int _{\\mathbb {R}}e^{z \\cdot y}d\\mathbb {} (y)$ is finite on some open interval containing zero in particular, this implies that $\\mathbb {} $ has light tails.. From this, we shall construct — basically in Section below — a sequence $(W_{n})_{n\\in \\mathbb {N}}$ of i.i.d.","copies of a random variable $W$ whose distribution (under $\\mathbb {\\Pi }$ ) is $\\mathbb {}$ (i.e.","$\\mathbb {\\Pi }[W \\in \\cdot \\, ] = \\mathbb {}[ \\, \\cdot \\,]$ ), from which the desired $(\\xi _{n})_{n\\in \\mathbb {N}}$ will be constructed.","Since $\\varphi $ attains its minimal value at the point 1, fit follows that $\\varphi ^{\\prime }(1)=0.$ By (REF ), for all $t$ in $int(dom(\\varphi ))$ , $\\varphi ^{\\prime }(t)$ is the reciprocal of $\\psi (z):=\\left(d/dz\\right) \\log MGF(z)$ at point $t$ , whence $\\psi (0)=1$ , which is to say that the expectation $E_\\mathbb {\\Pi }[W]=1$ .","The class of functions $\\varphi \\in \\widetilde{\\Upsilon } (]a,b[)$ satisfying the representability (REF ) will be denoted by $\\Upsilon (]a,b[)$ .","Remark 5 The condition $\\varphi \\in \\Upsilon (]a,b[)$ implies that $\\mathbb {}$ can not be a one-point distribution (Dirac mass) $\\delta _{y}$ at some point $y$ , since for such a situation one can straightforwardly deduce from (REF ) that $\\varphi (y)=0$ and $\\varphi (t)=\\infty $ for all $t \\ne y$ , which leads to $int(dom(\\varphi )) = \\emptyset $ and thus $\\varphi \\notin \\widetilde{\\Upsilon } (]a,b[)$ (in fact, our requirement $\\varphi (y)=0$ would narrow down to $y=1$ anyway).","Let us remark that the class $\\Upsilon (]a,b[)$ contains many divergence generators; this together with $\\varphi -$ construction principles will be developed at length in Section below.","Also, for the minimization problems considered in Section REF hereunder, we mostly modify the generator $\\varphi $ into $\\widetilde{c}\\cdot \\varphi $ for strictly positive scales $\\widetilde{c}$ .","At this point, for the sake of transparency, we only present a summarizing Table 1 of a selection of concrete examples which will be treated in detail below: Table: NO_CAPTION Table 1.","Selection of concrete examples treated in this paper, included with some important features (to be explained in the course of method build-up).","As already explained above, the representability (REF ) is the cornerstone for our approach, and opens the gate to make use of simulation methods in appropriate contexts.","We first develop this approach for deterministic minimization problems (cf.","Subsection REF ); thereafter, in Subsection REF , we “extend” this to the setup where $\\mathbf {P}$ is identified with an unknown probability vector in the simplex $\\mathbb {S}^{K}$ which is supposed to be the limit (as $n$ tends to infinity) of the empirical distribution pertaining to a collection of observations $\\mathbf {X}_{n}$ $:=\\left(X_{1},..,X_{n}\\right) $ ; in the classical statistical setting, this amounts to the estimation of $\\Phi _{\\mathbf {P}}\\left( \\Omega \\right) $ based on $\\mathbf {X}_{n}$ , leading to the important “minimization-distance estimation problem” in statistics, artificial intelligence and machine learning.","Finally, we end up this Section by shortly dealing with divergences between fuzzy sets (cf.","Subsection REF ) and basic belief assignments (cf.","Subsection REF )."],["Deterministic minimization problems","Problem 6 For pregiven $\\varphi \\in \\Upsilon (]a,b[)$ , positive-entries vector $\\mathbf {P}:=\\left( p_{1},..,p_{K}\\right) \\in \\mathbb {R}_{>0}^{K}$ (or from some subset thereof), and subset $\\mathbf {\\Omega } \\subset \\mathbb {R}^{K}$ (also denoted in boldface letters, with a slight abuse of notation) with regularity properties $cl(\\mathbf {\\Omega } )=cl\\left( int\\left( \\mathbf {\\Omega } \\right) \\right) , \\qquad int\\left( \\mathbf {\\Omega } \\right) \\ne \\emptyset ,$ find $\\Phi _{\\mathbf {P}}(\\mathbf {\\Omega }) := \\inf _{\\mathbf {Q}\\in \\mathbf {\\Omega } } D_{\\varphi }(\\mathbf {Q},\\mathbf {P}),$ provided that $\\inf _{\\mathbf {Q}\\in \\mathbf {\\Omega } } D_{\\varphi }(\\mathbf {Q},\\mathbf {P}) < \\infty .$ An immediate consequence thereof is — for pregiven $\\varphi \\in \\Upsilon (]a,b[)$ — the treatment of the more flexible problem $\\Phi _{\\mathbf {P},h}(\\mathbf {\\Omega }) := \\inf _{\\mathbf {Q}\\in \\mathbf {\\Omega } }h\\Big (D_{\\varphi }(\\mathbf {Q},\\mathbf {P}) \\Big )= h\\Big (\\inf _{\\mathbf {Q}\\in \\mathbf {\\Omega } } D_{\\varphi }(\\mathbf {Q},\\mathbf {P}) \\Big )$ for any continuous strictly increasing function $h: \\mathcal {H} \\, \\mapsto \\mathbb {R}$ with $\\mathcal {H} := [0,\\infty [$ and extension $h(\\infty ) := \\sup _{y \\in \\mathcal {H}}(y)$ (depending on the problem, a sufficiently large $\\mathcal {H} \\subset [0,\\infty [$ may be enough), respectively of $\\sup _{\\mathbf {Q}\\in \\mathbf {\\Omega } }h\\Big (D_{\\varphi }(\\mathbf {Q},\\mathbf {P}) \\Big )= h\\Big (\\inf _{\\mathbf {Q}\\in \\mathbf {\\Omega } } D_{\\varphi }(\\mathbf {Q},\\mathbf {P}) \\Big )$ for any continuous strictly decreasing function $h: \\mathcal {H} \\, \\mapsto \\mathbb {R}$ and extension $h(\\infty ) := \\inf _{y \\in \\mathcal {H}}(y)$ .","Remark 7 (a) By the basic properties of $\\varphi $ , it follows that for all $c>0$ the level sets $\\mathbf {\\varphi }_{c}:=\\left\\lbrace x\\in \\mathbb {R}:\\varphi (x)\\le c\\right\\rbrace $ are compact and so are the level sets of $\\mathbf {Q\\rightarrow }D_{\\varphi }(\\mathbf {Q},\\mathbf {P})$ $\\Gamma _{c}:=\\left\\lbrace \\mathbf {Q}\\in \\mathbb {R}^{K}:D_{\\varphi }(\\mathbf {Q},\\mathbf {P})\\le c\\right\\rbrace \\nonumber $ for all $c>0$ .","(b) When $\\mathbf {\\Omega }$ is not closed but merely satisfies (REF ), then the infimum in (REF ) may not be reached in $\\mathbf {\\Omega }$ although being finite; however we aim for finding the infimum/minimum in (REF ).","Finding the minimizers in (REF ) is another question.","For instance, this can be solved whenever, additionally, $\\mathbf {\\Omega }$ is a closed set which implies the existence of minimizers in $\\mathbf {\\Omega }$ .","In this case, and when the number of such minimizers is finite, those can be approximated by dichotomic search.","For the sake of brevity, this will not be addressed in this paper.","(c) The purpose of the condition (REF ) is to get rid of the $\\lim \\sup $ type and $\\lim \\inf $ type results in our below-mentioned “bare-simulation” approach and to obtain simple limit-statements which motivate our construction.","In practice, it is enough to verify $\\mathbf {\\Omega }\\subseteq cl\\left(int\\left( \\mathbf {\\Omega }\\right) \\right) $ , which is equivalent to the left-hand part of (REF ).","Clearly, any open set $\\mathbf {\\Omega }\\subset \\mathbb {R}^{K}$ satisfies the left-hand part of (REF ).","In the subsetup where $\\mathbf {\\Omega }$ is a closed convex set and $int(\\mathbf {\\Omega })\\ne \\emptyset $ , (REF ) is satisfied and the minimizer $\\mathbf {Q}_{min}\\in \\mathbf {\\Omega }$ in (REF ) is attained and even unique.","When $\\mathbf {\\Omega }$ is open and satisfies (REF ), then the infimum in (REF ) exists but is reached at some generalized projection of $\\mathbf {P}$ on $\\mathbf {\\Omega }$ (see Csiszar [97] for the definition in the Kullback-Leibler case of probability measures, which extends to any $\\varphi -$ divergence in our framework).","(d) Without further mentioning, the regularity condition (REF ) is supposed to hold in the full topology.","Of course, $int\\left( \\mathbb {S}^{K} \\right) = \\emptyset $ and thus, for the important probability-vector setup $\\mathbf {\\Omega } \\subset \\mathbb {S}^{K}$ the condition (REF ) is violated which requires extra refinements (cf.","Subsection REF below).","The same is needed for $\\mathbf {\\Omega } \\subset A \\cdot \\mathbb {S}^{K}$ for some $A \\ne 1$ , since obviously $int\\left( A \\cdot \\mathbb {S}^{K} \\right) = \\emptyset $ ; such a context appears naturally e.g.","in connection with mass transportation problems (cf.","(REF ) below) and with distributed energy management (cf.","the paragraph after (REF )).","(e) Often, $\\mathbf {\\Omega }$ will present a (discrete) model recall that an alternative naming also used in literature is to call $\\Omega $ a model class (rather than model), and each $P \\in \\Omega $ a model (rather than model element).","Since $\\mathbf {\\Omega }$ is assumed to have a non-void interior (cf.","the right-hand part of (REF )), this will exclude (parametric) models $\\mathbf {\\Omega }:=\\lbrace \\mathbf {Q}_{\\theta }:\\theta \\in \\Theta \\rbrace $ for some $\\Theta \\subset \\mathbb {R}^{d}$ ($d0$ and let $\\widetilde{{P}}:=\\mathbf {P}/M_{\\mathbf {P}},$ and for $\\mathbf {Q}$ in $\\mathbf {\\Omega }$ , let $\\widetilde{\\mathbf {Q}}:=\\mathbf {Q}/M_{\\mathbf {P}}$ (notice that $\\widetilde{\\mathbf {Q}}$ may be a non-probability vector).","With the function $\\widetilde{\\varphi } \\in \\Upsilon (]a,b[)$ defined through $\\widetilde{\\varphi }:=M_{\\mathbf {P}} \\cdot \\varphi $ , we obtain $D_{\\varphi }(\\mathbf {Q},\\mathbf {P})=\\sum _{k=1}^{K}p_{k}\\cdot \\varphi \\left( \\frac{q_{k}}{p_{k}}\\right) =\\sum _{k=1}^{K}M_{\\mathbf {P}}\\cdot \\widetilde{p_{k}}\\cdot \\frac{\\varphi \\left( \\frac{M_{\\mathbf {P}}\\cdot \\widetilde{q_{k}}}{M_{\\mathbf {P}}\\cdot \\widetilde{p_{k}}}\\right) }{M_{\\mathbf {P}}}=D_{\\widetilde{\\varphi }}(\\widetilde{\\mathbf {Q}},\\widetilde{{P}}).$ It follows that the solution of (REF ) coincides with the one of the problem of finding $\\widetilde{\\Phi }_{\\widetilde{{P}}}(\\widetilde{\\mathbf {\\Omega }}) := \\inf _{\\widetilde{\\mathbf {Q}}\\in \\widetilde{\\mathbf {\\Omega }} }D_{\\widetilde{\\varphi } }(\\widetilde{\\mathbf {Q}},\\widetilde{{P}}),\\qquad \\textrm {with } \\widetilde{\\mathbf {\\Omega }}:=\\mathbf {\\Omega } /M_{\\mathbf {P}};$ as a side remark, one can see that in such a situation the rescaling of the divergence generator $\\varphi $ is important, which is one incentive that we incorporate multiples of $\\varphi $ below.","As an important special case we get for the choice $\\mathbf {P} := (1, \\ldots , 1) := \\mathbf {1}$ that the “prominent/frequent” separable nonlinear optimization problem of finding the optimal value $\\inf _{\\mathbf {Q} \\in \\mathbf {\\Omega } } \\sum _{k=1}^{K}\\varphi (q_{k})$ — with objective (e.g.","cost, energy, purpose) function $\\varphi \\in \\Upsilon (]a,b[)$ and constraint set (choice set, search space) $\\mathbf {\\Omega }$ — can be imbedded into our BS-approach by $\\inf _{\\mathbf {Q} \\in \\mathbf {\\Omega } } \\sum _{k=1}^{K}\\varphi (q_{k}) =\\inf _{\\mathbf {Q} \\in \\mathbf {\\Omega } } D_{\\varphi }(\\mathbf {Q},\\mathbf {1}) =\\inf _{\\widetilde{\\mathbf {Q}} \\in \\mathbf {\\Omega }/K }D_{K \\cdot \\varphi }(\\widetilde{\\mathbf {Q}},{P}^{unif}),$ with ${P}^{unif} := (\\frac{1}{K}, \\ldots , \\frac{1}{K})$ being the probability vector of frequencies of the uniform distribution on $\\lbrace 1, \\ldots , K\\rbrace $ .","Notice that with our new BS approach one may even tackle more general optimization problems of the form $\\inf _{\\breve{Q} \\in \\breve{\\Omega }} \\sum _{k=1}^{K} \\breve{\\varphi } (\\breve{q}_{k})$ where $\\breve{\\varphi }$ is some function which is finite and convex in a non-empty neighborhood (say, $]t_{0} + a- 1, t_{0} + b- 1[$ with $a < 1 < b$ ) of some point $t_{0} \\in \\mathbb {R}$ as well as strictly convex in a non-empty sub-neighborhood of $t_{0}$ ; for this, the function $\\varphi (t) := \\breve{\\varphi }(t+t_{0}-1) - \\breve{\\varphi }^{\\prime }(t_{0})\\cdot \\Big ((t+t_{0}-1) - t_{0} \\Big ) - \\breve{\\varphi }(t_{0}), \\qquad t \\in ]a,b[,\\nonumber $ (which corresponds to shifting the argument and adding an affine-linear function) should be a member of $\\Upsilon (]a,b[)$ , and from the corresponding minimization problem $& &\\inf _{\\widetilde{\\mathbf {Q}} \\in \\mathbf {\\Omega }/K }D_{K \\cdot \\varphi }(\\widetilde{\\mathbf {Q}},{P}^{unif}) =\\inf _{\\mathbf {Q} \\in \\mathbf {\\Omega } } \\sum _{k=1}^{K} \\varphi (q_{k})= \\inf _{\\mathbf {Q} \\in \\mathbf {\\Omega } } \\sum _{k=1}^{K}\\Big (\\breve{\\varphi }(q_{k}+t_{0}-1) - \\breve{\\varphi }^{\\prime }(t_{0})\\cdot ((q_{k}+t_{0}-1) -t_{0}) - \\breve{\\varphi }(t_{0})\\Big )\\nonumber \\\\& & = \\inf _{\\breve{\\mathbf {Q}} \\in \\mathbf {\\Omega } + t_{0} -1} \\sum _{k=1}^{K} \\ \\Big (\\breve{\\varphi }(\\breve{q}_{k}) - \\breve{\\varphi }^{\\prime }(t_{0})\\cdot (\\breve{q}_{k} -t_{0}) - \\breve{\\varphi }(t_{0})\\Big )\\nonumber \\\\& & = K \\cdot \\Big (t_{0} \\cdot \\breve{\\varphi }^{\\prime }(t_{0}) - \\breve{\\varphi }(t_{0})\\Big )+ \\inf _{\\breve{\\mathbf {Q}} \\in \\breve{\\mathbf {\\Omega }} } \\left(\\sum _{k=1}^{K} \\breve{\\varphi }(\\breve{q}_{k})- \\breve{\\varphi }^{\\prime }(t_{0}) \\cdot \\sum _{k=1}^{K} \\breve{q}_{k}\\right),\\qquad \\textrm {with } \\breve{\\mathbf {\\Omega }} := \\mathbf {\\Omega } + t_{0} -1,$ the term $\\inf _{\\breve{Q} \\in \\breve{\\Omega }} \\sum _{k=1}^{K} \\breve{\\varphi } (\\breve{q}_{k})$ should be recoverable; for instance, later on we shall employ constraints sets $\\breve{\\mathbf {\\Omega }}$ which particularly include $\\sum _{k=1}^{K} \\breve{q}_{k} = A > 0$ , whereas another possibility would be to use a $\\breve{\\varphi }$ which satisfies $\\breve{\\varphi }^{\\prime }(t_{0}) =0$ .","As a different line of flexibilization of (REF ), we can also deal with the problem $\\inf _{\\mathbf {Q} \\in \\mathbf {\\Omega } } h\\Big (\\sum _{k=1}^{K}\\varphi (q_{k})\\Big )$ through $\\inf _{\\mathbf {Q} \\in \\mathbf {\\Omega } }h\\Big (\\sum _{k=1}^{K}\\varphi (q_{k})\\Big ) =h\\Big ( \\inf _{\\widetilde{\\mathbf {Q}} \\in \\mathbf {\\Omega }/K }D_{K \\cdot \\varphi }(\\widetilde{\\mathbf {Q}},{P}^{unif}) \\Big )$ for any $\\varphi \\in \\Upsilon (]a,b[)$ and any continuous strictly increasing function $h: \\mathcal {H} \\, \\mapsto \\mathbb {R}$ with $\\mathcal {H} := [0,\\infty [$ (or a sufficiently large subset thereof), and with the problem $\\sup _{\\mathbf {Q} \\in \\mathbf {\\Omega } } h\\Big (\\sum _{k=1}^{K}\\varphi (q_{k})\\Big )$ through $\\sup _{\\mathbf {Q} \\in \\mathbf {\\Omega } }h\\Big (\\sum _{k=1}^{K}\\varphi (q_{k})\\Big ) =h\\Big ( \\inf _{\\widetilde{\\mathbf {Q}} \\in \\mathbf {\\Omega }/K }D_{K \\cdot \\varphi }(\\widetilde{\\mathbf {Q}},{P}^{unif}) \\Big )$ for any $\\varphi \\in \\Upsilon (]a,b[)$ and any continuous strictly decreasing function $h: \\mathcal {H} \\, \\mapsto \\mathbb {R}$ .","Combining (REF ) with (REF ) (respectively, with (REF )) leads to a further flexibilization.","Of course, we can also apply our BS method to the maximization $\\sup _{\\mathbf {Q} \\in \\mathbf {\\Omega } } h\\Big (\\sum _{k=1}^{K}\\zeta (q_{k})\\Big )$ for any concave function $\\zeta $ with $-\\zeta \\in \\Upsilon (]a,b[)$ and any continuous strictly increasing function $h: \\mathcal {H} \\, \\mapsto \\mathbb {R}$ with $\\mathcal {H} := - [\\infty ,0]$ (or a sufficiently large subset thereof), via $\\sup _{\\mathbf {Q} \\in \\mathbf {\\Omega } }h\\Big (\\sum _{k=1}^{K}\\zeta (q_{k})\\Big ) =h\\Big ( - \\inf _{\\widetilde{\\mathbf {Q}} \\in \\mathbf {\\Omega }/K }D_{- K \\cdot \\zeta }(\\widetilde{\\mathbf {Q}},{P}^{unif}) \\Big ) ,$ and to $\\inf _{\\mathbf {Q} \\in \\mathbf {\\Omega } } h\\Big (\\sum _{k=1}^{K}\\zeta (q_{k})\\Big )$ for any concave function $\\zeta $ with $-\\zeta \\in \\Upsilon (]a,b[)$ and any continuous strictly decreasing function $h: \\mathcal {H} \\, \\mapsto \\mathbb {R}$ , via $\\inf _{\\mathbf {Q} \\in \\mathbf {\\Omega } }h\\Big (\\sum _{k=1}^{K}\\zeta (q_{k})\\Big ) =h\\Big ( - \\inf _{\\widetilde{\\mathbf {Q}} \\in \\mathbf {\\Omega }/K }D_{- K \\cdot \\zeta }(\\widetilde{\\mathbf {Q}},{P}^{unif}) \\Big ) .$ Moreover, we can tackle $\\sup _{\\breve{Q} \\in \\breve{\\Omega }} \\sum _{k=1}^{K} \\breve{\\zeta } (\\breve{q}_{k})$ where $\\breve{\\zeta }$ is some function which is finite and concave in a non-empty neighborhood $]t_{0} + a- 1, t_{0} + b- 1[$ (with $a < 1 < b$ ) of some point $t_{0} \\in \\mathbb {R}$ as well as strictly concave in a non-empty sub-neighborhood of $t_{0}$ ; for this, the function $- \\zeta (t) := - \\breve{\\zeta }(t+t_{0}-1) + \\breve{\\zeta }^{\\prime }(t_{0})\\cdot \\Big ((t+t_{0}-1) - t_{0} \\Big ) + \\breve{\\zeta }(t_{0}), \\qquad t \\in ]a,b[,\\nonumber $ should be a member of $\\Upsilon (]a,b[)$ , and from the corresponding minimization problem $& &- \\inf _{\\widetilde{\\mathbf {Q}} \\in \\mathbf {\\Omega }/K }D_{- K \\cdot \\zeta }(\\widetilde{\\mathbf {Q}},{P}^{unif})= \\sup _{\\mathbf {Q} \\in \\mathbf {\\Omega } } \\sum _{k=1}^{K} \\zeta (q_{k})= \\sup _{\\mathbf {Q} \\in \\mathbf {\\Omega } } \\sum _{k=1}^{K}\\Big (\\breve{\\zeta }(q_{k}+t_{0}-1) - \\breve{\\zeta }^{\\prime }(t_{0})\\cdot ((q_{k}+t_{0}-1) -t_{0}) - \\breve{\\zeta }(t_{0})\\Big )\\nonumber \\\\& & = \\sup _{\\breve{\\mathbf {Q}} \\in \\mathbf {\\Omega } + t_{0} -1} \\sum _{k=1}^{K} \\ \\Big (\\breve{\\zeta }(\\breve{q}_{k}) - \\breve{\\zeta }^{\\prime }(t_{0})\\cdot (\\breve{q}_{k} -t_{0}) - \\breve{\\zeta }(t_{0})\\Big )\\nonumber \\\\& & = K \\cdot \\Big (t_{0} \\cdot \\breve{\\zeta }^{\\prime }(t_{0}) - \\breve{\\zeta }(t_{0})\\Big )+ \\sup _{\\breve{\\mathbf {Q}} \\in \\breve{\\mathbf {\\Omega }} } \\left(\\sum _{k=1}^{K} \\breve{\\zeta }(\\breve{q}_{k})- \\breve{\\zeta }^{\\prime }(t_{0}) \\cdot \\sum _{k=1}^{K} \\breve{q}_{k}\\right),\\qquad \\textrm {with } \\breve{\\mathbf {\\Omega }} := \\mathbf {\\Omega } + t_{0} -1,$ the term $\\sup _{\\breve{Q} \\in \\breve{\\Omega }} \\sum _{k=1}^{K} \\breve{\\zeta } (\\breve{q}_{k})$ should be recoverable; the left-hand side of (REF ) corresponds to the special case $h(x) := x$ of the BS-minimizable (REF ).","A combination of (REF ) with (REF ) (respectively, with (REF )) leads to a further flexibilization.","Remark 8 (a) Since $\\mathbf {1}$ can be seen e.g.","as a reference vector with (normalized) equal components, the quantity $\\inf _{\\mathbf {Q} \\in \\mathbf {\\Omega } } D_{\\varphi }(\\mathbf {Q},\\mathbf {1})$ in (REF ) can be interpreted as an “index/degree of (in)equality of the set $\\mathbf {\\Omega }$ ”, respectively as an “index/degree of diversity of the set $\\mathbf {\\Omega }$ ”.","(b) The quantity $\\sum _{k=1}^{K}\\varphi (q_{k})$ in (REF ) can be interpreted as (non-probability extension of an) $\\varphi -$ entropy in the sense of Burbea & Rao [68] (see also Csiszar [95], Ben-Bassat [38], Ben-Tal & Teboulle [40], Kesavan & Kapur [187], Dacunha-Castelle & Gamboa [102], Teboulle & Vajda [357], Gamboa & Gassiat [132], Vajda & Zvarova [376]); for applications to scalar quantization for lossy coding of information sources see e.g.","György & Linder [149].","More generally, the quantity $h\\Big (\\sum _{k=1}^{K}\\varphi (q_{k})\\Big )$ in (REF ) can be seen as (non-probability extension of an) $(h,\\varphi )-$ entropy in the sense of Salicru et al.","[314] (see also e.g.","Pardo [282], Vajda & Vasek [375], as well as e.g.","Chen et al.","[78] for uses as supervised adaption criterion within stochastic information gradient algorithms and Ren et al.","[308] for applications to tracking in networked control systems).","Important special cases will be discussed in more detail, below.","Returning to the original distance-minimizing Problem REF , after the first step (REF ) and (REF ), we proceed as follows: Step 2: construct an appropriate sequence $(\\xi _{n})_{n\\in \\mathbb {N}}$ of $\\mathbb {R}^{K}-$ valued random variables/random vectors (cf.","(REF ) in Definition REF ): The following condition transposes the minimization problem (REF ) into a BS minimizable/amenable problem in the sense of Definition REF and it is required in order that Problem (REF ) is equivalent to Problem (REF ).","The connection of this condition with (REF ) will be discussed in Proposition REF and its surroundings, see Section .","Condition 9 With $M_{\\mathbf {P}} =\\sum _{i=1}^{K}p_{i}>0$ , the divergence generator $\\varphi $ in (REF ) (cf.","also (REF )) satisfies $\\widetilde{\\varphi } := M_{\\mathbf {P}} \\cdot \\varphi \\in \\Upsilon (]a,b[)$ , i.e.","$\\widetilde{\\varphi } \\in \\widetilde{\\Upsilon }(]a,b[)$ (which is equivalent to $\\varphi \\in \\widetilde{\\Upsilon }(]a,b[)$ ) and there holds the representation $\\widetilde{\\varphi }(t) =\\sup _{z \\in \\mathbb {R}} \\left( z\\cdot t - \\log \\int _{\\mathbb {R}} e^{zy} d\\widetilde{\\mathbb {}}(y) \\right),\\qquad t \\in \\mathbb {R},$ for some probability measure $\\widetilde{\\mathbb {}}$ on the real line such that the function $z \\mapsto MGF_{\\widetilde{\\mathbb {}}}(z) := \\int _{\\mathbb {R}} e^{zy} d\\widetilde{\\mathbb {}}(y)$ is finite on some open interval containing zero in particular, this implies that $\\int _{\\mathbb {R}} y d\\widetilde{\\mathbb {}}(y) =1$ (cf.","(G11i) below) and that $\\widetilde{\\mathbb {}}$ has light tails..","In the following, let us explain the above-mentioned Step 2 in detail: for any $n \\in \\mathbb {N}$ and any $k \\in \\left\\lbrace 1, \\ldots ,K\\right\\rbrace $ , let $n_{k}:=\\lfloor n \\cdot \\widetilde{p}_{k}\\rfloor $ where $\\lfloor x \\rfloor $ denotes the integer part of $x$ .","We assume $\\mathbf {P} \\in \\mathbb {R}_{> 0}^{K}$ , and since thus none of the $\\widetilde{p}_{k}$ 's is zero, one has $\\lim _{n\\rightarrow \\infty } \\frac{n_{k}}{n} = \\widetilde{p}_{k}.$ Moreover, we assume that $n \\in \\mathbb {N}$ is large enough, namely $n \\ge \\max _{k \\in \\lbrace 1, \\ldots , K\\rbrace } \\frac{1}{\\widetilde{p}_{k}}$ , and decompose the set $\\lbrace 1, \\ldots , n\\rbrace $ of all integers from 1 to $n$ into the following disjoint blocks: $I_{1}^{(n)}:=\\left\\lbrace 1,\\ldots ,n_{1}\\right\\rbrace $ , $I_{2}^{(n)}:=\\left\\lbrace n_{1}+1,\\ldots ,n_{1}+n_{2}\\right\\rbrace $ , and so on until the last block $I_{K}^{(n)} := \\lbrace \\sum _{k=1}^{K-1} n_{k} + 1, \\ldots , n \\rbrace $ which therefore contains all integers from $n_{1}+ \\ldots +n_{K-1}+1$ to $n$ .","Clearly, $I_{k}^{(n)}$ has $n_{k} \\ge 1$ elements (i.e.","$card(I_{k}^{(n)}) = n_{k}$ where $card(A)$ denotes the number of elements in a set $A$ ) for all $k \\in \\lbrace 1, \\ldots , K-1\\rbrace $ , and the last block $I_{K}^{(n)}$ has $n- \\sum _{k=1}^{K-1} n_{k} \\ge 1$ elements which anyhow satisfies $\\lim _{n\\rightarrow \\infty } card(I_{K}^{(n)})/n=\\widetilde{p}_{K}$ if all $\\widetilde{p}_{k}$ ($k=1,\\ldots ,K$ ) are rational numbers in $]0,1[$ with $\\sum _{k=1}^{K} \\widetilde{p}_{k} =1$ and $N$ is the (always existing) smallest integer such that all $N \\cdot \\widetilde{p}_{k}$ ($k=1,\\ldots ,K$ ) are integers (i.e.","$\\in \\mathbb {N}$ ), then for any multiple $n= \\ell \\cdot N$ ($\\ell \\in \\mathbb {N}$ ) one gets that all $n_{k} = n \\cdot \\widetilde{p}_{k}$ are integers and that $card(I_{K}^{(n)}) = n_{K}$ ..","Furthermore, consider a vector $\\mathbf {\\widetilde{W}}:=\\left( \\widetilde{W}_{1},\\ldots ,\\widetilde{W}_{n}\\right) $ where the $\\widetilde{W}_{i}$ 's are i.i.d.","copies of the random variable $\\widetilde{W}$ whose distribution is associated with the divergence-generator $\\widetilde{\\varphi }:=M_{\\textbf {P}} \\cdot \\varphi $ through (REF ), in the sense that $\\mathbb {\\Pi }[\\widetilde{W}\\in \\cdot \\,]=\\widetilde{\\mathbb {}}[\\,\\cdot \\,]$ .","We group the $\\widetilde{W}_{i}$ 's according the above-mentioned blocks and sum them up blockwise, in order to build the following $K-$ component random vector $\\xi _{n}^{\\mathbf {\\widetilde{W}}}:=\\Big (\\frac{1}{n}\\sum _{i\\in I_{1}^{(n)}}\\widetilde{W}_{i},\\ldots ,\\frac{1}{n}\\sum _{i\\in I_{K}^{(n)}}\\widetilde{W}_{i}\\Big );$ notice that the signs of its components may be negative, depending on the nature of the $\\widetilde{W}_{i}$ 's; moreover, the expectation of its $k-$ th component converges to $\\widetilde{p}_{k}$ as $n$ tends to infinity (since the expectation of $\\widetilde{W}_{1} $ is 1), whereas the $n-$ fold of the corresponding variance converges to $\\widetilde{p}_{k}$ times the variance of $\\widetilde{W}_{1}$ .","For such a context, we obtain the following assertion on BS-minimizability: Theorem 10 Let $\\mathbf {P} \\in \\mathbb {R}_{> 0}^{K}$ , $M_{\\mathbf {P}}:=\\sum _{i=1}^{K}p_{i}>0$ , and suppose that the divergence generator $\\varphi $ satisfies the Condition REF above, with $\\widetilde{\\mathbb {}}$ (cf.","(REF )).","Additionally, let $\\widetilde{W}:=(\\widetilde{W}_{i})_{i\\in \\mathbb {N}}$ be a sequence of random variables where the $\\widetilde{W}_{i}$ 's are i.i.d.","copies of the random variable $\\widetilde{W}$ whose distribution is $\\mathbb {\\Pi }[\\widetilde{W}\\in \\cdot \\,]=\\widetilde{\\mathbb {}}[\\,\\cdot \\,]$ and thus, $E_{\\mathbb {\\Pi }}[\\widetilde{W}_{i}]=1$.","Then, in terms of the random vectors $\\xi _{n}^{\\mathbf {\\widetilde{W}}}=\\Big (\\frac{1}{n}\\sum _{i\\in I_{1}^{(n)}}\\widetilde{W}_{i},\\ldots ,\\frac{1}{n}\\sum _{i\\in I_{K}^{(n)}}\\widetilde{W}_{i}\\Big )\\hspace{56.9055pt} \\text{(cf.","(\\ref {Xi_n^W vector}))}\\nonumber $ there holds $-\\lim _{n\\rightarrow \\infty }\\frac{1}{n}\\log \\,\\mathbb {\\Pi }\\left[\\xi _{n}^{\\mathbf {\\widetilde{W}}}\\in \\mathbf {\\Omega } /M_{\\mathbf {P}}\\right]=\\inf _{Q\\in \\mathbf {\\Omega }}D_{\\varphi }(\\mathbf {Q},\\mathbf {P}\\ )$ for any $\\mathbf {\\Omega }\\subset \\mathbb {R}^{K}$ with regularity properties (REF ) and finiteness property (REF ).","In particular, for each $\\mathbf {P} \\in \\mathbb {R}_{> 0}^{K}$ the function $\\Phi _{\\mathbf {P}} \\left( \\cdot \\right) := D_{\\varphi }( \\cdot , \\mathbf {P} )$ (cf.","(REF )) is bare-simulation minimizable (BS-minimizable, cf.","(REF ))) on any such $\\mathbf {\\Omega }\\subset \\mathbb {R}^{K}$ .","The proof of Theorem REF will be given in Appendix A.","Remark 11 (i) Whenever $int(\\mathbf {\\Omega }) \\ne \\emptyset $ , it clearly holds that $\\liminf _{n\\rightarrow \\infty }\\frac{1}{n}\\log \\,\\mathbb {\\Pi }\\left[\\xi _{n}^{\\mathbf {\\widetilde{W}}}\\in \\mathbf {\\Omega } /M_{\\mathbf {P}}\\right] > 0$ ; see the proof of Theorem REF .","Hence, the limit in (REF ) exists and is finite when $\\mathbf {\\Omega }$ satisfies (REF ).","(ii) For some contexts, one can explicitly give the distribution of each of the independent (non-deterministic parts of the) components $\\Big (\\sum _{i\\in I_{k}^{(n)}}\\widetilde{W}_{i}\\Big )_{k=1,\\ldots ,K}$ of the vector $\\xi _{n}^{\\mathbf {\\widetilde{W}}}$ ; this will ease the corresponding concrete simulations.","For instance, we shall give those in the Examples REF , REF , REF , REF and REF in Section below.","(iii) Let us emphasize that we have assumed $\\mathbf {P} \\in \\mathbb {R}_{> 0}^{K}$ in Theorem REF which excludes $\\mathbf {P}$ from having zero components.","However, in cases where $\\lim _{x\\rightarrow \\infty }\\left|\\frac{\\varphi \\left(x \\cdot sgn(q)\\right) }{x \\cdot sgn(q)}\\right|=+\\infty $ for $q\\ne 0$ then if $p_{k_{0}}=0$ for some $k_{0\\text{ }}$ it follows that $q_{k_{0}}=0$ , which proves that $\\mathbf {P}\\in \\mathbb {R}_{>0}^{K}$ imposes no restriction in Theorem 9, since the projection of $\\mathbf {P}$ in $\\mathbf {\\Omega }$ then belongs to the subspace of $\\mathbb {R}^{K}$ generated by the non-null components of $\\mathbf {P}$ ; such a situation appears e.g.","for power divergence generators $\\varphi _{\\gamma }$ with $\\gamma > 2$ .","So there is no loss of generality assuming $\\mathbf {P}\\in \\mathbb {R}_{>0}^{K}$ in this case.","As examples for the applicability of Theorem REF , one can e.g.","combine each of the divergence generators $\\varphi $ of Table 1 (except for the 9th row) with any of the optimization problems (REF ), (REF ), (REF ), (REF ), (REF ), (REF ); the needed distributions $\\mathbb {\\Pi }[\\widetilde{W}\\in \\cdot \\,]=\\widetilde{\\mathbb {}}[\\,\\cdot \\,]$ correspond to the entry in the second last column with the choice $\\widetilde{c} \\cdot M_{\\mathbf {P}}$ instead of $\\widetilde{c}$ .","By taking $\\zeta := - \\varphi $ instead, one can solve the corresponding problems (REF ) and (REF ).","Returning to the general context, the limit statement (REF ) provides the principle for the approximation of the solution of Problem REF .","Indeed, by replacing the left-hand side in (REF ) by its finite counterpart, we deduce for given large $n$ $- \\frac{1}{n}\\log \\mathbb {\\Pi } \\left[ \\xi _{n}^{\\mathbf {\\widetilde{W}}}\\in \\mathbf {\\Omega }/M_{\\mathbf {P}} \\right]\\approx \\inf _{Q\\in \\mathbf {\\Omega } }D_{\\varphi }(\\mathbf {Q},\\mathbf {P});$ it remains to estimate the left-hand side of (REF ).","The latter can be performed either by a naive estimator of the frequency of those replications of $\\xi _{n,\\mathbf {\\widetilde{x}}}^{\\mathbf {\\widetilde{W}}}$ which hit $\\Omega /M_{\\mathbf {P}}$ , or more efficiently by some improved estimator; this will be discussed in detail in Section below.","Remark 12 According to (REF ) of Theorem 9 as well as (REF ), we can principally tackle the (approximative) computation of the minimum value $\\inf _{Q\\in \\mathbf {\\Omega }}D_{\\varphi }(\\mathbf {Q},\\mathbf {P}\\ ) =\\inf _{Q\\in \\mathbf {\\Omega }}\\sum _{k=1}^{K}p_{k}\\cdot \\varphi \\left( \\frac{q_{k}}{p_{k}}\\right)\\nonumber $ and in particular of $\\inf _{\\mathbf {Q} \\in \\mathbf {\\Omega } } \\sum _{k=1}^{K}\\varphi (q_{k}) =\\inf _{\\mathbf {Q} \\in \\mathbf {\\Omega } } D_{\\varphi }(\\mathbf {Q},\\mathbf {1})\\qquad \\textrm {(cf.","(\\ref {min Pb one}))}\\nonumber $ by basically only employing a fast and accurate — pseudo, true, natural, quantum — random number generatorsee e.g.","Tucci [366], Teh et al.","[358], Aghamohammadi & Crutchfield [6], Herrero-Collantes & Garcia-Escartin [159], Balygin et al.","[31], Dang et al.","[103], Gong et al.","[138], Chandrasekaran et al.","[77], Drahi et al.","[117], Kollmitzer et al.","[194], Liu et al.","[228], Fischer & Gauthier [126], Kim et al.","[188], Stoller & Campbell [342], provided that the constraint set $\\mathbf {\\Omega }$ satisfies the mild assumptions (REF ) and (REF ).","Notice that (REF ) also covers (e.g.","high-dimensional) constraint sets $\\mathbf {\\Omega }$ which are non-convex and even highly disconnected, and for which other minimization methods (e.g.","pure enumeration, gradient or steepest descent methods, etc.","a detailed discussion and comparisons are beyond the scope of this paper, given its current length ) may be problematic or intractable.","For instance, (REF ) covers kind of “$K-$ dimensional (not necessarily regular) polka dot (leopard skin) pattern type” relaxations $\\mathbf {\\Omega } := \\dot{\\bigcup }_{i=1}^{N} \\mathcal {U}_{i}(Q_{i}^{dis})$ of finite discrete constraint sets $\\mathbf {\\Omega }^{dis} := \\lbrace Q_{1}^{dis}, \\ldots , Q_{N}^{dis}\\rbrace $ of high cardinality $N$ (e.g.","being exponential or factorial in a large $K$ ), where each $K-$ dimensional vector $Q_{i}^{dis}$ (e.g.","having pure integer components only) is surrounded by some small (in particular, non-overlapping/disjoint) neighborhood $\\mathcal {U}_{i}(Q_{i}^{dis})$ ; in such a context, e.g.","$\\inf _{\\mathbf {Q} \\in \\mathbf {\\Omega } } \\sum _{k=1}^{K}\\varphi (q_{k})$ can be regarded as a “BS-tractable” relaxation of the nonlinear discrete (e.g.","integer, combinatorial see e.g.","Schrijver [324], Bertsimas & Weismantel [45], Chen et al.","[83], Onn [279], Korte & Vygen [195], Wolsey [393] for comprehensive books on discrete, integer and combinatorial programming and their vast applications ) optimization program $\\inf _{\\mathbf {Q} \\in \\mathbf {\\Omega }^{dis} } \\sum _{k=1}^{K}\\varphi (q_{k})$ ."],["Minimum distance/risk estimation\n","In statistics of discrete data — and in the adjacent research fields of information theory, artificial intelligence and machine learning — one often encounters the following minimum distance estimation (MDE) problem which is often also named as estimation of the empirical risk: for index $i\\in \\mathbb {N}$ , let the generation of the $i-$ th (uncertainty-prone) data point be represented by the random variable $X_{i}$ which takes values in the discrete set $\\mathcal {Y}:=\\left\\lbrace d_{1},\\cdots ,d_{K}\\right\\rbrace $ of $K$ distinct values “of any kind”.","It is assumed that there exists a probability measure $\\mathbb {P}[\\cdot \\,]$ on $\\mathcal {Y}$ which is the a.s. limit of the empirical measures $\\mathbb {P}_{n}^{emp}$ defined by the collection of collected $\\left( X_{1},..,X_{n}\\right) $ as $n$ tends to infinity, in formula $\\lim _{n\\rightarrow \\infty }\\mathbb {P}_{n}^{emp}:=\\lim _{n\\rightarrow \\infty }\\frac{1}{n}\\sum _{i=1}^{n}\\delta _{X_{i}}=\\mathbb {P} \\qquad \\text{a.s.}$ where $\\delta _{y}$ denotes the one-point distribution (Dirac mass) at point $y$ notice that $\\mathbb {P}_{n}^{emp}$ a probability measure on the data space $\\mathcal {Y}$ , which is random due to its dependence on the $X_{i}$ ’s.","We will assume that none of the entries of $\\mathbb {P}$ bears zero mass so that $\\mathbb {P}$ is identified with a point in the interior of $\\mathbb {S}^{K}$ (see below).","The underlying probability space (say, $(\\mathfrak {X},\\mathcal {A},\\mathbb {\\Pi })$ ) where the above a.s. convergence holds, pertains to the random generation of the sequence $\\left( X_{n}\\right)_{n\\ge 1}$ , of which we do not need to know but for (REF ).","Examples include the i.i.d.","case (where the $X_{i}$ ’s are independent and have common distribution $\\mathbb {P}$ ), ergodic Markov chains on $\\mathcal {Y}$ with stationary distribution $\\mathbb {P}$ , more globally autoregressive chains with stationary measure $\\mathbb {P}$ , etc.","Let us briefly discuss our assumption (REF ) (resp.","its vector form (REF ) below) on the limit behavior of the empirical distribution of the observed sample $\\mathbf {X}_{n}:=\\left(X_{1},..,X_{n}\\right)$ as $n$ tends to infinity.","In the “basic” statistical context, the sample $\\mathbf {X}_{n}$ consists of i.i.d.","replications of a generic random variable $X$ with probability distribution ${P}$ .","However, our approach captures many other sampling schemes, where the distribution ${P}$ is defined implicitly through (REF ) for which we aim at some estimate of $\\Phi _{{P}}\\left(\\mathbb {\\Omega }\\right)$ of a family $\\mathbb {\\Omega }$ of probability distributions on $\\mathcal {Y}$ .","Sometimes the sequence of samples $\\left( \\mathbf {X}_{n}\\right) _{n\\ge 1}$ may stem from a triangular array so that $\\mathbf {X}_{n}=\\left(X_{1,n},..,X_{k_{n},n}\\right) $ with $k_{n}\\rightarrow \\infty $ and statement (REF ) is substituted by $\\lim _{n\\rightarrow \\infty }\\frac{1}{k_{n}}\\sum _{i=1}^{k_{n}}\\delta _{X_{i,n}} ={P} \\text{ \\ a.s.}$ which does not alter the results of this paper by any means.","given a model $\\mathbb {\\Omega }$ , i.e.","a family $\\mathbb {\\Omega }$ of probability distributions on $\\mathcal {Y}$ each of which serves as a potential description of the underlying (unknown) data-generating mechanism $\\mathbb {P}$ , one would like to find $\\Phi _{\\mathbb {P}}(\\mathbb {\\Omega }) := \\inf _{\\mathbb {Q}\\in \\mathbb {\\Omega }} D_{\\varphi }( \\mathbb {Q}, \\mathbb {P} )$ which quantifies the adequacy of the model $\\mathbb {\\Omega }$ for modeling $\\mathbb {P}$ , via the minimal distance/dissimilarity of $\\mathbb {\\Omega }$ to $\\mathbb {P}$ ; a lower $\\Phi _{\\mathbb {P}}-$ value means a better adequacy (in the sense of a lower departure between the model and the truth, cf.","Lindsay [222], Lindsay et al.","[223], Markatou & Sofikitou [248], Markatou & Chen [249]).","Hence, especially in the context of model selection within complex big-data contexts, for the search of appropriate models $\\mathbb {\\Omega }$ and model elements/members therein, the (fast and efficient) computation of $\\Phi _{\\mathbb {P}}(\\mathbb {\\Omega })$ constitutes a decisive first step, since if the latter is “too large” (respectively “much larger than” $\\Phi _{\\mathbb {P}}(\\overline{\\mathbb {\\Omega }})$ for some competing model $\\overline{\\mathbb {\\Omega }}$ ), then the model $\\mathbb {\\Omega }$ is “not adequate enough” (respectively “much less adequate than” $\\overline{\\mathbb {\\Omega }}$ ); in such a situation, the effort of computing the (not necessarily unique) best model element/member $\\arg \\inf _{\\mathbb {Q}\\in \\mathbb {\\Omega }} D_{\\varphi }( \\mathbb {Q}, \\mathbb {P} )$ within the model $\\mathbb {\\Omega }$ is “not very useful” and is thus a “waste of computational time”.","Because of such considerations, we concentrate ourselves to finding the infimum (REF ) rather than finding the corresponding minimizer(s).","Variants of (REF ) are of interest, too.","Since $int(\\mathbb {\\Omega })$ is supposed to be a non-empty set in the space of probability distribution on $\\mathcal {Y}$ , the present procedure is fitted for semi-parametric models $\\mathbb {\\Omega }$ , e.g.","such as defined through moment conditions (as extensions of the Empirical Likelihood paradigm, see e.g.","Broniatowski & Keziou [62]), or through L-moment conditions (i.e.","moment conditions pertaining to quantile measures, see Broniatowski & Decurninge [59]), or even more involved non-parametric models where the geometry of $\\mathbb {\\Omega }$ does not allow for ad-hoc procedures.","The measurement or the estimation of $\\Phi _{\\mathbb {P}}(\\mathbb {\\Omega })$ is a tool for the choice of pertinent putative models $\\mathbb {\\Omega }$ among a class of specifications.","The case when $\\Phi _{\\mathbb {P}}(\\mathbb {\\Omega })>0$ is interesting in its own, since it is quite common in engineering modelling to argue in favor of misspecified models (or (non-void) neighborhoods of such models for sake of robustness issues), due to quest for conservatism; the choice between them is a widely open field e.g.","in the practice of reliability.","This also opens the question of the choice of the divergence generator $\\varphi $ ; although this will not be discussed in this paper, as a motivating running example the reader may keep in mind the generator $\\varphi _{2}(x):=(x-1)^{2}/2$ which induces the divergence $D_{\\varphi _{2}}(\\mathbb {Q},\\mathbb {P})$ (see (REF ) below for details) which quantifies the expected square relative error when substituting the true distribution $\\mathbb {P}$ by the model $\\mathbb {Q}$ .","As examples of sets $\\mathbb {\\Omega }$ of probability distributions on $\\mathcal {Y}$ which obey (through their $K-$ vector of corresponding probability masses/frequencies) the global assumptions (REF ), one can consider semi-parametric models defined by moment conditions or defined through L-moment constraints (hence on the quantile functions), as well as more involved ones, for which no closed form of the divergence with respect to any probability distribution is available.","In the context of model selection, the choice of $\\mathbb {\\Omega }$ may be dictated by various considerations, and misspecification may be assumed as a requisite, for example for conservatism in reliability design.","An estimate of $\\Phi _{\\mathbb {P}}(\\mathbb {\\Omega })$ can be used as a statistics for some test of fit, and indeed the likelihood ratio test adapted to some semi-parametric models has been generalized to the divergence setting (see Broniatowski & Keziou [62]).","The statement of the limit distributions of our estimate, under the model and under misspecification, is postponed to future work.","In the following, we compute/approximate (REF ) — and some variants thereof — by our bare simulation (BS) method, by “mimicking” the deterministic minimization problem (REF ) respectively (REF ).","Let us first remark that, as usual, each probability distribution (probability measure) $\\mathbb {P}$ on $\\mathcal {Y}=\\left\\lbrace d_{1},\\ldots ,d_{K}\\right\\rbrace $ can be uniquely identified with the (row) vector ${P} := (p_{1}, \\ldots , p_{K}) \\in \\mathbb {S}^{K}$ of the corresponding probability masses (frequencies) $p_{k} = \\mathbb {P}[\\lbrace d_{k} \\rbrace ]$ via $\\mathbb {P}[A] = \\sum _{k=1}^{K} p_{k} \\cdot {1}_A(d_{k}) $ for each $A \\subset \\mathcal {Y}$ , where ${1}_A(\\cdot )$ denotes the indicator function on the set $A$ .","In particular, the probability distribution $\\mathbb {P}$ in (MDE1) can be identified with $(p_{1}, \\ldots , p_{K})$ in terms of $p_{k} = \\mathbb {P}[\\lbrace d_{k} \\rbrace ]$ (which in the i.i.d.","case turns into $p_{k} = \\mathbb {\\Pi }[X_{1} = d_{k}]$ ).","Along this line, the family $\\mathbb {\\Omega }$ of probability distributions in (MDE2) can be identified with a subset $\\Omega $$\\Omega \\subset \\mathbb {S}^{K}$ of probability vectors (viz.","of vectors of probability masses).","Analogously, each finite nonnegative measure $Q$ on $\\mathcal {Y}$ can be uniquely identified with a vector $\\mathbf {Q} := (q_{1}, \\ldots , q_{K}) \\in \\mathbb {R}_{\\ge 0}^{K}$ , and each finite signed measure $Q$ with a vector $\\mathbf {Q} := (q_{1}, \\ldots , q_{K}) \\in \\mathbb {R}^{K}$ .","The corresponding divergences between distributions/measures are then, as usual, defined through the divergences between their respective masses/frequencies: $D_{\\varphi }(Q,\\mathbb {P}) : = D_{\\varphi }(\\mathbf {Q},{P}).$ In particular, $\\mathbb {P}_{n}^{emp}$ can be identified with the vector ${P}_{n}^{emp} := (p_{n,1}^{emp}, \\ldots , p_{n,K}^{emp})$ where $p_{n,k}^{emp} \\ := \\ \\frac{1}{n} \\cdot n_{k} \\ := \\ \\frac{1}{n} \\cdot card(\\bigl \\lbrace i \\in \\lbrace 1, \\ldots , n\\rbrace : \\ X_{i} = d_{k} \\bigr \\rbrace )\\ =: \\ \\frac{1}{n} \\cdot card(I_{k}^{(n)}) , \\quad k \\in \\lbrace 1, \\ldots , K\\rbrace ,$ and accordingly the required limit behaviour (REF ) is equivalent to the vector-convergence $\\lim _{n\\rightarrow \\infty } \\Big ( \\frac{n_{1}}{n}, \\ldots , \\frac{n_{K}}{n} \\Big ) = (p_{1}, \\ldots , p_{K})\\qquad \\textrm {a.s.}$ Notice that, in contrast to the case handled in the above Subsection REF , the sets $I_{k}^{(n)}$ of indexes introduced in (REF ) and their numbers $n_{k} = card(I_{k}^{(n)})$ of elements are now random (due to their dependence on the $X_{i}$ ’s) and $M_{{P}_{n}^{emp}}=1$ .","In a batch procedure, when $D_{\\varphi }(\\textrm {$$\\hspace{-6.544pt}$$},{P}_{n}^{emp})$ is estimated once the sample $\\left(X_{1},..,X_{n}\\right)$ is observed, we may reorder this sample by putting the $n_{1}$ sample points $X_{i}$ which are equal to $d_{1}$ in the first places, and so on; accordingly one ends up with index sets $I_{k}^{(n)}$ as defined in Section REF .","When the online acquisition of the data $X_{i}$ ’s is required, then we usually do not reorder the sample, and the $I_{k}^{(n)}$ 's do not consist in consecutive indexes, which does not make any change with respect to the resulting construction nor to the estimator.","The above considerations open the gate to our desired “mimicking” of (REF ) and (REF ) to achieve (REF ) (and some variants thereof) by our bare simulation (BS) method.","To proceed, we employ a family of random variables $(W_{i})_{i \\in \\mathbb {N}}$ of independent and identically distributed $\\mathbb {R}-$ valued random variables with probability distribution $\\mathbb {}[ \\cdot \\, ] := \\mathbb {\\Pi }[W_{1} \\in \\cdot \\, ]$ (being connected with the divergence generator $\\varphi \\in \\Upsilon (]a,b[)$ via the representability (REF )), such that $(W_{i})_{i \\in \\mathbb {N}}$ is independent of $(X_{i})_{i \\in \\mathbb {N}}$ on the common underlying probability space $(\\mathfrak {X},\\mathcal {A},\\mathbb {\\Pi })$.","As a next step, notice that the “natural candidate” $\\xi _{n,\\mathbf {X}}^{\\mathbf {W}}:= \\frac{1}{n} \\cdot \\sum _{k=1}^{K}\\left(\\sum _{i \\in I_{k}^{(n)}}W_{i}\\right) \\cdot \\delta _{d_{k}}\\ = \\ \\frac{1}{n}\\sum _{i=1}^{n} W_{i} \\cdot \\delta _{X_{i}}$ is not a probability measure since its total mass is not 1 in general, since in terms of its equivalent vector version $\\xi _{n,\\mathbf {X}}^{\\mathbf {W}} := \\Big (\\frac{1}{n}\\sum _{i\\in I_{1}^{(n)}} W_{i}, \\ldots , \\frac{1}{n} \\sum _{i\\in I_{K}^{(n)}} W_{i} \\Big )$ the sum $\\sum _{k=1}^{K} \\frac{1}{n} \\sum _{i\\in I_{k}^{(n)}} W_{i} =\\frac{1}{n} \\sum _{j=1}^{n} W_{i}$ of the $K$ vector components of (REF ) is typically not equal to 1; this implies that no limit result of the form (REF ) with finite limit can hold, since $\\xi _{n,\\mathbf {X}}^{\\mathbf {W}}$ takes values in $\\mathbb {R}^{K}$ and $\\textrm {$$\\hspace{-6.544pt}$$}$ is a subset in the probability simplex $\\mathbb {S}^{K}$ which has void interior in $\\mathbb {R}^{K}$ causing a violation of condition (REF ) (cf.","Remark REF (c)); moreover, depending on the concrete form of the generator $\\varphi $ , the corresponding weights may take negative values.","Therefore, we need some “rescaling”.","Indeed, let us introduce the normalized weighted empirical measure $\\xi _{n,\\mathbf {X}}^{w\\mathbf {W}} &:=&{\\left\\lbrace \\begin{array}{ll}\\frac{1}{\\sum _{k=1}^{K}\\sum _{i \\in I_{k}^{(n)}}W_{i}} \\cdot \\sum _{k=1}^{K}\\left(\\sum _{i \\in I_{k}^{(n)}}W_{i}\\right) \\cdot \\delta _{d_{k}}= \\sum _{i=1}^{n} \\frac{W_{i}}{\\sum _{j=1}^{n} W_{j}} \\cdot \\delta _{X_{i}},\\qquad \\textrm {if } \\sum _{j=1}^{n} W_{j} \\ne 0, \\\\\\infty \\cdot \\sum _{k=1}^{K} \\delta _{d_{k}} =: \\underline{\\infty }, \\hspace{216.2411pt}\\textrm {if } \\sum _{j=1}^{n} W_{j} = 0,\\end{array}\\right.","}$ which will substitute $\\xi _{n,\\mathbf {X}}^{\\mathbf {W}}$ and which may belong to $\\textrm {$$\\hspace{-6.544pt}$$}$ with positive probability.","The equivalent vector version of $\\xi _{n,\\mathbf {X}}^{w\\mathbf {W}}$ is given by $\\xi _{n,\\mathbf {X}}^{w\\mathbf {W}} &:=&{\\left\\lbrace \\begin{array}{ll}\\left(\\frac{\\sum _{i \\in I_{1}^{(n)}}W_{i}}{\\sum _{k=1}^{K}\\sum _{i \\in I_{k}^{(n)}}W_{i}},\\ldots , \\frac{\\sum _{i \\in I_{K}^{(n)}}W_{i}}{\\sum _{k=1}^{K}\\sum _{i \\in I_{k}^{(n)}}W_{i}} \\right) ,\\qquad \\textrm {if } \\sum _{j=1}^{n} W_{j} \\ne 0, \\\\\\ (\\infty , \\ldots , \\infty ) =: \\infty , \\hspace{113.81102pt} \\textrm {if } \\sum _{j=1}^{n} W_{j} = 0,\\end{array}\\right.","}$ a point in the linear subset of $\\mathbb {R}^{K}$ spanned by $\\mathbb {S}^{K}$ at infinity.","Remark 13 (i) (Concerning e.g.","computer-program command availability) In case of $\\sum _{j=1}^{n}W_{j}=0$ , in (REF ) we may equivalently assign to $\\xi _{n,\\mathbf {X}}^{w\\mathbf {W}}$ instead of $\\underline{\\infty }$ any measure (e.g.","probability distribution) which does not belong to $\\mathbb {\\Omega }$ , respectively, in (REF ) we may equivalently choose for $\\xi _{n,\\mathbf {X}}^{w\\mathbf {W}}$ any vector outside of $\\Omega $$\\Omega $ instead of $\\infty $ .","(ii) By construction, in case of $\\sum _{j=1}^{n} W_{j} \\ne 0$ , the sum of the random $K$ vector components of (REF ) is now automatically equal to 1, but — as (depending on $\\varphi $ ) the $W_{i}$ ’s may take both positive and negative values — these random components may be negative (resp.","nonnegative) with probability strictly greater (resp.","smaller) than zero (resp.","one); in the framework of (REF ) this means that $\\xi _{n,\\mathbf {X}}^{w\\mathbf {W}}$ is in general a random signed measure with total mass 1, in case of $\\sum _{j=1}^{n} W_{j} \\ne 0$ .","However, $\\mathbb {\\Pi } [\\xi _{n,\\mathbf {X}}^{w\\mathbf {W}}\\in \\mathbb {S}_{>0}^{K}]>0$ since all the (identically distributed) random variables $W_{i}$ have expectation 1 (as a consequence of the assumed representability (REF )); in case of $\\mathbb {\\Pi }[W_{1}>0]=1$ one has even $\\mathbb {\\Pi }[\\xi _{n,\\mathbf {X}}^{w\\mathbf {W}}\\in \\mathbb {S}_{>0}^{K}]=1$ .","In the particular context of Example REF (c), one gets $\\mathbb {\\Pi }[\\xi _{n,\\mathbf {X}}^{w\\mathbf {W}}\\in \\mathbb {S}_{>0}^{K}]=\\left( \\mathbb {\\Pi }[ W_{1}>0]\\right)^{n}=\\left( \\int _{0}^{\\infty }\\sqrt{\\frac{\\widetilde{c}}{2\\pi }}\\cdot \\exp (-\\frac{\\widetilde{c}\\cdot (u-1)^{2}}{2})du\\right) ^{n}\\in ]0,1[$ .","Summing up things, the probability $\\mathbb {\\Pi }[\\xi _{n,\\mathbf {X}}^{w\\mathbf {W}}\\in \\Omega $$\\Omega ]$ is strictly positive and finite at least for large n, whenever $\\Phi _{{P}}(\\Omega $$\\Omega )= \\inf _{{Q}\\in \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega } }\\ D_{\\varphi }({Q},{P})$ is finite.","(iii) By generalizing the terminology of e.g.","Vajda [372], through the right-hand side of (REF ) one can interpret (for $\\sum _{j=1}^{n} W_{j} \\ne 0$ ) the normalized weighted empirical measure $\\xi _{n,\\mathbf {X}}^{w\\mathbf {W}}$ as response of an output neuron in a random perceptron consisting of random inputs $\\mathbf {X}$ , a layer with $n$ units having one-point-distribution-valued responses $\\delta _{X_{1}}, \\ldots , \\delta _{X_{n}}$ , and independent random synaptic weights $\\left(\\frac{W_{1}}{\\sum _{j=1}^{n} W_{j}}, \\ldots , \\frac{W_{n}}{\\sum _{j=1}^{n} W_{j}}\\right)$ .","With the above-mentioned ingredients, we are now in the position to tackle a variant of the distance minimization problem (REF ), by our bare simulation method through “mimicking” the deterministic minimization problem (REF ) respectively (REF ).","For this, we also employ the conditional distributions $\\mathbb {\\Pi }_{n}[\\, \\cdot \\, ] := \\mathbb {\\Pi }_{X_{1}^{n}}[\\, \\cdot \\, ] := \\mathbb {\\Pi }[ \\, \\cdot \\, | \\,X_{1}, \\ldots , X_{n} ]$ and obtain the following Theorem 14 Suppose that $(X_{i})_{i\\in \\mathbb {N}}$ is a sequence of random variables with values in $\\mathcal {Y}:=\\left\\lbrace d_{1},\\cdots ,d_{K}\\right\\rbrace $ such that (REF ) holds for some probability measure $\\mathbb {P}[\\cdot \\,]$ on $\\mathcal {Y}$ having no zero-mass frequencies (or equivalently, (REF ) holds for some probability vector ${P} \\in \\mathbb {S}_{> 0}^{K}$ ).","Moreover, let $(W_{i})_{i \\in \\mathbb {N}}$ be a family of independent and identically distributed $\\mathbb {R}-$ valued random variables with probability distribution $\\mathbb {}[ \\cdot \\, ] := \\mathbb {\\Pi }[W_{1} \\in \\cdot \\, ]$ being connected with the divergence generator $\\varphi \\in \\Upsilon (]a,b[)$ via the representability (REF ), such that $(W_{i})_{i \\in \\mathbb {N}}$ is independent of $(X_{i})_{i \\in \\mathbb {N}}$ .","Then there holds $-\\lim _{n\\rightarrow \\infty }\\frac{1}{n}\\log \\,\\mathbb {\\Pi }_{X_{1}^{n}}\\left[\\xi _{n,\\mathbf {X}}^{w\\mathbf {W}}\\in \\mathbb {\\Omega }\\right]& =\\inf _{\\mathbb {Q}\\in \\mathbb {\\Omega } }\\ \\inf _{m\\ne 0}D_{\\varphi }(m\\cdot \\mathbb {Q},\\mathbb {P})\\, \\\\& =\\inf _{m\\ne 0}\\ \\inf _{\\mathbb {Q}\\in \\mathbb {\\Omega } }D_{\\varphi }(m\\cdot \\mathbb {Q},\\mathbb {P})\\\\& =\\inf _{m\\ne 0}\\ \\inf _{{Q}\\in \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega } }D_{\\varphi }(m\\cdot {Q},{P})\\\\& = \\inf _{{Q}\\in \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega } }\\ \\inf _{m\\ne 0}D_{\\varphi }(m\\cdot {Q},{P})= -\\lim _{n\\rightarrow \\infty }\\frac{1}{n}\\log \\,\\mathbb {\\Pi }_{X_{1}^{n}}\\left[\\xi _{n,\\mathbf {X}}^{w\\mathbf {W}}\\in \\textrm {\\right.\\Omega \\hspace{-6.544pt}\\Omega }$ for all sets $\\mathbb {\\Omega }$ of probability distributions such that their equivalent probability-vector form $\\Omega $$\\Omega $ satisfies the regularity properties (REF ) in the relative topology and the finiteness property (REF ); notice that for the equality () we have used the “divergence link” (REF ).","In particular, for each ${P} \\in \\mathbb {S}_{>0}^{K}$ (respectively, its equivalent probability-distribution $\\mathbb {P}$ ) the function ${Q} \\mapsto \\inf _{m\\ne 0}D_{\\varphi }(m\\cdot {Q},{P}) $ (respectively, the function $\\mathbb {Q} \\mapsto \\inf _{m\\ne 0}D_{\\varphi }(m\\cdot \\mathbb {Q},\\mathbb {P})$ ) is BS-minimizable (cf.","(REF )) on all sets $\\textrm {$$\\hspace{-6.544pt}$$} \\subset \\mathbb {S}^{K}$ satisfying (7) in the relative topology and (9) (respectively, on their probability-distribution-equivalent $\\mathbb {\\Omega }$ ).","The proof of Theorem REF will be given in Appendix B. Analogously to Remark REF (iii), let us emphasize that we have assumed ${P} \\in \\mathbb {S}_{> 0}^{K}$ in Theorem REF .","Henceforth, for sets $\\textrm {$$\\hspace{-6.544pt}$$} \\subset \\mathbb {S}^{K}$ of probability vectors we deal with (REF ) only in the relative topology; thus, the latter will be unmentioned for the sake of brevity.","Remark REF (a),(b),(c),(e) applies accordingly.","Remark 15 (i) In strong contrast to Theorem REF , the above result does not provide a direct tool for the solution of Problem (REF ) since the limit in (REF ) bears no direct information on the minimum divergence $D_{\\varphi }\\left( \\mathbb {\\Omega },\\mathbb {P} \\right) :=\\inf _{\\mathbb {Q}\\in \\mathbb {\\Omega }} D_{\\varphi }( \\mathbb {Q}, \\mathbb {P})$ ; the link between the corresponding quantities can be emphasized and exploited e.g.","in the case of power type divergences, which leads to explicit minimization procedures as shown in the Subsection REF below.","For general divergences, Theorem REF allows for the estimation of upper and lower bounds of $D_{\\varphi }\\left( \\mathbb {\\Omega },\\mathbb {P} \\right)$ , as developed in the Subsection REF below.","(ii) Notice that $\\breve{D}_{\\varphi }(\\mathbb {Q},\\mathbb {P}):=\\inf _{m\\ne 0} D_{\\varphi }(m\\cdot \\mathbb {Q},\\mathbb {P})$ satisfies the axioms of a divergence, that is, $\\breve{D}_{\\varphi }(\\mathbb {Q},\\mathbb {P}) \\ge 0$ , as well as $\\breve{D}_{\\varphi }(\\mathbb {Q},\\mathbb {P}) = 0$ if and only if $\\mathbb {Q} = \\mathbb {P}$ (reflexivity).","Hence, in Theorem REF we are still within our framework of bare simulation of a divergence minimum w.r.t.","its first component (however, notice the difference to (i)).","(iii) Viewed from a “reverse” angle, Theorem REF gives a crude approximation for the probability for $\\xi _{n,\\mathbf {X}}^{w\\mathbf {W}}$ to belong to $\\mathbb {\\Omega }$ , conditionally upon $\\mathbf {X} = (X_{1}, \\ldots , X_{n})$ .","(iv) In the same spirit as Remark REF (ii), for some contexts one can explicitly give the distribution of each of the independent components $\\Big (\\sum _{i\\in I_{k}^{(n)}}W_{i}\\Big )_{k=1,\\ldots ,K}$ of the vector $\\xi _{n}^{w\\mathbf {W}}$ ; this will ease the corresponding concrete simulations in a batch procedure.","For instance, we shall give those in the Examples REF , REF , REF , REF and REF in the Section below.","(v) Consider the special “degenerate” case where all the data observations are certain and thus $(X_{i})_{i \\in \\mathbb {N}}$ is nothing but a purely deterministic sequence, say $(\\widetilde{x}_{i})_{i\\in \\mathbb {N}}$ , of elements $\\widetilde{x}_{i}$ from the arbitrary set $\\mathcal {Y}:=\\left\\lbrace d_{1},\\ldots ,d_{K}\\right\\rbrace $ of $K$ distinct values “of any kind” (e.g., $\\mathcal {Y}$ may consist of $K$ distinct numbers); then the corresponding empirical distribution $\\mathbb {P}_{n}^{emp}$ can be identified with the vector ${P}_{n}^{emp} := (p_{n,1}^{emp}, \\ldots , p_{n,K}^{emp})$ where $p_{n,k}^{emp} \\ := \\ \\frac{1}{n} \\cdot n_{k} \\ := \\ \\frac{1}{n} \\cdot card(\\bigl \\lbrace i \\in \\lbrace 1, \\ldots , n\\rbrace : \\ \\widetilde{x}_{i} = d_{k} \\bigr \\rbrace )\\ =: \\ \\frac{1}{n} \\cdot card(I_{k}^{(n)}) , \\quad k \\in \\lbrace 1, \\ldots , K\\rbrace ,$ and accordingly the required limit behaviour (REF ) is equivalent to the vector-convergence $\\lim _{n\\rightarrow \\infty } \\Big ( \\frac{n_{1}}{n}, \\ldots , \\frac{n_{K}}{n} \\Big ) = (p_{1}, \\ldots , p_{K})\\quad \\textrm {for some p_{1} >0, \\ldots , p_{K} >0 such that \\sum _{k=1}^{K} p_{k} = 1.","}$ Correspondingly, with the notations ${P} := (p_{1}, \\ldots , p_{K})$ and $\\mathbf {\\widetilde{x}} := (\\widetilde{x}_{1}, \\ldots , \\widetilde{x}_{n})$ , the vector-form part of the assertion (REF ) of Theorem REF becomes $-\\lim _{n\\rightarrow \\infty }\\frac{1}{n}\\log \\,\\mathbb {\\Pi }\\left[\\xi _{n,\\mathbf {\\widetilde{x}}}^{w\\mathbf {W}}\\in \\textrm {\\right.\\Omega \\hspace{-6.544pt}\\Omega }\\ = \\ \\inf _{{Q}\\in \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega } }\\ \\inf _{m\\ne 0}D_{\\varphi }(m\\cdot {Q},{P})\\ = \\ \\inf _{m\\ne 0}\\ \\inf _{{Q}\\in \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega } }D_{\\varphi }(m\\cdot {Q},{P})$ for all subsets $\\Omega $$\\Omega \\subset \\mathbb {S}^{K}$ satisfying the regularity properties (REF ) and the finiteness property (REF ); notice that the conditional probability $\\mathbb {\\Pi }_{X_{1}^{n}}[\\, \\cdot \\, ]$ has degenerated to the ordinary probability $\\mathbb {\\Pi }[\\, \\cdot \\, ]$ .","(vi) In a similar fashion to the proof of (the special degenerate case (v) of) Theorem REF , one can show $-\\lim _{n\\rightarrow \\infty }\\frac{1}{n}\\log \\,\\mathbb {\\Pi }\\left[\\xi _{n}^{w\\mathbf {W}}\\in \\textrm {\\right.\\Omega \\hspace{-6.544pt}\\Omega }\\ = \\ \\inf _{{Q}\\in \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega } }\\ \\inf _{m\\ne 0}D_{\\varphi }(m\\cdot {Q},{P})\\ = \\ \\inf _{m\\ne 0}\\ \\inf _{{Q}\\in \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega } }D_{\\varphi }(m\\cdot {Q},{P})$ for all subsets $\\Omega $$\\Omega \\subset \\mathbb {S}^{K}$ with regularity properties (REF ) and the finiteness property (REF ), where $\\xi _{n}^{w\\mathbf {W}} &:=&{\\left\\lbrace \\begin{array}{ll}\\left(\\frac{\\sum _{i \\in I_{1}^{(n)}}W_{i}}{\\sum _{k=1}^{K}\\sum _{i \\in I_{k}^{(n)}}W_{i}},\\ldots , \\frac{\\sum _{i \\in I_{K}^{(n)}}W_{i}}{\\sum _{k=1}^{K}\\sum _{i \\in I_{k}^{(n)}}W_{i}} \\right)= \\frac{n \\cdot \\xi _{n}^{\\mathbf {W}}}{\\sum _{i=1}^{n}W_{i}},\\qquad \\textrm {if } \\sum _{j=1}^{n} W_{j} \\ne 0, \\\\\\ (\\infty , \\ldots , \\infty ) =: \\infty , \\hspace{159.3356pt} \\textrm {if } \\sum _{j=1}^{n} W_{j} = 0,\\end{array}\\right.","}$ with $I_{1}^{(n)}:=\\left\\lbrace 1,\\ldots ,n_{1}\\right\\rbrace $ , $I_{2}^{(n)}:=\\left\\lbrace n_{1}+1,\\ldots ,n_{1}+n_{2}\\right\\rbrace $ , ..., $I_{K}^{(n)} := \\lbrace \\sum _{k=1}^{K-1} n_{k} + 1, \\ldots , n \\rbrace $ and $n_{k}:=\\lfloor n \\cdot p_{k}\\rfloor $ ($k \\in \\lbrace 1,\\ldots ,K\\rbrace $ ) for some pregiven known probability vector ${P} := (p_{1}, \\ldots , p_{K})$ .","Recall the definition of $\\xi _{n}^{\\mathbf {W}}$ in (REF ) (with $\\mathbf {W}$ instead of $\\mathbf {\\widetilde{W}}$ ).","The limit behaviour (REF ) contrasts to the one of Theorem REF , where $-\\lim _{n\\rightarrow \\infty }\\frac{1}{n}\\log \\,\\mathbb {\\Pi }\\left[\\xi _{n}^{\\mathbf {\\widetilde{W}}}\\in \\Omega /M_{\\mathbf {P}}\\right]=\\inf _{Q\\in \\mathbf {\\Omega }}D_{\\varphi }(\\mathbf {Q},\\mathbf {P})\\hspace{113.81102pt} \\text{(cf.","(\\ref {LDP Minimization}))}\\nonumber $ for any $\\mathbf {\\Omega }\\subset \\mathbb {R}^{K}$ with regularity properties (REF ) and the finiteness property (REF ); recall that $(\\widetilde{W}_{i})_{i \\in \\mathbb {N}}$ are i.i.d.","random variables with probability distribution $\\widetilde{\\mathbb {}}$ (being connected with the divergence generator $\\widetilde{\\varphi } := M_{\\mathbf {P}} \\cdot \\varphi $ via the representability (REF )), whereas $(W_{i})_{i \\in \\mathbb {N}}$ are i.i.d.","random variables with probability distribution $\\mathbb {}$ (being connected with the divergence generator $\\varphi $ via the representability (REF )).","Indeed, the construction leading to Theorem REF does not hold any longer when $\\mathbf {\\Omega } \\subset \\mathbb {S}^{K}$ is a set of vectors within the probability simplex $\\mathbb {S}^{K}$ and $\\mathbf {P}\\in \\mathbb {S}_{>0}^{K}$ is a known vector in this simplex with no zero entries.","In such a case, one has to use (REF ) and (REF ) instead.","Notice that for each constant $A >0$ , (REF ) can be rewritten as $& &\\hspace{-42.67912pt}-\\lim _{n\\rightarrow \\infty }\\frac{1}{n}\\log \\,\\mathbb {\\Pi }\\left[ \\xi _{n}^{w\\mathbf {W}}\\in \\textrm {\\right.\\Omega \\hspace{-6.544pt}\\Omega } = \\inf _{\\mathbf {Q}\\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega } }\\ \\inf _{m\\ne 0}D_{\\varphi }\\Big (\\frac{m}{A} \\cdot \\mathbf {Q},{P}\\Big )= \\inf _{\\mathbf {Q}\\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega } }\\ \\inf _{\\widetilde{m}\\ne 0}D_{\\varphi }(\\widetilde{m} \\cdot \\mathbf {Q},{P})= \\inf _{\\widetilde{m}\\ne 0}\\ \\inf _{\\mathbf {Q}\\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega } }D_{\\varphi }(\\widetilde{m} \\cdot \\mathbf {Q},{P}) ;$ therein, the constraint $\\mathbf {Q}\\in A \\cdot \\textrm {$$\\hspace{-6.544pt}$$}$ means geometrically that the vector $\\mathbf {Q}$ lives in a subset of a simplex which is parallel to the simplex $\\mathbb {S}^{K}$ of probability vectors and which is cut off at the edges of the first/positive orthant; in view of Remark REF (d) and (REF ), we can also handle such a situation.","Namely, in the light of the third expression in (REF ) in combination with (REF ) to (REF ) for the special case of $\\mathbf {\\Omega } := \\textrm {$$\\hspace{-6.544pt}$$}$ lying in the probability simplex, it makes sense to study e.g.","functional relationships between $\\inf _{\\widetilde{m}\\ne 0}D_{\\widetilde{c}\\cdot \\varphi }(\\widetilde{m} \\cdot \\mathbf {Q},{P})$ and $D_{\\widetilde{c}\\cdot \\varphi }(\\mathbf {Q},{P})$ ($\\widetilde{c} >0$ ) for $\\mathbf {Q} \\in A \\cdot \\mathbb {S}^{K}$ with arbitrary $A >0$ not necessarily being equal to 1 (i.e.","$\\mathbf {Q} = A \\cdot {Q}$ for some probability distribution ${Q}$ ).","Indeed, such a context appears naturally e.g.","in connection with mass transportation problems (cf.","(REF ) below) and with distributed energy management (cf.","the paragraph after (REF )); the special case $A=1/K$ of (REF ) will also be used below for the application of our BS method to solving (generalized) minimum/maximum entropy problems for probability vectors (and even for sub-/super-probability vectors) ${Q}$ with constraints.","Let us proceed with the main context.","As indicated in Remark REF (i), in a number of important cases the limit in the above Theorem REF can be stated in terms of an invertible function $G^{-1}$ (cf.","(REF )) of $\\inf _{{Q}\\in \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega }}D_{\\varphi }({Q},{P})$ by elimination of $m$ .","As explained above, for the degenerate case (cf.","Remark REF (v), (vi)) the search for $G^{-1}$ is even interesting for the more general infimum over non-probability vectors.","This is the scope of the following development."],["Construction principle for the estimation of the minimum\ndivergence, the power-type case ","Within the context of Theorem REF respectively Remark REF (v) and (vi), we obtain an explicit solution for the inner (i.e.","$m-$ concerning) minimization in (REF ) for the important case of power-divergence generators $\\varphi _{\\gamma } : \\mathbb {R} \\mapsto [0,\\infty ]$ defined by $\\varphi _{\\gamma }(t) \\hspace{-5.69046pt} &:=& \\hspace{-5.69046pt}{\\left\\lbrace \\begin{array}{ll}\\frac{t^\\gamma -\\gamma \\cdot t+ \\gamma - 1}{\\gamma \\cdot (\\gamma -1)}, \\hspace{170.71652pt} \\textrm {if } \\gamma \\in \\, ]-\\infty ,0[ \\ \\textrm {and } t \\in ]0,\\infty [,\\\\- \\log t + t - 1, \\hspace{156.49014pt} \\textrm {if } \\gamma = 0 \\ \\textrm {and } t \\in ]0,\\infty [,\\\\\\frac{t^\\gamma -\\gamma \\cdot t+ \\gamma - 1}{\\gamma \\cdot (\\gamma -1)}, \\hspace{170.71652pt} \\textrm {if } \\gamma \\in \\, ]0,1[ \\ \\textrm {and } t \\in [0,\\infty [,\\\\t \\cdot \\log t + 1 - t, \\hspace{156.49014pt} \\textrm {if } \\gamma = 1 \\ \\textrm {and } t \\in [0,\\infty [,\\\\\\frac{t^\\gamma -\\gamma \\cdot t+ \\gamma - 1}{\\gamma \\cdot (\\gamma -1)} \\cdot {1}_{]0,\\infty [}(t)+(\\frac{1}{\\gamma } - \\frac{t}{\\gamma -1}) \\cdot {1}_{]-\\infty ,0]}(t),\\hspace{28.45274pt} \\textrm {if } \\gamma \\in \\, ]1,2[ \\ \\textrm {and } t \\in \\, ]-\\infty ,\\infty [,\\\\\\frac{(t - 1)^2}{2}, \\hspace{193.47882pt} \\textrm {if } \\gamma = 2 \\ \\textrm {and } t \\in \\, ]-\\infty ,\\infty [,\\\\\\frac{t^\\gamma -\\gamma \\cdot t+ \\gamma - 1}{\\gamma \\cdot (\\gamma -1)} \\cdot {1}_{]0,\\infty [}(t)+(\\frac{1}{\\gamma } - \\frac{t}{\\gamma -1}) \\cdot {1}_{]-\\infty ,0]}(t),\\hspace{28.45274pt} \\textrm {if } \\gamma \\in \\, ]2,\\infty [ \\ \\textrm {and } t \\in \\, ]-\\infty ,\\infty [,\\\\\\infty , \\hspace{209.12791pt} \\textrm {else},\\end{array}\\right.","}$ which for arbitrary multiplier $\\widetilde{c} >0$ generate (the vector-valued form of) the generalized power divergences displayed in the first six rows of Table 1 (and beyond), i.e.","$D_{\\widetilde{c} \\cdot \\varphi _{\\gamma }}(\\mathbf {Q},\\mathbf {P}) \\hspace{-5.69046pt} &:=& \\hspace{-5.69046pt}{\\left\\lbrace \\begin{array}{ll}\\widetilde{c} \\cdot \\Big \\lbrace \\frac{ \\sum \\limits _{k=1}^{K} (q_{k})^{\\gamma } \\cdot (p_{k})^{1-\\gamma }}{\\gamma \\cdot (\\gamma -1)}- \\frac{1}{\\gamma -1} \\cdot \\sum \\limits _{k=1}^{K} q_{k} + \\frac{1}{\\gamma } \\cdot \\sum \\limits _{k=1}^{K} p_{k} \\Big \\rbrace ,\\hspace{56.9055pt} \\textrm {if } \\gamma \\in \\, ]-\\infty ,0[, \\ \\mathbf {P} \\in \\mathbb {R}_{\\ge 0}^{K} \\ \\textrm {and } \\mathbf {Q} \\in \\mathbb {R}_{> 0}^{K},\\\\\\widetilde{c} \\cdot \\Big \\lbrace \\sum \\limits _{k=1}^{K} p_{k} \\cdot \\log \\Big (\\frac{p_{k}}{q_{k}} \\Big )+ \\sum \\limits _{k=1}^{K} q_{k} - \\sum \\limits _{k=1}^{K} p_{k} \\Big \\rbrace ,\\hspace{88.2037pt} \\textrm {if } \\gamma = 0, \\ \\mathbf {P} \\in \\mathbb {R}_{\\ge 0}^{K} \\ \\textrm {and } \\mathbf {Q} \\in \\mathbb {R}_{> 0}^{K},\\\\\\widetilde{c} \\cdot \\Big \\lbrace \\frac{ \\sum \\limits _{k=1}^{K} (q_{k})^{\\gamma } \\cdot (p_{k})^{1-\\gamma }}{\\gamma \\cdot (\\gamma -1)}- \\frac{1}{\\gamma -1} \\cdot \\sum \\limits _{k=1}^{K} q_{k} + \\frac{1}{\\gamma } \\cdot \\sum \\limits _{k=1}^{K} p_{k} \\Big \\rbrace ,\\hspace{59.75095pt} \\textrm {if } \\gamma \\in \\, ]0,1[, \\ \\mathbf {P} \\in \\mathbb {R}_{\\ge 0}^{K} \\ \\textrm {and } \\mathbf {Q} \\in \\mathbb {R}_{\\ge 0}^{K},\\\\\\widetilde{c} \\cdot \\Big \\lbrace \\sum \\limits _{k=1}^{K} q_{k} \\cdot \\log \\Big (\\frac{q_{k}}{p_{k}} \\Big )- \\sum \\limits _{k=1}^{K} q_{k} + \\sum \\limits _{k=1}^{K} p_{k} \\Big \\rbrace ,\\hspace{91.04872pt} \\textrm {if } \\gamma = 1, \\ \\mathbf {P} \\in \\mathbb {R}_{> 0}^{K} \\ \\textrm {and } \\mathbf {Q} \\in \\mathbb {R}_{\\ge 0}^{K},\\\\\\widetilde{c} \\cdot \\Big \\lbrace \\sum \\limits _{k=1}^{K} \\frac{(q_{k})^{\\gamma } \\cdot (p_{k})^{1-\\gamma }}{\\gamma \\cdot (\\gamma -1)}\\cdot {1}_{[0,\\infty [}(q_{k})- \\frac{1}{\\gamma -1} \\cdot \\sum \\limits _{k=1}^{K} q_{k} + \\frac{1}{\\gamma } \\cdot \\sum \\limits _{k=1}^{K} p_{k} \\Big \\rbrace ,\\hspace{8.5359pt} \\textrm {if } \\gamma \\in \\, ]1,2[, \\ \\mathbf {P} \\in \\mathbb {R}_{>0}^{K} \\ \\textrm {and } \\mathbf {Q} \\in \\mathbb {R}^{K},\\\\\\widetilde{c}\\cdot \\sum \\limits _{k=1}^{K}\\frac{ (q_{k}-p_{k})^{2}}{2 \\cdot p_{k}} ,\\hspace{202.01474pt} \\textrm {if } \\gamma = 2, \\ \\mathbf {P} \\in \\mathbb {R}_{>0}^{K} \\ \\textrm {and } \\mathbf {Q} \\in \\mathbb {R}^{K},\\\\\\widetilde{c} \\cdot \\Big \\lbrace \\sum \\limits _{k=1}^{K} \\frac{(q_{k})^{\\gamma } \\cdot (p_{k})^{1-\\gamma }}{\\gamma \\cdot (\\gamma -1)}\\cdot {1}_{[0,\\infty [}(q_{k})- \\frac{1}{\\gamma -1} \\cdot \\sum \\limits _{k=1}^{K} q_{k} + \\frac{1}{\\gamma } \\cdot \\sum \\limits _{k=1}^{K} p_{k} \\Big \\rbrace ,\\hspace{11.38092pt} \\textrm {if } \\gamma \\in \\, ]2,\\infty [, \\ \\mathbf {P} \\in \\mathbb {R}_{>0}^{K} \\ \\textrm {and } \\mathbf {Q} \\in \\mathbb {R}^{K},\\\\\\infty , \\hspace{257.49751pt} \\textrm {else};\\end{array}\\right.","}$ notice that one has the straightforward relationship $D_{\\widetilde{c}\\cdot \\varphi _{\\gamma }}(\\cdot ,\\cdot )=\\widetilde{c}\\cdot D_{\\varphi _{\\gamma }}(\\cdot ,\\cdot )$ ; however, as a motivation for the introduction of $\\widetilde{c}>0$ , we shall show in the Examples REF , REF , REF , REF in Section below that the corresponding probability distribution $\\mathbb {\\mathbb {}}$ of the $W_{i}$ ’s depends on $\\widetilde{c}$ in a non-straightforward way (see also Remark REF (vi) for another motivation for $\\widetilde{c}$ ).","In the course of this, it turns out that $\\widetilde{c} \\cdot \\varphi _{\\gamma } \\in \\Upsilon (]a_{\\gamma },\\infty [)$ with $a_{\\gamma } =0$ for $\\gamma \\in ]-\\infty ,1]$ and $a_{\\gamma } = -\\infty $ for $\\gamma \\in [2,\\infty [$ .","For $\\widetilde{c}=1$ and probability vectors ${Q}$ , ${P}$ in $\\mathbb {S}^{K}$ respectively $\\mathbb {S}_{>0}^{K}$ , the divergences (REF ) simplify considerably, namely to the well-known power divergences $D_{\\varphi _{\\gamma }}({Q},{P})$ in the scaling of e.g.","Liese & Vajda [217] (in other scalings they are also called Rathie & Kannapan’s non-additive directed divergences of order $\\gamma $ [302], Cressie-Read divergences [93] [303], relative Tsallis entropies or Tsallis cross-entropies [364] (see also Shiino [331]), Amari’s alpha-divergences [12]); for some comprehensive overviews on power divergences $D_{\\varphi _{\\gamma }}({Q},{P})$ — including statistical applications to goodness-of-fit testing and minimum distance estimation — the reader is referred to the insightful books of e.g.","Liese & Vajda [217], Read & Cressie [303], Vajda [371], Stummer [344], Pardo [282], Liese & Miescke [216], the survey articles of e.g.","Liese & Vajda [218], Vajda & van der Meulen [374], and the references therein.","Prominent and widely used special cases of $D_{\\varphi _{\\gamma }}({Q},{P})$ are the omnipresent Kullback-Leibler information divergence (relative entropy) where $\\gamma =1$ , the equally important reverse Kullback-Leibler information divergence (reverse relative entropy) where $\\gamma =0$ , the Pearson chi-square divergence ($\\gamma =2$ ), the Neyman chi-square divergence ($\\gamma =-1$ ), the Hellinger divergence ($\\gamma =\\frac{1}{2}$ , also called squared Hellinger distance, squared Matusita distance [256] or squared Hellinger-Kakutani metric, see e.g.","Deza & Deza [113] in some literature, the (square root of the) Hellinger divergence (HD) is misleadingly called Bhattacharyya distance; however, the latter is basically some rescaled logarithm of HD, namely $R_{1/2}({Q},{P})$ (cf.","(REF ) with $\\gamma =1/2$ )).","Some exemplary (relatively) recent studies and applications of power divergences $D_{\\varphi _{\\gamma }}({Q},{P})$ — aside from the vast statistical literature (including in particular maximum likelihood estimation and Pearson’s chi-square test) — appear e.g.","in Matsuyama [253] for flexibilizations of the well-known expectation-maximization (EM) algorithm and their uses for big-data completion (cf.","[254]) and data credit computation in blockchain networks (cf.","[255]), Ku & Fine [199] in connection with blind source separation, Stummer & Vajda [348] as well as Stummer & Lao [347] for optimal decisions about some alternative financial models, Berend et al.","[41] for the derivation of a kind of reverse Pinsker’s inequality (with $\\gamma =1$ ), Verrelst et al.","[383] in geoscientific remote sensing via semiautomatic mapping of biophysical parameters from optical earth observations, Salem at al.","[315] for automatic alarm-triggering detection of events (e.g.","patient health degradations) from collected data by biomedical sensors, Fu et al.","[128] for the study of income distributions in China, Ha et al.","[151] for $x-$ ray spectrum reconstruction in computer tomography (CT) systems (with $\\gamma =1$ ), Iqbal & Seghouane [161] for robust sequential dictionary learning, Luppino et al.","[233] for unsupervised change detection in heterogeneous multi-temporal satellite images (with $\\gamma =\\frac{1}{2}$ ), Sason [320] in connection with data-processing and majorization inequalities, Krömer & Stummer [198] for the smoothing and error-correcting of crude mortality rates (where they even employ non-probability-type vectors), Bekhet & Ahmed [37] for effectiveness evaluations in video retrieval (with $\\gamma =-1$ , $\\gamma =\\frac{1}{2}$ ), Cai et al.","[71] for the stabilization of trainings of generative adversarial networks (GANs), Fu et al.","[129] for automatic molecule optimization, Görtler et al.","[139] for dimensionality reduction on uncertain data in visualization and computer graphics (with $\\gamma =\\frac{1}{2}$ ), Kammerer & Stummer [179] for optimal decision making in the presence of pandemics (e.g.","COVID-19), Kanapram et al.","[180] for the development of collective self-awareness in a network of connected and autonomous vehicles through agent-centered detection of abnormal situations (with $\\gamma =\\frac{1}{2}$ ), Kumbhakar [206] for modelling the streamwise velocity profile in open-channel flows, Sigmon et al.","[335] for the improvement of genetic quality control in mouse research for biomedical applications (with $\\gamma =2$ ), Zhang et al.","[420] for the design of a noise-adaptation adapted generative adversarial network for medical image analysis (with $\\gamma =\\frac{1}{2}$ ), Chen et al.","[79] for clustering high-dimensional microbial data from RNA sequencing (with $\\gamma =\\frac{1}{2}$ ), Dharmawan et al.","[114] for the development of improvements in long-term cell observations via semiconductor-chips-based lensless holographic microscopy, Liu & Sun [229] for analyzing approximate inferences in Bayesian neural networks, Rekavandi et al.","[307] for detections in functional magnetic resonance imaging (fMRI) as well as hyperspectral and synthetic aperture radar (SAR) data, Seghouane & Shokouhi [325] for adaptive learning within robust radial basis function networks (RBFN), and Wang et al.","[388] for recommender-system relevant collaborative filtering in sparse data.","For $\\widetilde{c}=1$ and nonnegative-component vectors $\\mathbf {Q}$ , $\\mathbf {P}$ in $\\mathbb {R}_{\\ge 0}^{K}$ respectively $\\mathbb {R}_{>0}^{K}$ , the generalized power divergences $D_{\\varphi _{\\gamma }}(\\mathbf {Q},\\mathbf {P})$ of (REF ) also (partially) simplify, and were treated by Stummer & Vajda [349] (for even more general probability measures, deriving e.g.","also generalized Pinsker’s inequalities); for a more general comprehensive technical treatment see also e.g.","Broniatowski & Stummer [64].","Returning to the general context, in Theorem REF we stated that for each ${P} \\in \\mathbb {S}_{>0}^{K}$ the function ${Q} \\mapsto \\inf _{m\\ne 0}D_{\\varphi }(m\\cdot {Q},{P}) $ is BS-minimizable (cf.","(REF )) on all sets $\\textrm {$$\\hspace{-6.544pt}$$} \\subset \\mathbb {S}^{K}$ satisfying (7) and (9).","The (corresponding subsetup of the) following Lemma REF is the cornerstone leading from this statement to BS-minimizability of the function ${Q} \\mapsto D_{\\varphi }({Q},{P}))$ on those same sets, for the special divergences in (REF ).","After giving the fundamental preparatory Lemma REF , we shall derive from it some BS-minimizability/BS-maximizability results for (extensions of) a variety of important, widely used, closely related divergences respectively entropy/diversity indices.","To achieve this in a transparent way, we employ the following three fundamental quantities $H_{\\gamma }(\\mathbf {Q},{P})$ , $I(\\mathbf {Q},{P})$ and $\\widetilde{I}(\\mathbf {Q},{P})$ .","To begin with, let $A >0$ be an arbitrary constant (notice that for the choice $A=1$ , all the following vectors $\\mathbf {Q}$ will turn into probability vectors ${Q}$ ).","Moreover — for any constellation $(\\gamma , {P},\\mathbf {Q}) \\in \\widetilde{\\Gamma }\\times \\widetilde{\\mathcal {M}}_{1}\\times \\widetilde{\\mathcal {M}}_{2}$ , where $\\widetilde{\\Gamma }\\times \\widetilde{\\mathcal {M}}_{1}\\times \\widetilde{\\mathcal {M}}_{2}:=]0,1[\\times \\mathbb {S}^{K}\\times A \\cdot \\mathbb {S}^{K}$ or $\\widetilde{\\Gamma }\\times \\widetilde{\\mathcal {M}}_{1}\\times \\widetilde{\\mathcal {M}}_{2}:=]-\\infty ,0[\\times \\mathbb {S}^{K}\\times A \\cdot \\mathbb {S}_{>0}^{K} $ or $\\widetilde{\\Gamma }\\times \\widetilde{\\mathcal {M}}_{1}\\times \\widetilde{\\mathcal {M}}_{2}:=]1,\\infty [\\times \\mathbb {S}_{>0}^{K}\\times A \\cdot \\mathbb {S}^{K}$ — let $0 < H_{\\gamma }(\\mathbf {Q},{P}) := \\sum \\displaylimits _{k=1}^{K} (q_{k})^{\\gamma } \\cdot (p_{k})^{1-\\gamma }\\ = \\ 1 + \\gamma \\cdot (A-1) +\\gamma \\cdot (\\gamma -1) \\cdot D_{\\varphi _{\\gamma }}(\\mathbf {Q}, {P}),\\qquad \\gamma \\in \\mathbb {R}\\backslash \\lbrace 0,1\\rbrace ,$ be the modified $\\gamma -$ order Hellinger integral of $\\mathbf {Q}$ and ${P}$ .","Furthermore, for any ${P} \\in \\mathbb {S}_{>0}^{K}$ , $\\mathbf {Q} \\in A \\cdot \\mathbb {S}^{K}$ , let $-1 \\ < \\ I(\\mathbf {Q},{P}) := \\sum \\displaylimits _{k=1}^{K} q_{k} \\cdot \\log \\left( \\frac{q_{k}}{p_{k}} \\right)\\ = \\ D_{\\varphi _{1}}(\\mathbf {Q}, {P}) + A - 1,$ be the modified Kullback-Leibler information (modified relative entropy).","Finally, for any ${P}\\in \\mathbb {S}^{K}$ , $\\mathbf {Q} \\in A \\cdot \\mathbb {S}_{>0}^{K}$ , let $1 - A \\ \\le \\ \\widetilde{I}(\\mathbf {Q},{P}) := \\sum \\displaylimits _{k=1}^{K} p_{k} \\cdot \\log \\left( \\frac{p_{k}}{q_{k}} \\right)\\ = \\ D_{\\varphi _{0}}(\\mathbf {Q}, {P}) + 1 - A,$ be the modified reverse Kullback-Leibler information (modified reverse relative entropy).","In terms of (REF ), (REF ) and (REF ) we obtain the following Lemma 16 Let $A >0$ be an arbitrary constant.","(a) Let $\\widetilde{c}>0$ be arbitrary and $(\\gamma , {P},\\mathbf {Q}) \\in \\widetilde{\\Gamma }\\times \\widetilde{\\mathcal {M}}_{1}\\times \\widetilde{\\mathcal {M}}_{2}$ as above.","Then one has $\\inf _{m\\ne 0}D_{\\widetilde{c}\\cdot \\varphi _{\\gamma }}(m \\cdot \\mathbf {Q},{P})=\\inf _{m>0}D_{\\widetilde{c}\\cdot \\varphi _{\\gamma }}(m\\cdot \\mathbf {Q},{P})&=&\\frac{\\widetilde{c}}{\\gamma }\\cdot \\left[ 1- A^{\\gamma /(\\gamma -1)} \\cdot \\left[ 1+ \\gamma \\cdot (A-1) + \\frac{\\gamma \\cdot \\left( \\gamma -1\\right) }{\\widetilde{c}}\\cdot D_{\\widetilde{c}\\cdot \\varphi _{\\gamma }}(\\mathbf {Q},{P})\\right]^{-1/\\left( \\gamma -1\\right) }\\right]\\nonumber \\\\& &\\\\&=&\\frac{\\widetilde{c}}{\\gamma }\\cdot \\left[ 1- A^{\\gamma /(\\gamma -1)} \\cdot H_{\\gamma }(\\mathbf {Q},{P})^{-1/\\left( \\gamma -1\\right) }\\right]\\nonumber $ and consequently for any subset $A \\cdot \\Omega $$\\Omega \\subset \\widetilde{\\mathcal {M}}_{2}$ $&&\\hspace{-19.91684pt}\\inf _{\\mathbf {Q}\\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega }}\\ \\inf _{m\\ne 0} D_{\\widetilde{c}\\cdot \\varphi _{\\gamma }}(m \\cdot \\mathbf {Q},{P})=\\frac{\\widetilde{c}}{\\gamma }\\cdot \\left[ 1-A^{\\gamma /(\\gamma -1)} \\cdot \\left[ 1+ \\gamma \\cdot (A-1) +\\frac{\\gamma \\cdot \\left( \\gamma -1\\right) }{\\widetilde{c}}\\cdot \\inf _{\\mathbf {Q}\\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega } }D_{\\widetilde{c}\\cdot \\varphi _{\\gamma }}(\\mathbf {Q},{P})\\right]^{-1/\\left( \\gamma -1\\right) }\\right] ,\\qquad \\ \\\\&&\\hspace{-19.91684pt}\\arg \\inf {}_{\\mathbf {Q}\\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega } }\\inf _{m\\ne 0}\\ D_{\\widetilde{c}\\cdot \\varphi _{\\gamma }}(m \\cdot \\mathbf {Q},{P})=\\arg \\inf {}_{\\mathbf {Q}\\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega } } \\ D_{\\widetilde{c}\\cdot \\varphi _{\\gamma }}(\\mathbf {Q},{P}),\\qquad \\ \\\\&&\\hspace{-19.91684pt}\\inf _{\\mathbf {Q}\\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega }}\\ \\inf _{m\\ne 0} D_{\\varphi _{\\gamma }}(m \\cdot \\mathbf {Q},{P})=\\frac{1}{\\gamma }\\cdot \\left[ 1-A^{\\gamma /(\\gamma -1)} \\cdot \\left[ \\inf _{\\mathbf {Q}\\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega } }H_{\\gamma }(\\mathbf {Q},{P})\\right]^{-1/\\left( \\gamma -1\\right) }\\right] ,\\qquad \\textrm {for \\gamma <0 and \\gamma >1}, \\\\&&\\hspace{-19.91684pt}\\arg \\inf {}_{\\mathbf {Q}\\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega } }\\inf _{m\\ne 0}\\ D_{\\varphi _{\\gamma }}(m \\cdot \\mathbf {Q},{P})=\\arg \\inf {}_{\\mathbf {Q}\\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega } } \\ H_{\\gamma }(\\mathbf {Q},{P}),\\hspace{96.73918pt} \\textrm {for \\gamma <0 and \\gamma >1}, \\\\&&\\hspace{-19.91684pt}\\inf _{\\mathbf {Q}\\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega }}\\ \\inf _{m\\ne 0} D_{\\varphi _{\\gamma }}(m \\cdot \\mathbf {Q},{P})=\\frac{1}{\\gamma }\\cdot \\left[ 1-A^{\\gamma /(\\gamma -1)} \\cdot \\left[ \\sup _{\\mathbf {Q}\\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega } }H_{\\gamma }(\\mathbf {Q},{P})\\right]^{-1/\\left( \\gamma -1\\right) }\\right] , \\hspace{22.76228pt}\\textrm {for \\gamma \\in ]0,1[}, \\\\&&\\hspace{-19.91684pt}\\arg \\inf {}_{\\mathbf {Q}\\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega } }\\inf _{m\\ne 0}\\ D_{\\varphi _{\\gamma }}(m \\cdot \\mathbf {Q},{P})=\\arg \\sup {}_{\\mathbf {Q}\\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega } } \\ H_{\\gamma }(\\mathbf {Q},{P}), \\hspace{96.73918pt} \\textrm {for \\gamma \\in ]0,1[},$ provided that the infimum on the right-hand side of (REF ) exists.","(b) For any ${P} \\in \\mathbb {S}_{>0}^{K}$ , $\\mathbf {Q} \\in A \\cdot \\mathbb {S}^{K}$ , $\\widetilde{c}>0$ one gets $\\inf _{m\\ne 0}D_{\\widetilde{c}\\cdot \\varphi _{1}}(m \\cdot \\mathbf {Q},{P})=\\inf _{m>0}D_{\\widetilde{c}\\cdot \\varphi _{1}}(m \\cdot \\mathbf {Q},{P})&=&\\widetilde{c}\\cdot \\left[ 1- A \\cdot \\exp \\left( -\\frac{1}{A \\cdot \\widetilde{c}}\\cdot D_{\\widetilde{c}\\cdot \\varphi _{1}}(\\mathbf {Q},{P})+ \\frac{1}{A} -1 \\right) \\right] \\\\&=&\\widetilde{c}\\cdot \\left[ 1- A \\cdot \\exp \\left( -\\frac{1}{A}\\cdot I(\\mathbf {Q},{P})\\right) \\right]\\nonumber $ and consequently for any subset $A \\cdot \\Omega $$\\Omega \\subset A \\cdot \\mathbb {S}^{K}$ $& &\\inf _{\\mathbf {Q}\\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega } } \\ \\inf _{m\\ne 0}D_{\\widetilde{c}\\cdot \\varphi _{1}}(m \\cdot \\mathbf {Q},{P})=\\widetilde{c}\\cdot \\left[ 1-A \\cdot \\exp \\left( -\\frac{1}{A \\cdot \\widetilde{c}}\\cdot \\inf _{Q\\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega } }D_{\\widetilde{c}\\cdot \\varphi _{1}}(\\mathbf {Q},{P}) + \\frac{1}{A} -1 \\right) \\right] , \\\\& & \\arg \\inf {}_{\\mathbf {Q}\\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega } } \\ \\inf _{m\\ne 0}D_{\\widetilde{c}\\cdot \\varphi _{1}}(m \\cdot \\mathbf {Q},{P}) =\\arg \\inf {}_{\\mathbf {Q}\\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega } } \\ D_{\\widetilde{c}\\cdot \\varphi _{1}}(Q,P),\\qquad \\ \\\\& &\\inf _{\\mathbf {Q}\\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega } } \\ \\inf _{m\\ne 0}D_{\\varphi _{1}}(m \\cdot \\mathbf {Q},{P})=\\left[ 1- A \\cdot \\exp \\left( -\\frac{1}{A}\\cdot \\inf _{Q\\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega } }I(\\mathbf {Q},{P}) \\right) \\right] , \\\\& & \\arg \\inf {}_{\\mathbf {Q}\\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega } } \\ \\inf _{m\\ne 0}D_{\\varphi _{1}}(m \\cdot \\mathbf {Q},{P}) =\\arg \\inf {}_{\\mathbf {Q}\\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega } } \\ I(\\mathbf {Q},{P}),\\qquad \\ $ provided that the infimum on the right-hand side of (REF ) exists.","(c) For any ${P}\\in \\mathbb {S}^{K}$ , $\\mathbf {Q} \\in A \\cdot \\mathbb {S}_{>0}^{K}$ , $\\widetilde{c}>0$ we obtain $\\inf _{m\\ne 0}D_{\\widetilde{c}\\cdot \\varphi _{0}}(m \\cdot \\mathbf {Q},{P})=\\inf _{m>0}D_{\\widetilde{c}\\cdot \\varphi _{0}}(m \\cdot \\mathbf {Q},{P})&=& D_{\\widetilde{c}\\cdot \\varphi _{0}}(\\mathbf {Q},{P})+ \\widetilde{c} \\cdot (1 - A + \\log A) \\\\&=& \\widetilde{c} \\cdot \\Big ( \\widetilde{I}(\\mathbf {Q},{P}) + \\log A \\Big )\\nonumber $ and consequently for any set subset $A \\cdot \\Omega $$\\Omega \\subset A \\cdot \\mathbb {S}_{>0}^{K}$ $&&\\hspace{-19.91684pt}\\inf _{\\mathbf {Q} \\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega }} \\ \\inf _{m\\ne 0} D_{\\widetilde{c}\\cdot \\varphi _{0}}(m \\cdot \\mathbf {Q},{P})= \\widetilde{c} \\cdot (1 - A + \\log A) +\\inf _{\\mathbf {Q} \\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega }} \\ D_{\\widetilde{c}\\cdot \\varphi _{0}}(\\mathbf {Q},{P}), \\\\&&\\hspace{-19.91684pt}\\arg \\inf {}_{\\mathbf {Q}\\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega } } \\ \\inf _{m\\ne 0}D_{\\widetilde{c}\\cdot \\varphi _{0}}(m \\cdot \\mathbf {Q},{P})=\\arg \\inf {}_{\\mathbf {Q}\\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega } } \\ D_{\\widetilde{c}\\cdot \\varphi _{0}}(\\mathbf {Q},{P}), \\qquad \\ \\\\&&\\hspace{-19.91684pt}\\inf _{\\mathbf {Q} \\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega }} \\ \\inf _{m\\ne 0} D_{\\varphi _{1}}(m \\cdot \\mathbf {Q},{P})= \\log A +\\inf _{\\mathbf {Q} \\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega }} \\ \\widetilde{I}(\\mathbf {Q},{P}), \\\\&&\\hspace{-19.91684pt}\\arg \\inf {}_{\\mathbf {Q}\\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega } } \\ \\inf _{m\\ne 0} D_{\\varphi _{0}}(m \\cdot \\mathbf {Q},{P})=\\arg \\inf {}_{\\mathbf {Q}\\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega } } \\ \\widetilde{I}(\\mathbf {Q},{P}), \\qquad \\ $ provided that the infimum on the right-hand side of (REF ) exists.","The proof of Lemma REF is given in Appendix C. Remark 17 Notice that for ${P} \\in \\mathbb {S}_{>0}^{K}$ and $\\mathbf {Q} \\in A \\cdot \\mathbb {S}^{K}$ , the modified Kullback-Leibler information has the property $I(\\mathbf {Q},{P}) \\ge 0$ if $A \\ge 1$ (cf.","(REF )); otherwise, $I(\\mathbf {Q},{P})$ may become negative, as can be easily seen from the case where ${P} := {P}^{unif} :=(\\frac{1}{K}, \\ldots , \\frac{1}{K})$ is the probability vector of frequencies of the uniform distribution on $\\lbrace 1, \\ldots , K\\rbrace $ , and $\\mathbf {Q} := (\\frac{1}{K+1}, 0, \\ldots , 0)$ .","Analogously, for ${P}\\in \\mathbb {S}^{K}$ and $\\mathbf {Q} \\in A \\cdot \\mathbb {S}_{>0}^{K}$ one gets $\\widetilde{I}(\\mathbf {Q},{P}) \\ge 0$ if $A \\le 1$ (cf.","(REF )); otherwise, $\\widetilde{I}(\\mathbf {Q},{P})$ may become negative (take e.g.","$\\mathbf {Q} = (\\frac{K+1}{K}, \\ldots , \\frac{K+1}{K})$ and ${P} := (1,0, \\ldots , 0)$ ).","Remark 18 (a) In the context of Remark REF (vi), according to (REF ) applied to $\\varphi := \\widetilde{c}\\cdot \\varphi _{\\gamma }$ , for all cases $\\gamma \\in \\, ]-\\infty ,0[ \\ \\cup \\ ]0,1[ \\ \\cup \\ [ \\ 2,\\infty [$ the left-hand side of each of (REF ), (), () is independent of $A >0$ and equal to $-\\lim _{n\\rightarrow \\infty }\\frac{1}{n}\\log \\,\\mathbb {\\Pi }\\left[\\xi _{n}^{w\\mathbf {W}}\\in \\textrm {\\right.$$\\hspace{-6.544pt}$$}$ where — as will be shown below — the corresponding $\\mathbf {W}$ 's have probability distribution $\\mathbb {}[\\cdot \\,]=\\mathbb {\\Pi }[W_{1}\\in \\cdot \\,]$ (cf.","(REF )) which varies “quite drastically” with $\\gamma $ (and the case $\\gamma \\in ]1,2[$ has to be even excluded for analytical difficulties because in this case there are some indications that the representation (REF ) only holds for some signed probability distribution $$ (e.g.","having a density with positive and negative values).).","Analogously, each of the left-hand sides of (REF ), (), (REF ), () is also independent of $A >0$ and equal to $-\\lim _{n\\rightarrow \\infty }\\frac{1}{n}\\log \\,\\mathbb {\\Pi }\\left[\\xi _{n}^{w\\mathbf {W}}\\in \\textrm {\\right.$$\\hspace{-6.544pt}$$}$ for some $\\mathbf {W}$ of respective distribution.","Hence, by inversion, all the extremum-describing target quantities $\\inf _{\\mathbf {Q}\\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega } }D_{\\widetilde{c}\\cdot \\varphi _{\\gamma }}(\\mathbf {Q},{P})$ ($\\gamma \\in \\mathbb {R}\\backslash ]1,2[$ ), $\\inf _{\\mathbf {Q}\\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega } }H_{\\gamma }(\\mathbf {Q},{P})$ ($\\gamma \\in ]-\\infty ,0[ \\, \\cup \\, [2,\\infty [$ ), $\\sup _{\\mathbf {Q}\\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega } }H_{\\gamma }(\\mathbf {Q},{P})$ ($\\gamma \\in ]0,1[$ ), $\\inf _{Q\\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega } }I(\\mathbf {Q},{P})$ and $\\inf _{Q\\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega } }\\widetilde{I}(\\mathbf {Q},{P})$ can be expressed as $G\\big (-\\lim _{n\\rightarrow \\infty }\\frac{1}{n}\\log \\,\\mathbb {\\Pi }\\left[\\xi _{n}^{w\\mathbf {W}}\\in \\textrm {\\right.$$\\hspace{-6.544pt}$$})\\, $ for some explicitly known ($A-$ dependent) function $G$ .","This means that — in the sense of Definition REF — all the corresponding four “cornerstone quantities” $D_{\\widetilde{c}\\cdot \\varphi _{\\gamma }}(\\mathbf {Q},{P})$ , $H_{\\gamma }(\\mathbf {Q},{P})$ , $I(\\mathbf {Q},{P})$ , $\\widetilde{I}(\\mathbf {Q},{P})$ are BS-minimizable, respectively BS-maximizable, on $\\mathbf {\\Omega } = A \\cdot \\textrm {$$\\hspace{-6.544pt}$$}$ .","The above-mentioned inversions (i.e.","constructions of $G(\\cdot )$ ) will be concretely carried out below — namely in the Propositions REF , REF , REF , REF , REF and REF .","In those, we also involve the BS-minimizability/maximizability of several other important closely related divergences and measures of entropy (measures of diversity, measures of heterogeneity/homogeneity, measures of concentration) which (i) are widely used in information theory and its applications to artificial intelligence, machine learning and physics, and which (ii) can be built from the above-mentioned four cornerstone quantities (power divergences, Hellinger integrals, Kullback-Leibler information divergences).","(b) The special case $\\varphi := \\widetilde{c}\\cdot \\varphi _{\\gamma }$ ($\\gamma \\in \\, ]-\\infty ,0[ \\ \\cup \\ ]0,1[ \\ \\cup \\ [ \\ 2,\\infty [$ ) of Theorem REF works analogously to (a), with the differences that we employ $A=1$ (instead of arbitrary $A>0$ ), () (instead of (REF )), $\\mathbb {\\Pi }_{X_{1}^{n}}[\\cdot ]$ (instead of $\\mathbb {\\Pi }[\\cdot ]$ ), and $\\xi _{n,\\mathbf {X}}^{w\\mathbf {W}}$ (instead of $\\xi _{n}^{w\\mathbf {W}}$ ).","(c) From the proof of Lemma REF in Appendix C below, one can see that for the important case $\\gamma =2$ the formulas (REF ) to () also hold for $A<0$ .","In the following, we further elaborate the three points (a),(b) and (c) of Remark REF “comprehensively and unifyingly”, where the expression “BS minimizable/maximizable” always has to be interpreted accordingly in terms of $-\\lim _{n\\rightarrow \\infty }\\frac{1}{n}\\log \\mathbb {\\Pi }[\\xi _{n}^{w\\mathbf {W}} \\in \\cdot \\ ]$ respectively $-\\lim _{n\\rightarrow \\infty }\\frac{1}{n}\\log \\mathbb {\\Pi }_{X_{1}^{n}}[\\xi _{n,\\mathbf {X}}^{w\\mathbf {W}}\\in \\cdot \\ ]$ (without explicit mentioning, for the sake of brevity).","Let us fix $\\widetilde{c}=1$ and an arbitrary triple $(\\gamma , {P},\\mathbf {Q})$ which satisfies the assumptions of Lemma REF (a) with $A := \\sum _{k=1}^{K} q_{k} >0$ .","For such a setup, we have obtained in (REF ) the $\\gamma -$ order Hellinger integral (of $\\mathbf {Q}$ and ${P}$ ) $H_{\\gamma }(\\mathbf {Q},{P}) > 0$ , which is not a divergence; as a terminology-concerning side remark, let us mention that $H_{\\gamma }(\\mathbf {Q},{P})$ ($\\gamma \\ge 1$ ) is called relative information generating function in Guiasu & Reischer [144], see e.g.","also Clark [90]; moreover, $H_{\\gamma }({Q},{P})$ is sometimes termed ($\\gamma -$ order) Chernoff coefficient being a component of the Chernoff distances/informations [85].","Torgersen [361] uses the name ($\\gamma -$ order) Hellinger transform.","Notice that the special case $\\gamma =\\frac{1}{2}$ is nothing but (a multiple of) the well-known important Bhattacharyya coefficient (cf.","[48],[49],[50]) $BC(\\mathbf {Q},{P}) :=H_{1/2}(\\mathbf {Q},{P}) =\\sum _{k=1}^{K} \\sqrt{q_{k} \\cdot p_{k}} =1 + \\frac{1}{2} \\cdot (A-1) + \\frac{1}{2} \\cdot (\\frac{1}{2}-1) \\cdot D_{\\varphi _{\\frac{1}{2}}}(\\mathbf {Q},{P})\\nonumber $ 0.3cm which is also known as affinity (cf.","Matusita [256], see e.g.","also Toussaint [362]) and (classic, non-quantum) fidelity similarity (cf.","e.g.","Deza & Deza [113]); for non-probability vectors $\\mathbf {P} \\in \\mathbb {R}_{\\ge 0}^{K}$ one can simply retransform ${P} := \\frac{\\mathbf {P}}{M_{\\mathbf {P}}}$ and thus imbed $BC(\\mathbf {Q},\\mathbf {P}) = \\sqrt{M_{\\mathbf {P}}} \\cdot BC(\\mathbf {Q},{P})$ into our BS context.","There is a vast literature on very recent applications of the Bhattacharyya coefficient, for instance it appears exemplarily in Peng & Li [289] for object tracking from successive video frames, Ayed et al.","[26] for efficient graph cut algorithms, Patra et al.","[287] for collaborative filtering in sparse data, El Merabet et al.","[119] for region classification in intelligent transport systems in order to compensate the lack of performance of Global Navigation Satellites Systems, Chiu et al.","[86] for the design of interactive mobile augmented reality systems, Noh et al.","[274] for dimension reduction in interacting fluid flow models, Bai et al.","[29] for material defect detection through ultrasonic array imaging, Dixit & Jain [115] for the design of recommender systems on highly sparse context aware datasets, Guan et al.","[143] for visible light positioning methods based on image sensors, Lin et al.","[220] for probabilistic representation of color image pixels, Chen et al.","[80] for distributed compressive video sensing, Jain et al.","[162] for the enhancement of multistage user-based collaborative filtering in recommendation systems, Pascuzzo et al.","[285] for brain-diffusion-MRI based early diagnosis of the sporadic Creutzfeldt–-Jakob disease, Sun et al.","[351] for the design of automatic detection methods multitemporal (e.g.","landslide) point clouds, Valpione et al.","[377] for the investigation of T cell dynamics in immunotherapy, Wang et al.","[387] for the tracking and prediction of downbursts from meteorological data, Xu et al.","[403] for adaptive distributed compressed video sensing for coal mine monitoring, Zhao et al.","[424] for the shared sparse machine learning of the affective content of images, Chen et al.","[82] for image segmentation and domain partitioning, De Oliveira et al.","[105] for the prediction of cell-penetrating peptides, Eshaghi et al.","[122] for the identification of multiple sclerosis subtypes through machine learning of brain MRI scans, Feng et al.","[125] for improvements of MRI-based detection of epilepsy-causing cortical malformations, Hanli et al.","[153] for designing pilot protection schemes for transmission lines, Jiang et al.","[170] for flow-assisted visual tracking through event cameras, Lysiak & Szmajda [235] for comparisons of selected feature quality evaluations, Joel & Sivakumar [172] for the despeckling enhancement of medical ultrasound image quality, Reising et al.","[305] for the design of security protection of Internet-of-Things (IoT) devices, Skrbic et al.","[338] for the uncovering of interplays between amino acid sequences and local structures in proteins, Tsiapoki et al.","[365] for the improvement of the detection performance of structural health monitoring frameworks, van Molle et al.","[381] for uncertainty quantification in deep neural networks, Yang et al.","[413] for the determination of the onset of transient signals, and Zhou & Yu [427] for the modelling of spatiotemporal human eye movements.","To proceed with the general context, for any $\\gamma \\in \\, ]-\\infty ,0[ \\ \\cup \\ ]0,1[ \\ \\cup \\ [ \\ 2,\\infty [$ let the function $h_{\\gamma } : \\, ]0,\\infty [ \\, \\mapsto \\, ]-\\infty ,\\infty [$ be such that $x \\mapsto h_{\\gamma }(1 + \\gamma \\cdot (A-1) + \\gamma \\cdot (\\gamma -1) \\cdot x)$ is continuous and strictly increasing (respectively, strictly decreasing) for all $x \\ge 0$ with $1 + \\gamma \\cdot (A-1) + \\gamma \\cdot (\\gamma -1) \\cdot x > 0$ ; since $D_{\\varphi _{\\gamma }}(\\mathbf {Q},{P})$ is BS-minimizable on $\\mathbf {\\Omega } = A \\cdot \\textrm {$$\\hspace{-6.544pt}$$}$ , then also the — not necessarily nonnegative — quantity $h_{\\gamma }\\Big (1 + \\gamma \\cdot (A-1) + \\gamma \\cdot (\\gamma -1) \\cdot D_{\\varphi _{\\gamma }}(\\mathbf {Q},{P})\\Big )= h_{\\gamma }\\Big (H_{\\gamma }(\\mathbf {Q},{P})\\Big )$ is BS-minimizable (respectively, BS-maximizable) on $\\mathbf {\\Omega } = A \\cdot \\textrm {$$\\hspace{-6.544pt}$$}$ .","If $h_{\\gamma }$ satisfies additionally $h_{\\gamma }(1)=0$ as well as $h_{\\gamma }(1 + \\gamma \\cdot (A-1) + \\gamma \\cdot (\\gamma -1) \\cdot x) \\ge 0$ for all $x \\ge 0$ with $1 + \\gamma \\cdot (A-1) + \\gamma \\cdot (\\gamma -1) \\cdot x > 0$ , then $D_{h_{\\gamma }}(\\mathbf {Q},{P}) :=h_{\\gamma }\\Big (1 + \\gamma \\cdot (A-1) + \\gamma \\cdot (\\gamma -1) \\cdot D_{\\varphi _{\\gamma }}(\\mathbf {Q},{P})\\Big )= h_{\\gamma }\\Big (H_{\\gamma }(\\mathbf {Q},{P})\\Big )$ constitutes a divergence in the usual sense that $D_{h_{\\gamma }}(\\mathbf {Q},{P}) \\ge 0 $ with equality iff ${Q}={P}$ .","which is BS-minimizable on $\\mathbf {\\Omega } = A \\cdot \\textrm {$$\\hspace{-6.544pt}$$}$ (respectively, BS-maximizable on $\\mathbf {\\Omega } = A \\cdot \\textrm {$$\\hspace{-6.544pt}$$}$ ).","Let us consider some important examples.","For the identity mapping $h_{\\gamma }^{Id}(y) := y$ ($y>0$ ) the function $x \\mapsto 1 + \\gamma \\cdot (A-1) + \\gamma \\cdot (\\gamma -1) \\cdot x$ is strictly increasing for $\\gamma <0$ and $\\gamma >1$ (on the required domain of $x$ ), and strictly decreasing for $\\gamma \\in ]0,1[$ .","Accordingly, $H_{\\gamma }(\\mathbf {Q},{P})$ is BS-minimizable on $\\mathbf {\\Omega } = A \\cdot \\textrm {$$\\hspace{-6.544pt}$$}$ for $\\gamma <0$ and $\\gamma \\ge 2$ and BS-maximizable on $\\mathbf {\\Omega } = A \\cdot \\textrm {$$\\hspace{-6.544pt}$$}$ for $\\gamma \\in \\, ]0,1[$ (this is consistent with (), ()); in particular, the Bhattacharyya coefficient $BC(\\mathbf {Q},{P})$ is BS-maximizable on $\\mathbf {\\Omega } = A \\cdot \\textrm {$$\\hspace{-6.544pt}$$}$ .","Some other important choices are $& & h_{\\gamma }(y) := h_{c_{1},c_{2},c_{3}}(y) := c_{1} \\cdot \\Big (y^{c_{2}} - c_{3} \\Big ),\\qquad y>0, \\, c_{1}, c_{2} \\in \\mathbb {R}\\backslash \\lbrace 0\\rbrace , \\,c_{3} \\in \\mathbb {R},\\\\& & h_{\\gamma }(y) := h_{c_{4},f}^{R}(y) := \\lim _{c_{2} \\rightarrow 0} h_{c_{4}/f(c_{2}),c_{2},1}(y) =\\frac{c_{4}}{f^{\\prime }(0)} \\cdot \\log (y),\\qquad y>0, \\, c_{4} \\in \\mathbb {R}\\backslash \\lbrace 0\\rbrace , \\,\\\\& & h_{\\gamma }(y) := h_{c_{5},c_{6}}^{GB2}(y) := c_{5} \\cdot (\\arccos (y))^{c_{6}},\\qquad \\gamma \\in \\,]0,1[, \\, y \\in \\, ]0,1], \\, c_{5}>0, \\, c_{6}>0,\\\\& & h_{\\gamma }(y) := h_{\\nu ,c_{7}}^{BB}(y) := c_{7} \\cdot \\frac{\\log (1-\\frac{1-y}{\\nu })}{\\log (1-\\frac{1}{\\nu })},\\qquad \\gamma \\in \\,]0,1[, \\, y \\in \\, ]0,1], \\, c_{7}>0, \\, \\nu \\in ]-\\infty ,0[ \\, \\cup \\, ]1,\\infty [,$ where the constants $c_{1}$ to $c_{7}$ may depend on $\\gamma $ , and $f$ is some (maybe $\\gamma -$ dependent) function which is differentiable in a neighborhood of 0 and satisfies $f(0)=0$ , $f^{\\prime }(0) \\ne 0$ (e.g.","$f(z)=c_{8} \\cdot z$ for some non-zero constant $c_{8}$ ).","Clearly, $h_{c_{1},c_{2},c_{3}}(\\cdot )$ is strictly increasing (respectively, strictly decreasing) if and only if $c_{1} \\cdot c_{2} >0$ (respectively, $c_{1} \\cdot c_{2} <0$ ).","Moreover, $h_{c_{4},f}^{R}(\\cdot )$ is strictly increasing (respectively, strictly decreasing) if and only if $\\frac{c_{4}}{f^{\\prime }(0)} >0$ (respectively, $\\frac{c_{4}}{f^{\\prime }(0)} < 0$ ).","Furthermore, both $h_{c_{5},c_{6}}^{GB2}(\\cdot )$ and $h_{\\nu ,c_{7}}^{GoBa}(\\cdot )$ are strictly decreasing.","For instance, the special case $h_{\\gamma }(y) = h_{c_{4},Id}^{R}(y)$ with $c_{4} := \\frac{1}{\\gamma \\cdot (\\gamma -1)}$ (recall that $\\gamma \\in \\, ]-\\infty ,0[ \\ \\cup \\ ]0,1[ \\ \\cup \\ [ \\ 2,\\infty [$ ) and identity function $f := Id$ leads to the quantities $\\hspace{-42.67912pt}R_{\\gamma }(\\mathbf {Q},{P}) &: =&D_{h_{c_{4},Id}^{R}}(\\mathbf {Q},{P}) =\\frac{\\log \\Big ( 1 + \\gamma \\cdot (A-1) +\\gamma \\cdot (\\gamma -1) \\cdot D_{\\varphi _{\\gamma }}(\\mathbf {Q},{P}) \\Big )}{\\gamma \\cdot (\\gamma -1)}= \\frac{\\log \\Big ( H_{\\gamma }(\\mathbf {Q},{P}) \\Big )}{\\gamma \\cdot (\\gamma -1)}\\nonumber \\\\&=& \\frac{\\log \\Big ( \\sum \\displaylimits _{k=1}^{K} (q_{k})^{\\gamma } \\cdot (p_{k})^{1-\\gamma } \\Big )}{\\gamma \\cdot (\\gamma -1)} \\, ,\\qquad \\gamma \\in \\, ]-\\infty ,0[ \\ \\cup \\ ]0,1[ \\ \\cup \\ [ \\ 2,\\infty [,$ (provided that all involved power divergences are finite), which are thus BS-minimizable on $\\mathbf {\\Omega } = A \\cdot \\textrm {$$\\hspace{-6.544pt}$$}$ ; notice that $R_{\\gamma }(\\mathbf {Q},{P}) \\ge 0$ if $\\gamma \\in \\ ]0,1[ \\ \\cup \\ [ \\ 2,\\infty [$ together with $A \\in [1,\\infty [$ , and if $\\gamma \\in \\, ]-\\infty ,0[$ together with $A \\in \\, ]0,1]$ .","The special subcase $A=1$ in (REF ) (and thus, $\\mathbf {Q}$ is a probability vector ${Q}$ ) corresponds to the prominent Renyi divergences/distances [309] (in the scaling of e.g.","Liese & Vajda [217] and in probability-vector form), see e.g.","van Erven & Harremoes [380] for a comprehensive study of their properties; as a side remark, $\\gamma \\cdot (\\gamma -1) \\cdot R_{\\gamma }({Q},{P})$ is also employed in the Chernoff distances/informations [85].","The special subcase $R_{1/2}({Q},{P})$ (i.e.","$\\gamma =1/2$ and $A=1$ in (REF )) corresponds to (a multiple of) the widely used Bhattacharyya distance (of type 1) between ${Q}$ and ${P}$ , cf.","[48] (see e.g.","also Kailath [178]).","Sometimes, $\\exp (R_{\\gamma }(\\mathbf {Q},{P}))$ is also called Renyi divergence/distance.","Some exemplary (relatively) recent studies and applications of Renyi divergences $R_{\\gamma }({Q},{P})$ (respectively, their multiple or exponential) — aside from the substantial statistical literature — appear e.g.","in Zhao et al.","[423] for the study of isomeric stability of fullerenes (which are e.g.","employed for state-of-the-art organic solar cells), in the papers of Sundaresan [353], Bunte & Lapidoth [67], Sason [318], Kumar et al.","[205] for (mismatch-cases of) coding and guessing as well as task partitioning, in the papers of Prest [296], Bai et al.","[30] for lattice-based cryptography, in He et al.","[158] for robot active olfaction search (by infotaxis) in turbulent flows, in Momeni et al.","[263] for the design of reprogrammable encrypted graphene-based coding metasurfaces, in Staszowska et al.","[340] for accurate and precise cluster analysis for super-resolution localization microscopy, in Yu & Tan [414] for distributed source simulation problems, in Zhang et al.","[419] for sensor control, in Yu & Tan [415] for the so-called random variable simulation problem, in Blanchet et al.","[55] for the robust treatment of extreme values in rainfall accumulation data, in Cai et al.","[70] for sensor tasking for search and catalog maintenance of geosynchronous space objects, in Gholami & Hodtani [134] for refinements of safety-and-security-targeted location verification systems in wireless communication networks (e.g in Intelligent Transportation Systems (ITSs) and vehicular technology), in Seweryn et al.","[326] for the assessment of similarity and diversity of expression profiles in single cell systems, in Zhou [426] for the study of secrecy constraints in key generation problems where side information might be present at untrusted users, in Makkawi et al.","[243] for the design of an automated decision-support framework for adaptive diagnosis of fault–tolerant multi–sensor data fusion for vehicle localization, in Mao et al.","[245] for privacy-preserving computation offloading for parallel deep neural networks training.","There is vast literature on recent applications of the above-mentioned special case $R_{1/2}({Q},{P})$ — that is, the Bhattacharyya distance (of type 1); for instance, it appears in Tarighati & Jalden [356] for rate balancing in wireless sensor networks, Bi et al.","[51], [52] for certain uncertainty quantifications respectively stochastic sensitivity analyses in mechanical systems and signal processing, Fu & He [130] for the design of multibit quantizers for cooperative spectrum sensing in cognitive radio networks, Cohen et al.","[91] for adaptive and causal random linear network coding with forward error correction for a point-to-point communication channel with delayed feedback, Xu et al.","[401] for cost minimization problems of big data analytics on geo-distributed data centers connected to renewable energy sources with unpredictable capacity, Xu et al.","[402] for community identification in networks, Arrigoni & Madsen [18] for automated discovering of low-energy defect configurations in materials, Fan et al.","[123] for region-merging-based methods for synthetic aperture radar (SAR) image segmentation, Mahfouz et al.","[242] for some refined ensemble classifications in microarray-based automated cancer diagnosis, Matchev & Shyamsundar [252] for some machine-learning based signal discovery in high energy physics (HEP) experiments, Wang et al.","[386] for the investigation of intratumoral heterogeneity (ITH) of some gastric cancer, Webster et al.","[389] for the characterization, identification, clustering and classification of disease, and Xiahou et al.","[396] for the prediction of remaining useful life (RUL) through fusion of expert knowledge and condition monitoring information.","As a further example, consider $\\mathcal {B}_{\\gamma ,c_{5},c_{6}}({Q},{P}) &: =&D_{h_{c_{5},c_{6}}^{GB2}}({Q},{P}) =c_{5} \\cdot \\Big (\\arccos \\Big (1 + \\gamma \\cdot (\\gamma -1) \\cdot D_{\\varphi _{\\gamma }}({Q},{P})\\Big ) \\, \\Big )^{c_{6}}= c_{5} \\cdot \\Big (\\arccos \\Big (H_{\\varphi _{\\gamma }}({Q},{P})\\Big ) \\, \\Big )^{c_{6}}\\nonumber \\\\&=& c_{5} \\cdot \\Big (\\arccos \\Big (\\sum \\displaylimits _{k=1}^{K} (q_{k})^{\\gamma } \\cdot (p_{k})^{1-\\gamma }\\Big ) \\, \\Big )^{c_{6}}\\ge 0 \\, ,\\qquad \\gamma \\in \\, ]0,1[, \\, c_5 >0, \\, c_{6} >0,\\nonumber $ which is BS-maximizable on $\\textrm {$$\\hspace{-6.544pt}$$}$ .","The case $\\mathcal {B}_{1/2,1,1}({Q},{P})$ corresponds to the well-known Bhattacharyya arccos distance (Bhattacharyya distance of type 2) in [50] (which is also called Wootters distance [395]), and $\\mathcal {B}_{1/2,1,2}({Q},{P})$ to its variant in [49]; the case $\\mathcal {B}_{1/2,2,1}({Q},{P})$ is known as Fisher distance or Rao distance or geodesic distance (see e.g.","Deza & Deza [113]); a nice graphical illustration of the geometric connection between the Fisher distance $\\mathcal {B}_{1/2,2,1}({Q},{P})$ and the Hellinger distance/metric $\\sqrt{\\frac{1}{2} \\cdot D_{\\varphi _{1/2}}({Q},{P})}$ can be found e.g.","on p.35 in Ay et al.","[25].","Some exemplary applications of the Bhattacharyya arccos distance $\\mathcal {B}_{1/2,1,1}({Q},{P})$ can be found e.g.","in Rao [301] and Juhasz [177] for cluster analysis of human populations, in Martin-Fernandez et al.","[251] for general hierarchical clustering, Greenacre [141] for metric scaling, and in Chen et al.","[79] for clustering high-dimensional microbial data from RNA sequencing.","Let us give another example, namely $\\widetilde{\\mathcal {B}}_{\\gamma ,\\nu ,c_{7}}({Q},{P}) &: =&D_{h_{\\nu ,c_{7}}^{BB}}({Q},{P}) =\\frac{c_{7}}{\\log (1-\\frac{1}{\\nu })} \\cdot \\log \\bigg (1-\\frac{1-\\Big ( 1 + \\gamma \\cdot (\\gamma -1) \\cdot D_{\\varphi _{\\gamma }}({Q},{P})\\Big )}{\\nu }\\bigg )\\nonumber \\\\&=& \\frac{c_{7}}{\\log (1-\\frac{1}{\\nu })} \\cdot \\log \\bigg (1-\\frac{1-H_{\\varphi _{\\gamma }}({Q},{P})}{\\nu }\\bigg )= \\frac{c_{7}}{\\log (1-\\frac{1}{\\nu })} \\cdot \\log \\bigg (1-\\frac{1-\\sum \\displaylimits _{k=1}^{K} (q_{k})^{\\gamma } \\cdot (p_{k})^{1-\\gamma }}{\\nu }\\bigg )\\in [0, c_{7}[ \\,, \\nonumber \\\\& & \\hspace{227.62204pt} \\gamma \\in \\,]0,1[, \\, \\, c_{7}>0, \\, \\nu \\in \\, ]-\\infty ,0[ \\, \\cup \\, ]1,\\infty [,\\nonumber $ which is BS-maximizable on $\\textrm {$$\\hspace{-6.544pt}$$}$ .","The case $\\widetilde{\\mathcal {B}}_{1/2,\\nu ,1}({Q},{P})$ corresponds to the Bounded Bhattacharyya Distance Measures of Jolad et al.","[174].","We can also employ divergences of the form $\\breve{R}_{\\gamma }({Q},{P}) :=R_{\\gamma }(T_{1}({Q}),T_{2}({P}))$ and analogously power divergences $\\breve{D}_{\\widetilde{c} \\cdot \\varphi _{\\gamma }}({Q},{P}) :=D_{\\widetilde{c} \\cdot \\varphi _{\\gamma }}(T_{1}({Q}),T_{2}({P}))$ etc.","where $T_{1}: \\mathcal {D}_{1} \\mapsto \\mathcal {R}_{1}$ , $T_{2}: \\mathcal {D}_{1} \\mapsto \\mathcal {R}_{2}$ are (say) invertible functions on appropriately chosen subsets $\\mathcal {D}_{1}, \\mathcal {D}_{2}, \\mathcal {R}_{1}, \\mathcal {R}_{2}$ of the probability-vector simplex $\\mathbb {S}^{K}$ .","For instance, consider the following special case (with a slight abuse of notation): $\\breve{R}_{\\gamma }({Q},{P}): = R_{\\gamma }(\\widetilde{{Q}},\\widetilde{{P}})=\\frac{1}{\\gamma \\cdot (\\gamma -1)} \\cdot \\log \\left( \\sum \\displaylimits _{k=1}^{K}\\left(\\frac{(q_{k})^{\\nu _{1}}}{\\sum _{j=1}^{K} (q_{j})^{\\nu _{1}}}\\right)^{\\gamma } \\cdot \\left(\\frac{(p_{k})^{\\nu _{2}}}{\\sum _{j=1}^{K} (p_{j})^{\\nu _{2}}}\\right)^{1-\\gamma }\\right)$ where (i) $\\widetilde{{Q}} := (\\widetilde{q}_{k})_{k=1}^{K}$ with $\\widetilde{q}_{k} := \\frac{(q_{k})^{\\nu _{1}}}{\\sum _{j=1}^{K} (q_{j})^{\\nu _{1}}}$ is the escort probability distribution (in vector form) associated with the probability distribution (in vector form) ${Q} := (q_{k})_{k=1}^{K} \\in \\mathbb {S}_{> 0}^{K}$ , and (ii) $\\widetilde{{P}} := (\\widetilde{p}_{k})_{k=1}^{K}$ with $\\widetilde{p}_{k} := \\frac{(p_{k})^{\\nu _{2}}}{\\sum _{j=1}^{K} (p_{j})^{\\nu _{2}}}$ is the escort probability distribution associated with the probability distribution ${P} := (p_{k})_{k=1}^{K} \\in \\mathbb {S}_{> 0}^{K}$ , in terms of some fixed escort parameters $\\nu _{1} >0$ , $\\nu _{2} >0$ .","In particular, for the special choice $\\nu _{1}= \\nu _{2} >0$ and $\\gamma := \\frac{\\nu }{\\nu _{1}}$ with $\\nu \\in ]0,\\nu _{1}[ \\, \\cup \\, [2\\nu _{1}, \\infty [$ we obtain from (REF ) $\\hspace{-28.45274pt}&& 0 \\le \\frac{\\nu }{\\nu _{1}} \\cdot R_{\\nu /\\nu _{1}}(\\widetilde{{Q}},\\widetilde{{P}})=\\frac{\\log \\Big ( \\sum \\displaylimits _{k=1}^{K} (\\widetilde{q}_{k})^{\\nu /\\nu _{1}} \\cdot (\\widetilde{p}_{k})^{1-(\\nu /\\nu _{1})} \\Big )}{\\frac{\\nu }{\\nu _{1}} -1}\\nonumber \\\\\\hspace{-28.45274pt}&& =\\frac{\\nu _{1}}{\\nu - \\nu _{1}} \\cdot \\log \\Big ( \\sum \\displaylimits _{k=1}^{K} (q_{k})^{\\nu } \\cdot (p_{k})^{\\nu _{1} - \\nu } \\Big )- \\frac{\\nu }{\\nu - \\nu _{1}} \\cdot \\log \\Big ( \\sum \\displaylimits _{k=1}^{K} (q_{k})^{\\nu _{1}} \\Big )+ \\log \\Big ( \\sum \\displaylimits _{k=1}^{K} (p_{k})^{\\nu _{1}} \\Big )=: \\breve{R}_{\\nu /\\nu _{1}}({Q},{P})$ which is BS-minimizable (in $\\widetilde{{Q}}$ ) on $\\textrm {$$\\hspace{-6.544pt}$$}$ .","Our divergence $\\breve{R}_{\\nu /\\nu _{1}}({Q},{P})$ in (REF ) is basically a multiple of a divergence which has been very recently used in Ghosh & Basu [136].","Moreover, $\\breve{R}_{1/\\nu _{1}}({Q},{P})$ (i.e.","the special case $\\nu =1$ in (REF )) is equal to Sundaresan’s divergence [352] [353] (see also Lutwak et al.","[234], Kumar & Sundaresan [203], [204], Yagli et al.","[408]); for our BS-approach, we need the restriction $\\nu _{1} \\in \\, ]0,\\frac{1}{2}] \\, \\cup \\, ]1,\\infty [$ .","Notice that Sundaresan’s divergence can be employed in mismatch-cases of (i) Campbell’s coding problem, (ii) Arikan’s guessing problem, (iii) memoryless guessing, and (iv) task partitioning problems; see e.g.","Sundaresan [353], Bunte & Lapidoth [67], Kumar et al.","[205].","Returning to the general context, functions of the modified Kullback-Leibler information $I(\\mathbf {Q},{P})$ and the modified reverse Kullback-Leibler information $\\widetilde{I}(\\mathbf {Q},{P})$ can be treated analogously.","For the sake of brevity, we only deal with the former and fix arbitrary ${P} \\in \\mathbb {S}_{>0}^{K}$ and $\\mathbf {Q} \\in A \\cdot \\mathbb {S}^{K}$ with $A := \\sum _{k=1}^{K} q_{k} >0$ .","For this, in (REF ) we have obtained $I(\\mathbf {Q},{P})$ which is generally not a divergence (cf.","Remark REF ).","In the following, let the function $h_{1} : \\, ]-1,\\infty [ \\, \\mapsto \\, ]-\\infty ,\\infty [$ be continuous and strictly increasing (respectively, strictly decreasing); since $D_{\\varphi _{1}}(\\mathbf {Q},{P})$ is BS-minimizable on $\\mathbf {\\Omega } = A \\cdot \\textrm {$$\\hspace{-6.544pt}$$}$ , also the quantity $h_{1}\\Big ( A - 1 + D_{\\varphi _{1}}(\\mathbf {Q},{P})\\Big )= h_{1}\\Big (I(\\mathbf {Q},{P})\\Big )$ is BS-minimizable on $\\mathbf {\\Omega } = A \\cdot \\textrm {$$\\hspace{-6.544pt}$$}$ (respectively, BS-maximizable on $\\mathbf {\\Omega } = A \\cdot \\textrm {$$\\hspace{-6.544pt}$$}$ ).","In particular, by using the negative identity mapping $h_{\\gamma }^{-Id}(y) := -y$ ($y>-1$ ) we get that $-I(\\mathbf {Q},{P})$ is BS-maximizable.","Another exemplary choice for $h_{1}$ is (cf.","Sharma & Mittal [330] in the scaling of e.g.","Morales et al.","[264]) $& & h_{1}(y) := h_{s}^{SM}(y) := \\frac{e^{(s-1) \\cdot y} -1}{s-1},\\qquad y \\in \\mathbb {R}, \\, s \\in \\, ]0,1[ \\, \\cup \\, ]1,\\infty [,$ which is strictly increasing; hence, $h_{s}^{SM}(I(\\mathbf {Q},{P}))$ (and also $h_{s}^{SM}(D_{\\varphi _{1}}(\\mathbf {Q},{P}))$ ) is BS-minimizable on $\\mathbf {\\Omega } = A \\cdot \\textrm {$$\\hspace{-6.544pt}$$}$ .","As another important application line, let us fix any $(\\gamma , \\mathbf {Q}) \\in (\\widetilde{\\Gamma }\\backslash ]1,2[) \\times \\widetilde{\\mathcal {M}}_{2}$ (cf.","Lemma REF (a)) with $A := \\sum _{k=1}^{K} q_{k} >0$ .","Moreover, we take ${P} := {P}^{unif} :=(\\frac{1}{K}, \\ldots , \\frac{1}{K})$ to be the probability vector of frequencies of the uniform distribution on $\\lbrace 1, \\ldots , K\\rbrace $ .","Then, for $\\gamma \\in \\, ]-\\infty ,0[ \\ \\cup \\ ]0,1[ \\ \\cup \\ [ \\ 2,\\infty [$ one gets $H_{\\gamma }(\\mathbf {Q},{P}^{unif}) =K^{\\gamma -1} \\cdot \\sum _{k=1}^{K} q_{k}^{\\gamma }$ .","One can rewrite $K^{1- \\gamma } \\cdot H_{\\gamma }(\\mathbf {Q},{P}^{unif})= \\sum _{k=1}^{K} q_{k}^{\\gamma }$ ; the latter is sometimes called heterogeneity index of type $\\gamma $, see e.g.","van der Lubbe [379], with $\\gamma =2$ being the Simpson-Herfindahl index which is also known as index of coincidence (cf.","Harremoes & Topsoe [155] and its generalization in Harremoes & Vajda [156]).","Alternatively, $\\sum _{k=1}^{K} q_{k}^{\\gamma }$ is also called Onicescu’s information energy in case of $\\gamma =2$ (cf.","Onicescu [278], see also Pardo & Taneja [283] for comprehensive investigations) and in general information energy of order $\\gamma $ (cf.","Theodorescu [359], see also e.g.","Pardo [281]); for exemplary applications to electron density functional theory (DFT) for quantum chemical reactivity, the reader may take (discretized versions of) e.g.","Liu et al.","[226], Lopez-Rosa et al.","[231] and Rong et al.","[311].","In some other literature (see e.g.","Clark [90]), $\\sum _{k=1}^{K} q_{k}^{\\gamma }$ is alternatively called Golomb’s [140] information generating function (of a probability distribution ${Q}$ ); yet another name is generalized information potential and for $\\gamma =2$ information potential (cf.","e.g.","Principe [297], Acu et al.","[4]).","From the above-mentioned investigations, we obtain that $\\sum _{k=1}^{K} q_{k}^{\\gamma }$ is BS-minimizable on $\\mathbf {\\Omega } = A \\cdot \\textrm {$$\\hspace{-6.544pt}$$}$ for $\\gamma <0$ and $\\gamma \\ge 2$ , and BS-maximizable on $\\mathbf {\\Omega } = A \\cdot \\textrm {$$\\hspace{-6.544pt}$$}$ for $\\gamma \\in ]0,1[$ .","More generally, by employing (REF ) and (), for the class of entropies (diversity indices) $\\mathcal {E}_{\\gamma ,c_{1},c_{2},c_{3}}(\\mathbf {Q}) &:=&h_{c_{1},c_{2},c_{3}}\\left(\\sum _{k=1}^{K} q_{k}^{\\gamma }\\right) =c_{1} \\cdot \\left(\\left(\\sum _{k=1}^{K} q_{k}^{\\gamma }\\right)^{c_{2}} - c_{3} \\right)\\nonumber \\\\&=&c_{1} \\cdot \\left(K^{c_{2} \\cdot (1- \\gamma )} \\cdot H_{\\gamma }(\\mathbf {Q},{P}^{unif})^{c_{2}} - c_{3} \\right),\\qquad c_{1}, c_{2} \\in \\mathbb {R}\\backslash \\lbrace 0\\rbrace , \\,c_{3} \\in \\mathbb {R},\\\\\\mathcal {E}_{c_{4},f}^{R}(\\mathbf {Q}) &:=&h_{c_{4},f}^{R}\\left(\\sum _{k=1}^{K} q_{k}^{\\gamma }\\right)= \\frac{c_{4}}{f^{\\prime }(0)} \\cdot \\log \\left(\\sum _{k=1}^{K} q_{k}^{\\gamma }\\right),\\nonumber \\\\&=& \\frac{c_{4}}{f^{\\prime }(0)} \\cdot \\Big ( \\log \\left(H_{\\gamma }(\\mathbf {Q},{P}^{unif})\\right) +(1- \\gamma ) \\cdot \\log (K)\\Big ),\\qquad c_{4} \\in \\mathbb {R}\\backslash \\lbrace 0\\rbrace , \\,$ (which is similar to the entropy-class of Morales et al.","[265] who use a different, more restrictive parametrization and probability distributions ${Q}$ ), one gets the following extremum-behaviour: $\\mathcal {E}_{\\gamma ,c_{1},c_{2},c_{3}}(\\mathbf {Q})$ is BS-minimziable if $\\gamma <0$ and $c_{1} \\cdot c_{2} >0$ ; $\\mathcal {E}_{\\gamma ,c_{1},c_{2},c_{3}}(\\mathbf {Q})$ is BS-minimizable if $\\gamma \\ge 2$ and $c_{1} \\cdot c_{2} >0$ ; $\\mathcal {E}_{\\gamma ,c_{1},c_{2},c_{3}}(\\mathbf {Q})$ is BS-minimizable if $\\gamma \\in \\, ]0,1[$ and $c_{1} \\cdot c_{2} < 0$ ; $\\mathcal {E}_{\\gamma ,c_{1},c_{2},c_{3}}(\\mathbf {Q})$ is BS-maximizable if $\\gamma <0$ and $c_{1} \\cdot c_{2} < 0$ ; $\\mathcal {E}_{\\gamma ,c_{1},c_{2},c_{3}}(\\mathbf {Q})$ is BS-maximizable if $\\gamma \\ge 2$ and $c_{1} \\cdot c_{2} < 0$ ; $\\mathcal {E}_{\\gamma ,c_{1},c_{2},c_{3}}(\\mathbf {Q})$ is BS-maximizable if $\\gamma \\in \\, ]0,1[$ and $c_{1} \\cdot c_{2} > 0$ ; $\\mathcal {E}_{c_{4},f}^{R}(\\mathbf {Q})$ is BS-minimizable if $\\gamma <0$ and $\\frac{c_{4}}{f^{\\prime }(0)} > 0$ ; $\\mathcal {E}_{c_{4},f}^{R}(\\mathbf {Q})$ is BS-minimizable if $\\gamma \\ge 2$ and $\\frac{c_{4}}{f^{\\prime }(0)} > 0$ ; $\\mathcal {E}_{c_{4},f}^{R}(\\mathbf {Q})$ is BS-minimizable if $\\gamma \\in \\, ]0,1[$ and $\\frac{c_{4}}{f^{\\prime }(0)} < 0$ ; $\\mathcal {E}_{c_{4},f}^{R}(\\mathbf {Q})$ is BS-maximizable if $\\gamma <0$ and $\\frac{c_{4}}{f^{\\prime }(0)} < 0$ ; $\\mathcal {E}_{c_{4},f}^{R}(\\mathbf {Q})$ is BS-maximizable if $\\gamma \\ge 2$ and $\\frac{c_{4}}{f^{\\prime }(0)} < 0$ ; $\\mathcal {E}_{c_{4},f}^{R}(\\mathbf {Q})$ is BS-maximizable if $\\gamma \\in \\, ]0,1[$ and $\\frac{c_{4}}{f^{\\prime }(0)} > 0$ .","From this, one can deduce that our new BS method works for the constrained minimization/maximization of the following well-known, prominently used measures of entropy respectively measures of diversity, and beyond: $c_{1}=1$ , $c_{2} = \\frac{1}{\\gamma }$ , $c_{3}=0$ : the Euclidean $\\gamma -$ norm (also known as $\\gamma -$ norm heterogeneity index, see e.g.","van der Lubbe [379]) $|| \\mathbf {Q} ||_{\\gamma } := \\Big (\\sum _{k=1}^{K} q_{k}^{\\gamma }\\Big )^{1/\\gamma }= K^{(1- \\gamma )/\\gamma } \\cdot \\Big (H_{\\gamma }(\\mathbf {Q},{P}^{unif})\\Big )^{1/\\gamma }$ is BS-minimizable on $\\mathbf {\\Omega } = A \\cdot \\textrm {$$\\hspace{-6.544pt}$$}$ for $\\gamma \\in ]0,1[$ and $\\gamma \\ge 2$ , and BS-maximizable on $\\mathbf {\\Omega } = A \\cdot \\textrm {$$\\hspace{-6.544pt}$$}$ for $\\gamma <0$ (note that $|| \\mathbf {Q} ||_{1} =A$ ) ; similarly, the $\\gamma -$ mean heterogeneity index (see e.g.","[379], as well as Jost [176] for its interpretation as “effective number of species” respectively as “numbers equivalent”) given by $\\mathcal {E}^{HI}(\\mathbf {Q}) := \\Big (\\sum _{k=1}^{K} q_{k}^{\\gamma }\\Big )^{1/(\\gamma -1)}= \\frac{1}{K} \\cdot \\Big (H_{\\gamma }(\\mathbf {Q},{P}^{unif})\\Big )^{1/(\\gamma -1)}$ is BS-minimizable on $\\mathbf {\\Omega } = A \\cdot \\textrm {$$\\hspace{-6.544pt}$$}$ for $\\gamma \\ge 2$ , and BS-maximizable on $\\mathbf {\\Omega } = A \\cdot \\textrm {$$\\hspace{-6.544pt}$$}$ for $\\gamma <0$ and $\\gamma \\in ]0,1[$ .","Alternatively, $\\mathcal {E}^{HI}(\\mathbf {Q})$ is also called ($\\gamma -$ order) Hill diversity index or Hill number [160], respectively ($\\gamma -$ order) Hannah-Kay index [154], respectively ($\\gamma -$ order) Renyi heterogeneity (cf.","Nunes et al.","[276]), respectively ($\\gamma -$ order) exponential Renyi entropy or exponential entropy (cf.","Campbell [72]) since it is equal to $\\exp (\\mathcal {E}^{gR}(\\mathbf {Q}))$ (cf.","(E6) below).","The $\\gamma -$ mean heterogeneity index (under one of the above-mentioned namings) was recently employed e.g.","by Greiff et al.","[142] for immunodiagnostic design of fingerprints of an individual’s ongoing immunological status (e.g., healthy, infected, vaccinated) — culminating in accurate and early detection of disease and infection, by Ma & Li [237] for the quantification of metagenome diversity and similarity, by Jasinska et al.","[164] for studying bacterial evolution — in particular evolution under sub-inhibitory antibiotic levels, by Ma et al.","[238] for the definition of individual-level genetic diversity and similarity profiles as well as their applications to datasets from the 1000-Genomes Project, and by Lassance & Vrins [209] for some optimal selection procedure of financial-asset portfolios.","$c_{1}=\\frac{1}{2^{1-\\gamma }-1}$ , $c_{2} = 1$ , $c_{3}=1$ : the entropy $\\mathcal {E}^{gHC}(\\mathbf {Q}) :=\\frac{1}{2^{1-\\gamma }-1} \\cdot \\left(\\sum _{k=1}^{K} q_{k}^{\\gamma } - 1 \\right)=\\frac{1}{2^{1-\\gamma }-1} \\cdot \\left(K^{1- \\gamma } \\cdot H_{\\gamma }(\\mathbf {Q},{P}^{unif}) - 1 \\right)$ is BS-minimizable on $\\mathbf {\\Omega } = A \\cdot \\textrm {$$\\hspace{-6.544pt}$$}$ for $\\gamma <0$ , and BS-maximizable on $\\mathbf {\\Omega } = A \\cdot \\textrm {$$\\hspace{-6.544pt}$$}$ for $\\gamma \\in ]0,1[$ and $\\gamma \\ge 2$ ; the special subcase $A=1$ in (REF ) (and thus, $\\mathbf {Q} ={Q}$ is a probability vector) corresponds to the $\\gamma -$ order entropy of Havrda-Charvat [157] (also called non-additive $\\gamma -$ order Tsallis entropy [363] in statistical physics) where the special case $\\gamma =2$ is (a multiple of) Vajda’s quadratic entropy [371] and Ahlswede’s identification entropy [7] (see also Ahlswede & Cai [8]).","Some exemplary (relatively) recent studies and applications of $\\mathcal {E}^{gHC}({Q})$ appear e.g.","in Peter & Rangarajan [291] for shape matching, in Liu et al.","[226] as well as in Rong et al.","[311] to electron density functional theory (DFT) for quantum chemical reactivity, in Yalcin & Beck [409] for the investigation of energy spectra of cosmic rays, in Wen & Jiang [390] for the quantification of complexity degrees in complex networks, in Bhandari [46] for fast multilevel thresholding for color image segmentation, in Erguzel et al.","[121] for the investigation of Electroencephalography (EEG) signals of subjects suffering from some psychiatric disorders, in Kang & Kim [182] for automatic synthetic aperture radar (SAR) image registration, in Namdari & Li [269] for the modelling of Lithium-Ion battery capacity fade, in Seweryn et al.","[326] for the assessment of similarity and diversity of expression profiles in single cell systems, in Zhang et al.","[418] for the search of functional relationships between groundwater depth and vegetation distribution, in Kumbhakar et al.","[207] for the modelling of streamwise velocity profiles in wide–open channel turbulent flows (e.g.","in rivers, streams, canals, ditches), and in Ramezani & Pourdarvish [300] for transfer learning for image classification of gravitational waves.","For the special case $\\gamma =2$ , a directly connected quantity is the measure of concentration (cf.","e.g.","De Wet et al.","[107]) $\\mathcal {E}^{gMC}({Q}) : = 1- \\frac{1}{K} - \\mathcal {E}^{gHC}({Q})= \\sum _{k=1}^{K} \\left(q_{k} - \\frac{1}{K} \\right)^{2}$ which (up to a multiple) was introduced by Brukner & Zeilinger [66] as an appropriate measure of information for quantum experiments.","$\\gamma := \\frac{1}{\\widetilde{\\gamma }}$ , $c_{1}=\\frac{1}{\\widetilde{\\gamma }-1}$ , $c_{2} = \\widetilde{\\gamma }$ , $c_{3}=1$ : the entropy $\\mathcal {E}^{gA}(\\mathbf {Q}) :=\\frac{1}{\\widetilde{\\gamma }-1} \\cdot \\left(\\left(\\sum _{k=1}^{K}q_{k}^{1/\\widetilde{\\gamma }}\\right)^{\\widetilde{\\gamma }} - 1 \\right)=\\frac{1}{\\widetilde{\\gamma }-1} \\cdot \\left(K^{\\widetilde{\\gamma } \\cdot (1- \\gamma )} \\cdot H_{1/\\widetilde{\\gamma }}(\\mathbf {Q},{P}^{unif})^{\\widetilde{\\gamma }} - 1 \\right)$ is BS-minimizable on $\\mathbf {\\Omega } = A \\cdot \\textrm {$$\\hspace{-6.544pt}$$}$ for $\\widetilde{\\gamma } <0$ and $\\widetilde{\\gamma } \\in \\, ]0,1[$ , and BS-maximizable on $\\mathbf {\\Omega } = A \\cdot \\textrm {$$\\hspace{-6.544pt}$$}$ for $\\gamma \\ge 2$ ; the special subcase $A=1$ in (REF ) (and thus, $\\mathbf {Q} ={Q}$ is a probability vector) corresponds to the $\\widetilde{\\gamma }-$ order entropy of Arimoto [16].","$s \\in \\mathbb {R}\\backslash \\lbrace 1\\rbrace $ , $c_{1}=\\frac{1}{1-s}$ , $c_{2} = \\frac{1-s}{1-\\gamma }$ , $c_{3}=1$ : the entropy $\\mathcal {E}^{gSM1}(\\mathbf {Q}):= \\frac{1}{1-s} \\cdot \\left(\\left(\\sum _{k=1}^{K} q_{k}^{\\gamma }\\right)^{(1-s)/(1-\\gamma )} - 1 \\right)= \\frac{1}{1-s} \\cdot \\left(K^{1- s} \\cdot H_{\\gamma }(\\mathbf {Q},{P}^{unif})^{(1-s)/(1-\\gamma )} - 1 \\right)$ is BS-minimizable on $\\mathbf {\\Omega } = A \\cdot \\textrm {$$\\hspace{-6.544pt}$$}$ for $\\gamma < 0$ and BS-maximizable on $\\mathbf {\\Omega } = A \\cdot \\textrm {$$\\hspace{-6.544pt}$$}$ for $\\gamma \\in \\, ]0,1[$ and $\\gamma \\ge 2$ ; the special subcase $A=1$ in (REF ) (and thus, $\\mathbf {Q} ={Q}$ is a probability vector) corresponds to the entropy of order $\\gamma $ and degree $s$ of Sharma & Mittal [329] in the scaling of e.g.","Salicru et al.","[314].","$s \\in \\mathbb {R}\\backslash \\lbrace 0\\rbrace $ , $\\gamma = s+1$ , $c_{1}= - \\frac{1}{s}$ , $c_{2} = 1$ , $c_{3}=1$ : the diversity index $\\mathcal {E}^{gPT}(\\mathbf {Q}):= - \\frac{1}{s} \\cdot \\left(\\sum _{k=1}^{K} q_{k}^{s+1} - 1 \\right)= - \\frac{1}{s} \\cdot \\left(K^{- s} \\cdot H_{s+1}(\\mathbf {Q},{P}^{unif}) - 1 \\right)$ is BS-minimizable on $\\mathbf {\\Omega } = A \\cdot \\textrm {$$\\hspace{-6.544pt}$$}$ for $s < -1$ and BS-maximizable on $\\mathbf {\\Omega } = A \\cdot \\textrm {$$\\hspace{-6.544pt}$$}$ for $s \\in \\, ]-1,0[$ and $s >0$ ; the special subcase $A=1$ in (REF ) (and thus, $\\mathbf {Q} ={Q}$ is a probability vector) corresponds to the diversity index of degree $s$ of Patil & Taillie [286]; the case $s=1$ for probability measures $\\mathbf {Q}={Q}$ gives the well-known Gini-Simpson diversity index.","$c_{4}=\\frac{1}{1-\\gamma }$ , $f(z) =z$ : the entropy $\\mathcal {E}^{gR}(\\mathbf {Q}):= \\frac{1}{1-\\gamma } \\cdot \\log \\left(\\sum _{k=1}^{K} q_{k}^{\\gamma }\\right)= \\frac{1}{1-\\gamma } \\cdot \\Big ( \\log \\left(H_{\\gamma }(\\mathbf {Q},{P}^{unif})\\right) + (1- \\gamma ) \\cdot \\log (K) \\Big )= \\frac{\\log 2}{1-\\gamma } \\cdot \\log _{2}\\left(\\sum _{k=1}^{K} q_{k}^{\\gamma }\\right)$ is BS-minimizable on $\\mathbf {\\Omega } = A \\cdot \\textrm {$$\\hspace{-6.544pt}$$}$ for $\\gamma <0$ , and BS-maximizable on $\\mathbf {\\Omega } = A \\cdot \\textrm {$$\\hspace{-6.544pt}$$}$ for $\\gamma \\in ]0,1[$ and $\\gamma \\ge 2$ ; the special subcase $A=1$ in (REF ) (and thus, $\\mathbf {Q} = {Q}$ is a probability vector) corresponds to the prominent (additive) $\\gamma -$ order Renyi entropy [309].","As well known, there is a vast literature on Renyi entropies $\\mathcal {E}^{gR}({Q})$ .","Some exemplary (mostly recent) studies and applications appear e.g.","in Nath [270] — as well as in Arikan [15], Sundaresan [353], Bunte & Lapidoth [67], Sason & Verdu [321], Kumar et al.","[205] — for coding and guessing, in Bennett et al.","[39] in connection with unconditionally secure secret-key agreement protocols and quantum cryptography, in Mayoral [257] for cluster sampling, in Aviyente et al.","[21] for information extraction in certain neurophysiological signals (so-called event-related potentials), in Tao et al.","[355] as well as in Jiao et al.","[171] for early defect/fault detection of rolling element bearings, in Pham et al.",": [292] for blind source separation, in Liu et al.","[226] as well as in Rong et al.","[311] to electron density functional theory (DFT) for quantum chemical reactivity, in Sason [319] for data compression, in Carravilla et al.","[73] for the recognition of HIV-1 antibodies through STED microscopy and the corresponding design of therapeutic interventions, in Joshi et al.","[175] for the identification and tracking of relevant T cell receptors for adoptive immunotherapy, in Erguzel et al.","[121] for the investigation of Electroencephalography (EEG) signals of subjects suffering from some psychiatric disorders, in German–Sallo [133] for fault–characteristics extraction from discrete signals in manufacturing systems, in Schober et al.","[323] for investigations of some evolutions of the T cell antigen receptor (TCR) repertoire, in Seweryn et al.","[326] for the assessment of similarity and diversity of expression profiles in single cell systems, in Amezquita-Sanchez [13] for the detection of incipient damage in high-rise buildings subjected to dynamic vibrations, in Barennes et al.","[32] for comparing the accuracy of current T cell receptor sequencing methods employed for the understanding of adaptive immune responses, in Kumar et al.","[201] for the segmentation of digital images through multilevel iterative variational mode decomposition (VMD), and in Pandey [280] for the quantification of cosmic homogeneity.","Remark 19 (i) For Renyi entropies there are also matrix versions $\\mathcal {E}^{gR}(X) :=\\frac{1}{1-\\gamma } \\cdot \\log \\left(\\sum _{i=1}^{K_{1}}\\sum _{j=1}^{K_{2}} x_{ij}^{\\gamma }\\right)$ where $X:=(x_{ij})_{i=1,\\ldots ,K_{1}}^{j=1,\\ldots ,K_{2}}$ is a $K_{1} \\times K_{2}-$ matrix whose elements $x_{ij}$ are (say) strictly positive and sum up to $A$ .","Such a setup with $A=1$ is e.g.","used in time-frequency analyses of signals where the $i$ ’s correspond to discrete time points, the $j$ ’s to discrete frequencies, and $x_{ij}$ to the probability that $(i,j)$ occurs; see e.g.","Popescu & Aiordachioaie [295] for change detection in seismic signals.","Another line of application is to use as $X$ the normalized communicability matrix of a directed network (respectively the upper triangular part of $X$ in case of an unweighted and undirected network).","Of course, the matrix version $\\mathcal {E}^{gR}(X)$ can be easily and equivalently rewritten in our vector version $\\mathcal {E}^{gR}(\\mathbf {Q})$ by setting $\\mathbf {Q} := (q_{1}, \\ldots , q_{K_{1} \\cdot K_{2}})$ such that $x_{ij} = q_{(i-1)\\cdot K_{2} +j}$ ($i=1,\\ldots ,K_{1}$ , $j=1,\\ldots ,K_{2}$ and hence $K:= K_{1} \\cdot K_{2}$ ; accordingly, we can apply our BS method.","(ii) The latter conversion works analogously also for matrix versions of all the other entropies, divergences, etc.","of this paper; more flexible versions where $i \\in \\lbrace 1,\\ldots ,K_{1}\\rbrace $ , $j \\in J_{i}$ for some $J_{i} \\subseteq \\lbrace 1,\\ldots ,K_{2}\\rbrace $ as well as multidimensional-array/tensor versions can be transformed in a similar book-keeping manner, too.","For instance, within the above-mentioned framework of unweighted and undirected networks, Chen et al.","[81] and Shi et al.","[332] employ communicability matrix versions of the Shannon entropy and the Jensen-Shannon divergence (JSD), e.g.","in order to derive a new complexity measure of such kind of networks; see also Bagrow and Bollt [28] for similar network applications of the JSD.","Moreover, Jena et al.","[167] use “3D versions” of Tsallis entropies for brain magnetic resonance (MR) image segmentation.","Remark 20 All the above cases which are BS-maximizable can be interpreted as bare-simulation approach to the solution of generalized maximum entropy problems on $\\mathbf {\\Omega } = A \\cdot \\textrm {$$\\hspace{-6.544pt}$$}$.","Remark 21 (i) If (all) the above- and below-mentioned entropies are used for probability vectors ${Q} \\in \\mathbf {S}^{K}$ — i.e.","one employs $\\mathcal {E}({Q})$ — then typically the components $q_{k}$ of ${Q}$ represent a genuine probability mass (frequency) $q_{k} = \\mathbb {\\Pi }[\\lbrace d_{k} \\rbrace ]$ of some data point (state) $d_{k}$ .","However, ${Q} \\in \\mathbf {S}^{K}$ may alternatively be artificially generated.","For instance, for the purpose of fault detections of mechanical drives, Boskoski & Juricic [57] use Renyi entropies where the $q_{k}$ ’s are normalized squared energy-describing coefficients of the wavelet packet transform of measured vibration records.","Another exemplary “artificial” operation is concatenation, see e.g.","Subsection REF below.","(ii) An analogous statement holds for the employment of (all) the above- and below-mentioned divergences $D({Q},{P})$ — and their transformations — between genuine respectively artificially generated probability vectors ${Q}, {P} \\in \\mathbf {S}^{K}$ .","The remaining parameter cases $\\gamma =0$ and $\\gamma =1$ can be treated analogously.","For the sake of brevity, we only deal with the latter.","For this, let $\\mathbf {Q} \\in A \\cdot \\mathbb {S}^{K}$ with $A := \\sum _{k=1}^{K} q_{k} >0$ and ${P} := {P}^{unif}$ .","Clearly, $I(\\mathbf {Q},{P}^{unif}) - \\frac{\\log K}{K}=\\sum _{k=1}^{K} q_{k} \\cdot \\log (q_{k})$ ; thus the latter is BS-minimizable on $\\mathbf {\\Omega } = A \\cdot \\textrm {$$\\hspace{-6.544pt}$$}$ .","More generally, for any continuous strictly increasing (respectively strictly decreasing) function $h_{1} : [- \\frac{K}{e}, 0[ \\, \\mapsto \\mathbb {R}$ , the quantity $h_{1}\\Big (\\sum _{k=1}^{K} q_{k} \\cdot \\log (q_{k})\\Big )$ is BS-minimizable on $\\mathbf {\\Omega } = A \\cdot \\textrm {$$\\hspace{-6.544pt}$$}$ (respectively BS-maximizable on $\\mathbf {\\Omega } = A \\cdot \\textrm {$$\\hspace{-6.544pt}$$}$ ).","Important special cases are: $h_{1}(y) := h_{1}^{-Id}(y) = -y$ : the entropy $\\mathcal {E}^{Sh}(\\mathbf {Q}) :=h_{1}^{-Id}\\Big (\\sum _{k=1}^{K} q_{k} \\cdot \\log (q_{k})\\Big )= - \\sum _{k=1}^{K} q_{k} \\cdot \\log (q_{k})$ is BS-maximizable on $\\mathbf {\\Omega } = A \\cdot \\textrm {$$\\hspace{-6.544pt}$$}$ ; the special subcase $A=1$ in (REF ) (and thus, $\\mathbf {Q} = {Q}$ is a probability vector) corresponds to the omnipresent Shannon entropy; hence, by our bare-simulation approach we can particularly tackle maximum entropy problems on almost arbitrary sets $\\textrm {$$\\hspace{-6.544pt}$$}$ of probability vectors.","Analogously, we can treat $\\frac{1}{\\log (K)} \\cdot \\mathcal {E}^{Sh}({Q})$ which is called Pielou’s evenness index [293], and $1-\\frac{1}{\\log (K)} \\cdot \\mathcal {E}^{Sh}({Q}) \\in [0,1]$ which is sometimes used as clonality (clonotype diversity) index (see e.g.","Gabriel et al.","[131] for applications to HIV-connected T cell receptor repertoires, and Bashford-Rogers et al.","[33] (with supplementary private communication) for its use for comparative analyses of the BCR repertoire in immune-mediated diseases, for the sake of understanding pathological mechanisms and designing treatment strategies).","As a further example for Remark REF , Lyubushin [236] uses $q_{k}$ ’s which are normalized squared coefficients of an orthogonal wavelet decomposition of some seismic noise, and accordingly, $\\frac{1}{\\log (K)} \\cdot \\mathcal {E}^{Sh}({Q})$ can be interpreted as the entropy of the distribution of energy of oscillations at various frequency and time scales.","Some further exemplary studies and applications of the maximization of $\\mathcal {E}^{Sh}({Q})$ — aside from the vast physics literature — appear e.g.","in De Santis et al.","[106] for cryptanalytic guessing problems for breaking ciphertexts with probabilistic brute-force attacks, Johansson & Sternad [173] for tackling certain resource allocation problems under uncertainty, Marano & Franceschetti [246] for ray propagation in percolating lattices, Miao et al.","[260] for unsupervised mixed-pixel decomposition in image processing, Rodrigues et al.","[310] for modelling biological species geographic distribution, Xiong et al.","[400] for capturing desirable phrasal and hierarchical segmentations within a statistical machine translation context, Chan et al.","[76] for alignment-free DNA sequence comparison, Mann & Garnett [244] for capturing some collective behaviours of intelligent agents in social interactions, Singh et al.","[336] for the study of finite buffer queueing systems, Baddeley [27] for geoscientifical prediction of the occurrence of mineral deposits on regional scales, Einicke et al.","[118] for feature selection within change classification during running, and Han et al.","[152] for substructure imaging of blood cells by means of maximum entropy tomography (MET).","$s \\in \\, ]0,1[ \\, \\cup \\, ]1,\\infty [$ , $h_{1}(y) := h_{s}^{SM2}(y) := \\frac{e^{(s-1) \\cdot y} -1}{1-s}$ (cf.","(REF )) with $y \\in \\mathbb {R}$ : the entropy $\\mathcal {E}^{SM2}(\\mathbf {Q}) :=h_{s}^{SM2}\\Big (\\sum _{k=1}^{K} q_{k} \\cdot \\log (q_{k})\\Big )= \\frac{1}{1-s} \\cdot \\left( \\exp \\Big \\lbrace (s-1) \\cdot \\sum _{k=1}^{K} q_{k} \\cdot \\log (q_{k})\\Big \\rbrace -1 \\right)$ is BS-maximizable on $\\mathbf {\\Omega } = A \\cdot \\textrm {$$\\hspace{-6.544pt}$$}$ ; the special subcase $A=1$ in (REF ) (and thus, $\\mathbf {Q} = {Q}$ is a probability vector) corresponds to the (second type) entropy of Sharma & Mittal [329] in the scaling of e.g.","Pardo [282] (p.20).","Returning to the general context, we now (as already indicated above) state explicitly the corresponding bare-simulation-minimizations (respectively maximizations) of the power divergences $\\inf _{\\mathbf {Q}\\in \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega } }D_{\\widetilde{c}\\cdot \\varphi _{\\gamma }}(\\mathbf {Q},{P})$ ($\\gamma \\in \\mathbb {R}$ ), the Renyi divergences $\\inf _{\\mathbf {Q}\\in \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega } }R_{\\gamma }(\\mathbf {Q},{P})$ ($\\gamma \\in \\mathbb {R}$ ), the Hellinger integrals $\\inf _{\\mathbf {Q}\\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega } }H_{\\gamma }(\\mathbf {Q},{P})$ ($\\gamma \\in ]-\\infty ,0[ \\, \\cup \\, ]1,\\infty [$ ), $\\sup _{\\mathbf {Q}\\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega } }H_{\\gamma }(\\mathbf {Q},{P})$ ($\\gamma \\in ]0,1[$ ), the modified Kullback-Leibler information $\\inf _{Q\\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega } }I(\\mathbf {Q},{P})$ , the modified reverse Kullback-Leibler information $\\inf _{Q\\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega } }\\widetilde{I}(\\mathbf {Q},{P})$ , as well as the above-mentioned measures of entropy (diversity).","Since the corresponding probability distribution $\\mathbb {}[\\cdot \\,]=\\mathbb {\\Pi }[W_{1}\\in \\cdot \\,]$ of the $W_{i}$ 's (cf.","the representability (REF )) varies “quite drastically” with $\\gamma $ , we split this issue into several pieces.","Proposition 22 (a) Consider the context of Remark REF (vi) for $\\varphi := \\widetilde{c}\\cdot \\varphi _{\\gamma }$ with $\\gamma <0$ , and let ${P} \\in \\mathbb {S}_{>0}^{K}$ as well as $\\widetilde{c}>0$ be arbitrary but fixed.","Furthermore, let $W:=(W_{i})_{i\\in \\mathbb {N}}$ be an i.i.d.","sequence of non-negative real-valued random variables having densityin the classical sense, with respect to Lebesgue measure $f_{W_{1}}(y)\\ :=\\ \\frac{\\exp \\lbrace -\\frac{y \\cdot \\widetilde{c}}{1-\\gamma }\\rbrace }{\\exp \\lbrace \\widetilde{c}/\\gamma \\rbrace }\\cdot f_{Z}(y) \\cdot {1}_{]-\\infty ,0[}(y), \\qquad y \\in \\mathbb {R},$ where $f_{Z}$ is the density of a random variable $Z$ which has stable law with parameter-quadruple $(\\frac{-\\gamma }{1-\\gamma },1,0,-\\frac{\\widetilde{c}^{1/(1-\\gamma )} \\cdot (1-\\gamma )^{-\\gamma /(1-\\gamma )}}{\\gamma })$ in terms of “form-B notation” in Zolotarev [428], p.12.","Then for all $A>0$ and all $\\Omega $$\\Omega \\subset \\mathbb {S}_{>0}^{K}$ with (REF ) there holds $\\hspace{-19.91684pt}-\\lim _{n\\rightarrow \\infty }\\frac{1}{n}\\log \\,\\mathbb {\\Pi }\\left[\\xi _{n}^{w\\mathbf {W}}\\in \\textrm {\\right.\\Omega \\hspace{-6.544pt}\\Omega } =\\inf _{\\mathbf {Q}\\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega }}\\frac{\\widetilde{c}}{\\gamma }\\cdot \\left[ 1-A^{\\gamma /(\\gamma -1)} \\cdot \\left[ 1+ \\gamma \\cdot (A-1) +\\frac{\\gamma \\cdot \\left( \\gamma -1\\right) }{\\widetilde{c}}\\cdot D_{\\widetilde{c}\\cdot \\varphi _{\\gamma }}(\\mathbf {Q},{P})\\right]^{-1/\\left( \\gamma -1\\right) }\\right]$ as well as the BS minimizabilities/maximizabilites (cf.","Definition REF ) $& & \\hspace{-25.6073pt}\\inf _{\\mathbf {Q}\\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega }}D_{\\widetilde{c}\\cdot \\varphi _{\\gamma }}(\\mathbf {Q},{P})=\\lim _{n\\rightarrow \\infty }\\frac{\\widetilde{c}}{\\gamma \\cdot \\left(\\gamma -1\\right) }\\cdot \\left\\lbrace A^{\\gamma } \\cdot \\left( 1+\\frac{\\gamma }{\\widetilde{c}}\\cdot \\frac{1}{n}\\cdot \\log \\,\\mathbb {\\Pi }\\left[\\xi _{n}^{w\\mathbf {W}}\\in \\textrm {\\right.\\right.\\right.\\Omega \\hspace{-6.544pt}\\Omega } ^{1-\\gamma } + \\gamma \\cdot (1-A) -1 ,\\qquad \\ \\\\& & \\hspace{-25.6073pt}\\inf _{\\mathbf {Q}\\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega }}H_{\\gamma }(\\mathbf {Q},{P})=\\lim _{n\\rightarrow \\infty } A^{\\gamma } \\cdot \\left( 1+ \\gamma \\cdot \\frac{1}{n}\\cdot \\log \\,\\mathbb {\\Pi }\\left[\\breve{\\xi }_{n}^{w\\mathbf {W}}\\in \\textrm {\\right.\\right.\\Omega \\hspace{-6.544pt}\\Omega } ^{1-\\gamma } ,\\qquad \\ \\\\& & \\hspace{-25.6073pt}\\inf _{\\mathbf {Q}\\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega }}c_{1} \\cdot \\Big (H_{\\gamma }(\\mathbf {Q},{P})^{c_{2}} - c_{3} \\Big )=\\lim _{n\\rightarrow \\infty } c_{1} \\cdot \\left\\lbrace A^{c_{2} \\cdot \\gamma } \\cdot \\left( 1+\\frac{\\gamma }{n}\\cdot \\log \\,\\mathbb {\\Pi }\\left[\\breve{\\xi }_{n}^{w\\mathbf {W}}\\in \\textrm {\\right.\\right.\\right.\\Omega \\hspace{-6.544pt}\\Omega } ^{c_{2} \\cdot (1-\\gamma )} - c_{3} ,\\ \\ \\textrm {if c_{1} \\cdot c_{2} >0, c_{3} \\in \\mathbb {R}}, \\quad \\ \\\\& & \\hspace{-25.6073pt}\\sup _{\\mathbf {Q}\\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega }}c_{1} \\cdot \\Big (H_{\\gamma }(\\mathbf {Q},{P})^{c_{2}} - c_{3} \\Big )=\\lim _{n\\rightarrow \\infty } c_{1} \\cdot \\left\\lbrace A^{c_{2} \\cdot \\gamma } \\cdot \\left( 1+\\frac{\\gamma }{n}\\cdot \\log \\,\\mathbb {\\Pi }\\left[\\breve{\\xi }_{n}^{w\\mathbf {W}}\\in \\textrm {\\right.\\right.\\right.\\Omega \\hspace{-6.544pt}\\Omega } ^{c_{2} \\cdot (1-\\gamma )} - c_{3} ,\\ \\ \\textrm {if c_{1} \\cdot c_{2} < 0, c_{3} \\in \\mathbb {R}}, \\quad \\ \\\\& & \\hspace{-25.6073pt}\\inf _{\\mathbf {Q}\\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega }}R_{\\gamma }(\\mathbf {Q},{P})= \\lim _{n\\rightarrow \\infty } \\frac{1}{\\gamma \\cdot (\\gamma -1)} \\cdot \\log \\Big ( A^{\\gamma } \\cdot \\Big ( 1+ \\gamma \\cdot \\frac{1}{n}\\cdot \\log \\,\\mathbb {\\Pi }\\left[\\breve{\\xi }_{n}^{w\\mathbf {W}}\\in \\textrm {\\right.\\Omega \\hspace{-6.544pt}\\Omega } \\Big ) ^{1-\\gamma } \\Big ),\\\\& & \\hspace{-25.6073pt}\\inf _{\\mathbf {Q}\\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega }}c_{1} \\cdot \\Big (\\Big (\\sum _{k=1}^{K} q_{k}^{\\gamma }\\Big )^{c_{2}} - c_{3} \\Big )=\\lim _{n\\rightarrow \\infty } c_{1} \\cdot \\left\\lbrace K^{c_{2}\\cdot (1-\\gamma )} \\cdot A^{c_{2} \\cdot \\gamma } \\cdot \\left( 1+\\frac{\\gamma }{n}\\cdot \\log \\,\\mathbb {\\Pi }\\left[\\breve{\\breve{\\xi }}_{n}^{w\\mathbf {W}}\\in \\textrm {\\right.\\right.\\right.\\Omega \\hspace{-6.544pt}\\Omega } ^{c_{2} \\cdot (1-\\gamma )} - c_{3} ,\\ \\ \\nonumber \\\\& & \\hspace{369.88582pt}\\textrm {if c_{1} \\cdot c_{2} >0, c_{3} \\in \\mathbb {R}}, \\quad \\ $ $& & \\hspace{-25.6073pt}\\sup _{\\mathbf {Q}\\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega }}c_{1} \\cdot \\Big (\\Big (\\sum _{k=1}^{K} q_{k}^{\\gamma }\\Big )^{c_{2}} - c_{3} \\Big )=\\lim _{n\\rightarrow \\infty } c_{1} \\cdot \\left\\lbrace K^{c_{2}\\cdot (1-\\gamma )} \\cdot A^{c_{2} \\cdot \\gamma } \\cdot \\left( 1+\\frac{\\gamma }{n}\\cdot \\log \\,\\mathbb {\\Pi }\\left[\\breve{\\breve{\\xi }}_{n}^{w\\mathbf {W}}\\in \\textrm {\\right.\\right.\\right.\\Omega \\hspace{-6.544pt}\\Omega } ^{c_{2} \\cdot (1-\\gamma )} - c_{3} ,\\ \\ \\nonumber \\\\& & \\hspace{369.88582pt}\\textrm {if c_{1} \\cdot c_{2} < 0, c_{3} \\in \\mathbb {R}}, \\quad \\ \\\\& & \\hspace{-25.6073pt}\\inf _{\\mathbf {Q}\\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega }}\\frac{1}{1-\\gamma } \\cdot \\log \\Big (\\sum _{k=1}^{K} q_{k}^{\\gamma }\\Big )= \\lim _{n\\rightarrow \\infty } \\frac{1}{\\gamma \\cdot (\\gamma -1)} \\cdot \\left[\\log \\Big ( A^{\\gamma } \\cdot \\Big ( 1+ \\gamma \\cdot \\frac{1}{n}\\cdot \\log \\,\\mathbb {\\Pi }\\left[\\breve{\\breve{\\xi }}_{n}^{w\\mathbf {W}}\\in \\textrm {\\right.\\right.\\Omega \\hspace{-6.544pt}\\Omega } \\Big ) ^{1-\\gamma } \\Big ) + (1-\\gamma ) \\cdot \\log (K) ,$ where $\\xi _{n}^{w\\mathbf {W}}$ is the normalized randomly weighted empirical measure given in (REF ), $\\breve{\\xi }_{n}^{w\\mathbf {W}}$ is its special case for $\\widetilde{c}=1$ , and $\\breve{\\breve{\\xi }}_{n}^{w\\mathbf {W}}$ is its special case for $\\widetilde{c}=1$ together with ${P} = {P}^{unif}$ the latter two notations will be also used in the following Propositions REF to REF.","From this, the BS-minimizability/maximizability of the important norms/entropies/diversity indices (E1) to (E6) follow immediately as special cases.","(b) The special case $\\varphi := \\widetilde{c}\\cdot \\varphi _{\\gamma }$ ($\\gamma <0$ ) of Theorem REF works analogously to (a), with the differences that we employ (i) additionally a sequence $(X_{i})_{i\\in \\mathbb {N}}$ of random variables being independent of $(W_{i})_{i\\in \\mathbb {N}}$ and satisfying condition (REF ) (resp.","(REF )), (ii) $A=1$ (instead of arbitrary $A>0$ ), (iii) $\\mathbb {\\Pi }_{X_{1}^{n}}[\\cdot ]$ (instead of $\\mathbb {\\Pi }[\\cdot ]$ ), (iv) $\\xi _{n,\\mathbf {X}}^{w\\mathbf {W}}$ (instead of $\\xi _{n}^{w\\mathbf {W}}$ ), (v) $\\breve{\\xi }_{n,\\mathbf {X}}^{w\\mathbf {W}}$ (instead of $\\breve{\\xi }_{n}^{w\\mathbf {W}}$ ), and (vi) $\\breve{\\breve{\\xi }}_{n,\\mathbf {X}}^{w\\mathbf {W}}$ (instead of $\\breve{\\breve{\\xi }}_{n}^{w\\mathbf {W}}$ ).","The assertions of Proposition REF can be deduced from Theorem REF , Remark REF (vi), Lemma REF (a), (REF ), (REF ), (REF ), () and the below-mentioned $\\mathbb {}-$ concerning Example REF (d).","Employing Lemma REF (c) and Example REF (a) instead, one ends up with the following proposition on the reverse Kullback-Leibler divergence: Proposition 23 (a) Consider the context of Remark REF (vi) for $\\varphi := \\widetilde{c}\\cdot \\varphi _{\\gamma }$ with $\\gamma =0$ , and let ${P} \\in \\mathbb {S}_{>0}^{K}$ as well as $\\widetilde{c}>0$ be arbitrary but fixed.","Furthermore, let $W:=(W_{i})_{i\\in \\mathbb {N}}$ be an i.i.d.","sequence of non-negative real-valued random variables with Gamma distribution $\\mathbb {} =GAM(\\widetilde{c},\\widetilde{c})$ here and henceforth, we use the notation that a Gamma distribution $GAM(\\alpha ,\\beta )$ with rate parameter (inverse scale parameter) $\\alpha >0$ and shape parameter $\\beta >0$ has (Lebesgue-)density $f(y) := \\frac{\\alpha ^{\\beta } \\cdot y^{\\beta -1} \\cdot e^{-\\alpha \\cdot y} }{\\Gamma (\\beta )}\\cdot {1}_{]0,\\infty [}(y)$ , $y \\in \\mathbb {R}$ ; its cumulant generating function is $\\Lambda (z) = \\beta \\cdot \\log (\\frac{\\alpha }{\\alpha -z})$ for $z \\in ]-\\infty ,\\alpha [$ (and $\\Lambda (z) = \\infty $ for $z \\ge \\alpha $ ).","(where the subcase $\\widetilde{c}=1$ is the exponential distribution $\\mathbb {} =EXP(1)$ with mean 1).","Then for all $A >0$ and all $\\Omega $$\\Omega \\subset \\mathbb {S}_{>0}^{K}$ with (REF ) there holds the BS minimizabilites (cf.","(REF )) $& & \\inf _{{Q}\\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega }}D_{\\widetilde{c}\\cdot \\varphi _{0}}({Q},{P}) =-\\lim _{n\\rightarrow \\infty }\\frac{1}{n} \\log \\,\\mathbb {\\Pi }\\left[\\xi _{n}^{w\\mathbf {W}}\\in \\textrm {\\right.\\Omega \\hspace{-6.544pt}\\Omega } + \\widetilde{c} \\cdot (A-1- \\log A),\\\\& & \\inf _{{Q}\\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega }}\\widetilde{I}({Q},{P}) =\\inf _{{Q}\\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega }}\\sum \\displaylimits _{k=1}^{K} p_{k} \\cdot \\log \\left( \\frac{p_{k}}{q_{k}} \\right)= -\\lim _{n\\rightarrow \\infty }\\frac{1}{n} \\log \\,\\mathbb {\\Pi }\\left[\\xi _{n}^{w\\mathbf {W}}\\in \\textrm {\\right.\\Omega \\hspace{-6.544pt}\\Omega } - \\log A .\\nonumber $ (b) The special case $\\varphi := \\widetilde{c}\\cdot \\varphi _{\\gamma }$ ($\\gamma =0$ ) of Theorem REF works analogously to (a), with the differences that we employ (i) additionally a sequence $(X_{i})_{i\\in \\mathbb {N}}$ of random variables being independent of $(W_{i})_{i\\in \\mathbb {N}}$ and satisfying condition (REF ) (resp.","(REF )), (ii) $A=1$ (instead of arbitrary $A>0$ ), (iii) $\\mathbb {\\Pi }_{X_{1}^{n}}[\\cdot ]$ (instead of $\\mathbb {\\Pi }[\\cdot ]$ ), and (iv) $\\xi _{n,\\mathbf {X}}^{w\\mathbf {W}}$ (instead of $\\xi _{n}^{w\\mathbf {W}}$ ).","Proposition 24 (a) Consider the context of Remark REF (vi) for $\\varphi := \\widetilde{c}\\cdot \\varphi _{\\gamma }$ with $\\gamma \\in ]0,1[$ , and let ${P} \\in \\mathbb {S}_{>0}^{K}$ as well as $\\widetilde{c}>0$ be arbitrary but fixed.","Furthermore, let $W:=(W_{i})_{i\\in \\mathbb {N}}$ be an i.i.d.","sequence of non-negative real-valued random variables with Compound-Poisson-Gamma distribution $\\mathbb {} =C(POI(\\theta ),GAM(\\alpha ,\\beta ))$ having parameters $\\theta =\\frac{\\widetilde{c}}{\\gamma }>0$ , $\\alpha =\\frac{\\widetilde{c}}{1-\\gamma }>0$ , $\\beta =\\frac{\\gamma }{1-\\gamma }>0$ ; in other words, the $W_{i}$ are independent copies of a random variable $W_{1}:=\\sum _{j=1}^{N}\\overline{W}_{j}$ with the usual convention $\\sum _{i=1}^{0}\\overline{W}_{i}:=0$ constituted of some i.i.d.","sequence $(\\overline{W}_{j})_{j\\in \\mathbb {N}}$ of $Gamma(\\alpha ,\\beta )-$ distributed random variables and some independent $POI(\\theta )-$ distributed random variable $N$ .","Then for all $A >0$ and all $\\Omega $$\\Omega \\subset \\mathbb {S}^{K}$ with (REF ) there hold (REF ), (REF ), (), () as well as $& & \\hspace{-25.6073pt}\\sup _{\\mathbf {Q}\\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega }}H_{\\gamma }(\\mathbf {Q},{P})=\\lim _{n\\rightarrow \\infty } A^{\\gamma } \\cdot \\left( 1+ \\gamma \\cdot \\frac{1}{n}\\cdot \\log \\,\\mathbb {\\Pi }\\left[\\breve{\\xi }_{n}^{w\\mathbf {W}}\\in \\textrm {\\right.\\right.\\Omega \\hspace{-6.544pt}\\Omega } ^{1-\\gamma } ,\\qquad \\ \\\\& & \\hspace{-25.6073pt}\\sup _{\\mathbf {Q}\\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega }}c_{1} \\cdot \\Big (H_{\\gamma }(\\mathbf {Q},{P})^{c_{2}} - c_{3} \\Big )=\\lim _{n\\rightarrow \\infty } c_{1} \\cdot \\left\\lbrace A^{c_{2} \\cdot \\gamma } \\cdot \\left( 1+\\frac{\\gamma }{n}\\cdot \\log \\,\\mathbb {\\Pi }\\left[\\breve{\\xi }_{n}^{w\\mathbf {W}}\\in \\textrm {\\right.\\right.\\right.\\Omega \\hspace{-6.544pt}\\Omega } ^{c_{2} \\cdot (1-\\gamma )} - c_{3} ,\\ \\ \\textrm {if c_{1} \\cdot c_{2} >0, c_{3} \\in \\mathbb {R}}, \\quad \\ \\\\& & \\hspace{-25.6073pt}\\inf _{\\mathbf {Q}\\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega }}c_{1} \\cdot \\Big (H_{\\gamma }(\\mathbf {Q},{P})^{c_{2}} - c_{3} \\Big )=\\lim _{n\\rightarrow \\infty } c_{1} \\cdot \\left\\lbrace A^{c_{2} \\cdot \\gamma } \\cdot \\left( 1+\\frac{\\gamma }{n}\\cdot \\log \\,\\mathbb {\\Pi }\\left[\\breve{\\xi }_{n}^{w\\mathbf {W}}\\in \\textrm {\\right.\\right.\\right.\\Omega \\hspace{-6.544pt}\\Omega } ^{c_{2} \\cdot (1-\\gamma )} - c_{3} ,\\ \\ \\textrm {if c_{1} \\cdot c_{2} < 0, c_{3} \\in \\mathbb {R}}, \\quad \\ \\\\& & \\hspace{-25.6073pt}\\sup _{\\mathbf {Q}\\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega }}c_{1} \\cdot \\Big (\\Big (\\sum _{k=1}^{K} q_{k}^{\\gamma }\\Big )^{c_{2}} - c_{3} \\Big )=\\lim _{n\\rightarrow \\infty } c_{1} \\cdot \\left\\lbrace K^{c_{2}\\cdot (1-\\gamma )} \\cdot A^{c_{2} \\cdot \\gamma } \\cdot \\left( 1+\\frac{\\gamma }{n}\\cdot \\log \\,\\mathbb {\\Pi }\\left[\\breve{\\breve{\\xi }}_{n}^{w\\mathbf {W}}\\in \\textrm {\\right.\\right.\\right.\\Omega \\hspace{-6.544pt}\\Omega } ^{c_{2} \\cdot (1-\\gamma )} - c_{3} ,\\ \\ \\nonumber \\\\& & \\hspace{375.57628pt}\\textrm {if c_{1} \\cdot c_{2} >0, c_{3} \\in \\mathbb {R}}, \\quad \\ $ $& & \\hspace{-25.6073pt}\\inf _{\\mathbf {Q}\\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega }}c_{1} \\cdot \\Big (\\Big (\\sum _{k=1}^{K} q_{k}^{\\gamma }\\Big )^{c_{2}} - c_{3} \\Big )=\\lim _{n\\rightarrow \\infty } c_{1} \\cdot \\left\\lbrace K^{c_{2}\\cdot (1-\\gamma )} \\cdot A^{c_{2} \\cdot \\gamma } \\cdot \\left( 1+\\frac{\\gamma }{n}\\cdot \\log \\,\\mathbb {\\Pi }\\left[\\breve{\\breve{\\xi }}_{n}^{w\\mathbf {W}}\\in \\textrm {\\right.\\right.\\right.\\Omega \\hspace{-6.544pt}\\Omega } ^{c_{2} \\cdot (1-\\gamma )} - c_{3} ,\\ \\ \\nonumber \\\\& & \\hspace{375.57628pt}\\textrm {if c_{1} \\cdot c_{2} < 0, c_{3} \\in \\mathbb {R}} .","\\quad \\ $ From this, the BS-minimizability/maximizability of the important norms/entropies/diversity indices (E1) to (E6) follows immediately as special cases.","(b) The special case $\\varphi := \\widetilde{c}\\cdot \\varphi _{\\gamma }$ ($\\gamma \\in ]0,1[$ ) of Theorem REF works analogously to (a), with the differences that we employ (i) additionally a sequence $(X_{i})_{i\\in \\mathbb {N}}$ of random variables being independent of $(W_{i})_{i\\in \\mathbb {N}}$ and satisfying condition (REF ) (resp.","(REF )), (ii) $A=1$ (instead of arbitrary $A>0$ ), (iii) $\\mathbb {\\Pi }_{X_{1}^{n}}[\\cdot ]$ (instead of $\\mathbb {\\Pi }[\\cdot ]$ ), (iv) $\\xi _{n,\\mathbf {X}}^{w\\mathbf {W}}$ (instead of $\\xi _{n}^{w\\mathbf {W}}$ ), (v) $\\breve{\\xi }_{n,\\mathbf {X}}^{w\\mathbf {W}}$ (instead of $\\breve{\\xi }_{n}^{w\\mathbf {W}}$ ), and (vi) $\\breve{\\breve{\\xi }}_{n,\\mathbf {X}}^{w\\mathbf {W}}$ (instead of $\\breve{\\breve{\\xi }}_{n}^{w\\mathbf {W}}$ ).","This follows from Theorem REF , Remark REF (vi), Lemma REF (a), (REF ), (REF ), (REF ), () and the below-mentioned $\\mathbb {}-$ concerning Example REF (b).","Employing Lemma REF (b) and Example REF (a) instead, one ends up with the following proposition on the Kullback-Leibler divergence: Proposition 25 (a) Consider the context of Remark REF (vi) for $\\varphi := \\widetilde{c}\\cdot \\varphi _{\\gamma }$ with $\\gamma =1$ , and let ${P} \\in \\mathbb {S}_{>0}^{K}$ as well as $\\widetilde{c}>0$ be arbitrary but fixed.","Furthermore, let $W:=(W_{i})_{i\\in \\mathbb {N}}$ be an i.i.d.","sequence of non-negative real-valued random variables with distribution $\\mathbb {} =\\frac{1}{\\widetilde{c}}\\cdot POI(\\widetilde{c})$ being the “$\\frac{1}{\\widetilde{c}}-$ fold Poisson distribution with mean $\\widetilde{c}$ ” , which means that $W_{1}=\\frac{1}{\\widetilde{c}}\\cdot Z$ for a Poissonian $POI(\\widetilde{c})-$ distributed random variable $Z$ with mean $\\widetilde{c}$ (where the subcase $\\widetilde{c}=1$ amounts to $\\mathbb {} =POI(1)$ ).","Then for all $A >0$ and all $\\Omega $$\\Omega \\subset \\mathbb {S}^{K}$ with (REF ) there holds $\\hspace{-19.91684pt}-\\lim _{n\\rightarrow \\infty }\\frac{1}{n} \\log \\,\\mathbb {\\Pi }\\left[\\xi _{n}^{w\\mathbf {W}}\\in \\textrm {\\right.\\Omega \\hspace{-6.544pt}\\Omega } &=& \\inf _{{Q}\\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega } }\\ \\widetilde{c}\\cdot \\left[ 1-A \\cdot \\exp \\left( -\\frac{1}{A \\cdot \\widetilde{c}}\\cdot D_{\\widetilde{c}\\cdot \\varphi _{1}}(\\mathbf {Q},{P}) + \\frac{1}{A} -1 \\right) \\right]$ and the BS minimizabilities/maximizabilites (cf.","Definition REF ) $& & \\hspace{-34.14322pt}\\inf _{\\mathbf {Q}\\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega } }D_{\\widetilde{c}\\cdot \\varphi _{1}}(\\mathbf {Q},{P})=\\lim _{n\\rightarrow \\infty } \\widetilde{c}\\cdot \\left\\lbrace 1 - A \\cdot \\left[ 1 + \\log \\left( \\frac{1}{A} \\cdot \\Big (1+\\frac{1}{\\widetilde{c}}\\cdot \\frac{1}{n}\\cdot \\log \\,\\mathbb {\\Pi }\\left[\\xi _{n}^{w\\mathbf {W}}\\in \\textrm {\\right.\\right.\\right.\\right.\\Omega \\hspace{-6.544pt}\\Omega } \\Big ) ,\\\\& & \\hspace{-34.14322pt}\\inf _{\\mathbf {Q}\\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega } } \\,I(\\mathbf {Q},{P})=\\inf _{\\mathbf {Q}\\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega } } \\,\\sum \\displaylimits _{k=1}^{K} q_{k} \\cdot \\log \\left( \\frac{q_{k}}{p_{k}} \\right)= - \\lim _{n\\rightarrow \\infty } A \\cdot \\log \\left( \\frac{1}{A} \\cdot \\Big (1+\\frac{1}{n}\\cdot \\log \\,\\mathbb {\\Pi }\\left[\\breve{\\xi }_{n}^{w\\mathbf {W}}\\in \\textrm {\\right.\\right.\\Omega \\hspace{-6.544pt}\\Omega } \\Big ) ,\\nonumber \\\\& & \\hspace{-34.14322pt}\\max _{\\mathbf {Q}\\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega } } \\,\\mathcal {E}^{Sh}(\\mathbf {Q})= \\max _{\\mathbf {Q}\\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega } } \\,(-1) \\cdot \\sum _{k=1}^{K} q_{k} \\cdot \\log (q_{k})= \\lim _{n\\rightarrow \\infty }\\frac{\\log K}{K} + A \\cdot \\log \\left( \\frac{1}{A} \\cdot \\Big (1+\\frac{1}{n}\\cdot \\log \\,\\mathbb {\\Pi }\\left[\\breve{\\breve{\\xi }}_{n}^{w\\mathbf {W}}\\in \\textrm {\\right.\\right.\\Omega \\hspace{-6.544pt}\\Omega } \\Big ), \\\\& & \\hspace{-34.14322pt}\\max _{\\mathbf {Q}\\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega } } \\,\\mathcal {E}^{gSM2}(\\mathbf {Q})= \\max _{\\mathbf {Q}\\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega } } \\,\\frac{1}{1-s} \\cdot \\exp \\Big \\lbrace (s-1) \\cdot \\sum _{k=1}^{K} q_{k} \\cdot \\log (q_{k}) -1\\Big \\rbrace \\nonumber \\\\& & \\hspace{-34.14322pt}= \\lim _{n\\rightarrow \\infty }\\frac{1}{1-s} \\cdot \\exp \\left\\lbrace (1-s) \\cdot \\left[\\frac{\\log K}{K} + A \\cdot \\log \\left( \\frac{1}{A} \\cdot \\Big (1+\\frac{1}{n}\\cdot \\log \\,\\mathbb {\\Pi }\\left[\\breve{\\breve{\\xi }}_{n}^{w\\mathbf {W}}\\in \\textrm {\\right.\\right.\\right.\\right.\\Omega \\hspace{-6.544pt}\\Omega } \\Big ) -1 ,\\ \\ s \\in \\, ]0,1[ \\, \\cup \\, ]1,\\infty [ .$ The special subcase $A=1$ in () (and thus, $\\mathbf {Q}$ is a probability vector) corresponds to the maximum entropy problem for the Shannon entropy $\\mathcal {E}^{Sh}(\\cdot )$ .","This can hence be tackled by our bare-simulation approach for almost arbitrary sets $\\textrm {$$\\hspace{-6.544pt}$$}$ of probability vectors.","(b) The special case $\\varphi := \\widetilde{c}\\cdot \\varphi _{\\gamma }$ ($\\gamma =1$ ) of Theorem REF works analogously to (a), with the differences that we employ (i) additionally a sequence $(X_{i})_{i\\in \\mathbb {N}}$ of random variables being independent of $(W_{i})_{i\\in \\mathbb {N}}$ and satisfying condition (REF ) (resp.","(REF )), (ii) $A=1$ (instead of arbitrary $A>0$ ), (iii) $\\mathbb {\\Pi }_{X_{1}^{n}}[\\cdot ]$ (instead of $\\mathbb {\\Pi }[\\cdot ]$ ), (iv) $\\xi _{n,\\mathbf {X}}^{w\\mathbf {W}}$ (instead of $\\xi _{n}^{w\\mathbf {W}}$ ), (v) $\\breve{\\xi }_{n,\\mathbf {X}}^{w\\mathbf {W}}$ (instead of $\\breve{\\xi }_{n}^{w\\mathbf {W}}$ ), and (vi) $\\breve{\\breve{\\xi }}_{n,\\mathbf {X}}^{w\\mathbf {W}}$ (instead of $\\breve{\\breve{\\xi }}_{n}^{w\\mathbf {W}}$ ).","For the sake of completeness, let us mention here that we do not deal with the case $\\gamma \\in ]1,2[$ , for which we conjecture that $\\mathbb {}$ becomes a signed finite measure with total mass 1, i.e.","it has a density (with respect to some dominating measure) with positive and negative values which “integrates to 1” ; accordingly, our BS method can not be applied to this situation.","To proceed with further $\\gamma -$ cases, a combination of Theorem REF respectively Remark REF (vi), Lemma REF (a), (REF ), (REF ), (REF ), () and the below-mentioned $\\mathbb {}-$ concerning Example REF (c) leads to the following Proposition 26 (a) Consider the context of Remark REF (vi) for $\\varphi := \\widetilde{c}\\cdot \\varphi _{\\gamma }$ with $\\gamma =2$ , and let ${P} \\in \\mathbb {S}_{>0}^{K}$ as well as $\\widetilde{c}>0$ be arbitrary but fixed.","Furthermore, let $W:=(W_{i})_{i\\in \\mathbb {N}}$ be an i.i.d.","sequence of real-valued random variables with probability distribution $\\mathbb {} =NOR(1,\\frac{1}{\\widetilde{c}})$ being the Normal (Gaussian) law with mean 1 and variance $\\frac{1}{\\widetilde{c}}$ .","Then for all $A >0$ and $\\Omega $$\\Omega \\subset \\mathbb {S}^{K}$ with (REF ) there hold all the BS-extremizabilites (REF ) to () as well as (REF ) (below) with plugging-in $\\gamma =2$ .","From this, the BS-minimizability/maximizability of the important norms/entropies/diversity indices (E1) to (E6) follow immediately as special cases.","By Remark REF (c), one can even take $A<0$ in (REF ) to () and (REF ) as well as in (E1), (E2), (E4) and (E6).","(b) The special case $\\varphi := \\widetilde{c}\\cdot \\varphi _{\\gamma }$ ($\\gamma =2$ ) of Theorem REF works analogously to (a), with the differences that we employ (i) additionally a sequence $(X_{i})_{i\\in \\mathbb {N}}$ of random variables being independent of $(W_{i})_{i\\in \\mathbb {N}}$ and satisfying condition (REF ) (resp.","(REF )), (ii) $A=1$ (instead of arbitrary $A>0$ ), (iii) $\\mathbb {\\Pi }_{X_{1}^{n}}[\\cdot ]$ (instead of $\\mathbb {\\Pi }[\\cdot ]$ ), (iv) $\\xi _{n,\\mathbf {X}}^{w\\mathbf {W}}$ (instead of $\\xi _{n}^{w\\mathbf {W}}$ ), (v) $\\breve{\\xi }_{n,\\mathbf {X}}^{w\\mathbf {W}}$ (instead of $\\breve{\\xi }_{n}^{w\\mathbf {W}}$ ), and (vi) $\\breve{\\breve{\\xi }}_{n,\\mathbf {X}}^{w\\mathbf {W}}$ (instead of $\\breve{\\breve{\\xi }}_{n}^{w\\mathbf {W}}$ ).","For instance, the BS minimizability () of Proposition REF (a) can be employed to solve the following discrete Monge-Kantorovich-type optimal mass transportation problem (optimal coupling problem) with side (i.e.","additional) constraints: given two nonnegative-entries vectors $\\mu :=(\\mu _{1}, \\ldots \\mu _{K_{1}} ) \\in [0,\\infty [^{K_{1}}$ and $\\nu := (\\nu _{1}, \\ldots \\nu _{K_{2}} ) \\in [0,\\infty [^{K_{2}}$ with equal total “mass” $\\sum _{k=1}^{K_{1}} \\mu _{k} =\\sum _{k=1}^{K_{2}} \\nu _{k} = A >0$ , compute $&& \\inf _{K_{1} \\times K_{2}-\\textrm {matrices} \\ \\pi }K_{1} \\cdot K_{2} \\cdot \\sum _{u=1}^{K_{1}} \\sum _{v=1}^{K_{2}}\\left(\\pi _{u,v} - \\frac{1}{K_{1} \\cdot K_{2}} \\right)^{2}\\\\&&\\textrm {subject to}\\nonumber \\\\&& \\sum _{v=1}^{K_{2}} \\pi _{u,v} = \\mu _{u} \\quad \\textrm {for all } u \\in \\lbrace 1,\\ldots ,K_{1}\\rbrace ,\\\\&& \\sum _{u=1}^{K_{1}} \\pi _{u,v} = \\nu _{v} \\quad \\textrm {for all } v \\in \\lbrace 1,\\ldots ,K_{2}\\rbrace ,\\\\&& \\pi _{u,v} \\in [0,A] \\quad \\textrm {for all } u \\in \\lbrace 1,\\ldots ,K_{1}\\rbrace , \\, v \\in \\lbrace 1,\\ldots ,K_{2}\\rbrace ,\\\\&& \\textrm {side constraints on \\pi , \\mu , \\nu }.$ Indeed, this problem can be equivalently rewritten in terms $K_{1} \\cdot K_{2}-$ dimensional vectors as follows: given two nonnegative-entries vectors $\\mu $ ,$\\nu $ as above, compute $&& \\inf _{\\mathbf {Q}\\in \\mathbf {\\Omega }}K_{1} \\cdot K_{2} \\cdot \\sum _{k=1}^{K_{1} \\cdot K_{2}}\\left(q_{k} - \\frac{1}{K_{1} \\cdot K_{2}} \\right)^{2}\\ = \\ \\inf _{\\mathbf {Q}\\in \\mathbf {\\Omega }}K_{1} \\cdot K_{2} \\cdot \\sum _{k=1}^{K_{1} \\cdot K_{2}}q_{k}^{2} + 1 - 2A \\\\&& \\textrm {where\\mathbf {\\Omega } \\subset \\mathbb {R}^{K_{1}\\cdot K_{2}} is the set of allvectors \\mathbf {Q} = (q_{1}, \\ldots , q_{K_{1} \\cdot K_{2}})which satisfy the constraints}\\nonumber \\\\&& \\sum _{j=1}^{K_{2}} q_{(i-1)\\cdot K_{2} +j} \\ = \\ \\mu _{i} \\quad \\textrm {for all } i \\in \\lbrace 1,\\ldots ,K_{1}\\rbrace , \\\\&& \\sum _{i=1}^{K_{1}} q_{(i-1)\\cdot K_{2} +j} \\ = \\ \\nu _{j} \\quad \\textrm {for all } j \\in \\lbrace 1,\\ldots ,K_{2}\\rbrace , \\\\&& q_{k} \\in [0,A] \\quad \\textrm {for all } k \\in \\lbrace 1,\\ldots ,K_{1} \\cdot K_{2} \\rbrace , \\\\&& \\textrm {side constraints on \\mathbf {Q}, \\mu , \\nu }.$ Clearly, via divisions by $A$ , one can equivalently rewrite $\\mathbf {\\Omega }= A \\cdot \\textrm {$$\\hspace{-6.544pt}$$}$ for some $\\textrm {$$\\hspace{-6.544pt}$$} \\subset \\mathbb {S}^{K_{1} \\cdot K_{2}}$ in the $K_{1}\\cdot K_{2}-$ dimensional probability simplex.","Hence, we can employ () with $c_{1}=K_{1} \\cdot K_{2}$ , $c_{2}=1$ and $c_{3} = 1-2A$ , provided that the side constraints () are such that $\\mathbf {\\Omega }$ satisfies the regularity property (REF ) and the finiteness property (REF ).","Notice that (REF ) is equal to $\\inf _{\\mathbf {Q}\\in \\mathbf {\\Omega }}D_{2 \\cdot \\varphi _{2}}(\\mathbf {Q},{P}^{unif})\\nonumber $ where ${P}^{unif} := (\\frac{1}{K_{1} \\cdot K_{2}},\\ldots , \\frac{1}{K_{1} \\cdot K_{2}})$ is the probability vector of frequencies of the uniform distribution on $\\lbrace 1, \\ldots , K_{1} \\cdot K_{2}\\rbrace $ , and $\\widetilde{c}=2$ .","The special case $A=1$ with side constraint () of the form $K_{1} \\cdot \\min _{i \\in \\lbrace 1,\\ldots , K_{1}\\rbrace } \\mu _{i} +K_{2} \\cdot \\min _{j \\in \\lbrace 1,\\ldots , K_{2}\\rbrace } \\nu _{j} \\ge 1$ was explicitly solved by e.g.","Bertrand et al.","[43], [44], who also give applications to cryptographic guessing problems (spy problems), task partitioning and graph clustering.","The importance of the case $\\gamma =2$ stems also from the fact that one can equivalently rewrite separable quadratic minimization problems as minimization problems of Pearson chi-square divergences.","Indeed, by straightforward calculations one can derive that $\\inf _{\\mathbf {\\breve{Q}}\\in \\mathbf {\\breve{\\Omega }}} \\,\\sum _{k=1}^{K} ( \\, c_{1,k} + c_{2,k} \\cdot \\breve{q}_{k} + c_{3,k} \\cdot \\breve{q}_{k}^2 \\, ) \\, ,\\qquad c_{1,k} \\in \\mathbb {R}, \\ c_{2,k} \\in \\mathbb {R}\\backslash \\lbrace 0\\rbrace , \\ c_{3,k} \\in \\, ]0,\\infty [,$ is equal to (recall that $\\varphi _{2}(t) := \\frac{(t - 1)^2}{2}$ , cf.","(REF )) $c_{4} + \\inf _{\\mathbf {Q}\\in \\mathbf {\\Omega }}D_{2 \\cdot \\varphi _{2}}(\\mathbf {Q},\\mathbf {P}) \\, ,$ where $\\mathbf {Q} := (q_{1}, \\ldots , q_{K})$ with $q_{k} := - c_{2,k} \\cdot \\breve{q}_{k}$ , $\\mathbf {P} := (p_{1}, \\ldots , p_{K})$ with $p_{k}:= \\frac{c_{2,k}^{2}}{2 \\cdot c_{3,k}} >0$ , $c_{4} := \\sum _{k=1}^{K} \\big ( \\, c_{1,k} - \\frac{c_{2,k}^{2}}{4 \\cdot c_{3,k}} \\, \\big )$ , and $\\mathbf {\\Omega }$ is the corresponding reformulation of the constraint set $\\mathbf {\\breve{\\Omega }}$ .","To achieve the applicability of our BS method, we further transform (REF ) into its equal form (cf.","(REF )) $c_{4} + \\inf _{\\widetilde{\\mathbf {Q}}\\in \\mathbf {\\Omega }/M_{P}}D_{2 M_{\\mathbf {P}} \\cdot \\varphi _{2}}(\\mathbf {Q},\\widetilde{{P}}) \\,$ with $M_{\\mathbf {P}}:=\\sum _{k=1}^{K} p_{k}>0$ and $\\widetilde{{P}}:=\\mathbf {P}/M_{\\mathbf {P}}$ .","If $\\mathbf {\\Omega }/M_{\\mathbf {P}}$ satisfies (REF ) and (REF ) (e.g.","it may be highly disconnected), then we can apply Theorem REF .","In contrast, if $\\mathbf {\\Omega }/M_{\\mathbf {P}} = A \\cdot \\textrm {$$\\hspace{-6.544pt}$$}$ for some $A \\in \\mathbb {R}\\backslash \\lbrace 0\\rbrace $ and some $\\Omega $$\\Omega \\subset \\mathbb {S}_{>0}^{K}$ satisfying (REF ), then we can apply Proposition REF (a) together with Remark REF (c); for instance, this may appear if $\\mathbf {\\breve{\\Omega }}$ contains (amongst others) the original constraint $\\sum _{k=1}^{K} \\breve{q}_{k} = C$ for some constant $C>0$ , and $c_{2,k} =c_{2}$ does not depend on $k$ , which leads to the choice $A = - \\frac{c_{2} \\cdot C}{M_{\\mathbf {P}}}$ .","Notice that $A<0$ if $c_{2}>0$ .","For example, optimization problems (REF ) with $c_{1,k} >0$ , $c_{2,k} >0$ , $c_{3,k} >0$ and constraints $\\sum _{k=1}^{K} \\breve{q}_{k} = C$ , $\\breve{q}_{k} \\in [\\underline{\\breve{q}}_{k},\\overline{\\breve{q}}_{k}]$ appear in distributed energy management as economic dispatch problems in smart grids of power generators, where $\\breve{q}_{k}$ is the active power generation of the $k-$ th generator, $C$ is the total power demand, $\\underline{\\breve{q}}_{k}$ resp.","$\\overline{\\breve{q}}_{k}$ represent the lower resp.","upper bound of the $k-$ th generator's output, and the cost of power generation is $c_{1,k} + c_{2,k} \\cdot \\breve{q}_{k} + c_{3,k} \\cdot \\breve{q}_{k}^2$ (cf.","e.g.","Yang et al.","[412], Loia & Vaccaro [230], Wood et al.","[394], Xu et al.","[404]).","Another important special case of (REF ) to (REF ) is the omnipresent $L_{2}-$ minimization; indeed, with the choices $c_{3,k}=1$ , $c_{2,k}= - 2 v_{k}$ , and $c_{1,k}= v_{k}^2$ for some $\\mathbf {V} =(v_{1},\\ldots ,v_{K})$ , the minimization problem (REF ) is nothing but $\\inf _{\\mathbf {\\breve{Q}}\\in \\mathbf {\\breve{\\Omega }}} \\, || \\mathbf {\\breve{Q}} - \\mathbf {V} ||_{2}^{2}$ ; if $\\mathbf {\\breve{\\Omega }}$ depends on a pregiven $L-$ dimensional vector $\\mathbf {x}$ (with $L < K$ ), this can be regarded as a non-parametric regression problem in a wide sense.","To continue with our general investigations, by combining Theorem REF respectively Remark REF (vi), Lemma REF (a), (REF ), (REF ), (REF ), () and the below-mentioned $\\mathbb {}-$ concerning Example REF (e), we arrive at the following Proposition 27 (a) Consider the context of Remark REF (vi) for $\\varphi := \\widetilde{c}\\cdot \\varphi _{\\gamma }$ with $\\gamma >2$ , and let ${P} \\in \\mathbb {S}_{>0}^{K}$ as well as $\\widetilde{c}>0$ be arbitrary but fixed.","Furthermore, let $W:=(W_{i})_{i\\in \\mathbb {N}}$ be an i.i.d.","sequence of real-valued random variables having densityin the classical sense, with respect to Lebesgue measure $\\frac{\\exp \\lbrace \\frac{y \\cdot \\widetilde{c}}{\\gamma -1}\\rbrace }{\\exp \\lbrace \\widetilde{c}/\\gamma \\rbrace }\\cdot f_{Z}(- y), \\qquad y\\in ]-\\infty ,\\infty [,$ where $f_{Z}$ is the density of a random variable $Z$ which has stable law with parameter-quadruple $(\\frac{\\gamma }{\\gamma -1},1,0,\\frac{\\widetilde{c}^{1/(1-\\gamma )} \\cdot (\\gamma -1)^{\\gamma /(\\gamma -1)}}{\\gamma })$ in terms of the above-mentioned “form-B notation” in Zolotarev [428].","Then for all $A >0$ and $\\Omega $$\\Omega \\subset \\mathbb {S}^{K}$ with (REF ) there hold all the BS-extremizabilites (REF ) to (REF ) as well as $& & \\hspace{-42.67912pt}\\sup _{\\mathbf {Q}\\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega }}\\frac{1}{1-\\gamma } \\cdot \\log \\Big (\\sum _{k=1}^{K} q_{k}^{\\gamma }\\Big )= \\lim _{n\\rightarrow \\infty } \\frac{1}{\\gamma \\cdot (\\gamma -1)} \\cdot \\left[\\log \\Big ( A^{\\gamma } \\cdot \\Big ( 1+ \\gamma \\cdot \\frac{1}{n}\\cdot \\log \\,\\mathbb {\\Pi }\\left[\\breve{\\breve{\\xi }}_{n}^{w\\mathbf {W}}\\in \\textrm {\\right.\\right.\\Omega \\hspace{-6.544pt}\\Omega } \\Big ) ^{1-\\gamma } \\Big ) + (1-\\gamma ) \\cdot \\log (K) .$ From this, the BS-minimizability/maximizability of the important norms/entropies/diversity indices (E1) to (E6) follow immediately as special cases.","(b) The special case $\\varphi := \\widetilde{c}\\cdot \\varphi _{\\gamma }$ ($\\gamma >2$ ) of Theorem REF works analogously to (a), with the differences that we employ (i) additionally a sequence $(X_{i})_{i\\in \\mathbb {N}}$ of random variables being independent of $(W_{i})_{i\\in \\mathbb {N}}$ and satisfying condition (REF ) (resp.","(REF )), (ii) $A=1$ (instead of arbitrary $A>0$ ), (iii) $\\mathbb {\\Pi }_{X_{1}^{n}}[\\cdot ]$ (instead of $\\mathbb {\\Pi }[\\cdot ]$ ), (iv) $\\xi _{n,\\mathbf {X}}^{w\\mathbf {W}}$ (instead of $\\xi _{n}^{w\\mathbf {W}}$ ), (v) $\\breve{\\xi }_{n,\\mathbf {X}}^{w\\mathbf {W}}$ (instead of $\\breve{\\xi }_{n}^{w\\mathbf {W}}$ ), and (vi) $\\breve{\\breve{\\xi }}_{n,\\mathbf {X}}^{w\\mathbf {W}}$ (instead of $\\breve{\\breve{\\xi }}_{n}^{w\\mathbf {W}}$ ).","As mentioned above, in the Propositions REF to REF we have combined Theorem REF respectively Remark REF (vi), Lemma REF and explicitly solved representations (REF ).","The latter, important step will be discussed in a structured, comprehensive manner in the Section below.","By retransformation, we can even deal with optimizations of nonnegative linear objective functions with constraint sets on Euclidean $\\gamma $ -norm spheres.","Indeed, for nonnegative $\\breve{\\mathbf {Q}} := (\\breve{q}_{1}, \\ldots , \\breve{q}_{K})$ and $\\breve{\\mathbf {P}} := (\\breve{p}_{1}, \\ldots , \\breve{p}_{K})$ one can rewrite their scalar product as $\\gamma -$ order Hellinger integrals $& & \\hspace{-36.98866pt}\\sum _{k=1}^{K} \\breve{q}_{k} \\cdot \\breve{p}_{k}\\ = \\ c_{1} \\cdot \\sum _{k=1}^{K} q_{k}^{\\gamma } \\cdot p_{k}^{1-\\gamma }\\ = \\ c_{1} \\cdot H_{\\gamma }(\\mathbf {Q},{P}) \\, \\hspace{14.22636pt} \\textrm {where}\\\\& & \\hspace{-39.83368pt}\\textrm {\\gamma \\in \\, ]0,1[ \\, \\cup \\, [2,\\infty [ \\, if \\, \\breve{\\mathbf {Q}} \\in [0,\\infty [^{K},\\breve{\\mathbf {P}} \\in \\, ]0,\\infty [^{K}\\hspace{11.38092pt} respectively \\hspace{11.38092pt}\\gamma \\in \\, ]-\\infty ,0[ \\, if \\, \\breve{\\mathbf {Q}} \\in \\, ]0,\\infty [^{K},\\breve{\\mathbf {P}} \\in \\, ]0,\\infty [^{K},}\\\\& & \\hspace{-36.98866pt}q_{k} \\ := \\ \\breve{q}_{k}^{1/\\gamma } ,\\quad p_{k} \\ := \\ \\frac{\\breve{p}_{k}^{1/(1-\\gamma )}}{\\sum _{i=1}^{K} \\breve{p}_{i}^{1/(1-\\gamma )}} ,\\quad c_{1}:= \\Big ( \\sum _{i=1}^{K} \\breve{p}_{i}^{1/(1-\\gamma )} \\Big )^{1-\\gamma } = :|| \\breve{\\mathbf {P}} ||_{1-\\gamma } \\ .$ The required constraint $\\sum _{k=1}^{K} q_{k} = A >0$ retransforms to $|| \\breve{\\mathbf {Q}} ||_{\\gamma }=A^{1/\\gamma }$ and thus, $\\breve{\\mathbf {Q}}$ must lie on (the positive/nonnegative part of) the $\\gamma -$ norm-sphere $\\partial B_{\\gamma }(0,A^{1/\\gamma })$ around the origin with radius $A^{1/\\gamma }$ .","Accordingly, for $\\gamma \\in \\, [2,\\infty [$ we have $\\inf _{\\breve{\\mathbf {Q}} \\in \\breve{\\mathbf {\\Omega }}} \\,\\sum _{k=1}^{K} \\breve{q}_{k} \\cdot \\breve{p}_{k}\\ = \\ c_{1} \\cdot \\inf _{\\mathbf {Q}\\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega } }H_{\\gamma }(\\mathbf {Q},{P})$ and we can apply () of Proposition REF (a) respectively Proposition REF (a) here and analogously henceforth, by this we mean the condition () as it appears in the Proposition REF (a) respectively Proposition REF (a), as long as the original constraint set $\\breve{\\mathbf {\\Omega }} \\in \\partial B_{\\gamma }(0,A^{1/\\gamma }) \\, \\cap \\, [0,\\infty [^{K}$ transforms (via $q_{k} = \\breve{q}_{k}^{1/\\gamma }$ ) into a constraint set $A \\cdot \\textrm {$$\\hspace{-6.544pt}$$}$ which satisfies the regularity assumption (REF ) in the relative topology (as a side remark, notice that $int(\\partial B_{\\gamma }(0,A^{1/\\gamma })) = \\emptyset $ in the full topology).","For the case $\\gamma \\in \\, ]-\\infty ,0[$ we also have (REF ) and apply () of Proposition REF (a) for any original constraint set $\\breve{\\mathbf {\\Omega }} \\in \\partial B_{\\gamma }(0,A^{1/\\gamma }) \\, \\cap \\, ]0,\\infty [^{K}$ which transforms into $A \\cdot \\textrm {$$\\hspace{-6.544pt}$$}$ satisfying (REF ) in the relative topology.","In contrast, for the case $\\gamma \\in \\, ]0,1[$ we get $\\sup _{\\breve{\\mathbf {Q}} \\in \\breve{\\mathbf {\\Omega }}} \\,\\sum _{k=1}^{K} \\breve{q}_{k} \\cdot \\breve{p}_{k}\\ = \\ c_{1} \\cdot \\sup _{\\mathbf {Q}\\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega } }H_{\\gamma }(\\mathbf {Q},{P})\\nonumber $ and apply (REF ) of Proposition REF (a) for any original constraint set $\\breve{\\mathbf {\\Omega }} \\in \\partial B_{\\gamma }(0,A^{1/\\gamma }) \\, \\cap \\, [0,\\infty [^{K}$ which transforms into $A \\cdot \\textrm {$$\\hspace{-6.544pt}$$}$ satisfying (REF ) in the relative topology.","As a continuation of Remark REF , we can principally tackle all the optimization problems of this Subsection REF by basically only employing a fast and accurate — pseudo, true, natural, quantum — random number generator, provided that the constraint set $\\textrm {$ A $\\hspace{-6.544pt}$$}$ satisfies the mild assumptions (REF ) (in the relative topology) and (REF ).","Recall that $A>0$ (and for $\\varphi _{2}$ even $A \\in \\mathbb {R}\\backslash \\lbrace 0\\rbrace $ ) and that $\\mathbf {Q} \\in \\textrm {$ A $\\hspace{-6.544pt}$$}$ implies in particular the constraint $\\sum _{k=1}^{K} q_{k} = A$ .","The regularity assumption (REF ) allows for e.g.","high-dimensional constraint sets $\\textrm {$ A $\\hspace{-6.544pt}$$}$ which are non-convex and even highly disconnected, and for which other minimization methods (e.g.","pure enumeration, gradient or steepest descent methods, etc.)","may be problematic or intractable.","For example, (REF ) covers kind of “$K-$ dimensional (not necessarily regular) polka dot pattern type” relaxations $\\textrm {$ A $\\hspace{-6.544pt}$$} :=\\dot{\\bigcup }_{i=1}^{N} \\mathcal {U}_{i}(Q_{i}^{dis})$ of finite discrete constraint sets $\\textrm {$ A $\\hspace{-6.544pt}$ dis$}:= \\lbrace Q_{1}^{dis}, \\ldots , Q_{N}^{dis}\\rbrace $ of high cardinality $N$ (e.g.","being exponential or factorial in a large $K$ ), where each $K-$ dimensional vector $Q_{i}^{dis}$ has total-sum-of-components equal to $A$ and is surrounded by some small (“flat”, i.e.","in the relative topology) neighborhood $\\mathcal {U}_{i}(Q_{i}^{dis})$ .","For the sake of brevity, in the following discussion we confine ourselves to the deterministic setup (e.g.","Proposition REF (a) rather than (b)) which particularly involves $\\mathbb {\\Pi }[\\cdot ]$ (rather than $\\mathbb {\\Pi }_{X_{1}^{n}}[\\cdot ]$ ) and $\\xi _{n}^{w\\mathbf {W}}$ (rather than $\\xi _{n,\\mathbf {X}}^{w\\mathbf {W}}$ ).","In such a context, all the optimization problems of this Subsection REF , subsumed as (cf.","(REF ) to (REF )) $\\inf _{\\mathbf {Q}\\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega } } \\,\\Phi (\\mathbf {Q})\\qquad \\textrm {respectively} \\qquad \\sup _{\\mathbf {Q}\\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega } } \\,\\Phi (\\mathbf {Q})\\nonumber $ can be regarded as a “BS-tractable” relaxations of the corresponding nonlinear discrete (e.g.","integer, combinatorial) programming problems $\\inf _{\\mathbf {Q}\\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega ^{dis}} } \\,\\Phi (\\mathbf {Q})\\qquad \\textrm {respectively} \\qquad \\sup _{\\mathbf {Q}\\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega ^{dis}} } \\,\\Phi (\\mathbf {Q}) \\, ;\\nonumber $ as examples take e.g.","$\\Phi (\\mathbf {Q}) =c_{1} \\cdot \\Big ( \\Big (\\sum _{k=1}^{K} q_{k}^{\\gamma }\\Big )^{c_{2}}- c_{3} \\Big )$ (with $\\gamma \\ne 0,1$ ) or $\\Phi (\\mathbf {Q}) = \\Phi _{{P}}(\\mathbf {Q}) =D_{\\widetilde{c}\\cdot \\varphi _{\\gamma }}(\\mathbf {Q},{P})$ .","For instance, $A \\cdot \\textrm {$$\\hspace{-6.544pt}$ dis$}$ may contain only $K-$ dimensional vectors $Q_{i}^{dis}$ ($i=1,\\ldots ,N$ ) whose components stem from a finite set $\\mathcal {B}$ of nonnegative integers and add up to $A$ .","If $\\mathcal {B} = \\lbrace 0,1\\rbrace $ , then we can even deal with nonnegative linear objective functions $\\Phi (\\mathbf {Q}) = \\sum _{k=1}^{K} \\breve{p}_{k} \\cdot q_{k}$ where $\\mathbf {Q} := (q_{1}, \\ldots , q_{K})$ with $q_{k} \\in \\lbrace 0,1\\rbrace $ and $\\breve{\\mathbf {P}} := (\\breve{p}_{1}, \\ldots , \\breve{p}_{K})$ has components $\\breve{p}_{k} >0$ which reflect e.g.","the cost associated with the $k-$ th state.","Indeed, by noticing that $q_{k}^{1/\\gamma }=q_{k}$ for $\\gamma \\in \\, ]0,1[ \\, \\cup \\, [2,\\infty [$ , we can employ (REF ) and () to end up with $& & \\hspace{-31.2982pt}\\inf _{\\mathbf {Q}\\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega ^{dis}} } \\,\\sum _{k=1}^{K} q_{k} \\cdot \\breve{p}_{k}= || \\breve{\\mathbf {P}} ||_{1-\\gamma } \\cdot \\inf _{\\mathbf {Q}\\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega ^{dis}} } \\,\\sum _{k=1}^{K} q_{k}^{\\gamma } \\cdot \\left(\\frac{\\breve{p}_{k}^{1/(1-\\gamma )}}{\\sum _{i=1}^{K} \\breve{p}_{i}^{1/(1-\\gamma )}}\\right)^{1-\\gamma }= || \\breve{\\mathbf {P}} ||_{1-\\gamma } \\cdot \\inf _{\\mathbf {Q}\\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega ^{dis}} }H_{\\gamma }(\\mathbf {Q},{P}), \\ \\ \\gamma \\in \\, [2,\\infty [, \\ \\ \\ \\\\& & \\hspace{-31.2982pt}\\sup _{\\mathbf {Q}\\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega ^{dis}} } \\,\\sum _{k=1}^{K} q_{k} \\cdot \\breve{p}_{k}= || \\breve{\\mathbf {P}} ||_{1-\\gamma } \\cdot \\sup _{\\mathbf {Q}\\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega ^{dis}} }H_{\\gamma }(\\mathbf {Q},{P}), \\ \\ \\gamma \\in \\, ]0,1[ \\, .$ The corresponding relaxations are $& & \\hspace{-48.36958pt}\\inf _{\\mathbf {Q}\\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega } } \\,\\sum _{k=1}^{K} q_{k} \\cdot \\breve{p}_{k}\\ = \\ || \\breve{\\mathbf {P}} ||_{1-\\gamma } \\cdot \\inf _{\\mathbf {Q}\\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega } }H_{\\gamma }(\\mathbf {Q},{P}), \\ \\ \\gamma \\in \\, [2,\\infty [ \\, ,\\\\& & \\hspace{-48.36958pt}\\sup _{\\mathbf {Q}\\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega } } \\,\\sum _{k=1}^{K} q_{k} \\cdot \\breve{p}_{k}\\ = \\ || \\breve{\\mathbf {P}} ||_{1-\\gamma } \\cdot \\sup _{\\mathbf {Q}\\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega } }H_{\\gamma }(\\mathbf {Q},{P}), \\ \\ \\gamma \\in \\, ]0,1[ \\, ;$ for (REF ) we can apply () of Proposition REF (a) respectively Proposition REF (a), whereas for () we apply (REF ) of Proposition REF (a) — as long as the relaxation constraint set $A \\cdot \\textrm {$$\\hspace{-6.544pt}$$}$ satisfies (REF ) in the relative topology.","For the sake of illustration, let us consider a sum-minimization-type linear assignment problem with side constraints (for a comprehensive book on assignment problems see e.g.","Burkard et al.","[69]).","Suppose that there are $K$ individuals (people, machines, etc.)","to carry out $K$ tasks (jobs, etc.).","Each individual is assigned to carry out exactly one task.","There is cost $c_{ij} >0$ if individual $i$ is assigned to (i.e., carries out) task $j$ .","We want to find the minimum total cost amongst all assignments.","There may be side constraints, e.g.","each assignment has a value $v_{ij} >0$ and the total value of the assignment should be above a pregiven threshold.","As usual, the problem can be formulated with the help of binary variables $x_{ij}$ where $x_{ij} =1$ if individual $i$ is assigned to task $j$ , and $x_{ij} =0$ otherwise.","Accordingly, we want to compute $&& \\inf _{K \\times K-\\textrm {matrices} \\ x=(x_{ij})} \\ \\sum _{i=1}^{K} \\sum _{j=1}^{K}c_{ij} \\cdot x_{ij}\\\\&&\\textrm {subject to}\\nonumber \\\\&& \\sum _{j=1}^{K} x_{ij} = 1 \\quad \\textrm {for all } i \\in \\lbrace 1,\\ldots ,K\\rbrace ,\\qquad \\textrm {(i.e.","each individual i does one task),}\\\\&& \\sum _{i=1}^{K} x_{ij} = 1 \\quad \\textrm {for all } j\\in \\lbrace 1,\\ldots ,K\\rbrace ,\\qquad \\textrm {(i.e.","each task j is done by one individual),}\\\\&& x_{ij} \\in \\lbrace 0,1\\rbrace \\quad \\textrm {for all } i \\in \\lbrace 1,\\ldots ,K\\rbrace , \\, j \\in \\lbrace 1,\\ldots ,K\\rbrace ,\\\\&& \\textrm {side (i.e.","additional) constraints on x=(x_{ij})_{i,j=1,\\ldots ,K}}.$ Analogously to (REF ), this problem can be equivalently rewritten in terms of $K^{2}-$ dimensional vectors as follows: let $\\mathbf {Q} := (q_{1}, \\ldots , q_{K^{2}})$ and $\\breve{\\mathbf {P}} := (\\breve{p}_{1}, \\ldots , \\breve{p}_{K^{2}})$ be such that $c_{ij}= \\breve{p}_{(i-1)\\cdot K +j}$ and $x_{ij}= q_{(i-1)\\cdot K +j}$ for $i,j \\in \\lbrace 1,\\ldots ,K\\rbrace $ and compute $&& \\inf _{\\mathbf {Q}\\in K \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega ^{dis}} } \\,\\sum _{k=1}^{K} q_{k} \\cdot \\breve{p}_{k} \\\\&& \\textrm {whereK \\cdot \\textrm {\\Omega \\hspace{-6.544pt}\\Omega ^{dis}}\\subset \\mathbb {R}^{K^{2}} is the set of allvectors \\mathbf {Q} = (q_{1}, \\ldots , q_{K^{2}}) which satisfy the constraints}\\nonumber \\\\&& \\sum _{j=1}^{K} q_{(i-1)\\cdot K +j} \\ = \\ 1 \\quad \\textrm {for all } i \\in \\lbrace 1,\\ldots ,K\\rbrace ,\\\\&& \\sum _{i=1}^{K} q_{(i-1)\\cdot K +j} \\ = \\ 1 \\quad \\textrm {for all } j \\in \\lbrace 1,\\ldots ,K\\rbrace ,\\\\&& q_{k} \\in \\lbrace 0,1\\rbrace \\quad \\textrm {for all } k \\in \\lbrace 1,\\ldots ,K^{2} \\rbrace ,\\\\&& \\textrm {side constraints on \\mathbf {Q}}.$ As seen above, this can be rewritten as $\\gamma -$ order Hellinger-integral minimization problem (REF ), with $\\gamma \\ge 2$ .","We can obtain a highly disconnected “non-void-interior-type” relaxation of the binary integer programming problem (REF ) to () by replacing () with $q_{k} \\in [0,\\varepsilon _{1}] \\, \\cup \\, [1-\\varepsilon _{2},1]\\quad \\textrm {for all } k \\in \\lbrace 1,\\ldots ,K^{2} \\rbrace ,$ for some (possibly arbitrarily) small $\\varepsilon _{1}, \\varepsilon _{2} >0$ with $\\varepsilon _{1} + \\varepsilon _{2} < 1$ .","We denote by $K \\cdot \\textrm {$$\\hspace{-6.544pt}$$}$ the outcoming set manifested by the constraints (), (), () and (REF ), and accordingly we end up with a minimization problem of type (REF ), which we can tackle by () of Proposition REF (a) respectively Proposition REF (a), as long as (REF ) (in the relative topology) is satisfied.","For instance, we can take $\\gamma =2$ and basically solve the corresponding optimization problem by basically simulating $K^{2}-$ dimensional Gaussian random variables (even though the cardinality of $K \\cdot \\textrm {$$\\hspace{-6.544pt}$ dis$}$ may be high).","As a side remark, let us mention that our relaxation (REF ) contrasts considerably to the frequently used continuous linear programming (LP) relaxation $q_{k} \\in [0,1]\\quad \\textrm {for all } k \\in \\lbrace 1,\\ldots ,K^{2} \\rbrace .\\nonumber $ Let us finally mention that an important special case of a minimization problem (REF ) to () is — the integer programming formulation of — the omnipresent (asymmetric) traveling salesman problem (TSP) with possible side constraints see e.g.","Applegate et al.","[14], Gutin & Punnen [147], Cook [92] for comprehensive books on TSP, its variations and its applications to logistics, machine scheduling, printed circuit board drilling, communication-network frequencing, genome sequencing, data clustering, and many others..","There, one has $K$ cities and the cost of traveling from city $i$ to city $j\\ne i$ is given by $c_{ij}>0$ .","Moreover, one sets $x_{ij} =1$ if the traveler goes directly from city $i$ to city $j$ (in that order), and $x_{ij} =0$ otherwise.","For technical reasons, for $i=j$ we attribute a cost $c_{ii}>0$ (e.g.","hotel costs), but we require that always $x_{ii}=0$ which we subsume as the first part of the constraints ().","Then, the constraint () means that the traveler leaves from city $i$ exactly once, whereas () reflects that the traveler arrives at city $j$ exactly once.","The goal is to find a directed tour — i.e.","a directed cycle/circuit that visits all $K$ cities once — of minimum cost.","Within this context, the second part of the constraints () should basically exclude solutions which consist of disconnected subtours (subtour elimination constraints (of e.g.","the seminal Dantzig et al.","[104]), connectivity constraints, cut-set constraints).","Here, we also allow for additional/side constraints which we subsume as the third part () of the constraints.","Hence, by the above-mentioned considerations we can principally tackle such kind of TSP problems with our BS method.","For sum-maximization-type linear assignment problems with side constraints, where e.g.","$c_{ij}$ is a profit (rather than a cost) and the ultimate goal is total profit maximization, we can proceed analogously, by employing () and () (instead of (REF ) and (REF )).","Let us end this subsection with a comparison: suppose that we have a (sufficiently large) number $n$ of concrete data observations $X_{i} = x_{i}$ ($i=1,\\ldots ,n$ ) from the unknown probability distribution ${P}$ (in vector form), and from these we want to approximate/estimate the unknown distance $\\inf _{{Q}\\in \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega }}D_{\\varphi _{\\gamma }}( {Q}, {P} )$ from a family of probability models (in vector form) $\\Omega $$\\Omega $ (e.g.","for model-adequacy evaluations, for goodness-of-fit testing purposes): by the above-mentioned Propositions REF to REF (and especially, by (REF ), (REF ), (REF )) one can use $G\\Big (-\\frac{1}{n} \\cdot \\log \\,\\mathbb {\\Pi }_{x_{1}^{n}}\\left[\\xi _{n,\\mathbf {x}}^{w\\mathbf {W}}\\in \\textrm {\\right.\\Omega \\hspace{-6.544pt}\\Omega } \\Big )$ where $\\mathbb {\\Pi }_{x_{1}^{n}}[\\, \\cdot \\, ] := \\mathbb {\\Pi }[ \\, \\cdot \\, | \\,X_{1}=x_{1}, \\ldots , X_{n} = x_{n}]$ , $\\mathbf {x} := (x_{1},\\ldots ,x_{n})$ , and $G$ (cf.","(REF )) is e.g.","chosen as follows: $G(z) := -\\frac{\\widetilde{c}}{\\gamma \\cdot \\left( \\gamma -1\\right) }\\cdot \\left\\lbrace 1-\\left( 1-\\frac{\\gamma }{\\widetilde{c}} \\cdot z \\right) ^{1-\\gamma } \\right\\rbrace $ for the three cases $\\gamma <0$ , $\\gamma \\in ]0,1[$ and $\\gamma \\ge 2$ , $G(z) := z$ for $\\gamma =0$ (reversed Kullback-Leibler divergence), and $G(z):= - \\widetilde{c} \\cdot \\log (1- \\frac{1}{\\widetilde{c}} \\cdot z)$ for $\\gamma =1$ (Kullback-Leibler divergence).","Notice that (REF ) contrasts to the alternative approximation (of $\\inf _{{Q}\\in \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega }}D_{\\varphi _{\\gamma }}( {Q}, {P} )$ ) given by $\\inf _{{Q}\\in \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega }}D_{\\varphi _{\\gamma }}({Q}, \\mathbb {P}_{n}^{emp, co} )$ which is used in the context of “classical” statistical minimum distance estimation (MDE) with power divergences; in (REF ), we have employed $\\mathbb {P}_{n}^{emp, co} = \\frac{1}{n} \\cdot \\sum _{i=1}^{n} \\delta _{x_{i}}$ to be the realization of the empirical distribution $\\mathbb {P}_{n}^{emp} = \\frac{1}{n} \\cdot \\sum _{i=1}^{n} \\delta _{X_{i}}$ .","Indeed, especially in complicated high-dimensional non-parametric or semi-parametric big-data contexts, we have substituted a quite difficult optimization problem (REF ) by a much easier solvable counting problem (REF ).","The same holds analogously for Renyi distances/divergences, etc."],["Construction principle for bounds of the minimum divergence\nin the general case ","Turning back to Theorem REF , we now consider the general case when the divergence $\\varphi \\in \\Upsilon (]a,b[)$ is not of the power type (REF ).","Recall from (REF ) the crucial terms (with ${P} \\in \\mathbb {S}_{>0}$ ) $\\inf _{m\\ne 0} D_{\\varphi }(m\\cdot \\textrm {\\Omega \\hspace{-6.544pt}\\Omega },{P}):= \\inf _{m\\ne 0}\\ \\inf _{{Q}\\in \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega } }D_{\\varphi }(m\\cdot {Q},{P})= \\inf _{{Q}\\in \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega } }\\ \\inf _{m\\ne 0}D_{\\varphi }(m\\cdot {Q},{P}) \\, < \\, \\infty $ for all sets $\\Omega $$\\Omega $ satisfying the regularity properties (REF ) and the convenient, more restrictive finiteness property $\\inf _{{Q}\\in \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega }}\\ \\inf _{k=1,\\ldots ,K}\\frac{q_{k}}{p_{k}}\\in dom(\\varphi ),\\quad \\sup _{{Q}\\in \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega }}\\ \\sup _{k=1,\\ldots ,K}\\frac{q_{k}}{p_{k}}\\in dom(\\varphi )$ which implies (REF ); notice that $\\inf _{k=1,\\ldots ,K}\\frac{q_{k}}{p_{k}} \\le 1$ , $\\sup _{k=1,\\ldots ,K}\\frac{q_{k}}{p_{k}} \\ge 1$ with equalities if and only if ${Q} = {P}$ .","Since $\\textrm {$$\\hspace{-6.544pt}$$} \\ne \\lbrace {P} \\rbrace $ (cf.","the right-hand side of (REF )), the double infimum (supremum) in (REF ) is strictly smaller (larger) than 1.","In general, the inner minimization $\\inf _{m\\ne 0} D_{\\varphi }(m\\cdot {Q},{P})$ in (REF ) can not be performed in explicit closed form, but e.g.","in the specific case of power divergences (cf.","(REF ), (REF )) the optimization $\\inf _{m\\ne 0} D_{\\widetilde{c} \\cdot \\varphi _{\\gamma }}(m\\cdot {Q},{P})$ produces an explicit form, which in turn leads to a simple one-to-one correspondence between $D_{\\widetilde{c} \\cdot \\varphi _{\\gamma }}(\\textrm {$$\\hspace{-6.544pt}$$},{P})$ and $\\inf _{m\\ne 0}\\ D_{\\widetilde{c} \\cdot \\varphi _{\\gamma }}(m\\cdot \\textrm {$$\\hspace{-6.544pt}$$},{P})$ (cf.","Lemma REF ).","Under (REF ) and (REF ) it clearly holds $\\inf _{m\\ne 0} D_{\\varphi }(m\\cdot \\textrm {\\Omega \\hspace{-6.544pt}\\Omega },{P})\\ \\le \\ D_{\\varphi }(\\textrm {\\Omega \\hspace{-6.544pt}\\Omega },{P})\\ \\le \\ D_{\\varphi }({Q},{P}) .$ For transparency, we first investigate the (widely useable) subsetup where $dom(\\varphi )= ]0,\\infty [$ (and thus, $int(dom(\\varphi )) = \\, ]a,b[ \\, = \\, ]0,\\infty [$ ) and $\\textrm {$$\\hspace{-6.544pt}$$} \\subset \\mathbb {S}_{> 0}^{K}$ .","Let us start with the lower bound $\\inf _{m\\ne 0} D_{\\varphi }(m\\cdot \\textrm {$$\\hspace{-6.544pt}$$},{P})$ .","It can be proved that the minimizer in $m$ is a well defined constant, which belongs to a compact set in $\\mathbb {R}_{> 0}$ .","To see this, let us first observe that, obviously, from (REF ) one can obtain $\\Big [\\ \\inf _{\\mathbf {Q}\\in \\mathbf {\\Omega }}\\ \\inf _{k=1,\\ldots ,K}\\frac{m \\cdot q_{k}}{p_{k}}\\in dom(\\varphi ),\\quad \\sup _{\\mathbf {Q}\\in \\mathbf {\\Omega }}\\ \\sup _{k=1,\\ldots ,K}\\frac{m \\cdot q_{k}}{p_{k}}\\in dom(\\varphi ) \\Big ]\\ \\Longleftrightarrow \\ m \\in ]0,\\infty [ .$ Moreover, for any fixed ${Q}$ in $\\Omega $$\\Omega $ there is a unique number $m=m({Q}) > 0$ which satisfies the first-order optimality condition for $m \\in \\ ]0,\\infty [$ $\\psi _{{Q}}(m):=\\frac{d}{dm} D_{\\varphi }\\left( m \\cdot {Q},{P}\\right)=\\sum _{k=1}^{K}q_{k} \\cdot \\varphi ^{\\prime }\\left( \\frac{m \\cdot q_{k}}{p_{k}}\\right) =0$ and thus $D_{\\varphi }\\left( m({Q}) \\cdot {Q},{P}\\right) =\\inf _{m \\ne 0} D_{\\varphi }\\left( m \\cdot {Q},{P}\\right) ;$ indeed, the mapping $ ]0,\\infty [ \\ \\ni m \\rightarrow D_{\\varphi }\\left( m \\cdot {Q},{P}\\right) $ is strictly convex and infinitely differentiable (which follows straightforwardly from (G5),(G6) in the below-mentioned Section together with (C7ii), (C7iii) in Appendix D), and the strictly increasing function $\\psi _{{Q}}$ is such that $\\psi _{{Q}}(m)$ is strictly negative for all $m \\in ]0,1[$ for which $\\sup _{k=1,\\ldots ,K}\\frac{m \\cdot q_{k}}{p_{k}} <1$ whereas $\\psi _{{Q}}(m)$ is strictly positive for all $m > 1$ for which $\\inf _{k=1,\\ldots ,K}\\frac{m \\cdot q_{k}}{p_{k}} >1$ (recall the note right after (REF ) and $\\varphi ^{\\prime }(1)=0$ ).","Hence, for any ${Q} \\in \\textrm {$$\\hspace{-6.544pt}$$}$ the unique zero $m({Q})$ of (REF ) (and hence, unique minimizer in (REF )) is in the compact interval $\\Big [\\frac{1}{\\sup _{k=1,\\ldots ,K}\\frac{q_{k}}{p_{k}}}, \\, \\frac{1}{\\inf _{k=1,\\ldots ,K}\\frac{q_{k}}{p_{k}}}\\Big ]\\ \\subseteq \\ \\Big [\\frac{1}{\\sup _{{Q}\\in \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega }}\\ \\sup _{k=1,\\ldots ,K}\\frac{q_{k}}{p_{k}}}, \\,\\frac{1}{\\inf _{{Q}\\in \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega }}\\ \\inf _{k=1,\\ldots ,K}\\frac{q_{k}}{p_{k}}}\\Big ]\\ \\subset \\ \\Big ]\\frac{1}{b}, \\frac{1}{a}\\Big [ \\ = \\ ]0,\\infty [ .$ When $\\textrm {$$\\hspace{-6.544pt}$$}$ is closed in $\\mathbb {S}^{K}$ , then by continuity of the function ${Q} \\mapsto D_{\\varphi }\\left( m\\left( {Q}\\right)\\cdot {Q} ,{P}\\right) $ there exists a ${Q}^{\\ast }$ in $\\textrm {$$\\hspace{-6.544pt}$$}$ which achieves the infimum on $\\textrm {$$\\hspace{-6.544pt}$$}$ .","When $\\textrm {$$\\hspace{-6.544pt}$$}$ is not closed but satisfies (7), then the infimum exists anyway, possibly on the boundary $\\partial \\textrm {$$\\hspace{-6.544pt}$$}$ .","Anyhow, for such ${Q}^{\\ast }$ there holds $D_{\\varphi }\\left( m({Q}^{\\ast }) \\cdot {Q}^{\\ast },{P}\\right)\\le D_{\\varphi }\\left(\\textrm {\\right.\\Omega \\hspace{-6.544pt}\\Omega } ,{P}\\le D_{\\varphi }\\left( {Q}^{\\ast },{P}\\right) ,$ where we use the continuity of ${Q}\\mapsto D_{\\varphi }\\left( {Q},{P}\\right) $ and (REF ) to obtain the last inequality above, even when ${Q}^{\\ast }\\in \\partial \\textrm {$$\\hspace{-6.544pt}$$}$ and ${Q}^{\\ast }\\notin \\textrm {$$\\hspace{-6.544pt}$$}$ .","That (REF ) provides sharp bounds can be seen through the case of power divergences.","Indeed, for the latter one basically gets (cf.","Appendix C) $m({Q}) = (1 + \\frac{\\gamma \\cdot (\\gamma -1)}{\\widetilde{c}} \\cdot D_{\\widetilde{c} \\cdot \\varphi _{\\gamma }}({Q}, {P}) \\, )^{1/(1-\\gamma )}$ and $D_{\\varphi _{\\gamma }}\\left( m({Q}) \\cdot {Q},{P}\\right) =\\frac{\\widetilde{c}}{\\gamma } (1- m({Q}))$ for the case $\\gamma \\in \\mathbb {R}\\backslash \\lbrace 0,1\\rbrace $ , respectively, $m({Q}) = \\exp (-\\frac{1}{\\widetilde{c}} \\cdot D_{\\widetilde{c}\\cdot \\varphi _{1}}({Q}, {P}))$ and $D_{\\widetilde{c} \\cdot \\varphi _{1}}\\left( m({Q}) \\cdot {Q},{P}\\right) =\\frac{\\widetilde{c}}{\\gamma } (1- m({Q}))$ for the case $\\gamma =1$ , respectively, $m({Q}) = 1$ and $D_{\\widetilde{c} \\cdot \\varphi _{0}}\\left( m({Q}) \\cdot {Q},{P}\\right) =D_{\\widetilde{c} \\cdot \\varphi _{0}}\\left({Q},{P}\\right)$ for the remaining case $\\gamma =0$ .","In all cases, $D_{\\varphi _{\\gamma }}\\left( m({Q}) \\cdot {Q},{P}\\right)$ is an increasing function of $D_{\\varphi _{\\gamma }}\\left({Q},{P}\\right)$ and therefore, ${Q}^{\\ast }\\in \\arg \\inf _{{Q}\\in \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega } } D_{\\varphi }\\left(m({Q}) \\cdot {Q},{P}\\right)$ also satisfies ${Q}^{\\ast }\\in \\arg \\inf _{{Q}\\in \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega }}D_{\\varphi }\\left( {Q},{P}\\right)$ .","Hence, the right-hand side and the left-hand side of (REF ) coincide.","Now due to (REF ), the LHS of (REF ) can be estimated since by Theorem REF for each ${P}\\in \\mathbb {S}_{>0}^{K}$ the divergence $\\inf _{m\\ne 0}D_{\\varphi }\\left( m \\cdot {Q},{P}\\right)$ is BS-minimizable on sets $\\textrm {$$\\hspace{-6.544pt}$$} \\subset \\mathbb {S}^{K}$ .","We shall propose in Section an algorithm to handle the estimation of the RHS of (REF ), whenever ${P}$ is known (as in Remark 13(v)) or when ${P}$ is approximated by the empirical distribution of the data set $\\left(X_{1},..,X_{n}\\right)$ .","Also note that (REF ) holds also for $\\textrm {$$\\hspace{-6.544pt}$$}$ substituted by $A \\cdot \\textrm {$$\\hspace{-6.544pt}$$}$ for any $A\\ne 0$ .","Other cases of interest include when $dom\\left( \\varphi \\right) $ is not $\\left] 0,\\infty \\right[$ .","We list two cases which extend the above discussion.","Firstly, consider $\\varphi $ with $dom\\left( \\varphi \\right) =\\left[ 0,\\infty \\right[$ .","Then — since $\\varphi ^{\\prime }(0)=-\\infty $ in order that (REF ) should hold (see (G10ii) in Section below) — we may extend (REF ) to cases when $\\textrm {$$\\hspace{-6.544pt}$$} \\subset \\mathbb {S}^{K}$ instead of $\\textrm {$$\\hspace{-6.544pt}$$} \\subset \\mathbb {S}_{>0}^{K}$ , hence allowing for possible null entries in $\\textrm {$$\\hspace{-6.544pt}$$}$ .","When $dom\\left( \\varphi \\right) =\\left] a,b\\right[ $ for some $a<0$ , then clearly the same argument leading to (REF ) holds; this case is of interest, for instance, when extending a statistical model to signed measures (see e.g.","Broniatowski et al.","[63] for the important task of testing the number of components in a parametric probability mixture model).","Example 28 Consider the (non-probability version of the) Jensen-Shannon divergence defined by $J(\\mathbf {Q},\\mathbf {P}):=I(\\mathbf {Q},(\\mathbf {Q}+\\mathbf {P})/2)+I(\\mathbf {P},(\\mathbf {Q}+\\mathbf {P})/2),\\qquad \\mathbf {P}, \\mathbf {Q} \\in \\mathbb {R}_{\\ge 0}^{K},$ where $I(\\mathbf {Q},\\mathbf {P})$ denotes the modified Kullback-Leibler information between $\\mathbf {Q}$ and $\\mathbf {P}$ (cf.","(REF ) with $\\mathbf {P}$ instead of ${P}$ ).","In (REF ) and (REF ) of Example REF below, we shall show that $J(\\mathbf {Q},\\mathbf {P}) = D_{\\varphi _{snKL}}(\\mathbf {Q},\\mathbf {P})$ with (basically) divergence generator $\\varphi _{snKL}(t) := \\, t \\cdot \\log t + (t+1) \\cdot \\log \\Big ( \\frac{2}{t+1} \\Big )$ for $t > 0$ .","It is known that $J^{2}$ is a metric.","We explore the sharpness of the bounds for $J(\\textrm {$$\\hspace{-6.544pt}$$} ,{P})$ as defined in (REF ).","For this, we consider a given probability distribution ${P}$ on $\\mathcal {Y}$ with strictly positive entries; the set $\\textrm {$$\\hspace{-6.544pt}$$}$ consists of all probability distributions ${Q}$ on $\\mathcal {Y}$ whose total variation distance $V({Q},{P}):=\\sum _{k=1}^{K}\\left|q_{k}-p_{k}\\right|$ notice that $V({Q},{P})$ always takes values in the interval $[0,2[$ to ${P}$ lies between $v$ and $v+h$ for $v>0$ and small $h$ and which also satisfies $\\sup \\left( \\sup _{k=1,\\ldots ,K}\\frac{p_{k}}{q_{k}},\\sup _{k=1,\\ldots ,K}\\frac{q_{k}}{p_{k}}\\right)\\le L$ for some strictly positive finite $L$ .","This set $\\textrm {$$\\hspace{-6.544pt}$$}$ defines a class of distributions ${Q}$ away from ${P}$ still keeping some regularity w.r.t.","${P}$ .","Also, $\\textrm {$$\\hspace{-6.544pt}$$}$ satisfies (7).","We will prove that the bounds in (120) are sharp in this case.","Notice that $J(m \\cdot {Q},{P}) = \\infty $ for $m<0$ and hence $\\inf _{{Q}\\in \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega }}\\inf _{m \\ne 0}J(m \\cdot {Q},{P}) =\\inf _{{Q}\\in \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega }}\\inf _{m>0}J(m \\cdot {Q},{P})$ .","We first provide a lower bound for the latter.","It holds for all $m>0$ and ${Q}$ in $\\textrm {$$\\hspace{-6.544pt}$$}$ $& & J(m{Q},{P}) =(m+1) \\cdot \\log \\left( 2 \\right)- (m+1) \\cdot \\log \\left( m+1\\right)+m\\log m+I^{\\alpha }({P},{Q})+m \\cdot I^{1-\\alpha }({Q},{P})\\nonumber $ where $\\alpha :=1/\\left( m+1\\right) $ and $I^{\\alpha }({P},{Q})$ is the $\\alpha -$ skewed Kullback-Leibler divergence between ${P}$ and ${Q}$ defined through $I^{\\alpha }({P},{Q}):=I({P},\\alpha {P}+(1-\\alpha ){Q}).$ By Inequality (27) in Yamano [410] $I^{\\alpha }({P},{Q})\\ge -\\log \\left( 1-\\frac{\\alpha ^{2}}{4} V({Q},{P})^{2}\\right).$ Since $(m+1) \\cdot \\log \\left(2\\right) -(m+1) \\cdot \\log (m+1)+ m \\cdot \\log m$ is non-negative for all $m>0$ and takes its minimal value 0 for $m=1$ , we obtain $\\inf _{m>0} J(m {Q},{P})\\ge \\inf _{m>0}K(m)$ where $K(m):=-\\log \\left( 1-\\frac{1}{4\\left( m+1\\right) ^{2}} \\cdot V({Q},{P})^{2}\\right) -m \\cdot \\log \\left( 1-\\frac{m^{2}}{4\\left( m+1\\right)^{2}}\\cdot V({Q},{P})^{2}\\right) .$ Since $-\\log (1-x)\\ge x$ for all $x<1$ and both $\\frac{1}{4\\left(m+1\\right) ^{2}} \\cdot V({Q},{P})^{2}$ and $\\frac{m^{2}}{4\\left( m+1\\right)^{2}}\\cdot V({Q},{P})^{2}$ are less than 1, it follows that $K(m)\\ge \\frac{V({Q},{P})^{2}}{4} \\cdot \\frac{m^{3} +1}{\\left(m+1\\right)^{2}}$ where the right-hand side attains its minimal value on $]0,\\infty [$ at $m^{+}= \\sqrt{3}-1 \\approx 0.73$ .","Hence, we obtain $\\inf _{m>0}J(m\\cdot Q,P)\\ge \\frac{V({Q},{P})^{2}}{4}\\cdot (2\\sqrt{3}-3) > 0.116 \\, v^2$ Now by (19) in Yamano [410], for any ${Q}$ $J({Q},{P})\\le \\frac{1}{4}\\underline{J}({Q},{P})$ where $\\underline{J}({Q},{P}):=I({Q},{P})+I({P},{Q})$ is the Jensen divergence (also called symmetrized Kullback-Leibler divergence) between ${Q}$ and ${P}$ .","Since (see Dragomir [116]) $I({P},{Q})\\le \\sum _{k=1}^{K}\\sqrt{\\frac{p_{k}}{q_{k}}}\\cdot \\left|q_{k}-q_{k}\\right|,$ it follows that $J({Q}^{\\ast },{P})\\le \\frac{1}{2}\\sqrt{L} \\cdot V({Q}^{\\ast },{P})$ which provides $0.116 \\, v^2\\ \\le \\ \\inf _{m >0}J(m \\cdot {Q}^{\\ast },{P})=J(m({Q}^{\\ast })\\cdot {Q}^{\\ast },{P}) \\ \\le \\ J(\\textrm {$ $\\hspace{-6.544pt}$$},{P})\\ \\le \\ J({Q}^{\\ast },{P}) \\ \\le \\ \\frac{1}{2}\\sqrt{L} \\cdot (v+h).$$For small $ v$ thedifference between the RHS and the LHS in the above display is $ cstv + o(v) + 12 L h$which proves that the bounds are sharp locally, with non-trivial lower bound.Other upper bounds can be adapted to sets $$\\Omega $$\\Omega $$ defined throughtighter conditions on $ Q$\\Omega $$\\Omega $ k=1,...,Kpkqk$and $ Q$\\Omega $$\\Omega $ k=1,...,Kqkpk$ (of e.g.","Dragomir \\cite {Dra:00}).$ 0.3cm"],["On the difference between minimization problems of\ndeterministic nature and risk minimization","In the context of minimization of the functional $\\Phi _{\\mathbf {P}}(\\mathbf {Q})$ over $\\mathbf {\\Omega } \\subset \\mathbb {R}^{K}$ for known vector $\\mathbf {P}$ , due to Theorem REF our bare simulation approach allows for the approximate solution for any divergence $D_{\\varphi }$ satisfying the basic representation (REF ).","Indeed, any proxy of $\\mathbb {\\Pi }\\big [\\xi _{n}^{\\mathbf {\\widetilde{W}}}\\in \\Omega /M_{\\mathbf {P}}\\big ]$ yields a corresponding proxy for $\\Phi _{\\mathbf {P}}\\left[\\mathbf {\\mathbf {\\Omega }}\\right]$ .","This paves the way to the solution of numerous optimization problems, where the divergence $D_{\\varphi }$ is specifically suited to the problem at hand.","In the statistical context, when the probability distribution (in its vector-form) ${P}$ is unknown up to some indirect information provided by sampling or by any mean providing a sequence $(X_{i})_{i\\in \\mathbb {N}}$ satisfying condition (REF ) (resp.","(REF )), Theorem REF adds a complementary step of complexity; indeed, the estimation of $\\Phi _{\\mathbf {P}}(\\textrm {$$\\hspace{-6.544pt}$$})$ over $\\textrm {$$\\hspace{-6.544pt}$$}\\subset \\mathbb {S}^{K}$ results as its subproduct through the optimization upon $m$ which can be performed explicitly only in a number of specific divergences $D_{\\varphi }$ , e.g.","the power divergences $D_{\\varphi _{\\gamma }}$ , and which carries over also to their monotone transformations such as e.g.","the Renyi divergences.","It is of relevance to mention that — as already indicated above — these divergences cover a very broad range of statistical criteria, indeed most of them, from the (maximum-likelihood estimation connected) likelihood divergence ($\\gamma =0$ ) to the Kullback-Leibler one ($\\gamma =1$ ), the two standard Chi-square distances ($\\gamma =2$ , $\\gamma =-1$ ), the Hellinger distance ($\\gamma =1/2$ ), etc.","; in contrast with deterministic minimization problems, the choice of a statistical criterion (or risk function) is not imposed by the modelling of the problem at hand, but is dictated by the need for sharp measures of fit.","Other divergences are more difficult to handle and our general results in Section REF should still prove some usefulness, since estimation of upper and lower bounds for risk is of common use.","As a “preparatory” remark, recall first that each probability distribution (probability measure) $\\mathbb {P}$ on $\\mathcal {Y}=\\left\\lbrace d_{1},\\ldots ,d_{K}\\right\\rbrace $ has been uniquely identified with the vector ${P} := (p_{1}, \\ldots , p_{K}) \\in \\mathbb {S}^{K}$ of the corresponding probability masses (frequencies) $p_{k} = \\mathbb {P}[\\lbrace d_{k} \\rbrace ]$ via $\\mathbb {P}[A] = \\sum _{k=1}^{K} p_{k} \\cdot {1}_A(d_{k}) $ for each $A \\subset \\mathcal {Y}$ ; from this, we have measured the distance/divergence between two probability distributions $\\mathbb {P}, \\mathbb {Q}$ through the distance/divergence between their frequency vectors ${P}, {Q}$ : $D_{\\varphi }(\\mathbb {Q},\\mathbb {P}) : = D_{\\varphi }({Q},{P})\\qquad \\textrm {(cf.","(\\ref {measure divergence}))} .\\nonumber $ However, it has been noted in Kißlinger & Stummer [190] in a context of even more general divergences $D(\\mathbf {Q}, \\mathbf {P})$ between vectors $\\mathbf {P}, \\mathbf {Q}$ that — alternatively — the latter two may consist of components $p_{k} = \\mathbb {P}[\\lbrace E_{k} \\rbrace ]$ , $q_{k} = \\mathbb {Q}[\\lbrace E_{k} \\rbrace ]$ which are probabilities of only some selected (e.g.","increasing) events $(E_{k})_{k \\in \\lbrace 1,\\ldots ,M\\rbrace }$ of application-based concrete interest (within not necessarily discrete probability models).","Of course, we can apply our BS method to such a vector context.","As other alternatives, in the following we deal with divergences between non-probabilistic uncertainty quantifications."],["Minimization problems with fuzzy sets","Our BS framework also covers the — imprecise/inexact/vague information describing — fuzzy sets (cf.","Zadeh [417]) and optimization problems on divergences between those.","Indeed, let $\\mathcal {Y}=\\left\\lbrace d_{1},\\ldots ,d_{K}\\right\\rbrace $ be a finite set (called the universe (of discourse)), $A \\subset \\mathcal {Y}$ and $M^{A}: \\mathcal {Y} \\mapsto [0,1]$ be a corresponding membership function, where $M^{A}(d_{k})$ represents the degree/grade of membership of the element $d_{k}$ to the set $A$ ; accordingly, the object $A^{\\ast } := \\lbrace \\langle x,M^{A}(x) \\rangle | x \\in \\mathcal {Y}\\rbrace $ is called a fuzzy set in $\\mathcal {Y}$ (or fuzzy subset of $ \\mathcal {Y}$ ).","Moreover, if $A \\subset \\mathcal {Y}$ and $B \\subset \\mathcal {Y}$ are unequal, then the corresponding membership functions $M^{A}$ and $M^{B}$ should be unequal.","Furthermore, we model the vector of membership degrees to $A$ by $\\mathbf {P}^{A} := \\left(p_{k}^{A} \\right)_{k=1,\\ldots ,K}:= \\left(M^{A}(d_{k}) \\right)_{k=1,\\ldots ,K}$ which satisfies the key constraint $0 \\le p_{k}^{A} \\le 1$ for all $k \\in \\lbrace 1,\\ldots ,K\\rbrace $ and, consequently, the aggregated key constraint $0 \\le \\sum _{k=1}^{K} p_{k}^{A} \\le K$ (as a side remark, $\\sum _{k=1}^{K} M^{A}(d_{k})$ is called power of the fuzzy set $A^{\\ast }$).","For divergence generators $\\varphi $ in $\\Upsilon (]a,b[)$ (resp.","$\\widetilde{\\Upsilon }(]a,b[)$ ) with — say — $0 \\le a<10$ .","As an example, for the special choice $\\varphi (t): = \\varphi _{1}(t) := t \\cdot \\log t + 1 - t \\in [0, \\infty [$ (cf.","REF ), we derive from (REF ) and (REF ) the divergence $& & 0 \\le D_{\\varphi _{1}}(\\breve{M}_{2}, \\breve{M}_{1}) =\\sum \\limits _{k=2}^{2^{K}} \\frac{M_{2}(A_{k})}{2^{|A_{k}|}-1} \\cdot \\log \\Big (\\frac{M_{2}(A_{k})}{M_{1}(A_{k})} \\Big )- \\sum \\limits _{k=2}^{2^{K}} \\frac{M_{2}(A_{k})}{2^{|A_{k}|}-1} + \\sum \\limits _{k=2}^{2^{K}} \\frac{M_{1}(A_{k})}{2^{|A_{k}|}-1}\\nonumber \\\\& & =: D^{SD}(M_{2}, M_{1}) -\\sum \\limits _{k=2}^{2^{K}} \\frac{M_{2}(A_{k})}{2^{|A_{k}|}-1} + \\sum \\limits _{k=2}^{2^{K}} \\frac{M_{1}(A_{k})}{2^{|A_{k}|}-1}\\nonumber $ where $D^{SD}(M_{2}, M_{1})$ has been recently developed by Song & Deng [339]; notice that $D^{SD}(M_{2}, M_{1})$ may be negative (cf.","Stummer & Vajda [349]) and then it is not a divergence anymore.","However, for applications to data fusion Song & Deng apply the symmetrization $\\frac{1}{2} \\cdot \\left(D^{SD}(M_{2}, M_{1}) + D^{SD}(M_{1}, M_{2}) \\right)$ which is equal to $\\frac{1}{2} \\cdot \\left( D_{\\varphi _{1}}(\\breve{M}_{2}, \\breve{M}_{1})+ D_{\\varphi _{1}}(\\breve{M}_{1}, \\breve{M}_{2}) \\right)$ and thus nonnegative.","Of course, we can also correspondingly adapt the transformations of $\\varphi -$ divergences (e.g.","Renyi divergences) and entropy-type special cases given in the sections above and below, to BBAs as well as crossover cases and rescalings, and apply our BS method for this.","Recall first that in Theorem REF , one crucial component is the sequence $(W_{n})_{n\\in \\mathbb {N}}$ of weights being i.i.d.","copies of a random variable $W$ whose probability distribution is $\\mathbb {}$ (i.e.","$\\mathbb {\\Pi }[W \\in \\cdot \\, ] = \\mathbb {}[ \\, \\cdot \\,]$ ), where the latter has to be connected with the divergence generator $\\varphi \\in \\Upsilon (]a,b[)$ through the representation $\\varphi (t)=\\sup _{z\\in \\mathbb {R}}\\left( z\\cdot t-\\log \\int _{\\mathbb {R}}e^{zy}d\\mathbb {} (y)\\right), \\qquad t\\in \\mathbb {R},\\ \\ \\hspace{28.45274pt} (cf.","\\ (\\ref {Phi Legendre of mgf(W)}))$ under the additional requirement that the function $z\\mapsto MGF_{\\mathbb {} }(z):=\\int _{\\mathbb {R}}e^{zy}d\\mathbb {} (y)$ is finite on some open interval containing zero (“light-tailedness”); for Theorem REF , we need the corresponding variant (REF ) for $M_{\\mathbf {P}} \\cdot \\varphi \\in \\Upsilon (]a,b[)$ (rather than $\\varphi $ ).","Hence, finding such “BS-associated pairs $(\\varphi ,\\mathbb {})$ ” is an important issue.","Subsequently, let us discuss the following direction: starting from a concrete optimization problem (REF ) — respectively (REF ) — with pregiven $\\varphi \\in \\widetilde{\\Upsilon } (]a,b[)$ (cf.","Definition REF ), as a first step one would like to verify whether indeed $\\varphi \\in \\Upsilon (]a,b[)$ (i.e.","it additionally satisfies (REF )) — respectively $M_{\\mathbf {P}} \\cdot \\varphi \\in \\Upsilon (]a,b[)$ ; as a second step, one would like to find the corresponding $\\mathbb {}$ explicitly.","As far as the above-mentioned first step is concerned, let us first present some fundamental properties of all $\\varphi \\in \\Upsilon (]a,b[)$ : Proposition 29 Let $\\varphi \\in \\Upsilon (]a,b[)$ .","Then the following assertions hold: $\\varphi : \\, ]-\\infty ,\\infty [ \\rightarrow [0,\\infty ]$ is lower semicontinuous and convex; $\\varphi (1)=0$ ; $int(dom(\\varphi )) = ]a,b[$ for some $-\\infty \\le a < 1 < b \\le \\infty $ ; $\\varphi $ is continuously differentiable on $]a,b[$ (i.e.","$\\varphi \\in C^{1}(]a,b[)$ ; $\\varphi $ is strictly convex only in a non-empty neighborhood $]t_{-}^{sc},t_{+}^{sc}[ \\subseteq ]a,b[$ of one ($t_{-}^{sc} < 1 < t_{+}^{sc}$ ); $\\varphi $ is infinitly differentiable on $]t_{-}^{sc},t_{+}^{sc}[$ (i.e.","$\\varphi \\in C^{\\infty }(]t_{-}^{sc},t_{+}^{sc}[$ ), and hence, $\\varphi ^{\\prime }(1) = 0$ , $\\varphi ^{\\prime \\prime }(t) > 0$ for all $t \\in ]t_{-}^{sc},t_{+}^{sc}[$ ; notice that the left-hand second derivative and the right-hand second derivative of $\\varphi $ may not coincide at $t_{-}^{sc}$ respectively at $t_{+}^{sc}$ (i.e.","possible non-second-differentiability at these two points); if $a > -\\infty $ , then $a=t_{-}^{sc}$ ; if $a = -\\infty $ , then either $t_{-}^{sc} = - \\infty $ or $\\varphi (t) = \\varphi (t_{-}^{sc}) + \\varphi ^{\\prime }(t_{-}^{sc}) \\cdot (t- t_{-}^{sc})$ for all $t \\in ]-\\infty , t_{-}^{sc}[$ (affine-linearity); notice that $\\varphi ^{\\prime }(t_{-}^{sc}) < 0$ ; if $b < \\infty $ , then $b=t_{+}^{sc}$ ; if $b = \\infty $ , then either $t_{+}^{sc} = \\infty $ or $\\varphi (t) = \\varphi (t_{+}^{sc}) + \\varphi ^{\\prime }(t_{+}^{sc}) \\cdot (t- t_{+}^{sc})$ for all $t \\in ]t_{+}^{sc},\\infty [$ (affine-linearity); notice that $\\varphi ^{\\prime }(t_{+}^{sc}) > 0$ ; the Fenchel-Legendre transform (also called convex conjugate) of $\\varphi $ $\\varphi ^{*} (z)=\\sup _{t\\in \\mathbb {R}}\\left( z\\cdot t- \\varphi (t) \\right)= \\sup _{t\\in ]a,b[}\\left( z\\cdot t- \\varphi (t) \\right), \\qquad z\\in \\mathbb {R},\\ \\ $ has the following properties: $int(dom(\\varphi ^{*})) = ]\\lambda _{-},\\lambda _{+}[$ , where $dom(\\varphi ^{*}) := \\lbrace z \\in \\mathbb {R} : -\\infty < \\varphi ^{*}(z) < \\infty \\rbrace $ , $\\lambda _{-} := \\inf _{t \\in ]a,b[} \\varphi ^{\\prime }(t) =\\lim _{t \\downarrow a} \\varphi ^{\\prime }(t) =: \\varphi ^{\\prime }(a) < 0$ and $\\lambda _{+} := \\sup _{t \\in ]a,b[} \\varphi ^{\\prime }(t) =\\lim _{t \\uparrow b} \\varphi ^{\\prime }(t) =: \\varphi ^{\\prime }(b) >0$ ; if $a >-\\infty $ , then $\\lambda _{-} = - \\infty $ ; the function $z \\mapsto e^{-a\\cdot z + \\varphi ^{*}(z)} =: M(z)$ is absolutely monotone on $]-\\infty ,0[$ , i.e.","all derivatives exist and satisfy $\\frac{\\partial ^{k}}{\\partial z^k} M(z) \\ge 0$ ($k\\in \\mathbb {N}_{0}$ , $z \\in ]-\\infty ,0[$ ); $\\lim _{z \\rightarrow 0-} M(z) =1$ ; if $b < \\infty $ , then $\\lambda _{+} = \\infty $ ; the function $z \\mapsto e^{b\\cdot z + \\varphi ^{*}(- z)} =: M(z)$ is absolutely monotone on $]-\\infty ,0[$ ; $\\lim _{z \\rightarrow 0-} M(z) =1$ ; if $a = -\\infty $ and $b = -\\infty $ , then the function $z \\mapsto e^{\\varphi ^{*}(z)} =: M(z)$ is exponentially convex on $]\\lambda _{-}, \\lambda _{+}[$ , i.e.","$M(\\cdot )$ is continuous and satisfies $\\sum _{i,j=1}^{n} c_{i} \\cdot c_{j} \\cdot M\\Big (\\frac{z_{i} + z_{j}}{2}\\Big ) \\ge 0 \\qquad \\textrm {for all n \\in \\mathbb {N},c_{i}, c_{j} \\in \\mathbb {R} and z_{i}, z_{j} \\in ]\\lambda _{-}, \\lambda _{+}[;}$ $\\lim _{z \\rightarrow 0-} M(z) =1$ ; as a side remark, notice the well-known fact that exponential-convexity is stronger than the usual log-convexity.","the endpoints of $int(dom(\\varphi )) =]a,b[$ have the following important “functioning” for the underlying probability distribution $\\mathbb {}$ (cf.","(REF )) respectively of an associated random variable $W$ with $\\mathbb {}[ \\cdot \\, ] := \\mathbb {\\Pi }[W \\in \\cdot \\, ]$ : $a = \\inf supp(\\mathbb {}) = \\inf supp(W)$ , $b = \\sup supp(\\mathbb {}) = \\sup supp(W)$ , where $supp(\\mathbb {})$ respectively $supp(W)$ denotes the support of $\\mathbb {}$ respectively $W$ ; consequently, $]a,b[ = int(conv(supp(\\mathbb {}))) = int(conv(supp(W)))$ where $conv(A)$ denotes the convex hull of a set $A$ ; if $a > -\\infty $ , then $\\varphi (a) = - \\log \\mathbb {}[\\lbrace a\\rbrace ]= - \\log \\mathbb {\\Pi }[W = a \\, ]$ ; consequently, there holds: $a = \\min supp(\\mathbb {}) = \\min supp(W)$ if and only if $\\mathbb {}[\\lbrace a\\rbrace ] = \\mathbb {\\Pi }[W = a \\, ] > 0$ if and only if $\\varphi (a) < \\infty $ if and only if $a \\in dom(\\varphi )$ ; if $b < \\infty $ , then $\\varphi (b) = - \\log \\mathbb {}[\\lbrace b\\rbrace ]= - \\log \\mathbb {\\Pi }[W = b \\, ]$ ; consequently, there holds: $b = \\max supp(\\mathbb {}) = \\max supp(W)$ if and only if $\\mathbb {}[\\lbrace b\\rbrace ] = \\mathbb {\\Pi }[W = b \\, ] > 0$ if and only if $\\varphi (b) < \\infty $ if and only if $b \\in dom(\\varphi )$ .","the first two derivatives of $\\varphi $ at the point 1 have the following important “functioning” for the underlying probability distribution $\\mathbb {}$ (cf.","(REF )) respectively of an associated random variable $W$ : $1 = \\varphi ^{\\prime -1}(0) = \\int _{\\mathbb {R}} y \\, d\\mathbb {} (y)= E_{\\mathbb {\\Pi }}[W]$ where $\\varphi ^{\\prime -1}(\\cdot )$ denotes the inverse of the first derivative $\\varphi ^{\\prime }(\\cdot )$ of $\\varphi (\\cdot )$ , $\\frac{1}{\\varphi ^{\\prime \\prime }(1)} =\\int _{\\mathbb {R}} \\Big (y - \\int _{\\mathbb {R}} \\widetilde{y} \\, d\\mathbb {} (\\widetilde{y})\\Big )^{2}\\, d\\mathbb {} (y)= E_{\\mathbb {\\Pi }}[W^{2}] - (E_{\\mathbb {\\Pi }}[W])^{2} = Var_{\\mathbb {\\Pi }}[W]$ ; in particular, scaling $\\widetilde{c} \\cdot \\varphi $ ($\\widetilde{c} >0$ ) does not change the mean 1 but the variance of $W$ .","The corresponding proof of Proposition REF will be given in Appendix D, except for the second items of (G9ii) and (G9iii) as well as the first item of (G9iv).","Those will be treated in the second next paragraph below, because the corresponding line of argumentation builds an insightful start for subsequently performed procedures to further track down the weight distribution $$ .","The properties (G1) to (G9iv) constitute necessary conditions for a pregiven function $\\varphi $ to belong to $\\Upsilon (]a,b[)$ ); accordingly, these should be verified first, in concrete situations where one aims to apply the BS approach.","An important role is played by the boundary points $a$ and $b$ of $int(dom(\\varphi ))$ through (G10i) to (G10iii), because their finiteness opens the gate to apply — via some straightforward transformations — a rich class of real-valued characterization theorems for probability distributions whose support lies in the positive real line $[0,\\infty [$ .","In contrast, there exist much less real-valued characterization theorems for probability distributions whose support is the whole real line $]-\\infty ,\\infty [$ ; typically, the involved conditions are also more difficult to verify.","Indeed, if $\\varphi \\in \\Upsilon (]a,b[)$ then one can deduce straightforwardly from the representation (REF ) that $e^{\\varphi ^{*}(z)} =\\int _{-\\infty }^{\\infty } e^{z\\cdot y} \\, \\mathrm {d}\\mathbb {} (y) =E_{\\mathbb {\\Pi }}[e^{z \\cdot W}] , \\quad z \\in ]\\lambda _{-}, \\lambda _{+}[,$ where $W$ is a random variable whose distribution is $\\mathbb {\\Pi }[W \\in \\cdot \\, ] = \\mathbb {}[ \\, \\cdot \\,]$ ; under the additional knowledge $a > -\\infty $ (and consequently $\\lambda _{-} = -\\infty $ ) employed together with (G10i) and thus $\\mathbb {\\Pi }[W \\ge a \\, ] =\\mathbb {}[ \\, [a,\\infty [ \\,] = 1$ , one arrives at $e^{\\varphi ^{*}(z) - a \\cdot z} =\\int _{a}^{\\infty } e^{z \\cdot (y-a)} \\, \\mathrm {d}\\mathbb {} (y) =\\int _{0}^{\\infty } e^{z\\cdot \\widetilde{y}} \\, \\mathrm {d}\\widetilde{\\widetilde{\\mathbb {}}} (\\widetilde{y})= E_{\\mathbb {\\Pi }}[e^{z \\cdot (W-a)}] , \\quad z \\in ]-\\infty , \\lambda _{+}[,$ where the probability distribution $\\widetilde{\\widetilde{\\mathbb {}}}[ \\, \\cdot \\, ] := \\mathbb {}[\\, \\cdot + a \\, ]$ is the $a-$ shifted companion of $\\mathbb {}$ ; recall that $\\lambda _{+} >0$ .","Put in other words, $\\mathbb {\\Pi }[\\widetilde{W} \\in \\cdot \\, ] = \\widetilde{\\widetilde{\\mathbb {}}}[ \\, \\cdot \\,]$ is the probability distribution of the (a.s.) nonnegative random variable $\\widetilde{W} := W-a$ .","Naturally, in this context, the interesting case is $-\\infty < a \\le 0$ .","Similarly, if $\\varphi \\in \\Upsilon (]a,b[)$ and $b < \\infty $ (and hence $\\lambda _{+} = \\infty $ ), one can derive from (G10i) and its consequence $\\mathbb {\\Pi }[W \\le b \\, ] =\\mathbb {}[ \\, ]-\\infty ,b] \\,]= 1$ that $e^{\\varphi ^{*}(-z) + b \\cdot z} =\\int _{-\\infty }^{b} e^{z \\cdot (b-y)} \\, \\mathrm {d}\\mathbb {} (y) =\\int _{0}^{\\infty } e^{z\\cdot \\widetilde{y}} \\, \\mathrm {d}\\widetilde{\\widetilde{\\mathbb {}}} (\\widetilde{y})= E_{\\mathbb {\\Pi }}[e^{z \\cdot (b-W)}] , \\quad z \\in ]- \\infty , - \\lambda _{-}[,$ where $- \\lambda _{-} >0$ and the probability distribution $\\widetilde{\\widetilde{\\mathbb {}}}[ \\, \\cdot \\, ] := \\mathbb {}[\\, b - \\, \\cdot \\, ]$ is the mirrored$-b-$ shifted companion of $\\mathbb {}$ .","This means that $\\mathbb {\\Pi }[\\widetilde{W} \\in \\cdot \\, ] = \\widetilde{\\widetilde{\\mathbb {}}}[ \\, \\cdot \\,]$ is the probability distribution of the (a.s.) nonnegative random variable $\\widetilde{W} := b-W$ .","Naturally, the interesting case is $0 < b \\le \\infty $ .","As already indicated above, the considerations (REF ) to (REF ) open the gate to the adaption of well-known real-valued (rather than complex-valued) characterizations from probability theory.","To begin with, the following assertions are very prominent: Theorem 30 (a) Let $M: ]-\\infty ,0] \\mapsto ]0,\\infty [$ be continuous on $]-\\infty ,0]$ with $M(0) =1$ .","Then one has $\\textrm {M is absolutely monotone on ]-\\infty ,0[}\\ \\Longleftrightarrow \\ \\exists \\ \\textrm {unique prob.","distr.","\\widetilde{\\widetilde{\\mathbb {}}} on [0,\\infty [s.t.}","\\ \\ M(z) = \\int _{0}^{\\infty } e^{z \\cdot y} \\mathrm {d}\\widetilde{\\widetilde{\\mathbb {}}}(y)\\ \\textrm {for all z \\in ]-\\infty ,0[.","}$ (b) Let $I$ be an open interval which contains 0, and $M: I \\mapsto [0,\\infty [$ be continuous with $M(0) =1$ .","Then one gets $\\textrm {M is exponentially convex}\\ \\Longleftrightarrow \\ \\exists \\ \\textrm {unique prob.","distr.","\\widetilde{\\widetilde{\\mathbb {}}} on ]-\\infty ,\\infty [such that} \\ \\ M(z) = \\int _{-\\infty }^{\\infty } e^{z \\cdot y} \\mathrm {d}\\widetilde{\\widetilde{\\mathbb {}}}(y)\\ \\ \\textrm {for all z \\in I.","}$ Assertion (a) of Theorem REF is known as (probability-version of) Bernstein’s theorem [42] (see e.g.","also Schilling et al.","[322]), whereas assertion (b) is known as (probability-version of) Widder’s theorem [391] for the relevant conversion between the involved Riemann-Stieltjes integral with nondecreasing (but not necessarily right-continuous) integrator into a measure integral, one can apply the general theory in e.g.","Chapter 6 of Chow & Teicher [88].","(see e.g.","also Widder [392], Akhiezer [9], Shucker [333], Jaksetic & Pecaric [163], Kotelina & Pevny [196]).","From Theorem REF (b) and (REF ), the first item in (G9iv) follows immediately by using the choice $I= ]\\lambda _{-}, \\lambda _{+}[$ .","Moreover, Theorem REF (a) together with (REF ) (respectively (REF )) implies the second item of (G9ii) (respectively of (G9iii)).","In fact, with the help of Theorem REF and some further considerations e.g.","on light-tailedness, one even gets assertions on the sufficiency of (G9ii), (G9iii) and (G9iv) for a “candidate generator” $\\varphi $ to belong to the BS-suitable class $\\Upsilon (]a,b[)$ .","More precisely, we obtain Proposition 31 Suppose that $\\varphi : ]-\\infty ,\\infty [ \\mapsto [0,\\infty ]$ satisfies (G1) to (G8), and recall the notations in (G9i).","Then, $\\varphi \\in \\Upsilon (]a,b[)$ if one of the following three conditions holds: (a) $a >-\\infty $ , $\\lambda _{-} = - \\infty $ , and the function $z \\mapsto e^{-a\\cdot z + \\varphi ^{*}(z)}$ is absolutely monotone on $]-\\infty ,0[$ , (b) $b < \\infty $ , $\\lambda _{+} = \\infty $ , and the function $z \\mapsto e^{b\\cdot z + \\varphi ^{*}(- z)}$ is absolutely monotone on $]-\\infty ,0[$ , (c) $a = -\\infty $ , $b = -\\infty $ , and the function $z \\mapsto e^{\\varphi ^{*}(z)}$ is exponentially convex on $]\\lambda _{-}, \\lambda _{+}[$ .","If one of the three conditions (a) to (c) holds, thenbasically by Theorem REF with $M(\\cdot )$ defined in G9(ii),(iii) or (iv); see Appendix D. the associated probability distribution $\\mathbb {}$ (cf.","(REF )) has expectation $\\int _{\\mathbb {R}} y d\\mathbb {}(y) =1$ and finite moments of all orders, i.e.","$\\int _{\\mathbb {R}} y^{j} d\\mathbb {}(y) < \\infty $ for all $j \\in \\mathbb {N}_{0}$ ; in terms of $\\mathbb {}[ \\cdot \\, ] := \\mathbb {\\Pi }[W \\in \\cdot \\, ]$ this means that $E_{\\mathbb {\\Pi }}[W] =1$ and $E_{\\mathbb {\\Pi }}[W^{j}] < \\infty $ .","The proof of Proposition REF will be given in Appendix D. As far as applicability is concerned, it is well known that, in general, verifying absolute monotonicity is typically more comfortable than verifying exponential convexity.","Fortunately, one can often use the former, since for many known divergence generators there holds $a > -\\infty $ (often $a=0$ ) or $b < \\infty $ or both, which by virtue of (G10i) is directly linked with the (endpoints of the) support of the potentially existing probability distribution $\\mathbb {}$ .","For a pregiven divergence generator $\\varphi $ , once its membership in $\\Upsilon (]a,b[)$ (and in particular, the representability (REF )) is verified, one would like to concretely find the underlying probability distribution $\\mathbb {}$ .","This may be quite challenging, but can be made more comfortable by systematically narrowing down the family of distributions where $\\mathbb {}$ belongs to.","In fact, we have already performed a first down-narrowing, in terms of identifying the endpoints of the support of $\\mathbb {}$ to be the endpoints of the effective domain of $\\varphi $ (cf.","(G10i)).","A further down-narrowing can be achieved from (REF ) to (REF ) in combination with real-valued characterization theorems which are more specific than Theorem REF .","This will be shown exemplarily for a few important sub-setups, in the following.","For the identification of light-tailed semi/half-lattice distributions, we obtain the following two sets of sufficient conditions, which even allow for the desired explicit determination of $\\mathbb {}$ : Proposition 32 Suppose that $\\varphi : ]-\\infty ,\\infty [ \\mapsto [0,\\infty ]$ satisfies (G1) to (G8), with some $a > -\\infty $ .","Furthermore, assume that there exists some constant $\\breve{c} >0$ as well as some function $H: [0,\\infty [ \\mapsto [0,\\infty [$ which is continuous on $[0,1]$ with $H(1)=1$ and absolutely monotone on $]0,1[$ , such that $e^{\\varphi ^{*}(\\frac{z}{\\breve{c}}) - a\\cdot \\frac{z}{\\breve{c}} } = H(e^{z}), \\qquad z \\in ]-\\infty , \\breve{c} \\cdot \\lambda _{+}[.$ Then one has $\\varphi \\in \\Upsilon (]a,b[)$ and $\\mathbb {} = \\sum _{n=0}^{\\infty } p_{n} \\cdot \\delta _{a + \\breve{c} \\cdot n}\\qquad \\textrm {with \\ } p_{n} := \\frac{1}{n !}","\\cdot \\frac{\\mathrm {d}^{n}H}{\\mathrm {d}t}(0),$ i.e.","$\\mathbb {\\Pi }[W = a + \\breve{c} \\cdot n \\, ] = p_{n}$ ($n \\in \\mathbb {N}_{0}$ ).","Proposition 33 Suppose that $\\varphi : ]-\\infty ,\\infty [ \\mapsto [0,\\infty ]$ satisfies (G1) to (G8), with some $b < \\infty $ .","Furthermore, assume that there exists some constant $\\breve{c} >0$ as well as some function $H: [0,\\infty [ \\mapsto [0,\\infty [$ which is continuous on $[0,1]$ with $H(1)=1$ and absolutely monotone on $]0,1[$ , such that $e^{\\varphi ^{*}(- \\frac{z}{\\breve{c}}) + b \\cdot \\frac{z}{\\breve{c}} } = H(e^{z}), \\qquad z \\in ]-\\infty , - \\breve{c} \\cdot \\lambda _{-}[.$ Then one has $\\varphi \\in \\Upsilon (]a,b[)$ and $\\mathbb {} = \\sum _{n=0}^{\\infty } p_{n} \\cdot \\delta _{b - \\breve{c} \\cdot n}\\qquad \\textrm {with \\ } p_{n} := \\frac{1}{n !}","\\cdot \\frac{\\mathrm {d}^{n}H}{\\mathrm {d}t}(0),$ i.e.","$\\mathbb {\\Pi }[W = b - \\breve{c} \\cdot n \\, ] = p_{n}$ ($n \\in \\mathbb {N}_{0}$ ).","The Propositions REF respectively REF follow from (REF ) respectively (REF ), some straightforward transformations, and a well-known characterization of probability generating functions $H$ (see e.g.","in Theorem 1.2.10 of Stroock [343]).","As an incentive for the following investigations, let us recall the discussion in the surroundings of Condition REF pertaining to the minimization problem (REF ), where we have addressed possible connections between the two representabilities (REF ) (needed e.g.","for Theorem REF ) and (REF ) (needed e.g.","for Theorem REF ); this strongly relates to the question, for which constants $\\widetilde{c} >0$ the validity $\\varphi \\in \\Upsilon (]a,b[)$ triggers the validity of $\\widetilde{c} \\cdot \\varphi \\in \\Upsilon (]a,b[)$ .","To begin with, it is straightforward to see that $\\varphi \\in \\Upsilon (]a,b[)$ always implies $\\widetilde{c} \\cdot \\varphi \\in \\Upsilon (]a,b[)$ for all integers $\\widetilde{c} \\in \\mathbb {N}$ ; indeed, if $\\varphi $ satisfies (REF ) for some $\\mathbb {} = \\mathbb {\\Pi }[ W \\in \\cdot \\, ]$ , then for each integer $\\widetilde{c} \\in \\mathbb {N}$ one gets that $\\widetilde{c} \\cdot \\varphi $ satisfies (REF ) for $\\widetilde{\\mathbb {}} = \\mathbb {\\Pi }[ \\sum _{j=1}^{\\widetilde{c}} \\frac{W_{j}}{\\widetilde{c}} \\in \\cdot \\, ]$ ; in the latter, the $W_{j}$ ’s are i.i.d.","copies from $W$ .","Clearly, $MGF_{\\widetilde{\\mathbb {}}}$ is then finite on some open interval containing zero (differing from the one for $MGF_{\\mathbb {}}$ only by a scaling with $1/\\widetilde{c}$ ).","For the following family of distributions, one can even trigger $\\widetilde{c} \\cdot \\varphi \\in \\Upsilon (]a,b[)$ for all $\\widetilde{c} >0$ : for the sake of a corresponding precise formulation, recall first the common knowledge that, generally speaking, a probability distribution $\\mathbb {}$ on $\\mathbb {R}$ with light tails — in the sense that its moment generating function $z\\mapsto MGF_{\\mathbb {} }(z):=\\int _{\\mathbb {R}}e^{z \\cdot y}d\\mathbb {} (y)$ is finite on some open interval $]\\lambda _{-},\\lambda _{+}[$ containing zero — is (said to be) infinitely divisible if there holds $\\textrm {for each n \\in \\mathbb {N} there existsa probability distribution \\mathbb {}_{n} on \\mathbb {R} such that}\\int _{-\\infty }^{\\infty } e^{z \\cdot y} d\\mathbb {} (y) =\\Big (\\int _{-\\infty }^{\\infty } e^{z \\cdot y} d\\mathbb {}_{n} (y) \\Big )^{n}, \\quad z \\in ]\\lambda _{-},\\lambda _{+}[ ;$ in fact, (REF ) means that the (light-tailed) moment generating function $MGF_{\\mathbb {} }$ is infinitely divisible in the sense that each $n-$ th root $(MGF_{\\mathbb {} })^{1/n}$ must be the moment generating function of some (light-tailed) probability distribution (denoted here by $\\mathbb {}_{n}$ ).","In particular, (REF ) implies that $\\mathbb {}_{n}$ is unique, and that $\\mathbb {}$ must necessarily have (one-sided or two-sided) unbounded support $supp(\\mathbb {})$ .","The latter may differ from $supp(\\mathbb {}_{n})$ .","In our BS context (REF ), (REF ) equivalently means that the associated random variable $W$ is infinitely divisible (with light-tailed distribution), in the sense that $\\textrm {for each n \\in \\mathbb {N} there exists a sequence of i.i.d.","random variables Y_{n,1}, \\cdots , Y_{n,n} such that }W \\stackrel{\\text{\\tiny d}}{=} Y_{n,1} + \\cdots + Y_{n,n} ,$ where $\\stackrel{\\text{\\tiny d}}{=}$ means “have equal probability distributions” and $\\mathbb {\\Pi }[W \\in \\cdot \\, ] = \\mathbb {}[ \\, \\cdot \\,]$ , $\\mathbb {\\Pi }[Y_{n,1} \\in \\cdot \\, ] = \\mathbb {}_{n}[ \\, \\cdot \\,]$ .","For the above-mentioned context, we obtain the useful Proposition 34 Suppose that $\\varphi \\in \\Upsilon (]a,b[)$ , with connected probability distribution $\\mathbb {}$ from (REF ) (recall that this implies that $\\mathbb {}$ is not a one-point distribution, cf.","Remark REF ).","Then there holds: $\\textrm {\\widetilde{c} \\cdot \\varphi \\in \\Upsilon (]a,b[)for all \\widetilde{c} > 0} \\ \\ \\Longleftrightarrow \\ \\ \\textrm {\\mathbb {} is infinitely divisible.","}$ The proof of Proposition REF is given in Appendix E. Notice that Proposition REF covers especially the important prominent power divergences (cf.","Examples REF and REF below) for which we provide the corresponding infinitely divisible distributions explicitly in the Examples REF and REF below, and for which the subsequent form of estimators (cf.","Chapter below) can be simplified.","For the identification of light-tailed infinitely divisible distributions, we obtain the following three sets of sufficient conditions: Proposition 35 Suppose that $\\varphi : ]-\\infty ,\\infty [ \\mapsto [0,\\infty ]$ satisfies (G1) to (G8), and recall the notations in (G9i) as well as $a = \\inf supp(\\mathbb {})$ , $b = \\sup supp(\\mathbb {})$ (cf.","(G10i)).","Then, $\\varphi \\in \\Upsilon (]a,b[)$ and the associated probability distribution $\\mathbb {}$ is infinitely divisible, if one of the following three conditions holds: (a) $a >-\\infty $ , $\\lambda _{-} = - \\infty $ , and the function $z \\mapsto \\varphi ^{* \\prime }(z) - a = (\\varphi ^{\\prime })^{-1}(z) - a $ is absolutely monotone on $]-\\infty ,0[$ , (b) $b < \\infty $ , $\\lambda _{+} = \\infty $ , and the function $z \\mapsto - \\varphi ^{* \\prime }(- z) + b= - (\\varphi ^{\\prime })^{-1}(-z) +b $ is absolutely monotone on $]-\\infty ,0[$ , (c) $a = -\\infty $ , $b = -\\infty $ , and the function $z \\mapsto \\frac{\\varphi ^{* \\prime \\prime }(z)}{\\varphi ^{* \\prime \\prime }(0)}= \\frac{\\varphi ^{\\prime \\prime }(1)}{\\varphi ^{\\prime \\prime }((\\varphi ^{\\prime })^{-1}(z))} $ is exponentially convex on $]\\lambda _{-}, \\lambda _{+}[$ .","In the first case (a) there automatically follows $b=\\infty $ , whereas in the second case (b) one automatically gets $a =- \\infty $ .","The proof of Proposition REF is given in Appendix E. So far, in the current section we have started from a given divergence generator $\\varphi \\in \\widetilde{\\Upsilon }(]a,b[)$ having some additional properties, switched to its Fenchel-Legendre transform $\\varphi ^{*}$ and some exponentially-linear transforms thereof, and presented some sufficient conditions for verifying that the outcome is a moment-generating function $MGF_{\\mathbb {}}$ of a unique probability distribution $\\mathbb {}$ which has light tails.","For finding the concrete $\\mathbb {}$ , one typically should know the explicit form of $\\varphi ^{*}$ .","However, it is well known that it can sometimes be hard to determine the explicit form of the Fenchel-Legendre transform of a convex function.","This issue also applies for the reverse direction of starting from a concrete probability distribution $\\mathbb {}$ with light tails, computing its log-moment-generating function (called cumulant-generating function) $z \\mapsto \\Lambda _{\\mathbb {}}(z) := \\log MGF_{\\mathbb {}}(z)$ and the corresponding Fenchel-Legendre transform $\\Lambda _{\\mathbb {}}^{*}$ which is nothing but the associated divergence generator $\\varphi $ (cf.","(REF )).","As will be illuminated in several examples below, the — “kind of intermediate” — construction method given in the below-mentioned Theorem REF can help to ease these two tasks.","To formulate this, we employ the class ${F}$ of functions $F: ]-\\infty ,\\infty [ \\mapsto [-\\infty ,\\infty ]$ with the following properties: $int(dom(F)) = ]a_{F},b_{F}[$ for some $-\\infty \\le a_{F} < 1 < b_{F} \\le \\infty $ ; $F$ is smooth (infinitely continuously differentiable) on $]a_{F},b_{F}[$ ; $F$ is strictly increasing on $]a_{F},b_{F}[$ .","Clearly, for any $F \\in {F}$ one gets the existence of $F(a_{F}) := \\lim _{t \\downarrow a_{F}} F(t) \\in [-\\infty , \\infty [$ and $F(b_{F}) := \\lim _{t \\uparrow b_{F}} F(t) \\in ]-\\infty , \\infty ]$ ; moreover, its inverse $F^{-1}: \\mathcal {R}(F) \\mapsto [a_{F},b_{F}]$ exists, where $\\mathcal {R}(F) := \\lbrace F(t): t \\in dom(F) \\rbrace $ .","Furthermore, $F^{-1}$ is strictly increasing and smooth (infinitely continuously differentiable) on the open interval $int(\\mathcal {R}(F)) = \\lbrace F(t): t \\in ]a_{F},b_{F}[ \\rbrace =]F(a_{F}),F(b_{F})[$ , and $F^{-1}(int(\\mathcal {R}(F))) = ]a_{F},b_{F}[$ .","Within such a context, we obtain Theorem 36 Let $F \\in {F}$ and fix an arbitrary point $c \\in int(\\mathcal {R}(F))$ .","Moreover, introduce the notationsfor the sake of brevity, we avoid here the more complete notation $\\lambda _{-}^{F,c}$ , $\\lambda _{+}^{F,c}$ , $t_{-}^{sc,F,c}$ , $t_{+}^{sc,F,c}$ indicating the dependence on $F$ and $c$ .","$]\\lambda _{-},\\lambda _{+}[ := int(\\mathcal {R}(F)) -c$ and $]t_{-}^{sc},t_{+}^{sc}[ := ]1+a_{F}-F^{-1}(c),1+b_{F}-F^{-1}(c)[$ (which implies $\\lambda _{-} < 0 < \\lambda _{+}$ and $t_{-}^{sc} < 1 < t_{+}^{sc}$ ).","Furthermore, define the functions $\\Lambda : \\ ]-\\infty ,\\infty [ \\ \\mapsto \\, [-\\infty , \\infty ]$ and $\\varphi : \\ ]-\\infty ,\\infty [ \\ \\mapsto \\, [0, \\infty ]$ by $\\hspace{-19.91684pt} \\Lambda (z) := \\Lambda ^{(c)}(z) \\hspace{-5.69046pt} &:=& \\hspace{-5.69046pt}{\\left\\lbrace \\begin{array}{ll}\\int \\displaylimits _{0}^{z} F^{-1}(u+c) \\, du+ z \\cdot (1-F^{-1}(c)) \\ \\in ]-\\infty , \\infty [,\\hspace{85.35826pt} \\textrm {if } z \\in ]\\lambda _{-},\\lambda _{+}[,\\\\\\int \\displaylimits _{0}^{\\lambda _{-}} F^{-1}(u+c) \\, du+ \\lambda _{-} \\cdot (1-F^{-1}(c))\\ \\in [-\\infty ,\\infty ],\\hspace{71.13188pt} \\textrm {if } z = \\lambda _{-} > -\\infty ,\\\\\\int \\displaylimits _{0}^{\\lambda _{+}} F^{-1}(u+c) \\, du+ \\lambda _{+} \\cdot (1-F^{-1}(c))\\ \\in \\ [-\\infty ,\\infty ],\\hspace{66.86414pt} \\textrm {if } z = \\lambda _{+} < \\infty ,\\\\\\infty , \\hspace{283.10483pt} \\textrm {else},\\end{array}\\right.","}$ where the second respectively third line are meant as $\\lim _{z \\downarrow \\lambda _{-}} \\big ( \\int \\displaylimits _{0}^{z} F^{-1}(u+c) \\, du+ z \\cdot (1-F^{-1}(c)) \\big )$ respectively $\\lim _{z \\uparrow \\lambda _{+}} \\big ( \\int \\displaylimits _{0}^{z} F^{-1}(u+c) \\, du+ z \\cdot (1-F^{-1}(c)) \\big )$ , and $\\varphi (t) := \\varphi ^{(c)}(t) \\hspace{-5.69046pt} &:=& \\hspace{-5.69046pt}{\\left\\lbrace \\begin{array}{ll}(t+F^{-1}(c)-1) \\cdot [ F\\left(t+F^{-1}(c)-1 \\right) - c ]-\\int \\displaylimits _{0}^{F\\left(t+F^{-1}(c)-1 \\right) - c} F^{-1}(u+c) du\\ \\in \\ [0,\\infty [,\\\\\\hspace{327.20668pt}\\quad \\textrm {if }t \\in ]t_{-}^{sc},t_{+}^{sc}[,\\\\(t_{-}^{sc}+F^{-1}(c)-1) \\cdot [ F\\left(t_{-}^{sc}+F^{-1}(c)-1 \\right) - c ]-\\int \\displaylimits _{0}^{F\\left(t_{-}^{sc}+F^{-1}(c)-1 \\right) - c} F^{-1}(u+c) du\\ \\in \\ ]0,\\infty ],\\\\\\hspace{327.20668pt}\\quad \\textrm {if }t = t_{-}^{sc} > -\\infty ,\\\\(t_{+}^{sc}+F^{-1}(c)-1) \\cdot [ F\\left(t_{+}^{sc}+F^{-1}(c)-1 \\right) - c ]-\\int \\displaylimits _{0}^{F\\left(t_{+}^{sc}+F^{-1}(c)-1 \\right) - c} F^{-1}(u+c) du\\ \\in \\ ]0,\\infty ],\\\\\\hspace{327.20668pt}\\quad \\textrm {if }t = t_{+}^{sc} < \\infty ,\\\\\\varphi (t_{-}^{sc}) +\\lambda _{-}\\cdot (t- t_{-}^{sc}) \\ \\in \\ ]0,\\infty ],\\hspace{99.58464pt} \\textrm {if }t_{-}^{sc} > - \\infty , \\ \\textrm {and} \\ t \\in \\ ]-\\infty , t_{-}^{sc}[,\\\\\\varphi (t_{+}^{sc}) +\\lambda _{+}\\cdot (t - t_{+}^{sc}) \\ \\in \\ ]0,\\infty ],\\hspace{99.58464pt} \\textrm {if }t_{+}^{sc} < \\infty , \\ \\textrm {and} \\ t \\in \\ ]t_{+}^{sc}, \\infty [,\\\\\\infty , \\hspace{321.51622pt} \\textrm {else},\\end{array}\\right.","}$ where the second respectively third line are again meant as lower respectively upper limit.","Then, $\\Lambda $ and $\\varphi $ have the following properties: (i) On $]\\lambda _{-},\\lambda _{+}[$ , the function $\\Lambda $ is smooth and strictly convex and consequently, $\\exp (\\Lambda )$ ) is smooth and strictly log-convex; moreover, there holds $\\Lambda (0) =0$ , $\\Lambda ^{\\prime }(0) =1$ ; (ii) $\\varphi \\in \\widetilde{\\Upsilon }(]a,b[)$ , where $a := t_{-}^{sc} \\cdot {1}_{ \\lbrace -\\infty \\rbrace }(\\lambda _{-}) - \\infty \\cdot {1}_{]-\\infty ,0[}(\\lambda _{-})$ , $b := t_{+}^{sc} \\cdot {1}_{ \\lbrace \\infty \\rbrace }(\\lambda _{+}) + \\infty \\cdot {1}_{]0,\\infty [}(\\lambda _{+})$ , and $\\varphi $ has the properties (G1) to (G8).","(iii) $\\varphi (t) =\\sup _{z \\in ]-\\infty ,\\infty [} \\left( z\\cdot t -\\Lambda (z)\\right) =\\sup _{z \\in ]\\lambda _{-},\\lambda _{+}[} \\left( z\\cdot t -\\Lambda (z)\\right)$ for all $t \\in \\mathbb {R}$ .","(iv) $\\Lambda (z) = \\varphi ^{*}(z) =\\sup _{t \\in ]-\\infty ,\\infty [} \\left( t\\cdot z -\\varphi (t)\\right) =\\sup _{t \\in ]a,b[} \\left( t\\cdot z -\\varphi (t) \\right)$ for all $z \\in \\mathbb {R}$ .","The proof of Theorem REF will be given Appendix F. Remark 37 Theorem REF indicates that the $F-$ constructed function $z \\mapsto \\exp (\\Lambda (z)) = \\exp (\\varphi ^{*}(z))$ is a good candidate for a moment generating function of a probability distribution $\\mathbb {}$ , and hence for the representability (REF ).","However, one still needs to verify one of the conditions (a) to (c) of Proposition REF .","This may go wrong, as the case of power divergences $\\varphi _{\\gamma }$ with $\\gamma \\in ]1,2[$ indicates (cf.","the conjecture of Example REF (f) below).","Notice that the newly constructed $\\Lambda $ and $\\varphi $ (cf.","(REF ), (REF )) depend on the choice of the anchor point $c$ ; this is e.g.","illustrated in Example REF (b) below.","Hence, as a side effect, by using whole families $(F_{\\vartheta })_{\\vartheta }$ together with different anchor points $c$ , via Theorem REF one can generate new classes (and new classifications) of $\\varphi -$ divergence generators — and thus of corresponding $\\varphi -$ divergences — which can be of great use, even in other contexts beyond our BS optimization framework.","If $F$ satisfies $F(1)=0$ and thus $F^{-1}(0)=1$ , then the natural choice $c:=0$ induces $]\\lambda _{-},\\lambda _{+}[ = int(\\mathcal {R}(F))$ and $]t_{-}^{sc},t_{+}^{sc}[ = ]a_{F},b_{F}[$ , and consequently (due to $F^{-1}(c)-1 =0$ ) leads to the simplification of “the first lines of” (REF ) and (REF ) to $& & \\hspace{-19.91684pt} \\Lambda (z) := \\Lambda ^{(0)}(z) :=\\int \\displaylimits _{0}^{z} F^{-1}(u) du,\\qquad z \\in int(\\mathcal {R}(F)),\\\\& & \\hspace{-19.91684pt} \\varphi (t) := \\varphi ^{(0)}(t) :=t \\cdot F\\left(t\\right)-\\int \\displaylimits _{0}^{F\\left(t\\right)} F^{-1}(u) du,\\qquad t \\in ]a_{F},b_{F}[ ;$ the simplifications of the respective other lines of (REF ) and (REF ) are straightforward.","Remark 38 Let $F \\in {F}$ with $a_{F} =0$ , $b_{F} = \\infty $ , $F(1)=0$ and hence, $int(\\mathcal {R}(F)) = \\, ]F(0),F(\\infty )[$ .","Then the transformation $\\widetilde{F}(t) &:=&{\\left\\lbrace \\begin{array}{ll}- \\int _{0}^{F(\\frac{1}{t})} F^{-1}(u) \\, du, \\qquad \\textrm {if } \\ t \\in ]0,\\infty [, \\\\- \\int _{0}^{F(\\infty )} F^{-1}(u) \\, du, \\qquad \\textrm {if } \\ t=0, \\\\-\\infty , \\qquad \\hspace{64.01869pt} \\textrm {if } \\ t \\in ]-\\infty , 0[,\\end{array}\\right.","}$ satisfies $\\widetilde{F} \\in {F}$ with $a_{\\widetilde{F}} =0$ , $b_{\\widetilde{F}} = \\infty $ , $\\widetilde{F}(1)=0$ and $int(\\mathcal {R}(\\widetilde{F})) =\\big ] - \\int _{0}^{F(\\infty )} F^{-1}(u) \\, du, - \\int _{0}^{F(0)} F^{-1}(u) \\, du \\, \\big [$ .","By choosing the natural anchor point $c=0$ (for both $F$ and $\\widetilde{F}$ ) and by using the relations $\\widetilde{F}(t) = - \\Lambda (F(\\frac{1}{t}))$ , $\\widetilde{F}^{-1}(z) = \\frac{1}{F^{-1}(\\Lambda ^{-1}(-z))}$ , as well as (REF ) in combination with (REF ) (for both contexts), it is straightforward to see that the corresponding quantities $\\widetilde{\\Lambda }$ and $\\widetilde{\\varphi }$ satisfy $\\widetilde{\\Lambda }(z) =- (-\\Lambda )^{-1}(z)$ ($z \\in int(\\mathcal {R}(\\widetilde{F}))$ ) and $\\widetilde{\\varphi }(t) = t \\cdot \\varphi (\\frac{1}{t})$ ($t \\in ]0,\\infty [$ ).","Hence, the corresponding divergences (cf.","(REF )) are “reciprocal to each other” in the sense that $D_{\\widetilde{\\varphi }}( \\mathbf {Q}, \\mathbf {P} ) =D_{\\varphi }( \\mathbf {P}, \\mathbf {Q} )$ for all $\\mathbf {P},\\mathbf {Q} \\in \\mathbb {S}_{>0}^{K}$ , and in case that $\\Lambda $ and $\\widetilde{\\Lambda }$ are indeed cumulant generating functions of some light-tailed distributions $\\mathbb {}$ and $\\widetilde{\\mathbb {}}$ (cf.","Remark REF ), then the latter two are said to be inverse to each other in the sense of Tweedie [368] (see also e.g.","Folks [127]).","As already indicated above, from Theorem REF one can comfortably generate various interesting examples, which we demonstrate in the following.","Example 39 (a) For $\\gamma \\in \\mathbb {R}\\backslash \\lbrace 1,2\\rbrace $ , $\\widetilde{c} \\in ]0,\\infty [$ and $]a_{F_{\\gamma ,\\widetilde{c}}},b_{F_{\\gamma ,\\widetilde{c}}}[ \\, = \\, ]0,\\infty [$ we define $F_{\\gamma ,\\widetilde{c}}(t) &:=&{\\left\\lbrace \\begin{array}{ll}\\frac{\\widetilde{c}}{\\gamma -1} \\cdot (t^{\\gamma -1}-1), \\qquad \\textrm {if } \\ t \\in \\, ]0,\\infty [, \\\\-\\frac{\\widetilde{c}}{\\gamma -1}, \\hspace{62.59596pt}\\textrm {if } \\ t=0 \\ \\textrm {and } \\ \\gamma \\in \\, ]1,2[ \\, \\cup \\, ]2,\\infty [, \\\\-\\infty , \\hspace{71.13188pt} \\textrm {if } \\ t=0 \\ \\textrm {and } \\ \\gamma < 1, \\\\- \\infty , \\hspace{71.13188pt} \\textrm {if } \\ t \\in \\, ]-\\infty ,0[,\\end{array}\\right.","}\\nonumber $ Clearly, $\\mathcal {R}(F_{\\gamma ,\\widetilde{c}})= \\big [-\\frac{\\widetilde{c}}{\\gamma -1},\\infty \\big [$ for $\\gamma \\in \\, ]1,2[ \\, \\cup \\, ]2,\\infty [$ , respectively $\\mathcal {R}(F_{\\gamma ,\\widetilde{c}})=\\big ]-\\infty , \\frac{\\widetilde{c}}{1-\\gamma }\\big [$ for $\\gamma <1$ ; notice that $0 \\in int(\\mathcal {R}(F_{\\gamma ,\\widetilde{c}}))$ for all $\\gamma \\in \\mathbb {R}\\backslash \\lbrace 1,2\\rbrace $ .","Furthermore, $F_{\\gamma ,\\widetilde{c}}(\\cdot )$ is strictly increasing and smooth on $]0,\\infty [$ , and thus, $F_{\\gamma ,\\widetilde{c}} \\in {F}$ .","Since $F_{\\gamma ,\\widetilde{c}}(1)=0$ , let us choose the natural anchor point $c:=0$ , which leads to $]\\lambda _{-},\\lambda _{+}[= int(\\mathcal {R}(F_{\\gamma ,\\widetilde{c}}))$ and $]t_{-}^{sc},t_{+}^{sc}[ = ]0,\\infty [$ .","By using $F_{\\gamma ,\\widetilde{c}}^{-1}(x) = (1+\\frac{(\\gamma -1) \\cdot x}{\\widetilde{c}})^{\\frac{1}{\\gamma -1}}$ for $x \\in int(\\mathcal {R}(F_{\\gamma ,\\widetilde{c}}))$ , we can derive from formula (REF ) (see also (REF )) for all $\\gamma \\in \\mathbb {R}\\backslash \\lbrace 0,1,2\\rbrace $ and $z \\in \\mathbb {R}$ $\\Lambda _{\\gamma ,\\widetilde{c}}(z) := \\Lambda _{\\gamma ,\\widetilde{c}}^{(0)}(z) &=&{\\left\\lbrace \\begin{array}{ll}\\frac{\\widetilde{c}}{\\gamma } \\cdot \\left\\lbrace \\left( \\frac{\\gamma -1}{\\widetilde{c}} \\cdot z +1 \\right)^{\\frac{\\gamma }{\\gamma -1}} -1 \\right\\rbrace , \\qquad \\textrm {if } \\ \\gamma \\in \\, ]1,2[ \\, \\cup \\, ]2,\\infty [\\ \\textrm {and } \\ z \\in \\big ]-\\frac{\\widetilde{c}}{\\gamma -1},\\infty \\big [ \\\\\\hspace{130.88284pt} \\textrm {or if } \\ \\gamma \\in ]-\\infty ,0[ \\cup ]0,1[ \\ \\textrm {and } \\ z \\in \\big ]-\\infty , \\frac{\\widetilde{c}}{1-\\gamma }\\big [, \\\\- \\frac{\\widetilde{c}}{\\gamma } < 0, \\hspace{105.2751pt} \\textrm {if } \\ \\gamma \\in \\, ]1,2[ \\, \\cup \\, ]2,\\infty [\\ \\textrm {and } \\ z = -\\frac{\\widetilde{c}}{\\gamma -1}, \\\\- \\frac{\\widetilde{c}}{\\gamma } >0, \\hspace{105.2751pt} \\textrm {if } \\ \\gamma < 0 \\ \\textrm {and } \\ z = \\frac{\\widetilde{c}}{1-\\gamma }, \\\\\\infty , \\hspace{128.0374pt} \\textrm {if } \\ \\gamma \\in \\, ]0,1[ \\ \\textrm {and } \\ z = \\frac{\\widetilde{c}}{1-\\gamma }, \\\\\\infty , \\hspace{128.0374pt} \\textrm {else} .\\end{array}\\right.","}$ Notice that $\\Lambda _{\\gamma ,\\widetilde{c}}(0) = 0$ for all $\\gamma \\in \\mathbb {R}\\backslash \\lbrace 0,1,2\\rbrace $ .","Moreover, for $\\gamma \\in \\, ]1,2[ \\, \\cup \\, ]2,\\infty [$ one has $\\Lambda _{\\gamma ,\\widetilde{c}}(\\infty ) = \\infty $ , $\\Lambda _{\\gamma ,\\widetilde{c}}^{\\prime }(-\\frac{\\widetilde{c}}{\\gamma -1}) = 0$ and $\\Lambda _{\\gamma ,\\widetilde{c}}^{\\prime }(\\infty ) = \\infty $ .","For $\\gamma < 0$ one gets $\\Lambda _{\\gamma ,\\widetilde{c}}(-\\infty ) = -\\infty $ , $\\Lambda _{\\gamma ,\\widetilde{c}}^{\\prime }(\\frac{\\widetilde{c}}{1-\\gamma }) = \\infty $ and $\\Lambda _{\\gamma ,\\widetilde{c}}^{\\prime }(-\\infty ) = 0$ .","In contrast, if $\\gamma \\in \\, ]0,1[$ then $\\Lambda _{\\gamma ,\\widetilde{c}}(-\\infty ) = - \\frac{\\widetilde{c}}{\\gamma } <0$ , $\\Lambda _{\\gamma ,\\widetilde{c}}^{\\prime }(\\frac{\\widetilde{c}}{1-\\gamma }) = \\infty $ and $\\Lambda _{\\gamma ,\\widetilde{c}}^{\\prime }(-\\infty ) = 0$ .","To proceed, from formula (REF ) (see also ()) we can deduce for all $\\gamma \\in \\mathbb {R}\\backslash \\lbrace 0,1,2\\rbrace $ and $t \\in \\mathbb {R}$ $\\varphi _{\\gamma ,\\widetilde{c}}(t) := \\varphi _{\\gamma ,\\widetilde{c}}^{(0)}(t)\\hspace{-5.69046pt} &=& \\hspace{-5.69046pt}{\\left\\lbrace \\begin{array}{ll}\\widetilde{c} \\cdot \\frac{t^\\gamma -\\gamma \\cdot t+ \\gamma - 1}{\\gamma \\cdot (\\gamma -1)}\\ \\in \\ [0,\\infty [,\\qquad \\quad \\textrm {if }t \\in ]0,\\infty [,\\\\\\frac{\\widetilde{c}}{\\gamma } > 0, \\hspace{108.12054pt} \\textrm {if } \\ \\gamma \\in \\, ]1,2[ \\, \\cup \\, ]2,\\infty [\\ \\textrm {and } \\ t = 0, \\\\\\infty , \\hspace{123.76965pt} \\textrm {if } \\ \\gamma < 0 \\ \\textrm {and } \\ t = 0, \\\\\\frac{\\widetilde{c}}{\\gamma } > 0, \\hspace{108.12054pt} \\textrm {if } \\ \\gamma \\in \\, ]0,1[ \\ \\textrm {and } \\ t = 0, \\\\\\frac{\\widetilde{c}}{\\gamma } -\\frac{\\widetilde{c}}{\\gamma -1} \\cdot t \\ \\in \\ ]0,\\infty [,\\hspace{45.52458pt} \\textrm {if } \\ \\gamma \\in \\, ]1,2[ \\, \\cup \\, ]2,\\infty [\\ \\textrm {and } \\ t < 0, \\\\\\infty , \\hspace{123.76965pt} \\textrm {else} ,\\end{array}\\right.","}$ which coincides with $\\widetilde{c} \\cdot \\varphi _{\\gamma }(t)$ for $\\varphi _{\\gamma }(t)$ from (REF ) and which generates the $\\gamma -$ corresponding power divergences given in (REF ); the first line in (REF ) can be proved by $& & \\hspace{-19.91684pt}\\varphi _{\\gamma ,\\widetilde{c}}(t) := \\varphi _{\\gamma ,\\widetilde{c}}^{(0)}(t) :=t \\cdot F_{\\gamma ,\\widetilde{c}}\\left(t\\right)-\\int \\displaylimits _{0}^{F_{\\gamma ,\\widetilde{c}}\\left(t\\right)} F_{\\gamma ,\\widetilde{c}}^{-1}(u) du\\nonumber \\\\& & \\hspace{-19.91684pt}= \\frac{t \\cdot \\widetilde{c}}{\\gamma -1} \\cdot (t^{\\gamma -1}-1)- \\frac{\\widetilde{c}}{\\gamma } \\cdot \\left\\lbrace \\left( \\frac{\\gamma -1}{\\widetilde{c}} \\cdot \\left[ \\frac{\\widetilde{c}}{\\gamma -1} \\cdot (t^{\\gamma -1}-1) \\right]+ 1 \\right)^{\\frac{\\gamma }{\\gamma -1}} -1 \\right\\rbrace \\nonumber \\\\& & \\hspace{-19.91684pt}= \\widetilde{c} \\cdot \\frac{t^\\gamma -\\gamma \\cdot t+ \\gamma - 1}{\\gamma \\cdot (\\gamma -1)},\\qquad t \\in ]0,\\infty [ \\, .$ Notice that for all $\\gamma \\in \\mathbb {R}\\backslash \\lbrace 0,1,2\\rbrace $ one has $\\varphi _{\\gamma ,\\widetilde{c}}(1) = 0$ , $\\varphi _{\\gamma ,\\widetilde{c}}^{\\prime }(1) = 0$ and $\\varphi _{\\gamma ,\\widetilde{c}}(\\infty ) = \\infty $ .","Moreover, for $\\gamma \\in \\, ]1,2[ \\, \\cup \\, ]2,\\infty [$ one has $\\varphi _{\\gamma ,\\widetilde{c}}^{\\prime }(0) = -\\frac{\\widetilde{c}}{\\gamma -1} < 0$ and $\\varphi _{\\gamma ,\\widetilde{c}}^{\\prime }(\\infty ) = \\infty $ .","In contrast, for $\\gamma < 0$ and $\\gamma \\in \\, ]0,1[$ one gets $\\varphi _{\\gamma ,\\widetilde{c}}^{\\prime }(0) = - \\infty $ and $\\varphi _{\\gamma ,\\widetilde{c}}^{\\prime }(\\infty ) = \\frac{\\widetilde{c}}{1-\\gamma } > 0$ .","(b) For $\\gamma =2$ , $\\widetilde{c} \\in ]0,\\infty [$ and $]a_{F_{\\gamma ,\\widetilde{c}}},b_{F_{\\gamma ,\\widetilde{c}}}[ \\, = \\, ]-\\infty ,\\infty [$ we define $F_{2,\\widetilde{c}}(t) &:=&\\widetilde{c} \\cdot (t-1), \\qquad t \\in \\, ]-\\infty ,\\infty [,\\nonumber $ Clearly, $\\mathcal {R}(F_{2,\\widetilde{c}})= \\, ]-\\infty , \\infty [$ , $0 \\in int(\\mathcal {R}(F_{2,\\widetilde{c}}))$ , and $F_{2,\\widetilde{c}}(\\cdot )$ is strictly increasing as well as smooth on $]-\\infty ,\\infty [$ .","Hence, $F_{2,\\widetilde{c}} \\in {F}$ .","Since $F_{2,\\widetilde{c}}(1)=0$ , let us choose the natural anchor point $c:=0$ , which leads to $]\\lambda _{-},\\lambda _{+}[= int(\\mathcal {R}(F_{2,\\widetilde{c}}))= \\, ]-\\infty , \\infty [$ and $]t_{-}^{sc},t_{+}^{sc}[ = \\, ]-\\infty , \\infty [$ .","By using $F_{2,\\widetilde{c}}^{-1}(x) = 1+\\frac{x}{\\widetilde{c}}$ for $x \\in int(\\mathcal {R}(F_{2,\\widetilde{c}}))$ , we can derive from formula (REF ) (see also (REF )) $\\Lambda _{2,\\widetilde{c}}(z) := \\Lambda _{2,\\widetilde{c}}^{(0)}(z) &=&\\frac{\\widetilde{c}}{2} \\cdot \\left\\lbrace \\left( \\frac{1}{\\widetilde{c}} \\cdot z +1 \\right)^{2} -1 \\right\\rbrace =\\frac{z^2}{2 \\widetilde{c}} + z,\\qquad z \\in \\, ]-\\infty ,\\infty [.$ Notice that $\\Lambda _{2,\\widetilde{c}}(0) = 0$ , $\\Lambda _{2,\\widetilde{c}}(-\\infty ) = \\Lambda _{2,\\widetilde{c}}(\\infty ) = \\infty $ , $\\Lambda _{2,\\widetilde{c}}^{\\prime }(-\\infty ) = -\\infty $ and $\\Lambda _{2,\\widetilde{c}}^{\\prime }(\\infty ) = \\infty $ .","From formula (REF ) (see also ()) we can deduce analogously to (REF ) $\\varphi _{2,\\widetilde{c}}(t) := \\varphi _{2,\\widetilde{c}}^{(0)}(t)&=&\\widetilde{c} \\cdot \\frac{(t-1)^{2}}{2}\\ \\in \\ [0,\\infty [,\\qquad t \\in \\, ]-\\infty ,\\infty [,$ which coincides with $\\widetilde{c} \\cdot \\varphi _{2}(t)$ for $\\varphi _{2}(t)$ from (REF ) which generates the corresponding power divergence given in the sixth line of (REF ).","Notice that $\\varphi _{2,\\widetilde{c}}(1) = 0$ , $\\varphi _{2,\\widetilde{c}}^{\\prime }(1) = 0$ and $\\varphi _{2,\\widetilde{c}}(-\\infty ) = \\varphi _{2,\\widetilde{c}}(\\infty ) = \\infty $ .","As an application of the reciprocity considerations of Remark REF , it is straightforward to see from the above-mentioned considerations (a) and (b) that for all $\\gamma \\in \\mathbb {R}\\backslash \\lbrace 0,1\\rbrace $ one has $\\widetilde{F}_{\\gamma ,\\widetilde{c}}(t) =- \\Lambda _{\\gamma ,\\widetilde{c}}(F_{\\gamma ,\\widetilde{c}}(\\frac{1}{t}))= F_{1-\\gamma ,\\widetilde{c}}(t)$ for all $t \\in ]0,\\infty [$ .","(c) Let us now continue with the remaining case $\\gamma =0$ (recall the natural anchor point $c:=0$ ).","By using $F_{0,\\widetilde{c}}^{-1}(x) =\\frac{1}{1- \\frac{x}{\\widetilde{c}}} $ for $x \\in int(\\mathcal {R}(F_{0,\\widetilde{c}})) = ]-\\infty , \\widetilde{c}[$ , we can derive from formula (REF ) (see also (REF )) $\\Lambda _{0,\\widetilde{c}}(z) := \\Lambda _{0,\\widetilde{c}}^{(0)}(z) &=&{\\left\\lbrace \\begin{array}{ll}- \\widetilde{c} \\cdot \\log \\left( 1 - \\frac{z}{\\widetilde{c}} \\right), \\qquad \\textrm {if } \\ z \\in \\big ]-\\infty ,\\widetilde{c} \\big [, \\\\\\infty , \\hspace{78.24507pt} \\textrm {if } \\ z \\in \\big [\\widetilde{c}, \\infty \\big [ .\\end{array}\\right.","}$ Notice that $\\Lambda _{0,\\widetilde{c}}(0) = 0$ , $\\Lambda _{0,\\widetilde{c}}(-\\infty ) = - \\infty $ , $\\Lambda _{0,\\widetilde{c}}^{\\prime }(\\widetilde{c}) = \\infty $ and $\\Lambda _{0,\\widetilde{c}}^{\\prime }(-\\infty ) = 0$ .","Moreover, from formula (REF ) (see also ()) we can deduce $\\varphi _{0,\\widetilde{c}}(t) := \\varphi _{0,\\widetilde{c}}^{(0)}(t)\\hspace{-5.69046pt} &=& \\hspace{-5.69046pt}{\\left\\lbrace \\begin{array}{ll}\\widetilde{c} \\cdot \\left(- \\log t + t -1 \\right)\\ \\in \\ [0,\\infty [,\\qquad \\quad \\textrm {if }t \\in \\, ]0,\\infty [,\\\\\\infty , \\hspace{145.10922pt} \\textrm {if } \\ t \\in \\, ]-\\infty , 0] ,\\end{array}\\right.","}$ which coincides with $\\widetilde{c} \\cdot \\varphi _{0}(t)$ for the generator $\\varphi _{0}(t)$ from (REF ) which generates the reverse Kullback-Leibler divergence (reverse relative entropy) given in (REF ) with $\\widetilde{c}=1$ ; the first line in (REF ) can be proved by $& & \\hspace{-19.91684pt}\\varphi _{0,\\widetilde{c}}(t) := \\varphi _{0,\\widetilde{c}}^{(0)}(t) :=t \\cdot F_{0,\\widetilde{c}}\\left(t\\right)-\\int \\displaylimits _{0}^{F_{0,\\widetilde{c}}\\left(t\\right)} F_{0,\\widetilde{c}}^{-1}(u) du\\nonumber \\\\& & \\hspace{-19.91684pt}= t \\cdot \\widetilde{c} \\cdot \\Big (1-\\frac{1}{t}\\Big )- (- \\widetilde{c}) \\cdot \\log \\left(1-\\frac{1}{\\widetilde{c}} \\cdot \\left[ \\widetilde{c} \\cdot \\Big ( 1-\\frac{1}{t}\\Big ) \\right]\\right)= \\widetilde{c} \\cdot \\left(- \\log t + t -1 \\right),\\qquad t \\in ]0,\\infty [ \\, .$ Notice that one has $\\varphi _{0,\\widetilde{c}}(1) = 0$ , $\\varphi _{0,\\widetilde{c}}(\\infty ) = \\infty $ , $\\varphi _{0,\\widetilde{c}}^{\\prime }(1) = 0$ , $\\varphi _{0,\\widetilde{c}}^{\\prime }(0) = -\\infty $ and $\\varphi _{0,\\widetilde{c}}^{\\prime }(\\infty ) = \\widetilde{c}$ .","Example 40 (a) For the remaining case $\\gamma =1$ , $\\widetilde{c} \\in ]0,\\infty [$ and $]a_{F_{1,\\widetilde{c}}},b_{F_{1,\\widetilde{c}}}[ = ]0,\\infty [$ we define $F_{1,\\widetilde{c}}(t) &:=&{\\left\\lbrace \\begin{array}{ll}\\widetilde{c} \\cdot \\log t =\\lim _{\\gamma \\rightarrow 1} F_{\\gamma ,\\widetilde{c}}(t), \\qquad \\textrm {if } \\ t \\in \\, ]0,\\infty [, \\\\-\\infty , \\hspace{108.12054pt} \\textrm {if } \\ t \\in ]-\\infty ,0].","\\\\\\end{array}\\right.","}\\nonumber $ Clearly, $\\mathcal {R}(F_{1,\\widetilde{c}})= ]-\\infty ,\\infty [$ .","Moreover, $F_{1,\\widetilde{c}}(\\cdot )$ is strictly increasing and smooth on $]0,\\infty [$ , and hence, $F_{\\gamma ,\\widetilde{c}} \\in {F}$ .","Since $F_{1,\\widetilde{c}}(1)=0$ , let us first choose the natural anchor point $c:=0$ , which leads to $]\\lambda _{-},\\lambda _{+}[= int(\\mathcal {R}(F_{1,\\widetilde{c}})) = ]-\\infty ,\\infty [$ and $]t_{-}^{sc},t_{+}^{sc}[ = ]0,\\infty [$ .","By using $F_{1,\\widetilde{c}}^{-1}(x) = \\exp (\\frac{x}{\\widetilde{c}})$ for $x \\in \\mathcal {R}(F_{1,\\widetilde{c}})$ , we can derive from formula (REF ) (see also (REF )) $& & \\hspace{-19.91684pt} \\Lambda _{1,\\widetilde{c}}(z) := \\Lambda _{1,\\widetilde{c}}^{(0)}(z) :=\\int \\displaylimits _{0}^{z} F_{1,\\widetilde{c}}^{-1}(u) \\, du= \\widetilde{c} \\cdot \\left( \\exp \\Big (\\frac{z}{\\widetilde{c}}\\Big ) - 1 \\right),\\ \\ z \\in ]-\\infty ,\\infty [ .$ Notice that $\\Lambda _{1,\\widetilde{c}}(0) = 0$ , $\\Lambda _{1,\\widetilde{c}}(-\\infty ) = - \\widetilde{c}$ , $\\Lambda _{1,\\widetilde{c}}(\\infty ) = \\infty $ , $\\Lambda _{1,\\widetilde{c}}^{\\prime }(-\\infty ) = 0$ and $\\Lambda _{0,\\widetilde{c}}^{\\prime }(\\infty ) = \\infty $ .","Moreover, from formula (REF ) (see also ()) we can deduce $\\varphi _{1,\\widetilde{c}}(t) := \\varphi _{1,\\widetilde{c}}^{(0)}(t)\\hspace{-5.69046pt} &:=& \\hspace{-5.69046pt}{\\left\\lbrace \\begin{array}{ll}\\widetilde{c} \\cdot \\left( t \\cdot \\log t + 1 - t \\right)\\ \\in \\ [0,\\infty [,\\qquad \\quad \\textrm {if }t \\in \\, ]0,\\infty [,\\\\1, \\hspace{150.79968pt} \\textrm {if } \\ t = 0, \\\\\\infty , \\hspace{145.10922pt} \\textrm {if } \\ t \\in \\, ]-\\infty ,0[ ,\\end{array}\\right.","}$ which coincides with $\\widetilde{c} \\cdot \\varphi _{1}(t)$ for the generator $\\varphi _{1}(t)$ from (REF ) which generates the Kullback-Leibler divergence (relative entropy) given in (REF ) with $\\widetilde{c}=1$ ; the first line in (REF ) can be proved by $& & \\hspace{-19.91684pt}\\varphi _{1,\\widetilde{c}}(t) := \\varphi _{1,\\widetilde{c}}^{(0)}(t) :=t \\cdot F_{1,\\widetilde{c}}\\left(t\\right)-\\int \\displaylimits _{0}^{F_{1,\\widetilde{c}}\\left(t\\right)} F_{1,\\widetilde{c}}^{-1}(u) du\\nonumber \\\\& & \\hspace{-19.91684pt}= t \\cdot \\widetilde{c} \\cdot \\log t- \\widetilde{c} \\cdot \\left( \\exp \\left(\\frac{1}{\\widetilde{c}} \\cdot \\left[ \\widetilde{c} \\cdot \\log t \\right]\\right) - 1 \\right)= \\widetilde{c} \\cdot \\left( t \\cdot \\log t + 1 - t \\right),\\qquad t \\in ]0,\\infty [ \\, ,$ Notice that one has $\\varphi _{1,\\widetilde{c}}(1) = 0$ , $\\varphi _{1,\\widetilde{c}}(\\infty ) = \\infty $ , $\\varphi _{1,\\widetilde{c}}^{\\prime }(1) = 0$ , $\\varphi _{1,\\widetilde{c}}^{\\prime }(0) = -\\infty $ and $\\varphi _{1,\\widetilde{c}}^{\\prime }(\\infty ) = \\infty $ .","As an application of the reciprocity considerations of Remark REF , it is straightforward to see that $\\widetilde{F}_{1,\\widetilde{c}}(t) =- \\Lambda _{1,\\widetilde{c}}(F_{1,\\widetilde{c}}(\\frac{1}{t}))= F_{0,\\widetilde{c}}(t)$ for all $t \\in ]0,\\infty [$ .","(b) For the choice $\\widetilde{c}=1$ , let us now fix a general anchor point $c \\in \\mathcal {R}(F_{1,\\widetilde{c}})= ]-\\infty ,\\infty [$ (rather than $c=0$ ), which leads to $]\\lambda _{-},\\lambda _{+}[= int(\\mathcal {R}(F_{1,1}))- c = ]-\\infty ,\\infty [$ and $]t_{-}^{sc},t_{+}^{sc}[ = ]1+a_{F_{1,1}}-F_{1,1}^{-1}(c),1+b_{F_{1,1}}-F_{1,1}^{-1}(c)[ \\,= \\, ]1-e^{c},\\infty [$ .","Accordingly, the formula (REF ) (see also (REF )) leads to $& & \\hspace{-19.91684pt} \\Lambda _{1,1}(z) := \\Lambda _{1,1}^{(c)}(z) :=\\int \\displaylimits _{0}^{z} F_{1,1}^{-1}(u+c) du+ z \\cdot (1-F_{1,1}^{-1}(c))\\nonumber \\\\& & \\hspace{59.75095pt}= e^{c} \\cdot \\left( e^{z} - 1 \\right) + z \\cdot (1- e^{c}),\\qquad z \\in ]-\\infty ,\\infty [, \\qquad \\ $ for which there holds $\\Lambda _{1,1}^{(c)}(0) = 0$ , $\\Lambda _{1,1}^{(c)}(-\\infty ) =\\infty \\cdot {1}_{]0,\\infty [}(c)- \\infty \\cdot {1}_{]-\\infty ,0[}(c)- 1 \\cdot {1}_{\\lbrace 0\\rbrace }(c)$ , $\\Lambda _{1,1}^{(c)}(\\infty ) = \\infty $ , $\\Lambda _{1,1}^{(c) \\prime }(-\\infty ) = 1- e^{c}$ and $\\Lambda _{1,1}^{(c) \\prime }(\\infty ) = \\infty $ .","Moreover, from formula (REF ) (see also ()) we can deduce $\\varphi _{1,1}(t) := \\varphi _{1,1}^{(c)}(t)\\hspace{-5.69046pt} &:=& \\hspace{-5.69046pt}{\\left\\lbrace \\begin{array}{ll}(t + e^{c} -1) \\cdot [\\log (t + e^{c} -1) -c] + 1 - t\\ \\in \\ [0,\\infty [,\\qquad \\quad \\textrm {if }t \\in \\, ]1-e^{c},\\infty [,\\\\e^{c}, \\hspace{240.42569pt} \\textrm {if } \\ t = 1-e^{c}, \\\\\\infty , \\hspace{239.00298pt} \\textrm {if } \\ t \\in \\, ]-\\infty ,1-e^{c}[ ;\\end{array}\\right.","}$ the first line in (REF ) can be proved by $& & \\hspace{-19.91684pt}\\varphi _{1,1}(t) := \\varphi _{1,1}^{(c)}(t) :=(t+F_{1,1}^{-1}(c)-1) \\cdot [ F_{1,1}\\left(t+F_{1,1}^{-1}(c)-1 \\right) - c ]-\\int \\displaylimits _{0}^{F_{1,1}\\left(t+F_{1,1}^{-1}(c)-1 \\right) - c}F_{1,1}^{-1}(u+c) du\\nonumber \\\\& & \\hspace{-19.91684pt}= (t + e^{c} -1) \\cdot [\\log (t + e^{c} -1) -c]- e^{c} \\cdot \\Big \\lbrace \\exp [\\log (t + e^{c} -1) -c] - 1 \\Big \\rbrace \\nonumber \\\\& & \\hspace{-19.91684pt}= (t + e^{c} -1) \\cdot [\\log (t + e^{c} -1) -c] + 1 - t ,\\qquad t \\in ]1-e^{c},\\infty [\\, .$ Clearly, one has $\\varphi _{1,1}^{(c)}(1) = 0$ , $\\varphi _{1,1}^{(c)}(\\infty ) = \\infty $ , $\\varphi _{1,1}^{(c) \\prime }(1) = 0$ , $\\varphi _{1,1}^{(c) \\prime }(1-e^{c}) = -\\infty $ and $\\varphi _{1,1}^{(c) \\prime }(\\infty ) = \\infty $ .","The corresponding divergence $D_{\\varphi _{1,1}^{(c)}}(\\mathbb {Q},\\mathbb {P})$ has been recently used in Broniatowski et al.","[63] for the important task of testing mixtures of probability distributions; in fact, in order to get considerable comfort in testing mixture-type hypotheses against corresponding marginal-type alternatives, they employ choices $c>0$ since then $\\varphi _{1,1}^{(c)}(t)$ is finite especially for some range of negative values $t<0$ .","The latter feature is also valid for the divergence generator $\\varphi _{bw,\\beta ,\\widetilde{c}}$ in the next example (cf.","(REF ) below).","Example 41 For $\\beta \\in \\, ]0,1]$ , $\\widetilde{c} \\in \\, ]0,\\infty [$ and $]a_{F_{bw,\\beta ,\\widetilde{c}}},b_{F_{bw,\\beta ,\\widetilde{c}}}[ \\,= \\, ]1-\\frac{1}{\\beta },\\infty [$ we define $F_{bw,\\beta ,\\widetilde{c}}(t) &:=&{\\left\\lbrace \\begin{array}{ll}\\frac{\\widetilde{c}}{2\\beta } \\cdot \\Big (1-\\frac{1}{(\\beta \\cdot t + 1 - \\beta )^2}\\Big ), \\qquad \\textrm {if } \\ t \\in \\, ]1-\\frac{1}{\\beta },\\infty [, \\\\- \\infty , \\hspace{93.89418pt} \\textrm {if } \\ t \\in \\, ]-\\infty ,1-\\frac{1}{\\beta }].\\end{array}\\right.","}\\nonumber $ Clearly, $\\mathcal {R}(F_{bw,\\beta ,\\widetilde{c}})= \\big ]-\\infty ,\\frac{\\widetilde{c}}{2\\beta }\\big [$ and $0 \\in int(\\mathcal {R}(F_{bw,\\beta ,\\widetilde{c}}))$ .","Moreover, $F_{bw,\\beta ,\\widetilde{c}}(\\cdot )$ is strictly increasing and smooth on $]1-\\frac{1}{\\beta },\\infty [$ , and thus, $F_{bw,\\beta ,\\widetilde{c}} \\in {F}$ .","Since $F_{bw,\\beta ,\\widetilde{c}}(1)=0$ , let us choose the natural anchor point $c:=0$ , which leads to $]\\lambda _{-},\\lambda _{+}[ \\, = int(\\mathcal {R}(F_{bw,\\beta ,\\widetilde{c}}))= \\big ]-\\infty ,\\frac{\\widetilde{c}}{2\\beta }\\big [$ and $]t_{-}^{sc},t_{+}^{sc}[ \\, = \\,]a_{F_{bw,\\beta ,\\widetilde{c}}},b_{F_{bw,\\beta ,\\widetilde{c}}}[\\, = \\, ]1-\\frac{1}{\\beta },\\infty [$ .","By using $F_{bw,\\beta ,\\widetilde{c}}^{-1}(x) =\\frac{1}{\\beta } \\cdot \\Big \\lbrace \\frac{1}{\\sqrt{1-2\\beta \\cdot x/\\widetilde{c}}} + \\beta -1 \\Big \\rbrace $ for $x \\in int(\\mathcal {R}(F_{bw,\\beta ,\\widetilde{c}}))$ , we can derive from formula (REF ) (see also (REF )) for all $\\beta \\in \\, ]0,1]$ and $z \\in \\mathbb {R}$ $\\Lambda _{bw,\\beta ,\\widetilde{c}}(z) :=\\Lambda _{bw,\\beta ,\\widetilde{c}}^{(0)}(z) &=&{\\left\\lbrace \\begin{array}{ll}-(\\frac{1}{\\beta }-1) \\cdot z + \\frac{\\widetilde{c}}{\\beta ^{2}} \\cdot \\Big \\lbrace 1 - \\sqrt{1-\\frac{2\\beta }{\\widetilde{c}} \\cdot z} \\ \\Big \\rbrace ,\\qquad \\textrm {if } z \\in \\big ]-\\infty ,\\frac{\\widetilde{c}}{2\\beta } \\big ], \\\\\\infty , \\hspace{176.407pt} \\textrm {else} .\\end{array}\\right.","}$ Notice that $\\Lambda _{bw,\\beta ,\\widetilde{c}}(0) = 0$ .","Moreover, $\\Lambda _{bw,\\beta ,\\widetilde{c}}(-\\infty ) = \\infty $ , $\\Lambda _{bw,\\beta ,\\widetilde{c}}(\\frac{\\widetilde{c}}{2\\beta })= \\frac{\\widetilde{c} \\cdot (\\beta +1)}{2 \\beta ^{2}}$ , $\\Lambda _{bw,\\beta ,\\widetilde{c}}^{\\prime }(-\\infty ) = - \\frac{1-\\beta }{\\beta } <0$ and $\\Lambda _{bw,\\beta ,\\widetilde{c}}^{\\prime }(\\frac{\\widetilde{c}}{2\\beta }) = \\infty $ .","Furthermore, from formula (REF ) (see also ()) we can straightforwardly deduce for all $t \\in \\mathbb {R}$ $\\varphi _{bw,\\beta ,\\widetilde{c}}(t) := \\varphi _{bw,\\beta ,\\widetilde{c}}^{(0)}(t)\\hspace{-5.69046pt} &:=& \\hspace{-5.69046pt}{\\left\\lbrace \\begin{array}{ll}\\widetilde{c} \\cdot \\frac{(t-1)^{2}}{2(\\beta \\cdot t +1 -\\beta )}\\ \\in \\ [0,\\infty [,\\qquad \\quad \\textrm {if }t \\in \\, ]1-\\frac{1}{\\beta },\\infty [,\\\\\\infty , \\hspace{120.92421pt} \\textrm {if } \\ t \\in \\, ]-\\infty ,1-\\frac{1}{\\beta }] .\\end{array}\\right.","}$ Note that $1-\\frac{1}{\\beta } <0$ so that negative $t$ are allowed here.","For $t\\ge 0$ , $\\varphi _{bw,\\beta ,\\widetilde{c}}(t)$ is known as Rukhin's generator (cf.","[312], see e.g.","also Marhuenda et al.","[247], Pardo [282]).","Obviously, one has $\\varphi _{bw,\\beta ,\\widetilde{c}}(1) = 0$ , $\\varphi _{bw,\\beta ,\\widetilde{c}}^{\\prime }(1) = 0$ , $\\varphi _{bw,\\beta ,\\widetilde{c}}^{\\prime }(1-\\frac{1}{\\beta }) = -\\infty $ and $\\varphi _{bw,\\beta ,\\widetilde{c}}^{\\prime }(\\infty ) = \\frac{\\widetilde{c}}{2\\beta }$ .","From the generator $\\varphi _{bw,\\beta ,\\widetilde{c}}$ given in (REF ), we build the corresponding divergence (cf.","(REF )) $& &\\hspace{-45.52458pt}D_{\\varphi _{bw,\\beta ,\\widetilde{c}}}(\\mathbf {Q},\\mathbf {P})= \\widetilde{c} \\cdot \\sum _{k=1}^{K} p_{k} \\cdot \\frac{(\\frac{q_{k}}{p_{k}}-1)^{2}}{2(\\beta \\cdot \\frac{q_{k}}{p_{k}} +1 -\\beta )}\\nonumber \\\\& & \\hspace{-45.52458pt}= \\frac{\\widetilde{c}}{2} \\cdot \\sum \\limits _{k=1}^{K}\\frac{(q_{k}-p_{k})^{2}}{\\beta \\cdot q_{k} + (1 -\\beta )\\cdot p_{k}},\\hspace{42.67912pt}\\textrm {if \\mathbf {P} \\in \\mathbb {R}^{K} and \\mathbf {Q} \\in \\mathbb {R}^{K}with \\mathbf {Q} \\in \\, ] \\, \\mathbf {P} \\cdot (1-\\frac{1}{\\beta }),\\infty [ component-wise;}$ for the special subcase $\\widetilde{c}=1$ and $\\mathbf {Q} \\in \\mathbb {R}_{> 0}^{K}$ , $D_{\\varphi _{bw,\\beta ,1}}(\\mathbf {Q},\\mathbf {P})$ can be interpreted as — “non-probability version” of — the well-known blended weight chi-square divergence of Lindsay [221] (see e.g.","also Basu & Lindsay [35], Györfy & Vajda [148], Basu et al.","[36]).","The special case $\\widetilde{c}=1$ and $\\beta =\\frac{1}{2}$ for probability vectors, i.e.","$D_{\\varphi _{bw,1/2,1}}({Q},{P})$ , is equal to (a multiple of the matrix-vector-converted (cf.","Remark REF )) Sanghvi’s genetic difference measure [316] and equal to the double of the so-called (squared) Vincze-Le Cam distance (cf.","Vincze [384], Le Cam [212], see also e.g.","Topsoe [360] — who used the alternative naming triangular discrimination — and Vajda [373]); this divergence $D_{\\varphi _{bw,1/2,1}}({Q},{P})$ has been used e.g.","in Liu et al.","[227] for a machine learning context of detecting salient objects, where ${Q}$ and ${P}$ are appropriate histograms of RGB color.","Remark 42 (a) By straightforward calculations, one can show that $\\varphi _{bw,1,\\widetilde{c}}$ (i.e.","with the choice $\\beta =1$ ) is equal to the $\\widetilde{c}-fold$ power-divergence generator $\\varphi _{\\gamma ,\\widetilde{c}} = \\widetilde{c} \\cdot \\varphi _{\\gamma }$ (cf.","(REF )) with $\\gamma =-1$ ; the corresponding divergence $D_{\\varphi _{bw,1,\\widetilde{c}}}(\\mathbf {Q},\\mathbf {P})$ is thus equal to the power divergence $D_{\\widetilde{c} \\cdot \\varphi _{-1}}(\\mathbf {Q},\\mathbf {P})$ (cf.","(REF )) which is nothing but the — “non-probability version” — of Neyman's chi-square divergence.","(b) For the case $\\beta =0$ — which has been excluded in Example REF for technical brevity — the divergence generator $\\varphi _{bw,0,\\widetilde{c}}$ corresponds to $\\widetilde{c}-fold$ power-divergence generator $\\varphi _{\\gamma ,\\widetilde{c}}$ with $\\gamma =2$ ; the corresponding divergence $D_{\\varphi _{bw,0,\\widetilde{c}}}(\\mathbf {Q},\\mathbf {P})$ is thus equal to the power divergence $D_{\\widetilde{c} \\cdot \\varphi _{2}}(\\mathbf {Q},\\mathbf {P})$ (cf.","(REF )) which is nothing but the — “non-probability version” — of Pearson's (i.e.","the classical) chi-square divergence.","Example 43 Let us give an interesting generalization of the Kullback-Leibler case of Example REF (a).","For $\\widetilde{c} >0$ and $\\alpha \\in \\, ]-1,0[ \\ \\cup \\ ]0,\\infty [$ let us define $F_{gKL,\\alpha ,\\widetilde{c}}(t) &:=&{\\left\\lbrace \\begin{array}{ll}\\widetilde{c} \\cdot \\log \\left( \\frac{(1+\\alpha ) \\cdot t}{1+ \\alpha \\cdot t} \\right), \\qquad \\textrm {if \\lbrace \\alpha \\in \\, ]0,\\infty [ and t \\in \\, ]0,\\infty [ \\rbrace or \\lbrace \\alpha \\in \\, ]-1,0[ and t \\in \\, ]0,-\\frac{1}{\\alpha }[ \\rbrace ,} \\\\- \\infty , \\hspace{71.13188pt} \\textrm {if \\alpha \\in \\, ]-1,0[ \\ \\cup \\ ]0,\\infty [ andt \\in \\, ]-\\infty ,0],} \\\\\\infty , \\hspace{78.52945pt} \\textrm {if \\alpha \\in \\, ]-1,0[ and t \\in \\, [-\\frac{1}{\\alpha },\\infty [,}\\end{array}\\right.","}\\nonumber $ (notice that $\\lim _{\\alpha \\rightarrow 0_{+}} F_{gKL,\\alpha ,\\widetilde{c}}(t)= F_{1,\\widetilde{c}}(t)$ , cf.","Example REF (a)).","Clearly, $]a_{F_{gKL,\\alpha ,\\widetilde{c}}},b_{F_{gKL,\\alpha ,\\widetilde{c}}}[ \\, := \\, ]0,\\infty [$ for $\\alpha \\in \\, ]0,\\infty [$ and $]a_{F_{gKL,\\alpha ,\\widetilde{c}}},b_{F_{gKL,\\alpha ,\\widetilde{c}}}[ \\, := \\, ]0,-\\frac{1}{\\alpha }[$ for $\\alpha \\in \\, ]-1,0[$ .","Moreover, $\\mathcal {R}(F_{gKL,\\alpha ,\\widetilde{c}})=\\, ]-\\infty , \\widetilde{c} \\cdot \\log (1+ \\frac{1}{\\alpha })[$ for $\\alpha \\in \\, ]0,\\infty [$ and $\\mathcal {R}(F_{gKL,\\alpha ,\\widetilde{c}})= \\, ]-\\infty ,\\infty [$ for $\\alpha \\in \\, ]-1,0[$ .","Furthermore, $F_{gKL,\\alpha ,\\widetilde{c}}(\\cdot )$ is strictly increasing and smooth on the respective $]a_{F_{gKL,\\alpha ,\\widetilde{c}}},b_{F_{gKL,\\alpha ,\\widetilde{c}}}[$ , and thus, $F_{gKL,\\alpha ,\\widetilde{c}} \\in {F}$ .","Since $F_{gKL,\\alpha ,\\widetilde{c}}(1)=0$ , let us choose the natural anchor point $c:=0$ , which leads to $]\\lambda _{-},\\lambda _{+}[ \\, = int(\\mathcal {R}(F_{gKL,\\alpha ,\\widetilde{c}})) = \\,]-\\infty , \\widetilde{c} \\cdot \\log (1+ \\frac{1}{\\alpha })[$ and $]t_{-}^{sc},t_{+}^{sc}[ \\, = \\, ]0,\\infty [$ for the case $\\alpha \\in \\, ]0,\\infty [$ , respectively, to $]\\lambda _{-},\\lambda _{+}[ \\, = int(\\mathcal {R}(F_{gKL,\\alpha ,\\widetilde{c}})) = \\, ]-\\infty ,\\infty [$ and $]t_{-}^{sc},t_{+}^{sc}[ \\, = \\, ]0,-\\frac{1}{\\alpha }[$ for the case $\\alpha \\in \\, ]-1,0[$ .","By employing $F_{gKL,\\alpha ,\\widetilde{c}}^{-1}(x) =\\frac{1}{(1+\\alpha ) \\cdot e^{-x/\\widetilde{c}} - \\alpha }$ for $x \\in ]\\lambda _{-},\\lambda _{+}[$ , one can deduce from formula (REF ) (see also (REF )) $& & \\Lambda _{gKL,\\alpha ,\\widetilde{c}}(z) := \\Lambda _{gKL,\\alpha ,\\widetilde{c}}^{(0)}(z)\\nonumber \\\\& & :={\\left\\lbrace \\begin{array}{ll}\\int \\displaylimits _{0}^{z} F_{gKL,\\alpha ,\\widetilde{c}}^{-1}(u) \\, du= - \\frac{\\widetilde{c}}{\\alpha } \\cdot \\log ((1+\\alpha ) - \\alpha \\cdot e^{z/\\widetilde{c}}),\\qquad \\textrm {if \\alpha \\in \\, ]0,\\infty [ and z \\in \\, ]-\\infty , \\widetilde{c} \\cdot \\log (1+ \\frac{1}{\\alpha })[}, \\\\\\int \\displaylimits _{0}^{z} F_{gKL,\\alpha ,\\widetilde{c}}^{-1}(u) \\, du= - \\frac{\\widetilde{c}}{\\alpha } \\cdot \\log ((1+\\alpha ) - \\alpha \\cdot e^{z/\\widetilde{c}}),\\qquad \\textrm {if \\alpha \\in \\, ]-1,0[ and z \\in \\, ]-\\infty , \\infty [}, \\\\\\infty , \\hspace{217.6634pt}\\textrm {if \\alpha \\in \\, ]0,\\infty [ and z \\in \\, [\\widetilde{c} \\cdot \\log (1+ \\frac{1}{\\alpha }), \\infty [}, \\\\\\end{array}\\right.","}$ for which there holds $\\Lambda _{gKL,\\alpha ,\\widetilde{c}}(0) = 0$ and $\\Lambda _{gKL,\\alpha ,\\widetilde{c}}(-\\infty ) = - \\frac{\\widetilde{c}}{\\alpha } \\cdot \\log (1+\\alpha )$ for $\\alpha \\in \\, ]-1,0[ \\ \\cup \\ ]0,\\infty [$ , as well as $\\Lambda _{gKL,\\alpha ,\\widetilde{c}}(\\widetilde{c} \\cdot \\log (1+ \\frac{1}{\\alpha })) = \\infty $ for $\\alpha \\in \\, ]0,\\infty [$ and $\\Lambda _{gKL,\\alpha ,\\widetilde{c}}(\\infty ) = \\infty $ for $\\alpha \\in \\, ]-1,0[$ .","The corresponding derivative satisfies $\\Lambda _{gKL,\\alpha ,\\widetilde{c}}^{\\prime }(- \\infty ) = 0$ for $\\alpha \\in \\, ]-1,0[ \\ \\cup \\ ]0,\\infty [$ , as well as $\\Lambda _{gKL,\\alpha ,\\widetilde{c}}^{\\prime }(\\widetilde{c}\\cdot \\log (1+ \\frac{1}{\\alpha })) = \\infty $ for $\\alpha \\in \\, ]0,\\infty [$ and $\\Lambda _{gKL,\\alpha ,\\widetilde{c}}^{\\prime }(\\infty ) = - \\frac{1}{\\alpha }$ for $\\alpha \\in \\, ]-1,0[$ .","Furthermore, from formula (REF ) (see also ()) one can derive $& & \\hspace{-19.91684pt} \\varphi _{gKL,\\alpha ,\\widetilde{c}}(t) := \\varphi _{gKL,\\alpha ,\\widetilde{c}}^{(0)}(t)\\nonumber \\\\& & \\hspace{-19.91684pt} :={\\left\\lbrace \\begin{array}{ll}\\widetilde{c} \\cdot \\left[ \\, t \\cdot \\log t + (t+\\frac{1}{\\alpha }) \\cdot \\log \\Big ( \\frac{1+\\alpha }{1+\\alpha \\cdot t} \\Big ) \\, \\right]\\ \\in \\ [0,\\infty [,\\quad \\textrm {if \\lbrace \\alpha \\in \\, ]0,\\infty [ and t \\in \\, ]0,\\infty [ \\rbrace or \\lbrace \\alpha \\in \\, ]-1,0[ and t \\in \\, ]0,-\\frac{1}{\\alpha }[ \\rbrace ,}\\\\\\frac{\\widetilde{c}}{\\alpha } \\cdot \\log (1+\\alpha )\\ \\in \\ ]0,\\infty [, \\hspace{106.69783pt}\\textrm {if \\alpha \\in \\, ]-1,0[ \\ \\cup \\ ]0,\\infty [ andt =0,} \\\\\\infty , \\hspace{197.74655pt}\\textrm {if \\alpha \\in \\, ]-1,0[ \\ \\cup \\ ]0,\\infty [ andt \\in \\, ]-\\infty ,0[,} \\\\\\infty , \\hspace{197.74655pt}\\textrm {if \\alpha \\in \\, ]-1,0[ andt \\in \\, [-\\frac{1}{\\alpha },\\infty [;}\\end{array}\\right.","}$ the first line in (REF ) can be proved by $& & \\hspace{-19.91684pt}\\varphi _{gKL,\\alpha ,\\widetilde{c}}(t) := \\varphi _{gKL,\\alpha ,\\widetilde{c}}^{(0)}(t) :=t \\cdot F_{gKL,\\alpha ,\\widetilde{c}}\\left(t\\right)-\\int \\displaylimits _{0}^{F_{gKL,\\alpha ,\\widetilde{c}}\\left(t\\right)}F_{gKL,\\alpha ,\\widetilde{c}}^{-1}(u) \\, du\\nonumber \\\\& & \\hspace{-19.91684pt}= \\widetilde{c} \\cdot t \\cdot \\log \\left( \\frac{(1+\\alpha ) \\cdot t}{1+ \\alpha \\cdot t} \\right)+ \\frac{\\widetilde{c}}{\\alpha } \\cdot \\log \\left( (1 +\\alpha ) -\\alpha \\cdot \\exp \\left[ \\log \\left( \\frac{(1+ \\alpha ) \\cdot t}{1+ \\alpha \\cdot t} \\right) \\right] \\right)\\nonumber \\\\& & \\hspace{-19.91684pt}= \\widetilde{c} \\cdot \\left[ \\, t \\cdot \\log t+ t \\cdot \\log \\Big ( \\frac{1+\\alpha }{1+ \\alpha \\cdot t} \\Big )+ \\frac{1}{\\alpha } \\cdot \\log \\Big ( \\frac{1+\\alpha }{1+ \\alpha \\cdot t} \\Big ) \\, \\right].$ Obviously, one has $\\varphi _{gKL,\\alpha ,\\widetilde{c}}(1) = 0$ , $\\varphi _{gKL,\\alpha ,\\widetilde{c}}^{\\prime }(1) = 0$ , $\\varphi _{gKL,\\alpha ,\\widetilde{c}}^{\\prime }(0) = -\\infty $ for $\\alpha \\in \\, ]-1,0[ \\ \\cup \\ ]0,\\infty [$ .","Moreover, for $\\alpha \\in \\, ]0,\\infty [$ there holds $\\varphi _{gKL,\\alpha ,\\widetilde{c}}(\\infty ) = \\infty $ , and $\\varphi _{gKL,\\alpha ,\\widetilde{c}}^{\\prime }(\\infty ) = \\widetilde{c} \\cdot \\log (1+\\frac{1}{\\alpha })$ , whereas for $\\alpha \\in \\, ]-1,0[$ we obtain $\\varphi _{gKL,\\alpha ,\\widetilde{c}}(-\\frac{1}{\\alpha }) = \\infty $ , and $\\varphi _{gKL,\\alpha ,\\widetilde{c}}^{\\prime }(-\\frac{1}{\\alpha }) = \\infty $ .","From the generator $\\varphi _{gKL,\\alpha ,\\widetilde{c}}$ given in (REF ), we build the corresponding divergence (cf.","(REF )) $& &\\hspace{-42.67912pt}D_{\\varphi _{gKL,\\alpha ,\\widetilde{c}}}(\\mathbf {Q},\\mathbf {P})= \\widetilde{c} \\cdot \\Big \\lbrace \\sum \\limits _{k=1}^{K}q_{k} \\cdot \\log \\Big (\\frac{q_{k}}{(1-\\frac{1}{1+\\alpha }) \\cdot q_{k} + \\frac{1}{1+\\alpha } \\cdot p_{k}} \\Big )+ \\frac{1}{\\alpha } \\cdot \\sum \\limits _{k=1}^{K} p_{k} \\cdot \\log \\Big (\\frac{p_{k}}{(1-\\frac{1}{1+\\alpha }) \\cdot q_{k} + \\frac{1}{1+\\alpha } \\cdot p_{k}} \\Big ) \\Big \\rbrace ,\\\\& & \\textrm {if \\lbrace \\alpha \\in \\, ]0,\\infty [,\\mathbf {P} \\in \\mathbb {R}_{> 0}^{K} and\\mathbf {Q} \\in \\mathbb {R}_{\\ge 0}^{K} \\rbrace or \\lbrace \\alpha \\in \\, ]-1,0[,\\mathbf {P} \\in \\mathbb {R}_{> 0}^{K} and\\mathbf {Q} \\in \\mathbb {R}_{\\ge 0}^{K}with \\mathbf {Q} \\le - \\frac{1}{\\alpha } \\cdot \\mathbf {P} \\rbrace .}","\\nonumber $ Notice that the symmetry $D_{\\varphi _{gKL,\\alpha ,\\widetilde{c}}}(\\mathbf {Q},\\mathbf {P})= D_{\\varphi _{gKL,\\alpha ,\\widetilde{c}}}(\\mathbf {P},\\mathbf {Q})$ generally holds only if $\\mathbf {P}, \\mathbf {Q} \\in \\mathbb {R}_{> 0}^{K}$ and $\\alpha =1$ ; indeed, this special case leads to $\\varphi _{snKL,\\widetilde{c}}(t):= \\varphi _{gKL,1,\\widetilde{c}}(t)\\hspace{-5.69046pt} &:=& \\hspace{-5.69046pt}{\\left\\lbrace \\begin{array}{ll}\\widetilde{c} \\cdot \\left[ \\, t \\cdot \\log t + (t+1) \\cdot \\log \\Big ( \\frac{2}{t+1} \\Big ) \\, \\right]\\ \\in \\ [0,\\infty [,\\qquad \\quad \\textrm {if }t \\in \\, ]0,\\infty [,\\\\\\widetilde{c} \\cdot \\log 2, \\hspace{187.78836pt} \\textrm {if } \\ t = 0, \\\\\\infty , \\hspace{209.12791pt} \\textrm {if } \\ t \\in \\, ]-\\infty ,0[ ,\\end{array}\\right.","}$ and $D_{\\varphi _{snKL,\\widetilde{c}}}(\\mathbf {Q},\\mathbf {P}):= D_{\\varphi _{gKL,1,\\widetilde{c}}}(\\mathbf {Q},\\mathbf {P})= \\widetilde{c} \\cdot \\Big \\lbrace \\sum \\limits _{k=1}^{K} q_{k} \\cdot \\log \\Big (\\frac{2 q_{k}}{q_{k} + p_{k}} \\Big )+ \\sum \\limits _{k=1}^{K} p_{k} \\cdot \\log \\Big (\\frac{2 p_{k}}{q_{k} + p_{k}} \\Big ) \\Big \\rbrace ,\\qquad \\mathbf {P} \\in \\mathbb {R}_{> 0}^{K}, \\mathbf {Q} \\in \\mathbb {R}_{\\ge 0}^{K}.$ For the special subcase that $\\widetilde{c} =1$ and that $\\mathbf {P} = {P}$ , $\\mathbf {Q} = {Q}$ are probability vectors, the divergence (REF ) can be rewritten as sum of two Kullback-Leibler divergences (cf.","(REF )) $D_{\\varphi _{snKL,1}}({Q},{P})= D_{\\varphi _{1}}({Q},({Q}+{P})/2)+ D_{\\varphi _{1}}({P},({Q}+{P})/2),\\qquad {P} \\in \\mathbb {S}_{> 0}^{K}, {Q} \\in \\mathbb {S}_{\\ge 0}^{K},$ which is the well-known (cf.","Burbea & Rao [68], Lin [219], Pardo & Vajda [284], Topsoe [360], Endres & Schindelin [120], Vajda [373], Sason [317]) Jensen-Shannon divergence (being also called symmetrized and normalized Kullback-Leibler divergence, symmetrized and normalized relative entropy, capacitory discrimination); this is equal to the $(2\\log 2)-$ fold of a special (namely, equally-weighted two-population) case of the Sibson information radius of order 1 (cf.","[334]) which has also been addressed e.g.","by Rao [301] for genetic cluster analysis.","By the way, for $\\alpha >0$ the divergence $D_{\\varphi _{gKL,\\alpha ,\\widetilde{c}}}({Q},{P})$ can also be interpreted as a multiple of a special non-equally-weighted Sibson information radius of order 1.","In a context of comparison of — not necessarily connected — networks where ${Q}$ , ${P}$ are probability vectors derived from matrices (cf.","Remark REF ) which are transforms of corresponding graph invariants (e.g.","network portraits), the (matrix-equivalent of the) Jensen-Shannon divergence $D_{\\varphi _{snKL,1}}({Q},{P})$ is also called the network portrait divergence, cf.","Bagrow and Bollt [28].","There is a vast literature on recent applications of the Jensen-Shannon divergence, for instance it appears exemplarily in Kvitsiani et al.","[208] for finding connections between the circuit-level function of different interneuron types in regulating the flow of information and the behavioural functions served by the cortical circuits, in Xu et al.","(2014) for browsing and exploration of video sequences, in Jenkinson et al.","[168] for the fundamental understanding of the epigenome that leads to a powerful approach for studying its role in disease and aging, in Martin et al.","[250] for the implementation of an evolutionary-based global localization filter for mobile robots, in Suo et al.","[354] for the revelation of critical regulators of cell identity in mice, in Abante et al.","[2] for the detection of biologically significant differences in DNA methylation between alleles associated with local changes in genetic sequences — for a better understanding of the mechanism of complex human diseases, in Afek et al.","[5] for revealing mechanisms by which mismatches can recruit transcription factors for modulating replication and repair activities in cells, in Alaiz-Rodriguez & Parnell [10] for the quantification of stability in feature selection and ranking algorithms, in Biau et al.","[53] for generative adversarial networks (GANs) in artificial intelligence and machine learning, in Carre et al.","[74] for the standardization of brain magnetic resonance (MR) images, in Chakraborty et al.","[75] for hierarchical clustering in foreign exchange FOREX markets (e.g.","in periods of major international crises), in Chong et al.","[87] as part of a web-based platform for comprehensive analysis of microbiome data outputs, in Cui et al.","[101] for modelling latent friend recommendation in online social media, in Gholami & Hodtani [134] for refinements of safety-and-security-targeted location verification systems in wireless communication networks (e.g in Intelligent Transportation Systems (ITSs) and vehicular technology), in Guo & Yuan [146] for accurate abnormality classification in semi-supervised Wireless Capsule Endoscopy (WCE) for digestive system cancer diagnosis, in Jiang et al.","[169] for the training of deep neural discriminative and generative networks used for designing and evaluating photonic devices, in Kartal et al.","[186] for uncovering the relationship between some genomic features and cell type-specific methylome diversity, in Laszlovszky et al.","[210] for investigating mechanisms of basal forebrain neurons which modulate synaptic plasticity,cortical processing, brain states and oscillations, in Lawson et al.","[211] for the improved understanding of some genetic circuits that allow cancer cells to evade destruction by the host immune system, in Li et al.","[215] for the search of causes of the progressive neurodevelopmental disorder Rett syndrome, in Machado et al.","[239] for discovering relations between distinct RNA viruses (including SARS-CoV-2), in Mohammadi et al.","[261] for the identification of cell states and their underlying topology, in Mohanty et al.","[262] for the design of implantable nanophotonic (i.e.","chip-scale optical circuit type) silicon probes for sub-millisecond deep-brain optical stimulation — e.g.","for the purpose of gaining a deeper understanding of the neural code, in Perera et al.","[290] for the quantification of the level of rationality in supply chain networks, in Pierri et al.","[294] for the study of growth of malicious/misleading information in some social media diffusion networks, in Rabadan et al.","[299] for the identification of gene mutations that lead to the genesis and progression of tumors, in Reiter et al.","[306] for quantifying metastatic phylogenetic diversity, in Van de Sande et al.","[378] as part of a computational toolbox for single-cell gene regulatory network analysis, in Skinnider et al.","[337] for the prediction of the chemical structures of genomically encoded antibiotics — in order to find means against the looming global crisis of antibiotic resistance, in Tuo et al.","[367] for the detection of high-order single nucleotide polymorphism (SNP) interactions, in Uttam et al.","[370] for predicting the risk of colorectal cancer recurrence and inferring associated tumor microenvironment networks, in Zhang et al.","[421] for incipient fault (namely, crack) detection, in Zhi et al.","[425] for the strengthening of information-centric networks against interest flooding attack (IFAs), in Acera Mateos et al.","[3] for deep-learning classification of SARS-CoV-2 and co-infecting RNA viruses, in Avsec et al.","[24] for uncovering the motifs and syntax of cis-regulatory sequences in genomics data, in Barennes et al.","[32] for comparing the accuracy of current T cell receptor sequencing methods employed for the understanding of adaptive immune responses, in Chen et al.","[79] for clustering high-dimensional microbial data from RNA sequencing, in Chen et al.","[84] for investigating key aspects of effective vocal social communication, in Koldobskiy et al.","[193] for investigations of genetic and epigenetic drivers of paediatric acute lymphoblastic leukaemia, in McGinnis et al.","[258] for evaluating RNA sequencing of pooled blood cell samples, in Mühlroth & Grottke [268] for the detection of emerging trends and technologies through artificial intelligence techniques, in Necci et al.","[272] for the assessment of protein intrinsic disorder predictions, in Okada et al.","[277] for the identification of genetic factors that cause individual differences in whole lymphocyte profiles and their changes after vaccination, and in Zhang et al.","[422] for the learning of functional magnetic resonance imaging (fMRI) time-series in a brain disease diagnosis context.","Remark 44 Let us transform $\\varphi _{gSH,\\alpha }(t) := \\frac{1-t}{\\alpha } \\cdot \\log (1+\\alpha )- \\varphi _{gKL,\\alpha ,1}(t)= - t \\cdot \\log t+ \\frac{1}{\\alpha } \\cdot (1 + \\alpha \\cdot t) \\cdot \\log (1 + \\alpha \\cdot t)- \\frac{1}{\\alpha } \\cdot (1+\\alpha ) \\cdot t \\cdot \\log (1+\\alpha )$ (for $t \\in [0,1]$ ).","The function $\\varphi _{gSH,\\alpha }(\\cdot )$ is strictly concave on $[0,1]$ with $\\varphi _{gSH,\\alpha }(0)= \\varphi _{gSH,\\alpha }(1)=0$ .","Hence, for probability vectors ${Q} =(q_{k})_{k=1,\\ldots ,K}$ , the $\\varphi -$ entropy $\\sum _{k=1}^{K}\\varphi _{gSH,\\alpha }(q_{k})$ is Kapur’s [183] generalization of the Shannon entropy (which corresponds to $\\alpha =0$ in the limit) whose maximization has been connected with generalizations of the Bose-Einstein statistics and the Fermi-Dirac statistics e.g.","in Kapur & Kesavan [185].","Example 45 Let us fix any $z_{1},z_{2} \\in \\mathbb {R}$ , $p \\in ]0,1[$ which satisfy $z_{1} < 1 < z_{2}$ and $z_{1} \\cdot p + z_{2} \\cdot (1-p) =1$ (and thus $p= \\frac{z_{2} -1}{z_{2} - z_{1}}$ ).","On $]a_{F_{twop}},b_{F_{twop}}[ := ]z_{1},z_{2}[$ we define $F_{twop}(t) &:=& \\frac{1}{z_{2}-z_{1}} \\cdot \\log \\left( \\frac{(t-z_{1})\\cdot p}{(z_{2}-t)\\cdot (1- p)} \\right)\\nonumber \\\\&=&\\frac{1}{z_{2}-z_{1}} \\cdot \\log \\left( \\frac{(t-z_{1})\\cdot (z_{2} -1)}{(z_{2}-t)\\cdot (1-z_{1})} \\right) \\, ,\\qquad t \\in \\, ]z_{1},z_{2}[ ,\\nonumber $ where for the last equality we have used the above constraint (in order to obtain a two-parameter representation).","Straightforwardly, we have $\\mathcal {R}(F_{twop})= ]-\\infty ,\\infty [$ .","Moreover, $F_{twop}(\\cdot )$ is strictly increasing and smooth on $]0,\\infty [$ , and thus, $F_{twop} \\in {F}$ .","Since $F_{twop}(1)=0$ , let us choose the natural anchor point $c:=0$ , which leads to $]\\lambda _{-},\\lambda _{+}[= int(\\mathcal {R}(F_{snKL,\\widetilde{c}})) = ]-\\infty , \\infty [$ and $]t_{-}^{sc},t_{+}^{sc}[ = ]z_{1},z_{2}[$ .","By using $F_{twop}^{-1}(x) = \\frac{p \\cdot z_{1} + (1- p) \\cdot z_{2} \\cdot e^{(z_{2}-z_{1}) \\cdot x}}{p + (1- p) \\cdot e^{(z_{2}-z_{1}) \\cdot x}} \\, , \\qquad x \\in ]-\\infty , \\infty [,$ we derive from formula (REF ) (see also (REF )) $& & \\hspace{-19.91684pt} \\Lambda _{twop}(z) := \\Lambda _{twop}^{(0)}(z) :=\\int \\displaylimits _{0}^{z} F_{twop}^{-1}(u) du= \\log \\Big ( p \\cdot e^{z_{1} \\cdot z} + (1- p) \\cdot e^{z_{2} \\cdot z} \\Big ),\\qquad z \\in ]-\\infty , \\infty [ ,$ which has the properties $\\Lambda _{twop}(0) = 0$ , $\\Lambda _{twop}(-\\infty ) =\\infty \\cdot {1}_{]-\\infty ,0[}(z_{1})- \\infty \\cdot {1}_{]0,\\infty [}(z_{1})+ \\log p \\cdot {1}_{\\lbrace 0\\rbrace }(z_{1})$ , $\\Lambda _{twop}(\\infty ) = \\infty $ , $\\Lambda _{twop}^{\\prime }(- \\infty ) = z_{1}$ and $\\Lambda _{twop}^{\\prime }(\\infty ) = z_{2}$ .","Furthermore, from formula (REF ) (see also ()) we deduce $\\varphi _{twop}(t) := \\varphi _{twop}^{(0)}(t)\\hspace{-5.69046pt} &:=& \\hspace{-5.69046pt}{\\left\\lbrace \\begin{array}{ll}\\frac{t-z_{1}}{z_{2}-z_{1}} \\cdot \\log \\left( \\frac{(t-z_{1})\\cdot (z_{2}-1)}{(z_{2}-t)\\cdot (1-z_{1})} \\right)- \\log \\left( \\frac{z_{2}-1}{z_{2}-t} \\right)\\ \\in \\ [0,\\infty [,\\qquad \\quad \\textrm {if }t \\in \\, ]0,\\infty [,\\\\\\log \\left( \\frac{z_{2} - z_{1}}{z_{2} -1} \\right), \\hspace{219.08612pt} \\textrm {if } \\ t = z_{1}, \\\\\\log \\left( \\frac{z_{2} - z_{1}}{1 - z_{1}} \\right), \\hspace{219.08612pt} \\textrm {if } \\ t = z_{2}, \\\\\\infty , \\hspace{233.3125pt} \\textrm {if } \\ t \\in \\, ]-\\infty ,z_{1}[ \\, \\cup \\, ]z_{2}, \\infty [ ;\\end{array}\\right.","}$ the first line in (REF ) can be proved by $& & \\hspace{-19.91684pt}\\varphi _{twop}(t) := \\varphi _{twop}^{(0)}(t) :=t \\cdot F_{twop}\\left(t\\right)-\\int \\displaylimits _{0}^{F_{twop}\\left(t\\right)} F_{twop}^{-1}(u) du\\nonumber \\\\& & \\hspace{-19.91684pt}= \\frac{t}{z_{2}-z_{1}} \\cdot \\log \\left( \\frac{(t-z_{1})\\cdot p}{(z_{2}-t)\\cdot (1- p)} \\right)\\nonumber \\\\& & \\hspace{-19.91684pt}- \\log \\left( p \\cdot \\left( \\frac{(t-z_{1})\\cdot p}{(z_{2}-t)\\cdot (1- p)} \\right)^{\\frac{z_{1}}{z_{2}-z_{1}}}+ (1- p) \\cdot \\left( \\frac{(t-z_{1})\\cdot p}{(z_{2}-t)\\cdot (1- p)} \\right)^{\\frac{z_{2}}{z_{2}-z_{1}}} \\right)\\nonumber \\\\& & \\hspace{-19.91684pt}= \\frac{t-z_{1}}{z_{2}-z_{1}} \\cdot \\log \\left( \\frac{(t-z_{1})\\cdot p}{(z_{2}-t)\\cdot (1- p)} \\right)- \\log \\left( \\frac{(z_{2}-z_{1})\\cdot p}{z_{2}-t} \\right)\\\\& & \\hspace{-19.91684pt}= \\frac{t-z_{1}}{z_{2}-z_{1}} \\cdot \\log \\left( \\frac{(t-z_{1})\\cdot (z_{2}-1)}{(z_{2}-t)\\cdot (1-z_{1})} \\right)- \\log \\left( \\frac{z_{2}-1}{z_{2}-t} \\right) , \\qquad t \\in \\, ]z_{1}, z_{2} [ \\, ,\\nonumber $ where for the last equality we have used the above constraint (to obtain a two-parameter representation).","Straightforwardly, one has $\\varphi _{twop}(1) = 0$ , $\\varphi _{twop}^{\\prime }(1) = 0$ , $\\varphi _{twop}^{\\prime }(z_{1}) = -\\infty $ and $\\varphi _{twop}^{\\prime }(z_{2}) = \\infty $ .","From the generator $\\varphi _{twop}$ given in (REF ), we build the corresponding divergence (cf.","(REF )) $D_{\\varphi _{twop}}(\\mathbf {Q},\\mathbf {P})=\\sum \\limits _{k=1}^{K} \\frac{q_{k} - z_{1} \\cdot p_{k}}{z_{2} - z_{1}} \\cdot \\log \\Big (\\frac{(z_{2}-1) \\cdot (q_{k} - z_{1} \\cdot p_{k})}{(1-z_{1}) \\cdot (z_{2} \\cdot p_{k} - q_{k})} \\Big )- \\sum \\limits _{k=1}^{K} p_{k} \\cdot \\log \\Big (\\frac{(z_{2}-1) \\cdot p_{k}}{z_{2} \\cdot p_{k} - q_{k}} \\Big ) .$ It is known that some types of robustness properties of minimum-divergence estimators are connected with the boundedness of the derivative $\\varphi ^{\\prime }$ of the divergence generator $\\varphi $ ; this property is satisfied for the next Example REF (and its $W-$ concerning continuation in Example REF ), which leads to the new classes of divergences (REF ), (REF ) and (REF ): Example 46 (a) For any parameter-quadrupel $\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c} \\in \\, ]0,\\infty [$ with $\\beta _{1} < \\beta _{2}$ , we choose $]a_{F},b_{F}[\\ \\, := \\ ]a_{F_{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}},b_{F_{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}}[ \\ \\, := \\ \\Big ]1 - \\alpha \\cdot \\frac{(\\beta _{1} - \\beta _{2})^{2} + \\beta _{1}^{2} + \\beta _{1} \\cdot \\beta _{2}}{2\\beta _{1}\\cdot \\beta _{2}\\cdot (\\beta _{2} - \\beta _{1} )}, \\,\\infty \\Big [ \\ \\ni 1\\nonumber $ and define with $\\breve{\\theta } := 1 + \\alpha \\cdot \\Big (\\frac{1}{\\beta _{2}}- \\frac{1}{\\beta _{1}} \\Big ) < 1$ $\\hspace{-17.07182pt}F_{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}(t)&:=& {\\left\\lbrace \\begin{array}{ll}\\widetilde{c} \\cdot \\frac{\\beta _{1}-\\beta _{2}}{2}+ \\frac{\\widetilde{c}}{\\frac{1-t}{\\alpha } + \\frac{1}{\\beta _{2}} - \\frac{1}{\\beta _{1}}}\\cdot \\Big (1 - \\frac{1}{2} \\cdot \\sqrt{4 + \\big (\\frac{1-t}{\\alpha } + \\frac{1}{\\beta _{2}} - \\frac{1}{\\beta _{1}}\\big )^{2} \\cdot (\\beta _{1}+\\beta _{2})^{2}}\\, \\Big ),\\quad \\textrm {if } \\ t \\in \\, ]a_{F},b_{F}[ \\backslash \\lbrace \\breve{\\theta }\\rbrace , \\\\\\widetilde{c} \\cdot \\frac{\\beta _{1}-\\beta _{2}}{2}, \\hspace{277.41437pt}\\textrm {if } \\ t= \\breve{\\theta } \\in \\, ]a_{F},b_{F}[, \\\\- \\widetilde{c} \\cdot \\beta _{1}, \\hspace{284.52756pt} \\textrm {if } \\ t=a_{F}, \\\\- \\infty , \\hspace{295.90848pt} \\textrm {if } \\ t \\in \\, ]-\\infty ,a_{F}[.\\end{array}\\right.","}$ Notice that $\\breve{\\theta } \\in \\, ]a_{F},b_{F}[$ if and only if $\\beta _{1} \\in \\,]\\frac{\\beta _{2}}{3},\\beta _{2}[$ ; if (say) the latter holds, then one has the continuity $\\lim _{t \\rightarrow \\breve{\\theta }}F_{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}(t) = \\widetilde{c} \\cdot \\frac{\\beta _{1}-\\beta _{2}}{2}$ .","For $\\beta _{1} \\le \\frac{\\beta _{2}}{3}$ one gets $]a_{F},b_{F}[ \\backslash \\lbrace \\breve{\\theta } \\rbrace = \\, ]a_{F},b_{F}[$ .","Returning to the general case, one can see in a straightforward way that $F_{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}(\\cdot )$ is strictly increasing and that $\\mathcal {R}(F_{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}})=[-\\widetilde{c}\\cdot \\beta _{1},\\widetilde{c}\\cdot \\beta _{1} [$ .","Furthermore, $F_{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}(\\cdot )$ is smooth on $]a_{F},b_{F}[$ , and thus $F_{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}} \\in {F}$ .","Since $F_{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}(1)=0$ , let us choose the natural anchor point $c:=0$ , which leads to $]\\lambda _{-},\\lambda _{+}[ \\,= int(\\mathcal {R}(F_{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}})= \\, ]-\\widetilde{c}\\cdot \\beta _{1},\\widetilde{c}\\cdot \\beta _{1} [$ and $]t_{-}^{sc},t_{+}^{sc}[ = \\, ]a_{F},b_{F}[$ .","Moreover, it is straightforward to see that the corresponding inverse is $F_{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}^{-1}(x) =1 + \\alpha \\cdot \\Big (\\frac{1}{\\beta _{2}}- \\frac{1}{\\beta _{1}} \\Big ) - \\alpha \\cdot \\frac{\\frac{1}{\\beta _{2}} - \\frac{1}{\\beta _{1}} -\\frac{2 x}{\\widetilde{c} \\cdot \\beta _{1} \\cdot \\beta _{2} }}{1+ \\frac{x}{\\widetilde{c}} \\cdot \\Big (\\frac{1}{\\beta _{2}} - \\frac{1}{\\beta _{1}} \\Big )- \\frac{x^{2}}{\\widetilde{c}^{2} \\cdot \\beta _{1} \\cdot \\beta _{2} }},\\qquad x \\in int(\\mathcal {R}(F_{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}})) ;$ from this, we can derive from formula (REF ) (see also (REF )) for all $z \\in \\mathbb {R}$ $\\hspace{-19.91684pt}\\Lambda _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}(z) :=\\Lambda _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}^{(0)}(z) &=&{\\left\\lbrace \\begin{array}{ll}\\breve{\\theta } \\cdot z - \\widetilde{c} \\cdot \\alpha \\cdot \\log \\Big (1+ \\frac{z}{\\widetilde{c}} \\cdot \\Big (\\frac{1}{\\beta _{2}} - \\frac{1}{\\beta _{1}} \\Big )- \\frac{z^{2}}{\\widetilde{c}^{2} \\cdot \\beta _{1} \\cdot \\beta _{2} }\\Big ),\\quad \\textrm {if } \\ z \\in \\, ]-\\widetilde{c}\\cdot \\beta _{1},\\widetilde{c}\\cdot \\beta _{1} [,\\\\- \\widetilde{c} \\cdot \\breve{\\theta } \\cdot \\beta _{1} -\\widetilde{c} \\cdot \\alpha \\cdot \\log \\Big (2 - 2 \\frac{\\beta _{1}}{\\beta _{2}}\\Big ),\\hspace{73.97733pt} \\textrm {if } \\ z = -\\widetilde{c}\\cdot \\beta _{1},\\\\\\infty , \\hspace{202.01474pt} \\textrm {else} .\\end{array}\\right.","}$ Notice that $\\Lambda _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}(0) = 0$ and $\\lim _{z \\rightarrow \\widetilde{c}\\cdot \\beta _{1}}\\Lambda _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}(z) = \\infty $ .","Moreover, $\\Lambda _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}^{\\prime }( -\\widetilde{c}\\cdot \\beta _{1})= a_{F}$ and $\\Lambda _{ -\\widetilde{c}\\cdot \\beta _{1}}^{\\prime }(\\widetilde{c}\\cdot \\beta _{1}) = \\infty = b_{F}$ (which have to be interpreted as limits, as usual).","To proceed, from formula (REF ) (see also ()) we can deduce for all $t \\in \\mathbb {R}$ $\\hspace{-5.69046pt}\\varphi _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}(t):= \\varphi _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}^{(0)}(t)\\hspace{-8.5359pt} &=& \\hspace{-8.5359pt}{\\left\\lbrace \\begin{array}{ll}\\widetilde{c} \\cdot \\alpha \\cdot \\Big \\lbrace \\frac{\\sqrt{4 + (\\beta _{1} + \\beta _{2})^{2}\\cdot (\\frac{1-t}{\\alpha } + \\frac{1}{\\beta _{2}} - \\frac{1}{\\beta _{1}})^2}\\, - \\, (\\frac{1-t}{\\alpha } + \\frac{1}{\\beta _{2}} - \\frac{1}{\\beta _{1}}) \\cdot (\\beta _{1} - \\beta _{2}) \\, - \\, 2}{2} \\\\+ \\log \\frac{\\sqrt{4 + (\\beta _{1} + \\beta _{2})^{2}\\cdot (\\frac{1-t}{\\alpha } + \\frac{1}{\\beta _{2}} \\, - \\, \\frac{1}{\\beta _{1}})^2} \\, - \\, 2}{\\beta _{1} \\beta _{2} \\cdot (\\frac{1-t}{\\alpha } + \\frac{1}{\\beta _{2}} \\, - \\, \\frac{1}{\\beta _{1}})^{2}} \\Big \\rbrace \\ \\in [0,\\infty [,\\hspace{71.13188pt} \\textrm {if }t \\in \\, ]a_{F},\\infty [,\\\\\\widetilde{c} \\cdot \\alpha \\cdot \\Big \\lbrace \\frac{3 \\beta _{1} - \\beta _{2}}{2 (\\beta _{2} - \\beta _{1})}+ \\log \\frac{2(\\beta _{2} - \\beta _{1})}{\\beta _{2}} \\Big \\rbrace - \\widetilde{c} \\cdot \\beta _{1} \\cdot (t - a_{F}) \\ \\in \\, ]0,\\infty [,\\hspace{19.91684pt} \\textrm {if } \\ t \\in ]-\\infty , a_{F}].\\end{array}\\right.","}$ The first subcase in (REF ) can be proved by $& & \\hspace{-19.91684pt}\\varphi _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}(t) :=\\varphi _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}^{(0)}(t) :=t \\cdot F_{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}\\left(t\\right)\\ - \\int \\displaylimits _{0}^{F_{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}\\left(t\\right)}F_{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}^{-1}(u) \\, du\\nonumber \\\\& & \\hspace{-19.91684pt}= (t - \\breve{\\theta }) \\cdot F_{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}\\left(t\\right)+ \\widetilde{c} \\cdot \\alpha \\cdot \\log \\Big (1+ \\frac{F_{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}\\left(t\\right)}{\\widetilde{c}} \\cdot \\Big (\\frac{1}{\\beta _{2}} - \\frac{1}{\\beta _{1}} \\Big )- \\frac{(F_{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}\\left(t\\right))^{2}}{\\widetilde{c}^{2} \\cdot \\beta _{1} \\cdot \\beta _{2} }\\Big )\\nonumber \\\\& & \\hspace{-19.91684pt}= \\widetilde{c} \\cdot \\alpha \\cdot \\frac{\\sqrt{4 + (\\beta _{1} + \\beta _{2})^{2}\\cdot (\\frac{1-t}{\\alpha } + \\frac{1}{\\beta _{2}} - \\frac{1}{\\beta _{1}})^2}\\, - \\, (\\frac{1-t}{\\alpha } + \\frac{1}{\\beta _{2}} - \\frac{1}{\\beta _{1}}) \\cdot (\\beta _{1} - \\beta _{2}) \\, - \\, 2}{2}\\nonumber \\\\& & \\hspace{-19.91684pt}\\ \\ \\ + \\ \\widetilde{c} \\cdot \\alpha \\cdot \\log \\bigg (1+ \\Big [\\frac{\\beta _{1}-\\beta _{2}}{2}+ \\frac{1}{\\frac{1-t}{\\alpha } + \\frac{1}{\\beta _{2}} - \\frac{1}{\\beta _{1}}}\\cdot \\Big (1 - \\frac{1}{2} \\cdot \\sqrt{4 + \\big (\\frac{1-t}{\\alpha } + \\frac{1}{\\beta _{2}} - \\frac{1}{\\beta _{1}}\\big )^{2} \\cdot (\\beta _{1}+\\beta _{2})^{2}}\\, \\Big ) \\Big ]\\cdot \\frac{\\beta _{1} - \\beta _{2}}{\\beta _{1} \\cdot \\beta _{2}}\\nonumber \\\\& & \\hspace{-19.91684pt}\\ \\ \\ - \\ \\Big [\\frac{\\beta _{1}-\\beta _{2}}{2}+ \\frac{1}{\\frac{1-t}{\\alpha } + \\frac{1}{\\beta _{2}} - \\frac{1}{\\beta _{1}}}\\cdot \\Big (1 - \\frac{1}{2} \\cdot \\sqrt{4 + \\big (\\frac{1-t}{\\alpha } + \\frac{1}{\\beta _{2}} - \\frac{1}{\\beta _{1}}\\big )^{2} \\cdot (\\beta _{1}+\\beta _{2})^{2}}\\, \\Big ) \\Big ]^{2} \\cdot \\frac{1}{\\beta _{1} \\cdot \\beta _{2} }\\bigg )\\nonumber $ and some straightforward calculations.","The second line in (REF ) follows by computing $\\varphi _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}(a_{F})=\\widetilde{c} \\cdot \\alpha \\cdot \\Big \\lbrace \\frac{3 \\beta _{1} - \\beta _{2}}{2 (\\beta _{2} - \\beta _{1})}+ \\log \\frac{2(\\beta _{2} - \\beta _{1})}{\\beta _{2}} \\Big \\rbrace $ .","Notice that $\\varphi _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}(1) = 0$ , $\\varphi _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}^{\\prime }(1) = 0$ , $\\varphi _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}(-\\infty ) = \\infty $ and $\\varphi _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}(\\infty ) = \\infty $ .","Moreover, $\\varphi _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}^{\\prime }(-\\infty ) =\\varphi _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}^{\\prime }(a_{F}) =-\\widetilde{c} \\cdot \\beta _{1}$ and $\\varphi _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}^{\\prime }(\\infty ) =\\widetilde{c} \\cdot \\beta _{1}$ .","From the generator $\\varphi _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}$ given in (REF ), we construct the corresponding divergence (cf.","(REF )) $& &D_{\\varphi _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}}(\\mathbf {Q},\\mathbf {P})= \\sum \\limits _{k=1}^{K} p_{k} \\cdot \\varphi _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}\\Big (\\frac{q_{k}}{p_{k}}\\Big )\\nonumber \\\\& &= \\sum \\limits _{k=1}^{K}p_{k} \\cdot \\bigg [{1}_{]a_{F},\\infty [}(\\frac{q_{k}}{p_{k}}) \\cdot \\widetilde{c} \\cdot \\alpha \\cdot \\Big \\lbrace \\frac{\\sqrt{4 + (\\beta _{1} + \\beta _{2})^{2}\\cdot (\\frac{1-\\frac{q_{k}}{p_{k}}}{\\alpha } + \\frac{1}{\\beta _{2}} - \\frac{1}{\\beta _{1}})^2}\\, - \\, (\\frac{1-\\frac{q_{k}}{p_{k}}}{\\alpha } + \\frac{1}{\\beta _{2}} - \\frac{1}{\\beta _{1}}) \\cdot (\\beta _{1} - \\beta _{2}) \\, - \\, 2}{2}\\nonumber \\\\& &+ \\log \\frac{\\sqrt{4 + (\\beta _{1} + \\beta _{2})^{2}\\cdot (\\frac{1-\\frac{q_{k}}{p_{k}}}{\\alpha } + \\frac{1}{\\beta _{2}} \\, - \\, \\frac{1}{\\beta _{1}})^2} \\, - \\, 2}{\\beta _{1} \\beta _{2} \\cdot (\\frac{1-\\frac{q_{k}}{p_{k}}}{\\alpha } + \\frac{1}{\\beta _{2}} \\, - \\, \\frac{1}{\\beta _{1}})^{2}} \\Big \\rbrace \\nonumber \\\\& &+ \\ {1}_{]-\\infty , a_{F}]}(\\frac{q_{k}}{p_{k}}) \\cdot \\widetilde{c} \\cdot \\Big \\lbrace \\alpha \\cdot \\Big \\lbrace \\frac{3 \\beta _{1} - \\beta _{2}}{2 (\\beta _{2} - \\beta _{1})}+ \\log \\frac{2(\\beta _{2} - \\beta _{1})}{\\beta _{2}} \\Big \\rbrace - \\beta _{1} \\cdot (\\frac{q_{k}}{p_{k}} - a_{F}) \\Big \\rbrace \\bigg ],\\qquad \\mathbf {P} \\in \\mathbb {R}_{\\ge 0}^{K}, \\mathbf {Q} \\in \\mathbb {R}^{K}.$ Notice that we can particularly include the case where $p_{k}=0$ in combination with $q_{k} \\ne 0$ , since $\\lim _{t \\rightarrow 0_{+}} t \\cdot \\varphi _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}(\\frac{1}{t})= \\widetilde{c} \\cdot \\beta _{1}$ and $\\lim _{t \\rightarrow 0_{-}} t \\cdot \\varphi _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}(\\frac{1}{t})= - \\widetilde{c} \\cdot \\beta _{1}$ are both finite, and hence $p_{k} \\cdot \\varphi _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}(\\frac{q_{k}}{p_{k}})= q_{k} \\cdot \\frac{p_{k}}{q_{k}} \\cdot \\varphi _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}(\\frac{q_{k}}{p_{k}}) $ stays finite as $p_{k}$ tends to zero.","(b) For any parameter-quadrupel $\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c} \\in \\, ]0,\\infty [$ with $\\beta _{1} > \\beta _{2}$ , one can proceed analogously to (a).","Let us start by choosing $]a_{F},b_{F}[\\ \\, := \\ ]a_{F_{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}},b_{F_{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}}[ \\ \\, := \\ \\Big ]-\\infty , \\, 1 + \\alpha \\cdot \\frac{(\\beta _{1} - \\beta _{2})^{2} + \\beta _{1} \\cdot \\beta _{2} + \\beta _{2}^{2}}{2\\beta _{1}\\cdot \\beta _{2}\\cdot (\\beta _{1} - \\beta _{2} )}, \\,\\Big [ \\ \\ni 1\\nonumber $ and defining with the same $\\breve{\\theta } := 1 + \\alpha \\cdot \\Big (\\frac{1}{\\beta _{2}}- \\frac{1}{\\beta _{1}} \\Big ) > 1$ $\\hspace{-17.07182pt}F_{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}(t)&:=& {\\left\\lbrace \\begin{array}{ll}\\widetilde{c} \\cdot \\frac{\\beta _{1}-\\beta _{2}}{2}+ \\frac{\\widetilde{c}}{\\frac{1-t}{\\alpha } + \\frac{1}{\\beta _{2}} - \\frac{1}{\\beta _{1}}}\\cdot \\Big (1 - \\frac{1}{2} \\cdot \\sqrt{4 + \\big (\\frac{1-t}{\\alpha } + \\frac{1}{\\beta _{2}} - \\frac{1}{\\beta _{1}}\\big )^{2} \\cdot (\\beta _{1}+\\beta _{2})^{2}}\\, \\Big ),\\quad \\textrm {if } \\ t \\in \\, ]a_{F},b_{F}[ \\backslash \\lbrace \\breve{\\theta }\\rbrace , \\\\\\widetilde{c} \\cdot \\frac{\\beta _{1}-\\beta _{2}}{2}, \\hspace{277.41437pt}\\textrm {if } \\ t= \\breve{\\theta } \\in \\, ]a_{F},b_{F}[, \\\\\\widetilde{c} \\cdot \\beta _{2}, \\hspace{293.06346pt} \\textrm {if } \\ t=b_{F}, \\\\\\infty , \\hspace{304.4444pt} \\textrm {if } \\ t \\in \\, ]b_{F}, \\infty [.\\end{array}\\right.","}$ Clearly, $\\breve{\\theta } \\in \\, ]a_{F},b_{F}[$ if and only if $\\beta _{1} \\in \\,]\\beta _{2}, 3\\beta _{2}[$ ; if (say) the latter holds, then one gets the continuity $\\lim _{t \\rightarrow \\breve{\\theta }}F_{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}(t) = \\widetilde{c} \\cdot \\frac{\\beta _{1}-\\beta _{2}}{2}$ .","For $\\beta _{1} \\le 3 \\beta _{2}$ there holds $]a_{F},b_{F}[ \\backslash \\lbrace \\breve{\\theta } \\rbrace = \\, ]a_{F},b_{F}[$ .","Returning to the general case, one can show comfortably that $F_{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}(\\cdot )$ is strictly increasing and that $\\mathcal {R}(F_{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}})=]-\\widetilde{c}\\cdot \\beta _{2},\\widetilde{c}\\cdot \\beta _{2} ]$ .","Moreover, $F_{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}(\\cdot )$ is smooth on $]a_{F},b_{F}[$ , and hence $F_{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}} \\in {F}$ .","In face of the validity of $F_{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}(1)=0$ , let us choose the natural anchor point $c:=0$ , which amounts to $]\\lambda _{-},\\lambda _{+}[ \\,= int(\\mathcal {R}(F_{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}})= \\, ]-\\widetilde{c}\\cdot \\beta _{2},\\widetilde{c}\\cdot \\beta _{2} [$ and $]t_{-}^{sc},t_{+}^{sc}[ = \\, ]a_{F},b_{F}[$ .","Since the first line in (REF ) coincides formally with that of (REF ) (with different $]a_{F},b_{F}[$ ), the corresponding inverse is formally the same as (REF ) (with different $]a_{F},b_{F}[$ ), and hence $\\hspace{-19.91684pt}\\Lambda _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}(z) :=\\Lambda _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}^{(0)}(z) &=&{\\left\\lbrace \\begin{array}{ll}\\breve{\\theta } \\cdot z - \\widetilde{c} \\cdot \\alpha \\cdot \\log \\Big (1+ \\frac{z}{\\widetilde{c}} \\cdot \\Big (\\frac{1}{\\beta _{2}} - \\frac{1}{\\beta _{1}} \\Big )- \\frac{z^{2}}{\\widetilde{c}^{2} \\cdot \\beta _{1} \\cdot \\beta _{2} }\\Big ),\\quad \\textrm {if } \\ z \\in \\, ]-\\widetilde{c}\\cdot \\beta _{2},\\widetilde{c}\\cdot \\beta _{2} [,\\\\\\widetilde{c} \\cdot \\breve{\\theta } \\cdot \\beta _{2} -\\widetilde{c} \\cdot \\alpha \\cdot \\log \\Big (2 - 2 \\frac{\\beta _{2}}{\\beta _{1}}\\Big ),\\hspace{73.97733pt} \\textrm {if } \\ z = \\widetilde{c}\\cdot \\beta _{2},\\\\\\infty , \\hspace{202.01474pt} \\textrm {else} .\\end{array}\\right.","}$ Notice that $\\Lambda _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}(0) = 0$ and $\\lim _{z \\rightarrow - \\widetilde{c}\\cdot \\beta _{2}}\\Lambda _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}(z) = -\\infty $ .","Furthermore, $\\Lambda _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}^{\\prime }( -\\widetilde{c}\\cdot \\beta _{2})= - \\infty = a_{F}$ and $\\Lambda _{ -\\widetilde{c}\\cdot \\beta _{1}}^{\\prime }(\\widetilde{c}\\cdot \\beta _{2})= b_{F}$ .","To proceed, from formula (REF ) (see also ()) we can derive — analogously to (REF ) — for all $t \\in \\mathbb {R}$ $\\hspace{-5.69046pt}\\varphi _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}(t):= \\varphi _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}^{(0)}(t)\\hspace{-8.5359pt} &=& \\hspace{-8.5359pt}{\\left\\lbrace \\begin{array}{ll}\\widetilde{c} \\cdot \\alpha \\cdot \\Big \\lbrace \\frac{\\sqrt{4 + (\\beta _{1} + \\beta _{2})^{2}\\cdot (\\frac{1-t}{\\alpha } + \\frac{1}{\\beta _{2}} - \\frac{1}{\\beta _{1}})^2}\\, - \\, (\\frac{1-t}{\\alpha } + \\frac{1}{\\beta _{2}} - \\frac{1}{\\beta _{1}}) \\cdot (\\beta _{1} - \\beta _{2}) \\, - \\, 2}{2} \\\\+ \\log \\frac{\\sqrt{4 + (\\beta _{1} + \\beta _{2})^{2}\\cdot (\\frac{1-t}{\\alpha } + \\frac{1}{\\beta _{2}} \\, - \\, \\frac{1}{\\beta _{1}})^2} \\, - \\, 2}{\\beta _{1} \\beta _{2} \\cdot (\\frac{1-t}{\\alpha } + \\frac{1}{\\beta _{2}} \\, - \\, \\frac{1}{\\beta _{1}})^{2}} \\Big \\rbrace \\ \\in [0,\\infty [,\\hspace{71.13188pt} \\textrm {if }t \\in \\, ]-\\infty , b_{F}[,\\\\\\widetilde{c} \\cdot \\alpha \\cdot \\Big \\lbrace \\frac{3 \\beta _{2} - \\beta _{1}}{2 (\\beta _{1} - \\beta _{2})}+ \\log \\frac{2(\\beta _{1} - \\beta _{2})}{\\beta _{1}} \\Big \\rbrace + \\widetilde{c} \\cdot \\beta _{2} \\cdot (t - b_{F}) \\ \\in \\, ]0,\\infty [,\\hspace{19.91684pt} \\textrm {if } \\ t \\in [b_{F}, \\infty [,\\end{array}\\right.","}$ where the last line in (REF ) follows by calculating $\\varphi _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}(b_{F})=\\widetilde{c} \\cdot \\alpha \\cdot \\Big \\lbrace \\frac{3 \\beta _{2} - \\beta _{1}}{2 (\\beta _{1} - \\beta _{2})}+ \\log \\frac{2(\\beta _{1} - \\beta _{2})}{\\beta _{1}} \\Big \\rbrace $ .","Notice that $\\varphi _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}(1) = 0$ , $\\varphi _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}^{\\prime }(1) = 0$ , $\\varphi _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}(-\\infty ) = \\infty $ and $\\varphi _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}(\\infty ) = \\infty $ .","Furthermore, $\\varphi _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}^{\\prime }(-\\infty ) =-\\widetilde{c} \\cdot \\beta _{2}$ and $\\varphi _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}^{\\prime }(\\infty ) =\\varphi _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}^{\\prime }(b_{F}) =\\widetilde{c} \\cdot \\beta _{2}$ .","From the generator $\\varphi _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}$ given in (REF ), we construct the corresponding divergence (cf.","(REF )) $& &D_{\\varphi _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}}(\\mathbf {Q},\\mathbf {P})= \\sum \\limits _{k=1}^{K} p_{k} \\cdot \\varphi _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}\\Big (\\frac{q_{k}}{p_{k}}\\Big )\\nonumber \\\\& &= \\sum \\limits _{k=1}^{K}p_{k} \\cdot \\bigg [{1}_{]-\\infty , b_{F}[}(\\frac{q_{k}}{p_{k}}) \\cdot \\widetilde{c} \\cdot \\alpha \\cdot \\Big \\lbrace \\frac{\\sqrt{4 + (\\beta _{1} + \\beta _{2})^{2}\\cdot (\\frac{1-\\frac{q_{k}}{p_{k}}}{\\alpha } + \\frac{1}{\\beta _{2}} - \\frac{1}{\\beta _{1}})^2}\\, - \\, (\\frac{1-\\frac{q_{k}}{p_{k}}}{\\alpha } + \\frac{1}{\\beta _{2}} - \\frac{1}{\\beta _{1}}) \\cdot (\\beta _{1} - \\beta _{2}) \\, - \\, 2}{2}\\nonumber \\\\& &+ \\log \\frac{\\sqrt{4 + (\\beta _{1} + \\beta _{2})^{2}\\cdot (\\frac{1-\\frac{q_{k}}{p_{k}}}{\\alpha } + \\frac{1}{\\beta _{2}} \\, - \\, \\frac{1}{\\beta _{1}})^2} \\, - \\, 2}{\\beta _{1} \\beta _{2} \\cdot (\\frac{1-\\frac{q_{k}}{p_{k}}}{\\alpha } + \\frac{1}{\\beta _{2}} \\, - \\, \\frac{1}{\\beta _{1}})^{2}} \\Big \\rbrace \\nonumber \\\\& &+ \\ {1}_{[b_{F}, \\infty [}(\\frac{q_{k}}{p_{k}}) \\cdot \\widetilde{c} \\cdot \\Big \\lbrace \\alpha \\cdot \\Big \\lbrace \\frac{3 \\beta _{2} - \\beta _{1}}{2 (\\beta _{1} - \\beta _{2})}+ \\log \\frac{2(\\beta _{1} - \\beta _{2})}{\\beta _{1}} \\Big \\rbrace + \\beta _{2} \\cdot (\\frac{q_{k}}{p_{k}} - b_{F}) \\Big \\rbrace \\bigg ],\\qquad \\mathbf {P} \\in \\mathbb {R}_{\\ge 0}^{K}, \\mathbf {Q} \\in \\mathbb {R}^{K}.$ As above, we can particularly include the case where $p_{k}=0$ in combination with $q_{k} \\ne 0$ , since $\\lim _{t \\rightarrow 0_{+}} t \\cdot \\varphi _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}(\\frac{1}{t})= \\widetilde{c} \\cdot \\beta _{2}$ and $\\lim _{t \\rightarrow 0_{-}} t \\cdot \\varphi _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}(\\frac{1}{t})= - \\widetilde{c} \\cdot \\beta _{2}$ are both finite.","(c) The analysis for the case $\\beta _{1} = \\beta _{2} =: \\beta $ can be obtained by taking $\\lim _{\\beta _1 \\rightarrow \\beta _{2}}$ in (a) respectively (b).","Alternatively, one can start afresh.","Due to its importance and its particularities, we nevertheless state the corresponding results explicitly.","To begin with, for any parameter-triple $\\alpha ,\\beta ,\\widetilde{c} \\in \\, ]0,\\infty [$ we choose $]a_{F},b_{F}[\\ \\, := \\ ]a_{F_{\\alpha ,\\beta ,\\widetilde{c}}},b_{F_{\\alpha ,\\beta ,\\widetilde{c}}}[ \\ \\, := \\ ]-\\infty , \\infty \\, [\\nonumber $ and define with $\\breve{\\theta } := 1$ $\\hspace{-17.07182pt}F_{\\alpha ,\\beta ,\\widetilde{c}}(t)&:=& {\\left\\lbrace \\begin{array}{ll}\\frac{\\widetilde{c} \\cdot \\alpha }{1-t}\\cdot \\Big (1 - \\sqrt{1 + \\big (\\frac{1-t}{\\alpha } \\big )^{2} \\cdot \\beta ^{2}}\\, \\Big ),\\qquad \\textrm {if } \\ t \\in \\, ]a_{F},b_{F}[ \\backslash \\lbrace \\breve{\\theta }\\rbrace , \\\\0, \\hspace{145.10922pt}\\textrm {if } \\ t= \\breve{\\theta }.\\end{array}\\right.","}$ Clearly, one has the continuity $\\lim _{t \\rightarrow \\breve{\\theta }}F_{\\alpha ,\\beta ,\\widetilde{c}}(t) = 0$ .","Moreover, one can see in a straightforward way that $F_{\\alpha ,\\beta ,\\widetilde{c}}(\\cdot )$ is strictly increasing and that $\\mathcal {R}(F_{\\alpha ,\\beta ,\\widetilde{c}})=]-\\widetilde{c}\\cdot \\beta ,\\widetilde{c}\\cdot \\beta [$ .","Furthermore, $F_{\\alpha ,\\beta ,\\widetilde{c}}(\\cdot )$ is smooth on $]a_{F},b_{F}[$ , and thus $F_{\\alpha ,\\beta ,\\widetilde{c}} \\in {F}$ .","Since $F_{\\alpha ,\\beta ,\\widetilde{c}}(1)=0$ , let us choose the natural anchor point $c:=0$ , which leads to the choice $]\\lambda _{-},\\lambda _{+}[ \\,= int(\\mathcal {R}(F_{\\alpha ,\\beta ,\\widetilde{c}})= \\, ]-\\widetilde{c}\\cdot \\beta ,\\widetilde{c}\\cdot \\beta [$ and $]t_{-}^{sc},t_{+}^{sc}[ = \\, ]a_{F},b_{F}[ = \\, ]-\\infty ,\\infty [$ .","The inverse in (REF ) collapses to $F_{\\alpha ,\\beta ,\\widetilde{c}}^{-1}(x) =1 + \\alpha \\cdot \\frac{\\frac{2 x}{\\widetilde{c} \\cdot \\beta ^{2} }}{1 - \\frac{x^{2}}{\\widetilde{c}^{2} \\cdot \\beta ^{2} }},\\qquad x \\in int(\\mathcal {R}(F_{\\alpha ,\\beta ,\\widetilde{c}})) ;$ from this, we can derive from formula (REF ) (see also (REF )) for all $z \\in \\mathbb {R}$ $\\hspace{-19.91684pt}\\Lambda _{\\alpha ,\\beta ,\\widetilde{c}}(z) :=\\Lambda _{\\alpha ,\\beta ,\\widetilde{c}}^{(0)}(z) &=&{\\left\\lbrace \\begin{array}{ll}\\breve{\\theta } \\cdot z - \\widetilde{c} \\cdot \\alpha \\cdot \\log \\Big (1 - \\frac{z^{2}}{\\widetilde{c}^{2} \\cdot \\beta ^{2} }\\Big ),\\qquad \\textrm {if } \\ z \\in \\, ]-\\widetilde{c}\\cdot \\beta ,\\widetilde{c}\\cdot \\beta [,\\\\\\infty , \\hspace{130.88284pt} \\textrm {else} .\\end{array}\\right.","}$ Notice that $\\Lambda _{\\alpha ,\\beta ,\\widetilde{c}}(0) = 0$ , $\\lim _{z \\rightarrow - \\widetilde{c}\\cdot \\beta }\\Lambda _{\\alpha ,\\beta ,\\widetilde{c}}(z) = -\\infty = $ , and $\\lim _{z \\rightarrow \\widetilde{c}\\cdot \\beta }\\Lambda _{\\alpha ,\\beta ,\\widetilde{c}}(z) = \\infty $ .","Furthermore, $\\lim _{z \\rightarrow - \\widetilde{c}\\cdot \\beta }\\Lambda _{\\alpha ,\\beta ,\\widetilde{c}}^{\\prime }(z) = -\\infty = a_{F}$ , and $\\lim _{z \\rightarrow \\widetilde{c}\\cdot \\beta }\\Lambda _{\\alpha ,\\beta ,\\widetilde{c}}^{\\prime }(z) = \\infty = b_{F}$ .","To proceed, the formula (REF ) (respectively, (REF )) collapses to $\\varphi _{\\alpha ,\\beta ,\\widetilde{c}}(t):=\\varphi _{\\alpha ,\\beta ,\\widetilde{c}}^{(0)}(t)= \\widetilde{c} \\cdot \\alpha \\cdot \\Big \\lbrace \\sqrt{1 + \\beta ^{2}\\cdot \\Big (\\frac{1-t}{\\alpha }\\Big )^2} \\, - \\, 1+ \\log \\frac{2 \\cdot \\Big (\\sqrt{1 + \\beta ^{2}\\cdot \\Big (\\frac{1-t}{\\alpha } \\Big )^2} \\, - \\, 1\\Big )}{\\beta ^{2} \\cdot \\Big (\\frac{1-t}{\\alpha }\\Big )^{2}} \\Big \\rbrace \\ \\in [0,\\infty [, \\ \\ t \\in \\, ]-\\infty , \\infty [ \\, = \\, ]a_{F}, b_{F}[.$ Notice that $\\varphi _{\\alpha ,\\beta ,\\widetilde{c}}(1) = 0$ , $\\varphi _{\\alpha ,\\beta }^{\\prime }(1) = 0$ , $\\varphi _{\\alpha ,\\beta ,\\widetilde{c}}(-\\infty ) = \\infty $ and $\\varphi _{\\alpha ,\\beta ,\\widetilde{c}}(\\infty ) = \\infty $ .","Moreover, $\\varphi _{\\alpha ,\\beta ,\\widetilde{c}}^{\\prime }(-\\infty ) =\\varphi _{\\alpha ,\\beta ,\\widetilde{c}}^{\\prime }(a_{F}) =-\\widetilde{c} \\cdot \\beta $ and $\\varphi _{\\alpha ,\\beta ,\\widetilde{c}}^{\\prime }(\\infty ) =\\varphi _{\\alpha ,\\beta ,\\widetilde{c}}^{\\prime }(b_{F}) =\\widetilde{c} \\cdot \\beta $ .","From the generator $\\varphi _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}$ given in (REF ), we construct the corresponding divergence (cf.","(REF )) $& & \\hspace{-19.91684pt}D_{\\varphi _{\\alpha ,\\beta ,\\widetilde{c}}}(\\mathbf {Q},\\mathbf {P})= \\sum \\limits _{k=1}^{K} p_{k} \\cdot \\varphi _{\\alpha ,\\beta ,\\widetilde{c}}\\Big (\\frac{q_{k}}{p_{k}}\\Big )\\nonumber \\\\& & \\hspace{-19.91684pt}= \\widetilde{c} \\cdot \\alpha \\cdot \\sum \\limits _{k=1}^{K}p_{k} \\cdot \\Big \\lbrace \\sqrt{1 + \\beta ^{2}\\cdot \\Big (\\frac{1-\\frac{q_{k}}{p_{k}}}{\\alpha }\\Big )^2} \\, - \\, 1+ \\log \\frac{2 \\cdot \\Big (\\sqrt{1 + \\beta ^{2}\\cdot \\Big (\\frac{1-\\frac{q_{k}}{p_{k}}}{\\alpha } \\Big )^2} \\, - \\, 1\\Big )}{\\beta ^{2} \\cdot \\Big (\\frac{1-\\frac{q_{k}}{p_{k}}}{\\alpha }\\Big )^{2}} \\Big \\rbrace ,\\qquad \\mathbf {P} \\in \\mathbb {R}_{\\ge 0}^{K}, \\mathbf {Q} \\in \\mathbb {R}^{K}.$ As above, we can particularly include the case where $p_{k}=0$ in combination with $q_{k} \\ne 0$ , since $\\lim _{t \\rightarrow 0_{+}} t \\cdot \\varphi _{\\alpha ,\\beta ,\\widetilde{c}}(\\frac{1}{t})= \\widetilde{c} \\cdot \\beta $ and $\\lim _{t \\rightarrow 0_{-}} t \\cdot \\varphi _{\\alpha ,\\beta ,\\widetilde{c}}(\\frac{1}{t})= - \\widetilde{c} \\cdot \\beta $ are both finite.","This ends the current Example REF .","As a side effect in the above-mentioned Example REF , for fixed $\\beta _{2}, \\alpha ,\\widetilde{c}$ notice the interesting behaviour (e.g.","with respect to $int(dom(F)) = ]a_{F},b_{F}[$ and the range of $\\varphi ^{\\prime }$ ) as $\\beta _{1}$ moves from $]0,\\beta _{2}[$ to $\\beta _{2}$ and further to $]\\beta _{2}, \\infty [$ .","Remark 47 The characterization of the probability distribution $$ in (REF ) which may result from Theorem REF — as seen through the above examples — considerably improves other approaches which make use of their identification through the concept of power variance functions of Natural Exponential Families, as developed by Tweedie [369], Morris [267], Letac & Mora [214], and others.","This approach has been used in Broniatowski [58] in a similar perspective as developed here, but can not be extended outside the range of power divergences, in contrast with the Examples REF , REF , REF and REF which can only be handled as a consequence of Theorem REF .","To continue with our general procedure, suppose now that for a divergence generator $\\varphi $ of interest we have concretely/explicitly found (e.g.","by direct calculations or via our $F-$ connection in Theorem REF , see also Remark REF ) its Fenchel-Legendre transform $\\Lambda = \\varphi ^{*}$ ; for this “candidate”, in order to achieve the desired representability (REF ) it remains to verify that $\\exp (\\Lambda (z)) = \\int _{\\mathbb {R}}e^{z \\cdot y} \\, d\\mathbb {} (y),\\qquad z \\in \\mathbb {R},$ for some probability distribution/measure $\\mathbb {} $ on the real line (the light-tailedness in the sense of finiteness on some open interval containing zero, will be typically guaranteed automatically by the assumptions on $\\varphi $ ); of course, this is equivalent to “the existence ” of a random variable $W$ whose moment generating function is equal to $\\exp (\\Lambda )$ (and thus, its cumulant generating function (log moment generating function) is $\\Lambda $ ), i.e.","$\\exp (\\Lambda (z)) = E_{\\mathbb {\\Pi }}[\\exp (z \\cdot W)]\\qquad z \\in \\mathbb {R},$ with $\\mathbb {\\Pi }[W \\in \\cdot \\, ] = \\mathbb {}[ \\, \\cdot \\,]$ ); recall that from this, we need to simulate a sequence $(W_{i})_{i\\in \\mathbb {N}}$ of i.i.d.","copies of $W$ which are the crucial building ingredients of $\\xi _{n}^{\\mathbf {W}}$ in Theorem REF , respectively, of $\\xi _{n,\\mathbf {X}}^{w\\mathbf {W}}$ in Theorem REF .","For the above-mentioned Examples REF to REF , we can give explicit solutions to the representabilities (REF ) respectively (REF ); this is achieved in the following Examples REF to REF (notice that the corresponding supports of $\\mathbb {}$ are explicitly mentioned in the summarizing Table 1 above): Example 48 for the power-divergence context of Example REF we obtain: (a) Case $\\gamma =0$ , $\\widetilde{c} >0$ : $\\Lambda _{0,\\widetilde{c}}(z)= - \\widetilde{c} \\cdot \\log \\left( 1 - \\frac{z}{\\widetilde{c}} \\right)$ (cf.","(REF )) is the cumulant generating function of the Gamma distribution $\\mathbb {} = GAM(\\widetilde{c},\\widetilde{c})$ with rate parameter (inverse scale parameter) $\\widetilde{c}$ and shape parameter $\\widetilde{c}$ ; hence, $\\varphi _{0,\\widetilde{c}} \\in \\Upsilon (]0,\\infty [)$ .","Prominent special case $\\widetilde{c} =1$ : $\\mathbb {} = GAM(1,1) = EXP(1)$ is the exponential distribution with mean 1.","Type: $\\mathbb {}$ is an infinitely divisible (cf.","Proposition REF ) continuous distribution with density $f(y) := \\frac{\\widetilde{c}^{\\widetilde{c}} \\cdot y^{\\widetilde{c}-1} \\cdot e^{-\\widetilde{c} \\cdot y} }{\\Gamma (\\widetilde{c})} \\cdot {1}_{]0,\\infty [}(y)$ ($y \\in \\mathbb {R}$ ).","Behaviour at zero: $\\mathbb {}[ \\, ]0,\\infty [ \\, ] = \\mathbb {\\Pi }[W>0]=1$ .","Corresponding generator: $\\varphi _{0,\\widetilde{c}} = \\widetilde{c} \\cdot \\varphi _{0}$ (cf.","(REF ), (REF )) of the $\\widetilde{c}-$ fold of the reversed Kullback-Leibler divergence (reversed relative entropy) given in the second line of (REF ).","Sums: for i.i.d.","copies $(W_{i})_{i \\in \\mathbb {N}}$ of $W$ , the probability distribution of $\\breve{W} := \\sum _{i\\in I_{k}^{(n)}} W_{i}$ (cf.","Remark REF (ii)) is $GAM(\\widetilde{c},\\widetilde{c} \\cdot card(I_{k}^{(n)}))$ .","(b) Case $\\gamma \\in \\, ]0,1[$ , $\\widetilde{c} >0$ : $\\Lambda _{\\gamma ,\\widetilde{c}}^{(0)}(z)= \\frac{\\widetilde{c}}{\\gamma } \\cdot \\Big \\lbrace \\left( \\frac{\\gamma -1}{\\widetilde{c}} \\cdot z + 1 \\right)^{\\frac{\\gamma }{\\gamma -1}} -1 \\Big \\rbrace $ (cf.","(REF )) is the cumulant generating function of the Compound-Poisson-Gamma distribution $\\mathbb {} = C(POI(\\theta ),GAM(\\alpha ,\\beta ))$ with $\\theta = \\frac{\\widetilde{c}}{\\gamma } > 0$ , rate parameter (inverse scale parameter) $\\alpha = \\frac{\\widetilde{c}}{1-\\gamma } >0$ , and shape parameter $\\beta = \\frac{\\gamma }{1-\\gamma } >0$ .","In other words, $W$ has the comfortably simulable form $W = \\sum _{i=1}^{N} \\widetilde{W}_{i}$ with the usual convention $\\sum _{i=1}^{0} \\widetilde{W}_{i} := 0$ for some i.i.d.","sequence $( \\widetilde{W}_{i})_{i\\in \\mathbb {N}}$ of Gamma $GAM(\\alpha ,\\beta )$ distributed random variables (with parameter-pair $(\\alpha ,\\beta )$ ) and some independent $POI(\\theta )-$ distributed random variable $N$ .","Hence, $\\varphi _{\\gamma ,\\widetilde{c}} \\in \\Upsilon (]0,\\infty [)$ .","Type: $\\mathbb {}$ is an infinitely divisible distribution (cf.","Proposition REF ), mixture of a one-point distribution at zero and a continuous distribution on $[0,\\infty [$ , with $\\mathbb {}[\\lbrace 0\\rbrace ] = \\mathbb {\\Pi }[W = 0]= e^{-\\theta }$ and $\\mathbb {}[B] = \\mathbb {\\Pi }[W \\in B] = \\int _{B} f_{\\widetilde{c},\\gamma }(u) \\, du$ for every (measurable) subset of $]0,\\infty [$ having density $& & \\hspace{-22.76228pt}f_{C(POI(\\theta ),GAM(\\alpha ,\\beta ))}(y): = \\frac{\\exp \\left(- \\alpha \\cdot y - \\theta \\right)}{y}\\cdot \\sum _{k=1}^{\\infty } \\frac{\\theta ^{k} \\cdot (\\alpha y)^{k\\beta }}{k!","\\cdot \\Gamma (k\\beta )}\\cdot {1}_{]0,\\infty [}(y)\\\\& & \\hspace{-22.76228pt}= \\frac{1}{y} \\cdot \\exp \\left(-\\widetilde{c} \\cdot \\left(\\frac{y}{1-\\gamma } + \\frac{1}{\\gamma } \\right) \\right)\\cdot \\sum _{k=1}^{\\infty } \\frac{a_{k}}{k!}","\\cdot \\widetilde{c}^{k/(1-\\gamma )} \\cdot \\gamma ^{-k}\\cdot (1-\\gamma )^{-k\\gamma /(1-\\gamma )} \\cdot y^{k\\gamma /(1-\\gamma )}\\cdot {1}_{[0,\\infty [}(y)=: f_{\\widetilde{c},\\gamma }(y),\\quad y \\in \\mathbb {R},\\nonumber $ where $a_{k} := 1/\\Gamma (\\frac{k \\cdot \\gamma }{1-\\gamma } )$ (see e.g.","Aalen [1] with a different parametrization).","Behaviour at zero: $\\mathbb {}[ \\, [0,\\infty [ \\, ] = \\mathbb {\\Pi }[W \\ge 0]=1$ , $\\mathbb {}[ \\, \\lbrace 0\\rbrace \\, ] = \\mathbb {\\Pi }[W = 0]= e^{-\\theta }$ .","Corresponding generator: $\\varphi _{\\gamma ,\\widetilde{c}}^{(0)} = \\widetilde{c} \\cdot \\varphi _{\\gamma }$ (cf.","(REF ), (REF )) of the power divergence given in the third line of (REF ); recall that the special case $\\gamma =\\frac{1}{2}$ corresponds to the prominent (multiple of the squared) Hellinger distance.","Sums: for i.i.d.","copies $(W_{i})_{i \\in \\mathbb {N}}$ of $W$ , the probability distribution of $\\breve{W} := \\sum _{i\\in I_{k}^{(n)}} W_{i}$ (cf.","Remark REF (ii)) is $C(POI(\\breve{\\theta }),GAM(\\alpha ,\\beta ))$ with $\\breve{\\theta } = \\frac{\\widetilde{c} \\cdot card(I_{k}^{(n)})}{\\gamma } > 0$ , $\\alpha = \\frac{\\widetilde{c}}{1-\\gamma } >0$ , $\\beta = \\frac{\\gamma }{1-\\gamma } >0$ .","(c) Case $\\gamma =2$ , $\\widetilde{c} >0$ : $\\Lambda _{2,\\widetilde{c}}^{(0)}(z)= \\frac{z^{2}}{2 \\widetilde{c}} + z$ (cf.","(REF ) ) is the well-known cumulant generating function of the Normal distribution (Gaussian distribution) $\\mathbb {} = N(1,\\frac{1}{\\widetilde{c}})$ with mean 1 and variance $\\frac{1}{\\widetilde{c}}$ .","Thus, $\\varphi _{2,\\widetilde{c}} \\in \\Upsilon (]-\\infty ,\\infty [)$ .","Type: $\\mathbb {}$ is an infinitely divisible (cf.","Proposition REF ) continuous distribution with density $f_{N(1,\\frac{1}{\\widetilde{c}})}(y) := \\sqrt{\\frac{\\widetilde{c}}{2 \\pi }} \\cdot \\exp (- \\frac{\\widetilde{c} \\cdot (y-1)^2}{2} )$ , ($y \\in \\mathbb {R}$ ).","Behaviour at zero: $\\mathbb {}[ \\, ]0,\\infty [ \\, ] = \\mathbb {\\Pi }[W > 0]=\\int _{0}^{\\infty } f_{N(1,\\frac{1}{\\widetilde{c}})}(u) \\, du \\in \\, ]0,1[$ , $\\mathbb {}[ \\, \\lbrace 0\\rbrace \\, ] = \\mathbb {\\Pi }[W = 0]= 0$ .","Corresponding generator: $\\varphi _{2,\\widetilde{c}}^{(0)} = \\widetilde{c} \\cdot \\varphi _{2}$ (cf.","(REF ), (REF )) is the generator of the $\\widetilde{c}-$ fold of the half Pearson-chisquare divergence given in the sixth line of (REF ).","Sums: for i.i.d.","copies $(W_{i})_{i \\in \\mathbb {N}}$ of $W$ , the probability distribution of $\\breve{W} := \\sum _{i\\in I_{k}^{(n)}} W_{i}$ (cf.","Remark REF (ii)) is $N(card(I_{k}^{(n)}),\\frac{card(I_{k}^{(n)})}{\\widetilde{c}})$ .","(d) Case $\\gamma <0$ , $\\widetilde{c} >0$ : $\\Lambda _{\\gamma ,\\widetilde{c}}^{(0)}(z) = \\frac{\\widetilde{c}}{\\gamma } \\cdot \\left\\lbrace \\left( \\frac{\\gamma -1}{\\widetilde{c}} \\cdot z +1 \\right)^{\\frac{\\gamma }{\\gamma -1}} -1 \\right\\rbrace $ (cf.","(REF )) is the cumulant generating function of a “tilted (i.e.","negatively distorted) stable distribution” $\\mathbb {}[ \\, \\cdot \\,] = \\mathbb {\\Pi }[W \\in \\cdot \\, ]$ of a random variable $W$ , which can be constructed as follows: let $Z$ be an auxiliary random variable (having density $f_{Z}$ and support $supp(Z) = [0,\\infty [$ ) of a stable law with parameter-quadruple $(\\frac{-\\gamma }{1-\\gamma },1,0,-\\frac{\\widetilde{c}^{1/(1-\\gamma )} \\cdot (1-\\gamma )^{-\\gamma /(1-\\gamma )}}{\\gamma })$ in terms of the “form-B notation” on p.12 in Zolotarev [428]; by applying a general Laplace-transform result on p.112 of the same text we can deduce $M_{Z}(z) := E_{\\mathbb {\\Pi }}[\\exp (z \\cdot Z)]= \\int _{0}^{\\infty } \\exp (z \\cdot y) \\cdot f_{Z}(y) \\, dy \\hspace{-5.69046pt} &=& \\hspace{-5.69046pt}{\\left\\lbrace \\begin{array}{ll}\\exp \\Big (\\frac{\\widetilde{c}^{1/(1-\\gamma )} \\cdot (1-\\gamma )^{-\\gamma /(1-\\gamma )}}{\\gamma }\\cdot (-z)^{\\alpha } \\Big ),\\quad \\textrm {if } \\ z \\in ]-\\infty ,0] , \\\\\\infty , \\hspace{150.79968pt} \\textrm {if } \\ z \\in \\, ]0,\\infty [, \\\\\\end{array}\\right.","}$ where $\\alpha := - \\frac{\\gamma }{1-\\gamma } \\in \\, ]0,1[$ .","Since $0 \\notin int(dom(M_{Z}))$ (and thus, $Z$ does not have light-tails) we have to tilt (dampen) the density in order to extend the effective domain.","Accordingly, let $W$ be a random variable having density $f_{W}(y)\\ :=\\ \\frac{\\exp \\lbrace -\\frac{y \\cdot \\widetilde{c}}{1-\\gamma }\\rbrace }{\\exp \\lbrace \\widetilde{c}/\\gamma \\rbrace }\\cdot f_{Z}(y)\\cdot {1}_{]0,\\infty [}(y),\\qquad y \\in \\mathbb {R},\\qquad \\text{(cf.","(\\ref {brostu3:fo.norweiemp7a}))}.\\nonumber $ Then one can straightforwardly deduce from (REF ) that $\\int _{0}^{\\infty } f_{W}(y) \\, dy =1$ and that $M_{W}(z) := E_{\\mathbb {\\Pi }}[\\exp (z \\cdot W)]= \\int _{0}^{\\infty } \\exp (z \\cdot y) \\cdot f_{W}(y) \\, dy \\hspace{-5.69046pt} &=& \\hspace{-5.69046pt}{\\left\\lbrace \\begin{array}{ll}\\exp \\left(\\frac{\\widetilde{c}}{\\gamma } \\cdot \\left\\lbrace \\left( \\frac{\\gamma -1}{\\widetilde{c}} \\cdot z +1 \\right)^{\\frac{\\gamma }{\\gamma -1}} -1 \\right\\rbrace \\right),\\qquad \\textrm {if } \\ z \\in ]-\\infty ,\\frac{\\widetilde{c}}{1-\\gamma }] , \\\\\\infty , \\hspace{156.49014pt} \\textrm {if } \\ z \\in \\, ]\\frac{\\widetilde{c}}{1-\\gamma },\\infty [ .","\\\\\\end{array}\\right.","}\\nonumber $ Hence, $\\varphi _{\\gamma ,\\widetilde{c}} \\in \\Upsilon (]0,\\infty [)$ .","Type: $\\mathbb {}$ is an infinitely divisible (cf.","Proposition REF ) continuous distribution with density $f_{W}$ .","Behaviour at zero: $\\mathbb {}[ \\, ] 0,\\infty [ \\, ] = \\mathbb {\\Pi }[W > 0]=1$ .","Corresponding generator: $\\varphi _{\\gamma ,\\widetilde{c}}^{(0)} = \\widetilde{c} \\cdot \\varphi _{\\gamma }$ (cf.","(REF ), (REF )) of the power divergence given in the first line of (REF ).","Sums: for i.i.d.","copies $(W_{i})_{i \\in \\mathbb {N}}$ of $W$ , the probability distribution of $\\breve{W} := \\sum _{i\\in I_{k}^{(n)}} W_{i}$ (cf.","Remark REF (ii)) has density $f_{\\breve{W}}(y)\\ :=\\ \\frac{\\exp \\lbrace -\\frac{y \\cdot \\widetilde{c}}{1-\\gamma }\\rbrace }{\\exp \\lbrace \\widetilde{c} \\cdot card(I_{k}^{(n)})/\\gamma \\rbrace }\\cdot f_{\\breve{Z}}(y)\\cdot {1}_{]0,\\infty [}(y),\\qquad y \\in \\mathbb {R},$ where $\\breve{Z}$ is a random variable with density $f_{\\breve{Z}}$ of a stable law with parameter-quadruple $(\\frac{-\\gamma }{1-\\gamma },1,0,-card(I_{k}^{(n)}) \\cdot \\frac{\\widetilde{c}^{1/(1-\\gamma )} \\cdot (1-\\gamma )^{-\\gamma /(1-\\gamma )}}{\\gamma })$ .","(e) Case $\\gamma >2$ , $\\widetilde{c} >0$ : $\\Lambda _{\\gamma ,\\widetilde{c}}^{(0)}(z) = \\frac{\\widetilde{c}}{\\gamma } \\cdot \\left\\lbrace \\left( \\frac{\\gamma -1}{\\widetilde{c}} \\cdot z +1 \\right)^{\\frac{\\gamma }{\\gamma -1}} -1 \\right\\rbrace $ (cf.","(REF )) is the cumulant generating function of a “distorted stable distribution” $\\mathbb {}[ \\, \\cdot \\,] = \\mathbb {\\Pi }[W \\in \\cdot \\, ]$ of a random variable $W$ , which can be constructed as follows: let $Z$ be an auxiliary random variable (having density $f_{Z}$ and support $supp(Z) = ]-\\infty ,\\infty ]$ ) of a stable law with parameter-quadruple $(\\frac{\\gamma }{\\gamma -1},1,0,\\frac{\\widetilde{c}^{1/(1-\\gamma )} \\cdot (\\gamma -1)^{\\gamma /(\\gamma -1)}}{\\gamma })$ in terms of the above-mentioned “form-B notation” ; by applying a general Laplace-transform result on p. 112 of Zolotarev [428], we can derive $M_{Z}(z) := E_{\\mathbb {\\Pi }}[\\exp (z \\cdot Z)]= \\int _{0}^{\\infty } \\exp (z \\cdot y) \\cdot f_{Z}(y) \\, dy \\hspace{-5.69046pt} &=& \\hspace{-5.69046pt}{\\left\\lbrace \\begin{array}{ll}\\exp \\Big (\\frac{\\widetilde{c}^{1/(1-\\gamma )} \\cdot (\\gamma -1 )^{\\gamma /(\\gamma -1)}}{\\gamma }\\cdot (-z)^{\\alpha } \\Big ),\\quad \\textrm {if } \\ z \\in ]-\\infty ,0] , \\\\\\infty , \\hspace{145.10922pt} \\textrm {if } \\ z \\in \\, ]0,\\infty [, \\\\\\end{array}\\right.","}$ where $\\alpha := \\frac{\\gamma }{\\gamma -1} \\in \\, ]1,2[$ .","Since $0 \\notin int(dom(M_{Z}))$ (and thus, $Z$ does not have light-tails) we have to distort the density in order to extend the effective domain.","Accordingly, let $W$ be a random variable having density $f_{W}(y)\\ :=\\ \\frac{\\exp \\lbrace \\frac{y \\cdot \\widetilde{c}}{\\gamma -1}\\rbrace }{\\exp \\lbrace \\widetilde{c}/\\gamma \\rbrace }\\cdot f_{Z}(- y), \\qquad y \\in \\mathbb {R},\\qquad \\text{(cf.","(\\ref {brostu3:fo.norweiemp7atwo}))}.\\nonumber $ Then one can straightforwardly deduce from (REF ) that $\\int _{-\\infty }^{\\infty } f_{W}(y) \\, dy =1$ and that $M_{W}(z) := E_{\\mathbb {\\Pi }}[\\exp (z \\cdot W)]= \\int _{-\\infty }^{\\infty } \\exp (z \\cdot y) \\cdot f_{W}(y) \\, dy \\hspace{-5.69046pt} &=& \\hspace{-5.69046pt}{\\left\\lbrace \\begin{array}{ll}\\exp \\left(\\frac{\\widetilde{c}}{\\gamma } \\cdot \\left\\lbrace \\left( \\frac{\\gamma -1}{\\widetilde{c}} \\cdot z +1 \\right)^{\\frac{\\gamma }{\\gamma -1}} -1 \\right\\rbrace \\right),\\qquad \\textrm {if } \\ z \\in [- \\frac{\\widetilde{c}}{\\gamma -1},\\infty [ , \\\\\\infty , \\hspace{156.49014pt} \\textrm {if } \\ z \\in \\, ]-\\infty , -\\frac{\\widetilde{c}}{\\gamma -1}[ .","\\\\\\end{array}\\right.","}\\nonumber $ Thus, $\\varphi _{\\gamma ,\\widetilde{c}} \\in \\Upsilon (]-\\infty ,\\infty [)$ .","Type: $\\mathbb {}$ is an infinitely divisible (cf.","Proposition REF ) continuous distribution with density $f_{W}$ .","Behaviour at zero: $\\mathbb {}[ \\, ]0,\\infty [ \\, ] = \\mathbb {\\Pi }[W > 0]=\\int _{0}^{\\infty } f_{W}(u) \\, du \\in \\, ]0,1[$ , $\\mathbb {}[ \\, \\lbrace 0\\rbrace \\, ] = \\mathbb {\\Pi }[W = 0]= 0$ .","Corresponding generator: $\\varphi _{\\gamma ,\\widetilde{c}}^{(0)} = \\widetilde{c} \\cdot \\varphi _{\\gamma }$ (cf.","(REF ), (REF )) of the power divergence given in the seventh line of (REF ).","Sums: for i.i.d.","copies $(W_{i})_{i \\in \\mathbb {N}}$ of $W$ , the probability distribution of $\\breve{W} := \\sum _{i\\in I_{k}^{(n)}} W_{i}$ (cf.","Remark REF (ii)) has density $f_{\\breve{W}}(y)\\ :=\\ \\frac{\\exp \\lbrace \\frac{y \\cdot \\widetilde{c}}{\\gamma -1}\\rbrace }{\\exp \\lbrace \\widetilde{c} \\cdot card(I_{k}^{(n)})/\\gamma \\rbrace }\\cdot f_{\\breve{Z}}(- y), \\qquad y \\in \\mathbb {R},\\nonumber $ where $\\breve{Z}$ is a random variable with density $f_{\\breve{Z}}$ of a stable law with parameter-quadruple $(\\frac{\\gamma }{\\gamma -1},1,0,card(I_{k}^{(n)})\\cdot \\frac{\\widetilde{c}^{1/(1-\\gamma )} \\cdot (\\gamma -1)^{\\gamma /(\\gamma -1)}}{\\gamma })$ .","(f) Case $\\gamma \\in ]1,2[$ , $\\widetilde{c} >0$ : one still has the (cumulant-generating-function) candidate $\\Lambda _{\\gamma ,\\widetilde{c}}^{(0)}(z) = \\frac{\\widetilde{c}}{\\gamma } \\cdot \\left\\lbrace \\left( \\frac{\\gamma -1}{\\widetilde{c}} \\cdot z +1 \\right)^{\\frac{\\gamma }{\\gamma -1}} -1 \\right\\rbrace $ (cf.","(REF )), but for the crucial exponent there holds $\\frac{\\gamma }{\\gamma -1} > 2$ .","From this, we conjecture that $\\mathbb {}$ becomes a signed finite measure with total mass 1, i.e.","it has a density (with respect to some dominating measure) with positive and negative values which “integrates to 1” ; accordingly, our BS method can not be applied to this situation.","Remark 49 As a continuation of Remark REF and the note in the third line after (REF ), we have shown as a side effect that for $\\gamma \\in \\, ]-\\infty ,-1] \\, \\cup \\, ]0,1[ \\, \\cup \\, [2,\\infty [$ the distributions $\\mathbb {}_{\\gamma }$ and $\\mathbb {}_{1-\\gamma }$ of Example REF (b)-(e) are inverse to each other.","Example 50 for the power-divergence context of Example REF we obtain: (a) Case $\\gamma =1$ , $\\widetilde{c} >0$ , anchor point $c=0$ : $\\Lambda _{1,\\widetilde{c}}(z) =\\widetilde{c} \\cdot \\left( \\exp (\\frac{z}{\\widetilde{c}}) - 1 \\right)$ (cf.","(REF )) is the cumulant generating function of $\\mathbb {} = \\frac{1}{\\widetilde{c}} \\cdot POI(\\widetilde{c})$ being the “$\\frac{1}{\\widetilde{c}}-$ fold Poisson distribution with mean $\\widetilde{c}$ ” which means that $W = \\frac{1}{\\widetilde{c}} \\cdot Z$ for a $POI(\\widetilde{c})-$ distributed random variable $Z$ .","Thus, $\\varphi _{1,\\widetilde{c}} \\in \\Upsilon (]0,\\infty [)$ .","Prominent special case $\\widetilde{c} =1$ : $\\mathbb {} = POI(1)$ is the Poisson distribution with mean 1.","Type: $\\mathbb {}$ is an infinitely divisible (cf.","Proposition REF ) discrete distribution with frequencies: $\\mathbb {\\Pi }[W=\\ell \\cdot \\frac{1}{\\widetilde{c}}]= \\exp (-\\widetilde{c}) \\cdot \\frac{ \\widetilde{c}^{\\ell }}{\\ell !","}$ for all nonnegative integers $\\ell \\in \\mathbb {N}_{0}$ (and zero elsewhere).","Behaviour at zero: $\\mathbb {\\Pi }[W\\ge 0]=1$ , $\\mathbb {\\Pi }[W=0]= \\exp (-\\widetilde{c})$ .","Corresponding generator: $\\varphi _{1,\\widetilde{c}} = \\widetilde{c} \\cdot \\varphi _{1}$ (cf.","(REF ), (REF )) of the $\\widetilde{c}-$ fold of the Kullback-Leibler divergence (relative entropy) given in the fourth line of (REF ).","Sums: for i.i.d.","copies $(W_{i})_{i \\in \\mathbb {N}}$ of $W$ , the probability distribution of $\\breve{W} := \\sum _{i\\in I_{k}^{(n)}} W_{i}$ (cf.","Remark REF (ii)) is $\\frac{1}{\\widetilde{c}} \\cdot POI(\\widetilde{c} \\cdot card(I_{k}^{(n)}))$ .","(b) Case $\\gamma =1$ , $\\widetilde{c} =1$ , anchor point $c \\in \\mathbb {R}$ : $\\Lambda _{1,1}^{(c)}(z)= e^{c} \\cdot \\left( e^{z} - 1 \\right) + z \\cdot (1- e^{c})$ (cf.","(REF )) is the well-known cumulant generating function of the “shifted Poisson distribution” $\\mathbb {} = POI(e^{c}) + 1-e^{c}$ , i.e.","$W := Z + 1-e^{c}$ with a $POI(e^{c})-$ distributed random variable $Z$ .","Hence, $\\varphi _{1,1}^{(c)} \\in \\Upsilon (]1- e^{c},\\infty [)$ .","Type: $\\mathbb {}$ is a discrete distribution with frequencies: $\\mathbb {\\Pi }[W=\\ell + 1- e^{c}]= \\exp (-e^{c}) \\cdot \\frac{ e^{c \\cdot \\ell }}{\\ell !","}$ for all $\\ell \\in \\mathbb {N}_{0}$ (and zero elsewhere).","Behaviour at zero: $\\mathbb {\\Pi }[W > 0]=1$ iff $c <0$ , $\\mathbb {\\Pi }[W < 0] >0$ iff $c >0$ , $\\mathbb {\\Pi }[W=0] \\ne 0$ iff “$c =\\log (1+k)$ for some $k \\in \\mathbb {N}_{0}$ ” .","Corresponding generator: $\\varphi _{1,1}^{(c)}$ (cf.","(REF )) of the divergence $& & D_{\\varphi _{1,1}^{(c)}}(\\mathbf {Q},\\mathbf {P}) :=\\sum \\limits _{k=1}^{K} \\Big (q_{k} + p_{k} \\cdot (e^{c}-1) \\Big )\\cdot \\Big \\lbrace \\log \\Big (\\frac{q_{k}}{p_{k}} + e^{c}-1 \\Big ) - c \\Big \\rbrace - \\sum \\limits _{k=1}^{K} q_{k} + \\sum \\limits _{k=1}^{K} p_{k} ,\\nonumber \\\\& & \\hspace{113.81102pt} \\textrm {if } \\ \\mathbf {P} \\in \\mathbb {R}_{\\ne 0}^{K} \\ \\textrm {and } \\mathbf {Q} \\in \\mathbb {R}^{K}\\textrm {with } \\mathbf {Q} \\in \\, [ (1-e^{c}) \\cdot \\mathbf {P}, \\mathbf {\\infty } [\\ \\textrm {component-wise},$ which for $c=0$ coincides with the Kullback-Leibler divergence (relative entropy) given in the fourth line of (REF ).","Sums: for i.i.d.","copies $(W_{i})_{i \\in \\mathbb {N}}$ of $W$ , the probability distribution of $\\breve{W} := \\sum _{i\\in I_{k}^{(n)}} W_{i}$ (cf.","Remark REF (ii)) is $POI(card(I_{k}^{(n)}) \\cdot e^{c}) + (1-e^{c}) \\cdot card(I_{k}^{(n)})$ .","Remark 51 (a) One can see from the Examples REF and REF the interesting effect that the “homogeneous” class of power-divergence generators $(\\varphi _{\\gamma })_{\\gamma \\in \\mathbb {R}}$ are connected to a “very inhomogeneous” family $(\\mathbb {}_{\\gamma })_{\\gamma \\in \\mathbb {R}}$ of $W-$ distributions: discrete, continuous, mixture of discrete and continuous, as the parameter $\\gamma $ varies.","Moreover, some cases satisfy $\\mathbb {\\Pi }[W=0] =0$ and some $\\mathbb {\\Pi }[W=0] >0$ , some $\\mathbb {\\Pi }[W>0]=1$ and some $\\mathbb {\\Pi }[W>0] \\in ]0,1[$ .","(b) As a continuation of Remark REF and the note in the last line of Example REF (a), we have shown as a side effect that for the the natural-anchor-point choice $c=0$ , the distributions $\\mathbb {}_{1}$ of of Example REF (a) and $\\mathbb {}_{0}$ of Example REF (a) are inverse to each other.","Example 52 for the context of Example REF we obtain: Case $\\widetilde{c} >0$ , anchor point $c=0$ : $\\Lambda _{bw,\\beta ,\\widetilde{c}}(z) =-(\\frac{1}{\\beta }-1) \\cdot z + \\frac{\\widetilde{c}}{\\beta ^{2}} \\cdot \\Big \\lbrace 1 - \\sqrt{1-\\frac{2\\beta }{\\widetilde{c}} \\cdot z} \\ \\Big \\rbrace $ (cf.","(REF )) is the cumulant generating function of a probability distribution $\\mathbb {}[ \\, \\cdot \\,] = \\mathbb {\\Pi }[\\check{W} \\in \\cdot \\, ]$ of a random variable $\\check{W}$ , which can be constructed as follows: $\\check{W} := \\frac{W}{\\beta } - (\\frac{1}{\\beta } - 1)$ , where $W$ is the random variable constructed in Example REF (d) with $\\gamma =-1$ and with $\\widetilde{c}$ replaced by $\\frac{\\widetilde{c}}{\\beta ^{2}}$ (recall that $W$ has a tilted stable distribution).","In other words, $\\mathbb {}$ is a special kind of modified tilted stable distribution.","Type: $\\mathbb {}$ is an infinitely divisible (cf.","Proposition REF ) continuous distribution with density $f_{\\check{W}}(u) := \\beta \\cdot f_{W}(\\beta \\cdot u + 1 -\\beta )\\cdot \\mathbf {1}_{]- (\\frac{1}{\\beta } -1),\\infty [}(u)$ ($u \\in \\mathbb {R}$ ), where $f_{W}(\\cdot )$ is given in (REF ) with $\\gamma =-1$ and with $\\widetilde{c}$ replaced by $\\frac{\\widetilde{c}}{\\beta ^{2}}$ .","Behaviour at zero: $\\mathbb {}[ \\, ] 0,\\infty [ \\, ] =\\mathbb {\\Pi }[\\check{W} > 0] >0$ .","Corresponding generator: $\\varphi _{bw,\\beta ,\\widetilde{c}}^{(0)}$ (cf.","(REF )) of the — “non-probability version” of — the well-known blended weight chi-square divergence given in (REF ).","Sums: for i.i.d.","copies $(\\check{W}_{i})_{i \\in \\mathbb {N}}$ of $\\check{W}$ , the probability distribution of $\\breve{\\check{W}} := \\sum _{i\\in I_{k}^{(n)}} \\check{W}_{i}= \\frac{1}{\\beta } \\cdot \\sum _{i\\in I_{k}^{(n)}} W_{i} - n_{k}\\cdot (\\frac{1}{\\beta } -1)$ (cf.","Remark REF (ii)) has density $f_{\\breve{\\check{W}}}(u) := \\beta \\cdot f_{\\breve{W}}(\\beta \\cdot u + (1 -\\beta )\\cdot n_{k})\\cdot \\mathbf {1}_{]- n_{k} \\cdot (\\frac{1}{\\beta } -1),\\infty [}(u)$ ($u \\in \\mathbb {R}$ ), where $f_{\\breve{W}}(\\cdot )$ is given in (REF ) (cf.","Example REF (d)) with $\\gamma =-1$ and with $\\widetilde{c}$ replaced by $\\frac{\\widetilde{c}}{\\beta ^{2}}$ .","Example 53 for the context of Example REF we obtain: (a) Case $\\alpha \\in \\, ]0,\\infty [$ , $\\widetilde{c} >0$ , anchor point $c=0$ : $\\Lambda _{gKL,\\alpha ,\\widetilde{c}}(z) =- \\frac{\\widetilde{c}}{\\alpha } \\cdot \\log ((1+\\alpha ) - \\alpha \\cdot e^{z/\\widetilde{c}})$ (cf.","(REF )) is the cumulant generating function of $\\mathbb {} = \\frac{1}{\\widetilde{c}} \\cdot NB(\\frac{\\widetilde{c}}{\\alpha },\\frac{1}{1+\\alpha })$ being the “$\\frac{1}{\\widetilde{c}}-$ fold Negative-Binomial distribution with parameters $\\frac{\\widetilde{c}}{\\alpha }$ and $\\frac{1}{1+\\alpha }$ ” which means that $W = \\frac{1}{\\widetilde{c}} \\cdot Z$ for a $NB(\\frac{\\widetilde{c}}{\\alpha },\\frac{1}{1+\\alpha })-$ distributed random variable $Z$ .","Thus, $\\varphi _{gKL,\\alpha ,\\widetilde{c}} \\in \\Upsilon (]0,\\infty [)$ .","Prominent special case $\\widetilde{c}=1$ , $\\alpha =1$ (see below): $\\mathbb {} = NB(1,\\frac{1}{2})$ is the Negative-Binomial distribution with parameters 1 and $\\frac{1}{2}$ .","Type: $\\mathbb {}$ is an infinitely divisible (cf.","Proposition REF ) discrete distribution with frequencies: $\\mathbb {\\Pi }[W=\\ell \\cdot \\frac{1}{\\widetilde{c}}]=(-1)^{\\ell } \\cdot \\binom{- \\frac{\\widetilde{c}}{\\alpha }}{\\ell }\\cdot {\\alpha }^{\\ell } \\cdot (1+\\alpha )^{-\\ell - \\widetilde{c}/\\alpha }$ for all nonnegative integers $\\ell \\in \\mathbb {N}_{0}$ (and zero elsewhere).","Behaviour at zero: $\\mathbb {\\Pi }[W\\ge 0]=1$ , $\\mathbb {\\Pi }[W=0]= \\frac{1}{(1+a)^{\\widetilde{c}/\\alpha }}$ .","Corresponding generator: $\\varphi _{gKL,\\alpha ,\\widetilde{c}}$ (cf.","(REF )) of the divergence (REF ); the special case $\\widetilde{c}=1$ , $\\alpha =1$ — i.e.","$\\varphi _{gKL,1,1} =: \\varphi _{snKL,1}$ (cf.","(REF )) — corresponds to the generator of the — “non-probability version” of the — Jensen-Shannon divergence (symmetrized and normalized Kullback-Leibler divergence, symmetrized and normalized relative entropy) given in (REF ).","Sums: for i.i.d.","copies $(W_{i})_{i \\in \\mathbb {N}}$ of $W$ , the probability distribution of $\\breve{W} := \\sum _{i\\in I_{k}^{(n)}} W_{i}$ (cf.","Remark REF (ii)) is $\\frac{1}{\\widetilde{c}} \\cdot NB(\\frac{\\widetilde{c}}{\\alpha }\\cdot card(I_{k}^{(n)}),\\frac{1}{1+\\alpha })$ .","(b) Case $\\alpha \\in \\, ]-1,0[$ , $\\widetilde{c} >0$ , anchor point $c=0$ : for any integer $m \\in \\mathbb {N}$ being strictly larger than $\\widetilde{c}$ and the choice $\\alpha = - \\frac{\\widetilde{c}}{m}$ , we obtain $\\Lambda _{gKL,-\\widetilde{c}/m,\\widetilde{c}}(z) =m \\cdot \\log ((1 - \\frac{\\widetilde{c}}{m}) + \\frac{\\widetilde{c}}{m} \\cdot e^{z/\\widetilde{c}})$ (cf.","(REF )) which is the cumulant generating function of $\\mathbb {} = \\frac{1}{\\widetilde{c}} \\cdot BIN(m,\\frac{\\widetilde{c}}{m})$ being the “$\\frac{1}{\\widetilde{c}}-$ fold Binomial distribution with parameters $m$ and $\\frac{\\widetilde{c}}{m}$ ” which means that $W = \\frac{1}{\\widetilde{c}} \\cdot Z$ for a $BIN(m,\\frac{\\widetilde{c}}{m})-$ distributed random variable $Z$ .","Thus, $\\varphi _{gKL,-\\widetilde{c}/m,\\widetilde{c}} \\in \\Upsilon (]0,\\infty [)$ .","Type: $\\mathbb {}$ is a non-infinitely divisible discrete distribution with frequencies: $\\mathbb {\\Pi }[W=\\ell \\cdot \\frac{1}{\\widetilde{c}}]=\\binom{m}{\\ell }\\cdot (\\frac{\\widetilde{c}}{m})^{\\ell } \\cdot (1-\\frac{\\widetilde{c}}{m})^{m-\\ell }$ for $\\ell \\in \\lbrace 0,1, \\ldots , m\\rbrace $ (and zero elsewhere).","Behaviour at zero: $\\mathbb {\\Pi }[W\\ge 0]=1$ , $\\mathbb {\\Pi }[W=0]= (1-\\frac{\\widetilde{c}}{m})^{m}$ .","Corresponding generator: $\\varphi _{gKL,\\alpha ,\\widetilde{c}}$ (cf.","(REF )) of the divergence (REF ).","Sums: for i.i.d.","copies $(W_{i})_{i \\in \\mathbb {N}}$ of $W$ , the probability distribution of $\\breve{W} := \\sum _{i\\in I_{k}^{(n)}} W_{i}$ (cf.","Remark REF (ii)) is $\\frac{1}{\\widetilde{c}} \\cdot BIN(m \\cdot card(I_{k}^{(n)}),\\frac{\\widetilde{c}}{m})$ .","Example 54 for the context of Example REF we obtain: Case of anchor point $c=0$ : $\\Lambda _{twop}(z)= \\log \\Big ( p \\cdot e^{z_{1} \\cdot z} + (1- p) \\cdot e^{z_{2} \\cdot z} \\Big )$ (cf.","(REF )) is the well-known cumulant generating function of the two-point probability distribution $\\mathbb {} = p \\cdot \\delta _{z_{1}} + (1-p) \\cdot \\delta _{z_{2}}$ , where $z_{1} < 1 < z_{2}$ and $p= \\frac{z_{2} -1}{z_{2} - z_{1}}$ .","Hence, $\\varphi _{twop} \\in \\Upsilon (]z_{1},z_{2}[)$ .","Type: $\\mathbb {}$ is a discrete distribution with frequencies: $\\mathbb {\\Pi }[W=z_{1}] = p$ , $\\mathbb {\\Pi }[W=z_{2}] = 1-p$ (and zero elsewhere).","Behaviour at zero: $\\mathbb {\\Pi }[W > 0]=1$ iff $z_{1} >0$ , $\\mathbb {\\Pi }[W=0] \\ne 0$ iff $z_{1}=0$ .","Corresponding generator: $\\varphi _{twop}$ (cf.","(REF )) of the divergence given in (REF ).","Sums: for i.i.d.","copies $(W_{i})_{i \\in \\mathbb {N}}$ of $W$ , the probability distribution of $\\breve{W} := \\sum _{i\\in I_{k}^{(n)}} W_{i}$ (cf.","Remark REF (ii)) is the distribution of the $card(I_{k}^{(n)})-$ th step of a generalized random walk starting at zero; this has a nice explicit (“binomial-type” ) expression in the special case $z_{1} = - z_{2}$ , namely $\\sum _{\\ell =0}^{card(I_{k}^{(n)})} \\binom{card(I_{k}^{(n)})}{\\ell }\\cdot p^{card(I_{k}^{(n)}) - \\ell } \\cdot (1-p)^{\\ell } \\cdot \\delta _{z_{2}\\cdot (2 \\ell - card(I_{k}^{(n)}))}$ .","Example 55 for the context of Example REF we obtain: Case $\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c} \\in \\, ]0,\\infty [$ , anchor point $c=0$ : by using $\\breve{\\theta } := 1 + \\alpha \\cdot \\Big (\\frac{1}{\\beta _{2}}- \\frac{1}{\\beta _{1}} \\Big )$ one can see that $\\Lambda _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}(z) =\\breve{\\theta } \\cdot z - \\widetilde{c} \\cdot \\alpha \\cdot \\log \\Big (1+ \\frac{z}{\\widetilde{c}} \\cdot \\Big (\\frac{1}{\\beta _{2}} - \\frac{1}{\\beta _{1}} \\Big )- \\frac{z^{2}}{\\widetilde{c}^{2} \\cdot \\beta _{1} \\cdot \\beta _{2} }\\Big )$ for $z \\in \\, ]- \\widetilde{c} \\cdot \\min \\lbrace \\beta _{1}, \\beta _{2}\\rbrace ,\\widetilde{c} \\cdot \\min \\lbrace \\beta _{1}, \\beta _{2}\\rbrace [$ — with different boundary behaviour for the three subcases $\\beta _{1} < \\beta _{2}$ resp.","$\\beta _{1} > \\beta _{2}$ resp.","$\\beta _{1} = \\beta _{2}$ (cf.","(REF ),(REF ),(REF )) — is the cumulant generating function of a generalized asymmetric Laplace distribution $\\mathbb {}[ \\, \\cdot \\,] = \\mathbb {\\Pi }[W \\in \\cdot \\, ]$ of a random variable $W:= \\breve{\\theta } + Z_{1} - Z_{2}$ , where $Z_{1}$ respectively $Z_{2}$ are auxiliary random variables which are independent and $GAM(\\widetilde{c} \\cdot \\beta _{1},\\widetilde{c} \\cdot \\alpha )-$ distributed respectively $GAM(\\widetilde{c} \\cdot \\beta _{2},\\widetilde{c} \\cdot \\alpha )-$ distributed.","In particular, $E_{\\mathbb {\\Pi }}[W] = \\breve{\\theta } + \\frac{\\widetilde{c} \\cdot \\alpha }{\\widetilde{c} \\cdot \\beta _{1}}- \\frac{\\widetilde{c} \\cdot \\alpha }{\\widetilde{c} \\cdot \\beta _{2}} =1$ (as required).","Thus, $\\varphi _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}\\in \\Upsilon (]-\\infty ,\\infty [)$ .","Prominent special case $\\widetilde{c} =1$ , $\\alpha =1$ , $\\beta _{1}=\\beta _{2} =: \\beta $ (and hence, $\\breve{\\theta }=1$ ): $\\mathbb {}$ is a classical Laplace distribution (two-tailed exponential distribution, bilateral exponential law) with location parameter 1 and scale parameter $\\frac{1}{\\beta }$ .","Type: $\\mathbb {}$ is an infinitely divisible (cf.","Proposition REF ) continuous distribution with density $f(u) := \\frac{\\sqrt{2} \\cdot \\exp \\lbrace \\frac{1}{\\sigma \\cdot \\sqrt{2}} \\cdot (\\frac{1}{\\kappa } - \\kappa ) \\cdot (u-\\theta ) \\rbrace }{\\sqrt{\\pi } \\cdot \\sigma ^{\\tau + 1/2} \\cdot \\Gamma (\\tau )} \\cdot \\left( \\frac{\\sqrt{2} \\cdot |u - \\theta |}{\\kappa + \\frac{1}{\\kappa }} \\right)^{\\tau - 1/2}\\cdot K_{\\tau - 1/2}\\left( \\frac{1}{\\sigma \\cdot \\sqrt{2}} \\cdot \\Big (\\kappa + \\frac{1}{\\kappa }\\Big ) \\cdot |u - \\theta | \\right),\\quad u \\ne \\theta ,$ where $(\\theta ,\\kappa ,\\sigma ,\\tau )$ is given in Remark REF below and $K_{\\lambda }$ is the modified Bessel function of the third kind with index $\\lambda $ .","For the above-mentioned special case of the classical Laplace distribution, this considerably simplifies to $f(u):= \\frac{\\beta }{2} \\exp \\lbrace - \\beta \\cdot |u -1 | \\rbrace $ .","Behaviour at zero: $\\mathbb {}[ \\, ]0,\\infty [ \\, ] = \\mathbb {\\Pi }[W > 0]=\\int _{0}^{\\infty } f(u) \\, du \\in \\, ]0,1[$ , $\\mathbb {}[ \\, \\lbrace 0\\rbrace \\, ] = \\mathbb {\\Pi }[W = 0]= 0$ .","Corresponding generator: $\\varphi _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}$ (cf.","(REF ) respectively (REF ) respectively (REF )) of the divergence given in (REF ) respectively (REF ) respectively (REF ).","Sums: for i.i.d.","copies $(W_{i})_{i \\in \\mathbb {N}}$ of $W$ , the probability distribution of $\\breve{W} := \\sum _{i\\in I_{k}^{(n)}} W_{i}$ (cf.","Remark REF (ii)) is the same as that of a random variable $\\breve{\\widetilde{W}} := \\breve{\\theta } \\cdot card(I_{k}^{(n)}) +\\breve{Z}_{1} - \\breve{Z}_{2}$ , where $\\breve{Z}_{1}$ respectively $\\breve{Z}_{2}$ are auxiliary random variables which are independent and $GAM(\\widetilde{c} \\cdot \\beta _{1},\\widetilde{c} \\cdot \\alpha \\cdot card(I_{k}^{(n)}))-$ distributed respectively $GAM(\\widetilde{c} \\cdot \\beta _{2},\\widetilde{c}\\cdot \\alpha \\cdot card(I_{k}^{(n)}))-$ distributed.","Remark 56 In the book of Kotz et al.","[197] one can find a very comprehensive study on generalized asymmetric Laplace distributions (also known as Bessel function distributions, McKay distributions), their close relatives (such as e.g.","the financial-econometric variance gamma model of Madan & Seneta [240]) as well as their applications; see also e.g.","Klar [192] for connections with some other Gamma difference distributions.","[197] use a different parametrization $(\\theta ,\\kappa ,\\sigma ,\\tau )$ which is one-to-one with our parametrization $(\\breve{\\theta },\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}=1)$ , as follows: $\\theta = \\breve{\\theta }$ , $\\tau = \\widetilde{c} \\cdot \\alpha $ , $\\sigma = \\frac{1}{\\widetilde{c}} \\cdot \\sqrt{\\frac{2}{\\beta _{1} \\cdot \\beta _{2}}}$ , $\\kappa = \\frac{\\sqrt{\\frac{4}{\\widetilde{c}^{2}} + (\\beta _{1} - \\beta _{2})^2} \\, + \\beta _{2}- \\beta _{1}}{2 \\cdot \\sqrt{\\beta _{1} \\cdot \\beta _{2}}} >0$ .","In particular, this implies that we cover all generalized asymmetric Laplace distributions with mean 1.","For better comparability, we have used the parametrization $(\\theta ,\\kappa ,\\sigma ,\\tau )$ in the above-mentioned representation (REF ) of the density (due to [197]).","Let us end this section by giving some further comments on the task of finding concretely the probability distribution (if existent) $\\mathbb {}[ \\, \\cdot \\,] = \\mathbb {\\Pi }[W \\in \\cdot \\, ]$ from the Fenchel-Legendre transform $\\Lambda = \\varphi ^{*}$ of a pregiven divergence generator $\\varphi $ , which should satisfy $\\exp (\\Lambda (z)) = \\int _{\\mathbb {R}}e^{z \\cdot y} \\, d\\mathbb {} (y)= E_{\\mathbb {\\Pi }}[\\exp (z \\cdot W)] ,\\qquad z \\in \\mathbb {R}, \\qquad \\text{(cf.","(\\ref {brostu3:repres lambda1}), (\\ref {brostu3:repres lambda2}))}.\\nonumber $ Recall that this is used for the simulation of the weights $(W_{i})_{i\\in \\mathbb {N}}$ which are i.i.d.","copies of $W$ and which are the crucial building ingredients of $\\xi _{n}^{\\mathbf {W}}$ in Theorem REF , respectively, of $\\xi _{n,\\mathbf {X}}^{w\\mathbf {W}}$ in Theorem REF .","The search for $\\mathbb {}$ can be done e.g.","by inversion of the moment generating function MGF, or by search in tables or computer software which list distributions and their MGF.","As already indicated above, we have eased/narrowed down this task by giving (additional) sufficient conditions for some deriving principal properties of $\\mathbb {}$ .","Also notice that $\\mathbb {}$ needs not necessarily to be explicitly known in full detail (e.g.","in terms of a computationally tractable density or frequency); for instance, as well known from insurance applications, for — comfortably straightforwardly simulable — doubly-random sums $W := \\sum _{i=1}^{N} A_{i}$ of nonnegative i.i.d.","random variables $(A_{i})_{i\\in \\mathbb {N}}$ with known law $\\Pi _A[\\, \\cdot \\, ]:= \\Pi [A \\in \\cdot \\, ]$ being independent of a counting-type random variable $N$ with known law $\\Pi _N$ , one can mostly compute explicitly $MGF_{\\mathbb {}}(z) = PGF_{\\Pi _N}(MGF_{\\Pi _A}(z))$ in terms of $\\mathbb {} := \\Pi _W$ and the probability generating function $PGF_{\\Pi _N}$ of $\\Pi _N$ , but the corresponding density/frequency of $\\mathbb {}$ may not be known explicitly in a tractable form.","The above-mentioned Example REF (b) of power divergences with generator $\\varphi _{\\gamma }$ ($\\gamma \\in ]0,1[$ ) manifests such a situation.","In the end, if no explicit distribution $\\mathbb {}$ and no comfortably simulable $W-$ construction are available, one can still try to simulate an i.i.d.","sequence $(W_{i})_{i\\in \\mathbb {N}}$ from the pregiven moment generating function (which is $\\exp (\\Lambda (z))$ here); see e.g.","McLeish [259] and references therein which also contains saddle point methods approximation techniques."],["Estimators","In the following, we demonstrate how one can principally implement our BS approach; a further, deeper analysis will be given in a follow-up paper."],["Estimators for the deterministic minimization problem","We address the minimization problem $D_{\\varphi }(\\mathbf {\\Omega },\\mathbf {P}):= \\inf _{\\mathbf {Q}\\in \\mathbf {\\Omega } } D_{\\varphi }(\\mathbf {Q},\\mathbf {P}) =\\inf _{\\widetilde{\\mathbf {Q}}\\in \\widetilde{\\mathbf {\\Omega }} }D_{\\widetilde{\\varphi } }(\\widetilde{\\mathbf {Q}},\\widetilde{{P}})=: D_{\\widetilde{\\varphi } }(\\widetilde{\\mathbf {\\Omega }},\\widetilde{{P}})\\qquad \\textrm {with } \\widetilde{\\mathbf {\\Omega }} :=\\mathbf {\\Omega } /M_{\\mathbf {P}}\\qquad \\textrm {(cf.","(\\ref {min Pb}) and (\\ref {min Pb prob2}))},$ whose numerical solution is based on Theorem REF which basically states that for large integer $n \\in \\mathbb {N}$ one has $\\inf _{\\mathbf {Q}\\in \\mathbf {\\Omega } } D_{\\varphi }(\\mathbf {Q},\\mathbf {P})\\approx -\\frac{1}{n}\\log \\,\\mathbb {\\Pi }\\left[\\xi _{n}^{\\mathbf {\\widetilde{W}}}\\in \\widetilde{\\mathbf {\\Omega }}\\right]$ in terms of $\\widetilde{\\varphi }:=M_{\\mathbf {P}} \\cdot \\varphi $ and the random vectors $\\xi _{n}^{\\mathbf {\\widetilde{W}}}=\\Big (\\frac{1}{n}\\sum _{i\\in I_{1}^{(n)}}\\widetilde{W}_{i},\\ldots ,\\frac{1}{n}\\sum _{i\\in I_{K}^{(n)}}\\widetilde{W}_{i}\\Big )\\hspace{56.9055pt} \\text{(cf.","(\\ref {Xi_n^W vector}))}\\nonumber $ with $n_{k}:=\\lfloor n \\cdot \\widetilde{p}_{k}\\rfloor $ leading to the disjoint index blocks $I_{1}^{(n)}:=\\left\\lbrace 1,\\ldots ,n_{1}\\right\\rbrace $ , $I_{2}^{(n)}:=\\left\\lbrace n_{1}+1,\\ldots ,n_{1}+n_{2}\\right\\rbrace $ , $\\ldots $ , $I_{K}^{(n)} := \\lbrace \\sum _{k=1}^{K-1} n_{k} + 1, \\ldots , n \\rbrace $ .","Recall that $\\mathbf {\\widetilde{W}} := (\\widetilde{W}_{1}, \\ldots , \\widetilde{W}_{n})$ is a random vector consisting of components $\\widetilde{W}_{i}$ which are i.i.d.","copies of the random variable $\\widetilde{W}$ whose distribution is $\\mathbb {\\Pi }[\\widetilde{W}\\in \\cdot \\,]=\\widetilde{\\mathbb {}}[\\,\\cdot \\,]$ obeying the representation $\\widetilde{\\varphi }(t) =\\sup _{z \\in \\mathbb {R}} \\left( z\\cdot t - \\log \\int _{\\mathbb {R}} e^{zy} d\\widetilde{\\mathbb {}}(y) \\right),\\qquad t \\in \\mathbb {R},\\qquad \\textrm {(cf.","(\\ref {brostu3:fo.link.var}))} .\\nonumber $ Hence, the estimation of $D_{\\varphi }(\\mathbf {\\Omega },\\mathbf {P})$ amounts to the estimation of $\\mathbb {\\Pi }\\left[\\xi _{n}^{\\mathbf {\\widetilde{W}}}\\in \\widetilde{\\mathbf {\\Omega }} \\right]$ .","For the rest of this subsection, we assume that $\\widetilde{{P}} \\in \\mathbb {S}_{> 0}^{K}$ , that $n$ is chosen such that all $n \\cdot \\widetilde{p}_{k}$ are integers (and hence, $n = \\sum _{k=1}^{K} n_{k}$ ), and that $\\widetilde{\\mathbf {\\Omega }} \\subset \\mathbb {R}^{K}$ satisfies the regularity property $cl(\\widetilde{\\mathbf {\\Omega }})=cl\\left( int\\left( \\widetilde{\\mathbf {\\Omega }} \\right) \\right) , \\qquad int\\left( \\widetilde{\\mathbf {\\Omega }} \\right) \\ne \\emptyset \\nonumber $ which implies that the same condition holds for $\\mathbf {\\Omega }$ ; moreover, we suppose that $D_{\\widetilde{\\varphi } }(\\widetilde{\\mathbf {\\Omega }},\\widetilde{{P}})$ is finite.","For the ease of the following discussions, we introduce the notations $T\\left( \\mathbf {x}\\right) :=\\left( \\frac{1}{n_{1}}\\sum _{i\\in I_{1}^{(n)}}x_{i},\\ldots ,\\frac{1}{n_{K}}\\sum _{i\\in I_{K}^{(n)}} x_{i}\\right)\\quad \\textrm {for any $ x:=( x1,..,xn) Rn$,}$$as well as $ D$ for the diagonal matrix with diagonal entries $ 1/p1,...,1/pK$and null entries off the diagonal.Accordingly, the probability in (\\ref {LDP Minimization approx}) becomes$$\\mathbb {\\Pi }\\left[\\xi _{n}^{\\mathbf {\\widetilde{W}}}\\in \\widetilde{\\mathbf {\\Omega }} \\right] \\ = \\ \\mathbb {\\Pi }\\left[ T(\\mathbf {\\widetilde{W}}) \\in \\mathbf {\\Lambda }\\right]$$where$$\\mathbf {\\Lambda } :=\\mathfrak {D} \\cdot \\widetilde{\\mathbf {\\Omega }}$$is a set of vectors in $ RK$ which is known/derived from the concrete context.The \\textit {naive estimator} $ Lnaive$ of$ [ nW ]$is constructedthrough the following procedure:simulate independently $ L$ copies$ W1,...,WL$ of the vector$ W:=( W1,...,Wn) $,with independent entries under $$, and define(with a slight abuse of notation)$$\\widehat{\\widetilde{\\Pi }}_{L}^{naive} :=\\frac{1}{L}\\sum _{\\ell =1}^{L} \\mathbf {1}_{\\mathbf {\\Lambda }}\\left( T\\left( \\mathbf {\\widetilde{W}}^{\\ell }\\right) \\right) ;$$however this procedure is time costly, since this estimate has a very bad hit rate.Thus, in the following, a so-called “efficient Importance Sampling (IS)”\\ scheme --- in the sense of Sadowsky\\& Bucklew \\cite {Sad:90} (denoted [SB] hereunder) ---is adapted for the sophisticated (i.e.","non-naive) estimation of$ [ nW ]$.The main property of IS schemes lays in the fact that the runtime for anestimate with a controlled relative error does \\textit {not} increase atexponential rate as $ n$ increases, in contrast to$ Lnaive$ which has exponential increase.In detail, let $ >0$ be agiven relative precision for an estimator $ PSLn$ of$ [ nW ]$, based on a number $ Ln$ of simulated samplesgenerated under some distribution $ S$, so that$$\\delta :=\\frac{var_{\\widetilde{S}}P_{\\widetilde{S}}^{L_{n}}}{\\left(\\mathbb {\\Pi }\\left[\\xi _{n}^{\\mathbf {\\widetilde{W}}}\\in \\widetilde{\\mathbf {\\Omega }}\\right]\\right)^{2}} \\, .$$Then $ Ln$ will grow exponentially as $ n$ tends to infinity if and only if$ S$ is not “asymptotically optimal”\\ ,the derivation of which is the scope of the current section.$ To start with the details, for the sake of brevity (to avoid certain substantial discussions on potential technical relaxations) we shall employ the following additional Assumption (OM) on the set $\\widetilde{\\mathbf {\\Omega }}$ : (OM) For any $\\widetilde{\\omega } \\in cl(\\widetilde{\\mathbf {\\Omega }})$ there exists a vector $\\mathbf {x} = \\left(x_{1},\\ldots ,x_{n}\\right) \\in \\left] t_{-}^{sc},t_{+}^{sc}\\right[^{n}$ such that $\\widetilde{\\omega } = \\left( \\frac{1}{n}\\sum _{i\\in I_{1}^{(n)}}x_{i},\\ldots ,\\frac{1}{n} \\sum _{i\\in I_{K}^{(n)}}x_{i}\\right)$ , or equivalently, for any $\\lambda \\in cl(\\mathbf {\\Lambda })$ there exists a vector $\\mathbf {x} = \\left(x_{1},\\ldots ,x_{n}\\right) \\in \\left] t_{-}^{sc},t_{+}^{sc}\\right[^{n}$ such that $\\lambda =T\\left(\\mathbf {x}\\right)$ .","For instance, in the common case $dom(\\widetilde{\\varphi })= dom(\\varphi ) = \\, ]a,b[ \\, =\\, \\left] t_{-}^{sc},t_{+}^{sc}\\right[ \\, = \\, ]0,\\infty [$ (e.g.","for the power-divergence generators $\\widetilde{\\varphi }= \\widetilde{c} \\cdot \\varphi _{\\gamma }$ , $\\gamma \\le 0$ , cf.","Example REF ) the Assumption (OM) is always feasible.","To proceed, for any distribution $\\widetilde{S}$ on $\\mathbb {R}^{n}$ with support included in the support of the product measure $\\widetilde{\\mathbb {}}^{\\otimes n}$ it holds $\\mathbb {\\Pi }\\left[\\xi _{n}^{\\mathbf {\\widetilde{W}}}\\in \\widetilde{\\mathbf {\\Omega }}\\right]=E_{\\widetilde{\\mathbb {}}^{\\otimes n}} \\left[\\mathbf {1}_{\\mathbf {\\Lambda } }( T( \\mathbf {\\widetilde{W}}))\\right] =E_{\\widetilde{S}}\\left[\\mathbf {1}_{\\mathbf {\\Lambda } }\\left( T\\left( \\mathbf {\\widetilde{V}}\\right) \\right) \\cdot \\frac{d\\widetilde{\\mathbb {}}^{\\otimes n}}{d\\widetilde{S}}\\left( \\mathbf {\\widetilde{V}}\\right) \\right]$ from where the improved IS estimator of $\\mathbb {\\Pi }\\left[\\xi _{n}^{\\mathbf {\\widetilde{W}}}\\in \\widetilde{\\mathbf {\\Omega }} \\right]$ is obtained by sampling $L$ i.i.d.","replications $\\mathbf {\\widetilde{V}}^{1},\\ldots ,\\mathbf {\\widetilde{V}}^{L}$ of the random vector $\\mathbf {\\widetilde{V}}$ with distribution $\\widetilde{S}$ and by defining $\\widehat{\\widetilde{\\Pi }}_{L}^{improved}:=\\frac{1}{L}\\sum _{\\ell =1}^{L}\\mathbf {1}_{\\mathbf {\\Lambda }}( T( \\mathbf {\\widetilde{V}}^{(\\ell )})) \\cdot \\frac{d\\widetilde{\\mathbb {}}^{\\otimes n}}{d\\widetilde{S}}\\left( \\mathbf {\\widetilde{V}}^{(\\ell )}\\right) \\ .$ The precise form of the efficient IS distribution $\\widetilde{S}^{opt}$ relies on the definition of a “dominating point” of $\\mathbf {\\Lambda }$ , which we recall now.","For $\\mathbf {x} := \\left( x_{1},..,x_{n}\\right)$ in $\\mathbb {R}^{n}$ we define $I_{\\mathbf {\\widetilde{W}}}(\\mathbf {x}):=\\sup _{\\mathbf {z} \\in \\mathbb {R}^{n}}\\left(\\left\\langle \\mathbf {z}, \\mathbf {x} \\right\\rangle -\\log E_{\\widetilde{\\mathbb {}}}[ \\, \\exp (\\langle \\mathbf {z},\\mathbf {\\widetilde{W}}\\rangle ) ]\\right) \\, ,$ and for $\\lambda $ in $\\mathbf {\\Lambda }$ we let $I(\\lambda ):=\\inf \\left\\lbrace I_{\\mathbf {\\widetilde{W}}}(\\mathbf {x}):T(\\mathbf {x)=\\lambda }\\right\\rbrace .$ Let $\\underline{\\lambda }:=\\left( \\underline{\\lambda }_{1},\\ldots ,\\underline{\\lambda }_{K}\\right) \\in \\partial \\mathbf {\\Lambda }$ .","We call $\\underline{\\lambda }$ a minimal rate point (mrp) of $\\mathbf {\\Lambda }$ if $I(\\underline{\\lambda })\\le I(\\lambda ) \\quad \\text{ for all }\\lambda \\in \\mathbf {\\Lambda } .$ A minimal rate point $\\underline{\\lambda }$ is called a dominating point of $\\mathbf {\\Lambda }$ if a) $\\underline{\\lambda }\\in \\partial \\mathbf {\\Lambda }$ , and b) $I\\left(\\underline{\\lambda }\\right)\\le I(\\lambda )$ for all $\\lambda \\in \\mathbf {\\Lambda }$ with attainment, namely there exists a vector $\\underline{\\mathbf {x}} \\in \\left] t_{-}^{sc},t_{+}^{sc}\\right[^{n}$ such that $I_{\\widetilde{W}}\\left( \\underline{\\mathbf {x}}\\right) =I\\left(\\underline{\\lambda }\\right)$ with $\\underline{\\lambda }=T\\left( \\underline{\\mathbf {x}}\\right)$ .","The characterization of the dominating point $\\underline{\\lambda }$ is settled in the following Lemma 57 Let $\\underline{\\lambda }$ be a mrp of $\\mathbf {\\Lambda }$ .","Then, under Assumption (OM), $\\underline{\\lambda }$ is a dominating point, and $\\inf \\left\\lbrace I_{\\mathbf {\\widetilde{W}}}\\left( \\mathbf {x}\\right) ,T(\\mathbf {x})=\\underline{\\lambda }\\right\\rbrace $ is reached at some vector $\\underline{\\mathbf {x}}$ in $\\left] t_{-}^{sc},t_{+}^{sc}\\right[^{n}$ such that for all $k \\in \\left\\lbrace 1,\\ldots ,K\\right\\rbrace $ and all $i \\in I_{k}^{(n)}$ there holds $\\underline{x}_{i}= \\underline{\\lambda }_{k}$ and $I_{\\mathbf {\\widetilde{W}}}(\\underline{\\mathbf {x}})=n \\cdot \\sum _{k=1}^{K}\\widetilde{p}_{k} \\cdot \\widetilde{\\varphi }\\left(\\underline{\\lambda }_{k}\\right)$ .","The proof Lemma REF is given in Appendix G. Notice that (OM) implies the existence of a dominating point $\\underline{\\lambda }$ , but uniqueness may not hold.","In the latter case, one can try to proceed as in Theorem 2 of [SB] and the discussion thereafter.","However, we assume now uniqueness of $\\underline{\\lambda }$ ; this allows for the identification of $\\widetilde{S}^{opt}$ .","By Theorem 1 of [SB] and Theorem 3.1 of Csiszar [96], the asymptotically optimal IS distribution $\\widetilde{S}^{opt}$ is obtained as the Kullback-Leibler projection of $\\widetilde{\\mathbb {}}^{n\\otimes }$ on the set of all probability distributions on $\\mathbb {R}^{n}$ centered at point $\\underline{\\mathbf {x}}$ , whose coordinates are — according to Lemma REF — functions of the coordinates of $\\underline{\\mathbf {\\widetilde{Q}}} := \\mathfrak {D}^{-1} \\underline{\\lambda }$ such that $T\\left( \\underline{\\mathbf {x}}\\right)= \\mathfrak {D} \\underline{\\mathbf {\\widetilde{Q}}}$ .","The above definition of $\\widetilde{S}^{opt}$ presumes the knowledge of $\\underline{\\lambda }$ , which cannot be assumed (otherwise the minimization problem is solved in advance).","The aim of the following construction is to provide a proxy $\\widetilde{S}$ to $\\widetilde{S}^{opt}$ , where $\\widetilde{S}$ is the Kullback-Leibler projection of $\\widetilde{\\mathbb {}}^{\\otimes n}$ on the set of all probability distributions on $\\mathbb {R}^{n}$ centered at some point $\\mathbf {x}^{\\ast }$ which is close to $\\underline{\\mathbf {x}}$ .","For this sake, we need to have at hand a proxy of $\\underline{\\lambda }$ or, equivalently, a preliminary guess $\\mathbf {\\widetilde{Q}}^{\\ast }$ of $\\underline{\\mathbf {\\widetilde{Q}}}:=\\arg \\inf _{\\mathbf {\\widetilde{Q}} \\in \\widetilde{\\mathbf {\\Omega }}}\\sum _{k=1}^{K} \\widetilde{p}_{k} \\cdot \\widetilde{\\varphi }(\\widetilde{q}_{k}/\\widetilde{p}_{k})$ .","This guess is by no means produced in order to provide a direct estimate of $D_{\\widetilde{\\varphi } }(\\widetilde{\\mathbf {\\Omega }},\\widetilde{{P}})$ but merely to provide the IS distribution $\\widetilde{S}$ which in turn leads to a sharp estimate of $D_{\\widetilde{\\varphi } }(\\widetilde{\\mathbf {\\Omega }},\\widetilde{{P}})$ .","Proxy method 1: in some cases we might have at hand some particular point $\\mathbf {\\widetilde{Q}}^{\\ast } :=\\left(\\widetilde{q}_{1}^{\\ast },..,\\widetilde{q}_{K}^{\\ast }\\right)$ in $\\widetilde{\\mathbf {\\Omega }}$ ; the resulting IS distribution $\\widetilde{S}$ with $\\mathbf {\\widetilde{Q}}$ substituted by $\\mathbf {\\widetilde{Q}^{\\ast }}$ is not optimal in the sense of [SB], but anyhow produces an estimator with good hitting rate, possibly with a loss in the variance.","A simple way to obtain such a point $\\mathbf {\\widetilde{Q}^{\\ast }}$ in $\\widetilde{\\mathbf {\\Omega }}$ is to simulate runs of (say) $M-$ variate i.i.d.","vectors $\\mathbf {\\widetilde{W}}$ under $\\widetilde{\\mathbb {}}^{\\otimes M}$ until the first time where $\\xi _{M}^{\\mathbf {\\widetilde{W}}}$ belongs to $\\widetilde{\\mathbf {\\Omega }}$ ; then we set $\\mathbf {\\widetilde{Q}^{\\ast }} := \\xi _{M}^{\\mathbf {\\widetilde{W}}}$ for the succeeding realization $\\mathbf {\\widetilde{W}}$ .","Before we proceed, it is useful to mention that the need for a drastic fall in the number of simulation runs pertains for cases when $D_{\\widetilde{\\varphi } }(\\widetilde{\\mathbf {\\Omega }},\\widetilde{{P}})$ is large.","The following construction is suited to this case, which is of relevance in applications both in optimization and in statistics when choosing between competing models none of which is assumed to represent the true one, but merely less inadequate ones.","Proxy method 2: when $D_{\\widetilde{\\varphi } }(\\widetilde{\\mathbf {\\Omega }},\\widetilde{{P}})$ is presumably large, we make use of asymptotic approximation to get a proxy of $\\underline{\\mathbf {\\widetilde{Q}}}$ .","For this, we define a sampling distribution on $\\mathbb {R}^{K}$ fitted to the divergence through $f(\\mathbf {\\widetilde{Q}}):=C \\cdot \\exp \\left( -\\sum _{k=1}^{K} \\widetilde{p}_{k} \\cdot \\widetilde{\\varphi }(\\widetilde{q}_{k}/\\widetilde{p}_{k})\\right) \\ = \\ C \\cdot \\exp \\left( -D_{\\widetilde{\\varphi }}\\left( \\mathbf {\\widetilde{Q}},\\widetilde{{P}}\\right)\\right)$ where $C$ is a normalizing constant.","Let $\\mathbf {T}$ be a $K-$ variate random variable with density $f$ .","The distribution of $\\mathbf {T}$ given $\\left( \\mathbf {T} \\in \\widetilde{\\mathbf {\\Omega }}\\right)$ concentrates around $\\arg \\inf _{\\widetilde{\\mathbf {Q}}\\in \\widetilde{\\mathbf {\\Omega }} }D_{\\widetilde{\\varphi } }(\\widetilde{\\mathbf {Q}},\\widetilde{{P}})$ when $D_{\\widetilde{\\varphi } }(\\widetilde{\\mathbf {\\Omega }},\\widetilde{{P}})$ is large.","Indeed, for any $\\widetilde{\\mathbf {Q}}\\in \\widetilde{\\mathbf {\\Omega }}$ denote by $\\mathbf {V}_{\\varepsilon }(\\widetilde{\\mathbf {Q}})$ a small neighborhood of $\\widetilde{\\mathbf {Q}}$ in $\\mathbb {R}^{K}$ with radius $\\varepsilon $ ; clearly, the probability of the event $\\left( \\mathbf {T} \\in \\mathbf {V}_{\\varepsilon }(\\widetilde{\\mathbf {Q}})\\right)$ when restricted to $\\widetilde{\\mathbf {Q}}\\in \\widetilde{\\mathbf {\\Omega }}$ is maximum when $\\widetilde{\\mathbf {Q}} = \\underline{\\widetilde{\\mathbf {Q}}}$ , where $\\underline{\\widetilde{\\mathbf {Q}}}$ is the “dominating point of $\\widetilde{\\mathbf {\\Omega }}$ ” in the sense that $\\underline{\\mathbf {\\widetilde{Q}}} := \\mathfrak {D}^{-1} \\underline{\\lambda }$ is the above-defined transform of the dominating point $\\underline{\\lambda }$ (assuming uniqueness); a precise argumentation under adequate conditions is postponed to Appendix G. Accordingly, we obtain a proxy $\\mathbf {\\widetilde{Q}}^{\\ast }$ of $\\underline{\\mathbf {\\widetilde{Q}}}$ by simulating a sequence of independent $K-$ variate random variables $\\mathbf {T}_{1},\\ldots $ with distribution (REF ) until (say) $\\mathbf {T}_{m}$ belongs to $\\widetilde{\\mathbf {\\Omega }}$ and set $\\mathbf {\\widetilde{Q}}^{\\ast }:=\\mathbf {T}_{m}$ .","To proceed with the derivation of the IS sampling distribution $\\widetilde{S}$ on $\\mathbb {R}^{n}$ , we fix $\\mathbf {\\widetilde{Q}}^{\\ast } :=\\left(\\widetilde{q}_{1}^{\\ast },..,\\widetilde{q}_{K}^{\\ast }\\right)$ to be a proxy of $\\underline{\\mathbf {\\widetilde{Q}}}$ or an initial guess in $\\widetilde{\\mathbf {\\Omega }}$ .","As an intermediate step, we construct the probability distribution $\\widetilde{U}_{k}$ on $\\mathbb {R}$ given by $d\\widetilde{U}_{k}(v) \\ := \\ \\exp \\left( \\tau _{k} \\cdot v -\\Lambda _{\\widetilde{\\mathbb {}}}(\\tau _{k})\\right) d\\widetilde{\\mathbb {}}(v)=\\frac{\\exp \\left(\\tau _{k} \\cdot v\\right)}{MGF_{\\widetilde{\\mathbb {}}}(\\tau _{k})} \\, d\\widetilde{\\mathbb {}}(v)$ where $\\tau _{k} \\in int(dom(MGF_{\\widetilde{\\mathbb {}}}))$ is the unique solution of the equation $\\Lambda _{\\widetilde{\\mathbb {}}}^{\\prime }\\left( \\tau _{k}\\right) =\\frac{\\widetilde{q}_{k}^{\\ast }}{\\widetilde{p}_{k}}$ and thus — by relation (REF ) of Appendix F — we can compute explicitly $\\tau _{k} = \\ \\widetilde{\\varphi }^{\\, \\prime }\\left(\\frac{\\widetilde{q}_{k}^{\\ast }}{\\widetilde{p}_{k}}\\right) \\ .$ Therefore, $\\widetilde{U}_{k}$ is the Kullback-Leibler projection of $\\widetilde{\\mathbb {}}$ on the class of all probability distributions on $\\mathbb {R}$ whose expectation is $\\widetilde{q}_{k}^{\\ast }$ .","As a side remark, notice that one possible way of obtaining an explicit form of the probability distribution $\\widetilde{U}_{k}$ may be by identification through its moment generating function $dom(MGF_{\\widetilde{\\mathbb {}}})-\\tau _{k} \\ \\ni \\ z \\ \\mapsto MGF_{\\widetilde{U}_{k}}(z) =\\frac{MGF_{\\widetilde{\\mathbb {}}}(z+\\tau _{k})}{MGF_{\\widetilde{\\mathbb {}}}(\\tau _{k})}$ of which all ingredients are principally available.","For instance, this will be used in Example REF below.","From (REF ), we define $\\widetilde{S}_{k}:=\\underbrace{\\widetilde{U}_{k} \\otimes \\cdots \\otimes \\widetilde{U}_{k}}_{n_{k}\\text{times}}\\qquad \\textrm {for all $ {1,...,K}$},$$whence\\begin{eqnarray}d\\widetilde{S}_{k}\\left( v_{k,1},\\ldots ,v_{k,n_{k}}\\right) =\\exp \\Big ( \\sum _{i\\in I_{k}^{(n)}}\\tau _{k} \\cdot v_{k,i}-n_{k} \\cdot \\Lambda _{\\widetilde{\\mathbb {}}}(\\tau _{k})\\Big ) \\ d\\widetilde{\\mathbb {}}\\left(v_{k,1}\\right)\\cdots d\\widetilde{\\mathbb {}}\\left(v_{k,n_{k}}\\right),\\\\\\end{eqnarray}which manifests $ Sk$ as the Kullback-Leibler projection of$ nktimes$on the class of all probability distributions on $ Rk$whose expectation vector is$ Q = (q1,...,qK) Rk$.","Let now\\begin{equation}\\widetilde{S} := \\widetilde{S}_{1} \\otimes \\cdots \\otimes \\widetilde{S}_{K} \\, ,\\end{equation}which therefore satisfies (recall that $ k=1K nk =n$)\\begin{equation}d\\widetilde{S}\\left( v_{1,1},\\ldots v_{1,n_{1}},\\ldots , v_{K,1},\\ldots v_{K,n_{K}}\\right)=\\exp \\Big (\\sum _{k=1}^{K}\\sum _{i\\in I_{k}^{(n)}} \\big ( \\tau _{k} \\cdot v_{k,i} - n_{k} \\cdot \\Lambda _{\\widetilde{\\mathbb {}}}(\\tau _{k}) \\big )\\Big ) \\,d\\widetilde{\\mathbb {}}^{\\otimes n}\\left( v_{1,1},\\ldots v_{1,n_{1}},\\ldots , v_{K,1},\\ldots v_{K,n_{K}}\\right).\\ \\end{equation}The same procedure with all $ qk$substituted by the coordinates $qk$of $Q$produces $ Sopt$.","Therefore, $ S$ is a substitutefor $ Sopt$ with the change in the centering fromthe unknown vector $Q$to its proxy $ Q$.$ As a straightforward consequence of (REF ) and (), we obtain the improved IS estimator of $\\mathbb {\\Pi }\\left[\\xi _{n}^{\\mathbf {\\widetilde{W}}}\\in \\widetilde{\\mathbf {\\Omega }}\\right]$ as $\\widehat{\\widetilde{\\Pi }}_{L}^{improved}\\ =\\ \\frac{1}{L}\\sum _{\\ell =1}^{L}\\mathbf {1}_{\\mathbf {\\Lambda }}( T( \\mathbf {\\widetilde{V}}^{(\\ell )})) \\cdot \\prod _{k=1}^{K} IS_{k}(\\mathbf {\\widetilde{V}}_{k}^{(\\ell )})$ where $\\mathbf {\\widetilde{V}}_{k}^{(\\ell )} := \\left( \\widetilde{V}_{i}^{(\\ell )} \\right)_{i \\in I_{k}^{(n)}}$ is the $k-$ th block of the $\\ell -$ th replication $\\mathbf {\\widetilde{V}}^{(\\ell )}$ of $\\mathbf {\\widetilde{V}}$ under $S$ , and the $k-$ th importance-sampling factor is $\\widetilde{IS}_{k}(v_{k,1},\\ldots ,v_{k,n_{k}}) \\ := \\ \\frac{d\\widetilde{\\mathbb {}}^{\\otimes n_{k}}}{d\\widetilde{S}_{k}}\\left( v_{k,1},\\ldots ,v_{k,n_{k}}\\right)= \\exp \\Big (n_{k} \\cdot \\Lambda _{\\widetilde{\\mathbb {}}}(\\tau _{k}) \\, - \\, \\tau _{k}\\cdot \\sum _{i=1}^{n_{k}} v_{k,i} \\Big )\\nonumber $ with $n_{k} = card(I_{k}^{(n)})$ .","Summing up things, we arrive at the following algorithm in case that $\\widetilde{\\mathbf {\\Omega }}$ has a unique dominating point (in the above-defined sense): Step D1 Exemplarily, we start with proxy method 2 (the other proxy method 1 works analogously): get a proxy $\\mathbf {\\widetilde{Q}}^{\\ast }$ of $\\underline{\\mathbf {\\widetilde{Q}}}$ by simulating a sequence of independent $K-$ variate random variables $\\mathbf {T}_{1},\\ldots $ with distribution (REF ) until (say) $\\mathbf {T}_{m}$ belongs to $\\widetilde{\\mathbf {\\Omega }}$ and set $\\mathbf {\\widetilde{Q}}^{\\ast }:=\\mathbf {T}_{m}$ .","Step D2 For all $k$ in $\\lbrace 1,\\ldots ,K\\rbrace $ compute $\\tau _{k} = \\ \\widetilde{\\varphi }^{\\, \\prime }\\left(\\frac{\\widetilde{q}_{k}^{\\ast }}{\\widetilde{p}_{k}}\\right)$ .","Step D3 For all $\\ell $ in $\\lbrace 1,\\ldots ,L\\rbrace $ perform a run of $\\mathbf {\\widetilde{V}}^{(\\ell )}$ under $\\widetilde{S}$ as follows: For all $k$ in $\\lbrace 1,\\ldots ,K\\rbrace $ simulate $n_{k}$ i.i.d.","random variables $\\widetilde{V}_{k_{1}}^{(\\ell )},\\ldots ,\\widetilde{V}_{k_{n_{k}}}^{(\\ell )}$ with common distribution $\\widetilde{U}_{k}$ defined in (REF ).","Set $\\mathbf {\\widetilde{V}}_{k}^{(\\ell )} :=(\\widetilde{V}_{k_{1}}^{(\\ell )},\\ldots ,\\widetilde{V}_{k_{n_{k}}}^{(\\ell )})$ to be the corresponding row vector.","Construct $\\mathbf {\\widetilde{V}}^{(\\ell )}$ as the row vector obtained by concatenating the $\\mathbf {\\widetilde{V}}_{k}^{(\\ell )}$ , i.e.","$\\mathbf {\\widetilde{V}}^{(\\ell )}:=\\left( \\mathbf {\\widetilde{V}}_{1}^{(\\ell )},\\ldots ,\\mathbf {\\widetilde{V}}_{K}^{(\\ell )}\\right) \\, ,$ and make use of $\\widehat{\\widetilde{\\Pi }}_{L}^{improved}$ given in (REF ) with the $\\tau _{k}$ 's obtained in Step D2 above to define (in the light of (REF ), (REF )) the BS minimum-distance estimator $\\widehat{D_{\\varphi }}(\\mathbf {\\Omega },\\mathbf {P})\\ := \\ \\widehat{D_{\\widetilde{\\varphi }}}(\\widetilde{\\mathbf {\\Omega }},\\widetilde{{P}})\\ := \\ - \\frac{1}{n}\\log \\widehat{\\widetilde{\\Pi }}_{L}^{improved} \\ .$ For many cases, the simulation burden needed for the computation of $\\widehat{\\widetilde{\\Pi }}_{L}^{improved}$ — and thus of $\\widehat{D_{\\varphi }}(\\mathbf {\\Omega },\\mathbf {P})$ — can be drastically reduced, especially for high dimensions $K$ and large sample size $n \\cdot L$ .","In fact, in terms of the notations $n_{k}:=card(I_{k}^{(n)})$ , $\\widehat{\\mathit {W}}_{k}^{(\\ell )}:=\\sum _{i\\in I_{k}^{(n)}}\\widetilde{V}_{i}^{(\\ell )}$ and $\\widetilde{ISF}_{k}(x) \\ := \\ \\frac{d\\widetilde{\\mathbb {}}^{\\ast n_{k}}}{d\\widetilde{U}_{k}^{\\ast n_{k}}}(x) \\ = \\ \\exp (n_{k} \\cdot \\Lambda _{\\widetilde{\\mathbb {}}}(\\tau _{k}) \\, - \\, x \\cdot \\tau _{k} )$ (where $\\widetilde{\\mathbb {}}^{\\ast n_{k}}$ is the $n_{k}-$ convolution of the measure $\\widetilde{\\mathbb {}}$ ), one can rewrite (REF ) as $\\widehat{\\widetilde{\\Pi }}_{L}^{improved}=\\frac{1}{L}\\sum _{\\ell =1}^{L}\\mathbf {1}_{\\mathbf {\\Lambda }}\\big ( \\,\\big (\\frac{1}{n_{1}} \\widehat{\\mathit {W}}_{1}^{(\\ell )},\\ldots , \\frac{1}{n_{K}} \\widehat{\\mathit {W}}_{K}^{(\\ell )}\\big ) \\, \\big ) \\cdot \\prod \\limits _{k=1}^{K}\\widetilde{ISF}_{k}(\\widehat{\\mathit {W}}_{k}^{(\\ell )}) .$ with $K-$ vector $\\big (\\frac{1}{n_{1}} \\widehat{\\mathit {W}}_{1}^{(\\ell )},\\ldots , \\frac{1}{n_{K}} \\widehat{\\mathit {W}}_{K}^{(\\ell )} \\big )$ .","Clearly, the random variable $\\widehat{\\mathit {W}}_{k}^{(\\ell )}$ ($k=1, \\ldots , K$ ) has distribution $\\widetilde{U}_{k}^{\\ast n_{k}}$ .","Hence, if $\\widetilde{U}_{k}^{\\ast n_{k}}$ can be explicitly constructed, then for the computation of $\\widehat{\\widetilde{\\Pi }}_{L}^{improved}$ it suffices to simulate the $K \\cdot L$ random variables $\\widehat{\\mathit {W}}_{k}^{(\\ell )}$ rather than the $n \\cdot L$ random variables $\\widetilde{V}_{i}^{(\\ell )}$ ; notice that according to the right-hand side of (REF ), one can explicitly compute $ISF_{k}\\left( \\cdot \\right)$ which can be interpreted as Importance Sampling Factor pertaining to the block $k$.","In the case that $\\widetilde{\\mathbb {}}$ is infinitely divisible, simulation issues may become especially comfortable.","In the following, we exemplarily demonstrate the tractability of this reduction effect, for the BS minimization of the important power divergences (for which the infinite divisibility holds): Example 58 Let $\\varphi _{\\gamma }$ ($\\gamma \\in \\mathbb {R}\\backslash ]1,2[$ ) be the power divergence generator from the Examples REF and REF , $\\mathbf {P} \\in \\mathbb {R}_{> 0}^{K}$ , $M_{\\mathbf {P}}:=\\sum _{i=1}^{K}p_{i}>0$ and $n_{k} = n \\cdot p_{k} \\in \\mathbb {N}$ where we have employed our notation $n_{k}= n \\cdot p_{k} \\in \\mathbb {N}$ for all $k \\in \\lbrace 1,\\ldots ,K\\rbrace $ .","Moreover, let $\\mathbf {\\widetilde{Q}}^{\\ast } :=\\left(\\widetilde{q}_{1}^{\\ast },\\ldots ,\\widetilde{q}_{K}^{\\ast }\\right)$ be a proxy obtained by either proxy method 1 or 2.","Case 1: Example REF (a): $\\gamma =0,\\widetilde{c}>0$ .","There holds $\\widetilde{U}_{k}^{\\ast n_{k}}=GAM\\left( \\widetilde{c} \\cdot M_{\\mathbf {P}} - \\tau _{k},n_{k}\\cdot \\widetilde{c} \\cdot M_{\\mathbf {P}}\\right)$ , with $\\tau _{k}=\\widetilde{c} \\cdot M_{\\mathbf {P}} \\cdot ( 1-\\frac{p_{k}}{M_{\\mathbf {P}} \\cdot \\widetilde{q}_{k}^{\\ast }})$ for $\\widetilde{q}_{k}^{\\ast } >0$ (the latter is equivalent to $\\tau _{k} < \\widetilde{c} \\cdot M_{\\mathbf {P}}$ ).","Moreover, for all $x >0$ one gets $\\widetilde{ISF}_{k}(x)=\\left( \\frac{\\widetilde{c} \\cdot M_{\\mathbf {P}}}{\\widetilde{c}\\cdot M_{\\mathbf {P}} - \\tau _{k}}\\right)^{n_{k}\\cdot \\widetilde{c}\\cdot M_{\\mathbf {P}}}\\cdot e^{-\\tau _{k} \\cdot x}$ .","Case 2: REF (b): $\\gamma \\in \\left( 0,1\\right) ,\\widetilde{c}>0$ .","We derive $\\widetilde{U}_{k}^{\\ast n_{k}}=C\\big ( POI( n_{k}\\cdot \\breve{\\theta }),GAM\\big (\\frac{\\widetilde{c} \\cdot M_{\\mathbf {P}}}{1-\\gamma } - \\tau _{k},\\frac{\\gamma }{1-\\gamma } \\big ) \\big )$ with $\\breve{\\theta }:= \\frac{\\widetilde{c} \\cdot M_{\\mathbf {P}}}{\\gamma }\\cdot \\big (\\frac{(\\gamma -1) \\cdot \\tau _{k}}{\\widetilde{c} \\cdot M_{\\mathbf {P}}}+1\\big )^{\\gamma /(\\gamma -1)}$ and $\\tau _{k} = \\widetilde{c} \\cdot M_{\\mathbf {P}} \\cdot \\frac{1-\\big (\\frac{\\widetilde{q}_{k}^{\\ast } \\cdot M_{\\mathbf {P}}}{p_{k}}\\big )^{\\gamma -1}}{1-\\gamma }$ for $\\widetilde{q}_{k}^{\\ast } >0$ .","Furthermore, $\\widetilde{ISF}_{k}(x)=e^{-\\tau _{k}x} \\cdot \\exp \\left( \\frac{n_{k}\\cdot \\widetilde{c} \\cdot M_{\\mathbf {P}}}{\\gamma }\\cdot \\left(\\left(1+ \\frac{\\gamma -1}{\\widetilde{c} \\cdot M_{\\mathbf {P}}} \\cdot \\tau _{k}\\right) ^{\\frac{\\gamma }{\\gamma -1}}-1 \\right)\\right), \\qquad x \\ge 0 ,$ (where $x=0$ covers the atom at zero).","Case 3: Example REF (c): $\\gamma =2,\\widetilde{c}>0$ .","One gets $\\widetilde{U}_{k}^{\\ast n_{k}}=N(n_{k}\\cdot (1+\\frac{\\tau _{k}}{\\widetilde{c} \\cdot M_{\\mathbf {P}}}),\\frac{n_{k}}{\\widetilde{c} \\cdot M_{\\mathbf {P}}})$ with $\\tau _{k}=\\widetilde{c} \\cdot M_{\\mathbf {P}} \\cdot ( \\frac{\\widetilde{q}_{k}^{\\ast } \\cdot M_{\\mathbf {P}}}{p_{k}} - 1 ) $ for $\\widetilde{q}_{k}^{\\ast } \\in \\mathbb {R}$ .","Moreover, for all $x \\in \\mathbb {R}$ one obtains $\\widetilde{ISF}_{k}(x)= \\exp \\big (\\frac{n_{k} \\cdot \\tau _{k}^2}{2\\widetilde{c} \\cdot M_{\\mathbf {P}}}- (x- n_{k}) \\cdot \\tau _{k} \\big )$ .","Case 4: Example REF (d): $\\gamma <0,\\widetilde{c}>0$ .","It holds that $\\widetilde{U}_{k}^{\\ast n_{k}}$ has the (Lebesgue-)density $f_{\\widetilde{U}_{k}^{\\ast n_{k}}}(x)\\ := \\ \\frac{\\exp ((\\tau _{k} - \\frac{\\widetilde{c}\\cdot M_{\\mathbf {P}}}{1-\\gamma })\\cdot x)}{\\exp \\left(n_{k} \\cdot \\frac{\\widetilde{c}\\cdot M_{\\mathbf {P}}}{\\gamma }\\cdot (1+\\frac{\\gamma -1}{\\widetilde{c} \\cdot M_{\\mathbf {P}}}\\cdot \\tau _{k})^{\\gamma /(\\gamma -1)}\\right)}\\cdot f_{\\breve{\\breve{Z}}}(x) \\cdot {1}_{]0,\\infty [}(x),\\qquad x \\in \\mathbb {R},$ where $\\tau _{k} = \\widetilde{c} \\cdot M_{\\mathbf {P}} \\cdot \\frac{1-\\big (\\frac{\\widetilde{q}_{k}^{\\ast } \\cdot M_{\\mathbf {P}}}{p_{k}}\\big )^{\\gamma -1}}{1-\\gamma }$ for $\\widetilde{q}_{k}^{\\ast } >0$ , and $\\breve{\\breve{Z}}$ is a random variable with density $f_{\\breve{\\breve{Z}}}$ of a stable law with parameter-quadruple $(\\frac{-\\gamma }{1-\\gamma },1,0,- n_{k} \\cdot \\frac{(\\widetilde{c}\\cdot M_{\\mathbf {P}})^{1/(1-\\gamma )} \\cdot (1-\\gamma )^{-\\gamma /(1-\\gamma )}}{\\gamma })$ (analogously to $\\breve{Z}$ of Example 40 (d) but with $\\widetilde{c}$ replaced by $\\widetilde{c}\\cdot M_{\\mathbf {P}}$ ).","Also, $\\widetilde{ISF}_{k}(x)=e^{-\\tau _{k}x} \\cdot \\exp \\left( \\frac{n_{k}\\cdot \\widetilde{c} \\cdot M_{\\mathbf {P}}}{\\gamma }\\cdot \\left(\\left(1+ \\frac{\\gamma -1}{\\widetilde{c} \\cdot M_{\\mathbf {P}}} \\cdot \\tau _{k}\\right) ^{\\frac{\\gamma }{\\gamma -1}}-1 \\right)\\right), \\qquad x >0.$ Case 5 : Example REF (e): $\\gamma >2,\\widetilde{c}>0$ .","We derive that $\\widetilde{U}_{k}^{\\ast n_{k}}$ has the (Lebesgue-)density $f_{\\widetilde{U}_{k}^{\\ast n_{k}}}(x)\\ := \\ \\frac{\\exp ((\\tau _{k} + \\frac{\\widetilde{c}\\cdot M_{\\mathbf {P}}}{\\gamma -1})\\cdot x)}{\\exp \\left(n_{k} \\cdot \\frac{\\widetilde{c}\\cdot M_{\\mathbf {P}}}{\\gamma }\\cdot (1+\\frac{\\gamma -1}{\\widetilde{c} \\cdot M_{\\mathbf {P}}}\\cdot \\tau _{k})^{\\gamma /(\\gamma -1)}\\right)}\\cdot f_{\\breve{\\breve{Z}}}(-x) ,\\qquad x \\in \\mathbb {R},$ where $\\tau _{k} = - \\frac{\\widetilde{c} \\cdot M_{\\mathbf {P}}}{\\gamma -1} \\cdot \\big (1- \\big (\\frac{\\widetilde{q}_{k}^{\\ast } \\cdot M_{\\mathbf {P}}}{p_{k}}\\big )^{\\gamma -1}\\cdot {1}_{]0,\\infty [}(\\widetilde{q}_{k}^{\\ast }) \\big )$ for $\\widetilde{q}_{k}^{\\ast } \\in \\mathbb {R}$ , and $\\breve{\\breve{Z}}$ is a random variable with density $f_{\\breve{\\breve{Z}}}$ of a stable law with parameter-quadruple $(\\frac{\\gamma }{\\gamma -1},1,0,n_{k} \\cdot \\frac{(\\widetilde{c}\\cdot M_{\\mathbf {P}})^{1/(1-\\gamma )} \\cdot (\\gamma -1)^{\\gamma /(\\gamma -1)}}{\\gamma })$ (analogously to $\\breve{Z}$ of Example 40 (e) but with $\\widetilde{c}$ replaced by $\\widetilde{c}\\cdot M_{\\mathbf {P}}$ ).","Furthermore, $\\widetilde{ISF}_{k}(x)=e^{-\\tau _{k}x} \\cdot \\exp \\left( \\frac{n_{k}\\cdot \\widetilde{c} \\cdot M_{\\mathbf {P}}}{\\gamma }\\cdot \\left(\\left(1+ \\frac{\\gamma -1}{\\widetilde{c} \\cdot M_{\\mathbf {P}}} \\cdot \\tau _{k}\\right) ^{\\frac{\\gamma }{\\gamma -1}}-1 \\right)\\right), \\qquad x \\in \\mathbb {R}.$ Case 6: Example REF (a): $\\gamma =1,\\widetilde{c}>0$ , anchor point $c=0$ .","It holds that $\\widetilde{U}_{k}^{\\ast n_{k}}$ is the probability distribution $\\frac{1}{\\widetilde{c} \\cdot M_{\\mathbf {P}}} \\cdot POI\\left(n_{k} \\cdot \\widetilde{c} \\cdot M_{\\mathbf {P}} \\cdot \\exp (\\frac{\\tau _{k}}{\\widetilde{c} \\cdot M_{\\mathbf {P}}})\\right) $ with support on the lattice $\\left\\lbrace \\frac{j}{\\widetilde{c} \\cdot M_{\\mathbf {P}}}, \\, j\\in \\mathbb {N}_{0}\\right\\rbrace $ , where $\\tau _{k}=\\widetilde{c} \\cdot \\log \\left( \\frac{M_{\\mathbf {P}} \\cdot \\widetilde{q}_{k}^{\\ast }}{p_{k}}\\right) $ for $\\widetilde{\\omega } _{k} >0$ .","Moreover, for all $j \\in \\mathbb {N}_{0}$ we obtain (by setting $x:= \\frac{j}{\\widetilde{c} \\cdot M_{\\mathbf {P}}}$ ) $\\widetilde{ISF}_{k}\\left(\\frac{j}{\\widetilde{c} \\cdot M_{\\mathbf {P}}}\\right)= \\exp \\left(n_{k} \\cdot \\widetilde{c} \\cdot M_{\\mathbf {P}} \\cdot \\left( \\exp \\left( \\frac{\\tau _{k}}{\\widetilde{c} \\cdot M_{\\mathbf {P}}}\\right) -1 \\right)- m \\cdot \\frac{\\tau _{k}}{\\widetilde{c} \\cdot M_{\\mathbf {P}}}\\right) .$ Case 7: Example REF (b): $\\gamma =1,\\widetilde{c}=1,$ anchor point $c\\in \\mathbb {R}$ .","For $M_{P}=1$ , $\\widetilde{U}_{k}^{\\ast n_{k}}$ is the shifted Poisson distribution $ POI\\left(n_{k} \\cdot e^{c +\\tau _{k}}\\right) + n_{k} \\cdot (1-e^{c})$ with support on the lattice $\\left\\lbrace j + n_{k} \\cdot (1-e^{c}), \\, j\\in \\mathbb {N}_{0}\\right\\rbrace $ , where $\\tau _{k}= \\log \\big (\\frac{\\widetilde{q}_{k}^{\\ast }}{p_{k}} + e^{c}-1\\big )-c$ for $\\widetilde{q}_{k}^{\\ast } > p_{k} \\cdot (1-e^{c})$ .","Furthermore, for all $j \\in \\mathbb {N}_{0}$ we obtain (by setting $x:= j + n_{k} \\cdot (1-e^{c})$ ) $\\widetilde{ISF}_{k}\\left(j + n_{k} \\cdot (1-e^{c})\\right)= \\exp \\left(n_{k} \\cdot e^{c} \\cdot (e^{\\tau _{k}} -1) - j \\cdot \\tau _{k} \\right).$ Notice that the mass of $\\widetilde{U}_{k}^{\\ast n_{k}}$ at zero depends on the value of the anchor point $c$ , since $\\widetilde{U}_{k}^{\\ast n_{k}}[\\lbrace 0\\rbrace ] >0$ if and only if $c=\\log (1+\\frac{\\ell }{n_{k}})$ for some $\\ell \\in \\mathbb {N}_{0}$ ; moreover, $\\widetilde{U}_{k}^{\\ast n_{k}}\\big [ \\ ]0,\\infty [ \\ \\big ] =1$ if $c <0$ and $\\widetilde{U}_{k}^{\\ast n_{k}}\\big [ \\ ]-\\infty ,0[ \\ \\big ] >0$ if $c >0$ .","Remark 59 (a) One can explicitly see in all cases of the above Example REF that all ingredients for computation are at hand.","(b) For both Cases 4 and 5 in the above Example REF , algorithms for simulation can be obtained by adapting e.g.","the works of Devroye [111] and Devroye & James [112].","In the previous Subsection REF , as a first step we have estimated $\\mathbb {\\Pi }\\left[\\xi _{n}^{\\mathbf {\\widetilde{W}}}\\in \\widetilde{\\mathbf {\\Omega }}\\right]$ in terms of the improved IS estimator $\\widehat{\\widetilde{\\Pi }}_{L}^{improved}$ .","From this, as a second step, we have derived — on the basis of Theorem REF — the estimator $\\widehat{D_{\\varphi }}(\\mathbf {\\Omega },\\mathbf {P})\\ := \\ - \\frac{1}{n}\\log \\widehat{\\widetilde{\\Pi }}_{L}^{improved} \\ \\nonumber \\qquad \\textrm {(cf.","(\\ref {estimator minimization}))}$ of the minimum distance $D_{\\varphi }(\\mathbf {\\Omega },\\mathbf {P}):= \\inf _{\\mathbf {Q}\\in \\mathbf {\\Omega } } D_{\\varphi }(\\mathbf {Q},\\mathbf {P})$ , where $\\mathbf {P} \\in \\mathbb {R}_{> 0}^{K}$ and $\\mathbf {\\Omega }\\subset \\mathbb {R}^{K}$ .","Recall that $\\widetilde{\\mathbf {\\Omega }} :=\\mathbf {\\Omega } /M_{\\mathbf {P}}$ with $M_{\\mathbf {P}}:=\\sum _{i=1}^{K}p_{i}>0$ , and that $\\xi _{n}^{\\mathbf {\\widetilde{W}}}=\\Big (\\frac{1}{n}\\sum _{i\\in I_{1}^{(n)}}\\widetilde{W}_{i},\\ldots ,\\frac{1}{n}\\sum _{i\\in I_{K}^{(n)}}\\widetilde{W}_{i}\\Big )\\hspace{56.9055pt} \\text{(cf.","(\\ref {Xi_n^W vector}))}\\nonumber $ where $\\mathbf {\\widetilde{W}} := (\\widetilde{W}_{1}, \\ldots , \\widetilde{W}_{n})$ is a random vector consisting of components $\\widetilde{W}_{i}$ which are i.i.d.","copies of the random variable $\\widetilde{W}$ whose distribution is $\\mathbb {\\Pi }[\\widetilde{W}\\in \\cdot \\,]=\\widetilde{\\mathbb {}}[\\,\\cdot \\,]$ obeying the representation (REF ).","In contrast, we now proceed as follows: as a first step, we derive an improved estimator $\\widehat{\\Pi }_{L}^{improved}$ of $\\mathbb {\\Pi }_{X_{1}^{n}}\\left[\\xi _{n,\\mathbf {X}}^{w\\mathbf {W}}\\in \\textrm {\\right.$ $\\hspace{-6.544pt}$$}$$where $$\\hspace{-6.544pt}$ SK$is a set of probability vectors which satisfies theregularity properties (\\ref {regularity}) and the finiteness property (\\ref {def fi wrt Omega}).Recall that\\begin{eqnarray}\\xi _{n,\\mathbf {X}}^{w\\mathbf {W}} &:=&{\\left\\lbrace \\begin{array}{ll}\\left(\\frac{\\sum _{i \\in I_{1}^{(n)}}W_{i}}{\\sum _{k=1}^{K}\\sum _{i \\in I_{k}^{(n)}}W_{i}},\\ldots , \\frac{\\sum _{i \\in I_{K}^{(n)}}W_{i}}{\\sum _{k=1}^{K}\\sum _{i \\in I_{k}^{(n)}}W_{i}} \\right) ,\\qquad \\textrm {if } \\sum _{j=1}^{n} W_{j} \\ne 0, \\\\\\ (\\infty , \\ldots , \\infty ) =: \\infty , \\hspace{113.81102pt} \\textrm {if } \\sum _{j=1}^{n} W_{j} = 0,\\end{array}\\right.","}\\hspace{42.67912pt}\\textrm {(cf.","(\\ref {brostu3:fo.norweiemp.vec}))}\\nonumber \\end{eqnarray}where $ (Xi)iN$ is a sequence of random variableswith values in $ Y:={ d1,,dK}$such that\\begin{equation}\\lim _{n\\rightarrow \\infty } \\Big ( \\frac{n_{1}}{n}, \\ldots , \\frac{n_{K}}{n} \\Big ) = (p_{1}, \\ldots , p_{K})\\qquad \\textrm {a.s.} \\qquad \\textrm {cf.","((\\ref {cv emp measure X to P vector}))}\\nonumber \\end{equation}holds for some probability vector $ P := (p1, ..., pK) S> 0K$,by employing the notation$$n_{k} \\ := \\ card(\\bigl \\lbrace i \\in \\lbrace 1, \\ldots , n\\rbrace : \\ X_{i} = d_{k} \\bigr \\rbrace )\\ =: \\ card(I_{k}^{(n)})\\qquad \\textrm {(cf.","(\\ref {I^(n)_k for stat case}));}$$hence, on the $ k$-th block of indexes $ Ik(n)$ all the $ Xi$^{\\prime }s share the same value $ dk$.Moreover, recall that $ (Wi)i N$ is a familyof independent and identically distributed $ R-$valued random variableswith probability distribution $ [ ] := [W1 ]$being connected with the divergence generator $ (]a,b[)$ via the representability(\\ref {Phi Legendre of mgf(W)}),such that $ (Wi)i N$ is independent of $ (Xi)i N$.$ As a second step (see Subsubsection REF below), for the important special case of the power-divergence generators $\\varphi _{\\gamma }$ (cf.","(REF )) we employ the Propositions REF to REF in order to deduce via the corresponding $\\widehat{\\Pi }_{L}^{improved}$ the estimators (e.g.","for $\\gamma <0$ ) $\\widehat{D_{\\widetilde{c}\\cdot \\varphi _{\\gamma }}(\\textrm {\\Omega \\hspace{-6.544pt}\\Omega },{P})} \\ := \\ \\frac{\\widetilde{c}}{\\gamma \\cdot \\left(\\gamma -1\\right) }\\cdot \\left\\lbrace \\left( 1+\\frac{\\gamma }{\\widetilde{c}}\\cdot \\frac{1}{n}\\cdot \\log \\,\\widehat{\\Pi }_{L}^{improved}\\right) ^{1-\\gamma } -1 \\right\\rbrace ,\\qquad \\ $ of the minimum power divergences $D_{\\widetilde{c}\\cdot \\varphi _{\\gamma }}(\\textrm {$ $\\hspace{-6.544pt}$$},{P}) :=\\inf _{{Q}\\in \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega }}D_{\\widetilde{c}\\cdot \\varphi _{\\gamma }}({Q},{P})$$as well as connected estimators of important deterministic transformations thereof.$ As a third step (see Subsubsection REF below), on the basis of Subsubsection REF we derive estimators of bounds of $D_{\\varphi }(\\textrm {$$\\hspace{-6.544pt}$$},{P})$ for more general divergence generators $\\varphi $ .","Let us start with the above-mentioned first step, by remarking that the development of the estimator $\\widehat{\\Pi }_{L}^{improved}$ works quite analogously to that of $\\widehat{\\widetilde{\\Pi }}_{L}^{improved}$ in the previous Subsection REF .","To make this even more transparent, we employ the notation $p_{n,k}^{emp}:=n_{k}/n$ (cf.","(REF )) and label all random vectors of length $n$ in the same way as above: we sort the already given and thus fixed data $X_{i}$ 's in such a way that the first $n_{1}$ of them share the same value $d_{1}$ , and so on, until the last block with length $n_{K}$ in which the data have common value $d_{K}$ .","In the light of the above considerations, we could achieve a naive estimate $\\widehat{\\Pi }_{L}^{naive}$ of $\\mathbb {\\Pi }_{X_{1}^{n}}[\\xi _{n,\\mathbf {X}}^{w\\mathbf {W}}\\in \\textrm {$$\\hspace{-6.544pt}$$}]$ through the following procedure.","We simulate independently $L$ replicates $\\mathbf {W}^{(1)},\\ldots ,\\mathbf {W}^{(L)}$ of the vector $\\mathbf {W}:=\\left( W_{1},\\ldots ,W_{n}\\right) $ , with independent entries under $\\mathbb {}$ (cf.","(REF )); those realizations do not depend on the $X_{i}$ 's.","Then we construct $\\widehat{\\Pi }_{L}^{naive} :=\\frac{1}{L}\\sum _{\\ell =1}^{L}\\mathbf {1}_{\\textrm {\\Omega \\hspace{-5.40608pt}\\Omega }}\\left(\\xi _{n,\\mathbf {X}}^{w\\mathbf {W}^{(\\ell )}}\\right) .$ However, this procedure is time costly, since the estimate given in (REF ) has a very bad hit rate.","Hence, analogously to Subsection REF we apply again an “efficient Importance Sampling (IS)” scheme in the sense of Sadowsky & Bucklew [313].","This will involve the simulation of $L$ independent $n-$ tuples $\\mathbf {V}^{(\\ell )}\\mathbf {:=}\\left(V_{n}^{(\\ell )},\\ldots ,V_{n}^{(\\ell )}\\right) $ with common distribution $S$ on $\\mathbb {R}^{n}$ , such that $\\mathbb {}^{\\otimes n}$ is (measure-)equivalent with respect to $S$ .","In fact, we rewrite $\\mathbb {\\Pi }_{X_{1}^{n}}[\\xi _{n,\\mathbf {X}}^{w\\mathbf {W}}\\in \\textrm {$$\\hspace{-6.544pt}$$}]$ as $\\mathbb {\\Pi }_{X_{1}^{n}}[\\xi _{n,\\mathbf {X}}^{w\\mathbf {W}}\\in \\textrm {\\Omega \\hspace{-6.544pt}\\Omega }]= E_{S}\\Big [\\frac{\\mathrm {d}\\mathbb {} ^{\\otimes n}}{\\mathrm {d}S}(V_{1},\\ldots ,V_{n})\\cdot \\mathbf {1}_{\\textrm {\\Omega \\hspace{-5.40608pt}\\Omega }}(\\xi _{n,\\mathbf {X}}^{w\\mathbf {V}})\\Big ]$ where $S$ designates any IS distribution of the vector $\\mathbf {V:=}(V_{1},\\ldots ,V_{n})$ , and $E_{S}[ \\, \\cdot \\, ]$ denotes the corresponding expectation operation.","Notice that $S$ is a random probability distribution on $\\mathbb {R}^{n}$ ; in fact, $S$ is a conditional probability distribution given $X_{1}^{n}$ , and thus it would be more precise to write $S | X_{1}^{n}$ instead of $S$ ; for the sake of brevity, we omit $| X_{1}^{n}$ .","As a consequence of (REF ), for adequately chosen $S$ , an improved estimator of $\\mathbb {\\Pi }_{X_{1}^{n}}[\\xi _{n,\\mathbf {X}}^{w\\mathbf {W}}\\in \\textrm {$$\\hspace{-6.544pt}$$}]$ is given by $\\widehat{\\Pi }_{L}^{improved}:= \\frac{1}{L}\\sum _{\\ell =1}^{L}\\frac{\\mathrm {d}\\mathbb {}^{\\otimes n}}{\\mathrm {d}S}(V_{1}^{(\\ell )},\\ldots ,V_{n}^{(\\ell )})\\cdot \\mathbf {1}_{\\textrm {\\Omega \\hspace{-5.40608pt}\\Omega }}(\\xi _{n,\\mathbf {X}}^{w\\mathbf {V}^{(\\ell )}}) \\, ,$ which also estimates $\\inf _{{Q}\\in \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega } }\\ \\inf _{m\\ne 0}D_{\\varphi }(m\\cdot {Q},{P})$ by the virtue of ().","Let us now deal with the concrete construction of a reasonable $S$ .","Given some (typically) large integer $M$ , we start with the realization $\\mathbf {W}^{\\ast }:=\\left( W_{1}^{\\ast },\\ldots ,W_{M}^{\\ast }\\right)$ such that $\\mathbf {Q}^{\\ast }:=\\xi _{M,\\mathbf {X}}^{w\\mathbf {W}^{\\ast }} \\in int(\\textrm {$$\\hspace{-6.544pt}$$})$ .","This may be given in advance or it may be achieved by drawing replicates $\\mathbf {W} = (W_{1}, \\ldots , W_{M})$ under $\\mathbb {}^{\\otimes M}$ until the first time where $\\xi _{M,\\mathbf {X}}^{w\\mathbf {W}}$ belongs to $int(\\textrm {$$\\hspace{-6.544pt}$$})$ .","Notice that by the nature of $\\textrm {$$\\hspace{-6.544pt}$$}$ , $\\mathbf {Q}^{\\ast }$ is a probability vector which has the $K$ components $q_{k}^{\\ast }:=\\sum _{i=1}^{M}\\frac{W_{i}^{\\ast }}{\\sum _{j=1}^{M}W_{j}^{\\ast }}\\mathbf {1}_{\\lbrace d_{k}\\rbrace }(X_{i}),\\hspace{42.67912pt} k=1,\\ldots ,K.$ Before we proceed, let us give the substantial remark that changing $\\left( V_{1},\\ldots ,V_{n}\\right) $ drawn under $S$ to $\\left(c \\cdot V_{1}, \\ldots ,c \\cdot V_{n}\\right) $ for any $c\\ne 0$ yields $\\xi _{n,\\mathbf {X}}^{w\\mathbf {V}}= \\xi _{n,\\mathbf {X}}^{w\\, c\\cdot \\mathbf {V}}$ so that the distribution $S$ is not uniquely determined.","Amongst all candidates, we choose the — uniquely determined — $S$ which is the Kullback-Leibler projection of $\\mathbb {}^{\\otimes n}$ on the set of all probability distributions on $\\mathbb {R}^{n}$ such that the $K$ “non-normalized” moment constraints $E_{S}[\\xi _{n,\\mathbf {X}}^{\\mathbf {V}}] =\\xi _{M,\\mathbf {X}}^{\\mathbf {W}^{\\ast }}$ (rather than the normalized $E_{S}[ \\xi _{n,\\mathbf {X}}^{w\\mathbf {V}}] =\\xi _{M,\\mathbf {X}}^{w\\mathbf {W}^{\\ast }}$ ) are satisfied, with the non-normalized vectors $\\xi _{M,\\mathbf {X}}^{\\mathbf {W}^{\\ast }} :=\\left( \\frac{1}{M}\\sum _{j=1}^{M}W_{j}^{\\ast }\\right) \\cdot Q^{\\ast } =: \\overline{W^{\\ast }} \\cdot Q^{\\ast },\\qquad \\xi _{n,\\mathbf {X}}^{\\mathbf {V}} :=\\left( \\frac{1}{n}\\sum _{j=1}^{n}V_{j}\\right) \\cdot \\xi _{n,\\mathbf {X}}^{w\\mathbf {V}} \\ .$ As already indicated above, this projection $S$ is a well-determined unique distribution on $\\mathbb {R}^{n}$ and — as we shall see in Proposition REF below — it is such that $\\xi _{n,\\mathbf {X}}^{w\\mathbf {V}}$ belongs to $\\textrm {$$\\hspace{-6.544pt}$$}$ with probability bounded away from 0 as $n$ increases, when $\\left( V_{1},\\ldots ,V_{n}\\right) $ are drawn under $S$ .","Therefore, this IS distribution produces an estimate of $\\mathbb {\\Pi }_{X_{1}^{n}}[\\xi _{n,\\mathbf {X}}^{w\\mathbf {W}}\\in \\textrm {$$\\hspace{-6.544pt}$$}]$ .","In order to justify the above construction of $S$ , we give the following result, which states that the IS sampling distribution $S$ yields a good hitting rate.","Its proof will be given in Appendix H. Proposition 60 With the above definition of $S$ , $\\lim \\inf _{n\\rightarrow \\infty }$ $S\\left[ \\xi _{n,\\mathbf {X}}^{w\\mathbf {V}} \\in \\textrm {\\right.$$\\hspace{-6.544pt}$$} $ is bounded away from 0.","We now come to the detailed construction of $S$ .","The constraints (REF ) can be written in explicit form as $E_{S}\\Big [ \\, \\frac{1}{n_{k}}\\sum _{i\\in I_{k}^{(n)}} V_{i} \\, \\Big ]\\ = \\ \\overline{W^{\\ast }} \\cdot \\frac{q_{k}^{\\ast }}{p_{n,k}^{emp}} \\, , \\qquad k=1,\\ldots ,K.$ The distribution $S$ can be obtained by blocks.","Indeed, let us define $S^{k}$ as the Kullback-Leibler (KL) projection of $\\mathbb {}^{\\otimes n_{k}}$ on the set of all distributions on $\\mathbb {R}^{n_{k}}$ such that (REF ) holds.","We define the resulting $S$ as the product distribution of those $S^{k}$ 's.","To obtain the latter, we start by defining $U_{k}$ as the KL projection of $\\mathbb {} $ on the set of all measures $Q$ on $\\mathbb {R}$ under (REF ).","Then, $dU_{k}(v)= \\exp (\\tau _{k}v-\\Lambda _{\\mathbb {} }\\left( \\tau _{k}\\right) )\\, d\\mathbb {} (v) \\, ,$ where $\\tau _{k} \\in int(dom(MGF_{\\mathbb {}}))$ is the unique solution of the equation $\\Lambda _{\\mathbb {} }^{\\prime }\\left( \\tau _{k}\\right) =\\overline{W^{\\ast }} \\cdot \\frac{q_{k}^{\\ast }}{p_{n,k}^{emp}}$ and thus — by relation (REF ) of Appendix F — we can compute explicitly $\\tau _{k} = \\varphi ^{\\prime }\\left( \\frac{\\overline{W^{\\ast }} \\cdot q_{k}^{\\ast }}{p_{n,k}^{emp}} \\right) .$ The distribution $S^{k}$ is then defined by $S^{k}:=\\underbrace{U_{k}\\otimes \\cdots \\otimes U_{k}}_{n_{k}\\text{times}}$ from which we obtain $S := S^{1}\\otimes \\cdots \\otimes S^{K}.$ With this construction, it holds $\\frac{dS}{d\\mathbb {} ^{\\otimes n}}(v_{1},\\ldots ,v_{n})=\\exp \\left(\\sum \\limits _{k=1}^{K}\\left( \\sum _{i\\in I_{k}^{(n)}}\\tau _{k} \\cdot v_{i}-\\Lambda _{\\mathbb {}}(\\tau _{k})\\right) \\right)$ which proves that $S$ is indeed the KL projection of $\\mathbb {} ^{\\otimes n}$ we aimed at.","Therefore, $\\mathbf {V}$ is composed of $K$ independent blocks of length $n_{k}$ each, and the $k-$ th subvector $\\mathbf {V}_{k}$ consists of all the random variables $V_{i}$ whose index $i$ satisfies $X_{i}=d_{k}.$ Within $\\mathbf {V}_{k}$ , all components are i.i.d.","with same distribution $U_{k}$ on $\\mathbb {R}$ defined through $\\frac{dU_{k}}{d\\mathbb {} }(u)=\\exp \\left\\lbrace \\tau _{k}\\cdot u-\\Lambda _{\\mathbb {}}(\\tau _{k})\\right\\rbrace =\\frac{\\exp \\left\\lbrace \\tau _{k}\\cdot u\\right\\rbrace }{MGF_{\\mathbb {} }(\\tau _{k})},$ which leads to the moment generating function $dom(MGF_{\\mathbb {}})-\\tau _{k} \\ \\ni \\ z \\ \\mapsto MGF_{U_{k}}(z):=\\int _{\\mathbb {R}}e^{zy}dU_{k}(y)=\\frac{MGF_{\\mathbb {}}(z+\\tau _{k})}{MGF_{\\mathbb {} }(\\tau _{k})}.$ Let us remark that $U_{k}$ can be interpreted as the distorted distribution of $\\mathbb {}$ with the distortion parameter $\\tau _{k}$ (in some cases, this distortion even becomes a tilting/dampening).","The estimator $\\widehat{\\Pi }_{L}^{improved}$ defined in (REF ) can be implemented through the following algorithm: Step S1 Choose some (typically large) $M$ and simulate repeatedly i.i.d.","vectors $\\left(W_{1},..,W_{M}\\right) $ — whose independent components have common distribution $\\mathbb {} $ — until $\\xi _{M,\\mathbf {X}}^{w\\mathbf {W}}$ belongs to $\\textrm {$$\\hspace{-6.544pt}$$}$ .","Call $\\left( W_{1}^{\\ast },..,W_{M}^{\\ast }\\right) $ the corresponding vector and $\\overline{W^{\\ast }}$ the arithmetic mean of its components.","Moreover, denote by $\\xi _{M,\\mathbf {X}}^{w\\mathbf {W}^{\\ast }}$ the corresponding normalized weighted empirical measure, identified with the $K-$ component vector $Q^{\\ast } := (q_{1}^{\\ast },..,q_{K}^{\\ast })$ with $q_{k}^{\\ast }$ defined in (REF ).","Step S2 For all $k \\in \\lbrace 1,\\ldots ,K\\rbrace $ compute $\\tau _{k} = \\varphi ^{\\prime }\\left( \\frac{\\overline{W^{\\ast }} \\cdot q_{k}^{\\ast }}{p_{n,k}^{emp}} \\right)$ .","Step S3 For all $\\ell \\in \\lbrace 1,\\ldots ,L\\rbrace $ simulate independently for all $k \\in \\lbrace 1,\\ldots ,K\\rbrace $ a row vector $\\mathbf {V}_{k}^{(\\ell )}$ $:=\\left(V_{k_{1}}^{(\\ell )},...,V_{k_{n_{k}}}^{(\\ell )}\\right) $ with independent components with common distribution $U_{k}$ defined in (REF ).","Concatenate these vectors to define the row vector $\\mathbf {V}^{(\\ell )}.$ Step S4 Compute the estimator $\\widehat{\\Pi }_{L}^{improved}$ by making use of the formula (REF ) which turns into the explicit form $& & \\widehat{\\Pi }_{L}^{improved} =\\frac{1}{L}\\sum _{\\ell =1}^{L}\\exp \\left(\\sum \\limits _{k=1}^{K}\\left(n_{k} \\cdot \\Lambda _{\\mathbb {} }(\\tau _{k}) -\\tau _{k} \\cdot \\sum _{i\\in I_{k}^{(n)}}V_{i}^{(\\ell )}\\right) \\right)\\cdot \\mathbf {1}_{\\textrm {\\Omega \\hspace{-5.40608pt}\\Omega }}\\left( \\xi _{n,\\mathbf {X}}^{w\\mathbf {V}^{(\\ell )}}\\right)$ Analogously to the paragraph right after (REF ) of the previous Subsection REF , in many cases we may improve the simulation burden needed for the computation of the estimator $\\widehat{\\Pi }_{L}^{improved}$ .","In fact, in terms of the notations $\\widehat{\\mathit {W}}_{k}^{(\\ell )}:=\\sum _{i\\in I_{k}^{(n)}}V_{i}^{(\\ell )}$ we can rewrite (REF ) as $\\widehat{\\Pi }_{L}^{improved} =\\frac{1}{L}\\sum _{\\ell =1}^{L}\\mathbf {1}_{\\textrm {\\Omega \\hspace{-5.40608pt}\\Omega }}\\left( \\xi _{n,\\mathbf {X}}^{w\\mathbf {V}^{(\\ell )}} \\right)\\cdot \\prod _{k=1}^{K}ISF_{k}\\left(\\widehat{\\mathit {W}}_{k}^{(\\ell )}\\right)$ with $ISF_{k}(x) \\ := \\ \\exp (n_{k} \\cdot \\Lambda _{\\mathbb {}}(\\tau _{k}) \\, - \\, x \\cdot \\tau _{k} )$ and $\\xi _{n,\\mathbf {X}}^{w\\mathbf {V}^{(\\ell )}} &=&{\\left\\lbrace \\begin{array}{ll}\\left(\\frac{\\widehat{\\mathit {W}}_{1}^{(\\ell )}}{\\sum _{k=1}^{K}\\widehat{\\mathit {W}}_{k}^{(\\ell )}},\\ldots , \\frac{\\widehat{\\mathit {W}}_{K}^{(\\ell )}}{\\sum _{k=1}^{K}\\widehat{\\mathit {W}}_{k}^{(\\ell )}} \\right) ,\\qquad \\textrm {if } \\sum _{k=1}^{K}\\widehat{\\mathit {W}}_{k}^{(\\ell )} \\ne 0, \\\\\\ (\\infty , \\ldots , \\infty ) =: \\infty , \\hspace{65.44142pt} \\textrm {if }\\sum _{k=1}^{K}\\widehat{\\mathit {W}}_{k}^{(\\ell )} = 0 \\, .\\end{array}\\right.","}$ Clearly, the random variable $\\widehat{\\mathit {W}}_{k}^{(\\ell )}$ ($k=1, \\ldots , K$ ) has distribution $U_{k}^{\\ast n_{k}}$ .","Hence, if $U_{k}^{\\ast n_{k}}$ can be explicitly constructed, then for the computation of $\\widehat{\\Pi }_{L}^{improved}$ it suffices to independently simulate the $K \\cdot L$ random variables $\\widehat{\\mathit {W}}_{k}^{(\\ell )}$ (rather than the $n \\cdot L$ random variables $V_{i}^{(\\ell )}$ ).","In the following subsubsection, we exemplarily demonstrate the tractability of this reduction effect."],["BS minimization of power divergences and related quantities","´ Consider the special case of power divergence generators $\\varphi := \\widetilde{c} \\cdot \\varphi _{\\gamma }$ ($\\gamma \\in \\mathbb {R}\\backslash ]1,2[$ ) of the Examples REF and REF .","The corresponding estimators $\\widehat{\\Pi }_{L}^{improved}$ can be obtained as follows: within the results of Example REF , set $M_{\\mathbf {P}}=1$ , and replace $\\widetilde{q}_{k}^{\\ast }$ by $\\overline{W^{\\ast }} \\cdot q_{k}^{\\ast }$ as well as $p_{k}$ by $p_{n,k}^{emp}$ ; accordingly, $\\widetilde{U}_{k}^{\\ast n_{k}}$ turns into $U_{k}^{\\ast n_{k}}$ and $\\widetilde{ISF}_{k}$ into $ISF_{k}$ ; simulate independently the random variables $\\widehat{\\mathit {W}}_{k}^{(\\ell )}$ from $U_{k}^{\\ast n_{k}}$ ($k \\in \\lbrace 1, \\ldots , K\\rbrace $ , $\\ell \\in \\lbrace 1,\\ldots ,L\\rbrace $ ); plug in the results of (i),(ii) into (REF ), (REF ), and (REF ) in order to concretely compute $\\widehat{\\Pi }_{L}^{improved}$ .","From this, we can easily generate improved estimators of the power divergences $\\inf _{\\mathbf {Q}\\in \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega }}D_{\\widetilde{c}\\cdot \\varphi _{\\gamma }}(\\mathbf {Q},{P})$ — and more generally, improved estimators of all the infimum-quantities (e.g.","Renyi divergences) respectively supremum-quantities in the parts (b) of the Propositions REF , REF , REF , REF , REF and REF with $A=1$ — by simply replacing $\\mathbb {\\Pi }_{X_{1}^{n}}[\\xi _{n}^{w\\mathbf {W}}\\in \\textrm {$$\\hspace{-6.544pt}$$} ]$ (respectively, its variants) by the corresponding estimator $\\widehat{\\Pi }_{L}^{improved}$ .","If — in the light of Remark REF (vi) — the ${P} = (p_{1}, \\ldots , p_{K})$ is a pregiven known probability vector e.g.","the uniform distribution ${P}^{unif}$ on $\\lbrace 1,\\ldots ,K\\rbrace $ (rather than the limit of the vector of empirical frequencies/masses of a sequence of random variables $X_{i}$ , cf.","(REF )), then we proceed analogously as above by replacing $p_{n,k}^{emp}$ with $p_{k}$ ; correspondingly, we obtain improved estimators of all the infimum-quantities respectively supremum-quantities (e.g.","Renyi entropies, diversity indices) in the parts (a) of the Propositions REF , REF , REF , REF , REF and REF with $A=1$ .","For the sake of brevity, in the following we only present explicitly the outcoming improved estimators for the power divergences (in the “$X_{i}-$ context” ).","Indeed, we simply replace the $\\mathbb {\\Pi }_{X_{1}^{n}}[\\xi _{n}^{w\\mathbf {W}}\\in \\textrm {$$\\hspace{-6.544pt}$$} ]$ in the formulas (REF ), (REF ), (REF ) (with $A=1$ ) by the improved estimator $\\widehat{\\Pi }_{L}^{improved}$ obtained through (i) to (iii); for arbitrarily fixed $\\widetilde{c} >0$ , this leads to the improved power-divergence estimators (BS estimators of power divergences) $&&\\hspace{-54.06006pt}\\widehat{D_{\\widetilde{c}\\cdot \\varphi _{\\gamma }}(\\textrm {\\Omega \\hspace{-6.544pt}\\Omega },{P})} \\ := \\ -\\frac{\\widetilde{c}}{\\gamma (\\gamma -1)}\\left\\lbrace 1-\\left( 1+\\frac{\\gamma }{\\widetilde{c}} \\cdot \\frac{1}{n} \\cdot \\log \\widehat{\\Pi }_{L}^{improved}\\right)^{1-\\gamma }\\right\\rbrace ,\\qquad \\gamma \\in \\, ]-\\infty ,0[ \\, \\cup \\, ]0,1[ \\, \\cup \\, [2,\\infty [,\\\\& &\\hspace{-54.06006pt}\\widehat{D_{\\widetilde{c}\\cdot \\varphi _{0}}(\\textrm {\\Omega \\hspace{-6.544pt}\\Omega },{P})} \\ := \\ -\\frac{1}{n}\\log \\widehat{\\Pi }_{L}^{improved} ,\\hspace{165.02606pt} \\gamma =0,\\\\& & \\hspace{-54.06006pt}\\widehat{D_{\\widetilde{c}\\cdot \\varphi _{1}}(\\textrm {\\Omega \\hspace{-6.544pt}\\Omega },{P})}\\ : = \\ - \\widetilde{c} \\cdot \\log \\left( 1+\\frac{1}{\\widetilde{c}} \\cdot \\frac{1}{n} \\cdot \\log \\widehat{\\Pi }_{L}^{improved}\\right) ,\\hspace{85.35826pt} \\gamma =1.$ Let us finally remark that from the above-mentioned Steps S1 to S4 (and analogously D1 to D4) one can see that for our BS method we basically need only a fast and accurate — pseudo, true, natural, quantum — random number generator.","The corresponding computations can be principally run in parallel, and require relatively moderate computer memory/storage; a detailed discussion is beyond the scope of this paper, given its current length."],["General case, part 2","The algorithm which is presented in this section aims at the evaluation of the bounds $\\inf _{m\\ne 0}D_{\\varphi }\\left( m \\cdot \\textrm {\\right.\\Omega \\hspace{-6.544pt}\\Omega },{P} =\\inf _{{Q}\\in \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega }}D_{\\varphi }\\left( m\\left( {Q}\\right) \\cdot {Q},{P}\\right)\\stackrel{(1)}{=} D_{\\varphi }\\left( m({Q}^{\\ast })\\cdot {Q}^{\\ast },{P}\\right) \\le D_{\\varphi }\\left( \\textrm {\\right.\\Omega \\hspace{-6.544pt}\\Omega },{P} \\le D_{\\varphi }\\left( {Q}^{\\ast },{P}\\right)$ obtained in Section REF , where ${Q}^{\\ast }$ satisfies the above equality $(1)$ .","The estimator of the lower bound in (REF ) is $\\widehat{D}:=-\\frac{1}{n}\\log \\widehat{\\text{ }\\Pi }_{L}^{improved}$ defined in (REF ).","We now turn to an estimate of the upper bound.","Consider for any fixed ${Q}:=\\left( q_{1},..,q_{K}\\right) $ in $\\mathbb {S}_{>0}^{K}$ the real number $m_{n}({Q})$ which satisfies $D_{\\varphi }\\left( m_{n}({Q}) \\cdot {Q},{P}_{n}^{emp} \\right) =\\inf _{m\\ne 0}D_{\\varphi }\\left( m\\cdot \\textrm {\\right.$ $\\hspace{-6.544pt}$$},{P}_{n}^{emp}$$where $ Pnemp$ was defined in the course of (\\ref {I^(n)_k for stat case}).Such $ mn(Q)$ is well defined for all $ Q$since it satisfies the equation (in $ m$)\\begin{equation}\\frac{d}{dm}D_{\\varphi }\\left( m \\cdot {Q},{P}_{n}^{emp}\\right) =\\sum _{k=1}^{K} q_{k} \\cdot \\varphi ^{\\prime }\\left( \\frac{m \\cdot q_{k}}{p_{n,k}^{emp}}\\right) =0 .\\end{equation}Since the mapping $ mD( m Q,P) $ is convexand differentiable, existence and uniqueness of $ mn(Q)$ hold; furthermore,$ mn(Q)] k pn,kemp/qk,k pn,kemp/qk[ $since $ ddmD( m Q,Pnemp) $is negative when $ m=k pn,kemp/qk$ and positive when $ m=k pn,kemp/qk$.$ An estimate of the distribution ${Q}^{\\ast }$ is required.","This can be achieved as follows: Estimate $\\inf _{m\\ne 0}D_{\\varphi }\\left( m \\cdot \\textrm {\\right.$$\\hspace{-6.544pt}$$}, {P}$ through $\\widehat{D} := -\\frac{1}{n}\\log \\widehat{\\Pi }_{L}^{improved}$ defined in (REF ).","Set $i=0.$ Get some ${Q}_{i} := (q_{i,1}, \\ldots , q_{i,K})$ in $\\textrm {$$\\hspace{-6.544pt}$$}$ ; this can be obtained by simulating runs of vectors $\\left( W_{1},..,W_{n}\\right) $ through i.i.d.","sampling under $\\mathbb {}$ .","Evaluate $m_{n}({Q}_{i})$ by solving () (with $q_{i,k}$ instead of $q_{k}$ ) for $m$ , which is a fast calculation by the bisection method.","If $D_{\\varphi }\\left( m_{n}({Q}_{i}) \\cdot {Q}_{i}, {P}_{n}^{emp} \\right) <\\widehat{D}+\\eta $ for some small $\\eta >0$ , then the proxy of ${Q}^{\\ast }$ is ${Q}_{i},$ denoted by $\\widehat{{Q}^{\\ast }}$ .","Else set $i\\leftarrow i+1$ and get ${Q}_{i}$ in $\\textrm {$$\\hspace{-6.544pt}$$}\\cap \\left\\lbrace {Q} \\, : \\, D_{\\varphi }\\left( {Q},{P}_{n}^{emp} \\right) 0$ we normalized $\\widetilde{{P}}:=\\mathbf {P}/M_{\\mathbf {P}},$ and $\\widetilde{\\mathbf {Q}}:=\\mathbf {Q}/M_{\\mathbf {P}}$ for $\\mathbf {Q}$ in $\\mathbf {\\Omega }$ .","With $\\widetilde{\\varphi } \\in \\Upsilon (]a,b[)$ defined through $\\widetilde{\\varphi }:=M_{\\mathbf {P}} \\cdot \\varphi $ , we have obtained $D_{\\varphi }(\\mathbf {Q},\\mathbf {P})=\\sum _{k=1}^{K}p_{k}\\cdot \\varphi \\left( \\frac{q_{k}}{p_{k}}\\right) =\\sum _{k=1}^{K}M_{\\mathbf {P}}\\cdot \\widetilde{p_{k}}\\cdot \\frac{\\varphi \\left( \\frac{M_{\\mathbf {P}}\\cdot \\widetilde{q_{k}}}{M_{\\mathbf {P}}\\cdot \\widetilde{p_{k}}}\\right) }{M_{\\mathbf {P}}}=D_{\\widetilde{\\varphi }}(\\widetilde{\\mathbf {Q}},\\widetilde{{P}})\\qquad \\textrm {(cf.","(\\ref {min Pb prob1}))}.$ It has followed that the solution of (REF ) coincides with the one of the problem of finding $\\widetilde{\\Phi }_{\\widetilde{{P}}}(\\widetilde{\\mathbf {\\Omega }}) := \\inf _{\\widetilde{\\mathbf {Q}}\\in \\widetilde{\\mathbf {\\Omega }} }D_{\\widetilde{\\varphi } }(\\widetilde{\\mathbf {Q}},\\widetilde{{P}}),\\qquad \\textrm {with } \\widetilde{\\mathbf {\\Omega }}:=\\mathbf {\\Omega } /M_{\\mathbf {P}}\\qquad \\textrm {(cf.","(\\ref {min Pb prob2}))}.$ So let us continue by tackling (REF ).","From the assumptions on $\\widetilde{\\varphi }$ and the requirement (REF ) one can see that $\\text{$\\widetilde{W}_{1}$ has moment generating function$t\\rightarrow E_{\\mathbb {\\Pi }}[e^{z \\cdot \\widetilde{W}_{1}}]= MGF_{\\widetilde{\\mathbb {}}}(z)$which is finite on a non-void neighborhood of $0$},$ $E_{\\mathbb {\\Pi }}[\\widetilde{W}_1] = 1 ,$ since $\\widetilde{\\varphi }(1)=0=\\widetilde{\\varphi }^{\\prime }(1)$ .","With the help of these, we obtain the following Proposition 61 Under the assumptions of Theorem REF , for any set $\\widetilde{\\mathbf {\\Omega }}\\subset \\mathcal {M} := \\mathbb {R}^{K}$ with (REF ) one has $-\\inf _{\\widetilde{\\mathbf {Q}} \\in int(\\widetilde{\\mathbf {\\Omega }}) }D_{\\varphi }\\left( \\widetilde{\\mathbf {Q}},\\widetilde{{P}}\\right) &\\le &\\lim \\inf _{n\\rightarrow \\infty }\\frac{1}{n}\\log \\mathbb {\\Pi }\\left[\\xi _{n}^{\\widetilde{\\textbf {W}}}\\in \\widetilde{\\mathbf {\\Omega }} \\right] \\nonumber \\\\&\\le &\\lim \\sup _{n\\rightarrow \\infty }\\frac{1}{n}\\log \\mathbb {\\Pi }\\left[\\xi _{n}^{\\widetilde{\\textbf {W}}}\\in \\widetilde{\\mathbf {\\Omega }} \\right]\\le -\\inf _{\\widetilde{\\mathbf {Q}}\\in cl(\\widetilde{\\mathbf {\\Omega }}) }D_{\\varphi }\\left( \\widetilde{\\mathbf {Q}},\\widetilde{{P}}\\right) .$ Proof of Proposition REF .","Recall from Remark REF (v) that $I_{k}^{(n)} := \\lbrace i \\in \\lbrace 1,\\ldots ,n\\rbrace : \\widetilde{x}_{i}=d_{k}\\rbrace $ and $n_{k} := card(I_{k}^{(n)})$ denotes the number of elements therein ($k \\in \\lbrace 1,\\ldots ,K\\rbrace $ ), i.e.","$n_{k}$ is the number of the $\\widetilde{x}_{i}$ ’s which equal $d_{k}$ .","We follow the line of proof of Theorem 2.2.30 in Dembo & Zeitouni [108], which states the large deviation principle (LDP) for the vector of partial sums of random vectors in $\\mathbb {R}^{K}$ , where we also use Corollary 6.1.6 in [108] in relation with condition (REF ).","Indeed, since the $k-$ th component of the vector $\\xi _{n}^{\\mathbf {\\widetilde{W}}}$ is the $1/n-$ fold of the sum of the $\\widetilde{W}_{i}$ ’s for which the corresponding $\\widetilde{x}_{i}$ ’s equal $d_{k}$ (i.e., $\\frac{1}{n} \\sum _{i\\in I_{k}^{(n)}} \\widetilde{W}_{i}$ ) the proof will follow from a similar treatment as for the standard Cramer LDP in $\\mathbb {R}^{K}.$ The only difference lies in two facts: the number of the summands for the coordinate $k$ is $n_{k}$ , the number of $\\widetilde{x}_{i}$ ’s which equal $d_{k}$ , instead of $n$ in the standard case.","Furthermore we will need to substitute $n_{k}$ by its equivalent $n \\cdot \\widetilde{p}_{k}$ , which adds an approximation step.","For the upper bound, the proof is based on the corresponding result for $B=B_{1}\\times \\cdots \\times B_{K}$ where the $B_{k}$ ’s are open bounded intervals on $\\mathbb {R}^{+}.$ Since the sequence $\\left( \\widetilde{x}_{1}, \\ldots \\right)$ satisfies $\\lim _{n\\rightarrow \\infty }\\frac{n_{k}}{n}= \\widetilde{p}_{k} ,\\qquad \\textrm {(cf.","(\\ref {fo.freqlim}))}$ there holds $&&\\frac{1}{n}\\log \\mathbb {\\Pi } \\left[\\xi _{n}^{\\mathbf {\\widetilde{W}}} \\in B\\right]=\\frac{1}{n}\\log \\mathbb {\\Pi } \\bigg [ \\bigcap \\limits _{k=1}^{K}\\bigg (\\frac{1}{n} \\sum _{i\\in I_{k}^{(n)}} \\widetilde{W}_{i}\\in B_{k}\\bigg ) \\bigg ]\\nonumber \\\\&=&\\frac{1}{n}\\sum \\limits _{k=1}^{K}\\log \\mathbb {\\Pi } \\bigg [ \\frac{1+o(1)}{n_{k}}\\sum _{i\\in I_{k}^{(n)}} \\widetilde{W}_{i}\\in \\frac{1}{\\widetilde{p}_{k}} B_{k}\\bigg ],$ and hence $\\lim \\sup _{n\\rightarrow \\infty }\\frac{1}{n}\\log \\mathbb {\\Pi } \\left[\\xi _{n}^{\\mathbf {\\widetilde{W}}}\\in B\\right]&\\le &\\sum \\limits _{k=1}^{K} \\widetilde{p}_{k} \\cdot \\lim \\sup _{n_{k}\\rightarrow \\infty }\\frac{1}{n_{k}}\\log \\mathbb {\\Pi } \\bigg [ \\frac{1}{n_{k}}\\sum _{i\\in I_{k}^{(n)}} \\widetilde{W}_{i}\\in \\frac{1}{p_{k}}B_{k}\\bigg ] \\nonumber \\\\&\\le &-\\sum \\limits _{k=1}^{K}\\inf _{x_{k}\\in cl(B_{k})} \\widetilde{p}_{k} \\cdot \\varphi \\left(\\frac{x_{k}}{\\widetilde{p}_{k}}\\right) .$ To deduce (REF ) from (REF ), we have used (i) the fact that for all $k$ the random variables $\\frac{1}{n_{k}}\\left( 1+o(1))\\right) \\cdot \\sum _{i\\in I_{k}^{(n)}} \\widetilde{W}_{i}$ and $\\frac{1}{n_{k}}\\sum _{i\\in I_{k}^{(n)}} \\widetilde{W}_{i}$ are exponentially equivalent in the sense that their difference $\\Delta _{n_{k}}$ satisfies $\\lim \\sup _{n_{k}\\rightarrow \\infty }\\frac{1}{n_{k}}\\log \\mathbb {\\Pi }\\left[ \\,\\left|\\Delta _{n_{k}}\\right|>\\eta \\, \\right] =-\\infty ,$ making use of the Chernoff inequality for all positive $\\eta $ , as well as (ii) Theorem 4.2.13 in [108].","Now the summation and the inf-operations can be permuted in (REF ) which proves the claim for the rectangle $B$ .","As in [108], for a compact set $\\Omega $ we consider its finite covering by such open sets $B$ and conclude; for $\\Omega $ being a closed set, a tightness argument holds, following [108] Theorem 2.2.30 verbatim.","For the lower bound consider the same rectangle $B$ .","The argument which locates the tilted distribution at the center of $B$ , together with the use of the LLN for the corresponding r.v’s as in [108], in combination with the same approximations as above to handle the approximation of $n_{k}$ by $n \\cdot \\widetilde{p}_{k}$ , complete the proof of Proposition REF .","We omit the details.","$\\blacksquare $ Let us continue with the proof of Theorem REF , by giving the following two helpful lemmas for $\\Phi _{{P}}(\\mathbf {A}) := \\inf _{\\mathbf {Q} \\in \\mathbf {A}} D_{\\varphi }\\left(\\mathbf {Q},{P}\\right),\\qquad \\mathbf {A} \\subset \\mathcal {M} := \\mathbb {R}^{K},$ Lemma 62 For any open set $\\mathbf {A} \\subset \\mathcal {M} := \\mathbb {R}^{K}$ one has $\\Phi _{{P}}(\\mathbf {A}) = \\Phi _{{P}}(cl(\\mathbf {A}))$ .","This is clear from the continuity of $\\Phi _{{P}}$ .","Lemma 63 For any $\\mathbf {A} \\subset \\mathcal {M} := \\mathbb {R}^{K}$ satisfying (REF ) one has $\\Phi _{{P}}(cl(\\mathbf {A})) = \\Phi _{{P}}(\\mathbf {A}) = \\Phi _{{P}}(int(\\mathbf {A}))$ .","Proof of Lemma REF .","Assume first that $\\Phi _{{P}}(\\mathbf {A})$ is finite.","Then suppose that $\\mathbf {A}$ satisfies (REF ) and $\\Phi _{{P}}(cl(\\mathbf {A})) < \\Phi _{{P}}(int(\\mathbf {A}))$ .","The latter implies the existence of a point $a \\in cl(\\mathbf {A})$ such that $a \\notin int(\\mathbf {A})$ and $\\Phi _{{P}}(a)= \\Phi _{{P}}(cl(\\mathbf {A}))$ .","But then, by Lemma REF and (REF ) one gets $\\Phi _{{P}}(int(\\mathbf {A})) = \\Phi _{{P}}(cl(int(\\mathbf {A}))) = \\Phi _{{P}}(cl(\\mathbf {A})) = \\Phi _{{P}}(a)$ which leads to a contradiction.","When $\\Phi _{{P}}(\\mathbf {A}) = \\infty $ then $\\Phi _{{P}} (cl(\\mathbf {A}))=\\Phi _{{P}} (int(\\mathbf {A}))=\\Phi _{{P}} (\\mathbf {A})=\\infty $ .","$\\blacksquare $ Putting things together, the required asymptotic assertion (REF ) follows from (REF ), (REF ) and Lemma REF .","This completes the proof of Theorem REF .","$\\blacksquare $"],["Proofs — Part 2","Before we tackle the proof of Theorem REF , let us introduce the following Lemma 64 If $\\Omega $$\\Omega \\subset \\mathbb {S}^{K}$ satisfies condition (REF ), then $\\widetilde{\\widetilde{\\textrm {\\Omega \\hspace{-6.544pt}\\Omega }}}:= \\bigcup \\limits _{m\\ne 0}cl(m \\cdot \\textrm {$$\\hspace{-6.544pt}$$})$ has the property (REF ).","This can be deduced in a straightforward way: the assumption implies that $cl(\\textrm {$$\\hspace{-6.544pt}$$})$ satisfies (REF ), and thus also $m \\cdot cl(\\textrm {$$\\hspace{-6.544pt}$$})$ satisfies (REF ).","But this implies the validity of (REF ) for the “cone” $\\bigcup \\limits _{m\\ne 0} m \\cdot cl(\\textrm {$$\\hspace{-6.544pt}$$})$ which is nothing but $\\bigcup \\limits _{m\\ne 0} cl(m \\cdot \\textrm {$$\\hspace{-6.544pt}$$})$ .","Proof of Theorem REF .","Recall the interpretations of the two vectors $\\xi _{n,\\mathbf {X}}^{\\mathbf {W}}$ respectively $\\xi _{n,\\mathbf {X}}^{w\\mathbf {W}}$ given in (REF ) respectively (REF ), and that the sum of their $k$ components are $\\sum _{k=1}^{K} \\frac{1}{n} \\sum _{i\\in I_{k}^{(n)}} W_{i} =\\frac{1}{n}\\sum _{i=1}^{n} W_{i}$ respectively $\\sum _{k=1}^{K}\\frac{\\sum _{i \\in I_{k}^{(n)}}W_{i}}{\\sum _{k=1}^{K}\\sum _{i \\in I_{k}^{(n)}}W_{i}}=1$ (in case of $\\sum _{i=1}^{n}W_{i} \\ne 0$ ).","In the light of these, for $\\textrm {$$\\hspace{-6.544pt}$$} \\subset \\mathbb {S}^{K}$ one gets the set identification $\\left\\lbrace \\xi _{n,\\mathbf {X}}^{w\\mathbf {W}} \\in \\textrm {\\right.$ $\\hspace{-6.544pt}$$} =\\bigcup \\limits _{m\\ne 0}\\left\\lbrace \\xi _{n,\\mathbf {X}}^{\\mathbf {W}}\\in m\\cdot \\textrm {\\right.$$\\hspace{-6.544pt}$$} ,\\frac{1}{n}\\sum _{i=1}^{n} W_{i} = m$$since $ { i=1nWi=0} $ amounts to $ m=0$, which cannothold when $ { n,XwW $\\Omega $$\\Omega $ }$.","Now\\begin{eqnarray*}\\mathbb {\\Pi }_{X_{1}^{n}}\\left[\\xi _{n,\\mathbf {X}}^{w\\mathbf {W}}\\in \\textrm {\\right.\\Omega \\hspace{-6.544pt}\\Omega } &=&\\mathbb {\\Pi }_{X_{1}^{n}} \\bigg [ \\bigcup \\limits _{m\\ne 0}\\bigg \\lbrace \\xi _{n,\\mathbf {X}}^{\\mathbf {W}} \\in m \\cdot \\textrm {\\Omega \\hspace{-6.544pt}\\Omega },\\frac{1}{n}\\sum _{i=1}^{n} W_{i} =m \\bigg \\rbrace \\bigg ] \\\\&=&\\mathbb {\\Pi }_{X_{1}^{n}}\\bigg [ \\bigcup \\limits _{m\\ne 0}\\bigg \\lbrace \\xi _{n,\\mathbf {X}}^{\\mathbf {W}} \\in m\\cdot \\textrm {\\Omega \\hspace{-6.544pt}\\Omega }\\big \\rbrace \\bigg ]= \\mathbb {\\Pi }_{X_{1}^{n}}\\bigg [ \\xi _{n,\\mathbf {X}}^{\\mathbf {W}}\\in \\bigcup \\limits _{m\\ne 0} m\\cdot \\textrm {\\Omega \\hspace{-6.544pt}\\Omega } \\bigg ]\\end{eqnarray*}since $ { n,XW m$\\Omega $$\\Omega $ } { 1ni=1n Wi =m }$.","Therefore\\begin{equation}\\frac{1}{n}\\log \\mathbb {\\Pi }_{X_{1}^{n}}\\left[\\xi _{n,\\mathbf {X}}^{w\\mathbf {W}}\\in \\textrm {\\right.\\Omega \\hspace{-6.544pt}\\Omega } =\\frac{1}{n}\\log \\mathbb {\\Pi }_{X_{1}^{n}}\\bigg [ \\xi _{n,\\mathbf {X}}^{\\mathbf {W}} \\in \\bigcup \\limits _{m\\ne 0}m\\cdot \\textrm {\\Omega \\hspace{-6.544pt}\\Omega } \\bigg ] .\\end{equation}Because of Proposition \\ref {PropLDPWEM-finitease} --- applied to$$\\Omega $$\\Omega $ := m0m $\\Omega $$\\Omega $$ --- one getsin terms of (\\ref {phishort})$ $-\\Phi _{{P}}\\Big (int\\Big (\\bigcup \\limits _{m\\ne 0}m\\cdot \\textrm {\\Omega \\hspace{-6.544pt}\\Omega }\\Big )\\Big )&\\le &\\lim \\inf _{n\\rightarrow \\infty }\\frac{1}{n}\\log \\mathbb {\\Pi }_{X_{1}^{n}}\\bigg [ \\xi _{n,\\mathbf {X}}^{\\mathbf {W}} \\in \\bigcup \\limits _{m\\ne 0}m\\cdot \\textrm {\\Omega \\hspace{-6.544pt}\\Omega } \\bigg ] \\nonumber \\\\&\\le &\\lim \\sup _{n\\rightarrow \\infty }\\frac{1}{n}\\log \\mathbb {\\Pi }_{X_{1}^{n}}\\bigg [\\xi _{n,\\mathbf {X}}^{\\mathbf {W}}\\in \\bigcup \\limits _{m\\ne 0}m\\cdot \\textrm {\\Omega \\hspace{-6.544pt}\\Omega } \\bigg ] \\le -\\Phi _{{P}}\\Big (cl\\Big (\\bigcup \\limits _{m\\ne 0}m\\cdot \\textrm {\\Omega \\hspace{-6.544pt}\\Omega }\\Big )\\Big ) .$ $\\hspace{-176.407pt}\\text{But}& & \\Phi _{{P}}\\Big (int\\Big (\\bigcup \\limits _{m\\ne 0}m\\cdot \\textrm {\\Omega \\hspace{-6.544pt}\\Omega }\\Big )\\Big ) \\le \\Phi _{{P}}\\Big (\\bigcup \\limits _{m\\ne 0}int\\Big (m\\cdot \\textrm {\\Omega \\hspace{-6.544pt}\\Omega }\\Big )\\Big ) =\\inf _{m\\ne 0} \\Phi _{{P}}(int(m\\cdot \\textrm {\\Omega \\hspace{-6.544pt}\\Omega }))\\\\\\hspace{-176.407pt}\\text{and}& & \\Phi _{{P}}\\Big (cl\\Big (\\bigcup \\limits _{m\\ne 0}m\\cdot \\textrm {\\Omega \\hspace{-6.544pt}\\Omega }\\Big )\\Big ) \\ge \\Phi _{{P}}\\Big (\\bigcup \\limits _{m\\ne 0}cl\\Big (m\\cdot \\textrm {\\Omega \\hspace{-6.544pt}\\Omega }\\Big )\\Big ) =\\inf _{m\\ne 0} \\Phi _{{P}}(cl(m\\cdot \\textrm {\\Omega \\hspace{-6.544pt}\\Omega })) .$ In fact, the inequality in (REF ) is straightforward because of $\\bigcup \\limits _{m\\ne 0}int(m\\cdot \\textrm {$$\\hspace{-6.544pt}$$})\\subset int(\\bigcup \\limits _{m\\ne 0}m\\cdot \\textrm {$$\\hspace{-6.544pt}$$})$ (since the latter is the largest open set contained in $\\bigcup \\limits _{m\\ne 0}m\\cdot \\textrm {$$\\hspace{-6.544pt}$$}$ ); the inequality in () follows from $& & \\Phi _{{P}}\\Big (cl\\Big (\\bigcup \\limits _{m\\ne 0}m\\cdot \\textrm {\\Omega \\hspace{-6.544pt}\\Omega }\\Big )\\Big ) \\ge \\Phi _{{P}}\\Big (cl\\Big (\\bigcup \\limits _{m\\ne 0}cl\\Big (m\\cdot \\textrm {\\Omega \\hspace{-6.544pt}\\Omega }\\Big )\\Big )\\Big ) =\\Phi _{{P}}\\Big (\\bigcup \\limits _{m\\ne 0}cl\\Big (m\\cdot \\textrm {\\Omega \\hspace{-6.544pt}\\Omega }\\Big )\\Big )\\nonumber $ An application of Lemma REF yields $\\Phi _{{P}}(int(m\\cdot \\textrm {$$\\hspace{-6.544pt}$$})) =\\Phi _{{P}}(m\\cdot \\textrm {$$\\hspace{-6.544pt}$$}) =\\Phi _{{P}}(cl(m\\cdot \\textrm {$$\\hspace{-6.544pt}$$}))$ for all $m \\ne 0$ , and hence $\\inf _{m\\ne 0} \\Phi _{{P}}(int(m\\cdot \\textrm {\\Omega \\hspace{-6.544pt}\\Omega })) =\\inf _{m\\ne 0} \\Phi _{{P}}(m\\cdot \\textrm {\\Omega \\hspace{-6.544pt}\\Omega })= \\inf _{m\\ne 0} \\Phi _{{P}}(cl(m\\cdot \\textrm {\\Omega \\hspace{-6.544pt}\\Omega })) .$ By combining (), (REF ), (REF ), () and (REF ), one arrives at $&&\\lim _{n \\rightarrow \\infty } \\frac{1}{n}\\log \\mathbb {\\Pi }_{X_{1}^{n}} \\left[\\xi _{n,\\mathbf {X}}^{w\\mathbf {W}}\\in \\textrm {\\right.\\Omega \\hspace{-6.544pt}\\Omega } = \\lim _{n \\rightarrow \\infty } \\frac{1}{n}\\log \\mathbb {\\Pi }_{X_{1}^{n}} \\bigg [ \\xi _{n,\\mathbf {X}}^{\\mathbf {W}}\\in \\bigcup \\limits _{m\\ne 0}m\\cdot \\textrm {\\Omega \\hspace{-6.544pt}\\Omega } \\bigg ]\\nonumber \\\\&& = - \\inf _{m\\ne 0} \\Phi _{{P}}(m\\cdot \\textrm {\\Omega \\hspace{-6.544pt}\\Omega })= - \\inf _{m\\ne 0} \\inf _{\\mathbf {Q} \\in m\\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega }}D_{\\varphi }\\left(\\mathbf {Q},{P}\\right)= - \\inf _{m\\ne 0} \\inf _{{Q} \\in \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega }}D_{\\varphi }\\left(m\\cdot {Q}, {P}\\right) ,\\nonumber $ where in the second last equality we have “reverted” the notation (REF ).","Note that we did not assume (REF ) for $\\bigcup \\limits _{m\\ne 0} m \\cdot \\textrm {$$\\hspace{-6.544pt}$$}$ .","$\\blacksquare $"],["Proofs — Part 3","Proof of Lemma REF .","From (REF ) one gets straightforwardly for arbitrary $\\widetilde{c} >0$ $D_{\\widetilde{c} \\cdot \\varphi _{\\gamma }}(m \\cdot \\mathbf {Q},{P}) \\hspace{-5.69046pt} &:=& \\hspace{-5.69046pt}{\\left\\lbrace \\begin{array}{ll}\\frac{\\widetilde{c} \\cdot \\left(m^{\\gamma } \\cdot H_{\\gamma } - m \\cdot A \\cdot \\gamma + \\gamma -1 \\right)}{\\gamma \\cdot (\\gamma -1)},\\hspace{75.39963pt} \\textrm {if } \\gamma \\in \\, ]-\\infty ,0[, \\ {P} \\in \\mathbb {S}_{\\ge 0}^{K}, \\ \\mathbf {Q} \\in A \\cdot \\mathbb {S}_{> 0}^{K} \\ \\ \\textrm {and } m >0, \\\\\\widetilde{c} \\cdot (- \\log m + \\widetilde{I} -1 + m \\cdot A),\\hspace{42.67912pt} \\textrm {if } \\gamma = 0, \\ {P} \\in \\mathbb {S}_{\\ge 0}^{K}, \\ A \\cdot \\mathbf {Q} \\in \\mathbb {S}_{> 0}^{K} \\ \\ \\textrm {and } m >0, \\\\\\frac{\\widetilde{c} \\cdot \\left(m^{\\gamma } \\cdot H_{\\gamma } - m \\cdot A \\cdot \\gamma + \\gamma -1 \\right)}{\\gamma \\cdot (\\gamma -1)},\\hspace{75.39963pt} \\textrm {if } \\gamma \\in \\, ]0,1[, \\ {P} \\in \\mathbb {S}_{\\ge 0}^{K}, \\ \\mathbf {Q} \\in A \\cdot \\mathbb {S}_{\\ge 0}^{K} \\ \\ \\textrm {and } m \\ge 0, \\\\\\widetilde{c} \\cdot (A \\cdot m \\cdot \\log m + m \\cdot (I-A) +1),\\hspace{14.22636pt} \\textrm {if } \\gamma = 1, \\ {P} \\in \\mathbb {S}_{> 0}^{K}, \\ \\mathbf {Q} \\in A \\cdot \\mathbb {S}_{\\ge 0}^{K} \\ \\ \\textrm {and } m \\ge 0, \\\\\\frac{\\widetilde{c} \\cdot \\left(m^{\\gamma } \\cdot H_{\\gamma } \\cdot {1}_{[0,\\infty [}(m)- m \\cdot A \\cdot \\gamma + \\gamma -1 \\right)}{\\gamma \\cdot (\\gamma -1)},\\hspace{39.83368pt} \\textrm {if } \\gamma \\in \\, ]1,2[, \\ {P} \\in \\mathbb {S}_{>0}^{K}, \\ \\mathbf {Q} \\in A \\cdot \\mathbb {S}^{K} \\ \\ \\textrm {and } m \\in ]-\\infty ,\\infty [,\\\\\\frac{\\widetilde{c} \\cdot \\left(m^{2} \\cdot H_{2} - m \\cdot A \\cdot 2 + 2 -1 \\right)}{2 \\cdot (2-1)},\\hspace{76.82234pt} \\textrm {if } \\gamma = 2, \\ {P} \\in \\mathbb {S}_{>0}^{K}, \\ \\mathbf {Q} \\in A \\cdot \\mathbb {S}^{K} \\ \\ \\textrm {and } m \\in ]-\\infty ,\\infty [,\\\\\\frac{\\widetilde{c} \\cdot \\left(m^{\\gamma } \\cdot H_{\\gamma } \\cdot {1}_{[0,\\infty [}(m)- m \\cdot A \\cdot \\gamma + \\gamma -1 \\right)}{\\gamma \\cdot (\\gamma -1)},\\hspace{39.83368pt} \\textrm {if } \\gamma \\in \\, ]2,\\infty [, \\ {P} \\in \\mathbb {S}_{>0}^{K}, \\ \\mathbf {Q} \\in A \\cdot \\mathbb {S}^{K} \\ \\ \\textrm {and } m \\in ]-\\infty ,\\infty [,\\\\\\infty , \\hspace{156.49014pt} \\textrm {else},\\end{array}\\right.","}$ where we have used the three $m-$ independent abbreviations $H_{\\gamma } := \\sum \\displaylimits _{k=1}^{K} (q_{k})^{\\gamma } \\cdot (p_{k})^{1-\\gamma }\\ = \\ 1 + \\gamma \\cdot (A-1) +\\frac{\\gamma \\cdot (\\gamma -1)}{\\widetilde{c}} \\cdot D_{\\widetilde{c} \\cdot \\varphi _{\\gamma }}(\\mathbf {Q}, {P}),\\qquad \\text{(cf.","(\\ref {brostu3:fo.divpow.hellinger1}))}\\nonumber $ $I:= \\sum \\displaylimits _{k=1}^{K} q_{k} \\cdot \\log \\left( \\frac{q_{k}}{p_{k}} \\right)\\ = \\ \\frac{1}{\\widetilde{c}} \\cdot D_{\\widetilde{c} \\cdot \\varphi _{1}}(\\mathbf {Q}, {P}) + A - 1,\\qquad \\text{(cf.","(\\ref {brostu3:fo.divpow.Kull1}))}\\nonumber $ $\\widetilde{I} := \\sum \\displaylimits _{k=1}^{K} p_{k} \\cdot \\log \\left( \\frac{p_{k}}{q_{k}} \\right)\\ = \\ \\frac{1}{\\widetilde{c}} \\cdot D_{\\widetilde{c} \\cdot \\varphi _{0}}(\\mathbf {Q}, {P}) + 1 - A.\\qquad \\text{(cf.","(\\ref {brostu3:fo.divpow.RevKull1}))}\\nonumber $ To proceed, let us fix an arbitrary constant $\\widetilde{c} >0$ .","(i) Case $\\gamma \\cdot (1-\\gamma ) \\ne 0$ .","(ia) Let us start with the subcase $\\gamma \\in ]-\\infty ,0[$ .","From the first and the last line of (REF ), it is clear that the corresponding $m-$ infimum can not be achieved for $m \\le 0$ ; since $H_{\\gamma } > 0$ one gets the unique minimizer $m_{min} = \\Big (\\frac{H_{\\gamma }}{A}\\Big )^{1/(1-\\gamma )} >0$ and the minimum $D_{\\widetilde{c} \\cdot \\varphi _{\\gamma }}(m_{min} \\cdot \\mathbf {Q},{P})=\\frac{\\widetilde{c}}{\\gamma } \\cdot (1- \\frac{H^{1/(1-\\gamma )}}{A^{\\gamma /(1-\\gamma )}} )$ .","Hence, (REF ) is established.","The assertions (REF ) and () follow immediately by monotonicity inspection of $x \\rightarrow \\frac{\\widetilde{c}}{\\gamma } \\cdot \\left[ 1-\\frac{1}{A^{\\gamma /(1-\\gamma )}} \\cdot \\left[ 1 + \\gamma \\cdot (A-1) + \\frac{\\gamma \\cdot \\left( \\gamma -1\\right)}{\\widetilde{c}}\\cdot x\\right] ^{-1/\\left( \\gamma -1\\right) }\\right]$ for $x \\ge 0$ such that $1 + \\gamma \\cdot (A-1) + \\frac{\\gamma \\cdot \\left( \\gamma -1\\right)}{\\widetilde{c}}\\cdot x \\ge 0$ .","(ib) The subcase $\\gamma \\in ]0,1[$ (cf.","the third line of (REF )) works analogously if $H_{\\gamma } >0$ ; furthermore, if $H_{\\gamma } =0$ — which can only appear when ${P}$ , $\\mathbf {Q}$ have disjoint supports (singularity)— then $\\inf _{m>0} D_{\\widetilde{c} \\cdot \\varphi _{\\gamma }}(m \\cdot \\mathbf {Q},{P}) =\\frac{\\widetilde{c}}{\\gamma }$ which is (the corresponding special case of) (REF ).","(ic) In the subcase $\\gamma \\in ]1,\\infty [$ (cf.","the fifth, sixth and seventh line of (REF )) it is straightforward to see that the desired infimum can not be achieved for $m < 0$ .","Hence, one can proceed analogously to subcase (ia).","(id) The assertions () to () are straightforward.","(ii) Case $\\gamma =1$ .","From the fourth line of (REF ), one obtains the unique minimizer $m_{min} = \\exp \\lbrace -I/A\\rbrace $ and the minimum $D_{\\widetilde{c} \\cdot \\varphi _{1}}(m_{min} \\cdot \\mathbf {Q},{P})= \\widetilde{c} \\cdot (1- A \\cdot m_{min})$ , which leads to (REF ).","The monotonicity of $x \\rightarrow \\widetilde{c} \\cdot (1- \\exp \\lbrace -x/\\widetilde{c}\\rbrace )$ for $x\\ge 0$ implies immediately (REF ) and (); moreover, () and () are immediate.","(iii) Case $\\gamma =0$ .","The second line of (REF ) implies the unique minimizer $m_{min} = 1/A$ , the minimum $D_{\\widetilde{c} \\cdot \\varphi _{0}}(m_{min} \\cdot \\mathbf {Q},{P})= \\widetilde{c} \\cdot (\\widetilde{I} + \\log A)$ , and hence (REF ).","The assertions (REF ) to () are obvious.","$\\blacksquare $"],["Proofs — Part 4","Proof of Proposition REF .","Clearly, (G1) and (G2) are part of the definition of $\\widetilde{\\Upsilon }(]a,b[)$ .","Recall our required representability (REF ).","The therein involved Laplace-Stieltjes transform (Laplace-Lebesgue transform) $z\\mapsto MGF_{\\mathbb {}}(z):=\\int _{\\mathbb {R}}e^{z \\cdot y} \\, d\\mathbb {} (y)= E_{\\mathbb {\\Pi }}[e^{z\\cdot W}]$ of a probability measure $\\mathbb {}$ on the real line respectively of an associated random variable $W$ (with $\\mathbb {}[ \\cdot \\, ] := \\mathbb {\\Pi }[W \\in \\cdot \\, ]$ ) has the following fundamental properties, according to well-known general theory: $MGF_{\\mathbb {}}$ takes values in $]0,\\infty ]$ ; the effective domain $dom(MGF_{\\mathbb {}})$ is an interval which contains 0 and which may be degenerated or even the whole real line; correspondingly, we denote its interior by $]\\lambda _{-},\\lambda _{+}[ := int(dom(MGF_{\\mathbb {}}))$ which may be the empty set (in case that $dom(MGF_{\\mathbb {}})=\\lbrace 0\\rbrace $ , i.e.","$\\lambda _{-}=\\lambda _{+}=0$ ); clearly, there holds $\\lambda _{-} \\in [-\\infty ,0]$ and $\\lambda _{+} \\in [0,\\infty ]$ ; $MGF_{\\mathbb {}}$ is continuous on $dom(MGF_{\\mathbb {}})$ and lower semicontinuous on $\\mathbb {R}$ ; if $\\lambda _{-} \\ne \\lambda _{+}$ , then $MGF_{\\mathbb {}}$ is real analytic and thus infinitely differentiable on $]\\lambda _{-},\\lambda _{+}[$ ; if $MGF_{\\mathbb {}}$ is finite in a neighborhood of zero, i.e.","$0 \\in ]\\lambda _{-},\\lambda _{+}[$ , then for all $k \\in \\mathbb {N}_{0}$ the $k-$ th moment of $\\mathbb {}$ respectively $W$ exists and is finite and can be computed in terms of the $k-$ th derivative $MGF_{\\mathbb {}}^{(k)}$ as $MGF_{\\mathbb {}}^{(k)}(0) =\\int _{\\mathbb {R}} y^{k} \\, d\\mathbb {} (y)= E_{\\mathbb {\\Pi }}[W^{k}],$ which, by the way, then allows the interpretation of $MGF_{\\mathbb {}}$ as “moment generating function of $\\mathbb {}$ resp.","$W$ ”since we assume $0 \\in ]\\lambda _{-},\\lambda _{+}[$ , we have already used the meaningful abbreviation $MGF$ (rather than LST) in (REF ); if $\\lambda _{-} \\ne \\lambda _{+}$ , then $MGF_{\\mathbb {}}$ is strictly convex on $]\\lambda _{-},\\lambda _{+}[$ .","Hence, the logarithm of the Laplace-Stieltjes transform $z\\mapsto \\Lambda _{\\mathbb {}}(z) := \\log MGF_{\\mathbb {} }(z):=\\log \\int _{\\mathbb {R}}e^{z \\cdot y} \\, d\\mathbb {} (y)= \\log E_{\\mathbb {\\Pi }}[e^{z\\cdot W}]$ (which in case of $0 \\in ]\\lambda _{-},\\lambda _{+}[$ can be interpreted as cumulant generating function) “carries over” (M1) to (M6), which partially can be even refined: $\\Lambda _{\\mathbb {}}$ takes values in $]-\\infty ,\\infty ]$ ; $dom(\\Lambda _{\\mathbb {}}) = dom(MGF_{\\mathbb {}})$ and thus $int(dom(\\Lambda _{\\mathbb {} })) = \\, ]\\lambda _{-},\\lambda _{+}[$ ; $\\Lambda _{\\mathbb {} }$ is continuous on $dom(\\Lambda _{\\mathbb {}})$ and lower semicontinuous on $\\mathbb {R}$ ; if $\\lambda _{-} \\ne \\lambda _{+}$ , then $\\lambda _{\\mathbb {}}$ is infinitely differentiable on $]\\lambda _{-},\\lambda _{+}[$ ; if $0 \\in ]\\lambda _{-},\\lambda _{+}[$ , then $& & \\Lambda _{\\mathbb {} }(0) = 0, \\quad \\Lambda _{\\mathbb {} }^{\\prime }(0) = \\int _{\\mathbb {R}} y \\, d\\mathbb {} (y)= E_{\\mathbb {\\Pi }}[W],\\\\& & \\Lambda _{\\mathbb {} }^{\\prime \\prime }(0) =\\int _{\\mathbb {R}} \\Big (y - \\int _{\\mathbb {R}} \\widetilde{y} \\, d\\mathbb {} (\\widetilde{y})\\Big )^{2}\\, d\\mathbb {} (y)= E_{\\mathbb {\\Pi }}[W^{2}] - (E_{\\mathbb {\\Pi }}[W])^{2} = Var_{\\mathbb {\\Pi }}[W];$ under the assumption $\\lambda _{-} \\ne \\lambda _{+}$ there holds: $\\Lambda _{\\mathbb {}}$ is strictly convex on $]\\lambda _{-},\\lambda _{+}[$ if and only if $\\mathbb {}$ is not a one-point distribution (Dirac mass) if and only if $W$ is not a.s. constant; otherwise, $\\Lambda _{\\mathbb {}}$ is linear; under the assumption that $\\mathbb {}$ is not a one-point distribution (Dirac mass) — with the notations $a := \\inf supp(\\mathbb {}) = \\inf supp(W)$ , $b := \\sup supp(\\mathbb {}) = \\sup supp(W)$ , $t_{-}^{sc} := \\inf \\lbrace \\Lambda _{\\mathbb {}}^{\\prime }(z) \\, : \\, z \\in ]\\lambda _{-},\\lambda _{+}[ \\rbrace = \\lim _{z \\downarrow \\lambda _{-}} \\Lambda _{\\mathbb {}}^{\\prime }(z)$ and $t_{+}^{sc} := \\sup \\lbrace \\Lambda _{\\mathbb {}}^{\\prime }(z) \\, : \\, z \\in ]\\lambda _{-},\\lambda _{+}[ \\rbrace = \\lim _{z \\uparrow \\lambda _{+}} \\Lambda _{\\mathbb {}}^{\\prime }(z)$ — one gets the following assertions: $]t_{-}^{sc},t_{+}^{sc}[ \\ \\subseteq \\ ]a,b[$ ; if $a > - \\infty $ , then $\\lambda _{-} = - \\infty $ , $t_{-}^{sc} = \\lim _{z \\rightarrow -\\infty } \\Lambda _{\\mathbb {}}^{\\prime }(z)= \\lim _{z \\rightarrow -\\infty } \\frac{\\Lambda _{\\mathbb {}}(z)}{z} = a$ ; if $b < \\infty $ , then $\\lambda _{+} = \\infty $ , $t_{+}^{sc} = \\lim _{z \\rightarrow \\infty } \\Lambda _{\\mathbb {}}^{\\prime }(z)= \\lim _{z \\rightarrow \\infty } \\frac{\\Lambda _{\\mathbb {}}(z)}{z} = b$ ; if $a = - \\infty $ and $\\lambda _{-} = - \\infty $ , then $t_{-}^{sc} = \\lim _{z \\rightarrow -\\infty } \\Lambda _{\\mathbb {}}^{\\prime }(z)= -\\infty = a$ ; if $b = \\infty $ and $\\lambda _{+} = \\infty $ , then $t_{+}^{sc} = \\lim _{z \\rightarrow \\infty } \\Lambda _{\\mathbb {}}^{\\prime }(z)= \\infty = b$ ; if $\\lambda _{-} \\in \\, ]-\\infty ,0[$ and $t_{-}^{sc} > -\\infty $ , then $a = - \\infty $ , $\\Lambda _{\\mathbb {}}(\\lambda _{-}) \\in \\, ]-\\infty ,\\infty [$ , $\\Lambda _{\\mathbb {}}(z) = \\infty $ for all $z < \\lambda _{-}$ , $\\Lambda _{\\mathbb {}}^{\\prime }(\\lambda _{-}) \\in \\, ]-\\infty ,\\infty [$ ; if $\\lambda _{+} \\in \\, ]0,\\infty [$ and $t_{+}^{sc} < \\infty $ , then $b = \\infty $ , $\\Lambda _{\\mathbb {}}(\\lambda _{+}) \\in \\, ]-\\infty ,\\infty [$ , $\\Lambda _{\\mathbb {}}(z) = \\infty $ for all $z > \\lambda _{+}$ , $\\Lambda _{\\mathbb {}}^{\\prime }(\\lambda _{+})\\in \\, ]-\\infty ,\\infty [$ ; if $\\lambda _{-} \\in \\, ]-\\infty ,0[$ and $t_{-}^{sc} = -\\infty $ , then $a= - \\infty $ ; if $\\lambda _{+} \\in \\, ]0,\\infty [$ and $t_{+}^{sc} = \\infty $ , then $b= \\infty $ .","Notice that (C7ii) to (C7ix) cover all possible constellations.","For a proof of (C7ii) to (C7vii) as well as further details, see e.g.","Section 9.1 in Borovkov [56].","By contradiction, (C7viii) follows from (C7ii) and (C7ix) follows from (C7iii).","Moreover, (C7i) is a consequence (C7ii) to (C7ix).","As a side remark, notice that (C6) refines (M6).","According to the representability requirement (REF ), one has $\\varphi (t)=\\sup _{z\\in \\mathbb {R}}\\left( z\\cdot t- \\Lambda _{\\mathbb {}}(z)\\right)=: \\Lambda _{\\mathbb {}}^{*}(t), \\qquad t\\in \\mathbb {R},\\ \\ $ (i.e.","the divergence generator $\\varphi $ must be equal to the Fenchel-Legendre transform $\\Lambda _{\\mathbb {}}^{*}$ of a cumulant generating function $\\Lambda _{\\mathbb {}}$ ) of some probability distribution $\\mathbb {}$ , such that $\\lambda _{-} < 0 < \\lambda _{+}$ holds.","Moreover, $\\varphi $ should satisfy $\\varphi (1)=0$ , and should be finite as well as strictly convex in a non-empty neighborhood $]t_{-}^{sc},t_{+}^{sc}[$ of 1 (cf.","the definition of $\\widetilde{\\Upsilon }(]a,b[)$ ).","The latter rules out that $\\mathbb {}$ is any one-point distribution (Dirac distribution), say $\\mathbb {} = \\delta _{y_{0}}$ for some $y_{0} \\in \\mathbb {R}$ , since in such a situation one gets $\\Lambda _{\\mathbb {}}(z) = z \\cdot y_{0}$ , and thus $\\varphi (t) = \\Lambda _{\\mathbb {}}^{*}(t) = 0$ for $t=y_{0}$ and $\\varphi (t) = \\Lambda _{\\mathbb {}}^{*}(t) = \\infty $ for all $t \\in \\mathbb {R}\\backslash \\lbrace y_{0}\\rbrace $ (even in the case $y_{0} =1$ for which $\\varphi (1)=0$ is satisfied).","Consequently, $\\Lambda _{\\mathbb {}}$ is strictly convex on $]\\lambda _{-},\\lambda _{+}[ \\ = int(dom(\\Lambda _{\\mathbb {} }))$ (cf.","(C6)) and (C7) applies.","Clearly, by continuity one gets $\\Lambda _{\\mathbb {}}^{*}(t) =\\sup _{z\\in ]\\lambda _{-},\\lambda _{+}[}\\left( t \\cdot z - \\Lambda _{\\mathbb {}}(z)\\right),\\qquad t\\in \\mathbb {R}.","\\ \\ $ For $t \\in ]t_{-}^{sc},t_{+}^{sc}[$ , the optimization problem (REF ) can be solved explicitly by the well-known “pure/original” Legendre transform, namely $\\Lambda _{\\mathbb {}}^{*}(t) = t \\cdot \\Lambda _{\\mathbb {}}^{\\prime -1}(t) -\\Lambda _{\\mathbb {}}\\Big (\\Lambda _{\\mathbb {}}^{\\prime -1}(t)\\Big ),\\qquad t \\in ]t_{-}^{sc},t_{+}^{sc}[.$ Let us inspect the further cases $t \\le t_{-}^{sc}$ .","In the contexts of (C7iv) and (C7viii), this is obsolete since $t_{-}^{sc} = a = -\\infty $ .","For (C7ii), where $t_{-}^{sc} = a > -\\infty $ , one can show $\\Lambda _{\\mathbb {}}^{*}(a) = - \\log \\mathbb {}[\\lbrace a\\rbrace ]= - \\log \\mathbb {\\Pi }[W = a \\, ]$ which together with (REF ) proves (G10ii); moreover, $\\Lambda _{\\mathbb {}}^{*}(t) = \\infty $ for all $t < a$ (see e.g.","Section 9.1 of Borovkov [56]).","In the setup (C7vi), where $t_{-}^{sc} > a = - \\infty $ it is clear that $\\Lambda _{\\mathbb {}}^{*}(t_{-}^{sc}) =t_{-}^{sc} \\cdot \\Lambda _{\\mathbb {}}^{\\prime -1}(t_{-}^{sc}) -\\Lambda _{\\mathbb {}}\\Big (\\Lambda _{\\mathbb {}}^{\\prime -1}(t_{-}^{sc})\\Big )= t_{-}^{sc} \\cdot \\lambda _{-} - \\Lambda _{\\mathbb {}}(\\lambda _{-})$ and $\\Lambda _{\\mathbb {}}^{*}(t)= t \\cdot \\lambda _{-} - \\Lambda _{\\mathbb {}}(\\lambda _{-})= \\Lambda _{\\mathbb {}}^{*}(t_{-}^{sc}) + \\lambda _{-} \\cdot (t- t_{-}^{sc})\\quad \\text{for all $t \\in ]-\\infty ,t_{-}^{sc}[$}.$ As far as the cases $t \\ge t_{+}^{sc}$ is concerned, in the situations of (C7v) and (C7ix), this is obsolete since $t_{+}^{sc} = b = \\infty $ .","For (C7iii), where $t_{+}^{sc} = b < \\infty $ , one can show $\\Lambda _{\\mathbb {}}^{*}(b) = - \\log \\mathbb {}[\\lbrace b\\rbrace ]= - \\log \\mathbb {\\Pi }[W = b \\, ]$ which together with (REF ) proves (G10iii); moreover, $\\Lambda _{\\mathbb {}}^{*}(t) = \\infty $ for all $t > b$ (see e.g.","Section 9.1 of Borovkov [56]).","In the setup (C7vii), where $t_{+}^{sc} < b = \\infty $ it is clear that $\\Lambda _{\\mathbb {}}^{*}(t_{+}^{sc}) =t_{+}^{sc} \\cdot \\Lambda _{\\mathbb {}}^{\\prime -1}(t_{+}^{sc}) -\\Lambda _{\\mathbb {}}\\Big (\\Lambda _{\\mathbb {}}^{\\prime -1}(t_{+}^{sc})\\Big )= t_{+}^{sc} \\cdot \\lambda _{+} - \\Lambda _{\\mathbb {}}(\\lambda _{+})$ and $\\Lambda _{\\mathbb {}}^{*}(t)= t \\cdot \\lambda _{+} - \\Lambda _{\\mathbb {}}(\\lambda _{+})= \\Lambda _{\\mathbb {}}^{*}(t_{+}^{sc}) + \\lambda _{+} \\cdot (t- t_{+}^{sc})\\quad \\text{for all $t \\in ]t_{+}^{sc}, \\infty [$}.$ As a side effect, we have thus also proved (G10i) and (G3) (notice that in (G3) we have started with $a, b$ to be the endpoints of the support of $\\mathbb {}$ respectively $W$ , in contrast to Definition REF where $a$ , $b$ are defined as the endpoints of the effective domain of $\\varphi $ ).","To proceed, from (REF ) and (REF ) we obtain $\\varphi ^{\\prime }(t) = (\\Lambda _{\\mathbb {}}^{*})^{\\prime }(t) = \\Lambda _{\\mathbb {}}^{\\prime -1}(t),\\qquad \\varphi ^{\\prime \\prime }(t) = (\\Lambda _{\\mathbb {}}^{*})^{\\prime \\prime }(t) =\\frac{1}{\\Lambda _{\\mathbb {}}^{\\prime \\prime }\\big (\\Lambda _{\\mathbb {}}^{\\prime -1}(t)\\big )} > 0,\\qquad t \\in ]t_{-}^{sc},t_{+}^{sc}[,$ which — together with the investigations below (REF ) — provides (G4) and (G5); moreover, (G6) is immediate since the infinite differentiability is straightforward and $\\varphi ^{\\prime }(1) =0$ because we have required both the nonnegativity of $\\varphi $ and (G2) (cf.","the definition of $\\widetilde{\\Upsilon }(]a,b[)$ ).","The property (G7) follows from (C7ii), (C7iv), (C7viii), (REF ), (REF ) and $\\varphi ^{\\prime }(t_{-}^{sc}) = \\Lambda _{\\mathbb {}}^{\\prime -1}(t_{-}^{sc}) = \\lambda _{-}$ .","Analogously, we get (G8) from (C7iii), (C7v), (C7ix), (REF ), (REF ) and $\\varphi ^{\\prime }(t_{+}^{sc}) = \\Lambda _{\\mathbb {}}^{\\prime -1}(t_{+}^{sc}) = \\lambda _{+}$ .","Let us continue with (G9).","By applying the general theory of double Fenchel-Legendre transforms (bi-conjugates), (REF ) turns into $\\varphi ^{*} (z) = \\Lambda _{\\mathbb {}}(z), \\qquad z\\in \\mathbb {R},\\ \\ $ which deduces (G9i).","The properties (G9ii), (G9iii) and (G9iv) follow from Theorem REF (cf.","the discussion thereafter).","Finally, we obtain (G11i) and (G11ii) from (REF ), (REF ) and ().","$\\blacksquare $ Proof of Proposition REF .","The assertions follow immediately from (REF ), (REF ), (REF ), Theorem REF , (REF ) (and the discussion thereafter) as well as (M5).","$\\blacksquare $"],["Proofs — Part 5","Proof of Proposition REF .","The assertion follows straightforwardly from the following two facts: (i) a moment generating function $MGF$ is infinitely divisible if and only if $MGF^{c}$ is a moment generating function for all $c >0$ (cf.","e.g.","(the MGF-version of) Prop.","IV.2.5 of Steutel & van Harn [341]).","(ii) $z \\mapsto MGF(z)$ is a moment generating function if and only if $z \\mapsto MGF(\\breve{c} \\cdot z) =: MGF_{\\breve{c}}(z)$ is a moment generating function for all $\\breve{c} >0$ .","Notice that for each $c >0$ , $\\breve{c} >0$ one has $int(dom(MGF)) = int(dom(MGF^{c}))$ and $int(dom(MGF_{\\breve{c}})) = \\frac{1}{\\breve{c}} \\cdot int(dom(MGF))$ , and hence the light-tailedness remains unchanged: $0 \\in int(dom(MGF))$ if and only if $0 \\in int(dom(MGF^{c}))$ if and only if $0 \\in int(dom(MGF_{\\breve{c}}))$ .","Since $\\varphi \\in \\Upsilon (]a,b[)$ , we have $\\varphi (t)=\\sup _{z\\in ]\\lambda _{-}, \\lambda _{+}[}\\left( z \\cdot t-\\log \\Big (\\int _{\\mathbb {R}} e^{z\\cdot y }\\, d\\mathbb {} (y) \\Big ) \\right), \\quad t\\in ]a,b[ \\, , \\ \\ $ and thus for the exponential of its Fenchel-Legendre transform $\\int _{\\mathbb {R}} e^{z \\cdot y} d\\mathbb {} (y) , \\qquad z \\in ]\\lambda _{-}, \\lambda _{+}[ .$ Now, let $\\widetilde{\\varphi } := \\widetilde{c} \\cdot \\varphi \\in \\Upsilon (]a,b[)$ for arbitrarily fixed $\\widetilde{c} >0$ .","From the application of (REF ) to $\\widetilde{\\varphi }$ we obtain $\\widetilde{\\varphi } (t)=\\sup _{\\widetilde{z} \\in ]\\widetilde{\\lambda }_{-}, \\widetilde{\\lambda }_{+}[}\\left( \\widetilde{z} \\cdot t-\\log \\int _{\\mathbb {R}}e^{\\widetilde{z} \\cdot \\widetilde{y}} d\\widetilde{\\mathbb {}}_{\\widetilde{c}} (\\widetilde{y})\\right), \\qquad t\\in ]a,b[ \\, , \\ \\ $ for some unique probability distribution $\\widetilde{\\mathbb {}}_{\\widetilde{c}}$ on $\\mathbb {R}$ .","Here, according to (G9i) for $\\widetilde{\\varphi }$ we have used $\\widetilde{\\lambda }_{-} := \\inf _{t \\in ]a,b[} \\widetilde{\\varphi }^{\\prime }(t) = \\widetilde{c} \\cdot \\lambda _{-}$ and $\\widetilde{\\lambda }_{+} := \\sup _{t \\in ]a,b[} \\widetilde{\\varphi }^{\\prime }(t) = \\widetilde{c} \\cdot \\lambda _{+}$ .","Dividing (REF ) by $\\widetilde{c}$ , we arrive at $\\varphi (t) = \\frac{\\widetilde{\\varphi } (t)}{\\widetilde{c}}&=&\\sup _{\\widetilde{z} \\in ]\\widetilde{c} \\cdot \\lambda _{-}, \\widetilde{c} \\cdot \\lambda _{+}[}\\left( \\frac{\\widetilde{z}}{\\widetilde{c}} \\cdot t-\\log \\Big (\\int _{\\mathbb {R}}e^{\\frac{\\widetilde{z}}{\\widetilde{c}} \\cdot \\widetilde{y} \\cdot \\widetilde{c}} d\\widetilde{\\mathbb {}}_{\\widetilde{c}} (\\widetilde{y})\\Big )^{1/\\widetilde{c}}\\right),\\nonumber \\\\&=&\\sup _{z \\in ]\\lambda _{-}, \\lambda _{+}[}\\left( z \\cdot t-\\log \\Big (\\int _{\\mathbb {R}}e^{z \\cdot \\widetilde{y} \\cdot \\widetilde{c}}d\\widetilde{\\mathbb {}}_{\\widetilde{c}} (\\widetilde{y})\\Big )^{1/\\widetilde{c}}\\right),\\qquad t\\in ]a,b[ \\, , \\ \\ $ and hence for the exponential of its Fenchel-Legendre transform $e^{\\varphi _{*}(z)} =\\Big ( \\int _{\\mathbb {R}} e^{z \\cdot \\widetilde{y} \\cdot \\widetilde{c}}d\\widetilde{\\mathbb {}}_{\\widetilde{c}} (\\widetilde{y}) \\Big )^{1/\\widetilde{c}}, \\qquad z \\in ]\\lambda _{-}, \\lambda _{+}[ .$ Here, according to (G9i) for $\\widetilde{\\varphi }$ we have used $\\widetilde{\\lambda }_{-} := \\inf _{t \\in ]a,b[} \\widetilde{\\varphi }^{\\prime }(t) = \\widetilde{c} \\cdot \\lambda _{-}$ and $\\widetilde{\\lambda }_{+} := \\sup _{t \\in ]a,b[} \\widetilde{\\varphi }^{\\prime }(t) = \\widetilde{c} \\cdot \\lambda _{+}$ .","From (REF ) and (REF ) we deduce for $\\widetilde{c} := \\frac{1}{n}$ the relation $MGF_{\\mathbb {}}(z) = (MGF_{\\widetilde{\\mathbb {}}_{1/n}}(\\frac{z}{n}))^{n}$ for all $n \\in \\mathbb {N}$ which (with the help of (ii)) implies the infinitely divisibility of $\\mathbb {}$ .","For the reverse direction, let us assume that $\\varphi \\in \\Upsilon (]a,b[)$ and that the corresponding $\\mathbb {}$ is infinitely divisible.","Recall that $]a,b[ = int(dom(\\varphi ))$ .","Moreover, we fix an arbitrary constant $\\widetilde{c} >0$ .","Of course, there holds $\\widetilde{c} \\cdot \\varphi \\in \\widetilde{\\Upsilon }(]a,b[)$ and $dom(\\widetilde{c} \\cdot \\varphi ) = dom(\\varphi )$ .","Furthermore, by multiplying (REF ) with $\\widetilde{c} >0$ and by employing (i), (ii) we get $\\widetilde{c} \\cdot \\varphi (t) &=&\\sup _{z\\in ]\\lambda _{-}, \\lambda _{+}[}\\left( \\widetilde{c} \\cdot z \\cdot t-\\log \\Big (\\int _{\\mathbb {R}} e^{\\widetilde{c} \\cdot z\\cdot \\frac{y}{\\widetilde{c}}} d\\mathbb {} (y) \\Big )^{\\widetilde{c}} \\right)= \\sup _{\\widetilde{z} \\in ]\\widetilde{c} \\cdot \\lambda _{-}, \\widetilde{c} \\cdot \\lambda _{+}[}\\left( \\widetilde{z} \\cdot t-\\log \\Big (\\int _{\\mathbb {R}}e^{\\frac{\\widetilde{z}}{\\widetilde{c}} \\cdot y} \\, d\\mathbb {} (y) \\Big )^{\\widetilde{c}} \\right)\\nonumber \\\\&=&\\sup _{\\widetilde{z} \\in ]\\widetilde{c} \\cdot \\lambda _{-}, \\widetilde{c} \\cdot \\lambda _{+}[}\\left( \\widetilde{z} \\cdot t-\\log \\Big ( \\int _{\\mathbb {R}}e^{\\widetilde{z} \\cdot y} d\\mathbb {}_{\\widetilde{c}} (y) \\Big ) \\right), \\quad t\\in ]a,b[; \\ \\ $ for some probability distribution $\\mathbb {}_{\\widetilde{c}}$ on $\\mathbb {R}$ .","$\\blacksquare $ Proof of Proposition REF .","It is well known that a candidate function $M: ]-\\infty ,0[ \\mapsto ]0,\\infty [$ is the moment-generating function of an infinitely divisible probability distribution if and only if $(\\log M)^{\\prime }$ is absolutely monotone (see e.g.","Theorem 5.11 of Schilling et al.","[322]).","By applying this to $M(z) := e^{-a\\cdot z + \\varphi ^{*}(z)}$ respectively $M(z) := e^{b\\cdot z + \\varphi ^{*}(- z)}$ , one gets straightforwardly the assertion (a) respectively (b); notice that the light-tailedness follows then from (G1) to (G8), and $b=\\infty $ respectively $a =- \\infty $ can be deduced from the fact that the support of an infinitely distribution is always (one-sided or two-sided) unbounded.","For the third case $a= - \\infty $ , $b= \\infty $ one can use the assertion (cf.","e.g.","Morris [267], p.73) that a candidate function $M: \\, ]\\lambda _{-}, \\lambda _{+}[ \\, \\mapsto \\, ]0,\\infty [$ is the moment-generating function of an infinitely divisible probability distribution if the connected function $z \\mapsto (\\log M)^{\\prime \\prime }(z)/(\\log M)^{\\prime \\prime }(0)$ is the moment-generating function of some auxiliary probability distribution; but the latter is equivalent to exponentially convexity (cf.","Theorem REF (b)).","By applying this to $M(z) := e^{\\varphi ^{*}(z)}$ , one ends up with (c).","$\\blacksquare $"],["Proofs — Part 6","Proof of Theorem REF .","(i) Clearly, on $]\\lambda _{-},\\lambda _{+}[$ the function $\\Lambda $ is differentiable with strictly increasing derivative $\\Lambda ^{\\prime }(z) = F^{-1}(z+c) + 1- F^{-1}(c) ,\\qquad z \\in ]\\lambda _{-},\\lambda _{+}[.$ Hence, $\\Lambda $ is strictly convex and smooth (because of the smoothness of $F^{-1}$ ), and satisfies $\\Lambda (0) =0$ as well as $\\Lambda ^{\\prime }(0) =1$ .","Also, the corresponding extensions of $\\Lambda $ to $z=\\lambda _{-}$ and $z=\\lambda _{+}$ are continuous.","(ii) It is straightforward to see that on $]t_{-}^{sc},t_{+}^{sc}[$ the function $\\varphi $ is differentiable with strictly increasing derivative $\\varphi ^{\\prime }(t) = F(t+ F^{-1}(c) - 1) - c ,\\qquad t \\in ]t_{-}^{sc},t_{+}^{sc}[.$ Hence, $\\varphi $ is strictly convex and smooth (because of the smoothness of $F$ ), and satisfies $\\varphi (1) =0$ as well as $\\varphi ^{\\prime }(1) =0$ .","Also, the corresponding extensions of $\\varphi $ to $t=t_{-}^{sc}$ and $t=t_{-}^{sc}$ are continuous.","Hence (G1), (G2), (G5) and (G6) hold.","To prove (G3) (and hence (G1)), let us first notice that obviously there holds $a \\le t_{-}^{sc}$ and $t_{+}^{sc} \\le b$ .","Moreover, the validity of $\\varphi (t) < \\infty $ for all $t \\in ]t_{-}^{sc},t_{+}^{sc}[$ is clear from (REF ) since $t+F^{-1}(c)-1 \\in ]a_{F},b_{F}[ = int(dom(F))$ and the involved integral over the continuous function $F^{-1}$ is taken over a compact interval.","For the subcase $t_{-}^{sc} =- \\infty = a$ we have thus shown $dom(\\varphi ) \\cap ]-\\infty ,1] = ]-\\infty ,1]= ]a,1]$ , whereas for the subcase $t_{+}^{sc} = \\infty = b$ we have verified $dom(\\varphi ) \\cap [1,\\infty [ = [1,\\infty [ = [1,b[$ .","Let us next examine the subcase “$t_{-}^{sc} > - \\infty $ and $\\varphi (t_{-}^{sc}) < \\infty $ ”: if $\\lambda _{-} > - \\infty $ then $a = -\\infty $ and (REF ) implies $\\varphi (t) = \\varphi (t_{-}^{sc}) +\\lambda _{-} \\cdot (t- t_{-}^{sc}) < \\infty $ for all $t \\in \\ ]-\\infty , t_{-}^{sc}] = ]a,t_{-}^{sc}]$ , which leads to $dom(\\varphi ) \\cap ]-\\infty ,1] = ]-\\infty ,1] = ]a,1]$ ; in contrast, if $\\lambda _{-} = - \\infty $ then $a = t_{-}^{sc}$ and (REF ) implies $\\varphi (t) = \\varphi (t_{-}^{sc}) +\\lambda _{-} \\cdot (t- t_{-}^{sc}) = \\infty $ for all $t \\in \\ ]-\\infty , t_{-}^{sc}[ = ]-\\infty ,a[$ , which leads to $dom(\\varphi ) \\cap ]-\\infty ,1] = [a,1]$ .","In the subcase “$t_{-}^{sc} > - \\infty $ and $\\varphi (t_{-}^{sc}) = \\infty $ ”, due to the strict convexity of $\\varphi $ one always has $\\lim _{t \\downarrow t_{-}^{sc}} \\varphi ^{\\prime }(t) = - \\infty $ ; this implies, by the below-mentioned (REF ), that $\\lambda _{-} = - \\infty $ and thus $a = t_{-}^{sc}$ ; from (REF ) we derive $\\varphi (t) = \\varphi (t_{-}^{sc}) +\\lambda _{-} \\cdot (t- t_{-}^{sc}) = \\infty $ for all $t \\in \\ ]-\\infty , t_{-}^{sc}[ = ]-\\infty ,a[$ , which leads to $dom(\\varphi ) \\cap ]-\\infty ,1] = ]a,1]$ .","As a further step, we deal with the subcase “$t_{+}^{sc} < \\infty $ and $\\varphi (t_{+}^{sc}) < \\infty $ ”: if $\\lambda _{+} < \\infty $ then $b = \\infty $ and (REF ) implies $\\varphi (t) = \\varphi (t_{+}^{sc}) +\\lambda _{+} \\cdot (t- t_{+}^{sc}) < \\infty $ for all $t \\in \\ [t_{+}^{sc},\\infty [ = [t_{+}^{sc},b[$ , which leads to $dom(\\varphi ) \\cap [1, \\infty [ = [1, \\infty [ = [1, b[$ ; in contrast, if $\\lambda _{+} = \\infty $ then $b = t_{+}^{sc}$ and (REF ) implies $\\varphi (t) = \\varphi (t_{+}^{sc}) +\\lambda _{+} \\cdot (t- t_{+}^{sc}) = \\infty $ for all $t \\in \\ ]t_{-}^{sc}, \\infty [ = ]b,\\infty [$ , which leads to $dom(\\varphi ) \\cap [1,\\infty [ = [1,b]$ .","In the subcase “$t_{+}^{sc} < + \\infty $ and $\\varphi (t_{+}^{sc}) = \\infty $ ”, due to the strict convexity of $\\varphi $ one always gets $\\lim _{t \\uparrow t_{+}^{sc}} \\varphi ^{\\prime }(t) = \\infty $ ; this implies, by the below-mentioned (), that $\\lambda _{+} = \\infty $ and thus $b = t_{+}^{sc}$ ; from (REF ) we deduce $\\varphi (t) = \\varphi (t_{+}^{sc}) +\\lambda _{+} \\cdot (t- t_{+}^{sc}) = \\infty $ for all $t \\in \\ ]t_{+}^{sc}, \\infty [ = ]b, \\infty [$ , which leads to $dom(\\varphi ) \\cap [1,\\infty [ = [1,b[$ .","Putting things together, we have proved (G3).","The property (G4) follows straightforwardly from (REF ), the continuity of $F$ and from $\\lim _{t \\downarrow t_{-}^{sc}} \\varphi ^{\\prime }(t) = \\lambda _{-}$ , $\\lim _{t \\uparrow t_{+}^{sc}} \\varphi ^{\\prime }(t) = \\lambda _{+}$ .","To see the latter two, from (REF ) we obtain $& & \\lim _{t \\downarrow t_{-}^{sc}} \\varphi ^{\\prime }(t) = \\lim _{t \\downarrow t_{-}^{sc}} F(t+ F^{-1}(c) - 1) - c= \\lim _{t \\downarrow t_{-}^{sc}} F(t+ a_{F} - t_{-}^{sc}) - c= \\inf \\lbrace F(\\widetilde{t}) - c: \\widetilde{t} \\in ]a_{F},b_{F}[ \\rbrace = \\lambda _{-},\\\\& & \\lim _{t \\uparrow t_{+}^{sc}} \\varphi ^{\\prime }(t) = \\lim _{t \\uparrow t_{+}^{sc}} F(t+ F^{-1}(c) - 1) - c= \\lim _{t \\uparrow t_{+}^{sc}} F(t+ b_{F} - t_{+}^{sc}) - c= \\sup \\lbrace F(\\widetilde{t}) - c: \\widetilde{t} \\in ]a_{F},b_{F}[ \\rbrace = \\lambda _{+}.$ The two properties (G7) and (G8) are clear form the above considerations.","(iii) From (REF ) and (REF ) one gets easily $\\Lambda ^{\\prime -1}(t) = F\\left(t+F^{-1}(c)-1 \\right) - c = \\varphi ^{\\prime }(t), \\qquad t \\in ]t_{-}^{sc},t_{+}^{sc}[,$ as well as $\\Lambda ^{\\prime -1}(1)=0$ .","From this, we derive $&& t \\cdot \\Lambda ^{\\prime -1}(t) - \\Lambda \\left(\\Lambda ^{\\prime -1}(t)\\right)\\nonumber \\\\&& = t \\cdot [F\\left(t+F^{-1}(c)-1 \\right) - c]+ [F^{-1}(c)-1] \\cdot [F\\left(t+F^{-1}(c)-1 \\right) - c]\\nonumber \\\\& & -\\int \\displaylimits _{0}^{F\\left(t+F^{-1}(c)-1 \\right) - c} F^{-1}(u+c) du\\nonumber \\\\& & = \\varphi (t), \\qquad t \\in ]t_{-}^{sc},t_{+}^{sc}[,$ and hence, with the help of (REF ) in combination with (REF ), () $\\varphi (t) = \\max _{z \\in ]\\lambda _{-},\\lambda _{+}[} \\left( z\\cdot t -\\Lambda (z)\\right), \\qquad t \\in ]t_{-}^{sc},t_{+}^{sc}[ ,$ i.e.","on $]t_{-}^{sc},t_{+}^{sc}[$ the divergence generator $\\varphi $ is the classical Legendre transform of the restriction of $\\Lambda $ to $]\\lambda _{-},\\lambda _{+}[$ .","If “$\\lambda _{-} > -\\infty $ , $\\Lambda (\\lambda _{-}) \\in ]-\\infty ,\\infty [$ and $\\Lambda ^{\\prime }(\\lambda _{-}) \\in ]-\\infty ,\\infty [$ ” respectively “$\\lambda _{+} < -\\infty $ , $\\Lambda (\\lambda _{+}) \\in ]-\\infty ,\\infty [$ and $\\Lambda ^{\\prime }(\\lambda _{+}) \\in ]-\\infty ,\\infty [$ ”, then one can apply classical facts of Fenchel-Legendre transformation to get the corresponding left-hand respectively right-hand linear extensions of $\\varphi $ on the complement of $]t_{-}^{sc},t_{+}^{sc}[$ , in order to obtain the desired $\\varphi (t) =\\sup _{z \\in ]-\\infty ,\\infty [} \\left( z\\cdot t - \\Lambda (z)\\right) ,\\qquad t \\in \\mathbb {R};$ notice that $t_{-}^{sc} = \\lim _{z \\downarrow \\lambda _{-}} \\Lambda ^{\\prime }(z)$ and $t_{+}^{sc} = \\lim _{z \\uparrow \\lambda _{+}} \\Lambda ^{\\prime }(z)$ .","(iv) This is just the inverse of (iii), by applying standard Fenchel-Legendre-transformation theory.","$\\blacksquare $"],["Further details and proofs for\nSubsection ","Proof of Lemma REF.","By Assumption (OM), one gets for all $\\lambda \\in cl(\\mathbf {\\Lambda })$ that $\\lbrace \\mathbf {x} \\in (dom(\\widetilde{\\varphi })^{n} : T(\\mathbf {x}) = \\lambda \\rbrace \\, \\cap \\, ]t_{-}^{sc},t_{+}^{sc}[^{n} \\ne \\emptyset $ .","Moreover, for any $\\mathbf {x}=\\left( x_{1},..,x_{n}\\right) $ in $\\mathbb {R}^{n}$ , by the independence of the components of $\\mathbf {\\widetilde{W}}$ as well as (REF ) and (REF ), we have $& & I_{\\mathbf {\\widetilde{W}}}(\\mathbf {x})=\\sup _{\\mathbf {z}=\\left( z_{1},\\ldots ,z_{n}\\right) \\in \\mathbb {R}^{n}}\\left( \\left\\langle \\mathbf {x},\\mathbf {z}\\right\\rangle -\\sum _{i=1}^{n}\\Lambda _{\\widetilde{\\mathbb {}}}(z_{i}) \\right)= \\sup _{\\mathbf {z}\\in ]\\lambda _{-},\\lambda _{+}[^{n}}\\left( \\sum _{i=1}^{n} \\left( x_{i} \\cdot z_{i}-\\Lambda _{\\widetilde{\\mathbb {}}}(z_{i}) \\right) \\right)\\nonumber \\\\& & = \\sum _{i=1}^{n} \\left( \\sup _{z_{i}\\in ]\\lambda _{-},\\lambda _{+}[}\\left( x_{i}\\cdot z_{i} - \\Lambda _{\\widetilde{\\mathbb {}}}(z_{i})\\right) \\right)= \\sum _{i=1}^{n} \\widetilde{\\varphi }(x_{i})=\\sum _{k=1}^{K}\\sum _{i\\in I_{k}^{(n)}} \\widetilde{\\varphi }(x_{i})$ which is finite if and only if $\\mathbf {x} \\in (dom(\\widetilde{\\varphi }))^{n}$ (recall that $\\widetilde{\\varphi }$ is a nonnegative function).","Hence, for each $\\lambda \\in \\mathbf {\\Lambda }$ we obtain $I(\\lambda ) &:=&\\inf _{\\mathbf {x} \\in \\mathbb {R}^{n}: \\, T(\\mathbf {x})=\\lambda }I_{\\mathbf {\\widetilde{W}}}(\\mathbf {x})= \\inf _{\\mathbf {x} \\in (dom(\\widetilde{\\varphi }))^{n}: \\, T(\\mathbf {x})=\\lambda }I_{\\mathbf {\\widetilde{W}}}(\\mathbf {x}) =\\inf _{\\mathbf {x} \\in (dom(\\widetilde{\\varphi }))^{n}: \\, T(\\mathbf {x})=\\lambda } \\ \\sum _{k=1}^{K}\\sum _{i\\in I_{k}^{(n)}} \\widetilde{\\varphi }(x_{i})\\\\&=& \\sum _{k=1}^{K} n_{k} \\cdot \\widetilde{\\varphi }(\\lambda _{k})\\ = \\ n \\cdot \\sum _{k=1}^{K} \\widetilde{p}_{k} \\cdot \\widetilde{\\varphi }(\\lambda _{k})= \\inf _{\\mathbf {x} \\in ]t_{-}^{sc},t_{+}^{sc}[^{n} \\, : \\, T(\\mathbf {x})=\\lambda }I_{\\mathbf {\\widetilde{W}}}(\\mathbf {x}) \\, ;$ here, we have employed the following facts: (i) the right-most infimum in (REF ) is achieved by minimizing each of the $K$ terms $\\sum _{i\\in I_{k}^{(n)}}\\widetilde{\\varphi }(x_{i})$ under the linear constraint $\\frac{1}{n_{k}} \\cdot \\sum _{i\\in I_{k}^{(n)}} x_{i}= \\lambda _{k}$ , and by the strict convexity of $\\widetilde{\\varphi }$ on $]t_{-}^{sc},t_{+}^{sc}[$ (cf.","(G5)) the minimum of this generic term is attained when all components $x_{i}$ are equal to $\\lambda _{k}$ , and (ii) the outcoming minimum does not depend on the particular (generally non-unique) choice of the $x_{i}$ ’s.","Notice that we have used the relation $n_{k} = n \\cdot \\widetilde{p}_{k}$ as well.","To proceed, let $\\underline{\\lambda }$ be a minimal rate point of $\\mathbf {\\Lambda }$ , which means that $\\underline{\\lambda } \\in \\partial \\mathbf {\\Lambda }$ and $I(\\underline{\\lambda }) \\le I(\\lambda )$ for all $\\lambda \\in \\mathbf {\\Lambda }$ .","By Assumption (OM) one can run all the steps in (REF ) and () with $\\underline{\\lambda }$ instead of $\\lambda $ , and hence $I(\\underline{\\lambda }) \\ = \\ \\inf _{\\mathbf {x} \\in \\mathbb {R}^{n}: \\, T(\\mathbf {x})=\\underline{\\lambda }}I_{\\mathbf {\\widetilde{W}}}(\\mathbf {x})\\ = \\inf _{\\mathbf {x} \\in ]t_{-}^{sc},t_{+}^{sc}[^{n} \\, : \\, T(\\mathbf {x})=\\underline{\\lambda }}I_{\\mathbf {\\widetilde{W}}}(\\mathbf {x})\\ = \\ n \\cdot \\sum _{k=1}^{K} \\widetilde{p}_{k} \\cdot \\widetilde{\\varphi }(\\underline{\\lambda }_{k})\\ = \\ n \\cdot \\sum _{k=1}^{K} \\widetilde{p}_{k} \\cdot \\widetilde{\\varphi }(\\underline{\\widetilde{q}}_{k}/\\widetilde{p}_{k})$ where for the last equality we have employed the vector $\\underline{\\mathbf {\\widetilde{Q}}} := \\mathfrak {D}^{-1} \\underline{\\lambda }$ which we have called the “dominating point of $\\widetilde{\\mathbf {\\Omega }}$ ”.","Also we have proved $I(\\underline{\\lambda })\\ = \\ n \\cdot \\inf _{\\mathbf {\\widetilde{Q}} \\in \\widetilde{\\mathbf {\\Omega }}}\\sum _{k=1}^{K} \\widetilde{p}_{k} \\cdot \\widetilde{\\varphi }(\\widetilde{q}_{k}/\\widetilde{p}_{k}).","\\qquad \\qquad \\blacksquare $ On the obtainment of proxies of minimal rate points by proxy method 2: For the rest of this section, besides (OM) we assume that $dom(\\widetilde{\\varphi }) = \\, ]a,b[ \\, = \\, ]t_{-}^{sc},t_{+}^{sc}[$ , and that in case of $a=-\\infty $ or $b=+\\infty $ the divergence generator $\\widetilde{\\varphi }$ is regularly varying at $a$ or $b$ accordingly, with positive index $\\beta $ , i.e.","(with a slight abuse of notation) if $a=-\\infty $ , then for all $\\lambda >0$ there holds $\\lim _{u\\rightarrow -\\infty }\\frac{\\widetilde{\\varphi } \\left( \\lambda u\\right) }{\\widetilde{\\varphi }\\left( u\\right) }=\\lambda ^{\\beta },$ if $b=+\\infty $ , then for all $\\lambda >0$ there holds $\\lim _{u\\rightarrow +\\infty }\\frac{\\widetilde{\\varphi } \\left( \\lambda u\\right) }{\\widetilde{\\varphi }\\left( u\\right) }=\\lambda ^{\\beta };$ this assumption is denoted by (H$\\widetilde{\\varphi }$ ).","A proxy of $\\underline{\\mathbf {\\widetilde{Q}}}$ can be obtained by sampling from a distribution on $\\mathbb {R}^{K}$ defined through $f(\\mathbf {\\widetilde{Q}}):=C \\cdot \\exp \\left( -\\sum _{k=1}^{K} \\widetilde{p}_{k} \\cdot \\widetilde{\\varphi }(\\widetilde{q}_{k}/\\widetilde{p}_{k})\\right) \\ = \\ C \\cdot \\exp \\left( -D_{\\widetilde{\\varphi }}\\left( \\mathbf {\\widetilde{Q}},\\widetilde{{P}}\\right)\\right)\\hspace{51.21504pt} \\textrm {(cf.","(\\ref {simul distr}))}\\nonumber $ where $C$ is a normalizing constant; strict convexity (cf.","(G5)) of $\\widetilde{\\varphi }$ together with (H$\\widetilde{\\varphi }$ ) prove that $f$ is a well-defined (Lebesgue-) density for a random variable $\\mathbf {T}$ on $\\mathbb {R}^{K}$ .","We denote by $\\mathbb {F}(\\cdot ) := \\mathbb {\\Pi }[\\mathbf {T} \\in \\cdot \\, ]$ the corresponding distribution on $\\mathbb {R}^{K}$ having density $f$ .","The distribution of $\\mathbf {T}$ given $\\left( \\mathbf {T} \\in \\widetilde{\\mathbf {\\Omega }} \\right)$ concentrates on the points in $\\widetilde{\\mathbf {\\Omega }}$ which minimize $D_{\\widetilde{\\varphi }}\\left(\\mathbf {\\widetilde{Q}},\\widetilde{{P}}\\right) $ as $\\mathbf {\\widetilde{Q}}$ runs in $\\widetilde{\\mathbf {\\Omega }}$ , when $D_{\\widetilde{\\varphi }}(\\widetilde{\\mathbf {\\Omega }},\\widetilde{{P}})$ is large.","This can be argued as follows.","We will consider the case when $\\widetilde{\\mathbf {\\Omega }}$ is a compact subset in $\\mathbb {R}_{> 0}^{K}$ and $\\widetilde{\\varphi }$ satisfies (H$\\widetilde{\\varphi }$ ) with $b=+\\infty $ .","For the case when $\\widetilde{\\mathbf {\\Omega }}$ is not compact, or belongs to $\\mathbb {R}^{K}/\\left\\lbrace \\mathbf {0}\\right\\rbrace $ , see the Remark REF hereunder.","Consider a compact set $\\mathbf {\\Gamma }$ in $\\widetilde{\\mathbf {\\Omega }}$ and let $\\mathbf {\\Gamma }_{t}$ be defined as deduced from $\\mathbf {\\Gamma } $ in a way that makes $D_{\\widetilde{\\varphi }}\\left( \\mathbf {\\Gamma }_{t},\\widetilde{{P}}\\right) $ increase with $t$ for sufficiently large $t$ .","For instance, set $\\mathbf {\\Gamma }_{t}:=t \\cdot \\mathbf {\\Gamma } .$ Hence, in case of $b=+\\infty $ the divergence $D_{\\widetilde{\\varphi }}\\left( \\mathbf {\\Gamma }_{t},\\widetilde{{P}}\\right) =\\inf _{g_{t}\\in \\mathbf {\\Gamma }_{t}}\\sum _{k=1}^{K} \\widetilde{p}_{k}\\cdot \\widetilde{\\varphi }\\left( \\frac{\\left(g_{t}\\right)_{k}}{\\widetilde{p}_{k}}\\right) =\\inf _{g \\in \\mathbf {\\Gamma } }\\sum _{k=1}^{K} \\widetilde{p}_{k}\\cdot \\widetilde{\\varphi }\\left( \\frac{t \\cdot g_{k}}{\\widetilde{p}_{k}}\\right)$ tends to infinity as $t \\rightarrow \\infty $ ; the case $a=-\\infty $ works analogously with $t\\rightarrow -\\infty $ .","In case of $b < \\infty $ we may consider $\\mathbf {\\Gamma } _{t}:=\\left\\lbrace b-g /t;g \\in \\mathbf {\\Gamma } \\right\\rbrace $ and indeed $D_{\\widetilde{\\varphi }}\\left( \\mathbf {\\Gamma } _{t},\\widetilde{{P}}\\right) \\rightarrow \\infty $ as $t\\rightarrow \\infty $ , with a similar statement when $a>-\\infty $ .","Assume that $\\mathbf {\\Gamma }$ has a dominating point $\\underline{g}$ .","Then $\\mathbf {\\Gamma }_{t}$ has dominating point $\\underline{g}_{t} := t \\cdot \\underline{g}$ .","We prove that $\\mathbf {T}$ with distribution (REF ) cannot be too far away (depending on $t$ ) from $\\underline{g}_{t}$ whenever $\\mathbf {T}$ belongs to $\\mathbf {\\Gamma } _{t}.$ This argument is valid in the present description of some asymptotics which makes $\\mathbf {\\Gamma } _{t}$ as a model for $\\widetilde{\\mathbf {\\Omega }}$ for large $t$ ; considering the case when $D_{\\widetilde{\\varphi } }\\left( \\widetilde{\\mathbf {\\Omega }} ,\\widetilde{{P}}\\right) $ is large is captured through the asymptotic statement $\\lim _{t\\rightarrow \\infty }D_{\\widetilde{\\varphi } }\\left( \\mathbf {\\Gamma } _{t},\\widetilde{{P}}\\right) =+\\infty .$ There holds the following Proposition 65 With the above notation and under condition (H$\\widetilde{\\varphi }$ ), denote by $\\mathbf {B}$ a neighborhood of $\\underline{g}$ and $\\mathbf {B}_{t}:=t \\cdot \\mathbf {B}$ .","Then $\\mathbb {F}[\\mathbf {\\Gamma }_{t}\\cap \\mathbf {B}_{t}^{c} \\, \\vert \\, \\mathbf {\\Gamma }_{t} \\, ]= \\mathbb {\\Pi }\\left[ \\left.","\\mathbf {T}\\in \\mathbf {\\Gamma }_{t}\\cap \\mathbf {B}_{t}^{c}\\, \\right|\\,\\mathbf {T}\\in \\mathbf {\\Gamma }_{t}\\right] \\rightarrow 0$ as $t\\rightarrow \\infty $ , which proves that simulations under (REF ) produce proxies of the dominating points $\\underline{g}_{t}$ in $\\mathbf {\\Gamma }_{t}$ .","Before we start with the proof of Proposition REF , we first quote the following Lemma 66 Let $\\widetilde{\\varphi } $ satisfy (H$\\widetilde{\\varphi }$ ) with $b=+\\infty $ .","Then for all $\\mathbf {A}$ in $\\mathbb {R}^{K}$ such that $\\breve{\\alpha } :=D_{\\widetilde{\\varphi }}(\\mathbf {A},\\widetilde{{P}}):=\\inf _{\\mathbf {v}\\in \\mathbf {A}}\\sum _{k=1}^{K} \\widetilde{p}_{k} \\cdot \\widetilde{\\varphi } \\left(\\frac{v_{k}}{\\widetilde{p}_{k}}\\right)$ is finite there holds $\\lim _{t\\rightarrow \\infty }\\frac{1}{t}\\log \\int _{\\mathbf {A}}\\exp \\left(-t\\sum _{k=1}^{K} \\widetilde{p}_{k} \\cdot \\widetilde{\\varphi }\\left( \\frac{v_{k}}{\\widetilde{p}_{k}}\\right) \\right) \\, dv_{1} \\ldots dv_{k}= - D_{\\widetilde{\\varphi }}\\left(\\mathbf {A},\\widetilde{{P}}\\right) .$ Proof of Lemma REF .","Let us first remark that according to the geometry of the set $\\mathbf {A}$ , various combinations for the asymptotics (REF ) or (REF ) may occur; for sake of brevity, we only handle the simplest ones, since all turn to be amenable through the same arguments.","Denote for positive $r$ $\\mathbf {B}(r):=\\left\\lbrace \\mathbf {v}\\in \\mathbb {R}^{K}:\\sum _{k=1}^{K} \\widetilde{p}_{k}\\widetilde{\\varphi } \\left( \\frac{v_{k}}{\\widetilde{p}_{k}}\\right) >r\\right\\rbrace .$ It holds, by making the change of variable $r=t\\cdot \\breve{\\alpha } +t \\cdot s$ , $& & \\int _{\\mathbf {A}}\\exp \\left( -t\\sum _{k=1}^{K} \\widetilde{p}_{k} \\cdot \\widetilde{\\varphi } \\left( \\frac{v_{k}}{\\widetilde{p}_{k}}\\right) \\right) \\,dv_{1} \\ldots dv_{k}=\\int \\cdots \\int 1_{\\mathbb {R}^{+}}(r)\\cdot 1_{\\mathbf {A}}(\\mathbf {v})\\cdot 1_{\\left]t\\sum _{k=1}^{K} \\widetilde{p}_{k}\\widetilde{\\varphi } \\left( \\frac{v_{k}}{\\widetilde{p}_{k}}\\right) ,\\infty \\right[}(r)\\cdot e^{-r} \\, dr \\, dv_{1}\\ldots dv_{K} \\\\& & = te^{-t\\breve{\\alpha } }\\int \\cdots \\int 1_{]- \\breve{\\alpha },\\infty [}(s) \\cdot 1_{\\mathbf {A}}(\\mathbf {v}) \\cdot 1_{\\mathbf {B}^{c}(\\breve{\\alpha }+s)}(\\mathbf {v})\\cdot e^{-ts} \\, ds \\, dv_{1}\\ldots dv_{K} \\ = \\ te^{-t\\breve{\\alpha } }\\int _{-\\breve{\\alpha }}^{\\infty }Vol\\left( \\mathbf {A}\\cap \\mathbf {B}^{c}(\\breve{\\alpha }+s)\\right) \\cdot e^{-ts} \\, ds.$ Let $I_{t}:=t \\cdot \\int _{0}^{\\infty }Vol\\left(\\mathbf {A}\\cap \\mathbf {B}^{c}(\\breve{\\alpha } +s)\\right) e^{-ts}ds $ .","We prove that $\\lim _{t\\rightarrow \\infty }\\frac{1}{t}\\log I_{t}=0.","$ When $a=-\\infty $ or $b=+\\infty $ , since $\\widetilde{\\varphi }$ satisfies (H$\\widetilde{\\varphi }$ ) there exists a polynomial $P$ such that $Vol\\left( \\mathbf {A}\\cap \\mathbf {B}^{c}(\\breve{\\alpha } +s)\\right) \\le P(s) \\, ;$ whence, assuming without loss of generality that $dom(\\widetilde{\\varphi }) =\\mathbb {R}^{+}$ , we obtain $\\frac{1}{t}\\log I_{t}\\le \\frac{1}{t}\\log \\int _{0}^{\\infty }P\\left(\\frac{u}{t}\\right) te^{-u}du$ which yields that for large $t$ $\\frac{1}{t}\\log I_{t}<0.$ When dealing with a context where $a$ or $b$ have finite value and the corresponding sets $\\mathbf {\\Gamma }_{t}$ are “far away” from $\\mathbf {\\Gamma } $ in terms of the distance measure $D_{\\widetilde{\\varphi }}\\left( \\cdot ,\\widetilde{{P}}\\right)$ , then $Vol\\left( \\mathbf {A}\\cap \\mathbf {B}^{c}(\\breve{\\alpha } +s)\\right) $ is bounded.","Hence, $\\lim \\sup _{t\\rightarrow \\infty }\\frac{1}{t}\\log I_{t}\\le 0$ .","Now fix $\\varepsilon >0.$ Then, since $Vol\\left( \\mathbf {A}\\cap \\mathbf {B}^{c}(a+s)\\right) $ is increasing in $s$ , we get $I_{t} &\\ge &t\\int _{\\varepsilon }^{\\infty }Vol\\left( \\mathbf {A}\\cap \\mathbf {B}^{c}(\\breve{\\alpha }+s)\\right) e^{-ts}ds \\\\&\\ge &Vol\\left( \\mathbf {A}\\cap \\mathbf {B}^{c}(\\breve{\\alpha } +\\varepsilon )\\right) e^{-t\\varepsilon }.$ Hence $\\frac{1}{t}\\log I_{t}\\ge \\frac{1}{t}\\log Vol\\left( \\mathbf {A}\\cap \\mathbf {B}^{c}(\\breve{\\alpha }+\\varepsilon )\\right) -\\varepsilon $ which yields $\\lim \\inf _{t\\rightarrow \\infty }\\frac{1}{t}\\log I_{t}\\ge 0.$ Therefore (REF ) holds, which concludes the proof.","$\\blacksquare $ We now turn to the Proof of Proposition REF .","Without loss of generality, let $b=+\\infty $ , $\\mathbf {\\Gamma }_{t}$ as in (REF ) and Condition (H$\\widetilde{\\varphi }$ ) hold.","Moreover, consider an arbitrary neighborhood $\\mathbf {B}$ of $\\underline{g}$ and the corresponding neighborhoods $\\mathbf {B}_{t} : =t \\cdot \\mathbf {B}$ of $\\underline{g}_{t} = t \\cdot \\underline{g}$ .","There holds $\\frac{1}{\\widetilde{\\varphi }(t)}\\log \\mathbb {\\Pi }\\left[ \\mathbf {T}\\in \\mathbf {\\Gamma }_{t}\\right] &=&\\frac{C}{\\widetilde{\\varphi }(t)}\\log \\int _{\\mathbf {\\Gamma } _{t}}\\exp \\left(-\\sum _{k=1}^{K}\\widetilde{p}_{k} \\cdot \\widetilde{\\varphi }\\left(\\frac{w_{k}}{\\widetilde{p}_{k}}\\right)\\right) \\, dw_{1}\\ldots dw_{K} \\\\&\\stackrel{(1)}{=}&\\frac{CK}{\\widetilde{\\varphi }(t)}\\log t+\\frac{C}{\\widetilde{\\varphi }(t)}\\log \\int _{\\mathbf {\\Gamma }}\\exp \\left(-t^{\\beta } \\cdot \\sum _{k=1}^{K}\\widetilde{p}_{k} \\cdot \\left( \\widetilde{\\varphi }\\left( \\frac{v_{k}}{\\widetilde{p}_{k}}\\right) \\cdot (1+o(1))\\right)\\right) \\, dv_{1} \\ldots dv_{K} \\\\&\\stackrel{(2)}{=}& \\frac{CK}{\\widetilde{\\varphi }(t)}\\log t+\\frac{C}{\\left( \\widetilde{\\varphi }(t)/t^{\\beta }\\right) } \\cdot \\frac{1}{t^{\\beta }}\\log \\left( (1+o(1)) \\cdot \\int _{\\mathbf {\\Gamma } }\\exp \\left( -t^{\\beta }\\sum _{k=1}^{K}\\widetilde{p}_{k} \\cdot \\widetilde{\\varphi }\\left( \\frac{v_{k}}{\\widetilde{p}_{k}}\\right) \\right)dv_{1} \\ldots dv_{K} \\right)\\\\&\\stackrel{(3)}{=}& -\\frac{Ct^{\\beta }}{\\widetilde{\\varphi }(t)}\\cdot D_{\\widetilde{\\varphi }}(\\mathbf {\\Gamma } ,\\widetilde{{P}}) \\cdot (1+o(1)) \\\\&\\stackrel{(4)}{=}& - \\breve{l}(t) \\cdot D_{\\widetilde{\\varphi }}(\\mathbf {\\Gamma } ,\\widetilde{{P}}) \\cdot (1+o(1))$ as $t$ tends to infinity.","In the above display, $(1)$ follows from $\\widetilde{\\varphi }(tx)=\\left( tx\\right)^{\\beta } \\cdot \\ell (tx)=t^{\\beta } \\cdot x^{\\beta } \\cdot \\ell (x) \\cdot \\frac{\\ell (tx)}{\\ell (x)} =t^{\\beta } \\cdot \\widetilde{\\varphi } (x) \\cdot \\left( 1+o(1)\\right) $ as $t$ tends to infinity and $x$ lies in a compact subset of $]0,\\infty [$ , where $\\ell $ is a slowly varying function.","The equality $(2)$ follows from compactness of $\\mathbf {\\Gamma }$ together with the fact that $\\widetilde{\\varphi }$ is a regularly varying function with index $\\beta $ , so that $\\lim _{t\\rightarrow \\infty }\\frac{\\widetilde{\\varphi }(tv)}{\\widetilde{\\varphi }(t)}=v^{\\beta }$ uniformly upon $v$ on compact sets in $]0,\\infty [$ .","The remaining equalities $(3)$ and $(4)$ follow from classical properties of regularly varying functions, where $\\breve{\\ell } := 1/\\ell $ is a slowly varying function at infinity, together with standard Laplace-Integral approximation.","In the same way we can show $\\frac{1}{\\widetilde{\\varphi }(t)}\\log \\mathbb {\\Pi } \\left[ \\, \\mathbf {T}\\in \\mathbf {\\Gamma } _{t}\\cap \\mathbf {B}_{t}^{c} \\, \\right]=- \\breve{l}(t) \\cdot D_{\\widetilde{\\varphi }}(\\mathbf {\\Gamma }\\cap \\mathbf {B}^{c},\\widetilde{{P}}) \\cdot (1+o(1))$ as $t$ tends to infinity.","Since $\\mathbf {B}$ is a neighborhood of the unique dominating point $\\underline{g}$ of $\\mathbf {\\Gamma }$ , one gets that $D_{\\widetilde{\\varphi }}(\\mathbf {\\Gamma }\\cap \\mathbf {B}^{c},\\widetilde{{P}}) > D_{\\widetilde{\\varphi }}(\\mathbf {\\Gamma } ,\\widetilde{{P}})$ .","This implies that $\\mathbb {\\Pi } \\left[ \\, \\mathbf {T}\\in \\mathbf {\\Gamma }_{t}\\cap \\mathbf {B}_{t}^{c} \\, \\big \\vert \\, \\mathbf {T}\\in \\mathbf {\\Gamma }_{t} \\, \\right] \\rightarrow 0\\qquad \\textrm {as $ $.","\\qquad $$}$$$ Remark 67 Firstly, let us quote that the case when $\\widetilde{\\mathbf {\\Omega }}$ is an unbounded subset in $\\mathbb {R}^{K}/\\left\\lbrace \\mathbf {0}\\right\\rbrace $ is somewhat immaterial for applications.","Anyhow, if compactness of $\\mathbf {\\Gamma } $ is lost, then in order to use the same line of arguments as above, it is necessary to strengthen the assumptions (H$\\widetilde{\\varphi }$ ) e.g.","as follows: when $b=+\\infty $ then $\\widetilde{\\varphi }$ has to be asymptotically homogeneous with degree $\\beta >0$ , in the sense that $\\widetilde{\\varphi }(tx)=t^{\\beta }\\widetilde{\\varphi }(x)\\cdot (1+o(1))$ as $t\\rightarrow \\infty $ ; for the subcase $a=-\\infty $ one employs an analogous assumption as $t\\rightarrow -\\infty $ .","The case when $\\widetilde{\\mathbf {\\Omega }}$ is a compact set in $\\mathbb {R}^{K}\\backslash \\lbrace \\mathbf {0}\\rbrace $ can be treated as above, by combining the asymptotics in $t$ in the neighborhood of $a$ and $b$ accordingly."],["Proof for Subsection ","Proof of Proposition REF.","Recall the weighted empirical measure $\\xi _{n,\\mathbf {X}}^{\\mathbf {V}}:=\\left( \\frac{1}{n}\\sum _{i\\in I_{1}^{(n)}} V_{i},\\ldots ,\\frac{1}{n}\\sum _{i\\in I_{K}^{(n)}} V_{i}\\right)$ which satisfies the $K$ linear constraints defined in (REF ) through $E_{S}[\\xi _{n,\\mathbf {X}}^{\\mathbf {V}}] =\\xi _{M,\\mathbf {X}}^{\\mathbf {W}^{\\ast }}= \\overline{W^{\\ast }} \\cdot \\xi _{M,\\mathbf {X}}^{w\\mathbf {W}^{\\ast }}$ where $\\mathbf {Q}^{\\ast }:= \\left(q_{1}^{\\ast }, \\ldots , q_{K}^{\\ast }\\right)= \\xi _{M,\\mathbf {X}}^{w\\mathbf {W}^{\\ast }}\\in int(\\textrm {$$\\hspace{-6.544pt}$$})$ and $\\overline{W^{\\ast }} = \\frac{1}{M}\\sum _{j=1}^{M}W_{j}^{\\ast }$ .","The probability distribution $S$ defined on $\\mathbb {R}^{n}$ is the Kullback-Leibler projection of $\\mathbb {}^{\\otimes n}$ on the class of all probability distributions on $\\mathbb {R}^{n}$ which satisfy (REF ).","We prove that $\\lim \\inf _{n\\rightarrow \\infty }S\\left[\\xi _{n,\\mathbf {X}}^{w\\mathbf {V}} \\in \\textrm {\\right.$$\\hspace{-6.544pt}$$}> 0$ .","To start with, we define for strictly positive $\\delta $ the set $A_{n,\\delta } := \\left\\lbrace \\left|\\frac{1}{n}\\sum _{i=1}^{n}V_{i}-\\overline{W^{\\ast }}\\right|\\le \\delta \\right\\rbrace $ and write $S\\left[\\xi _{n,\\mathbf {X}}^{w\\mathbf {V}} \\in \\textrm {\\right.$ $\\hspace{-6.544pt}$$}=S\\left[\\lbrace \\xi _{n,\\mathbf {X}}^{w\\mathbf {V}} \\in \\textrm {\\right.$$\\hspace{-6.544pt}$$} \\rbrace \\cap A_{n,\\delta }+ S\\left[\\lbrace \\xi _{n,\\mathbf {X}}^{w\\mathbf {V}} \\in \\textrm {\\right.$$\\hspace{-6.544pt}$$} \\rbrace \\cap A_{n,\\delta }^{c}=:I+II.$$By the law of large numbers, the second term $ II$ tends to $ 0$ as $ n$ tends to infinity.Moreover, one can rewrite$$I=S\\bigg [ \\bigcup \\limits _{m\\in \\left[ \\overline{W^{\\ast }}-\\delta ,\\overline{W^{\\ast }}+\\delta \\right] }\\left\\lbrace \\xi _{n,\\mathbf {X}}^{\\mathbf {V}}\\in m \\cdot \\textrm {\\right.$$\\hspace{-6.544pt}$$} \\bigg ]$$which entails$$I \\ \\ge \\ S\\bigg [ \\frac{1}{n_{k}}\\sum _{i\\in I_{k}^{(n)}}V_{i}\\in \\mathcal {V}_{\\eta }\\left( \\overline{W^{\\ast }}\\frac{q_{k}^{\\ast }}{p_{k}}\\right) \\text{ for all }k \\in \\lbrace 1,\\ldots ,K\\rbrace \\bigg ] ,$$where $ V( W qkpk) $ denotes a neighborhood of$Wqkpk$ with radius $$ being smallwhen $$ is small, for large enough $ n$, making use of the a.s. convergenceof $ nk/n$ to $ pk$.","Now, for any $ k {1,...,K}$ one has\\begin{equation}S\\bigg [ \\frac{1}{n_{k}}\\sum _{i\\in I_{k}^{(n)}}V_{i}\\notin \\mathcal {V}_{\\eta }\\left( \\overline{W^{\\ast }}\\frac{q_{k}^{\\ast }}{p_{k}}\\right)\\bigg ]\\ \\le \\ \\exp \\bigg (-n_{k} \\cdot \\inf _{x\\in \\mathcal {V}_{\\eta }\\left(\\overline{W^{\\ast }}\\frac{q_{k}^{\\ast }}{p_{k}}\\right)^{c}} \\,\\varphi \\left( x\\right) \\bigg )\\end{equation}since any margin of $ S$ with index in $ Ik(n)$ is a correspondingKullback-Leibler projection of $$ on the set of all distributionson $ R$ with expectation $W qkpM,kemp$ ---where $ pM,kemp$ denotes the fraction ofthe $ Xi$’s (within $ X1,...,XM$) which are equal to $ dk$(cf.","(\\ref {I^(n)_k for stat case})) ---and therefore has amoment generating function which is finite in a non-void neighborhood of $ 0$,which yields (\\ref {ineg}) by the Markov Inequality.","Note that the event$ { M,XwWint( $\\Omega $$\\Omega $ ) }$ is regenerative, so that $ M$ can be chosen large enough to make$ pM,kemp$ close to $ pk$ for all $ k {1,...,K}$.","This proves the claim.\\hspace{28.45274pt} $$$"],["Acknowledgment","W. Stummer is grateful to the Sorbonne Université Paris for its multiple partial financial support and especially the LPSM for its multiple great hospitality.","M. Broniatowski thanks very much the FAU Erlangen-Nürnberg for its partial financial support and hospitality.","Moreover, W. Stummer would like to thank Rene Schilling for an interesting discussion on complex-valued foundations of the Bernstein-Widder theorem."],["Finding/Constructing/On the distribution of the weights ","Recall first that in Theorem REF , one crucial component is the sequence $(W_{n})_{n\\in \\mathbb {N}}$ of weights being i.i.d.","copies of a random variable $W$ whose probability distribution is $\\mathbb {}$ (i.e.","$\\mathbb {\\Pi }[W \\in \\cdot \\, ] = \\mathbb {}[ \\, \\cdot \\,]$ ), where the latter has to be connected with the divergence generator $\\varphi \\in \\Upsilon (]a,b[)$ through the representation $\\varphi (t)=\\sup _{z\\in \\mathbb {R}}\\left( z\\cdot t-\\log \\int _{\\mathbb {R}}e^{zy}d\\mathbb {} (y)\\right), \\qquad t\\in \\mathbb {R},\\ \\ \\hspace{28.45274pt} (cf.","\\ (\\ref {Phi Legendre of mgf(W)}))$ under the additional requirement that the function $z\\mapsto MGF_{\\mathbb {} }(z):=\\int _{\\mathbb {R}}e^{zy}d\\mathbb {} (y)$ is finite on some open interval containing zero (“light-tailedness”); for Theorem REF , we need the corresponding variant (REF ) for $M_{\\mathbf {P}} \\cdot \\varphi \\in \\Upsilon (]a,b[)$ (rather than $\\varphi $ ).","Hence, finding such “BS-associated pairs $(\\varphi ,\\mathbb {})$ ” is an important issue.","Subsequently, let us discuss the following direction: starting from a concrete optimization problem (REF ) — respectively (REF ) — with pregiven $\\varphi \\in \\widetilde{\\Upsilon } (]a,b[)$ (cf.","Definition REF ), as a first step one would like to verify whether indeed $\\varphi \\in \\Upsilon (]a,b[)$ (i.e.","it additionally satisfies (REF )) — respectively $M_{\\mathbf {P}} \\cdot \\varphi \\in \\Upsilon (]a,b[)$ ; as a second step, one would like to find the corresponding $\\mathbb {}$ explicitly.","As far as the above-mentioned first step is concerned, let us first present some fundamental properties of all $\\varphi \\in \\Upsilon (]a,b[)$ : Proposition 29 Let $\\varphi \\in \\Upsilon (]a,b[)$ .","Then the following assertions hold: $\\varphi : \\, ]-\\infty ,\\infty [ \\rightarrow [0,\\infty ]$ is lower semicontinuous and convex; $\\varphi (1)=0$ ; $int(dom(\\varphi )) = ]a,b[$ for some $-\\infty \\le a < 1 < b \\le \\infty $ ; $\\varphi $ is continuously differentiable on $]a,b[$ (i.e.","$\\varphi \\in C^{1}(]a,b[)$ ; $\\varphi $ is strictly convex only in a non-empty neighborhood $]t_{-}^{sc},t_{+}^{sc}[ \\subseteq ]a,b[$ of one ($t_{-}^{sc} < 1 < t_{+}^{sc}$ ); $\\varphi $ is infinitly differentiable on $]t_{-}^{sc},t_{+}^{sc}[$ (i.e.","$\\varphi \\in C^{\\infty }(]t_{-}^{sc},t_{+}^{sc}[$ ), and hence, $\\varphi ^{\\prime }(1) = 0$ , $\\varphi ^{\\prime \\prime }(t) > 0$ for all $t \\in ]t_{-}^{sc},t_{+}^{sc}[$ ; notice that the left-hand second derivative and the right-hand second derivative of $\\varphi $ may not coincide at $t_{-}^{sc}$ respectively at $t_{+}^{sc}$ (i.e.","possible non-second-differentiability at these two points); if $a > -\\infty $ , then $a=t_{-}^{sc}$ ; if $a = -\\infty $ , then either $t_{-}^{sc} = - \\infty $ or $\\varphi (t) = \\varphi (t_{-}^{sc}) + \\varphi ^{\\prime }(t_{-}^{sc}) \\cdot (t- t_{-}^{sc})$ for all $t \\in ]-\\infty , t_{-}^{sc}[$ (affine-linearity); notice that $\\varphi ^{\\prime }(t_{-}^{sc}) < 0$ ; if $b < \\infty $ , then $b=t_{+}^{sc}$ ; if $b = \\infty $ , then either $t_{+}^{sc} = \\infty $ or $\\varphi (t) = \\varphi (t_{+}^{sc}) + \\varphi ^{\\prime }(t_{+}^{sc}) \\cdot (t- t_{+}^{sc})$ for all $t \\in ]t_{+}^{sc},\\infty [$ (affine-linearity); notice that $\\varphi ^{\\prime }(t_{+}^{sc}) > 0$ ; the Fenchel-Legendre transform (also called convex conjugate) of $\\varphi $ $\\varphi ^{*} (z)=\\sup _{t\\in \\mathbb {R}}\\left( z\\cdot t- \\varphi (t) \\right)= \\sup _{t\\in ]a,b[}\\left( z\\cdot t- \\varphi (t) \\right), \\qquad z\\in \\mathbb {R},\\ \\ $ has the following properties: $int(dom(\\varphi ^{*})) = ]\\lambda _{-},\\lambda _{+}[$ , where $dom(\\varphi ^{*}) := \\lbrace z \\in \\mathbb {R} : -\\infty < \\varphi ^{*}(z) < \\infty \\rbrace $ , $\\lambda _{-} := \\inf _{t \\in ]a,b[} \\varphi ^{\\prime }(t) =\\lim _{t \\downarrow a} \\varphi ^{\\prime }(t) =: \\varphi ^{\\prime }(a) < 0$ and $\\lambda _{+} := \\sup _{t \\in ]a,b[} \\varphi ^{\\prime }(t) =\\lim _{t \\uparrow b} \\varphi ^{\\prime }(t) =: \\varphi ^{\\prime }(b) >0$ ; if $a >-\\infty $ , then $\\lambda _{-} = - \\infty $ ; the function $z \\mapsto e^{-a\\cdot z + \\varphi ^{*}(z)} =: M(z)$ is absolutely monotone on $]-\\infty ,0[$ , i.e.","all derivatives exist and satisfy $\\frac{\\partial ^{k}}{\\partial z^k} M(z) \\ge 0$ ($k\\in \\mathbb {N}_{0}$ , $z \\in ]-\\infty ,0[$ ); $\\lim _{z \\rightarrow 0-} M(z) =1$ ; if $b < \\infty $ , then $\\lambda _{+} = \\infty $ ; the function $z \\mapsto e^{b\\cdot z + \\varphi ^{*}(- z)} =: M(z)$ is absolutely monotone on $]-\\infty ,0[$ ; $\\lim _{z \\rightarrow 0-} M(z) =1$ ; if $a = -\\infty $ and $b = -\\infty $ , then the function $z \\mapsto e^{\\varphi ^{*}(z)} =: M(z)$ is exponentially convex on $]\\lambda _{-}, \\lambda _{+}[$ , i.e.","$M(\\cdot )$ is continuous and satisfies $\\sum _{i,j=1}^{n} c_{i} \\cdot c_{j} \\cdot M\\Big (\\frac{z_{i} + z_{j}}{2}\\Big ) \\ge 0 \\qquad \\textrm {for all n \\in \\mathbb {N},c_{i}, c_{j} \\in \\mathbb {R} and z_{i}, z_{j} \\in ]\\lambda _{-}, \\lambda _{+}[;}$ $\\lim _{z \\rightarrow 0-} M(z) =1$ ; as a side remark, notice the well-known fact that exponential-convexity is stronger than the usual log-convexity.","the endpoints of $int(dom(\\varphi )) =]a,b[$ have the following important “functioning” for the underlying probability distribution $\\mathbb {}$ (cf.","(REF )) respectively of an associated random variable $W$ with $\\mathbb {}[ \\cdot \\, ] := \\mathbb {\\Pi }[W \\in \\cdot \\, ]$ : $a = \\inf supp(\\mathbb {}) = \\inf supp(W)$ , $b = \\sup supp(\\mathbb {}) = \\sup supp(W)$ , where $supp(\\mathbb {})$ respectively $supp(W)$ denotes the support of $\\mathbb {}$ respectively $W$ ; consequently, $]a,b[ = int(conv(supp(\\mathbb {}))) = int(conv(supp(W)))$ where $conv(A)$ denotes the convex hull of a set $A$ ; if $a > -\\infty $ , then $\\varphi (a) = - \\log \\mathbb {}[\\lbrace a\\rbrace ]= - \\log \\mathbb {\\Pi }[W = a \\, ]$ ; consequently, there holds: $a = \\min supp(\\mathbb {}) = \\min supp(W)$ if and only if $\\mathbb {}[\\lbrace a\\rbrace ] = \\mathbb {\\Pi }[W = a \\, ] > 0$ if and only if $\\varphi (a) < \\infty $ if and only if $a \\in dom(\\varphi )$ ; if $b < \\infty $ , then $\\varphi (b) = - \\log \\mathbb {}[\\lbrace b\\rbrace ]= - \\log \\mathbb {\\Pi }[W = b \\, ]$ ; consequently, there holds: $b = \\max supp(\\mathbb {}) = \\max supp(W)$ if and only if $\\mathbb {}[\\lbrace b\\rbrace ] = \\mathbb {\\Pi }[W = b \\, ] > 0$ if and only if $\\varphi (b) < \\infty $ if and only if $b \\in dom(\\varphi )$ .","the first two derivatives of $\\varphi $ at the point 1 have the following important “functioning” for the underlying probability distribution $\\mathbb {}$ (cf.","(REF )) respectively of an associated random variable $W$ : $1 = \\varphi ^{\\prime -1}(0) = \\int _{\\mathbb {R}} y \\, d\\mathbb {} (y)= E_{\\mathbb {\\Pi }}[W]$ where $\\varphi ^{\\prime -1}(\\cdot )$ denotes the inverse of the first derivative $\\varphi ^{\\prime }(\\cdot )$ of $\\varphi (\\cdot )$ , $\\frac{1}{\\varphi ^{\\prime \\prime }(1)} =\\int _{\\mathbb {R}} \\Big (y - \\int _{\\mathbb {R}} \\widetilde{y} \\, d\\mathbb {} (\\widetilde{y})\\Big )^{2}\\, d\\mathbb {} (y)= E_{\\mathbb {\\Pi }}[W^{2}] - (E_{\\mathbb {\\Pi }}[W])^{2} = Var_{\\mathbb {\\Pi }}[W]$ ; in particular, scaling $\\widetilde{c} \\cdot \\varphi $ ($\\widetilde{c} >0$ ) does not change the mean 1 but the variance of $W$ .","The corresponding proof of Proposition REF will be given in Appendix D, except for the second items of (G9ii) and (G9iii) as well as the first item of (G9iv).","Those will be treated in the second next paragraph below, because the corresponding line of argumentation builds an insightful start for subsequently performed procedures to further track down the weight distribution $$ .","The properties (G1) to (G9iv) constitute necessary conditions for a pregiven function $\\varphi $ to belong to $\\Upsilon (]a,b[)$ ); accordingly, these should be verified first, in concrete situations where one aims to apply the BS approach.","An important role is played by the boundary points $a$ and $b$ of $int(dom(\\varphi ))$ through (G10i) to (G10iii), because their finiteness opens the gate to apply — via some straightforward transformations — a rich class of real-valued characterization theorems for probability distributions whose support lies in the positive real line $[0,\\infty [$ .","In contrast, there exist much less real-valued characterization theorems for probability distributions whose support is the whole real line $]-\\infty ,\\infty [$ ; typically, the involved conditions are also more difficult to verify.","Indeed, if $\\varphi \\in \\Upsilon (]a,b[)$ then one can deduce straightforwardly from the representation (REF ) that $e^{\\varphi ^{*}(z)} =\\int _{-\\infty }^{\\infty } e^{z\\cdot y} \\, \\mathrm {d}\\mathbb {} (y) =E_{\\mathbb {\\Pi }}[e^{z \\cdot W}] , \\quad z \\in ]\\lambda _{-}, \\lambda _{+}[,$ where $W$ is a random variable whose distribution is $\\mathbb {\\Pi }[W \\in \\cdot \\, ] = \\mathbb {}[ \\, \\cdot \\,]$ ; under the additional knowledge $a > -\\infty $ (and consequently $\\lambda _{-} = -\\infty $ ) employed together with (G10i) and thus $\\mathbb {\\Pi }[W \\ge a \\, ] =\\mathbb {}[ \\, [a,\\infty [ \\,] = 1$ , one arrives at $e^{\\varphi ^{*}(z) - a \\cdot z} =\\int _{a}^{\\infty } e^{z \\cdot (y-a)} \\, \\mathrm {d}\\mathbb {} (y) =\\int _{0}^{\\infty } e^{z\\cdot \\widetilde{y}} \\, \\mathrm {d}\\widetilde{\\widetilde{\\mathbb {}}} (\\widetilde{y})= E_{\\mathbb {\\Pi }}[e^{z \\cdot (W-a)}] , \\quad z \\in ]-\\infty , \\lambda _{+}[,$ where the probability distribution $\\widetilde{\\widetilde{\\mathbb {}}}[ \\, \\cdot \\, ] := \\mathbb {}[\\, \\cdot + a \\, ]$ is the $a-$ shifted companion of $\\mathbb {}$ ; recall that $\\lambda _{+} >0$ .","Put in other words, $\\mathbb {\\Pi }[\\widetilde{W} \\in \\cdot \\, ] = \\widetilde{\\widetilde{\\mathbb {}}}[ \\, \\cdot \\,]$ is the probability distribution of the (a.s.) nonnegative random variable $\\widetilde{W} := W-a$ .","Naturally, in this context, the interesting case is $-\\infty < a \\le 0$ .","Similarly, if $\\varphi \\in \\Upsilon (]a,b[)$ and $b < \\infty $ (and hence $\\lambda _{+} = \\infty $ ), one can derive from (G10i) and its consequence $\\mathbb {\\Pi }[W \\le b \\, ] =\\mathbb {}[ \\, ]-\\infty ,b] \\,]= 1$ that $e^{\\varphi ^{*}(-z) + b \\cdot z} =\\int _{-\\infty }^{b} e^{z \\cdot (b-y)} \\, \\mathrm {d}\\mathbb {} (y) =\\int _{0}^{\\infty } e^{z\\cdot \\widetilde{y}} \\, \\mathrm {d}\\widetilde{\\widetilde{\\mathbb {}}} (\\widetilde{y})= E_{\\mathbb {\\Pi }}[e^{z \\cdot (b-W)}] , \\quad z \\in ]- \\infty , - \\lambda _{-}[,$ where $- \\lambda _{-} >0$ and the probability distribution $\\widetilde{\\widetilde{\\mathbb {}}}[ \\, \\cdot \\, ] := \\mathbb {}[\\, b - \\, \\cdot \\, ]$ is the mirrored$-b-$ shifted companion of $\\mathbb {}$ .","This means that $\\mathbb {\\Pi }[\\widetilde{W} \\in \\cdot \\, ] = \\widetilde{\\widetilde{\\mathbb {}}}[ \\, \\cdot \\,]$ is the probability distribution of the (a.s.) nonnegative random variable $\\widetilde{W} := b-W$ .","Naturally, the interesting case is $0 < b \\le \\infty $ .","As already indicated above, the considerations (REF ) to (REF ) open the gate to the adaption of well-known real-valued (rather than complex-valued) characterizations from probability theory.","To begin with, the following assertions are very prominent: Theorem 30 (a) Let $M: ]-\\infty ,0] \\mapsto ]0,\\infty [$ be continuous on $]-\\infty ,0]$ with $M(0) =1$ .","Then one has $\\textrm {M is absolutely monotone on ]-\\infty ,0[}\\ \\Longleftrightarrow \\ \\exists \\ \\textrm {unique prob.","distr.","\\widetilde{\\widetilde{\\mathbb {}}} on [0,\\infty [s.t.}","\\ \\ M(z) = \\int _{0}^{\\infty } e^{z \\cdot y} \\mathrm {d}\\widetilde{\\widetilde{\\mathbb {}}}(y)\\ \\textrm {for all z \\in ]-\\infty ,0[.","}$ (b) Let $I$ be an open interval which contains 0, and $M: I \\mapsto [0,\\infty [$ be continuous with $M(0) =1$ .","Then one gets $\\textrm {M is exponentially convex}\\ \\Longleftrightarrow \\ \\exists \\ \\textrm {unique prob.","distr.","\\widetilde{\\widetilde{\\mathbb {}}} on ]-\\infty ,\\infty [such that} \\ \\ M(z) = \\int _{-\\infty }^{\\infty } e^{z \\cdot y} \\mathrm {d}\\widetilde{\\widetilde{\\mathbb {}}}(y)\\ \\ \\textrm {for all z \\in I.","}$ Assertion (a) of Theorem REF is known as (probability-version of) Bernstein’s theorem [42] (see e.g.","also Schilling et al.","[322]), whereas assertion (b) is known as (probability-version of) Widder’s theorem [391] for the relevant conversion between the involved Riemann-Stieltjes integral with nondecreasing (but not necessarily right-continuous) integrator into a measure integral, one can apply the general theory in e.g.","Chapter 6 of Chow & Teicher [88].","(see e.g.","also Widder [392], Akhiezer [9], Shucker [333], Jaksetic & Pecaric [163], Kotelina & Pevny [196]).","From Theorem REF (b) and (REF ), the first item in (G9iv) follows immediately by using the choice $I= ]\\lambda _{-}, \\lambda _{+}[$ .","Moreover, Theorem REF (a) together with (REF ) (respectively (REF )) implies the second item of (G9ii) (respectively of (G9iii)).","In fact, with the help of Theorem REF and some further considerations e.g.","on light-tailedness, one even gets assertions on the sufficiency of (G9ii), (G9iii) and (G9iv) for a “candidate generator” $\\varphi $ to belong to the BS-suitable class $\\Upsilon (]a,b[)$ .","More precisely, we obtain Proposition 31 Suppose that $\\varphi : ]-\\infty ,\\infty [ \\mapsto [0,\\infty ]$ satisfies (G1) to (G8), and recall the notations in (G9i).","Then, $\\varphi \\in \\Upsilon (]a,b[)$ if one of the following three conditions holds: (a) $a >-\\infty $ , $\\lambda _{-} = - \\infty $ , and the function $z \\mapsto e^{-a\\cdot z + \\varphi ^{*}(z)}$ is absolutely monotone on $]-\\infty ,0[$ , (b) $b < \\infty $ , $\\lambda _{+} = \\infty $ , and the function $z \\mapsto e^{b\\cdot z + \\varphi ^{*}(- z)}$ is absolutely monotone on $]-\\infty ,0[$ , (c) $a = -\\infty $ , $b = -\\infty $ , and the function $z \\mapsto e^{\\varphi ^{*}(z)}$ is exponentially convex on $]\\lambda _{-}, \\lambda _{+}[$ .","If one of the three conditions (a) to (c) holds, thenbasically by Theorem REF with $M(\\cdot )$ defined in G9(ii),(iii) or (iv); see Appendix D. the associated probability distribution $\\mathbb {}$ (cf.","(REF )) has expectation $\\int _{\\mathbb {R}} y d\\mathbb {}(y) =1$ and finite moments of all orders, i.e.","$\\int _{\\mathbb {R}} y^{j} d\\mathbb {}(y) < \\infty $ for all $j \\in \\mathbb {N}_{0}$ ; in terms of $\\mathbb {}[ \\cdot \\, ] := \\mathbb {\\Pi }[W \\in \\cdot \\, ]$ this means that $E_{\\mathbb {\\Pi }}[W] =1$ and $E_{\\mathbb {\\Pi }}[W^{j}] < \\infty $ .","The proof of Proposition REF will be given in Appendix D. As far as applicability is concerned, it is well known that, in general, verifying absolute monotonicity is typically more comfortable than verifying exponential convexity.","Fortunately, one can often use the former, since for many known divergence generators there holds $a > -\\infty $ (often $a=0$ ) or $b < \\infty $ or both, which by virtue of (G10i) is directly linked with the (endpoints of the) support of the potentially existing probability distribution $\\mathbb {}$ .","For a pregiven divergence generator $\\varphi $ , once its membership in $\\Upsilon (]a,b[)$ (and in particular, the representability (REF )) is verified, one would like to concretely find the underlying probability distribution $\\mathbb {}$ .","This may be quite challenging, but can be made more comfortable by systematically narrowing down the family of distributions where $\\mathbb {}$ belongs to.","In fact, we have already performed a first down-narrowing, in terms of identifying the endpoints of the support of $\\mathbb {}$ to be the endpoints of the effective domain of $\\varphi $ (cf.","(G10i)).","A further down-narrowing can be achieved from (REF ) to (REF ) in combination with real-valued characterization theorems which are more specific than Theorem REF .","This will be shown exemplarily for a few important sub-setups, in the following.","For the identification of light-tailed semi/half-lattice distributions, we obtain the following two sets of sufficient conditions, which even allow for the desired explicit determination of $\\mathbb {}$ : Proposition 32 Suppose that $\\varphi : ]-\\infty ,\\infty [ \\mapsto [0,\\infty ]$ satisfies (G1) to (G8), with some $a > -\\infty $ .","Furthermore, assume that there exists some constant $\\breve{c} >0$ as well as some function $H: [0,\\infty [ \\mapsto [0,\\infty [$ which is continuous on $[0,1]$ with $H(1)=1$ and absolutely monotone on $]0,1[$ , such that $e^{\\varphi ^{*}(\\frac{z}{\\breve{c}}) - a\\cdot \\frac{z}{\\breve{c}} } = H(e^{z}), \\qquad z \\in ]-\\infty , \\breve{c} \\cdot \\lambda _{+}[.$ Then one has $\\varphi \\in \\Upsilon (]a,b[)$ and $\\mathbb {} = \\sum _{n=0}^{\\infty } p_{n} \\cdot \\delta _{a + \\breve{c} \\cdot n}\\qquad \\textrm {with \\ } p_{n} := \\frac{1}{n !}","\\cdot \\frac{\\mathrm {d}^{n}H}{\\mathrm {d}t}(0),$ i.e.","$\\mathbb {\\Pi }[W = a + \\breve{c} \\cdot n \\, ] = p_{n}$ ($n \\in \\mathbb {N}_{0}$ ).","Proposition 33 Suppose that $\\varphi : ]-\\infty ,\\infty [ \\mapsto [0,\\infty ]$ satisfies (G1) to (G8), with some $b < \\infty $ .","Furthermore, assume that there exists some constant $\\breve{c} >0$ as well as some function $H: [0,\\infty [ \\mapsto [0,\\infty [$ which is continuous on $[0,1]$ with $H(1)=1$ and absolutely monotone on $]0,1[$ , such that $e^{\\varphi ^{*}(- \\frac{z}{\\breve{c}}) + b \\cdot \\frac{z}{\\breve{c}} } = H(e^{z}), \\qquad z \\in ]-\\infty , - \\breve{c} \\cdot \\lambda _{-}[.$ Then one has $\\varphi \\in \\Upsilon (]a,b[)$ and $\\mathbb {} = \\sum _{n=0}^{\\infty } p_{n} \\cdot \\delta _{b - \\breve{c} \\cdot n}\\qquad \\textrm {with \\ } p_{n} := \\frac{1}{n !}","\\cdot \\frac{\\mathrm {d}^{n}H}{\\mathrm {d}t}(0),$ i.e.","$\\mathbb {\\Pi }[W = b - \\breve{c} \\cdot n \\, ] = p_{n}$ ($n \\in \\mathbb {N}_{0}$ ).","The Propositions REF respectively REF follow from (REF ) respectively (REF ), some straightforward transformations, and a well-known characterization of probability generating functions $H$ (see e.g.","in Theorem 1.2.10 of Stroock [343]).","As an incentive for the following investigations, let us recall the discussion in the surroundings of Condition REF pertaining to the minimization problem (REF ), where we have addressed possible connections between the two representabilities (REF ) (needed e.g.","for Theorem REF ) and (REF ) (needed e.g.","for Theorem REF ); this strongly relates to the question, for which constants $\\widetilde{c} >0$ the validity $\\varphi \\in \\Upsilon (]a,b[)$ triggers the validity of $\\widetilde{c} \\cdot \\varphi \\in \\Upsilon (]a,b[)$ .","To begin with, it is straightforward to see that $\\varphi \\in \\Upsilon (]a,b[)$ always implies $\\widetilde{c} \\cdot \\varphi \\in \\Upsilon (]a,b[)$ for all integers $\\widetilde{c} \\in \\mathbb {N}$ ; indeed, if $\\varphi $ satisfies (REF ) for some $\\mathbb {} = \\mathbb {\\Pi }[ W \\in \\cdot \\, ]$ , then for each integer $\\widetilde{c} \\in \\mathbb {N}$ one gets that $\\widetilde{c} \\cdot \\varphi $ satisfies (REF ) for $\\widetilde{\\mathbb {}} = \\mathbb {\\Pi }[ \\sum _{j=1}^{\\widetilde{c}} \\frac{W_{j}}{\\widetilde{c}} \\in \\cdot \\, ]$ ; in the latter, the $W_{j}$ ’s are i.i.d.","copies from $W$ .","Clearly, $MGF_{\\widetilde{\\mathbb {}}}$ is then finite on some open interval containing zero (differing from the one for $MGF_{\\mathbb {}}$ only by a scaling with $1/\\widetilde{c}$ ).","For the following family of distributions, one can even trigger $\\widetilde{c} \\cdot \\varphi \\in \\Upsilon (]a,b[)$ for all $\\widetilde{c} >0$ : for the sake of a corresponding precise formulation, recall first the common knowledge that, generally speaking, a probability distribution $\\mathbb {}$ on $\\mathbb {R}$ with light tails — in the sense that its moment generating function $z\\mapsto MGF_{\\mathbb {} }(z):=\\int _{\\mathbb {R}}e^{z \\cdot y}d\\mathbb {} (y)$ is finite on some open interval $]\\lambda _{-},\\lambda _{+}[$ containing zero — is (said to be) infinitely divisible if there holds $\\textrm {for each n \\in \\mathbb {N} there existsa probability distribution \\mathbb {}_{n} on \\mathbb {R} such that}\\int _{-\\infty }^{\\infty } e^{z \\cdot y} d\\mathbb {} (y) =\\Big (\\int _{-\\infty }^{\\infty } e^{z \\cdot y} d\\mathbb {}_{n} (y) \\Big )^{n}, \\quad z \\in ]\\lambda _{-},\\lambda _{+}[ ;$ in fact, (REF ) means that the (light-tailed) moment generating function $MGF_{\\mathbb {} }$ is infinitely divisible in the sense that each $n-$ th root $(MGF_{\\mathbb {} })^{1/n}$ must be the moment generating function of some (light-tailed) probability distribution (denoted here by $\\mathbb {}_{n}$ ).","In particular, (REF ) implies that $\\mathbb {}_{n}$ is unique, and that $\\mathbb {}$ must necessarily have (one-sided or two-sided) unbounded support $supp(\\mathbb {})$ .","The latter may differ from $supp(\\mathbb {}_{n})$ .","In our BS context (REF ), (REF ) equivalently means that the associated random variable $W$ is infinitely divisible (with light-tailed distribution), in the sense that $\\textrm {for each n \\in \\mathbb {N} there exists a sequence of i.i.d.","random variables Y_{n,1}, \\cdots , Y_{n,n} such that }W \\stackrel{\\text{\\tiny d}}{=} Y_{n,1} + \\cdots + Y_{n,n} ,$ where $\\stackrel{\\text{\\tiny d}}{=}$ means “have equal probability distributions” and $\\mathbb {\\Pi }[W \\in \\cdot \\, ] = \\mathbb {}[ \\, \\cdot \\,]$ , $\\mathbb {\\Pi }[Y_{n,1} \\in \\cdot \\, ] = \\mathbb {}_{n}[ \\, \\cdot \\,]$ .","For the above-mentioned context, we obtain the useful Proposition 34 Suppose that $\\varphi \\in \\Upsilon (]a,b[)$ , with connected probability distribution $\\mathbb {}$ from (REF ) (recall that this implies that $\\mathbb {}$ is not a one-point distribution, cf.","Remark REF ).","Then there holds: $\\textrm {\\widetilde{c} \\cdot \\varphi \\in \\Upsilon (]a,b[)for all \\widetilde{c} > 0} \\ \\ \\Longleftrightarrow \\ \\ \\textrm {\\mathbb {} is infinitely divisible.","}$ The proof of Proposition REF is given in Appendix E. Notice that Proposition REF covers especially the important prominent power divergences (cf.","Examples REF and REF below) for which we provide the corresponding infinitely divisible distributions explicitly in the Examples REF and REF below, and for which the subsequent form of estimators (cf.","Chapter below) can be simplified.","For the identification of light-tailed infinitely divisible distributions, we obtain the following three sets of sufficient conditions: Proposition 35 Suppose that $\\varphi : ]-\\infty ,\\infty [ \\mapsto [0,\\infty ]$ satisfies (G1) to (G8), and recall the notations in (G9i) as well as $a = \\inf supp(\\mathbb {})$ , $b = \\sup supp(\\mathbb {})$ (cf.","(G10i)).","Then, $\\varphi \\in \\Upsilon (]a,b[)$ and the associated probability distribution $\\mathbb {}$ is infinitely divisible, if one of the following three conditions holds: (a) $a >-\\infty $ , $\\lambda _{-} = - \\infty $ , and the function $z \\mapsto \\varphi ^{* \\prime }(z) - a = (\\varphi ^{\\prime })^{-1}(z) - a $ is absolutely monotone on $]-\\infty ,0[$ , (b) $b < \\infty $ , $\\lambda _{+} = \\infty $ , and the function $z \\mapsto - \\varphi ^{* \\prime }(- z) + b= - (\\varphi ^{\\prime })^{-1}(-z) +b $ is absolutely monotone on $]-\\infty ,0[$ , (c) $a = -\\infty $ , $b = -\\infty $ , and the function $z \\mapsto \\frac{\\varphi ^{* \\prime \\prime }(z)}{\\varphi ^{* \\prime \\prime }(0)}= \\frac{\\varphi ^{\\prime \\prime }(1)}{\\varphi ^{\\prime \\prime }((\\varphi ^{\\prime })^{-1}(z))} $ is exponentially convex on $]\\lambda _{-}, \\lambda _{+}[$ .","In the first case (a) there automatically follows $b=\\infty $ , whereas in the second case (b) one automatically gets $a =- \\infty $ .","The proof of Proposition REF is given in Appendix E. So far, in the current section we have started from a given divergence generator $\\varphi \\in \\widetilde{\\Upsilon }(]a,b[)$ having some additional properties, switched to its Fenchel-Legendre transform $\\varphi ^{*}$ and some exponentially-linear transforms thereof, and presented some sufficient conditions for verifying that the outcome is a moment-generating function $MGF_{\\mathbb {}}$ of a unique probability distribution $\\mathbb {}$ which has light tails.","For finding the concrete $\\mathbb {}$ , one typically should know the explicit form of $\\varphi ^{*}$ .","However, it is well known that it can sometimes be hard to determine the explicit form of the Fenchel-Legendre transform of a convex function.","This issue also applies for the reverse direction of starting from a concrete probability distribution $\\mathbb {}$ with light tails, computing its log-moment-generating function (called cumulant-generating function) $z \\mapsto \\Lambda _{\\mathbb {}}(z) := \\log MGF_{\\mathbb {}}(z)$ and the corresponding Fenchel-Legendre transform $\\Lambda _{\\mathbb {}}^{*}$ which is nothing but the associated divergence generator $\\varphi $ (cf.","(REF )).","As will be illuminated in several examples below, the — “kind of intermediate” — construction method given in the below-mentioned Theorem REF can help to ease these two tasks.","To formulate this, we employ the class ${F}$ of functions $F: ]-\\infty ,\\infty [ \\mapsto [-\\infty ,\\infty ]$ with the following properties: $int(dom(F)) = ]a_{F},b_{F}[$ for some $-\\infty \\le a_{F} < 1 < b_{F} \\le \\infty $ ; $F$ is smooth (infinitely continuously differentiable) on $]a_{F},b_{F}[$ ; $F$ is strictly increasing on $]a_{F},b_{F}[$ .","Clearly, for any $F \\in {F}$ one gets the existence of $F(a_{F}) := \\lim _{t \\downarrow a_{F}} F(t) \\in [-\\infty , \\infty [$ and $F(b_{F}) := \\lim _{t \\uparrow b_{F}} F(t) \\in ]-\\infty , \\infty ]$ ; moreover, its inverse $F^{-1}: \\mathcal {R}(F) \\mapsto [a_{F},b_{F}]$ exists, where $\\mathcal {R}(F) := \\lbrace F(t): t \\in dom(F) \\rbrace $ .","Furthermore, $F^{-1}$ is strictly increasing and smooth (infinitely continuously differentiable) on the open interval $int(\\mathcal {R}(F)) = \\lbrace F(t): t \\in ]a_{F},b_{F}[ \\rbrace =]F(a_{F}),F(b_{F})[$ , and $F^{-1}(int(\\mathcal {R}(F))) = ]a_{F},b_{F}[$ .","Within such a context, we obtain Theorem 36 Let $F \\in {F}$ and fix an arbitrary point $c \\in int(\\mathcal {R}(F))$ .","Moreover, introduce the notationsfor the sake of brevity, we avoid here the more complete notation $\\lambda _{-}^{F,c}$ , $\\lambda _{+}^{F,c}$ , $t_{-}^{sc,F,c}$ , $t_{+}^{sc,F,c}$ indicating the dependence on $F$ and $c$ .","$]\\lambda _{-},\\lambda _{+}[ := int(\\mathcal {R}(F)) -c$ and $]t_{-}^{sc},t_{+}^{sc}[ := ]1+a_{F}-F^{-1}(c),1+b_{F}-F^{-1}(c)[$ (which implies $\\lambda _{-} < 0 < \\lambda _{+}$ and $t_{-}^{sc} < 1 < t_{+}^{sc}$ ).","Furthermore, define the functions $\\Lambda : \\ ]-\\infty ,\\infty [ \\ \\mapsto \\, [-\\infty , \\infty ]$ and $\\varphi : \\ ]-\\infty ,\\infty [ \\ \\mapsto \\, [0, \\infty ]$ by $\\hspace{-19.91684pt} \\Lambda (z) := \\Lambda ^{(c)}(z) \\hspace{-5.69046pt} &:=& \\hspace{-5.69046pt}{\\left\\lbrace \\begin{array}{ll}\\int \\displaylimits _{0}^{z} F^{-1}(u+c) \\, du+ z \\cdot (1-F^{-1}(c)) \\ \\in ]-\\infty , \\infty [,\\hspace{85.35826pt} \\textrm {if } z \\in ]\\lambda _{-},\\lambda _{+}[,\\\\\\int \\displaylimits _{0}^{\\lambda _{-}} F^{-1}(u+c) \\, du+ \\lambda _{-} \\cdot (1-F^{-1}(c))\\ \\in [-\\infty ,\\infty ],\\hspace{71.13188pt} \\textrm {if } z = \\lambda _{-} > -\\infty ,\\\\\\int \\displaylimits _{0}^{\\lambda _{+}} F^{-1}(u+c) \\, du+ \\lambda _{+} \\cdot (1-F^{-1}(c))\\ \\in \\ [-\\infty ,\\infty ],\\hspace{66.86414pt} \\textrm {if } z = \\lambda _{+} < \\infty ,\\\\\\infty , \\hspace{283.10483pt} \\textrm {else},\\end{array}\\right.","}$ where the second respectively third line are meant as $\\lim _{z \\downarrow \\lambda _{-}} \\big ( \\int \\displaylimits _{0}^{z} F^{-1}(u+c) \\, du+ z \\cdot (1-F^{-1}(c)) \\big )$ respectively $\\lim _{z \\uparrow \\lambda _{+}} \\big ( \\int \\displaylimits _{0}^{z} F^{-1}(u+c) \\, du+ z \\cdot (1-F^{-1}(c)) \\big )$ , and $\\varphi (t) := \\varphi ^{(c)}(t) \\hspace{-5.69046pt} &:=& \\hspace{-5.69046pt}{\\left\\lbrace \\begin{array}{ll}(t+F^{-1}(c)-1) \\cdot [ F\\left(t+F^{-1}(c)-1 \\right) - c ]-\\int \\displaylimits _{0}^{F\\left(t+F^{-1}(c)-1 \\right) - c} F^{-1}(u+c) du\\ \\in \\ [0,\\infty [,\\\\\\hspace{327.20668pt}\\quad \\textrm {if }t \\in ]t_{-}^{sc},t_{+}^{sc}[,\\\\(t_{-}^{sc}+F^{-1}(c)-1) \\cdot [ F\\left(t_{-}^{sc}+F^{-1}(c)-1 \\right) - c ]-\\int \\displaylimits _{0}^{F\\left(t_{-}^{sc}+F^{-1}(c)-1 \\right) - c} F^{-1}(u+c) du\\ \\in \\ ]0,\\infty ],\\\\\\hspace{327.20668pt}\\quad \\textrm {if }t = t_{-}^{sc} > -\\infty ,\\\\(t_{+}^{sc}+F^{-1}(c)-1) \\cdot [ F\\left(t_{+}^{sc}+F^{-1}(c)-1 \\right) - c ]-\\int \\displaylimits _{0}^{F\\left(t_{+}^{sc}+F^{-1}(c)-1 \\right) - c} F^{-1}(u+c) du\\ \\in \\ ]0,\\infty ],\\\\\\hspace{327.20668pt}\\quad \\textrm {if }t = t_{+}^{sc} < \\infty ,\\\\\\varphi (t_{-}^{sc}) +\\lambda _{-}\\cdot (t- t_{-}^{sc}) \\ \\in \\ ]0,\\infty ],\\hspace{99.58464pt} \\textrm {if }t_{-}^{sc} > - \\infty , \\ \\textrm {and} \\ t \\in \\ ]-\\infty , t_{-}^{sc}[,\\\\\\varphi (t_{+}^{sc}) +\\lambda _{+}\\cdot (t - t_{+}^{sc}) \\ \\in \\ ]0,\\infty ],\\hspace{99.58464pt} \\textrm {if }t_{+}^{sc} < \\infty , \\ \\textrm {and} \\ t \\in \\ ]t_{+}^{sc}, \\infty [,\\\\\\infty , \\hspace{321.51622pt} \\textrm {else},\\end{array}\\right.","}$ where the second respectively third line are again meant as lower respectively upper limit.","Then, $\\Lambda $ and $\\varphi $ have the following properties: (i) On $]\\lambda _{-},\\lambda _{+}[$ , the function $\\Lambda $ is smooth and strictly convex and consequently, $\\exp (\\Lambda )$ ) is smooth and strictly log-convex; moreover, there holds $\\Lambda (0) =0$ , $\\Lambda ^{\\prime }(0) =1$ ; (ii) $\\varphi \\in \\widetilde{\\Upsilon }(]a,b[)$ , where $a := t_{-}^{sc} \\cdot {1}_{ \\lbrace -\\infty \\rbrace }(\\lambda _{-}) - \\infty \\cdot {1}_{]-\\infty ,0[}(\\lambda _{-})$ , $b := t_{+}^{sc} \\cdot {1}_{ \\lbrace \\infty \\rbrace }(\\lambda _{+}) + \\infty \\cdot {1}_{]0,\\infty [}(\\lambda _{+})$ , and $\\varphi $ has the properties (G1) to (G8).","(iii) $\\varphi (t) =\\sup _{z \\in ]-\\infty ,\\infty [} \\left( z\\cdot t -\\Lambda (z)\\right) =\\sup _{z \\in ]\\lambda _{-},\\lambda _{+}[} \\left( z\\cdot t -\\Lambda (z)\\right)$ for all $t \\in \\mathbb {R}$ .","(iv) $\\Lambda (z) = \\varphi ^{*}(z) =\\sup _{t \\in ]-\\infty ,\\infty [} \\left( t\\cdot z -\\varphi (t)\\right) =\\sup _{t \\in ]a,b[} \\left( t\\cdot z -\\varphi (t) \\right)$ for all $z \\in \\mathbb {R}$ .","The proof of Theorem REF will be given Appendix F. Remark 37 Theorem REF indicates that the $F-$ constructed function $z \\mapsto \\exp (\\Lambda (z)) = \\exp (\\varphi ^{*}(z))$ is a good candidate for a moment generating function of a probability distribution $\\mathbb {}$ , and hence for the representability (REF ).","However, one still needs to verify one of the conditions (a) to (c) of Proposition REF .","This may go wrong, as the case of power divergences $\\varphi _{\\gamma }$ with $\\gamma \\in ]1,2[$ indicates (cf.","the conjecture of Example REF (f) below).","Notice that the newly constructed $\\Lambda $ and $\\varphi $ (cf.","(REF ), (REF )) depend on the choice of the anchor point $c$ ; this is e.g.","illustrated in Example REF (b) below.","Hence, as a side effect, by using whole families $(F_{\\vartheta })_{\\vartheta }$ together with different anchor points $c$ , via Theorem REF one can generate new classes (and new classifications) of $\\varphi -$ divergence generators — and thus of corresponding $\\varphi -$ divergences — which can be of great use, even in other contexts beyond our BS optimization framework.","If $F$ satisfies $F(1)=0$ and thus $F^{-1}(0)=1$ , then the natural choice $c:=0$ induces $]\\lambda _{-},\\lambda _{+}[ = int(\\mathcal {R}(F))$ and $]t_{-}^{sc},t_{+}^{sc}[ = ]a_{F},b_{F}[$ , and consequently (due to $F^{-1}(c)-1 =0$ ) leads to the simplification of “the first lines of” (REF ) and (REF ) to $& & \\hspace{-19.91684pt} \\Lambda (z) := \\Lambda ^{(0)}(z) :=\\int \\displaylimits _{0}^{z} F^{-1}(u) du,\\qquad z \\in int(\\mathcal {R}(F)),\\\\& & \\hspace{-19.91684pt} \\varphi (t) := \\varphi ^{(0)}(t) :=t \\cdot F\\left(t\\right)-\\int \\displaylimits _{0}^{F\\left(t\\right)} F^{-1}(u) du,\\qquad t \\in ]a_{F},b_{F}[ ;$ the simplifications of the respective other lines of (REF ) and (REF ) are straightforward.","Remark 38 Let $F \\in {F}$ with $a_{F} =0$ , $b_{F} = \\infty $ , $F(1)=0$ and hence, $int(\\mathcal {R}(F)) = \\, ]F(0),F(\\infty )[$ .","Then the transformation $\\widetilde{F}(t) &:=&{\\left\\lbrace \\begin{array}{ll}- \\int _{0}^{F(\\frac{1}{t})} F^{-1}(u) \\, du, \\qquad \\textrm {if } \\ t \\in ]0,\\infty [, \\\\- \\int _{0}^{F(\\infty )} F^{-1}(u) \\, du, \\qquad \\textrm {if } \\ t=0, \\\\-\\infty , \\qquad \\hspace{64.01869pt} \\textrm {if } \\ t \\in ]-\\infty , 0[,\\end{array}\\right.","}$ satisfies $\\widetilde{F} \\in {F}$ with $a_{\\widetilde{F}} =0$ , $b_{\\widetilde{F}} = \\infty $ , $\\widetilde{F}(1)=0$ and $int(\\mathcal {R}(\\widetilde{F})) =\\big ] - \\int _{0}^{F(\\infty )} F^{-1}(u) \\, du, - \\int _{0}^{F(0)} F^{-1}(u) \\, du \\, \\big [$ .","By choosing the natural anchor point $c=0$ (for both $F$ and $\\widetilde{F}$ ) and by using the relations $\\widetilde{F}(t) = - \\Lambda (F(\\frac{1}{t}))$ , $\\widetilde{F}^{-1}(z) = \\frac{1}{F^{-1}(\\Lambda ^{-1}(-z))}$ , as well as (REF ) in combination with (REF ) (for both contexts), it is straightforward to see that the corresponding quantities $\\widetilde{\\Lambda }$ and $\\widetilde{\\varphi }$ satisfy $\\widetilde{\\Lambda }(z) =- (-\\Lambda )^{-1}(z)$ ($z \\in int(\\mathcal {R}(\\widetilde{F}))$ ) and $\\widetilde{\\varphi }(t) = t \\cdot \\varphi (\\frac{1}{t})$ ($t \\in ]0,\\infty [$ ).","Hence, the corresponding divergences (cf.","(REF )) are “reciprocal to each other” in the sense that $D_{\\widetilde{\\varphi }}( \\mathbf {Q}, \\mathbf {P} ) =D_{\\varphi }( \\mathbf {P}, \\mathbf {Q} )$ for all $\\mathbf {P},\\mathbf {Q} \\in \\mathbb {S}_{>0}^{K}$ , and in case that $\\Lambda $ and $\\widetilde{\\Lambda }$ are indeed cumulant generating functions of some light-tailed distributions $\\mathbb {}$ and $\\widetilde{\\mathbb {}}$ (cf.","Remark REF ), then the latter two are said to be inverse to each other in the sense of Tweedie [368] (see also e.g.","Folks [127]).","As already indicated above, from Theorem REF one can comfortably generate various interesting examples, which we demonstrate in the following.","Example 39 (a) For $\\gamma \\in \\mathbb {R}\\backslash \\lbrace 1,2\\rbrace $ , $\\widetilde{c} \\in ]0,\\infty [$ and $]a_{F_{\\gamma ,\\widetilde{c}}},b_{F_{\\gamma ,\\widetilde{c}}}[ \\, = \\, ]0,\\infty [$ we define $F_{\\gamma ,\\widetilde{c}}(t) &:=&{\\left\\lbrace \\begin{array}{ll}\\frac{\\widetilde{c}}{\\gamma -1} \\cdot (t^{\\gamma -1}-1), \\qquad \\textrm {if } \\ t \\in \\, ]0,\\infty [, \\\\-\\frac{\\widetilde{c}}{\\gamma -1}, \\hspace{62.59596pt}\\textrm {if } \\ t=0 \\ \\textrm {and } \\ \\gamma \\in \\, ]1,2[ \\, \\cup \\, ]2,\\infty [, \\\\-\\infty , \\hspace{71.13188pt} \\textrm {if } \\ t=0 \\ \\textrm {and } \\ \\gamma < 1, \\\\- \\infty , \\hspace{71.13188pt} \\textrm {if } \\ t \\in \\, ]-\\infty ,0[,\\end{array}\\right.","}\\nonumber $ Clearly, $\\mathcal {R}(F_{\\gamma ,\\widetilde{c}})= \\big [-\\frac{\\widetilde{c}}{\\gamma -1},\\infty \\big [$ for $\\gamma \\in \\, ]1,2[ \\, \\cup \\, ]2,\\infty [$ , respectively $\\mathcal {R}(F_{\\gamma ,\\widetilde{c}})=\\big ]-\\infty , \\frac{\\widetilde{c}}{1-\\gamma }\\big [$ for $\\gamma <1$ ; notice that $0 \\in int(\\mathcal {R}(F_{\\gamma ,\\widetilde{c}}))$ for all $\\gamma \\in \\mathbb {R}\\backslash \\lbrace 1,2\\rbrace $ .","Furthermore, $F_{\\gamma ,\\widetilde{c}}(\\cdot )$ is strictly increasing and smooth on $]0,\\infty [$ , and thus, $F_{\\gamma ,\\widetilde{c}} \\in {F}$ .","Since $F_{\\gamma ,\\widetilde{c}}(1)=0$ , let us choose the natural anchor point $c:=0$ , which leads to $]\\lambda _{-},\\lambda _{+}[= int(\\mathcal {R}(F_{\\gamma ,\\widetilde{c}}))$ and $]t_{-}^{sc},t_{+}^{sc}[ = ]0,\\infty [$ .","By using $F_{\\gamma ,\\widetilde{c}}^{-1}(x) = (1+\\frac{(\\gamma -1) \\cdot x}{\\widetilde{c}})^{\\frac{1}{\\gamma -1}}$ for $x \\in int(\\mathcal {R}(F_{\\gamma ,\\widetilde{c}}))$ , we can derive from formula (REF ) (see also (REF )) for all $\\gamma \\in \\mathbb {R}\\backslash \\lbrace 0,1,2\\rbrace $ and $z \\in \\mathbb {R}$ $\\Lambda _{\\gamma ,\\widetilde{c}}(z) := \\Lambda _{\\gamma ,\\widetilde{c}}^{(0)}(z) &=&{\\left\\lbrace \\begin{array}{ll}\\frac{\\widetilde{c}}{\\gamma } \\cdot \\left\\lbrace \\left( \\frac{\\gamma -1}{\\widetilde{c}} \\cdot z +1 \\right)^{\\frac{\\gamma }{\\gamma -1}} -1 \\right\\rbrace , \\qquad \\textrm {if } \\ \\gamma \\in \\, ]1,2[ \\, \\cup \\, ]2,\\infty [\\ \\textrm {and } \\ z \\in \\big ]-\\frac{\\widetilde{c}}{\\gamma -1},\\infty \\big [ \\\\\\hspace{130.88284pt} \\textrm {or if } \\ \\gamma \\in ]-\\infty ,0[ \\cup ]0,1[ \\ \\textrm {and } \\ z \\in \\big ]-\\infty , \\frac{\\widetilde{c}}{1-\\gamma }\\big [, \\\\- \\frac{\\widetilde{c}}{\\gamma } < 0, \\hspace{105.2751pt} \\textrm {if } \\ \\gamma \\in \\, ]1,2[ \\, \\cup \\, ]2,\\infty [\\ \\textrm {and } \\ z = -\\frac{\\widetilde{c}}{\\gamma -1}, \\\\- \\frac{\\widetilde{c}}{\\gamma } >0, \\hspace{105.2751pt} \\textrm {if } \\ \\gamma < 0 \\ \\textrm {and } \\ z = \\frac{\\widetilde{c}}{1-\\gamma }, \\\\\\infty , \\hspace{128.0374pt} \\textrm {if } \\ \\gamma \\in \\, ]0,1[ \\ \\textrm {and } \\ z = \\frac{\\widetilde{c}}{1-\\gamma }, \\\\\\infty , \\hspace{128.0374pt} \\textrm {else} .\\end{array}\\right.","}$ Notice that $\\Lambda _{\\gamma ,\\widetilde{c}}(0) = 0$ for all $\\gamma \\in \\mathbb {R}\\backslash \\lbrace 0,1,2\\rbrace $ .","Moreover, for $\\gamma \\in \\, ]1,2[ \\, \\cup \\, ]2,\\infty [$ one has $\\Lambda _{\\gamma ,\\widetilde{c}}(\\infty ) = \\infty $ , $\\Lambda _{\\gamma ,\\widetilde{c}}^{\\prime }(-\\frac{\\widetilde{c}}{\\gamma -1}) = 0$ and $\\Lambda _{\\gamma ,\\widetilde{c}}^{\\prime }(\\infty ) = \\infty $ .","For $\\gamma < 0$ one gets $\\Lambda _{\\gamma ,\\widetilde{c}}(-\\infty ) = -\\infty $ , $\\Lambda _{\\gamma ,\\widetilde{c}}^{\\prime }(\\frac{\\widetilde{c}}{1-\\gamma }) = \\infty $ and $\\Lambda _{\\gamma ,\\widetilde{c}}^{\\prime }(-\\infty ) = 0$ .","In contrast, if $\\gamma \\in \\, ]0,1[$ then $\\Lambda _{\\gamma ,\\widetilde{c}}(-\\infty ) = - \\frac{\\widetilde{c}}{\\gamma } <0$ , $\\Lambda _{\\gamma ,\\widetilde{c}}^{\\prime }(\\frac{\\widetilde{c}}{1-\\gamma }) = \\infty $ and $\\Lambda _{\\gamma ,\\widetilde{c}}^{\\prime }(-\\infty ) = 0$ .","To proceed, from formula (REF ) (see also ()) we can deduce for all $\\gamma \\in \\mathbb {R}\\backslash \\lbrace 0,1,2\\rbrace $ and $t \\in \\mathbb {R}$ $\\varphi _{\\gamma ,\\widetilde{c}}(t) := \\varphi _{\\gamma ,\\widetilde{c}}^{(0)}(t)\\hspace{-5.69046pt} &=& \\hspace{-5.69046pt}{\\left\\lbrace \\begin{array}{ll}\\widetilde{c} \\cdot \\frac{t^\\gamma -\\gamma \\cdot t+ \\gamma - 1}{\\gamma \\cdot (\\gamma -1)}\\ \\in \\ [0,\\infty [,\\qquad \\quad \\textrm {if }t \\in ]0,\\infty [,\\\\\\frac{\\widetilde{c}}{\\gamma } > 0, \\hspace{108.12054pt} \\textrm {if } \\ \\gamma \\in \\, ]1,2[ \\, \\cup \\, ]2,\\infty [\\ \\textrm {and } \\ t = 0, \\\\\\infty , \\hspace{123.76965pt} \\textrm {if } \\ \\gamma < 0 \\ \\textrm {and } \\ t = 0, \\\\\\frac{\\widetilde{c}}{\\gamma } > 0, \\hspace{108.12054pt} \\textrm {if } \\ \\gamma \\in \\, ]0,1[ \\ \\textrm {and } \\ t = 0, \\\\\\frac{\\widetilde{c}}{\\gamma } -\\frac{\\widetilde{c}}{\\gamma -1} \\cdot t \\ \\in \\ ]0,\\infty [,\\hspace{45.52458pt} \\textrm {if } \\ \\gamma \\in \\, ]1,2[ \\, \\cup \\, ]2,\\infty [\\ \\textrm {and } \\ t < 0, \\\\\\infty , \\hspace{123.76965pt} \\textrm {else} ,\\end{array}\\right.","}$ which coincides with $\\widetilde{c} \\cdot \\varphi _{\\gamma }(t)$ for $\\varphi _{\\gamma }(t)$ from (REF ) and which generates the $\\gamma -$ corresponding power divergences given in (REF ); the first line in (REF ) can be proved by $& & \\hspace{-19.91684pt}\\varphi _{\\gamma ,\\widetilde{c}}(t) := \\varphi _{\\gamma ,\\widetilde{c}}^{(0)}(t) :=t \\cdot F_{\\gamma ,\\widetilde{c}}\\left(t\\right)-\\int \\displaylimits _{0}^{F_{\\gamma ,\\widetilde{c}}\\left(t\\right)} F_{\\gamma ,\\widetilde{c}}^{-1}(u) du\\nonumber \\\\& & \\hspace{-19.91684pt}= \\frac{t \\cdot \\widetilde{c}}{\\gamma -1} \\cdot (t^{\\gamma -1}-1)- \\frac{\\widetilde{c}}{\\gamma } \\cdot \\left\\lbrace \\left( \\frac{\\gamma -1}{\\widetilde{c}} \\cdot \\left[ \\frac{\\widetilde{c}}{\\gamma -1} \\cdot (t^{\\gamma -1}-1) \\right]+ 1 \\right)^{\\frac{\\gamma }{\\gamma -1}} -1 \\right\\rbrace \\nonumber \\\\& & \\hspace{-19.91684pt}= \\widetilde{c} \\cdot \\frac{t^\\gamma -\\gamma \\cdot t+ \\gamma - 1}{\\gamma \\cdot (\\gamma -1)},\\qquad t \\in ]0,\\infty [ \\, .$ Notice that for all $\\gamma \\in \\mathbb {R}\\backslash \\lbrace 0,1,2\\rbrace $ one has $\\varphi _{\\gamma ,\\widetilde{c}}(1) = 0$ , $\\varphi _{\\gamma ,\\widetilde{c}}^{\\prime }(1) = 0$ and $\\varphi _{\\gamma ,\\widetilde{c}}(\\infty ) = \\infty $ .","Moreover, for $\\gamma \\in \\, ]1,2[ \\, \\cup \\, ]2,\\infty [$ one has $\\varphi _{\\gamma ,\\widetilde{c}}^{\\prime }(0) = -\\frac{\\widetilde{c}}{\\gamma -1} < 0$ and $\\varphi _{\\gamma ,\\widetilde{c}}^{\\prime }(\\infty ) = \\infty $ .","In contrast, for $\\gamma < 0$ and $\\gamma \\in \\, ]0,1[$ one gets $\\varphi _{\\gamma ,\\widetilde{c}}^{\\prime }(0) = - \\infty $ and $\\varphi _{\\gamma ,\\widetilde{c}}^{\\prime }(\\infty ) = \\frac{\\widetilde{c}}{1-\\gamma } > 0$ .","(b) For $\\gamma =2$ , $\\widetilde{c} \\in ]0,\\infty [$ and $]a_{F_{\\gamma ,\\widetilde{c}}},b_{F_{\\gamma ,\\widetilde{c}}}[ \\, = \\, ]-\\infty ,\\infty [$ we define $F_{2,\\widetilde{c}}(t) &:=&\\widetilde{c} \\cdot (t-1), \\qquad t \\in \\, ]-\\infty ,\\infty [,\\nonumber $ Clearly, $\\mathcal {R}(F_{2,\\widetilde{c}})= \\, ]-\\infty , \\infty [$ , $0 \\in int(\\mathcal {R}(F_{2,\\widetilde{c}}))$ , and $F_{2,\\widetilde{c}}(\\cdot )$ is strictly increasing as well as smooth on $]-\\infty ,\\infty [$ .","Hence, $F_{2,\\widetilde{c}} \\in {F}$ .","Since $F_{2,\\widetilde{c}}(1)=0$ , let us choose the natural anchor point $c:=0$ , which leads to $]\\lambda _{-},\\lambda _{+}[= int(\\mathcal {R}(F_{2,\\widetilde{c}}))= \\, ]-\\infty , \\infty [$ and $]t_{-}^{sc},t_{+}^{sc}[ = \\, ]-\\infty , \\infty [$ .","By using $F_{2,\\widetilde{c}}^{-1}(x) = 1+\\frac{x}{\\widetilde{c}}$ for $x \\in int(\\mathcal {R}(F_{2,\\widetilde{c}}))$ , we can derive from formula (REF ) (see also (REF )) $\\Lambda _{2,\\widetilde{c}}(z) := \\Lambda _{2,\\widetilde{c}}^{(0)}(z) &=&\\frac{\\widetilde{c}}{2} \\cdot \\left\\lbrace \\left( \\frac{1}{\\widetilde{c}} \\cdot z +1 \\right)^{2} -1 \\right\\rbrace =\\frac{z^2}{2 \\widetilde{c}} + z,\\qquad z \\in \\, ]-\\infty ,\\infty [.$ Notice that $\\Lambda _{2,\\widetilde{c}}(0) = 0$ , $\\Lambda _{2,\\widetilde{c}}(-\\infty ) = \\Lambda _{2,\\widetilde{c}}(\\infty ) = \\infty $ , $\\Lambda _{2,\\widetilde{c}}^{\\prime }(-\\infty ) = -\\infty $ and $\\Lambda _{2,\\widetilde{c}}^{\\prime }(\\infty ) = \\infty $ .","From formula (REF ) (see also ()) we can deduce analogously to (REF ) $\\varphi _{2,\\widetilde{c}}(t) := \\varphi _{2,\\widetilde{c}}^{(0)}(t)&=&\\widetilde{c} \\cdot \\frac{(t-1)^{2}}{2}\\ \\in \\ [0,\\infty [,\\qquad t \\in \\, ]-\\infty ,\\infty [,$ which coincides with $\\widetilde{c} \\cdot \\varphi _{2}(t)$ for $\\varphi _{2}(t)$ from (REF ) which generates the corresponding power divergence given in the sixth line of (REF ).","Notice that $\\varphi _{2,\\widetilde{c}}(1) = 0$ , $\\varphi _{2,\\widetilde{c}}^{\\prime }(1) = 0$ and $\\varphi _{2,\\widetilde{c}}(-\\infty ) = \\varphi _{2,\\widetilde{c}}(\\infty ) = \\infty $ .","As an application of the reciprocity considerations of Remark REF , it is straightforward to see from the above-mentioned considerations (a) and (b) that for all $\\gamma \\in \\mathbb {R}\\backslash \\lbrace 0,1\\rbrace $ one has $\\widetilde{F}_{\\gamma ,\\widetilde{c}}(t) =- \\Lambda _{\\gamma ,\\widetilde{c}}(F_{\\gamma ,\\widetilde{c}}(\\frac{1}{t}))= F_{1-\\gamma ,\\widetilde{c}}(t)$ for all $t \\in ]0,\\infty [$ .","(c) Let us now continue with the remaining case $\\gamma =0$ (recall the natural anchor point $c:=0$ ).","By using $F_{0,\\widetilde{c}}^{-1}(x) =\\frac{1}{1- \\frac{x}{\\widetilde{c}}} $ for $x \\in int(\\mathcal {R}(F_{0,\\widetilde{c}})) = ]-\\infty , \\widetilde{c}[$ , we can derive from formula (REF ) (see also (REF )) $\\Lambda _{0,\\widetilde{c}}(z) := \\Lambda _{0,\\widetilde{c}}^{(0)}(z) &=&{\\left\\lbrace \\begin{array}{ll}- \\widetilde{c} \\cdot \\log \\left( 1 - \\frac{z}{\\widetilde{c}} \\right), \\qquad \\textrm {if } \\ z \\in \\big ]-\\infty ,\\widetilde{c} \\big [, \\\\\\infty , \\hspace{78.24507pt} \\textrm {if } \\ z \\in \\big [\\widetilde{c}, \\infty \\big [ .\\end{array}\\right.","}$ Notice that $\\Lambda _{0,\\widetilde{c}}(0) = 0$ , $\\Lambda _{0,\\widetilde{c}}(-\\infty ) = - \\infty $ , $\\Lambda _{0,\\widetilde{c}}^{\\prime }(\\widetilde{c}) = \\infty $ and $\\Lambda _{0,\\widetilde{c}}^{\\prime }(-\\infty ) = 0$ .","Moreover, from formula (REF ) (see also ()) we can deduce $\\varphi _{0,\\widetilde{c}}(t) := \\varphi _{0,\\widetilde{c}}^{(0)}(t)\\hspace{-5.69046pt} &=& \\hspace{-5.69046pt}{\\left\\lbrace \\begin{array}{ll}\\widetilde{c} \\cdot \\left(- \\log t + t -1 \\right)\\ \\in \\ [0,\\infty [,\\qquad \\quad \\textrm {if }t \\in \\, ]0,\\infty [,\\\\\\infty , \\hspace{145.10922pt} \\textrm {if } \\ t \\in \\, ]-\\infty , 0] ,\\end{array}\\right.","}$ which coincides with $\\widetilde{c} \\cdot \\varphi _{0}(t)$ for the generator $\\varphi _{0}(t)$ from (REF ) which generates the reverse Kullback-Leibler divergence (reverse relative entropy) given in (REF ) with $\\widetilde{c}=1$ ; the first line in (REF ) can be proved by $& & \\hspace{-19.91684pt}\\varphi _{0,\\widetilde{c}}(t) := \\varphi _{0,\\widetilde{c}}^{(0)}(t) :=t \\cdot F_{0,\\widetilde{c}}\\left(t\\right)-\\int \\displaylimits _{0}^{F_{0,\\widetilde{c}}\\left(t\\right)} F_{0,\\widetilde{c}}^{-1}(u) du\\nonumber \\\\& & \\hspace{-19.91684pt}= t \\cdot \\widetilde{c} \\cdot \\Big (1-\\frac{1}{t}\\Big )- (- \\widetilde{c}) \\cdot \\log \\left(1-\\frac{1}{\\widetilde{c}} \\cdot \\left[ \\widetilde{c} \\cdot \\Big ( 1-\\frac{1}{t}\\Big ) \\right]\\right)= \\widetilde{c} \\cdot \\left(- \\log t + t -1 \\right),\\qquad t \\in ]0,\\infty [ \\, .$ Notice that one has $\\varphi _{0,\\widetilde{c}}(1) = 0$ , $\\varphi _{0,\\widetilde{c}}(\\infty ) = \\infty $ , $\\varphi _{0,\\widetilde{c}}^{\\prime }(1) = 0$ , $\\varphi _{0,\\widetilde{c}}^{\\prime }(0) = -\\infty $ and $\\varphi _{0,\\widetilde{c}}^{\\prime }(\\infty ) = \\widetilde{c}$ .","Example 40 (a) For the remaining case $\\gamma =1$ , $\\widetilde{c} \\in ]0,\\infty [$ and $]a_{F_{1,\\widetilde{c}}},b_{F_{1,\\widetilde{c}}}[ = ]0,\\infty [$ we define $F_{1,\\widetilde{c}}(t) &:=&{\\left\\lbrace \\begin{array}{ll}\\widetilde{c} \\cdot \\log t =\\lim _{\\gamma \\rightarrow 1} F_{\\gamma ,\\widetilde{c}}(t), \\qquad \\textrm {if } \\ t \\in \\, ]0,\\infty [, \\\\-\\infty , \\hspace{108.12054pt} \\textrm {if } \\ t \\in ]-\\infty ,0].","\\\\\\end{array}\\right.","}\\nonumber $ Clearly, $\\mathcal {R}(F_{1,\\widetilde{c}})= ]-\\infty ,\\infty [$ .","Moreover, $F_{1,\\widetilde{c}}(\\cdot )$ is strictly increasing and smooth on $]0,\\infty [$ , and hence, $F_{\\gamma ,\\widetilde{c}} \\in {F}$ .","Since $F_{1,\\widetilde{c}}(1)=0$ , let us first choose the natural anchor point $c:=0$ , which leads to $]\\lambda _{-},\\lambda _{+}[= int(\\mathcal {R}(F_{1,\\widetilde{c}})) = ]-\\infty ,\\infty [$ and $]t_{-}^{sc},t_{+}^{sc}[ = ]0,\\infty [$ .","By using $F_{1,\\widetilde{c}}^{-1}(x) = \\exp (\\frac{x}{\\widetilde{c}})$ for $x \\in \\mathcal {R}(F_{1,\\widetilde{c}})$ , we can derive from formula (REF ) (see also (REF )) $& & \\hspace{-19.91684pt} \\Lambda _{1,\\widetilde{c}}(z) := \\Lambda _{1,\\widetilde{c}}^{(0)}(z) :=\\int \\displaylimits _{0}^{z} F_{1,\\widetilde{c}}^{-1}(u) \\, du= \\widetilde{c} \\cdot \\left( \\exp \\Big (\\frac{z}{\\widetilde{c}}\\Big ) - 1 \\right),\\ \\ z \\in ]-\\infty ,\\infty [ .$ Notice that $\\Lambda _{1,\\widetilde{c}}(0) = 0$ , $\\Lambda _{1,\\widetilde{c}}(-\\infty ) = - \\widetilde{c}$ , $\\Lambda _{1,\\widetilde{c}}(\\infty ) = \\infty $ , $\\Lambda _{1,\\widetilde{c}}^{\\prime }(-\\infty ) = 0$ and $\\Lambda _{0,\\widetilde{c}}^{\\prime }(\\infty ) = \\infty $ .","Moreover, from formula (REF ) (see also ()) we can deduce $\\varphi _{1,\\widetilde{c}}(t) := \\varphi _{1,\\widetilde{c}}^{(0)}(t)\\hspace{-5.69046pt} &:=& \\hspace{-5.69046pt}{\\left\\lbrace \\begin{array}{ll}\\widetilde{c} \\cdot \\left( t \\cdot \\log t + 1 - t \\right)\\ \\in \\ [0,\\infty [,\\qquad \\quad \\textrm {if }t \\in \\, ]0,\\infty [,\\\\1, \\hspace{150.79968pt} \\textrm {if } \\ t = 0, \\\\\\infty , \\hspace{145.10922pt} \\textrm {if } \\ t \\in \\, ]-\\infty ,0[ ,\\end{array}\\right.","}$ which coincides with $\\widetilde{c} \\cdot \\varphi _{1}(t)$ for the generator $\\varphi _{1}(t)$ from (REF ) which generates the Kullback-Leibler divergence (relative entropy) given in (REF ) with $\\widetilde{c}=1$ ; the first line in (REF ) can be proved by $& & \\hspace{-19.91684pt}\\varphi _{1,\\widetilde{c}}(t) := \\varphi _{1,\\widetilde{c}}^{(0)}(t) :=t \\cdot F_{1,\\widetilde{c}}\\left(t\\right)-\\int \\displaylimits _{0}^{F_{1,\\widetilde{c}}\\left(t\\right)} F_{1,\\widetilde{c}}^{-1}(u) du\\nonumber \\\\& & \\hspace{-19.91684pt}= t \\cdot \\widetilde{c} \\cdot \\log t- \\widetilde{c} \\cdot \\left( \\exp \\left(\\frac{1}{\\widetilde{c}} \\cdot \\left[ \\widetilde{c} \\cdot \\log t \\right]\\right) - 1 \\right)= \\widetilde{c} \\cdot \\left( t \\cdot \\log t + 1 - t \\right),\\qquad t \\in ]0,\\infty [ \\, ,$ Notice that one has $\\varphi _{1,\\widetilde{c}}(1) = 0$ , $\\varphi _{1,\\widetilde{c}}(\\infty ) = \\infty $ , $\\varphi _{1,\\widetilde{c}}^{\\prime }(1) = 0$ , $\\varphi _{1,\\widetilde{c}}^{\\prime }(0) = -\\infty $ and $\\varphi _{1,\\widetilde{c}}^{\\prime }(\\infty ) = \\infty $ .","As an application of the reciprocity considerations of Remark REF , it is straightforward to see that $\\widetilde{F}_{1,\\widetilde{c}}(t) =- \\Lambda _{1,\\widetilde{c}}(F_{1,\\widetilde{c}}(\\frac{1}{t}))= F_{0,\\widetilde{c}}(t)$ for all $t \\in ]0,\\infty [$ .","(b) For the choice $\\widetilde{c}=1$ , let us now fix a general anchor point $c \\in \\mathcal {R}(F_{1,\\widetilde{c}})= ]-\\infty ,\\infty [$ (rather than $c=0$ ), which leads to $]\\lambda _{-},\\lambda _{+}[= int(\\mathcal {R}(F_{1,1}))- c = ]-\\infty ,\\infty [$ and $]t_{-}^{sc},t_{+}^{sc}[ = ]1+a_{F_{1,1}}-F_{1,1}^{-1}(c),1+b_{F_{1,1}}-F_{1,1}^{-1}(c)[ \\,= \\, ]1-e^{c},\\infty [$ .","Accordingly, the formula (REF ) (see also (REF )) leads to $& & \\hspace{-19.91684pt} \\Lambda _{1,1}(z) := \\Lambda _{1,1}^{(c)}(z) :=\\int \\displaylimits _{0}^{z} F_{1,1}^{-1}(u+c) du+ z \\cdot (1-F_{1,1}^{-1}(c))\\nonumber \\\\& & \\hspace{59.75095pt}= e^{c} \\cdot \\left( e^{z} - 1 \\right) + z \\cdot (1- e^{c}),\\qquad z \\in ]-\\infty ,\\infty [, \\qquad \\ $ for which there holds $\\Lambda _{1,1}^{(c)}(0) = 0$ , $\\Lambda _{1,1}^{(c)}(-\\infty ) =\\infty \\cdot {1}_{]0,\\infty [}(c)- \\infty \\cdot {1}_{]-\\infty ,0[}(c)- 1 \\cdot {1}_{\\lbrace 0\\rbrace }(c)$ , $\\Lambda _{1,1}^{(c)}(\\infty ) = \\infty $ , $\\Lambda _{1,1}^{(c) \\prime }(-\\infty ) = 1- e^{c}$ and $\\Lambda _{1,1}^{(c) \\prime }(\\infty ) = \\infty $ .","Moreover, from formula (REF ) (see also ()) we can deduce $\\varphi _{1,1}(t) := \\varphi _{1,1}^{(c)}(t)\\hspace{-5.69046pt} &:=& \\hspace{-5.69046pt}{\\left\\lbrace \\begin{array}{ll}(t + e^{c} -1) \\cdot [\\log (t + e^{c} -1) -c] + 1 - t\\ \\in \\ [0,\\infty [,\\qquad \\quad \\textrm {if }t \\in \\, ]1-e^{c},\\infty [,\\\\e^{c}, \\hspace{240.42569pt} \\textrm {if } \\ t = 1-e^{c}, \\\\\\infty , \\hspace{239.00298pt} \\textrm {if } \\ t \\in \\, ]-\\infty ,1-e^{c}[ ;\\end{array}\\right.","}$ the first line in (REF ) can be proved by $& & \\hspace{-19.91684pt}\\varphi _{1,1}(t) := \\varphi _{1,1}^{(c)}(t) :=(t+F_{1,1}^{-1}(c)-1) \\cdot [ F_{1,1}\\left(t+F_{1,1}^{-1}(c)-1 \\right) - c ]-\\int \\displaylimits _{0}^{F_{1,1}\\left(t+F_{1,1}^{-1}(c)-1 \\right) - c}F_{1,1}^{-1}(u+c) du\\nonumber \\\\& & \\hspace{-19.91684pt}= (t + e^{c} -1) \\cdot [\\log (t + e^{c} -1) -c]- e^{c} \\cdot \\Big \\lbrace \\exp [\\log (t + e^{c} -1) -c] - 1 \\Big \\rbrace \\nonumber \\\\& & \\hspace{-19.91684pt}= (t + e^{c} -1) \\cdot [\\log (t + e^{c} -1) -c] + 1 - t ,\\qquad t \\in ]1-e^{c},\\infty [\\, .$ Clearly, one has $\\varphi _{1,1}^{(c)}(1) = 0$ , $\\varphi _{1,1}^{(c)}(\\infty ) = \\infty $ , $\\varphi _{1,1}^{(c) \\prime }(1) = 0$ , $\\varphi _{1,1}^{(c) \\prime }(1-e^{c}) = -\\infty $ and $\\varphi _{1,1}^{(c) \\prime }(\\infty ) = \\infty $ .","The corresponding divergence $D_{\\varphi _{1,1}^{(c)}}(\\mathbb {Q},\\mathbb {P})$ has been recently used in Broniatowski et al.","[63] for the important task of testing mixtures of probability distributions; in fact, in order to get considerable comfort in testing mixture-type hypotheses against corresponding marginal-type alternatives, they employ choices $c>0$ since then $\\varphi _{1,1}^{(c)}(t)$ is finite especially for some range of negative values $t<0$ .","The latter feature is also valid for the divergence generator $\\varphi _{bw,\\beta ,\\widetilde{c}}$ in the next example (cf.","(REF ) below).","Example 41 For $\\beta \\in \\, ]0,1]$ , $\\widetilde{c} \\in \\, ]0,\\infty [$ and $]a_{F_{bw,\\beta ,\\widetilde{c}}},b_{F_{bw,\\beta ,\\widetilde{c}}}[ \\,= \\, ]1-\\frac{1}{\\beta },\\infty [$ we define $F_{bw,\\beta ,\\widetilde{c}}(t) &:=&{\\left\\lbrace \\begin{array}{ll}\\frac{\\widetilde{c}}{2\\beta } \\cdot \\Big (1-\\frac{1}{(\\beta \\cdot t + 1 - \\beta )^2}\\Big ), \\qquad \\textrm {if } \\ t \\in \\, ]1-\\frac{1}{\\beta },\\infty [, \\\\- \\infty , \\hspace{93.89418pt} \\textrm {if } \\ t \\in \\, ]-\\infty ,1-\\frac{1}{\\beta }].\\end{array}\\right.","}\\nonumber $ Clearly, $\\mathcal {R}(F_{bw,\\beta ,\\widetilde{c}})= \\big ]-\\infty ,\\frac{\\widetilde{c}}{2\\beta }\\big [$ and $0 \\in int(\\mathcal {R}(F_{bw,\\beta ,\\widetilde{c}}))$ .","Moreover, $F_{bw,\\beta ,\\widetilde{c}}(\\cdot )$ is strictly increasing and smooth on $]1-\\frac{1}{\\beta },\\infty [$ , and thus, $F_{bw,\\beta ,\\widetilde{c}} \\in {F}$ .","Since $F_{bw,\\beta ,\\widetilde{c}}(1)=0$ , let us choose the natural anchor point $c:=0$ , which leads to $]\\lambda _{-},\\lambda _{+}[ \\, = int(\\mathcal {R}(F_{bw,\\beta ,\\widetilde{c}}))= \\big ]-\\infty ,\\frac{\\widetilde{c}}{2\\beta }\\big [$ and $]t_{-}^{sc},t_{+}^{sc}[ \\, = \\,]a_{F_{bw,\\beta ,\\widetilde{c}}},b_{F_{bw,\\beta ,\\widetilde{c}}}[\\, = \\, ]1-\\frac{1}{\\beta },\\infty [$ .","By using $F_{bw,\\beta ,\\widetilde{c}}^{-1}(x) =\\frac{1}{\\beta } \\cdot \\Big \\lbrace \\frac{1}{\\sqrt{1-2\\beta \\cdot x/\\widetilde{c}}} + \\beta -1 \\Big \\rbrace $ for $x \\in int(\\mathcal {R}(F_{bw,\\beta ,\\widetilde{c}}))$ , we can derive from formula (REF ) (see also (REF )) for all $\\beta \\in \\, ]0,1]$ and $z \\in \\mathbb {R}$ $\\Lambda _{bw,\\beta ,\\widetilde{c}}(z) :=\\Lambda _{bw,\\beta ,\\widetilde{c}}^{(0)}(z) &=&{\\left\\lbrace \\begin{array}{ll}-(\\frac{1}{\\beta }-1) \\cdot z + \\frac{\\widetilde{c}}{\\beta ^{2}} \\cdot \\Big \\lbrace 1 - \\sqrt{1-\\frac{2\\beta }{\\widetilde{c}} \\cdot z} \\ \\Big \\rbrace ,\\qquad \\textrm {if } z \\in \\big ]-\\infty ,\\frac{\\widetilde{c}}{2\\beta } \\big ], \\\\\\infty , \\hspace{176.407pt} \\textrm {else} .\\end{array}\\right.","}$ Notice that $\\Lambda _{bw,\\beta ,\\widetilde{c}}(0) = 0$ .","Moreover, $\\Lambda _{bw,\\beta ,\\widetilde{c}}(-\\infty ) = \\infty $ , $\\Lambda _{bw,\\beta ,\\widetilde{c}}(\\frac{\\widetilde{c}}{2\\beta })= \\frac{\\widetilde{c} \\cdot (\\beta +1)}{2 \\beta ^{2}}$ , $\\Lambda _{bw,\\beta ,\\widetilde{c}}^{\\prime }(-\\infty ) = - \\frac{1-\\beta }{\\beta } <0$ and $\\Lambda _{bw,\\beta ,\\widetilde{c}}^{\\prime }(\\frac{\\widetilde{c}}{2\\beta }) = \\infty $ .","Furthermore, from formula (REF ) (see also ()) we can straightforwardly deduce for all $t \\in \\mathbb {R}$ $\\varphi _{bw,\\beta ,\\widetilde{c}}(t) := \\varphi _{bw,\\beta ,\\widetilde{c}}^{(0)}(t)\\hspace{-5.69046pt} &:=& \\hspace{-5.69046pt}{\\left\\lbrace \\begin{array}{ll}\\widetilde{c} \\cdot \\frac{(t-1)^{2}}{2(\\beta \\cdot t +1 -\\beta )}\\ \\in \\ [0,\\infty [,\\qquad \\quad \\textrm {if }t \\in \\, ]1-\\frac{1}{\\beta },\\infty [,\\\\\\infty , \\hspace{120.92421pt} \\textrm {if } \\ t \\in \\, ]-\\infty ,1-\\frac{1}{\\beta }] .\\end{array}\\right.","}$ Note that $1-\\frac{1}{\\beta } <0$ so that negative $t$ are allowed here.","For $t\\ge 0$ , $\\varphi _{bw,\\beta ,\\widetilde{c}}(t)$ is known as Rukhin's generator (cf.","[312], see e.g.","also Marhuenda et al.","[247], Pardo [282]).","Obviously, one has $\\varphi _{bw,\\beta ,\\widetilde{c}}(1) = 0$ , $\\varphi _{bw,\\beta ,\\widetilde{c}}^{\\prime }(1) = 0$ , $\\varphi _{bw,\\beta ,\\widetilde{c}}^{\\prime }(1-\\frac{1}{\\beta }) = -\\infty $ and $\\varphi _{bw,\\beta ,\\widetilde{c}}^{\\prime }(\\infty ) = \\frac{\\widetilde{c}}{2\\beta }$ .","From the generator $\\varphi _{bw,\\beta ,\\widetilde{c}}$ given in (REF ), we build the corresponding divergence (cf.","(REF )) $& &\\hspace{-45.52458pt}D_{\\varphi _{bw,\\beta ,\\widetilde{c}}}(\\mathbf {Q},\\mathbf {P})= \\widetilde{c} \\cdot \\sum _{k=1}^{K} p_{k} \\cdot \\frac{(\\frac{q_{k}}{p_{k}}-1)^{2}}{2(\\beta \\cdot \\frac{q_{k}}{p_{k}} +1 -\\beta )}\\nonumber \\\\& & \\hspace{-45.52458pt}= \\frac{\\widetilde{c}}{2} \\cdot \\sum \\limits _{k=1}^{K}\\frac{(q_{k}-p_{k})^{2}}{\\beta \\cdot q_{k} + (1 -\\beta )\\cdot p_{k}},\\hspace{42.67912pt}\\textrm {if \\mathbf {P} \\in \\mathbb {R}^{K} and \\mathbf {Q} \\in \\mathbb {R}^{K}with \\mathbf {Q} \\in \\, ] \\, \\mathbf {P} \\cdot (1-\\frac{1}{\\beta }),\\infty [ component-wise;}$ for the special subcase $\\widetilde{c}=1$ and $\\mathbf {Q} \\in \\mathbb {R}_{> 0}^{K}$ , $D_{\\varphi _{bw,\\beta ,1}}(\\mathbf {Q},\\mathbf {P})$ can be interpreted as — “non-probability version” of — the well-known blended weight chi-square divergence of Lindsay [221] (see e.g.","also Basu & Lindsay [35], Györfy & Vajda [148], Basu et al.","[36]).","The special case $\\widetilde{c}=1$ and $\\beta =\\frac{1}{2}$ for probability vectors, i.e.","$D_{\\varphi _{bw,1/2,1}}({Q},{P})$ , is equal to (a multiple of the matrix-vector-converted (cf.","Remark REF )) Sanghvi’s genetic difference measure [316] and equal to the double of the so-called (squared) Vincze-Le Cam distance (cf.","Vincze [384], Le Cam [212], see also e.g.","Topsoe [360] — who used the alternative naming triangular discrimination — and Vajda [373]); this divergence $D_{\\varphi _{bw,1/2,1}}({Q},{P})$ has been used e.g.","in Liu et al.","[227] for a machine learning context of detecting salient objects, where ${Q}$ and ${P}$ are appropriate histograms of RGB color.","Remark 42 (a) By straightforward calculations, one can show that $\\varphi _{bw,1,\\widetilde{c}}$ (i.e.","with the choice $\\beta =1$ ) is equal to the $\\widetilde{c}-fold$ power-divergence generator $\\varphi _{\\gamma ,\\widetilde{c}} = \\widetilde{c} \\cdot \\varphi _{\\gamma }$ (cf.","(REF )) with $\\gamma =-1$ ; the corresponding divergence $D_{\\varphi _{bw,1,\\widetilde{c}}}(\\mathbf {Q},\\mathbf {P})$ is thus equal to the power divergence $D_{\\widetilde{c} \\cdot \\varphi _{-1}}(\\mathbf {Q},\\mathbf {P})$ (cf.","(REF )) which is nothing but the — “non-probability version” — of Neyman's chi-square divergence.","(b) For the case $\\beta =0$ — which has been excluded in Example REF for technical brevity — the divergence generator $\\varphi _{bw,0,\\widetilde{c}}$ corresponds to $\\widetilde{c}-fold$ power-divergence generator $\\varphi _{\\gamma ,\\widetilde{c}}$ with $\\gamma =2$ ; the corresponding divergence $D_{\\varphi _{bw,0,\\widetilde{c}}}(\\mathbf {Q},\\mathbf {P})$ is thus equal to the power divergence $D_{\\widetilde{c} \\cdot \\varphi _{2}}(\\mathbf {Q},\\mathbf {P})$ (cf.","(REF )) which is nothing but the — “non-probability version” — of Pearson's (i.e.","the classical) chi-square divergence.","Example 43 Let us give an interesting generalization of the Kullback-Leibler case of Example REF (a).","For $\\widetilde{c} >0$ and $\\alpha \\in \\, ]-1,0[ \\ \\cup \\ ]0,\\infty [$ let us define $F_{gKL,\\alpha ,\\widetilde{c}}(t) &:=&{\\left\\lbrace \\begin{array}{ll}\\widetilde{c} \\cdot \\log \\left( \\frac{(1+\\alpha ) \\cdot t}{1+ \\alpha \\cdot t} \\right), \\qquad \\textrm {if \\lbrace \\alpha \\in \\, ]0,\\infty [ and t \\in \\, ]0,\\infty [ \\rbrace or \\lbrace \\alpha \\in \\, ]-1,0[ and t \\in \\, ]0,-\\frac{1}{\\alpha }[ \\rbrace ,} \\\\- \\infty , \\hspace{71.13188pt} \\textrm {if \\alpha \\in \\, ]-1,0[ \\ \\cup \\ ]0,\\infty [ andt \\in \\, ]-\\infty ,0],} \\\\\\infty , \\hspace{78.52945pt} \\textrm {if \\alpha \\in \\, ]-1,0[ and t \\in \\, [-\\frac{1}{\\alpha },\\infty [,}\\end{array}\\right.","}\\nonumber $ (notice that $\\lim _{\\alpha \\rightarrow 0_{+}} F_{gKL,\\alpha ,\\widetilde{c}}(t)= F_{1,\\widetilde{c}}(t)$ , cf.","Example REF (a)).","Clearly, $]a_{F_{gKL,\\alpha ,\\widetilde{c}}},b_{F_{gKL,\\alpha ,\\widetilde{c}}}[ \\, := \\, ]0,\\infty [$ for $\\alpha \\in \\, ]0,\\infty [$ and $]a_{F_{gKL,\\alpha ,\\widetilde{c}}},b_{F_{gKL,\\alpha ,\\widetilde{c}}}[ \\, := \\, ]0,-\\frac{1}{\\alpha }[$ for $\\alpha \\in \\, ]-1,0[$ .","Moreover, $\\mathcal {R}(F_{gKL,\\alpha ,\\widetilde{c}})=\\, ]-\\infty , \\widetilde{c} \\cdot \\log (1+ \\frac{1}{\\alpha })[$ for $\\alpha \\in \\, ]0,\\infty [$ and $\\mathcal {R}(F_{gKL,\\alpha ,\\widetilde{c}})= \\, ]-\\infty ,\\infty [$ for $\\alpha \\in \\, ]-1,0[$ .","Furthermore, $F_{gKL,\\alpha ,\\widetilde{c}}(\\cdot )$ is strictly increasing and smooth on the respective $]a_{F_{gKL,\\alpha ,\\widetilde{c}}},b_{F_{gKL,\\alpha ,\\widetilde{c}}}[$ , and thus, $F_{gKL,\\alpha ,\\widetilde{c}} \\in {F}$ .","Since $F_{gKL,\\alpha ,\\widetilde{c}}(1)=0$ , let us choose the natural anchor point $c:=0$ , which leads to $]\\lambda _{-},\\lambda _{+}[ \\, = int(\\mathcal {R}(F_{gKL,\\alpha ,\\widetilde{c}})) = \\,]-\\infty , \\widetilde{c} \\cdot \\log (1+ \\frac{1}{\\alpha })[$ and $]t_{-}^{sc},t_{+}^{sc}[ \\, = \\, ]0,\\infty [$ for the case $\\alpha \\in \\, ]0,\\infty [$ , respectively, to $]\\lambda _{-},\\lambda _{+}[ \\, = int(\\mathcal {R}(F_{gKL,\\alpha ,\\widetilde{c}})) = \\, ]-\\infty ,\\infty [$ and $]t_{-}^{sc},t_{+}^{sc}[ \\, = \\, ]0,-\\frac{1}{\\alpha }[$ for the case $\\alpha \\in \\, ]-1,0[$ .","By employing $F_{gKL,\\alpha ,\\widetilde{c}}^{-1}(x) =\\frac{1}{(1+\\alpha ) \\cdot e^{-x/\\widetilde{c}} - \\alpha }$ for $x \\in ]\\lambda _{-},\\lambda _{+}[$ , one can deduce from formula (REF ) (see also (REF )) $& & \\Lambda _{gKL,\\alpha ,\\widetilde{c}}(z) := \\Lambda _{gKL,\\alpha ,\\widetilde{c}}^{(0)}(z)\\nonumber \\\\& & :={\\left\\lbrace \\begin{array}{ll}\\int \\displaylimits _{0}^{z} F_{gKL,\\alpha ,\\widetilde{c}}^{-1}(u) \\, du= - \\frac{\\widetilde{c}}{\\alpha } \\cdot \\log ((1+\\alpha ) - \\alpha \\cdot e^{z/\\widetilde{c}}),\\qquad \\textrm {if \\alpha \\in \\, ]0,\\infty [ and z \\in \\, ]-\\infty , \\widetilde{c} \\cdot \\log (1+ \\frac{1}{\\alpha })[}, \\\\\\int \\displaylimits _{0}^{z} F_{gKL,\\alpha ,\\widetilde{c}}^{-1}(u) \\, du= - \\frac{\\widetilde{c}}{\\alpha } \\cdot \\log ((1+\\alpha ) - \\alpha \\cdot e^{z/\\widetilde{c}}),\\qquad \\textrm {if \\alpha \\in \\, ]-1,0[ and z \\in \\, ]-\\infty , \\infty [}, \\\\\\infty , \\hspace{217.6634pt}\\textrm {if \\alpha \\in \\, ]0,\\infty [ and z \\in \\, [\\widetilde{c} \\cdot \\log (1+ \\frac{1}{\\alpha }), \\infty [}, \\\\\\end{array}\\right.","}$ for which there holds $\\Lambda _{gKL,\\alpha ,\\widetilde{c}}(0) = 0$ and $\\Lambda _{gKL,\\alpha ,\\widetilde{c}}(-\\infty ) = - \\frac{\\widetilde{c}}{\\alpha } \\cdot \\log (1+\\alpha )$ for $\\alpha \\in \\, ]-1,0[ \\ \\cup \\ ]0,\\infty [$ , as well as $\\Lambda _{gKL,\\alpha ,\\widetilde{c}}(\\widetilde{c} \\cdot \\log (1+ \\frac{1}{\\alpha })) = \\infty $ for $\\alpha \\in \\, ]0,\\infty [$ and $\\Lambda _{gKL,\\alpha ,\\widetilde{c}}(\\infty ) = \\infty $ for $\\alpha \\in \\, ]-1,0[$ .","The corresponding derivative satisfies $\\Lambda _{gKL,\\alpha ,\\widetilde{c}}^{\\prime }(- \\infty ) = 0$ for $\\alpha \\in \\, ]-1,0[ \\ \\cup \\ ]0,\\infty [$ , as well as $\\Lambda _{gKL,\\alpha ,\\widetilde{c}}^{\\prime }(\\widetilde{c}\\cdot \\log (1+ \\frac{1}{\\alpha })) = \\infty $ for $\\alpha \\in \\, ]0,\\infty [$ and $\\Lambda _{gKL,\\alpha ,\\widetilde{c}}^{\\prime }(\\infty ) = - \\frac{1}{\\alpha }$ for $\\alpha \\in \\, ]-1,0[$ .","Furthermore, from formula (REF ) (see also ()) one can derive $& & \\hspace{-19.91684pt} \\varphi _{gKL,\\alpha ,\\widetilde{c}}(t) := \\varphi _{gKL,\\alpha ,\\widetilde{c}}^{(0)}(t)\\nonumber \\\\& & \\hspace{-19.91684pt} :={\\left\\lbrace \\begin{array}{ll}\\widetilde{c} \\cdot \\left[ \\, t \\cdot \\log t + (t+\\frac{1}{\\alpha }) \\cdot \\log \\Big ( \\frac{1+\\alpha }{1+\\alpha \\cdot t} \\Big ) \\, \\right]\\ \\in \\ [0,\\infty [,\\quad \\textrm {if \\lbrace \\alpha \\in \\, ]0,\\infty [ and t \\in \\, ]0,\\infty [ \\rbrace or \\lbrace \\alpha \\in \\, ]-1,0[ and t \\in \\, ]0,-\\frac{1}{\\alpha }[ \\rbrace ,}\\\\\\frac{\\widetilde{c}}{\\alpha } \\cdot \\log (1+\\alpha )\\ \\in \\ ]0,\\infty [, \\hspace{106.69783pt}\\textrm {if \\alpha \\in \\, ]-1,0[ \\ \\cup \\ ]0,\\infty [ andt =0,} \\\\\\infty , \\hspace{197.74655pt}\\textrm {if \\alpha \\in \\, ]-1,0[ \\ \\cup \\ ]0,\\infty [ andt \\in \\, ]-\\infty ,0[,} \\\\\\infty , \\hspace{197.74655pt}\\textrm {if \\alpha \\in \\, ]-1,0[ andt \\in \\, [-\\frac{1}{\\alpha },\\infty [;}\\end{array}\\right.","}$ the first line in (REF ) can be proved by $& & \\hspace{-19.91684pt}\\varphi _{gKL,\\alpha ,\\widetilde{c}}(t) := \\varphi _{gKL,\\alpha ,\\widetilde{c}}^{(0)}(t) :=t \\cdot F_{gKL,\\alpha ,\\widetilde{c}}\\left(t\\right)-\\int \\displaylimits _{0}^{F_{gKL,\\alpha ,\\widetilde{c}}\\left(t\\right)}F_{gKL,\\alpha ,\\widetilde{c}}^{-1}(u) \\, du\\nonumber \\\\& & \\hspace{-19.91684pt}= \\widetilde{c} \\cdot t \\cdot \\log \\left( \\frac{(1+\\alpha ) \\cdot t}{1+ \\alpha \\cdot t} \\right)+ \\frac{\\widetilde{c}}{\\alpha } \\cdot \\log \\left( (1 +\\alpha ) -\\alpha \\cdot \\exp \\left[ \\log \\left( \\frac{(1+ \\alpha ) \\cdot t}{1+ \\alpha \\cdot t} \\right) \\right] \\right)\\nonumber \\\\& & \\hspace{-19.91684pt}= \\widetilde{c} \\cdot \\left[ \\, t \\cdot \\log t+ t \\cdot \\log \\Big ( \\frac{1+\\alpha }{1+ \\alpha \\cdot t} \\Big )+ \\frac{1}{\\alpha } \\cdot \\log \\Big ( \\frac{1+\\alpha }{1+ \\alpha \\cdot t} \\Big ) \\, \\right].$ Obviously, one has $\\varphi _{gKL,\\alpha ,\\widetilde{c}}(1) = 0$ , $\\varphi _{gKL,\\alpha ,\\widetilde{c}}^{\\prime }(1) = 0$ , $\\varphi _{gKL,\\alpha ,\\widetilde{c}}^{\\prime }(0) = -\\infty $ for $\\alpha \\in \\, ]-1,0[ \\ \\cup \\ ]0,\\infty [$ .","Moreover, for $\\alpha \\in \\, ]0,\\infty [$ there holds $\\varphi _{gKL,\\alpha ,\\widetilde{c}}(\\infty ) = \\infty $ , and $\\varphi _{gKL,\\alpha ,\\widetilde{c}}^{\\prime }(\\infty ) = \\widetilde{c} \\cdot \\log (1+\\frac{1}{\\alpha })$ , whereas for $\\alpha \\in \\, ]-1,0[$ we obtain $\\varphi _{gKL,\\alpha ,\\widetilde{c}}(-\\frac{1}{\\alpha }) = \\infty $ , and $\\varphi _{gKL,\\alpha ,\\widetilde{c}}^{\\prime }(-\\frac{1}{\\alpha }) = \\infty $ .","From the generator $\\varphi _{gKL,\\alpha ,\\widetilde{c}}$ given in (REF ), we build the corresponding divergence (cf.","(REF )) $& &\\hspace{-42.67912pt}D_{\\varphi _{gKL,\\alpha ,\\widetilde{c}}}(\\mathbf {Q},\\mathbf {P})= \\widetilde{c} \\cdot \\Big \\lbrace \\sum \\limits _{k=1}^{K}q_{k} \\cdot \\log \\Big (\\frac{q_{k}}{(1-\\frac{1}{1+\\alpha }) \\cdot q_{k} + \\frac{1}{1+\\alpha } \\cdot p_{k}} \\Big )+ \\frac{1}{\\alpha } \\cdot \\sum \\limits _{k=1}^{K} p_{k} \\cdot \\log \\Big (\\frac{p_{k}}{(1-\\frac{1}{1+\\alpha }) \\cdot q_{k} + \\frac{1}{1+\\alpha } \\cdot p_{k}} \\Big ) \\Big \\rbrace ,\\\\& & \\textrm {if \\lbrace \\alpha \\in \\, ]0,\\infty [,\\mathbf {P} \\in \\mathbb {R}_{> 0}^{K} and\\mathbf {Q} \\in \\mathbb {R}_{\\ge 0}^{K} \\rbrace or \\lbrace \\alpha \\in \\, ]-1,0[,\\mathbf {P} \\in \\mathbb {R}_{> 0}^{K} and\\mathbf {Q} \\in \\mathbb {R}_{\\ge 0}^{K}with \\mathbf {Q} \\le - \\frac{1}{\\alpha } \\cdot \\mathbf {P} \\rbrace .}","\\nonumber $ Notice that the symmetry $D_{\\varphi _{gKL,\\alpha ,\\widetilde{c}}}(\\mathbf {Q},\\mathbf {P})= D_{\\varphi _{gKL,\\alpha ,\\widetilde{c}}}(\\mathbf {P},\\mathbf {Q})$ generally holds only if $\\mathbf {P}, \\mathbf {Q} \\in \\mathbb {R}_{> 0}^{K}$ and $\\alpha =1$ ; indeed, this special case leads to $\\varphi _{snKL,\\widetilde{c}}(t):= \\varphi _{gKL,1,\\widetilde{c}}(t)\\hspace{-5.69046pt} &:=& \\hspace{-5.69046pt}{\\left\\lbrace \\begin{array}{ll}\\widetilde{c} \\cdot \\left[ \\, t \\cdot \\log t + (t+1) \\cdot \\log \\Big ( \\frac{2}{t+1} \\Big ) \\, \\right]\\ \\in \\ [0,\\infty [,\\qquad \\quad \\textrm {if }t \\in \\, ]0,\\infty [,\\\\\\widetilde{c} \\cdot \\log 2, \\hspace{187.78836pt} \\textrm {if } \\ t = 0, \\\\\\infty , \\hspace{209.12791pt} \\textrm {if } \\ t \\in \\, ]-\\infty ,0[ ,\\end{array}\\right.","}$ and $D_{\\varphi _{snKL,\\widetilde{c}}}(\\mathbf {Q},\\mathbf {P}):= D_{\\varphi _{gKL,1,\\widetilde{c}}}(\\mathbf {Q},\\mathbf {P})= \\widetilde{c} \\cdot \\Big \\lbrace \\sum \\limits _{k=1}^{K} q_{k} \\cdot \\log \\Big (\\frac{2 q_{k}}{q_{k} + p_{k}} \\Big )+ \\sum \\limits _{k=1}^{K} p_{k} \\cdot \\log \\Big (\\frac{2 p_{k}}{q_{k} + p_{k}} \\Big ) \\Big \\rbrace ,\\qquad \\mathbf {P} \\in \\mathbb {R}_{> 0}^{K}, \\mathbf {Q} \\in \\mathbb {R}_{\\ge 0}^{K}.$ For the special subcase that $\\widetilde{c} =1$ and that $\\mathbf {P} = {P}$ , $\\mathbf {Q} = {Q}$ are probability vectors, the divergence (REF ) can be rewritten as sum of two Kullback-Leibler divergences (cf.","(REF )) $D_{\\varphi _{snKL,1}}({Q},{P})= D_{\\varphi _{1}}({Q},({Q}+{P})/2)+ D_{\\varphi _{1}}({P},({Q}+{P})/2),\\qquad {P} \\in \\mathbb {S}_{> 0}^{K}, {Q} \\in \\mathbb {S}_{\\ge 0}^{K},$ which is the well-known (cf.","Burbea & Rao [68], Lin [219], Pardo & Vajda [284], Topsoe [360], Endres & Schindelin [120], Vajda [373], Sason [317]) Jensen-Shannon divergence (being also called symmetrized and normalized Kullback-Leibler divergence, symmetrized and normalized relative entropy, capacitory discrimination); this is equal to the $(2\\log 2)-$ fold of a special (namely, equally-weighted two-population) case of the Sibson information radius of order 1 (cf.","[334]) which has also been addressed e.g.","by Rao [301] for genetic cluster analysis.","By the way, for $\\alpha >0$ the divergence $D_{\\varphi _{gKL,\\alpha ,\\widetilde{c}}}({Q},{P})$ can also be interpreted as a multiple of a special non-equally-weighted Sibson information radius of order 1.","In a context of comparison of — not necessarily connected — networks where ${Q}$ , ${P}$ are probability vectors derived from matrices (cf.","Remark REF ) which are transforms of corresponding graph invariants (e.g.","network portraits), the (matrix-equivalent of the) Jensen-Shannon divergence $D_{\\varphi _{snKL,1}}({Q},{P})$ is also called the network portrait divergence, cf.","Bagrow and Bollt [28].","There is a vast literature on recent applications of the Jensen-Shannon divergence, for instance it appears exemplarily in Kvitsiani et al.","[208] for finding connections between the circuit-level function of different interneuron types in regulating the flow of information and the behavioural functions served by the cortical circuits, in Xu et al.","(2014) for browsing and exploration of video sequences, in Jenkinson et al.","[168] for the fundamental understanding of the epigenome that leads to a powerful approach for studying its role in disease and aging, in Martin et al.","[250] for the implementation of an evolutionary-based global localization filter for mobile robots, in Suo et al.","[354] for the revelation of critical regulators of cell identity in mice, in Abante et al.","[2] for the detection of biologically significant differences in DNA methylation between alleles associated with local changes in genetic sequences — for a better understanding of the mechanism of complex human diseases, in Afek et al.","[5] for revealing mechanisms by which mismatches can recruit transcription factors for modulating replication and repair activities in cells, in Alaiz-Rodriguez & Parnell [10] for the quantification of stability in feature selection and ranking algorithms, in Biau et al.","[53] for generative adversarial networks (GANs) in artificial intelligence and machine learning, in Carre et al.","[74] for the standardization of brain magnetic resonance (MR) images, in Chakraborty et al.","[75] for hierarchical clustering in foreign exchange FOREX markets (e.g.","in periods of major international crises), in Chong et al.","[87] as part of a web-based platform for comprehensive analysis of microbiome data outputs, in Cui et al.","[101] for modelling latent friend recommendation in online social media, in Gholami & Hodtani [134] for refinements of safety-and-security-targeted location verification systems in wireless communication networks (e.g in Intelligent Transportation Systems (ITSs) and vehicular technology), in Guo & Yuan [146] for accurate abnormality classification in semi-supervised Wireless Capsule Endoscopy (WCE) for digestive system cancer diagnosis, in Jiang et al.","[169] for the training of deep neural discriminative and generative networks used for designing and evaluating photonic devices, in Kartal et al.","[186] for uncovering the relationship between some genomic features and cell type-specific methylome diversity, in Laszlovszky et al.","[210] for investigating mechanisms of basal forebrain neurons which modulate synaptic plasticity,cortical processing, brain states and oscillations, in Lawson et al.","[211] for the improved understanding of some genetic circuits that allow cancer cells to evade destruction by the host immune system, in Li et al.","[215] for the search of causes of the progressive neurodevelopmental disorder Rett syndrome, in Machado et al.","[239] for discovering relations between distinct RNA viruses (including SARS-CoV-2), in Mohammadi et al.","[261] for the identification of cell states and their underlying topology, in Mohanty et al.","[262] for the design of implantable nanophotonic (i.e.","chip-scale optical circuit type) silicon probes for sub-millisecond deep-brain optical stimulation — e.g.","for the purpose of gaining a deeper understanding of the neural code, in Perera et al.","[290] for the quantification of the level of rationality in supply chain networks, in Pierri et al.","[294] for the study of growth of malicious/misleading information in some social media diffusion networks, in Rabadan et al.","[299] for the identification of gene mutations that lead to the genesis and progression of tumors, in Reiter et al.","[306] for quantifying metastatic phylogenetic diversity, in Van de Sande et al.","[378] as part of a computational toolbox for single-cell gene regulatory network analysis, in Skinnider et al.","[337] for the prediction of the chemical structures of genomically encoded antibiotics — in order to find means against the looming global crisis of antibiotic resistance, in Tuo et al.","[367] for the detection of high-order single nucleotide polymorphism (SNP) interactions, in Uttam et al.","[370] for predicting the risk of colorectal cancer recurrence and inferring associated tumor microenvironment networks, in Zhang et al.","[421] for incipient fault (namely, crack) detection, in Zhi et al.","[425] for the strengthening of information-centric networks against interest flooding attack (IFAs), in Acera Mateos et al.","[3] for deep-learning classification of SARS-CoV-2 and co-infecting RNA viruses, in Avsec et al.","[24] for uncovering the motifs and syntax of cis-regulatory sequences in genomics data, in Barennes et al.","[32] for comparing the accuracy of current T cell receptor sequencing methods employed for the understanding of adaptive immune responses, in Chen et al.","[79] for clustering high-dimensional microbial data from RNA sequencing, in Chen et al.","[84] for investigating key aspects of effective vocal social communication, in Koldobskiy et al.","[193] for investigations of genetic and epigenetic drivers of paediatric acute lymphoblastic leukaemia, in McGinnis et al.","[258] for evaluating RNA sequencing of pooled blood cell samples, in Mühlroth & Grottke [268] for the detection of emerging trends and technologies through artificial intelligence techniques, in Necci et al.","[272] for the assessment of protein intrinsic disorder predictions, in Okada et al.","[277] for the identification of genetic factors that cause individual differences in whole lymphocyte profiles and their changes after vaccination, and in Zhang et al.","[422] for the learning of functional magnetic resonance imaging (fMRI) time-series in a brain disease diagnosis context.","Remark 44 Let us transform $\\varphi _{gSH,\\alpha }(t) := \\frac{1-t}{\\alpha } \\cdot \\log (1+\\alpha )- \\varphi _{gKL,\\alpha ,1}(t)= - t \\cdot \\log t+ \\frac{1}{\\alpha } \\cdot (1 + \\alpha \\cdot t) \\cdot \\log (1 + \\alpha \\cdot t)- \\frac{1}{\\alpha } \\cdot (1+\\alpha ) \\cdot t \\cdot \\log (1+\\alpha )$ (for $t \\in [0,1]$ ).","The function $\\varphi _{gSH,\\alpha }(\\cdot )$ is strictly concave on $[0,1]$ with $\\varphi _{gSH,\\alpha }(0)= \\varphi _{gSH,\\alpha }(1)=0$ .","Hence, for probability vectors ${Q} =(q_{k})_{k=1,\\ldots ,K}$ , the $\\varphi -$ entropy $\\sum _{k=1}^{K}\\varphi _{gSH,\\alpha }(q_{k})$ is Kapur’s [183] generalization of the Shannon entropy (which corresponds to $\\alpha =0$ in the limit) whose maximization has been connected with generalizations of the Bose-Einstein statistics and the Fermi-Dirac statistics e.g.","in Kapur & Kesavan [185].","Example 45 Let us fix any $z_{1},z_{2} \\in \\mathbb {R}$ , $p \\in ]0,1[$ which satisfy $z_{1} < 1 < z_{2}$ and $z_{1} \\cdot p + z_{2} \\cdot (1-p) =1$ (and thus $p= \\frac{z_{2} -1}{z_{2} - z_{1}}$ ).","On $]a_{F_{twop}},b_{F_{twop}}[ := ]z_{1},z_{2}[$ we define $F_{twop}(t) &:=& \\frac{1}{z_{2}-z_{1}} \\cdot \\log \\left( \\frac{(t-z_{1})\\cdot p}{(z_{2}-t)\\cdot (1- p)} \\right)\\nonumber \\\\&=&\\frac{1}{z_{2}-z_{1}} \\cdot \\log \\left( \\frac{(t-z_{1})\\cdot (z_{2} -1)}{(z_{2}-t)\\cdot (1-z_{1})} \\right) \\, ,\\qquad t \\in \\, ]z_{1},z_{2}[ ,\\nonumber $ where for the last equality we have used the above constraint (in order to obtain a two-parameter representation).","Straightforwardly, we have $\\mathcal {R}(F_{twop})= ]-\\infty ,\\infty [$ .","Moreover, $F_{twop}(\\cdot )$ is strictly increasing and smooth on $]0,\\infty [$ , and thus, $F_{twop} \\in {F}$ .","Since $F_{twop}(1)=0$ , let us choose the natural anchor point $c:=0$ , which leads to $]\\lambda _{-},\\lambda _{+}[= int(\\mathcal {R}(F_{snKL,\\widetilde{c}})) = ]-\\infty , \\infty [$ and $]t_{-}^{sc},t_{+}^{sc}[ = ]z_{1},z_{2}[$ .","By using $F_{twop}^{-1}(x) = \\frac{p \\cdot z_{1} + (1- p) \\cdot z_{2} \\cdot e^{(z_{2}-z_{1}) \\cdot x}}{p + (1- p) \\cdot e^{(z_{2}-z_{1}) \\cdot x}} \\, , \\qquad x \\in ]-\\infty , \\infty [,$ we derive from formula (REF ) (see also (REF )) $& & \\hspace{-19.91684pt} \\Lambda _{twop}(z) := \\Lambda _{twop}^{(0)}(z) :=\\int \\displaylimits _{0}^{z} F_{twop}^{-1}(u) du= \\log \\Big ( p \\cdot e^{z_{1} \\cdot z} + (1- p) \\cdot e^{z_{2} \\cdot z} \\Big ),\\qquad z \\in ]-\\infty , \\infty [ ,$ which has the properties $\\Lambda _{twop}(0) = 0$ , $\\Lambda _{twop}(-\\infty ) =\\infty \\cdot {1}_{]-\\infty ,0[}(z_{1})- \\infty \\cdot {1}_{]0,\\infty [}(z_{1})+ \\log p \\cdot {1}_{\\lbrace 0\\rbrace }(z_{1})$ , $\\Lambda _{twop}(\\infty ) = \\infty $ , $\\Lambda _{twop}^{\\prime }(- \\infty ) = z_{1}$ and $\\Lambda _{twop}^{\\prime }(\\infty ) = z_{2}$ .","Furthermore, from formula (REF ) (see also ()) we deduce $\\varphi _{twop}(t) := \\varphi _{twop}^{(0)}(t)\\hspace{-5.69046pt} &:=& \\hspace{-5.69046pt}{\\left\\lbrace \\begin{array}{ll}\\frac{t-z_{1}}{z_{2}-z_{1}} \\cdot \\log \\left( \\frac{(t-z_{1})\\cdot (z_{2}-1)}{(z_{2}-t)\\cdot (1-z_{1})} \\right)- \\log \\left( \\frac{z_{2}-1}{z_{2}-t} \\right)\\ \\in \\ [0,\\infty [,\\qquad \\quad \\textrm {if }t \\in \\, ]0,\\infty [,\\\\\\log \\left( \\frac{z_{2} - z_{1}}{z_{2} -1} \\right), \\hspace{219.08612pt} \\textrm {if } \\ t = z_{1}, \\\\\\log \\left( \\frac{z_{2} - z_{1}}{1 - z_{1}} \\right), \\hspace{219.08612pt} \\textrm {if } \\ t = z_{2}, \\\\\\infty , \\hspace{233.3125pt} \\textrm {if } \\ t \\in \\, ]-\\infty ,z_{1}[ \\, \\cup \\, ]z_{2}, \\infty [ ;\\end{array}\\right.","}$ the first line in (REF ) can be proved by $& & \\hspace{-19.91684pt}\\varphi _{twop}(t) := \\varphi _{twop}^{(0)}(t) :=t \\cdot F_{twop}\\left(t\\right)-\\int \\displaylimits _{0}^{F_{twop}\\left(t\\right)} F_{twop}^{-1}(u) du\\nonumber \\\\& & \\hspace{-19.91684pt}= \\frac{t}{z_{2}-z_{1}} \\cdot \\log \\left( \\frac{(t-z_{1})\\cdot p}{(z_{2}-t)\\cdot (1- p)} \\right)\\nonumber \\\\& & \\hspace{-19.91684pt}- \\log \\left( p \\cdot \\left( \\frac{(t-z_{1})\\cdot p}{(z_{2}-t)\\cdot (1- p)} \\right)^{\\frac{z_{1}}{z_{2}-z_{1}}}+ (1- p) \\cdot \\left( \\frac{(t-z_{1})\\cdot p}{(z_{2}-t)\\cdot (1- p)} \\right)^{\\frac{z_{2}}{z_{2}-z_{1}}} \\right)\\nonumber \\\\& & \\hspace{-19.91684pt}= \\frac{t-z_{1}}{z_{2}-z_{1}} \\cdot \\log \\left( \\frac{(t-z_{1})\\cdot p}{(z_{2}-t)\\cdot (1- p)} \\right)- \\log \\left( \\frac{(z_{2}-z_{1})\\cdot p}{z_{2}-t} \\right)\\\\& & \\hspace{-19.91684pt}= \\frac{t-z_{1}}{z_{2}-z_{1}} \\cdot \\log \\left( \\frac{(t-z_{1})\\cdot (z_{2}-1)}{(z_{2}-t)\\cdot (1-z_{1})} \\right)- \\log \\left( \\frac{z_{2}-1}{z_{2}-t} \\right) , \\qquad t \\in \\, ]z_{1}, z_{2} [ \\, ,\\nonumber $ where for the last equality we have used the above constraint (to obtain a two-parameter representation).","Straightforwardly, one has $\\varphi _{twop}(1) = 0$ , $\\varphi _{twop}^{\\prime }(1) = 0$ , $\\varphi _{twop}^{\\prime }(z_{1}) = -\\infty $ and $\\varphi _{twop}^{\\prime }(z_{2}) = \\infty $ .","From the generator $\\varphi _{twop}$ given in (REF ), we build the corresponding divergence (cf.","(REF )) $D_{\\varphi _{twop}}(\\mathbf {Q},\\mathbf {P})=\\sum \\limits _{k=1}^{K} \\frac{q_{k} - z_{1} \\cdot p_{k}}{z_{2} - z_{1}} \\cdot \\log \\Big (\\frac{(z_{2}-1) \\cdot (q_{k} - z_{1} \\cdot p_{k})}{(1-z_{1}) \\cdot (z_{2} \\cdot p_{k} - q_{k})} \\Big )- \\sum \\limits _{k=1}^{K} p_{k} \\cdot \\log \\Big (\\frac{(z_{2}-1) \\cdot p_{k}}{z_{2} \\cdot p_{k} - q_{k}} \\Big ) .$ It is known that some types of robustness properties of minimum-divergence estimators are connected with the boundedness of the derivative $\\varphi ^{\\prime }$ of the divergence generator $\\varphi $ ; this property is satisfied for the next Example REF (and its $W-$ concerning continuation in Example REF ), which leads to the new classes of divergences (REF ), (REF ) and (REF ): Example 46 (a) For any parameter-quadrupel $\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c} \\in \\, ]0,\\infty [$ with $\\beta _{1} < \\beta _{2}$ , we choose $]a_{F},b_{F}[\\ \\, := \\ ]a_{F_{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}},b_{F_{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}}[ \\ \\, := \\ \\Big ]1 - \\alpha \\cdot \\frac{(\\beta _{1} - \\beta _{2})^{2} + \\beta _{1}^{2} + \\beta _{1} \\cdot \\beta _{2}}{2\\beta _{1}\\cdot \\beta _{2}\\cdot (\\beta _{2} - \\beta _{1} )}, \\,\\infty \\Big [ \\ \\ni 1\\nonumber $ and define with $\\breve{\\theta } := 1 + \\alpha \\cdot \\Big (\\frac{1}{\\beta _{2}}- \\frac{1}{\\beta _{1}} \\Big ) < 1$ $\\hspace{-17.07182pt}F_{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}(t)&:=& {\\left\\lbrace \\begin{array}{ll}\\widetilde{c} \\cdot \\frac{\\beta _{1}-\\beta _{2}}{2}+ \\frac{\\widetilde{c}}{\\frac{1-t}{\\alpha } + \\frac{1}{\\beta _{2}} - \\frac{1}{\\beta _{1}}}\\cdot \\Big (1 - \\frac{1}{2} \\cdot \\sqrt{4 + \\big (\\frac{1-t}{\\alpha } + \\frac{1}{\\beta _{2}} - \\frac{1}{\\beta _{1}}\\big )^{2} \\cdot (\\beta _{1}+\\beta _{2})^{2}}\\, \\Big ),\\quad \\textrm {if } \\ t \\in \\, ]a_{F},b_{F}[ \\backslash \\lbrace \\breve{\\theta }\\rbrace , \\\\\\widetilde{c} \\cdot \\frac{\\beta _{1}-\\beta _{2}}{2}, \\hspace{277.41437pt}\\textrm {if } \\ t= \\breve{\\theta } \\in \\, ]a_{F},b_{F}[, \\\\- \\widetilde{c} \\cdot \\beta _{1}, \\hspace{284.52756pt} \\textrm {if } \\ t=a_{F}, \\\\- \\infty , \\hspace{295.90848pt} \\textrm {if } \\ t \\in \\, ]-\\infty ,a_{F}[.\\end{array}\\right.","}$ Notice that $\\breve{\\theta } \\in \\, ]a_{F},b_{F}[$ if and only if $\\beta _{1} \\in \\,]\\frac{\\beta _{2}}{3},\\beta _{2}[$ ; if (say) the latter holds, then one has the continuity $\\lim _{t \\rightarrow \\breve{\\theta }}F_{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}(t) = \\widetilde{c} \\cdot \\frac{\\beta _{1}-\\beta _{2}}{2}$ .","For $\\beta _{1} \\le \\frac{\\beta _{2}}{3}$ one gets $]a_{F},b_{F}[ \\backslash \\lbrace \\breve{\\theta } \\rbrace = \\, ]a_{F},b_{F}[$ .","Returning to the general case, one can see in a straightforward way that $F_{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}(\\cdot )$ is strictly increasing and that $\\mathcal {R}(F_{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}})=[-\\widetilde{c}\\cdot \\beta _{1},\\widetilde{c}\\cdot \\beta _{1} [$ .","Furthermore, $F_{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}(\\cdot )$ is smooth on $]a_{F},b_{F}[$ , and thus $F_{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}} \\in {F}$ .","Since $F_{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}(1)=0$ , let us choose the natural anchor point $c:=0$ , which leads to $]\\lambda _{-},\\lambda _{+}[ \\,= int(\\mathcal {R}(F_{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}})= \\, ]-\\widetilde{c}\\cdot \\beta _{1},\\widetilde{c}\\cdot \\beta _{1} [$ and $]t_{-}^{sc},t_{+}^{sc}[ = \\, ]a_{F},b_{F}[$ .","Moreover, it is straightforward to see that the corresponding inverse is $F_{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}^{-1}(x) =1 + \\alpha \\cdot \\Big (\\frac{1}{\\beta _{2}}- \\frac{1}{\\beta _{1}} \\Big ) - \\alpha \\cdot \\frac{\\frac{1}{\\beta _{2}} - \\frac{1}{\\beta _{1}} -\\frac{2 x}{\\widetilde{c} \\cdot \\beta _{1} \\cdot \\beta _{2} }}{1+ \\frac{x}{\\widetilde{c}} \\cdot \\Big (\\frac{1}{\\beta _{2}} - \\frac{1}{\\beta _{1}} \\Big )- \\frac{x^{2}}{\\widetilde{c}^{2} \\cdot \\beta _{1} \\cdot \\beta _{2} }},\\qquad x \\in int(\\mathcal {R}(F_{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}})) ;$ from this, we can derive from formula (REF ) (see also (REF )) for all $z \\in \\mathbb {R}$ $\\hspace{-19.91684pt}\\Lambda _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}(z) :=\\Lambda _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}^{(0)}(z) &=&{\\left\\lbrace \\begin{array}{ll}\\breve{\\theta } \\cdot z - \\widetilde{c} \\cdot \\alpha \\cdot \\log \\Big (1+ \\frac{z}{\\widetilde{c}} \\cdot \\Big (\\frac{1}{\\beta _{2}} - \\frac{1}{\\beta _{1}} \\Big )- \\frac{z^{2}}{\\widetilde{c}^{2} \\cdot \\beta _{1} \\cdot \\beta _{2} }\\Big ),\\quad \\textrm {if } \\ z \\in \\, ]-\\widetilde{c}\\cdot \\beta _{1},\\widetilde{c}\\cdot \\beta _{1} [,\\\\- \\widetilde{c} \\cdot \\breve{\\theta } \\cdot \\beta _{1} -\\widetilde{c} \\cdot \\alpha \\cdot \\log \\Big (2 - 2 \\frac{\\beta _{1}}{\\beta _{2}}\\Big ),\\hspace{73.97733pt} \\textrm {if } \\ z = -\\widetilde{c}\\cdot \\beta _{1},\\\\\\infty , \\hspace{202.01474pt} \\textrm {else} .\\end{array}\\right.","}$ Notice that $\\Lambda _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}(0) = 0$ and $\\lim _{z \\rightarrow \\widetilde{c}\\cdot \\beta _{1}}\\Lambda _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}(z) = \\infty $ .","Moreover, $\\Lambda _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}^{\\prime }( -\\widetilde{c}\\cdot \\beta _{1})= a_{F}$ and $\\Lambda _{ -\\widetilde{c}\\cdot \\beta _{1}}^{\\prime }(\\widetilde{c}\\cdot \\beta _{1}) = \\infty = b_{F}$ (which have to be interpreted as limits, as usual).","To proceed, from formula (REF ) (see also ()) we can deduce for all $t \\in \\mathbb {R}$ $\\hspace{-5.69046pt}\\varphi _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}(t):= \\varphi _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}^{(0)}(t)\\hspace{-8.5359pt} &=& \\hspace{-8.5359pt}{\\left\\lbrace \\begin{array}{ll}\\widetilde{c} \\cdot \\alpha \\cdot \\Big \\lbrace \\frac{\\sqrt{4 + (\\beta _{1} + \\beta _{2})^{2}\\cdot (\\frac{1-t}{\\alpha } + \\frac{1}{\\beta _{2}} - \\frac{1}{\\beta _{1}})^2}\\, - \\, (\\frac{1-t}{\\alpha } + \\frac{1}{\\beta _{2}} - \\frac{1}{\\beta _{1}}) \\cdot (\\beta _{1} - \\beta _{2}) \\, - \\, 2}{2} \\\\+ \\log \\frac{\\sqrt{4 + (\\beta _{1} + \\beta _{2})^{2}\\cdot (\\frac{1-t}{\\alpha } + \\frac{1}{\\beta _{2}} \\, - \\, \\frac{1}{\\beta _{1}})^2} \\, - \\, 2}{\\beta _{1} \\beta _{2} \\cdot (\\frac{1-t}{\\alpha } + \\frac{1}{\\beta _{2}} \\, - \\, \\frac{1}{\\beta _{1}})^{2}} \\Big \\rbrace \\ \\in [0,\\infty [,\\hspace{71.13188pt} \\textrm {if }t \\in \\, ]a_{F},\\infty [,\\\\\\widetilde{c} \\cdot \\alpha \\cdot \\Big \\lbrace \\frac{3 \\beta _{1} - \\beta _{2}}{2 (\\beta _{2} - \\beta _{1})}+ \\log \\frac{2(\\beta _{2} - \\beta _{1})}{\\beta _{2}} \\Big \\rbrace - \\widetilde{c} \\cdot \\beta _{1} \\cdot (t - a_{F}) \\ \\in \\, ]0,\\infty [,\\hspace{19.91684pt} \\textrm {if } \\ t \\in ]-\\infty , a_{F}].\\end{array}\\right.","}$ The first subcase in (REF ) can be proved by $& & \\hspace{-19.91684pt}\\varphi _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}(t) :=\\varphi _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}^{(0)}(t) :=t \\cdot F_{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}\\left(t\\right)\\ - \\int \\displaylimits _{0}^{F_{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}\\left(t\\right)}F_{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}^{-1}(u) \\, du\\nonumber \\\\& & \\hspace{-19.91684pt}= (t - \\breve{\\theta }) \\cdot F_{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}\\left(t\\right)+ \\widetilde{c} \\cdot \\alpha \\cdot \\log \\Big (1+ \\frac{F_{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}\\left(t\\right)}{\\widetilde{c}} \\cdot \\Big (\\frac{1}{\\beta _{2}} - \\frac{1}{\\beta _{1}} \\Big )- \\frac{(F_{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}\\left(t\\right))^{2}}{\\widetilde{c}^{2} \\cdot \\beta _{1} \\cdot \\beta _{2} }\\Big )\\nonumber \\\\& & \\hspace{-19.91684pt}= \\widetilde{c} \\cdot \\alpha \\cdot \\frac{\\sqrt{4 + (\\beta _{1} + \\beta _{2})^{2}\\cdot (\\frac{1-t}{\\alpha } + \\frac{1}{\\beta _{2}} - \\frac{1}{\\beta _{1}})^2}\\, - \\, (\\frac{1-t}{\\alpha } + \\frac{1}{\\beta _{2}} - \\frac{1}{\\beta _{1}}) \\cdot (\\beta _{1} - \\beta _{2}) \\, - \\, 2}{2}\\nonumber \\\\& & \\hspace{-19.91684pt}\\ \\ \\ + \\ \\widetilde{c} \\cdot \\alpha \\cdot \\log \\bigg (1+ \\Big [\\frac{\\beta _{1}-\\beta _{2}}{2}+ \\frac{1}{\\frac{1-t}{\\alpha } + \\frac{1}{\\beta _{2}} - \\frac{1}{\\beta _{1}}}\\cdot \\Big (1 - \\frac{1}{2} \\cdot \\sqrt{4 + \\big (\\frac{1-t}{\\alpha } + \\frac{1}{\\beta _{2}} - \\frac{1}{\\beta _{1}}\\big )^{2} \\cdot (\\beta _{1}+\\beta _{2})^{2}}\\, \\Big ) \\Big ]\\cdot \\frac{\\beta _{1} - \\beta _{2}}{\\beta _{1} \\cdot \\beta _{2}}\\nonumber \\\\& & \\hspace{-19.91684pt}\\ \\ \\ - \\ \\Big [\\frac{\\beta _{1}-\\beta _{2}}{2}+ \\frac{1}{\\frac{1-t}{\\alpha } + \\frac{1}{\\beta _{2}} - \\frac{1}{\\beta _{1}}}\\cdot \\Big (1 - \\frac{1}{2} \\cdot \\sqrt{4 + \\big (\\frac{1-t}{\\alpha } + \\frac{1}{\\beta _{2}} - \\frac{1}{\\beta _{1}}\\big )^{2} \\cdot (\\beta _{1}+\\beta _{2})^{2}}\\, \\Big ) \\Big ]^{2} \\cdot \\frac{1}{\\beta _{1} \\cdot \\beta _{2} }\\bigg )\\nonumber $ and some straightforward calculations.","The second line in (REF ) follows by computing $\\varphi _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}(a_{F})=\\widetilde{c} \\cdot \\alpha \\cdot \\Big \\lbrace \\frac{3 \\beta _{1} - \\beta _{2}}{2 (\\beta _{2} - \\beta _{1})}+ \\log \\frac{2(\\beta _{2} - \\beta _{1})}{\\beta _{2}} \\Big \\rbrace $ .","Notice that $\\varphi _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}(1) = 0$ , $\\varphi _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}^{\\prime }(1) = 0$ , $\\varphi _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}(-\\infty ) = \\infty $ and $\\varphi _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}(\\infty ) = \\infty $ .","Moreover, $\\varphi _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}^{\\prime }(-\\infty ) =\\varphi _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}^{\\prime }(a_{F}) =-\\widetilde{c} \\cdot \\beta _{1}$ and $\\varphi _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}^{\\prime }(\\infty ) =\\widetilde{c} \\cdot \\beta _{1}$ .","From the generator $\\varphi _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}$ given in (REF ), we construct the corresponding divergence (cf.","(REF )) $& &D_{\\varphi _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}}(\\mathbf {Q},\\mathbf {P})= \\sum \\limits _{k=1}^{K} p_{k} \\cdot \\varphi _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}\\Big (\\frac{q_{k}}{p_{k}}\\Big )\\nonumber \\\\& &= \\sum \\limits _{k=1}^{K}p_{k} \\cdot \\bigg [{1}_{]a_{F},\\infty [}(\\frac{q_{k}}{p_{k}}) \\cdot \\widetilde{c} \\cdot \\alpha \\cdot \\Big \\lbrace \\frac{\\sqrt{4 + (\\beta _{1} + \\beta _{2})^{2}\\cdot (\\frac{1-\\frac{q_{k}}{p_{k}}}{\\alpha } + \\frac{1}{\\beta _{2}} - \\frac{1}{\\beta _{1}})^2}\\, - \\, (\\frac{1-\\frac{q_{k}}{p_{k}}}{\\alpha } + \\frac{1}{\\beta _{2}} - \\frac{1}{\\beta _{1}}) \\cdot (\\beta _{1} - \\beta _{2}) \\, - \\, 2}{2}\\nonumber \\\\& &+ \\log \\frac{\\sqrt{4 + (\\beta _{1} + \\beta _{2})^{2}\\cdot (\\frac{1-\\frac{q_{k}}{p_{k}}}{\\alpha } + \\frac{1}{\\beta _{2}} \\, - \\, \\frac{1}{\\beta _{1}})^2} \\, - \\, 2}{\\beta _{1} \\beta _{2} \\cdot (\\frac{1-\\frac{q_{k}}{p_{k}}}{\\alpha } + \\frac{1}{\\beta _{2}} \\, - \\, \\frac{1}{\\beta _{1}})^{2}} \\Big \\rbrace \\nonumber \\\\& &+ \\ {1}_{]-\\infty , a_{F}]}(\\frac{q_{k}}{p_{k}}) \\cdot \\widetilde{c} \\cdot \\Big \\lbrace \\alpha \\cdot \\Big \\lbrace \\frac{3 \\beta _{1} - \\beta _{2}}{2 (\\beta _{2} - \\beta _{1})}+ \\log \\frac{2(\\beta _{2} - \\beta _{1})}{\\beta _{2}} \\Big \\rbrace - \\beta _{1} \\cdot (\\frac{q_{k}}{p_{k}} - a_{F}) \\Big \\rbrace \\bigg ],\\qquad \\mathbf {P} \\in \\mathbb {R}_{\\ge 0}^{K}, \\mathbf {Q} \\in \\mathbb {R}^{K}.$ Notice that we can particularly include the case where $p_{k}=0$ in combination with $q_{k} \\ne 0$ , since $\\lim _{t \\rightarrow 0_{+}} t \\cdot \\varphi _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}(\\frac{1}{t})= \\widetilde{c} \\cdot \\beta _{1}$ and $\\lim _{t \\rightarrow 0_{-}} t \\cdot \\varphi _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}(\\frac{1}{t})= - \\widetilde{c} \\cdot \\beta _{1}$ are both finite, and hence $p_{k} \\cdot \\varphi _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}(\\frac{q_{k}}{p_{k}})= q_{k} \\cdot \\frac{p_{k}}{q_{k}} \\cdot \\varphi _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}(\\frac{q_{k}}{p_{k}}) $ stays finite as $p_{k}$ tends to zero.","(b) For any parameter-quadrupel $\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c} \\in \\, ]0,\\infty [$ with $\\beta _{1} > \\beta _{2}$ , one can proceed analogously to (a).","Let us start by choosing $]a_{F},b_{F}[\\ \\, := \\ ]a_{F_{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}},b_{F_{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}}[ \\ \\, := \\ \\Big ]-\\infty , \\, 1 + \\alpha \\cdot \\frac{(\\beta _{1} - \\beta _{2})^{2} + \\beta _{1} \\cdot \\beta _{2} + \\beta _{2}^{2}}{2\\beta _{1}\\cdot \\beta _{2}\\cdot (\\beta _{1} - \\beta _{2} )}, \\,\\Big [ \\ \\ni 1\\nonumber $ and defining with the same $\\breve{\\theta } := 1 + \\alpha \\cdot \\Big (\\frac{1}{\\beta _{2}}- \\frac{1}{\\beta _{1}} \\Big ) > 1$ $\\hspace{-17.07182pt}F_{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}(t)&:=& {\\left\\lbrace \\begin{array}{ll}\\widetilde{c} \\cdot \\frac{\\beta _{1}-\\beta _{2}}{2}+ \\frac{\\widetilde{c}}{\\frac{1-t}{\\alpha } + \\frac{1}{\\beta _{2}} - \\frac{1}{\\beta _{1}}}\\cdot \\Big (1 - \\frac{1}{2} \\cdot \\sqrt{4 + \\big (\\frac{1-t}{\\alpha } + \\frac{1}{\\beta _{2}} - \\frac{1}{\\beta _{1}}\\big )^{2} \\cdot (\\beta _{1}+\\beta _{2})^{2}}\\, \\Big ),\\quad \\textrm {if } \\ t \\in \\, ]a_{F},b_{F}[ \\backslash \\lbrace \\breve{\\theta }\\rbrace , \\\\\\widetilde{c} \\cdot \\frac{\\beta _{1}-\\beta _{2}}{2}, \\hspace{277.41437pt}\\textrm {if } \\ t= \\breve{\\theta } \\in \\, ]a_{F},b_{F}[, \\\\\\widetilde{c} \\cdot \\beta _{2}, \\hspace{293.06346pt} \\textrm {if } \\ t=b_{F}, \\\\\\infty , \\hspace{304.4444pt} \\textrm {if } \\ t \\in \\, ]b_{F}, \\infty [.\\end{array}\\right.","}$ Clearly, $\\breve{\\theta } \\in \\, ]a_{F},b_{F}[$ if and only if $\\beta _{1} \\in \\,]\\beta _{2}, 3\\beta _{2}[$ ; if (say) the latter holds, then one gets the continuity $\\lim _{t \\rightarrow \\breve{\\theta }}F_{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}(t) = \\widetilde{c} \\cdot \\frac{\\beta _{1}-\\beta _{2}}{2}$ .","For $\\beta _{1} \\le 3 \\beta _{2}$ there holds $]a_{F},b_{F}[ \\backslash \\lbrace \\breve{\\theta } \\rbrace = \\, ]a_{F},b_{F}[$ .","Returning to the general case, one can show comfortably that $F_{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}(\\cdot )$ is strictly increasing and that $\\mathcal {R}(F_{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}})=]-\\widetilde{c}\\cdot \\beta _{2},\\widetilde{c}\\cdot \\beta _{2} ]$ .","Moreover, $F_{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}(\\cdot )$ is smooth on $]a_{F},b_{F}[$ , and hence $F_{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}} \\in {F}$ .","In face of the validity of $F_{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}(1)=0$ , let us choose the natural anchor point $c:=0$ , which amounts to $]\\lambda _{-},\\lambda _{+}[ \\,= int(\\mathcal {R}(F_{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}})= \\, ]-\\widetilde{c}\\cdot \\beta _{2},\\widetilde{c}\\cdot \\beta _{2} [$ and $]t_{-}^{sc},t_{+}^{sc}[ = \\, ]a_{F},b_{F}[$ .","Since the first line in (REF ) coincides formally with that of (REF ) (with different $]a_{F},b_{F}[$ ), the corresponding inverse is formally the same as (REF ) (with different $]a_{F},b_{F}[$ ), and hence $\\hspace{-19.91684pt}\\Lambda _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}(z) :=\\Lambda _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}^{(0)}(z) &=&{\\left\\lbrace \\begin{array}{ll}\\breve{\\theta } \\cdot z - \\widetilde{c} \\cdot \\alpha \\cdot \\log \\Big (1+ \\frac{z}{\\widetilde{c}} \\cdot \\Big (\\frac{1}{\\beta _{2}} - \\frac{1}{\\beta _{1}} \\Big )- \\frac{z^{2}}{\\widetilde{c}^{2} \\cdot \\beta _{1} \\cdot \\beta _{2} }\\Big ),\\quad \\textrm {if } \\ z \\in \\, ]-\\widetilde{c}\\cdot \\beta _{2},\\widetilde{c}\\cdot \\beta _{2} [,\\\\\\widetilde{c} \\cdot \\breve{\\theta } \\cdot \\beta _{2} -\\widetilde{c} \\cdot \\alpha \\cdot \\log \\Big (2 - 2 \\frac{\\beta _{2}}{\\beta _{1}}\\Big ),\\hspace{73.97733pt} \\textrm {if } \\ z = \\widetilde{c}\\cdot \\beta _{2},\\\\\\infty , \\hspace{202.01474pt} \\textrm {else} .\\end{array}\\right.","}$ Notice that $\\Lambda _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}(0) = 0$ and $\\lim _{z \\rightarrow - \\widetilde{c}\\cdot \\beta _{2}}\\Lambda _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}(z) = -\\infty $ .","Furthermore, $\\Lambda _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}^{\\prime }( -\\widetilde{c}\\cdot \\beta _{2})= - \\infty = a_{F}$ and $\\Lambda _{ -\\widetilde{c}\\cdot \\beta _{1}}^{\\prime }(\\widetilde{c}\\cdot \\beta _{2})= b_{F}$ .","To proceed, from formula (REF ) (see also ()) we can derive — analogously to (REF ) — for all $t \\in \\mathbb {R}$ $\\hspace{-5.69046pt}\\varphi _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}(t):= \\varphi _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}^{(0)}(t)\\hspace{-8.5359pt} &=& \\hspace{-8.5359pt}{\\left\\lbrace \\begin{array}{ll}\\widetilde{c} \\cdot \\alpha \\cdot \\Big \\lbrace \\frac{\\sqrt{4 + (\\beta _{1} + \\beta _{2})^{2}\\cdot (\\frac{1-t}{\\alpha } + \\frac{1}{\\beta _{2}} - \\frac{1}{\\beta _{1}})^2}\\, - \\, (\\frac{1-t}{\\alpha } + \\frac{1}{\\beta _{2}} - \\frac{1}{\\beta _{1}}) \\cdot (\\beta _{1} - \\beta _{2}) \\, - \\, 2}{2} \\\\+ \\log \\frac{\\sqrt{4 + (\\beta _{1} + \\beta _{2})^{2}\\cdot (\\frac{1-t}{\\alpha } + \\frac{1}{\\beta _{2}} \\, - \\, \\frac{1}{\\beta _{1}})^2} \\, - \\, 2}{\\beta _{1} \\beta _{2} \\cdot (\\frac{1-t}{\\alpha } + \\frac{1}{\\beta _{2}} \\, - \\, \\frac{1}{\\beta _{1}})^{2}} \\Big \\rbrace \\ \\in [0,\\infty [,\\hspace{71.13188pt} \\textrm {if }t \\in \\, ]-\\infty , b_{F}[,\\\\\\widetilde{c} \\cdot \\alpha \\cdot \\Big \\lbrace \\frac{3 \\beta _{2} - \\beta _{1}}{2 (\\beta _{1} - \\beta _{2})}+ \\log \\frac{2(\\beta _{1} - \\beta _{2})}{\\beta _{1}} \\Big \\rbrace + \\widetilde{c} \\cdot \\beta _{2} \\cdot (t - b_{F}) \\ \\in \\, ]0,\\infty [,\\hspace{19.91684pt} \\textrm {if } \\ t \\in [b_{F}, \\infty [,\\end{array}\\right.","}$ where the last line in (REF ) follows by calculating $\\varphi _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}(b_{F})=\\widetilde{c} \\cdot \\alpha \\cdot \\Big \\lbrace \\frac{3 \\beta _{2} - \\beta _{1}}{2 (\\beta _{1} - \\beta _{2})}+ \\log \\frac{2(\\beta _{1} - \\beta _{2})}{\\beta _{1}} \\Big \\rbrace $ .","Notice that $\\varphi _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}(1) = 0$ , $\\varphi _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}^{\\prime }(1) = 0$ , $\\varphi _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}(-\\infty ) = \\infty $ and $\\varphi _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}(\\infty ) = \\infty $ .","Furthermore, $\\varphi _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}^{\\prime }(-\\infty ) =-\\widetilde{c} \\cdot \\beta _{2}$ and $\\varphi _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}^{\\prime }(\\infty ) =\\varphi _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}^{\\prime }(b_{F}) =\\widetilde{c} \\cdot \\beta _{2}$ .","From the generator $\\varphi _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}$ given in (REF ), we construct the corresponding divergence (cf.","(REF )) $& &D_{\\varphi _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}}(\\mathbf {Q},\\mathbf {P})= \\sum \\limits _{k=1}^{K} p_{k} \\cdot \\varphi _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}\\Big (\\frac{q_{k}}{p_{k}}\\Big )\\nonumber \\\\& &= \\sum \\limits _{k=1}^{K}p_{k} \\cdot \\bigg [{1}_{]-\\infty , b_{F}[}(\\frac{q_{k}}{p_{k}}) \\cdot \\widetilde{c} \\cdot \\alpha \\cdot \\Big \\lbrace \\frac{\\sqrt{4 + (\\beta _{1} + \\beta _{2})^{2}\\cdot (\\frac{1-\\frac{q_{k}}{p_{k}}}{\\alpha } + \\frac{1}{\\beta _{2}} - \\frac{1}{\\beta _{1}})^2}\\, - \\, (\\frac{1-\\frac{q_{k}}{p_{k}}}{\\alpha } + \\frac{1}{\\beta _{2}} - \\frac{1}{\\beta _{1}}) \\cdot (\\beta _{1} - \\beta _{2}) \\, - \\, 2}{2}\\nonumber \\\\& &+ \\log \\frac{\\sqrt{4 + (\\beta _{1} + \\beta _{2})^{2}\\cdot (\\frac{1-\\frac{q_{k}}{p_{k}}}{\\alpha } + \\frac{1}{\\beta _{2}} \\, - \\, \\frac{1}{\\beta _{1}})^2} \\, - \\, 2}{\\beta _{1} \\beta _{2} \\cdot (\\frac{1-\\frac{q_{k}}{p_{k}}}{\\alpha } + \\frac{1}{\\beta _{2}} \\, - \\, \\frac{1}{\\beta _{1}})^{2}} \\Big \\rbrace \\nonumber \\\\& &+ \\ {1}_{[b_{F}, \\infty [}(\\frac{q_{k}}{p_{k}}) \\cdot \\widetilde{c} \\cdot \\Big \\lbrace \\alpha \\cdot \\Big \\lbrace \\frac{3 \\beta _{2} - \\beta _{1}}{2 (\\beta _{1} - \\beta _{2})}+ \\log \\frac{2(\\beta _{1} - \\beta _{2})}{\\beta _{1}} \\Big \\rbrace + \\beta _{2} \\cdot (\\frac{q_{k}}{p_{k}} - b_{F}) \\Big \\rbrace \\bigg ],\\qquad \\mathbf {P} \\in \\mathbb {R}_{\\ge 0}^{K}, \\mathbf {Q} \\in \\mathbb {R}^{K}.$ As above, we can particularly include the case where $p_{k}=0$ in combination with $q_{k} \\ne 0$ , since $\\lim _{t \\rightarrow 0_{+}} t \\cdot \\varphi _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}(\\frac{1}{t})= \\widetilde{c} \\cdot \\beta _{2}$ and $\\lim _{t \\rightarrow 0_{-}} t \\cdot \\varphi _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}(\\frac{1}{t})= - \\widetilde{c} \\cdot \\beta _{2}$ are both finite.","(c) The analysis for the case $\\beta _{1} = \\beta _{2} =: \\beta $ can be obtained by taking $\\lim _{\\beta _1 \\rightarrow \\beta _{2}}$ in (a) respectively (b).","Alternatively, one can start afresh.","Due to its importance and its particularities, we nevertheless state the corresponding results explicitly.","To begin with, for any parameter-triple $\\alpha ,\\beta ,\\widetilde{c} \\in \\, ]0,\\infty [$ we choose $]a_{F},b_{F}[\\ \\, := \\ ]a_{F_{\\alpha ,\\beta ,\\widetilde{c}}},b_{F_{\\alpha ,\\beta ,\\widetilde{c}}}[ \\ \\, := \\ ]-\\infty , \\infty \\, [\\nonumber $ and define with $\\breve{\\theta } := 1$ $\\hspace{-17.07182pt}F_{\\alpha ,\\beta ,\\widetilde{c}}(t)&:=& {\\left\\lbrace \\begin{array}{ll}\\frac{\\widetilde{c} \\cdot \\alpha }{1-t}\\cdot \\Big (1 - \\sqrt{1 + \\big (\\frac{1-t}{\\alpha } \\big )^{2} \\cdot \\beta ^{2}}\\, \\Big ),\\qquad \\textrm {if } \\ t \\in \\, ]a_{F},b_{F}[ \\backslash \\lbrace \\breve{\\theta }\\rbrace , \\\\0, \\hspace{145.10922pt}\\textrm {if } \\ t= \\breve{\\theta }.\\end{array}\\right.","}$ Clearly, one has the continuity $\\lim _{t \\rightarrow \\breve{\\theta }}F_{\\alpha ,\\beta ,\\widetilde{c}}(t) = 0$ .","Moreover, one can see in a straightforward way that $F_{\\alpha ,\\beta ,\\widetilde{c}}(\\cdot )$ is strictly increasing and that $\\mathcal {R}(F_{\\alpha ,\\beta ,\\widetilde{c}})=]-\\widetilde{c}\\cdot \\beta ,\\widetilde{c}\\cdot \\beta [$ .","Furthermore, $F_{\\alpha ,\\beta ,\\widetilde{c}}(\\cdot )$ is smooth on $]a_{F},b_{F}[$ , and thus $F_{\\alpha ,\\beta ,\\widetilde{c}} \\in {F}$ .","Since $F_{\\alpha ,\\beta ,\\widetilde{c}}(1)=0$ , let us choose the natural anchor point $c:=0$ , which leads to the choice $]\\lambda _{-},\\lambda _{+}[ \\,= int(\\mathcal {R}(F_{\\alpha ,\\beta ,\\widetilde{c}})= \\, ]-\\widetilde{c}\\cdot \\beta ,\\widetilde{c}\\cdot \\beta [$ and $]t_{-}^{sc},t_{+}^{sc}[ = \\, ]a_{F},b_{F}[ = \\, ]-\\infty ,\\infty [$ .","The inverse in (REF ) collapses to $F_{\\alpha ,\\beta ,\\widetilde{c}}^{-1}(x) =1 + \\alpha \\cdot \\frac{\\frac{2 x}{\\widetilde{c} \\cdot \\beta ^{2} }}{1 - \\frac{x^{2}}{\\widetilde{c}^{2} \\cdot \\beta ^{2} }},\\qquad x \\in int(\\mathcal {R}(F_{\\alpha ,\\beta ,\\widetilde{c}})) ;$ from this, we can derive from formula (REF ) (see also (REF )) for all $z \\in \\mathbb {R}$ $\\hspace{-19.91684pt}\\Lambda _{\\alpha ,\\beta ,\\widetilde{c}}(z) :=\\Lambda _{\\alpha ,\\beta ,\\widetilde{c}}^{(0)}(z) &=&{\\left\\lbrace \\begin{array}{ll}\\breve{\\theta } \\cdot z - \\widetilde{c} \\cdot \\alpha \\cdot \\log \\Big (1 - \\frac{z^{2}}{\\widetilde{c}^{2} \\cdot \\beta ^{2} }\\Big ),\\qquad \\textrm {if } \\ z \\in \\, ]-\\widetilde{c}\\cdot \\beta ,\\widetilde{c}\\cdot \\beta [,\\\\\\infty , \\hspace{130.88284pt} \\textrm {else} .\\end{array}\\right.","}$ Notice that $\\Lambda _{\\alpha ,\\beta ,\\widetilde{c}}(0) = 0$ , $\\lim _{z \\rightarrow - \\widetilde{c}\\cdot \\beta }\\Lambda _{\\alpha ,\\beta ,\\widetilde{c}}(z) = -\\infty = $ , and $\\lim _{z \\rightarrow \\widetilde{c}\\cdot \\beta }\\Lambda _{\\alpha ,\\beta ,\\widetilde{c}}(z) = \\infty $ .","Furthermore, $\\lim _{z \\rightarrow - \\widetilde{c}\\cdot \\beta }\\Lambda _{\\alpha ,\\beta ,\\widetilde{c}}^{\\prime }(z) = -\\infty = a_{F}$ , and $\\lim _{z \\rightarrow \\widetilde{c}\\cdot \\beta }\\Lambda _{\\alpha ,\\beta ,\\widetilde{c}}^{\\prime }(z) = \\infty = b_{F}$ .","To proceed, the formula (REF ) (respectively, (REF )) collapses to $\\varphi _{\\alpha ,\\beta ,\\widetilde{c}}(t):=\\varphi _{\\alpha ,\\beta ,\\widetilde{c}}^{(0)}(t)= \\widetilde{c} \\cdot \\alpha \\cdot \\Big \\lbrace \\sqrt{1 + \\beta ^{2}\\cdot \\Big (\\frac{1-t}{\\alpha }\\Big )^2} \\, - \\, 1+ \\log \\frac{2 \\cdot \\Big (\\sqrt{1 + \\beta ^{2}\\cdot \\Big (\\frac{1-t}{\\alpha } \\Big )^2} \\, - \\, 1\\Big )}{\\beta ^{2} \\cdot \\Big (\\frac{1-t}{\\alpha }\\Big )^{2}} \\Big \\rbrace \\ \\in [0,\\infty [, \\ \\ t \\in \\, ]-\\infty , \\infty [ \\, = \\, ]a_{F}, b_{F}[.$ Notice that $\\varphi _{\\alpha ,\\beta ,\\widetilde{c}}(1) = 0$ , $\\varphi _{\\alpha ,\\beta }^{\\prime }(1) = 0$ , $\\varphi _{\\alpha ,\\beta ,\\widetilde{c}}(-\\infty ) = \\infty $ and $\\varphi _{\\alpha ,\\beta ,\\widetilde{c}}(\\infty ) = \\infty $ .","Moreover, $\\varphi _{\\alpha ,\\beta ,\\widetilde{c}}^{\\prime }(-\\infty ) =\\varphi _{\\alpha ,\\beta ,\\widetilde{c}}^{\\prime }(a_{F}) =-\\widetilde{c} \\cdot \\beta $ and $\\varphi _{\\alpha ,\\beta ,\\widetilde{c}}^{\\prime }(\\infty ) =\\varphi _{\\alpha ,\\beta ,\\widetilde{c}}^{\\prime }(b_{F}) =\\widetilde{c} \\cdot \\beta $ .","From the generator $\\varphi _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}$ given in (REF ), we construct the corresponding divergence (cf.","(REF )) $& & \\hspace{-19.91684pt}D_{\\varphi _{\\alpha ,\\beta ,\\widetilde{c}}}(\\mathbf {Q},\\mathbf {P})= \\sum \\limits _{k=1}^{K} p_{k} \\cdot \\varphi _{\\alpha ,\\beta ,\\widetilde{c}}\\Big (\\frac{q_{k}}{p_{k}}\\Big )\\nonumber \\\\& & \\hspace{-19.91684pt}= \\widetilde{c} \\cdot \\alpha \\cdot \\sum \\limits _{k=1}^{K}p_{k} \\cdot \\Big \\lbrace \\sqrt{1 + \\beta ^{2}\\cdot \\Big (\\frac{1-\\frac{q_{k}}{p_{k}}}{\\alpha }\\Big )^2} \\, - \\, 1+ \\log \\frac{2 \\cdot \\Big (\\sqrt{1 + \\beta ^{2}\\cdot \\Big (\\frac{1-\\frac{q_{k}}{p_{k}}}{\\alpha } \\Big )^2} \\, - \\, 1\\Big )}{\\beta ^{2} \\cdot \\Big (\\frac{1-\\frac{q_{k}}{p_{k}}}{\\alpha }\\Big )^{2}} \\Big \\rbrace ,\\qquad \\mathbf {P} \\in \\mathbb {R}_{\\ge 0}^{K}, \\mathbf {Q} \\in \\mathbb {R}^{K}.$ As above, we can particularly include the case where $p_{k}=0$ in combination with $q_{k} \\ne 0$ , since $\\lim _{t \\rightarrow 0_{+}} t \\cdot \\varphi _{\\alpha ,\\beta ,\\widetilde{c}}(\\frac{1}{t})= \\widetilde{c} \\cdot \\beta $ and $\\lim _{t \\rightarrow 0_{-}} t \\cdot \\varphi _{\\alpha ,\\beta ,\\widetilde{c}}(\\frac{1}{t})= - \\widetilde{c} \\cdot \\beta $ are both finite.","This ends the current Example REF .","As a side effect in the above-mentioned Example REF , for fixed $\\beta _{2}, \\alpha ,\\widetilde{c}$ notice the interesting behaviour (e.g.","with respect to $int(dom(F)) = ]a_{F},b_{F}[$ and the range of $\\varphi ^{\\prime }$ ) as $\\beta _{1}$ moves from $]0,\\beta _{2}[$ to $\\beta _{2}$ and further to $]\\beta _{2}, \\infty [$ .","Remark 47 The characterization of the probability distribution $$ in (REF ) which may result from Theorem REF — as seen through the above examples — considerably improves other approaches which make use of their identification through the concept of power variance functions of Natural Exponential Families, as developed by Tweedie [369], Morris [267], Letac & Mora [214], and others.","This approach has been used in Broniatowski [58] in a similar perspective as developed here, but can not be extended outside the range of power divergences, in contrast with the Examples REF , REF , REF and REF which can only be handled as a consequence of Theorem REF .","To continue with our general procedure, suppose now that for a divergence generator $\\varphi $ of interest we have concretely/explicitly found (e.g.","by direct calculations or via our $F-$ connection in Theorem REF , see also Remark REF ) its Fenchel-Legendre transform $\\Lambda = \\varphi ^{*}$ ; for this “candidate”, in order to achieve the desired representability (REF ) it remains to verify that $\\exp (\\Lambda (z)) = \\int _{\\mathbb {R}}e^{z \\cdot y} \\, d\\mathbb {} (y),\\qquad z \\in \\mathbb {R},$ for some probability distribution/measure $\\mathbb {} $ on the real line (the light-tailedness in the sense of finiteness on some open interval containing zero, will be typically guaranteed automatically by the assumptions on $\\varphi $ ); of course, this is equivalent to “the existence ” of a random variable $W$ whose moment generating function is equal to $\\exp (\\Lambda )$ (and thus, its cumulant generating function (log moment generating function) is $\\Lambda $ ), i.e.","$\\exp (\\Lambda (z)) = E_{\\mathbb {\\Pi }}[\\exp (z \\cdot W)]\\qquad z \\in \\mathbb {R},$ with $\\mathbb {\\Pi }[W \\in \\cdot \\, ] = \\mathbb {}[ \\, \\cdot \\,]$ ); recall that from this, we need to simulate a sequence $(W_{i})_{i\\in \\mathbb {N}}$ of i.i.d.","copies of $W$ which are the crucial building ingredients of $\\xi _{n}^{\\mathbf {W}}$ in Theorem REF , respectively, of $\\xi _{n,\\mathbf {X}}^{w\\mathbf {W}}$ in Theorem REF .","For the above-mentioned Examples REF to REF , we can give explicit solutions to the representabilities (REF ) respectively (REF ); this is achieved in the following Examples REF to REF (notice that the corresponding supports of $\\mathbb {}$ are explicitly mentioned in the summarizing Table 1 above): Example 48 for the power-divergence context of Example REF we obtain: (a) Case $\\gamma =0$ , $\\widetilde{c} >0$ : $\\Lambda _{0,\\widetilde{c}}(z)= - \\widetilde{c} \\cdot \\log \\left( 1 - \\frac{z}{\\widetilde{c}} \\right)$ (cf.","(REF )) is the cumulant generating function of the Gamma distribution $\\mathbb {} = GAM(\\widetilde{c},\\widetilde{c})$ with rate parameter (inverse scale parameter) $\\widetilde{c}$ and shape parameter $\\widetilde{c}$ ; hence, $\\varphi _{0,\\widetilde{c}} \\in \\Upsilon (]0,\\infty [)$ .","Prominent special case $\\widetilde{c} =1$ : $\\mathbb {} = GAM(1,1) = EXP(1)$ is the exponential distribution with mean 1.","Type: $\\mathbb {}$ is an infinitely divisible (cf.","Proposition REF ) continuous distribution with density $f(y) := \\frac{\\widetilde{c}^{\\widetilde{c}} \\cdot y^{\\widetilde{c}-1} \\cdot e^{-\\widetilde{c} \\cdot y} }{\\Gamma (\\widetilde{c})} \\cdot {1}_{]0,\\infty [}(y)$ ($y \\in \\mathbb {R}$ ).","Behaviour at zero: $\\mathbb {}[ \\, ]0,\\infty [ \\, ] = \\mathbb {\\Pi }[W>0]=1$ .","Corresponding generator: $\\varphi _{0,\\widetilde{c}} = \\widetilde{c} \\cdot \\varphi _{0}$ (cf.","(REF ), (REF )) of the $\\widetilde{c}-$ fold of the reversed Kullback-Leibler divergence (reversed relative entropy) given in the second line of (REF ).","Sums: for i.i.d.","copies $(W_{i})_{i \\in \\mathbb {N}}$ of $W$ , the probability distribution of $\\breve{W} := \\sum _{i\\in I_{k}^{(n)}} W_{i}$ (cf.","Remark REF (ii)) is $GAM(\\widetilde{c},\\widetilde{c} \\cdot card(I_{k}^{(n)}))$ .","(b) Case $\\gamma \\in \\, ]0,1[$ , $\\widetilde{c} >0$ : $\\Lambda _{\\gamma ,\\widetilde{c}}^{(0)}(z)= \\frac{\\widetilde{c}}{\\gamma } \\cdot \\Big \\lbrace \\left( \\frac{\\gamma -1}{\\widetilde{c}} \\cdot z + 1 \\right)^{\\frac{\\gamma }{\\gamma -1}} -1 \\Big \\rbrace $ (cf.","(REF )) is the cumulant generating function of the Compound-Poisson-Gamma distribution $\\mathbb {} = C(POI(\\theta ),GAM(\\alpha ,\\beta ))$ with $\\theta = \\frac{\\widetilde{c}}{\\gamma } > 0$ , rate parameter (inverse scale parameter) $\\alpha = \\frac{\\widetilde{c}}{1-\\gamma } >0$ , and shape parameter $\\beta = \\frac{\\gamma }{1-\\gamma } >0$ .","In other words, $W$ has the comfortably simulable form $W = \\sum _{i=1}^{N} \\widetilde{W}_{i}$ with the usual convention $\\sum _{i=1}^{0} \\widetilde{W}_{i} := 0$ for some i.i.d.","sequence $( \\widetilde{W}_{i})_{i\\in \\mathbb {N}}$ of Gamma $GAM(\\alpha ,\\beta )$ distributed random variables (with parameter-pair $(\\alpha ,\\beta )$ ) and some independent $POI(\\theta )-$ distributed random variable $N$ .","Hence, $\\varphi _{\\gamma ,\\widetilde{c}} \\in \\Upsilon (]0,\\infty [)$ .","Type: $\\mathbb {}$ is an infinitely divisible distribution (cf.","Proposition REF ), mixture of a one-point distribution at zero and a continuous distribution on $[0,\\infty [$ , with $\\mathbb {}[\\lbrace 0\\rbrace ] = \\mathbb {\\Pi }[W = 0]= e^{-\\theta }$ and $\\mathbb {}[B] = \\mathbb {\\Pi }[W \\in B] = \\int _{B} f_{\\widetilde{c},\\gamma }(u) \\, du$ for every (measurable) subset of $]0,\\infty [$ having density $& & \\hspace{-22.76228pt}f_{C(POI(\\theta ),GAM(\\alpha ,\\beta ))}(y): = \\frac{\\exp \\left(- \\alpha \\cdot y - \\theta \\right)}{y}\\cdot \\sum _{k=1}^{\\infty } \\frac{\\theta ^{k} \\cdot (\\alpha y)^{k\\beta }}{k!","\\cdot \\Gamma (k\\beta )}\\cdot {1}_{]0,\\infty [}(y)\\\\& & \\hspace{-22.76228pt}= \\frac{1}{y} \\cdot \\exp \\left(-\\widetilde{c} \\cdot \\left(\\frac{y}{1-\\gamma } + \\frac{1}{\\gamma } \\right) \\right)\\cdot \\sum _{k=1}^{\\infty } \\frac{a_{k}}{k!}","\\cdot \\widetilde{c}^{k/(1-\\gamma )} \\cdot \\gamma ^{-k}\\cdot (1-\\gamma )^{-k\\gamma /(1-\\gamma )} \\cdot y^{k\\gamma /(1-\\gamma )}\\cdot {1}_{[0,\\infty [}(y)=: f_{\\widetilde{c},\\gamma }(y),\\quad y \\in \\mathbb {R},\\nonumber $ where $a_{k} := 1/\\Gamma (\\frac{k \\cdot \\gamma }{1-\\gamma } )$ (see e.g.","Aalen [1] with a different parametrization).","Behaviour at zero: $\\mathbb {}[ \\, [0,\\infty [ \\, ] = \\mathbb {\\Pi }[W \\ge 0]=1$ , $\\mathbb {}[ \\, \\lbrace 0\\rbrace \\, ] = \\mathbb {\\Pi }[W = 0]= e^{-\\theta }$ .","Corresponding generator: $\\varphi _{\\gamma ,\\widetilde{c}}^{(0)} = \\widetilde{c} \\cdot \\varphi _{\\gamma }$ (cf.","(REF ), (REF )) of the power divergence given in the third line of (REF ); recall that the special case $\\gamma =\\frac{1}{2}$ corresponds to the prominent (multiple of the squared) Hellinger distance.","Sums: for i.i.d.","copies $(W_{i})_{i \\in \\mathbb {N}}$ of $W$ , the probability distribution of $\\breve{W} := \\sum _{i\\in I_{k}^{(n)}} W_{i}$ (cf.","Remark REF (ii)) is $C(POI(\\breve{\\theta }),GAM(\\alpha ,\\beta ))$ with $\\breve{\\theta } = \\frac{\\widetilde{c} \\cdot card(I_{k}^{(n)})}{\\gamma } > 0$ , $\\alpha = \\frac{\\widetilde{c}}{1-\\gamma } >0$ , $\\beta = \\frac{\\gamma }{1-\\gamma } >0$ .","(c) Case $\\gamma =2$ , $\\widetilde{c} >0$ : $\\Lambda _{2,\\widetilde{c}}^{(0)}(z)= \\frac{z^{2}}{2 \\widetilde{c}} + z$ (cf.","(REF ) ) is the well-known cumulant generating function of the Normal distribution (Gaussian distribution) $\\mathbb {} = N(1,\\frac{1}{\\widetilde{c}})$ with mean 1 and variance $\\frac{1}{\\widetilde{c}}$ .","Thus, $\\varphi _{2,\\widetilde{c}} \\in \\Upsilon (]-\\infty ,\\infty [)$ .","Type: $\\mathbb {}$ is an infinitely divisible (cf.","Proposition REF ) continuous distribution with density $f_{N(1,\\frac{1}{\\widetilde{c}})}(y) := \\sqrt{\\frac{\\widetilde{c}}{2 \\pi }} \\cdot \\exp (- \\frac{\\widetilde{c} \\cdot (y-1)^2}{2} )$ , ($y \\in \\mathbb {R}$ ).","Behaviour at zero: $\\mathbb {}[ \\, ]0,\\infty [ \\, ] = \\mathbb {\\Pi }[W > 0]=\\int _{0}^{\\infty } f_{N(1,\\frac{1}{\\widetilde{c}})}(u) \\, du \\in \\, ]0,1[$ , $\\mathbb {}[ \\, \\lbrace 0\\rbrace \\, ] = \\mathbb {\\Pi }[W = 0]= 0$ .","Corresponding generator: $\\varphi _{2,\\widetilde{c}}^{(0)} = \\widetilde{c} \\cdot \\varphi _{2}$ (cf.","(REF ), (REF )) is the generator of the $\\widetilde{c}-$ fold of the half Pearson-chisquare divergence given in the sixth line of (REF ).","Sums: for i.i.d.","copies $(W_{i})_{i \\in \\mathbb {N}}$ of $W$ , the probability distribution of $\\breve{W} := \\sum _{i\\in I_{k}^{(n)}} W_{i}$ (cf.","Remark REF (ii)) is $N(card(I_{k}^{(n)}),\\frac{card(I_{k}^{(n)})}{\\widetilde{c}})$ .","(d) Case $\\gamma <0$ , $\\widetilde{c} >0$ : $\\Lambda _{\\gamma ,\\widetilde{c}}^{(0)}(z) = \\frac{\\widetilde{c}}{\\gamma } \\cdot \\left\\lbrace \\left( \\frac{\\gamma -1}{\\widetilde{c}} \\cdot z +1 \\right)^{\\frac{\\gamma }{\\gamma -1}} -1 \\right\\rbrace $ (cf.","(REF )) is the cumulant generating function of a “tilted (i.e.","negatively distorted) stable distribution” $\\mathbb {}[ \\, \\cdot \\,] = \\mathbb {\\Pi }[W \\in \\cdot \\, ]$ of a random variable $W$ , which can be constructed as follows: let $Z$ be an auxiliary random variable (having density $f_{Z}$ and support $supp(Z) = [0,\\infty [$ ) of a stable law with parameter-quadruple $(\\frac{-\\gamma }{1-\\gamma },1,0,-\\frac{\\widetilde{c}^{1/(1-\\gamma )} \\cdot (1-\\gamma )^{-\\gamma /(1-\\gamma )}}{\\gamma })$ in terms of the “form-B notation” on p.12 in Zolotarev [428]; by applying a general Laplace-transform result on p.112 of the same text we can deduce $M_{Z}(z) := E_{\\mathbb {\\Pi }}[\\exp (z \\cdot Z)]= \\int _{0}^{\\infty } \\exp (z \\cdot y) \\cdot f_{Z}(y) \\, dy \\hspace{-5.69046pt} &=& \\hspace{-5.69046pt}{\\left\\lbrace \\begin{array}{ll}\\exp \\Big (\\frac{\\widetilde{c}^{1/(1-\\gamma )} \\cdot (1-\\gamma )^{-\\gamma /(1-\\gamma )}}{\\gamma }\\cdot (-z)^{\\alpha } \\Big ),\\quad \\textrm {if } \\ z \\in ]-\\infty ,0] , \\\\\\infty , \\hspace{150.79968pt} \\textrm {if } \\ z \\in \\, ]0,\\infty [, \\\\\\end{array}\\right.","}$ where $\\alpha := - \\frac{\\gamma }{1-\\gamma } \\in \\, ]0,1[$ .","Since $0 \\notin int(dom(M_{Z}))$ (and thus, $Z$ does not have light-tails) we have to tilt (dampen) the density in order to extend the effective domain.","Accordingly, let $W$ be a random variable having density $f_{W}(y)\\ :=\\ \\frac{\\exp \\lbrace -\\frac{y \\cdot \\widetilde{c}}{1-\\gamma }\\rbrace }{\\exp \\lbrace \\widetilde{c}/\\gamma \\rbrace }\\cdot f_{Z}(y)\\cdot {1}_{]0,\\infty [}(y),\\qquad y \\in \\mathbb {R},\\qquad \\text{(cf.","(\\ref {brostu3:fo.norweiemp7a}))}.\\nonumber $ Then one can straightforwardly deduce from (REF ) that $\\int _{0}^{\\infty } f_{W}(y) \\, dy =1$ and that $M_{W}(z) := E_{\\mathbb {\\Pi }}[\\exp (z \\cdot W)]= \\int _{0}^{\\infty } \\exp (z \\cdot y) \\cdot f_{W}(y) \\, dy \\hspace{-5.69046pt} &=& \\hspace{-5.69046pt}{\\left\\lbrace \\begin{array}{ll}\\exp \\left(\\frac{\\widetilde{c}}{\\gamma } \\cdot \\left\\lbrace \\left( \\frac{\\gamma -1}{\\widetilde{c}} \\cdot z +1 \\right)^{\\frac{\\gamma }{\\gamma -1}} -1 \\right\\rbrace \\right),\\qquad \\textrm {if } \\ z \\in ]-\\infty ,\\frac{\\widetilde{c}}{1-\\gamma }] , \\\\\\infty , \\hspace{156.49014pt} \\textrm {if } \\ z \\in \\, ]\\frac{\\widetilde{c}}{1-\\gamma },\\infty [ .","\\\\\\end{array}\\right.","}\\nonumber $ Hence, $\\varphi _{\\gamma ,\\widetilde{c}} \\in \\Upsilon (]0,\\infty [)$ .","Type: $\\mathbb {}$ is an infinitely divisible (cf.","Proposition REF ) continuous distribution with density $f_{W}$ .","Behaviour at zero: $\\mathbb {}[ \\, ] 0,\\infty [ \\, ] = \\mathbb {\\Pi }[W > 0]=1$ .","Corresponding generator: $\\varphi _{\\gamma ,\\widetilde{c}}^{(0)} = \\widetilde{c} \\cdot \\varphi _{\\gamma }$ (cf.","(REF ), (REF )) of the power divergence given in the first line of (REF ).","Sums: for i.i.d.","copies $(W_{i})_{i \\in \\mathbb {N}}$ of $W$ , the probability distribution of $\\breve{W} := \\sum _{i\\in I_{k}^{(n)}} W_{i}$ (cf.","Remark REF (ii)) has density $f_{\\breve{W}}(y)\\ :=\\ \\frac{\\exp \\lbrace -\\frac{y \\cdot \\widetilde{c}}{1-\\gamma }\\rbrace }{\\exp \\lbrace \\widetilde{c} \\cdot card(I_{k}^{(n)})/\\gamma \\rbrace }\\cdot f_{\\breve{Z}}(y)\\cdot {1}_{]0,\\infty [}(y),\\qquad y \\in \\mathbb {R},$ where $\\breve{Z}$ is a random variable with density $f_{\\breve{Z}}$ of a stable law with parameter-quadruple $(\\frac{-\\gamma }{1-\\gamma },1,0,-card(I_{k}^{(n)}) \\cdot \\frac{\\widetilde{c}^{1/(1-\\gamma )} \\cdot (1-\\gamma )^{-\\gamma /(1-\\gamma )}}{\\gamma })$ .","(e) Case $\\gamma >2$ , $\\widetilde{c} >0$ : $\\Lambda _{\\gamma ,\\widetilde{c}}^{(0)}(z) = \\frac{\\widetilde{c}}{\\gamma } \\cdot \\left\\lbrace \\left( \\frac{\\gamma -1}{\\widetilde{c}} \\cdot z +1 \\right)^{\\frac{\\gamma }{\\gamma -1}} -1 \\right\\rbrace $ (cf.","(REF )) is the cumulant generating function of a “distorted stable distribution” $\\mathbb {}[ \\, \\cdot \\,] = \\mathbb {\\Pi }[W \\in \\cdot \\, ]$ of a random variable $W$ , which can be constructed as follows: let $Z$ be an auxiliary random variable (having density $f_{Z}$ and support $supp(Z) = ]-\\infty ,\\infty ]$ ) of a stable law with parameter-quadruple $(\\frac{\\gamma }{\\gamma -1},1,0,\\frac{\\widetilde{c}^{1/(1-\\gamma )} \\cdot (\\gamma -1)^{\\gamma /(\\gamma -1)}}{\\gamma })$ in terms of the above-mentioned “form-B notation” ; by applying a general Laplace-transform result on p. 112 of Zolotarev [428], we can derive $M_{Z}(z) := E_{\\mathbb {\\Pi }}[\\exp (z \\cdot Z)]= \\int _{0}^{\\infty } \\exp (z \\cdot y) \\cdot f_{Z}(y) \\, dy \\hspace{-5.69046pt} &=& \\hspace{-5.69046pt}{\\left\\lbrace \\begin{array}{ll}\\exp \\Big (\\frac{\\widetilde{c}^{1/(1-\\gamma )} \\cdot (\\gamma -1 )^{\\gamma /(\\gamma -1)}}{\\gamma }\\cdot (-z)^{\\alpha } \\Big ),\\quad \\textrm {if } \\ z \\in ]-\\infty ,0] , \\\\\\infty , \\hspace{145.10922pt} \\textrm {if } \\ z \\in \\, ]0,\\infty [, \\\\\\end{array}\\right.","}$ where $\\alpha := \\frac{\\gamma }{\\gamma -1} \\in \\, ]1,2[$ .","Since $0 \\notin int(dom(M_{Z}))$ (and thus, $Z$ does not have light-tails) we have to distort the density in order to extend the effective domain.","Accordingly, let $W$ be a random variable having density $f_{W}(y)\\ :=\\ \\frac{\\exp \\lbrace \\frac{y \\cdot \\widetilde{c}}{\\gamma -1}\\rbrace }{\\exp \\lbrace \\widetilde{c}/\\gamma \\rbrace }\\cdot f_{Z}(- y), \\qquad y \\in \\mathbb {R},\\qquad \\text{(cf.","(\\ref {brostu3:fo.norweiemp7atwo}))}.\\nonumber $ Then one can straightforwardly deduce from (REF ) that $\\int _{-\\infty }^{\\infty } f_{W}(y) \\, dy =1$ and that $M_{W}(z) := E_{\\mathbb {\\Pi }}[\\exp (z \\cdot W)]= \\int _{-\\infty }^{\\infty } \\exp (z \\cdot y) \\cdot f_{W}(y) \\, dy \\hspace{-5.69046pt} &=& \\hspace{-5.69046pt}{\\left\\lbrace \\begin{array}{ll}\\exp \\left(\\frac{\\widetilde{c}}{\\gamma } \\cdot \\left\\lbrace \\left( \\frac{\\gamma -1}{\\widetilde{c}} \\cdot z +1 \\right)^{\\frac{\\gamma }{\\gamma -1}} -1 \\right\\rbrace \\right),\\qquad \\textrm {if } \\ z \\in [- \\frac{\\widetilde{c}}{\\gamma -1},\\infty [ , \\\\\\infty , \\hspace{156.49014pt} \\textrm {if } \\ z \\in \\, ]-\\infty , -\\frac{\\widetilde{c}}{\\gamma -1}[ .","\\\\\\end{array}\\right.","}\\nonumber $ Thus, $\\varphi _{\\gamma ,\\widetilde{c}} \\in \\Upsilon (]-\\infty ,\\infty [)$ .","Type: $\\mathbb {}$ is an infinitely divisible (cf.","Proposition REF ) continuous distribution with density $f_{W}$ .","Behaviour at zero: $\\mathbb {}[ \\, ]0,\\infty [ \\, ] = \\mathbb {\\Pi }[W > 0]=\\int _{0}^{\\infty } f_{W}(u) \\, du \\in \\, ]0,1[$ , $\\mathbb {}[ \\, \\lbrace 0\\rbrace \\, ] = \\mathbb {\\Pi }[W = 0]= 0$ .","Corresponding generator: $\\varphi _{\\gamma ,\\widetilde{c}}^{(0)} = \\widetilde{c} \\cdot \\varphi _{\\gamma }$ (cf.","(REF ), (REF )) of the power divergence given in the seventh line of (REF ).","Sums: for i.i.d.","copies $(W_{i})_{i \\in \\mathbb {N}}$ of $W$ , the probability distribution of $\\breve{W} := \\sum _{i\\in I_{k}^{(n)}} W_{i}$ (cf.","Remark REF (ii)) has density $f_{\\breve{W}}(y)\\ :=\\ \\frac{\\exp \\lbrace \\frac{y \\cdot \\widetilde{c}}{\\gamma -1}\\rbrace }{\\exp \\lbrace \\widetilde{c} \\cdot card(I_{k}^{(n)})/\\gamma \\rbrace }\\cdot f_{\\breve{Z}}(- y), \\qquad y \\in \\mathbb {R},\\nonumber $ where $\\breve{Z}$ is a random variable with density $f_{\\breve{Z}}$ of a stable law with parameter-quadruple $(\\frac{\\gamma }{\\gamma -1},1,0,card(I_{k}^{(n)})\\cdot \\frac{\\widetilde{c}^{1/(1-\\gamma )} \\cdot (\\gamma -1)^{\\gamma /(\\gamma -1)}}{\\gamma })$ .","(f) Case $\\gamma \\in ]1,2[$ , $\\widetilde{c} >0$ : one still has the (cumulant-generating-function) candidate $\\Lambda _{\\gamma ,\\widetilde{c}}^{(0)}(z) = \\frac{\\widetilde{c}}{\\gamma } \\cdot \\left\\lbrace \\left( \\frac{\\gamma -1}{\\widetilde{c}} \\cdot z +1 \\right)^{\\frac{\\gamma }{\\gamma -1}} -1 \\right\\rbrace $ (cf.","(REF )), but for the crucial exponent there holds $\\frac{\\gamma }{\\gamma -1} > 2$ .","From this, we conjecture that $\\mathbb {}$ becomes a signed finite measure with total mass 1, i.e.","it has a density (with respect to some dominating measure) with positive and negative values which “integrates to 1” ; accordingly, our BS method can not be applied to this situation.","Remark 49 As a continuation of Remark REF and the note in the third line after (REF ), we have shown as a side effect that for $\\gamma \\in \\, ]-\\infty ,-1] \\, \\cup \\, ]0,1[ \\, \\cup \\, [2,\\infty [$ the distributions $\\mathbb {}_{\\gamma }$ and $\\mathbb {}_{1-\\gamma }$ of Example REF (b)-(e) are inverse to each other.","Example 50 for the power-divergence context of Example REF we obtain: (a) Case $\\gamma =1$ , $\\widetilde{c} >0$ , anchor point $c=0$ : $\\Lambda _{1,\\widetilde{c}}(z) =\\widetilde{c} \\cdot \\left( \\exp (\\frac{z}{\\widetilde{c}}) - 1 \\right)$ (cf.","(REF )) is the cumulant generating function of $\\mathbb {} = \\frac{1}{\\widetilde{c}} \\cdot POI(\\widetilde{c})$ being the “$\\frac{1}{\\widetilde{c}}-$ fold Poisson distribution with mean $\\widetilde{c}$ ” which means that $W = \\frac{1}{\\widetilde{c}} \\cdot Z$ for a $POI(\\widetilde{c})-$ distributed random variable $Z$ .","Thus, $\\varphi _{1,\\widetilde{c}} \\in \\Upsilon (]0,\\infty [)$ .","Prominent special case $\\widetilde{c} =1$ : $\\mathbb {} = POI(1)$ is the Poisson distribution with mean 1.","Type: $\\mathbb {}$ is an infinitely divisible (cf.","Proposition REF ) discrete distribution with frequencies: $\\mathbb {\\Pi }[W=\\ell \\cdot \\frac{1}{\\widetilde{c}}]= \\exp (-\\widetilde{c}) \\cdot \\frac{ \\widetilde{c}^{\\ell }}{\\ell !","}$ for all nonnegative integers $\\ell \\in \\mathbb {N}_{0}$ (and zero elsewhere).","Behaviour at zero: $\\mathbb {\\Pi }[W\\ge 0]=1$ , $\\mathbb {\\Pi }[W=0]= \\exp (-\\widetilde{c})$ .","Corresponding generator: $\\varphi _{1,\\widetilde{c}} = \\widetilde{c} \\cdot \\varphi _{1}$ (cf.","(REF ), (REF )) of the $\\widetilde{c}-$ fold of the Kullback-Leibler divergence (relative entropy) given in the fourth line of (REF ).","Sums: for i.i.d.","copies $(W_{i})_{i \\in \\mathbb {N}}$ of $W$ , the probability distribution of $\\breve{W} := \\sum _{i\\in I_{k}^{(n)}} W_{i}$ (cf.","Remark REF (ii)) is $\\frac{1}{\\widetilde{c}} \\cdot POI(\\widetilde{c} \\cdot card(I_{k}^{(n)}))$ .","(b) Case $\\gamma =1$ , $\\widetilde{c} =1$ , anchor point $c \\in \\mathbb {R}$ : $\\Lambda _{1,1}^{(c)}(z)= e^{c} \\cdot \\left( e^{z} - 1 \\right) + z \\cdot (1- e^{c})$ (cf.","(REF )) is the well-known cumulant generating function of the “shifted Poisson distribution” $\\mathbb {} = POI(e^{c}) + 1-e^{c}$ , i.e.","$W := Z + 1-e^{c}$ with a $POI(e^{c})-$ distributed random variable $Z$ .","Hence, $\\varphi _{1,1}^{(c)} \\in \\Upsilon (]1- e^{c},\\infty [)$ .","Type: $\\mathbb {}$ is a discrete distribution with frequencies: $\\mathbb {\\Pi }[W=\\ell + 1- e^{c}]= \\exp (-e^{c}) \\cdot \\frac{ e^{c \\cdot \\ell }}{\\ell !","}$ for all $\\ell \\in \\mathbb {N}_{0}$ (and zero elsewhere).","Behaviour at zero: $\\mathbb {\\Pi }[W > 0]=1$ iff $c <0$ , $\\mathbb {\\Pi }[W < 0] >0$ iff $c >0$ , $\\mathbb {\\Pi }[W=0] \\ne 0$ iff “$c =\\log (1+k)$ for some $k \\in \\mathbb {N}_{0}$ ” .","Corresponding generator: $\\varphi _{1,1}^{(c)}$ (cf.","(REF )) of the divergence $& & D_{\\varphi _{1,1}^{(c)}}(\\mathbf {Q},\\mathbf {P}) :=\\sum \\limits _{k=1}^{K} \\Big (q_{k} + p_{k} \\cdot (e^{c}-1) \\Big )\\cdot \\Big \\lbrace \\log \\Big (\\frac{q_{k}}{p_{k}} + e^{c}-1 \\Big ) - c \\Big \\rbrace - \\sum \\limits _{k=1}^{K} q_{k} + \\sum \\limits _{k=1}^{K} p_{k} ,\\nonumber \\\\& & \\hspace{113.81102pt} \\textrm {if } \\ \\mathbf {P} \\in \\mathbb {R}_{\\ne 0}^{K} \\ \\textrm {and } \\mathbf {Q} \\in \\mathbb {R}^{K}\\textrm {with } \\mathbf {Q} \\in \\, [ (1-e^{c}) \\cdot \\mathbf {P}, \\mathbf {\\infty } [\\ \\textrm {component-wise},$ which for $c=0$ coincides with the Kullback-Leibler divergence (relative entropy) given in the fourth line of (REF ).","Sums: for i.i.d.","copies $(W_{i})_{i \\in \\mathbb {N}}$ of $W$ , the probability distribution of $\\breve{W} := \\sum _{i\\in I_{k}^{(n)}} W_{i}$ (cf.","Remark REF (ii)) is $POI(card(I_{k}^{(n)}) \\cdot e^{c}) + (1-e^{c}) \\cdot card(I_{k}^{(n)})$ .","Remark 51 (a) One can see from the Examples REF and REF the interesting effect that the “homogeneous” class of power-divergence generators $(\\varphi _{\\gamma })_{\\gamma \\in \\mathbb {R}}$ are connected to a “very inhomogeneous” family $(\\mathbb {}_{\\gamma })_{\\gamma \\in \\mathbb {R}}$ of $W-$ distributions: discrete, continuous, mixture of discrete and continuous, as the parameter $\\gamma $ varies.","Moreover, some cases satisfy $\\mathbb {\\Pi }[W=0] =0$ and some $\\mathbb {\\Pi }[W=0] >0$ , some $\\mathbb {\\Pi }[W>0]=1$ and some $\\mathbb {\\Pi }[W>0] \\in ]0,1[$ .","(b) As a continuation of Remark REF and the note in the last line of Example REF (a), we have shown as a side effect that for the the natural-anchor-point choice $c=0$ , the distributions $\\mathbb {}_{1}$ of of Example REF (a) and $\\mathbb {}_{0}$ of Example REF (a) are inverse to each other.","Example 52 for the context of Example REF we obtain: Case $\\widetilde{c} >0$ , anchor point $c=0$ : $\\Lambda _{bw,\\beta ,\\widetilde{c}}(z) =-(\\frac{1}{\\beta }-1) \\cdot z + \\frac{\\widetilde{c}}{\\beta ^{2}} \\cdot \\Big \\lbrace 1 - \\sqrt{1-\\frac{2\\beta }{\\widetilde{c}} \\cdot z} \\ \\Big \\rbrace $ (cf.","(REF )) is the cumulant generating function of a probability distribution $\\mathbb {}[ \\, \\cdot \\,] = \\mathbb {\\Pi }[\\check{W} \\in \\cdot \\, ]$ of a random variable $\\check{W}$ , which can be constructed as follows: $\\check{W} := \\frac{W}{\\beta } - (\\frac{1}{\\beta } - 1)$ , where $W$ is the random variable constructed in Example REF (d) with $\\gamma =-1$ and with $\\widetilde{c}$ replaced by $\\frac{\\widetilde{c}}{\\beta ^{2}}$ (recall that $W$ has a tilted stable distribution).","In other words, $\\mathbb {}$ is a special kind of modified tilted stable distribution.","Type: $\\mathbb {}$ is an infinitely divisible (cf.","Proposition REF ) continuous distribution with density $f_{\\check{W}}(u) := \\beta \\cdot f_{W}(\\beta \\cdot u + 1 -\\beta )\\cdot \\mathbf {1}_{]- (\\frac{1}{\\beta } -1),\\infty [}(u)$ ($u \\in \\mathbb {R}$ ), where $f_{W}(\\cdot )$ is given in (REF ) with $\\gamma =-1$ and with $\\widetilde{c}$ replaced by $\\frac{\\widetilde{c}}{\\beta ^{2}}$ .","Behaviour at zero: $\\mathbb {}[ \\, ] 0,\\infty [ \\, ] =\\mathbb {\\Pi }[\\check{W} > 0] >0$ .","Corresponding generator: $\\varphi _{bw,\\beta ,\\widetilde{c}}^{(0)}$ (cf.","(REF )) of the — “non-probability version” of — the well-known blended weight chi-square divergence given in (REF ).","Sums: for i.i.d.","copies $(\\check{W}_{i})_{i \\in \\mathbb {N}}$ of $\\check{W}$ , the probability distribution of $\\breve{\\check{W}} := \\sum _{i\\in I_{k}^{(n)}} \\check{W}_{i}= \\frac{1}{\\beta } \\cdot \\sum _{i\\in I_{k}^{(n)}} W_{i} - n_{k}\\cdot (\\frac{1}{\\beta } -1)$ (cf.","Remark REF (ii)) has density $f_{\\breve{\\check{W}}}(u) := \\beta \\cdot f_{\\breve{W}}(\\beta \\cdot u + (1 -\\beta )\\cdot n_{k})\\cdot \\mathbf {1}_{]- n_{k} \\cdot (\\frac{1}{\\beta } -1),\\infty [}(u)$ ($u \\in \\mathbb {R}$ ), where $f_{\\breve{W}}(\\cdot )$ is given in (REF ) (cf.","Example REF (d)) with $\\gamma =-1$ and with $\\widetilde{c}$ replaced by $\\frac{\\widetilde{c}}{\\beta ^{2}}$ .","Example 53 for the context of Example REF we obtain: (a) Case $\\alpha \\in \\, ]0,\\infty [$ , $\\widetilde{c} >0$ , anchor point $c=0$ : $\\Lambda _{gKL,\\alpha ,\\widetilde{c}}(z) =- \\frac{\\widetilde{c}}{\\alpha } \\cdot \\log ((1+\\alpha ) - \\alpha \\cdot e^{z/\\widetilde{c}})$ (cf.","(REF )) is the cumulant generating function of $\\mathbb {} = \\frac{1}{\\widetilde{c}} \\cdot NB(\\frac{\\widetilde{c}}{\\alpha },\\frac{1}{1+\\alpha })$ being the “$\\frac{1}{\\widetilde{c}}-$ fold Negative-Binomial distribution with parameters $\\frac{\\widetilde{c}}{\\alpha }$ and $\\frac{1}{1+\\alpha }$ ” which means that $W = \\frac{1}{\\widetilde{c}} \\cdot Z$ for a $NB(\\frac{\\widetilde{c}}{\\alpha },\\frac{1}{1+\\alpha })-$ distributed random variable $Z$ .","Thus, $\\varphi _{gKL,\\alpha ,\\widetilde{c}} \\in \\Upsilon (]0,\\infty [)$ .","Prominent special case $\\widetilde{c}=1$ , $\\alpha =1$ (see below): $\\mathbb {} = NB(1,\\frac{1}{2})$ is the Negative-Binomial distribution with parameters 1 and $\\frac{1}{2}$ .","Type: $\\mathbb {}$ is an infinitely divisible (cf.","Proposition REF ) discrete distribution with frequencies: $\\mathbb {\\Pi }[W=\\ell \\cdot \\frac{1}{\\widetilde{c}}]=(-1)^{\\ell } \\cdot \\binom{- \\frac{\\widetilde{c}}{\\alpha }}{\\ell }\\cdot {\\alpha }^{\\ell } \\cdot (1+\\alpha )^{-\\ell - \\widetilde{c}/\\alpha }$ for all nonnegative integers $\\ell \\in \\mathbb {N}_{0}$ (and zero elsewhere).","Behaviour at zero: $\\mathbb {\\Pi }[W\\ge 0]=1$ , $\\mathbb {\\Pi }[W=0]= \\frac{1}{(1+a)^{\\widetilde{c}/\\alpha }}$ .","Corresponding generator: $\\varphi _{gKL,\\alpha ,\\widetilde{c}}$ (cf.","(REF )) of the divergence (REF ); the special case $\\widetilde{c}=1$ , $\\alpha =1$ — i.e.","$\\varphi _{gKL,1,1} =: \\varphi _{snKL,1}$ (cf.","(REF )) — corresponds to the generator of the — “non-probability version” of the — Jensen-Shannon divergence (symmetrized and normalized Kullback-Leibler divergence, symmetrized and normalized relative entropy) given in (REF ).","Sums: for i.i.d.","copies $(W_{i})_{i \\in \\mathbb {N}}$ of $W$ , the probability distribution of $\\breve{W} := \\sum _{i\\in I_{k}^{(n)}} W_{i}$ (cf.","Remark REF (ii)) is $\\frac{1}{\\widetilde{c}} \\cdot NB(\\frac{\\widetilde{c}}{\\alpha }\\cdot card(I_{k}^{(n)}),\\frac{1}{1+\\alpha })$ .","(b) Case $\\alpha \\in \\, ]-1,0[$ , $\\widetilde{c} >0$ , anchor point $c=0$ : for any integer $m \\in \\mathbb {N}$ being strictly larger than $\\widetilde{c}$ and the choice $\\alpha = - \\frac{\\widetilde{c}}{m}$ , we obtain $\\Lambda _{gKL,-\\widetilde{c}/m,\\widetilde{c}}(z) =m \\cdot \\log ((1 - \\frac{\\widetilde{c}}{m}) + \\frac{\\widetilde{c}}{m} \\cdot e^{z/\\widetilde{c}})$ (cf.","(REF )) which is the cumulant generating function of $\\mathbb {} = \\frac{1}{\\widetilde{c}} \\cdot BIN(m,\\frac{\\widetilde{c}}{m})$ being the “$\\frac{1}{\\widetilde{c}}-$ fold Binomial distribution with parameters $m$ and $\\frac{\\widetilde{c}}{m}$ ” which means that $W = \\frac{1}{\\widetilde{c}} \\cdot Z$ for a $BIN(m,\\frac{\\widetilde{c}}{m})-$ distributed random variable $Z$ .","Thus, $\\varphi _{gKL,-\\widetilde{c}/m,\\widetilde{c}} \\in \\Upsilon (]0,\\infty [)$ .","Type: $\\mathbb {}$ is a non-infinitely divisible discrete distribution with frequencies: $\\mathbb {\\Pi }[W=\\ell \\cdot \\frac{1}{\\widetilde{c}}]=\\binom{m}{\\ell }\\cdot (\\frac{\\widetilde{c}}{m})^{\\ell } \\cdot (1-\\frac{\\widetilde{c}}{m})^{m-\\ell }$ for $\\ell \\in \\lbrace 0,1, \\ldots , m\\rbrace $ (and zero elsewhere).","Behaviour at zero: $\\mathbb {\\Pi }[W\\ge 0]=1$ , $\\mathbb {\\Pi }[W=0]= (1-\\frac{\\widetilde{c}}{m})^{m}$ .","Corresponding generator: $\\varphi _{gKL,\\alpha ,\\widetilde{c}}$ (cf.","(REF )) of the divergence (REF ).","Sums: for i.i.d.","copies $(W_{i})_{i \\in \\mathbb {N}}$ of $W$ , the probability distribution of $\\breve{W} := \\sum _{i\\in I_{k}^{(n)}} W_{i}$ (cf.","Remark REF (ii)) is $\\frac{1}{\\widetilde{c}} \\cdot BIN(m \\cdot card(I_{k}^{(n)}),\\frac{\\widetilde{c}}{m})$ .","Example 54 for the context of Example REF we obtain: Case of anchor point $c=0$ : $\\Lambda _{twop}(z)= \\log \\Big ( p \\cdot e^{z_{1} \\cdot z} + (1- p) \\cdot e^{z_{2} \\cdot z} \\Big )$ (cf.","(REF )) is the well-known cumulant generating function of the two-point probability distribution $\\mathbb {} = p \\cdot \\delta _{z_{1}} + (1-p) \\cdot \\delta _{z_{2}}$ , where $z_{1} < 1 < z_{2}$ and $p= \\frac{z_{2} -1}{z_{2} - z_{1}}$ .","Hence, $\\varphi _{twop} \\in \\Upsilon (]z_{1},z_{2}[)$ .","Type: $\\mathbb {}$ is a discrete distribution with frequencies: $\\mathbb {\\Pi }[W=z_{1}] = p$ , $\\mathbb {\\Pi }[W=z_{2}] = 1-p$ (and zero elsewhere).","Behaviour at zero: $\\mathbb {\\Pi }[W > 0]=1$ iff $z_{1} >0$ , $\\mathbb {\\Pi }[W=0] \\ne 0$ iff $z_{1}=0$ .","Corresponding generator: $\\varphi _{twop}$ (cf.","(REF )) of the divergence given in (REF ).","Sums: for i.i.d.","copies $(W_{i})_{i \\in \\mathbb {N}}$ of $W$ , the probability distribution of $\\breve{W} := \\sum _{i\\in I_{k}^{(n)}} W_{i}$ (cf.","Remark REF (ii)) is the distribution of the $card(I_{k}^{(n)})-$ th step of a generalized random walk starting at zero; this has a nice explicit (“binomial-type” ) expression in the special case $z_{1} = - z_{2}$ , namely $\\sum _{\\ell =0}^{card(I_{k}^{(n)})} \\binom{card(I_{k}^{(n)})}{\\ell }\\cdot p^{card(I_{k}^{(n)}) - \\ell } \\cdot (1-p)^{\\ell } \\cdot \\delta _{z_{2}\\cdot (2 \\ell - card(I_{k}^{(n)}))}$ .","Example 55 for the context of Example REF we obtain: Case $\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c} \\in \\, ]0,\\infty [$ , anchor point $c=0$ : by using $\\breve{\\theta } := 1 + \\alpha \\cdot \\Big (\\frac{1}{\\beta _{2}}- \\frac{1}{\\beta _{1}} \\Big )$ one can see that $\\Lambda _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}(z) =\\breve{\\theta } \\cdot z - \\widetilde{c} \\cdot \\alpha \\cdot \\log \\Big (1+ \\frac{z}{\\widetilde{c}} \\cdot \\Big (\\frac{1}{\\beta _{2}} - \\frac{1}{\\beta _{1}} \\Big )- \\frac{z^{2}}{\\widetilde{c}^{2} \\cdot \\beta _{1} \\cdot \\beta _{2} }\\Big )$ for $z \\in \\, ]- \\widetilde{c} \\cdot \\min \\lbrace \\beta _{1}, \\beta _{2}\\rbrace ,\\widetilde{c} \\cdot \\min \\lbrace \\beta _{1}, \\beta _{2}\\rbrace [$ — with different boundary behaviour for the three subcases $\\beta _{1} < \\beta _{2}$ resp.","$\\beta _{1} > \\beta _{2}$ resp.","$\\beta _{1} = \\beta _{2}$ (cf.","(REF ),(REF ),(REF )) — is the cumulant generating function of a generalized asymmetric Laplace distribution $\\mathbb {}[ \\, \\cdot \\,] = \\mathbb {\\Pi }[W \\in \\cdot \\, ]$ of a random variable $W:= \\breve{\\theta } + Z_{1} - Z_{2}$ , where $Z_{1}$ respectively $Z_{2}$ are auxiliary random variables which are independent and $GAM(\\widetilde{c} \\cdot \\beta _{1},\\widetilde{c} \\cdot \\alpha )-$ distributed respectively $GAM(\\widetilde{c} \\cdot \\beta _{2},\\widetilde{c} \\cdot \\alpha )-$ distributed.","In particular, $E_{\\mathbb {\\Pi }}[W] = \\breve{\\theta } + \\frac{\\widetilde{c} \\cdot \\alpha }{\\widetilde{c} \\cdot \\beta _{1}}- \\frac{\\widetilde{c} \\cdot \\alpha }{\\widetilde{c} \\cdot \\beta _{2}} =1$ (as required).","Thus, $\\varphi _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}\\in \\Upsilon (]-\\infty ,\\infty [)$ .","Prominent special case $\\widetilde{c} =1$ , $\\alpha =1$ , $\\beta _{1}=\\beta _{2} =: \\beta $ (and hence, $\\breve{\\theta }=1$ ): $\\mathbb {}$ is a classical Laplace distribution (two-tailed exponential distribution, bilateral exponential law) with location parameter 1 and scale parameter $\\frac{1}{\\beta }$ .","Type: $\\mathbb {}$ is an infinitely divisible (cf.","Proposition REF ) continuous distribution with density $f(u) := \\frac{\\sqrt{2} \\cdot \\exp \\lbrace \\frac{1}{\\sigma \\cdot \\sqrt{2}} \\cdot (\\frac{1}{\\kappa } - \\kappa ) \\cdot (u-\\theta ) \\rbrace }{\\sqrt{\\pi } \\cdot \\sigma ^{\\tau + 1/2} \\cdot \\Gamma (\\tau )} \\cdot \\left( \\frac{\\sqrt{2} \\cdot |u - \\theta |}{\\kappa + \\frac{1}{\\kappa }} \\right)^{\\tau - 1/2}\\cdot K_{\\tau - 1/2}\\left( \\frac{1}{\\sigma \\cdot \\sqrt{2}} \\cdot \\Big (\\kappa + \\frac{1}{\\kappa }\\Big ) \\cdot |u - \\theta | \\right),\\quad u \\ne \\theta ,$ where $(\\theta ,\\kappa ,\\sigma ,\\tau )$ is given in Remark REF below and $K_{\\lambda }$ is the modified Bessel function of the third kind with index $\\lambda $ .","For the above-mentioned special case of the classical Laplace distribution, this considerably simplifies to $f(u):= \\frac{\\beta }{2} \\exp \\lbrace - \\beta \\cdot |u -1 | \\rbrace $ .","Behaviour at zero: $\\mathbb {}[ \\, ]0,\\infty [ \\, ] = \\mathbb {\\Pi }[W > 0]=\\int _{0}^{\\infty } f(u) \\, du \\in \\, ]0,1[$ , $\\mathbb {}[ \\, \\lbrace 0\\rbrace \\, ] = \\mathbb {\\Pi }[W = 0]= 0$ .","Corresponding generator: $\\varphi _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}$ (cf.","(REF ) respectively (REF ) respectively (REF )) of the divergence given in (REF ) respectively (REF ) respectively (REF ).","Sums: for i.i.d.","copies $(W_{i})_{i \\in \\mathbb {N}}$ of $W$ , the probability distribution of $\\breve{W} := \\sum _{i\\in I_{k}^{(n)}} W_{i}$ (cf.","Remark REF (ii)) is the same as that of a random variable $\\breve{\\widetilde{W}} := \\breve{\\theta } \\cdot card(I_{k}^{(n)}) +\\breve{Z}_{1} - \\breve{Z}_{2}$ , where $\\breve{Z}_{1}$ respectively $\\breve{Z}_{2}$ are auxiliary random variables which are independent and $GAM(\\widetilde{c} \\cdot \\beta _{1},\\widetilde{c} \\cdot \\alpha \\cdot card(I_{k}^{(n)}))-$ distributed respectively $GAM(\\widetilde{c} \\cdot \\beta _{2},\\widetilde{c}\\cdot \\alpha \\cdot card(I_{k}^{(n)}))-$ distributed.","Remark 56 In the book of Kotz et al.","[197] one can find a very comprehensive study on generalized asymmetric Laplace distributions (also known as Bessel function distributions, McKay distributions), their close relatives (such as e.g.","the financial-econometric variance gamma model of Madan & Seneta [240]) as well as their applications; see also e.g.","Klar [192] for connections with some other Gamma difference distributions.","[197] use a different parametrization $(\\theta ,\\kappa ,\\sigma ,\\tau )$ which is one-to-one with our parametrization $(\\breve{\\theta },\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}=1)$ , as follows: $\\theta = \\breve{\\theta }$ , $\\tau = \\widetilde{c} \\cdot \\alpha $ , $\\sigma = \\frac{1}{\\widetilde{c}} \\cdot \\sqrt{\\frac{2}{\\beta _{1} \\cdot \\beta _{2}}}$ , $\\kappa = \\frac{\\sqrt{\\frac{4}{\\widetilde{c}^{2}} + (\\beta _{1} - \\beta _{2})^2} \\, + \\beta _{2}- \\beta _{1}}{2 \\cdot \\sqrt{\\beta _{1} \\cdot \\beta _{2}}} >0$ .","In particular, this implies that we cover all generalized asymmetric Laplace distributions with mean 1.","For better comparability, we have used the parametrization $(\\theta ,\\kappa ,\\sigma ,\\tau )$ in the above-mentioned representation (REF ) of the density (due to [197]).","Let us end this section by giving some further comments on the task of finding concretely the probability distribution (if existent) $\\mathbb {}[ \\, \\cdot \\,] = \\mathbb {\\Pi }[W \\in \\cdot \\, ]$ from the Fenchel-Legendre transform $\\Lambda = \\varphi ^{*}$ of a pregiven divergence generator $\\varphi $ , which should satisfy $\\exp (\\Lambda (z)) = \\int _{\\mathbb {R}}e^{z \\cdot y} \\, d\\mathbb {} (y)= E_{\\mathbb {\\Pi }}[\\exp (z \\cdot W)] ,\\qquad z \\in \\mathbb {R}, \\qquad \\text{(cf.","(\\ref {brostu3:repres lambda1}), (\\ref {brostu3:repres lambda2}))}.\\nonumber $ Recall that this is used for the simulation of the weights $(W_{i})_{i\\in \\mathbb {N}}$ which are i.i.d.","copies of $W$ and which are the crucial building ingredients of $\\xi _{n}^{\\mathbf {W}}$ in Theorem REF , respectively, of $\\xi _{n,\\mathbf {X}}^{w\\mathbf {W}}$ in Theorem REF .","The search for $\\mathbb {}$ can be done e.g.","by inversion of the moment generating function MGF, or by search in tables or computer software which list distributions and their MGF.","As already indicated above, we have eased/narrowed down this task by giving (additional) sufficient conditions for some deriving principal properties of $\\mathbb {}$ .","Also notice that $\\mathbb {}$ needs not necessarily to be explicitly known in full detail (e.g.","in terms of a computationally tractable density or frequency); for instance, as well known from insurance applications, for — comfortably straightforwardly simulable — doubly-random sums $W := \\sum _{i=1}^{N} A_{i}$ of nonnegative i.i.d.","random variables $(A_{i})_{i\\in \\mathbb {N}}$ with known law $\\Pi _A[\\, \\cdot \\, ]:= \\Pi [A \\in \\cdot \\, ]$ being independent of a counting-type random variable $N$ with known law $\\Pi _N$ , one can mostly compute explicitly $MGF_{\\mathbb {}}(z) = PGF_{\\Pi _N}(MGF_{\\Pi _A}(z))$ in terms of $\\mathbb {} := \\Pi _W$ and the probability generating function $PGF_{\\Pi _N}$ of $\\Pi _N$ , but the corresponding density/frequency of $\\mathbb {}$ may not be known explicitly in a tractable form.","The above-mentioned Example REF (b) of power divergences with generator $\\varphi _{\\gamma }$ ($\\gamma \\in ]0,1[$ ) manifests such a situation.","In the end, if no explicit distribution $\\mathbb {}$ and no comfortably simulable $W-$ construction are available, one can still try to simulate an i.i.d.","sequence $(W_{i})_{i\\in \\mathbb {N}}$ from the pregiven moment generating function (which is $\\exp (\\Lambda (z))$ here); see e.g.","McLeish [259] and references therein which also contains saddle point methods approximation techniques."],["Estimators","In the following, we demonstrate how one can principally implement our BS approach; a further, deeper analysis will be given in a follow-up paper."],["Estimators for the deterministic minimization problem","We address the minimization problem $D_{\\varphi }(\\mathbf {\\Omega },\\mathbf {P}):= \\inf _{\\mathbf {Q}\\in \\mathbf {\\Omega } } D_{\\varphi }(\\mathbf {Q},\\mathbf {P}) =\\inf _{\\widetilde{\\mathbf {Q}}\\in \\widetilde{\\mathbf {\\Omega }} }D_{\\widetilde{\\varphi } }(\\widetilde{\\mathbf {Q}},\\widetilde{{P}})=: D_{\\widetilde{\\varphi } }(\\widetilde{\\mathbf {\\Omega }},\\widetilde{{P}})\\qquad \\textrm {with } \\widetilde{\\mathbf {\\Omega }} :=\\mathbf {\\Omega } /M_{\\mathbf {P}}\\qquad \\textrm {(cf.","(\\ref {min Pb}) and (\\ref {min Pb prob2}))},$ whose numerical solution is based on Theorem REF which basically states that for large integer $n \\in \\mathbb {N}$ one has $\\inf _{\\mathbf {Q}\\in \\mathbf {\\Omega } } D_{\\varphi }(\\mathbf {Q},\\mathbf {P})\\approx -\\frac{1}{n}\\log \\,\\mathbb {\\Pi }\\left[\\xi _{n}^{\\mathbf {\\widetilde{W}}}\\in \\widetilde{\\mathbf {\\Omega }}\\right]$ in terms of $\\widetilde{\\varphi }:=M_{\\mathbf {P}} \\cdot \\varphi $ and the random vectors $\\xi _{n}^{\\mathbf {\\widetilde{W}}}=\\Big (\\frac{1}{n}\\sum _{i\\in I_{1}^{(n)}}\\widetilde{W}_{i},\\ldots ,\\frac{1}{n}\\sum _{i\\in I_{K}^{(n)}}\\widetilde{W}_{i}\\Big )\\hspace{56.9055pt} \\text{(cf.","(\\ref {Xi_n^W vector}))}\\nonumber $ with $n_{k}:=\\lfloor n \\cdot \\widetilde{p}_{k}\\rfloor $ leading to the disjoint index blocks $I_{1}^{(n)}:=\\left\\lbrace 1,\\ldots ,n_{1}\\right\\rbrace $ , $I_{2}^{(n)}:=\\left\\lbrace n_{1}+1,\\ldots ,n_{1}+n_{2}\\right\\rbrace $ , $\\ldots $ , $I_{K}^{(n)} := \\lbrace \\sum _{k=1}^{K-1} n_{k} + 1, \\ldots , n \\rbrace $ .","Recall that $\\mathbf {\\widetilde{W}} := (\\widetilde{W}_{1}, \\ldots , \\widetilde{W}_{n})$ is a random vector consisting of components $\\widetilde{W}_{i}$ which are i.i.d.","copies of the random variable $\\widetilde{W}$ whose distribution is $\\mathbb {\\Pi }[\\widetilde{W}\\in \\cdot \\,]=\\widetilde{\\mathbb {}}[\\,\\cdot \\,]$ obeying the representation $\\widetilde{\\varphi }(t) =\\sup _{z \\in \\mathbb {R}} \\left( z\\cdot t - \\log \\int _{\\mathbb {R}} e^{zy} d\\widetilde{\\mathbb {}}(y) \\right),\\qquad t \\in \\mathbb {R},\\qquad \\textrm {(cf.","(\\ref {brostu3:fo.link.var}))} .\\nonumber $ Hence, the estimation of $D_{\\varphi }(\\mathbf {\\Omega },\\mathbf {P})$ amounts to the estimation of $\\mathbb {\\Pi }\\left[\\xi _{n}^{\\mathbf {\\widetilde{W}}}\\in \\widetilde{\\mathbf {\\Omega }} \\right]$ .","For the rest of this subsection, we assume that $\\widetilde{{P}} \\in \\mathbb {S}_{> 0}^{K}$ , that $n$ is chosen such that all $n \\cdot \\widetilde{p}_{k}$ are integers (and hence, $n = \\sum _{k=1}^{K} n_{k}$ ), and that $\\widetilde{\\mathbf {\\Omega }} \\subset \\mathbb {R}^{K}$ satisfies the regularity property $cl(\\widetilde{\\mathbf {\\Omega }})=cl\\left( int\\left( \\widetilde{\\mathbf {\\Omega }} \\right) \\right) , \\qquad int\\left( \\widetilde{\\mathbf {\\Omega }} \\right) \\ne \\emptyset \\nonumber $ which implies that the same condition holds for $\\mathbf {\\Omega }$ ; moreover, we suppose that $D_{\\widetilde{\\varphi } }(\\widetilde{\\mathbf {\\Omega }},\\widetilde{{P}})$ is finite.","For the ease of the following discussions, we introduce the notations $T\\left( \\mathbf {x}\\right) :=\\left( \\frac{1}{n_{1}}\\sum _{i\\in I_{1}^{(n)}}x_{i},\\ldots ,\\frac{1}{n_{K}}\\sum _{i\\in I_{K}^{(n)}} x_{i}\\right)\\quad \\textrm {for any $ x:=( x1,..,xn) Rn$,}$$as well as $ D$ for the diagonal matrix with diagonal entries $ 1/p1,...,1/pK$and null entries off the diagonal.Accordingly, the probability in (\\ref {LDP Minimization approx}) becomes$$\\mathbb {\\Pi }\\left[\\xi _{n}^{\\mathbf {\\widetilde{W}}}\\in \\widetilde{\\mathbf {\\Omega }} \\right] \\ = \\ \\mathbb {\\Pi }\\left[ T(\\mathbf {\\widetilde{W}}) \\in \\mathbf {\\Lambda }\\right]$$where$$\\mathbf {\\Lambda } :=\\mathfrak {D} \\cdot \\widetilde{\\mathbf {\\Omega }}$$is a set of vectors in $ RK$ which is known/derived from the concrete context.The \\textit {naive estimator} $ Lnaive$ of$ [ nW ]$is constructedthrough the following procedure:simulate independently $ L$ copies$ W1,...,WL$ of the vector$ W:=( W1,...,Wn) $,with independent entries under $$, and define(with a slight abuse of notation)$$\\widehat{\\widetilde{\\Pi }}_{L}^{naive} :=\\frac{1}{L}\\sum _{\\ell =1}^{L} \\mathbf {1}_{\\mathbf {\\Lambda }}\\left( T\\left( \\mathbf {\\widetilde{W}}^{\\ell }\\right) \\right) ;$$however this procedure is time costly, since this estimate has a very bad hit rate.Thus, in the following, a so-called “efficient Importance Sampling (IS)”\\ scheme --- in the sense of Sadowsky\\& Bucklew \\cite {Sad:90} (denoted [SB] hereunder) ---is adapted for the sophisticated (i.e.","non-naive) estimation of$ [ nW ]$.The main property of IS schemes lays in the fact that the runtime for anestimate with a controlled relative error does \\textit {not} increase atexponential rate as $ n$ increases, in contrast to$ Lnaive$ which has exponential increase.In detail, let $ >0$ be agiven relative precision for an estimator $ PSLn$ of$ [ nW ]$, based on a number $ Ln$ of simulated samplesgenerated under some distribution $ S$, so that$$\\delta :=\\frac{var_{\\widetilde{S}}P_{\\widetilde{S}}^{L_{n}}}{\\left(\\mathbb {\\Pi }\\left[\\xi _{n}^{\\mathbf {\\widetilde{W}}}\\in \\widetilde{\\mathbf {\\Omega }}\\right]\\right)^{2}} \\, .$$Then $ Ln$ will grow exponentially as $ n$ tends to infinity if and only if$ S$ is not “asymptotically optimal”\\ ,the derivation of which is the scope of the current section.$ To start with the details, for the sake of brevity (to avoid certain substantial discussions on potential technical relaxations) we shall employ the following additional Assumption (OM) on the set $\\widetilde{\\mathbf {\\Omega }}$ : (OM) For any $\\widetilde{\\omega } \\in cl(\\widetilde{\\mathbf {\\Omega }})$ there exists a vector $\\mathbf {x} = \\left(x_{1},\\ldots ,x_{n}\\right) \\in \\left] t_{-}^{sc},t_{+}^{sc}\\right[^{n}$ such that $\\widetilde{\\omega } = \\left( \\frac{1}{n}\\sum _{i\\in I_{1}^{(n)}}x_{i},\\ldots ,\\frac{1}{n} \\sum _{i\\in I_{K}^{(n)}}x_{i}\\right)$ , or equivalently, for any $\\lambda \\in cl(\\mathbf {\\Lambda })$ there exists a vector $\\mathbf {x} = \\left(x_{1},\\ldots ,x_{n}\\right) \\in \\left] t_{-}^{sc},t_{+}^{sc}\\right[^{n}$ such that $\\lambda =T\\left(\\mathbf {x}\\right)$ .","For instance, in the common case $dom(\\widetilde{\\varphi })= dom(\\varphi ) = \\, ]a,b[ \\, =\\, \\left] t_{-}^{sc},t_{+}^{sc}\\right[ \\, = \\, ]0,\\infty [$ (e.g.","for the power-divergence generators $\\widetilde{\\varphi }= \\widetilde{c} \\cdot \\varphi _{\\gamma }$ , $\\gamma \\le 0$ , cf.","Example REF ) the Assumption (OM) is always feasible.","To proceed, for any distribution $\\widetilde{S}$ on $\\mathbb {R}^{n}$ with support included in the support of the product measure $\\widetilde{\\mathbb {}}^{\\otimes n}$ it holds $\\mathbb {\\Pi }\\left[\\xi _{n}^{\\mathbf {\\widetilde{W}}}\\in \\widetilde{\\mathbf {\\Omega }}\\right]=E_{\\widetilde{\\mathbb {}}^{\\otimes n}} \\left[\\mathbf {1}_{\\mathbf {\\Lambda } }( T( \\mathbf {\\widetilde{W}}))\\right] =E_{\\widetilde{S}}\\left[\\mathbf {1}_{\\mathbf {\\Lambda } }\\left( T\\left( \\mathbf {\\widetilde{V}}\\right) \\right) \\cdot \\frac{d\\widetilde{\\mathbb {}}^{\\otimes n}}{d\\widetilde{S}}\\left( \\mathbf {\\widetilde{V}}\\right) \\right]$ from where the improved IS estimator of $\\mathbb {\\Pi }\\left[\\xi _{n}^{\\mathbf {\\widetilde{W}}}\\in \\widetilde{\\mathbf {\\Omega }} \\right]$ is obtained by sampling $L$ i.i.d.","replications $\\mathbf {\\widetilde{V}}^{1},\\ldots ,\\mathbf {\\widetilde{V}}^{L}$ of the random vector $\\mathbf {\\widetilde{V}}$ with distribution $\\widetilde{S}$ and by defining $\\widehat{\\widetilde{\\Pi }}_{L}^{improved}:=\\frac{1}{L}\\sum _{\\ell =1}^{L}\\mathbf {1}_{\\mathbf {\\Lambda }}( T( \\mathbf {\\widetilde{V}}^{(\\ell )})) \\cdot \\frac{d\\widetilde{\\mathbb {}}^{\\otimes n}}{d\\widetilde{S}}\\left( \\mathbf {\\widetilde{V}}^{(\\ell )}\\right) \\ .$ The precise form of the efficient IS distribution $\\widetilde{S}^{opt}$ relies on the definition of a “dominating point” of $\\mathbf {\\Lambda }$ , which we recall now.","For $\\mathbf {x} := \\left( x_{1},..,x_{n}\\right)$ in $\\mathbb {R}^{n}$ we define $I_{\\mathbf {\\widetilde{W}}}(\\mathbf {x}):=\\sup _{\\mathbf {z} \\in \\mathbb {R}^{n}}\\left(\\left\\langle \\mathbf {z}, \\mathbf {x} \\right\\rangle -\\log E_{\\widetilde{\\mathbb {}}}[ \\, \\exp (\\langle \\mathbf {z},\\mathbf {\\widetilde{W}}\\rangle ) ]\\right) \\, ,$ and for $\\lambda $ in $\\mathbf {\\Lambda }$ we let $I(\\lambda ):=\\inf \\left\\lbrace I_{\\mathbf {\\widetilde{W}}}(\\mathbf {x}):T(\\mathbf {x)=\\lambda }\\right\\rbrace .$ Let $\\underline{\\lambda }:=\\left( \\underline{\\lambda }_{1},\\ldots ,\\underline{\\lambda }_{K}\\right) \\in \\partial \\mathbf {\\Lambda }$ .","We call $\\underline{\\lambda }$ a minimal rate point (mrp) of $\\mathbf {\\Lambda }$ if $I(\\underline{\\lambda })\\le I(\\lambda ) \\quad \\text{ for all }\\lambda \\in \\mathbf {\\Lambda } .$ A minimal rate point $\\underline{\\lambda }$ is called a dominating point of $\\mathbf {\\Lambda }$ if a) $\\underline{\\lambda }\\in \\partial \\mathbf {\\Lambda }$ , and b) $I\\left(\\underline{\\lambda }\\right)\\le I(\\lambda )$ for all $\\lambda \\in \\mathbf {\\Lambda }$ with attainment, namely there exists a vector $\\underline{\\mathbf {x}} \\in \\left] t_{-}^{sc},t_{+}^{sc}\\right[^{n}$ such that $I_{\\widetilde{W}}\\left( \\underline{\\mathbf {x}}\\right) =I\\left(\\underline{\\lambda }\\right)$ with $\\underline{\\lambda }=T\\left( \\underline{\\mathbf {x}}\\right)$ .","The characterization of the dominating point $\\underline{\\lambda }$ is settled in the following Lemma 57 Let $\\underline{\\lambda }$ be a mrp of $\\mathbf {\\Lambda }$ .","Then, under Assumption (OM), $\\underline{\\lambda }$ is a dominating point, and $\\inf \\left\\lbrace I_{\\mathbf {\\widetilde{W}}}\\left( \\mathbf {x}\\right) ,T(\\mathbf {x})=\\underline{\\lambda }\\right\\rbrace $ is reached at some vector $\\underline{\\mathbf {x}}$ in $\\left] t_{-}^{sc},t_{+}^{sc}\\right[^{n}$ such that for all $k \\in \\left\\lbrace 1,\\ldots ,K\\right\\rbrace $ and all $i \\in I_{k}^{(n)}$ there holds $\\underline{x}_{i}= \\underline{\\lambda }_{k}$ and $I_{\\mathbf {\\widetilde{W}}}(\\underline{\\mathbf {x}})=n \\cdot \\sum _{k=1}^{K}\\widetilde{p}_{k} \\cdot \\widetilde{\\varphi }\\left(\\underline{\\lambda }_{k}\\right)$ .","The proof Lemma REF is given in Appendix G. Notice that (OM) implies the existence of a dominating point $\\underline{\\lambda }$ , but uniqueness may not hold.","In the latter case, one can try to proceed as in Theorem 2 of [SB] and the discussion thereafter.","However, we assume now uniqueness of $\\underline{\\lambda }$ ; this allows for the identification of $\\widetilde{S}^{opt}$ .","By Theorem 1 of [SB] and Theorem 3.1 of Csiszar [96], the asymptotically optimal IS distribution $\\widetilde{S}^{opt}$ is obtained as the Kullback-Leibler projection of $\\widetilde{\\mathbb {}}^{n\\otimes }$ on the set of all probability distributions on $\\mathbb {R}^{n}$ centered at point $\\underline{\\mathbf {x}}$ , whose coordinates are — according to Lemma REF — functions of the coordinates of $\\underline{\\mathbf {\\widetilde{Q}}} := \\mathfrak {D}^{-1} \\underline{\\lambda }$ such that $T\\left( \\underline{\\mathbf {x}}\\right)= \\mathfrak {D} \\underline{\\mathbf {\\widetilde{Q}}}$ .","The above definition of $\\widetilde{S}^{opt}$ presumes the knowledge of $\\underline{\\lambda }$ , which cannot be assumed (otherwise the minimization problem is solved in advance).","The aim of the following construction is to provide a proxy $\\widetilde{S}$ to $\\widetilde{S}^{opt}$ , where $\\widetilde{S}$ is the Kullback-Leibler projection of $\\widetilde{\\mathbb {}}^{\\otimes n}$ on the set of all probability distributions on $\\mathbb {R}^{n}$ centered at some point $\\mathbf {x}^{\\ast }$ which is close to $\\underline{\\mathbf {x}}$ .","For this sake, we need to have at hand a proxy of $\\underline{\\lambda }$ or, equivalently, a preliminary guess $\\mathbf {\\widetilde{Q}}^{\\ast }$ of $\\underline{\\mathbf {\\widetilde{Q}}}:=\\arg \\inf _{\\mathbf {\\widetilde{Q}} \\in \\widetilde{\\mathbf {\\Omega }}}\\sum _{k=1}^{K} \\widetilde{p}_{k} \\cdot \\widetilde{\\varphi }(\\widetilde{q}_{k}/\\widetilde{p}_{k})$ .","This guess is by no means produced in order to provide a direct estimate of $D_{\\widetilde{\\varphi } }(\\widetilde{\\mathbf {\\Omega }},\\widetilde{{P}})$ but merely to provide the IS distribution $\\widetilde{S}$ which in turn leads to a sharp estimate of $D_{\\widetilde{\\varphi } }(\\widetilde{\\mathbf {\\Omega }},\\widetilde{{P}})$ .","Proxy method 1: in some cases we might have at hand some particular point $\\mathbf {\\widetilde{Q}}^{\\ast } :=\\left(\\widetilde{q}_{1}^{\\ast },..,\\widetilde{q}_{K}^{\\ast }\\right)$ in $\\widetilde{\\mathbf {\\Omega }}$ ; the resulting IS distribution $\\widetilde{S}$ with $\\mathbf {\\widetilde{Q}}$ substituted by $\\mathbf {\\widetilde{Q}^{\\ast }}$ is not optimal in the sense of [SB], but anyhow produces an estimator with good hitting rate, possibly with a loss in the variance.","A simple way to obtain such a point $\\mathbf {\\widetilde{Q}^{\\ast }}$ in $\\widetilde{\\mathbf {\\Omega }}$ is to simulate runs of (say) $M-$ variate i.i.d.","vectors $\\mathbf {\\widetilde{W}}$ under $\\widetilde{\\mathbb {}}^{\\otimes M}$ until the first time where $\\xi _{M}^{\\mathbf {\\widetilde{W}}}$ belongs to $\\widetilde{\\mathbf {\\Omega }}$ ; then we set $\\mathbf {\\widetilde{Q}^{\\ast }} := \\xi _{M}^{\\mathbf {\\widetilde{W}}}$ for the succeeding realization $\\mathbf {\\widetilde{W}}$ .","Before we proceed, it is useful to mention that the need for a drastic fall in the number of simulation runs pertains for cases when $D_{\\widetilde{\\varphi } }(\\widetilde{\\mathbf {\\Omega }},\\widetilde{{P}})$ is large.","The following construction is suited to this case, which is of relevance in applications both in optimization and in statistics when choosing between competing models none of which is assumed to represent the true one, but merely less inadequate ones.","Proxy method 2: when $D_{\\widetilde{\\varphi } }(\\widetilde{\\mathbf {\\Omega }},\\widetilde{{P}})$ is presumably large, we make use of asymptotic approximation to get a proxy of $\\underline{\\mathbf {\\widetilde{Q}}}$ .","For this, we define a sampling distribution on $\\mathbb {R}^{K}$ fitted to the divergence through $f(\\mathbf {\\widetilde{Q}}):=C \\cdot \\exp \\left( -\\sum _{k=1}^{K} \\widetilde{p}_{k} \\cdot \\widetilde{\\varphi }(\\widetilde{q}_{k}/\\widetilde{p}_{k})\\right) \\ = \\ C \\cdot \\exp \\left( -D_{\\widetilde{\\varphi }}\\left( \\mathbf {\\widetilde{Q}},\\widetilde{{P}}\\right)\\right)$ where $C$ is a normalizing constant.","Let $\\mathbf {T}$ be a $K-$ variate random variable with density $f$ .","The distribution of $\\mathbf {T}$ given $\\left( \\mathbf {T} \\in \\widetilde{\\mathbf {\\Omega }}\\right)$ concentrates around $\\arg \\inf _{\\widetilde{\\mathbf {Q}}\\in \\widetilde{\\mathbf {\\Omega }} }D_{\\widetilde{\\varphi } }(\\widetilde{\\mathbf {Q}},\\widetilde{{P}})$ when $D_{\\widetilde{\\varphi } }(\\widetilde{\\mathbf {\\Omega }},\\widetilde{{P}})$ is large.","Indeed, for any $\\widetilde{\\mathbf {Q}}\\in \\widetilde{\\mathbf {\\Omega }}$ denote by $\\mathbf {V}_{\\varepsilon }(\\widetilde{\\mathbf {Q}})$ a small neighborhood of $\\widetilde{\\mathbf {Q}}$ in $\\mathbb {R}^{K}$ with radius $\\varepsilon $ ; clearly, the probability of the event $\\left( \\mathbf {T} \\in \\mathbf {V}_{\\varepsilon }(\\widetilde{\\mathbf {Q}})\\right)$ when restricted to $\\widetilde{\\mathbf {Q}}\\in \\widetilde{\\mathbf {\\Omega }}$ is maximum when $\\widetilde{\\mathbf {Q}} = \\underline{\\widetilde{\\mathbf {Q}}}$ , where $\\underline{\\widetilde{\\mathbf {Q}}}$ is the “dominating point of $\\widetilde{\\mathbf {\\Omega }}$ ” in the sense that $\\underline{\\mathbf {\\widetilde{Q}}} := \\mathfrak {D}^{-1} \\underline{\\lambda }$ is the above-defined transform of the dominating point $\\underline{\\lambda }$ (assuming uniqueness); a precise argumentation under adequate conditions is postponed to Appendix G. Accordingly, we obtain a proxy $\\mathbf {\\widetilde{Q}}^{\\ast }$ of $\\underline{\\mathbf {\\widetilde{Q}}}$ by simulating a sequence of independent $K-$ variate random variables $\\mathbf {T}_{1},\\ldots $ with distribution (REF ) until (say) $\\mathbf {T}_{m}$ belongs to $\\widetilde{\\mathbf {\\Omega }}$ and set $\\mathbf {\\widetilde{Q}}^{\\ast }:=\\mathbf {T}_{m}$ .","To proceed with the derivation of the IS sampling distribution $\\widetilde{S}$ on $\\mathbb {R}^{n}$ , we fix $\\mathbf {\\widetilde{Q}}^{\\ast } :=\\left(\\widetilde{q}_{1}^{\\ast },..,\\widetilde{q}_{K}^{\\ast }\\right)$ to be a proxy of $\\underline{\\mathbf {\\widetilde{Q}}}$ or an initial guess in $\\widetilde{\\mathbf {\\Omega }}$ .","As an intermediate step, we construct the probability distribution $\\widetilde{U}_{k}$ on $\\mathbb {R}$ given by $d\\widetilde{U}_{k}(v) \\ := \\ \\exp \\left( \\tau _{k} \\cdot v -\\Lambda _{\\widetilde{\\mathbb {}}}(\\tau _{k})\\right) d\\widetilde{\\mathbb {}}(v)=\\frac{\\exp \\left(\\tau _{k} \\cdot v\\right)}{MGF_{\\widetilde{\\mathbb {}}}(\\tau _{k})} \\, d\\widetilde{\\mathbb {}}(v)$ where $\\tau _{k} \\in int(dom(MGF_{\\widetilde{\\mathbb {}}}))$ is the unique solution of the equation $\\Lambda _{\\widetilde{\\mathbb {}}}^{\\prime }\\left( \\tau _{k}\\right) =\\frac{\\widetilde{q}_{k}^{\\ast }}{\\widetilde{p}_{k}}$ and thus — by relation (REF ) of Appendix F — we can compute explicitly $\\tau _{k} = \\ \\widetilde{\\varphi }^{\\, \\prime }\\left(\\frac{\\widetilde{q}_{k}^{\\ast }}{\\widetilde{p}_{k}}\\right) \\ .$ Therefore, $\\widetilde{U}_{k}$ is the Kullback-Leibler projection of $\\widetilde{\\mathbb {}}$ on the class of all probability distributions on $\\mathbb {R}$ whose expectation is $\\widetilde{q}_{k}^{\\ast }$ .","As a side remark, notice that one possible way of obtaining an explicit form of the probability distribution $\\widetilde{U}_{k}$ may be by identification through its moment generating function $dom(MGF_{\\widetilde{\\mathbb {}}})-\\tau _{k} \\ \\ni \\ z \\ \\mapsto MGF_{\\widetilde{U}_{k}}(z) =\\frac{MGF_{\\widetilde{\\mathbb {}}}(z+\\tau _{k})}{MGF_{\\widetilde{\\mathbb {}}}(\\tau _{k})}$ of which all ingredients are principally available.","For instance, this will be used in Example REF below.","From (REF ), we define $\\widetilde{S}_{k}:=\\underbrace{\\widetilde{U}_{k} \\otimes \\cdots \\otimes \\widetilde{U}_{k}}_{n_{k}\\text{times}}\\qquad \\textrm {for all $ {1,...,K}$},$$whence\\begin{eqnarray}d\\widetilde{S}_{k}\\left( v_{k,1},\\ldots ,v_{k,n_{k}}\\right) =\\exp \\Big ( \\sum _{i\\in I_{k}^{(n)}}\\tau _{k} \\cdot v_{k,i}-n_{k} \\cdot \\Lambda _{\\widetilde{\\mathbb {}}}(\\tau _{k})\\Big ) \\ d\\widetilde{\\mathbb {}}\\left(v_{k,1}\\right)\\cdots d\\widetilde{\\mathbb {}}\\left(v_{k,n_{k}}\\right),\\\\\\end{eqnarray}which manifests $ Sk$ as the Kullback-Leibler projection of$ nktimes$on the class of all probability distributions on $ Rk$whose expectation vector is$ Q = (q1,...,qK) Rk$.","Let now\\begin{equation}\\widetilde{S} := \\widetilde{S}_{1} \\otimes \\cdots \\otimes \\widetilde{S}_{K} \\, ,\\end{equation}which therefore satisfies (recall that $ k=1K nk =n$)\\begin{equation}d\\widetilde{S}\\left( v_{1,1},\\ldots v_{1,n_{1}},\\ldots , v_{K,1},\\ldots v_{K,n_{K}}\\right)=\\exp \\Big (\\sum _{k=1}^{K}\\sum _{i\\in I_{k}^{(n)}} \\big ( \\tau _{k} \\cdot v_{k,i} - n_{k} \\cdot \\Lambda _{\\widetilde{\\mathbb {}}}(\\tau _{k}) \\big )\\Big ) \\,d\\widetilde{\\mathbb {}}^{\\otimes n}\\left( v_{1,1},\\ldots v_{1,n_{1}},\\ldots , v_{K,1},\\ldots v_{K,n_{K}}\\right).\\ \\end{equation}The same procedure with all $ qk$substituted by the coordinates $qk$of $Q$produces $ Sopt$.","Therefore, $ S$ is a substitutefor $ Sopt$ with the change in the centering fromthe unknown vector $Q$to its proxy $ Q$.$ As a straightforward consequence of (REF ) and (), we obtain the improved IS estimator of $\\mathbb {\\Pi }\\left[\\xi _{n}^{\\mathbf {\\widetilde{W}}}\\in \\widetilde{\\mathbf {\\Omega }}\\right]$ as $\\widehat{\\widetilde{\\Pi }}_{L}^{improved}\\ =\\ \\frac{1}{L}\\sum _{\\ell =1}^{L}\\mathbf {1}_{\\mathbf {\\Lambda }}( T( \\mathbf {\\widetilde{V}}^{(\\ell )})) \\cdot \\prod _{k=1}^{K} IS_{k}(\\mathbf {\\widetilde{V}}_{k}^{(\\ell )})$ where $\\mathbf {\\widetilde{V}}_{k}^{(\\ell )} := \\left( \\widetilde{V}_{i}^{(\\ell )} \\right)_{i \\in I_{k}^{(n)}}$ is the $k-$ th block of the $\\ell -$ th replication $\\mathbf {\\widetilde{V}}^{(\\ell )}$ of $\\mathbf {\\widetilde{V}}$ under $S$ , and the $k-$ th importance-sampling factor is $\\widetilde{IS}_{k}(v_{k,1},\\ldots ,v_{k,n_{k}}) \\ := \\ \\frac{d\\widetilde{\\mathbb {}}^{\\otimes n_{k}}}{d\\widetilde{S}_{k}}\\left( v_{k,1},\\ldots ,v_{k,n_{k}}\\right)= \\exp \\Big (n_{k} \\cdot \\Lambda _{\\widetilde{\\mathbb {}}}(\\tau _{k}) \\, - \\, \\tau _{k}\\cdot \\sum _{i=1}^{n_{k}} v_{k,i} \\Big )\\nonumber $ with $n_{k} = card(I_{k}^{(n)})$ .","Summing up things, we arrive at the following algorithm in case that $\\widetilde{\\mathbf {\\Omega }}$ has a unique dominating point (in the above-defined sense): Step D1 Exemplarily, we start with proxy method 2 (the other proxy method 1 works analogously): get a proxy $\\mathbf {\\widetilde{Q}}^{\\ast }$ of $\\underline{\\mathbf {\\widetilde{Q}}}$ by simulating a sequence of independent $K-$ variate random variables $\\mathbf {T}_{1},\\ldots $ with distribution (REF ) until (say) $\\mathbf {T}_{m}$ belongs to $\\widetilde{\\mathbf {\\Omega }}$ and set $\\mathbf {\\widetilde{Q}}^{\\ast }:=\\mathbf {T}_{m}$ .","Step D2 For all $k$ in $\\lbrace 1,\\ldots ,K\\rbrace $ compute $\\tau _{k} = \\ \\widetilde{\\varphi }^{\\, \\prime }\\left(\\frac{\\widetilde{q}_{k}^{\\ast }}{\\widetilde{p}_{k}}\\right)$ .","Step D3 For all $\\ell $ in $\\lbrace 1,\\ldots ,L\\rbrace $ perform a run of $\\mathbf {\\widetilde{V}}^{(\\ell )}$ under $\\widetilde{S}$ as follows: For all $k$ in $\\lbrace 1,\\ldots ,K\\rbrace $ simulate $n_{k}$ i.i.d.","random variables $\\widetilde{V}_{k_{1}}^{(\\ell )},\\ldots ,\\widetilde{V}_{k_{n_{k}}}^{(\\ell )}$ with common distribution $\\widetilde{U}_{k}$ defined in (REF ).","Set $\\mathbf {\\widetilde{V}}_{k}^{(\\ell )} :=(\\widetilde{V}_{k_{1}}^{(\\ell )},\\ldots ,\\widetilde{V}_{k_{n_{k}}}^{(\\ell )})$ to be the corresponding row vector.","Construct $\\mathbf {\\widetilde{V}}^{(\\ell )}$ as the row vector obtained by concatenating the $\\mathbf {\\widetilde{V}}_{k}^{(\\ell )}$ , i.e.","$\\mathbf {\\widetilde{V}}^{(\\ell )}:=\\left( \\mathbf {\\widetilde{V}}_{1}^{(\\ell )},\\ldots ,\\mathbf {\\widetilde{V}}_{K}^{(\\ell )}\\right) \\, ,$ and make use of $\\widehat{\\widetilde{\\Pi }}_{L}^{improved}$ given in (REF ) with the $\\tau _{k}$ 's obtained in Step D2 above to define (in the light of (REF ), (REF )) the BS minimum-distance estimator $\\widehat{D_{\\varphi }}(\\mathbf {\\Omega },\\mathbf {P})\\ := \\ \\widehat{D_{\\widetilde{\\varphi }}}(\\widetilde{\\mathbf {\\Omega }},\\widetilde{{P}})\\ := \\ - \\frac{1}{n}\\log \\widehat{\\widetilde{\\Pi }}_{L}^{improved} \\ .$ For many cases, the simulation burden needed for the computation of $\\widehat{\\widetilde{\\Pi }}_{L}^{improved}$ — and thus of $\\widehat{D_{\\varphi }}(\\mathbf {\\Omega },\\mathbf {P})$ — can be drastically reduced, especially for high dimensions $K$ and large sample size $n \\cdot L$ .","In fact, in terms of the notations $n_{k}:=card(I_{k}^{(n)})$ , $\\widehat{\\mathit {W}}_{k}^{(\\ell )}:=\\sum _{i\\in I_{k}^{(n)}}\\widetilde{V}_{i}^{(\\ell )}$ and $\\widetilde{ISF}_{k}(x) \\ := \\ \\frac{d\\widetilde{\\mathbb {}}^{\\ast n_{k}}}{d\\widetilde{U}_{k}^{\\ast n_{k}}}(x) \\ = \\ \\exp (n_{k} \\cdot \\Lambda _{\\widetilde{\\mathbb {}}}(\\tau _{k}) \\, - \\, x \\cdot \\tau _{k} )$ (where $\\widetilde{\\mathbb {}}^{\\ast n_{k}}$ is the $n_{k}-$ convolution of the measure $\\widetilde{\\mathbb {}}$ ), one can rewrite (REF ) as $\\widehat{\\widetilde{\\Pi }}_{L}^{improved}=\\frac{1}{L}\\sum _{\\ell =1}^{L}\\mathbf {1}_{\\mathbf {\\Lambda }}\\big ( \\,\\big (\\frac{1}{n_{1}} \\widehat{\\mathit {W}}_{1}^{(\\ell )},\\ldots , \\frac{1}{n_{K}} \\widehat{\\mathit {W}}_{K}^{(\\ell )}\\big ) \\, \\big ) \\cdot \\prod \\limits _{k=1}^{K}\\widetilde{ISF}_{k}(\\widehat{\\mathit {W}}_{k}^{(\\ell )}) .$ with $K-$ vector $\\big (\\frac{1}{n_{1}} \\widehat{\\mathit {W}}_{1}^{(\\ell )},\\ldots , \\frac{1}{n_{K}} \\widehat{\\mathit {W}}_{K}^{(\\ell )} \\big )$ .","Clearly, the random variable $\\widehat{\\mathit {W}}_{k}^{(\\ell )}$ ($k=1, \\ldots , K$ ) has distribution $\\widetilde{U}_{k}^{\\ast n_{k}}$ .","Hence, if $\\widetilde{U}_{k}^{\\ast n_{k}}$ can be explicitly constructed, then for the computation of $\\widehat{\\widetilde{\\Pi }}_{L}^{improved}$ it suffices to simulate the $K \\cdot L$ random variables $\\widehat{\\mathit {W}}_{k}^{(\\ell )}$ rather than the $n \\cdot L$ random variables $\\widetilde{V}_{i}^{(\\ell )}$ ; notice that according to the right-hand side of (REF ), one can explicitly compute $ISF_{k}\\left( \\cdot \\right)$ which can be interpreted as Importance Sampling Factor pertaining to the block $k$.","In the case that $\\widetilde{\\mathbb {}}$ is infinitely divisible, simulation issues may become especially comfortable.","In the following, we exemplarily demonstrate the tractability of this reduction effect, for the BS minimization of the important power divergences (for which the infinite divisibility holds): Example 58 Let $\\varphi _{\\gamma }$ ($\\gamma \\in \\mathbb {R}\\backslash ]1,2[$ ) be the power divergence generator from the Examples REF and REF , $\\mathbf {P} \\in \\mathbb {R}_{> 0}^{K}$ , $M_{\\mathbf {P}}:=\\sum _{i=1}^{K}p_{i}>0$ and $n_{k} = n \\cdot p_{k} \\in \\mathbb {N}$ where we have employed our notation $n_{k}= n \\cdot p_{k} \\in \\mathbb {N}$ for all $k \\in \\lbrace 1,\\ldots ,K\\rbrace $ .","Moreover, let $\\mathbf {\\widetilde{Q}}^{\\ast } :=\\left(\\widetilde{q}_{1}^{\\ast },\\ldots ,\\widetilde{q}_{K}^{\\ast }\\right)$ be a proxy obtained by either proxy method 1 or 2.","Case 1: Example REF (a): $\\gamma =0,\\widetilde{c}>0$ .","There holds $\\widetilde{U}_{k}^{\\ast n_{k}}=GAM\\left( \\widetilde{c} \\cdot M_{\\mathbf {P}} - \\tau _{k},n_{k}\\cdot \\widetilde{c} \\cdot M_{\\mathbf {P}}\\right)$ , with $\\tau _{k}=\\widetilde{c} \\cdot M_{\\mathbf {P}} \\cdot ( 1-\\frac{p_{k}}{M_{\\mathbf {P}} \\cdot \\widetilde{q}_{k}^{\\ast }})$ for $\\widetilde{q}_{k}^{\\ast } >0$ (the latter is equivalent to $\\tau _{k} < \\widetilde{c} \\cdot M_{\\mathbf {P}}$ ).","Moreover, for all $x >0$ one gets $\\widetilde{ISF}_{k}(x)=\\left( \\frac{\\widetilde{c} \\cdot M_{\\mathbf {P}}}{\\widetilde{c}\\cdot M_{\\mathbf {P}} - \\tau _{k}}\\right)^{n_{k}\\cdot \\widetilde{c}\\cdot M_{\\mathbf {P}}}\\cdot e^{-\\tau _{k} \\cdot x}$ .","Case 2: REF (b): $\\gamma \\in \\left( 0,1\\right) ,\\widetilde{c}>0$ .","We derive $\\widetilde{U}_{k}^{\\ast n_{k}}=C\\big ( POI( n_{k}\\cdot \\breve{\\theta }),GAM\\big (\\frac{\\widetilde{c} \\cdot M_{\\mathbf {P}}}{1-\\gamma } - \\tau _{k},\\frac{\\gamma }{1-\\gamma } \\big ) \\big )$ with $\\breve{\\theta }:= \\frac{\\widetilde{c} \\cdot M_{\\mathbf {P}}}{\\gamma }\\cdot \\big (\\frac{(\\gamma -1) \\cdot \\tau _{k}}{\\widetilde{c} \\cdot M_{\\mathbf {P}}}+1\\big )^{\\gamma /(\\gamma -1)}$ and $\\tau _{k} = \\widetilde{c} \\cdot M_{\\mathbf {P}} \\cdot \\frac{1-\\big (\\frac{\\widetilde{q}_{k}^{\\ast } \\cdot M_{\\mathbf {P}}}{p_{k}}\\big )^{\\gamma -1}}{1-\\gamma }$ for $\\widetilde{q}_{k}^{\\ast } >0$ .","Furthermore, $\\widetilde{ISF}_{k}(x)=e^{-\\tau _{k}x} \\cdot \\exp \\left( \\frac{n_{k}\\cdot \\widetilde{c} \\cdot M_{\\mathbf {P}}}{\\gamma }\\cdot \\left(\\left(1+ \\frac{\\gamma -1}{\\widetilde{c} \\cdot M_{\\mathbf {P}}} \\cdot \\tau _{k}\\right) ^{\\frac{\\gamma }{\\gamma -1}}-1 \\right)\\right), \\qquad x \\ge 0 ,$ (where $x=0$ covers the atom at zero).","Case 3: Example REF (c): $\\gamma =2,\\widetilde{c}>0$ .","One gets $\\widetilde{U}_{k}^{\\ast n_{k}}=N(n_{k}\\cdot (1+\\frac{\\tau _{k}}{\\widetilde{c} \\cdot M_{\\mathbf {P}}}),\\frac{n_{k}}{\\widetilde{c} \\cdot M_{\\mathbf {P}}})$ with $\\tau _{k}=\\widetilde{c} \\cdot M_{\\mathbf {P}} \\cdot ( \\frac{\\widetilde{q}_{k}^{\\ast } \\cdot M_{\\mathbf {P}}}{p_{k}} - 1 ) $ for $\\widetilde{q}_{k}^{\\ast } \\in \\mathbb {R}$ .","Moreover, for all $x \\in \\mathbb {R}$ one obtains $\\widetilde{ISF}_{k}(x)= \\exp \\big (\\frac{n_{k} \\cdot \\tau _{k}^2}{2\\widetilde{c} \\cdot M_{\\mathbf {P}}}- (x- n_{k}) \\cdot \\tau _{k} \\big )$ .","Case 4: Example REF (d): $\\gamma <0,\\widetilde{c}>0$ .","It holds that $\\widetilde{U}_{k}^{\\ast n_{k}}$ has the (Lebesgue-)density $f_{\\widetilde{U}_{k}^{\\ast n_{k}}}(x)\\ := \\ \\frac{\\exp ((\\tau _{k} - \\frac{\\widetilde{c}\\cdot M_{\\mathbf {P}}}{1-\\gamma })\\cdot x)}{\\exp \\left(n_{k} \\cdot \\frac{\\widetilde{c}\\cdot M_{\\mathbf {P}}}{\\gamma }\\cdot (1+\\frac{\\gamma -1}{\\widetilde{c} \\cdot M_{\\mathbf {P}}}\\cdot \\tau _{k})^{\\gamma /(\\gamma -1)}\\right)}\\cdot f_{\\breve{\\breve{Z}}}(x) \\cdot {1}_{]0,\\infty [}(x),\\qquad x \\in \\mathbb {R},$ where $\\tau _{k} = \\widetilde{c} \\cdot M_{\\mathbf {P}} \\cdot \\frac{1-\\big (\\frac{\\widetilde{q}_{k}^{\\ast } \\cdot M_{\\mathbf {P}}}{p_{k}}\\big )^{\\gamma -1}}{1-\\gamma }$ for $\\widetilde{q}_{k}^{\\ast } >0$ , and $\\breve{\\breve{Z}}$ is a random variable with density $f_{\\breve{\\breve{Z}}}$ of a stable law with parameter-quadruple $(\\frac{-\\gamma }{1-\\gamma },1,0,- n_{k} \\cdot \\frac{(\\widetilde{c}\\cdot M_{\\mathbf {P}})^{1/(1-\\gamma )} \\cdot (1-\\gamma )^{-\\gamma /(1-\\gamma )}}{\\gamma })$ (analogously to $\\breve{Z}$ of Example 40 (d) but with $\\widetilde{c}$ replaced by $\\widetilde{c}\\cdot M_{\\mathbf {P}}$ ).","Also, $\\widetilde{ISF}_{k}(x)=e^{-\\tau _{k}x} \\cdot \\exp \\left( \\frac{n_{k}\\cdot \\widetilde{c} \\cdot M_{\\mathbf {P}}}{\\gamma }\\cdot \\left(\\left(1+ \\frac{\\gamma -1}{\\widetilde{c} \\cdot M_{\\mathbf {P}}} \\cdot \\tau _{k}\\right) ^{\\frac{\\gamma }{\\gamma -1}}-1 \\right)\\right), \\qquad x >0.$ Case 5 : Example REF (e): $\\gamma >2,\\widetilde{c}>0$ .","We derive that $\\widetilde{U}_{k}^{\\ast n_{k}}$ has the (Lebesgue-)density $f_{\\widetilde{U}_{k}^{\\ast n_{k}}}(x)\\ := \\ \\frac{\\exp ((\\tau _{k} + \\frac{\\widetilde{c}\\cdot M_{\\mathbf {P}}}{\\gamma -1})\\cdot x)}{\\exp \\left(n_{k} \\cdot \\frac{\\widetilde{c}\\cdot M_{\\mathbf {P}}}{\\gamma }\\cdot (1+\\frac{\\gamma -1}{\\widetilde{c} \\cdot M_{\\mathbf {P}}}\\cdot \\tau _{k})^{\\gamma /(\\gamma -1)}\\right)}\\cdot f_{\\breve{\\breve{Z}}}(-x) ,\\qquad x \\in \\mathbb {R},$ where $\\tau _{k} = - \\frac{\\widetilde{c} \\cdot M_{\\mathbf {P}}}{\\gamma -1} \\cdot \\big (1- \\big (\\frac{\\widetilde{q}_{k}^{\\ast } \\cdot M_{\\mathbf {P}}}{p_{k}}\\big )^{\\gamma -1}\\cdot {1}_{]0,\\infty [}(\\widetilde{q}_{k}^{\\ast }) \\big )$ for $\\widetilde{q}_{k}^{\\ast } \\in \\mathbb {R}$ , and $\\breve{\\breve{Z}}$ is a random variable with density $f_{\\breve{\\breve{Z}}}$ of a stable law with parameter-quadruple $(\\frac{\\gamma }{\\gamma -1},1,0,n_{k} \\cdot \\frac{(\\widetilde{c}\\cdot M_{\\mathbf {P}})^{1/(1-\\gamma )} \\cdot (\\gamma -1)^{\\gamma /(\\gamma -1)}}{\\gamma })$ (analogously to $\\breve{Z}$ of Example 40 (e) but with $\\widetilde{c}$ replaced by $\\widetilde{c}\\cdot M_{\\mathbf {P}}$ ).","Furthermore, $\\widetilde{ISF}_{k}(x)=e^{-\\tau _{k}x} \\cdot \\exp \\left( \\frac{n_{k}\\cdot \\widetilde{c} \\cdot M_{\\mathbf {P}}}{\\gamma }\\cdot \\left(\\left(1+ \\frac{\\gamma -1}{\\widetilde{c} \\cdot M_{\\mathbf {P}}} \\cdot \\tau _{k}\\right) ^{\\frac{\\gamma }{\\gamma -1}}-1 \\right)\\right), \\qquad x \\in \\mathbb {R}.$ Case 6: Example REF (a): $\\gamma =1,\\widetilde{c}>0$ , anchor point $c=0$ .","It holds that $\\widetilde{U}_{k}^{\\ast n_{k}}$ is the probability distribution $\\frac{1}{\\widetilde{c} \\cdot M_{\\mathbf {P}}} \\cdot POI\\left(n_{k} \\cdot \\widetilde{c} \\cdot M_{\\mathbf {P}} \\cdot \\exp (\\frac{\\tau _{k}}{\\widetilde{c} \\cdot M_{\\mathbf {P}}})\\right) $ with support on the lattice $\\left\\lbrace \\frac{j}{\\widetilde{c} \\cdot M_{\\mathbf {P}}}, \\, j\\in \\mathbb {N}_{0}\\right\\rbrace $ , where $\\tau _{k}=\\widetilde{c} \\cdot \\log \\left( \\frac{M_{\\mathbf {P}} \\cdot \\widetilde{q}_{k}^{\\ast }}{p_{k}}\\right) $ for $\\widetilde{\\omega } _{k} >0$ .","Moreover, for all $j \\in \\mathbb {N}_{0}$ we obtain (by setting $x:= \\frac{j}{\\widetilde{c} \\cdot M_{\\mathbf {P}}}$ ) $\\widetilde{ISF}_{k}\\left(\\frac{j}{\\widetilde{c} \\cdot M_{\\mathbf {P}}}\\right)= \\exp \\left(n_{k} \\cdot \\widetilde{c} \\cdot M_{\\mathbf {P}} \\cdot \\left( \\exp \\left( \\frac{\\tau _{k}}{\\widetilde{c} \\cdot M_{\\mathbf {P}}}\\right) -1 \\right)- m \\cdot \\frac{\\tau _{k}}{\\widetilde{c} \\cdot M_{\\mathbf {P}}}\\right) .$ Case 7: Example REF (b): $\\gamma =1,\\widetilde{c}=1,$ anchor point $c\\in \\mathbb {R}$ .","For $M_{P}=1$ , $\\widetilde{U}_{k}^{\\ast n_{k}}$ is the shifted Poisson distribution $ POI\\left(n_{k} \\cdot e^{c +\\tau _{k}}\\right) + n_{k} \\cdot (1-e^{c})$ with support on the lattice $\\left\\lbrace j + n_{k} \\cdot (1-e^{c}), \\, j\\in \\mathbb {N}_{0}\\right\\rbrace $ , where $\\tau _{k}= \\log \\big (\\frac{\\widetilde{q}_{k}^{\\ast }}{p_{k}} + e^{c}-1\\big )-c$ for $\\widetilde{q}_{k}^{\\ast } > p_{k} \\cdot (1-e^{c})$ .","Furthermore, for all $j \\in \\mathbb {N}_{0}$ we obtain (by setting $x:= j + n_{k} \\cdot (1-e^{c})$ ) $\\widetilde{ISF}_{k}\\left(j + n_{k} \\cdot (1-e^{c})\\right)= \\exp \\left(n_{k} \\cdot e^{c} \\cdot (e^{\\tau _{k}} -1) - j \\cdot \\tau _{k} \\right).$ Notice that the mass of $\\widetilde{U}_{k}^{\\ast n_{k}}$ at zero depends on the value of the anchor point $c$ , since $\\widetilde{U}_{k}^{\\ast n_{k}}[\\lbrace 0\\rbrace ] >0$ if and only if $c=\\log (1+\\frac{\\ell }{n_{k}})$ for some $\\ell \\in \\mathbb {N}_{0}$ ; moreover, $\\widetilde{U}_{k}^{\\ast n_{k}}\\big [ \\ ]0,\\infty [ \\ \\big ] =1$ if $c <0$ and $\\widetilde{U}_{k}^{\\ast n_{k}}\\big [ \\ ]-\\infty ,0[ \\ \\big ] >0$ if $c >0$ .","Remark 59 (a) One can explicitly see in all cases of the above Example REF that all ingredients for computation are at hand.","(b) For both Cases 4 and 5 in the above Example REF , algorithms for simulation can be obtained by adapting e.g.","the works of Devroye [111] and Devroye & James [112].","In the previous Subsection REF , as a first step we have estimated $\\mathbb {\\Pi }\\left[\\xi _{n}^{\\mathbf {\\widetilde{W}}}\\in \\widetilde{\\mathbf {\\Omega }}\\right]$ in terms of the improved IS estimator $\\widehat{\\widetilde{\\Pi }}_{L}^{improved}$ .","From this, as a second step, we have derived — on the basis of Theorem REF — the estimator $\\widehat{D_{\\varphi }}(\\mathbf {\\Omega },\\mathbf {P})\\ := \\ - \\frac{1}{n}\\log \\widehat{\\widetilde{\\Pi }}_{L}^{improved} \\ \\nonumber \\qquad \\textrm {(cf.","(\\ref {estimator minimization}))}$ of the minimum distance $D_{\\varphi }(\\mathbf {\\Omega },\\mathbf {P}):= \\inf _{\\mathbf {Q}\\in \\mathbf {\\Omega } } D_{\\varphi }(\\mathbf {Q},\\mathbf {P})$ , where $\\mathbf {P} \\in \\mathbb {R}_{> 0}^{K}$ and $\\mathbf {\\Omega }\\subset \\mathbb {R}^{K}$ .","Recall that $\\widetilde{\\mathbf {\\Omega }} :=\\mathbf {\\Omega } /M_{\\mathbf {P}}$ with $M_{\\mathbf {P}}:=\\sum _{i=1}^{K}p_{i}>0$ , and that $\\xi _{n}^{\\mathbf {\\widetilde{W}}}=\\Big (\\frac{1}{n}\\sum _{i\\in I_{1}^{(n)}}\\widetilde{W}_{i},\\ldots ,\\frac{1}{n}\\sum _{i\\in I_{K}^{(n)}}\\widetilde{W}_{i}\\Big )\\hspace{56.9055pt} \\text{(cf.","(\\ref {Xi_n^W vector}))}\\nonumber $ where $\\mathbf {\\widetilde{W}} := (\\widetilde{W}_{1}, \\ldots , \\widetilde{W}_{n})$ is a random vector consisting of components $\\widetilde{W}_{i}$ which are i.i.d.","copies of the random variable $\\widetilde{W}$ whose distribution is $\\mathbb {\\Pi }[\\widetilde{W}\\in \\cdot \\,]=\\widetilde{\\mathbb {}}[\\,\\cdot \\,]$ obeying the representation (REF ).","In contrast, we now proceed as follows: as a first step, we derive an improved estimator $\\widehat{\\Pi }_{L}^{improved}$ of $\\mathbb {\\Pi }_{X_{1}^{n}}\\left[\\xi _{n,\\mathbf {X}}^{w\\mathbf {W}}\\in \\textrm {\\right.$ $\\hspace{-6.544pt}$$}$$where $$\\hspace{-6.544pt}$ SK$is a set of probability vectors which satisfies theregularity properties (\\ref {regularity}) and the finiteness property (\\ref {def fi wrt Omega}).Recall that\\begin{eqnarray}\\xi _{n,\\mathbf {X}}^{w\\mathbf {W}} &:=&{\\left\\lbrace \\begin{array}{ll}\\left(\\frac{\\sum _{i \\in I_{1}^{(n)}}W_{i}}{\\sum _{k=1}^{K}\\sum _{i \\in I_{k}^{(n)}}W_{i}},\\ldots , \\frac{\\sum _{i \\in I_{K}^{(n)}}W_{i}}{\\sum _{k=1}^{K}\\sum _{i \\in I_{k}^{(n)}}W_{i}} \\right) ,\\qquad \\textrm {if } \\sum _{j=1}^{n} W_{j} \\ne 0, \\\\\\ (\\infty , \\ldots , \\infty ) =: \\infty , \\hspace{113.81102pt} \\textrm {if } \\sum _{j=1}^{n} W_{j} = 0,\\end{array}\\right.","}\\hspace{42.67912pt}\\textrm {(cf.","(\\ref {brostu3:fo.norweiemp.vec}))}\\nonumber \\end{eqnarray}where $ (Xi)iN$ is a sequence of random variableswith values in $ Y:={ d1,,dK}$such that\\begin{equation}\\lim _{n\\rightarrow \\infty } \\Big ( \\frac{n_{1}}{n}, \\ldots , \\frac{n_{K}}{n} \\Big ) = (p_{1}, \\ldots , p_{K})\\qquad \\textrm {a.s.} \\qquad \\textrm {cf.","((\\ref {cv emp measure X to P vector}))}\\nonumber \\end{equation}holds for some probability vector $ P := (p1, ..., pK) S> 0K$,by employing the notation$$n_{k} \\ := \\ card(\\bigl \\lbrace i \\in \\lbrace 1, \\ldots , n\\rbrace : \\ X_{i} = d_{k} \\bigr \\rbrace )\\ =: \\ card(I_{k}^{(n)})\\qquad \\textrm {(cf.","(\\ref {I^(n)_k for stat case}));}$$hence, on the $ k$-th block of indexes $ Ik(n)$ all the $ Xi$^{\\prime }s share the same value $ dk$.Moreover, recall that $ (Wi)i N$ is a familyof independent and identically distributed $ R-$valued random variableswith probability distribution $ [ ] := [W1 ]$being connected with the divergence generator $ (]a,b[)$ via the representability(\\ref {Phi Legendre of mgf(W)}),such that $ (Wi)i N$ is independent of $ (Xi)i N$.$ As a second step (see Subsubsection REF below), for the important special case of the power-divergence generators $\\varphi _{\\gamma }$ (cf.","(REF )) we employ the Propositions REF to REF in order to deduce via the corresponding $\\widehat{\\Pi }_{L}^{improved}$ the estimators (e.g.","for $\\gamma <0$ ) $\\widehat{D_{\\widetilde{c}\\cdot \\varphi _{\\gamma }}(\\textrm {\\Omega \\hspace{-6.544pt}\\Omega },{P})} \\ := \\ \\frac{\\widetilde{c}}{\\gamma \\cdot \\left(\\gamma -1\\right) }\\cdot \\left\\lbrace \\left( 1+\\frac{\\gamma }{\\widetilde{c}}\\cdot \\frac{1}{n}\\cdot \\log \\,\\widehat{\\Pi }_{L}^{improved}\\right) ^{1-\\gamma } -1 \\right\\rbrace ,\\qquad \\ $ of the minimum power divergences $D_{\\widetilde{c}\\cdot \\varphi _{\\gamma }}(\\textrm {$ $\\hspace{-6.544pt}$$},{P}) :=\\inf _{{Q}\\in \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega }}D_{\\widetilde{c}\\cdot \\varphi _{\\gamma }}({Q},{P})$$as well as connected estimators of important deterministic transformations thereof.$ As a third step (see Subsubsection REF below), on the basis of Subsubsection REF we derive estimators of bounds of $D_{\\varphi }(\\textrm {$$\\hspace{-6.544pt}$$},{P})$ for more general divergence generators $\\varphi $ .","Let us start with the above-mentioned first step, by remarking that the development of the estimator $\\widehat{\\Pi }_{L}^{improved}$ works quite analogously to that of $\\widehat{\\widetilde{\\Pi }}_{L}^{improved}$ in the previous Subsection REF .","To make this even more transparent, we employ the notation $p_{n,k}^{emp}:=n_{k}/n$ (cf.","(REF )) and label all random vectors of length $n$ in the same way as above: we sort the already given and thus fixed data $X_{i}$ 's in such a way that the first $n_{1}$ of them share the same value $d_{1}$ , and so on, until the last block with length $n_{K}$ in which the data have common value $d_{K}$ .","In the light of the above considerations, we could achieve a naive estimate $\\widehat{\\Pi }_{L}^{naive}$ of $\\mathbb {\\Pi }_{X_{1}^{n}}[\\xi _{n,\\mathbf {X}}^{w\\mathbf {W}}\\in \\textrm {$$\\hspace{-6.544pt}$$}]$ through the following procedure.","We simulate independently $L$ replicates $\\mathbf {W}^{(1)},\\ldots ,\\mathbf {W}^{(L)}$ of the vector $\\mathbf {W}:=\\left( W_{1},\\ldots ,W_{n}\\right) $ , with independent entries under $\\mathbb {}$ (cf.","(REF )); those realizations do not depend on the $X_{i}$ 's.","Then we construct $\\widehat{\\Pi }_{L}^{naive} :=\\frac{1}{L}\\sum _{\\ell =1}^{L}\\mathbf {1}_{\\textrm {\\Omega \\hspace{-5.40608pt}\\Omega }}\\left(\\xi _{n,\\mathbf {X}}^{w\\mathbf {W}^{(\\ell )}}\\right) .$ However, this procedure is time costly, since the estimate given in (REF ) has a very bad hit rate.","Hence, analogously to Subsection REF we apply again an “efficient Importance Sampling (IS)” scheme in the sense of Sadowsky & Bucklew [313].","This will involve the simulation of $L$ independent $n-$ tuples $\\mathbf {V}^{(\\ell )}\\mathbf {:=}\\left(V_{n}^{(\\ell )},\\ldots ,V_{n}^{(\\ell )}\\right) $ with common distribution $S$ on $\\mathbb {R}^{n}$ , such that $\\mathbb {}^{\\otimes n}$ is (measure-)equivalent with respect to $S$ .","In fact, we rewrite $\\mathbb {\\Pi }_{X_{1}^{n}}[\\xi _{n,\\mathbf {X}}^{w\\mathbf {W}}\\in \\textrm {$$\\hspace{-6.544pt}$$}]$ as $\\mathbb {\\Pi }_{X_{1}^{n}}[\\xi _{n,\\mathbf {X}}^{w\\mathbf {W}}\\in \\textrm {\\Omega \\hspace{-6.544pt}\\Omega }]= E_{S}\\Big [\\frac{\\mathrm {d}\\mathbb {} ^{\\otimes n}}{\\mathrm {d}S}(V_{1},\\ldots ,V_{n})\\cdot \\mathbf {1}_{\\textrm {\\Omega \\hspace{-5.40608pt}\\Omega }}(\\xi _{n,\\mathbf {X}}^{w\\mathbf {V}})\\Big ]$ where $S$ designates any IS distribution of the vector $\\mathbf {V:=}(V_{1},\\ldots ,V_{n})$ , and $E_{S}[ \\, \\cdot \\, ]$ denotes the corresponding expectation operation.","Notice that $S$ is a random probability distribution on $\\mathbb {R}^{n}$ ; in fact, $S$ is a conditional probability distribution given $X_{1}^{n}$ , and thus it would be more precise to write $S | X_{1}^{n}$ instead of $S$ ; for the sake of brevity, we omit $| X_{1}^{n}$ .","As a consequence of (REF ), for adequately chosen $S$ , an improved estimator of $\\mathbb {\\Pi }_{X_{1}^{n}}[\\xi _{n,\\mathbf {X}}^{w\\mathbf {W}}\\in \\textrm {$$\\hspace{-6.544pt}$$}]$ is given by $\\widehat{\\Pi }_{L}^{improved}:= \\frac{1}{L}\\sum _{\\ell =1}^{L}\\frac{\\mathrm {d}\\mathbb {}^{\\otimes n}}{\\mathrm {d}S}(V_{1}^{(\\ell )},\\ldots ,V_{n}^{(\\ell )})\\cdot \\mathbf {1}_{\\textrm {\\Omega \\hspace{-5.40608pt}\\Omega }}(\\xi _{n,\\mathbf {X}}^{w\\mathbf {V}^{(\\ell )}}) \\, ,$ which also estimates $\\inf _{{Q}\\in \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega } }\\ \\inf _{m\\ne 0}D_{\\varphi }(m\\cdot {Q},{P})$ by the virtue of ().","Let us now deal with the concrete construction of a reasonable $S$ .","Given some (typically) large integer $M$ , we start with the realization $\\mathbf {W}^{\\ast }:=\\left( W_{1}^{\\ast },\\ldots ,W_{M}^{\\ast }\\right)$ such that $\\mathbf {Q}^{\\ast }:=\\xi _{M,\\mathbf {X}}^{w\\mathbf {W}^{\\ast }} \\in int(\\textrm {$$\\hspace{-6.544pt}$$})$ .","This may be given in advance or it may be achieved by drawing replicates $\\mathbf {W} = (W_{1}, \\ldots , W_{M})$ under $\\mathbb {}^{\\otimes M}$ until the first time where $\\xi _{M,\\mathbf {X}}^{w\\mathbf {W}}$ belongs to $int(\\textrm {$$\\hspace{-6.544pt}$$})$ .","Notice that by the nature of $\\textrm {$$\\hspace{-6.544pt}$$}$ , $\\mathbf {Q}^{\\ast }$ is a probability vector which has the $K$ components $q_{k}^{\\ast }:=\\sum _{i=1}^{M}\\frac{W_{i}^{\\ast }}{\\sum _{j=1}^{M}W_{j}^{\\ast }}\\mathbf {1}_{\\lbrace d_{k}\\rbrace }(X_{i}),\\hspace{42.67912pt} k=1,\\ldots ,K.$ Before we proceed, let us give the substantial remark that changing $\\left( V_{1},\\ldots ,V_{n}\\right) $ drawn under $S$ to $\\left(c \\cdot V_{1}, \\ldots ,c \\cdot V_{n}\\right) $ for any $c\\ne 0$ yields $\\xi _{n,\\mathbf {X}}^{w\\mathbf {V}}= \\xi _{n,\\mathbf {X}}^{w\\, c\\cdot \\mathbf {V}}$ so that the distribution $S$ is not uniquely determined.","Amongst all candidates, we choose the — uniquely determined — $S$ which is the Kullback-Leibler projection of $\\mathbb {}^{\\otimes n}$ on the set of all probability distributions on $\\mathbb {R}^{n}$ such that the $K$ “non-normalized” moment constraints $E_{S}[\\xi _{n,\\mathbf {X}}^{\\mathbf {V}}] =\\xi _{M,\\mathbf {X}}^{\\mathbf {W}^{\\ast }}$ (rather than the normalized $E_{S}[ \\xi _{n,\\mathbf {X}}^{w\\mathbf {V}}] =\\xi _{M,\\mathbf {X}}^{w\\mathbf {W}^{\\ast }}$ ) are satisfied, with the non-normalized vectors $\\xi _{M,\\mathbf {X}}^{\\mathbf {W}^{\\ast }} :=\\left( \\frac{1}{M}\\sum _{j=1}^{M}W_{j}^{\\ast }\\right) \\cdot Q^{\\ast } =: \\overline{W^{\\ast }} \\cdot Q^{\\ast },\\qquad \\xi _{n,\\mathbf {X}}^{\\mathbf {V}} :=\\left( \\frac{1}{n}\\sum _{j=1}^{n}V_{j}\\right) \\cdot \\xi _{n,\\mathbf {X}}^{w\\mathbf {V}} \\ .$ As already indicated above, this projection $S$ is a well-determined unique distribution on $\\mathbb {R}^{n}$ and — as we shall see in Proposition REF below — it is such that $\\xi _{n,\\mathbf {X}}^{w\\mathbf {V}}$ belongs to $\\textrm {$$\\hspace{-6.544pt}$$}$ with probability bounded away from 0 as $n$ increases, when $\\left( V_{1},\\ldots ,V_{n}\\right) $ are drawn under $S$ .","Therefore, this IS distribution produces an estimate of $\\mathbb {\\Pi }_{X_{1}^{n}}[\\xi _{n,\\mathbf {X}}^{w\\mathbf {W}}\\in \\textrm {$$\\hspace{-6.544pt}$$}]$ .","In order to justify the above construction of $S$ , we give the following result, which states that the IS sampling distribution $S$ yields a good hitting rate.","Its proof will be given in Appendix H. Proposition 60 With the above definition of $S$ , $\\lim \\inf _{n\\rightarrow \\infty }$ $S\\left[ \\xi _{n,\\mathbf {X}}^{w\\mathbf {V}} \\in \\textrm {\\right.$$\\hspace{-6.544pt}$$} $ is bounded away from 0.","We now come to the detailed construction of $S$ .","The constraints (REF ) can be written in explicit form as $E_{S}\\Big [ \\, \\frac{1}{n_{k}}\\sum _{i\\in I_{k}^{(n)}} V_{i} \\, \\Big ]\\ = \\ \\overline{W^{\\ast }} \\cdot \\frac{q_{k}^{\\ast }}{p_{n,k}^{emp}} \\, , \\qquad k=1,\\ldots ,K.$ The distribution $S$ can be obtained by blocks.","Indeed, let us define $S^{k}$ as the Kullback-Leibler (KL) projection of $\\mathbb {}^{\\otimes n_{k}}$ on the set of all distributions on $\\mathbb {R}^{n_{k}}$ such that (REF ) holds.","We define the resulting $S$ as the product distribution of those $S^{k}$ 's.","To obtain the latter, we start by defining $U_{k}$ as the KL projection of $\\mathbb {} $ on the set of all measures $Q$ on $\\mathbb {R}$ under (REF ).","Then, $dU_{k}(v)= \\exp (\\tau _{k}v-\\Lambda _{\\mathbb {} }\\left( \\tau _{k}\\right) )\\, d\\mathbb {} (v) \\, ,$ where $\\tau _{k} \\in int(dom(MGF_{\\mathbb {}}))$ is the unique solution of the equation $\\Lambda _{\\mathbb {} }^{\\prime }\\left( \\tau _{k}\\right) =\\overline{W^{\\ast }} \\cdot \\frac{q_{k}^{\\ast }}{p_{n,k}^{emp}}$ and thus — by relation (REF ) of Appendix F — we can compute explicitly $\\tau _{k} = \\varphi ^{\\prime }\\left( \\frac{\\overline{W^{\\ast }} \\cdot q_{k}^{\\ast }}{p_{n,k}^{emp}} \\right) .$ The distribution $S^{k}$ is then defined by $S^{k}:=\\underbrace{U_{k}\\otimes \\cdots \\otimes U_{k}}_{n_{k}\\text{times}}$ from which we obtain $S := S^{1}\\otimes \\cdots \\otimes S^{K}.$ With this construction, it holds $\\frac{dS}{d\\mathbb {} ^{\\otimes n}}(v_{1},\\ldots ,v_{n})=\\exp \\left(\\sum \\limits _{k=1}^{K}\\left( \\sum _{i\\in I_{k}^{(n)}}\\tau _{k} \\cdot v_{i}-\\Lambda _{\\mathbb {}}(\\tau _{k})\\right) \\right)$ which proves that $S$ is indeed the KL projection of $\\mathbb {} ^{\\otimes n}$ we aimed at.","Therefore, $\\mathbf {V}$ is composed of $K$ independent blocks of length $n_{k}$ each, and the $k-$ th subvector $\\mathbf {V}_{k}$ consists of all the random variables $V_{i}$ whose index $i$ satisfies $X_{i}=d_{k}.$ Within $\\mathbf {V}_{k}$ , all components are i.i.d.","with same distribution $U_{k}$ on $\\mathbb {R}$ defined through $\\frac{dU_{k}}{d\\mathbb {} }(u)=\\exp \\left\\lbrace \\tau _{k}\\cdot u-\\Lambda _{\\mathbb {}}(\\tau _{k})\\right\\rbrace =\\frac{\\exp \\left\\lbrace \\tau _{k}\\cdot u\\right\\rbrace }{MGF_{\\mathbb {} }(\\tau _{k})},$ which leads to the moment generating function $dom(MGF_{\\mathbb {}})-\\tau _{k} \\ \\ni \\ z \\ \\mapsto MGF_{U_{k}}(z):=\\int _{\\mathbb {R}}e^{zy}dU_{k}(y)=\\frac{MGF_{\\mathbb {}}(z+\\tau _{k})}{MGF_{\\mathbb {} }(\\tau _{k})}.$ Let us remark that $U_{k}$ can be interpreted as the distorted distribution of $\\mathbb {}$ with the distortion parameter $\\tau _{k}$ (in some cases, this distortion even becomes a tilting/dampening).","The estimator $\\widehat{\\Pi }_{L}^{improved}$ defined in (REF ) can be implemented through the following algorithm: Step S1 Choose some (typically large) $M$ and simulate repeatedly i.i.d.","vectors $\\left(W_{1},..,W_{M}\\right) $ — whose independent components have common distribution $\\mathbb {} $ — until $\\xi _{M,\\mathbf {X}}^{w\\mathbf {W}}$ belongs to $\\textrm {$$\\hspace{-6.544pt}$$}$ .","Call $\\left( W_{1}^{\\ast },..,W_{M}^{\\ast }\\right) $ the corresponding vector and $\\overline{W^{\\ast }}$ the arithmetic mean of its components.","Moreover, denote by $\\xi _{M,\\mathbf {X}}^{w\\mathbf {W}^{\\ast }}$ the corresponding normalized weighted empirical measure, identified with the $K-$ component vector $Q^{\\ast } := (q_{1}^{\\ast },..,q_{K}^{\\ast })$ with $q_{k}^{\\ast }$ defined in (REF ).","Step S2 For all $k \\in \\lbrace 1,\\ldots ,K\\rbrace $ compute $\\tau _{k} = \\varphi ^{\\prime }\\left( \\frac{\\overline{W^{\\ast }} \\cdot q_{k}^{\\ast }}{p_{n,k}^{emp}} \\right)$ .","Step S3 For all $\\ell \\in \\lbrace 1,\\ldots ,L\\rbrace $ simulate independently for all $k \\in \\lbrace 1,\\ldots ,K\\rbrace $ a row vector $\\mathbf {V}_{k}^{(\\ell )}$ $:=\\left(V_{k_{1}}^{(\\ell )},...,V_{k_{n_{k}}}^{(\\ell )}\\right) $ with independent components with common distribution $U_{k}$ defined in (REF ).","Concatenate these vectors to define the row vector $\\mathbf {V}^{(\\ell )}.$ Step S4 Compute the estimator $\\widehat{\\Pi }_{L}^{improved}$ by making use of the formula (REF ) which turns into the explicit form $& & \\widehat{\\Pi }_{L}^{improved} =\\frac{1}{L}\\sum _{\\ell =1}^{L}\\exp \\left(\\sum \\limits _{k=1}^{K}\\left(n_{k} \\cdot \\Lambda _{\\mathbb {} }(\\tau _{k}) -\\tau _{k} \\cdot \\sum _{i\\in I_{k}^{(n)}}V_{i}^{(\\ell )}\\right) \\right)\\cdot \\mathbf {1}_{\\textrm {\\Omega \\hspace{-5.40608pt}\\Omega }}\\left( \\xi _{n,\\mathbf {X}}^{w\\mathbf {V}^{(\\ell )}}\\right)$ Analogously to the paragraph right after (REF ) of the previous Subsection REF , in many cases we may improve the simulation burden needed for the computation of the estimator $\\widehat{\\Pi }_{L}^{improved}$ .","In fact, in terms of the notations $\\widehat{\\mathit {W}}_{k}^{(\\ell )}:=\\sum _{i\\in I_{k}^{(n)}}V_{i}^{(\\ell )}$ we can rewrite (REF ) as $\\widehat{\\Pi }_{L}^{improved} =\\frac{1}{L}\\sum _{\\ell =1}^{L}\\mathbf {1}_{\\textrm {\\Omega \\hspace{-5.40608pt}\\Omega }}\\left( \\xi _{n,\\mathbf {X}}^{w\\mathbf {V}^{(\\ell )}} \\right)\\cdot \\prod _{k=1}^{K}ISF_{k}\\left(\\widehat{\\mathit {W}}_{k}^{(\\ell )}\\right)$ with $ISF_{k}(x) \\ := \\ \\exp (n_{k} \\cdot \\Lambda _{\\mathbb {}}(\\tau _{k}) \\, - \\, x \\cdot \\tau _{k} )$ and $\\xi _{n,\\mathbf {X}}^{w\\mathbf {V}^{(\\ell )}} &=&{\\left\\lbrace \\begin{array}{ll}\\left(\\frac{\\widehat{\\mathit {W}}_{1}^{(\\ell )}}{\\sum _{k=1}^{K}\\widehat{\\mathit {W}}_{k}^{(\\ell )}},\\ldots , \\frac{\\widehat{\\mathit {W}}_{K}^{(\\ell )}}{\\sum _{k=1}^{K}\\widehat{\\mathit {W}}_{k}^{(\\ell )}} \\right) ,\\qquad \\textrm {if } \\sum _{k=1}^{K}\\widehat{\\mathit {W}}_{k}^{(\\ell )} \\ne 0, \\\\\\ (\\infty , \\ldots , \\infty ) =: \\infty , \\hspace{65.44142pt} \\textrm {if }\\sum _{k=1}^{K}\\widehat{\\mathit {W}}_{k}^{(\\ell )} = 0 \\, .\\end{array}\\right.","}$ Clearly, the random variable $\\widehat{\\mathit {W}}_{k}^{(\\ell )}$ ($k=1, \\ldots , K$ ) has distribution $U_{k}^{\\ast n_{k}}$ .","Hence, if $U_{k}^{\\ast n_{k}}$ can be explicitly constructed, then for the computation of $\\widehat{\\Pi }_{L}^{improved}$ it suffices to independently simulate the $K \\cdot L$ random variables $\\widehat{\\mathit {W}}_{k}^{(\\ell )}$ (rather than the $n \\cdot L$ random variables $V_{i}^{(\\ell )}$ ).","In the following subsubsection, we exemplarily demonstrate the tractability of this reduction effect."],["BS minimization of power divergences and related quantities","´ Consider the special case of power divergence generators $\\varphi := \\widetilde{c} \\cdot \\varphi _{\\gamma }$ ($\\gamma \\in \\mathbb {R}\\backslash ]1,2[$ ) of the Examples REF and REF .","The corresponding estimators $\\widehat{\\Pi }_{L}^{improved}$ can be obtained as follows: within the results of Example REF , set $M_{\\mathbf {P}}=1$ , and replace $\\widetilde{q}_{k}^{\\ast }$ by $\\overline{W^{\\ast }} \\cdot q_{k}^{\\ast }$ as well as $p_{k}$ by $p_{n,k}^{emp}$ ; accordingly, $\\widetilde{U}_{k}^{\\ast n_{k}}$ turns into $U_{k}^{\\ast n_{k}}$ and $\\widetilde{ISF}_{k}$ into $ISF_{k}$ ; simulate independently the random variables $\\widehat{\\mathit {W}}_{k}^{(\\ell )}$ from $U_{k}^{\\ast n_{k}}$ ($k \\in \\lbrace 1, \\ldots , K\\rbrace $ , $\\ell \\in \\lbrace 1,\\ldots ,L\\rbrace $ ); plug in the results of (i),(ii) into (REF ), (REF ), and (REF ) in order to concretely compute $\\widehat{\\Pi }_{L}^{improved}$ .","From this, we can easily generate improved estimators of the power divergences $\\inf _{\\mathbf {Q}\\in \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega }}D_{\\widetilde{c}\\cdot \\varphi _{\\gamma }}(\\mathbf {Q},{P})$ — and more generally, improved estimators of all the infimum-quantities (e.g.","Renyi divergences) respectively supremum-quantities in the parts (b) of the Propositions REF , REF , REF , REF , REF and REF with $A=1$ — by simply replacing $\\mathbb {\\Pi }_{X_{1}^{n}}[\\xi _{n}^{w\\mathbf {W}}\\in \\textrm {$$\\hspace{-6.544pt}$$} ]$ (respectively, its variants) by the corresponding estimator $\\widehat{\\Pi }_{L}^{improved}$ .","If — in the light of Remark REF (vi) — the ${P} = (p_{1}, \\ldots , p_{K})$ is a pregiven known probability vector e.g.","the uniform distribution ${P}^{unif}$ on $\\lbrace 1,\\ldots ,K\\rbrace $ (rather than the limit of the vector of empirical frequencies/masses of a sequence of random variables $X_{i}$ , cf.","(REF )), then we proceed analogously as above by replacing $p_{n,k}^{emp}$ with $p_{k}$ ; correspondingly, we obtain improved estimators of all the infimum-quantities respectively supremum-quantities (e.g.","Renyi entropies, diversity indices) in the parts (a) of the Propositions REF , REF , REF , REF , REF and REF with $A=1$ .","For the sake of brevity, in the following we only present explicitly the outcoming improved estimators for the power divergences (in the “$X_{i}-$ context” ).","Indeed, we simply replace the $\\mathbb {\\Pi }_{X_{1}^{n}}[\\xi _{n}^{w\\mathbf {W}}\\in \\textrm {$$\\hspace{-6.544pt}$$} ]$ in the formulas (REF ), (REF ), (REF ) (with $A=1$ ) by the improved estimator $\\widehat{\\Pi }_{L}^{improved}$ obtained through (i) to (iii); for arbitrarily fixed $\\widetilde{c} >0$ , this leads to the improved power-divergence estimators (BS estimators of power divergences) $&&\\hspace{-54.06006pt}\\widehat{D_{\\widetilde{c}\\cdot \\varphi _{\\gamma }}(\\textrm {\\Omega \\hspace{-6.544pt}\\Omega },{P})} \\ := \\ -\\frac{\\widetilde{c}}{\\gamma (\\gamma -1)}\\left\\lbrace 1-\\left( 1+\\frac{\\gamma }{\\widetilde{c}} \\cdot \\frac{1}{n} \\cdot \\log \\widehat{\\Pi }_{L}^{improved}\\right)^{1-\\gamma }\\right\\rbrace ,\\qquad \\gamma \\in \\, ]-\\infty ,0[ \\, \\cup \\, ]0,1[ \\, \\cup \\, [2,\\infty [,\\\\& &\\hspace{-54.06006pt}\\widehat{D_{\\widetilde{c}\\cdot \\varphi _{0}}(\\textrm {\\Omega \\hspace{-6.544pt}\\Omega },{P})} \\ := \\ -\\frac{1}{n}\\log \\widehat{\\Pi }_{L}^{improved} ,\\hspace{165.02606pt} \\gamma =0,\\\\& & \\hspace{-54.06006pt}\\widehat{D_{\\widetilde{c}\\cdot \\varphi _{1}}(\\textrm {\\Omega \\hspace{-6.544pt}\\Omega },{P})}\\ : = \\ - \\widetilde{c} \\cdot \\log \\left( 1+\\frac{1}{\\widetilde{c}} \\cdot \\frac{1}{n} \\cdot \\log \\widehat{\\Pi }_{L}^{improved}\\right) ,\\hspace{85.35826pt} \\gamma =1.$ Let us finally remark that from the above-mentioned Steps S1 to S4 (and analogously D1 to D4) one can see that for our BS method we basically need only a fast and accurate — pseudo, true, natural, quantum — random number generator.","The corresponding computations can be principally run in parallel, and require relatively moderate computer memory/storage; a detailed discussion is beyond the scope of this paper, given its current length."],["General case, part 2","The algorithm which is presented in this section aims at the evaluation of the bounds $\\inf _{m\\ne 0}D_{\\varphi }\\left( m \\cdot \\textrm {\\right.\\Omega \\hspace{-6.544pt}\\Omega },{P} =\\inf _{{Q}\\in \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega }}D_{\\varphi }\\left( m\\left( {Q}\\right) \\cdot {Q},{P}\\right)\\stackrel{(1)}{=} D_{\\varphi }\\left( m({Q}^{\\ast })\\cdot {Q}^{\\ast },{P}\\right) \\le D_{\\varphi }\\left( \\textrm {\\right.\\Omega \\hspace{-6.544pt}\\Omega },{P} \\le D_{\\varphi }\\left( {Q}^{\\ast },{P}\\right)$ obtained in Section REF , where ${Q}^{\\ast }$ satisfies the above equality $(1)$ .","The estimator of the lower bound in (REF ) is $\\widehat{D}:=-\\frac{1}{n}\\log \\widehat{\\text{ }\\Pi }_{L}^{improved}$ defined in (REF ).","We now turn to an estimate of the upper bound.","Consider for any fixed ${Q}:=\\left( q_{1},..,q_{K}\\right) $ in $\\mathbb {S}_{>0}^{K}$ the real number $m_{n}({Q})$ which satisfies $D_{\\varphi }\\left( m_{n}({Q}) \\cdot {Q},{P}_{n}^{emp} \\right) =\\inf _{m\\ne 0}D_{\\varphi }\\left( m\\cdot \\textrm {\\right.$ $\\hspace{-6.544pt}$$},{P}_{n}^{emp}$$where $ Pnemp$ was defined in the course of (\\ref {I^(n)_k for stat case}).Such $ mn(Q)$ is well defined for all $ Q$since it satisfies the equation (in $ m$)\\begin{equation}\\frac{d}{dm}D_{\\varphi }\\left( m \\cdot {Q},{P}_{n}^{emp}\\right) =\\sum _{k=1}^{K} q_{k} \\cdot \\varphi ^{\\prime }\\left( \\frac{m \\cdot q_{k}}{p_{n,k}^{emp}}\\right) =0 .\\end{equation}Since the mapping $ mD( m Q,P) $ is convexand differentiable, existence and uniqueness of $ mn(Q)$ hold; furthermore,$ mn(Q)] k pn,kemp/qk,k pn,kemp/qk[ $since $ ddmD( m Q,Pnemp) $is negative when $ m=k pn,kemp/qk$ and positive when $ m=k pn,kemp/qk$.$ An estimate of the distribution ${Q}^{\\ast }$ is required.","This can be achieved as follows: Estimate $\\inf _{m\\ne 0}D_{\\varphi }\\left( m \\cdot \\textrm {\\right.$$\\hspace{-6.544pt}$$}, {P}$ through $\\widehat{D} := -\\frac{1}{n}\\log \\widehat{\\Pi }_{L}^{improved}$ defined in (REF ).","Set $i=0.$ Get some ${Q}_{i} := (q_{i,1}, \\ldots , q_{i,K})$ in $\\textrm {$$\\hspace{-6.544pt}$$}$ ; this can be obtained by simulating runs of vectors $\\left( W_{1},..,W_{n}\\right) $ through i.i.d.","sampling under $\\mathbb {}$ .","Evaluate $m_{n}({Q}_{i})$ by solving () (with $q_{i,k}$ instead of $q_{k}$ ) for $m$ , which is a fast calculation by the bisection method.","If $D_{\\varphi }\\left( m_{n}({Q}_{i}) \\cdot {Q}_{i}, {P}_{n}^{emp} \\right) <\\widehat{D}+\\eta $ for some small $\\eta >0$ , then the proxy of ${Q}^{\\ast }$ is ${Q}_{i},$ denoted by $\\widehat{{Q}^{\\ast }}$ .","Else set $i\\leftarrow i+1$ and get ${Q}_{i}$ in $\\textrm {$$\\hspace{-6.544pt}$$}\\cap \\left\\lbrace {Q} \\, : \\, D_{\\varphi }\\left( {Q},{P}_{n}^{emp} \\right) 0$ we normalized $\\widetilde{{P}}:=\\mathbf {P}/M_{\\mathbf {P}},$ and $\\widetilde{\\mathbf {Q}}:=\\mathbf {Q}/M_{\\mathbf {P}}$ for $\\mathbf {Q}$ in $\\mathbf {\\Omega }$ .","With $\\widetilde{\\varphi } \\in \\Upsilon (]a,b[)$ defined through $\\widetilde{\\varphi }:=M_{\\mathbf {P}} \\cdot \\varphi $ , we have obtained $D_{\\varphi }(\\mathbf {Q},\\mathbf {P})=\\sum _{k=1}^{K}p_{k}\\cdot \\varphi \\left( \\frac{q_{k}}{p_{k}}\\right) =\\sum _{k=1}^{K}M_{\\mathbf {P}}\\cdot \\widetilde{p_{k}}\\cdot \\frac{\\varphi \\left( \\frac{M_{\\mathbf {P}}\\cdot \\widetilde{q_{k}}}{M_{\\mathbf {P}}\\cdot \\widetilde{p_{k}}}\\right) }{M_{\\mathbf {P}}}=D_{\\widetilde{\\varphi }}(\\widetilde{\\mathbf {Q}},\\widetilde{{P}})\\qquad \\textrm {(cf.","(\\ref {min Pb prob1}))}.$ It has followed that the solution of (REF ) coincides with the one of the problem of finding $\\widetilde{\\Phi }_{\\widetilde{{P}}}(\\widetilde{\\mathbf {\\Omega }}) := \\inf _{\\widetilde{\\mathbf {Q}}\\in \\widetilde{\\mathbf {\\Omega }} }D_{\\widetilde{\\varphi } }(\\widetilde{\\mathbf {Q}},\\widetilde{{P}}),\\qquad \\textrm {with } \\widetilde{\\mathbf {\\Omega }}:=\\mathbf {\\Omega } /M_{\\mathbf {P}}\\qquad \\textrm {(cf.","(\\ref {min Pb prob2}))}.$ So let us continue by tackling (REF ).","From the assumptions on $\\widetilde{\\varphi }$ and the requirement (REF ) one can see that $\\text{$\\widetilde{W}_{1}$ has moment generating function$t\\rightarrow E_{\\mathbb {\\Pi }}[e^{z \\cdot \\widetilde{W}_{1}}]= MGF_{\\widetilde{\\mathbb {}}}(z)$which is finite on a non-void neighborhood of $0$},$ $E_{\\mathbb {\\Pi }}[\\widetilde{W}_1] = 1 ,$ since $\\widetilde{\\varphi }(1)=0=\\widetilde{\\varphi }^{\\prime }(1)$ .","With the help of these, we obtain the following Proposition 61 Under the assumptions of Theorem REF , for any set $\\widetilde{\\mathbf {\\Omega }}\\subset \\mathcal {M} := \\mathbb {R}^{K}$ with (REF ) one has $-\\inf _{\\widetilde{\\mathbf {Q}} \\in int(\\widetilde{\\mathbf {\\Omega }}) }D_{\\varphi }\\left( \\widetilde{\\mathbf {Q}},\\widetilde{{P}}\\right) &\\le &\\lim \\inf _{n\\rightarrow \\infty }\\frac{1}{n}\\log \\mathbb {\\Pi }\\left[\\xi _{n}^{\\widetilde{\\textbf {W}}}\\in \\widetilde{\\mathbf {\\Omega }} \\right] \\nonumber \\\\&\\le &\\lim \\sup _{n\\rightarrow \\infty }\\frac{1}{n}\\log \\mathbb {\\Pi }\\left[\\xi _{n}^{\\widetilde{\\textbf {W}}}\\in \\widetilde{\\mathbf {\\Omega }} \\right]\\le -\\inf _{\\widetilde{\\mathbf {Q}}\\in cl(\\widetilde{\\mathbf {\\Omega }}) }D_{\\varphi }\\left( \\widetilde{\\mathbf {Q}},\\widetilde{{P}}\\right) .$ Proof of Proposition REF .","Recall from Remark REF (v) that $I_{k}^{(n)} := \\lbrace i \\in \\lbrace 1,\\ldots ,n\\rbrace : \\widetilde{x}_{i}=d_{k}\\rbrace $ and $n_{k} := card(I_{k}^{(n)})$ denotes the number of elements therein ($k \\in \\lbrace 1,\\ldots ,K\\rbrace $ ), i.e.","$n_{k}$ is the number of the $\\widetilde{x}_{i}$ ’s which equal $d_{k}$ .","We follow the line of proof of Theorem 2.2.30 in Dembo & Zeitouni [108], which states the large deviation principle (LDP) for the vector of partial sums of random vectors in $\\mathbb {R}^{K}$ , where we also use Corollary 6.1.6 in [108] in relation with condition (REF ).","Indeed, since the $k-$ th component of the vector $\\xi _{n}^{\\mathbf {\\widetilde{W}}}$ is the $1/n-$ fold of the sum of the $\\widetilde{W}_{i}$ ’s for which the corresponding $\\widetilde{x}_{i}$ ’s equal $d_{k}$ (i.e., $\\frac{1}{n} \\sum _{i\\in I_{k}^{(n)}} \\widetilde{W}_{i}$ ) the proof will follow from a similar treatment as for the standard Cramer LDP in $\\mathbb {R}^{K}.$ The only difference lies in two facts: the number of the summands for the coordinate $k$ is $n_{k}$ , the number of $\\widetilde{x}_{i}$ ’s which equal $d_{k}$ , instead of $n$ in the standard case.","Furthermore we will need to substitute $n_{k}$ by its equivalent $n \\cdot \\widetilde{p}_{k}$ , which adds an approximation step.","For the upper bound, the proof is based on the corresponding result for $B=B_{1}\\times \\cdots \\times B_{K}$ where the $B_{k}$ ’s are open bounded intervals on $\\mathbb {R}^{+}.$ Since the sequence $\\left( \\widetilde{x}_{1}, \\ldots \\right)$ satisfies $\\lim _{n\\rightarrow \\infty }\\frac{n_{k}}{n}= \\widetilde{p}_{k} ,\\qquad \\textrm {(cf.","(\\ref {fo.freqlim}))}$ there holds $&&\\frac{1}{n}\\log \\mathbb {\\Pi } \\left[\\xi _{n}^{\\mathbf {\\widetilde{W}}} \\in B\\right]=\\frac{1}{n}\\log \\mathbb {\\Pi } \\bigg [ \\bigcap \\limits _{k=1}^{K}\\bigg (\\frac{1}{n} \\sum _{i\\in I_{k}^{(n)}} \\widetilde{W}_{i}\\in B_{k}\\bigg ) \\bigg ]\\nonumber \\\\&=&\\frac{1}{n}\\sum \\limits _{k=1}^{K}\\log \\mathbb {\\Pi } \\bigg [ \\frac{1+o(1)}{n_{k}}\\sum _{i\\in I_{k}^{(n)}} \\widetilde{W}_{i}\\in \\frac{1}{\\widetilde{p}_{k}} B_{k}\\bigg ],$ and hence $\\lim \\sup _{n\\rightarrow \\infty }\\frac{1}{n}\\log \\mathbb {\\Pi } \\left[\\xi _{n}^{\\mathbf {\\widetilde{W}}}\\in B\\right]&\\le &\\sum \\limits _{k=1}^{K} \\widetilde{p}_{k} \\cdot \\lim \\sup _{n_{k}\\rightarrow \\infty }\\frac{1}{n_{k}}\\log \\mathbb {\\Pi } \\bigg [ \\frac{1}{n_{k}}\\sum _{i\\in I_{k}^{(n)}} \\widetilde{W}_{i}\\in \\frac{1}{p_{k}}B_{k}\\bigg ] \\nonumber \\\\&\\le &-\\sum \\limits _{k=1}^{K}\\inf _{x_{k}\\in cl(B_{k})} \\widetilde{p}_{k} \\cdot \\varphi \\left(\\frac{x_{k}}{\\widetilde{p}_{k}}\\right) .$ To deduce (REF ) from (REF ), we have used (i) the fact that for all $k$ the random variables $\\frac{1}{n_{k}}\\left( 1+o(1))\\right) \\cdot \\sum _{i\\in I_{k}^{(n)}} \\widetilde{W}_{i}$ and $\\frac{1}{n_{k}}\\sum _{i\\in I_{k}^{(n)}} \\widetilde{W}_{i}$ are exponentially equivalent in the sense that their difference $\\Delta _{n_{k}}$ satisfies $\\lim \\sup _{n_{k}\\rightarrow \\infty }\\frac{1}{n_{k}}\\log \\mathbb {\\Pi }\\left[ \\,\\left|\\Delta _{n_{k}}\\right|>\\eta \\, \\right] =-\\infty ,$ making use of the Chernoff inequality for all positive $\\eta $ , as well as (ii) Theorem 4.2.13 in [108].","Now the summation and the inf-operations can be permuted in (REF ) which proves the claim for the rectangle $B$ .","As in [108], for a compact set $\\Omega $ we consider its finite covering by such open sets $B$ and conclude; for $\\Omega $ being a closed set, a tightness argument holds, following [108] Theorem 2.2.30 verbatim.","For the lower bound consider the same rectangle $B$ .","The argument which locates the tilted distribution at the center of $B$ , together with the use of the LLN for the corresponding r.v’s as in [108], in combination with the same approximations as above to handle the approximation of $n_{k}$ by $n \\cdot \\widetilde{p}_{k}$ , complete the proof of Proposition REF .","We omit the details.","$\\blacksquare $ Let us continue with the proof of Theorem REF , by giving the following two helpful lemmas for $\\Phi _{{P}}(\\mathbf {A}) := \\inf _{\\mathbf {Q} \\in \\mathbf {A}} D_{\\varphi }\\left(\\mathbf {Q},{P}\\right),\\qquad \\mathbf {A} \\subset \\mathcal {M} := \\mathbb {R}^{K},$ Lemma 62 For any open set $\\mathbf {A} \\subset \\mathcal {M} := \\mathbb {R}^{K}$ one has $\\Phi _{{P}}(\\mathbf {A}) = \\Phi _{{P}}(cl(\\mathbf {A}))$ .","This is clear from the continuity of $\\Phi _{{P}}$ .","Lemma 63 For any $\\mathbf {A} \\subset \\mathcal {M} := \\mathbb {R}^{K}$ satisfying (REF ) one has $\\Phi _{{P}}(cl(\\mathbf {A})) = \\Phi _{{P}}(\\mathbf {A}) = \\Phi _{{P}}(int(\\mathbf {A}))$ .","Proof of Lemma REF .","Assume first that $\\Phi _{{P}}(\\mathbf {A})$ is finite.","Then suppose that $\\mathbf {A}$ satisfies (REF ) and $\\Phi _{{P}}(cl(\\mathbf {A})) < \\Phi _{{P}}(int(\\mathbf {A}))$ .","The latter implies the existence of a point $a \\in cl(\\mathbf {A})$ such that $a \\notin int(\\mathbf {A})$ and $\\Phi _{{P}}(a)= \\Phi _{{P}}(cl(\\mathbf {A}))$ .","But then, by Lemma REF and (REF ) one gets $\\Phi _{{P}}(int(\\mathbf {A})) = \\Phi _{{P}}(cl(int(\\mathbf {A}))) = \\Phi _{{P}}(cl(\\mathbf {A})) = \\Phi _{{P}}(a)$ which leads to a contradiction.","When $\\Phi _{{P}}(\\mathbf {A}) = \\infty $ then $\\Phi _{{P}} (cl(\\mathbf {A}))=\\Phi _{{P}} (int(\\mathbf {A}))=\\Phi _{{P}} (\\mathbf {A})=\\infty $ .","$\\blacksquare $ Putting things together, the required asymptotic assertion (REF ) follows from (REF ), (REF ) and Lemma REF .","This completes the proof of Theorem REF .","$\\blacksquare $"],["Proofs — Part 2","Before we tackle the proof of Theorem REF , let us introduce the following Lemma 64 If $\\Omega $$\\Omega \\subset \\mathbb {S}^{K}$ satisfies condition (REF ), then $\\widetilde{\\widetilde{\\textrm {\\Omega \\hspace{-6.544pt}\\Omega }}}:= \\bigcup \\limits _{m\\ne 0}cl(m \\cdot \\textrm {$$\\hspace{-6.544pt}$$})$ has the property (REF ).","This can be deduced in a straightforward way: the assumption implies that $cl(\\textrm {$$\\hspace{-6.544pt}$$})$ satisfies (REF ), and thus also $m \\cdot cl(\\textrm {$$\\hspace{-6.544pt}$$})$ satisfies (REF ).","But this implies the validity of (REF ) for the “cone” $\\bigcup \\limits _{m\\ne 0} m \\cdot cl(\\textrm {$$\\hspace{-6.544pt}$$})$ which is nothing but $\\bigcup \\limits _{m\\ne 0} cl(m \\cdot \\textrm {$$\\hspace{-6.544pt}$$})$ .","Proof of Theorem REF .","Recall the interpretations of the two vectors $\\xi _{n,\\mathbf {X}}^{\\mathbf {W}}$ respectively $\\xi _{n,\\mathbf {X}}^{w\\mathbf {W}}$ given in (REF ) respectively (REF ), and that the sum of their $k$ components are $\\sum _{k=1}^{K} \\frac{1}{n} \\sum _{i\\in I_{k}^{(n)}} W_{i} =\\frac{1}{n}\\sum _{i=1}^{n} W_{i}$ respectively $\\sum _{k=1}^{K}\\frac{\\sum _{i \\in I_{k}^{(n)}}W_{i}}{\\sum _{k=1}^{K}\\sum _{i \\in I_{k}^{(n)}}W_{i}}=1$ (in case of $\\sum _{i=1}^{n}W_{i} \\ne 0$ ).","In the light of these, for $\\textrm {$$\\hspace{-6.544pt}$$} \\subset \\mathbb {S}^{K}$ one gets the set identification $\\left\\lbrace \\xi _{n,\\mathbf {X}}^{w\\mathbf {W}} \\in \\textrm {\\right.$ $\\hspace{-6.544pt}$$} =\\bigcup \\limits _{m\\ne 0}\\left\\lbrace \\xi _{n,\\mathbf {X}}^{\\mathbf {W}}\\in m\\cdot \\textrm {\\right.$$\\hspace{-6.544pt}$$} ,\\frac{1}{n}\\sum _{i=1}^{n} W_{i} = m$$since $ { i=1nWi=0} $ amounts to $ m=0$, which cannothold when $ { n,XwW $\\Omega $$\\Omega $ }$.","Now\\begin{eqnarray*}\\mathbb {\\Pi }_{X_{1}^{n}}\\left[\\xi _{n,\\mathbf {X}}^{w\\mathbf {W}}\\in \\textrm {\\right.\\Omega \\hspace{-6.544pt}\\Omega } &=&\\mathbb {\\Pi }_{X_{1}^{n}} \\bigg [ \\bigcup \\limits _{m\\ne 0}\\bigg \\lbrace \\xi _{n,\\mathbf {X}}^{\\mathbf {W}} \\in m \\cdot \\textrm {\\Omega \\hspace{-6.544pt}\\Omega },\\frac{1}{n}\\sum _{i=1}^{n} W_{i} =m \\bigg \\rbrace \\bigg ] \\\\&=&\\mathbb {\\Pi }_{X_{1}^{n}}\\bigg [ \\bigcup \\limits _{m\\ne 0}\\bigg \\lbrace \\xi _{n,\\mathbf {X}}^{\\mathbf {W}} \\in m\\cdot \\textrm {\\Omega \\hspace{-6.544pt}\\Omega }\\big \\rbrace \\bigg ]= \\mathbb {\\Pi }_{X_{1}^{n}}\\bigg [ \\xi _{n,\\mathbf {X}}^{\\mathbf {W}}\\in \\bigcup \\limits _{m\\ne 0} m\\cdot \\textrm {\\Omega \\hspace{-6.544pt}\\Omega } \\bigg ]\\end{eqnarray*}since $ { n,XW m$\\Omega $$\\Omega $ } { 1ni=1n Wi =m }$.","Therefore\\begin{equation}\\frac{1}{n}\\log \\mathbb {\\Pi }_{X_{1}^{n}}\\left[\\xi _{n,\\mathbf {X}}^{w\\mathbf {W}}\\in \\textrm {\\right.\\Omega \\hspace{-6.544pt}\\Omega } =\\frac{1}{n}\\log \\mathbb {\\Pi }_{X_{1}^{n}}\\bigg [ \\xi _{n,\\mathbf {X}}^{\\mathbf {W}} \\in \\bigcup \\limits _{m\\ne 0}m\\cdot \\textrm {\\Omega \\hspace{-6.544pt}\\Omega } \\bigg ] .\\end{equation}Because of Proposition \\ref {PropLDPWEM-finitease} --- applied to$$\\Omega $$\\Omega $ := m0m $\\Omega $$\\Omega $$ --- one getsin terms of (\\ref {phishort})$ $-\\Phi _{{P}}\\Big (int\\Big (\\bigcup \\limits _{m\\ne 0}m\\cdot \\textrm {\\Omega \\hspace{-6.544pt}\\Omega }\\Big )\\Big )&\\le &\\lim \\inf _{n\\rightarrow \\infty }\\frac{1}{n}\\log \\mathbb {\\Pi }_{X_{1}^{n}}\\bigg [ \\xi _{n,\\mathbf {X}}^{\\mathbf {W}} \\in \\bigcup \\limits _{m\\ne 0}m\\cdot \\textrm {\\Omega \\hspace{-6.544pt}\\Omega } \\bigg ] \\nonumber \\\\&\\le &\\lim \\sup _{n\\rightarrow \\infty }\\frac{1}{n}\\log \\mathbb {\\Pi }_{X_{1}^{n}}\\bigg [\\xi _{n,\\mathbf {X}}^{\\mathbf {W}}\\in \\bigcup \\limits _{m\\ne 0}m\\cdot \\textrm {\\Omega \\hspace{-6.544pt}\\Omega } \\bigg ] \\le -\\Phi _{{P}}\\Big (cl\\Big (\\bigcup \\limits _{m\\ne 0}m\\cdot \\textrm {\\Omega \\hspace{-6.544pt}\\Omega }\\Big )\\Big ) .$ $\\hspace{-176.407pt}\\text{But}& & \\Phi _{{P}}\\Big (int\\Big (\\bigcup \\limits _{m\\ne 0}m\\cdot \\textrm {\\Omega \\hspace{-6.544pt}\\Omega }\\Big )\\Big ) \\le \\Phi _{{P}}\\Big (\\bigcup \\limits _{m\\ne 0}int\\Big (m\\cdot \\textrm {\\Omega \\hspace{-6.544pt}\\Omega }\\Big )\\Big ) =\\inf _{m\\ne 0} \\Phi _{{P}}(int(m\\cdot \\textrm {\\Omega \\hspace{-6.544pt}\\Omega }))\\\\\\hspace{-176.407pt}\\text{and}& & \\Phi _{{P}}\\Big (cl\\Big (\\bigcup \\limits _{m\\ne 0}m\\cdot \\textrm {\\Omega \\hspace{-6.544pt}\\Omega }\\Big )\\Big ) \\ge \\Phi _{{P}}\\Big (\\bigcup \\limits _{m\\ne 0}cl\\Big (m\\cdot \\textrm {\\Omega \\hspace{-6.544pt}\\Omega }\\Big )\\Big ) =\\inf _{m\\ne 0} \\Phi _{{P}}(cl(m\\cdot \\textrm {\\Omega \\hspace{-6.544pt}\\Omega })) .$ In fact, the inequality in (REF ) is straightforward because of $\\bigcup \\limits _{m\\ne 0}int(m\\cdot \\textrm {$$\\hspace{-6.544pt}$$})\\subset int(\\bigcup \\limits _{m\\ne 0}m\\cdot \\textrm {$$\\hspace{-6.544pt}$$})$ (since the latter is the largest open set contained in $\\bigcup \\limits _{m\\ne 0}m\\cdot \\textrm {$$\\hspace{-6.544pt}$$}$ ); the inequality in () follows from $& & \\Phi _{{P}}\\Big (cl\\Big (\\bigcup \\limits _{m\\ne 0}m\\cdot \\textrm {\\Omega \\hspace{-6.544pt}\\Omega }\\Big )\\Big ) \\ge \\Phi _{{P}}\\Big (cl\\Big (\\bigcup \\limits _{m\\ne 0}cl\\Big (m\\cdot \\textrm {\\Omega \\hspace{-6.544pt}\\Omega }\\Big )\\Big )\\Big ) =\\Phi _{{P}}\\Big (\\bigcup \\limits _{m\\ne 0}cl\\Big (m\\cdot \\textrm {\\Omega \\hspace{-6.544pt}\\Omega }\\Big )\\Big )\\nonumber $ An application of Lemma REF yields $\\Phi _{{P}}(int(m\\cdot \\textrm {$$\\hspace{-6.544pt}$$})) =\\Phi _{{P}}(m\\cdot \\textrm {$$\\hspace{-6.544pt}$$}) =\\Phi _{{P}}(cl(m\\cdot \\textrm {$$\\hspace{-6.544pt}$$}))$ for all $m \\ne 0$ , and hence $\\inf _{m\\ne 0} \\Phi _{{P}}(int(m\\cdot \\textrm {\\Omega \\hspace{-6.544pt}\\Omega })) =\\inf _{m\\ne 0} \\Phi _{{P}}(m\\cdot \\textrm {\\Omega \\hspace{-6.544pt}\\Omega })= \\inf _{m\\ne 0} \\Phi _{{P}}(cl(m\\cdot \\textrm {\\Omega \\hspace{-6.544pt}\\Omega })) .$ By combining (), (REF ), (REF ), () and (REF ), one arrives at $&&\\lim _{n \\rightarrow \\infty } \\frac{1}{n}\\log \\mathbb {\\Pi }_{X_{1}^{n}} \\left[\\xi _{n,\\mathbf {X}}^{w\\mathbf {W}}\\in \\textrm {\\right.\\Omega \\hspace{-6.544pt}\\Omega } = \\lim _{n \\rightarrow \\infty } \\frac{1}{n}\\log \\mathbb {\\Pi }_{X_{1}^{n}} \\bigg [ \\xi _{n,\\mathbf {X}}^{\\mathbf {W}}\\in \\bigcup \\limits _{m\\ne 0}m\\cdot \\textrm {\\Omega \\hspace{-6.544pt}\\Omega } \\bigg ]\\nonumber \\\\&& = - \\inf _{m\\ne 0} \\Phi _{{P}}(m\\cdot \\textrm {\\Omega \\hspace{-6.544pt}\\Omega })= - \\inf _{m\\ne 0} \\inf _{\\mathbf {Q} \\in m\\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega }}D_{\\varphi }\\left(\\mathbf {Q},{P}\\right)= - \\inf _{m\\ne 0} \\inf _{{Q} \\in \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega }}D_{\\varphi }\\left(m\\cdot {Q}, {P}\\right) ,\\nonumber $ where in the second last equality we have “reverted” the notation (REF ).","Note that we did not assume (REF ) for $\\bigcup \\limits _{m\\ne 0} m \\cdot \\textrm {$$\\hspace{-6.544pt}$$}$ .","$\\blacksquare $"],["Proofs — Part 3","Proof of Lemma REF .","From (REF ) one gets straightforwardly for arbitrary $\\widetilde{c} >0$ $D_{\\widetilde{c} \\cdot \\varphi _{\\gamma }}(m \\cdot \\mathbf {Q},{P}) \\hspace{-5.69046pt} &:=& \\hspace{-5.69046pt}{\\left\\lbrace \\begin{array}{ll}\\frac{\\widetilde{c} \\cdot \\left(m^{\\gamma } \\cdot H_{\\gamma } - m \\cdot A \\cdot \\gamma + \\gamma -1 \\right)}{\\gamma \\cdot (\\gamma -1)},\\hspace{75.39963pt} \\textrm {if } \\gamma \\in \\, ]-\\infty ,0[, \\ {P} \\in \\mathbb {S}_{\\ge 0}^{K}, \\ \\mathbf {Q} \\in A \\cdot \\mathbb {S}_{> 0}^{K} \\ \\ \\textrm {and } m >0, \\\\\\widetilde{c} \\cdot (- \\log m + \\widetilde{I} -1 + m \\cdot A),\\hspace{42.67912pt} \\textrm {if } \\gamma = 0, \\ {P} \\in \\mathbb {S}_{\\ge 0}^{K}, \\ A \\cdot \\mathbf {Q} \\in \\mathbb {S}_{> 0}^{K} \\ \\ \\textrm {and } m >0, \\\\\\frac{\\widetilde{c} \\cdot \\left(m^{\\gamma } \\cdot H_{\\gamma } - m \\cdot A \\cdot \\gamma + \\gamma -1 \\right)}{\\gamma \\cdot (\\gamma -1)},\\hspace{75.39963pt} \\textrm {if } \\gamma \\in \\, ]0,1[, \\ {P} \\in \\mathbb {S}_{\\ge 0}^{K}, \\ \\mathbf {Q} \\in A \\cdot \\mathbb {S}_{\\ge 0}^{K} \\ \\ \\textrm {and } m \\ge 0, \\\\\\widetilde{c} \\cdot (A \\cdot m \\cdot \\log m + m \\cdot (I-A) +1),\\hspace{14.22636pt} \\textrm {if } \\gamma = 1, \\ {P} \\in \\mathbb {S}_{> 0}^{K}, \\ \\mathbf {Q} \\in A \\cdot \\mathbb {S}_{\\ge 0}^{K} \\ \\ \\textrm {and } m \\ge 0, \\\\\\frac{\\widetilde{c} \\cdot \\left(m^{\\gamma } \\cdot H_{\\gamma } \\cdot {1}_{[0,\\infty [}(m)- m \\cdot A \\cdot \\gamma + \\gamma -1 \\right)}{\\gamma \\cdot (\\gamma -1)},\\hspace{39.83368pt} \\textrm {if } \\gamma \\in \\, ]1,2[, \\ {P} \\in \\mathbb {S}_{>0}^{K}, \\ \\mathbf {Q} \\in A \\cdot \\mathbb {S}^{K} \\ \\ \\textrm {and } m \\in ]-\\infty ,\\infty [,\\\\\\frac{\\widetilde{c} \\cdot \\left(m^{2} \\cdot H_{2} - m \\cdot A \\cdot 2 + 2 -1 \\right)}{2 \\cdot (2-1)},\\hspace{76.82234pt} \\textrm {if } \\gamma = 2, \\ {P} \\in \\mathbb {S}_{>0}^{K}, \\ \\mathbf {Q} \\in A \\cdot \\mathbb {S}^{K} \\ \\ \\textrm {and } m \\in ]-\\infty ,\\infty [,\\\\\\frac{\\widetilde{c} \\cdot \\left(m^{\\gamma } \\cdot H_{\\gamma } \\cdot {1}_{[0,\\infty [}(m)- m \\cdot A \\cdot \\gamma + \\gamma -1 \\right)}{\\gamma \\cdot (\\gamma -1)},\\hspace{39.83368pt} \\textrm {if } \\gamma \\in \\, ]2,\\infty [, \\ {P} \\in \\mathbb {S}_{>0}^{K}, \\ \\mathbf {Q} \\in A \\cdot \\mathbb {S}^{K} \\ \\ \\textrm {and } m \\in ]-\\infty ,\\infty [,\\\\\\infty , \\hspace{156.49014pt} \\textrm {else},\\end{array}\\right.","}$ where we have used the three $m-$ independent abbreviations $H_{\\gamma } := \\sum \\displaylimits _{k=1}^{K} (q_{k})^{\\gamma } \\cdot (p_{k})^{1-\\gamma }\\ = \\ 1 + \\gamma \\cdot (A-1) +\\frac{\\gamma \\cdot (\\gamma -1)}{\\widetilde{c}} \\cdot D_{\\widetilde{c} \\cdot \\varphi _{\\gamma }}(\\mathbf {Q}, {P}),\\qquad \\text{(cf.","(\\ref {brostu3:fo.divpow.hellinger1}))}\\nonumber $ $I:= \\sum \\displaylimits _{k=1}^{K} q_{k} \\cdot \\log \\left( \\frac{q_{k}}{p_{k}} \\right)\\ = \\ \\frac{1}{\\widetilde{c}} \\cdot D_{\\widetilde{c} \\cdot \\varphi _{1}}(\\mathbf {Q}, {P}) + A - 1,\\qquad \\text{(cf.","(\\ref {brostu3:fo.divpow.Kull1}))}\\nonumber $ $\\widetilde{I} := \\sum \\displaylimits _{k=1}^{K} p_{k} \\cdot \\log \\left( \\frac{p_{k}}{q_{k}} \\right)\\ = \\ \\frac{1}{\\widetilde{c}} \\cdot D_{\\widetilde{c} \\cdot \\varphi _{0}}(\\mathbf {Q}, {P}) + 1 - A.\\qquad \\text{(cf.","(\\ref {brostu3:fo.divpow.RevKull1}))}\\nonumber $ To proceed, let us fix an arbitrary constant $\\widetilde{c} >0$ .","(i) Case $\\gamma \\cdot (1-\\gamma ) \\ne 0$ .","(ia) Let us start with the subcase $\\gamma \\in ]-\\infty ,0[$ .","From the first and the last line of (REF ), it is clear that the corresponding $m-$ infimum can not be achieved for $m \\le 0$ ; since $H_{\\gamma } > 0$ one gets the unique minimizer $m_{min} = \\Big (\\frac{H_{\\gamma }}{A}\\Big )^{1/(1-\\gamma )} >0$ and the minimum $D_{\\widetilde{c} \\cdot \\varphi _{\\gamma }}(m_{min} \\cdot \\mathbf {Q},{P})=\\frac{\\widetilde{c}}{\\gamma } \\cdot (1- \\frac{H^{1/(1-\\gamma )}}{A^{\\gamma /(1-\\gamma )}} )$ .","Hence, (REF ) is established.","The assertions (REF ) and () follow immediately by monotonicity inspection of $x \\rightarrow \\frac{\\widetilde{c}}{\\gamma } \\cdot \\left[ 1-\\frac{1}{A^{\\gamma /(1-\\gamma )}} \\cdot \\left[ 1 + \\gamma \\cdot (A-1) + \\frac{\\gamma \\cdot \\left( \\gamma -1\\right)}{\\widetilde{c}}\\cdot x\\right] ^{-1/\\left( \\gamma -1\\right) }\\right]$ for $x \\ge 0$ such that $1 + \\gamma \\cdot (A-1) + \\frac{\\gamma \\cdot \\left( \\gamma -1\\right)}{\\widetilde{c}}\\cdot x \\ge 0$ .","(ib) The subcase $\\gamma \\in ]0,1[$ (cf.","the third line of (REF )) works analogously if $H_{\\gamma } >0$ ; furthermore, if $H_{\\gamma } =0$ — which can only appear when ${P}$ , $\\mathbf {Q}$ have disjoint supports (singularity)— then $\\inf _{m>0} D_{\\widetilde{c} \\cdot \\varphi _{\\gamma }}(m \\cdot \\mathbf {Q},{P}) =\\frac{\\widetilde{c}}{\\gamma }$ which is (the corresponding special case of) (REF ).","(ic) In the subcase $\\gamma \\in ]1,\\infty [$ (cf.","the fifth, sixth and seventh line of (REF )) it is straightforward to see that the desired infimum can not be achieved for $m < 0$ .","Hence, one can proceed analogously to subcase (ia).","(id) The assertions () to () are straightforward.","(ii) Case $\\gamma =1$ .","From the fourth line of (REF ), one obtains the unique minimizer $m_{min} = \\exp \\lbrace -I/A\\rbrace $ and the minimum $D_{\\widetilde{c} \\cdot \\varphi _{1}}(m_{min} \\cdot \\mathbf {Q},{P})= \\widetilde{c} \\cdot (1- A \\cdot m_{min})$ , which leads to (REF ).","The monotonicity of $x \\rightarrow \\widetilde{c} \\cdot (1- \\exp \\lbrace -x/\\widetilde{c}\\rbrace )$ for $x\\ge 0$ implies immediately (REF ) and (); moreover, () and () are immediate.","(iii) Case $\\gamma =0$ .","The second line of (REF ) implies the unique minimizer $m_{min} = 1/A$ , the minimum $D_{\\widetilde{c} \\cdot \\varphi _{0}}(m_{min} \\cdot \\mathbf {Q},{P})= \\widetilde{c} \\cdot (\\widetilde{I} + \\log A)$ , and hence (REF ).","The assertions (REF ) to () are obvious.","$\\blacksquare $"],["Proofs — Part 4","Proof of Proposition REF .","Clearly, (G1) and (G2) are part of the definition of $\\widetilde{\\Upsilon }(]a,b[)$ .","Recall our required representability (REF ).","The therein involved Laplace-Stieltjes transform (Laplace-Lebesgue transform) $z\\mapsto MGF_{\\mathbb {}}(z):=\\int _{\\mathbb {R}}e^{z \\cdot y} \\, d\\mathbb {} (y)= E_{\\mathbb {\\Pi }}[e^{z\\cdot W}]$ of a probability measure $\\mathbb {}$ on the real line respectively of an associated random variable $W$ (with $\\mathbb {}[ \\cdot \\, ] := \\mathbb {\\Pi }[W \\in \\cdot \\, ]$ ) has the following fundamental properties, according to well-known general theory: $MGF_{\\mathbb {}}$ takes values in $]0,\\infty ]$ ; the effective domain $dom(MGF_{\\mathbb {}})$ is an interval which contains 0 and which may be degenerated or even the whole real line; correspondingly, we denote its interior by $]\\lambda _{-},\\lambda _{+}[ := int(dom(MGF_{\\mathbb {}}))$ which may be the empty set (in case that $dom(MGF_{\\mathbb {}})=\\lbrace 0\\rbrace $ , i.e.","$\\lambda _{-}=\\lambda _{+}=0$ ); clearly, there holds $\\lambda _{-} \\in [-\\infty ,0]$ and $\\lambda _{+} \\in [0,\\infty ]$ ; $MGF_{\\mathbb {}}$ is continuous on $dom(MGF_{\\mathbb {}})$ and lower semicontinuous on $\\mathbb {R}$ ; if $\\lambda _{-} \\ne \\lambda _{+}$ , then $MGF_{\\mathbb {}}$ is real analytic and thus infinitely differentiable on $]\\lambda _{-},\\lambda _{+}[$ ; if $MGF_{\\mathbb {}}$ is finite in a neighborhood of zero, i.e.","$0 \\in ]\\lambda _{-},\\lambda _{+}[$ , then for all $k \\in \\mathbb {N}_{0}$ the $k-$ th moment of $\\mathbb {}$ respectively $W$ exists and is finite and can be computed in terms of the $k-$ th derivative $MGF_{\\mathbb {}}^{(k)}$ as $MGF_{\\mathbb {}}^{(k)}(0) =\\int _{\\mathbb {R}} y^{k} \\, d\\mathbb {} (y)= E_{\\mathbb {\\Pi }}[W^{k}],$ which, by the way, then allows the interpretation of $MGF_{\\mathbb {}}$ as “moment generating function of $\\mathbb {}$ resp.","$W$ ”since we assume $0 \\in ]\\lambda _{-},\\lambda _{+}[$ , we have already used the meaningful abbreviation $MGF$ (rather than LST) in (REF ); if $\\lambda _{-} \\ne \\lambda _{+}$ , then $MGF_{\\mathbb {}}$ is strictly convex on $]\\lambda _{-},\\lambda _{+}[$ .","Hence, the logarithm of the Laplace-Stieltjes transform $z\\mapsto \\Lambda _{\\mathbb {}}(z) := \\log MGF_{\\mathbb {} }(z):=\\log \\int _{\\mathbb {R}}e^{z \\cdot y} \\, d\\mathbb {} (y)= \\log E_{\\mathbb {\\Pi }}[e^{z\\cdot W}]$ (which in case of $0 \\in ]\\lambda _{-},\\lambda _{+}[$ can be interpreted as cumulant generating function) “carries over” (M1) to (M6), which partially can be even refined: $\\Lambda _{\\mathbb {}}$ takes values in $]-\\infty ,\\infty ]$ ; $dom(\\Lambda _{\\mathbb {}}) = dom(MGF_{\\mathbb {}})$ and thus $int(dom(\\Lambda _{\\mathbb {} })) = \\, ]\\lambda _{-},\\lambda _{+}[$ ; $\\Lambda _{\\mathbb {} }$ is continuous on $dom(\\Lambda _{\\mathbb {}})$ and lower semicontinuous on $\\mathbb {R}$ ; if $\\lambda _{-} \\ne \\lambda _{+}$ , then $\\lambda _{\\mathbb {}}$ is infinitely differentiable on $]\\lambda _{-},\\lambda _{+}[$ ; if $0 \\in ]\\lambda _{-},\\lambda _{+}[$ , then $& & \\Lambda _{\\mathbb {} }(0) = 0, \\quad \\Lambda _{\\mathbb {} }^{\\prime }(0) = \\int _{\\mathbb {R}} y \\, d\\mathbb {} (y)= E_{\\mathbb {\\Pi }}[W],\\\\& & \\Lambda _{\\mathbb {} }^{\\prime \\prime }(0) =\\int _{\\mathbb {R}} \\Big (y - \\int _{\\mathbb {R}} \\widetilde{y} \\, d\\mathbb {} (\\widetilde{y})\\Big )^{2}\\, d\\mathbb {} (y)= E_{\\mathbb {\\Pi }}[W^{2}] - (E_{\\mathbb {\\Pi }}[W])^{2} = Var_{\\mathbb {\\Pi }}[W];$ under the assumption $\\lambda _{-} \\ne \\lambda _{+}$ there holds: $\\Lambda _{\\mathbb {}}$ is strictly convex on $]\\lambda _{-},\\lambda _{+}[$ if and only if $\\mathbb {}$ is not a one-point distribution (Dirac mass) if and only if $W$ is not a.s. constant; otherwise, $\\Lambda _{\\mathbb {}}$ is linear; under the assumption that $\\mathbb {}$ is not a one-point distribution (Dirac mass) — with the notations $a := \\inf supp(\\mathbb {}) = \\inf supp(W)$ , $b := \\sup supp(\\mathbb {}) = \\sup supp(W)$ , $t_{-}^{sc} := \\inf \\lbrace \\Lambda _{\\mathbb {}}^{\\prime }(z) \\, : \\, z \\in ]\\lambda _{-},\\lambda _{+}[ \\rbrace = \\lim _{z \\downarrow \\lambda _{-}} \\Lambda _{\\mathbb {}}^{\\prime }(z)$ and $t_{+}^{sc} := \\sup \\lbrace \\Lambda _{\\mathbb {}}^{\\prime }(z) \\, : \\, z \\in ]\\lambda _{-},\\lambda _{+}[ \\rbrace = \\lim _{z \\uparrow \\lambda _{+}} \\Lambda _{\\mathbb {}}^{\\prime }(z)$ — one gets the following assertions: $]t_{-}^{sc},t_{+}^{sc}[ \\ \\subseteq \\ ]a,b[$ ; if $a > - \\infty $ , then $\\lambda _{-} = - \\infty $ , $t_{-}^{sc} = \\lim _{z \\rightarrow -\\infty } \\Lambda _{\\mathbb {}}^{\\prime }(z)= \\lim _{z \\rightarrow -\\infty } \\frac{\\Lambda _{\\mathbb {}}(z)}{z} = a$ ; if $b < \\infty $ , then $\\lambda _{+} = \\infty $ , $t_{+}^{sc} = \\lim _{z \\rightarrow \\infty } \\Lambda _{\\mathbb {}}^{\\prime }(z)= \\lim _{z \\rightarrow \\infty } \\frac{\\Lambda _{\\mathbb {}}(z)}{z} = b$ ; if $a = - \\infty $ and $\\lambda _{-} = - \\infty $ , then $t_{-}^{sc} = \\lim _{z \\rightarrow -\\infty } \\Lambda _{\\mathbb {}}^{\\prime }(z)= -\\infty = a$ ; if $b = \\infty $ and $\\lambda _{+} = \\infty $ , then $t_{+}^{sc} = \\lim _{z \\rightarrow \\infty } \\Lambda _{\\mathbb {}}^{\\prime }(z)= \\infty = b$ ; if $\\lambda _{-} \\in \\, ]-\\infty ,0[$ and $t_{-}^{sc} > -\\infty $ , then $a = - \\infty $ , $\\Lambda _{\\mathbb {}}(\\lambda _{-}) \\in \\, ]-\\infty ,\\infty [$ , $\\Lambda _{\\mathbb {}}(z) = \\infty $ for all $z < \\lambda _{-}$ , $\\Lambda _{\\mathbb {}}^{\\prime }(\\lambda _{-}) \\in \\, ]-\\infty ,\\infty [$ ; if $\\lambda _{+} \\in \\, ]0,\\infty [$ and $t_{+}^{sc} < \\infty $ , then $b = \\infty $ , $\\Lambda _{\\mathbb {}}(\\lambda _{+}) \\in \\, ]-\\infty ,\\infty [$ , $\\Lambda _{\\mathbb {}}(z) = \\infty $ for all $z > \\lambda _{+}$ , $\\Lambda _{\\mathbb {}}^{\\prime }(\\lambda _{+})\\in \\, ]-\\infty ,\\infty [$ ; if $\\lambda _{-} \\in \\, ]-\\infty ,0[$ and $t_{-}^{sc} = -\\infty $ , then $a= - \\infty $ ; if $\\lambda _{+} \\in \\, ]0,\\infty [$ and $t_{+}^{sc} = \\infty $ , then $b= \\infty $ .","Notice that (C7ii) to (C7ix) cover all possible constellations.","For a proof of (C7ii) to (C7vii) as well as further details, see e.g.","Section 9.1 in Borovkov [56].","By contradiction, (C7viii) follows from (C7ii) and (C7ix) follows from (C7iii).","Moreover, (C7i) is a consequence (C7ii) to (C7ix).","As a side remark, notice that (C6) refines (M6).","According to the representability requirement (REF ), one has $\\varphi (t)=\\sup _{z\\in \\mathbb {R}}\\left( z\\cdot t- \\Lambda _{\\mathbb {}}(z)\\right)=: \\Lambda _{\\mathbb {}}^{*}(t), \\qquad t\\in \\mathbb {R},\\ \\ $ (i.e.","the divergence generator $\\varphi $ must be equal to the Fenchel-Legendre transform $\\Lambda _{\\mathbb {}}^{*}$ of a cumulant generating function $\\Lambda _{\\mathbb {}}$ ) of some probability distribution $\\mathbb {}$ , such that $\\lambda _{-} < 0 < \\lambda _{+}$ holds.","Moreover, $\\varphi $ should satisfy $\\varphi (1)=0$ , and should be finite as well as strictly convex in a non-empty neighborhood $]t_{-}^{sc},t_{+}^{sc}[$ of 1 (cf.","the definition of $\\widetilde{\\Upsilon }(]a,b[)$ ).","The latter rules out that $\\mathbb {}$ is any one-point distribution (Dirac distribution), say $\\mathbb {} = \\delta _{y_{0}}$ for some $y_{0} \\in \\mathbb {R}$ , since in such a situation one gets $\\Lambda _{\\mathbb {}}(z) = z \\cdot y_{0}$ , and thus $\\varphi (t) = \\Lambda _{\\mathbb {}}^{*}(t) = 0$ for $t=y_{0}$ and $\\varphi (t) = \\Lambda _{\\mathbb {}}^{*}(t) = \\infty $ for all $t \\in \\mathbb {R}\\backslash \\lbrace y_{0}\\rbrace $ (even in the case $y_{0} =1$ for which $\\varphi (1)=0$ is satisfied).","Consequently, $\\Lambda _{\\mathbb {}}$ is strictly convex on $]\\lambda _{-},\\lambda _{+}[ \\ = int(dom(\\Lambda _{\\mathbb {} }))$ (cf.","(C6)) and (C7) applies.","Clearly, by continuity one gets $\\Lambda _{\\mathbb {}}^{*}(t) =\\sup _{z\\in ]\\lambda _{-},\\lambda _{+}[}\\left( t \\cdot z - \\Lambda _{\\mathbb {}}(z)\\right),\\qquad t\\in \\mathbb {R}.","\\ \\ $ For $t \\in ]t_{-}^{sc},t_{+}^{sc}[$ , the optimization problem (REF ) can be solved explicitly by the well-known “pure/original” Legendre transform, namely $\\Lambda _{\\mathbb {}}^{*}(t) = t \\cdot \\Lambda _{\\mathbb {}}^{\\prime -1}(t) -\\Lambda _{\\mathbb {}}\\Big (\\Lambda _{\\mathbb {}}^{\\prime -1}(t)\\Big ),\\qquad t \\in ]t_{-}^{sc},t_{+}^{sc}[.$ Let us inspect the further cases $t \\le t_{-}^{sc}$ .","In the contexts of (C7iv) and (C7viii), this is obsolete since $t_{-}^{sc} = a = -\\infty $ .","For (C7ii), where $t_{-}^{sc} = a > -\\infty $ , one can show $\\Lambda _{\\mathbb {}}^{*}(a) = - \\log \\mathbb {}[\\lbrace a\\rbrace ]= - \\log \\mathbb {\\Pi }[W = a \\, ]$ which together with (REF ) proves (G10ii); moreover, $\\Lambda _{\\mathbb {}}^{*}(t) = \\infty $ for all $t < a$ (see e.g.","Section 9.1 of Borovkov [56]).","In the setup (C7vi), where $t_{-}^{sc} > a = - \\infty $ it is clear that $\\Lambda _{\\mathbb {}}^{*}(t_{-}^{sc}) =t_{-}^{sc} \\cdot \\Lambda _{\\mathbb {}}^{\\prime -1}(t_{-}^{sc}) -\\Lambda _{\\mathbb {}}\\Big (\\Lambda _{\\mathbb {}}^{\\prime -1}(t_{-}^{sc})\\Big )= t_{-}^{sc} \\cdot \\lambda _{-} - \\Lambda _{\\mathbb {}}(\\lambda _{-})$ and $\\Lambda _{\\mathbb {}}^{*}(t)= t \\cdot \\lambda _{-} - \\Lambda _{\\mathbb {}}(\\lambda _{-})= \\Lambda _{\\mathbb {}}^{*}(t_{-}^{sc}) + \\lambda _{-} \\cdot (t- t_{-}^{sc})\\quad \\text{for all $t \\in ]-\\infty ,t_{-}^{sc}[$}.$ As far as the cases $t \\ge t_{+}^{sc}$ is concerned, in the situations of (C7v) and (C7ix), this is obsolete since $t_{+}^{sc} = b = \\infty $ .","For (C7iii), where $t_{+}^{sc} = b < \\infty $ , one can show $\\Lambda _{\\mathbb {}}^{*}(b) = - \\log \\mathbb {}[\\lbrace b\\rbrace ]= - \\log \\mathbb {\\Pi }[W = b \\, ]$ which together with (REF ) proves (G10iii); moreover, $\\Lambda _{\\mathbb {}}^{*}(t) = \\infty $ for all $t > b$ (see e.g.","Section 9.1 of Borovkov [56]).","In the setup (C7vii), where $t_{+}^{sc} < b = \\infty $ it is clear that $\\Lambda _{\\mathbb {}}^{*}(t_{+}^{sc}) =t_{+}^{sc} \\cdot \\Lambda _{\\mathbb {}}^{\\prime -1}(t_{+}^{sc}) -\\Lambda _{\\mathbb {}}\\Big (\\Lambda _{\\mathbb {}}^{\\prime -1}(t_{+}^{sc})\\Big )= t_{+}^{sc} \\cdot \\lambda _{+} - \\Lambda _{\\mathbb {}}(\\lambda _{+})$ and $\\Lambda _{\\mathbb {}}^{*}(t)= t \\cdot \\lambda _{+} - \\Lambda _{\\mathbb {}}(\\lambda _{+})= \\Lambda _{\\mathbb {}}^{*}(t_{+}^{sc}) + \\lambda _{+} \\cdot (t- t_{+}^{sc})\\quad \\text{for all $t \\in ]t_{+}^{sc}, \\infty [$}.$ As a side effect, we have thus also proved (G10i) and (G3) (notice that in (G3) we have started with $a, b$ to be the endpoints of the support of $\\mathbb {}$ respectively $W$ , in contrast to Definition REF where $a$ , $b$ are defined as the endpoints of the effective domain of $\\varphi $ ).","To proceed, from (REF ) and (REF ) we obtain $\\varphi ^{\\prime }(t) = (\\Lambda _{\\mathbb {}}^{*})^{\\prime }(t) = \\Lambda _{\\mathbb {}}^{\\prime -1}(t),\\qquad \\varphi ^{\\prime \\prime }(t) = (\\Lambda _{\\mathbb {}}^{*})^{\\prime \\prime }(t) =\\frac{1}{\\Lambda _{\\mathbb {}}^{\\prime \\prime }\\big (\\Lambda _{\\mathbb {}}^{\\prime -1}(t)\\big )} > 0,\\qquad t \\in ]t_{-}^{sc},t_{+}^{sc}[,$ which — together with the investigations below (REF ) — provides (G4) and (G5); moreover, (G6) is immediate since the infinite differentiability is straightforward and $\\varphi ^{\\prime }(1) =0$ because we have required both the nonnegativity of $\\varphi $ and (G2) (cf.","the definition of $\\widetilde{\\Upsilon }(]a,b[)$ ).","The property (G7) follows from (C7ii), (C7iv), (C7viii), (REF ), (REF ) and $\\varphi ^{\\prime }(t_{-}^{sc}) = \\Lambda _{\\mathbb {}}^{\\prime -1}(t_{-}^{sc}) = \\lambda _{-}$ .","Analogously, we get (G8) from (C7iii), (C7v), (C7ix), (REF ), (REF ) and $\\varphi ^{\\prime }(t_{+}^{sc}) = \\Lambda _{\\mathbb {}}^{\\prime -1}(t_{+}^{sc}) = \\lambda _{+}$ .","Let us continue with (G9).","By applying the general theory of double Fenchel-Legendre transforms (bi-conjugates), (REF ) turns into $\\varphi ^{*} (z) = \\Lambda _{\\mathbb {}}(z), \\qquad z\\in \\mathbb {R},\\ \\ $ which deduces (G9i).","The properties (G9ii), (G9iii) and (G9iv) follow from Theorem REF (cf.","the discussion thereafter).","Finally, we obtain (G11i) and (G11ii) from (REF ), (REF ) and ().","$\\blacksquare $ Proof of Proposition REF .","The assertions follow immediately from (REF ), (REF ), (REF ), Theorem REF , (REF ) (and the discussion thereafter) as well as (M5).","$\\blacksquare $"],["Proofs — Part 5","Proof of Proposition REF .","The assertion follows straightforwardly from the following two facts: (i) a moment generating function $MGF$ is infinitely divisible if and only if $MGF^{c}$ is a moment generating function for all $c >0$ (cf.","e.g.","(the MGF-version of) Prop.","IV.2.5 of Steutel & van Harn [341]).","(ii) $z \\mapsto MGF(z)$ is a moment generating function if and only if $z \\mapsto MGF(\\breve{c} \\cdot z) =: MGF_{\\breve{c}}(z)$ is a moment generating function for all $\\breve{c} >0$ .","Notice that for each $c >0$ , $\\breve{c} >0$ one has $int(dom(MGF)) = int(dom(MGF^{c}))$ and $int(dom(MGF_{\\breve{c}})) = \\frac{1}{\\breve{c}} \\cdot int(dom(MGF))$ , and hence the light-tailedness remains unchanged: $0 \\in int(dom(MGF))$ if and only if $0 \\in int(dom(MGF^{c}))$ if and only if $0 \\in int(dom(MGF_{\\breve{c}}))$ .","Since $\\varphi \\in \\Upsilon (]a,b[)$ , we have $\\varphi (t)=\\sup _{z\\in ]\\lambda _{-}, \\lambda _{+}[}\\left( z \\cdot t-\\log \\Big (\\int _{\\mathbb {R}} e^{z\\cdot y }\\, d\\mathbb {} (y) \\Big ) \\right), \\quad t\\in ]a,b[ \\, , \\ \\ $ and thus for the exponential of its Fenchel-Legendre transform $\\int _{\\mathbb {R}} e^{z \\cdot y} d\\mathbb {} (y) , \\qquad z \\in ]\\lambda _{-}, \\lambda _{+}[ .$ Now, let $\\widetilde{\\varphi } := \\widetilde{c} \\cdot \\varphi \\in \\Upsilon (]a,b[)$ for arbitrarily fixed $\\widetilde{c} >0$ .","From the application of (REF ) to $\\widetilde{\\varphi }$ we obtain $\\widetilde{\\varphi } (t)=\\sup _{\\widetilde{z} \\in ]\\widetilde{\\lambda }_{-}, \\widetilde{\\lambda }_{+}[}\\left( \\widetilde{z} \\cdot t-\\log \\int _{\\mathbb {R}}e^{\\widetilde{z} \\cdot \\widetilde{y}} d\\widetilde{\\mathbb {}}_{\\widetilde{c}} (\\widetilde{y})\\right), \\qquad t\\in ]a,b[ \\, , \\ \\ $ for some unique probability distribution $\\widetilde{\\mathbb {}}_{\\widetilde{c}}$ on $\\mathbb {R}$ .","Here, according to (G9i) for $\\widetilde{\\varphi }$ we have used $\\widetilde{\\lambda }_{-} := \\inf _{t \\in ]a,b[} \\widetilde{\\varphi }^{\\prime }(t) = \\widetilde{c} \\cdot \\lambda _{-}$ and $\\widetilde{\\lambda }_{+} := \\sup _{t \\in ]a,b[} \\widetilde{\\varphi }^{\\prime }(t) = \\widetilde{c} \\cdot \\lambda _{+}$ .","Dividing (REF ) by $\\widetilde{c}$ , we arrive at $\\varphi (t) = \\frac{\\widetilde{\\varphi } (t)}{\\widetilde{c}}&=&\\sup _{\\widetilde{z} \\in ]\\widetilde{c} \\cdot \\lambda _{-}, \\widetilde{c} \\cdot \\lambda _{+}[}\\left( \\frac{\\widetilde{z}}{\\widetilde{c}} \\cdot t-\\log \\Big (\\int _{\\mathbb {R}}e^{\\frac{\\widetilde{z}}{\\widetilde{c}} \\cdot \\widetilde{y} \\cdot \\widetilde{c}} d\\widetilde{\\mathbb {}}_{\\widetilde{c}} (\\widetilde{y})\\Big )^{1/\\widetilde{c}}\\right),\\nonumber \\\\&=&\\sup _{z \\in ]\\lambda _{-}, \\lambda _{+}[}\\left( z \\cdot t-\\log \\Big (\\int _{\\mathbb {R}}e^{z \\cdot \\widetilde{y} \\cdot \\widetilde{c}}d\\widetilde{\\mathbb {}}_{\\widetilde{c}} (\\widetilde{y})\\Big )^{1/\\widetilde{c}}\\right),\\qquad t\\in ]a,b[ \\, , \\ \\ $ and hence for the exponential of its Fenchel-Legendre transform $e^{\\varphi _{*}(z)} =\\Big ( \\int _{\\mathbb {R}} e^{z \\cdot \\widetilde{y} \\cdot \\widetilde{c}}d\\widetilde{\\mathbb {}}_{\\widetilde{c}} (\\widetilde{y}) \\Big )^{1/\\widetilde{c}}, \\qquad z \\in ]\\lambda _{-}, \\lambda _{+}[ .$ Here, according to (G9i) for $\\widetilde{\\varphi }$ we have used $\\widetilde{\\lambda }_{-} := \\inf _{t \\in ]a,b[} \\widetilde{\\varphi }^{\\prime }(t) = \\widetilde{c} \\cdot \\lambda _{-}$ and $\\widetilde{\\lambda }_{+} := \\sup _{t \\in ]a,b[} \\widetilde{\\varphi }^{\\prime }(t) = \\widetilde{c} \\cdot \\lambda _{+}$ .","From (REF ) and (REF ) we deduce for $\\widetilde{c} := \\frac{1}{n}$ the relation $MGF_{\\mathbb {}}(z) = (MGF_{\\widetilde{\\mathbb {}}_{1/n}}(\\frac{z}{n}))^{n}$ for all $n \\in \\mathbb {N}$ which (with the help of (ii)) implies the infinitely divisibility of $\\mathbb {}$ .","For the reverse direction, let us assume that $\\varphi \\in \\Upsilon (]a,b[)$ and that the corresponding $\\mathbb {}$ is infinitely divisible.","Recall that $]a,b[ = int(dom(\\varphi ))$ .","Moreover, we fix an arbitrary constant $\\widetilde{c} >0$ .","Of course, there holds $\\widetilde{c} \\cdot \\varphi \\in \\widetilde{\\Upsilon }(]a,b[)$ and $dom(\\widetilde{c} \\cdot \\varphi ) = dom(\\varphi )$ .","Furthermore, by multiplying (REF ) with $\\widetilde{c} >0$ and by employing (i), (ii) we get $\\widetilde{c} \\cdot \\varphi (t) &=&\\sup _{z\\in ]\\lambda _{-}, \\lambda _{+}[}\\left( \\widetilde{c} \\cdot z \\cdot t-\\log \\Big (\\int _{\\mathbb {R}} e^{\\widetilde{c} \\cdot z\\cdot \\frac{y}{\\widetilde{c}}} d\\mathbb {} (y) \\Big )^{\\widetilde{c}} \\right)= \\sup _{\\widetilde{z} \\in ]\\widetilde{c} \\cdot \\lambda _{-}, \\widetilde{c} \\cdot \\lambda _{+}[}\\left( \\widetilde{z} \\cdot t-\\log \\Big (\\int _{\\mathbb {R}}e^{\\frac{\\widetilde{z}}{\\widetilde{c}} \\cdot y} \\, d\\mathbb {} (y) \\Big )^{\\widetilde{c}} \\right)\\nonumber \\\\&=&\\sup _{\\widetilde{z} \\in ]\\widetilde{c} \\cdot \\lambda _{-}, \\widetilde{c} \\cdot \\lambda _{+}[}\\left( \\widetilde{z} \\cdot t-\\log \\Big ( \\int _{\\mathbb {R}}e^{\\widetilde{z} \\cdot y} d\\mathbb {}_{\\widetilde{c}} (y) \\Big ) \\right), \\quad t\\in ]a,b[; \\ \\ $ for some probability distribution $\\mathbb {}_{\\widetilde{c}}$ on $\\mathbb {R}$ .","$\\blacksquare $ Proof of Proposition REF .","It is well known that a candidate function $M: ]-\\infty ,0[ \\mapsto ]0,\\infty [$ is the moment-generating function of an infinitely divisible probability distribution if and only if $(\\log M)^{\\prime }$ is absolutely monotone (see e.g.","Theorem 5.11 of Schilling et al.","[322]).","By applying this to $M(z) := e^{-a\\cdot z + \\varphi ^{*}(z)}$ respectively $M(z) := e^{b\\cdot z + \\varphi ^{*}(- z)}$ , one gets straightforwardly the assertion (a) respectively (b); notice that the light-tailedness follows then from (G1) to (G8), and $b=\\infty $ respectively $a =- \\infty $ can be deduced from the fact that the support of an infinitely distribution is always (one-sided or two-sided) unbounded.","For the third case $a= - \\infty $ , $b= \\infty $ one can use the assertion (cf.","e.g.","Morris [267], p.73) that a candidate function $M: \\, ]\\lambda _{-}, \\lambda _{+}[ \\, \\mapsto \\, ]0,\\infty [$ is the moment-generating function of an infinitely divisible probability distribution if the connected function $z \\mapsto (\\log M)^{\\prime \\prime }(z)/(\\log M)^{\\prime \\prime }(0)$ is the moment-generating function of some auxiliary probability distribution; but the latter is equivalent to exponentially convexity (cf.","Theorem REF (b)).","By applying this to $M(z) := e^{\\varphi ^{*}(z)}$ , one ends up with (c).","$\\blacksquare $"],["Proofs — Part 6","Proof of Theorem REF .","(i) Clearly, on $]\\lambda _{-},\\lambda _{+}[$ the function $\\Lambda $ is differentiable with strictly increasing derivative $\\Lambda ^{\\prime }(z) = F^{-1}(z+c) + 1- F^{-1}(c) ,\\qquad z \\in ]\\lambda _{-},\\lambda _{+}[.$ Hence, $\\Lambda $ is strictly convex and smooth (because of the smoothness of $F^{-1}$ ), and satisfies $\\Lambda (0) =0$ as well as $\\Lambda ^{\\prime }(0) =1$ .","Also, the corresponding extensions of $\\Lambda $ to $z=\\lambda _{-}$ and $z=\\lambda _{+}$ are continuous.","(ii) It is straightforward to see that on $]t_{-}^{sc},t_{+}^{sc}[$ the function $\\varphi $ is differentiable with strictly increasing derivative $\\varphi ^{\\prime }(t) = F(t+ F^{-1}(c) - 1) - c ,\\qquad t \\in ]t_{-}^{sc},t_{+}^{sc}[.$ Hence, $\\varphi $ is strictly convex and smooth (because of the smoothness of $F$ ), and satisfies $\\varphi (1) =0$ as well as $\\varphi ^{\\prime }(1) =0$ .","Also, the corresponding extensions of $\\varphi $ to $t=t_{-}^{sc}$ and $t=t_{-}^{sc}$ are continuous.","Hence (G1), (G2), (G5) and (G6) hold.","To prove (G3) (and hence (G1)), let us first notice that obviously there holds $a \\le t_{-}^{sc}$ and $t_{+}^{sc} \\le b$ .","Moreover, the validity of $\\varphi (t) < \\infty $ for all $t \\in ]t_{-}^{sc},t_{+}^{sc}[$ is clear from (REF ) since $t+F^{-1}(c)-1 \\in ]a_{F},b_{F}[ = int(dom(F))$ and the involved integral over the continuous function $F^{-1}$ is taken over a compact interval.","For the subcase $t_{-}^{sc} =- \\infty = a$ we have thus shown $dom(\\varphi ) \\cap ]-\\infty ,1] = ]-\\infty ,1]= ]a,1]$ , whereas for the subcase $t_{+}^{sc} = \\infty = b$ we have verified $dom(\\varphi ) \\cap [1,\\infty [ = [1,\\infty [ = [1,b[$ .","Let us next examine the subcase “$t_{-}^{sc} > - \\infty $ and $\\varphi (t_{-}^{sc}) < \\infty $ ”: if $\\lambda _{-} > - \\infty $ then $a = -\\infty $ and (REF ) implies $\\varphi (t) = \\varphi (t_{-}^{sc}) +\\lambda _{-} \\cdot (t- t_{-}^{sc}) < \\infty $ for all $t \\in \\ ]-\\infty , t_{-}^{sc}] = ]a,t_{-}^{sc}]$ , which leads to $dom(\\varphi ) \\cap ]-\\infty ,1] = ]-\\infty ,1] = ]a,1]$ ; in contrast, if $\\lambda _{-} = - \\infty $ then $a = t_{-}^{sc}$ and (REF ) implies $\\varphi (t) = \\varphi (t_{-}^{sc}) +\\lambda _{-} \\cdot (t- t_{-}^{sc}) = \\infty $ for all $t \\in \\ ]-\\infty , t_{-}^{sc}[ = ]-\\infty ,a[$ , which leads to $dom(\\varphi ) \\cap ]-\\infty ,1] = [a,1]$ .","In the subcase “$t_{-}^{sc} > - \\infty $ and $\\varphi (t_{-}^{sc}) = \\infty $ ”, due to the strict convexity of $\\varphi $ one always has $\\lim _{t \\downarrow t_{-}^{sc}} \\varphi ^{\\prime }(t) = - \\infty $ ; this implies, by the below-mentioned (REF ), that $\\lambda _{-} = - \\infty $ and thus $a = t_{-}^{sc}$ ; from (REF ) we derive $\\varphi (t) = \\varphi (t_{-}^{sc}) +\\lambda _{-} \\cdot (t- t_{-}^{sc}) = \\infty $ for all $t \\in \\ ]-\\infty , t_{-}^{sc}[ = ]-\\infty ,a[$ , which leads to $dom(\\varphi ) \\cap ]-\\infty ,1] = ]a,1]$ .","As a further step, we deal with the subcase “$t_{+}^{sc} < \\infty $ and $\\varphi (t_{+}^{sc}) < \\infty $ ”: if $\\lambda _{+} < \\infty $ then $b = \\infty $ and (REF ) implies $\\varphi (t) = \\varphi (t_{+}^{sc}) +\\lambda _{+} \\cdot (t- t_{+}^{sc}) < \\infty $ for all $t \\in \\ [t_{+}^{sc},\\infty [ = [t_{+}^{sc},b[$ , which leads to $dom(\\varphi ) \\cap [1, \\infty [ = [1, \\infty [ = [1, b[$ ; in contrast, if $\\lambda _{+} = \\infty $ then $b = t_{+}^{sc}$ and (REF ) implies $\\varphi (t) = \\varphi (t_{+}^{sc}) +\\lambda _{+} \\cdot (t- t_{+}^{sc}) = \\infty $ for all $t \\in \\ ]t_{-}^{sc}, \\infty [ = ]b,\\infty [$ , which leads to $dom(\\varphi ) \\cap [1,\\infty [ = [1,b]$ .","In the subcase “$t_{+}^{sc} < + \\infty $ and $\\varphi (t_{+}^{sc}) = \\infty $ ”, due to the strict convexity of $\\varphi $ one always gets $\\lim _{t \\uparrow t_{+}^{sc}} \\varphi ^{\\prime }(t) = \\infty $ ; this implies, by the below-mentioned (), that $\\lambda _{+} = \\infty $ and thus $b = t_{+}^{sc}$ ; from (REF ) we deduce $\\varphi (t) = \\varphi (t_{+}^{sc}) +\\lambda _{+} \\cdot (t- t_{+}^{sc}) = \\infty $ for all $t \\in \\ ]t_{+}^{sc}, \\infty [ = ]b, \\infty [$ , which leads to $dom(\\varphi ) \\cap [1,\\infty [ = [1,b[$ .","Putting things together, we have proved (G3).","The property (G4) follows straightforwardly from (REF ), the continuity of $F$ and from $\\lim _{t \\downarrow t_{-}^{sc}} \\varphi ^{\\prime }(t) = \\lambda _{-}$ , $\\lim _{t \\uparrow t_{+}^{sc}} \\varphi ^{\\prime }(t) = \\lambda _{+}$ .","To see the latter two, from (REF ) we obtain $& & \\lim _{t \\downarrow t_{-}^{sc}} \\varphi ^{\\prime }(t) = \\lim _{t \\downarrow t_{-}^{sc}} F(t+ F^{-1}(c) - 1) - c= \\lim _{t \\downarrow t_{-}^{sc}} F(t+ a_{F} - t_{-}^{sc}) - c= \\inf \\lbrace F(\\widetilde{t}) - c: \\widetilde{t} \\in ]a_{F},b_{F}[ \\rbrace = \\lambda _{-},\\\\& & \\lim _{t \\uparrow t_{+}^{sc}} \\varphi ^{\\prime }(t) = \\lim _{t \\uparrow t_{+}^{sc}} F(t+ F^{-1}(c) - 1) - c= \\lim _{t \\uparrow t_{+}^{sc}} F(t+ b_{F} - t_{+}^{sc}) - c= \\sup \\lbrace F(\\widetilde{t}) - c: \\widetilde{t} \\in ]a_{F},b_{F}[ \\rbrace = \\lambda _{+}.$ The two properties (G7) and (G8) are clear form the above considerations.","(iii) From (REF ) and (REF ) one gets easily $\\Lambda ^{\\prime -1}(t) = F\\left(t+F^{-1}(c)-1 \\right) - c = \\varphi ^{\\prime }(t), \\qquad t \\in ]t_{-}^{sc},t_{+}^{sc}[,$ as well as $\\Lambda ^{\\prime -1}(1)=0$ .","From this, we derive $&& t \\cdot \\Lambda ^{\\prime -1}(t) - \\Lambda \\left(\\Lambda ^{\\prime -1}(t)\\right)\\nonumber \\\\&& = t \\cdot [F\\left(t+F^{-1}(c)-1 \\right) - c]+ [F^{-1}(c)-1] \\cdot [F\\left(t+F^{-1}(c)-1 \\right) - c]\\nonumber \\\\& & -\\int \\displaylimits _{0}^{F\\left(t+F^{-1}(c)-1 \\right) - c} F^{-1}(u+c) du\\nonumber \\\\& & = \\varphi (t), \\qquad t \\in ]t_{-}^{sc},t_{+}^{sc}[,$ and hence, with the help of (REF ) in combination with (REF ), () $\\varphi (t) = \\max _{z \\in ]\\lambda _{-},\\lambda _{+}[} \\left( z\\cdot t -\\Lambda (z)\\right), \\qquad t \\in ]t_{-}^{sc},t_{+}^{sc}[ ,$ i.e.","on $]t_{-}^{sc},t_{+}^{sc}[$ the divergence generator $\\varphi $ is the classical Legendre transform of the restriction of $\\Lambda $ to $]\\lambda _{-},\\lambda _{+}[$ .","If “$\\lambda _{-} > -\\infty $ , $\\Lambda (\\lambda _{-}) \\in ]-\\infty ,\\infty [$ and $\\Lambda ^{\\prime }(\\lambda _{-}) \\in ]-\\infty ,\\infty [$ ” respectively “$\\lambda _{+} < -\\infty $ , $\\Lambda (\\lambda _{+}) \\in ]-\\infty ,\\infty [$ and $\\Lambda ^{\\prime }(\\lambda _{+}) \\in ]-\\infty ,\\infty [$ ”, then one can apply classical facts of Fenchel-Legendre transformation to get the corresponding left-hand respectively right-hand linear extensions of $\\varphi $ on the complement of $]t_{-}^{sc},t_{+}^{sc}[$ , in order to obtain the desired $\\varphi (t) =\\sup _{z \\in ]-\\infty ,\\infty [} \\left( z\\cdot t - \\Lambda (z)\\right) ,\\qquad t \\in \\mathbb {R};$ notice that $t_{-}^{sc} = \\lim _{z \\downarrow \\lambda _{-}} \\Lambda ^{\\prime }(z)$ and $t_{+}^{sc} = \\lim _{z \\uparrow \\lambda _{+}} \\Lambda ^{\\prime }(z)$ .","(iv) This is just the inverse of (iii), by applying standard Fenchel-Legendre-transformation theory.","$\\blacksquare $"],["Further details and proofs for\nSubsection ","Proof of Lemma REF.","By Assumption (OM), one gets for all $\\lambda \\in cl(\\mathbf {\\Lambda })$ that $\\lbrace \\mathbf {x} \\in (dom(\\widetilde{\\varphi })^{n} : T(\\mathbf {x}) = \\lambda \\rbrace \\, \\cap \\, ]t_{-}^{sc},t_{+}^{sc}[^{n} \\ne \\emptyset $ .","Moreover, for any $\\mathbf {x}=\\left( x_{1},..,x_{n}\\right) $ in $\\mathbb {R}^{n}$ , by the independence of the components of $\\mathbf {\\widetilde{W}}$ as well as (REF ) and (REF ), we have $& & I_{\\mathbf {\\widetilde{W}}}(\\mathbf {x})=\\sup _{\\mathbf {z}=\\left( z_{1},\\ldots ,z_{n}\\right) \\in \\mathbb {R}^{n}}\\left( \\left\\langle \\mathbf {x},\\mathbf {z}\\right\\rangle -\\sum _{i=1}^{n}\\Lambda _{\\widetilde{\\mathbb {}}}(z_{i}) \\right)= \\sup _{\\mathbf {z}\\in ]\\lambda _{-},\\lambda _{+}[^{n}}\\left( \\sum _{i=1}^{n} \\left( x_{i} \\cdot z_{i}-\\Lambda _{\\widetilde{\\mathbb {}}}(z_{i}) \\right) \\right)\\nonumber \\\\& & = \\sum _{i=1}^{n} \\left( \\sup _{z_{i}\\in ]\\lambda _{-},\\lambda _{+}[}\\left( x_{i}\\cdot z_{i} - \\Lambda _{\\widetilde{\\mathbb {}}}(z_{i})\\right) \\right)= \\sum _{i=1}^{n} \\widetilde{\\varphi }(x_{i})=\\sum _{k=1}^{K}\\sum _{i\\in I_{k}^{(n)}} \\widetilde{\\varphi }(x_{i})$ which is finite if and only if $\\mathbf {x} \\in (dom(\\widetilde{\\varphi }))^{n}$ (recall that $\\widetilde{\\varphi }$ is a nonnegative function).","Hence, for each $\\lambda \\in \\mathbf {\\Lambda }$ we obtain $I(\\lambda ) &:=&\\inf _{\\mathbf {x} \\in \\mathbb {R}^{n}: \\, T(\\mathbf {x})=\\lambda }I_{\\mathbf {\\widetilde{W}}}(\\mathbf {x})= \\inf _{\\mathbf {x} \\in (dom(\\widetilde{\\varphi }))^{n}: \\, T(\\mathbf {x})=\\lambda }I_{\\mathbf {\\widetilde{W}}}(\\mathbf {x}) =\\inf _{\\mathbf {x} \\in (dom(\\widetilde{\\varphi }))^{n}: \\, T(\\mathbf {x})=\\lambda } \\ \\sum _{k=1}^{K}\\sum _{i\\in I_{k}^{(n)}} \\widetilde{\\varphi }(x_{i})\\\\&=& \\sum _{k=1}^{K} n_{k} \\cdot \\widetilde{\\varphi }(\\lambda _{k})\\ = \\ n \\cdot \\sum _{k=1}^{K} \\widetilde{p}_{k} \\cdot \\widetilde{\\varphi }(\\lambda _{k})= \\inf _{\\mathbf {x} \\in ]t_{-}^{sc},t_{+}^{sc}[^{n} \\, : \\, T(\\mathbf {x})=\\lambda }I_{\\mathbf {\\widetilde{W}}}(\\mathbf {x}) \\, ;$ here, we have employed the following facts: (i) the right-most infimum in (REF ) is achieved by minimizing each of the $K$ terms $\\sum _{i\\in I_{k}^{(n)}}\\widetilde{\\varphi }(x_{i})$ under the linear constraint $\\frac{1}{n_{k}} \\cdot \\sum _{i\\in I_{k}^{(n)}} x_{i}= \\lambda _{k}$ , and by the strict convexity of $\\widetilde{\\varphi }$ on $]t_{-}^{sc},t_{+}^{sc}[$ (cf.","(G5)) the minimum of this generic term is attained when all components $x_{i}$ are equal to $\\lambda _{k}$ , and (ii) the outcoming minimum does not depend on the particular (generally non-unique) choice of the $x_{i}$ ’s.","Notice that we have used the relation $n_{k} = n \\cdot \\widetilde{p}_{k}$ as well.","To proceed, let $\\underline{\\lambda }$ be a minimal rate point of $\\mathbf {\\Lambda }$ , which means that $\\underline{\\lambda } \\in \\partial \\mathbf {\\Lambda }$ and $I(\\underline{\\lambda }) \\le I(\\lambda )$ for all $\\lambda \\in \\mathbf {\\Lambda }$ .","By Assumption (OM) one can run all the steps in (REF ) and () with $\\underline{\\lambda }$ instead of $\\lambda $ , and hence $I(\\underline{\\lambda }) \\ = \\ \\inf _{\\mathbf {x} \\in \\mathbb {R}^{n}: \\, T(\\mathbf {x})=\\underline{\\lambda }}I_{\\mathbf {\\widetilde{W}}}(\\mathbf {x})\\ = \\inf _{\\mathbf {x} \\in ]t_{-}^{sc},t_{+}^{sc}[^{n} \\, : \\, T(\\mathbf {x})=\\underline{\\lambda }}I_{\\mathbf {\\widetilde{W}}}(\\mathbf {x})\\ = \\ n \\cdot \\sum _{k=1}^{K} \\widetilde{p}_{k} \\cdot \\widetilde{\\varphi }(\\underline{\\lambda }_{k})\\ = \\ n \\cdot \\sum _{k=1}^{K} \\widetilde{p}_{k} \\cdot \\widetilde{\\varphi }(\\underline{\\widetilde{q}}_{k}/\\widetilde{p}_{k})$ where for the last equality we have employed the vector $\\underline{\\mathbf {\\widetilde{Q}}} := \\mathfrak {D}^{-1} \\underline{\\lambda }$ which we have called the “dominating point of $\\widetilde{\\mathbf {\\Omega }}$ ”.","Also we have proved $I(\\underline{\\lambda })\\ = \\ n \\cdot \\inf _{\\mathbf {\\widetilde{Q}} \\in \\widetilde{\\mathbf {\\Omega }}}\\sum _{k=1}^{K} \\widetilde{p}_{k} \\cdot \\widetilde{\\varphi }(\\widetilde{q}_{k}/\\widetilde{p}_{k}).","\\qquad \\qquad \\blacksquare $ On the obtainment of proxies of minimal rate points by proxy method 2: For the rest of this section, besides (OM) we assume that $dom(\\widetilde{\\varphi }) = \\, ]a,b[ \\, = \\, ]t_{-}^{sc},t_{+}^{sc}[$ , and that in case of $a=-\\infty $ or $b=+\\infty $ the divergence generator $\\widetilde{\\varphi }$ is regularly varying at $a$ or $b$ accordingly, with positive index $\\beta $ , i.e.","(with a slight abuse of notation) if $a=-\\infty $ , then for all $\\lambda >0$ there holds $\\lim _{u\\rightarrow -\\infty }\\frac{\\widetilde{\\varphi } \\left( \\lambda u\\right) }{\\widetilde{\\varphi }\\left( u\\right) }=\\lambda ^{\\beta },$ if $b=+\\infty $ , then for all $\\lambda >0$ there holds $\\lim _{u\\rightarrow +\\infty }\\frac{\\widetilde{\\varphi } \\left( \\lambda u\\right) }{\\widetilde{\\varphi }\\left( u\\right) }=\\lambda ^{\\beta };$ this assumption is denoted by (H$\\widetilde{\\varphi }$ ).","A proxy of $\\underline{\\mathbf {\\widetilde{Q}}}$ can be obtained by sampling from a distribution on $\\mathbb {R}^{K}$ defined through $f(\\mathbf {\\widetilde{Q}}):=C \\cdot \\exp \\left( -\\sum _{k=1}^{K} \\widetilde{p}_{k} \\cdot \\widetilde{\\varphi }(\\widetilde{q}_{k}/\\widetilde{p}_{k})\\right) \\ = \\ C \\cdot \\exp \\left( -D_{\\widetilde{\\varphi }}\\left( \\mathbf {\\widetilde{Q}},\\widetilde{{P}}\\right)\\right)\\hspace{51.21504pt} \\textrm {(cf.","(\\ref {simul distr}))}\\nonumber $ where $C$ is a normalizing constant; strict convexity (cf.","(G5)) of $\\widetilde{\\varphi }$ together with (H$\\widetilde{\\varphi }$ ) prove that $f$ is a well-defined (Lebesgue-) density for a random variable $\\mathbf {T}$ on $\\mathbb {R}^{K}$ .","We denote by $\\mathbb {F}(\\cdot ) := \\mathbb {\\Pi }[\\mathbf {T} \\in \\cdot \\, ]$ the corresponding distribution on $\\mathbb {R}^{K}$ having density $f$ .","The distribution of $\\mathbf {T}$ given $\\left( \\mathbf {T} \\in \\widetilde{\\mathbf {\\Omega }} \\right)$ concentrates on the points in $\\widetilde{\\mathbf {\\Omega }}$ which minimize $D_{\\widetilde{\\varphi }}\\left(\\mathbf {\\widetilde{Q}},\\widetilde{{P}}\\right) $ as $\\mathbf {\\widetilde{Q}}$ runs in $\\widetilde{\\mathbf {\\Omega }}$ , when $D_{\\widetilde{\\varphi }}(\\widetilde{\\mathbf {\\Omega }},\\widetilde{{P}})$ is large.","This can be argued as follows.","We will consider the case when $\\widetilde{\\mathbf {\\Omega }}$ is a compact subset in $\\mathbb {R}_{> 0}^{K}$ and $\\widetilde{\\varphi }$ satisfies (H$\\widetilde{\\varphi }$ ) with $b=+\\infty $ .","For the case when $\\widetilde{\\mathbf {\\Omega }}$ is not compact, or belongs to $\\mathbb {R}^{K}/\\left\\lbrace \\mathbf {0}\\right\\rbrace $ , see the Remark REF hereunder.","Consider a compact set $\\mathbf {\\Gamma }$ in $\\widetilde{\\mathbf {\\Omega }}$ and let $\\mathbf {\\Gamma }_{t}$ be defined as deduced from $\\mathbf {\\Gamma } $ in a way that makes $D_{\\widetilde{\\varphi }}\\left( \\mathbf {\\Gamma }_{t},\\widetilde{{P}}\\right) $ increase with $t$ for sufficiently large $t$ .","For instance, set $\\mathbf {\\Gamma }_{t}:=t \\cdot \\mathbf {\\Gamma } .$ Hence, in case of $b=+\\infty $ the divergence $D_{\\widetilde{\\varphi }}\\left( \\mathbf {\\Gamma }_{t},\\widetilde{{P}}\\right) =\\inf _{g_{t}\\in \\mathbf {\\Gamma }_{t}}\\sum _{k=1}^{K} \\widetilde{p}_{k}\\cdot \\widetilde{\\varphi }\\left( \\frac{\\left(g_{t}\\right)_{k}}{\\widetilde{p}_{k}}\\right) =\\inf _{g \\in \\mathbf {\\Gamma } }\\sum _{k=1}^{K} \\widetilde{p}_{k}\\cdot \\widetilde{\\varphi }\\left( \\frac{t \\cdot g_{k}}{\\widetilde{p}_{k}}\\right)$ tends to infinity as $t \\rightarrow \\infty $ ; the case $a=-\\infty $ works analogously with $t\\rightarrow -\\infty $ .","In case of $b < \\infty $ we may consider $\\mathbf {\\Gamma } _{t}:=\\left\\lbrace b-g /t;g \\in \\mathbf {\\Gamma } \\right\\rbrace $ and indeed $D_{\\widetilde{\\varphi }}\\left( \\mathbf {\\Gamma } _{t},\\widetilde{{P}}\\right) \\rightarrow \\infty $ as $t\\rightarrow \\infty $ , with a similar statement when $a>-\\infty $ .","Assume that $\\mathbf {\\Gamma }$ has a dominating point $\\underline{g}$ .","Then $\\mathbf {\\Gamma }_{t}$ has dominating point $\\underline{g}_{t} := t \\cdot \\underline{g}$ .","We prove that $\\mathbf {T}$ with distribution (REF ) cannot be too far away (depending on $t$ ) from $\\underline{g}_{t}$ whenever $\\mathbf {T}$ belongs to $\\mathbf {\\Gamma } _{t}.$ This argument is valid in the present description of some asymptotics which makes $\\mathbf {\\Gamma } _{t}$ as a model for $\\widetilde{\\mathbf {\\Omega }}$ for large $t$ ; considering the case when $D_{\\widetilde{\\varphi } }\\left( \\widetilde{\\mathbf {\\Omega }} ,\\widetilde{{P}}\\right) $ is large is captured through the asymptotic statement $\\lim _{t\\rightarrow \\infty }D_{\\widetilde{\\varphi } }\\left( \\mathbf {\\Gamma } _{t},\\widetilde{{P}}\\right) =+\\infty .$ There holds the following Proposition 65 With the above notation and under condition (H$\\widetilde{\\varphi }$ ), denote by $\\mathbf {B}$ a neighborhood of $\\underline{g}$ and $\\mathbf {B}_{t}:=t \\cdot \\mathbf {B}$ .","Then $\\mathbb {F}[\\mathbf {\\Gamma }_{t}\\cap \\mathbf {B}_{t}^{c} \\, \\vert \\, \\mathbf {\\Gamma }_{t} \\, ]= \\mathbb {\\Pi }\\left[ \\left.","\\mathbf {T}\\in \\mathbf {\\Gamma }_{t}\\cap \\mathbf {B}_{t}^{c}\\, \\right|\\,\\mathbf {T}\\in \\mathbf {\\Gamma }_{t}\\right] \\rightarrow 0$ as $t\\rightarrow \\infty $ , which proves that simulations under (REF ) produce proxies of the dominating points $\\underline{g}_{t}$ in $\\mathbf {\\Gamma }_{t}$ .","Before we start with the proof of Proposition REF , we first quote the following Lemma 66 Let $\\widetilde{\\varphi } $ satisfy (H$\\widetilde{\\varphi }$ ) with $b=+\\infty $ .","Then for all $\\mathbf {A}$ in $\\mathbb {R}^{K}$ such that $\\breve{\\alpha } :=D_{\\widetilde{\\varphi }}(\\mathbf {A},\\widetilde{{P}}):=\\inf _{\\mathbf {v}\\in \\mathbf {A}}\\sum _{k=1}^{K} \\widetilde{p}_{k} \\cdot \\widetilde{\\varphi } \\left(\\frac{v_{k}}{\\widetilde{p}_{k}}\\right)$ is finite there holds $\\lim _{t\\rightarrow \\infty }\\frac{1}{t}\\log \\int _{\\mathbf {A}}\\exp \\left(-t\\sum _{k=1}^{K} \\widetilde{p}_{k} \\cdot \\widetilde{\\varphi }\\left( \\frac{v_{k}}{\\widetilde{p}_{k}}\\right) \\right) \\, dv_{1} \\ldots dv_{k}= - D_{\\widetilde{\\varphi }}\\left(\\mathbf {A},\\widetilde{{P}}\\right) .$ Proof of Lemma REF .","Let us first remark that according to the geometry of the set $\\mathbf {A}$ , various combinations for the asymptotics (REF ) or (REF ) may occur; for sake of brevity, we only handle the simplest ones, since all turn to be amenable through the same arguments.","Denote for positive $r$ $\\mathbf {B}(r):=\\left\\lbrace \\mathbf {v}\\in \\mathbb {R}^{K}:\\sum _{k=1}^{K} \\widetilde{p}_{k}\\widetilde{\\varphi } \\left( \\frac{v_{k}}{\\widetilde{p}_{k}}\\right) >r\\right\\rbrace .$ It holds, by making the change of variable $r=t\\cdot \\breve{\\alpha } +t \\cdot s$ , $& & \\int _{\\mathbf {A}}\\exp \\left( -t\\sum _{k=1}^{K} \\widetilde{p}_{k} \\cdot \\widetilde{\\varphi } \\left( \\frac{v_{k}}{\\widetilde{p}_{k}}\\right) \\right) \\,dv_{1} \\ldots dv_{k}=\\int \\cdots \\int 1_{\\mathbb {R}^{+}}(r)\\cdot 1_{\\mathbf {A}}(\\mathbf {v})\\cdot 1_{\\left]t\\sum _{k=1}^{K} \\widetilde{p}_{k}\\widetilde{\\varphi } \\left( \\frac{v_{k}}{\\widetilde{p}_{k}}\\right) ,\\infty \\right[}(r)\\cdot e^{-r} \\, dr \\, dv_{1}\\ldots dv_{K} \\\\& & = te^{-t\\breve{\\alpha } }\\int \\cdots \\int 1_{]- \\breve{\\alpha },\\infty [}(s) \\cdot 1_{\\mathbf {A}}(\\mathbf {v}) \\cdot 1_{\\mathbf {B}^{c}(\\breve{\\alpha }+s)}(\\mathbf {v})\\cdot e^{-ts} \\, ds \\, dv_{1}\\ldots dv_{K} \\ = \\ te^{-t\\breve{\\alpha } }\\int _{-\\breve{\\alpha }}^{\\infty }Vol\\left( \\mathbf {A}\\cap \\mathbf {B}^{c}(\\breve{\\alpha }+s)\\right) \\cdot e^{-ts} \\, ds.$ Let $I_{t}:=t \\cdot \\int _{0}^{\\infty }Vol\\left(\\mathbf {A}\\cap \\mathbf {B}^{c}(\\breve{\\alpha } +s)\\right) e^{-ts}ds $ .","We prove that $\\lim _{t\\rightarrow \\infty }\\frac{1}{t}\\log I_{t}=0.","$ When $a=-\\infty $ or $b=+\\infty $ , since $\\widetilde{\\varphi }$ satisfies (H$\\widetilde{\\varphi }$ ) there exists a polynomial $P$ such that $Vol\\left( \\mathbf {A}\\cap \\mathbf {B}^{c}(\\breve{\\alpha } +s)\\right) \\le P(s) \\, ;$ whence, assuming without loss of generality that $dom(\\widetilde{\\varphi }) =\\mathbb {R}^{+}$ , we obtain $\\frac{1}{t}\\log I_{t}\\le \\frac{1}{t}\\log \\int _{0}^{\\infty }P\\left(\\frac{u}{t}\\right) te^{-u}du$ which yields that for large $t$ $\\frac{1}{t}\\log I_{t}<0.$ When dealing with a context where $a$ or $b$ have finite value and the corresponding sets $\\mathbf {\\Gamma }_{t}$ are “far away” from $\\mathbf {\\Gamma } $ in terms of the distance measure $D_{\\widetilde{\\varphi }}\\left( \\cdot ,\\widetilde{{P}}\\right)$ , then $Vol\\left( \\mathbf {A}\\cap \\mathbf {B}^{c}(\\breve{\\alpha } +s)\\right) $ is bounded.","Hence, $\\lim \\sup _{t\\rightarrow \\infty }\\frac{1}{t}\\log I_{t}\\le 0$ .","Now fix $\\varepsilon >0.$ Then, since $Vol\\left( \\mathbf {A}\\cap \\mathbf {B}^{c}(a+s)\\right) $ is increasing in $s$ , we get $I_{t} &\\ge &t\\int _{\\varepsilon }^{\\infty }Vol\\left( \\mathbf {A}\\cap \\mathbf {B}^{c}(\\breve{\\alpha }+s)\\right) e^{-ts}ds \\\\&\\ge &Vol\\left( \\mathbf {A}\\cap \\mathbf {B}^{c}(\\breve{\\alpha } +\\varepsilon )\\right) e^{-t\\varepsilon }.$ Hence $\\frac{1}{t}\\log I_{t}\\ge \\frac{1}{t}\\log Vol\\left( \\mathbf {A}\\cap \\mathbf {B}^{c}(\\breve{\\alpha }+\\varepsilon )\\right) -\\varepsilon $ which yields $\\lim \\inf _{t\\rightarrow \\infty }\\frac{1}{t}\\log I_{t}\\ge 0.$ Therefore (REF ) holds, which concludes the proof.","$\\blacksquare $ We now turn to the Proof of Proposition REF .","Without loss of generality, let $b=+\\infty $ , $\\mathbf {\\Gamma }_{t}$ as in (REF ) and Condition (H$\\widetilde{\\varphi }$ ) hold.","Moreover, consider an arbitrary neighborhood $\\mathbf {B}$ of $\\underline{g}$ and the corresponding neighborhoods $\\mathbf {B}_{t} : =t \\cdot \\mathbf {B}$ of $\\underline{g}_{t} = t \\cdot \\underline{g}$ .","There holds $\\frac{1}{\\widetilde{\\varphi }(t)}\\log \\mathbb {\\Pi }\\left[ \\mathbf {T}\\in \\mathbf {\\Gamma }_{t}\\right] &=&\\frac{C}{\\widetilde{\\varphi }(t)}\\log \\int _{\\mathbf {\\Gamma } _{t}}\\exp \\left(-\\sum _{k=1}^{K}\\widetilde{p}_{k} \\cdot \\widetilde{\\varphi }\\left(\\frac{w_{k}}{\\widetilde{p}_{k}}\\right)\\right) \\, dw_{1}\\ldots dw_{K} \\\\&\\stackrel{(1)}{=}&\\frac{CK}{\\widetilde{\\varphi }(t)}\\log t+\\frac{C}{\\widetilde{\\varphi }(t)}\\log \\int _{\\mathbf {\\Gamma }}\\exp \\left(-t^{\\beta } \\cdot \\sum _{k=1}^{K}\\widetilde{p}_{k} \\cdot \\left( \\widetilde{\\varphi }\\left( \\frac{v_{k}}{\\widetilde{p}_{k}}\\right) \\cdot (1+o(1))\\right)\\right) \\, dv_{1} \\ldots dv_{K} \\\\&\\stackrel{(2)}{=}& \\frac{CK}{\\widetilde{\\varphi }(t)}\\log t+\\frac{C}{\\left( \\widetilde{\\varphi }(t)/t^{\\beta }\\right) } \\cdot \\frac{1}{t^{\\beta }}\\log \\left( (1+o(1)) \\cdot \\int _{\\mathbf {\\Gamma } }\\exp \\left( -t^{\\beta }\\sum _{k=1}^{K}\\widetilde{p}_{k} \\cdot \\widetilde{\\varphi }\\left( \\frac{v_{k}}{\\widetilde{p}_{k}}\\right) \\right)dv_{1} \\ldots dv_{K} \\right)\\\\&\\stackrel{(3)}{=}& -\\frac{Ct^{\\beta }}{\\widetilde{\\varphi }(t)}\\cdot D_{\\widetilde{\\varphi }}(\\mathbf {\\Gamma } ,\\widetilde{{P}}) \\cdot (1+o(1)) \\\\&\\stackrel{(4)}{=}& - \\breve{l}(t) \\cdot D_{\\widetilde{\\varphi }}(\\mathbf {\\Gamma } ,\\widetilde{{P}}) \\cdot (1+o(1))$ as $t$ tends to infinity.","In the above display, $(1)$ follows from $\\widetilde{\\varphi }(tx)=\\left( tx\\right)^{\\beta } \\cdot \\ell (tx)=t^{\\beta } \\cdot x^{\\beta } \\cdot \\ell (x) \\cdot \\frac{\\ell (tx)}{\\ell (x)} =t^{\\beta } \\cdot \\widetilde{\\varphi } (x) \\cdot \\left( 1+o(1)\\right) $ as $t$ tends to infinity and $x$ lies in a compact subset of $]0,\\infty [$ , where $\\ell $ is a slowly varying function.","The equality $(2)$ follows from compactness of $\\mathbf {\\Gamma }$ together with the fact that $\\widetilde{\\varphi }$ is a regularly varying function with index $\\beta $ , so that $\\lim _{t\\rightarrow \\infty }\\frac{\\widetilde{\\varphi }(tv)}{\\widetilde{\\varphi }(t)}=v^{\\beta }$ uniformly upon $v$ on compact sets in $]0,\\infty [$ .","The remaining equalities $(3)$ and $(4)$ follow from classical properties of regularly varying functions, where $\\breve{\\ell } := 1/\\ell $ is a slowly varying function at infinity, together with standard Laplace-Integral approximation.","In the same way we can show $\\frac{1}{\\widetilde{\\varphi }(t)}\\log \\mathbb {\\Pi } \\left[ \\, \\mathbf {T}\\in \\mathbf {\\Gamma } _{t}\\cap \\mathbf {B}_{t}^{c} \\, \\right]=- \\breve{l}(t) \\cdot D_{\\widetilde{\\varphi }}(\\mathbf {\\Gamma }\\cap \\mathbf {B}^{c},\\widetilde{{P}}) \\cdot (1+o(1))$ as $t$ tends to infinity.","Since $\\mathbf {B}$ is a neighborhood of the unique dominating point $\\underline{g}$ of $\\mathbf {\\Gamma }$ , one gets that $D_{\\widetilde{\\varphi }}(\\mathbf {\\Gamma }\\cap \\mathbf {B}^{c},\\widetilde{{P}}) > D_{\\widetilde{\\varphi }}(\\mathbf {\\Gamma } ,\\widetilde{{P}})$ .","This implies that $\\mathbb {\\Pi } \\left[ \\, \\mathbf {T}\\in \\mathbf {\\Gamma }_{t}\\cap \\mathbf {B}_{t}^{c} \\, \\big \\vert \\, \\mathbf {T}\\in \\mathbf {\\Gamma }_{t} \\, \\right] \\rightarrow 0\\qquad \\textrm {as $ $.","\\qquad $$}$$$ Remark 67 Firstly, let us quote that the case when $\\widetilde{\\mathbf {\\Omega }}$ is an unbounded subset in $\\mathbb {R}^{K}/\\left\\lbrace \\mathbf {0}\\right\\rbrace $ is somewhat immaterial for applications.","Anyhow, if compactness of $\\mathbf {\\Gamma } $ is lost, then in order to use the same line of arguments as above, it is necessary to strengthen the assumptions (H$\\widetilde{\\varphi }$ ) e.g.","as follows: when $b=+\\infty $ then $\\widetilde{\\varphi }$ has to be asymptotically homogeneous with degree $\\beta >0$ , in the sense that $\\widetilde{\\varphi }(tx)=t^{\\beta }\\widetilde{\\varphi }(x)\\cdot (1+o(1))$ as $t\\rightarrow \\infty $ ; for the subcase $a=-\\infty $ one employs an analogous assumption as $t\\rightarrow -\\infty $ .","The case when $\\widetilde{\\mathbf {\\Omega }}$ is a compact set in $\\mathbb {R}^{K}\\backslash \\lbrace \\mathbf {0}\\rbrace $ can be treated as above, by combining the asymptotics in $t$ in the neighborhood of $a$ and $b$ accordingly."],["Proof for Subsection ","Proof of Proposition REF.","Recall the weighted empirical measure $\\xi _{n,\\mathbf {X}}^{\\mathbf {V}}:=\\left( \\frac{1}{n}\\sum _{i\\in I_{1}^{(n)}} V_{i},\\ldots ,\\frac{1}{n}\\sum _{i\\in I_{K}^{(n)}} V_{i}\\right)$ which satisfies the $K$ linear constraints defined in (REF ) through $E_{S}[\\xi _{n,\\mathbf {X}}^{\\mathbf {V}}] =\\xi _{M,\\mathbf {X}}^{\\mathbf {W}^{\\ast }}= \\overline{W^{\\ast }} \\cdot \\xi _{M,\\mathbf {X}}^{w\\mathbf {W}^{\\ast }}$ where $\\mathbf {Q}^{\\ast }:= \\left(q_{1}^{\\ast }, \\ldots , q_{K}^{\\ast }\\right)= \\xi _{M,\\mathbf {X}}^{w\\mathbf {W}^{\\ast }}\\in int(\\textrm {$$\\hspace{-6.544pt}$$})$ and $\\overline{W^{\\ast }} = \\frac{1}{M}\\sum _{j=1}^{M}W_{j}^{\\ast }$ .","The probability distribution $S$ defined on $\\mathbb {R}^{n}$ is the Kullback-Leibler projection of $\\mathbb {}^{\\otimes n}$ on the class of all probability distributions on $\\mathbb {R}^{n}$ which satisfy (REF ).","We prove that $\\lim \\inf _{n\\rightarrow \\infty }S\\left[\\xi _{n,\\mathbf {X}}^{w\\mathbf {V}} \\in \\textrm {\\right.$$\\hspace{-6.544pt}$$}> 0$ .","To start with, we define for strictly positive $\\delta $ the set $A_{n,\\delta } := \\left\\lbrace \\left|\\frac{1}{n}\\sum _{i=1}^{n}V_{i}-\\overline{W^{\\ast }}\\right|\\le \\delta \\right\\rbrace $ and write $S\\left[\\xi _{n,\\mathbf {X}}^{w\\mathbf {V}} \\in \\textrm {\\right.$ $\\hspace{-6.544pt}$$}=S\\left[\\lbrace \\xi _{n,\\mathbf {X}}^{w\\mathbf {V}} \\in \\textrm {\\right.$$\\hspace{-6.544pt}$$} \\rbrace \\cap A_{n,\\delta }+ S\\left[\\lbrace \\xi _{n,\\mathbf {X}}^{w\\mathbf {V}} \\in \\textrm {\\right.$$\\hspace{-6.544pt}$$} \\rbrace \\cap A_{n,\\delta }^{c}=:I+II.$$By the law of large numbers, the second term $ II$ tends to $ 0$ as $ n$ tends to infinity.Moreover, one can rewrite$$I=S\\bigg [ \\bigcup \\limits _{m\\in \\left[ \\overline{W^{\\ast }}-\\delta ,\\overline{W^{\\ast }}+\\delta \\right] }\\left\\lbrace \\xi _{n,\\mathbf {X}}^{\\mathbf {V}}\\in m \\cdot \\textrm {\\right.$$\\hspace{-6.544pt}$$} \\bigg ]$$which entails$$I \\ \\ge \\ S\\bigg [ \\frac{1}{n_{k}}\\sum _{i\\in I_{k}^{(n)}}V_{i}\\in \\mathcal {V}_{\\eta }\\left( \\overline{W^{\\ast }}\\frac{q_{k}^{\\ast }}{p_{k}}\\right) \\text{ for all }k \\in \\lbrace 1,\\ldots ,K\\rbrace \\bigg ] ,$$where $ V( W qkpk) $ denotes a neighborhood of$Wqkpk$ with radius $$ being smallwhen $$ is small, for large enough $ n$, making use of the a.s. convergenceof $ nk/n$ to $ pk$.","Now, for any $ k {1,...,K}$ one has\\begin{equation}S\\bigg [ \\frac{1}{n_{k}}\\sum _{i\\in I_{k}^{(n)}}V_{i}\\notin \\mathcal {V}_{\\eta }\\left( \\overline{W^{\\ast }}\\frac{q_{k}^{\\ast }}{p_{k}}\\right)\\bigg ]\\ \\le \\ \\exp \\bigg (-n_{k} \\cdot \\inf _{x\\in \\mathcal {V}_{\\eta }\\left(\\overline{W^{\\ast }}\\frac{q_{k}^{\\ast }}{p_{k}}\\right)^{c}} \\,\\varphi \\left( x\\right) \\bigg )\\end{equation}since any margin of $ S$ with index in $ Ik(n)$ is a correspondingKullback-Leibler projection of $$ on the set of all distributionson $ R$ with expectation $W qkpM,kemp$ ---where $ pM,kemp$ denotes the fraction ofthe $ Xi$’s (within $ X1,...,XM$) which are equal to $ dk$(cf.","(\\ref {I^(n)_k for stat case})) ---and therefore has amoment generating function which is finite in a non-void neighborhood of $ 0$,which yields (\\ref {ineg}) by the Markov Inequality.","Note that the event$ { M,XwWint( $\\Omega $$\\Omega $ ) }$ is regenerative, so that $ M$ can be chosen large enough to make$ pM,kemp$ close to $ pk$ for all $ k {1,...,K}$.","This proves the claim.\\hspace{28.45274pt} $$$"],["Acknowledgment","W. Stummer is grateful to the Sorbonne Université Paris for its multiple partial financial support and especially the LPSM for its multiple great hospitality.","M. Broniatowski thanks very much the FAU Erlangen-Nürnberg for its partial financial support and hospitality.","Moreover, W. Stummer would like to thank Rene Schilling for an interesting discussion on complex-valued foundations of the Bernstein-Widder theorem."]],"string":"[\n [\n \"A precise bare simulation approach to the minimization of some\\n distances. Foundations\"\n ],\n [\n \"Abstract In information theory -- as well as in the adjacent fields of statistics, machine learning, artificial intelligence, signal processing and pattern recognition -- many flexibilizations of the omnipresent Kullback-Leibler information distance (relative entropy) and of the closely related Shannon entropy have become frequently used tools.\",\n \"To tackle corresponding constrained minimization (respectively maximization) problems by a newly developed dimension-free bare (pure) simulation method, is the main goal of this paper.\",\n \"Almost no assumptions (like convexity) on the set of constraints are needed, within our discrete setup of arbitrary dimension, and our method is precise (i.e., converges in the limit).\",\n \"As a side effect, we also derive an innovative way of constructing new useful distances/divergences.\",\n \"To illustrate the core of our approach, we present numerous examples.\",\n \"The potential for widespread applicability is indicated, too; in particular, we deliver many recent references for uses of the involved distances/divergences and entropies in various different research fields (which may also serve as an interdisciplinary interface).\"\n ],\n [\n \"Introduction\",\n \"Directed (i.e., not necessarily symmetric) distances $D(\\\\mathbf {P},\\\\mathbf {Q})$ between two finite discretefor reasons of technicality, in this paper we only deal with such kind of distributions; for instance, these can be also achieved from more involved systems by quantizations of observations represented by finite partitions of the observation/data space, or by making use of the dual representation for CASM $\\\\varphi -$ divergences (cf.\",\n \"Liese & Vajda [218], Broniatowski & Keziou [61]).\",\n \"(probability) distributions $\\\\mathbf {P},\\\\mathbf {Q}$ or between two general Euclidean vectors $\\\\mathbf {P},\\\\mathbf {Q}$ are known as divergences; they serve as important (dis)similarity measures, proximity measures and discrepancy measures in various different research areas such as information theory, statistics, artificial intelligence, machine learning, signal processing, pattern recognition, physics, finance, etc.since there exists a vast literature on divergences and connected entropies in these fields, for the sake of brevity we will give in this introduction only some basic references; many corresponding concrete applications will be mentioned in the following sections, in the course of the method-illuminating examples..\",\n \"Besides Bregman distances/divergences (with which we do not deal here), another major class are the $\\\\varphi -$ divergences $D_{\\\\varphi }( \\\\mathbf {P},\\\\mathbf {Q})$ of Csiszar-Ali-Silvey-Morimoto (in short CASM $\\\\varphi -$ divergences, cf.\",\n \"[94], [11], [266]).\",\n \"The latter covers — with corresponding choices of $\\\\varphi $ — e.g.\",\n \"the omnipresent Kullback-Leibler information distance/divergence [200] (also known as relative entropy), the Jensen-Shannon distance/divergence, as well as the power divergences (also known as alpha-divergences, Cressie-Read measures, and Tsallis cross-entropies).\",\n \"For some comprehensive overviews on CASM $\\\\varphi -$ divergences, the reader is referred to the insightful books of e.g.\",\n \"Liese & Vajda [217], Read & Cressie [303], Vajda [371], Csiszar & Shields [99], Stummer [344], Pardo [282], Liese & Miescke [216], the survey articles of e.g.\",\n \"Liese & Vajda [218], Vajda & van der Meulen [374], Reid & Williamson [304], Basseville [34], and the references therein; an imbedding of CASM $\\\\varphi -$ divergences to more general frameworks can be found e.g.\",\n \"in Stummer & Vajda [350], Broniatowski & Vajda [65], Stummer & Kißlinger [346] and Broniatowski & Stummer [64].\",\n \"0.5cm Frequently used special cases of the above-mentioned power divergences are e.g.\",\n \"the (squared) Hellinger distance, the Pearson chi-square divergence, and the Neyman chi-square divergence.\",\n \"Moreover, several deterministic transformations of power divergences are also prominently used in research, most notably the Bhattacharyya distance [48] and the more general Renyi divergences [309] (also known as Renyi cross-entropies); a comprehensive exposition of the latter is given e.g.\",\n \"in van Erven & Harremoes [380].\",\n \"Some other important deterministic transformations of power divergences include the Bhattacharyya coefficient (cf.\",\n \"[48],[49],[50]) — which is also called affinity (cf.\",\n \"Matusita [256]) and fidelity similarity (cf.\",\n \"e.g.\",\n \"Deza & Deza [113]) — as well as the Bhattacharyya arccos distance (cf.\",\n \"[50]) and the Fisher distance (also known as Rao distance, geodesic distance, cf.\",\n \"e.g.\",\n \"Deza & Deza [113]).\",\n \"As shown below, by further explicit transformations we can also recover Sundaresan’s divergence [352] [353].\",\n \"The minimization $\\\\inf _{\\\\mathbf {Q}\\\\in \\\\mathbf {\\\\Omega }} D( \\\\mathbf {Q}, \\\\mathbf {P} )$ of divergences from one distribution (respectively, its equivalent vector of frequencies) $\\\\mathbf {P}$ to an appropriate set $\\\\mathbf {\\\\Omega }$ of distributions (frequency vectors) appears in a natural way in various different contexts, as indicated in the following.\",\n \"For instance, let $\\\\mathbf {P} = \\\\mathbf {P}_{true}$ be the true distribution of a mechanism which generates non-deterministic data and $\\\\mathbf {\\\\Omega }$ is a pregiven model in the sense of a (parametric or non-parametric) family of distributions which serves as an “approximation” (in fact, a collection of approximations) of the “truth” $\\\\mathbf {P}_{true}$ .\",\n \"If $\\\\mathbf {P}_{true} \\\\notin \\\\mathbf {\\\\Omega }$ — e.g.\",\n \"since $\\\\mathbf {\\\\Omega }$ reflects some simplifications of $\\\\mathbf {P}_{true}$ which is in line with the general scientific procedure — then the positive quantity $\\\\Phi _{\\\\mathbf {P}_{true}}(\\\\mathbf {\\\\Omega }) := \\\\inf _{\\\\mathbf {Q}\\\\in \\\\mathbf {\\\\Omega }}D( \\\\mathbf {Q}, \\\\mathbf {P}_{true} )$ can be used as an index of model adequacy in the sense of a degree of departure between the model and the truth (cf.\",\n \"Lindsay [222], see also e.g.\",\n \"Lindsay et al.\",\n \"[223], Markatou & Sofikitou [248], Markatou & Chen [249]); small index values should indicate high adequacy.\",\n \"If $\\\\mathbf {P}_{true} \\\\in \\\\mathbf {\\\\Omega }$ , then $\\\\Phi _{\\\\mathbf {P}_{true}}(\\\\mathbf {\\\\Omega }) = 0$ which corresponds to full adequacy.\",\n \"This index of model adequacy $\\\\Phi _{\\\\mathbf {P}_{true}}(\\\\mathbf {\\\\Omega })$ can also be seen as index of goodness/quality of approximation to the truth or as model misspecification error, and it can be used for model assessment as well as for model search (model selection, model hunting) by comparing the indices $\\\\Phi _{\\\\mathbf {P}_{true}}(\\\\mathbf {\\\\Omega _{1}}),\\\\Phi _{\\\\mathbf {P}_{true}}(\\\\mathbf {\\\\Omega _{2}}), \\\\ldots $ of competing models $\\\\mathbf {\\\\Omega _{1}}, \\\\mathbf {\\\\Omega _{2}}, \\\\ldots $ and choosing the one with the smallest index; this idea can be also used for classification (e.g.\",\n \"analogously to Bilik & Khomchuk [54] who deal with continuous (rather than discrete) distributions) where the $\\\\mathbf {\\\\Omega _{i}}$ are interpreted as (possibly data-derived but fixed) classes which are disjoint and non-exhaustive.\",\n \"Typically, in statistical analyses the true distribution $\\\\mathbf {P}_{true}$ is unknown and is either replaced by a hypothesis-distribution $\\\\mathbf {P}_{hyp}$ or by a distribution $\\\\mathbf {P}_{data}$ derived from data (generated by $\\\\mathbf {P}_{true}$ ) which converges to $\\\\mathbf {P}_{true}$ as the data/sample size tends to infinity (e.g.\",\n \"$\\\\mathbf {P}_{data}$ may be the well-known empirical distribution or a conditional distribution).\",\n \"Correspondingly, $\\\\Phi _{\\\\mathbf {P}_{data}}(\\\\mathbf {\\\\Omega })$ reflects a data-derived approximation (estimate) of the index of model adequacy (resp.\",\n \"of the model misspecification error) from which one can cast corresponding model-adequacy tests and related goodness-of-fit tests.\",\n \"Moreover, for the case of i.i.d.\",\n \"data-generation and $\\\\mathbf {P}_{data}$ to be the corresponding empirical distribution, the (not necessarily existent or unique) best-model-member/element choice $\\\\arg \\\\min _{\\\\mathbf {Q}\\\\in \\\\mathbf {\\\\Omega }} D( \\\\mathbf {Q}, \\\\mathbf {P}_{data})$ amounts to the well-known minimum distance estimator which for the Kullback-Leibler information divergence $D$ is equal to the omnipresent maximum likelihood estimator; for comprehensive surveys on divergence-based statistical testing and estimation, the reader is referred to e.g.\",\n \"the references in the second half of the first paragraph in the current introduction.\",\n \"Most of the above-mentioned considerations also hold for deterministic (rather than non-deterministic) frameworks where $\\\\mathbf {P}$ is a general Euclidean vector (rather than a probability-distribution describing frequency vector in the probability simplex) and $\\\\mathbf {\\\\Omega }$ is a model in the sense of a family of general Euclidean vectors (which may be encodings of more complicated context descriptions).\",\n \"Returning to the general context, let us mention that from CASM $\\\\varphi -$ divergences one can also derive the widely used $\\\\varphi -$ entropies $\\\\mathcal {E}_{\\\\varphi }(\\\\mathbf {Q})$ of a distribution $\\\\mathbf {Q}$ (and non-probability versions thereof) in the sense of Burbea & Rao [68] (see also Csiszar [95], Ben-Bassat [38], Ben-Tal & Teboulle [40], Kesavan & Kapur [187], Dacunha-Castelle & Gamboa [102], Teboulle & Vajda [357], Gamboa & Gassiat [132], Vajda & Zvarova [376]); these entropies can e.g.\",\n \"be basically constructed from $D_{\\\\varphi }(\\\\mathbf {Q},\\\\mathbf {P}^{unif})$ where $\\\\mathbf {P}^{unif}$ denotes the uniform distribution.\",\n \"Moreover, by use of certain deterministic transformations $h$ one can also deduce the more general $(h,\\\\varphi )-$ entropies (and non-probability versions thereof) in the sense of Salicru et al.\",\n \"[314] (see also e.g.\",\n \"Pardo [282]).\",\n \"As will be worked out in detail in Subsection REF below, from this one can deduce as special cases a variety of prominently used quantities in research, such as for instance: the omnipresent Shannon entropy [328], the $\\\\gamma -$ order Renyi entropy [309], the $\\\\gamma -$ order entropy of Havrda-Charvat [157] (also called non-additive $\\\\gamma -$ order Tsallis entropy [363] in statistical physics), the $\\\\widetilde{\\\\gamma }-$ order entropy of Arimoto [16], Vajda's quadratic entropy [371], Sharma-Mittal entropies [329], the Euclidean $\\\\gamma -$ norms, as well as measures of diversity, heterogeneity and unevenness, like the Gini-Simpson diversity index, the diversity index of Hill [160], the Simpson-Herfindahl index (which is also known as index of coincidence, cf.\",\n \"Harremoes & Topsoe [155] and its generalization in Harremoes & Vajda [156]), the diversity index of Patil & Taillie [286], the $\\\\gamma -$ mean heterogeneity index (see e.g.\",\n \"van der Lubbe [379]); see also Nayak [271] and Jost [176] for some interrelations with the above-mentioned entropies.\",\n \"Given that the constraint set $\\\\mathbf {\\\\Omega }$ reflects some incomplete/partial information about a system (e.g.\",\n \"moment constraints), the maximization over $\\\\mathbf {Q}\\\\in \\\\mathbf {\\\\Omega }$ of the above-mentioned entropies, norms and diversity indices (and the more general $(h,\\\\varphi )-$ entropies) is important for many research topics, most notably manifested in Jaynes’s [165],[166] omnipresent, “universally applicable” maximum entropy principle (which employs the Shannon entropy), and its generalizations (see e.g.\",\n \"the books of Kapur [184], Kapur & Kesavan [185], Arndt [17], and Gzyl et al.\",\n \"[150] for comprehensive surveys).\",\n \"Besides the above-mentioned principal overview, let us now briefly discuss some existing technical issues for the minimization of CASM $\\\\varphi -$ divergences $\\\\Phi _{\\\\mathbf {P}}(\\\\mathbf {\\\\Omega }) :=\\\\inf _{\\\\mathbf {Q}\\\\in \\\\mathbf {\\\\Omega }} D_{\\\\varphi }( \\\\mathbf {Q}, \\\\mathbf {P} )$ .\",\n \"For (not necessarily discrete) probability distributions/measures $\\\\mathbf {P}$ and sets $\\\\mathbf {\\\\Omega }$ of probability distributions/measures satisfying a finite set of linear equality constraints, $\\\\Phi _{\\\\mathbf {P}}(\\\\mathbf {\\\\Omega })$ has been characterized in Csiszar [96] and more recently by Csiszar & Matus [98], Broniatowski & Keziou [60], Leonard [213], and Pelletier [288] among others, in various contexts; those results extend to inequality constraints.\",\n \"Minimizations of $\\\\gamma -$ order Renyi divergences on $\\\\gamma -$ convex sets $\\\\mathbf {\\\\Omega }$ are studied e.g.\",\n \"in Kumar & Sason [202], whereas Kumar & Sundaresan [203] [204] investigate minimizations of Sundaresan’s divergence on certain convex sets $\\\\mathbf {\\\\Omega }$ .\",\n \"To our knowledge, no general representation for $\\\\Phi _{\\\\mathbf {P}}(\\\\mathbf {\\\\Omega })$ for a positive distribution/measure $\\\\mathbf {P}$ (respectively, for an Euclidean vector with positive components) and a general set $\\\\mathbf {\\\\Omega }$ of signed measures (respectively, of Euclidean vector with components of arbitrary sign) exists.\",\n \"At the contrary, many algorithmic approaches for such minimization problems have been proposed; they mostly aim at finding minimizers more than at the evaluation of the minimum divergence itself, which is obtained as a by-product.\",\n \"Moreover, it is well-known that such kind of CASM $\\\\varphi -$ divergence minimization problems may be hard to tackle or even intractable via usual methods such as the omnipresent gradient descent method and versions thereof, especially for non-parametric or semi-parametric $\\\\mathbf {\\\\Omega }$ in sufficiently high-dimensional situations.\",\n \"For instance, $\\\\mathbf {\\\\Omega }$ may consist (only) of constraints on moments or on L-moments (see e.g.\",\n \"Broniatowski & Decurninge [59]); alternatively, $\\\\mathbf {\\\\Omega }$ may be e.g.\",\n \"a tubular neighborhood of a parametric model (see e.g.\",\n \"Liu & Lindsay [225], Ghosh & Basu [135]).\",\n \"The same intractability problem holds for the above-mentioned $(h,\\\\varphi )-$ entropy maximization problems.\",\n \"In the light of this, the goals of this paper are: to solve constrained minimization problems of a large range of CASM $\\\\varphi -$ divergences and deterministic transformations thereof (respectively constrained maximization problems of $(h,\\\\varphi )-$ entropies including Euclidean norms and diversity indices), by means of a newly developed dimension-free bare (pure) simulation method which is precise (i.e., converges in the limit) and which needs almost no assumptions (like convexity) on the set $\\\\mathbf {\\\\Omega }$ of constraints; in doing so, for the sake of brevity we concentrate on finding/computing the minimum divergences themselves rather than the corresponding minimizers (to achieve the latter, e.g.\",\n \"dichotomous search could be used in a subsequent step, however); to derive a method of constructing new useful distances/divergences; to present numerous examples in order to illuminate our method and its potential for wide-spread applicability; as we go along, we also deliver many recent references for uses of the outcoming distances/divergences and entropies (covering in particular all the above-mentioned ones).\",\n \"This agenda is achieved in the following way.\",\n \"In the next Section , we briefly introduce the principal idea of our new bare-simulation optimization paradigm.\",\n \"After manifesting the fundamentally employed class of CASM $\\\\varphi -$ divergences in Section , we give in Section the main cornerstones, construction principles and theorems, for deterministic as well as for statistical divergence-minimization problems; the maximization of generalized entropies is addressed, too.\",\n \"Section deals with the concrete determination of the involved simulation-weights, as well as with the interrelated issue of creating associated CASM $\\\\varphi -$ divergences.\",\n \"Some sampling-concerning details for the principal implementation of our bare-simulation optimization approach are worked out in Section .\",\n \"The main proofs are presented in the appendices.\",\n \"A first simulation-based algorithm in vein with the present proposal has been developed by Broniatowski [58], in the restricted setup of risk estimation for power divergences.\",\n \"The present paper extends this considerably by considering general CASM $\\\\varphi -$ divergences and related entropies, and by dealing with corresponding general optimization problems, of both deterministic respectively stochastic type.\"\n ],\n [\n \"A new minimization paradigm \",\n \"We concern with minimization problems of the following type, where $\\\\mathcal {M}$ is a topological space and $\\\\mathcal {T}$ is the Borel $\\\\sigma -$ field over a given base on $\\\\mathcal {M}$ ; e.g.\",\n \"take $\\\\mathcal {M} = \\\\mathbb {R}^{K}$ to be the $K-$ dimensional Euclidean space equipped with the Borel $\\\\sigma -$ field $\\\\mathcal {T}$ .\",\n \"Definition 1 A measurable function $\\\\Phi : \\\\mathcal {M} \\\\mapsto [0,\\\\infty ]$ and measurable set $\\\\Omega \\\\subset \\\\mathcal {M}$ i.e.\",\n \"$\\\\Omega \\\\in \\\\mathcal {T}$ are called “bare-simulation minimizable” (BS-minimizable) respectively “bare-simulation maximizable” (BS-maximizable) if for $\\\\Phi (\\\\Omega ):=\\\\inf _{Q\\\\in \\\\Omega }\\\\left\\\\lbrace \\\\Phi (Q) \\\\right\\\\rbrace < \\\\infty \\\\qquad \\\\textrm {respectively} \\\\qquad \\\\Phi (\\\\Omega ):=\\\\sup _{Q\\\\in \\\\Omega }\\\\left\\\\lbrace \\\\Phi (Q) \\\\right\\\\rbrace < \\\\infty $ there exists a measurable function $G: [0,\\\\infty [ \\\\mapsto [0,\\\\infty [$ as well as a sequence $\\\\left((\\\\mathfrak {X}_{n},\\\\mathcal {A}_{n},\\\\mathbb {\\\\Pi }_{n})\\\\right)_{n \\\\in \\\\mathbb {N}}$ of probability spaces and on them a sequence $(\\\\xi _{n})_{n\\\\in \\\\mathbb {N}}$ in order to emphasize the dependence on $\\\\Phi $ , one should use the notations $(\\\\xi _{\\\\Phi ,n})_{n\\\\in \\\\mathbb {N}}$ , $\\\\mathbb {\\\\Pi }_{\\\\Phi ,n}$ , etc.\",\n \"; this is avoided for the sake of a better readability.\",\n \"of $\\\\mathcal {M}-$ valued random variables such that $G\\\\left(-\\\\lim _{n\\\\rightarrow \\\\infty }\\\\frac{1}{n}\\\\log \\\\mathbb {\\\\Pi }_{n}\\\\left[\\\\xi _{n}\\\\in \\\\Omega \\\\right] \\\\right) =\\\\inf _{Q\\\\in \\\\Omega }\\\\Phi (Q)=\\\\Phi (\\\\Omega )$ respectively $G\\\\left(-\\\\lim _{n\\\\rightarrow \\\\infty }\\\\frac{1}{n}\\\\log \\\\mathbb {\\\\Pi }_{n}\\\\left[\\\\xi _{n}\\\\in \\\\Omega \\\\right] \\\\right) =\\\\sup _{Q\\\\in \\\\Omega }\\\\Phi (Q) =\\\\Phi (\\\\Omega );$ in situations where $\\\\Phi $ is fixed and different $\\\\Omega $ ’s are considered, we say that “$\\\\Phi $ is bare-simulation minimizable (BS-minimizable) on $\\\\Omega $ ” respectively “$\\\\Phi $ is bare-simulation maximizable (BS-maximizable) on $\\\\Omega $ ”.\",\n \"Remark 2 (a) Even in situations where one can uniformly choose $(\\\\mathfrak {X}_{n},\\\\mathcal {A}_{n},\\\\mathbb {\\\\Pi }_{n}) \\\\equiv (\\\\widetilde{\\\\mathfrak {X}},\\\\widetilde{\\\\mathcal {A}},\\\\widetilde{\\\\mathbb {\\\\Pi }})$ , the sequence $(\\\\xi _{n})_{n\\\\in \\\\mathbb {N}}$ may be not “independent and identically distributed” .\",\n \"(b) Throughout the paper, we shall mainly deal with $BS-$ minimizability.\",\n \"The basic idea/incentive of this new approach is: if a minimization problem (REF ) has no explicit solution and is computationally intractable (or unfeasible) but can be shown to be BS-minimizable with concretely constructable $(\\\\xi _{n})_{n\\\\in \\\\mathbb {N}}$ and $(\\\\mathbb {\\\\Pi }_{n})_{n\\\\in \\\\mathbb {N}}$ , then one can basically simulate the log-probabilities $-\\\\frac{1}{n} \\\\mathbb {\\\\Pi }_{n}\\\\left[\\\\xi _{n}\\\\in \\\\Omega \\\\right]$ for large enough integer $n \\\\in \\\\mathbb {N}$ to obtain an approximation of (REF ) without having to evaluate the corresponding (not necessarily unique) minimizer, where the latter is typically time-costly.\",\n \"Finding minimizers can be performed through dichotomic search, once an algorithm leading to the minimal value of the divergence on adequate families of sets $\\\\Omega $ is at hand; for the sake of brevity, this is omitted in the current paper.\",\n \"For reasons of transparency, we start to demonstrate this approach for the following important/prominent class of minimization problems with the following components: $\\\\mathcal {M}$ is the $K-$ dimensional Euclidean space $\\\\mathbb {R}^{K}$ , i.e.\",\n \"$\\\\Omega $ is a set of vectors $Q$ with a number of $K$ components (where $K$ may be huge, as it is e.g.\",\n \"the case in big data contexts); $\\\\Phi (\\\\cdot ) := \\\\Phi _{P}(\\\\cdot )$ depends on some known vector $P$ in $\\\\mathbb {R}^{K}$ with $K$ nonnegative components; $\\\\Phi _{P}(\\\\cdot )$ is a “directed distance” (divergence) from $P$ into $\\\\Omega $ in the sense of $\\\\Omega \\\\ni Q \\\\mapsto \\\\Phi _{P}(Q):= D(Q, P)$ , where $D(\\\\cdot ,\\\\cdot )$ has the the two properties “$D(Q, P)\\\\ge 0$ ” and “$D(Q, P) = 0$ if and only if $Q=P$ ”.\",\n \"In particular, $D(\\\\cdot ,\\\\cdot )$ needs neither satisfy the symmetry $D(Q,P)= D(P,Q)$ nor the triangular inequality.\",\n \"In other words, (1) together with (i)-(iii) constitutes a distance/divergence-minimization problem; we design a “universal” method to solve such problems by constructing appropriate (cf.\",\n \"(REF )) sequences $(\\\\xi _{n})_{n\\\\in \\\\mathbb {N}}$ of $\\\\mathbb {R}^{K}-$ valued random variables, for all directed distances $D(\\\\cdot ,\\\\cdot )$ from a large subclass of the important omnipresent Csiszar-Ali-Silvey-Morimoto CASM $\\\\varphi -$ divergences (also called $f-$ divergences).\",\n \"As a second demonstration for the workability of our paradigm, we “extend” (i) to (iii) to the setup where $P$ is a random element of the simplex $\\\\mathbb {S}^{K}$ of $K-$ component probability (frequency) vectors (cf.\",\n \"the exact definition below) and $\\\\Omega \\\\subset \\\\mathbb {S}^{K}$ ; for the sub-setup where $P$ corresponds to a data-observation-dependent probability distribution and $\\\\Omega $ corresponds to a pregiven model in the sense of a family of probability distributions, the formula (REF ) amounts to the corresponding (discrete) “minimization-distance estimation problem (MDEP)” of choosing the best model element/member under given data an alternative naming also used in literature is to call $\\\\Omega $ a model class (rather than model), and each $P \\\\in \\\\Omega $ a model (rather than model element).\",\n \"This is important/prominent in statistics and in the adjacent research fields of artificial intelligence and machine learning; the concrete solving of the MDEP is especially “hard” for nonparametric respectively semiparametric problems, and our BS method is predestined for such kind of contexts.\"\n ],\n [\n \"Directed distances \",\n \"In detail, concerning the above-mentioned point (i) we take the $K-$ dimensional Euclidean space $\\\\mathcal {M}=\\\\mathbb {R}^{K}$ , denote from now on — as usual — its elements (i.e.\",\n \"vectors) in boldface letters, and also employ the subsets $&& \\\\mathbb {R}_{\\\\ne 0}^{K} := \\\\lbrace \\\\mathbf {Q}:= (q_{1},\\\\ldots ,q_{K}) \\\\in \\\\mathbb {R}^K: \\\\,q_{i} \\\\ne 0 \\\\ \\\\text{for all} \\\\ i=1,\\\\ldots ,K \\\\rbrace , \\\\\\\\&& \\\\mathbb {R}_{> 0}^{K} := \\\\lbrace \\\\mathbf {Q}:= (q_{1},\\\\ldots ,q_{K}) \\\\in \\\\mathbb {R}^K: \\\\,q_{i} > 0 \\\\ \\\\text{for all} \\\\ i=1,\\\\ldots ,K \\\\rbrace , \\\\\\\\&& \\\\mathbb {R}_{\\\\ge 0}^{K} := \\\\lbrace \\\\mathbf {Q}:= (q_{1},\\\\ldots ,q_{K}) \\\\in \\\\mathbb {R}^K: \\\\, q_{i} \\\\ge 0 \\\\ \\\\text{for all} \\\\ i=1,\\\\ldots ,K \\\\rbrace , \\\\\\\\&& \\\\mathbb {R}_{\\\\le 0}^{K} := \\\\lbrace \\\\mathbf {Q}:= (q_{1},\\\\ldots ,q_{K}) \\\\in \\\\mathbb {R}^K: \\\\, q_{i} \\\\le 0 \\\\ \\\\text{for all} \\\\ i=1,\\\\ldots ,K \\\\rbrace , \\\\\\\\&& \\\\mathbb {S}^{K} := \\\\lbrace \\\\mathbf {Q} := (q_{1},\\\\ldots ,q_{K}) \\\\in \\\\mathbb {R}_{\\\\ge 0}^{K}:\\\\, \\\\sum _{i=1}^{K} q_{i} =1 \\\\rbrace \\\\quad \\\\text{(simplex of probability vectors)},\\\\\\\\&& \\\\mathbb {S}_{> 0}^{K} := \\\\lbrace \\\\mathbf {Q}:= (q_{1},\\\\ldots ,q_{K}) \\\\in \\\\mathbb {R}_{>0}^{K}: \\\\, \\\\sum _{i=1}^{K} q_{i} =1 \\\\rbrace .\",\n \"$ Concerning the directed distances $D(\\\\cdot ,\\\\cdot )$ in (ii) and (iii), we deal with the important omnipresent Csiszar-Ali-Silvey-Morimoto $\\\\varphi -$ divergences (CASM $\\\\varphi -$ divergences) — adapted to our context: Definition 3 (a) Let the “divergence-generator” be a lower semicontinuous convex function $\\\\varphi : \\\\, ]-\\\\infty ,\\\\infty [ \\\\rightarrow [0,\\\\infty ]$ satisfying $\\\\varphi (1)=0$ .\",\n \"Furthermore, for the effective domain $dom(\\\\varphi ) := \\\\lbrace t \\\\in \\\\mathbb {R} : \\\\varphi (t) < \\\\infty \\\\rbrace $ we assume that its interior $int(dom(\\\\varphi ))$ is non-empty which implies that $int(dom(\\\\varphi )) = ]a,b[$ for some $-\\\\infty \\\\le a < 1 < b \\\\le \\\\infty $ .\",\n \"Moreover, we suppose that $\\\\varphi $ is strictly convex in a non-empty neighborhood $]t_{-}^{sc},t_{+}^{sc}[ \\\\subseteq ]a,b[$ of one ($t_{-}^{sc} < 1 < t_{+}^{sc}$ ).\",\n \"Also, we set $\\\\varphi (a) := \\\\lim _{t \\\\downarrow a} \\\\varphi (t)$ and $\\\\varphi (b) := \\\\lim _{t \\\\uparrow b} \\\\varphi (t)$ (these limits always exist).\",\n \"The class of all such functions $\\\\varphi $ will be denoted by $\\\\widetilde{\\\\Upsilon }(]a,b[)$ .\",\n \"A frequent choice is e.g.\",\n \"$]a,b[=]0,\\\\infty [$ or $]a,b[=]-\\\\infty ,\\\\infty [$ .\",\n \"(b) For $\\\\varphi \\\\in \\\\widetilde{\\\\Upsilon }(]a,b[)$ , $\\\\mathbf {P} := (p_{1},\\\\ldots ,p_{K}) \\\\in \\\\mathbb {R}_{\\\\ge 0}^{K}$ and $\\\\mathbf {Q} := (q_{1},\\\\ldots ,q_{K}) \\\\in \\\\mathbf {\\\\Omega } \\\\subset \\\\mathbb {R}^{K}$ , we define the Csiszar-Ali-Silvey-Morimoto $\\\\varphi -$ divergence $\\\\Phi _{\\\\mathbf {P}} \\\\left(\\\\mathbf {Q}\\\\right) := D_{\\\\varphi }( \\\\mathbf {Q}, \\\\mathbf {P} ) :=\\\\sum _{k=1}^{K} p_{k} \\\\cdot \\\\varphi \\\\left( \\\\frac{q_{k}}{p_{k}}\\\\right) \\\\, \\\\ge 0.$ As usual, in (REF ) we employ the three conventions that $p\\\\cdot \\\\varphi \\\\left( \\\\frac{0}{p}\\\\right) = p \\\\cdot \\\\varphi (0) >0$ for all $p > 0$ , and $0 \\\\cdot \\\\varphi \\\\left( \\\\frac{q}{0}\\\\right) = q \\\\cdot \\\\lim _{x\\\\rightarrow \\\\infty } \\\\frac{\\\\varphi (x \\\\cdot \\\\textrm {sgn}(q))}{x\\\\cdot \\\\textrm {sgn}(q)} >0$ for $q \\\\ne 0$ (employing the sign of $q$ ), and $0 \\\\cdot \\\\varphi \\\\left( \\\\frac{0}{0}\\\\right) :=0$ .\",\n \"Throughout the paper, we only consider constellations $(\\\\varphi ,\\\\mathbf {P},\\\\mathbf {\\\\Omega })$ for which the very mild condition $\\\\Phi _{\\\\mathbf {P}}(\\\\Omega ) := \\\\inf _{\\\\mathbf {Q}\\\\in \\\\mathbf {\\\\Omega }}D_{\\\\varphi }( \\\\mathbf {Q}, \\\\mathbf {P} ) \\\\ne \\\\infty \\\\ \\\\ \\\\footnote {i.e.\",\n \"dom(\\\\varphi ) covers (at least) a non-empty part of \\\\lbrace 1\\\\rbrace \\\\cup \\\\mathcal {R}\\\\big (\\\\frac{\\\\mathbf {\\\\Omega }}{\\\\mathbf {P}}\\\\big ), where \\\\mathcal {R}\\\\big (\\\\frac{\\\\mathbf {\\\\Omega }}{\\\\mathbf {P}}\\\\big ) := \\\\big \\\\lbrace \\\\frac{q_{k}}{p_{k}}: \\\\, k \\\\in \\\\lbrace 1,\\\\ldots ,K\\\\rbrace , \\\\mathbf {Q}:=(q_{1},\\\\ldots ,q_{K}) \\\\in \\\\Omega \\\\big \\\\rbrace is the range of all possibleentry-ratios.\",\n \"}\\\\nonumber $ holds.\",\n \"For probability vectors ${P}$ and ${Q}$ in $\\\\mathbb {S}^{K}$ , the $\\\\varphi -$ divergences $D_{\\\\varphi }( {Q}, {P} )$ were introduced by Csiszar [94], Ali & Silvey [11] and Morimoto [266] (where the first two references even deal with more general probability distributions); for some comprehensive overviews — including statistical applications to goodness-of-fit testing and minimum distance estimation — the reader is referred to the insightful books of e.g.\",\n \"Liese & Vajda [217], Read & Cressie [303], Vajda [371], Csiszar & Shields [99], Stummer [344], Pardo [282], Liese & Miescke [216], the survey articles of e.g.\",\n \"Liese & Vajda [218], Vajda & van der Meulen [374], Reid & Williamson [304], Basseville [34], and the references therein.\",\n \"Some exemplary recent studies and applications of CASM $\\\\varphi -$ divergences appear e.g.\",\n \"in Qiao & Minematsu [298] for invariances in speech recognition, Nguyen et al.\",\n \"[273] in connection with empirical risk optimization, Feixas et al.\",\n \"[124] for various different image processing tasks, Luo et al.\",\n \"[232] for video clip segmentation and key frame generation, Kißlinger & Stummer [189] for model preselection (structure detection) in the context of nonlinear recursive models with additional exogenous inputs, Mahboubi & Kochenderfer [241] within a context of traffic-pattern learning from flight tracks, Guo et al.\",\n \"[145] for local contrastive descriptors in image classification through e.g.\",\n \"regional color distributions, Csiszar & Breuer [100] for modelling generalized-ball type constraints in expectation minimization problems, Kißlinger & Stummer [191] for the detection of distributional changes in random data (streams and clouds), Noh et al.\",\n \"[275] within a context of generative local metric learning for nearest neighbor classification, Yu et al.\",\n \"[416] for adversarial learning within oil spill segmentation, Arslan [19] for automated active reconfiguration in mobile sensor networks, Sason [320] in connection with with data-processing and majorization inequalities, Ciftci et al.\",\n \"[89] for the optimization of multienergy microgrids in energy infrastructure systems, and Stummer [345] for solving some new optimal transport (OT) problems which flexibilize some Wasserstein-distance based OTs.\",\n \"For the setup of $D_{\\\\varphi }( \\\\mathbf {Q}, \\\\mathbf {P} )$ for vectors $\\\\mathbf {P}$ , $\\\\mathbf {Q}$ with non-negative components the reader is referred to e.g.\",\n \"Stummer & Vajda [349] (who deal with even more general nonnegative measures and giving some statistical as well as information-theoretic applications) and Gietl & Reffel [137] (including applications to iterative proportional fitting).\",\n \"The case of $\\\\varphi -$ divergences for vectors with arbitrary components can be extracted from e.g.\",\n \"Broniatowski & Keziou [60] who actually deal with finite signed measures.\",\n \"For a comprehensive technical treatment, see also Broniatowski & Stummer [64].\",\n \"Clearly, from (REF ) it is obvious that in general $D_{\\\\varphi }( \\\\mathbf {Q}, \\\\mathbf {P} ) \\\\ne D_{\\\\varphi }(\\\\mathbf {P}, \\\\mathbf {Q})$ (non-symmetry).\",\n \"Moreover, it is straightforward to deduce that $D_{\\\\varphi }(\\\\mathbf {Q}, \\\\mathbf {P}) = 0$ if and only if $\\\\mathbf {Q}=\\\\mathbf {P}$ (reflexivity).\",\n \"Very prominent and important examples of CASM $\\\\varphi -$ divergences are the power divergences in the scaling of e.g.\",\n \"Liese & Vajda [217] (in other scalings also called Rathie & Kannapan’s non-additive directed divergences of order $\\\\gamma $ [302], Cressie-Read divergences [93] [303], relative Tsallis entropies or Tsallis cross-entropies [364] (see also Shiino [331]), Amari’s alpha-divergences [12]) where basically (up to technicalities) $\\\\varphi (t): = \\\\varphi _{\\\\gamma }(t) :=\\\\frac{t^\\\\gamma -\\\\gamma \\\\cdot t+ \\\\gamma - 1}{\\\\gamma \\\\cdot (\\\\gamma -1)}$ ($\\\\gamma \\\\in \\\\mathbb {R}\\\\backslash \\\\lbrace 0,1\\\\rbrace $ ), $\\\\varphi (t): = \\\\varphi _{0}(t) :=\\\\lim _{\\\\gamma \\\\rightarrow 0} \\\\varphi _{\\\\gamma }(t) = - \\\\log t + t - 1$ , $\\\\varphi (t): = \\\\varphi _{1}(t) :=\\\\lim _{\\\\gamma \\\\rightarrow 1} \\\\varphi _{\\\\gamma }(t) = t \\\\cdot \\\\log t + 1 - t$ .\",\n \"Usually, in the literature one takes $t \\\\in \\\\, ]0,\\\\infty [$ (and the limit as $t \\\\rightarrow 0$ ), except for the case $\\\\gamma =2$ where one handles $t \\\\in \\\\, ]-\\\\infty ,\\\\infty [$ ; for our purposes, we have to essentially extend these divergence generators $\\\\varphi _{\\\\gamma }$ for $t<0$ , which will be carried out and discussed in detail below, namely in (REF ), (REF ) (see also Table 1), as well as at several other places in this paper.\",\n \"Notice that $D_{\\\\varphi _{1}}(\\\\mathbf {Q},\\\\mathbf {P})$ basically corresponds to the (extended form of) the omnipresent Kullback-Leibler information resp.\",\n \"relative entropy.\",\n \"Below, we shall also consider the minimization/maximization of important transforms of power divergences such as Renyi divergences/entropies, Sundaresan’s divergence, etc., which are frequently used in information theory and its applications to e.g.\",\n \"artificial intelligence, machine learning, and physics.\",\n \"Remark 4 Since, in general, our methods work also for non-probability vectors $\\\\mathbf {Q},\\\\mathbf {P}$ , we can also deal with — plain versions and transformations of — weighted $\\\\varphi -$ divergences of the form $D_{\\\\varphi }^{wei}( \\\\mathbf {Q}, \\\\mathbf {P} ) :=\\\\sum _{k=1}^{K} c_{k} \\\\cdot p_{k} \\\\cdot \\\\varphi \\\\left( \\\\frac{q_{k}}{p_{k}}\\\\right) \\\\, \\\\ge 0$ where $c_{k} >0$ ($k=1,\\\\ldots ,K$ ) are weights which not necessarily add up to one.\",\n \"Indeed, by means of (REF ) we formally end up with $\\\\inf _{\\\\mathbf {Q}\\\\in \\\\mathbf {\\\\Omega }}D_{\\\\varphi }^{wei}( \\\\mathbf {Q}, \\\\mathbf {P} ) =\\\\inf _{\\\\mathbf {Q}^{wei}\\\\in \\\\mathbf {\\\\Omega }^{wei}}D_{\\\\varphi }( \\\\mathbf {Q}^{wei}, \\\\mathbf {P}^{wei} )\\\\nonumber $ where $\\\\mathbf {P}^{wei} := (c_{1} \\\\cdot p_{1},\\\\ldots ,c_{K} \\\\cdot p_{K})$ , $\\\\mathbf {Q}^{wei} := (c_{1} \\\\cdot q_{1},\\\\ldots ,c_{K} \\\\cdot q_{K})$ and $\\\\mathbf {\\\\Omega }^{wei}$ is the corresponding rescaling of $\\\\mathbf {\\\\Omega }$ .\",\n \"Of course, all the necessary technicalities for the $\\\\varphi -$ divergences (see below) have to be adapted to the weighted $\\\\varphi -$ divergences; for the sake of brevity, this will not be discussed in detail.\",\n \"Notice that $\\\\mathbf {P}^{wei}$ , $\\\\mathbf {Q}^{wei}$ are generally not probability vectors anymore, even if $\\\\mathbf {Q},\\\\mathbf {P}$ are probability vectors.\",\n \"In the latter case, and under the assumption $\\\\sum _{k=1}^{K} c_{k} = 1$ , the divergences (REF ) coincide with the discrete versions of the ($\\\\mathbf {c}-$ )local divergences of Avlogiaris et al.\",\n \"[22], [23] who also give absolutely-continuous versions and beyond (see also Broniatowski & Stummer [64] for an imbedding in a general divergence framework).\"\n ],\n [\n \"The cornerstone \",\n \"In this Section , we show that a number of deterministic optimization problems and and problems in statistical minimum risk based approaches pertaining to non- or semi-parametric contexts are BS-minimizable/amenable in the sense of Definition REF .\",\n \"The below-mentioned Sections and will draw conclusions, proposing effective solutions.\",\n \"For the construction of the desired sequence $(\\\\xi _{n})_{n\\\\in \\\\mathbb {N}}$ of $\\\\mathbb {R}^{K}-$ valued random variables (viz.\",\n \"random vectors) and a corresponding probability distribution $\\\\mathbb {\\\\Pi }$ (which will not depend on $n$ ), we will assume that the divergence generator $\\\\varphi \\\\in \\\\widetilde{\\\\Upsilon } (]a,b[)$ has the additional property that it can be represented as $\\\\varphi (t)=\\\\sup _{z\\\\in \\\\mathbb {R}}\\\\Big ( z\\\\cdot t-\\\\log \\\\int _{\\\\mathbb {R}}e^{z \\\\cdot y}d\\\\mathbb {} (y)\\\\Big ), \\\\qquad t\\\\in \\\\mathbb {R},\\\\ \\\\ $ for some probability distribution/measure $\\\\mathbb {} $ on the real line such that the function $z\\\\mapsto MGF_{\\\\mathbb {} }(z):=\\\\int _{\\\\mathbb {R}}e^{z \\\\cdot y}d\\\\mathbb {} (y)$ is finite on some open interval containing zero in particular, this implies that $\\\\mathbb {} $ has light tails.. From this, we shall construct — basically in Section below — a sequence $(W_{n})_{n\\\\in \\\\mathbb {N}}$ of i.i.d.\",\n \"copies of a random variable $W$ whose distribution (under $\\\\mathbb {\\\\Pi }$ ) is $\\\\mathbb {}$ (i.e.\",\n \"$\\\\mathbb {\\\\Pi }[W \\\\in \\\\cdot \\\\, ] = \\\\mathbb {}[ \\\\, \\\\cdot \\\\,]$ ), from which the desired $(\\\\xi _{n})_{n\\\\in \\\\mathbb {N}}$ will be constructed.\",\n \"Since $\\\\varphi $ attains its minimal value at the point 1, fit follows that $\\\\varphi ^{\\\\prime }(1)=0.$ By (REF ), for all $t$ in $int(dom(\\\\varphi ))$ , $\\\\varphi ^{\\\\prime }(t)$ is the reciprocal of $\\\\psi (z):=\\\\left(d/dz\\\\right) \\\\log MGF(z)$ at point $t$ , whence $\\\\psi (0)=1$ , which is to say that the expectation $E_\\\\mathbb {\\\\Pi }[W]=1$ .\",\n \"The class of functions $\\\\varphi \\\\in \\\\widetilde{\\\\Upsilon } (]a,b[)$ satisfying the representability (REF ) will be denoted by $\\\\Upsilon (]a,b[)$ .\",\n \"Remark 5 The condition $\\\\varphi \\\\in \\\\Upsilon (]a,b[)$ implies that $\\\\mathbb {}$ can not be a one-point distribution (Dirac mass) $\\\\delta _{y}$ at some point $y$ , since for such a situation one can straightforwardly deduce from (REF ) that $\\\\varphi (y)=0$ and $\\\\varphi (t)=\\\\infty $ for all $t \\\\ne y$ , which leads to $int(dom(\\\\varphi )) = \\\\emptyset $ and thus $\\\\varphi \\\\notin \\\\widetilde{\\\\Upsilon } (]a,b[)$ (in fact, our requirement $\\\\varphi (y)=0$ would narrow down to $y=1$ anyway).\",\n \"Let us remark that the class $\\\\Upsilon (]a,b[)$ contains many divergence generators; this together with $\\\\varphi -$ construction principles will be developed at length in Section below.\",\n \"Also, for the minimization problems considered in Section REF hereunder, we mostly modify the generator $\\\\varphi $ into $\\\\widetilde{c}\\\\cdot \\\\varphi $ for strictly positive scales $\\\\widetilde{c}$ .\",\n \"At this point, for the sake of transparency, we only present a summarizing Table 1 of a selection of concrete examples which will be treated in detail below: Table: NO_CAPTION Table 1.\",\n \"Selection of concrete examples treated in this paper, included with some important features (to be explained in the course of method build-up).\",\n \"As already explained above, the representability (REF ) is the cornerstone for our approach, and opens the gate to make use of simulation methods in appropriate contexts.\",\n \"We first develop this approach for deterministic minimization problems (cf.\",\n \"Subsection REF ); thereafter, in Subsection REF , we “extend” this to the setup where $\\\\mathbf {P}$ is identified with an unknown probability vector in the simplex $\\\\mathbb {S}^{K}$ which is supposed to be the limit (as $n$ tends to infinity) of the empirical distribution pertaining to a collection of observations $\\\\mathbf {X}_{n}$ $:=\\\\left(X_{1},..,X_{n}\\\\right) $ ; in the classical statistical setting, this amounts to the estimation of $\\\\Phi _{\\\\mathbf {P}}\\\\left( \\\\Omega \\\\right) $ based on $\\\\mathbf {X}_{n}$ , leading to the important “minimization-distance estimation problem” in statistics, artificial intelligence and machine learning.\",\n \"Finally, we end up this Section by shortly dealing with divergences between fuzzy sets (cf.\",\n \"Subsection REF ) and basic belief assignments (cf.\",\n \"Subsection REF ).\"\n ],\n [\n \"Deterministic minimization problems\",\n \"Problem 6 For pregiven $\\\\varphi \\\\in \\\\Upsilon (]a,b[)$ , positive-entries vector $\\\\mathbf {P}:=\\\\left( p_{1},..,p_{K}\\\\right) \\\\in \\\\mathbb {R}_{>0}^{K}$ (or from some subset thereof), and subset $\\\\mathbf {\\\\Omega } \\\\subset \\\\mathbb {R}^{K}$ (also denoted in boldface letters, with a slight abuse of notation) with regularity properties $cl(\\\\mathbf {\\\\Omega } )=cl\\\\left( int\\\\left( \\\\mathbf {\\\\Omega } \\\\right) \\\\right) , \\\\qquad int\\\\left( \\\\mathbf {\\\\Omega } \\\\right) \\\\ne \\\\emptyset ,$ find $\\\\Phi _{\\\\mathbf {P}}(\\\\mathbf {\\\\Omega }) := \\\\inf _{\\\\mathbf {Q}\\\\in \\\\mathbf {\\\\Omega } } D_{\\\\varphi }(\\\\mathbf {Q},\\\\mathbf {P}),$ provided that $\\\\inf _{\\\\mathbf {Q}\\\\in \\\\mathbf {\\\\Omega } } D_{\\\\varphi }(\\\\mathbf {Q},\\\\mathbf {P}) < \\\\infty .$ An immediate consequence thereof is — for pregiven $\\\\varphi \\\\in \\\\Upsilon (]a,b[)$ — the treatment of the more flexible problem $\\\\Phi _{\\\\mathbf {P},h}(\\\\mathbf {\\\\Omega }) := \\\\inf _{\\\\mathbf {Q}\\\\in \\\\mathbf {\\\\Omega } }h\\\\Big (D_{\\\\varphi }(\\\\mathbf {Q},\\\\mathbf {P}) \\\\Big )= h\\\\Big (\\\\inf _{\\\\mathbf {Q}\\\\in \\\\mathbf {\\\\Omega } } D_{\\\\varphi }(\\\\mathbf {Q},\\\\mathbf {P}) \\\\Big )$ for any continuous strictly increasing function $h: \\\\mathcal {H} \\\\, \\\\mapsto \\\\mathbb {R}$ with $\\\\mathcal {H} := [0,\\\\infty [$ and extension $h(\\\\infty ) := \\\\sup _{y \\\\in \\\\mathcal {H}}(y)$ (depending on the problem, a sufficiently large $\\\\mathcal {H} \\\\subset [0,\\\\infty [$ may be enough), respectively of $\\\\sup _{\\\\mathbf {Q}\\\\in \\\\mathbf {\\\\Omega } }h\\\\Big (D_{\\\\varphi }(\\\\mathbf {Q},\\\\mathbf {P}) \\\\Big )= h\\\\Big (\\\\inf _{\\\\mathbf {Q}\\\\in \\\\mathbf {\\\\Omega } } D_{\\\\varphi }(\\\\mathbf {Q},\\\\mathbf {P}) \\\\Big )$ for any continuous strictly decreasing function $h: \\\\mathcal {H} \\\\, \\\\mapsto \\\\mathbb {R}$ and extension $h(\\\\infty ) := \\\\inf _{y \\\\in \\\\mathcal {H}}(y)$ .\",\n \"Remark 7 (a) By the basic properties of $\\\\varphi $ , it follows that for all $c>0$ the level sets $\\\\mathbf {\\\\varphi }_{c}:=\\\\left\\\\lbrace x\\\\in \\\\mathbb {R}:\\\\varphi (x)\\\\le c\\\\right\\\\rbrace $ are compact and so are the level sets of $\\\\mathbf {Q\\\\rightarrow }D_{\\\\varphi }(\\\\mathbf {Q},\\\\mathbf {P})$ $\\\\Gamma _{c}:=\\\\left\\\\lbrace \\\\mathbf {Q}\\\\in \\\\mathbb {R}^{K}:D_{\\\\varphi }(\\\\mathbf {Q},\\\\mathbf {P})\\\\le c\\\\right\\\\rbrace \\\\nonumber $ for all $c>0$ .\",\n \"(b) When $\\\\mathbf {\\\\Omega }$ is not closed but merely satisfies (REF ), then the infimum in (REF ) may not be reached in $\\\\mathbf {\\\\Omega }$ although being finite; however we aim for finding the infimum/minimum in (REF ).\",\n \"Finding the minimizers in (REF ) is another question.\",\n \"For instance, this can be solved whenever, additionally, $\\\\mathbf {\\\\Omega }$ is a closed set which implies the existence of minimizers in $\\\\mathbf {\\\\Omega }$ .\",\n \"In this case, and when the number of such minimizers is finite, those can be approximated by dichotomic search.\",\n \"For the sake of brevity, this will not be addressed in this paper.\",\n \"(c) The purpose of the condition (REF ) is to get rid of the $\\\\lim \\\\sup $ type and $\\\\lim \\\\inf $ type results in our below-mentioned “bare-simulation” approach and to obtain simple limit-statements which motivate our construction.\",\n \"In practice, it is enough to verify $\\\\mathbf {\\\\Omega }\\\\subseteq cl\\\\left(int\\\\left( \\\\mathbf {\\\\Omega }\\\\right) \\\\right) $ , which is equivalent to the left-hand part of (REF ).\",\n \"Clearly, any open set $\\\\mathbf {\\\\Omega }\\\\subset \\\\mathbb {R}^{K}$ satisfies the left-hand part of (REF ).\",\n \"In the subsetup where $\\\\mathbf {\\\\Omega }$ is a closed convex set and $int(\\\\mathbf {\\\\Omega })\\\\ne \\\\emptyset $ , (REF ) is satisfied and the minimizer $\\\\mathbf {Q}_{min}\\\\in \\\\mathbf {\\\\Omega }$ in (REF ) is attained and even unique.\",\n \"When $\\\\mathbf {\\\\Omega }$ is open and satisfies (REF ), then the infimum in (REF ) exists but is reached at some generalized projection of $\\\\mathbf {P}$ on $\\\\mathbf {\\\\Omega }$ (see Csiszar [97] for the definition in the Kullback-Leibler case of probability measures, which extends to any $\\\\varphi -$ divergence in our framework).\",\n \"(d) Without further mentioning, the regularity condition (REF ) is supposed to hold in the full topology.\",\n \"Of course, $int\\\\left( \\\\mathbb {S}^{K} \\\\right) = \\\\emptyset $ and thus, for the important probability-vector setup $\\\\mathbf {\\\\Omega } \\\\subset \\\\mathbb {S}^{K}$ the condition (REF ) is violated which requires extra refinements (cf.\",\n \"Subsection REF below).\",\n \"The same is needed for $\\\\mathbf {\\\\Omega } \\\\subset A \\\\cdot \\\\mathbb {S}^{K}$ for some $A \\\\ne 1$ , since obviously $int\\\\left( A \\\\cdot \\\\mathbb {S}^{K} \\\\right) = \\\\emptyset $ ; such a context appears naturally e.g.\",\n \"in connection with mass transportation problems (cf.\",\n \"(REF ) below) and with distributed energy management (cf.\",\n \"the paragraph after (REF )).\",\n \"(e) Often, $\\\\mathbf {\\\\Omega }$ will present a (discrete) model recall that an alternative naming also used in literature is to call $\\\\Omega $ a model class (rather than model), and each $P \\\\in \\\\Omega $ a model (rather than model element).\",\n \"Since $\\\\mathbf {\\\\Omega }$ is assumed to have a non-void interior (cf.\",\n \"the right-hand part of (REF )), this will exclude (parametric) models $\\\\mathbf {\\\\Omega }:=\\\\lbrace \\\\mathbf {Q}_{\\\\theta }:\\\\theta \\\\in \\\\Theta \\\\rbrace $ for some $\\\\Theta \\\\subset \\\\mathbb {R}^{d}$ ($d0$ and let $\\\\widetilde{{P}}:=\\\\mathbf {P}/M_{\\\\mathbf {P}},$ and for $\\\\mathbf {Q}$ in $\\\\mathbf {\\\\Omega }$ , let $\\\\widetilde{\\\\mathbf {Q}}:=\\\\mathbf {Q}/M_{\\\\mathbf {P}}$ (notice that $\\\\widetilde{\\\\mathbf {Q}}$ may be a non-probability vector).\",\n \"With the function $\\\\widetilde{\\\\varphi } \\\\in \\\\Upsilon (]a,b[)$ defined through $\\\\widetilde{\\\\varphi }:=M_{\\\\mathbf {P}} \\\\cdot \\\\varphi $ , we obtain $D_{\\\\varphi }(\\\\mathbf {Q},\\\\mathbf {P})=\\\\sum _{k=1}^{K}p_{k}\\\\cdot \\\\varphi \\\\left( \\\\frac{q_{k}}{p_{k}}\\\\right) =\\\\sum _{k=1}^{K}M_{\\\\mathbf {P}}\\\\cdot \\\\widetilde{p_{k}}\\\\cdot \\\\frac{\\\\varphi \\\\left( \\\\frac{M_{\\\\mathbf {P}}\\\\cdot \\\\widetilde{q_{k}}}{M_{\\\\mathbf {P}}\\\\cdot \\\\widetilde{p_{k}}}\\\\right) }{M_{\\\\mathbf {P}}}=D_{\\\\widetilde{\\\\varphi }}(\\\\widetilde{\\\\mathbf {Q}},\\\\widetilde{{P}}).$ It follows that the solution of (REF ) coincides with the one of the problem of finding $\\\\widetilde{\\\\Phi }_{\\\\widetilde{{P}}}(\\\\widetilde{\\\\mathbf {\\\\Omega }}) := \\\\inf _{\\\\widetilde{\\\\mathbf {Q}}\\\\in \\\\widetilde{\\\\mathbf {\\\\Omega }} }D_{\\\\widetilde{\\\\varphi } }(\\\\widetilde{\\\\mathbf {Q}},\\\\widetilde{{P}}),\\\\qquad \\\\textrm {with } \\\\widetilde{\\\\mathbf {\\\\Omega }}:=\\\\mathbf {\\\\Omega } /M_{\\\\mathbf {P}};$ as a side remark, one can see that in such a situation the rescaling of the divergence generator $\\\\varphi $ is important, which is one incentive that we incorporate multiples of $\\\\varphi $ below.\",\n \"As an important special case we get for the choice $\\\\mathbf {P} := (1, \\\\ldots , 1) := \\\\mathbf {1}$ that the “prominent/frequent” separable nonlinear optimization problem of finding the optimal value $\\\\inf _{\\\\mathbf {Q} \\\\in \\\\mathbf {\\\\Omega } } \\\\sum _{k=1}^{K}\\\\varphi (q_{k})$ — with objective (e.g.\",\n \"cost, energy, purpose) function $\\\\varphi \\\\in \\\\Upsilon (]a,b[)$ and constraint set (choice set, search space) $\\\\mathbf {\\\\Omega }$ — can be imbedded into our BS-approach by $\\\\inf _{\\\\mathbf {Q} \\\\in \\\\mathbf {\\\\Omega } } \\\\sum _{k=1}^{K}\\\\varphi (q_{k}) =\\\\inf _{\\\\mathbf {Q} \\\\in \\\\mathbf {\\\\Omega } } D_{\\\\varphi }(\\\\mathbf {Q},\\\\mathbf {1}) =\\\\inf _{\\\\widetilde{\\\\mathbf {Q}} \\\\in \\\\mathbf {\\\\Omega }/K }D_{K \\\\cdot \\\\varphi }(\\\\widetilde{\\\\mathbf {Q}},{P}^{unif}),$ with ${P}^{unif} := (\\\\frac{1}{K}, \\\\ldots , \\\\frac{1}{K})$ being the probability vector of frequencies of the uniform distribution on $\\\\lbrace 1, \\\\ldots , K\\\\rbrace $ .\",\n \"Notice that with our new BS approach one may even tackle more general optimization problems of the form $\\\\inf _{\\\\breve{Q} \\\\in \\\\breve{\\\\Omega }} \\\\sum _{k=1}^{K} \\\\breve{\\\\varphi } (\\\\breve{q}_{k})$ where $\\\\breve{\\\\varphi }$ is some function which is finite and convex in a non-empty neighborhood (say, $]t_{0} + a- 1, t_{0} + b- 1[$ with $a < 1 < b$ ) of some point $t_{0} \\\\in \\\\mathbb {R}$ as well as strictly convex in a non-empty sub-neighborhood of $t_{0}$ ; for this, the function $\\\\varphi (t) := \\\\breve{\\\\varphi }(t+t_{0}-1) - \\\\breve{\\\\varphi }^{\\\\prime }(t_{0})\\\\cdot \\\\Big ((t+t_{0}-1) - t_{0} \\\\Big ) - \\\\breve{\\\\varphi }(t_{0}), \\\\qquad t \\\\in ]a,b[,\\\\nonumber $ (which corresponds to shifting the argument and adding an affine-linear function) should be a member of $\\\\Upsilon (]a,b[)$ , and from the corresponding minimization problem $& &\\\\inf _{\\\\widetilde{\\\\mathbf {Q}} \\\\in \\\\mathbf {\\\\Omega }/K }D_{K \\\\cdot \\\\varphi }(\\\\widetilde{\\\\mathbf {Q}},{P}^{unif}) =\\\\inf _{\\\\mathbf {Q} \\\\in \\\\mathbf {\\\\Omega } } \\\\sum _{k=1}^{K} \\\\varphi (q_{k})= \\\\inf _{\\\\mathbf {Q} \\\\in \\\\mathbf {\\\\Omega } } \\\\sum _{k=1}^{K}\\\\Big (\\\\breve{\\\\varphi }(q_{k}+t_{0}-1) - \\\\breve{\\\\varphi }^{\\\\prime }(t_{0})\\\\cdot ((q_{k}+t_{0}-1) -t_{0}) - \\\\breve{\\\\varphi }(t_{0})\\\\Big )\\\\nonumber \\\\\\\\& & = \\\\inf _{\\\\breve{\\\\mathbf {Q}} \\\\in \\\\mathbf {\\\\Omega } + t_{0} -1} \\\\sum _{k=1}^{K} \\\\ \\\\Big (\\\\breve{\\\\varphi }(\\\\breve{q}_{k}) - \\\\breve{\\\\varphi }^{\\\\prime }(t_{0})\\\\cdot (\\\\breve{q}_{k} -t_{0}) - \\\\breve{\\\\varphi }(t_{0})\\\\Big )\\\\nonumber \\\\\\\\& & = K \\\\cdot \\\\Big (t_{0} \\\\cdot \\\\breve{\\\\varphi }^{\\\\prime }(t_{0}) - \\\\breve{\\\\varphi }(t_{0})\\\\Big )+ \\\\inf _{\\\\breve{\\\\mathbf {Q}} \\\\in \\\\breve{\\\\mathbf {\\\\Omega }} } \\\\left(\\\\sum _{k=1}^{K} \\\\breve{\\\\varphi }(\\\\breve{q}_{k})- \\\\breve{\\\\varphi }^{\\\\prime }(t_{0}) \\\\cdot \\\\sum _{k=1}^{K} \\\\breve{q}_{k}\\\\right),\\\\qquad \\\\textrm {with } \\\\breve{\\\\mathbf {\\\\Omega }} := \\\\mathbf {\\\\Omega } + t_{0} -1,$ the term $\\\\inf _{\\\\breve{Q} \\\\in \\\\breve{\\\\Omega }} \\\\sum _{k=1}^{K} \\\\breve{\\\\varphi } (\\\\breve{q}_{k})$ should be recoverable; for instance, later on we shall employ constraints sets $\\\\breve{\\\\mathbf {\\\\Omega }}$ which particularly include $\\\\sum _{k=1}^{K} \\\\breve{q}_{k} = A > 0$ , whereas another possibility would be to use a $\\\\breve{\\\\varphi }$ which satisfies $\\\\breve{\\\\varphi }^{\\\\prime }(t_{0}) =0$ .\",\n \"As a different line of flexibilization of (REF ), we can also deal with the problem $\\\\inf _{\\\\mathbf {Q} \\\\in \\\\mathbf {\\\\Omega } } h\\\\Big (\\\\sum _{k=1}^{K}\\\\varphi (q_{k})\\\\Big )$ through $\\\\inf _{\\\\mathbf {Q} \\\\in \\\\mathbf {\\\\Omega } }h\\\\Big (\\\\sum _{k=1}^{K}\\\\varphi (q_{k})\\\\Big ) =h\\\\Big ( \\\\inf _{\\\\widetilde{\\\\mathbf {Q}} \\\\in \\\\mathbf {\\\\Omega }/K }D_{K \\\\cdot \\\\varphi }(\\\\widetilde{\\\\mathbf {Q}},{P}^{unif}) \\\\Big )$ for any $\\\\varphi \\\\in \\\\Upsilon (]a,b[)$ and any continuous strictly increasing function $h: \\\\mathcal {H} \\\\, \\\\mapsto \\\\mathbb {R}$ with $\\\\mathcal {H} := [0,\\\\infty [$ (or a sufficiently large subset thereof), and with the problem $\\\\sup _{\\\\mathbf {Q} \\\\in \\\\mathbf {\\\\Omega } } h\\\\Big (\\\\sum _{k=1}^{K}\\\\varphi (q_{k})\\\\Big )$ through $\\\\sup _{\\\\mathbf {Q} \\\\in \\\\mathbf {\\\\Omega } }h\\\\Big (\\\\sum _{k=1}^{K}\\\\varphi (q_{k})\\\\Big ) =h\\\\Big ( \\\\inf _{\\\\widetilde{\\\\mathbf {Q}} \\\\in \\\\mathbf {\\\\Omega }/K }D_{K \\\\cdot \\\\varphi }(\\\\widetilde{\\\\mathbf {Q}},{P}^{unif}) \\\\Big )$ for any $\\\\varphi \\\\in \\\\Upsilon (]a,b[)$ and any continuous strictly decreasing function $h: \\\\mathcal {H} \\\\, \\\\mapsto \\\\mathbb {R}$ .\",\n \"Combining (REF ) with (REF ) (respectively, with (REF )) leads to a further flexibilization.\",\n \"Of course, we can also apply our BS method to the maximization $\\\\sup _{\\\\mathbf {Q} \\\\in \\\\mathbf {\\\\Omega } } h\\\\Big (\\\\sum _{k=1}^{K}\\\\zeta (q_{k})\\\\Big )$ for any concave function $\\\\zeta $ with $-\\\\zeta \\\\in \\\\Upsilon (]a,b[)$ and any continuous strictly increasing function $h: \\\\mathcal {H} \\\\, \\\\mapsto \\\\mathbb {R}$ with $\\\\mathcal {H} := - [\\\\infty ,0]$ (or a sufficiently large subset thereof), via $\\\\sup _{\\\\mathbf {Q} \\\\in \\\\mathbf {\\\\Omega } }h\\\\Big (\\\\sum _{k=1}^{K}\\\\zeta (q_{k})\\\\Big ) =h\\\\Big ( - \\\\inf _{\\\\widetilde{\\\\mathbf {Q}} \\\\in \\\\mathbf {\\\\Omega }/K }D_{- K \\\\cdot \\\\zeta }(\\\\widetilde{\\\\mathbf {Q}},{P}^{unif}) \\\\Big ) ,$ and to $\\\\inf _{\\\\mathbf {Q} \\\\in \\\\mathbf {\\\\Omega } } h\\\\Big (\\\\sum _{k=1}^{K}\\\\zeta (q_{k})\\\\Big )$ for any concave function $\\\\zeta $ with $-\\\\zeta \\\\in \\\\Upsilon (]a,b[)$ and any continuous strictly decreasing function $h: \\\\mathcal {H} \\\\, \\\\mapsto \\\\mathbb {R}$ , via $\\\\inf _{\\\\mathbf {Q} \\\\in \\\\mathbf {\\\\Omega } }h\\\\Big (\\\\sum _{k=1}^{K}\\\\zeta (q_{k})\\\\Big ) =h\\\\Big ( - \\\\inf _{\\\\widetilde{\\\\mathbf {Q}} \\\\in \\\\mathbf {\\\\Omega }/K }D_{- K \\\\cdot \\\\zeta }(\\\\widetilde{\\\\mathbf {Q}},{P}^{unif}) \\\\Big ) .$ Moreover, we can tackle $\\\\sup _{\\\\breve{Q} \\\\in \\\\breve{\\\\Omega }} \\\\sum _{k=1}^{K} \\\\breve{\\\\zeta } (\\\\breve{q}_{k})$ where $\\\\breve{\\\\zeta }$ is some function which is finite and concave in a non-empty neighborhood $]t_{0} + a- 1, t_{0} + b- 1[$ (with $a < 1 < b$ ) of some point $t_{0} \\\\in \\\\mathbb {R}$ as well as strictly concave in a non-empty sub-neighborhood of $t_{0}$ ; for this, the function $- \\\\zeta (t) := - \\\\breve{\\\\zeta }(t+t_{0}-1) + \\\\breve{\\\\zeta }^{\\\\prime }(t_{0})\\\\cdot \\\\Big ((t+t_{0}-1) - t_{0} \\\\Big ) + \\\\breve{\\\\zeta }(t_{0}), \\\\qquad t \\\\in ]a,b[,\\\\nonumber $ should be a member of $\\\\Upsilon (]a,b[)$ , and from the corresponding minimization problem $& &- \\\\inf _{\\\\widetilde{\\\\mathbf {Q}} \\\\in \\\\mathbf {\\\\Omega }/K }D_{- K \\\\cdot \\\\zeta }(\\\\widetilde{\\\\mathbf {Q}},{P}^{unif})= \\\\sup _{\\\\mathbf {Q} \\\\in \\\\mathbf {\\\\Omega } } \\\\sum _{k=1}^{K} \\\\zeta (q_{k})= \\\\sup _{\\\\mathbf {Q} \\\\in \\\\mathbf {\\\\Omega } } \\\\sum _{k=1}^{K}\\\\Big (\\\\breve{\\\\zeta }(q_{k}+t_{0}-1) - \\\\breve{\\\\zeta }^{\\\\prime }(t_{0})\\\\cdot ((q_{k}+t_{0}-1) -t_{0}) - \\\\breve{\\\\zeta }(t_{0})\\\\Big )\\\\nonumber \\\\\\\\& & = \\\\sup _{\\\\breve{\\\\mathbf {Q}} \\\\in \\\\mathbf {\\\\Omega } + t_{0} -1} \\\\sum _{k=1}^{K} \\\\ \\\\Big (\\\\breve{\\\\zeta }(\\\\breve{q}_{k}) - \\\\breve{\\\\zeta }^{\\\\prime }(t_{0})\\\\cdot (\\\\breve{q}_{k} -t_{0}) - \\\\breve{\\\\zeta }(t_{0})\\\\Big )\\\\nonumber \\\\\\\\& & = K \\\\cdot \\\\Big (t_{0} \\\\cdot \\\\breve{\\\\zeta }^{\\\\prime }(t_{0}) - \\\\breve{\\\\zeta }(t_{0})\\\\Big )+ \\\\sup _{\\\\breve{\\\\mathbf {Q}} \\\\in \\\\breve{\\\\mathbf {\\\\Omega }} } \\\\left(\\\\sum _{k=1}^{K} \\\\breve{\\\\zeta }(\\\\breve{q}_{k})- \\\\breve{\\\\zeta }^{\\\\prime }(t_{0}) \\\\cdot \\\\sum _{k=1}^{K} \\\\breve{q}_{k}\\\\right),\\\\qquad \\\\textrm {with } \\\\breve{\\\\mathbf {\\\\Omega }} := \\\\mathbf {\\\\Omega } + t_{0} -1,$ the term $\\\\sup _{\\\\breve{Q} \\\\in \\\\breve{\\\\Omega }} \\\\sum _{k=1}^{K} \\\\breve{\\\\zeta } (\\\\breve{q}_{k})$ should be recoverable; the left-hand side of (REF ) corresponds to the special case $h(x) := x$ of the BS-minimizable (REF ).\",\n \"A combination of (REF ) with (REF ) (respectively, with (REF )) leads to a further flexibilization.\",\n \"Remark 8 (a) Since $\\\\mathbf {1}$ can be seen e.g.\",\n \"as a reference vector with (normalized) equal components, the quantity $\\\\inf _{\\\\mathbf {Q} \\\\in \\\\mathbf {\\\\Omega } } D_{\\\\varphi }(\\\\mathbf {Q},\\\\mathbf {1})$ in (REF ) can be interpreted as an “index/degree of (in)equality of the set $\\\\mathbf {\\\\Omega }$ ”, respectively as an “index/degree of diversity of the set $\\\\mathbf {\\\\Omega }$ ”.\",\n \"(b) The quantity $\\\\sum _{k=1}^{K}\\\\varphi (q_{k})$ in (REF ) can be interpreted as (non-probability extension of an) $\\\\varphi -$ entropy in the sense of Burbea & Rao [68] (see also Csiszar [95], Ben-Bassat [38], Ben-Tal & Teboulle [40], Kesavan & Kapur [187], Dacunha-Castelle & Gamboa [102], Teboulle & Vajda [357], Gamboa & Gassiat [132], Vajda & Zvarova [376]); for applications to scalar quantization for lossy coding of information sources see e.g.\",\n \"György & Linder [149].\",\n \"More generally, the quantity $h\\\\Big (\\\\sum _{k=1}^{K}\\\\varphi (q_{k})\\\\Big )$ in (REF ) can be seen as (non-probability extension of an) $(h,\\\\varphi )-$ entropy in the sense of Salicru et al.\",\n \"[314] (see also e.g.\",\n \"Pardo [282], Vajda & Vasek [375], as well as e.g.\",\n \"Chen et al.\",\n \"[78] for uses as supervised adaption criterion within stochastic information gradient algorithms and Ren et al.\",\n \"[308] for applications to tracking in networked control systems).\",\n \"Important special cases will be discussed in more detail, below.\",\n \"Returning to the original distance-minimizing Problem REF , after the first step (REF ) and (REF ), we proceed as follows: Step 2: construct an appropriate sequence $(\\\\xi _{n})_{n\\\\in \\\\mathbb {N}}$ of $\\\\mathbb {R}^{K}-$ valued random variables/random vectors (cf.\",\n \"(REF ) in Definition REF ): The following condition transposes the minimization problem (REF ) into a BS minimizable/amenable problem in the sense of Definition REF and it is required in order that Problem (REF ) is equivalent to Problem (REF ).\",\n \"The connection of this condition with (REF ) will be discussed in Proposition REF and its surroundings, see Section .\",\n \"Condition 9 With $M_{\\\\mathbf {P}} =\\\\sum _{i=1}^{K}p_{i}>0$ , the divergence generator $\\\\varphi $ in (REF ) (cf.\",\n \"also (REF )) satisfies $\\\\widetilde{\\\\varphi } := M_{\\\\mathbf {P}} \\\\cdot \\\\varphi \\\\in \\\\Upsilon (]a,b[)$ , i.e.\",\n \"$\\\\widetilde{\\\\varphi } \\\\in \\\\widetilde{\\\\Upsilon }(]a,b[)$ (which is equivalent to $\\\\varphi \\\\in \\\\widetilde{\\\\Upsilon }(]a,b[)$ ) and there holds the representation $\\\\widetilde{\\\\varphi }(t) =\\\\sup _{z \\\\in \\\\mathbb {R}} \\\\left( z\\\\cdot t - \\\\log \\\\int _{\\\\mathbb {R}} e^{zy} d\\\\widetilde{\\\\mathbb {}}(y) \\\\right),\\\\qquad t \\\\in \\\\mathbb {R},$ for some probability measure $\\\\widetilde{\\\\mathbb {}}$ on the real line such that the function $z \\\\mapsto MGF_{\\\\widetilde{\\\\mathbb {}}}(z) := \\\\int _{\\\\mathbb {R}} e^{zy} d\\\\widetilde{\\\\mathbb {}}(y)$ is finite on some open interval containing zero in particular, this implies that $\\\\int _{\\\\mathbb {R}} y d\\\\widetilde{\\\\mathbb {}}(y) =1$ (cf.\",\n \"(G11i) below) and that $\\\\widetilde{\\\\mathbb {}}$ has light tails..\",\n \"In the following, let us explain the above-mentioned Step 2 in detail: for any $n \\\\in \\\\mathbb {N}$ and any $k \\\\in \\\\left\\\\lbrace 1, \\\\ldots ,K\\\\right\\\\rbrace $ , let $n_{k}:=\\\\lfloor n \\\\cdot \\\\widetilde{p}_{k}\\\\rfloor $ where $\\\\lfloor x \\\\rfloor $ denotes the integer part of $x$ .\",\n \"We assume $\\\\mathbf {P} \\\\in \\\\mathbb {R}_{> 0}^{K}$ , and since thus none of the $\\\\widetilde{p}_{k}$ 's is zero, one has $\\\\lim _{n\\\\rightarrow \\\\infty } \\\\frac{n_{k}}{n} = \\\\widetilde{p}_{k}.$ Moreover, we assume that $n \\\\in \\\\mathbb {N}$ is large enough, namely $n \\\\ge \\\\max _{k \\\\in \\\\lbrace 1, \\\\ldots , K\\\\rbrace } \\\\frac{1}{\\\\widetilde{p}_{k}}$ , and decompose the set $\\\\lbrace 1, \\\\ldots , n\\\\rbrace $ of all integers from 1 to $n$ into the following disjoint blocks: $I_{1}^{(n)}:=\\\\left\\\\lbrace 1,\\\\ldots ,n_{1}\\\\right\\\\rbrace $ , $I_{2}^{(n)}:=\\\\left\\\\lbrace n_{1}+1,\\\\ldots ,n_{1}+n_{2}\\\\right\\\\rbrace $ , and so on until the last block $I_{K}^{(n)} := \\\\lbrace \\\\sum _{k=1}^{K-1} n_{k} + 1, \\\\ldots , n \\\\rbrace $ which therefore contains all integers from $n_{1}+ \\\\ldots +n_{K-1}+1$ to $n$ .\",\n \"Clearly, $I_{k}^{(n)}$ has $n_{k} \\\\ge 1$ elements (i.e.\",\n \"$card(I_{k}^{(n)}) = n_{k}$ where $card(A)$ denotes the number of elements in a set $A$ ) for all $k \\\\in \\\\lbrace 1, \\\\ldots , K-1\\\\rbrace $ , and the last block $I_{K}^{(n)}$ has $n- \\\\sum _{k=1}^{K-1} n_{k} \\\\ge 1$ elements which anyhow satisfies $\\\\lim _{n\\\\rightarrow \\\\infty } card(I_{K}^{(n)})/n=\\\\widetilde{p}_{K}$ if all $\\\\widetilde{p}_{k}$ ($k=1,\\\\ldots ,K$ ) are rational numbers in $]0,1[$ with $\\\\sum _{k=1}^{K} \\\\widetilde{p}_{k} =1$ and $N$ is the (always existing) smallest integer such that all $N \\\\cdot \\\\widetilde{p}_{k}$ ($k=1,\\\\ldots ,K$ ) are integers (i.e.\",\n \"$\\\\in \\\\mathbb {N}$ ), then for any multiple $n= \\\\ell \\\\cdot N$ ($\\\\ell \\\\in \\\\mathbb {N}$ ) one gets that all $n_{k} = n \\\\cdot \\\\widetilde{p}_{k}$ are integers and that $card(I_{K}^{(n)}) = n_{K}$ ..\",\n \"Furthermore, consider a vector $\\\\mathbf {\\\\widetilde{W}}:=\\\\left( \\\\widetilde{W}_{1},\\\\ldots ,\\\\widetilde{W}_{n}\\\\right) $ where the $\\\\widetilde{W}_{i}$ 's are i.i.d.\",\n \"copies of the random variable $\\\\widetilde{W}$ whose distribution is associated with the divergence-generator $\\\\widetilde{\\\\varphi }:=M_{\\\\textbf {P}} \\\\cdot \\\\varphi $ through (REF ), in the sense that $\\\\mathbb {\\\\Pi }[\\\\widetilde{W}\\\\in \\\\cdot \\\\,]=\\\\widetilde{\\\\mathbb {}}[\\\\,\\\\cdot \\\\,]$ .\",\n \"We group the $\\\\widetilde{W}_{i}$ 's according the above-mentioned blocks and sum them up blockwise, in order to build the following $K-$ component random vector $\\\\xi _{n}^{\\\\mathbf {\\\\widetilde{W}}}:=\\\\Big (\\\\frac{1}{n}\\\\sum _{i\\\\in I_{1}^{(n)}}\\\\widetilde{W}_{i},\\\\ldots ,\\\\frac{1}{n}\\\\sum _{i\\\\in I_{K}^{(n)}}\\\\widetilde{W}_{i}\\\\Big );$ notice that the signs of its components may be negative, depending on the nature of the $\\\\widetilde{W}_{i}$ 's; moreover, the expectation of its $k-$ th component converges to $\\\\widetilde{p}_{k}$ as $n$ tends to infinity (since the expectation of $\\\\widetilde{W}_{1} $ is 1), whereas the $n-$ fold of the corresponding variance converges to $\\\\widetilde{p}_{k}$ times the variance of $\\\\widetilde{W}_{1}$ .\",\n \"For such a context, we obtain the following assertion on BS-minimizability: Theorem 10 Let $\\\\mathbf {P} \\\\in \\\\mathbb {R}_{> 0}^{K}$ , $M_{\\\\mathbf {P}}:=\\\\sum _{i=1}^{K}p_{i}>0$ , and suppose that the divergence generator $\\\\varphi $ satisfies the Condition REF above, with $\\\\widetilde{\\\\mathbb {}}$ (cf.\",\n \"(REF )).\",\n \"Additionally, let $\\\\widetilde{W}:=(\\\\widetilde{W}_{i})_{i\\\\in \\\\mathbb {N}}$ be a sequence of random variables where the $\\\\widetilde{W}_{i}$ 's are i.i.d.\",\n \"copies of the random variable $\\\\widetilde{W}$ whose distribution is $\\\\mathbb {\\\\Pi }[\\\\widetilde{W}\\\\in \\\\cdot \\\\,]=\\\\widetilde{\\\\mathbb {}}[\\\\,\\\\cdot \\\\,]$ and thus, $E_{\\\\mathbb {\\\\Pi }}[\\\\widetilde{W}_{i}]=1$.\",\n \"Then, in terms of the random vectors $\\\\xi _{n}^{\\\\mathbf {\\\\widetilde{W}}}=\\\\Big (\\\\frac{1}{n}\\\\sum _{i\\\\in I_{1}^{(n)}}\\\\widetilde{W}_{i},\\\\ldots ,\\\\frac{1}{n}\\\\sum _{i\\\\in I_{K}^{(n)}}\\\\widetilde{W}_{i}\\\\Big )\\\\hspace{56.9055pt} \\\\text{(cf.\",\n \"(\\\\ref {Xi_n^W vector}))}\\\\nonumber $ there holds $-\\\\lim _{n\\\\rightarrow \\\\infty }\\\\frac{1}{n}\\\\log \\\\,\\\\mathbb {\\\\Pi }\\\\left[\\\\xi _{n}^{\\\\mathbf {\\\\widetilde{W}}}\\\\in \\\\mathbf {\\\\Omega } /M_{\\\\mathbf {P}}\\\\right]=\\\\inf _{Q\\\\in \\\\mathbf {\\\\Omega }}D_{\\\\varphi }(\\\\mathbf {Q},\\\\mathbf {P}\\\\ )$ for any $\\\\mathbf {\\\\Omega }\\\\subset \\\\mathbb {R}^{K}$ with regularity properties (REF ) and finiteness property (REF ).\",\n \"In particular, for each $\\\\mathbf {P} \\\\in \\\\mathbb {R}_{> 0}^{K}$ the function $\\\\Phi _{\\\\mathbf {P}} \\\\left( \\\\cdot \\\\right) := D_{\\\\varphi }( \\\\cdot , \\\\mathbf {P} )$ (cf.\",\n \"(REF )) is bare-simulation minimizable (BS-minimizable, cf.\",\n \"(REF ))) on any such $\\\\mathbf {\\\\Omega }\\\\subset \\\\mathbb {R}^{K}$ .\",\n \"The proof of Theorem REF will be given in Appendix A.\",\n \"Remark 11 (i) Whenever $int(\\\\mathbf {\\\\Omega }) \\\\ne \\\\emptyset $ , it clearly holds that $\\\\liminf _{n\\\\rightarrow \\\\infty }\\\\frac{1}{n}\\\\log \\\\,\\\\mathbb {\\\\Pi }\\\\left[\\\\xi _{n}^{\\\\mathbf {\\\\widetilde{W}}}\\\\in \\\\mathbf {\\\\Omega } /M_{\\\\mathbf {P}}\\\\right] > 0$ ; see the proof of Theorem REF .\",\n \"Hence, the limit in (REF ) exists and is finite when $\\\\mathbf {\\\\Omega }$ satisfies (REF ).\",\n \"(ii) For some contexts, one can explicitly give the distribution of each of the independent (non-deterministic parts of the) components $\\\\Big (\\\\sum _{i\\\\in I_{k}^{(n)}}\\\\widetilde{W}_{i}\\\\Big )_{k=1,\\\\ldots ,K}$ of the vector $\\\\xi _{n}^{\\\\mathbf {\\\\widetilde{W}}}$ ; this will ease the corresponding concrete simulations.\",\n \"For instance, we shall give those in the Examples REF , REF , REF , REF and REF in Section below.\",\n \"(iii) Let us emphasize that we have assumed $\\\\mathbf {P} \\\\in \\\\mathbb {R}_{> 0}^{K}$ in Theorem REF which excludes $\\\\mathbf {P}$ from having zero components.\",\n \"However, in cases where $\\\\lim _{x\\\\rightarrow \\\\infty }\\\\left|\\\\frac{\\\\varphi \\\\left(x \\\\cdot sgn(q)\\\\right) }{x \\\\cdot sgn(q)}\\\\right|=+\\\\infty $ for $q\\\\ne 0$ then if $p_{k_{0}}=0$ for some $k_{0\\\\text{ }}$ it follows that $q_{k_{0}}=0$ , which proves that $\\\\mathbf {P}\\\\in \\\\mathbb {R}_{>0}^{K}$ imposes no restriction in Theorem 9, since the projection of $\\\\mathbf {P}$ in $\\\\mathbf {\\\\Omega }$ then belongs to the subspace of $\\\\mathbb {R}^{K}$ generated by the non-null components of $\\\\mathbf {P}$ ; such a situation appears e.g.\",\n \"for power divergence generators $\\\\varphi _{\\\\gamma }$ with $\\\\gamma > 2$ .\",\n \"So there is no loss of generality assuming $\\\\mathbf {P}\\\\in \\\\mathbb {R}_{>0}^{K}$ in this case.\",\n \"As examples for the applicability of Theorem REF , one can e.g.\",\n \"combine each of the divergence generators $\\\\varphi $ of Table 1 (except for the 9th row) with any of the optimization problems (REF ), (REF ), (REF ), (REF ), (REF ), (REF ); the needed distributions $\\\\mathbb {\\\\Pi }[\\\\widetilde{W}\\\\in \\\\cdot \\\\,]=\\\\widetilde{\\\\mathbb {}}[\\\\,\\\\cdot \\\\,]$ correspond to the entry in the second last column with the choice $\\\\widetilde{c} \\\\cdot M_{\\\\mathbf {P}}$ instead of $\\\\widetilde{c}$ .\",\n \"By taking $\\\\zeta := - \\\\varphi $ instead, one can solve the corresponding problems (REF ) and (REF ).\",\n \"Returning to the general context, the limit statement (REF ) provides the principle for the approximation of the solution of Problem REF .\",\n \"Indeed, by replacing the left-hand side in (REF ) by its finite counterpart, we deduce for given large $n$ $- \\\\frac{1}{n}\\\\log \\\\mathbb {\\\\Pi } \\\\left[ \\\\xi _{n}^{\\\\mathbf {\\\\widetilde{W}}}\\\\in \\\\mathbf {\\\\Omega }/M_{\\\\mathbf {P}} \\\\right]\\\\approx \\\\inf _{Q\\\\in \\\\mathbf {\\\\Omega } }D_{\\\\varphi }(\\\\mathbf {Q},\\\\mathbf {P});$ it remains to estimate the left-hand side of (REF ).\",\n \"The latter can be performed either by a naive estimator of the frequency of those replications of $\\\\xi _{n,\\\\mathbf {\\\\widetilde{x}}}^{\\\\mathbf {\\\\widetilde{W}}}$ which hit $\\\\Omega /M_{\\\\mathbf {P}}$ , or more efficiently by some improved estimator; this will be discussed in detail in Section below.\",\n \"Remark 12 According to (REF ) of Theorem 9 as well as (REF ), we can principally tackle the (approximative) computation of the minimum value $\\\\inf _{Q\\\\in \\\\mathbf {\\\\Omega }}D_{\\\\varphi }(\\\\mathbf {Q},\\\\mathbf {P}\\\\ ) =\\\\inf _{Q\\\\in \\\\mathbf {\\\\Omega }}\\\\sum _{k=1}^{K}p_{k}\\\\cdot \\\\varphi \\\\left( \\\\frac{q_{k}}{p_{k}}\\\\right)\\\\nonumber $ and in particular of $\\\\inf _{\\\\mathbf {Q} \\\\in \\\\mathbf {\\\\Omega } } \\\\sum _{k=1}^{K}\\\\varphi (q_{k}) =\\\\inf _{\\\\mathbf {Q} \\\\in \\\\mathbf {\\\\Omega } } D_{\\\\varphi }(\\\\mathbf {Q},\\\\mathbf {1})\\\\qquad \\\\textrm {(cf.\",\n \"(\\\\ref {min Pb one}))}\\\\nonumber $ by basically only employing a fast and accurate — pseudo, true, natural, quantum — random number generatorsee e.g.\",\n \"Tucci [366], Teh et al.\",\n \"[358], Aghamohammadi & Crutchfield [6], Herrero-Collantes & Garcia-Escartin [159], Balygin et al.\",\n \"[31], Dang et al.\",\n \"[103], Gong et al.\",\n \"[138], Chandrasekaran et al.\",\n \"[77], Drahi et al.\",\n \"[117], Kollmitzer et al.\",\n \"[194], Liu et al.\",\n \"[228], Fischer & Gauthier [126], Kim et al.\",\n \"[188], Stoller & Campbell [342], provided that the constraint set $\\\\mathbf {\\\\Omega }$ satisfies the mild assumptions (REF ) and (REF ).\",\n \"Notice that (REF ) also covers (e.g.\",\n \"high-dimensional) constraint sets $\\\\mathbf {\\\\Omega }$ which are non-convex and even highly disconnected, and for which other minimization methods (e.g.\",\n \"pure enumeration, gradient or steepest descent methods, etc.\",\n \"a detailed discussion and comparisons are beyond the scope of this paper, given its current length ) may be problematic or intractable.\",\n \"For instance, (REF ) covers kind of “$K-$ dimensional (not necessarily regular) polka dot (leopard skin) pattern type” relaxations $\\\\mathbf {\\\\Omega } := \\\\dot{\\\\bigcup }_{i=1}^{N} \\\\mathcal {U}_{i}(Q_{i}^{dis})$ of finite discrete constraint sets $\\\\mathbf {\\\\Omega }^{dis} := \\\\lbrace Q_{1}^{dis}, \\\\ldots , Q_{N}^{dis}\\\\rbrace $ of high cardinality $N$ (e.g.\",\n \"being exponential or factorial in a large $K$ ), where each $K-$ dimensional vector $Q_{i}^{dis}$ (e.g.\",\n \"having pure integer components only) is surrounded by some small (in particular, non-overlapping/disjoint) neighborhood $\\\\mathcal {U}_{i}(Q_{i}^{dis})$ ; in such a context, e.g.\",\n \"$\\\\inf _{\\\\mathbf {Q} \\\\in \\\\mathbf {\\\\Omega } } \\\\sum _{k=1}^{K}\\\\varphi (q_{k})$ can be regarded as a “BS-tractable” relaxation of the nonlinear discrete (e.g.\",\n \"integer, combinatorial see e.g.\",\n \"Schrijver [324], Bertsimas & Weismantel [45], Chen et al.\",\n \"[83], Onn [279], Korte & Vygen [195], Wolsey [393] for comprehensive books on discrete, integer and combinatorial programming and their vast applications ) optimization program $\\\\inf _{\\\\mathbf {Q} \\\\in \\\\mathbf {\\\\Omega }^{dis} } \\\\sum _{k=1}^{K}\\\\varphi (q_{k})$ .\"\n ],\n [\n \"Minimum distance/risk estimation\\n\",\n \"In statistics of discrete data — and in the adjacent research fields of information theory, artificial intelligence and machine learning — one often encounters the following minimum distance estimation (MDE) problem which is often also named as estimation of the empirical risk: for index $i\\\\in \\\\mathbb {N}$ , let the generation of the $i-$ th (uncertainty-prone) data point be represented by the random variable $X_{i}$ which takes values in the discrete set $\\\\mathcal {Y}:=\\\\left\\\\lbrace d_{1},\\\\cdots ,d_{K}\\\\right\\\\rbrace $ of $K$ distinct values “of any kind”.\",\n \"It is assumed that there exists a probability measure $\\\\mathbb {P}[\\\\cdot \\\\,]$ on $\\\\mathcal {Y}$ which is the a.s. limit of the empirical measures $\\\\mathbb {P}_{n}^{emp}$ defined by the collection of collected $\\\\left( X_{1},..,X_{n}\\\\right) $ as $n$ tends to infinity, in formula $\\\\lim _{n\\\\rightarrow \\\\infty }\\\\mathbb {P}_{n}^{emp}:=\\\\lim _{n\\\\rightarrow \\\\infty }\\\\frac{1}{n}\\\\sum _{i=1}^{n}\\\\delta _{X_{i}}=\\\\mathbb {P} \\\\qquad \\\\text{a.s.}$ where $\\\\delta _{y}$ denotes the one-point distribution (Dirac mass) at point $y$ notice that $\\\\mathbb {P}_{n}^{emp}$ a probability measure on the data space $\\\\mathcal {Y}$ , which is random due to its dependence on the $X_{i}$ ’s.\",\n \"We will assume that none of the entries of $\\\\mathbb {P}$ bears zero mass so that $\\\\mathbb {P}$ is identified with a point in the interior of $\\\\mathbb {S}^{K}$ (see below).\",\n \"The underlying probability space (say, $(\\\\mathfrak {X},\\\\mathcal {A},\\\\mathbb {\\\\Pi })$ ) where the above a.s. convergence holds, pertains to the random generation of the sequence $\\\\left( X_{n}\\\\right)_{n\\\\ge 1}$ , of which we do not need to know but for (REF ).\",\n \"Examples include the i.i.d.\",\n \"case (where the $X_{i}$ ’s are independent and have common distribution $\\\\mathbb {P}$ ), ergodic Markov chains on $\\\\mathcal {Y}$ with stationary distribution $\\\\mathbb {P}$ , more globally autoregressive chains with stationary measure $\\\\mathbb {P}$ , etc.\",\n \"Let us briefly discuss our assumption (REF ) (resp.\",\n \"its vector form (REF ) below) on the limit behavior of the empirical distribution of the observed sample $\\\\mathbf {X}_{n}:=\\\\left(X_{1},..,X_{n}\\\\right)$ as $n$ tends to infinity.\",\n \"In the “basic” statistical context, the sample $\\\\mathbf {X}_{n}$ consists of i.i.d.\",\n \"replications of a generic random variable $X$ with probability distribution ${P}$ .\",\n \"However, our approach captures many other sampling schemes, where the distribution ${P}$ is defined implicitly through (REF ) for which we aim at some estimate of $\\\\Phi _{{P}}\\\\left(\\\\mathbb {\\\\Omega }\\\\right)$ of a family $\\\\mathbb {\\\\Omega }$ of probability distributions on $\\\\mathcal {Y}$ .\",\n \"Sometimes the sequence of samples $\\\\left( \\\\mathbf {X}_{n}\\\\right) _{n\\\\ge 1}$ may stem from a triangular array so that $\\\\mathbf {X}_{n}=\\\\left(X_{1,n},..,X_{k_{n},n}\\\\right) $ with $k_{n}\\\\rightarrow \\\\infty $ and statement (REF ) is substituted by $\\\\lim _{n\\\\rightarrow \\\\infty }\\\\frac{1}{k_{n}}\\\\sum _{i=1}^{k_{n}}\\\\delta _{X_{i,n}} ={P} \\\\text{ \\\\ a.s.}$ which does not alter the results of this paper by any means.\",\n \"given a model $\\\\mathbb {\\\\Omega }$ , i.e.\",\n \"a family $\\\\mathbb {\\\\Omega }$ of probability distributions on $\\\\mathcal {Y}$ each of which serves as a potential description of the underlying (unknown) data-generating mechanism $\\\\mathbb {P}$ , one would like to find $\\\\Phi _{\\\\mathbb {P}}(\\\\mathbb {\\\\Omega }) := \\\\inf _{\\\\mathbb {Q}\\\\in \\\\mathbb {\\\\Omega }} D_{\\\\varphi }( \\\\mathbb {Q}, \\\\mathbb {P} )$ which quantifies the adequacy of the model $\\\\mathbb {\\\\Omega }$ for modeling $\\\\mathbb {P}$ , via the minimal distance/dissimilarity of $\\\\mathbb {\\\\Omega }$ to $\\\\mathbb {P}$ ; a lower $\\\\Phi _{\\\\mathbb {P}}-$ value means a better adequacy (in the sense of a lower departure between the model and the truth, cf.\",\n \"Lindsay [222], Lindsay et al.\",\n \"[223], Markatou & Sofikitou [248], Markatou & Chen [249]).\",\n \"Hence, especially in the context of model selection within complex big-data contexts, for the search of appropriate models $\\\\mathbb {\\\\Omega }$ and model elements/members therein, the (fast and efficient) computation of $\\\\Phi _{\\\\mathbb {P}}(\\\\mathbb {\\\\Omega })$ constitutes a decisive first step, since if the latter is “too large” (respectively “much larger than” $\\\\Phi _{\\\\mathbb {P}}(\\\\overline{\\\\mathbb {\\\\Omega }})$ for some competing model $\\\\overline{\\\\mathbb {\\\\Omega }}$ ), then the model $\\\\mathbb {\\\\Omega }$ is “not adequate enough” (respectively “much less adequate than” $\\\\overline{\\\\mathbb {\\\\Omega }}$ ); in such a situation, the effort of computing the (not necessarily unique) best model element/member $\\\\arg \\\\inf _{\\\\mathbb {Q}\\\\in \\\\mathbb {\\\\Omega }} D_{\\\\varphi }( \\\\mathbb {Q}, \\\\mathbb {P} )$ within the model $\\\\mathbb {\\\\Omega }$ is “not very useful” and is thus a “waste of computational time”.\",\n \"Because of such considerations, we concentrate ourselves to finding the infimum (REF ) rather than finding the corresponding minimizer(s).\",\n \"Variants of (REF ) are of interest, too.\",\n \"Since $int(\\\\mathbb {\\\\Omega })$ is supposed to be a non-empty set in the space of probability distribution on $\\\\mathcal {Y}$ , the present procedure is fitted for semi-parametric models $\\\\mathbb {\\\\Omega }$ , e.g.\",\n \"such as defined through moment conditions (as extensions of the Empirical Likelihood paradigm, see e.g.\",\n \"Broniatowski & Keziou [62]), or through L-moment conditions (i.e.\",\n \"moment conditions pertaining to quantile measures, see Broniatowski & Decurninge [59]), or even more involved non-parametric models where the geometry of $\\\\mathbb {\\\\Omega }$ does not allow for ad-hoc procedures.\",\n \"The measurement or the estimation of $\\\\Phi _{\\\\mathbb {P}}(\\\\mathbb {\\\\Omega })$ is a tool for the choice of pertinent putative models $\\\\mathbb {\\\\Omega }$ among a class of specifications.\",\n \"The case when $\\\\Phi _{\\\\mathbb {P}}(\\\\mathbb {\\\\Omega })>0$ is interesting in its own, since it is quite common in engineering modelling to argue in favor of misspecified models (or (non-void) neighborhoods of such models for sake of robustness issues), due to quest for conservatism; the choice between them is a widely open field e.g.\",\n \"in the practice of reliability.\",\n \"This also opens the question of the choice of the divergence generator $\\\\varphi $ ; although this will not be discussed in this paper, as a motivating running example the reader may keep in mind the generator $\\\\varphi _{2}(x):=(x-1)^{2}/2$ which induces the divergence $D_{\\\\varphi _{2}}(\\\\mathbb {Q},\\\\mathbb {P})$ (see (REF ) below for details) which quantifies the expected square relative error when substituting the true distribution $\\\\mathbb {P}$ by the model $\\\\mathbb {Q}$ .\",\n \"As examples of sets $\\\\mathbb {\\\\Omega }$ of probability distributions on $\\\\mathcal {Y}$ which obey (through their $K-$ vector of corresponding probability masses/frequencies) the global assumptions (REF ), one can consider semi-parametric models defined by moment conditions or defined through L-moment constraints (hence on the quantile functions), as well as more involved ones, for which no closed form of the divergence with respect to any probability distribution is available.\",\n \"In the context of model selection, the choice of $\\\\mathbb {\\\\Omega }$ may be dictated by various considerations, and misspecification may be assumed as a requisite, for example for conservatism in reliability design.\",\n \"An estimate of $\\\\Phi _{\\\\mathbb {P}}(\\\\mathbb {\\\\Omega })$ can be used as a statistics for some test of fit, and indeed the likelihood ratio test adapted to some semi-parametric models has been generalized to the divergence setting (see Broniatowski & Keziou [62]).\",\n \"The statement of the limit distributions of our estimate, under the model and under misspecification, is postponed to future work.\",\n \"In the following, we compute/approximate (REF ) — and some variants thereof — by our bare simulation (BS) method, by “mimicking” the deterministic minimization problem (REF ) respectively (REF ).\",\n \"Let us first remark that, as usual, each probability distribution (probability measure) $\\\\mathbb {P}$ on $\\\\mathcal {Y}=\\\\left\\\\lbrace d_{1},\\\\ldots ,d_{K}\\\\right\\\\rbrace $ can be uniquely identified with the (row) vector ${P} := (p_{1}, \\\\ldots , p_{K}) \\\\in \\\\mathbb {S}^{K}$ of the corresponding probability masses (frequencies) $p_{k} = \\\\mathbb {P}[\\\\lbrace d_{k} \\\\rbrace ]$ via $\\\\mathbb {P}[A] = \\\\sum _{k=1}^{K} p_{k} \\\\cdot {1}_A(d_{k}) $ for each $A \\\\subset \\\\mathcal {Y}$ , where ${1}_A(\\\\cdot )$ denotes the indicator function on the set $A$ .\",\n \"In particular, the probability distribution $\\\\mathbb {P}$ in (MDE1) can be identified with $(p_{1}, \\\\ldots , p_{K})$ in terms of $p_{k} = \\\\mathbb {P}[\\\\lbrace d_{k} \\\\rbrace ]$ (which in the i.i.d.\",\n \"case turns into $p_{k} = \\\\mathbb {\\\\Pi }[X_{1} = d_{k}]$ ).\",\n \"Along this line, the family $\\\\mathbb {\\\\Omega }$ of probability distributions in (MDE2) can be identified with a subset $\\\\Omega $$\\\\Omega \\\\subset \\\\mathbb {S}^{K}$ of probability vectors (viz.\",\n \"of vectors of probability masses).\",\n \"Analogously, each finite nonnegative measure $Q$ on $\\\\mathcal {Y}$ can be uniquely identified with a vector $\\\\mathbf {Q} := (q_{1}, \\\\ldots , q_{K}) \\\\in \\\\mathbb {R}_{\\\\ge 0}^{K}$ , and each finite signed measure $Q$ with a vector $\\\\mathbf {Q} := (q_{1}, \\\\ldots , q_{K}) \\\\in \\\\mathbb {R}^{K}$ .\",\n \"The corresponding divergences between distributions/measures are then, as usual, defined through the divergences between their respective masses/frequencies: $D_{\\\\varphi }(Q,\\\\mathbb {P}) : = D_{\\\\varphi }(\\\\mathbf {Q},{P}).$ In particular, $\\\\mathbb {P}_{n}^{emp}$ can be identified with the vector ${P}_{n}^{emp} := (p_{n,1}^{emp}, \\\\ldots , p_{n,K}^{emp})$ where $p_{n,k}^{emp} \\\\ := \\\\ \\\\frac{1}{n} \\\\cdot n_{k} \\\\ := \\\\ \\\\frac{1}{n} \\\\cdot card(\\\\bigl \\\\lbrace i \\\\in \\\\lbrace 1, \\\\ldots , n\\\\rbrace : \\\\ X_{i} = d_{k} \\\\bigr \\\\rbrace )\\\\ =: \\\\ \\\\frac{1}{n} \\\\cdot card(I_{k}^{(n)}) , \\\\quad k \\\\in \\\\lbrace 1, \\\\ldots , K\\\\rbrace ,$ and accordingly the required limit behaviour (REF ) is equivalent to the vector-convergence $\\\\lim _{n\\\\rightarrow \\\\infty } \\\\Big ( \\\\frac{n_{1}}{n}, \\\\ldots , \\\\frac{n_{K}}{n} \\\\Big ) = (p_{1}, \\\\ldots , p_{K})\\\\qquad \\\\textrm {a.s.}$ Notice that, in contrast to the case handled in the above Subsection REF , the sets $I_{k}^{(n)}$ of indexes introduced in (REF ) and their numbers $n_{k} = card(I_{k}^{(n)})$ of elements are now random (due to their dependence on the $X_{i}$ ’s) and $M_{{P}_{n}^{emp}}=1$ .\",\n \"In a batch procedure, when $D_{\\\\varphi }(\\\\textrm {$$\\\\hspace{-6.544pt}$$},{P}_{n}^{emp})$ is estimated once the sample $\\\\left(X_{1},..,X_{n}\\\\right)$ is observed, we may reorder this sample by putting the $n_{1}$ sample points $X_{i}$ which are equal to $d_{1}$ in the first places, and so on; accordingly one ends up with index sets $I_{k}^{(n)}$ as defined in Section REF .\",\n \"When the online acquisition of the data $X_{i}$ ’s is required, then we usually do not reorder the sample, and the $I_{k}^{(n)}$ 's do not consist in consecutive indexes, which does not make any change with respect to the resulting construction nor to the estimator.\",\n \"The above considerations open the gate to our desired “mimicking” of (REF ) and (REF ) to achieve (REF ) (and some variants thereof) by our bare simulation (BS) method.\",\n \"To proceed, we employ a family of random variables $(W_{i})_{i \\\\in \\\\mathbb {N}}$ of independent and identically distributed $\\\\mathbb {R}-$ valued random variables with probability distribution $\\\\mathbb {}[ \\\\cdot \\\\, ] := \\\\mathbb {\\\\Pi }[W_{1} \\\\in \\\\cdot \\\\, ]$ (being connected with the divergence generator $\\\\varphi \\\\in \\\\Upsilon (]a,b[)$ via the representability (REF )), such that $(W_{i})_{i \\\\in \\\\mathbb {N}}$ is independent of $(X_{i})_{i \\\\in \\\\mathbb {N}}$ on the common underlying probability space $(\\\\mathfrak {X},\\\\mathcal {A},\\\\mathbb {\\\\Pi })$.\",\n \"As a next step, notice that the “natural candidate” $\\\\xi _{n,\\\\mathbf {X}}^{\\\\mathbf {W}}:= \\\\frac{1}{n} \\\\cdot \\\\sum _{k=1}^{K}\\\\left(\\\\sum _{i \\\\in I_{k}^{(n)}}W_{i}\\\\right) \\\\cdot \\\\delta _{d_{k}}\\\\ = \\\\ \\\\frac{1}{n}\\\\sum _{i=1}^{n} W_{i} \\\\cdot \\\\delta _{X_{i}}$ is not a probability measure since its total mass is not 1 in general, since in terms of its equivalent vector version $\\\\xi _{n,\\\\mathbf {X}}^{\\\\mathbf {W}} := \\\\Big (\\\\frac{1}{n}\\\\sum _{i\\\\in I_{1}^{(n)}} W_{i}, \\\\ldots , \\\\frac{1}{n} \\\\sum _{i\\\\in I_{K}^{(n)}} W_{i} \\\\Big )$ the sum $\\\\sum _{k=1}^{K} \\\\frac{1}{n} \\\\sum _{i\\\\in I_{k}^{(n)}} W_{i} =\\\\frac{1}{n} \\\\sum _{j=1}^{n} W_{i}$ of the $K$ vector components of (REF ) is typically not equal to 1; this implies that no limit result of the form (REF ) with finite limit can hold, since $\\\\xi _{n,\\\\mathbf {X}}^{\\\\mathbf {W}}$ takes values in $\\\\mathbb {R}^{K}$ and $\\\\textrm {$$\\\\hspace{-6.544pt}$$}$ is a subset in the probability simplex $\\\\mathbb {S}^{K}$ which has void interior in $\\\\mathbb {R}^{K}$ causing a violation of condition (REF ) (cf.\",\n \"Remark REF (c)); moreover, depending on the concrete form of the generator $\\\\varphi $ , the corresponding weights may take negative values.\",\n \"Therefore, we need some “rescaling”.\",\n \"Indeed, let us introduce the normalized weighted empirical measure $\\\\xi _{n,\\\\mathbf {X}}^{w\\\\mathbf {W}} &:=&{\\\\left\\\\lbrace \\\\begin{array}{ll}\\\\frac{1}{\\\\sum _{k=1}^{K}\\\\sum _{i \\\\in I_{k}^{(n)}}W_{i}} \\\\cdot \\\\sum _{k=1}^{K}\\\\left(\\\\sum _{i \\\\in I_{k}^{(n)}}W_{i}\\\\right) \\\\cdot \\\\delta _{d_{k}}= \\\\sum _{i=1}^{n} \\\\frac{W_{i}}{\\\\sum _{j=1}^{n} W_{j}} \\\\cdot \\\\delta _{X_{i}},\\\\qquad \\\\textrm {if } \\\\sum _{j=1}^{n} W_{j} \\\\ne 0, \\\\\\\\\\\\infty \\\\cdot \\\\sum _{k=1}^{K} \\\\delta _{d_{k}} =: \\\\underline{\\\\infty }, \\\\hspace{216.2411pt}\\\\textrm {if } \\\\sum _{j=1}^{n} W_{j} = 0,\\\\end{array}\\\\right.\",\n \"}$ which will substitute $\\\\xi _{n,\\\\mathbf {X}}^{\\\\mathbf {W}}$ and which may belong to $\\\\textrm {$$\\\\hspace{-6.544pt}$$}$ with positive probability.\",\n \"The equivalent vector version of $\\\\xi _{n,\\\\mathbf {X}}^{w\\\\mathbf {W}}$ is given by $\\\\xi _{n,\\\\mathbf {X}}^{w\\\\mathbf {W}} &:=&{\\\\left\\\\lbrace \\\\begin{array}{ll}\\\\left(\\\\frac{\\\\sum _{i \\\\in I_{1}^{(n)}}W_{i}}{\\\\sum _{k=1}^{K}\\\\sum _{i \\\\in I_{k}^{(n)}}W_{i}},\\\\ldots , \\\\frac{\\\\sum _{i \\\\in I_{K}^{(n)}}W_{i}}{\\\\sum _{k=1}^{K}\\\\sum _{i \\\\in I_{k}^{(n)}}W_{i}} \\\\right) ,\\\\qquad \\\\textrm {if } \\\\sum _{j=1}^{n} W_{j} \\\\ne 0, \\\\\\\\\\\\ (\\\\infty , \\\\ldots , \\\\infty ) =: \\\\infty , \\\\hspace{113.81102pt} \\\\textrm {if } \\\\sum _{j=1}^{n} W_{j} = 0,\\\\end{array}\\\\right.\",\n \"}$ a point in the linear subset of $\\\\mathbb {R}^{K}$ spanned by $\\\\mathbb {S}^{K}$ at infinity.\",\n \"Remark 13 (i) (Concerning e.g.\",\n \"computer-program command availability) In case of $\\\\sum _{j=1}^{n}W_{j}=0$ , in (REF ) we may equivalently assign to $\\\\xi _{n,\\\\mathbf {X}}^{w\\\\mathbf {W}}$ instead of $\\\\underline{\\\\infty }$ any measure (e.g.\",\n \"probability distribution) which does not belong to $\\\\mathbb {\\\\Omega }$ , respectively, in (REF ) we may equivalently choose for $\\\\xi _{n,\\\\mathbf {X}}^{w\\\\mathbf {W}}$ any vector outside of $\\\\Omega $$\\\\Omega $ instead of $\\\\infty $ .\",\n \"(ii) By construction, in case of $\\\\sum _{j=1}^{n} W_{j} \\\\ne 0$ , the sum of the random $K$ vector components of (REF ) is now automatically equal to 1, but — as (depending on $\\\\varphi $ ) the $W_{i}$ ’s may take both positive and negative values — these random components may be negative (resp.\",\n \"nonnegative) with probability strictly greater (resp.\",\n \"smaller) than zero (resp.\",\n \"one); in the framework of (REF ) this means that $\\\\xi _{n,\\\\mathbf {X}}^{w\\\\mathbf {W}}$ is in general a random signed measure with total mass 1, in case of $\\\\sum _{j=1}^{n} W_{j} \\\\ne 0$ .\",\n \"However, $\\\\mathbb {\\\\Pi } [\\\\xi _{n,\\\\mathbf {X}}^{w\\\\mathbf {W}}\\\\in \\\\mathbb {S}_{>0}^{K}]>0$ since all the (identically distributed) random variables $W_{i}$ have expectation 1 (as a consequence of the assumed representability (REF )); in case of $\\\\mathbb {\\\\Pi }[W_{1}>0]=1$ one has even $\\\\mathbb {\\\\Pi }[\\\\xi _{n,\\\\mathbf {X}}^{w\\\\mathbf {W}}\\\\in \\\\mathbb {S}_{>0}^{K}]=1$ .\",\n \"In the particular context of Example REF (c), one gets $\\\\mathbb {\\\\Pi }[\\\\xi _{n,\\\\mathbf {X}}^{w\\\\mathbf {W}}\\\\in \\\\mathbb {S}_{>0}^{K}]=\\\\left( \\\\mathbb {\\\\Pi }[ W_{1}>0]\\\\right)^{n}=\\\\left( \\\\int _{0}^{\\\\infty }\\\\sqrt{\\\\frac{\\\\widetilde{c}}{2\\\\pi }}\\\\cdot \\\\exp (-\\\\frac{\\\\widetilde{c}\\\\cdot (u-1)^{2}}{2})du\\\\right) ^{n}\\\\in ]0,1[$ .\",\n \"Summing up things, the probability $\\\\mathbb {\\\\Pi }[\\\\xi _{n,\\\\mathbf {X}}^{w\\\\mathbf {W}}\\\\in \\\\Omega $$\\\\Omega ]$ is strictly positive and finite at least for large n, whenever $\\\\Phi _{{P}}(\\\\Omega $$\\\\Omega )= \\\\inf _{{Q}\\\\in \\\\textrm {\\\\Omega \\\\hspace{-5.40608pt}\\\\Omega } }\\\\ D_{\\\\varphi }({Q},{P})$ is finite.\",\n \"(iii) By generalizing the terminology of e.g.\",\n \"Vajda [372], through the right-hand side of (REF ) one can interpret (for $\\\\sum _{j=1}^{n} W_{j} \\\\ne 0$ ) the normalized weighted empirical measure $\\\\xi _{n,\\\\mathbf {X}}^{w\\\\mathbf {W}}$ as response of an output neuron in a random perceptron consisting of random inputs $\\\\mathbf {X}$ , a layer with $n$ units having one-point-distribution-valued responses $\\\\delta _{X_{1}}, \\\\ldots , \\\\delta _{X_{n}}$ , and independent random synaptic weights $\\\\left(\\\\frac{W_{1}}{\\\\sum _{j=1}^{n} W_{j}}, \\\\ldots , \\\\frac{W_{n}}{\\\\sum _{j=1}^{n} W_{j}}\\\\right)$ .\",\n \"With the above-mentioned ingredients, we are now in the position to tackle a variant of the distance minimization problem (REF ), by our bare simulation method through “mimicking” the deterministic minimization problem (REF ) respectively (REF ).\",\n \"For this, we also employ the conditional distributions $\\\\mathbb {\\\\Pi }_{n}[\\\\, \\\\cdot \\\\, ] := \\\\mathbb {\\\\Pi }_{X_{1}^{n}}[\\\\, \\\\cdot \\\\, ] := \\\\mathbb {\\\\Pi }[ \\\\, \\\\cdot \\\\, | \\\\,X_{1}, \\\\ldots , X_{n} ]$ and obtain the following Theorem 14 Suppose that $(X_{i})_{i\\\\in \\\\mathbb {N}}$ is a sequence of random variables with values in $\\\\mathcal {Y}:=\\\\left\\\\lbrace d_{1},\\\\cdots ,d_{K}\\\\right\\\\rbrace $ such that (REF ) holds for some probability measure $\\\\mathbb {P}[\\\\cdot \\\\,]$ on $\\\\mathcal {Y}$ having no zero-mass frequencies (or equivalently, (REF ) holds for some probability vector ${P} \\\\in \\\\mathbb {S}_{> 0}^{K}$ ).\",\n \"Moreover, let $(W_{i})_{i \\\\in \\\\mathbb {N}}$ be a family of independent and identically distributed $\\\\mathbb {R}-$ valued random variables with probability distribution $\\\\mathbb {}[ \\\\cdot \\\\, ] := \\\\mathbb {\\\\Pi }[W_{1} \\\\in \\\\cdot \\\\, ]$ being connected with the divergence generator $\\\\varphi \\\\in \\\\Upsilon (]a,b[)$ via the representability (REF ), such that $(W_{i})_{i \\\\in \\\\mathbb {N}}$ is independent of $(X_{i})_{i \\\\in \\\\mathbb {N}}$ .\",\n \"Then there holds $-\\\\lim _{n\\\\rightarrow \\\\infty }\\\\frac{1}{n}\\\\log \\\\,\\\\mathbb {\\\\Pi }_{X_{1}^{n}}\\\\left[\\\\xi _{n,\\\\mathbf {X}}^{w\\\\mathbf {W}}\\\\in \\\\mathbb {\\\\Omega }\\\\right]& =\\\\inf _{\\\\mathbb {Q}\\\\in \\\\mathbb {\\\\Omega } }\\\\ \\\\inf _{m\\\\ne 0}D_{\\\\varphi }(m\\\\cdot \\\\mathbb {Q},\\\\mathbb {P})\\\\, \\\\\\\\& =\\\\inf _{m\\\\ne 0}\\\\ \\\\inf _{\\\\mathbb {Q}\\\\in \\\\mathbb {\\\\Omega } }D_{\\\\varphi }(m\\\\cdot \\\\mathbb {Q},\\\\mathbb {P})\\\\\\\\& =\\\\inf _{m\\\\ne 0}\\\\ \\\\inf _{{Q}\\\\in \\\\textrm {\\\\Omega \\\\hspace{-5.40608pt}\\\\Omega } }D_{\\\\varphi }(m\\\\cdot {Q},{P})\\\\\\\\& = \\\\inf _{{Q}\\\\in \\\\textrm {\\\\Omega \\\\hspace{-5.40608pt}\\\\Omega } }\\\\ \\\\inf _{m\\\\ne 0}D_{\\\\varphi }(m\\\\cdot {Q},{P})= -\\\\lim _{n\\\\rightarrow \\\\infty }\\\\frac{1}{n}\\\\log \\\\,\\\\mathbb {\\\\Pi }_{X_{1}^{n}}\\\\left[\\\\xi _{n,\\\\mathbf {X}}^{w\\\\mathbf {W}}\\\\in \\\\textrm {\\\\right.\\\\Omega \\\\hspace{-6.544pt}\\\\Omega }$ for all sets $\\\\mathbb {\\\\Omega }$ of probability distributions such that their equivalent probability-vector form $\\\\Omega $$\\\\Omega $ satisfies the regularity properties (REF ) in the relative topology and the finiteness property (REF ); notice that for the equality () we have used the “divergence link” (REF ).\",\n \"In particular, for each ${P} \\\\in \\\\mathbb {S}_{>0}^{K}$ (respectively, its equivalent probability-distribution $\\\\mathbb {P}$ ) the function ${Q} \\\\mapsto \\\\inf _{m\\\\ne 0}D_{\\\\varphi }(m\\\\cdot {Q},{P}) $ (respectively, the function $\\\\mathbb {Q} \\\\mapsto \\\\inf _{m\\\\ne 0}D_{\\\\varphi }(m\\\\cdot \\\\mathbb {Q},\\\\mathbb {P})$ ) is BS-minimizable (cf.\",\n \"(REF )) on all sets $\\\\textrm {$$\\\\hspace{-6.544pt}$$} \\\\subset \\\\mathbb {S}^{K}$ satisfying (7) in the relative topology and (9) (respectively, on their probability-distribution-equivalent $\\\\mathbb {\\\\Omega }$ ).\",\n \"The proof of Theorem REF will be given in Appendix B. Analogously to Remark REF (iii), let us emphasize that we have assumed ${P} \\\\in \\\\mathbb {S}_{> 0}^{K}$ in Theorem REF .\",\n \"Henceforth, for sets $\\\\textrm {$$\\\\hspace{-6.544pt}$$} \\\\subset \\\\mathbb {S}^{K}$ of probability vectors we deal with (REF ) only in the relative topology; thus, the latter will be unmentioned for the sake of brevity.\",\n \"Remark REF (a),(b),(c),(e) applies accordingly.\",\n \"Remark 15 (i) In strong contrast to Theorem REF , the above result does not provide a direct tool for the solution of Problem (REF ) since the limit in (REF ) bears no direct information on the minimum divergence $D_{\\\\varphi }\\\\left( \\\\mathbb {\\\\Omega },\\\\mathbb {P} \\\\right) :=\\\\inf _{\\\\mathbb {Q}\\\\in \\\\mathbb {\\\\Omega }} D_{\\\\varphi }( \\\\mathbb {Q}, \\\\mathbb {P})$ ; the link between the corresponding quantities can be emphasized and exploited e.g.\",\n \"in the case of power type divergences, which leads to explicit minimization procedures as shown in the Subsection REF below.\",\n \"For general divergences, Theorem REF allows for the estimation of upper and lower bounds of $D_{\\\\varphi }\\\\left( \\\\mathbb {\\\\Omega },\\\\mathbb {P} \\\\right)$ , as developed in the Subsection REF below.\",\n \"(ii) Notice that $\\\\breve{D}_{\\\\varphi }(\\\\mathbb {Q},\\\\mathbb {P}):=\\\\inf _{m\\\\ne 0} D_{\\\\varphi }(m\\\\cdot \\\\mathbb {Q},\\\\mathbb {P})$ satisfies the axioms of a divergence, that is, $\\\\breve{D}_{\\\\varphi }(\\\\mathbb {Q},\\\\mathbb {P}) \\\\ge 0$ , as well as $\\\\breve{D}_{\\\\varphi }(\\\\mathbb {Q},\\\\mathbb {P}) = 0$ if and only if $\\\\mathbb {Q} = \\\\mathbb {P}$ (reflexivity).\",\n \"Hence, in Theorem REF we are still within our framework of bare simulation of a divergence minimum w.r.t.\",\n \"its first component (however, notice the difference to (i)).\",\n \"(iii) Viewed from a “reverse” angle, Theorem REF gives a crude approximation for the probability for $\\\\xi _{n,\\\\mathbf {X}}^{w\\\\mathbf {W}}$ to belong to $\\\\mathbb {\\\\Omega }$ , conditionally upon $\\\\mathbf {X} = (X_{1}, \\\\ldots , X_{n})$ .\",\n \"(iv) In the same spirit as Remark REF (ii), for some contexts one can explicitly give the distribution of each of the independent components $\\\\Big (\\\\sum _{i\\\\in I_{k}^{(n)}}W_{i}\\\\Big )_{k=1,\\\\ldots ,K}$ of the vector $\\\\xi _{n}^{w\\\\mathbf {W}}$ ; this will ease the corresponding concrete simulations in a batch procedure.\",\n \"For instance, we shall give those in the Examples REF , REF , REF , REF and REF in the Section below.\",\n \"(v) Consider the special “degenerate” case where all the data observations are certain and thus $(X_{i})_{i \\\\in \\\\mathbb {N}}$ is nothing but a purely deterministic sequence, say $(\\\\widetilde{x}_{i})_{i\\\\in \\\\mathbb {N}}$ , of elements $\\\\widetilde{x}_{i}$ from the arbitrary set $\\\\mathcal {Y}:=\\\\left\\\\lbrace d_{1},\\\\ldots ,d_{K}\\\\right\\\\rbrace $ of $K$ distinct values “of any kind” (e.g., $\\\\mathcal {Y}$ may consist of $K$ distinct numbers); then the corresponding empirical distribution $\\\\mathbb {P}_{n}^{emp}$ can be identified with the vector ${P}_{n}^{emp} := (p_{n,1}^{emp}, \\\\ldots , p_{n,K}^{emp})$ where $p_{n,k}^{emp} \\\\ := \\\\ \\\\frac{1}{n} \\\\cdot n_{k} \\\\ := \\\\ \\\\frac{1}{n} \\\\cdot card(\\\\bigl \\\\lbrace i \\\\in \\\\lbrace 1, \\\\ldots , n\\\\rbrace : \\\\ \\\\widetilde{x}_{i} = d_{k} \\\\bigr \\\\rbrace )\\\\ =: \\\\ \\\\frac{1}{n} \\\\cdot card(I_{k}^{(n)}) , \\\\quad k \\\\in \\\\lbrace 1, \\\\ldots , K\\\\rbrace ,$ and accordingly the required limit behaviour (REF ) is equivalent to the vector-convergence $\\\\lim _{n\\\\rightarrow \\\\infty } \\\\Big ( \\\\frac{n_{1}}{n}, \\\\ldots , \\\\frac{n_{K}}{n} \\\\Big ) = (p_{1}, \\\\ldots , p_{K})\\\\quad \\\\textrm {for some p_{1} >0, \\\\ldots , p_{K} >0 such that \\\\sum _{k=1}^{K} p_{k} = 1.\",\n \"}$ Correspondingly, with the notations ${P} := (p_{1}, \\\\ldots , p_{K})$ and $\\\\mathbf {\\\\widetilde{x}} := (\\\\widetilde{x}_{1}, \\\\ldots , \\\\widetilde{x}_{n})$ , the vector-form part of the assertion (REF ) of Theorem REF becomes $-\\\\lim _{n\\\\rightarrow \\\\infty }\\\\frac{1}{n}\\\\log \\\\,\\\\mathbb {\\\\Pi }\\\\left[\\\\xi _{n,\\\\mathbf {\\\\widetilde{x}}}^{w\\\\mathbf {W}}\\\\in \\\\textrm {\\\\right.\\\\Omega \\\\hspace{-6.544pt}\\\\Omega }\\\\ = \\\\ \\\\inf _{{Q}\\\\in \\\\textrm {\\\\Omega \\\\hspace{-5.40608pt}\\\\Omega } }\\\\ \\\\inf _{m\\\\ne 0}D_{\\\\varphi }(m\\\\cdot {Q},{P})\\\\ = \\\\ \\\\inf _{m\\\\ne 0}\\\\ \\\\inf _{{Q}\\\\in \\\\textrm {\\\\Omega \\\\hspace{-5.40608pt}\\\\Omega } }D_{\\\\varphi }(m\\\\cdot {Q},{P})$ for all subsets $\\\\Omega $$\\\\Omega \\\\subset \\\\mathbb {S}^{K}$ satisfying the regularity properties (REF ) and the finiteness property (REF ); notice that the conditional probability $\\\\mathbb {\\\\Pi }_{X_{1}^{n}}[\\\\, \\\\cdot \\\\, ]$ has degenerated to the ordinary probability $\\\\mathbb {\\\\Pi }[\\\\, \\\\cdot \\\\, ]$ .\",\n \"(vi) In a similar fashion to the proof of (the special degenerate case (v) of) Theorem REF , one can show $-\\\\lim _{n\\\\rightarrow \\\\infty }\\\\frac{1}{n}\\\\log \\\\,\\\\mathbb {\\\\Pi }\\\\left[\\\\xi _{n}^{w\\\\mathbf {W}}\\\\in \\\\textrm {\\\\right.\\\\Omega \\\\hspace{-6.544pt}\\\\Omega }\\\\ = \\\\ \\\\inf _{{Q}\\\\in \\\\textrm {\\\\Omega \\\\hspace{-5.40608pt}\\\\Omega } }\\\\ \\\\inf _{m\\\\ne 0}D_{\\\\varphi }(m\\\\cdot {Q},{P})\\\\ = \\\\ \\\\inf _{m\\\\ne 0}\\\\ \\\\inf _{{Q}\\\\in \\\\textrm {\\\\Omega \\\\hspace{-5.40608pt}\\\\Omega } }D_{\\\\varphi }(m\\\\cdot {Q},{P})$ for all subsets $\\\\Omega $$\\\\Omega \\\\subset \\\\mathbb {S}^{K}$ with regularity properties (REF ) and the finiteness property (REF ), where $\\\\xi _{n}^{w\\\\mathbf {W}} &:=&{\\\\left\\\\lbrace \\\\begin{array}{ll}\\\\left(\\\\frac{\\\\sum _{i \\\\in I_{1}^{(n)}}W_{i}}{\\\\sum _{k=1}^{K}\\\\sum _{i \\\\in I_{k}^{(n)}}W_{i}},\\\\ldots , \\\\frac{\\\\sum _{i \\\\in I_{K}^{(n)}}W_{i}}{\\\\sum _{k=1}^{K}\\\\sum _{i \\\\in I_{k}^{(n)}}W_{i}} \\\\right)= \\\\frac{n \\\\cdot \\\\xi _{n}^{\\\\mathbf {W}}}{\\\\sum _{i=1}^{n}W_{i}},\\\\qquad \\\\textrm {if } \\\\sum _{j=1}^{n} W_{j} \\\\ne 0, \\\\\\\\\\\\ (\\\\infty , \\\\ldots , \\\\infty ) =: \\\\infty , \\\\hspace{159.3356pt} \\\\textrm {if } \\\\sum _{j=1}^{n} W_{j} = 0,\\\\end{array}\\\\right.\",\n \"}$ with $I_{1}^{(n)}:=\\\\left\\\\lbrace 1,\\\\ldots ,n_{1}\\\\right\\\\rbrace $ , $I_{2}^{(n)}:=\\\\left\\\\lbrace n_{1}+1,\\\\ldots ,n_{1}+n_{2}\\\\right\\\\rbrace $ , ..., $I_{K}^{(n)} := \\\\lbrace \\\\sum _{k=1}^{K-1} n_{k} + 1, \\\\ldots , n \\\\rbrace $ and $n_{k}:=\\\\lfloor n \\\\cdot p_{k}\\\\rfloor $ ($k \\\\in \\\\lbrace 1,\\\\ldots ,K\\\\rbrace $ ) for some pregiven known probability vector ${P} := (p_{1}, \\\\ldots , p_{K})$ .\",\n \"Recall the definition of $\\\\xi _{n}^{\\\\mathbf {W}}$ in (REF ) (with $\\\\mathbf {W}$ instead of $\\\\mathbf {\\\\widetilde{W}}$ ).\",\n \"The limit behaviour (REF ) contrasts to the one of Theorem REF , where $-\\\\lim _{n\\\\rightarrow \\\\infty }\\\\frac{1}{n}\\\\log \\\\,\\\\mathbb {\\\\Pi }\\\\left[\\\\xi _{n}^{\\\\mathbf {\\\\widetilde{W}}}\\\\in \\\\Omega /M_{\\\\mathbf {P}}\\\\right]=\\\\inf _{Q\\\\in \\\\mathbf {\\\\Omega }}D_{\\\\varphi }(\\\\mathbf {Q},\\\\mathbf {P})\\\\hspace{113.81102pt} \\\\text{(cf.\",\n \"(\\\\ref {LDP Minimization}))}\\\\nonumber $ for any $\\\\mathbf {\\\\Omega }\\\\subset \\\\mathbb {R}^{K}$ with regularity properties (REF ) and the finiteness property (REF ); recall that $(\\\\widetilde{W}_{i})_{i \\\\in \\\\mathbb {N}}$ are i.i.d.\",\n \"random variables with probability distribution $\\\\widetilde{\\\\mathbb {}}$ (being connected with the divergence generator $\\\\widetilde{\\\\varphi } := M_{\\\\mathbf {P}} \\\\cdot \\\\varphi $ via the representability (REF )), whereas $(W_{i})_{i \\\\in \\\\mathbb {N}}$ are i.i.d.\",\n \"random variables with probability distribution $\\\\mathbb {}$ (being connected with the divergence generator $\\\\varphi $ via the representability (REF )).\",\n \"Indeed, the construction leading to Theorem REF does not hold any longer when $\\\\mathbf {\\\\Omega } \\\\subset \\\\mathbb {S}^{K}$ is a set of vectors within the probability simplex $\\\\mathbb {S}^{K}$ and $\\\\mathbf {P}\\\\in \\\\mathbb {S}_{>0}^{K}$ is a known vector in this simplex with no zero entries.\",\n \"In such a case, one has to use (REF ) and (REF ) instead.\",\n \"Notice that for each constant $A >0$ , (REF ) can be rewritten as $& &\\\\hspace{-42.67912pt}-\\\\lim _{n\\\\rightarrow \\\\infty }\\\\frac{1}{n}\\\\log \\\\,\\\\mathbb {\\\\Pi }\\\\left[ \\\\xi _{n}^{w\\\\mathbf {W}}\\\\in \\\\textrm {\\\\right.\\\\Omega \\\\hspace{-6.544pt}\\\\Omega } = \\\\inf _{\\\\mathbf {Q}\\\\in A \\\\cdot \\\\textrm {\\\\Omega \\\\hspace{-5.40608pt}\\\\Omega } }\\\\ \\\\inf _{m\\\\ne 0}D_{\\\\varphi }\\\\Big (\\\\frac{m}{A} \\\\cdot \\\\mathbf {Q},{P}\\\\Big )= \\\\inf _{\\\\mathbf {Q}\\\\in A \\\\cdot \\\\textrm {\\\\Omega \\\\hspace{-5.40608pt}\\\\Omega } }\\\\ \\\\inf _{\\\\widetilde{m}\\\\ne 0}D_{\\\\varphi }(\\\\widetilde{m} \\\\cdot \\\\mathbf {Q},{P})= \\\\inf _{\\\\widetilde{m}\\\\ne 0}\\\\ \\\\inf _{\\\\mathbf {Q}\\\\in A \\\\cdot \\\\textrm {\\\\Omega \\\\hspace{-5.40608pt}\\\\Omega } }D_{\\\\varphi }(\\\\widetilde{m} \\\\cdot \\\\mathbf {Q},{P}) ;$ therein, the constraint $\\\\mathbf {Q}\\\\in A \\\\cdot \\\\textrm {$$\\\\hspace{-6.544pt}$$}$ means geometrically that the vector $\\\\mathbf {Q}$ lives in a subset of a simplex which is parallel to the simplex $\\\\mathbb {S}^{K}$ of probability vectors and which is cut off at the edges of the first/positive orthant; in view of Remark REF (d) and (REF ), we can also handle such a situation.\",\n \"Namely, in the light of the third expression in (REF ) in combination with (REF ) to (REF ) for the special case of $\\\\mathbf {\\\\Omega } := \\\\textrm {$$\\\\hspace{-6.544pt}$$}$ lying in the probability simplex, it makes sense to study e.g.\",\n \"functional relationships between $\\\\inf _{\\\\widetilde{m}\\\\ne 0}D_{\\\\widetilde{c}\\\\cdot \\\\varphi }(\\\\widetilde{m} \\\\cdot \\\\mathbf {Q},{P})$ and $D_{\\\\widetilde{c}\\\\cdot \\\\varphi }(\\\\mathbf {Q},{P})$ ($\\\\widetilde{c} >0$ ) for $\\\\mathbf {Q} \\\\in A \\\\cdot \\\\mathbb {S}^{K}$ with arbitrary $A >0$ not necessarily being equal to 1 (i.e.\",\n \"$\\\\mathbf {Q} = A \\\\cdot {Q}$ for some probability distribution ${Q}$ ).\",\n \"Indeed, such a context appears naturally e.g.\",\n \"in connection with mass transportation problems (cf.\",\n \"(REF ) below) and with distributed energy management (cf.\",\n \"the paragraph after (REF )); the special case $A=1/K$ of (REF ) will also be used below for the application of our BS method to solving (generalized) minimum/maximum entropy problems for probability vectors (and even for sub-/super-probability vectors) ${Q}$ with constraints.\",\n \"Let us proceed with the main context.\",\n \"As indicated in Remark REF (i), in a number of important cases the limit in the above Theorem REF can be stated in terms of an invertible function $G^{-1}$ (cf.\",\n \"(REF )) of $\\\\inf _{{Q}\\\\in \\\\textrm {\\\\Omega \\\\hspace{-5.40608pt}\\\\Omega }}D_{\\\\varphi }({Q},{P})$ by elimination of $m$ .\",\n \"As explained above, for the degenerate case (cf.\",\n \"Remark REF (v), (vi)) the search for $G^{-1}$ is even interesting for the more general infimum over non-probability vectors.\",\n \"This is the scope of the following development.\"\n ],\n [\n \"Construction principle for the estimation of the minimum\\ndivergence, the power-type case \",\n \"Within the context of Theorem REF respectively Remark REF (v) and (vi), we obtain an explicit solution for the inner (i.e.\",\n \"$m-$ concerning) minimization in (REF ) for the important case of power-divergence generators $\\\\varphi _{\\\\gamma } : \\\\mathbb {R} \\\\mapsto [0,\\\\infty ]$ defined by $\\\\varphi _{\\\\gamma }(t) \\\\hspace{-5.69046pt} &:=& \\\\hspace{-5.69046pt}{\\\\left\\\\lbrace \\\\begin{array}{ll}\\\\frac{t^\\\\gamma -\\\\gamma \\\\cdot t+ \\\\gamma - 1}{\\\\gamma \\\\cdot (\\\\gamma -1)}, \\\\hspace{170.71652pt} \\\\textrm {if } \\\\gamma \\\\in \\\\, ]-\\\\infty ,0[ \\\\ \\\\textrm {and } t \\\\in ]0,\\\\infty [,\\\\\\\\- \\\\log t + t - 1, \\\\hspace{156.49014pt} \\\\textrm {if } \\\\gamma = 0 \\\\ \\\\textrm {and } t \\\\in ]0,\\\\infty [,\\\\\\\\\\\\frac{t^\\\\gamma -\\\\gamma \\\\cdot t+ \\\\gamma - 1}{\\\\gamma \\\\cdot (\\\\gamma -1)}, \\\\hspace{170.71652pt} \\\\textrm {if } \\\\gamma \\\\in \\\\, ]0,1[ \\\\ \\\\textrm {and } t \\\\in [0,\\\\infty [,\\\\\\\\t \\\\cdot \\\\log t + 1 - t, \\\\hspace{156.49014pt} \\\\textrm {if } \\\\gamma = 1 \\\\ \\\\textrm {and } t \\\\in [0,\\\\infty [,\\\\\\\\\\\\frac{t^\\\\gamma -\\\\gamma \\\\cdot t+ \\\\gamma - 1}{\\\\gamma \\\\cdot (\\\\gamma -1)} \\\\cdot {1}_{]0,\\\\infty [}(t)+(\\\\frac{1}{\\\\gamma } - \\\\frac{t}{\\\\gamma -1}) \\\\cdot {1}_{]-\\\\infty ,0]}(t),\\\\hspace{28.45274pt} \\\\textrm {if } \\\\gamma \\\\in \\\\, ]1,2[ \\\\ \\\\textrm {and } t \\\\in \\\\, ]-\\\\infty ,\\\\infty [,\\\\\\\\\\\\frac{(t - 1)^2}{2}, \\\\hspace{193.47882pt} \\\\textrm {if } \\\\gamma = 2 \\\\ \\\\textrm {and } t \\\\in \\\\, ]-\\\\infty ,\\\\infty [,\\\\\\\\\\\\frac{t^\\\\gamma -\\\\gamma \\\\cdot t+ \\\\gamma - 1}{\\\\gamma \\\\cdot (\\\\gamma -1)} \\\\cdot {1}_{]0,\\\\infty [}(t)+(\\\\frac{1}{\\\\gamma } - \\\\frac{t}{\\\\gamma -1}) \\\\cdot {1}_{]-\\\\infty ,0]}(t),\\\\hspace{28.45274pt} \\\\textrm {if } \\\\gamma \\\\in \\\\, ]2,\\\\infty [ \\\\ \\\\textrm {and } t \\\\in \\\\, ]-\\\\infty ,\\\\infty [,\\\\\\\\\\\\infty , \\\\hspace{209.12791pt} \\\\textrm {else},\\\\end{array}\\\\right.\",\n \"}$ which for arbitrary multiplier $\\\\widetilde{c} >0$ generate (the vector-valued form of) the generalized power divergences displayed in the first six rows of Table 1 (and beyond), i.e.\",\n \"$D_{\\\\widetilde{c} \\\\cdot \\\\varphi _{\\\\gamma }}(\\\\mathbf {Q},\\\\mathbf {P}) \\\\hspace{-5.69046pt} &:=& \\\\hspace{-5.69046pt}{\\\\left\\\\lbrace \\\\begin{array}{ll}\\\\widetilde{c} \\\\cdot \\\\Big \\\\lbrace \\\\frac{ \\\\sum \\\\limits _{k=1}^{K} (q_{k})^{\\\\gamma } \\\\cdot (p_{k})^{1-\\\\gamma }}{\\\\gamma \\\\cdot (\\\\gamma -1)}- \\\\frac{1}{\\\\gamma -1} \\\\cdot \\\\sum \\\\limits _{k=1}^{K} q_{k} + \\\\frac{1}{\\\\gamma } \\\\cdot \\\\sum \\\\limits _{k=1}^{K} p_{k} \\\\Big \\\\rbrace ,\\\\hspace{56.9055pt} \\\\textrm {if } \\\\gamma \\\\in \\\\, ]-\\\\infty ,0[, \\\\ \\\\mathbf {P} \\\\in \\\\mathbb {R}_{\\\\ge 0}^{K} \\\\ \\\\textrm {and } \\\\mathbf {Q} \\\\in \\\\mathbb {R}_{> 0}^{K},\\\\\\\\\\\\widetilde{c} \\\\cdot \\\\Big \\\\lbrace \\\\sum \\\\limits _{k=1}^{K} p_{k} \\\\cdot \\\\log \\\\Big (\\\\frac{p_{k}}{q_{k}} \\\\Big )+ \\\\sum \\\\limits _{k=1}^{K} q_{k} - \\\\sum \\\\limits _{k=1}^{K} p_{k} \\\\Big \\\\rbrace ,\\\\hspace{88.2037pt} \\\\textrm {if } \\\\gamma = 0, \\\\ \\\\mathbf {P} \\\\in \\\\mathbb {R}_{\\\\ge 0}^{K} \\\\ \\\\textrm {and } \\\\mathbf {Q} \\\\in \\\\mathbb {R}_{> 0}^{K},\\\\\\\\\\\\widetilde{c} \\\\cdot \\\\Big \\\\lbrace \\\\frac{ \\\\sum \\\\limits _{k=1}^{K} (q_{k})^{\\\\gamma } \\\\cdot (p_{k})^{1-\\\\gamma }}{\\\\gamma \\\\cdot (\\\\gamma -1)}- \\\\frac{1}{\\\\gamma -1} \\\\cdot \\\\sum \\\\limits _{k=1}^{K} q_{k} + \\\\frac{1}{\\\\gamma } \\\\cdot \\\\sum \\\\limits _{k=1}^{K} p_{k} \\\\Big \\\\rbrace ,\\\\hspace{59.75095pt} \\\\textrm {if } \\\\gamma \\\\in \\\\, ]0,1[, \\\\ \\\\mathbf {P} \\\\in \\\\mathbb {R}_{\\\\ge 0}^{K} \\\\ \\\\textrm {and } \\\\mathbf {Q} \\\\in \\\\mathbb {R}_{\\\\ge 0}^{K},\\\\\\\\\\\\widetilde{c} \\\\cdot \\\\Big \\\\lbrace \\\\sum \\\\limits _{k=1}^{K} q_{k} \\\\cdot \\\\log \\\\Big (\\\\frac{q_{k}}{p_{k}} \\\\Big )- \\\\sum \\\\limits _{k=1}^{K} q_{k} + \\\\sum \\\\limits _{k=1}^{K} p_{k} \\\\Big \\\\rbrace ,\\\\hspace{91.04872pt} \\\\textrm {if } \\\\gamma = 1, \\\\ \\\\mathbf {P} \\\\in \\\\mathbb {R}_{> 0}^{K} \\\\ \\\\textrm {and } \\\\mathbf {Q} \\\\in \\\\mathbb {R}_{\\\\ge 0}^{K},\\\\\\\\\\\\widetilde{c} \\\\cdot \\\\Big \\\\lbrace \\\\sum \\\\limits _{k=1}^{K} \\\\frac{(q_{k})^{\\\\gamma } \\\\cdot (p_{k})^{1-\\\\gamma }}{\\\\gamma \\\\cdot (\\\\gamma -1)}\\\\cdot {1}_{[0,\\\\infty [}(q_{k})- \\\\frac{1}{\\\\gamma -1} \\\\cdot \\\\sum \\\\limits _{k=1}^{K} q_{k} + \\\\frac{1}{\\\\gamma } \\\\cdot \\\\sum \\\\limits _{k=1}^{K} p_{k} \\\\Big \\\\rbrace ,\\\\hspace{8.5359pt} \\\\textrm {if } \\\\gamma \\\\in \\\\, ]1,2[, \\\\ \\\\mathbf {P} \\\\in \\\\mathbb {R}_{>0}^{K} \\\\ \\\\textrm {and } \\\\mathbf {Q} \\\\in \\\\mathbb {R}^{K},\\\\\\\\\\\\widetilde{c}\\\\cdot \\\\sum \\\\limits _{k=1}^{K}\\\\frac{ (q_{k}-p_{k})^{2}}{2 \\\\cdot p_{k}} ,\\\\hspace{202.01474pt} \\\\textrm {if } \\\\gamma = 2, \\\\ \\\\mathbf {P} \\\\in \\\\mathbb {R}_{>0}^{K} \\\\ \\\\textrm {and } \\\\mathbf {Q} \\\\in \\\\mathbb {R}^{K},\\\\\\\\\\\\widetilde{c} \\\\cdot \\\\Big \\\\lbrace \\\\sum \\\\limits _{k=1}^{K} \\\\frac{(q_{k})^{\\\\gamma } \\\\cdot (p_{k})^{1-\\\\gamma }}{\\\\gamma \\\\cdot (\\\\gamma -1)}\\\\cdot {1}_{[0,\\\\infty [}(q_{k})- \\\\frac{1}{\\\\gamma -1} \\\\cdot \\\\sum \\\\limits _{k=1}^{K} q_{k} + \\\\frac{1}{\\\\gamma } \\\\cdot \\\\sum \\\\limits _{k=1}^{K} p_{k} \\\\Big \\\\rbrace ,\\\\hspace{11.38092pt} \\\\textrm {if } \\\\gamma \\\\in \\\\, ]2,\\\\infty [, \\\\ \\\\mathbf {P} \\\\in \\\\mathbb {R}_{>0}^{K} \\\\ \\\\textrm {and } \\\\mathbf {Q} \\\\in \\\\mathbb {R}^{K},\\\\\\\\\\\\infty , \\\\hspace{257.49751pt} \\\\textrm {else};\\\\end{array}\\\\right.\",\n \"}$ notice that one has the straightforward relationship $D_{\\\\widetilde{c}\\\\cdot \\\\varphi _{\\\\gamma }}(\\\\cdot ,\\\\cdot )=\\\\widetilde{c}\\\\cdot D_{\\\\varphi _{\\\\gamma }}(\\\\cdot ,\\\\cdot )$ ; however, as a motivation for the introduction of $\\\\widetilde{c}>0$ , we shall show in the Examples REF , REF , REF , REF in Section below that the corresponding probability distribution $\\\\mathbb {\\\\mathbb {}}$ of the $W_{i}$ ’s depends on $\\\\widetilde{c}$ in a non-straightforward way (see also Remark REF (vi) for another motivation for $\\\\widetilde{c}$ ).\",\n \"In the course of this, it turns out that $\\\\widetilde{c} \\\\cdot \\\\varphi _{\\\\gamma } \\\\in \\\\Upsilon (]a_{\\\\gamma },\\\\infty [)$ with $a_{\\\\gamma } =0$ for $\\\\gamma \\\\in ]-\\\\infty ,1]$ and $a_{\\\\gamma } = -\\\\infty $ for $\\\\gamma \\\\in [2,\\\\infty [$ .\",\n \"For $\\\\widetilde{c}=1$ and probability vectors ${Q}$ , ${P}$ in $\\\\mathbb {S}^{K}$ respectively $\\\\mathbb {S}_{>0}^{K}$ , the divergences (REF ) simplify considerably, namely to the well-known power divergences $D_{\\\\varphi _{\\\\gamma }}({Q},{P})$ in the scaling of e.g.\",\n \"Liese & Vajda [217] (in other scalings they are also called Rathie & Kannapan’s non-additive directed divergences of order $\\\\gamma $ [302], Cressie-Read divergences [93] [303], relative Tsallis entropies or Tsallis cross-entropies [364] (see also Shiino [331]), Amari’s alpha-divergences [12]); for some comprehensive overviews on power divergences $D_{\\\\varphi _{\\\\gamma }}({Q},{P})$ — including statistical applications to goodness-of-fit testing and minimum distance estimation — the reader is referred to the insightful books of e.g.\",\n \"Liese & Vajda [217], Read & Cressie [303], Vajda [371], Stummer [344], Pardo [282], Liese & Miescke [216], the survey articles of e.g.\",\n \"Liese & Vajda [218], Vajda & van der Meulen [374], and the references therein.\",\n \"Prominent and widely used special cases of $D_{\\\\varphi _{\\\\gamma }}({Q},{P})$ are the omnipresent Kullback-Leibler information divergence (relative entropy) where $\\\\gamma =1$ , the equally important reverse Kullback-Leibler information divergence (reverse relative entropy) where $\\\\gamma =0$ , the Pearson chi-square divergence ($\\\\gamma =2$ ), the Neyman chi-square divergence ($\\\\gamma =-1$ ), the Hellinger divergence ($\\\\gamma =\\\\frac{1}{2}$ , also called squared Hellinger distance, squared Matusita distance [256] or squared Hellinger-Kakutani metric, see e.g.\",\n \"Deza & Deza [113] in some literature, the (square root of the) Hellinger divergence (HD) is misleadingly called Bhattacharyya distance; however, the latter is basically some rescaled logarithm of HD, namely $R_{1/2}({Q},{P})$ (cf.\",\n \"(REF ) with $\\\\gamma =1/2$ )).\",\n \"Some exemplary (relatively) recent studies and applications of power divergences $D_{\\\\varphi _{\\\\gamma }}({Q},{P})$ — aside from the vast statistical literature (including in particular maximum likelihood estimation and Pearson’s chi-square test) — appear e.g.\",\n \"in Matsuyama [253] for flexibilizations of the well-known expectation-maximization (EM) algorithm and their uses for big-data completion (cf.\",\n \"[254]) and data credit computation in blockchain networks (cf.\",\n \"[255]), Ku & Fine [199] in connection with blind source separation, Stummer & Vajda [348] as well as Stummer & Lao [347] for optimal decisions about some alternative financial models, Berend et al.\",\n \"[41] for the derivation of a kind of reverse Pinsker’s inequality (with $\\\\gamma =1$ ), Verrelst et al.\",\n \"[383] in geoscientific remote sensing via semiautomatic mapping of biophysical parameters from optical earth observations, Salem at al.\",\n \"[315] for automatic alarm-triggering detection of events (e.g.\",\n \"patient health degradations) from collected data by biomedical sensors, Fu et al.\",\n \"[128] for the study of income distributions in China, Ha et al.\",\n \"[151] for $x-$ ray spectrum reconstruction in computer tomography (CT) systems (with $\\\\gamma =1$ ), Iqbal & Seghouane [161] for robust sequential dictionary learning, Luppino et al.\",\n \"[233] for unsupervised change detection in heterogeneous multi-temporal satellite images (with $\\\\gamma =\\\\frac{1}{2}$ ), Sason [320] in connection with data-processing and majorization inequalities, Krömer & Stummer [198] for the smoothing and error-correcting of crude mortality rates (where they even employ non-probability-type vectors), Bekhet & Ahmed [37] for effectiveness evaluations in video retrieval (with $\\\\gamma =-1$ , $\\\\gamma =\\\\frac{1}{2}$ ), Cai et al.\",\n \"[71] for the stabilization of trainings of generative adversarial networks (GANs), Fu et al.\",\n \"[129] for automatic molecule optimization, Görtler et al.\",\n \"[139] for dimensionality reduction on uncertain data in visualization and computer graphics (with $\\\\gamma =\\\\frac{1}{2}$ ), Kammerer & Stummer [179] for optimal decision making in the presence of pandemics (e.g.\",\n \"COVID-19), Kanapram et al.\",\n \"[180] for the development of collective self-awareness in a network of connected and autonomous vehicles through agent-centered detection of abnormal situations (with $\\\\gamma =\\\\frac{1}{2}$ ), Kumbhakar [206] for modelling the streamwise velocity profile in open-channel flows, Sigmon et al.\",\n \"[335] for the improvement of genetic quality control in mouse research for biomedical applications (with $\\\\gamma =2$ ), Zhang et al.\",\n \"[420] for the design of a noise-adaptation adapted generative adversarial network for medical image analysis (with $\\\\gamma =\\\\frac{1}{2}$ ), Chen et al.\",\n \"[79] for clustering high-dimensional microbial data from RNA sequencing (with $\\\\gamma =\\\\frac{1}{2}$ ), Dharmawan et al.\",\n \"[114] for the development of improvements in long-term cell observations via semiconductor-chips-based lensless holographic microscopy, Liu & Sun [229] for analyzing approximate inferences in Bayesian neural networks, Rekavandi et al.\",\n \"[307] for detections in functional magnetic resonance imaging (fMRI) as well as hyperspectral and synthetic aperture radar (SAR) data, Seghouane & Shokouhi [325] for adaptive learning within robust radial basis function networks (RBFN), and Wang et al.\",\n \"[388] for recommender-system relevant collaborative filtering in sparse data.\",\n \"For $\\\\widetilde{c}=1$ and nonnegative-component vectors $\\\\mathbf {Q}$ , $\\\\mathbf {P}$ in $\\\\mathbb {R}_{\\\\ge 0}^{K}$ respectively $\\\\mathbb {R}_{>0}^{K}$ , the generalized power divergences $D_{\\\\varphi _{\\\\gamma }}(\\\\mathbf {Q},\\\\mathbf {P})$ of (REF ) also (partially) simplify, and were treated by Stummer & Vajda [349] (for even more general probability measures, deriving e.g.\",\n \"also generalized Pinsker’s inequalities); for a more general comprehensive technical treatment see also e.g.\",\n \"Broniatowski & Stummer [64].\",\n \"Returning to the general context, in Theorem REF we stated that for each ${P} \\\\in \\\\mathbb {S}_{>0}^{K}$ the function ${Q} \\\\mapsto \\\\inf _{m\\\\ne 0}D_{\\\\varphi }(m\\\\cdot {Q},{P}) $ is BS-minimizable (cf.\",\n \"(REF )) on all sets $\\\\textrm {$$\\\\hspace{-6.544pt}$$} \\\\subset \\\\mathbb {S}^{K}$ satisfying (7) and (9).\",\n \"The (corresponding subsetup of the) following Lemma REF is the cornerstone leading from this statement to BS-minimizability of the function ${Q} \\\\mapsto D_{\\\\varphi }({Q},{P}))$ on those same sets, for the special divergences in (REF ).\",\n \"After giving the fundamental preparatory Lemma REF , we shall derive from it some BS-minimizability/BS-maximizability results for (extensions of) a variety of important, widely used, closely related divergences respectively entropy/diversity indices.\",\n \"To achieve this in a transparent way, we employ the following three fundamental quantities $H_{\\\\gamma }(\\\\mathbf {Q},{P})$ , $I(\\\\mathbf {Q},{P})$ and $\\\\widetilde{I}(\\\\mathbf {Q},{P})$ .\",\n \"To begin with, let $A >0$ be an arbitrary constant (notice that for the choice $A=1$ , all the following vectors $\\\\mathbf {Q}$ will turn into probability vectors ${Q}$ ).\",\n \"Moreover — for any constellation $(\\\\gamma , {P},\\\\mathbf {Q}) \\\\in \\\\widetilde{\\\\Gamma }\\\\times \\\\widetilde{\\\\mathcal {M}}_{1}\\\\times \\\\widetilde{\\\\mathcal {M}}_{2}$ , where $\\\\widetilde{\\\\Gamma }\\\\times \\\\widetilde{\\\\mathcal {M}}_{1}\\\\times \\\\widetilde{\\\\mathcal {M}}_{2}:=]0,1[\\\\times \\\\mathbb {S}^{K}\\\\times A \\\\cdot \\\\mathbb {S}^{K}$ or $\\\\widetilde{\\\\Gamma }\\\\times \\\\widetilde{\\\\mathcal {M}}_{1}\\\\times \\\\widetilde{\\\\mathcal {M}}_{2}:=]-\\\\infty ,0[\\\\times \\\\mathbb {S}^{K}\\\\times A \\\\cdot \\\\mathbb {S}_{>0}^{K} $ or $\\\\widetilde{\\\\Gamma }\\\\times \\\\widetilde{\\\\mathcal {M}}_{1}\\\\times \\\\widetilde{\\\\mathcal {M}}_{2}:=]1,\\\\infty [\\\\times \\\\mathbb {S}_{>0}^{K}\\\\times A \\\\cdot \\\\mathbb {S}^{K}$ — let $0 < H_{\\\\gamma }(\\\\mathbf {Q},{P}) := \\\\sum \\\\displaylimits _{k=1}^{K} (q_{k})^{\\\\gamma } \\\\cdot (p_{k})^{1-\\\\gamma }\\\\ = \\\\ 1 + \\\\gamma \\\\cdot (A-1) +\\\\gamma \\\\cdot (\\\\gamma -1) \\\\cdot D_{\\\\varphi _{\\\\gamma }}(\\\\mathbf {Q}, {P}),\\\\qquad \\\\gamma \\\\in \\\\mathbb {R}\\\\backslash \\\\lbrace 0,1\\\\rbrace ,$ be the modified $\\\\gamma -$ order Hellinger integral of $\\\\mathbf {Q}$ and ${P}$ .\",\n \"Furthermore, for any ${P} \\\\in \\\\mathbb {S}_{>0}^{K}$ , $\\\\mathbf {Q} \\\\in A \\\\cdot \\\\mathbb {S}^{K}$ , let $-1 \\\\ < \\\\ I(\\\\mathbf {Q},{P}) := \\\\sum \\\\displaylimits _{k=1}^{K} q_{k} \\\\cdot \\\\log \\\\left( \\\\frac{q_{k}}{p_{k}} \\\\right)\\\\ = \\\\ D_{\\\\varphi _{1}}(\\\\mathbf {Q}, {P}) + A - 1,$ be the modified Kullback-Leibler information (modified relative entropy).\",\n \"Finally, for any ${P}\\\\in \\\\mathbb {S}^{K}$ , $\\\\mathbf {Q} \\\\in A \\\\cdot \\\\mathbb {S}_{>0}^{K}$ , let $1 - A \\\\ \\\\le \\\\ \\\\widetilde{I}(\\\\mathbf {Q},{P}) := \\\\sum \\\\displaylimits _{k=1}^{K} p_{k} \\\\cdot \\\\log \\\\left( \\\\frac{p_{k}}{q_{k}} \\\\right)\\\\ = \\\\ D_{\\\\varphi _{0}}(\\\\mathbf {Q}, {P}) + 1 - A,$ be the modified reverse Kullback-Leibler information (modified reverse relative entropy).\",\n \"In terms of (REF ), (REF ) and (REF ) we obtain the following Lemma 16 Let $A >0$ be an arbitrary constant.\",\n \"(a) Let $\\\\widetilde{c}>0$ be arbitrary and $(\\\\gamma , {P},\\\\mathbf {Q}) \\\\in \\\\widetilde{\\\\Gamma }\\\\times \\\\widetilde{\\\\mathcal {M}}_{1}\\\\times \\\\widetilde{\\\\mathcal {M}}_{2}$ as above.\",\n \"Then one has $\\\\inf _{m\\\\ne 0}D_{\\\\widetilde{c}\\\\cdot \\\\varphi _{\\\\gamma }}(m \\\\cdot \\\\mathbf {Q},{P})=\\\\inf _{m>0}D_{\\\\widetilde{c}\\\\cdot \\\\varphi _{\\\\gamma }}(m\\\\cdot \\\\mathbf {Q},{P})&=&\\\\frac{\\\\widetilde{c}}{\\\\gamma }\\\\cdot \\\\left[ 1- A^{\\\\gamma /(\\\\gamma -1)} \\\\cdot \\\\left[ 1+ \\\\gamma \\\\cdot (A-1) + \\\\frac{\\\\gamma \\\\cdot \\\\left( \\\\gamma -1\\\\right) }{\\\\widetilde{c}}\\\\cdot D_{\\\\widetilde{c}\\\\cdot \\\\varphi _{\\\\gamma }}(\\\\mathbf {Q},{P})\\\\right]^{-1/\\\\left( \\\\gamma -1\\\\right) }\\\\right]\\\\nonumber \\\\\\\\& &\\\\\\\\&=&\\\\frac{\\\\widetilde{c}}{\\\\gamma }\\\\cdot \\\\left[ 1- A^{\\\\gamma /(\\\\gamma -1)} \\\\cdot H_{\\\\gamma }(\\\\mathbf {Q},{P})^{-1/\\\\left( \\\\gamma -1\\\\right) }\\\\right]\\\\nonumber $ and consequently for any subset $A \\\\cdot \\\\Omega $$\\\\Omega \\\\subset \\\\widetilde{\\\\mathcal {M}}_{2}$ $&&\\\\hspace{-19.91684pt}\\\\inf _{\\\\mathbf {Q}\\\\in A \\\\cdot \\\\textrm {\\\\Omega \\\\hspace{-5.40608pt}\\\\Omega }}\\\\ \\\\inf _{m\\\\ne 0} D_{\\\\widetilde{c}\\\\cdot \\\\varphi _{\\\\gamma }}(m \\\\cdot \\\\mathbf {Q},{P})=\\\\frac{\\\\widetilde{c}}{\\\\gamma }\\\\cdot \\\\left[ 1-A^{\\\\gamma /(\\\\gamma -1)} \\\\cdot \\\\left[ 1+ \\\\gamma \\\\cdot (A-1) +\\\\frac{\\\\gamma \\\\cdot \\\\left( \\\\gamma -1\\\\right) }{\\\\widetilde{c}}\\\\cdot \\\\inf _{\\\\mathbf {Q}\\\\in A \\\\cdot \\\\textrm {\\\\Omega \\\\hspace{-5.40608pt}\\\\Omega } }D_{\\\\widetilde{c}\\\\cdot \\\\varphi _{\\\\gamma }}(\\\\mathbf {Q},{P})\\\\right]^{-1/\\\\left( \\\\gamma -1\\\\right) }\\\\right] ,\\\\qquad \\\\ \\\\\\\\&&\\\\hspace{-19.91684pt}\\\\arg \\\\inf {}_{\\\\mathbf {Q}\\\\in A \\\\cdot \\\\textrm {\\\\Omega \\\\hspace{-5.40608pt}\\\\Omega } }\\\\inf _{m\\\\ne 0}\\\\ D_{\\\\widetilde{c}\\\\cdot \\\\varphi _{\\\\gamma }}(m \\\\cdot \\\\mathbf {Q},{P})=\\\\arg \\\\inf {}_{\\\\mathbf {Q}\\\\in A \\\\cdot \\\\textrm {\\\\Omega \\\\hspace{-5.40608pt}\\\\Omega } } \\\\ D_{\\\\widetilde{c}\\\\cdot \\\\varphi _{\\\\gamma }}(\\\\mathbf {Q},{P}),\\\\qquad \\\\ \\\\\\\\&&\\\\hspace{-19.91684pt}\\\\inf _{\\\\mathbf {Q}\\\\in A \\\\cdot \\\\textrm {\\\\Omega \\\\hspace{-5.40608pt}\\\\Omega }}\\\\ \\\\inf _{m\\\\ne 0} D_{\\\\varphi _{\\\\gamma }}(m \\\\cdot \\\\mathbf {Q},{P})=\\\\frac{1}{\\\\gamma }\\\\cdot \\\\left[ 1-A^{\\\\gamma /(\\\\gamma -1)} \\\\cdot \\\\left[ \\\\inf _{\\\\mathbf {Q}\\\\in A \\\\cdot \\\\textrm {\\\\Omega \\\\hspace{-5.40608pt}\\\\Omega } }H_{\\\\gamma }(\\\\mathbf {Q},{P})\\\\right]^{-1/\\\\left( \\\\gamma -1\\\\right) }\\\\right] ,\\\\qquad \\\\textrm {for \\\\gamma <0 and \\\\gamma >1}, \\\\\\\\&&\\\\hspace{-19.91684pt}\\\\arg \\\\inf {}_{\\\\mathbf {Q}\\\\in A \\\\cdot \\\\textrm {\\\\Omega \\\\hspace{-5.40608pt}\\\\Omega } }\\\\inf _{m\\\\ne 0}\\\\ D_{\\\\varphi _{\\\\gamma }}(m \\\\cdot \\\\mathbf {Q},{P})=\\\\arg \\\\inf {}_{\\\\mathbf {Q}\\\\in A \\\\cdot \\\\textrm {\\\\Omega \\\\hspace{-5.40608pt}\\\\Omega } } \\\\ H_{\\\\gamma }(\\\\mathbf {Q},{P}),\\\\hspace{96.73918pt} \\\\textrm {for \\\\gamma <0 and \\\\gamma >1}, \\\\\\\\&&\\\\hspace{-19.91684pt}\\\\inf _{\\\\mathbf {Q}\\\\in A \\\\cdot \\\\textrm {\\\\Omega \\\\hspace{-5.40608pt}\\\\Omega }}\\\\ \\\\inf _{m\\\\ne 0} D_{\\\\varphi _{\\\\gamma }}(m \\\\cdot \\\\mathbf {Q},{P})=\\\\frac{1}{\\\\gamma }\\\\cdot \\\\left[ 1-A^{\\\\gamma /(\\\\gamma -1)} \\\\cdot \\\\left[ \\\\sup _{\\\\mathbf {Q}\\\\in A \\\\cdot \\\\textrm {\\\\Omega \\\\hspace{-5.40608pt}\\\\Omega } }H_{\\\\gamma }(\\\\mathbf {Q},{P})\\\\right]^{-1/\\\\left( \\\\gamma -1\\\\right) }\\\\right] , \\\\hspace{22.76228pt}\\\\textrm {for \\\\gamma \\\\in ]0,1[}, \\\\\\\\&&\\\\hspace{-19.91684pt}\\\\arg \\\\inf {}_{\\\\mathbf {Q}\\\\in A \\\\cdot \\\\textrm {\\\\Omega \\\\hspace{-5.40608pt}\\\\Omega } }\\\\inf _{m\\\\ne 0}\\\\ D_{\\\\varphi _{\\\\gamma }}(m \\\\cdot \\\\mathbf {Q},{P})=\\\\arg \\\\sup {}_{\\\\mathbf {Q}\\\\in A \\\\cdot \\\\textrm {\\\\Omega \\\\hspace{-5.40608pt}\\\\Omega } } \\\\ H_{\\\\gamma }(\\\\mathbf {Q},{P}), \\\\hspace{96.73918pt} \\\\textrm {for \\\\gamma \\\\in ]0,1[},$ provided that the infimum on the right-hand side of (REF ) exists.\",\n \"(b) For any ${P} \\\\in \\\\mathbb {S}_{>0}^{K}$ , $\\\\mathbf {Q} \\\\in A \\\\cdot \\\\mathbb {S}^{K}$ , $\\\\widetilde{c}>0$ one gets $\\\\inf _{m\\\\ne 0}D_{\\\\widetilde{c}\\\\cdot \\\\varphi _{1}}(m \\\\cdot \\\\mathbf {Q},{P})=\\\\inf _{m>0}D_{\\\\widetilde{c}\\\\cdot \\\\varphi _{1}}(m \\\\cdot \\\\mathbf {Q},{P})&=&\\\\widetilde{c}\\\\cdot \\\\left[ 1- A \\\\cdot \\\\exp \\\\left( -\\\\frac{1}{A \\\\cdot \\\\widetilde{c}}\\\\cdot D_{\\\\widetilde{c}\\\\cdot \\\\varphi _{1}}(\\\\mathbf {Q},{P})+ \\\\frac{1}{A} -1 \\\\right) \\\\right] \\\\\\\\&=&\\\\widetilde{c}\\\\cdot \\\\left[ 1- A \\\\cdot \\\\exp \\\\left( -\\\\frac{1}{A}\\\\cdot I(\\\\mathbf {Q},{P})\\\\right) \\\\right]\\\\nonumber $ and consequently for any subset $A \\\\cdot \\\\Omega $$\\\\Omega \\\\subset A \\\\cdot \\\\mathbb {S}^{K}$ $& &\\\\inf _{\\\\mathbf {Q}\\\\in A \\\\cdot \\\\textrm {\\\\Omega \\\\hspace{-5.40608pt}\\\\Omega } } \\\\ \\\\inf _{m\\\\ne 0}D_{\\\\widetilde{c}\\\\cdot \\\\varphi _{1}}(m \\\\cdot \\\\mathbf {Q},{P})=\\\\widetilde{c}\\\\cdot \\\\left[ 1-A \\\\cdot \\\\exp \\\\left( -\\\\frac{1}{A \\\\cdot \\\\widetilde{c}}\\\\cdot \\\\inf _{Q\\\\in A \\\\cdot \\\\textrm {\\\\Omega \\\\hspace{-5.40608pt}\\\\Omega } }D_{\\\\widetilde{c}\\\\cdot \\\\varphi _{1}}(\\\\mathbf {Q},{P}) + \\\\frac{1}{A} -1 \\\\right) \\\\right] , \\\\\\\\& & \\\\arg \\\\inf {}_{\\\\mathbf {Q}\\\\in A \\\\cdot \\\\textrm {\\\\Omega \\\\hspace{-5.40608pt}\\\\Omega } } \\\\ \\\\inf _{m\\\\ne 0}D_{\\\\widetilde{c}\\\\cdot \\\\varphi _{1}}(m \\\\cdot \\\\mathbf {Q},{P}) =\\\\arg \\\\inf {}_{\\\\mathbf {Q}\\\\in A \\\\cdot \\\\textrm {\\\\Omega \\\\hspace{-5.40608pt}\\\\Omega } } \\\\ D_{\\\\widetilde{c}\\\\cdot \\\\varphi _{1}}(Q,P),\\\\qquad \\\\ \\\\\\\\& &\\\\inf _{\\\\mathbf {Q}\\\\in A \\\\cdot \\\\textrm {\\\\Omega \\\\hspace{-5.40608pt}\\\\Omega } } \\\\ \\\\inf _{m\\\\ne 0}D_{\\\\varphi _{1}}(m \\\\cdot \\\\mathbf {Q},{P})=\\\\left[ 1- A \\\\cdot \\\\exp \\\\left( -\\\\frac{1}{A}\\\\cdot \\\\inf _{Q\\\\in A \\\\cdot \\\\textrm {\\\\Omega \\\\hspace{-5.40608pt}\\\\Omega } }I(\\\\mathbf {Q},{P}) \\\\right) \\\\right] , \\\\\\\\& & \\\\arg \\\\inf {}_{\\\\mathbf {Q}\\\\in A \\\\cdot \\\\textrm {\\\\Omega \\\\hspace{-5.40608pt}\\\\Omega } } \\\\ \\\\inf _{m\\\\ne 0}D_{\\\\varphi _{1}}(m \\\\cdot \\\\mathbf {Q},{P}) =\\\\arg \\\\inf {}_{\\\\mathbf {Q}\\\\in A \\\\cdot \\\\textrm {\\\\Omega \\\\hspace{-5.40608pt}\\\\Omega } } \\\\ I(\\\\mathbf {Q},{P}),\\\\qquad \\\\ $ provided that the infimum on the right-hand side of (REF ) exists.\",\n \"(c) For any ${P}\\\\in \\\\mathbb {S}^{K}$ , $\\\\mathbf {Q} \\\\in A \\\\cdot \\\\mathbb {S}_{>0}^{K}$ , $\\\\widetilde{c}>0$ we obtain $\\\\inf _{m\\\\ne 0}D_{\\\\widetilde{c}\\\\cdot \\\\varphi _{0}}(m \\\\cdot \\\\mathbf {Q},{P})=\\\\inf _{m>0}D_{\\\\widetilde{c}\\\\cdot \\\\varphi _{0}}(m \\\\cdot \\\\mathbf {Q},{P})&=& D_{\\\\widetilde{c}\\\\cdot \\\\varphi _{0}}(\\\\mathbf {Q},{P})+ \\\\widetilde{c} \\\\cdot (1 - A + \\\\log A) \\\\\\\\&=& \\\\widetilde{c} \\\\cdot \\\\Big ( \\\\widetilde{I}(\\\\mathbf {Q},{P}) + \\\\log A \\\\Big )\\\\nonumber $ and consequently for any set subset $A \\\\cdot \\\\Omega $$\\\\Omega \\\\subset A \\\\cdot \\\\mathbb {S}_{>0}^{K}$ $&&\\\\hspace{-19.91684pt}\\\\inf _{\\\\mathbf {Q} \\\\in A \\\\cdot \\\\textrm {\\\\Omega \\\\hspace{-5.40608pt}\\\\Omega }} \\\\ \\\\inf _{m\\\\ne 0} D_{\\\\widetilde{c}\\\\cdot \\\\varphi _{0}}(m \\\\cdot \\\\mathbf {Q},{P})= \\\\widetilde{c} \\\\cdot (1 - A + \\\\log A) +\\\\inf _{\\\\mathbf {Q} \\\\in A \\\\cdot \\\\textrm {\\\\Omega \\\\hspace{-5.40608pt}\\\\Omega }} \\\\ D_{\\\\widetilde{c}\\\\cdot \\\\varphi _{0}}(\\\\mathbf {Q},{P}), \\\\\\\\&&\\\\hspace{-19.91684pt}\\\\arg \\\\inf {}_{\\\\mathbf {Q}\\\\in A \\\\cdot \\\\textrm {\\\\Omega \\\\hspace{-5.40608pt}\\\\Omega } } \\\\ \\\\inf _{m\\\\ne 0}D_{\\\\widetilde{c}\\\\cdot \\\\varphi _{0}}(m \\\\cdot \\\\mathbf {Q},{P})=\\\\arg \\\\inf {}_{\\\\mathbf {Q}\\\\in A \\\\cdot \\\\textrm {\\\\Omega \\\\hspace{-5.40608pt}\\\\Omega } } \\\\ D_{\\\\widetilde{c}\\\\cdot \\\\varphi _{0}}(\\\\mathbf {Q},{P}), \\\\qquad \\\\ \\\\\\\\&&\\\\hspace{-19.91684pt}\\\\inf _{\\\\mathbf {Q} \\\\in A \\\\cdot \\\\textrm {\\\\Omega \\\\hspace{-5.40608pt}\\\\Omega }} \\\\ \\\\inf _{m\\\\ne 0} D_{\\\\varphi _{1}}(m \\\\cdot \\\\mathbf {Q},{P})= \\\\log A +\\\\inf _{\\\\mathbf {Q} \\\\in A \\\\cdot \\\\textrm {\\\\Omega \\\\hspace{-5.40608pt}\\\\Omega }} \\\\ \\\\widetilde{I}(\\\\mathbf {Q},{P}), \\\\\\\\&&\\\\hspace{-19.91684pt}\\\\arg \\\\inf {}_{\\\\mathbf {Q}\\\\in A \\\\cdot \\\\textrm {\\\\Omega \\\\hspace{-5.40608pt}\\\\Omega } } \\\\ \\\\inf _{m\\\\ne 0} D_{\\\\varphi _{0}}(m \\\\cdot \\\\mathbf {Q},{P})=\\\\arg \\\\inf {}_{\\\\mathbf {Q}\\\\in A \\\\cdot \\\\textrm {\\\\Omega \\\\hspace{-5.40608pt}\\\\Omega } } \\\\ \\\\widetilde{I}(\\\\mathbf {Q},{P}), \\\\qquad \\\\ $ provided that the infimum on the right-hand side of (REF ) exists.\",\n \"The proof of Lemma REF is given in Appendix C. Remark 17 Notice that for ${P} \\\\in \\\\mathbb {S}_{>0}^{K}$ and $\\\\mathbf {Q} \\\\in A \\\\cdot \\\\mathbb {S}^{K}$ , the modified Kullback-Leibler information has the property $I(\\\\mathbf {Q},{P}) \\\\ge 0$ if $A \\\\ge 1$ (cf.\",\n \"(REF )); otherwise, $I(\\\\mathbf {Q},{P})$ may become negative, as can be easily seen from the case where ${P} := {P}^{unif} :=(\\\\frac{1}{K}, \\\\ldots , \\\\frac{1}{K})$ is the probability vector of frequencies of the uniform distribution on $\\\\lbrace 1, \\\\ldots , K\\\\rbrace $ , and $\\\\mathbf {Q} := (\\\\frac{1}{K+1}, 0, \\\\ldots , 0)$ .\",\n \"Analogously, for ${P}\\\\in \\\\mathbb {S}^{K}$ and $\\\\mathbf {Q} \\\\in A \\\\cdot \\\\mathbb {S}_{>0}^{K}$ one gets $\\\\widetilde{I}(\\\\mathbf {Q},{P}) \\\\ge 0$ if $A \\\\le 1$ (cf.\",\n \"(REF )); otherwise, $\\\\widetilde{I}(\\\\mathbf {Q},{P})$ may become negative (take e.g.\",\n \"$\\\\mathbf {Q} = (\\\\frac{K+1}{K}, \\\\ldots , \\\\frac{K+1}{K})$ and ${P} := (1,0, \\\\ldots , 0)$ ).\",\n \"Remark 18 (a) In the context of Remark REF (vi), according to (REF ) applied to $\\\\varphi := \\\\widetilde{c}\\\\cdot \\\\varphi _{\\\\gamma }$ , for all cases $\\\\gamma \\\\in \\\\, ]-\\\\infty ,0[ \\\\ \\\\cup \\\\ ]0,1[ \\\\ \\\\cup \\\\ [ \\\\ 2,\\\\infty [$ the left-hand side of each of (REF ), (), () is independent of $A >0$ and equal to $-\\\\lim _{n\\\\rightarrow \\\\infty }\\\\frac{1}{n}\\\\log \\\\,\\\\mathbb {\\\\Pi }\\\\left[\\\\xi _{n}^{w\\\\mathbf {W}}\\\\in \\\\textrm {\\\\right.$$\\\\hspace{-6.544pt}$$}$ where — as will be shown below — the corresponding $\\\\mathbf {W}$ 's have probability distribution $\\\\mathbb {}[\\\\cdot \\\\,]=\\\\mathbb {\\\\Pi }[W_{1}\\\\in \\\\cdot \\\\,]$ (cf.\",\n \"(REF )) which varies “quite drastically” with $\\\\gamma $ (and the case $\\\\gamma \\\\in ]1,2[$ has to be even excluded for analytical difficulties because in this case there are some indications that the representation (REF ) only holds for some signed probability distribution $$ (e.g.\",\n \"having a density with positive and negative values).).\",\n \"Analogously, each of the left-hand sides of (REF ), (), (REF ), () is also independent of $A >0$ and equal to $-\\\\lim _{n\\\\rightarrow \\\\infty }\\\\frac{1}{n}\\\\log \\\\,\\\\mathbb {\\\\Pi }\\\\left[\\\\xi _{n}^{w\\\\mathbf {W}}\\\\in \\\\textrm {\\\\right.$$\\\\hspace{-6.544pt}$$}$ for some $\\\\mathbf {W}$ of respective distribution.\",\n \"Hence, by inversion, all the extremum-describing target quantities $\\\\inf _{\\\\mathbf {Q}\\\\in A \\\\cdot \\\\textrm {\\\\Omega \\\\hspace{-5.40608pt}\\\\Omega } }D_{\\\\widetilde{c}\\\\cdot \\\\varphi _{\\\\gamma }}(\\\\mathbf {Q},{P})$ ($\\\\gamma \\\\in \\\\mathbb {R}\\\\backslash ]1,2[$ ), $\\\\inf _{\\\\mathbf {Q}\\\\in A \\\\cdot \\\\textrm {\\\\Omega \\\\hspace{-5.40608pt}\\\\Omega } }H_{\\\\gamma }(\\\\mathbf {Q},{P})$ ($\\\\gamma \\\\in ]-\\\\infty ,0[ \\\\, \\\\cup \\\\, [2,\\\\infty [$ ), $\\\\sup _{\\\\mathbf {Q}\\\\in A \\\\cdot \\\\textrm {\\\\Omega \\\\hspace{-5.40608pt}\\\\Omega } }H_{\\\\gamma }(\\\\mathbf {Q},{P})$ ($\\\\gamma \\\\in ]0,1[$ ), $\\\\inf _{Q\\\\in A \\\\cdot \\\\textrm {\\\\Omega \\\\hspace{-5.40608pt}\\\\Omega } }I(\\\\mathbf {Q},{P})$ and $\\\\inf _{Q\\\\in A \\\\cdot \\\\textrm {\\\\Omega \\\\hspace{-5.40608pt}\\\\Omega } }\\\\widetilde{I}(\\\\mathbf {Q},{P})$ can be expressed as $G\\\\big (-\\\\lim _{n\\\\rightarrow \\\\infty }\\\\frac{1}{n}\\\\log \\\\,\\\\mathbb {\\\\Pi }\\\\left[\\\\xi _{n}^{w\\\\mathbf {W}}\\\\in \\\\textrm {\\\\right.$$\\\\hspace{-6.544pt}$$})\\\\, $ for some explicitly known ($A-$ dependent) function $G$ .\",\n \"This means that — in the sense of Definition REF — all the corresponding four “cornerstone quantities” $D_{\\\\widetilde{c}\\\\cdot \\\\varphi _{\\\\gamma }}(\\\\mathbf {Q},{P})$ , $H_{\\\\gamma }(\\\\mathbf {Q},{P})$ , $I(\\\\mathbf {Q},{P})$ , $\\\\widetilde{I}(\\\\mathbf {Q},{P})$ are BS-minimizable, respectively BS-maximizable, on $\\\\mathbf {\\\\Omega } = A \\\\cdot \\\\textrm {$$\\\\hspace{-6.544pt}$$}$ .\",\n \"The above-mentioned inversions (i.e.\",\n \"constructions of $G(\\\\cdot )$ ) will be concretely carried out below — namely in the Propositions REF , REF , REF , REF , REF and REF .\",\n \"In those, we also involve the BS-minimizability/maximizability of several other important closely related divergences and measures of entropy (measures of diversity, measures of heterogeneity/homogeneity, measures of concentration) which (i) are widely used in information theory and its applications to artificial intelligence, machine learning and physics, and which (ii) can be built from the above-mentioned four cornerstone quantities (power divergences, Hellinger integrals, Kullback-Leibler information divergences).\",\n \"(b) The special case $\\\\varphi := \\\\widetilde{c}\\\\cdot \\\\varphi _{\\\\gamma }$ ($\\\\gamma \\\\in \\\\, ]-\\\\infty ,0[ \\\\ \\\\cup \\\\ ]0,1[ \\\\ \\\\cup \\\\ [ \\\\ 2,\\\\infty [$ ) of Theorem REF works analogously to (a), with the differences that we employ $A=1$ (instead of arbitrary $A>0$ ), () (instead of (REF )), $\\\\mathbb {\\\\Pi }_{X_{1}^{n}}[\\\\cdot ]$ (instead of $\\\\mathbb {\\\\Pi }[\\\\cdot ]$ ), and $\\\\xi _{n,\\\\mathbf {X}}^{w\\\\mathbf {W}}$ (instead of $\\\\xi _{n}^{w\\\\mathbf {W}}$ ).\",\n \"(c) From the proof of Lemma REF in Appendix C below, one can see that for the important case $\\\\gamma =2$ the formulas (REF ) to () also hold for $A<0$ .\",\n \"In the following, we further elaborate the three points (a),(b) and (c) of Remark REF “comprehensively and unifyingly”, where the expression “BS minimizable/maximizable” always has to be interpreted accordingly in terms of $-\\\\lim _{n\\\\rightarrow \\\\infty }\\\\frac{1}{n}\\\\log \\\\mathbb {\\\\Pi }[\\\\xi _{n}^{w\\\\mathbf {W}} \\\\in \\\\cdot \\\\ ]$ respectively $-\\\\lim _{n\\\\rightarrow \\\\infty }\\\\frac{1}{n}\\\\log \\\\mathbb {\\\\Pi }_{X_{1}^{n}}[\\\\xi _{n,\\\\mathbf {X}}^{w\\\\mathbf {W}}\\\\in \\\\cdot \\\\ ]$ (without explicit mentioning, for the sake of brevity).\",\n \"Let us fix $\\\\widetilde{c}=1$ and an arbitrary triple $(\\\\gamma , {P},\\\\mathbf {Q})$ which satisfies the assumptions of Lemma REF (a) with $A := \\\\sum _{k=1}^{K} q_{k} >0$ .\",\n \"For such a setup, we have obtained in (REF ) the $\\\\gamma -$ order Hellinger integral (of $\\\\mathbf {Q}$ and ${P}$ ) $H_{\\\\gamma }(\\\\mathbf {Q},{P}) > 0$ , which is not a divergence; as a terminology-concerning side remark, let us mention that $H_{\\\\gamma }(\\\\mathbf {Q},{P})$ ($\\\\gamma \\\\ge 1$ ) is called relative information generating function in Guiasu & Reischer [144], see e.g.\",\n \"also Clark [90]; moreover, $H_{\\\\gamma }({Q},{P})$ is sometimes termed ($\\\\gamma -$ order) Chernoff coefficient being a component of the Chernoff distances/informations [85].\",\n \"Torgersen [361] uses the name ($\\\\gamma -$ order) Hellinger transform.\",\n \"Notice that the special case $\\\\gamma =\\\\frac{1}{2}$ is nothing but (a multiple of) the well-known important Bhattacharyya coefficient (cf.\",\n \"[48],[49],[50]) $BC(\\\\mathbf {Q},{P}) :=H_{1/2}(\\\\mathbf {Q},{P}) =\\\\sum _{k=1}^{K} \\\\sqrt{q_{k} \\\\cdot p_{k}} =1 + \\\\frac{1}{2} \\\\cdot (A-1) + \\\\frac{1}{2} \\\\cdot (\\\\frac{1}{2}-1) \\\\cdot D_{\\\\varphi _{\\\\frac{1}{2}}}(\\\\mathbf {Q},{P})\\\\nonumber $ 0.3cm which is also known as affinity (cf.\",\n \"Matusita [256], see e.g.\",\n \"also Toussaint [362]) and (classic, non-quantum) fidelity similarity (cf.\",\n \"e.g.\",\n \"Deza & Deza [113]); for non-probability vectors $\\\\mathbf {P} \\\\in \\\\mathbb {R}_{\\\\ge 0}^{K}$ one can simply retransform ${P} := \\\\frac{\\\\mathbf {P}}{M_{\\\\mathbf {P}}}$ and thus imbed $BC(\\\\mathbf {Q},\\\\mathbf {P}) = \\\\sqrt{M_{\\\\mathbf {P}}} \\\\cdot BC(\\\\mathbf {Q},{P})$ into our BS context.\",\n \"There is a vast literature on very recent applications of the Bhattacharyya coefficient, for instance it appears exemplarily in Peng & Li [289] for object tracking from successive video frames, Ayed et al.\",\n \"[26] for efficient graph cut algorithms, Patra et al.\",\n \"[287] for collaborative filtering in sparse data, El Merabet et al.\",\n \"[119] for region classification in intelligent transport systems in order to compensate the lack of performance of Global Navigation Satellites Systems, Chiu et al.\",\n \"[86] for the design of interactive mobile augmented reality systems, Noh et al.\",\n \"[274] for dimension reduction in interacting fluid flow models, Bai et al.\",\n \"[29] for material defect detection through ultrasonic array imaging, Dixit & Jain [115] for the design of recommender systems on highly sparse context aware datasets, Guan et al.\",\n \"[143] for visible light positioning methods based on image sensors, Lin et al.\",\n \"[220] for probabilistic representation of color image pixels, Chen et al.\",\n \"[80] for distributed compressive video sensing, Jain et al.\",\n \"[162] for the enhancement of multistage user-based collaborative filtering in recommendation systems, Pascuzzo et al.\",\n \"[285] for brain-diffusion-MRI based early diagnosis of the sporadic Creutzfeldt–-Jakob disease, Sun et al.\",\n \"[351] for the design of automatic detection methods multitemporal (e.g.\",\n \"landslide) point clouds, Valpione et al.\",\n \"[377] for the investigation of T cell dynamics in immunotherapy, Wang et al.\",\n \"[387] for the tracking and prediction of downbursts from meteorological data, Xu et al.\",\n \"[403] for adaptive distributed compressed video sensing for coal mine monitoring, Zhao et al.\",\n \"[424] for the shared sparse machine learning of the affective content of images, Chen et al.\",\n \"[82] for image segmentation and domain partitioning, De Oliveira et al.\",\n \"[105] for the prediction of cell-penetrating peptides, Eshaghi et al.\",\n \"[122] for the identification of multiple sclerosis subtypes through machine learning of brain MRI scans, Feng et al.\",\n \"[125] for improvements of MRI-based detection of epilepsy-causing cortical malformations, Hanli et al.\",\n \"[153] for designing pilot protection schemes for transmission lines, Jiang et al.\",\n \"[170] for flow-assisted visual tracking through event cameras, Lysiak & Szmajda [235] for comparisons of selected feature quality evaluations, Joel & Sivakumar [172] for the despeckling enhancement of medical ultrasound image quality, Reising et al.\",\n \"[305] for the design of security protection of Internet-of-Things (IoT) devices, Skrbic et al.\",\n \"[338] for the uncovering of interplays between amino acid sequences and local structures in proteins, Tsiapoki et al.\",\n \"[365] for the improvement of the detection performance of structural health monitoring frameworks, van Molle et al.\",\n \"[381] for uncertainty quantification in deep neural networks, Yang et al.\",\n \"[413] for the determination of the onset of transient signals, and Zhou & Yu [427] for the modelling of spatiotemporal human eye movements.\",\n \"To proceed with the general context, for any $\\\\gamma \\\\in \\\\, ]-\\\\infty ,0[ \\\\ \\\\cup \\\\ ]0,1[ \\\\ \\\\cup \\\\ [ \\\\ 2,\\\\infty [$ let the function $h_{\\\\gamma } : \\\\, ]0,\\\\infty [ \\\\, \\\\mapsto \\\\, ]-\\\\infty ,\\\\infty [$ be such that $x \\\\mapsto h_{\\\\gamma }(1 + \\\\gamma \\\\cdot (A-1) + \\\\gamma \\\\cdot (\\\\gamma -1) \\\\cdot x)$ is continuous and strictly increasing (respectively, strictly decreasing) for all $x \\\\ge 0$ with $1 + \\\\gamma \\\\cdot (A-1) + \\\\gamma \\\\cdot (\\\\gamma -1) \\\\cdot x > 0$ ; since $D_{\\\\varphi _{\\\\gamma }}(\\\\mathbf {Q},{P})$ is BS-minimizable on $\\\\mathbf {\\\\Omega } = A \\\\cdot \\\\textrm {$$\\\\hspace{-6.544pt}$$}$ , then also the — not necessarily nonnegative — quantity $h_{\\\\gamma }\\\\Big (1 + \\\\gamma \\\\cdot (A-1) + \\\\gamma \\\\cdot (\\\\gamma -1) \\\\cdot D_{\\\\varphi _{\\\\gamma }}(\\\\mathbf {Q},{P})\\\\Big )= h_{\\\\gamma }\\\\Big (H_{\\\\gamma }(\\\\mathbf {Q},{P})\\\\Big )$ is BS-minimizable (respectively, BS-maximizable) on $\\\\mathbf {\\\\Omega } = A \\\\cdot \\\\textrm {$$\\\\hspace{-6.544pt}$$}$ .\",\n \"If $h_{\\\\gamma }$ satisfies additionally $h_{\\\\gamma }(1)=0$ as well as $h_{\\\\gamma }(1 + \\\\gamma \\\\cdot (A-1) + \\\\gamma \\\\cdot (\\\\gamma -1) \\\\cdot x) \\\\ge 0$ for all $x \\\\ge 0$ with $1 + \\\\gamma \\\\cdot (A-1) + \\\\gamma \\\\cdot (\\\\gamma -1) \\\\cdot x > 0$ , then $D_{h_{\\\\gamma }}(\\\\mathbf {Q},{P}) :=h_{\\\\gamma }\\\\Big (1 + \\\\gamma \\\\cdot (A-1) + \\\\gamma \\\\cdot (\\\\gamma -1) \\\\cdot D_{\\\\varphi _{\\\\gamma }}(\\\\mathbf {Q},{P})\\\\Big )= h_{\\\\gamma }\\\\Big (H_{\\\\gamma }(\\\\mathbf {Q},{P})\\\\Big )$ constitutes a divergence in the usual sense that $D_{h_{\\\\gamma }}(\\\\mathbf {Q},{P}) \\\\ge 0 $ with equality iff ${Q}={P}$ .\",\n \"which is BS-minimizable on $\\\\mathbf {\\\\Omega } = A \\\\cdot \\\\textrm {$$\\\\hspace{-6.544pt}$$}$ (respectively, BS-maximizable on $\\\\mathbf {\\\\Omega } = A \\\\cdot \\\\textrm {$$\\\\hspace{-6.544pt}$$}$ ).\",\n \"Let us consider some important examples.\",\n \"For the identity mapping $h_{\\\\gamma }^{Id}(y) := y$ ($y>0$ ) the function $x \\\\mapsto 1 + \\\\gamma \\\\cdot (A-1) + \\\\gamma \\\\cdot (\\\\gamma -1) \\\\cdot x$ is strictly increasing for $\\\\gamma <0$ and $\\\\gamma >1$ (on the required domain of $x$ ), and strictly decreasing for $\\\\gamma \\\\in ]0,1[$ .\",\n \"Accordingly, $H_{\\\\gamma }(\\\\mathbf {Q},{P})$ is BS-minimizable on $\\\\mathbf {\\\\Omega } = A \\\\cdot \\\\textrm {$$\\\\hspace{-6.544pt}$$}$ for $\\\\gamma <0$ and $\\\\gamma \\\\ge 2$ and BS-maximizable on $\\\\mathbf {\\\\Omega } = A \\\\cdot \\\\textrm {$$\\\\hspace{-6.544pt}$$}$ for $\\\\gamma \\\\in \\\\, ]0,1[$ (this is consistent with (), ()); in particular, the Bhattacharyya coefficient $BC(\\\\mathbf {Q},{P})$ is BS-maximizable on $\\\\mathbf {\\\\Omega } = A \\\\cdot \\\\textrm {$$\\\\hspace{-6.544pt}$$}$ .\",\n \"Some other important choices are $& & h_{\\\\gamma }(y) := h_{c_{1},c_{2},c_{3}}(y) := c_{1} \\\\cdot \\\\Big (y^{c_{2}} - c_{3} \\\\Big ),\\\\qquad y>0, \\\\, c_{1}, c_{2} \\\\in \\\\mathbb {R}\\\\backslash \\\\lbrace 0\\\\rbrace , \\\\,c_{3} \\\\in \\\\mathbb {R},\\\\\\\\& & h_{\\\\gamma }(y) := h_{c_{4},f}^{R}(y) := \\\\lim _{c_{2} \\\\rightarrow 0} h_{c_{4}/f(c_{2}),c_{2},1}(y) =\\\\frac{c_{4}}{f^{\\\\prime }(0)} \\\\cdot \\\\log (y),\\\\qquad y>0, \\\\, c_{4} \\\\in \\\\mathbb {R}\\\\backslash \\\\lbrace 0\\\\rbrace , \\\\,\\\\\\\\& & h_{\\\\gamma }(y) := h_{c_{5},c_{6}}^{GB2}(y) := c_{5} \\\\cdot (\\\\arccos (y))^{c_{6}},\\\\qquad \\\\gamma \\\\in \\\\,]0,1[, \\\\, y \\\\in \\\\, ]0,1], \\\\, c_{5}>0, \\\\, c_{6}>0,\\\\\\\\& & h_{\\\\gamma }(y) := h_{\\\\nu ,c_{7}}^{BB}(y) := c_{7} \\\\cdot \\\\frac{\\\\log (1-\\\\frac{1-y}{\\\\nu })}{\\\\log (1-\\\\frac{1}{\\\\nu })},\\\\qquad \\\\gamma \\\\in \\\\,]0,1[, \\\\, y \\\\in \\\\, ]0,1], \\\\, c_{7}>0, \\\\, \\\\nu \\\\in ]-\\\\infty ,0[ \\\\, \\\\cup \\\\, ]1,\\\\infty [,$ where the constants $c_{1}$ to $c_{7}$ may depend on $\\\\gamma $ , and $f$ is some (maybe $\\\\gamma -$ dependent) function which is differentiable in a neighborhood of 0 and satisfies $f(0)=0$ , $f^{\\\\prime }(0) \\\\ne 0$ (e.g.\",\n \"$f(z)=c_{8} \\\\cdot z$ for some non-zero constant $c_{8}$ ).\",\n \"Clearly, $h_{c_{1},c_{2},c_{3}}(\\\\cdot )$ is strictly increasing (respectively, strictly decreasing) if and only if $c_{1} \\\\cdot c_{2} >0$ (respectively, $c_{1} \\\\cdot c_{2} <0$ ).\",\n \"Moreover, $h_{c_{4},f}^{R}(\\\\cdot )$ is strictly increasing (respectively, strictly decreasing) if and only if $\\\\frac{c_{4}}{f^{\\\\prime }(0)} >0$ (respectively, $\\\\frac{c_{4}}{f^{\\\\prime }(0)} < 0$ ).\",\n \"Furthermore, both $h_{c_{5},c_{6}}^{GB2}(\\\\cdot )$ and $h_{\\\\nu ,c_{7}}^{GoBa}(\\\\cdot )$ are strictly decreasing.\",\n \"For instance, the special case $h_{\\\\gamma }(y) = h_{c_{4},Id}^{R}(y)$ with $c_{4} := \\\\frac{1}{\\\\gamma \\\\cdot (\\\\gamma -1)}$ (recall that $\\\\gamma \\\\in \\\\, ]-\\\\infty ,0[ \\\\ \\\\cup \\\\ ]0,1[ \\\\ \\\\cup \\\\ [ \\\\ 2,\\\\infty [$ ) and identity function $f := Id$ leads to the quantities $\\\\hspace{-42.67912pt}R_{\\\\gamma }(\\\\mathbf {Q},{P}) &: =&D_{h_{c_{4},Id}^{R}}(\\\\mathbf {Q},{P}) =\\\\frac{\\\\log \\\\Big ( 1 + \\\\gamma \\\\cdot (A-1) +\\\\gamma \\\\cdot (\\\\gamma -1) \\\\cdot D_{\\\\varphi _{\\\\gamma }}(\\\\mathbf {Q},{P}) \\\\Big )}{\\\\gamma \\\\cdot (\\\\gamma -1)}= \\\\frac{\\\\log \\\\Big ( H_{\\\\gamma }(\\\\mathbf {Q},{P}) \\\\Big )}{\\\\gamma \\\\cdot (\\\\gamma -1)}\\\\nonumber \\\\\\\\&=& \\\\frac{\\\\log \\\\Big ( \\\\sum \\\\displaylimits _{k=1}^{K} (q_{k})^{\\\\gamma } \\\\cdot (p_{k})^{1-\\\\gamma } \\\\Big )}{\\\\gamma \\\\cdot (\\\\gamma -1)} \\\\, ,\\\\qquad \\\\gamma \\\\in \\\\, ]-\\\\infty ,0[ \\\\ \\\\cup \\\\ ]0,1[ \\\\ \\\\cup \\\\ [ \\\\ 2,\\\\infty [,$ (provided that all involved power divergences are finite), which are thus BS-minimizable on $\\\\mathbf {\\\\Omega } = A \\\\cdot \\\\textrm {$$\\\\hspace{-6.544pt}$$}$ ; notice that $R_{\\\\gamma }(\\\\mathbf {Q},{P}) \\\\ge 0$ if $\\\\gamma \\\\in \\\\ ]0,1[ \\\\ \\\\cup \\\\ [ \\\\ 2,\\\\infty [$ together with $A \\\\in [1,\\\\infty [$ , and if $\\\\gamma \\\\in \\\\, ]-\\\\infty ,0[$ together with $A \\\\in \\\\, ]0,1]$ .\",\n \"The special subcase $A=1$ in (REF ) (and thus, $\\\\mathbf {Q}$ is a probability vector ${Q}$ ) corresponds to the prominent Renyi divergences/distances [309] (in the scaling of e.g.\",\n \"Liese & Vajda [217] and in probability-vector form), see e.g.\",\n \"van Erven & Harremoes [380] for a comprehensive study of their properties; as a side remark, $\\\\gamma \\\\cdot (\\\\gamma -1) \\\\cdot R_{\\\\gamma }({Q},{P})$ is also employed in the Chernoff distances/informations [85].\",\n \"The special subcase $R_{1/2}({Q},{P})$ (i.e.\",\n \"$\\\\gamma =1/2$ and $A=1$ in (REF )) corresponds to (a multiple of) the widely used Bhattacharyya distance (of type 1) between ${Q}$ and ${P}$ , cf.\",\n \"[48] (see e.g.\",\n \"also Kailath [178]).\",\n \"Sometimes, $\\\\exp (R_{\\\\gamma }(\\\\mathbf {Q},{P}))$ is also called Renyi divergence/distance.\",\n \"Some exemplary (relatively) recent studies and applications of Renyi divergences $R_{\\\\gamma }({Q},{P})$ (respectively, their multiple or exponential) — aside from the substantial statistical literature — appear e.g.\",\n \"in Zhao et al.\",\n \"[423] for the study of isomeric stability of fullerenes (which are e.g.\",\n \"employed for state-of-the-art organic solar cells), in the papers of Sundaresan [353], Bunte & Lapidoth [67], Sason [318], Kumar et al.\",\n \"[205] for (mismatch-cases of) coding and guessing as well as task partitioning, in the papers of Prest [296], Bai et al.\",\n \"[30] for lattice-based cryptography, in He et al.\",\n \"[158] for robot active olfaction search (by infotaxis) in turbulent flows, in Momeni et al.\",\n \"[263] for the design of reprogrammable encrypted graphene-based coding metasurfaces, in Staszowska et al.\",\n \"[340] for accurate and precise cluster analysis for super-resolution localization microscopy, in Yu & Tan [414] for distributed source simulation problems, in Zhang et al.\",\n \"[419] for sensor control, in Yu & Tan [415] for the so-called random variable simulation problem, in Blanchet et al.\",\n \"[55] for the robust treatment of extreme values in rainfall accumulation data, in Cai et al.\",\n \"[70] for sensor tasking for search and catalog maintenance of geosynchronous space objects, in Gholami & Hodtani [134] for refinements of safety-and-security-targeted location verification systems in wireless communication networks (e.g in Intelligent Transportation Systems (ITSs) and vehicular technology), in Seweryn et al.\",\n \"[326] for the assessment of similarity and diversity of expression profiles in single cell systems, in Zhou [426] for the study of secrecy constraints in key generation problems where side information might be present at untrusted users, in Makkawi et al.\",\n \"[243] for the design of an automated decision-support framework for adaptive diagnosis of fault–tolerant multi–sensor data fusion for vehicle localization, in Mao et al.\",\n \"[245] for privacy-preserving computation offloading for parallel deep neural networks training.\",\n \"There is vast literature on recent applications of the above-mentioned special case $R_{1/2}({Q},{P})$ — that is, the Bhattacharyya distance (of type 1); for instance, it appears in Tarighati & Jalden [356] for rate balancing in wireless sensor networks, Bi et al.\",\n \"[51], [52] for certain uncertainty quantifications respectively stochastic sensitivity analyses in mechanical systems and signal processing, Fu & He [130] for the design of multibit quantizers for cooperative spectrum sensing in cognitive radio networks, Cohen et al.\",\n \"[91] for adaptive and causal random linear network coding with forward error correction for a point-to-point communication channel with delayed feedback, Xu et al.\",\n \"[401] for cost minimization problems of big data analytics on geo-distributed data centers connected to renewable energy sources with unpredictable capacity, Xu et al.\",\n \"[402] for community identification in networks, Arrigoni & Madsen [18] for automated discovering of low-energy defect configurations in materials, Fan et al.\",\n \"[123] for region-merging-based methods for synthetic aperture radar (SAR) image segmentation, Mahfouz et al.\",\n \"[242] for some refined ensemble classifications in microarray-based automated cancer diagnosis, Matchev & Shyamsundar [252] for some machine-learning based signal discovery in high energy physics (HEP) experiments, Wang et al.\",\n \"[386] for the investigation of intratumoral heterogeneity (ITH) of some gastric cancer, Webster et al.\",\n \"[389] for the characterization, identification, clustering and classification of disease, and Xiahou et al.\",\n \"[396] for the prediction of remaining useful life (RUL) through fusion of expert knowledge and condition monitoring information.\",\n \"As a further example, consider $\\\\mathcal {B}_{\\\\gamma ,c_{5},c_{6}}({Q},{P}) &: =&D_{h_{c_{5},c_{6}}^{GB2}}({Q},{P}) =c_{5} \\\\cdot \\\\Big (\\\\arccos \\\\Big (1 + \\\\gamma \\\\cdot (\\\\gamma -1) \\\\cdot D_{\\\\varphi _{\\\\gamma }}({Q},{P})\\\\Big ) \\\\, \\\\Big )^{c_{6}}= c_{5} \\\\cdot \\\\Big (\\\\arccos \\\\Big (H_{\\\\varphi _{\\\\gamma }}({Q},{P})\\\\Big ) \\\\, \\\\Big )^{c_{6}}\\\\nonumber \\\\\\\\&=& c_{5} \\\\cdot \\\\Big (\\\\arccos \\\\Big (\\\\sum \\\\displaylimits _{k=1}^{K} (q_{k})^{\\\\gamma } \\\\cdot (p_{k})^{1-\\\\gamma }\\\\Big ) \\\\, \\\\Big )^{c_{6}}\\\\ge 0 \\\\, ,\\\\qquad \\\\gamma \\\\in \\\\, ]0,1[, \\\\, c_5 >0, \\\\, c_{6} >0,\\\\nonumber $ which is BS-maximizable on $\\\\textrm {$$\\\\hspace{-6.544pt}$$}$ .\",\n \"The case $\\\\mathcal {B}_{1/2,1,1}({Q},{P})$ corresponds to the well-known Bhattacharyya arccos distance (Bhattacharyya distance of type 2) in [50] (which is also called Wootters distance [395]), and $\\\\mathcal {B}_{1/2,1,2}({Q},{P})$ to its variant in [49]; the case $\\\\mathcal {B}_{1/2,2,1}({Q},{P})$ is known as Fisher distance or Rao distance or geodesic distance (see e.g.\",\n \"Deza & Deza [113]); a nice graphical illustration of the geometric connection between the Fisher distance $\\\\mathcal {B}_{1/2,2,1}({Q},{P})$ and the Hellinger distance/metric $\\\\sqrt{\\\\frac{1}{2} \\\\cdot D_{\\\\varphi _{1/2}}({Q},{P})}$ can be found e.g.\",\n \"on p.35 in Ay et al.\",\n \"[25].\",\n \"Some exemplary applications of the Bhattacharyya arccos distance $\\\\mathcal {B}_{1/2,1,1}({Q},{P})$ can be found e.g.\",\n \"in Rao [301] and Juhasz [177] for cluster analysis of human populations, in Martin-Fernandez et al.\",\n \"[251] for general hierarchical clustering, Greenacre [141] for metric scaling, and in Chen et al.\",\n \"[79] for clustering high-dimensional microbial data from RNA sequencing.\",\n \"Let us give another example, namely $\\\\widetilde{\\\\mathcal {B}}_{\\\\gamma ,\\\\nu ,c_{7}}({Q},{P}) &: =&D_{h_{\\\\nu ,c_{7}}^{BB}}({Q},{P}) =\\\\frac{c_{7}}{\\\\log (1-\\\\frac{1}{\\\\nu })} \\\\cdot \\\\log \\\\bigg (1-\\\\frac{1-\\\\Big ( 1 + \\\\gamma \\\\cdot (\\\\gamma -1) \\\\cdot D_{\\\\varphi _{\\\\gamma }}({Q},{P})\\\\Big )}{\\\\nu }\\\\bigg )\\\\nonumber \\\\\\\\&=& \\\\frac{c_{7}}{\\\\log (1-\\\\frac{1}{\\\\nu })} \\\\cdot \\\\log \\\\bigg (1-\\\\frac{1-H_{\\\\varphi _{\\\\gamma }}({Q},{P})}{\\\\nu }\\\\bigg )= \\\\frac{c_{7}}{\\\\log (1-\\\\frac{1}{\\\\nu })} \\\\cdot \\\\log \\\\bigg (1-\\\\frac{1-\\\\sum \\\\displaylimits _{k=1}^{K} (q_{k})^{\\\\gamma } \\\\cdot (p_{k})^{1-\\\\gamma }}{\\\\nu }\\\\bigg )\\\\in [0, c_{7}[ \\\\,, \\\\nonumber \\\\\\\\& & \\\\hspace{227.62204pt} \\\\gamma \\\\in \\\\,]0,1[, \\\\, \\\\, c_{7}>0, \\\\, \\\\nu \\\\in \\\\, ]-\\\\infty ,0[ \\\\, \\\\cup \\\\, ]1,\\\\infty [,\\\\nonumber $ which is BS-maximizable on $\\\\textrm {$$\\\\hspace{-6.544pt}$$}$ .\",\n \"The case $\\\\widetilde{\\\\mathcal {B}}_{1/2,\\\\nu ,1}({Q},{P})$ corresponds to the Bounded Bhattacharyya Distance Measures of Jolad et al.\",\n \"[174].\",\n \"We can also employ divergences of the form $\\\\breve{R}_{\\\\gamma }({Q},{P}) :=R_{\\\\gamma }(T_{1}({Q}),T_{2}({P}))$ and analogously power divergences $\\\\breve{D}_{\\\\widetilde{c} \\\\cdot \\\\varphi _{\\\\gamma }}({Q},{P}) :=D_{\\\\widetilde{c} \\\\cdot \\\\varphi _{\\\\gamma }}(T_{1}({Q}),T_{2}({P}))$ etc.\",\n \"where $T_{1}: \\\\mathcal {D}_{1} \\\\mapsto \\\\mathcal {R}_{1}$ , $T_{2}: \\\\mathcal {D}_{1} \\\\mapsto \\\\mathcal {R}_{2}$ are (say) invertible functions on appropriately chosen subsets $\\\\mathcal {D}_{1}, \\\\mathcal {D}_{2}, \\\\mathcal {R}_{1}, \\\\mathcal {R}_{2}$ of the probability-vector simplex $\\\\mathbb {S}^{K}$ .\",\n \"For instance, consider the following special case (with a slight abuse of notation): $\\\\breve{R}_{\\\\gamma }({Q},{P}): = R_{\\\\gamma }(\\\\widetilde{{Q}},\\\\widetilde{{P}})=\\\\frac{1}{\\\\gamma \\\\cdot (\\\\gamma -1)} \\\\cdot \\\\log \\\\left( \\\\sum \\\\displaylimits _{k=1}^{K}\\\\left(\\\\frac{(q_{k})^{\\\\nu _{1}}}{\\\\sum _{j=1}^{K} (q_{j})^{\\\\nu _{1}}}\\\\right)^{\\\\gamma } \\\\cdot \\\\left(\\\\frac{(p_{k})^{\\\\nu _{2}}}{\\\\sum _{j=1}^{K} (p_{j})^{\\\\nu _{2}}}\\\\right)^{1-\\\\gamma }\\\\right)$ where (i) $\\\\widetilde{{Q}} := (\\\\widetilde{q}_{k})_{k=1}^{K}$ with $\\\\widetilde{q}_{k} := \\\\frac{(q_{k})^{\\\\nu _{1}}}{\\\\sum _{j=1}^{K} (q_{j})^{\\\\nu _{1}}}$ is the escort probability distribution (in vector form) associated with the probability distribution (in vector form) ${Q} := (q_{k})_{k=1}^{K} \\\\in \\\\mathbb {S}_{> 0}^{K}$ , and (ii) $\\\\widetilde{{P}} := (\\\\widetilde{p}_{k})_{k=1}^{K}$ with $\\\\widetilde{p}_{k} := \\\\frac{(p_{k})^{\\\\nu _{2}}}{\\\\sum _{j=1}^{K} (p_{j})^{\\\\nu _{2}}}$ is the escort probability distribution associated with the probability distribution ${P} := (p_{k})_{k=1}^{K} \\\\in \\\\mathbb {S}_{> 0}^{K}$ , in terms of some fixed escort parameters $\\\\nu _{1} >0$ , $\\\\nu _{2} >0$ .\",\n \"In particular, for the special choice $\\\\nu _{1}= \\\\nu _{2} >0$ and $\\\\gamma := \\\\frac{\\\\nu }{\\\\nu _{1}}$ with $\\\\nu \\\\in ]0,\\\\nu _{1}[ \\\\, \\\\cup \\\\, [2\\\\nu _{1}, \\\\infty [$ we obtain from (REF ) $\\\\hspace{-28.45274pt}&& 0 \\\\le \\\\frac{\\\\nu }{\\\\nu _{1}} \\\\cdot R_{\\\\nu /\\\\nu _{1}}(\\\\widetilde{{Q}},\\\\widetilde{{P}})=\\\\frac{\\\\log \\\\Big ( \\\\sum \\\\displaylimits _{k=1}^{K} (\\\\widetilde{q}_{k})^{\\\\nu /\\\\nu _{1}} \\\\cdot (\\\\widetilde{p}_{k})^{1-(\\\\nu /\\\\nu _{1})} \\\\Big )}{\\\\frac{\\\\nu }{\\\\nu _{1}} -1}\\\\nonumber \\\\\\\\\\\\hspace{-28.45274pt}&& =\\\\frac{\\\\nu _{1}}{\\\\nu - \\\\nu _{1}} \\\\cdot \\\\log \\\\Big ( \\\\sum \\\\displaylimits _{k=1}^{K} (q_{k})^{\\\\nu } \\\\cdot (p_{k})^{\\\\nu _{1} - \\\\nu } \\\\Big )- \\\\frac{\\\\nu }{\\\\nu - \\\\nu _{1}} \\\\cdot \\\\log \\\\Big ( \\\\sum \\\\displaylimits _{k=1}^{K} (q_{k})^{\\\\nu _{1}} \\\\Big )+ \\\\log \\\\Big ( \\\\sum \\\\displaylimits _{k=1}^{K} (p_{k})^{\\\\nu _{1}} \\\\Big )=: \\\\breve{R}_{\\\\nu /\\\\nu _{1}}({Q},{P})$ which is BS-minimizable (in $\\\\widetilde{{Q}}$ ) on $\\\\textrm {$$\\\\hspace{-6.544pt}$$}$ .\",\n \"Our divergence $\\\\breve{R}_{\\\\nu /\\\\nu _{1}}({Q},{P})$ in (REF ) is basically a multiple of a divergence which has been very recently used in Ghosh & Basu [136].\",\n \"Moreover, $\\\\breve{R}_{1/\\\\nu _{1}}({Q},{P})$ (i.e.\",\n \"the special case $\\\\nu =1$ in (REF )) is equal to Sundaresan’s divergence [352] [353] (see also Lutwak et al.\",\n \"[234], Kumar & Sundaresan [203], [204], Yagli et al.\",\n \"[408]); for our BS-approach, we need the restriction $\\\\nu _{1} \\\\in \\\\, ]0,\\\\frac{1}{2}] \\\\, \\\\cup \\\\, ]1,\\\\infty [$ .\",\n \"Notice that Sundaresan’s divergence can be employed in mismatch-cases of (i) Campbell’s coding problem, (ii) Arikan’s guessing problem, (iii) memoryless guessing, and (iv) task partitioning problems; see e.g.\",\n \"Sundaresan [353], Bunte & Lapidoth [67], Kumar et al.\",\n \"[205].\",\n \"Returning to the general context, functions of the modified Kullback-Leibler information $I(\\\\mathbf {Q},{P})$ and the modified reverse Kullback-Leibler information $\\\\widetilde{I}(\\\\mathbf {Q},{P})$ can be treated analogously.\",\n \"For the sake of brevity, we only deal with the former and fix arbitrary ${P} \\\\in \\\\mathbb {S}_{>0}^{K}$ and $\\\\mathbf {Q} \\\\in A \\\\cdot \\\\mathbb {S}^{K}$ with $A := \\\\sum _{k=1}^{K} q_{k} >0$ .\",\n \"For this, in (REF ) we have obtained $I(\\\\mathbf {Q},{P})$ which is generally not a divergence (cf.\",\n \"Remark REF ).\",\n \"In the following, let the function $h_{1} : \\\\, ]-1,\\\\infty [ \\\\, \\\\mapsto \\\\, ]-\\\\infty ,\\\\infty [$ be continuous and strictly increasing (respectively, strictly decreasing); since $D_{\\\\varphi _{1}}(\\\\mathbf {Q},{P})$ is BS-minimizable on $\\\\mathbf {\\\\Omega } = A \\\\cdot \\\\textrm {$$\\\\hspace{-6.544pt}$$}$ , also the quantity $h_{1}\\\\Big ( A - 1 + D_{\\\\varphi _{1}}(\\\\mathbf {Q},{P})\\\\Big )= h_{1}\\\\Big (I(\\\\mathbf {Q},{P})\\\\Big )$ is BS-minimizable on $\\\\mathbf {\\\\Omega } = A \\\\cdot \\\\textrm {$$\\\\hspace{-6.544pt}$$}$ (respectively, BS-maximizable on $\\\\mathbf {\\\\Omega } = A \\\\cdot \\\\textrm {$$\\\\hspace{-6.544pt}$$}$ ).\",\n \"In particular, by using the negative identity mapping $h_{\\\\gamma }^{-Id}(y) := -y$ ($y>-1$ ) we get that $-I(\\\\mathbf {Q},{P})$ is BS-maximizable.\",\n \"Another exemplary choice for $h_{1}$ is (cf.\",\n \"Sharma & Mittal [330] in the scaling of e.g.\",\n \"Morales et al.\",\n \"[264]) $& & h_{1}(y) := h_{s}^{SM}(y) := \\\\frac{e^{(s-1) \\\\cdot y} -1}{s-1},\\\\qquad y \\\\in \\\\mathbb {R}, \\\\, s \\\\in \\\\, ]0,1[ \\\\, \\\\cup \\\\, ]1,\\\\infty [,$ which is strictly increasing; hence, $h_{s}^{SM}(I(\\\\mathbf {Q},{P}))$ (and also $h_{s}^{SM}(D_{\\\\varphi _{1}}(\\\\mathbf {Q},{P}))$ ) is BS-minimizable on $\\\\mathbf {\\\\Omega } = A \\\\cdot \\\\textrm {$$\\\\hspace{-6.544pt}$$}$ .\",\n \"As another important application line, let us fix any $(\\\\gamma , \\\\mathbf {Q}) \\\\in (\\\\widetilde{\\\\Gamma }\\\\backslash ]1,2[) \\\\times \\\\widetilde{\\\\mathcal {M}}_{2}$ (cf.\",\n \"Lemma REF (a)) with $A := \\\\sum _{k=1}^{K} q_{k} >0$ .\",\n \"Moreover, we take ${P} := {P}^{unif} :=(\\\\frac{1}{K}, \\\\ldots , \\\\frac{1}{K})$ to be the probability vector of frequencies of the uniform distribution on $\\\\lbrace 1, \\\\ldots , K\\\\rbrace $ .\",\n \"Then, for $\\\\gamma \\\\in \\\\, ]-\\\\infty ,0[ \\\\ \\\\cup \\\\ ]0,1[ \\\\ \\\\cup \\\\ [ \\\\ 2,\\\\infty [$ one gets $H_{\\\\gamma }(\\\\mathbf {Q},{P}^{unif}) =K^{\\\\gamma -1} \\\\cdot \\\\sum _{k=1}^{K} q_{k}^{\\\\gamma }$ .\",\n \"One can rewrite $K^{1- \\\\gamma } \\\\cdot H_{\\\\gamma }(\\\\mathbf {Q},{P}^{unif})= \\\\sum _{k=1}^{K} q_{k}^{\\\\gamma }$ ; the latter is sometimes called heterogeneity index of type $\\\\gamma $, see e.g.\",\n \"van der Lubbe [379], with $\\\\gamma =2$ being the Simpson-Herfindahl index which is also known as index of coincidence (cf.\",\n \"Harremoes & Topsoe [155] and its generalization in Harremoes & Vajda [156]).\",\n \"Alternatively, $\\\\sum _{k=1}^{K} q_{k}^{\\\\gamma }$ is also called Onicescu’s information energy in case of $\\\\gamma =2$ (cf.\",\n \"Onicescu [278], see also Pardo & Taneja [283] for comprehensive investigations) and in general information energy of order $\\\\gamma $ (cf.\",\n \"Theodorescu [359], see also e.g.\",\n \"Pardo [281]); for exemplary applications to electron density functional theory (DFT) for quantum chemical reactivity, the reader may take (discretized versions of) e.g.\",\n \"Liu et al.\",\n \"[226], Lopez-Rosa et al.\",\n \"[231] and Rong et al.\",\n \"[311].\",\n \"In some other literature (see e.g.\",\n \"Clark [90]), $\\\\sum _{k=1}^{K} q_{k}^{\\\\gamma }$ is alternatively called Golomb’s [140] information generating function (of a probability distribution ${Q}$ ); yet another name is generalized information potential and for $\\\\gamma =2$ information potential (cf.\",\n \"e.g.\",\n \"Principe [297], Acu et al.\",\n \"[4]).\",\n \"From the above-mentioned investigations, we obtain that $\\\\sum _{k=1}^{K} q_{k}^{\\\\gamma }$ is BS-minimizable on $\\\\mathbf {\\\\Omega } = A \\\\cdot \\\\textrm {$$\\\\hspace{-6.544pt}$$}$ for $\\\\gamma <0$ and $\\\\gamma \\\\ge 2$ , and BS-maximizable on $\\\\mathbf {\\\\Omega } = A \\\\cdot \\\\textrm {$$\\\\hspace{-6.544pt}$$}$ for $\\\\gamma \\\\in ]0,1[$ .\",\n \"More generally, by employing (REF ) and (), for the class of entropies (diversity indices) $\\\\mathcal {E}_{\\\\gamma ,c_{1},c_{2},c_{3}}(\\\\mathbf {Q}) &:=&h_{c_{1},c_{2},c_{3}}\\\\left(\\\\sum _{k=1}^{K} q_{k}^{\\\\gamma }\\\\right) =c_{1} \\\\cdot \\\\left(\\\\left(\\\\sum _{k=1}^{K} q_{k}^{\\\\gamma }\\\\right)^{c_{2}} - c_{3} \\\\right)\\\\nonumber \\\\\\\\&=&c_{1} \\\\cdot \\\\left(K^{c_{2} \\\\cdot (1- \\\\gamma )} \\\\cdot H_{\\\\gamma }(\\\\mathbf {Q},{P}^{unif})^{c_{2}} - c_{3} \\\\right),\\\\qquad c_{1}, c_{2} \\\\in \\\\mathbb {R}\\\\backslash \\\\lbrace 0\\\\rbrace , \\\\,c_{3} \\\\in \\\\mathbb {R},\\\\\\\\\\\\mathcal {E}_{c_{4},f}^{R}(\\\\mathbf {Q}) &:=&h_{c_{4},f}^{R}\\\\left(\\\\sum _{k=1}^{K} q_{k}^{\\\\gamma }\\\\right)= \\\\frac{c_{4}}{f^{\\\\prime }(0)} \\\\cdot \\\\log \\\\left(\\\\sum _{k=1}^{K} q_{k}^{\\\\gamma }\\\\right),\\\\nonumber \\\\\\\\&=& \\\\frac{c_{4}}{f^{\\\\prime }(0)} \\\\cdot \\\\Big ( \\\\log \\\\left(H_{\\\\gamma }(\\\\mathbf {Q},{P}^{unif})\\\\right) +(1- \\\\gamma ) \\\\cdot \\\\log (K)\\\\Big ),\\\\qquad c_{4} \\\\in \\\\mathbb {R}\\\\backslash \\\\lbrace 0\\\\rbrace , \\\\,$ (which is similar to the entropy-class of Morales et al.\",\n \"[265] who use a different, more restrictive parametrization and probability distributions ${Q}$ ), one gets the following extremum-behaviour: $\\\\mathcal {E}_{\\\\gamma ,c_{1},c_{2},c_{3}}(\\\\mathbf {Q})$ is BS-minimziable if $\\\\gamma <0$ and $c_{1} \\\\cdot c_{2} >0$ ; $\\\\mathcal {E}_{\\\\gamma ,c_{1},c_{2},c_{3}}(\\\\mathbf {Q})$ is BS-minimizable if $\\\\gamma \\\\ge 2$ and $c_{1} \\\\cdot c_{2} >0$ ; $\\\\mathcal {E}_{\\\\gamma ,c_{1},c_{2},c_{3}}(\\\\mathbf {Q})$ is BS-minimizable if $\\\\gamma \\\\in \\\\, ]0,1[$ and $c_{1} \\\\cdot c_{2} < 0$ ; $\\\\mathcal {E}_{\\\\gamma ,c_{1},c_{2},c_{3}}(\\\\mathbf {Q})$ is BS-maximizable if $\\\\gamma <0$ and $c_{1} \\\\cdot c_{2} < 0$ ; $\\\\mathcal {E}_{\\\\gamma ,c_{1},c_{2},c_{3}}(\\\\mathbf {Q})$ is BS-maximizable if $\\\\gamma \\\\ge 2$ and $c_{1} \\\\cdot c_{2} < 0$ ; $\\\\mathcal {E}_{\\\\gamma ,c_{1},c_{2},c_{3}}(\\\\mathbf {Q})$ is BS-maximizable if $\\\\gamma \\\\in \\\\, ]0,1[$ and $c_{1} \\\\cdot c_{2} > 0$ ; $\\\\mathcal {E}_{c_{4},f}^{R}(\\\\mathbf {Q})$ is BS-minimizable if $\\\\gamma <0$ and $\\\\frac{c_{4}}{f^{\\\\prime }(0)} > 0$ ; $\\\\mathcal {E}_{c_{4},f}^{R}(\\\\mathbf {Q})$ is BS-minimizable if $\\\\gamma \\\\ge 2$ and $\\\\frac{c_{4}}{f^{\\\\prime }(0)} > 0$ ; $\\\\mathcal {E}_{c_{4},f}^{R}(\\\\mathbf {Q})$ is BS-minimizable if $\\\\gamma \\\\in \\\\, ]0,1[$ and $\\\\frac{c_{4}}{f^{\\\\prime }(0)} < 0$ ; $\\\\mathcal {E}_{c_{4},f}^{R}(\\\\mathbf {Q})$ is BS-maximizable if $\\\\gamma <0$ and $\\\\frac{c_{4}}{f^{\\\\prime }(0)} < 0$ ; $\\\\mathcal {E}_{c_{4},f}^{R}(\\\\mathbf {Q})$ is BS-maximizable if $\\\\gamma \\\\ge 2$ and $\\\\frac{c_{4}}{f^{\\\\prime }(0)} < 0$ ; $\\\\mathcal {E}_{c_{4},f}^{R}(\\\\mathbf {Q})$ is BS-maximizable if $\\\\gamma \\\\in \\\\, ]0,1[$ and $\\\\frac{c_{4}}{f^{\\\\prime }(0)} > 0$ .\",\n \"From this, one can deduce that our new BS method works for the constrained minimization/maximization of the following well-known, prominently used measures of entropy respectively measures of diversity, and beyond: $c_{1}=1$ , $c_{2} = \\\\frac{1}{\\\\gamma }$ , $c_{3}=0$ : the Euclidean $\\\\gamma -$ norm (also known as $\\\\gamma -$ norm heterogeneity index, see e.g.\",\n \"van der Lubbe [379]) $|| \\\\mathbf {Q} ||_{\\\\gamma } := \\\\Big (\\\\sum _{k=1}^{K} q_{k}^{\\\\gamma }\\\\Big )^{1/\\\\gamma }= K^{(1- \\\\gamma )/\\\\gamma } \\\\cdot \\\\Big (H_{\\\\gamma }(\\\\mathbf {Q},{P}^{unif})\\\\Big )^{1/\\\\gamma }$ is BS-minimizable on $\\\\mathbf {\\\\Omega } = A \\\\cdot \\\\textrm {$$\\\\hspace{-6.544pt}$$}$ for $\\\\gamma \\\\in ]0,1[$ and $\\\\gamma \\\\ge 2$ , and BS-maximizable on $\\\\mathbf {\\\\Omega } = A \\\\cdot \\\\textrm {$$\\\\hspace{-6.544pt}$$}$ for $\\\\gamma <0$ (note that $|| \\\\mathbf {Q} ||_{1} =A$ ) ; similarly, the $\\\\gamma -$ mean heterogeneity index (see e.g.\",\n \"[379], as well as Jost [176] for its interpretation as “effective number of species” respectively as “numbers equivalent”) given by $\\\\mathcal {E}^{HI}(\\\\mathbf {Q}) := \\\\Big (\\\\sum _{k=1}^{K} q_{k}^{\\\\gamma }\\\\Big )^{1/(\\\\gamma -1)}= \\\\frac{1}{K} \\\\cdot \\\\Big (H_{\\\\gamma }(\\\\mathbf {Q},{P}^{unif})\\\\Big )^{1/(\\\\gamma -1)}$ is BS-minimizable on $\\\\mathbf {\\\\Omega } = A \\\\cdot \\\\textrm {$$\\\\hspace{-6.544pt}$$}$ for $\\\\gamma \\\\ge 2$ , and BS-maximizable on $\\\\mathbf {\\\\Omega } = A \\\\cdot \\\\textrm {$$\\\\hspace{-6.544pt}$$}$ for $\\\\gamma <0$ and $\\\\gamma \\\\in ]0,1[$ .\",\n \"Alternatively, $\\\\mathcal {E}^{HI}(\\\\mathbf {Q})$ is also called ($\\\\gamma -$ order) Hill diversity index or Hill number [160], respectively ($\\\\gamma -$ order) Hannah-Kay index [154], respectively ($\\\\gamma -$ order) Renyi heterogeneity (cf.\",\n \"Nunes et al.\",\n \"[276]), respectively ($\\\\gamma -$ order) exponential Renyi entropy or exponential entropy (cf.\",\n \"Campbell [72]) since it is equal to $\\\\exp (\\\\mathcal {E}^{gR}(\\\\mathbf {Q}))$ (cf.\",\n \"(E6) below).\",\n \"The $\\\\gamma -$ mean heterogeneity index (under one of the above-mentioned namings) was recently employed e.g.\",\n \"by Greiff et al.\",\n \"[142] for immunodiagnostic design of fingerprints of an individual’s ongoing immunological status (e.g., healthy, infected, vaccinated) — culminating in accurate and early detection of disease and infection, by Ma & Li [237] for the quantification of metagenome diversity and similarity, by Jasinska et al.\",\n \"[164] for studying bacterial evolution — in particular evolution under sub-inhibitory antibiotic levels, by Ma et al.\",\n \"[238] for the definition of individual-level genetic diversity and similarity profiles as well as their applications to datasets from the 1000-Genomes Project, and by Lassance & Vrins [209] for some optimal selection procedure of financial-asset portfolios.\",\n \"$c_{1}=\\\\frac{1}{2^{1-\\\\gamma }-1}$ , $c_{2} = 1$ , $c_{3}=1$ : the entropy $\\\\mathcal {E}^{gHC}(\\\\mathbf {Q}) :=\\\\frac{1}{2^{1-\\\\gamma }-1} \\\\cdot \\\\left(\\\\sum _{k=1}^{K} q_{k}^{\\\\gamma } - 1 \\\\right)=\\\\frac{1}{2^{1-\\\\gamma }-1} \\\\cdot \\\\left(K^{1- \\\\gamma } \\\\cdot H_{\\\\gamma }(\\\\mathbf {Q},{P}^{unif}) - 1 \\\\right)$ is BS-minimizable on $\\\\mathbf {\\\\Omega } = A \\\\cdot \\\\textrm {$$\\\\hspace{-6.544pt}$$}$ for $\\\\gamma <0$ , and BS-maximizable on $\\\\mathbf {\\\\Omega } = A \\\\cdot \\\\textrm {$$\\\\hspace{-6.544pt}$$}$ for $\\\\gamma \\\\in ]0,1[$ and $\\\\gamma \\\\ge 2$ ; the special subcase $A=1$ in (REF ) (and thus, $\\\\mathbf {Q} ={Q}$ is a probability vector) corresponds to the $\\\\gamma -$ order entropy of Havrda-Charvat [157] (also called non-additive $\\\\gamma -$ order Tsallis entropy [363] in statistical physics) where the special case $\\\\gamma =2$ is (a multiple of) Vajda’s quadratic entropy [371] and Ahlswede’s identification entropy [7] (see also Ahlswede & Cai [8]).\",\n \"Some exemplary (relatively) recent studies and applications of $\\\\mathcal {E}^{gHC}({Q})$ appear e.g.\",\n \"in Peter & Rangarajan [291] for shape matching, in Liu et al.\",\n \"[226] as well as in Rong et al.\",\n \"[311] to electron density functional theory (DFT) for quantum chemical reactivity, in Yalcin & Beck [409] for the investigation of energy spectra of cosmic rays, in Wen & Jiang [390] for the quantification of complexity degrees in complex networks, in Bhandari [46] for fast multilevel thresholding for color image segmentation, in Erguzel et al.\",\n \"[121] for the investigation of Electroencephalography (EEG) signals of subjects suffering from some psychiatric disorders, in Kang & Kim [182] for automatic synthetic aperture radar (SAR) image registration, in Namdari & Li [269] for the modelling of Lithium-Ion battery capacity fade, in Seweryn et al.\",\n \"[326] for the assessment of similarity and diversity of expression profiles in single cell systems, in Zhang et al.\",\n \"[418] for the search of functional relationships between groundwater depth and vegetation distribution, in Kumbhakar et al.\",\n \"[207] for the modelling of streamwise velocity profiles in wide–open channel turbulent flows (e.g.\",\n \"in rivers, streams, canals, ditches), and in Ramezani & Pourdarvish [300] for transfer learning for image classification of gravitational waves.\",\n \"For the special case $\\\\gamma =2$ , a directly connected quantity is the measure of concentration (cf.\",\n \"e.g.\",\n \"De Wet et al.\",\n \"[107]) $\\\\mathcal {E}^{gMC}({Q}) : = 1- \\\\frac{1}{K} - \\\\mathcal {E}^{gHC}({Q})= \\\\sum _{k=1}^{K} \\\\left(q_{k} - \\\\frac{1}{K} \\\\right)^{2}$ which (up to a multiple) was introduced by Brukner & Zeilinger [66] as an appropriate measure of information for quantum experiments.\",\n \"$\\\\gamma := \\\\frac{1}{\\\\widetilde{\\\\gamma }}$ , $c_{1}=\\\\frac{1}{\\\\widetilde{\\\\gamma }-1}$ , $c_{2} = \\\\widetilde{\\\\gamma }$ , $c_{3}=1$ : the entropy $\\\\mathcal {E}^{gA}(\\\\mathbf {Q}) :=\\\\frac{1}{\\\\widetilde{\\\\gamma }-1} \\\\cdot \\\\left(\\\\left(\\\\sum _{k=1}^{K}q_{k}^{1/\\\\widetilde{\\\\gamma }}\\\\right)^{\\\\widetilde{\\\\gamma }} - 1 \\\\right)=\\\\frac{1}{\\\\widetilde{\\\\gamma }-1} \\\\cdot \\\\left(K^{\\\\widetilde{\\\\gamma } \\\\cdot (1- \\\\gamma )} \\\\cdot H_{1/\\\\widetilde{\\\\gamma }}(\\\\mathbf {Q},{P}^{unif})^{\\\\widetilde{\\\\gamma }} - 1 \\\\right)$ is BS-minimizable on $\\\\mathbf {\\\\Omega } = A \\\\cdot \\\\textrm {$$\\\\hspace{-6.544pt}$$}$ for $\\\\widetilde{\\\\gamma } <0$ and $\\\\widetilde{\\\\gamma } \\\\in \\\\, ]0,1[$ , and BS-maximizable on $\\\\mathbf {\\\\Omega } = A \\\\cdot \\\\textrm {$$\\\\hspace{-6.544pt}$$}$ for $\\\\gamma \\\\ge 2$ ; the special subcase $A=1$ in (REF ) (and thus, $\\\\mathbf {Q} ={Q}$ is a probability vector) corresponds to the $\\\\widetilde{\\\\gamma }-$ order entropy of Arimoto [16].\",\n \"$s \\\\in \\\\mathbb {R}\\\\backslash \\\\lbrace 1\\\\rbrace $ , $c_{1}=\\\\frac{1}{1-s}$ , $c_{2} = \\\\frac{1-s}{1-\\\\gamma }$ , $c_{3}=1$ : the entropy $\\\\mathcal {E}^{gSM1}(\\\\mathbf {Q}):= \\\\frac{1}{1-s} \\\\cdot \\\\left(\\\\left(\\\\sum _{k=1}^{K} q_{k}^{\\\\gamma }\\\\right)^{(1-s)/(1-\\\\gamma )} - 1 \\\\right)= \\\\frac{1}{1-s} \\\\cdot \\\\left(K^{1- s} \\\\cdot H_{\\\\gamma }(\\\\mathbf {Q},{P}^{unif})^{(1-s)/(1-\\\\gamma )} - 1 \\\\right)$ is BS-minimizable on $\\\\mathbf {\\\\Omega } = A \\\\cdot \\\\textrm {$$\\\\hspace{-6.544pt}$$}$ for $\\\\gamma < 0$ and BS-maximizable on $\\\\mathbf {\\\\Omega } = A \\\\cdot \\\\textrm {$$\\\\hspace{-6.544pt}$$}$ for $\\\\gamma \\\\in \\\\, ]0,1[$ and $\\\\gamma \\\\ge 2$ ; the special subcase $A=1$ in (REF ) (and thus, $\\\\mathbf {Q} ={Q}$ is a probability vector) corresponds to the entropy of order $\\\\gamma $ and degree $s$ of Sharma & Mittal [329] in the scaling of e.g.\",\n \"Salicru et al.\",\n \"[314].\",\n \"$s \\\\in \\\\mathbb {R}\\\\backslash \\\\lbrace 0\\\\rbrace $ , $\\\\gamma = s+1$ , $c_{1}= - \\\\frac{1}{s}$ , $c_{2} = 1$ , $c_{3}=1$ : the diversity index $\\\\mathcal {E}^{gPT}(\\\\mathbf {Q}):= - \\\\frac{1}{s} \\\\cdot \\\\left(\\\\sum _{k=1}^{K} q_{k}^{s+1} - 1 \\\\right)= - \\\\frac{1}{s} \\\\cdot \\\\left(K^{- s} \\\\cdot H_{s+1}(\\\\mathbf {Q},{P}^{unif}) - 1 \\\\right)$ is BS-minimizable on $\\\\mathbf {\\\\Omega } = A \\\\cdot \\\\textrm {$$\\\\hspace{-6.544pt}$$}$ for $s < -1$ and BS-maximizable on $\\\\mathbf {\\\\Omega } = A \\\\cdot \\\\textrm {$$\\\\hspace{-6.544pt}$$}$ for $s \\\\in \\\\, ]-1,0[$ and $s >0$ ; the special subcase $A=1$ in (REF ) (and thus, $\\\\mathbf {Q} ={Q}$ is a probability vector) corresponds to the diversity index of degree $s$ of Patil & Taillie [286]; the case $s=1$ for probability measures $\\\\mathbf {Q}={Q}$ gives the well-known Gini-Simpson diversity index.\",\n \"$c_{4}=\\\\frac{1}{1-\\\\gamma }$ , $f(z) =z$ : the entropy $\\\\mathcal {E}^{gR}(\\\\mathbf {Q}):= \\\\frac{1}{1-\\\\gamma } \\\\cdot \\\\log \\\\left(\\\\sum _{k=1}^{K} q_{k}^{\\\\gamma }\\\\right)= \\\\frac{1}{1-\\\\gamma } \\\\cdot \\\\Big ( \\\\log \\\\left(H_{\\\\gamma }(\\\\mathbf {Q},{P}^{unif})\\\\right) + (1- \\\\gamma ) \\\\cdot \\\\log (K) \\\\Big )= \\\\frac{\\\\log 2}{1-\\\\gamma } \\\\cdot \\\\log _{2}\\\\left(\\\\sum _{k=1}^{K} q_{k}^{\\\\gamma }\\\\right)$ is BS-minimizable on $\\\\mathbf {\\\\Omega } = A \\\\cdot \\\\textrm {$$\\\\hspace{-6.544pt}$$}$ for $\\\\gamma <0$ , and BS-maximizable on $\\\\mathbf {\\\\Omega } = A \\\\cdot \\\\textrm {$$\\\\hspace{-6.544pt}$$}$ for $\\\\gamma \\\\in ]0,1[$ and $\\\\gamma \\\\ge 2$ ; the special subcase $A=1$ in (REF ) (and thus, $\\\\mathbf {Q} = {Q}$ is a probability vector) corresponds to the prominent (additive) $\\\\gamma -$ order Renyi entropy [309].\",\n \"As well known, there is a vast literature on Renyi entropies $\\\\mathcal {E}^{gR}({Q})$ .\",\n \"Some exemplary (mostly recent) studies and applications appear e.g.\",\n \"in Nath [270] — as well as in Arikan [15], Sundaresan [353], Bunte & Lapidoth [67], Sason & Verdu [321], Kumar et al.\",\n \"[205] — for coding and guessing, in Bennett et al.\",\n \"[39] in connection with unconditionally secure secret-key agreement protocols and quantum cryptography, in Mayoral [257] for cluster sampling, in Aviyente et al.\",\n \"[21] for information extraction in certain neurophysiological signals (so-called event-related potentials), in Tao et al.\",\n \"[355] as well as in Jiao et al.\",\n \"[171] for early defect/fault detection of rolling element bearings, in Pham et al.\",\n \": [292] for blind source separation, in Liu et al.\",\n \"[226] as well as in Rong et al.\",\n \"[311] to electron density functional theory (DFT) for quantum chemical reactivity, in Sason [319] for data compression, in Carravilla et al.\",\n \"[73] for the recognition of HIV-1 antibodies through STED microscopy and the corresponding design of therapeutic interventions, in Joshi et al.\",\n \"[175] for the identification and tracking of relevant T cell receptors for adoptive immunotherapy, in Erguzel et al.\",\n \"[121] for the investigation of Electroencephalography (EEG) signals of subjects suffering from some psychiatric disorders, in German–Sallo [133] for fault–characteristics extraction from discrete signals in manufacturing systems, in Schober et al.\",\n \"[323] for investigations of some evolutions of the T cell antigen receptor (TCR) repertoire, in Seweryn et al.\",\n \"[326] for the assessment of similarity and diversity of expression profiles in single cell systems, in Amezquita-Sanchez [13] for the detection of incipient damage in high-rise buildings subjected to dynamic vibrations, in Barennes et al.\",\n \"[32] for comparing the accuracy of current T cell receptor sequencing methods employed for the understanding of adaptive immune responses, in Kumar et al.\",\n \"[201] for the segmentation of digital images through multilevel iterative variational mode decomposition (VMD), and in Pandey [280] for the quantification of cosmic homogeneity.\",\n \"Remark 19 (i) For Renyi entropies there are also matrix versions $\\\\mathcal {E}^{gR}(X) :=\\\\frac{1}{1-\\\\gamma } \\\\cdot \\\\log \\\\left(\\\\sum _{i=1}^{K_{1}}\\\\sum _{j=1}^{K_{2}} x_{ij}^{\\\\gamma }\\\\right)$ where $X:=(x_{ij})_{i=1,\\\\ldots ,K_{1}}^{j=1,\\\\ldots ,K_{2}}$ is a $K_{1} \\\\times K_{2}-$ matrix whose elements $x_{ij}$ are (say) strictly positive and sum up to $A$ .\",\n \"Such a setup with $A=1$ is e.g.\",\n \"used in time-frequency analyses of signals where the $i$ ’s correspond to discrete time points, the $j$ ’s to discrete frequencies, and $x_{ij}$ to the probability that $(i,j)$ occurs; see e.g.\",\n \"Popescu & Aiordachioaie [295] for change detection in seismic signals.\",\n \"Another line of application is to use as $X$ the normalized communicability matrix of a directed network (respectively the upper triangular part of $X$ in case of an unweighted and undirected network).\",\n \"Of course, the matrix version $\\\\mathcal {E}^{gR}(X)$ can be easily and equivalently rewritten in our vector version $\\\\mathcal {E}^{gR}(\\\\mathbf {Q})$ by setting $\\\\mathbf {Q} := (q_{1}, \\\\ldots , q_{K_{1} \\\\cdot K_{2}})$ such that $x_{ij} = q_{(i-1)\\\\cdot K_{2} +j}$ ($i=1,\\\\ldots ,K_{1}$ , $j=1,\\\\ldots ,K_{2}$ and hence $K:= K_{1} \\\\cdot K_{2}$ ; accordingly, we can apply our BS method.\",\n \"(ii) The latter conversion works analogously also for matrix versions of all the other entropies, divergences, etc.\",\n \"of this paper; more flexible versions where $i \\\\in \\\\lbrace 1,\\\\ldots ,K_{1}\\\\rbrace $ , $j \\\\in J_{i}$ for some $J_{i} \\\\subseteq \\\\lbrace 1,\\\\ldots ,K_{2}\\\\rbrace $ as well as multidimensional-array/tensor versions can be transformed in a similar book-keeping manner, too.\",\n \"For instance, within the above-mentioned framework of unweighted and undirected networks, Chen et al.\",\n \"[81] and Shi et al.\",\n \"[332] employ communicability matrix versions of the Shannon entropy and the Jensen-Shannon divergence (JSD), e.g.\",\n \"in order to derive a new complexity measure of such kind of networks; see also Bagrow and Bollt [28] for similar network applications of the JSD.\",\n \"Moreover, Jena et al.\",\n \"[167] use “3D versions” of Tsallis entropies for brain magnetic resonance (MR) image segmentation.\",\n \"Remark 20 All the above cases which are BS-maximizable can be interpreted as bare-simulation approach to the solution of generalized maximum entropy problems on $\\\\mathbf {\\\\Omega } = A \\\\cdot \\\\textrm {$$\\\\hspace{-6.544pt}$$}$.\",\n \"Remark 21 (i) If (all) the above- and below-mentioned entropies are used for probability vectors ${Q} \\\\in \\\\mathbf {S}^{K}$ — i.e.\",\n \"one employs $\\\\mathcal {E}({Q})$ — then typically the components $q_{k}$ of ${Q}$ represent a genuine probability mass (frequency) $q_{k} = \\\\mathbb {\\\\Pi }[\\\\lbrace d_{k} \\\\rbrace ]$ of some data point (state) $d_{k}$ .\",\n \"However, ${Q} \\\\in \\\\mathbf {S}^{K}$ may alternatively be artificially generated.\",\n \"For instance, for the purpose of fault detections of mechanical drives, Boskoski & Juricic [57] use Renyi entropies where the $q_{k}$ ’s are normalized squared energy-describing coefficients of the wavelet packet transform of measured vibration records.\",\n \"Another exemplary “artificial” operation is concatenation, see e.g.\",\n \"Subsection REF below.\",\n \"(ii) An analogous statement holds for the employment of (all) the above- and below-mentioned divergences $D({Q},{P})$ — and their transformations — between genuine respectively artificially generated probability vectors ${Q}, {P} \\\\in \\\\mathbf {S}^{K}$ .\",\n \"The remaining parameter cases $\\\\gamma =0$ and $\\\\gamma =1$ can be treated analogously.\",\n \"For the sake of brevity, we only deal with the latter.\",\n \"For this, let $\\\\mathbf {Q} \\\\in A \\\\cdot \\\\mathbb {S}^{K}$ with $A := \\\\sum _{k=1}^{K} q_{k} >0$ and ${P} := {P}^{unif}$ .\",\n \"Clearly, $I(\\\\mathbf {Q},{P}^{unif}) - \\\\frac{\\\\log K}{K}=\\\\sum _{k=1}^{K} q_{k} \\\\cdot \\\\log (q_{k})$ ; thus the latter is BS-minimizable on $\\\\mathbf {\\\\Omega } = A \\\\cdot \\\\textrm {$$\\\\hspace{-6.544pt}$$}$ .\",\n \"More generally, for any continuous strictly increasing (respectively strictly decreasing) function $h_{1} : [- \\\\frac{K}{e}, 0[ \\\\, \\\\mapsto \\\\mathbb {R}$ , the quantity $h_{1}\\\\Big (\\\\sum _{k=1}^{K} q_{k} \\\\cdot \\\\log (q_{k})\\\\Big )$ is BS-minimizable on $\\\\mathbf {\\\\Omega } = A \\\\cdot \\\\textrm {$$\\\\hspace{-6.544pt}$$}$ (respectively BS-maximizable on $\\\\mathbf {\\\\Omega } = A \\\\cdot \\\\textrm {$$\\\\hspace{-6.544pt}$$}$ ).\",\n \"Important special cases are: $h_{1}(y) := h_{1}^{-Id}(y) = -y$ : the entropy $\\\\mathcal {E}^{Sh}(\\\\mathbf {Q}) :=h_{1}^{-Id}\\\\Big (\\\\sum _{k=1}^{K} q_{k} \\\\cdot \\\\log (q_{k})\\\\Big )= - \\\\sum _{k=1}^{K} q_{k} \\\\cdot \\\\log (q_{k})$ is BS-maximizable on $\\\\mathbf {\\\\Omega } = A \\\\cdot \\\\textrm {$$\\\\hspace{-6.544pt}$$}$ ; the special subcase $A=1$ in (REF ) (and thus, $\\\\mathbf {Q} = {Q}$ is a probability vector) corresponds to the omnipresent Shannon entropy; hence, by our bare-simulation approach we can particularly tackle maximum entropy problems on almost arbitrary sets $\\\\textrm {$$\\\\hspace{-6.544pt}$$}$ of probability vectors.\",\n \"Analogously, we can treat $\\\\frac{1}{\\\\log (K)} \\\\cdot \\\\mathcal {E}^{Sh}({Q})$ which is called Pielou’s evenness index [293], and $1-\\\\frac{1}{\\\\log (K)} \\\\cdot \\\\mathcal {E}^{Sh}({Q}) \\\\in [0,1]$ which is sometimes used as clonality (clonotype diversity) index (see e.g.\",\n \"Gabriel et al.\",\n \"[131] for applications to HIV-connected T cell receptor repertoires, and Bashford-Rogers et al.\",\n \"[33] (with supplementary private communication) for its use for comparative analyses of the BCR repertoire in immune-mediated diseases, for the sake of understanding pathological mechanisms and designing treatment strategies).\",\n \"As a further example for Remark REF , Lyubushin [236] uses $q_{k}$ ’s which are normalized squared coefficients of an orthogonal wavelet decomposition of some seismic noise, and accordingly, $\\\\frac{1}{\\\\log (K)} \\\\cdot \\\\mathcal {E}^{Sh}({Q})$ can be interpreted as the entropy of the distribution of energy of oscillations at various frequency and time scales.\",\n \"Some further exemplary studies and applications of the maximization of $\\\\mathcal {E}^{Sh}({Q})$ — aside from the vast physics literature — appear e.g.\",\n \"in De Santis et al.\",\n \"[106] for cryptanalytic guessing problems for breaking ciphertexts with probabilistic brute-force attacks, Johansson & Sternad [173] for tackling certain resource allocation problems under uncertainty, Marano & Franceschetti [246] for ray propagation in percolating lattices, Miao et al.\",\n \"[260] for unsupervised mixed-pixel decomposition in image processing, Rodrigues et al.\",\n \"[310] for modelling biological species geographic distribution, Xiong et al.\",\n \"[400] for capturing desirable phrasal and hierarchical segmentations within a statistical machine translation context, Chan et al.\",\n \"[76] for alignment-free DNA sequence comparison, Mann & Garnett [244] for capturing some collective behaviours of intelligent agents in social interactions, Singh et al.\",\n \"[336] for the study of finite buffer queueing systems, Baddeley [27] for geoscientifical prediction of the occurrence of mineral deposits on regional scales, Einicke et al.\",\n \"[118] for feature selection within change classification during running, and Han et al.\",\n \"[152] for substructure imaging of blood cells by means of maximum entropy tomography (MET).\",\n \"$s \\\\in \\\\, ]0,1[ \\\\, \\\\cup \\\\, ]1,\\\\infty [$ , $h_{1}(y) := h_{s}^{SM2}(y) := \\\\frac{e^{(s-1) \\\\cdot y} -1}{1-s}$ (cf.\",\n \"(REF )) with $y \\\\in \\\\mathbb {R}$ : the entropy $\\\\mathcal {E}^{SM2}(\\\\mathbf {Q}) :=h_{s}^{SM2}\\\\Big (\\\\sum _{k=1}^{K} q_{k} \\\\cdot \\\\log (q_{k})\\\\Big )= \\\\frac{1}{1-s} \\\\cdot \\\\left( \\\\exp \\\\Big \\\\lbrace (s-1) \\\\cdot \\\\sum _{k=1}^{K} q_{k} \\\\cdot \\\\log (q_{k})\\\\Big \\\\rbrace -1 \\\\right)$ is BS-maximizable on $\\\\mathbf {\\\\Omega } = A \\\\cdot \\\\textrm {$$\\\\hspace{-6.544pt}$$}$ ; the special subcase $A=1$ in (REF ) (and thus, $\\\\mathbf {Q} = {Q}$ is a probability vector) corresponds to the (second type) entropy of Sharma & Mittal [329] in the scaling of e.g.\",\n \"Pardo [282] (p.20).\",\n \"Returning to the general context, we now (as already indicated above) state explicitly the corresponding bare-simulation-minimizations (respectively maximizations) of the power divergences $\\\\inf _{\\\\mathbf {Q}\\\\in \\\\textrm {\\\\Omega \\\\hspace{-5.40608pt}\\\\Omega } }D_{\\\\widetilde{c}\\\\cdot \\\\varphi _{\\\\gamma }}(\\\\mathbf {Q},{P})$ ($\\\\gamma \\\\in \\\\mathbb {R}$ ), the Renyi divergences $\\\\inf _{\\\\mathbf {Q}\\\\in \\\\textrm {\\\\Omega \\\\hspace{-5.40608pt}\\\\Omega } }R_{\\\\gamma }(\\\\mathbf {Q},{P})$ ($\\\\gamma \\\\in \\\\mathbb {R}$ ), the Hellinger integrals $\\\\inf _{\\\\mathbf {Q}\\\\in A \\\\cdot \\\\textrm {\\\\Omega \\\\hspace{-5.40608pt}\\\\Omega } }H_{\\\\gamma }(\\\\mathbf {Q},{P})$ ($\\\\gamma \\\\in ]-\\\\infty ,0[ \\\\, \\\\cup \\\\, ]1,\\\\infty [$ ), $\\\\sup _{\\\\mathbf {Q}\\\\in A \\\\cdot \\\\textrm {\\\\Omega \\\\hspace{-5.40608pt}\\\\Omega } }H_{\\\\gamma }(\\\\mathbf {Q},{P})$ ($\\\\gamma \\\\in ]0,1[$ ), the modified Kullback-Leibler information $\\\\inf _{Q\\\\in A \\\\cdot \\\\textrm {\\\\Omega \\\\hspace{-5.40608pt}\\\\Omega } }I(\\\\mathbf {Q},{P})$ , the modified reverse Kullback-Leibler information $\\\\inf _{Q\\\\in A \\\\cdot \\\\textrm {\\\\Omega \\\\hspace{-5.40608pt}\\\\Omega } }\\\\widetilde{I}(\\\\mathbf {Q},{P})$ , as well as the above-mentioned measures of entropy (diversity).\",\n \"Since the corresponding probability distribution $\\\\mathbb {}[\\\\cdot \\\\,]=\\\\mathbb {\\\\Pi }[W_{1}\\\\in \\\\cdot \\\\,]$ of the $W_{i}$ 's (cf.\",\n \"the representability (REF )) varies “quite drastically” with $\\\\gamma $ , we split this issue into several pieces.\",\n \"Proposition 22 (a) Consider the context of Remark REF (vi) for $\\\\varphi := \\\\widetilde{c}\\\\cdot \\\\varphi _{\\\\gamma }$ with $\\\\gamma <0$ , and let ${P} \\\\in \\\\mathbb {S}_{>0}^{K}$ as well as $\\\\widetilde{c}>0$ be arbitrary but fixed.\",\n \"Furthermore, let $W:=(W_{i})_{i\\\\in \\\\mathbb {N}}$ be an i.i.d.\",\n \"sequence of non-negative real-valued random variables having densityin the classical sense, with respect to Lebesgue measure $f_{W_{1}}(y)\\\\ :=\\\\ \\\\frac{\\\\exp \\\\lbrace -\\\\frac{y \\\\cdot \\\\widetilde{c}}{1-\\\\gamma }\\\\rbrace }{\\\\exp \\\\lbrace \\\\widetilde{c}/\\\\gamma \\\\rbrace }\\\\cdot f_{Z}(y) \\\\cdot {1}_{]-\\\\infty ,0[}(y), \\\\qquad y \\\\in \\\\mathbb {R},$ where $f_{Z}$ is the density of a random variable $Z$ which has stable law with parameter-quadruple $(\\\\frac{-\\\\gamma }{1-\\\\gamma },1,0,-\\\\frac{\\\\widetilde{c}^{1/(1-\\\\gamma )} \\\\cdot (1-\\\\gamma )^{-\\\\gamma /(1-\\\\gamma )}}{\\\\gamma })$ in terms of “form-B notation” in Zolotarev [428], p.12.\",\n \"Then for all $A>0$ and all $\\\\Omega $$\\\\Omega \\\\subset \\\\mathbb {S}_{>0}^{K}$ with (REF ) there holds $\\\\hspace{-19.91684pt}-\\\\lim _{n\\\\rightarrow \\\\infty }\\\\frac{1}{n}\\\\log \\\\,\\\\mathbb {\\\\Pi }\\\\left[\\\\xi _{n}^{w\\\\mathbf {W}}\\\\in \\\\textrm {\\\\right.\\\\Omega \\\\hspace{-6.544pt}\\\\Omega } =\\\\inf _{\\\\mathbf {Q}\\\\in A \\\\cdot \\\\textrm {\\\\Omega \\\\hspace{-5.40608pt}\\\\Omega }}\\\\frac{\\\\widetilde{c}}{\\\\gamma }\\\\cdot \\\\left[ 1-A^{\\\\gamma /(\\\\gamma -1)} \\\\cdot \\\\left[ 1+ \\\\gamma \\\\cdot (A-1) +\\\\frac{\\\\gamma \\\\cdot \\\\left( \\\\gamma -1\\\\right) }{\\\\widetilde{c}}\\\\cdot D_{\\\\widetilde{c}\\\\cdot \\\\varphi _{\\\\gamma }}(\\\\mathbf {Q},{P})\\\\right]^{-1/\\\\left( \\\\gamma -1\\\\right) }\\\\right]$ as well as the BS minimizabilities/maximizabilites (cf.\",\n \"Definition REF ) $& & \\\\hspace{-25.6073pt}\\\\inf _{\\\\mathbf {Q}\\\\in A \\\\cdot \\\\textrm {\\\\Omega \\\\hspace{-5.40608pt}\\\\Omega }}D_{\\\\widetilde{c}\\\\cdot \\\\varphi _{\\\\gamma }}(\\\\mathbf {Q},{P})=\\\\lim _{n\\\\rightarrow \\\\infty }\\\\frac{\\\\widetilde{c}}{\\\\gamma \\\\cdot \\\\left(\\\\gamma -1\\\\right) }\\\\cdot \\\\left\\\\lbrace A^{\\\\gamma } \\\\cdot \\\\left( 1+\\\\frac{\\\\gamma }{\\\\widetilde{c}}\\\\cdot \\\\frac{1}{n}\\\\cdot \\\\log \\\\,\\\\mathbb {\\\\Pi }\\\\left[\\\\xi _{n}^{w\\\\mathbf {W}}\\\\in \\\\textrm {\\\\right.\\\\right.\\\\right.\\\\Omega \\\\hspace{-6.544pt}\\\\Omega } ^{1-\\\\gamma } + \\\\gamma \\\\cdot (1-A) -1 ,\\\\qquad \\\\ \\\\\\\\& & \\\\hspace{-25.6073pt}\\\\inf _{\\\\mathbf {Q}\\\\in A \\\\cdot \\\\textrm {\\\\Omega \\\\hspace{-5.40608pt}\\\\Omega }}H_{\\\\gamma }(\\\\mathbf {Q},{P})=\\\\lim _{n\\\\rightarrow \\\\infty } A^{\\\\gamma } \\\\cdot \\\\left( 1+ \\\\gamma \\\\cdot \\\\frac{1}{n}\\\\cdot \\\\log \\\\,\\\\mathbb {\\\\Pi }\\\\left[\\\\breve{\\\\xi }_{n}^{w\\\\mathbf {W}}\\\\in \\\\textrm {\\\\right.\\\\right.\\\\Omega \\\\hspace{-6.544pt}\\\\Omega } ^{1-\\\\gamma } ,\\\\qquad \\\\ \\\\\\\\& & \\\\hspace{-25.6073pt}\\\\inf _{\\\\mathbf {Q}\\\\in A \\\\cdot \\\\textrm {\\\\Omega \\\\hspace{-5.40608pt}\\\\Omega }}c_{1} \\\\cdot \\\\Big (H_{\\\\gamma }(\\\\mathbf {Q},{P})^{c_{2}} - c_{3} \\\\Big )=\\\\lim _{n\\\\rightarrow \\\\infty } c_{1} \\\\cdot \\\\left\\\\lbrace A^{c_{2} \\\\cdot \\\\gamma } \\\\cdot \\\\left( 1+\\\\frac{\\\\gamma }{n}\\\\cdot \\\\log \\\\,\\\\mathbb {\\\\Pi }\\\\left[\\\\breve{\\\\xi }_{n}^{w\\\\mathbf {W}}\\\\in \\\\textrm {\\\\right.\\\\right.\\\\right.\\\\Omega \\\\hspace{-6.544pt}\\\\Omega } ^{c_{2} \\\\cdot (1-\\\\gamma )} - c_{3} ,\\\\ \\\\ \\\\textrm {if c_{1} \\\\cdot c_{2} >0, c_{3} \\\\in \\\\mathbb {R}}, \\\\quad \\\\ \\\\\\\\& & \\\\hspace{-25.6073pt}\\\\sup _{\\\\mathbf {Q}\\\\in A \\\\cdot \\\\textrm {\\\\Omega \\\\hspace{-5.40608pt}\\\\Omega }}c_{1} \\\\cdot \\\\Big (H_{\\\\gamma }(\\\\mathbf {Q},{P})^{c_{2}} - c_{3} \\\\Big )=\\\\lim _{n\\\\rightarrow \\\\infty } c_{1} \\\\cdot \\\\left\\\\lbrace A^{c_{2} \\\\cdot \\\\gamma } \\\\cdot \\\\left( 1+\\\\frac{\\\\gamma }{n}\\\\cdot \\\\log \\\\,\\\\mathbb {\\\\Pi }\\\\left[\\\\breve{\\\\xi }_{n}^{w\\\\mathbf {W}}\\\\in \\\\textrm {\\\\right.\\\\right.\\\\right.\\\\Omega \\\\hspace{-6.544pt}\\\\Omega } ^{c_{2} \\\\cdot (1-\\\\gamma )} - c_{3} ,\\\\ \\\\ \\\\textrm {if c_{1} \\\\cdot c_{2} < 0, c_{3} \\\\in \\\\mathbb {R}}, \\\\quad \\\\ \\\\\\\\& & \\\\hspace{-25.6073pt}\\\\inf _{\\\\mathbf {Q}\\\\in A \\\\cdot \\\\textrm {\\\\Omega \\\\hspace{-5.40608pt}\\\\Omega }}R_{\\\\gamma }(\\\\mathbf {Q},{P})= \\\\lim _{n\\\\rightarrow \\\\infty } \\\\frac{1}{\\\\gamma \\\\cdot (\\\\gamma -1)} \\\\cdot \\\\log \\\\Big ( A^{\\\\gamma } \\\\cdot \\\\Big ( 1+ \\\\gamma \\\\cdot \\\\frac{1}{n}\\\\cdot \\\\log \\\\,\\\\mathbb {\\\\Pi }\\\\left[\\\\breve{\\\\xi }_{n}^{w\\\\mathbf {W}}\\\\in \\\\textrm {\\\\right.\\\\Omega \\\\hspace{-6.544pt}\\\\Omega } \\\\Big ) ^{1-\\\\gamma } \\\\Big ),\\\\\\\\& & \\\\hspace{-25.6073pt}\\\\inf _{\\\\mathbf {Q}\\\\in A \\\\cdot \\\\textrm {\\\\Omega \\\\hspace{-5.40608pt}\\\\Omega }}c_{1} \\\\cdot \\\\Big (\\\\Big (\\\\sum _{k=1}^{K} q_{k}^{\\\\gamma }\\\\Big )^{c_{2}} - c_{3} \\\\Big )=\\\\lim _{n\\\\rightarrow \\\\infty } c_{1} \\\\cdot \\\\left\\\\lbrace K^{c_{2}\\\\cdot (1-\\\\gamma )} \\\\cdot A^{c_{2} \\\\cdot \\\\gamma } \\\\cdot \\\\left( 1+\\\\frac{\\\\gamma }{n}\\\\cdot \\\\log \\\\,\\\\mathbb {\\\\Pi }\\\\left[\\\\breve{\\\\breve{\\\\xi }}_{n}^{w\\\\mathbf {W}}\\\\in \\\\textrm {\\\\right.\\\\right.\\\\right.\\\\Omega \\\\hspace{-6.544pt}\\\\Omega } ^{c_{2} \\\\cdot (1-\\\\gamma )} - c_{3} ,\\\\ \\\\ \\\\nonumber \\\\\\\\& & \\\\hspace{369.88582pt}\\\\textrm {if c_{1} \\\\cdot c_{2} >0, c_{3} \\\\in \\\\mathbb {R}}, \\\\quad \\\\ $ $& & \\\\hspace{-25.6073pt}\\\\sup _{\\\\mathbf {Q}\\\\in A \\\\cdot \\\\textrm {\\\\Omega \\\\hspace{-5.40608pt}\\\\Omega }}c_{1} \\\\cdot \\\\Big (\\\\Big (\\\\sum _{k=1}^{K} q_{k}^{\\\\gamma }\\\\Big )^{c_{2}} - c_{3} \\\\Big )=\\\\lim _{n\\\\rightarrow \\\\infty } c_{1} \\\\cdot \\\\left\\\\lbrace K^{c_{2}\\\\cdot (1-\\\\gamma )} \\\\cdot A^{c_{2} \\\\cdot \\\\gamma } \\\\cdot \\\\left( 1+\\\\frac{\\\\gamma }{n}\\\\cdot \\\\log \\\\,\\\\mathbb {\\\\Pi }\\\\left[\\\\breve{\\\\breve{\\\\xi }}_{n}^{w\\\\mathbf {W}}\\\\in \\\\textrm {\\\\right.\\\\right.\\\\right.\\\\Omega \\\\hspace{-6.544pt}\\\\Omega } ^{c_{2} \\\\cdot (1-\\\\gamma )} - c_{3} ,\\\\ \\\\ \\\\nonumber \\\\\\\\& & \\\\hspace{369.88582pt}\\\\textrm {if c_{1} \\\\cdot c_{2} < 0, c_{3} \\\\in \\\\mathbb {R}}, \\\\quad \\\\ \\\\\\\\& & \\\\hspace{-25.6073pt}\\\\inf _{\\\\mathbf {Q}\\\\in A \\\\cdot \\\\textrm {\\\\Omega \\\\hspace{-5.40608pt}\\\\Omega }}\\\\frac{1}{1-\\\\gamma } \\\\cdot \\\\log \\\\Big (\\\\sum _{k=1}^{K} q_{k}^{\\\\gamma }\\\\Big )= \\\\lim _{n\\\\rightarrow \\\\infty } \\\\frac{1}{\\\\gamma \\\\cdot (\\\\gamma -1)} \\\\cdot \\\\left[\\\\log \\\\Big ( A^{\\\\gamma } \\\\cdot \\\\Big ( 1+ \\\\gamma \\\\cdot \\\\frac{1}{n}\\\\cdot \\\\log \\\\,\\\\mathbb {\\\\Pi }\\\\left[\\\\breve{\\\\breve{\\\\xi }}_{n}^{w\\\\mathbf {W}}\\\\in \\\\textrm {\\\\right.\\\\right.\\\\Omega \\\\hspace{-6.544pt}\\\\Omega } \\\\Big ) ^{1-\\\\gamma } \\\\Big ) + (1-\\\\gamma ) \\\\cdot \\\\log (K) ,$ where $\\\\xi _{n}^{w\\\\mathbf {W}}$ is the normalized randomly weighted empirical measure given in (REF ), $\\\\breve{\\\\xi }_{n}^{w\\\\mathbf {W}}$ is its special case for $\\\\widetilde{c}=1$ , and $\\\\breve{\\\\breve{\\\\xi }}_{n}^{w\\\\mathbf {W}}$ is its special case for $\\\\widetilde{c}=1$ together with ${P} = {P}^{unif}$ the latter two notations will be also used in the following Propositions REF to REF.\",\n \"From this, the BS-minimizability/maximizability of the important norms/entropies/diversity indices (E1) to (E6) follow immediately as special cases.\",\n \"(b) The special case $\\\\varphi := \\\\widetilde{c}\\\\cdot \\\\varphi _{\\\\gamma }$ ($\\\\gamma <0$ ) of Theorem REF works analogously to (a), with the differences that we employ (i) additionally a sequence $(X_{i})_{i\\\\in \\\\mathbb {N}}$ of random variables being independent of $(W_{i})_{i\\\\in \\\\mathbb {N}}$ and satisfying condition (REF ) (resp.\",\n \"(REF )), (ii) $A=1$ (instead of arbitrary $A>0$ ), (iii) $\\\\mathbb {\\\\Pi }_{X_{1}^{n}}[\\\\cdot ]$ (instead of $\\\\mathbb {\\\\Pi }[\\\\cdot ]$ ), (iv) $\\\\xi _{n,\\\\mathbf {X}}^{w\\\\mathbf {W}}$ (instead of $\\\\xi _{n}^{w\\\\mathbf {W}}$ ), (v) $\\\\breve{\\\\xi }_{n,\\\\mathbf {X}}^{w\\\\mathbf {W}}$ (instead of $\\\\breve{\\\\xi }_{n}^{w\\\\mathbf {W}}$ ), and (vi) $\\\\breve{\\\\breve{\\\\xi }}_{n,\\\\mathbf {X}}^{w\\\\mathbf {W}}$ (instead of $\\\\breve{\\\\breve{\\\\xi }}_{n}^{w\\\\mathbf {W}}$ ).\",\n \"The assertions of Proposition REF can be deduced from Theorem REF , Remark REF (vi), Lemma REF (a), (REF ), (REF ), (REF ), () and the below-mentioned $\\\\mathbb {}-$ concerning Example REF (d).\",\n \"Employing Lemma REF (c) and Example REF (a) instead, one ends up with the following proposition on the reverse Kullback-Leibler divergence: Proposition 23 (a) Consider the context of Remark REF (vi) for $\\\\varphi := \\\\widetilde{c}\\\\cdot \\\\varphi _{\\\\gamma }$ with $\\\\gamma =0$ , and let ${P} \\\\in \\\\mathbb {S}_{>0}^{K}$ as well as $\\\\widetilde{c}>0$ be arbitrary but fixed.\",\n \"Furthermore, let $W:=(W_{i})_{i\\\\in \\\\mathbb {N}}$ be an i.i.d.\",\n \"sequence of non-negative real-valued random variables with Gamma distribution $\\\\mathbb {} =GAM(\\\\widetilde{c},\\\\widetilde{c})$ here and henceforth, we use the notation that a Gamma distribution $GAM(\\\\alpha ,\\\\beta )$ with rate parameter (inverse scale parameter) $\\\\alpha >0$ and shape parameter $\\\\beta >0$ has (Lebesgue-)density $f(y) := \\\\frac{\\\\alpha ^{\\\\beta } \\\\cdot y^{\\\\beta -1} \\\\cdot e^{-\\\\alpha \\\\cdot y} }{\\\\Gamma (\\\\beta )}\\\\cdot {1}_{]0,\\\\infty [}(y)$ , $y \\\\in \\\\mathbb {R}$ ; its cumulant generating function is $\\\\Lambda (z) = \\\\beta \\\\cdot \\\\log (\\\\frac{\\\\alpha }{\\\\alpha -z})$ for $z \\\\in ]-\\\\infty ,\\\\alpha [$ (and $\\\\Lambda (z) = \\\\infty $ for $z \\\\ge \\\\alpha $ ).\",\n \"(where the subcase $\\\\widetilde{c}=1$ is the exponential distribution $\\\\mathbb {} =EXP(1)$ with mean 1).\",\n \"Then for all $A >0$ and all $\\\\Omega $$\\\\Omega \\\\subset \\\\mathbb {S}_{>0}^{K}$ with (REF ) there holds the BS minimizabilites (cf.\",\n \"(REF )) $& & \\\\inf _{{Q}\\\\in A \\\\cdot \\\\textrm {\\\\Omega \\\\hspace{-5.40608pt}\\\\Omega }}D_{\\\\widetilde{c}\\\\cdot \\\\varphi _{0}}({Q},{P}) =-\\\\lim _{n\\\\rightarrow \\\\infty }\\\\frac{1}{n} \\\\log \\\\,\\\\mathbb {\\\\Pi }\\\\left[\\\\xi _{n}^{w\\\\mathbf {W}}\\\\in \\\\textrm {\\\\right.\\\\Omega \\\\hspace{-6.544pt}\\\\Omega } + \\\\widetilde{c} \\\\cdot (A-1- \\\\log A),\\\\\\\\& & \\\\inf _{{Q}\\\\in A \\\\cdot \\\\textrm {\\\\Omega \\\\hspace{-5.40608pt}\\\\Omega }}\\\\widetilde{I}({Q},{P}) =\\\\inf _{{Q}\\\\in A \\\\cdot \\\\textrm {\\\\Omega \\\\hspace{-5.40608pt}\\\\Omega }}\\\\sum \\\\displaylimits _{k=1}^{K} p_{k} \\\\cdot \\\\log \\\\left( \\\\frac{p_{k}}{q_{k}} \\\\right)= -\\\\lim _{n\\\\rightarrow \\\\infty }\\\\frac{1}{n} \\\\log \\\\,\\\\mathbb {\\\\Pi }\\\\left[\\\\xi _{n}^{w\\\\mathbf {W}}\\\\in \\\\textrm {\\\\right.\\\\Omega \\\\hspace{-6.544pt}\\\\Omega } - \\\\log A .\\\\nonumber $ (b) The special case $\\\\varphi := \\\\widetilde{c}\\\\cdot \\\\varphi _{\\\\gamma }$ ($\\\\gamma =0$ ) of Theorem REF works analogously to (a), with the differences that we employ (i) additionally a sequence $(X_{i})_{i\\\\in \\\\mathbb {N}}$ of random variables being independent of $(W_{i})_{i\\\\in \\\\mathbb {N}}$ and satisfying condition (REF ) (resp.\",\n \"(REF )), (ii) $A=1$ (instead of arbitrary $A>0$ ), (iii) $\\\\mathbb {\\\\Pi }_{X_{1}^{n}}[\\\\cdot ]$ (instead of $\\\\mathbb {\\\\Pi }[\\\\cdot ]$ ), and (iv) $\\\\xi _{n,\\\\mathbf {X}}^{w\\\\mathbf {W}}$ (instead of $\\\\xi _{n}^{w\\\\mathbf {W}}$ ).\",\n \"Proposition 24 (a) Consider the context of Remark REF (vi) for $\\\\varphi := \\\\widetilde{c}\\\\cdot \\\\varphi _{\\\\gamma }$ with $\\\\gamma \\\\in ]0,1[$ , and let ${P} \\\\in \\\\mathbb {S}_{>0}^{K}$ as well as $\\\\widetilde{c}>0$ be arbitrary but fixed.\",\n \"Furthermore, let $W:=(W_{i})_{i\\\\in \\\\mathbb {N}}$ be an i.i.d.\",\n \"sequence of non-negative real-valued random variables with Compound-Poisson-Gamma distribution $\\\\mathbb {} =C(POI(\\\\theta ),GAM(\\\\alpha ,\\\\beta ))$ having parameters $\\\\theta =\\\\frac{\\\\widetilde{c}}{\\\\gamma }>0$ , $\\\\alpha =\\\\frac{\\\\widetilde{c}}{1-\\\\gamma }>0$ , $\\\\beta =\\\\frac{\\\\gamma }{1-\\\\gamma }>0$ ; in other words, the $W_{i}$ are independent copies of a random variable $W_{1}:=\\\\sum _{j=1}^{N}\\\\overline{W}_{j}$ with the usual convention $\\\\sum _{i=1}^{0}\\\\overline{W}_{i}:=0$ constituted of some i.i.d.\",\n \"sequence $(\\\\overline{W}_{j})_{j\\\\in \\\\mathbb {N}}$ of $Gamma(\\\\alpha ,\\\\beta )-$ distributed random variables and some independent $POI(\\\\theta )-$ distributed random variable $N$ .\",\n \"Then for all $A >0$ and all $\\\\Omega $$\\\\Omega \\\\subset \\\\mathbb {S}^{K}$ with (REF ) there hold (REF ), (REF ), (), () as well as $& & \\\\hspace{-25.6073pt}\\\\sup _{\\\\mathbf {Q}\\\\in A \\\\cdot \\\\textrm {\\\\Omega \\\\hspace{-5.40608pt}\\\\Omega }}H_{\\\\gamma }(\\\\mathbf {Q},{P})=\\\\lim _{n\\\\rightarrow \\\\infty } A^{\\\\gamma } \\\\cdot \\\\left( 1+ \\\\gamma \\\\cdot \\\\frac{1}{n}\\\\cdot \\\\log \\\\,\\\\mathbb {\\\\Pi }\\\\left[\\\\breve{\\\\xi }_{n}^{w\\\\mathbf {W}}\\\\in \\\\textrm {\\\\right.\\\\right.\\\\Omega \\\\hspace{-6.544pt}\\\\Omega } ^{1-\\\\gamma } ,\\\\qquad \\\\ \\\\\\\\& & \\\\hspace{-25.6073pt}\\\\sup _{\\\\mathbf {Q}\\\\in A \\\\cdot \\\\textrm {\\\\Omega \\\\hspace{-5.40608pt}\\\\Omega }}c_{1} \\\\cdot \\\\Big (H_{\\\\gamma }(\\\\mathbf {Q},{P})^{c_{2}} - c_{3} \\\\Big )=\\\\lim _{n\\\\rightarrow \\\\infty } c_{1} \\\\cdot \\\\left\\\\lbrace A^{c_{2} \\\\cdot \\\\gamma } \\\\cdot \\\\left( 1+\\\\frac{\\\\gamma }{n}\\\\cdot \\\\log \\\\,\\\\mathbb {\\\\Pi }\\\\left[\\\\breve{\\\\xi }_{n}^{w\\\\mathbf {W}}\\\\in \\\\textrm {\\\\right.\\\\right.\\\\right.\\\\Omega \\\\hspace{-6.544pt}\\\\Omega } ^{c_{2} \\\\cdot (1-\\\\gamma )} - c_{3} ,\\\\ \\\\ \\\\textrm {if c_{1} \\\\cdot c_{2} >0, c_{3} \\\\in \\\\mathbb {R}}, \\\\quad \\\\ \\\\\\\\& & \\\\hspace{-25.6073pt}\\\\inf _{\\\\mathbf {Q}\\\\in A \\\\cdot \\\\textrm {\\\\Omega \\\\hspace{-5.40608pt}\\\\Omega }}c_{1} \\\\cdot \\\\Big (H_{\\\\gamma }(\\\\mathbf {Q},{P})^{c_{2}} - c_{3} \\\\Big )=\\\\lim _{n\\\\rightarrow \\\\infty } c_{1} \\\\cdot \\\\left\\\\lbrace A^{c_{2} \\\\cdot \\\\gamma } \\\\cdot \\\\left( 1+\\\\frac{\\\\gamma }{n}\\\\cdot \\\\log \\\\,\\\\mathbb {\\\\Pi }\\\\left[\\\\breve{\\\\xi }_{n}^{w\\\\mathbf {W}}\\\\in \\\\textrm {\\\\right.\\\\right.\\\\right.\\\\Omega \\\\hspace{-6.544pt}\\\\Omega } ^{c_{2} \\\\cdot (1-\\\\gamma )} - c_{3} ,\\\\ \\\\ \\\\textrm {if c_{1} \\\\cdot c_{2} < 0, c_{3} \\\\in \\\\mathbb {R}}, \\\\quad \\\\ \\\\\\\\& & \\\\hspace{-25.6073pt}\\\\sup _{\\\\mathbf {Q}\\\\in A \\\\cdot \\\\textrm {\\\\Omega \\\\hspace{-5.40608pt}\\\\Omega }}c_{1} \\\\cdot \\\\Big (\\\\Big (\\\\sum _{k=1}^{K} q_{k}^{\\\\gamma }\\\\Big )^{c_{2}} - c_{3} \\\\Big )=\\\\lim _{n\\\\rightarrow \\\\infty } c_{1} \\\\cdot \\\\left\\\\lbrace K^{c_{2}\\\\cdot (1-\\\\gamma )} \\\\cdot A^{c_{2} \\\\cdot \\\\gamma } \\\\cdot \\\\left( 1+\\\\frac{\\\\gamma }{n}\\\\cdot \\\\log \\\\,\\\\mathbb {\\\\Pi }\\\\left[\\\\breve{\\\\breve{\\\\xi }}_{n}^{w\\\\mathbf {W}}\\\\in \\\\textrm {\\\\right.\\\\right.\\\\right.\\\\Omega \\\\hspace{-6.544pt}\\\\Omega } ^{c_{2} \\\\cdot (1-\\\\gamma )} - c_{3} ,\\\\ \\\\ \\\\nonumber \\\\\\\\& & \\\\hspace{375.57628pt}\\\\textrm {if c_{1} \\\\cdot c_{2} >0, c_{3} \\\\in \\\\mathbb {R}}, \\\\quad \\\\ $ $& & \\\\hspace{-25.6073pt}\\\\inf _{\\\\mathbf {Q}\\\\in A \\\\cdot \\\\textrm {\\\\Omega \\\\hspace{-5.40608pt}\\\\Omega }}c_{1} \\\\cdot \\\\Big (\\\\Big (\\\\sum _{k=1}^{K} q_{k}^{\\\\gamma }\\\\Big )^{c_{2}} - c_{3} \\\\Big )=\\\\lim _{n\\\\rightarrow \\\\infty } c_{1} \\\\cdot \\\\left\\\\lbrace K^{c_{2}\\\\cdot (1-\\\\gamma )} \\\\cdot A^{c_{2} \\\\cdot \\\\gamma } \\\\cdot \\\\left( 1+\\\\frac{\\\\gamma }{n}\\\\cdot \\\\log \\\\,\\\\mathbb {\\\\Pi }\\\\left[\\\\breve{\\\\breve{\\\\xi }}_{n}^{w\\\\mathbf {W}}\\\\in \\\\textrm {\\\\right.\\\\right.\\\\right.\\\\Omega \\\\hspace{-6.544pt}\\\\Omega } ^{c_{2} \\\\cdot (1-\\\\gamma )} - c_{3} ,\\\\ \\\\ \\\\nonumber \\\\\\\\& & \\\\hspace{375.57628pt}\\\\textrm {if c_{1} \\\\cdot c_{2} < 0, c_{3} \\\\in \\\\mathbb {R}} .\",\n \"\\\\quad \\\\ $ From this, the BS-minimizability/maximizability of the important norms/entropies/diversity indices (E1) to (E6) follows immediately as special cases.\",\n \"(b) The special case $\\\\varphi := \\\\widetilde{c}\\\\cdot \\\\varphi _{\\\\gamma }$ ($\\\\gamma \\\\in ]0,1[$ ) of Theorem REF works analogously to (a), with the differences that we employ (i) additionally a sequence $(X_{i})_{i\\\\in \\\\mathbb {N}}$ of random variables being independent of $(W_{i})_{i\\\\in \\\\mathbb {N}}$ and satisfying condition (REF ) (resp.\",\n \"(REF )), (ii) $A=1$ (instead of arbitrary $A>0$ ), (iii) $\\\\mathbb {\\\\Pi }_{X_{1}^{n}}[\\\\cdot ]$ (instead of $\\\\mathbb {\\\\Pi }[\\\\cdot ]$ ), (iv) $\\\\xi _{n,\\\\mathbf {X}}^{w\\\\mathbf {W}}$ (instead of $\\\\xi _{n}^{w\\\\mathbf {W}}$ ), (v) $\\\\breve{\\\\xi }_{n,\\\\mathbf {X}}^{w\\\\mathbf {W}}$ (instead of $\\\\breve{\\\\xi }_{n}^{w\\\\mathbf {W}}$ ), and (vi) $\\\\breve{\\\\breve{\\\\xi }}_{n,\\\\mathbf {X}}^{w\\\\mathbf {W}}$ (instead of $\\\\breve{\\\\breve{\\\\xi }}_{n}^{w\\\\mathbf {W}}$ ).\",\n \"This follows from Theorem REF , Remark REF (vi), Lemma REF (a), (REF ), (REF ), (REF ), () and the below-mentioned $\\\\mathbb {}-$ concerning Example REF (b).\",\n \"Employing Lemma REF (b) and Example REF (a) instead, one ends up with the following proposition on the Kullback-Leibler divergence: Proposition 25 (a) Consider the context of Remark REF (vi) for $\\\\varphi := \\\\widetilde{c}\\\\cdot \\\\varphi _{\\\\gamma }$ with $\\\\gamma =1$ , and let ${P} \\\\in \\\\mathbb {S}_{>0}^{K}$ as well as $\\\\widetilde{c}>0$ be arbitrary but fixed.\",\n \"Furthermore, let $W:=(W_{i})_{i\\\\in \\\\mathbb {N}}$ be an i.i.d.\",\n \"sequence of non-negative real-valued random variables with distribution $\\\\mathbb {} =\\\\frac{1}{\\\\widetilde{c}}\\\\cdot POI(\\\\widetilde{c})$ being the “$\\\\frac{1}{\\\\widetilde{c}}-$ fold Poisson distribution with mean $\\\\widetilde{c}$ ” , which means that $W_{1}=\\\\frac{1}{\\\\widetilde{c}}\\\\cdot Z$ for a Poissonian $POI(\\\\widetilde{c})-$ distributed random variable $Z$ with mean $\\\\widetilde{c}$ (where the subcase $\\\\widetilde{c}=1$ amounts to $\\\\mathbb {} =POI(1)$ ).\",\n \"Then for all $A >0$ and all $\\\\Omega $$\\\\Omega \\\\subset \\\\mathbb {S}^{K}$ with (REF ) there holds $\\\\hspace{-19.91684pt}-\\\\lim _{n\\\\rightarrow \\\\infty }\\\\frac{1}{n} \\\\log \\\\,\\\\mathbb {\\\\Pi }\\\\left[\\\\xi _{n}^{w\\\\mathbf {W}}\\\\in \\\\textrm {\\\\right.\\\\Omega \\\\hspace{-6.544pt}\\\\Omega } &=& \\\\inf _{{Q}\\\\in A \\\\cdot \\\\textrm {\\\\Omega \\\\hspace{-5.40608pt}\\\\Omega } }\\\\ \\\\widetilde{c}\\\\cdot \\\\left[ 1-A \\\\cdot \\\\exp \\\\left( -\\\\frac{1}{A \\\\cdot \\\\widetilde{c}}\\\\cdot D_{\\\\widetilde{c}\\\\cdot \\\\varphi _{1}}(\\\\mathbf {Q},{P}) + \\\\frac{1}{A} -1 \\\\right) \\\\right]$ and the BS minimizabilities/maximizabilites (cf.\",\n \"Definition REF ) $& & \\\\hspace{-34.14322pt}\\\\inf _{\\\\mathbf {Q}\\\\in A \\\\cdot \\\\textrm {\\\\Omega \\\\hspace{-5.40608pt}\\\\Omega } }D_{\\\\widetilde{c}\\\\cdot \\\\varphi _{1}}(\\\\mathbf {Q},{P})=\\\\lim _{n\\\\rightarrow \\\\infty } \\\\widetilde{c}\\\\cdot \\\\left\\\\lbrace 1 - A \\\\cdot \\\\left[ 1 + \\\\log \\\\left( \\\\frac{1}{A} \\\\cdot \\\\Big (1+\\\\frac{1}{\\\\widetilde{c}}\\\\cdot \\\\frac{1}{n}\\\\cdot \\\\log \\\\,\\\\mathbb {\\\\Pi }\\\\left[\\\\xi _{n}^{w\\\\mathbf {W}}\\\\in \\\\textrm {\\\\right.\\\\right.\\\\right.\\\\right.\\\\Omega \\\\hspace{-6.544pt}\\\\Omega } \\\\Big ) ,\\\\\\\\& & \\\\hspace{-34.14322pt}\\\\inf _{\\\\mathbf {Q}\\\\in A \\\\cdot \\\\textrm {\\\\Omega \\\\hspace{-5.40608pt}\\\\Omega } } \\\\,I(\\\\mathbf {Q},{P})=\\\\inf _{\\\\mathbf {Q}\\\\in A \\\\cdot \\\\textrm {\\\\Omega \\\\hspace{-5.40608pt}\\\\Omega } } \\\\,\\\\sum \\\\displaylimits _{k=1}^{K} q_{k} \\\\cdot \\\\log \\\\left( \\\\frac{q_{k}}{p_{k}} \\\\right)= - \\\\lim _{n\\\\rightarrow \\\\infty } A \\\\cdot \\\\log \\\\left( \\\\frac{1}{A} \\\\cdot \\\\Big (1+\\\\frac{1}{n}\\\\cdot \\\\log \\\\,\\\\mathbb {\\\\Pi }\\\\left[\\\\breve{\\\\xi }_{n}^{w\\\\mathbf {W}}\\\\in \\\\textrm {\\\\right.\\\\right.\\\\Omega \\\\hspace{-6.544pt}\\\\Omega } \\\\Big ) ,\\\\nonumber \\\\\\\\& & \\\\hspace{-34.14322pt}\\\\max _{\\\\mathbf {Q}\\\\in A \\\\cdot \\\\textrm {\\\\Omega \\\\hspace{-5.40608pt}\\\\Omega } } \\\\,\\\\mathcal {E}^{Sh}(\\\\mathbf {Q})= \\\\max _{\\\\mathbf {Q}\\\\in A \\\\cdot \\\\textrm {\\\\Omega \\\\hspace{-5.40608pt}\\\\Omega } } \\\\,(-1) \\\\cdot \\\\sum _{k=1}^{K} q_{k} \\\\cdot \\\\log (q_{k})= \\\\lim _{n\\\\rightarrow \\\\infty }\\\\frac{\\\\log K}{K} + A \\\\cdot \\\\log \\\\left( \\\\frac{1}{A} \\\\cdot \\\\Big (1+\\\\frac{1}{n}\\\\cdot \\\\log \\\\,\\\\mathbb {\\\\Pi }\\\\left[\\\\breve{\\\\breve{\\\\xi }}_{n}^{w\\\\mathbf {W}}\\\\in \\\\textrm {\\\\right.\\\\right.\\\\Omega \\\\hspace{-6.544pt}\\\\Omega } \\\\Big ), \\\\\\\\& & \\\\hspace{-34.14322pt}\\\\max _{\\\\mathbf {Q}\\\\in A \\\\cdot \\\\textrm {\\\\Omega \\\\hspace{-5.40608pt}\\\\Omega } } \\\\,\\\\mathcal {E}^{gSM2}(\\\\mathbf {Q})= \\\\max _{\\\\mathbf {Q}\\\\in A \\\\cdot \\\\textrm {\\\\Omega \\\\hspace{-5.40608pt}\\\\Omega } } \\\\,\\\\frac{1}{1-s} \\\\cdot \\\\exp \\\\Big \\\\lbrace (s-1) \\\\cdot \\\\sum _{k=1}^{K} q_{k} \\\\cdot \\\\log (q_{k}) -1\\\\Big \\\\rbrace \\\\nonumber \\\\\\\\& & \\\\hspace{-34.14322pt}= \\\\lim _{n\\\\rightarrow \\\\infty }\\\\frac{1}{1-s} \\\\cdot \\\\exp \\\\left\\\\lbrace (1-s) \\\\cdot \\\\left[\\\\frac{\\\\log K}{K} + A \\\\cdot \\\\log \\\\left( \\\\frac{1}{A} \\\\cdot \\\\Big (1+\\\\frac{1}{n}\\\\cdot \\\\log \\\\,\\\\mathbb {\\\\Pi }\\\\left[\\\\breve{\\\\breve{\\\\xi }}_{n}^{w\\\\mathbf {W}}\\\\in \\\\textrm {\\\\right.\\\\right.\\\\right.\\\\right.\\\\Omega \\\\hspace{-6.544pt}\\\\Omega } \\\\Big ) -1 ,\\\\ \\\\ s \\\\in \\\\, ]0,1[ \\\\, \\\\cup \\\\, ]1,\\\\infty [ .$ The special subcase $A=1$ in () (and thus, $\\\\mathbf {Q}$ is a probability vector) corresponds to the maximum entropy problem for the Shannon entropy $\\\\mathcal {E}^{Sh}(\\\\cdot )$ .\",\n \"This can hence be tackled by our bare-simulation approach for almost arbitrary sets $\\\\textrm {$$\\\\hspace{-6.544pt}$$}$ of probability vectors.\",\n \"(b) The special case $\\\\varphi := \\\\widetilde{c}\\\\cdot \\\\varphi _{\\\\gamma }$ ($\\\\gamma =1$ ) of Theorem REF works analogously to (a), with the differences that we employ (i) additionally a sequence $(X_{i})_{i\\\\in \\\\mathbb {N}}$ of random variables being independent of $(W_{i})_{i\\\\in \\\\mathbb {N}}$ and satisfying condition (REF ) (resp.\",\n \"(REF )), (ii) $A=1$ (instead of arbitrary $A>0$ ), (iii) $\\\\mathbb {\\\\Pi }_{X_{1}^{n}}[\\\\cdot ]$ (instead of $\\\\mathbb {\\\\Pi }[\\\\cdot ]$ ), (iv) $\\\\xi _{n,\\\\mathbf {X}}^{w\\\\mathbf {W}}$ (instead of $\\\\xi _{n}^{w\\\\mathbf {W}}$ ), (v) $\\\\breve{\\\\xi }_{n,\\\\mathbf {X}}^{w\\\\mathbf {W}}$ (instead of $\\\\breve{\\\\xi }_{n}^{w\\\\mathbf {W}}$ ), and (vi) $\\\\breve{\\\\breve{\\\\xi }}_{n,\\\\mathbf {X}}^{w\\\\mathbf {W}}$ (instead of $\\\\breve{\\\\breve{\\\\xi }}_{n}^{w\\\\mathbf {W}}$ ).\",\n \"For the sake of completeness, let us mention here that we do not deal with the case $\\\\gamma \\\\in ]1,2[$ , for which we conjecture that $\\\\mathbb {}$ becomes a signed finite measure with total mass 1, i.e.\",\n \"it has a density (with respect to some dominating measure) with positive and negative values which “integrates to 1” ; accordingly, our BS method can not be applied to this situation.\",\n \"To proceed with further $\\\\gamma -$ cases, a combination of Theorem REF respectively Remark REF (vi), Lemma REF (a), (REF ), (REF ), (REF ), () and the below-mentioned $\\\\mathbb {}-$ concerning Example REF (c) leads to the following Proposition 26 (a) Consider the context of Remark REF (vi) for $\\\\varphi := \\\\widetilde{c}\\\\cdot \\\\varphi _{\\\\gamma }$ with $\\\\gamma =2$ , and let ${P} \\\\in \\\\mathbb {S}_{>0}^{K}$ as well as $\\\\widetilde{c}>0$ be arbitrary but fixed.\",\n \"Furthermore, let $W:=(W_{i})_{i\\\\in \\\\mathbb {N}}$ be an i.i.d.\",\n \"sequence of real-valued random variables with probability distribution $\\\\mathbb {} =NOR(1,\\\\frac{1}{\\\\widetilde{c}})$ being the Normal (Gaussian) law with mean 1 and variance $\\\\frac{1}{\\\\widetilde{c}}$ .\",\n \"Then for all $A >0$ and $\\\\Omega $$\\\\Omega \\\\subset \\\\mathbb {S}^{K}$ with (REF ) there hold all the BS-extremizabilites (REF ) to () as well as (REF ) (below) with plugging-in $\\\\gamma =2$ .\",\n \"From this, the BS-minimizability/maximizability of the important norms/entropies/diversity indices (E1) to (E6) follow immediately as special cases.\",\n \"By Remark REF (c), one can even take $A<0$ in (REF ) to () and (REF ) as well as in (E1), (E2), (E4) and (E6).\",\n \"(b) The special case $\\\\varphi := \\\\widetilde{c}\\\\cdot \\\\varphi _{\\\\gamma }$ ($\\\\gamma =2$ ) of Theorem REF works analogously to (a), with the differences that we employ (i) additionally a sequence $(X_{i})_{i\\\\in \\\\mathbb {N}}$ of random variables being independent of $(W_{i})_{i\\\\in \\\\mathbb {N}}$ and satisfying condition (REF ) (resp.\",\n \"(REF )), (ii) $A=1$ (instead of arbitrary $A>0$ ), (iii) $\\\\mathbb {\\\\Pi }_{X_{1}^{n}}[\\\\cdot ]$ (instead of $\\\\mathbb {\\\\Pi }[\\\\cdot ]$ ), (iv) $\\\\xi _{n,\\\\mathbf {X}}^{w\\\\mathbf {W}}$ (instead of $\\\\xi _{n}^{w\\\\mathbf {W}}$ ), (v) $\\\\breve{\\\\xi }_{n,\\\\mathbf {X}}^{w\\\\mathbf {W}}$ (instead of $\\\\breve{\\\\xi }_{n}^{w\\\\mathbf {W}}$ ), and (vi) $\\\\breve{\\\\breve{\\\\xi }}_{n,\\\\mathbf {X}}^{w\\\\mathbf {W}}$ (instead of $\\\\breve{\\\\breve{\\\\xi }}_{n}^{w\\\\mathbf {W}}$ ).\",\n \"For instance, the BS minimizability () of Proposition REF (a) can be employed to solve the following discrete Monge-Kantorovich-type optimal mass transportation problem (optimal coupling problem) with side (i.e.\",\n \"additional) constraints: given two nonnegative-entries vectors $\\\\mu :=(\\\\mu _{1}, \\\\ldots \\\\mu _{K_{1}} ) \\\\in [0,\\\\infty [^{K_{1}}$ and $\\\\nu := (\\\\nu _{1}, \\\\ldots \\\\nu _{K_{2}} ) \\\\in [0,\\\\infty [^{K_{2}}$ with equal total “mass” $\\\\sum _{k=1}^{K_{1}} \\\\mu _{k} =\\\\sum _{k=1}^{K_{2}} \\\\nu _{k} = A >0$ , compute $&& \\\\inf _{K_{1} \\\\times K_{2}-\\\\textrm {matrices} \\\\ \\\\pi }K_{1} \\\\cdot K_{2} \\\\cdot \\\\sum _{u=1}^{K_{1}} \\\\sum _{v=1}^{K_{2}}\\\\left(\\\\pi _{u,v} - \\\\frac{1}{K_{1} \\\\cdot K_{2}} \\\\right)^{2}\\\\\\\\&&\\\\textrm {subject to}\\\\nonumber \\\\\\\\&& \\\\sum _{v=1}^{K_{2}} \\\\pi _{u,v} = \\\\mu _{u} \\\\quad \\\\textrm {for all } u \\\\in \\\\lbrace 1,\\\\ldots ,K_{1}\\\\rbrace ,\\\\\\\\&& \\\\sum _{u=1}^{K_{1}} \\\\pi _{u,v} = \\\\nu _{v} \\\\quad \\\\textrm {for all } v \\\\in \\\\lbrace 1,\\\\ldots ,K_{2}\\\\rbrace ,\\\\\\\\&& \\\\pi _{u,v} \\\\in [0,A] \\\\quad \\\\textrm {for all } u \\\\in \\\\lbrace 1,\\\\ldots ,K_{1}\\\\rbrace , \\\\, v \\\\in \\\\lbrace 1,\\\\ldots ,K_{2}\\\\rbrace ,\\\\\\\\&& \\\\textrm {side constraints on \\\\pi , \\\\mu , \\\\nu }.$ Indeed, this problem can be equivalently rewritten in terms $K_{1} \\\\cdot K_{2}-$ dimensional vectors as follows: given two nonnegative-entries vectors $\\\\mu $ ,$\\\\nu $ as above, compute $&& \\\\inf _{\\\\mathbf {Q}\\\\in \\\\mathbf {\\\\Omega }}K_{1} \\\\cdot K_{2} \\\\cdot \\\\sum _{k=1}^{K_{1} \\\\cdot K_{2}}\\\\left(q_{k} - \\\\frac{1}{K_{1} \\\\cdot K_{2}} \\\\right)^{2}\\\\ = \\\\ \\\\inf _{\\\\mathbf {Q}\\\\in \\\\mathbf {\\\\Omega }}K_{1} \\\\cdot K_{2} \\\\cdot \\\\sum _{k=1}^{K_{1} \\\\cdot K_{2}}q_{k}^{2} + 1 - 2A \\\\\\\\&& \\\\textrm {where\\\\mathbf {\\\\Omega } \\\\subset \\\\mathbb {R}^{K_{1}\\\\cdot K_{2}} is the set of allvectors \\\\mathbf {Q} = (q_{1}, \\\\ldots , q_{K_{1} \\\\cdot K_{2}})which satisfy the constraints}\\\\nonumber \\\\\\\\&& \\\\sum _{j=1}^{K_{2}} q_{(i-1)\\\\cdot K_{2} +j} \\\\ = \\\\ \\\\mu _{i} \\\\quad \\\\textrm {for all } i \\\\in \\\\lbrace 1,\\\\ldots ,K_{1}\\\\rbrace , \\\\\\\\&& \\\\sum _{i=1}^{K_{1}} q_{(i-1)\\\\cdot K_{2} +j} \\\\ = \\\\ \\\\nu _{j} \\\\quad \\\\textrm {for all } j \\\\in \\\\lbrace 1,\\\\ldots ,K_{2}\\\\rbrace , \\\\\\\\&& q_{k} \\\\in [0,A] \\\\quad \\\\textrm {for all } k \\\\in \\\\lbrace 1,\\\\ldots ,K_{1} \\\\cdot K_{2} \\\\rbrace , \\\\\\\\&& \\\\textrm {side constraints on \\\\mathbf {Q}, \\\\mu , \\\\nu }.$ Clearly, via divisions by $A$ , one can equivalently rewrite $\\\\mathbf {\\\\Omega }= A \\\\cdot \\\\textrm {$$\\\\hspace{-6.544pt}$$}$ for some $\\\\textrm {$$\\\\hspace{-6.544pt}$$} \\\\subset \\\\mathbb {S}^{K_{1} \\\\cdot K_{2}}$ in the $K_{1}\\\\cdot K_{2}-$ dimensional probability simplex.\",\n \"Hence, we can employ () with $c_{1}=K_{1} \\\\cdot K_{2}$ , $c_{2}=1$ and $c_{3} = 1-2A$ , provided that the side constraints () are such that $\\\\mathbf {\\\\Omega }$ satisfies the regularity property (REF ) and the finiteness property (REF ).\",\n \"Notice that (REF ) is equal to $\\\\inf _{\\\\mathbf {Q}\\\\in \\\\mathbf {\\\\Omega }}D_{2 \\\\cdot \\\\varphi _{2}}(\\\\mathbf {Q},{P}^{unif})\\\\nonumber $ where ${P}^{unif} := (\\\\frac{1}{K_{1} \\\\cdot K_{2}},\\\\ldots , \\\\frac{1}{K_{1} \\\\cdot K_{2}})$ is the probability vector of frequencies of the uniform distribution on $\\\\lbrace 1, \\\\ldots , K_{1} \\\\cdot K_{2}\\\\rbrace $ , and $\\\\widetilde{c}=2$ .\",\n \"The special case $A=1$ with side constraint () of the form $K_{1} \\\\cdot \\\\min _{i \\\\in \\\\lbrace 1,\\\\ldots , K_{1}\\\\rbrace } \\\\mu _{i} +K_{2} \\\\cdot \\\\min _{j \\\\in \\\\lbrace 1,\\\\ldots , K_{2}\\\\rbrace } \\\\nu _{j} \\\\ge 1$ was explicitly solved by e.g.\",\n \"Bertrand et al.\",\n \"[43], [44], who also give applications to cryptographic guessing problems (spy problems), task partitioning and graph clustering.\",\n \"The importance of the case $\\\\gamma =2$ stems also from the fact that one can equivalently rewrite separable quadratic minimization problems as minimization problems of Pearson chi-square divergences.\",\n \"Indeed, by straightforward calculations one can derive that $\\\\inf _{\\\\mathbf {\\\\breve{Q}}\\\\in \\\\mathbf {\\\\breve{\\\\Omega }}} \\\\,\\\\sum _{k=1}^{K} ( \\\\, c_{1,k} + c_{2,k} \\\\cdot \\\\breve{q}_{k} + c_{3,k} \\\\cdot \\\\breve{q}_{k}^2 \\\\, ) \\\\, ,\\\\qquad c_{1,k} \\\\in \\\\mathbb {R}, \\\\ c_{2,k} \\\\in \\\\mathbb {R}\\\\backslash \\\\lbrace 0\\\\rbrace , \\\\ c_{3,k} \\\\in \\\\, ]0,\\\\infty [,$ is equal to (recall that $\\\\varphi _{2}(t) := \\\\frac{(t - 1)^2}{2}$ , cf.\",\n \"(REF )) $c_{4} + \\\\inf _{\\\\mathbf {Q}\\\\in \\\\mathbf {\\\\Omega }}D_{2 \\\\cdot \\\\varphi _{2}}(\\\\mathbf {Q},\\\\mathbf {P}) \\\\, ,$ where $\\\\mathbf {Q} := (q_{1}, \\\\ldots , q_{K})$ with $q_{k} := - c_{2,k} \\\\cdot \\\\breve{q}_{k}$ , $\\\\mathbf {P} := (p_{1}, \\\\ldots , p_{K})$ with $p_{k}:= \\\\frac{c_{2,k}^{2}}{2 \\\\cdot c_{3,k}} >0$ , $c_{4} := \\\\sum _{k=1}^{K} \\\\big ( \\\\, c_{1,k} - \\\\frac{c_{2,k}^{2}}{4 \\\\cdot c_{3,k}} \\\\, \\\\big )$ , and $\\\\mathbf {\\\\Omega }$ is the corresponding reformulation of the constraint set $\\\\mathbf {\\\\breve{\\\\Omega }}$ .\",\n \"To achieve the applicability of our BS method, we further transform (REF ) into its equal form (cf.\",\n \"(REF )) $c_{4} + \\\\inf _{\\\\widetilde{\\\\mathbf {Q}}\\\\in \\\\mathbf {\\\\Omega }/M_{P}}D_{2 M_{\\\\mathbf {P}} \\\\cdot \\\\varphi _{2}}(\\\\mathbf {Q},\\\\widetilde{{P}}) \\\\,$ with $M_{\\\\mathbf {P}}:=\\\\sum _{k=1}^{K} p_{k}>0$ and $\\\\widetilde{{P}}:=\\\\mathbf {P}/M_{\\\\mathbf {P}}$ .\",\n \"If $\\\\mathbf {\\\\Omega }/M_{\\\\mathbf {P}}$ satisfies (REF ) and (REF ) (e.g.\",\n \"it may be highly disconnected), then we can apply Theorem REF .\",\n \"In contrast, if $\\\\mathbf {\\\\Omega }/M_{\\\\mathbf {P}} = A \\\\cdot \\\\textrm {$$\\\\hspace{-6.544pt}$$}$ for some $A \\\\in \\\\mathbb {R}\\\\backslash \\\\lbrace 0\\\\rbrace $ and some $\\\\Omega $$\\\\Omega \\\\subset \\\\mathbb {S}_{>0}^{K}$ satisfying (REF ), then we can apply Proposition REF (a) together with Remark REF (c); for instance, this may appear if $\\\\mathbf {\\\\breve{\\\\Omega }}$ contains (amongst others) the original constraint $\\\\sum _{k=1}^{K} \\\\breve{q}_{k} = C$ for some constant $C>0$ , and $c_{2,k} =c_{2}$ does not depend on $k$ , which leads to the choice $A = - \\\\frac{c_{2} \\\\cdot C}{M_{\\\\mathbf {P}}}$ .\",\n \"Notice that $A<0$ if $c_{2}>0$ .\",\n \"For example, optimization problems (REF ) with $c_{1,k} >0$ , $c_{2,k} >0$ , $c_{3,k} >0$ and constraints $\\\\sum _{k=1}^{K} \\\\breve{q}_{k} = C$ , $\\\\breve{q}_{k} \\\\in [\\\\underline{\\\\breve{q}}_{k},\\\\overline{\\\\breve{q}}_{k}]$ appear in distributed energy management as economic dispatch problems in smart grids of power generators, where $\\\\breve{q}_{k}$ is the active power generation of the $k-$ th generator, $C$ is the total power demand, $\\\\underline{\\\\breve{q}}_{k}$ resp.\",\n \"$\\\\overline{\\\\breve{q}}_{k}$ represent the lower resp.\",\n \"upper bound of the $k-$ th generator's output, and the cost of power generation is $c_{1,k} + c_{2,k} \\\\cdot \\\\breve{q}_{k} + c_{3,k} \\\\cdot \\\\breve{q}_{k}^2$ (cf.\",\n \"e.g.\",\n \"Yang et al.\",\n \"[412], Loia & Vaccaro [230], Wood et al.\",\n \"[394], Xu et al.\",\n \"[404]).\",\n \"Another important special case of (REF ) to (REF ) is the omnipresent $L_{2}-$ minimization; indeed, with the choices $c_{3,k}=1$ , $c_{2,k}= - 2 v_{k}$ , and $c_{1,k}= v_{k}^2$ for some $\\\\mathbf {V} =(v_{1},\\\\ldots ,v_{K})$ , the minimization problem (REF ) is nothing but $\\\\inf _{\\\\mathbf {\\\\breve{Q}}\\\\in \\\\mathbf {\\\\breve{\\\\Omega }}} \\\\, || \\\\mathbf {\\\\breve{Q}} - \\\\mathbf {V} ||_{2}^{2}$ ; if $\\\\mathbf {\\\\breve{\\\\Omega }}$ depends on a pregiven $L-$ dimensional vector $\\\\mathbf {x}$ (with $L < K$ ), this can be regarded as a non-parametric regression problem in a wide sense.\",\n \"To continue with our general investigations, by combining Theorem REF respectively Remark REF (vi), Lemma REF (a), (REF ), (REF ), (REF ), () and the below-mentioned $\\\\mathbb {}-$ concerning Example REF (e), we arrive at the following Proposition 27 (a) Consider the context of Remark REF (vi) for $\\\\varphi := \\\\widetilde{c}\\\\cdot \\\\varphi _{\\\\gamma }$ with $\\\\gamma >2$ , and let ${P} \\\\in \\\\mathbb {S}_{>0}^{K}$ as well as $\\\\widetilde{c}>0$ be arbitrary but fixed.\",\n \"Furthermore, let $W:=(W_{i})_{i\\\\in \\\\mathbb {N}}$ be an i.i.d.\",\n \"sequence of real-valued random variables having densityin the classical sense, with respect to Lebesgue measure $\\\\frac{\\\\exp \\\\lbrace \\\\frac{y \\\\cdot \\\\widetilde{c}}{\\\\gamma -1}\\\\rbrace }{\\\\exp \\\\lbrace \\\\widetilde{c}/\\\\gamma \\\\rbrace }\\\\cdot f_{Z}(- y), \\\\qquad y\\\\in ]-\\\\infty ,\\\\infty [,$ where $f_{Z}$ is the density of a random variable $Z$ which has stable law with parameter-quadruple $(\\\\frac{\\\\gamma }{\\\\gamma -1},1,0,\\\\frac{\\\\widetilde{c}^{1/(1-\\\\gamma )} \\\\cdot (\\\\gamma -1)^{\\\\gamma /(\\\\gamma -1)}}{\\\\gamma })$ in terms of the above-mentioned “form-B notation” in Zolotarev [428].\",\n \"Then for all $A >0$ and $\\\\Omega $$\\\\Omega \\\\subset \\\\mathbb {S}^{K}$ with (REF ) there hold all the BS-extremizabilites (REF ) to (REF ) as well as $& & \\\\hspace{-42.67912pt}\\\\sup _{\\\\mathbf {Q}\\\\in A \\\\cdot \\\\textrm {\\\\Omega \\\\hspace{-5.40608pt}\\\\Omega }}\\\\frac{1}{1-\\\\gamma } \\\\cdot \\\\log \\\\Big (\\\\sum _{k=1}^{K} q_{k}^{\\\\gamma }\\\\Big )= \\\\lim _{n\\\\rightarrow \\\\infty } \\\\frac{1}{\\\\gamma \\\\cdot (\\\\gamma -1)} \\\\cdot \\\\left[\\\\log \\\\Big ( A^{\\\\gamma } \\\\cdot \\\\Big ( 1+ \\\\gamma \\\\cdot \\\\frac{1}{n}\\\\cdot \\\\log \\\\,\\\\mathbb {\\\\Pi }\\\\left[\\\\breve{\\\\breve{\\\\xi }}_{n}^{w\\\\mathbf {W}}\\\\in \\\\textrm {\\\\right.\\\\right.\\\\Omega \\\\hspace{-6.544pt}\\\\Omega } \\\\Big ) ^{1-\\\\gamma } \\\\Big ) + (1-\\\\gamma ) \\\\cdot \\\\log (K) .$ From this, the BS-minimizability/maximizability of the important norms/entropies/diversity indices (E1) to (E6) follow immediately as special cases.\",\n \"(b) The special case $\\\\varphi := \\\\widetilde{c}\\\\cdot \\\\varphi _{\\\\gamma }$ ($\\\\gamma >2$ ) of Theorem REF works analogously to (a), with the differences that we employ (i) additionally a sequence $(X_{i})_{i\\\\in \\\\mathbb {N}}$ of random variables being independent of $(W_{i})_{i\\\\in \\\\mathbb {N}}$ and satisfying condition (REF ) (resp.\",\n \"(REF )), (ii) $A=1$ (instead of arbitrary $A>0$ ), (iii) $\\\\mathbb {\\\\Pi }_{X_{1}^{n}}[\\\\cdot ]$ (instead of $\\\\mathbb {\\\\Pi }[\\\\cdot ]$ ), (iv) $\\\\xi _{n,\\\\mathbf {X}}^{w\\\\mathbf {W}}$ (instead of $\\\\xi _{n}^{w\\\\mathbf {W}}$ ), (v) $\\\\breve{\\\\xi }_{n,\\\\mathbf {X}}^{w\\\\mathbf {W}}$ (instead of $\\\\breve{\\\\xi }_{n}^{w\\\\mathbf {W}}$ ), and (vi) $\\\\breve{\\\\breve{\\\\xi }}_{n,\\\\mathbf {X}}^{w\\\\mathbf {W}}$ (instead of $\\\\breve{\\\\breve{\\\\xi }}_{n}^{w\\\\mathbf {W}}$ ).\",\n \"As mentioned above, in the Propositions REF to REF we have combined Theorem REF respectively Remark REF (vi), Lemma REF and explicitly solved representations (REF ).\",\n \"The latter, important step will be discussed in a structured, comprehensive manner in the Section below.\",\n \"By retransformation, we can even deal with optimizations of nonnegative linear objective functions with constraint sets on Euclidean $\\\\gamma $ -norm spheres.\",\n \"Indeed, for nonnegative $\\\\breve{\\\\mathbf {Q}} := (\\\\breve{q}_{1}, \\\\ldots , \\\\breve{q}_{K})$ and $\\\\breve{\\\\mathbf {P}} := (\\\\breve{p}_{1}, \\\\ldots , \\\\breve{p}_{K})$ one can rewrite their scalar product as $\\\\gamma -$ order Hellinger integrals $& & \\\\hspace{-36.98866pt}\\\\sum _{k=1}^{K} \\\\breve{q}_{k} \\\\cdot \\\\breve{p}_{k}\\\\ = \\\\ c_{1} \\\\cdot \\\\sum _{k=1}^{K} q_{k}^{\\\\gamma } \\\\cdot p_{k}^{1-\\\\gamma }\\\\ = \\\\ c_{1} \\\\cdot H_{\\\\gamma }(\\\\mathbf {Q},{P}) \\\\, \\\\hspace{14.22636pt} \\\\textrm {where}\\\\\\\\& & \\\\hspace{-39.83368pt}\\\\textrm {\\\\gamma \\\\in \\\\, ]0,1[ \\\\, \\\\cup \\\\, [2,\\\\infty [ \\\\, if \\\\, \\\\breve{\\\\mathbf {Q}} \\\\in [0,\\\\infty [^{K},\\\\breve{\\\\mathbf {P}} \\\\in \\\\, ]0,\\\\infty [^{K}\\\\hspace{11.38092pt} respectively \\\\hspace{11.38092pt}\\\\gamma \\\\in \\\\, ]-\\\\infty ,0[ \\\\, if \\\\, \\\\breve{\\\\mathbf {Q}} \\\\in \\\\, ]0,\\\\infty [^{K},\\\\breve{\\\\mathbf {P}} \\\\in \\\\, ]0,\\\\infty [^{K},}\\\\\\\\& & \\\\hspace{-36.98866pt}q_{k} \\\\ := \\\\ \\\\breve{q}_{k}^{1/\\\\gamma } ,\\\\quad p_{k} \\\\ := \\\\ \\\\frac{\\\\breve{p}_{k}^{1/(1-\\\\gamma )}}{\\\\sum _{i=1}^{K} \\\\breve{p}_{i}^{1/(1-\\\\gamma )}} ,\\\\quad c_{1}:= \\\\Big ( \\\\sum _{i=1}^{K} \\\\breve{p}_{i}^{1/(1-\\\\gamma )} \\\\Big )^{1-\\\\gamma } = :|| \\\\breve{\\\\mathbf {P}} ||_{1-\\\\gamma } \\\\ .$ The required constraint $\\\\sum _{k=1}^{K} q_{k} = A >0$ retransforms to $|| \\\\breve{\\\\mathbf {Q}} ||_{\\\\gamma }=A^{1/\\\\gamma }$ and thus, $\\\\breve{\\\\mathbf {Q}}$ must lie on (the positive/nonnegative part of) the $\\\\gamma -$ norm-sphere $\\\\partial B_{\\\\gamma }(0,A^{1/\\\\gamma })$ around the origin with radius $A^{1/\\\\gamma }$ .\",\n \"Accordingly, for $\\\\gamma \\\\in \\\\, [2,\\\\infty [$ we have $\\\\inf _{\\\\breve{\\\\mathbf {Q}} \\\\in \\\\breve{\\\\mathbf {\\\\Omega }}} \\\\,\\\\sum _{k=1}^{K} \\\\breve{q}_{k} \\\\cdot \\\\breve{p}_{k}\\\\ = \\\\ c_{1} \\\\cdot \\\\inf _{\\\\mathbf {Q}\\\\in A \\\\cdot \\\\textrm {\\\\Omega \\\\hspace{-5.40608pt}\\\\Omega } }H_{\\\\gamma }(\\\\mathbf {Q},{P})$ and we can apply () of Proposition REF (a) respectively Proposition REF (a) here and analogously henceforth, by this we mean the condition () as it appears in the Proposition REF (a) respectively Proposition REF (a), as long as the original constraint set $\\\\breve{\\\\mathbf {\\\\Omega }} \\\\in \\\\partial B_{\\\\gamma }(0,A^{1/\\\\gamma }) \\\\, \\\\cap \\\\, [0,\\\\infty [^{K}$ transforms (via $q_{k} = \\\\breve{q}_{k}^{1/\\\\gamma }$ ) into a constraint set $A \\\\cdot \\\\textrm {$$\\\\hspace{-6.544pt}$$}$ which satisfies the regularity assumption (REF ) in the relative topology (as a side remark, notice that $int(\\\\partial B_{\\\\gamma }(0,A^{1/\\\\gamma })) = \\\\emptyset $ in the full topology).\",\n \"For the case $\\\\gamma \\\\in \\\\, ]-\\\\infty ,0[$ we also have (REF ) and apply () of Proposition REF (a) for any original constraint set $\\\\breve{\\\\mathbf {\\\\Omega }} \\\\in \\\\partial B_{\\\\gamma }(0,A^{1/\\\\gamma }) \\\\, \\\\cap \\\\, ]0,\\\\infty [^{K}$ which transforms into $A \\\\cdot \\\\textrm {$$\\\\hspace{-6.544pt}$$}$ satisfying (REF ) in the relative topology.\",\n \"In contrast, for the case $\\\\gamma \\\\in \\\\, ]0,1[$ we get $\\\\sup _{\\\\breve{\\\\mathbf {Q}} \\\\in \\\\breve{\\\\mathbf {\\\\Omega }}} \\\\,\\\\sum _{k=1}^{K} \\\\breve{q}_{k} \\\\cdot \\\\breve{p}_{k}\\\\ = \\\\ c_{1} \\\\cdot \\\\sup _{\\\\mathbf {Q}\\\\in A \\\\cdot \\\\textrm {\\\\Omega \\\\hspace{-5.40608pt}\\\\Omega } }H_{\\\\gamma }(\\\\mathbf {Q},{P})\\\\nonumber $ and apply (REF ) of Proposition REF (a) for any original constraint set $\\\\breve{\\\\mathbf {\\\\Omega }} \\\\in \\\\partial B_{\\\\gamma }(0,A^{1/\\\\gamma }) \\\\, \\\\cap \\\\, [0,\\\\infty [^{K}$ which transforms into $A \\\\cdot \\\\textrm {$$\\\\hspace{-6.544pt}$$}$ satisfying (REF ) in the relative topology.\",\n \"As a continuation of Remark REF , we can principally tackle all the optimization problems of this Subsection REF by basically only employing a fast and accurate — pseudo, true, natural, quantum — random number generator, provided that the constraint set $\\\\textrm {$ A $\\\\hspace{-6.544pt}$$}$ satisfies the mild assumptions (REF ) (in the relative topology) and (REF ).\",\n \"Recall that $A>0$ (and for $\\\\varphi _{2}$ even $A \\\\in \\\\mathbb {R}\\\\backslash \\\\lbrace 0\\\\rbrace $ ) and that $\\\\mathbf {Q} \\\\in \\\\textrm {$ A $\\\\hspace{-6.544pt}$$}$ implies in particular the constraint $\\\\sum _{k=1}^{K} q_{k} = A$ .\",\n \"The regularity assumption (REF ) allows for e.g.\",\n \"high-dimensional constraint sets $\\\\textrm {$ A $\\\\hspace{-6.544pt}$$}$ which are non-convex and even highly disconnected, and for which other minimization methods (e.g.\",\n \"pure enumeration, gradient or steepest descent methods, etc.)\",\n \"may be problematic or intractable.\",\n \"For example, (REF ) covers kind of “$K-$ dimensional (not necessarily regular) polka dot pattern type” relaxations $\\\\textrm {$ A $\\\\hspace{-6.544pt}$$} :=\\\\dot{\\\\bigcup }_{i=1}^{N} \\\\mathcal {U}_{i}(Q_{i}^{dis})$ of finite discrete constraint sets $\\\\textrm {$ A $\\\\hspace{-6.544pt}$ dis$}:= \\\\lbrace Q_{1}^{dis}, \\\\ldots , Q_{N}^{dis}\\\\rbrace $ of high cardinality $N$ (e.g.\",\n \"being exponential or factorial in a large $K$ ), where each $K-$ dimensional vector $Q_{i}^{dis}$ has total-sum-of-components equal to $A$ and is surrounded by some small (“flat”, i.e.\",\n \"in the relative topology) neighborhood $\\\\mathcal {U}_{i}(Q_{i}^{dis})$ .\",\n \"For the sake of brevity, in the following discussion we confine ourselves to the deterministic setup (e.g.\",\n \"Proposition REF (a) rather than (b)) which particularly involves $\\\\mathbb {\\\\Pi }[\\\\cdot ]$ (rather than $\\\\mathbb {\\\\Pi }_{X_{1}^{n}}[\\\\cdot ]$ ) and $\\\\xi _{n}^{w\\\\mathbf {W}}$ (rather than $\\\\xi _{n,\\\\mathbf {X}}^{w\\\\mathbf {W}}$ ).\",\n \"In such a context, all the optimization problems of this Subsection REF , subsumed as (cf.\",\n \"(REF ) to (REF )) $\\\\inf _{\\\\mathbf {Q}\\\\in A \\\\cdot \\\\textrm {\\\\Omega \\\\hspace{-5.40608pt}\\\\Omega } } \\\\,\\\\Phi (\\\\mathbf {Q})\\\\qquad \\\\textrm {respectively} \\\\qquad \\\\sup _{\\\\mathbf {Q}\\\\in A \\\\cdot \\\\textrm {\\\\Omega \\\\hspace{-5.40608pt}\\\\Omega } } \\\\,\\\\Phi (\\\\mathbf {Q})\\\\nonumber $ can be regarded as a “BS-tractable” relaxations of the corresponding nonlinear discrete (e.g.\",\n \"integer, combinatorial) programming problems $\\\\inf _{\\\\mathbf {Q}\\\\in A \\\\cdot \\\\textrm {\\\\Omega \\\\hspace{-5.40608pt}\\\\Omega ^{dis}} } \\\\,\\\\Phi (\\\\mathbf {Q})\\\\qquad \\\\textrm {respectively} \\\\qquad \\\\sup _{\\\\mathbf {Q}\\\\in A \\\\cdot \\\\textrm {\\\\Omega \\\\hspace{-5.40608pt}\\\\Omega ^{dis}} } \\\\,\\\\Phi (\\\\mathbf {Q}) \\\\, ;\\\\nonumber $ as examples take e.g.\",\n \"$\\\\Phi (\\\\mathbf {Q}) =c_{1} \\\\cdot \\\\Big ( \\\\Big (\\\\sum _{k=1}^{K} q_{k}^{\\\\gamma }\\\\Big )^{c_{2}}- c_{3} \\\\Big )$ (with $\\\\gamma \\\\ne 0,1$ ) or $\\\\Phi (\\\\mathbf {Q}) = \\\\Phi _{{P}}(\\\\mathbf {Q}) =D_{\\\\widetilde{c}\\\\cdot \\\\varphi _{\\\\gamma }}(\\\\mathbf {Q},{P})$ .\",\n \"For instance, $A \\\\cdot \\\\textrm {$$\\\\hspace{-6.544pt}$ dis$}$ may contain only $K-$ dimensional vectors $Q_{i}^{dis}$ ($i=1,\\\\ldots ,N$ ) whose components stem from a finite set $\\\\mathcal {B}$ of nonnegative integers and add up to $A$ .\",\n \"If $\\\\mathcal {B} = \\\\lbrace 0,1\\\\rbrace $ , then we can even deal with nonnegative linear objective functions $\\\\Phi (\\\\mathbf {Q}) = \\\\sum _{k=1}^{K} \\\\breve{p}_{k} \\\\cdot q_{k}$ where $\\\\mathbf {Q} := (q_{1}, \\\\ldots , q_{K})$ with $q_{k} \\\\in \\\\lbrace 0,1\\\\rbrace $ and $\\\\breve{\\\\mathbf {P}} := (\\\\breve{p}_{1}, \\\\ldots , \\\\breve{p}_{K})$ has components $\\\\breve{p}_{k} >0$ which reflect e.g.\",\n \"the cost associated with the $k-$ th state.\",\n \"Indeed, by noticing that $q_{k}^{1/\\\\gamma }=q_{k}$ for $\\\\gamma \\\\in \\\\, ]0,1[ \\\\, \\\\cup \\\\, [2,\\\\infty [$ , we can employ (REF ) and () to end up with $& & \\\\hspace{-31.2982pt}\\\\inf _{\\\\mathbf {Q}\\\\in A \\\\cdot \\\\textrm {\\\\Omega \\\\hspace{-5.40608pt}\\\\Omega ^{dis}} } \\\\,\\\\sum _{k=1}^{K} q_{k} \\\\cdot \\\\breve{p}_{k}= || \\\\breve{\\\\mathbf {P}} ||_{1-\\\\gamma } \\\\cdot \\\\inf _{\\\\mathbf {Q}\\\\in A \\\\cdot \\\\textrm {\\\\Omega \\\\hspace{-5.40608pt}\\\\Omega ^{dis}} } \\\\,\\\\sum _{k=1}^{K} q_{k}^{\\\\gamma } \\\\cdot \\\\left(\\\\frac{\\\\breve{p}_{k}^{1/(1-\\\\gamma )}}{\\\\sum _{i=1}^{K} \\\\breve{p}_{i}^{1/(1-\\\\gamma )}}\\\\right)^{1-\\\\gamma }= || \\\\breve{\\\\mathbf {P}} ||_{1-\\\\gamma } \\\\cdot \\\\inf _{\\\\mathbf {Q}\\\\in A \\\\cdot \\\\textrm {\\\\Omega \\\\hspace{-5.40608pt}\\\\Omega ^{dis}} }H_{\\\\gamma }(\\\\mathbf {Q},{P}), \\\\ \\\\ \\\\gamma \\\\in \\\\, [2,\\\\infty [, \\\\ \\\\ \\\\ \\\\\\\\& & \\\\hspace{-31.2982pt}\\\\sup _{\\\\mathbf {Q}\\\\in A \\\\cdot \\\\textrm {\\\\Omega \\\\hspace{-5.40608pt}\\\\Omega ^{dis}} } \\\\,\\\\sum _{k=1}^{K} q_{k} \\\\cdot \\\\breve{p}_{k}= || \\\\breve{\\\\mathbf {P}} ||_{1-\\\\gamma } \\\\cdot \\\\sup _{\\\\mathbf {Q}\\\\in A \\\\cdot \\\\textrm {\\\\Omega \\\\hspace{-5.40608pt}\\\\Omega ^{dis}} }H_{\\\\gamma }(\\\\mathbf {Q},{P}), \\\\ \\\\ \\\\gamma \\\\in \\\\, ]0,1[ \\\\, .$ The corresponding relaxations are $& & \\\\hspace{-48.36958pt}\\\\inf _{\\\\mathbf {Q}\\\\in A \\\\cdot \\\\textrm {\\\\Omega \\\\hspace{-5.40608pt}\\\\Omega } } \\\\,\\\\sum _{k=1}^{K} q_{k} \\\\cdot \\\\breve{p}_{k}\\\\ = \\\\ || \\\\breve{\\\\mathbf {P}} ||_{1-\\\\gamma } \\\\cdot \\\\inf _{\\\\mathbf {Q}\\\\in A \\\\cdot \\\\textrm {\\\\Omega \\\\hspace{-5.40608pt}\\\\Omega } }H_{\\\\gamma }(\\\\mathbf {Q},{P}), \\\\ \\\\ \\\\gamma \\\\in \\\\, [2,\\\\infty [ \\\\, ,\\\\\\\\& & \\\\hspace{-48.36958pt}\\\\sup _{\\\\mathbf {Q}\\\\in A \\\\cdot \\\\textrm {\\\\Omega \\\\hspace{-5.40608pt}\\\\Omega } } \\\\,\\\\sum _{k=1}^{K} q_{k} \\\\cdot \\\\breve{p}_{k}\\\\ = \\\\ || \\\\breve{\\\\mathbf {P}} ||_{1-\\\\gamma } \\\\cdot \\\\sup _{\\\\mathbf {Q}\\\\in A \\\\cdot \\\\textrm {\\\\Omega \\\\hspace{-5.40608pt}\\\\Omega } }H_{\\\\gamma }(\\\\mathbf {Q},{P}), \\\\ \\\\ \\\\gamma \\\\in \\\\, ]0,1[ \\\\, ;$ for (REF ) we can apply () of Proposition REF (a) respectively Proposition REF (a), whereas for () we apply (REF ) of Proposition REF (a) — as long as the relaxation constraint set $A \\\\cdot \\\\textrm {$$\\\\hspace{-6.544pt}$$}$ satisfies (REF ) in the relative topology.\",\n \"For the sake of illustration, let us consider a sum-minimization-type linear assignment problem with side constraints (for a comprehensive book on assignment problems see e.g.\",\n \"Burkard et al.\",\n \"[69]).\",\n \"Suppose that there are $K$ individuals (people, machines, etc.)\",\n \"to carry out $K$ tasks (jobs, etc.).\",\n \"Each individual is assigned to carry out exactly one task.\",\n \"There is cost $c_{ij} >0$ if individual $i$ is assigned to (i.e., carries out) task $j$ .\",\n \"We want to find the minimum total cost amongst all assignments.\",\n \"There may be side constraints, e.g.\",\n \"each assignment has a value $v_{ij} >0$ and the total value of the assignment should be above a pregiven threshold.\",\n \"As usual, the problem can be formulated with the help of binary variables $x_{ij}$ where $x_{ij} =1$ if individual $i$ is assigned to task $j$ , and $x_{ij} =0$ otherwise.\",\n \"Accordingly, we want to compute $&& \\\\inf _{K \\\\times K-\\\\textrm {matrices} \\\\ x=(x_{ij})} \\\\ \\\\sum _{i=1}^{K} \\\\sum _{j=1}^{K}c_{ij} \\\\cdot x_{ij}\\\\\\\\&&\\\\textrm {subject to}\\\\nonumber \\\\\\\\&& \\\\sum _{j=1}^{K} x_{ij} = 1 \\\\quad \\\\textrm {for all } i \\\\in \\\\lbrace 1,\\\\ldots ,K\\\\rbrace ,\\\\qquad \\\\textrm {(i.e.\",\n \"each individual i does one task),}\\\\\\\\&& \\\\sum _{i=1}^{K} x_{ij} = 1 \\\\quad \\\\textrm {for all } j\\\\in \\\\lbrace 1,\\\\ldots ,K\\\\rbrace ,\\\\qquad \\\\textrm {(i.e.\",\n \"each task j is done by one individual),}\\\\\\\\&& x_{ij} \\\\in \\\\lbrace 0,1\\\\rbrace \\\\quad \\\\textrm {for all } i \\\\in \\\\lbrace 1,\\\\ldots ,K\\\\rbrace , \\\\, j \\\\in \\\\lbrace 1,\\\\ldots ,K\\\\rbrace ,\\\\\\\\&& \\\\textrm {side (i.e.\",\n \"additional) constraints on x=(x_{ij})_{i,j=1,\\\\ldots ,K}}.$ Analogously to (REF ), this problem can be equivalently rewritten in terms of $K^{2}-$ dimensional vectors as follows: let $\\\\mathbf {Q} := (q_{1}, \\\\ldots , q_{K^{2}})$ and $\\\\breve{\\\\mathbf {P}} := (\\\\breve{p}_{1}, \\\\ldots , \\\\breve{p}_{K^{2}})$ be such that $c_{ij}= \\\\breve{p}_{(i-1)\\\\cdot K +j}$ and $x_{ij}= q_{(i-1)\\\\cdot K +j}$ for $i,j \\\\in \\\\lbrace 1,\\\\ldots ,K\\\\rbrace $ and compute $&& \\\\inf _{\\\\mathbf {Q}\\\\in K \\\\cdot \\\\textrm {\\\\Omega \\\\hspace{-5.40608pt}\\\\Omega ^{dis}} } \\\\,\\\\sum _{k=1}^{K} q_{k} \\\\cdot \\\\breve{p}_{k} \\\\\\\\&& \\\\textrm {whereK \\\\cdot \\\\textrm {\\\\Omega \\\\hspace{-6.544pt}\\\\Omega ^{dis}}\\\\subset \\\\mathbb {R}^{K^{2}} is the set of allvectors \\\\mathbf {Q} = (q_{1}, \\\\ldots , q_{K^{2}}) which satisfy the constraints}\\\\nonumber \\\\\\\\&& \\\\sum _{j=1}^{K} q_{(i-1)\\\\cdot K +j} \\\\ = \\\\ 1 \\\\quad \\\\textrm {for all } i \\\\in \\\\lbrace 1,\\\\ldots ,K\\\\rbrace ,\\\\\\\\&& \\\\sum _{i=1}^{K} q_{(i-1)\\\\cdot K +j} \\\\ = \\\\ 1 \\\\quad \\\\textrm {for all } j \\\\in \\\\lbrace 1,\\\\ldots ,K\\\\rbrace ,\\\\\\\\&& q_{k} \\\\in \\\\lbrace 0,1\\\\rbrace \\\\quad \\\\textrm {for all } k \\\\in \\\\lbrace 1,\\\\ldots ,K^{2} \\\\rbrace ,\\\\\\\\&& \\\\textrm {side constraints on \\\\mathbf {Q}}.$ As seen above, this can be rewritten as $\\\\gamma -$ order Hellinger-integral minimization problem (REF ), with $\\\\gamma \\\\ge 2$ .\",\n \"We can obtain a highly disconnected “non-void-interior-type” relaxation of the binary integer programming problem (REF ) to () by replacing () with $q_{k} \\\\in [0,\\\\varepsilon _{1}] \\\\, \\\\cup \\\\, [1-\\\\varepsilon _{2},1]\\\\quad \\\\textrm {for all } k \\\\in \\\\lbrace 1,\\\\ldots ,K^{2} \\\\rbrace ,$ for some (possibly arbitrarily) small $\\\\varepsilon _{1}, \\\\varepsilon _{2} >0$ with $\\\\varepsilon _{1} + \\\\varepsilon _{2} < 1$ .\",\n \"We denote by $K \\\\cdot \\\\textrm {$$\\\\hspace{-6.544pt}$$}$ the outcoming set manifested by the constraints (), (), () and (REF ), and accordingly we end up with a minimization problem of type (REF ), which we can tackle by () of Proposition REF (a) respectively Proposition REF (a), as long as (REF ) (in the relative topology) is satisfied.\",\n \"For instance, we can take $\\\\gamma =2$ and basically solve the corresponding optimization problem by basically simulating $K^{2}-$ dimensional Gaussian random variables (even though the cardinality of $K \\\\cdot \\\\textrm {$$\\\\hspace{-6.544pt}$ dis$}$ may be high).\",\n \"As a side remark, let us mention that our relaxation (REF ) contrasts considerably to the frequently used continuous linear programming (LP) relaxation $q_{k} \\\\in [0,1]\\\\quad \\\\textrm {for all } k \\\\in \\\\lbrace 1,\\\\ldots ,K^{2} \\\\rbrace .\\\\nonumber $ Let us finally mention that an important special case of a minimization problem (REF ) to () is — the integer programming formulation of — the omnipresent (asymmetric) traveling salesman problem (TSP) with possible side constraints see e.g.\",\n \"Applegate et al.\",\n \"[14], Gutin & Punnen [147], Cook [92] for comprehensive books on TSP, its variations and its applications to logistics, machine scheduling, printed circuit board drilling, communication-network frequencing, genome sequencing, data clustering, and many others..\",\n \"There, one has $K$ cities and the cost of traveling from city $i$ to city $j\\\\ne i$ is given by $c_{ij}>0$ .\",\n \"Moreover, one sets $x_{ij} =1$ if the traveler goes directly from city $i$ to city $j$ (in that order), and $x_{ij} =0$ otherwise.\",\n \"For technical reasons, for $i=j$ we attribute a cost $c_{ii}>0$ (e.g.\",\n \"hotel costs), but we require that always $x_{ii}=0$ which we subsume as the first part of the constraints ().\",\n \"Then, the constraint () means that the traveler leaves from city $i$ exactly once, whereas () reflects that the traveler arrives at city $j$ exactly once.\",\n \"The goal is to find a directed tour — i.e.\",\n \"a directed cycle/circuit that visits all $K$ cities once — of minimum cost.\",\n \"Within this context, the second part of the constraints () should basically exclude solutions which consist of disconnected subtours (subtour elimination constraints (of e.g.\",\n \"the seminal Dantzig et al.\",\n \"[104]), connectivity constraints, cut-set constraints).\",\n \"Here, we also allow for additional/side constraints which we subsume as the third part () of the constraints.\",\n \"Hence, by the above-mentioned considerations we can principally tackle such kind of TSP problems with our BS method.\",\n \"For sum-maximization-type linear assignment problems with side constraints, where e.g.\",\n \"$c_{ij}$ is a profit (rather than a cost) and the ultimate goal is total profit maximization, we can proceed analogously, by employing () and () (instead of (REF ) and (REF )).\",\n \"Let us end this subsection with a comparison: suppose that we have a (sufficiently large) number $n$ of concrete data observations $X_{i} = x_{i}$ ($i=1,\\\\ldots ,n$ ) from the unknown probability distribution ${P}$ (in vector form), and from these we want to approximate/estimate the unknown distance $\\\\inf _{{Q}\\\\in \\\\textrm {\\\\Omega \\\\hspace{-5.40608pt}\\\\Omega }}D_{\\\\varphi _{\\\\gamma }}( {Q}, {P} )$ from a family of probability models (in vector form) $\\\\Omega $$\\\\Omega $ (e.g.\",\n \"for model-adequacy evaluations, for goodness-of-fit testing purposes): by the above-mentioned Propositions REF to REF (and especially, by (REF ), (REF ), (REF )) one can use $G\\\\Big (-\\\\frac{1}{n} \\\\cdot \\\\log \\\\,\\\\mathbb {\\\\Pi }_{x_{1}^{n}}\\\\left[\\\\xi _{n,\\\\mathbf {x}}^{w\\\\mathbf {W}}\\\\in \\\\textrm {\\\\right.\\\\Omega \\\\hspace{-6.544pt}\\\\Omega } \\\\Big )$ where $\\\\mathbb {\\\\Pi }_{x_{1}^{n}}[\\\\, \\\\cdot \\\\, ] := \\\\mathbb {\\\\Pi }[ \\\\, \\\\cdot \\\\, | \\\\,X_{1}=x_{1}, \\\\ldots , X_{n} = x_{n}]$ , $\\\\mathbf {x} := (x_{1},\\\\ldots ,x_{n})$ , and $G$ (cf.\",\n \"(REF )) is e.g.\",\n \"chosen as follows: $G(z) := -\\\\frac{\\\\widetilde{c}}{\\\\gamma \\\\cdot \\\\left( \\\\gamma -1\\\\right) }\\\\cdot \\\\left\\\\lbrace 1-\\\\left( 1-\\\\frac{\\\\gamma }{\\\\widetilde{c}} \\\\cdot z \\\\right) ^{1-\\\\gamma } \\\\right\\\\rbrace $ for the three cases $\\\\gamma <0$ , $\\\\gamma \\\\in ]0,1[$ and $\\\\gamma \\\\ge 2$ , $G(z) := z$ for $\\\\gamma =0$ (reversed Kullback-Leibler divergence), and $G(z):= - \\\\widetilde{c} \\\\cdot \\\\log (1- \\\\frac{1}{\\\\widetilde{c}} \\\\cdot z)$ for $\\\\gamma =1$ (Kullback-Leibler divergence).\",\n \"Notice that (REF ) contrasts to the alternative approximation (of $\\\\inf _{{Q}\\\\in \\\\textrm {\\\\Omega \\\\hspace{-5.40608pt}\\\\Omega }}D_{\\\\varphi _{\\\\gamma }}( {Q}, {P} )$ ) given by $\\\\inf _{{Q}\\\\in \\\\textrm {\\\\Omega \\\\hspace{-5.40608pt}\\\\Omega }}D_{\\\\varphi _{\\\\gamma }}({Q}, \\\\mathbb {P}_{n}^{emp, co} )$ which is used in the context of “classical” statistical minimum distance estimation (MDE) with power divergences; in (REF ), we have employed $\\\\mathbb {P}_{n}^{emp, co} = \\\\frac{1}{n} \\\\cdot \\\\sum _{i=1}^{n} \\\\delta _{x_{i}}$ to be the realization of the empirical distribution $\\\\mathbb {P}_{n}^{emp} = \\\\frac{1}{n} \\\\cdot \\\\sum _{i=1}^{n} \\\\delta _{X_{i}}$ .\",\n \"Indeed, especially in complicated high-dimensional non-parametric or semi-parametric big-data contexts, we have substituted a quite difficult optimization problem (REF ) by a much easier solvable counting problem (REF ).\",\n \"The same holds analogously for Renyi distances/divergences, etc.\"\n ],\n [\n \"Construction principle for bounds of the minimum divergence\\nin the general case \",\n \"Turning back to Theorem REF , we now consider the general case when the divergence $\\\\varphi \\\\in \\\\Upsilon (]a,b[)$ is not of the power type (REF ).\",\n \"Recall from (REF ) the crucial terms (with ${P} \\\\in \\\\mathbb {S}_{>0}$ ) $\\\\inf _{m\\\\ne 0} D_{\\\\varphi }(m\\\\cdot \\\\textrm {\\\\Omega \\\\hspace{-6.544pt}\\\\Omega },{P}):= \\\\inf _{m\\\\ne 0}\\\\ \\\\inf _{{Q}\\\\in \\\\textrm {\\\\Omega \\\\hspace{-5.40608pt}\\\\Omega } }D_{\\\\varphi }(m\\\\cdot {Q},{P})= \\\\inf _{{Q}\\\\in \\\\textrm {\\\\Omega \\\\hspace{-5.40608pt}\\\\Omega } }\\\\ \\\\inf _{m\\\\ne 0}D_{\\\\varphi }(m\\\\cdot {Q},{P}) \\\\, < \\\\, \\\\infty $ for all sets $\\\\Omega $$\\\\Omega $ satisfying the regularity properties (REF ) and the convenient, more restrictive finiteness property $\\\\inf _{{Q}\\\\in \\\\textrm {\\\\Omega \\\\hspace{-5.40608pt}\\\\Omega }}\\\\ \\\\inf _{k=1,\\\\ldots ,K}\\\\frac{q_{k}}{p_{k}}\\\\in dom(\\\\varphi ),\\\\quad \\\\sup _{{Q}\\\\in \\\\textrm {\\\\Omega \\\\hspace{-5.40608pt}\\\\Omega }}\\\\ \\\\sup _{k=1,\\\\ldots ,K}\\\\frac{q_{k}}{p_{k}}\\\\in dom(\\\\varphi )$ which implies (REF ); notice that $\\\\inf _{k=1,\\\\ldots ,K}\\\\frac{q_{k}}{p_{k}} \\\\le 1$ , $\\\\sup _{k=1,\\\\ldots ,K}\\\\frac{q_{k}}{p_{k}} \\\\ge 1$ with equalities if and only if ${Q} = {P}$ .\",\n \"Since $\\\\textrm {$$\\\\hspace{-6.544pt}$$} \\\\ne \\\\lbrace {P} \\\\rbrace $ (cf.\",\n \"the right-hand side of (REF )), the double infimum (supremum) in (REF ) is strictly smaller (larger) than 1.\",\n \"In general, the inner minimization $\\\\inf _{m\\\\ne 0} D_{\\\\varphi }(m\\\\cdot {Q},{P})$ in (REF ) can not be performed in explicit closed form, but e.g.\",\n \"in the specific case of power divergences (cf.\",\n \"(REF ), (REF )) the optimization $\\\\inf _{m\\\\ne 0} D_{\\\\widetilde{c} \\\\cdot \\\\varphi _{\\\\gamma }}(m\\\\cdot {Q},{P})$ produces an explicit form, which in turn leads to a simple one-to-one correspondence between $D_{\\\\widetilde{c} \\\\cdot \\\\varphi _{\\\\gamma }}(\\\\textrm {$$\\\\hspace{-6.544pt}$$},{P})$ and $\\\\inf _{m\\\\ne 0}\\\\ D_{\\\\widetilde{c} \\\\cdot \\\\varphi _{\\\\gamma }}(m\\\\cdot \\\\textrm {$$\\\\hspace{-6.544pt}$$},{P})$ (cf.\",\n \"Lemma REF ).\",\n \"Under (REF ) and (REF ) it clearly holds $\\\\inf _{m\\\\ne 0} D_{\\\\varphi }(m\\\\cdot \\\\textrm {\\\\Omega \\\\hspace{-6.544pt}\\\\Omega },{P})\\\\ \\\\le \\\\ D_{\\\\varphi }(\\\\textrm {\\\\Omega \\\\hspace{-6.544pt}\\\\Omega },{P})\\\\ \\\\le \\\\ D_{\\\\varphi }({Q},{P}) .$ For transparency, we first investigate the (widely useable) subsetup where $dom(\\\\varphi )= ]0,\\\\infty [$ (and thus, $int(dom(\\\\varphi )) = \\\\, ]a,b[ \\\\, = \\\\, ]0,\\\\infty [$ ) and $\\\\textrm {$$\\\\hspace{-6.544pt}$$} \\\\subset \\\\mathbb {S}_{> 0}^{K}$ .\",\n \"Let us start with the lower bound $\\\\inf _{m\\\\ne 0} D_{\\\\varphi }(m\\\\cdot \\\\textrm {$$\\\\hspace{-6.544pt}$$},{P})$ .\",\n \"It can be proved that the minimizer in $m$ is a well defined constant, which belongs to a compact set in $\\\\mathbb {R}_{> 0}$ .\",\n \"To see this, let us first observe that, obviously, from (REF ) one can obtain $\\\\Big [\\\\ \\\\inf _{\\\\mathbf {Q}\\\\in \\\\mathbf {\\\\Omega }}\\\\ \\\\inf _{k=1,\\\\ldots ,K}\\\\frac{m \\\\cdot q_{k}}{p_{k}}\\\\in dom(\\\\varphi ),\\\\quad \\\\sup _{\\\\mathbf {Q}\\\\in \\\\mathbf {\\\\Omega }}\\\\ \\\\sup _{k=1,\\\\ldots ,K}\\\\frac{m \\\\cdot q_{k}}{p_{k}}\\\\in dom(\\\\varphi ) \\\\Big ]\\\\ \\\\Longleftrightarrow \\\\ m \\\\in ]0,\\\\infty [ .$ Moreover, for any fixed ${Q}$ in $\\\\Omega $$\\\\Omega $ there is a unique number $m=m({Q}) > 0$ which satisfies the first-order optimality condition for $m \\\\in \\\\ ]0,\\\\infty [$ $\\\\psi _{{Q}}(m):=\\\\frac{d}{dm} D_{\\\\varphi }\\\\left( m \\\\cdot {Q},{P}\\\\right)=\\\\sum _{k=1}^{K}q_{k} \\\\cdot \\\\varphi ^{\\\\prime }\\\\left( \\\\frac{m \\\\cdot q_{k}}{p_{k}}\\\\right) =0$ and thus $D_{\\\\varphi }\\\\left( m({Q}) \\\\cdot {Q},{P}\\\\right) =\\\\inf _{m \\\\ne 0} D_{\\\\varphi }\\\\left( m \\\\cdot {Q},{P}\\\\right) ;$ indeed, the mapping $ ]0,\\\\infty [ \\\\ \\\\ni m \\\\rightarrow D_{\\\\varphi }\\\\left( m \\\\cdot {Q},{P}\\\\right) $ is strictly convex and infinitely differentiable (which follows straightforwardly from (G5),(G6) in the below-mentioned Section together with (C7ii), (C7iii) in Appendix D), and the strictly increasing function $\\\\psi _{{Q}}$ is such that $\\\\psi _{{Q}}(m)$ is strictly negative for all $m \\\\in ]0,1[$ for which $\\\\sup _{k=1,\\\\ldots ,K}\\\\frac{m \\\\cdot q_{k}}{p_{k}} <1$ whereas $\\\\psi _{{Q}}(m)$ is strictly positive for all $m > 1$ for which $\\\\inf _{k=1,\\\\ldots ,K}\\\\frac{m \\\\cdot q_{k}}{p_{k}} >1$ (recall the note right after (REF ) and $\\\\varphi ^{\\\\prime }(1)=0$ ).\",\n \"Hence, for any ${Q} \\\\in \\\\textrm {$$\\\\hspace{-6.544pt}$$}$ the unique zero $m({Q})$ of (REF ) (and hence, unique minimizer in (REF )) is in the compact interval $\\\\Big [\\\\frac{1}{\\\\sup _{k=1,\\\\ldots ,K}\\\\frac{q_{k}}{p_{k}}}, \\\\, \\\\frac{1}{\\\\inf _{k=1,\\\\ldots ,K}\\\\frac{q_{k}}{p_{k}}}\\\\Big ]\\\\ \\\\subseteq \\\\ \\\\Big [\\\\frac{1}{\\\\sup _{{Q}\\\\in \\\\textrm {\\\\Omega \\\\hspace{-5.40608pt}\\\\Omega }}\\\\ \\\\sup _{k=1,\\\\ldots ,K}\\\\frac{q_{k}}{p_{k}}}, \\\\,\\\\frac{1}{\\\\inf _{{Q}\\\\in \\\\textrm {\\\\Omega \\\\hspace{-5.40608pt}\\\\Omega }}\\\\ \\\\inf _{k=1,\\\\ldots ,K}\\\\frac{q_{k}}{p_{k}}}\\\\Big ]\\\\ \\\\subset \\\\ \\\\Big ]\\\\frac{1}{b}, \\\\frac{1}{a}\\\\Big [ \\\\ = \\\\ ]0,\\\\infty [ .$ When $\\\\textrm {$$\\\\hspace{-6.544pt}$$}$ is closed in $\\\\mathbb {S}^{K}$ , then by continuity of the function ${Q} \\\\mapsto D_{\\\\varphi }\\\\left( m\\\\left( {Q}\\\\right)\\\\cdot {Q} ,{P}\\\\right) $ there exists a ${Q}^{\\\\ast }$ in $\\\\textrm {$$\\\\hspace{-6.544pt}$$}$ which achieves the infimum on $\\\\textrm {$$\\\\hspace{-6.544pt}$$}$ .\",\n \"When $\\\\textrm {$$\\\\hspace{-6.544pt}$$}$ is not closed but satisfies (7), then the infimum exists anyway, possibly on the boundary $\\\\partial \\\\textrm {$$\\\\hspace{-6.544pt}$$}$ .\",\n \"Anyhow, for such ${Q}^{\\\\ast }$ there holds $D_{\\\\varphi }\\\\left( m({Q}^{\\\\ast }) \\\\cdot {Q}^{\\\\ast },{P}\\\\right)\\\\le D_{\\\\varphi }\\\\left(\\\\textrm {\\\\right.\\\\Omega \\\\hspace{-6.544pt}\\\\Omega } ,{P}\\\\le D_{\\\\varphi }\\\\left( {Q}^{\\\\ast },{P}\\\\right) ,$ where we use the continuity of ${Q}\\\\mapsto D_{\\\\varphi }\\\\left( {Q},{P}\\\\right) $ and (REF ) to obtain the last inequality above, even when ${Q}^{\\\\ast }\\\\in \\\\partial \\\\textrm {$$\\\\hspace{-6.544pt}$$}$ and ${Q}^{\\\\ast }\\\\notin \\\\textrm {$$\\\\hspace{-6.544pt}$$}$ .\",\n \"That (REF ) provides sharp bounds can be seen through the case of power divergences.\",\n \"Indeed, for the latter one basically gets (cf.\",\n \"Appendix C) $m({Q}) = (1 + \\\\frac{\\\\gamma \\\\cdot (\\\\gamma -1)}{\\\\widetilde{c}} \\\\cdot D_{\\\\widetilde{c} \\\\cdot \\\\varphi _{\\\\gamma }}({Q}, {P}) \\\\, )^{1/(1-\\\\gamma )}$ and $D_{\\\\varphi _{\\\\gamma }}\\\\left( m({Q}) \\\\cdot {Q},{P}\\\\right) =\\\\frac{\\\\widetilde{c}}{\\\\gamma } (1- m({Q}))$ for the case $\\\\gamma \\\\in \\\\mathbb {R}\\\\backslash \\\\lbrace 0,1\\\\rbrace $ , respectively, $m({Q}) = \\\\exp (-\\\\frac{1}{\\\\widetilde{c}} \\\\cdot D_{\\\\widetilde{c}\\\\cdot \\\\varphi _{1}}({Q}, {P}))$ and $D_{\\\\widetilde{c} \\\\cdot \\\\varphi _{1}}\\\\left( m({Q}) \\\\cdot {Q},{P}\\\\right) =\\\\frac{\\\\widetilde{c}}{\\\\gamma } (1- m({Q}))$ for the case $\\\\gamma =1$ , respectively, $m({Q}) = 1$ and $D_{\\\\widetilde{c} \\\\cdot \\\\varphi _{0}}\\\\left( m({Q}) \\\\cdot {Q},{P}\\\\right) =D_{\\\\widetilde{c} \\\\cdot \\\\varphi _{0}}\\\\left({Q},{P}\\\\right)$ for the remaining case $\\\\gamma =0$ .\",\n \"In all cases, $D_{\\\\varphi _{\\\\gamma }}\\\\left( m({Q}) \\\\cdot {Q},{P}\\\\right)$ is an increasing function of $D_{\\\\varphi _{\\\\gamma }}\\\\left({Q},{P}\\\\right)$ and therefore, ${Q}^{\\\\ast }\\\\in \\\\arg \\\\inf _{{Q}\\\\in \\\\textrm {\\\\Omega \\\\hspace{-5.40608pt}\\\\Omega } } D_{\\\\varphi }\\\\left(m({Q}) \\\\cdot {Q},{P}\\\\right)$ also satisfies ${Q}^{\\\\ast }\\\\in \\\\arg \\\\inf _{{Q}\\\\in \\\\textrm {\\\\Omega \\\\hspace{-5.40608pt}\\\\Omega }}D_{\\\\varphi }\\\\left( {Q},{P}\\\\right)$ .\",\n \"Hence, the right-hand side and the left-hand side of (REF ) coincide.\",\n \"Now due to (REF ), the LHS of (REF ) can be estimated since by Theorem REF for each ${P}\\\\in \\\\mathbb {S}_{>0}^{K}$ the divergence $\\\\inf _{m\\\\ne 0}D_{\\\\varphi }\\\\left( m \\\\cdot {Q},{P}\\\\right)$ is BS-minimizable on sets $\\\\textrm {$$\\\\hspace{-6.544pt}$$} \\\\subset \\\\mathbb {S}^{K}$ .\",\n \"We shall propose in Section an algorithm to handle the estimation of the RHS of (REF ), whenever ${P}$ is known (as in Remark 13(v)) or when ${P}$ is approximated by the empirical distribution of the data set $\\\\left(X_{1},..,X_{n}\\\\right)$ .\",\n \"Also note that (REF ) holds also for $\\\\textrm {$$\\\\hspace{-6.544pt}$$}$ substituted by $A \\\\cdot \\\\textrm {$$\\\\hspace{-6.544pt}$$}$ for any $A\\\\ne 0$ .\",\n \"Other cases of interest include when $dom\\\\left( \\\\varphi \\\\right) $ is not $\\\\left] 0,\\\\infty \\\\right[$ .\",\n \"We list two cases which extend the above discussion.\",\n \"Firstly, consider $\\\\varphi $ with $dom\\\\left( \\\\varphi \\\\right) =\\\\left[ 0,\\\\infty \\\\right[$ .\",\n \"Then — since $\\\\varphi ^{\\\\prime }(0)=-\\\\infty $ in order that (REF ) should hold (see (G10ii) in Section below) — we may extend (REF ) to cases when $\\\\textrm {$$\\\\hspace{-6.544pt}$$} \\\\subset \\\\mathbb {S}^{K}$ instead of $\\\\textrm {$$\\\\hspace{-6.544pt}$$} \\\\subset \\\\mathbb {S}_{>0}^{K}$ , hence allowing for possible null entries in $\\\\textrm {$$\\\\hspace{-6.544pt}$$}$ .\",\n \"When $dom\\\\left( \\\\varphi \\\\right) =\\\\left] a,b\\\\right[ $ for some $a<0$ , then clearly the same argument leading to (REF ) holds; this case is of interest, for instance, when extending a statistical model to signed measures (see e.g.\",\n \"Broniatowski et al.\",\n \"[63] for the important task of testing the number of components in a parametric probability mixture model).\",\n \"Example 28 Consider the (non-probability version of the) Jensen-Shannon divergence defined by $J(\\\\mathbf {Q},\\\\mathbf {P}):=I(\\\\mathbf {Q},(\\\\mathbf {Q}+\\\\mathbf {P})/2)+I(\\\\mathbf {P},(\\\\mathbf {Q}+\\\\mathbf {P})/2),\\\\qquad \\\\mathbf {P}, \\\\mathbf {Q} \\\\in \\\\mathbb {R}_{\\\\ge 0}^{K},$ where $I(\\\\mathbf {Q},\\\\mathbf {P})$ denotes the modified Kullback-Leibler information between $\\\\mathbf {Q}$ and $\\\\mathbf {P}$ (cf.\",\n \"(REF ) with $\\\\mathbf {P}$ instead of ${P}$ ).\",\n \"In (REF ) and (REF ) of Example REF below, we shall show that $J(\\\\mathbf {Q},\\\\mathbf {P}) = D_{\\\\varphi _{snKL}}(\\\\mathbf {Q},\\\\mathbf {P})$ with (basically) divergence generator $\\\\varphi _{snKL}(t) := \\\\, t \\\\cdot \\\\log t + (t+1) \\\\cdot \\\\log \\\\Big ( \\\\frac{2}{t+1} \\\\Big )$ for $t > 0$ .\",\n \"It is known that $J^{2}$ is a metric.\",\n \"We explore the sharpness of the bounds for $J(\\\\textrm {$$\\\\hspace{-6.544pt}$$} ,{P})$ as defined in (REF ).\",\n \"For this, we consider a given probability distribution ${P}$ on $\\\\mathcal {Y}$ with strictly positive entries; the set $\\\\textrm {$$\\\\hspace{-6.544pt}$$}$ consists of all probability distributions ${Q}$ on $\\\\mathcal {Y}$ whose total variation distance $V({Q},{P}):=\\\\sum _{k=1}^{K}\\\\left|q_{k}-p_{k}\\\\right|$ notice that $V({Q},{P})$ always takes values in the interval $[0,2[$ to ${P}$ lies between $v$ and $v+h$ for $v>0$ and small $h$ and which also satisfies $\\\\sup \\\\left( \\\\sup _{k=1,\\\\ldots ,K}\\\\frac{p_{k}}{q_{k}},\\\\sup _{k=1,\\\\ldots ,K}\\\\frac{q_{k}}{p_{k}}\\\\right)\\\\le L$ for some strictly positive finite $L$ .\",\n \"This set $\\\\textrm {$$\\\\hspace{-6.544pt}$$}$ defines a class of distributions ${Q}$ away from ${P}$ still keeping some regularity w.r.t.\",\n \"${P}$ .\",\n \"Also, $\\\\textrm {$$\\\\hspace{-6.544pt}$$}$ satisfies (7).\",\n \"We will prove that the bounds in (120) are sharp in this case.\",\n \"Notice that $J(m \\\\cdot {Q},{P}) = \\\\infty $ for $m<0$ and hence $\\\\inf _{{Q}\\\\in \\\\textrm {\\\\Omega \\\\hspace{-5.40608pt}\\\\Omega }}\\\\inf _{m \\\\ne 0}J(m \\\\cdot {Q},{P}) =\\\\inf _{{Q}\\\\in \\\\textrm {\\\\Omega \\\\hspace{-5.40608pt}\\\\Omega }}\\\\inf _{m>0}J(m \\\\cdot {Q},{P})$ .\",\n \"We first provide a lower bound for the latter.\",\n \"It holds for all $m>0$ and ${Q}$ in $\\\\textrm {$$\\\\hspace{-6.544pt}$$}$ $& & J(m{Q},{P}) =(m+1) \\\\cdot \\\\log \\\\left( 2 \\\\right)- (m+1) \\\\cdot \\\\log \\\\left( m+1\\\\right)+m\\\\log m+I^{\\\\alpha }({P},{Q})+m \\\\cdot I^{1-\\\\alpha }({Q},{P})\\\\nonumber $ where $\\\\alpha :=1/\\\\left( m+1\\\\right) $ and $I^{\\\\alpha }({P},{Q})$ is the $\\\\alpha -$ skewed Kullback-Leibler divergence between ${P}$ and ${Q}$ defined through $I^{\\\\alpha }({P},{Q}):=I({P},\\\\alpha {P}+(1-\\\\alpha ){Q}).$ By Inequality (27) in Yamano [410] $I^{\\\\alpha }({P},{Q})\\\\ge -\\\\log \\\\left( 1-\\\\frac{\\\\alpha ^{2}}{4} V({Q},{P})^{2}\\\\right).$ Since $(m+1) \\\\cdot \\\\log \\\\left(2\\\\right) -(m+1) \\\\cdot \\\\log (m+1)+ m \\\\cdot \\\\log m$ is non-negative for all $m>0$ and takes its minimal value 0 for $m=1$ , we obtain $\\\\inf _{m>0} J(m {Q},{P})\\\\ge \\\\inf _{m>0}K(m)$ where $K(m):=-\\\\log \\\\left( 1-\\\\frac{1}{4\\\\left( m+1\\\\right) ^{2}} \\\\cdot V({Q},{P})^{2}\\\\right) -m \\\\cdot \\\\log \\\\left( 1-\\\\frac{m^{2}}{4\\\\left( m+1\\\\right)^{2}}\\\\cdot V({Q},{P})^{2}\\\\right) .$ Since $-\\\\log (1-x)\\\\ge x$ for all $x<1$ and both $\\\\frac{1}{4\\\\left(m+1\\\\right) ^{2}} \\\\cdot V({Q},{P})^{2}$ and $\\\\frac{m^{2}}{4\\\\left( m+1\\\\right)^{2}}\\\\cdot V({Q},{P})^{2}$ are less than 1, it follows that $K(m)\\\\ge \\\\frac{V({Q},{P})^{2}}{4} \\\\cdot \\\\frac{m^{3} +1}{\\\\left(m+1\\\\right)^{2}}$ where the right-hand side attains its minimal value on $]0,\\\\infty [$ at $m^{+}= \\\\sqrt{3}-1 \\\\approx 0.73$ .\",\n \"Hence, we obtain $\\\\inf _{m>0}J(m\\\\cdot Q,P)\\\\ge \\\\frac{V({Q},{P})^{2}}{4}\\\\cdot (2\\\\sqrt{3}-3) > 0.116 \\\\, v^2$ Now by (19) in Yamano [410], for any ${Q}$ $J({Q},{P})\\\\le \\\\frac{1}{4}\\\\underline{J}({Q},{P})$ where $\\\\underline{J}({Q},{P}):=I({Q},{P})+I({P},{Q})$ is the Jensen divergence (also called symmetrized Kullback-Leibler divergence) between ${Q}$ and ${P}$ .\",\n \"Since (see Dragomir [116]) $I({P},{Q})\\\\le \\\\sum _{k=1}^{K}\\\\sqrt{\\\\frac{p_{k}}{q_{k}}}\\\\cdot \\\\left|q_{k}-q_{k}\\\\right|,$ it follows that $J({Q}^{\\\\ast },{P})\\\\le \\\\frac{1}{2}\\\\sqrt{L} \\\\cdot V({Q}^{\\\\ast },{P})$ which provides $0.116 \\\\, v^2\\\\ \\\\le \\\\ \\\\inf _{m >0}J(m \\\\cdot {Q}^{\\\\ast },{P})=J(m({Q}^{\\\\ast })\\\\cdot {Q}^{\\\\ast },{P}) \\\\ \\\\le \\\\ J(\\\\textrm {$ $\\\\hspace{-6.544pt}$$},{P})\\\\ \\\\le \\\\ J({Q}^{\\\\ast },{P}) \\\\ \\\\le \\\\ \\\\frac{1}{2}\\\\sqrt{L} \\\\cdot (v+h).$$For small $ v$ thedifference between the RHS and the LHS in the above display is $ cstv + o(v) + 12 L h$which proves that the bounds are sharp locally, with non-trivial lower bound.Other upper bounds can be adapted to sets $$\\\\Omega $$\\\\Omega $$ defined throughtighter conditions on $ Q$\\\\Omega $$\\\\Omega $ k=1,...,Kpkqk$and $ Q$\\\\Omega $$\\\\Omega $ k=1,...,Kqkpk$ (of e.g.\",\n \"Dragomir \\\\cite {Dra:00}).$ 0.3cm\"\n ],\n [\n \"On the difference between minimization problems of\\ndeterministic nature and risk minimization\",\n \"In the context of minimization of the functional $\\\\Phi _{\\\\mathbf {P}}(\\\\mathbf {Q})$ over $\\\\mathbf {\\\\Omega } \\\\subset \\\\mathbb {R}^{K}$ for known vector $\\\\mathbf {P}$ , due to Theorem REF our bare simulation approach allows for the approximate solution for any divergence $D_{\\\\varphi }$ satisfying the basic representation (REF ).\",\n \"Indeed, any proxy of $\\\\mathbb {\\\\Pi }\\\\big [\\\\xi _{n}^{\\\\mathbf {\\\\widetilde{W}}}\\\\in \\\\Omega /M_{\\\\mathbf {P}}\\\\big ]$ yields a corresponding proxy for $\\\\Phi _{\\\\mathbf {P}}\\\\left[\\\\mathbf {\\\\mathbf {\\\\Omega }}\\\\right]$ .\",\n \"This paves the way to the solution of numerous optimization problems, where the divergence $D_{\\\\varphi }$ is specifically suited to the problem at hand.\",\n \"In the statistical context, when the probability distribution (in its vector-form) ${P}$ is unknown up to some indirect information provided by sampling or by any mean providing a sequence $(X_{i})_{i\\\\in \\\\mathbb {N}}$ satisfying condition (REF ) (resp.\",\n \"(REF )), Theorem REF adds a complementary step of complexity; indeed, the estimation of $\\\\Phi _{\\\\mathbf {P}}(\\\\textrm {$$\\\\hspace{-6.544pt}$$})$ over $\\\\textrm {$$\\\\hspace{-6.544pt}$$}\\\\subset \\\\mathbb {S}^{K}$ results as its subproduct through the optimization upon $m$ which can be performed explicitly only in a number of specific divergences $D_{\\\\varphi }$ , e.g.\",\n \"the power divergences $D_{\\\\varphi _{\\\\gamma }}$ , and which carries over also to their monotone transformations such as e.g.\",\n \"the Renyi divergences.\",\n \"It is of relevance to mention that — as already indicated above — these divergences cover a very broad range of statistical criteria, indeed most of them, from the (maximum-likelihood estimation connected) likelihood divergence ($\\\\gamma =0$ ) to the Kullback-Leibler one ($\\\\gamma =1$ ), the two standard Chi-square distances ($\\\\gamma =2$ , $\\\\gamma =-1$ ), the Hellinger distance ($\\\\gamma =1/2$ ), etc.\",\n \"; in contrast with deterministic minimization problems, the choice of a statistical criterion (or risk function) is not imposed by the modelling of the problem at hand, but is dictated by the need for sharp measures of fit.\",\n \"Other divergences are more difficult to handle and our general results in Section REF should still prove some usefulness, since estimation of upper and lower bounds for risk is of common use.\",\n \"As a “preparatory” remark, recall first that each probability distribution (probability measure) $\\\\mathbb {P}$ on $\\\\mathcal {Y}=\\\\left\\\\lbrace d_{1},\\\\ldots ,d_{K}\\\\right\\\\rbrace $ has been uniquely identified with the vector ${P} := (p_{1}, \\\\ldots , p_{K}) \\\\in \\\\mathbb {S}^{K}$ of the corresponding probability masses (frequencies) $p_{k} = \\\\mathbb {P}[\\\\lbrace d_{k} \\\\rbrace ]$ via $\\\\mathbb {P}[A] = \\\\sum _{k=1}^{K} p_{k} \\\\cdot {1}_A(d_{k}) $ for each $A \\\\subset \\\\mathcal {Y}$ ; from this, we have measured the distance/divergence between two probability distributions $\\\\mathbb {P}, \\\\mathbb {Q}$ through the distance/divergence between their frequency vectors ${P}, {Q}$ : $D_{\\\\varphi }(\\\\mathbb {Q},\\\\mathbb {P}) : = D_{\\\\varphi }({Q},{P})\\\\qquad \\\\textrm {(cf.\",\n \"(\\\\ref {measure divergence}))} .\\\\nonumber $ However, it has been noted in Kißlinger & Stummer [190] in a context of even more general divergences $D(\\\\mathbf {Q}, \\\\mathbf {P})$ between vectors $\\\\mathbf {P}, \\\\mathbf {Q}$ that — alternatively — the latter two may consist of components $p_{k} = \\\\mathbb {P}[\\\\lbrace E_{k} \\\\rbrace ]$ , $q_{k} = \\\\mathbb {Q}[\\\\lbrace E_{k} \\\\rbrace ]$ which are probabilities of only some selected (e.g.\",\n \"increasing) events $(E_{k})_{k \\\\in \\\\lbrace 1,\\\\ldots ,M\\\\rbrace }$ of application-based concrete interest (within not necessarily discrete probability models).\",\n \"Of course, we can apply our BS method to such a vector context.\",\n \"As other alternatives, in the following we deal with divergences between non-probabilistic uncertainty quantifications.\"\n ],\n [\n \"Minimization problems with fuzzy sets\",\n \"Our BS framework also covers the — imprecise/inexact/vague information describing — fuzzy sets (cf.\",\n \"Zadeh [417]) and optimization problems on divergences between those.\",\n \"Indeed, let $\\\\mathcal {Y}=\\\\left\\\\lbrace d_{1},\\\\ldots ,d_{K}\\\\right\\\\rbrace $ be a finite set (called the universe (of discourse)), $A \\\\subset \\\\mathcal {Y}$ and $M^{A}: \\\\mathcal {Y} \\\\mapsto [0,1]$ be a corresponding membership function, where $M^{A}(d_{k})$ represents the degree/grade of membership of the element $d_{k}$ to the set $A$ ; accordingly, the object $A^{\\\\ast } := \\\\lbrace \\\\langle x,M^{A}(x) \\\\rangle | x \\\\in \\\\mathcal {Y}\\\\rbrace $ is called a fuzzy set in $\\\\mathcal {Y}$ (or fuzzy subset of $ \\\\mathcal {Y}$ ).\",\n \"Moreover, if $A \\\\subset \\\\mathcal {Y}$ and $B \\\\subset \\\\mathcal {Y}$ are unequal, then the corresponding membership functions $M^{A}$ and $M^{B}$ should be unequal.\",\n \"Furthermore, we model the vector of membership degrees to $A$ by $\\\\mathbf {P}^{A} := \\\\left(p_{k}^{A} \\\\right)_{k=1,\\\\ldots ,K}:= \\\\left(M^{A}(d_{k}) \\\\right)_{k=1,\\\\ldots ,K}$ which satisfies the key constraint $0 \\\\le p_{k}^{A} \\\\le 1$ for all $k \\\\in \\\\lbrace 1,\\\\ldots ,K\\\\rbrace $ and, consequently, the aggregated key constraint $0 \\\\le \\\\sum _{k=1}^{K} p_{k}^{A} \\\\le K$ (as a side remark, $\\\\sum _{k=1}^{K} M^{A}(d_{k})$ is called power of the fuzzy set $A^{\\\\ast }$).\",\n \"For divergence generators $\\\\varphi $ in $\\\\Upsilon (]a,b[)$ (resp.\",\n \"$\\\\widetilde{\\\\Upsilon }(]a,b[)$ ) with — say — $0 \\\\le a<10$ .\",\n \"As an example, for the special choice $\\\\varphi (t): = \\\\varphi _{1}(t) := t \\\\cdot \\\\log t + 1 - t \\\\in [0, \\\\infty [$ (cf.\",\n \"REF ), we derive from (REF ) and (REF ) the divergence $& & 0 \\\\le D_{\\\\varphi _{1}}(\\\\breve{M}_{2}, \\\\breve{M}_{1}) =\\\\sum \\\\limits _{k=2}^{2^{K}} \\\\frac{M_{2}(A_{k})}{2^{|A_{k}|}-1} \\\\cdot \\\\log \\\\Big (\\\\frac{M_{2}(A_{k})}{M_{1}(A_{k})} \\\\Big )- \\\\sum \\\\limits _{k=2}^{2^{K}} \\\\frac{M_{2}(A_{k})}{2^{|A_{k}|}-1} + \\\\sum \\\\limits _{k=2}^{2^{K}} \\\\frac{M_{1}(A_{k})}{2^{|A_{k}|}-1}\\\\nonumber \\\\\\\\& & =: D^{SD}(M_{2}, M_{1}) -\\\\sum \\\\limits _{k=2}^{2^{K}} \\\\frac{M_{2}(A_{k})}{2^{|A_{k}|}-1} + \\\\sum \\\\limits _{k=2}^{2^{K}} \\\\frac{M_{1}(A_{k})}{2^{|A_{k}|}-1}\\\\nonumber $ where $D^{SD}(M_{2}, M_{1})$ has been recently developed by Song & Deng [339]; notice that $D^{SD}(M_{2}, M_{1})$ may be negative (cf.\",\n \"Stummer & Vajda [349]) and then it is not a divergence anymore.\",\n \"However, for applications to data fusion Song & Deng apply the symmetrization $\\\\frac{1}{2} \\\\cdot \\\\left(D^{SD}(M_{2}, M_{1}) + D^{SD}(M_{1}, M_{2}) \\\\right)$ which is equal to $\\\\frac{1}{2} \\\\cdot \\\\left( D_{\\\\varphi _{1}}(\\\\breve{M}_{2}, \\\\breve{M}_{1})+ D_{\\\\varphi _{1}}(\\\\breve{M}_{1}, \\\\breve{M}_{2}) \\\\right)$ and thus nonnegative.\",\n \"Of course, we can also correspondingly adapt the transformations of $\\\\varphi -$ divergences (e.g.\",\n \"Renyi divergences) and entropy-type special cases given in the sections above and below, to BBAs as well as crossover cases and rescalings, and apply our BS method for this.\",\n \"Recall first that in Theorem REF , one crucial component is the sequence $(W_{n})_{n\\\\in \\\\mathbb {N}}$ of weights being i.i.d.\",\n \"copies of a random variable $W$ whose probability distribution is $\\\\mathbb {}$ (i.e.\",\n \"$\\\\mathbb {\\\\Pi }[W \\\\in \\\\cdot \\\\, ] = \\\\mathbb {}[ \\\\, \\\\cdot \\\\,]$ ), where the latter has to be connected with the divergence generator $\\\\varphi \\\\in \\\\Upsilon (]a,b[)$ through the representation $\\\\varphi (t)=\\\\sup _{z\\\\in \\\\mathbb {R}}\\\\left( z\\\\cdot t-\\\\log \\\\int _{\\\\mathbb {R}}e^{zy}d\\\\mathbb {} (y)\\\\right), \\\\qquad t\\\\in \\\\mathbb {R},\\\\ \\\\ \\\\hspace{28.45274pt} (cf.\",\n \"\\\\ (\\\\ref {Phi Legendre of mgf(W)}))$ under the additional requirement that the function $z\\\\mapsto MGF_{\\\\mathbb {} }(z):=\\\\int _{\\\\mathbb {R}}e^{zy}d\\\\mathbb {} (y)$ is finite on some open interval containing zero (“light-tailedness”); for Theorem REF , we need the corresponding variant (REF ) for $M_{\\\\mathbf {P}} \\\\cdot \\\\varphi \\\\in \\\\Upsilon (]a,b[)$ (rather than $\\\\varphi $ ).\",\n \"Hence, finding such “BS-associated pairs $(\\\\varphi ,\\\\mathbb {})$ ” is an important issue.\",\n \"Subsequently, let us discuss the following direction: starting from a concrete optimization problem (REF ) — respectively (REF ) — with pregiven $\\\\varphi \\\\in \\\\widetilde{\\\\Upsilon } (]a,b[)$ (cf.\",\n \"Definition REF ), as a first step one would like to verify whether indeed $\\\\varphi \\\\in \\\\Upsilon (]a,b[)$ (i.e.\",\n \"it additionally satisfies (REF )) — respectively $M_{\\\\mathbf {P}} \\\\cdot \\\\varphi \\\\in \\\\Upsilon (]a,b[)$ ; as a second step, one would like to find the corresponding $\\\\mathbb {}$ explicitly.\",\n \"As far as the above-mentioned first step is concerned, let us first present some fundamental properties of all $\\\\varphi \\\\in \\\\Upsilon (]a,b[)$ : Proposition 29 Let $\\\\varphi \\\\in \\\\Upsilon (]a,b[)$ .\",\n \"Then the following assertions hold: $\\\\varphi : \\\\, ]-\\\\infty ,\\\\infty [ \\\\rightarrow [0,\\\\infty ]$ is lower semicontinuous and convex; $\\\\varphi (1)=0$ ; $int(dom(\\\\varphi )) = ]a,b[$ for some $-\\\\infty \\\\le a < 1 < b \\\\le \\\\infty $ ; $\\\\varphi $ is continuously differentiable on $]a,b[$ (i.e.\",\n \"$\\\\varphi \\\\in C^{1}(]a,b[)$ ; $\\\\varphi $ is strictly convex only in a non-empty neighborhood $]t_{-}^{sc},t_{+}^{sc}[ \\\\subseteq ]a,b[$ of one ($t_{-}^{sc} < 1 < t_{+}^{sc}$ ); $\\\\varphi $ is infinitly differentiable on $]t_{-}^{sc},t_{+}^{sc}[$ (i.e.\",\n \"$\\\\varphi \\\\in C^{\\\\infty }(]t_{-}^{sc},t_{+}^{sc}[$ ), and hence, $\\\\varphi ^{\\\\prime }(1) = 0$ , $\\\\varphi ^{\\\\prime \\\\prime }(t) > 0$ for all $t \\\\in ]t_{-}^{sc},t_{+}^{sc}[$ ; notice that the left-hand second derivative and the right-hand second derivative of $\\\\varphi $ may not coincide at $t_{-}^{sc}$ respectively at $t_{+}^{sc}$ (i.e.\",\n \"possible non-second-differentiability at these two points); if $a > -\\\\infty $ , then $a=t_{-}^{sc}$ ; if $a = -\\\\infty $ , then either $t_{-}^{sc} = - \\\\infty $ or $\\\\varphi (t) = \\\\varphi (t_{-}^{sc}) + \\\\varphi ^{\\\\prime }(t_{-}^{sc}) \\\\cdot (t- t_{-}^{sc})$ for all $t \\\\in ]-\\\\infty , t_{-}^{sc}[$ (affine-linearity); notice that $\\\\varphi ^{\\\\prime }(t_{-}^{sc}) < 0$ ; if $b < \\\\infty $ , then $b=t_{+}^{sc}$ ; if $b = \\\\infty $ , then either $t_{+}^{sc} = \\\\infty $ or $\\\\varphi (t) = \\\\varphi (t_{+}^{sc}) + \\\\varphi ^{\\\\prime }(t_{+}^{sc}) \\\\cdot (t- t_{+}^{sc})$ for all $t \\\\in ]t_{+}^{sc},\\\\infty [$ (affine-linearity); notice that $\\\\varphi ^{\\\\prime }(t_{+}^{sc}) > 0$ ; the Fenchel-Legendre transform (also called convex conjugate) of $\\\\varphi $ $\\\\varphi ^{*} (z)=\\\\sup _{t\\\\in \\\\mathbb {R}}\\\\left( z\\\\cdot t- \\\\varphi (t) \\\\right)= \\\\sup _{t\\\\in ]a,b[}\\\\left( z\\\\cdot t- \\\\varphi (t) \\\\right), \\\\qquad z\\\\in \\\\mathbb {R},\\\\ \\\\ $ has the following properties: $int(dom(\\\\varphi ^{*})) = ]\\\\lambda _{-},\\\\lambda _{+}[$ , where $dom(\\\\varphi ^{*}) := \\\\lbrace z \\\\in \\\\mathbb {R} : -\\\\infty < \\\\varphi ^{*}(z) < \\\\infty \\\\rbrace $ , $\\\\lambda _{-} := \\\\inf _{t \\\\in ]a,b[} \\\\varphi ^{\\\\prime }(t) =\\\\lim _{t \\\\downarrow a} \\\\varphi ^{\\\\prime }(t) =: \\\\varphi ^{\\\\prime }(a) < 0$ and $\\\\lambda _{+} := \\\\sup _{t \\\\in ]a,b[} \\\\varphi ^{\\\\prime }(t) =\\\\lim _{t \\\\uparrow b} \\\\varphi ^{\\\\prime }(t) =: \\\\varphi ^{\\\\prime }(b) >0$ ; if $a >-\\\\infty $ , then $\\\\lambda _{-} = - \\\\infty $ ; the function $z \\\\mapsto e^{-a\\\\cdot z + \\\\varphi ^{*}(z)} =: M(z)$ is absolutely monotone on $]-\\\\infty ,0[$ , i.e.\",\n \"all derivatives exist and satisfy $\\\\frac{\\\\partial ^{k}}{\\\\partial z^k} M(z) \\\\ge 0$ ($k\\\\in \\\\mathbb {N}_{0}$ , $z \\\\in ]-\\\\infty ,0[$ ); $\\\\lim _{z \\\\rightarrow 0-} M(z) =1$ ; if $b < \\\\infty $ , then $\\\\lambda _{+} = \\\\infty $ ; the function $z \\\\mapsto e^{b\\\\cdot z + \\\\varphi ^{*}(- z)} =: M(z)$ is absolutely monotone on $]-\\\\infty ,0[$ ; $\\\\lim _{z \\\\rightarrow 0-} M(z) =1$ ; if $a = -\\\\infty $ and $b = -\\\\infty $ , then the function $z \\\\mapsto e^{\\\\varphi ^{*}(z)} =: M(z)$ is exponentially convex on $]\\\\lambda _{-}, \\\\lambda _{+}[$ , i.e.\",\n \"$M(\\\\cdot )$ is continuous and satisfies $\\\\sum _{i,j=1}^{n} c_{i} \\\\cdot c_{j} \\\\cdot M\\\\Big (\\\\frac{z_{i} + z_{j}}{2}\\\\Big ) \\\\ge 0 \\\\qquad \\\\textrm {for all n \\\\in \\\\mathbb {N},c_{i}, c_{j} \\\\in \\\\mathbb {R} and z_{i}, z_{j} \\\\in ]\\\\lambda _{-}, \\\\lambda _{+}[;}$ $\\\\lim _{z \\\\rightarrow 0-} M(z) =1$ ; as a side remark, notice the well-known fact that exponential-convexity is stronger than the usual log-convexity.\",\n \"the endpoints of $int(dom(\\\\varphi )) =]a,b[$ have the following important “functioning” for the underlying probability distribution $\\\\mathbb {}$ (cf.\",\n \"(REF )) respectively of an associated random variable $W$ with $\\\\mathbb {}[ \\\\cdot \\\\, ] := \\\\mathbb {\\\\Pi }[W \\\\in \\\\cdot \\\\, ]$ : $a = \\\\inf supp(\\\\mathbb {}) = \\\\inf supp(W)$ , $b = \\\\sup supp(\\\\mathbb {}) = \\\\sup supp(W)$ , where $supp(\\\\mathbb {})$ respectively $supp(W)$ denotes the support of $\\\\mathbb {}$ respectively $W$ ; consequently, $]a,b[ = int(conv(supp(\\\\mathbb {}))) = int(conv(supp(W)))$ where $conv(A)$ denotes the convex hull of a set $A$ ; if $a > -\\\\infty $ , then $\\\\varphi (a) = - \\\\log \\\\mathbb {}[\\\\lbrace a\\\\rbrace ]= - \\\\log \\\\mathbb {\\\\Pi }[W = a \\\\, ]$ ; consequently, there holds: $a = \\\\min supp(\\\\mathbb {}) = \\\\min supp(W)$ if and only if $\\\\mathbb {}[\\\\lbrace a\\\\rbrace ] = \\\\mathbb {\\\\Pi }[W = a \\\\, ] > 0$ if and only if $\\\\varphi (a) < \\\\infty $ if and only if $a \\\\in dom(\\\\varphi )$ ; if $b < \\\\infty $ , then $\\\\varphi (b) = - \\\\log \\\\mathbb {}[\\\\lbrace b\\\\rbrace ]= - \\\\log \\\\mathbb {\\\\Pi }[W = b \\\\, ]$ ; consequently, there holds: $b = \\\\max supp(\\\\mathbb {}) = \\\\max supp(W)$ if and only if $\\\\mathbb {}[\\\\lbrace b\\\\rbrace ] = \\\\mathbb {\\\\Pi }[W = b \\\\, ] > 0$ if and only if $\\\\varphi (b) < \\\\infty $ if and only if $b \\\\in dom(\\\\varphi )$ .\",\n \"the first two derivatives of $\\\\varphi $ at the point 1 have the following important “functioning” for the underlying probability distribution $\\\\mathbb {}$ (cf.\",\n \"(REF )) respectively of an associated random variable $W$ : $1 = \\\\varphi ^{\\\\prime -1}(0) = \\\\int _{\\\\mathbb {R}} y \\\\, d\\\\mathbb {} (y)= E_{\\\\mathbb {\\\\Pi }}[W]$ where $\\\\varphi ^{\\\\prime -1}(\\\\cdot )$ denotes the inverse of the first derivative $\\\\varphi ^{\\\\prime }(\\\\cdot )$ of $\\\\varphi (\\\\cdot )$ , $\\\\frac{1}{\\\\varphi ^{\\\\prime \\\\prime }(1)} =\\\\int _{\\\\mathbb {R}} \\\\Big (y - \\\\int _{\\\\mathbb {R}} \\\\widetilde{y} \\\\, d\\\\mathbb {} (\\\\widetilde{y})\\\\Big )^{2}\\\\, d\\\\mathbb {} (y)= E_{\\\\mathbb {\\\\Pi }}[W^{2}] - (E_{\\\\mathbb {\\\\Pi }}[W])^{2} = Var_{\\\\mathbb {\\\\Pi }}[W]$ ; in particular, scaling $\\\\widetilde{c} \\\\cdot \\\\varphi $ ($\\\\widetilde{c} >0$ ) does not change the mean 1 but the variance of $W$ .\",\n \"The corresponding proof of Proposition REF will be given in Appendix D, except for the second items of (G9ii) and (G9iii) as well as the first item of (G9iv).\",\n \"Those will be treated in the second next paragraph below, because the corresponding line of argumentation builds an insightful start for subsequently performed procedures to further track down the weight distribution $$ .\",\n \"The properties (G1) to (G9iv) constitute necessary conditions for a pregiven function $\\\\varphi $ to belong to $\\\\Upsilon (]a,b[)$ ); accordingly, these should be verified first, in concrete situations where one aims to apply the BS approach.\",\n \"An important role is played by the boundary points $a$ and $b$ of $int(dom(\\\\varphi ))$ through (G10i) to (G10iii), because their finiteness opens the gate to apply — via some straightforward transformations — a rich class of real-valued characterization theorems for probability distributions whose support lies in the positive real line $[0,\\\\infty [$ .\",\n \"In contrast, there exist much less real-valued characterization theorems for probability distributions whose support is the whole real line $]-\\\\infty ,\\\\infty [$ ; typically, the involved conditions are also more difficult to verify.\",\n \"Indeed, if $\\\\varphi \\\\in \\\\Upsilon (]a,b[)$ then one can deduce straightforwardly from the representation (REF ) that $e^{\\\\varphi ^{*}(z)} =\\\\int _{-\\\\infty }^{\\\\infty } e^{z\\\\cdot y} \\\\, \\\\mathrm {d}\\\\mathbb {} (y) =E_{\\\\mathbb {\\\\Pi }}[e^{z \\\\cdot W}] , \\\\quad z \\\\in ]\\\\lambda _{-}, \\\\lambda _{+}[,$ where $W$ is a random variable whose distribution is $\\\\mathbb {\\\\Pi }[W \\\\in \\\\cdot \\\\, ] = \\\\mathbb {}[ \\\\, \\\\cdot \\\\,]$ ; under the additional knowledge $a > -\\\\infty $ (and consequently $\\\\lambda _{-} = -\\\\infty $ ) employed together with (G10i) and thus $\\\\mathbb {\\\\Pi }[W \\\\ge a \\\\, ] =\\\\mathbb {}[ \\\\, [a,\\\\infty [ \\\\,] = 1$ , one arrives at $e^{\\\\varphi ^{*}(z) - a \\\\cdot z} =\\\\int _{a}^{\\\\infty } e^{z \\\\cdot (y-a)} \\\\, \\\\mathrm {d}\\\\mathbb {} (y) =\\\\int _{0}^{\\\\infty } e^{z\\\\cdot \\\\widetilde{y}} \\\\, \\\\mathrm {d}\\\\widetilde{\\\\widetilde{\\\\mathbb {}}} (\\\\widetilde{y})= E_{\\\\mathbb {\\\\Pi }}[e^{z \\\\cdot (W-a)}] , \\\\quad z \\\\in ]-\\\\infty , \\\\lambda _{+}[,$ where the probability distribution $\\\\widetilde{\\\\widetilde{\\\\mathbb {}}}[ \\\\, \\\\cdot \\\\, ] := \\\\mathbb {}[\\\\, \\\\cdot + a \\\\, ]$ is the $a-$ shifted companion of $\\\\mathbb {}$ ; recall that $\\\\lambda _{+} >0$ .\",\n \"Put in other words, $\\\\mathbb {\\\\Pi }[\\\\widetilde{W} \\\\in \\\\cdot \\\\, ] = \\\\widetilde{\\\\widetilde{\\\\mathbb {}}}[ \\\\, \\\\cdot \\\\,]$ is the probability distribution of the (a.s.) nonnegative random variable $\\\\widetilde{W} := W-a$ .\",\n \"Naturally, in this context, the interesting case is $-\\\\infty < a \\\\le 0$ .\",\n \"Similarly, if $\\\\varphi \\\\in \\\\Upsilon (]a,b[)$ and $b < \\\\infty $ (and hence $\\\\lambda _{+} = \\\\infty $ ), one can derive from (G10i) and its consequence $\\\\mathbb {\\\\Pi }[W \\\\le b \\\\, ] =\\\\mathbb {}[ \\\\, ]-\\\\infty ,b] \\\\,]= 1$ that $e^{\\\\varphi ^{*}(-z) + b \\\\cdot z} =\\\\int _{-\\\\infty }^{b} e^{z \\\\cdot (b-y)} \\\\, \\\\mathrm {d}\\\\mathbb {} (y) =\\\\int _{0}^{\\\\infty } e^{z\\\\cdot \\\\widetilde{y}} \\\\, \\\\mathrm {d}\\\\widetilde{\\\\widetilde{\\\\mathbb {}}} (\\\\widetilde{y})= E_{\\\\mathbb {\\\\Pi }}[e^{z \\\\cdot (b-W)}] , \\\\quad z \\\\in ]- \\\\infty , - \\\\lambda _{-}[,$ where $- \\\\lambda _{-} >0$ and the probability distribution $\\\\widetilde{\\\\widetilde{\\\\mathbb {}}}[ \\\\, \\\\cdot \\\\, ] := \\\\mathbb {}[\\\\, b - \\\\, \\\\cdot \\\\, ]$ is the mirrored$-b-$ shifted companion of $\\\\mathbb {}$ .\",\n \"This means that $\\\\mathbb {\\\\Pi }[\\\\widetilde{W} \\\\in \\\\cdot \\\\, ] = \\\\widetilde{\\\\widetilde{\\\\mathbb {}}}[ \\\\, \\\\cdot \\\\,]$ is the probability distribution of the (a.s.) nonnegative random variable $\\\\widetilde{W} := b-W$ .\",\n \"Naturally, the interesting case is $0 < b \\\\le \\\\infty $ .\",\n \"As already indicated above, the considerations (REF ) to (REF ) open the gate to the adaption of well-known real-valued (rather than complex-valued) characterizations from probability theory.\",\n \"To begin with, the following assertions are very prominent: Theorem 30 (a) Let $M: ]-\\\\infty ,0] \\\\mapsto ]0,\\\\infty [$ be continuous on $]-\\\\infty ,0]$ with $M(0) =1$ .\",\n \"Then one has $\\\\textrm {M is absolutely monotone on ]-\\\\infty ,0[}\\\\ \\\\Longleftrightarrow \\\\ \\\\exists \\\\ \\\\textrm {unique prob.\",\n \"distr.\",\n \"\\\\widetilde{\\\\widetilde{\\\\mathbb {}}} on [0,\\\\infty [s.t.}\",\n \"\\\\ \\\\ M(z) = \\\\int _{0}^{\\\\infty } e^{z \\\\cdot y} \\\\mathrm {d}\\\\widetilde{\\\\widetilde{\\\\mathbb {}}}(y)\\\\ \\\\textrm {for all z \\\\in ]-\\\\infty ,0[.\",\n \"}$ (b) Let $I$ be an open interval which contains 0, and $M: I \\\\mapsto [0,\\\\infty [$ be continuous with $M(0) =1$ .\",\n \"Then one gets $\\\\textrm {M is exponentially convex}\\\\ \\\\Longleftrightarrow \\\\ \\\\exists \\\\ \\\\textrm {unique prob.\",\n \"distr.\",\n \"\\\\widetilde{\\\\widetilde{\\\\mathbb {}}} on ]-\\\\infty ,\\\\infty [such that} \\\\ \\\\ M(z) = \\\\int _{-\\\\infty }^{\\\\infty } e^{z \\\\cdot y} \\\\mathrm {d}\\\\widetilde{\\\\widetilde{\\\\mathbb {}}}(y)\\\\ \\\\ \\\\textrm {for all z \\\\in I.\",\n \"}$ Assertion (a) of Theorem REF is known as (probability-version of) Bernstein’s theorem [42] (see e.g.\",\n \"also Schilling et al.\",\n \"[322]), whereas assertion (b) is known as (probability-version of) Widder’s theorem [391] for the relevant conversion between the involved Riemann-Stieltjes integral with nondecreasing (but not necessarily right-continuous) integrator into a measure integral, one can apply the general theory in e.g.\",\n \"Chapter 6 of Chow & Teicher [88].\",\n \"(see e.g.\",\n \"also Widder [392], Akhiezer [9], Shucker [333], Jaksetic & Pecaric [163], Kotelina & Pevny [196]).\",\n \"From Theorem REF (b) and (REF ), the first item in (G9iv) follows immediately by using the choice $I= ]\\\\lambda _{-}, \\\\lambda _{+}[$ .\",\n \"Moreover, Theorem REF (a) together with (REF ) (respectively (REF )) implies the second item of (G9ii) (respectively of (G9iii)).\",\n \"In fact, with the help of Theorem REF and some further considerations e.g.\",\n \"on light-tailedness, one even gets assertions on the sufficiency of (G9ii), (G9iii) and (G9iv) for a “candidate generator” $\\\\varphi $ to belong to the BS-suitable class $\\\\Upsilon (]a,b[)$ .\",\n \"More precisely, we obtain Proposition 31 Suppose that $\\\\varphi : ]-\\\\infty ,\\\\infty [ \\\\mapsto [0,\\\\infty ]$ satisfies (G1) to (G8), and recall the notations in (G9i).\",\n \"Then, $\\\\varphi \\\\in \\\\Upsilon (]a,b[)$ if one of the following three conditions holds: (a) $a >-\\\\infty $ , $\\\\lambda _{-} = - \\\\infty $ , and the function $z \\\\mapsto e^{-a\\\\cdot z + \\\\varphi ^{*}(z)}$ is absolutely monotone on $]-\\\\infty ,0[$ , (b) $b < \\\\infty $ , $\\\\lambda _{+} = \\\\infty $ , and the function $z \\\\mapsto e^{b\\\\cdot z + \\\\varphi ^{*}(- z)}$ is absolutely monotone on $]-\\\\infty ,0[$ , (c) $a = -\\\\infty $ , $b = -\\\\infty $ , and the function $z \\\\mapsto e^{\\\\varphi ^{*}(z)}$ is exponentially convex on $]\\\\lambda _{-}, \\\\lambda _{+}[$ .\",\n \"If one of the three conditions (a) to (c) holds, thenbasically by Theorem REF with $M(\\\\cdot )$ defined in G9(ii),(iii) or (iv); see Appendix D. the associated probability distribution $\\\\mathbb {}$ (cf.\",\n \"(REF )) has expectation $\\\\int _{\\\\mathbb {R}} y d\\\\mathbb {}(y) =1$ and finite moments of all orders, i.e.\",\n \"$\\\\int _{\\\\mathbb {R}} y^{j} d\\\\mathbb {}(y) < \\\\infty $ for all $j \\\\in \\\\mathbb {N}_{0}$ ; in terms of $\\\\mathbb {}[ \\\\cdot \\\\, ] := \\\\mathbb {\\\\Pi }[W \\\\in \\\\cdot \\\\, ]$ this means that $E_{\\\\mathbb {\\\\Pi }}[W] =1$ and $E_{\\\\mathbb {\\\\Pi }}[W^{j}] < \\\\infty $ .\",\n \"The proof of Proposition REF will be given in Appendix D. As far as applicability is concerned, it is well known that, in general, verifying absolute monotonicity is typically more comfortable than verifying exponential convexity.\",\n \"Fortunately, one can often use the former, since for many known divergence generators there holds $a > -\\\\infty $ (often $a=0$ ) or $b < \\\\infty $ or both, which by virtue of (G10i) is directly linked with the (endpoints of the) support of the potentially existing probability distribution $\\\\mathbb {}$ .\",\n \"For a pregiven divergence generator $\\\\varphi $ , once its membership in $\\\\Upsilon (]a,b[)$ (and in particular, the representability (REF )) is verified, one would like to concretely find the underlying probability distribution $\\\\mathbb {}$ .\",\n \"This may be quite challenging, but can be made more comfortable by systematically narrowing down the family of distributions where $\\\\mathbb {}$ belongs to.\",\n \"In fact, we have already performed a first down-narrowing, in terms of identifying the endpoints of the support of $\\\\mathbb {}$ to be the endpoints of the effective domain of $\\\\varphi $ (cf.\",\n \"(G10i)).\",\n \"A further down-narrowing can be achieved from (REF ) to (REF ) in combination with real-valued characterization theorems which are more specific than Theorem REF .\",\n \"This will be shown exemplarily for a few important sub-setups, in the following.\",\n \"For the identification of light-tailed semi/half-lattice distributions, we obtain the following two sets of sufficient conditions, which even allow for the desired explicit determination of $\\\\mathbb {}$ : Proposition 32 Suppose that $\\\\varphi : ]-\\\\infty ,\\\\infty [ \\\\mapsto [0,\\\\infty ]$ satisfies (G1) to (G8), with some $a > -\\\\infty $ .\",\n \"Furthermore, assume that there exists some constant $\\\\breve{c} >0$ as well as some function $H: [0,\\\\infty [ \\\\mapsto [0,\\\\infty [$ which is continuous on $[0,1]$ with $H(1)=1$ and absolutely monotone on $]0,1[$ , such that $e^{\\\\varphi ^{*}(\\\\frac{z}{\\\\breve{c}}) - a\\\\cdot \\\\frac{z}{\\\\breve{c}} } = H(e^{z}), \\\\qquad z \\\\in ]-\\\\infty , \\\\breve{c} \\\\cdot \\\\lambda _{+}[.$ Then one has $\\\\varphi \\\\in \\\\Upsilon (]a,b[)$ and $\\\\mathbb {} = \\\\sum _{n=0}^{\\\\infty } p_{n} \\\\cdot \\\\delta _{a + \\\\breve{c} \\\\cdot n}\\\\qquad \\\\textrm {with \\\\ } p_{n} := \\\\frac{1}{n !}\",\n \"\\\\cdot \\\\frac{\\\\mathrm {d}^{n}H}{\\\\mathrm {d}t}(0),$ i.e.\",\n \"$\\\\mathbb {\\\\Pi }[W = a + \\\\breve{c} \\\\cdot n \\\\, ] = p_{n}$ ($n \\\\in \\\\mathbb {N}_{0}$ ).\",\n \"Proposition 33 Suppose that $\\\\varphi : ]-\\\\infty ,\\\\infty [ \\\\mapsto [0,\\\\infty ]$ satisfies (G1) to (G8), with some $b < \\\\infty $ .\",\n \"Furthermore, assume that there exists some constant $\\\\breve{c} >0$ as well as some function $H: [0,\\\\infty [ \\\\mapsto [0,\\\\infty [$ which is continuous on $[0,1]$ with $H(1)=1$ and absolutely monotone on $]0,1[$ , such that $e^{\\\\varphi ^{*}(- \\\\frac{z}{\\\\breve{c}}) + b \\\\cdot \\\\frac{z}{\\\\breve{c}} } = H(e^{z}), \\\\qquad z \\\\in ]-\\\\infty , - \\\\breve{c} \\\\cdot \\\\lambda _{-}[.$ Then one has $\\\\varphi \\\\in \\\\Upsilon (]a,b[)$ and $\\\\mathbb {} = \\\\sum _{n=0}^{\\\\infty } p_{n} \\\\cdot \\\\delta _{b - \\\\breve{c} \\\\cdot n}\\\\qquad \\\\textrm {with \\\\ } p_{n} := \\\\frac{1}{n !}\",\n \"\\\\cdot \\\\frac{\\\\mathrm {d}^{n}H}{\\\\mathrm {d}t}(0),$ i.e.\",\n \"$\\\\mathbb {\\\\Pi }[W = b - \\\\breve{c} \\\\cdot n \\\\, ] = p_{n}$ ($n \\\\in \\\\mathbb {N}_{0}$ ).\",\n \"The Propositions REF respectively REF follow from (REF ) respectively (REF ), some straightforward transformations, and a well-known characterization of probability generating functions $H$ (see e.g.\",\n \"in Theorem 1.2.10 of Stroock [343]).\",\n \"As an incentive for the following investigations, let us recall the discussion in the surroundings of Condition REF pertaining to the minimization problem (REF ), where we have addressed possible connections between the two representabilities (REF ) (needed e.g.\",\n \"for Theorem REF ) and (REF ) (needed e.g.\",\n \"for Theorem REF ); this strongly relates to the question, for which constants $\\\\widetilde{c} >0$ the validity $\\\\varphi \\\\in \\\\Upsilon (]a,b[)$ triggers the validity of $\\\\widetilde{c} \\\\cdot \\\\varphi \\\\in \\\\Upsilon (]a,b[)$ .\",\n \"To begin with, it is straightforward to see that $\\\\varphi \\\\in \\\\Upsilon (]a,b[)$ always implies $\\\\widetilde{c} \\\\cdot \\\\varphi \\\\in \\\\Upsilon (]a,b[)$ for all integers $\\\\widetilde{c} \\\\in \\\\mathbb {N}$ ; indeed, if $\\\\varphi $ satisfies (REF ) for some $\\\\mathbb {} = \\\\mathbb {\\\\Pi }[ W \\\\in \\\\cdot \\\\, ]$ , then for each integer $\\\\widetilde{c} \\\\in \\\\mathbb {N}$ one gets that $\\\\widetilde{c} \\\\cdot \\\\varphi $ satisfies (REF ) for $\\\\widetilde{\\\\mathbb {}} = \\\\mathbb {\\\\Pi }[ \\\\sum _{j=1}^{\\\\widetilde{c}} \\\\frac{W_{j}}{\\\\widetilde{c}} \\\\in \\\\cdot \\\\, ]$ ; in the latter, the $W_{j}$ ’s are i.i.d.\",\n \"copies from $W$ .\",\n \"Clearly, $MGF_{\\\\widetilde{\\\\mathbb {}}}$ is then finite on some open interval containing zero (differing from the one for $MGF_{\\\\mathbb {}}$ only by a scaling with $1/\\\\widetilde{c}$ ).\",\n \"For the following family of distributions, one can even trigger $\\\\widetilde{c} \\\\cdot \\\\varphi \\\\in \\\\Upsilon (]a,b[)$ for all $\\\\widetilde{c} >0$ : for the sake of a corresponding precise formulation, recall first the common knowledge that, generally speaking, a probability distribution $\\\\mathbb {}$ on $\\\\mathbb {R}$ with light tails — in the sense that its moment generating function $z\\\\mapsto MGF_{\\\\mathbb {} }(z):=\\\\int _{\\\\mathbb {R}}e^{z \\\\cdot y}d\\\\mathbb {} (y)$ is finite on some open interval $]\\\\lambda _{-},\\\\lambda _{+}[$ containing zero — is (said to be) infinitely divisible if there holds $\\\\textrm {for each n \\\\in \\\\mathbb {N} there existsa probability distribution \\\\mathbb {}_{n} on \\\\mathbb {R} such that}\\\\int _{-\\\\infty }^{\\\\infty } e^{z \\\\cdot y} d\\\\mathbb {} (y) =\\\\Big (\\\\int _{-\\\\infty }^{\\\\infty } e^{z \\\\cdot y} d\\\\mathbb {}_{n} (y) \\\\Big )^{n}, \\\\quad z \\\\in ]\\\\lambda _{-},\\\\lambda _{+}[ ;$ in fact, (REF ) means that the (light-tailed) moment generating function $MGF_{\\\\mathbb {} }$ is infinitely divisible in the sense that each $n-$ th root $(MGF_{\\\\mathbb {} })^{1/n}$ must be the moment generating function of some (light-tailed) probability distribution (denoted here by $\\\\mathbb {}_{n}$ ).\",\n \"In particular, (REF ) implies that $\\\\mathbb {}_{n}$ is unique, and that $\\\\mathbb {}$ must necessarily have (one-sided or two-sided) unbounded support $supp(\\\\mathbb {})$ .\",\n \"The latter may differ from $supp(\\\\mathbb {}_{n})$ .\",\n \"In our BS context (REF ), (REF ) equivalently means that the associated random variable $W$ is infinitely divisible (with light-tailed distribution), in the sense that $\\\\textrm {for each n \\\\in \\\\mathbb {N} there exists a sequence of i.i.d.\",\n \"random variables Y_{n,1}, \\\\cdots , Y_{n,n} such that }W \\\\stackrel{\\\\text{\\\\tiny d}}{=} Y_{n,1} + \\\\cdots + Y_{n,n} ,$ where $\\\\stackrel{\\\\text{\\\\tiny d}}{=}$ means “have equal probability distributions” and $\\\\mathbb {\\\\Pi }[W \\\\in \\\\cdot \\\\, ] = \\\\mathbb {}[ \\\\, \\\\cdot \\\\,]$ , $\\\\mathbb {\\\\Pi }[Y_{n,1} \\\\in \\\\cdot \\\\, ] = \\\\mathbb {}_{n}[ \\\\, \\\\cdot \\\\,]$ .\",\n \"For the above-mentioned context, we obtain the useful Proposition 34 Suppose that $\\\\varphi \\\\in \\\\Upsilon (]a,b[)$ , with connected probability distribution $\\\\mathbb {}$ from (REF ) (recall that this implies that $\\\\mathbb {}$ is not a one-point distribution, cf.\",\n \"Remark REF ).\",\n \"Then there holds: $\\\\textrm {\\\\widetilde{c} \\\\cdot \\\\varphi \\\\in \\\\Upsilon (]a,b[)for all \\\\widetilde{c} > 0} \\\\ \\\\ \\\\Longleftrightarrow \\\\ \\\\ \\\\textrm {\\\\mathbb {} is infinitely divisible.\",\n \"}$ The proof of Proposition REF is given in Appendix E. Notice that Proposition REF covers especially the important prominent power divergences (cf.\",\n \"Examples REF and REF below) for which we provide the corresponding infinitely divisible distributions explicitly in the Examples REF and REF below, and for which the subsequent form of estimators (cf.\",\n \"Chapter below) can be simplified.\",\n \"For the identification of light-tailed infinitely divisible distributions, we obtain the following three sets of sufficient conditions: Proposition 35 Suppose that $\\\\varphi : ]-\\\\infty ,\\\\infty [ \\\\mapsto [0,\\\\infty ]$ satisfies (G1) to (G8), and recall the notations in (G9i) as well as $a = \\\\inf supp(\\\\mathbb {})$ , $b = \\\\sup supp(\\\\mathbb {})$ (cf.\",\n \"(G10i)).\",\n \"Then, $\\\\varphi \\\\in \\\\Upsilon (]a,b[)$ and the associated probability distribution $\\\\mathbb {}$ is infinitely divisible, if one of the following three conditions holds: (a) $a >-\\\\infty $ , $\\\\lambda _{-} = - \\\\infty $ , and the function $z \\\\mapsto \\\\varphi ^{* \\\\prime }(z) - a = (\\\\varphi ^{\\\\prime })^{-1}(z) - a $ is absolutely monotone on $]-\\\\infty ,0[$ , (b) $b < \\\\infty $ , $\\\\lambda _{+} = \\\\infty $ , and the function $z \\\\mapsto - \\\\varphi ^{* \\\\prime }(- z) + b= - (\\\\varphi ^{\\\\prime })^{-1}(-z) +b $ is absolutely monotone on $]-\\\\infty ,0[$ , (c) $a = -\\\\infty $ , $b = -\\\\infty $ , and the function $z \\\\mapsto \\\\frac{\\\\varphi ^{* \\\\prime \\\\prime }(z)}{\\\\varphi ^{* \\\\prime \\\\prime }(0)}= \\\\frac{\\\\varphi ^{\\\\prime \\\\prime }(1)}{\\\\varphi ^{\\\\prime \\\\prime }((\\\\varphi ^{\\\\prime })^{-1}(z))} $ is exponentially convex on $]\\\\lambda _{-}, \\\\lambda _{+}[$ .\",\n \"In the first case (a) there automatically follows $b=\\\\infty $ , whereas in the second case (b) one automatically gets $a =- \\\\infty $ .\",\n \"The proof of Proposition REF is given in Appendix E. So far, in the current section we have started from a given divergence generator $\\\\varphi \\\\in \\\\widetilde{\\\\Upsilon }(]a,b[)$ having some additional properties, switched to its Fenchel-Legendre transform $\\\\varphi ^{*}$ and some exponentially-linear transforms thereof, and presented some sufficient conditions for verifying that the outcome is a moment-generating function $MGF_{\\\\mathbb {}}$ of a unique probability distribution $\\\\mathbb {}$ which has light tails.\",\n \"For finding the concrete $\\\\mathbb {}$ , one typically should know the explicit form of $\\\\varphi ^{*}$ .\",\n \"However, it is well known that it can sometimes be hard to determine the explicit form of the Fenchel-Legendre transform of a convex function.\",\n \"This issue also applies for the reverse direction of starting from a concrete probability distribution $\\\\mathbb {}$ with light tails, computing its log-moment-generating function (called cumulant-generating function) $z \\\\mapsto \\\\Lambda _{\\\\mathbb {}}(z) := \\\\log MGF_{\\\\mathbb {}}(z)$ and the corresponding Fenchel-Legendre transform $\\\\Lambda _{\\\\mathbb {}}^{*}$ which is nothing but the associated divergence generator $\\\\varphi $ (cf.\",\n \"(REF )).\",\n \"As will be illuminated in several examples below, the — “kind of intermediate” — construction method given in the below-mentioned Theorem REF can help to ease these two tasks.\",\n \"To formulate this, we employ the class ${F}$ of functions $F: ]-\\\\infty ,\\\\infty [ \\\\mapsto [-\\\\infty ,\\\\infty ]$ with the following properties: $int(dom(F)) = ]a_{F},b_{F}[$ for some $-\\\\infty \\\\le a_{F} < 1 < b_{F} \\\\le \\\\infty $ ; $F$ is smooth (infinitely continuously differentiable) on $]a_{F},b_{F}[$ ; $F$ is strictly increasing on $]a_{F},b_{F}[$ .\",\n \"Clearly, for any $F \\\\in {F}$ one gets the existence of $F(a_{F}) := \\\\lim _{t \\\\downarrow a_{F}} F(t) \\\\in [-\\\\infty , \\\\infty [$ and $F(b_{F}) := \\\\lim _{t \\\\uparrow b_{F}} F(t) \\\\in ]-\\\\infty , \\\\infty ]$ ; moreover, its inverse $F^{-1}: \\\\mathcal {R}(F) \\\\mapsto [a_{F},b_{F}]$ exists, where $\\\\mathcal {R}(F) := \\\\lbrace F(t): t \\\\in dom(F) \\\\rbrace $ .\",\n \"Furthermore, $F^{-1}$ is strictly increasing and smooth (infinitely continuously differentiable) on the open interval $int(\\\\mathcal {R}(F)) = \\\\lbrace F(t): t \\\\in ]a_{F},b_{F}[ \\\\rbrace =]F(a_{F}),F(b_{F})[$ , and $F^{-1}(int(\\\\mathcal {R}(F))) = ]a_{F},b_{F}[$ .\",\n \"Within such a context, we obtain Theorem 36 Let $F \\\\in {F}$ and fix an arbitrary point $c \\\\in int(\\\\mathcal {R}(F))$ .\",\n \"Moreover, introduce the notationsfor the sake of brevity, we avoid here the more complete notation $\\\\lambda _{-}^{F,c}$ , $\\\\lambda _{+}^{F,c}$ , $t_{-}^{sc,F,c}$ , $t_{+}^{sc,F,c}$ indicating the dependence on $F$ and $c$ .\",\n \"$]\\\\lambda _{-},\\\\lambda _{+}[ := int(\\\\mathcal {R}(F)) -c$ and $]t_{-}^{sc},t_{+}^{sc}[ := ]1+a_{F}-F^{-1}(c),1+b_{F}-F^{-1}(c)[$ (which implies $\\\\lambda _{-} < 0 < \\\\lambda _{+}$ and $t_{-}^{sc} < 1 < t_{+}^{sc}$ ).\",\n \"Furthermore, define the functions $\\\\Lambda : \\\\ ]-\\\\infty ,\\\\infty [ \\\\ \\\\mapsto \\\\, [-\\\\infty , \\\\infty ]$ and $\\\\varphi : \\\\ ]-\\\\infty ,\\\\infty [ \\\\ \\\\mapsto \\\\, [0, \\\\infty ]$ by $\\\\hspace{-19.91684pt} \\\\Lambda (z) := \\\\Lambda ^{(c)}(z) \\\\hspace{-5.69046pt} &:=& \\\\hspace{-5.69046pt}{\\\\left\\\\lbrace \\\\begin{array}{ll}\\\\int \\\\displaylimits _{0}^{z} F^{-1}(u+c) \\\\, du+ z \\\\cdot (1-F^{-1}(c)) \\\\ \\\\in ]-\\\\infty , \\\\infty [,\\\\hspace{85.35826pt} \\\\textrm {if } z \\\\in ]\\\\lambda _{-},\\\\lambda _{+}[,\\\\\\\\\\\\int \\\\displaylimits _{0}^{\\\\lambda _{-}} F^{-1}(u+c) \\\\, du+ \\\\lambda _{-} \\\\cdot (1-F^{-1}(c))\\\\ \\\\in [-\\\\infty ,\\\\infty ],\\\\hspace{71.13188pt} \\\\textrm {if } z = \\\\lambda _{-} > -\\\\infty ,\\\\\\\\\\\\int \\\\displaylimits _{0}^{\\\\lambda _{+}} F^{-1}(u+c) \\\\, du+ \\\\lambda _{+} \\\\cdot (1-F^{-1}(c))\\\\ \\\\in \\\\ [-\\\\infty ,\\\\infty ],\\\\hspace{66.86414pt} \\\\textrm {if } z = \\\\lambda _{+} < \\\\infty ,\\\\\\\\\\\\infty , \\\\hspace{283.10483pt} \\\\textrm {else},\\\\end{array}\\\\right.\",\n \"}$ where the second respectively third line are meant as $\\\\lim _{z \\\\downarrow \\\\lambda _{-}} \\\\big ( \\\\int \\\\displaylimits _{0}^{z} F^{-1}(u+c) \\\\, du+ z \\\\cdot (1-F^{-1}(c)) \\\\big )$ respectively $\\\\lim _{z \\\\uparrow \\\\lambda _{+}} \\\\big ( \\\\int \\\\displaylimits _{0}^{z} F^{-1}(u+c) \\\\, du+ z \\\\cdot (1-F^{-1}(c)) \\\\big )$ , and $\\\\varphi (t) := \\\\varphi ^{(c)}(t) \\\\hspace{-5.69046pt} &:=& \\\\hspace{-5.69046pt}{\\\\left\\\\lbrace \\\\begin{array}{ll}(t+F^{-1}(c)-1) \\\\cdot [ F\\\\left(t+F^{-1}(c)-1 \\\\right) - c ]-\\\\int \\\\displaylimits _{0}^{F\\\\left(t+F^{-1}(c)-1 \\\\right) - c} F^{-1}(u+c) du\\\\ \\\\in \\\\ [0,\\\\infty [,\\\\\\\\\\\\hspace{327.20668pt}\\\\quad \\\\textrm {if }t \\\\in ]t_{-}^{sc},t_{+}^{sc}[,\\\\\\\\(t_{-}^{sc}+F^{-1}(c)-1) \\\\cdot [ F\\\\left(t_{-}^{sc}+F^{-1}(c)-1 \\\\right) - c ]-\\\\int \\\\displaylimits _{0}^{F\\\\left(t_{-}^{sc}+F^{-1}(c)-1 \\\\right) - c} F^{-1}(u+c) du\\\\ \\\\in \\\\ ]0,\\\\infty ],\\\\\\\\\\\\hspace{327.20668pt}\\\\quad \\\\textrm {if }t = t_{-}^{sc} > -\\\\infty ,\\\\\\\\(t_{+}^{sc}+F^{-1}(c)-1) \\\\cdot [ F\\\\left(t_{+}^{sc}+F^{-1}(c)-1 \\\\right) - c ]-\\\\int \\\\displaylimits _{0}^{F\\\\left(t_{+}^{sc}+F^{-1}(c)-1 \\\\right) - c} F^{-1}(u+c) du\\\\ \\\\in \\\\ ]0,\\\\infty ],\\\\\\\\\\\\hspace{327.20668pt}\\\\quad \\\\textrm {if }t = t_{+}^{sc} < \\\\infty ,\\\\\\\\\\\\varphi (t_{-}^{sc}) +\\\\lambda _{-}\\\\cdot (t- t_{-}^{sc}) \\\\ \\\\in \\\\ ]0,\\\\infty ],\\\\hspace{99.58464pt} \\\\textrm {if }t_{-}^{sc} > - \\\\infty , \\\\ \\\\textrm {and} \\\\ t \\\\in \\\\ ]-\\\\infty , t_{-}^{sc}[,\\\\\\\\\\\\varphi (t_{+}^{sc}) +\\\\lambda _{+}\\\\cdot (t - t_{+}^{sc}) \\\\ \\\\in \\\\ ]0,\\\\infty ],\\\\hspace{99.58464pt} \\\\textrm {if }t_{+}^{sc} < \\\\infty , \\\\ \\\\textrm {and} \\\\ t \\\\in \\\\ ]t_{+}^{sc}, \\\\infty [,\\\\\\\\\\\\infty , \\\\hspace{321.51622pt} \\\\textrm {else},\\\\end{array}\\\\right.\",\n \"}$ where the second respectively third line are again meant as lower respectively upper limit.\",\n \"Then, $\\\\Lambda $ and $\\\\varphi $ have the following properties: (i) On $]\\\\lambda _{-},\\\\lambda _{+}[$ , the function $\\\\Lambda $ is smooth and strictly convex and consequently, $\\\\exp (\\\\Lambda )$ ) is smooth and strictly log-convex; moreover, there holds $\\\\Lambda (0) =0$ , $\\\\Lambda ^{\\\\prime }(0) =1$ ; (ii) $\\\\varphi \\\\in \\\\widetilde{\\\\Upsilon }(]a,b[)$ , where $a := t_{-}^{sc} \\\\cdot {1}_{ \\\\lbrace -\\\\infty \\\\rbrace }(\\\\lambda _{-}) - \\\\infty \\\\cdot {1}_{]-\\\\infty ,0[}(\\\\lambda _{-})$ , $b := t_{+}^{sc} \\\\cdot {1}_{ \\\\lbrace \\\\infty \\\\rbrace }(\\\\lambda _{+}) + \\\\infty \\\\cdot {1}_{]0,\\\\infty [}(\\\\lambda _{+})$ , and $\\\\varphi $ has the properties (G1) to (G8).\",\n \"(iii) $\\\\varphi (t) =\\\\sup _{z \\\\in ]-\\\\infty ,\\\\infty [} \\\\left( z\\\\cdot t -\\\\Lambda (z)\\\\right) =\\\\sup _{z \\\\in ]\\\\lambda _{-},\\\\lambda _{+}[} \\\\left( z\\\\cdot t -\\\\Lambda (z)\\\\right)$ for all $t \\\\in \\\\mathbb {R}$ .\",\n \"(iv) $\\\\Lambda (z) = \\\\varphi ^{*}(z) =\\\\sup _{t \\\\in ]-\\\\infty ,\\\\infty [} \\\\left( t\\\\cdot z -\\\\varphi (t)\\\\right) =\\\\sup _{t \\\\in ]a,b[} \\\\left( t\\\\cdot z -\\\\varphi (t) \\\\right)$ for all $z \\\\in \\\\mathbb {R}$ .\",\n \"The proof of Theorem REF will be given Appendix F. Remark 37 Theorem REF indicates that the $F-$ constructed function $z \\\\mapsto \\\\exp (\\\\Lambda (z)) = \\\\exp (\\\\varphi ^{*}(z))$ is a good candidate for a moment generating function of a probability distribution $\\\\mathbb {}$ , and hence for the representability (REF ).\",\n \"However, one still needs to verify one of the conditions (a) to (c) of Proposition REF .\",\n \"This may go wrong, as the case of power divergences $\\\\varphi _{\\\\gamma }$ with $\\\\gamma \\\\in ]1,2[$ indicates (cf.\",\n \"the conjecture of Example REF (f) below).\",\n \"Notice that the newly constructed $\\\\Lambda $ and $\\\\varphi $ (cf.\",\n \"(REF ), (REF )) depend on the choice of the anchor point $c$ ; this is e.g.\",\n \"illustrated in Example REF (b) below.\",\n \"Hence, as a side effect, by using whole families $(F_{\\\\vartheta })_{\\\\vartheta }$ together with different anchor points $c$ , via Theorem REF one can generate new classes (and new classifications) of $\\\\varphi -$ divergence generators — and thus of corresponding $\\\\varphi -$ divergences — which can be of great use, even in other contexts beyond our BS optimization framework.\",\n \"If $F$ satisfies $F(1)=0$ and thus $F^{-1}(0)=1$ , then the natural choice $c:=0$ induces $]\\\\lambda _{-},\\\\lambda _{+}[ = int(\\\\mathcal {R}(F))$ and $]t_{-}^{sc},t_{+}^{sc}[ = ]a_{F},b_{F}[$ , and consequently (due to $F^{-1}(c)-1 =0$ ) leads to the simplification of “the first lines of” (REF ) and (REF ) to $& & \\\\hspace{-19.91684pt} \\\\Lambda (z) := \\\\Lambda ^{(0)}(z) :=\\\\int \\\\displaylimits _{0}^{z} F^{-1}(u) du,\\\\qquad z \\\\in int(\\\\mathcal {R}(F)),\\\\\\\\& & \\\\hspace{-19.91684pt} \\\\varphi (t) := \\\\varphi ^{(0)}(t) :=t \\\\cdot F\\\\left(t\\\\right)-\\\\int \\\\displaylimits _{0}^{F\\\\left(t\\\\right)} F^{-1}(u) du,\\\\qquad t \\\\in ]a_{F},b_{F}[ ;$ the simplifications of the respective other lines of (REF ) and (REF ) are straightforward.\",\n \"Remark 38 Let $F \\\\in {F}$ with $a_{F} =0$ , $b_{F} = \\\\infty $ , $F(1)=0$ and hence, $int(\\\\mathcal {R}(F)) = \\\\, ]F(0),F(\\\\infty )[$ .\",\n \"Then the transformation $\\\\widetilde{F}(t) &:=&{\\\\left\\\\lbrace \\\\begin{array}{ll}- \\\\int _{0}^{F(\\\\frac{1}{t})} F^{-1}(u) \\\\, du, \\\\qquad \\\\textrm {if } \\\\ t \\\\in ]0,\\\\infty [, \\\\\\\\- \\\\int _{0}^{F(\\\\infty )} F^{-1}(u) \\\\, du, \\\\qquad \\\\textrm {if } \\\\ t=0, \\\\\\\\-\\\\infty , \\\\qquad \\\\hspace{64.01869pt} \\\\textrm {if } \\\\ t \\\\in ]-\\\\infty , 0[,\\\\end{array}\\\\right.\",\n \"}$ satisfies $\\\\widetilde{F} \\\\in {F}$ with $a_{\\\\widetilde{F}} =0$ , $b_{\\\\widetilde{F}} = \\\\infty $ , $\\\\widetilde{F}(1)=0$ and $int(\\\\mathcal {R}(\\\\widetilde{F})) =\\\\big ] - \\\\int _{0}^{F(\\\\infty )} F^{-1}(u) \\\\, du, - \\\\int _{0}^{F(0)} F^{-1}(u) \\\\, du \\\\, \\\\big [$ .\",\n \"By choosing the natural anchor point $c=0$ (for both $F$ and $\\\\widetilde{F}$ ) and by using the relations $\\\\widetilde{F}(t) = - \\\\Lambda (F(\\\\frac{1}{t}))$ , $\\\\widetilde{F}^{-1}(z) = \\\\frac{1}{F^{-1}(\\\\Lambda ^{-1}(-z))}$ , as well as (REF ) in combination with (REF ) (for both contexts), it is straightforward to see that the corresponding quantities $\\\\widetilde{\\\\Lambda }$ and $\\\\widetilde{\\\\varphi }$ satisfy $\\\\widetilde{\\\\Lambda }(z) =- (-\\\\Lambda )^{-1}(z)$ ($z \\\\in int(\\\\mathcal {R}(\\\\widetilde{F}))$ ) and $\\\\widetilde{\\\\varphi }(t) = t \\\\cdot \\\\varphi (\\\\frac{1}{t})$ ($t \\\\in ]0,\\\\infty [$ ).\",\n \"Hence, the corresponding divergences (cf.\",\n \"(REF )) are “reciprocal to each other” in the sense that $D_{\\\\widetilde{\\\\varphi }}( \\\\mathbf {Q}, \\\\mathbf {P} ) =D_{\\\\varphi }( \\\\mathbf {P}, \\\\mathbf {Q} )$ for all $\\\\mathbf {P},\\\\mathbf {Q} \\\\in \\\\mathbb {S}_{>0}^{K}$ , and in case that $\\\\Lambda $ and $\\\\widetilde{\\\\Lambda }$ are indeed cumulant generating functions of some light-tailed distributions $\\\\mathbb {}$ and $\\\\widetilde{\\\\mathbb {}}$ (cf.\",\n \"Remark REF ), then the latter two are said to be inverse to each other in the sense of Tweedie [368] (see also e.g.\",\n \"Folks [127]).\",\n \"As already indicated above, from Theorem REF one can comfortably generate various interesting examples, which we demonstrate in the following.\",\n \"Example 39 (a) For $\\\\gamma \\\\in \\\\mathbb {R}\\\\backslash \\\\lbrace 1,2\\\\rbrace $ , $\\\\widetilde{c} \\\\in ]0,\\\\infty [$ and $]a_{F_{\\\\gamma ,\\\\widetilde{c}}},b_{F_{\\\\gamma ,\\\\widetilde{c}}}[ \\\\, = \\\\, ]0,\\\\infty [$ we define $F_{\\\\gamma ,\\\\widetilde{c}}(t) &:=&{\\\\left\\\\lbrace \\\\begin{array}{ll}\\\\frac{\\\\widetilde{c}}{\\\\gamma -1} \\\\cdot (t^{\\\\gamma -1}-1), \\\\qquad \\\\textrm {if } \\\\ t \\\\in \\\\, ]0,\\\\infty [, \\\\\\\\-\\\\frac{\\\\widetilde{c}}{\\\\gamma -1}, \\\\hspace{62.59596pt}\\\\textrm {if } \\\\ t=0 \\\\ \\\\textrm {and } \\\\ \\\\gamma \\\\in \\\\, ]1,2[ \\\\, \\\\cup \\\\, ]2,\\\\infty [, \\\\\\\\-\\\\infty , \\\\hspace{71.13188pt} \\\\textrm {if } \\\\ t=0 \\\\ \\\\textrm {and } \\\\ \\\\gamma < 1, \\\\\\\\- \\\\infty , \\\\hspace{71.13188pt} \\\\textrm {if } \\\\ t \\\\in \\\\, ]-\\\\infty ,0[,\\\\end{array}\\\\right.\",\n \"}\\\\nonumber $ Clearly, $\\\\mathcal {R}(F_{\\\\gamma ,\\\\widetilde{c}})= \\\\big [-\\\\frac{\\\\widetilde{c}}{\\\\gamma -1},\\\\infty \\\\big [$ for $\\\\gamma \\\\in \\\\, ]1,2[ \\\\, \\\\cup \\\\, ]2,\\\\infty [$ , respectively $\\\\mathcal {R}(F_{\\\\gamma ,\\\\widetilde{c}})=\\\\big ]-\\\\infty , \\\\frac{\\\\widetilde{c}}{1-\\\\gamma }\\\\big [$ for $\\\\gamma <1$ ; notice that $0 \\\\in int(\\\\mathcal {R}(F_{\\\\gamma ,\\\\widetilde{c}}))$ for all $\\\\gamma \\\\in \\\\mathbb {R}\\\\backslash \\\\lbrace 1,2\\\\rbrace $ .\",\n \"Furthermore, $F_{\\\\gamma ,\\\\widetilde{c}}(\\\\cdot )$ is strictly increasing and smooth on $]0,\\\\infty [$ , and thus, $F_{\\\\gamma ,\\\\widetilde{c}} \\\\in {F}$ .\",\n \"Since $F_{\\\\gamma ,\\\\widetilde{c}}(1)=0$ , let us choose the natural anchor point $c:=0$ , which leads to $]\\\\lambda _{-},\\\\lambda _{+}[= int(\\\\mathcal {R}(F_{\\\\gamma ,\\\\widetilde{c}}))$ and $]t_{-}^{sc},t_{+}^{sc}[ = ]0,\\\\infty [$ .\",\n \"By using $F_{\\\\gamma ,\\\\widetilde{c}}^{-1}(x) = (1+\\\\frac{(\\\\gamma -1) \\\\cdot x}{\\\\widetilde{c}})^{\\\\frac{1}{\\\\gamma -1}}$ for $x \\\\in int(\\\\mathcal {R}(F_{\\\\gamma ,\\\\widetilde{c}}))$ , we can derive from formula (REF ) (see also (REF )) for all $\\\\gamma \\\\in \\\\mathbb {R}\\\\backslash \\\\lbrace 0,1,2\\\\rbrace $ and $z \\\\in \\\\mathbb {R}$ $\\\\Lambda _{\\\\gamma ,\\\\widetilde{c}}(z) := \\\\Lambda _{\\\\gamma ,\\\\widetilde{c}}^{(0)}(z) &=&{\\\\left\\\\lbrace \\\\begin{array}{ll}\\\\frac{\\\\widetilde{c}}{\\\\gamma } \\\\cdot \\\\left\\\\lbrace \\\\left( \\\\frac{\\\\gamma -1}{\\\\widetilde{c}} \\\\cdot z +1 \\\\right)^{\\\\frac{\\\\gamma }{\\\\gamma -1}} -1 \\\\right\\\\rbrace , \\\\qquad \\\\textrm {if } \\\\ \\\\gamma \\\\in \\\\, ]1,2[ \\\\, \\\\cup \\\\, ]2,\\\\infty [\\\\ \\\\textrm {and } \\\\ z \\\\in \\\\big ]-\\\\frac{\\\\widetilde{c}}{\\\\gamma -1},\\\\infty \\\\big [ \\\\\\\\\\\\hspace{130.88284pt} \\\\textrm {or if } \\\\ \\\\gamma \\\\in ]-\\\\infty ,0[ \\\\cup ]0,1[ \\\\ \\\\textrm {and } \\\\ z \\\\in \\\\big ]-\\\\infty , \\\\frac{\\\\widetilde{c}}{1-\\\\gamma }\\\\big [, \\\\\\\\- \\\\frac{\\\\widetilde{c}}{\\\\gamma } < 0, \\\\hspace{105.2751pt} \\\\textrm {if } \\\\ \\\\gamma \\\\in \\\\, ]1,2[ \\\\, \\\\cup \\\\, ]2,\\\\infty [\\\\ \\\\textrm {and } \\\\ z = -\\\\frac{\\\\widetilde{c}}{\\\\gamma -1}, \\\\\\\\- \\\\frac{\\\\widetilde{c}}{\\\\gamma } >0, \\\\hspace{105.2751pt} \\\\textrm {if } \\\\ \\\\gamma < 0 \\\\ \\\\textrm {and } \\\\ z = \\\\frac{\\\\widetilde{c}}{1-\\\\gamma }, \\\\\\\\\\\\infty , \\\\hspace{128.0374pt} \\\\textrm {if } \\\\ \\\\gamma \\\\in \\\\, ]0,1[ \\\\ \\\\textrm {and } \\\\ z = \\\\frac{\\\\widetilde{c}}{1-\\\\gamma }, \\\\\\\\\\\\infty , \\\\hspace{128.0374pt} \\\\textrm {else} .\\\\end{array}\\\\right.\",\n \"}$ Notice that $\\\\Lambda _{\\\\gamma ,\\\\widetilde{c}}(0) = 0$ for all $\\\\gamma \\\\in \\\\mathbb {R}\\\\backslash \\\\lbrace 0,1,2\\\\rbrace $ .\",\n \"Moreover, for $\\\\gamma \\\\in \\\\, ]1,2[ \\\\, \\\\cup \\\\, ]2,\\\\infty [$ one has $\\\\Lambda _{\\\\gamma ,\\\\widetilde{c}}(\\\\infty ) = \\\\infty $ , $\\\\Lambda _{\\\\gamma ,\\\\widetilde{c}}^{\\\\prime }(-\\\\frac{\\\\widetilde{c}}{\\\\gamma -1}) = 0$ and $\\\\Lambda _{\\\\gamma ,\\\\widetilde{c}}^{\\\\prime }(\\\\infty ) = \\\\infty $ .\",\n \"For $\\\\gamma < 0$ one gets $\\\\Lambda _{\\\\gamma ,\\\\widetilde{c}}(-\\\\infty ) = -\\\\infty $ , $\\\\Lambda _{\\\\gamma ,\\\\widetilde{c}}^{\\\\prime }(\\\\frac{\\\\widetilde{c}}{1-\\\\gamma }) = \\\\infty $ and $\\\\Lambda _{\\\\gamma ,\\\\widetilde{c}}^{\\\\prime }(-\\\\infty ) = 0$ .\",\n \"In contrast, if $\\\\gamma \\\\in \\\\, ]0,1[$ then $\\\\Lambda _{\\\\gamma ,\\\\widetilde{c}}(-\\\\infty ) = - \\\\frac{\\\\widetilde{c}}{\\\\gamma } <0$ , $\\\\Lambda _{\\\\gamma ,\\\\widetilde{c}}^{\\\\prime }(\\\\frac{\\\\widetilde{c}}{1-\\\\gamma }) = \\\\infty $ and $\\\\Lambda _{\\\\gamma ,\\\\widetilde{c}}^{\\\\prime }(-\\\\infty ) = 0$ .\",\n \"To proceed, from formula (REF ) (see also ()) we can deduce for all $\\\\gamma \\\\in \\\\mathbb {R}\\\\backslash \\\\lbrace 0,1,2\\\\rbrace $ and $t \\\\in \\\\mathbb {R}$ $\\\\varphi _{\\\\gamma ,\\\\widetilde{c}}(t) := \\\\varphi _{\\\\gamma ,\\\\widetilde{c}}^{(0)}(t)\\\\hspace{-5.69046pt} &=& \\\\hspace{-5.69046pt}{\\\\left\\\\lbrace \\\\begin{array}{ll}\\\\widetilde{c} \\\\cdot \\\\frac{t^\\\\gamma -\\\\gamma \\\\cdot t+ \\\\gamma - 1}{\\\\gamma \\\\cdot (\\\\gamma -1)}\\\\ \\\\in \\\\ [0,\\\\infty [,\\\\qquad \\\\quad \\\\textrm {if }t \\\\in ]0,\\\\infty [,\\\\\\\\\\\\frac{\\\\widetilde{c}}{\\\\gamma } > 0, \\\\hspace{108.12054pt} \\\\textrm {if } \\\\ \\\\gamma \\\\in \\\\, ]1,2[ \\\\, \\\\cup \\\\, ]2,\\\\infty [\\\\ \\\\textrm {and } \\\\ t = 0, \\\\\\\\\\\\infty , \\\\hspace{123.76965pt} \\\\textrm {if } \\\\ \\\\gamma < 0 \\\\ \\\\textrm {and } \\\\ t = 0, \\\\\\\\\\\\frac{\\\\widetilde{c}}{\\\\gamma } > 0, \\\\hspace{108.12054pt} \\\\textrm {if } \\\\ \\\\gamma \\\\in \\\\, ]0,1[ \\\\ \\\\textrm {and } \\\\ t = 0, \\\\\\\\\\\\frac{\\\\widetilde{c}}{\\\\gamma } -\\\\frac{\\\\widetilde{c}}{\\\\gamma -1} \\\\cdot t \\\\ \\\\in \\\\ ]0,\\\\infty [,\\\\hspace{45.52458pt} \\\\textrm {if } \\\\ \\\\gamma \\\\in \\\\, ]1,2[ \\\\, \\\\cup \\\\, ]2,\\\\infty [\\\\ \\\\textrm {and } \\\\ t < 0, \\\\\\\\\\\\infty , \\\\hspace{123.76965pt} \\\\textrm {else} ,\\\\end{array}\\\\right.\",\n \"}$ which coincides with $\\\\widetilde{c} \\\\cdot \\\\varphi _{\\\\gamma }(t)$ for $\\\\varphi _{\\\\gamma }(t)$ from (REF ) and which generates the $\\\\gamma -$ corresponding power divergences given in (REF ); the first line in (REF ) can be proved by $& & \\\\hspace{-19.91684pt}\\\\varphi _{\\\\gamma ,\\\\widetilde{c}}(t) := \\\\varphi _{\\\\gamma ,\\\\widetilde{c}}^{(0)}(t) :=t \\\\cdot F_{\\\\gamma ,\\\\widetilde{c}}\\\\left(t\\\\right)-\\\\int \\\\displaylimits _{0}^{F_{\\\\gamma ,\\\\widetilde{c}}\\\\left(t\\\\right)} F_{\\\\gamma ,\\\\widetilde{c}}^{-1}(u) du\\\\nonumber \\\\\\\\& & \\\\hspace{-19.91684pt}= \\\\frac{t \\\\cdot \\\\widetilde{c}}{\\\\gamma -1} \\\\cdot (t^{\\\\gamma -1}-1)- \\\\frac{\\\\widetilde{c}}{\\\\gamma } \\\\cdot \\\\left\\\\lbrace \\\\left( \\\\frac{\\\\gamma -1}{\\\\widetilde{c}} \\\\cdot \\\\left[ \\\\frac{\\\\widetilde{c}}{\\\\gamma -1} \\\\cdot (t^{\\\\gamma -1}-1) \\\\right]+ 1 \\\\right)^{\\\\frac{\\\\gamma }{\\\\gamma -1}} -1 \\\\right\\\\rbrace \\\\nonumber \\\\\\\\& & \\\\hspace{-19.91684pt}= \\\\widetilde{c} \\\\cdot \\\\frac{t^\\\\gamma -\\\\gamma \\\\cdot t+ \\\\gamma - 1}{\\\\gamma \\\\cdot (\\\\gamma -1)},\\\\qquad t \\\\in ]0,\\\\infty [ \\\\, .$ Notice that for all $\\\\gamma \\\\in \\\\mathbb {R}\\\\backslash \\\\lbrace 0,1,2\\\\rbrace $ one has $\\\\varphi _{\\\\gamma ,\\\\widetilde{c}}(1) = 0$ , $\\\\varphi _{\\\\gamma ,\\\\widetilde{c}}^{\\\\prime }(1) = 0$ and $\\\\varphi _{\\\\gamma ,\\\\widetilde{c}}(\\\\infty ) = \\\\infty $ .\",\n \"Moreover, for $\\\\gamma \\\\in \\\\, ]1,2[ \\\\, \\\\cup \\\\, ]2,\\\\infty [$ one has $\\\\varphi _{\\\\gamma ,\\\\widetilde{c}}^{\\\\prime }(0) = -\\\\frac{\\\\widetilde{c}}{\\\\gamma -1} < 0$ and $\\\\varphi _{\\\\gamma ,\\\\widetilde{c}}^{\\\\prime }(\\\\infty ) = \\\\infty $ .\",\n \"In contrast, for $\\\\gamma < 0$ and $\\\\gamma \\\\in \\\\, ]0,1[$ one gets $\\\\varphi _{\\\\gamma ,\\\\widetilde{c}}^{\\\\prime }(0) = - \\\\infty $ and $\\\\varphi _{\\\\gamma ,\\\\widetilde{c}}^{\\\\prime }(\\\\infty ) = \\\\frac{\\\\widetilde{c}}{1-\\\\gamma } > 0$ .\",\n \"(b) For $\\\\gamma =2$ , $\\\\widetilde{c} \\\\in ]0,\\\\infty [$ and $]a_{F_{\\\\gamma ,\\\\widetilde{c}}},b_{F_{\\\\gamma ,\\\\widetilde{c}}}[ \\\\, = \\\\, ]-\\\\infty ,\\\\infty [$ we define $F_{2,\\\\widetilde{c}}(t) &:=&\\\\widetilde{c} \\\\cdot (t-1), \\\\qquad t \\\\in \\\\, ]-\\\\infty ,\\\\infty [,\\\\nonumber $ Clearly, $\\\\mathcal {R}(F_{2,\\\\widetilde{c}})= \\\\, ]-\\\\infty , \\\\infty [$ , $0 \\\\in int(\\\\mathcal {R}(F_{2,\\\\widetilde{c}}))$ , and $F_{2,\\\\widetilde{c}}(\\\\cdot )$ is strictly increasing as well as smooth on $]-\\\\infty ,\\\\infty [$ .\",\n \"Hence, $F_{2,\\\\widetilde{c}} \\\\in {F}$ .\",\n \"Since $F_{2,\\\\widetilde{c}}(1)=0$ , let us choose the natural anchor point $c:=0$ , which leads to $]\\\\lambda _{-},\\\\lambda _{+}[= int(\\\\mathcal {R}(F_{2,\\\\widetilde{c}}))= \\\\, ]-\\\\infty , \\\\infty [$ and $]t_{-}^{sc},t_{+}^{sc}[ = \\\\, ]-\\\\infty , \\\\infty [$ .\",\n \"By using $F_{2,\\\\widetilde{c}}^{-1}(x) = 1+\\\\frac{x}{\\\\widetilde{c}}$ for $x \\\\in int(\\\\mathcal {R}(F_{2,\\\\widetilde{c}}))$ , we can derive from formula (REF ) (see also (REF )) $\\\\Lambda _{2,\\\\widetilde{c}}(z) := \\\\Lambda _{2,\\\\widetilde{c}}^{(0)}(z) &=&\\\\frac{\\\\widetilde{c}}{2} \\\\cdot \\\\left\\\\lbrace \\\\left( \\\\frac{1}{\\\\widetilde{c}} \\\\cdot z +1 \\\\right)^{2} -1 \\\\right\\\\rbrace =\\\\frac{z^2}{2 \\\\widetilde{c}} + z,\\\\qquad z \\\\in \\\\, ]-\\\\infty ,\\\\infty [.$ Notice that $\\\\Lambda _{2,\\\\widetilde{c}}(0) = 0$ , $\\\\Lambda _{2,\\\\widetilde{c}}(-\\\\infty ) = \\\\Lambda _{2,\\\\widetilde{c}}(\\\\infty ) = \\\\infty $ , $\\\\Lambda _{2,\\\\widetilde{c}}^{\\\\prime }(-\\\\infty ) = -\\\\infty $ and $\\\\Lambda _{2,\\\\widetilde{c}}^{\\\\prime }(\\\\infty ) = \\\\infty $ .\",\n \"From formula (REF ) (see also ()) we can deduce analogously to (REF ) $\\\\varphi _{2,\\\\widetilde{c}}(t) := \\\\varphi _{2,\\\\widetilde{c}}^{(0)}(t)&=&\\\\widetilde{c} \\\\cdot \\\\frac{(t-1)^{2}}{2}\\\\ \\\\in \\\\ [0,\\\\infty [,\\\\qquad t \\\\in \\\\, ]-\\\\infty ,\\\\infty [,$ which coincides with $\\\\widetilde{c} \\\\cdot \\\\varphi _{2}(t)$ for $\\\\varphi _{2}(t)$ from (REF ) which generates the corresponding power divergence given in the sixth line of (REF ).\",\n \"Notice that $\\\\varphi _{2,\\\\widetilde{c}}(1) = 0$ , $\\\\varphi _{2,\\\\widetilde{c}}^{\\\\prime }(1) = 0$ and $\\\\varphi _{2,\\\\widetilde{c}}(-\\\\infty ) = \\\\varphi _{2,\\\\widetilde{c}}(\\\\infty ) = \\\\infty $ .\",\n \"As an application of the reciprocity considerations of Remark REF , it is straightforward to see from the above-mentioned considerations (a) and (b) that for all $\\\\gamma \\\\in \\\\mathbb {R}\\\\backslash \\\\lbrace 0,1\\\\rbrace $ one has $\\\\widetilde{F}_{\\\\gamma ,\\\\widetilde{c}}(t) =- \\\\Lambda _{\\\\gamma ,\\\\widetilde{c}}(F_{\\\\gamma ,\\\\widetilde{c}}(\\\\frac{1}{t}))= F_{1-\\\\gamma ,\\\\widetilde{c}}(t)$ for all $t \\\\in ]0,\\\\infty [$ .\",\n \"(c) Let us now continue with the remaining case $\\\\gamma =0$ (recall the natural anchor point $c:=0$ ).\",\n \"By using $F_{0,\\\\widetilde{c}}^{-1}(x) =\\\\frac{1}{1- \\\\frac{x}{\\\\widetilde{c}}} $ for $x \\\\in int(\\\\mathcal {R}(F_{0,\\\\widetilde{c}})) = ]-\\\\infty , \\\\widetilde{c}[$ , we can derive from formula (REF ) (see also (REF )) $\\\\Lambda _{0,\\\\widetilde{c}}(z) := \\\\Lambda _{0,\\\\widetilde{c}}^{(0)}(z) &=&{\\\\left\\\\lbrace \\\\begin{array}{ll}- \\\\widetilde{c} \\\\cdot \\\\log \\\\left( 1 - \\\\frac{z}{\\\\widetilde{c}} \\\\right), \\\\qquad \\\\textrm {if } \\\\ z \\\\in \\\\big ]-\\\\infty ,\\\\widetilde{c} \\\\big [, \\\\\\\\\\\\infty , \\\\hspace{78.24507pt} \\\\textrm {if } \\\\ z \\\\in \\\\big [\\\\widetilde{c}, \\\\infty \\\\big [ .\\\\end{array}\\\\right.\",\n \"}$ Notice that $\\\\Lambda _{0,\\\\widetilde{c}}(0) = 0$ , $\\\\Lambda _{0,\\\\widetilde{c}}(-\\\\infty ) = - \\\\infty $ , $\\\\Lambda _{0,\\\\widetilde{c}}^{\\\\prime }(\\\\widetilde{c}) = \\\\infty $ and $\\\\Lambda _{0,\\\\widetilde{c}}^{\\\\prime }(-\\\\infty ) = 0$ .\",\n \"Moreover, from formula (REF ) (see also ()) we can deduce $\\\\varphi _{0,\\\\widetilde{c}}(t) := \\\\varphi _{0,\\\\widetilde{c}}^{(0)}(t)\\\\hspace{-5.69046pt} &=& \\\\hspace{-5.69046pt}{\\\\left\\\\lbrace \\\\begin{array}{ll}\\\\widetilde{c} \\\\cdot \\\\left(- \\\\log t + t -1 \\\\right)\\\\ \\\\in \\\\ [0,\\\\infty [,\\\\qquad \\\\quad \\\\textrm {if }t \\\\in \\\\, ]0,\\\\infty [,\\\\\\\\\\\\infty , \\\\hspace{145.10922pt} \\\\textrm {if } \\\\ t \\\\in \\\\, ]-\\\\infty , 0] ,\\\\end{array}\\\\right.\",\n \"}$ which coincides with $\\\\widetilde{c} \\\\cdot \\\\varphi _{0}(t)$ for the generator $\\\\varphi _{0}(t)$ from (REF ) which generates the reverse Kullback-Leibler divergence (reverse relative entropy) given in (REF ) with $\\\\widetilde{c}=1$ ; the first line in (REF ) can be proved by $& & \\\\hspace{-19.91684pt}\\\\varphi _{0,\\\\widetilde{c}}(t) := \\\\varphi _{0,\\\\widetilde{c}}^{(0)}(t) :=t \\\\cdot F_{0,\\\\widetilde{c}}\\\\left(t\\\\right)-\\\\int \\\\displaylimits _{0}^{F_{0,\\\\widetilde{c}}\\\\left(t\\\\right)} F_{0,\\\\widetilde{c}}^{-1}(u) du\\\\nonumber \\\\\\\\& & \\\\hspace{-19.91684pt}= t \\\\cdot \\\\widetilde{c} \\\\cdot \\\\Big (1-\\\\frac{1}{t}\\\\Big )- (- \\\\widetilde{c}) \\\\cdot \\\\log \\\\left(1-\\\\frac{1}{\\\\widetilde{c}} \\\\cdot \\\\left[ \\\\widetilde{c} \\\\cdot \\\\Big ( 1-\\\\frac{1}{t}\\\\Big ) \\\\right]\\\\right)= \\\\widetilde{c} \\\\cdot \\\\left(- \\\\log t + t -1 \\\\right),\\\\qquad t \\\\in ]0,\\\\infty [ \\\\, .$ Notice that one has $\\\\varphi _{0,\\\\widetilde{c}}(1) = 0$ , $\\\\varphi _{0,\\\\widetilde{c}}(\\\\infty ) = \\\\infty $ , $\\\\varphi _{0,\\\\widetilde{c}}^{\\\\prime }(1) = 0$ , $\\\\varphi _{0,\\\\widetilde{c}}^{\\\\prime }(0) = -\\\\infty $ and $\\\\varphi _{0,\\\\widetilde{c}}^{\\\\prime }(\\\\infty ) = \\\\widetilde{c}$ .\",\n \"Example 40 (a) For the remaining case $\\\\gamma =1$ , $\\\\widetilde{c} \\\\in ]0,\\\\infty [$ and $]a_{F_{1,\\\\widetilde{c}}},b_{F_{1,\\\\widetilde{c}}}[ = ]0,\\\\infty [$ we define $F_{1,\\\\widetilde{c}}(t) &:=&{\\\\left\\\\lbrace \\\\begin{array}{ll}\\\\widetilde{c} \\\\cdot \\\\log t =\\\\lim _{\\\\gamma \\\\rightarrow 1} F_{\\\\gamma ,\\\\widetilde{c}}(t), \\\\qquad \\\\textrm {if } \\\\ t \\\\in \\\\, ]0,\\\\infty [, \\\\\\\\-\\\\infty , \\\\hspace{108.12054pt} \\\\textrm {if } \\\\ t \\\\in ]-\\\\infty ,0].\",\n \"\\\\\\\\\\\\end{array}\\\\right.\",\n \"}\\\\nonumber $ Clearly, $\\\\mathcal {R}(F_{1,\\\\widetilde{c}})= ]-\\\\infty ,\\\\infty [$ .\",\n \"Moreover, $F_{1,\\\\widetilde{c}}(\\\\cdot )$ is strictly increasing and smooth on $]0,\\\\infty [$ , and hence, $F_{\\\\gamma ,\\\\widetilde{c}} \\\\in {F}$ .\",\n \"Since $F_{1,\\\\widetilde{c}}(1)=0$ , let us first choose the natural anchor point $c:=0$ , which leads to $]\\\\lambda _{-},\\\\lambda _{+}[= int(\\\\mathcal {R}(F_{1,\\\\widetilde{c}})) = ]-\\\\infty ,\\\\infty [$ and $]t_{-}^{sc},t_{+}^{sc}[ = ]0,\\\\infty [$ .\",\n \"By using $F_{1,\\\\widetilde{c}}^{-1}(x) = \\\\exp (\\\\frac{x}{\\\\widetilde{c}})$ for $x \\\\in \\\\mathcal {R}(F_{1,\\\\widetilde{c}})$ , we can derive from formula (REF ) (see also (REF )) $& & \\\\hspace{-19.91684pt} \\\\Lambda _{1,\\\\widetilde{c}}(z) := \\\\Lambda _{1,\\\\widetilde{c}}^{(0)}(z) :=\\\\int \\\\displaylimits _{0}^{z} F_{1,\\\\widetilde{c}}^{-1}(u) \\\\, du= \\\\widetilde{c} \\\\cdot \\\\left( \\\\exp \\\\Big (\\\\frac{z}{\\\\widetilde{c}}\\\\Big ) - 1 \\\\right),\\\\ \\\\ z \\\\in ]-\\\\infty ,\\\\infty [ .$ Notice that $\\\\Lambda _{1,\\\\widetilde{c}}(0) = 0$ , $\\\\Lambda _{1,\\\\widetilde{c}}(-\\\\infty ) = - \\\\widetilde{c}$ , $\\\\Lambda _{1,\\\\widetilde{c}}(\\\\infty ) = \\\\infty $ , $\\\\Lambda _{1,\\\\widetilde{c}}^{\\\\prime }(-\\\\infty ) = 0$ and $\\\\Lambda _{0,\\\\widetilde{c}}^{\\\\prime }(\\\\infty ) = \\\\infty $ .\",\n \"Moreover, from formula (REF ) (see also ()) we can deduce $\\\\varphi _{1,\\\\widetilde{c}}(t) := \\\\varphi _{1,\\\\widetilde{c}}^{(0)}(t)\\\\hspace{-5.69046pt} &:=& \\\\hspace{-5.69046pt}{\\\\left\\\\lbrace \\\\begin{array}{ll}\\\\widetilde{c} \\\\cdot \\\\left( t \\\\cdot \\\\log t + 1 - t \\\\right)\\\\ \\\\in \\\\ [0,\\\\infty [,\\\\qquad \\\\quad \\\\textrm {if }t \\\\in \\\\, ]0,\\\\infty [,\\\\\\\\1, \\\\hspace{150.79968pt} \\\\textrm {if } \\\\ t = 0, \\\\\\\\\\\\infty , \\\\hspace{145.10922pt} \\\\textrm {if } \\\\ t \\\\in \\\\, ]-\\\\infty ,0[ ,\\\\end{array}\\\\right.\",\n \"}$ which coincides with $\\\\widetilde{c} \\\\cdot \\\\varphi _{1}(t)$ for the generator $\\\\varphi _{1}(t)$ from (REF ) which generates the Kullback-Leibler divergence (relative entropy) given in (REF ) with $\\\\widetilde{c}=1$ ; the first line in (REF ) can be proved by $& & \\\\hspace{-19.91684pt}\\\\varphi _{1,\\\\widetilde{c}}(t) := \\\\varphi _{1,\\\\widetilde{c}}^{(0)}(t) :=t \\\\cdot F_{1,\\\\widetilde{c}}\\\\left(t\\\\right)-\\\\int \\\\displaylimits _{0}^{F_{1,\\\\widetilde{c}}\\\\left(t\\\\right)} F_{1,\\\\widetilde{c}}^{-1}(u) du\\\\nonumber \\\\\\\\& & \\\\hspace{-19.91684pt}= t \\\\cdot \\\\widetilde{c} \\\\cdot \\\\log t- \\\\widetilde{c} \\\\cdot \\\\left( \\\\exp \\\\left(\\\\frac{1}{\\\\widetilde{c}} \\\\cdot \\\\left[ \\\\widetilde{c} \\\\cdot \\\\log t \\\\right]\\\\right) - 1 \\\\right)= \\\\widetilde{c} \\\\cdot \\\\left( t \\\\cdot \\\\log t + 1 - t \\\\right),\\\\qquad t \\\\in ]0,\\\\infty [ \\\\, ,$ Notice that one has $\\\\varphi _{1,\\\\widetilde{c}}(1) = 0$ , $\\\\varphi _{1,\\\\widetilde{c}}(\\\\infty ) = \\\\infty $ , $\\\\varphi _{1,\\\\widetilde{c}}^{\\\\prime }(1) = 0$ , $\\\\varphi _{1,\\\\widetilde{c}}^{\\\\prime }(0) = -\\\\infty $ and $\\\\varphi _{1,\\\\widetilde{c}}^{\\\\prime }(\\\\infty ) = \\\\infty $ .\",\n \"As an application of the reciprocity considerations of Remark REF , it is straightforward to see that $\\\\widetilde{F}_{1,\\\\widetilde{c}}(t) =- \\\\Lambda _{1,\\\\widetilde{c}}(F_{1,\\\\widetilde{c}}(\\\\frac{1}{t}))= F_{0,\\\\widetilde{c}}(t)$ for all $t \\\\in ]0,\\\\infty [$ .\",\n \"(b) For the choice $\\\\widetilde{c}=1$ , let us now fix a general anchor point $c \\\\in \\\\mathcal {R}(F_{1,\\\\widetilde{c}})= ]-\\\\infty ,\\\\infty [$ (rather than $c=0$ ), which leads to $]\\\\lambda _{-},\\\\lambda _{+}[= int(\\\\mathcal {R}(F_{1,1}))- c = ]-\\\\infty ,\\\\infty [$ and $]t_{-}^{sc},t_{+}^{sc}[ = ]1+a_{F_{1,1}}-F_{1,1}^{-1}(c),1+b_{F_{1,1}}-F_{1,1}^{-1}(c)[ \\\\,= \\\\, ]1-e^{c},\\\\infty [$ .\",\n \"Accordingly, the formula (REF ) (see also (REF )) leads to $& & \\\\hspace{-19.91684pt} \\\\Lambda _{1,1}(z) := \\\\Lambda _{1,1}^{(c)}(z) :=\\\\int \\\\displaylimits _{0}^{z} F_{1,1}^{-1}(u+c) du+ z \\\\cdot (1-F_{1,1}^{-1}(c))\\\\nonumber \\\\\\\\& & \\\\hspace{59.75095pt}= e^{c} \\\\cdot \\\\left( e^{z} - 1 \\\\right) + z \\\\cdot (1- e^{c}),\\\\qquad z \\\\in ]-\\\\infty ,\\\\infty [, \\\\qquad \\\\ $ for which there holds $\\\\Lambda _{1,1}^{(c)}(0) = 0$ , $\\\\Lambda _{1,1}^{(c)}(-\\\\infty ) =\\\\infty \\\\cdot {1}_{]0,\\\\infty [}(c)- \\\\infty \\\\cdot {1}_{]-\\\\infty ,0[}(c)- 1 \\\\cdot {1}_{\\\\lbrace 0\\\\rbrace }(c)$ , $\\\\Lambda _{1,1}^{(c)}(\\\\infty ) = \\\\infty $ , $\\\\Lambda _{1,1}^{(c) \\\\prime }(-\\\\infty ) = 1- e^{c}$ and $\\\\Lambda _{1,1}^{(c) \\\\prime }(\\\\infty ) = \\\\infty $ .\",\n \"Moreover, from formula (REF ) (see also ()) we can deduce $\\\\varphi _{1,1}(t) := \\\\varphi _{1,1}^{(c)}(t)\\\\hspace{-5.69046pt} &:=& \\\\hspace{-5.69046pt}{\\\\left\\\\lbrace \\\\begin{array}{ll}(t + e^{c} -1) \\\\cdot [\\\\log (t + e^{c} -1) -c] + 1 - t\\\\ \\\\in \\\\ [0,\\\\infty [,\\\\qquad \\\\quad \\\\textrm {if }t \\\\in \\\\, ]1-e^{c},\\\\infty [,\\\\\\\\e^{c}, \\\\hspace{240.42569pt} \\\\textrm {if } \\\\ t = 1-e^{c}, \\\\\\\\\\\\infty , \\\\hspace{239.00298pt} \\\\textrm {if } \\\\ t \\\\in \\\\, ]-\\\\infty ,1-e^{c}[ ;\\\\end{array}\\\\right.\",\n \"}$ the first line in (REF ) can be proved by $& & \\\\hspace{-19.91684pt}\\\\varphi _{1,1}(t) := \\\\varphi _{1,1}^{(c)}(t) :=(t+F_{1,1}^{-1}(c)-1) \\\\cdot [ F_{1,1}\\\\left(t+F_{1,1}^{-1}(c)-1 \\\\right) - c ]-\\\\int \\\\displaylimits _{0}^{F_{1,1}\\\\left(t+F_{1,1}^{-1}(c)-1 \\\\right) - c}F_{1,1}^{-1}(u+c) du\\\\nonumber \\\\\\\\& & \\\\hspace{-19.91684pt}= (t + e^{c} -1) \\\\cdot [\\\\log (t + e^{c} -1) -c]- e^{c} \\\\cdot \\\\Big \\\\lbrace \\\\exp [\\\\log (t + e^{c} -1) -c] - 1 \\\\Big \\\\rbrace \\\\nonumber \\\\\\\\& & \\\\hspace{-19.91684pt}= (t + e^{c} -1) \\\\cdot [\\\\log (t + e^{c} -1) -c] + 1 - t ,\\\\qquad t \\\\in ]1-e^{c},\\\\infty [\\\\, .$ Clearly, one has $\\\\varphi _{1,1}^{(c)}(1) = 0$ , $\\\\varphi _{1,1}^{(c)}(\\\\infty ) = \\\\infty $ , $\\\\varphi _{1,1}^{(c) \\\\prime }(1) = 0$ , $\\\\varphi _{1,1}^{(c) \\\\prime }(1-e^{c}) = -\\\\infty $ and $\\\\varphi _{1,1}^{(c) \\\\prime }(\\\\infty ) = \\\\infty $ .\",\n \"The corresponding divergence $D_{\\\\varphi _{1,1}^{(c)}}(\\\\mathbb {Q},\\\\mathbb {P})$ has been recently used in Broniatowski et al.\",\n \"[63] for the important task of testing mixtures of probability distributions; in fact, in order to get considerable comfort in testing mixture-type hypotheses against corresponding marginal-type alternatives, they employ choices $c>0$ since then $\\\\varphi _{1,1}^{(c)}(t)$ is finite especially for some range of negative values $t<0$ .\",\n \"The latter feature is also valid for the divergence generator $\\\\varphi _{bw,\\\\beta ,\\\\widetilde{c}}$ in the next example (cf.\",\n \"(REF ) below).\",\n \"Example 41 For $\\\\beta \\\\in \\\\, ]0,1]$ , $\\\\widetilde{c} \\\\in \\\\, ]0,\\\\infty [$ and $]a_{F_{bw,\\\\beta ,\\\\widetilde{c}}},b_{F_{bw,\\\\beta ,\\\\widetilde{c}}}[ \\\\,= \\\\, ]1-\\\\frac{1}{\\\\beta },\\\\infty [$ we define $F_{bw,\\\\beta ,\\\\widetilde{c}}(t) &:=&{\\\\left\\\\lbrace \\\\begin{array}{ll}\\\\frac{\\\\widetilde{c}}{2\\\\beta } \\\\cdot \\\\Big (1-\\\\frac{1}{(\\\\beta \\\\cdot t + 1 - \\\\beta )^2}\\\\Big ), \\\\qquad \\\\textrm {if } \\\\ t \\\\in \\\\, ]1-\\\\frac{1}{\\\\beta },\\\\infty [, \\\\\\\\- \\\\infty , \\\\hspace{93.89418pt} \\\\textrm {if } \\\\ t \\\\in \\\\, ]-\\\\infty ,1-\\\\frac{1}{\\\\beta }].\\\\end{array}\\\\right.\",\n \"}\\\\nonumber $ Clearly, $\\\\mathcal {R}(F_{bw,\\\\beta ,\\\\widetilde{c}})= \\\\big ]-\\\\infty ,\\\\frac{\\\\widetilde{c}}{2\\\\beta }\\\\big [$ and $0 \\\\in int(\\\\mathcal {R}(F_{bw,\\\\beta ,\\\\widetilde{c}}))$ .\",\n \"Moreover, $F_{bw,\\\\beta ,\\\\widetilde{c}}(\\\\cdot )$ is strictly increasing and smooth on $]1-\\\\frac{1}{\\\\beta },\\\\infty [$ , and thus, $F_{bw,\\\\beta ,\\\\widetilde{c}} \\\\in {F}$ .\",\n \"Since $F_{bw,\\\\beta ,\\\\widetilde{c}}(1)=0$ , let us choose the natural anchor point $c:=0$ , which leads to $]\\\\lambda _{-},\\\\lambda _{+}[ \\\\, = int(\\\\mathcal {R}(F_{bw,\\\\beta ,\\\\widetilde{c}}))= \\\\big ]-\\\\infty ,\\\\frac{\\\\widetilde{c}}{2\\\\beta }\\\\big [$ and $]t_{-}^{sc},t_{+}^{sc}[ \\\\, = \\\\,]a_{F_{bw,\\\\beta ,\\\\widetilde{c}}},b_{F_{bw,\\\\beta ,\\\\widetilde{c}}}[\\\\, = \\\\, ]1-\\\\frac{1}{\\\\beta },\\\\infty [$ .\",\n \"By using $F_{bw,\\\\beta ,\\\\widetilde{c}}^{-1}(x) =\\\\frac{1}{\\\\beta } \\\\cdot \\\\Big \\\\lbrace \\\\frac{1}{\\\\sqrt{1-2\\\\beta \\\\cdot x/\\\\widetilde{c}}} + \\\\beta -1 \\\\Big \\\\rbrace $ for $x \\\\in int(\\\\mathcal {R}(F_{bw,\\\\beta ,\\\\widetilde{c}}))$ , we can derive from formula (REF ) (see also (REF )) for all $\\\\beta \\\\in \\\\, ]0,1]$ and $z \\\\in \\\\mathbb {R}$ $\\\\Lambda _{bw,\\\\beta ,\\\\widetilde{c}}(z) :=\\\\Lambda _{bw,\\\\beta ,\\\\widetilde{c}}^{(0)}(z) &=&{\\\\left\\\\lbrace \\\\begin{array}{ll}-(\\\\frac{1}{\\\\beta }-1) \\\\cdot z + \\\\frac{\\\\widetilde{c}}{\\\\beta ^{2}} \\\\cdot \\\\Big \\\\lbrace 1 - \\\\sqrt{1-\\\\frac{2\\\\beta }{\\\\widetilde{c}} \\\\cdot z} \\\\ \\\\Big \\\\rbrace ,\\\\qquad \\\\textrm {if } z \\\\in \\\\big ]-\\\\infty ,\\\\frac{\\\\widetilde{c}}{2\\\\beta } \\\\big ], \\\\\\\\\\\\infty , \\\\hspace{176.407pt} \\\\textrm {else} .\\\\end{array}\\\\right.\",\n \"}$ Notice that $\\\\Lambda _{bw,\\\\beta ,\\\\widetilde{c}}(0) = 0$ .\",\n \"Moreover, $\\\\Lambda _{bw,\\\\beta ,\\\\widetilde{c}}(-\\\\infty ) = \\\\infty $ , $\\\\Lambda _{bw,\\\\beta ,\\\\widetilde{c}}(\\\\frac{\\\\widetilde{c}}{2\\\\beta })= \\\\frac{\\\\widetilde{c} \\\\cdot (\\\\beta +1)}{2 \\\\beta ^{2}}$ , $\\\\Lambda _{bw,\\\\beta ,\\\\widetilde{c}}^{\\\\prime }(-\\\\infty ) = - \\\\frac{1-\\\\beta }{\\\\beta } <0$ and $\\\\Lambda _{bw,\\\\beta ,\\\\widetilde{c}}^{\\\\prime }(\\\\frac{\\\\widetilde{c}}{2\\\\beta }) = \\\\infty $ .\",\n \"Furthermore, from formula (REF ) (see also ()) we can straightforwardly deduce for all $t \\\\in \\\\mathbb {R}$ $\\\\varphi _{bw,\\\\beta ,\\\\widetilde{c}}(t) := \\\\varphi _{bw,\\\\beta ,\\\\widetilde{c}}^{(0)}(t)\\\\hspace{-5.69046pt} &:=& \\\\hspace{-5.69046pt}{\\\\left\\\\lbrace \\\\begin{array}{ll}\\\\widetilde{c} \\\\cdot \\\\frac{(t-1)^{2}}{2(\\\\beta \\\\cdot t +1 -\\\\beta )}\\\\ \\\\in \\\\ [0,\\\\infty [,\\\\qquad \\\\quad \\\\textrm {if }t \\\\in \\\\, ]1-\\\\frac{1}{\\\\beta },\\\\infty [,\\\\\\\\\\\\infty , \\\\hspace{120.92421pt} \\\\textrm {if } \\\\ t \\\\in \\\\, ]-\\\\infty ,1-\\\\frac{1}{\\\\beta }] .\\\\end{array}\\\\right.\",\n \"}$ Note that $1-\\\\frac{1}{\\\\beta } <0$ so that negative $t$ are allowed here.\",\n \"For $t\\\\ge 0$ , $\\\\varphi _{bw,\\\\beta ,\\\\widetilde{c}}(t)$ is known as Rukhin's generator (cf.\",\n \"[312], see e.g.\",\n \"also Marhuenda et al.\",\n \"[247], Pardo [282]).\",\n \"Obviously, one has $\\\\varphi _{bw,\\\\beta ,\\\\widetilde{c}}(1) = 0$ , $\\\\varphi _{bw,\\\\beta ,\\\\widetilde{c}}^{\\\\prime }(1) = 0$ , $\\\\varphi _{bw,\\\\beta ,\\\\widetilde{c}}^{\\\\prime }(1-\\\\frac{1}{\\\\beta }) = -\\\\infty $ and $\\\\varphi _{bw,\\\\beta ,\\\\widetilde{c}}^{\\\\prime }(\\\\infty ) = \\\\frac{\\\\widetilde{c}}{2\\\\beta }$ .\",\n \"From the generator $\\\\varphi _{bw,\\\\beta ,\\\\widetilde{c}}$ given in (REF ), we build the corresponding divergence (cf.\",\n \"(REF )) $& &\\\\hspace{-45.52458pt}D_{\\\\varphi _{bw,\\\\beta ,\\\\widetilde{c}}}(\\\\mathbf {Q},\\\\mathbf {P})= \\\\widetilde{c} \\\\cdot \\\\sum _{k=1}^{K} p_{k} \\\\cdot \\\\frac{(\\\\frac{q_{k}}{p_{k}}-1)^{2}}{2(\\\\beta \\\\cdot \\\\frac{q_{k}}{p_{k}} +1 -\\\\beta )}\\\\nonumber \\\\\\\\& & \\\\hspace{-45.52458pt}= \\\\frac{\\\\widetilde{c}}{2} \\\\cdot \\\\sum \\\\limits _{k=1}^{K}\\\\frac{(q_{k}-p_{k})^{2}}{\\\\beta \\\\cdot q_{k} + (1 -\\\\beta )\\\\cdot p_{k}},\\\\hspace{42.67912pt}\\\\textrm {if \\\\mathbf {P} \\\\in \\\\mathbb {R}^{K} and \\\\mathbf {Q} \\\\in \\\\mathbb {R}^{K}with \\\\mathbf {Q} \\\\in \\\\, ] \\\\, \\\\mathbf {P} \\\\cdot (1-\\\\frac{1}{\\\\beta }),\\\\infty [ component-wise;}$ for the special subcase $\\\\widetilde{c}=1$ and $\\\\mathbf {Q} \\\\in \\\\mathbb {R}_{> 0}^{K}$ , $D_{\\\\varphi _{bw,\\\\beta ,1}}(\\\\mathbf {Q},\\\\mathbf {P})$ can be interpreted as — “non-probability version” of — the well-known blended weight chi-square divergence of Lindsay [221] (see e.g.\",\n \"also Basu & Lindsay [35], Györfy & Vajda [148], Basu et al.\",\n \"[36]).\",\n \"The special case $\\\\widetilde{c}=1$ and $\\\\beta =\\\\frac{1}{2}$ for probability vectors, i.e.\",\n \"$D_{\\\\varphi _{bw,1/2,1}}({Q},{P})$ , is equal to (a multiple of the matrix-vector-converted (cf.\",\n \"Remark REF )) Sanghvi’s genetic difference measure [316] and equal to the double of the so-called (squared) Vincze-Le Cam distance (cf.\",\n \"Vincze [384], Le Cam [212], see also e.g.\",\n \"Topsoe [360] — who used the alternative naming triangular discrimination — and Vajda [373]); this divergence $D_{\\\\varphi _{bw,1/2,1}}({Q},{P})$ has been used e.g.\",\n \"in Liu et al.\",\n \"[227] for a machine learning context of detecting salient objects, where ${Q}$ and ${P}$ are appropriate histograms of RGB color.\",\n \"Remark 42 (a) By straightforward calculations, one can show that $\\\\varphi _{bw,1,\\\\widetilde{c}}$ (i.e.\",\n \"with the choice $\\\\beta =1$ ) is equal to the $\\\\widetilde{c}-fold$ power-divergence generator $\\\\varphi _{\\\\gamma ,\\\\widetilde{c}} = \\\\widetilde{c} \\\\cdot \\\\varphi _{\\\\gamma }$ (cf.\",\n \"(REF )) with $\\\\gamma =-1$ ; the corresponding divergence $D_{\\\\varphi _{bw,1,\\\\widetilde{c}}}(\\\\mathbf {Q},\\\\mathbf {P})$ is thus equal to the power divergence $D_{\\\\widetilde{c} \\\\cdot \\\\varphi _{-1}}(\\\\mathbf {Q},\\\\mathbf {P})$ (cf.\",\n \"(REF )) which is nothing but the — “non-probability version” — of Neyman's chi-square divergence.\",\n \"(b) For the case $\\\\beta =0$ — which has been excluded in Example REF for technical brevity — the divergence generator $\\\\varphi _{bw,0,\\\\widetilde{c}}$ corresponds to $\\\\widetilde{c}-fold$ power-divergence generator $\\\\varphi _{\\\\gamma ,\\\\widetilde{c}}$ with $\\\\gamma =2$ ; the corresponding divergence $D_{\\\\varphi _{bw,0,\\\\widetilde{c}}}(\\\\mathbf {Q},\\\\mathbf {P})$ is thus equal to the power divergence $D_{\\\\widetilde{c} \\\\cdot \\\\varphi _{2}}(\\\\mathbf {Q},\\\\mathbf {P})$ (cf.\",\n \"(REF )) which is nothing but the — “non-probability version” — of Pearson's (i.e.\",\n \"the classical) chi-square divergence.\",\n \"Example 43 Let us give an interesting generalization of the Kullback-Leibler case of Example REF (a).\",\n \"For $\\\\widetilde{c} >0$ and $\\\\alpha \\\\in \\\\, ]-1,0[ \\\\ \\\\cup \\\\ ]0,\\\\infty [$ let us define $F_{gKL,\\\\alpha ,\\\\widetilde{c}}(t) &:=&{\\\\left\\\\lbrace \\\\begin{array}{ll}\\\\widetilde{c} \\\\cdot \\\\log \\\\left( \\\\frac{(1+\\\\alpha ) \\\\cdot t}{1+ \\\\alpha \\\\cdot t} \\\\right), \\\\qquad \\\\textrm {if \\\\lbrace \\\\alpha \\\\in \\\\, ]0,\\\\infty [ and t \\\\in \\\\, ]0,\\\\infty [ \\\\rbrace or \\\\lbrace \\\\alpha \\\\in \\\\, ]-1,0[ and t \\\\in \\\\, ]0,-\\\\frac{1}{\\\\alpha }[ \\\\rbrace ,} \\\\\\\\- \\\\infty , \\\\hspace{71.13188pt} \\\\textrm {if \\\\alpha \\\\in \\\\, ]-1,0[ \\\\ \\\\cup \\\\ ]0,\\\\infty [ andt \\\\in \\\\, ]-\\\\infty ,0],} \\\\\\\\\\\\infty , \\\\hspace{78.52945pt} \\\\textrm {if \\\\alpha \\\\in \\\\, ]-1,0[ and t \\\\in \\\\, [-\\\\frac{1}{\\\\alpha },\\\\infty [,}\\\\end{array}\\\\right.\",\n \"}\\\\nonumber $ (notice that $\\\\lim _{\\\\alpha \\\\rightarrow 0_{+}} F_{gKL,\\\\alpha ,\\\\widetilde{c}}(t)= F_{1,\\\\widetilde{c}}(t)$ , cf.\",\n \"Example REF (a)).\",\n \"Clearly, $]a_{F_{gKL,\\\\alpha ,\\\\widetilde{c}}},b_{F_{gKL,\\\\alpha ,\\\\widetilde{c}}}[ \\\\, := \\\\, ]0,\\\\infty [$ for $\\\\alpha \\\\in \\\\, ]0,\\\\infty [$ and $]a_{F_{gKL,\\\\alpha ,\\\\widetilde{c}}},b_{F_{gKL,\\\\alpha ,\\\\widetilde{c}}}[ \\\\, := \\\\, ]0,-\\\\frac{1}{\\\\alpha }[$ for $\\\\alpha \\\\in \\\\, ]-1,0[$ .\",\n \"Moreover, $\\\\mathcal {R}(F_{gKL,\\\\alpha ,\\\\widetilde{c}})=\\\\, ]-\\\\infty , \\\\widetilde{c} \\\\cdot \\\\log (1+ \\\\frac{1}{\\\\alpha })[$ for $\\\\alpha \\\\in \\\\, ]0,\\\\infty [$ and $\\\\mathcal {R}(F_{gKL,\\\\alpha ,\\\\widetilde{c}})= \\\\, ]-\\\\infty ,\\\\infty [$ for $\\\\alpha \\\\in \\\\, ]-1,0[$ .\",\n \"Furthermore, $F_{gKL,\\\\alpha ,\\\\widetilde{c}}(\\\\cdot )$ is strictly increasing and smooth on the respective $]a_{F_{gKL,\\\\alpha ,\\\\widetilde{c}}},b_{F_{gKL,\\\\alpha ,\\\\widetilde{c}}}[$ , and thus, $F_{gKL,\\\\alpha ,\\\\widetilde{c}} \\\\in {F}$ .\",\n \"Since $F_{gKL,\\\\alpha ,\\\\widetilde{c}}(1)=0$ , let us choose the natural anchor point $c:=0$ , which leads to $]\\\\lambda _{-},\\\\lambda _{+}[ \\\\, = int(\\\\mathcal {R}(F_{gKL,\\\\alpha ,\\\\widetilde{c}})) = \\\\,]-\\\\infty , \\\\widetilde{c} \\\\cdot \\\\log (1+ \\\\frac{1}{\\\\alpha })[$ and $]t_{-}^{sc},t_{+}^{sc}[ \\\\, = \\\\, ]0,\\\\infty [$ for the case $\\\\alpha \\\\in \\\\, ]0,\\\\infty [$ , respectively, to $]\\\\lambda _{-},\\\\lambda _{+}[ \\\\, = int(\\\\mathcal {R}(F_{gKL,\\\\alpha ,\\\\widetilde{c}})) = \\\\, ]-\\\\infty ,\\\\infty [$ and $]t_{-}^{sc},t_{+}^{sc}[ \\\\, = \\\\, ]0,-\\\\frac{1}{\\\\alpha }[$ for the case $\\\\alpha \\\\in \\\\, ]-1,0[$ .\",\n \"By employing $F_{gKL,\\\\alpha ,\\\\widetilde{c}}^{-1}(x) =\\\\frac{1}{(1+\\\\alpha ) \\\\cdot e^{-x/\\\\widetilde{c}} - \\\\alpha }$ for $x \\\\in ]\\\\lambda _{-},\\\\lambda _{+}[$ , one can deduce from formula (REF ) (see also (REF )) $& & \\\\Lambda _{gKL,\\\\alpha ,\\\\widetilde{c}}(z) := \\\\Lambda _{gKL,\\\\alpha ,\\\\widetilde{c}}^{(0)}(z)\\\\nonumber \\\\\\\\& & :={\\\\left\\\\lbrace \\\\begin{array}{ll}\\\\int \\\\displaylimits _{0}^{z} F_{gKL,\\\\alpha ,\\\\widetilde{c}}^{-1}(u) \\\\, du= - \\\\frac{\\\\widetilde{c}}{\\\\alpha } \\\\cdot \\\\log ((1+\\\\alpha ) - \\\\alpha \\\\cdot e^{z/\\\\widetilde{c}}),\\\\qquad \\\\textrm {if \\\\alpha \\\\in \\\\, ]0,\\\\infty [ and z \\\\in \\\\, ]-\\\\infty , \\\\widetilde{c} \\\\cdot \\\\log (1+ \\\\frac{1}{\\\\alpha })[}, \\\\\\\\\\\\int \\\\displaylimits _{0}^{z} F_{gKL,\\\\alpha ,\\\\widetilde{c}}^{-1}(u) \\\\, du= - \\\\frac{\\\\widetilde{c}}{\\\\alpha } \\\\cdot \\\\log ((1+\\\\alpha ) - \\\\alpha \\\\cdot e^{z/\\\\widetilde{c}}),\\\\qquad \\\\textrm {if \\\\alpha \\\\in \\\\, ]-1,0[ and z \\\\in \\\\, ]-\\\\infty , \\\\infty [}, \\\\\\\\\\\\infty , \\\\hspace{217.6634pt}\\\\textrm {if \\\\alpha \\\\in \\\\, ]0,\\\\infty [ and z \\\\in \\\\, [\\\\widetilde{c} \\\\cdot \\\\log (1+ \\\\frac{1}{\\\\alpha }), \\\\infty [}, \\\\\\\\\\\\end{array}\\\\right.\",\n \"}$ for which there holds $\\\\Lambda _{gKL,\\\\alpha ,\\\\widetilde{c}}(0) = 0$ and $\\\\Lambda _{gKL,\\\\alpha ,\\\\widetilde{c}}(-\\\\infty ) = - \\\\frac{\\\\widetilde{c}}{\\\\alpha } \\\\cdot \\\\log (1+\\\\alpha )$ for $\\\\alpha \\\\in \\\\, ]-1,0[ \\\\ \\\\cup \\\\ ]0,\\\\infty [$ , as well as $\\\\Lambda _{gKL,\\\\alpha ,\\\\widetilde{c}}(\\\\widetilde{c} \\\\cdot \\\\log (1+ \\\\frac{1}{\\\\alpha })) = \\\\infty $ for $\\\\alpha \\\\in \\\\, ]0,\\\\infty [$ and $\\\\Lambda _{gKL,\\\\alpha ,\\\\widetilde{c}}(\\\\infty ) = \\\\infty $ for $\\\\alpha \\\\in \\\\, ]-1,0[$ .\",\n \"The corresponding derivative satisfies $\\\\Lambda _{gKL,\\\\alpha ,\\\\widetilde{c}}^{\\\\prime }(- \\\\infty ) = 0$ for $\\\\alpha \\\\in \\\\, ]-1,0[ \\\\ \\\\cup \\\\ ]0,\\\\infty [$ , as well as $\\\\Lambda _{gKL,\\\\alpha ,\\\\widetilde{c}}^{\\\\prime }(\\\\widetilde{c}\\\\cdot \\\\log (1+ \\\\frac{1}{\\\\alpha })) = \\\\infty $ for $\\\\alpha \\\\in \\\\, ]0,\\\\infty [$ and $\\\\Lambda _{gKL,\\\\alpha ,\\\\widetilde{c}}^{\\\\prime }(\\\\infty ) = - \\\\frac{1}{\\\\alpha }$ for $\\\\alpha \\\\in \\\\, ]-1,0[$ .\",\n \"Furthermore, from formula (REF ) (see also ()) one can derive $& & \\\\hspace{-19.91684pt} \\\\varphi _{gKL,\\\\alpha ,\\\\widetilde{c}}(t) := \\\\varphi _{gKL,\\\\alpha ,\\\\widetilde{c}}^{(0)}(t)\\\\nonumber \\\\\\\\& & \\\\hspace{-19.91684pt} :={\\\\left\\\\lbrace \\\\begin{array}{ll}\\\\widetilde{c} \\\\cdot \\\\left[ \\\\, t \\\\cdot \\\\log t + (t+\\\\frac{1}{\\\\alpha }) \\\\cdot \\\\log \\\\Big ( \\\\frac{1+\\\\alpha }{1+\\\\alpha \\\\cdot t} \\\\Big ) \\\\, \\\\right]\\\\ \\\\in \\\\ [0,\\\\infty [,\\\\quad \\\\textrm {if \\\\lbrace \\\\alpha \\\\in \\\\, ]0,\\\\infty [ and t \\\\in \\\\, ]0,\\\\infty [ \\\\rbrace or \\\\lbrace \\\\alpha \\\\in \\\\, ]-1,0[ and t \\\\in \\\\, ]0,-\\\\frac{1}{\\\\alpha }[ \\\\rbrace ,}\\\\\\\\\\\\frac{\\\\widetilde{c}}{\\\\alpha } \\\\cdot \\\\log (1+\\\\alpha )\\\\ \\\\in \\\\ ]0,\\\\infty [, \\\\hspace{106.69783pt}\\\\textrm {if \\\\alpha \\\\in \\\\, ]-1,0[ \\\\ \\\\cup \\\\ ]0,\\\\infty [ andt =0,} \\\\\\\\\\\\infty , \\\\hspace{197.74655pt}\\\\textrm {if \\\\alpha \\\\in \\\\, ]-1,0[ \\\\ \\\\cup \\\\ ]0,\\\\infty [ andt \\\\in \\\\, ]-\\\\infty ,0[,} \\\\\\\\\\\\infty , \\\\hspace{197.74655pt}\\\\textrm {if \\\\alpha \\\\in \\\\, ]-1,0[ andt \\\\in \\\\, [-\\\\frac{1}{\\\\alpha },\\\\infty [;}\\\\end{array}\\\\right.\",\n \"}$ the first line in (REF ) can be proved by $& & \\\\hspace{-19.91684pt}\\\\varphi _{gKL,\\\\alpha ,\\\\widetilde{c}}(t) := \\\\varphi _{gKL,\\\\alpha ,\\\\widetilde{c}}^{(0)}(t) :=t \\\\cdot F_{gKL,\\\\alpha ,\\\\widetilde{c}}\\\\left(t\\\\right)-\\\\int \\\\displaylimits _{0}^{F_{gKL,\\\\alpha ,\\\\widetilde{c}}\\\\left(t\\\\right)}F_{gKL,\\\\alpha ,\\\\widetilde{c}}^{-1}(u) \\\\, du\\\\nonumber \\\\\\\\& & \\\\hspace{-19.91684pt}= \\\\widetilde{c} \\\\cdot t \\\\cdot \\\\log \\\\left( \\\\frac{(1+\\\\alpha ) \\\\cdot t}{1+ \\\\alpha \\\\cdot t} \\\\right)+ \\\\frac{\\\\widetilde{c}}{\\\\alpha } \\\\cdot \\\\log \\\\left( (1 +\\\\alpha ) -\\\\alpha \\\\cdot \\\\exp \\\\left[ \\\\log \\\\left( \\\\frac{(1+ \\\\alpha ) \\\\cdot t}{1+ \\\\alpha \\\\cdot t} \\\\right) \\\\right] \\\\right)\\\\nonumber \\\\\\\\& & \\\\hspace{-19.91684pt}= \\\\widetilde{c} \\\\cdot \\\\left[ \\\\, t \\\\cdot \\\\log t+ t \\\\cdot \\\\log \\\\Big ( \\\\frac{1+\\\\alpha }{1+ \\\\alpha \\\\cdot t} \\\\Big )+ \\\\frac{1}{\\\\alpha } \\\\cdot \\\\log \\\\Big ( \\\\frac{1+\\\\alpha }{1+ \\\\alpha \\\\cdot t} \\\\Big ) \\\\, \\\\right].$ Obviously, one has $\\\\varphi _{gKL,\\\\alpha ,\\\\widetilde{c}}(1) = 0$ , $\\\\varphi _{gKL,\\\\alpha ,\\\\widetilde{c}}^{\\\\prime }(1) = 0$ , $\\\\varphi _{gKL,\\\\alpha ,\\\\widetilde{c}}^{\\\\prime }(0) = -\\\\infty $ for $\\\\alpha \\\\in \\\\, ]-1,0[ \\\\ \\\\cup \\\\ ]0,\\\\infty [$ .\",\n \"Moreover, for $\\\\alpha \\\\in \\\\, ]0,\\\\infty [$ there holds $\\\\varphi _{gKL,\\\\alpha ,\\\\widetilde{c}}(\\\\infty ) = \\\\infty $ , and $\\\\varphi _{gKL,\\\\alpha ,\\\\widetilde{c}}^{\\\\prime }(\\\\infty ) = \\\\widetilde{c} \\\\cdot \\\\log (1+\\\\frac{1}{\\\\alpha })$ , whereas for $\\\\alpha \\\\in \\\\, ]-1,0[$ we obtain $\\\\varphi _{gKL,\\\\alpha ,\\\\widetilde{c}}(-\\\\frac{1}{\\\\alpha }) = \\\\infty $ , and $\\\\varphi _{gKL,\\\\alpha ,\\\\widetilde{c}}^{\\\\prime }(-\\\\frac{1}{\\\\alpha }) = \\\\infty $ .\",\n \"From the generator $\\\\varphi _{gKL,\\\\alpha ,\\\\widetilde{c}}$ given in (REF ), we build the corresponding divergence (cf.\",\n \"(REF )) $& &\\\\hspace{-42.67912pt}D_{\\\\varphi _{gKL,\\\\alpha ,\\\\widetilde{c}}}(\\\\mathbf {Q},\\\\mathbf {P})= \\\\widetilde{c} \\\\cdot \\\\Big \\\\lbrace \\\\sum \\\\limits _{k=1}^{K}q_{k} \\\\cdot \\\\log \\\\Big (\\\\frac{q_{k}}{(1-\\\\frac{1}{1+\\\\alpha }) \\\\cdot q_{k} + \\\\frac{1}{1+\\\\alpha } \\\\cdot p_{k}} \\\\Big )+ \\\\frac{1}{\\\\alpha } \\\\cdot \\\\sum \\\\limits _{k=1}^{K} p_{k} \\\\cdot \\\\log \\\\Big (\\\\frac{p_{k}}{(1-\\\\frac{1}{1+\\\\alpha }) \\\\cdot q_{k} + \\\\frac{1}{1+\\\\alpha } \\\\cdot p_{k}} \\\\Big ) \\\\Big \\\\rbrace ,\\\\\\\\& & \\\\textrm {if \\\\lbrace \\\\alpha \\\\in \\\\, ]0,\\\\infty [,\\\\mathbf {P} \\\\in \\\\mathbb {R}_{> 0}^{K} and\\\\mathbf {Q} \\\\in \\\\mathbb {R}_{\\\\ge 0}^{K} \\\\rbrace or \\\\lbrace \\\\alpha \\\\in \\\\, ]-1,0[,\\\\mathbf {P} \\\\in \\\\mathbb {R}_{> 0}^{K} and\\\\mathbf {Q} \\\\in \\\\mathbb {R}_{\\\\ge 0}^{K}with \\\\mathbf {Q} \\\\le - \\\\frac{1}{\\\\alpha } \\\\cdot \\\\mathbf {P} \\\\rbrace .}\",\n \"\\\\nonumber $ Notice that the symmetry $D_{\\\\varphi _{gKL,\\\\alpha ,\\\\widetilde{c}}}(\\\\mathbf {Q},\\\\mathbf {P})= D_{\\\\varphi _{gKL,\\\\alpha ,\\\\widetilde{c}}}(\\\\mathbf {P},\\\\mathbf {Q})$ generally holds only if $\\\\mathbf {P}, \\\\mathbf {Q} \\\\in \\\\mathbb {R}_{> 0}^{K}$ and $\\\\alpha =1$ ; indeed, this special case leads to $\\\\varphi _{snKL,\\\\widetilde{c}}(t):= \\\\varphi _{gKL,1,\\\\widetilde{c}}(t)\\\\hspace{-5.69046pt} &:=& \\\\hspace{-5.69046pt}{\\\\left\\\\lbrace \\\\begin{array}{ll}\\\\widetilde{c} \\\\cdot \\\\left[ \\\\, t \\\\cdot \\\\log t + (t+1) \\\\cdot \\\\log \\\\Big ( \\\\frac{2}{t+1} \\\\Big ) \\\\, \\\\right]\\\\ \\\\in \\\\ [0,\\\\infty [,\\\\qquad \\\\quad \\\\textrm {if }t \\\\in \\\\, ]0,\\\\infty [,\\\\\\\\\\\\widetilde{c} \\\\cdot \\\\log 2, \\\\hspace{187.78836pt} \\\\textrm {if } \\\\ t = 0, \\\\\\\\\\\\infty , \\\\hspace{209.12791pt} \\\\textrm {if } \\\\ t \\\\in \\\\, ]-\\\\infty ,0[ ,\\\\end{array}\\\\right.\",\n \"}$ and $D_{\\\\varphi _{snKL,\\\\widetilde{c}}}(\\\\mathbf {Q},\\\\mathbf {P}):= D_{\\\\varphi _{gKL,1,\\\\widetilde{c}}}(\\\\mathbf {Q},\\\\mathbf {P})= \\\\widetilde{c} \\\\cdot \\\\Big \\\\lbrace \\\\sum \\\\limits _{k=1}^{K} q_{k} \\\\cdot \\\\log \\\\Big (\\\\frac{2 q_{k}}{q_{k} + p_{k}} \\\\Big )+ \\\\sum \\\\limits _{k=1}^{K} p_{k} \\\\cdot \\\\log \\\\Big (\\\\frac{2 p_{k}}{q_{k} + p_{k}} \\\\Big ) \\\\Big \\\\rbrace ,\\\\qquad \\\\mathbf {P} \\\\in \\\\mathbb {R}_{> 0}^{K}, \\\\mathbf {Q} \\\\in \\\\mathbb {R}_{\\\\ge 0}^{K}.$ For the special subcase that $\\\\widetilde{c} =1$ and that $\\\\mathbf {P} = {P}$ , $\\\\mathbf {Q} = {Q}$ are probability vectors, the divergence (REF ) can be rewritten as sum of two Kullback-Leibler divergences (cf.\",\n \"(REF )) $D_{\\\\varphi _{snKL,1}}({Q},{P})= D_{\\\\varphi _{1}}({Q},({Q}+{P})/2)+ D_{\\\\varphi _{1}}({P},({Q}+{P})/2),\\\\qquad {P} \\\\in \\\\mathbb {S}_{> 0}^{K}, {Q} \\\\in \\\\mathbb {S}_{\\\\ge 0}^{K},$ which is the well-known (cf.\",\n \"Burbea & Rao [68], Lin [219], Pardo & Vajda [284], Topsoe [360], Endres & Schindelin [120], Vajda [373], Sason [317]) Jensen-Shannon divergence (being also called symmetrized and normalized Kullback-Leibler divergence, symmetrized and normalized relative entropy, capacitory discrimination); this is equal to the $(2\\\\log 2)-$ fold of a special (namely, equally-weighted two-population) case of the Sibson information radius of order 1 (cf.\",\n \"[334]) which has also been addressed e.g.\",\n \"by Rao [301] for genetic cluster analysis.\",\n \"By the way, for $\\\\alpha >0$ the divergence $D_{\\\\varphi _{gKL,\\\\alpha ,\\\\widetilde{c}}}({Q},{P})$ can also be interpreted as a multiple of a special non-equally-weighted Sibson information radius of order 1.\",\n \"In a context of comparison of — not necessarily connected — networks where ${Q}$ , ${P}$ are probability vectors derived from matrices (cf.\",\n \"Remark REF ) which are transforms of corresponding graph invariants (e.g.\",\n \"network portraits), the (matrix-equivalent of the) Jensen-Shannon divergence $D_{\\\\varphi _{snKL,1}}({Q},{P})$ is also called the network portrait divergence, cf.\",\n \"Bagrow and Bollt [28].\",\n \"There is a vast literature on recent applications of the Jensen-Shannon divergence, for instance it appears exemplarily in Kvitsiani et al.\",\n \"[208] for finding connections between the circuit-level function of different interneuron types in regulating the flow of information and the behavioural functions served by the cortical circuits, in Xu et al.\",\n \"(2014) for browsing and exploration of video sequences, in Jenkinson et al.\",\n \"[168] for the fundamental understanding of the epigenome that leads to a powerful approach for studying its role in disease and aging, in Martin et al.\",\n \"[250] for the implementation of an evolutionary-based global localization filter for mobile robots, in Suo et al.\",\n \"[354] for the revelation of critical regulators of cell identity in mice, in Abante et al.\",\n \"[2] for the detection of biologically significant differences in DNA methylation between alleles associated with local changes in genetic sequences — for a better understanding of the mechanism of complex human diseases, in Afek et al.\",\n \"[5] for revealing mechanisms by which mismatches can recruit transcription factors for modulating replication and repair activities in cells, in Alaiz-Rodriguez & Parnell [10] for the quantification of stability in feature selection and ranking algorithms, in Biau et al.\",\n \"[53] for generative adversarial networks (GANs) in artificial intelligence and machine learning, in Carre et al.\",\n \"[74] for the standardization of brain magnetic resonance (MR) images, in Chakraborty et al.\",\n \"[75] for hierarchical clustering in foreign exchange FOREX markets (e.g.\",\n \"in periods of major international crises), in Chong et al.\",\n \"[87] as part of a web-based platform for comprehensive analysis of microbiome data outputs, in Cui et al.\",\n \"[101] for modelling latent friend recommendation in online social media, in Gholami & Hodtani [134] for refinements of safety-and-security-targeted location verification systems in wireless communication networks (e.g in Intelligent Transportation Systems (ITSs) and vehicular technology), in Guo & Yuan [146] for accurate abnormality classification in semi-supervised Wireless Capsule Endoscopy (WCE) for digestive system cancer diagnosis, in Jiang et al.\",\n \"[169] for the training of deep neural discriminative and generative networks used for designing and evaluating photonic devices, in Kartal et al.\",\n \"[186] for uncovering the relationship between some genomic features and cell type-specific methylome diversity, in Laszlovszky et al.\",\n \"[210] for investigating mechanisms of basal forebrain neurons which modulate synaptic plasticity,cortical processing, brain states and oscillations, in Lawson et al.\",\n \"[211] for the improved understanding of some genetic circuits that allow cancer cells to evade destruction by the host immune system, in Li et al.\",\n \"[215] for the search of causes of the progressive neurodevelopmental disorder Rett syndrome, in Machado et al.\",\n \"[239] for discovering relations between distinct RNA viruses (including SARS-CoV-2), in Mohammadi et al.\",\n \"[261] for the identification of cell states and their underlying topology, in Mohanty et al.\",\n \"[262] for the design of implantable nanophotonic (i.e.\",\n \"chip-scale optical circuit type) silicon probes for sub-millisecond deep-brain optical stimulation — e.g.\",\n \"for the purpose of gaining a deeper understanding of the neural code, in Perera et al.\",\n \"[290] for the quantification of the level of rationality in supply chain networks, in Pierri et al.\",\n \"[294] for the study of growth of malicious/misleading information in some social media diffusion networks, in Rabadan et al.\",\n \"[299] for the identification of gene mutations that lead to the genesis and progression of tumors, in Reiter et al.\",\n \"[306] for quantifying metastatic phylogenetic diversity, in Van de Sande et al.\",\n \"[378] as part of a computational toolbox for single-cell gene regulatory network analysis, in Skinnider et al.\",\n \"[337] for the prediction of the chemical structures of genomically encoded antibiotics — in order to find means against the looming global crisis of antibiotic resistance, in Tuo et al.\",\n \"[367] for the detection of high-order single nucleotide polymorphism (SNP) interactions, in Uttam et al.\",\n \"[370] for predicting the risk of colorectal cancer recurrence and inferring associated tumor microenvironment networks, in Zhang et al.\",\n \"[421] for incipient fault (namely, crack) detection, in Zhi et al.\",\n \"[425] for the strengthening of information-centric networks against interest flooding attack (IFAs), in Acera Mateos et al.\",\n \"[3] for deep-learning classification of SARS-CoV-2 and co-infecting RNA viruses, in Avsec et al.\",\n \"[24] for uncovering the motifs and syntax of cis-regulatory sequences in genomics data, in Barennes et al.\",\n \"[32] for comparing the accuracy of current T cell receptor sequencing methods employed for the understanding of adaptive immune responses, in Chen et al.\",\n \"[79] for clustering high-dimensional microbial data from RNA sequencing, in Chen et al.\",\n \"[84] for investigating key aspects of effective vocal social communication, in Koldobskiy et al.\",\n \"[193] for investigations of genetic and epigenetic drivers of paediatric acute lymphoblastic leukaemia, in McGinnis et al.\",\n \"[258] for evaluating RNA sequencing of pooled blood cell samples, in Mühlroth & Grottke [268] for the detection of emerging trends and technologies through artificial intelligence techniques, in Necci et al.\",\n \"[272] for the assessment of protein intrinsic disorder predictions, in Okada et al.\",\n \"[277] for the identification of genetic factors that cause individual differences in whole lymphocyte profiles and their changes after vaccination, and in Zhang et al.\",\n \"[422] for the learning of functional magnetic resonance imaging (fMRI) time-series in a brain disease diagnosis context.\",\n \"Remark 44 Let us transform $\\\\varphi _{gSH,\\\\alpha }(t) := \\\\frac{1-t}{\\\\alpha } \\\\cdot \\\\log (1+\\\\alpha )- \\\\varphi _{gKL,\\\\alpha ,1}(t)= - t \\\\cdot \\\\log t+ \\\\frac{1}{\\\\alpha } \\\\cdot (1 + \\\\alpha \\\\cdot t) \\\\cdot \\\\log (1 + \\\\alpha \\\\cdot t)- \\\\frac{1}{\\\\alpha } \\\\cdot (1+\\\\alpha ) \\\\cdot t \\\\cdot \\\\log (1+\\\\alpha )$ (for $t \\\\in [0,1]$ ).\",\n \"The function $\\\\varphi _{gSH,\\\\alpha }(\\\\cdot )$ is strictly concave on $[0,1]$ with $\\\\varphi _{gSH,\\\\alpha }(0)= \\\\varphi _{gSH,\\\\alpha }(1)=0$ .\",\n \"Hence, for probability vectors ${Q} =(q_{k})_{k=1,\\\\ldots ,K}$ , the $\\\\varphi -$ entropy $\\\\sum _{k=1}^{K}\\\\varphi _{gSH,\\\\alpha }(q_{k})$ is Kapur’s [183] generalization of the Shannon entropy (which corresponds to $\\\\alpha =0$ in the limit) whose maximization has been connected with generalizations of the Bose-Einstein statistics and the Fermi-Dirac statistics e.g.\",\n \"in Kapur & Kesavan [185].\",\n \"Example 45 Let us fix any $z_{1},z_{2} \\\\in \\\\mathbb {R}$ , $p \\\\in ]0,1[$ which satisfy $z_{1} < 1 < z_{2}$ and $z_{1} \\\\cdot p + z_{2} \\\\cdot (1-p) =1$ (and thus $p= \\\\frac{z_{2} -1}{z_{2} - z_{1}}$ ).\",\n \"On $]a_{F_{twop}},b_{F_{twop}}[ := ]z_{1},z_{2}[$ we define $F_{twop}(t) &:=& \\\\frac{1}{z_{2}-z_{1}} \\\\cdot \\\\log \\\\left( \\\\frac{(t-z_{1})\\\\cdot p}{(z_{2}-t)\\\\cdot (1- p)} \\\\right)\\\\nonumber \\\\\\\\&=&\\\\frac{1}{z_{2}-z_{1}} \\\\cdot \\\\log \\\\left( \\\\frac{(t-z_{1})\\\\cdot (z_{2} -1)}{(z_{2}-t)\\\\cdot (1-z_{1})} \\\\right) \\\\, ,\\\\qquad t \\\\in \\\\, ]z_{1},z_{2}[ ,\\\\nonumber $ where for the last equality we have used the above constraint (in order to obtain a two-parameter representation).\",\n \"Straightforwardly, we have $\\\\mathcal {R}(F_{twop})= ]-\\\\infty ,\\\\infty [$ .\",\n \"Moreover, $F_{twop}(\\\\cdot )$ is strictly increasing and smooth on $]0,\\\\infty [$ , and thus, $F_{twop} \\\\in {F}$ .\",\n \"Since $F_{twop}(1)=0$ , let us choose the natural anchor point $c:=0$ , which leads to $]\\\\lambda _{-},\\\\lambda _{+}[= int(\\\\mathcal {R}(F_{snKL,\\\\widetilde{c}})) = ]-\\\\infty , \\\\infty [$ and $]t_{-}^{sc},t_{+}^{sc}[ = ]z_{1},z_{2}[$ .\",\n \"By using $F_{twop}^{-1}(x) = \\\\frac{p \\\\cdot z_{1} + (1- p) \\\\cdot z_{2} \\\\cdot e^{(z_{2}-z_{1}) \\\\cdot x}}{p + (1- p) \\\\cdot e^{(z_{2}-z_{1}) \\\\cdot x}} \\\\, , \\\\qquad x \\\\in ]-\\\\infty , \\\\infty [,$ we derive from formula (REF ) (see also (REF )) $& & \\\\hspace{-19.91684pt} \\\\Lambda _{twop}(z) := \\\\Lambda _{twop}^{(0)}(z) :=\\\\int \\\\displaylimits _{0}^{z} F_{twop}^{-1}(u) du= \\\\log \\\\Big ( p \\\\cdot e^{z_{1} \\\\cdot z} + (1- p) \\\\cdot e^{z_{2} \\\\cdot z} \\\\Big ),\\\\qquad z \\\\in ]-\\\\infty , \\\\infty [ ,$ which has the properties $\\\\Lambda _{twop}(0) = 0$ , $\\\\Lambda _{twop}(-\\\\infty ) =\\\\infty \\\\cdot {1}_{]-\\\\infty ,0[}(z_{1})- \\\\infty \\\\cdot {1}_{]0,\\\\infty [}(z_{1})+ \\\\log p \\\\cdot {1}_{\\\\lbrace 0\\\\rbrace }(z_{1})$ , $\\\\Lambda _{twop}(\\\\infty ) = \\\\infty $ , $\\\\Lambda _{twop}^{\\\\prime }(- \\\\infty ) = z_{1}$ and $\\\\Lambda _{twop}^{\\\\prime }(\\\\infty ) = z_{2}$ .\",\n \"Furthermore, from formula (REF ) (see also ()) we deduce $\\\\varphi _{twop}(t) := \\\\varphi _{twop}^{(0)}(t)\\\\hspace{-5.69046pt} &:=& \\\\hspace{-5.69046pt}{\\\\left\\\\lbrace \\\\begin{array}{ll}\\\\frac{t-z_{1}}{z_{2}-z_{1}} \\\\cdot \\\\log \\\\left( \\\\frac{(t-z_{1})\\\\cdot (z_{2}-1)}{(z_{2}-t)\\\\cdot (1-z_{1})} \\\\right)- \\\\log \\\\left( \\\\frac{z_{2}-1}{z_{2}-t} \\\\right)\\\\ \\\\in \\\\ [0,\\\\infty [,\\\\qquad \\\\quad \\\\textrm {if }t \\\\in \\\\, ]0,\\\\infty [,\\\\\\\\\\\\log \\\\left( \\\\frac{z_{2} - z_{1}}{z_{2} -1} \\\\right), \\\\hspace{219.08612pt} \\\\textrm {if } \\\\ t = z_{1}, \\\\\\\\\\\\log \\\\left( \\\\frac{z_{2} - z_{1}}{1 - z_{1}} \\\\right), \\\\hspace{219.08612pt} \\\\textrm {if } \\\\ t = z_{2}, \\\\\\\\\\\\infty , \\\\hspace{233.3125pt} \\\\textrm {if } \\\\ t \\\\in \\\\, ]-\\\\infty ,z_{1}[ \\\\, \\\\cup \\\\, ]z_{2}, \\\\infty [ ;\\\\end{array}\\\\right.\",\n \"}$ the first line in (REF ) can be proved by $& & \\\\hspace{-19.91684pt}\\\\varphi _{twop}(t) := \\\\varphi _{twop}^{(0)}(t) :=t \\\\cdot F_{twop}\\\\left(t\\\\right)-\\\\int \\\\displaylimits _{0}^{F_{twop}\\\\left(t\\\\right)} F_{twop}^{-1}(u) du\\\\nonumber \\\\\\\\& & \\\\hspace{-19.91684pt}= \\\\frac{t}{z_{2}-z_{1}} \\\\cdot \\\\log \\\\left( \\\\frac{(t-z_{1})\\\\cdot p}{(z_{2}-t)\\\\cdot (1- p)} \\\\right)\\\\nonumber \\\\\\\\& & \\\\hspace{-19.91684pt}- \\\\log \\\\left( p \\\\cdot \\\\left( \\\\frac{(t-z_{1})\\\\cdot p}{(z_{2}-t)\\\\cdot (1- p)} \\\\right)^{\\\\frac{z_{1}}{z_{2}-z_{1}}}+ (1- p) \\\\cdot \\\\left( \\\\frac{(t-z_{1})\\\\cdot p}{(z_{2}-t)\\\\cdot (1- p)} \\\\right)^{\\\\frac{z_{2}}{z_{2}-z_{1}}} \\\\right)\\\\nonumber \\\\\\\\& & \\\\hspace{-19.91684pt}= \\\\frac{t-z_{1}}{z_{2}-z_{1}} \\\\cdot \\\\log \\\\left( \\\\frac{(t-z_{1})\\\\cdot p}{(z_{2}-t)\\\\cdot (1- p)} \\\\right)- \\\\log \\\\left( \\\\frac{(z_{2}-z_{1})\\\\cdot p}{z_{2}-t} \\\\right)\\\\\\\\& & \\\\hspace{-19.91684pt}= \\\\frac{t-z_{1}}{z_{2}-z_{1}} \\\\cdot \\\\log \\\\left( \\\\frac{(t-z_{1})\\\\cdot (z_{2}-1)}{(z_{2}-t)\\\\cdot (1-z_{1})} \\\\right)- \\\\log \\\\left( \\\\frac{z_{2}-1}{z_{2}-t} \\\\right) , \\\\qquad t \\\\in \\\\, ]z_{1}, z_{2} [ \\\\, ,\\\\nonumber $ where for the last equality we have used the above constraint (to obtain a two-parameter representation).\",\n \"Straightforwardly, one has $\\\\varphi _{twop}(1) = 0$ , $\\\\varphi _{twop}^{\\\\prime }(1) = 0$ , $\\\\varphi _{twop}^{\\\\prime }(z_{1}) = -\\\\infty $ and $\\\\varphi _{twop}^{\\\\prime }(z_{2}) = \\\\infty $ .\",\n \"From the generator $\\\\varphi _{twop}$ given in (REF ), we build the corresponding divergence (cf.\",\n \"(REF )) $D_{\\\\varphi _{twop}}(\\\\mathbf {Q},\\\\mathbf {P})=\\\\sum \\\\limits _{k=1}^{K} \\\\frac{q_{k} - z_{1} \\\\cdot p_{k}}{z_{2} - z_{1}} \\\\cdot \\\\log \\\\Big (\\\\frac{(z_{2}-1) \\\\cdot (q_{k} - z_{1} \\\\cdot p_{k})}{(1-z_{1}) \\\\cdot (z_{2} \\\\cdot p_{k} - q_{k})} \\\\Big )- \\\\sum \\\\limits _{k=1}^{K} p_{k} \\\\cdot \\\\log \\\\Big (\\\\frac{(z_{2}-1) \\\\cdot p_{k}}{z_{2} \\\\cdot p_{k} - q_{k}} \\\\Big ) .$ It is known that some types of robustness properties of minimum-divergence estimators are connected with the boundedness of the derivative $\\\\varphi ^{\\\\prime }$ of the divergence generator $\\\\varphi $ ; this property is satisfied for the next Example REF (and its $W-$ concerning continuation in Example REF ), which leads to the new classes of divergences (REF ), (REF ) and (REF ): Example 46 (a) For any parameter-quadrupel $\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c} \\\\in \\\\, ]0,\\\\infty [$ with $\\\\beta _{1} < \\\\beta _{2}$ , we choose $]a_{F},b_{F}[\\\\ \\\\, := \\\\ ]a_{F_{\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c}}},b_{F_{\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c}}}[ \\\\ \\\\, := \\\\ \\\\Big ]1 - \\\\alpha \\\\cdot \\\\frac{(\\\\beta _{1} - \\\\beta _{2})^{2} + \\\\beta _{1}^{2} + \\\\beta _{1} \\\\cdot \\\\beta _{2}}{2\\\\beta _{1}\\\\cdot \\\\beta _{2}\\\\cdot (\\\\beta _{2} - \\\\beta _{1} )}, \\\\,\\\\infty \\\\Big [ \\\\ \\\\ni 1\\\\nonumber $ and define with $\\\\breve{\\\\theta } := 1 + \\\\alpha \\\\cdot \\\\Big (\\\\frac{1}{\\\\beta _{2}}- \\\\frac{1}{\\\\beta _{1}} \\\\Big ) < 1$ $\\\\hspace{-17.07182pt}F_{\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c}}(t)&:=& {\\\\left\\\\lbrace \\\\begin{array}{ll}\\\\widetilde{c} \\\\cdot \\\\frac{\\\\beta _{1}-\\\\beta _{2}}{2}+ \\\\frac{\\\\widetilde{c}}{\\\\frac{1-t}{\\\\alpha } + \\\\frac{1}{\\\\beta _{2}} - \\\\frac{1}{\\\\beta _{1}}}\\\\cdot \\\\Big (1 - \\\\frac{1}{2} \\\\cdot \\\\sqrt{4 + \\\\big (\\\\frac{1-t}{\\\\alpha } + \\\\frac{1}{\\\\beta _{2}} - \\\\frac{1}{\\\\beta _{1}}\\\\big )^{2} \\\\cdot (\\\\beta _{1}+\\\\beta _{2})^{2}}\\\\, \\\\Big ),\\\\quad \\\\textrm {if } \\\\ t \\\\in \\\\, ]a_{F},b_{F}[ \\\\backslash \\\\lbrace \\\\breve{\\\\theta }\\\\rbrace , \\\\\\\\\\\\widetilde{c} \\\\cdot \\\\frac{\\\\beta _{1}-\\\\beta _{2}}{2}, \\\\hspace{277.41437pt}\\\\textrm {if } \\\\ t= \\\\breve{\\\\theta } \\\\in \\\\, ]a_{F},b_{F}[, \\\\\\\\- \\\\widetilde{c} \\\\cdot \\\\beta _{1}, \\\\hspace{284.52756pt} \\\\textrm {if } \\\\ t=a_{F}, \\\\\\\\- \\\\infty , \\\\hspace{295.90848pt} \\\\textrm {if } \\\\ t \\\\in \\\\, ]-\\\\infty ,a_{F}[.\\\\end{array}\\\\right.\",\n \"}$ Notice that $\\\\breve{\\\\theta } \\\\in \\\\, ]a_{F},b_{F}[$ if and only if $\\\\beta _{1} \\\\in \\\\,]\\\\frac{\\\\beta _{2}}{3},\\\\beta _{2}[$ ; if (say) the latter holds, then one has the continuity $\\\\lim _{t \\\\rightarrow \\\\breve{\\\\theta }}F_{\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c}}(t) = \\\\widetilde{c} \\\\cdot \\\\frac{\\\\beta _{1}-\\\\beta _{2}}{2}$ .\",\n \"For $\\\\beta _{1} \\\\le \\\\frac{\\\\beta _{2}}{3}$ one gets $]a_{F},b_{F}[ \\\\backslash \\\\lbrace \\\\breve{\\\\theta } \\\\rbrace = \\\\, ]a_{F},b_{F}[$ .\",\n \"Returning to the general case, one can see in a straightforward way that $F_{\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c}}(\\\\cdot )$ is strictly increasing and that $\\\\mathcal {R}(F_{\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c}})=[-\\\\widetilde{c}\\\\cdot \\\\beta _{1},\\\\widetilde{c}\\\\cdot \\\\beta _{1} [$ .\",\n \"Furthermore, $F_{\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c}}(\\\\cdot )$ is smooth on $]a_{F},b_{F}[$ , and thus $F_{\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c}} \\\\in {F}$ .\",\n \"Since $F_{\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c}}(1)=0$ , let us choose the natural anchor point $c:=0$ , which leads to $]\\\\lambda _{-},\\\\lambda _{+}[ \\\\,= int(\\\\mathcal {R}(F_{\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c}})= \\\\, ]-\\\\widetilde{c}\\\\cdot \\\\beta _{1},\\\\widetilde{c}\\\\cdot \\\\beta _{1} [$ and $]t_{-}^{sc},t_{+}^{sc}[ = \\\\, ]a_{F},b_{F}[$ .\",\n \"Moreover, it is straightforward to see that the corresponding inverse is $F_{\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c}}^{-1}(x) =1 + \\\\alpha \\\\cdot \\\\Big (\\\\frac{1}{\\\\beta _{2}}- \\\\frac{1}{\\\\beta _{1}} \\\\Big ) - \\\\alpha \\\\cdot \\\\frac{\\\\frac{1}{\\\\beta _{2}} - \\\\frac{1}{\\\\beta _{1}} -\\\\frac{2 x}{\\\\widetilde{c} \\\\cdot \\\\beta _{1} \\\\cdot \\\\beta _{2} }}{1+ \\\\frac{x}{\\\\widetilde{c}} \\\\cdot \\\\Big (\\\\frac{1}{\\\\beta _{2}} - \\\\frac{1}{\\\\beta _{1}} \\\\Big )- \\\\frac{x^{2}}{\\\\widetilde{c}^{2} \\\\cdot \\\\beta _{1} \\\\cdot \\\\beta _{2} }},\\\\qquad x \\\\in int(\\\\mathcal {R}(F_{\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c}})) ;$ from this, we can derive from formula (REF ) (see also (REF )) for all $z \\\\in \\\\mathbb {R}$ $\\\\hspace{-19.91684pt}\\\\Lambda _{\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c}}(z) :=\\\\Lambda _{\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c}}^{(0)}(z) &=&{\\\\left\\\\lbrace \\\\begin{array}{ll}\\\\breve{\\\\theta } \\\\cdot z - \\\\widetilde{c} \\\\cdot \\\\alpha \\\\cdot \\\\log \\\\Big (1+ \\\\frac{z}{\\\\widetilde{c}} \\\\cdot \\\\Big (\\\\frac{1}{\\\\beta _{2}} - \\\\frac{1}{\\\\beta _{1}} \\\\Big )- \\\\frac{z^{2}}{\\\\widetilde{c}^{2} \\\\cdot \\\\beta _{1} \\\\cdot \\\\beta _{2} }\\\\Big ),\\\\quad \\\\textrm {if } \\\\ z \\\\in \\\\, ]-\\\\widetilde{c}\\\\cdot \\\\beta _{1},\\\\widetilde{c}\\\\cdot \\\\beta _{1} [,\\\\\\\\- \\\\widetilde{c} \\\\cdot \\\\breve{\\\\theta } \\\\cdot \\\\beta _{1} -\\\\widetilde{c} \\\\cdot \\\\alpha \\\\cdot \\\\log \\\\Big (2 - 2 \\\\frac{\\\\beta _{1}}{\\\\beta _{2}}\\\\Big ),\\\\hspace{73.97733pt} \\\\textrm {if } \\\\ z = -\\\\widetilde{c}\\\\cdot \\\\beta _{1},\\\\\\\\\\\\infty , \\\\hspace{202.01474pt} \\\\textrm {else} .\\\\end{array}\\\\right.\",\n \"}$ Notice that $\\\\Lambda _{\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c}}(0) = 0$ and $\\\\lim _{z \\\\rightarrow \\\\widetilde{c}\\\\cdot \\\\beta _{1}}\\\\Lambda _{\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c}}(z) = \\\\infty $ .\",\n \"Moreover, $\\\\Lambda _{\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c}}^{\\\\prime }( -\\\\widetilde{c}\\\\cdot \\\\beta _{1})= a_{F}$ and $\\\\Lambda _{ -\\\\widetilde{c}\\\\cdot \\\\beta _{1}}^{\\\\prime }(\\\\widetilde{c}\\\\cdot \\\\beta _{1}) = \\\\infty = b_{F}$ (which have to be interpreted as limits, as usual).\",\n \"To proceed, from formula (REF ) (see also ()) we can deduce for all $t \\\\in \\\\mathbb {R}$ $\\\\hspace{-5.69046pt}\\\\varphi _{\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c}}(t):= \\\\varphi _{\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c}}^{(0)}(t)\\\\hspace{-8.5359pt} &=& \\\\hspace{-8.5359pt}{\\\\left\\\\lbrace \\\\begin{array}{ll}\\\\widetilde{c} \\\\cdot \\\\alpha \\\\cdot \\\\Big \\\\lbrace \\\\frac{\\\\sqrt{4 + (\\\\beta _{1} + \\\\beta _{2})^{2}\\\\cdot (\\\\frac{1-t}{\\\\alpha } + \\\\frac{1}{\\\\beta _{2}} - \\\\frac{1}{\\\\beta _{1}})^2}\\\\, - \\\\, (\\\\frac{1-t}{\\\\alpha } + \\\\frac{1}{\\\\beta _{2}} - \\\\frac{1}{\\\\beta _{1}}) \\\\cdot (\\\\beta _{1} - \\\\beta _{2}) \\\\, - \\\\, 2}{2} \\\\\\\\+ \\\\log \\\\frac{\\\\sqrt{4 + (\\\\beta _{1} + \\\\beta _{2})^{2}\\\\cdot (\\\\frac{1-t}{\\\\alpha } + \\\\frac{1}{\\\\beta _{2}} \\\\, - \\\\, \\\\frac{1}{\\\\beta _{1}})^2} \\\\, - \\\\, 2}{\\\\beta _{1} \\\\beta _{2} \\\\cdot (\\\\frac{1-t}{\\\\alpha } + \\\\frac{1}{\\\\beta _{2}} \\\\, - \\\\, \\\\frac{1}{\\\\beta _{1}})^{2}} \\\\Big \\\\rbrace \\\\ \\\\in [0,\\\\infty [,\\\\hspace{71.13188pt} \\\\textrm {if }t \\\\in \\\\, ]a_{F},\\\\infty [,\\\\\\\\\\\\widetilde{c} \\\\cdot \\\\alpha \\\\cdot \\\\Big \\\\lbrace \\\\frac{3 \\\\beta _{1} - \\\\beta _{2}}{2 (\\\\beta _{2} - \\\\beta _{1})}+ \\\\log \\\\frac{2(\\\\beta _{2} - \\\\beta _{1})}{\\\\beta _{2}} \\\\Big \\\\rbrace - \\\\widetilde{c} \\\\cdot \\\\beta _{1} \\\\cdot (t - a_{F}) \\\\ \\\\in \\\\, ]0,\\\\infty [,\\\\hspace{19.91684pt} \\\\textrm {if } \\\\ t \\\\in ]-\\\\infty , a_{F}].\\\\end{array}\\\\right.\",\n \"}$ The first subcase in (REF ) can be proved by $& & \\\\hspace{-19.91684pt}\\\\varphi _{\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c}}(t) :=\\\\varphi _{\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c}}^{(0)}(t) :=t \\\\cdot F_{\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c}}\\\\left(t\\\\right)\\\\ - \\\\int \\\\displaylimits _{0}^{F_{\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c}}\\\\left(t\\\\right)}F_{\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c}}^{-1}(u) \\\\, du\\\\nonumber \\\\\\\\& & \\\\hspace{-19.91684pt}= (t - \\\\breve{\\\\theta }) \\\\cdot F_{\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c}}\\\\left(t\\\\right)+ \\\\widetilde{c} \\\\cdot \\\\alpha \\\\cdot \\\\log \\\\Big (1+ \\\\frac{F_{\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c}}\\\\left(t\\\\right)}{\\\\widetilde{c}} \\\\cdot \\\\Big (\\\\frac{1}{\\\\beta _{2}} - \\\\frac{1}{\\\\beta _{1}} \\\\Big )- \\\\frac{(F_{\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c}}\\\\left(t\\\\right))^{2}}{\\\\widetilde{c}^{2} \\\\cdot \\\\beta _{1} \\\\cdot \\\\beta _{2} }\\\\Big )\\\\nonumber \\\\\\\\& & \\\\hspace{-19.91684pt}= \\\\widetilde{c} \\\\cdot \\\\alpha \\\\cdot \\\\frac{\\\\sqrt{4 + (\\\\beta _{1} + \\\\beta _{2})^{2}\\\\cdot (\\\\frac{1-t}{\\\\alpha } + \\\\frac{1}{\\\\beta _{2}} - \\\\frac{1}{\\\\beta _{1}})^2}\\\\, - \\\\, (\\\\frac{1-t}{\\\\alpha } + \\\\frac{1}{\\\\beta _{2}} - \\\\frac{1}{\\\\beta _{1}}) \\\\cdot (\\\\beta _{1} - \\\\beta _{2}) \\\\, - \\\\, 2}{2}\\\\nonumber \\\\\\\\& & \\\\hspace{-19.91684pt}\\\\ \\\\ \\\\ + \\\\ \\\\widetilde{c} \\\\cdot \\\\alpha \\\\cdot \\\\log \\\\bigg (1+ \\\\Big [\\\\frac{\\\\beta _{1}-\\\\beta _{2}}{2}+ \\\\frac{1}{\\\\frac{1-t}{\\\\alpha } + \\\\frac{1}{\\\\beta _{2}} - \\\\frac{1}{\\\\beta _{1}}}\\\\cdot \\\\Big (1 - \\\\frac{1}{2} \\\\cdot \\\\sqrt{4 + \\\\big (\\\\frac{1-t}{\\\\alpha } + \\\\frac{1}{\\\\beta _{2}} - \\\\frac{1}{\\\\beta _{1}}\\\\big )^{2} \\\\cdot (\\\\beta _{1}+\\\\beta _{2})^{2}}\\\\, \\\\Big ) \\\\Big ]\\\\cdot \\\\frac{\\\\beta _{1} - \\\\beta _{2}}{\\\\beta _{1} \\\\cdot \\\\beta _{2}}\\\\nonumber \\\\\\\\& & \\\\hspace{-19.91684pt}\\\\ \\\\ \\\\ - \\\\ \\\\Big [\\\\frac{\\\\beta _{1}-\\\\beta _{2}}{2}+ \\\\frac{1}{\\\\frac{1-t}{\\\\alpha } + \\\\frac{1}{\\\\beta _{2}} - \\\\frac{1}{\\\\beta _{1}}}\\\\cdot \\\\Big (1 - \\\\frac{1}{2} \\\\cdot \\\\sqrt{4 + \\\\big (\\\\frac{1-t}{\\\\alpha } + \\\\frac{1}{\\\\beta _{2}} - \\\\frac{1}{\\\\beta _{1}}\\\\big )^{2} \\\\cdot (\\\\beta _{1}+\\\\beta _{2})^{2}}\\\\, \\\\Big ) \\\\Big ]^{2} \\\\cdot \\\\frac{1}{\\\\beta _{1} \\\\cdot \\\\beta _{2} }\\\\bigg )\\\\nonumber $ and some straightforward calculations.\",\n \"The second line in (REF ) follows by computing $\\\\varphi _{\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c}}(a_{F})=\\\\widetilde{c} \\\\cdot \\\\alpha \\\\cdot \\\\Big \\\\lbrace \\\\frac{3 \\\\beta _{1} - \\\\beta _{2}}{2 (\\\\beta _{2} - \\\\beta _{1})}+ \\\\log \\\\frac{2(\\\\beta _{2} - \\\\beta _{1})}{\\\\beta _{2}} \\\\Big \\\\rbrace $ .\",\n \"Notice that $\\\\varphi _{\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c}}(1) = 0$ , $\\\\varphi _{\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c}}^{\\\\prime }(1) = 0$ , $\\\\varphi _{\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c}}(-\\\\infty ) = \\\\infty $ and $\\\\varphi _{\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c}}(\\\\infty ) = \\\\infty $ .\",\n \"Moreover, $\\\\varphi _{\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c}}^{\\\\prime }(-\\\\infty ) =\\\\varphi _{\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c}}^{\\\\prime }(a_{F}) =-\\\\widetilde{c} \\\\cdot \\\\beta _{1}$ and $\\\\varphi _{\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c}}^{\\\\prime }(\\\\infty ) =\\\\widetilde{c} \\\\cdot \\\\beta _{1}$ .\",\n \"From the generator $\\\\varphi _{\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c}}$ given in (REF ), we construct the corresponding divergence (cf.\",\n \"(REF )) $& &D_{\\\\varphi _{\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c}}}(\\\\mathbf {Q},\\\\mathbf {P})= \\\\sum \\\\limits _{k=1}^{K} p_{k} \\\\cdot \\\\varphi _{\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c}}\\\\Big (\\\\frac{q_{k}}{p_{k}}\\\\Big )\\\\nonumber \\\\\\\\& &= \\\\sum \\\\limits _{k=1}^{K}p_{k} \\\\cdot \\\\bigg [{1}_{]a_{F},\\\\infty [}(\\\\frac{q_{k}}{p_{k}}) \\\\cdot \\\\widetilde{c} \\\\cdot \\\\alpha \\\\cdot \\\\Big \\\\lbrace \\\\frac{\\\\sqrt{4 + (\\\\beta _{1} + \\\\beta _{2})^{2}\\\\cdot (\\\\frac{1-\\\\frac{q_{k}}{p_{k}}}{\\\\alpha } + \\\\frac{1}{\\\\beta _{2}} - \\\\frac{1}{\\\\beta _{1}})^2}\\\\, - \\\\, (\\\\frac{1-\\\\frac{q_{k}}{p_{k}}}{\\\\alpha } + \\\\frac{1}{\\\\beta _{2}} - \\\\frac{1}{\\\\beta _{1}}) \\\\cdot (\\\\beta _{1} - \\\\beta _{2}) \\\\, - \\\\, 2}{2}\\\\nonumber \\\\\\\\& &+ \\\\log \\\\frac{\\\\sqrt{4 + (\\\\beta _{1} + \\\\beta _{2})^{2}\\\\cdot (\\\\frac{1-\\\\frac{q_{k}}{p_{k}}}{\\\\alpha } + \\\\frac{1}{\\\\beta _{2}} \\\\, - \\\\, \\\\frac{1}{\\\\beta _{1}})^2} \\\\, - \\\\, 2}{\\\\beta _{1} \\\\beta _{2} \\\\cdot (\\\\frac{1-\\\\frac{q_{k}}{p_{k}}}{\\\\alpha } + \\\\frac{1}{\\\\beta _{2}} \\\\, - \\\\, \\\\frac{1}{\\\\beta _{1}})^{2}} \\\\Big \\\\rbrace \\\\nonumber \\\\\\\\& &+ \\\\ {1}_{]-\\\\infty , a_{F}]}(\\\\frac{q_{k}}{p_{k}}) \\\\cdot \\\\widetilde{c} \\\\cdot \\\\Big \\\\lbrace \\\\alpha \\\\cdot \\\\Big \\\\lbrace \\\\frac{3 \\\\beta _{1} - \\\\beta _{2}}{2 (\\\\beta _{2} - \\\\beta _{1})}+ \\\\log \\\\frac{2(\\\\beta _{2} - \\\\beta _{1})}{\\\\beta _{2}} \\\\Big \\\\rbrace - \\\\beta _{1} \\\\cdot (\\\\frac{q_{k}}{p_{k}} - a_{F}) \\\\Big \\\\rbrace \\\\bigg ],\\\\qquad \\\\mathbf {P} \\\\in \\\\mathbb {R}_{\\\\ge 0}^{K}, \\\\mathbf {Q} \\\\in \\\\mathbb {R}^{K}.$ Notice that we can particularly include the case where $p_{k}=0$ in combination with $q_{k} \\\\ne 0$ , since $\\\\lim _{t \\\\rightarrow 0_{+}} t \\\\cdot \\\\varphi _{\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c}}(\\\\frac{1}{t})= \\\\widetilde{c} \\\\cdot \\\\beta _{1}$ and $\\\\lim _{t \\\\rightarrow 0_{-}} t \\\\cdot \\\\varphi _{\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c}}(\\\\frac{1}{t})= - \\\\widetilde{c} \\\\cdot \\\\beta _{1}$ are both finite, and hence $p_{k} \\\\cdot \\\\varphi _{\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c}}(\\\\frac{q_{k}}{p_{k}})= q_{k} \\\\cdot \\\\frac{p_{k}}{q_{k}} \\\\cdot \\\\varphi _{\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c}}(\\\\frac{q_{k}}{p_{k}}) $ stays finite as $p_{k}$ tends to zero.\",\n \"(b) For any parameter-quadrupel $\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c} \\\\in \\\\, ]0,\\\\infty [$ with $\\\\beta _{1} > \\\\beta _{2}$ , one can proceed analogously to (a).\",\n \"Let us start by choosing $]a_{F},b_{F}[\\\\ \\\\, := \\\\ ]a_{F_{\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c}}},b_{F_{\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c}}}[ \\\\ \\\\, := \\\\ \\\\Big ]-\\\\infty , \\\\, 1 + \\\\alpha \\\\cdot \\\\frac{(\\\\beta _{1} - \\\\beta _{2})^{2} + \\\\beta _{1} \\\\cdot \\\\beta _{2} + \\\\beta _{2}^{2}}{2\\\\beta _{1}\\\\cdot \\\\beta _{2}\\\\cdot (\\\\beta _{1} - \\\\beta _{2} )}, \\\\,\\\\Big [ \\\\ \\\\ni 1\\\\nonumber $ and defining with the same $\\\\breve{\\\\theta } := 1 + \\\\alpha \\\\cdot \\\\Big (\\\\frac{1}{\\\\beta _{2}}- \\\\frac{1}{\\\\beta _{1}} \\\\Big ) > 1$ $\\\\hspace{-17.07182pt}F_{\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c}}(t)&:=& {\\\\left\\\\lbrace \\\\begin{array}{ll}\\\\widetilde{c} \\\\cdot \\\\frac{\\\\beta _{1}-\\\\beta _{2}}{2}+ \\\\frac{\\\\widetilde{c}}{\\\\frac{1-t}{\\\\alpha } + \\\\frac{1}{\\\\beta _{2}} - \\\\frac{1}{\\\\beta _{1}}}\\\\cdot \\\\Big (1 - \\\\frac{1}{2} \\\\cdot \\\\sqrt{4 + \\\\big (\\\\frac{1-t}{\\\\alpha } + \\\\frac{1}{\\\\beta _{2}} - \\\\frac{1}{\\\\beta _{1}}\\\\big )^{2} \\\\cdot (\\\\beta _{1}+\\\\beta _{2})^{2}}\\\\, \\\\Big ),\\\\quad \\\\textrm {if } \\\\ t \\\\in \\\\, ]a_{F},b_{F}[ \\\\backslash \\\\lbrace \\\\breve{\\\\theta }\\\\rbrace , \\\\\\\\\\\\widetilde{c} \\\\cdot \\\\frac{\\\\beta _{1}-\\\\beta _{2}}{2}, \\\\hspace{277.41437pt}\\\\textrm {if } \\\\ t= \\\\breve{\\\\theta } \\\\in \\\\, ]a_{F},b_{F}[, \\\\\\\\\\\\widetilde{c} \\\\cdot \\\\beta _{2}, \\\\hspace{293.06346pt} \\\\textrm {if } \\\\ t=b_{F}, \\\\\\\\\\\\infty , \\\\hspace{304.4444pt} \\\\textrm {if } \\\\ t \\\\in \\\\, ]b_{F}, \\\\infty [.\\\\end{array}\\\\right.\",\n \"}$ Clearly, $\\\\breve{\\\\theta } \\\\in \\\\, ]a_{F},b_{F}[$ if and only if $\\\\beta _{1} \\\\in \\\\,]\\\\beta _{2}, 3\\\\beta _{2}[$ ; if (say) the latter holds, then one gets the continuity $\\\\lim _{t \\\\rightarrow \\\\breve{\\\\theta }}F_{\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c}}(t) = \\\\widetilde{c} \\\\cdot \\\\frac{\\\\beta _{1}-\\\\beta _{2}}{2}$ .\",\n \"For $\\\\beta _{1} \\\\le 3 \\\\beta _{2}$ there holds $]a_{F},b_{F}[ \\\\backslash \\\\lbrace \\\\breve{\\\\theta } \\\\rbrace = \\\\, ]a_{F},b_{F}[$ .\",\n \"Returning to the general case, one can show comfortably that $F_{\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c}}(\\\\cdot )$ is strictly increasing and that $\\\\mathcal {R}(F_{\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c}})=]-\\\\widetilde{c}\\\\cdot \\\\beta _{2},\\\\widetilde{c}\\\\cdot \\\\beta _{2} ]$ .\",\n \"Moreover, $F_{\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c}}(\\\\cdot )$ is smooth on $]a_{F},b_{F}[$ , and hence $F_{\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c}} \\\\in {F}$ .\",\n \"In face of the validity of $F_{\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c}}(1)=0$ , let us choose the natural anchor point $c:=0$ , which amounts to $]\\\\lambda _{-},\\\\lambda _{+}[ \\\\,= int(\\\\mathcal {R}(F_{\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c}})= \\\\, ]-\\\\widetilde{c}\\\\cdot \\\\beta _{2},\\\\widetilde{c}\\\\cdot \\\\beta _{2} [$ and $]t_{-}^{sc},t_{+}^{sc}[ = \\\\, ]a_{F},b_{F}[$ .\",\n \"Since the first line in (REF ) coincides formally with that of (REF ) (with different $]a_{F},b_{F}[$ ), the corresponding inverse is formally the same as (REF ) (with different $]a_{F},b_{F}[$ ), and hence $\\\\hspace{-19.91684pt}\\\\Lambda _{\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c}}(z) :=\\\\Lambda _{\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c}}^{(0)}(z) &=&{\\\\left\\\\lbrace \\\\begin{array}{ll}\\\\breve{\\\\theta } \\\\cdot z - \\\\widetilde{c} \\\\cdot \\\\alpha \\\\cdot \\\\log \\\\Big (1+ \\\\frac{z}{\\\\widetilde{c}} \\\\cdot \\\\Big (\\\\frac{1}{\\\\beta _{2}} - \\\\frac{1}{\\\\beta _{1}} \\\\Big )- \\\\frac{z^{2}}{\\\\widetilde{c}^{2} \\\\cdot \\\\beta _{1} \\\\cdot \\\\beta _{2} }\\\\Big ),\\\\quad \\\\textrm {if } \\\\ z \\\\in \\\\, ]-\\\\widetilde{c}\\\\cdot \\\\beta _{2},\\\\widetilde{c}\\\\cdot \\\\beta _{2} [,\\\\\\\\\\\\widetilde{c} \\\\cdot \\\\breve{\\\\theta } \\\\cdot \\\\beta _{2} -\\\\widetilde{c} \\\\cdot \\\\alpha \\\\cdot \\\\log \\\\Big (2 - 2 \\\\frac{\\\\beta _{2}}{\\\\beta _{1}}\\\\Big ),\\\\hspace{73.97733pt} \\\\textrm {if } \\\\ z = \\\\widetilde{c}\\\\cdot \\\\beta _{2},\\\\\\\\\\\\infty , \\\\hspace{202.01474pt} \\\\textrm {else} .\\\\end{array}\\\\right.\",\n \"}$ Notice that $\\\\Lambda _{\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c}}(0) = 0$ and $\\\\lim _{z \\\\rightarrow - \\\\widetilde{c}\\\\cdot \\\\beta _{2}}\\\\Lambda _{\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c}}(z) = -\\\\infty $ .\",\n \"Furthermore, $\\\\Lambda _{\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c}}^{\\\\prime }( -\\\\widetilde{c}\\\\cdot \\\\beta _{2})= - \\\\infty = a_{F}$ and $\\\\Lambda _{ -\\\\widetilde{c}\\\\cdot \\\\beta _{1}}^{\\\\prime }(\\\\widetilde{c}\\\\cdot \\\\beta _{2})= b_{F}$ .\",\n \"To proceed, from formula (REF ) (see also ()) we can derive — analogously to (REF ) — for all $t \\\\in \\\\mathbb {R}$ $\\\\hspace{-5.69046pt}\\\\varphi _{\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c}}(t):= \\\\varphi _{\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c}}^{(0)}(t)\\\\hspace{-8.5359pt} &=& \\\\hspace{-8.5359pt}{\\\\left\\\\lbrace \\\\begin{array}{ll}\\\\widetilde{c} \\\\cdot \\\\alpha \\\\cdot \\\\Big \\\\lbrace \\\\frac{\\\\sqrt{4 + (\\\\beta _{1} + \\\\beta _{2})^{2}\\\\cdot (\\\\frac{1-t}{\\\\alpha } + \\\\frac{1}{\\\\beta _{2}} - \\\\frac{1}{\\\\beta _{1}})^2}\\\\, - \\\\, (\\\\frac{1-t}{\\\\alpha } + \\\\frac{1}{\\\\beta _{2}} - \\\\frac{1}{\\\\beta _{1}}) \\\\cdot (\\\\beta _{1} - \\\\beta _{2}) \\\\, - \\\\, 2}{2} \\\\\\\\+ \\\\log \\\\frac{\\\\sqrt{4 + (\\\\beta _{1} + \\\\beta _{2})^{2}\\\\cdot (\\\\frac{1-t}{\\\\alpha } + \\\\frac{1}{\\\\beta _{2}} \\\\, - \\\\, \\\\frac{1}{\\\\beta _{1}})^2} \\\\, - \\\\, 2}{\\\\beta _{1} \\\\beta _{2} \\\\cdot (\\\\frac{1-t}{\\\\alpha } + \\\\frac{1}{\\\\beta _{2}} \\\\, - \\\\, \\\\frac{1}{\\\\beta _{1}})^{2}} \\\\Big \\\\rbrace \\\\ \\\\in [0,\\\\infty [,\\\\hspace{71.13188pt} \\\\textrm {if }t \\\\in \\\\, ]-\\\\infty , b_{F}[,\\\\\\\\\\\\widetilde{c} \\\\cdot \\\\alpha \\\\cdot \\\\Big \\\\lbrace \\\\frac{3 \\\\beta _{2} - \\\\beta _{1}}{2 (\\\\beta _{1} - \\\\beta _{2})}+ \\\\log \\\\frac{2(\\\\beta _{1} - \\\\beta _{2})}{\\\\beta _{1}} \\\\Big \\\\rbrace + \\\\widetilde{c} \\\\cdot \\\\beta _{2} \\\\cdot (t - b_{F}) \\\\ \\\\in \\\\, ]0,\\\\infty [,\\\\hspace{19.91684pt} \\\\textrm {if } \\\\ t \\\\in [b_{F}, \\\\infty [,\\\\end{array}\\\\right.\",\n \"}$ where the last line in (REF ) follows by calculating $\\\\varphi _{\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c}}(b_{F})=\\\\widetilde{c} \\\\cdot \\\\alpha \\\\cdot \\\\Big \\\\lbrace \\\\frac{3 \\\\beta _{2} - \\\\beta _{1}}{2 (\\\\beta _{1} - \\\\beta _{2})}+ \\\\log \\\\frac{2(\\\\beta _{1} - \\\\beta _{2})}{\\\\beta _{1}} \\\\Big \\\\rbrace $ .\",\n \"Notice that $\\\\varphi _{\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c}}(1) = 0$ , $\\\\varphi _{\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c}}^{\\\\prime }(1) = 0$ , $\\\\varphi _{\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c}}(-\\\\infty ) = \\\\infty $ and $\\\\varphi _{\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c}}(\\\\infty ) = \\\\infty $ .\",\n \"Furthermore, $\\\\varphi _{\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c}}^{\\\\prime }(-\\\\infty ) =-\\\\widetilde{c} \\\\cdot \\\\beta _{2}$ and $\\\\varphi _{\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c}}^{\\\\prime }(\\\\infty ) =\\\\varphi _{\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c}}^{\\\\prime }(b_{F}) =\\\\widetilde{c} \\\\cdot \\\\beta _{2}$ .\",\n \"From the generator $\\\\varphi _{\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c}}$ given in (REF ), we construct the corresponding divergence (cf.\",\n \"(REF )) $& &D_{\\\\varphi _{\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c}}}(\\\\mathbf {Q},\\\\mathbf {P})= \\\\sum \\\\limits _{k=1}^{K} p_{k} \\\\cdot \\\\varphi _{\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c}}\\\\Big (\\\\frac{q_{k}}{p_{k}}\\\\Big )\\\\nonumber \\\\\\\\& &= \\\\sum \\\\limits _{k=1}^{K}p_{k} \\\\cdot \\\\bigg [{1}_{]-\\\\infty , b_{F}[}(\\\\frac{q_{k}}{p_{k}}) \\\\cdot \\\\widetilde{c} \\\\cdot \\\\alpha \\\\cdot \\\\Big \\\\lbrace \\\\frac{\\\\sqrt{4 + (\\\\beta _{1} + \\\\beta _{2})^{2}\\\\cdot (\\\\frac{1-\\\\frac{q_{k}}{p_{k}}}{\\\\alpha } + \\\\frac{1}{\\\\beta _{2}} - \\\\frac{1}{\\\\beta _{1}})^2}\\\\, - \\\\, (\\\\frac{1-\\\\frac{q_{k}}{p_{k}}}{\\\\alpha } + \\\\frac{1}{\\\\beta _{2}} - \\\\frac{1}{\\\\beta _{1}}) \\\\cdot (\\\\beta _{1} - \\\\beta _{2}) \\\\, - \\\\, 2}{2}\\\\nonumber \\\\\\\\& &+ \\\\log \\\\frac{\\\\sqrt{4 + (\\\\beta _{1} + \\\\beta _{2})^{2}\\\\cdot (\\\\frac{1-\\\\frac{q_{k}}{p_{k}}}{\\\\alpha } + \\\\frac{1}{\\\\beta _{2}} \\\\, - \\\\, \\\\frac{1}{\\\\beta _{1}})^2} \\\\, - \\\\, 2}{\\\\beta _{1} \\\\beta _{2} \\\\cdot (\\\\frac{1-\\\\frac{q_{k}}{p_{k}}}{\\\\alpha } + \\\\frac{1}{\\\\beta _{2}} \\\\, - \\\\, \\\\frac{1}{\\\\beta _{1}})^{2}} \\\\Big \\\\rbrace \\\\nonumber \\\\\\\\& &+ \\\\ {1}_{[b_{F}, \\\\infty [}(\\\\frac{q_{k}}{p_{k}}) \\\\cdot \\\\widetilde{c} \\\\cdot \\\\Big \\\\lbrace \\\\alpha \\\\cdot \\\\Big \\\\lbrace \\\\frac{3 \\\\beta _{2} - \\\\beta _{1}}{2 (\\\\beta _{1} - \\\\beta _{2})}+ \\\\log \\\\frac{2(\\\\beta _{1} - \\\\beta _{2})}{\\\\beta _{1}} \\\\Big \\\\rbrace + \\\\beta _{2} \\\\cdot (\\\\frac{q_{k}}{p_{k}} - b_{F}) \\\\Big \\\\rbrace \\\\bigg ],\\\\qquad \\\\mathbf {P} \\\\in \\\\mathbb {R}_{\\\\ge 0}^{K}, \\\\mathbf {Q} \\\\in \\\\mathbb {R}^{K}.$ As above, we can particularly include the case where $p_{k}=0$ in combination with $q_{k} \\\\ne 0$ , since $\\\\lim _{t \\\\rightarrow 0_{+}} t \\\\cdot \\\\varphi _{\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c}}(\\\\frac{1}{t})= \\\\widetilde{c} \\\\cdot \\\\beta _{2}$ and $\\\\lim _{t \\\\rightarrow 0_{-}} t \\\\cdot \\\\varphi _{\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c}}(\\\\frac{1}{t})= - \\\\widetilde{c} \\\\cdot \\\\beta _{2}$ are both finite.\",\n \"(c) The analysis for the case $\\\\beta _{1} = \\\\beta _{2} =: \\\\beta $ can be obtained by taking $\\\\lim _{\\\\beta _1 \\\\rightarrow \\\\beta _{2}}$ in (a) respectively (b).\",\n \"Alternatively, one can start afresh.\",\n \"Due to its importance and its particularities, we nevertheless state the corresponding results explicitly.\",\n \"To begin with, for any parameter-triple $\\\\alpha ,\\\\beta ,\\\\widetilde{c} \\\\in \\\\, ]0,\\\\infty [$ we choose $]a_{F},b_{F}[\\\\ \\\\, := \\\\ ]a_{F_{\\\\alpha ,\\\\beta ,\\\\widetilde{c}}},b_{F_{\\\\alpha ,\\\\beta ,\\\\widetilde{c}}}[ \\\\ \\\\, := \\\\ ]-\\\\infty , \\\\infty \\\\, [\\\\nonumber $ and define with $\\\\breve{\\\\theta } := 1$ $\\\\hspace{-17.07182pt}F_{\\\\alpha ,\\\\beta ,\\\\widetilde{c}}(t)&:=& {\\\\left\\\\lbrace \\\\begin{array}{ll}\\\\frac{\\\\widetilde{c} \\\\cdot \\\\alpha }{1-t}\\\\cdot \\\\Big (1 - \\\\sqrt{1 + \\\\big (\\\\frac{1-t}{\\\\alpha } \\\\big )^{2} \\\\cdot \\\\beta ^{2}}\\\\, \\\\Big ),\\\\qquad \\\\textrm {if } \\\\ t \\\\in \\\\, ]a_{F},b_{F}[ \\\\backslash \\\\lbrace \\\\breve{\\\\theta }\\\\rbrace , \\\\\\\\0, \\\\hspace{145.10922pt}\\\\textrm {if } \\\\ t= \\\\breve{\\\\theta }.\\\\end{array}\\\\right.\",\n \"}$ Clearly, one has the continuity $\\\\lim _{t \\\\rightarrow \\\\breve{\\\\theta }}F_{\\\\alpha ,\\\\beta ,\\\\widetilde{c}}(t) = 0$ .\",\n \"Moreover, one can see in a straightforward way that $F_{\\\\alpha ,\\\\beta ,\\\\widetilde{c}}(\\\\cdot )$ is strictly increasing and that $\\\\mathcal {R}(F_{\\\\alpha ,\\\\beta ,\\\\widetilde{c}})=]-\\\\widetilde{c}\\\\cdot \\\\beta ,\\\\widetilde{c}\\\\cdot \\\\beta [$ .\",\n \"Furthermore, $F_{\\\\alpha ,\\\\beta ,\\\\widetilde{c}}(\\\\cdot )$ is smooth on $]a_{F},b_{F}[$ , and thus $F_{\\\\alpha ,\\\\beta ,\\\\widetilde{c}} \\\\in {F}$ .\",\n \"Since $F_{\\\\alpha ,\\\\beta ,\\\\widetilde{c}}(1)=0$ , let us choose the natural anchor point $c:=0$ , which leads to the choice $]\\\\lambda _{-},\\\\lambda _{+}[ \\\\,= int(\\\\mathcal {R}(F_{\\\\alpha ,\\\\beta ,\\\\widetilde{c}})= \\\\, ]-\\\\widetilde{c}\\\\cdot \\\\beta ,\\\\widetilde{c}\\\\cdot \\\\beta [$ and $]t_{-}^{sc},t_{+}^{sc}[ = \\\\, ]a_{F},b_{F}[ = \\\\, ]-\\\\infty ,\\\\infty [$ .\",\n \"The inverse in (REF ) collapses to $F_{\\\\alpha ,\\\\beta ,\\\\widetilde{c}}^{-1}(x) =1 + \\\\alpha \\\\cdot \\\\frac{\\\\frac{2 x}{\\\\widetilde{c} \\\\cdot \\\\beta ^{2} }}{1 - \\\\frac{x^{2}}{\\\\widetilde{c}^{2} \\\\cdot \\\\beta ^{2} }},\\\\qquad x \\\\in int(\\\\mathcal {R}(F_{\\\\alpha ,\\\\beta ,\\\\widetilde{c}})) ;$ from this, we can derive from formula (REF ) (see also (REF )) for all $z \\\\in \\\\mathbb {R}$ $\\\\hspace{-19.91684pt}\\\\Lambda _{\\\\alpha ,\\\\beta ,\\\\widetilde{c}}(z) :=\\\\Lambda _{\\\\alpha ,\\\\beta ,\\\\widetilde{c}}^{(0)}(z) &=&{\\\\left\\\\lbrace \\\\begin{array}{ll}\\\\breve{\\\\theta } \\\\cdot z - \\\\widetilde{c} \\\\cdot \\\\alpha \\\\cdot \\\\log \\\\Big (1 - \\\\frac{z^{2}}{\\\\widetilde{c}^{2} \\\\cdot \\\\beta ^{2} }\\\\Big ),\\\\qquad \\\\textrm {if } \\\\ z \\\\in \\\\, ]-\\\\widetilde{c}\\\\cdot \\\\beta ,\\\\widetilde{c}\\\\cdot \\\\beta [,\\\\\\\\\\\\infty , \\\\hspace{130.88284pt} \\\\textrm {else} .\\\\end{array}\\\\right.\",\n \"}$ Notice that $\\\\Lambda _{\\\\alpha ,\\\\beta ,\\\\widetilde{c}}(0) = 0$ , $\\\\lim _{z \\\\rightarrow - \\\\widetilde{c}\\\\cdot \\\\beta }\\\\Lambda _{\\\\alpha ,\\\\beta ,\\\\widetilde{c}}(z) = -\\\\infty = $ , and $\\\\lim _{z \\\\rightarrow \\\\widetilde{c}\\\\cdot \\\\beta }\\\\Lambda _{\\\\alpha ,\\\\beta ,\\\\widetilde{c}}(z) = \\\\infty $ .\",\n \"Furthermore, $\\\\lim _{z \\\\rightarrow - \\\\widetilde{c}\\\\cdot \\\\beta }\\\\Lambda _{\\\\alpha ,\\\\beta ,\\\\widetilde{c}}^{\\\\prime }(z) = -\\\\infty = a_{F}$ , and $\\\\lim _{z \\\\rightarrow \\\\widetilde{c}\\\\cdot \\\\beta }\\\\Lambda _{\\\\alpha ,\\\\beta ,\\\\widetilde{c}}^{\\\\prime }(z) = \\\\infty = b_{F}$ .\",\n \"To proceed, the formula (REF ) (respectively, (REF )) collapses to $\\\\varphi _{\\\\alpha ,\\\\beta ,\\\\widetilde{c}}(t):=\\\\varphi _{\\\\alpha ,\\\\beta ,\\\\widetilde{c}}^{(0)}(t)= \\\\widetilde{c} \\\\cdot \\\\alpha \\\\cdot \\\\Big \\\\lbrace \\\\sqrt{1 + \\\\beta ^{2}\\\\cdot \\\\Big (\\\\frac{1-t}{\\\\alpha }\\\\Big )^2} \\\\, - \\\\, 1+ \\\\log \\\\frac{2 \\\\cdot \\\\Big (\\\\sqrt{1 + \\\\beta ^{2}\\\\cdot \\\\Big (\\\\frac{1-t}{\\\\alpha } \\\\Big )^2} \\\\, - \\\\, 1\\\\Big )}{\\\\beta ^{2} \\\\cdot \\\\Big (\\\\frac{1-t}{\\\\alpha }\\\\Big )^{2}} \\\\Big \\\\rbrace \\\\ \\\\in [0,\\\\infty [, \\\\ \\\\ t \\\\in \\\\, ]-\\\\infty , \\\\infty [ \\\\, = \\\\, ]a_{F}, b_{F}[.$ Notice that $\\\\varphi _{\\\\alpha ,\\\\beta ,\\\\widetilde{c}}(1) = 0$ , $\\\\varphi _{\\\\alpha ,\\\\beta }^{\\\\prime }(1) = 0$ , $\\\\varphi _{\\\\alpha ,\\\\beta ,\\\\widetilde{c}}(-\\\\infty ) = \\\\infty $ and $\\\\varphi _{\\\\alpha ,\\\\beta ,\\\\widetilde{c}}(\\\\infty ) = \\\\infty $ .\",\n \"Moreover, $\\\\varphi _{\\\\alpha ,\\\\beta ,\\\\widetilde{c}}^{\\\\prime }(-\\\\infty ) =\\\\varphi _{\\\\alpha ,\\\\beta ,\\\\widetilde{c}}^{\\\\prime }(a_{F}) =-\\\\widetilde{c} \\\\cdot \\\\beta $ and $\\\\varphi _{\\\\alpha ,\\\\beta ,\\\\widetilde{c}}^{\\\\prime }(\\\\infty ) =\\\\varphi _{\\\\alpha ,\\\\beta ,\\\\widetilde{c}}^{\\\\prime }(b_{F}) =\\\\widetilde{c} \\\\cdot \\\\beta $ .\",\n \"From the generator $\\\\varphi _{\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c}}$ given in (REF ), we construct the corresponding divergence (cf.\",\n \"(REF )) $& & \\\\hspace{-19.91684pt}D_{\\\\varphi _{\\\\alpha ,\\\\beta ,\\\\widetilde{c}}}(\\\\mathbf {Q},\\\\mathbf {P})= \\\\sum \\\\limits _{k=1}^{K} p_{k} \\\\cdot \\\\varphi _{\\\\alpha ,\\\\beta ,\\\\widetilde{c}}\\\\Big (\\\\frac{q_{k}}{p_{k}}\\\\Big )\\\\nonumber \\\\\\\\& & \\\\hspace{-19.91684pt}= \\\\widetilde{c} \\\\cdot \\\\alpha \\\\cdot \\\\sum \\\\limits _{k=1}^{K}p_{k} \\\\cdot \\\\Big \\\\lbrace \\\\sqrt{1 + \\\\beta ^{2}\\\\cdot \\\\Big (\\\\frac{1-\\\\frac{q_{k}}{p_{k}}}{\\\\alpha }\\\\Big )^2} \\\\, - \\\\, 1+ \\\\log \\\\frac{2 \\\\cdot \\\\Big (\\\\sqrt{1 + \\\\beta ^{2}\\\\cdot \\\\Big (\\\\frac{1-\\\\frac{q_{k}}{p_{k}}}{\\\\alpha } \\\\Big )^2} \\\\, - \\\\, 1\\\\Big )}{\\\\beta ^{2} \\\\cdot \\\\Big (\\\\frac{1-\\\\frac{q_{k}}{p_{k}}}{\\\\alpha }\\\\Big )^{2}} \\\\Big \\\\rbrace ,\\\\qquad \\\\mathbf {P} \\\\in \\\\mathbb {R}_{\\\\ge 0}^{K}, \\\\mathbf {Q} \\\\in \\\\mathbb {R}^{K}.$ As above, we can particularly include the case where $p_{k}=0$ in combination with $q_{k} \\\\ne 0$ , since $\\\\lim _{t \\\\rightarrow 0_{+}} t \\\\cdot \\\\varphi _{\\\\alpha ,\\\\beta ,\\\\widetilde{c}}(\\\\frac{1}{t})= \\\\widetilde{c} \\\\cdot \\\\beta $ and $\\\\lim _{t \\\\rightarrow 0_{-}} t \\\\cdot \\\\varphi _{\\\\alpha ,\\\\beta ,\\\\widetilde{c}}(\\\\frac{1}{t})= - \\\\widetilde{c} \\\\cdot \\\\beta $ are both finite.\",\n \"This ends the current Example REF .\",\n \"As a side effect in the above-mentioned Example REF , for fixed $\\\\beta _{2}, \\\\alpha ,\\\\widetilde{c}$ notice the interesting behaviour (e.g.\",\n \"with respect to $int(dom(F)) = ]a_{F},b_{F}[$ and the range of $\\\\varphi ^{\\\\prime }$ ) as $\\\\beta _{1}$ moves from $]0,\\\\beta _{2}[$ to $\\\\beta _{2}$ and further to $]\\\\beta _{2}, \\\\infty [$ .\",\n \"Remark 47 The characterization of the probability distribution $$ in (REF ) which may result from Theorem REF — as seen through the above examples — considerably improves other approaches which make use of their identification through the concept of power variance functions of Natural Exponential Families, as developed by Tweedie [369], Morris [267], Letac & Mora [214], and others.\",\n \"This approach has been used in Broniatowski [58] in a similar perspective as developed here, but can not be extended outside the range of power divergences, in contrast with the Examples REF , REF , REF and REF which can only be handled as a consequence of Theorem REF .\",\n \"To continue with our general procedure, suppose now that for a divergence generator $\\\\varphi $ of interest we have concretely/explicitly found (e.g.\",\n \"by direct calculations or via our $F-$ connection in Theorem REF , see also Remark REF ) its Fenchel-Legendre transform $\\\\Lambda = \\\\varphi ^{*}$ ; for this “candidate”, in order to achieve the desired representability (REF ) it remains to verify that $\\\\exp (\\\\Lambda (z)) = \\\\int _{\\\\mathbb {R}}e^{z \\\\cdot y} \\\\, d\\\\mathbb {} (y),\\\\qquad z \\\\in \\\\mathbb {R},$ for some probability distribution/measure $\\\\mathbb {} $ on the real line (the light-tailedness in the sense of finiteness on some open interval containing zero, will be typically guaranteed automatically by the assumptions on $\\\\varphi $ ); of course, this is equivalent to “the existence ” of a random variable $W$ whose moment generating function is equal to $\\\\exp (\\\\Lambda )$ (and thus, its cumulant generating function (log moment generating function) is $\\\\Lambda $ ), i.e.\",\n \"$\\\\exp (\\\\Lambda (z)) = E_{\\\\mathbb {\\\\Pi }}[\\\\exp (z \\\\cdot W)]\\\\qquad z \\\\in \\\\mathbb {R},$ with $\\\\mathbb {\\\\Pi }[W \\\\in \\\\cdot \\\\, ] = \\\\mathbb {}[ \\\\, \\\\cdot \\\\,]$ ); recall that from this, we need to simulate a sequence $(W_{i})_{i\\\\in \\\\mathbb {N}}$ of i.i.d.\",\n \"copies of $W$ which are the crucial building ingredients of $\\\\xi _{n}^{\\\\mathbf {W}}$ in Theorem REF , respectively, of $\\\\xi _{n,\\\\mathbf {X}}^{w\\\\mathbf {W}}$ in Theorem REF .\",\n \"For the above-mentioned Examples REF to REF , we can give explicit solutions to the representabilities (REF ) respectively (REF ); this is achieved in the following Examples REF to REF (notice that the corresponding supports of $\\\\mathbb {}$ are explicitly mentioned in the summarizing Table 1 above): Example 48 for the power-divergence context of Example REF we obtain: (a) Case $\\\\gamma =0$ , $\\\\widetilde{c} >0$ : $\\\\Lambda _{0,\\\\widetilde{c}}(z)= - \\\\widetilde{c} \\\\cdot \\\\log \\\\left( 1 - \\\\frac{z}{\\\\widetilde{c}} \\\\right)$ (cf.\",\n \"(REF )) is the cumulant generating function of the Gamma distribution $\\\\mathbb {} = GAM(\\\\widetilde{c},\\\\widetilde{c})$ with rate parameter (inverse scale parameter) $\\\\widetilde{c}$ and shape parameter $\\\\widetilde{c}$ ; hence, $\\\\varphi _{0,\\\\widetilde{c}} \\\\in \\\\Upsilon (]0,\\\\infty [)$ .\",\n \"Prominent special case $\\\\widetilde{c} =1$ : $\\\\mathbb {} = GAM(1,1) = EXP(1)$ is the exponential distribution with mean 1.\",\n \"Type: $\\\\mathbb {}$ is an infinitely divisible (cf.\",\n \"Proposition REF ) continuous distribution with density $f(y) := \\\\frac{\\\\widetilde{c}^{\\\\widetilde{c}} \\\\cdot y^{\\\\widetilde{c}-1} \\\\cdot e^{-\\\\widetilde{c} \\\\cdot y} }{\\\\Gamma (\\\\widetilde{c})} \\\\cdot {1}_{]0,\\\\infty [}(y)$ ($y \\\\in \\\\mathbb {R}$ ).\",\n \"Behaviour at zero: $\\\\mathbb {}[ \\\\, ]0,\\\\infty [ \\\\, ] = \\\\mathbb {\\\\Pi }[W>0]=1$ .\",\n \"Corresponding generator: $\\\\varphi _{0,\\\\widetilde{c}} = \\\\widetilde{c} \\\\cdot \\\\varphi _{0}$ (cf.\",\n \"(REF ), (REF )) of the $\\\\widetilde{c}-$ fold of the reversed Kullback-Leibler divergence (reversed relative entropy) given in the second line of (REF ).\",\n \"Sums: for i.i.d.\",\n \"copies $(W_{i})_{i \\\\in \\\\mathbb {N}}$ of $W$ , the probability distribution of $\\\\breve{W} := \\\\sum _{i\\\\in I_{k}^{(n)}} W_{i}$ (cf.\",\n \"Remark REF (ii)) is $GAM(\\\\widetilde{c},\\\\widetilde{c} \\\\cdot card(I_{k}^{(n)}))$ .\",\n \"(b) Case $\\\\gamma \\\\in \\\\, ]0,1[$ , $\\\\widetilde{c} >0$ : $\\\\Lambda _{\\\\gamma ,\\\\widetilde{c}}^{(0)}(z)= \\\\frac{\\\\widetilde{c}}{\\\\gamma } \\\\cdot \\\\Big \\\\lbrace \\\\left( \\\\frac{\\\\gamma -1}{\\\\widetilde{c}} \\\\cdot z + 1 \\\\right)^{\\\\frac{\\\\gamma }{\\\\gamma -1}} -1 \\\\Big \\\\rbrace $ (cf.\",\n \"(REF )) is the cumulant generating function of the Compound-Poisson-Gamma distribution $\\\\mathbb {} = C(POI(\\\\theta ),GAM(\\\\alpha ,\\\\beta ))$ with $\\\\theta = \\\\frac{\\\\widetilde{c}}{\\\\gamma } > 0$ , rate parameter (inverse scale parameter) $\\\\alpha = \\\\frac{\\\\widetilde{c}}{1-\\\\gamma } >0$ , and shape parameter $\\\\beta = \\\\frac{\\\\gamma }{1-\\\\gamma } >0$ .\",\n \"In other words, $W$ has the comfortably simulable form $W = \\\\sum _{i=1}^{N} \\\\widetilde{W}_{i}$ with the usual convention $\\\\sum _{i=1}^{0} \\\\widetilde{W}_{i} := 0$ for some i.i.d.\",\n \"sequence $( \\\\widetilde{W}_{i})_{i\\\\in \\\\mathbb {N}}$ of Gamma $GAM(\\\\alpha ,\\\\beta )$ distributed random variables (with parameter-pair $(\\\\alpha ,\\\\beta )$ ) and some independent $POI(\\\\theta )-$ distributed random variable $N$ .\",\n \"Hence, $\\\\varphi _{\\\\gamma ,\\\\widetilde{c}} \\\\in \\\\Upsilon (]0,\\\\infty [)$ .\",\n \"Type: $\\\\mathbb {}$ is an infinitely divisible distribution (cf.\",\n \"Proposition REF ), mixture of a one-point distribution at zero and a continuous distribution on $[0,\\\\infty [$ , with $\\\\mathbb {}[\\\\lbrace 0\\\\rbrace ] = \\\\mathbb {\\\\Pi }[W = 0]= e^{-\\\\theta }$ and $\\\\mathbb {}[B] = \\\\mathbb {\\\\Pi }[W \\\\in B] = \\\\int _{B} f_{\\\\widetilde{c},\\\\gamma }(u) \\\\, du$ for every (measurable) subset of $]0,\\\\infty [$ having density $& & \\\\hspace{-22.76228pt}f_{C(POI(\\\\theta ),GAM(\\\\alpha ,\\\\beta ))}(y): = \\\\frac{\\\\exp \\\\left(- \\\\alpha \\\\cdot y - \\\\theta \\\\right)}{y}\\\\cdot \\\\sum _{k=1}^{\\\\infty } \\\\frac{\\\\theta ^{k} \\\\cdot (\\\\alpha y)^{k\\\\beta }}{k!\",\n \"\\\\cdot \\\\Gamma (k\\\\beta )}\\\\cdot {1}_{]0,\\\\infty [}(y)\\\\\\\\& & \\\\hspace{-22.76228pt}= \\\\frac{1}{y} \\\\cdot \\\\exp \\\\left(-\\\\widetilde{c} \\\\cdot \\\\left(\\\\frac{y}{1-\\\\gamma } + \\\\frac{1}{\\\\gamma } \\\\right) \\\\right)\\\\cdot \\\\sum _{k=1}^{\\\\infty } \\\\frac{a_{k}}{k!}\",\n \"\\\\cdot \\\\widetilde{c}^{k/(1-\\\\gamma )} \\\\cdot \\\\gamma ^{-k}\\\\cdot (1-\\\\gamma )^{-k\\\\gamma /(1-\\\\gamma )} \\\\cdot y^{k\\\\gamma /(1-\\\\gamma )}\\\\cdot {1}_{[0,\\\\infty [}(y)=: f_{\\\\widetilde{c},\\\\gamma }(y),\\\\quad y \\\\in \\\\mathbb {R},\\\\nonumber $ where $a_{k} := 1/\\\\Gamma (\\\\frac{k \\\\cdot \\\\gamma }{1-\\\\gamma } )$ (see e.g.\",\n \"Aalen [1] with a different parametrization).\",\n \"Behaviour at zero: $\\\\mathbb {}[ \\\\, [0,\\\\infty [ \\\\, ] = \\\\mathbb {\\\\Pi }[W \\\\ge 0]=1$ , $\\\\mathbb {}[ \\\\, \\\\lbrace 0\\\\rbrace \\\\, ] = \\\\mathbb {\\\\Pi }[W = 0]= e^{-\\\\theta }$ .\",\n \"Corresponding generator: $\\\\varphi _{\\\\gamma ,\\\\widetilde{c}}^{(0)} = \\\\widetilde{c} \\\\cdot \\\\varphi _{\\\\gamma }$ (cf.\",\n \"(REF ), (REF )) of the power divergence given in the third line of (REF ); recall that the special case $\\\\gamma =\\\\frac{1}{2}$ corresponds to the prominent (multiple of the squared) Hellinger distance.\",\n \"Sums: for i.i.d.\",\n \"copies $(W_{i})_{i \\\\in \\\\mathbb {N}}$ of $W$ , the probability distribution of $\\\\breve{W} := \\\\sum _{i\\\\in I_{k}^{(n)}} W_{i}$ (cf.\",\n \"Remark REF (ii)) is $C(POI(\\\\breve{\\\\theta }),GAM(\\\\alpha ,\\\\beta ))$ with $\\\\breve{\\\\theta } = \\\\frac{\\\\widetilde{c} \\\\cdot card(I_{k}^{(n)})}{\\\\gamma } > 0$ , $\\\\alpha = \\\\frac{\\\\widetilde{c}}{1-\\\\gamma } >0$ , $\\\\beta = \\\\frac{\\\\gamma }{1-\\\\gamma } >0$ .\",\n \"(c) Case $\\\\gamma =2$ , $\\\\widetilde{c} >0$ : $\\\\Lambda _{2,\\\\widetilde{c}}^{(0)}(z)= \\\\frac{z^{2}}{2 \\\\widetilde{c}} + z$ (cf.\",\n \"(REF ) ) is the well-known cumulant generating function of the Normal distribution (Gaussian distribution) $\\\\mathbb {} = N(1,\\\\frac{1}{\\\\widetilde{c}})$ with mean 1 and variance $\\\\frac{1}{\\\\widetilde{c}}$ .\",\n \"Thus, $\\\\varphi _{2,\\\\widetilde{c}} \\\\in \\\\Upsilon (]-\\\\infty ,\\\\infty [)$ .\",\n \"Type: $\\\\mathbb {}$ is an infinitely divisible (cf.\",\n \"Proposition REF ) continuous distribution with density $f_{N(1,\\\\frac{1}{\\\\widetilde{c}})}(y) := \\\\sqrt{\\\\frac{\\\\widetilde{c}}{2 \\\\pi }} \\\\cdot \\\\exp (- \\\\frac{\\\\widetilde{c} \\\\cdot (y-1)^2}{2} )$ , ($y \\\\in \\\\mathbb {R}$ ).\",\n \"Behaviour at zero: $\\\\mathbb {}[ \\\\, ]0,\\\\infty [ \\\\, ] = \\\\mathbb {\\\\Pi }[W > 0]=\\\\int _{0}^{\\\\infty } f_{N(1,\\\\frac{1}{\\\\widetilde{c}})}(u) \\\\, du \\\\in \\\\, ]0,1[$ , $\\\\mathbb {}[ \\\\, \\\\lbrace 0\\\\rbrace \\\\, ] = \\\\mathbb {\\\\Pi }[W = 0]= 0$ .\",\n \"Corresponding generator: $\\\\varphi _{2,\\\\widetilde{c}}^{(0)} = \\\\widetilde{c} \\\\cdot \\\\varphi _{2}$ (cf.\",\n \"(REF ), (REF )) is the generator of the $\\\\widetilde{c}-$ fold of the half Pearson-chisquare divergence given in the sixth line of (REF ).\",\n \"Sums: for i.i.d.\",\n \"copies $(W_{i})_{i \\\\in \\\\mathbb {N}}$ of $W$ , the probability distribution of $\\\\breve{W} := \\\\sum _{i\\\\in I_{k}^{(n)}} W_{i}$ (cf.\",\n \"Remark REF (ii)) is $N(card(I_{k}^{(n)}),\\\\frac{card(I_{k}^{(n)})}{\\\\widetilde{c}})$ .\",\n \"(d) Case $\\\\gamma <0$ , $\\\\widetilde{c} >0$ : $\\\\Lambda _{\\\\gamma ,\\\\widetilde{c}}^{(0)}(z) = \\\\frac{\\\\widetilde{c}}{\\\\gamma } \\\\cdot \\\\left\\\\lbrace \\\\left( \\\\frac{\\\\gamma -1}{\\\\widetilde{c}} \\\\cdot z +1 \\\\right)^{\\\\frac{\\\\gamma }{\\\\gamma -1}} -1 \\\\right\\\\rbrace $ (cf.\",\n \"(REF )) is the cumulant generating function of a “tilted (i.e.\",\n \"negatively distorted) stable distribution” $\\\\mathbb {}[ \\\\, \\\\cdot \\\\,] = \\\\mathbb {\\\\Pi }[W \\\\in \\\\cdot \\\\, ]$ of a random variable $W$ , which can be constructed as follows: let $Z$ be an auxiliary random variable (having density $f_{Z}$ and support $supp(Z) = [0,\\\\infty [$ ) of a stable law with parameter-quadruple $(\\\\frac{-\\\\gamma }{1-\\\\gamma },1,0,-\\\\frac{\\\\widetilde{c}^{1/(1-\\\\gamma )} \\\\cdot (1-\\\\gamma )^{-\\\\gamma /(1-\\\\gamma )}}{\\\\gamma })$ in terms of the “form-B notation” on p.12 in Zolotarev [428]; by applying a general Laplace-transform result on p.112 of the same text we can deduce $M_{Z}(z) := E_{\\\\mathbb {\\\\Pi }}[\\\\exp (z \\\\cdot Z)]= \\\\int _{0}^{\\\\infty } \\\\exp (z \\\\cdot y) \\\\cdot f_{Z}(y) \\\\, dy \\\\hspace{-5.69046pt} &=& \\\\hspace{-5.69046pt}{\\\\left\\\\lbrace \\\\begin{array}{ll}\\\\exp \\\\Big (\\\\frac{\\\\widetilde{c}^{1/(1-\\\\gamma )} \\\\cdot (1-\\\\gamma )^{-\\\\gamma /(1-\\\\gamma )}}{\\\\gamma }\\\\cdot (-z)^{\\\\alpha } \\\\Big ),\\\\quad \\\\textrm {if } \\\\ z \\\\in ]-\\\\infty ,0] , \\\\\\\\\\\\infty , \\\\hspace{150.79968pt} \\\\textrm {if } \\\\ z \\\\in \\\\, ]0,\\\\infty [, \\\\\\\\\\\\end{array}\\\\right.\",\n \"}$ where $\\\\alpha := - \\\\frac{\\\\gamma }{1-\\\\gamma } \\\\in \\\\, ]0,1[$ .\",\n \"Since $0 \\\\notin int(dom(M_{Z}))$ (and thus, $Z$ does not have light-tails) we have to tilt (dampen) the density in order to extend the effective domain.\",\n \"Accordingly, let $W$ be a random variable having density $f_{W}(y)\\\\ :=\\\\ \\\\frac{\\\\exp \\\\lbrace -\\\\frac{y \\\\cdot \\\\widetilde{c}}{1-\\\\gamma }\\\\rbrace }{\\\\exp \\\\lbrace \\\\widetilde{c}/\\\\gamma \\\\rbrace }\\\\cdot f_{Z}(y)\\\\cdot {1}_{]0,\\\\infty [}(y),\\\\qquad y \\\\in \\\\mathbb {R},\\\\qquad \\\\text{(cf.\",\n \"(\\\\ref {brostu3:fo.norweiemp7a}))}.\\\\nonumber $ Then one can straightforwardly deduce from (REF ) that $\\\\int _{0}^{\\\\infty } f_{W}(y) \\\\, dy =1$ and that $M_{W}(z) := E_{\\\\mathbb {\\\\Pi }}[\\\\exp (z \\\\cdot W)]= \\\\int _{0}^{\\\\infty } \\\\exp (z \\\\cdot y) \\\\cdot f_{W}(y) \\\\, dy \\\\hspace{-5.69046pt} &=& \\\\hspace{-5.69046pt}{\\\\left\\\\lbrace \\\\begin{array}{ll}\\\\exp \\\\left(\\\\frac{\\\\widetilde{c}}{\\\\gamma } \\\\cdot \\\\left\\\\lbrace \\\\left( \\\\frac{\\\\gamma -1}{\\\\widetilde{c}} \\\\cdot z +1 \\\\right)^{\\\\frac{\\\\gamma }{\\\\gamma -1}} -1 \\\\right\\\\rbrace \\\\right),\\\\qquad \\\\textrm {if } \\\\ z \\\\in ]-\\\\infty ,\\\\frac{\\\\widetilde{c}}{1-\\\\gamma }] , \\\\\\\\\\\\infty , \\\\hspace{156.49014pt} \\\\textrm {if } \\\\ z \\\\in \\\\, ]\\\\frac{\\\\widetilde{c}}{1-\\\\gamma },\\\\infty [ .\",\n \"\\\\\\\\\\\\end{array}\\\\right.\",\n \"}\\\\nonumber $ Hence, $\\\\varphi _{\\\\gamma ,\\\\widetilde{c}} \\\\in \\\\Upsilon (]0,\\\\infty [)$ .\",\n \"Type: $\\\\mathbb {}$ is an infinitely divisible (cf.\",\n \"Proposition REF ) continuous distribution with density $f_{W}$ .\",\n \"Behaviour at zero: $\\\\mathbb {}[ \\\\, ] 0,\\\\infty [ \\\\, ] = \\\\mathbb {\\\\Pi }[W > 0]=1$ .\",\n \"Corresponding generator: $\\\\varphi _{\\\\gamma ,\\\\widetilde{c}}^{(0)} = \\\\widetilde{c} \\\\cdot \\\\varphi _{\\\\gamma }$ (cf.\",\n \"(REF ), (REF )) of the power divergence given in the first line of (REF ).\",\n \"Sums: for i.i.d.\",\n \"copies $(W_{i})_{i \\\\in \\\\mathbb {N}}$ of $W$ , the probability distribution of $\\\\breve{W} := \\\\sum _{i\\\\in I_{k}^{(n)}} W_{i}$ (cf.\",\n \"Remark REF (ii)) has density $f_{\\\\breve{W}}(y)\\\\ :=\\\\ \\\\frac{\\\\exp \\\\lbrace -\\\\frac{y \\\\cdot \\\\widetilde{c}}{1-\\\\gamma }\\\\rbrace }{\\\\exp \\\\lbrace \\\\widetilde{c} \\\\cdot card(I_{k}^{(n)})/\\\\gamma \\\\rbrace }\\\\cdot f_{\\\\breve{Z}}(y)\\\\cdot {1}_{]0,\\\\infty [}(y),\\\\qquad y \\\\in \\\\mathbb {R},$ where $\\\\breve{Z}$ is a random variable with density $f_{\\\\breve{Z}}$ of a stable law with parameter-quadruple $(\\\\frac{-\\\\gamma }{1-\\\\gamma },1,0,-card(I_{k}^{(n)}) \\\\cdot \\\\frac{\\\\widetilde{c}^{1/(1-\\\\gamma )} \\\\cdot (1-\\\\gamma )^{-\\\\gamma /(1-\\\\gamma )}}{\\\\gamma })$ .\",\n \"(e) Case $\\\\gamma >2$ , $\\\\widetilde{c} >0$ : $\\\\Lambda _{\\\\gamma ,\\\\widetilde{c}}^{(0)}(z) = \\\\frac{\\\\widetilde{c}}{\\\\gamma } \\\\cdot \\\\left\\\\lbrace \\\\left( \\\\frac{\\\\gamma -1}{\\\\widetilde{c}} \\\\cdot z +1 \\\\right)^{\\\\frac{\\\\gamma }{\\\\gamma -1}} -1 \\\\right\\\\rbrace $ (cf.\",\n \"(REF )) is the cumulant generating function of a “distorted stable distribution” $\\\\mathbb {}[ \\\\, \\\\cdot \\\\,] = \\\\mathbb {\\\\Pi }[W \\\\in \\\\cdot \\\\, ]$ of a random variable $W$ , which can be constructed as follows: let $Z$ be an auxiliary random variable (having density $f_{Z}$ and support $supp(Z) = ]-\\\\infty ,\\\\infty ]$ ) of a stable law with parameter-quadruple $(\\\\frac{\\\\gamma }{\\\\gamma -1},1,0,\\\\frac{\\\\widetilde{c}^{1/(1-\\\\gamma )} \\\\cdot (\\\\gamma -1)^{\\\\gamma /(\\\\gamma -1)}}{\\\\gamma })$ in terms of the above-mentioned “form-B notation” ; by applying a general Laplace-transform result on p. 112 of Zolotarev [428], we can derive $M_{Z}(z) := E_{\\\\mathbb {\\\\Pi }}[\\\\exp (z \\\\cdot Z)]= \\\\int _{0}^{\\\\infty } \\\\exp (z \\\\cdot y) \\\\cdot f_{Z}(y) \\\\, dy \\\\hspace{-5.69046pt} &=& \\\\hspace{-5.69046pt}{\\\\left\\\\lbrace \\\\begin{array}{ll}\\\\exp \\\\Big (\\\\frac{\\\\widetilde{c}^{1/(1-\\\\gamma )} \\\\cdot (\\\\gamma -1 )^{\\\\gamma /(\\\\gamma -1)}}{\\\\gamma }\\\\cdot (-z)^{\\\\alpha } \\\\Big ),\\\\quad \\\\textrm {if } \\\\ z \\\\in ]-\\\\infty ,0] , \\\\\\\\\\\\infty , \\\\hspace{145.10922pt} \\\\textrm {if } \\\\ z \\\\in \\\\, ]0,\\\\infty [, \\\\\\\\\\\\end{array}\\\\right.\",\n \"}$ where $\\\\alpha := \\\\frac{\\\\gamma }{\\\\gamma -1} \\\\in \\\\, ]1,2[$ .\",\n \"Since $0 \\\\notin int(dom(M_{Z}))$ (and thus, $Z$ does not have light-tails) we have to distort the density in order to extend the effective domain.\",\n \"Accordingly, let $W$ be a random variable having density $f_{W}(y)\\\\ :=\\\\ \\\\frac{\\\\exp \\\\lbrace \\\\frac{y \\\\cdot \\\\widetilde{c}}{\\\\gamma -1}\\\\rbrace }{\\\\exp \\\\lbrace \\\\widetilde{c}/\\\\gamma \\\\rbrace }\\\\cdot f_{Z}(- y), \\\\qquad y \\\\in \\\\mathbb {R},\\\\qquad \\\\text{(cf.\",\n \"(\\\\ref {brostu3:fo.norweiemp7atwo}))}.\\\\nonumber $ Then one can straightforwardly deduce from (REF ) that $\\\\int _{-\\\\infty }^{\\\\infty } f_{W}(y) \\\\, dy =1$ and that $M_{W}(z) := E_{\\\\mathbb {\\\\Pi }}[\\\\exp (z \\\\cdot W)]= \\\\int _{-\\\\infty }^{\\\\infty } \\\\exp (z \\\\cdot y) \\\\cdot f_{W}(y) \\\\, dy \\\\hspace{-5.69046pt} &=& \\\\hspace{-5.69046pt}{\\\\left\\\\lbrace \\\\begin{array}{ll}\\\\exp \\\\left(\\\\frac{\\\\widetilde{c}}{\\\\gamma } \\\\cdot \\\\left\\\\lbrace \\\\left( \\\\frac{\\\\gamma -1}{\\\\widetilde{c}} \\\\cdot z +1 \\\\right)^{\\\\frac{\\\\gamma }{\\\\gamma -1}} -1 \\\\right\\\\rbrace \\\\right),\\\\qquad \\\\textrm {if } \\\\ z \\\\in [- \\\\frac{\\\\widetilde{c}}{\\\\gamma -1},\\\\infty [ , \\\\\\\\\\\\infty , \\\\hspace{156.49014pt} \\\\textrm {if } \\\\ z \\\\in \\\\, ]-\\\\infty , -\\\\frac{\\\\widetilde{c}}{\\\\gamma -1}[ .\",\n \"\\\\\\\\\\\\end{array}\\\\right.\",\n \"}\\\\nonumber $ Thus, $\\\\varphi _{\\\\gamma ,\\\\widetilde{c}} \\\\in \\\\Upsilon (]-\\\\infty ,\\\\infty [)$ .\",\n \"Type: $\\\\mathbb {}$ is an infinitely divisible (cf.\",\n \"Proposition REF ) continuous distribution with density $f_{W}$ .\",\n \"Behaviour at zero: $\\\\mathbb {}[ \\\\, ]0,\\\\infty [ \\\\, ] = \\\\mathbb {\\\\Pi }[W > 0]=\\\\int _{0}^{\\\\infty } f_{W}(u) \\\\, du \\\\in \\\\, ]0,1[$ , $\\\\mathbb {}[ \\\\, \\\\lbrace 0\\\\rbrace \\\\, ] = \\\\mathbb {\\\\Pi }[W = 0]= 0$ .\",\n \"Corresponding generator: $\\\\varphi _{\\\\gamma ,\\\\widetilde{c}}^{(0)} = \\\\widetilde{c} \\\\cdot \\\\varphi _{\\\\gamma }$ (cf.\",\n \"(REF ), (REF )) of the power divergence given in the seventh line of (REF ).\",\n \"Sums: for i.i.d.\",\n \"copies $(W_{i})_{i \\\\in \\\\mathbb {N}}$ of $W$ , the probability distribution of $\\\\breve{W} := \\\\sum _{i\\\\in I_{k}^{(n)}} W_{i}$ (cf.\",\n \"Remark REF (ii)) has density $f_{\\\\breve{W}}(y)\\\\ :=\\\\ \\\\frac{\\\\exp \\\\lbrace \\\\frac{y \\\\cdot \\\\widetilde{c}}{\\\\gamma -1}\\\\rbrace }{\\\\exp \\\\lbrace \\\\widetilde{c} \\\\cdot card(I_{k}^{(n)})/\\\\gamma \\\\rbrace }\\\\cdot f_{\\\\breve{Z}}(- y), \\\\qquad y \\\\in \\\\mathbb {R},\\\\nonumber $ where $\\\\breve{Z}$ is a random variable with density $f_{\\\\breve{Z}}$ of a stable law with parameter-quadruple $(\\\\frac{\\\\gamma }{\\\\gamma -1},1,0,card(I_{k}^{(n)})\\\\cdot \\\\frac{\\\\widetilde{c}^{1/(1-\\\\gamma )} \\\\cdot (\\\\gamma -1)^{\\\\gamma /(\\\\gamma -1)}}{\\\\gamma })$ .\",\n \"(f) Case $\\\\gamma \\\\in ]1,2[$ , $\\\\widetilde{c} >0$ : one still has the (cumulant-generating-function) candidate $\\\\Lambda _{\\\\gamma ,\\\\widetilde{c}}^{(0)}(z) = \\\\frac{\\\\widetilde{c}}{\\\\gamma } \\\\cdot \\\\left\\\\lbrace \\\\left( \\\\frac{\\\\gamma -1}{\\\\widetilde{c}} \\\\cdot z +1 \\\\right)^{\\\\frac{\\\\gamma }{\\\\gamma -1}} -1 \\\\right\\\\rbrace $ (cf.\",\n \"(REF )), but for the crucial exponent there holds $\\\\frac{\\\\gamma }{\\\\gamma -1} > 2$ .\",\n \"From this, we conjecture that $\\\\mathbb {}$ becomes a signed finite measure with total mass 1, i.e.\",\n \"it has a density (with respect to some dominating measure) with positive and negative values which “integrates to 1” ; accordingly, our BS method can not be applied to this situation.\",\n \"Remark 49 As a continuation of Remark REF and the note in the third line after (REF ), we have shown as a side effect that for $\\\\gamma \\\\in \\\\, ]-\\\\infty ,-1] \\\\, \\\\cup \\\\, ]0,1[ \\\\, \\\\cup \\\\, [2,\\\\infty [$ the distributions $\\\\mathbb {}_{\\\\gamma }$ and $\\\\mathbb {}_{1-\\\\gamma }$ of Example REF (b)-(e) are inverse to each other.\",\n \"Example 50 for the power-divergence context of Example REF we obtain: (a) Case $\\\\gamma =1$ , $\\\\widetilde{c} >0$ , anchor point $c=0$ : $\\\\Lambda _{1,\\\\widetilde{c}}(z) =\\\\widetilde{c} \\\\cdot \\\\left( \\\\exp (\\\\frac{z}{\\\\widetilde{c}}) - 1 \\\\right)$ (cf.\",\n \"(REF )) is the cumulant generating function of $\\\\mathbb {} = \\\\frac{1}{\\\\widetilde{c}} \\\\cdot POI(\\\\widetilde{c})$ being the “$\\\\frac{1}{\\\\widetilde{c}}-$ fold Poisson distribution with mean $\\\\widetilde{c}$ ” which means that $W = \\\\frac{1}{\\\\widetilde{c}} \\\\cdot Z$ for a $POI(\\\\widetilde{c})-$ distributed random variable $Z$ .\",\n \"Thus, $\\\\varphi _{1,\\\\widetilde{c}} \\\\in \\\\Upsilon (]0,\\\\infty [)$ .\",\n \"Prominent special case $\\\\widetilde{c} =1$ : $\\\\mathbb {} = POI(1)$ is the Poisson distribution with mean 1.\",\n \"Type: $\\\\mathbb {}$ is an infinitely divisible (cf.\",\n \"Proposition REF ) discrete distribution with frequencies: $\\\\mathbb {\\\\Pi }[W=\\\\ell \\\\cdot \\\\frac{1}{\\\\widetilde{c}}]= \\\\exp (-\\\\widetilde{c}) \\\\cdot \\\\frac{ \\\\widetilde{c}^{\\\\ell }}{\\\\ell !\",\n \"}$ for all nonnegative integers $\\\\ell \\\\in \\\\mathbb {N}_{0}$ (and zero elsewhere).\",\n \"Behaviour at zero: $\\\\mathbb {\\\\Pi }[W\\\\ge 0]=1$ , $\\\\mathbb {\\\\Pi }[W=0]= \\\\exp (-\\\\widetilde{c})$ .\",\n \"Corresponding generator: $\\\\varphi _{1,\\\\widetilde{c}} = \\\\widetilde{c} \\\\cdot \\\\varphi _{1}$ (cf.\",\n \"(REF ), (REF )) of the $\\\\widetilde{c}-$ fold of the Kullback-Leibler divergence (relative entropy) given in the fourth line of (REF ).\",\n \"Sums: for i.i.d.\",\n \"copies $(W_{i})_{i \\\\in \\\\mathbb {N}}$ of $W$ , the probability distribution of $\\\\breve{W} := \\\\sum _{i\\\\in I_{k}^{(n)}} W_{i}$ (cf.\",\n \"Remark REF (ii)) is $\\\\frac{1}{\\\\widetilde{c}} \\\\cdot POI(\\\\widetilde{c} \\\\cdot card(I_{k}^{(n)}))$ .\",\n \"(b) Case $\\\\gamma =1$ , $\\\\widetilde{c} =1$ , anchor point $c \\\\in \\\\mathbb {R}$ : $\\\\Lambda _{1,1}^{(c)}(z)= e^{c} \\\\cdot \\\\left( e^{z} - 1 \\\\right) + z \\\\cdot (1- e^{c})$ (cf.\",\n \"(REF )) is the well-known cumulant generating function of the “shifted Poisson distribution” $\\\\mathbb {} = POI(e^{c}) + 1-e^{c}$ , i.e.\",\n \"$W := Z + 1-e^{c}$ with a $POI(e^{c})-$ distributed random variable $Z$ .\",\n \"Hence, $\\\\varphi _{1,1}^{(c)} \\\\in \\\\Upsilon (]1- e^{c},\\\\infty [)$ .\",\n \"Type: $\\\\mathbb {}$ is a discrete distribution with frequencies: $\\\\mathbb {\\\\Pi }[W=\\\\ell + 1- e^{c}]= \\\\exp (-e^{c}) \\\\cdot \\\\frac{ e^{c \\\\cdot \\\\ell }}{\\\\ell !\",\n \"}$ for all $\\\\ell \\\\in \\\\mathbb {N}_{0}$ (and zero elsewhere).\",\n \"Behaviour at zero: $\\\\mathbb {\\\\Pi }[W > 0]=1$ iff $c <0$ , $\\\\mathbb {\\\\Pi }[W < 0] >0$ iff $c >0$ , $\\\\mathbb {\\\\Pi }[W=0] \\\\ne 0$ iff “$c =\\\\log (1+k)$ for some $k \\\\in \\\\mathbb {N}_{0}$ ” .\",\n \"Corresponding generator: $\\\\varphi _{1,1}^{(c)}$ (cf.\",\n \"(REF )) of the divergence $& & D_{\\\\varphi _{1,1}^{(c)}}(\\\\mathbf {Q},\\\\mathbf {P}) :=\\\\sum \\\\limits _{k=1}^{K} \\\\Big (q_{k} + p_{k} \\\\cdot (e^{c}-1) \\\\Big )\\\\cdot \\\\Big \\\\lbrace \\\\log \\\\Big (\\\\frac{q_{k}}{p_{k}} + e^{c}-1 \\\\Big ) - c \\\\Big \\\\rbrace - \\\\sum \\\\limits _{k=1}^{K} q_{k} + \\\\sum \\\\limits _{k=1}^{K} p_{k} ,\\\\nonumber \\\\\\\\& & \\\\hspace{113.81102pt} \\\\textrm {if } \\\\ \\\\mathbf {P} \\\\in \\\\mathbb {R}_{\\\\ne 0}^{K} \\\\ \\\\textrm {and } \\\\mathbf {Q} \\\\in \\\\mathbb {R}^{K}\\\\textrm {with } \\\\mathbf {Q} \\\\in \\\\, [ (1-e^{c}) \\\\cdot \\\\mathbf {P}, \\\\mathbf {\\\\infty } [\\\\ \\\\textrm {component-wise},$ which for $c=0$ coincides with the Kullback-Leibler divergence (relative entropy) given in the fourth line of (REF ).\",\n \"Sums: for i.i.d.\",\n \"copies $(W_{i})_{i \\\\in \\\\mathbb {N}}$ of $W$ , the probability distribution of $\\\\breve{W} := \\\\sum _{i\\\\in I_{k}^{(n)}} W_{i}$ (cf.\",\n \"Remark REF (ii)) is $POI(card(I_{k}^{(n)}) \\\\cdot e^{c}) + (1-e^{c}) \\\\cdot card(I_{k}^{(n)})$ .\",\n \"Remark 51 (a) One can see from the Examples REF and REF the interesting effect that the “homogeneous” class of power-divergence generators $(\\\\varphi _{\\\\gamma })_{\\\\gamma \\\\in \\\\mathbb {R}}$ are connected to a “very inhomogeneous” family $(\\\\mathbb {}_{\\\\gamma })_{\\\\gamma \\\\in \\\\mathbb {R}}$ of $W-$ distributions: discrete, continuous, mixture of discrete and continuous, as the parameter $\\\\gamma $ varies.\",\n \"Moreover, some cases satisfy $\\\\mathbb {\\\\Pi }[W=0] =0$ and some $\\\\mathbb {\\\\Pi }[W=0] >0$ , some $\\\\mathbb {\\\\Pi }[W>0]=1$ and some $\\\\mathbb {\\\\Pi }[W>0] \\\\in ]0,1[$ .\",\n \"(b) As a continuation of Remark REF and the note in the last line of Example REF (a), we have shown as a side effect that for the the natural-anchor-point choice $c=0$ , the distributions $\\\\mathbb {}_{1}$ of of Example REF (a) and $\\\\mathbb {}_{0}$ of Example REF (a) are inverse to each other.\",\n \"Example 52 for the context of Example REF we obtain: Case $\\\\widetilde{c} >0$ , anchor point $c=0$ : $\\\\Lambda _{bw,\\\\beta ,\\\\widetilde{c}}(z) =-(\\\\frac{1}{\\\\beta }-1) \\\\cdot z + \\\\frac{\\\\widetilde{c}}{\\\\beta ^{2}} \\\\cdot \\\\Big \\\\lbrace 1 - \\\\sqrt{1-\\\\frac{2\\\\beta }{\\\\widetilde{c}} \\\\cdot z} \\\\ \\\\Big \\\\rbrace $ (cf.\",\n \"(REF )) is the cumulant generating function of a probability distribution $\\\\mathbb {}[ \\\\, \\\\cdot \\\\,] = \\\\mathbb {\\\\Pi }[\\\\check{W} \\\\in \\\\cdot \\\\, ]$ of a random variable $\\\\check{W}$ , which can be constructed as follows: $\\\\check{W} := \\\\frac{W}{\\\\beta } - (\\\\frac{1}{\\\\beta } - 1)$ , where $W$ is the random variable constructed in Example REF (d) with $\\\\gamma =-1$ and with $\\\\widetilde{c}$ replaced by $\\\\frac{\\\\widetilde{c}}{\\\\beta ^{2}}$ (recall that $W$ has a tilted stable distribution).\",\n \"In other words, $\\\\mathbb {}$ is a special kind of modified tilted stable distribution.\",\n \"Type: $\\\\mathbb {}$ is an infinitely divisible (cf.\",\n \"Proposition REF ) continuous distribution with density $f_{\\\\check{W}}(u) := \\\\beta \\\\cdot f_{W}(\\\\beta \\\\cdot u + 1 -\\\\beta )\\\\cdot \\\\mathbf {1}_{]- (\\\\frac{1}{\\\\beta } -1),\\\\infty [}(u)$ ($u \\\\in \\\\mathbb {R}$ ), where $f_{W}(\\\\cdot )$ is given in (REF ) with $\\\\gamma =-1$ and with $\\\\widetilde{c}$ replaced by $\\\\frac{\\\\widetilde{c}}{\\\\beta ^{2}}$ .\",\n \"Behaviour at zero: $\\\\mathbb {}[ \\\\, ] 0,\\\\infty [ \\\\, ] =\\\\mathbb {\\\\Pi }[\\\\check{W} > 0] >0$ .\",\n \"Corresponding generator: $\\\\varphi _{bw,\\\\beta ,\\\\widetilde{c}}^{(0)}$ (cf.\",\n \"(REF )) of the — “non-probability version” of — the well-known blended weight chi-square divergence given in (REF ).\",\n \"Sums: for i.i.d.\",\n \"copies $(\\\\check{W}_{i})_{i \\\\in \\\\mathbb {N}}$ of $\\\\check{W}$ , the probability distribution of $\\\\breve{\\\\check{W}} := \\\\sum _{i\\\\in I_{k}^{(n)}} \\\\check{W}_{i}= \\\\frac{1}{\\\\beta } \\\\cdot \\\\sum _{i\\\\in I_{k}^{(n)}} W_{i} - n_{k}\\\\cdot (\\\\frac{1}{\\\\beta } -1)$ (cf.\",\n \"Remark REF (ii)) has density $f_{\\\\breve{\\\\check{W}}}(u) := \\\\beta \\\\cdot f_{\\\\breve{W}}(\\\\beta \\\\cdot u + (1 -\\\\beta )\\\\cdot n_{k})\\\\cdot \\\\mathbf {1}_{]- n_{k} \\\\cdot (\\\\frac{1}{\\\\beta } -1),\\\\infty [}(u)$ ($u \\\\in \\\\mathbb {R}$ ), where $f_{\\\\breve{W}}(\\\\cdot )$ is given in (REF ) (cf.\",\n \"Example REF (d)) with $\\\\gamma =-1$ and with $\\\\widetilde{c}$ replaced by $\\\\frac{\\\\widetilde{c}}{\\\\beta ^{2}}$ .\",\n \"Example 53 for the context of Example REF we obtain: (a) Case $\\\\alpha \\\\in \\\\, ]0,\\\\infty [$ , $\\\\widetilde{c} >0$ , anchor point $c=0$ : $\\\\Lambda _{gKL,\\\\alpha ,\\\\widetilde{c}}(z) =- \\\\frac{\\\\widetilde{c}}{\\\\alpha } \\\\cdot \\\\log ((1+\\\\alpha ) - \\\\alpha \\\\cdot e^{z/\\\\widetilde{c}})$ (cf.\",\n \"(REF )) is the cumulant generating function of $\\\\mathbb {} = \\\\frac{1}{\\\\widetilde{c}} \\\\cdot NB(\\\\frac{\\\\widetilde{c}}{\\\\alpha },\\\\frac{1}{1+\\\\alpha })$ being the “$\\\\frac{1}{\\\\widetilde{c}}-$ fold Negative-Binomial distribution with parameters $\\\\frac{\\\\widetilde{c}}{\\\\alpha }$ and $\\\\frac{1}{1+\\\\alpha }$ ” which means that $W = \\\\frac{1}{\\\\widetilde{c}} \\\\cdot Z$ for a $NB(\\\\frac{\\\\widetilde{c}}{\\\\alpha },\\\\frac{1}{1+\\\\alpha })-$ distributed random variable $Z$ .\",\n \"Thus, $\\\\varphi _{gKL,\\\\alpha ,\\\\widetilde{c}} \\\\in \\\\Upsilon (]0,\\\\infty [)$ .\",\n \"Prominent special case $\\\\widetilde{c}=1$ , $\\\\alpha =1$ (see below): $\\\\mathbb {} = NB(1,\\\\frac{1}{2})$ is the Negative-Binomial distribution with parameters 1 and $\\\\frac{1}{2}$ .\",\n \"Type: $\\\\mathbb {}$ is an infinitely divisible (cf.\",\n \"Proposition REF ) discrete distribution with frequencies: $\\\\mathbb {\\\\Pi }[W=\\\\ell \\\\cdot \\\\frac{1}{\\\\widetilde{c}}]=(-1)^{\\\\ell } \\\\cdot \\\\binom{- \\\\frac{\\\\widetilde{c}}{\\\\alpha }}{\\\\ell }\\\\cdot {\\\\alpha }^{\\\\ell } \\\\cdot (1+\\\\alpha )^{-\\\\ell - \\\\widetilde{c}/\\\\alpha }$ for all nonnegative integers $\\\\ell \\\\in \\\\mathbb {N}_{0}$ (and zero elsewhere).\",\n \"Behaviour at zero: $\\\\mathbb {\\\\Pi }[W\\\\ge 0]=1$ , $\\\\mathbb {\\\\Pi }[W=0]= \\\\frac{1}{(1+a)^{\\\\widetilde{c}/\\\\alpha }}$ .\",\n \"Corresponding generator: $\\\\varphi _{gKL,\\\\alpha ,\\\\widetilde{c}}$ (cf.\",\n \"(REF )) of the divergence (REF ); the special case $\\\\widetilde{c}=1$ , $\\\\alpha =1$ — i.e.\",\n \"$\\\\varphi _{gKL,1,1} =: \\\\varphi _{snKL,1}$ (cf.\",\n \"(REF )) — corresponds to the generator of the — “non-probability version” of the — Jensen-Shannon divergence (symmetrized and normalized Kullback-Leibler divergence, symmetrized and normalized relative entropy) given in (REF ).\",\n \"Sums: for i.i.d.\",\n \"copies $(W_{i})_{i \\\\in \\\\mathbb {N}}$ of $W$ , the probability distribution of $\\\\breve{W} := \\\\sum _{i\\\\in I_{k}^{(n)}} W_{i}$ (cf.\",\n \"Remark REF (ii)) is $\\\\frac{1}{\\\\widetilde{c}} \\\\cdot NB(\\\\frac{\\\\widetilde{c}}{\\\\alpha }\\\\cdot card(I_{k}^{(n)}),\\\\frac{1}{1+\\\\alpha })$ .\",\n \"(b) Case $\\\\alpha \\\\in \\\\, ]-1,0[$ , $\\\\widetilde{c} >0$ , anchor point $c=0$ : for any integer $m \\\\in \\\\mathbb {N}$ being strictly larger than $\\\\widetilde{c}$ and the choice $\\\\alpha = - \\\\frac{\\\\widetilde{c}}{m}$ , we obtain $\\\\Lambda _{gKL,-\\\\widetilde{c}/m,\\\\widetilde{c}}(z) =m \\\\cdot \\\\log ((1 - \\\\frac{\\\\widetilde{c}}{m}) + \\\\frac{\\\\widetilde{c}}{m} \\\\cdot e^{z/\\\\widetilde{c}})$ (cf.\",\n \"(REF )) which is the cumulant generating function of $\\\\mathbb {} = \\\\frac{1}{\\\\widetilde{c}} \\\\cdot BIN(m,\\\\frac{\\\\widetilde{c}}{m})$ being the “$\\\\frac{1}{\\\\widetilde{c}}-$ fold Binomial distribution with parameters $m$ and $\\\\frac{\\\\widetilde{c}}{m}$ ” which means that $W = \\\\frac{1}{\\\\widetilde{c}} \\\\cdot Z$ for a $BIN(m,\\\\frac{\\\\widetilde{c}}{m})-$ distributed random variable $Z$ .\",\n \"Thus, $\\\\varphi _{gKL,-\\\\widetilde{c}/m,\\\\widetilde{c}} \\\\in \\\\Upsilon (]0,\\\\infty [)$ .\",\n \"Type: $\\\\mathbb {}$ is a non-infinitely divisible discrete distribution with frequencies: $\\\\mathbb {\\\\Pi }[W=\\\\ell \\\\cdot \\\\frac{1}{\\\\widetilde{c}}]=\\\\binom{m}{\\\\ell }\\\\cdot (\\\\frac{\\\\widetilde{c}}{m})^{\\\\ell } \\\\cdot (1-\\\\frac{\\\\widetilde{c}}{m})^{m-\\\\ell }$ for $\\\\ell \\\\in \\\\lbrace 0,1, \\\\ldots , m\\\\rbrace $ (and zero elsewhere).\",\n \"Behaviour at zero: $\\\\mathbb {\\\\Pi }[W\\\\ge 0]=1$ , $\\\\mathbb {\\\\Pi }[W=0]= (1-\\\\frac{\\\\widetilde{c}}{m})^{m}$ .\",\n \"Corresponding generator: $\\\\varphi _{gKL,\\\\alpha ,\\\\widetilde{c}}$ (cf.\",\n \"(REF )) of the divergence (REF ).\",\n \"Sums: for i.i.d.\",\n \"copies $(W_{i})_{i \\\\in \\\\mathbb {N}}$ of $W$ , the probability distribution of $\\\\breve{W} := \\\\sum _{i\\\\in I_{k}^{(n)}} W_{i}$ (cf.\",\n \"Remark REF (ii)) is $\\\\frac{1}{\\\\widetilde{c}} \\\\cdot BIN(m \\\\cdot card(I_{k}^{(n)}),\\\\frac{\\\\widetilde{c}}{m})$ .\",\n \"Example 54 for the context of Example REF we obtain: Case of anchor point $c=0$ : $\\\\Lambda _{twop}(z)= \\\\log \\\\Big ( p \\\\cdot e^{z_{1} \\\\cdot z} + (1- p) \\\\cdot e^{z_{2} \\\\cdot z} \\\\Big )$ (cf.\",\n \"(REF )) is the well-known cumulant generating function of the two-point probability distribution $\\\\mathbb {} = p \\\\cdot \\\\delta _{z_{1}} + (1-p) \\\\cdot \\\\delta _{z_{2}}$ , where $z_{1} < 1 < z_{2}$ and $p= \\\\frac{z_{2} -1}{z_{2} - z_{1}}$ .\",\n \"Hence, $\\\\varphi _{twop} \\\\in \\\\Upsilon (]z_{1},z_{2}[)$ .\",\n \"Type: $\\\\mathbb {}$ is a discrete distribution with frequencies: $\\\\mathbb {\\\\Pi }[W=z_{1}] = p$ , $\\\\mathbb {\\\\Pi }[W=z_{2}] = 1-p$ (and zero elsewhere).\",\n \"Behaviour at zero: $\\\\mathbb {\\\\Pi }[W > 0]=1$ iff $z_{1} >0$ , $\\\\mathbb {\\\\Pi }[W=0] \\\\ne 0$ iff $z_{1}=0$ .\",\n \"Corresponding generator: $\\\\varphi _{twop}$ (cf.\",\n \"(REF )) of the divergence given in (REF ).\",\n \"Sums: for i.i.d.\",\n \"copies $(W_{i})_{i \\\\in \\\\mathbb {N}}$ of $W$ , the probability distribution of $\\\\breve{W} := \\\\sum _{i\\\\in I_{k}^{(n)}} W_{i}$ (cf.\",\n \"Remark REF (ii)) is the distribution of the $card(I_{k}^{(n)})-$ th step of a generalized random walk starting at zero; this has a nice explicit (“binomial-type” ) expression in the special case $z_{1} = - z_{2}$ , namely $\\\\sum _{\\\\ell =0}^{card(I_{k}^{(n)})} \\\\binom{card(I_{k}^{(n)})}{\\\\ell }\\\\cdot p^{card(I_{k}^{(n)}) - \\\\ell } \\\\cdot (1-p)^{\\\\ell } \\\\cdot \\\\delta _{z_{2}\\\\cdot (2 \\\\ell - card(I_{k}^{(n)}))}$ .\",\n \"Example 55 for the context of Example REF we obtain: Case $\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c} \\\\in \\\\, ]0,\\\\infty [$ , anchor point $c=0$ : by using $\\\\breve{\\\\theta } := 1 + \\\\alpha \\\\cdot \\\\Big (\\\\frac{1}{\\\\beta _{2}}- \\\\frac{1}{\\\\beta _{1}} \\\\Big )$ one can see that $\\\\Lambda _{\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c}}(z) =\\\\breve{\\\\theta } \\\\cdot z - \\\\widetilde{c} \\\\cdot \\\\alpha \\\\cdot \\\\log \\\\Big (1+ \\\\frac{z}{\\\\widetilde{c}} \\\\cdot \\\\Big (\\\\frac{1}{\\\\beta _{2}} - \\\\frac{1}{\\\\beta _{1}} \\\\Big )- \\\\frac{z^{2}}{\\\\widetilde{c}^{2} \\\\cdot \\\\beta _{1} \\\\cdot \\\\beta _{2} }\\\\Big )$ for $z \\\\in \\\\, ]- \\\\widetilde{c} \\\\cdot \\\\min \\\\lbrace \\\\beta _{1}, \\\\beta _{2}\\\\rbrace ,\\\\widetilde{c} \\\\cdot \\\\min \\\\lbrace \\\\beta _{1}, \\\\beta _{2}\\\\rbrace [$ — with different boundary behaviour for the three subcases $\\\\beta _{1} < \\\\beta _{2}$ resp.\",\n \"$\\\\beta _{1} > \\\\beta _{2}$ resp.\",\n \"$\\\\beta _{1} = \\\\beta _{2}$ (cf.\",\n \"(REF ),(REF ),(REF )) — is the cumulant generating function of a generalized asymmetric Laplace distribution $\\\\mathbb {}[ \\\\, \\\\cdot \\\\,] = \\\\mathbb {\\\\Pi }[W \\\\in \\\\cdot \\\\, ]$ of a random variable $W:= \\\\breve{\\\\theta } + Z_{1} - Z_{2}$ , where $Z_{1}$ respectively $Z_{2}$ are auxiliary random variables which are independent and $GAM(\\\\widetilde{c} \\\\cdot \\\\beta _{1},\\\\widetilde{c} \\\\cdot \\\\alpha )-$ distributed respectively $GAM(\\\\widetilde{c} \\\\cdot \\\\beta _{2},\\\\widetilde{c} \\\\cdot \\\\alpha )-$ distributed.\",\n \"In particular, $E_{\\\\mathbb {\\\\Pi }}[W] = \\\\breve{\\\\theta } + \\\\frac{\\\\widetilde{c} \\\\cdot \\\\alpha }{\\\\widetilde{c} \\\\cdot \\\\beta _{1}}- \\\\frac{\\\\widetilde{c} \\\\cdot \\\\alpha }{\\\\widetilde{c} \\\\cdot \\\\beta _{2}} =1$ (as required).\",\n \"Thus, $\\\\varphi _{\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c}}\\\\in \\\\Upsilon (]-\\\\infty ,\\\\infty [)$ .\",\n \"Prominent special case $\\\\widetilde{c} =1$ , $\\\\alpha =1$ , $\\\\beta _{1}=\\\\beta _{2} =: \\\\beta $ (and hence, $\\\\breve{\\\\theta }=1$ ): $\\\\mathbb {}$ is a classical Laplace distribution (two-tailed exponential distribution, bilateral exponential law) with location parameter 1 and scale parameter $\\\\frac{1}{\\\\beta }$ .\",\n \"Type: $\\\\mathbb {}$ is an infinitely divisible (cf.\",\n \"Proposition REF ) continuous distribution with density $f(u) := \\\\frac{\\\\sqrt{2} \\\\cdot \\\\exp \\\\lbrace \\\\frac{1}{\\\\sigma \\\\cdot \\\\sqrt{2}} \\\\cdot (\\\\frac{1}{\\\\kappa } - \\\\kappa ) \\\\cdot (u-\\\\theta ) \\\\rbrace }{\\\\sqrt{\\\\pi } \\\\cdot \\\\sigma ^{\\\\tau + 1/2} \\\\cdot \\\\Gamma (\\\\tau )} \\\\cdot \\\\left( \\\\frac{\\\\sqrt{2} \\\\cdot |u - \\\\theta |}{\\\\kappa + \\\\frac{1}{\\\\kappa }} \\\\right)^{\\\\tau - 1/2}\\\\cdot K_{\\\\tau - 1/2}\\\\left( \\\\frac{1}{\\\\sigma \\\\cdot \\\\sqrt{2}} \\\\cdot \\\\Big (\\\\kappa + \\\\frac{1}{\\\\kappa }\\\\Big ) \\\\cdot |u - \\\\theta | \\\\right),\\\\quad u \\\\ne \\\\theta ,$ where $(\\\\theta ,\\\\kappa ,\\\\sigma ,\\\\tau )$ is given in Remark REF below and $K_{\\\\lambda }$ is the modified Bessel function of the third kind with index $\\\\lambda $ .\",\n \"For the above-mentioned special case of the classical Laplace distribution, this considerably simplifies to $f(u):= \\\\frac{\\\\beta }{2} \\\\exp \\\\lbrace - \\\\beta \\\\cdot |u -1 | \\\\rbrace $ .\",\n \"Behaviour at zero: $\\\\mathbb {}[ \\\\, ]0,\\\\infty [ \\\\, ] = \\\\mathbb {\\\\Pi }[W > 0]=\\\\int _{0}^{\\\\infty } f(u) \\\\, du \\\\in \\\\, ]0,1[$ , $\\\\mathbb {}[ \\\\, \\\\lbrace 0\\\\rbrace \\\\, ] = \\\\mathbb {\\\\Pi }[W = 0]= 0$ .\",\n \"Corresponding generator: $\\\\varphi _{\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c}}$ (cf.\",\n \"(REF ) respectively (REF ) respectively (REF )) of the divergence given in (REF ) respectively (REF ) respectively (REF ).\",\n \"Sums: for i.i.d.\",\n \"copies $(W_{i})_{i \\\\in \\\\mathbb {N}}$ of $W$ , the probability distribution of $\\\\breve{W} := \\\\sum _{i\\\\in I_{k}^{(n)}} W_{i}$ (cf.\",\n \"Remark REF (ii)) is the same as that of a random variable $\\\\breve{\\\\widetilde{W}} := \\\\breve{\\\\theta } \\\\cdot card(I_{k}^{(n)}) +\\\\breve{Z}_{1} - \\\\breve{Z}_{2}$ , where $\\\\breve{Z}_{1}$ respectively $\\\\breve{Z}_{2}$ are auxiliary random variables which are independent and $GAM(\\\\widetilde{c} \\\\cdot \\\\beta _{1},\\\\widetilde{c} \\\\cdot \\\\alpha \\\\cdot card(I_{k}^{(n)}))-$ distributed respectively $GAM(\\\\widetilde{c} \\\\cdot \\\\beta _{2},\\\\widetilde{c}\\\\cdot \\\\alpha \\\\cdot card(I_{k}^{(n)}))-$ distributed.\",\n \"Remark 56 In the book of Kotz et al.\",\n \"[197] one can find a very comprehensive study on generalized asymmetric Laplace distributions (also known as Bessel function distributions, McKay distributions), their close relatives (such as e.g.\",\n \"the financial-econometric variance gamma model of Madan & Seneta [240]) as well as their applications; see also e.g.\",\n \"Klar [192] for connections with some other Gamma difference distributions.\",\n \"[197] use a different parametrization $(\\\\theta ,\\\\kappa ,\\\\sigma ,\\\\tau )$ which is one-to-one with our parametrization $(\\\\breve{\\\\theta },\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c}=1)$ , as follows: $\\\\theta = \\\\breve{\\\\theta }$ , $\\\\tau = \\\\widetilde{c} \\\\cdot \\\\alpha $ , $\\\\sigma = \\\\frac{1}{\\\\widetilde{c}} \\\\cdot \\\\sqrt{\\\\frac{2}{\\\\beta _{1} \\\\cdot \\\\beta _{2}}}$ , $\\\\kappa = \\\\frac{\\\\sqrt{\\\\frac{4}{\\\\widetilde{c}^{2}} + (\\\\beta _{1} - \\\\beta _{2})^2} \\\\, + \\\\beta _{2}- \\\\beta _{1}}{2 \\\\cdot \\\\sqrt{\\\\beta _{1} \\\\cdot \\\\beta _{2}}} >0$ .\",\n \"In particular, this implies that we cover all generalized asymmetric Laplace distributions with mean 1.\",\n \"For better comparability, we have used the parametrization $(\\\\theta ,\\\\kappa ,\\\\sigma ,\\\\tau )$ in the above-mentioned representation (REF ) of the density (due to [197]).\",\n \"Let us end this section by giving some further comments on the task of finding concretely the probability distribution (if existent) $\\\\mathbb {}[ \\\\, \\\\cdot \\\\,] = \\\\mathbb {\\\\Pi }[W \\\\in \\\\cdot \\\\, ]$ from the Fenchel-Legendre transform $\\\\Lambda = \\\\varphi ^{*}$ of a pregiven divergence generator $\\\\varphi $ , which should satisfy $\\\\exp (\\\\Lambda (z)) = \\\\int _{\\\\mathbb {R}}e^{z \\\\cdot y} \\\\, d\\\\mathbb {} (y)= E_{\\\\mathbb {\\\\Pi }}[\\\\exp (z \\\\cdot W)] ,\\\\qquad z \\\\in \\\\mathbb {R}, \\\\qquad \\\\text{(cf.\",\n \"(\\\\ref {brostu3:repres lambda1}), (\\\\ref {brostu3:repres lambda2}))}.\\\\nonumber $ Recall that this is used for the simulation of the weights $(W_{i})_{i\\\\in \\\\mathbb {N}}$ which are i.i.d.\",\n \"copies of $W$ and which are the crucial building ingredients of $\\\\xi _{n}^{\\\\mathbf {W}}$ in Theorem REF , respectively, of $\\\\xi _{n,\\\\mathbf {X}}^{w\\\\mathbf {W}}$ in Theorem REF .\",\n \"The search for $\\\\mathbb {}$ can be done e.g.\",\n \"by inversion of the moment generating function MGF, or by search in tables or computer software which list distributions and their MGF.\",\n \"As already indicated above, we have eased/narrowed down this task by giving (additional) sufficient conditions for some deriving principal properties of $\\\\mathbb {}$ .\",\n \"Also notice that $\\\\mathbb {}$ needs not necessarily to be explicitly known in full detail (e.g.\",\n \"in terms of a computationally tractable density or frequency); for instance, as well known from insurance applications, for — comfortably straightforwardly simulable — doubly-random sums $W := \\\\sum _{i=1}^{N} A_{i}$ of nonnegative i.i.d.\",\n \"random variables $(A_{i})_{i\\\\in \\\\mathbb {N}}$ with known law $\\\\Pi _A[\\\\, \\\\cdot \\\\, ]:= \\\\Pi [A \\\\in \\\\cdot \\\\, ]$ being independent of a counting-type random variable $N$ with known law $\\\\Pi _N$ , one can mostly compute explicitly $MGF_{\\\\mathbb {}}(z) = PGF_{\\\\Pi _N}(MGF_{\\\\Pi _A}(z))$ in terms of $\\\\mathbb {} := \\\\Pi _W$ and the probability generating function $PGF_{\\\\Pi _N}$ of $\\\\Pi _N$ , but the corresponding density/frequency of $\\\\mathbb {}$ may not be known explicitly in a tractable form.\",\n \"The above-mentioned Example REF (b) of power divergences with generator $\\\\varphi _{\\\\gamma }$ ($\\\\gamma \\\\in ]0,1[$ ) manifests such a situation.\",\n \"In the end, if no explicit distribution $\\\\mathbb {}$ and no comfortably simulable $W-$ construction are available, one can still try to simulate an i.i.d.\",\n \"sequence $(W_{i})_{i\\\\in \\\\mathbb {N}}$ from the pregiven moment generating function (which is $\\\\exp (\\\\Lambda (z))$ here); see e.g.\",\n \"McLeish [259] and references therein which also contains saddle point methods approximation techniques.\"\n ],\n [\n \"Estimators\",\n \"In the following, we demonstrate how one can principally implement our BS approach; a further, deeper analysis will be given in a follow-up paper.\"\n ],\n [\n \"Estimators for the deterministic minimization problem\",\n \"We address the minimization problem $D_{\\\\varphi }(\\\\mathbf {\\\\Omega },\\\\mathbf {P}):= \\\\inf _{\\\\mathbf {Q}\\\\in \\\\mathbf {\\\\Omega } } D_{\\\\varphi }(\\\\mathbf {Q},\\\\mathbf {P}) =\\\\inf _{\\\\widetilde{\\\\mathbf {Q}}\\\\in \\\\widetilde{\\\\mathbf {\\\\Omega }} }D_{\\\\widetilde{\\\\varphi } }(\\\\widetilde{\\\\mathbf {Q}},\\\\widetilde{{P}})=: D_{\\\\widetilde{\\\\varphi } }(\\\\widetilde{\\\\mathbf {\\\\Omega }},\\\\widetilde{{P}})\\\\qquad \\\\textrm {with } \\\\widetilde{\\\\mathbf {\\\\Omega }} :=\\\\mathbf {\\\\Omega } /M_{\\\\mathbf {P}}\\\\qquad \\\\textrm {(cf.\",\n \"(\\\\ref {min Pb}) and (\\\\ref {min Pb prob2}))},$ whose numerical solution is based on Theorem REF which basically states that for large integer $n \\\\in \\\\mathbb {N}$ one has $\\\\inf _{\\\\mathbf {Q}\\\\in \\\\mathbf {\\\\Omega } } D_{\\\\varphi }(\\\\mathbf {Q},\\\\mathbf {P})\\\\approx -\\\\frac{1}{n}\\\\log \\\\,\\\\mathbb {\\\\Pi }\\\\left[\\\\xi _{n}^{\\\\mathbf {\\\\widetilde{W}}}\\\\in \\\\widetilde{\\\\mathbf {\\\\Omega }}\\\\right]$ in terms of $\\\\widetilde{\\\\varphi }:=M_{\\\\mathbf {P}} \\\\cdot \\\\varphi $ and the random vectors $\\\\xi _{n}^{\\\\mathbf {\\\\widetilde{W}}}=\\\\Big (\\\\frac{1}{n}\\\\sum _{i\\\\in I_{1}^{(n)}}\\\\widetilde{W}_{i},\\\\ldots ,\\\\frac{1}{n}\\\\sum _{i\\\\in I_{K}^{(n)}}\\\\widetilde{W}_{i}\\\\Big )\\\\hspace{56.9055pt} \\\\text{(cf.\",\n \"(\\\\ref {Xi_n^W vector}))}\\\\nonumber $ with $n_{k}:=\\\\lfloor n \\\\cdot \\\\widetilde{p}_{k}\\\\rfloor $ leading to the disjoint index blocks $I_{1}^{(n)}:=\\\\left\\\\lbrace 1,\\\\ldots ,n_{1}\\\\right\\\\rbrace $ , $I_{2}^{(n)}:=\\\\left\\\\lbrace n_{1}+1,\\\\ldots ,n_{1}+n_{2}\\\\right\\\\rbrace $ , $\\\\ldots $ , $I_{K}^{(n)} := \\\\lbrace \\\\sum _{k=1}^{K-1} n_{k} + 1, \\\\ldots , n \\\\rbrace $ .\",\n \"Recall that $\\\\mathbf {\\\\widetilde{W}} := (\\\\widetilde{W}_{1}, \\\\ldots , \\\\widetilde{W}_{n})$ is a random vector consisting of components $\\\\widetilde{W}_{i}$ which are i.i.d.\",\n \"copies of the random variable $\\\\widetilde{W}$ whose distribution is $\\\\mathbb {\\\\Pi }[\\\\widetilde{W}\\\\in \\\\cdot \\\\,]=\\\\widetilde{\\\\mathbb {}}[\\\\,\\\\cdot \\\\,]$ obeying the representation $\\\\widetilde{\\\\varphi }(t) =\\\\sup _{z \\\\in \\\\mathbb {R}} \\\\left( z\\\\cdot t - \\\\log \\\\int _{\\\\mathbb {R}} e^{zy} d\\\\widetilde{\\\\mathbb {}}(y) \\\\right),\\\\qquad t \\\\in \\\\mathbb {R},\\\\qquad \\\\textrm {(cf.\",\n \"(\\\\ref {brostu3:fo.link.var}))} .\\\\nonumber $ Hence, the estimation of $D_{\\\\varphi }(\\\\mathbf {\\\\Omega },\\\\mathbf {P})$ amounts to the estimation of $\\\\mathbb {\\\\Pi }\\\\left[\\\\xi _{n}^{\\\\mathbf {\\\\widetilde{W}}}\\\\in \\\\widetilde{\\\\mathbf {\\\\Omega }} \\\\right]$ .\",\n \"For the rest of this subsection, we assume that $\\\\widetilde{{P}} \\\\in \\\\mathbb {S}_{> 0}^{K}$ , that $n$ is chosen such that all $n \\\\cdot \\\\widetilde{p}_{k}$ are integers (and hence, $n = \\\\sum _{k=1}^{K} n_{k}$ ), and that $\\\\widetilde{\\\\mathbf {\\\\Omega }} \\\\subset \\\\mathbb {R}^{K}$ satisfies the regularity property $cl(\\\\widetilde{\\\\mathbf {\\\\Omega }})=cl\\\\left( int\\\\left( \\\\widetilde{\\\\mathbf {\\\\Omega }} \\\\right) \\\\right) , \\\\qquad int\\\\left( \\\\widetilde{\\\\mathbf {\\\\Omega }} \\\\right) \\\\ne \\\\emptyset \\\\nonumber $ which implies that the same condition holds for $\\\\mathbf {\\\\Omega }$ ; moreover, we suppose that $D_{\\\\widetilde{\\\\varphi } }(\\\\widetilde{\\\\mathbf {\\\\Omega }},\\\\widetilde{{P}})$ is finite.\",\n \"For the ease of the following discussions, we introduce the notations $T\\\\left( \\\\mathbf {x}\\\\right) :=\\\\left( \\\\frac{1}{n_{1}}\\\\sum _{i\\\\in I_{1}^{(n)}}x_{i},\\\\ldots ,\\\\frac{1}{n_{K}}\\\\sum _{i\\\\in I_{K}^{(n)}} x_{i}\\\\right)\\\\quad \\\\textrm {for any $ x:=( x1,..,xn) Rn$,}$$as well as $ D$ for the diagonal matrix with diagonal entries $ 1/p1,...,1/pK$and null entries off the diagonal.Accordingly, the probability in (\\\\ref {LDP Minimization approx}) becomes$$\\\\mathbb {\\\\Pi }\\\\left[\\\\xi _{n}^{\\\\mathbf {\\\\widetilde{W}}}\\\\in \\\\widetilde{\\\\mathbf {\\\\Omega }} \\\\right] \\\\ = \\\\ \\\\mathbb {\\\\Pi }\\\\left[ T(\\\\mathbf {\\\\widetilde{W}}) \\\\in \\\\mathbf {\\\\Lambda }\\\\right]$$where$$\\\\mathbf {\\\\Lambda } :=\\\\mathfrak {D} \\\\cdot \\\\widetilde{\\\\mathbf {\\\\Omega }}$$is a set of vectors in $ RK$ which is known/derived from the concrete context.The \\\\textit {naive estimator} $ Lnaive$ of$ [ nW ]$is constructedthrough the following procedure:simulate independently $ L$ copies$ W1,...,WL$ of the vector$ W:=( W1,...,Wn) $,with independent entries under $$, and define(with a slight abuse of notation)$$\\\\widehat{\\\\widetilde{\\\\Pi }}_{L}^{naive} :=\\\\frac{1}{L}\\\\sum _{\\\\ell =1}^{L} \\\\mathbf {1}_{\\\\mathbf {\\\\Lambda }}\\\\left( T\\\\left( \\\\mathbf {\\\\widetilde{W}}^{\\\\ell }\\\\right) \\\\right) ;$$however this procedure is time costly, since this estimate has a very bad hit rate.Thus, in the following, a so-called “efficient Importance Sampling (IS)”\\\\ scheme --- in the sense of Sadowsky\\\\& Bucklew \\\\cite {Sad:90} (denoted [SB] hereunder) ---is adapted for the sophisticated (i.e.\",\n \"non-naive) estimation of$ [ nW ]$.The main property of IS schemes lays in the fact that the runtime for anestimate with a controlled relative error does \\\\textit {not} increase atexponential rate as $ n$ increases, in contrast to$ Lnaive$ which has exponential increase.In detail, let $ >0$ be agiven relative precision for an estimator $ PSLn$ of$ [ nW ]$, based on a number $ Ln$ of simulated samplesgenerated under some distribution $ S$, so that$$\\\\delta :=\\\\frac{var_{\\\\widetilde{S}}P_{\\\\widetilde{S}}^{L_{n}}}{\\\\left(\\\\mathbb {\\\\Pi }\\\\left[\\\\xi _{n}^{\\\\mathbf {\\\\widetilde{W}}}\\\\in \\\\widetilde{\\\\mathbf {\\\\Omega }}\\\\right]\\\\right)^{2}} \\\\, .$$Then $ Ln$ will grow exponentially as $ n$ tends to infinity if and only if$ S$ is not “asymptotically optimal”\\\\ ,the derivation of which is the scope of the current section.$ To start with the details, for the sake of brevity (to avoid certain substantial discussions on potential technical relaxations) we shall employ the following additional Assumption (OM) on the set $\\\\widetilde{\\\\mathbf {\\\\Omega }}$ : (OM) For any $\\\\widetilde{\\\\omega } \\\\in cl(\\\\widetilde{\\\\mathbf {\\\\Omega }})$ there exists a vector $\\\\mathbf {x} = \\\\left(x_{1},\\\\ldots ,x_{n}\\\\right) \\\\in \\\\left] t_{-}^{sc},t_{+}^{sc}\\\\right[^{n}$ such that $\\\\widetilde{\\\\omega } = \\\\left( \\\\frac{1}{n}\\\\sum _{i\\\\in I_{1}^{(n)}}x_{i},\\\\ldots ,\\\\frac{1}{n} \\\\sum _{i\\\\in I_{K}^{(n)}}x_{i}\\\\right)$ , or equivalently, for any $\\\\lambda \\\\in cl(\\\\mathbf {\\\\Lambda })$ there exists a vector $\\\\mathbf {x} = \\\\left(x_{1},\\\\ldots ,x_{n}\\\\right) \\\\in \\\\left] t_{-}^{sc},t_{+}^{sc}\\\\right[^{n}$ such that $\\\\lambda =T\\\\left(\\\\mathbf {x}\\\\right)$ .\",\n \"For instance, in the common case $dom(\\\\widetilde{\\\\varphi })= dom(\\\\varphi ) = \\\\, ]a,b[ \\\\, =\\\\, \\\\left] t_{-}^{sc},t_{+}^{sc}\\\\right[ \\\\, = \\\\, ]0,\\\\infty [$ (e.g.\",\n \"for the power-divergence generators $\\\\widetilde{\\\\varphi }= \\\\widetilde{c} \\\\cdot \\\\varphi _{\\\\gamma }$ , $\\\\gamma \\\\le 0$ , cf.\",\n \"Example REF ) the Assumption (OM) is always feasible.\",\n \"To proceed, for any distribution $\\\\widetilde{S}$ on $\\\\mathbb {R}^{n}$ with support included in the support of the product measure $\\\\widetilde{\\\\mathbb {}}^{\\\\otimes n}$ it holds $\\\\mathbb {\\\\Pi }\\\\left[\\\\xi _{n}^{\\\\mathbf {\\\\widetilde{W}}}\\\\in \\\\widetilde{\\\\mathbf {\\\\Omega }}\\\\right]=E_{\\\\widetilde{\\\\mathbb {}}^{\\\\otimes n}} \\\\left[\\\\mathbf {1}_{\\\\mathbf {\\\\Lambda } }( T( \\\\mathbf {\\\\widetilde{W}}))\\\\right] =E_{\\\\widetilde{S}}\\\\left[\\\\mathbf {1}_{\\\\mathbf {\\\\Lambda } }\\\\left( T\\\\left( \\\\mathbf {\\\\widetilde{V}}\\\\right) \\\\right) \\\\cdot \\\\frac{d\\\\widetilde{\\\\mathbb {}}^{\\\\otimes n}}{d\\\\widetilde{S}}\\\\left( \\\\mathbf {\\\\widetilde{V}}\\\\right) \\\\right]$ from where the improved IS estimator of $\\\\mathbb {\\\\Pi }\\\\left[\\\\xi _{n}^{\\\\mathbf {\\\\widetilde{W}}}\\\\in \\\\widetilde{\\\\mathbf {\\\\Omega }} \\\\right]$ is obtained by sampling $L$ i.i.d.\",\n \"replications $\\\\mathbf {\\\\widetilde{V}}^{1},\\\\ldots ,\\\\mathbf {\\\\widetilde{V}}^{L}$ of the random vector $\\\\mathbf {\\\\widetilde{V}}$ with distribution $\\\\widetilde{S}$ and by defining $\\\\widehat{\\\\widetilde{\\\\Pi }}_{L}^{improved}:=\\\\frac{1}{L}\\\\sum _{\\\\ell =1}^{L}\\\\mathbf {1}_{\\\\mathbf {\\\\Lambda }}( T( \\\\mathbf {\\\\widetilde{V}}^{(\\\\ell )})) \\\\cdot \\\\frac{d\\\\widetilde{\\\\mathbb {}}^{\\\\otimes n}}{d\\\\widetilde{S}}\\\\left( \\\\mathbf {\\\\widetilde{V}}^{(\\\\ell )}\\\\right) \\\\ .$ The precise form of the efficient IS distribution $\\\\widetilde{S}^{opt}$ relies on the definition of a “dominating point” of $\\\\mathbf {\\\\Lambda }$ , which we recall now.\",\n \"For $\\\\mathbf {x} := \\\\left( x_{1},..,x_{n}\\\\right)$ in $\\\\mathbb {R}^{n}$ we define $I_{\\\\mathbf {\\\\widetilde{W}}}(\\\\mathbf {x}):=\\\\sup _{\\\\mathbf {z} \\\\in \\\\mathbb {R}^{n}}\\\\left(\\\\left\\\\langle \\\\mathbf {z}, \\\\mathbf {x} \\\\right\\\\rangle -\\\\log E_{\\\\widetilde{\\\\mathbb {}}}[ \\\\, \\\\exp (\\\\langle \\\\mathbf {z},\\\\mathbf {\\\\widetilde{W}}\\\\rangle ) ]\\\\right) \\\\, ,$ and for $\\\\lambda $ in $\\\\mathbf {\\\\Lambda }$ we let $I(\\\\lambda ):=\\\\inf \\\\left\\\\lbrace I_{\\\\mathbf {\\\\widetilde{W}}}(\\\\mathbf {x}):T(\\\\mathbf {x)=\\\\lambda }\\\\right\\\\rbrace .$ Let $\\\\underline{\\\\lambda }:=\\\\left( \\\\underline{\\\\lambda }_{1},\\\\ldots ,\\\\underline{\\\\lambda }_{K}\\\\right) \\\\in \\\\partial \\\\mathbf {\\\\Lambda }$ .\",\n \"We call $\\\\underline{\\\\lambda }$ a minimal rate point (mrp) of $\\\\mathbf {\\\\Lambda }$ if $I(\\\\underline{\\\\lambda })\\\\le I(\\\\lambda ) \\\\quad \\\\text{ for all }\\\\lambda \\\\in \\\\mathbf {\\\\Lambda } .$ A minimal rate point $\\\\underline{\\\\lambda }$ is called a dominating point of $\\\\mathbf {\\\\Lambda }$ if a) $\\\\underline{\\\\lambda }\\\\in \\\\partial \\\\mathbf {\\\\Lambda }$ , and b) $I\\\\left(\\\\underline{\\\\lambda }\\\\right)\\\\le I(\\\\lambda )$ for all $\\\\lambda \\\\in \\\\mathbf {\\\\Lambda }$ with attainment, namely there exists a vector $\\\\underline{\\\\mathbf {x}} \\\\in \\\\left] t_{-}^{sc},t_{+}^{sc}\\\\right[^{n}$ such that $I_{\\\\widetilde{W}}\\\\left( \\\\underline{\\\\mathbf {x}}\\\\right) =I\\\\left(\\\\underline{\\\\lambda }\\\\right)$ with $\\\\underline{\\\\lambda }=T\\\\left( \\\\underline{\\\\mathbf {x}}\\\\right)$ .\",\n \"The characterization of the dominating point $\\\\underline{\\\\lambda }$ is settled in the following Lemma 57 Let $\\\\underline{\\\\lambda }$ be a mrp of $\\\\mathbf {\\\\Lambda }$ .\",\n \"Then, under Assumption (OM), $\\\\underline{\\\\lambda }$ is a dominating point, and $\\\\inf \\\\left\\\\lbrace I_{\\\\mathbf {\\\\widetilde{W}}}\\\\left( \\\\mathbf {x}\\\\right) ,T(\\\\mathbf {x})=\\\\underline{\\\\lambda }\\\\right\\\\rbrace $ is reached at some vector $\\\\underline{\\\\mathbf {x}}$ in $\\\\left] t_{-}^{sc},t_{+}^{sc}\\\\right[^{n}$ such that for all $k \\\\in \\\\left\\\\lbrace 1,\\\\ldots ,K\\\\right\\\\rbrace $ and all $i \\\\in I_{k}^{(n)}$ there holds $\\\\underline{x}_{i}= \\\\underline{\\\\lambda }_{k}$ and $I_{\\\\mathbf {\\\\widetilde{W}}}(\\\\underline{\\\\mathbf {x}})=n \\\\cdot \\\\sum _{k=1}^{K}\\\\widetilde{p}_{k} \\\\cdot \\\\widetilde{\\\\varphi }\\\\left(\\\\underline{\\\\lambda }_{k}\\\\right)$ .\",\n \"The proof Lemma REF is given in Appendix G. Notice that (OM) implies the existence of a dominating point $\\\\underline{\\\\lambda }$ , but uniqueness may not hold.\",\n \"In the latter case, one can try to proceed as in Theorem 2 of [SB] and the discussion thereafter.\",\n \"However, we assume now uniqueness of $\\\\underline{\\\\lambda }$ ; this allows for the identification of $\\\\widetilde{S}^{opt}$ .\",\n \"By Theorem 1 of [SB] and Theorem 3.1 of Csiszar [96], the asymptotically optimal IS distribution $\\\\widetilde{S}^{opt}$ is obtained as the Kullback-Leibler projection of $\\\\widetilde{\\\\mathbb {}}^{n\\\\otimes }$ on the set of all probability distributions on $\\\\mathbb {R}^{n}$ centered at point $\\\\underline{\\\\mathbf {x}}$ , whose coordinates are — according to Lemma REF — functions of the coordinates of $\\\\underline{\\\\mathbf {\\\\widetilde{Q}}} := \\\\mathfrak {D}^{-1} \\\\underline{\\\\lambda }$ such that $T\\\\left( \\\\underline{\\\\mathbf {x}}\\\\right)= \\\\mathfrak {D} \\\\underline{\\\\mathbf {\\\\widetilde{Q}}}$ .\",\n \"The above definition of $\\\\widetilde{S}^{opt}$ presumes the knowledge of $\\\\underline{\\\\lambda }$ , which cannot be assumed (otherwise the minimization problem is solved in advance).\",\n \"The aim of the following construction is to provide a proxy $\\\\widetilde{S}$ to $\\\\widetilde{S}^{opt}$ , where $\\\\widetilde{S}$ is the Kullback-Leibler projection of $\\\\widetilde{\\\\mathbb {}}^{\\\\otimes n}$ on the set of all probability distributions on $\\\\mathbb {R}^{n}$ centered at some point $\\\\mathbf {x}^{\\\\ast }$ which is close to $\\\\underline{\\\\mathbf {x}}$ .\",\n \"For this sake, we need to have at hand a proxy of $\\\\underline{\\\\lambda }$ or, equivalently, a preliminary guess $\\\\mathbf {\\\\widetilde{Q}}^{\\\\ast }$ of $\\\\underline{\\\\mathbf {\\\\widetilde{Q}}}:=\\\\arg \\\\inf _{\\\\mathbf {\\\\widetilde{Q}} \\\\in \\\\widetilde{\\\\mathbf {\\\\Omega }}}\\\\sum _{k=1}^{K} \\\\widetilde{p}_{k} \\\\cdot \\\\widetilde{\\\\varphi }(\\\\widetilde{q}_{k}/\\\\widetilde{p}_{k})$ .\",\n \"This guess is by no means produced in order to provide a direct estimate of $D_{\\\\widetilde{\\\\varphi } }(\\\\widetilde{\\\\mathbf {\\\\Omega }},\\\\widetilde{{P}})$ but merely to provide the IS distribution $\\\\widetilde{S}$ which in turn leads to a sharp estimate of $D_{\\\\widetilde{\\\\varphi } }(\\\\widetilde{\\\\mathbf {\\\\Omega }},\\\\widetilde{{P}})$ .\",\n \"Proxy method 1: in some cases we might have at hand some particular point $\\\\mathbf {\\\\widetilde{Q}}^{\\\\ast } :=\\\\left(\\\\widetilde{q}_{1}^{\\\\ast },..,\\\\widetilde{q}_{K}^{\\\\ast }\\\\right)$ in $\\\\widetilde{\\\\mathbf {\\\\Omega }}$ ; the resulting IS distribution $\\\\widetilde{S}$ with $\\\\mathbf {\\\\widetilde{Q}}$ substituted by $\\\\mathbf {\\\\widetilde{Q}^{\\\\ast }}$ is not optimal in the sense of [SB], but anyhow produces an estimator with good hitting rate, possibly with a loss in the variance.\",\n \"A simple way to obtain such a point $\\\\mathbf {\\\\widetilde{Q}^{\\\\ast }}$ in $\\\\widetilde{\\\\mathbf {\\\\Omega }}$ is to simulate runs of (say) $M-$ variate i.i.d.\",\n \"vectors $\\\\mathbf {\\\\widetilde{W}}$ under $\\\\widetilde{\\\\mathbb {}}^{\\\\otimes M}$ until the first time where $\\\\xi _{M}^{\\\\mathbf {\\\\widetilde{W}}}$ belongs to $\\\\widetilde{\\\\mathbf {\\\\Omega }}$ ; then we set $\\\\mathbf {\\\\widetilde{Q}^{\\\\ast }} := \\\\xi _{M}^{\\\\mathbf {\\\\widetilde{W}}}$ for the succeeding realization $\\\\mathbf {\\\\widetilde{W}}$ .\",\n \"Before we proceed, it is useful to mention that the need for a drastic fall in the number of simulation runs pertains for cases when $D_{\\\\widetilde{\\\\varphi } }(\\\\widetilde{\\\\mathbf {\\\\Omega }},\\\\widetilde{{P}})$ is large.\",\n \"The following construction is suited to this case, which is of relevance in applications both in optimization and in statistics when choosing between competing models none of which is assumed to represent the true one, but merely less inadequate ones.\",\n \"Proxy method 2: when $D_{\\\\widetilde{\\\\varphi } }(\\\\widetilde{\\\\mathbf {\\\\Omega }},\\\\widetilde{{P}})$ is presumably large, we make use of asymptotic approximation to get a proxy of $\\\\underline{\\\\mathbf {\\\\widetilde{Q}}}$ .\",\n \"For this, we define a sampling distribution on $\\\\mathbb {R}^{K}$ fitted to the divergence through $f(\\\\mathbf {\\\\widetilde{Q}}):=C \\\\cdot \\\\exp \\\\left( -\\\\sum _{k=1}^{K} \\\\widetilde{p}_{k} \\\\cdot \\\\widetilde{\\\\varphi }(\\\\widetilde{q}_{k}/\\\\widetilde{p}_{k})\\\\right) \\\\ = \\\\ C \\\\cdot \\\\exp \\\\left( -D_{\\\\widetilde{\\\\varphi }}\\\\left( \\\\mathbf {\\\\widetilde{Q}},\\\\widetilde{{P}}\\\\right)\\\\right)$ where $C$ is a normalizing constant.\",\n \"Let $\\\\mathbf {T}$ be a $K-$ variate random variable with density $f$ .\",\n \"The distribution of $\\\\mathbf {T}$ given $\\\\left( \\\\mathbf {T} \\\\in \\\\widetilde{\\\\mathbf {\\\\Omega }}\\\\right)$ concentrates around $\\\\arg \\\\inf _{\\\\widetilde{\\\\mathbf {Q}}\\\\in \\\\widetilde{\\\\mathbf {\\\\Omega }} }D_{\\\\widetilde{\\\\varphi } }(\\\\widetilde{\\\\mathbf {Q}},\\\\widetilde{{P}})$ when $D_{\\\\widetilde{\\\\varphi } }(\\\\widetilde{\\\\mathbf {\\\\Omega }},\\\\widetilde{{P}})$ is large.\",\n \"Indeed, for any $\\\\widetilde{\\\\mathbf {Q}}\\\\in \\\\widetilde{\\\\mathbf {\\\\Omega }}$ denote by $\\\\mathbf {V}_{\\\\varepsilon }(\\\\widetilde{\\\\mathbf {Q}})$ a small neighborhood of $\\\\widetilde{\\\\mathbf {Q}}$ in $\\\\mathbb {R}^{K}$ with radius $\\\\varepsilon $ ; clearly, the probability of the event $\\\\left( \\\\mathbf {T} \\\\in \\\\mathbf {V}_{\\\\varepsilon }(\\\\widetilde{\\\\mathbf {Q}})\\\\right)$ when restricted to $\\\\widetilde{\\\\mathbf {Q}}\\\\in \\\\widetilde{\\\\mathbf {\\\\Omega }}$ is maximum when $\\\\widetilde{\\\\mathbf {Q}} = \\\\underline{\\\\widetilde{\\\\mathbf {Q}}}$ , where $\\\\underline{\\\\widetilde{\\\\mathbf {Q}}}$ is the “dominating point of $\\\\widetilde{\\\\mathbf {\\\\Omega }}$ ” in the sense that $\\\\underline{\\\\mathbf {\\\\widetilde{Q}}} := \\\\mathfrak {D}^{-1} \\\\underline{\\\\lambda }$ is the above-defined transform of the dominating point $\\\\underline{\\\\lambda }$ (assuming uniqueness); a precise argumentation under adequate conditions is postponed to Appendix G. Accordingly, we obtain a proxy $\\\\mathbf {\\\\widetilde{Q}}^{\\\\ast }$ of $\\\\underline{\\\\mathbf {\\\\widetilde{Q}}}$ by simulating a sequence of independent $K-$ variate random variables $\\\\mathbf {T}_{1},\\\\ldots $ with distribution (REF ) until (say) $\\\\mathbf {T}_{m}$ belongs to $\\\\widetilde{\\\\mathbf {\\\\Omega }}$ and set $\\\\mathbf {\\\\widetilde{Q}}^{\\\\ast }:=\\\\mathbf {T}_{m}$ .\",\n \"To proceed with the derivation of the IS sampling distribution $\\\\widetilde{S}$ on $\\\\mathbb {R}^{n}$ , we fix $\\\\mathbf {\\\\widetilde{Q}}^{\\\\ast } :=\\\\left(\\\\widetilde{q}_{1}^{\\\\ast },..,\\\\widetilde{q}_{K}^{\\\\ast }\\\\right)$ to be a proxy of $\\\\underline{\\\\mathbf {\\\\widetilde{Q}}}$ or an initial guess in $\\\\widetilde{\\\\mathbf {\\\\Omega }}$ .\",\n \"As an intermediate step, we construct the probability distribution $\\\\widetilde{U}_{k}$ on $\\\\mathbb {R}$ given by $d\\\\widetilde{U}_{k}(v) \\\\ := \\\\ \\\\exp \\\\left( \\\\tau _{k} \\\\cdot v -\\\\Lambda _{\\\\widetilde{\\\\mathbb {}}}(\\\\tau _{k})\\\\right) d\\\\widetilde{\\\\mathbb {}}(v)=\\\\frac{\\\\exp \\\\left(\\\\tau _{k} \\\\cdot v\\\\right)}{MGF_{\\\\widetilde{\\\\mathbb {}}}(\\\\tau _{k})} \\\\, d\\\\widetilde{\\\\mathbb {}}(v)$ where $\\\\tau _{k} \\\\in int(dom(MGF_{\\\\widetilde{\\\\mathbb {}}}))$ is the unique solution of the equation $\\\\Lambda _{\\\\widetilde{\\\\mathbb {}}}^{\\\\prime }\\\\left( \\\\tau _{k}\\\\right) =\\\\frac{\\\\widetilde{q}_{k}^{\\\\ast }}{\\\\widetilde{p}_{k}}$ and thus — by relation (REF ) of Appendix F — we can compute explicitly $\\\\tau _{k} = \\\\ \\\\widetilde{\\\\varphi }^{\\\\, \\\\prime }\\\\left(\\\\frac{\\\\widetilde{q}_{k}^{\\\\ast }}{\\\\widetilde{p}_{k}}\\\\right) \\\\ .$ Therefore, $\\\\widetilde{U}_{k}$ is the Kullback-Leibler projection of $\\\\widetilde{\\\\mathbb {}}$ on the class of all probability distributions on $\\\\mathbb {R}$ whose expectation is $\\\\widetilde{q}_{k}^{\\\\ast }$ .\",\n \"As a side remark, notice that one possible way of obtaining an explicit form of the probability distribution $\\\\widetilde{U}_{k}$ may be by identification through its moment generating function $dom(MGF_{\\\\widetilde{\\\\mathbb {}}})-\\\\tau _{k} \\\\ \\\\ni \\\\ z \\\\ \\\\mapsto MGF_{\\\\widetilde{U}_{k}}(z) =\\\\frac{MGF_{\\\\widetilde{\\\\mathbb {}}}(z+\\\\tau _{k})}{MGF_{\\\\widetilde{\\\\mathbb {}}}(\\\\tau _{k})}$ of which all ingredients are principally available.\",\n \"For instance, this will be used in Example REF below.\",\n \"From (REF ), we define $\\\\widetilde{S}_{k}:=\\\\underbrace{\\\\widetilde{U}_{k} \\\\otimes \\\\cdots \\\\otimes \\\\widetilde{U}_{k}}_{n_{k}\\\\text{times}}\\\\qquad \\\\textrm {for all $ {1,...,K}$},$$whence\\\\begin{eqnarray}d\\\\widetilde{S}_{k}\\\\left( v_{k,1},\\\\ldots ,v_{k,n_{k}}\\\\right) =\\\\exp \\\\Big ( \\\\sum _{i\\\\in I_{k}^{(n)}}\\\\tau _{k} \\\\cdot v_{k,i}-n_{k} \\\\cdot \\\\Lambda _{\\\\widetilde{\\\\mathbb {}}}(\\\\tau _{k})\\\\Big ) \\\\ d\\\\widetilde{\\\\mathbb {}}\\\\left(v_{k,1}\\\\right)\\\\cdots d\\\\widetilde{\\\\mathbb {}}\\\\left(v_{k,n_{k}}\\\\right),\\\\\\\\\\\\end{eqnarray}which manifests $ Sk$ as the Kullback-Leibler projection of$ nktimes$on the class of all probability distributions on $ Rk$whose expectation vector is$ Q = (q1,...,qK) Rk$.\",\n \"Let now\\\\begin{equation}\\\\widetilde{S} := \\\\widetilde{S}_{1} \\\\otimes \\\\cdots \\\\otimes \\\\widetilde{S}_{K} \\\\, ,\\\\end{equation}which therefore satisfies (recall that $ k=1K nk =n$)\\\\begin{equation}d\\\\widetilde{S}\\\\left( v_{1,1},\\\\ldots v_{1,n_{1}},\\\\ldots , v_{K,1},\\\\ldots v_{K,n_{K}}\\\\right)=\\\\exp \\\\Big (\\\\sum _{k=1}^{K}\\\\sum _{i\\\\in I_{k}^{(n)}} \\\\big ( \\\\tau _{k} \\\\cdot v_{k,i} - n_{k} \\\\cdot \\\\Lambda _{\\\\widetilde{\\\\mathbb {}}}(\\\\tau _{k}) \\\\big )\\\\Big ) \\\\,d\\\\widetilde{\\\\mathbb {}}^{\\\\otimes n}\\\\left( v_{1,1},\\\\ldots v_{1,n_{1}},\\\\ldots , v_{K,1},\\\\ldots v_{K,n_{K}}\\\\right).\\\\ \\\\end{equation}The same procedure with all $ qk$substituted by the coordinates $qk$of $Q$produces $ Sopt$.\",\n \"Therefore, $ S$ is a substitutefor $ Sopt$ with the change in the centering fromthe unknown vector $Q$to its proxy $ Q$.$ As a straightforward consequence of (REF ) and (), we obtain the improved IS estimator of $\\\\mathbb {\\\\Pi }\\\\left[\\\\xi _{n}^{\\\\mathbf {\\\\widetilde{W}}}\\\\in \\\\widetilde{\\\\mathbf {\\\\Omega }}\\\\right]$ as $\\\\widehat{\\\\widetilde{\\\\Pi }}_{L}^{improved}\\\\ =\\\\ \\\\frac{1}{L}\\\\sum _{\\\\ell =1}^{L}\\\\mathbf {1}_{\\\\mathbf {\\\\Lambda }}( T( \\\\mathbf {\\\\widetilde{V}}^{(\\\\ell )})) \\\\cdot \\\\prod _{k=1}^{K} IS_{k}(\\\\mathbf {\\\\widetilde{V}}_{k}^{(\\\\ell )})$ where $\\\\mathbf {\\\\widetilde{V}}_{k}^{(\\\\ell )} := \\\\left( \\\\widetilde{V}_{i}^{(\\\\ell )} \\\\right)_{i \\\\in I_{k}^{(n)}}$ is the $k-$ th block of the $\\\\ell -$ th replication $\\\\mathbf {\\\\widetilde{V}}^{(\\\\ell )}$ of $\\\\mathbf {\\\\widetilde{V}}$ under $S$ , and the $k-$ th importance-sampling factor is $\\\\widetilde{IS}_{k}(v_{k,1},\\\\ldots ,v_{k,n_{k}}) \\\\ := \\\\ \\\\frac{d\\\\widetilde{\\\\mathbb {}}^{\\\\otimes n_{k}}}{d\\\\widetilde{S}_{k}}\\\\left( v_{k,1},\\\\ldots ,v_{k,n_{k}}\\\\right)= \\\\exp \\\\Big (n_{k} \\\\cdot \\\\Lambda _{\\\\widetilde{\\\\mathbb {}}}(\\\\tau _{k}) \\\\, - \\\\, \\\\tau _{k}\\\\cdot \\\\sum _{i=1}^{n_{k}} v_{k,i} \\\\Big )\\\\nonumber $ with $n_{k} = card(I_{k}^{(n)})$ .\",\n \"Summing up things, we arrive at the following algorithm in case that $\\\\widetilde{\\\\mathbf {\\\\Omega }}$ has a unique dominating point (in the above-defined sense): Step D1 Exemplarily, we start with proxy method 2 (the other proxy method 1 works analogously): get a proxy $\\\\mathbf {\\\\widetilde{Q}}^{\\\\ast }$ of $\\\\underline{\\\\mathbf {\\\\widetilde{Q}}}$ by simulating a sequence of independent $K-$ variate random variables $\\\\mathbf {T}_{1},\\\\ldots $ with distribution (REF ) until (say) $\\\\mathbf {T}_{m}$ belongs to $\\\\widetilde{\\\\mathbf {\\\\Omega }}$ and set $\\\\mathbf {\\\\widetilde{Q}}^{\\\\ast }:=\\\\mathbf {T}_{m}$ .\",\n \"Step D2 For all $k$ in $\\\\lbrace 1,\\\\ldots ,K\\\\rbrace $ compute $\\\\tau _{k} = \\\\ \\\\widetilde{\\\\varphi }^{\\\\, \\\\prime }\\\\left(\\\\frac{\\\\widetilde{q}_{k}^{\\\\ast }}{\\\\widetilde{p}_{k}}\\\\right)$ .\",\n \"Step D3 For all $\\\\ell $ in $\\\\lbrace 1,\\\\ldots ,L\\\\rbrace $ perform a run of $\\\\mathbf {\\\\widetilde{V}}^{(\\\\ell )}$ under $\\\\widetilde{S}$ as follows: For all $k$ in $\\\\lbrace 1,\\\\ldots ,K\\\\rbrace $ simulate $n_{k}$ i.i.d.\",\n \"random variables $\\\\widetilde{V}_{k_{1}}^{(\\\\ell )},\\\\ldots ,\\\\widetilde{V}_{k_{n_{k}}}^{(\\\\ell )}$ with common distribution $\\\\widetilde{U}_{k}$ defined in (REF ).\",\n \"Set $\\\\mathbf {\\\\widetilde{V}}_{k}^{(\\\\ell )} :=(\\\\widetilde{V}_{k_{1}}^{(\\\\ell )},\\\\ldots ,\\\\widetilde{V}_{k_{n_{k}}}^{(\\\\ell )})$ to be the corresponding row vector.\",\n \"Construct $\\\\mathbf {\\\\widetilde{V}}^{(\\\\ell )}$ as the row vector obtained by concatenating the $\\\\mathbf {\\\\widetilde{V}}_{k}^{(\\\\ell )}$ , i.e.\",\n \"$\\\\mathbf {\\\\widetilde{V}}^{(\\\\ell )}:=\\\\left( \\\\mathbf {\\\\widetilde{V}}_{1}^{(\\\\ell )},\\\\ldots ,\\\\mathbf {\\\\widetilde{V}}_{K}^{(\\\\ell )}\\\\right) \\\\, ,$ and make use of $\\\\widehat{\\\\widetilde{\\\\Pi }}_{L}^{improved}$ given in (REF ) with the $\\\\tau _{k}$ 's obtained in Step D2 above to define (in the light of (REF ), (REF )) the BS minimum-distance estimator $\\\\widehat{D_{\\\\varphi }}(\\\\mathbf {\\\\Omega },\\\\mathbf {P})\\\\ := \\\\ \\\\widehat{D_{\\\\widetilde{\\\\varphi }}}(\\\\widetilde{\\\\mathbf {\\\\Omega }},\\\\widetilde{{P}})\\\\ := \\\\ - \\\\frac{1}{n}\\\\log \\\\widehat{\\\\widetilde{\\\\Pi }}_{L}^{improved} \\\\ .$ For many cases, the simulation burden needed for the computation of $\\\\widehat{\\\\widetilde{\\\\Pi }}_{L}^{improved}$ — and thus of $\\\\widehat{D_{\\\\varphi }}(\\\\mathbf {\\\\Omega },\\\\mathbf {P})$ — can be drastically reduced, especially for high dimensions $K$ and large sample size $n \\\\cdot L$ .\",\n \"In fact, in terms of the notations $n_{k}:=card(I_{k}^{(n)})$ , $\\\\widehat{\\\\mathit {W}}_{k}^{(\\\\ell )}:=\\\\sum _{i\\\\in I_{k}^{(n)}}\\\\widetilde{V}_{i}^{(\\\\ell )}$ and $\\\\widetilde{ISF}_{k}(x) \\\\ := \\\\ \\\\frac{d\\\\widetilde{\\\\mathbb {}}^{\\\\ast n_{k}}}{d\\\\widetilde{U}_{k}^{\\\\ast n_{k}}}(x) \\\\ = \\\\ \\\\exp (n_{k} \\\\cdot \\\\Lambda _{\\\\widetilde{\\\\mathbb {}}}(\\\\tau _{k}) \\\\, - \\\\, x \\\\cdot \\\\tau _{k} )$ (where $\\\\widetilde{\\\\mathbb {}}^{\\\\ast n_{k}}$ is the $n_{k}-$ convolution of the measure $\\\\widetilde{\\\\mathbb {}}$ ), one can rewrite (REF ) as $\\\\widehat{\\\\widetilde{\\\\Pi }}_{L}^{improved}=\\\\frac{1}{L}\\\\sum _{\\\\ell =1}^{L}\\\\mathbf {1}_{\\\\mathbf {\\\\Lambda }}\\\\big ( \\\\,\\\\big (\\\\frac{1}{n_{1}} \\\\widehat{\\\\mathit {W}}_{1}^{(\\\\ell )},\\\\ldots , \\\\frac{1}{n_{K}} \\\\widehat{\\\\mathit {W}}_{K}^{(\\\\ell )}\\\\big ) \\\\, \\\\big ) \\\\cdot \\\\prod \\\\limits _{k=1}^{K}\\\\widetilde{ISF}_{k}(\\\\widehat{\\\\mathit {W}}_{k}^{(\\\\ell )}) .$ with $K-$ vector $\\\\big (\\\\frac{1}{n_{1}} \\\\widehat{\\\\mathit {W}}_{1}^{(\\\\ell )},\\\\ldots , \\\\frac{1}{n_{K}} \\\\widehat{\\\\mathit {W}}_{K}^{(\\\\ell )} \\\\big )$ .\",\n \"Clearly, the random variable $\\\\widehat{\\\\mathit {W}}_{k}^{(\\\\ell )}$ ($k=1, \\\\ldots , K$ ) has distribution $\\\\widetilde{U}_{k}^{\\\\ast n_{k}}$ .\",\n \"Hence, if $\\\\widetilde{U}_{k}^{\\\\ast n_{k}}$ can be explicitly constructed, then for the computation of $\\\\widehat{\\\\widetilde{\\\\Pi }}_{L}^{improved}$ it suffices to simulate the $K \\\\cdot L$ random variables $\\\\widehat{\\\\mathit {W}}_{k}^{(\\\\ell )}$ rather than the $n \\\\cdot L$ random variables $\\\\widetilde{V}_{i}^{(\\\\ell )}$ ; notice that according to the right-hand side of (REF ), one can explicitly compute $ISF_{k}\\\\left( \\\\cdot \\\\right)$ which can be interpreted as Importance Sampling Factor pertaining to the block $k$.\",\n \"In the case that $\\\\widetilde{\\\\mathbb {}}$ is infinitely divisible, simulation issues may become especially comfortable.\",\n \"In the following, we exemplarily demonstrate the tractability of this reduction effect, for the BS minimization of the important power divergences (for which the infinite divisibility holds): Example 58 Let $\\\\varphi _{\\\\gamma }$ ($\\\\gamma \\\\in \\\\mathbb {R}\\\\backslash ]1,2[$ ) be the power divergence generator from the Examples REF and REF , $\\\\mathbf {P} \\\\in \\\\mathbb {R}_{> 0}^{K}$ , $M_{\\\\mathbf {P}}:=\\\\sum _{i=1}^{K}p_{i}>0$ and $n_{k} = n \\\\cdot p_{k} \\\\in \\\\mathbb {N}$ where we have employed our notation $n_{k}= n \\\\cdot p_{k} \\\\in \\\\mathbb {N}$ for all $k \\\\in \\\\lbrace 1,\\\\ldots ,K\\\\rbrace $ .\",\n \"Moreover, let $\\\\mathbf {\\\\widetilde{Q}}^{\\\\ast } :=\\\\left(\\\\widetilde{q}_{1}^{\\\\ast },\\\\ldots ,\\\\widetilde{q}_{K}^{\\\\ast }\\\\right)$ be a proxy obtained by either proxy method 1 or 2.\",\n \"Case 1: Example REF (a): $\\\\gamma =0,\\\\widetilde{c}>0$ .\",\n \"There holds $\\\\widetilde{U}_{k}^{\\\\ast n_{k}}=GAM\\\\left( \\\\widetilde{c} \\\\cdot M_{\\\\mathbf {P}} - \\\\tau _{k},n_{k}\\\\cdot \\\\widetilde{c} \\\\cdot M_{\\\\mathbf {P}}\\\\right)$ , with $\\\\tau _{k}=\\\\widetilde{c} \\\\cdot M_{\\\\mathbf {P}} \\\\cdot ( 1-\\\\frac{p_{k}}{M_{\\\\mathbf {P}} \\\\cdot \\\\widetilde{q}_{k}^{\\\\ast }})$ for $\\\\widetilde{q}_{k}^{\\\\ast } >0$ (the latter is equivalent to $\\\\tau _{k} < \\\\widetilde{c} \\\\cdot M_{\\\\mathbf {P}}$ ).\",\n \"Moreover, for all $x >0$ one gets $\\\\widetilde{ISF}_{k}(x)=\\\\left( \\\\frac{\\\\widetilde{c} \\\\cdot M_{\\\\mathbf {P}}}{\\\\widetilde{c}\\\\cdot M_{\\\\mathbf {P}} - \\\\tau _{k}}\\\\right)^{n_{k}\\\\cdot \\\\widetilde{c}\\\\cdot M_{\\\\mathbf {P}}}\\\\cdot e^{-\\\\tau _{k} \\\\cdot x}$ .\",\n \"Case 2: REF (b): $\\\\gamma \\\\in \\\\left( 0,1\\\\right) ,\\\\widetilde{c}>0$ .\",\n \"We derive $\\\\widetilde{U}_{k}^{\\\\ast n_{k}}=C\\\\big ( POI( n_{k}\\\\cdot \\\\breve{\\\\theta }),GAM\\\\big (\\\\frac{\\\\widetilde{c} \\\\cdot M_{\\\\mathbf {P}}}{1-\\\\gamma } - \\\\tau _{k},\\\\frac{\\\\gamma }{1-\\\\gamma } \\\\big ) \\\\big )$ with $\\\\breve{\\\\theta }:= \\\\frac{\\\\widetilde{c} \\\\cdot M_{\\\\mathbf {P}}}{\\\\gamma }\\\\cdot \\\\big (\\\\frac{(\\\\gamma -1) \\\\cdot \\\\tau _{k}}{\\\\widetilde{c} \\\\cdot M_{\\\\mathbf {P}}}+1\\\\big )^{\\\\gamma /(\\\\gamma -1)}$ and $\\\\tau _{k} = \\\\widetilde{c} \\\\cdot M_{\\\\mathbf {P}} \\\\cdot \\\\frac{1-\\\\big (\\\\frac{\\\\widetilde{q}_{k}^{\\\\ast } \\\\cdot M_{\\\\mathbf {P}}}{p_{k}}\\\\big )^{\\\\gamma -1}}{1-\\\\gamma }$ for $\\\\widetilde{q}_{k}^{\\\\ast } >0$ .\",\n \"Furthermore, $\\\\widetilde{ISF}_{k}(x)=e^{-\\\\tau _{k}x} \\\\cdot \\\\exp \\\\left( \\\\frac{n_{k}\\\\cdot \\\\widetilde{c} \\\\cdot M_{\\\\mathbf {P}}}{\\\\gamma }\\\\cdot \\\\left(\\\\left(1+ \\\\frac{\\\\gamma -1}{\\\\widetilde{c} \\\\cdot M_{\\\\mathbf {P}}} \\\\cdot \\\\tau _{k}\\\\right) ^{\\\\frac{\\\\gamma }{\\\\gamma -1}}-1 \\\\right)\\\\right), \\\\qquad x \\\\ge 0 ,$ (where $x=0$ covers the atom at zero).\",\n \"Case 3: Example REF (c): $\\\\gamma =2,\\\\widetilde{c}>0$ .\",\n \"One gets $\\\\widetilde{U}_{k}^{\\\\ast n_{k}}=N(n_{k}\\\\cdot (1+\\\\frac{\\\\tau _{k}}{\\\\widetilde{c} \\\\cdot M_{\\\\mathbf {P}}}),\\\\frac{n_{k}}{\\\\widetilde{c} \\\\cdot M_{\\\\mathbf {P}}})$ with $\\\\tau _{k}=\\\\widetilde{c} \\\\cdot M_{\\\\mathbf {P}} \\\\cdot ( \\\\frac{\\\\widetilde{q}_{k}^{\\\\ast } \\\\cdot M_{\\\\mathbf {P}}}{p_{k}} - 1 ) $ for $\\\\widetilde{q}_{k}^{\\\\ast } \\\\in \\\\mathbb {R}$ .\",\n \"Moreover, for all $x \\\\in \\\\mathbb {R}$ one obtains $\\\\widetilde{ISF}_{k}(x)= \\\\exp \\\\big (\\\\frac{n_{k} \\\\cdot \\\\tau _{k}^2}{2\\\\widetilde{c} \\\\cdot M_{\\\\mathbf {P}}}- (x- n_{k}) \\\\cdot \\\\tau _{k} \\\\big )$ .\",\n \"Case 4: Example REF (d): $\\\\gamma <0,\\\\widetilde{c}>0$ .\",\n \"It holds that $\\\\widetilde{U}_{k}^{\\\\ast n_{k}}$ has the (Lebesgue-)density $f_{\\\\widetilde{U}_{k}^{\\\\ast n_{k}}}(x)\\\\ := \\\\ \\\\frac{\\\\exp ((\\\\tau _{k} - \\\\frac{\\\\widetilde{c}\\\\cdot M_{\\\\mathbf {P}}}{1-\\\\gamma })\\\\cdot x)}{\\\\exp \\\\left(n_{k} \\\\cdot \\\\frac{\\\\widetilde{c}\\\\cdot M_{\\\\mathbf {P}}}{\\\\gamma }\\\\cdot (1+\\\\frac{\\\\gamma -1}{\\\\widetilde{c} \\\\cdot M_{\\\\mathbf {P}}}\\\\cdot \\\\tau _{k})^{\\\\gamma /(\\\\gamma -1)}\\\\right)}\\\\cdot f_{\\\\breve{\\\\breve{Z}}}(x) \\\\cdot {1}_{]0,\\\\infty [}(x),\\\\qquad x \\\\in \\\\mathbb {R},$ where $\\\\tau _{k} = \\\\widetilde{c} \\\\cdot M_{\\\\mathbf {P}} \\\\cdot \\\\frac{1-\\\\big (\\\\frac{\\\\widetilde{q}_{k}^{\\\\ast } \\\\cdot M_{\\\\mathbf {P}}}{p_{k}}\\\\big )^{\\\\gamma -1}}{1-\\\\gamma }$ for $\\\\widetilde{q}_{k}^{\\\\ast } >0$ , and $\\\\breve{\\\\breve{Z}}$ is a random variable with density $f_{\\\\breve{\\\\breve{Z}}}$ of a stable law with parameter-quadruple $(\\\\frac{-\\\\gamma }{1-\\\\gamma },1,0,- n_{k} \\\\cdot \\\\frac{(\\\\widetilde{c}\\\\cdot M_{\\\\mathbf {P}})^{1/(1-\\\\gamma )} \\\\cdot (1-\\\\gamma )^{-\\\\gamma /(1-\\\\gamma )}}{\\\\gamma })$ (analogously to $\\\\breve{Z}$ of Example 40 (d) but with $\\\\widetilde{c}$ replaced by $\\\\widetilde{c}\\\\cdot M_{\\\\mathbf {P}}$ ).\",\n \"Also, $\\\\widetilde{ISF}_{k}(x)=e^{-\\\\tau _{k}x} \\\\cdot \\\\exp \\\\left( \\\\frac{n_{k}\\\\cdot \\\\widetilde{c} \\\\cdot M_{\\\\mathbf {P}}}{\\\\gamma }\\\\cdot \\\\left(\\\\left(1+ \\\\frac{\\\\gamma -1}{\\\\widetilde{c} \\\\cdot M_{\\\\mathbf {P}}} \\\\cdot \\\\tau _{k}\\\\right) ^{\\\\frac{\\\\gamma }{\\\\gamma -1}}-1 \\\\right)\\\\right), \\\\qquad x >0.$ Case 5 : Example REF (e): $\\\\gamma >2,\\\\widetilde{c}>0$ .\",\n \"We derive that $\\\\widetilde{U}_{k}^{\\\\ast n_{k}}$ has the (Lebesgue-)density $f_{\\\\widetilde{U}_{k}^{\\\\ast n_{k}}}(x)\\\\ := \\\\ \\\\frac{\\\\exp ((\\\\tau _{k} + \\\\frac{\\\\widetilde{c}\\\\cdot M_{\\\\mathbf {P}}}{\\\\gamma -1})\\\\cdot x)}{\\\\exp \\\\left(n_{k} \\\\cdot \\\\frac{\\\\widetilde{c}\\\\cdot M_{\\\\mathbf {P}}}{\\\\gamma }\\\\cdot (1+\\\\frac{\\\\gamma -1}{\\\\widetilde{c} \\\\cdot M_{\\\\mathbf {P}}}\\\\cdot \\\\tau _{k})^{\\\\gamma /(\\\\gamma -1)}\\\\right)}\\\\cdot f_{\\\\breve{\\\\breve{Z}}}(-x) ,\\\\qquad x \\\\in \\\\mathbb {R},$ where $\\\\tau _{k} = - \\\\frac{\\\\widetilde{c} \\\\cdot M_{\\\\mathbf {P}}}{\\\\gamma -1} \\\\cdot \\\\big (1- \\\\big (\\\\frac{\\\\widetilde{q}_{k}^{\\\\ast } \\\\cdot M_{\\\\mathbf {P}}}{p_{k}}\\\\big )^{\\\\gamma -1}\\\\cdot {1}_{]0,\\\\infty [}(\\\\widetilde{q}_{k}^{\\\\ast }) \\\\big )$ for $\\\\widetilde{q}_{k}^{\\\\ast } \\\\in \\\\mathbb {R}$ , and $\\\\breve{\\\\breve{Z}}$ is a random variable with density $f_{\\\\breve{\\\\breve{Z}}}$ of a stable law with parameter-quadruple $(\\\\frac{\\\\gamma }{\\\\gamma -1},1,0,n_{k} \\\\cdot \\\\frac{(\\\\widetilde{c}\\\\cdot M_{\\\\mathbf {P}})^{1/(1-\\\\gamma )} \\\\cdot (\\\\gamma -1)^{\\\\gamma /(\\\\gamma -1)}}{\\\\gamma })$ (analogously to $\\\\breve{Z}$ of Example 40 (e) but with $\\\\widetilde{c}$ replaced by $\\\\widetilde{c}\\\\cdot M_{\\\\mathbf {P}}$ ).\",\n \"Furthermore, $\\\\widetilde{ISF}_{k}(x)=e^{-\\\\tau _{k}x} \\\\cdot \\\\exp \\\\left( \\\\frac{n_{k}\\\\cdot \\\\widetilde{c} \\\\cdot M_{\\\\mathbf {P}}}{\\\\gamma }\\\\cdot \\\\left(\\\\left(1+ \\\\frac{\\\\gamma -1}{\\\\widetilde{c} \\\\cdot M_{\\\\mathbf {P}}} \\\\cdot \\\\tau _{k}\\\\right) ^{\\\\frac{\\\\gamma }{\\\\gamma -1}}-1 \\\\right)\\\\right), \\\\qquad x \\\\in \\\\mathbb {R}.$ Case 6: Example REF (a): $\\\\gamma =1,\\\\widetilde{c}>0$ , anchor point $c=0$ .\",\n \"It holds that $\\\\widetilde{U}_{k}^{\\\\ast n_{k}}$ is the probability distribution $\\\\frac{1}{\\\\widetilde{c} \\\\cdot M_{\\\\mathbf {P}}} \\\\cdot POI\\\\left(n_{k} \\\\cdot \\\\widetilde{c} \\\\cdot M_{\\\\mathbf {P}} \\\\cdot \\\\exp (\\\\frac{\\\\tau _{k}}{\\\\widetilde{c} \\\\cdot M_{\\\\mathbf {P}}})\\\\right) $ with support on the lattice $\\\\left\\\\lbrace \\\\frac{j}{\\\\widetilde{c} \\\\cdot M_{\\\\mathbf {P}}}, \\\\, j\\\\in \\\\mathbb {N}_{0}\\\\right\\\\rbrace $ , where $\\\\tau _{k}=\\\\widetilde{c} \\\\cdot \\\\log \\\\left( \\\\frac{M_{\\\\mathbf {P}} \\\\cdot \\\\widetilde{q}_{k}^{\\\\ast }}{p_{k}}\\\\right) $ for $\\\\widetilde{\\\\omega } _{k} >0$ .\",\n \"Moreover, for all $j \\\\in \\\\mathbb {N}_{0}$ we obtain (by setting $x:= \\\\frac{j}{\\\\widetilde{c} \\\\cdot M_{\\\\mathbf {P}}}$ ) $\\\\widetilde{ISF}_{k}\\\\left(\\\\frac{j}{\\\\widetilde{c} \\\\cdot M_{\\\\mathbf {P}}}\\\\right)= \\\\exp \\\\left(n_{k} \\\\cdot \\\\widetilde{c} \\\\cdot M_{\\\\mathbf {P}} \\\\cdot \\\\left( \\\\exp \\\\left( \\\\frac{\\\\tau _{k}}{\\\\widetilde{c} \\\\cdot M_{\\\\mathbf {P}}}\\\\right) -1 \\\\right)- m \\\\cdot \\\\frac{\\\\tau _{k}}{\\\\widetilde{c} \\\\cdot M_{\\\\mathbf {P}}}\\\\right) .$ Case 7: Example REF (b): $\\\\gamma =1,\\\\widetilde{c}=1,$ anchor point $c\\\\in \\\\mathbb {R}$ .\",\n \"For $M_{P}=1$ , $\\\\widetilde{U}_{k}^{\\\\ast n_{k}}$ is the shifted Poisson distribution $ POI\\\\left(n_{k} \\\\cdot e^{c +\\\\tau _{k}}\\\\right) + n_{k} \\\\cdot (1-e^{c})$ with support on the lattice $\\\\left\\\\lbrace j + n_{k} \\\\cdot (1-e^{c}), \\\\, j\\\\in \\\\mathbb {N}_{0}\\\\right\\\\rbrace $ , where $\\\\tau _{k}= \\\\log \\\\big (\\\\frac{\\\\widetilde{q}_{k}^{\\\\ast }}{p_{k}} + e^{c}-1\\\\big )-c$ for $\\\\widetilde{q}_{k}^{\\\\ast } > p_{k} \\\\cdot (1-e^{c})$ .\",\n \"Furthermore, for all $j \\\\in \\\\mathbb {N}_{0}$ we obtain (by setting $x:= j + n_{k} \\\\cdot (1-e^{c})$ ) $\\\\widetilde{ISF}_{k}\\\\left(j + n_{k} \\\\cdot (1-e^{c})\\\\right)= \\\\exp \\\\left(n_{k} \\\\cdot e^{c} \\\\cdot (e^{\\\\tau _{k}} -1) - j \\\\cdot \\\\tau _{k} \\\\right).$ Notice that the mass of $\\\\widetilde{U}_{k}^{\\\\ast n_{k}}$ at zero depends on the value of the anchor point $c$ , since $\\\\widetilde{U}_{k}^{\\\\ast n_{k}}[\\\\lbrace 0\\\\rbrace ] >0$ if and only if $c=\\\\log (1+\\\\frac{\\\\ell }{n_{k}})$ for some $\\\\ell \\\\in \\\\mathbb {N}_{0}$ ; moreover, $\\\\widetilde{U}_{k}^{\\\\ast n_{k}}\\\\big [ \\\\ ]0,\\\\infty [ \\\\ \\\\big ] =1$ if $c <0$ and $\\\\widetilde{U}_{k}^{\\\\ast n_{k}}\\\\big [ \\\\ ]-\\\\infty ,0[ \\\\ \\\\big ] >0$ if $c >0$ .\",\n \"Remark 59 (a) One can explicitly see in all cases of the above Example REF that all ingredients for computation are at hand.\",\n \"(b) For both Cases 4 and 5 in the above Example REF , algorithms for simulation can be obtained by adapting e.g.\",\n \"the works of Devroye [111] and Devroye & James [112].\",\n \"In the previous Subsection REF , as a first step we have estimated $\\\\mathbb {\\\\Pi }\\\\left[\\\\xi _{n}^{\\\\mathbf {\\\\widetilde{W}}}\\\\in \\\\widetilde{\\\\mathbf {\\\\Omega }}\\\\right]$ in terms of the improved IS estimator $\\\\widehat{\\\\widetilde{\\\\Pi }}_{L}^{improved}$ .\",\n \"From this, as a second step, we have derived — on the basis of Theorem REF — the estimator $\\\\widehat{D_{\\\\varphi }}(\\\\mathbf {\\\\Omega },\\\\mathbf {P})\\\\ := \\\\ - \\\\frac{1}{n}\\\\log \\\\widehat{\\\\widetilde{\\\\Pi }}_{L}^{improved} \\\\ \\\\nonumber \\\\qquad \\\\textrm {(cf.\",\n \"(\\\\ref {estimator minimization}))}$ of the minimum distance $D_{\\\\varphi }(\\\\mathbf {\\\\Omega },\\\\mathbf {P}):= \\\\inf _{\\\\mathbf {Q}\\\\in \\\\mathbf {\\\\Omega } } D_{\\\\varphi }(\\\\mathbf {Q},\\\\mathbf {P})$ , where $\\\\mathbf {P} \\\\in \\\\mathbb {R}_{> 0}^{K}$ and $\\\\mathbf {\\\\Omega }\\\\subset \\\\mathbb {R}^{K}$ .\",\n \"Recall that $\\\\widetilde{\\\\mathbf {\\\\Omega }} :=\\\\mathbf {\\\\Omega } /M_{\\\\mathbf {P}}$ with $M_{\\\\mathbf {P}}:=\\\\sum _{i=1}^{K}p_{i}>0$ , and that $\\\\xi _{n}^{\\\\mathbf {\\\\widetilde{W}}}=\\\\Big (\\\\frac{1}{n}\\\\sum _{i\\\\in I_{1}^{(n)}}\\\\widetilde{W}_{i},\\\\ldots ,\\\\frac{1}{n}\\\\sum _{i\\\\in I_{K}^{(n)}}\\\\widetilde{W}_{i}\\\\Big )\\\\hspace{56.9055pt} \\\\text{(cf.\",\n \"(\\\\ref {Xi_n^W vector}))}\\\\nonumber $ where $\\\\mathbf {\\\\widetilde{W}} := (\\\\widetilde{W}_{1}, \\\\ldots , \\\\widetilde{W}_{n})$ is a random vector consisting of components $\\\\widetilde{W}_{i}$ which are i.i.d.\",\n \"copies of the random variable $\\\\widetilde{W}$ whose distribution is $\\\\mathbb {\\\\Pi }[\\\\widetilde{W}\\\\in \\\\cdot \\\\,]=\\\\widetilde{\\\\mathbb {}}[\\\\,\\\\cdot \\\\,]$ obeying the representation (REF ).\",\n \"In contrast, we now proceed as follows: as a first step, we derive an improved estimator $\\\\widehat{\\\\Pi }_{L}^{improved}$ of $\\\\mathbb {\\\\Pi }_{X_{1}^{n}}\\\\left[\\\\xi _{n,\\\\mathbf {X}}^{w\\\\mathbf {W}}\\\\in \\\\textrm {\\\\right.$ $\\\\hspace{-6.544pt}$$}$$where $$\\\\hspace{-6.544pt}$ SK$is a set of probability vectors which satisfies theregularity properties (\\\\ref {regularity}) and the finiteness property (\\\\ref {def fi wrt Omega}).Recall that\\\\begin{eqnarray}\\\\xi _{n,\\\\mathbf {X}}^{w\\\\mathbf {W}} &:=&{\\\\left\\\\lbrace \\\\begin{array}{ll}\\\\left(\\\\frac{\\\\sum _{i \\\\in I_{1}^{(n)}}W_{i}}{\\\\sum _{k=1}^{K}\\\\sum _{i \\\\in I_{k}^{(n)}}W_{i}},\\\\ldots , \\\\frac{\\\\sum _{i \\\\in I_{K}^{(n)}}W_{i}}{\\\\sum _{k=1}^{K}\\\\sum _{i \\\\in I_{k}^{(n)}}W_{i}} \\\\right) ,\\\\qquad \\\\textrm {if } \\\\sum _{j=1}^{n} W_{j} \\\\ne 0, \\\\\\\\\\\\ (\\\\infty , \\\\ldots , \\\\infty ) =: \\\\infty , \\\\hspace{113.81102pt} \\\\textrm {if } \\\\sum _{j=1}^{n} W_{j} = 0,\\\\end{array}\\\\right.\",\n \"}\\\\hspace{42.67912pt}\\\\textrm {(cf.\",\n \"(\\\\ref {brostu3:fo.norweiemp.vec}))}\\\\nonumber \\\\end{eqnarray}where $ (Xi)iN$ is a sequence of random variableswith values in $ Y:={ d1,,dK}$such that\\\\begin{equation}\\\\lim _{n\\\\rightarrow \\\\infty } \\\\Big ( \\\\frac{n_{1}}{n}, \\\\ldots , \\\\frac{n_{K}}{n} \\\\Big ) = (p_{1}, \\\\ldots , p_{K})\\\\qquad \\\\textrm {a.s.} \\\\qquad \\\\textrm {cf.\",\n \"((\\\\ref {cv emp measure X to P vector}))}\\\\nonumber \\\\end{equation}holds for some probability vector $ P := (p1, ..., pK) S> 0K$,by employing the notation$$n_{k} \\\\ := \\\\ card(\\\\bigl \\\\lbrace i \\\\in \\\\lbrace 1, \\\\ldots , n\\\\rbrace : \\\\ X_{i} = d_{k} \\\\bigr \\\\rbrace )\\\\ =: \\\\ card(I_{k}^{(n)})\\\\qquad \\\\textrm {(cf.\",\n \"(\\\\ref {I^(n)_k for stat case}));}$$hence, on the $ k$-th block of indexes $ Ik(n)$ all the $ Xi$^{\\\\prime }s share the same value $ dk$.Moreover, recall that $ (Wi)i N$ is a familyof independent and identically distributed $ R-$valued random variableswith probability distribution $ [ ] := [W1 ]$being connected with the divergence generator $ (]a,b[)$ via the representability(\\\\ref {Phi Legendre of mgf(W)}),such that $ (Wi)i N$ is independent of $ (Xi)i N$.$ As a second step (see Subsubsection REF below), for the important special case of the power-divergence generators $\\\\varphi _{\\\\gamma }$ (cf.\",\n \"(REF )) we employ the Propositions REF to REF in order to deduce via the corresponding $\\\\widehat{\\\\Pi }_{L}^{improved}$ the estimators (e.g.\",\n \"for $\\\\gamma <0$ ) $\\\\widehat{D_{\\\\widetilde{c}\\\\cdot \\\\varphi _{\\\\gamma }}(\\\\textrm {\\\\Omega \\\\hspace{-6.544pt}\\\\Omega },{P})} \\\\ := \\\\ \\\\frac{\\\\widetilde{c}}{\\\\gamma \\\\cdot \\\\left(\\\\gamma -1\\\\right) }\\\\cdot \\\\left\\\\lbrace \\\\left( 1+\\\\frac{\\\\gamma }{\\\\widetilde{c}}\\\\cdot \\\\frac{1}{n}\\\\cdot \\\\log \\\\,\\\\widehat{\\\\Pi }_{L}^{improved}\\\\right) ^{1-\\\\gamma } -1 \\\\right\\\\rbrace ,\\\\qquad \\\\ $ of the minimum power divergences $D_{\\\\widetilde{c}\\\\cdot \\\\varphi _{\\\\gamma }}(\\\\textrm {$ $\\\\hspace{-6.544pt}$$},{P}) :=\\\\inf _{{Q}\\\\in \\\\textrm {\\\\Omega \\\\hspace{-5.40608pt}\\\\Omega }}D_{\\\\widetilde{c}\\\\cdot \\\\varphi _{\\\\gamma }}({Q},{P})$$as well as connected estimators of important deterministic transformations thereof.$ As a third step (see Subsubsection REF below), on the basis of Subsubsection REF we derive estimators of bounds of $D_{\\\\varphi }(\\\\textrm {$$\\\\hspace{-6.544pt}$$},{P})$ for more general divergence generators $\\\\varphi $ .\",\n \"Let us start with the above-mentioned first step, by remarking that the development of the estimator $\\\\widehat{\\\\Pi }_{L}^{improved}$ works quite analogously to that of $\\\\widehat{\\\\widetilde{\\\\Pi }}_{L}^{improved}$ in the previous Subsection REF .\",\n \"To make this even more transparent, we employ the notation $p_{n,k}^{emp}:=n_{k}/n$ (cf.\",\n \"(REF )) and label all random vectors of length $n$ in the same way as above: we sort the already given and thus fixed data $X_{i}$ 's in such a way that the first $n_{1}$ of them share the same value $d_{1}$ , and so on, until the last block with length $n_{K}$ in which the data have common value $d_{K}$ .\",\n \"In the light of the above considerations, we could achieve a naive estimate $\\\\widehat{\\\\Pi }_{L}^{naive}$ of $\\\\mathbb {\\\\Pi }_{X_{1}^{n}}[\\\\xi _{n,\\\\mathbf {X}}^{w\\\\mathbf {W}}\\\\in \\\\textrm {$$\\\\hspace{-6.544pt}$$}]$ through the following procedure.\",\n \"We simulate independently $L$ replicates $\\\\mathbf {W}^{(1)},\\\\ldots ,\\\\mathbf {W}^{(L)}$ of the vector $\\\\mathbf {W}:=\\\\left( W_{1},\\\\ldots ,W_{n}\\\\right) $ , with independent entries under $\\\\mathbb {}$ (cf.\",\n \"(REF )); those realizations do not depend on the $X_{i}$ 's.\",\n \"Then we construct $\\\\widehat{\\\\Pi }_{L}^{naive} :=\\\\frac{1}{L}\\\\sum _{\\\\ell =1}^{L}\\\\mathbf {1}_{\\\\textrm {\\\\Omega \\\\hspace{-5.40608pt}\\\\Omega }}\\\\left(\\\\xi _{n,\\\\mathbf {X}}^{w\\\\mathbf {W}^{(\\\\ell )}}\\\\right) .$ However, this procedure is time costly, since the estimate given in (REF ) has a very bad hit rate.\",\n \"Hence, analogously to Subsection REF we apply again an “efficient Importance Sampling (IS)” scheme in the sense of Sadowsky & Bucklew [313].\",\n \"This will involve the simulation of $L$ independent $n-$ tuples $\\\\mathbf {V}^{(\\\\ell )}\\\\mathbf {:=}\\\\left(V_{n}^{(\\\\ell )},\\\\ldots ,V_{n}^{(\\\\ell )}\\\\right) $ with common distribution $S$ on $\\\\mathbb {R}^{n}$ , such that $\\\\mathbb {}^{\\\\otimes n}$ is (measure-)equivalent with respect to $S$ .\",\n \"In fact, we rewrite $\\\\mathbb {\\\\Pi }_{X_{1}^{n}}[\\\\xi _{n,\\\\mathbf {X}}^{w\\\\mathbf {W}}\\\\in \\\\textrm {$$\\\\hspace{-6.544pt}$$}]$ as $\\\\mathbb {\\\\Pi }_{X_{1}^{n}}[\\\\xi _{n,\\\\mathbf {X}}^{w\\\\mathbf {W}}\\\\in \\\\textrm {\\\\Omega \\\\hspace{-6.544pt}\\\\Omega }]= E_{S}\\\\Big [\\\\frac{\\\\mathrm {d}\\\\mathbb {} ^{\\\\otimes n}}{\\\\mathrm {d}S}(V_{1},\\\\ldots ,V_{n})\\\\cdot \\\\mathbf {1}_{\\\\textrm {\\\\Omega \\\\hspace{-5.40608pt}\\\\Omega }}(\\\\xi _{n,\\\\mathbf {X}}^{w\\\\mathbf {V}})\\\\Big ]$ where $S$ designates any IS distribution of the vector $\\\\mathbf {V:=}(V_{1},\\\\ldots ,V_{n})$ , and $E_{S}[ \\\\, \\\\cdot \\\\, ]$ denotes the corresponding expectation operation.\",\n \"Notice that $S$ is a random probability distribution on $\\\\mathbb {R}^{n}$ ; in fact, $S$ is a conditional probability distribution given $X_{1}^{n}$ , and thus it would be more precise to write $S | X_{1}^{n}$ instead of $S$ ; for the sake of brevity, we omit $| X_{1}^{n}$ .\",\n \"As a consequence of (REF ), for adequately chosen $S$ , an improved estimator of $\\\\mathbb {\\\\Pi }_{X_{1}^{n}}[\\\\xi _{n,\\\\mathbf {X}}^{w\\\\mathbf {W}}\\\\in \\\\textrm {$$\\\\hspace{-6.544pt}$$}]$ is given by $\\\\widehat{\\\\Pi }_{L}^{improved}:= \\\\frac{1}{L}\\\\sum _{\\\\ell =1}^{L}\\\\frac{\\\\mathrm {d}\\\\mathbb {}^{\\\\otimes n}}{\\\\mathrm {d}S}(V_{1}^{(\\\\ell )},\\\\ldots ,V_{n}^{(\\\\ell )})\\\\cdot \\\\mathbf {1}_{\\\\textrm {\\\\Omega \\\\hspace{-5.40608pt}\\\\Omega }}(\\\\xi _{n,\\\\mathbf {X}}^{w\\\\mathbf {V}^{(\\\\ell )}}) \\\\, ,$ which also estimates $\\\\inf _{{Q}\\\\in \\\\textrm {\\\\Omega \\\\hspace{-5.40608pt}\\\\Omega } }\\\\ \\\\inf _{m\\\\ne 0}D_{\\\\varphi }(m\\\\cdot {Q},{P})$ by the virtue of ().\",\n \"Let us now deal with the concrete construction of a reasonable $S$ .\",\n \"Given some (typically) large integer $M$ , we start with the realization $\\\\mathbf {W}^{\\\\ast }:=\\\\left( W_{1}^{\\\\ast },\\\\ldots ,W_{M}^{\\\\ast }\\\\right)$ such that $\\\\mathbf {Q}^{\\\\ast }:=\\\\xi _{M,\\\\mathbf {X}}^{w\\\\mathbf {W}^{\\\\ast }} \\\\in int(\\\\textrm {$$\\\\hspace{-6.544pt}$$})$ .\",\n \"This may be given in advance or it may be achieved by drawing replicates $\\\\mathbf {W} = (W_{1}, \\\\ldots , W_{M})$ under $\\\\mathbb {}^{\\\\otimes M}$ until the first time where $\\\\xi _{M,\\\\mathbf {X}}^{w\\\\mathbf {W}}$ belongs to $int(\\\\textrm {$$\\\\hspace{-6.544pt}$$})$ .\",\n \"Notice that by the nature of $\\\\textrm {$$\\\\hspace{-6.544pt}$$}$ , $\\\\mathbf {Q}^{\\\\ast }$ is a probability vector which has the $K$ components $q_{k}^{\\\\ast }:=\\\\sum _{i=1}^{M}\\\\frac{W_{i}^{\\\\ast }}{\\\\sum _{j=1}^{M}W_{j}^{\\\\ast }}\\\\mathbf {1}_{\\\\lbrace d_{k}\\\\rbrace }(X_{i}),\\\\hspace{42.67912pt} k=1,\\\\ldots ,K.$ Before we proceed, let us give the substantial remark that changing $\\\\left( V_{1},\\\\ldots ,V_{n}\\\\right) $ drawn under $S$ to $\\\\left(c \\\\cdot V_{1}, \\\\ldots ,c \\\\cdot V_{n}\\\\right) $ for any $c\\\\ne 0$ yields $\\\\xi _{n,\\\\mathbf {X}}^{w\\\\mathbf {V}}= \\\\xi _{n,\\\\mathbf {X}}^{w\\\\, c\\\\cdot \\\\mathbf {V}}$ so that the distribution $S$ is not uniquely determined.\",\n \"Amongst all candidates, we choose the — uniquely determined — $S$ which is the Kullback-Leibler projection of $\\\\mathbb {}^{\\\\otimes n}$ on the set of all probability distributions on $\\\\mathbb {R}^{n}$ such that the $K$ “non-normalized” moment constraints $E_{S}[\\\\xi _{n,\\\\mathbf {X}}^{\\\\mathbf {V}}] =\\\\xi _{M,\\\\mathbf {X}}^{\\\\mathbf {W}^{\\\\ast }}$ (rather than the normalized $E_{S}[ \\\\xi _{n,\\\\mathbf {X}}^{w\\\\mathbf {V}}] =\\\\xi _{M,\\\\mathbf {X}}^{w\\\\mathbf {W}^{\\\\ast }}$ ) are satisfied, with the non-normalized vectors $\\\\xi _{M,\\\\mathbf {X}}^{\\\\mathbf {W}^{\\\\ast }} :=\\\\left( \\\\frac{1}{M}\\\\sum _{j=1}^{M}W_{j}^{\\\\ast }\\\\right) \\\\cdot Q^{\\\\ast } =: \\\\overline{W^{\\\\ast }} \\\\cdot Q^{\\\\ast },\\\\qquad \\\\xi _{n,\\\\mathbf {X}}^{\\\\mathbf {V}} :=\\\\left( \\\\frac{1}{n}\\\\sum _{j=1}^{n}V_{j}\\\\right) \\\\cdot \\\\xi _{n,\\\\mathbf {X}}^{w\\\\mathbf {V}} \\\\ .$ As already indicated above, this projection $S$ is a well-determined unique distribution on $\\\\mathbb {R}^{n}$ and — as we shall see in Proposition REF below — it is such that $\\\\xi _{n,\\\\mathbf {X}}^{w\\\\mathbf {V}}$ belongs to $\\\\textrm {$$\\\\hspace{-6.544pt}$$}$ with probability bounded away from 0 as $n$ increases, when $\\\\left( V_{1},\\\\ldots ,V_{n}\\\\right) $ are drawn under $S$ .\",\n \"Therefore, this IS distribution produces an estimate of $\\\\mathbb {\\\\Pi }_{X_{1}^{n}}[\\\\xi _{n,\\\\mathbf {X}}^{w\\\\mathbf {W}}\\\\in \\\\textrm {$$\\\\hspace{-6.544pt}$$}]$ .\",\n \"In order to justify the above construction of $S$ , we give the following result, which states that the IS sampling distribution $S$ yields a good hitting rate.\",\n \"Its proof will be given in Appendix H. Proposition 60 With the above definition of $S$ , $\\\\lim \\\\inf _{n\\\\rightarrow \\\\infty }$ $S\\\\left[ \\\\xi _{n,\\\\mathbf {X}}^{w\\\\mathbf {V}} \\\\in \\\\textrm {\\\\right.$$\\\\hspace{-6.544pt}$$} $ is bounded away from 0.\",\n \"We now come to the detailed construction of $S$ .\",\n \"The constraints (REF ) can be written in explicit form as $E_{S}\\\\Big [ \\\\, \\\\frac{1}{n_{k}}\\\\sum _{i\\\\in I_{k}^{(n)}} V_{i} \\\\, \\\\Big ]\\\\ = \\\\ \\\\overline{W^{\\\\ast }} \\\\cdot \\\\frac{q_{k}^{\\\\ast }}{p_{n,k}^{emp}} \\\\, , \\\\qquad k=1,\\\\ldots ,K.$ The distribution $S$ can be obtained by blocks.\",\n \"Indeed, let us define $S^{k}$ as the Kullback-Leibler (KL) projection of $\\\\mathbb {}^{\\\\otimes n_{k}}$ on the set of all distributions on $\\\\mathbb {R}^{n_{k}}$ such that (REF ) holds.\",\n \"We define the resulting $S$ as the product distribution of those $S^{k}$ 's.\",\n \"To obtain the latter, we start by defining $U_{k}$ as the KL projection of $\\\\mathbb {} $ on the set of all measures $Q$ on $\\\\mathbb {R}$ under (REF ).\",\n \"Then, $dU_{k}(v)= \\\\exp (\\\\tau _{k}v-\\\\Lambda _{\\\\mathbb {} }\\\\left( \\\\tau _{k}\\\\right) )\\\\, d\\\\mathbb {} (v) \\\\, ,$ where $\\\\tau _{k} \\\\in int(dom(MGF_{\\\\mathbb {}}))$ is the unique solution of the equation $\\\\Lambda _{\\\\mathbb {} }^{\\\\prime }\\\\left( \\\\tau _{k}\\\\right) =\\\\overline{W^{\\\\ast }} \\\\cdot \\\\frac{q_{k}^{\\\\ast }}{p_{n,k}^{emp}}$ and thus — by relation (REF ) of Appendix F — we can compute explicitly $\\\\tau _{k} = \\\\varphi ^{\\\\prime }\\\\left( \\\\frac{\\\\overline{W^{\\\\ast }} \\\\cdot q_{k}^{\\\\ast }}{p_{n,k}^{emp}} \\\\right) .$ The distribution $S^{k}$ is then defined by $S^{k}:=\\\\underbrace{U_{k}\\\\otimes \\\\cdots \\\\otimes U_{k}}_{n_{k}\\\\text{times}}$ from which we obtain $S := S^{1}\\\\otimes \\\\cdots \\\\otimes S^{K}.$ With this construction, it holds $\\\\frac{dS}{d\\\\mathbb {} ^{\\\\otimes n}}(v_{1},\\\\ldots ,v_{n})=\\\\exp \\\\left(\\\\sum \\\\limits _{k=1}^{K}\\\\left( \\\\sum _{i\\\\in I_{k}^{(n)}}\\\\tau _{k} \\\\cdot v_{i}-\\\\Lambda _{\\\\mathbb {}}(\\\\tau _{k})\\\\right) \\\\right)$ which proves that $S$ is indeed the KL projection of $\\\\mathbb {} ^{\\\\otimes n}$ we aimed at.\",\n \"Therefore, $\\\\mathbf {V}$ is composed of $K$ independent blocks of length $n_{k}$ each, and the $k-$ th subvector $\\\\mathbf {V}_{k}$ consists of all the random variables $V_{i}$ whose index $i$ satisfies $X_{i}=d_{k}.$ Within $\\\\mathbf {V}_{k}$ , all components are i.i.d.\",\n \"with same distribution $U_{k}$ on $\\\\mathbb {R}$ defined through $\\\\frac{dU_{k}}{d\\\\mathbb {} }(u)=\\\\exp \\\\left\\\\lbrace \\\\tau _{k}\\\\cdot u-\\\\Lambda _{\\\\mathbb {}}(\\\\tau _{k})\\\\right\\\\rbrace =\\\\frac{\\\\exp \\\\left\\\\lbrace \\\\tau _{k}\\\\cdot u\\\\right\\\\rbrace }{MGF_{\\\\mathbb {} }(\\\\tau _{k})},$ which leads to the moment generating function $dom(MGF_{\\\\mathbb {}})-\\\\tau _{k} \\\\ \\\\ni \\\\ z \\\\ \\\\mapsto MGF_{U_{k}}(z):=\\\\int _{\\\\mathbb {R}}e^{zy}dU_{k}(y)=\\\\frac{MGF_{\\\\mathbb {}}(z+\\\\tau _{k})}{MGF_{\\\\mathbb {} }(\\\\tau _{k})}.$ Let us remark that $U_{k}$ can be interpreted as the distorted distribution of $\\\\mathbb {}$ with the distortion parameter $\\\\tau _{k}$ (in some cases, this distortion even becomes a tilting/dampening).\",\n \"The estimator $\\\\widehat{\\\\Pi }_{L}^{improved}$ defined in (REF ) can be implemented through the following algorithm: Step S1 Choose some (typically large) $M$ and simulate repeatedly i.i.d.\",\n \"vectors $\\\\left(W_{1},..,W_{M}\\\\right) $ — whose independent components have common distribution $\\\\mathbb {} $ — until $\\\\xi _{M,\\\\mathbf {X}}^{w\\\\mathbf {W}}$ belongs to $\\\\textrm {$$\\\\hspace{-6.544pt}$$}$ .\",\n \"Call $\\\\left( W_{1}^{\\\\ast },..,W_{M}^{\\\\ast }\\\\right) $ the corresponding vector and $\\\\overline{W^{\\\\ast }}$ the arithmetic mean of its components.\",\n \"Moreover, denote by $\\\\xi _{M,\\\\mathbf {X}}^{w\\\\mathbf {W}^{\\\\ast }}$ the corresponding normalized weighted empirical measure, identified with the $K-$ component vector $Q^{\\\\ast } := (q_{1}^{\\\\ast },..,q_{K}^{\\\\ast })$ with $q_{k}^{\\\\ast }$ defined in (REF ).\",\n \"Step S2 For all $k \\\\in \\\\lbrace 1,\\\\ldots ,K\\\\rbrace $ compute $\\\\tau _{k} = \\\\varphi ^{\\\\prime }\\\\left( \\\\frac{\\\\overline{W^{\\\\ast }} \\\\cdot q_{k}^{\\\\ast }}{p_{n,k}^{emp}} \\\\right)$ .\",\n \"Step S3 For all $\\\\ell \\\\in \\\\lbrace 1,\\\\ldots ,L\\\\rbrace $ simulate independently for all $k \\\\in \\\\lbrace 1,\\\\ldots ,K\\\\rbrace $ a row vector $\\\\mathbf {V}_{k}^{(\\\\ell )}$ $:=\\\\left(V_{k_{1}}^{(\\\\ell )},...,V_{k_{n_{k}}}^{(\\\\ell )}\\\\right) $ with independent components with common distribution $U_{k}$ defined in (REF ).\",\n \"Concatenate these vectors to define the row vector $\\\\mathbf {V}^{(\\\\ell )}.$ Step S4 Compute the estimator $\\\\widehat{\\\\Pi }_{L}^{improved}$ by making use of the formula (REF ) which turns into the explicit form $& & \\\\widehat{\\\\Pi }_{L}^{improved} =\\\\frac{1}{L}\\\\sum _{\\\\ell =1}^{L}\\\\exp \\\\left(\\\\sum \\\\limits _{k=1}^{K}\\\\left(n_{k} \\\\cdot \\\\Lambda _{\\\\mathbb {} }(\\\\tau _{k}) -\\\\tau _{k} \\\\cdot \\\\sum _{i\\\\in I_{k}^{(n)}}V_{i}^{(\\\\ell )}\\\\right) \\\\right)\\\\cdot \\\\mathbf {1}_{\\\\textrm {\\\\Omega \\\\hspace{-5.40608pt}\\\\Omega }}\\\\left( \\\\xi _{n,\\\\mathbf {X}}^{w\\\\mathbf {V}^{(\\\\ell )}}\\\\right)$ Analogously to the paragraph right after (REF ) of the previous Subsection REF , in many cases we may improve the simulation burden needed for the computation of the estimator $\\\\widehat{\\\\Pi }_{L}^{improved}$ .\",\n \"In fact, in terms of the notations $\\\\widehat{\\\\mathit {W}}_{k}^{(\\\\ell )}:=\\\\sum _{i\\\\in I_{k}^{(n)}}V_{i}^{(\\\\ell )}$ we can rewrite (REF ) as $\\\\widehat{\\\\Pi }_{L}^{improved} =\\\\frac{1}{L}\\\\sum _{\\\\ell =1}^{L}\\\\mathbf {1}_{\\\\textrm {\\\\Omega \\\\hspace{-5.40608pt}\\\\Omega }}\\\\left( \\\\xi _{n,\\\\mathbf {X}}^{w\\\\mathbf {V}^{(\\\\ell )}} \\\\right)\\\\cdot \\\\prod _{k=1}^{K}ISF_{k}\\\\left(\\\\widehat{\\\\mathit {W}}_{k}^{(\\\\ell )}\\\\right)$ with $ISF_{k}(x) \\\\ := \\\\ \\\\exp (n_{k} \\\\cdot \\\\Lambda _{\\\\mathbb {}}(\\\\tau _{k}) \\\\, - \\\\, x \\\\cdot \\\\tau _{k} )$ and $\\\\xi _{n,\\\\mathbf {X}}^{w\\\\mathbf {V}^{(\\\\ell )}} &=&{\\\\left\\\\lbrace \\\\begin{array}{ll}\\\\left(\\\\frac{\\\\widehat{\\\\mathit {W}}_{1}^{(\\\\ell )}}{\\\\sum _{k=1}^{K}\\\\widehat{\\\\mathit {W}}_{k}^{(\\\\ell )}},\\\\ldots , \\\\frac{\\\\widehat{\\\\mathit {W}}_{K}^{(\\\\ell )}}{\\\\sum _{k=1}^{K}\\\\widehat{\\\\mathit {W}}_{k}^{(\\\\ell )}} \\\\right) ,\\\\qquad \\\\textrm {if } \\\\sum _{k=1}^{K}\\\\widehat{\\\\mathit {W}}_{k}^{(\\\\ell )} \\\\ne 0, \\\\\\\\\\\\ (\\\\infty , \\\\ldots , \\\\infty ) =: \\\\infty , \\\\hspace{65.44142pt} \\\\textrm {if }\\\\sum _{k=1}^{K}\\\\widehat{\\\\mathit {W}}_{k}^{(\\\\ell )} = 0 \\\\, .\\\\end{array}\\\\right.\",\n \"}$ Clearly, the random variable $\\\\widehat{\\\\mathit {W}}_{k}^{(\\\\ell )}$ ($k=1, \\\\ldots , K$ ) has distribution $U_{k}^{\\\\ast n_{k}}$ .\",\n \"Hence, if $U_{k}^{\\\\ast n_{k}}$ can be explicitly constructed, then for the computation of $\\\\widehat{\\\\Pi }_{L}^{improved}$ it suffices to independently simulate the $K \\\\cdot L$ random variables $\\\\widehat{\\\\mathit {W}}_{k}^{(\\\\ell )}$ (rather than the $n \\\\cdot L$ random variables $V_{i}^{(\\\\ell )}$ ).\",\n \"In the following subsubsection, we exemplarily demonstrate the tractability of this reduction effect.\"\n ],\n [\n \"BS minimization of power divergences and related quantities\",\n \"´ Consider the special case of power divergence generators $\\\\varphi := \\\\widetilde{c} \\\\cdot \\\\varphi _{\\\\gamma }$ ($\\\\gamma \\\\in \\\\mathbb {R}\\\\backslash ]1,2[$ ) of the Examples REF and REF .\",\n \"The corresponding estimators $\\\\widehat{\\\\Pi }_{L}^{improved}$ can be obtained as follows: within the results of Example REF , set $M_{\\\\mathbf {P}}=1$ , and replace $\\\\widetilde{q}_{k}^{\\\\ast }$ by $\\\\overline{W^{\\\\ast }} \\\\cdot q_{k}^{\\\\ast }$ as well as $p_{k}$ by $p_{n,k}^{emp}$ ; accordingly, $\\\\widetilde{U}_{k}^{\\\\ast n_{k}}$ turns into $U_{k}^{\\\\ast n_{k}}$ and $\\\\widetilde{ISF}_{k}$ into $ISF_{k}$ ; simulate independently the random variables $\\\\widehat{\\\\mathit {W}}_{k}^{(\\\\ell )}$ from $U_{k}^{\\\\ast n_{k}}$ ($k \\\\in \\\\lbrace 1, \\\\ldots , K\\\\rbrace $ , $\\\\ell \\\\in \\\\lbrace 1,\\\\ldots ,L\\\\rbrace $ ); plug in the results of (i),(ii) into (REF ), (REF ), and (REF ) in order to concretely compute $\\\\widehat{\\\\Pi }_{L}^{improved}$ .\",\n \"From this, we can easily generate improved estimators of the power divergences $\\\\inf _{\\\\mathbf {Q}\\\\in \\\\textrm {\\\\Omega \\\\hspace{-5.40608pt}\\\\Omega }}D_{\\\\widetilde{c}\\\\cdot \\\\varphi _{\\\\gamma }}(\\\\mathbf {Q},{P})$ — and more generally, improved estimators of all the infimum-quantities (e.g.\",\n \"Renyi divergences) respectively supremum-quantities in the parts (b) of the Propositions REF , REF , REF , REF , REF and REF with $A=1$ — by simply replacing $\\\\mathbb {\\\\Pi }_{X_{1}^{n}}[\\\\xi _{n}^{w\\\\mathbf {W}}\\\\in \\\\textrm {$$\\\\hspace{-6.544pt}$$} ]$ (respectively, its variants) by the corresponding estimator $\\\\widehat{\\\\Pi }_{L}^{improved}$ .\",\n \"If — in the light of Remark REF (vi) — the ${P} = (p_{1}, \\\\ldots , p_{K})$ is a pregiven known probability vector e.g.\",\n \"the uniform distribution ${P}^{unif}$ on $\\\\lbrace 1,\\\\ldots ,K\\\\rbrace $ (rather than the limit of the vector of empirical frequencies/masses of a sequence of random variables $X_{i}$ , cf.\",\n \"(REF )), then we proceed analogously as above by replacing $p_{n,k}^{emp}$ with $p_{k}$ ; correspondingly, we obtain improved estimators of all the infimum-quantities respectively supremum-quantities (e.g.\",\n \"Renyi entropies, diversity indices) in the parts (a) of the Propositions REF , REF , REF , REF , REF and REF with $A=1$ .\",\n \"For the sake of brevity, in the following we only present explicitly the outcoming improved estimators for the power divergences (in the “$X_{i}-$ context” ).\",\n \"Indeed, we simply replace the $\\\\mathbb {\\\\Pi }_{X_{1}^{n}}[\\\\xi _{n}^{w\\\\mathbf {W}}\\\\in \\\\textrm {$$\\\\hspace{-6.544pt}$$} ]$ in the formulas (REF ), (REF ), (REF ) (with $A=1$ ) by the improved estimator $\\\\widehat{\\\\Pi }_{L}^{improved}$ obtained through (i) to (iii); for arbitrarily fixed $\\\\widetilde{c} >0$ , this leads to the improved power-divergence estimators (BS estimators of power divergences) $&&\\\\hspace{-54.06006pt}\\\\widehat{D_{\\\\widetilde{c}\\\\cdot \\\\varphi _{\\\\gamma }}(\\\\textrm {\\\\Omega \\\\hspace{-6.544pt}\\\\Omega },{P})} \\\\ := \\\\ -\\\\frac{\\\\widetilde{c}}{\\\\gamma (\\\\gamma -1)}\\\\left\\\\lbrace 1-\\\\left( 1+\\\\frac{\\\\gamma }{\\\\widetilde{c}} \\\\cdot \\\\frac{1}{n} \\\\cdot \\\\log \\\\widehat{\\\\Pi }_{L}^{improved}\\\\right)^{1-\\\\gamma }\\\\right\\\\rbrace ,\\\\qquad \\\\gamma \\\\in \\\\, ]-\\\\infty ,0[ \\\\, \\\\cup \\\\, ]0,1[ \\\\, \\\\cup \\\\, [2,\\\\infty [,\\\\\\\\& &\\\\hspace{-54.06006pt}\\\\widehat{D_{\\\\widetilde{c}\\\\cdot \\\\varphi _{0}}(\\\\textrm {\\\\Omega \\\\hspace{-6.544pt}\\\\Omega },{P})} \\\\ := \\\\ -\\\\frac{1}{n}\\\\log \\\\widehat{\\\\Pi }_{L}^{improved} ,\\\\hspace{165.02606pt} \\\\gamma =0,\\\\\\\\& & \\\\hspace{-54.06006pt}\\\\widehat{D_{\\\\widetilde{c}\\\\cdot \\\\varphi _{1}}(\\\\textrm {\\\\Omega \\\\hspace{-6.544pt}\\\\Omega },{P})}\\\\ : = \\\\ - \\\\widetilde{c} \\\\cdot \\\\log \\\\left( 1+\\\\frac{1}{\\\\widetilde{c}} \\\\cdot \\\\frac{1}{n} \\\\cdot \\\\log \\\\widehat{\\\\Pi }_{L}^{improved}\\\\right) ,\\\\hspace{85.35826pt} \\\\gamma =1.$ Let us finally remark that from the above-mentioned Steps S1 to S4 (and analogously D1 to D4) one can see that for our BS method we basically need only a fast and accurate — pseudo, true, natural, quantum — random number generator.\",\n \"The corresponding computations can be principally run in parallel, and require relatively moderate computer memory/storage; a detailed discussion is beyond the scope of this paper, given its current length.\"\n ],\n [\n \"General case, part 2\",\n \"The algorithm which is presented in this section aims at the evaluation of the bounds $\\\\inf _{m\\\\ne 0}D_{\\\\varphi }\\\\left( m \\\\cdot \\\\textrm {\\\\right.\\\\Omega \\\\hspace{-6.544pt}\\\\Omega },{P} =\\\\inf _{{Q}\\\\in \\\\textrm {\\\\Omega \\\\hspace{-5.40608pt}\\\\Omega }}D_{\\\\varphi }\\\\left( m\\\\left( {Q}\\\\right) \\\\cdot {Q},{P}\\\\right)\\\\stackrel{(1)}{=} D_{\\\\varphi }\\\\left( m({Q}^{\\\\ast })\\\\cdot {Q}^{\\\\ast },{P}\\\\right) \\\\le D_{\\\\varphi }\\\\left( \\\\textrm {\\\\right.\\\\Omega \\\\hspace{-6.544pt}\\\\Omega },{P} \\\\le D_{\\\\varphi }\\\\left( {Q}^{\\\\ast },{P}\\\\right)$ obtained in Section REF , where ${Q}^{\\\\ast }$ satisfies the above equality $(1)$ .\",\n \"The estimator of the lower bound in (REF ) is $\\\\widehat{D}:=-\\\\frac{1}{n}\\\\log \\\\widehat{\\\\text{ }\\\\Pi }_{L}^{improved}$ defined in (REF ).\",\n \"We now turn to an estimate of the upper bound.\",\n \"Consider for any fixed ${Q}:=\\\\left( q_{1},..,q_{K}\\\\right) $ in $\\\\mathbb {S}_{>0}^{K}$ the real number $m_{n}({Q})$ which satisfies $D_{\\\\varphi }\\\\left( m_{n}({Q}) \\\\cdot {Q},{P}_{n}^{emp} \\\\right) =\\\\inf _{m\\\\ne 0}D_{\\\\varphi }\\\\left( m\\\\cdot \\\\textrm {\\\\right.$ $\\\\hspace{-6.544pt}$$},{P}_{n}^{emp}$$where $ Pnemp$ was defined in the course of (\\\\ref {I^(n)_k for stat case}).Such $ mn(Q)$ is well defined for all $ Q$since it satisfies the equation (in $ m$)\\\\begin{equation}\\\\frac{d}{dm}D_{\\\\varphi }\\\\left( m \\\\cdot {Q},{P}_{n}^{emp}\\\\right) =\\\\sum _{k=1}^{K} q_{k} \\\\cdot \\\\varphi ^{\\\\prime }\\\\left( \\\\frac{m \\\\cdot q_{k}}{p_{n,k}^{emp}}\\\\right) =0 .\\\\end{equation}Since the mapping $ mD( m Q,P) $ is convexand differentiable, existence and uniqueness of $ mn(Q)$ hold; furthermore,$ mn(Q)] k pn,kemp/qk,k pn,kemp/qk[ $since $ ddmD( m Q,Pnemp) $is negative when $ m=k pn,kemp/qk$ and positive when $ m=k pn,kemp/qk$.$ An estimate of the distribution ${Q}^{\\\\ast }$ is required.\",\n \"This can be achieved as follows: Estimate $\\\\inf _{m\\\\ne 0}D_{\\\\varphi }\\\\left( m \\\\cdot \\\\textrm {\\\\right.$$\\\\hspace{-6.544pt}$$}, {P}$ through $\\\\widehat{D} := -\\\\frac{1}{n}\\\\log \\\\widehat{\\\\Pi }_{L}^{improved}$ defined in (REF ).\",\n \"Set $i=0.$ Get some ${Q}_{i} := (q_{i,1}, \\\\ldots , q_{i,K})$ in $\\\\textrm {$$\\\\hspace{-6.544pt}$$}$ ; this can be obtained by simulating runs of vectors $\\\\left( W_{1},..,W_{n}\\\\right) $ through i.i.d.\",\n \"sampling under $\\\\mathbb {}$ .\",\n \"Evaluate $m_{n}({Q}_{i})$ by solving () (with $q_{i,k}$ instead of $q_{k}$ ) for $m$ , which is a fast calculation by the bisection method.\",\n \"If $D_{\\\\varphi }\\\\left( m_{n}({Q}_{i}) \\\\cdot {Q}_{i}, {P}_{n}^{emp} \\\\right) <\\\\widehat{D}+\\\\eta $ for some small $\\\\eta >0$ , then the proxy of ${Q}^{\\\\ast }$ is ${Q}_{i},$ denoted by $\\\\widehat{{Q}^{\\\\ast }}$ .\",\n \"Else set $i\\\\leftarrow i+1$ and get ${Q}_{i}$ in $\\\\textrm {$$\\\\hspace{-6.544pt}$$}\\\\cap \\\\left\\\\lbrace {Q} \\\\, : \\\\, D_{\\\\varphi }\\\\left( {Q},{P}_{n}^{emp} \\\\right) 0$ we normalized $\\\\widetilde{{P}}:=\\\\mathbf {P}/M_{\\\\mathbf {P}},$ and $\\\\widetilde{\\\\mathbf {Q}}:=\\\\mathbf {Q}/M_{\\\\mathbf {P}}$ for $\\\\mathbf {Q}$ in $\\\\mathbf {\\\\Omega }$ .\",\n \"With $\\\\widetilde{\\\\varphi } \\\\in \\\\Upsilon (]a,b[)$ defined through $\\\\widetilde{\\\\varphi }:=M_{\\\\mathbf {P}} \\\\cdot \\\\varphi $ , we have obtained $D_{\\\\varphi }(\\\\mathbf {Q},\\\\mathbf {P})=\\\\sum _{k=1}^{K}p_{k}\\\\cdot \\\\varphi \\\\left( \\\\frac{q_{k}}{p_{k}}\\\\right) =\\\\sum _{k=1}^{K}M_{\\\\mathbf {P}}\\\\cdot \\\\widetilde{p_{k}}\\\\cdot \\\\frac{\\\\varphi \\\\left( \\\\frac{M_{\\\\mathbf {P}}\\\\cdot \\\\widetilde{q_{k}}}{M_{\\\\mathbf {P}}\\\\cdot \\\\widetilde{p_{k}}}\\\\right) }{M_{\\\\mathbf {P}}}=D_{\\\\widetilde{\\\\varphi }}(\\\\widetilde{\\\\mathbf {Q}},\\\\widetilde{{P}})\\\\qquad \\\\textrm {(cf.\",\n \"(\\\\ref {min Pb prob1}))}.$ It has followed that the solution of (REF ) coincides with the one of the problem of finding $\\\\widetilde{\\\\Phi }_{\\\\widetilde{{P}}}(\\\\widetilde{\\\\mathbf {\\\\Omega }}) := \\\\inf _{\\\\widetilde{\\\\mathbf {Q}}\\\\in \\\\widetilde{\\\\mathbf {\\\\Omega }} }D_{\\\\widetilde{\\\\varphi } }(\\\\widetilde{\\\\mathbf {Q}},\\\\widetilde{{P}}),\\\\qquad \\\\textrm {with } \\\\widetilde{\\\\mathbf {\\\\Omega }}:=\\\\mathbf {\\\\Omega } /M_{\\\\mathbf {P}}\\\\qquad \\\\textrm {(cf.\",\n \"(\\\\ref {min Pb prob2}))}.$ So let us continue by tackling (REF ).\",\n \"From the assumptions on $\\\\widetilde{\\\\varphi }$ and the requirement (REF ) one can see that $\\\\text{$\\\\widetilde{W}_{1}$ has moment generating function$t\\\\rightarrow E_{\\\\mathbb {\\\\Pi }}[e^{z \\\\cdot \\\\widetilde{W}_{1}}]= MGF_{\\\\widetilde{\\\\mathbb {}}}(z)$which is finite on a non-void neighborhood of $0$},$ $E_{\\\\mathbb {\\\\Pi }}[\\\\widetilde{W}_1] = 1 ,$ since $\\\\widetilde{\\\\varphi }(1)=0=\\\\widetilde{\\\\varphi }^{\\\\prime }(1)$ .\",\n \"With the help of these, we obtain the following Proposition 61 Under the assumptions of Theorem REF , for any set $\\\\widetilde{\\\\mathbf {\\\\Omega }}\\\\subset \\\\mathcal {M} := \\\\mathbb {R}^{K}$ with (REF ) one has $-\\\\inf _{\\\\widetilde{\\\\mathbf {Q}} \\\\in int(\\\\widetilde{\\\\mathbf {\\\\Omega }}) }D_{\\\\varphi }\\\\left( \\\\widetilde{\\\\mathbf {Q}},\\\\widetilde{{P}}\\\\right) &\\\\le &\\\\lim \\\\inf _{n\\\\rightarrow \\\\infty }\\\\frac{1}{n}\\\\log \\\\mathbb {\\\\Pi }\\\\left[\\\\xi _{n}^{\\\\widetilde{\\\\textbf {W}}}\\\\in \\\\widetilde{\\\\mathbf {\\\\Omega }} \\\\right] \\\\nonumber \\\\\\\\&\\\\le &\\\\lim \\\\sup _{n\\\\rightarrow \\\\infty }\\\\frac{1}{n}\\\\log \\\\mathbb {\\\\Pi }\\\\left[\\\\xi _{n}^{\\\\widetilde{\\\\textbf {W}}}\\\\in \\\\widetilde{\\\\mathbf {\\\\Omega }} \\\\right]\\\\le -\\\\inf _{\\\\widetilde{\\\\mathbf {Q}}\\\\in cl(\\\\widetilde{\\\\mathbf {\\\\Omega }}) }D_{\\\\varphi }\\\\left( \\\\widetilde{\\\\mathbf {Q}},\\\\widetilde{{P}}\\\\right) .$ Proof of Proposition REF .\",\n \"Recall from Remark REF (v) that $I_{k}^{(n)} := \\\\lbrace i \\\\in \\\\lbrace 1,\\\\ldots ,n\\\\rbrace : \\\\widetilde{x}_{i}=d_{k}\\\\rbrace $ and $n_{k} := card(I_{k}^{(n)})$ denotes the number of elements therein ($k \\\\in \\\\lbrace 1,\\\\ldots ,K\\\\rbrace $ ), i.e.\",\n \"$n_{k}$ is the number of the $\\\\widetilde{x}_{i}$ ’s which equal $d_{k}$ .\",\n \"We follow the line of proof of Theorem 2.2.30 in Dembo & Zeitouni [108], which states the large deviation principle (LDP) for the vector of partial sums of random vectors in $\\\\mathbb {R}^{K}$ , where we also use Corollary 6.1.6 in [108] in relation with condition (REF ).\",\n \"Indeed, since the $k-$ th component of the vector $\\\\xi _{n}^{\\\\mathbf {\\\\widetilde{W}}}$ is the $1/n-$ fold of the sum of the $\\\\widetilde{W}_{i}$ ’s for which the corresponding $\\\\widetilde{x}_{i}$ ’s equal $d_{k}$ (i.e., $\\\\frac{1}{n} \\\\sum _{i\\\\in I_{k}^{(n)}} \\\\widetilde{W}_{i}$ ) the proof will follow from a similar treatment as for the standard Cramer LDP in $\\\\mathbb {R}^{K}.$ The only difference lies in two facts: the number of the summands for the coordinate $k$ is $n_{k}$ , the number of $\\\\widetilde{x}_{i}$ ’s which equal $d_{k}$ , instead of $n$ in the standard case.\",\n \"Furthermore we will need to substitute $n_{k}$ by its equivalent $n \\\\cdot \\\\widetilde{p}_{k}$ , which adds an approximation step.\",\n \"For the upper bound, the proof is based on the corresponding result for $B=B_{1}\\\\times \\\\cdots \\\\times B_{K}$ where the $B_{k}$ ’s are open bounded intervals on $\\\\mathbb {R}^{+}.$ Since the sequence $\\\\left( \\\\widetilde{x}_{1}, \\\\ldots \\\\right)$ satisfies $\\\\lim _{n\\\\rightarrow \\\\infty }\\\\frac{n_{k}}{n}= \\\\widetilde{p}_{k} ,\\\\qquad \\\\textrm {(cf.\",\n \"(\\\\ref {fo.freqlim}))}$ there holds $&&\\\\frac{1}{n}\\\\log \\\\mathbb {\\\\Pi } \\\\left[\\\\xi _{n}^{\\\\mathbf {\\\\widetilde{W}}} \\\\in B\\\\right]=\\\\frac{1}{n}\\\\log \\\\mathbb {\\\\Pi } \\\\bigg [ \\\\bigcap \\\\limits _{k=1}^{K}\\\\bigg (\\\\frac{1}{n} \\\\sum _{i\\\\in I_{k}^{(n)}} \\\\widetilde{W}_{i}\\\\in B_{k}\\\\bigg ) \\\\bigg ]\\\\nonumber \\\\\\\\&=&\\\\frac{1}{n}\\\\sum \\\\limits _{k=1}^{K}\\\\log \\\\mathbb {\\\\Pi } \\\\bigg [ \\\\frac{1+o(1)}{n_{k}}\\\\sum _{i\\\\in I_{k}^{(n)}} \\\\widetilde{W}_{i}\\\\in \\\\frac{1}{\\\\widetilde{p}_{k}} B_{k}\\\\bigg ],$ and hence $\\\\lim \\\\sup _{n\\\\rightarrow \\\\infty }\\\\frac{1}{n}\\\\log \\\\mathbb {\\\\Pi } \\\\left[\\\\xi _{n}^{\\\\mathbf {\\\\widetilde{W}}}\\\\in B\\\\right]&\\\\le &\\\\sum \\\\limits _{k=1}^{K} \\\\widetilde{p}_{k} \\\\cdot \\\\lim \\\\sup _{n_{k}\\\\rightarrow \\\\infty }\\\\frac{1}{n_{k}}\\\\log \\\\mathbb {\\\\Pi } \\\\bigg [ \\\\frac{1}{n_{k}}\\\\sum _{i\\\\in I_{k}^{(n)}} \\\\widetilde{W}_{i}\\\\in \\\\frac{1}{p_{k}}B_{k}\\\\bigg ] \\\\nonumber \\\\\\\\&\\\\le &-\\\\sum \\\\limits _{k=1}^{K}\\\\inf _{x_{k}\\\\in cl(B_{k})} \\\\widetilde{p}_{k} \\\\cdot \\\\varphi \\\\left(\\\\frac{x_{k}}{\\\\widetilde{p}_{k}}\\\\right) .$ To deduce (REF ) from (REF ), we have used (i) the fact that for all $k$ the random variables $\\\\frac{1}{n_{k}}\\\\left( 1+o(1))\\\\right) \\\\cdot \\\\sum _{i\\\\in I_{k}^{(n)}} \\\\widetilde{W}_{i}$ and $\\\\frac{1}{n_{k}}\\\\sum _{i\\\\in I_{k}^{(n)}} \\\\widetilde{W}_{i}$ are exponentially equivalent in the sense that their difference $\\\\Delta _{n_{k}}$ satisfies $\\\\lim \\\\sup _{n_{k}\\\\rightarrow \\\\infty }\\\\frac{1}{n_{k}}\\\\log \\\\mathbb {\\\\Pi }\\\\left[ \\\\,\\\\left|\\\\Delta _{n_{k}}\\\\right|>\\\\eta \\\\, \\\\right] =-\\\\infty ,$ making use of the Chernoff inequality for all positive $\\\\eta $ , as well as (ii) Theorem 4.2.13 in [108].\",\n \"Now the summation and the inf-operations can be permuted in (REF ) which proves the claim for the rectangle $B$ .\",\n \"As in [108], for a compact set $\\\\Omega $ we consider its finite covering by such open sets $B$ and conclude; for $\\\\Omega $ being a closed set, a tightness argument holds, following [108] Theorem 2.2.30 verbatim.\",\n \"For the lower bound consider the same rectangle $B$ .\",\n \"The argument which locates the tilted distribution at the center of $B$ , together with the use of the LLN for the corresponding r.v’s as in [108], in combination with the same approximations as above to handle the approximation of $n_{k}$ by $n \\\\cdot \\\\widetilde{p}_{k}$ , complete the proof of Proposition REF .\",\n \"We omit the details.\",\n \"$\\\\blacksquare $ Let us continue with the proof of Theorem REF , by giving the following two helpful lemmas for $\\\\Phi _{{P}}(\\\\mathbf {A}) := \\\\inf _{\\\\mathbf {Q} \\\\in \\\\mathbf {A}} D_{\\\\varphi }\\\\left(\\\\mathbf {Q},{P}\\\\right),\\\\qquad \\\\mathbf {A} \\\\subset \\\\mathcal {M} := \\\\mathbb {R}^{K},$ Lemma 62 For any open set $\\\\mathbf {A} \\\\subset \\\\mathcal {M} := \\\\mathbb {R}^{K}$ one has $\\\\Phi _{{P}}(\\\\mathbf {A}) = \\\\Phi _{{P}}(cl(\\\\mathbf {A}))$ .\",\n \"This is clear from the continuity of $\\\\Phi _{{P}}$ .\",\n \"Lemma 63 For any $\\\\mathbf {A} \\\\subset \\\\mathcal {M} := \\\\mathbb {R}^{K}$ satisfying (REF ) one has $\\\\Phi _{{P}}(cl(\\\\mathbf {A})) = \\\\Phi _{{P}}(\\\\mathbf {A}) = \\\\Phi _{{P}}(int(\\\\mathbf {A}))$ .\",\n \"Proof of Lemma REF .\",\n \"Assume first that $\\\\Phi _{{P}}(\\\\mathbf {A})$ is finite.\",\n \"Then suppose that $\\\\mathbf {A}$ satisfies (REF ) and $\\\\Phi _{{P}}(cl(\\\\mathbf {A})) < \\\\Phi _{{P}}(int(\\\\mathbf {A}))$ .\",\n \"The latter implies the existence of a point $a \\\\in cl(\\\\mathbf {A})$ such that $a \\\\notin int(\\\\mathbf {A})$ and $\\\\Phi _{{P}}(a)= \\\\Phi _{{P}}(cl(\\\\mathbf {A}))$ .\",\n \"But then, by Lemma REF and (REF ) one gets $\\\\Phi _{{P}}(int(\\\\mathbf {A})) = \\\\Phi _{{P}}(cl(int(\\\\mathbf {A}))) = \\\\Phi _{{P}}(cl(\\\\mathbf {A})) = \\\\Phi _{{P}}(a)$ which leads to a contradiction.\",\n \"When $\\\\Phi _{{P}}(\\\\mathbf {A}) = \\\\infty $ then $\\\\Phi _{{P}} (cl(\\\\mathbf {A}))=\\\\Phi _{{P}} (int(\\\\mathbf {A}))=\\\\Phi _{{P}} (\\\\mathbf {A})=\\\\infty $ .\",\n \"$\\\\blacksquare $ Putting things together, the required asymptotic assertion (REF ) follows from (REF ), (REF ) and Lemma REF .\",\n \"This completes the proof of Theorem REF .\",\n \"$\\\\blacksquare $\"\n ],\n [\n \"Proofs — Part 2\",\n \"Before we tackle the proof of Theorem REF , let us introduce the following Lemma 64 If $\\\\Omega $$\\\\Omega \\\\subset \\\\mathbb {S}^{K}$ satisfies condition (REF ), then $\\\\widetilde{\\\\widetilde{\\\\textrm {\\\\Omega \\\\hspace{-6.544pt}\\\\Omega }}}:= \\\\bigcup \\\\limits _{m\\\\ne 0}cl(m \\\\cdot \\\\textrm {$$\\\\hspace{-6.544pt}$$})$ has the property (REF ).\",\n \"This can be deduced in a straightforward way: the assumption implies that $cl(\\\\textrm {$$\\\\hspace{-6.544pt}$$})$ satisfies (REF ), and thus also $m \\\\cdot cl(\\\\textrm {$$\\\\hspace{-6.544pt}$$})$ satisfies (REF ).\",\n \"But this implies the validity of (REF ) for the “cone” $\\\\bigcup \\\\limits _{m\\\\ne 0} m \\\\cdot cl(\\\\textrm {$$\\\\hspace{-6.544pt}$$})$ which is nothing but $\\\\bigcup \\\\limits _{m\\\\ne 0} cl(m \\\\cdot \\\\textrm {$$\\\\hspace{-6.544pt}$$})$ .\",\n \"Proof of Theorem REF .\",\n \"Recall the interpretations of the two vectors $\\\\xi _{n,\\\\mathbf {X}}^{\\\\mathbf {W}}$ respectively $\\\\xi _{n,\\\\mathbf {X}}^{w\\\\mathbf {W}}$ given in (REF ) respectively (REF ), and that the sum of their $k$ components are $\\\\sum _{k=1}^{K} \\\\frac{1}{n} \\\\sum _{i\\\\in I_{k}^{(n)}} W_{i} =\\\\frac{1}{n}\\\\sum _{i=1}^{n} W_{i}$ respectively $\\\\sum _{k=1}^{K}\\\\frac{\\\\sum _{i \\\\in I_{k}^{(n)}}W_{i}}{\\\\sum _{k=1}^{K}\\\\sum _{i \\\\in I_{k}^{(n)}}W_{i}}=1$ (in case of $\\\\sum _{i=1}^{n}W_{i} \\\\ne 0$ ).\",\n \"In the light of these, for $\\\\textrm {$$\\\\hspace{-6.544pt}$$} \\\\subset \\\\mathbb {S}^{K}$ one gets the set identification $\\\\left\\\\lbrace \\\\xi _{n,\\\\mathbf {X}}^{w\\\\mathbf {W}} \\\\in \\\\textrm {\\\\right.$ $\\\\hspace{-6.544pt}$$} =\\\\bigcup \\\\limits _{m\\\\ne 0}\\\\left\\\\lbrace \\\\xi _{n,\\\\mathbf {X}}^{\\\\mathbf {W}}\\\\in m\\\\cdot \\\\textrm {\\\\right.$$\\\\hspace{-6.544pt}$$} ,\\\\frac{1}{n}\\\\sum _{i=1}^{n} W_{i} = m$$since $ { i=1nWi=0} $ amounts to $ m=0$, which cannothold when $ { n,XwW $\\\\Omega $$\\\\Omega $ }$.\",\n \"Now\\\\begin{eqnarray*}\\\\mathbb {\\\\Pi }_{X_{1}^{n}}\\\\left[\\\\xi _{n,\\\\mathbf {X}}^{w\\\\mathbf {W}}\\\\in \\\\textrm {\\\\right.\\\\Omega \\\\hspace{-6.544pt}\\\\Omega } &=&\\\\mathbb {\\\\Pi }_{X_{1}^{n}} \\\\bigg [ \\\\bigcup \\\\limits _{m\\\\ne 0}\\\\bigg \\\\lbrace \\\\xi _{n,\\\\mathbf {X}}^{\\\\mathbf {W}} \\\\in m \\\\cdot \\\\textrm {\\\\Omega \\\\hspace{-6.544pt}\\\\Omega },\\\\frac{1}{n}\\\\sum _{i=1}^{n} W_{i} =m \\\\bigg \\\\rbrace \\\\bigg ] \\\\\\\\&=&\\\\mathbb {\\\\Pi }_{X_{1}^{n}}\\\\bigg [ \\\\bigcup \\\\limits _{m\\\\ne 0}\\\\bigg \\\\lbrace \\\\xi _{n,\\\\mathbf {X}}^{\\\\mathbf {W}} \\\\in m\\\\cdot \\\\textrm {\\\\Omega \\\\hspace{-6.544pt}\\\\Omega }\\\\big \\\\rbrace \\\\bigg ]= \\\\mathbb {\\\\Pi }_{X_{1}^{n}}\\\\bigg [ \\\\xi _{n,\\\\mathbf {X}}^{\\\\mathbf {W}}\\\\in \\\\bigcup \\\\limits _{m\\\\ne 0} m\\\\cdot \\\\textrm {\\\\Omega \\\\hspace{-6.544pt}\\\\Omega } \\\\bigg ]\\\\end{eqnarray*}since $ { n,XW m$\\\\Omega $$\\\\Omega $ } { 1ni=1n Wi =m }$.\",\n \"Therefore\\\\begin{equation}\\\\frac{1}{n}\\\\log \\\\mathbb {\\\\Pi }_{X_{1}^{n}}\\\\left[\\\\xi _{n,\\\\mathbf {X}}^{w\\\\mathbf {W}}\\\\in \\\\textrm {\\\\right.\\\\Omega \\\\hspace{-6.544pt}\\\\Omega } =\\\\frac{1}{n}\\\\log \\\\mathbb {\\\\Pi }_{X_{1}^{n}}\\\\bigg [ \\\\xi _{n,\\\\mathbf {X}}^{\\\\mathbf {W}} \\\\in \\\\bigcup \\\\limits _{m\\\\ne 0}m\\\\cdot \\\\textrm {\\\\Omega \\\\hspace{-6.544pt}\\\\Omega } \\\\bigg ] .\\\\end{equation}Because of Proposition \\\\ref {PropLDPWEM-finitease} --- applied to$$\\\\Omega $$\\\\Omega $ := m0m $\\\\Omega $$\\\\Omega $$ --- one getsin terms of (\\\\ref {phishort})$ $-\\\\Phi _{{P}}\\\\Big (int\\\\Big (\\\\bigcup \\\\limits _{m\\\\ne 0}m\\\\cdot \\\\textrm {\\\\Omega \\\\hspace{-6.544pt}\\\\Omega }\\\\Big )\\\\Big )&\\\\le &\\\\lim \\\\inf _{n\\\\rightarrow \\\\infty }\\\\frac{1}{n}\\\\log \\\\mathbb {\\\\Pi }_{X_{1}^{n}}\\\\bigg [ \\\\xi _{n,\\\\mathbf {X}}^{\\\\mathbf {W}} \\\\in \\\\bigcup \\\\limits _{m\\\\ne 0}m\\\\cdot \\\\textrm {\\\\Omega \\\\hspace{-6.544pt}\\\\Omega } \\\\bigg ] \\\\nonumber \\\\\\\\&\\\\le &\\\\lim \\\\sup _{n\\\\rightarrow \\\\infty }\\\\frac{1}{n}\\\\log \\\\mathbb {\\\\Pi }_{X_{1}^{n}}\\\\bigg [\\\\xi _{n,\\\\mathbf {X}}^{\\\\mathbf {W}}\\\\in \\\\bigcup \\\\limits _{m\\\\ne 0}m\\\\cdot \\\\textrm {\\\\Omega \\\\hspace{-6.544pt}\\\\Omega } \\\\bigg ] \\\\le -\\\\Phi _{{P}}\\\\Big (cl\\\\Big (\\\\bigcup \\\\limits _{m\\\\ne 0}m\\\\cdot \\\\textrm {\\\\Omega \\\\hspace{-6.544pt}\\\\Omega }\\\\Big )\\\\Big ) .$ $\\\\hspace{-176.407pt}\\\\text{But}& & \\\\Phi _{{P}}\\\\Big (int\\\\Big (\\\\bigcup \\\\limits _{m\\\\ne 0}m\\\\cdot \\\\textrm {\\\\Omega \\\\hspace{-6.544pt}\\\\Omega }\\\\Big )\\\\Big ) \\\\le \\\\Phi _{{P}}\\\\Big (\\\\bigcup \\\\limits _{m\\\\ne 0}int\\\\Big (m\\\\cdot \\\\textrm {\\\\Omega \\\\hspace{-6.544pt}\\\\Omega }\\\\Big )\\\\Big ) =\\\\inf _{m\\\\ne 0} \\\\Phi _{{P}}(int(m\\\\cdot \\\\textrm {\\\\Omega \\\\hspace{-6.544pt}\\\\Omega }))\\\\\\\\\\\\hspace{-176.407pt}\\\\text{and}& & \\\\Phi _{{P}}\\\\Big (cl\\\\Big (\\\\bigcup \\\\limits _{m\\\\ne 0}m\\\\cdot \\\\textrm {\\\\Omega \\\\hspace{-6.544pt}\\\\Omega }\\\\Big )\\\\Big ) \\\\ge \\\\Phi _{{P}}\\\\Big (\\\\bigcup \\\\limits _{m\\\\ne 0}cl\\\\Big (m\\\\cdot \\\\textrm {\\\\Omega \\\\hspace{-6.544pt}\\\\Omega }\\\\Big )\\\\Big ) =\\\\inf _{m\\\\ne 0} \\\\Phi _{{P}}(cl(m\\\\cdot \\\\textrm {\\\\Omega \\\\hspace{-6.544pt}\\\\Omega })) .$ In fact, the inequality in (REF ) is straightforward because of $\\\\bigcup \\\\limits _{m\\\\ne 0}int(m\\\\cdot \\\\textrm {$$\\\\hspace{-6.544pt}$$})\\\\subset int(\\\\bigcup \\\\limits _{m\\\\ne 0}m\\\\cdot \\\\textrm {$$\\\\hspace{-6.544pt}$$})$ (since the latter is the largest open set contained in $\\\\bigcup \\\\limits _{m\\\\ne 0}m\\\\cdot \\\\textrm {$$\\\\hspace{-6.544pt}$$}$ ); the inequality in () follows from $& & \\\\Phi _{{P}}\\\\Big (cl\\\\Big (\\\\bigcup \\\\limits _{m\\\\ne 0}m\\\\cdot \\\\textrm {\\\\Omega \\\\hspace{-6.544pt}\\\\Omega }\\\\Big )\\\\Big ) \\\\ge \\\\Phi _{{P}}\\\\Big (cl\\\\Big (\\\\bigcup \\\\limits _{m\\\\ne 0}cl\\\\Big (m\\\\cdot \\\\textrm {\\\\Omega \\\\hspace{-6.544pt}\\\\Omega }\\\\Big )\\\\Big )\\\\Big ) =\\\\Phi _{{P}}\\\\Big (\\\\bigcup \\\\limits _{m\\\\ne 0}cl\\\\Big (m\\\\cdot \\\\textrm {\\\\Omega \\\\hspace{-6.544pt}\\\\Omega }\\\\Big )\\\\Big )\\\\nonumber $ An application of Lemma REF yields $\\\\Phi _{{P}}(int(m\\\\cdot \\\\textrm {$$\\\\hspace{-6.544pt}$$})) =\\\\Phi _{{P}}(m\\\\cdot \\\\textrm {$$\\\\hspace{-6.544pt}$$}) =\\\\Phi _{{P}}(cl(m\\\\cdot \\\\textrm {$$\\\\hspace{-6.544pt}$$}))$ for all $m \\\\ne 0$ , and hence $\\\\inf _{m\\\\ne 0} \\\\Phi _{{P}}(int(m\\\\cdot \\\\textrm {\\\\Omega \\\\hspace{-6.544pt}\\\\Omega })) =\\\\inf _{m\\\\ne 0} \\\\Phi _{{P}}(m\\\\cdot \\\\textrm {\\\\Omega \\\\hspace{-6.544pt}\\\\Omega })= \\\\inf _{m\\\\ne 0} \\\\Phi _{{P}}(cl(m\\\\cdot \\\\textrm {\\\\Omega \\\\hspace{-6.544pt}\\\\Omega })) .$ By combining (), (REF ), (REF ), () and (REF ), one arrives at $&&\\\\lim _{n \\\\rightarrow \\\\infty } \\\\frac{1}{n}\\\\log \\\\mathbb {\\\\Pi }_{X_{1}^{n}} \\\\left[\\\\xi _{n,\\\\mathbf {X}}^{w\\\\mathbf {W}}\\\\in \\\\textrm {\\\\right.\\\\Omega \\\\hspace{-6.544pt}\\\\Omega } = \\\\lim _{n \\\\rightarrow \\\\infty } \\\\frac{1}{n}\\\\log \\\\mathbb {\\\\Pi }_{X_{1}^{n}} \\\\bigg [ \\\\xi _{n,\\\\mathbf {X}}^{\\\\mathbf {W}}\\\\in \\\\bigcup \\\\limits _{m\\\\ne 0}m\\\\cdot \\\\textrm {\\\\Omega \\\\hspace{-6.544pt}\\\\Omega } \\\\bigg ]\\\\nonumber \\\\\\\\&& = - \\\\inf _{m\\\\ne 0} \\\\Phi _{{P}}(m\\\\cdot \\\\textrm {\\\\Omega \\\\hspace{-6.544pt}\\\\Omega })= - \\\\inf _{m\\\\ne 0} \\\\inf _{\\\\mathbf {Q} \\\\in m\\\\cdot \\\\textrm {\\\\Omega \\\\hspace{-5.40608pt}\\\\Omega }}D_{\\\\varphi }\\\\left(\\\\mathbf {Q},{P}\\\\right)= - \\\\inf _{m\\\\ne 0} \\\\inf _{{Q} \\\\in \\\\textrm {\\\\Omega \\\\hspace{-5.40608pt}\\\\Omega }}D_{\\\\varphi }\\\\left(m\\\\cdot {Q}, {P}\\\\right) ,\\\\nonumber $ where in the second last equality we have “reverted” the notation (REF ).\",\n \"Note that we did not assume (REF ) for $\\\\bigcup \\\\limits _{m\\\\ne 0} m \\\\cdot \\\\textrm {$$\\\\hspace{-6.544pt}$$}$ .\",\n \"$\\\\blacksquare $\"\n ],\n [\n \"Proofs — Part 3\",\n \"Proof of Lemma REF .\",\n \"From (REF ) one gets straightforwardly for arbitrary $\\\\widetilde{c} >0$ $D_{\\\\widetilde{c} \\\\cdot \\\\varphi _{\\\\gamma }}(m \\\\cdot \\\\mathbf {Q},{P}) \\\\hspace{-5.69046pt} &:=& \\\\hspace{-5.69046pt}{\\\\left\\\\lbrace \\\\begin{array}{ll}\\\\frac{\\\\widetilde{c} \\\\cdot \\\\left(m^{\\\\gamma } \\\\cdot H_{\\\\gamma } - m \\\\cdot A \\\\cdot \\\\gamma + \\\\gamma -1 \\\\right)}{\\\\gamma \\\\cdot (\\\\gamma -1)},\\\\hspace{75.39963pt} \\\\textrm {if } \\\\gamma \\\\in \\\\, ]-\\\\infty ,0[, \\\\ {P} \\\\in \\\\mathbb {S}_{\\\\ge 0}^{K}, \\\\ \\\\mathbf {Q} \\\\in A \\\\cdot \\\\mathbb {S}_{> 0}^{K} \\\\ \\\\ \\\\textrm {and } m >0, \\\\\\\\\\\\widetilde{c} \\\\cdot (- \\\\log m + \\\\widetilde{I} -1 + m \\\\cdot A),\\\\hspace{42.67912pt} \\\\textrm {if } \\\\gamma = 0, \\\\ {P} \\\\in \\\\mathbb {S}_{\\\\ge 0}^{K}, \\\\ A \\\\cdot \\\\mathbf {Q} \\\\in \\\\mathbb {S}_{> 0}^{K} \\\\ \\\\ \\\\textrm {and } m >0, \\\\\\\\\\\\frac{\\\\widetilde{c} \\\\cdot \\\\left(m^{\\\\gamma } \\\\cdot H_{\\\\gamma } - m \\\\cdot A \\\\cdot \\\\gamma + \\\\gamma -1 \\\\right)}{\\\\gamma \\\\cdot (\\\\gamma -1)},\\\\hspace{75.39963pt} \\\\textrm {if } \\\\gamma \\\\in \\\\, ]0,1[, \\\\ {P} \\\\in \\\\mathbb {S}_{\\\\ge 0}^{K}, \\\\ \\\\mathbf {Q} \\\\in A \\\\cdot \\\\mathbb {S}_{\\\\ge 0}^{K} \\\\ \\\\ \\\\textrm {and } m \\\\ge 0, \\\\\\\\\\\\widetilde{c} \\\\cdot (A \\\\cdot m \\\\cdot \\\\log m + m \\\\cdot (I-A) +1),\\\\hspace{14.22636pt} \\\\textrm {if } \\\\gamma = 1, \\\\ {P} \\\\in \\\\mathbb {S}_{> 0}^{K}, \\\\ \\\\mathbf {Q} \\\\in A \\\\cdot \\\\mathbb {S}_{\\\\ge 0}^{K} \\\\ \\\\ \\\\textrm {and } m \\\\ge 0, \\\\\\\\\\\\frac{\\\\widetilde{c} \\\\cdot \\\\left(m^{\\\\gamma } \\\\cdot H_{\\\\gamma } \\\\cdot {1}_{[0,\\\\infty [}(m)- m \\\\cdot A \\\\cdot \\\\gamma + \\\\gamma -1 \\\\right)}{\\\\gamma \\\\cdot (\\\\gamma -1)},\\\\hspace{39.83368pt} \\\\textrm {if } \\\\gamma \\\\in \\\\, ]1,2[, \\\\ {P} \\\\in \\\\mathbb {S}_{>0}^{K}, \\\\ \\\\mathbf {Q} \\\\in A \\\\cdot \\\\mathbb {S}^{K} \\\\ \\\\ \\\\textrm {and } m \\\\in ]-\\\\infty ,\\\\infty [,\\\\\\\\\\\\frac{\\\\widetilde{c} \\\\cdot \\\\left(m^{2} \\\\cdot H_{2} - m \\\\cdot A \\\\cdot 2 + 2 -1 \\\\right)}{2 \\\\cdot (2-1)},\\\\hspace{76.82234pt} \\\\textrm {if } \\\\gamma = 2, \\\\ {P} \\\\in \\\\mathbb {S}_{>0}^{K}, \\\\ \\\\mathbf {Q} \\\\in A \\\\cdot \\\\mathbb {S}^{K} \\\\ \\\\ \\\\textrm {and } m \\\\in ]-\\\\infty ,\\\\infty [,\\\\\\\\\\\\frac{\\\\widetilde{c} \\\\cdot \\\\left(m^{\\\\gamma } \\\\cdot H_{\\\\gamma } \\\\cdot {1}_{[0,\\\\infty [}(m)- m \\\\cdot A \\\\cdot \\\\gamma + \\\\gamma -1 \\\\right)}{\\\\gamma \\\\cdot (\\\\gamma -1)},\\\\hspace{39.83368pt} \\\\textrm {if } \\\\gamma \\\\in \\\\, ]2,\\\\infty [, \\\\ {P} \\\\in \\\\mathbb {S}_{>0}^{K}, \\\\ \\\\mathbf {Q} \\\\in A \\\\cdot \\\\mathbb {S}^{K} \\\\ \\\\ \\\\textrm {and } m \\\\in ]-\\\\infty ,\\\\infty [,\\\\\\\\\\\\infty , \\\\hspace{156.49014pt} \\\\textrm {else},\\\\end{array}\\\\right.\",\n \"}$ where we have used the three $m-$ independent abbreviations $H_{\\\\gamma } := \\\\sum \\\\displaylimits _{k=1}^{K} (q_{k})^{\\\\gamma } \\\\cdot (p_{k})^{1-\\\\gamma }\\\\ = \\\\ 1 + \\\\gamma \\\\cdot (A-1) +\\\\frac{\\\\gamma \\\\cdot (\\\\gamma -1)}{\\\\widetilde{c}} \\\\cdot D_{\\\\widetilde{c} \\\\cdot \\\\varphi _{\\\\gamma }}(\\\\mathbf {Q}, {P}),\\\\qquad \\\\text{(cf.\",\n \"(\\\\ref {brostu3:fo.divpow.hellinger1}))}\\\\nonumber $ $I:= \\\\sum \\\\displaylimits _{k=1}^{K} q_{k} \\\\cdot \\\\log \\\\left( \\\\frac{q_{k}}{p_{k}} \\\\right)\\\\ = \\\\ \\\\frac{1}{\\\\widetilde{c}} \\\\cdot D_{\\\\widetilde{c} \\\\cdot \\\\varphi _{1}}(\\\\mathbf {Q}, {P}) + A - 1,\\\\qquad \\\\text{(cf.\",\n \"(\\\\ref {brostu3:fo.divpow.Kull1}))}\\\\nonumber $ $\\\\widetilde{I} := \\\\sum \\\\displaylimits _{k=1}^{K} p_{k} \\\\cdot \\\\log \\\\left( \\\\frac{p_{k}}{q_{k}} \\\\right)\\\\ = \\\\ \\\\frac{1}{\\\\widetilde{c}} \\\\cdot D_{\\\\widetilde{c} \\\\cdot \\\\varphi _{0}}(\\\\mathbf {Q}, {P}) + 1 - A.\\\\qquad \\\\text{(cf.\",\n \"(\\\\ref {brostu3:fo.divpow.RevKull1}))}\\\\nonumber $ To proceed, let us fix an arbitrary constant $\\\\widetilde{c} >0$ .\",\n \"(i) Case $\\\\gamma \\\\cdot (1-\\\\gamma ) \\\\ne 0$ .\",\n \"(ia) Let us start with the subcase $\\\\gamma \\\\in ]-\\\\infty ,0[$ .\",\n \"From the first and the last line of (REF ), it is clear that the corresponding $m-$ infimum can not be achieved for $m \\\\le 0$ ; since $H_{\\\\gamma } > 0$ one gets the unique minimizer $m_{min} = \\\\Big (\\\\frac{H_{\\\\gamma }}{A}\\\\Big )^{1/(1-\\\\gamma )} >0$ and the minimum $D_{\\\\widetilde{c} \\\\cdot \\\\varphi _{\\\\gamma }}(m_{min} \\\\cdot \\\\mathbf {Q},{P})=\\\\frac{\\\\widetilde{c}}{\\\\gamma } \\\\cdot (1- \\\\frac{H^{1/(1-\\\\gamma )}}{A^{\\\\gamma /(1-\\\\gamma )}} )$ .\",\n \"Hence, (REF ) is established.\",\n \"The assertions (REF ) and () follow immediately by monotonicity inspection of $x \\\\rightarrow \\\\frac{\\\\widetilde{c}}{\\\\gamma } \\\\cdot \\\\left[ 1-\\\\frac{1}{A^{\\\\gamma /(1-\\\\gamma )}} \\\\cdot \\\\left[ 1 + \\\\gamma \\\\cdot (A-1) + \\\\frac{\\\\gamma \\\\cdot \\\\left( \\\\gamma -1\\\\right)}{\\\\widetilde{c}}\\\\cdot x\\\\right] ^{-1/\\\\left( \\\\gamma -1\\\\right) }\\\\right]$ for $x \\\\ge 0$ such that $1 + \\\\gamma \\\\cdot (A-1) + \\\\frac{\\\\gamma \\\\cdot \\\\left( \\\\gamma -1\\\\right)}{\\\\widetilde{c}}\\\\cdot x \\\\ge 0$ .\",\n \"(ib) The subcase $\\\\gamma \\\\in ]0,1[$ (cf.\",\n \"the third line of (REF )) works analogously if $H_{\\\\gamma } >0$ ; furthermore, if $H_{\\\\gamma } =0$ — which can only appear when ${P}$ , $\\\\mathbf {Q}$ have disjoint supports (singularity)— then $\\\\inf _{m>0} D_{\\\\widetilde{c} \\\\cdot \\\\varphi _{\\\\gamma }}(m \\\\cdot \\\\mathbf {Q},{P}) =\\\\frac{\\\\widetilde{c}}{\\\\gamma }$ which is (the corresponding special case of) (REF ).\",\n \"(ic) In the subcase $\\\\gamma \\\\in ]1,\\\\infty [$ (cf.\",\n \"the fifth, sixth and seventh line of (REF )) it is straightforward to see that the desired infimum can not be achieved for $m < 0$ .\",\n \"Hence, one can proceed analogously to subcase (ia).\",\n \"(id) The assertions () to () are straightforward.\",\n \"(ii) Case $\\\\gamma =1$ .\",\n \"From the fourth line of (REF ), one obtains the unique minimizer $m_{min} = \\\\exp \\\\lbrace -I/A\\\\rbrace $ and the minimum $D_{\\\\widetilde{c} \\\\cdot \\\\varphi _{1}}(m_{min} \\\\cdot \\\\mathbf {Q},{P})= \\\\widetilde{c} \\\\cdot (1- A \\\\cdot m_{min})$ , which leads to (REF ).\",\n \"The monotonicity of $x \\\\rightarrow \\\\widetilde{c} \\\\cdot (1- \\\\exp \\\\lbrace -x/\\\\widetilde{c}\\\\rbrace )$ for $x\\\\ge 0$ implies immediately (REF ) and (); moreover, () and () are immediate.\",\n \"(iii) Case $\\\\gamma =0$ .\",\n \"The second line of (REF ) implies the unique minimizer $m_{min} = 1/A$ , the minimum $D_{\\\\widetilde{c} \\\\cdot \\\\varphi _{0}}(m_{min} \\\\cdot \\\\mathbf {Q},{P})= \\\\widetilde{c} \\\\cdot (\\\\widetilde{I} + \\\\log A)$ , and hence (REF ).\",\n \"The assertions (REF ) to () are obvious.\",\n \"$\\\\blacksquare $\"\n ],\n [\n \"Proofs — Part 4\",\n \"Proof of Proposition REF .\",\n \"Clearly, (G1) and (G2) are part of the definition of $\\\\widetilde{\\\\Upsilon }(]a,b[)$ .\",\n \"Recall our required representability (REF ).\",\n \"The therein involved Laplace-Stieltjes transform (Laplace-Lebesgue transform) $z\\\\mapsto MGF_{\\\\mathbb {}}(z):=\\\\int _{\\\\mathbb {R}}e^{z \\\\cdot y} \\\\, d\\\\mathbb {} (y)= E_{\\\\mathbb {\\\\Pi }}[e^{z\\\\cdot W}]$ of a probability measure $\\\\mathbb {}$ on the real line respectively of an associated random variable $W$ (with $\\\\mathbb {}[ \\\\cdot \\\\, ] := \\\\mathbb {\\\\Pi }[W \\\\in \\\\cdot \\\\, ]$ ) has the following fundamental properties, according to well-known general theory: $MGF_{\\\\mathbb {}}$ takes values in $]0,\\\\infty ]$ ; the effective domain $dom(MGF_{\\\\mathbb {}})$ is an interval which contains 0 and which may be degenerated or even the whole real line; correspondingly, we denote its interior by $]\\\\lambda _{-},\\\\lambda _{+}[ := int(dom(MGF_{\\\\mathbb {}}))$ which may be the empty set (in case that $dom(MGF_{\\\\mathbb {}})=\\\\lbrace 0\\\\rbrace $ , i.e.\",\n \"$\\\\lambda _{-}=\\\\lambda _{+}=0$ ); clearly, there holds $\\\\lambda _{-} \\\\in [-\\\\infty ,0]$ and $\\\\lambda _{+} \\\\in [0,\\\\infty ]$ ; $MGF_{\\\\mathbb {}}$ is continuous on $dom(MGF_{\\\\mathbb {}})$ and lower semicontinuous on $\\\\mathbb {R}$ ; if $\\\\lambda _{-} \\\\ne \\\\lambda _{+}$ , then $MGF_{\\\\mathbb {}}$ is real analytic and thus infinitely differentiable on $]\\\\lambda _{-},\\\\lambda _{+}[$ ; if $MGF_{\\\\mathbb {}}$ is finite in a neighborhood of zero, i.e.\",\n \"$0 \\\\in ]\\\\lambda _{-},\\\\lambda _{+}[$ , then for all $k \\\\in \\\\mathbb {N}_{0}$ the $k-$ th moment of $\\\\mathbb {}$ respectively $W$ exists and is finite and can be computed in terms of the $k-$ th derivative $MGF_{\\\\mathbb {}}^{(k)}$ as $MGF_{\\\\mathbb {}}^{(k)}(0) =\\\\int _{\\\\mathbb {R}} y^{k} \\\\, d\\\\mathbb {} (y)= E_{\\\\mathbb {\\\\Pi }}[W^{k}],$ which, by the way, then allows the interpretation of $MGF_{\\\\mathbb {}}$ as “moment generating function of $\\\\mathbb {}$ resp.\",\n \"$W$ ”since we assume $0 \\\\in ]\\\\lambda _{-},\\\\lambda _{+}[$ , we have already used the meaningful abbreviation $MGF$ (rather than LST) in (REF ); if $\\\\lambda _{-} \\\\ne \\\\lambda _{+}$ , then $MGF_{\\\\mathbb {}}$ is strictly convex on $]\\\\lambda _{-},\\\\lambda _{+}[$ .\",\n \"Hence, the logarithm of the Laplace-Stieltjes transform $z\\\\mapsto \\\\Lambda _{\\\\mathbb {}}(z) := \\\\log MGF_{\\\\mathbb {} }(z):=\\\\log \\\\int _{\\\\mathbb {R}}e^{z \\\\cdot y} \\\\, d\\\\mathbb {} (y)= \\\\log E_{\\\\mathbb {\\\\Pi }}[e^{z\\\\cdot W}]$ (which in case of $0 \\\\in ]\\\\lambda _{-},\\\\lambda _{+}[$ can be interpreted as cumulant generating function) “carries over” (M1) to (M6), which partially can be even refined: $\\\\Lambda _{\\\\mathbb {}}$ takes values in $]-\\\\infty ,\\\\infty ]$ ; $dom(\\\\Lambda _{\\\\mathbb {}}) = dom(MGF_{\\\\mathbb {}})$ and thus $int(dom(\\\\Lambda _{\\\\mathbb {} })) = \\\\, ]\\\\lambda _{-},\\\\lambda _{+}[$ ; $\\\\Lambda _{\\\\mathbb {} }$ is continuous on $dom(\\\\Lambda _{\\\\mathbb {}})$ and lower semicontinuous on $\\\\mathbb {R}$ ; if $\\\\lambda _{-} \\\\ne \\\\lambda _{+}$ , then $\\\\lambda _{\\\\mathbb {}}$ is infinitely differentiable on $]\\\\lambda _{-},\\\\lambda _{+}[$ ; if $0 \\\\in ]\\\\lambda _{-},\\\\lambda _{+}[$ , then $& & \\\\Lambda _{\\\\mathbb {} }(0) = 0, \\\\quad \\\\Lambda _{\\\\mathbb {} }^{\\\\prime }(0) = \\\\int _{\\\\mathbb {R}} y \\\\, d\\\\mathbb {} (y)= E_{\\\\mathbb {\\\\Pi }}[W],\\\\\\\\& & \\\\Lambda _{\\\\mathbb {} }^{\\\\prime \\\\prime }(0) =\\\\int _{\\\\mathbb {R}} \\\\Big (y - \\\\int _{\\\\mathbb {R}} \\\\widetilde{y} \\\\, d\\\\mathbb {} (\\\\widetilde{y})\\\\Big )^{2}\\\\, d\\\\mathbb {} (y)= E_{\\\\mathbb {\\\\Pi }}[W^{2}] - (E_{\\\\mathbb {\\\\Pi }}[W])^{2} = Var_{\\\\mathbb {\\\\Pi }}[W];$ under the assumption $\\\\lambda _{-} \\\\ne \\\\lambda _{+}$ there holds: $\\\\Lambda _{\\\\mathbb {}}$ is strictly convex on $]\\\\lambda _{-},\\\\lambda _{+}[$ if and only if $\\\\mathbb {}$ is not a one-point distribution (Dirac mass) if and only if $W$ is not a.s. constant; otherwise, $\\\\Lambda _{\\\\mathbb {}}$ is linear; under the assumption that $\\\\mathbb {}$ is not a one-point distribution (Dirac mass) — with the notations $a := \\\\inf supp(\\\\mathbb {}) = \\\\inf supp(W)$ , $b := \\\\sup supp(\\\\mathbb {}) = \\\\sup supp(W)$ , $t_{-}^{sc} := \\\\inf \\\\lbrace \\\\Lambda _{\\\\mathbb {}}^{\\\\prime }(z) \\\\, : \\\\, z \\\\in ]\\\\lambda _{-},\\\\lambda _{+}[ \\\\rbrace = \\\\lim _{z \\\\downarrow \\\\lambda _{-}} \\\\Lambda _{\\\\mathbb {}}^{\\\\prime }(z)$ and $t_{+}^{sc} := \\\\sup \\\\lbrace \\\\Lambda _{\\\\mathbb {}}^{\\\\prime }(z) \\\\, : \\\\, z \\\\in ]\\\\lambda _{-},\\\\lambda _{+}[ \\\\rbrace = \\\\lim _{z \\\\uparrow \\\\lambda _{+}} \\\\Lambda _{\\\\mathbb {}}^{\\\\prime }(z)$ — one gets the following assertions: $]t_{-}^{sc},t_{+}^{sc}[ \\\\ \\\\subseteq \\\\ ]a,b[$ ; if $a > - \\\\infty $ , then $\\\\lambda _{-} = - \\\\infty $ , $t_{-}^{sc} = \\\\lim _{z \\\\rightarrow -\\\\infty } \\\\Lambda _{\\\\mathbb {}}^{\\\\prime }(z)= \\\\lim _{z \\\\rightarrow -\\\\infty } \\\\frac{\\\\Lambda _{\\\\mathbb {}}(z)}{z} = a$ ; if $b < \\\\infty $ , then $\\\\lambda _{+} = \\\\infty $ , $t_{+}^{sc} = \\\\lim _{z \\\\rightarrow \\\\infty } \\\\Lambda _{\\\\mathbb {}}^{\\\\prime }(z)= \\\\lim _{z \\\\rightarrow \\\\infty } \\\\frac{\\\\Lambda _{\\\\mathbb {}}(z)}{z} = b$ ; if $a = - \\\\infty $ and $\\\\lambda _{-} = - \\\\infty $ , then $t_{-}^{sc} = \\\\lim _{z \\\\rightarrow -\\\\infty } \\\\Lambda _{\\\\mathbb {}}^{\\\\prime }(z)= -\\\\infty = a$ ; if $b = \\\\infty $ and $\\\\lambda _{+} = \\\\infty $ , then $t_{+}^{sc} = \\\\lim _{z \\\\rightarrow \\\\infty } \\\\Lambda _{\\\\mathbb {}}^{\\\\prime }(z)= \\\\infty = b$ ; if $\\\\lambda _{-} \\\\in \\\\, ]-\\\\infty ,0[$ and $t_{-}^{sc} > -\\\\infty $ , then $a = - \\\\infty $ , $\\\\Lambda _{\\\\mathbb {}}(\\\\lambda _{-}) \\\\in \\\\, ]-\\\\infty ,\\\\infty [$ , $\\\\Lambda _{\\\\mathbb {}}(z) = \\\\infty $ for all $z < \\\\lambda _{-}$ , $\\\\Lambda _{\\\\mathbb {}}^{\\\\prime }(\\\\lambda _{-}) \\\\in \\\\, ]-\\\\infty ,\\\\infty [$ ; if $\\\\lambda _{+} \\\\in \\\\, ]0,\\\\infty [$ and $t_{+}^{sc} < \\\\infty $ , then $b = \\\\infty $ , $\\\\Lambda _{\\\\mathbb {}}(\\\\lambda _{+}) \\\\in \\\\, ]-\\\\infty ,\\\\infty [$ , $\\\\Lambda _{\\\\mathbb {}}(z) = \\\\infty $ for all $z > \\\\lambda _{+}$ , $\\\\Lambda _{\\\\mathbb {}}^{\\\\prime }(\\\\lambda _{+})\\\\in \\\\, ]-\\\\infty ,\\\\infty [$ ; if $\\\\lambda _{-} \\\\in \\\\, ]-\\\\infty ,0[$ and $t_{-}^{sc} = -\\\\infty $ , then $a= - \\\\infty $ ; if $\\\\lambda _{+} \\\\in \\\\, ]0,\\\\infty [$ and $t_{+}^{sc} = \\\\infty $ , then $b= \\\\infty $ .\",\n \"Notice that (C7ii) to (C7ix) cover all possible constellations.\",\n \"For a proof of (C7ii) to (C7vii) as well as further details, see e.g.\",\n \"Section 9.1 in Borovkov [56].\",\n \"By contradiction, (C7viii) follows from (C7ii) and (C7ix) follows from (C7iii).\",\n \"Moreover, (C7i) is a consequence (C7ii) to (C7ix).\",\n \"As a side remark, notice that (C6) refines (M6).\",\n \"According to the representability requirement (REF ), one has $\\\\varphi (t)=\\\\sup _{z\\\\in \\\\mathbb {R}}\\\\left( z\\\\cdot t- \\\\Lambda _{\\\\mathbb {}}(z)\\\\right)=: \\\\Lambda _{\\\\mathbb {}}^{*}(t), \\\\qquad t\\\\in \\\\mathbb {R},\\\\ \\\\ $ (i.e.\",\n \"the divergence generator $\\\\varphi $ must be equal to the Fenchel-Legendre transform $\\\\Lambda _{\\\\mathbb {}}^{*}$ of a cumulant generating function $\\\\Lambda _{\\\\mathbb {}}$ ) of some probability distribution $\\\\mathbb {}$ , such that $\\\\lambda _{-} < 0 < \\\\lambda _{+}$ holds.\",\n \"Moreover, $\\\\varphi $ should satisfy $\\\\varphi (1)=0$ , and should be finite as well as strictly convex in a non-empty neighborhood $]t_{-}^{sc},t_{+}^{sc}[$ of 1 (cf.\",\n \"the definition of $\\\\widetilde{\\\\Upsilon }(]a,b[)$ ).\",\n \"The latter rules out that $\\\\mathbb {}$ is any one-point distribution (Dirac distribution), say $\\\\mathbb {} = \\\\delta _{y_{0}}$ for some $y_{0} \\\\in \\\\mathbb {R}$ , since in such a situation one gets $\\\\Lambda _{\\\\mathbb {}}(z) = z \\\\cdot y_{0}$ , and thus $\\\\varphi (t) = \\\\Lambda _{\\\\mathbb {}}^{*}(t) = 0$ for $t=y_{0}$ and $\\\\varphi (t) = \\\\Lambda _{\\\\mathbb {}}^{*}(t) = \\\\infty $ for all $t \\\\in \\\\mathbb {R}\\\\backslash \\\\lbrace y_{0}\\\\rbrace $ (even in the case $y_{0} =1$ for which $\\\\varphi (1)=0$ is satisfied).\",\n \"Consequently, $\\\\Lambda _{\\\\mathbb {}}$ is strictly convex on $]\\\\lambda _{-},\\\\lambda _{+}[ \\\\ = int(dom(\\\\Lambda _{\\\\mathbb {} }))$ (cf.\",\n \"(C6)) and (C7) applies.\",\n \"Clearly, by continuity one gets $\\\\Lambda _{\\\\mathbb {}}^{*}(t) =\\\\sup _{z\\\\in ]\\\\lambda _{-},\\\\lambda _{+}[}\\\\left( t \\\\cdot z - \\\\Lambda _{\\\\mathbb {}}(z)\\\\right),\\\\qquad t\\\\in \\\\mathbb {R}.\",\n \"\\\\ \\\\ $ For $t \\\\in ]t_{-}^{sc},t_{+}^{sc}[$ , the optimization problem (REF ) can be solved explicitly by the well-known “pure/original” Legendre transform, namely $\\\\Lambda _{\\\\mathbb {}}^{*}(t) = t \\\\cdot \\\\Lambda _{\\\\mathbb {}}^{\\\\prime -1}(t) -\\\\Lambda _{\\\\mathbb {}}\\\\Big (\\\\Lambda _{\\\\mathbb {}}^{\\\\prime -1}(t)\\\\Big ),\\\\qquad t \\\\in ]t_{-}^{sc},t_{+}^{sc}[.$ Let us inspect the further cases $t \\\\le t_{-}^{sc}$ .\",\n \"In the contexts of (C7iv) and (C7viii), this is obsolete since $t_{-}^{sc} = a = -\\\\infty $ .\",\n \"For (C7ii), where $t_{-}^{sc} = a > -\\\\infty $ , one can show $\\\\Lambda _{\\\\mathbb {}}^{*}(a) = - \\\\log \\\\mathbb {}[\\\\lbrace a\\\\rbrace ]= - \\\\log \\\\mathbb {\\\\Pi }[W = a \\\\, ]$ which together with (REF ) proves (G10ii); moreover, $\\\\Lambda _{\\\\mathbb {}}^{*}(t) = \\\\infty $ for all $t < a$ (see e.g.\",\n \"Section 9.1 of Borovkov [56]).\",\n \"In the setup (C7vi), where $t_{-}^{sc} > a = - \\\\infty $ it is clear that $\\\\Lambda _{\\\\mathbb {}}^{*}(t_{-}^{sc}) =t_{-}^{sc} \\\\cdot \\\\Lambda _{\\\\mathbb {}}^{\\\\prime -1}(t_{-}^{sc}) -\\\\Lambda _{\\\\mathbb {}}\\\\Big (\\\\Lambda _{\\\\mathbb {}}^{\\\\prime -1}(t_{-}^{sc})\\\\Big )= t_{-}^{sc} \\\\cdot \\\\lambda _{-} - \\\\Lambda _{\\\\mathbb {}}(\\\\lambda _{-})$ and $\\\\Lambda _{\\\\mathbb {}}^{*}(t)= t \\\\cdot \\\\lambda _{-} - \\\\Lambda _{\\\\mathbb {}}(\\\\lambda _{-})= \\\\Lambda _{\\\\mathbb {}}^{*}(t_{-}^{sc}) + \\\\lambda _{-} \\\\cdot (t- t_{-}^{sc})\\\\quad \\\\text{for all $t \\\\in ]-\\\\infty ,t_{-}^{sc}[$}.$ As far as the cases $t \\\\ge t_{+}^{sc}$ is concerned, in the situations of (C7v) and (C7ix), this is obsolete since $t_{+}^{sc} = b = \\\\infty $ .\",\n \"For (C7iii), where $t_{+}^{sc} = b < \\\\infty $ , one can show $\\\\Lambda _{\\\\mathbb {}}^{*}(b) = - \\\\log \\\\mathbb {}[\\\\lbrace b\\\\rbrace ]= - \\\\log \\\\mathbb {\\\\Pi }[W = b \\\\, ]$ which together with (REF ) proves (G10iii); moreover, $\\\\Lambda _{\\\\mathbb {}}^{*}(t) = \\\\infty $ for all $t > b$ (see e.g.\",\n \"Section 9.1 of Borovkov [56]).\",\n \"In the setup (C7vii), where $t_{+}^{sc} < b = \\\\infty $ it is clear that $\\\\Lambda _{\\\\mathbb {}}^{*}(t_{+}^{sc}) =t_{+}^{sc} \\\\cdot \\\\Lambda _{\\\\mathbb {}}^{\\\\prime -1}(t_{+}^{sc}) -\\\\Lambda _{\\\\mathbb {}}\\\\Big (\\\\Lambda _{\\\\mathbb {}}^{\\\\prime -1}(t_{+}^{sc})\\\\Big )= t_{+}^{sc} \\\\cdot \\\\lambda _{+} - \\\\Lambda _{\\\\mathbb {}}(\\\\lambda _{+})$ and $\\\\Lambda _{\\\\mathbb {}}^{*}(t)= t \\\\cdot \\\\lambda _{+} - \\\\Lambda _{\\\\mathbb {}}(\\\\lambda _{+})= \\\\Lambda _{\\\\mathbb {}}^{*}(t_{+}^{sc}) + \\\\lambda _{+} \\\\cdot (t- t_{+}^{sc})\\\\quad \\\\text{for all $t \\\\in ]t_{+}^{sc}, \\\\infty [$}.$ As a side effect, we have thus also proved (G10i) and (G3) (notice that in (G3) we have started with $a, b$ to be the endpoints of the support of $\\\\mathbb {}$ respectively $W$ , in contrast to Definition REF where $a$ , $b$ are defined as the endpoints of the effective domain of $\\\\varphi $ ).\",\n \"To proceed, from (REF ) and (REF ) we obtain $\\\\varphi ^{\\\\prime }(t) = (\\\\Lambda _{\\\\mathbb {}}^{*})^{\\\\prime }(t) = \\\\Lambda _{\\\\mathbb {}}^{\\\\prime -1}(t),\\\\qquad \\\\varphi ^{\\\\prime \\\\prime }(t) = (\\\\Lambda _{\\\\mathbb {}}^{*})^{\\\\prime \\\\prime }(t) =\\\\frac{1}{\\\\Lambda _{\\\\mathbb {}}^{\\\\prime \\\\prime }\\\\big (\\\\Lambda _{\\\\mathbb {}}^{\\\\prime -1}(t)\\\\big )} > 0,\\\\qquad t \\\\in ]t_{-}^{sc},t_{+}^{sc}[,$ which — together with the investigations below (REF ) — provides (G4) and (G5); moreover, (G6) is immediate since the infinite differentiability is straightforward and $\\\\varphi ^{\\\\prime }(1) =0$ because we have required both the nonnegativity of $\\\\varphi $ and (G2) (cf.\",\n \"the definition of $\\\\widetilde{\\\\Upsilon }(]a,b[)$ ).\",\n \"The property (G7) follows from (C7ii), (C7iv), (C7viii), (REF ), (REF ) and $\\\\varphi ^{\\\\prime }(t_{-}^{sc}) = \\\\Lambda _{\\\\mathbb {}}^{\\\\prime -1}(t_{-}^{sc}) = \\\\lambda _{-}$ .\",\n \"Analogously, we get (G8) from (C7iii), (C7v), (C7ix), (REF ), (REF ) and $\\\\varphi ^{\\\\prime }(t_{+}^{sc}) = \\\\Lambda _{\\\\mathbb {}}^{\\\\prime -1}(t_{+}^{sc}) = \\\\lambda _{+}$ .\",\n \"Let us continue with (G9).\",\n \"By applying the general theory of double Fenchel-Legendre transforms (bi-conjugates), (REF ) turns into $\\\\varphi ^{*} (z) = \\\\Lambda _{\\\\mathbb {}}(z), \\\\qquad z\\\\in \\\\mathbb {R},\\\\ \\\\ $ which deduces (G9i).\",\n \"The properties (G9ii), (G9iii) and (G9iv) follow from Theorem REF (cf.\",\n \"the discussion thereafter).\",\n \"Finally, we obtain (G11i) and (G11ii) from (REF ), (REF ) and ().\",\n \"$\\\\blacksquare $ Proof of Proposition REF .\",\n \"The assertions follow immediately from (REF ), (REF ), (REF ), Theorem REF , (REF ) (and the discussion thereafter) as well as (M5).\",\n \"$\\\\blacksquare $\"\n ],\n [\n \"Proofs — Part 5\",\n \"Proof of Proposition REF .\",\n \"The assertion follows straightforwardly from the following two facts: (i) a moment generating function $MGF$ is infinitely divisible if and only if $MGF^{c}$ is a moment generating function for all $c >0$ (cf.\",\n \"e.g.\",\n \"(the MGF-version of) Prop.\",\n \"IV.2.5 of Steutel & van Harn [341]).\",\n \"(ii) $z \\\\mapsto MGF(z)$ is a moment generating function if and only if $z \\\\mapsto MGF(\\\\breve{c} \\\\cdot z) =: MGF_{\\\\breve{c}}(z)$ is a moment generating function for all $\\\\breve{c} >0$ .\",\n \"Notice that for each $c >0$ , $\\\\breve{c} >0$ one has $int(dom(MGF)) = int(dom(MGF^{c}))$ and $int(dom(MGF_{\\\\breve{c}})) = \\\\frac{1}{\\\\breve{c}} \\\\cdot int(dom(MGF))$ , and hence the light-tailedness remains unchanged: $0 \\\\in int(dom(MGF))$ if and only if $0 \\\\in int(dom(MGF^{c}))$ if and only if $0 \\\\in int(dom(MGF_{\\\\breve{c}}))$ .\",\n \"Since $\\\\varphi \\\\in \\\\Upsilon (]a,b[)$ , we have $\\\\varphi (t)=\\\\sup _{z\\\\in ]\\\\lambda _{-}, \\\\lambda _{+}[}\\\\left( z \\\\cdot t-\\\\log \\\\Big (\\\\int _{\\\\mathbb {R}} e^{z\\\\cdot y }\\\\, d\\\\mathbb {} (y) \\\\Big ) \\\\right), \\\\quad t\\\\in ]a,b[ \\\\, , \\\\ \\\\ $ and thus for the exponential of its Fenchel-Legendre transform $\\\\int _{\\\\mathbb {R}} e^{z \\\\cdot y} d\\\\mathbb {} (y) , \\\\qquad z \\\\in ]\\\\lambda _{-}, \\\\lambda _{+}[ .$ Now, let $\\\\widetilde{\\\\varphi } := \\\\widetilde{c} \\\\cdot \\\\varphi \\\\in \\\\Upsilon (]a,b[)$ for arbitrarily fixed $\\\\widetilde{c} >0$ .\",\n \"From the application of (REF ) to $\\\\widetilde{\\\\varphi }$ we obtain $\\\\widetilde{\\\\varphi } (t)=\\\\sup _{\\\\widetilde{z} \\\\in ]\\\\widetilde{\\\\lambda }_{-}, \\\\widetilde{\\\\lambda }_{+}[}\\\\left( \\\\widetilde{z} \\\\cdot t-\\\\log \\\\int _{\\\\mathbb {R}}e^{\\\\widetilde{z} \\\\cdot \\\\widetilde{y}} d\\\\widetilde{\\\\mathbb {}}_{\\\\widetilde{c}} (\\\\widetilde{y})\\\\right), \\\\qquad t\\\\in ]a,b[ \\\\, , \\\\ \\\\ $ for some unique probability distribution $\\\\widetilde{\\\\mathbb {}}_{\\\\widetilde{c}}$ on $\\\\mathbb {R}$ .\",\n \"Here, according to (G9i) for $\\\\widetilde{\\\\varphi }$ we have used $\\\\widetilde{\\\\lambda }_{-} := \\\\inf _{t \\\\in ]a,b[} \\\\widetilde{\\\\varphi }^{\\\\prime }(t) = \\\\widetilde{c} \\\\cdot \\\\lambda _{-}$ and $\\\\widetilde{\\\\lambda }_{+} := \\\\sup _{t \\\\in ]a,b[} \\\\widetilde{\\\\varphi }^{\\\\prime }(t) = \\\\widetilde{c} \\\\cdot \\\\lambda _{+}$ .\",\n \"Dividing (REF ) by $\\\\widetilde{c}$ , we arrive at $\\\\varphi (t) = \\\\frac{\\\\widetilde{\\\\varphi } (t)}{\\\\widetilde{c}}&=&\\\\sup _{\\\\widetilde{z} \\\\in ]\\\\widetilde{c} \\\\cdot \\\\lambda _{-}, \\\\widetilde{c} \\\\cdot \\\\lambda _{+}[}\\\\left( \\\\frac{\\\\widetilde{z}}{\\\\widetilde{c}} \\\\cdot t-\\\\log \\\\Big (\\\\int _{\\\\mathbb {R}}e^{\\\\frac{\\\\widetilde{z}}{\\\\widetilde{c}} \\\\cdot \\\\widetilde{y} \\\\cdot \\\\widetilde{c}} d\\\\widetilde{\\\\mathbb {}}_{\\\\widetilde{c}} (\\\\widetilde{y})\\\\Big )^{1/\\\\widetilde{c}}\\\\right),\\\\nonumber \\\\\\\\&=&\\\\sup _{z \\\\in ]\\\\lambda _{-}, \\\\lambda _{+}[}\\\\left( z \\\\cdot t-\\\\log \\\\Big (\\\\int _{\\\\mathbb {R}}e^{z \\\\cdot \\\\widetilde{y} \\\\cdot \\\\widetilde{c}}d\\\\widetilde{\\\\mathbb {}}_{\\\\widetilde{c}} (\\\\widetilde{y})\\\\Big )^{1/\\\\widetilde{c}}\\\\right),\\\\qquad t\\\\in ]a,b[ \\\\, , \\\\ \\\\ $ and hence for the exponential of its Fenchel-Legendre transform $e^{\\\\varphi _{*}(z)} =\\\\Big ( \\\\int _{\\\\mathbb {R}} e^{z \\\\cdot \\\\widetilde{y} \\\\cdot \\\\widetilde{c}}d\\\\widetilde{\\\\mathbb {}}_{\\\\widetilde{c}} (\\\\widetilde{y}) \\\\Big )^{1/\\\\widetilde{c}}, \\\\qquad z \\\\in ]\\\\lambda _{-}, \\\\lambda _{+}[ .$ Here, according to (G9i) for $\\\\widetilde{\\\\varphi }$ we have used $\\\\widetilde{\\\\lambda }_{-} := \\\\inf _{t \\\\in ]a,b[} \\\\widetilde{\\\\varphi }^{\\\\prime }(t) = \\\\widetilde{c} \\\\cdot \\\\lambda _{-}$ and $\\\\widetilde{\\\\lambda }_{+} := \\\\sup _{t \\\\in ]a,b[} \\\\widetilde{\\\\varphi }^{\\\\prime }(t) = \\\\widetilde{c} \\\\cdot \\\\lambda _{+}$ .\",\n \"From (REF ) and (REF ) we deduce for $\\\\widetilde{c} := \\\\frac{1}{n}$ the relation $MGF_{\\\\mathbb {}}(z) = (MGF_{\\\\widetilde{\\\\mathbb {}}_{1/n}}(\\\\frac{z}{n}))^{n}$ for all $n \\\\in \\\\mathbb {N}$ which (with the help of (ii)) implies the infinitely divisibility of $\\\\mathbb {}$ .\",\n \"For the reverse direction, let us assume that $\\\\varphi \\\\in \\\\Upsilon (]a,b[)$ and that the corresponding $\\\\mathbb {}$ is infinitely divisible.\",\n \"Recall that $]a,b[ = int(dom(\\\\varphi ))$ .\",\n \"Moreover, we fix an arbitrary constant $\\\\widetilde{c} >0$ .\",\n \"Of course, there holds $\\\\widetilde{c} \\\\cdot \\\\varphi \\\\in \\\\widetilde{\\\\Upsilon }(]a,b[)$ and $dom(\\\\widetilde{c} \\\\cdot \\\\varphi ) = dom(\\\\varphi )$ .\",\n \"Furthermore, by multiplying (REF ) with $\\\\widetilde{c} >0$ and by employing (i), (ii) we get $\\\\widetilde{c} \\\\cdot \\\\varphi (t) &=&\\\\sup _{z\\\\in ]\\\\lambda _{-}, \\\\lambda _{+}[}\\\\left( \\\\widetilde{c} \\\\cdot z \\\\cdot t-\\\\log \\\\Big (\\\\int _{\\\\mathbb {R}} e^{\\\\widetilde{c} \\\\cdot z\\\\cdot \\\\frac{y}{\\\\widetilde{c}}} d\\\\mathbb {} (y) \\\\Big )^{\\\\widetilde{c}} \\\\right)= \\\\sup _{\\\\widetilde{z} \\\\in ]\\\\widetilde{c} \\\\cdot \\\\lambda _{-}, \\\\widetilde{c} \\\\cdot \\\\lambda _{+}[}\\\\left( \\\\widetilde{z} \\\\cdot t-\\\\log \\\\Big (\\\\int _{\\\\mathbb {R}}e^{\\\\frac{\\\\widetilde{z}}{\\\\widetilde{c}} \\\\cdot y} \\\\, d\\\\mathbb {} (y) \\\\Big )^{\\\\widetilde{c}} \\\\right)\\\\nonumber \\\\\\\\&=&\\\\sup _{\\\\widetilde{z} \\\\in ]\\\\widetilde{c} \\\\cdot \\\\lambda _{-}, \\\\widetilde{c} \\\\cdot \\\\lambda _{+}[}\\\\left( \\\\widetilde{z} \\\\cdot t-\\\\log \\\\Big ( \\\\int _{\\\\mathbb {R}}e^{\\\\widetilde{z} \\\\cdot y} d\\\\mathbb {}_{\\\\widetilde{c}} (y) \\\\Big ) \\\\right), \\\\quad t\\\\in ]a,b[; \\\\ \\\\ $ for some probability distribution $\\\\mathbb {}_{\\\\widetilde{c}}$ on $\\\\mathbb {R}$ .\",\n \"$\\\\blacksquare $ Proof of Proposition REF .\",\n \"It is well known that a candidate function $M: ]-\\\\infty ,0[ \\\\mapsto ]0,\\\\infty [$ is the moment-generating function of an infinitely divisible probability distribution if and only if $(\\\\log M)^{\\\\prime }$ is absolutely monotone (see e.g.\",\n \"Theorem 5.11 of Schilling et al.\",\n \"[322]).\",\n \"By applying this to $M(z) := e^{-a\\\\cdot z + \\\\varphi ^{*}(z)}$ respectively $M(z) := e^{b\\\\cdot z + \\\\varphi ^{*}(- z)}$ , one gets straightforwardly the assertion (a) respectively (b); notice that the light-tailedness follows then from (G1) to (G8), and $b=\\\\infty $ respectively $a =- \\\\infty $ can be deduced from the fact that the support of an infinitely distribution is always (one-sided or two-sided) unbounded.\",\n \"For the third case $a= - \\\\infty $ , $b= \\\\infty $ one can use the assertion (cf.\",\n \"e.g.\",\n \"Morris [267], p.73) that a candidate function $M: \\\\, ]\\\\lambda _{-}, \\\\lambda _{+}[ \\\\, \\\\mapsto \\\\, ]0,\\\\infty [$ is the moment-generating function of an infinitely divisible probability distribution if the connected function $z \\\\mapsto (\\\\log M)^{\\\\prime \\\\prime }(z)/(\\\\log M)^{\\\\prime \\\\prime }(0)$ is the moment-generating function of some auxiliary probability distribution; but the latter is equivalent to exponentially convexity (cf.\",\n \"Theorem REF (b)).\",\n \"By applying this to $M(z) := e^{\\\\varphi ^{*}(z)}$ , one ends up with (c).\",\n \"$\\\\blacksquare $\"\n ],\n [\n \"Proofs — Part 6\",\n \"Proof of Theorem REF .\",\n \"(i) Clearly, on $]\\\\lambda _{-},\\\\lambda _{+}[$ the function $\\\\Lambda $ is differentiable with strictly increasing derivative $\\\\Lambda ^{\\\\prime }(z) = F^{-1}(z+c) + 1- F^{-1}(c) ,\\\\qquad z \\\\in ]\\\\lambda _{-},\\\\lambda _{+}[.$ Hence, $\\\\Lambda $ is strictly convex and smooth (because of the smoothness of $F^{-1}$ ), and satisfies $\\\\Lambda (0) =0$ as well as $\\\\Lambda ^{\\\\prime }(0) =1$ .\",\n \"Also, the corresponding extensions of $\\\\Lambda $ to $z=\\\\lambda _{-}$ and $z=\\\\lambda _{+}$ are continuous.\",\n \"(ii) It is straightforward to see that on $]t_{-}^{sc},t_{+}^{sc}[$ the function $\\\\varphi $ is differentiable with strictly increasing derivative $\\\\varphi ^{\\\\prime }(t) = F(t+ F^{-1}(c) - 1) - c ,\\\\qquad t \\\\in ]t_{-}^{sc},t_{+}^{sc}[.$ Hence, $\\\\varphi $ is strictly convex and smooth (because of the smoothness of $F$ ), and satisfies $\\\\varphi (1) =0$ as well as $\\\\varphi ^{\\\\prime }(1) =0$ .\",\n \"Also, the corresponding extensions of $\\\\varphi $ to $t=t_{-}^{sc}$ and $t=t_{-}^{sc}$ are continuous.\",\n \"Hence (G1), (G2), (G5) and (G6) hold.\",\n \"To prove (G3) (and hence (G1)), let us first notice that obviously there holds $a \\\\le t_{-}^{sc}$ and $t_{+}^{sc} \\\\le b$ .\",\n \"Moreover, the validity of $\\\\varphi (t) < \\\\infty $ for all $t \\\\in ]t_{-}^{sc},t_{+}^{sc}[$ is clear from (REF ) since $t+F^{-1}(c)-1 \\\\in ]a_{F},b_{F}[ = int(dom(F))$ and the involved integral over the continuous function $F^{-1}$ is taken over a compact interval.\",\n \"For the subcase $t_{-}^{sc} =- \\\\infty = a$ we have thus shown $dom(\\\\varphi ) \\\\cap ]-\\\\infty ,1] = ]-\\\\infty ,1]= ]a,1]$ , whereas for the subcase $t_{+}^{sc} = \\\\infty = b$ we have verified $dom(\\\\varphi ) \\\\cap [1,\\\\infty [ = [1,\\\\infty [ = [1,b[$ .\",\n \"Let us next examine the subcase “$t_{-}^{sc} > - \\\\infty $ and $\\\\varphi (t_{-}^{sc}) < \\\\infty $ ”: if $\\\\lambda _{-} > - \\\\infty $ then $a = -\\\\infty $ and (REF ) implies $\\\\varphi (t) = \\\\varphi (t_{-}^{sc}) +\\\\lambda _{-} \\\\cdot (t- t_{-}^{sc}) < \\\\infty $ for all $t \\\\in \\\\ ]-\\\\infty , t_{-}^{sc}] = ]a,t_{-}^{sc}]$ , which leads to $dom(\\\\varphi ) \\\\cap ]-\\\\infty ,1] = ]-\\\\infty ,1] = ]a,1]$ ; in contrast, if $\\\\lambda _{-} = - \\\\infty $ then $a = t_{-}^{sc}$ and (REF ) implies $\\\\varphi (t) = \\\\varphi (t_{-}^{sc}) +\\\\lambda _{-} \\\\cdot (t- t_{-}^{sc}) = \\\\infty $ for all $t \\\\in \\\\ ]-\\\\infty , t_{-}^{sc}[ = ]-\\\\infty ,a[$ , which leads to $dom(\\\\varphi ) \\\\cap ]-\\\\infty ,1] = [a,1]$ .\",\n \"In the subcase “$t_{-}^{sc} > - \\\\infty $ and $\\\\varphi (t_{-}^{sc}) = \\\\infty $ ”, due to the strict convexity of $\\\\varphi $ one always has $\\\\lim _{t \\\\downarrow t_{-}^{sc}} \\\\varphi ^{\\\\prime }(t) = - \\\\infty $ ; this implies, by the below-mentioned (REF ), that $\\\\lambda _{-} = - \\\\infty $ and thus $a = t_{-}^{sc}$ ; from (REF ) we derive $\\\\varphi (t) = \\\\varphi (t_{-}^{sc}) +\\\\lambda _{-} \\\\cdot (t- t_{-}^{sc}) = \\\\infty $ for all $t \\\\in \\\\ ]-\\\\infty , t_{-}^{sc}[ = ]-\\\\infty ,a[$ , which leads to $dom(\\\\varphi ) \\\\cap ]-\\\\infty ,1] = ]a,1]$ .\",\n \"As a further step, we deal with the subcase “$t_{+}^{sc} < \\\\infty $ and $\\\\varphi (t_{+}^{sc}) < \\\\infty $ ”: if $\\\\lambda _{+} < \\\\infty $ then $b = \\\\infty $ and (REF ) implies $\\\\varphi (t) = \\\\varphi (t_{+}^{sc}) +\\\\lambda _{+} \\\\cdot (t- t_{+}^{sc}) < \\\\infty $ for all $t \\\\in \\\\ [t_{+}^{sc},\\\\infty [ = [t_{+}^{sc},b[$ , which leads to $dom(\\\\varphi ) \\\\cap [1, \\\\infty [ = [1, \\\\infty [ = [1, b[$ ; in contrast, if $\\\\lambda _{+} = \\\\infty $ then $b = t_{+}^{sc}$ and (REF ) implies $\\\\varphi (t) = \\\\varphi (t_{+}^{sc}) +\\\\lambda _{+} \\\\cdot (t- t_{+}^{sc}) = \\\\infty $ for all $t \\\\in \\\\ ]t_{-}^{sc}, \\\\infty [ = ]b,\\\\infty [$ , which leads to $dom(\\\\varphi ) \\\\cap [1,\\\\infty [ = [1,b]$ .\",\n \"In the subcase “$t_{+}^{sc} < + \\\\infty $ and $\\\\varphi (t_{+}^{sc}) = \\\\infty $ ”, due to the strict convexity of $\\\\varphi $ one always gets $\\\\lim _{t \\\\uparrow t_{+}^{sc}} \\\\varphi ^{\\\\prime }(t) = \\\\infty $ ; this implies, by the below-mentioned (), that $\\\\lambda _{+} = \\\\infty $ and thus $b = t_{+}^{sc}$ ; from (REF ) we deduce $\\\\varphi (t) = \\\\varphi (t_{+}^{sc}) +\\\\lambda _{+} \\\\cdot (t- t_{+}^{sc}) = \\\\infty $ for all $t \\\\in \\\\ ]t_{+}^{sc}, \\\\infty [ = ]b, \\\\infty [$ , which leads to $dom(\\\\varphi ) \\\\cap [1,\\\\infty [ = [1,b[$ .\",\n \"Putting things together, we have proved (G3).\",\n \"The property (G4) follows straightforwardly from (REF ), the continuity of $F$ and from $\\\\lim _{t \\\\downarrow t_{-}^{sc}} \\\\varphi ^{\\\\prime }(t) = \\\\lambda _{-}$ , $\\\\lim _{t \\\\uparrow t_{+}^{sc}} \\\\varphi ^{\\\\prime }(t) = \\\\lambda _{+}$ .\",\n \"To see the latter two, from (REF ) we obtain $& & \\\\lim _{t \\\\downarrow t_{-}^{sc}} \\\\varphi ^{\\\\prime }(t) = \\\\lim _{t \\\\downarrow t_{-}^{sc}} F(t+ F^{-1}(c) - 1) - c= \\\\lim _{t \\\\downarrow t_{-}^{sc}} F(t+ a_{F} - t_{-}^{sc}) - c= \\\\inf \\\\lbrace F(\\\\widetilde{t}) - c: \\\\widetilde{t} \\\\in ]a_{F},b_{F}[ \\\\rbrace = \\\\lambda _{-},\\\\\\\\& & \\\\lim _{t \\\\uparrow t_{+}^{sc}} \\\\varphi ^{\\\\prime }(t) = \\\\lim _{t \\\\uparrow t_{+}^{sc}} F(t+ F^{-1}(c) - 1) - c= \\\\lim _{t \\\\uparrow t_{+}^{sc}} F(t+ b_{F} - t_{+}^{sc}) - c= \\\\sup \\\\lbrace F(\\\\widetilde{t}) - c: \\\\widetilde{t} \\\\in ]a_{F},b_{F}[ \\\\rbrace = \\\\lambda _{+}.$ The two properties (G7) and (G8) are clear form the above considerations.\",\n \"(iii) From (REF ) and (REF ) one gets easily $\\\\Lambda ^{\\\\prime -1}(t) = F\\\\left(t+F^{-1}(c)-1 \\\\right) - c = \\\\varphi ^{\\\\prime }(t), \\\\qquad t \\\\in ]t_{-}^{sc},t_{+}^{sc}[,$ as well as $\\\\Lambda ^{\\\\prime -1}(1)=0$ .\",\n \"From this, we derive $&& t \\\\cdot \\\\Lambda ^{\\\\prime -1}(t) - \\\\Lambda \\\\left(\\\\Lambda ^{\\\\prime -1}(t)\\\\right)\\\\nonumber \\\\\\\\&& = t \\\\cdot [F\\\\left(t+F^{-1}(c)-1 \\\\right) - c]+ [F^{-1}(c)-1] \\\\cdot [F\\\\left(t+F^{-1}(c)-1 \\\\right) - c]\\\\nonumber \\\\\\\\& & -\\\\int \\\\displaylimits _{0}^{F\\\\left(t+F^{-1}(c)-1 \\\\right) - c} F^{-1}(u+c) du\\\\nonumber \\\\\\\\& & = \\\\varphi (t), \\\\qquad t \\\\in ]t_{-}^{sc},t_{+}^{sc}[,$ and hence, with the help of (REF ) in combination with (REF ), () $\\\\varphi (t) = \\\\max _{z \\\\in ]\\\\lambda _{-},\\\\lambda _{+}[} \\\\left( z\\\\cdot t -\\\\Lambda (z)\\\\right), \\\\qquad t \\\\in ]t_{-}^{sc},t_{+}^{sc}[ ,$ i.e.\",\n \"on $]t_{-}^{sc},t_{+}^{sc}[$ the divergence generator $\\\\varphi $ is the classical Legendre transform of the restriction of $\\\\Lambda $ to $]\\\\lambda _{-},\\\\lambda _{+}[$ .\",\n \"If “$\\\\lambda _{-} > -\\\\infty $ , $\\\\Lambda (\\\\lambda _{-}) \\\\in ]-\\\\infty ,\\\\infty [$ and $\\\\Lambda ^{\\\\prime }(\\\\lambda _{-}) \\\\in ]-\\\\infty ,\\\\infty [$ ” respectively “$\\\\lambda _{+} < -\\\\infty $ , $\\\\Lambda (\\\\lambda _{+}) \\\\in ]-\\\\infty ,\\\\infty [$ and $\\\\Lambda ^{\\\\prime }(\\\\lambda _{+}) \\\\in ]-\\\\infty ,\\\\infty [$ ”, then one can apply classical facts of Fenchel-Legendre transformation to get the corresponding left-hand respectively right-hand linear extensions of $\\\\varphi $ on the complement of $]t_{-}^{sc},t_{+}^{sc}[$ , in order to obtain the desired $\\\\varphi (t) =\\\\sup _{z \\\\in ]-\\\\infty ,\\\\infty [} \\\\left( z\\\\cdot t - \\\\Lambda (z)\\\\right) ,\\\\qquad t \\\\in \\\\mathbb {R};$ notice that $t_{-}^{sc} = \\\\lim _{z \\\\downarrow \\\\lambda _{-}} \\\\Lambda ^{\\\\prime }(z)$ and $t_{+}^{sc} = \\\\lim _{z \\\\uparrow \\\\lambda _{+}} \\\\Lambda ^{\\\\prime }(z)$ .\",\n \"(iv) This is just the inverse of (iii), by applying standard Fenchel-Legendre-transformation theory.\",\n \"$\\\\blacksquare $\"\n ],\n [\n \"Further details and proofs for\\nSubsection \",\n \"Proof of Lemma REF.\",\n \"By Assumption (OM), one gets for all $\\\\lambda \\\\in cl(\\\\mathbf {\\\\Lambda })$ that $\\\\lbrace \\\\mathbf {x} \\\\in (dom(\\\\widetilde{\\\\varphi })^{n} : T(\\\\mathbf {x}) = \\\\lambda \\\\rbrace \\\\, \\\\cap \\\\, ]t_{-}^{sc},t_{+}^{sc}[^{n} \\\\ne \\\\emptyset $ .\",\n \"Moreover, for any $\\\\mathbf {x}=\\\\left( x_{1},..,x_{n}\\\\right) $ in $\\\\mathbb {R}^{n}$ , by the independence of the components of $\\\\mathbf {\\\\widetilde{W}}$ as well as (REF ) and (REF ), we have $& & I_{\\\\mathbf {\\\\widetilde{W}}}(\\\\mathbf {x})=\\\\sup _{\\\\mathbf {z}=\\\\left( z_{1},\\\\ldots ,z_{n}\\\\right) \\\\in \\\\mathbb {R}^{n}}\\\\left( \\\\left\\\\langle \\\\mathbf {x},\\\\mathbf {z}\\\\right\\\\rangle -\\\\sum _{i=1}^{n}\\\\Lambda _{\\\\widetilde{\\\\mathbb {}}}(z_{i}) \\\\right)= \\\\sup _{\\\\mathbf {z}\\\\in ]\\\\lambda _{-},\\\\lambda _{+}[^{n}}\\\\left( \\\\sum _{i=1}^{n} \\\\left( x_{i} \\\\cdot z_{i}-\\\\Lambda _{\\\\widetilde{\\\\mathbb {}}}(z_{i}) \\\\right) \\\\right)\\\\nonumber \\\\\\\\& & = \\\\sum _{i=1}^{n} \\\\left( \\\\sup _{z_{i}\\\\in ]\\\\lambda _{-},\\\\lambda _{+}[}\\\\left( x_{i}\\\\cdot z_{i} - \\\\Lambda _{\\\\widetilde{\\\\mathbb {}}}(z_{i})\\\\right) \\\\right)= \\\\sum _{i=1}^{n} \\\\widetilde{\\\\varphi }(x_{i})=\\\\sum _{k=1}^{K}\\\\sum _{i\\\\in I_{k}^{(n)}} \\\\widetilde{\\\\varphi }(x_{i})$ which is finite if and only if $\\\\mathbf {x} \\\\in (dom(\\\\widetilde{\\\\varphi }))^{n}$ (recall that $\\\\widetilde{\\\\varphi }$ is a nonnegative function).\",\n \"Hence, for each $\\\\lambda \\\\in \\\\mathbf {\\\\Lambda }$ we obtain $I(\\\\lambda ) &:=&\\\\inf _{\\\\mathbf {x} \\\\in \\\\mathbb {R}^{n}: \\\\, T(\\\\mathbf {x})=\\\\lambda }I_{\\\\mathbf {\\\\widetilde{W}}}(\\\\mathbf {x})= \\\\inf _{\\\\mathbf {x} \\\\in (dom(\\\\widetilde{\\\\varphi }))^{n}: \\\\, T(\\\\mathbf {x})=\\\\lambda }I_{\\\\mathbf {\\\\widetilde{W}}}(\\\\mathbf {x}) =\\\\inf _{\\\\mathbf {x} \\\\in (dom(\\\\widetilde{\\\\varphi }))^{n}: \\\\, T(\\\\mathbf {x})=\\\\lambda } \\\\ \\\\sum _{k=1}^{K}\\\\sum _{i\\\\in I_{k}^{(n)}} \\\\widetilde{\\\\varphi }(x_{i})\\\\\\\\&=& \\\\sum _{k=1}^{K} n_{k} \\\\cdot \\\\widetilde{\\\\varphi }(\\\\lambda _{k})\\\\ = \\\\ n \\\\cdot \\\\sum _{k=1}^{K} \\\\widetilde{p}_{k} \\\\cdot \\\\widetilde{\\\\varphi }(\\\\lambda _{k})= \\\\inf _{\\\\mathbf {x} \\\\in ]t_{-}^{sc},t_{+}^{sc}[^{n} \\\\, : \\\\, T(\\\\mathbf {x})=\\\\lambda }I_{\\\\mathbf {\\\\widetilde{W}}}(\\\\mathbf {x}) \\\\, ;$ here, we have employed the following facts: (i) the right-most infimum in (REF ) is achieved by minimizing each of the $K$ terms $\\\\sum _{i\\\\in I_{k}^{(n)}}\\\\widetilde{\\\\varphi }(x_{i})$ under the linear constraint $\\\\frac{1}{n_{k}} \\\\cdot \\\\sum _{i\\\\in I_{k}^{(n)}} x_{i}= \\\\lambda _{k}$ , and by the strict convexity of $\\\\widetilde{\\\\varphi }$ on $]t_{-}^{sc},t_{+}^{sc}[$ (cf.\",\n \"(G5)) the minimum of this generic term is attained when all components $x_{i}$ are equal to $\\\\lambda _{k}$ , and (ii) the outcoming minimum does not depend on the particular (generally non-unique) choice of the $x_{i}$ ’s.\",\n \"Notice that we have used the relation $n_{k} = n \\\\cdot \\\\widetilde{p}_{k}$ as well.\",\n \"To proceed, let $\\\\underline{\\\\lambda }$ be a minimal rate point of $\\\\mathbf {\\\\Lambda }$ , which means that $\\\\underline{\\\\lambda } \\\\in \\\\partial \\\\mathbf {\\\\Lambda }$ and $I(\\\\underline{\\\\lambda }) \\\\le I(\\\\lambda )$ for all $\\\\lambda \\\\in \\\\mathbf {\\\\Lambda }$ .\",\n \"By Assumption (OM) one can run all the steps in (REF ) and () with $\\\\underline{\\\\lambda }$ instead of $\\\\lambda $ , and hence $I(\\\\underline{\\\\lambda }) \\\\ = \\\\ \\\\inf _{\\\\mathbf {x} \\\\in \\\\mathbb {R}^{n}: \\\\, T(\\\\mathbf {x})=\\\\underline{\\\\lambda }}I_{\\\\mathbf {\\\\widetilde{W}}}(\\\\mathbf {x})\\\\ = \\\\inf _{\\\\mathbf {x} \\\\in ]t_{-}^{sc},t_{+}^{sc}[^{n} \\\\, : \\\\, T(\\\\mathbf {x})=\\\\underline{\\\\lambda }}I_{\\\\mathbf {\\\\widetilde{W}}}(\\\\mathbf {x})\\\\ = \\\\ n \\\\cdot \\\\sum _{k=1}^{K} \\\\widetilde{p}_{k} \\\\cdot \\\\widetilde{\\\\varphi }(\\\\underline{\\\\lambda }_{k})\\\\ = \\\\ n \\\\cdot \\\\sum _{k=1}^{K} \\\\widetilde{p}_{k} \\\\cdot \\\\widetilde{\\\\varphi }(\\\\underline{\\\\widetilde{q}}_{k}/\\\\widetilde{p}_{k})$ where for the last equality we have employed the vector $\\\\underline{\\\\mathbf {\\\\widetilde{Q}}} := \\\\mathfrak {D}^{-1} \\\\underline{\\\\lambda }$ which we have called the “dominating point of $\\\\widetilde{\\\\mathbf {\\\\Omega }}$ ”.\",\n \"Also we have proved $I(\\\\underline{\\\\lambda })\\\\ = \\\\ n \\\\cdot \\\\inf _{\\\\mathbf {\\\\widetilde{Q}} \\\\in \\\\widetilde{\\\\mathbf {\\\\Omega }}}\\\\sum _{k=1}^{K} \\\\widetilde{p}_{k} \\\\cdot \\\\widetilde{\\\\varphi }(\\\\widetilde{q}_{k}/\\\\widetilde{p}_{k}).\",\n \"\\\\qquad \\\\qquad \\\\blacksquare $ On the obtainment of proxies of minimal rate points by proxy method 2: For the rest of this section, besides (OM) we assume that $dom(\\\\widetilde{\\\\varphi }) = \\\\, ]a,b[ \\\\, = \\\\, ]t_{-}^{sc},t_{+}^{sc}[$ , and that in case of $a=-\\\\infty $ or $b=+\\\\infty $ the divergence generator $\\\\widetilde{\\\\varphi }$ is regularly varying at $a$ or $b$ accordingly, with positive index $\\\\beta $ , i.e.\",\n \"(with a slight abuse of notation) if $a=-\\\\infty $ , then for all $\\\\lambda >0$ there holds $\\\\lim _{u\\\\rightarrow -\\\\infty }\\\\frac{\\\\widetilde{\\\\varphi } \\\\left( \\\\lambda u\\\\right) }{\\\\widetilde{\\\\varphi }\\\\left( u\\\\right) }=\\\\lambda ^{\\\\beta },$ if $b=+\\\\infty $ , then for all $\\\\lambda >0$ there holds $\\\\lim _{u\\\\rightarrow +\\\\infty }\\\\frac{\\\\widetilde{\\\\varphi } \\\\left( \\\\lambda u\\\\right) }{\\\\widetilde{\\\\varphi }\\\\left( u\\\\right) }=\\\\lambda ^{\\\\beta };$ this assumption is denoted by (H$\\\\widetilde{\\\\varphi }$ ).\",\n \"A proxy of $\\\\underline{\\\\mathbf {\\\\widetilde{Q}}}$ can be obtained by sampling from a distribution on $\\\\mathbb {R}^{K}$ defined through $f(\\\\mathbf {\\\\widetilde{Q}}):=C \\\\cdot \\\\exp \\\\left( -\\\\sum _{k=1}^{K} \\\\widetilde{p}_{k} \\\\cdot \\\\widetilde{\\\\varphi }(\\\\widetilde{q}_{k}/\\\\widetilde{p}_{k})\\\\right) \\\\ = \\\\ C \\\\cdot \\\\exp \\\\left( -D_{\\\\widetilde{\\\\varphi }}\\\\left( \\\\mathbf {\\\\widetilde{Q}},\\\\widetilde{{P}}\\\\right)\\\\right)\\\\hspace{51.21504pt} \\\\textrm {(cf.\",\n \"(\\\\ref {simul distr}))}\\\\nonumber $ where $C$ is a normalizing constant; strict convexity (cf.\",\n \"(G5)) of $\\\\widetilde{\\\\varphi }$ together with (H$\\\\widetilde{\\\\varphi }$ ) prove that $f$ is a well-defined (Lebesgue-) density for a random variable $\\\\mathbf {T}$ on $\\\\mathbb {R}^{K}$ .\",\n \"We denote by $\\\\mathbb {F}(\\\\cdot ) := \\\\mathbb {\\\\Pi }[\\\\mathbf {T} \\\\in \\\\cdot \\\\, ]$ the corresponding distribution on $\\\\mathbb {R}^{K}$ having density $f$ .\",\n \"The distribution of $\\\\mathbf {T}$ given $\\\\left( \\\\mathbf {T} \\\\in \\\\widetilde{\\\\mathbf {\\\\Omega }} \\\\right)$ concentrates on the points in $\\\\widetilde{\\\\mathbf {\\\\Omega }}$ which minimize $D_{\\\\widetilde{\\\\varphi }}\\\\left(\\\\mathbf {\\\\widetilde{Q}},\\\\widetilde{{P}}\\\\right) $ as $\\\\mathbf {\\\\widetilde{Q}}$ runs in $\\\\widetilde{\\\\mathbf {\\\\Omega }}$ , when $D_{\\\\widetilde{\\\\varphi }}(\\\\widetilde{\\\\mathbf {\\\\Omega }},\\\\widetilde{{P}})$ is large.\",\n \"This can be argued as follows.\",\n \"We will consider the case when $\\\\widetilde{\\\\mathbf {\\\\Omega }}$ is a compact subset in $\\\\mathbb {R}_{> 0}^{K}$ and $\\\\widetilde{\\\\varphi }$ satisfies (H$\\\\widetilde{\\\\varphi }$ ) with $b=+\\\\infty $ .\",\n \"For the case when $\\\\widetilde{\\\\mathbf {\\\\Omega }}$ is not compact, or belongs to $\\\\mathbb {R}^{K}/\\\\left\\\\lbrace \\\\mathbf {0}\\\\right\\\\rbrace $ , see the Remark REF hereunder.\",\n \"Consider a compact set $\\\\mathbf {\\\\Gamma }$ in $\\\\widetilde{\\\\mathbf {\\\\Omega }}$ and let $\\\\mathbf {\\\\Gamma }_{t}$ be defined as deduced from $\\\\mathbf {\\\\Gamma } $ in a way that makes $D_{\\\\widetilde{\\\\varphi }}\\\\left( \\\\mathbf {\\\\Gamma }_{t},\\\\widetilde{{P}}\\\\right) $ increase with $t$ for sufficiently large $t$ .\",\n \"For instance, set $\\\\mathbf {\\\\Gamma }_{t}:=t \\\\cdot \\\\mathbf {\\\\Gamma } .$ Hence, in case of $b=+\\\\infty $ the divergence $D_{\\\\widetilde{\\\\varphi }}\\\\left( \\\\mathbf {\\\\Gamma }_{t},\\\\widetilde{{P}}\\\\right) =\\\\inf _{g_{t}\\\\in \\\\mathbf {\\\\Gamma }_{t}}\\\\sum _{k=1}^{K} \\\\widetilde{p}_{k}\\\\cdot \\\\widetilde{\\\\varphi }\\\\left( \\\\frac{\\\\left(g_{t}\\\\right)_{k}}{\\\\widetilde{p}_{k}}\\\\right) =\\\\inf _{g \\\\in \\\\mathbf {\\\\Gamma } }\\\\sum _{k=1}^{K} \\\\widetilde{p}_{k}\\\\cdot \\\\widetilde{\\\\varphi }\\\\left( \\\\frac{t \\\\cdot g_{k}}{\\\\widetilde{p}_{k}}\\\\right)$ tends to infinity as $t \\\\rightarrow \\\\infty $ ; the case $a=-\\\\infty $ works analogously with $t\\\\rightarrow -\\\\infty $ .\",\n \"In case of $b < \\\\infty $ we may consider $\\\\mathbf {\\\\Gamma } _{t}:=\\\\left\\\\lbrace b-g /t;g \\\\in \\\\mathbf {\\\\Gamma } \\\\right\\\\rbrace $ and indeed $D_{\\\\widetilde{\\\\varphi }}\\\\left( \\\\mathbf {\\\\Gamma } _{t},\\\\widetilde{{P}}\\\\right) \\\\rightarrow \\\\infty $ as $t\\\\rightarrow \\\\infty $ , with a similar statement when $a>-\\\\infty $ .\",\n \"Assume that $\\\\mathbf {\\\\Gamma }$ has a dominating point $\\\\underline{g}$ .\",\n \"Then $\\\\mathbf {\\\\Gamma }_{t}$ has dominating point $\\\\underline{g}_{t} := t \\\\cdot \\\\underline{g}$ .\",\n \"We prove that $\\\\mathbf {T}$ with distribution (REF ) cannot be too far away (depending on $t$ ) from $\\\\underline{g}_{t}$ whenever $\\\\mathbf {T}$ belongs to $\\\\mathbf {\\\\Gamma } _{t}.$ This argument is valid in the present description of some asymptotics which makes $\\\\mathbf {\\\\Gamma } _{t}$ as a model for $\\\\widetilde{\\\\mathbf {\\\\Omega }}$ for large $t$ ; considering the case when $D_{\\\\widetilde{\\\\varphi } }\\\\left( \\\\widetilde{\\\\mathbf {\\\\Omega }} ,\\\\widetilde{{P}}\\\\right) $ is large is captured through the asymptotic statement $\\\\lim _{t\\\\rightarrow \\\\infty }D_{\\\\widetilde{\\\\varphi } }\\\\left( \\\\mathbf {\\\\Gamma } _{t},\\\\widetilde{{P}}\\\\right) =+\\\\infty .$ There holds the following Proposition 65 With the above notation and under condition (H$\\\\widetilde{\\\\varphi }$ ), denote by $\\\\mathbf {B}$ a neighborhood of $\\\\underline{g}$ and $\\\\mathbf {B}_{t}:=t \\\\cdot \\\\mathbf {B}$ .\",\n \"Then $\\\\mathbb {F}[\\\\mathbf {\\\\Gamma }_{t}\\\\cap \\\\mathbf {B}_{t}^{c} \\\\, \\\\vert \\\\, \\\\mathbf {\\\\Gamma }_{t} \\\\, ]= \\\\mathbb {\\\\Pi }\\\\left[ \\\\left.\",\n \"\\\\mathbf {T}\\\\in \\\\mathbf {\\\\Gamma }_{t}\\\\cap \\\\mathbf {B}_{t}^{c}\\\\, \\\\right|\\\\,\\\\mathbf {T}\\\\in \\\\mathbf {\\\\Gamma }_{t}\\\\right] \\\\rightarrow 0$ as $t\\\\rightarrow \\\\infty $ , which proves that simulations under (REF ) produce proxies of the dominating points $\\\\underline{g}_{t}$ in $\\\\mathbf {\\\\Gamma }_{t}$ .\",\n \"Before we start with the proof of Proposition REF , we first quote the following Lemma 66 Let $\\\\widetilde{\\\\varphi } $ satisfy (H$\\\\widetilde{\\\\varphi }$ ) with $b=+\\\\infty $ .\",\n \"Then for all $\\\\mathbf {A}$ in $\\\\mathbb {R}^{K}$ such that $\\\\breve{\\\\alpha } :=D_{\\\\widetilde{\\\\varphi }}(\\\\mathbf {A},\\\\widetilde{{P}}):=\\\\inf _{\\\\mathbf {v}\\\\in \\\\mathbf {A}}\\\\sum _{k=1}^{K} \\\\widetilde{p}_{k} \\\\cdot \\\\widetilde{\\\\varphi } \\\\left(\\\\frac{v_{k}}{\\\\widetilde{p}_{k}}\\\\right)$ is finite there holds $\\\\lim _{t\\\\rightarrow \\\\infty }\\\\frac{1}{t}\\\\log \\\\int _{\\\\mathbf {A}}\\\\exp \\\\left(-t\\\\sum _{k=1}^{K} \\\\widetilde{p}_{k} \\\\cdot \\\\widetilde{\\\\varphi }\\\\left( \\\\frac{v_{k}}{\\\\widetilde{p}_{k}}\\\\right) \\\\right) \\\\, dv_{1} \\\\ldots dv_{k}= - D_{\\\\widetilde{\\\\varphi }}\\\\left(\\\\mathbf {A},\\\\widetilde{{P}}\\\\right) .$ Proof of Lemma REF .\",\n \"Let us first remark that according to the geometry of the set $\\\\mathbf {A}$ , various combinations for the asymptotics (REF ) or (REF ) may occur; for sake of brevity, we only handle the simplest ones, since all turn to be amenable through the same arguments.\",\n \"Denote for positive $r$ $\\\\mathbf {B}(r):=\\\\left\\\\lbrace \\\\mathbf {v}\\\\in \\\\mathbb {R}^{K}:\\\\sum _{k=1}^{K} \\\\widetilde{p}_{k}\\\\widetilde{\\\\varphi } \\\\left( \\\\frac{v_{k}}{\\\\widetilde{p}_{k}}\\\\right) >r\\\\right\\\\rbrace .$ It holds, by making the change of variable $r=t\\\\cdot \\\\breve{\\\\alpha } +t \\\\cdot s$ , $& & \\\\int _{\\\\mathbf {A}}\\\\exp \\\\left( -t\\\\sum _{k=1}^{K} \\\\widetilde{p}_{k} \\\\cdot \\\\widetilde{\\\\varphi } \\\\left( \\\\frac{v_{k}}{\\\\widetilde{p}_{k}}\\\\right) \\\\right) \\\\,dv_{1} \\\\ldots dv_{k}=\\\\int \\\\cdots \\\\int 1_{\\\\mathbb {R}^{+}}(r)\\\\cdot 1_{\\\\mathbf {A}}(\\\\mathbf {v})\\\\cdot 1_{\\\\left]t\\\\sum _{k=1}^{K} \\\\widetilde{p}_{k}\\\\widetilde{\\\\varphi } \\\\left( \\\\frac{v_{k}}{\\\\widetilde{p}_{k}}\\\\right) ,\\\\infty \\\\right[}(r)\\\\cdot e^{-r} \\\\, dr \\\\, dv_{1}\\\\ldots dv_{K} \\\\\\\\& & = te^{-t\\\\breve{\\\\alpha } }\\\\int \\\\cdots \\\\int 1_{]- \\\\breve{\\\\alpha },\\\\infty [}(s) \\\\cdot 1_{\\\\mathbf {A}}(\\\\mathbf {v}) \\\\cdot 1_{\\\\mathbf {B}^{c}(\\\\breve{\\\\alpha }+s)}(\\\\mathbf {v})\\\\cdot e^{-ts} \\\\, ds \\\\, dv_{1}\\\\ldots dv_{K} \\\\ = \\\\ te^{-t\\\\breve{\\\\alpha } }\\\\int _{-\\\\breve{\\\\alpha }}^{\\\\infty }Vol\\\\left( \\\\mathbf {A}\\\\cap \\\\mathbf {B}^{c}(\\\\breve{\\\\alpha }+s)\\\\right) \\\\cdot e^{-ts} \\\\, ds.$ Let $I_{t}:=t \\\\cdot \\\\int _{0}^{\\\\infty }Vol\\\\left(\\\\mathbf {A}\\\\cap \\\\mathbf {B}^{c}(\\\\breve{\\\\alpha } +s)\\\\right) e^{-ts}ds $ .\",\n \"We prove that $\\\\lim _{t\\\\rightarrow \\\\infty }\\\\frac{1}{t}\\\\log I_{t}=0.\",\n \"$ When $a=-\\\\infty $ or $b=+\\\\infty $ , since $\\\\widetilde{\\\\varphi }$ satisfies (H$\\\\widetilde{\\\\varphi }$ ) there exists a polynomial $P$ such that $Vol\\\\left( \\\\mathbf {A}\\\\cap \\\\mathbf {B}^{c}(\\\\breve{\\\\alpha } +s)\\\\right) \\\\le P(s) \\\\, ;$ whence, assuming without loss of generality that $dom(\\\\widetilde{\\\\varphi }) =\\\\mathbb {R}^{+}$ , we obtain $\\\\frac{1}{t}\\\\log I_{t}\\\\le \\\\frac{1}{t}\\\\log \\\\int _{0}^{\\\\infty }P\\\\left(\\\\frac{u}{t}\\\\right) te^{-u}du$ which yields that for large $t$ $\\\\frac{1}{t}\\\\log I_{t}<0.$ When dealing with a context where $a$ or $b$ have finite value and the corresponding sets $\\\\mathbf {\\\\Gamma }_{t}$ are “far away” from $\\\\mathbf {\\\\Gamma } $ in terms of the distance measure $D_{\\\\widetilde{\\\\varphi }}\\\\left( \\\\cdot ,\\\\widetilde{{P}}\\\\right)$ , then $Vol\\\\left( \\\\mathbf {A}\\\\cap \\\\mathbf {B}^{c}(\\\\breve{\\\\alpha } +s)\\\\right) $ is bounded.\",\n \"Hence, $\\\\lim \\\\sup _{t\\\\rightarrow \\\\infty }\\\\frac{1}{t}\\\\log I_{t}\\\\le 0$ .\",\n \"Now fix $\\\\varepsilon >0.$ Then, since $Vol\\\\left( \\\\mathbf {A}\\\\cap \\\\mathbf {B}^{c}(a+s)\\\\right) $ is increasing in $s$ , we get $I_{t} &\\\\ge &t\\\\int _{\\\\varepsilon }^{\\\\infty }Vol\\\\left( \\\\mathbf {A}\\\\cap \\\\mathbf {B}^{c}(\\\\breve{\\\\alpha }+s)\\\\right) e^{-ts}ds \\\\\\\\&\\\\ge &Vol\\\\left( \\\\mathbf {A}\\\\cap \\\\mathbf {B}^{c}(\\\\breve{\\\\alpha } +\\\\varepsilon )\\\\right) e^{-t\\\\varepsilon }.$ Hence $\\\\frac{1}{t}\\\\log I_{t}\\\\ge \\\\frac{1}{t}\\\\log Vol\\\\left( \\\\mathbf {A}\\\\cap \\\\mathbf {B}^{c}(\\\\breve{\\\\alpha }+\\\\varepsilon )\\\\right) -\\\\varepsilon $ which yields $\\\\lim \\\\inf _{t\\\\rightarrow \\\\infty }\\\\frac{1}{t}\\\\log I_{t}\\\\ge 0.$ Therefore (REF ) holds, which concludes the proof.\",\n \"$\\\\blacksquare $ We now turn to the Proof of Proposition REF .\",\n \"Without loss of generality, let $b=+\\\\infty $ , $\\\\mathbf {\\\\Gamma }_{t}$ as in (REF ) and Condition (H$\\\\widetilde{\\\\varphi }$ ) hold.\",\n \"Moreover, consider an arbitrary neighborhood $\\\\mathbf {B}$ of $\\\\underline{g}$ and the corresponding neighborhoods $\\\\mathbf {B}_{t} : =t \\\\cdot \\\\mathbf {B}$ of $\\\\underline{g}_{t} = t \\\\cdot \\\\underline{g}$ .\",\n \"There holds $\\\\frac{1}{\\\\widetilde{\\\\varphi }(t)}\\\\log \\\\mathbb {\\\\Pi }\\\\left[ \\\\mathbf {T}\\\\in \\\\mathbf {\\\\Gamma }_{t}\\\\right] &=&\\\\frac{C}{\\\\widetilde{\\\\varphi }(t)}\\\\log \\\\int _{\\\\mathbf {\\\\Gamma } _{t}}\\\\exp \\\\left(-\\\\sum _{k=1}^{K}\\\\widetilde{p}_{k} \\\\cdot \\\\widetilde{\\\\varphi }\\\\left(\\\\frac{w_{k}}{\\\\widetilde{p}_{k}}\\\\right)\\\\right) \\\\, dw_{1}\\\\ldots dw_{K} \\\\\\\\&\\\\stackrel{(1)}{=}&\\\\frac{CK}{\\\\widetilde{\\\\varphi }(t)}\\\\log t+\\\\frac{C}{\\\\widetilde{\\\\varphi }(t)}\\\\log \\\\int _{\\\\mathbf {\\\\Gamma }}\\\\exp \\\\left(-t^{\\\\beta } \\\\cdot \\\\sum _{k=1}^{K}\\\\widetilde{p}_{k} \\\\cdot \\\\left( \\\\widetilde{\\\\varphi }\\\\left( \\\\frac{v_{k}}{\\\\widetilde{p}_{k}}\\\\right) \\\\cdot (1+o(1))\\\\right)\\\\right) \\\\, dv_{1} \\\\ldots dv_{K} \\\\\\\\&\\\\stackrel{(2)}{=}& \\\\frac{CK}{\\\\widetilde{\\\\varphi }(t)}\\\\log t+\\\\frac{C}{\\\\left( \\\\widetilde{\\\\varphi }(t)/t^{\\\\beta }\\\\right) } \\\\cdot \\\\frac{1}{t^{\\\\beta }}\\\\log \\\\left( (1+o(1)) \\\\cdot \\\\int _{\\\\mathbf {\\\\Gamma } }\\\\exp \\\\left( -t^{\\\\beta }\\\\sum _{k=1}^{K}\\\\widetilde{p}_{k} \\\\cdot \\\\widetilde{\\\\varphi }\\\\left( \\\\frac{v_{k}}{\\\\widetilde{p}_{k}}\\\\right) \\\\right)dv_{1} \\\\ldots dv_{K} \\\\right)\\\\\\\\&\\\\stackrel{(3)}{=}& -\\\\frac{Ct^{\\\\beta }}{\\\\widetilde{\\\\varphi }(t)}\\\\cdot D_{\\\\widetilde{\\\\varphi }}(\\\\mathbf {\\\\Gamma } ,\\\\widetilde{{P}}) \\\\cdot (1+o(1)) \\\\\\\\&\\\\stackrel{(4)}{=}& - \\\\breve{l}(t) \\\\cdot D_{\\\\widetilde{\\\\varphi }}(\\\\mathbf {\\\\Gamma } ,\\\\widetilde{{P}}) \\\\cdot (1+o(1))$ as $t$ tends to infinity.\",\n \"In the above display, $(1)$ follows from $\\\\widetilde{\\\\varphi }(tx)=\\\\left( tx\\\\right)^{\\\\beta } \\\\cdot \\\\ell (tx)=t^{\\\\beta } \\\\cdot x^{\\\\beta } \\\\cdot \\\\ell (x) \\\\cdot \\\\frac{\\\\ell (tx)}{\\\\ell (x)} =t^{\\\\beta } \\\\cdot \\\\widetilde{\\\\varphi } (x) \\\\cdot \\\\left( 1+o(1)\\\\right) $ as $t$ tends to infinity and $x$ lies in a compact subset of $]0,\\\\infty [$ , where $\\\\ell $ is a slowly varying function.\",\n \"The equality $(2)$ follows from compactness of $\\\\mathbf {\\\\Gamma }$ together with the fact that $\\\\widetilde{\\\\varphi }$ is a regularly varying function with index $\\\\beta $ , so that $\\\\lim _{t\\\\rightarrow \\\\infty }\\\\frac{\\\\widetilde{\\\\varphi }(tv)}{\\\\widetilde{\\\\varphi }(t)}=v^{\\\\beta }$ uniformly upon $v$ on compact sets in $]0,\\\\infty [$ .\",\n \"The remaining equalities $(3)$ and $(4)$ follow from classical properties of regularly varying functions, where $\\\\breve{\\\\ell } := 1/\\\\ell $ is a slowly varying function at infinity, together with standard Laplace-Integral approximation.\",\n \"In the same way we can show $\\\\frac{1}{\\\\widetilde{\\\\varphi }(t)}\\\\log \\\\mathbb {\\\\Pi } \\\\left[ \\\\, \\\\mathbf {T}\\\\in \\\\mathbf {\\\\Gamma } _{t}\\\\cap \\\\mathbf {B}_{t}^{c} \\\\, \\\\right]=- \\\\breve{l}(t) \\\\cdot D_{\\\\widetilde{\\\\varphi }}(\\\\mathbf {\\\\Gamma }\\\\cap \\\\mathbf {B}^{c},\\\\widetilde{{P}}) \\\\cdot (1+o(1))$ as $t$ tends to infinity.\",\n \"Since $\\\\mathbf {B}$ is a neighborhood of the unique dominating point $\\\\underline{g}$ of $\\\\mathbf {\\\\Gamma }$ , one gets that $D_{\\\\widetilde{\\\\varphi }}(\\\\mathbf {\\\\Gamma }\\\\cap \\\\mathbf {B}^{c},\\\\widetilde{{P}}) > D_{\\\\widetilde{\\\\varphi }}(\\\\mathbf {\\\\Gamma } ,\\\\widetilde{{P}})$ .\",\n \"This implies that $\\\\mathbb {\\\\Pi } \\\\left[ \\\\, \\\\mathbf {T}\\\\in \\\\mathbf {\\\\Gamma }_{t}\\\\cap \\\\mathbf {B}_{t}^{c} \\\\, \\\\big \\\\vert \\\\, \\\\mathbf {T}\\\\in \\\\mathbf {\\\\Gamma }_{t} \\\\, \\\\right] \\\\rightarrow 0\\\\qquad \\\\textrm {as $ $.\",\n \"\\\\qquad $$}$$$ Remark 67 Firstly, let us quote that the case when $\\\\widetilde{\\\\mathbf {\\\\Omega }}$ is an unbounded subset in $\\\\mathbb {R}^{K}/\\\\left\\\\lbrace \\\\mathbf {0}\\\\right\\\\rbrace $ is somewhat immaterial for applications.\",\n \"Anyhow, if compactness of $\\\\mathbf {\\\\Gamma } $ is lost, then in order to use the same line of arguments as above, it is necessary to strengthen the assumptions (H$\\\\widetilde{\\\\varphi }$ ) e.g.\",\n \"as follows: when $b=+\\\\infty $ then $\\\\widetilde{\\\\varphi }$ has to be asymptotically homogeneous with degree $\\\\beta >0$ , in the sense that $\\\\widetilde{\\\\varphi }(tx)=t^{\\\\beta }\\\\widetilde{\\\\varphi }(x)\\\\cdot (1+o(1))$ as $t\\\\rightarrow \\\\infty $ ; for the subcase $a=-\\\\infty $ one employs an analogous assumption as $t\\\\rightarrow -\\\\infty $ .\",\n \"The case when $\\\\widetilde{\\\\mathbf {\\\\Omega }}$ is a compact set in $\\\\mathbb {R}^{K}\\\\backslash \\\\lbrace \\\\mathbf {0}\\\\rbrace $ can be treated as above, by combining the asymptotics in $t$ in the neighborhood of $a$ and $b$ accordingly.\"\n ],\n [\n \"Proof for Subsection \",\n \"Proof of Proposition REF.\",\n \"Recall the weighted empirical measure $\\\\xi _{n,\\\\mathbf {X}}^{\\\\mathbf {V}}:=\\\\left( \\\\frac{1}{n}\\\\sum _{i\\\\in I_{1}^{(n)}} V_{i},\\\\ldots ,\\\\frac{1}{n}\\\\sum _{i\\\\in I_{K}^{(n)}} V_{i}\\\\right)$ which satisfies the $K$ linear constraints defined in (REF ) through $E_{S}[\\\\xi _{n,\\\\mathbf {X}}^{\\\\mathbf {V}}] =\\\\xi _{M,\\\\mathbf {X}}^{\\\\mathbf {W}^{\\\\ast }}= \\\\overline{W^{\\\\ast }} \\\\cdot \\\\xi _{M,\\\\mathbf {X}}^{w\\\\mathbf {W}^{\\\\ast }}$ where $\\\\mathbf {Q}^{\\\\ast }:= \\\\left(q_{1}^{\\\\ast }, \\\\ldots , q_{K}^{\\\\ast }\\\\right)= \\\\xi _{M,\\\\mathbf {X}}^{w\\\\mathbf {W}^{\\\\ast }}\\\\in int(\\\\textrm {$$\\\\hspace{-6.544pt}$$})$ and $\\\\overline{W^{\\\\ast }} = \\\\frac{1}{M}\\\\sum _{j=1}^{M}W_{j}^{\\\\ast }$ .\",\n \"The probability distribution $S$ defined on $\\\\mathbb {R}^{n}$ is the Kullback-Leibler projection of $\\\\mathbb {}^{\\\\otimes n}$ on the class of all probability distributions on $\\\\mathbb {R}^{n}$ which satisfy (REF ).\",\n \"We prove that $\\\\lim \\\\inf _{n\\\\rightarrow \\\\infty }S\\\\left[\\\\xi _{n,\\\\mathbf {X}}^{w\\\\mathbf {V}} \\\\in \\\\textrm {\\\\right.$$\\\\hspace{-6.544pt}$$}> 0$ .\",\n \"To start with, we define for strictly positive $\\\\delta $ the set $A_{n,\\\\delta } := \\\\left\\\\lbrace \\\\left|\\\\frac{1}{n}\\\\sum _{i=1}^{n}V_{i}-\\\\overline{W^{\\\\ast }}\\\\right|\\\\le \\\\delta \\\\right\\\\rbrace $ and write $S\\\\left[\\\\xi _{n,\\\\mathbf {X}}^{w\\\\mathbf {V}} \\\\in \\\\textrm {\\\\right.$ $\\\\hspace{-6.544pt}$$}=S\\\\left[\\\\lbrace \\\\xi _{n,\\\\mathbf {X}}^{w\\\\mathbf {V}} \\\\in \\\\textrm {\\\\right.$$\\\\hspace{-6.544pt}$$} \\\\rbrace \\\\cap A_{n,\\\\delta }+ S\\\\left[\\\\lbrace \\\\xi _{n,\\\\mathbf {X}}^{w\\\\mathbf {V}} \\\\in \\\\textrm {\\\\right.$$\\\\hspace{-6.544pt}$$} \\\\rbrace \\\\cap A_{n,\\\\delta }^{c}=:I+II.$$By the law of large numbers, the second term $ II$ tends to $ 0$ as $ n$ tends to infinity.Moreover, one can rewrite$$I=S\\\\bigg [ \\\\bigcup \\\\limits _{m\\\\in \\\\left[ \\\\overline{W^{\\\\ast }}-\\\\delta ,\\\\overline{W^{\\\\ast }}+\\\\delta \\\\right] }\\\\left\\\\lbrace \\\\xi _{n,\\\\mathbf {X}}^{\\\\mathbf {V}}\\\\in m \\\\cdot \\\\textrm {\\\\right.$$\\\\hspace{-6.544pt}$$} \\\\bigg ]$$which entails$$I \\\\ \\\\ge \\\\ S\\\\bigg [ \\\\frac{1}{n_{k}}\\\\sum _{i\\\\in I_{k}^{(n)}}V_{i}\\\\in \\\\mathcal {V}_{\\\\eta }\\\\left( \\\\overline{W^{\\\\ast }}\\\\frac{q_{k}^{\\\\ast }}{p_{k}}\\\\right) \\\\text{ for all }k \\\\in \\\\lbrace 1,\\\\ldots ,K\\\\rbrace \\\\bigg ] ,$$where $ V( W qkpk) $ denotes a neighborhood of$Wqkpk$ with radius $$ being smallwhen $$ is small, for large enough $ n$, making use of the a.s. convergenceof $ nk/n$ to $ pk$.\",\n \"Now, for any $ k {1,...,K}$ one has\\\\begin{equation}S\\\\bigg [ \\\\frac{1}{n_{k}}\\\\sum _{i\\\\in I_{k}^{(n)}}V_{i}\\\\notin \\\\mathcal {V}_{\\\\eta }\\\\left( \\\\overline{W^{\\\\ast }}\\\\frac{q_{k}^{\\\\ast }}{p_{k}}\\\\right)\\\\bigg ]\\\\ \\\\le \\\\ \\\\exp \\\\bigg (-n_{k} \\\\cdot \\\\inf _{x\\\\in \\\\mathcal {V}_{\\\\eta }\\\\left(\\\\overline{W^{\\\\ast }}\\\\frac{q_{k}^{\\\\ast }}{p_{k}}\\\\right)^{c}} \\\\,\\\\varphi \\\\left( x\\\\right) \\\\bigg )\\\\end{equation}since any margin of $ S$ with index in $ Ik(n)$ is a correspondingKullback-Leibler projection of $$ on the set of all distributionson $ R$ with expectation $W qkpM,kemp$ ---where $ pM,kemp$ denotes the fraction ofthe $ Xi$’s (within $ X1,...,XM$) which are equal to $ dk$(cf.\",\n \"(\\\\ref {I^(n)_k for stat case})) ---and therefore has amoment generating function which is finite in a non-void neighborhood of $ 0$,which yields (\\\\ref {ineg}) by the Markov Inequality.\",\n \"Note that the event$ { M,XwWint( $\\\\Omega $$\\\\Omega $ ) }$ is regenerative, so that $ M$ can be chosen large enough to make$ pM,kemp$ close to $ pk$ for all $ k {1,...,K}$.\",\n \"This proves the claim.\\\\hspace{28.45274pt} $$$\"\n ],\n [\n \"Acknowledgment\",\n \"W. Stummer is grateful to the Sorbonne Université Paris for its multiple partial financial support and especially the LPSM for its multiple great hospitality.\",\n \"M. Broniatowski thanks very much the FAU Erlangen-Nürnberg for its partial financial support and hospitality.\",\n \"Moreover, W. Stummer would like to thank Rene Schilling for an interesting discussion on complex-valued foundations of the Bernstein-Widder theorem.\"\n ],\n [\n \"Finding/Constructing/On the distribution of the weights \",\n \"Recall first that in Theorem REF , one crucial component is the sequence $(W_{n})_{n\\\\in \\\\mathbb {N}}$ of weights being i.i.d.\",\n \"copies of a random variable $W$ whose probability distribution is $\\\\mathbb {}$ (i.e.\",\n \"$\\\\mathbb {\\\\Pi }[W \\\\in \\\\cdot \\\\, ] = \\\\mathbb {}[ \\\\, \\\\cdot \\\\,]$ ), where the latter has to be connected with the divergence generator $\\\\varphi \\\\in \\\\Upsilon (]a,b[)$ through the representation $\\\\varphi (t)=\\\\sup _{z\\\\in \\\\mathbb {R}}\\\\left( z\\\\cdot t-\\\\log \\\\int _{\\\\mathbb {R}}e^{zy}d\\\\mathbb {} (y)\\\\right), \\\\qquad t\\\\in \\\\mathbb {R},\\\\ \\\\ \\\\hspace{28.45274pt} (cf.\",\n \"\\\\ (\\\\ref {Phi Legendre of mgf(W)}))$ under the additional requirement that the function $z\\\\mapsto MGF_{\\\\mathbb {} }(z):=\\\\int _{\\\\mathbb {R}}e^{zy}d\\\\mathbb {} (y)$ is finite on some open interval containing zero (“light-tailedness”); for Theorem REF , we need the corresponding variant (REF ) for $M_{\\\\mathbf {P}} \\\\cdot \\\\varphi \\\\in \\\\Upsilon (]a,b[)$ (rather than $\\\\varphi $ ).\",\n \"Hence, finding such “BS-associated pairs $(\\\\varphi ,\\\\mathbb {})$ ” is an important issue.\",\n \"Subsequently, let us discuss the following direction: starting from a concrete optimization problem (REF ) — respectively (REF ) — with pregiven $\\\\varphi \\\\in \\\\widetilde{\\\\Upsilon } (]a,b[)$ (cf.\",\n \"Definition REF ), as a first step one would like to verify whether indeed $\\\\varphi \\\\in \\\\Upsilon (]a,b[)$ (i.e.\",\n \"it additionally satisfies (REF )) — respectively $M_{\\\\mathbf {P}} \\\\cdot \\\\varphi \\\\in \\\\Upsilon (]a,b[)$ ; as a second step, one would like to find the corresponding $\\\\mathbb {}$ explicitly.\",\n \"As far as the above-mentioned first step is concerned, let us first present some fundamental properties of all $\\\\varphi \\\\in \\\\Upsilon (]a,b[)$ : Proposition 29 Let $\\\\varphi \\\\in \\\\Upsilon (]a,b[)$ .\",\n \"Then the following assertions hold: $\\\\varphi : \\\\, ]-\\\\infty ,\\\\infty [ \\\\rightarrow [0,\\\\infty ]$ is lower semicontinuous and convex; $\\\\varphi (1)=0$ ; $int(dom(\\\\varphi )) = ]a,b[$ for some $-\\\\infty \\\\le a < 1 < b \\\\le \\\\infty $ ; $\\\\varphi $ is continuously differentiable on $]a,b[$ (i.e.\",\n \"$\\\\varphi \\\\in C^{1}(]a,b[)$ ; $\\\\varphi $ is strictly convex only in a non-empty neighborhood $]t_{-}^{sc},t_{+}^{sc}[ \\\\subseteq ]a,b[$ of one ($t_{-}^{sc} < 1 < t_{+}^{sc}$ ); $\\\\varphi $ is infinitly differentiable on $]t_{-}^{sc},t_{+}^{sc}[$ (i.e.\",\n \"$\\\\varphi \\\\in C^{\\\\infty }(]t_{-}^{sc},t_{+}^{sc}[$ ), and hence, $\\\\varphi ^{\\\\prime }(1) = 0$ , $\\\\varphi ^{\\\\prime \\\\prime }(t) > 0$ for all $t \\\\in ]t_{-}^{sc},t_{+}^{sc}[$ ; notice that the left-hand second derivative and the right-hand second derivative of $\\\\varphi $ may not coincide at $t_{-}^{sc}$ respectively at $t_{+}^{sc}$ (i.e.\",\n \"possible non-second-differentiability at these two points); if $a > -\\\\infty $ , then $a=t_{-}^{sc}$ ; if $a = -\\\\infty $ , then either $t_{-}^{sc} = - \\\\infty $ or $\\\\varphi (t) = \\\\varphi (t_{-}^{sc}) + \\\\varphi ^{\\\\prime }(t_{-}^{sc}) \\\\cdot (t- t_{-}^{sc})$ for all $t \\\\in ]-\\\\infty , t_{-}^{sc}[$ (affine-linearity); notice that $\\\\varphi ^{\\\\prime }(t_{-}^{sc}) < 0$ ; if $b < \\\\infty $ , then $b=t_{+}^{sc}$ ; if $b = \\\\infty $ , then either $t_{+}^{sc} = \\\\infty $ or $\\\\varphi (t) = \\\\varphi (t_{+}^{sc}) + \\\\varphi ^{\\\\prime }(t_{+}^{sc}) \\\\cdot (t- t_{+}^{sc})$ for all $t \\\\in ]t_{+}^{sc},\\\\infty [$ (affine-linearity); notice that $\\\\varphi ^{\\\\prime }(t_{+}^{sc}) > 0$ ; the Fenchel-Legendre transform (also called convex conjugate) of $\\\\varphi $ $\\\\varphi ^{*} (z)=\\\\sup _{t\\\\in \\\\mathbb {R}}\\\\left( z\\\\cdot t- \\\\varphi (t) \\\\right)= \\\\sup _{t\\\\in ]a,b[}\\\\left( z\\\\cdot t- \\\\varphi (t) \\\\right), \\\\qquad z\\\\in \\\\mathbb {R},\\\\ \\\\ $ has the following properties: $int(dom(\\\\varphi ^{*})) = ]\\\\lambda _{-},\\\\lambda _{+}[$ , where $dom(\\\\varphi ^{*}) := \\\\lbrace z \\\\in \\\\mathbb {R} : -\\\\infty < \\\\varphi ^{*}(z) < \\\\infty \\\\rbrace $ , $\\\\lambda _{-} := \\\\inf _{t \\\\in ]a,b[} \\\\varphi ^{\\\\prime }(t) =\\\\lim _{t \\\\downarrow a} \\\\varphi ^{\\\\prime }(t) =: \\\\varphi ^{\\\\prime }(a) < 0$ and $\\\\lambda _{+} := \\\\sup _{t \\\\in ]a,b[} \\\\varphi ^{\\\\prime }(t) =\\\\lim _{t \\\\uparrow b} \\\\varphi ^{\\\\prime }(t) =: \\\\varphi ^{\\\\prime }(b) >0$ ; if $a >-\\\\infty $ , then $\\\\lambda _{-} = - \\\\infty $ ; the function $z \\\\mapsto e^{-a\\\\cdot z + \\\\varphi ^{*}(z)} =: M(z)$ is absolutely monotone on $]-\\\\infty ,0[$ , i.e.\",\n \"all derivatives exist and satisfy $\\\\frac{\\\\partial ^{k}}{\\\\partial z^k} M(z) \\\\ge 0$ ($k\\\\in \\\\mathbb {N}_{0}$ , $z \\\\in ]-\\\\infty ,0[$ ); $\\\\lim _{z \\\\rightarrow 0-} M(z) =1$ ; if $b < \\\\infty $ , then $\\\\lambda _{+} = \\\\infty $ ; the function $z \\\\mapsto e^{b\\\\cdot z + \\\\varphi ^{*}(- z)} =: M(z)$ is absolutely monotone on $]-\\\\infty ,0[$ ; $\\\\lim _{z \\\\rightarrow 0-} M(z) =1$ ; if $a = -\\\\infty $ and $b = -\\\\infty $ , then the function $z \\\\mapsto e^{\\\\varphi ^{*}(z)} =: M(z)$ is exponentially convex on $]\\\\lambda _{-}, \\\\lambda _{+}[$ , i.e.\",\n \"$M(\\\\cdot )$ is continuous and satisfies $\\\\sum _{i,j=1}^{n} c_{i} \\\\cdot c_{j} \\\\cdot M\\\\Big (\\\\frac{z_{i} + z_{j}}{2}\\\\Big ) \\\\ge 0 \\\\qquad \\\\textrm {for all n \\\\in \\\\mathbb {N},c_{i}, c_{j} \\\\in \\\\mathbb {R} and z_{i}, z_{j} \\\\in ]\\\\lambda _{-}, \\\\lambda _{+}[;}$ $\\\\lim _{z \\\\rightarrow 0-} M(z) =1$ ; as a side remark, notice the well-known fact that exponential-convexity is stronger than the usual log-convexity.\",\n \"the endpoints of $int(dom(\\\\varphi )) =]a,b[$ have the following important “functioning” for the underlying probability distribution $\\\\mathbb {}$ (cf.\",\n \"(REF )) respectively of an associated random variable $W$ with $\\\\mathbb {}[ \\\\cdot \\\\, ] := \\\\mathbb {\\\\Pi }[W \\\\in \\\\cdot \\\\, ]$ : $a = \\\\inf supp(\\\\mathbb {}) = \\\\inf supp(W)$ , $b = \\\\sup supp(\\\\mathbb {}) = \\\\sup supp(W)$ , where $supp(\\\\mathbb {})$ respectively $supp(W)$ denotes the support of $\\\\mathbb {}$ respectively $W$ ; consequently, $]a,b[ = int(conv(supp(\\\\mathbb {}))) = int(conv(supp(W)))$ where $conv(A)$ denotes the convex hull of a set $A$ ; if $a > -\\\\infty $ , then $\\\\varphi (a) = - \\\\log \\\\mathbb {}[\\\\lbrace a\\\\rbrace ]= - \\\\log \\\\mathbb {\\\\Pi }[W = a \\\\, ]$ ; consequently, there holds: $a = \\\\min supp(\\\\mathbb {}) = \\\\min supp(W)$ if and only if $\\\\mathbb {}[\\\\lbrace a\\\\rbrace ] = \\\\mathbb {\\\\Pi }[W = a \\\\, ] > 0$ if and only if $\\\\varphi (a) < \\\\infty $ if and only if $a \\\\in dom(\\\\varphi )$ ; if $b < \\\\infty $ , then $\\\\varphi (b) = - \\\\log \\\\mathbb {}[\\\\lbrace b\\\\rbrace ]= - \\\\log \\\\mathbb {\\\\Pi }[W = b \\\\, ]$ ; consequently, there holds: $b = \\\\max supp(\\\\mathbb {}) = \\\\max supp(W)$ if and only if $\\\\mathbb {}[\\\\lbrace b\\\\rbrace ] = \\\\mathbb {\\\\Pi }[W = b \\\\, ] > 0$ if and only if $\\\\varphi (b) < \\\\infty $ if and only if $b \\\\in dom(\\\\varphi )$ .\",\n \"the first two derivatives of $\\\\varphi $ at the point 1 have the following important “functioning” for the underlying probability distribution $\\\\mathbb {}$ (cf.\",\n \"(REF )) respectively of an associated random variable $W$ : $1 = \\\\varphi ^{\\\\prime -1}(0) = \\\\int _{\\\\mathbb {R}} y \\\\, d\\\\mathbb {} (y)= E_{\\\\mathbb {\\\\Pi }}[W]$ where $\\\\varphi ^{\\\\prime -1}(\\\\cdot )$ denotes the inverse of the first derivative $\\\\varphi ^{\\\\prime }(\\\\cdot )$ of $\\\\varphi (\\\\cdot )$ , $\\\\frac{1}{\\\\varphi ^{\\\\prime \\\\prime }(1)} =\\\\int _{\\\\mathbb {R}} \\\\Big (y - \\\\int _{\\\\mathbb {R}} \\\\widetilde{y} \\\\, d\\\\mathbb {} (\\\\widetilde{y})\\\\Big )^{2}\\\\, d\\\\mathbb {} (y)= E_{\\\\mathbb {\\\\Pi }}[W^{2}] - (E_{\\\\mathbb {\\\\Pi }}[W])^{2} = Var_{\\\\mathbb {\\\\Pi }}[W]$ ; in particular, scaling $\\\\widetilde{c} \\\\cdot \\\\varphi $ ($\\\\widetilde{c} >0$ ) does not change the mean 1 but the variance of $W$ .\",\n \"The corresponding proof of Proposition REF will be given in Appendix D, except for the second items of (G9ii) and (G9iii) as well as the first item of (G9iv).\",\n \"Those will be treated in the second next paragraph below, because the corresponding line of argumentation builds an insightful start for subsequently performed procedures to further track down the weight distribution $$ .\",\n \"The properties (G1) to (G9iv) constitute necessary conditions for a pregiven function $\\\\varphi $ to belong to $\\\\Upsilon (]a,b[)$ ); accordingly, these should be verified first, in concrete situations where one aims to apply the BS approach.\",\n \"An important role is played by the boundary points $a$ and $b$ of $int(dom(\\\\varphi ))$ through (G10i) to (G10iii), because their finiteness opens the gate to apply — via some straightforward transformations — a rich class of real-valued characterization theorems for probability distributions whose support lies in the positive real line $[0,\\\\infty [$ .\",\n \"In contrast, there exist much less real-valued characterization theorems for probability distributions whose support is the whole real line $]-\\\\infty ,\\\\infty [$ ; typically, the involved conditions are also more difficult to verify.\",\n \"Indeed, if $\\\\varphi \\\\in \\\\Upsilon (]a,b[)$ then one can deduce straightforwardly from the representation (REF ) that $e^{\\\\varphi ^{*}(z)} =\\\\int _{-\\\\infty }^{\\\\infty } e^{z\\\\cdot y} \\\\, \\\\mathrm {d}\\\\mathbb {} (y) =E_{\\\\mathbb {\\\\Pi }}[e^{z \\\\cdot W}] , \\\\quad z \\\\in ]\\\\lambda _{-}, \\\\lambda _{+}[,$ where $W$ is a random variable whose distribution is $\\\\mathbb {\\\\Pi }[W \\\\in \\\\cdot \\\\, ] = \\\\mathbb {}[ \\\\, \\\\cdot \\\\,]$ ; under the additional knowledge $a > -\\\\infty $ (and consequently $\\\\lambda _{-} = -\\\\infty $ ) employed together with (G10i) and thus $\\\\mathbb {\\\\Pi }[W \\\\ge a \\\\, ] =\\\\mathbb {}[ \\\\, [a,\\\\infty [ \\\\,] = 1$ , one arrives at $e^{\\\\varphi ^{*}(z) - a \\\\cdot z} =\\\\int _{a}^{\\\\infty } e^{z \\\\cdot (y-a)} \\\\, \\\\mathrm {d}\\\\mathbb {} (y) =\\\\int _{0}^{\\\\infty } e^{z\\\\cdot \\\\widetilde{y}} \\\\, \\\\mathrm {d}\\\\widetilde{\\\\widetilde{\\\\mathbb {}}} (\\\\widetilde{y})= E_{\\\\mathbb {\\\\Pi }}[e^{z \\\\cdot (W-a)}] , \\\\quad z \\\\in ]-\\\\infty , \\\\lambda _{+}[,$ where the probability distribution $\\\\widetilde{\\\\widetilde{\\\\mathbb {}}}[ \\\\, \\\\cdot \\\\, ] := \\\\mathbb {}[\\\\, \\\\cdot + a \\\\, ]$ is the $a-$ shifted companion of $\\\\mathbb {}$ ; recall that $\\\\lambda _{+} >0$ .\",\n \"Put in other words, $\\\\mathbb {\\\\Pi }[\\\\widetilde{W} \\\\in \\\\cdot \\\\, ] = \\\\widetilde{\\\\widetilde{\\\\mathbb {}}}[ \\\\, \\\\cdot \\\\,]$ is the probability distribution of the (a.s.) nonnegative random variable $\\\\widetilde{W} := W-a$ .\",\n \"Naturally, in this context, the interesting case is $-\\\\infty < a \\\\le 0$ .\",\n \"Similarly, if $\\\\varphi \\\\in \\\\Upsilon (]a,b[)$ and $b < \\\\infty $ (and hence $\\\\lambda _{+} = \\\\infty $ ), one can derive from (G10i) and its consequence $\\\\mathbb {\\\\Pi }[W \\\\le b \\\\, ] =\\\\mathbb {}[ \\\\, ]-\\\\infty ,b] \\\\,]= 1$ that $e^{\\\\varphi ^{*}(-z) + b \\\\cdot z} =\\\\int _{-\\\\infty }^{b} e^{z \\\\cdot (b-y)} \\\\, \\\\mathrm {d}\\\\mathbb {} (y) =\\\\int _{0}^{\\\\infty } e^{z\\\\cdot \\\\widetilde{y}} \\\\, \\\\mathrm {d}\\\\widetilde{\\\\widetilde{\\\\mathbb {}}} (\\\\widetilde{y})= E_{\\\\mathbb {\\\\Pi }}[e^{z \\\\cdot (b-W)}] , \\\\quad z \\\\in ]- \\\\infty , - \\\\lambda _{-}[,$ where $- \\\\lambda _{-} >0$ and the probability distribution $\\\\widetilde{\\\\widetilde{\\\\mathbb {}}}[ \\\\, \\\\cdot \\\\, ] := \\\\mathbb {}[\\\\, b - \\\\, \\\\cdot \\\\, ]$ is the mirrored$-b-$ shifted companion of $\\\\mathbb {}$ .\",\n \"This means that $\\\\mathbb {\\\\Pi }[\\\\widetilde{W} \\\\in \\\\cdot \\\\, ] = \\\\widetilde{\\\\widetilde{\\\\mathbb {}}}[ \\\\, \\\\cdot \\\\,]$ is the probability distribution of the (a.s.) nonnegative random variable $\\\\widetilde{W} := b-W$ .\",\n \"Naturally, the interesting case is $0 < b \\\\le \\\\infty $ .\",\n \"As already indicated above, the considerations (REF ) to (REF ) open the gate to the adaption of well-known real-valued (rather than complex-valued) characterizations from probability theory.\",\n \"To begin with, the following assertions are very prominent: Theorem 30 (a) Let $M: ]-\\\\infty ,0] \\\\mapsto ]0,\\\\infty [$ be continuous on $]-\\\\infty ,0]$ with $M(0) =1$ .\",\n \"Then one has $\\\\textrm {M is absolutely monotone on ]-\\\\infty ,0[}\\\\ \\\\Longleftrightarrow \\\\ \\\\exists \\\\ \\\\textrm {unique prob.\",\n \"distr.\",\n \"\\\\widetilde{\\\\widetilde{\\\\mathbb {}}} on [0,\\\\infty [s.t.}\",\n \"\\\\ \\\\ M(z) = \\\\int _{0}^{\\\\infty } e^{z \\\\cdot y} \\\\mathrm {d}\\\\widetilde{\\\\widetilde{\\\\mathbb {}}}(y)\\\\ \\\\textrm {for all z \\\\in ]-\\\\infty ,0[.\",\n \"}$ (b) Let $I$ be an open interval which contains 0, and $M: I \\\\mapsto [0,\\\\infty [$ be continuous with $M(0) =1$ .\",\n \"Then one gets $\\\\textrm {M is exponentially convex}\\\\ \\\\Longleftrightarrow \\\\ \\\\exists \\\\ \\\\textrm {unique prob.\",\n \"distr.\",\n \"\\\\widetilde{\\\\widetilde{\\\\mathbb {}}} on ]-\\\\infty ,\\\\infty [such that} \\\\ \\\\ M(z) = \\\\int _{-\\\\infty }^{\\\\infty } e^{z \\\\cdot y} \\\\mathrm {d}\\\\widetilde{\\\\widetilde{\\\\mathbb {}}}(y)\\\\ \\\\ \\\\textrm {for all z \\\\in I.\",\n \"}$ Assertion (a) of Theorem REF is known as (probability-version of) Bernstein’s theorem [42] (see e.g.\",\n \"also Schilling et al.\",\n \"[322]), whereas assertion (b) is known as (probability-version of) Widder’s theorem [391] for the relevant conversion between the involved Riemann-Stieltjes integral with nondecreasing (but not necessarily right-continuous) integrator into a measure integral, one can apply the general theory in e.g.\",\n \"Chapter 6 of Chow & Teicher [88].\",\n \"(see e.g.\",\n \"also Widder [392], Akhiezer [9], Shucker [333], Jaksetic & Pecaric [163], Kotelina & Pevny [196]).\",\n \"From Theorem REF (b) and (REF ), the first item in (G9iv) follows immediately by using the choice $I= ]\\\\lambda _{-}, \\\\lambda _{+}[$ .\",\n \"Moreover, Theorem REF (a) together with (REF ) (respectively (REF )) implies the second item of (G9ii) (respectively of (G9iii)).\",\n \"In fact, with the help of Theorem REF and some further considerations e.g.\",\n \"on light-tailedness, one even gets assertions on the sufficiency of (G9ii), (G9iii) and (G9iv) for a “candidate generator” $\\\\varphi $ to belong to the BS-suitable class $\\\\Upsilon (]a,b[)$ .\",\n \"More precisely, we obtain Proposition 31 Suppose that $\\\\varphi : ]-\\\\infty ,\\\\infty [ \\\\mapsto [0,\\\\infty ]$ satisfies (G1) to (G8), and recall the notations in (G9i).\",\n \"Then, $\\\\varphi \\\\in \\\\Upsilon (]a,b[)$ if one of the following three conditions holds: (a) $a >-\\\\infty $ , $\\\\lambda _{-} = - \\\\infty $ , and the function $z \\\\mapsto e^{-a\\\\cdot z + \\\\varphi ^{*}(z)}$ is absolutely monotone on $]-\\\\infty ,0[$ , (b) $b < \\\\infty $ , $\\\\lambda _{+} = \\\\infty $ , and the function $z \\\\mapsto e^{b\\\\cdot z + \\\\varphi ^{*}(- z)}$ is absolutely monotone on $]-\\\\infty ,0[$ , (c) $a = -\\\\infty $ , $b = -\\\\infty $ , and the function $z \\\\mapsto e^{\\\\varphi ^{*}(z)}$ is exponentially convex on $]\\\\lambda _{-}, \\\\lambda _{+}[$ .\",\n \"If one of the three conditions (a) to (c) holds, thenbasically by Theorem REF with $M(\\\\cdot )$ defined in G9(ii),(iii) or (iv); see Appendix D. the associated probability distribution $\\\\mathbb {}$ (cf.\",\n \"(REF )) has expectation $\\\\int _{\\\\mathbb {R}} y d\\\\mathbb {}(y) =1$ and finite moments of all orders, i.e.\",\n \"$\\\\int _{\\\\mathbb {R}} y^{j} d\\\\mathbb {}(y) < \\\\infty $ for all $j \\\\in \\\\mathbb {N}_{0}$ ; in terms of $\\\\mathbb {}[ \\\\cdot \\\\, ] := \\\\mathbb {\\\\Pi }[W \\\\in \\\\cdot \\\\, ]$ this means that $E_{\\\\mathbb {\\\\Pi }}[W] =1$ and $E_{\\\\mathbb {\\\\Pi }}[W^{j}] < \\\\infty $ .\",\n \"The proof of Proposition REF will be given in Appendix D. As far as applicability is concerned, it is well known that, in general, verifying absolute monotonicity is typically more comfortable than verifying exponential convexity.\",\n \"Fortunately, one can often use the former, since for many known divergence generators there holds $a > -\\\\infty $ (often $a=0$ ) or $b < \\\\infty $ or both, which by virtue of (G10i) is directly linked with the (endpoints of the) support of the potentially existing probability distribution $\\\\mathbb {}$ .\",\n \"For a pregiven divergence generator $\\\\varphi $ , once its membership in $\\\\Upsilon (]a,b[)$ (and in particular, the representability (REF )) is verified, one would like to concretely find the underlying probability distribution $\\\\mathbb {}$ .\",\n \"This may be quite challenging, but can be made more comfortable by systematically narrowing down the family of distributions where $\\\\mathbb {}$ belongs to.\",\n \"In fact, we have already performed a first down-narrowing, in terms of identifying the endpoints of the support of $\\\\mathbb {}$ to be the endpoints of the effective domain of $\\\\varphi $ (cf.\",\n \"(G10i)).\",\n \"A further down-narrowing can be achieved from (REF ) to (REF ) in combination with real-valued characterization theorems which are more specific than Theorem REF .\",\n \"This will be shown exemplarily for a few important sub-setups, in the following.\",\n \"For the identification of light-tailed semi/half-lattice distributions, we obtain the following two sets of sufficient conditions, which even allow for the desired explicit determination of $\\\\mathbb {}$ : Proposition 32 Suppose that $\\\\varphi : ]-\\\\infty ,\\\\infty [ \\\\mapsto [0,\\\\infty ]$ satisfies (G1) to (G8), with some $a > -\\\\infty $ .\",\n \"Furthermore, assume that there exists some constant $\\\\breve{c} >0$ as well as some function $H: [0,\\\\infty [ \\\\mapsto [0,\\\\infty [$ which is continuous on $[0,1]$ with $H(1)=1$ and absolutely monotone on $]0,1[$ , such that $e^{\\\\varphi ^{*}(\\\\frac{z}{\\\\breve{c}}) - a\\\\cdot \\\\frac{z}{\\\\breve{c}} } = H(e^{z}), \\\\qquad z \\\\in ]-\\\\infty , \\\\breve{c} \\\\cdot \\\\lambda _{+}[.$ Then one has $\\\\varphi \\\\in \\\\Upsilon (]a,b[)$ and $\\\\mathbb {} = \\\\sum _{n=0}^{\\\\infty } p_{n} \\\\cdot \\\\delta _{a + \\\\breve{c} \\\\cdot n}\\\\qquad \\\\textrm {with \\\\ } p_{n} := \\\\frac{1}{n !}\",\n \"\\\\cdot \\\\frac{\\\\mathrm {d}^{n}H}{\\\\mathrm {d}t}(0),$ i.e.\",\n \"$\\\\mathbb {\\\\Pi }[W = a + \\\\breve{c} \\\\cdot n \\\\, ] = p_{n}$ ($n \\\\in \\\\mathbb {N}_{0}$ ).\",\n \"Proposition 33 Suppose that $\\\\varphi : ]-\\\\infty ,\\\\infty [ \\\\mapsto [0,\\\\infty ]$ satisfies (G1) to (G8), with some $b < \\\\infty $ .\",\n \"Furthermore, assume that there exists some constant $\\\\breve{c} >0$ as well as some function $H: [0,\\\\infty [ \\\\mapsto [0,\\\\infty [$ which is continuous on $[0,1]$ with $H(1)=1$ and absolutely monotone on $]0,1[$ , such that $e^{\\\\varphi ^{*}(- \\\\frac{z}{\\\\breve{c}}) + b \\\\cdot \\\\frac{z}{\\\\breve{c}} } = H(e^{z}), \\\\qquad z \\\\in ]-\\\\infty , - \\\\breve{c} \\\\cdot \\\\lambda _{-}[.$ Then one has $\\\\varphi \\\\in \\\\Upsilon (]a,b[)$ and $\\\\mathbb {} = \\\\sum _{n=0}^{\\\\infty } p_{n} \\\\cdot \\\\delta _{b - \\\\breve{c} \\\\cdot n}\\\\qquad \\\\textrm {with \\\\ } p_{n} := \\\\frac{1}{n !}\",\n \"\\\\cdot \\\\frac{\\\\mathrm {d}^{n}H}{\\\\mathrm {d}t}(0),$ i.e.\",\n \"$\\\\mathbb {\\\\Pi }[W = b - \\\\breve{c} \\\\cdot n \\\\, ] = p_{n}$ ($n \\\\in \\\\mathbb {N}_{0}$ ).\",\n \"The Propositions REF respectively REF follow from (REF ) respectively (REF ), some straightforward transformations, and a well-known characterization of probability generating functions $H$ (see e.g.\",\n \"in Theorem 1.2.10 of Stroock [343]).\",\n \"As an incentive for the following investigations, let us recall the discussion in the surroundings of Condition REF pertaining to the minimization problem (REF ), where we have addressed possible connections between the two representabilities (REF ) (needed e.g.\",\n \"for Theorem REF ) and (REF ) (needed e.g.\",\n \"for Theorem REF ); this strongly relates to the question, for which constants $\\\\widetilde{c} >0$ the validity $\\\\varphi \\\\in \\\\Upsilon (]a,b[)$ triggers the validity of $\\\\widetilde{c} \\\\cdot \\\\varphi \\\\in \\\\Upsilon (]a,b[)$ .\",\n \"To begin with, it is straightforward to see that $\\\\varphi \\\\in \\\\Upsilon (]a,b[)$ always implies $\\\\widetilde{c} \\\\cdot \\\\varphi \\\\in \\\\Upsilon (]a,b[)$ for all integers $\\\\widetilde{c} \\\\in \\\\mathbb {N}$ ; indeed, if $\\\\varphi $ satisfies (REF ) for some $\\\\mathbb {} = \\\\mathbb {\\\\Pi }[ W \\\\in \\\\cdot \\\\, ]$ , then for each integer $\\\\widetilde{c} \\\\in \\\\mathbb {N}$ one gets that $\\\\widetilde{c} \\\\cdot \\\\varphi $ satisfies (REF ) for $\\\\widetilde{\\\\mathbb {}} = \\\\mathbb {\\\\Pi }[ \\\\sum _{j=1}^{\\\\widetilde{c}} \\\\frac{W_{j}}{\\\\widetilde{c}} \\\\in \\\\cdot \\\\, ]$ ; in the latter, the $W_{j}$ ’s are i.i.d.\",\n \"copies from $W$ .\",\n \"Clearly, $MGF_{\\\\widetilde{\\\\mathbb {}}}$ is then finite on some open interval containing zero (differing from the one for $MGF_{\\\\mathbb {}}$ only by a scaling with $1/\\\\widetilde{c}$ ).\",\n \"For the following family of distributions, one can even trigger $\\\\widetilde{c} \\\\cdot \\\\varphi \\\\in \\\\Upsilon (]a,b[)$ for all $\\\\widetilde{c} >0$ : for the sake of a corresponding precise formulation, recall first the common knowledge that, generally speaking, a probability distribution $\\\\mathbb {}$ on $\\\\mathbb {R}$ with light tails — in the sense that its moment generating function $z\\\\mapsto MGF_{\\\\mathbb {} }(z):=\\\\int _{\\\\mathbb {R}}e^{z \\\\cdot y}d\\\\mathbb {} (y)$ is finite on some open interval $]\\\\lambda _{-},\\\\lambda _{+}[$ containing zero — is (said to be) infinitely divisible if there holds $\\\\textrm {for each n \\\\in \\\\mathbb {N} there existsa probability distribution \\\\mathbb {}_{n} on \\\\mathbb {R} such that}\\\\int _{-\\\\infty }^{\\\\infty } e^{z \\\\cdot y} d\\\\mathbb {} (y) =\\\\Big (\\\\int _{-\\\\infty }^{\\\\infty } e^{z \\\\cdot y} d\\\\mathbb {}_{n} (y) \\\\Big )^{n}, \\\\quad z \\\\in ]\\\\lambda _{-},\\\\lambda _{+}[ ;$ in fact, (REF ) means that the (light-tailed) moment generating function $MGF_{\\\\mathbb {} }$ is infinitely divisible in the sense that each $n-$ th root $(MGF_{\\\\mathbb {} })^{1/n}$ must be the moment generating function of some (light-tailed) probability distribution (denoted here by $\\\\mathbb {}_{n}$ ).\",\n \"In particular, (REF ) implies that $\\\\mathbb {}_{n}$ is unique, and that $\\\\mathbb {}$ must necessarily have (one-sided or two-sided) unbounded support $supp(\\\\mathbb {})$ .\",\n \"The latter may differ from $supp(\\\\mathbb {}_{n})$ .\",\n \"In our BS context (REF ), (REF ) equivalently means that the associated random variable $W$ is infinitely divisible (with light-tailed distribution), in the sense that $\\\\textrm {for each n \\\\in \\\\mathbb {N} there exists a sequence of i.i.d.\",\n \"random variables Y_{n,1}, \\\\cdots , Y_{n,n} such that }W \\\\stackrel{\\\\text{\\\\tiny d}}{=} Y_{n,1} + \\\\cdots + Y_{n,n} ,$ where $\\\\stackrel{\\\\text{\\\\tiny d}}{=}$ means “have equal probability distributions” and $\\\\mathbb {\\\\Pi }[W \\\\in \\\\cdot \\\\, ] = \\\\mathbb {}[ \\\\, \\\\cdot \\\\,]$ , $\\\\mathbb {\\\\Pi }[Y_{n,1} \\\\in \\\\cdot \\\\, ] = \\\\mathbb {}_{n}[ \\\\, \\\\cdot \\\\,]$ .\",\n \"For the above-mentioned context, we obtain the useful Proposition 34 Suppose that $\\\\varphi \\\\in \\\\Upsilon (]a,b[)$ , with connected probability distribution $\\\\mathbb {}$ from (REF ) (recall that this implies that $\\\\mathbb {}$ is not a one-point distribution, cf.\",\n \"Remark REF ).\",\n \"Then there holds: $\\\\textrm {\\\\widetilde{c} \\\\cdot \\\\varphi \\\\in \\\\Upsilon (]a,b[)for all \\\\widetilde{c} > 0} \\\\ \\\\ \\\\Longleftrightarrow \\\\ \\\\ \\\\textrm {\\\\mathbb {} is infinitely divisible.\",\n \"}$ The proof of Proposition REF is given in Appendix E. Notice that Proposition REF covers especially the important prominent power divergences (cf.\",\n \"Examples REF and REF below) for which we provide the corresponding infinitely divisible distributions explicitly in the Examples REF and REF below, and for which the subsequent form of estimators (cf.\",\n \"Chapter below) can be simplified.\",\n \"For the identification of light-tailed infinitely divisible distributions, we obtain the following three sets of sufficient conditions: Proposition 35 Suppose that $\\\\varphi : ]-\\\\infty ,\\\\infty [ \\\\mapsto [0,\\\\infty ]$ satisfies (G1) to (G8), and recall the notations in (G9i) as well as $a = \\\\inf supp(\\\\mathbb {})$ , $b = \\\\sup supp(\\\\mathbb {})$ (cf.\",\n \"(G10i)).\",\n \"Then, $\\\\varphi \\\\in \\\\Upsilon (]a,b[)$ and the associated probability distribution $\\\\mathbb {}$ is infinitely divisible, if one of the following three conditions holds: (a) $a >-\\\\infty $ , $\\\\lambda _{-} = - \\\\infty $ , and the function $z \\\\mapsto \\\\varphi ^{* \\\\prime }(z) - a = (\\\\varphi ^{\\\\prime })^{-1}(z) - a $ is absolutely monotone on $]-\\\\infty ,0[$ , (b) $b < \\\\infty $ , $\\\\lambda _{+} = \\\\infty $ , and the function $z \\\\mapsto - \\\\varphi ^{* \\\\prime }(- z) + b= - (\\\\varphi ^{\\\\prime })^{-1}(-z) +b $ is absolutely monotone on $]-\\\\infty ,0[$ , (c) $a = -\\\\infty $ , $b = -\\\\infty $ , and the function $z \\\\mapsto \\\\frac{\\\\varphi ^{* \\\\prime \\\\prime }(z)}{\\\\varphi ^{* \\\\prime \\\\prime }(0)}= \\\\frac{\\\\varphi ^{\\\\prime \\\\prime }(1)}{\\\\varphi ^{\\\\prime \\\\prime }((\\\\varphi ^{\\\\prime })^{-1}(z))} $ is exponentially convex on $]\\\\lambda _{-}, \\\\lambda _{+}[$ .\",\n \"In the first case (a) there automatically follows $b=\\\\infty $ , whereas in the second case (b) one automatically gets $a =- \\\\infty $ .\",\n \"The proof of Proposition REF is given in Appendix E. So far, in the current section we have started from a given divergence generator $\\\\varphi \\\\in \\\\widetilde{\\\\Upsilon }(]a,b[)$ having some additional properties, switched to its Fenchel-Legendre transform $\\\\varphi ^{*}$ and some exponentially-linear transforms thereof, and presented some sufficient conditions for verifying that the outcome is a moment-generating function $MGF_{\\\\mathbb {}}$ of a unique probability distribution $\\\\mathbb {}$ which has light tails.\",\n \"For finding the concrete $\\\\mathbb {}$ , one typically should know the explicit form of $\\\\varphi ^{*}$ .\",\n \"However, it is well known that it can sometimes be hard to determine the explicit form of the Fenchel-Legendre transform of a convex function.\",\n \"This issue also applies for the reverse direction of starting from a concrete probability distribution $\\\\mathbb {}$ with light tails, computing its log-moment-generating function (called cumulant-generating function) $z \\\\mapsto \\\\Lambda _{\\\\mathbb {}}(z) := \\\\log MGF_{\\\\mathbb {}}(z)$ and the corresponding Fenchel-Legendre transform $\\\\Lambda _{\\\\mathbb {}}^{*}$ which is nothing but the associated divergence generator $\\\\varphi $ (cf.\",\n \"(REF )).\",\n \"As will be illuminated in several examples below, the — “kind of intermediate” — construction method given in the below-mentioned Theorem REF can help to ease these two tasks.\",\n \"To formulate this, we employ the class ${F}$ of functions $F: ]-\\\\infty ,\\\\infty [ \\\\mapsto [-\\\\infty ,\\\\infty ]$ with the following properties: $int(dom(F)) = ]a_{F},b_{F}[$ for some $-\\\\infty \\\\le a_{F} < 1 < b_{F} \\\\le \\\\infty $ ; $F$ is smooth (infinitely continuously differentiable) on $]a_{F},b_{F}[$ ; $F$ is strictly increasing on $]a_{F},b_{F}[$ .\",\n \"Clearly, for any $F \\\\in {F}$ one gets the existence of $F(a_{F}) := \\\\lim _{t \\\\downarrow a_{F}} F(t) \\\\in [-\\\\infty , \\\\infty [$ and $F(b_{F}) := \\\\lim _{t \\\\uparrow b_{F}} F(t) \\\\in ]-\\\\infty , \\\\infty ]$ ; moreover, its inverse $F^{-1}: \\\\mathcal {R}(F) \\\\mapsto [a_{F},b_{F}]$ exists, where $\\\\mathcal {R}(F) := \\\\lbrace F(t): t \\\\in dom(F) \\\\rbrace $ .\",\n \"Furthermore, $F^{-1}$ is strictly increasing and smooth (infinitely continuously differentiable) on the open interval $int(\\\\mathcal {R}(F)) = \\\\lbrace F(t): t \\\\in ]a_{F},b_{F}[ \\\\rbrace =]F(a_{F}),F(b_{F})[$ , and $F^{-1}(int(\\\\mathcal {R}(F))) = ]a_{F},b_{F}[$ .\",\n \"Within such a context, we obtain Theorem 36 Let $F \\\\in {F}$ and fix an arbitrary point $c \\\\in int(\\\\mathcal {R}(F))$ .\",\n \"Moreover, introduce the notationsfor the sake of brevity, we avoid here the more complete notation $\\\\lambda _{-}^{F,c}$ , $\\\\lambda _{+}^{F,c}$ , $t_{-}^{sc,F,c}$ , $t_{+}^{sc,F,c}$ indicating the dependence on $F$ and $c$ .\",\n \"$]\\\\lambda _{-},\\\\lambda _{+}[ := int(\\\\mathcal {R}(F)) -c$ and $]t_{-}^{sc},t_{+}^{sc}[ := ]1+a_{F}-F^{-1}(c),1+b_{F}-F^{-1}(c)[$ (which implies $\\\\lambda _{-} < 0 < \\\\lambda _{+}$ and $t_{-}^{sc} < 1 < t_{+}^{sc}$ ).\",\n \"Furthermore, define the functions $\\\\Lambda : \\\\ ]-\\\\infty ,\\\\infty [ \\\\ \\\\mapsto \\\\, [-\\\\infty , \\\\infty ]$ and $\\\\varphi : \\\\ ]-\\\\infty ,\\\\infty [ \\\\ \\\\mapsto \\\\, [0, \\\\infty ]$ by $\\\\hspace{-19.91684pt} \\\\Lambda (z) := \\\\Lambda ^{(c)}(z) \\\\hspace{-5.69046pt} &:=& \\\\hspace{-5.69046pt}{\\\\left\\\\lbrace \\\\begin{array}{ll}\\\\int \\\\displaylimits _{0}^{z} F^{-1}(u+c) \\\\, du+ z \\\\cdot (1-F^{-1}(c)) \\\\ \\\\in ]-\\\\infty , \\\\infty [,\\\\hspace{85.35826pt} \\\\textrm {if } z \\\\in ]\\\\lambda _{-},\\\\lambda _{+}[,\\\\\\\\\\\\int \\\\displaylimits _{0}^{\\\\lambda _{-}} F^{-1}(u+c) \\\\, du+ \\\\lambda _{-} \\\\cdot (1-F^{-1}(c))\\\\ \\\\in [-\\\\infty ,\\\\infty ],\\\\hspace{71.13188pt} \\\\textrm {if } z = \\\\lambda _{-} > -\\\\infty ,\\\\\\\\\\\\int \\\\displaylimits _{0}^{\\\\lambda _{+}} F^{-1}(u+c) \\\\, du+ \\\\lambda _{+} \\\\cdot (1-F^{-1}(c))\\\\ \\\\in \\\\ [-\\\\infty ,\\\\infty ],\\\\hspace{66.86414pt} \\\\textrm {if } z = \\\\lambda _{+} < \\\\infty ,\\\\\\\\\\\\infty , \\\\hspace{283.10483pt} \\\\textrm {else},\\\\end{array}\\\\right.\",\n \"}$ where the second respectively third line are meant as $\\\\lim _{z \\\\downarrow \\\\lambda _{-}} \\\\big ( \\\\int \\\\displaylimits _{0}^{z} F^{-1}(u+c) \\\\, du+ z \\\\cdot (1-F^{-1}(c)) \\\\big )$ respectively $\\\\lim _{z \\\\uparrow \\\\lambda _{+}} \\\\big ( \\\\int \\\\displaylimits _{0}^{z} F^{-1}(u+c) \\\\, du+ z \\\\cdot (1-F^{-1}(c)) \\\\big )$ , and $\\\\varphi (t) := \\\\varphi ^{(c)}(t) \\\\hspace{-5.69046pt} &:=& \\\\hspace{-5.69046pt}{\\\\left\\\\lbrace \\\\begin{array}{ll}(t+F^{-1}(c)-1) \\\\cdot [ F\\\\left(t+F^{-1}(c)-1 \\\\right) - c ]-\\\\int \\\\displaylimits _{0}^{F\\\\left(t+F^{-1}(c)-1 \\\\right) - c} F^{-1}(u+c) du\\\\ \\\\in \\\\ [0,\\\\infty [,\\\\\\\\\\\\hspace{327.20668pt}\\\\quad \\\\textrm {if }t \\\\in ]t_{-}^{sc},t_{+}^{sc}[,\\\\\\\\(t_{-}^{sc}+F^{-1}(c)-1) \\\\cdot [ F\\\\left(t_{-}^{sc}+F^{-1}(c)-1 \\\\right) - c ]-\\\\int \\\\displaylimits _{0}^{F\\\\left(t_{-}^{sc}+F^{-1}(c)-1 \\\\right) - c} F^{-1}(u+c) du\\\\ \\\\in \\\\ ]0,\\\\infty ],\\\\\\\\\\\\hspace{327.20668pt}\\\\quad \\\\textrm {if }t = t_{-}^{sc} > -\\\\infty ,\\\\\\\\(t_{+}^{sc}+F^{-1}(c)-1) \\\\cdot [ F\\\\left(t_{+}^{sc}+F^{-1}(c)-1 \\\\right) - c ]-\\\\int \\\\displaylimits _{0}^{F\\\\left(t_{+}^{sc}+F^{-1}(c)-1 \\\\right) - c} F^{-1}(u+c) du\\\\ \\\\in \\\\ ]0,\\\\infty ],\\\\\\\\\\\\hspace{327.20668pt}\\\\quad \\\\textrm {if }t = t_{+}^{sc} < \\\\infty ,\\\\\\\\\\\\varphi (t_{-}^{sc}) +\\\\lambda _{-}\\\\cdot (t- t_{-}^{sc}) \\\\ \\\\in \\\\ ]0,\\\\infty ],\\\\hspace{99.58464pt} \\\\textrm {if }t_{-}^{sc} > - \\\\infty , \\\\ \\\\textrm {and} \\\\ t \\\\in \\\\ ]-\\\\infty , t_{-}^{sc}[,\\\\\\\\\\\\varphi (t_{+}^{sc}) +\\\\lambda _{+}\\\\cdot (t - t_{+}^{sc}) \\\\ \\\\in \\\\ ]0,\\\\infty ],\\\\hspace{99.58464pt} \\\\textrm {if }t_{+}^{sc} < \\\\infty , \\\\ \\\\textrm {and} \\\\ t \\\\in \\\\ ]t_{+}^{sc}, \\\\infty [,\\\\\\\\\\\\infty , \\\\hspace{321.51622pt} \\\\textrm {else},\\\\end{array}\\\\right.\",\n \"}$ where the second respectively third line are again meant as lower respectively upper limit.\",\n \"Then, $\\\\Lambda $ and $\\\\varphi $ have the following properties: (i) On $]\\\\lambda _{-},\\\\lambda _{+}[$ , the function $\\\\Lambda $ is smooth and strictly convex and consequently, $\\\\exp (\\\\Lambda )$ ) is smooth and strictly log-convex; moreover, there holds $\\\\Lambda (0) =0$ , $\\\\Lambda ^{\\\\prime }(0) =1$ ; (ii) $\\\\varphi \\\\in \\\\widetilde{\\\\Upsilon }(]a,b[)$ , where $a := t_{-}^{sc} \\\\cdot {1}_{ \\\\lbrace -\\\\infty \\\\rbrace }(\\\\lambda _{-}) - \\\\infty \\\\cdot {1}_{]-\\\\infty ,0[}(\\\\lambda _{-})$ , $b := t_{+}^{sc} \\\\cdot {1}_{ \\\\lbrace \\\\infty \\\\rbrace }(\\\\lambda _{+}) + \\\\infty \\\\cdot {1}_{]0,\\\\infty [}(\\\\lambda _{+})$ , and $\\\\varphi $ has the properties (G1) to (G8).\",\n \"(iii) $\\\\varphi (t) =\\\\sup _{z \\\\in ]-\\\\infty ,\\\\infty [} \\\\left( z\\\\cdot t -\\\\Lambda (z)\\\\right) =\\\\sup _{z \\\\in ]\\\\lambda _{-},\\\\lambda _{+}[} \\\\left( z\\\\cdot t -\\\\Lambda (z)\\\\right)$ for all $t \\\\in \\\\mathbb {R}$ .\",\n \"(iv) $\\\\Lambda (z) = \\\\varphi ^{*}(z) =\\\\sup _{t \\\\in ]-\\\\infty ,\\\\infty [} \\\\left( t\\\\cdot z -\\\\varphi (t)\\\\right) =\\\\sup _{t \\\\in ]a,b[} \\\\left( t\\\\cdot z -\\\\varphi (t) \\\\right)$ for all $z \\\\in \\\\mathbb {R}$ .\",\n \"The proof of Theorem REF will be given Appendix F. Remark 37 Theorem REF indicates that the $F-$ constructed function $z \\\\mapsto \\\\exp (\\\\Lambda (z)) = \\\\exp (\\\\varphi ^{*}(z))$ is a good candidate for a moment generating function of a probability distribution $\\\\mathbb {}$ , and hence for the representability (REF ).\",\n \"However, one still needs to verify one of the conditions (a) to (c) of Proposition REF .\",\n \"This may go wrong, as the case of power divergences $\\\\varphi _{\\\\gamma }$ with $\\\\gamma \\\\in ]1,2[$ indicates (cf.\",\n \"the conjecture of Example REF (f) below).\",\n \"Notice that the newly constructed $\\\\Lambda $ and $\\\\varphi $ (cf.\",\n \"(REF ), (REF )) depend on the choice of the anchor point $c$ ; this is e.g.\",\n \"illustrated in Example REF (b) below.\",\n \"Hence, as a side effect, by using whole families $(F_{\\\\vartheta })_{\\\\vartheta }$ together with different anchor points $c$ , via Theorem REF one can generate new classes (and new classifications) of $\\\\varphi -$ divergence generators — and thus of corresponding $\\\\varphi -$ divergences — which can be of great use, even in other contexts beyond our BS optimization framework.\",\n \"If $F$ satisfies $F(1)=0$ and thus $F^{-1}(0)=1$ , then the natural choice $c:=0$ induces $]\\\\lambda _{-},\\\\lambda _{+}[ = int(\\\\mathcal {R}(F))$ and $]t_{-}^{sc},t_{+}^{sc}[ = ]a_{F},b_{F}[$ , and consequently (due to $F^{-1}(c)-1 =0$ ) leads to the simplification of “the first lines of” (REF ) and (REF ) to $& & \\\\hspace{-19.91684pt} \\\\Lambda (z) := \\\\Lambda ^{(0)}(z) :=\\\\int \\\\displaylimits _{0}^{z} F^{-1}(u) du,\\\\qquad z \\\\in int(\\\\mathcal {R}(F)),\\\\\\\\& & \\\\hspace{-19.91684pt} \\\\varphi (t) := \\\\varphi ^{(0)}(t) :=t \\\\cdot F\\\\left(t\\\\right)-\\\\int \\\\displaylimits _{0}^{F\\\\left(t\\\\right)} F^{-1}(u) du,\\\\qquad t \\\\in ]a_{F},b_{F}[ ;$ the simplifications of the respective other lines of (REF ) and (REF ) are straightforward.\",\n \"Remark 38 Let $F \\\\in {F}$ with $a_{F} =0$ , $b_{F} = \\\\infty $ , $F(1)=0$ and hence, $int(\\\\mathcal {R}(F)) = \\\\, ]F(0),F(\\\\infty )[$ .\",\n \"Then the transformation $\\\\widetilde{F}(t) &:=&{\\\\left\\\\lbrace \\\\begin{array}{ll}- \\\\int _{0}^{F(\\\\frac{1}{t})} F^{-1}(u) \\\\, du, \\\\qquad \\\\textrm {if } \\\\ t \\\\in ]0,\\\\infty [, \\\\\\\\- \\\\int _{0}^{F(\\\\infty )} F^{-1}(u) \\\\, du, \\\\qquad \\\\textrm {if } \\\\ t=0, \\\\\\\\-\\\\infty , \\\\qquad \\\\hspace{64.01869pt} \\\\textrm {if } \\\\ t \\\\in ]-\\\\infty , 0[,\\\\end{array}\\\\right.\",\n \"}$ satisfies $\\\\widetilde{F} \\\\in {F}$ with $a_{\\\\widetilde{F}} =0$ , $b_{\\\\widetilde{F}} = \\\\infty $ , $\\\\widetilde{F}(1)=0$ and $int(\\\\mathcal {R}(\\\\widetilde{F})) =\\\\big ] - \\\\int _{0}^{F(\\\\infty )} F^{-1}(u) \\\\, du, - \\\\int _{0}^{F(0)} F^{-1}(u) \\\\, du \\\\, \\\\big [$ .\",\n \"By choosing the natural anchor point $c=0$ (for both $F$ and $\\\\widetilde{F}$ ) and by using the relations $\\\\widetilde{F}(t) = - \\\\Lambda (F(\\\\frac{1}{t}))$ , $\\\\widetilde{F}^{-1}(z) = \\\\frac{1}{F^{-1}(\\\\Lambda ^{-1}(-z))}$ , as well as (REF ) in combination with (REF ) (for both contexts), it is straightforward to see that the corresponding quantities $\\\\widetilde{\\\\Lambda }$ and $\\\\widetilde{\\\\varphi }$ satisfy $\\\\widetilde{\\\\Lambda }(z) =- (-\\\\Lambda )^{-1}(z)$ ($z \\\\in int(\\\\mathcal {R}(\\\\widetilde{F}))$ ) and $\\\\widetilde{\\\\varphi }(t) = t \\\\cdot \\\\varphi (\\\\frac{1}{t})$ ($t \\\\in ]0,\\\\infty [$ ).\",\n \"Hence, the corresponding divergences (cf.\",\n \"(REF )) are “reciprocal to each other” in the sense that $D_{\\\\widetilde{\\\\varphi }}( \\\\mathbf {Q}, \\\\mathbf {P} ) =D_{\\\\varphi }( \\\\mathbf {P}, \\\\mathbf {Q} )$ for all $\\\\mathbf {P},\\\\mathbf {Q} \\\\in \\\\mathbb {S}_{>0}^{K}$ , and in case that $\\\\Lambda $ and $\\\\widetilde{\\\\Lambda }$ are indeed cumulant generating functions of some light-tailed distributions $\\\\mathbb {}$ and $\\\\widetilde{\\\\mathbb {}}$ (cf.\",\n \"Remark REF ), then the latter two are said to be inverse to each other in the sense of Tweedie [368] (see also e.g.\",\n \"Folks [127]).\",\n \"As already indicated above, from Theorem REF one can comfortably generate various interesting examples, which we demonstrate in the following.\",\n \"Example 39 (a) For $\\\\gamma \\\\in \\\\mathbb {R}\\\\backslash \\\\lbrace 1,2\\\\rbrace $ , $\\\\widetilde{c} \\\\in ]0,\\\\infty [$ and $]a_{F_{\\\\gamma ,\\\\widetilde{c}}},b_{F_{\\\\gamma ,\\\\widetilde{c}}}[ \\\\, = \\\\, ]0,\\\\infty [$ we define $F_{\\\\gamma ,\\\\widetilde{c}}(t) &:=&{\\\\left\\\\lbrace \\\\begin{array}{ll}\\\\frac{\\\\widetilde{c}}{\\\\gamma -1} \\\\cdot (t^{\\\\gamma -1}-1), \\\\qquad \\\\textrm {if } \\\\ t \\\\in \\\\, ]0,\\\\infty [, \\\\\\\\-\\\\frac{\\\\widetilde{c}}{\\\\gamma -1}, \\\\hspace{62.59596pt}\\\\textrm {if } \\\\ t=0 \\\\ \\\\textrm {and } \\\\ \\\\gamma \\\\in \\\\, ]1,2[ \\\\, \\\\cup \\\\, ]2,\\\\infty [, \\\\\\\\-\\\\infty , \\\\hspace{71.13188pt} \\\\textrm {if } \\\\ t=0 \\\\ \\\\textrm {and } \\\\ \\\\gamma < 1, \\\\\\\\- \\\\infty , \\\\hspace{71.13188pt} \\\\textrm {if } \\\\ t \\\\in \\\\, ]-\\\\infty ,0[,\\\\end{array}\\\\right.\",\n \"}\\\\nonumber $ Clearly, $\\\\mathcal {R}(F_{\\\\gamma ,\\\\widetilde{c}})= \\\\big [-\\\\frac{\\\\widetilde{c}}{\\\\gamma -1},\\\\infty \\\\big [$ for $\\\\gamma \\\\in \\\\, ]1,2[ \\\\, \\\\cup \\\\, ]2,\\\\infty [$ , respectively $\\\\mathcal {R}(F_{\\\\gamma ,\\\\widetilde{c}})=\\\\big ]-\\\\infty , \\\\frac{\\\\widetilde{c}}{1-\\\\gamma }\\\\big [$ for $\\\\gamma <1$ ; notice that $0 \\\\in int(\\\\mathcal {R}(F_{\\\\gamma ,\\\\widetilde{c}}))$ for all $\\\\gamma \\\\in \\\\mathbb {R}\\\\backslash \\\\lbrace 1,2\\\\rbrace $ .\",\n \"Furthermore, $F_{\\\\gamma ,\\\\widetilde{c}}(\\\\cdot )$ is strictly increasing and smooth on $]0,\\\\infty [$ , and thus, $F_{\\\\gamma ,\\\\widetilde{c}} \\\\in {F}$ .\",\n \"Since $F_{\\\\gamma ,\\\\widetilde{c}}(1)=0$ , let us choose the natural anchor point $c:=0$ , which leads to $]\\\\lambda _{-},\\\\lambda _{+}[= int(\\\\mathcal {R}(F_{\\\\gamma ,\\\\widetilde{c}}))$ and $]t_{-}^{sc},t_{+}^{sc}[ = ]0,\\\\infty [$ .\",\n \"By using $F_{\\\\gamma ,\\\\widetilde{c}}^{-1}(x) = (1+\\\\frac{(\\\\gamma -1) \\\\cdot x}{\\\\widetilde{c}})^{\\\\frac{1}{\\\\gamma -1}}$ for $x \\\\in int(\\\\mathcal {R}(F_{\\\\gamma ,\\\\widetilde{c}}))$ , we can derive from formula (REF ) (see also (REF )) for all $\\\\gamma \\\\in \\\\mathbb {R}\\\\backslash \\\\lbrace 0,1,2\\\\rbrace $ and $z \\\\in \\\\mathbb {R}$ $\\\\Lambda _{\\\\gamma ,\\\\widetilde{c}}(z) := \\\\Lambda _{\\\\gamma ,\\\\widetilde{c}}^{(0)}(z) &=&{\\\\left\\\\lbrace \\\\begin{array}{ll}\\\\frac{\\\\widetilde{c}}{\\\\gamma } \\\\cdot \\\\left\\\\lbrace \\\\left( \\\\frac{\\\\gamma -1}{\\\\widetilde{c}} \\\\cdot z +1 \\\\right)^{\\\\frac{\\\\gamma }{\\\\gamma -1}} -1 \\\\right\\\\rbrace , \\\\qquad \\\\textrm {if } \\\\ \\\\gamma \\\\in \\\\, ]1,2[ \\\\, \\\\cup \\\\, ]2,\\\\infty [\\\\ \\\\textrm {and } \\\\ z \\\\in \\\\big ]-\\\\frac{\\\\widetilde{c}}{\\\\gamma -1},\\\\infty \\\\big [ \\\\\\\\\\\\hspace{130.88284pt} \\\\textrm {or if } \\\\ \\\\gamma \\\\in ]-\\\\infty ,0[ \\\\cup ]0,1[ \\\\ \\\\textrm {and } \\\\ z \\\\in \\\\big ]-\\\\infty , \\\\frac{\\\\widetilde{c}}{1-\\\\gamma }\\\\big [, \\\\\\\\- \\\\frac{\\\\widetilde{c}}{\\\\gamma } < 0, \\\\hspace{105.2751pt} \\\\textrm {if } \\\\ \\\\gamma \\\\in \\\\, ]1,2[ \\\\, \\\\cup \\\\, ]2,\\\\infty [\\\\ \\\\textrm {and } \\\\ z = -\\\\frac{\\\\widetilde{c}}{\\\\gamma -1}, \\\\\\\\- \\\\frac{\\\\widetilde{c}}{\\\\gamma } >0, \\\\hspace{105.2751pt} \\\\textrm {if } \\\\ \\\\gamma < 0 \\\\ \\\\textrm {and } \\\\ z = \\\\frac{\\\\widetilde{c}}{1-\\\\gamma }, \\\\\\\\\\\\infty , \\\\hspace{128.0374pt} \\\\textrm {if } \\\\ \\\\gamma \\\\in \\\\, ]0,1[ \\\\ \\\\textrm {and } \\\\ z = \\\\frac{\\\\widetilde{c}}{1-\\\\gamma }, \\\\\\\\\\\\infty , \\\\hspace{128.0374pt} \\\\textrm {else} .\\\\end{array}\\\\right.\",\n \"}$ Notice that $\\\\Lambda _{\\\\gamma ,\\\\widetilde{c}}(0) = 0$ for all $\\\\gamma \\\\in \\\\mathbb {R}\\\\backslash \\\\lbrace 0,1,2\\\\rbrace $ .\",\n \"Moreover, for $\\\\gamma \\\\in \\\\, ]1,2[ \\\\, \\\\cup \\\\, ]2,\\\\infty [$ one has $\\\\Lambda _{\\\\gamma ,\\\\widetilde{c}}(\\\\infty ) = \\\\infty $ , $\\\\Lambda _{\\\\gamma ,\\\\widetilde{c}}^{\\\\prime }(-\\\\frac{\\\\widetilde{c}}{\\\\gamma -1}) = 0$ and $\\\\Lambda _{\\\\gamma ,\\\\widetilde{c}}^{\\\\prime }(\\\\infty ) = \\\\infty $ .\",\n \"For $\\\\gamma < 0$ one gets $\\\\Lambda _{\\\\gamma ,\\\\widetilde{c}}(-\\\\infty ) = -\\\\infty $ , $\\\\Lambda _{\\\\gamma ,\\\\widetilde{c}}^{\\\\prime }(\\\\frac{\\\\widetilde{c}}{1-\\\\gamma }) = \\\\infty $ and $\\\\Lambda _{\\\\gamma ,\\\\widetilde{c}}^{\\\\prime }(-\\\\infty ) = 0$ .\",\n \"In contrast, if $\\\\gamma \\\\in \\\\, ]0,1[$ then $\\\\Lambda _{\\\\gamma ,\\\\widetilde{c}}(-\\\\infty ) = - \\\\frac{\\\\widetilde{c}}{\\\\gamma } <0$ , $\\\\Lambda _{\\\\gamma ,\\\\widetilde{c}}^{\\\\prime }(\\\\frac{\\\\widetilde{c}}{1-\\\\gamma }) = \\\\infty $ and $\\\\Lambda _{\\\\gamma ,\\\\widetilde{c}}^{\\\\prime }(-\\\\infty ) = 0$ .\",\n \"To proceed, from formula (REF ) (see also ()) we can deduce for all $\\\\gamma \\\\in \\\\mathbb {R}\\\\backslash \\\\lbrace 0,1,2\\\\rbrace $ and $t \\\\in \\\\mathbb {R}$ $\\\\varphi _{\\\\gamma ,\\\\widetilde{c}}(t) := \\\\varphi _{\\\\gamma ,\\\\widetilde{c}}^{(0)}(t)\\\\hspace{-5.69046pt} &=& \\\\hspace{-5.69046pt}{\\\\left\\\\lbrace \\\\begin{array}{ll}\\\\widetilde{c} \\\\cdot \\\\frac{t^\\\\gamma -\\\\gamma \\\\cdot t+ \\\\gamma - 1}{\\\\gamma \\\\cdot (\\\\gamma -1)}\\\\ \\\\in \\\\ [0,\\\\infty [,\\\\qquad \\\\quad \\\\textrm {if }t \\\\in ]0,\\\\infty [,\\\\\\\\\\\\frac{\\\\widetilde{c}}{\\\\gamma } > 0, \\\\hspace{108.12054pt} \\\\textrm {if } \\\\ \\\\gamma \\\\in \\\\, ]1,2[ \\\\, \\\\cup \\\\, ]2,\\\\infty [\\\\ \\\\textrm {and } \\\\ t = 0, \\\\\\\\\\\\infty , \\\\hspace{123.76965pt} \\\\textrm {if } \\\\ \\\\gamma < 0 \\\\ \\\\textrm {and } \\\\ t = 0, \\\\\\\\\\\\frac{\\\\widetilde{c}}{\\\\gamma } > 0, \\\\hspace{108.12054pt} \\\\textrm {if } \\\\ \\\\gamma \\\\in \\\\, ]0,1[ \\\\ \\\\textrm {and } \\\\ t = 0, \\\\\\\\\\\\frac{\\\\widetilde{c}}{\\\\gamma } -\\\\frac{\\\\widetilde{c}}{\\\\gamma -1} \\\\cdot t \\\\ \\\\in \\\\ ]0,\\\\infty [,\\\\hspace{45.52458pt} \\\\textrm {if } \\\\ \\\\gamma \\\\in \\\\, ]1,2[ \\\\, \\\\cup \\\\, ]2,\\\\infty [\\\\ \\\\textrm {and } \\\\ t < 0, \\\\\\\\\\\\infty , \\\\hspace{123.76965pt} \\\\textrm {else} ,\\\\end{array}\\\\right.\",\n \"}$ which coincides with $\\\\widetilde{c} \\\\cdot \\\\varphi _{\\\\gamma }(t)$ for $\\\\varphi _{\\\\gamma }(t)$ from (REF ) and which generates the $\\\\gamma -$ corresponding power divergences given in (REF ); the first line in (REF ) can be proved by $& & \\\\hspace{-19.91684pt}\\\\varphi _{\\\\gamma ,\\\\widetilde{c}}(t) := \\\\varphi _{\\\\gamma ,\\\\widetilde{c}}^{(0)}(t) :=t \\\\cdot F_{\\\\gamma ,\\\\widetilde{c}}\\\\left(t\\\\right)-\\\\int \\\\displaylimits _{0}^{F_{\\\\gamma ,\\\\widetilde{c}}\\\\left(t\\\\right)} F_{\\\\gamma ,\\\\widetilde{c}}^{-1}(u) du\\\\nonumber \\\\\\\\& & \\\\hspace{-19.91684pt}= \\\\frac{t \\\\cdot \\\\widetilde{c}}{\\\\gamma -1} \\\\cdot (t^{\\\\gamma -1}-1)- \\\\frac{\\\\widetilde{c}}{\\\\gamma } \\\\cdot \\\\left\\\\lbrace \\\\left( \\\\frac{\\\\gamma -1}{\\\\widetilde{c}} \\\\cdot \\\\left[ \\\\frac{\\\\widetilde{c}}{\\\\gamma -1} \\\\cdot (t^{\\\\gamma -1}-1) \\\\right]+ 1 \\\\right)^{\\\\frac{\\\\gamma }{\\\\gamma -1}} -1 \\\\right\\\\rbrace \\\\nonumber \\\\\\\\& & \\\\hspace{-19.91684pt}= \\\\widetilde{c} \\\\cdot \\\\frac{t^\\\\gamma -\\\\gamma \\\\cdot t+ \\\\gamma - 1}{\\\\gamma \\\\cdot (\\\\gamma -1)},\\\\qquad t \\\\in ]0,\\\\infty [ \\\\, .$ Notice that for all $\\\\gamma \\\\in \\\\mathbb {R}\\\\backslash \\\\lbrace 0,1,2\\\\rbrace $ one has $\\\\varphi _{\\\\gamma ,\\\\widetilde{c}}(1) = 0$ , $\\\\varphi _{\\\\gamma ,\\\\widetilde{c}}^{\\\\prime }(1) = 0$ and $\\\\varphi _{\\\\gamma ,\\\\widetilde{c}}(\\\\infty ) = \\\\infty $ .\",\n \"Moreover, for $\\\\gamma \\\\in \\\\, ]1,2[ \\\\, \\\\cup \\\\, ]2,\\\\infty [$ one has $\\\\varphi _{\\\\gamma ,\\\\widetilde{c}}^{\\\\prime }(0) = -\\\\frac{\\\\widetilde{c}}{\\\\gamma -1} < 0$ and $\\\\varphi _{\\\\gamma ,\\\\widetilde{c}}^{\\\\prime }(\\\\infty ) = \\\\infty $ .\",\n \"In contrast, for $\\\\gamma < 0$ and $\\\\gamma \\\\in \\\\, ]0,1[$ one gets $\\\\varphi _{\\\\gamma ,\\\\widetilde{c}}^{\\\\prime }(0) = - \\\\infty $ and $\\\\varphi _{\\\\gamma ,\\\\widetilde{c}}^{\\\\prime }(\\\\infty ) = \\\\frac{\\\\widetilde{c}}{1-\\\\gamma } > 0$ .\",\n \"(b) For $\\\\gamma =2$ , $\\\\widetilde{c} \\\\in ]0,\\\\infty [$ and $]a_{F_{\\\\gamma ,\\\\widetilde{c}}},b_{F_{\\\\gamma ,\\\\widetilde{c}}}[ \\\\, = \\\\, ]-\\\\infty ,\\\\infty [$ we define $F_{2,\\\\widetilde{c}}(t) &:=&\\\\widetilde{c} \\\\cdot (t-1), \\\\qquad t \\\\in \\\\, ]-\\\\infty ,\\\\infty [,\\\\nonumber $ Clearly, $\\\\mathcal {R}(F_{2,\\\\widetilde{c}})= \\\\, ]-\\\\infty , \\\\infty [$ , $0 \\\\in int(\\\\mathcal {R}(F_{2,\\\\widetilde{c}}))$ , and $F_{2,\\\\widetilde{c}}(\\\\cdot )$ is strictly increasing as well as smooth on $]-\\\\infty ,\\\\infty [$ .\",\n \"Hence, $F_{2,\\\\widetilde{c}} \\\\in {F}$ .\",\n \"Since $F_{2,\\\\widetilde{c}}(1)=0$ , let us choose the natural anchor point $c:=0$ , which leads to $]\\\\lambda _{-},\\\\lambda _{+}[= int(\\\\mathcal {R}(F_{2,\\\\widetilde{c}}))= \\\\, ]-\\\\infty , \\\\infty [$ and $]t_{-}^{sc},t_{+}^{sc}[ = \\\\, ]-\\\\infty , \\\\infty [$ .\",\n \"By using $F_{2,\\\\widetilde{c}}^{-1}(x) = 1+\\\\frac{x}{\\\\widetilde{c}}$ for $x \\\\in int(\\\\mathcal {R}(F_{2,\\\\widetilde{c}}))$ , we can derive from formula (REF ) (see also (REF )) $\\\\Lambda _{2,\\\\widetilde{c}}(z) := \\\\Lambda _{2,\\\\widetilde{c}}^{(0)}(z) &=&\\\\frac{\\\\widetilde{c}}{2} \\\\cdot \\\\left\\\\lbrace \\\\left( \\\\frac{1}{\\\\widetilde{c}} \\\\cdot z +1 \\\\right)^{2} -1 \\\\right\\\\rbrace =\\\\frac{z^2}{2 \\\\widetilde{c}} + z,\\\\qquad z \\\\in \\\\, ]-\\\\infty ,\\\\infty [.$ Notice that $\\\\Lambda _{2,\\\\widetilde{c}}(0) = 0$ , $\\\\Lambda _{2,\\\\widetilde{c}}(-\\\\infty ) = \\\\Lambda _{2,\\\\widetilde{c}}(\\\\infty ) = \\\\infty $ , $\\\\Lambda _{2,\\\\widetilde{c}}^{\\\\prime }(-\\\\infty ) = -\\\\infty $ and $\\\\Lambda _{2,\\\\widetilde{c}}^{\\\\prime }(\\\\infty ) = \\\\infty $ .\",\n \"From formula (REF ) (see also ()) we can deduce analogously to (REF ) $\\\\varphi _{2,\\\\widetilde{c}}(t) := \\\\varphi _{2,\\\\widetilde{c}}^{(0)}(t)&=&\\\\widetilde{c} \\\\cdot \\\\frac{(t-1)^{2}}{2}\\\\ \\\\in \\\\ [0,\\\\infty [,\\\\qquad t \\\\in \\\\, ]-\\\\infty ,\\\\infty [,$ which coincides with $\\\\widetilde{c} \\\\cdot \\\\varphi _{2}(t)$ for $\\\\varphi _{2}(t)$ from (REF ) which generates the corresponding power divergence given in the sixth line of (REF ).\",\n \"Notice that $\\\\varphi _{2,\\\\widetilde{c}}(1) = 0$ , $\\\\varphi _{2,\\\\widetilde{c}}^{\\\\prime }(1) = 0$ and $\\\\varphi _{2,\\\\widetilde{c}}(-\\\\infty ) = \\\\varphi _{2,\\\\widetilde{c}}(\\\\infty ) = \\\\infty $ .\",\n \"As an application of the reciprocity considerations of Remark REF , it is straightforward to see from the above-mentioned considerations (a) and (b) that for all $\\\\gamma \\\\in \\\\mathbb {R}\\\\backslash \\\\lbrace 0,1\\\\rbrace $ one has $\\\\widetilde{F}_{\\\\gamma ,\\\\widetilde{c}}(t) =- \\\\Lambda _{\\\\gamma ,\\\\widetilde{c}}(F_{\\\\gamma ,\\\\widetilde{c}}(\\\\frac{1}{t}))= F_{1-\\\\gamma ,\\\\widetilde{c}}(t)$ for all $t \\\\in ]0,\\\\infty [$ .\",\n \"(c) Let us now continue with the remaining case $\\\\gamma =0$ (recall the natural anchor point $c:=0$ ).\",\n \"By using $F_{0,\\\\widetilde{c}}^{-1}(x) =\\\\frac{1}{1- \\\\frac{x}{\\\\widetilde{c}}} $ for $x \\\\in int(\\\\mathcal {R}(F_{0,\\\\widetilde{c}})) = ]-\\\\infty , \\\\widetilde{c}[$ , we can derive from formula (REF ) (see also (REF )) $\\\\Lambda _{0,\\\\widetilde{c}}(z) := \\\\Lambda _{0,\\\\widetilde{c}}^{(0)}(z) &=&{\\\\left\\\\lbrace \\\\begin{array}{ll}- \\\\widetilde{c} \\\\cdot \\\\log \\\\left( 1 - \\\\frac{z}{\\\\widetilde{c}} \\\\right), \\\\qquad \\\\textrm {if } \\\\ z \\\\in \\\\big ]-\\\\infty ,\\\\widetilde{c} \\\\big [, \\\\\\\\\\\\infty , \\\\hspace{78.24507pt} \\\\textrm {if } \\\\ z \\\\in \\\\big [\\\\widetilde{c}, \\\\infty \\\\big [ .\\\\end{array}\\\\right.\",\n \"}$ Notice that $\\\\Lambda _{0,\\\\widetilde{c}}(0) = 0$ , $\\\\Lambda _{0,\\\\widetilde{c}}(-\\\\infty ) = - \\\\infty $ , $\\\\Lambda _{0,\\\\widetilde{c}}^{\\\\prime }(\\\\widetilde{c}) = \\\\infty $ and $\\\\Lambda _{0,\\\\widetilde{c}}^{\\\\prime }(-\\\\infty ) = 0$ .\",\n \"Moreover, from formula (REF ) (see also ()) we can deduce $\\\\varphi _{0,\\\\widetilde{c}}(t) := \\\\varphi _{0,\\\\widetilde{c}}^{(0)}(t)\\\\hspace{-5.69046pt} &=& \\\\hspace{-5.69046pt}{\\\\left\\\\lbrace \\\\begin{array}{ll}\\\\widetilde{c} \\\\cdot \\\\left(- \\\\log t + t -1 \\\\right)\\\\ \\\\in \\\\ [0,\\\\infty [,\\\\qquad \\\\quad \\\\textrm {if }t \\\\in \\\\, ]0,\\\\infty [,\\\\\\\\\\\\infty , \\\\hspace{145.10922pt} \\\\textrm {if } \\\\ t \\\\in \\\\, ]-\\\\infty , 0] ,\\\\end{array}\\\\right.\",\n \"}$ which coincides with $\\\\widetilde{c} \\\\cdot \\\\varphi _{0}(t)$ for the generator $\\\\varphi _{0}(t)$ from (REF ) which generates the reverse Kullback-Leibler divergence (reverse relative entropy) given in (REF ) with $\\\\widetilde{c}=1$ ; the first line in (REF ) can be proved by $& & \\\\hspace{-19.91684pt}\\\\varphi _{0,\\\\widetilde{c}}(t) := \\\\varphi _{0,\\\\widetilde{c}}^{(0)}(t) :=t \\\\cdot F_{0,\\\\widetilde{c}}\\\\left(t\\\\right)-\\\\int \\\\displaylimits _{0}^{F_{0,\\\\widetilde{c}}\\\\left(t\\\\right)} F_{0,\\\\widetilde{c}}^{-1}(u) du\\\\nonumber \\\\\\\\& & \\\\hspace{-19.91684pt}= t \\\\cdot \\\\widetilde{c} \\\\cdot \\\\Big (1-\\\\frac{1}{t}\\\\Big )- (- \\\\widetilde{c}) \\\\cdot \\\\log \\\\left(1-\\\\frac{1}{\\\\widetilde{c}} \\\\cdot \\\\left[ \\\\widetilde{c} \\\\cdot \\\\Big ( 1-\\\\frac{1}{t}\\\\Big ) \\\\right]\\\\right)= \\\\widetilde{c} \\\\cdot \\\\left(- \\\\log t + t -1 \\\\right),\\\\qquad t \\\\in ]0,\\\\infty [ \\\\, .$ Notice that one has $\\\\varphi _{0,\\\\widetilde{c}}(1) = 0$ , $\\\\varphi _{0,\\\\widetilde{c}}(\\\\infty ) = \\\\infty $ , $\\\\varphi _{0,\\\\widetilde{c}}^{\\\\prime }(1) = 0$ , $\\\\varphi _{0,\\\\widetilde{c}}^{\\\\prime }(0) = -\\\\infty $ and $\\\\varphi _{0,\\\\widetilde{c}}^{\\\\prime }(\\\\infty ) = \\\\widetilde{c}$ .\",\n \"Example 40 (a) For the remaining case $\\\\gamma =1$ , $\\\\widetilde{c} \\\\in ]0,\\\\infty [$ and $]a_{F_{1,\\\\widetilde{c}}},b_{F_{1,\\\\widetilde{c}}}[ = ]0,\\\\infty [$ we define $F_{1,\\\\widetilde{c}}(t) &:=&{\\\\left\\\\lbrace \\\\begin{array}{ll}\\\\widetilde{c} \\\\cdot \\\\log t =\\\\lim _{\\\\gamma \\\\rightarrow 1} F_{\\\\gamma ,\\\\widetilde{c}}(t), \\\\qquad \\\\textrm {if } \\\\ t \\\\in \\\\, ]0,\\\\infty [, \\\\\\\\-\\\\infty , \\\\hspace{108.12054pt} \\\\textrm {if } \\\\ t \\\\in ]-\\\\infty ,0].\",\n \"\\\\\\\\\\\\end{array}\\\\right.\",\n \"}\\\\nonumber $ Clearly, $\\\\mathcal {R}(F_{1,\\\\widetilde{c}})= ]-\\\\infty ,\\\\infty [$ .\",\n \"Moreover, $F_{1,\\\\widetilde{c}}(\\\\cdot )$ is strictly increasing and smooth on $]0,\\\\infty [$ , and hence, $F_{\\\\gamma ,\\\\widetilde{c}} \\\\in {F}$ .\",\n \"Since $F_{1,\\\\widetilde{c}}(1)=0$ , let us first choose the natural anchor point $c:=0$ , which leads to $]\\\\lambda _{-},\\\\lambda _{+}[= int(\\\\mathcal {R}(F_{1,\\\\widetilde{c}})) = ]-\\\\infty ,\\\\infty [$ and $]t_{-}^{sc},t_{+}^{sc}[ = ]0,\\\\infty [$ .\",\n \"By using $F_{1,\\\\widetilde{c}}^{-1}(x) = \\\\exp (\\\\frac{x}{\\\\widetilde{c}})$ for $x \\\\in \\\\mathcal {R}(F_{1,\\\\widetilde{c}})$ , we can derive from formula (REF ) (see also (REF )) $& & \\\\hspace{-19.91684pt} \\\\Lambda _{1,\\\\widetilde{c}}(z) := \\\\Lambda _{1,\\\\widetilde{c}}^{(0)}(z) :=\\\\int \\\\displaylimits _{0}^{z} F_{1,\\\\widetilde{c}}^{-1}(u) \\\\, du= \\\\widetilde{c} \\\\cdot \\\\left( \\\\exp \\\\Big (\\\\frac{z}{\\\\widetilde{c}}\\\\Big ) - 1 \\\\right),\\\\ \\\\ z \\\\in ]-\\\\infty ,\\\\infty [ .$ Notice that $\\\\Lambda _{1,\\\\widetilde{c}}(0) = 0$ , $\\\\Lambda _{1,\\\\widetilde{c}}(-\\\\infty ) = - \\\\widetilde{c}$ , $\\\\Lambda _{1,\\\\widetilde{c}}(\\\\infty ) = \\\\infty $ , $\\\\Lambda _{1,\\\\widetilde{c}}^{\\\\prime }(-\\\\infty ) = 0$ and $\\\\Lambda _{0,\\\\widetilde{c}}^{\\\\prime }(\\\\infty ) = \\\\infty $ .\",\n \"Moreover, from formula (REF ) (see also ()) we can deduce $\\\\varphi _{1,\\\\widetilde{c}}(t) := \\\\varphi _{1,\\\\widetilde{c}}^{(0)}(t)\\\\hspace{-5.69046pt} &:=& \\\\hspace{-5.69046pt}{\\\\left\\\\lbrace \\\\begin{array}{ll}\\\\widetilde{c} \\\\cdot \\\\left( t \\\\cdot \\\\log t + 1 - t \\\\right)\\\\ \\\\in \\\\ [0,\\\\infty [,\\\\qquad \\\\quad \\\\textrm {if }t \\\\in \\\\, ]0,\\\\infty [,\\\\\\\\1, \\\\hspace{150.79968pt} \\\\textrm {if } \\\\ t = 0, \\\\\\\\\\\\infty , \\\\hspace{145.10922pt} \\\\textrm {if } \\\\ t \\\\in \\\\, ]-\\\\infty ,0[ ,\\\\end{array}\\\\right.\",\n \"}$ which coincides with $\\\\widetilde{c} \\\\cdot \\\\varphi _{1}(t)$ for the generator $\\\\varphi _{1}(t)$ from (REF ) which generates the Kullback-Leibler divergence (relative entropy) given in (REF ) with $\\\\widetilde{c}=1$ ; the first line in (REF ) can be proved by $& & \\\\hspace{-19.91684pt}\\\\varphi _{1,\\\\widetilde{c}}(t) := \\\\varphi _{1,\\\\widetilde{c}}^{(0)}(t) :=t \\\\cdot F_{1,\\\\widetilde{c}}\\\\left(t\\\\right)-\\\\int \\\\displaylimits _{0}^{F_{1,\\\\widetilde{c}}\\\\left(t\\\\right)} F_{1,\\\\widetilde{c}}^{-1}(u) du\\\\nonumber \\\\\\\\& & \\\\hspace{-19.91684pt}= t \\\\cdot \\\\widetilde{c} \\\\cdot \\\\log t- \\\\widetilde{c} \\\\cdot \\\\left( \\\\exp \\\\left(\\\\frac{1}{\\\\widetilde{c}} \\\\cdot \\\\left[ \\\\widetilde{c} \\\\cdot \\\\log t \\\\right]\\\\right) - 1 \\\\right)= \\\\widetilde{c} \\\\cdot \\\\left( t \\\\cdot \\\\log t + 1 - t \\\\right),\\\\qquad t \\\\in ]0,\\\\infty [ \\\\, ,$ Notice that one has $\\\\varphi _{1,\\\\widetilde{c}}(1) = 0$ , $\\\\varphi _{1,\\\\widetilde{c}}(\\\\infty ) = \\\\infty $ , $\\\\varphi _{1,\\\\widetilde{c}}^{\\\\prime }(1) = 0$ , $\\\\varphi _{1,\\\\widetilde{c}}^{\\\\prime }(0) = -\\\\infty $ and $\\\\varphi _{1,\\\\widetilde{c}}^{\\\\prime }(\\\\infty ) = \\\\infty $ .\",\n \"As an application of the reciprocity considerations of Remark REF , it is straightforward to see that $\\\\widetilde{F}_{1,\\\\widetilde{c}}(t) =- \\\\Lambda _{1,\\\\widetilde{c}}(F_{1,\\\\widetilde{c}}(\\\\frac{1}{t}))= F_{0,\\\\widetilde{c}}(t)$ for all $t \\\\in ]0,\\\\infty [$ .\",\n \"(b) For the choice $\\\\widetilde{c}=1$ , let us now fix a general anchor point $c \\\\in \\\\mathcal {R}(F_{1,\\\\widetilde{c}})= ]-\\\\infty ,\\\\infty [$ (rather than $c=0$ ), which leads to $]\\\\lambda _{-},\\\\lambda _{+}[= int(\\\\mathcal {R}(F_{1,1}))- c = ]-\\\\infty ,\\\\infty [$ and $]t_{-}^{sc},t_{+}^{sc}[ = ]1+a_{F_{1,1}}-F_{1,1}^{-1}(c),1+b_{F_{1,1}}-F_{1,1}^{-1}(c)[ \\\\,= \\\\, ]1-e^{c},\\\\infty [$ .\",\n \"Accordingly, the formula (REF ) (see also (REF )) leads to $& & \\\\hspace{-19.91684pt} \\\\Lambda _{1,1}(z) := \\\\Lambda _{1,1}^{(c)}(z) :=\\\\int \\\\displaylimits _{0}^{z} F_{1,1}^{-1}(u+c) du+ z \\\\cdot (1-F_{1,1}^{-1}(c))\\\\nonumber \\\\\\\\& & \\\\hspace{59.75095pt}= e^{c} \\\\cdot \\\\left( e^{z} - 1 \\\\right) + z \\\\cdot (1- e^{c}),\\\\qquad z \\\\in ]-\\\\infty ,\\\\infty [, \\\\qquad \\\\ $ for which there holds $\\\\Lambda _{1,1}^{(c)}(0) = 0$ , $\\\\Lambda _{1,1}^{(c)}(-\\\\infty ) =\\\\infty \\\\cdot {1}_{]0,\\\\infty [}(c)- \\\\infty \\\\cdot {1}_{]-\\\\infty ,0[}(c)- 1 \\\\cdot {1}_{\\\\lbrace 0\\\\rbrace }(c)$ , $\\\\Lambda _{1,1}^{(c)}(\\\\infty ) = \\\\infty $ , $\\\\Lambda _{1,1}^{(c) \\\\prime }(-\\\\infty ) = 1- e^{c}$ and $\\\\Lambda _{1,1}^{(c) \\\\prime }(\\\\infty ) = \\\\infty $ .\",\n \"Moreover, from formula (REF ) (see also ()) we can deduce $\\\\varphi _{1,1}(t) := \\\\varphi _{1,1}^{(c)}(t)\\\\hspace{-5.69046pt} &:=& \\\\hspace{-5.69046pt}{\\\\left\\\\lbrace \\\\begin{array}{ll}(t + e^{c} -1) \\\\cdot [\\\\log (t + e^{c} -1) -c] + 1 - t\\\\ \\\\in \\\\ [0,\\\\infty [,\\\\qquad \\\\quad \\\\textrm {if }t \\\\in \\\\, ]1-e^{c},\\\\infty [,\\\\\\\\e^{c}, \\\\hspace{240.42569pt} \\\\textrm {if } \\\\ t = 1-e^{c}, \\\\\\\\\\\\infty , \\\\hspace{239.00298pt} \\\\textrm {if } \\\\ t \\\\in \\\\, ]-\\\\infty ,1-e^{c}[ ;\\\\end{array}\\\\right.\",\n \"}$ the first line in (REF ) can be proved by $& & \\\\hspace{-19.91684pt}\\\\varphi _{1,1}(t) := \\\\varphi _{1,1}^{(c)}(t) :=(t+F_{1,1}^{-1}(c)-1) \\\\cdot [ F_{1,1}\\\\left(t+F_{1,1}^{-1}(c)-1 \\\\right) - c ]-\\\\int \\\\displaylimits _{0}^{F_{1,1}\\\\left(t+F_{1,1}^{-1}(c)-1 \\\\right) - c}F_{1,1}^{-1}(u+c) du\\\\nonumber \\\\\\\\& & \\\\hspace{-19.91684pt}= (t + e^{c} -1) \\\\cdot [\\\\log (t + e^{c} -1) -c]- e^{c} \\\\cdot \\\\Big \\\\lbrace \\\\exp [\\\\log (t + e^{c} -1) -c] - 1 \\\\Big \\\\rbrace \\\\nonumber \\\\\\\\& & \\\\hspace{-19.91684pt}= (t + e^{c} -1) \\\\cdot [\\\\log (t + e^{c} -1) -c] + 1 - t ,\\\\qquad t \\\\in ]1-e^{c},\\\\infty [\\\\, .$ Clearly, one has $\\\\varphi _{1,1}^{(c)}(1) = 0$ , $\\\\varphi _{1,1}^{(c)}(\\\\infty ) = \\\\infty $ , $\\\\varphi _{1,1}^{(c) \\\\prime }(1) = 0$ , $\\\\varphi _{1,1}^{(c) \\\\prime }(1-e^{c}) = -\\\\infty $ and $\\\\varphi _{1,1}^{(c) \\\\prime }(\\\\infty ) = \\\\infty $ .\",\n \"The corresponding divergence $D_{\\\\varphi _{1,1}^{(c)}}(\\\\mathbb {Q},\\\\mathbb {P})$ has been recently used in Broniatowski et al.\",\n \"[63] for the important task of testing mixtures of probability distributions; in fact, in order to get considerable comfort in testing mixture-type hypotheses against corresponding marginal-type alternatives, they employ choices $c>0$ since then $\\\\varphi _{1,1}^{(c)}(t)$ is finite especially for some range of negative values $t<0$ .\",\n \"The latter feature is also valid for the divergence generator $\\\\varphi _{bw,\\\\beta ,\\\\widetilde{c}}$ in the next example (cf.\",\n \"(REF ) below).\",\n \"Example 41 For $\\\\beta \\\\in \\\\, ]0,1]$ , $\\\\widetilde{c} \\\\in \\\\, ]0,\\\\infty [$ and $]a_{F_{bw,\\\\beta ,\\\\widetilde{c}}},b_{F_{bw,\\\\beta ,\\\\widetilde{c}}}[ \\\\,= \\\\, ]1-\\\\frac{1}{\\\\beta },\\\\infty [$ we define $F_{bw,\\\\beta ,\\\\widetilde{c}}(t) &:=&{\\\\left\\\\lbrace \\\\begin{array}{ll}\\\\frac{\\\\widetilde{c}}{2\\\\beta } \\\\cdot \\\\Big (1-\\\\frac{1}{(\\\\beta \\\\cdot t + 1 - \\\\beta )^2}\\\\Big ), \\\\qquad \\\\textrm {if } \\\\ t \\\\in \\\\, ]1-\\\\frac{1}{\\\\beta },\\\\infty [, \\\\\\\\- \\\\infty , \\\\hspace{93.89418pt} \\\\textrm {if } \\\\ t \\\\in \\\\, ]-\\\\infty ,1-\\\\frac{1}{\\\\beta }].\\\\end{array}\\\\right.\",\n \"}\\\\nonumber $ Clearly, $\\\\mathcal {R}(F_{bw,\\\\beta ,\\\\widetilde{c}})= \\\\big ]-\\\\infty ,\\\\frac{\\\\widetilde{c}}{2\\\\beta }\\\\big [$ and $0 \\\\in int(\\\\mathcal {R}(F_{bw,\\\\beta ,\\\\widetilde{c}}))$ .\",\n \"Moreover, $F_{bw,\\\\beta ,\\\\widetilde{c}}(\\\\cdot )$ is strictly increasing and smooth on $]1-\\\\frac{1}{\\\\beta },\\\\infty [$ , and thus, $F_{bw,\\\\beta ,\\\\widetilde{c}} \\\\in {F}$ .\",\n \"Since $F_{bw,\\\\beta ,\\\\widetilde{c}}(1)=0$ , let us choose the natural anchor point $c:=0$ , which leads to $]\\\\lambda _{-},\\\\lambda _{+}[ \\\\, = int(\\\\mathcal {R}(F_{bw,\\\\beta ,\\\\widetilde{c}}))= \\\\big ]-\\\\infty ,\\\\frac{\\\\widetilde{c}}{2\\\\beta }\\\\big [$ and $]t_{-}^{sc},t_{+}^{sc}[ \\\\, = \\\\,]a_{F_{bw,\\\\beta ,\\\\widetilde{c}}},b_{F_{bw,\\\\beta ,\\\\widetilde{c}}}[\\\\, = \\\\, ]1-\\\\frac{1}{\\\\beta },\\\\infty [$ .\",\n \"By using $F_{bw,\\\\beta ,\\\\widetilde{c}}^{-1}(x) =\\\\frac{1}{\\\\beta } \\\\cdot \\\\Big \\\\lbrace \\\\frac{1}{\\\\sqrt{1-2\\\\beta \\\\cdot x/\\\\widetilde{c}}} + \\\\beta -1 \\\\Big \\\\rbrace $ for $x \\\\in int(\\\\mathcal {R}(F_{bw,\\\\beta ,\\\\widetilde{c}}))$ , we can derive from formula (REF ) (see also (REF )) for all $\\\\beta \\\\in \\\\, ]0,1]$ and $z \\\\in \\\\mathbb {R}$ $\\\\Lambda _{bw,\\\\beta ,\\\\widetilde{c}}(z) :=\\\\Lambda _{bw,\\\\beta ,\\\\widetilde{c}}^{(0)}(z) &=&{\\\\left\\\\lbrace \\\\begin{array}{ll}-(\\\\frac{1}{\\\\beta }-1) \\\\cdot z + \\\\frac{\\\\widetilde{c}}{\\\\beta ^{2}} \\\\cdot \\\\Big \\\\lbrace 1 - \\\\sqrt{1-\\\\frac{2\\\\beta }{\\\\widetilde{c}} \\\\cdot z} \\\\ \\\\Big \\\\rbrace ,\\\\qquad \\\\textrm {if } z \\\\in \\\\big ]-\\\\infty ,\\\\frac{\\\\widetilde{c}}{2\\\\beta } \\\\big ], \\\\\\\\\\\\infty , \\\\hspace{176.407pt} \\\\textrm {else} .\\\\end{array}\\\\right.\",\n \"}$ Notice that $\\\\Lambda _{bw,\\\\beta ,\\\\widetilde{c}}(0) = 0$ .\",\n \"Moreover, $\\\\Lambda _{bw,\\\\beta ,\\\\widetilde{c}}(-\\\\infty ) = \\\\infty $ , $\\\\Lambda _{bw,\\\\beta ,\\\\widetilde{c}}(\\\\frac{\\\\widetilde{c}}{2\\\\beta })= \\\\frac{\\\\widetilde{c} \\\\cdot (\\\\beta +1)}{2 \\\\beta ^{2}}$ , $\\\\Lambda _{bw,\\\\beta ,\\\\widetilde{c}}^{\\\\prime }(-\\\\infty ) = - \\\\frac{1-\\\\beta }{\\\\beta } <0$ and $\\\\Lambda _{bw,\\\\beta ,\\\\widetilde{c}}^{\\\\prime }(\\\\frac{\\\\widetilde{c}}{2\\\\beta }) = \\\\infty $ .\",\n \"Furthermore, from formula (REF ) (see also ()) we can straightforwardly deduce for all $t \\\\in \\\\mathbb {R}$ $\\\\varphi _{bw,\\\\beta ,\\\\widetilde{c}}(t) := \\\\varphi _{bw,\\\\beta ,\\\\widetilde{c}}^{(0)}(t)\\\\hspace{-5.69046pt} &:=& \\\\hspace{-5.69046pt}{\\\\left\\\\lbrace \\\\begin{array}{ll}\\\\widetilde{c} \\\\cdot \\\\frac{(t-1)^{2}}{2(\\\\beta \\\\cdot t +1 -\\\\beta )}\\\\ \\\\in \\\\ [0,\\\\infty [,\\\\qquad \\\\quad \\\\textrm {if }t \\\\in \\\\, ]1-\\\\frac{1}{\\\\beta },\\\\infty [,\\\\\\\\\\\\infty , \\\\hspace{120.92421pt} \\\\textrm {if } \\\\ t \\\\in \\\\, ]-\\\\infty ,1-\\\\frac{1}{\\\\beta }] .\\\\end{array}\\\\right.\",\n \"}$ Note that $1-\\\\frac{1}{\\\\beta } <0$ so that negative $t$ are allowed here.\",\n \"For $t\\\\ge 0$ , $\\\\varphi _{bw,\\\\beta ,\\\\widetilde{c}}(t)$ is known as Rukhin's generator (cf.\",\n \"[312], see e.g.\",\n \"also Marhuenda et al.\",\n \"[247], Pardo [282]).\",\n \"Obviously, one has $\\\\varphi _{bw,\\\\beta ,\\\\widetilde{c}}(1) = 0$ , $\\\\varphi _{bw,\\\\beta ,\\\\widetilde{c}}^{\\\\prime }(1) = 0$ , $\\\\varphi _{bw,\\\\beta ,\\\\widetilde{c}}^{\\\\prime }(1-\\\\frac{1}{\\\\beta }) = -\\\\infty $ and $\\\\varphi _{bw,\\\\beta ,\\\\widetilde{c}}^{\\\\prime }(\\\\infty ) = \\\\frac{\\\\widetilde{c}}{2\\\\beta }$ .\",\n \"From the generator $\\\\varphi _{bw,\\\\beta ,\\\\widetilde{c}}$ given in (REF ), we build the corresponding divergence (cf.\",\n \"(REF )) $& &\\\\hspace{-45.52458pt}D_{\\\\varphi _{bw,\\\\beta ,\\\\widetilde{c}}}(\\\\mathbf {Q},\\\\mathbf {P})= \\\\widetilde{c} \\\\cdot \\\\sum _{k=1}^{K} p_{k} \\\\cdot \\\\frac{(\\\\frac{q_{k}}{p_{k}}-1)^{2}}{2(\\\\beta \\\\cdot \\\\frac{q_{k}}{p_{k}} +1 -\\\\beta )}\\\\nonumber \\\\\\\\& & \\\\hspace{-45.52458pt}= \\\\frac{\\\\widetilde{c}}{2} \\\\cdot \\\\sum \\\\limits _{k=1}^{K}\\\\frac{(q_{k}-p_{k})^{2}}{\\\\beta \\\\cdot q_{k} + (1 -\\\\beta )\\\\cdot p_{k}},\\\\hspace{42.67912pt}\\\\textrm {if \\\\mathbf {P} \\\\in \\\\mathbb {R}^{K} and \\\\mathbf {Q} \\\\in \\\\mathbb {R}^{K}with \\\\mathbf {Q} \\\\in \\\\, ] \\\\, \\\\mathbf {P} \\\\cdot (1-\\\\frac{1}{\\\\beta }),\\\\infty [ component-wise;}$ for the special subcase $\\\\widetilde{c}=1$ and $\\\\mathbf {Q} \\\\in \\\\mathbb {R}_{> 0}^{K}$ , $D_{\\\\varphi _{bw,\\\\beta ,1}}(\\\\mathbf {Q},\\\\mathbf {P})$ can be interpreted as — “non-probability version” of — the well-known blended weight chi-square divergence of Lindsay [221] (see e.g.\",\n \"also Basu & Lindsay [35], Györfy & Vajda [148], Basu et al.\",\n \"[36]).\",\n \"The special case $\\\\widetilde{c}=1$ and $\\\\beta =\\\\frac{1}{2}$ for probability vectors, i.e.\",\n \"$D_{\\\\varphi _{bw,1/2,1}}({Q},{P})$ , is equal to (a multiple of the matrix-vector-converted (cf.\",\n \"Remark REF )) Sanghvi’s genetic difference measure [316] and equal to the double of the so-called (squared) Vincze-Le Cam distance (cf.\",\n \"Vincze [384], Le Cam [212], see also e.g.\",\n \"Topsoe [360] — who used the alternative naming triangular discrimination — and Vajda [373]); this divergence $D_{\\\\varphi _{bw,1/2,1}}({Q},{P})$ has been used e.g.\",\n \"in Liu et al.\",\n \"[227] for a machine learning context of detecting salient objects, where ${Q}$ and ${P}$ are appropriate histograms of RGB color.\",\n \"Remark 42 (a) By straightforward calculations, one can show that $\\\\varphi _{bw,1,\\\\widetilde{c}}$ (i.e.\",\n \"with the choice $\\\\beta =1$ ) is equal to the $\\\\widetilde{c}-fold$ power-divergence generator $\\\\varphi _{\\\\gamma ,\\\\widetilde{c}} = \\\\widetilde{c} \\\\cdot \\\\varphi _{\\\\gamma }$ (cf.\",\n \"(REF )) with $\\\\gamma =-1$ ; the corresponding divergence $D_{\\\\varphi _{bw,1,\\\\widetilde{c}}}(\\\\mathbf {Q},\\\\mathbf {P})$ is thus equal to the power divergence $D_{\\\\widetilde{c} \\\\cdot \\\\varphi _{-1}}(\\\\mathbf {Q},\\\\mathbf {P})$ (cf.\",\n \"(REF )) which is nothing but the — “non-probability version” — of Neyman's chi-square divergence.\",\n \"(b) For the case $\\\\beta =0$ — which has been excluded in Example REF for technical brevity — the divergence generator $\\\\varphi _{bw,0,\\\\widetilde{c}}$ corresponds to $\\\\widetilde{c}-fold$ power-divergence generator $\\\\varphi _{\\\\gamma ,\\\\widetilde{c}}$ with $\\\\gamma =2$ ; the corresponding divergence $D_{\\\\varphi _{bw,0,\\\\widetilde{c}}}(\\\\mathbf {Q},\\\\mathbf {P})$ is thus equal to the power divergence $D_{\\\\widetilde{c} \\\\cdot \\\\varphi _{2}}(\\\\mathbf {Q},\\\\mathbf {P})$ (cf.\",\n \"(REF )) which is nothing but the — “non-probability version” — of Pearson's (i.e.\",\n \"the classical) chi-square divergence.\",\n \"Example 43 Let us give an interesting generalization of the Kullback-Leibler case of Example REF (a).\",\n \"For $\\\\widetilde{c} >0$ and $\\\\alpha \\\\in \\\\, ]-1,0[ \\\\ \\\\cup \\\\ ]0,\\\\infty [$ let us define $F_{gKL,\\\\alpha ,\\\\widetilde{c}}(t) &:=&{\\\\left\\\\lbrace \\\\begin{array}{ll}\\\\widetilde{c} \\\\cdot \\\\log \\\\left( \\\\frac{(1+\\\\alpha ) \\\\cdot t}{1+ \\\\alpha \\\\cdot t} \\\\right), \\\\qquad \\\\textrm {if \\\\lbrace \\\\alpha \\\\in \\\\, ]0,\\\\infty [ and t \\\\in \\\\, ]0,\\\\infty [ \\\\rbrace or \\\\lbrace \\\\alpha \\\\in \\\\, ]-1,0[ and t \\\\in \\\\, ]0,-\\\\frac{1}{\\\\alpha }[ \\\\rbrace ,} \\\\\\\\- \\\\infty , \\\\hspace{71.13188pt} \\\\textrm {if \\\\alpha \\\\in \\\\, ]-1,0[ \\\\ \\\\cup \\\\ ]0,\\\\infty [ andt \\\\in \\\\, ]-\\\\infty ,0],} \\\\\\\\\\\\infty , \\\\hspace{78.52945pt} \\\\textrm {if \\\\alpha \\\\in \\\\, ]-1,0[ and t \\\\in \\\\, [-\\\\frac{1}{\\\\alpha },\\\\infty [,}\\\\end{array}\\\\right.\",\n \"}\\\\nonumber $ (notice that $\\\\lim _{\\\\alpha \\\\rightarrow 0_{+}} F_{gKL,\\\\alpha ,\\\\widetilde{c}}(t)= F_{1,\\\\widetilde{c}}(t)$ , cf.\",\n \"Example REF (a)).\",\n \"Clearly, $]a_{F_{gKL,\\\\alpha ,\\\\widetilde{c}}},b_{F_{gKL,\\\\alpha ,\\\\widetilde{c}}}[ \\\\, := \\\\, ]0,\\\\infty [$ for $\\\\alpha \\\\in \\\\, ]0,\\\\infty [$ and $]a_{F_{gKL,\\\\alpha ,\\\\widetilde{c}}},b_{F_{gKL,\\\\alpha ,\\\\widetilde{c}}}[ \\\\, := \\\\, ]0,-\\\\frac{1}{\\\\alpha }[$ for $\\\\alpha \\\\in \\\\, ]-1,0[$ .\",\n \"Moreover, $\\\\mathcal {R}(F_{gKL,\\\\alpha ,\\\\widetilde{c}})=\\\\, ]-\\\\infty , \\\\widetilde{c} \\\\cdot \\\\log (1+ \\\\frac{1}{\\\\alpha })[$ for $\\\\alpha \\\\in \\\\, ]0,\\\\infty [$ and $\\\\mathcal {R}(F_{gKL,\\\\alpha ,\\\\widetilde{c}})= \\\\, ]-\\\\infty ,\\\\infty [$ for $\\\\alpha \\\\in \\\\, ]-1,0[$ .\",\n \"Furthermore, $F_{gKL,\\\\alpha ,\\\\widetilde{c}}(\\\\cdot )$ is strictly increasing and smooth on the respective $]a_{F_{gKL,\\\\alpha ,\\\\widetilde{c}}},b_{F_{gKL,\\\\alpha ,\\\\widetilde{c}}}[$ , and thus, $F_{gKL,\\\\alpha ,\\\\widetilde{c}} \\\\in {F}$ .\",\n \"Since $F_{gKL,\\\\alpha ,\\\\widetilde{c}}(1)=0$ , let us choose the natural anchor point $c:=0$ , which leads to $]\\\\lambda _{-},\\\\lambda _{+}[ \\\\, = int(\\\\mathcal {R}(F_{gKL,\\\\alpha ,\\\\widetilde{c}})) = \\\\,]-\\\\infty , \\\\widetilde{c} \\\\cdot \\\\log (1+ \\\\frac{1}{\\\\alpha })[$ and $]t_{-}^{sc},t_{+}^{sc}[ \\\\, = \\\\, ]0,\\\\infty [$ for the case $\\\\alpha \\\\in \\\\, ]0,\\\\infty [$ , respectively, to $]\\\\lambda _{-},\\\\lambda _{+}[ \\\\, = int(\\\\mathcal {R}(F_{gKL,\\\\alpha ,\\\\widetilde{c}})) = \\\\, ]-\\\\infty ,\\\\infty [$ and $]t_{-}^{sc},t_{+}^{sc}[ \\\\, = \\\\, ]0,-\\\\frac{1}{\\\\alpha }[$ for the case $\\\\alpha \\\\in \\\\, ]-1,0[$ .\",\n \"By employing $F_{gKL,\\\\alpha ,\\\\widetilde{c}}^{-1}(x) =\\\\frac{1}{(1+\\\\alpha ) \\\\cdot e^{-x/\\\\widetilde{c}} - \\\\alpha }$ for $x \\\\in ]\\\\lambda _{-},\\\\lambda _{+}[$ , one can deduce from formula (REF ) (see also (REF )) $& & \\\\Lambda _{gKL,\\\\alpha ,\\\\widetilde{c}}(z) := \\\\Lambda _{gKL,\\\\alpha ,\\\\widetilde{c}}^{(0)}(z)\\\\nonumber \\\\\\\\& & :={\\\\left\\\\lbrace \\\\begin{array}{ll}\\\\int \\\\displaylimits _{0}^{z} F_{gKL,\\\\alpha ,\\\\widetilde{c}}^{-1}(u) \\\\, du= - \\\\frac{\\\\widetilde{c}}{\\\\alpha } \\\\cdot \\\\log ((1+\\\\alpha ) - \\\\alpha \\\\cdot e^{z/\\\\widetilde{c}}),\\\\qquad \\\\textrm {if \\\\alpha \\\\in \\\\, ]0,\\\\infty [ and z \\\\in \\\\, ]-\\\\infty , \\\\widetilde{c} \\\\cdot \\\\log (1+ \\\\frac{1}{\\\\alpha })[}, \\\\\\\\\\\\int \\\\displaylimits _{0}^{z} F_{gKL,\\\\alpha ,\\\\widetilde{c}}^{-1}(u) \\\\, du= - \\\\frac{\\\\widetilde{c}}{\\\\alpha } \\\\cdot \\\\log ((1+\\\\alpha ) - \\\\alpha \\\\cdot e^{z/\\\\widetilde{c}}),\\\\qquad \\\\textrm {if \\\\alpha \\\\in \\\\, ]-1,0[ and z \\\\in \\\\, ]-\\\\infty , \\\\infty [}, \\\\\\\\\\\\infty , \\\\hspace{217.6634pt}\\\\textrm {if \\\\alpha \\\\in \\\\, ]0,\\\\infty [ and z \\\\in \\\\, [\\\\widetilde{c} \\\\cdot \\\\log (1+ \\\\frac{1}{\\\\alpha }), \\\\infty [}, \\\\\\\\\\\\end{array}\\\\right.\",\n \"}$ for which there holds $\\\\Lambda _{gKL,\\\\alpha ,\\\\widetilde{c}}(0) = 0$ and $\\\\Lambda _{gKL,\\\\alpha ,\\\\widetilde{c}}(-\\\\infty ) = - \\\\frac{\\\\widetilde{c}}{\\\\alpha } \\\\cdot \\\\log (1+\\\\alpha )$ for $\\\\alpha \\\\in \\\\, ]-1,0[ \\\\ \\\\cup \\\\ ]0,\\\\infty [$ , as well as $\\\\Lambda _{gKL,\\\\alpha ,\\\\widetilde{c}}(\\\\widetilde{c} \\\\cdot \\\\log (1+ \\\\frac{1}{\\\\alpha })) = \\\\infty $ for $\\\\alpha \\\\in \\\\, ]0,\\\\infty [$ and $\\\\Lambda _{gKL,\\\\alpha ,\\\\widetilde{c}}(\\\\infty ) = \\\\infty $ for $\\\\alpha \\\\in \\\\, ]-1,0[$ .\",\n \"The corresponding derivative satisfies $\\\\Lambda _{gKL,\\\\alpha ,\\\\widetilde{c}}^{\\\\prime }(- \\\\infty ) = 0$ for $\\\\alpha \\\\in \\\\, ]-1,0[ \\\\ \\\\cup \\\\ ]0,\\\\infty [$ , as well as $\\\\Lambda _{gKL,\\\\alpha ,\\\\widetilde{c}}^{\\\\prime }(\\\\widetilde{c}\\\\cdot \\\\log (1+ \\\\frac{1}{\\\\alpha })) = \\\\infty $ for $\\\\alpha \\\\in \\\\, ]0,\\\\infty [$ and $\\\\Lambda _{gKL,\\\\alpha ,\\\\widetilde{c}}^{\\\\prime }(\\\\infty ) = - \\\\frac{1}{\\\\alpha }$ for $\\\\alpha \\\\in \\\\, ]-1,0[$ .\",\n \"Furthermore, from formula (REF ) (see also ()) one can derive $& & \\\\hspace{-19.91684pt} \\\\varphi _{gKL,\\\\alpha ,\\\\widetilde{c}}(t) := \\\\varphi _{gKL,\\\\alpha ,\\\\widetilde{c}}^{(0)}(t)\\\\nonumber \\\\\\\\& & \\\\hspace{-19.91684pt} :={\\\\left\\\\lbrace \\\\begin{array}{ll}\\\\widetilde{c} \\\\cdot \\\\left[ \\\\, t \\\\cdot \\\\log t + (t+\\\\frac{1}{\\\\alpha }) \\\\cdot \\\\log \\\\Big ( \\\\frac{1+\\\\alpha }{1+\\\\alpha \\\\cdot t} \\\\Big ) \\\\, \\\\right]\\\\ \\\\in \\\\ [0,\\\\infty [,\\\\quad \\\\textrm {if \\\\lbrace \\\\alpha \\\\in \\\\, ]0,\\\\infty [ and t \\\\in \\\\, ]0,\\\\infty [ \\\\rbrace or \\\\lbrace \\\\alpha \\\\in \\\\, ]-1,0[ and t \\\\in \\\\, ]0,-\\\\frac{1}{\\\\alpha }[ \\\\rbrace ,}\\\\\\\\\\\\frac{\\\\widetilde{c}}{\\\\alpha } \\\\cdot \\\\log (1+\\\\alpha )\\\\ \\\\in \\\\ ]0,\\\\infty [, \\\\hspace{106.69783pt}\\\\textrm {if \\\\alpha \\\\in \\\\, ]-1,0[ \\\\ \\\\cup \\\\ ]0,\\\\infty [ andt =0,} \\\\\\\\\\\\infty , \\\\hspace{197.74655pt}\\\\textrm {if \\\\alpha \\\\in \\\\, ]-1,0[ \\\\ \\\\cup \\\\ ]0,\\\\infty [ andt \\\\in \\\\, ]-\\\\infty ,0[,} \\\\\\\\\\\\infty , \\\\hspace{197.74655pt}\\\\textrm {if \\\\alpha \\\\in \\\\, ]-1,0[ andt \\\\in \\\\, [-\\\\frac{1}{\\\\alpha },\\\\infty [;}\\\\end{array}\\\\right.\",\n \"}$ the first line in (REF ) can be proved by $& & \\\\hspace{-19.91684pt}\\\\varphi _{gKL,\\\\alpha ,\\\\widetilde{c}}(t) := \\\\varphi _{gKL,\\\\alpha ,\\\\widetilde{c}}^{(0)}(t) :=t \\\\cdot F_{gKL,\\\\alpha ,\\\\widetilde{c}}\\\\left(t\\\\right)-\\\\int \\\\displaylimits _{0}^{F_{gKL,\\\\alpha ,\\\\widetilde{c}}\\\\left(t\\\\right)}F_{gKL,\\\\alpha ,\\\\widetilde{c}}^{-1}(u) \\\\, du\\\\nonumber \\\\\\\\& & \\\\hspace{-19.91684pt}= \\\\widetilde{c} \\\\cdot t \\\\cdot \\\\log \\\\left( \\\\frac{(1+\\\\alpha ) \\\\cdot t}{1+ \\\\alpha \\\\cdot t} \\\\right)+ \\\\frac{\\\\widetilde{c}}{\\\\alpha } \\\\cdot \\\\log \\\\left( (1 +\\\\alpha ) -\\\\alpha \\\\cdot \\\\exp \\\\left[ \\\\log \\\\left( \\\\frac{(1+ \\\\alpha ) \\\\cdot t}{1+ \\\\alpha \\\\cdot t} \\\\right) \\\\right] \\\\right)\\\\nonumber \\\\\\\\& & \\\\hspace{-19.91684pt}= \\\\widetilde{c} \\\\cdot \\\\left[ \\\\, t \\\\cdot \\\\log t+ t \\\\cdot \\\\log \\\\Big ( \\\\frac{1+\\\\alpha }{1+ \\\\alpha \\\\cdot t} \\\\Big )+ \\\\frac{1}{\\\\alpha } \\\\cdot \\\\log \\\\Big ( \\\\frac{1+\\\\alpha }{1+ \\\\alpha \\\\cdot t} \\\\Big ) \\\\, \\\\right].$ Obviously, one has $\\\\varphi _{gKL,\\\\alpha ,\\\\widetilde{c}}(1) = 0$ , $\\\\varphi _{gKL,\\\\alpha ,\\\\widetilde{c}}^{\\\\prime }(1) = 0$ , $\\\\varphi _{gKL,\\\\alpha ,\\\\widetilde{c}}^{\\\\prime }(0) = -\\\\infty $ for $\\\\alpha \\\\in \\\\, ]-1,0[ \\\\ \\\\cup \\\\ ]0,\\\\infty [$ .\",\n \"Moreover, for $\\\\alpha \\\\in \\\\, ]0,\\\\infty [$ there holds $\\\\varphi _{gKL,\\\\alpha ,\\\\widetilde{c}}(\\\\infty ) = \\\\infty $ , and $\\\\varphi _{gKL,\\\\alpha ,\\\\widetilde{c}}^{\\\\prime }(\\\\infty ) = \\\\widetilde{c} \\\\cdot \\\\log (1+\\\\frac{1}{\\\\alpha })$ , whereas for $\\\\alpha \\\\in \\\\, ]-1,0[$ we obtain $\\\\varphi _{gKL,\\\\alpha ,\\\\widetilde{c}}(-\\\\frac{1}{\\\\alpha }) = \\\\infty $ , and $\\\\varphi _{gKL,\\\\alpha ,\\\\widetilde{c}}^{\\\\prime }(-\\\\frac{1}{\\\\alpha }) = \\\\infty $ .\",\n \"From the generator $\\\\varphi _{gKL,\\\\alpha ,\\\\widetilde{c}}$ given in (REF ), we build the corresponding divergence (cf.\",\n \"(REF )) $& &\\\\hspace{-42.67912pt}D_{\\\\varphi _{gKL,\\\\alpha ,\\\\widetilde{c}}}(\\\\mathbf {Q},\\\\mathbf {P})= \\\\widetilde{c} \\\\cdot \\\\Big \\\\lbrace \\\\sum \\\\limits _{k=1}^{K}q_{k} \\\\cdot \\\\log \\\\Big (\\\\frac{q_{k}}{(1-\\\\frac{1}{1+\\\\alpha }) \\\\cdot q_{k} + \\\\frac{1}{1+\\\\alpha } \\\\cdot p_{k}} \\\\Big )+ \\\\frac{1}{\\\\alpha } \\\\cdot \\\\sum \\\\limits _{k=1}^{K} p_{k} \\\\cdot \\\\log \\\\Big (\\\\frac{p_{k}}{(1-\\\\frac{1}{1+\\\\alpha }) \\\\cdot q_{k} + \\\\frac{1}{1+\\\\alpha } \\\\cdot p_{k}} \\\\Big ) \\\\Big \\\\rbrace ,\\\\\\\\& & \\\\textrm {if \\\\lbrace \\\\alpha \\\\in \\\\, ]0,\\\\infty [,\\\\mathbf {P} \\\\in \\\\mathbb {R}_{> 0}^{K} and\\\\mathbf {Q} \\\\in \\\\mathbb {R}_{\\\\ge 0}^{K} \\\\rbrace or \\\\lbrace \\\\alpha \\\\in \\\\, ]-1,0[,\\\\mathbf {P} \\\\in \\\\mathbb {R}_{> 0}^{K} and\\\\mathbf {Q} \\\\in \\\\mathbb {R}_{\\\\ge 0}^{K}with \\\\mathbf {Q} \\\\le - \\\\frac{1}{\\\\alpha } \\\\cdot \\\\mathbf {P} \\\\rbrace .}\",\n \"\\\\nonumber $ Notice that the symmetry $D_{\\\\varphi _{gKL,\\\\alpha ,\\\\widetilde{c}}}(\\\\mathbf {Q},\\\\mathbf {P})= D_{\\\\varphi _{gKL,\\\\alpha ,\\\\widetilde{c}}}(\\\\mathbf {P},\\\\mathbf {Q})$ generally holds only if $\\\\mathbf {P}, \\\\mathbf {Q} \\\\in \\\\mathbb {R}_{> 0}^{K}$ and $\\\\alpha =1$ ; indeed, this special case leads to $\\\\varphi _{snKL,\\\\widetilde{c}}(t):= \\\\varphi _{gKL,1,\\\\widetilde{c}}(t)\\\\hspace{-5.69046pt} &:=& \\\\hspace{-5.69046pt}{\\\\left\\\\lbrace \\\\begin{array}{ll}\\\\widetilde{c} \\\\cdot \\\\left[ \\\\, t \\\\cdot \\\\log t + (t+1) \\\\cdot \\\\log \\\\Big ( \\\\frac{2}{t+1} \\\\Big ) \\\\, \\\\right]\\\\ \\\\in \\\\ [0,\\\\infty [,\\\\qquad \\\\quad \\\\textrm {if }t \\\\in \\\\, ]0,\\\\infty [,\\\\\\\\\\\\widetilde{c} \\\\cdot \\\\log 2, \\\\hspace{187.78836pt} \\\\textrm {if } \\\\ t = 0, \\\\\\\\\\\\infty , \\\\hspace{209.12791pt} \\\\textrm {if } \\\\ t \\\\in \\\\, ]-\\\\infty ,0[ ,\\\\end{array}\\\\right.\",\n \"}$ and $D_{\\\\varphi _{snKL,\\\\widetilde{c}}}(\\\\mathbf {Q},\\\\mathbf {P}):= D_{\\\\varphi _{gKL,1,\\\\widetilde{c}}}(\\\\mathbf {Q},\\\\mathbf {P})= \\\\widetilde{c} \\\\cdot \\\\Big \\\\lbrace \\\\sum \\\\limits _{k=1}^{K} q_{k} \\\\cdot \\\\log \\\\Big (\\\\frac{2 q_{k}}{q_{k} + p_{k}} \\\\Big )+ \\\\sum \\\\limits _{k=1}^{K} p_{k} \\\\cdot \\\\log \\\\Big (\\\\frac{2 p_{k}}{q_{k} + p_{k}} \\\\Big ) \\\\Big \\\\rbrace ,\\\\qquad \\\\mathbf {P} \\\\in \\\\mathbb {R}_{> 0}^{K}, \\\\mathbf {Q} \\\\in \\\\mathbb {R}_{\\\\ge 0}^{K}.$ For the special subcase that $\\\\widetilde{c} =1$ and that $\\\\mathbf {P} = {P}$ , $\\\\mathbf {Q} = {Q}$ are probability vectors, the divergence (REF ) can be rewritten as sum of two Kullback-Leibler divergences (cf.\",\n \"(REF )) $D_{\\\\varphi _{snKL,1}}({Q},{P})= D_{\\\\varphi _{1}}({Q},({Q}+{P})/2)+ D_{\\\\varphi _{1}}({P},({Q}+{P})/2),\\\\qquad {P} \\\\in \\\\mathbb {S}_{> 0}^{K}, {Q} \\\\in \\\\mathbb {S}_{\\\\ge 0}^{K},$ which is the well-known (cf.\",\n \"Burbea & Rao [68], Lin [219], Pardo & Vajda [284], Topsoe [360], Endres & Schindelin [120], Vajda [373], Sason [317]) Jensen-Shannon divergence (being also called symmetrized and normalized Kullback-Leibler divergence, symmetrized and normalized relative entropy, capacitory discrimination); this is equal to the $(2\\\\log 2)-$ fold of a special (namely, equally-weighted two-population) case of the Sibson information radius of order 1 (cf.\",\n \"[334]) which has also been addressed e.g.\",\n \"by Rao [301] for genetic cluster analysis.\",\n \"By the way, for $\\\\alpha >0$ the divergence $D_{\\\\varphi _{gKL,\\\\alpha ,\\\\widetilde{c}}}({Q},{P})$ can also be interpreted as a multiple of a special non-equally-weighted Sibson information radius of order 1.\",\n \"In a context of comparison of — not necessarily connected — networks where ${Q}$ , ${P}$ are probability vectors derived from matrices (cf.\",\n \"Remark REF ) which are transforms of corresponding graph invariants (e.g.\",\n \"network portraits), the (matrix-equivalent of the) Jensen-Shannon divergence $D_{\\\\varphi _{snKL,1}}({Q},{P})$ is also called the network portrait divergence, cf.\",\n \"Bagrow and Bollt [28].\",\n \"There is a vast literature on recent applications of the Jensen-Shannon divergence, for instance it appears exemplarily in Kvitsiani et al.\",\n \"[208] for finding connections between the circuit-level function of different interneuron types in regulating the flow of information and the behavioural functions served by the cortical circuits, in Xu et al.\",\n \"(2014) for browsing and exploration of video sequences, in Jenkinson et al.\",\n \"[168] for the fundamental understanding of the epigenome that leads to a powerful approach for studying its role in disease and aging, in Martin et al.\",\n \"[250] for the implementation of an evolutionary-based global localization filter for mobile robots, in Suo et al.\",\n \"[354] for the revelation of critical regulators of cell identity in mice, in Abante et al.\",\n \"[2] for the detection of biologically significant differences in DNA methylation between alleles associated with local changes in genetic sequences — for a better understanding of the mechanism of complex human diseases, in Afek et al.\",\n \"[5] for revealing mechanisms by which mismatches can recruit transcription factors for modulating replication and repair activities in cells, in Alaiz-Rodriguez & Parnell [10] for the quantification of stability in feature selection and ranking algorithms, in Biau et al.\",\n \"[53] for generative adversarial networks (GANs) in artificial intelligence and machine learning, in Carre et al.\",\n \"[74] for the standardization of brain magnetic resonance (MR) images, in Chakraborty et al.\",\n \"[75] for hierarchical clustering in foreign exchange FOREX markets (e.g.\",\n \"in periods of major international crises), in Chong et al.\",\n \"[87] as part of a web-based platform for comprehensive analysis of microbiome data outputs, in Cui et al.\",\n \"[101] for modelling latent friend recommendation in online social media, in Gholami & Hodtani [134] for refinements of safety-and-security-targeted location verification systems in wireless communication networks (e.g in Intelligent Transportation Systems (ITSs) and vehicular technology), in Guo & Yuan [146] for accurate abnormality classification in semi-supervised Wireless Capsule Endoscopy (WCE) for digestive system cancer diagnosis, in Jiang et al.\",\n \"[169] for the training of deep neural discriminative and generative networks used for designing and evaluating photonic devices, in Kartal et al.\",\n \"[186] for uncovering the relationship between some genomic features and cell type-specific methylome diversity, in Laszlovszky et al.\",\n \"[210] for investigating mechanisms of basal forebrain neurons which modulate synaptic plasticity,cortical processing, brain states and oscillations, in Lawson et al.\",\n \"[211] for the improved understanding of some genetic circuits that allow cancer cells to evade destruction by the host immune system, in Li et al.\",\n \"[215] for the search of causes of the progressive neurodevelopmental disorder Rett syndrome, in Machado et al.\",\n \"[239] for discovering relations between distinct RNA viruses (including SARS-CoV-2), in Mohammadi et al.\",\n \"[261] for the identification of cell states and their underlying topology, in Mohanty et al.\",\n \"[262] for the design of implantable nanophotonic (i.e.\",\n \"chip-scale optical circuit type) silicon probes for sub-millisecond deep-brain optical stimulation — e.g.\",\n \"for the purpose of gaining a deeper understanding of the neural code, in Perera et al.\",\n \"[290] for the quantification of the level of rationality in supply chain networks, in Pierri et al.\",\n \"[294] for the study of growth of malicious/misleading information in some social media diffusion networks, in Rabadan et al.\",\n \"[299] for the identification of gene mutations that lead to the genesis and progression of tumors, in Reiter et al.\",\n \"[306] for quantifying metastatic phylogenetic diversity, in Van de Sande et al.\",\n \"[378] as part of a computational toolbox for single-cell gene regulatory network analysis, in Skinnider et al.\",\n \"[337] for the prediction of the chemical structures of genomically encoded antibiotics — in order to find means against the looming global crisis of antibiotic resistance, in Tuo et al.\",\n \"[367] for the detection of high-order single nucleotide polymorphism (SNP) interactions, in Uttam et al.\",\n \"[370] for predicting the risk of colorectal cancer recurrence and inferring associated tumor microenvironment networks, in Zhang et al.\",\n \"[421] for incipient fault (namely, crack) detection, in Zhi et al.\",\n \"[425] for the strengthening of information-centric networks against interest flooding attack (IFAs), in Acera Mateos et al.\",\n \"[3] for deep-learning classification of SARS-CoV-2 and co-infecting RNA viruses, in Avsec et al.\",\n \"[24] for uncovering the motifs and syntax of cis-regulatory sequences in genomics data, in Barennes et al.\",\n \"[32] for comparing the accuracy of current T cell receptor sequencing methods employed for the understanding of adaptive immune responses, in Chen et al.\",\n \"[79] for clustering high-dimensional microbial data from RNA sequencing, in Chen et al.\",\n \"[84] for investigating key aspects of effective vocal social communication, in Koldobskiy et al.\",\n \"[193] for investigations of genetic and epigenetic drivers of paediatric acute lymphoblastic leukaemia, in McGinnis et al.\",\n \"[258] for evaluating RNA sequencing of pooled blood cell samples, in Mühlroth & Grottke [268] for the detection of emerging trends and technologies through artificial intelligence techniques, in Necci et al.\",\n \"[272] for the assessment of protein intrinsic disorder predictions, in Okada et al.\",\n \"[277] for the identification of genetic factors that cause individual differences in whole lymphocyte profiles and their changes after vaccination, and in Zhang et al.\",\n \"[422] for the learning of functional magnetic resonance imaging (fMRI) time-series in a brain disease diagnosis context.\",\n \"Remark 44 Let us transform $\\\\varphi _{gSH,\\\\alpha }(t) := \\\\frac{1-t}{\\\\alpha } \\\\cdot \\\\log (1+\\\\alpha )- \\\\varphi _{gKL,\\\\alpha ,1}(t)= - t \\\\cdot \\\\log t+ \\\\frac{1}{\\\\alpha } \\\\cdot (1 + \\\\alpha \\\\cdot t) \\\\cdot \\\\log (1 + \\\\alpha \\\\cdot t)- \\\\frac{1}{\\\\alpha } \\\\cdot (1+\\\\alpha ) \\\\cdot t \\\\cdot \\\\log (1+\\\\alpha )$ (for $t \\\\in [0,1]$ ).\",\n \"The function $\\\\varphi _{gSH,\\\\alpha }(\\\\cdot )$ is strictly concave on $[0,1]$ with $\\\\varphi _{gSH,\\\\alpha }(0)= \\\\varphi _{gSH,\\\\alpha }(1)=0$ .\",\n \"Hence, for probability vectors ${Q} =(q_{k})_{k=1,\\\\ldots ,K}$ , the $\\\\varphi -$ entropy $\\\\sum _{k=1}^{K}\\\\varphi _{gSH,\\\\alpha }(q_{k})$ is Kapur’s [183] generalization of the Shannon entropy (which corresponds to $\\\\alpha =0$ in the limit) whose maximization has been connected with generalizations of the Bose-Einstein statistics and the Fermi-Dirac statistics e.g.\",\n \"in Kapur & Kesavan [185].\",\n \"Example 45 Let us fix any $z_{1},z_{2} \\\\in \\\\mathbb {R}$ , $p \\\\in ]0,1[$ which satisfy $z_{1} < 1 < z_{2}$ and $z_{1} \\\\cdot p + z_{2} \\\\cdot (1-p) =1$ (and thus $p= \\\\frac{z_{2} -1}{z_{2} - z_{1}}$ ).\",\n \"On $]a_{F_{twop}},b_{F_{twop}}[ := ]z_{1},z_{2}[$ we define $F_{twop}(t) &:=& \\\\frac{1}{z_{2}-z_{1}} \\\\cdot \\\\log \\\\left( \\\\frac{(t-z_{1})\\\\cdot p}{(z_{2}-t)\\\\cdot (1- p)} \\\\right)\\\\nonumber \\\\\\\\&=&\\\\frac{1}{z_{2}-z_{1}} \\\\cdot \\\\log \\\\left( \\\\frac{(t-z_{1})\\\\cdot (z_{2} -1)}{(z_{2}-t)\\\\cdot (1-z_{1})} \\\\right) \\\\, ,\\\\qquad t \\\\in \\\\, ]z_{1},z_{2}[ ,\\\\nonumber $ where for the last equality we have used the above constraint (in order to obtain a two-parameter representation).\",\n \"Straightforwardly, we have $\\\\mathcal {R}(F_{twop})= ]-\\\\infty ,\\\\infty [$ .\",\n \"Moreover, $F_{twop}(\\\\cdot )$ is strictly increasing and smooth on $]0,\\\\infty [$ , and thus, $F_{twop} \\\\in {F}$ .\",\n \"Since $F_{twop}(1)=0$ , let us choose the natural anchor point $c:=0$ , which leads to $]\\\\lambda _{-},\\\\lambda _{+}[= int(\\\\mathcal {R}(F_{snKL,\\\\widetilde{c}})) = ]-\\\\infty , \\\\infty [$ and $]t_{-}^{sc},t_{+}^{sc}[ = ]z_{1},z_{2}[$ .\",\n \"By using $F_{twop}^{-1}(x) = \\\\frac{p \\\\cdot z_{1} + (1- p) \\\\cdot z_{2} \\\\cdot e^{(z_{2}-z_{1}) \\\\cdot x}}{p + (1- p) \\\\cdot e^{(z_{2}-z_{1}) \\\\cdot x}} \\\\, , \\\\qquad x \\\\in ]-\\\\infty , \\\\infty [,$ we derive from formula (REF ) (see also (REF )) $& & \\\\hspace{-19.91684pt} \\\\Lambda _{twop}(z) := \\\\Lambda _{twop}^{(0)}(z) :=\\\\int \\\\displaylimits _{0}^{z} F_{twop}^{-1}(u) du= \\\\log \\\\Big ( p \\\\cdot e^{z_{1} \\\\cdot z} + (1- p) \\\\cdot e^{z_{2} \\\\cdot z} \\\\Big ),\\\\qquad z \\\\in ]-\\\\infty , \\\\infty [ ,$ which has the properties $\\\\Lambda _{twop}(0) = 0$ , $\\\\Lambda _{twop}(-\\\\infty ) =\\\\infty \\\\cdot {1}_{]-\\\\infty ,0[}(z_{1})- \\\\infty \\\\cdot {1}_{]0,\\\\infty [}(z_{1})+ \\\\log p \\\\cdot {1}_{\\\\lbrace 0\\\\rbrace }(z_{1})$ , $\\\\Lambda _{twop}(\\\\infty ) = \\\\infty $ , $\\\\Lambda _{twop}^{\\\\prime }(- \\\\infty ) = z_{1}$ and $\\\\Lambda _{twop}^{\\\\prime }(\\\\infty ) = z_{2}$ .\",\n \"Furthermore, from formula (REF ) (see also ()) we deduce $\\\\varphi _{twop}(t) := \\\\varphi _{twop}^{(0)}(t)\\\\hspace{-5.69046pt} &:=& \\\\hspace{-5.69046pt}{\\\\left\\\\lbrace \\\\begin{array}{ll}\\\\frac{t-z_{1}}{z_{2}-z_{1}} \\\\cdot \\\\log \\\\left( \\\\frac{(t-z_{1})\\\\cdot (z_{2}-1)}{(z_{2}-t)\\\\cdot (1-z_{1})} \\\\right)- \\\\log \\\\left( \\\\frac{z_{2}-1}{z_{2}-t} \\\\right)\\\\ \\\\in \\\\ [0,\\\\infty [,\\\\qquad \\\\quad \\\\textrm {if }t \\\\in \\\\, ]0,\\\\infty [,\\\\\\\\\\\\log \\\\left( \\\\frac{z_{2} - z_{1}}{z_{2} -1} \\\\right), \\\\hspace{219.08612pt} \\\\textrm {if } \\\\ t = z_{1}, \\\\\\\\\\\\log \\\\left( \\\\frac{z_{2} - z_{1}}{1 - z_{1}} \\\\right), \\\\hspace{219.08612pt} \\\\textrm {if } \\\\ t = z_{2}, \\\\\\\\\\\\infty , \\\\hspace{233.3125pt} \\\\textrm {if } \\\\ t \\\\in \\\\, ]-\\\\infty ,z_{1}[ \\\\, \\\\cup \\\\, ]z_{2}, \\\\infty [ ;\\\\end{array}\\\\right.\",\n \"}$ the first line in (REF ) can be proved by $& & \\\\hspace{-19.91684pt}\\\\varphi _{twop}(t) := \\\\varphi _{twop}^{(0)}(t) :=t \\\\cdot F_{twop}\\\\left(t\\\\right)-\\\\int \\\\displaylimits _{0}^{F_{twop}\\\\left(t\\\\right)} F_{twop}^{-1}(u) du\\\\nonumber \\\\\\\\& & \\\\hspace{-19.91684pt}= \\\\frac{t}{z_{2}-z_{1}} \\\\cdot \\\\log \\\\left( \\\\frac{(t-z_{1})\\\\cdot p}{(z_{2}-t)\\\\cdot (1- p)} \\\\right)\\\\nonumber \\\\\\\\& & \\\\hspace{-19.91684pt}- \\\\log \\\\left( p \\\\cdot \\\\left( \\\\frac{(t-z_{1})\\\\cdot p}{(z_{2}-t)\\\\cdot (1- p)} \\\\right)^{\\\\frac{z_{1}}{z_{2}-z_{1}}}+ (1- p) \\\\cdot \\\\left( \\\\frac{(t-z_{1})\\\\cdot p}{(z_{2}-t)\\\\cdot (1- p)} \\\\right)^{\\\\frac{z_{2}}{z_{2}-z_{1}}} \\\\right)\\\\nonumber \\\\\\\\& & \\\\hspace{-19.91684pt}= \\\\frac{t-z_{1}}{z_{2}-z_{1}} \\\\cdot \\\\log \\\\left( \\\\frac{(t-z_{1})\\\\cdot p}{(z_{2}-t)\\\\cdot (1- p)} \\\\right)- \\\\log \\\\left( \\\\frac{(z_{2}-z_{1})\\\\cdot p}{z_{2}-t} \\\\right)\\\\\\\\& & \\\\hspace{-19.91684pt}= \\\\frac{t-z_{1}}{z_{2}-z_{1}} \\\\cdot \\\\log \\\\left( \\\\frac{(t-z_{1})\\\\cdot (z_{2}-1)}{(z_{2}-t)\\\\cdot (1-z_{1})} \\\\right)- \\\\log \\\\left( \\\\frac{z_{2}-1}{z_{2}-t} \\\\right) , \\\\qquad t \\\\in \\\\, ]z_{1}, z_{2} [ \\\\, ,\\\\nonumber $ where for the last equality we have used the above constraint (to obtain a two-parameter representation).\",\n \"Straightforwardly, one has $\\\\varphi _{twop}(1) = 0$ , $\\\\varphi _{twop}^{\\\\prime }(1) = 0$ , $\\\\varphi _{twop}^{\\\\prime }(z_{1}) = -\\\\infty $ and $\\\\varphi _{twop}^{\\\\prime }(z_{2}) = \\\\infty $ .\",\n \"From the generator $\\\\varphi _{twop}$ given in (REF ), we build the corresponding divergence (cf.\",\n \"(REF )) $D_{\\\\varphi _{twop}}(\\\\mathbf {Q},\\\\mathbf {P})=\\\\sum \\\\limits _{k=1}^{K} \\\\frac{q_{k} - z_{1} \\\\cdot p_{k}}{z_{2} - z_{1}} \\\\cdot \\\\log \\\\Big (\\\\frac{(z_{2}-1) \\\\cdot (q_{k} - z_{1} \\\\cdot p_{k})}{(1-z_{1}) \\\\cdot (z_{2} \\\\cdot p_{k} - q_{k})} \\\\Big )- \\\\sum \\\\limits _{k=1}^{K} p_{k} \\\\cdot \\\\log \\\\Big (\\\\frac{(z_{2}-1) \\\\cdot p_{k}}{z_{2} \\\\cdot p_{k} - q_{k}} \\\\Big ) .$ It is known that some types of robustness properties of minimum-divergence estimators are connected with the boundedness of the derivative $\\\\varphi ^{\\\\prime }$ of the divergence generator $\\\\varphi $ ; this property is satisfied for the next Example REF (and its $W-$ concerning continuation in Example REF ), which leads to the new classes of divergences (REF ), (REF ) and (REF ): Example 46 (a) For any parameter-quadrupel $\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c} \\\\in \\\\, ]0,\\\\infty [$ with $\\\\beta _{1} < \\\\beta _{2}$ , we choose $]a_{F},b_{F}[\\\\ \\\\, := \\\\ ]a_{F_{\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c}}},b_{F_{\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c}}}[ \\\\ \\\\, := \\\\ \\\\Big ]1 - \\\\alpha \\\\cdot \\\\frac{(\\\\beta _{1} - \\\\beta _{2})^{2} + \\\\beta _{1}^{2} + \\\\beta _{1} \\\\cdot \\\\beta _{2}}{2\\\\beta _{1}\\\\cdot \\\\beta _{2}\\\\cdot (\\\\beta _{2} - \\\\beta _{1} )}, \\\\,\\\\infty \\\\Big [ \\\\ \\\\ni 1\\\\nonumber $ and define with $\\\\breve{\\\\theta } := 1 + \\\\alpha \\\\cdot \\\\Big (\\\\frac{1}{\\\\beta _{2}}- \\\\frac{1}{\\\\beta _{1}} \\\\Big ) < 1$ $\\\\hspace{-17.07182pt}F_{\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c}}(t)&:=& {\\\\left\\\\lbrace \\\\begin{array}{ll}\\\\widetilde{c} \\\\cdot \\\\frac{\\\\beta _{1}-\\\\beta _{2}}{2}+ \\\\frac{\\\\widetilde{c}}{\\\\frac{1-t}{\\\\alpha } + \\\\frac{1}{\\\\beta _{2}} - \\\\frac{1}{\\\\beta _{1}}}\\\\cdot \\\\Big (1 - \\\\frac{1}{2} \\\\cdot \\\\sqrt{4 + \\\\big (\\\\frac{1-t}{\\\\alpha } + \\\\frac{1}{\\\\beta _{2}} - \\\\frac{1}{\\\\beta _{1}}\\\\big )^{2} \\\\cdot (\\\\beta _{1}+\\\\beta _{2})^{2}}\\\\, \\\\Big ),\\\\quad \\\\textrm {if } \\\\ t \\\\in \\\\, ]a_{F},b_{F}[ \\\\backslash \\\\lbrace \\\\breve{\\\\theta }\\\\rbrace , \\\\\\\\\\\\widetilde{c} \\\\cdot \\\\frac{\\\\beta _{1}-\\\\beta _{2}}{2}, \\\\hspace{277.41437pt}\\\\textrm {if } \\\\ t= \\\\breve{\\\\theta } \\\\in \\\\, ]a_{F},b_{F}[, \\\\\\\\- \\\\widetilde{c} \\\\cdot \\\\beta _{1}, \\\\hspace{284.52756pt} \\\\textrm {if } \\\\ t=a_{F}, \\\\\\\\- \\\\infty , \\\\hspace{295.90848pt} \\\\textrm {if } \\\\ t \\\\in \\\\, ]-\\\\infty ,a_{F}[.\\\\end{array}\\\\right.\",\n \"}$ Notice that $\\\\breve{\\\\theta } \\\\in \\\\, ]a_{F},b_{F}[$ if and only if $\\\\beta _{1} \\\\in \\\\,]\\\\frac{\\\\beta _{2}}{3},\\\\beta _{2}[$ ; if (say) the latter holds, then one has the continuity $\\\\lim _{t \\\\rightarrow \\\\breve{\\\\theta }}F_{\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c}}(t) = \\\\widetilde{c} \\\\cdot \\\\frac{\\\\beta _{1}-\\\\beta _{2}}{2}$ .\",\n \"For $\\\\beta _{1} \\\\le \\\\frac{\\\\beta _{2}}{3}$ one gets $]a_{F},b_{F}[ \\\\backslash \\\\lbrace \\\\breve{\\\\theta } \\\\rbrace = \\\\, ]a_{F},b_{F}[$ .\",\n \"Returning to the general case, one can see in a straightforward way that $F_{\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c}}(\\\\cdot )$ is strictly increasing and that $\\\\mathcal {R}(F_{\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c}})=[-\\\\widetilde{c}\\\\cdot \\\\beta _{1},\\\\widetilde{c}\\\\cdot \\\\beta _{1} [$ .\",\n \"Furthermore, $F_{\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c}}(\\\\cdot )$ is smooth on $]a_{F},b_{F}[$ , and thus $F_{\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c}} \\\\in {F}$ .\",\n \"Since $F_{\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c}}(1)=0$ , let us choose the natural anchor point $c:=0$ , which leads to $]\\\\lambda _{-},\\\\lambda _{+}[ \\\\,= int(\\\\mathcal {R}(F_{\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c}})= \\\\, ]-\\\\widetilde{c}\\\\cdot \\\\beta _{1},\\\\widetilde{c}\\\\cdot \\\\beta _{1} [$ and $]t_{-}^{sc},t_{+}^{sc}[ = \\\\, ]a_{F},b_{F}[$ .\",\n \"Moreover, it is straightforward to see that the corresponding inverse is $F_{\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c}}^{-1}(x) =1 + \\\\alpha \\\\cdot \\\\Big (\\\\frac{1}{\\\\beta _{2}}- \\\\frac{1}{\\\\beta _{1}} \\\\Big ) - \\\\alpha \\\\cdot \\\\frac{\\\\frac{1}{\\\\beta _{2}} - \\\\frac{1}{\\\\beta _{1}} -\\\\frac{2 x}{\\\\widetilde{c} \\\\cdot \\\\beta _{1} \\\\cdot \\\\beta _{2} }}{1+ \\\\frac{x}{\\\\widetilde{c}} \\\\cdot \\\\Big (\\\\frac{1}{\\\\beta _{2}} - \\\\frac{1}{\\\\beta _{1}} \\\\Big )- \\\\frac{x^{2}}{\\\\widetilde{c}^{2} \\\\cdot \\\\beta _{1} \\\\cdot \\\\beta _{2} }},\\\\qquad x \\\\in int(\\\\mathcal {R}(F_{\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c}})) ;$ from this, we can derive from formula (REF ) (see also (REF )) for all $z \\\\in \\\\mathbb {R}$ $\\\\hspace{-19.91684pt}\\\\Lambda _{\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c}}(z) :=\\\\Lambda _{\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c}}^{(0)}(z) &=&{\\\\left\\\\lbrace \\\\begin{array}{ll}\\\\breve{\\\\theta } \\\\cdot z - \\\\widetilde{c} \\\\cdot \\\\alpha \\\\cdot \\\\log \\\\Big (1+ \\\\frac{z}{\\\\widetilde{c}} \\\\cdot \\\\Big (\\\\frac{1}{\\\\beta _{2}} - \\\\frac{1}{\\\\beta _{1}} \\\\Big )- \\\\frac{z^{2}}{\\\\widetilde{c}^{2} \\\\cdot \\\\beta _{1} \\\\cdot \\\\beta _{2} }\\\\Big ),\\\\quad \\\\textrm {if } \\\\ z \\\\in \\\\, ]-\\\\widetilde{c}\\\\cdot \\\\beta _{1},\\\\widetilde{c}\\\\cdot \\\\beta _{1} [,\\\\\\\\- \\\\widetilde{c} \\\\cdot \\\\breve{\\\\theta } \\\\cdot \\\\beta _{1} -\\\\widetilde{c} \\\\cdot \\\\alpha \\\\cdot \\\\log \\\\Big (2 - 2 \\\\frac{\\\\beta _{1}}{\\\\beta _{2}}\\\\Big ),\\\\hspace{73.97733pt} \\\\textrm {if } \\\\ z = -\\\\widetilde{c}\\\\cdot \\\\beta _{1},\\\\\\\\\\\\infty , \\\\hspace{202.01474pt} \\\\textrm {else} .\\\\end{array}\\\\right.\",\n \"}$ Notice that $\\\\Lambda _{\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c}}(0) = 0$ and $\\\\lim _{z \\\\rightarrow \\\\widetilde{c}\\\\cdot \\\\beta _{1}}\\\\Lambda _{\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c}}(z) = \\\\infty $ .\",\n \"Moreover, $\\\\Lambda _{\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c}}^{\\\\prime }( -\\\\widetilde{c}\\\\cdot \\\\beta _{1})= a_{F}$ and $\\\\Lambda _{ -\\\\widetilde{c}\\\\cdot \\\\beta _{1}}^{\\\\prime }(\\\\widetilde{c}\\\\cdot \\\\beta _{1}) = \\\\infty = b_{F}$ (which have to be interpreted as limits, as usual).\",\n \"To proceed, from formula (REF ) (see also ()) we can deduce for all $t \\\\in \\\\mathbb {R}$ $\\\\hspace{-5.69046pt}\\\\varphi _{\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c}}(t):= \\\\varphi _{\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c}}^{(0)}(t)\\\\hspace{-8.5359pt} &=& \\\\hspace{-8.5359pt}{\\\\left\\\\lbrace \\\\begin{array}{ll}\\\\widetilde{c} \\\\cdot \\\\alpha \\\\cdot \\\\Big \\\\lbrace \\\\frac{\\\\sqrt{4 + (\\\\beta _{1} + \\\\beta _{2})^{2}\\\\cdot (\\\\frac{1-t}{\\\\alpha } + \\\\frac{1}{\\\\beta _{2}} - \\\\frac{1}{\\\\beta _{1}})^2}\\\\, - \\\\, (\\\\frac{1-t}{\\\\alpha } + \\\\frac{1}{\\\\beta _{2}} - \\\\frac{1}{\\\\beta _{1}}) \\\\cdot (\\\\beta _{1} - \\\\beta _{2}) \\\\, - \\\\, 2}{2} \\\\\\\\+ \\\\log \\\\frac{\\\\sqrt{4 + (\\\\beta _{1} + \\\\beta _{2})^{2}\\\\cdot (\\\\frac{1-t}{\\\\alpha } + \\\\frac{1}{\\\\beta _{2}} \\\\, - \\\\, \\\\frac{1}{\\\\beta _{1}})^2} \\\\, - \\\\, 2}{\\\\beta _{1} \\\\beta _{2} \\\\cdot (\\\\frac{1-t}{\\\\alpha } + \\\\frac{1}{\\\\beta _{2}} \\\\, - \\\\, \\\\frac{1}{\\\\beta _{1}})^{2}} \\\\Big \\\\rbrace \\\\ \\\\in [0,\\\\infty [,\\\\hspace{71.13188pt} \\\\textrm {if }t \\\\in \\\\, ]a_{F},\\\\infty [,\\\\\\\\\\\\widetilde{c} \\\\cdot \\\\alpha \\\\cdot \\\\Big \\\\lbrace \\\\frac{3 \\\\beta _{1} - \\\\beta _{2}}{2 (\\\\beta _{2} - \\\\beta _{1})}+ \\\\log \\\\frac{2(\\\\beta _{2} - \\\\beta _{1})}{\\\\beta _{2}} \\\\Big \\\\rbrace - \\\\widetilde{c} \\\\cdot \\\\beta _{1} \\\\cdot (t - a_{F}) \\\\ \\\\in \\\\, ]0,\\\\infty [,\\\\hspace{19.91684pt} \\\\textrm {if } \\\\ t \\\\in ]-\\\\infty , a_{F}].\\\\end{array}\\\\right.\",\n \"}$ The first subcase in (REF ) can be proved by $& & \\\\hspace{-19.91684pt}\\\\varphi _{\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c}}(t) :=\\\\varphi _{\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c}}^{(0)}(t) :=t \\\\cdot F_{\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c}}\\\\left(t\\\\right)\\\\ - \\\\int \\\\displaylimits _{0}^{F_{\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c}}\\\\left(t\\\\right)}F_{\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c}}^{-1}(u) \\\\, du\\\\nonumber \\\\\\\\& & \\\\hspace{-19.91684pt}= (t - \\\\breve{\\\\theta }) \\\\cdot F_{\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c}}\\\\left(t\\\\right)+ \\\\widetilde{c} \\\\cdot \\\\alpha \\\\cdot \\\\log \\\\Big (1+ \\\\frac{F_{\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c}}\\\\left(t\\\\right)}{\\\\widetilde{c}} \\\\cdot \\\\Big (\\\\frac{1}{\\\\beta _{2}} - \\\\frac{1}{\\\\beta _{1}} \\\\Big )- \\\\frac{(F_{\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c}}\\\\left(t\\\\right))^{2}}{\\\\widetilde{c}^{2} \\\\cdot \\\\beta _{1} \\\\cdot \\\\beta _{2} }\\\\Big )\\\\nonumber \\\\\\\\& & \\\\hspace{-19.91684pt}= \\\\widetilde{c} \\\\cdot \\\\alpha \\\\cdot \\\\frac{\\\\sqrt{4 + (\\\\beta _{1} + \\\\beta _{2})^{2}\\\\cdot (\\\\frac{1-t}{\\\\alpha } + \\\\frac{1}{\\\\beta _{2}} - \\\\frac{1}{\\\\beta _{1}})^2}\\\\, - \\\\, (\\\\frac{1-t}{\\\\alpha } + \\\\frac{1}{\\\\beta _{2}} - \\\\frac{1}{\\\\beta _{1}}) \\\\cdot (\\\\beta _{1} - \\\\beta _{2}) \\\\, - \\\\, 2}{2}\\\\nonumber \\\\\\\\& & \\\\hspace{-19.91684pt}\\\\ \\\\ \\\\ + \\\\ \\\\widetilde{c} \\\\cdot \\\\alpha \\\\cdot \\\\log \\\\bigg (1+ \\\\Big [\\\\frac{\\\\beta _{1}-\\\\beta _{2}}{2}+ \\\\frac{1}{\\\\frac{1-t}{\\\\alpha } + \\\\frac{1}{\\\\beta _{2}} - \\\\frac{1}{\\\\beta _{1}}}\\\\cdot \\\\Big (1 - \\\\frac{1}{2} \\\\cdot \\\\sqrt{4 + \\\\big (\\\\frac{1-t}{\\\\alpha } + \\\\frac{1}{\\\\beta _{2}} - \\\\frac{1}{\\\\beta _{1}}\\\\big )^{2} \\\\cdot (\\\\beta _{1}+\\\\beta _{2})^{2}}\\\\, \\\\Big ) \\\\Big ]\\\\cdot \\\\frac{\\\\beta _{1} - \\\\beta _{2}}{\\\\beta _{1} \\\\cdot \\\\beta _{2}}\\\\nonumber \\\\\\\\& & \\\\hspace{-19.91684pt}\\\\ \\\\ \\\\ - \\\\ \\\\Big [\\\\frac{\\\\beta _{1}-\\\\beta _{2}}{2}+ \\\\frac{1}{\\\\frac{1-t}{\\\\alpha } + \\\\frac{1}{\\\\beta _{2}} - \\\\frac{1}{\\\\beta _{1}}}\\\\cdot \\\\Big (1 - \\\\frac{1}{2} \\\\cdot \\\\sqrt{4 + \\\\big (\\\\frac{1-t}{\\\\alpha } + \\\\frac{1}{\\\\beta _{2}} - \\\\frac{1}{\\\\beta _{1}}\\\\big )^{2} \\\\cdot (\\\\beta _{1}+\\\\beta _{2})^{2}}\\\\, \\\\Big ) \\\\Big ]^{2} \\\\cdot \\\\frac{1}{\\\\beta _{1} \\\\cdot \\\\beta _{2} }\\\\bigg )\\\\nonumber $ and some straightforward calculations.\",\n \"The second line in (REF ) follows by computing $\\\\varphi _{\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c}}(a_{F})=\\\\widetilde{c} \\\\cdot \\\\alpha \\\\cdot \\\\Big \\\\lbrace \\\\frac{3 \\\\beta _{1} - \\\\beta _{2}}{2 (\\\\beta _{2} - \\\\beta _{1})}+ \\\\log \\\\frac{2(\\\\beta _{2} - \\\\beta _{1})}{\\\\beta _{2}} \\\\Big \\\\rbrace $ .\",\n \"Notice that $\\\\varphi _{\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c}}(1) = 0$ , $\\\\varphi _{\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c}}^{\\\\prime }(1) = 0$ , $\\\\varphi _{\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c}}(-\\\\infty ) = \\\\infty $ and $\\\\varphi _{\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c}}(\\\\infty ) = \\\\infty $ .\",\n \"Moreover, $\\\\varphi _{\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c}}^{\\\\prime }(-\\\\infty ) =\\\\varphi _{\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c}}^{\\\\prime }(a_{F}) =-\\\\widetilde{c} \\\\cdot \\\\beta _{1}$ and $\\\\varphi _{\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c}}^{\\\\prime }(\\\\infty ) =\\\\widetilde{c} \\\\cdot \\\\beta _{1}$ .\",\n \"From the generator $\\\\varphi _{\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c}}$ given in (REF ), we construct the corresponding divergence (cf.\",\n \"(REF )) $& &D_{\\\\varphi _{\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c}}}(\\\\mathbf {Q},\\\\mathbf {P})= \\\\sum \\\\limits _{k=1}^{K} p_{k} \\\\cdot \\\\varphi _{\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c}}\\\\Big (\\\\frac{q_{k}}{p_{k}}\\\\Big )\\\\nonumber \\\\\\\\& &= \\\\sum \\\\limits _{k=1}^{K}p_{k} \\\\cdot \\\\bigg [{1}_{]a_{F},\\\\infty [}(\\\\frac{q_{k}}{p_{k}}) \\\\cdot \\\\widetilde{c} \\\\cdot \\\\alpha \\\\cdot \\\\Big \\\\lbrace \\\\frac{\\\\sqrt{4 + (\\\\beta _{1} + \\\\beta _{2})^{2}\\\\cdot (\\\\frac{1-\\\\frac{q_{k}}{p_{k}}}{\\\\alpha } + \\\\frac{1}{\\\\beta _{2}} - \\\\frac{1}{\\\\beta _{1}})^2}\\\\, - \\\\, (\\\\frac{1-\\\\frac{q_{k}}{p_{k}}}{\\\\alpha } + \\\\frac{1}{\\\\beta _{2}} - \\\\frac{1}{\\\\beta _{1}}) \\\\cdot (\\\\beta _{1} - \\\\beta _{2}) \\\\, - \\\\, 2}{2}\\\\nonumber \\\\\\\\& &+ \\\\log \\\\frac{\\\\sqrt{4 + (\\\\beta _{1} + \\\\beta _{2})^{2}\\\\cdot (\\\\frac{1-\\\\frac{q_{k}}{p_{k}}}{\\\\alpha } + \\\\frac{1}{\\\\beta _{2}} \\\\, - \\\\, \\\\frac{1}{\\\\beta _{1}})^2} \\\\, - \\\\, 2}{\\\\beta _{1} \\\\beta _{2} \\\\cdot (\\\\frac{1-\\\\frac{q_{k}}{p_{k}}}{\\\\alpha } + \\\\frac{1}{\\\\beta _{2}} \\\\, - \\\\, \\\\frac{1}{\\\\beta _{1}})^{2}} \\\\Big \\\\rbrace \\\\nonumber \\\\\\\\& &+ \\\\ {1}_{]-\\\\infty , a_{F}]}(\\\\frac{q_{k}}{p_{k}}) \\\\cdot \\\\widetilde{c} \\\\cdot \\\\Big \\\\lbrace \\\\alpha \\\\cdot \\\\Big \\\\lbrace \\\\frac{3 \\\\beta _{1} - \\\\beta _{2}}{2 (\\\\beta _{2} - \\\\beta _{1})}+ \\\\log \\\\frac{2(\\\\beta _{2} - \\\\beta _{1})}{\\\\beta _{2}} \\\\Big \\\\rbrace - \\\\beta _{1} \\\\cdot (\\\\frac{q_{k}}{p_{k}} - a_{F}) \\\\Big \\\\rbrace \\\\bigg ],\\\\qquad \\\\mathbf {P} \\\\in \\\\mathbb {R}_{\\\\ge 0}^{K}, \\\\mathbf {Q} \\\\in \\\\mathbb {R}^{K}.$ Notice that we can particularly include the case where $p_{k}=0$ in combination with $q_{k} \\\\ne 0$ , since $\\\\lim _{t \\\\rightarrow 0_{+}} t \\\\cdot \\\\varphi _{\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c}}(\\\\frac{1}{t})= \\\\widetilde{c} \\\\cdot \\\\beta _{1}$ and $\\\\lim _{t \\\\rightarrow 0_{-}} t \\\\cdot \\\\varphi _{\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c}}(\\\\frac{1}{t})= - \\\\widetilde{c} \\\\cdot \\\\beta _{1}$ are both finite, and hence $p_{k} \\\\cdot \\\\varphi _{\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c}}(\\\\frac{q_{k}}{p_{k}})= q_{k} \\\\cdot \\\\frac{p_{k}}{q_{k}} \\\\cdot \\\\varphi _{\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c}}(\\\\frac{q_{k}}{p_{k}}) $ stays finite as $p_{k}$ tends to zero.\",\n \"(b) For any parameter-quadrupel $\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c} \\\\in \\\\, ]0,\\\\infty [$ with $\\\\beta _{1} > \\\\beta _{2}$ , one can proceed analogously to (a).\",\n \"Let us start by choosing $]a_{F},b_{F}[\\\\ \\\\, := \\\\ ]a_{F_{\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c}}},b_{F_{\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c}}}[ \\\\ \\\\, := \\\\ \\\\Big ]-\\\\infty , \\\\, 1 + \\\\alpha \\\\cdot \\\\frac{(\\\\beta _{1} - \\\\beta _{2})^{2} + \\\\beta _{1} \\\\cdot \\\\beta _{2} + \\\\beta _{2}^{2}}{2\\\\beta _{1}\\\\cdot \\\\beta _{2}\\\\cdot (\\\\beta _{1} - \\\\beta _{2} )}, \\\\,\\\\Big [ \\\\ \\\\ni 1\\\\nonumber $ and defining with the same $\\\\breve{\\\\theta } := 1 + \\\\alpha \\\\cdot \\\\Big (\\\\frac{1}{\\\\beta _{2}}- \\\\frac{1}{\\\\beta _{1}} \\\\Big ) > 1$ $\\\\hspace{-17.07182pt}F_{\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c}}(t)&:=& {\\\\left\\\\lbrace \\\\begin{array}{ll}\\\\widetilde{c} \\\\cdot \\\\frac{\\\\beta _{1}-\\\\beta _{2}}{2}+ \\\\frac{\\\\widetilde{c}}{\\\\frac{1-t}{\\\\alpha } + \\\\frac{1}{\\\\beta _{2}} - \\\\frac{1}{\\\\beta _{1}}}\\\\cdot \\\\Big (1 - \\\\frac{1}{2} \\\\cdot \\\\sqrt{4 + \\\\big (\\\\frac{1-t}{\\\\alpha } + \\\\frac{1}{\\\\beta _{2}} - \\\\frac{1}{\\\\beta _{1}}\\\\big )^{2} \\\\cdot (\\\\beta _{1}+\\\\beta _{2})^{2}}\\\\, \\\\Big ),\\\\quad \\\\textrm {if } \\\\ t \\\\in \\\\, ]a_{F},b_{F}[ \\\\backslash \\\\lbrace \\\\breve{\\\\theta }\\\\rbrace , \\\\\\\\\\\\widetilde{c} \\\\cdot \\\\frac{\\\\beta _{1}-\\\\beta _{2}}{2}, \\\\hspace{277.41437pt}\\\\textrm {if } \\\\ t= \\\\breve{\\\\theta } \\\\in \\\\, ]a_{F},b_{F}[, \\\\\\\\\\\\widetilde{c} \\\\cdot \\\\beta _{2}, \\\\hspace{293.06346pt} \\\\textrm {if } \\\\ t=b_{F}, \\\\\\\\\\\\infty , \\\\hspace{304.4444pt} \\\\textrm {if } \\\\ t \\\\in \\\\, ]b_{F}, \\\\infty [.\\\\end{array}\\\\right.\",\n \"}$ Clearly, $\\\\breve{\\\\theta } \\\\in \\\\, ]a_{F},b_{F}[$ if and only if $\\\\beta _{1} \\\\in \\\\,]\\\\beta _{2}, 3\\\\beta _{2}[$ ; if (say) the latter holds, then one gets the continuity $\\\\lim _{t \\\\rightarrow \\\\breve{\\\\theta }}F_{\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c}}(t) = \\\\widetilde{c} \\\\cdot \\\\frac{\\\\beta _{1}-\\\\beta _{2}}{2}$ .\",\n \"For $\\\\beta _{1} \\\\le 3 \\\\beta _{2}$ there holds $]a_{F},b_{F}[ \\\\backslash \\\\lbrace \\\\breve{\\\\theta } \\\\rbrace = \\\\, ]a_{F},b_{F}[$ .\",\n \"Returning to the general case, one can show comfortably that $F_{\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c}}(\\\\cdot )$ is strictly increasing and that $\\\\mathcal {R}(F_{\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c}})=]-\\\\widetilde{c}\\\\cdot \\\\beta _{2},\\\\widetilde{c}\\\\cdot \\\\beta _{2} ]$ .\",\n \"Moreover, $F_{\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c}}(\\\\cdot )$ is smooth on $]a_{F},b_{F}[$ , and hence $F_{\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c}} \\\\in {F}$ .\",\n \"In face of the validity of $F_{\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c}}(1)=0$ , let us choose the natural anchor point $c:=0$ , which amounts to $]\\\\lambda _{-},\\\\lambda _{+}[ \\\\,= int(\\\\mathcal {R}(F_{\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c}})= \\\\, ]-\\\\widetilde{c}\\\\cdot \\\\beta _{2},\\\\widetilde{c}\\\\cdot \\\\beta _{2} [$ and $]t_{-}^{sc},t_{+}^{sc}[ = \\\\, ]a_{F},b_{F}[$ .\",\n \"Since the first line in (REF ) coincides formally with that of (REF ) (with different $]a_{F},b_{F}[$ ), the corresponding inverse is formally the same as (REF ) (with different $]a_{F},b_{F}[$ ), and hence $\\\\hspace{-19.91684pt}\\\\Lambda _{\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c}}(z) :=\\\\Lambda _{\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c}}^{(0)}(z) &=&{\\\\left\\\\lbrace \\\\begin{array}{ll}\\\\breve{\\\\theta } \\\\cdot z - \\\\widetilde{c} \\\\cdot \\\\alpha \\\\cdot \\\\log \\\\Big (1+ \\\\frac{z}{\\\\widetilde{c}} \\\\cdot \\\\Big (\\\\frac{1}{\\\\beta _{2}} - \\\\frac{1}{\\\\beta _{1}} \\\\Big )- \\\\frac{z^{2}}{\\\\widetilde{c}^{2} \\\\cdot \\\\beta _{1} \\\\cdot \\\\beta _{2} }\\\\Big ),\\\\quad \\\\textrm {if } \\\\ z \\\\in \\\\, ]-\\\\widetilde{c}\\\\cdot \\\\beta _{2},\\\\widetilde{c}\\\\cdot \\\\beta _{2} [,\\\\\\\\\\\\widetilde{c} \\\\cdot \\\\breve{\\\\theta } \\\\cdot \\\\beta _{2} -\\\\widetilde{c} \\\\cdot \\\\alpha \\\\cdot \\\\log \\\\Big (2 - 2 \\\\frac{\\\\beta _{2}}{\\\\beta _{1}}\\\\Big ),\\\\hspace{73.97733pt} \\\\textrm {if } \\\\ z = \\\\widetilde{c}\\\\cdot \\\\beta _{2},\\\\\\\\\\\\infty , \\\\hspace{202.01474pt} \\\\textrm {else} .\\\\end{array}\\\\right.\",\n \"}$ Notice that $\\\\Lambda _{\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c}}(0) = 0$ and $\\\\lim _{z \\\\rightarrow - \\\\widetilde{c}\\\\cdot \\\\beta _{2}}\\\\Lambda _{\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c}}(z) = -\\\\infty $ .\",\n \"Furthermore, $\\\\Lambda _{\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c}}^{\\\\prime }( -\\\\widetilde{c}\\\\cdot \\\\beta _{2})= - \\\\infty = a_{F}$ and $\\\\Lambda _{ -\\\\widetilde{c}\\\\cdot \\\\beta _{1}}^{\\\\prime }(\\\\widetilde{c}\\\\cdot \\\\beta _{2})= b_{F}$ .\",\n \"To proceed, from formula (REF ) (see also ()) we can derive — analogously to (REF ) — for all $t \\\\in \\\\mathbb {R}$ $\\\\hspace{-5.69046pt}\\\\varphi _{\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c}}(t):= \\\\varphi _{\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c}}^{(0)}(t)\\\\hspace{-8.5359pt} &=& \\\\hspace{-8.5359pt}{\\\\left\\\\lbrace \\\\begin{array}{ll}\\\\widetilde{c} \\\\cdot \\\\alpha \\\\cdot \\\\Big \\\\lbrace \\\\frac{\\\\sqrt{4 + (\\\\beta _{1} + \\\\beta _{2})^{2}\\\\cdot (\\\\frac{1-t}{\\\\alpha } + \\\\frac{1}{\\\\beta _{2}} - \\\\frac{1}{\\\\beta _{1}})^2}\\\\, - \\\\, (\\\\frac{1-t}{\\\\alpha } + \\\\frac{1}{\\\\beta _{2}} - \\\\frac{1}{\\\\beta _{1}}) \\\\cdot (\\\\beta _{1} - \\\\beta _{2}) \\\\, - \\\\, 2}{2} \\\\\\\\+ \\\\log \\\\frac{\\\\sqrt{4 + (\\\\beta _{1} + \\\\beta _{2})^{2}\\\\cdot (\\\\frac{1-t}{\\\\alpha } + \\\\frac{1}{\\\\beta _{2}} \\\\, - \\\\, \\\\frac{1}{\\\\beta _{1}})^2} \\\\, - \\\\, 2}{\\\\beta _{1} \\\\beta _{2} \\\\cdot (\\\\frac{1-t}{\\\\alpha } + \\\\frac{1}{\\\\beta _{2}} \\\\, - \\\\, \\\\frac{1}{\\\\beta _{1}})^{2}} \\\\Big \\\\rbrace \\\\ \\\\in [0,\\\\infty [,\\\\hspace{71.13188pt} \\\\textrm {if }t \\\\in \\\\, ]-\\\\infty , b_{F}[,\\\\\\\\\\\\widetilde{c} \\\\cdot \\\\alpha \\\\cdot \\\\Big \\\\lbrace \\\\frac{3 \\\\beta _{2} - \\\\beta _{1}}{2 (\\\\beta _{1} - \\\\beta _{2})}+ \\\\log \\\\frac{2(\\\\beta _{1} - \\\\beta _{2})}{\\\\beta _{1}} \\\\Big \\\\rbrace + \\\\widetilde{c} \\\\cdot \\\\beta _{2} \\\\cdot (t - b_{F}) \\\\ \\\\in \\\\, ]0,\\\\infty [,\\\\hspace{19.91684pt} \\\\textrm {if } \\\\ t \\\\in [b_{F}, \\\\infty [,\\\\end{array}\\\\right.\",\n \"}$ where the last line in (REF ) follows by calculating $\\\\varphi _{\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c}}(b_{F})=\\\\widetilde{c} \\\\cdot \\\\alpha \\\\cdot \\\\Big \\\\lbrace \\\\frac{3 \\\\beta _{2} - \\\\beta _{1}}{2 (\\\\beta _{1} - \\\\beta _{2})}+ \\\\log \\\\frac{2(\\\\beta _{1} - \\\\beta _{2})}{\\\\beta _{1}} \\\\Big \\\\rbrace $ .\",\n \"Notice that $\\\\varphi _{\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c}}(1) = 0$ , $\\\\varphi _{\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c}}^{\\\\prime }(1) = 0$ , $\\\\varphi _{\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c}}(-\\\\infty ) = \\\\infty $ and $\\\\varphi _{\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c}}(\\\\infty ) = \\\\infty $ .\",\n \"Furthermore, $\\\\varphi _{\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c}}^{\\\\prime }(-\\\\infty ) =-\\\\widetilde{c} \\\\cdot \\\\beta _{2}$ and $\\\\varphi _{\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c}}^{\\\\prime }(\\\\infty ) =\\\\varphi _{\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c}}^{\\\\prime }(b_{F}) =\\\\widetilde{c} \\\\cdot \\\\beta _{2}$ .\",\n \"From the generator $\\\\varphi _{\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c}}$ given in (REF ), we construct the corresponding divergence (cf.\",\n \"(REF )) $& &D_{\\\\varphi _{\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c}}}(\\\\mathbf {Q},\\\\mathbf {P})= \\\\sum \\\\limits _{k=1}^{K} p_{k} \\\\cdot \\\\varphi _{\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c}}\\\\Big (\\\\frac{q_{k}}{p_{k}}\\\\Big )\\\\nonumber \\\\\\\\& &= \\\\sum \\\\limits _{k=1}^{K}p_{k} \\\\cdot \\\\bigg [{1}_{]-\\\\infty , b_{F}[}(\\\\frac{q_{k}}{p_{k}}) \\\\cdot \\\\widetilde{c} \\\\cdot \\\\alpha \\\\cdot \\\\Big \\\\lbrace \\\\frac{\\\\sqrt{4 + (\\\\beta _{1} + \\\\beta _{2})^{2}\\\\cdot (\\\\frac{1-\\\\frac{q_{k}}{p_{k}}}{\\\\alpha } + \\\\frac{1}{\\\\beta _{2}} - \\\\frac{1}{\\\\beta _{1}})^2}\\\\, - \\\\, (\\\\frac{1-\\\\frac{q_{k}}{p_{k}}}{\\\\alpha } + \\\\frac{1}{\\\\beta _{2}} - \\\\frac{1}{\\\\beta _{1}}) \\\\cdot (\\\\beta _{1} - \\\\beta _{2}) \\\\, - \\\\, 2}{2}\\\\nonumber \\\\\\\\& &+ \\\\log \\\\frac{\\\\sqrt{4 + (\\\\beta _{1} + \\\\beta _{2})^{2}\\\\cdot (\\\\frac{1-\\\\frac{q_{k}}{p_{k}}}{\\\\alpha } + \\\\frac{1}{\\\\beta _{2}} \\\\, - \\\\, \\\\frac{1}{\\\\beta _{1}})^2} \\\\, - \\\\, 2}{\\\\beta _{1} \\\\beta _{2} \\\\cdot (\\\\frac{1-\\\\frac{q_{k}}{p_{k}}}{\\\\alpha } + \\\\frac{1}{\\\\beta _{2}} \\\\, - \\\\, \\\\frac{1}{\\\\beta _{1}})^{2}} \\\\Big \\\\rbrace \\\\nonumber \\\\\\\\& &+ \\\\ {1}_{[b_{F}, \\\\infty [}(\\\\frac{q_{k}}{p_{k}}) \\\\cdot \\\\widetilde{c} \\\\cdot \\\\Big \\\\lbrace \\\\alpha \\\\cdot \\\\Big \\\\lbrace \\\\frac{3 \\\\beta _{2} - \\\\beta _{1}}{2 (\\\\beta _{1} - \\\\beta _{2})}+ \\\\log \\\\frac{2(\\\\beta _{1} - \\\\beta _{2})}{\\\\beta _{1}} \\\\Big \\\\rbrace + \\\\beta _{2} \\\\cdot (\\\\frac{q_{k}}{p_{k}} - b_{F}) \\\\Big \\\\rbrace \\\\bigg ],\\\\qquad \\\\mathbf {P} \\\\in \\\\mathbb {R}_{\\\\ge 0}^{K}, \\\\mathbf {Q} \\\\in \\\\mathbb {R}^{K}.$ As above, we can particularly include the case where $p_{k}=0$ in combination with $q_{k} \\\\ne 0$ , since $\\\\lim _{t \\\\rightarrow 0_{+}} t \\\\cdot \\\\varphi _{\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c}}(\\\\frac{1}{t})= \\\\widetilde{c} \\\\cdot \\\\beta _{2}$ and $\\\\lim _{t \\\\rightarrow 0_{-}} t \\\\cdot \\\\varphi _{\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c}}(\\\\frac{1}{t})= - \\\\widetilde{c} \\\\cdot \\\\beta _{2}$ are both finite.\",\n \"(c) The analysis for the case $\\\\beta _{1} = \\\\beta _{2} =: \\\\beta $ can be obtained by taking $\\\\lim _{\\\\beta _1 \\\\rightarrow \\\\beta _{2}}$ in (a) respectively (b).\",\n \"Alternatively, one can start afresh.\",\n \"Due to its importance and its particularities, we nevertheless state the corresponding results explicitly.\",\n \"To begin with, for any parameter-triple $\\\\alpha ,\\\\beta ,\\\\widetilde{c} \\\\in \\\\, ]0,\\\\infty [$ we choose $]a_{F},b_{F}[\\\\ \\\\, := \\\\ ]a_{F_{\\\\alpha ,\\\\beta ,\\\\widetilde{c}}},b_{F_{\\\\alpha ,\\\\beta ,\\\\widetilde{c}}}[ \\\\ \\\\, := \\\\ ]-\\\\infty , \\\\infty \\\\, [\\\\nonumber $ and define with $\\\\breve{\\\\theta } := 1$ $\\\\hspace{-17.07182pt}F_{\\\\alpha ,\\\\beta ,\\\\widetilde{c}}(t)&:=& {\\\\left\\\\lbrace \\\\begin{array}{ll}\\\\frac{\\\\widetilde{c} \\\\cdot \\\\alpha }{1-t}\\\\cdot \\\\Big (1 - \\\\sqrt{1 + \\\\big (\\\\frac{1-t}{\\\\alpha } \\\\big )^{2} \\\\cdot \\\\beta ^{2}}\\\\, \\\\Big ),\\\\qquad \\\\textrm {if } \\\\ t \\\\in \\\\, ]a_{F},b_{F}[ \\\\backslash \\\\lbrace \\\\breve{\\\\theta }\\\\rbrace , \\\\\\\\0, \\\\hspace{145.10922pt}\\\\textrm {if } \\\\ t= \\\\breve{\\\\theta }.\\\\end{array}\\\\right.\",\n \"}$ Clearly, one has the continuity $\\\\lim _{t \\\\rightarrow \\\\breve{\\\\theta }}F_{\\\\alpha ,\\\\beta ,\\\\widetilde{c}}(t) = 0$ .\",\n \"Moreover, one can see in a straightforward way that $F_{\\\\alpha ,\\\\beta ,\\\\widetilde{c}}(\\\\cdot )$ is strictly increasing and that $\\\\mathcal {R}(F_{\\\\alpha ,\\\\beta ,\\\\widetilde{c}})=]-\\\\widetilde{c}\\\\cdot \\\\beta ,\\\\widetilde{c}\\\\cdot \\\\beta [$ .\",\n \"Furthermore, $F_{\\\\alpha ,\\\\beta ,\\\\widetilde{c}}(\\\\cdot )$ is smooth on $]a_{F},b_{F}[$ , and thus $F_{\\\\alpha ,\\\\beta ,\\\\widetilde{c}} \\\\in {F}$ .\",\n \"Since $F_{\\\\alpha ,\\\\beta ,\\\\widetilde{c}}(1)=0$ , let us choose the natural anchor point $c:=0$ , which leads to the choice $]\\\\lambda _{-},\\\\lambda _{+}[ \\\\,= int(\\\\mathcal {R}(F_{\\\\alpha ,\\\\beta ,\\\\widetilde{c}})= \\\\, ]-\\\\widetilde{c}\\\\cdot \\\\beta ,\\\\widetilde{c}\\\\cdot \\\\beta [$ and $]t_{-}^{sc},t_{+}^{sc}[ = \\\\, ]a_{F},b_{F}[ = \\\\, ]-\\\\infty ,\\\\infty [$ .\",\n \"The inverse in (REF ) collapses to $F_{\\\\alpha ,\\\\beta ,\\\\widetilde{c}}^{-1}(x) =1 + \\\\alpha \\\\cdot \\\\frac{\\\\frac{2 x}{\\\\widetilde{c} \\\\cdot \\\\beta ^{2} }}{1 - \\\\frac{x^{2}}{\\\\widetilde{c}^{2} \\\\cdot \\\\beta ^{2} }},\\\\qquad x \\\\in int(\\\\mathcal {R}(F_{\\\\alpha ,\\\\beta ,\\\\widetilde{c}})) ;$ from this, we can derive from formula (REF ) (see also (REF )) for all $z \\\\in \\\\mathbb {R}$ $\\\\hspace{-19.91684pt}\\\\Lambda _{\\\\alpha ,\\\\beta ,\\\\widetilde{c}}(z) :=\\\\Lambda _{\\\\alpha ,\\\\beta ,\\\\widetilde{c}}^{(0)}(z) &=&{\\\\left\\\\lbrace \\\\begin{array}{ll}\\\\breve{\\\\theta } \\\\cdot z - \\\\widetilde{c} \\\\cdot \\\\alpha \\\\cdot \\\\log \\\\Big (1 - \\\\frac{z^{2}}{\\\\widetilde{c}^{2} \\\\cdot \\\\beta ^{2} }\\\\Big ),\\\\qquad \\\\textrm {if } \\\\ z \\\\in \\\\, ]-\\\\widetilde{c}\\\\cdot \\\\beta ,\\\\widetilde{c}\\\\cdot \\\\beta [,\\\\\\\\\\\\infty , \\\\hspace{130.88284pt} \\\\textrm {else} .\\\\end{array}\\\\right.\",\n \"}$ Notice that $\\\\Lambda _{\\\\alpha ,\\\\beta ,\\\\widetilde{c}}(0) = 0$ , $\\\\lim _{z \\\\rightarrow - \\\\widetilde{c}\\\\cdot \\\\beta }\\\\Lambda _{\\\\alpha ,\\\\beta ,\\\\widetilde{c}}(z) = -\\\\infty = $ , and $\\\\lim _{z \\\\rightarrow \\\\widetilde{c}\\\\cdot \\\\beta }\\\\Lambda _{\\\\alpha ,\\\\beta ,\\\\widetilde{c}}(z) = \\\\infty $ .\",\n \"Furthermore, $\\\\lim _{z \\\\rightarrow - \\\\widetilde{c}\\\\cdot \\\\beta }\\\\Lambda _{\\\\alpha ,\\\\beta ,\\\\widetilde{c}}^{\\\\prime }(z) = -\\\\infty = a_{F}$ , and $\\\\lim _{z \\\\rightarrow \\\\widetilde{c}\\\\cdot \\\\beta }\\\\Lambda _{\\\\alpha ,\\\\beta ,\\\\widetilde{c}}^{\\\\prime }(z) = \\\\infty = b_{F}$ .\",\n \"To proceed, the formula (REF ) (respectively, (REF )) collapses to $\\\\varphi _{\\\\alpha ,\\\\beta ,\\\\widetilde{c}}(t):=\\\\varphi _{\\\\alpha ,\\\\beta ,\\\\widetilde{c}}^{(0)}(t)= \\\\widetilde{c} \\\\cdot \\\\alpha \\\\cdot \\\\Big \\\\lbrace \\\\sqrt{1 + \\\\beta ^{2}\\\\cdot \\\\Big (\\\\frac{1-t}{\\\\alpha }\\\\Big )^2} \\\\, - \\\\, 1+ \\\\log \\\\frac{2 \\\\cdot \\\\Big (\\\\sqrt{1 + \\\\beta ^{2}\\\\cdot \\\\Big (\\\\frac{1-t}{\\\\alpha } \\\\Big )^2} \\\\, - \\\\, 1\\\\Big )}{\\\\beta ^{2} \\\\cdot \\\\Big (\\\\frac{1-t}{\\\\alpha }\\\\Big )^{2}} \\\\Big \\\\rbrace \\\\ \\\\in [0,\\\\infty [, \\\\ \\\\ t \\\\in \\\\, ]-\\\\infty , \\\\infty [ \\\\, = \\\\, ]a_{F}, b_{F}[.$ Notice that $\\\\varphi _{\\\\alpha ,\\\\beta ,\\\\widetilde{c}}(1) = 0$ , $\\\\varphi _{\\\\alpha ,\\\\beta }^{\\\\prime }(1) = 0$ , $\\\\varphi _{\\\\alpha ,\\\\beta ,\\\\widetilde{c}}(-\\\\infty ) = \\\\infty $ and $\\\\varphi _{\\\\alpha ,\\\\beta ,\\\\widetilde{c}}(\\\\infty ) = \\\\infty $ .\",\n \"Moreover, $\\\\varphi _{\\\\alpha ,\\\\beta ,\\\\widetilde{c}}^{\\\\prime }(-\\\\infty ) =\\\\varphi _{\\\\alpha ,\\\\beta ,\\\\widetilde{c}}^{\\\\prime }(a_{F}) =-\\\\widetilde{c} \\\\cdot \\\\beta $ and $\\\\varphi _{\\\\alpha ,\\\\beta ,\\\\widetilde{c}}^{\\\\prime }(\\\\infty ) =\\\\varphi _{\\\\alpha ,\\\\beta ,\\\\widetilde{c}}^{\\\\prime }(b_{F}) =\\\\widetilde{c} \\\\cdot \\\\beta $ .\",\n \"From the generator $\\\\varphi _{\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c}}$ given in (REF ), we construct the corresponding divergence (cf.\",\n \"(REF )) $& & \\\\hspace{-19.91684pt}D_{\\\\varphi _{\\\\alpha ,\\\\beta ,\\\\widetilde{c}}}(\\\\mathbf {Q},\\\\mathbf {P})= \\\\sum \\\\limits _{k=1}^{K} p_{k} \\\\cdot \\\\varphi _{\\\\alpha ,\\\\beta ,\\\\widetilde{c}}\\\\Big (\\\\frac{q_{k}}{p_{k}}\\\\Big )\\\\nonumber \\\\\\\\& & \\\\hspace{-19.91684pt}= \\\\widetilde{c} \\\\cdot \\\\alpha \\\\cdot \\\\sum \\\\limits _{k=1}^{K}p_{k} \\\\cdot \\\\Big \\\\lbrace \\\\sqrt{1 + \\\\beta ^{2}\\\\cdot \\\\Big (\\\\frac{1-\\\\frac{q_{k}}{p_{k}}}{\\\\alpha }\\\\Big )^2} \\\\, - \\\\, 1+ \\\\log \\\\frac{2 \\\\cdot \\\\Big (\\\\sqrt{1 + \\\\beta ^{2}\\\\cdot \\\\Big (\\\\frac{1-\\\\frac{q_{k}}{p_{k}}}{\\\\alpha } \\\\Big )^2} \\\\, - \\\\, 1\\\\Big )}{\\\\beta ^{2} \\\\cdot \\\\Big (\\\\frac{1-\\\\frac{q_{k}}{p_{k}}}{\\\\alpha }\\\\Big )^{2}} \\\\Big \\\\rbrace ,\\\\qquad \\\\mathbf {P} \\\\in \\\\mathbb {R}_{\\\\ge 0}^{K}, \\\\mathbf {Q} \\\\in \\\\mathbb {R}^{K}.$ As above, we can particularly include the case where $p_{k}=0$ in combination with $q_{k} \\\\ne 0$ , since $\\\\lim _{t \\\\rightarrow 0_{+}} t \\\\cdot \\\\varphi _{\\\\alpha ,\\\\beta ,\\\\widetilde{c}}(\\\\frac{1}{t})= \\\\widetilde{c} \\\\cdot \\\\beta $ and $\\\\lim _{t \\\\rightarrow 0_{-}} t \\\\cdot \\\\varphi _{\\\\alpha ,\\\\beta ,\\\\widetilde{c}}(\\\\frac{1}{t})= - \\\\widetilde{c} \\\\cdot \\\\beta $ are both finite.\",\n \"This ends the current Example REF .\",\n \"As a side effect in the above-mentioned Example REF , for fixed $\\\\beta _{2}, \\\\alpha ,\\\\widetilde{c}$ notice the interesting behaviour (e.g.\",\n \"with respect to $int(dom(F)) = ]a_{F},b_{F}[$ and the range of $\\\\varphi ^{\\\\prime }$ ) as $\\\\beta _{1}$ moves from $]0,\\\\beta _{2}[$ to $\\\\beta _{2}$ and further to $]\\\\beta _{2}, \\\\infty [$ .\",\n \"Remark 47 The characterization of the probability distribution $$ in (REF ) which may result from Theorem REF — as seen through the above examples — considerably improves other approaches which make use of their identification through the concept of power variance functions of Natural Exponential Families, as developed by Tweedie [369], Morris [267], Letac & Mora [214], and others.\",\n \"This approach has been used in Broniatowski [58] in a similar perspective as developed here, but can not be extended outside the range of power divergences, in contrast with the Examples REF , REF , REF and REF which can only be handled as a consequence of Theorem REF .\",\n \"To continue with our general procedure, suppose now that for a divergence generator $\\\\varphi $ of interest we have concretely/explicitly found (e.g.\",\n \"by direct calculations or via our $F-$ connection in Theorem REF , see also Remark REF ) its Fenchel-Legendre transform $\\\\Lambda = \\\\varphi ^{*}$ ; for this “candidate”, in order to achieve the desired representability (REF ) it remains to verify that $\\\\exp (\\\\Lambda (z)) = \\\\int _{\\\\mathbb {R}}e^{z \\\\cdot y} \\\\, d\\\\mathbb {} (y),\\\\qquad z \\\\in \\\\mathbb {R},$ for some probability distribution/measure $\\\\mathbb {} $ on the real line (the light-tailedness in the sense of finiteness on some open interval containing zero, will be typically guaranteed automatically by the assumptions on $\\\\varphi $ ); of course, this is equivalent to “the existence ” of a random variable $W$ whose moment generating function is equal to $\\\\exp (\\\\Lambda )$ (and thus, its cumulant generating function (log moment generating function) is $\\\\Lambda $ ), i.e.\",\n \"$\\\\exp (\\\\Lambda (z)) = E_{\\\\mathbb {\\\\Pi }}[\\\\exp (z \\\\cdot W)]\\\\qquad z \\\\in \\\\mathbb {R},$ with $\\\\mathbb {\\\\Pi }[W \\\\in \\\\cdot \\\\, ] = \\\\mathbb {}[ \\\\, \\\\cdot \\\\,]$ ); recall that from this, we need to simulate a sequence $(W_{i})_{i\\\\in \\\\mathbb {N}}$ of i.i.d.\",\n \"copies of $W$ which are the crucial building ingredients of $\\\\xi _{n}^{\\\\mathbf {W}}$ in Theorem REF , respectively, of $\\\\xi _{n,\\\\mathbf {X}}^{w\\\\mathbf {W}}$ in Theorem REF .\",\n \"For the above-mentioned Examples REF to REF , we can give explicit solutions to the representabilities (REF ) respectively (REF ); this is achieved in the following Examples REF to REF (notice that the corresponding supports of $\\\\mathbb {}$ are explicitly mentioned in the summarizing Table 1 above): Example 48 for the power-divergence context of Example REF we obtain: (a) Case $\\\\gamma =0$ , $\\\\widetilde{c} >0$ : $\\\\Lambda _{0,\\\\widetilde{c}}(z)= - \\\\widetilde{c} \\\\cdot \\\\log \\\\left( 1 - \\\\frac{z}{\\\\widetilde{c}} \\\\right)$ (cf.\",\n \"(REF )) is the cumulant generating function of the Gamma distribution $\\\\mathbb {} = GAM(\\\\widetilde{c},\\\\widetilde{c})$ with rate parameter (inverse scale parameter) $\\\\widetilde{c}$ and shape parameter $\\\\widetilde{c}$ ; hence, $\\\\varphi _{0,\\\\widetilde{c}} \\\\in \\\\Upsilon (]0,\\\\infty [)$ .\",\n \"Prominent special case $\\\\widetilde{c} =1$ : $\\\\mathbb {} = GAM(1,1) = EXP(1)$ is the exponential distribution with mean 1.\",\n \"Type: $\\\\mathbb {}$ is an infinitely divisible (cf.\",\n \"Proposition REF ) continuous distribution with density $f(y) := \\\\frac{\\\\widetilde{c}^{\\\\widetilde{c}} \\\\cdot y^{\\\\widetilde{c}-1} \\\\cdot e^{-\\\\widetilde{c} \\\\cdot y} }{\\\\Gamma (\\\\widetilde{c})} \\\\cdot {1}_{]0,\\\\infty [}(y)$ ($y \\\\in \\\\mathbb {R}$ ).\",\n \"Behaviour at zero: $\\\\mathbb {}[ \\\\, ]0,\\\\infty [ \\\\, ] = \\\\mathbb {\\\\Pi }[W>0]=1$ .\",\n \"Corresponding generator: $\\\\varphi _{0,\\\\widetilde{c}} = \\\\widetilde{c} \\\\cdot \\\\varphi _{0}$ (cf.\",\n \"(REF ), (REF )) of the $\\\\widetilde{c}-$ fold of the reversed Kullback-Leibler divergence (reversed relative entropy) given in the second line of (REF ).\",\n \"Sums: for i.i.d.\",\n \"copies $(W_{i})_{i \\\\in \\\\mathbb {N}}$ of $W$ , the probability distribution of $\\\\breve{W} := \\\\sum _{i\\\\in I_{k}^{(n)}} W_{i}$ (cf.\",\n \"Remark REF (ii)) is $GAM(\\\\widetilde{c},\\\\widetilde{c} \\\\cdot card(I_{k}^{(n)}))$ .\",\n \"(b) Case $\\\\gamma \\\\in \\\\, ]0,1[$ , $\\\\widetilde{c} >0$ : $\\\\Lambda _{\\\\gamma ,\\\\widetilde{c}}^{(0)}(z)= \\\\frac{\\\\widetilde{c}}{\\\\gamma } \\\\cdot \\\\Big \\\\lbrace \\\\left( \\\\frac{\\\\gamma -1}{\\\\widetilde{c}} \\\\cdot z + 1 \\\\right)^{\\\\frac{\\\\gamma }{\\\\gamma -1}} -1 \\\\Big \\\\rbrace $ (cf.\",\n \"(REF )) is the cumulant generating function of the Compound-Poisson-Gamma distribution $\\\\mathbb {} = C(POI(\\\\theta ),GAM(\\\\alpha ,\\\\beta ))$ with $\\\\theta = \\\\frac{\\\\widetilde{c}}{\\\\gamma } > 0$ , rate parameter (inverse scale parameter) $\\\\alpha = \\\\frac{\\\\widetilde{c}}{1-\\\\gamma } >0$ , and shape parameter $\\\\beta = \\\\frac{\\\\gamma }{1-\\\\gamma } >0$ .\",\n \"In other words, $W$ has the comfortably simulable form $W = \\\\sum _{i=1}^{N} \\\\widetilde{W}_{i}$ with the usual convention $\\\\sum _{i=1}^{0} \\\\widetilde{W}_{i} := 0$ for some i.i.d.\",\n \"sequence $( \\\\widetilde{W}_{i})_{i\\\\in \\\\mathbb {N}}$ of Gamma $GAM(\\\\alpha ,\\\\beta )$ distributed random variables (with parameter-pair $(\\\\alpha ,\\\\beta )$ ) and some independent $POI(\\\\theta )-$ distributed random variable $N$ .\",\n \"Hence, $\\\\varphi _{\\\\gamma ,\\\\widetilde{c}} \\\\in \\\\Upsilon (]0,\\\\infty [)$ .\",\n \"Type: $\\\\mathbb {}$ is an infinitely divisible distribution (cf.\",\n \"Proposition REF ), mixture of a one-point distribution at zero and a continuous distribution on $[0,\\\\infty [$ , with $\\\\mathbb {}[\\\\lbrace 0\\\\rbrace ] = \\\\mathbb {\\\\Pi }[W = 0]= e^{-\\\\theta }$ and $\\\\mathbb {}[B] = \\\\mathbb {\\\\Pi }[W \\\\in B] = \\\\int _{B} f_{\\\\widetilde{c},\\\\gamma }(u) \\\\, du$ for every (measurable) subset of $]0,\\\\infty [$ having density $& & \\\\hspace{-22.76228pt}f_{C(POI(\\\\theta ),GAM(\\\\alpha ,\\\\beta ))}(y): = \\\\frac{\\\\exp \\\\left(- \\\\alpha \\\\cdot y - \\\\theta \\\\right)}{y}\\\\cdot \\\\sum _{k=1}^{\\\\infty } \\\\frac{\\\\theta ^{k} \\\\cdot (\\\\alpha y)^{k\\\\beta }}{k!\",\n \"\\\\cdot \\\\Gamma (k\\\\beta )}\\\\cdot {1}_{]0,\\\\infty [}(y)\\\\\\\\& & \\\\hspace{-22.76228pt}= \\\\frac{1}{y} \\\\cdot \\\\exp \\\\left(-\\\\widetilde{c} \\\\cdot \\\\left(\\\\frac{y}{1-\\\\gamma } + \\\\frac{1}{\\\\gamma } \\\\right) \\\\right)\\\\cdot \\\\sum _{k=1}^{\\\\infty } \\\\frac{a_{k}}{k!}\",\n \"\\\\cdot \\\\widetilde{c}^{k/(1-\\\\gamma )} \\\\cdot \\\\gamma ^{-k}\\\\cdot (1-\\\\gamma )^{-k\\\\gamma /(1-\\\\gamma )} \\\\cdot y^{k\\\\gamma /(1-\\\\gamma )}\\\\cdot {1}_{[0,\\\\infty [}(y)=: f_{\\\\widetilde{c},\\\\gamma }(y),\\\\quad y \\\\in \\\\mathbb {R},\\\\nonumber $ where $a_{k} := 1/\\\\Gamma (\\\\frac{k \\\\cdot \\\\gamma }{1-\\\\gamma } )$ (see e.g.\",\n \"Aalen [1] with a different parametrization).\",\n \"Behaviour at zero: $\\\\mathbb {}[ \\\\, [0,\\\\infty [ \\\\, ] = \\\\mathbb {\\\\Pi }[W \\\\ge 0]=1$ , $\\\\mathbb {}[ \\\\, \\\\lbrace 0\\\\rbrace \\\\, ] = \\\\mathbb {\\\\Pi }[W = 0]= e^{-\\\\theta }$ .\",\n \"Corresponding generator: $\\\\varphi _{\\\\gamma ,\\\\widetilde{c}}^{(0)} = \\\\widetilde{c} \\\\cdot \\\\varphi _{\\\\gamma }$ (cf.\",\n \"(REF ), (REF )) of the power divergence given in the third line of (REF ); recall that the special case $\\\\gamma =\\\\frac{1}{2}$ corresponds to the prominent (multiple of the squared) Hellinger distance.\",\n \"Sums: for i.i.d.\",\n \"copies $(W_{i})_{i \\\\in \\\\mathbb {N}}$ of $W$ , the probability distribution of $\\\\breve{W} := \\\\sum _{i\\\\in I_{k}^{(n)}} W_{i}$ (cf.\",\n \"Remark REF (ii)) is $C(POI(\\\\breve{\\\\theta }),GAM(\\\\alpha ,\\\\beta ))$ with $\\\\breve{\\\\theta } = \\\\frac{\\\\widetilde{c} \\\\cdot card(I_{k}^{(n)})}{\\\\gamma } > 0$ , $\\\\alpha = \\\\frac{\\\\widetilde{c}}{1-\\\\gamma } >0$ , $\\\\beta = \\\\frac{\\\\gamma }{1-\\\\gamma } >0$ .\",\n \"(c) Case $\\\\gamma =2$ , $\\\\widetilde{c} >0$ : $\\\\Lambda _{2,\\\\widetilde{c}}^{(0)}(z)= \\\\frac{z^{2}}{2 \\\\widetilde{c}} + z$ (cf.\",\n \"(REF ) ) is the well-known cumulant generating function of the Normal distribution (Gaussian distribution) $\\\\mathbb {} = N(1,\\\\frac{1}{\\\\widetilde{c}})$ with mean 1 and variance $\\\\frac{1}{\\\\widetilde{c}}$ .\",\n \"Thus, $\\\\varphi _{2,\\\\widetilde{c}} \\\\in \\\\Upsilon (]-\\\\infty ,\\\\infty [)$ .\",\n \"Type: $\\\\mathbb {}$ is an infinitely divisible (cf.\",\n \"Proposition REF ) continuous distribution with density $f_{N(1,\\\\frac{1}{\\\\widetilde{c}})}(y) := \\\\sqrt{\\\\frac{\\\\widetilde{c}}{2 \\\\pi }} \\\\cdot \\\\exp (- \\\\frac{\\\\widetilde{c} \\\\cdot (y-1)^2}{2} )$ , ($y \\\\in \\\\mathbb {R}$ ).\",\n \"Behaviour at zero: $\\\\mathbb {}[ \\\\, ]0,\\\\infty [ \\\\, ] = \\\\mathbb {\\\\Pi }[W > 0]=\\\\int _{0}^{\\\\infty } f_{N(1,\\\\frac{1}{\\\\widetilde{c}})}(u) \\\\, du \\\\in \\\\, ]0,1[$ , $\\\\mathbb {}[ \\\\, \\\\lbrace 0\\\\rbrace \\\\, ] = \\\\mathbb {\\\\Pi }[W = 0]= 0$ .\",\n \"Corresponding generator: $\\\\varphi _{2,\\\\widetilde{c}}^{(0)} = \\\\widetilde{c} \\\\cdot \\\\varphi _{2}$ (cf.\",\n \"(REF ), (REF )) is the generator of the $\\\\widetilde{c}-$ fold of the half Pearson-chisquare divergence given in the sixth line of (REF ).\",\n \"Sums: for i.i.d.\",\n \"copies $(W_{i})_{i \\\\in \\\\mathbb {N}}$ of $W$ , the probability distribution of $\\\\breve{W} := \\\\sum _{i\\\\in I_{k}^{(n)}} W_{i}$ (cf.\",\n \"Remark REF (ii)) is $N(card(I_{k}^{(n)}),\\\\frac{card(I_{k}^{(n)})}{\\\\widetilde{c}})$ .\",\n \"(d) Case $\\\\gamma <0$ , $\\\\widetilde{c} >0$ : $\\\\Lambda _{\\\\gamma ,\\\\widetilde{c}}^{(0)}(z) = \\\\frac{\\\\widetilde{c}}{\\\\gamma } \\\\cdot \\\\left\\\\lbrace \\\\left( \\\\frac{\\\\gamma -1}{\\\\widetilde{c}} \\\\cdot z +1 \\\\right)^{\\\\frac{\\\\gamma }{\\\\gamma -1}} -1 \\\\right\\\\rbrace $ (cf.\",\n \"(REF )) is the cumulant generating function of a “tilted (i.e.\",\n \"negatively distorted) stable distribution” $\\\\mathbb {}[ \\\\, \\\\cdot \\\\,] = \\\\mathbb {\\\\Pi }[W \\\\in \\\\cdot \\\\, ]$ of a random variable $W$ , which can be constructed as follows: let $Z$ be an auxiliary random variable (having density $f_{Z}$ and support $supp(Z) = [0,\\\\infty [$ ) of a stable law with parameter-quadruple $(\\\\frac{-\\\\gamma }{1-\\\\gamma },1,0,-\\\\frac{\\\\widetilde{c}^{1/(1-\\\\gamma )} \\\\cdot (1-\\\\gamma )^{-\\\\gamma /(1-\\\\gamma )}}{\\\\gamma })$ in terms of the “form-B notation” on p.12 in Zolotarev [428]; by applying a general Laplace-transform result on p.112 of the same text we can deduce $M_{Z}(z) := E_{\\\\mathbb {\\\\Pi }}[\\\\exp (z \\\\cdot Z)]= \\\\int _{0}^{\\\\infty } \\\\exp (z \\\\cdot y) \\\\cdot f_{Z}(y) \\\\, dy \\\\hspace{-5.69046pt} &=& \\\\hspace{-5.69046pt}{\\\\left\\\\lbrace \\\\begin{array}{ll}\\\\exp \\\\Big (\\\\frac{\\\\widetilde{c}^{1/(1-\\\\gamma )} \\\\cdot (1-\\\\gamma )^{-\\\\gamma /(1-\\\\gamma )}}{\\\\gamma }\\\\cdot (-z)^{\\\\alpha } \\\\Big ),\\\\quad \\\\textrm {if } \\\\ z \\\\in ]-\\\\infty ,0] , \\\\\\\\\\\\infty , \\\\hspace{150.79968pt} \\\\textrm {if } \\\\ z \\\\in \\\\, ]0,\\\\infty [, \\\\\\\\\\\\end{array}\\\\right.\",\n \"}$ where $\\\\alpha := - \\\\frac{\\\\gamma }{1-\\\\gamma } \\\\in \\\\, ]0,1[$ .\",\n \"Since $0 \\\\notin int(dom(M_{Z}))$ (and thus, $Z$ does not have light-tails) we have to tilt (dampen) the density in order to extend the effective domain.\",\n \"Accordingly, let $W$ be a random variable having density $f_{W}(y)\\\\ :=\\\\ \\\\frac{\\\\exp \\\\lbrace -\\\\frac{y \\\\cdot \\\\widetilde{c}}{1-\\\\gamma }\\\\rbrace }{\\\\exp \\\\lbrace \\\\widetilde{c}/\\\\gamma \\\\rbrace }\\\\cdot f_{Z}(y)\\\\cdot {1}_{]0,\\\\infty [}(y),\\\\qquad y \\\\in \\\\mathbb {R},\\\\qquad \\\\text{(cf.\",\n \"(\\\\ref {brostu3:fo.norweiemp7a}))}.\\\\nonumber $ Then one can straightforwardly deduce from (REF ) that $\\\\int _{0}^{\\\\infty } f_{W}(y) \\\\, dy =1$ and that $M_{W}(z) := E_{\\\\mathbb {\\\\Pi }}[\\\\exp (z \\\\cdot W)]= \\\\int _{0}^{\\\\infty } \\\\exp (z \\\\cdot y) \\\\cdot f_{W}(y) \\\\, dy \\\\hspace{-5.69046pt} &=& \\\\hspace{-5.69046pt}{\\\\left\\\\lbrace \\\\begin{array}{ll}\\\\exp \\\\left(\\\\frac{\\\\widetilde{c}}{\\\\gamma } \\\\cdot \\\\left\\\\lbrace \\\\left( \\\\frac{\\\\gamma -1}{\\\\widetilde{c}} \\\\cdot z +1 \\\\right)^{\\\\frac{\\\\gamma }{\\\\gamma -1}} -1 \\\\right\\\\rbrace \\\\right),\\\\qquad \\\\textrm {if } \\\\ z \\\\in ]-\\\\infty ,\\\\frac{\\\\widetilde{c}}{1-\\\\gamma }] , \\\\\\\\\\\\infty , \\\\hspace{156.49014pt} \\\\textrm {if } \\\\ z \\\\in \\\\, ]\\\\frac{\\\\widetilde{c}}{1-\\\\gamma },\\\\infty [ .\",\n \"\\\\\\\\\\\\end{array}\\\\right.\",\n \"}\\\\nonumber $ Hence, $\\\\varphi _{\\\\gamma ,\\\\widetilde{c}} \\\\in \\\\Upsilon (]0,\\\\infty [)$ .\",\n \"Type: $\\\\mathbb {}$ is an infinitely divisible (cf.\",\n \"Proposition REF ) continuous distribution with density $f_{W}$ .\",\n \"Behaviour at zero: $\\\\mathbb {}[ \\\\, ] 0,\\\\infty [ \\\\, ] = \\\\mathbb {\\\\Pi }[W > 0]=1$ .\",\n \"Corresponding generator: $\\\\varphi _{\\\\gamma ,\\\\widetilde{c}}^{(0)} = \\\\widetilde{c} \\\\cdot \\\\varphi _{\\\\gamma }$ (cf.\",\n \"(REF ), (REF )) of the power divergence given in the first line of (REF ).\",\n \"Sums: for i.i.d.\",\n \"copies $(W_{i})_{i \\\\in \\\\mathbb {N}}$ of $W$ , the probability distribution of $\\\\breve{W} := \\\\sum _{i\\\\in I_{k}^{(n)}} W_{i}$ (cf.\",\n \"Remark REF (ii)) has density $f_{\\\\breve{W}}(y)\\\\ :=\\\\ \\\\frac{\\\\exp \\\\lbrace -\\\\frac{y \\\\cdot \\\\widetilde{c}}{1-\\\\gamma }\\\\rbrace }{\\\\exp \\\\lbrace \\\\widetilde{c} \\\\cdot card(I_{k}^{(n)})/\\\\gamma \\\\rbrace }\\\\cdot f_{\\\\breve{Z}}(y)\\\\cdot {1}_{]0,\\\\infty [}(y),\\\\qquad y \\\\in \\\\mathbb {R},$ where $\\\\breve{Z}$ is a random variable with density $f_{\\\\breve{Z}}$ of a stable law with parameter-quadruple $(\\\\frac{-\\\\gamma }{1-\\\\gamma },1,0,-card(I_{k}^{(n)}) \\\\cdot \\\\frac{\\\\widetilde{c}^{1/(1-\\\\gamma )} \\\\cdot (1-\\\\gamma )^{-\\\\gamma /(1-\\\\gamma )}}{\\\\gamma })$ .\",\n \"(e) Case $\\\\gamma >2$ , $\\\\widetilde{c} >0$ : $\\\\Lambda _{\\\\gamma ,\\\\widetilde{c}}^{(0)}(z) = \\\\frac{\\\\widetilde{c}}{\\\\gamma } \\\\cdot \\\\left\\\\lbrace \\\\left( \\\\frac{\\\\gamma -1}{\\\\widetilde{c}} \\\\cdot z +1 \\\\right)^{\\\\frac{\\\\gamma }{\\\\gamma -1}} -1 \\\\right\\\\rbrace $ (cf.\",\n \"(REF )) is the cumulant generating function of a “distorted stable distribution” $\\\\mathbb {}[ \\\\, \\\\cdot \\\\,] = \\\\mathbb {\\\\Pi }[W \\\\in \\\\cdot \\\\, ]$ of a random variable $W$ , which can be constructed as follows: let $Z$ be an auxiliary random variable (having density $f_{Z}$ and support $supp(Z) = ]-\\\\infty ,\\\\infty ]$ ) of a stable law with parameter-quadruple $(\\\\frac{\\\\gamma }{\\\\gamma -1},1,0,\\\\frac{\\\\widetilde{c}^{1/(1-\\\\gamma )} \\\\cdot (\\\\gamma -1)^{\\\\gamma /(\\\\gamma -1)}}{\\\\gamma })$ in terms of the above-mentioned “form-B notation” ; by applying a general Laplace-transform result on p. 112 of Zolotarev [428], we can derive $M_{Z}(z) := E_{\\\\mathbb {\\\\Pi }}[\\\\exp (z \\\\cdot Z)]= \\\\int _{0}^{\\\\infty } \\\\exp (z \\\\cdot y) \\\\cdot f_{Z}(y) \\\\, dy \\\\hspace{-5.69046pt} &=& \\\\hspace{-5.69046pt}{\\\\left\\\\lbrace \\\\begin{array}{ll}\\\\exp \\\\Big (\\\\frac{\\\\widetilde{c}^{1/(1-\\\\gamma )} \\\\cdot (\\\\gamma -1 )^{\\\\gamma /(\\\\gamma -1)}}{\\\\gamma }\\\\cdot (-z)^{\\\\alpha } \\\\Big ),\\\\quad \\\\textrm {if } \\\\ z \\\\in ]-\\\\infty ,0] , \\\\\\\\\\\\infty , \\\\hspace{145.10922pt} \\\\textrm {if } \\\\ z \\\\in \\\\, ]0,\\\\infty [, \\\\\\\\\\\\end{array}\\\\right.\",\n \"}$ where $\\\\alpha := \\\\frac{\\\\gamma }{\\\\gamma -1} \\\\in \\\\, ]1,2[$ .\",\n \"Since $0 \\\\notin int(dom(M_{Z}))$ (and thus, $Z$ does not have light-tails) we have to distort the density in order to extend the effective domain.\",\n \"Accordingly, let $W$ be a random variable having density $f_{W}(y)\\\\ :=\\\\ \\\\frac{\\\\exp \\\\lbrace \\\\frac{y \\\\cdot \\\\widetilde{c}}{\\\\gamma -1}\\\\rbrace }{\\\\exp \\\\lbrace \\\\widetilde{c}/\\\\gamma \\\\rbrace }\\\\cdot f_{Z}(- y), \\\\qquad y \\\\in \\\\mathbb {R},\\\\qquad \\\\text{(cf.\",\n \"(\\\\ref {brostu3:fo.norweiemp7atwo}))}.\\\\nonumber $ Then one can straightforwardly deduce from (REF ) that $\\\\int _{-\\\\infty }^{\\\\infty } f_{W}(y) \\\\, dy =1$ and that $M_{W}(z) := E_{\\\\mathbb {\\\\Pi }}[\\\\exp (z \\\\cdot W)]= \\\\int _{-\\\\infty }^{\\\\infty } \\\\exp (z \\\\cdot y) \\\\cdot f_{W}(y) \\\\, dy \\\\hspace{-5.69046pt} &=& \\\\hspace{-5.69046pt}{\\\\left\\\\lbrace \\\\begin{array}{ll}\\\\exp \\\\left(\\\\frac{\\\\widetilde{c}}{\\\\gamma } \\\\cdot \\\\left\\\\lbrace \\\\left( \\\\frac{\\\\gamma -1}{\\\\widetilde{c}} \\\\cdot z +1 \\\\right)^{\\\\frac{\\\\gamma }{\\\\gamma -1}} -1 \\\\right\\\\rbrace \\\\right),\\\\qquad \\\\textrm {if } \\\\ z \\\\in [- \\\\frac{\\\\widetilde{c}}{\\\\gamma -1},\\\\infty [ , \\\\\\\\\\\\infty , \\\\hspace{156.49014pt} \\\\textrm {if } \\\\ z \\\\in \\\\, ]-\\\\infty , -\\\\frac{\\\\widetilde{c}}{\\\\gamma -1}[ .\",\n \"\\\\\\\\\\\\end{array}\\\\right.\",\n \"}\\\\nonumber $ Thus, $\\\\varphi _{\\\\gamma ,\\\\widetilde{c}} \\\\in \\\\Upsilon (]-\\\\infty ,\\\\infty [)$ .\",\n \"Type: $\\\\mathbb {}$ is an infinitely divisible (cf.\",\n \"Proposition REF ) continuous distribution with density $f_{W}$ .\",\n \"Behaviour at zero: $\\\\mathbb {}[ \\\\, ]0,\\\\infty [ \\\\, ] = \\\\mathbb {\\\\Pi }[W > 0]=\\\\int _{0}^{\\\\infty } f_{W}(u) \\\\, du \\\\in \\\\, ]0,1[$ , $\\\\mathbb {}[ \\\\, \\\\lbrace 0\\\\rbrace \\\\, ] = \\\\mathbb {\\\\Pi }[W = 0]= 0$ .\",\n \"Corresponding generator: $\\\\varphi _{\\\\gamma ,\\\\widetilde{c}}^{(0)} = \\\\widetilde{c} \\\\cdot \\\\varphi _{\\\\gamma }$ (cf.\",\n \"(REF ), (REF )) of the power divergence given in the seventh line of (REF ).\",\n \"Sums: for i.i.d.\",\n \"copies $(W_{i})_{i \\\\in \\\\mathbb {N}}$ of $W$ , the probability distribution of $\\\\breve{W} := \\\\sum _{i\\\\in I_{k}^{(n)}} W_{i}$ (cf.\",\n \"Remark REF (ii)) has density $f_{\\\\breve{W}}(y)\\\\ :=\\\\ \\\\frac{\\\\exp \\\\lbrace \\\\frac{y \\\\cdot \\\\widetilde{c}}{\\\\gamma -1}\\\\rbrace }{\\\\exp \\\\lbrace \\\\widetilde{c} \\\\cdot card(I_{k}^{(n)})/\\\\gamma \\\\rbrace }\\\\cdot f_{\\\\breve{Z}}(- y), \\\\qquad y \\\\in \\\\mathbb {R},\\\\nonumber $ where $\\\\breve{Z}$ is a random variable with density $f_{\\\\breve{Z}}$ of a stable law with parameter-quadruple $(\\\\frac{\\\\gamma }{\\\\gamma -1},1,0,card(I_{k}^{(n)})\\\\cdot \\\\frac{\\\\widetilde{c}^{1/(1-\\\\gamma )} \\\\cdot (\\\\gamma -1)^{\\\\gamma /(\\\\gamma -1)}}{\\\\gamma })$ .\",\n \"(f) Case $\\\\gamma \\\\in ]1,2[$ , $\\\\widetilde{c} >0$ : one still has the (cumulant-generating-function) candidate $\\\\Lambda _{\\\\gamma ,\\\\widetilde{c}}^{(0)}(z) = \\\\frac{\\\\widetilde{c}}{\\\\gamma } \\\\cdot \\\\left\\\\lbrace \\\\left( \\\\frac{\\\\gamma -1}{\\\\widetilde{c}} \\\\cdot z +1 \\\\right)^{\\\\frac{\\\\gamma }{\\\\gamma -1}} -1 \\\\right\\\\rbrace $ (cf.\",\n \"(REF )), but for the crucial exponent there holds $\\\\frac{\\\\gamma }{\\\\gamma -1} > 2$ .\",\n \"From this, we conjecture that $\\\\mathbb {}$ becomes a signed finite measure with total mass 1, i.e.\",\n \"it has a density (with respect to some dominating measure) with positive and negative values which “integrates to 1” ; accordingly, our BS method can not be applied to this situation.\",\n \"Remark 49 As a continuation of Remark REF and the note in the third line after (REF ), we have shown as a side effect that for $\\\\gamma \\\\in \\\\, ]-\\\\infty ,-1] \\\\, \\\\cup \\\\, ]0,1[ \\\\, \\\\cup \\\\, [2,\\\\infty [$ the distributions $\\\\mathbb {}_{\\\\gamma }$ and $\\\\mathbb {}_{1-\\\\gamma }$ of Example REF (b)-(e) are inverse to each other.\",\n \"Example 50 for the power-divergence context of Example REF we obtain: (a) Case $\\\\gamma =1$ , $\\\\widetilde{c} >0$ , anchor point $c=0$ : $\\\\Lambda _{1,\\\\widetilde{c}}(z) =\\\\widetilde{c} \\\\cdot \\\\left( \\\\exp (\\\\frac{z}{\\\\widetilde{c}}) - 1 \\\\right)$ (cf.\",\n \"(REF )) is the cumulant generating function of $\\\\mathbb {} = \\\\frac{1}{\\\\widetilde{c}} \\\\cdot POI(\\\\widetilde{c})$ being the “$\\\\frac{1}{\\\\widetilde{c}}-$ fold Poisson distribution with mean $\\\\widetilde{c}$ ” which means that $W = \\\\frac{1}{\\\\widetilde{c}} \\\\cdot Z$ for a $POI(\\\\widetilde{c})-$ distributed random variable $Z$ .\",\n \"Thus, $\\\\varphi _{1,\\\\widetilde{c}} \\\\in \\\\Upsilon (]0,\\\\infty [)$ .\",\n \"Prominent special case $\\\\widetilde{c} =1$ : $\\\\mathbb {} = POI(1)$ is the Poisson distribution with mean 1.\",\n \"Type: $\\\\mathbb {}$ is an infinitely divisible (cf.\",\n \"Proposition REF ) discrete distribution with frequencies: $\\\\mathbb {\\\\Pi }[W=\\\\ell \\\\cdot \\\\frac{1}{\\\\widetilde{c}}]= \\\\exp (-\\\\widetilde{c}) \\\\cdot \\\\frac{ \\\\widetilde{c}^{\\\\ell }}{\\\\ell !\",\n \"}$ for all nonnegative integers $\\\\ell \\\\in \\\\mathbb {N}_{0}$ (and zero elsewhere).\",\n \"Behaviour at zero: $\\\\mathbb {\\\\Pi }[W\\\\ge 0]=1$ , $\\\\mathbb {\\\\Pi }[W=0]= \\\\exp (-\\\\widetilde{c})$ .\",\n \"Corresponding generator: $\\\\varphi _{1,\\\\widetilde{c}} = \\\\widetilde{c} \\\\cdot \\\\varphi _{1}$ (cf.\",\n \"(REF ), (REF )) of the $\\\\widetilde{c}-$ fold of the Kullback-Leibler divergence (relative entropy) given in the fourth line of (REF ).\",\n \"Sums: for i.i.d.\",\n \"copies $(W_{i})_{i \\\\in \\\\mathbb {N}}$ of $W$ , the probability distribution of $\\\\breve{W} := \\\\sum _{i\\\\in I_{k}^{(n)}} W_{i}$ (cf.\",\n \"Remark REF (ii)) is $\\\\frac{1}{\\\\widetilde{c}} \\\\cdot POI(\\\\widetilde{c} \\\\cdot card(I_{k}^{(n)}))$ .\",\n \"(b) Case $\\\\gamma =1$ , $\\\\widetilde{c} =1$ , anchor point $c \\\\in \\\\mathbb {R}$ : $\\\\Lambda _{1,1}^{(c)}(z)= e^{c} \\\\cdot \\\\left( e^{z} - 1 \\\\right) + z \\\\cdot (1- e^{c})$ (cf.\",\n \"(REF )) is the well-known cumulant generating function of the “shifted Poisson distribution” $\\\\mathbb {} = POI(e^{c}) + 1-e^{c}$ , i.e.\",\n \"$W := Z + 1-e^{c}$ with a $POI(e^{c})-$ distributed random variable $Z$ .\",\n \"Hence, $\\\\varphi _{1,1}^{(c)} \\\\in \\\\Upsilon (]1- e^{c},\\\\infty [)$ .\",\n \"Type: $\\\\mathbb {}$ is a discrete distribution with frequencies: $\\\\mathbb {\\\\Pi }[W=\\\\ell + 1- e^{c}]= \\\\exp (-e^{c}) \\\\cdot \\\\frac{ e^{c \\\\cdot \\\\ell }}{\\\\ell !\",\n \"}$ for all $\\\\ell \\\\in \\\\mathbb {N}_{0}$ (and zero elsewhere).\",\n \"Behaviour at zero: $\\\\mathbb {\\\\Pi }[W > 0]=1$ iff $c <0$ , $\\\\mathbb {\\\\Pi }[W < 0] >0$ iff $c >0$ , $\\\\mathbb {\\\\Pi }[W=0] \\\\ne 0$ iff “$c =\\\\log (1+k)$ for some $k \\\\in \\\\mathbb {N}_{0}$ ” .\",\n \"Corresponding generator: $\\\\varphi _{1,1}^{(c)}$ (cf.\",\n \"(REF )) of the divergence $& & D_{\\\\varphi _{1,1}^{(c)}}(\\\\mathbf {Q},\\\\mathbf {P}) :=\\\\sum \\\\limits _{k=1}^{K} \\\\Big (q_{k} + p_{k} \\\\cdot (e^{c}-1) \\\\Big )\\\\cdot \\\\Big \\\\lbrace \\\\log \\\\Big (\\\\frac{q_{k}}{p_{k}} + e^{c}-1 \\\\Big ) - c \\\\Big \\\\rbrace - \\\\sum \\\\limits _{k=1}^{K} q_{k} + \\\\sum \\\\limits _{k=1}^{K} p_{k} ,\\\\nonumber \\\\\\\\& & \\\\hspace{113.81102pt} \\\\textrm {if } \\\\ \\\\mathbf {P} \\\\in \\\\mathbb {R}_{\\\\ne 0}^{K} \\\\ \\\\textrm {and } \\\\mathbf {Q} \\\\in \\\\mathbb {R}^{K}\\\\textrm {with } \\\\mathbf {Q} \\\\in \\\\, [ (1-e^{c}) \\\\cdot \\\\mathbf {P}, \\\\mathbf {\\\\infty } [\\\\ \\\\textrm {component-wise},$ which for $c=0$ coincides with the Kullback-Leibler divergence (relative entropy) given in the fourth line of (REF ).\",\n \"Sums: for i.i.d.\",\n \"copies $(W_{i})_{i \\\\in \\\\mathbb {N}}$ of $W$ , the probability distribution of $\\\\breve{W} := \\\\sum _{i\\\\in I_{k}^{(n)}} W_{i}$ (cf.\",\n \"Remark REF (ii)) is $POI(card(I_{k}^{(n)}) \\\\cdot e^{c}) + (1-e^{c}) \\\\cdot card(I_{k}^{(n)})$ .\",\n \"Remark 51 (a) One can see from the Examples REF and REF the interesting effect that the “homogeneous” class of power-divergence generators $(\\\\varphi _{\\\\gamma })_{\\\\gamma \\\\in \\\\mathbb {R}}$ are connected to a “very inhomogeneous” family $(\\\\mathbb {}_{\\\\gamma })_{\\\\gamma \\\\in \\\\mathbb {R}}$ of $W-$ distributions: discrete, continuous, mixture of discrete and continuous, as the parameter $\\\\gamma $ varies.\",\n \"Moreover, some cases satisfy $\\\\mathbb {\\\\Pi }[W=0] =0$ and some $\\\\mathbb {\\\\Pi }[W=0] >0$ , some $\\\\mathbb {\\\\Pi }[W>0]=1$ and some $\\\\mathbb {\\\\Pi }[W>0] \\\\in ]0,1[$ .\",\n \"(b) As a continuation of Remark REF and the note in the last line of Example REF (a), we have shown as a side effect that for the the natural-anchor-point choice $c=0$ , the distributions $\\\\mathbb {}_{1}$ of of Example REF (a) and $\\\\mathbb {}_{0}$ of Example REF (a) are inverse to each other.\",\n \"Example 52 for the context of Example REF we obtain: Case $\\\\widetilde{c} >0$ , anchor point $c=0$ : $\\\\Lambda _{bw,\\\\beta ,\\\\widetilde{c}}(z) =-(\\\\frac{1}{\\\\beta }-1) \\\\cdot z + \\\\frac{\\\\widetilde{c}}{\\\\beta ^{2}} \\\\cdot \\\\Big \\\\lbrace 1 - \\\\sqrt{1-\\\\frac{2\\\\beta }{\\\\widetilde{c}} \\\\cdot z} \\\\ \\\\Big \\\\rbrace $ (cf.\",\n \"(REF )) is the cumulant generating function of a probability distribution $\\\\mathbb {}[ \\\\, \\\\cdot \\\\,] = \\\\mathbb {\\\\Pi }[\\\\check{W} \\\\in \\\\cdot \\\\, ]$ of a random variable $\\\\check{W}$ , which can be constructed as follows: $\\\\check{W} := \\\\frac{W}{\\\\beta } - (\\\\frac{1}{\\\\beta } - 1)$ , where $W$ is the random variable constructed in Example REF (d) with $\\\\gamma =-1$ and with $\\\\widetilde{c}$ replaced by $\\\\frac{\\\\widetilde{c}}{\\\\beta ^{2}}$ (recall that $W$ has a tilted stable distribution).\",\n \"In other words, $\\\\mathbb {}$ is a special kind of modified tilted stable distribution.\",\n \"Type: $\\\\mathbb {}$ is an infinitely divisible (cf.\",\n \"Proposition REF ) continuous distribution with density $f_{\\\\check{W}}(u) := \\\\beta \\\\cdot f_{W}(\\\\beta \\\\cdot u + 1 -\\\\beta )\\\\cdot \\\\mathbf {1}_{]- (\\\\frac{1}{\\\\beta } -1),\\\\infty [}(u)$ ($u \\\\in \\\\mathbb {R}$ ), where $f_{W}(\\\\cdot )$ is given in (REF ) with $\\\\gamma =-1$ and with $\\\\widetilde{c}$ replaced by $\\\\frac{\\\\widetilde{c}}{\\\\beta ^{2}}$ .\",\n \"Behaviour at zero: $\\\\mathbb {}[ \\\\, ] 0,\\\\infty [ \\\\, ] =\\\\mathbb {\\\\Pi }[\\\\check{W} > 0] >0$ .\",\n \"Corresponding generator: $\\\\varphi _{bw,\\\\beta ,\\\\widetilde{c}}^{(0)}$ (cf.\",\n \"(REF )) of the — “non-probability version” of — the well-known blended weight chi-square divergence given in (REF ).\",\n \"Sums: for i.i.d.\",\n \"copies $(\\\\check{W}_{i})_{i \\\\in \\\\mathbb {N}}$ of $\\\\check{W}$ , the probability distribution of $\\\\breve{\\\\check{W}} := \\\\sum _{i\\\\in I_{k}^{(n)}} \\\\check{W}_{i}= \\\\frac{1}{\\\\beta } \\\\cdot \\\\sum _{i\\\\in I_{k}^{(n)}} W_{i} - n_{k}\\\\cdot (\\\\frac{1}{\\\\beta } -1)$ (cf.\",\n \"Remark REF (ii)) has density $f_{\\\\breve{\\\\check{W}}}(u) := \\\\beta \\\\cdot f_{\\\\breve{W}}(\\\\beta \\\\cdot u + (1 -\\\\beta )\\\\cdot n_{k})\\\\cdot \\\\mathbf {1}_{]- n_{k} \\\\cdot (\\\\frac{1}{\\\\beta } -1),\\\\infty [}(u)$ ($u \\\\in \\\\mathbb {R}$ ), where $f_{\\\\breve{W}}(\\\\cdot )$ is given in (REF ) (cf.\",\n \"Example REF (d)) with $\\\\gamma =-1$ and with $\\\\widetilde{c}$ replaced by $\\\\frac{\\\\widetilde{c}}{\\\\beta ^{2}}$ .\",\n \"Example 53 for the context of Example REF we obtain: (a) Case $\\\\alpha \\\\in \\\\, ]0,\\\\infty [$ , $\\\\widetilde{c} >0$ , anchor point $c=0$ : $\\\\Lambda _{gKL,\\\\alpha ,\\\\widetilde{c}}(z) =- \\\\frac{\\\\widetilde{c}}{\\\\alpha } \\\\cdot \\\\log ((1+\\\\alpha ) - \\\\alpha \\\\cdot e^{z/\\\\widetilde{c}})$ (cf.\",\n \"(REF )) is the cumulant generating function of $\\\\mathbb {} = \\\\frac{1}{\\\\widetilde{c}} \\\\cdot NB(\\\\frac{\\\\widetilde{c}}{\\\\alpha },\\\\frac{1}{1+\\\\alpha })$ being the “$\\\\frac{1}{\\\\widetilde{c}}-$ fold Negative-Binomial distribution with parameters $\\\\frac{\\\\widetilde{c}}{\\\\alpha }$ and $\\\\frac{1}{1+\\\\alpha }$ ” which means that $W = \\\\frac{1}{\\\\widetilde{c}} \\\\cdot Z$ for a $NB(\\\\frac{\\\\widetilde{c}}{\\\\alpha },\\\\frac{1}{1+\\\\alpha })-$ distributed random variable $Z$ .\",\n \"Thus, $\\\\varphi _{gKL,\\\\alpha ,\\\\widetilde{c}} \\\\in \\\\Upsilon (]0,\\\\infty [)$ .\",\n \"Prominent special case $\\\\widetilde{c}=1$ , $\\\\alpha =1$ (see below): $\\\\mathbb {} = NB(1,\\\\frac{1}{2})$ is the Negative-Binomial distribution with parameters 1 and $\\\\frac{1}{2}$ .\",\n \"Type: $\\\\mathbb {}$ is an infinitely divisible (cf.\",\n \"Proposition REF ) discrete distribution with frequencies: $\\\\mathbb {\\\\Pi }[W=\\\\ell \\\\cdot \\\\frac{1}{\\\\widetilde{c}}]=(-1)^{\\\\ell } \\\\cdot \\\\binom{- \\\\frac{\\\\widetilde{c}}{\\\\alpha }}{\\\\ell }\\\\cdot {\\\\alpha }^{\\\\ell } \\\\cdot (1+\\\\alpha )^{-\\\\ell - \\\\widetilde{c}/\\\\alpha }$ for all nonnegative integers $\\\\ell \\\\in \\\\mathbb {N}_{0}$ (and zero elsewhere).\",\n \"Behaviour at zero: $\\\\mathbb {\\\\Pi }[W\\\\ge 0]=1$ , $\\\\mathbb {\\\\Pi }[W=0]= \\\\frac{1}{(1+a)^{\\\\widetilde{c}/\\\\alpha }}$ .\",\n \"Corresponding generator: $\\\\varphi _{gKL,\\\\alpha ,\\\\widetilde{c}}$ (cf.\",\n \"(REF )) of the divergence (REF ); the special case $\\\\widetilde{c}=1$ , $\\\\alpha =1$ — i.e.\",\n \"$\\\\varphi _{gKL,1,1} =: \\\\varphi _{snKL,1}$ (cf.\",\n \"(REF )) — corresponds to the generator of the — “non-probability version” of the — Jensen-Shannon divergence (symmetrized and normalized Kullback-Leibler divergence, symmetrized and normalized relative entropy) given in (REF ).\",\n \"Sums: for i.i.d.\",\n \"copies $(W_{i})_{i \\\\in \\\\mathbb {N}}$ of $W$ , the probability distribution of $\\\\breve{W} := \\\\sum _{i\\\\in I_{k}^{(n)}} W_{i}$ (cf.\",\n \"Remark REF (ii)) is $\\\\frac{1}{\\\\widetilde{c}} \\\\cdot NB(\\\\frac{\\\\widetilde{c}}{\\\\alpha }\\\\cdot card(I_{k}^{(n)}),\\\\frac{1}{1+\\\\alpha })$ .\",\n \"(b) Case $\\\\alpha \\\\in \\\\, ]-1,0[$ , $\\\\widetilde{c} >0$ , anchor point $c=0$ : for any integer $m \\\\in \\\\mathbb {N}$ being strictly larger than $\\\\widetilde{c}$ and the choice $\\\\alpha = - \\\\frac{\\\\widetilde{c}}{m}$ , we obtain $\\\\Lambda _{gKL,-\\\\widetilde{c}/m,\\\\widetilde{c}}(z) =m \\\\cdot \\\\log ((1 - \\\\frac{\\\\widetilde{c}}{m}) + \\\\frac{\\\\widetilde{c}}{m} \\\\cdot e^{z/\\\\widetilde{c}})$ (cf.\",\n \"(REF )) which is the cumulant generating function of $\\\\mathbb {} = \\\\frac{1}{\\\\widetilde{c}} \\\\cdot BIN(m,\\\\frac{\\\\widetilde{c}}{m})$ being the “$\\\\frac{1}{\\\\widetilde{c}}-$ fold Binomial distribution with parameters $m$ and $\\\\frac{\\\\widetilde{c}}{m}$ ” which means that $W = \\\\frac{1}{\\\\widetilde{c}} \\\\cdot Z$ for a $BIN(m,\\\\frac{\\\\widetilde{c}}{m})-$ distributed random variable $Z$ .\",\n \"Thus, $\\\\varphi _{gKL,-\\\\widetilde{c}/m,\\\\widetilde{c}} \\\\in \\\\Upsilon (]0,\\\\infty [)$ .\",\n \"Type: $\\\\mathbb {}$ is a non-infinitely divisible discrete distribution with frequencies: $\\\\mathbb {\\\\Pi }[W=\\\\ell \\\\cdot \\\\frac{1}{\\\\widetilde{c}}]=\\\\binom{m}{\\\\ell }\\\\cdot (\\\\frac{\\\\widetilde{c}}{m})^{\\\\ell } \\\\cdot (1-\\\\frac{\\\\widetilde{c}}{m})^{m-\\\\ell }$ for $\\\\ell \\\\in \\\\lbrace 0,1, \\\\ldots , m\\\\rbrace $ (and zero elsewhere).\",\n \"Behaviour at zero: $\\\\mathbb {\\\\Pi }[W\\\\ge 0]=1$ , $\\\\mathbb {\\\\Pi }[W=0]= (1-\\\\frac{\\\\widetilde{c}}{m})^{m}$ .\",\n \"Corresponding generator: $\\\\varphi _{gKL,\\\\alpha ,\\\\widetilde{c}}$ (cf.\",\n \"(REF )) of the divergence (REF ).\",\n \"Sums: for i.i.d.\",\n \"copies $(W_{i})_{i \\\\in \\\\mathbb {N}}$ of $W$ , the probability distribution of $\\\\breve{W} := \\\\sum _{i\\\\in I_{k}^{(n)}} W_{i}$ (cf.\",\n \"Remark REF (ii)) is $\\\\frac{1}{\\\\widetilde{c}} \\\\cdot BIN(m \\\\cdot card(I_{k}^{(n)}),\\\\frac{\\\\widetilde{c}}{m})$ .\",\n \"Example 54 for the context of Example REF we obtain: Case of anchor point $c=0$ : $\\\\Lambda _{twop}(z)= \\\\log \\\\Big ( p \\\\cdot e^{z_{1} \\\\cdot z} + (1- p) \\\\cdot e^{z_{2} \\\\cdot z} \\\\Big )$ (cf.\",\n \"(REF )) is the well-known cumulant generating function of the two-point probability distribution $\\\\mathbb {} = p \\\\cdot \\\\delta _{z_{1}} + (1-p) \\\\cdot \\\\delta _{z_{2}}$ , where $z_{1} < 1 < z_{2}$ and $p= \\\\frac{z_{2} -1}{z_{2} - z_{1}}$ .\",\n \"Hence, $\\\\varphi _{twop} \\\\in \\\\Upsilon (]z_{1},z_{2}[)$ .\",\n \"Type: $\\\\mathbb {}$ is a discrete distribution with frequencies: $\\\\mathbb {\\\\Pi }[W=z_{1}] = p$ , $\\\\mathbb {\\\\Pi }[W=z_{2}] = 1-p$ (and zero elsewhere).\",\n \"Behaviour at zero: $\\\\mathbb {\\\\Pi }[W > 0]=1$ iff $z_{1} >0$ , $\\\\mathbb {\\\\Pi }[W=0] \\\\ne 0$ iff $z_{1}=0$ .\",\n \"Corresponding generator: $\\\\varphi _{twop}$ (cf.\",\n \"(REF )) of the divergence given in (REF ).\",\n \"Sums: for i.i.d.\",\n \"copies $(W_{i})_{i \\\\in \\\\mathbb {N}}$ of $W$ , the probability distribution of $\\\\breve{W} := \\\\sum _{i\\\\in I_{k}^{(n)}} W_{i}$ (cf.\",\n \"Remark REF (ii)) is the distribution of the $card(I_{k}^{(n)})-$ th step of a generalized random walk starting at zero; this has a nice explicit (“binomial-type” ) expression in the special case $z_{1} = - z_{2}$ , namely $\\\\sum _{\\\\ell =0}^{card(I_{k}^{(n)})} \\\\binom{card(I_{k}^{(n)})}{\\\\ell }\\\\cdot p^{card(I_{k}^{(n)}) - \\\\ell } \\\\cdot (1-p)^{\\\\ell } \\\\cdot \\\\delta _{z_{2}\\\\cdot (2 \\\\ell - card(I_{k}^{(n)}))}$ .\",\n \"Example 55 for the context of Example REF we obtain: Case $\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c} \\\\in \\\\, ]0,\\\\infty [$ , anchor point $c=0$ : by using $\\\\breve{\\\\theta } := 1 + \\\\alpha \\\\cdot \\\\Big (\\\\frac{1}{\\\\beta _{2}}- \\\\frac{1}{\\\\beta _{1}} \\\\Big )$ one can see that $\\\\Lambda _{\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c}}(z) =\\\\breve{\\\\theta } \\\\cdot z - \\\\widetilde{c} \\\\cdot \\\\alpha \\\\cdot \\\\log \\\\Big (1+ \\\\frac{z}{\\\\widetilde{c}} \\\\cdot \\\\Big (\\\\frac{1}{\\\\beta _{2}} - \\\\frac{1}{\\\\beta _{1}} \\\\Big )- \\\\frac{z^{2}}{\\\\widetilde{c}^{2} \\\\cdot \\\\beta _{1} \\\\cdot \\\\beta _{2} }\\\\Big )$ for $z \\\\in \\\\, ]- \\\\widetilde{c} \\\\cdot \\\\min \\\\lbrace \\\\beta _{1}, \\\\beta _{2}\\\\rbrace ,\\\\widetilde{c} \\\\cdot \\\\min \\\\lbrace \\\\beta _{1}, \\\\beta _{2}\\\\rbrace [$ — with different boundary behaviour for the three subcases $\\\\beta _{1} < \\\\beta _{2}$ resp.\",\n \"$\\\\beta _{1} > \\\\beta _{2}$ resp.\",\n \"$\\\\beta _{1} = \\\\beta _{2}$ (cf.\",\n \"(REF ),(REF ),(REF )) — is the cumulant generating function of a generalized asymmetric Laplace distribution $\\\\mathbb {}[ \\\\, \\\\cdot \\\\,] = \\\\mathbb {\\\\Pi }[W \\\\in \\\\cdot \\\\, ]$ of a random variable $W:= \\\\breve{\\\\theta } + Z_{1} - Z_{2}$ , where $Z_{1}$ respectively $Z_{2}$ are auxiliary random variables which are independent and $GAM(\\\\widetilde{c} \\\\cdot \\\\beta _{1},\\\\widetilde{c} \\\\cdot \\\\alpha )-$ distributed respectively $GAM(\\\\widetilde{c} \\\\cdot \\\\beta _{2},\\\\widetilde{c} \\\\cdot \\\\alpha )-$ distributed.\",\n \"In particular, $E_{\\\\mathbb {\\\\Pi }}[W] = \\\\breve{\\\\theta } + \\\\frac{\\\\widetilde{c} \\\\cdot \\\\alpha }{\\\\widetilde{c} \\\\cdot \\\\beta _{1}}- \\\\frac{\\\\widetilde{c} \\\\cdot \\\\alpha }{\\\\widetilde{c} \\\\cdot \\\\beta _{2}} =1$ (as required).\",\n \"Thus, $\\\\varphi _{\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c}}\\\\in \\\\Upsilon (]-\\\\infty ,\\\\infty [)$ .\",\n \"Prominent special case $\\\\widetilde{c} =1$ , $\\\\alpha =1$ , $\\\\beta _{1}=\\\\beta _{2} =: \\\\beta $ (and hence, $\\\\breve{\\\\theta }=1$ ): $\\\\mathbb {}$ is a classical Laplace distribution (two-tailed exponential distribution, bilateral exponential law) with location parameter 1 and scale parameter $\\\\frac{1}{\\\\beta }$ .\",\n \"Type: $\\\\mathbb {}$ is an infinitely divisible (cf.\",\n \"Proposition REF ) continuous distribution with density $f(u) := \\\\frac{\\\\sqrt{2} \\\\cdot \\\\exp \\\\lbrace \\\\frac{1}{\\\\sigma \\\\cdot \\\\sqrt{2}} \\\\cdot (\\\\frac{1}{\\\\kappa } - \\\\kappa ) \\\\cdot (u-\\\\theta ) \\\\rbrace }{\\\\sqrt{\\\\pi } \\\\cdot \\\\sigma ^{\\\\tau + 1/2} \\\\cdot \\\\Gamma (\\\\tau )} \\\\cdot \\\\left( \\\\frac{\\\\sqrt{2} \\\\cdot |u - \\\\theta |}{\\\\kappa + \\\\frac{1}{\\\\kappa }} \\\\right)^{\\\\tau - 1/2}\\\\cdot K_{\\\\tau - 1/2}\\\\left( \\\\frac{1}{\\\\sigma \\\\cdot \\\\sqrt{2}} \\\\cdot \\\\Big (\\\\kappa + \\\\frac{1}{\\\\kappa }\\\\Big ) \\\\cdot |u - \\\\theta | \\\\right),\\\\quad u \\\\ne \\\\theta ,$ where $(\\\\theta ,\\\\kappa ,\\\\sigma ,\\\\tau )$ is given in Remark REF below and $K_{\\\\lambda }$ is the modified Bessel function of the third kind with index $\\\\lambda $ .\",\n \"For the above-mentioned special case of the classical Laplace distribution, this considerably simplifies to $f(u):= \\\\frac{\\\\beta }{2} \\\\exp \\\\lbrace - \\\\beta \\\\cdot |u -1 | \\\\rbrace $ .\",\n \"Behaviour at zero: $\\\\mathbb {}[ \\\\, ]0,\\\\infty [ \\\\, ] = \\\\mathbb {\\\\Pi }[W > 0]=\\\\int _{0}^{\\\\infty } f(u) \\\\, du \\\\in \\\\, ]0,1[$ , $\\\\mathbb {}[ \\\\, \\\\lbrace 0\\\\rbrace \\\\, ] = \\\\mathbb {\\\\Pi }[W = 0]= 0$ .\",\n \"Corresponding generator: $\\\\varphi _{\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c}}$ (cf.\",\n \"(REF ) respectively (REF ) respectively (REF )) of the divergence given in (REF ) respectively (REF ) respectively (REF ).\",\n \"Sums: for i.i.d.\",\n \"copies $(W_{i})_{i \\\\in \\\\mathbb {N}}$ of $W$ , the probability distribution of $\\\\breve{W} := \\\\sum _{i\\\\in I_{k}^{(n)}} W_{i}$ (cf.\",\n \"Remark REF (ii)) is the same as that of a random variable $\\\\breve{\\\\widetilde{W}} := \\\\breve{\\\\theta } \\\\cdot card(I_{k}^{(n)}) +\\\\breve{Z}_{1} - \\\\breve{Z}_{2}$ , where $\\\\breve{Z}_{1}$ respectively $\\\\breve{Z}_{2}$ are auxiliary random variables which are independent and $GAM(\\\\widetilde{c} \\\\cdot \\\\beta _{1},\\\\widetilde{c} \\\\cdot \\\\alpha \\\\cdot card(I_{k}^{(n)}))-$ distributed respectively $GAM(\\\\widetilde{c} \\\\cdot \\\\beta _{2},\\\\widetilde{c}\\\\cdot \\\\alpha \\\\cdot card(I_{k}^{(n)}))-$ distributed.\",\n \"Remark 56 In the book of Kotz et al.\",\n \"[197] one can find a very comprehensive study on generalized asymmetric Laplace distributions (also known as Bessel function distributions, McKay distributions), their close relatives (such as e.g.\",\n \"the financial-econometric variance gamma model of Madan & Seneta [240]) as well as their applications; see also e.g.\",\n \"Klar [192] for connections with some other Gamma difference distributions.\",\n \"[197] use a different parametrization $(\\\\theta ,\\\\kappa ,\\\\sigma ,\\\\tau )$ which is one-to-one with our parametrization $(\\\\breve{\\\\theta },\\\\alpha ,\\\\beta _{1},\\\\beta _{2},\\\\widetilde{c}=1)$ , as follows: $\\\\theta = \\\\breve{\\\\theta }$ , $\\\\tau = \\\\widetilde{c} \\\\cdot \\\\alpha $ , $\\\\sigma = \\\\frac{1}{\\\\widetilde{c}} \\\\cdot \\\\sqrt{\\\\frac{2}{\\\\beta _{1} \\\\cdot \\\\beta _{2}}}$ , $\\\\kappa = \\\\frac{\\\\sqrt{\\\\frac{4}{\\\\widetilde{c}^{2}} + (\\\\beta _{1} - \\\\beta _{2})^2} \\\\, + \\\\beta _{2}- \\\\beta _{1}}{2 \\\\cdot \\\\sqrt{\\\\beta _{1} \\\\cdot \\\\beta _{2}}} >0$ .\",\n \"In particular, this implies that we cover all generalized asymmetric Laplace distributions with mean 1.\",\n \"For better comparability, we have used the parametrization $(\\\\theta ,\\\\kappa ,\\\\sigma ,\\\\tau )$ in the above-mentioned representation (REF ) of the density (due to [197]).\",\n \"Let us end this section by giving some further comments on the task of finding concretely the probability distribution (if existent) $\\\\mathbb {}[ \\\\, \\\\cdot \\\\,] = \\\\mathbb {\\\\Pi }[W \\\\in \\\\cdot \\\\, ]$ from the Fenchel-Legendre transform $\\\\Lambda = \\\\varphi ^{*}$ of a pregiven divergence generator $\\\\varphi $ , which should satisfy $\\\\exp (\\\\Lambda (z)) = \\\\int _{\\\\mathbb {R}}e^{z \\\\cdot y} \\\\, d\\\\mathbb {} (y)= E_{\\\\mathbb {\\\\Pi }}[\\\\exp (z \\\\cdot W)] ,\\\\qquad z \\\\in \\\\mathbb {R}, \\\\qquad \\\\text{(cf.\",\n \"(\\\\ref {brostu3:repres lambda1}), (\\\\ref {brostu3:repres lambda2}))}.\\\\nonumber $ Recall that this is used for the simulation of the weights $(W_{i})_{i\\\\in \\\\mathbb {N}}$ which are i.i.d.\",\n \"copies of $W$ and which are the crucial building ingredients of $\\\\xi _{n}^{\\\\mathbf {W}}$ in Theorem REF , respectively, of $\\\\xi _{n,\\\\mathbf {X}}^{w\\\\mathbf {W}}$ in Theorem REF .\",\n \"The search for $\\\\mathbb {}$ can be done e.g.\",\n \"by inversion of the moment generating function MGF, or by search in tables or computer software which list distributions and their MGF.\",\n \"As already indicated above, we have eased/narrowed down this task by giving (additional) sufficient conditions for some deriving principal properties of $\\\\mathbb {}$ .\",\n \"Also notice that $\\\\mathbb {}$ needs not necessarily to be explicitly known in full detail (e.g.\",\n \"in terms of a computationally tractable density or frequency); for instance, as well known from insurance applications, for — comfortably straightforwardly simulable — doubly-random sums $W := \\\\sum _{i=1}^{N} A_{i}$ of nonnegative i.i.d.\",\n \"random variables $(A_{i})_{i\\\\in \\\\mathbb {N}}$ with known law $\\\\Pi _A[\\\\, \\\\cdot \\\\, ]:= \\\\Pi [A \\\\in \\\\cdot \\\\, ]$ being independent of a counting-type random variable $N$ with known law $\\\\Pi _N$ , one can mostly compute explicitly $MGF_{\\\\mathbb {}}(z) = PGF_{\\\\Pi _N}(MGF_{\\\\Pi _A}(z))$ in terms of $\\\\mathbb {} := \\\\Pi _W$ and the probability generating function $PGF_{\\\\Pi _N}$ of $\\\\Pi _N$ , but the corresponding density/frequency of $\\\\mathbb {}$ may not be known explicitly in a tractable form.\",\n \"The above-mentioned Example REF (b) of power divergences with generator $\\\\varphi _{\\\\gamma }$ ($\\\\gamma \\\\in ]0,1[$ ) manifests such a situation.\",\n \"In the end, if no explicit distribution $\\\\mathbb {}$ and no comfortably simulable $W-$ construction are available, one can still try to simulate an i.i.d.\",\n \"sequence $(W_{i})_{i\\\\in \\\\mathbb {N}}$ from the pregiven moment generating function (which is $\\\\exp (\\\\Lambda (z))$ here); see e.g.\",\n \"McLeish [259] and references therein which also contains saddle point methods approximation techniques.\"\n ],\n [\n \"Estimators\",\n \"In the following, we demonstrate how one can principally implement our BS approach; a further, deeper analysis will be given in a follow-up paper.\"\n ],\n [\n \"Estimators for the deterministic minimization problem\",\n \"We address the minimization problem $D_{\\\\varphi }(\\\\mathbf {\\\\Omega },\\\\mathbf {P}):= \\\\inf _{\\\\mathbf {Q}\\\\in \\\\mathbf {\\\\Omega } } D_{\\\\varphi }(\\\\mathbf {Q},\\\\mathbf {P}) =\\\\inf _{\\\\widetilde{\\\\mathbf {Q}}\\\\in \\\\widetilde{\\\\mathbf {\\\\Omega }} }D_{\\\\widetilde{\\\\varphi } }(\\\\widetilde{\\\\mathbf {Q}},\\\\widetilde{{P}})=: D_{\\\\widetilde{\\\\varphi } }(\\\\widetilde{\\\\mathbf {\\\\Omega }},\\\\widetilde{{P}})\\\\qquad \\\\textrm {with } \\\\widetilde{\\\\mathbf {\\\\Omega }} :=\\\\mathbf {\\\\Omega } /M_{\\\\mathbf {P}}\\\\qquad \\\\textrm {(cf.\",\n \"(\\\\ref {min Pb}) and (\\\\ref {min Pb prob2}))},$ whose numerical solution is based on Theorem REF which basically states that for large integer $n \\\\in \\\\mathbb {N}$ one has $\\\\inf _{\\\\mathbf {Q}\\\\in \\\\mathbf {\\\\Omega } } D_{\\\\varphi }(\\\\mathbf {Q},\\\\mathbf {P})\\\\approx -\\\\frac{1}{n}\\\\log \\\\,\\\\mathbb {\\\\Pi }\\\\left[\\\\xi _{n}^{\\\\mathbf {\\\\widetilde{W}}}\\\\in \\\\widetilde{\\\\mathbf {\\\\Omega }}\\\\right]$ in terms of $\\\\widetilde{\\\\varphi }:=M_{\\\\mathbf {P}} \\\\cdot \\\\varphi $ and the random vectors $\\\\xi _{n}^{\\\\mathbf {\\\\widetilde{W}}}=\\\\Big (\\\\frac{1}{n}\\\\sum _{i\\\\in I_{1}^{(n)}}\\\\widetilde{W}_{i},\\\\ldots ,\\\\frac{1}{n}\\\\sum _{i\\\\in I_{K}^{(n)}}\\\\widetilde{W}_{i}\\\\Big )\\\\hspace{56.9055pt} \\\\text{(cf.\",\n \"(\\\\ref {Xi_n^W vector}))}\\\\nonumber $ with $n_{k}:=\\\\lfloor n \\\\cdot \\\\widetilde{p}_{k}\\\\rfloor $ leading to the disjoint index blocks $I_{1}^{(n)}:=\\\\left\\\\lbrace 1,\\\\ldots ,n_{1}\\\\right\\\\rbrace $ , $I_{2}^{(n)}:=\\\\left\\\\lbrace n_{1}+1,\\\\ldots ,n_{1}+n_{2}\\\\right\\\\rbrace $ , $\\\\ldots $ , $I_{K}^{(n)} := \\\\lbrace \\\\sum _{k=1}^{K-1} n_{k} + 1, \\\\ldots , n \\\\rbrace $ .\",\n \"Recall that $\\\\mathbf {\\\\widetilde{W}} := (\\\\widetilde{W}_{1}, \\\\ldots , \\\\widetilde{W}_{n})$ is a random vector consisting of components $\\\\widetilde{W}_{i}$ which are i.i.d.\",\n \"copies of the random variable $\\\\widetilde{W}$ whose distribution is $\\\\mathbb {\\\\Pi }[\\\\widetilde{W}\\\\in \\\\cdot \\\\,]=\\\\widetilde{\\\\mathbb {}}[\\\\,\\\\cdot \\\\,]$ obeying the representation $\\\\widetilde{\\\\varphi }(t) =\\\\sup _{z \\\\in \\\\mathbb {R}} \\\\left( z\\\\cdot t - \\\\log \\\\int _{\\\\mathbb {R}} e^{zy} d\\\\widetilde{\\\\mathbb {}}(y) \\\\right),\\\\qquad t \\\\in \\\\mathbb {R},\\\\qquad \\\\textrm {(cf.\",\n \"(\\\\ref {brostu3:fo.link.var}))} .\\\\nonumber $ Hence, the estimation of $D_{\\\\varphi }(\\\\mathbf {\\\\Omega },\\\\mathbf {P})$ amounts to the estimation of $\\\\mathbb {\\\\Pi }\\\\left[\\\\xi _{n}^{\\\\mathbf {\\\\widetilde{W}}}\\\\in \\\\widetilde{\\\\mathbf {\\\\Omega }} \\\\right]$ .\",\n \"For the rest of this subsection, we assume that $\\\\widetilde{{P}} \\\\in \\\\mathbb {S}_{> 0}^{K}$ , that $n$ is chosen such that all $n \\\\cdot \\\\widetilde{p}_{k}$ are integers (and hence, $n = \\\\sum _{k=1}^{K} n_{k}$ ), and that $\\\\widetilde{\\\\mathbf {\\\\Omega }} \\\\subset \\\\mathbb {R}^{K}$ satisfies the regularity property $cl(\\\\widetilde{\\\\mathbf {\\\\Omega }})=cl\\\\left( int\\\\left( \\\\widetilde{\\\\mathbf {\\\\Omega }} \\\\right) \\\\right) , \\\\qquad int\\\\left( \\\\widetilde{\\\\mathbf {\\\\Omega }} \\\\right) \\\\ne \\\\emptyset \\\\nonumber $ which implies that the same condition holds for $\\\\mathbf {\\\\Omega }$ ; moreover, we suppose that $D_{\\\\widetilde{\\\\varphi } }(\\\\widetilde{\\\\mathbf {\\\\Omega }},\\\\widetilde{{P}})$ is finite.\",\n \"For the ease of the following discussions, we introduce the notations $T\\\\left( \\\\mathbf {x}\\\\right) :=\\\\left( \\\\frac{1}{n_{1}}\\\\sum _{i\\\\in I_{1}^{(n)}}x_{i},\\\\ldots ,\\\\frac{1}{n_{K}}\\\\sum _{i\\\\in I_{K}^{(n)}} x_{i}\\\\right)\\\\quad \\\\textrm {for any $ x:=( x1,..,xn) Rn$,}$$as well as $ D$ for the diagonal matrix with diagonal entries $ 1/p1,...,1/pK$and null entries off the diagonal.Accordingly, the probability in (\\\\ref {LDP Minimization approx}) becomes$$\\\\mathbb {\\\\Pi }\\\\left[\\\\xi _{n}^{\\\\mathbf {\\\\widetilde{W}}}\\\\in \\\\widetilde{\\\\mathbf {\\\\Omega }} \\\\right] \\\\ = \\\\ \\\\mathbb {\\\\Pi }\\\\left[ T(\\\\mathbf {\\\\widetilde{W}}) \\\\in \\\\mathbf {\\\\Lambda }\\\\right]$$where$$\\\\mathbf {\\\\Lambda } :=\\\\mathfrak {D} \\\\cdot \\\\widetilde{\\\\mathbf {\\\\Omega }}$$is a set of vectors in $ RK$ which is known/derived from the concrete context.The \\\\textit {naive estimator} $ Lnaive$ of$ [ nW ]$is constructedthrough the following procedure:simulate independently $ L$ copies$ W1,...,WL$ of the vector$ W:=( W1,...,Wn) $,with independent entries under $$, and define(with a slight abuse of notation)$$\\\\widehat{\\\\widetilde{\\\\Pi }}_{L}^{naive} :=\\\\frac{1}{L}\\\\sum _{\\\\ell =1}^{L} \\\\mathbf {1}_{\\\\mathbf {\\\\Lambda }}\\\\left( T\\\\left( \\\\mathbf {\\\\widetilde{W}}^{\\\\ell }\\\\right) \\\\right) ;$$however this procedure is time costly, since this estimate has a very bad hit rate.Thus, in the following, a so-called “efficient Importance Sampling (IS)”\\\\ scheme --- in the sense of Sadowsky\\\\& Bucklew \\\\cite {Sad:90} (denoted [SB] hereunder) ---is adapted for the sophisticated (i.e.\",\n \"non-naive) estimation of$ [ nW ]$.The main property of IS schemes lays in the fact that the runtime for anestimate with a controlled relative error does \\\\textit {not} increase atexponential rate as $ n$ increases, in contrast to$ Lnaive$ which has exponential increase.In detail, let $ >0$ be agiven relative precision for an estimator $ PSLn$ of$ [ nW ]$, based on a number $ Ln$ of simulated samplesgenerated under some distribution $ S$, so that$$\\\\delta :=\\\\frac{var_{\\\\widetilde{S}}P_{\\\\widetilde{S}}^{L_{n}}}{\\\\left(\\\\mathbb {\\\\Pi }\\\\left[\\\\xi _{n}^{\\\\mathbf {\\\\widetilde{W}}}\\\\in \\\\widetilde{\\\\mathbf {\\\\Omega }}\\\\right]\\\\right)^{2}} \\\\, .$$Then $ Ln$ will grow exponentially as $ n$ tends to infinity if and only if$ S$ is not “asymptotically optimal”\\\\ ,the derivation of which is the scope of the current section.$ To start with the details, for the sake of brevity (to avoid certain substantial discussions on potential technical relaxations) we shall employ the following additional Assumption (OM) on the set $\\\\widetilde{\\\\mathbf {\\\\Omega }}$ : (OM) For any $\\\\widetilde{\\\\omega } \\\\in cl(\\\\widetilde{\\\\mathbf {\\\\Omega }})$ there exists a vector $\\\\mathbf {x} = \\\\left(x_{1},\\\\ldots ,x_{n}\\\\right) \\\\in \\\\left] t_{-}^{sc},t_{+}^{sc}\\\\right[^{n}$ such that $\\\\widetilde{\\\\omega } = \\\\left( \\\\frac{1}{n}\\\\sum _{i\\\\in I_{1}^{(n)}}x_{i},\\\\ldots ,\\\\frac{1}{n} \\\\sum _{i\\\\in I_{K}^{(n)}}x_{i}\\\\right)$ , or equivalently, for any $\\\\lambda \\\\in cl(\\\\mathbf {\\\\Lambda })$ there exists a vector $\\\\mathbf {x} = \\\\left(x_{1},\\\\ldots ,x_{n}\\\\right) \\\\in \\\\left] t_{-}^{sc},t_{+}^{sc}\\\\right[^{n}$ such that $\\\\lambda =T\\\\left(\\\\mathbf {x}\\\\right)$ .\",\n \"For instance, in the common case $dom(\\\\widetilde{\\\\varphi })= dom(\\\\varphi ) = \\\\, ]a,b[ \\\\, =\\\\, \\\\left] t_{-}^{sc},t_{+}^{sc}\\\\right[ \\\\, = \\\\, ]0,\\\\infty [$ (e.g.\",\n \"for the power-divergence generators $\\\\widetilde{\\\\varphi }= \\\\widetilde{c} \\\\cdot \\\\varphi _{\\\\gamma }$ , $\\\\gamma \\\\le 0$ , cf.\",\n \"Example REF ) the Assumption (OM) is always feasible.\",\n \"To proceed, for any distribution $\\\\widetilde{S}$ on $\\\\mathbb {R}^{n}$ with support included in the support of the product measure $\\\\widetilde{\\\\mathbb {}}^{\\\\otimes n}$ it holds $\\\\mathbb {\\\\Pi }\\\\left[\\\\xi _{n}^{\\\\mathbf {\\\\widetilde{W}}}\\\\in \\\\widetilde{\\\\mathbf {\\\\Omega }}\\\\right]=E_{\\\\widetilde{\\\\mathbb {}}^{\\\\otimes n}} \\\\left[\\\\mathbf {1}_{\\\\mathbf {\\\\Lambda } }( T( \\\\mathbf {\\\\widetilde{W}}))\\\\right] =E_{\\\\widetilde{S}}\\\\left[\\\\mathbf {1}_{\\\\mathbf {\\\\Lambda } }\\\\left( T\\\\left( \\\\mathbf {\\\\widetilde{V}}\\\\right) \\\\right) \\\\cdot \\\\frac{d\\\\widetilde{\\\\mathbb {}}^{\\\\otimes n}}{d\\\\widetilde{S}}\\\\left( \\\\mathbf {\\\\widetilde{V}}\\\\right) \\\\right]$ from where the improved IS estimator of $\\\\mathbb {\\\\Pi }\\\\left[\\\\xi _{n}^{\\\\mathbf {\\\\widetilde{W}}}\\\\in \\\\widetilde{\\\\mathbf {\\\\Omega }} \\\\right]$ is obtained by sampling $L$ i.i.d.\",\n \"replications $\\\\mathbf {\\\\widetilde{V}}^{1},\\\\ldots ,\\\\mathbf {\\\\widetilde{V}}^{L}$ of the random vector $\\\\mathbf {\\\\widetilde{V}}$ with distribution $\\\\widetilde{S}$ and by defining $\\\\widehat{\\\\widetilde{\\\\Pi }}_{L}^{improved}:=\\\\frac{1}{L}\\\\sum _{\\\\ell =1}^{L}\\\\mathbf {1}_{\\\\mathbf {\\\\Lambda }}( T( \\\\mathbf {\\\\widetilde{V}}^{(\\\\ell )})) \\\\cdot \\\\frac{d\\\\widetilde{\\\\mathbb {}}^{\\\\otimes n}}{d\\\\widetilde{S}}\\\\left( \\\\mathbf {\\\\widetilde{V}}^{(\\\\ell )}\\\\right) \\\\ .$ The precise form of the efficient IS distribution $\\\\widetilde{S}^{opt}$ relies on the definition of a “dominating point” of $\\\\mathbf {\\\\Lambda }$ , which we recall now.\",\n \"For $\\\\mathbf {x} := \\\\left( x_{1},..,x_{n}\\\\right)$ in $\\\\mathbb {R}^{n}$ we define $I_{\\\\mathbf {\\\\widetilde{W}}}(\\\\mathbf {x}):=\\\\sup _{\\\\mathbf {z} \\\\in \\\\mathbb {R}^{n}}\\\\left(\\\\left\\\\langle \\\\mathbf {z}, \\\\mathbf {x} \\\\right\\\\rangle -\\\\log E_{\\\\widetilde{\\\\mathbb {}}}[ \\\\, \\\\exp (\\\\langle \\\\mathbf {z},\\\\mathbf {\\\\widetilde{W}}\\\\rangle ) ]\\\\right) \\\\, ,$ and for $\\\\lambda $ in $\\\\mathbf {\\\\Lambda }$ we let $I(\\\\lambda ):=\\\\inf \\\\left\\\\lbrace I_{\\\\mathbf {\\\\widetilde{W}}}(\\\\mathbf {x}):T(\\\\mathbf {x)=\\\\lambda }\\\\right\\\\rbrace .$ Let $\\\\underline{\\\\lambda }:=\\\\left( \\\\underline{\\\\lambda }_{1},\\\\ldots ,\\\\underline{\\\\lambda }_{K}\\\\right) \\\\in \\\\partial \\\\mathbf {\\\\Lambda }$ .\",\n \"We call $\\\\underline{\\\\lambda }$ a minimal rate point (mrp) of $\\\\mathbf {\\\\Lambda }$ if $I(\\\\underline{\\\\lambda })\\\\le I(\\\\lambda ) \\\\quad \\\\text{ for all }\\\\lambda \\\\in \\\\mathbf {\\\\Lambda } .$ A minimal rate point $\\\\underline{\\\\lambda }$ is called a dominating point of $\\\\mathbf {\\\\Lambda }$ if a) $\\\\underline{\\\\lambda }\\\\in \\\\partial \\\\mathbf {\\\\Lambda }$ , and b) $I\\\\left(\\\\underline{\\\\lambda }\\\\right)\\\\le I(\\\\lambda )$ for all $\\\\lambda \\\\in \\\\mathbf {\\\\Lambda }$ with attainment, namely there exists a vector $\\\\underline{\\\\mathbf {x}} \\\\in \\\\left] t_{-}^{sc},t_{+}^{sc}\\\\right[^{n}$ such that $I_{\\\\widetilde{W}}\\\\left( \\\\underline{\\\\mathbf {x}}\\\\right) =I\\\\left(\\\\underline{\\\\lambda }\\\\right)$ with $\\\\underline{\\\\lambda }=T\\\\left( \\\\underline{\\\\mathbf {x}}\\\\right)$ .\",\n \"The characterization of the dominating point $\\\\underline{\\\\lambda }$ is settled in the following Lemma 57 Let $\\\\underline{\\\\lambda }$ be a mrp of $\\\\mathbf {\\\\Lambda }$ .\",\n \"Then, under Assumption (OM), $\\\\underline{\\\\lambda }$ is a dominating point, and $\\\\inf \\\\left\\\\lbrace I_{\\\\mathbf {\\\\widetilde{W}}}\\\\left( \\\\mathbf {x}\\\\right) ,T(\\\\mathbf {x})=\\\\underline{\\\\lambda }\\\\right\\\\rbrace $ is reached at some vector $\\\\underline{\\\\mathbf {x}}$ in $\\\\left] t_{-}^{sc},t_{+}^{sc}\\\\right[^{n}$ such that for all $k \\\\in \\\\left\\\\lbrace 1,\\\\ldots ,K\\\\right\\\\rbrace $ and all $i \\\\in I_{k}^{(n)}$ there holds $\\\\underline{x}_{i}= \\\\underline{\\\\lambda }_{k}$ and $I_{\\\\mathbf {\\\\widetilde{W}}}(\\\\underline{\\\\mathbf {x}})=n \\\\cdot \\\\sum _{k=1}^{K}\\\\widetilde{p}_{k} \\\\cdot \\\\widetilde{\\\\varphi }\\\\left(\\\\underline{\\\\lambda }_{k}\\\\right)$ .\",\n \"The proof Lemma REF is given in Appendix G. Notice that (OM) implies the existence of a dominating point $\\\\underline{\\\\lambda }$ , but uniqueness may not hold.\",\n \"In the latter case, one can try to proceed as in Theorem 2 of [SB] and the discussion thereafter.\",\n \"However, we assume now uniqueness of $\\\\underline{\\\\lambda }$ ; this allows for the identification of $\\\\widetilde{S}^{opt}$ .\",\n \"By Theorem 1 of [SB] and Theorem 3.1 of Csiszar [96], the asymptotically optimal IS distribution $\\\\widetilde{S}^{opt}$ is obtained as the Kullback-Leibler projection of $\\\\widetilde{\\\\mathbb {}}^{n\\\\otimes }$ on the set of all probability distributions on $\\\\mathbb {R}^{n}$ centered at point $\\\\underline{\\\\mathbf {x}}$ , whose coordinates are — according to Lemma REF — functions of the coordinates of $\\\\underline{\\\\mathbf {\\\\widetilde{Q}}} := \\\\mathfrak {D}^{-1} \\\\underline{\\\\lambda }$ such that $T\\\\left( \\\\underline{\\\\mathbf {x}}\\\\right)= \\\\mathfrak {D} \\\\underline{\\\\mathbf {\\\\widetilde{Q}}}$ .\",\n \"The above definition of $\\\\widetilde{S}^{opt}$ presumes the knowledge of $\\\\underline{\\\\lambda }$ , which cannot be assumed (otherwise the minimization problem is solved in advance).\",\n \"The aim of the following construction is to provide a proxy $\\\\widetilde{S}$ to $\\\\widetilde{S}^{opt}$ , where $\\\\widetilde{S}$ is the Kullback-Leibler projection of $\\\\widetilde{\\\\mathbb {}}^{\\\\otimes n}$ on the set of all probability distributions on $\\\\mathbb {R}^{n}$ centered at some point $\\\\mathbf {x}^{\\\\ast }$ which is close to $\\\\underline{\\\\mathbf {x}}$ .\",\n \"For this sake, we need to have at hand a proxy of $\\\\underline{\\\\lambda }$ or, equivalently, a preliminary guess $\\\\mathbf {\\\\widetilde{Q}}^{\\\\ast }$ of $\\\\underline{\\\\mathbf {\\\\widetilde{Q}}}:=\\\\arg \\\\inf _{\\\\mathbf {\\\\widetilde{Q}} \\\\in \\\\widetilde{\\\\mathbf {\\\\Omega }}}\\\\sum _{k=1}^{K} \\\\widetilde{p}_{k} \\\\cdot \\\\widetilde{\\\\varphi }(\\\\widetilde{q}_{k}/\\\\widetilde{p}_{k})$ .\",\n \"This guess is by no means produced in order to provide a direct estimate of $D_{\\\\widetilde{\\\\varphi } }(\\\\widetilde{\\\\mathbf {\\\\Omega }},\\\\widetilde{{P}})$ but merely to provide the IS distribution $\\\\widetilde{S}$ which in turn leads to a sharp estimate of $D_{\\\\widetilde{\\\\varphi } }(\\\\widetilde{\\\\mathbf {\\\\Omega }},\\\\widetilde{{P}})$ .\",\n \"Proxy method 1: in some cases we might have at hand some particular point $\\\\mathbf {\\\\widetilde{Q}}^{\\\\ast } :=\\\\left(\\\\widetilde{q}_{1}^{\\\\ast },..,\\\\widetilde{q}_{K}^{\\\\ast }\\\\right)$ in $\\\\widetilde{\\\\mathbf {\\\\Omega }}$ ; the resulting IS distribution $\\\\widetilde{S}$ with $\\\\mathbf {\\\\widetilde{Q}}$ substituted by $\\\\mathbf {\\\\widetilde{Q}^{\\\\ast }}$ is not optimal in the sense of [SB], but anyhow produces an estimator with good hitting rate, possibly with a loss in the variance.\",\n \"A simple way to obtain such a point $\\\\mathbf {\\\\widetilde{Q}^{\\\\ast }}$ in $\\\\widetilde{\\\\mathbf {\\\\Omega }}$ is to simulate runs of (say) $M-$ variate i.i.d.\",\n \"vectors $\\\\mathbf {\\\\widetilde{W}}$ under $\\\\widetilde{\\\\mathbb {}}^{\\\\otimes M}$ until the first time where $\\\\xi _{M}^{\\\\mathbf {\\\\widetilde{W}}}$ belongs to $\\\\widetilde{\\\\mathbf {\\\\Omega }}$ ; then we set $\\\\mathbf {\\\\widetilde{Q}^{\\\\ast }} := \\\\xi _{M}^{\\\\mathbf {\\\\widetilde{W}}}$ for the succeeding realization $\\\\mathbf {\\\\widetilde{W}}$ .\",\n \"Before we proceed, it is useful to mention that the need for a drastic fall in the number of simulation runs pertains for cases when $D_{\\\\widetilde{\\\\varphi } }(\\\\widetilde{\\\\mathbf {\\\\Omega }},\\\\widetilde{{P}})$ is large.\",\n \"The following construction is suited to this case, which is of relevance in applications both in optimization and in statistics when choosing between competing models none of which is assumed to represent the true one, but merely less inadequate ones.\",\n \"Proxy method 2: when $D_{\\\\widetilde{\\\\varphi } }(\\\\widetilde{\\\\mathbf {\\\\Omega }},\\\\widetilde{{P}})$ is presumably large, we make use of asymptotic approximation to get a proxy of $\\\\underline{\\\\mathbf {\\\\widetilde{Q}}}$ .\",\n \"For this, we define a sampling distribution on $\\\\mathbb {R}^{K}$ fitted to the divergence through $f(\\\\mathbf {\\\\widetilde{Q}}):=C \\\\cdot \\\\exp \\\\left( -\\\\sum _{k=1}^{K} \\\\widetilde{p}_{k} \\\\cdot \\\\widetilde{\\\\varphi }(\\\\widetilde{q}_{k}/\\\\widetilde{p}_{k})\\\\right) \\\\ = \\\\ C \\\\cdot \\\\exp \\\\left( -D_{\\\\widetilde{\\\\varphi }}\\\\left( \\\\mathbf {\\\\widetilde{Q}},\\\\widetilde{{P}}\\\\right)\\\\right)$ where $C$ is a normalizing constant.\",\n \"Let $\\\\mathbf {T}$ be a $K-$ variate random variable with density $f$ .\",\n \"The distribution of $\\\\mathbf {T}$ given $\\\\left( \\\\mathbf {T} \\\\in \\\\widetilde{\\\\mathbf {\\\\Omega }}\\\\right)$ concentrates around $\\\\arg \\\\inf _{\\\\widetilde{\\\\mathbf {Q}}\\\\in \\\\widetilde{\\\\mathbf {\\\\Omega }} }D_{\\\\widetilde{\\\\varphi } }(\\\\widetilde{\\\\mathbf {Q}},\\\\widetilde{{P}})$ when $D_{\\\\widetilde{\\\\varphi } }(\\\\widetilde{\\\\mathbf {\\\\Omega }},\\\\widetilde{{P}})$ is large.\",\n \"Indeed, for any $\\\\widetilde{\\\\mathbf {Q}}\\\\in \\\\widetilde{\\\\mathbf {\\\\Omega }}$ denote by $\\\\mathbf {V}_{\\\\varepsilon }(\\\\widetilde{\\\\mathbf {Q}})$ a small neighborhood of $\\\\widetilde{\\\\mathbf {Q}}$ in $\\\\mathbb {R}^{K}$ with radius $\\\\varepsilon $ ; clearly, the probability of the event $\\\\left( \\\\mathbf {T} \\\\in \\\\mathbf {V}_{\\\\varepsilon }(\\\\widetilde{\\\\mathbf {Q}})\\\\right)$ when restricted to $\\\\widetilde{\\\\mathbf {Q}}\\\\in \\\\widetilde{\\\\mathbf {\\\\Omega }}$ is maximum when $\\\\widetilde{\\\\mathbf {Q}} = \\\\underline{\\\\widetilde{\\\\mathbf {Q}}}$ , where $\\\\underline{\\\\widetilde{\\\\mathbf {Q}}}$ is the “dominating point of $\\\\widetilde{\\\\mathbf {\\\\Omega }}$ ” in the sense that $\\\\underline{\\\\mathbf {\\\\widetilde{Q}}} := \\\\mathfrak {D}^{-1} \\\\underline{\\\\lambda }$ is the above-defined transform of the dominating point $\\\\underline{\\\\lambda }$ (assuming uniqueness); a precise argumentation under adequate conditions is postponed to Appendix G. Accordingly, we obtain a proxy $\\\\mathbf {\\\\widetilde{Q}}^{\\\\ast }$ of $\\\\underline{\\\\mathbf {\\\\widetilde{Q}}}$ by simulating a sequence of independent $K-$ variate random variables $\\\\mathbf {T}_{1},\\\\ldots $ with distribution (REF ) until (say) $\\\\mathbf {T}_{m}$ belongs to $\\\\widetilde{\\\\mathbf {\\\\Omega }}$ and set $\\\\mathbf {\\\\widetilde{Q}}^{\\\\ast }:=\\\\mathbf {T}_{m}$ .\",\n \"To proceed with the derivation of the IS sampling distribution $\\\\widetilde{S}$ on $\\\\mathbb {R}^{n}$ , we fix $\\\\mathbf {\\\\widetilde{Q}}^{\\\\ast } :=\\\\left(\\\\widetilde{q}_{1}^{\\\\ast },..,\\\\widetilde{q}_{K}^{\\\\ast }\\\\right)$ to be a proxy of $\\\\underline{\\\\mathbf {\\\\widetilde{Q}}}$ or an initial guess in $\\\\widetilde{\\\\mathbf {\\\\Omega }}$ .\",\n \"As an intermediate step, we construct the probability distribution $\\\\widetilde{U}_{k}$ on $\\\\mathbb {R}$ given by $d\\\\widetilde{U}_{k}(v) \\\\ := \\\\ \\\\exp \\\\left( \\\\tau _{k} \\\\cdot v -\\\\Lambda _{\\\\widetilde{\\\\mathbb {}}}(\\\\tau _{k})\\\\right) d\\\\widetilde{\\\\mathbb {}}(v)=\\\\frac{\\\\exp \\\\left(\\\\tau _{k} \\\\cdot v\\\\right)}{MGF_{\\\\widetilde{\\\\mathbb {}}}(\\\\tau _{k})} \\\\, d\\\\widetilde{\\\\mathbb {}}(v)$ where $\\\\tau _{k} \\\\in int(dom(MGF_{\\\\widetilde{\\\\mathbb {}}}))$ is the unique solution of the equation $\\\\Lambda _{\\\\widetilde{\\\\mathbb {}}}^{\\\\prime }\\\\left( \\\\tau _{k}\\\\right) =\\\\frac{\\\\widetilde{q}_{k}^{\\\\ast }}{\\\\widetilde{p}_{k}}$ and thus — by relation (REF ) of Appendix F — we can compute explicitly $\\\\tau _{k} = \\\\ \\\\widetilde{\\\\varphi }^{\\\\, \\\\prime }\\\\left(\\\\frac{\\\\widetilde{q}_{k}^{\\\\ast }}{\\\\widetilde{p}_{k}}\\\\right) \\\\ .$ Therefore, $\\\\widetilde{U}_{k}$ is the Kullback-Leibler projection of $\\\\widetilde{\\\\mathbb {}}$ on the class of all probability distributions on $\\\\mathbb {R}$ whose expectation is $\\\\widetilde{q}_{k}^{\\\\ast }$ .\",\n \"As a side remark, notice that one possible way of obtaining an explicit form of the probability distribution $\\\\widetilde{U}_{k}$ may be by identification through its moment generating function $dom(MGF_{\\\\widetilde{\\\\mathbb {}}})-\\\\tau _{k} \\\\ \\\\ni \\\\ z \\\\ \\\\mapsto MGF_{\\\\widetilde{U}_{k}}(z) =\\\\frac{MGF_{\\\\widetilde{\\\\mathbb {}}}(z+\\\\tau _{k})}{MGF_{\\\\widetilde{\\\\mathbb {}}}(\\\\tau _{k})}$ of which all ingredients are principally available.\",\n \"For instance, this will be used in Example REF below.\",\n \"From (REF ), we define $\\\\widetilde{S}_{k}:=\\\\underbrace{\\\\widetilde{U}_{k} \\\\otimes \\\\cdots \\\\otimes \\\\widetilde{U}_{k}}_{n_{k}\\\\text{times}}\\\\qquad \\\\textrm {for all $ {1,...,K}$},$$whence\\\\begin{eqnarray}d\\\\widetilde{S}_{k}\\\\left( v_{k,1},\\\\ldots ,v_{k,n_{k}}\\\\right) =\\\\exp \\\\Big ( \\\\sum _{i\\\\in I_{k}^{(n)}}\\\\tau _{k} \\\\cdot v_{k,i}-n_{k} \\\\cdot \\\\Lambda _{\\\\widetilde{\\\\mathbb {}}}(\\\\tau _{k})\\\\Big ) \\\\ d\\\\widetilde{\\\\mathbb {}}\\\\left(v_{k,1}\\\\right)\\\\cdots d\\\\widetilde{\\\\mathbb {}}\\\\left(v_{k,n_{k}}\\\\right),\\\\\\\\\\\\end{eqnarray}which manifests $ Sk$ as the Kullback-Leibler projection of$ nktimes$on the class of all probability distributions on $ Rk$whose expectation vector is$ Q = (q1,...,qK) Rk$.\",\n \"Let now\\\\begin{equation}\\\\widetilde{S} := \\\\widetilde{S}_{1} \\\\otimes \\\\cdots \\\\otimes \\\\widetilde{S}_{K} \\\\, ,\\\\end{equation}which therefore satisfies (recall that $ k=1K nk =n$)\\\\begin{equation}d\\\\widetilde{S}\\\\left( v_{1,1},\\\\ldots v_{1,n_{1}},\\\\ldots , v_{K,1},\\\\ldots v_{K,n_{K}}\\\\right)=\\\\exp \\\\Big (\\\\sum _{k=1}^{K}\\\\sum _{i\\\\in I_{k}^{(n)}} \\\\big ( \\\\tau _{k} \\\\cdot v_{k,i} - n_{k} \\\\cdot \\\\Lambda _{\\\\widetilde{\\\\mathbb {}}}(\\\\tau _{k}) \\\\big )\\\\Big ) \\\\,d\\\\widetilde{\\\\mathbb {}}^{\\\\otimes n}\\\\left( v_{1,1},\\\\ldots v_{1,n_{1}},\\\\ldots , v_{K,1},\\\\ldots v_{K,n_{K}}\\\\right).\\\\ \\\\end{equation}The same procedure with all $ qk$substituted by the coordinates $qk$of $Q$produces $ Sopt$.\",\n \"Therefore, $ S$ is a substitutefor $ Sopt$ with the change in the centering fromthe unknown vector $Q$to its proxy $ Q$.$ As a straightforward consequence of (REF ) and (), we obtain the improved IS estimator of $\\\\mathbb {\\\\Pi }\\\\left[\\\\xi _{n}^{\\\\mathbf {\\\\widetilde{W}}}\\\\in \\\\widetilde{\\\\mathbf {\\\\Omega }}\\\\right]$ as $\\\\widehat{\\\\widetilde{\\\\Pi }}_{L}^{improved}\\\\ =\\\\ \\\\frac{1}{L}\\\\sum _{\\\\ell =1}^{L}\\\\mathbf {1}_{\\\\mathbf {\\\\Lambda }}( T( \\\\mathbf {\\\\widetilde{V}}^{(\\\\ell )})) \\\\cdot \\\\prod _{k=1}^{K} IS_{k}(\\\\mathbf {\\\\widetilde{V}}_{k}^{(\\\\ell )})$ where $\\\\mathbf {\\\\widetilde{V}}_{k}^{(\\\\ell )} := \\\\left( \\\\widetilde{V}_{i}^{(\\\\ell )} \\\\right)_{i \\\\in I_{k}^{(n)}}$ is the $k-$ th block of the $\\\\ell -$ th replication $\\\\mathbf {\\\\widetilde{V}}^{(\\\\ell )}$ of $\\\\mathbf {\\\\widetilde{V}}$ under $S$ , and the $k-$ th importance-sampling factor is $\\\\widetilde{IS}_{k}(v_{k,1},\\\\ldots ,v_{k,n_{k}}) \\\\ := \\\\ \\\\frac{d\\\\widetilde{\\\\mathbb {}}^{\\\\otimes n_{k}}}{d\\\\widetilde{S}_{k}}\\\\left( v_{k,1},\\\\ldots ,v_{k,n_{k}}\\\\right)= \\\\exp \\\\Big (n_{k} \\\\cdot \\\\Lambda _{\\\\widetilde{\\\\mathbb {}}}(\\\\tau _{k}) \\\\, - \\\\, \\\\tau _{k}\\\\cdot \\\\sum _{i=1}^{n_{k}} v_{k,i} \\\\Big )\\\\nonumber $ with $n_{k} = card(I_{k}^{(n)})$ .\",\n \"Summing up things, we arrive at the following algorithm in case that $\\\\widetilde{\\\\mathbf {\\\\Omega }}$ has a unique dominating point (in the above-defined sense): Step D1 Exemplarily, we start with proxy method 2 (the other proxy method 1 works analogously): get a proxy $\\\\mathbf {\\\\widetilde{Q}}^{\\\\ast }$ of $\\\\underline{\\\\mathbf {\\\\widetilde{Q}}}$ by simulating a sequence of independent $K-$ variate random variables $\\\\mathbf {T}_{1},\\\\ldots $ with distribution (REF ) until (say) $\\\\mathbf {T}_{m}$ belongs to $\\\\widetilde{\\\\mathbf {\\\\Omega }}$ and set $\\\\mathbf {\\\\widetilde{Q}}^{\\\\ast }:=\\\\mathbf {T}_{m}$ .\",\n \"Step D2 For all $k$ in $\\\\lbrace 1,\\\\ldots ,K\\\\rbrace $ compute $\\\\tau _{k} = \\\\ \\\\widetilde{\\\\varphi }^{\\\\, \\\\prime }\\\\left(\\\\frac{\\\\widetilde{q}_{k}^{\\\\ast }}{\\\\widetilde{p}_{k}}\\\\right)$ .\",\n \"Step D3 For all $\\\\ell $ in $\\\\lbrace 1,\\\\ldots ,L\\\\rbrace $ perform a run of $\\\\mathbf {\\\\widetilde{V}}^{(\\\\ell )}$ under $\\\\widetilde{S}$ as follows: For all $k$ in $\\\\lbrace 1,\\\\ldots ,K\\\\rbrace $ simulate $n_{k}$ i.i.d.\",\n \"random variables $\\\\widetilde{V}_{k_{1}}^{(\\\\ell )},\\\\ldots ,\\\\widetilde{V}_{k_{n_{k}}}^{(\\\\ell )}$ with common distribution $\\\\widetilde{U}_{k}$ defined in (REF ).\",\n \"Set $\\\\mathbf {\\\\widetilde{V}}_{k}^{(\\\\ell )} :=(\\\\widetilde{V}_{k_{1}}^{(\\\\ell )},\\\\ldots ,\\\\widetilde{V}_{k_{n_{k}}}^{(\\\\ell )})$ to be the corresponding row vector.\",\n \"Construct $\\\\mathbf {\\\\widetilde{V}}^{(\\\\ell )}$ as the row vector obtained by concatenating the $\\\\mathbf {\\\\widetilde{V}}_{k}^{(\\\\ell )}$ , i.e.\",\n \"$\\\\mathbf {\\\\widetilde{V}}^{(\\\\ell )}:=\\\\left( \\\\mathbf {\\\\widetilde{V}}_{1}^{(\\\\ell )},\\\\ldots ,\\\\mathbf {\\\\widetilde{V}}_{K}^{(\\\\ell )}\\\\right) \\\\, ,$ and make use of $\\\\widehat{\\\\widetilde{\\\\Pi }}_{L}^{improved}$ given in (REF ) with the $\\\\tau _{k}$ 's obtained in Step D2 above to define (in the light of (REF ), (REF )) the BS minimum-distance estimator $\\\\widehat{D_{\\\\varphi }}(\\\\mathbf {\\\\Omega },\\\\mathbf {P})\\\\ := \\\\ \\\\widehat{D_{\\\\widetilde{\\\\varphi }}}(\\\\widetilde{\\\\mathbf {\\\\Omega }},\\\\widetilde{{P}})\\\\ := \\\\ - \\\\frac{1}{n}\\\\log \\\\widehat{\\\\widetilde{\\\\Pi }}_{L}^{improved} \\\\ .$ For many cases, the simulation burden needed for the computation of $\\\\widehat{\\\\widetilde{\\\\Pi }}_{L}^{improved}$ — and thus of $\\\\widehat{D_{\\\\varphi }}(\\\\mathbf {\\\\Omega },\\\\mathbf {P})$ — can be drastically reduced, especially for high dimensions $K$ and large sample size $n \\\\cdot L$ .\",\n \"In fact, in terms of the notations $n_{k}:=card(I_{k}^{(n)})$ , $\\\\widehat{\\\\mathit {W}}_{k}^{(\\\\ell )}:=\\\\sum _{i\\\\in I_{k}^{(n)}}\\\\widetilde{V}_{i}^{(\\\\ell )}$ and $\\\\widetilde{ISF}_{k}(x) \\\\ := \\\\ \\\\frac{d\\\\widetilde{\\\\mathbb {}}^{\\\\ast n_{k}}}{d\\\\widetilde{U}_{k}^{\\\\ast n_{k}}}(x) \\\\ = \\\\ \\\\exp (n_{k} \\\\cdot \\\\Lambda _{\\\\widetilde{\\\\mathbb {}}}(\\\\tau _{k}) \\\\, - \\\\, x \\\\cdot \\\\tau _{k} )$ (where $\\\\widetilde{\\\\mathbb {}}^{\\\\ast n_{k}}$ is the $n_{k}-$ convolution of the measure $\\\\widetilde{\\\\mathbb {}}$ ), one can rewrite (REF ) as $\\\\widehat{\\\\widetilde{\\\\Pi }}_{L}^{improved}=\\\\frac{1}{L}\\\\sum _{\\\\ell =1}^{L}\\\\mathbf {1}_{\\\\mathbf {\\\\Lambda }}\\\\big ( \\\\,\\\\big (\\\\frac{1}{n_{1}} \\\\widehat{\\\\mathit {W}}_{1}^{(\\\\ell )},\\\\ldots , \\\\frac{1}{n_{K}} \\\\widehat{\\\\mathit {W}}_{K}^{(\\\\ell )}\\\\big ) \\\\, \\\\big ) \\\\cdot \\\\prod \\\\limits _{k=1}^{K}\\\\widetilde{ISF}_{k}(\\\\widehat{\\\\mathit {W}}_{k}^{(\\\\ell )}) .$ with $K-$ vector $\\\\big (\\\\frac{1}{n_{1}} \\\\widehat{\\\\mathit {W}}_{1}^{(\\\\ell )},\\\\ldots , \\\\frac{1}{n_{K}} \\\\widehat{\\\\mathit {W}}_{K}^{(\\\\ell )} \\\\big )$ .\",\n \"Clearly, the random variable $\\\\widehat{\\\\mathit {W}}_{k}^{(\\\\ell )}$ ($k=1, \\\\ldots , K$ ) has distribution $\\\\widetilde{U}_{k}^{\\\\ast n_{k}}$ .\",\n \"Hence, if $\\\\widetilde{U}_{k}^{\\\\ast n_{k}}$ can be explicitly constructed, then for the computation of $\\\\widehat{\\\\widetilde{\\\\Pi }}_{L}^{improved}$ it suffices to simulate the $K \\\\cdot L$ random variables $\\\\widehat{\\\\mathit {W}}_{k}^{(\\\\ell )}$ rather than the $n \\\\cdot L$ random variables $\\\\widetilde{V}_{i}^{(\\\\ell )}$ ; notice that according to the right-hand side of (REF ), one can explicitly compute $ISF_{k}\\\\left( \\\\cdot \\\\right)$ which can be interpreted as Importance Sampling Factor pertaining to the block $k$.\",\n \"In the case that $\\\\widetilde{\\\\mathbb {}}$ is infinitely divisible, simulation issues may become especially comfortable.\",\n \"In the following, we exemplarily demonstrate the tractability of this reduction effect, for the BS minimization of the important power divergences (for which the infinite divisibility holds): Example 58 Let $\\\\varphi _{\\\\gamma }$ ($\\\\gamma \\\\in \\\\mathbb {R}\\\\backslash ]1,2[$ ) be the power divergence generator from the Examples REF and REF , $\\\\mathbf {P} \\\\in \\\\mathbb {R}_{> 0}^{K}$ , $M_{\\\\mathbf {P}}:=\\\\sum _{i=1}^{K}p_{i}>0$ and $n_{k} = n \\\\cdot p_{k} \\\\in \\\\mathbb {N}$ where we have employed our notation $n_{k}= n \\\\cdot p_{k} \\\\in \\\\mathbb {N}$ for all $k \\\\in \\\\lbrace 1,\\\\ldots ,K\\\\rbrace $ .\",\n \"Moreover, let $\\\\mathbf {\\\\widetilde{Q}}^{\\\\ast } :=\\\\left(\\\\widetilde{q}_{1}^{\\\\ast },\\\\ldots ,\\\\widetilde{q}_{K}^{\\\\ast }\\\\right)$ be a proxy obtained by either proxy method 1 or 2.\",\n \"Case 1: Example REF (a): $\\\\gamma =0,\\\\widetilde{c}>0$ .\",\n \"There holds $\\\\widetilde{U}_{k}^{\\\\ast n_{k}}=GAM\\\\left( \\\\widetilde{c} \\\\cdot M_{\\\\mathbf {P}} - \\\\tau _{k},n_{k}\\\\cdot \\\\widetilde{c} \\\\cdot M_{\\\\mathbf {P}}\\\\right)$ , with $\\\\tau _{k}=\\\\widetilde{c} \\\\cdot M_{\\\\mathbf {P}} \\\\cdot ( 1-\\\\frac{p_{k}}{M_{\\\\mathbf {P}} \\\\cdot \\\\widetilde{q}_{k}^{\\\\ast }})$ for $\\\\widetilde{q}_{k}^{\\\\ast } >0$ (the latter is equivalent to $\\\\tau _{k} < \\\\widetilde{c} \\\\cdot M_{\\\\mathbf {P}}$ ).\",\n \"Moreover, for all $x >0$ one gets $\\\\widetilde{ISF}_{k}(x)=\\\\left( \\\\frac{\\\\widetilde{c} \\\\cdot M_{\\\\mathbf {P}}}{\\\\widetilde{c}\\\\cdot M_{\\\\mathbf {P}} - \\\\tau _{k}}\\\\right)^{n_{k}\\\\cdot \\\\widetilde{c}\\\\cdot M_{\\\\mathbf {P}}}\\\\cdot e^{-\\\\tau _{k} \\\\cdot x}$ .\",\n \"Case 2: REF (b): $\\\\gamma \\\\in \\\\left( 0,1\\\\right) ,\\\\widetilde{c}>0$ .\",\n \"We derive $\\\\widetilde{U}_{k}^{\\\\ast n_{k}}=C\\\\big ( POI( n_{k}\\\\cdot \\\\breve{\\\\theta }),GAM\\\\big (\\\\frac{\\\\widetilde{c} \\\\cdot M_{\\\\mathbf {P}}}{1-\\\\gamma } - \\\\tau _{k},\\\\frac{\\\\gamma }{1-\\\\gamma } \\\\big ) \\\\big )$ with $\\\\breve{\\\\theta }:= \\\\frac{\\\\widetilde{c} \\\\cdot M_{\\\\mathbf {P}}}{\\\\gamma }\\\\cdot \\\\big (\\\\frac{(\\\\gamma -1) \\\\cdot \\\\tau _{k}}{\\\\widetilde{c} \\\\cdot M_{\\\\mathbf {P}}}+1\\\\big )^{\\\\gamma /(\\\\gamma -1)}$ and $\\\\tau _{k} = \\\\widetilde{c} \\\\cdot M_{\\\\mathbf {P}} \\\\cdot \\\\frac{1-\\\\big (\\\\frac{\\\\widetilde{q}_{k}^{\\\\ast } \\\\cdot M_{\\\\mathbf {P}}}{p_{k}}\\\\big )^{\\\\gamma -1}}{1-\\\\gamma }$ for $\\\\widetilde{q}_{k}^{\\\\ast } >0$ .\",\n \"Furthermore, $\\\\widetilde{ISF}_{k}(x)=e^{-\\\\tau _{k}x} \\\\cdot \\\\exp \\\\left( \\\\frac{n_{k}\\\\cdot \\\\widetilde{c} \\\\cdot M_{\\\\mathbf {P}}}{\\\\gamma }\\\\cdot \\\\left(\\\\left(1+ \\\\frac{\\\\gamma -1}{\\\\widetilde{c} \\\\cdot M_{\\\\mathbf {P}}} \\\\cdot \\\\tau _{k}\\\\right) ^{\\\\frac{\\\\gamma }{\\\\gamma -1}}-1 \\\\right)\\\\right), \\\\qquad x \\\\ge 0 ,$ (where $x=0$ covers the atom at zero).\",\n \"Case 3: Example REF (c): $\\\\gamma =2,\\\\widetilde{c}>0$ .\",\n \"One gets $\\\\widetilde{U}_{k}^{\\\\ast n_{k}}=N(n_{k}\\\\cdot (1+\\\\frac{\\\\tau _{k}}{\\\\widetilde{c} \\\\cdot M_{\\\\mathbf {P}}}),\\\\frac{n_{k}}{\\\\widetilde{c} \\\\cdot M_{\\\\mathbf {P}}})$ with $\\\\tau _{k}=\\\\widetilde{c} \\\\cdot M_{\\\\mathbf {P}} \\\\cdot ( \\\\frac{\\\\widetilde{q}_{k}^{\\\\ast } \\\\cdot M_{\\\\mathbf {P}}}{p_{k}} - 1 ) $ for $\\\\widetilde{q}_{k}^{\\\\ast } \\\\in \\\\mathbb {R}$ .\",\n \"Moreover, for all $x \\\\in \\\\mathbb {R}$ one obtains $\\\\widetilde{ISF}_{k}(x)= \\\\exp \\\\big (\\\\frac{n_{k} \\\\cdot \\\\tau _{k}^2}{2\\\\widetilde{c} \\\\cdot M_{\\\\mathbf {P}}}- (x- n_{k}) \\\\cdot \\\\tau _{k} \\\\big )$ .\",\n \"Case 4: Example REF (d): $\\\\gamma <0,\\\\widetilde{c}>0$ .\",\n \"It holds that $\\\\widetilde{U}_{k}^{\\\\ast n_{k}}$ has the (Lebesgue-)density $f_{\\\\widetilde{U}_{k}^{\\\\ast n_{k}}}(x)\\\\ := \\\\ \\\\frac{\\\\exp ((\\\\tau _{k} - \\\\frac{\\\\widetilde{c}\\\\cdot M_{\\\\mathbf {P}}}{1-\\\\gamma })\\\\cdot x)}{\\\\exp \\\\left(n_{k} \\\\cdot \\\\frac{\\\\widetilde{c}\\\\cdot M_{\\\\mathbf {P}}}{\\\\gamma }\\\\cdot (1+\\\\frac{\\\\gamma -1}{\\\\widetilde{c} \\\\cdot M_{\\\\mathbf {P}}}\\\\cdot \\\\tau _{k})^{\\\\gamma /(\\\\gamma -1)}\\\\right)}\\\\cdot f_{\\\\breve{\\\\breve{Z}}}(x) \\\\cdot {1}_{]0,\\\\infty [}(x),\\\\qquad x \\\\in \\\\mathbb {R},$ where $\\\\tau _{k} = \\\\widetilde{c} \\\\cdot M_{\\\\mathbf {P}} \\\\cdot \\\\frac{1-\\\\big (\\\\frac{\\\\widetilde{q}_{k}^{\\\\ast } \\\\cdot M_{\\\\mathbf {P}}}{p_{k}}\\\\big )^{\\\\gamma -1}}{1-\\\\gamma }$ for $\\\\widetilde{q}_{k}^{\\\\ast } >0$ , and $\\\\breve{\\\\breve{Z}}$ is a random variable with density $f_{\\\\breve{\\\\breve{Z}}}$ of a stable law with parameter-quadruple $(\\\\frac{-\\\\gamma }{1-\\\\gamma },1,0,- n_{k} \\\\cdot \\\\frac{(\\\\widetilde{c}\\\\cdot M_{\\\\mathbf {P}})^{1/(1-\\\\gamma )} \\\\cdot (1-\\\\gamma )^{-\\\\gamma /(1-\\\\gamma )}}{\\\\gamma })$ (analogously to $\\\\breve{Z}$ of Example 40 (d) but with $\\\\widetilde{c}$ replaced by $\\\\widetilde{c}\\\\cdot M_{\\\\mathbf {P}}$ ).\",\n \"Also, $\\\\widetilde{ISF}_{k}(x)=e^{-\\\\tau _{k}x} \\\\cdot \\\\exp \\\\left( \\\\frac{n_{k}\\\\cdot \\\\widetilde{c} \\\\cdot M_{\\\\mathbf {P}}}{\\\\gamma }\\\\cdot \\\\left(\\\\left(1+ \\\\frac{\\\\gamma -1}{\\\\widetilde{c} \\\\cdot M_{\\\\mathbf {P}}} \\\\cdot \\\\tau _{k}\\\\right) ^{\\\\frac{\\\\gamma }{\\\\gamma -1}}-1 \\\\right)\\\\right), \\\\qquad x >0.$ Case 5 : Example REF (e): $\\\\gamma >2,\\\\widetilde{c}>0$ .\",\n \"We derive that $\\\\widetilde{U}_{k}^{\\\\ast n_{k}}$ has the (Lebesgue-)density $f_{\\\\widetilde{U}_{k}^{\\\\ast n_{k}}}(x)\\\\ := \\\\ \\\\frac{\\\\exp ((\\\\tau _{k} + \\\\frac{\\\\widetilde{c}\\\\cdot M_{\\\\mathbf {P}}}{\\\\gamma -1})\\\\cdot x)}{\\\\exp \\\\left(n_{k} \\\\cdot \\\\frac{\\\\widetilde{c}\\\\cdot M_{\\\\mathbf {P}}}{\\\\gamma }\\\\cdot (1+\\\\frac{\\\\gamma -1}{\\\\widetilde{c} \\\\cdot M_{\\\\mathbf {P}}}\\\\cdot \\\\tau _{k})^{\\\\gamma /(\\\\gamma -1)}\\\\right)}\\\\cdot f_{\\\\breve{\\\\breve{Z}}}(-x) ,\\\\qquad x \\\\in \\\\mathbb {R},$ where $\\\\tau _{k} = - \\\\frac{\\\\widetilde{c} \\\\cdot M_{\\\\mathbf {P}}}{\\\\gamma -1} \\\\cdot \\\\big (1- \\\\big (\\\\frac{\\\\widetilde{q}_{k}^{\\\\ast } \\\\cdot M_{\\\\mathbf {P}}}{p_{k}}\\\\big )^{\\\\gamma -1}\\\\cdot {1}_{]0,\\\\infty [}(\\\\widetilde{q}_{k}^{\\\\ast }) \\\\big )$ for $\\\\widetilde{q}_{k}^{\\\\ast } \\\\in \\\\mathbb {R}$ , and $\\\\breve{\\\\breve{Z}}$ is a random variable with density $f_{\\\\breve{\\\\breve{Z}}}$ of a stable law with parameter-quadruple $(\\\\frac{\\\\gamma }{\\\\gamma -1},1,0,n_{k} \\\\cdot \\\\frac{(\\\\widetilde{c}\\\\cdot M_{\\\\mathbf {P}})^{1/(1-\\\\gamma )} \\\\cdot (\\\\gamma -1)^{\\\\gamma /(\\\\gamma -1)}}{\\\\gamma })$ (analogously to $\\\\breve{Z}$ of Example 40 (e) but with $\\\\widetilde{c}$ replaced by $\\\\widetilde{c}\\\\cdot M_{\\\\mathbf {P}}$ ).\",\n \"Furthermore, $\\\\widetilde{ISF}_{k}(x)=e^{-\\\\tau _{k}x} \\\\cdot \\\\exp \\\\left( \\\\frac{n_{k}\\\\cdot \\\\widetilde{c} \\\\cdot M_{\\\\mathbf {P}}}{\\\\gamma }\\\\cdot \\\\left(\\\\left(1+ \\\\frac{\\\\gamma -1}{\\\\widetilde{c} \\\\cdot M_{\\\\mathbf {P}}} \\\\cdot \\\\tau _{k}\\\\right) ^{\\\\frac{\\\\gamma }{\\\\gamma -1}}-1 \\\\right)\\\\right), \\\\qquad x \\\\in \\\\mathbb {R}.$ Case 6: Example REF (a): $\\\\gamma =1,\\\\widetilde{c}>0$ , anchor point $c=0$ .\",\n \"It holds that $\\\\widetilde{U}_{k}^{\\\\ast n_{k}}$ is the probability distribution $\\\\frac{1}{\\\\widetilde{c} \\\\cdot M_{\\\\mathbf {P}}} \\\\cdot POI\\\\left(n_{k} \\\\cdot \\\\widetilde{c} \\\\cdot M_{\\\\mathbf {P}} \\\\cdot \\\\exp (\\\\frac{\\\\tau _{k}}{\\\\widetilde{c} \\\\cdot M_{\\\\mathbf {P}}})\\\\right) $ with support on the lattice $\\\\left\\\\lbrace \\\\frac{j}{\\\\widetilde{c} \\\\cdot M_{\\\\mathbf {P}}}, \\\\, j\\\\in \\\\mathbb {N}_{0}\\\\right\\\\rbrace $ , where $\\\\tau _{k}=\\\\widetilde{c} \\\\cdot \\\\log \\\\left( \\\\frac{M_{\\\\mathbf {P}} \\\\cdot \\\\widetilde{q}_{k}^{\\\\ast }}{p_{k}}\\\\right) $ for $\\\\widetilde{\\\\omega } _{k} >0$ .\",\n \"Moreover, for all $j \\\\in \\\\mathbb {N}_{0}$ we obtain (by setting $x:= \\\\frac{j}{\\\\widetilde{c} \\\\cdot M_{\\\\mathbf {P}}}$ ) $\\\\widetilde{ISF}_{k}\\\\left(\\\\frac{j}{\\\\widetilde{c} \\\\cdot M_{\\\\mathbf {P}}}\\\\right)= \\\\exp \\\\left(n_{k} \\\\cdot \\\\widetilde{c} \\\\cdot M_{\\\\mathbf {P}} \\\\cdot \\\\left( \\\\exp \\\\left( \\\\frac{\\\\tau _{k}}{\\\\widetilde{c} \\\\cdot M_{\\\\mathbf {P}}}\\\\right) -1 \\\\right)- m \\\\cdot \\\\frac{\\\\tau _{k}}{\\\\widetilde{c} \\\\cdot M_{\\\\mathbf {P}}}\\\\right) .$ Case 7: Example REF (b): $\\\\gamma =1,\\\\widetilde{c}=1,$ anchor point $c\\\\in \\\\mathbb {R}$ .\",\n \"For $M_{P}=1$ , $\\\\widetilde{U}_{k}^{\\\\ast n_{k}}$ is the shifted Poisson distribution $ POI\\\\left(n_{k} \\\\cdot e^{c +\\\\tau _{k}}\\\\right) + n_{k} \\\\cdot (1-e^{c})$ with support on the lattice $\\\\left\\\\lbrace j + n_{k} \\\\cdot (1-e^{c}), \\\\, j\\\\in \\\\mathbb {N}_{0}\\\\right\\\\rbrace $ , where $\\\\tau _{k}= \\\\log \\\\big (\\\\frac{\\\\widetilde{q}_{k}^{\\\\ast }}{p_{k}} + e^{c}-1\\\\big )-c$ for $\\\\widetilde{q}_{k}^{\\\\ast } > p_{k} \\\\cdot (1-e^{c})$ .\",\n \"Furthermore, for all $j \\\\in \\\\mathbb {N}_{0}$ we obtain (by setting $x:= j + n_{k} \\\\cdot (1-e^{c})$ ) $\\\\widetilde{ISF}_{k}\\\\left(j + n_{k} \\\\cdot (1-e^{c})\\\\right)= \\\\exp \\\\left(n_{k} \\\\cdot e^{c} \\\\cdot (e^{\\\\tau _{k}} -1) - j \\\\cdot \\\\tau _{k} \\\\right).$ Notice that the mass of $\\\\widetilde{U}_{k}^{\\\\ast n_{k}}$ at zero depends on the value of the anchor point $c$ , since $\\\\widetilde{U}_{k}^{\\\\ast n_{k}}[\\\\lbrace 0\\\\rbrace ] >0$ if and only if $c=\\\\log (1+\\\\frac{\\\\ell }{n_{k}})$ for some $\\\\ell \\\\in \\\\mathbb {N}_{0}$ ; moreover, $\\\\widetilde{U}_{k}^{\\\\ast n_{k}}\\\\big [ \\\\ ]0,\\\\infty [ \\\\ \\\\big ] =1$ if $c <0$ and $\\\\widetilde{U}_{k}^{\\\\ast n_{k}}\\\\big [ \\\\ ]-\\\\infty ,0[ \\\\ \\\\big ] >0$ if $c >0$ .\",\n \"Remark 59 (a) One can explicitly see in all cases of the above Example REF that all ingredients for computation are at hand.\",\n \"(b) For both Cases 4 and 5 in the above Example REF , algorithms for simulation can be obtained by adapting e.g.\",\n \"the works of Devroye [111] and Devroye & James [112].\",\n \"In the previous Subsection REF , as a first step we have estimated $\\\\mathbb {\\\\Pi }\\\\left[\\\\xi _{n}^{\\\\mathbf {\\\\widetilde{W}}}\\\\in \\\\widetilde{\\\\mathbf {\\\\Omega }}\\\\right]$ in terms of the improved IS estimator $\\\\widehat{\\\\widetilde{\\\\Pi }}_{L}^{improved}$ .\",\n \"From this, as a second step, we have derived — on the basis of Theorem REF — the estimator $\\\\widehat{D_{\\\\varphi }}(\\\\mathbf {\\\\Omega },\\\\mathbf {P})\\\\ := \\\\ - \\\\frac{1}{n}\\\\log \\\\widehat{\\\\widetilde{\\\\Pi }}_{L}^{improved} \\\\ \\\\nonumber \\\\qquad \\\\textrm {(cf.\",\n \"(\\\\ref {estimator minimization}))}$ of the minimum distance $D_{\\\\varphi }(\\\\mathbf {\\\\Omega },\\\\mathbf {P}):= \\\\inf _{\\\\mathbf {Q}\\\\in \\\\mathbf {\\\\Omega } } D_{\\\\varphi }(\\\\mathbf {Q},\\\\mathbf {P})$ , where $\\\\mathbf {P} \\\\in \\\\mathbb {R}_{> 0}^{K}$ and $\\\\mathbf {\\\\Omega }\\\\subset \\\\mathbb {R}^{K}$ .\",\n \"Recall that $\\\\widetilde{\\\\mathbf {\\\\Omega }} :=\\\\mathbf {\\\\Omega } /M_{\\\\mathbf {P}}$ with $M_{\\\\mathbf {P}}:=\\\\sum _{i=1}^{K}p_{i}>0$ , and that $\\\\xi _{n}^{\\\\mathbf {\\\\widetilde{W}}}=\\\\Big (\\\\frac{1}{n}\\\\sum _{i\\\\in I_{1}^{(n)}}\\\\widetilde{W}_{i},\\\\ldots ,\\\\frac{1}{n}\\\\sum _{i\\\\in I_{K}^{(n)}}\\\\widetilde{W}_{i}\\\\Big )\\\\hspace{56.9055pt} \\\\text{(cf.\",\n \"(\\\\ref {Xi_n^W vector}))}\\\\nonumber $ where $\\\\mathbf {\\\\widetilde{W}} := (\\\\widetilde{W}_{1}, \\\\ldots , \\\\widetilde{W}_{n})$ is a random vector consisting of components $\\\\widetilde{W}_{i}$ which are i.i.d.\",\n \"copies of the random variable $\\\\widetilde{W}$ whose distribution is $\\\\mathbb {\\\\Pi }[\\\\widetilde{W}\\\\in \\\\cdot \\\\,]=\\\\widetilde{\\\\mathbb {}}[\\\\,\\\\cdot \\\\,]$ obeying the representation (REF ).\",\n \"In contrast, we now proceed as follows: as a first step, we derive an improved estimator $\\\\widehat{\\\\Pi }_{L}^{improved}$ of $\\\\mathbb {\\\\Pi }_{X_{1}^{n}}\\\\left[\\\\xi _{n,\\\\mathbf {X}}^{w\\\\mathbf {W}}\\\\in \\\\textrm {\\\\right.$ $\\\\hspace{-6.544pt}$$}$$where $$\\\\hspace{-6.544pt}$ SK$is a set of probability vectors which satisfies theregularity properties (\\\\ref {regularity}) and the finiteness property (\\\\ref {def fi wrt Omega}).Recall that\\\\begin{eqnarray}\\\\xi _{n,\\\\mathbf {X}}^{w\\\\mathbf {W}} &:=&{\\\\left\\\\lbrace \\\\begin{array}{ll}\\\\left(\\\\frac{\\\\sum _{i \\\\in I_{1}^{(n)}}W_{i}}{\\\\sum _{k=1}^{K}\\\\sum _{i \\\\in I_{k}^{(n)}}W_{i}},\\\\ldots , \\\\frac{\\\\sum _{i \\\\in I_{K}^{(n)}}W_{i}}{\\\\sum _{k=1}^{K}\\\\sum _{i \\\\in I_{k}^{(n)}}W_{i}} \\\\right) ,\\\\qquad \\\\textrm {if } \\\\sum _{j=1}^{n} W_{j} \\\\ne 0, \\\\\\\\\\\\ (\\\\infty , \\\\ldots , \\\\infty ) =: \\\\infty , \\\\hspace{113.81102pt} \\\\textrm {if } \\\\sum _{j=1}^{n} W_{j} = 0,\\\\end{array}\\\\right.\",\n \"}\\\\hspace{42.67912pt}\\\\textrm {(cf.\",\n \"(\\\\ref {brostu3:fo.norweiemp.vec}))}\\\\nonumber \\\\end{eqnarray}where $ (Xi)iN$ is a sequence of random variableswith values in $ Y:={ d1,,dK}$such that\\\\begin{equation}\\\\lim _{n\\\\rightarrow \\\\infty } \\\\Big ( \\\\frac{n_{1}}{n}, \\\\ldots , \\\\frac{n_{K}}{n} \\\\Big ) = (p_{1}, \\\\ldots , p_{K})\\\\qquad \\\\textrm {a.s.} \\\\qquad \\\\textrm {cf.\",\n \"((\\\\ref {cv emp measure X to P vector}))}\\\\nonumber \\\\end{equation}holds for some probability vector $ P := (p1, ..., pK) S> 0K$,by employing the notation$$n_{k} \\\\ := \\\\ card(\\\\bigl \\\\lbrace i \\\\in \\\\lbrace 1, \\\\ldots , n\\\\rbrace : \\\\ X_{i} = d_{k} \\\\bigr \\\\rbrace )\\\\ =: \\\\ card(I_{k}^{(n)})\\\\qquad \\\\textrm {(cf.\",\n \"(\\\\ref {I^(n)_k for stat case}));}$$hence, on the $ k$-th block of indexes $ Ik(n)$ all the $ Xi$^{\\\\prime }s share the same value $ dk$.Moreover, recall that $ (Wi)i N$ is a familyof independent and identically distributed $ R-$valued random variableswith probability distribution $ [ ] := [W1 ]$being connected with the divergence generator $ (]a,b[)$ via the representability(\\\\ref {Phi Legendre of mgf(W)}),such that $ (Wi)i N$ is independent of $ (Xi)i N$.$ As a second step (see Subsubsection REF below), for the important special case of the power-divergence generators $\\\\varphi _{\\\\gamma }$ (cf.\",\n \"(REF )) we employ the Propositions REF to REF in order to deduce via the corresponding $\\\\widehat{\\\\Pi }_{L}^{improved}$ the estimators (e.g.\",\n \"for $\\\\gamma <0$ ) $\\\\widehat{D_{\\\\widetilde{c}\\\\cdot \\\\varphi _{\\\\gamma }}(\\\\textrm {\\\\Omega \\\\hspace{-6.544pt}\\\\Omega },{P})} \\\\ := \\\\ \\\\frac{\\\\widetilde{c}}{\\\\gamma \\\\cdot \\\\left(\\\\gamma -1\\\\right) }\\\\cdot \\\\left\\\\lbrace \\\\left( 1+\\\\frac{\\\\gamma }{\\\\widetilde{c}}\\\\cdot \\\\frac{1}{n}\\\\cdot \\\\log \\\\,\\\\widehat{\\\\Pi }_{L}^{improved}\\\\right) ^{1-\\\\gamma } -1 \\\\right\\\\rbrace ,\\\\qquad \\\\ $ of the minimum power divergences $D_{\\\\widetilde{c}\\\\cdot \\\\varphi _{\\\\gamma }}(\\\\textrm {$ $\\\\hspace{-6.544pt}$$},{P}) :=\\\\inf _{{Q}\\\\in \\\\textrm {\\\\Omega \\\\hspace{-5.40608pt}\\\\Omega }}D_{\\\\widetilde{c}\\\\cdot \\\\varphi _{\\\\gamma }}({Q},{P})$$as well as connected estimators of important deterministic transformations thereof.$ As a third step (see Subsubsection REF below), on the basis of Subsubsection REF we derive estimators of bounds of $D_{\\\\varphi }(\\\\textrm {$$\\\\hspace{-6.544pt}$$},{P})$ for more general divergence generators $\\\\varphi $ .\",\n \"Let us start with the above-mentioned first step, by remarking that the development of the estimator $\\\\widehat{\\\\Pi }_{L}^{improved}$ works quite analogously to that of $\\\\widehat{\\\\widetilde{\\\\Pi }}_{L}^{improved}$ in the previous Subsection REF .\",\n \"To make this even more transparent, we employ the notation $p_{n,k}^{emp}:=n_{k}/n$ (cf.\",\n \"(REF )) and label all random vectors of length $n$ in the same way as above: we sort the already given and thus fixed data $X_{i}$ 's in such a way that the first $n_{1}$ of them share the same value $d_{1}$ , and so on, until the last block with length $n_{K}$ in which the data have common value $d_{K}$ .\",\n \"In the light of the above considerations, we could achieve a naive estimate $\\\\widehat{\\\\Pi }_{L}^{naive}$ of $\\\\mathbb {\\\\Pi }_{X_{1}^{n}}[\\\\xi _{n,\\\\mathbf {X}}^{w\\\\mathbf {W}}\\\\in \\\\textrm {$$\\\\hspace{-6.544pt}$$}]$ through the following procedure.\",\n \"We simulate independently $L$ replicates $\\\\mathbf {W}^{(1)},\\\\ldots ,\\\\mathbf {W}^{(L)}$ of the vector $\\\\mathbf {W}:=\\\\left( W_{1},\\\\ldots ,W_{n}\\\\right) $ , with independent entries under $\\\\mathbb {}$ (cf.\",\n \"(REF )); those realizations do not depend on the $X_{i}$ 's.\",\n \"Then we construct $\\\\widehat{\\\\Pi }_{L}^{naive} :=\\\\frac{1}{L}\\\\sum _{\\\\ell =1}^{L}\\\\mathbf {1}_{\\\\textrm {\\\\Omega \\\\hspace{-5.40608pt}\\\\Omega }}\\\\left(\\\\xi _{n,\\\\mathbf {X}}^{w\\\\mathbf {W}^{(\\\\ell )}}\\\\right) .$ However, this procedure is time costly, since the estimate given in (REF ) has a very bad hit rate.\",\n \"Hence, analogously to Subsection REF we apply again an “efficient Importance Sampling (IS)” scheme in the sense of Sadowsky & Bucklew [313].\",\n \"This will involve the simulation of $L$ independent $n-$ tuples $\\\\mathbf {V}^{(\\\\ell )}\\\\mathbf {:=}\\\\left(V_{n}^{(\\\\ell )},\\\\ldots ,V_{n}^{(\\\\ell )}\\\\right) $ with common distribution $S$ on $\\\\mathbb {R}^{n}$ , such that $\\\\mathbb {}^{\\\\otimes n}$ is (measure-)equivalent with respect to $S$ .\",\n \"In fact, we rewrite $\\\\mathbb {\\\\Pi }_{X_{1}^{n}}[\\\\xi _{n,\\\\mathbf {X}}^{w\\\\mathbf {W}}\\\\in \\\\textrm {$$\\\\hspace{-6.544pt}$$}]$ as $\\\\mathbb {\\\\Pi }_{X_{1}^{n}}[\\\\xi _{n,\\\\mathbf {X}}^{w\\\\mathbf {W}}\\\\in \\\\textrm {\\\\Omega \\\\hspace{-6.544pt}\\\\Omega }]= E_{S}\\\\Big [\\\\frac{\\\\mathrm {d}\\\\mathbb {} ^{\\\\otimes n}}{\\\\mathrm {d}S}(V_{1},\\\\ldots ,V_{n})\\\\cdot \\\\mathbf {1}_{\\\\textrm {\\\\Omega \\\\hspace{-5.40608pt}\\\\Omega }}(\\\\xi _{n,\\\\mathbf {X}}^{w\\\\mathbf {V}})\\\\Big ]$ where $S$ designates any IS distribution of the vector $\\\\mathbf {V:=}(V_{1},\\\\ldots ,V_{n})$ , and $E_{S}[ \\\\, \\\\cdot \\\\, ]$ denotes the corresponding expectation operation.\",\n \"Notice that $S$ is a random probability distribution on $\\\\mathbb {R}^{n}$ ; in fact, $S$ is a conditional probability distribution given $X_{1}^{n}$ , and thus it would be more precise to write $S | X_{1}^{n}$ instead of $S$ ; for the sake of brevity, we omit $| X_{1}^{n}$ .\",\n \"As a consequence of (REF ), for adequately chosen $S$ , an improved estimator of $\\\\mathbb {\\\\Pi }_{X_{1}^{n}}[\\\\xi _{n,\\\\mathbf {X}}^{w\\\\mathbf {W}}\\\\in \\\\textrm {$$\\\\hspace{-6.544pt}$$}]$ is given by $\\\\widehat{\\\\Pi }_{L}^{improved}:= \\\\frac{1}{L}\\\\sum _{\\\\ell =1}^{L}\\\\frac{\\\\mathrm {d}\\\\mathbb {}^{\\\\otimes n}}{\\\\mathrm {d}S}(V_{1}^{(\\\\ell )},\\\\ldots ,V_{n}^{(\\\\ell )})\\\\cdot \\\\mathbf {1}_{\\\\textrm {\\\\Omega \\\\hspace{-5.40608pt}\\\\Omega }}(\\\\xi _{n,\\\\mathbf {X}}^{w\\\\mathbf {V}^{(\\\\ell )}}) \\\\, ,$ which also estimates $\\\\inf _{{Q}\\\\in \\\\textrm {\\\\Omega \\\\hspace{-5.40608pt}\\\\Omega } }\\\\ \\\\inf _{m\\\\ne 0}D_{\\\\varphi }(m\\\\cdot {Q},{P})$ by the virtue of ().\",\n \"Let us now deal with the concrete construction of a reasonable $S$ .\",\n \"Given some (typically) large integer $M$ , we start with the realization $\\\\mathbf {W}^{\\\\ast }:=\\\\left( W_{1}^{\\\\ast },\\\\ldots ,W_{M}^{\\\\ast }\\\\right)$ such that $\\\\mathbf {Q}^{\\\\ast }:=\\\\xi _{M,\\\\mathbf {X}}^{w\\\\mathbf {W}^{\\\\ast }} \\\\in int(\\\\textrm {$$\\\\hspace{-6.544pt}$$})$ .\",\n \"This may be given in advance or it may be achieved by drawing replicates $\\\\mathbf {W} = (W_{1}, \\\\ldots , W_{M})$ under $\\\\mathbb {}^{\\\\otimes M}$ until the first time where $\\\\xi _{M,\\\\mathbf {X}}^{w\\\\mathbf {W}}$ belongs to $int(\\\\textrm {$$\\\\hspace{-6.544pt}$$})$ .\",\n \"Notice that by the nature of $\\\\textrm {$$\\\\hspace{-6.544pt}$$}$ , $\\\\mathbf {Q}^{\\\\ast }$ is a probability vector which has the $K$ components $q_{k}^{\\\\ast }:=\\\\sum _{i=1}^{M}\\\\frac{W_{i}^{\\\\ast }}{\\\\sum _{j=1}^{M}W_{j}^{\\\\ast }}\\\\mathbf {1}_{\\\\lbrace d_{k}\\\\rbrace }(X_{i}),\\\\hspace{42.67912pt} k=1,\\\\ldots ,K.$ Before we proceed, let us give the substantial remark that changing $\\\\left( V_{1},\\\\ldots ,V_{n}\\\\right) $ drawn under $S$ to $\\\\left(c \\\\cdot V_{1}, \\\\ldots ,c \\\\cdot V_{n}\\\\right) $ for any $c\\\\ne 0$ yields $\\\\xi _{n,\\\\mathbf {X}}^{w\\\\mathbf {V}}= \\\\xi _{n,\\\\mathbf {X}}^{w\\\\, c\\\\cdot \\\\mathbf {V}}$ so that the distribution $S$ is not uniquely determined.\",\n \"Amongst all candidates, we choose the — uniquely determined — $S$ which is the Kullback-Leibler projection of $\\\\mathbb {}^{\\\\otimes n}$ on the set of all probability distributions on $\\\\mathbb {R}^{n}$ such that the $K$ “non-normalized” moment constraints $E_{S}[\\\\xi _{n,\\\\mathbf {X}}^{\\\\mathbf {V}}] =\\\\xi _{M,\\\\mathbf {X}}^{\\\\mathbf {W}^{\\\\ast }}$ (rather than the normalized $E_{S}[ \\\\xi _{n,\\\\mathbf {X}}^{w\\\\mathbf {V}}] =\\\\xi _{M,\\\\mathbf {X}}^{w\\\\mathbf {W}^{\\\\ast }}$ ) are satisfied, with the non-normalized vectors $\\\\xi _{M,\\\\mathbf {X}}^{\\\\mathbf {W}^{\\\\ast }} :=\\\\left( \\\\frac{1}{M}\\\\sum _{j=1}^{M}W_{j}^{\\\\ast }\\\\right) \\\\cdot Q^{\\\\ast } =: \\\\overline{W^{\\\\ast }} \\\\cdot Q^{\\\\ast },\\\\qquad \\\\xi _{n,\\\\mathbf {X}}^{\\\\mathbf {V}} :=\\\\left( \\\\frac{1}{n}\\\\sum _{j=1}^{n}V_{j}\\\\right) \\\\cdot \\\\xi _{n,\\\\mathbf {X}}^{w\\\\mathbf {V}} \\\\ .$ As already indicated above, this projection $S$ is a well-determined unique distribution on $\\\\mathbb {R}^{n}$ and — as we shall see in Proposition REF below — it is such that $\\\\xi _{n,\\\\mathbf {X}}^{w\\\\mathbf {V}}$ belongs to $\\\\textrm {$$\\\\hspace{-6.544pt}$$}$ with probability bounded away from 0 as $n$ increases, when $\\\\left( V_{1},\\\\ldots ,V_{n}\\\\right) $ are drawn under $S$ .\",\n \"Therefore, this IS distribution produces an estimate of $\\\\mathbb {\\\\Pi }_{X_{1}^{n}}[\\\\xi _{n,\\\\mathbf {X}}^{w\\\\mathbf {W}}\\\\in \\\\textrm {$$\\\\hspace{-6.544pt}$$}]$ .\",\n \"In order to justify the above construction of $S$ , we give the following result, which states that the IS sampling distribution $S$ yields a good hitting rate.\",\n \"Its proof will be given in Appendix H. Proposition 60 With the above definition of $S$ , $\\\\lim \\\\inf _{n\\\\rightarrow \\\\infty }$ $S\\\\left[ \\\\xi _{n,\\\\mathbf {X}}^{w\\\\mathbf {V}} \\\\in \\\\textrm {\\\\right.$$\\\\hspace{-6.544pt}$$} $ is bounded away from 0.\",\n \"We now come to the detailed construction of $S$ .\",\n \"The constraints (REF ) can be written in explicit form as $E_{S}\\\\Big [ \\\\, \\\\frac{1}{n_{k}}\\\\sum _{i\\\\in I_{k}^{(n)}} V_{i} \\\\, \\\\Big ]\\\\ = \\\\ \\\\overline{W^{\\\\ast }} \\\\cdot \\\\frac{q_{k}^{\\\\ast }}{p_{n,k}^{emp}} \\\\, , \\\\qquad k=1,\\\\ldots ,K.$ The distribution $S$ can be obtained by blocks.\",\n \"Indeed, let us define $S^{k}$ as the Kullback-Leibler (KL) projection of $\\\\mathbb {}^{\\\\otimes n_{k}}$ on the set of all distributions on $\\\\mathbb {R}^{n_{k}}$ such that (REF ) holds.\",\n \"We define the resulting $S$ as the product distribution of those $S^{k}$ 's.\",\n \"To obtain the latter, we start by defining $U_{k}$ as the KL projection of $\\\\mathbb {} $ on the set of all measures $Q$ on $\\\\mathbb {R}$ under (REF ).\",\n \"Then, $dU_{k}(v)= \\\\exp (\\\\tau _{k}v-\\\\Lambda _{\\\\mathbb {} }\\\\left( \\\\tau _{k}\\\\right) )\\\\, d\\\\mathbb {} (v) \\\\, ,$ where $\\\\tau _{k} \\\\in int(dom(MGF_{\\\\mathbb {}}))$ is the unique solution of the equation $\\\\Lambda _{\\\\mathbb {} }^{\\\\prime }\\\\left( \\\\tau _{k}\\\\right) =\\\\overline{W^{\\\\ast }} \\\\cdot \\\\frac{q_{k}^{\\\\ast }}{p_{n,k}^{emp}}$ and thus — by relation (REF ) of Appendix F — we can compute explicitly $\\\\tau _{k} = \\\\varphi ^{\\\\prime }\\\\left( \\\\frac{\\\\overline{W^{\\\\ast }} \\\\cdot q_{k}^{\\\\ast }}{p_{n,k}^{emp}} \\\\right) .$ The distribution $S^{k}$ is then defined by $S^{k}:=\\\\underbrace{U_{k}\\\\otimes \\\\cdots \\\\otimes U_{k}}_{n_{k}\\\\text{times}}$ from which we obtain $S := S^{1}\\\\otimes \\\\cdots \\\\otimes S^{K}.$ With this construction, it holds $\\\\frac{dS}{d\\\\mathbb {} ^{\\\\otimes n}}(v_{1},\\\\ldots ,v_{n})=\\\\exp \\\\left(\\\\sum \\\\limits _{k=1}^{K}\\\\left( \\\\sum _{i\\\\in I_{k}^{(n)}}\\\\tau _{k} \\\\cdot v_{i}-\\\\Lambda _{\\\\mathbb {}}(\\\\tau _{k})\\\\right) \\\\right)$ which proves that $S$ is indeed the KL projection of $\\\\mathbb {} ^{\\\\otimes n}$ we aimed at.\",\n \"Therefore, $\\\\mathbf {V}$ is composed of $K$ independent blocks of length $n_{k}$ each, and the $k-$ th subvector $\\\\mathbf {V}_{k}$ consists of all the random variables $V_{i}$ whose index $i$ satisfies $X_{i}=d_{k}.$ Within $\\\\mathbf {V}_{k}$ , all components are i.i.d.\",\n \"with same distribution $U_{k}$ on $\\\\mathbb {R}$ defined through $\\\\frac{dU_{k}}{d\\\\mathbb {} }(u)=\\\\exp \\\\left\\\\lbrace \\\\tau _{k}\\\\cdot u-\\\\Lambda _{\\\\mathbb {}}(\\\\tau _{k})\\\\right\\\\rbrace =\\\\frac{\\\\exp \\\\left\\\\lbrace \\\\tau _{k}\\\\cdot u\\\\right\\\\rbrace }{MGF_{\\\\mathbb {} }(\\\\tau _{k})},$ which leads to the moment generating function $dom(MGF_{\\\\mathbb {}})-\\\\tau _{k} \\\\ \\\\ni \\\\ z \\\\ \\\\mapsto MGF_{U_{k}}(z):=\\\\int _{\\\\mathbb {R}}e^{zy}dU_{k}(y)=\\\\frac{MGF_{\\\\mathbb {}}(z+\\\\tau _{k})}{MGF_{\\\\mathbb {} }(\\\\tau _{k})}.$ Let us remark that $U_{k}$ can be interpreted as the distorted distribution of $\\\\mathbb {}$ with the distortion parameter $\\\\tau _{k}$ (in some cases, this distortion even becomes a tilting/dampening).\",\n \"The estimator $\\\\widehat{\\\\Pi }_{L}^{improved}$ defined in (REF ) can be implemented through the following algorithm: Step S1 Choose some (typically large) $M$ and simulate repeatedly i.i.d.\",\n \"vectors $\\\\left(W_{1},..,W_{M}\\\\right) $ — whose independent components have common distribution $\\\\mathbb {} $ — until $\\\\xi _{M,\\\\mathbf {X}}^{w\\\\mathbf {W}}$ belongs to $\\\\textrm {$$\\\\hspace{-6.544pt}$$}$ .\",\n \"Call $\\\\left( W_{1}^{\\\\ast },..,W_{M}^{\\\\ast }\\\\right) $ the corresponding vector and $\\\\overline{W^{\\\\ast }}$ the arithmetic mean of its components.\",\n \"Moreover, denote by $\\\\xi _{M,\\\\mathbf {X}}^{w\\\\mathbf {W}^{\\\\ast }}$ the corresponding normalized weighted empirical measure, identified with the $K-$ component vector $Q^{\\\\ast } := (q_{1}^{\\\\ast },..,q_{K}^{\\\\ast })$ with $q_{k}^{\\\\ast }$ defined in (REF ).\",\n \"Step S2 For all $k \\\\in \\\\lbrace 1,\\\\ldots ,K\\\\rbrace $ compute $\\\\tau _{k} = \\\\varphi ^{\\\\prime }\\\\left( \\\\frac{\\\\overline{W^{\\\\ast }} \\\\cdot q_{k}^{\\\\ast }}{p_{n,k}^{emp}} \\\\right)$ .\",\n \"Step S3 For all $\\\\ell \\\\in \\\\lbrace 1,\\\\ldots ,L\\\\rbrace $ simulate independently for all $k \\\\in \\\\lbrace 1,\\\\ldots ,K\\\\rbrace $ a row vector $\\\\mathbf {V}_{k}^{(\\\\ell )}$ $:=\\\\left(V_{k_{1}}^{(\\\\ell )},...,V_{k_{n_{k}}}^{(\\\\ell )}\\\\right) $ with independent components with common distribution $U_{k}$ defined in (REF ).\",\n \"Concatenate these vectors to define the row vector $\\\\mathbf {V}^{(\\\\ell )}.$ Step S4 Compute the estimator $\\\\widehat{\\\\Pi }_{L}^{improved}$ by making use of the formula (REF ) which turns into the explicit form $& & \\\\widehat{\\\\Pi }_{L}^{improved} =\\\\frac{1}{L}\\\\sum _{\\\\ell =1}^{L}\\\\exp \\\\left(\\\\sum \\\\limits _{k=1}^{K}\\\\left(n_{k} \\\\cdot \\\\Lambda _{\\\\mathbb {} }(\\\\tau _{k}) -\\\\tau _{k} \\\\cdot \\\\sum _{i\\\\in I_{k}^{(n)}}V_{i}^{(\\\\ell )}\\\\right) \\\\right)\\\\cdot \\\\mathbf {1}_{\\\\textrm {\\\\Omega \\\\hspace{-5.40608pt}\\\\Omega }}\\\\left( \\\\xi _{n,\\\\mathbf {X}}^{w\\\\mathbf {V}^{(\\\\ell )}}\\\\right)$ Analogously to the paragraph right after (REF ) of the previous Subsection REF , in many cases we may improve the simulation burden needed for the computation of the estimator $\\\\widehat{\\\\Pi }_{L}^{improved}$ .\",\n \"In fact, in terms of the notations $\\\\widehat{\\\\mathit {W}}_{k}^{(\\\\ell )}:=\\\\sum _{i\\\\in I_{k}^{(n)}}V_{i}^{(\\\\ell )}$ we can rewrite (REF ) as $\\\\widehat{\\\\Pi }_{L}^{improved} =\\\\frac{1}{L}\\\\sum _{\\\\ell =1}^{L}\\\\mathbf {1}_{\\\\textrm {\\\\Omega \\\\hspace{-5.40608pt}\\\\Omega }}\\\\left( \\\\xi _{n,\\\\mathbf {X}}^{w\\\\mathbf {V}^{(\\\\ell )}} \\\\right)\\\\cdot \\\\prod _{k=1}^{K}ISF_{k}\\\\left(\\\\widehat{\\\\mathit {W}}_{k}^{(\\\\ell )}\\\\right)$ with $ISF_{k}(x) \\\\ := \\\\ \\\\exp (n_{k} \\\\cdot \\\\Lambda _{\\\\mathbb {}}(\\\\tau _{k}) \\\\, - \\\\, x \\\\cdot \\\\tau _{k} )$ and $\\\\xi _{n,\\\\mathbf {X}}^{w\\\\mathbf {V}^{(\\\\ell )}} &=&{\\\\left\\\\lbrace \\\\begin{array}{ll}\\\\left(\\\\frac{\\\\widehat{\\\\mathit {W}}_{1}^{(\\\\ell )}}{\\\\sum _{k=1}^{K}\\\\widehat{\\\\mathit {W}}_{k}^{(\\\\ell )}},\\\\ldots , \\\\frac{\\\\widehat{\\\\mathit {W}}_{K}^{(\\\\ell )}}{\\\\sum _{k=1}^{K}\\\\widehat{\\\\mathit {W}}_{k}^{(\\\\ell )}} \\\\right) ,\\\\qquad \\\\textrm {if } \\\\sum _{k=1}^{K}\\\\widehat{\\\\mathit {W}}_{k}^{(\\\\ell )} \\\\ne 0, \\\\\\\\\\\\ (\\\\infty , \\\\ldots , \\\\infty ) =: \\\\infty , \\\\hspace{65.44142pt} \\\\textrm {if }\\\\sum _{k=1}^{K}\\\\widehat{\\\\mathit {W}}_{k}^{(\\\\ell )} = 0 \\\\, .\\\\end{array}\\\\right.\",\n \"}$ Clearly, the random variable $\\\\widehat{\\\\mathit {W}}_{k}^{(\\\\ell )}$ ($k=1, \\\\ldots , K$ ) has distribution $U_{k}^{\\\\ast n_{k}}$ .\",\n \"Hence, if $U_{k}^{\\\\ast n_{k}}$ can be explicitly constructed, then for the computation of $\\\\widehat{\\\\Pi }_{L}^{improved}$ it suffices to independently simulate the $K \\\\cdot L$ random variables $\\\\widehat{\\\\mathit {W}}_{k}^{(\\\\ell )}$ (rather than the $n \\\\cdot L$ random variables $V_{i}^{(\\\\ell )}$ ).\",\n \"In the following subsubsection, we exemplarily demonstrate the tractability of this reduction effect.\"\n ],\n [\n \"BS minimization of power divergences and related quantities\",\n \"´ Consider the special case of power divergence generators $\\\\varphi := \\\\widetilde{c} \\\\cdot \\\\varphi _{\\\\gamma }$ ($\\\\gamma \\\\in \\\\mathbb {R}\\\\backslash ]1,2[$ ) of the Examples REF and REF .\",\n \"The corresponding estimators $\\\\widehat{\\\\Pi }_{L}^{improved}$ can be obtained as follows: within the results of Example REF , set $M_{\\\\mathbf {P}}=1$ , and replace $\\\\widetilde{q}_{k}^{\\\\ast }$ by $\\\\overline{W^{\\\\ast }} \\\\cdot q_{k}^{\\\\ast }$ as well as $p_{k}$ by $p_{n,k}^{emp}$ ; accordingly, $\\\\widetilde{U}_{k}^{\\\\ast n_{k}}$ turns into $U_{k}^{\\\\ast n_{k}}$ and $\\\\widetilde{ISF}_{k}$ into $ISF_{k}$ ; simulate independently the random variables $\\\\widehat{\\\\mathit {W}}_{k}^{(\\\\ell )}$ from $U_{k}^{\\\\ast n_{k}}$ ($k \\\\in \\\\lbrace 1, \\\\ldots , K\\\\rbrace $ , $\\\\ell \\\\in \\\\lbrace 1,\\\\ldots ,L\\\\rbrace $ ); plug in the results of (i),(ii) into (REF ), (REF ), and (REF ) in order to concretely compute $\\\\widehat{\\\\Pi }_{L}^{improved}$ .\",\n \"From this, we can easily generate improved estimators of the power divergences $\\\\inf _{\\\\mathbf {Q}\\\\in \\\\textrm {\\\\Omega \\\\hspace{-5.40608pt}\\\\Omega }}D_{\\\\widetilde{c}\\\\cdot \\\\varphi _{\\\\gamma }}(\\\\mathbf {Q},{P})$ — and more generally, improved estimators of all the infimum-quantities (e.g.\",\n \"Renyi divergences) respectively supremum-quantities in the parts (b) of the Propositions REF , REF , REF , REF , REF and REF with $A=1$ — by simply replacing $\\\\mathbb {\\\\Pi }_{X_{1}^{n}}[\\\\xi _{n}^{w\\\\mathbf {W}}\\\\in \\\\textrm {$$\\\\hspace{-6.544pt}$$} ]$ (respectively, its variants) by the corresponding estimator $\\\\widehat{\\\\Pi }_{L}^{improved}$ .\",\n \"If — in the light of Remark REF (vi) — the ${P} = (p_{1}, \\\\ldots , p_{K})$ is a pregiven known probability vector e.g.\",\n \"the uniform distribution ${P}^{unif}$ on $\\\\lbrace 1,\\\\ldots ,K\\\\rbrace $ (rather than the limit of the vector of empirical frequencies/masses of a sequence of random variables $X_{i}$ , cf.\",\n \"(REF )), then we proceed analogously as above by replacing $p_{n,k}^{emp}$ with $p_{k}$ ; correspondingly, we obtain improved estimators of all the infimum-quantities respectively supremum-quantities (e.g.\",\n \"Renyi entropies, diversity indices) in the parts (a) of the Propositions REF , REF , REF , REF , REF and REF with $A=1$ .\",\n \"For the sake of brevity, in the following we only present explicitly the outcoming improved estimators for the power divergences (in the “$X_{i}-$ context” ).\",\n \"Indeed, we simply replace the $\\\\mathbb {\\\\Pi }_{X_{1}^{n}}[\\\\xi _{n}^{w\\\\mathbf {W}}\\\\in \\\\textrm {$$\\\\hspace{-6.544pt}$$} ]$ in the formulas (REF ), (REF ), (REF ) (with $A=1$ ) by the improved estimator $\\\\widehat{\\\\Pi }_{L}^{improved}$ obtained through (i) to (iii); for arbitrarily fixed $\\\\widetilde{c} >0$ , this leads to the improved power-divergence estimators (BS estimators of power divergences) $&&\\\\hspace{-54.06006pt}\\\\widehat{D_{\\\\widetilde{c}\\\\cdot \\\\varphi _{\\\\gamma }}(\\\\textrm {\\\\Omega \\\\hspace{-6.544pt}\\\\Omega },{P})} \\\\ := \\\\ -\\\\frac{\\\\widetilde{c}}{\\\\gamma (\\\\gamma -1)}\\\\left\\\\lbrace 1-\\\\left( 1+\\\\frac{\\\\gamma }{\\\\widetilde{c}} \\\\cdot \\\\frac{1}{n} \\\\cdot \\\\log \\\\widehat{\\\\Pi }_{L}^{improved}\\\\right)^{1-\\\\gamma }\\\\right\\\\rbrace ,\\\\qquad \\\\gamma \\\\in \\\\, ]-\\\\infty ,0[ \\\\, \\\\cup \\\\, ]0,1[ \\\\, \\\\cup \\\\, [2,\\\\infty [,\\\\\\\\& &\\\\hspace{-54.06006pt}\\\\widehat{D_{\\\\widetilde{c}\\\\cdot \\\\varphi _{0}}(\\\\textrm {\\\\Omega \\\\hspace{-6.544pt}\\\\Omega },{P})} \\\\ := \\\\ -\\\\frac{1}{n}\\\\log \\\\widehat{\\\\Pi }_{L}^{improved} ,\\\\hspace{165.02606pt} \\\\gamma =0,\\\\\\\\& & \\\\hspace{-54.06006pt}\\\\widehat{D_{\\\\widetilde{c}\\\\cdot \\\\varphi _{1}}(\\\\textrm {\\\\Omega \\\\hspace{-6.544pt}\\\\Omega },{P})}\\\\ : = \\\\ - \\\\widetilde{c} \\\\cdot \\\\log \\\\left( 1+\\\\frac{1}{\\\\widetilde{c}} \\\\cdot \\\\frac{1}{n} \\\\cdot \\\\log \\\\widehat{\\\\Pi }_{L}^{improved}\\\\right) ,\\\\hspace{85.35826pt} \\\\gamma =1.$ Let us finally remark that from the above-mentioned Steps S1 to S4 (and analogously D1 to D4) one can see that for our BS method we basically need only a fast and accurate — pseudo, true, natural, quantum — random number generator.\",\n \"The corresponding computations can be principally run in parallel, and require relatively moderate computer memory/storage; a detailed discussion is beyond the scope of this paper, given its current length.\"\n ],\n [\n \"General case, part 2\",\n \"The algorithm which is presented in this section aims at the evaluation of the bounds $\\\\inf _{m\\\\ne 0}D_{\\\\varphi }\\\\left( m \\\\cdot \\\\textrm {\\\\right.\\\\Omega \\\\hspace{-6.544pt}\\\\Omega },{P} =\\\\inf _{{Q}\\\\in \\\\textrm {\\\\Omega \\\\hspace{-5.40608pt}\\\\Omega }}D_{\\\\varphi }\\\\left( m\\\\left( {Q}\\\\right) \\\\cdot {Q},{P}\\\\right)\\\\stackrel{(1)}{=} D_{\\\\varphi }\\\\left( m({Q}^{\\\\ast })\\\\cdot {Q}^{\\\\ast },{P}\\\\right) \\\\le D_{\\\\varphi }\\\\left( \\\\textrm {\\\\right.\\\\Omega \\\\hspace{-6.544pt}\\\\Omega },{P} \\\\le D_{\\\\varphi }\\\\left( {Q}^{\\\\ast },{P}\\\\right)$ obtained in Section REF , where ${Q}^{\\\\ast }$ satisfies the above equality $(1)$ .\",\n \"The estimator of the lower bound in (REF ) is $\\\\widehat{D}:=-\\\\frac{1}{n}\\\\log \\\\widehat{\\\\text{ }\\\\Pi }_{L}^{improved}$ defined in (REF ).\",\n \"We now turn to an estimate of the upper bound.\",\n \"Consider for any fixed ${Q}:=\\\\left( q_{1},..,q_{K}\\\\right) $ in $\\\\mathbb {S}_{>0}^{K}$ the real number $m_{n}({Q})$ which satisfies $D_{\\\\varphi }\\\\left( m_{n}({Q}) \\\\cdot {Q},{P}_{n}^{emp} \\\\right) =\\\\inf _{m\\\\ne 0}D_{\\\\varphi }\\\\left( m\\\\cdot \\\\textrm {\\\\right.$ $\\\\hspace{-6.544pt}$$},{P}_{n}^{emp}$$where $ Pnemp$ was defined in the course of (\\\\ref {I^(n)_k for stat case}).Such $ mn(Q)$ is well defined for all $ Q$since it satisfies the equation (in $ m$)\\\\begin{equation}\\\\frac{d}{dm}D_{\\\\varphi }\\\\left( m \\\\cdot {Q},{P}_{n}^{emp}\\\\right) =\\\\sum _{k=1}^{K} q_{k} \\\\cdot \\\\varphi ^{\\\\prime }\\\\left( \\\\frac{m \\\\cdot q_{k}}{p_{n,k}^{emp}}\\\\right) =0 .\\\\end{equation}Since the mapping $ mD( m Q,P) $ is convexand differentiable, existence and uniqueness of $ mn(Q)$ hold; furthermore,$ mn(Q)] k pn,kemp/qk,k pn,kemp/qk[ $since $ ddmD( m Q,Pnemp) $is negative when $ m=k pn,kemp/qk$ and positive when $ m=k pn,kemp/qk$.$ An estimate of the distribution ${Q}^{\\\\ast }$ is required.\",\n \"This can be achieved as follows: Estimate $\\\\inf _{m\\\\ne 0}D_{\\\\varphi }\\\\left( m \\\\cdot \\\\textrm {\\\\right.$$\\\\hspace{-6.544pt}$$}, {P}$ through $\\\\widehat{D} := -\\\\frac{1}{n}\\\\log \\\\widehat{\\\\Pi }_{L}^{improved}$ defined in (REF ).\",\n \"Set $i=0.$ Get some ${Q}_{i} := (q_{i,1}, \\\\ldots , q_{i,K})$ in $\\\\textrm {$$\\\\hspace{-6.544pt}$$}$ ; this can be obtained by simulating runs of vectors $\\\\left( W_{1},..,W_{n}\\\\right) $ through i.i.d.\",\n \"sampling under $\\\\mathbb {}$ .\",\n \"Evaluate $m_{n}({Q}_{i})$ by solving () (with $q_{i,k}$ instead of $q_{k}$ ) for $m$ , which is a fast calculation by the bisection method.\",\n \"If $D_{\\\\varphi }\\\\left( m_{n}({Q}_{i}) \\\\cdot {Q}_{i}, {P}_{n}^{emp} \\\\right) <\\\\widehat{D}+\\\\eta $ for some small $\\\\eta >0$ , then the proxy of ${Q}^{\\\\ast }$ is ${Q}_{i},$ denoted by $\\\\widehat{{Q}^{\\\\ast }}$ .\",\n \"Else set $i\\\\leftarrow i+1$ and get ${Q}_{i}$ in $\\\\textrm {$$\\\\hspace{-6.544pt}$$}\\\\cap \\\\left\\\\lbrace {Q} \\\\, : \\\\, D_{\\\\varphi }\\\\left( {Q},{P}_{n}^{emp} \\\\right) 0$ we normalized $\\\\widetilde{{P}}:=\\\\mathbf {P}/M_{\\\\mathbf {P}},$ and $\\\\widetilde{\\\\mathbf {Q}}:=\\\\mathbf {Q}/M_{\\\\mathbf {P}}$ for $\\\\mathbf {Q}$ in $\\\\mathbf {\\\\Omega }$ .\",\n \"With $\\\\widetilde{\\\\varphi } \\\\in \\\\Upsilon (]a,b[)$ defined through $\\\\widetilde{\\\\varphi }:=M_{\\\\mathbf {P}} \\\\cdot \\\\varphi $ , we have obtained $D_{\\\\varphi }(\\\\mathbf {Q},\\\\mathbf {P})=\\\\sum _{k=1}^{K}p_{k}\\\\cdot \\\\varphi \\\\left( \\\\frac{q_{k}}{p_{k}}\\\\right) =\\\\sum _{k=1}^{K}M_{\\\\mathbf {P}}\\\\cdot \\\\widetilde{p_{k}}\\\\cdot \\\\frac{\\\\varphi \\\\left( \\\\frac{M_{\\\\mathbf {P}}\\\\cdot \\\\widetilde{q_{k}}}{M_{\\\\mathbf {P}}\\\\cdot \\\\widetilde{p_{k}}}\\\\right) }{M_{\\\\mathbf {P}}}=D_{\\\\widetilde{\\\\varphi }}(\\\\widetilde{\\\\mathbf {Q}},\\\\widetilde{{P}})\\\\qquad \\\\textrm {(cf.\",\n \"(\\\\ref {min Pb prob1}))}.$ It has followed that the solution of (REF ) coincides with the one of the problem of finding $\\\\widetilde{\\\\Phi }_{\\\\widetilde{{P}}}(\\\\widetilde{\\\\mathbf {\\\\Omega }}) := \\\\inf _{\\\\widetilde{\\\\mathbf {Q}}\\\\in \\\\widetilde{\\\\mathbf {\\\\Omega }} }D_{\\\\widetilde{\\\\varphi } }(\\\\widetilde{\\\\mathbf {Q}},\\\\widetilde{{P}}),\\\\qquad \\\\textrm {with } \\\\widetilde{\\\\mathbf {\\\\Omega }}:=\\\\mathbf {\\\\Omega } /M_{\\\\mathbf {P}}\\\\qquad \\\\textrm {(cf.\",\n \"(\\\\ref {min Pb prob2}))}.$ So let us continue by tackling (REF ).\",\n \"From the assumptions on $\\\\widetilde{\\\\varphi }$ and the requirement (REF ) one can see that $\\\\text{$\\\\widetilde{W}_{1}$ has moment generating function$t\\\\rightarrow E_{\\\\mathbb {\\\\Pi }}[e^{z \\\\cdot \\\\widetilde{W}_{1}}]= MGF_{\\\\widetilde{\\\\mathbb {}}}(z)$which is finite on a non-void neighborhood of $0$},$ $E_{\\\\mathbb {\\\\Pi }}[\\\\widetilde{W}_1] = 1 ,$ since $\\\\widetilde{\\\\varphi }(1)=0=\\\\widetilde{\\\\varphi }^{\\\\prime }(1)$ .\",\n \"With the help of these, we obtain the following Proposition 61 Under the assumptions of Theorem REF , for any set $\\\\widetilde{\\\\mathbf {\\\\Omega }}\\\\subset \\\\mathcal {M} := \\\\mathbb {R}^{K}$ with (REF ) one has $-\\\\inf _{\\\\widetilde{\\\\mathbf {Q}} \\\\in int(\\\\widetilde{\\\\mathbf {\\\\Omega }}) }D_{\\\\varphi }\\\\left( \\\\widetilde{\\\\mathbf {Q}},\\\\widetilde{{P}}\\\\right) &\\\\le &\\\\lim \\\\inf _{n\\\\rightarrow \\\\infty }\\\\frac{1}{n}\\\\log \\\\mathbb {\\\\Pi }\\\\left[\\\\xi _{n}^{\\\\widetilde{\\\\textbf {W}}}\\\\in \\\\widetilde{\\\\mathbf {\\\\Omega }} \\\\right] \\\\nonumber \\\\\\\\&\\\\le &\\\\lim \\\\sup _{n\\\\rightarrow \\\\infty }\\\\frac{1}{n}\\\\log \\\\mathbb {\\\\Pi }\\\\left[\\\\xi _{n}^{\\\\widetilde{\\\\textbf {W}}}\\\\in \\\\widetilde{\\\\mathbf {\\\\Omega }} \\\\right]\\\\le -\\\\inf _{\\\\widetilde{\\\\mathbf {Q}}\\\\in cl(\\\\widetilde{\\\\mathbf {\\\\Omega }}) }D_{\\\\varphi }\\\\left( \\\\widetilde{\\\\mathbf {Q}},\\\\widetilde{{P}}\\\\right) .$ Proof of Proposition REF .\",\n \"Recall from Remark REF (v) that $I_{k}^{(n)} := \\\\lbrace i \\\\in \\\\lbrace 1,\\\\ldots ,n\\\\rbrace : \\\\widetilde{x}_{i}=d_{k}\\\\rbrace $ and $n_{k} := card(I_{k}^{(n)})$ denotes the number of elements therein ($k \\\\in \\\\lbrace 1,\\\\ldots ,K\\\\rbrace $ ), i.e.\",\n \"$n_{k}$ is the number of the $\\\\widetilde{x}_{i}$ ’s which equal $d_{k}$ .\",\n \"We follow the line of proof of Theorem 2.2.30 in Dembo & Zeitouni [108], which states the large deviation principle (LDP) for the vector of partial sums of random vectors in $\\\\mathbb {R}^{K}$ , where we also use Corollary 6.1.6 in [108] in relation with condition (REF ).\",\n \"Indeed, since the $k-$ th component of the vector $\\\\xi _{n}^{\\\\mathbf {\\\\widetilde{W}}}$ is the $1/n-$ fold of the sum of the $\\\\widetilde{W}_{i}$ ’s for which the corresponding $\\\\widetilde{x}_{i}$ ’s equal $d_{k}$ (i.e., $\\\\frac{1}{n} \\\\sum _{i\\\\in I_{k}^{(n)}} \\\\widetilde{W}_{i}$ ) the proof will follow from a similar treatment as for the standard Cramer LDP in $\\\\mathbb {R}^{K}.$ The only difference lies in two facts: the number of the summands for the coordinate $k$ is $n_{k}$ , the number of $\\\\widetilde{x}_{i}$ ’s which equal $d_{k}$ , instead of $n$ in the standard case.\",\n \"Furthermore we will need to substitute $n_{k}$ by its equivalent $n \\\\cdot \\\\widetilde{p}_{k}$ , which adds an approximation step.\",\n \"For the upper bound, the proof is based on the corresponding result for $B=B_{1}\\\\times \\\\cdots \\\\times B_{K}$ where the $B_{k}$ ’s are open bounded intervals on $\\\\mathbb {R}^{+}.$ Since the sequence $\\\\left( \\\\widetilde{x}_{1}, \\\\ldots \\\\right)$ satisfies $\\\\lim _{n\\\\rightarrow \\\\infty }\\\\frac{n_{k}}{n}= \\\\widetilde{p}_{k} ,\\\\qquad \\\\textrm {(cf.\",\n \"(\\\\ref {fo.freqlim}))}$ there holds $&&\\\\frac{1}{n}\\\\log \\\\mathbb {\\\\Pi } \\\\left[\\\\xi _{n}^{\\\\mathbf {\\\\widetilde{W}}} \\\\in B\\\\right]=\\\\frac{1}{n}\\\\log \\\\mathbb {\\\\Pi } \\\\bigg [ \\\\bigcap \\\\limits _{k=1}^{K}\\\\bigg (\\\\frac{1}{n} \\\\sum _{i\\\\in I_{k}^{(n)}} \\\\widetilde{W}_{i}\\\\in B_{k}\\\\bigg ) \\\\bigg ]\\\\nonumber \\\\\\\\&=&\\\\frac{1}{n}\\\\sum \\\\limits _{k=1}^{K}\\\\log \\\\mathbb {\\\\Pi } \\\\bigg [ \\\\frac{1+o(1)}{n_{k}}\\\\sum _{i\\\\in I_{k}^{(n)}} \\\\widetilde{W}_{i}\\\\in \\\\frac{1}{\\\\widetilde{p}_{k}} B_{k}\\\\bigg ],$ and hence $\\\\lim \\\\sup _{n\\\\rightarrow \\\\infty }\\\\frac{1}{n}\\\\log \\\\mathbb {\\\\Pi } \\\\left[\\\\xi _{n}^{\\\\mathbf {\\\\widetilde{W}}}\\\\in B\\\\right]&\\\\le &\\\\sum \\\\limits _{k=1}^{K} \\\\widetilde{p}_{k} \\\\cdot \\\\lim \\\\sup _{n_{k}\\\\rightarrow \\\\infty }\\\\frac{1}{n_{k}}\\\\log \\\\mathbb {\\\\Pi } \\\\bigg [ \\\\frac{1}{n_{k}}\\\\sum _{i\\\\in I_{k}^{(n)}} \\\\widetilde{W}_{i}\\\\in \\\\frac{1}{p_{k}}B_{k}\\\\bigg ] \\\\nonumber \\\\\\\\&\\\\le &-\\\\sum \\\\limits _{k=1}^{K}\\\\inf _{x_{k}\\\\in cl(B_{k})} \\\\widetilde{p}_{k} \\\\cdot \\\\varphi \\\\left(\\\\frac{x_{k}}{\\\\widetilde{p}_{k}}\\\\right) .$ To deduce (REF ) from (REF ), we have used (i) the fact that for all $k$ the random variables $\\\\frac{1}{n_{k}}\\\\left( 1+o(1))\\\\right) \\\\cdot \\\\sum _{i\\\\in I_{k}^{(n)}} \\\\widetilde{W}_{i}$ and $\\\\frac{1}{n_{k}}\\\\sum _{i\\\\in I_{k}^{(n)}} \\\\widetilde{W}_{i}$ are exponentially equivalent in the sense that their difference $\\\\Delta _{n_{k}}$ satisfies $\\\\lim \\\\sup _{n_{k}\\\\rightarrow \\\\infty }\\\\frac{1}{n_{k}}\\\\log \\\\mathbb {\\\\Pi }\\\\left[ \\\\,\\\\left|\\\\Delta _{n_{k}}\\\\right|>\\\\eta \\\\, \\\\right] =-\\\\infty ,$ making use of the Chernoff inequality for all positive $\\\\eta $ , as well as (ii) Theorem 4.2.13 in [108].\",\n \"Now the summation and the inf-operations can be permuted in (REF ) which proves the claim for the rectangle $B$ .\",\n \"As in [108], for a compact set $\\\\Omega $ we consider its finite covering by such open sets $B$ and conclude; for $\\\\Omega $ being a closed set, a tightness argument holds, following [108] Theorem 2.2.30 verbatim.\",\n \"For the lower bound consider the same rectangle $B$ .\",\n \"The argument which locates the tilted distribution at the center of $B$ , together with the use of the LLN for the corresponding r.v’s as in [108], in combination with the same approximations as above to handle the approximation of $n_{k}$ by $n \\\\cdot \\\\widetilde{p}_{k}$ , complete the proof of Proposition REF .\",\n \"We omit the details.\",\n \"$\\\\blacksquare $ Let us continue with the proof of Theorem REF , by giving the following two helpful lemmas for $\\\\Phi _{{P}}(\\\\mathbf {A}) := \\\\inf _{\\\\mathbf {Q} \\\\in \\\\mathbf {A}} D_{\\\\varphi }\\\\left(\\\\mathbf {Q},{P}\\\\right),\\\\qquad \\\\mathbf {A} \\\\subset \\\\mathcal {M} := \\\\mathbb {R}^{K},$ Lemma 62 For any open set $\\\\mathbf {A} \\\\subset \\\\mathcal {M} := \\\\mathbb {R}^{K}$ one has $\\\\Phi _{{P}}(\\\\mathbf {A}) = \\\\Phi _{{P}}(cl(\\\\mathbf {A}))$ .\",\n \"This is clear from the continuity of $\\\\Phi _{{P}}$ .\",\n \"Lemma 63 For any $\\\\mathbf {A} \\\\subset \\\\mathcal {M} := \\\\mathbb {R}^{K}$ satisfying (REF ) one has $\\\\Phi _{{P}}(cl(\\\\mathbf {A})) = \\\\Phi _{{P}}(\\\\mathbf {A}) = \\\\Phi _{{P}}(int(\\\\mathbf {A}))$ .\",\n \"Proof of Lemma REF .\",\n \"Assume first that $\\\\Phi _{{P}}(\\\\mathbf {A})$ is finite.\",\n \"Then suppose that $\\\\mathbf {A}$ satisfies (REF ) and $\\\\Phi _{{P}}(cl(\\\\mathbf {A})) < \\\\Phi _{{P}}(int(\\\\mathbf {A}))$ .\",\n \"The latter implies the existence of a point $a \\\\in cl(\\\\mathbf {A})$ such that $a \\\\notin int(\\\\mathbf {A})$ and $\\\\Phi _{{P}}(a)= \\\\Phi _{{P}}(cl(\\\\mathbf {A}))$ .\",\n \"But then, by Lemma REF and (REF ) one gets $\\\\Phi _{{P}}(int(\\\\mathbf {A})) = \\\\Phi _{{P}}(cl(int(\\\\mathbf {A}))) = \\\\Phi _{{P}}(cl(\\\\mathbf {A})) = \\\\Phi _{{P}}(a)$ which leads to a contradiction.\",\n \"When $\\\\Phi _{{P}}(\\\\mathbf {A}) = \\\\infty $ then $\\\\Phi _{{P}} (cl(\\\\mathbf {A}))=\\\\Phi _{{P}} (int(\\\\mathbf {A}))=\\\\Phi _{{P}} (\\\\mathbf {A})=\\\\infty $ .\",\n \"$\\\\blacksquare $ Putting things together, the required asymptotic assertion (REF ) follows from (REF ), (REF ) and Lemma REF .\",\n \"This completes the proof of Theorem REF .\",\n \"$\\\\blacksquare $\"\n ],\n [\n \"Proofs — Part 2\",\n \"Before we tackle the proof of Theorem REF , let us introduce the following Lemma 64 If $\\\\Omega $$\\\\Omega \\\\subset \\\\mathbb {S}^{K}$ satisfies condition (REF ), then $\\\\widetilde{\\\\widetilde{\\\\textrm {\\\\Omega \\\\hspace{-6.544pt}\\\\Omega }}}:= \\\\bigcup \\\\limits _{m\\\\ne 0}cl(m \\\\cdot \\\\textrm {$$\\\\hspace{-6.544pt}$$})$ has the property (REF ).\",\n \"This can be deduced in a straightforward way: the assumption implies that $cl(\\\\textrm {$$\\\\hspace{-6.544pt}$$})$ satisfies (REF ), and thus also $m \\\\cdot cl(\\\\textrm {$$\\\\hspace{-6.544pt}$$})$ satisfies (REF ).\",\n \"But this implies the validity of (REF ) for the “cone” $\\\\bigcup \\\\limits _{m\\\\ne 0} m \\\\cdot cl(\\\\textrm {$$\\\\hspace{-6.544pt}$$})$ which is nothing but $\\\\bigcup \\\\limits _{m\\\\ne 0} cl(m \\\\cdot \\\\textrm {$$\\\\hspace{-6.544pt}$$})$ .\",\n \"Proof of Theorem REF .\",\n \"Recall the interpretations of the two vectors $\\\\xi _{n,\\\\mathbf {X}}^{\\\\mathbf {W}}$ respectively $\\\\xi _{n,\\\\mathbf {X}}^{w\\\\mathbf {W}}$ given in (REF ) respectively (REF ), and that the sum of their $k$ components are $\\\\sum _{k=1}^{K} \\\\frac{1}{n} \\\\sum _{i\\\\in I_{k}^{(n)}} W_{i} =\\\\frac{1}{n}\\\\sum _{i=1}^{n} W_{i}$ respectively $\\\\sum _{k=1}^{K}\\\\frac{\\\\sum _{i \\\\in I_{k}^{(n)}}W_{i}}{\\\\sum _{k=1}^{K}\\\\sum _{i \\\\in I_{k}^{(n)}}W_{i}}=1$ (in case of $\\\\sum _{i=1}^{n}W_{i} \\\\ne 0$ ).\",\n \"In the light of these, for $\\\\textrm {$$\\\\hspace{-6.544pt}$$} \\\\subset \\\\mathbb {S}^{K}$ one gets the set identification $\\\\left\\\\lbrace \\\\xi _{n,\\\\mathbf {X}}^{w\\\\mathbf {W}} \\\\in \\\\textrm {\\\\right.$ $\\\\hspace{-6.544pt}$$} =\\\\bigcup \\\\limits _{m\\\\ne 0}\\\\left\\\\lbrace \\\\xi _{n,\\\\mathbf {X}}^{\\\\mathbf {W}}\\\\in m\\\\cdot \\\\textrm {\\\\right.$$\\\\hspace{-6.544pt}$$} ,\\\\frac{1}{n}\\\\sum _{i=1}^{n} W_{i} = m$$since $ { i=1nWi=0} $ amounts to $ m=0$, which cannothold when $ { n,XwW $\\\\Omega $$\\\\Omega $ }$.\",\n \"Now\\\\begin{eqnarray*}\\\\mathbb {\\\\Pi }_{X_{1}^{n}}\\\\left[\\\\xi _{n,\\\\mathbf {X}}^{w\\\\mathbf {W}}\\\\in \\\\textrm {\\\\right.\\\\Omega \\\\hspace{-6.544pt}\\\\Omega } &=&\\\\mathbb {\\\\Pi }_{X_{1}^{n}} \\\\bigg [ \\\\bigcup \\\\limits _{m\\\\ne 0}\\\\bigg \\\\lbrace \\\\xi _{n,\\\\mathbf {X}}^{\\\\mathbf {W}} \\\\in m \\\\cdot \\\\textrm {\\\\Omega \\\\hspace{-6.544pt}\\\\Omega },\\\\frac{1}{n}\\\\sum _{i=1}^{n} W_{i} =m \\\\bigg \\\\rbrace \\\\bigg ] \\\\\\\\&=&\\\\mathbb {\\\\Pi }_{X_{1}^{n}}\\\\bigg [ \\\\bigcup \\\\limits _{m\\\\ne 0}\\\\bigg \\\\lbrace \\\\xi _{n,\\\\mathbf {X}}^{\\\\mathbf {W}} \\\\in m\\\\cdot \\\\textrm {\\\\Omega \\\\hspace{-6.544pt}\\\\Omega }\\\\big \\\\rbrace \\\\bigg ]= \\\\mathbb {\\\\Pi }_{X_{1}^{n}}\\\\bigg [ \\\\xi _{n,\\\\mathbf {X}}^{\\\\mathbf {W}}\\\\in \\\\bigcup \\\\limits _{m\\\\ne 0} m\\\\cdot \\\\textrm {\\\\Omega \\\\hspace{-6.544pt}\\\\Omega } \\\\bigg ]\\\\end{eqnarray*}since $ { n,XW m$\\\\Omega $$\\\\Omega $ } { 1ni=1n Wi =m }$.\",\n \"Therefore\\\\begin{equation}\\\\frac{1}{n}\\\\log \\\\mathbb {\\\\Pi }_{X_{1}^{n}}\\\\left[\\\\xi _{n,\\\\mathbf {X}}^{w\\\\mathbf {W}}\\\\in \\\\textrm {\\\\right.\\\\Omega \\\\hspace{-6.544pt}\\\\Omega } =\\\\frac{1}{n}\\\\log \\\\mathbb {\\\\Pi }_{X_{1}^{n}}\\\\bigg [ \\\\xi _{n,\\\\mathbf {X}}^{\\\\mathbf {W}} \\\\in \\\\bigcup \\\\limits _{m\\\\ne 0}m\\\\cdot \\\\textrm {\\\\Omega \\\\hspace{-6.544pt}\\\\Omega } \\\\bigg ] .\\\\end{equation}Because of Proposition \\\\ref {PropLDPWEM-finitease} --- applied to$$\\\\Omega $$\\\\Omega $ := m0m $\\\\Omega $$\\\\Omega $$ --- one getsin terms of (\\\\ref {phishort})$ $-\\\\Phi _{{P}}\\\\Big (int\\\\Big (\\\\bigcup \\\\limits _{m\\\\ne 0}m\\\\cdot \\\\textrm {\\\\Omega \\\\hspace{-6.544pt}\\\\Omega }\\\\Big )\\\\Big )&\\\\le &\\\\lim \\\\inf _{n\\\\rightarrow \\\\infty }\\\\frac{1}{n}\\\\log \\\\mathbb {\\\\Pi }_{X_{1}^{n}}\\\\bigg [ \\\\xi _{n,\\\\mathbf {X}}^{\\\\mathbf {W}} \\\\in \\\\bigcup \\\\limits _{m\\\\ne 0}m\\\\cdot \\\\textrm {\\\\Omega \\\\hspace{-6.544pt}\\\\Omega } \\\\bigg ] \\\\nonumber \\\\\\\\&\\\\le &\\\\lim \\\\sup _{n\\\\rightarrow \\\\infty }\\\\frac{1}{n}\\\\log \\\\mathbb {\\\\Pi }_{X_{1}^{n}}\\\\bigg [\\\\xi _{n,\\\\mathbf {X}}^{\\\\mathbf {W}}\\\\in \\\\bigcup \\\\limits _{m\\\\ne 0}m\\\\cdot \\\\textrm {\\\\Omega \\\\hspace{-6.544pt}\\\\Omega } \\\\bigg ] \\\\le -\\\\Phi _{{P}}\\\\Big (cl\\\\Big (\\\\bigcup \\\\limits _{m\\\\ne 0}m\\\\cdot \\\\textrm {\\\\Omega \\\\hspace{-6.544pt}\\\\Omega }\\\\Big )\\\\Big ) .$ $\\\\hspace{-176.407pt}\\\\text{But}& & \\\\Phi _{{P}}\\\\Big (int\\\\Big (\\\\bigcup \\\\limits _{m\\\\ne 0}m\\\\cdot \\\\textrm {\\\\Omega \\\\hspace{-6.544pt}\\\\Omega }\\\\Big )\\\\Big ) \\\\le \\\\Phi _{{P}}\\\\Big (\\\\bigcup \\\\limits _{m\\\\ne 0}int\\\\Big (m\\\\cdot \\\\textrm {\\\\Omega \\\\hspace{-6.544pt}\\\\Omega }\\\\Big )\\\\Big ) =\\\\inf _{m\\\\ne 0} \\\\Phi _{{P}}(int(m\\\\cdot \\\\textrm {\\\\Omega \\\\hspace{-6.544pt}\\\\Omega }))\\\\\\\\\\\\hspace{-176.407pt}\\\\text{and}& & \\\\Phi _{{P}}\\\\Big (cl\\\\Big (\\\\bigcup \\\\limits _{m\\\\ne 0}m\\\\cdot \\\\textrm {\\\\Omega \\\\hspace{-6.544pt}\\\\Omega }\\\\Big )\\\\Big ) \\\\ge \\\\Phi _{{P}}\\\\Big (\\\\bigcup \\\\limits _{m\\\\ne 0}cl\\\\Big (m\\\\cdot \\\\textrm {\\\\Omega \\\\hspace{-6.544pt}\\\\Omega }\\\\Big )\\\\Big ) =\\\\inf _{m\\\\ne 0} \\\\Phi _{{P}}(cl(m\\\\cdot \\\\textrm {\\\\Omega \\\\hspace{-6.544pt}\\\\Omega })) .$ In fact, the inequality in (REF ) is straightforward because of $\\\\bigcup \\\\limits _{m\\\\ne 0}int(m\\\\cdot \\\\textrm {$$\\\\hspace{-6.544pt}$$})\\\\subset int(\\\\bigcup \\\\limits _{m\\\\ne 0}m\\\\cdot \\\\textrm {$$\\\\hspace{-6.544pt}$$})$ (since the latter is the largest open set contained in $\\\\bigcup \\\\limits _{m\\\\ne 0}m\\\\cdot \\\\textrm {$$\\\\hspace{-6.544pt}$$}$ ); the inequality in () follows from $& & \\\\Phi _{{P}}\\\\Big (cl\\\\Big (\\\\bigcup \\\\limits _{m\\\\ne 0}m\\\\cdot \\\\textrm {\\\\Omega \\\\hspace{-6.544pt}\\\\Omega }\\\\Big )\\\\Big ) \\\\ge \\\\Phi _{{P}}\\\\Big (cl\\\\Big (\\\\bigcup \\\\limits _{m\\\\ne 0}cl\\\\Big (m\\\\cdot \\\\textrm {\\\\Omega \\\\hspace{-6.544pt}\\\\Omega }\\\\Big )\\\\Big )\\\\Big ) =\\\\Phi _{{P}}\\\\Big (\\\\bigcup \\\\limits _{m\\\\ne 0}cl\\\\Big (m\\\\cdot \\\\textrm {\\\\Omega \\\\hspace{-6.544pt}\\\\Omega }\\\\Big )\\\\Big )\\\\nonumber $ An application of Lemma REF yields $\\\\Phi _{{P}}(int(m\\\\cdot \\\\textrm {$$\\\\hspace{-6.544pt}$$})) =\\\\Phi _{{P}}(m\\\\cdot \\\\textrm {$$\\\\hspace{-6.544pt}$$}) =\\\\Phi _{{P}}(cl(m\\\\cdot \\\\textrm {$$\\\\hspace{-6.544pt}$$}))$ for all $m \\\\ne 0$ , and hence $\\\\inf _{m\\\\ne 0} \\\\Phi _{{P}}(int(m\\\\cdot \\\\textrm {\\\\Omega \\\\hspace{-6.544pt}\\\\Omega })) =\\\\inf _{m\\\\ne 0} \\\\Phi _{{P}}(m\\\\cdot \\\\textrm {\\\\Omega \\\\hspace{-6.544pt}\\\\Omega })= \\\\inf _{m\\\\ne 0} \\\\Phi _{{P}}(cl(m\\\\cdot \\\\textrm {\\\\Omega \\\\hspace{-6.544pt}\\\\Omega })) .$ By combining (), (REF ), (REF ), () and (REF ), one arrives at $&&\\\\lim _{n \\\\rightarrow \\\\infty } \\\\frac{1}{n}\\\\log \\\\mathbb {\\\\Pi }_{X_{1}^{n}} \\\\left[\\\\xi _{n,\\\\mathbf {X}}^{w\\\\mathbf {W}}\\\\in \\\\textrm {\\\\right.\\\\Omega \\\\hspace{-6.544pt}\\\\Omega } = \\\\lim _{n \\\\rightarrow \\\\infty } \\\\frac{1}{n}\\\\log \\\\mathbb {\\\\Pi }_{X_{1}^{n}} \\\\bigg [ \\\\xi _{n,\\\\mathbf {X}}^{\\\\mathbf {W}}\\\\in \\\\bigcup \\\\limits _{m\\\\ne 0}m\\\\cdot \\\\textrm {\\\\Omega \\\\hspace{-6.544pt}\\\\Omega } \\\\bigg ]\\\\nonumber \\\\\\\\&& = - \\\\inf _{m\\\\ne 0} \\\\Phi _{{P}}(m\\\\cdot \\\\textrm {\\\\Omega \\\\hspace{-6.544pt}\\\\Omega })= - \\\\inf _{m\\\\ne 0} \\\\inf _{\\\\mathbf {Q} \\\\in m\\\\cdot \\\\textrm {\\\\Omega \\\\hspace{-5.40608pt}\\\\Omega }}D_{\\\\varphi }\\\\left(\\\\mathbf {Q},{P}\\\\right)= - \\\\inf _{m\\\\ne 0} \\\\inf _{{Q} \\\\in \\\\textrm {\\\\Omega \\\\hspace{-5.40608pt}\\\\Omega }}D_{\\\\varphi }\\\\left(m\\\\cdot {Q}, {P}\\\\right) ,\\\\nonumber $ where in the second last equality we have “reverted” the notation (REF ).\",\n \"Note that we did not assume (REF ) for $\\\\bigcup \\\\limits _{m\\\\ne 0} m \\\\cdot \\\\textrm {$$\\\\hspace{-6.544pt}$$}$ .\",\n \"$\\\\blacksquare $\"\n ],\n [\n \"Proofs — Part 3\",\n \"Proof of Lemma REF .\",\n \"From (REF ) one gets straightforwardly for arbitrary $\\\\widetilde{c} >0$ $D_{\\\\widetilde{c} \\\\cdot \\\\varphi _{\\\\gamma }}(m \\\\cdot \\\\mathbf {Q},{P}) \\\\hspace{-5.69046pt} &:=& \\\\hspace{-5.69046pt}{\\\\left\\\\lbrace \\\\begin{array}{ll}\\\\frac{\\\\widetilde{c} \\\\cdot \\\\left(m^{\\\\gamma } \\\\cdot H_{\\\\gamma } - m \\\\cdot A \\\\cdot \\\\gamma + \\\\gamma -1 \\\\right)}{\\\\gamma \\\\cdot (\\\\gamma -1)},\\\\hspace{75.39963pt} \\\\textrm {if } \\\\gamma \\\\in \\\\, ]-\\\\infty ,0[, \\\\ {P} \\\\in \\\\mathbb {S}_{\\\\ge 0}^{K}, \\\\ \\\\mathbf {Q} \\\\in A \\\\cdot \\\\mathbb {S}_{> 0}^{K} \\\\ \\\\ \\\\textrm {and } m >0, \\\\\\\\\\\\widetilde{c} \\\\cdot (- \\\\log m + \\\\widetilde{I} -1 + m \\\\cdot A),\\\\hspace{42.67912pt} \\\\textrm {if } \\\\gamma = 0, \\\\ {P} \\\\in \\\\mathbb {S}_{\\\\ge 0}^{K}, \\\\ A \\\\cdot \\\\mathbf {Q} \\\\in \\\\mathbb {S}_{> 0}^{K} \\\\ \\\\ \\\\textrm {and } m >0, \\\\\\\\\\\\frac{\\\\widetilde{c} \\\\cdot \\\\left(m^{\\\\gamma } \\\\cdot H_{\\\\gamma } - m \\\\cdot A \\\\cdot \\\\gamma + \\\\gamma -1 \\\\right)}{\\\\gamma \\\\cdot (\\\\gamma -1)},\\\\hspace{75.39963pt} \\\\textrm {if } \\\\gamma \\\\in \\\\, ]0,1[, \\\\ {P} \\\\in \\\\mathbb {S}_{\\\\ge 0}^{K}, \\\\ \\\\mathbf {Q} \\\\in A \\\\cdot \\\\mathbb {S}_{\\\\ge 0}^{K} \\\\ \\\\ \\\\textrm {and } m \\\\ge 0, \\\\\\\\\\\\widetilde{c} \\\\cdot (A \\\\cdot m \\\\cdot \\\\log m + m \\\\cdot (I-A) +1),\\\\hspace{14.22636pt} \\\\textrm {if } \\\\gamma = 1, \\\\ {P} \\\\in \\\\mathbb {S}_{> 0}^{K}, \\\\ \\\\mathbf {Q} \\\\in A \\\\cdot \\\\mathbb {S}_{\\\\ge 0}^{K} \\\\ \\\\ \\\\textrm {and } m \\\\ge 0, \\\\\\\\\\\\frac{\\\\widetilde{c} \\\\cdot \\\\left(m^{\\\\gamma } \\\\cdot H_{\\\\gamma } \\\\cdot {1}_{[0,\\\\infty [}(m)- m \\\\cdot A \\\\cdot \\\\gamma + \\\\gamma -1 \\\\right)}{\\\\gamma \\\\cdot (\\\\gamma -1)},\\\\hspace{39.83368pt} \\\\textrm {if } \\\\gamma \\\\in \\\\, ]1,2[, \\\\ {P} \\\\in \\\\mathbb {S}_{>0}^{K}, \\\\ \\\\mathbf {Q} \\\\in A \\\\cdot \\\\mathbb {S}^{K} \\\\ \\\\ \\\\textrm {and } m \\\\in ]-\\\\infty ,\\\\infty [,\\\\\\\\\\\\frac{\\\\widetilde{c} \\\\cdot \\\\left(m^{2} \\\\cdot H_{2} - m \\\\cdot A \\\\cdot 2 + 2 -1 \\\\right)}{2 \\\\cdot (2-1)},\\\\hspace{76.82234pt} \\\\textrm {if } \\\\gamma = 2, \\\\ {P} \\\\in \\\\mathbb {S}_{>0}^{K}, \\\\ \\\\mathbf {Q} \\\\in A \\\\cdot \\\\mathbb {S}^{K} \\\\ \\\\ \\\\textrm {and } m \\\\in ]-\\\\infty ,\\\\infty [,\\\\\\\\\\\\frac{\\\\widetilde{c} \\\\cdot \\\\left(m^{\\\\gamma } \\\\cdot H_{\\\\gamma } \\\\cdot {1}_{[0,\\\\infty [}(m)- m \\\\cdot A \\\\cdot \\\\gamma + \\\\gamma -1 \\\\right)}{\\\\gamma \\\\cdot (\\\\gamma -1)},\\\\hspace{39.83368pt} \\\\textrm {if } \\\\gamma \\\\in \\\\, ]2,\\\\infty [, \\\\ {P} \\\\in \\\\mathbb {S}_{>0}^{K}, \\\\ \\\\mathbf {Q} \\\\in A \\\\cdot \\\\mathbb {S}^{K} \\\\ \\\\ \\\\textrm {and } m \\\\in ]-\\\\infty ,\\\\infty [,\\\\\\\\\\\\infty , \\\\hspace{156.49014pt} \\\\textrm {else},\\\\end{array}\\\\right.\",\n \"}$ where we have used the three $m-$ independent abbreviations $H_{\\\\gamma } := \\\\sum \\\\displaylimits _{k=1}^{K} (q_{k})^{\\\\gamma } \\\\cdot (p_{k})^{1-\\\\gamma }\\\\ = \\\\ 1 + \\\\gamma \\\\cdot (A-1) +\\\\frac{\\\\gamma \\\\cdot (\\\\gamma -1)}{\\\\widetilde{c}} \\\\cdot D_{\\\\widetilde{c} \\\\cdot \\\\varphi _{\\\\gamma }}(\\\\mathbf {Q}, {P}),\\\\qquad \\\\text{(cf.\",\n \"(\\\\ref {brostu3:fo.divpow.hellinger1}))}\\\\nonumber $ $I:= \\\\sum \\\\displaylimits _{k=1}^{K} q_{k} \\\\cdot \\\\log \\\\left( \\\\frac{q_{k}}{p_{k}} \\\\right)\\\\ = \\\\ \\\\frac{1}{\\\\widetilde{c}} \\\\cdot D_{\\\\widetilde{c} \\\\cdot \\\\varphi _{1}}(\\\\mathbf {Q}, {P}) + A - 1,\\\\qquad \\\\text{(cf.\",\n \"(\\\\ref {brostu3:fo.divpow.Kull1}))}\\\\nonumber $ $\\\\widetilde{I} := \\\\sum \\\\displaylimits _{k=1}^{K} p_{k} \\\\cdot \\\\log \\\\left( \\\\frac{p_{k}}{q_{k}} \\\\right)\\\\ = \\\\ \\\\frac{1}{\\\\widetilde{c}} \\\\cdot D_{\\\\widetilde{c} \\\\cdot \\\\varphi _{0}}(\\\\mathbf {Q}, {P}) + 1 - A.\\\\qquad \\\\text{(cf.\",\n \"(\\\\ref {brostu3:fo.divpow.RevKull1}))}\\\\nonumber $ To proceed, let us fix an arbitrary constant $\\\\widetilde{c} >0$ .\",\n \"(i) Case $\\\\gamma \\\\cdot (1-\\\\gamma ) \\\\ne 0$ .\",\n \"(ia) Let us start with the subcase $\\\\gamma \\\\in ]-\\\\infty ,0[$ .\",\n \"From the first and the last line of (REF ), it is clear that the corresponding $m-$ infimum can not be achieved for $m \\\\le 0$ ; since $H_{\\\\gamma } > 0$ one gets the unique minimizer $m_{min} = \\\\Big (\\\\frac{H_{\\\\gamma }}{A}\\\\Big )^{1/(1-\\\\gamma )} >0$ and the minimum $D_{\\\\widetilde{c} \\\\cdot \\\\varphi _{\\\\gamma }}(m_{min} \\\\cdot \\\\mathbf {Q},{P})=\\\\frac{\\\\widetilde{c}}{\\\\gamma } \\\\cdot (1- \\\\frac{H^{1/(1-\\\\gamma )}}{A^{\\\\gamma /(1-\\\\gamma )}} )$ .\",\n \"Hence, (REF ) is established.\",\n \"The assertions (REF ) and () follow immediately by monotonicity inspection of $x \\\\rightarrow \\\\frac{\\\\widetilde{c}}{\\\\gamma } \\\\cdot \\\\left[ 1-\\\\frac{1}{A^{\\\\gamma /(1-\\\\gamma )}} \\\\cdot \\\\left[ 1 + \\\\gamma \\\\cdot (A-1) + \\\\frac{\\\\gamma \\\\cdot \\\\left( \\\\gamma -1\\\\right)}{\\\\widetilde{c}}\\\\cdot x\\\\right] ^{-1/\\\\left( \\\\gamma -1\\\\right) }\\\\right]$ for $x \\\\ge 0$ such that $1 + \\\\gamma \\\\cdot (A-1) + \\\\frac{\\\\gamma \\\\cdot \\\\left( \\\\gamma -1\\\\right)}{\\\\widetilde{c}}\\\\cdot x \\\\ge 0$ .\",\n \"(ib) The subcase $\\\\gamma \\\\in ]0,1[$ (cf.\",\n \"the third line of (REF )) works analogously if $H_{\\\\gamma } >0$ ; furthermore, if $H_{\\\\gamma } =0$ — which can only appear when ${P}$ , $\\\\mathbf {Q}$ have disjoint supports (singularity)— then $\\\\inf _{m>0} D_{\\\\widetilde{c} \\\\cdot \\\\varphi _{\\\\gamma }}(m \\\\cdot \\\\mathbf {Q},{P}) =\\\\frac{\\\\widetilde{c}}{\\\\gamma }$ which is (the corresponding special case of) (REF ).\",\n \"(ic) In the subcase $\\\\gamma \\\\in ]1,\\\\infty [$ (cf.\",\n \"the fifth, sixth and seventh line of (REF )) it is straightforward to see that the desired infimum can not be achieved for $m < 0$ .\",\n \"Hence, one can proceed analogously to subcase (ia).\",\n \"(id) The assertions () to () are straightforward.\",\n \"(ii) Case $\\\\gamma =1$ .\",\n \"From the fourth line of (REF ), one obtains the unique minimizer $m_{min} = \\\\exp \\\\lbrace -I/A\\\\rbrace $ and the minimum $D_{\\\\widetilde{c} \\\\cdot \\\\varphi _{1}}(m_{min} \\\\cdot \\\\mathbf {Q},{P})= \\\\widetilde{c} \\\\cdot (1- A \\\\cdot m_{min})$ , which leads to (REF ).\",\n \"The monotonicity of $x \\\\rightarrow \\\\widetilde{c} \\\\cdot (1- \\\\exp \\\\lbrace -x/\\\\widetilde{c}\\\\rbrace )$ for $x\\\\ge 0$ implies immediately (REF ) and (); moreover, () and () are immediate.\",\n \"(iii) Case $\\\\gamma =0$ .\",\n \"The second line of (REF ) implies the unique minimizer $m_{min} = 1/A$ , the minimum $D_{\\\\widetilde{c} \\\\cdot \\\\varphi _{0}}(m_{min} \\\\cdot \\\\mathbf {Q},{P})= \\\\widetilde{c} \\\\cdot (\\\\widetilde{I} + \\\\log A)$ , and hence (REF ).\",\n \"The assertions (REF ) to () are obvious.\",\n \"$\\\\blacksquare $\"\n ],\n [\n \"Proofs — Part 4\",\n \"Proof of Proposition REF .\",\n \"Clearly, (G1) and (G2) are part of the definition of $\\\\widetilde{\\\\Upsilon }(]a,b[)$ .\",\n \"Recall our required representability (REF ).\",\n \"The therein involved Laplace-Stieltjes transform (Laplace-Lebesgue transform) $z\\\\mapsto MGF_{\\\\mathbb {}}(z):=\\\\int _{\\\\mathbb {R}}e^{z \\\\cdot y} \\\\, d\\\\mathbb {} (y)= E_{\\\\mathbb {\\\\Pi }}[e^{z\\\\cdot W}]$ of a probability measure $\\\\mathbb {}$ on the real line respectively of an associated random variable $W$ (with $\\\\mathbb {}[ \\\\cdot \\\\, ] := \\\\mathbb {\\\\Pi }[W \\\\in \\\\cdot \\\\, ]$ ) has the following fundamental properties, according to well-known general theory: $MGF_{\\\\mathbb {}}$ takes values in $]0,\\\\infty ]$ ; the effective domain $dom(MGF_{\\\\mathbb {}})$ is an interval which contains 0 and which may be degenerated or even the whole real line; correspondingly, we denote its interior by $]\\\\lambda _{-},\\\\lambda _{+}[ := int(dom(MGF_{\\\\mathbb {}}))$ which may be the empty set (in case that $dom(MGF_{\\\\mathbb {}})=\\\\lbrace 0\\\\rbrace $ , i.e.\",\n \"$\\\\lambda _{-}=\\\\lambda _{+}=0$ ); clearly, there holds $\\\\lambda _{-} \\\\in [-\\\\infty ,0]$ and $\\\\lambda _{+} \\\\in [0,\\\\infty ]$ ; $MGF_{\\\\mathbb {}}$ is continuous on $dom(MGF_{\\\\mathbb {}})$ and lower semicontinuous on $\\\\mathbb {R}$ ; if $\\\\lambda _{-} \\\\ne \\\\lambda _{+}$ , then $MGF_{\\\\mathbb {}}$ is real analytic and thus infinitely differentiable on $]\\\\lambda _{-},\\\\lambda _{+}[$ ; if $MGF_{\\\\mathbb {}}$ is finite in a neighborhood of zero, i.e.\",\n \"$0 \\\\in ]\\\\lambda _{-},\\\\lambda _{+}[$ , then for all $k \\\\in \\\\mathbb {N}_{0}$ the $k-$ th moment of $\\\\mathbb {}$ respectively $W$ exists and is finite and can be computed in terms of the $k-$ th derivative $MGF_{\\\\mathbb {}}^{(k)}$ as $MGF_{\\\\mathbb {}}^{(k)}(0) =\\\\int _{\\\\mathbb {R}} y^{k} \\\\, d\\\\mathbb {} (y)= E_{\\\\mathbb {\\\\Pi }}[W^{k}],$ which, by the way, then allows the interpretation of $MGF_{\\\\mathbb {}}$ as “moment generating function of $\\\\mathbb {}$ resp.\",\n \"$W$ ”since we assume $0 \\\\in ]\\\\lambda _{-},\\\\lambda _{+}[$ , we have already used the meaningful abbreviation $MGF$ (rather than LST) in (REF ); if $\\\\lambda _{-} \\\\ne \\\\lambda _{+}$ , then $MGF_{\\\\mathbb {}}$ is strictly convex on $]\\\\lambda _{-},\\\\lambda _{+}[$ .\",\n \"Hence, the logarithm of the Laplace-Stieltjes transform $z\\\\mapsto \\\\Lambda _{\\\\mathbb {}}(z) := \\\\log MGF_{\\\\mathbb {} }(z):=\\\\log \\\\int _{\\\\mathbb {R}}e^{z \\\\cdot y} \\\\, d\\\\mathbb {} (y)= \\\\log E_{\\\\mathbb {\\\\Pi }}[e^{z\\\\cdot W}]$ (which in case of $0 \\\\in ]\\\\lambda _{-},\\\\lambda _{+}[$ can be interpreted as cumulant generating function) “carries over” (M1) to (M6), which partially can be even refined: $\\\\Lambda _{\\\\mathbb {}}$ takes values in $]-\\\\infty ,\\\\infty ]$ ; $dom(\\\\Lambda _{\\\\mathbb {}}) = dom(MGF_{\\\\mathbb {}})$ and thus $int(dom(\\\\Lambda _{\\\\mathbb {} })) = \\\\, ]\\\\lambda _{-},\\\\lambda _{+}[$ ; $\\\\Lambda _{\\\\mathbb {} }$ is continuous on $dom(\\\\Lambda _{\\\\mathbb {}})$ and lower semicontinuous on $\\\\mathbb {R}$ ; if $\\\\lambda _{-} \\\\ne \\\\lambda _{+}$ , then $\\\\lambda _{\\\\mathbb {}}$ is infinitely differentiable on $]\\\\lambda _{-},\\\\lambda _{+}[$ ; if $0 \\\\in ]\\\\lambda _{-},\\\\lambda _{+}[$ , then $& & \\\\Lambda _{\\\\mathbb {} }(0) = 0, \\\\quad \\\\Lambda _{\\\\mathbb {} }^{\\\\prime }(0) = \\\\int _{\\\\mathbb {R}} y \\\\, d\\\\mathbb {} (y)= E_{\\\\mathbb {\\\\Pi }}[W],\\\\\\\\& & \\\\Lambda _{\\\\mathbb {} }^{\\\\prime \\\\prime }(0) =\\\\int _{\\\\mathbb {R}} \\\\Big (y - \\\\int _{\\\\mathbb {R}} \\\\widetilde{y} \\\\, d\\\\mathbb {} (\\\\widetilde{y})\\\\Big )^{2}\\\\, d\\\\mathbb {} (y)= E_{\\\\mathbb {\\\\Pi }}[W^{2}] - (E_{\\\\mathbb {\\\\Pi }}[W])^{2} = Var_{\\\\mathbb {\\\\Pi }}[W];$ under the assumption $\\\\lambda _{-} \\\\ne \\\\lambda _{+}$ there holds: $\\\\Lambda _{\\\\mathbb {}}$ is strictly convex on $]\\\\lambda _{-},\\\\lambda _{+}[$ if and only if $\\\\mathbb {}$ is not a one-point distribution (Dirac mass) if and only if $W$ is not a.s. constant; otherwise, $\\\\Lambda _{\\\\mathbb {}}$ is linear; under the assumption that $\\\\mathbb {}$ is not a one-point distribution (Dirac mass) — with the notations $a := \\\\inf supp(\\\\mathbb {}) = \\\\inf supp(W)$ , $b := \\\\sup supp(\\\\mathbb {}) = \\\\sup supp(W)$ , $t_{-}^{sc} := \\\\inf \\\\lbrace \\\\Lambda _{\\\\mathbb {}}^{\\\\prime }(z) \\\\, : \\\\, z \\\\in ]\\\\lambda _{-},\\\\lambda _{+}[ \\\\rbrace = \\\\lim _{z \\\\downarrow \\\\lambda _{-}} \\\\Lambda _{\\\\mathbb {}}^{\\\\prime }(z)$ and $t_{+}^{sc} := \\\\sup \\\\lbrace \\\\Lambda _{\\\\mathbb {}}^{\\\\prime }(z) \\\\, : \\\\, z \\\\in ]\\\\lambda _{-},\\\\lambda _{+}[ \\\\rbrace = \\\\lim _{z \\\\uparrow \\\\lambda _{+}} \\\\Lambda _{\\\\mathbb {}}^{\\\\prime }(z)$ — one gets the following assertions: $]t_{-}^{sc},t_{+}^{sc}[ \\\\ \\\\subseteq \\\\ ]a,b[$ ; if $a > - \\\\infty $ , then $\\\\lambda _{-} = - \\\\infty $ , $t_{-}^{sc} = \\\\lim _{z \\\\rightarrow -\\\\infty } \\\\Lambda _{\\\\mathbb {}}^{\\\\prime }(z)= \\\\lim _{z \\\\rightarrow -\\\\infty } \\\\frac{\\\\Lambda _{\\\\mathbb {}}(z)}{z} = a$ ; if $b < \\\\infty $ , then $\\\\lambda _{+} = \\\\infty $ , $t_{+}^{sc} = \\\\lim _{z \\\\rightarrow \\\\infty } \\\\Lambda _{\\\\mathbb {}}^{\\\\prime }(z)= \\\\lim _{z \\\\rightarrow \\\\infty } \\\\frac{\\\\Lambda _{\\\\mathbb {}}(z)}{z} = b$ ; if $a = - \\\\infty $ and $\\\\lambda _{-} = - \\\\infty $ , then $t_{-}^{sc} = \\\\lim _{z \\\\rightarrow -\\\\infty } \\\\Lambda _{\\\\mathbb {}}^{\\\\prime }(z)= -\\\\infty = a$ ; if $b = \\\\infty $ and $\\\\lambda _{+} = \\\\infty $ , then $t_{+}^{sc} = \\\\lim _{z \\\\rightarrow \\\\infty } \\\\Lambda _{\\\\mathbb {}}^{\\\\prime }(z)= \\\\infty = b$ ; if $\\\\lambda _{-} \\\\in \\\\, ]-\\\\infty ,0[$ and $t_{-}^{sc} > -\\\\infty $ , then $a = - \\\\infty $ , $\\\\Lambda _{\\\\mathbb {}}(\\\\lambda _{-}) \\\\in \\\\, ]-\\\\infty ,\\\\infty [$ , $\\\\Lambda _{\\\\mathbb {}}(z) = \\\\infty $ for all $z < \\\\lambda _{-}$ , $\\\\Lambda _{\\\\mathbb {}}^{\\\\prime }(\\\\lambda _{-}) \\\\in \\\\, ]-\\\\infty ,\\\\infty [$ ; if $\\\\lambda _{+} \\\\in \\\\, ]0,\\\\infty [$ and $t_{+}^{sc} < \\\\infty $ , then $b = \\\\infty $ , $\\\\Lambda _{\\\\mathbb {}}(\\\\lambda _{+}) \\\\in \\\\, ]-\\\\infty ,\\\\infty [$ , $\\\\Lambda _{\\\\mathbb {}}(z) = \\\\infty $ for all $z > \\\\lambda _{+}$ , $\\\\Lambda _{\\\\mathbb {}}^{\\\\prime }(\\\\lambda _{+})\\\\in \\\\, ]-\\\\infty ,\\\\infty [$ ; if $\\\\lambda _{-} \\\\in \\\\, ]-\\\\infty ,0[$ and $t_{-}^{sc} = -\\\\infty $ , then $a= - \\\\infty $ ; if $\\\\lambda _{+} \\\\in \\\\, ]0,\\\\infty [$ and $t_{+}^{sc} = \\\\infty $ , then $b= \\\\infty $ .\",\n \"Notice that (C7ii) to (C7ix) cover all possible constellations.\",\n \"For a proof of (C7ii) to (C7vii) as well as further details, see e.g.\",\n \"Section 9.1 in Borovkov [56].\",\n \"By contradiction, (C7viii) follows from (C7ii) and (C7ix) follows from (C7iii).\",\n \"Moreover, (C7i) is a consequence (C7ii) to (C7ix).\",\n \"As a side remark, notice that (C6) refines (M6).\",\n \"According to the representability requirement (REF ), one has $\\\\varphi (t)=\\\\sup _{z\\\\in \\\\mathbb {R}}\\\\left( z\\\\cdot t- \\\\Lambda _{\\\\mathbb {}}(z)\\\\right)=: \\\\Lambda _{\\\\mathbb {}}^{*}(t), \\\\qquad t\\\\in \\\\mathbb {R},\\\\ \\\\ $ (i.e.\",\n \"the divergence generator $\\\\varphi $ must be equal to the Fenchel-Legendre transform $\\\\Lambda _{\\\\mathbb {}}^{*}$ of a cumulant generating function $\\\\Lambda _{\\\\mathbb {}}$ ) of some probability distribution $\\\\mathbb {}$ , such that $\\\\lambda _{-} < 0 < \\\\lambda _{+}$ holds.\",\n \"Moreover, $\\\\varphi $ should satisfy $\\\\varphi (1)=0$ , and should be finite as well as strictly convex in a non-empty neighborhood $]t_{-}^{sc},t_{+}^{sc}[$ of 1 (cf.\",\n \"the definition of $\\\\widetilde{\\\\Upsilon }(]a,b[)$ ).\",\n \"The latter rules out that $\\\\mathbb {}$ is any one-point distribution (Dirac distribution), say $\\\\mathbb {} = \\\\delta _{y_{0}}$ for some $y_{0} \\\\in \\\\mathbb {R}$ , since in such a situation one gets $\\\\Lambda _{\\\\mathbb {}}(z) = z \\\\cdot y_{0}$ , and thus $\\\\varphi (t) = \\\\Lambda _{\\\\mathbb {}}^{*}(t) = 0$ for $t=y_{0}$ and $\\\\varphi (t) = \\\\Lambda _{\\\\mathbb {}}^{*}(t) = \\\\infty $ for all $t \\\\in \\\\mathbb {R}\\\\backslash \\\\lbrace y_{0}\\\\rbrace $ (even in the case $y_{0} =1$ for which $\\\\varphi (1)=0$ is satisfied).\",\n \"Consequently, $\\\\Lambda _{\\\\mathbb {}}$ is strictly convex on $]\\\\lambda _{-},\\\\lambda _{+}[ \\\\ = int(dom(\\\\Lambda _{\\\\mathbb {} }))$ (cf.\",\n \"(C6)) and (C7) applies.\",\n \"Clearly, by continuity one gets $\\\\Lambda _{\\\\mathbb {}}^{*}(t) =\\\\sup _{z\\\\in ]\\\\lambda _{-},\\\\lambda _{+}[}\\\\left( t \\\\cdot z - \\\\Lambda _{\\\\mathbb {}}(z)\\\\right),\\\\qquad t\\\\in \\\\mathbb {R}.\",\n \"\\\\ \\\\ $ For $t \\\\in ]t_{-}^{sc},t_{+}^{sc}[$ , the optimization problem (REF ) can be solved explicitly by the well-known “pure/original” Legendre transform, namely $\\\\Lambda _{\\\\mathbb {}}^{*}(t) = t \\\\cdot \\\\Lambda _{\\\\mathbb {}}^{\\\\prime -1}(t) -\\\\Lambda _{\\\\mathbb {}}\\\\Big (\\\\Lambda _{\\\\mathbb {}}^{\\\\prime -1}(t)\\\\Big ),\\\\qquad t \\\\in ]t_{-}^{sc},t_{+}^{sc}[.$ Let us inspect the further cases $t \\\\le t_{-}^{sc}$ .\",\n \"In the contexts of (C7iv) and (C7viii), this is obsolete since $t_{-}^{sc} = a = -\\\\infty $ .\",\n \"For (C7ii), where $t_{-}^{sc} = a > -\\\\infty $ , one can show $\\\\Lambda _{\\\\mathbb {}}^{*}(a) = - \\\\log \\\\mathbb {}[\\\\lbrace a\\\\rbrace ]= - \\\\log \\\\mathbb {\\\\Pi }[W = a \\\\, ]$ which together with (REF ) proves (G10ii); moreover, $\\\\Lambda _{\\\\mathbb {}}^{*}(t) = \\\\infty $ for all $t < a$ (see e.g.\",\n \"Section 9.1 of Borovkov [56]).\",\n \"In the setup (C7vi), where $t_{-}^{sc} > a = - \\\\infty $ it is clear that $\\\\Lambda _{\\\\mathbb {}}^{*}(t_{-}^{sc}) =t_{-}^{sc} \\\\cdot \\\\Lambda _{\\\\mathbb {}}^{\\\\prime -1}(t_{-}^{sc}) -\\\\Lambda _{\\\\mathbb {}}\\\\Big (\\\\Lambda _{\\\\mathbb {}}^{\\\\prime -1}(t_{-}^{sc})\\\\Big )= t_{-}^{sc} \\\\cdot \\\\lambda _{-} - \\\\Lambda _{\\\\mathbb {}}(\\\\lambda _{-})$ and $\\\\Lambda _{\\\\mathbb {}}^{*}(t)= t \\\\cdot \\\\lambda _{-} - \\\\Lambda _{\\\\mathbb {}}(\\\\lambda _{-})= \\\\Lambda _{\\\\mathbb {}}^{*}(t_{-}^{sc}) + \\\\lambda _{-} \\\\cdot (t- t_{-}^{sc})\\\\quad \\\\text{for all $t \\\\in ]-\\\\infty ,t_{-}^{sc}[$}.$ As far as the cases $t \\\\ge t_{+}^{sc}$ is concerned, in the situations of (C7v) and (C7ix), this is obsolete since $t_{+}^{sc} = b = \\\\infty $ .\",\n \"For (C7iii), where $t_{+}^{sc} = b < \\\\infty $ , one can show $\\\\Lambda _{\\\\mathbb {}}^{*}(b) = - \\\\log \\\\mathbb {}[\\\\lbrace b\\\\rbrace ]= - \\\\log \\\\mathbb {\\\\Pi }[W = b \\\\, ]$ which together with (REF ) proves (G10iii); moreover, $\\\\Lambda _{\\\\mathbb {}}^{*}(t) = \\\\infty $ for all $t > b$ (see e.g.\",\n \"Section 9.1 of Borovkov [56]).\",\n \"In the setup (C7vii), where $t_{+}^{sc} < b = \\\\infty $ it is clear that $\\\\Lambda _{\\\\mathbb {}}^{*}(t_{+}^{sc}) =t_{+}^{sc} \\\\cdot \\\\Lambda _{\\\\mathbb {}}^{\\\\prime -1}(t_{+}^{sc}) -\\\\Lambda _{\\\\mathbb {}}\\\\Big (\\\\Lambda _{\\\\mathbb {}}^{\\\\prime -1}(t_{+}^{sc})\\\\Big )= t_{+}^{sc} \\\\cdot \\\\lambda _{+} - \\\\Lambda _{\\\\mathbb {}}(\\\\lambda _{+})$ and $\\\\Lambda _{\\\\mathbb {}}^{*}(t)= t \\\\cdot \\\\lambda _{+} - \\\\Lambda _{\\\\mathbb {}}(\\\\lambda _{+})= \\\\Lambda _{\\\\mathbb {}}^{*}(t_{+}^{sc}) + \\\\lambda _{+} \\\\cdot (t- t_{+}^{sc})\\\\quad \\\\text{for all $t \\\\in ]t_{+}^{sc}, \\\\infty [$}.$ As a side effect, we have thus also proved (G10i) and (G3) (notice that in (G3) we have started with $a, b$ to be the endpoints of the support of $\\\\mathbb {}$ respectively $W$ , in contrast to Definition REF where $a$ , $b$ are defined as the endpoints of the effective domain of $\\\\varphi $ ).\",\n \"To proceed, from (REF ) and (REF ) we obtain $\\\\varphi ^{\\\\prime }(t) = (\\\\Lambda _{\\\\mathbb {}}^{*})^{\\\\prime }(t) = \\\\Lambda _{\\\\mathbb {}}^{\\\\prime -1}(t),\\\\qquad \\\\varphi ^{\\\\prime \\\\prime }(t) = (\\\\Lambda _{\\\\mathbb {}}^{*})^{\\\\prime \\\\prime }(t) =\\\\frac{1}{\\\\Lambda _{\\\\mathbb {}}^{\\\\prime \\\\prime }\\\\big (\\\\Lambda _{\\\\mathbb {}}^{\\\\prime -1}(t)\\\\big )} > 0,\\\\qquad t \\\\in ]t_{-}^{sc},t_{+}^{sc}[,$ which — together with the investigations below (REF ) — provides (G4) and (G5); moreover, (G6) is immediate since the infinite differentiability is straightforward and $\\\\varphi ^{\\\\prime }(1) =0$ because we have required both the nonnegativity of $\\\\varphi $ and (G2) (cf.\",\n \"the definition of $\\\\widetilde{\\\\Upsilon }(]a,b[)$ ).\",\n \"The property (G7) follows from (C7ii), (C7iv), (C7viii), (REF ), (REF ) and $\\\\varphi ^{\\\\prime }(t_{-}^{sc}) = \\\\Lambda _{\\\\mathbb {}}^{\\\\prime -1}(t_{-}^{sc}) = \\\\lambda _{-}$ .\",\n \"Analogously, we get (G8) from (C7iii), (C7v), (C7ix), (REF ), (REF ) and $\\\\varphi ^{\\\\prime }(t_{+}^{sc}) = \\\\Lambda _{\\\\mathbb {}}^{\\\\prime -1}(t_{+}^{sc}) = \\\\lambda _{+}$ .\",\n \"Let us continue with (G9).\",\n \"By applying the general theory of double Fenchel-Legendre transforms (bi-conjugates), (REF ) turns into $\\\\varphi ^{*} (z) = \\\\Lambda _{\\\\mathbb {}}(z), \\\\qquad z\\\\in \\\\mathbb {R},\\\\ \\\\ $ which deduces (G9i).\",\n \"The properties (G9ii), (G9iii) and (G9iv) follow from Theorem REF (cf.\",\n \"the discussion thereafter).\",\n \"Finally, we obtain (G11i) and (G11ii) from (REF ), (REF ) and ().\",\n \"$\\\\blacksquare $ Proof of Proposition REF .\",\n \"The assertions follow immediately from (REF ), (REF ), (REF ), Theorem REF , (REF ) (and the discussion thereafter) as well as (M5).\",\n \"$\\\\blacksquare $\"\n ],\n [\n \"Proofs — Part 5\",\n \"Proof of Proposition REF .\",\n \"The assertion follows straightforwardly from the following two facts: (i) a moment generating function $MGF$ is infinitely divisible if and only if $MGF^{c}$ is a moment generating function for all $c >0$ (cf.\",\n \"e.g.\",\n \"(the MGF-version of) Prop.\",\n \"IV.2.5 of Steutel & van Harn [341]).\",\n \"(ii) $z \\\\mapsto MGF(z)$ is a moment generating function if and only if $z \\\\mapsto MGF(\\\\breve{c} \\\\cdot z) =: MGF_{\\\\breve{c}}(z)$ is a moment generating function for all $\\\\breve{c} >0$ .\",\n \"Notice that for each $c >0$ , $\\\\breve{c} >0$ one has $int(dom(MGF)) = int(dom(MGF^{c}))$ and $int(dom(MGF_{\\\\breve{c}})) = \\\\frac{1}{\\\\breve{c}} \\\\cdot int(dom(MGF))$ , and hence the light-tailedness remains unchanged: $0 \\\\in int(dom(MGF))$ if and only if $0 \\\\in int(dom(MGF^{c}))$ if and only if $0 \\\\in int(dom(MGF_{\\\\breve{c}}))$ .\",\n \"Since $\\\\varphi \\\\in \\\\Upsilon (]a,b[)$ , we have $\\\\varphi (t)=\\\\sup _{z\\\\in ]\\\\lambda _{-}, \\\\lambda _{+}[}\\\\left( z \\\\cdot t-\\\\log \\\\Big (\\\\int _{\\\\mathbb {R}} e^{z\\\\cdot y }\\\\, d\\\\mathbb {} (y) \\\\Big ) \\\\right), \\\\quad t\\\\in ]a,b[ \\\\, , \\\\ \\\\ $ and thus for the exponential of its Fenchel-Legendre transform $\\\\int _{\\\\mathbb {R}} e^{z \\\\cdot y} d\\\\mathbb {} (y) , \\\\qquad z \\\\in ]\\\\lambda _{-}, \\\\lambda _{+}[ .$ Now, let $\\\\widetilde{\\\\varphi } := \\\\widetilde{c} \\\\cdot \\\\varphi \\\\in \\\\Upsilon (]a,b[)$ for arbitrarily fixed $\\\\widetilde{c} >0$ .\",\n \"From the application of (REF ) to $\\\\widetilde{\\\\varphi }$ we obtain $\\\\widetilde{\\\\varphi } (t)=\\\\sup _{\\\\widetilde{z} \\\\in ]\\\\widetilde{\\\\lambda }_{-}, \\\\widetilde{\\\\lambda }_{+}[}\\\\left( \\\\widetilde{z} \\\\cdot t-\\\\log \\\\int _{\\\\mathbb {R}}e^{\\\\widetilde{z} \\\\cdot \\\\widetilde{y}} d\\\\widetilde{\\\\mathbb {}}_{\\\\widetilde{c}} (\\\\widetilde{y})\\\\right), \\\\qquad t\\\\in ]a,b[ \\\\, , \\\\ \\\\ $ for some unique probability distribution $\\\\widetilde{\\\\mathbb {}}_{\\\\widetilde{c}}$ on $\\\\mathbb {R}$ .\",\n \"Here, according to (G9i) for $\\\\widetilde{\\\\varphi }$ we have used $\\\\widetilde{\\\\lambda }_{-} := \\\\inf _{t \\\\in ]a,b[} \\\\widetilde{\\\\varphi }^{\\\\prime }(t) = \\\\widetilde{c} \\\\cdot \\\\lambda _{-}$ and $\\\\widetilde{\\\\lambda }_{+} := \\\\sup _{t \\\\in ]a,b[} \\\\widetilde{\\\\varphi }^{\\\\prime }(t) = \\\\widetilde{c} \\\\cdot \\\\lambda _{+}$ .\",\n \"Dividing (REF ) by $\\\\widetilde{c}$ , we arrive at $\\\\varphi (t) = \\\\frac{\\\\widetilde{\\\\varphi } (t)}{\\\\widetilde{c}}&=&\\\\sup _{\\\\widetilde{z} \\\\in ]\\\\widetilde{c} \\\\cdot \\\\lambda _{-}, \\\\widetilde{c} \\\\cdot \\\\lambda _{+}[}\\\\left( \\\\frac{\\\\widetilde{z}}{\\\\widetilde{c}} \\\\cdot t-\\\\log \\\\Big (\\\\int _{\\\\mathbb {R}}e^{\\\\frac{\\\\widetilde{z}}{\\\\widetilde{c}} \\\\cdot \\\\widetilde{y} \\\\cdot \\\\widetilde{c}} d\\\\widetilde{\\\\mathbb {}}_{\\\\widetilde{c}} (\\\\widetilde{y})\\\\Big )^{1/\\\\widetilde{c}}\\\\right),\\\\nonumber \\\\\\\\&=&\\\\sup _{z \\\\in ]\\\\lambda _{-}, \\\\lambda _{+}[}\\\\left( z \\\\cdot t-\\\\log \\\\Big (\\\\int _{\\\\mathbb {R}}e^{z \\\\cdot \\\\widetilde{y} \\\\cdot \\\\widetilde{c}}d\\\\widetilde{\\\\mathbb {}}_{\\\\widetilde{c}} (\\\\widetilde{y})\\\\Big )^{1/\\\\widetilde{c}}\\\\right),\\\\qquad t\\\\in ]a,b[ \\\\, , \\\\ \\\\ $ and hence for the exponential of its Fenchel-Legendre transform $e^{\\\\varphi _{*}(z)} =\\\\Big ( \\\\int _{\\\\mathbb {R}} e^{z \\\\cdot \\\\widetilde{y} \\\\cdot \\\\widetilde{c}}d\\\\widetilde{\\\\mathbb {}}_{\\\\widetilde{c}} (\\\\widetilde{y}) \\\\Big )^{1/\\\\widetilde{c}}, \\\\qquad z \\\\in ]\\\\lambda _{-}, \\\\lambda _{+}[ .$ Here, according to (G9i) for $\\\\widetilde{\\\\varphi }$ we have used $\\\\widetilde{\\\\lambda }_{-} := \\\\inf _{t \\\\in ]a,b[} \\\\widetilde{\\\\varphi }^{\\\\prime }(t) = \\\\widetilde{c} \\\\cdot \\\\lambda _{-}$ and $\\\\widetilde{\\\\lambda }_{+} := \\\\sup _{t \\\\in ]a,b[} \\\\widetilde{\\\\varphi }^{\\\\prime }(t) = \\\\widetilde{c} \\\\cdot \\\\lambda _{+}$ .\",\n \"From (REF ) and (REF ) we deduce for $\\\\widetilde{c} := \\\\frac{1}{n}$ the relation $MGF_{\\\\mathbb {}}(z) = (MGF_{\\\\widetilde{\\\\mathbb {}}_{1/n}}(\\\\frac{z}{n}))^{n}$ for all $n \\\\in \\\\mathbb {N}$ which (with the help of (ii)) implies the infinitely divisibility of $\\\\mathbb {}$ .\",\n \"For the reverse direction, let us assume that $\\\\varphi \\\\in \\\\Upsilon (]a,b[)$ and that the corresponding $\\\\mathbb {}$ is infinitely divisible.\",\n \"Recall that $]a,b[ = int(dom(\\\\varphi ))$ .\",\n \"Moreover, we fix an arbitrary constant $\\\\widetilde{c} >0$ .\",\n \"Of course, there holds $\\\\widetilde{c} \\\\cdot \\\\varphi \\\\in \\\\widetilde{\\\\Upsilon }(]a,b[)$ and $dom(\\\\widetilde{c} \\\\cdot \\\\varphi ) = dom(\\\\varphi )$ .\",\n \"Furthermore, by multiplying (REF ) with $\\\\widetilde{c} >0$ and by employing (i), (ii) we get $\\\\widetilde{c} \\\\cdot \\\\varphi (t) &=&\\\\sup _{z\\\\in ]\\\\lambda _{-}, \\\\lambda _{+}[}\\\\left( \\\\widetilde{c} \\\\cdot z \\\\cdot t-\\\\log \\\\Big (\\\\int _{\\\\mathbb {R}} e^{\\\\widetilde{c} \\\\cdot z\\\\cdot \\\\frac{y}{\\\\widetilde{c}}} d\\\\mathbb {} (y) \\\\Big )^{\\\\widetilde{c}} \\\\right)= \\\\sup _{\\\\widetilde{z} \\\\in ]\\\\widetilde{c} \\\\cdot \\\\lambda _{-}, \\\\widetilde{c} \\\\cdot \\\\lambda _{+}[}\\\\left( \\\\widetilde{z} \\\\cdot t-\\\\log \\\\Big (\\\\int _{\\\\mathbb {R}}e^{\\\\frac{\\\\widetilde{z}}{\\\\widetilde{c}} \\\\cdot y} \\\\, d\\\\mathbb {} (y) \\\\Big )^{\\\\widetilde{c}} \\\\right)\\\\nonumber \\\\\\\\&=&\\\\sup _{\\\\widetilde{z} \\\\in ]\\\\widetilde{c} \\\\cdot \\\\lambda _{-}, \\\\widetilde{c} \\\\cdot \\\\lambda _{+}[}\\\\left( \\\\widetilde{z} \\\\cdot t-\\\\log \\\\Big ( \\\\int _{\\\\mathbb {R}}e^{\\\\widetilde{z} \\\\cdot y} d\\\\mathbb {}_{\\\\widetilde{c}} (y) \\\\Big ) \\\\right), \\\\quad t\\\\in ]a,b[; \\\\ \\\\ $ for some probability distribution $\\\\mathbb {}_{\\\\widetilde{c}}$ on $\\\\mathbb {R}$ .\",\n \"$\\\\blacksquare $ Proof of Proposition REF .\",\n \"It is well known that a candidate function $M: ]-\\\\infty ,0[ \\\\mapsto ]0,\\\\infty [$ is the moment-generating function of an infinitely divisible probability distribution if and only if $(\\\\log M)^{\\\\prime }$ is absolutely monotone (see e.g.\",\n \"Theorem 5.11 of Schilling et al.\",\n \"[322]).\",\n \"By applying this to $M(z) := e^{-a\\\\cdot z + \\\\varphi ^{*}(z)}$ respectively $M(z) := e^{b\\\\cdot z + \\\\varphi ^{*}(- z)}$ , one gets straightforwardly the assertion (a) respectively (b); notice that the light-tailedness follows then from (G1) to (G8), and $b=\\\\infty $ respectively $a =- \\\\infty $ can be deduced from the fact that the support of an infinitely distribution is always (one-sided or two-sided) unbounded.\",\n \"For the third case $a= - \\\\infty $ , $b= \\\\infty $ one can use the assertion (cf.\",\n \"e.g.\",\n \"Morris [267], p.73) that a candidate function $M: \\\\, ]\\\\lambda _{-}, \\\\lambda _{+}[ \\\\, \\\\mapsto \\\\, ]0,\\\\infty [$ is the moment-generating function of an infinitely divisible probability distribution if the connected function $z \\\\mapsto (\\\\log M)^{\\\\prime \\\\prime }(z)/(\\\\log M)^{\\\\prime \\\\prime }(0)$ is the moment-generating function of some auxiliary probability distribution; but the latter is equivalent to exponentially convexity (cf.\",\n \"Theorem REF (b)).\",\n \"By applying this to $M(z) := e^{\\\\varphi ^{*}(z)}$ , one ends up with (c).\",\n \"$\\\\blacksquare $\"\n ],\n [\n \"Proofs — Part 6\",\n \"Proof of Theorem REF .\",\n \"(i) Clearly, on $]\\\\lambda _{-},\\\\lambda _{+}[$ the function $\\\\Lambda $ is differentiable with strictly increasing derivative $\\\\Lambda ^{\\\\prime }(z) = F^{-1}(z+c) + 1- F^{-1}(c) ,\\\\qquad z \\\\in ]\\\\lambda _{-},\\\\lambda _{+}[.$ Hence, $\\\\Lambda $ is strictly convex and smooth (because of the smoothness of $F^{-1}$ ), and satisfies $\\\\Lambda (0) =0$ as well as $\\\\Lambda ^{\\\\prime }(0) =1$ .\",\n \"Also, the corresponding extensions of $\\\\Lambda $ to $z=\\\\lambda _{-}$ and $z=\\\\lambda _{+}$ are continuous.\",\n \"(ii) It is straightforward to see that on $]t_{-}^{sc},t_{+}^{sc}[$ the function $\\\\varphi $ is differentiable with strictly increasing derivative $\\\\varphi ^{\\\\prime }(t) = F(t+ F^{-1}(c) - 1) - c ,\\\\qquad t \\\\in ]t_{-}^{sc},t_{+}^{sc}[.$ Hence, $\\\\varphi $ is strictly convex and smooth (because of the smoothness of $F$ ), and satisfies $\\\\varphi (1) =0$ as well as $\\\\varphi ^{\\\\prime }(1) =0$ .\",\n \"Also, the corresponding extensions of $\\\\varphi $ to $t=t_{-}^{sc}$ and $t=t_{-}^{sc}$ are continuous.\",\n \"Hence (G1), (G2), (G5) and (G6) hold.\",\n \"To prove (G3) (and hence (G1)), let us first notice that obviously there holds $a \\\\le t_{-}^{sc}$ and $t_{+}^{sc} \\\\le b$ .\",\n \"Moreover, the validity of $\\\\varphi (t) < \\\\infty $ for all $t \\\\in ]t_{-}^{sc},t_{+}^{sc}[$ is clear from (REF ) since $t+F^{-1}(c)-1 \\\\in ]a_{F},b_{F}[ = int(dom(F))$ and the involved integral over the continuous function $F^{-1}$ is taken over a compact interval.\",\n \"For the subcase $t_{-}^{sc} =- \\\\infty = a$ we have thus shown $dom(\\\\varphi ) \\\\cap ]-\\\\infty ,1] = ]-\\\\infty ,1]= ]a,1]$ , whereas for the subcase $t_{+}^{sc} = \\\\infty = b$ we have verified $dom(\\\\varphi ) \\\\cap [1,\\\\infty [ = [1,\\\\infty [ = [1,b[$ .\",\n \"Let us next examine the subcase “$t_{-}^{sc} > - \\\\infty $ and $\\\\varphi (t_{-}^{sc}) < \\\\infty $ ”: if $\\\\lambda _{-} > - \\\\infty $ then $a = -\\\\infty $ and (REF ) implies $\\\\varphi (t) = \\\\varphi (t_{-}^{sc}) +\\\\lambda _{-} \\\\cdot (t- t_{-}^{sc}) < \\\\infty $ for all $t \\\\in \\\\ ]-\\\\infty , t_{-}^{sc}] = ]a,t_{-}^{sc}]$ , which leads to $dom(\\\\varphi ) \\\\cap ]-\\\\infty ,1] = ]-\\\\infty ,1] = ]a,1]$ ; in contrast, if $\\\\lambda _{-} = - \\\\infty $ then $a = t_{-}^{sc}$ and (REF ) implies $\\\\varphi (t) = \\\\varphi (t_{-}^{sc}) +\\\\lambda _{-} \\\\cdot (t- t_{-}^{sc}) = \\\\infty $ for all $t \\\\in \\\\ ]-\\\\infty , t_{-}^{sc}[ = ]-\\\\infty ,a[$ , which leads to $dom(\\\\varphi ) \\\\cap ]-\\\\infty ,1] = [a,1]$ .\",\n \"In the subcase “$t_{-}^{sc} > - \\\\infty $ and $\\\\varphi (t_{-}^{sc}) = \\\\infty $ ”, due to the strict convexity of $\\\\varphi $ one always has $\\\\lim _{t \\\\downarrow t_{-}^{sc}} \\\\varphi ^{\\\\prime }(t) = - \\\\infty $ ; this implies, by the below-mentioned (REF ), that $\\\\lambda _{-} = - \\\\infty $ and thus $a = t_{-}^{sc}$ ; from (REF ) we derive $\\\\varphi (t) = \\\\varphi (t_{-}^{sc}) +\\\\lambda _{-} \\\\cdot (t- t_{-}^{sc}) = \\\\infty $ for all $t \\\\in \\\\ ]-\\\\infty , t_{-}^{sc}[ = ]-\\\\infty ,a[$ , which leads to $dom(\\\\varphi ) \\\\cap ]-\\\\infty ,1] = ]a,1]$ .\",\n \"As a further step, we deal with the subcase “$t_{+}^{sc} < \\\\infty $ and $\\\\varphi (t_{+}^{sc}) < \\\\infty $ ”: if $\\\\lambda _{+} < \\\\infty $ then $b = \\\\infty $ and (REF ) implies $\\\\varphi (t) = \\\\varphi (t_{+}^{sc}) +\\\\lambda _{+} \\\\cdot (t- t_{+}^{sc}) < \\\\infty $ for all $t \\\\in \\\\ [t_{+}^{sc},\\\\infty [ = [t_{+}^{sc},b[$ , which leads to $dom(\\\\varphi ) \\\\cap [1, \\\\infty [ = [1, \\\\infty [ = [1, b[$ ; in contrast, if $\\\\lambda _{+} = \\\\infty $ then $b = t_{+}^{sc}$ and (REF ) implies $\\\\varphi (t) = \\\\varphi (t_{+}^{sc}) +\\\\lambda _{+} \\\\cdot (t- t_{+}^{sc}) = \\\\infty $ for all $t \\\\in \\\\ ]t_{-}^{sc}, \\\\infty [ = ]b,\\\\infty [$ , which leads to $dom(\\\\varphi ) \\\\cap [1,\\\\infty [ = [1,b]$ .\",\n \"In the subcase “$t_{+}^{sc} < + \\\\infty $ and $\\\\varphi (t_{+}^{sc}) = \\\\infty $ ”, due to the strict convexity of $\\\\varphi $ one always gets $\\\\lim _{t \\\\uparrow t_{+}^{sc}} \\\\varphi ^{\\\\prime }(t) = \\\\infty $ ; this implies, by the below-mentioned (), that $\\\\lambda _{+} = \\\\infty $ and thus $b = t_{+}^{sc}$ ; from (REF ) we deduce $\\\\varphi (t) = \\\\varphi (t_{+}^{sc}) +\\\\lambda _{+} \\\\cdot (t- t_{+}^{sc}) = \\\\infty $ for all $t \\\\in \\\\ ]t_{+}^{sc}, \\\\infty [ = ]b, \\\\infty [$ , which leads to $dom(\\\\varphi ) \\\\cap [1,\\\\infty [ = [1,b[$ .\",\n \"Putting things together, we have proved (G3).\",\n \"The property (G4) follows straightforwardly from (REF ), the continuity of $F$ and from $\\\\lim _{t \\\\downarrow t_{-}^{sc}} \\\\varphi ^{\\\\prime }(t) = \\\\lambda _{-}$ , $\\\\lim _{t \\\\uparrow t_{+}^{sc}} \\\\varphi ^{\\\\prime }(t) = \\\\lambda _{+}$ .\",\n \"To see the latter two, from (REF ) we obtain $& & \\\\lim _{t \\\\downarrow t_{-}^{sc}} \\\\varphi ^{\\\\prime }(t) = \\\\lim _{t \\\\downarrow t_{-}^{sc}} F(t+ F^{-1}(c) - 1) - c= \\\\lim _{t \\\\downarrow t_{-}^{sc}} F(t+ a_{F} - t_{-}^{sc}) - c= \\\\inf \\\\lbrace F(\\\\widetilde{t}) - c: \\\\widetilde{t} \\\\in ]a_{F},b_{F}[ \\\\rbrace = \\\\lambda _{-},\\\\\\\\& & \\\\lim _{t \\\\uparrow t_{+}^{sc}} \\\\varphi ^{\\\\prime }(t) = \\\\lim _{t \\\\uparrow t_{+}^{sc}} F(t+ F^{-1}(c) - 1) - c= \\\\lim _{t \\\\uparrow t_{+}^{sc}} F(t+ b_{F} - t_{+}^{sc}) - c= \\\\sup \\\\lbrace F(\\\\widetilde{t}) - c: \\\\widetilde{t} \\\\in ]a_{F},b_{F}[ \\\\rbrace = \\\\lambda _{+}.$ The two properties (G7) and (G8) are clear form the above considerations.\",\n \"(iii) From (REF ) and (REF ) one gets easily $\\\\Lambda ^{\\\\prime -1}(t) = F\\\\left(t+F^{-1}(c)-1 \\\\right) - c = \\\\varphi ^{\\\\prime }(t), \\\\qquad t \\\\in ]t_{-}^{sc},t_{+}^{sc}[,$ as well as $\\\\Lambda ^{\\\\prime -1}(1)=0$ .\",\n \"From this, we derive $&& t \\\\cdot \\\\Lambda ^{\\\\prime -1}(t) - \\\\Lambda \\\\left(\\\\Lambda ^{\\\\prime -1}(t)\\\\right)\\\\nonumber \\\\\\\\&& = t \\\\cdot [F\\\\left(t+F^{-1}(c)-1 \\\\right) - c]+ [F^{-1}(c)-1] \\\\cdot [F\\\\left(t+F^{-1}(c)-1 \\\\right) - c]\\\\nonumber \\\\\\\\& & -\\\\int \\\\displaylimits _{0}^{F\\\\left(t+F^{-1}(c)-1 \\\\right) - c} F^{-1}(u+c) du\\\\nonumber \\\\\\\\& & = \\\\varphi (t), \\\\qquad t \\\\in ]t_{-}^{sc},t_{+}^{sc}[,$ and hence, with the help of (REF ) in combination with (REF ), () $\\\\varphi (t) = \\\\max _{z \\\\in ]\\\\lambda _{-},\\\\lambda _{+}[} \\\\left( z\\\\cdot t -\\\\Lambda (z)\\\\right), \\\\qquad t \\\\in ]t_{-}^{sc},t_{+}^{sc}[ ,$ i.e.\",\n \"on $]t_{-}^{sc},t_{+}^{sc}[$ the divergence generator $\\\\varphi $ is the classical Legendre transform of the restriction of $\\\\Lambda $ to $]\\\\lambda _{-},\\\\lambda _{+}[$ .\",\n \"If “$\\\\lambda _{-} > -\\\\infty $ , $\\\\Lambda (\\\\lambda _{-}) \\\\in ]-\\\\infty ,\\\\infty [$ and $\\\\Lambda ^{\\\\prime }(\\\\lambda _{-}) \\\\in ]-\\\\infty ,\\\\infty [$ ” respectively “$\\\\lambda _{+} < -\\\\infty $ , $\\\\Lambda (\\\\lambda _{+}) \\\\in ]-\\\\infty ,\\\\infty [$ and $\\\\Lambda ^{\\\\prime }(\\\\lambda _{+}) \\\\in ]-\\\\infty ,\\\\infty [$ ”, then one can apply classical facts of Fenchel-Legendre transformation to get the corresponding left-hand respectively right-hand linear extensions of $\\\\varphi $ on the complement of $]t_{-}^{sc},t_{+}^{sc}[$ , in order to obtain the desired $\\\\varphi (t) =\\\\sup _{z \\\\in ]-\\\\infty ,\\\\infty [} \\\\left( z\\\\cdot t - \\\\Lambda (z)\\\\right) ,\\\\qquad t \\\\in \\\\mathbb {R};$ notice that $t_{-}^{sc} = \\\\lim _{z \\\\downarrow \\\\lambda _{-}} \\\\Lambda ^{\\\\prime }(z)$ and $t_{+}^{sc} = \\\\lim _{z \\\\uparrow \\\\lambda _{+}} \\\\Lambda ^{\\\\prime }(z)$ .\",\n \"(iv) This is just the inverse of (iii), by applying standard Fenchel-Legendre-transformation theory.\",\n \"$\\\\blacksquare $\"\n ],\n [\n \"Further details and proofs for\\nSubsection \",\n \"Proof of Lemma REF.\",\n \"By Assumption (OM), one gets for all $\\\\lambda \\\\in cl(\\\\mathbf {\\\\Lambda })$ that $\\\\lbrace \\\\mathbf {x} \\\\in (dom(\\\\widetilde{\\\\varphi })^{n} : T(\\\\mathbf {x}) = \\\\lambda \\\\rbrace \\\\, \\\\cap \\\\, ]t_{-}^{sc},t_{+}^{sc}[^{n} \\\\ne \\\\emptyset $ .\",\n \"Moreover, for any $\\\\mathbf {x}=\\\\left( x_{1},..,x_{n}\\\\right) $ in $\\\\mathbb {R}^{n}$ , by the independence of the components of $\\\\mathbf {\\\\widetilde{W}}$ as well as (REF ) and (REF ), we have $& & I_{\\\\mathbf {\\\\widetilde{W}}}(\\\\mathbf {x})=\\\\sup _{\\\\mathbf {z}=\\\\left( z_{1},\\\\ldots ,z_{n}\\\\right) \\\\in \\\\mathbb {R}^{n}}\\\\left( \\\\left\\\\langle \\\\mathbf {x},\\\\mathbf {z}\\\\right\\\\rangle -\\\\sum _{i=1}^{n}\\\\Lambda _{\\\\widetilde{\\\\mathbb {}}}(z_{i}) \\\\right)= \\\\sup _{\\\\mathbf {z}\\\\in ]\\\\lambda _{-},\\\\lambda _{+}[^{n}}\\\\left( \\\\sum _{i=1}^{n} \\\\left( x_{i} \\\\cdot z_{i}-\\\\Lambda _{\\\\widetilde{\\\\mathbb {}}}(z_{i}) \\\\right) \\\\right)\\\\nonumber \\\\\\\\& & = \\\\sum _{i=1}^{n} \\\\left( \\\\sup _{z_{i}\\\\in ]\\\\lambda _{-},\\\\lambda _{+}[}\\\\left( x_{i}\\\\cdot z_{i} - \\\\Lambda _{\\\\widetilde{\\\\mathbb {}}}(z_{i})\\\\right) \\\\right)= \\\\sum _{i=1}^{n} \\\\widetilde{\\\\varphi }(x_{i})=\\\\sum _{k=1}^{K}\\\\sum _{i\\\\in I_{k}^{(n)}} \\\\widetilde{\\\\varphi }(x_{i})$ which is finite if and only if $\\\\mathbf {x} \\\\in (dom(\\\\widetilde{\\\\varphi }))^{n}$ (recall that $\\\\widetilde{\\\\varphi }$ is a nonnegative function).\",\n \"Hence, for each $\\\\lambda \\\\in \\\\mathbf {\\\\Lambda }$ we obtain $I(\\\\lambda ) &:=&\\\\inf _{\\\\mathbf {x} \\\\in \\\\mathbb {R}^{n}: \\\\, T(\\\\mathbf {x})=\\\\lambda }I_{\\\\mathbf {\\\\widetilde{W}}}(\\\\mathbf {x})= \\\\inf _{\\\\mathbf {x} \\\\in (dom(\\\\widetilde{\\\\varphi }))^{n}: \\\\, T(\\\\mathbf {x})=\\\\lambda }I_{\\\\mathbf {\\\\widetilde{W}}}(\\\\mathbf {x}) =\\\\inf _{\\\\mathbf {x} \\\\in (dom(\\\\widetilde{\\\\varphi }))^{n}: \\\\, T(\\\\mathbf {x})=\\\\lambda } \\\\ \\\\sum _{k=1}^{K}\\\\sum _{i\\\\in I_{k}^{(n)}} \\\\widetilde{\\\\varphi }(x_{i})\\\\\\\\&=& \\\\sum _{k=1}^{K} n_{k} \\\\cdot \\\\widetilde{\\\\varphi }(\\\\lambda _{k})\\\\ = \\\\ n \\\\cdot \\\\sum _{k=1}^{K} \\\\widetilde{p}_{k} \\\\cdot \\\\widetilde{\\\\varphi }(\\\\lambda _{k})= \\\\inf _{\\\\mathbf {x} \\\\in ]t_{-}^{sc},t_{+}^{sc}[^{n} \\\\, : \\\\, T(\\\\mathbf {x})=\\\\lambda }I_{\\\\mathbf {\\\\widetilde{W}}}(\\\\mathbf {x}) \\\\, ;$ here, we have employed the following facts: (i) the right-most infimum in (REF ) is achieved by minimizing each of the $K$ terms $\\\\sum _{i\\\\in I_{k}^{(n)}}\\\\widetilde{\\\\varphi }(x_{i})$ under the linear constraint $\\\\frac{1}{n_{k}} \\\\cdot \\\\sum _{i\\\\in I_{k}^{(n)}} x_{i}= \\\\lambda _{k}$ , and by the strict convexity of $\\\\widetilde{\\\\varphi }$ on $]t_{-}^{sc},t_{+}^{sc}[$ (cf.\",\n \"(G5)) the minimum of this generic term is attained when all components $x_{i}$ are equal to $\\\\lambda _{k}$ , and (ii) the outcoming minimum does not depend on the particular (generally non-unique) choice of the $x_{i}$ ’s.\",\n \"Notice that we have used the relation $n_{k} = n \\\\cdot \\\\widetilde{p}_{k}$ as well.\",\n \"To proceed, let $\\\\underline{\\\\lambda }$ be a minimal rate point of $\\\\mathbf {\\\\Lambda }$ , which means that $\\\\underline{\\\\lambda } \\\\in \\\\partial \\\\mathbf {\\\\Lambda }$ and $I(\\\\underline{\\\\lambda }) \\\\le I(\\\\lambda )$ for all $\\\\lambda \\\\in \\\\mathbf {\\\\Lambda }$ .\",\n \"By Assumption (OM) one can run all the steps in (REF ) and () with $\\\\underline{\\\\lambda }$ instead of $\\\\lambda $ , and hence $I(\\\\underline{\\\\lambda }) \\\\ = \\\\ \\\\inf _{\\\\mathbf {x} \\\\in \\\\mathbb {R}^{n}: \\\\, T(\\\\mathbf {x})=\\\\underline{\\\\lambda }}I_{\\\\mathbf {\\\\widetilde{W}}}(\\\\mathbf {x})\\\\ = \\\\inf _{\\\\mathbf {x} \\\\in ]t_{-}^{sc},t_{+}^{sc}[^{n} \\\\, : \\\\, T(\\\\mathbf {x})=\\\\underline{\\\\lambda }}I_{\\\\mathbf {\\\\widetilde{W}}}(\\\\mathbf {x})\\\\ = \\\\ n \\\\cdot \\\\sum _{k=1}^{K} \\\\widetilde{p}_{k} \\\\cdot \\\\widetilde{\\\\varphi }(\\\\underline{\\\\lambda }_{k})\\\\ = \\\\ n \\\\cdot \\\\sum _{k=1}^{K} \\\\widetilde{p}_{k} \\\\cdot \\\\widetilde{\\\\varphi }(\\\\underline{\\\\widetilde{q}}_{k}/\\\\widetilde{p}_{k})$ where for the last equality we have employed the vector $\\\\underline{\\\\mathbf {\\\\widetilde{Q}}} := \\\\mathfrak {D}^{-1} \\\\underline{\\\\lambda }$ which we have called the “dominating point of $\\\\widetilde{\\\\mathbf {\\\\Omega }}$ ”.\",\n \"Also we have proved $I(\\\\underline{\\\\lambda })\\\\ = \\\\ n \\\\cdot \\\\inf _{\\\\mathbf {\\\\widetilde{Q}} \\\\in \\\\widetilde{\\\\mathbf {\\\\Omega }}}\\\\sum _{k=1}^{K} \\\\widetilde{p}_{k} \\\\cdot \\\\widetilde{\\\\varphi }(\\\\widetilde{q}_{k}/\\\\widetilde{p}_{k}).\",\n \"\\\\qquad \\\\qquad \\\\blacksquare $ On the obtainment of proxies of minimal rate points by proxy method 2: For the rest of this section, besides (OM) we assume that $dom(\\\\widetilde{\\\\varphi }) = \\\\, ]a,b[ \\\\, = \\\\, ]t_{-}^{sc},t_{+}^{sc}[$ , and that in case of $a=-\\\\infty $ or $b=+\\\\infty $ the divergence generator $\\\\widetilde{\\\\varphi }$ is regularly varying at $a$ or $b$ accordingly, with positive index $\\\\beta $ , i.e.\",\n \"(with a slight abuse of notation) if $a=-\\\\infty $ , then for all $\\\\lambda >0$ there holds $\\\\lim _{u\\\\rightarrow -\\\\infty }\\\\frac{\\\\widetilde{\\\\varphi } \\\\left( \\\\lambda u\\\\right) }{\\\\widetilde{\\\\varphi }\\\\left( u\\\\right) }=\\\\lambda ^{\\\\beta },$ if $b=+\\\\infty $ , then for all $\\\\lambda >0$ there holds $\\\\lim _{u\\\\rightarrow +\\\\infty }\\\\frac{\\\\widetilde{\\\\varphi } \\\\left( \\\\lambda u\\\\right) }{\\\\widetilde{\\\\varphi }\\\\left( u\\\\right) }=\\\\lambda ^{\\\\beta };$ this assumption is denoted by (H$\\\\widetilde{\\\\varphi }$ ).\",\n \"A proxy of $\\\\underline{\\\\mathbf {\\\\widetilde{Q}}}$ can be obtained by sampling from a distribution on $\\\\mathbb {R}^{K}$ defined through $f(\\\\mathbf {\\\\widetilde{Q}}):=C \\\\cdot \\\\exp \\\\left( -\\\\sum _{k=1}^{K} \\\\widetilde{p}_{k} \\\\cdot \\\\widetilde{\\\\varphi }(\\\\widetilde{q}_{k}/\\\\widetilde{p}_{k})\\\\right) \\\\ = \\\\ C \\\\cdot \\\\exp \\\\left( -D_{\\\\widetilde{\\\\varphi }}\\\\left( \\\\mathbf {\\\\widetilde{Q}},\\\\widetilde{{P}}\\\\right)\\\\right)\\\\hspace{51.21504pt} \\\\textrm {(cf.\",\n \"(\\\\ref {simul distr}))}\\\\nonumber $ where $C$ is a normalizing constant; strict convexity (cf.\",\n \"(G5)) of $\\\\widetilde{\\\\varphi }$ together with (H$\\\\widetilde{\\\\varphi }$ ) prove that $f$ is a well-defined (Lebesgue-) density for a random variable $\\\\mathbf {T}$ on $\\\\mathbb {R}^{K}$ .\",\n \"We denote by $\\\\mathbb {F}(\\\\cdot ) := \\\\mathbb {\\\\Pi }[\\\\mathbf {T} \\\\in \\\\cdot \\\\, ]$ the corresponding distribution on $\\\\mathbb {R}^{K}$ having density $f$ .\",\n \"The distribution of $\\\\mathbf {T}$ given $\\\\left( \\\\mathbf {T} \\\\in \\\\widetilde{\\\\mathbf {\\\\Omega }} \\\\right)$ concentrates on the points in $\\\\widetilde{\\\\mathbf {\\\\Omega }}$ which minimize $D_{\\\\widetilde{\\\\varphi }}\\\\left(\\\\mathbf {\\\\widetilde{Q}},\\\\widetilde{{P}}\\\\right) $ as $\\\\mathbf {\\\\widetilde{Q}}$ runs in $\\\\widetilde{\\\\mathbf {\\\\Omega }}$ , when $D_{\\\\widetilde{\\\\varphi }}(\\\\widetilde{\\\\mathbf {\\\\Omega }},\\\\widetilde{{P}})$ is large.\",\n \"This can be argued as follows.\",\n \"We will consider the case when $\\\\widetilde{\\\\mathbf {\\\\Omega }}$ is a compact subset in $\\\\mathbb {R}_{> 0}^{K}$ and $\\\\widetilde{\\\\varphi }$ satisfies (H$\\\\widetilde{\\\\varphi }$ ) with $b=+\\\\infty $ .\",\n \"For the case when $\\\\widetilde{\\\\mathbf {\\\\Omega }}$ is not compact, or belongs to $\\\\mathbb {R}^{K}/\\\\left\\\\lbrace \\\\mathbf {0}\\\\right\\\\rbrace $ , see the Remark REF hereunder.\",\n \"Consider a compact set $\\\\mathbf {\\\\Gamma }$ in $\\\\widetilde{\\\\mathbf {\\\\Omega }}$ and let $\\\\mathbf {\\\\Gamma }_{t}$ be defined as deduced from $\\\\mathbf {\\\\Gamma } $ in a way that makes $D_{\\\\widetilde{\\\\varphi }}\\\\left( \\\\mathbf {\\\\Gamma }_{t},\\\\widetilde{{P}}\\\\right) $ increase with $t$ for sufficiently large $t$ .\",\n \"For instance, set $\\\\mathbf {\\\\Gamma }_{t}:=t \\\\cdot \\\\mathbf {\\\\Gamma } .$ Hence, in case of $b=+\\\\infty $ the divergence $D_{\\\\widetilde{\\\\varphi }}\\\\left( \\\\mathbf {\\\\Gamma }_{t},\\\\widetilde{{P}}\\\\right) =\\\\inf _{g_{t}\\\\in \\\\mathbf {\\\\Gamma }_{t}}\\\\sum _{k=1}^{K} \\\\widetilde{p}_{k}\\\\cdot \\\\widetilde{\\\\varphi }\\\\left( \\\\frac{\\\\left(g_{t}\\\\right)_{k}}{\\\\widetilde{p}_{k}}\\\\right) =\\\\inf _{g \\\\in \\\\mathbf {\\\\Gamma } }\\\\sum _{k=1}^{K} \\\\widetilde{p}_{k}\\\\cdot \\\\widetilde{\\\\varphi }\\\\left( \\\\frac{t \\\\cdot g_{k}}{\\\\widetilde{p}_{k}}\\\\right)$ tends to infinity as $t \\\\rightarrow \\\\infty $ ; the case $a=-\\\\infty $ works analogously with $t\\\\rightarrow -\\\\infty $ .\",\n \"In case of $b < \\\\infty $ we may consider $\\\\mathbf {\\\\Gamma } _{t}:=\\\\left\\\\lbrace b-g /t;g \\\\in \\\\mathbf {\\\\Gamma } \\\\right\\\\rbrace $ and indeed $D_{\\\\widetilde{\\\\varphi }}\\\\left( \\\\mathbf {\\\\Gamma } _{t},\\\\widetilde{{P}}\\\\right) \\\\rightarrow \\\\infty $ as $t\\\\rightarrow \\\\infty $ , with a similar statement when $a>-\\\\infty $ .\",\n \"Assume that $\\\\mathbf {\\\\Gamma }$ has a dominating point $\\\\underline{g}$ .\",\n \"Then $\\\\mathbf {\\\\Gamma }_{t}$ has dominating point $\\\\underline{g}_{t} := t \\\\cdot \\\\underline{g}$ .\",\n \"We prove that $\\\\mathbf {T}$ with distribution (REF ) cannot be too far away (depending on $t$ ) from $\\\\underline{g}_{t}$ whenever $\\\\mathbf {T}$ belongs to $\\\\mathbf {\\\\Gamma } _{t}.$ This argument is valid in the present description of some asymptotics which makes $\\\\mathbf {\\\\Gamma } _{t}$ as a model for $\\\\widetilde{\\\\mathbf {\\\\Omega }}$ for large $t$ ; considering the case when $D_{\\\\widetilde{\\\\varphi } }\\\\left( \\\\widetilde{\\\\mathbf {\\\\Omega }} ,\\\\widetilde{{P}}\\\\right) $ is large is captured through the asymptotic statement $\\\\lim _{t\\\\rightarrow \\\\infty }D_{\\\\widetilde{\\\\varphi } }\\\\left( \\\\mathbf {\\\\Gamma } _{t},\\\\widetilde{{P}}\\\\right) =+\\\\infty .$ There holds the following Proposition 65 With the above notation and under condition (H$\\\\widetilde{\\\\varphi }$ ), denote by $\\\\mathbf {B}$ a neighborhood of $\\\\underline{g}$ and $\\\\mathbf {B}_{t}:=t \\\\cdot \\\\mathbf {B}$ .\",\n \"Then $\\\\mathbb {F}[\\\\mathbf {\\\\Gamma }_{t}\\\\cap \\\\mathbf {B}_{t}^{c} \\\\, \\\\vert \\\\, \\\\mathbf {\\\\Gamma }_{t} \\\\, ]= \\\\mathbb {\\\\Pi }\\\\left[ \\\\left.\",\n \"\\\\mathbf {T}\\\\in \\\\mathbf {\\\\Gamma }_{t}\\\\cap \\\\mathbf {B}_{t}^{c}\\\\, \\\\right|\\\\,\\\\mathbf {T}\\\\in \\\\mathbf {\\\\Gamma }_{t}\\\\right] \\\\rightarrow 0$ as $t\\\\rightarrow \\\\infty $ , which proves that simulations under (REF ) produce proxies of the dominating points $\\\\underline{g}_{t}$ in $\\\\mathbf {\\\\Gamma }_{t}$ .\",\n \"Before we start with the proof of Proposition REF , we first quote the following Lemma 66 Let $\\\\widetilde{\\\\varphi } $ satisfy (H$\\\\widetilde{\\\\varphi }$ ) with $b=+\\\\infty $ .\",\n \"Then for all $\\\\mathbf {A}$ in $\\\\mathbb {R}^{K}$ such that $\\\\breve{\\\\alpha } :=D_{\\\\widetilde{\\\\varphi }}(\\\\mathbf {A},\\\\widetilde{{P}}):=\\\\inf _{\\\\mathbf {v}\\\\in \\\\mathbf {A}}\\\\sum _{k=1}^{K} \\\\widetilde{p}_{k} \\\\cdot \\\\widetilde{\\\\varphi } \\\\left(\\\\frac{v_{k}}{\\\\widetilde{p}_{k}}\\\\right)$ is finite there holds $\\\\lim _{t\\\\rightarrow \\\\infty }\\\\frac{1}{t}\\\\log \\\\int _{\\\\mathbf {A}}\\\\exp \\\\left(-t\\\\sum _{k=1}^{K} \\\\widetilde{p}_{k} \\\\cdot \\\\widetilde{\\\\varphi }\\\\left( \\\\frac{v_{k}}{\\\\widetilde{p}_{k}}\\\\right) \\\\right) \\\\, dv_{1} \\\\ldots dv_{k}= - D_{\\\\widetilde{\\\\varphi }}\\\\left(\\\\mathbf {A},\\\\widetilde{{P}}\\\\right) .$ Proof of Lemma REF .\",\n \"Let us first remark that according to the geometry of the set $\\\\mathbf {A}$ , various combinations for the asymptotics (REF ) or (REF ) may occur; for sake of brevity, we only handle the simplest ones, since all turn to be amenable through the same arguments.\",\n \"Denote for positive $r$ $\\\\mathbf {B}(r):=\\\\left\\\\lbrace \\\\mathbf {v}\\\\in \\\\mathbb {R}^{K}:\\\\sum _{k=1}^{K} \\\\widetilde{p}_{k}\\\\widetilde{\\\\varphi } \\\\left( \\\\frac{v_{k}}{\\\\widetilde{p}_{k}}\\\\right) >r\\\\right\\\\rbrace .$ It holds, by making the change of variable $r=t\\\\cdot \\\\breve{\\\\alpha } +t \\\\cdot s$ , $& & \\\\int _{\\\\mathbf {A}}\\\\exp \\\\left( -t\\\\sum _{k=1}^{K} \\\\widetilde{p}_{k} \\\\cdot \\\\widetilde{\\\\varphi } \\\\left( \\\\frac{v_{k}}{\\\\widetilde{p}_{k}}\\\\right) \\\\right) \\\\,dv_{1} \\\\ldots dv_{k}=\\\\int \\\\cdots \\\\int 1_{\\\\mathbb {R}^{+}}(r)\\\\cdot 1_{\\\\mathbf {A}}(\\\\mathbf {v})\\\\cdot 1_{\\\\left]t\\\\sum _{k=1}^{K} \\\\widetilde{p}_{k}\\\\widetilde{\\\\varphi } \\\\left( \\\\frac{v_{k}}{\\\\widetilde{p}_{k}}\\\\right) ,\\\\infty \\\\right[}(r)\\\\cdot e^{-r} \\\\, dr \\\\, dv_{1}\\\\ldots dv_{K} \\\\\\\\& & = te^{-t\\\\breve{\\\\alpha } }\\\\int \\\\cdots \\\\int 1_{]- \\\\breve{\\\\alpha },\\\\infty [}(s) \\\\cdot 1_{\\\\mathbf {A}}(\\\\mathbf {v}) \\\\cdot 1_{\\\\mathbf {B}^{c}(\\\\breve{\\\\alpha }+s)}(\\\\mathbf {v})\\\\cdot e^{-ts} \\\\, ds \\\\, dv_{1}\\\\ldots dv_{K} \\\\ = \\\\ te^{-t\\\\breve{\\\\alpha } }\\\\int _{-\\\\breve{\\\\alpha }}^{\\\\infty }Vol\\\\left( \\\\mathbf {A}\\\\cap \\\\mathbf {B}^{c}(\\\\breve{\\\\alpha }+s)\\\\right) \\\\cdot e^{-ts} \\\\, ds.$ Let $I_{t}:=t \\\\cdot \\\\int _{0}^{\\\\infty }Vol\\\\left(\\\\mathbf {A}\\\\cap \\\\mathbf {B}^{c}(\\\\breve{\\\\alpha } +s)\\\\right) e^{-ts}ds $ .\",\n \"We prove that $\\\\lim _{t\\\\rightarrow \\\\infty }\\\\frac{1}{t}\\\\log I_{t}=0.\",\n \"$ When $a=-\\\\infty $ or $b=+\\\\infty $ , since $\\\\widetilde{\\\\varphi }$ satisfies (H$\\\\widetilde{\\\\varphi }$ ) there exists a polynomial $P$ such that $Vol\\\\left( \\\\mathbf {A}\\\\cap \\\\mathbf {B}^{c}(\\\\breve{\\\\alpha } +s)\\\\right) \\\\le P(s) \\\\, ;$ whence, assuming without loss of generality that $dom(\\\\widetilde{\\\\varphi }) =\\\\mathbb {R}^{+}$ , we obtain $\\\\frac{1}{t}\\\\log I_{t}\\\\le \\\\frac{1}{t}\\\\log \\\\int _{0}^{\\\\infty }P\\\\left(\\\\frac{u}{t}\\\\right) te^{-u}du$ which yields that for large $t$ $\\\\frac{1}{t}\\\\log I_{t}<0.$ When dealing with a context where $a$ or $b$ have finite value and the corresponding sets $\\\\mathbf {\\\\Gamma }_{t}$ are “far away” from $\\\\mathbf {\\\\Gamma } $ in terms of the distance measure $D_{\\\\widetilde{\\\\varphi }}\\\\left( \\\\cdot ,\\\\widetilde{{P}}\\\\right)$ , then $Vol\\\\left( \\\\mathbf {A}\\\\cap \\\\mathbf {B}^{c}(\\\\breve{\\\\alpha } +s)\\\\right) $ is bounded.\",\n \"Hence, $\\\\lim \\\\sup _{t\\\\rightarrow \\\\infty }\\\\frac{1}{t}\\\\log I_{t}\\\\le 0$ .\",\n \"Now fix $\\\\varepsilon >0.$ Then, since $Vol\\\\left( \\\\mathbf {A}\\\\cap \\\\mathbf {B}^{c}(a+s)\\\\right) $ is increasing in $s$ , we get $I_{t} &\\\\ge &t\\\\int _{\\\\varepsilon }^{\\\\infty }Vol\\\\left( \\\\mathbf {A}\\\\cap \\\\mathbf {B}^{c}(\\\\breve{\\\\alpha }+s)\\\\right) e^{-ts}ds \\\\\\\\&\\\\ge &Vol\\\\left( \\\\mathbf {A}\\\\cap \\\\mathbf {B}^{c}(\\\\breve{\\\\alpha } +\\\\varepsilon )\\\\right) e^{-t\\\\varepsilon }.$ Hence $\\\\frac{1}{t}\\\\log I_{t}\\\\ge \\\\frac{1}{t}\\\\log Vol\\\\left( \\\\mathbf {A}\\\\cap \\\\mathbf {B}^{c}(\\\\breve{\\\\alpha }+\\\\varepsilon )\\\\right) -\\\\varepsilon $ which yields $\\\\lim \\\\inf _{t\\\\rightarrow \\\\infty }\\\\frac{1}{t}\\\\log I_{t}\\\\ge 0.$ Therefore (REF ) holds, which concludes the proof.\",\n \"$\\\\blacksquare $ We now turn to the Proof of Proposition REF .\",\n \"Without loss of generality, let $b=+\\\\infty $ , $\\\\mathbf {\\\\Gamma }_{t}$ as in (REF ) and Condition (H$\\\\widetilde{\\\\varphi }$ ) hold.\",\n \"Moreover, consider an arbitrary neighborhood $\\\\mathbf {B}$ of $\\\\underline{g}$ and the corresponding neighborhoods $\\\\mathbf {B}_{t} : =t \\\\cdot \\\\mathbf {B}$ of $\\\\underline{g}_{t} = t \\\\cdot \\\\underline{g}$ .\",\n \"There holds $\\\\frac{1}{\\\\widetilde{\\\\varphi }(t)}\\\\log \\\\mathbb {\\\\Pi }\\\\left[ \\\\mathbf {T}\\\\in \\\\mathbf {\\\\Gamma }_{t}\\\\right] &=&\\\\frac{C}{\\\\widetilde{\\\\varphi }(t)}\\\\log \\\\int _{\\\\mathbf {\\\\Gamma } _{t}}\\\\exp \\\\left(-\\\\sum _{k=1}^{K}\\\\widetilde{p}_{k} \\\\cdot \\\\widetilde{\\\\varphi }\\\\left(\\\\frac{w_{k}}{\\\\widetilde{p}_{k}}\\\\right)\\\\right) \\\\, dw_{1}\\\\ldots dw_{K} \\\\\\\\&\\\\stackrel{(1)}{=}&\\\\frac{CK}{\\\\widetilde{\\\\varphi }(t)}\\\\log t+\\\\frac{C}{\\\\widetilde{\\\\varphi }(t)}\\\\log \\\\int _{\\\\mathbf {\\\\Gamma }}\\\\exp \\\\left(-t^{\\\\beta } \\\\cdot \\\\sum _{k=1}^{K}\\\\widetilde{p}_{k} \\\\cdot \\\\left( \\\\widetilde{\\\\varphi }\\\\left( \\\\frac{v_{k}}{\\\\widetilde{p}_{k}}\\\\right) \\\\cdot (1+o(1))\\\\right)\\\\right) \\\\, dv_{1} \\\\ldots dv_{K} \\\\\\\\&\\\\stackrel{(2)}{=}& \\\\frac{CK}{\\\\widetilde{\\\\varphi }(t)}\\\\log t+\\\\frac{C}{\\\\left( \\\\widetilde{\\\\varphi }(t)/t^{\\\\beta }\\\\right) } \\\\cdot \\\\frac{1}{t^{\\\\beta }}\\\\log \\\\left( (1+o(1)) \\\\cdot \\\\int _{\\\\mathbf {\\\\Gamma } }\\\\exp \\\\left( -t^{\\\\beta }\\\\sum _{k=1}^{K}\\\\widetilde{p}_{k} \\\\cdot \\\\widetilde{\\\\varphi }\\\\left( \\\\frac{v_{k}}{\\\\widetilde{p}_{k}}\\\\right) \\\\right)dv_{1} \\\\ldots dv_{K} \\\\right)\\\\\\\\&\\\\stackrel{(3)}{=}& -\\\\frac{Ct^{\\\\beta }}{\\\\widetilde{\\\\varphi }(t)}\\\\cdot D_{\\\\widetilde{\\\\varphi }}(\\\\mathbf {\\\\Gamma } ,\\\\widetilde{{P}}) \\\\cdot (1+o(1)) \\\\\\\\&\\\\stackrel{(4)}{=}& - \\\\breve{l}(t) \\\\cdot D_{\\\\widetilde{\\\\varphi }}(\\\\mathbf {\\\\Gamma } ,\\\\widetilde{{P}}) \\\\cdot (1+o(1))$ as $t$ tends to infinity.\",\n \"In the above display, $(1)$ follows from $\\\\widetilde{\\\\varphi }(tx)=\\\\left( tx\\\\right)^{\\\\beta } \\\\cdot \\\\ell (tx)=t^{\\\\beta } \\\\cdot x^{\\\\beta } \\\\cdot \\\\ell (x) \\\\cdot \\\\frac{\\\\ell (tx)}{\\\\ell (x)} =t^{\\\\beta } \\\\cdot \\\\widetilde{\\\\varphi } (x) \\\\cdot \\\\left( 1+o(1)\\\\right) $ as $t$ tends to infinity and $x$ lies in a compact subset of $]0,\\\\infty [$ , where $\\\\ell $ is a slowly varying function.\",\n \"The equality $(2)$ follows from compactness of $\\\\mathbf {\\\\Gamma }$ together with the fact that $\\\\widetilde{\\\\varphi }$ is a regularly varying function with index $\\\\beta $ , so that $\\\\lim _{t\\\\rightarrow \\\\infty }\\\\frac{\\\\widetilde{\\\\varphi }(tv)}{\\\\widetilde{\\\\varphi }(t)}=v^{\\\\beta }$ uniformly upon $v$ on compact sets in $]0,\\\\infty [$ .\",\n \"The remaining equalities $(3)$ and $(4)$ follow from classical properties of regularly varying functions, where $\\\\breve{\\\\ell } := 1/\\\\ell $ is a slowly varying function at infinity, together with standard Laplace-Integral approximation.\",\n \"In the same way we can show $\\\\frac{1}{\\\\widetilde{\\\\varphi }(t)}\\\\log \\\\mathbb {\\\\Pi } \\\\left[ \\\\, \\\\mathbf {T}\\\\in \\\\mathbf {\\\\Gamma } _{t}\\\\cap \\\\mathbf {B}_{t}^{c} \\\\, \\\\right]=- \\\\breve{l}(t) \\\\cdot D_{\\\\widetilde{\\\\varphi }}(\\\\mathbf {\\\\Gamma }\\\\cap \\\\mathbf {B}^{c},\\\\widetilde{{P}}) \\\\cdot (1+o(1))$ as $t$ tends to infinity.\",\n \"Since $\\\\mathbf {B}$ is a neighborhood of the unique dominating point $\\\\underline{g}$ of $\\\\mathbf {\\\\Gamma }$ , one gets that $D_{\\\\widetilde{\\\\varphi }}(\\\\mathbf {\\\\Gamma }\\\\cap \\\\mathbf {B}^{c},\\\\widetilde{{P}}) > D_{\\\\widetilde{\\\\varphi }}(\\\\mathbf {\\\\Gamma } ,\\\\widetilde{{P}})$ .\",\n \"This implies that $\\\\mathbb {\\\\Pi } \\\\left[ \\\\, \\\\mathbf {T}\\\\in \\\\mathbf {\\\\Gamma }_{t}\\\\cap \\\\mathbf {B}_{t}^{c} \\\\, \\\\big \\\\vert \\\\, \\\\mathbf {T}\\\\in \\\\mathbf {\\\\Gamma }_{t} \\\\, \\\\right] \\\\rightarrow 0\\\\qquad \\\\textrm {as $ $.\",\n \"\\\\qquad $$}$$$ Remark 67 Firstly, let us quote that the case when $\\\\widetilde{\\\\mathbf {\\\\Omega }}$ is an unbounded subset in $\\\\mathbb {R}^{K}/\\\\left\\\\lbrace \\\\mathbf {0}\\\\right\\\\rbrace $ is somewhat immaterial for applications.\",\n \"Anyhow, if compactness of $\\\\mathbf {\\\\Gamma } $ is lost, then in order to use the same line of arguments as above, it is necessary to strengthen the assumptions (H$\\\\widetilde{\\\\varphi }$ ) e.g.\",\n \"as follows: when $b=+\\\\infty $ then $\\\\widetilde{\\\\varphi }$ has to be asymptotically homogeneous with degree $\\\\beta >0$ , in the sense that $\\\\widetilde{\\\\varphi }(tx)=t^{\\\\beta }\\\\widetilde{\\\\varphi }(x)\\\\cdot (1+o(1))$ as $t\\\\rightarrow \\\\infty $ ; for the subcase $a=-\\\\infty $ one employs an analogous assumption as $t\\\\rightarrow -\\\\infty $ .\",\n \"The case when $\\\\widetilde{\\\\mathbf {\\\\Omega }}$ is a compact set in $\\\\mathbb {R}^{K}\\\\backslash \\\\lbrace \\\\mathbf {0}\\\\rbrace $ can be treated as above, by combining the asymptotics in $t$ in the neighborhood of $a$ and $b$ accordingly.\"\n ],\n [\n \"Proof for Subsection \",\n \"Proof of Proposition REF.\",\n \"Recall the weighted empirical measure $\\\\xi _{n,\\\\mathbf {X}}^{\\\\mathbf {V}}:=\\\\left( \\\\frac{1}{n}\\\\sum _{i\\\\in I_{1}^{(n)}} V_{i},\\\\ldots ,\\\\frac{1}{n}\\\\sum _{i\\\\in I_{K}^{(n)}} V_{i}\\\\right)$ which satisfies the $K$ linear constraints defined in (REF ) through $E_{S}[\\\\xi _{n,\\\\mathbf {X}}^{\\\\mathbf {V}}] =\\\\xi _{M,\\\\mathbf {X}}^{\\\\mathbf {W}^{\\\\ast }}= \\\\overline{W^{\\\\ast }} \\\\cdot \\\\xi _{M,\\\\mathbf {X}}^{w\\\\mathbf {W}^{\\\\ast }}$ where $\\\\mathbf {Q}^{\\\\ast }:= \\\\left(q_{1}^{\\\\ast }, \\\\ldots , q_{K}^{\\\\ast }\\\\right)= \\\\xi _{M,\\\\mathbf {X}}^{w\\\\mathbf {W}^{\\\\ast }}\\\\in int(\\\\textrm {$$\\\\hspace{-6.544pt}$$})$ and $\\\\overline{W^{\\\\ast }} = \\\\frac{1}{M}\\\\sum _{j=1}^{M}W_{j}^{\\\\ast }$ .\",\n \"The probability distribution $S$ defined on $\\\\mathbb {R}^{n}$ is the Kullback-Leibler projection of $\\\\mathbb {}^{\\\\otimes n}$ on the class of all probability distributions on $\\\\mathbb {R}^{n}$ which satisfy (REF ).\",\n \"We prove that $\\\\lim \\\\inf _{n\\\\rightarrow \\\\infty }S\\\\left[\\\\xi _{n,\\\\mathbf {X}}^{w\\\\mathbf {V}} \\\\in \\\\textrm {\\\\right.$$\\\\hspace{-6.544pt}$$}> 0$ .\",\n \"To start with, we define for strictly positive $\\\\delta $ the set $A_{n,\\\\delta } := \\\\left\\\\lbrace \\\\left|\\\\frac{1}{n}\\\\sum _{i=1}^{n}V_{i}-\\\\overline{W^{\\\\ast }}\\\\right|\\\\le \\\\delta \\\\right\\\\rbrace $ and write $S\\\\left[\\\\xi _{n,\\\\mathbf {X}}^{w\\\\mathbf {V}} \\\\in \\\\textrm {\\\\right.$ $\\\\hspace{-6.544pt}$$}=S\\\\left[\\\\lbrace \\\\xi _{n,\\\\mathbf {X}}^{w\\\\mathbf {V}} \\\\in \\\\textrm {\\\\right.$$\\\\hspace{-6.544pt}$$} \\\\rbrace \\\\cap A_{n,\\\\delta }+ S\\\\left[\\\\lbrace \\\\xi _{n,\\\\mathbf {X}}^{w\\\\mathbf {V}} \\\\in \\\\textrm {\\\\right.$$\\\\hspace{-6.544pt}$$} \\\\rbrace \\\\cap A_{n,\\\\delta }^{c}=:I+II.$$By the law of large numbers, the second term $ II$ tends to $ 0$ as $ n$ tends to infinity.Moreover, one can rewrite$$I=S\\\\bigg [ \\\\bigcup \\\\limits _{m\\\\in \\\\left[ \\\\overline{W^{\\\\ast }}-\\\\delta ,\\\\overline{W^{\\\\ast }}+\\\\delta \\\\right] }\\\\left\\\\lbrace \\\\xi _{n,\\\\mathbf {X}}^{\\\\mathbf {V}}\\\\in m \\\\cdot \\\\textrm {\\\\right.$$\\\\hspace{-6.544pt}$$} \\\\bigg ]$$which entails$$I \\\\ \\\\ge \\\\ S\\\\bigg [ \\\\frac{1}{n_{k}}\\\\sum _{i\\\\in I_{k}^{(n)}}V_{i}\\\\in \\\\mathcal {V}_{\\\\eta }\\\\left( \\\\overline{W^{\\\\ast }}\\\\frac{q_{k}^{\\\\ast }}{p_{k}}\\\\right) \\\\text{ for all }k \\\\in \\\\lbrace 1,\\\\ldots ,K\\\\rbrace \\\\bigg ] ,$$where $ V( W qkpk) $ denotes a neighborhood of$Wqkpk$ with radius $$ being smallwhen $$ is small, for large enough $ n$, making use of the a.s. convergenceof $ nk/n$ to $ pk$.\",\n \"Now, for any $ k {1,...,K}$ one has\\\\begin{equation}S\\\\bigg [ \\\\frac{1}{n_{k}}\\\\sum _{i\\\\in I_{k}^{(n)}}V_{i}\\\\notin \\\\mathcal {V}_{\\\\eta }\\\\left( \\\\overline{W^{\\\\ast }}\\\\frac{q_{k}^{\\\\ast }}{p_{k}}\\\\right)\\\\bigg ]\\\\ \\\\le \\\\ \\\\exp \\\\bigg (-n_{k} \\\\cdot \\\\inf _{x\\\\in \\\\mathcal {V}_{\\\\eta }\\\\left(\\\\overline{W^{\\\\ast }}\\\\frac{q_{k}^{\\\\ast }}{p_{k}}\\\\right)^{c}} \\\\,\\\\varphi \\\\left( x\\\\right) \\\\bigg )\\\\end{equation}since any margin of $ S$ with index in $ Ik(n)$ is a correspondingKullback-Leibler projection of $$ on the set of all distributionson $ R$ with expectation $W qkpM,kemp$ ---where $ pM,kemp$ denotes the fraction ofthe $ Xi$’s (within $ X1,...,XM$) which are equal to $ dk$(cf.\",\n \"(\\\\ref {I^(n)_k for stat case})) ---and therefore has amoment generating function which is finite in a non-void neighborhood of $ 0$,which yields (\\\\ref {ineg}) by the Markov Inequality.\",\n \"Note that the event$ { M,XwWint( $\\\\Omega $$\\\\Omega $ ) }$ is regenerative, so that $ M$ can be chosen large enough to make$ pM,kemp$ close to $ pk$ for all $ k {1,...,K}$.\",\n \"This proves the claim.\\\\hspace{28.45274pt} $$$\"\n ],\n [\n \"Acknowledgment\",\n \"W. Stummer is grateful to the Sorbonne Université Paris for its multiple partial financial support and especially the LPSM for its multiple great hospitality.\",\n \"M. Broniatowski thanks very much the FAU Erlangen-Nürnberg for its partial financial support and hospitality.\",\n \"Moreover, W. Stummer would like to thank Rene Schilling for an interesting discussion on complex-valued foundations of the Bernstein-Widder theorem.\"\n ]\n]"},"id":{"kind":"string","value":"2107.01693"}}},{"rowIdx":1585,"cells":{"text":{"kind":"list like","value":[["Boosting Transferability of Targeted Adversarial Examples via\n Hierarchical Generative Networks"],["Abstract Transfer-based adversarial attacks can evaluate model robustness in the black-box setting.","Several methods have demonstrated impressive untargeted transferability, however, it is still challenging to efficiently produce targeted transferability.","To this end, we develop a simple yet effective framework to craft targeted transfer-based adversarial examples, applying a hierarchical generative network.","In particular, we contribute to amortized designs that well adapt to multi-class targeted attacks.","Extensive experiments on ImageNet show that our method improves the success rates of targeted black-box attacks by a significant margin over the existing methods -- it reaches an average success rate of 29.1\\% against six diverse models based only on one substitute white-box model, which significantly outperforms the state-of-the-art gradient-based attack methods.","Moreover, the proposed method is also more efficient beyond an order of magnitude than gradient-based methods."],["Introduction","Recent progress in adversarial machine learning demonstrates that deep neural networks (DNNs) are highly vulnerable to adversarial examples [42], [13], which are maliciously generated to mislead a model to produce incorrect predictions.","It has been demonstrated that adversarial examples possess an intriguing property of transferability [28], [45], [18], [5], [49] — the adversarial examples crafted for a white-box model can also mislead other unknown models, making black-box attacks feasible.","The threats of adversarial examples have raised concerns in numerous security-sensitive applications, such as autonomous driving [11] and face recognition [36], [48].","Figure: The targeted adversarial examples crafted by MIM and the conditional generative semantic pattern (C-GSP) crafted by our method for the Inception-v3 model given the target class Viaduct with perturbation budget 16 under the ℓ ∞ \\ell _{\\infty } norm constraint.","We also show the predicted labels and probabilities of these images by the black-box model DenseNet-201 .Tremendous efforts have been made to develop more effective black-box attack methods based on transferability since they can serve as an important surrogate to evaluate the model robustness in real-world scenarios [28], [9].","The current methods have achieved impressive performance of untargeted black-box attacks, intending to cause misclassification of the black-box models.","However, the targeted black-box attacks, aiming at misleading the black-box models by outputting the adversary-desired target class, perform unsatisfactorily [8] and have not been extensively explored [52].","The difficulty of fooling a black-box model by the existing targeted adversarial attacks could result in an over-estimation of model robustness under the challenging targeted black-box attack setting.","Existing efforts on targeted black-box attacks can be categorized as instance-specific and instance-agnostic attacks.","Specifically, the instance-specific attack methods [12], [30], [22], [9] craft adversarial examples by performing gradient updates iteratively, which achieve unsatisfactory performance for targeted black-box attacks due to easy overfitting to a white-box model [9], [46].","On the other hand, the instance-agnostic attack methods learn a universal adversarial perturbation [52] or a universal function [38], [31] on the data distribution independent of specific instances.","They can promote more general and transferable adversarial examples since the universal perturbation or function can alleviate the data-specific overfitting problem by training on an unlabeled dataset.","CD-AP [31], as one of the effective instance-agnostic methods, adopts a generative model as a universal function to obtain an acceptable performance when facing one specified target class.","However, CD-AP needs to learn a generative model for each target class while performing multi-target attack [14], , crafting adversarial examples targeted at different classes.","Thus it is not scalable to the increasing number of targets such as hundreds of classes, limiting practical efficiency.","To address the aforementioned issues and develop a targeted black-box attack in the practical scenario, in this paper we propose a conditional generative model as the universal adversarial function to craft adversarial perturbations.","Thus we can craft adversarial perturbations targeted at different classes, using a single model backbone with different class embeddings.","The proposed generative method is simple yet practical to obtain superior performance of targeted black-box attacks, meanwhile with two technical improvements including smooth projection mechanism that better helps the generator to probe targeted semantic knowledge from the classifier and adaptive Gaussian Smoothing with the focus of making generated results obtain adaptive ability against adversarially trained models.","The previous CD-AP requires costly training $N$ models while performing a multi-target attack with $N$ classes.","However, ours only trains one model and reaches an average success rate of 51.1% against six naturally trained models and 36.4% against three adversarially trained models based only on one substitute white-box model in NeurIPS ImageNet dataset, which outperforms CD-AP by a large margin of 6.0% and 31.3%, respectively.","While handling plenty of classes (, 1,000 classes in ImageNet), the effectiveness of generating targeted adversarial examples will be affected by a single generative model due to the difficulty of loss convergence in adversarial learning [47], [1].","Thus we train a feasible number of models (, 10$\\sim $ 20 models on ImageNet) to further promote the effectiveness beyond the single model backbone.","Specifically, each model is learned from a subset of classes specified by a designed hierarchical partition mechanism by considering the diversity property among subsets, for seeking a balance between effectiveness and scalability.","It reaches an average success rate of 29.6% against six different models, outperforming the state-of-the-art methods with an average success rate of $<$ 2% by a large margin, based only on one substitute white-box model in the NeurIPS 2017 competition.","Moreover, the proposed method achieves substantial speedup over gradient-based methods.","Furthermore, these adversarial perturbations generated by the proposed Conditional Generative models can arise as a result of strong Semantic Pattern (C-GSP) as shown in Fig.","REF .","We experimentally find that the generated adversarial semantic pattern itself achieves well-generalizing performance among the different models and is robust to the influence of data in Sec.","REF , which is very instructive for the understanding of adversarial examples.","Our main contributions can be summarized as follows: We present a systematical study on targeted black-box attacks involving instance-specific and instance-agnostic methods in ImageNet dataset with plenty of classes and face recognition.","We propose a simple yet practical conditional generative targeted attack method with a designed hierarchical partition mechanism, which can generate targeted adversarial examples without tuning the parameters.","Extensive experiments demonstrate that our method significantly improves the success rates of targeted black-box attacks over the existing methods."],["Related Work","In this section, we review related work on adversarial attacks belonging to different types.","Instance-specific attacks.","Some recent works [12], [30] adopt gradient-based optimization methods to generate the data-dependent perturbations.","MIM [9] introduces the momentum term into the iterative attack process to improve the black-box transferability.","DIM [46] and TIM [10] aim to achieve the better transferability by input or gradient diversity.","Instance-specific methods require iterative optimization for every instance, thus easily overfitting the current data point [9].","In contrast, we improve the transferability simultaneously with the inference-time efficiency.","Figure: An overview of our proposed generative method for crafting C-GSP, which includes modules of conditional generator and classifier.","The generator integrates the image and conditional class vector from Map network into a hidden incorporation.","Note that only the generator is trained in the whole pipeline to probe the target boundaries of the classifier.Instance-agnostic attacks.","Compared with instance-specific attacks, instance-agnostic attacks belong to image-independent (universal) methods.","The first pipeline is to learn a universal perturbation.","UAP [29] proposes to fool a model by adding a learned universal noise vector.","Another pipeline of attacks introduces learned generative models to craft adversarial examples.","GAP [33] and AAA [34] craft adversarial perturbations in a similar way based on target data directly and compress impressions, respectively.","Previous methods, including universal perturbation and function, require costly training the same number of models for multiple target classes.","Our method is capable of simultaneously generating adversarial samples for specifying multiple targets with better attack performance.","Multi-target attacks.","Instance-specific attacks have the ability for specifying any target in the optimization phase.","As elaborated in the introduction, these methods have degraded transferability and time-consuming iterative procedures.","Related MAN [14] trains a generative model in the ImageNet under the constraint of $\\ell _{2}$ norm to explore the targeted attacks, which specifies all 1,000 categories from ImageNet for seeking extreme speed and storage.","However, MAN does not demonstrate the effectiveness in terms of multi-target transferability against black-box models than previous instance-specific or instance-agnostic attacks, and the authors also claim that too many categories make it hard to transfer to another model.","Recent UAE [52] reveals better single-target transferability by learning universal perturbation, whereas they require to train multiple times while specifying multiple targets.","As a comparison, our method can generate adversarial samples for specifying multiple targets, meanwhile generated strong semantic patterns can outperform existing attacks by a significant margin."],["Method"," In this section, we introduce a conditional generative model to learn a universal adversarial function, which can achieve effective multi-target black-box attacks.","While handing plenty of classes, we design a hierarchical partition mechanism to make the generative model capable of specifying any target class under a feasible number of models, regarding both the effectiveness and scalability."],["Problem Formulation","We use $\\mathbf {x}_{s}$ to denote an input image belonging to an unlabeled training set $\\mathcal {X}_{s} \\subset \\mathbb {R}^{d}$ , and use $c \\in \\mathcal {C}$ to denote a specific target class.","Let $\\mathcal {F}_{\\phi }: \\mathcal {X}_{s} \\rightarrow \\mathbb {R}^{K}$ denote a classification network that outputs a class probability vector with $K$ classes.","To craft a targeted adversarial example $\\mathbf {x}_{s}^*$ from a real example $\\mathbf {x}_{s}$ , the targeted attack aims to fool the classifier $\\mathcal {F}_{\\phi }$ by outputting a specific label $c$ as $\\operatornamewithlimits{arg\\,max}_{i\\in \\mathcal {C}}{\\mathcal {F}_{\\phi }(\\mathbf {x}_{s}^{*})}_{i} = c$ , meanwhile the $\\ell _{\\infty }$ norm of the adversarial perturbation is required to no more than value $\\epsilon $ as $\\Vert \\mathbf {x}_{s}^{*}-\\mathbf {x}_{s}\\Vert _{\\infty }\\le \\epsilon $ .","Although some generative methods [33], [31] can learn targeted adversarial perturbation, it does not take into account the effectiveness of multi-target generation, thus leading to inconvenience.","To make the generative model learn how to specify multiple targets, we propose a conditional generative network $\\mathcal {G}_{\\theta }$ that effectively crafts multi-target adversarial perturbations by modeling class-conditional distribution.","Different from previous single-target methods [31], [33], the target label $c$ is regarded as a discrete variable rather than a constant.","As illustrated in Fig.","REF , our model contains a conditional generator $\\mathcal {G}_{\\theta }$ and a classification network $\\mathcal {F}_{\\phi }$ parameterized by $\\theta $ and $\\phi $ , respectively.","The conditional generative model $\\mathcal {G}_{\\theta }: (\\mathcal {X}_{s}, \\mathcal {C}) \\rightarrow \\mathcal {P}$ learns a perturbation $\\mathbf {\\delta } = \\mathcal {G}_{\\theta }(\\mathbf {x}_{s}, c) \\in \\mathcal {P} \\subset \\mathbb {R}^{d}$ on the training data.","The output $\\mathbf {\\delta }$ of $\\mathcal {G}_{\\theta }$ is projected within the fixed $\\ell _{\\infty }$ norm, thus generating the perturbed image $\\mathbf {x}_{s}^{*} = \\mathbf {x}_{s} + \\mathbf {\\delta }$ .","Given a pretrained network $\\mathcal {F}_{\\phi }$ parameterized by $\\phi $ , we propose to generate the targeted adversarial perturbations by solving $\\begin{split}\\min \\limits _{\\theta }&\\mathbb {E}_{(\\mathbf {x}_{s}\\sim \\mathcal {X}_{s}, c\\sim \\mathcal {C})}[{\\mathbb {CE}\\big (\\mathcal {F}_{\\phi }(\\mathcal {G}_{\\theta }(\\mathbf {x}_{s}, c) + \\mathbf {x}_{s}\\big ), c)}], \\\\& \\text{s.t. }","\\Vert \\mathcal {G}_{\\theta }(\\mathbf {x}_{s}, c) \\Vert _{\\infty } \\le \\epsilon .\\end{split}$ By solving problem (REF ), we can obtain a targeted conditional generator by minimizing the loss of specific target class in the unlabeled training dataset.","Note that we only optimize the parameter $\\theta $ of the generator $\\mathcal {G}_{\\theta }$ using the training data $\\mathcal {X}_{s}$ , then the targeted adversarial example $\\mathbf {x}_{t}^{*}$ can be crafted by $\\mathbf {x}_{t}^{*} = \\mathbf {x}_{t} + \\mathcal {G}_{\\theta }(\\mathbf {x}_{t}, c) $ for any given image $\\mathbf {x}_{t}$ in the test data $\\mathcal {X}_{t}$ , which only requires an inference for this targeted image $\\mathbf {x}_{t}$ .","We experimentally find that the objective (REF ) can enforce the transferability for the generated perturbation $\\mathbf {\\delta }$ .","A reasonable explanation is that $\\mathbf {\\delta }$ can arise as a result of strong and well-generalizing semantic pattern inherent to the target class, which is robust to the influence of any training data.","In Sec.","REF , we illustrate and corroborate our claim by directly feeding scaled adversarial perturbationsThe perturbation is linearly scaled from [-$\\epsilon $ , $\\epsilon $ ] to [0, 255].","from different methods into the classifier.","Indeed, we find that our semantic pattern can be classified as the target class with a high degree of confidence while the perturbation from MIM [9] performs like the noise, meanwhile the scaled semantic pattern performs well transferability in different black-box models."],["Network Architecture","We now present the details of the conditional generative model for targeted attack, as illustrated in Fig.","REF .","Specifically, we design a mapping network to generate a target-specific vector in the implicit space of each target and train conditional generator $\\mathcal {G}_{\\theta }$ to reflect this vector by constantly misleading the classifier $\\mathcal {F}_{\\phi }$ .","Mapping network.","Given an one-hot class encoding $\\mathbb {1}_{c} \\in \\mathbb {R}^{K}$ from target class $c$ , the mapping network aims to generate the targeted latent vector $\\mathbf {w} = \\mathcal {W}(\\mathbb {1}_{c})$ , where $\\mathbf {w} \\in \\mathbb {R}^{M}$ and $\\mathcal {W}(\\cdot )$ consists of a multi-layer perceptron (MLP) and a normalization layer, which can construct diverse targeted vectors $\\mathbf {w}$ for a given target class $c$ .","Thus $\\mathcal {W}$ is capable of learning effective targeted latent vectors by randomly sampling different classes $c \\in \\mathcal {C}$ in training phase.","Generator.","Given an input image $\\mathbf {x}_{s}$ , the encoder first calculates the feature map $\\mathbf {F} \\in \\mathbb {R}^{N\\times H \\times W}$ , where $N$ , $H$ and $W$ refer to the number of channels, height and width of the feature map, respectively.","The target latent vector $\\mathbf {w}$ , derived from the mapping network $\\mathcal {W}$ by introducing a specific target class $c$ , is expanded along height and width directions to obtain the label feature map $\\mathbf {w}_s\\in \\mathbb {R}^{M\\times H \\times W}$ .","Then the above two feature maps are concatenated along the channels to obtain $\\mathbf {F}^{\\prime } \\in \\mathbb {R}^{(N+M)\\times H \\times W}$ .","The obtained mixed feature map is then fed to the subsequent network.","Therefore, our generator $\\mathcal {G}_{\\theta }$ translates an input image $\\mathbf {x}_{s}$ and latent target vector $\\mathbf {w}$ into an output image $\\mathcal {G}_{\\theta }(\\mathbf {x}_{s}, \\mathbf {w})$ , which enables $\\mathcal {G}_{\\theta }$ to synthesize adversarial images of a series of targets.","For the output of feature map $\\mathbf {f} \\in \\mathbb {R}^{d}$ in the decoder, we adopt a smooth projection $P(\\cdot )$ to perform a change of variables over $\\mathbf {f}$ rather than directly minimizing its $\\ell _2$ norm as [14] or clipping values outside the fixed norm [31], which can be denoted as $\\mathbf {\\delta } = P(\\mathbf {f}) = \\epsilon \\cdot \\mathrm {tanh}(\\mathbf {f}),$ where $\\epsilon $ is the strength of perturbation.","Since $-1 \\le \\mathrm {tanh}(\\mathbf {f}) \\le 1$ , $\\mathbf {\\delta }$ can automatically satisfy the $\\ell _{\\infty }$ -ball bound with perturbation budget $\\epsilon $ .","This transformation can be regarded as a better smoothing of gradient than directly clipping values outside the fixed norm, which is also instrumental for $\\mathcal {G}_{\\theta }$ to probe and learn the targeted semantic knowledge from $\\mathcal {F}_{\\phi }$ .","Training objectives.","The training objectives seek to minimize the classification error on the perturbed image of the generator as $\\vspace{-0.42502pt}\\theta ^{*} \\leftarrow \\operatornamewithlimits{arg\\,min}_{\\theta }{\\mathbb {CE} \\Big (F_{\\phi }\\big (\\mathbf {x}_{s} + \\mathcal {G}_{\\theta }(\\mathbf {x}_{s}, \\mathcal {W}(\\mathbb {1}_{c}))\\big ), c\\Big )},$ which adopts an end-to-end training paradigm with the goal of generating adversarial images to mislead the classifier the target label, and $\\mathbb {CE}$ is the cross entropy loss.","Previous studies attempt different classification losses in their works [52], [31], and we found that cross-entropy loss works well in our settings.","The detailed optimization procedure is summarized in Algorithm REF .","[t] Training Algorithm for the Conditional Generative Attack Training Data $\\mathcal {D}_{s}$ ; a generative network $\\mathcal {G}_{\\theta }$ ; a classification network $\\mathcal {F}_{\\phi }$ ; a mapping network $\\mathcal {W}$ .","Adversarial perturbations $\\mathbf {\\theta }$ .","iter in MaxIterations T Randomly sample $B$ images $\\lbrace \\mathbf {x}_{s_{i}}\\rbrace _{i=1}^{B}$ Randomly sample $B$ target classes $\\lbrace {c}_{i}\\rbrace _{i=1}^{B}$ Forward pass ${c}_{i}$ into $\\mathcal {W}$ to compute the targeted latent vectors $\\mathbf {w}_{i}$ Obtain the perturbed images by $\\mathbf {x}_{s_{i}}^{*} = \\epsilon \\cdot \\mathrm {tanh}(\\mathcal {G}(\\mathbf {x}_{s_{i}}, \\mathbf {w}_{i})) + \\mathbf {x}_{s_{i}}$ Forward pass $\\mathbf {x}_{s_{i}}^{*}$ to $\\mathcal {F}_{\\phi }$ and compute loss in Eq.","(REF ) Backward pass and update the $\\mathcal {G}_{\\theta }$ Table: Transferability comparison for multi-target attacks on ImageNet NeurIPS validation set (1k images) with the perturbation budget of ℓ ∞ ≤16\\ell _{\\infty } \\le 16.","The results are averaged on 8 different target classes.","Note that CD-AP † ^{\\dagger } indicates that training 8 models can obtain results, while our method only train one conditional generative model.","* indicates white-box attacks."],["Hierarchical Partition for Classes"," While handling plenty of classes, the effectiveness of a conditional generative model will decrease as illustrated in Fig.","REF , because the representative capacity is limited with a single generator.","Therefore, we propose to divide all classes into a feasible number of subsets to train models when the class number $K$ is large, , 1,000 classes in ImageNet, with the aim of seeking the effectiveness of targeted black-box attack.","To obtain a good partition, we introduce a representative target class space, which is nearly equivalent to the original class space $\\mathcal {C}$ .","Specifically, we utilize the weights $\\phi _{cls} \\in \\mathbb {R}^{D\\times C}$ in the classifier layer for the classification network $\\mathcal {F}_{\\phi }$ .","Therefore, $\\phi _{cls}$ can be regarded as the alternative class space since the weight vector $\\mathbf {d}_{c} \\in \\mathbb {R}^{D}$ from $\\phi _{cls}$ can represent a class center of the feature embeddings of input images with same class $c$ .","Note that once those subsets with closer metric distance (, larger cosine similarity) in the target class space $\\phi _{cls}$ are regarded as conditional inputs of generative network, they obtain worse loss convergence and transferability than diverse them due to mutual influence among these input conditions, as illustrated in Fig.","REF .","Thus we focus on selecting target classes that do not tend to overlap or be close to each other as accessible subsets.","To capture more diverse examples in a given sampling space, we adopt K-determinantal point processes (DPP) [21], [20] to achieve a hierarchical partition, which can take advantage of the diversity property among subsets by assigning subset probabilities proportional to determinants of a kernel matrix.","First, we compute the RBF kernel matrix $L$ of $\\phi _{cls}$ and eigendecomposition of $L$ , and a random subset $V$ of the eigenvectors is chosen by regarding the eigenvalues as sampling probability.","Second, we select a new class $c_{i}$ to add to the set and update $V$ in a manner that de-emphaseizes items similar to the one selected.","Each successive point is selected and $V$ is updated by Gram-Schmidt orthogonalization, and the distribution shifts to avoid points near those already chosen.","The details are presented in Appendix A."],["Experiments"," In this section, we present extensive experiments to demonstrate the effectiveness of proposed method for targeted black-box attacks."],["Experimental Settings"," Datasets.","We consider the following datasets for training, including a widely used object detection dataset MS-COCO [26] and ImageNet training set [6].","We focus on standard and comprehensive testing settings, thus inference is performed on ImageNet validation set (50k samples), a subset (5k) of ImageNet proposed by [24] and ImageNet-NeurIPS (1k) proposed by [32].","Figure: Comparison of different projection functions and modes of Gaussian Smoothing.","Results are reported with Inc-v3 networkon ImageNet NeurIPS validation set.Networks.","We consider some naturally trained networks, , Inception-v3 (Inc-v3) [41], Inception-v4 (Inc-v4) [39], Resnet-v2-152 (Res-152) [15] and Inception-Resnet-v2 (IncRes-v2) [39], which are widely used for evaluating transferability.","Besides, we supplement DenseNet-201 (Dense-201) [16], GoogleNet [40] and VGG-16 [37] to fully evaluate the transferability.","Adversarially trained networks [43] are also selected to evaluate the performance, , ens3-adv-Inception-v3 ($\\textrm {Inc-v3}_\\textrm {ens3}$ ), ens4-adv-Inception-v3 ($\\textrm {Inc-v3}_\\textrm {ens4}$ ) and ens-adv-Inception-ResNet-v2 ($\\textrm {IncRes-v2}_\\textrm {ens}$ ).","All networks are publicly availablehttps://github.com/tensorflow/models/tree/master/research/slim https://github.com/tensorflow/models/tree/master/research/adv_imagenet_models https://github.com/pytorch/vision/tree/master/torchvision/models.","Implementation details.","We choose the same ResNet autoencoder architecture in [19], [31] as the basic generator networks, which consists of downsampling, residual and upsampling layers.","We initialize the learning rate as 2e-5 and set the mini-batch size as 32.","Smoothing mechanism is proposed to improve the transferability against adversarially trained models [10].","Instead of adopting smoothing for generated perturbation while the training is completed as CD-AP [31], we introduce adaptive Gaussian smoothing kernel to compute $\\mathbf {\\delta }$ from Eq.","(REF ) in the training phase, named adaptive Gaussian smoothing, with the focus of making generated results obtain adaptive ability.","More implementation details and discussion with other networks (, BigGAN [3]) are illustrated in Appendix B.","Table: Transferability results for targeted attacks on ImageNet validation set (50k images) with the perturbation budget of ℓ ∞ ≤10\\ell _{\\infty } \\le 10.","The attack is performed in same setting with the target class 'sea lion' and the training dataset MS-COCO."],["Transferability Evaluation","We consider 8 different target classes from [52] to form the multi-target black-box attack testing protocol with 8k times in 1k ImageNet NeurIPS set.","Efficiency of multi-target black-box attack.","Among comparable methods, instance-specific methods, , MIM, TI-DIM, DIM and TI-DIM, require iterative mechanism with $M$ steps by computing gradients to obtain adversarial examples.","Given the cost $t_{C}^{FP}$ and $t_{C}^{BP}$ of forward and backward passing the classifier, computing cost $T^{IS}$ of single data can be defined as $T^{IS} = t_{C}^{FP} * M + t_{C}^{BP} * M$ in Tab.","REF .","Instance-agnostic methods only require the inference cost from the trained generator as $T^{IA} = t_{G}^{FP}$ , thus possessing the priority for those attack scenarios within limited time.","However, instance-specific methods require to train 8 models to obtain all predictions from 8 different classes.","Due to time-consuming training and more storage, we only reproduce previous state-of-the-art generative method CD-AP [31] as a baseline, which already fully demonstrate the superior performance than other generative methods such as GAP [33] in their work.","As a comparison, our conditional generative method only trains one model to inference the results and outperforms in the aspect of efficiency.","Effectiveness of multi-target black-box attack.","Tab.","REF shows the transferability comparison of different methods on both naturally and adversarially trained models.","The success rate of instance-specific attacks are lower than 3%, possibly explained by the data-point overfitting that makes it hard to transfer another model.","The instance-agnostic attack CD-AP obtains acceptable performance, yet inferior to proposed method w.r.t black-box transferability.","The primary reason for such a trend lies in some distinctions as 1) direct clip projection in CD-AP and our smooth projection in Eq.","(REF ) and 2) their Gaussian Smoothing and our adaptive Gaussian Smoothing, as described in Sec.","REF and Appendix B.","Fig.","REF empirically shows the comparison results of single-target black-box attacks based on the CD-AP framework.","Thus proposed conditional generative method can be a reliable baseline w.r.t targeted black-box attacks, regarding both effectiveness and efficiency.","Results of single-target black-box attack.","Recent related work [52] has tried to solve the single-target transferable problem based on universal adversarial perturbation, and report an excellent single-target black-box performance.","We obtain single-target degraded version of our model by specifying an input target label during the training process.","We show the performance of black-box attack in terms of targeted attack success rate in Tab.","REF .","The promising results show generative semantic pattern from our method benefits black-box transferability than universal adversarial perturbations.","Some other instance-agnostic adversarial methods, , UAP [29], GAP [33] and RHP [24], have tendency towards the untargeted black-box problem.","Despite this, we follow the corresponding untargeted setting and compare different methods in Appendix C. Our method is steadily improved under untargeted black-box manner.","Table: Transferability comparison on NeurIPS 2017 competition with the perturbation budget of ℓ ∞ ≤16\\ell _{\\infty } \\le 16.","White-box substitute model is Inc-v3 for all attacks, following the standard protocol with 1,000 stochastic target classes.Table: The success rate of black-box impersonation attacks on face verification with the perturbation budget of ℓ ∞ ≤16\\ell _{\\infty } \\le 16.","ArcFace is chosen as white-box model.Figure: Generative examples of adversarial images with perturbation budget of ℓ ∞ ≤16\\ell _{\\infty }\\le 16.","We separately adopt the ImageNet and MS-COCO dataset as the training dataset to implement the generation of targeted perturbations.","Our method can generate semantic pattern independent of training dataset.Figure: Asr vs. numbers of conditional targets curve against Inc-v3 and VGG-16 models."],["Effectiveness on NeurIPS 2017 Competition"," To illustrate the effectiveness of our proposed attack methods in practical 1,000 classification, we here follow the official setting from NeurIPS 2017 adversarial competition [23] for testing targeted black-box transferability.","Considering limited resource, previous instance-agnostic attacks are not required as comparable methods due to training 1,000 models, thus we focus on the instance-specific attacks, which are official top attack methods in NeurIPS 2017 adversarial competition.","Our hierarchical partition mechanism can make conditional generative networks be capable of specifying any target class via a feasible number of models for the scalability.","We consider 20 models, with each specifying 50 diverse classes from k-DPP hierarchical partition in this setting, to implement targeted attack by only once inference for each target image.","Our method outperforms all other baseline methods in Tab.","REF .","The results demonstrate that this method can be reliable in practical targeted attacks, regarding both effectiveness and efficiency."],["Effectiveness on Realistic Face Recognition"," Adversarial perturbations added to original face have ability to evade being recognized or impersonate another individual [36], [50].","In this section, we consider the transferability of impersonation attack to further illustrate the generalization of our method, which is also corresponding to targeted attack in image classification.","Dataset and models.","We conduct the experiments on Labeled Faces in the Wild (LFW) [17] and introduce two test protocols.","For Protocol I defined as single-target impersonation attack, we choose 1 target identity and 1k source face images belonging with different identities from LFW as the attackers, thus forming 1k pairs.","For Protocol II named multi-target impersonation attack, 5 target identities and 1k source face images are selected to form 1k attack pairs, meaning that we need to implement 5k attacks.","We involve some excellent face recognition models for conducting black-box testing, including Sphereface [27], CosFace [44], FaceNet [35] and MobileFace [4].","These models lie in different model architectures and training objectives.","In all experiments, we only use one model ArcFace [7] as substitute model to craft adversarial samples, and test attack performance against other unknown models.","Figure: Comparison of loss convergence and transferability between diverse and close conditional subset with 10 target classes.Figure: Plots of logit vectors from the adversarial image L img L_{img} and scaled crafted perturbation L adv L_{adv} of MIM and proposed generative method, with their respective PCC values.Evaluation metrics.","We first compute the optimal threshold of every face recognition models from LFW dataset by following standard protocols.","If the similarity of a pair of images exceeds the threshold, we regard them as same identity, otherwise different identities.","Black-box attack results.","We adjust the optimization object function to adapt face recognition for chosen attack methods (detailed in Appendix C), and report the success rate of black-box impersonation attacks in Tab.","REF , which illustrates that our method can achieve nearly two times of the success rates than DIM in Protocol I and Protocol II.","The results indicate that our method is superior to other methods not only in image classification."],["Comparison Study about Target Classes"," We conduct an extensive study to investigate two key points about target classes.","Different numbers of target classes.","We conduct effectiveness for different numbers of target classes in Fig.","REF .","It can be seen that the results perform well within a feasible number of targets, whereas to a certain extent effectiveness tend to decay.","Therefore, the effectiveness of conditional generative networks is influenced by the number of conditional classes, due to the representative capacity of single generator.","We aim to divide all classes into a feasible number of set while handling plenty of classes.","Comparison of different multi-target conditions.","We select closer conditional classes with larger cosine similarity in the target class space $\\phi _{cls}$ and diverse conditional classes from k-DPP method.","In Fig.","REF , closer conditional classes have worse loss convergence and transferability than diverse them due to mutual influence among conditions."],["More Analyses"," Targeted adversarial samples from proposed generative method can produce semantic pattern inherent to the target class in Fig.","REF .","Why does generative semantic pattern work?","First, generative methods can produce strong targeted semantic pattern that is robust to the influence of data, which is obtained by minimizing the loss of specific target class in the training phase.","To corroborate our claim, we directly feed scaled crafted perturbations by instance-specific attack MIM and our generative method into the classifier.","Indeed, we find that our generative perturbation is considered as target class with a high degree of confidence whereas the perturbation from MIM performs like the noise, as shown in Fig.","REF .","We plot the logit relationship from scaled crafted perturbation and adversarial image in Fig.","REF , which is also consistent with previous claim.","Second, the generated adversarial semantic pattern achieves well-generalizing performance among the different models.","We feed 1k images from ImageNet test set into the generator trained by Inc-v3 model to obtain 1k semantic patterns, which are scaled to image pixel space and then fed into different classifiers.","We compute the mean confidence of $\\mathbf {0.46}$ for Dense-201, $\\mathbf {0.44}$ for Inc-v4, and $\\mathbf {0.35}$ for Res-152, whereas the perturbation from MIM is lower than $0.01$ .","The results show that our scaled semantic pattern can directly achieve well-generalizing performance among models, possibly explained by utilizing similar feature knowledge from the same class on different classifiers trained on same training data distribution.","Thus similar pattern can be instrumental for transferability among models."],["Discussion and Conclusion"," Transferability of targeted black-box attack is simultaneously affected by data and model.","Therefore, instance-specific methods easily overfit the data point and white-box model, resulting in weak transferability.","As a comparison, proposed generative method with powerful learning capacity reduces the dependency for data point by adopting the unlabeled training data, thus enabling the model to learn semantic pattern and improve the transferability of targeted black-box attack.","Extensive experiments demonstrate that proposed generative method can significantly improve the success rates of targeted black-box attacks against naturally and adversarial trained models.","Th us we hope that crafting C-GSP can be regarded as a new reliable baseline method in terms of targeted black-box attacks."],["Sampling Algorithm","We summarize the overall sampling procedure based on k-DPP [20] in Algorithm .","Compute the RBF kernel matrix $L$ of $\\phi _{cls}$ and eigendecomposition of $L$ .","A random subset $V$ of the eigenvectors is chosen by regarding the eigenvalues as sampling probability.","Select a new class $c_{i}$ to add to the set and update $V$ in a manner that de-emphaseizes items similar to the one selected.","Update $V$ by Gram-Schmidt orthogonalization, and the distribution shifts to avoid points near those already chosen.","By performing the Algorithm , we can obtain a subset with $k$ size.","Thus while handling the conditional classes with K, we can hierarchically adopt this algorithm to get the final $K/k$ subsets, which are regarded as conditional variables of generative models to craft adversarial examples."],["Some Implementation Details","The study of smoothing mechanism.","Smoothing mechanism has been proved to improve the transferability against adversarially trained models.","CD-AP [31] uses direct clip projection to have a fixed norm $\\epsilon $ , and adopts smoothing for generated perturbation while the generator $\\mathcal {G}$ is trained, , ${\\begin{array}{c}\\textbf {Train: }\\mathbf {x}_{s_{i}}^{*} = \\mathrm {Clip}_{\\epsilon }(\\mathcal {G}(\\mathbf {x}_{s_{i}}),\\\\\\textbf {Test: }\\mathbf {x}_{s_{i}}^{*} = \\mathrm {W} * \\mathrm {Clip}_{\\epsilon }(\\mathcal {G}(\\mathbf {x}_{s_{i}}),\\end{array}}$ where $\\mathrm {W}$ indicates Gaussian smoothing of kernel size of 3, $*$ indicates the convolution operation, and Clip$_{\\epsilon }$ means clipping values outside the fixed norm $\\epsilon $ .","As a comparison, we introduce adaptive Gaussian smoothing kernel to compute adversarial images $\\mathbf {x}_{s_{i}}^{*}$ from in the training phase, named adaptive Gaussian smoothing as $\\textbf {Train \\& Test: } \\mathbf {x}_{s_{i}}^{*} = \\epsilon \\cdot \\mathrm {W} * \\mathrm {tanh}(\\mathcal {G}(\\mathbf {x}_{s_{i}}) + \\mathbf {x}_{s_{i}},$ which can make generated results obtain adaptive ability in the training phase.","We perform training in ImageNet dataset to report all results including comparable baselines.","[t] Sampling Algorithm by kDPP Weight Vector $\\mathbf {\\theta }_{cls}$ ; Subset size $k$ .","A subset $C$ .","Compute RBF kernel matrix $L$ of $\\mathbf {\\theta }_{cls}$ Compute eigenvector/value $\\lbrace v_{n}, \\lambda _{n}\\rbrace _{n=1}^N$ pairs of $L$ // Phase I: $J\\leftarrow \\phi $ , $e_{k}\\left(\\lambda _{1}, \\ldots , \\lambda _{N}\\right)=\\sum _{|J|=k} \\prod _{n \\in J} \\lambda _{n}$ n = N, ..., 1 $u\\sim U[0,1] < \\lambda _{n} \\frac{e^{n-1}_{k-1}}{e^{n}_{k}}$ and $k>0$ $J\\leftarrow J \\cup \\lbrace n\\rbrace $ ; $k\\leftarrow k-1$ // Phase II: $V \\leftarrow \\left\\lbrace v_{n}\\right\\rbrace _{n \\in J}, Y\\leftarrow \\phi $ $|V| > 0$ Select $c_i$ from $\\mathcal {C}$ with $\\operatorname{P}\\left(c_{i}\\right)=\\frac{1}{|V|} \\sum _{v \\in V}\\left(v^{\\top } e_{i}\\right)^{2}$ $C \\leftarrow C \\cup \\left\\lbrace c_{i}\\right\\rbrace $ $V \\leftarrow V_{\\perp },$ an orthonormal basis for the subspace of $V$ orthogonal to $e_{i}$ Figure: Some examples of adversarial images with perturbation budget of ℓ ∞ ≤16\\ell _{\\infty }\\le 16.","We separately adopt the ImageNet, MS-COCO and Comics dataset as the training dataset to implement the generation of targeted perturbations.Figure: Some examples of adversarial images with perturbation budget of ℓ ∞ ≤16\\ell _{\\infty }\\le 16.","We separately adopt the ImageNet, MS-COCO and Comics dataset as the training dataset to implement the generation of targeted perturbations.Network architecture of generator.","We adopt the same autoencoder architecture in [31] as the basic generator networks.","Besides, we also explore BigGAN [3] as conditional generator network.","An very weak testing performance is obtained even in the white-box attack scenario, possibly explained by the weak diversity of latent variable with the Gaussian distribution from BigGAN in the training phase, whereas autoencoder can take full advantage of large-scale training dataset, , ImageNet.","Furthermore, we also train the autoencoder with Gaussian noise as the training dataset and obtain similar inferior performance in the white-box attack scenario, indicating that a large-scale training dataset is very significant for generating transferable targeted adversarial examples.","Some details.","In our experiments of testing time, we apply NVIDIA 1080Ti GPUs.","Instance-specific methods, , MIM, TI-DIM, DIM and TI-DIM, adopt iterative steps $M = 20$ and follow their reported hyperparameters."],["Additional Experimental Results","Results on different datasets.","We craft adversarial examples on different datasets, including ImageNet training set, MS-COCO and Comics dataset [2], which consist of 1.2M, 82k and 50K images, respectively.","MS-COCO dataset can be applied to large-scale object detection and segmentation, and those images from Comics dataset are regarded as other domains different from normal ones in ImageNet.","Despite this diverse training types, we still find the common property of crafted adversarial examples by our method.","Specifically, we craft some examples of adversarial images with perturbation budget of $\\ell _{\\infty }\\le 16$ , and separately adopt the ImageNet, MS-COCO and Comics dataset as the training dataset to implement the generation of targeted perturbations.","As illustrated in Fig.","REF , we produce semantic pattern independent of any training dataset.","Table: Comparison results of targeted black-box attacks on different datasets.","Incv3 is the substitute model.We also report the success rate of targeted black-box attack, as shown in Tab.","REF .","We experimentally find that semantic pattern derived from ImageNet dataset achieves better performance of black-box performance, possibly explained by instructional effectiveness from more diverse data in ImageNet dataset.","Results of untargeted black-box attack.","We evaluate our method and other generative methods including UAP [29], GAP [33] and RHP [24].","Untargeted transferability from naturally trained models to adversarially trained models occurs due to differences in model sources, data types and other factors, thus enabling challenging comparison.","As illustrated in Tab.","REF , we report the untargeted attacks increase in error rate of adversarial and clean images to evaluate different methods.","Our method is steadily improved in different black-box models under untargeted black-box manner.","Table: Transferability results for untargeted attacks increase in error rate after attack on subset of ImageNet (5k images) with the perturbation budget of ℓ ∞ ≤16/32\\ell _{\\infty } \\le 16/32."],["Impersonation Attack of Face Recognition","We list attack methods of face recognition as follows.","Given an input $\\mathbf {x}$ and an image $\\mathbf {x}^r$ belonging with another identity, an attack method can generate an adversarial example $\\mathbf {x}^{adv}$ with perturbation budget $\\epsilon $ under the $\\ell _p$ norm ($\\Vert \\mathbf {x}^{adv} - \\mathbf {x}\\Vert _p \\le \\epsilon $ ).","Therefore, impersonation attack aims to perform this objective of $\\mathcal {C} (\\mathbf {x}^{adv}, \\mathbf {x}^{r}) = \\mathbb {I} (\\mathcal {D}_f (\\mathbf {x}^{adv}, \\mathbf {x}^{r}) < \\delta ),$ where $\\mathbb {I}$ is the indicator function, $\\delta $ is a threshold, and $\\mathcal {D}_f(\\mathbf {x}^{adv}, \\mathbf {x}^{r}) = \\Vert f(\\mathbf {x}^{adv}) - f(\\mathbf {x}^{r})\\Vert _2^2.$ Basic Iterative Method (BIM) [22] extends FGSM by iteratively taking multiple small gradient updates as $\\mathbf {x}_{t+1}^{adv} = \\mathrm {clip}_{\\mathbf {x},\\epsilon } \\big (\\mathbf {x}_t^{adv} - \\alpha \\cdot \\mathrm {sign}(\\nabla _{\\mathbf {x}}\\mathcal {D}_f(\\mathbf {x}_t^{adv},\\mathbf {x}^{r}))\\big ),$ where $\\mathrm {clip}_{\\mathbf {x},\\epsilon }$ projects the adversarial example to satisfy the $\\ell _{\\infty }$ constrain and $\\alpha $ is the step size.","Momentum Iterative Method (MIM) [9] introduces a momentum term into BIM for improving the transferability of adversarial examples as ${\\begin{array}{c}\\mathbf {g}_{t+1} = \\mu \\cdot \\mathbf {g}_t + \\frac{\\nabla _{\\mathbf {x}}\\mathcal {D}_f(\\mathbf {x}_t^{adv},\\mathbf {x}^{r})}{\\Vert \\nabla _{\\mathbf {x}}\\mathcal {D}_f(\\mathbf {x}_t^{adv},\\mathbf {x}^{r})\\Vert _1}; \\\\ \\mathbf {x}^{adv}_{t+1}=\\mathrm {clip}_{\\mathbf {x},\\epsilon }(\\mathbf {x}^{adv}_t - \\alpha \\cdot \\mathrm {sign}(\\mathbf {g}_{t+1})).\\end{array}}$ The training objectives of our generative method seek to minimize the classification error on the perturbed image of the generator as $\\vspace{-0.42502pt}\\min _{\\theta }\\mathbb {E}_{(\\mathbf {x}\\sim \\mathcal {X}, c\\sim \\mathcal {C})}[{\\mathcal {D}_f \\big (\\mathbf {x} + \\mathcal {G}_{\\theta }(\\mathbf {x}, c), \\mathbf {x}^{r}_{c}\\big )}],$ where $\\mathbf {x}^{r}_{c}$ refers to $\\mathbf {x}^{r}$ with the corresponding identity $c$ .","In the training phase, we randomly select $1,000$ identities from CASIA-WebFace [51] as training dataset to craft adversarial examples.","Therefore, our method can be applied not only in image classification."],["More Examples","We also show more semantic patterns from different target models, as illustrated in Fig.","REF ."]],"string":"[\n [\n \"Boosting Transferability of Targeted Adversarial Examples via\\n Hierarchical Generative Networks\"\n ],\n [\n \"Abstract Transfer-based adversarial attacks can evaluate model robustness in the black-box setting.\",\n \"Several methods have demonstrated impressive untargeted transferability, however, it is still challenging to efficiently produce targeted transferability.\",\n \"To this end, we develop a simple yet effective framework to craft targeted transfer-based adversarial examples, applying a hierarchical generative network.\",\n \"In particular, we contribute to amortized designs that well adapt to multi-class targeted attacks.\",\n \"Extensive experiments on ImageNet show that our method improves the success rates of targeted black-box attacks by a significant margin over the existing methods -- it reaches an average success rate of 29.1\\\\% against six diverse models based only on one substitute white-box model, which significantly outperforms the state-of-the-art gradient-based attack methods.\",\n \"Moreover, the proposed method is also more efficient beyond an order of magnitude than gradient-based methods.\"\n ],\n [\n \"Introduction\",\n \"Recent progress in adversarial machine learning demonstrates that deep neural networks (DNNs) are highly vulnerable to adversarial examples [42], [13], which are maliciously generated to mislead a model to produce incorrect predictions.\",\n \"It has been demonstrated that adversarial examples possess an intriguing property of transferability [28], [45], [18], [5], [49] — the adversarial examples crafted for a white-box model can also mislead other unknown models, making black-box attacks feasible.\",\n \"The threats of adversarial examples have raised concerns in numerous security-sensitive applications, such as autonomous driving [11] and face recognition [36], [48].\",\n \"Figure: The targeted adversarial examples crafted by MIM and the conditional generative semantic pattern (C-GSP) crafted by our method for the Inception-v3 model given the target class Viaduct with perturbation budget 16 under the ℓ ∞ \\\\ell _{\\\\infty } norm constraint.\",\n \"We also show the predicted labels and probabilities of these images by the black-box model DenseNet-201 .Tremendous efforts have been made to develop more effective black-box attack methods based on transferability since they can serve as an important surrogate to evaluate the model robustness in real-world scenarios [28], [9].\",\n \"The current methods have achieved impressive performance of untargeted black-box attacks, intending to cause misclassification of the black-box models.\",\n \"However, the targeted black-box attacks, aiming at misleading the black-box models by outputting the adversary-desired target class, perform unsatisfactorily [8] and have not been extensively explored [52].\",\n \"The difficulty of fooling a black-box model by the existing targeted adversarial attacks could result in an over-estimation of model robustness under the challenging targeted black-box attack setting.\",\n \"Existing efforts on targeted black-box attacks can be categorized as instance-specific and instance-agnostic attacks.\",\n \"Specifically, the instance-specific attack methods [12], [30], [22], [9] craft adversarial examples by performing gradient updates iteratively, which achieve unsatisfactory performance for targeted black-box attacks due to easy overfitting to a white-box model [9], [46].\",\n \"On the other hand, the instance-agnostic attack methods learn a universal adversarial perturbation [52] or a universal function [38], [31] on the data distribution independent of specific instances.\",\n \"They can promote more general and transferable adversarial examples since the universal perturbation or function can alleviate the data-specific overfitting problem by training on an unlabeled dataset.\",\n \"CD-AP [31], as one of the effective instance-agnostic methods, adopts a generative model as a universal function to obtain an acceptable performance when facing one specified target class.\",\n \"However, CD-AP needs to learn a generative model for each target class while performing multi-target attack [14], , crafting adversarial examples targeted at different classes.\",\n \"Thus it is not scalable to the increasing number of targets such as hundreds of classes, limiting practical efficiency.\",\n \"To address the aforementioned issues and develop a targeted black-box attack in the practical scenario, in this paper we propose a conditional generative model as the universal adversarial function to craft adversarial perturbations.\",\n \"Thus we can craft adversarial perturbations targeted at different classes, using a single model backbone with different class embeddings.\",\n \"The proposed generative method is simple yet practical to obtain superior performance of targeted black-box attacks, meanwhile with two technical improvements including smooth projection mechanism that better helps the generator to probe targeted semantic knowledge from the classifier and adaptive Gaussian Smoothing with the focus of making generated results obtain adaptive ability against adversarially trained models.\",\n \"The previous CD-AP requires costly training $N$ models while performing a multi-target attack with $N$ classes.\",\n \"However, ours only trains one model and reaches an average success rate of 51.1% against six naturally trained models and 36.4% against three adversarially trained models based only on one substitute white-box model in NeurIPS ImageNet dataset, which outperforms CD-AP by a large margin of 6.0% and 31.3%, respectively.\",\n \"While handling plenty of classes (, 1,000 classes in ImageNet), the effectiveness of generating targeted adversarial examples will be affected by a single generative model due to the difficulty of loss convergence in adversarial learning [47], [1].\",\n \"Thus we train a feasible number of models (, 10$\\\\sim $ 20 models on ImageNet) to further promote the effectiveness beyond the single model backbone.\",\n \"Specifically, each model is learned from a subset of classes specified by a designed hierarchical partition mechanism by considering the diversity property among subsets, for seeking a balance between effectiveness and scalability.\",\n \"It reaches an average success rate of 29.6% against six different models, outperforming the state-of-the-art methods with an average success rate of $<$ 2% by a large margin, based only on one substitute white-box model in the NeurIPS 2017 competition.\",\n \"Moreover, the proposed method achieves substantial speedup over gradient-based methods.\",\n \"Furthermore, these adversarial perturbations generated by the proposed Conditional Generative models can arise as a result of strong Semantic Pattern (C-GSP) as shown in Fig.\",\n \"REF .\",\n \"We experimentally find that the generated adversarial semantic pattern itself achieves well-generalizing performance among the different models and is robust to the influence of data in Sec.\",\n \"REF , which is very instructive for the understanding of adversarial examples.\",\n \"Our main contributions can be summarized as follows: We present a systematical study on targeted black-box attacks involving instance-specific and instance-agnostic methods in ImageNet dataset with plenty of classes and face recognition.\",\n \"We propose a simple yet practical conditional generative targeted attack method with a designed hierarchical partition mechanism, which can generate targeted adversarial examples without tuning the parameters.\",\n \"Extensive experiments demonstrate that our method significantly improves the success rates of targeted black-box attacks over the existing methods.\"\n ],\n [\n \"Related Work\",\n \"In this section, we review related work on adversarial attacks belonging to different types.\",\n \"Instance-specific attacks.\",\n \"Some recent works [12], [30] adopt gradient-based optimization methods to generate the data-dependent perturbations.\",\n \"MIM [9] introduces the momentum term into the iterative attack process to improve the black-box transferability.\",\n \"DIM [46] and TIM [10] aim to achieve the better transferability by input or gradient diversity.\",\n \"Instance-specific methods require iterative optimization for every instance, thus easily overfitting the current data point [9].\",\n \"In contrast, we improve the transferability simultaneously with the inference-time efficiency.\",\n \"Figure: An overview of our proposed generative method for crafting C-GSP, which includes modules of conditional generator and classifier.\",\n \"The generator integrates the image and conditional class vector from Map network into a hidden incorporation.\",\n \"Note that only the generator is trained in the whole pipeline to probe the target boundaries of the classifier.Instance-agnostic attacks.\",\n \"Compared with instance-specific attacks, instance-agnostic attacks belong to image-independent (universal) methods.\",\n \"The first pipeline is to learn a universal perturbation.\",\n \"UAP [29] proposes to fool a model by adding a learned universal noise vector.\",\n \"Another pipeline of attacks introduces learned generative models to craft adversarial examples.\",\n \"GAP [33] and AAA [34] craft adversarial perturbations in a similar way based on target data directly and compress impressions, respectively.\",\n \"Previous methods, including universal perturbation and function, require costly training the same number of models for multiple target classes.\",\n \"Our method is capable of simultaneously generating adversarial samples for specifying multiple targets with better attack performance.\",\n \"Multi-target attacks.\",\n \"Instance-specific attacks have the ability for specifying any target in the optimization phase.\",\n \"As elaborated in the introduction, these methods have degraded transferability and time-consuming iterative procedures.\",\n \"Related MAN [14] trains a generative model in the ImageNet under the constraint of $\\\\ell _{2}$ norm to explore the targeted attacks, which specifies all 1,000 categories from ImageNet for seeking extreme speed and storage.\",\n \"However, MAN does not demonstrate the effectiveness in terms of multi-target transferability against black-box models than previous instance-specific or instance-agnostic attacks, and the authors also claim that too many categories make it hard to transfer to another model.\",\n \"Recent UAE [52] reveals better single-target transferability by learning universal perturbation, whereas they require to train multiple times while specifying multiple targets.\",\n \"As a comparison, our method can generate adversarial samples for specifying multiple targets, meanwhile generated strong semantic patterns can outperform existing attacks by a significant margin.\"\n ],\n [\n \"Method\",\n \" In this section, we introduce a conditional generative model to learn a universal adversarial function, which can achieve effective multi-target black-box attacks.\",\n \"While handing plenty of classes, we design a hierarchical partition mechanism to make the generative model capable of specifying any target class under a feasible number of models, regarding both the effectiveness and scalability.\"\n ],\n [\n \"Problem Formulation\",\n \"We use $\\\\mathbf {x}_{s}$ to denote an input image belonging to an unlabeled training set $\\\\mathcal {X}_{s} \\\\subset \\\\mathbb {R}^{d}$ , and use $c \\\\in \\\\mathcal {C}$ to denote a specific target class.\",\n \"Let $\\\\mathcal {F}_{\\\\phi }: \\\\mathcal {X}_{s} \\\\rightarrow \\\\mathbb {R}^{K}$ denote a classification network that outputs a class probability vector with $K$ classes.\",\n \"To craft a targeted adversarial example $\\\\mathbf {x}_{s}^*$ from a real example $\\\\mathbf {x}_{s}$ , the targeted attack aims to fool the classifier $\\\\mathcal {F}_{\\\\phi }$ by outputting a specific label $c$ as $\\\\operatornamewithlimits{arg\\\\,max}_{i\\\\in \\\\mathcal {C}}{\\\\mathcal {F}_{\\\\phi }(\\\\mathbf {x}_{s}^{*})}_{i} = c$ , meanwhile the $\\\\ell _{\\\\infty }$ norm of the adversarial perturbation is required to no more than value $\\\\epsilon $ as $\\\\Vert \\\\mathbf {x}_{s}^{*}-\\\\mathbf {x}_{s}\\\\Vert _{\\\\infty }\\\\le \\\\epsilon $ .\",\n \"Although some generative methods [33], [31] can learn targeted adversarial perturbation, it does not take into account the effectiveness of multi-target generation, thus leading to inconvenience.\",\n \"To make the generative model learn how to specify multiple targets, we propose a conditional generative network $\\\\mathcal {G}_{\\\\theta }$ that effectively crafts multi-target adversarial perturbations by modeling class-conditional distribution.\",\n \"Different from previous single-target methods [31], [33], the target label $c$ is regarded as a discrete variable rather than a constant.\",\n \"As illustrated in Fig.\",\n \"REF , our model contains a conditional generator $\\\\mathcal {G}_{\\\\theta }$ and a classification network $\\\\mathcal {F}_{\\\\phi }$ parameterized by $\\\\theta $ and $\\\\phi $ , respectively.\",\n \"The conditional generative model $\\\\mathcal {G}_{\\\\theta }: (\\\\mathcal {X}_{s}, \\\\mathcal {C}) \\\\rightarrow \\\\mathcal {P}$ learns a perturbation $\\\\mathbf {\\\\delta } = \\\\mathcal {G}_{\\\\theta }(\\\\mathbf {x}_{s}, c) \\\\in \\\\mathcal {P} \\\\subset \\\\mathbb {R}^{d}$ on the training data.\",\n \"The output $\\\\mathbf {\\\\delta }$ of $\\\\mathcal {G}_{\\\\theta }$ is projected within the fixed $\\\\ell _{\\\\infty }$ norm, thus generating the perturbed image $\\\\mathbf {x}_{s}^{*} = \\\\mathbf {x}_{s} + \\\\mathbf {\\\\delta }$ .\",\n \"Given a pretrained network $\\\\mathcal {F}_{\\\\phi }$ parameterized by $\\\\phi $ , we propose to generate the targeted adversarial perturbations by solving $\\\\begin{split}\\\\min \\\\limits _{\\\\theta }&\\\\mathbb {E}_{(\\\\mathbf {x}_{s}\\\\sim \\\\mathcal {X}_{s}, c\\\\sim \\\\mathcal {C})}[{\\\\mathbb {CE}\\\\big (\\\\mathcal {F}_{\\\\phi }(\\\\mathcal {G}_{\\\\theta }(\\\\mathbf {x}_{s}, c) + \\\\mathbf {x}_{s}\\\\big ), c)}], \\\\\\\\& \\\\text{s.t. }\",\n \"\\\\Vert \\\\mathcal {G}_{\\\\theta }(\\\\mathbf {x}_{s}, c) \\\\Vert _{\\\\infty } \\\\le \\\\epsilon .\\\\end{split}$ By solving problem (REF ), we can obtain a targeted conditional generator by minimizing the loss of specific target class in the unlabeled training dataset.\",\n \"Note that we only optimize the parameter $\\\\theta $ of the generator $\\\\mathcal {G}_{\\\\theta }$ using the training data $\\\\mathcal {X}_{s}$ , then the targeted adversarial example $\\\\mathbf {x}_{t}^{*}$ can be crafted by $\\\\mathbf {x}_{t}^{*} = \\\\mathbf {x}_{t} + \\\\mathcal {G}_{\\\\theta }(\\\\mathbf {x}_{t}, c) $ for any given image $\\\\mathbf {x}_{t}$ in the test data $\\\\mathcal {X}_{t}$ , which only requires an inference for this targeted image $\\\\mathbf {x}_{t}$ .\",\n \"We experimentally find that the objective (REF ) can enforce the transferability for the generated perturbation $\\\\mathbf {\\\\delta }$ .\",\n \"A reasonable explanation is that $\\\\mathbf {\\\\delta }$ can arise as a result of strong and well-generalizing semantic pattern inherent to the target class, which is robust to the influence of any training data.\",\n \"In Sec.\",\n \"REF , we illustrate and corroborate our claim by directly feeding scaled adversarial perturbationsThe perturbation is linearly scaled from [-$\\\\epsilon $ , $\\\\epsilon $ ] to [0, 255].\",\n \"from different methods into the classifier.\",\n \"Indeed, we find that our semantic pattern can be classified as the target class with a high degree of confidence while the perturbation from MIM [9] performs like the noise, meanwhile the scaled semantic pattern performs well transferability in different black-box models.\"\n ],\n [\n \"Network Architecture\",\n \"We now present the details of the conditional generative model for targeted attack, as illustrated in Fig.\",\n \"REF .\",\n \"Specifically, we design a mapping network to generate a target-specific vector in the implicit space of each target and train conditional generator $\\\\mathcal {G}_{\\\\theta }$ to reflect this vector by constantly misleading the classifier $\\\\mathcal {F}_{\\\\phi }$ .\",\n \"Mapping network.\",\n \"Given an one-hot class encoding $\\\\mathbb {1}_{c} \\\\in \\\\mathbb {R}^{K}$ from target class $c$ , the mapping network aims to generate the targeted latent vector $\\\\mathbf {w} = \\\\mathcal {W}(\\\\mathbb {1}_{c})$ , where $\\\\mathbf {w} \\\\in \\\\mathbb {R}^{M}$ and $\\\\mathcal {W}(\\\\cdot )$ consists of a multi-layer perceptron (MLP) and a normalization layer, which can construct diverse targeted vectors $\\\\mathbf {w}$ for a given target class $c$ .\",\n \"Thus $\\\\mathcal {W}$ is capable of learning effective targeted latent vectors by randomly sampling different classes $c \\\\in \\\\mathcal {C}$ in training phase.\",\n \"Generator.\",\n \"Given an input image $\\\\mathbf {x}_{s}$ , the encoder first calculates the feature map $\\\\mathbf {F} \\\\in \\\\mathbb {R}^{N\\\\times H \\\\times W}$ , where $N$ , $H$ and $W$ refer to the number of channels, height and width of the feature map, respectively.\",\n \"The target latent vector $\\\\mathbf {w}$ , derived from the mapping network $\\\\mathcal {W}$ by introducing a specific target class $c$ , is expanded along height and width directions to obtain the label feature map $\\\\mathbf {w}_s\\\\in \\\\mathbb {R}^{M\\\\times H \\\\times W}$ .\",\n \"Then the above two feature maps are concatenated along the channels to obtain $\\\\mathbf {F}^{\\\\prime } \\\\in \\\\mathbb {R}^{(N+M)\\\\times H \\\\times W}$ .\",\n \"The obtained mixed feature map is then fed to the subsequent network.\",\n \"Therefore, our generator $\\\\mathcal {G}_{\\\\theta }$ translates an input image $\\\\mathbf {x}_{s}$ and latent target vector $\\\\mathbf {w}$ into an output image $\\\\mathcal {G}_{\\\\theta }(\\\\mathbf {x}_{s}, \\\\mathbf {w})$ , which enables $\\\\mathcal {G}_{\\\\theta }$ to synthesize adversarial images of a series of targets.\",\n \"For the output of feature map $\\\\mathbf {f} \\\\in \\\\mathbb {R}^{d}$ in the decoder, we adopt a smooth projection $P(\\\\cdot )$ to perform a change of variables over $\\\\mathbf {f}$ rather than directly minimizing its $\\\\ell _2$ norm as [14] or clipping values outside the fixed norm [31], which can be denoted as $\\\\mathbf {\\\\delta } = P(\\\\mathbf {f}) = \\\\epsilon \\\\cdot \\\\mathrm {tanh}(\\\\mathbf {f}),$ where $\\\\epsilon $ is the strength of perturbation.\",\n \"Since $-1 \\\\le \\\\mathrm {tanh}(\\\\mathbf {f}) \\\\le 1$ , $\\\\mathbf {\\\\delta }$ can automatically satisfy the $\\\\ell _{\\\\infty }$ -ball bound with perturbation budget $\\\\epsilon $ .\",\n \"This transformation can be regarded as a better smoothing of gradient than directly clipping values outside the fixed norm, which is also instrumental for $\\\\mathcal {G}_{\\\\theta }$ to probe and learn the targeted semantic knowledge from $\\\\mathcal {F}_{\\\\phi }$ .\",\n \"Training objectives.\",\n \"The training objectives seek to minimize the classification error on the perturbed image of the generator as $\\\\vspace{-0.42502pt}\\\\theta ^{*} \\\\leftarrow \\\\operatornamewithlimits{arg\\\\,min}_{\\\\theta }{\\\\mathbb {CE} \\\\Big (F_{\\\\phi }\\\\big (\\\\mathbf {x}_{s} + \\\\mathcal {G}_{\\\\theta }(\\\\mathbf {x}_{s}, \\\\mathcal {W}(\\\\mathbb {1}_{c}))\\\\big ), c\\\\Big )},$ which adopts an end-to-end training paradigm with the goal of generating adversarial images to mislead the classifier the target label, and $\\\\mathbb {CE}$ is the cross entropy loss.\",\n \"Previous studies attempt different classification losses in their works [52], [31], and we found that cross-entropy loss works well in our settings.\",\n \"The detailed optimization procedure is summarized in Algorithm REF .\",\n \"[t] Training Algorithm for the Conditional Generative Attack Training Data $\\\\mathcal {D}_{s}$ ; a generative network $\\\\mathcal {G}_{\\\\theta }$ ; a classification network $\\\\mathcal {F}_{\\\\phi }$ ; a mapping network $\\\\mathcal {W}$ .\",\n \"Adversarial perturbations $\\\\mathbf {\\\\theta }$ .\",\n \"iter in MaxIterations T Randomly sample $B$ images $\\\\lbrace \\\\mathbf {x}_{s_{i}}\\\\rbrace _{i=1}^{B}$ Randomly sample $B$ target classes $\\\\lbrace {c}_{i}\\\\rbrace _{i=1}^{B}$ Forward pass ${c}_{i}$ into $\\\\mathcal {W}$ to compute the targeted latent vectors $\\\\mathbf {w}_{i}$ Obtain the perturbed images by $\\\\mathbf {x}_{s_{i}}^{*} = \\\\epsilon \\\\cdot \\\\mathrm {tanh}(\\\\mathcal {G}(\\\\mathbf {x}_{s_{i}}, \\\\mathbf {w}_{i})) + \\\\mathbf {x}_{s_{i}}$ Forward pass $\\\\mathbf {x}_{s_{i}}^{*}$ to $\\\\mathcal {F}_{\\\\phi }$ and compute loss in Eq.\",\n \"(REF ) Backward pass and update the $\\\\mathcal {G}_{\\\\theta }$ Table: Transferability comparison for multi-target attacks on ImageNet NeurIPS validation set (1k images) with the perturbation budget of ℓ ∞ ≤16\\\\ell _{\\\\infty } \\\\le 16.\",\n \"The results are averaged on 8 different target classes.\",\n \"Note that CD-AP † ^{\\\\dagger } indicates that training 8 models can obtain results, while our method only train one conditional generative model.\",\n \"* indicates white-box attacks.\"\n ],\n [\n \"Hierarchical Partition for Classes\",\n \" While handling plenty of classes, the effectiveness of a conditional generative model will decrease as illustrated in Fig.\",\n \"REF , because the representative capacity is limited with a single generator.\",\n \"Therefore, we propose to divide all classes into a feasible number of subsets to train models when the class number $K$ is large, , 1,000 classes in ImageNet, with the aim of seeking the effectiveness of targeted black-box attack.\",\n \"To obtain a good partition, we introduce a representative target class space, which is nearly equivalent to the original class space $\\\\mathcal {C}$ .\",\n \"Specifically, we utilize the weights $\\\\phi _{cls} \\\\in \\\\mathbb {R}^{D\\\\times C}$ in the classifier layer for the classification network $\\\\mathcal {F}_{\\\\phi }$ .\",\n \"Therefore, $\\\\phi _{cls}$ can be regarded as the alternative class space since the weight vector $\\\\mathbf {d}_{c} \\\\in \\\\mathbb {R}^{D}$ from $\\\\phi _{cls}$ can represent a class center of the feature embeddings of input images with same class $c$ .\",\n \"Note that once those subsets with closer metric distance (, larger cosine similarity) in the target class space $\\\\phi _{cls}$ are regarded as conditional inputs of generative network, they obtain worse loss convergence and transferability than diverse them due to mutual influence among these input conditions, as illustrated in Fig.\",\n \"REF .\",\n \"Thus we focus on selecting target classes that do not tend to overlap or be close to each other as accessible subsets.\",\n \"To capture more diverse examples in a given sampling space, we adopt K-determinantal point processes (DPP) [21], [20] to achieve a hierarchical partition, which can take advantage of the diversity property among subsets by assigning subset probabilities proportional to determinants of a kernel matrix.\",\n \"First, we compute the RBF kernel matrix $L$ of $\\\\phi _{cls}$ and eigendecomposition of $L$ , and a random subset $V$ of the eigenvectors is chosen by regarding the eigenvalues as sampling probability.\",\n \"Second, we select a new class $c_{i}$ to add to the set and update $V$ in a manner that de-emphaseizes items similar to the one selected.\",\n \"Each successive point is selected and $V$ is updated by Gram-Schmidt orthogonalization, and the distribution shifts to avoid points near those already chosen.\",\n \"The details are presented in Appendix A.\"\n ],\n [\n \"Experiments\",\n \" In this section, we present extensive experiments to demonstrate the effectiveness of proposed method for targeted black-box attacks.\"\n ],\n [\n \"Experimental Settings\",\n \" Datasets.\",\n \"We consider the following datasets for training, including a widely used object detection dataset MS-COCO [26] and ImageNet training set [6].\",\n \"We focus on standard and comprehensive testing settings, thus inference is performed on ImageNet validation set (50k samples), a subset (5k) of ImageNet proposed by [24] and ImageNet-NeurIPS (1k) proposed by [32].\",\n \"Figure: Comparison of different projection functions and modes of Gaussian Smoothing.\",\n \"Results are reported with Inc-v3 networkon ImageNet NeurIPS validation set.Networks.\",\n \"We consider some naturally trained networks, , Inception-v3 (Inc-v3) [41], Inception-v4 (Inc-v4) [39], Resnet-v2-152 (Res-152) [15] and Inception-Resnet-v2 (IncRes-v2) [39], which are widely used for evaluating transferability.\",\n \"Besides, we supplement DenseNet-201 (Dense-201) [16], GoogleNet [40] and VGG-16 [37] to fully evaluate the transferability.\",\n \"Adversarially trained networks [43] are also selected to evaluate the performance, , ens3-adv-Inception-v3 ($\\\\textrm {Inc-v3}_\\\\textrm {ens3}$ ), ens4-adv-Inception-v3 ($\\\\textrm {Inc-v3}_\\\\textrm {ens4}$ ) and ens-adv-Inception-ResNet-v2 ($\\\\textrm {IncRes-v2}_\\\\textrm {ens}$ ).\",\n \"All networks are publicly availablehttps://github.com/tensorflow/models/tree/master/research/slim https://github.com/tensorflow/models/tree/master/research/adv_imagenet_models https://github.com/pytorch/vision/tree/master/torchvision/models.\",\n \"Implementation details.\",\n \"We choose the same ResNet autoencoder architecture in [19], [31] as the basic generator networks, which consists of downsampling, residual and upsampling layers.\",\n \"We initialize the learning rate as 2e-5 and set the mini-batch size as 32.\",\n \"Smoothing mechanism is proposed to improve the transferability against adversarially trained models [10].\",\n \"Instead of adopting smoothing for generated perturbation while the training is completed as CD-AP [31], we introduce adaptive Gaussian smoothing kernel to compute $\\\\mathbf {\\\\delta }$ from Eq.\",\n \"(REF ) in the training phase, named adaptive Gaussian smoothing, with the focus of making generated results obtain adaptive ability.\",\n \"More implementation details and discussion with other networks (, BigGAN [3]) are illustrated in Appendix B.\",\n \"Table: Transferability results for targeted attacks on ImageNet validation set (50k images) with the perturbation budget of ℓ ∞ ≤10\\\\ell _{\\\\infty } \\\\le 10.\",\n \"The attack is performed in same setting with the target class 'sea lion' and the training dataset MS-COCO.\"\n ],\n [\n \"Transferability Evaluation\",\n \"We consider 8 different target classes from [52] to form the multi-target black-box attack testing protocol with 8k times in 1k ImageNet NeurIPS set.\",\n \"Efficiency of multi-target black-box attack.\",\n \"Among comparable methods, instance-specific methods, , MIM, TI-DIM, DIM and TI-DIM, require iterative mechanism with $M$ steps by computing gradients to obtain adversarial examples.\",\n \"Given the cost $t_{C}^{FP}$ and $t_{C}^{BP}$ of forward and backward passing the classifier, computing cost $T^{IS}$ of single data can be defined as $T^{IS} = t_{C}^{FP} * M + t_{C}^{BP} * M$ in Tab.\",\n \"REF .\",\n \"Instance-agnostic methods only require the inference cost from the trained generator as $T^{IA} = t_{G}^{FP}$ , thus possessing the priority for those attack scenarios within limited time.\",\n \"However, instance-specific methods require to train 8 models to obtain all predictions from 8 different classes.\",\n \"Due to time-consuming training and more storage, we only reproduce previous state-of-the-art generative method CD-AP [31] as a baseline, which already fully demonstrate the superior performance than other generative methods such as GAP [33] in their work.\",\n \"As a comparison, our conditional generative method only trains one model to inference the results and outperforms in the aspect of efficiency.\",\n \"Effectiveness of multi-target black-box attack.\",\n \"Tab.\",\n \"REF shows the transferability comparison of different methods on both naturally and adversarially trained models.\",\n \"The success rate of instance-specific attacks are lower than 3%, possibly explained by the data-point overfitting that makes it hard to transfer another model.\",\n \"The instance-agnostic attack CD-AP obtains acceptable performance, yet inferior to proposed method w.r.t black-box transferability.\",\n \"The primary reason for such a trend lies in some distinctions as 1) direct clip projection in CD-AP and our smooth projection in Eq.\",\n \"(REF ) and 2) their Gaussian Smoothing and our adaptive Gaussian Smoothing, as described in Sec.\",\n \"REF and Appendix B.\",\n \"Fig.\",\n \"REF empirically shows the comparison results of single-target black-box attacks based on the CD-AP framework.\",\n \"Thus proposed conditional generative method can be a reliable baseline w.r.t targeted black-box attacks, regarding both effectiveness and efficiency.\",\n \"Results of single-target black-box attack.\",\n \"Recent related work [52] has tried to solve the single-target transferable problem based on universal adversarial perturbation, and report an excellent single-target black-box performance.\",\n \"We obtain single-target degraded version of our model by specifying an input target label during the training process.\",\n \"We show the performance of black-box attack in terms of targeted attack success rate in Tab.\",\n \"REF .\",\n \"The promising results show generative semantic pattern from our method benefits black-box transferability than universal adversarial perturbations.\",\n \"Some other instance-agnostic adversarial methods, , UAP [29], GAP [33] and RHP [24], have tendency towards the untargeted black-box problem.\",\n \"Despite this, we follow the corresponding untargeted setting and compare different methods in Appendix C. Our method is steadily improved under untargeted black-box manner.\",\n \"Table: Transferability comparison on NeurIPS 2017 competition with the perturbation budget of ℓ ∞ ≤16\\\\ell _{\\\\infty } \\\\le 16.\",\n \"White-box substitute model is Inc-v3 for all attacks, following the standard protocol with 1,000 stochastic target classes.Table: The success rate of black-box impersonation attacks on face verification with the perturbation budget of ℓ ∞ ≤16\\\\ell _{\\\\infty } \\\\le 16.\",\n \"ArcFace is chosen as white-box model.Figure: Generative examples of adversarial images with perturbation budget of ℓ ∞ ≤16\\\\ell _{\\\\infty }\\\\le 16.\",\n \"We separately adopt the ImageNet and MS-COCO dataset as the training dataset to implement the generation of targeted perturbations.\",\n \"Our method can generate semantic pattern independent of training dataset.Figure: Asr vs. numbers of conditional targets curve against Inc-v3 and VGG-16 models.\"\n ],\n [\n \"Effectiveness on NeurIPS 2017 Competition\",\n \" To illustrate the effectiveness of our proposed attack methods in practical 1,000 classification, we here follow the official setting from NeurIPS 2017 adversarial competition [23] for testing targeted black-box transferability.\",\n \"Considering limited resource, previous instance-agnostic attacks are not required as comparable methods due to training 1,000 models, thus we focus on the instance-specific attacks, which are official top attack methods in NeurIPS 2017 adversarial competition.\",\n \"Our hierarchical partition mechanism can make conditional generative networks be capable of specifying any target class via a feasible number of models for the scalability.\",\n \"We consider 20 models, with each specifying 50 diverse classes from k-DPP hierarchical partition in this setting, to implement targeted attack by only once inference for each target image.\",\n \"Our method outperforms all other baseline methods in Tab.\",\n \"REF .\",\n \"The results demonstrate that this method can be reliable in practical targeted attacks, regarding both effectiveness and efficiency.\"\n ],\n [\n \"Effectiveness on Realistic Face Recognition\",\n \" Adversarial perturbations added to original face have ability to evade being recognized or impersonate another individual [36], [50].\",\n \"In this section, we consider the transferability of impersonation attack to further illustrate the generalization of our method, which is also corresponding to targeted attack in image classification.\",\n \"Dataset and models.\",\n \"We conduct the experiments on Labeled Faces in the Wild (LFW) [17] and introduce two test protocols.\",\n \"For Protocol I defined as single-target impersonation attack, we choose 1 target identity and 1k source face images belonging with different identities from LFW as the attackers, thus forming 1k pairs.\",\n \"For Protocol II named multi-target impersonation attack, 5 target identities and 1k source face images are selected to form 1k attack pairs, meaning that we need to implement 5k attacks.\",\n \"We involve some excellent face recognition models for conducting black-box testing, including Sphereface [27], CosFace [44], FaceNet [35] and MobileFace [4].\",\n \"These models lie in different model architectures and training objectives.\",\n \"In all experiments, we only use one model ArcFace [7] as substitute model to craft adversarial samples, and test attack performance against other unknown models.\",\n \"Figure: Comparison of loss convergence and transferability between diverse and close conditional subset with 10 target classes.Figure: Plots of logit vectors from the adversarial image L img L_{img} and scaled crafted perturbation L adv L_{adv} of MIM and proposed generative method, with their respective PCC values.Evaluation metrics.\",\n \"We first compute the optimal threshold of every face recognition models from LFW dataset by following standard protocols.\",\n \"If the similarity of a pair of images exceeds the threshold, we regard them as same identity, otherwise different identities.\",\n \"Black-box attack results.\",\n \"We adjust the optimization object function to adapt face recognition for chosen attack methods (detailed in Appendix C), and report the success rate of black-box impersonation attacks in Tab.\",\n \"REF , which illustrates that our method can achieve nearly two times of the success rates than DIM in Protocol I and Protocol II.\",\n \"The results indicate that our method is superior to other methods not only in image classification.\"\n ],\n [\n \"Comparison Study about Target Classes\",\n \" We conduct an extensive study to investigate two key points about target classes.\",\n \"Different numbers of target classes.\",\n \"We conduct effectiveness for different numbers of target classes in Fig.\",\n \"REF .\",\n \"It can be seen that the results perform well within a feasible number of targets, whereas to a certain extent effectiveness tend to decay.\",\n \"Therefore, the effectiveness of conditional generative networks is influenced by the number of conditional classes, due to the representative capacity of single generator.\",\n \"We aim to divide all classes into a feasible number of set while handling plenty of classes.\",\n \"Comparison of different multi-target conditions.\",\n \"We select closer conditional classes with larger cosine similarity in the target class space $\\\\phi _{cls}$ and diverse conditional classes from k-DPP method.\",\n \"In Fig.\",\n \"REF , closer conditional classes have worse loss convergence and transferability than diverse them due to mutual influence among conditions.\"\n ],\n [\n \"More Analyses\",\n \" Targeted adversarial samples from proposed generative method can produce semantic pattern inherent to the target class in Fig.\",\n \"REF .\",\n \"Why does generative semantic pattern work?\",\n \"First, generative methods can produce strong targeted semantic pattern that is robust to the influence of data, which is obtained by minimizing the loss of specific target class in the training phase.\",\n \"To corroborate our claim, we directly feed scaled crafted perturbations by instance-specific attack MIM and our generative method into the classifier.\",\n \"Indeed, we find that our generative perturbation is considered as target class with a high degree of confidence whereas the perturbation from MIM performs like the noise, as shown in Fig.\",\n \"REF .\",\n \"We plot the logit relationship from scaled crafted perturbation and adversarial image in Fig.\",\n \"REF , which is also consistent with previous claim.\",\n \"Second, the generated adversarial semantic pattern achieves well-generalizing performance among the different models.\",\n \"We feed 1k images from ImageNet test set into the generator trained by Inc-v3 model to obtain 1k semantic patterns, which are scaled to image pixel space and then fed into different classifiers.\",\n \"We compute the mean confidence of $\\\\mathbf {0.46}$ for Dense-201, $\\\\mathbf {0.44}$ for Inc-v4, and $\\\\mathbf {0.35}$ for Res-152, whereas the perturbation from MIM is lower than $0.01$ .\",\n \"The results show that our scaled semantic pattern can directly achieve well-generalizing performance among models, possibly explained by utilizing similar feature knowledge from the same class on different classifiers trained on same training data distribution.\",\n \"Thus similar pattern can be instrumental for transferability among models.\"\n ],\n [\n \"Discussion and Conclusion\",\n \" Transferability of targeted black-box attack is simultaneously affected by data and model.\",\n \"Therefore, instance-specific methods easily overfit the data point and white-box model, resulting in weak transferability.\",\n \"As a comparison, proposed generative method with powerful learning capacity reduces the dependency for data point by adopting the unlabeled training data, thus enabling the model to learn semantic pattern and improve the transferability of targeted black-box attack.\",\n \"Extensive experiments demonstrate that proposed generative method can significantly improve the success rates of targeted black-box attacks against naturally and adversarial trained models.\",\n \"Th us we hope that crafting C-GSP can be regarded as a new reliable baseline method in terms of targeted black-box attacks.\"\n ],\n [\n \"Sampling Algorithm\",\n \"We summarize the overall sampling procedure based on k-DPP [20] in Algorithm .\",\n \"Compute the RBF kernel matrix $L$ of $\\\\phi _{cls}$ and eigendecomposition of $L$ .\",\n \"A random subset $V$ of the eigenvectors is chosen by regarding the eigenvalues as sampling probability.\",\n \"Select a new class $c_{i}$ to add to the set and update $V$ in a manner that de-emphaseizes items similar to the one selected.\",\n \"Update $V$ by Gram-Schmidt orthogonalization, and the distribution shifts to avoid points near those already chosen.\",\n \"By performing the Algorithm , we can obtain a subset with $k$ size.\",\n \"Thus while handling the conditional classes with K, we can hierarchically adopt this algorithm to get the final $K/k$ subsets, which are regarded as conditional variables of generative models to craft adversarial examples.\"\n ],\n [\n \"Some Implementation Details\",\n \"The study of smoothing mechanism.\",\n \"Smoothing mechanism has been proved to improve the transferability against adversarially trained models.\",\n \"CD-AP [31] uses direct clip projection to have a fixed norm $\\\\epsilon $ , and adopts smoothing for generated perturbation while the generator $\\\\mathcal {G}$ is trained, , ${\\\\begin{array}{c}\\\\textbf {Train: }\\\\mathbf {x}_{s_{i}}^{*} = \\\\mathrm {Clip}_{\\\\epsilon }(\\\\mathcal {G}(\\\\mathbf {x}_{s_{i}}),\\\\\\\\\\\\textbf {Test: }\\\\mathbf {x}_{s_{i}}^{*} = \\\\mathrm {W} * \\\\mathrm {Clip}_{\\\\epsilon }(\\\\mathcal {G}(\\\\mathbf {x}_{s_{i}}),\\\\end{array}}$ where $\\\\mathrm {W}$ indicates Gaussian smoothing of kernel size of 3, $*$ indicates the convolution operation, and Clip$_{\\\\epsilon }$ means clipping values outside the fixed norm $\\\\epsilon $ .\",\n \"As a comparison, we introduce adaptive Gaussian smoothing kernel to compute adversarial images $\\\\mathbf {x}_{s_{i}}^{*}$ from in the training phase, named adaptive Gaussian smoothing as $\\\\textbf {Train \\\\& Test: } \\\\mathbf {x}_{s_{i}}^{*} = \\\\epsilon \\\\cdot \\\\mathrm {W} * \\\\mathrm {tanh}(\\\\mathcal {G}(\\\\mathbf {x}_{s_{i}}) + \\\\mathbf {x}_{s_{i}},$ which can make generated results obtain adaptive ability in the training phase.\",\n \"We perform training in ImageNet dataset to report all results including comparable baselines.\",\n \"[t] Sampling Algorithm by kDPP Weight Vector $\\\\mathbf {\\\\theta }_{cls}$ ; Subset size $k$ .\",\n \"A subset $C$ .\",\n \"Compute RBF kernel matrix $L$ of $\\\\mathbf {\\\\theta }_{cls}$ Compute eigenvector/value $\\\\lbrace v_{n}, \\\\lambda _{n}\\\\rbrace _{n=1}^N$ pairs of $L$ // Phase I: $J\\\\leftarrow \\\\phi $ , $e_{k}\\\\left(\\\\lambda _{1}, \\\\ldots , \\\\lambda _{N}\\\\right)=\\\\sum _{|J|=k} \\\\prod _{n \\\\in J} \\\\lambda _{n}$ n = N, ..., 1 $u\\\\sim U[0,1] < \\\\lambda _{n} \\\\frac{e^{n-1}_{k-1}}{e^{n}_{k}}$ and $k>0$ $J\\\\leftarrow J \\\\cup \\\\lbrace n\\\\rbrace $ ; $k\\\\leftarrow k-1$ // Phase II: $V \\\\leftarrow \\\\left\\\\lbrace v_{n}\\\\right\\\\rbrace _{n \\\\in J}, Y\\\\leftarrow \\\\phi $ $|V| > 0$ Select $c_i$ from $\\\\mathcal {C}$ with $\\\\operatorname{P}\\\\left(c_{i}\\\\right)=\\\\frac{1}{|V|} \\\\sum _{v \\\\in V}\\\\left(v^{\\\\top } e_{i}\\\\right)^{2}$ $C \\\\leftarrow C \\\\cup \\\\left\\\\lbrace c_{i}\\\\right\\\\rbrace $ $V \\\\leftarrow V_{\\\\perp },$ an orthonormal basis for the subspace of $V$ orthogonal to $e_{i}$ Figure: Some examples of adversarial images with perturbation budget of ℓ ∞ ≤16\\\\ell _{\\\\infty }\\\\le 16.\",\n \"We separately adopt the ImageNet, MS-COCO and Comics dataset as the training dataset to implement the generation of targeted perturbations.Figure: Some examples of adversarial images with perturbation budget of ℓ ∞ ≤16\\\\ell _{\\\\infty }\\\\le 16.\",\n \"We separately adopt the ImageNet, MS-COCO and Comics dataset as the training dataset to implement the generation of targeted perturbations.Network architecture of generator.\",\n \"We adopt the same autoencoder architecture in [31] as the basic generator networks.\",\n \"Besides, we also explore BigGAN [3] as conditional generator network.\",\n \"An very weak testing performance is obtained even in the white-box attack scenario, possibly explained by the weak diversity of latent variable with the Gaussian distribution from BigGAN in the training phase, whereas autoencoder can take full advantage of large-scale training dataset, , ImageNet.\",\n \"Furthermore, we also train the autoencoder with Gaussian noise as the training dataset and obtain similar inferior performance in the white-box attack scenario, indicating that a large-scale training dataset is very significant for generating transferable targeted adversarial examples.\",\n \"Some details.\",\n \"In our experiments of testing time, we apply NVIDIA 1080Ti GPUs.\",\n \"Instance-specific methods, , MIM, TI-DIM, DIM and TI-DIM, adopt iterative steps $M = 20$ and follow their reported hyperparameters.\"\n ],\n [\n \"Additional Experimental Results\",\n \"Results on different datasets.\",\n \"We craft adversarial examples on different datasets, including ImageNet training set, MS-COCO and Comics dataset [2], which consist of 1.2M, 82k and 50K images, respectively.\",\n \"MS-COCO dataset can be applied to large-scale object detection and segmentation, and those images from Comics dataset are regarded as other domains different from normal ones in ImageNet.\",\n \"Despite this diverse training types, we still find the common property of crafted adversarial examples by our method.\",\n \"Specifically, we craft some examples of adversarial images with perturbation budget of $\\\\ell _{\\\\infty }\\\\le 16$ , and separately adopt the ImageNet, MS-COCO and Comics dataset as the training dataset to implement the generation of targeted perturbations.\",\n \"As illustrated in Fig.\",\n \"REF , we produce semantic pattern independent of any training dataset.\",\n \"Table: Comparison results of targeted black-box attacks on different datasets.\",\n \"Incv3 is the substitute model.We also report the success rate of targeted black-box attack, as shown in Tab.\",\n \"REF .\",\n \"We experimentally find that semantic pattern derived from ImageNet dataset achieves better performance of black-box performance, possibly explained by instructional effectiveness from more diverse data in ImageNet dataset.\",\n \"Results of untargeted black-box attack.\",\n \"We evaluate our method and other generative methods including UAP [29], GAP [33] and RHP [24].\",\n \"Untargeted transferability from naturally trained models to adversarially trained models occurs due to differences in model sources, data types and other factors, thus enabling challenging comparison.\",\n \"As illustrated in Tab.\",\n \"REF , we report the untargeted attacks increase in error rate of adversarial and clean images to evaluate different methods.\",\n \"Our method is steadily improved in different black-box models under untargeted black-box manner.\",\n \"Table: Transferability results for untargeted attacks increase in error rate after attack on subset of ImageNet (5k images) with the perturbation budget of ℓ ∞ ≤16/32\\\\ell _{\\\\infty } \\\\le 16/32.\"\n ],\n [\n \"Impersonation Attack of Face Recognition\",\n \"We list attack methods of face recognition as follows.\",\n \"Given an input $\\\\mathbf {x}$ and an image $\\\\mathbf {x}^r$ belonging with another identity, an attack method can generate an adversarial example $\\\\mathbf {x}^{adv}$ with perturbation budget $\\\\epsilon $ under the $\\\\ell _p$ norm ($\\\\Vert \\\\mathbf {x}^{adv} - \\\\mathbf {x}\\\\Vert _p \\\\le \\\\epsilon $ ).\",\n \"Therefore, impersonation attack aims to perform this objective of $\\\\mathcal {C} (\\\\mathbf {x}^{adv}, \\\\mathbf {x}^{r}) = \\\\mathbb {I} (\\\\mathcal {D}_f (\\\\mathbf {x}^{adv}, \\\\mathbf {x}^{r}) < \\\\delta ),$ where $\\\\mathbb {I}$ is the indicator function, $\\\\delta $ is a threshold, and $\\\\mathcal {D}_f(\\\\mathbf {x}^{adv}, \\\\mathbf {x}^{r}) = \\\\Vert f(\\\\mathbf {x}^{adv}) - f(\\\\mathbf {x}^{r})\\\\Vert _2^2.$ Basic Iterative Method (BIM) [22] extends FGSM by iteratively taking multiple small gradient updates as $\\\\mathbf {x}_{t+1}^{adv} = \\\\mathrm {clip}_{\\\\mathbf {x},\\\\epsilon } \\\\big (\\\\mathbf {x}_t^{adv} - \\\\alpha \\\\cdot \\\\mathrm {sign}(\\\\nabla _{\\\\mathbf {x}}\\\\mathcal {D}_f(\\\\mathbf {x}_t^{adv},\\\\mathbf {x}^{r}))\\\\big ),$ where $\\\\mathrm {clip}_{\\\\mathbf {x},\\\\epsilon }$ projects the adversarial example to satisfy the $\\\\ell _{\\\\infty }$ constrain and $\\\\alpha $ is the step size.\",\n \"Momentum Iterative Method (MIM) [9] introduces a momentum term into BIM for improving the transferability of adversarial examples as ${\\\\begin{array}{c}\\\\mathbf {g}_{t+1} = \\\\mu \\\\cdot \\\\mathbf {g}_t + \\\\frac{\\\\nabla _{\\\\mathbf {x}}\\\\mathcal {D}_f(\\\\mathbf {x}_t^{adv},\\\\mathbf {x}^{r})}{\\\\Vert \\\\nabla _{\\\\mathbf {x}}\\\\mathcal {D}_f(\\\\mathbf {x}_t^{adv},\\\\mathbf {x}^{r})\\\\Vert _1}; \\\\\\\\ \\\\mathbf {x}^{adv}_{t+1}=\\\\mathrm {clip}_{\\\\mathbf {x},\\\\epsilon }(\\\\mathbf {x}^{adv}_t - \\\\alpha \\\\cdot \\\\mathrm {sign}(\\\\mathbf {g}_{t+1})).\\\\end{array}}$ The training objectives of our generative method seek to minimize the classification error on the perturbed image of the generator as $\\\\vspace{-0.42502pt}\\\\min _{\\\\theta }\\\\mathbb {E}_{(\\\\mathbf {x}\\\\sim \\\\mathcal {X}, c\\\\sim \\\\mathcal {C})}[{\\\\mathcal {D}_f \\\\big (\\\\mathbf {x} + \\\\mathcal {G}_{\\\\theta }(\\\\mathbf {x}, c), \\\\mathbf {x}^{r}_{c}\\\\big )}],$ where $\\\\mathbf {x}^{r}_{c}$ refers to $\\\\mathbf {x}^{r}$ with the corresponding identity $c$ .\",\n \"In the training phase, we randomly select $1,000$ identities from CASIA-WebFace [51] as training dataset to craft adversarial examples.\",\n \"Therefore, our method can be applied not only in image classification.\"\n ],\n [\n \"More Examples\",\n \"We also show more semantic patterns from different target models, as illustrated in Fig.\",\n \"REF .\"\n ]\n]"},"id":{"kind":"string","value":"2107.01809"}}},{"rowIdx":1586,"cells":{"text":{"kind":"list like","value":[["Autoencoder based Randomized Learning of Feedforward Neural Networks for\n Regression"],["Abstract Feedforward neural networks are widely used as universal predictive models to fit data distribution.","Common gradient-based learning, however, suffers from many drawbacks making the training process ineffective and time-consuming.","Alternative randomized learning does not use gradients but selects hidden node parameters randomly.","This makes the training process extremely fast.","However, the problem in randomized learning is how to determine the random parameters.","A recently proposed method uses autoencoders for unsupervised parameter learning.","This method showed superior performance on classification tasks.","In this work, we apply this method to regression problems, and, finding that it has some drawbacks, we show how to improve it.","We propose a learning method of autoencoders that controls the produced random weights.","We also propose how to determine the biases of hidden nodes.","We empirically compare autoencoder based learning with other randomized learning methods proposed recently for regression and find that despite the proposed improvement of the autoencoder based learning, it does not outperform its competitors in fitting accuracy.","Moreover, the method is much more complex than its competitors."],["Introduction","Feedforward neural networks (FNNs) have attracted a great deal of interest due to their excellence predictive performance and universal approximation capabilities.","The most popular learning methods involve some kind of gradient descent algorithm to learn FNN weights iteratively.","However, gradient-based methods suffer from many drawbacks making the learning process ineffective and time-consuming.","This is because they are sensitive to the initial values of the parameters.","The learning trajectory, starting with different initial parameters, leads to local minima of the loss function.","Thus, globally optimal parameters are not guaranteed.","Moreover, the learning process is time-consuming for complex target functions (TFs), big data sets and large FNN architectures.","Randomized learning, such as a random vector functional link (RVFL) network [1], has been proposed as an alternative to conventional FNNs iterative learning using gradients.","In randomized learning, the parameters of the hidden nodes are selected randomly and stay fixed.","They do not need to be tuned during the training stage.","The only parameters that need to be learned are the output weights.","This makes the optimization problem convex [2].","As such, it can be solved easily and quickly using a standard least-squares method.","Despite this simplification, randomized FNN learning still possesses universal approximation capabilities, provided there are a sufficient number of nonlinear hidden nodes [1], [3].","Many simulation studies reported in the literature show the high performance of the randomized FNN when compared to fully adaptable FNNs.","Randomization, which is cheaper than optimization, ensures simplicity of implementation and faster training.","However, a challenging and still open question in randomized learning is how to choose appropriate weights and biases for the hidden nodes to ensure best model performance [4], [5].","To deal with this problem, various methods of generating hidden node parameters have been proposed.","The simplest and most popular solution is to select both weights and biases from a uniform distribution over some symmetric interval $U=[-u, u]$ .","Usually, this interval is assigned as fixed, typically $[-1, 1]$ , regardless of the data, TF, and type of activation functions (AFs).","The independence of the hidden nodes from data is seen as an asset.","This overly-simplistic approach was criticized as illogical and misleading [6].","Therefore, to improve its performance, optimization of interval $U$ for a specified application is recommended.","For example, in [7], a supervisory mechanism which randomly assigns hidden node parameters from an adaptively selected interval, was proposed.","This paper clearly reveals that the selection of random parameters should be data dependent to ensure the universal approximation property of the resulting randomized FNN.","Data dependent random parameters were also recommended in [8].","The authors of this work noticed that if the hidden nodes are chosen at random and not subsequently trained, they are usually not placed in accordance with the density of the input data.","In such a case, training of linear parameters is less ineffective at reducing errors.","Therefore, in order to improve learning performance, the authors advise unsupervised placement of hidden nodes according to the input data density.","This recommendation was implemented in the methods proposed in [9] and [10].","In these works, it was noticed that as the weights and biases of hidden nodes have different functions, they should not be selected from the same interval.","The weights decide about AF slopes and should reflect TF complexity, while the biases decide about the placement of AF in the input space.","The biases should ensure the introduction of the most nonlinear fragments of AFs into the input hypercube.","These fragments are most useful for modeling TF fluctuations.","According to the methods described in [9] and [10], we first select the proper interval for the weights based on the AF features and TF properties.","Then, the biases are calculated based on the weights and data distribution.","In [11], a data-driven method was proposed to improve further the FNN randomized learning.","This method introduces the AFs into randomly selected regions of the input space and adjusts the slopes of individual AFs to the TF slopes in these regions.","As a result, the AFs mimic the TF locally and their linear combination approximates smoothly the entire TF.","An interesting method of generating parameters for RVFL was proposed recently in [12].","For this, the authors employ support-vector machines which in a supervised manner, by solving their corresponding optimization problems, generate the pre-trained weights.","These weights are used to initialize the hidden layer of the proposed RVFL architecture.","An alternative approach to generating hidden nodes in FNNs is unsupervised parameter learning using autoencoders (AEs), which was first introduced in [13].","AE, in the encoding phase, transforms input data into a meaningful feature representation obtained from the hidden layer.","Then, in the decoding phase, this feature representation is converted to the original inputs.","The information hidden in original data can be explored and encoded into the output weights of AE [14].","These output weights are then introduced to FNN as hidden node weights instead of randomly generated weights [13].","This approach was applied in [15] for classification tasks.","Here, RVFL uses a sparse AE with $\\ell _1$ -norm regularization to adaptively learn superior hidden node parameters for specific learning tasks.","The authors claim that the learned network parameters in their sparse pre-trained RVFL are embedded with the valuable information about input data, which alleviates the randomly generated parameter issue and improves algorithmic performance.","Another classifier based on RVFL with unsupervised parameter learning was proposed in [16].","In this solution, randomization based stacked AEs with a denoising criterion are used to extract better, higher-level representations.","Each randomization based AE acts as an independent feature extractor and a deep network is obtained by stacking several such AEs.","The network is built hierarchically with high level feature extraction followed by a final classification layer, which is RVFL with direct links.","The authors of both works on AE based randomized learning, [15] and [16], report experimental results on many real-world classification data sets from different domains.","The results confirm the excellent effectiveness of the proposed solutions.","Encouraged by these state-of-the-art results for FNN classifiers trained in an unsupervised manner using AEs, in this study, we analyze unsupervised parameter learning of FNN for regression.","We compare this approach with alternative methods, which were proposed recently in [10].","This paper makes the following contributions: We analyze AE based unsupervised parameter learning for a FNN regression model and find that this method has some drawbacks.","We show how to improve it.","We propose a learning method of AE that controls the produced random weights for FNN.","We also propose how to determine the biases for FNN.","We empirically compare AE based learning with other randomized learning methods proposed recently for regression and find that AE based learning does not outperform its competitors in fitting accuracy, and, in fact, it is much more complex.","The remainder of this paper is structured as follows.","In Section II, we describe randomized FNN learning and methods for generating random parameters.","AE based generating of random parameters is presented and critically analyzed from the perspective of AF distribution and shaping in Section III.","In Section IV, we analyze the complexity of randomized AE.","The performance of AE based randomized learning is evaluated in Section V. Finally, in Section VI, we conclude the work."],["Randomized Learning of FNN","Let us consider a shallow FNN architecture with $n$ inputs, a single-hidden layer including $m$ nonlinear nodes, and a single output.","AFs of hidden nodes, $h_i(\\mathbf {x})$ , map nonlinearly input vectors $\\mathbf {x}=[x_1, x_2,..., x_n]^T$ into $m$ -dimensional feature space.","An output node combines linearly $m$ nonlinear transformations of the inputs.","FNN expresses a function in the form: $\\varphi (\\mathbf {x}) = \\sum _{i=1}^{m}\\beta _ih_i(\\mathbf {x})$ where $\\beta _i$ is the output weight between the $i$ -th hidden node and the output node.","Such shallow architecture has a universal approximation property, even when the hidden layer parameters are not trained but generated randomly from the proper distribution [1], [3].","The output weights $ \\beta = [\\beta _1, \\beta _2, ..., \\beta _m]^T$ can be determined by solving the following linear problem: $\\mathbf {H}\\beta = \\mathbf {Y}$ , where $\\mathbf {H} = [\\mathbf {h}(\\mathbf {x}_1), \\mathbf {h}(\\mathbf {x}_2), ..., \\mathbf {h}(\\mathbf {x}_N)]^T \\in \\mathbb {R}^{N \\times m}$ is the hidden layer output matrix, and $ \\mathbf {Y} = [y_1, y_2, ..., y_N]^T $ is a vector of target outputs.","Using Moore-Penrose pseudoinverse, the optimal solution is given by: $\\beta = \\mathbf {H}^+\\mathbf {Y}$ where $ \\mathbf {H}^+ $ denotes the Moore–Penrose generalized inverse of matrix $ \\mathbf {H} $ .","The hidden node parameters, i.e.","weights $ \\mathbf {a} = [ a_{1}, a_{2}, ..., a_{n}]^T$ and bias $b$ , control AF slope, orientation and position in the input space.","For a sigmoid AF given by the formula: $h(\\mathbf {x}) = \\frac{1}{1 + \\exp \\left(-\\left(\\mathbf {a}^T\\mathbf {x} + b\\right)\\right)}$ weight $a_j$ expresses the sigmoid slope in the $j$ -th direction and bias $b$ decides about the sigmoid shift along a hyperplane containing all $x$ -axes.","An appropriate selection of the slopes and shifts of all sigmoids determine the approximation properties of the model.","As it was shown in [9] and [10], the standard way of selecting both the hidden node weights and biases randomly, from the same interval, $a_{i,j}, b_i \\sim U(-u, u)$ , is misguided.","This is because the optimal interval for weights, which determine the AF slope range, is not the optimal interval for biases, which represent an AF shift.","And vice versa.","With this in mind, in [9] and [10] separate methods for determining weights and biases were proposed.","The approach described in [9] first selects the weights $a_{i,j}$ from $U(-u, u)$ .","The bounds of the interval, $u$ , are adjusted to TF complexity.","For flat TFs we expect lower bounds, while for strongly fluctuating TFs we expect higher bounds.","Once the weights have been selected, the biases are determined in such a way as to ensure the steepest fragments of the sigmoids (which are around their inflection points) are introduced into input hypercube $ H = [x_{1,\\min }, x_{1,\\max }]\\times ... \\times [x_{n,\\min }, x_{n,\\max }] $ .","The resulting equation for the $i$ -th hidden node bias is as follows: $b_i = -\\mathbf {a}_i^T\\mathbf {x}_i^*$ where $\\mathbf {x}_i^*=[x_{i,1}^*, ..., x_{i,n}^*]$ is a point from $H$ where the $i$ -th sigmoid has one of its inflection points.","As you can see from (REF ), the biases are dependent on the weights.","Point $\\mathbf {x}_i^*$ can be selected as follows: this can be some point randomly selected from $H$ : $\\mathbf {x}_i^* \\sim U(H)$ .","This method is suitable when the input points are evenly distributed in $ H $ .","this can be some randomly selected training point: $\\mathbf {x}_i^* = \\mathbf {x}_\\xi \\in \\Phi $ , where $\\xi \\sim U\\lbrace 1, ..., N\\rbrace $ , and $\\Phi $ is a training set.","This method distributes the sigmoids according to the data density, avoiding empty regions.","this can be a prototype of the training point cluster: $\\mathbf {x}_i^* = \\mathbf {p}_i $ , where $ \\mathbf {p}_i $ is a prototype of the $ i $ -th cluster.","This method requires the clustering of training points into $ m =$ #nodes clusters.","It was noticed in [10] that the relationship between weights $a$ and the slope angles of sigmoids $\\alpha $ is highly nonlinear.","The standard interval for $a$ , $U=[-1, 1]$ corresponds to the interval $U_\\alpha =[-14^\\circ , 14^\\circ ]$ for $\\alpha $ , so only flat sigmoids are obtainable in such a case.","To get steep sigmoids, with $\\alpha $ near $90^\\circ $ , the bounds for $U$ should be $u>100$ .","For narrow $U$ , such as $[-1, 1]$ , the distribution of $\\alpha $ is similar to a uniform one.","When the interval for $a$ is extended, the $\\alpha $ distribution changes such that larger angles, near the bounds of $U_\\alpha $ , are more probable than smaller ones.","When $a \\in [-100, 100]$ , more than $77\\%$ of sigmoids are inclined at an angle greater than $80^\\circ $ , so they are very steep.","In such a case, there is a real threat of overfitting.","To generate sigmoids with uniformly distributed slope angles, first, we select randomly $|\\alpha _{i,j}| \\sim U(\\alpha _{\\min }, \\alpha _{\\max })$ , where the bound angles, $\\alpha _{\\min } \\in (0^\\circ , 90^\\circ )$ and $\\alpha _{\\max } \\in (\\alpha _{\\min }, 90^\\circ )$ , are adjusted to the TF complexity.","Then, we calculate the weights from: $a_{i,j}=4 \\tan \\alpha _{i,j}$ Finally, to introduce the sigmoids into the input hypercube $H$ , the biases are calculated from (REF ).","Both methods of generating random parameters of hidden nodes described above, we use in the experimental part of the work as comparative methods for AE based method.","We denote them as R$a$ M, i.e.","a random weights $a$ method, and R$\\alpha $ M, i.e.","a random slope angles $\\alpha $ method.","Fig.","REF illustrates an approximation of a highly nonlinear TF by FNN trained using R$\\alpha $ M (similar results were obtained for R$a$ M).","TF, shown as the dashed line in the left panel, is fitted accurately by the function built by FNN (red line).","This fitted function is composed of 25 hidden node sigmoids, which are shown in the right panel.","Note that the steepest fragments of the sigmoids are introduced by R$\\alpha $ M into the input interval, which is shown by the gray field in the panel on the right.","This fragments are the most useful for modeling the TF fluctuations.","The saturated AF fragments in the input interval are avoided.","The interval for sigmoid slope angles, $U_\\alpha =[0^\\circ , 83^\\circ ]$ , was adjusted to the TF complexity.","This gives a perfect fitting.","Figure: Fitted curve (left panel) and hidden node sigmoids (right panel) for Rα\\alpha M with |α|∼U(0 ∘ ,83 ∘ )|\\alpha | \\sim U(0^\\circ , 83^\\circ )."],["Autoencoder based Generation of Hidden Nodes Parameters","Unsupervised parameter learning using AEs was proposed in [13].","In this work, randomization based autoencoders (RAE) are used for unsupervised feature extraction for a multilayer FNN classifier.","RAE consists of two parts, an encoder and a decoder.","The encoder maps the input data randomly into some latent representation in such a way that the decoder is able to reconstruct the original input data form this representation.","RAE is a single hidden layer FNN with $n$ inputs, $n$ outputs, and $m$ nonlinear hidden nodes.","The sigmoid AFs, $g(\\mathbf {x})$ , are used for the hidden layer and linear AFs are used for the output layer.","The sigmoid AF is given by: $g(\\mathbf {x}) = \\frac{1}{1 + \\exp \\left(-\\left(\\mathbf {w}^T\\mathbf {x} + c\\right)\\right)}$ where $\\mathbf {w} = [ w_{1}, w_{2}, ..., w_{n}]^T $ are the hidden node weights and $ c$ is its bias.","In the first step, the learning method using RAE (RAEM) selects randomly the hidden node parameters for RAE, i.e.","weights $\\mathbf {w}_i$ and biases $ c_i, i = 1, 2, ..., m $ .","Typically, both are taken from the uniform distribution and interval $[-1,1]$ .","In the second step, the hidden layer output matrix is calculated: $\\mathbf {G} = [\\mathbf {g}(\\mathbf {x}_1), \\mathbf {g}(\\mathbf {x}_2), ..., \\mathbf {g}(\\mathbf {x}_N)]^T \\in \\mathbb {R}^{N \\times m}$ , where $\\mathbf {g}(\\mathbf {x}_l)= [g_1(\\mathbf {x}_l), g_2(\\mathbf {x}_l), ..., g_m(\\mathbf {x}_l)] $ is a vector of hidden node outputs for input pattern $\\mathbf {x}_l$ .","Finally, the output weight matrix, $\\mathbf {V} = [ \\mathbf {v}_1, \\mathbf {v}_2, ..., \\mathbf {v}_m]^T\\in \\mathbb {R}^{m \\times n}$ , where $ \\mathbf {v}_i=[v_{i,1}, v_{i,2}, ..., v_{i,n}]$ , is calculated from: $\\mathbf {V} = \\mathbf {G}^+\\mathbf {X}$ where $ \\mathbf {G}^+ $ denotes the Moore–Penrose generalized inverse of matrix $ \\mathbf {G} $ and $\\mathbf {X} \\in \\mathbb {R}^{N \\times n} $ is an input matrix.","Due to randomized learning of RAE, the output weights, $\\mathbf {V}$ , can be obtained easily using a standard linear least-squares method.","These weights are considered to be the latent features of input data [13], and are used as the hidden node weights for FNN instead of random weights.","Thus, $\\mathbf {A} = \\mathbf {V}^T$ , i.e.","$a_{j,i}=v_{i,j}$ .","As for the biases of hidden nodes $b_i$ , it is hard to find in the literature, how they are selected in RAE based FNN learning.","The exception is [15], where the authors outline the way of which the biases were determined.","They calculate the bias for the $i$ -th node as the mean value of this node weights: $b_i=1/n\\sum _{j=1}^{n}a_{j,i}$ .","Let us look at this case from the perspective of AF distribution in the input space.","When the bias is defined as the weight average, the inflection point of the sigmoid for the one-dimensional case, which is for $h(x)=0.5$ , can be obtained from: $h(x) = \\frac{1}{1 + \\exp \\left(-\\left(ax + b\\right)\\right)}=0.5 \\rightarrow x = -\\frac{b}{a} = -1$ where we substituted $b=a$ assuming that a bias is the mean value of the weights.","Thus, the inflection points of all sigmoids are in $x=-1$ .","This case is visualized in Fig.","REF .","Note that the steepest fragments of the sigmoids, which are around point $x=-1$ , are far from the input interval.","This interval includes many saturated fragments of the sigmoids, which results in poor fitting.","The accuracy does not improve with the number of hidden nodes.","Figure: Fitted curve (left panel) and hidden node sigmoids (right panel) for RAEM with b i =a i b_i=a_i.For the $n$ -dimensional case, the sigmoid value in the inflection points is: $h(\\mathbf {x}) = \\frac{1}{1 + \\exp \\left(-\\left(\\mathbf {a}^T\\mathbf {x} + b\\right)\\right)}=0.5 $ Substituting the mean value of weights for $b$ in this equation, after transformations we obtain: $\\mathbf {a}^T\\mathbf {x} + \\overline{a} = 0\\\\$ where $\\overline{a}=\\frac{1}{n}\\sum _{j=1}^{n}a_j$ .","This equation expresses the inflection hyperplane of the sigmoid.","As we can see, this hyperplane is totally dependent on weights $a_j$ .","So, the weights generated by the RAE determine all sigmoid features, slopes in all directions and shift.","The interdependence of the slopes and shift is an undoubted disadvantage of this approach.","It is unjustified and limits the model's flexibility.","The slopes should correspond to the TF complexity and the shift should be related to data distribution.","Unfortunately, RAEM proposed in [15] cannot control separately the slopes and shifts of the sigmoids when it generates them.","When ignoring biases and assuming $b_i=0$ for all hidden nodes, all sigmoids pass through $\\mathbf {x} = (0, ..., 0)$ .","This is also an unacceptable solution as it limits the approximation properties of the model.","In an attempt to improve the RAEM performance, we use the same solution for biases as in R$a$ M and R$\\alpha $ M. We calculate them from (REF ) on the basis of weights $\\mathbf {a}_i$ produced by RAE and randomly selected points $\\mathbf {x}^*$ .","Thus, we distribute the sigmoids in $H$ according to the data distribution.","The result for the one-dimensional case is shown in Fig.","REF .","As we can see, the accuracy did not improve significantly.","Moreover, when we add new hidden nodes (up to 2000), the RMSE remains unacceptable, i.e.","above 0.1.","This means that the problem is not only due to the mis-determination of the biases, but also due to the too flat sigmoids that do not correspond to TF complexity.","Figure: Fitted curve (left panel) and hidden node sigmoids (right panel) for RAEM with b i =-a i x i * b_i=-a_ix^*_i.The sigmoid slopes in FNN are determined by the RAE output weights, $v$ .","These weights are dependent on the RAE random projection $\\mathbf {G}$ , which in turn is dependent on the RAE hidden node random parameters: $w$ and $c$ .","In the above described simulations, these parameters were both selected from the standard interval $U_{AE}=[-1, 1]$ .","As we shown in Section , this is an incorrect approach.","Thus, to improve the regression model performance, we propose to use R$a$ M for generating $w$ and $c$ .","According to this method, we optimize the interval for $w$ , $U_{AE}=[-u_{AE}, u_{AE}]$ , and calculate the biases $c$ analogously to (REF ): $c = -\\mathbf {w}^T\\mathbf {x}^*$ As a result of this modified RAEM learning, we expect that RAE will produce the appropriate weights $\\mathbf {A} = \\mathbf {V}^T$ , which provide FNN sigmoids with the slopes adjusted to the TF complexity.","Fig.","REF shows the effect of the interval $U_{AE}$ bounds on the median of absolute values of weights $v$ (left panel) and on the fitting error (right panel) for TF shown in Figs.","REF -REF (the number of hidden nodes was $m=25$ , results are averaged over 100 runs).","Note that median of $|v|$ decreases quickly with $u_{AE}$ .","RMSE reaches its minimum for $u_{AE} \\in (0.05, 0.2)$ .","For such $U_{AE}$ bounds, RAE produces weights $v$ whose $median|v|$ is in the range from around 5 to 14 (for comparison, when weights $w$ were selected from the standard interval $U_{AE}=[-1, 1]$ , $median|v|$ was around 1.5).","For such a case, RMSE reaches an acceptable level of 0.005, which is similar to those obtained by R$a$ M and R$\\alpha $ M for the same number of hidden nodes.","Fig.","REF depicts fitting results for $U_{AE}=[-0.1, 0.1]$ .","Note steeper sigmoids than in the cases presented in Fig.","REF , resulting in good fitting.","It should be noted that the optimal interval $U_{AE}$ depends on the number of hidden nodes.","When, instead of 25, we used 200 hidden nodes, the optimal values of the interval bounds were $u_{AE} \\in (0.005, 0.022)$ .","In such a case $median|v|$ was from around 4 to 17.","Compare these values with $median(|v|) \\approx 0.19$ which we obtain for $U_{AE}=[-1, 1]$ .","Figure: The effect of the interval U AE U_{AE} bounds on the median of absolute values of weights vv (left panel) and on the fitting error (right panel).Figure: Fitted curve (left panel) and hidden node sigmoids (right panel) for RAEM with w i ∈[-0.1,0.1]w_i \\in [-0.1, 0.1] and c i =-w i x i * c_i=-w_ix^*_i.In the above analysis, RAE was trained without regularization.","However, $\\ell _1$ , $\\ell _2$ and elastic-net regularization are widely used in RAE to prevent overfitting [13], [15], [16].","Regularization in RAE decreases weights $v$ to improve the generalization property of the model, i.e.","generalization of mapping $\\mathbf {x}$ to themselves.","From the point of view of the regression FNN model trained using RAEM, regularisation in RAE is an unfavorable operation because it further flattens FNN sigmoids.","As we showed above, it is a disadvantage for strongly fluctuating TFs.","Note that RAEM in its standard version (without optimization of $U_{AE}$ ) produces weights $v$ dependent only on the distribution of points $\\mathbf {x}$ in the input space and independently of the TF.","So, for two TFs with different complexity (one of them flat and the second one with strong fluctuations), having the same distributed x-points, RAE can produce exactly the same weights $v$ .","This must be considered a serious drawback.","These findings on RAEM can be summarised in tree points: the output weights $v$ produced by RAE determine the sigmoid slopes in the FNN regression model and are crucial for accurate TF approximation.","These weights are dependent on the RAE hidden nodes parameters, i.e.","weights $w$ and biases $c$ .","The standard way of generating both these parameters from the same interval $U_{AE}$ is unjustified and misleading.","We recommend optimizing this interval for weights $w$ and determining biases $c$ from (REF ).","the optimal interval $U_{AE}$ for $w$ depends on the number of hidden nodes $m$ .","Thus, for each value of $m$ considered, interval $U_{AE}$ should be optimized.","the hidden nodes biases of the FNN regression model, $b$ , determine the sigmoid placement in the input space.","They should introduce the steepest fragments of the sigmoids into the input hypercube.","Incorrectly selected, such as $b_i=\\overline{a}_i$ or $b_i=0$ , they lead to the placement of the saturation parts of the sigmoids into $H$ .","To avoid this, we recommend determining biases $b$ from (REF )."],["Complexity of RAE","To generate weights $v$ , RAE can use linear least squares regression, lasso, ridge regression or elastic net algorithms [16].","In all these cases the total time complexity is $O(Nm^2+m^3)$ (this is true for lasso when least-angle regression (LARS) is used for fitting linear regression models [17]).","For $N>m$ the RAE runs in linear time with the size of the training set, $N$ , and in quadratic time with the number of nodes, $m$ .","For $N\\le m$ , it runs in cubic time with $m$ , although in [18] it was shown that for ridge regression this cost can be reduced to $O(N^2m)$ .","Taking into account the optimization process, i.e.","the selection of hyperparameters, the time complexity is as follows.","In RAE without regularization the number of hidden nodes $m$ and interval $U_{AE}$ need to be found using cross-validation.","Thus, the computational load increases linearly with the number of data splits used in cross-validation, $s$ , and also with the number of points of the grid which is $l_ml_u$ , where $l_m$ and $l_u$ are the number of searching values for $m$ and $u_{AE}$ , respectively.","So, the total complexity will be $O(sl_ml_uNm^2+sl_ml_um^3)$ .","In RAE with $\\ell _1$ and $\\ell _2$ regularization, the regularization parameter $\\lambda $ should also be selected in cross-validation.","So, the complexity of RAE with lasso or ridge regression is $O(sl_ml_ul_{\\lambda }Nm^2+sl_ml_ul_{\\lambda }m^3)$ , where $l_{\\lambda }$ is the number of searching points for $\\lambda $ .","Elastic net combines ridge and lasso, with the tuning parameter $\\alpha $ that balances the weights of ridge against lasso.","If the number of searching points for $\\alpha $ is $l_{\\alpha }$ , the total complexity of RAE with elastic net regularization is $O(sl_ml_ul_{\\lambda }l_{\\alpha }Nm^2+sl_ml_ul_{\\lambda }l_{\\alpha }m^3)$ ."],["Simulation Study","In this section, to demonstrate the fitting properties of RAEM, we report some simulation results over several regression problems.","They include an approximation of extremely nonlinear TFs: TF1 $g(\\mathbf {x}) = \\sum _{j=1}^{n}\\sin \\left(20\\cdot \\exp x_j\\right)\\cdot x_j^2, \\, x_i \\in [0, 1]$ TF2 $g(\\mathbf {x}) = -{\\sum _{i=1}^{n} \\sin (x_i) \\sin ^{20} \\left(\\frac{ix_i^2}{\\pi } \\right)}, \\, x_i \\in [0, \\pi ]$ TF3 $g(\\mathbf {x}) = 418.9829n -{\\sum _{i=1}^{n} x_i \\sin (\\sqrt{|x_i|})}, \\\\ x_i \\in [-500, 500]$ We considered these functions with $n=2, 5$ and 10 arguments.","The sizes of the training and test sets depended on the number of arguments.","They were 5000 for $n=2$ , 20,000 for $n=5$ , and 50,000 for $n=10$ .","All arguments for TF3-TF5 were normalized to $[0, 1]$ , and the function values were normalized to $[-1, 1]$ .","Two-argument functions TF1-TF3 are shown in Fig.","REF .","Note that TF1 combines flat regions with strongly fluctuating regions, TF2 expresses flat regions with perpendicular grooves, and TF3 fluctuates strongly, showing the greatest amplitude at the borders.","Figure: Target functions TF1-TF3 for n=2n=2.We use a modified version of RAEM.","That is, the interval for the hidden nodes in RAE, $U_{AE}$ , was optimized for each number of hidden nodes.","This is because, as we show in Section , the optimal interval is dependent on the hidden node number.","To introduce the sigmoids of RAE into input hypercube $H$ , biases $c$ were calculated from (REF ).","And analogously, to introduce the sigmoids of the FNN regression model into $H$ , we calculate biases $b$ from (REF ).","For comparison, we use R$a$ M and R$\\alpha $ M. For each experiment, we run 100 independent training sessions.","In Table REF , we present the fitting test errors (RMSE) of each method for each TF.","The optimal bounds of the intervals from which the model parameters ($w$ , $a$ and $\\alpha $ , respectively) were randomly selected are also shown.","In R$\\alpha $ M, we set fixed lower bounds, $\\alpha _{\\min }=0^\\circ $ , while the upper bounds, $\\alpha _{\\max }$ , were selected for each TF as $90^\\circ $ .","For RAEM and R$a$ M, we observe the difference in the optimal interval sizes between 2-argument and more than 2-argument TFs.","R$a$ M provides wide intervals, i.e.","steep sigmoids, for 2-argument TFs, and narrow intervals, i.e.","flat sigmoids, for 5- and 10-argument TFs.","Similarly RAEM provides steep sigmoids for 2-argument TFs (narrow $U_{AE}$ produces higher weights $a$ ), and flat sigmoids for 5- and 10-argument TFs.","This could be explained by the change in the TF landscape, which flattens with an increasing number of dimensions.","Interestingly, R$\\alpha $ M is insensitive to this phenomenon, giving the same broad interval for $\\alpha $ regardless of the number of arguments.","As can be seen from Table REF , R$\\alpha $ M demonstrates the highest fitting accuracy for all TFs.","This was confirmed by a Wilcoxon signed-rank test with $\\alpha =0.05$ .","Note that results for RAEM and R$a$ M are very similar.","Figs.","REF -REF depict fitting test errors depending on the number of hidden nodes $m$ .","Shaded regions are 10th and 90th percentiles, measured over 100 trials.","As can be seen from these figures, the confidence intervals of RAEM and R$a$ M overlap for higher $m$ .","This means that these two methods generate similar weights $a$ , which provide a similar set of basis functions for FNN.","To study this issue further, we show in Figs.","REF -REF the histograms of weights $a$ generated by RAEM for all TFs.","It is evident from these figures that weight $a$ distributions deviate from the uniform distribution, especially for $n=5$ and 10.","Thus, RAEM, unlike R$a$ M, does not generate uniformly distributed weights $a$ .","The weight distributions are unimodal, symmetrical, bell-shaped, and centered at 0.","Note that the intervals for $a$ observed in Figs.","REF -REF in most cases correspond to the intervals $U$ selected as optimal by R$a$ M (see hyperparameter $u$ for R$a$ M in Table REF ).","Table: Results for TF1-TF3.Figure: RMSE depending on the hidden node numbers for 2-argument TFs.Figure: RMSE depending on the hidden node numbers for 5-argument TFs.Figure: RMSE depending on the hidden node numbers for 10-argument TFs.Figure: Histograms of weights aa generated by RAEM for 2-argument TFs.Figure: Histograms of weights aa generated by RAEM for 5-argument TFs.Figure: Histograms of weights aa generated by RAEM for 10-argument TFs.It is clear from Table REF and Figs.","REF -REF that the best fitting was achieved for R$\\alpha $ M. Interestingly, for each TF this method generated weights $a$ from the same distribution, which is shown in Fig.","REF .","Figure: Histograms of weights aa generated by Rα\\alpha M for all TFs.To compare the performance of RAEM, R$a$ M and R$\\alpha $ M on real-world regression problems, we performed experiments on several data sets from different domains.","Data sets were collected from the KEEL repository, http://www.keel.es/ (stock, laser, treasury, dee, and machineCPU data sets) and from Delve repository, https://www.cs.toronto.edu/$\\sim $ delve/data/kin/desc.html (kin8nm data set).","The number of samples and arguments in each data set are shown in Table REF .","The input and output variables were normalized into $[0, 1]$ .","The data sets were divided into training sets containing 75% of the samples selected randomly, and test sets containing the remaining samples.","The optimal values of hyperparameters, i.e.","hidden node numbers and sizes of intervals for random parameters, were selected by 5-fold cross-validation.","Results are shown in Table REF .","The bold values indicate the lowest errors while the values in italics indicate the highest errors (Wilcoxon signed-rank test with $\\alpha =0.05$ was used to compare errors).","Note that RAEM demonstrate the highest errors for four data sets.","The best method is R$\\alpha $ M, which yielded the lowest errors for five out of six data sets.","Table: Results for real-word regression problems.Fig.","REF compares variants of RAEM: RAEM1 the improved RAEM proposed in this study (optimized interval for weights $w$ , $U_{AE}=[-u_{AE}, u_{AE}]$ ; biases $c$ and $b$ determined from (REF ) and (REF ), respectivelly), RAEM2 variant with the fixed interval for $w$ , $U_{AE}=[-1, 1]$ ; biases $c$ and $b$ determined from (REF ) and (REF ), respectivelly, RAEM3 variant with random selection of both $w$ and $c$ from the fixed interval $[-1, 1]$ ; biases $b$ determined from (REF ), RAEM4 variant with random selection of $w$ , $c$ and $b$ from the fixed interval $[-1, 1]$ , RAEM5 variant with random selection of both $w$ and $c$ from the fixed interval $[-1, 1]$ ; biases $b$ determined as mean values of weights, $b_i = \\overline{a}_i$ .","Note that the proposed modification of RAE in most cases gave the lowest errors compared to other RAE variants.","The laser and treasury data sets were insensitive to the method of generating weights and biases.","For these data sets, the errors for all RAE variants were similar.","Figure: Comparison of RAEM variants."],["Conclusion","In this work, we showed that RAE based randomized learning of FNN suffers from several drawbacks.","First, RAE produces the random weights for the FNN predictive model by taking into account just the input data.","This approach is questionable because the input data does not contain information about TF complexity.","TFs with strong fluctuations need higher weights than flat TFs to be modeled accurately.","Unfortunately, RAE for both these cases can generate similar sets of weights ignoring completely TF complexity.","Second, RAE does not generate the biases for the FNN predictive model.","These biases, which determine the distribution of the activation functions in the input space, are crucial for the approximation properties of the predictive model.","In this study, we propose improved, unsupervised parameter learning using RAEs.","First, we introduce the possibility of controlling the magnitude of the random weights produced by RAE.","This is realized by appropriately generating the RAE hidden node parameters.","Second, we determine the biases for the FNN predictive model so that the sigmoids have their steepest fragments introduced into an input hypercube.","These fragments are the most useful for modeling TF fluctuations.","The proposed modifications make the RAE method more flexible, more data dependent and more dependent on the complexity of the solved problem.","The experimental part of the work does not provide evidence that the improved RAEM outperforms in fitting accuracy other new methods of generating random parameters.","Moreover, its complexity is much greater as it requires additional learning of RAE.","Therefore, applying it to regression problems, rather than simpler and faster methods, may be questionable."]],"string":"[\n [\n \"Autoencoder based Randomized Learning of Feedforward Neural Networks for\\n Regression\"\n ],\n [\n \"Abstract Feedforward neural networks are widely used as universal predictive models to fit data distribution.\",\n \"Common gradient-based learning, however, suffers from many drawbacks making the training process ineffective and time-consuming.\",\n \"Alternative randomized learning does not use gradients but selects hidden node parameters randomly.\",\n \"This makes the training process extremely fast.\",\n \"However, the problem in randomized learning is how to determine the random parameters.\",\n \"A recently proposed method uses autoencoders for unsupervised parameter learning.\",\n \"This method showed superior performance on classification tasks.\",\n \"In this work, we apply this method to regression problems, and, finding that it has some drawbacks, we show how to improve it.\",\n \"We propose a learning method of autoencoders that controls the produced random weights.\",\n \"We also propose how to determine the biases of hidden nodes.\",\n \"We empirically compare autoencoder based learning with other randomized learning methods proposed recently for regression and find that despite the proposed improvement of the autoencoder based learning, it does not outperform its competitors in fitting accuracy.\",\n \"Moreover, the method is much more complex than its competitors.\"\n ],\n [\n \"Introduction\",\n \"Feedforward neural networks (FNNs) have attracted a great deal of interest due to their excellence predictive performance and universal approximation capabilities.\",\n \"The most popular learning methods involve some kind of gradient descent algorithm to learn FNN weights iteratively.\",\n \"However, gradient-based methods suffer from many drawbacks making the learning process ineffective and time-consuming.\",\n \"This is because they are sensitive to the initial values of the parameters.\",\n \"The learning trajectory, starting with different initial parameters, leads to local minima of the loss function.\",\n \"Thus, globally optimal parameters are not guaranteed.\",\n \"Moreover, the learning process is time-consuming for complex target functions (TFs), big data sets and large FNN architectures.\",\n \"Randomized learning, such as a random vector functional link (RVFL) network [1], has been proposed as an alternative to conventional FNNs iterative learning using gradients.\",\n \"In randomized learning, the parameters of the hidden nodes are selected randomly and stay fixed.\",\n \"They do not need to be tuned during the training stage.\",\n \"The only parameters that need to be learned are the output weights.\",\n \"This makes the optimization problem convex [2].\",\n \"As such, it can be solved easily and quickly using a standard least-squares method.\",\n \"Despite this simplification, randomized FNN learning still possesses universal approximation capabilities, provided there are a sufficient number of nonlinear hidden nodes [1], [3].\",\n \"Many simulation studies reported in the literature show the high performance of the randomized FNN when compared to fully adaptable FNNs.\",\n \"Randomization, which is cheaper than optimization, ensures simplicity of implementation and faster training.\",\n \"However, a challenging and still open question in randomized learning is how to choose appropriate weights and biases for the hidden nodes to ensure best model performance [4], [5].\",\n \"To deal with this problem, various methods of generating hidden node parameters have been proposed.\",\n \"The simplest and most popular solution is to select both weights and biases from a uniform distribution over some symmetric interval $U=[-u, u]$ .\",\n \"Usually, this interval is assigned as fixed, typically $[-1, 1]$ , regardless of the data, TF, and type of activation functions (AFs).\",\n \"The independence of the hidden nodes from data is seen as an asset.\",\n \"This overly-simplistic approach was criticized as illogical and misleading [6].\",\n \"Therefore, to improve its performance, optimization of interval $U$ for a specified application is recommended.\",\n \"For example, in [7], a supervisory mechanism which randomly assigns hidden node parameters from an adaptively selected interval, was proposed.\",\n \"This paper clearly reveals that the selection of random parameters should be data dependent to ensure the universal approximation property of the resulting randomized FNN.\",\n \"Data dependent random parameters were also recommended in [8].\",\n \"The authors of this work noticed that if the hidden nodes are chosen at random and not subsequently trained, they are usually not placed in accordance with the density of the input data.\",\n \"In such a case, training of linear parameters is less ineffective at reducing errors.\",\n \"Therefore, in order to improve learning performance, the authors advise unsupervised placement of hidden nodes according to the input data density.\",\n \"This recommendation was implemented in the methods proposed in [9] and [10].\",\n \"In these works, it was noticed that as the weights and biases of hidden nodes have different functions, they should not be selected from the same interval.\",\n \"The weights decide about AF slopes and should reflect TF complexity, while the biases decide about the placement of AF in the input space.\",\n \"The biases should ensure the introduction of the most nonlinear fragments of AFs into the input hypercube.\",\n \"These fragments are most useful for modeling TF fluctuations.\",\n \"According to the methods described in [9] and [10], we first select the proper interval for the weights based on the AF features and TF properties.\",\n \"Then, the biases are calculated based on the weights and data distribution.\",\n \"In [11], a data-driven method was proposed to improve further the FNN randomized learning.\",\n \"This method introduces the AFs into randomly selected regions of the input space and adjusts the slopes of individual AFs to the TF slopes in these regions.\",\n \"As a result, the AFs mimic the TF locally and their linear combination approximates smoothly the entire TF.\",\n \"An interesting method of generating parameters for RVFL was proposed recently in [12].\",\n \"For this, the authors employ support-vector machines which in a supervised manner, by solving their corresponding optimization problems, generate the pre-trained weights.\",\n \"These weights are used to initialize the hidden layer of the proposed RVFL architecture.\",\n \"An alternative approach to generating hidden nodes in FNNs is unsupervised parameter learning using autoencoders (AEs), which was first introduced in [13].\",\n \"AE, in the encoding phase, transforms input data into a meaningful feature representation obtained from the hidden layer.\",\n \"Then, in the decoding phase, this feature representation is converted to the original inputs.\",\n \"The information hidden in original data can be explored and encoded into the output weights of AE [14].\",\n \"These output weights are then introduced to FNN as hidden node weights instead of randomly generated weights [13].\",\n \"This approach was applied in [15] for classification tasks.\",\n \"Here, RVFL uses a sparse AE with $\\\\ell _1$ -norm regularization to adaptively learn superior hidden node parameters for specific learning tasks.\",\n \"The authors claim that the learned network parameters in their sparse pre-trained RVFL are embedded with the valuable information about input data, which alleviates the randomly generated parameter issue and improves algorithmic performance.\",\n \"Another classifier based on RVFL with unsupervised parameter learning was proposed in [16].\",\n \"In this solution, randomization based stacked AEs with a denoising criterion are used to extract better, higher-level representations.\",\n \"Each randomization based AE acts as an independent feature extractor and a deep network is obtained by stacking several such AEs.\",\n \"The network is built hierarchically with high level feature extraction followed by a final classification layer, which is RVFL with direct links.\",\n \"The authors of both works on AE based randomized learning, [15] and [16], report experimental results on many real-world classification data sets from different domains.\",\n \"The results confirm the excellent effectiveness of the proposed solutions.\",\n \"Encouraged by these state-of-the-art results for FNN classifiers trained in an unsupervised manner using AEs, in this study, we analyze unsupervised parameter learning of FNN for regression.\",\n \"We compare this approach with alternative methods, which were proposed recently in [10].\",\n \"This paper makes the following contributions: We analyze AE based unsupervised parameter learning for a FNN regression model and find that this method has some drawbacks.\",\n \"We show how to improve it.\",\n \"We propose a learning method of AE that controls the produced random weights for FNN.\",\n \"We also propose how to determine the biases for FNN.\",\n \"We empirically compare AE based learning with other randomized learning methods proposed recently for regression and find that AE based learning does not outperform its competitors in fitting accuracy, and, in fact, it is much more complex.\",\n \"The remainder of this paper is structured as follows.\",\n \"In Section II, we describe randomized FNN learning and methods for generating random parameters.\",\n \"AE based generating of random parameters is presented and critically analyzed from the perspective of AF distribution and shaping in Section III.\",\n \"In Section IV, we analyze the complexity of randomized AE.\",\n \"The performance of AE based randomized learning is evaluated in Section V. Finally, in Section VI, we conclude the work.\"\n ],\n [\n \"Randomized Learning of FNN\",\n \"Let us consider a shallow FNN architecture with $n$ inputs, a single-hidden layer including $m$ nonlinear nodes, and a single output.\",\n \"AFs of hidden nodes, $h_i(\\\\mathbf {x})$ , map nonlinearly input vectors $\\\\mathbf {x}=[x_1, x_2,..., x_n]^T$ into $m$ -dimensional feature space.\",\n \"An output node combines linearly $m$ nonlinear transformations of the inputs.\",\n \"FNN expresses a function in the form: $\\\\varphi (\\\\mathbf {x}) = \\\\sum _{i=1}^{m}\\\\beta _ih_i(\\\\mathbf {x})$ where $\\\\beta _i$ is the output weight between the $i$ -th hidden node and the output node.\",\n \"Such shallow architecture has a universal approximation property, even when the hidden layer parameters are not trained but generated randomly from the proper distribution [1], [3].\",\n \"The output weights $ \\\\beta = [\\\\beta _1, \\\\beta _2, ..., \\\\beta _m]^T$ can be determined by solving the following linear problem: $\\\\mathbf {H}\\\\beta = \\\\mathbf {Y}$ , where $\\\\mathbf {H} = [\\\\mathbf {h}(\\\\mathbf {x}_1), \\\\mathbf {h}(\\\\mathbf {x}_2), ..., \\\\mathbf {h}(\\\\mathbf {x}_N)]^T \\\\in \\\\mathbb {R}^{N \\\\times m}$ is the hidden layer output matrix, and $ \\\\mathbf {Y} = [y_1, y_2, ..., y_N]^T $ is a vector of target outputs.\",\n \"Using Moore-Penrose pseudoinverse, the optimal solution is given by: $\\\\beta = \\\\mathbf {H}^+\\\\mathbf {Y}$ where $ \\\\mathbf {H}^+ $ denotes the Moore–Penrose generalized inverse of matrix $ \\\\mathbf {H} $ .\",\n \"The hidden node parameters, i.e.\",\n \"weights $ \\\\mathbf {a} = [ a_{1}, a_{2}, ..., a_{n}]^T$ and bias $b$ , control AF slope, orientation and position in the input space.\",\n \"For a sigmoid AF given by the formula: $h(\\\\mathbf {x}) = \\\\frac{1}{1 + \\\\exp \\\\left(-\\\\left(\\\\mathbf {a}^T\\\\mathbf {x} + b\\\\right)\\\\right)}$ weight $a_j$ expresses the sigmoid slope in the $j$ -th direction and bias $b$ decides about the sigmoid shift along a hyperplane containing all $x$ -axes.\",\n \"An appropriate selection of the slopes and shifts of all sigmoids determine the approximation properties of the model.\",\n \"As it was shown in [9] and [10], the standard way of selecting both the hidden node weights and biases randomly, from the same interval, $a_{i,j}, b_i \\\\sim U(-u, u)$ , is misguided.\",\n \"This is because the optimal interval for weights, which determine the AF slope range, is not the optimal interval for biases, which represent an AF shift.\",\n \"And vice versa.\",\n \"With this in mind, in [9] and [10] separate methods for determining weights and biases were proposed.\",\n \"The approach described in [9] first selects the weights $a_{i,j}$ from $U(-u, u)$ .\",\n \"The bounds of the interval, $u$ , are adjusted to TF complexity.\",\n \"For flat TFs we expect lower bounds, while for strongly fluctuating TFs we expect higher bounds.\",\n \"Once the weights have been selected, the biases are determined in such a way as to ensure the steepest fragments of the sigmoids (which are around their inflection points) are introduced into input hypercube $ H = [x_{1,\\\\min }, x_{1,\\\\max }]\\\\times ... \\\\times [x_{n,\\\\min }, x_{n,\\\\max }] $ .\",\n \"The resulting equation for the $i$ -th hidden node bias is as follows: $b_i = -\\\\mathbf {a}_i^T\\\\mathbf {x}_i^*$ where $\\\\mathbf {x}_i^*=[x_{i,1}^*, ..., x_{i,n}^*]$ is a point from $H$ where the $i$ -th sigmoid has one of its inflection points.\",\n \"As you can see from (REF ), the biases are dependent on the weights.\",\n \"Point $\\\\mathbf {x}_i^*$ can be selected as follows: this can be some point randomly selected from $H$ : $\\\\mathbf {x}_i^* \\\\sim U(H)$ .\",\n \"This method is suitable when the input points are evenly distributed in $ H $ .\",\n \"this can be some randomly selected training point: $\\\\mathbf {x}_i^* = \\\\mathbf {x}_\\\\xi \\\\in \\\\Phi $ , where $\\\\xi \\\\sim U\\\\lbrace 1, ..., N\\\\rbrace $ , and $\\\\Phi $ is a training set.\",\n \"This method distributes the sigmoids according to the data density, avoiding empty regions.\",\n \"this can be a prototype of the training point cluster: $\\\\mathbf {x}_i^* = \\\\mathbf {p}_i $ , where $ \\\\mathbf {p}_i $ is a prototype of the $ i $ -th cluster.\",\n \"This method requires the clustering of training points into $ m =$ #nodes clusters.\",\n \"It was noticed in [10] that the relationship between weights $a$ and the slope angles of sigmoids $\\\\alpha $ is highly nonlinear.\",\n \"The standard interval for $a$ , $U=[-1, 1]$ corresponds to the interval $U_\\\\alpha =[-14^\\\\circ , 14^\\\\circ ]$ for $\\\\alpha $ , so only flat sigmoids are obtainable in such a case.\",\n \"To get steep sigmoids, with $\\\\alpha $ near $90^\\\\circ $ , the bounds for $U$ should be $u>100$ .\",\n \"For narrow $U$ , such as $[-1, 1]$ , the distribution of $\\\\alpha $ is similar to a uniform one.\",\n \"When the interval for $a$ is extended, the $\\\\alpha $ distribution changes such that larger angles, near the bounds of $U_\\\\alpha $ , are more probable than smaller ones.\",\n \"When $a \\\\in [-100, 100]$ , more than $77\\\\%$ of sigmoids are inclined at an angle greater than $80^\\\\circ $ , so they are very steep.\",\n \"In such a case, there is a real threat of overfitting.\",\n \"To generate sigmoids with uniformly distributed slope angles, first, we select randomly $|\\\\alpha _{i,j}| \\\\sim U(\\\\alpha _{\\\\min }, \\\\alpha _{\\\\max })$ , where the bound angles, $\\\\alpha _{\\\\min } \\\\in (0^\\\\circ , 90^\\\\circ )$ and $\\\\alpha _{\\\\max } \\\\in (\\\\alpha _{\\\\min }, 90^\\\\circ )$ , are adjusted to the TF complexity.\",\n \"Then, we calculate the weights from: $a_{i,j}=4 \\\\tan \\\\alpha _{i,j}$ Finally, to introduce the sigmoids into the input hypercube $H$ , the biases are calculated from (REF ).\",\n \"Both methods of generating random parameters of hidden nodes described above, we use in the experimental part of the work as comparative methods for AE based method.\",\n \"We denote them as R$a$ M, i.e.\",\n \"a random weights $a$ method, and R$\\\\alpha $ M, i.e.\",\n \"a random slope angles $\\\\alpha $ method.\",\n \"Fig.\",\n \"REF illustrates an approximation of a highly nonlinear TF by FNN trained using R$\\\\alpha $ M (similar results were obtained for R$a$ M).\",\n \"TF, shown as the dashed line in the left panel, is fitted accurately by the function built by FNN (red line).\",\n \"This fitted function is composed of 25 hidden node sigmoids, which are shown in the right panel.\",\n \"Note that the steepest fragments of the sigmoids are introduced by R$\\\\alpha $ M into the input interval, which is shown by the gray field in the panel on the right.\",\n \"This fragments are the most useful for modeling the TF fluctuations.\",\n \"The saturated AF fragments in the input interval are avoided.\",\n \"The interval for sigmoid slope angles, $U_\\\\alpha =[0^\\\\circ , 83^\\\\circ ]$ , was adjusted to the TF complexity.\",\n \"This gives a perfect fitting.\",\n \"Figure: Fitted curve (left panel) and hidden node sigmoids (right panel) for Rα\\\\alpha M with |α|∼U(0 ∘ ,83 ∘ )|\\\\alpha | \\\\sim U(0^\\\\circ , 83^\\\\circ ).\"\n ],\n [\n \"Autoencoder based Generation of Hidden Nodes Parameters\",\n \"Unsupervised parameter learning using AEs was proposed in [13].\",\n \"In this work, randomization based autoencoders (RAE) are used for unsupervised feature extraction for a multilayer FNN classifier.\",\n \"RAE consists of two parts, an encoder and a decoder.\",\n \"The encoder maps the input data randomly into some latent representation in such a way that the decoder is able to reconstruct the original input data form this representation.\",\n \"RAE is a single hidden layer FNN with $n$ inputs, $n$ outputs, and $m$ nonlinear hidden nodes.\",\n \"The sigmoid AFs, $g(\\\\mathbf {x})$ , are used for the hidden layer and linear AFs are used for the output layer.\",\n \"The sigmoid AF is given by: $g(\\\\mathbf {x}) = \\\\frac{1}{1 + \\\\exp \\\\left(-\\\\left(\\\\mathbf {w}^T\\\\mathbf {x} + c\\\\right)\\\\right)}$ where $\\\\mathbf {w} = [ w_{1}, w_{2}, ..., w_{n}]^T $ are the hidden node weights and $ c$ is its bias.\",\n \"In the first step, the learning method using RAE (RAEM) selects randomly the hidden node parameters for RAE, i.e.\",\n \"weights $\\\\mathbf {w}_i$ and biases $ c_i, i = 1, 2, ..., m $ .\",\n \"Typically, both are taken from the uniform distribution and interval $[-1,1]$ .\",\n \"In the second step, the hidden layer output matrix is calculated: $\\\\mathbf {G} = [\\\\mathbf {g}(\\\\mathbf {x}_1), \\\\mathbf {g}(\\\\mathbf {x}_2), ..., \\\\mathbf {g}(\\\\mathbf {x}_N)]^T \\\\in \\\\mathbb {R}^{N \\\\times m}$ , where $\\\\mathbf {g}(\\\\mathbf {x}_l)= [g_1(\\\\mathbf {x}_l), g_2(\\\\mathbf {x}_l), ..., g_m(\\\\mathbf {x}_l)] $ is a vector of hidden node outputs for input pattern $\\\\mathbf {x}_l$ .\",\n \"Finally, the output weight matrix, $\\\\mathbf {V} = [ \\\\mathbf {v}_1, \\\\mathbf {v}_2, ..., \\\\mathbf {v}_m]^T\\\\in \\\\mathbb {R}^{m \\\\times n}$ , where $ \\\\mathbf {v}_i=[v_{i,1}, v_{i,2}, ..., v_{i,n}]$ , is calculated from: $\\\\mathbf {V} = \\\\mathbf {G}^+\\\\mathbf {X}$ where $ \\\\mathbf {G}^+ $ denotes the Moore–Penrose generalized inverse of matrix $ \\\\mathbf {G} $ and $\\\\mathbf {X} \\\\in \\\\mathbb {R}^{N \\\\times n} $ is an input matrix.\",\n \"Due to randomized learning of RAE, the output weights, $\\\\mathbf {V}$ , can be obtained easily using a standard linear least-squares method.\",\n \"These weights are considered to be the latent features of input data [13], and are used as the hidden node weights for FNN instead of random weights.\",\n \"Thus, $\\\\mathbf {A} = \\\\mathbf {V}^T$ , i.e.\",\n \"$a_{j,i}=v_{i,j}$ .\",\n \"As for the biases of hidden nodes $b_i$ , it is hard to find in the literature, how they are selected in RAE based FNN learning.\",\n \"The exception is [15], where the authors outline the way of which the biases were determined.\",\n \"They calculate the bias for the $i$ -th node as the mean value of this node weights: $b_i=1/n\\\\sum _{j=1}^{n}a_{j,i}$ .\",\n \"Let us look at this case from the perspective of AF distribution in the input space.\",\n \"When the bias is defined as the weight average, the inflection point of the sigmoid for the one-dimensional case, which is for $h(x)=0.5$ , can be obtained from: $h(x) = \\\\frac{1}{1 + \\\\exp \\\\left(-\\\\left(ax + b\\\\right)\\\\right)}=0.5 \\\\rightarrow x = -\\\\frac{b}{a} = -1$ where we substituted $b=a$ assuming that a bias is the mean value of the weights.\",\n \"Thus, the inflection points of all sigmoids are in $x=-1$ .\",\n \"This case is visualized in Fig.\",\n \"REF .\",\n \"Note that the steepest fragments of the sigmoids, which are around point $x=-1$ , are far from the input interval.\",\n \"This interval includes many saturated fragments of the sigmoids, which results in poor fitting.\",\n \"The accuracy does not improve with the number of hidden nodes.\",\n \"Figure: Fitted curve (left panel) and hidden node sigmoids (right panel) for RAEM with b i =a i b_i=a_i.For the $n$ -dimensional case, the sigmoid value in the inflection points is: $h(\\\\mathbf {x}) = \\\\frac{1}{1 + \\\\exp \\\\left(-\\\\left(\\\\mathbf {a}^T\\\\mathbf {x} + b\\\\right)\\\\right)}=0.5 $ Substituting the mean value of weights for $b$ in this equation, after transformations we obtain: $\\\\mathbf {a}^T\\\\mathbf {x} + \\\\overline{a} = 0\\\\\\\\$ where $\\\\overline{a}=\\\\frac{1}{n}\\\\sum _{j=1}^{n}a_j$ .\",\n \"This equation expresses the inflection hyperplane of the sigmoid.\",\n \"As we can see, this hyperplane is totally dependent on weights $a_j$ .\",\n \"So, the weights generated by the RAE determine all sigmoid features, slopes in all directions and shift.\",\n \"The interdependence of the slopes and shift is an undoubted disadvantage of this approach.\",\n \"It is unjustified and limits the model's flexibility.\",\n \"The slopes should correspond to the TF complexity and the shift should be related to data distribution.\",\n \"Unfortunately, RAEM proposed in [15] cannot control separately the slopes and shifts of the sigmoids when it generates them.\",\n \"When ignoring biases and assuming $b_i=0$ for all hidden nodes, all sigmoids pass through $\\\\mathbf {x} = (0, ..., 0)$ .\",\n \"This is also an unacceptable solution as it limits the approximation properties of the model.\",\n \"In an attempt to improve the RAEM performance, we use the same solution for biases as in R$a$ M and R$\\\\alpha $ M. We calculate them from (REF ) on the basis of weights $\\\\mathbf {a}_i$ produced by RAE and randomly selected points $\\\\mathbf {x}^*$ .\",\n \"Thus, we distribute the sigmoids in $H$ according to the data distribution.\",\n \"The result for the one-dimensional case is shown in Fig.\",\n \"REF .\",\n \"As we can see, the accuracy did not improve significantly.\",\n \"Moreover, when we add new hidden nodes (up to 2000), the RMSE remains unacceptable, i.e.\",\n \"above 0.1.\",\n \"This means that the problem is not only due to the mis-determination of the biases, but also due to the too flat sigmoids that do not correspond to TF complexity.\",\n \"Figure: Fitted curve (left panel) and hidden node sigmoids (right panel) for RAEM with b i =-a i x i * b_i=-a_ix^*_i.The sigmoid slopes in FNN are determined by the RAE output weights, $v$ .\",\n \"These weights are dependent on the RAE random projection $\\\\mathbf {G}$ , which in turn is dependent on the RAE hidden node random parameters: $w$ and $c$ .\",\n \"In the above described simulations, these parameters were both selected from the standard interval $U_{AE}=[-1, 1]$ .\",\n \"As we shown in Section , this is an incorrect approach.\",\n \"Thus, to improve the regression model performance, we propose to use R$a$ M for generating $w$ and $c$ .\",\n \"According to this method, we optimize the interval for $w$ , $U_{AE}=[-u_{AE}, u_{AE}]$ , and calculate the biases $c$ analogously to (REF ): $c = -\\\\mathbf {w}^T\\\\mathbf {x}^*$ As a result of this modified RAEM learning, we expect that RAE will produce the appropriate weights $\\\\mathbf {A} = \\\\mathbf {V}^T$ , which provide FNN sigmoids with the slopes adjusted to the TF complexity.\",\n \"Fig.\",\n \"REF shows the effect of the interval $U_{AE}$ bounds on the median of absolute values of weights $v$ (left panel) and on the fitting error (right panel) for TF shown in Figs.\",\n \"REF -REF (the number of hidden nodes was $m=25$ , results are averaged over 100 runs).\",\n \"Note that median of $|v|$ decreases quickly with $u_{AE}$ .\",\n \"RMSE reaches its minimum for $u_{AE} \\\\in (0.05, 0.2)$ .\",\n \"For such $U_{AE}$ bounds, RAE produces weights $v$ whose $median|v|$ is in the range from around 5 to 14 (for comparison, when weights $w$ were selected from the standard interval $U_{AE}=[-1, 1]$ , $median|v|$ was around 1.5).\",\n \"For such a case, RMSE reaches an acceptable level of 0.005, which is similar to those obtained by R$a$ M and R$\\\\alpha $ M for the same number of hidden nodes.\",\n \"Fig.\",\n \"REF depicts fitting results for $U_{AE}=[-0.1, 0.1]$ .\",\n \"Note steeper sigmoids than in the cases presented in Fig.\",\n \"REF , resulting in good fitting.\",\n \"It should be noted that the optimal interval $U_{AE}$ depends on the number of hidden nodes.\",\n \"When, instead of 25, we used 200 hidden nodes, the optimal values of the interval bounds were $u_{AE} \\\\in (0.005, 0.022)$ .\",\n \"In such a case $median|v|$ was from around 4 to 17.\",\n \"Compare these values with $median(|v|) \\\\approx 0.19$ which we obtain for $U_{AE}=[-1, 1]$ .\",\n \"Figure: The effect of the interval U AE U_{AE} bounds on the median of absolute values of weights vv (left panel) and on the fitting error (right panel).Figure: Fitted curve (left panel) and hidden node sigmoids (right panel) for RAEM with w i ∈[-0.1,0.1]w_i \\\\in [-0.1, 0.1] and c i =-w i x i * c_i=-w_ix^*_i.In the above analysis, RAE was trained without regularization.\",\n \"However, $\\\\ell _1$ , $\\\\ell _2$ and elastic-net regularization are widely used in RAE to prevent overfitting [13], [15], [16].\",\n \"Regularization in RAE decreases weights $v$ to improve the generalization property of the model, i.e.\",\n \"generalization of mapping $\\\\mathbf {x}$ to themselves.\",\n \"From the point of view of the regression FNN model trained using RAEM, regularisation in RAE is an unfavorable operation because it further flattens FNN sigmoids.\",\n \"As we showed above, it is a disadvantage for strongly fluctuating TFs.\",\n \"Note that RAEM in its standard version (without optimization of $U_{AE}$ ) produces weights $v$ dependent only on the distribution of points $\\\\mathbf {x}$ in the input space and independently of the TF.\",\n \"So, for two TFs with different complexity (one of them flat and the second one with strong fluctuations), having the same distributed x-points, RAE can produce exactly the same weights $v$ .\",\n \"This must be considered a serious drawback.\",\n \"These findings on RAEM can be summarised in tree points: the output weights $v$ produced by RAE determine the sigmoid slopes in the FNN regression model and are crucial for accurate TF approximation.\",\n \"These weights are dependent on the RAE hidden nodes parameters, i.e.\",\n \"weights $w$ and biases $c$ .\",\n \"The standard way of generating both these parameters from the same interval $U_{AE}$ is unjustified and misleading.\",\n \"We recommend optimizing this interval for weights $w$ and determining biases $c$ from (REF ).\",\n \"the optimal interval $U_{AE}$ for $w$ depends on the number of hidden nodes $m$ .\",\n \"Thus, for each value of $m$ considered, interval $U_{AE}$ should be optimized.\",\n \"the hidden nodes biases of the FNN regression model, $b$ , determine the sigmoid placement in the input space.\",\n \"They should introduce the steepest fragments of the sigmoids into the input hypercube.\",\n \"Incorrectly selected, such as $b_i=\\\\overline{a}_i$ or $b_i=0$ , they lead to the placement of the saturation parts of the sigmoids into $H$ .\",\n \"To avoid this, we recommend determining biases $b$ from (REF ).\"\n ],\n [\n \"Complexity of RAE\",\n \"To generate weights $v$ , RAE can use linear least squares regression, lasso, ridge regression or elastic net algorithms [16].\",\n \"In all these cases the total time complexity is $O(Nm^2+m^3)$ (this is true for lasso when least-angle regression (LARS) is used for fitting linear regression models [17]).\",\n \"For $N>m$ the RAE runs in linear time with the size of the training set, $N$ , and in quadratic time with the number of nodes, $m$ .\",\n \"For $N\\\\le m$ , it runs in cubic time with $m$ , although in [18] it was shown that for ridge regression this cost can be reduced to $O(N^2m)$ .\",\n \"Taking into account the optimization process, i.e.\",\n \"the selection of hyperparameters, the time complexity is as follows.\",\n \"In RAE without regularization the number of hidden nodes $m$ and interval $U_{AE}$ need to be found using cross-validation.\",\n \"Thus, the computational load increases linearly with the number of data splits used in cross-validation, $s$ , and also with the number of points of the grid which is $l_ml_u$ , where $l_m$ and $l_u$ are the number of searching values for $m$ and $u_{AE}$ , respectively.\",\n \"So, the total complexity will be $O(sl_ml_uNm^2+sl_ml_um^3)$ .\",\n \"In RAE with $\\\\ell _1$ and $\\\\ell _2$ regularization, the regularization parameter $\\\\lambda $ should also be selected in cross-validation.\",\n \"So, the complexity of RAE with lasso or ridge regression is $O(sl_ml_ul_{\\\\lambda }Nm^2+sl_ml_ul_{\\\\lambda }m^3)$ , where $l_{\\\\lambda }$ is the number of searching points for $\\\\lambda $ .\",\n \"Elastic net combines ridge and lasso, with the tuning parameter $\\\\alpha $ that balances the weights of ridge against lasso.\",\n \"If the number of searching points for $\\\\alpha $ is $l_{\\\\alpha }$ , the total complexity of RAE with elastic net regularization is $O(sl_ml_ul_{\\\\lambda }l_{\\\\alpha }Nm^2+sl_ml_ul_{\\\\lambda }l_{\\\\alpha }m^3)$ .\"\n ],\n [\n \"Simulation Study\",\n \"In this section, to demonstrate the fitting properties of RAEM, we report some simulation results over several regression problems.\",\n \"They include an approximation of extremely nonlinear TFs: TF1 $g(\\\\mathbf {x}) = \\\\sum _{j=1}^{n}\\\\sin \\\\left(20\\\\cdot \\\\exp x_j\\\\right)\\\\cdot x_j^2, \\\\, x_i \\\\in [0, 1]$ TF2 $g(\\\\mathbf {x}) = -{\\\\sum _{i=1}^{n} \\\\sin (x_i) \\\\sin ^{20} \\\\left(\\\\frac{ix_i^2}{\\\\pi } \\\\right)}, \\\\, x_i \\\\in [0, \\\\pi ]$ TF3 $g(\\\\mathbf {x}) = 418.9829n -{\\\\sum _{i=1}^{n} x_i \\\\sin (\\\\sqrt{|x_i|})}, \\\\\\\\ x_i \\\\in [-500, 500]$ We considered these functions with $n=2, 5$ and 10 arguments.\",\n \"The sizes of the training and test sets depended on the number of arguments.\",\n \"They were 5000 for $n=2$ , 20,000 for $n=5$ , and 50,000 for $n=10$ .\",\n \"All arguments for TF3-TF5 were normalized to $[0, 1]$ , and the function values were normalized to $[-1, 1]$ .\",\n \"Two-argument functions TF1-TF3 are shown in Fig.\",\n \"REF .\",\n \"Note that TF1 combines flat regions with strongly fluctuating regions, TF2 expresses flat regions with perpendicular grooves, and TF3 fluctuates strongly, showing the greatest amplitude at the borders.\",\n \"Figure: Target functions TF1-TF3 for n=2n=2.We use a modified version of RAEM.\",\n \"That is, the interval for the hidden nodes in RAE, $U_{AE}$ , was optimized for each number of hidden nodes.\",\n \"This is because, as we show in Section , the optimal interval is dependent on the hidden node number.\",\n \"To introduce the sigmoids of RAE into input hypercube $H$ , biases $c$ were calculated from (REF ).\",\n \"And analogously, to introduce the sigmoids of the FNN regression model into $H$ , we calculate biases $b$ from (REF ).\",\n \"For comparison, we use R$a$ M and R$\\\\alpha $ M. For each experiment, we run 100 independent training sessions.\",\n \"In Table REF , we present the fitting test errors (RMSE) of each method for each TF.\",\n \"The optimal bounds of the intervals from which the model parameters ($w$ , $a$ and $\\\\alpha $ , respectively) were randomly selected are also shown.\",\n \"In R$\\\\alpha $ M, we set fixed lower bounds, $\\\\alpha _{\\\\min }=0^\\\\circ $ , while the upper bounds, $\\\\alpha _{\\\\max }$ , were selected for each TF as $90^\\\\circ $ .\",\n \"For RAEM and R$a$ M, we observe the difference in the optimal interval sizes between 2-argument and more than 2-argument TFs.\",\n \"R$a$ M provides wide intervals, i.e.\",\n \"steep sigmoids, for 2-argument TFs, and narrow intervals, i.e.\",\n \"flat sigmoids, for 5- and 10-argument TFs.\",\n \"Similarly RAEM provides steep sigmoids for 2-argument TFs (narrow $U_{AE}$ produces higher weights $a$ ), and flat sigmoids for 5- and 10-argument TFs.\",\n \"This could be explained by the change in the TF landscape, which flattens with an increasing number of dimensions.\",\n \"Interestingly, R$\\\\alpha $ M is insensitive to this phenomenon, giving the same broad interval for $\\\\alpha $ regardless of the number of arguments.\",\n \"As can be seen from Table REF , R$\\\\alpha $ M demonstrates the highest fitting accuracy for all TFs.\",\n \"This was confirmed by a Wilcoxon signed-rank test with $\\\\alpha =0.05$ .\",\n \"Note that results for RAEM and R$a$ M are very similar.\",\n \"Figs.\",\n \"REF -REF depict fitting test errors depending on the number of hidden nodes $m$ .\",\n \"Shaded regions are 10th and 90th percentiles, measured over 100 trials.\",\n \"As can be seen from these figures, the confidence intervals of RAEM and R$a$ M overlap for higher $m$ .\",\n \"This means that these two methods generate similar weights $a$ , which provide a similar set of basis functions for FNN.\",\n \"To study this issue further, we show in Figs.\",\n \"REF -REF the histograms of weights $a$ generated by RAEM for all TFs.\",\n \"It is evident from these figures that weight $a$ distributions deviate from the uniform distribution, especially for $n=5$ and 10.\",\n \"Thus, RAEM, unlike R$a$ M, does not generate uniformly distributed weights $a$ .\",\n \"The weight distributions are unimodal, symmetrical, bell-shaped, and centered at 0.\",\n \"Note that the intervals for $a$ observed in Figs.\",\n \"REF -REF in most cases correspond to the intervals $U$ selected as optimal by R$a$ M (see hyperparameter $u$ for R$a$ M in Table REF ).\",\n \"Table: Results for TF1-TF3.Figure: RMSE depending on the hidden node numbers for 2-argument TFs.Figure: RMSE depending on the hidden node numbers for 5-argument TFs.Figure: RMSE depending on the hidden node numbers for 10-argument TFs.Figure: Histograms of weights aa generated by RAEM for 2-argument TFs.Figure: Histograms of weights aa generated by RAEM for 5-argument TFs.Figure: Histograms of weights aa generated by RAEM for 10-argument TFs.It is clear from Table REF and Figs.\",\n \"REF -REF that the best fitting was achieved for R$\\\\alpha $ M. Interestingly, for each TF this method generated weights $a$ from the same distribution, which is shown in Fig.\",\n \"REF .\",\n \"Figure: Histograms of weights aa generated by Rα\\\\alpha M for all TFs.To compare the performance of RAEM, R$a$ M and R$\\\\alpha $ M on real-world regression problems, we performed experiments on several data sets from different domains.\",\n \"Data sets were collected from the KEEL repository, http://www.keel.es/ (stock, laser, treasury, dee, and machineCPU data sets) and from Delve repository, https://www.cs.toronto.edu/$\\\\sim $ delve/data/kin/desc.html (kin8nm data set).\",\n \"The number of samples and arguments in each data set are shown in Table REF .\",\n \"The input and output variables were normalized into $[0, 1]$ .\",\n \"The data sets were divided into training sets containing 75% of the samples selected randomly, and test sets containing the remaining samples.\",\n \"The optimal values of hyperparameters, i.e.\",\n \"hidden node numbers and sizes of intervals for random parameters, were selected by 5-fold cross-validation.\",\n \"Results are shown in Table REF .\",\n \"The bold values indicate the lowest errors while the values in italics indicate the highest errors (Wilcoxon signed-rank test with $\\\\alpha =0.05$ was used to compare errors).\",\n \"Note that RAEM demonstrate the highest errors for four data sets.\",\n \"The best method is R$\\\\alpha $ M, which yielded the lowest errors for five out of six data sets.\",\n \"Table: Results for real-word regression problems.Fig.\",\n \"REF compares variants of RAEM: RAEM1 the improved RAEM proposed in this study (optimized interval for weights $w$ , $U_{AE}=[-u_{AE}, u_{AE}]$ ; biases $c$ and $b$ determined from (REF ) and (REF ), respectivelly), RAEM2 variant with the fixed interval for $w$ , $U_{AE}=[-1, 1]$ ; biases $c$ and $b$ determined from (REF ) and (REF ), respectivelly, RAEM3 variant with random selection of both $w$ and $c$ from the fixed interval $[-1, 1]$ ; biases $b$ determined from (REF ), RAEM4 variant with random selection of $w$ , $c$ and $b$ from the fixed interval $[-1, 1]$ , RAEM5 variant with random selection of both $w$ and $c$ from the fixed interval $[-1, 1]$ ; biases $b$ determined as mean values of weights, $b_i = \\\\overline{a}_i$ .\",\n \"Note that the proposed modification of RAE in most cases gave the lowest errors compared to other RAE variants.\",\n \"The laser and treasury data sets were insensitive to the method of generating weights and biases.\",\n \"For these data sets, the errors for all RAE variants were similar.\",\n \"Figure: Comparison of RAEM variants.\"\n ],\n [\n \"Conclusion\",\n \"In this work, we showed that RAE based randomized learning of FNN suffers from several drawbacks.\",\n \"First, RAE produces the random weights for the FNN predictive model by taking into account just the input data.\",\n \"This approach is questionable because the input data does not contain information about TF complexity.\",\n \"TFs with strong fluctuations need higher weights than flat TFs to be modeled accurately.\",\n \"Unfortunately, RAE for both these cases can generate similar sets of weights ignoring completely TF complexity.\",\n \"Second, RAE does not generate the biases for the FNN predictive model.\",\n \"These biases, which determine the distribution of the activation functions in the input space, are crucial for the approximation properties of the predictive model.\",\n \"In this study, we propose improved, unsupervised parameter learning using RAEs.\",\n \"First, we introduce the possibility of controlling the magnitude of the random weights produced by RAE.\",\n \"This is realized by appropriately generating the RAE hidden node parameters.\",\n \"Second, we determine the biases for the FNN predictive model so that the sigmoids have their steepest fragments introduced into an input hypercube.\",\n \"These fragments are the most useful for modeling TF fluctuations.\",\n \"The proposed modifications make the RAE method more flexible, more data dependent and more dependent on the complexity of the solved problem.\",\n \"The experimental part of the work does not provide evidence that the improved RAEM outperforms in fitting accuracy other new methods of generating random parameters.\",\n \"Moreover, its complexity is much greater as it requires additional learning of RAE.\",\n \"Therefore, applying it to regression problems, rather than simpler and faster methods, may be questionable.\"\n ]\n]"},"id":{"kind":"string","value":"2107.01711"}}},{"rowIdx":1587,"cells":{"text":{"kind":"list like","value":[["Convex optimization of bioprocesses"],["Abstract We optimize a general model of bioprocesses, which is nonconvex due to the microbial growth in the biochemical reactors.","We formulate a convex relaxation and give conditions guaranteeing its exactness in both the transient and steady state cases.","When the growth kinetics are modeled by the Monod function under constant biomass or the Contois function, the relaxation is a second-order cone program, which can be solved efficiently at large scales.","We implement the model on a numerical example based on a wastewater treatment system."],["Introduction","We optimize a model of dynamical bioprocesses consisting of a set of biochemical reactors interconnected by diffusion and mass flow.","The objectives include minimizing substrate outflow, maximizing biogas production, and tracking setpoints.","Within each reactor, several microbial reactions convert any number of biotic or abiotic reactants into biomass and/or products.","This setup describes a variety of physical systems such as wastewater treatment networks, the production of various chemicals, and compartmental approximations of bioprocesses in continuous media.","Here we focus on the case when the kinetics can be represented by a second-order cone (SOC) constraint, as recently shown in [1] for the Monod [2] and Contois [3] growth rates.","The most closely related topics to ours are chemical process optimization, control and optimization of wastewater systems, and control of bioprocesses.","Most existing approaches to process optimization [4] and wastewater [5] do not explicitly model the microbial growth, and often use either linear programming or general nonlinear solvers.","There have been many applications of nonlinear control [6] and optimization [7] to bioprocesses, but not convex relaxations or second-order cone programming (SOCP).","Our main results are generalizations of those in [1], which focused on the gradostat with a single reaction [8].","Here we allow for any number of substrates and biomasses, general convex objectives, and multiple biochemical reactions.","Our original theoretical contributions are as follows.","In Section , we formulate a convex relaxation for optimizing the trajectory and steady state solution of a general bioprocess.","In Section , we give conditions under which the relaxations are guaranteed to be exact in both the transient and steady state cases.","To streamline exposition, the only external inputs to the model are the influent concentrations, e.g., biochemical oxygen demand and ammonia.","Our main exactness results straightforwardly apply when the flow rates are also variable, but the resulting bilinearities make the problem nonconvex.","This can be handled using techniques like disjunctive programming, as in [1], or further convex relaxation, which we discuss at the end of Section REF .","We apply our results in two examples.","In Section REF , we show that our exactness conditions simplify to those in [1] when specialized to the gradostat.","In Section REF , we optimize the allocation of sewage to three wastewater treatment plants over two weeks.","The relaxation is exact, and takes roughly twenty minutes to solve using SOCP [9].","The system consists of $s$ well-mixed tanks interconnected by mass flow and diffusion.","We denote the set of tanks $\\mathcal {S}$ .","$V\\in \\mathbb {R}^{s\\times s}$ is a diagonal matrix in which $V_{ii}$ is the volume of tank $i$ .","Tank $i$ has water inlet flow rate $Q_i^{\\textrm {in}}$ and outlet flow rate $Q_i^{\\textrm {out}}$ .","We let $Q_{ij}$ denote the flow from tank $i$ to tank $j$ .","Let $d_{ij}$ denote the diffusion between tanks $i$ and $j$ , where $d_{ij}=d_{ji}$ .","Let $C=\\textrm {diag}\\left[Q_i^{\\textrm {in}}\\right]$ , $M_{ij}&=\\left\\lbrace \\begin{array}{ll}Q_{ji}, & i\\ne j\\\\-Q_i^{\\textrm {out}}-\\sum _{k\\in {\\mathcal {S}}}Q_{ik},& i= j\\end{array}\\right., \\quad L_{ij}&=\\left\\lbrace \\begin{array}{ll}d_{ij}, & i\\ne j\\\\-\\sum _{k\\in {\\mathcal {S}}}d_{ik}, & i= j\\end{array}\\right.,$ and $N=M+L$ .","$M$ and $L$ are respectively compartmental and Laplacian matrices.","$M$ is invertible if the network is outflow connected, which is to say that there is a directed path from every tank to some tank with outflow [10].","Because $L$ is negative semidefinite, $N$ is also invertible if $M$ is outflow connected, and potentially even if $M$ is not outflow connected."],["Microbial growth","We model the microbial growth in the tanks using the notation of Section 1.5 of [6].","There are $m$ substrates and biomasses in each perfectly mixed tank.","$\\xi _i\\in \\mathbb {R}^m_+$ is the process state vector of tank $i\\in \\mathcal {S}$ , which contains the concentrations of the substrates and biomasses, and $\\xi _i^{\\textrm {in}}\\in \\mathbb {R}^m_+$ is the corresponding influent concentration vector.","This model is minimal in that $\\xi $ includes intermediary products, e.g., substrates produced by one reaction and consumed by another, but not final products such as the $\\textrm {CH}_4$ ultimately produced by anaerobic digestion.","There are $r$ different types of reactions that convert substrates to other substrates and biomasses.","$\\phi _i(\\xi _i)\\in \\mathbb {R}^r_+$ is a vector of the reaction kinetics in tank $i$ .","We are interested in the case where the elements of $\\phi _i(\\xi _i)$ are concave functions, and in particular representable as SOC constraints.","We show how to do this for Monod and Contois kinetics later in Examples REF and REF .","Let $\\kappa _i\\in \\mathbb {R}^{m\\times r}$ be the stoichiometric matrix relating the reaction vector, $\\phi _i(\\xi _i)$ , to the evolution of the process state in tank $i$ .","The dynamics in tank $i\\in \\mathcal {S}$ are $V_{ii}\\dot{\\xi }_i=V_{ii}\\kappa _i\\phi _i(\\xi _i) -Q_{i}^{\\textrm {out}}\\xi _i- \\sum _{j\\in \\mathcal {S}}(Q_{ij}+d_{ij})\\xi _i+Q_{i}^{\\textrm {in}}\\xi _i^{\\textrm {in}}+\\sum _{j\\in \\mathcal {S}}(Q_{ji}+d_{ij})\\xi _j.$ The following example illustrates $\\xi _i$ and $\\kappa _i$ .","Example 1 (Two-step anaerobic digestion) There are two substrates, $S_i^a$ and $S_i^b$ , and two biomasses, $X_i^a$ and $X_i^b$ .","We let $\\xi _i=\\left[S_i^a,S_i^b,X_i^a,X_i^b\\right]^{\\top }$ .","$S_i^a$ is converted to both $X_i^a$ and $S_i^b$ at the rate $\\mu ^a\\left(S_i^a\\right)X_i^a$ .","$S_i^b$ is converted to $X_i^b$ at the rate $\\mu ^b\\left(S_i^b\\right)X_i^b$ .","Therefore, $\\phi _i(\\xi _i)=\\left[\\begin{array}{c}\\mu ^a\\left(S_i^a\\right)X_i^a\\\\\\mu ^b\\left(S_i^b\\right)X_i^b\\end{array}\\right]\\;\\textrm {and}\\;\\kappa _i=\\left[\\begin{array}{cc}-1&0\\\\1&-1\\\\1&0\\\\0&1\\end{array}\\right].$ We now write the dynamics in vector form.","We suppress subscripts to represent stacked vectors, i.e., $\\xi =[\\xi _1,...,\\xi _s]^{\\top }$ and $\\phi (\\xi )=[\\phi _1(\\xi _1),...,\\phi _s(\\xi _s)]^{\\top }$ .","Let $A\\otimes B$ denote the Kronecker product of $A$ and $B$ , $I_\\alpha \\in \\mathbb {R}^{\\alpha \\times \\alpha }$ the identity matrix, and $\\hat{A}=A\\otimes I_m$ .","Let $K$ be a block diagonal matrix with $\\kappa _1,...,\\kappa _s$ on its main diagonal.","If $\\kappa _i=\\kappa $ for all $i\\in \\mathcal {S}$ , then $K=I_s\\otimes \\kappa $ .","The dynamics of the full system are given in vector form by $\\hat{V}\\dot{\\xi }=\\hat{V}K \\phi (\\xi )+\\hat{N}\\xi +\\hat{C}\\xi ^{\\textrm {in}}.$ We allow the dynamics to be non-autonomous, in which case $\\hat{N}$ , $\\hat{C}$ , and $\\xi ^{in}$ can be time-varying."],["Discretization in time","To make (REF ) compatible with finite-dimensional optimization, we replace the derivatives with a numerical approximation, which we denote $\\mathcal {D}_n$ .","For example, in the case of the implicit Euler method with time step $\\Delta $ , $\\mathcal {D}_n[\\xi (\\cdot )]=(\\xi (n)-\\xi (n-1))/\\Delta $ .","$\\mathcal {D}_n$ could also be a more sophisticated approximation such as a Runge-Kutta scheme [11].","The time periods are indexed $n\\in \\mathcal {N}=\\lbrace 1,...,\\tau \\rbrace $ .","We have $\\hat{V}\\mathcal {D}_n[\\xi (\\cdot )]=\\hat{V}K \\phi (\\xi (n))+\\hat{N}(n)\\xi (n)+\\hat{C}(n)\\xi ^{\\textrm {in}}(n)$ for $n\\in \\mathcal {N}$ .","The initial condition is $\\xi (0)=\\xi _0$ ."],["Objectives","We consider objectives of the form $\\mathcal {F}(\\xi ,T)=\\sum _{n\\in \\mathcal {N}} \\mathcal {F}_{\\xi }(\\xi (n))+\\mathcal {F}_{\\phi }(T(n)),$ where $\\mathcal {F}_{\\xi }$ and $\\mathcal {F}_{\\phi }$ are convex and $T(n)=\\phi (\\xi (n))$ .","The following are examples.","Minimizing the outflow of substrates, $\\mathcal {F}_{\\xi }(\\xi (n))=\\sum _{i\\in \\mathcal {S}}Q_{i}^{\\textrm {out}}\\eta _i^{\\top }\\xi _i(n),$ where $\\eta _i$ is a vector that selects the entries of $\\xi _i(n)$ corresponding to pollutants.","The production of biogas in a tank is proportional to the kinetics that convert substrates to biomass.","Let $\\sigma _i\\in \\mathbb {R}^m_+$ be a vector that is only nonzero for entries of $T_i(n)$ corresponding to biogas production.","Let $\\mathcal {M}\\subseteq \\mathcal {S}$ be the subset of tanks that can capture biogas from anaerobic digestion.","We maximize biogas through the objective $\\mathcal {F}_{\\phi }(T(n))=-\\sum _{i\\in \\mathcal {M}}V_{ii}\\sigma _i^{\\top }T_i(n).$ Setpoint tracking, $\\mathcal {F}_{\\xi }(\\xi (n))=\\left(\\xi (n)-\\bar{\\xi }\\right)^{\\top }A\\left(\\xi (n)-\\bar{\\xi }\\right),$ where $A\\succeq 0$ and $\\bar{\\xi }$ is a desired operating point."],["Problem statement","We aim to solve the following optimization problem.","$\\mathcal {P}\\quad &\\min \\;\\mathcal {F}(\\xi ,T) \\\\\\textrm {such that}\\quad &T(n)=\\phi (\\xi (n)),\\quad n\\in \\mathcal {N}\\\\&\\hat{V}\\mathcal {D}_n[\\xi (\\cdot )]=\\hat{V}K T(n)+\\hat{N}(n)\\xi (n)+\\hat{C}(n)\\xi ^{\\textrm {in}}(n),\\quad n\\in \\mathcal {N}\\\\&\\left(\\xi ,\\xi ^{\\textrm {in}},T\\right)\\in \\Omega .$ $\\mathcal {P}$ models the optimization of a broad range of bioprocesses such as wastewater treatment.","We refer the reader to [6] for broad coverage of this topic.","A solution of $\\mathcal {P}$ is a trajectory $\\left(\\xi (n),\\xi ^{\\textrm {in}}(n),T(n)\\right)$ , $n\\in \\mathcal {N}$ .","The flows and diffusions between tanks, encoded by the matrices $\\hat{N}(n)$ and $\\hat{C}(n)$ , are not decision variables.","For this reason constraint () is linear.","The set $\\Omega $ in () consists of linear constraints such as the initial condition, total input matter, and maximum substrate concentrations; several other examples are given in [1].","Note that $\\Omega $ can constrain $T(\\cdot )$ so as to allow constraints on the growth without adding nonlinearities.","The only nonconvexity is therefore (), the growth constraint; this is the focus of the next two sections.","We note that in many applications, the flow rates are important decision variables.","They are parameters here because our focus is on incorporating the growth kinetics in a convex fashion.","In the case that the flow rates are variable, () becomes bilinear and hence nonconvex.","There are several ways to handle the bilinearity, including McCormick [12] and lift-and-project [13] relaxations; disjunctive programming reformulations if the flow variables are binary, as in [1]; and finding a local minimum via nonlinear programming, e.g., exploiting the biconvex structure with the Alternating Direction Method of Multipliers [14].","All three of the above techniques are viable because, as described in the next section, we have a tractable way to represent the growth constraint, ()."],["Convex relaxation","$\\mathcal {P}$ is nonconvex because constraint () is a nonlinear equality.","One way around this difficulty is to instead solve a convex relaxation of $\\mathcal {P}$ , as in [1].","If all elements of the vector $\\phi (\\cdot )$ are concave functions, we obtain a convex relaxation by replacing () with the inequality $T(n)\\le \\phi (\\xi (n)),\\quad n\\in \\mathcal {N}.$ We refer to the resulting optimization as $\\mathcal {P}_{\\textrm {R}}$ .","As mentioned earlier, we are interested in the case where (REF ) is concave and, ideally, representable as an SOC constraint.","Several such examples are given below.","Example 2 (Contois growth) Suppose that there is a substrate of concentration $S$ , a biomass of concentration $X$ , and the growth rate is Contois [3].","Then constraint (REF ) takes the form $T(n)\\le \\frac{\\mu S(n)X(n)}{k_{\\textrm {C}}X(n)+S(n)},$ where $\\mu $ and $k_{\\textrm {C}}$ are constant parameters.","As shown in [1], this is concave and can be written as the SOC constraint $\\left\\Vert \\left[\\begin{array}{c}\\mu S(n)\\\\k_{\\textrm {C}}T(n)\\\\\\mu kX(n)\\end{array}\\right]\\right\\Vert \\le \\mu k_{\\textrm {C}}X(n) + \\mu S(n) - kT(n).$ Example 3 (Monod growth with constant biomass) Consider Example REF , but now suppose that the growth rate is Monod [2].","Then constraint (REF ) takes the form $T(n)\\le \\frac{\\mu S(n)X(n)}{k_{\\textrm {M}}+S(n)}.$ This constraint is quasiconcave.","It becomes concave if we assume that the biomass in each time period is not an optimization variable, but an exogenous parameter, i.e., $X(n)=\\bar{X}(n)$ for $n\\in \\mathcal {N}$ .","This approximation is often valid because the biomass concentration is typically larger and varies more slowly than the substrate concentrations, and is therefore relatively insensitive to the substrates.","In this case, as shown in [1], it can be written as the SOC constraint $\\left\\Vert \\left[\\begin{array}{c}\\mu S(n) \\bar{X}(n)\\\\k_{\\textrm {M}}T(n)\\\\\\mu k\\bar{X}(n)\\end{array}\\right]\\right\\Vert \\le \\mu k_{\\textrm {M}}\\bar{X}(n) + \\mu S(n)\\bar{X}(n) - kT(n).$ Example 4 (Interactive and non-interactive growth) Suppose the growth of the biomass depends on two rates, $\\mu ^a(S(n),X(n))$ and $\\mu ^b(S(n),X(n))$ , and the individual kinetics constraints, $T^a(n)\\le \\mu ^a(S(n),X(n))X(n)$ and $T^a(n)\\le \\mu ^b(S(n),X(n))X(n)$ , both have SOC representations.","The dependency often takes one of two forms: non-interactive, $\\min \\lbrace \\mu ^a(S(n),X(n)),\\mu ^b(S(n),X(n))\\rbrace $ , which in ecological modeling is known as Liebig's Law, and interactive, $\\mu ^a(S(n),X(n))\\mu ^b(S(n),X(n))$ , which is common in models of bioprocesses.","The non-interactive case is enforced by the two individual kinetics constraints along with $T(n)\\le T^a(n)$ and $T(n)\\le T^b(n)$ .","The interactive case is in general nonconvex, and does not have an SOC representation.","However, the geometric mean of the growth rates, $\\sqrt{\\mu ^a(S(n),X(n))\\mu ^b(S(n),X(n))}$ , does lead to an SOC representation.","It is enforced by the two individual kinetics constraints along with $T(n)^2\\le T^a(n)T^b(n)$ .","The latter is hyperbolic, a type of SOC constraint [9]."],["Exactness","When (REF ) is satisfied with equality, $\\mathcal {P}_{\\textrm {R}}$ is exact, i.e., has an optimal solution that also solves $\\mathcal {P}$ ; this is the ideal outcome.","When (REF ) is not satisfied with equality, $\\mathcal {P}_{\\textrm {R}}$ might still provide a close approximation of $\\mathcal {P}$ , but this is hard to guarantee.","It is therefore useful to have conditions, even if narrow, under which the exactness of $\\mathcal {P}_{\\textrm {R}}$ is guaranteed.","For the rest of this section we let $\\mathcal {D}_n[\\xi (\\cdot )]=(\\xi (n)-\\xi (n-1))/\\Delta ,$ which corresponds to the implicit Euler step.","We assume that the only constraint specified by $\\Omega $ in () is an initial condition, $\\xi (0)=\\xi _0$ , and that $\\xi ^{\\textrm {in}}(\\cdot )$ is fixed; note that as long as exactness holds for all feasible values of $\\xi ^{\\textrm {in}}(\\cdot )$ , it holds when $\\xi ^{\\textrm {in}}(\\cdot )$ is a variable.","We also assume that strong duality holds for $\\mathcal {P}_{\\textrm {R}}$ ; this is a mild assumption that, e.g., holds as long as there is a feasible solution in which $\\xi (n)>0$ for all $n\\in \\mathcal {N}$ .","Let $\\mathcal {J}(\\xi (n))\\in \\mathbb {R}^{rs\\times ms}$ denote the Jacobian matrix of $\\phi (\\cdot )$ at $\\xi (n)$ .","For convenience, we define the following quantities: $\\Gamma (n)&=\\frac{1}{\\Delta }\\left(\\hat{V}/\\Delta -\\hat{N}(n)^{\\top }-\\mathcal {J}(\\xi (n))^{\\top }K^{\\top }\\hat{V}\\right)^{-1}\\hat{V}\\\\&=\\left(I_{ms}-\\Delta \\hat{V}^{-1}\\left(\\hat{N}(n)^{\\top }+\\mathcal {J}(\\xi (n))^{\\top }K^{\\top }\\hat{V}\\right)\\right)^{-1}\\\\\\Omega (n)&=-\\nabla \\mathcal {F}_{\\phi }(T(n))-\\Delta K^{\\top }\\hat{V}\\sum _{k=n}^{\\tau }\\left(\\prod _{l=n}^{k}\\Gamma (l)\\right)\\hat{V}^{-1}\\left(\\nabla \\mathcal {F}_{\\xi }(\\xi (k))+\\mathcal {J}(\\xi (k))^{\\top }\\nabla \\mathcal {F}_{\\phi }(T(k))\\right).$ Observe that if $\\Delta $ is small enough, $\\Gamma (n)$ is positive definite and close to the identify matrix.","Theorem 1 $\\mathcal {P}_{\\textrm {R}}$ is exact if at an optimal solution, $\\Omega (n)>0$ for all $n\\in \\mathcal {N}$ .","Let $\\lambda (n)\\in \\mathbb {R}^{ms}$ and $\\rho (n)\\in \\mathbb {R}^{rs}$ be the respective dual multipliers of constraints () and (REF ) for $n\\in \\mathcal {N}$ .","The Lagrangian of $\\mathcal {P}_{\\textrm {R}}$ is $\\mathcal {L}&=\\mathcal {F}(\\xi ,T) +\\sum _{n\\in \\mathcal {N}}\\rho (n)^{\\top }\\left(T(n)-\\phi (\\xi (n))\\right)\\\\&\\quad +\\lambda (n)^{\\top }\\left(\\hat{V}(\\xi (n-1)-\\xi (n))/\\Delta + \\hat{V}KT(n)+\\hat{N}(n)\\xi (n) + \\hat{C}(n)\\xi ^{\\textrm {in}}(n) \\right).$ Differentiating the Lagrangian by $T(n)$ and $\\xi (n)$ and setting it to zero gives $-\\rho (n) &=\\nabla \\mathcal {F}_{\\phi }(T(n))+ K^{\\top }\\hat{V}\\lambda (n),\\quad n\\in \\mathcal {N},\\\\\\mathcal {J}(\\xi (n))^{\\top }\\rho (n)&=\\nabla \\mathcal {F}_{\\xi }(\\xi (n)) -\\left(\\hat{V}/\\Delta -\\hat{N}(n)^{\\top }\\right)\\lambda (n)+\\hat{V}\\lambda (n+1)/\\Delta ,\\quad n\\in \\mathcal {N}\\setminus \\tau \\\\\\mathcal {J}(\\xi (\\tau ))^{\\top }\\rho (\\tau )&=\\nabla \\mathcal {F}_{\\xi }(\\xi (\\tau )) -\\left(\\hat{V}/\\Delta -\\hat{N}(n)^{\\top }\\right)\\lambda (\\tau ).$ We now solve for $\\rho (n)$ .","Premultiplying (REF ) by $\\mathcal {J}(\\xi (n))^{\\top }$ and summing with () and () gives $\\lambda (\\tau )=\\Delta \\Gamma (\\tau )\\hat{V}^{-1}\\left(\\nabla \\mathcal {F}_{\\xi }(\\xi (\\tau ))+\\mathcal {J}(\\xi (\\tau ))^{\\top }\\nabla \\mathcal {F}_{\\phi }(T(\\tau ))\\right),$ and, for $n\\in \\mathcal {N}\\setminus \\tau $ , $\\lambda (n)&=\\Delta \\Gamma (n)\\hat{V}^{-1}\\left(\\nabla \\mathcal {F}_{\\xi }(\\xi (n))+\\mathcal {J}(\\xi (n))^{\\top }\\nabla \\mathcal {F}_{\\phi }(T(n))\\right)+\\Gamma (n)\\lambda (n+1).$ Expanding the recursion yields $\\lambda (n)&=\\Delta \\sum _{k=n}^{\\tau }\\left(\\prod _{l=n}^{k}\\Gamma (l)\\right)\\hat{V}^{-1}\\left(\\nabla \\mathcal {F}_{\\xi }(\\xi (k))+\\mathcal {J}(\\xi (k))^{\\top }\\nabla \\mathcal {F}_{\\phi }(T(k))\\right).$ We now substitute this into (REF ) to obtain $\\rho (n)&=-\\nabla \\mathcal {F}_{\\phi }(T(n))-\\Delta K^{\\top }\\hat{V}\\sum _{k=n}^{\\tau }\\left(\\prod _{l=n}^{k}\\Gamma (l)\\right)\\hat{V}^{-1}\\left(\\nabla \\mathcal {F}_{\\xi }(\\xi (k))+\\mathcal {J}(\\xi (k))^{\\top }\\nabla \\mathcal {F}_{\\phi }(T(k))\\right)\\\\&=\\Omega (n).$ From here we can see that the conditions of the theorem guarantee that $\\rho (n)>0$ for all $n\\in \\mathcal {N}$ .","Theorem REF is of limited immediate use because we must know the optimal solution of $\\mathcal {P}_{\\textrm {R}}$ to test if $\\Omega (n)>0$ .","It can however be used to derive sufficient conditions for exactness that are easy to test.","We now derive two such conditions that do not require knowledge of the optimal solution.","For the rest of this section, assume that $\\mathcal {F}_{\\xi }$ and $\\mathcal {F}_{\\phi }$ are linear with gradients $f_{\\xi }\\in \\mathbb {R}^{ms}$ and $f_{\\phi }\\in \\mathbb {R}^{rs}$ .","In this case, we can write $&\\Omega (n)=\\\\&\\quad -\\Delta K^{\\top }\\hat{V}\\left(\\sum _{k=n}^{\\tau }\\prod _{l=n}^{k}\\Gamma (l)\\right)\\hat{V}^{-1}f_{\\xi }-\\left(I_{ms} +\\Delta K^{\\top }\\hat{V}\\sum _{k=n}^{\\tau }\\left(\\prod _{l=n}^{k}\\Gamma (l)\\right)\\hat{V}^{-1}\\mathcal {J}(\\xi (k))^{\\top } \\right)f_{\\phi }.$ We also assume that each element of the vector of reaction kinetics has bounded slope, so that all entries of $\\mathcal {J}(\\xi (n))$ , $n\\in \\mathcal {N}$ , are bounded.","Let $\\Psi (n)=\\hat{V}^{-1}\\hat{N}(n)^{\\top }+\\hat{V}^{-1}\\mathcal {J}(\\xi (n))^{\\top }K^{\\top }\\hat{V}.$ The first term of $\\Psi (n)$ is negative semidefinite because $N(n)$ is compartmental [10].","The latter term is bounded by assumption, and as we will see in the examples, usually negative semidefinite—we assume that this is the case.","We therefore assume that the eigenvalues of $\\Psi (n)$ are in the range $[-\\bar{\\psi },0]$ , where $\\bar{\\psi }>0$ is an upper bound on the magnitude.","We can use the push-through identity to write $\\Gamma (n)&=I_{ms}+\\Delta \\Psi (n)(I_{ms}-\\Delta \\Psi (n))^{-1}.$ The eigenvalues of the second term are in the range $[-\\Delta \\bar{\\psi },0]$ , and the eigenvalues of $\\Gamma (n)$ are in the range of $[1-\\Delta \\bar{\\psi },1]$ .","Corollary 1 If $f_{\\phi }=0$ and $K^{\\top }f_{\\xi }<0$ , then there exists a $\\Delta >0$ for which $\\mathcal {P}_{\\textrm {R}}$ is exact.","(Sketch) Observe that $\\prod _{l=n}^{k}\\Gamma (l)$ is equal to $I_{ms}$ plus terms that are norm-bounded by positive powers of $\\Delta \\bar{\\psi }$ .","Similarly, $\\sum _{k=n}^{\\tau }\\prod _{l=n}^{k}\\Gamma (l)$ is equal to $(\\tau -n+1)I_{ms}$ plus terms that are norm-bounded by positive powers of $\\Delta \\bar{\\psi }$ .","We can make these terms arbitrarily small by choosing $\\Delta $ small.","We therefore write $\\sum _{k=n}^{\\tau }\\prod _{l=n}^{k}\\Gamma (l)\\approx (\\tau -n+1)I_{ms}.$ Because $f_{\\phi }=0$ , we have that $\\Omega (n)\\approx &-(\\tau -n+1)\\Delta K^{\\top }f_{\\xi }.$ This is strictly positive for all $n\\in \\mathcal {N}$ if $K^{\\top }f_{\\xi }<0$ .","Corollary 2 If $f_{\\xi }=0$ and $f_{\\phi }>0$ , then there exists a $\\Delta >0$ for which $\\mathcal {P}_{\\textrm {R}}$ is exact.","(Sketch) Following the same logic as Corollary REF , we can choose $\\Delta >0$ such that $\\Omega (n)\\approx &\\left(I_{ms}-\\Delta K^{\\top }\\sum _{k=n}^{\\tau }\\mathcal {J}(\\xi (k))^{\\top }\\right)f_{\\phi }.$ The second term in the parentheses can be made arbitrarily small by choosing $\\Delta $ to be small.","In this case $\\Omega (n)\\approx f_{\\phi }$ , which is positive if $f_{\\phi }>0$ .","A shortcoming of Corollaries REF and REF is that they can be limited to short time intervals.","This is because if an interval is to remain constant, the number of time periods, $\\tau $ , must increase as the step, $\\Delta $ , decreases, and the approximations in the proofs of the corollaries do not hold for increasing $\\tau $ .","On the other hand, these are conservative sufficient conditions.","For example, the second term in the parentheses of (REF ) will often be positive semidefinite or nearly so.","This is because it is typical for $K$ to be lower triangular with a negative diagonal and for $\\mathcal {J}(\\xi (n))$ to be nonnegative and nearly diagonal.","We therefore expect that exactness will sometimes hold for larger $\\Delta $ over longer time intervals.","We view these theoretical results not as a complete characterization of when exactness is guaranteed, but rather as evidence that $\\mathcal {P}_\\textrm {R}$ is exact for a meaningful set of problems, and as guidance as to how to identify them.","While there are certainly problems of interest for which their conditions do not hold, $\\mathcal {P}_\\textrm {R}$ may nonetheless provide a useful and sometimes perfect approximation."],["Steady state","It may be of interest to optimize (REF ) in steady state, e.g., when the solution does not change significantly on the timescale of interest, to find the best operating point, or to reduce the number of variables.","We obtain a steady state optimization by dropping the time index and replacing the finite difference in () with zero.","The resulting (relaxed) optimization is: $\\mathcal {P}_{\\textrm {RS}}\\quad &\\min \\;\\mathcal {F}(\\xi ,T) \\\\\\textrm {such that}\\quad &T\\le \\phi (\\xi )\\\\&0=\\hat{V}K T+\\hat{N}\\xi +\\hat{C}\\xi ^{\\textrm {in}}\\\\&\\left(\\xi ,\\xi ^{\\textrm {in}},T\\right)\\in \\Omega .$ We remark that, in general, a solution to $\\mathcal {P}_{\\textrm {RS}}$ is not guaranteed to be an equilibrium of (REF ).","One special case in which guarantees do exist is the gradostat, which we discuss in Example REF .","We refer the reader to [1] for a brief summary.","Theorem 2 $\\mathcal {P}_{\\textrm {RS}}$ is exact if the network is outflow connected and at the optimal solution, $0<&\\left(I_{ms}+K^{\\top }\\hat{V}\\left(\\hat{N}^{\\top }\\right)^{-1}\\mathcal {J}(\\xi )^{\\top }\\right)^{-1}\\left(K^{\\top }\\hat{V}\\left(\\hat{N}^{\\top }\\right)^{-1}\\nabla \\mathcal {F}_{\\xi }(\\xi )-\\nabla \\mathcal {F}_{\\phi }(T)\\right).$ The Lagrangian of $\\mathcal {P}_{\\textrm {RS}}$ is $\\mathcal {L}&=\\mathcal {F}(\\xi ,T) +\\rho ^{\\top }\\left(T-\\phi (\\xi )\\right)+\\lambda ^{\\top }\\left( \\hat{V}KT+\\hat{N}\\xi + \\hat{C}\\xi ^{\\textrm {in}} \\right).$ Differentiating the Lagrangian and setting it to zero gives $0&=\\nabla \\mathcal {F}_{\\phi }(T) +\\rho + K^{\\top }\\hat{V}\\lambda \\\\\\mathcal {J}(\\xi )^{\\top }\\rho &=\\nabla \\mathcal {F}_{\\xi }(\\xi ) + \\hat{N}^{\\top }\\lambda .$ We now solve for $\\rho $ .","$\\hat{N}$ is invertible due to outflow-connectedness.","Then $\\lambda &=\\left(\\hat{N}^{\\top }\\right)^{-1}\\left(\\mathcal {J}(\\xi )^{\\top }\\rho - \\nabla \\mathcal {F}_{\\xi }(\\xi ) \\right),$ and $0&=\\nabla \\mathcal {F}_{\\phi }(T) +\\rho + K^{\\top }\\hat{V}\\left(\\hat{N}^{\\top }\\right)^{-1}\\left(\\mathcal {J}(\\xi )^{\\top }\\rho - \\nabla \\mathcal {F}_{\\xi }(\\xi ) \\right).$ Solving, we have $\\rho &=\\left(I_{ms}+K^{\\top }\\hat{V}\\left(\\hat{N}^{\\top }\\right)^{-1}\\mathcal {J}(\\xi )^{\\top }\\right)^{-1}\\left(K^{\\top }\\hat{V}\\left(\\hat{N}^{\\top }\\right)^{-1}\\nabla \\mathcal {F}_{\\xi }(\\xi )-\\nabla \\mathcal {F}_{\\phi }(T)\\right).$ By complementary slackness, $\\mathcal {P}_{\\textrm {RS}}$ is exact when $\\rho >0$ .","As in the latter part of the previous section, we now assume that $\\mathcal {F}_{\\xi }$ and $\\mathcal {F}_{\\phi }$ are linear with gradients $f_{\\xi }\\in \\mathbb {R}^{ms}$ and $f_{\\phi }\\in \\mathbb {R}^{rs}$ .","Corollary 3 Suppose that for all $\\xi \\ge 0$ , $\\left(I_{ms}+K^{\\top }\\hat{V}\\left(\\hat{N}^{\\top }\\right)^{-1}\\mathcal {J}(\\xi )^{\\top }\\right)^{-1}\\ge 0,$ and $K^{\\top }\\hat{V}\\left(\\hat{N}^{\\top }\\right)^{-1}f_{\\xi }-f_{\\phi }\\ge 0.$ If at least one of the inequalities is strict, then $\\mathcal {P}_{\\textrm {RS}}$ is exact.","Like Theorem REF , Corollary REF depends on the optimal solution, but to a lesser extent.","Whereas Theorem REF depends on $\\xi $ and $T$ through the objective and $\\mathcal {J}(\\xi )$ , Corollary REF only depends on $\\xi $ through $\\mathcal {J}(\\xi )$ .","This is more manageable because $\\mathcal {J}(\\xi )$ is often positive and nearly diagonal.","For example, if we assume assume that biomass is constant and all growth rates are Monod, as in Example REF , then $\\mathcal {J}(\\xi )$ is positive on the diagonal and zero elsewhere.","The gradostat is a special case of (REF ) where in each tank $i\\in \\mathcal {S}$ , a single substrate, $S_i$ , is converted to a single type of biomass, $X_i$ .","The conversion occurs at the rate $\\phi _i(S_i,X_i)/y$ , where $y$ is the yield.","Then $\\xi _i=[S_i,X_i]^{\\top }$ and $\\kappa _i=[-1/y,1]^{\\top }$ .","In [1], several examples are given in which the gradostat is optimized over time and in steady state.","Here we first apply Corollary REF to the gradostat.","Suppose $\\mathcal {F}_{\\xi }(\\xi (n))=f_S^{\\top }S(n)+f_X^{\\top }X(n)$ .","In this case, Corollary REF holds if $-f_S/y+f_X<0$ .","When dealing with the decontamination of undesirable substrates, we do not seek to minimize biomass, in which case $f_X=0$ and the condition is satisfied if $f_S>0$ .","We now apply Theorem REF to the gradostat in steady state.","Let $\\mathcal {J}_S(\\xi )\\in \\mathbb {R}^{r\\times s}$ be the Jacobian matrix of $\\phi (\\xi )$ with respect to $S$ , and let $\\mathcal {J}_X(\\xi )\\in \\mathbb {R}^{r\\times s}$ be the Jacobian matrix of $\\phi (\\xi )$ with respect to $X$ .","$\\mathcal {P}_{\\textrm {RS}}$ is exact if $&\\left(I_s+V\\left(N^{\\top }\\right)^{-1}\\left(-\\mathcal {J}_S(\\xi )^{\\top }/y+\\mathcal {J}_X(\\xi )^{\\top }\\right)\\right)^{-1}\\left(K^{\\top }V\\left(N^{\\top }\\right)^{-1}\\nabla \\mathcal {F}_{\\xi }(\\xi )-\\nabla \\mathcal {F}_{\\phi }(T)\\right)>0 .$ It is straightforward to verify that when $\\mathcal {F}_{\\xi }(\\xi )=0$ , this condition directly implies Theorem 1 in [1].","Similarly, we obtain Corollary 1 in [1] if we specialize Corollary REF to the gradostat."],["Wastewater treatment","In this example, we optimize an idealized wastewater treatment system consisting of three wastewater treatment plants of the city of Paris and its suburbs.","The model is based on that in [15], [16], and the influent data from the Inf_rain_2006 dataset of [17].","We implemented the model using the parser CVX [18] and the solver Gurobi [19] on a personal computer from 2014 with a 1.4 GHz dual-core processor.","In the present study, the flow rates are considered constant and given by $Q^{\\textrm {in}}_1=0.1\\;m^3/s$ , $Q^{\\textrm {in}}_2=0.4\\;m^3/s$ , and $Q^{\\textrm {in}}_3=0.2\\;m^3/s$ .","All tanks have volume $1000\\;m^3$ .","In each plant $i\\in \\mathcal {S}$ , the state vector $\\xi _i=\\left[\\xi ^{\\textrm {BOD}}_i,\\xi ^{\\textrm {NH}_4^+}_i,\\xi ^{\\textrm {NO}_2^-}_i,\\xi ^{\\textrm {NO}_3^-}_i\\right]^{\\top }\\in \\mathbb {R}^4$ consists of biochemical oxygen demand, ammonia nitrogen, nitrite, and nitrate.","The biomass in each time period is assumed to be an exogenous parameter, and therefore not a component of the process state.","This is an admissible assumption in the sense that the biomass concentration is typically much slower than that of the other process components, and has a larger amplitude.","In each tank $i\\in \\mathcal {S}$ , the elements of the process state have kinetics $\\phi _i^{\\textrm {BOD}},\\phi _i^{\\textrm {NH}_4^+},\\phi _i^{\\textrm {NO}_2^-}$ , and $\\phi _i^{\\textrm {NO}_3^-}$ .","All assume Monod growth rates with parameters given in Table REF , which comes from Table 1 in [20].","Because the biomass is constant, the growth kinetics can be represented as SOC constraints in $\\mathcal {P}_{\\textrm {R}}$ .","Table: Growth function parametersThe stochiometric matrix for each plant $i\\in \\mathcal {S}$ is $\\kappa _i=\\left[\\begin{array}{cccc}-1&0 & 0 & 0\\\\0 & -1 & 0 & 0\\\\0 & 1/y_i^{\\textrm {NH}_4^+,\\textrm {NO}_2^-} & -1 & 0\\\\0 & 0 & 1/y_i^{\\textrm {NO}_2^-,\\textrm {NO}_3^-} & -1\\end{array}\\right].$ We used the implicit Euler method, $\\mathcal {D}_n[\\xi (\\cdot )]=(\\xi (n)-\\xi (n-1))/\\Delta $ , with the stepsize $\\Delta =1$ , which corresponds to 15 minutes.","There are $\\tau =1345$ time periods, so that the total time is two weeks.","The boundary condition is $\\xi (0)=\\xi (\\tau )$ .","This could represent periodic operation, or exogenous conditions that are similar from week to week.","The process state must satisfy $\\xi ^{\\textrm {BOD}}_i(n)\\le 150$ mg/L and $\\xi ^{\\textrm {NH}_4^+}_i(n)\\le 60$ mg/L for each $i\\in \\mathcal {S}$ and $n\\in \\mathcal {N}$ .","Without these constraints, most of the substrate would be directed to Plant 1, which is more efficient than the others.","This constraint could represent, for example, regulatory limits on the pollution released by the plants.","In each time period, fixed quantities of $\\textrm {BOD}$ and $\\textrm {NH}_4^+$ , $\\Xi ^{\\textrm {BOD}}(n)$ and $\\Xi ^{\\textrm {NH}_4^+}(n)$ , must be allocated over the treatment plants.","These quantities are based on the Inf_rain_2006 dataset of [17], which covers 1345 15-minute intervals.","$\\Xi ^{\\textrm {BOD}}(\\cdot )$ is set to $S_{\\textrm {S}}$ (readily biodegradable substrate), and $\\Xi ^{\\textrm {NH}_4^+}(\\cdot )$ to $S_{\\textrm {NH}}$ ($\\textrm {NH}_4^+$ and $\\textrm {NH}_3$ nitrogen) in [17].","The allocation is represented by the linear constraints $\\Xi ^{\\textrm {BOD}}(n)&=\\sum _{i=1}^3Q^{\\textrm {in}}_i(n)\\xi ^{\\textrm {BOD},\\textrm {in}}_i(n)\\\\\\Xi ^{\\textrm {NH}_4^+}(n)&=\\sum _{i=1}^3Q^{\\textrm {in}}_i(n)\\xi ^{\\textrm {NH}_4^+,\\textrm {in}}_i(n),$ for each $n\\in \\mathcal {N}$ .","The other influent concentrations are $\\xi ^{\\textrm {NO}_2^-,\\textrm {in}}_i(n)=3$ and $\\xi ^{\\textrm {NO}_3^-,\\textrm {in}}_i(n)=10$ for $i\\in \\mathcal {S}$ and $n\\in \\mathcal {N}$ .","The plant biomass concentrations are set to $\\bar{X}_i(n)=100\\left(1+(-1)^i\\sin (10\\pi n/\\tau )\\right)$ for $i\\in \\mathcal {S}$ and $n\\in \\mathcal {N}$ .","Observe that this has larger magnitude and varies slower than the other influents.","For each $i\\in \\mathcal {S}$ and $n\\in \\mathcal {N}$ , the optimization variables are $\\xi _i(n)$ , $\\xi ^{\\textrm {BOD},\\textrm {in}}_i(n)$ , and $\\xi ^{\\textrm {NH}4,\\textrm {in}}_i(n)$ .","The objective is to minimize the untreated wastewater released by the plants, given by $\\sum _{n\\in \\mathcal {N}}\\sum _{i\\in \\mathcal {S}}Q^{\\textrm {out}}_i\\eta _i^{\\top }\\xi _i(n),$ where we note that $Q^{\\textrm {out}}_i=Q^{\\textrm {in}}_i$ , and $\\eta _i=[2,2,0.3,0.1,0]^{\\top }$ .","This objective was chosen to satisfy Corollary REF .","Note, however, that the result does not fully apply due to the boundary condition, $\\xi (0)=\\xi (\\tau )$ , and the upper limit on the process state.","The convex relaxation $\\mathcal {P}_{\\textrm {R}}$ contains 145272 variables (in standard form) and took 17 minutes to solve.","The 18 solver iterations accounted for only four seconds, and the rest of the time was used for preprocessing.","Despite not fully satisfying Corollary REF , the solution was exact in all time periods.","Figures REF and REF respectively show the optimal influent allocation, $\\xi ^{\\textrm {in}}(\\cdot )$ , and the resulting plant effluent concentrations, $\\xi (\\cdot )$ , between hours 175 and 275.","Figure: ξ in (·)\\xi ^{\\textrm {in}}(\\cdot ) from hour 175 to 275.","The units are mg/L.Figure: ξ(·)\\xi (\\cdot ) from hour 175 to 275.","The units are mg/L.The slower variation of the biomass dominates that of the diurnal variation in the Inf_rain_2006 dataset, leading to concentration increases whenever the biomass influent into each plant is high.","There is a spike in $\\Xi ^{\\textrm {BOD}}(\\cdot )$ and $\\Xi ^{\\textrm {NH}_4^+}(\\cdot )$ around hour 250.","This causes $\\xi ^{\\textrm {BOD}}_i(\\cdot )$ , and $\\xi ^{\\textrm {NH}4}_i(n)$ to hit their concentration limits in Plants 1 and 3.","When this happens, the remainder is allocated to Plant 2, which has little biomass at the time, and hence cannot transform the substrates as efficiently."],["Conclusion","We have formulated a convex relaxation for optimizing a broad class of bioprocesses.","We proved that the relaxation is exact under simple conditions, and implemented it on a wastewater treatment example with over one hundred thousand variables.","We believe that a wide range of problems can be approached in this manner due to the generality of the model and the tractability of SOCP.","One direction of future work is dealing with inexactness.","Two options are deriving general convex underestimators to limit the relaxation gap, as in [1], and finding local minima of the non-relaxed problem.","In particular, the concave-convex procedure [21] is well-suited to the nonconvexity encountered here and would entail solving a sequence of SOCPs.","We also intend to incorporate new elements into the model such as biomass death and recirculation, and to apply the relaxation in other contexts such as enzymes, where Michaelis-Menten kinetics [22] have the same form as Monod kinetics."],["Acknowledgments","Funding is acknowledged from the Natural Sciences and Engineering Research Council of Canada and the French LabEx NUMEV (Project ANR-10 LABX-20), incorporated into the I-Site MUSE, which partially funded the sabbatical of J.A.","Taylor at MISTEA lab."]],"string":"[\n [\n \"Convex optimization of bioprocesses\"\n ],\n [\n \"Abstract We optimize a general model of bioprocesses, which is nonconvex due to the microbial growth in the biochemical reactors.\",\n \"We formulate a convex relaxation and give conditions guaranteeing its exactness in both the transient and steady state cases.\",\n \"When the growth kinetics are modeled by the Monod function under constant biomass or the Contois function, the relaxation is a second-order cone program, which can be solved efficiently at large scales.\",\n \"We implement the model on a numerical example based on a wastewater treatment system.\"\n ],\n [\n \"Introduction\",\n \"We optimize a model of dynamical bioprocesses consisting of a set of biochemical reactors interconnected by diffusion and mass flow.\",\n \"The objectives include minimizing substrate outflow, maximizing biogas production, and tracking setpoints.\",\n \"Within each reactor, several microbial reactions convert any number of biotic or abiotic reactants into biomass and/or products.\",\n \"This setup describes a variety of physical systems such as wastewater treatment networks, the production of various chemicals, and compartmental approximations of bioprocesses in continuous media.\",\n \"Here we focus on the case when the kinetics can be represented by a second-order cone (SOC) constraint, as recently shown in [1] for the Monod [2] and Contois [3] growth rates.\",\n \"The most closely related topics to ours are chemical process optimization, control and optimization of wastewater systems, and control of bioprocesses.\",\n \"Most existing approaches to process optimization [4] and wastewater [5] do not explicitly model the microbial growth, and often use either linear programming or general nonlinear solvers.\",\n \"There have been many applications of nonlinear control [6] and optimization [7] to bioprocesses, but not convex relaxations or second-order cone programming (SOCP).\",\n \"Our main results are generalizations of those in [1], which focused on the gradostat with a single reaction [8].\",\n \"Here we allow for any number of substrates and biomasses, general convex objectives, and multiple biochemical reactions.\",\n \"Our original theoretical contributions are as follows.\",\n \"In Section , we formulate a convex relaxation for optimizing the trajectory and steady state solution of a general bioprocess.\",\n \"In Section , we give conditions under which the relaxations are guaranteed to be exact in both the transient and steady state cases.\",\n \"To streamline exposition, the only external inputs to the model are the influent concentrations, e.g., biochemical oxygen demand and ammonia.\",\n \"Our main exactness results straightforwardly apply when the flow rates are also variable, but the resulting bilinearities make the problem nonconvex.\",\n \"This can be handled using techniques like disjunctive programming, as in [1], or further convex relaxation, which we discuss at the end of Section REF .\",\n \"We apply our results in two examples.\",\n \"In Section REF , we show that our exactness conditions simplify to those in [1] when specialized to the gradostat.\",\n \"In Section REF , we optimize the allocation of sewage to three wastewater treatment plants over two weeks.\",\n \"The relaxation is exact, and takes roughly twenty minutes to solve using SOCP [9].\",\n \"The system consists of $s$ well-mixed tanks interconnected by mass flow and diffusion.\",\n \"We denote the set of tanks $\\\\mathcal {S}$ .\",\n \"$V\\\\in \\\\mathbb {R}^{s\\\\times s}$ is a diagonal matrix in which $V_{ii}$ is the volume of tank $i$ .\",\n \"Tank $i$ has water inlet flow rate $Q_i^{\\\\textrm {in}}$ and outlet flow rate $Q_i^{\\\\textrm {out}}$ .\",\n \"We let $Q_{ij}$ denote the flow from tank $i$ to tank $j$ .\",\n \"Let $d_{ij}$ denote the diffusion between tanks $i$ and $j$ , where $d_{ij}=d_{ji}$ .\",\n \"Let $C=\\\\textrm {diag}\\\\left[Q_i^{\\\\textrm {in}}\\\\right]$ , $M_{ij}&=\\\\left\\\\lbrace \\\\begin{array}{ll}Q_{ji}, & i\\\\ne j\\\\\\\\-Q_i^{\\\\textrm {out}}-\\\\sum _{k\\\\in {\\\\mathcal {S}}}Q_{ik},& i= j\\\\end{array}\\\\right., \\\\quad L_{ij}&=\\\\left\\\\lbrace \\\\begin{array}{ll}d_{ij}, & i\\\\ne j\\\\\\\\-\\\\sum _{k\\\\in {\\\\mathcal {S}}}d_{ik}, & i= j\\\\end{array}\\\\right.,$ and $N=M+L$ .\",\n \"$M$ and $L$ are respectively compartmental and Laplacian matrices.\",\n \"$M$ is invertible if the network is outflow connected, which is to say that there is a directed path from every tank to some tank with outflow [10].\",\n \"Because $L$ is negative semidefinite, $N$ is also invertible if $M$ is outflow connected, and potentially even if $M$ is not outflow connected.\"\n ],\n [\n \"Microbial growth\",\n \"We model the microbial growth in the tanks using the notation of Section 1.5 of [6].\",\n \"There are $m$ substrates and biomasses in each perfectly mixed tank.\",\n \"$\\\\xi _i\\\\in \\\\mathbb {R}^m_+$ is the process state vector of tank $i\\\\in \\\\mathcal {S}$ , which contains the concentrations of the substrates and biomasses, and $\\\\xi _i^{\\\\textrm {in}}\\\\in \\\\mathbb {R}^m_+$ is the corresponding influent concentration vector.\",\n \"This model is minimal in that $\\\\xi $ includes intermediary products, e.g., substrates produced by one reaction and consumed by another, but not final products such as the $\\\\textrm {CH}_4$ ultimately produced by anaerobic digestion.\",\n \"There are $r$ different types of reactions that convert substrates to other substrates and biomasses.\",\n \"$\\\\phi _i(\\\\xi _i)\\\\in \\\\mathbb {R}^r_+$ is a vector of the reaction kinetics in tank $i$ .\",\n \"We are interested in the case where the elements of $\\\\phi _i(\\\\xi _i)$ are concave functions, and in particular representable as SOC constraints.\",\n \"We show how to do this for Monod and Contois kinetics later in Examples REF and REF .\",\n \"Let $\\\\kappa _i\\\\in \\\\mathbb {R}^{m\\\\times r}$ be the stoichiometric matrix relating the reaction vector, $\\\\phi _i(\\\\xi _i)$ , to the evolution of the process state in tank $i$ .\",\n \"The dynamics in tank $i\\\\in \\\\mathcal {S}$ are $V_{ii}\\\\dot{\\\\xi }_i=V_{ii}\\\\kappa _i\\\\phi _i(\\\\xi _i) -Q_{i}^{\\\\textrm {out}}\\\\xi _i- \\\\sum _{j\\\\in \\\\mathcal {S}}(Q_{ij}+d_{ij})\\\\xi _i+Q_{i}^{\\\\textrm {in}}\\\\xi _i^{\\\\textrm {in}}+\\\\sum _{j\\\\in \\\\mathcal {S}}(Q_{ji}+d_{ij})\\\\xi _j.$ The following example illustrates $\\\\xi _i$ and $\\\\kappa _i$ .\",\n \"Example 1 (Two-step anaerobic digestion) There are two substrates, $S_i^a$ and $S_i^b$ , and two biomasses, $X_i^a$ and $X_i^b$ .\",\n \"We let $\\\\xi _i=\\\\left[S_i^a,S_i^b,X_i^a,X_i^b\\\\right]^{\\\\top }$ .\",\n \"$S_i^a$ is converted to both $X_i^a$ and $S_i^b$ at the rate $\\\\mu ^a\\\\left(S_i^a\\\\right)X_i^a$ .\",\n \"$S_i^b$ is converted to $X_i^b$ at the rate $\\\\mu ^b\\\\left(S_i^b\\\\right)X_i^b$ .\",\n \"Therefore, $\\\\phi _i(\\\\xi _i)=\\\\left[\\\\begin{array}{c}\\\\mu ^a\\\\left(S_i^a\\\\right)X_i^a\\\\\\\\\\\\mu ^b\\\\left(S_i^b\\\\right)X_i^b\\\\end{array}\\\\right]\\\\;\\\\textrm {and}\\\\;\\\\kappa _i=\\\\left[\\\\begin{array}{cc}-1&0\\\\\\\\1&-1\\\\\\\\1&0\\\\\\\\0&1\\\\end{array}\\\\right].$ We now write the dynamics in vector form.\",\n \"We suppress subscripts to represent stacked vectors, i.e., $\\\\xi =[\\\\xi _1,...,\\\\xi _s]^{\\\\top }$ and $\\\\phi (\\\\xi )=[\\\\phi _1(\\\\xi _1),...,\\\\phi _s(\\\\xi _s)]^{\\\\top }$ .\",\n \"Let $A\\\\otimes B$ denote the Kronecker product of $A$ and $B$ , $I_\\\\alpha \\\\in \\\\mathbb {R}^{\\\\alpha \\\\times \\\\alpha }$ the identity matrix, and $\\\\hat{A}=A\\\\otimes I_m$ .\",\n \"Let $K$ be a block diagonal matrix with $\\\\kappa _1,...,\\\\kappa _s$ on its main diagonal.\",\n \"If $\\\\kappa _i=\\\\kappa $ for all $i\\\\in \\\\mathcal {S}$ , then $K=I_s\\\\otimes \\\\kappa $ .\",\n \"The dynamics of the full system are given in vector form by $\\\\hat{V}\\\\dot{\\\\xi }=\\\\hat{V}K \\\\phi (\\\\xi )+\\\\hat{N}\\\\xi +\\\\hat{C}\\\\xi ^{\\\\textrm {in}}.$ We allow the dynamics to be non-autonomous, in which case $\\\\hat{N}$ , $\\\\hat{C}$ , and $\\\\xi ^{in}$ can be time-varying.\"\n ],\n [\n \"Discretization in time\",\n \"To make (REF ) compatible with finite-dimensional optimization, we replace the derivatives with a numerical approximation, which we denote $\\\\mathcal {D}_n$ .\",\n \"For example, in the case of the implicit Euler method with time step $\\\\Delta $ , $\\\\mathcal {D}_n[\\\\xi (\\\\cdot )]=(\\\\xi (n)-\\\\xi (n-1))/\\\\Delta $ .\",\n \"$\\\\mathcal {D}_n$ could also be a more sophisticated approximation such as a Runge-Kutta scheme [11].\",\n \"The time periods are indexed $n\\\\in \\\\mathcal {N}=\\\\lbrace 1,...,\\\\tau \\\\rbrace $ .\",\n \"We have $\\\\hat{V}\\\\mathcal {D}_n[\\\\xi (\\\\cdot )]=\\\\hat{V}K \\\\phi (\\\\xi (n))+\\\\hat{N}(n)\\\\xi (n)+\\\\hat{C}(n)\\\\xi ^{\\\\textrm {in}}(n)$ for $n\\\\in \\\\mathcal {N}$ .\",\n \"The initial condition is $\\\\xi (0)=\\\\xi _0$ .\"\n ],\n [\n \"Objectives\",\n \"We consider objectives of the form $\\\\mathcal {F}(\\\\xi ,T)=\\\\sum _{n\\\\in \\\\mathcal {N}} \\\\mathcal {F}_{\\\\xi }(\\\\xi (n))+\\\\mathcal {F}_{\\\\phi }(T(n)),$ where $\\\\mathcal {F}_{\\\\xi }$ and $\\\\mathcal {F}_{\\\\phi }$ are convex and $T(n)=\\\\phi (\\\\xi (n))$ .\",\n \"The following are examples.\",\n \"Minimizing the outflow of substrates, $\\\\mathcal {F}_{\\\\xi }(\\\\xi (n))=\\\\sum _{i\\\\in \\\\mathcal {S}}Q_{i}^{\\\\textrm {out}}\\\\eta _i^{\\\\top }\\\\xi _i(n),$ where $\\\\eta _i$ is a vector that selects the entries of $\\\\xi _i(n)$ corresponding to pollutants.\",\n \"The production of biogas in a tank is proportional to the kinetics that convert substrates to biomass.\",\n \"Let $\\\\sigma _i\\\\in \\\\mathbb {R}^m_+$ be a vector that is only nonzero for entries of $T_i(n)$ corresponding to biogas production.\",\n \"Let $\\\\mathcal {M}\\\\subseteq \\\\mathcal {S}$ be the subset of tanks that can capture biogas from anaerobic digestion.\",\n \"We maximize biogas through the objective $\\\\mathcal {F}_{\\\\phi }(T(n))=-\\\\sum _{i\\\\in \\\\mathcal {M}}V_{ii}\\\\sigma _i^{\\\\top }T_i(n).$ Setpoint tracking, $\\\\mathcal {F}_{\\\\xi }(\\\\xi (n))=\\\\left(\\\\xi (n)-\\\\bar{\\\\xi }\\\\right)^{\\\\top }A\\\\left(\\\\xi (n)-\\\\bar{\\\\xi }\\\\right),$ where $A\\\\succeq 0$ and $\\\\bar{\\\\xi }$ is a desired operating point.\"\n ],\n [\n \"Problem statement\",\n \"We aim to solve the following optimization problem.\",\n \"$\\\\mathcal {P}\\\\quad &\\\\min \\\\;\\\\mathcal {F}(\\\\xi ,T) \\\\\\\\\\\\textrm {such that}\\\\quad &T(n)=\\\\phi (\\\\xi (n)),\\\\quad n\\\\in \\\\mathcal {N}\\\\\\\\&\\\\hat{V}\\\\mathcal {D}_n[\\\\xi (\\\\cdot )]=\\\\hat{V}K T(n)+\\\\hat{N}(n)\\\\xi (n)+\\\\hat{C}(n)\\\\xi ^{\\\\textrm {in}}(n),\\\\quad n\\\\in \\\\mathcal {N}\\\\\\\\&\\\\left(\\\\xi ,\\\\xi ^{\\\\textrm {in}},T\\\\right)\\\\in \\\\Omega .$ $\\\\mathcal {P}$ models the optimization of a broad range of bioprocesses such as wastewater treatment.\",\n \"We refer the reader to [6] for broad coverage of this topic.\",\n \"A solution of $\\\\mathcal {P}$ is a trajectory $\\\\left(\\\\xi (n),\\\\xi ^{\\\\textrm {in}}(n),T(n)\\\\right)$ , $n\\\\in \\\\mathcal {N}$ .\",\n \"The flows and diffusions between tanks, encoded by the matrices $\\\\hat{N}(n)$ and $\\\\hat{C}(n)$ , are not decision variables.\",\n \"For this reason constraint () is linear.\",\n \"The set $\\\\Omega $ in () consists of linear constraints such as the initial condition, total input matter, and maximum substrate concentrations; several other examples are given in [1].\",\n \"Note that $\\\\Omega $ can constrain $T(\\\\cdot )$ so as to allow constraints on the growth without adding nonlinearities.\",\n \"The only nonconvexity is therefore (), the growth constraint; this is the focus of the next two sections.\",\n \"We note that in many applications, the flow rates are important decision variables.\",\n \"They are parameters here because our focus is on incorporating the growth kinetics in a convex fashion.\",\n \"In the case that the flow rates are variable, () becomes bilinear and hence nonconvex.\",\n \"There are several ways to handle the bilinearity, including McCormick [12] and lift-and-project [13] relaxations; disjunctive programming reformulations if the flow variables are binary, as in [1]; and finding a local minimum via nonlinear programming, e.g., exploiting the biconvex structure with the Alternating Direction Method of Multipliers [14].\",\n \"All three of the above techniques are viable because, as described in the next section, we have a tractable way to represent the growth constraint, ().\"\n ],\n [\n \"Convex relaxation\",\n \"$\\\\mathcal {P}$ is nonconvex because constraint () is a nonlinear equality.\",\n \"One way around this difficulty is to instead solve a convex relaxation of $\\\\mathcal {P}$ , as in [1].\",\n \"If all elements of the vector $\\\\phi (\\\\cdot )$ are concave functions, we obtain a convex relaxation by replacing () with the inequality $T(n)\\\\le \\\\phi (\\\\xi (n)),\\\\quad n\\\\in \\\\mathcal {N}.$ We refer to the resulting optimization as $\\\\mathcal {P}_{\\\\textrm {R}}$ .\",\n \"As mentioned earlier, we are interested in the case where (REF ) is concave and, ideally, representable as an SOC constraint.\",\n \"Several such examples are given below.\",\n \"Example 2 (Contois growth) Suppose that there is a substrate of concentration $S$ , a biomass of concentration $X$ , and the growth rate is Contois [3].\",\n \"Then constraint (REF ) takes the form $T(n)\\\\le \\\\frac{\\\\mu S(n)X(n)}{k_{\\\\textrm {C}}X(n)+S(n)},$ where $\\\\mu $ and $k_{\\\\textrm {C}}$ are constant parameters.\",\n \"As shown in [1], this is concave and can be written as the SOC constraint $\\\\left\\\\Vert \\\\left[\\\\begin{array}{c}\\\\mu S(n)\\\\\\\\k_{\\\\textrm {C}}T(n)\\\\\\\\\\\\mu kX(n)\\\\end{array}\\\\right]\\\\right\\\\Vert \\\\le \\\\mu k_{\\\\textrm {C}}X(n) + \\\\mu S(n) - kT(n).$ Example 3 (Monod growth with constant biomass) Consider Example REF , but now suppose that the growth rate is Monod [2].\",\n \"Then constraint (REF ) takes the form $T(n)\\\\le \\\\frac{\\\\mu S(n)X(n)}{k_{\\\\textrm {M}}+S(n)}.$ This constraint is quasiconcave.\",\n \"It becomes concave if we assume that the biomass in each time period is not an optimization variable, but an exogenous parameter, i.e., $X(n)=\\\\bar{X}(n)$ for $n\\\\in \\\\mathcal {N}$ .\",\n \"This approximation is often valid because the biomass concentration is typically larger and varies more slowly than the substrate concentrations, and is therefore relatively insensitive to the substrates.\",\n \"In this case, as shown in [1], it can be written as the SOC constraint $\\\\left\\\\Vert \\\\left[\\\\begin{array}{c}\\\\mu S(n) \\\\bar{X}(n)\\\\\\\\k_{\\\\textrm {M}}T(n)\\\\\\\\\\\\mu k\\\\bar{X}(n)\\\\end{array}\\\\right]\\\\right\\\\Vert \\\\le \\\\mu k_{\\\\textrm {M}}\\\\bar{X}(n) + \\\\mu S(n)\\\\bar{X}(n) - kT(n).$ Example 4 (Interactive and non-interactive growth) Suppose the growth of the biomass depends on two rates, $\\\\mu ^a(S(n),X(n))$ and $\\\\mu ^b(S(n),X(n))$ , and the individual kinetics constraints, $T^a(n)\\\\le \\\\mu ^a(S(n),X(n))X(n)$ and $T^a(n)\\\\le \\\\mu ^b(S(n),X(n))X(n)$ , both have SOC representations.\",\n \"The dependency often takes one of two forms: non-interactive, $\\\\min \\\\lbrace \\\\mu ^a(S(n),X(n)),\\\\mu ^b(S(n),X(n))\\\\rbrace $ , which in ecological modeling is known as Liebig's Law, and interactive, $\\\\mu ^a(S(n),X(n))\\\\mu ^b(S(n),X(n))$ , which is common in models of bioprocesses.\",\n \"The non-interactive case is enforced by the two individual kinetics constraints along with $T(n)\\\\le T^a(n)$ and $T(n)\\\\le T^b(n)$ .\",\n \"The interactive case is in general nonconvex, and does not have an SOC representation.\",\n \"However, the geometric mean of the growth rates, $\\\\sqrt{\\\\mu ^a(S(n),X(n))\\\\mu ^b(S(n),X(n))}$ , does lead to an SOC representation.\",\n \"It is enforced by the two individual kinetics constraints along with $T(n)^2\\\\le T^a(n)T^b(n)$ .\",\n \"The latter is hyperbolic, a type of SOC constraint [9].\"\n ],\n [\n \"Exactness\",\n \"When (REF ) is satisfied with equality, $\\\\mathcal {P}_{\\\\textrm {R}}$ is exact, i.e., has an optimal solution that also solves $\\\\mathcal {P}$ ; this is the ideal outcome.\",\n \"When (REF ) is not satisfied with equality, $\\\\mathcal {P}_{\\\\textrm {R}}$ might still provide a close approximation of $\\\\mathcal {P}$ , but this is hard to guarantee.\",\n \"It is therefore useful to have conditions, even if narrow, under which the exactness of $\\\\mathcal {P}_{\\\\textrm {R}}$ is guaranteed.\",\n \"For the rest of this section we let $\\\\mathcal {D}_n[\\\\xi (\\\\cdot )]=(\\\\xi (n)-\\\\xi (n-1))/\\\\Delta ,$ which corresponds to the implicit Euler step.\",\n \"We assume that the only constraint specified by $\\\\Omega $ in () is an initial condition, $\\\\xi (0)=\\\\xi _0$ , and that $\\\\xi ^{\\\\textrm {in}}(\\\\cdot )$ is fixed; note that as long as exactness holds for all feasible values of $\\\\xi ^{\\\\textrm {in}}(\\\\cdot )$ , it holds when $\\\\xi ^{\\\\textrm {in}}(\\\\cdot )$ is a variable.\",\n \"We also assume that strong duality holds for $\\\\mathcal {P}_{\\\\textrm {R}}$ ; this is a mild assumption that, e.g., holds as long as there is a feasible solution in which $\\\\xi (n)>0$ for all $n\\\\in \\\\mathcal {N}$ .\",\n \"Let $\\\\mathcal {J}(\\\\xi (n))\\\\in \\\\mathbb {R}^{rs\\\\times ms}$ denote the Jacobian matrix of $\\\\phi (\\\\cdot )$ at $\\\\xi (n)$ .\",\n \"For convenience, we define the following quantities: $\\\\Gamma (n)&=\\\\frac{1}{\\\\Delta }\\\\left(\\\\hat{V}/\\\\Delta -\\\\hat{N}(n)^{\\\\top }-\\\\mathcal {J}(\\\\xi (n))^{\\\\top }K^{\\\\top }\\\\hat{V}\\\\right)^{-1}\\\\hat{V}\\\\\\\\&=\\\\left(I_{ms}-\\\\Delta \\\\hat{V}^{-1}\\\\left(\\\\hat{N}(n)^{\\\\top }+\\\\mathcal {J}(\\\\xi (n))^{\\\\top }K^{\\\\top }\\\\hat{V}\\\\right)\\\\right)^{-1}\\\\\\\\\\\\Omega (n)&=-\\\\nabla \\\\mathcal {F}_{\\\\phi }(T(n))-\\\\Delta K^{\\\\top }\\\\hat{V}\\\\sum _{k=n}^{\\\\tau }\\\\left(\\\\prod _{l=n}^{k}\\\\Gamma (l)\\\\right)\\\\hat{V}^{-1}\\\\left(\\\\nabla \\\\mathcal {F}_{\\\\xi }(\\\\xi (k))+\\\\mathcal {J}(\\\\xi (k))^{\\\\top }\\\\nabla \\\\mathcal {F}_{\\\\phi }(T(k))\\\\right).$ Observe that if $\\\\Delta $ is small enough, $\\\\Gamma (n)$ is positive definite and close to the identify matrix.\",\n \"Theorem 1 $\\\\mathcal {P}_{\\\\textrm {R}}$ is exact if at an optimal solution, $\\\\Omega (n)>0$ for all $n\\\\in \\\\mathcal {N}$ .\",\n \"Let $\\\\lambda (n)\\\\in \\\\mathbb {R}^{ms}$ and $\\\\rho (n)\\\\in \\\\mathbb {R}^{rs}$ be the respective dual multipliers of constraints () and (REF ) for $n\\\\in \\\\mathcal {N}$ .\",\n \"The Lagrangian of $\\\\mathcal {P}_{\\\\textrm {R}}$ is $\\\\mathcal {L}&=\\\\mathcal {F}(\\\\xi ,T) +\\\\sum _{n\\\\in \\\\mathcal {N}}\\\\rho (n)^{\\\\top }\\\\left(T(n)-\\\\phi (\\\\xi (n))\\\\right)\\\\\\\\&\\\\quad +\\\\lambda (n)^{\\\\top }\\\\left(\\\\hat{V}(\\\\xi (n-1)-\\\\xi (n))/\\\\Delta + \\\\hat{V}KT(n)+\\\\hat{N}(n)\\\\xi (n) + \\\\hat{C}(n)\\\\xi ^{\\\\textrm {in}}(n) \\\\right).$ Differentiating the Lagrangian by $T(n)$ and $\\\\xi (n)$ and setting it to zero gives $-\\\\rho (n) &=\\\\nabla \\\\mathcal {F}_{\\\\phi }(T(n))+ K^{\\\\top }\\\\hat{V}\\\\lambda (n),\\\\quad n\\\\in \\\\mathcal {N},\\\\\\\\\\\\mathcal {J}(\\\\xi (n))^{\\\\top }\\\\rho (n)&=\\\\nabla \\\\mathcal {F}_{\\\\xi }(\\\\xi (n)) -\\\\left(\\\\hat{V}/\\\\Delta -\\\\hat{N}(n)^{\\\\top }\\\\right)\\\\lambda (n)+\\\\hat{V}\\\\lambda (n+1)/\\\\Delta ,\\\\quad n\\\\in \\\\mathcal {N}\\\\setminus \\\\tau \\\\\\\\\\\\mathcal {J}(\\\\xi (\\\\tau ))^{\\\\top }\\\\rho (\\\\tau )&=\\\\nabla \\\\mathcal {F}_{\\\\xi }(\\\\xi (\\\\tau )) -\\\\left(\\\\hat{V}/\\\\Delta -\\\\hat{N}(n)^{\\\\top }\\\\right)\\\\lambda (\\\\tau ).$ We now solve for $\\\\rho (n)$ .\",\n \"Premultiplying (REF ) by $\\\\mathcal {J}(\\\\xi (n))^{\\\\top }$ and summing with () and () gives $\\\\lambda (\\\\tau )=\\\\Delta \\\\Gamma (\\\\tau )\\\\hat{V}^{-1}\\\\left(\\\\nabla \\\\mathcal {F}_{\\\\xi }(\\\\xi (\\\\tau ))+\\\\mathcal {J}(\\\\xi (\\\\tau ))^{\\\\top }\\\\nabla \\\\mathcal {F}_{\\\\phi }(T(\\\\tau ))\\\\right),$ and, for $n\\\\in \\\\mathcal {N}\\\\setminus \\\\tau $ , $\\\\lambda (n)&=\\\\Delta \\\\Gamma (n)\\\\hat{V}^{-1}\\\\left(\\\\nabla \\\\mathcal {F}_{\\\\xi }(\\\\xi (n))+\\\\mathcal {J}(\\\\xi (n))^{\\\\top }\\\\nabla \\\\mathcal {F}_{\\\\phi }(T(n))\\\\right)+\\\\Gamma (n)\\\\lambda (n+1).$ Expanding the recursion yields $\\\\lambda (n)&=\\\\Delta \\\\sum _{k=n}^{\\\\tau }\\\\left(\\\\prod _{l=n}^{k}\\\\Gamma (l)\\\\right)\\\\hat{V}^{-1}\\\\left(\\\\nabla \\\\mathcal {F}_{\\\\xi }(\\\\xi (k))+\\\\mathcal {J}(\\\\xi (k))^{\\\\top }\\\\nabla \\\\mathcal {F}_{\\\\phi }(T(k))\\\\right).$ We now substitute this into (REF ) to obtain $\\\\rho (n)&=-\\\\nabla \\\\mathcal {F}_{\\\\phi }(T(n))-\\\\Delta K^{\\\\top }\\\\hat{V}\\\\sum _{k=n}^{\\\\tau }\\\\left(\\\\prod _{l=n}^{k}\\\\Gamma (l)\\\\right)\\\\hat{V}^{-1}\\\\left(\\\\nabla \\\\mathcal {F}_{\\\\xi }(\\\\xi (k))+\\\\mathcal {J}(\\\\xi (k))^{\\\\top }\\\\nabla \\\\mathcal {F}_{\\\\phi }(T(k))\\\\right)\\\\\\\\&=\\\\Omega (n).$ From here we can see that the conditions of the theorem guarantee that $\\\\rho (n)>0$ for all $n\\\\in \\\\mathcal {N}$ .\",\n \"Theorem REF is of limited immediate use because we must know the optimal solution of $\\\\mathcal {P}_{\\\\textrm {R}}$ to test if $\\\\Omega (n)>0$ .\",\n \"It can however be used to derive sufficient conditions for exactness that are easy to test.\",\n \"We now derive two such conditions that do not require knowledge of the optimal solution.\",\n \"For the rest of this section, assume that $\\\\mathcal {F}_{\\\\xi }$ and $\\\\mathcal {F}_{\\\\phi }$ are linear with gradients $f_{\\\\xi }\\\\in \\\\mathbb {R}^{ms}$ and $f_{\\\\phi }\\\\in \\\\mathbb {R}^{rs}$ .\",\n \"In this case, we can write $&\\\\Omega (n)=\\\\\\\\&\\\\quad -\\\\Delta K^{\\\\top }\\\\hat{V}\\\\left(\\\\sum _{k=n}^{\\\\tau }\\\\prod _{l=n}^{k}\\\\Gamma (l)\\\\right)\\\\hat{V}^{-1}f_{\\\\xi }-\\\\left(I_{ms} +\\\\Delta K^{\\\\top }\\\\hat{V}\\\\sum _{k=n}^{\\\\tau }\\\\left(\\\\prod _{l=n}^{k}\\\\Gamma (l)\\\\right)\\\\hat{V}^{-1}\\\\mathcal {J}(\\\\xi (k))^{\\\\top } \\\\right)f_{\\\\phi }.$ We also assume that each element of the vector of reaction kinetics has bounded slope, so that all entries of $\\\\mathcal {J}(\\\\xi (n))$ , $n\\\\in \\\\mathcal {N}$ , are bounded.\",\n \"Let $\\\\Psi (n)=\\\\hat{V}^{-1}\\\\hat{N}(n)^{\\\\top }+\\\\hat{V}^{-1}\\\\mathcal {J}(\\\\xi (n))^{\\\\top }K^{\\\\top }\\\\hat{V}.$ The first term of $\\\\Psi (n)$ is negative semidefinite because $N(n)$ is compartmental [10].\",\n \"The latter term is bounded by assumption, and as we will see in the examples, usually negative semidefinite—we assume that this is the case.\",\n \"We therefore assume that the eigenvalues of $\\\\Psi (n)$ are in the range $[-\\\\bar{\\\\psi },0]$ , where $\\\\bar{\\\\psi }>0$ is an upper bound on the magnitude.\",\n \"We can use the push-through identity to write $\\\\Gamma (n)&=I_{ms}+\\\\Delta \\\\Psi (n)(I_{ms}-\\\\Delta \\\\Psi (n))^{-1}.$ The eigenvalues of the second term are in the range $[-\\\\Delta \\\\bar{\\\\psi },0]$ , and the eigenvalues of $\\\\Gamma (n)$ are in the range of $[1-\\\\Delta \\\\bar{\\\\psi },1]$ .\",\n \"Corollary 1 If $f_{\\\\phi }=0$ and $K^{\\\\top }f_{\\\\xi }<0$ , then there exists a $\\\\Delta >0$ for which $\\\\mathcal {P}_{\\\\textrm {R}}$ is exact.\",\n \"(Sketch) Observe that $\\\\prod _{l=n}^{k}\\\\Gamma (l)$ is equal to $I_{ms}$ plus terms that are norm-bounded by positive powers of $\\\\Delta \\\\bar{\\\\psi }$ .\",\n \"Similarly, $\\\\sum _{k=n}^{\\\\tau }\\\\prod _{l=n}^{k}\\\\Gamma (l)$ is equal to $(\\\\tau -n+1)I_{ms}$ plus terms that are norm-bounded by positive powers of $\\\\Delta \\\\bar{\\\\psi }$ .\",\n \"We can make these terms arbitrarily small by choosing $\\\\Delta $ small.\",\n \"We therefore write $\\\\sum _{k=n}^{\\\\tau }\\\\prod _{l=n}^{k}\\\\Gamma (l)\\\\approx (\\\\tau -n+1)I_{ms}.$ Because $f_{\\\\phi }=0$ , we have that $\\\\Omega (n)\\\\approx &-(\\\\tau -n+1)\\\\Delta K^{\\\\top }f_{\\\\xi }.$ This is strictly positive for all $n\\\\in \\\\mathcal {N}$ if $K^{\\\\top }f_{\\\\xi }<0$ .\",\n \"Corollary 2 If $f_{\\\\xi }=0$ and $f_{\\\\phi }>0$ , then there exists a $\\\\Delta >0$ for which $\\\\mathcal {P}_{\\\\textrm {R}}$ is exact.\",\n \"(Sketch) Following the same logic as Corollary REF , we can choose $\\\\Delta >0$ such that $\\\\Omega (n)\\\\approx &\\\\left(I_{ms}-\\\\Delta K^{\\\\top }\\\\sum _{k=n}^{\\\\tau }\\\\mathcal {J}(\\\\xi (k))^{\\\\top }\\\\right)f_{\\\\phi }.$ The second term in the parentheses can be made arbitrarily small by choosing $\\\\Delta $ to be small.\",\n \"In this case $\\\\Omega (n)\\\\approx f_{\\\\phi }$ , which is positive if $f_{\\\\phi }>0$ .\",\n \"A shortcoming of Corollaries REF and REF is that they can be limited to short time intervals.\",\n \"This is because if an interval is to remain constant, the number of time periods, $\\\\tau $ , must increase as the step, $\\\\Delta $ , decreases, and the approximations in the proofs of the corollaries do not hold for increasing $\\\\tau $ .\",\n \"On the other hand, these are conservative sufficient conditions.\",\n \"For example, the second term in the parentheses of (REF ) will often be positive semidefinite or nearly so.\",\n \"This is because it is typical for $K$ to be lower triangular with a negative diagonal and for $\\\\mathcal {J}(\\\\xi (n))$ to be nonnegative and nearly diagonal.\",\n \"We therefore expect that exactness will sometimes hold for larger $\\\\Delta $ over longer time intervals.\",\n \"We view these theoretical results not as a complete characterization of when exactness is guaranteed, but rather as evidence that $\\\\mathcal {P}_\\\\textrm {R}$ is exact for a meaningful set of problems, and as guidance as to how to identify them.\",\n \"While there are certainly problems of interest for which their conditions do not hold, $\\\\mathcal {P}_\\\\textrm {R}$ may nonetheless provide a useful and sometimes perfect approximation.\"\n ],\n [\n \"Steady state\",\n \"It may be of interest to optimize (REF ) in steady state, e.g., when the solution does not change significantly on the timescale of interest, to find the best operating point, or to reduce the number of variables.\",\n \"We obtain a steady state optimization by dropping the time index and replacing the finite difference in () with zero.\",\n \"The resulting (relaxed) optimization is: $\\\\mathcal {P}_{\\\\textrm {RS}}\\\\quad &\\\\min \\\\;\\\\mathcal {F}(\\\\xi ,T) \\\\\\\\\\\\textrm {such that}\\\\quad &T\\\\le \\\\phi (\\\\xi )\\\\\\\\&0=\\\\hat{V}K T+\\\\hat{N}\\\\xi +\\\\hat{C}\\\\xi ^{\\\\textrm {in}}\\\\\\\\&\\\\left(\\\\xi ,\\\\xi ^{\\\\textrm {in}},T\\\\right)\\\\in \\\\Omega .$ We remark that, in general, a solution to $\\\\mathcal {P}_{\\\\textrm {RS}}$ is not guaranteed to be an equilibrium of (REF ).\",\n \"One special case in which guarantees do exist is the gradostat, which we discuss in Example REF .\",\n \"We refer the reader to [1] for a brief summary.\",\n \"Theorem 2 $\\\\mathcal {P}_{\\\\textrm {RS}}$ is exact if the network is outflow connected and at the optimal solution, $0<&\\\\left(I_{ms}+K^{\\\\top }\\\\hat{V}\\\\left(\\\\hat{N}^{\\\\top }\\\\right)^{-1}\\\\mathcal {J}(\\\\xi )^{\\\\top }\\\\right)^{-1}\\\\left(K^{\\\\top }\\\\hat{V}\\\\left(\\\\hat{N}^{\\\\top }\\\\right)^{-1}\\\\nabla \\\\mathcal {F}_{\\\\xi }(\\\\xi )-\\\\nabla \\\\mathcal {F}_{\\\\phi }(T)\\\\right).$ The Lagrangian of $\\\\mathcal {P}_{\\\\textrm {RS}}$ is $\\\\mathcal {L}&=\\\\mathcal {F}(\\\\xi ,T) +\\\\rho ^{\\\\top }\\\\left(T-\\\\phi (\\\\xi )\\\\right)+\\\\lambda ^{\\\\top }\\\\left( \\\\hat{V}KT+\\\\hat{N}\\\\xi + \\\\hat{C}\\\\xi ^{\\\\textrm {in}} \\\\right).$ Differentiating the Lagrangian and setting it to zero gives $0&=\\\\nabla \\\\mathcal {F}_{\\\\phi }(T) +\\\\rho + K^{\\\\top }\\\\hat{V}\\\\lambda \\\\\\\\\\\\mathcal {J}(\\\\xi )^{\\\\top }\\\\rho &=\\\\nabla \\\\mathcal {F}_{\\\\xi }(\\\\xi ) + \\\\hat{N}^{\\\\top }\\\\lambda .$ We now solve for $\\\\rho $ .\",\n \"$\\\\hat{N}$ is invertible due to outflow-connectedness.\",\n \"Then $\\\\lambda &=\\\\left(\\\\hat{N}^{\\\\top }\\\\right)^{-1}\\\\left(\\\\mathcal {J}(\\\\xi )^{\\\\top }\\\\rho - \\\\nabla \\\\mathcal {F}_{\\\\xi }(\\\\xi ) \\\\right),$ and $0&=\\\\nabla \\\\mathcal {F}_{\\\\phi }(T) +\\\\rho + K^{\\\\top }\\\\hat{V}\\\\left(\\\\hat{N}^{\\\\top }\\\\right)^{-1}\\\\left(\\\\mathcal {J}(\\\\xi )^{\\\\top }\\\\rho - \\\\nabla \\\\mathcal {F}_{\\\\xi }(\\\\xi ) \\\\right).$ Solving, we have $\\\\rho &=\\\\left(I_{ms}+K^{\\\\top }\\\\hat{V}\\\\left(\\\\hat{N}^{\\\\top }\\\\right)^{-1}\\\\mathcal {J}(\\\\xi )^{\\\\top }\\\\right)^{-1}\\\\left(K^{\\\\top }\\\\hat{V}\\\\left(\\\\hat{N}^{\\\\top }\\\\right)^{-1}\\\\nabla \\\\mathcal {F}_{\\\\xi }(\\\\xi )-\\\\nabla \\\\mathcal {F}_{\\\\phi }(T)\\\\right).$ By complementary slackness, $\\\\mathcal {P}_{\\\\textrm {RS}}$ is exact when $\\\\rho >0$ .\",\n \"As in the latter part of the previous section, we now assume that $\\\\mathcal {F}_{\\\\xi }$ and $\\\\mathcal {F}_{\\\\phi }$ are linear with gradients $f_{\\\\xi }\\\\in \\\\mathbb {R}^{ms}$ and $f_{\\\\phi }\\\\in \\\\mathbb {R}^{rs}$ .\",\n \"Corollary 3 Suppose that for all $\\\\xi \\\\ge 0$ , $\\\\left(I_{ms}+K^{\\\\top }\\\\hat{V}\\\\left(\\\\hat{N}^{\\\\top }\\\\right)^{-1}\\\\mathcal {J}(\\\\xi )^{\\\\top }\\\\right)^{-1}\\\\ge 0,$ and $K^{\\\\top }\\\\hat{V}\\\\left(\\\\hat{N}^{\\\\top }\\\\right)^{-1}f_{\\\\xi }-f_{\\\\phi }\\\\ge 0.$ If at least one of the inequalities is strict, then $\\\\mathcal {P}_{\\\\textrm {RS}}$ is exact.\",\n \"Like Theorem REF , Corollary REF depends on the optimal solution, but to a lesser extent.\",\n \"Whereas Theorem REF depends on $\\\\xi $ and $T$ through the objective and $\\\\mathcal {J}(\\\\xi )$ , Corollary REF only depends on $\\\\xi $ through $\\\\mathcal {J}(\\\\xi )$ .\",\n \"This is more manageable because $\\\\mathcal {J}(\\\\xi )$ is often positive and nearly diagonal.\",\n \"For example, if we assume assume that biomass is constant and all growth rates are Monod, as in Example REF , then $\\\\mathcal {J}(\\\\xi )$ is positive on the diagonal and zero elsewhere.\",\n \"The gradostat is a special case of (REF ) where in each tank $i\\\\in \\\\mathcal {S}$ , a single substrate, $S_i$ , is converted to a single type of biomass, $X_i$ .\",\n \"The conversion occurs at the rate $\\\\phi _i(S_i,X_i)/y$ , where $y$ is the yield.\",\n \"Then $\\\\xi _i=[S_i,X_i]^{\\\\top }$ and $\\\\kappa _i=[-1/y,1]^{\\\\top }$ .\",\n \"In [1], several examples are given in which the gradostat is optimized over time and in steady state.\",\n \"Here we first apply Corollary REF to the gradostat.\",\n \"Suppose $\\\\mathcal {F}_{\\\\xi }(\\\\xi (n))=f_S^{\\\\top }S(n)+f_X^{\\\\top }X(n)$ .\",\n \"In this case, Corollary REF holds if $-f_S/y+f_X<0$ .\",\n \"When dealing with the decontamination of undesirable substrates, we do not seek to minimize biomass, in which case $f_X=0$ and the condition is satisfied if $f_S>0$ .\",\n \"We now apply Theorem REF to the gradostat in steady state.\",\n \"Let $\\\\mathcal {J}_S(\\\\xi )\\\\in \\\\mathbb {R}^{r\\\\times s}$ be the Jacobian matrix of $\\\\phi (\\\\xi )$ with respect to $S$ , and let $\\\\mathcal {J}_X(\\\\xi )\\\\in \\\\mathbb {R}^{r\\\\times s}$ be the Jacobian matrix of $\\\\phi (\\\\xi )$ with respect to $X$ .\",\n \"$\\\\mathcal {P}_{\\\\textrm {RS}}$ is exact if $&\\\\left(I_s+V\\\\left(N^{\\\\top }\\\\right)^{-1}\\\\left(-\\\\mathcal {J}_S(\\\\xi )^{\\\\top }/y+\\\\mathcal {J}_X(\\\\xi )^{\\\\top }\\\\right)\\\\right)^{-1}\\\\left(K^{\\\\top }V\\\\left(N^{\\\\top }\\\\right)^{-1}\\\\nabla \\\\mathcal {F}_{\\\\xi }(\\\\xi )-\\\\nabla \\\\mathcal {F}_{\\\\phi }(T)\\\\right)>0 .$ It is straightforward to verify that when $\\\\mathcal {F}_{\\\\xi }(\\\\xi )=0$ , this condition directly implies Theorem 1 in [1].\",\n \"Similarly, we obtain Corollary 1 in [1] if we specialize Corollary REF to the gradostat.\"\n ],\n [\n \"Wastewater treatment\",\n \"In this example, we optimize an idealized wastewater treatment system consisting of three wastewater treatment plants of the city of Paris and its suburbs.\",\n \"The model is based on that in [15], [16], and the influent data from the Inf_rain_2006 dataset of [17].\",\n \"We implemented the model using the parser CVX [18] and the solver Gurobi [19] on a personal computer from 2014 with a 1.4 GHz dual-core processor.\",\n \"In the present study, the flow rates are considered constant and given by $Q^{\\\\textrm {in}}_1=0.1\\\\;m^3/s$ , $Q^{\\\\textrm {in}}_2=0.4\\\\;m^3/s$ , and $Q^{\\\\textrm {in}}_3=0.2\\\\;m^3/s$ .\",\n \"All tanks have volume $1000\\\\;m^3$ .\",\n \"In each plant $i\\\\in \\\\mathcal {S}$ , the state vector $\\\\xi _i=\\\\left[\\\\xi ^{\\\\textrm {BOD}}_i,\\\\xi ^{\\\\textrm {NH}_4^+}_i,\\\\xi ^{\\\\textrm {NO}_2^-}_i,\\\\xi ^{\\\\textrm {NO}_3^-}_i\\\\right]^{\\\\top }\\\\in \\\\mathbb {R}^4$ consists of biochemical oxygen demand, ammonia nitrogen, nitrite, and nitrate.\",\n \"The biomass in each time period is assumed to be an exogenous parameter, and therefore not a component of the process state.\",\n \"This is an admissible assumption in the sense that the biomass concentration is typically much slower than that of the other process components, and has a larger amplitude.\",\n \"In each tank $i\\\\in \\\\mathcal {S}$ , the elements of the process state have kinetics $\\\\phi _i^{\\\\textrm {BOD}},\\\\phi _i^{\\\\textrm {NH}_4^+},\\\\phi _i^{\\\\textrm {NO}_2^-}$ , and $\\\\phi _i^{\\\\textrm {NO}_3^-}$ .\",\n \"All assume Monod growth rates with parameters given in Table REF , which comes from Table 1 in [20].\",\n \"Because the biomass is constant, the growth kinetics can be represented as SOC constraints in $\\\\mathcal {P}_{\\\\textrm {R}}$ .\",\n \"Table: Growth function parametersThe stochiometric matrix for each plant $i\\\\in \\\\mathcal {S}$ is $\\\\kappa _i=\\\\left[\\\\begin{array}{cccc}-1&0 & 0 & 0\\\\\\\\0 & -1 & 0 & 0\\\\\\\\0 & 1/y_i^{\\\\textrm {NH}_4^+,\\\\textrm {NO}_2^-} & -1 & 0\\\\\\\\0 & 0 & 1/y_i^{\\\\textrm {NO}_2^-,\\\\textrm {NO}_3^-} & -1\\\\end{array}\\\\right].$ We used the implicit Euler method, $\\\\mathcal {D}_n[\\\\xi (\\\\cdot )]=(\\\\xi (n)-\\\\xi (n-1))/\\\\Delta $ , with the stepsize $\\\\Delta =1$ , which corresponds to 15 minutes.\",\n \"There are $\\\\tau =1345$ time periods, so that the total time is two weeks.\",\n \"The boundary condition is $\\\\xi (0)=\\\\xi (\\\\tau )$ .\",\n \"This could represent periodic operation, or exogenous conditions that are similar from week to week.\",\n \"The process state must satisfy $\\\\xi ^{\\\\textrm {BOD}}_i(n)\\\\le 150$ mg/L and $\\\\xi ^{\\\\textrm {NH}_4^+}_i(n)\\\\le 60$ mg/L for each $i\\\\in \\\\mathcal {S}$ and $n\\\\in \\\\mathcal {N}$ .\",\n \"Without these constraints, most of the substrate would be directed to Plant 1, which is more efficient than the others.\",\n \"This constraint could represent, for example, regulatory limits on the pollution released by the plants.\",\n \"In each time period, fixed quantities of $\\\\textrm {BOD}$ and $\\\\textrm {NH}_4^+$ , $\\\\Xi ^{\\\\textrm {BOD}}(n)$ and $\\\\Xi ^{\\\\textrm {NH}_4^+}(n)$ , must be allocated over the treatment plants.\",\n \"These quantities are based on the Inf_rain_2006 dataset of [17], which covers 1345 15-minute intervals.\",\n \"$\\\\Xi ^{\\\\textrm {BOD}}(\\\\cdot )$ is set to $S_{\\\\textrm {S}}$ (readily biodegradable substrate), and $\\\\Xi ^{\\\\textrm {NH}_4^+}(\\\\cdot )$ to $S_{\\\\textrm {NH}}$ ($\\\\textrm {NH}_4^+$ and $\\\\textrm {NH}_3$ nitrogen) in [17].\",\n \"The allocation is represented by the linear constraints $\\\\Xi ^{\\\\textrm {BOD}}(n)&=\\\\sum _{i=1}^3Q^{\\\\textrm {in}}_i(n)\\\\xi ^{\\\\textrm {BOD},\\\\textrm {in}}_i(n)\\\\\\\\\\\\Xi ^{\\\\textrm {NH}_4^+}(n)&=\\\\sum _{i=1}^3Q^{\\\\textrm {in}}_i(n)\\\\xi ^{\\\\textrm {NH}_4^+,\\\\textrm {in}}_i(n),$ for each $n\\\\in \\\\mathcal {N}$ .\",\n \"The other influent concentrations are $\\\\xi ^{\\\\textrm {NO}_2^-,\\\\textrm {in}}_i(n)=3$ and $\\\\xi ^{\\\\textrm {NO}_3^-,\\\\textrm {in}}_i(n)=10$ for $i\\\\in \\\\mathcal {S}$ and $n\\\\in \\\\mathcal {N}$ .\",\n \"The plant biomass concentrations are set to $\\\\bar{X}_i(n)=100\\\\left(1+(-1)^i\\\\sin (10\\\\pi n/\\\\tau )\\\\right)$ for $i\\\\in \\\\mathcal {S}$ and $n\\\\in \\\\mathcal {N}$ .\",\n \"Observe that this has larger magnitude and varies slower than the other influents.\",\n \"For each $i\\\\in \\\\mathcal {S}$ and $n\\\\in \\\\mathcal {N}$ , the optimization variables are $\\\\xi _i(n)$ , $\\\\xi ^{\\\\textrm {BOD},\\\\textrm {in}}_i(n)$ , and $\\\\xi ^{\\\\textrm {NH}4,\\\\textrm {in}}_i(n)$ .\",\n \"The objective is to minimize the untreated wastewater released by the plants, given by $\\\\sum _{n\\\\in \\\\mathcal {N}}\\\\sum _{i\\\\in \\\\mathcal {S}}Q^{\\\\textrm {out}}_i\\\\eta _i^{\\\\top }\\\\xi _i(n),$ where we note that $Q^{\\\\textrm {out}}_i=Q^{\\\\textrm {in}}_i$ , and $\\\\eta _i=[2,2,0.3,0.1,0]^{\\\\top }$ .\",\n \"This objective was chosen to satisfy Corollary REF .\",\n \"Note, however, that the result does not fully apply due to the boundary condition, $\\\\xi (0)=\\\\xi (\\\\tau )$ , and the upper limit on the process state.\",\n \"The convex relaxation $\\\\mathcal {P}_{\\\\textrm {R}}$ contains 145272 variables (in standard form) and took 17 minutes to solve.\",\n \"The 18 solver iterations accounted for only four seconds, and the rest of the time was used for preprocessing.\",\n \"Despite not fully satisfying Corollary REF , the solution was exact in all time periods.\",\n \"Figures REF and REF respectively show the optimal influent allocation, $\\\\xi ^{\\\\textrm {in}}(\\\\cdot )$ , and the resulting plant effluent concentrations, $\\\\xi (\\\\cdot )$ , between hours 175 and 275.\",\n \"Figure: ξ in (·)\\\\xi ^{\\\\textrm {in}}(\\\\cdot ) from hour 175 to 275.\",\n \"The units are mg/L.Figure: ξ(·)\\\\xi (\\\\cdot ) from hour 175 to 275.\",\n \"The units are mg/L.The slower variation of the biomass dominates that of the diurnal variation in the Inf_rain_2006 dataset, leading to concentration increases whenever the biomass influent into each plant is high.\",\n \"There is a spike in $\\\\Xi ^{\\\\textrm {BOD}}(\\\\cdot )$ and $\\\\Xi ^{\\\\textrm {NH}_4^+}(\\\\cdot )$ around hour 250.\",\n \"This causes $\\\\xi ^{\\\\textrm {BOD}}_i(\\\\cdot )$ , and $\\\\xi ^{\\\\textrm {NH}4}_i(n)$ to hit their concentration limits in Plants 1 and 3.\",\n \"When this happens, the remainder is allocated to Plant 2, which has little biomass at the time, and hence cannot transform the substrates as efficiently.\"\n ],\n [\n \"Conclusion\",\n \"We have formulated a convex relaxation for optimizing a broad class of bioprocesses.\",\n \"We proved that the relaxation is exact under simple conditions, and implemented it on a wastewater treatment example with over one hundred thousand variables.\",\n \"We believe that a wide range of problems can be approached in this manner due to the generality of the model and the tractability of SOCP.\",\n \"One direction of future work is dealing with inexactness.\",\n \"Two options are deriving general convex underestimators to limit the relaxation gap, as in [1], and finding local minima of the non-relaxed problem.\",\n \"In particular, the concave-convex procedure [21] is well-suited to the nonconvexity encountered here and would entail solving a sequence of SOCPs.\",\n \"We also intend to incorporate new elements into the model such as biomass death and recirculation, and to apply the relaxation in other contexts such as enzymes, where Michaelis-Menten kinetics [22] have the same form as Monod kinetics.\"\n ],\n [\n \"Acknowledgments\",\n \"Funding is acknowledged from the Natural Sciences and Engineering Research Council of Canada and the French LabEx NUMEV (Project ANR-10 LABX-20), incorporated into the I-Site MUSE, which partially funded the sabbatical of J.A.\",\n \"Taylor at MISTEA lab.\"\n ]\n]"},"id":{"kind":"string","value":"2107.01843"}}},{"rowIdx":1588,"cells":{"text":{"kind":"list like","value":[["Exergy of passive states: Waste energy after ergotropy extraction"],["Abstract Work extraction protocol is always a significant issue in the context of quantum batteries, in which the notion of ergotropy is used to quantify a particular amount of energy that can be extracted through unitary processes.","Given the total amount of energy stored in a quantum system, quantifying wasted energy after the ergotropy extraction is a question to be considered when undesired coupling with thermal reservoirs is taken into account.","In this paper, we show that some amount of energy can be lost when we extract ergotropy from a quantum system and quantified by the exergy of passive states.","Through a particular example, one shows that ergotropy extraction can be done by preserving the quantum correlations of a quantum system.","Our study opens the perspective for new advances in open system quantum batteries able to explore exergy stored as quantum correlations."],["Introduction","The dynamics of quantum systems that are in contact with an external environment, the so-called open system, arises from the interaction between the quantum degrees of freedom and the time evolution of coherence and system-environment coupling strength.","Recently, efforts have been made to design the environment-mediated charging process of quantum batteries by different scenarios [1], [2], [3], [4], [5], [6], [7], which is a path to the realization of new quantum batteries.","With the advent of quantum thermodynamics, studies on quantum devices able to use quantum advantages to store and extract useful energy from physical systems [8], [9], [10], [11], [12], [13], called quantum batteries (QBs), have allowed to define new physical quantities.","For example, the maximal work that can be extracted from a quantum system by unitary operations is called ergotropy [14].","However, when we take into account non-unitary effects on the QB due to the coupling of the system with external thermal baths [1], [2], the battery performance leads to ask how much energy is lost due to thermal effects on the system.","In particular, there are situations in which ergotropy can be stored [15], but some part of the total energy available in the system is not stored as ergotropy and, consequently, part of the total energy cannot be extracted from cyclic unitary transformations.","Under this point of view, it is worth considering the amount of residual energy that cannot be extracted as useful work from open quantum batteries by unitary processes.","In this direction, in this paper we explore the amount of energy lost due to the constraint of unitary processes for work extraction.","To this end, we show that the system exergy, the maximum amount of work extracted from the system bringing it into the equilibrium with a thermal bath, can be decomposed into two quantities: ergotropy and residual energy.","Such residual energy cannot be extracted through a unitary process, then it refers to the non-optimal performance of a cyclic thermodynamics process for work extraction from QBs.","In a general way, we show that the waste energy is quantified by the amount of exergy of the passive state (a state in which no energy can be extracted as ergotropy).","We then apply our discussion to Werner states, where we study how entanglement and discord are associated with ergotropy and exergy of Werner passive states.","We can quantify the extractable energy from a quantum system using the definition of ergotropy (for unitary process) and exergy (for non-unitary process).","This is motivated by the definition of the amount of energy that is accessible due to the presence of the thermal bath.","So, let us consider the following scenario: we have a quantum system of interest, the system is in an arbitrary non-equilibrium state $\\rho _{0}$ with a Hamiltonian $H_{S}$ .","One may regard only unitary evolution, where the system evolves through a cyclic process.","In this protocol, the Hamiltonian of the system is the same at beginning and at the end of the process, so that ergotropy is the maximum work that can be extracted from the battery under such a process as the following form [14] $\\mathcal {E}_{S}&=(H_{S}\\rho _{S})-\\min _{U\\in \\mathcal {U}}(H_{S}U\\rho _{S}U^{\\dagger })=(H\\rho _{0}) - (H\\varrho _{\\rho _{0}}),$ where $\\mathcal {U}$ covers the set of unitary operations derived from Hamiltonian $H_{S}$ , and $\\varrho _{\\rho _{0}}$ is the called passive state [14].","In addition, in situations where the system-reservoir interaction is taken into account, energy can be lost by thermalization.","To take into account such interaction, let us consider the thermalization process of the system, initially in state $\\rho _{0}$ , with a thermal bath at inverse temperature $\\beta $ .","It is known that during the thermalization process an amount of energy is exchanged between the system and the reservoir given by the variation of the free energy [16] $\\Sigma ^{\\rho _{0}\\rightarrow \\rho _{\\beta }} = \\mathcal {F}(\\rho _{0})- \\mathcal {F}(\\rho _{\\beta }) , $ where $\\mathcal {F}(\\rho _{0})\\!=\\!\\lbrace H\\rho _{0}\\rbrace -\\beta ^{-1}S(\\rho _{0})$ is the nonequilibrium free energy of the battery at the initial stage, $\\mathcal {F}(\\rho _{\\beta })$ being the free energy of the final equilibrium state $\\rho _{\\beta }=e^{-\\beta H}/Z$ , with the von Neumann entropy and the partition function given by $S(\\rho )\\!=\\!-{\\rho \\ln \\rho }$ and $Z=tr(e^{-\\beta H})$ , respectively.","Since the quantity $\\Sigma ^{\\rho _{0}\\rightarrow \\rho _{\\beta }}$ is the amount of extractable work from a system through a process that brings the system into the equilibrium with a thermal reservoir, it is worth mentioning here that we can call it exergy.","We use it in analogy with the thermodynamics classical definition of exergy, as defined by Zoran Rant [17], from Greek `ex' [$\\varepsilon \\xi $ ] and `ergon' [$\\varepsilon \\rho \\gamma o\\nu $ ].","Figure: Schematic representation of the three states addressed in our discussion.","The ergotropy extraction process, which does not change the system entropy, is followed by the thermalization process that brings the system into equilibrium with the thermal reservoir at temperature β\\beta ."],["Residual energy after ergotropy extraction","It is known that after the ergotropy extraction, the internal energy of the system is not zero [14], [18], so that a residual amount of energy is yet available in the system.","In order to quantify the amount of extractable energy after ergotropy extraction, we consider the dynamics as depicted in Fig.","REF .","The available work in an initial state $\\rho _{0}$ can be extracted in two ways: (i) a unitary process which brings the system into a passive state $\\varrho _{\\rho _{0}}$ , so that the extractable work is quantified by ergotropy [14], and (ii) a non-unitary process that leads the system to the thermal equilibrium state $\\rho _{\\beta }$ , quantified by the variation of the free energy [16].","As sketched in Fig.","REF , it is possible to identify some amount of energy $\\Sigma _{\\text{ex}}$ so that the exergy $\\Sigma (\\rho _{0}\\rightarrow \\rho _{\\beta })$ extracted in the process $\\rho _{0}\\rightarrow \\rho _{\\beta }$ , and the extractable ergotropy $\\mathcal {E}$ in the process $\\rho _{0}\\rightarrow \\varrho _{\\rho _{0}}$ , satisfy the balance equation $\\Sigma (\\rho _{0}\\rightarrow \\rho _{\\beta }) = \\mathcal {E} + \\Sigma _{\\text{ex}}$ , where $\\Sigma _{\\text{ex}}$ is an available work that cannot be extracted by unitary processes.","In fact, consider Eq.","(REF ) as $\\Sigma ^{\\rho _{0}\\rightarrow \\rho _{\\beta }} = \\lbrace H\\rho _{0}\\rbrace -\\beta ^{-1}S(\\rho _{0}) - [\\lbrace H\\rho _{\\beta }\\rbrace -\\beta ^{-1}S(\\rho _{\\beta })]$ where we can assume the sum-zero equation given by $\\lbrace H^{(0)}\\varrho _{\\rho _{0}}\\rbrace - \\lbrace H^{(0)}\\varrho _{\\rho _{0}}\\rbrace $ in order to write $\\Sigma ^{\\rho _{0}\\rightarrow \\rho _{\\beta }} &= \\lbrace H\\rho _{0}\\rbrace - \\lbrace H\\varrho _{\\rho _{0}}\\rbrace - [\\lbrace H\\rho _{\\beta }\\rbrace -\\beta ^{-1}S(\\rho _{\\beta })] \\nonumber \\\\ &+ \\lbrace H\\varrho _{\\rho _{0}}\\rbrace - \\beta ^{-1}S(\\rho _{0}) .$ Now, notice that the first two terms are associated with the ergotropy of the initial state $\\rho _{0}$ , so that $\\Sigma ^{\\rho _{0}\\rightarrow \\rho _{\\beta }} &= \\mathcal {E}(\\rho _{0}) - \\left[\\lbrace H\\rho _{\\beta }\\rbrace -\\beta ^{-1}S(\\rho _{\\beta })\\right] \\nonumber \\\\ &+ \\lbrace H\\varrho _{\\rho _{0}}\\rbrace - \\beta ^{-1}S(\\rho _{0}) .$ In addition, since the process that brings the system from state $\\rho _{0}$ to the passive state $\\varrho _{\\rho _{0}}$ is unitary, we also can write $S(\\rho _{0})\\!=\\!S(\\varrho _{\\rho _{0}})$ , and Eq.","(REF ) becomes $\\Sigma ^{\\rho _{0}\\rightarrow \\rho _{\\beta }} &= \\mathcal {E}(\\rho _{0}) + \\left[\\mathcal {F}(\\varrho _{\\rho _{0}})- \\mathcal {F}(\\rho _{\\beta })\\right] .$ In conclusion, as sketched in Fig.","REF , the additional amount of energy $\\Sigma _{\\text{ex}}$ is given by the free energy of the passive state associated to $\\rho _{0}$ , which is equivalent to exergy stored in the passive state $\\varrho _{\\rho _{0}}$ , mathematically $\\Sigma _{\\text{ex}} = \\Sigma ^{\\varrho _{\\rho _{0}}\\rightarrow \\rho _{\\beta }} = \\mathcal {F}(\\varrho _{\\rho _{0}})- \\mathcal {F}(\\rho _{\\beta }) .$ This result can be understood in two different ways.","The first interpretation refers to the uniqueness of an energetically efficient initial state for store ergotropy.","In fact, given a system that interacts with a reservoir, the initial ergotropy is stored in a non-pure state $\\rho _0$ .","Then, for a short time interval, we can drive the system in order to extract ergotropy through the optimal unitary operation $U_\\text{opt}$ .","By adequately choosing the initial state so that $U_\\text{opt}\\rho _{0}U_\\text{opt}^{\\dagger }\\rightarrow \\rho _{\\beta }$ , it is possible to see that $\\Sigma ^{\\varrho _{\\rho _{0}}\\rightarrow \\rho _{\\beta }}\\!=\\!0$ , leading to $\\Sigma ^{\\rho _{0}\\rightarrow \\rho _{\\beta }} \\!=\\!\\mathcal {E}(\\rho _{0})$ .","In conclusion, all available energy of the system can be extracted as ergotropy.","So, given the uniqueness of the thermal state $\\rho _{\\beta }$ , for the optimal unitary operation $U_\\text{opt}$ we have the uniqueness of $\\rho _{0}$ .","The second case refers to the way to efficiently extract energy.","In addition, given that the quantity $\\Sigma ^{\\varrho _{\\rho _{0}}\\rightarrow \\rho _{\\beta }}$ cannot be a negative number, this means that energy lost during the ergotropy extraction is expected as a natural process due to the entropy production.","This second case can be understood as an immediate application of the second law of thermodynamics to quantum batteries.","In fact, for $\\Sigma ^{\\varrho _{\\rho _{0}}\\rightarrow \\rho _{\\beta }}\\!>\\!0$ , the Eq.","(REF ) gives $\\Delta S_{\\text{ex}} > \\beta \\left( \\lbrace H\\rho _{\\beta }\\rbrace - \\lbrace H\\varrho _{\\rho _{0}}\\rbrace \\right) ,$ where $\\Delta S_{\\text{ex}}$ is the entropy production required to extract the exergy stored in the system.","Then, given that during the thermalization process the heat exchanged between the system and reservoir is given by the internal energy variation of the system [19], we conclude that $\\Delta S_{\\text{ex}}\\!>\\!\\beta Q$ ."],["Residual energy as quantum correlations","In this section, we show that quantum correlations after ergotropy extraction can preserve a non-zero amount of energy.","To this end, we consider a two-spin Werner state given by $\\rho _{\\text{w}} = \\frac{1-\\varepsilon }{4} \\mathbb {1} + \\varepsilon {\\beta }{\\beta } , $ with the Bell state ${\\beta }\\!=\\!","({\\uparrow \\uparrow }+{\\downarrow \\downarrow })/\\sqrt{2}$ .","The above state is adequate for our study because we can adequately choose $\\varepsilon $ to control the amount of quantum correlation of $\\rho _{\\text{w}}$ .","In particular, here we consider concurrence [20] as the measurement of entanglement, and quantum discord [21] as correlations beyond entanglement.","The concurrence is given by $\\mathcal {C}(\\rho )\\!=\\!\\mathrm {max} \\lbrace 0 , \\lambda _{1} - \\lambda _{2} - \\lambda _{3} - \\lambda _{4} \\rbrace $ , where $\\lambda _{n}$ are the eigenvalues of the operator $\\sqrt{\\rho ^{1/2}\\tilde{\\rho }\\rho ^{1/2}}$ in decreasing order, with $\\tilde{\\rho }=(\\sigma _{y}\\otimes \\sigma _{y})\\rho ^{\\ast }(\\sigma _{y}\\otimes \\sigma _{y})$ , being the complex conjugate $\\rho ^{\\ast }$ of $\\rho $ taken in the basis 0 and 1 of the system.","Quantum discord is computed from the one-sided trace distance discord (TDD) $\\mathcal {D}(\\rho )\\!=\\!D^{(\\rightarrow )}(\\rho )$ for $X$ -states as $\\mathcal {D}(\\rho ) = \\frac{1}{2}\\sqrt{\\frac{\\gamma ^{2}_{1} \\max \\lbrace \\gamma _{3}^{2}, \\gamma _{2}^{2} + x^2\\rbrace - \\gamma ^{2}_{2} \\max \\lbrace \\gamma _{3}^{2}, \\gamma _{1}^{2}\\rbrace }{\\max \\lbrace \\gamma _{3}^{2}, \\gamma _{2}^{2} + x^2\\rbrace - \\min \\lbrace \\gamma _{3}^{2}, \\gamma _{1}^{2}\\rbrace + \\gamma ^{2}_{1} - \\gamma ^{2}_{2}}} ,$ where $\\gamma _{1}\\!=\\!2(\\rho _{32}+\\rho _{41})$ , $\\gamma _{2}\\!=\\!2(\\rho _{32}-\\rho _{41}$ , $\\gamma _{3}\\!=\\!1-2(\\rho _{22}+\\rho _{33})$ , and $x\\!=\\!2 (\\rho _{11}+\\rho _{22})- 1$ .","Then, for the state in Eq.","(REF ), one gets $\\mathcal {D}(\\rho _{\\text{w}}) = \\frac{\\varepsilon }{2} , ~~ \\mathcal {C}(\\rho _{\\text{w}}) = \\max \\left[ 0 , \\frac{3\\varepsilon -1}{2} \\right] .","$ As we shall see, both quantum discord and entanglement of the passive state depend on the Hamiltonian of the system.","In general, the passive state of a system depends on the reference Hamiltonian used to set the system ergotropy.","In order to illustrate this situation, we consider here two different scenarios: the Ising and Heisenberg models.","As we shall see, while the energy eigenstates of the Ising Hamiltonian do not present entanglement, quantum correlations are observed for the energy eigenstates of the Heisenberg Hamiltonian."],["Ising Hamiltonian","First, we focus on the Ising model, whose reference Hamiltonian reads as $H_{Is}=H_{0}+H^{\\mathrm {Is}}_{int},$ with $H_{0}=\\sum _{i=1,2}\\hbar \\omega \\sigma ^{i}_{z}$ and $H^{\\mathrm {Is}}_{int}=J\\hbar \\sigma ^{1}_{z}\\sigma ^{2}_{z}$ being the free and interaction Hamiltonians of the coupled qubit system, respectively.","Where $\\sigma ^{i}_{z}~(i=1,2)$ belongs to the set of standard Pauli matrices $\\sigma _{j}$ with $j\\in \\lbrace x,~y,~z\\rbrace $ for each subsystem and, $J$ is the strength of two-body interaction.","The eigenenergies can be found by the direct diagonalization of the reference Hamiltonian $H_{Is}$ in following set $\\text{spectrum} = \\left\\lbrace -J,~-J,~J-2\\omega ,~J+2\\omega \\right\\rbrace ,$ in which the ordering depends on the values of $J$ and $\\omega $ , associated with eigenstates ${\\downarrow \\uparrow },~{\\uparrow \\downarrow },~{\\downarrow \\downarrow },~{\\uparrow \\uparrow }$ , respectively.","Here, ${\\downarrow }$ is the ground state and ${\\uparrow }$ is the excited state of a single qubit.","Consequently, we can be considered two feasible situations $2\\omega \\ge J$ and $2\\omega < J$ , to achieve an increasing order of energy levels.","In the former case, one can obtain $\\mathcal {E}(\\rho _{w})=2\\varepsilon \\omega $ , where the passive state is given by $\\varrho _{\\rho _{\\text{w}}}^{\\text{Is}}(\\omega \\!\\ge \\!J) = \\frac{1-\\varepsilon }{4} \\mathbb {1} + \\varepsilon {\\downarrow \\downarrow }{\\downarrow \\downarrow } ,$ and for the latter, we have $\\mathcal {E}(\\rho _{w})=2\\varepsilon J$ with the corresponding passive state $\\varrho _{\\rho _{\\text{w}}}^{\\text{Is}}(2\\omega \\!<\\!J) = \\frac{1-\\varepsilon }{4} \\mathbb {1} + \\varepsilon {\\downarrow \\uparrow }{\\downarrow \\uparrow } .$ Therefore, the maximum charge of the battery fulfill the following form $\\mathcal {E}^{\\text{Is}}(\\rho _{\\text{w}}) = \\mathcal {E}^{\\text{Is}}_{0} \\varepsilon , ~~ \\mathcal {E}^{\\text{Is}}_{0} = \\left\\lbrace \\begin{matrix}2 \\omega \\hbar & \\omega \\ge J\\\\2 J\\hbar & \\omega < J\\end{matrix} \\right.",",$ for any desired value $\\varepsilon $ .","By comparing Eqs.","(REF ) and (REF ), we see that the ergotropy can be written in terms of the quantum discord as $\\mathcal {E}^{\\text{Is}}(\\rho _{\\text{w}}) = 2 \\mathcal {D}(\\rho _{\\text{w}}) \\mathcal {E}^{\\text{Is}}_{0}$ .","In addition, we can see that $\\mathcal {C}(\\rho _{\\text{w}})\\!=\\!0$ and $\\mathcal {E}(\\rho _{\\text{w}})\\!\\ne \\!0$ for $0\\!\\le \\!\\varepsilon \\!\\le \\!1/3$ .","This proves that, in general, entanglement is not an accessible resource for maximum work extraction compared to quantum discord in such a model.","On the other hand, it is straightforward to investigate that $\\mathcal {C}(\\varrho _{\\rho _{\\text{w}}}^{\\text{Is}})$ and $\\mathcal {D}(\\varrho _{\\rho _{\\text{w}}}^{\\text{Is}})$ are zero for both $2\\omega \\!<\\!J$ and $\\omega \\!\\ge \\!J$ .","So it can be argued that ergotropy is entirely stored in quantum discord.","According to Eq.","(REF ), after the ergotropy extraction procedure, we have an amount of energy that cannot be extracted through a coherent interaction with an external field, which leads to a unitary process.","Such residual amount of energy can be obtained for the Ising model as follow $\\Sigma ^{\\varrho _{\\rho _{0}}\\rightarrow \\rho _{\\beta }} = \\Sigma _{0} \\varepsilon + \\beta ^{-1}C , ~~ \\Sigma _{0} = \\left\\lbrace \\begin{matrix}(J-2 \\omega )\\hbar & \\omega \\ge J\\\\-J\\hbar & \\omega < J\\end{matrix} \\right.",",$ where $C_{\\text{Is}}\\!=\\!\\text{ln}(Z_{\\text{Is}})-S(\\varrho _{\\rho _{\\text{w}}}^{\\text{Is}})$ , with $Z_{\\text{Is}}\\!=\\!2(e^{\\beta J}+e^{-\\beta J}\\cosh [2\\beta \\omega ])$ the partition function of thermal state $\\rho _{\\beta }$ and $S(\\varrho _{\\rho _{\\text{w}}}^{\\text{Is}})$ the entropy of the passive state given by $S(\\varrho _{\\rho _{\\text{w}}}^{\\text{Is}})=2 - \\left[\\frac{3(1-\\varepsilon )}{4}\\log _{2}(1-\\varepsilon )+\\frac{(1+3\\varepsilon )}{4}\\log _{2}(1+3\\varepsilon )\\right] .","$ It is important to mention that, besides the exergy depends on the parameter $\\varepsilon $ , it is not associated with any amount of correlation, since the passive state $\\varrho _{\\rho _{\\text{w}}}^{\\text{Is}}$ , does not present any quantum correlation.","Then, exergy of passive states for the Ising Hamiltonian is not stored as correlations."],["Heisenberg Hamiltonian","The Heisenberg Hamiltonian describes a system where the interaction part is given by $H_{\\text{int}}^{\\text{H}} = \\frac{J\\hbar }{\\sqrt{2}} \\left( \\sigma _{x}^{1} \\sigma _{x}^{2} + \\sigma _{y}^{1} \\sigma _{y}^{2} \\right) ,$ where factor $\\sqrt{2}$ is considered here to give a fair comparison with the Ising model, so that $||H_{\\text{int}}^{\\text{H}}||\\!=\\!||H_{\\text{int}}^{\\text{H}}||$ , being $||A||\\!=\\!","(\\text{tr} ( A A^{\\dagger } ) )^{1/2}$ the Hilbert-Schmidt norm of the operator $A$ .","This quantity allows for quantifying the thermodynamic cost of implementing the dynamics driven by an arbitrary Hamiltonian [22].","By computing the spectrum of the reference Hamiltonian $H_{\\text{H}}\\!=\\!","H_{0} + H_{\\text{int}}^{\\text{H}}$ , we find $\\text{spectrum} = \\left\\lbrace -\\sqrt{2} J, -2 \\omega , \\sqrt{2} J, 2 \\omega \\right\\rbrace ,$ with eigenstates $({\\downarrow \\uparrow }-{\\uparrow \\downarrow })/\\sqrt{2}$ , ${\\downarrow \\downarrow }$ , $({\\downarrow \\uparrow }+{\\uparrow \\downarrow })/\\sqrt{2}$ and ${\\uparrow \\uparrow }$ , respectively.","We stress that, for this case, whenever we have $|J|\\!>\\!\\!\\sqrt{2}|\\omega |$ , the ground state of the system is a maximally entangled state.","After a detailed analysis, it is possible to show that the ergotropy can be adequately written as $\\mathcal {E}^{\\text{H}}(\\rho _{\\text{w}}) = \\mathcal {E}^{\\text{H}}_{0} \\varepsilon , ~~ \\mathcal {E}^{\\text{H}}_{0} = \\left\\lbrace \\begin{matrix}2 \\omega \\hbar & \\omega \\ge J/\\sqrt{2}\\\\\\sqrt{2} J\\hbar & \\omega < J/\\sqrt{2}\\end{matrix} \\right.",",$ Similar to the Ising case, given an arbitrary value for $\\varepsilon $ , it is not possible to write $\\mathcal {E}(\\rho _{\\text{w}})\\!\\propto \\!\\mathcal {C}(\\rho _{\\text{w}})$ , however, we can write $\\mathcal {E}^{\\text{H}}(\\rho _{\\text{w}})\\!=\\!2 \\mathcal {D}(\\rho _{\\text{w}}) \\mathcal {E}^{\\text{H}}_{0}$ .","Therefore, we conclude that the ergotropy stored in the Werner state is not stored as entanglement, but ergotropy is stored quantum discord.","Then, given recent results on discord-based ergotropy [23], we understand that entanglement is not the main quantum resource of quantum batteries in general and confirms the results presented in [11].","Then, after ergotropy extraction, the passive state for the Heisenberg Hamiltonian reads as $\\varrho _{\\rho _{\\text{w}}}^{\\text{H}}(\\omega \\!\\ge \\!J/\\sqrt{2}) = \\frac{1-\\varepsilon }{4} \\mathbb {1} + \\varepsilon {\\downarrow \\downarrow }{\\downarrow \\downarrow } ,$ for the regime where $\\omega \\!\\ge \\!J/\\sqrt{2}$ .","In this case, we can see the amount of energy $\\mathcal {E}(\\rho _{\\text{w}})\\!=\\!2\\varepsilon \\hbar \\omega $ is extracted from the system so that the passive state has no amount of correlation.","In fact, from concurrence and, discord we find $\\mathcal {C}(\\varrho _{\\rho _{\\text{w}}}^{\\text{H}})\\!=\\!\\mathcal {D}(\\varrho _{\\rho _{\\text{w}}}^{\\text{H}})\\!=\\!0$ .","Therefore, the maximum work extraction comes with fully “correlation extraction\".","On the other hand, by considering the case with $\\omega \\!<\\!J/\\sqrt{2}$ we get the passive state $\\varrho _{\\rho _{\\text{w}}}^{\\text{H}}(\\omega \\!<\\!J/\\sqrt{2}) = \\frac{1-\\varepsilon }{4} \\mathbb {1} + \\varepsilon {\\beta _{-}}{\\beta _{-}} ,$ where ${\\beta _{-}}\\!=\\!","({\\uparrow \\downarrow }-{\\downarrow \\uparrow })/\\sqrt{2}$ is the singlet-state of two-qubit, one of the Bell states.","It is worth highlighting this case for a particular motivation: the ergotropy extraction can be done without changing the amount of correlation in the system.","To the best of our knowledge, such a result was not observed so far and it has an implication different from our previous discussion.","Because the states $\\varrho _{\\rho _{\\text{w}}}^{\\text{H}}(\\omega \\!<\\!J/\\sqrt{2})$ and $\\rho _{\\text{w}}$ have the same amount of correlation, since we can obtain $\\rho _{\\text{w}}$ up to local rotations in $\\varrho _{\\rho _{\\text{w}}}^{\\text{H}}(\\omega \\!<\\!J/\\sqrt{2})$ , we understand that the ergotropy of the system with Heisenberg Hamiltonian, where $\\omega \\!<\\!J/\\sqrt{2}$ , is not stored as correlations.","It means that by increasing the interaction strength of the system, the ergotropy of Werner states can be extracted from the system without destroying correlations in the system, leading then to a Werner passive state.","According to the final interpretations of the previous section, it turns out that the passive state exergy can be a physical justification for such an event.","To clarify this idea underlying the waste energy, we find the exergy of passive state in the following form $\\Sigma ^{\\varrho _{\\rho _{0}}\\rightarrow \\rho _{\\beta }} = \\Sigma _{0} \\varepsilon + \\beta ^{-1}C_{\\text{H}} , ~~ \\Sigma _{0} = \\left\\lbrace \\begin{matrix}-2 \\omega \\hbar & \\omega \\ge \\frac{J}{\\sqrt{2}}\\\\-\\sqrt{2}J\\hbar & \\omega < \\frac{J}{\\sqrt{2}}\\end{matrix} \\right.",",$ with $C_{\\text{H}}\\!=\\!\\text{ln}(Z_{\\text{H}})-S(\\varrho _{\\rho _{\\text{w}}}^{\\text{H}})$ , with $Z_{\\text{H}}\\!=\\!2(\\cosh [\\sqrt{2}\\beta J]+\\cosh [2\\beta \\omega ])$ and $S(\\varrho _{\\rho _{\\text{w}}}^{\\text{H}})$ given by Eq.","(REF ), since the passive state $\\varrho _{\\rho _{\\text{w}}}^{\\text{H}}$ can be obtained from $\\varrho _{\\rho _{\\text{w}}}^{\\text{Is}}$ through unitary operations for any $J$ and $\\omega $ .","As the main result, it is possible to see that part of the exergy is stored as discord, since we can rewrite the above equation as $\\Sigma ^{\\varrho _{\\rho _{0}}\\rightarrow \\rho _{\\beta }} = 2\\Sigma _{0}\\mathcal {D}(\\rho _{\\text{w}}) + \\beta ^{-1}C_{\\text{H}}$ ."],["conclusion","This paper dealt with the loss of energy from a (cyclic) unitary work extraction from a quantum system, where such an amount of energy is identified as the exergy of the quantum passive state.","We showed that, in general, in a real scenario the ergotropy leads to loss of energy due to the limitation of the unitary process.","In addition, given the system-bath interaction that leads to the thermalization process, we discussed the existence and uniqueness of an optimal passive state for ergotropy and exergy extraction.","From the point of view of the second thermodynamics law, we explain our main result as a natural consequence of the entropy production of the thermalization process for exergy extraction.","As an application of our results, it is possible to identify a family of ergotropy and exergy extraction where the total amount of quantum correlations (as quantified by the quantum discord) of the system is conserved.","Then, it implies that the exergy of a quantum passive state can be stored as quantum correlations.","Since exergy is the amount of energy extractable through a thermalization process, a new prospect is opening up for exploring protocols of operational open quantum batteries.","This work has been supported by the University of Kurdistan.","F. H. Kamin and S. Salimi thank Vice Chancellorship of Research and Technology, University of Kurdistan.","A.C.S.","acknowledges the financial support of the São Paulo Research Foundation (FAPESP) (Grant No.","2019/22685-1)."]],"string":"[\n [\n \"Exergy of passive states: Waste energy after ergotropy extraction\"\n ],\n [\n \"Abstract Work extraction protocol is always a significant issue in the context of quantum batteries, in which the notion of ergotropy is used to quantify a particular amount of energy that can be extracted through unitary processes.\",\n \"Given the total amount of energy stored in a quantum system, quantifying wasted energy after the ergotropy extraction is a question to be considered when undesired coupling with thermal reservoirs is taken into account.\",\n \"In this paper, we show that some amount of energy can be lost when we extract ergotropy from a quantum system and quantified by the exergy of passive states.\",\n \"Through a particular example, one shows that ergotropy extraction can be done by preserving the quantum correlations of a quantum system.\",\n \"Our study opens the perspective for new advances in open system quantum batteries able to explore exergy stored as quantum correlations.\"\n ],\n [\n \"Introduction\",\n \"The dynamics of quantum systems that are in contact with an external environment, the so-called open system, arises from the interaction between the quantum degrees of freedom and the time evolution of coherence and system-environment coupling strength.\",\n \"Recently, efforts have been made to design the environment-mediated charging process of quantum batteries by different scenarios [1], [2], [3], [4], [5], [6], [7], which is a path to the realization of new quantum batteries.\",\n \"With the advent of quantum thermodynamics, studies on quantum devices able to use quantum advantages to store and extract useful energy from physical systems [8], [9], [10], [11], [12], [13], called quantum batteries (QBs), have allowed to define new physical quantities.\",\n \"For example, the maximal work that can be extracted from a quantum system by unitary operations is called ergotropy [14].\",\n \"However, when we take into account non-unitary effects on the QB due to the coupling of the system with external thermal baths [1], [2], the battery performance leads to ask how much energy is lost due to thermal effects on the system.\",\n \"In particular, there are situations in which ergotropy can be stored [15], but some part of the total energy available in the system is not stored as ergotropy and, consequently, part of the total energy cannot be extracted from cyclic unitary transformations.\",\n \"Under this point of view, it is worth considering the amount of residual energy that cannot be extracted as useful work from open quantum batteries by unitary processes.\",\n \"In this direction, in this paper we explore the amount of energy lost due to the constraint of unitary processes for work extraction.\",\n \"To this end, we show that the system exergy, the maximum amount of work extracted from the system bringing it into the equilibrium with a thermal bath, can be decomposed into two quantities: ergotropy and residual energy.\",\n \"Such residual energy cannot be extracted through a unitary process, then it refers to the non-optimal performance of a cyclic thermodynamics process for work extraction from QBs.\",\n \"In a general way, we show that the waste energy is quantified by the amount of exergy of the passive state (a state in which no energy can be extracted as ergotropy).\",\n \"We then apply our discussion to Werner states, where we study how entanglement and discord are associated with ergotropy and exergy of Werner passive states.\",\n \"We can quantify the extractable energy from a quantum system using the definition of ergotropy (for unitary process) and exergy (for non-unitary process).\",\n \"This is motivated by the definition of the amount of energy that is accessible due to the presence of the thermal bath.\",\n \"So, let us consider the following scenario: we have a quantum system of interest, the system is in an arbitrary non-equilibrium state $\\\\rho _{0}$ with a Hamiltonian $H_{S}$ .\",\n \"One may regard only unitary evolution, where the system evolves through a cyclic process.\",\n \"In this protocol, the Hamiltonian of the system is the same at beginning and at the end of the process, so that ergotropy is the maximum work that can be extracted from the battery under such a process as the following form [14] $\\\\mathcal {E}_{S}&=(H_{S}\\\\rho _{S})-\\\\min _{U\\\\in \\\\mathcal {U}}(H_{S}U\\\\rho _{S}U^{\\\\dagger })=(H\\\\rho _{0}) - (H\\\\varrho _{\\\\rho _{0}}),$ where $\\\\mathcal {U}$ covers the set of unitary operations derived from Hamiltonian $H_{S}$ , and $\\\\varrho _{\\\\rho _{0}}$ is the called passive state [14].\",\n \"In addition, in situations where the system-reservoir interaction is taken into account, energy can be lost by thermalization.\",\n \"To take into account such interaction, let us consider the thermalization process of the system, initially in state $\\\\rho _{0}$ , with a thermal bath at inverse temperature $\\\\beta $ .\",\n \"It is known that during the thermalization process an amount of energy is exchanged between the system and the reservoir given by the variation of the free energy [16] $\\\\Sigma ^{\\\\rho _{0}\\\\rightarrow \\\\rho _{\\\\beta }} = \\\\mathcal {F}(\\\\rho _{0})- \\\\mathcal {F}(\\\\rho _{\\\\beta }) , $ where $\\\\mathcal {F}(\\\\rho _{0})\\\\!=\\\\!\\\\lbrace H\\\\rho _{0}\\\\rbrace -\\\\beta ^{-1}S(\\\\rho _{0})$ is the nonequilibrium free energy of the battery at the initial stage, $\\\\mathcal {F}(\\\\rho _{\\\\beta })$ being the free energy of the final equilibrium state $\\\\rho _{\\\\beta }=e^{-\\\\beta H}/Z$ , with the von Neumann entropy and the partition function given by $S(\\\\rho )\\\\!=\\\\!-{\\\\rho \\\\ln \\\\rho }$ and $Z=tr(e^{-\\\\beta H})$ , respectively.\",\n \"Since the quantity $\\\\Sigma ^{\\\\rho _{0}\\\\rightarrow \\\\rho _{\\\\beta }}$ is the amount of extractable work from a system through a process that brings the system into the equilibrium with a thermal reservoir, it is worth mentioning here that we can call it exergy.\",\n \"We use it in analogy with the thermodynamics classical definition of exergy, as defined by Zoran Rant [17], from Greek `ex' [$\\\\varepsilon \\\\xi $ ] and `ergon' [$\\\\varepsilon \\\\rho \\\\gamma o\\\\nu $ ].\",\n \"Figure: Schematic representation of the three states addressed in our discussion.\",\n \"The ergotropy extraction process, which does not change the system entropy, is followed by the thermalization process that brings the system into equilibrium with the thermal reservoir at temperature β\\\\beta .\"\n ],\n [\n \"Residual energy after ergotropy extraction\",\n \"It is known that after the ergotropy extraction, the internal energy of the system is not zero [14], [18], so that a residual amount of energy is yet available in the system.\",\n \"In order to quantify the amount of extractable energy after ergotropy extraction, we consider the dynamics as depicted in Fig.\",\n \"REF .\",\n \"The available work in an initial state $\\\\rho _{0}$ can be extracted in two ways: (i) a unitary process which brings the system into a passive state $\\\\varrho _{\\\\rho _{0}}$ , so that the extractable work is quantified by ergotropy [14], and (ii) a non-unitary process that leads the system to the thermal equilibrium state $\\\\rho _{\\\\beta }$ , quantified by the variation of the free energy [16].\",\n \"As sketched in Fig.\",\n \"REF , it is possible to identify some amount of energy $\\\\Sigma _{\\\\text{ex}}$ so that the exergy $\\\\Sigma (\\\\rho _{0}\\\\rightarrow \\\\rho _{\\\\beta })$ extracted in the process $\\\\rho _{0}\\\\rightarrow \\\\rho _{\\\\beta }$ , and the extractable ergotropy $\\\\mathcal {E}$ in the process $\\\\rho _{0}\\\\rightarrow \\\\varrho _{\\\\rho _{0}}$ , satisfy the balance equation $\\\\Sigma (\\\\rho _{0}\\\\rightarrow \\\\rho _{\\\\beta }) = \\\\mathcal {E} + \\\\Sigma _{\\\\text{ex}}$ , where $\\\\Sigma _{\\\\text{ex}}$ is an available work that cannot be extracted by unitary processes.\",\n \"In fact, consider Eq.\",\n \"(REF ) as $\\\\Sigma ^{\\\\rho _{0}\\\\rightarrow \\\\rho _{\\\\beta }} = \\\\lbrace H\\\\rho _{0}\\\\rbrace -\\\\beta ^{-1}S(\\\\rho _{0}) - [\\\\lbrace H\\\\rho _{\\\\beta }\\\\rbrace -\\\\beta ^{-1}S(\\\\rho _{\\\\beta })]$ where we can assume the sum-zero equation given by $\\\\lbrace H^{(0)}\\\\varrho _{\\\\rho _{0}}\\\\rbrace - \\\\lbrace H^{(0)}\\\\varrho _{\\\\rho _{0}}\\\\rbrace $ in order to write $\\\\Sigma ^{\\\\rho _{0}\\\\rightarrow \\\\rho _{\\\\beta }} &= \\\\lbrace H\\\\rho _{0}\\\\rbrace - \\\\lbrace H\\\\varrho _{\\\\rho _{0}}\\\\rbrace - [\\\\lbrace H\\\\rho _{\\\\beta }\\\\rbrace -\\\\beta ^{-1}S(\\\\rho _{\\\\beta })] \\\\nonumber \\\\\\\\ &+ \\\\lbrace H\\\\varrho _{\\\\rho _{0}}\\\\rbrace - \\\\beta ^{-1}S(\\\\rho _{0}) .$ Now, notice that the first two terms are associated with the ergotropy of the initial state $\\\\rho _{0}$ , so that $\\\\Sigma ^{\\\\rho _{0}\\\\rightarrow \\\\rho _{\\\\beta }} &= \\\\mathcal {E}(\\\\rho _{0}) - \\\\left[\\\\lbrace H\\\\rho _{\\\\beta }\\\\rbrace -\\\\beta ^{-1}S(\\\\rho _{\\\\beta })\\\\right] \\\\nonumber \\\\\\\\ &+ \\\\lbrace H\\\\varrho _{\\\\rho _{0}}\\\\rbrace - \\\\beta ^{-1}S(\\\\rho _{0}) .$ In addition, since the process that brings the system from state $\\\\rho _{0}$ to the passive state $\\\\varrho _{\\\\rho _{0}}$ is unitary, we also can write $S(\\\\rho _{0})\\\\!=\\\\!S(\\\\varrho _{\\\\rho _{0}})$ , and Eq.\",\n \"(REF ) becomes $\\\\Sigma ^{\\\\rho _{0}\\\\rightarrow \\\\rho _{\\\\beta }} &= \\\\mathcal {E}(\\\\rho _{0}) + \\\\left[\\\\mathcal {F}(\\\\varrho _{\\\\rho _{0}})- \\\\mathcal {F}(\\\\rho _{\\\\beta })\\\\right] .$ In conclusion, as sketched in Fig.\",\n \"REF , the additional amount of energy $\\\\Sigma _{\\\\text{ex}}$ is given by the free energy of the passive state associated to $\\\\rho _{0}$ , which is equivalent to exergy stored in the passive state $\\\\varrho _{\\\\rho _{0}}$ , mathematically $\\\\Sigma _{\\\\text{ex}} = \\\\Sigma ^{\\\\varrho _{\\\\rho _{0}}\\\\rightarrow \\\\rho _{\\\\beta }} = \\\\mathcal {F}(\\\\varrho _{\\\\rho _{0}})- \\\\mathcal {F}(\\\\rho _{\\\\beta }) .$ This result can be understood in two different ways.\",\n \"The first interpretation refers to the uniqueness of an energetically efficient initial state for store ergotropy.\",\n \"In fact, given a system that interacts with a reservoir, the initial ergotropy is stored in a non-pure state $\\\\rho _0$ .\",\n \"Then, for a short time interval, we can drive the system in order to extract ergotropy through the optimal unitary operation $U_\\\\text{opt}$ .\",\n \"By adequately choosing the initial state so that $U_\\\\text{opt}\\\\rho _{0}U_\\\\text{opt}^{\\\\dagger }\\\\rightarrow \\\\rho _{\\\\beta }$ , it is possible to see that $\\\\Sigma ^{\\\\varrho _{\\\\rho _{0}}\\\\rightarrow \\\\rho _{\\\\beta }}\\\\!=\\\\!0$ , leading to $\\\\Sigma ^{\\\\rho _{0}\\\\rightarrow \\\\rho _{\\\\beta }} \\\\!=\\\\!\\\\mathcal {E}(\\\\rho _{0})$ .\",\n \"In conclusion, all available energy of the system can be extracted as ergotropy.\",\n \"So, given the uniqueness of the thermal state $\\\\rho _{\\\\beta }$ , for the optimal unitary operation $U_\\\\text{opt}$ we have the uniqueness of $\\\\rho _{0}$ .\",\n \"The second case refers to the way to efficiently extract energy.\",\n \"In addition, given that the quantity $\\\\Sigma ^{\\\\varrho _{\\\\rho _{0}}\\\\rightarrow \\\\rho _{\\\\beta }}$ cannot be a negative number, this means that energy lost during the ergotropy extraction is expected as a natural process due to the entropy production.\",\n \"This second case can be understood as an immediate application of the second law of thermodynamics to quantum batteries.\",\n \"In fact, for $\\\\Sigma ^{\\\\varrho _{\\\\rho _{0}}\\\\rightarrow \\\\rho _{\\\\beta }}\\\\!>\\\\!0$ , the Eq.\",\n \"(REF ) gives $\\\\Delta S_{\\\\text{ex}} > \\\\beta \\\\left( \\\\lbrace H\\\\rho _{\\\\beta }\\\\rbrace - \\\\lbrace H\\\\varrho _{\\\\rho _{0}}\\\\rbrace \\\\right) ,$ where $\\\\Delta S_{\\\\text{ex}}$ is the entropy production required to extract the exergy stored in the system.\",\n \"Then, given that during the thermalization process the heat exchanged between the system and reservoir is given by the internal energy variation of the system [19], we conclude that $\\\\Delta S_{\\\\text{ex}}\\\\!>\\\\!\\\\beta Q$ .\"\n ],\n [\n \"Residual energy as quantum correlations\",\n \"In this section, we show that quantum correlations after ergotropy extraction can preserve a non-zero amount of energy.\",\n \"To this end, we consider a two-spin Werner state given by $\\\\rho _{\\\\text{w}} = \\\\frac{1-\\\\varepsilon }{4} \\\\mathbb {1} + \\\\varepsilon {\\\\beta }{\\\\beta } , $ with the Bell state ${\\\\beta }\\\\!=\\\\!\",\n \"({\\\\uparrow \\\\uparrow }+{\\\\downarrow \\\\downarrow })/\\\\sqrt{2}$ .\",\n \"The above state is adequate for our study because we can adequately choose $\\\\varepsilon $ to control the amount of quantum correlation of $\\\\rho _{\\\\text{w}}$ .\",\n \"In particular, here we consider concurrence [20] as the measurement of entanglement, and quantum discord [21] as correlations beyond entanglement.\",\n \"The concurrence is given by $\\\\mathcal {C}(\\\\rho )\\\\!=\\\\!\\\\mathrm {max} \\\\lbrace 0 , \\\\lambda _{1} - \\\\lambda _{2} - \\\\lambda _{3} - \\\\lambda _{4} \\\\rbrace $ , where $\\\\lambda _{n}$ are the eigenvalues of the operator $\\\\sqrt{\\\\rho ^{1/2}\\\\tilde{\\\\rho }\\\\rho ^{1/2}}$ in decreasing order, with $\\\\tilde{\\\\rho }=(\\\\sigma _{y}\\\\otimes \\\\sigma _{y})\\\\rho ^{\\\\ast }(\\\\sigma _{y}\\\\otimes \\\\sigma _{y})$ , being the complex conjugate $\\\\rho ^{\\\\ast }$ of $\\\\rho $ taken in the basis 0 and 1 of the system.\",\n \"Quantum discord is computed from the one-sided trace distance discord (TDD) $\\\\mathcal {D}(\\\\rho )\\\\!=\\\\!D^{(\\\\rightarrow )}(\\\\rho )$ for $X$ -states as $\\\\mathcal {D}(\\\\rho ) = \\\\frac{1}{2}\\\\sqrt{\\\\frac{\\\\gamma ^{2}_{1} \\\\max \\\\lbrace \\\\gamma _{3}^{2}, \\\\gamma _{2}^{2} + x^2\\\\rbrace - \\\\gamma ^{2}_{2} \\\\max \\\\lbrace \\\\gamma _{3}^{2}, \\\\gamma _{1}^{2}\\\\rbrace }{\\\\max \\\\lbrace \\\\gamma _{3}^{2}, \\\\gamma _{2}^{2} + x^2\\\\rbrace - \\\\min \\\\lbrace \\\\gamma _{3}^{2}, \\\\gamma _{1}^{2}\\\\rbrace + \\\\gamma ^{2}_{1} - \\\\gamma ^{2}_{2}}} ,$ where $\\\\gamma _{1}\\\\!=\\\\!2(\\\\rho _{32}+\\\\rho _{41})$ , $\\\\gamma _{2}\\\\!=\\\\!2(\\\\rho _{32}-\\\\rho _{41}$ , $\\\\gamma _{3}\\\\!=\\\\!1-2(\\\\rho _{22}+\\\\rho _{33})$ , and $x\\\\!=\\\\!2 (\\\\rho _{11}+\\\\rho _{22})- 1$ .\",\n \"Then, for the state in Eq.\",\n \"(REF ), one gets $\\\\mathcal {D}(\\\\rho _{\\\\text{w}}) = \\\\frac{\\\\varepsilon }{2} , ~~ \\\\mathcal {C}(\\\\rho _{\\\\text{w}}) = \\\\max \\\\left[ 0 , \\\\frac{3\\\\varepsilon -1}{2} \\\\right] .\",\n \"$ As we shall see, both quantum discord and entanglement of the passive state depend on the Hamiltonian of the system.\",\n \"In general, the passive state of a system depends on the reference Hamiltonian used to set the system ergotropy.\",\n \"In order to illustrate this situation, we consider here two different scenarios: the Ising and Heisenberg models.\",\n \"As we shall see, while the energy eigenstates of the Ising Hamiltonian do not present entanglement, quantum correlations are observed for the energy eigenstates of the Heisenberg Hamiltonian.\"\n ],\n [\n \"Ising Hamiltonian\",\n \"First, we focus on the Ising model, whose reference Hamiltonian reads as $H_{Is}=H_{0}+H^{\\\\mathrm {Is}}_{int},$ with $H_{0}=\\\\sum _{i=1,2}\\\\hbar \\\\omega \\\\sigma ^{i}_{z}$ and $H^{\\\\mathrm {Is}}_{int}=J\\\\hbar \\\\sigma ^{1}_{z}\\\\sigma ^{2}_{z}$ being the free and interaction Hamiltonians of the coupled qubit system, respectively.\",\n \"Where $\\\\sigma ^{i}_{z}~(i=1,2)$ belongs to the set of standard Pauli matrices $\\\\sigma _{j}$ with $j\\\\in \\\\lbrace x,~y,~z\\\\rbrace $ for each subsystem and, $J$ is the strength of two-body interaction.\",\n \"The eigenenergies can be found by the direct diagonalization of the reference Hamiltonian $H_{Is}$ in following set $\\\\text{spectrum} = \\\\left\\\\lbrace -J,~-J,~J-2\\\\omega ,~J+2\\\\omega \\\\right\\\\rbrace ,$ in which the ordering depends on the values of $J$ and $\\\\omega $ , associated with eigenstates ${\\\\downarrow \\\\uparrow },~{\\\\uparrow \\\\downarrow },~{\\\\downarrow \\\\downarrow },~{\\\\uparrow \\\\uparrow }$ , respectively.\",\n \"Here, ${\\\\downarrow }$ is the ground state and ${\\\\uparrow }$ is the excited state of a single qubit.\",\n \"Consequently, we can be considered two feasible situations $2\\\\omega \\\\ge J$ and $2\\\\omega < J$ , to achieve an increasing order of energy levels.\",\n \"In the former case, one can obtain $\\\\mathcal {E}(\\\\rho _{w})=2\\\\varepsilon \\\\omega $ , where the passive state is given by $\\\\varrho _{\\\\rho _{\\\\text{w}}}^{\\\\text{Is}}(\\\\omega \\\\!\\\\ge \\\\!J) = \\\\frac{1-\\\\varepsilon }{4} \\\\mathbb {1} + \\\\varepsilon {\\\\downarrow \\\\downarrow }{\\\\downarrow \\\\downarrow } ,$ and for the latter, we have $\\\\mathcal {E}(\\\\rho _{w})=2\\\\varepsilon J$ with the corresponding passive state $\\\\varrho _{\\\\rho _{\\\\text{w}}}^{\\\\text{Is}}(2\\\\omega \\\\!<\\\\!J) = \\\\frac{1-\\\\varepsilon }{4} \\\\mathbb {1} + \\\\varepsilon {\\\\downarrow \\\\uparrow }{\\\\downarrow \\\\uparrow } .$ Therefore, the maximum charge of the battery fulfill the following form $\\\\mathcal {E}^{\\\\text{Is}}(\\\\rho _{\\\\text{w}}) = \\\\mathcal {E}^{\\\\text{Is}}_{0} \\\\varepsilon , ~~ \\\\mathcal {E}^{\\\\text{Is}}_{0} = \\\\left\\\\lbrace \\\\begin{matrix}2 \\\\omega \\\\hbar & \\\\omega \\\\ge J\\\\\\\\2 J\\\\hbar & \\\\omega < J\\\\end{matrix} \\\\right.\",\n \",$ for any desired value $\\\\varepsilon $ .\",\n \"By comparing Eqs.\",\n \"(REF ) and (REF ), we see that the ergotropy can be written in terms of the quantum discord as $\\\\mathcal {E}^{\\\\text{Is}}(\\\\rho _{\\\\text{w}}) = 2 \\\\mathcal {D}(\\\\rho _{\\\\text{w}}) \\\\mathcal {E}^{\\\\text{Is}}_{0}$ .\",\n \"In addition, we can see that $\\\\mathcal {C}(\\\\rho _{\\\\text{w}})\\\\!=\\\\!0$ and $\\\\mathcal {E}(\\\\rho _{\\\\text{w}})\\\\!\\\\ne \\\\!0$ for $0\\\\!\\\\le \\\\!\\\\varepsilon \\\\!\\\\le \\\\!1/3$ .\",\n \"This proves that, in general, entanglement is not an accessible resource for maximum work extraction compared to quantum discord in such a model.\",\n \"On the other hand, it is straightforward to investigate that $\\\\mathcal {C}(\\\\varrho _{\\\\rho _{\\\\text{w}}}^{\\\\text{Is}})$ and $\\\\mathcal {D}(\\\\varrho _{\\\\rho _{\\\\text{w}}}^{\\\\text{Is}})$ are zero for both $2\\\\omega \\\\!<\\\\!J$ and $\\\\omega \\\\!\\\\ge \\\\!J$ .\",\n \"So it can be argued that ergotropy is entirely stored in quantum discord.\",\n \"According to Eq.\",\n \"(REF ), after the ergotropy extraction procedure, we have an amount of energy that cannot be extracted through a coherent interaction with an external field, which leads to a unitary process.\",\n \"Such residual amount of energy can be obtained for the Ising model as follow $\\\\Sigma ^{\\\\varrho _{\\\\rho _{0}}\\\\rightarrow \\\\rho _{\\\\beta }} = \\\\Sigma _{0} \\\\varepsilon + \\\\beta ^{-1}C , ~~ \\\\Sigma _{0} = \\\\left\\\\lbrace \\\\begin{matrix}(J-2 \\\\omega )\\\\hbar & \\\\omega \\\\ge J\\\\\\\\-J\\\\hbar & \\\\omega < J\\\\end{matrix} \\\\right.\",\n \",$ where $C_{\\\\text{Is}}\\\\!=\\\\!\\\\text{ln}(Z_{\\\\text{Is}})-S(\\\\varrho _{\\\\rho _{\\\\text{w}}}^{\\\\text{Is}})$ , with $Z_{\\\\text{Is}}\\\\!=\\\\!2(e^{\\\\beta J}+e^{-\\\\beta J}\\\\cosh [2\\\\beta \\\\omega ])$ the partition function of thermal state $\\\\rho _{\\\\beta }$ and $S(\\\\varrho _{\\\\rho _{\\\\text{w}}}^{\\\\text{Is}})$ the entropy of the passive state given by $S(\\\\varrho _{\\\\rho _{\\\\text{w}}}^{\\\\text{Is}})=2 - \\\\left[\\\\frac{3(1-\\\\varepsilon )}{4}\\\\log _{2}(1-\\\\varepsilon )+\\\\frac{(1+3\\\\varepsilon )}{4}\\\\log _{2}(1+3\\\\varepsilon )\\\\right] .\",\n \"$ It is important to mention that, besides the exergy depends on the parameter $\\\\varepsilon $ , it is not associated with any amount of correlation, since the passive state $\\\\varrho _{\\\\rho _{\\\\text{w}}}^{\\\\text{Is}}$ , does not present any quantum correlation.\",\n \"Then, exergy of passive states for the Ising Hamiltonian is not stored as correlations.\"\n ],\n [\n \"Heisenberg Hamiltonian\",\n \"The Heisenberg Hamiltonian describes a system where the interaction part is given by $H_{\\\\text{int}}^{\\\\text{H}} = \\\\frac{J\\\\hbar }{\\\\sqrt{2}} \\\\left( \\\\sigma _{x}^{1} \\\\sigma _{x}^{2} + \\\\sigma _{y}^{1} \\\\sigma _{y}^{2} \\\\right) ,$ where factor $\\\\sqrt{2}$ is considered here to give a fair comparison with the Ising model, so that $||H_{\\\\text{int}}^{\\\\text{H}}||\\\\!=\\\\!||H_{\\\\text{int}}^{\\\\text{H}}||$ , being $||A||\\\\!=\\\\!\",\n \"(\\\\text{tr} ( A A^{\\\\dagger } ) )^{1/2}$ the Hilbert-Schmidt norm of the operator $A$ .\",\n \"This quantity allows for quantifying the thermodynamic cost of implementing the dynamics driven by an arbitrary Hamiltonian [22].\",\n \"By computing the spectrum of the reference Hamiltonian $H_{\\\\text{H}}\\\\!=\\\\!\",\n \"H_{0} + H_{\\\\text{int}}^{\\\\text{H}}$ , we find $\\\\text{spectrum} = \\\\left\\\\lbrace -\\\\sqrt{2} J, -2 \\\\omega , \\\\sqrt{2} J, 2 \\\\omega \\\\right\\\\rbrace ,$ with eigenstates $({\\\\downarrow \\\\uparrow }-{\\\\uparrow \\\\downarrow })/\\\\sqrt{2}$ , ${\\\\downarrow \\\\downarrow }$ , $({\\\\downarrow \\\\uparrow }+{\\\\uparrow \\\\downarrow })/\\\\sqrt{2}$ and ${\\\\uparrow \\\\uparrow }$ , respectively.\",\n \"We stress that, for this case, whenever we have $|J|\\\\!>\\\\!\\\\!\\\\sqrt{2}|\\\\omega |$ , the ground state of the system is a maximally entangled state.\",\n \"After a detailed analysis, it is possible to show that the ergotropy can be adequately written as $\\\\mathcal {E}^{\\\\text{H}}(\\\\rho _{\\\\text{w}}) = \\\\mathcal {E}^{\\\\text{H}}_{0} \\\\varepsilon , ~~ \\\\mathcal {E}^{\\\\text{H}}_{0} = \\\\left\\\\lbrace \\\\begin{matrix}2 \\\\omega \\\\hbar & \\\\omega \\\\ge J/\\\\sqrt{2}\\\\\\\\\\\\sqrt{2} J\\\\hbar & \\\\omega < J/\\\\sqrt{2}\\\\end{matrix} \\\\right.\",\n \",$ Similar to the Ising case, given an arbitrary value for $\\\\varepsilon $ , it is not possible to write $\\\\mathcal {E}(\\\\rho _{\\\\text{w}})\\\\!\\\\propto \\\\!\\\\mathcal {C}(\\\\rho _{\\\\text{w}})$ , however, we can write $\\\\mathcal {E}^{\\\\text{H}}(\\\\rho _{\\\\text{w}})\\\\!=\\\\!2 \\\\mathcal {D}(\\\\rho _{\\\\text{w}}) \\\\mathcal {E}^{\\\\text{H}}_{0}$ .\",\n \"Therefore, we conclude that the ergotropy stored in the Werner state is not stored as entanglement, but ergotropy is stored quantum discord.\",\n \"Then, given recent results on discord-based ergotropy [23], we understand that entanglement is not the main quantum resource of quantum batteries in general and confirms the results presented in [11].\",\n \"Then, after ergotropy extraction, the passive state for the Heisenberg Hamiltonian reads as $\\\\varrho _{\\\\rho _{\\\\text{w}}}^{\\\\text{H}}(\\\\omega \\\\!\\\\ge \\\\!J/\\\\sqrt{2}) = \\\\frac{1-\\\\varepsilon }{4} \\\\mathbb {1} + \\\\varepsilon {\\\\downarrow \\\\downarrow }{\\\\downarrow \\\\downarrow } ,$ for the regime where $\\\\omega \\\\!\\\\ge \\\\!J/\\\\sqrt{2}$ .\",\n \"In this case, we can see the amount of energy $\\\\mathcal {E}(\\\\rho _{\\\\text{w}})\\\\!=\\\\!2\\\\varepsilon \\\\hbar \\\\omega $ is extracted from the system so that the passive state has no amount of correlation.\",\n \"In fact, from concurrence and, discord we find $\\\\mathcal {C}(\\\\varrho _{\\\\rho _{\\\\text{w}}}^{\\\\text{H}})\\\\!=\\\\!\\\\mathcal {D}(\\\\varrho _{\\\\rho _{\\\\text{w}}}^{\\\\text{H}})\\\\!=\\\\!0$ .\",\n \"Therefore, the maximum work extraction comes with fully “correlation extraction\\\".\",\n \"On the other hand, by considering the case with $\\\\omega \\\\!<\\\\!J/\\\\sqrt{2}$ we get the passive state $\\\\varrho _{\\\\rho _{\\\\text{w}}}^{\\\\text{H}}(\\\\omega \\\\!<\\\\!J/\\\\sqrt{2}) = \\\\frac{1-\\\\varepsilon }{4} \\\\mathbb {1} + \\\\varepsilon {\\\\beta _{-}}{\\\\beta _{-}} ,$ where ${\\\\beta _{-}}\\\\!=\\\\!\",\n \"({\\\\uparrow \\\\downarrow }-{\\\\downarrow \\\\uparrow })/\\\\sqrt{2}$ is the singlet-state of two-qubit, one of the Bell states.\",\n \"It is worth highlighting this case for a particular motivation: the ergotropy extraction can be done without changing the amount of correlation in the system.\",\n \"To the best of our knowledge, such a result was not observed so far and it has an implication different from our previous discussion.\",\n \"Because the states $\\\\varrho _{\\\\rho _{\\\\text{w}}}^{\\\\text{H}}(\\\\omega \\\\!<\\\\!J/\\\\sqrt{2})$ and $\\\\rho _{\\\\text{w}}$ have the same amount of correlation, since we can obtain $\\\\rho _{\\\\text{w}}$ up to local rotations in $\\\\varrho _{\\\\rho _{\\\\text{w}}}^{\\\\text{H}}(\\\\omega \\\\!<\\\\!J/\\\\sqrt{2})$ , we understand that the ergotropy of the system with Heisenberg Hamiltonian, where $\\\\omega \\\\!<\\\\!J/\\\\sqrt{2}$ , is not stored as correlations.\",\n \"It means that by increasing the interaction strength of the system, the ergotropy of Werner states can be extracted from the system without destroying correlations in the system, leading then to a Werner passive state.\",\n \"According to the final interpretations of the previous section, it turns out that the passive state exergy can be a physical justification for such an event.\",\n \"To clarify this idea underlying the waste energy, we find the exergy of passive state in the following form $\\\\Sigma ^{\\\\varrho _{\\\\rho _{0}}\\\\rightarrow \\\\rho _{\\\\beta }} = \\\\Sigma _{0} \\\\varepsilon + \\\\beta ^{-1}C_{\\\\text{H}} , ~~ \\\\Sigma _{0} = \\\\left\\\\lbrace \\\\begin{matrix}-2 \\\\omega \\\\hbar & \\\\omega \\\\ge \\\\frac{J}{\\\\sqrt{2}}\\\\\\\\-\\\\sqrt{2}J\\\\hbar & \\\\omega < \\\\frac{J}{\\\\sqrt{2}}\\\\end{matrix} \\\\right.\",\n \",$ with $C_{\\\\text{H}}\\\\!=\\\\!\\\\text{ln}(Z_{\\\\text{H}})-S(\\\\varrho _{\\\\rho _{\\\\text{w}}}^{\\\\text{H}})$ , with $Z_{\\\\text{H}}\\\\!=\\\\!2(\\\\cosh [\\\\sqrt{2}\\\\beta J]+\\\\cosh [2\\\\beta \\\\omega ])$ and $S(\\\\varrho _{\\\\rho _{\\\\text{w}}}^{\\\\text{H}})$ given by Eq.\",\n \"(REF ), since the passive state $\\\\varrho _{\\\\rho _{\\\\text{w}}}^{\\\\text{H}}$ can be obtained from $\\\\varrho _{\\\\rho _{\\\\text{w}}}^{\\\\text{Is}}$ through unitary operations for any $J$ and $\\\\omega $ .\",\n \"As the main result, it is possible to see that part of the exergy is stored as discord, since we can rewrite the above equation as $\\\\Sigma ^{\\\\varrho _{\\\\rho _{0}}\\\\rightarrow \\\\rho _{\\\\beta }} = 2\\\\Sigma _{0}\\\\mathcal {D}(\\\\rho _{\\\\text{w}}) + \\\\beta ^{-1}C_{\\\\text{H}}$ .\"\n ],\n [\n \"conclusion\",\n \"This paper dealt with the loss of energy from a (cyclic) unitary work extraction from a quantum system, where such an amount of energy is identified as the exergy of the quantum passive state.\",\n \"We showed that, in general, in a real scenario the ergotropy leads to loss of energy due to the limitation of the unitary process.\",\n \"In addition, given the system-bath interaction that leads to the thermalization process, we discussed the existence and uniqueness of an optimal passive state for ergotropy and exergy extraction.\",\n \"From the point of view of the second thermodynamics law, we explain our main result as a natural consequence of the entropy production of the thermalization process for exergy extraction.\",\n \"As an application of our results, it is possible to identify a family of ergotropy and exergy extraction where the total amount of quantum correlations (as quantified by the quantum discord) of the system is conserved.\",\n \"Then, it implies that the exergy of a quantum passive state can be stored as quantum correlations.\",\n \"Since exergy is the amount of energy extractable through a thermalization process, a new prospect is opening up for exploring protocols of operational open quantum batteries.\",\n \"This work has been supported by the University of Kurdistan.\",\n \"F. H. Kamin and S. Salimi thank Vice Chancellorship of Research and Technology, University of Kurdistan.\",\n \"A.C.S.\",\n \"acknowledges the financial support of the São Paulo Research Foundation (FAPESP) (Grant No.\",\n \"2019/22685-1).\"\n ]\n]"},"id":{"kind":"string","value":"2107.01828"}}},{"rowIdx":1589,"cells":{"text":{"kind":"list like","value":[["Rates of Estimation of Optimal Transport Maps using Plug-in Estimators\n via Barycentric Projections"],["Abstract Optimal transport maps between two probability distributions $\\mu$ and $\\nu$ on $\\mathbb{R}^d$ have found extensive applications in both machine learning and statistics.","In practice, these maps need to be estimated from data sampled according to $\\mu$ and $\\nu$.","Plug-in estimators are perhaps most popular in estimating transport maps in the field of computational optimal transport.","In this paper, we provide a comprehensive analysis of the rates of convergences for general plug-in estimators defined via barycentric projections.","Our main contribution is a new stability estimate for barycentric projections which proceeds under minimal smoothness assumptions and can be used to analyze general plug-in estimators.","We illustrate the usefulness of this stability estimate by first providing rates of convergence for the natural discrete-discrete and semi-discrete estimators of optimal transport maps.","We then use the same stability estimate to show that, under additional smoothness assumptions of Besov type or Sobolev type, wavelet based or kernel smoothed plug-in estimators respectively speed up the rates of convergence and significantly mitigate the curse of dimensionality suffered by the natural discrete-discrete/semi-discrete estimators.","As a by-product of our analysis, we also obtain faster rates of convergence for plug-in estimators of $W_2(\\mu,\\nu)$, the Wasserstein distance between $\\mu$ and $\\nu$, under the aforementioned smoothness assumptions, thereby complementing recent results in Chizat et al.","(2020).","Finally, we illustrate the applicability of our results in obtaining rates of convergence for Wasserstein barycenters between two probability distributions and obtaining asymptotic detection thresholds for some recent optimal-transport based tests of independence."],["Introduction","Given two random variables $X\\sim \\mu $ and $Y\\sim \\nu $ , where $\\mu ,\\nu $ are probability measures on $\\mathbb {R}^d$ , $d\\ge 1$ , the problem of finding a “nice\" map $T_0(\\cdot )$ such that $T_0(X)\\sim \\nu $ has numerous applications in machine learning such as domain adaptation and data integration [67], [54], [38], [37], [41], [122], dimension reduction [72], [13], [98], generative models [66], [89], [96], [120], to name a few.","Of particular interest is the case when $T_0(\\cdot )$ is obtained by minimizing a cost function, a line of work initiated by Gaspard Monge [106] in 1781 (see (REF ) below), in which case $T_0(\\cdot )$ is termed an optimal transport (OT) map and has applications in shape matching/transfer problems [52], [131], [32], [117], Bayesian statistics [118], [51], [83], [88], econometrics [60], [16], [31], [56], [50], nonparametric statistical inference [44], [123], [124], [43], [42]; also see [139], [140], [121] for book-length treatments on the subject.","In this paper, we will focus on the OT map obtained using the standard Euclidean cost function, i.e., $T_0:=\\operatornamewithlimits{\\arg \\!\\min }\\limits _{T:T\\#\\mu =\\nu }\\mathbb {E}\\Vert X-T(X)\\Vert ^2,$ where $T\\#\\mu =\\nu $ means $T(X)\\sim \\nu $ for $X\\sim \\mu $ .","The estimation of $T_0$ has attracted a lot of interest in recent years due to its myriad applications (as stated above) and interesting geometrical properties (see [100], [62], [21] and def:otmpm below).","In practice, the main hurdle in constructing estimators for $T_0$ is that the explicit forms of the measures $\\mu ,\\nu $ are unknown; instead only random samples $X_1,\\ldots ,X_m\\sim \\mu \\qquad \\quad \\mbox{and} \\qquad \\quad Y_1,\\ldots ,Y_n\\sim \\nu $ are available.","A natural strategy in this scenario is to estimate $T_0$ using $\\widetilde{T}_{m,n}$ , where $\\widetilde{T}_{m,n}$ is computed as in (REF ) with $\\mu $ and $\\nu $ replaced by $\\widetilde{\\mu }_m$ and $\\widetilde{\\nu }_n$ which are empirical approximations of $\\mu $ and $\\nu $ based on $X_1,\\ldots ,X_m$ and $Y_1,\\ldots ,Y_n$ respectively (see def:bcenterproj).","Such estimators are often called plug-in estimators and have been used extensively, see [126], [102], [103], [8], [111], [73], [33].","The main goal of this paper is to study the rates of convergence of general plug-in estimators of $T_0$ under a unified framework.","We show that when $\\widetilde{\\mu }_m$ and $\\widetilde{\\nu }_n$ are chosen as $\\widehat{\\mu }_m$ and $\\widehat{\\nu }_n$ respectively, where $\\widehat{\\mu }_m$ and $\\widehat{\\nu }_n$ are the standard empirical distributions supported on $m$ and $n$ atoms, i.e., $\\widehat{\\mu }_m:=\\frac{1}{m}\\sum _{i=1}^m \\delta _{X_i}\\qquad \\quad \\mbox{and} \\qquad \\quad \\widehat{\\nu }_n:=\\frac{1}{n}\\sum _{j=1}^n \\delta _{Y_j},$ $\\widetilde{T}_{m,n}$ (appropriately defined using def:bcenterproj) converges at a rate of $m^{-2/d}+n^{-2/d}$ for $d\\ge 4$ .","This rate happens to be minimax optimal under minimal smoothness assumptions (see [80]) but suffers from the curse of dimensionality.","We next show that, if $\\mu $ and $\\nu $ are known to admit sufficiently smooth densities, it is possible to apply wavelet or kernel based smoothing techniques on $\\widehat{\\mu }_m$ and $\\widehat{\\nu }_n$ to obtain plug-in estimators that mitigate the aforementioned curse of dimensionality.","Our next contribution pertains to the estimation of $W_2^2(\\mu ,\\nu )$ (the squared Wasserstein distance), see (REF ) below, a quantity of independent interest in statistics and machine learning with applications in structured prediction [57], [97], image analysis [65], [19], nonparametric testing [17], [116], generative modeling [105], [11], etc.","In this paper, we also obtain rates of convergence for plug-in estimators $W_2^2(\\widetilde{\\mu }_m,\\widetilde{\\nu }_n)$ of $W_2^2(\\mu ,\\nu )$ .","We show that kernel smoothing $\\widehat{\\mu }_m$ and $\\widehat{\\nu }_n$ can be used to obtain plug-in estimators of $W_2^2(\\mu ,\\nu )$ that mitigate the curse of dimensionality as opposed to a direct plug-in approach using $\\widehat{\\mu }_m$ and $\\widehat{\\nu }_n$ (as used in [33]).","This provides an answer to the open question of estimating $W_2^2(\\mu ,\\nu )$ when $\\mu $ , $\\nu $ admit smooth densities laid out in [33].","In this section, we present some basic concepts and results associated with the OT problem that will play a crucial role in the sequel.","Let $\\mathcal {P}_{\\mathrm {ac}}(\\mathbb {R}^d)$ denote the set of all Lebesgue absolutely continuous probability measures on $\\mathbb {R}^d$ and $\\mathcal {P}_2(\\mathbb {R}^d)$ be the set of probability measures with finite second moments.","Then the 2-Wasserstein distance (squared) between $\\mu ,\\nu \\in \\mathcal {P}_2(\\mathbb {R}^d)$ is defined as: $W_2^2(\\mu ,\\nu ):=\\min \\limits _{\\pi \\in \\Pi (\\mu ,\\nu )}\\int \\Vert x-y\\Vert ^2\\,d\\pi (x,y),$ where $\\Pi (\\mu ,\\nu )$ is the set of probability measures on $\\mathbb {R}^d\\times \\mathbb {R}^d$ with marginals $\\mu $ and $\\nu $ .","The optimization problem in (REF ) is often called the Kantorovich relaxation (see [84], [85]) of the optimization problem in (REF ).","The existence of a minimizer in (REF ) follows from [140].","Proposition 1.1 (Brenier-McCann polar factorization theorem, see [139], [100]) Suppose $\\mu ,\\nu \\in \\mathcal {P}_{\\mathrm {ac}}(\\mathbb {R}^d)$ .","Then there exists a $\\mu $ -a.e.","(almost everywhere) unique function $T_0(\\cdot ):\\mathbb {R}^d\\rightarrow \\mathbb {R}^d$ , which is the gradient of a real-valued $d$ -variate convex function, say $\\varphi _0(\\cdot ):\\mathbb {R}^d\\rightarrow \\mathbb {R}$ , such that $T_0\\#\\mu =\\nu $ .","Further, the distribution defined as $\\pi (A\\times B)=\\mu (A\\cap (T_0)^{-1}(B))$ for all Borel sets $A,B\\subseteq \\mathbb {R}^d$ is the unique minimizer in (REF ) provided $\\mu ,\\nu \\in \\mathcal {P}_{\\mathrm {ac}}(\\mathbb {R}^d)\\cap \\mathcal {P}_2(\\mathbb {R}^d)$ .","[OT map and potential function] The function $T_0:\\mathbb {R}^d\\rightarrow \\mathbb {R}^d$ in prop:bmopt which satisfies $T_0\\#\\mu =\\nu $ will be called the OT map from $\\mu $ to $\\nu $ .","The convex function $\\varphi _0(\\cdot )$ in prop:bmopt satisfying $\\nabla \\varphi _0=T_0$ will be termed the OT potential.","The next and final important ingredient is the alternate dual representation of (REF ) which gives: $\\frac{1}{2}W_2^2(\\mu ,\\nu )&=\\frac{1}{2}\\int \\Vert x\\Vert ^2\\,d\\mu (x)+\\frac{1}{2}\\int \\Vert y\\Vert ^2\\,d\\nu (y)-\\min _{f\\in \\mathcal {F}} \\mathcal {S}_{\\mu ,\\nu }(f),\\,\\qquad \\mbox{where} \\\\ \\mathcal {S}_{\\mu ,\\nu }(f)&=\\int f\\,d\\mu +\\int f^*\\,d\\nu .$ Here $\\mathcal {F}$ denotes the space of convex functions on $\\mathbb {R}^d$ which are also elements of $L^1(\\mu )$ and $f^*(\\cdot )$ is the standard Legendre-Fenchel dual defined as: $f^*(x) := \\sup _{y \\in \\mathbb {R}^d} [y^\\top x - f(y)], \\qquad \\mbox{for } x \\in \\mbox{dom}(f).$"],["Estimating OT map via barycentric projection","Recall the setting from the Introduction.","Let $\\widetilde{\\mu }_m,\\widetilde{\\nu }_n\\in \\mathcal {P}_2(\\mathbb {R}^d)$ .","Here $\\widetilde{\\mu }_m,\\widetilde{\\nu }_n$ need not be absolutely continuous and can be very general.","Intuitively, $\\widetilde{\\mu }_m$ and $\\widetilde{\\nu }_n$ should be viewed as some empirical approximation of $\\mu $ and $\\nu $ respectively.","[Simple choices of $\\widetilde{\\mu }_m$ and $\\widetilde{\\nu }_n$ ] Let $X_1,\\ldots ,X_m\\overset{i.i.d.","}{\\sim }\\mu $ and $Y_1,\\ldots ,Y_n\\overset{i.i.d.","}{\\sim }\\nu $ ; in which case a natural choice would be to set $\\widetilde{\\mu }_m=\\widehat{\\mu }_m$ and $\\widetilde{\\nu }_n=\\widehat{\\nu }_n$ where $\\widehat{\\mu }_m$ and $\\widehat{\\nu }_n$ are the empirical distributions on $X_1,\\ldots ,X_m$ and $Y_1,\\ldots ,Y_n$ respectively, as defined in (REF ).","This is the standard choice adopted in the discrete-discrete Kantorovich relaxation; see [113].","Another popular choice is $\\widetilde{\\mu }_m=\\widehat{\\mu }_m$ , $\\widetilde{\\nu }_n=\\nu $ or $\\widetilde{\\mu }_m=\\mu $ , $\\widetilde{\\nu }_n=\\widehat{\\nu }_n$ .","This is the semi-discrete Kantorovich problem and is popular when one of the measures is fully specified; see [29], [61].","A natural way to estimate $T_0(\\cdot )$ , as defined in (REF ), would be to approximate it using the OT map from $\\widetilde{\\mu }_m$ to $\\widetilde{\\nu }_n$ .","However as $\\widetilde{\\mu }_m$ and $\\widetilde{\\nu }_n$ may not be elements of $\\mathcal {P}_{\\mathrm {ac}}(\\mathbb {R}^d)$ , prop:bmopt does not apply and an OT map may not exist from $\\widetilde{\\mu }_m$ to $\\widetilde{\\nu }_n$ .","Such is the case in ex:baseplug in the discrete-discrete case when $m\\ne n$ .","To circumvent this issue, we leverage the notion of barycentric projections (see [4]) defined below: [Barycentric projection] Define the set $\\widetilde{\\Gamma }_{\\mathrm {min}}:=\\operatornamewithlimits{\\arg \\!\\min }\\limits _{\\pi \\in \\Pi (\\widetilde{\\mu }_m,\\widetilde{\\nu }_n)} \\int \\Vert x-y\\Vert ^2\\,d\\pi (x,y).$ The optimization problem above is the plug-in analog of the optimization problem on the right hand side of (REF ).","Given any $\\gamma \\in \\widetilde{\\Gamma }_{\\mathrm {min}}$ , define the barycentric projection of $\\gamma $ as the conditional mean of $y$ given $x$ under $\\gamma $ , i.e., $\\widetilde{T}_{m,n}(x)\\equiv \\widetilde{T}_{m,n}^{\\gamma }(x):=\\frac{\\int _{y} y\\,d\\gamma (x,y)}{\\int _{y} d\\gamma (x,y)},\\qquad \\mbox{for}\\ x\\in \\mbox{supp}\\left({\\widetilde{\\mu }_m}\\right).$ In general, $\\widetilde{\\Gamma }_{\\mathrm {min}}$ need not be a singleton which is why we index the barycentric projection $\\widetilde{T}_{m,n}^{\\gamma }(\\cdot )$ by $\\gamma \\in \\widetilde{\\Gamma }_{\\mathrm {min}}$ .","Note that $\\widetilde{T}_{m,n}^{\\gamma }(\\cdot )$ need not be a transport map; however, if an OT map exists then it must be equal to $\\widetilde{T}_{m,n}^{\\gamma }(\\cdot )$ ($\\widetilde{\\mu }_m$ -a.e.).","Our goal is to obtain stochastic upper bounds for $\\sup \\limits _{\\gamma \\in \\widetilde{\\Gamma }_{\\mathrm {min}}} \\int \\big \\Vert \\widetilde{T}_{m,n}^{\\gamma }(x)-T_0(x)\\big \\Vert ^2\\,d\\widetilde{\\mu }_m(x).$ In addition, our proof techniques also yield rates of convergence for $\\big |W_2^2(\\widetilde{\\mu }_m,\\widetilde{\\nu }_n)-W_2^2(\\mu ,\\nu )\\big |.$ In this paper, we will focus on $d\\ge 2$ .","Due to the canonical ordering of $\\mathbb {R}$ , the case $d=1$ can be handled easily using the classical Hungarian embedding theorem [90]."],["Contributions"," We provide a new and flexible stability estimate thm:newubd which yields a unified approach to obtaining rates of convergence for general plug-in estimators of the OT map $T_0(\\cdot )$ .","Unlike existing stability estimates, thm:newubd holds for the barycentric projection (which is the same as the OT map when it exists) and does not require any smoothness assumptions on $\\widetilde{\\mu }_m$ , $\\widetilde{\\nu }_n$ or $\\widetilde{T}_{m,n}^{\\gamma }(\\cdot )$ ; also see rem:comstab for a comparison with the existing literature.","in Sections REF and REF , we use thm:newubd to bound (REF ) and (REF ): In sec:nsmooth, we show that in both the discrete-discrete and semi-discrete Kantorovich relaxation problems (see ex:baseplug), the rate of convergence of (REF ) is $m^{-2/d}+n^{-2/d}$ for $d\\ge 4$ when $T_0$ is assumed to be Lipschitz (see thm:nsmooth), which is the minimax rate (see [80]).","To the best of our knowledge, rates of convergence for these natural estimators weren't previously established in the literature.","In sec:smooth, we show that the curse of dimensionality in the above rates can be mitigated provided $\\mu $ and $\\nu $ admit Besov smooth densities (see sec:wavelet) or (uniform) Sobolev smooth densities (see sec:kernel).","In sec:wavelet, our plug-in estimator is obtained using natural wavelet based density estimators.","The rate of convergence in (REF ) turns out to be $n^{-\\frac{1+s}{d+2s}}$ where $s$ denotes the degree of Besov smoothness (see thm:smoothwav).","Note that by choosing $s$ large enough, the exponent in the rate can be made arbitrarily close to $1/2$ , thereby reducing the curse of dimensionality.","In sec:kernel, our plug-in estimator is obtained by choosing $\\widetilde{\\mu }_m$ (and $\\widetilde{\\nu }_n$ ) as the convolution of $\\widehat{\\mu }_m$ (and $\\widehat{\\nu }_n$ ) and a smooth kernel with an appropriate bandwidth.","Under this choice, the rate of convergence in (REF ) is $m^{-\\left(\\frac{s+2}{d}\\wedge \\frac{1}{2}\\right)}+n^{-\\left(\\frac{s+2}{d}\\wedge \\frac{1}{2}\\right)}$ , where $s$ denotes the degree of Sobolev smoothness (see thm:smooth).","Clearly, if $2(s+2)\\ge d$ , the rate of convergence becomes dimension-free and mitigates the curse of dimensionality.","We also show the same rates of convergence mentioned above also hold for (REF ) (see e.g., prop:wassrate) which makes a strong case in favor of incorporating smoothness in the construction of plug-in estimators as was conjectured in [33].","In sec:dismooth, we use a discretization technique from [143] to construct discrete approximations to the smoothed $\\widetilde{\\mu }_m$ and $\\widetilde{\\nu }_n$ from the previous paragraph that in turn yield computable plug-in estimators for $T_0$ (provided one can sample from $\\widetilde{\\mu }_m$ and $\\widetilde{\\nu }_n$ ) that also achieve the same statistical guarantees as the smoothed plug-in estimator from sec:smooth (see thm:dismooth).","However the number of atoms required in the discretizations and correspondingly the computational complexity increases with the degree of smoothness; this highlights a statistical and computational trade off.","We provide implications of our results in popular applications of OT such as estimating the barycenter of two multivariate probability distributions (see thm:barate in sec:bar) and in nonparametric independence testing (see thm:dethresh in sec:nonptest)."],["Related work","Many recent works have focused on obtaining consistent estimators of $T_0$ using the plug-in principle, see [29], [61] (in the semi-discrete problem) and [75], [144], [44] (in the discrete-discrete problem).","In [61], the authors have studied the rate of convergence of the semi-discrete optimal transport map from $\\nu $ (absolutely continuous) to $\\widehat{\\mu }_m$ .","This paper complements the aforementioned papers by studying the rates of convergence for general plug-in estimators in a unified fashion.","In two other papers [10] and [95], the authors use a “Voronoi tessellation\" approach to estimate $T_0$ , however the rates obtained in this paper, even in the absence of smoothness, are strictly better than those in [10], [95].","Perhaps the most closely related paper to ours would be [73].","In [73], the author uses variational techniques to arrive at stability estimates while we exploit the Lipschitz nature of the OT map (see def:otmpm).","Further the rates in this paper have exponents $\\frac{s+2}{d}\\wedge \\frac{1}{2}$ which are strictly better than the exponents $\\frac{s+2}{2(s+2)+d}$ obtained in [73] under the same smoothness assumptions (Sobolev type of order $s$ , see def:holder).","In another line of work [80], the authors use theoretical wavelet based estimators (not of the plug-in type) of $T_0$ to obtain nearly minimax optimal rates of convergence.","However these estimators, by themselves, are not transport maps between two probability measures, which makes them harder to interpret.","In contrast, our focus is on obtaining rates of convergence for plug-in estimators, which are transport maps between natural aprroximations of $\\mu $ and $\\nu $ .","Such plug-in type strategies are a lot more popular in computational OT [126], [102], [103], [8], [111], [73], [33].","In terms of obtaining rates of convergence for (REF ), some attempts include [126], [119] where parametric rates are obtained when $\\mu ,\\nu $ are known to be finitely supported or are both Gaussian.","In a related problem, bounds for $W_2^2(\\widehat{\\mu }_m,\\mu )$ were obtained in [133], [7], [46], [55], [109], [143].","Using these bounds, it is easy to get a $n^{-1/d}$ rate of convergence for (REF ).","This rate was recently improved to $n^{-2/d}$ in [33] under no smoothness assumptions.","Our rates coincide with the $n^{-2/d}$ rate from [33] under no smoothness assumptions.","But further, we show in this paper that the curse of dimensionality in the above rate can be mitigated by incorporating smoothness into the plug-in procedure."],["Main results","Recall the definition of $\\varphi _0(\\cdot )$ from def:otmpm.","The following is our main result.","Theorem 2.1 (Stability estimate) Suppose that $\\mu ,\\nu \\in \\mathcal {P}_{\\mathrm {ac}}(\\mathbb {R}^d)\\cap \\mathcal {P}_2(\\mathbb {R}^d)$ and $\\widetilde{\\mu }_m,\\widetilde{\\nu }_n\\in \\mathcal {P}_2(\\mathbb {R}^d)$ .","Assume that $T_0(\\cdot )$ (as defined in (REF )) is $L$ -Lipschitz ($L >0$ ).","Then, $\\sup \\limits _{\\gamma \\in \\widetilde{\\Gamma }_{\\mathrm {min}}} \\int \\Vert \\widetilde{T}_{m,n}^{\\gamma }(x)-T_0(x)\\Vert ^2& \\,d\\widetilde{\\mu }_m(x)\\le L \\max \\left\\lbrace \\bigg |\\int \\Psi _{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n}^* \\,d(\\widetilde{\\nu }_n-\\overline{\\nu }_m)\\bigg |,\\bigg |\\int \\Psi _{\\widetilde{\\mu }_m,\\overline{\\nu }_m}^* \\,d(\\widetilde{\\nu }_n-\\overline{\\nu }_m)\\bigg |\\right\\rbrace \\nonumber \\\\ &+2L\\int \\varphi _0^*(y)\\,d(\\widetilde{\\nu }_n-\\overline{\\nu }_m)(y),$ where $\\overline{\\nu }_m=T_0\\#\\widetilde{\\mu }_m$ , $\\varphi _0^*(\\cdot )$ is defined as in (REF ), and with $\\mathcal {S}_{\\cdot ,\\cdot }(\\cdot )$ defined as in (), $\\Psi _{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n}(\\cdot ):=\\operatornamewithlimits{\\arg \\!\\min }_{f\\in \\mathcal {F}} \\mathcal {S}_{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n}(f),\\quad \\Psi _{\\widetilde{\\mu }_m,\\overline{\\nu }_m}(\\cdot ):=\\operatornamewithlimits{\\arg \\!\\min }_{f\\in \\mathcal {F}} \\mathcal {S}_{\\widetilde{\\mu }_m,\\overline{\\nu }_m}(f)$ .","The proof of thm:newubd (see sec:mainrespf) starts along the same lines as the proof of the curvature estimate in [62].","This is followed by some careful manipulations of $W_2^2(\\cdot ,\\cdot )$ (as in (REF )) and an application of the conditional version of Jensen's inequality, see (REF ).","The final step of the proof uses the dual representation in (REF ) with techniques similar to some intermediate steps in the proof of [101] and [33].","Remark 2.1 (Comparison with other stability estimates) thm:newubd provides some important advantages to existing stability estimates in the literature.","One of the earliest results in this direction can be found in [62] but their bound involves a push-forward constraint which makes it hard to use for rate of convergence analysis.","A bound similar to thm:newubd is presented in [61] but there the authors assume the existence of an OT map from $\\widetilde{\\mu }_m$ to $\\widetilde{\\nu }_n$ .","Therefore, it does not apply to the discrete-discrete problem where $\\widetilde{\\mu }_m=\\widehat{\\mu }_m$ and $\\widetilde{\\nu }_n=\\widehat{\\nu }_n$ with $m\\ne n$ .","Overcoming all these limitations is an important contribution of thm:newubd and allows us to deal with popular plug-in estimators all in one go.","The stability estimate in [80] on the other hand requires $\\widetilde{\\mu }_m$ , $\\widetilde{\\nu }_n$ to be sufficiently smooth and hence it does not hold for discrete-discrete or semi-discrete plug-in estimators (see ex:baseplug).","Further their result requires all the measures involved to be compactly supported unlike the much milder requirements of thm:newubd.","However, a shortcoming of thm:newubd is that it is hard to obtain rates faster than $n^{-1/2}$ using it directly, whereas [80] can obtain rates arbitrarily close to $n^{-1}$ .","This is a price we pay for analyzing natural and popular plug-in estimators as opposed to the (more intractable) wavelet based estimators in [80].","Remark 2.2 (How to use thm:newubd to obtain rates of convergence?)","Note that the second term on the right hand side of (REF ), under appropriate moment assumptions, is $O_p(m^{-1/2}+n^{-1/2})$ (free of dimension) by a direct application of Markov's inequality.","We therefore focus on the first term.","By (), $\\Psi _{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n}^*(\\cdot )$ , $\\Psi _{\\widetilde{\\mu }_m,\\overline{\\nu }_m}^*(\\cdot )\\in \\mathcal {F}$ .","Further, by Caffarelli's regularity theory [23], [24], [25], depending on the “smoothness\" of $\\widetilde{\\mu }_m$ , $\\widetilde{\\nu }_n$ , it can be shown that there exists a further class of functions $\\mathcal {F}_s$ (see Remarks REF and REF ) such that $\\Psi _{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n}^*(\\cdot )$ , $\\Psi _{\\widetilde{\\mu }_m,\\overline{\\nu }_m}^*(\\cdot )\\in \\mathcal {F}\\cap \\mathcal {F}_s$ .","Thus, we can bound the first term on the right hand side of (REF ) as: $\\max \\left\\lbrace \\bigg |\\int \\Psi _{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n}^* \\,d(\\widetilde{\\nu }_n-\\overline{\\nu }_m)\\bigg |,\\bigg |\\int \\Psi _{\\widetilde{\\mu }_m,\\overline{\\nu }_m}^* \\,d(\\widetilde{\\nu }_n-\\overline{\\nu }_m)\\bigg |\\right\\rbrace \\le \\sup _{f\\in \\mathcal {F}\\cap \\mathcal {F}_s}\\bigg |\\int f\\,d(\\widetilde{\\nu }_n-\\overline{\\nu }_m)\\bigg |.$ The right hand side of (REF ) can now be bounded using the corresponding Dudley's entropy integral bounds using empirical process techniques, see [137].","To conclude, the two main steps in our strategy are identifying the family of functions $\\mathcal {F}_s$ and computing Dudley's entropy integral.","Further, the more the smoothness of $\\widetilde{\\mu }_m$ , $\\widetilde{\\nu }_n$ , the smaller is the class of functions $\\mathcal {F}_s$ and smaller the supremum on the right hand side of (REF ).","This shows why better rates can be expected under smoothness assumptions."],["Natural non-smooth plug-in estimator","In this case, we discuss the rates of convergence for the discrete-discrete problem and the semi-discrete problem, where no smoothness is available on $\\widetilde{\\mu }_m$ and $\\widetilde{\\nu }_n$ .","Theorem 2.2 Suppose that $T_0(\\cdot )$ is $L$ -Lipschitz, $\\nu $ is compactly supported and $\\mathbb {E}\\exp (t\\Vert X_1\\Vert ^{\\alpha })<\\infty $ for some $t>0$ , $\\alpha >0$ .","(Discrete-discrete): Set $\\widetilde{\\mu }_m=\\widehat{\\mu }_m$ and $\\widetilde{\\nu }_n=\\widehat{\\nu }_n$ .","Then the following holds: $\\sup \\limits _{\\gamma \\in \\widetilde{\\Gamma }_{\\mathrm {min}}} \\int \\Vert \\widetilde{T}_{m,n}^{\\gamma }(x)-T_0(x)\\Vert ^2\\,d\\widetilde{\\mu }_m(x)=O_p\\left(r^{(m,n)}_d\\times (\\log {(1+\\max \\lbrace m,n\\rbrace )})^{t_{d,\\alpha }}\\right),$ $\\mbox{where}\\quad r^{(m,n)}_d:={\\left\\lbrace \\begin{array}{ll} m^{-1/2}+n^{-1/2} & \\mbox{for}\\ d=2,3,\\\\ m^{-1/2}\\log {(1+m)}+n^{-1/2}\\log {(1+n)} & \\mbox{for}\\ d=4,\\\\ m^{-2/d}+n^{-2/d} & \\mbox{for}\\ d\\ge 5, \\end{array}\\right.","}$ and $t_{d,\\alpha }:={\\left\\lbrace \\begin{array}{ll} (4\\alpha )^{-1}(4+((2\\alpha +2d\\alpha -d)\\vee 0)) & \\mbox{for}\\ d<4,\\\\ (\\alpha ^{-1}\\vee 7/2)-1 & \\mbox{for}\\ d=4,\\\\ 2(1+d^{-1}) & \\mbox{for}\\ d>4.\\end{array}\\right.","}$ The same bound holds for $|W_2^2(\\widetilde{\\mu }_m,\\widetilde{\\nu }_n)-W_2^2(\\mu ,\\nu )|$ without assuming $T_0(\\cdot )$ is Lipschitz.","(Semi-discrete): Set $\\widetilde{\\mu }_m=\\mu $ , $\\widetilde{\\nu }_n=\\widehat{\\nu }_n$ or $\\widetilde{\\mu }_m=\\widehat{\\mu }_m$ , $\\widetilde{\\nu }_n=\\nu $ .","Then the left hand side of (REF ) is $O_p(r_d^{(n,n)}\\times (\\log {(1+n)})^{t_{d,\\alpha }})$ or $O_p(r_d^{(m,m)}\\times (\\log {(1+m)})^{t_{d,\\alpha }})$ respectively.","A stronger result can be proved if both $\\mu $ and $\\nu $ are compactly supported.","Corollary 2.3 Consider the setting from thm:nsmooth and assume further that $\\mu $ is compactly supported.","Then, with $r_d^{(m,n)}$ defined as in (REF ), we have: $\\mathbb {E}\\left[\\sup \\limits _{\\gamma \\in \\widetilde{\\Gamma }_{\\mathrm {min}}} \\int \\Vert \\widetilde{T}_{m,n}^{\\gamma }(x)-T_0(x)\\Vert ^2\\,d\\widetilde{\\mu }_m(x)\\right]\\le C r^{(m,n)}_d,$ for some constant $C>0$ , in both the discrete-discrete and semi-discrete settings from thm:nsmooth.","A brief description of the proof technique of thm:nsmooth using thm:newubd is provided in rem:pfnstech below, and the actual proof is presented in sec:mainrespf.","Remark 2.3 (Proof technique) The proof of thm:nsmooth proceeds using the strategy outlined in rem:pftech.","We first show that $\\mathcal {F}_s$ (see rem:pftech) can be chosen as a certain class of convex functions which are in $L^2(\\nu )$ .","We then use Dudley's entropy integral type bounds which in turn requires the bracketing entropy [137] of $\\mathcal {F}_s$ , recently proved in [91].","This strategy is slightly different from that used in the proof of [33], where the authors assume that $\\mu $ is compactly supported whereas we only assume the finiteness of $\\mathbb {E}\\exp (t\\Vert X_1\\Vert ^{\\alpha })$ for some $t>0$ , $\\alpha >0$ .","The compactness assumption on $\\mu $ allows one to further restrict $\\mathcal {F}_s$ to the class of Lipschitz functions.","This additional restriction does not seem to be immediate without the compactness assumption.","As discussed in sec:contrib, the exponents obtained in thm:nsmooth are minimax optimal under bare minimal smoothness assumptions (see [80]).","To the best of our knowledge, rates for the discrete-discrete case for $m\\ne n$ and those for the semi-discrete case were not known previously in the literature.","Our rates are also strictly better than those (for different estimators, based on space tessellations) obtained in [10], [95] and require less stringent assumptions than those in [33].","In the next section, we show how smoothness assumptions can be leveraged to mitigate the curse of dimensionality in thm:nsmooth."],["Smooth plug-in estimator: mitigating the curse of dimensionality","In this section, we focus on two types of plug-in estimators for the densities associated with the probability measures $\\mu $ and $\\nu $ : (a) wavelet based estimators (see [143], [135], [47], [86], [142]) in sec:wavelet, and (b) kernel based estimators (see [63], [64], [112], [125], [108]) in sec:kernel.","In both these cases, we will show, using thm:newubd, that the corresponding estimators of $T_0(\\cdot )$ achieve (near) dimension-free rates under sufficient smoothness assumptions."],["Wavelet based estimators","We begin this subsection by defining the Besov class of functions which will play a pivotal role in the sequel.","[Besov class of functions] We describe Besov classes following the notation from [143].","Suppose $s>0$ and let $n>s$ be a positive integer.","Given $\\Omega \\subseteq \\mathbb {R}^d$ , $h\\in \\mathbb {R}^d$ and $f(\\cdot ):\\mathbb {R}^d\\rightarrow \\mathbb {R}^d$ , set $\\Delta _h^1 f(x):=f(x+h)-f(x),$ $\\Delta _h^k f(x):= \\Delta _h^1\\left(\\Delta _h^{k-1} f\\right)(x), \\quad \\forall \\ 2\\le k\\le n,$ where these functions are defined on $\\Omega _{h,n}:=\\lbrace x\\in \\Omega :x+nh\\in \\Omega \\rbrace $ .","For $t>0$ , we then define $\\omega _n(f,t):=\\sup _{\\Vert h\\Vert \\le t} \\Vert \\Delta _h^n f\\Vert _{L^2(\\Omega _{h,n})}.$ Finally, we define the space $\\mathcal {B}^s(\\Omega )$ to be the set of functions for which the quantity $\\Vert f\\Vert _{\\mathcal {B}^s(\\Omega )}:=\\Vert f\\Vert _{L^2(\\Omega )}+\\sum _{j\\ge 0} 2^{sj}\\omega _n(f,2^{-j})$ is finite.","The above expression can also be used to define Besov spaces (and norms) for $s<0$ ; see [36].","In this subsection, we assume that $\\mu $ and $\\nu $ admit Besov smooth densities $f_{\\mu }(\\cdot )$ and $f_{\\nu }(\\cdot )$ (see [36] and def:besov above for details).","Given $\\Omega \\subseteq \\mathbb {R}^d$ and $s>0$ , let $\\mathcal {B}^s(\\Omega )$ denote the set of Besov smooth functions on $\\Omega $ of order $s$ .","Assumption (A1) (Regularity of the densities) Suppose that: $f_{\\mu }$ and $f_{\\nu }$ are supported on compact and convex subsets of $\\mathbb {R}^d$ , say $\\mathcal {X}$ and $\\mathcal {Y}$ respectively.","There exists $s,M> 0$ such that $\\Vert f_{\\mu }\\Vert _{\\mathcal {B}^{s}(\\mathcal {X})}\\le M$ , $\\Vert f_{\\nu }\\Vert _{\\mathcal {B}^{s}(\\mathcal {Y})}\\le M$ and $f_{\\mu }(x),f_{\\nu }(y)\\ge M^{-1}$ for all $x\\in \\mathcal {X}$ , $y\\in \\mathcal {Y}$ .","We now present our wavelet based estimators for $f_{\\mu }(\\cdot )$ and $f_{\\nu }(\\cdot )$ .","Towards this direction, we begin with sets of functions in $L^2(\\mathcal {X})$ (set of square integrable functions on $\\mathcal {X}$ ), $\\Phi $ and $\\lbrace \\Psi _j\\rbrace _{j\\ge 0}$ , which form an orthonormal basis of $L^2(\\mathcal {X})$ and satisfy the standard regularity assumptions for a wavelet basis (see [104], [77], [143]).","We defer a formal discussion on these assumptions to def:wavelet in the Appendix so as not to impede the flow of the paper.","For the moment, it is worth noting that such sets of functions (e.g., Haar wavelets, Daubechies wavelets) are readily available in standard statistical softwares, see e.g., the R package wavelets.","Next, fix $J_m\\in \\mathbb {N}$ (a truncation parameter to be chosen later depending on the sample size $m$ ).","Consider the following: $\\widehat{f}_{\\mu }(x):=\\sum _{\\phi \\in \\Phi } a_{\\phi }\\phi (x)+\\sum _{j=0}^{J_m} \\sum _{\\psi \\in \\Psi _j} b_{\\psi }\\psi (x),$ where $a_{\\phi }:=\\frac{1}{m}\\sum _{i=1}^m \\phi (X_i), \\qquad b_{\\psi }:=\\frac{1}{m}\\sum _{i=1}^m \\psi (X_i).$ Unfortunately $\\widehat{f}_{\\mu }(\\cdot )$ as defined in (REF ) may not be a probability density and consequently cannot be used to obtain plug-in estimators for $\\widetilde{T}_{m,n}^{\\gamma }(\\cdot )$ .","We therefore take the same route as in [143] to define the following estimator for $f_{\\mu }(\\cdot )$ : $\\widetilde{f}_{\\mu }:=\\min _{g\\in \\mathcal {D}(\\mathcal {X})}\\Vert g-\\widehat{f}_{\\mu }\\Vert _{\\mathcal {B}^{-1}(\\mathcal {X})},$ where $\\mathcal {D}(\\mathcal {X})$ is the space of probability density functions on $\\mathcal {X}$ and $\\mathcal {B}^{-1}(\\mathcal {X})$ is the Besov norm on $\\mathcal {X}$ of order $-1$ as stated in def:besov.","We can define $\\widetilde{f}_{\\nu }(\\cdot )$ similarly.","Computing both $\\widetilde{f}_{\\mu }(\\cdot )$ and $\\widehat{f}_{\\mu }(\\cdot )$ (as it involves infinite sums) is challenging and we would refer the interested reader to [143] and the references therein, for details.","Further discussion of this aspect is beyond the scope of this paper.","We are now in a position to present the main theorem of this subsection.","Theorem 2.4 Suppose that $T_0(\\cdot )$ is $L$ -Lipschitz, and $\\widetilde{\\mu }_m$ and $\\widetilde{\\nu }_n$ are the probability measures corresponding to the probability densities $\\widetilde{f}_{\\mu }(\\cdot )$ and $\\widetilde{f}_{\\nu }(\\cdot )$ with $m^{\\frac{1}{d+2s}}\\le 2^{J_m}\\le m^{\\frac{1}{d}}$ and $n^{\\frac{1}{d+2s}}\\le 2^{J_n}\\le n^{\\frac{1}{d}}$ , then the following holds for some constant $C>0$ : $\\mathbb {E}\\left[\\sup \\limits _{\\gamma \\in \\widetilde{\\Gamma }_{\\mathrm {min}}} \\int \\Vert \\widetilde{T}_{m,n}^{\\gamma }(x)-T_0(x)\\Vert ^2\\,d\\widetilde{\\mu }_m(x)\\right]\\le C \\widetilde{r}^{(m,n)}_{d,s},$ $\\mbox{where}\\quad \\widetilde{r}^{(m,n)}_{d,s}:={\\left\\lbrace \\begin{array}{ll} m^{-1/2}\\log {(1+m)}+n^{-1/2}\\log {(1+n)} & \\mbox{for}\\ d=2,\\\\ m^{-\\frac{1+s}{d+2s}}+n^{-\\frac{1+s}{d+2s}} & \\mbox{for}\\ d\\ge 3, \\end{array}\\right.","}$ The same bound also holds for $\\mathbb {E}|W_2^2(\\widetilde{\\mu }_m,\\widetilde{\\nu }_n)-W_2^2(\\mu ,\\nu )|$ .","Note that $\\frac{1+s}{d+2s}\\rightarrow \\frac{1}{2}$ as $s\\rightarrow \\infty $ .","Therefore thm:smoothwav shows that, when $m=n$ , the rate of convergence for the wavelet based estimator is “close\" to $n^{-1/2}$ provided $s$ is large enough for each fixed $d$ .","This shows that $T_0(\\cdot )$ obtained using the wavelet estimators for $f_{\\mu }(\\cdot )$ and $f_{\\nu }(\\cdot )$ mitigates the curse of dimensionality, contrast this with the estimator in thm:barate.","To avoid repetition, we defer further discussions on the rates observed in thm:smoothwav to rem:curdim where a holistic comparison is drawn with two other “smooth” plug-in estimators."],["Kernel based estimators","We first introduce the Sobolev class of functions which we will exploit in this subsection to construct estimators that achieve rates of convergence which mitigate the curse of dimensionality under sufficient smoothness.","[Uniform Sobolev class of functions] Let $\\Omega \\subseteq \\mathbb {R}^d$ and $f(\\cdot )$ be uniformly continuous on $\\Omega $ and admits uniformly continuous derivatives up to order $s$ on $\\Omega $ for some $s\\in \\mathbb {N}$ .","For any $\\mathfrak {m}:=(m_1,\\ldots ,m_d)\\in \\mathbb {N}^d$ , let $\\partial ^{\\mathfrak {m}} f:=\\frac{\\partial }{\\partial _{x_1}^{m_1}}\\ldots \\frac{\\partial }{\\partial _{x_d}^{m_d}} f,\\quad |\\mathfrak {m}|:=\\sum _{i=1}^d m_i.$ For any $k\\le s$ , we further define, $\\Vert f\\Vert _{C^k(\\Omega )}:=\\sum _{|\\mathfrak {m}|\\le k} \\Vert \\partial ^{\\mathfrak {m}} f\\Vert _{L^{\\infty }(\\Omega )}.$ The space $C^{s}(\\Omega )$ is defined as the set of functions $f(\\cdot )$ for which $\\Vert f\\Vert _{C^k(\\Omega )}<\\infty $ for all $k\\le s$ .","For this subsection, assume that $\\mu $ and $\\nu $ admit Sobolev smooth densities $f_{\\mu }(\\cdot )$ and $f_{\\nu }$ in the uniform norm (see def:holder above).","Given $\\Omega \\subseteq \\mathbb {R}^d$ and $s\\in \\mathbb {N}$ , let $C^s(\\Omega )$ denote the set of Sobolev smooth functions on $\\Omega $ of order $s$ .","Assumption (A2) (Regularity of the densities) Suppose that $f_{\\mu }$ and $f_{\\nu }$ are supported on compact and convex subsets of $\\mathbb {R}^d$ , say $\\mathcal {X}$ and $\\mathcal {Y}$ respectively.","There exists $s,M> 0$ such that $f_{\\mu }(\\cdot )\\in C^{s}(\\mathcal {X};M)$ and $f_{\\nu }(\\cdot )\\in C^{s}(\\mathcal {Y};M)$ where $C^{s}(\\mathcal {X};M)$ is the space of real valued functions supported on $\\mathcal {X}$ such that for all $f(\\cdot )\\in C^s(\\mathcal {X};M)$ , we have $M^{-1}\\le f(x)\\le M$ for all $x\\in \\mathcal {X}$ and $\\Vert f\\Vert _{C^s(\\mathcal {X})}\\le M$ .","Here $\\Vert \\cdot \\Vert _{C^s(\\mathcal {X})}$ is the standard uniform Sobolev norm as defined in def:holder.","The space $C^s(\\mathcal {Y};M)$ is defined analogously.","We now define our estimators for $f_{\\mu }(\\cdot )$ and $f_{\\nu }(\\cdot )$ using the standard kernel density estimation technique (see [135]).","Set $\\widehat{f}_{\\mu }(x):=\\frac{1}{mh_m^d}\\sum _{i=1}^m K_d\\left(\\frac{X_i-x}{h_m}\\right),$ for some bandwidth parameter $h_m>0$ and $d$ -variate kernel $K_d(\\cdot )$ .","We assume that $K_d(\\cdot )$ is the $d$ -fold product of univariate kernels, i.e., there exists a kernel $K(\\cdot )$ such that for $u=(u_1,\\ldots ,u_d)\\in \\mathbb {R}^d$ , $K_d(u)=\\prod _{i=1}^d K(u_i)$ .","We define $\\widehat{f}_{\\nu }(\\cdot )$ similarly with the same univariate kernel and bandwidth.","Assumption (A3) (Regularity of the kernel) Assume that $K(\\cdot )$ is a symmetric, bounded, $s+1$ times differentiable kernel on $\\mathbb {R}^d$ with all $s+1$ derivatives bounded and integrable.","Further, suppose that $K(\\cdot )$ is of order $2s+2$ , i.e., $\\int u^j K(u)\\,du=\\mathbb {1}(j=0),\\qquad \\mbox{for }\\ j=\\lbrace 0,1,2,\\ldots ,2s+1\\rbrace ,\\qquad \\mbox{and} \\ \\int |u|^{2s+2}|K(u)|\\,du<\\infty .$ The above assumptions on $K(\\cdot )$ are standard for estimating smooth densities and their derivatives of different orders in the kernel density estimation literature; see e.g.","[76], [5], [63], [135], [64].","There are several natural ways to construct kernels satisfying as:kernel, see [135]; an example is also provided in ex:ordker below.","[Example of a kernel satisfying as:kernel] Let $\\psi _m(\\cdot )$ be the $m$ -th Hermite polynomial on $\\mathbb {R}$ (see [92]).","Then the kernel function defined as $K(u):=\\sum _{m=0}^{2s+2} \\psi _{m}(0)\\psi _m(u)\\exp (-u^2/2)$ satisfies as:kernel.","It is evident from as:kernel that $K(\\cdot )$ may take some negative values, in which case, $\\widehat{f}_{\\mu }(\\cdot )$ (respectively $\\widehat{f}_{\\nu }(\\cdot )$ ) may not be a probability density.","Consequently the barycentric projection (see def:bcenterproj) between $\\widehat{f}_{\\mu }(\\cdot )$ and $\\widehat{f}_{\\nu }(\\cdot )$ is not well-defined.","We get around this by resorting to the same (approximate) projection technique we used in (REF ) for the wavelet based estimators.","In this case however, instead of (approximately) projecting using an appropriate Besov norm as in (REF ), we use a integral probability metric (see def:ipm; also see [127], [107], [114] for examples, computational procedures and applications of such metrics).","The corresponding measure is defined below: [Integral probability metric] Given a class $\\mathcal {F}$ of bounded functions on $\\mathbb {R}^d$ and two probability densities $g_1(\\cdot )$ and $g_2(\\cdot )$ on $\\mathbb {R}^d$ , the integral probability metric/distance between $g_1(\\cdot )$ and $g_2(\\cdot )$ with respect to $\\mathcal {F}$ is defined as $d_{\\mathrm {IP}}(g_1,g_2;\\mathcal {F}):=\\sup _{\\psi (\\cdot )\\in \\mathcal {F}} \\bigg |\\int \\psi (x)(g_1(x)-g_2(x))\\,dx\\bigg |.$ Sufficient conditions on $\\mathcal {F}$ for $d_{\\mathrm {IP}}(\\cdot ,\\cdot ;\\mathcal {F})$ to be a metric on the space of probability measures (not on the space of probability densities as they can be altered on set of Lebesgue measure 0 without altering the underlying probability measures) on $\\mathbb {R}^d$ have been discussed in [107].","Observe that the measure $d_{\\mathrm {IP}}(g_1,g_2;\\mathcal {F})$ is well defined even when $g_1(\\cdot )$ and $g_2(\\cdot )$ are not probability densities.","In thm:smooth below, we use $\\mathcal {F}=C^{s+2}(\\mathcal {X},M^{\\prime })$ .","Note that any function in $C^{s+2}(\\mathcal {X},M^{\\prime })$ can be extended to a function in $C^{s+2}(\\mathbb {R}^d;M^{\\prime })$ (see [80] and [134]).","The fact that this choice of $\\mathcal {F}$ results in a metric follows from the argument in [107].","We are now in a position to describe the projection estimators for $f_{\\mu }(\\cdot )$ and $f_{\\nu }(\\cdot )$ , and the rates achieved by the corresponding plug-in estimator.","Theorem 2.5 Assume that $T_0(\\cdot )$ is $L$ -Lipschitz and $f_{\\mu }$ , $f_{\\nu }$ are Lebesgue densities satisfying as:smdensities.","Also suppose that $K(\\cdot )$ satisfies as:kernel.","Define $h_m:=m^{-\\frac{1}{d+2s}}\\log {m}$ , $h_n:=n^{-\\frac{1}{d+2s}}\\log {n}$ and $T:=\\int |K_d(u)|\\,du+1$ .","Fix any $M^{\\prime }>0$ .","Consider any probability density $\\widetilde{f}_{\\mu }^{M^{\\prime }}(\\cdot )\\in C^s(\\mathcal {X};TM)$ (where $M$ is defined as in as:smdensities) which satisfies $d_{\\mathrm {IP}}\\left(\\widetilde{f}_{\\mu }^{M^{\\prime }},\\widehat{f}_{\\mu };C^{s+2}(\\mathcal {X};M^{\\prime })\\right)\\le \\inf _{\\begin{array}{c}f(\\cdot )\\in C^s(\\mathcal {X};TM)\\\\ f\\ge 0,\\ \\int f=1\\end{array}} d_{\\mathrm {IP}}\\left(\\widehat{f}_{\\mu },f;C^{s+2}(\\mathcal {X};M^{\\prime })\\right)+r_{d,s}^{(m,n)}$ where $r_{d,s}^{(m,n)}$ is defined as in (REF ) and $d_{\\mathrm {IP}}(\\cdot ,\\cdot ;C^{s+2}(\\mathcal {X};M^{\\prime }))$ is the integral probability metric defined in def:ipm.","We define $\\widetilde{f}_{\\nu }^{M^{\\prime }}(\\cdot )$ analogously as in (REF ) with $\\mathcal {X}$ , $\\widehat{f}_{\\mu }(\\cdot )$ replaced by $\\mathcal {Y}$ , $\\widehat{f}_{\\nu }(\\cdot )$ .","Then the following conclusions hold.","There exists $M^{\\prime }>0$ (depending on $M$ ) such that, if, $\\widetilde{\\mu }_m$ and $\\widetilde{\\nu }_n$ are the probability measures corresponding to the probability densities $\\widetilde{f}_{\\mu }^{M^{\\prime }}(\\cdot )$ and $\\widetilde{f}_{\\nu }^{M^{\\prime }}(\\cdot )$ , then the following holds for some constant $C>0$ : $\\mathbb {E}\\left[\\sup \\limits _{\\gamma \\in \\widetilde{\\Gamma }_{\\mathrm {min}}} \\int \\Vert \\widetilde{T}_{m,n}^{\\gamma }(x)-T_0(x)\\Vert ^2\\,d\\widetilde{\\mu }_m(x)\\right]\\le C r_{d,s}^{(m,n)},$ $ \\mbox{where}\\quad r_{d,s}^{(m,n)} := {\\left\\lbrace \\begin{array}{ll} m^{-1/2}+n^{-1/2} & \\mbox{for}\\ d<2(s+2),\\\\ m^{-1/2}\\left(\\log {(1+m)}\\right)^d+n^{-1/2}\\left(\\log {(1+n)}\\right)^d & \\mbox{for}\\ d=2(s+2),\\\\ m^{-\\frac{s+2}{d}}+n^{-\\frac{s+2}{d}} & \\mbox{for}\\ d\\ge 2(s+2).","\\end{array}\\right.","}$ The same bound also holds for $\\mathbb {E}|W_2^2(\\widetilde{\\mu }_m,\\widetilde{\\nu }_n)-W_2^2(\\mu ,\\nu )|$ .","$\\widehat{f}_{\\mu }(\\cdot )$ satisfies $\\lim \\limits _{n\\rightarrow \\infty } \\max \\left\\lbrace \\mathbb {P}\\left(\\Vert \\widehat{f}_{\\mu }\\Vert _{C^s(\\widetilde{\\mathcal {X}})}\\ge TM\\right),\\mathbb {P}\\left(\\sup _{x\\in \\widetilde{\\mathcal {X}}} |\\widehat{f}_{\\mu }(x)-f_{\\mu }(x)|\\ge \\varepsilon \\right)\\right\\rbrace =0$ for any $\\varepsilon >0$ , where $\\widetilde{\\mathcal {X}}$ is any compact subset of $\\mathcal {X}^o$ .","The same conclusion holds for $\\widehat{f}_{\\nu }(\\cdot )$ with $\\mathcal {X}$ replaced by $\\mathcal {Y}$ .","In thm:smooth, $\\widetilde{f}_{\\mu }^{M^{\\prime }}(\\cdot )$ can be viewed as an approximate minimizer of $d_{\\mathrm {IP}}(\\widehat{f}_{\\mu },\\cdot ;C^{s+2}(\\mathcal {X},M^{\\prime }))$ over an appropriate class of Sobolev smooth probability densities.","This is carried out because $\\widehat{f}_{\\mu }(\\cdot )$ by itself may not be a probability density.","Further note that $\\widetilde{\\mu }_m,\\widetilde{\\nu }_n$ as specified in thm:smooth are both smooth, and consequently $\\widetilde{\\Gamma }_{\\mathrm {min}}$ is a singleton and the supremum in thm:smooth can be dropped.","A brief description of the proof technique for thm:smooth is presented in rem:pfstech below and the actual proof is given in sec:mainrespf.","Remark 2.4 (Proof technique) The proof of thm:smooth proceeds along the same lines as rem:pfnstech.","We first show that $\\mathcal {F}_s$ (see rem:pftech) can be chosen as a certain subset of $C^{s+2}(\\mathcal {Y}^{\\circ })$ .","We then use Dudley's entropy integral type bounds which in turn requires the bracketing entropy [137] of the class of compactly supported Sobolev smooth functions which can be found in [138].","We now explain the implications of both the parts of thm:smooth in the following two remarks.","Remark 2.5 (Mitigating the curse of dimensionality) thm:smooth shows that, under enough smoothness, i.e., when $2(s+2)>d$ , both the upper bounds for (REF ) and (REF ) are $O_p(n^{-1/2})$ .","This shows that, for large dimensions, provided $\\mu $ and $\\nu $ admit smooth enough densities, it is possible to construct plug-in estimators that mitigate the curse of dimensionality.","Note that a similar estimator was analyzed in [73] when $m=n$ .","However, the rates obtained in thm:smooth are strictly better than those in [73].","For $m=n$ , when $d<2(s+2)$ , [73] obtained a rate of $n^{-\\frac{s+2}{2(s+2)+d}}$ which is worse than $n^{-1/2}$ obtained in thm:smooth.","For the other regimes, [73] obtains rates (up to log factors) of $n^{-1/4}$ and $n^{-\\frac{1}{(s+2)(d+2(s+2))}}$ which are both worse than the respective rates of $n^{-1/2}$ and $n^{-\\frac{s+2}{d}}$ in thm:smooth.","In fact, the rates obtained in thm:smoothwav are also strictly better than the rates obtained in [73] (for every fixed $d$ ) described above, but they are strictly worse than the rates obtained in thm:smooth.","When the degree of smoothness $s$ is large, both Theorems REF and REF lead to rates of (approximately) $n^{-1/2}$ .","However when $s$ is small, the rate in thm:smooth is much faster than that in thm:smoothwav, e.g., if $s$ is close to 0, the rate in thm:smoothwav is approximately $n^{-\\frac{1}{d}}$ whereas that in thm:smooth is the faster rate of $n^{-\\frac{2}{d}}$ .","It must be noted however that the smoothness assumptions are different in Theorems REF and REF .","Remark 2.6 (Computational aspects of thm:smooth) In thm:smooth, we have shown that the plug-in estimator for $T_0(\\cdot )$ using $\\widetilde{f}_{\\mu }^{M^{\\prime }}(\\cdot )$ and $\\widetilde{f}_{\\nu }^{M^{\\prime }}(\\cdot )$ achieve rates that mitigate the curse of dimensionality under sufficient smoothness.","However, as is evident, $\\widetilde{f}_{\\mu }^{M^{\\prime }}(\\cdot )$ is hard to compute whereas $\\widehat{f}_{\\mu }(\\cdot )$ is computable easily in linear time.","Note that if $\\widehat{f}_{\\mu }(\\cdot )$ itself were a probability density in $C^s(\\mathcal {X};TM)$ , then we would have $\\widehat{f}_{\\mu }=\\widetilde{f}_{\\mu }^{M^{\\prime }}$ .","While thm:smooth does not establish that, it does come close in part 2, from which we can easily derive the following: $\\lim _{n\\rightarrow \\infty }\\mathbb {P}(\\widehat{f}_{\\mu }(\\cdot )\\notin C^s(\\widetilde{\\mathcal {X}};TM))=0.$ The above shows that $\\widehat{f}_{\\mu }(\\cdot )$ is indeed bounded below by $(TM)^{-1}$ on $\\widetilde{\\mathcal {X}}$ (any compact subset of the interior of $\\mathcal {X}$ ), and additionally belongs to $C^s(\\widetilde{\\mathcal {X}};TM)$ with probability converging to 1.","This leads us to conjecture that the natural density version of $\\widehat{f}_{\\mu }(\\cdot )$ , i.e., $\\frac{\\max \\lbrace \\widehat{f}_{\\mu }(\\cdot ),0\\rbrace }{\\int \\max \\lbrace \\widehat{f}_{\\mu }(x),0\\rbrace \\,dx}$ should serve as a good proxy for $\\widetilde{f}_{\\mu }^{M^{\\prime }}(\\cdot )$ and lead to rates of convergence that mitigate the curse of dimensionality.","From a computational perspective, the density specified above is easy to simulate from using an accept-reject algorithm without computing the integral in the denominator (see [110]).","However, our current proof technique does not provide rates of convergence for the above density estimator based on $\\widehat{f}_{\\mu }(\\cdot )$ .","Another important implication of thm:smooth is the bound obtained on $|W_2(\\widetilde{\\mu }_m,\\widetilde{\\nu }_n)-W_2(\\mu ,\\nu )|$ when $\\mu \\ne \\nu $ .","We first present the result and then describe the implication.","Proposition 2.6 Consider the setting in thm:smooth.","Then, provided $\\mu \\ne \\nu $ , the following holds: $|W_2(\\widetilde{\\mu }_m,\\widetilde{\\nu }_n)-W_2(\\mu ,\\nu )|=O_p(r_{d,s}^{(m,n)}).$ prop:wassrate (see sec:mainrespf for a proof) shows an interesting distinction between the $\\mu \\ne \\nu $ case and the $\\mu =\\nu $ case.","For $\\mu =\\nu $ , the best possible exponent is $n^{-\\frac{1+s}{2s+d}}$ for $d\\ge 3$ (see [143] where the result was established under more general Besov smoothness assumptions).","On the contrary, when $\\mu \\ne \\nu $ , prop:wassrate establishes a rate of $n^{-\\frac{s+2}{d}}$ for the Wasserstein distance which is strictly better than the minimax achievable rate mentioned above when $\\mu =\\nu $ .","This observation complements [33] where the authors make a similar observation for the special case of $s=0$ ."],["Discretized plug-in estimator under smoothness assumptions","In sec:nsmooth, we discussed how smoothness can be incorporated into the plug-in procedure to get faster rates of convergence.","Such plug-in estimators are popular in the computational OT literature (see [8], [9], [28], [39]).","However, even after $\\widetilde{f}_{\\mu }(\\cdot )$ , $\\widetilde{f}_{\\nu }(\\cdot )$ are calculated, $\\widetilde{T}_{m,n}^{\\gamma }$ as in thm:smooth cannot be computed explicitly from data if $\\widetilde{f}_{\\mu }(\\cdot )$ and $\\widetilde{f}_{\\nu }(\\cdot )$ are continuous densities.","This is in contrast to $\\widetilde{T}_{m,n}^{\\gamma }$ from thm:nsmooth in the discrete-discrete case which is explicitly computable using a standard linear program, but achieves worse rates of convergence.","This is not unexpected.","Thanks to the no free lunch principle, better statistical accuracy is naturally accompanied by heavier computational challenges.","Therefore, our goal here is to construct estimators, under smoothness assumptions as in sec:smooth, which are computable in polynomial time (with complexity increasing with smoothness) provided $\\widetilde{f}_{\\mu }(\\cdot )$ and $\\widetilde{f}_{\\nu }(\\cdot )$ can be sampled from, and also attain rates that mitigate the curse of dimensionality.","Construction: We will illustrate the discretized estimator using the kernel based estimator from sec:kernel.","Similar results also hold for the wavelet based estimator from sec:wavelet.","Recall the kernel density estimators $\\widetilde{f}_{\\mu }(\\cdot )$ and $\\widetilde{f}_{\\nu }(\\cdot )$ (see (REF )).","Sample $M\\ge 1$ random points from both $\\widetilde{f}_{\\mu }(\\cdot )$ and $\\widetilde{f}_{\\nu }(\\cdot )$ .","Let $\\widehat{\\mu }_{m,M}$ and $\\widehat{\\nu }_{n,M}$ denote the standard empirical measures on the $M$ points sampled from $\\widetilde{f}_{\\mu }(\\cdot )$ and $\\widetilde{f}_{\\nu }(\\cdot )$ respectively.","Finally construct $\\widetilde{T}_{m,n}\\equiv \\widetilde{T}_{m,n}^{\\gamma }$ as in def:bcenterproj with $\\widetilde{\\mu }_m=\\widehat{\\mu }_{m,M}$ and $\\widetilde{\\nu }_n=\\widehat{\\nu }_{n,M}$ .","It should be pointed out that a similar construction was also used in [143] for estimating probability densities under the Wasserstein loss.","Based on this construction, the main result of this section is as follows: Theorem 2.7 Consider the setting in thm:smooth and the same construction of $\\widetilde{T}_{m,n}^{\\gamma }$ as above.","For simplicity, let's also assume $m=n$ .","Accordingly set $M=n^{\\frac{s+2}{2}}$ .","Then $\\widetilde{\\Gamma }_{\\mathrm {min}}$ is a singleton and consequently the following conclusion holds for some constant $C>0$ : $\\mathbb {E}\\left[\\int \\Vert \\widetilde{T}_{m,n}(x)-T_0(x)\\Vert ^2\\,d\\widetilde{\\mu }_m(x)\\right]\\le C r_{d,s}^{(n,n)}.$ The same rates also hold for $\\mathbb {E}|W_2^2(\\widetilde{\\mu }_m,\\widetilde{\\nu }_n)-W_2^2(\\mu ,\\nu )|$ .","The proof of thm:dismooth is given in sec:mainrespf.","Once the empirical measures $\\widehat{\\mu }_{m,M}$ and $\\widehat{\\nu }_{n,M}$ have been obtained, an explicit computation of $\\widetilde{T}_{m,n}$ as described above requires $O(M^3)=O(n^{\\frac{3(s+2)}{2}})$ steps using the Hungarian algorithm, see [81].","This highlights the statistical versus computational trade off, i.e., in order to mitigate the curse of dimensionality in convergence rates by exploiting smoothness, the computational complexity gets progressively worse by polynomial factors in $n$ .","It should be mentioned that (approximate) algorithms faster than the Hungarian algorithm stated above, can be found in [59], [1], [39] to name a few.","Due to space constraints, we avoid a detailed discussion on this.","In the above construction, sampling from the smoothed kernel densities $\\widetilde{f}_{\\mu }(\\cdot )$ and $\\widetilde{f}_{\\nu }(\\cdot )$ is crucial.","If we would simply draw $M$ bootstrap samples from the empirical distributions $\\widehat{\\mu }_m$ and $\\widehat{\\nu }_n$ , the rates of convergence wouldn't improve from those observed in thm:nsmooth no matter how large $M$ is chosen.","In this section, we will apply our results to two popular problems, namely — estimating the Wasserstein barycenter between two probability distributions (see [2], [40], [15], [27]) in sec:bar, and obtaining detection thresholds in some recent optimal transport based independence testing procedures (see [44], [43], [61], [123], [124]) in sec:nonptest."],["Wasserstein barycenter estimation","Let $\\mu ,\\nu \\in \\mathcal {P}_2(\\mathbb {R}^d)$ .","The Wasserstein barycenter between $\\mu $ and $\\nu $ is then given by: $\\rho _0:=\\min \\limits _{\\rho \\in \\mathcal {P}_{\\mathrm {ac}}(\\mathbb {R}^d)} \\left(\\frac{1}{2}W_2^2(\\mu ,\\rho )+\\frac{1}{2}W_2^2(\\rho ,\\nu )\\right).$ In fact, by prop:bmopt, there exists an optimal transport map $T_0$ from $\\mu $ to $\\nu $ and by [2], [15], [18], an alternative characterization of $\\rho _0$ is as follows: $\\rho _0=\\left(\\frac{1}{2}\\mbox{Id}+\\frac{1}{2}T_0\\right)\\#\\mu ,\\qquad \\mbox{where}\\qquad \\mbox{Id}(x)=x.$ Estimating $\\rho _0$ as in (REF ) has attracted significant attention over the past few years in economics [31], [26], Bayesian learning [130], [129], dynamic formulations [35], [30], algorithmic fairness [68], [34], etc.","The most natural strategy employed in estimating $\\rho _0$ is to use the empirical plug-in estimator, i.e., replacing $\\mu ,\\nu $ in (REF ) with $\\widehat{\\mu }_m,\\widehat{\\nu }_n$ .","This strategy has been used, approximated and analyzed extensively in e.g., [40], [27], [93], [18].","Based on (REF ), the natural plug-in estimator of $\\rho _0$ would be: $\\widehat{\\rho }_0^{\\gamma }=\\left(\\frac{1}{2}\\mbox{Id}+\\frac{1}{2}\\widetilde{T}_{m,n}^{\\gamma }\\right)\\#\\widetilde{\\mu }_m$ where $\\widetilde{T}_{m,n}^{\\gamma }$ is the plug-in estimator of $T_0$ obtained by solving (REF ), with $\\mu $ and $\\nu $ replaced by $\\widetilde{\\mu }_m$ and $\\widehat{\\nu }_n$ respectively and $\\gamma \\in \\widetilde{\\Gamma }_{\\mathrm {min}}$ .","While the consistency of $\\widehat{\\rho }_0^{\\gamma }$ has been analyzed for $m=n$ in [93] and rates have been obtained for $d=1$ in [14], the more general question of obtaining rates of convergence for $\\widehat{\\rho }_0^{\\gamma }$ for general dimensions $d\\ge 1$ is yet unanswered.","We address this question in the following result (see sec:Appf for a proof).","Theorem 3.1 Suppose that the same assumptions from thm:nsmooth hold.","Then, with $\\widehat{\\rho }_0^{\\gamma }$ as defined in (REF ) and $r_d^{(m,n)}$ , $t_{d,\\alpha }$ defined in thm:nsmooth, the following holds: $\\sup \\limits _{\\gamma \\in \\widetilde{\\Gamma }_{\\mathrm {min}}}W_2^2(\\widehat{\\rho }_0^{\\gamma },\\rho _0)=O_p\\left(r^{(m,n)}_d\\times (\\log {(1+\\max \\lbrace m,n\\rbrace )})^{t_{d,\\alpha }}\\right).$"],["Nonparametric independence testing: Optimal transport based Hilbert-Schmidt independence criterion","Let $(X_1,Y_1),\\ldots ,(X_n,Y_n)\\overset{i.i.d.","}{\\sim }\\pi $ , a probability measure on $\\mathbb {R}^{d_1+d_2}$ , with marginals $\\mu \\in \\mathcal {P}_{\\mathrm {ac}}(\\mathbb {R}^{d_1})$ and $\\nu \\in \\mathcal {P}_{\\mathrm {ac}}(\\mathbb {R}^{d_2})$ .","Our problem of interest is the following hypothesis testing problem, given as: $\\mathrm {H}_0:\\pi =\\mu \\otimes \\nu \\qquad \\mbox{versus} \\qquad \\mathrm {H}_1:\\pi \\ne \\mu \\otimes \\nu .$ This is the classical nonparametric independence testing problem which has received a lot of attention in the statistics and machine learning literature (see [132], [71], [78], [12], and [48], [82] for a review).","In keeping with the overall theme of this paper, our focus here will be on a large class of OT based independence testing procedures, introduced first in [44] followed by recent developments in [123], [124], [43].","These tests bear resemblance to the Hilbert-Schmidt independence criterion (HSIC); see [71], [69], [70] and have attractive properties such as distribution-freeness (see prop:testprop), consistency without moment assumptions and robustness against heavy-tailed distributions and against contamination [44], [123].","Below, we describe this class of tests, see (REF ) and (REF ).","Our main theoretical contribution of this section will be to provide detection thresholds of these OT based tests.","Construction: Suppose $\\upsilon _1$ , $\\upsilon _2$ be two compactly supported probability distributions on $\\mathbb {R}^{d_1}$ and $\\mathbb {R}^{d_2}$ respectively (e.g., $\\upsilon _1\\equiv \\mathrm {Unif}[0,1]^{d_1}$ , $\\upsilon _2\\equiv \\mathrm {Unif}[0,1]^{d_2}$ ).","Let $U_1,\\ldots ,U_n\\overset{i.i.d.","}{\\sim }\\upsilon _1$ , $V_1,\\ldots ,V_n\\overset{i.i.d.","}{\\sim }\\upsilon _2$ , $\\widehat{u}_n:=n^{-1}\\sum _{i=1}^n \\delta _{U_i}$ and $\\widehat{v}_n:=n^{-1}\\sum _{j=1}^n \\delta _{V_j}$ .","Recall the definitions of $\\widehat{\\mu }_n$ (with $m=n$ ) and $\\widehat{\\nu }_n$ from (REF ).","Let $\\widehat{T}_{1,n}$ ($\\widehat{T}_{2,n}$ ) be obtained by solving (REF ), with $\\mu $ and $\\nu $ replaced by $\\widehat{\\mu }_n$ and $\\widehat{u}_n$ ($\\widehat{\\nu }_n$ and $\\widehat{v}_n$ ) respectively.","Consider two non negative definite, continuous, characteristic kernels (see [58], [128] for definitions) $K_1(\\cdot ,\\cdot )$ and $K_2(\\cdot ,\\cdot )$ on $(\\mbox{supp}(\\upsilon _1))^2$ and $(\\mbox{supp}(\\upsilon _2))^2$ .","Set $\\widehat{x}_{ij}:=K_1(\\widehat{T}_{1,n}(X_i),\\widehat{T}_{1,n}(X_j))$ and $\\widehat{y}_{ij}:=K_2(\\widehat{T}_{2,n}(Y_i),\\widehat{T}_{2,n}(Y_j))$ .","Our test statistic is as follows: $\\widehat{\\mathrm {rHSIC}}:=n^{-2}\\sum _{i,j}\\widehat{x}_{ij}\\widehat{y}_{ij}+n^{-4}\\sum _{i,j,r,s}\\widehat{x}_{ij}\\widehat{y}_{rs}-2n^{-3}\\sum _{i,j,r} \\widehat{x}_{ij}\\widehat{y}_{ir}.$ Proposition 3.2 (See [44], [124]) (Distribution-freeness) When $X_1$ and $Y_1$ are independent, the distribution of $n\\times \\widehat{\\mathrm {rHSIC}}$ is universal, i.e., it does not depend on $\\mu $ and $\\nu $ for every fixed $n$ .","(Consistency against fixed alternatives) Let $c_{n,\\alpha }$ be the upper $(1-\\alpha )$ -th quantile from the universal distribution in part 1 above.","Then $\\widehat{\\mathrm {rHSIC}}\\overset{P}{\\longrightarrow }\\mathrm {rHSIC}(\\pi |\\mu \\otimes \\nu )$ where $&\\mathrm {rHSIC}(\\pi | \\mu \\otimes \\nu ):=\\mathbb {E}[K_1(T_1(X_1),T_1(X_2))K_2(T_2(Y_1),T_2(Y_2))]+\\mathbb {E}[K_1(T_1(X_1),T_1(X_2))]\\nonumber \\\\&\\times \\mathbb {E}[K_2(T_2(Y_1),T_2(Y_2))]-2\\mathbb {E}[K_1(T_1(X_1),T_1(X_2))K_2(T_2(Y_1),T_2(Y_3))],$ where $T_{1}(\\cdot )$ (respectively $T_2(\\cdot )$ ) is the optimal transport map from $\\mu $ ($\\nu $ ) to $\\upsilon _1$ ($\\upsilon _2$ ); see def:otmpm.","Further $\\mathrm {rHSIC}(\\pi |\\mu \\otimes \\nu )=0$ if and only if $\\pi =\\mu \\otimes \\nu $ .","Define the following test function: $\\phi _{n,\\alpha }:=\\mathbb {1}(n\\times \\widehat{\\mathrm {rHSIC}}\\ge c_{n,\\alpha }).$ Then $\\mathbb {E}[\\phi _{n,\\alpha }]\\rightarrow 1$ as $n\\rightarrow \\infty $ under $\\mathrm {H}_1$ , i.e., when $\\pi \\ne \\mu \\otimes \\nu $ .","prop:testprop shows that the test based on $\\widehat{\\mathrm {rHSIC}}$ (see (REF )), i.e., $\\phi _{n,\\alpha }$ (see (REF )), can be carried out without resorting to the permutation principle as is necessary for the usual HSIC based test (see [69]).","Further, when the sampling distribution is fixed, prop:testprop shows that $\\widehat{\\mathrm {rHSIC}}$ consistently estimates $\\mathrm {rHSIC}(\\pi |\\mu \\otimes \\nu )$ , a quantity which equals 0 if and only if $\\pi =\\mu \\otimes \\nu $ (this yields the consistency of $\\phi _{n,\\alpha })$ against fixed alternatives.","While consistency against fixed alternatives is an attractive feature of $\\phi _{n,\\alpha }$ , a more intricate question of statistical interest is to understand the local power of $\\phi _{n,\\alpha }$ under “changing sequence of alternatives converging to the null\" as $n\\rightarrow \\infty $ .","To study the local power of $\\phi _{n,\\alpha }$ , we need to consider a triangular array setting, where the data distribution changes with $n$ , i.e., $(X_1,Y_1),\\ldots ,(X_n,Y_n)\\overset{i.i.d.","}{\\sim }\\pi ^{(n)}$ , a probability measure on $\\mathbb {R}^{d_1+d_2}$ , with marginals $\\mu ^{(n)}\\in \\mathcal {P}_{\\mathrm {ac}}(\\mathbb {R}^{d_1})$ and $\\nu ^{(n)}\\in \\mathcal {P}_{\\mathrm {ac}}(\\mathbb {R}^{d_2})$ .","As $\\mathrm {rHSIC}(\\cdot |\\cdot )$ characterizes independence, a mathematical formulation of “alternatives converging to null\" would be to say $\\mathrm {rHSIC}(\\pi ^{(n)}|\\mu ^{(n)}\\otimes \\nu ^{(n)})\\rightarrow 0$ as $n\\rightarrow \\infty $ .","Similar questions have attracted a lot of attention in modern statistics, featuring measures (other than $\\mathrm {rHSIC}(\\cdot |\\cdot )$ ) which characterize independence, see e.g., [12], [87], [94], [6].","In the following result (see sec:Appf for a proof), we show that if $\\mathrm {rHSIC}(\\pi ^{(n)}|\\mu ^{(n)}\\otimes \\nu ^{(n)})\\rightarrow 0$ slowly enough with $n$ , then $\\phi _{n,\\alpha }$ yields a consistent sequence of tests for problem (REF ).","Theorem 3.3 Consider problem (REF ) with $\\pi ^{(n)}$ , $\\mu ^{(n)}$ , $\\nu ^{(n)}$ (changing with $n$ ) and suppose $T_{1,n}(\\cdot )$ and $T_{2,n}(\\cdot )$ are both $L$ -Lipschitz ($L$ is free of $n$ ).","Also assume $K_1(\\cdot )$ , $K_2(\\cdot )$ are Lipschitz, $\\mu ^{(n)}$ , $\\nu ^{(n)}$ are supported on fixed compact sets (supports are free of $n$ ).","Set $r_{d_1,d_2}^{(n,n)}:=r_{d_1}^{(n,n)}+r_{d_2}^{(n,n)}$ where $r_{d_1}^{(n,n)},r_{d_2}^{(n,n)}$ is defined via (REF ).","Then, $\\mathbb {E}[\\phi _{n,\\alpha }]\\rightarrow 1\\qquad \\mbox{if} \\qquad (r_{d_1,d_2}^{(n,n)})^{-1/2}\\times \\mathrm {rHSIC}(\\pi ^{(n)}|\\mu ^{(n)}\\otimes \\nu ^{(n)})\\rightarrow \\infty ,$"],["Appendix","This section is devoted to proving our main results and is organized as follows: In sec:mainrespf, we present the proofs of results from sec:mainres and in sec:Appf, we present the proofs from sec:App.","Throughout this section, we will use the $\\lesssim $ sign to hide constants that are free of $m,n$ ."],["Proofs from sec:mainres","[Proof of thm:newubd] We begin the proof by observing that $\\varphi _0^*(\\cdot )$ is convex and finite on $\\mbox{supp}(\\nu )$ , and hence differentiable $\\nu $ almost everywhere (a.e.).","Further by lem:gradual, we also have: $\\nabla \\varphi _0^*(T_0(x))=x\\qquad \\mbox{$\\mu $-a.e.~$x$.","}$ Fix any arbitrary $\\gamma \\in \\widetilde{\\Gamma }_{\\mathrm {min}}$ and suppose that $\\gamma (y|x)$ denotes the conditional distribution of $y$ given $x$ under $\\gamma $ .","Define, $D_1:= \\int \\varphi _0^*(y)\\,d\\widetilde{\\nu }_n(y) - \\int \\varphi _0^*(y) d \\overline{\\nu }_m(y).$ As $\\gamma $ has marginals $\\widetilde{\\mu }_m$ and $\\widetilde{\\nu }_n$ , we have: $D_1=\\int _{x,y} \\varphi _0^*(y)\\,d\\gamma (y|x)\\,d\\widetilde{\\mu }_m(x)-\\int _{x} \\varphi _0^*(T_0(x))\\,d\\widetilde{\\mu }_m(x).$ Next, by applying the conditional version of Jensen's inequality, $\\int _{x} \\left(\\int _{y} \\varphi _0^*(y)\\,d\\gamma (y|x)\\right)\\,d\\widetilde{\\mu }_m(x)&\\ge \\int _{x}\\varphi _0^*\\left(\\int _{y} y\\,d\\gamma (y|x)\\right)\\,d\\widetilde{\\mu }_m(x)\\nonumber \\\\&=\\int _{x}\\varphi _0^*(\\widetilde{T}_{m,n}^{\\gamma }(x))\\,d\\widetilde{\\mu }_m(x).$ Using (REF ) with (REF ) yields, $D_1&\\ge \\int [\\varphi _0^*(\\widetilde{T}_{m,n}^{\\gamma }(x)) - \\varphi _0^*(T_0(x))] d \\widetilde{\\mu }_m(x) \\nonumber \\\\& \\overset{(a)}{\\ge } \\int \\left\\lbrace \\nabla \\varphi _0^*(T_0(x))^\\top (\\widetilde{T}_{m,n}^{\\gamma }(x) - T_0(x)) + \\frac{1}{2L} \\Vert \\widetilde{T}_{m,n}^{\\gamma }(x)-T_0(x)\\Vert ^2 \\right\\rbrace \\,d {\\widetilde{\\mu }_m}(x) \\nonumber \\\\& \\overset{(b)}{=} \\underbrace{\\int x^\\top (\\widetilde{T}_{m,n}^{\\gamma }(x) - T_0(x)) \\,d {\\widetilde{\\mu }_m}(x)}_{D_2} + \\frac{1}{2L} \\int \\Vert \\widetilde{T}_{m,n}^{\\gamma }(x)-T_0(x)\\Vert ^2 \\, d {\\widetilde{\\mu }_m}(x).$ Here (a) follows from the strong convexity of $\\varphi _0^*(\\cdot )$ with parameter $(1/L)$ (see lem:assm) and (b) follows from (REF ).","Next, we will simplify the term $D_2$ .","Towards this direction, observe that for every $\\gamma \\in \\widetilde{\\Gamma }_{\\mathrm {min}}$ , $W_2^2(\\widetilde{\\mu }_m,\\widetilde{\\nu }_n)&=\\int \\Vert x-y\\Vert ^2\\,d\\gamma (x,y)\\nonumber \\\\ &=\\int \\Vert x\\Vert ^2\\,d\\widetilde{\\mu }_m(x)+\\int \\Vert y\\Vert ^2\\,d\\widetilde{\\nu }_n(y)-2\\int _{x} \\left(x^{\\top }\\int _{y} y\\,d\\gamma (y|x)\\right)\\,d\\widetilde{\\mu }_m(x)\\nonumber \\\\&=\\int \\Vert x\\Vert ^2\\,d\\widetilde{\\mu }_m(x)+\\int \\Vert y\\Vert ^2\\,d\\widetilde{\\nu }_n(y)-2\\int _{x} x^{\\top }\\widetilde{T}_{m,n}^{\\gamma }(x)\\,d\\widetilde{\\mu }_m(x).$ Also, as $T_0$ is the gradient of a convex function, it is also an OT map from $\\widetilde{\\mu }_m$ to $\\overline{\\nu }_m$ (see [3]), we have: $W_2^2(\\widetilde{\\mu }_m,\\overline{\\nu }_m)&=\\int \\Vert x-T_0(x)\\Vert ^2\\,d\\widetilde{\\mu }_m(x)\\nonumber \\\\ &=\\int \\Vert x\\Vert ^2\\,d\\widetilde{\\mu }_m(x)+\\int \\Vert y\\Vert ^2\\,d\\overline{\\nu }_m(y)-2\\int _{x} x^{\\top }T_0(x)\\,d\\widetilde{\\mu }_m(x).$ Now (REF ) and (REF ) imply $D_2=\\frac{1}{2}\\big (W_2^2(\\widetilde{\\mu }_m,\\overline{\\nu }_m)-W_2^2(\\widetilde{\\mu }_m,\\widetilde{\\nu }_n)\\big )+\\frac{1}{2}\\int \\Vert y\\Vert ^2\\,d(\\widetilde{\\nu }_n-\\overline{\\nu }_m)(y).$ Finally by combining (REF ) and (REF ), we get: $&\\;\\;\\;\\;\\frac{1}{2L}\\int \\Vert \\widetilde{T}_{m,n}^{\\gamma }(x)-T_0(x)\\Vert ^2\\,d\\widetilde{\\mu }_m(x)\\nonumber \\\\ &\\le \\frac{1}{2}\\big (W_2^2(\\widetilde{\\mu }_m,\\widetilde{\\nu }_n)-W_2^2(\\widetilde{\\mu }_m,\\overline{\\nu }_m)\\big )+\\int (\\varphi _0^*(y)-(1/2)\\Vert y\\Vert ^2)\\,d(\\widetilde{\\nu }_n-\\overline{\\nu }_m)(y).$ Now note that the bound on the right hand side of the above display is free of the particular choice of $\\gamma \\in \\widetilde{\\Gamma }_{\\mathrm {min}}$ .","Therefore, the same bound holds if we take a supremum over $\\gamma \\in \\widetilde{\\Gamma }_{\\mathrm {min}}$ on the left hand side.","We will now provide an upper bound for the right hand side of (REF ).","The remainder of the proof proceeds as in the proof of [101].","By the dual representation presented in (REF ) and (), and the definitions of $\\Psi _{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n}(\\cdot )$ and $\\Psi _{\\widetilde{\\mu }_m,\\overline{\\nu }_m}(\\cdot )$ in the statement of thm:newubd, we have $\\frac{1}{2}W_2^2(\\widetilde{\\mu }_m,\\widetilde{\\nu }_n)=\\frac{1}{2}\\int \\Vert x\\Vert ^2\\,d\\widetilde{\\mu }_m(x)+\\frac{1}{2}\\int \\Vert y\\Vert ^2\\,d\\widetilde{\\nu }_n(y)-\\mathcal {S}_{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n}(\\Psi _{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n}),$ $\\mbox{and}\\qquad \\frac{1}{2}W_2^2(\\widetilde{\\mu }_m,\\overline{\\nu }_m)=\\frac{1}{2}\\int \\Vert x\\Vert ^2\\,d\\widetilde{\\mu }_m(x)+\\frac{1}{2}\\int \\Vert y\\Vert ^2\\,d\\overline{\\nu }_m(y)-\\mathcal {S}_{\\widetilde{\\mu }_m,\\overline{\\nu }_m}(\\Psi _{\\widetilde{\\mu }_m,\\overline{\\nu }_m}).$ By subtracting the two equations above, we get: $\\frac{1}{2}W_2^2(\\widetilde{\\mu }_m,\\widetilde{\\nu }_n)-\\frac{1}{2}W_2^2(\\widetilde{\\mu }_m,\\overline{\\nu }_m)=\\frac{1}{2}\\int \\Vert y\\Vert ^2\\,d(\\widetilde{\\nu }_n-\\overline{\\nu }_m)-\\mathcal {S}_{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n}(\\Psi _{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n})+\\mathcal {S}_{\\widetilde{\\mu }_m,\\overline{\\nu }_m}(\\Psi _{\\widetilde{\\mu }_m,\\overline{\\nu }_m}).$ Next, we use () to make the following observations: $\\mathcal {S}_{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n}(\\Psi _{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n})\\le \\mathcal {S}_{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n}(\\Psi _{\\widetilde{\\mu }_m,\\overline{\\nu }_m}),\\qquad \\mathcal {S}_{\\widetilde{\\mu }_m,\\overline{\\nu }_m}(\\Psi _{\\widetilde{\\mu }_m,\\overline{\\nu }_m})\\le \\mathcal {S}_{\\widetilde{\\mu }_m,\\overline{\\nu }_m}(\\Psi _{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n}).$ Note that (REF ) immediately yields the following conclusions: $\\mathcal {S}_{\\widetilde{\\mu }_m,\\overline{\\nu }_m}(\\Psi _{\\widetilde{\\mu }_m,\\overline{\\nu }_m})-\\mathcal {S}_{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n}(\\Psi _{\\widetilde{\\mu }_m,\\overline{\\nu }_m})\\le \\mathcal {S}_{\\widetilde{\\mu }_m,\\overline{\\nu }_m}(\\Psi _{\\widetilde{\\mu }_m,\\overline{\\nu }_m})-\\mathcal {S}_{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n}(\\Psi _{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n}),$ and $\\mathcal {S}_{\\widetilde{\\mu }_m,\\overline{\\nu }_m}(\\Psi _{\\widetilde{\\mu }_m,\\overline{\\nu }_m})-\\mathcal {S}_{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n}(\\Psi _{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n})\\le \\mathcal {S}_{\\widetilde{\\mu }_m,\\overline{\\nu }_m}(\\Psi _{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n})-\\mathcal {S}_{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n}(\\Psi _{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n}).$ By combining the above two displays, we have: $&\\;\\;\\;\\;\\left|\\mathcal {S}_{\\widetilde{\\mu }_m,\\overline{\\nu }_m}(\\Psi _{\\widetilde{\\mu }_m,\\overline{\\nu }_m})-\\mathcal {S}_{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n}(\\Psi _{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n})\\right|\\nonumber \\\\ &\\le \\max \\left\\lbrace \\left|\\mathcal {S}_{\\widetilde{\\mu }_m,\\overline{\\nu }_m}(\\Psi _{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n})-\\mathcal {S}_{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n}(\\Psi _{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n})\\right|,\\left|\\mathcal {S}_{\\widetilde{\\mu }_m,\\overline{\\nu }_m}(\\Psi _{\\widetilde{\\mu }_m,\\overline{\\nu }_m})-\\mathcal {S}_{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n}(\\Psi _{\\widetilde{\\mu }_m,\\overline{\\nu }_m})\\right|\\right\\rbrace .$ By () and some simple algebra, the following holds: $ \\left|\\mathcal {S}_{\\widetilde{\\mu }_m,\\overline{\\nu }_m}(\\Psi _{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n})-\\mathcal {S}_{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n}(\\Psi _{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n})\\right|=\\left|\\int \\Psi _{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n}^*\\,d(\\overline{\\nu }_m-\\widetilde{\\nu }_n)\\right|.$ A similar expression holds for $|\\mathcal {S}_{\\widetilde{\\mu }_m,\\overline{\\nu }_m}(\\Psi _{\\widetilde{\\mu }_m,\\overline{\\nu }_m})-\\mathcal {S}_{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n}(\\Psi _{\\widetilde{\\mu }_m,\\overline{\\nu }_m})|$ .","Using the above observation in (REF ), we get: $\\left|\\mathcal {S}_{\\widetilde{\\mu }_m,\\overline{\\nu }_m}(\\Psi _{\\widetilde{\\mu }_m,\\overline{\\nu }_m})-\\mathcal {S}_{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n}(\\Psi _{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n})\\right|\\le \\max \\left\\lbrace \\bigg |\\int \\Psi _{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n}^* \\,d(\\widetilde{\\nu }_n-\\overline{\\nu }_m)\\bigg |,\\bigg |\\int \\Psi _{\\widetilde{\\mu }_m,\\overline{\\nu }_m}^* \\,d(\\widetilde{\\nu }_n-\\overline{\\nu }_m)\\bigg |\\right\\rbrace .$ Combining the above display with (REF ), we further have: $&\\;\\;\\;\\;\\left|\\frac{1}{2}W_2^2(\\widetilde{\\mu }_m,\\widetilde{\\nu }_n)-\\frac{1}{2}W_2^2(\\widetilde{\\mu }_m,\\overline{\\nu }_m)-\\left(\\frac{1}{2}\\int \\Vert y\\Vert ^2\\,d(\\widetilde{\\nu }_n-\\overline{\\nu }_m)\\right)\\right|\\nonumber \\\\ &\\le \\max \\left\\lbrace \\bigg |\\int \\Psi _{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n}^* \\,d(\\widetilde{\\nu }_n-\\overline{\\nu }_m)\\bigg |,\\bigg |\\int \\Psi _{\\widetilde{\\mu }_m,\\overline{\\nu }_m}^* \\,d(\\widetilde{\\nu }_n-\\overline{\\nu }_m)\\bigg |\\right\\rbrace .$ Combining (REF ) with (REF ) then completes the proof.","[Proof of thm:nsmooth] First observe that $\\limsup _{M\\rightarrow \\infty }\\limsup _{m,n\\rightarrow \\infty } \\mathbb {P}\\left(\\Big |\\int \\varphi _0^*\\,d(\\widehat{\\nu }_n-\\overline{\\nu }_m)\\big |\\ge M \\left(r_d^{(m,m)}+r_d^{(n,n)}\\right)\\right)=0$ by the weak law of large numbers as $(r_{d}^{(n,n)})^{-1}n^{-1/2}=O(1)$ and $(r_{d}^{(m,m)})^{-1}m^{-1/2}=O(1)$ .","Combining the above observation with thm:newubd, we have: $&\\;\\;\\;\\;\\limsup _{M\\rightarrow \\infty }\\limsup _{m,n\\rightarrow \\infty }\\mathbb {P}\\left(\\sup \\limits _{\\gamma \\in \\widetilde{\\Gamma }_{\\mathrm {min}}} \\int \\Vert \\widetilde{T}_{m,n}^{\\gamma }(x)-T_0(x)\\Vert ^2 \\,d\\widetilde{\\mu }_m(x)\\ge Mr_d^{(m,n)}\\right)\\nonumber \\\\ &\\le \\limsup _{M\\rightarrow \\infty }\\limsup _{m,n\\rightarrow \\infty }\\mathbb {P}\\left(\\max \\left\\lbrace \\bigg |\\int \\Psi _{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n}^* \\,d(\\widetilde{\\nu }_n-\\overline{\\nu }_m)\\bigg |,\\bigg |\\int \\Psi _{\\widetilde{\\mu }_m,\\overline{\\nu }_m}^* \\,d(\\widetilde{\\nu }_n-\\overline{\\nu }_m)\\bigg |\\right\\rbrace \\ge \\frac{M}{2} r_d^{(m,n)}\\right)\\nonumber \\\\ &\\le \\limsup _{M\\rightarrow \\infty }\\limsup _{m,n\\rightarrow \\infty }\\Bigg [\\mathbb {P}\\left(\\left|\\int \\Psi _{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n}^* \\,d(\\widetilde{\\nu }_n-\\nu )\\right|\\ge \\frac{M}{2}r_d^{(n,n)}\\right)+\\mathbb {P}\\left(\\left|\\int \\Psi _{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n}^* \\,d(\\overline{\\nu }_m-\\nu )\\right|\\ge \\frac{M}{2}r_d^{(m,m)}\\right)\\nonumber \\\\&+\\mathbb {P}\\left(\\left|\\int \\Psi _{\\widetilde{\\mu }_m,\\overline{\\nu }_m}^* \\,d(\\widetilde{\\nu }_n-\\nu )\\right|\\ge \\frac{M}{2} r_d^{(n,n)}\\right)+\\mathbb {P}\\left(\\left|\\int \\Psi _{\\widetilde{\\mu }_m,\\overline{\\nu }_m}^* \\,d(\\overline{\\nu }_m-\\nu )\\right|\\ge \\frac{M}{2} r_d^{(m,m)}\\right)\\Bigg ].$ In the sequel, we will only discuss how to bound the first term on the right hand side of (REF ).","Once that is understood, the other terms can be bounded similarly.","Therefore, our focus is on bounding $\\limsup _{M\\rightarrow \\infty }\\limsup _{m,n\\rightarrow \\infty }\\mathbb {P}\\left(\\left|\\int \\Psi _{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n}^* \\,d(\\widetilde{\\nu }_n-\\nu )\\right|\\ge \\frac{M}{2}r_d^{(n,n)}(\\log {(1+\\max \\lbrace m,n\\rbrace )})^{t_{d,\\alpha }}\\right).$ For the next part, to simplify notation, let us begin with some notation.","Set $\\mathcal {Y}:=\\mbox{supp}(\\nu )$ and $\\mathcal {X}_{n,\\mu }$ denote the closure of the convex hull of $X_1,\\ldots ,X_n$ .","Note that if we replace $\\Psi _{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n}(\\cdot )$ by $\\Psi _{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n}(\\cdot )-C$ for some constant $C>0$ , then $\\Psi _{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n}^*(\\cdot )\\mapsto \\Psi _{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n}^*+C$ .","However replacing $\\Psi _{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n}^*(\\cdot )$ by $\\Psi _{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n}^*(\\cdot )+C$ in (REF ) doesn't change its value as $\\widetilde{\\nu }_n$ and $\\nu $ are both probability measures.","Therefore, without loss of generality, we can assume that $\\Psi _{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n}(X_1)=0$ for all $m,n$ .","We will stick to this convention for the rest of the proof.","Also note that $\\Psi _{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n}(\\cdot )$ is only determined at the data points $X_1,\\ldots ,X_n$ .","Without loss of generality, we extend $\\Psi _{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n}(\\cdot )$ to the whole of $\\mathbb {R}^d$ by linear interpolation for any $x\\in \\mathcal {X}_{n,\\mu }$ and setting $\\Psi _{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n}(x)=\\infty $ for $x\\in \\mathcal {X}_{n,\\mu }^c$ .","The proof now proceeds using the following steps: Step I: There exists a constant $C_1>0$ and $y_n\\in \\mbox{supp}(\\nu )=\\mathcal {Y}$ such that $|\\Psi _{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n}^*(y_n)|\\le \\max _{1\\le i\\le m}\\Vert X_i\\Vert .$ [Proof of step I] By Kantorovich duality, there exists $y_n$ such that $\\Psi _{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n}^*(y_n)+\\Psi _{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n}(X_1)=\\langle X_1,y_n\\rangle \\quad \\Rightarrow \\quad |\\Psi _{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n}^*(y_n)|\\le C_1\\Vert X_1\\Vert \\le C_1\\max _{1\\le i\\le m}\\Vert X_i\\Vert ,$ where $C_1:=\\sup \\lbrace \\Vert y\\Vert :\\ y\\in \\mathcal {Y}\\rbrace $ .","Step II: There exists a constant $C_2>0$ such that the following holds: $\\Vert \\Psi _{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n}^*\\Vert _{\\infty ,\\mathcal {Y}}\\le C_2\\max _{1\\le i\\le n} \\Vert X_i\\Vert ,$ where $\\Vert \\cdot \\Vert _{\\infty ,\\mathcal {Y}}$ is the uniform norm on the support of $\\nu $ .","[Proof of step II] As $\\Psi _{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n}(x)=\\infty $ for $x\\in \\mathcal {X}_{n,\\mu }^c$ , using (REF ), we can write $\\Psi _{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n}^*(y)=\\max _{x\\in \\mathcal {X}_{n,\\mu }} (\\langle x,y\\rangle -\\Psi _{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n}(x))$ for all $y\\in \\mathcal {Y}$ .","For any $y_0\\in \\mathcal {Y}$ , let $x_0\\in \\mathcal {X}_{n,\\mu }$ be such that $\\Psi _{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n}^*(y_0)=\\langle x_0,y_0\\rangle -\\Psi _{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n}(x_0)$ .","Then, for any $y\\in \\mathcal {X}$ , we have: $&{\\left\\lbrace \\begin{array}{ll} \\Psi _{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n}^*(y_0)=\\langle x_0,y_0\\rangle -\\Psi _{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n}(x_0)\\\\ \\Psi _{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n}^*(y)\\ge \\langle x_0,y\\rangle -\\Psi _{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n}(x_0)\\end{array}\\right.","}\\\\ \\Rightarrow &|\\Psi _{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n}^*(y_0)-\\Psi _{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n}^*(y)|\\le |\\langle x_0,y_0-y\\rangle |\\le \\left(\\max _{1\\le i\\le m}\\Vert X_i\\Vert \\right)\\Vert y_0-y\\Vert .$ where the last line uses the fact that $y_0,y$ are arbitrary.","In particular, by setting $y_0:=y_n$ from step I, we get: $\\Vert \\Psi _{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n}^*\\Vert _{\\infty ,\\mathcal {Y}}\\le |\\Psi _{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n}^*(y_n)|+ \\left(\\max _{1\\le i\\le m}\\Vert X_i\\Vert \\right)\\sup _{y\\in \\mathcal {Y}}\\Vert y_n-y\\Vert \\le C_2\\left(\\max _{1\\le i\\le m}\\Vert X_i\\Vert \\right),$ where $C_2:=3C_1$ with $C_1$ defined as specified in the proof of step I.","The above lemma allows us to bound (with high probability) the $L^{\\infty }$ -norm of $\\Psi _{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n}^*(\\cdot )$ on $\\mathcal {Y}$ , using the tail assumption $\\mathbb {E}\\exp (t\\Vert X_1\\Vert ^{\\alpha })<\\infty $ for some $t>0$ and $\\alpha >0$ .","This is the focus of the next step.","Step III: For $K>0$ , define the following two sets: $A_{m,n,K}:=\\left\\lbrace \\int (\\Psi _{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n}^*(u))^2\\,d\\nu (u)\\ge K\\right\\rbrace ,\\quad \\mathrm {and},$ $\\widetilde{A}_{m,n,K}:=\\left\\lbrace \\Vert \\Psi _{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n}^*\\Vert _{\\infty ,\\mathcal {Y}}\\ge K\\big (\\log {n}\\big )^{1/\\alpha }\\right\\rbrace .$ Then there exists $K_0>0$ such that for any $K\\ge K_0$ , we have: $\\lim _{m,n\\rightarrow \\infty }\\mathbb {P}(\\widetilde{A}_{m,n,K})=0.$ and $\\lim _{m,n\\rightarrow \\infty }\\mathbb {P}(A_{m,n,K})=0.$ [Proof of step III] By using the exponential Markov's inequality coupled with the standard union bound, we have: $\\mathbb {P}\\left(\\max _{1\\le i\\le m} \\Vert X_i\\Vert \\ge K(\\log {m})^{1/\\alpha }\\right)&\\le m\\mathbb {P}\\left(\\Vert X_1\\Vert \\ge K(\\log {m})^{1/\\alpha }\\right)\\\\ &\\le m\\exp (-tK^{\\alpha }(\\log {m}))\\mathbb {E}\\exp (t\\Vert X_1\\Vert ^{\\alpha })\\overset{m\\rightarrow \\infty }{\\longrightarrow }0$ provided $K>t^{-\\alpha }$ .","Using the above observation coupled with step II, (REF ) follows by choosing $K_0>C_2 t^{-\\alpha }$ .","For the next part, we define another set: $B_{m,n,\\varepsilon }:=\\left\\lbrace \\int \\big |\\Psi _{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n}^*(u)-\\Psi _{\\mu ,\\nu }^*(u)|^2\\,d\\nu (u)\\ge \\varepsilon \\right\\rbrace $ for $\\varepsilon >0$ , where, as in (), we have: $W_2^2(\\mu ,\\nu )=\\int \\Vert x\\Vert ^2\\,d\\mu (x)+\\int \\Vert y\\Vert ^2\\,d\\nu (y)-2\\left(\\int \\Psi _{\\mu ,\\nu }(x)\\,d\\mu (x)+\\int \\Psi _{\\mu ,\\nu }^*(y)\\,d\\nu (y)\\right).$ Now by using [45], we have $\\mathbb {P}(B_{m,n,\\varepsilon })\\rightarrow 0$ as $m,n\\rightarrow \\infty $ for all $\\varepsilon >0$ .","As $\\int (\\Psi _{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n}^*(u))^2\\,d\\nu (u)\\le 2\\int \\big |\\Psi _{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n}^*(u)-\\Psi _{\\mu ,\\nu }^*(u)|^2\\,d\\nu (u)+2\\int (\\Psi _{\\mu ,\\nu }^*(u))^2\\,d\\nu (u),$ (REF ) follows with $K_0>2\\int (\\Psi _{\\mu ,\\nu }^*(u))^2\\,d\\nu (u)+1$ if we choose $\\epsilon =1/2$ .","We are now in a position to complete the proof of thm:nsmooth using steps I-III.","Towards this direction, set $K^{\\prime }:=2K_0$ where $K_0$ is defined as in the proof of step III and observe that for any $M>0$ , $&\\;\\;\\;\\limsup _{M\\rightarrow \\infty }\\limsup _{m,n\\rightarrow \\infty }\\mathbb {P}\\left(\\Bigg |\\int \\Psi _{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n}^*(u)\\,d(\\widetilde{\\nu }_n-\\nu )\\Bigg |\\ge Mr_d^{(n,n)}(\\log {(1+m)})^{t_{d,\\alpha }}\\right)\\nonumber \\\\&\\le \\limsup _{M\\rightarrow \\infty }\\limsup _{m,n\\rightarrow \\infty }\\mathbb {P}\\left(\\Bigg |\\int \\Psi _{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n}^*(u)\\,d(\\widetilde{\\nu }_n-\\nu )\\Bigg |\\ge Mr_d^{(n,n)}(\\log {(1+m)})^{t_{d,\\alpha }}, A_{m,n,K^{\\prime }}^c\\cap \\widetilde{A}_{m,n,K^{\\prime }}^c\\right)\\nonumber \\\\ &+\\limsup \\limits _{n\\rightarrow \\infty }\\mathbb {P}(\\widetilde{A}_{m,n,K^{\\prime }})+\\limsup \\limits _{m,n\\rightarrow \\infty }\\mathbb {P}(A_{m,n,K^{\\prime }})\\nonumber \\\\&\\le \\limsup _{M\\rightarrow \\infty }\\limsup \\limits _{m,n\\rightarrow \\infty }\\mathbb {P}\\left(\\Bigg |\\int \\Psi _{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n}^*(u)\\,d(\\widetilde{\\nu }_n-\\nu )\\Bigg |\\ge Mr_d^{(n,n)}(\\log {(1+m)})^{t_{d,\\alpha }},A_{m,n,K^{\\prime }}^c\\cap \\widetilde{A}_{m,n,K^{\\prime }}^c\\right),$ where the last step follows from step III.","Observe that the left hand side of (REF ) is the same as (REF ).","Therefore, it is now enough to bound the right hand side of (REF ).","In order to achieve the above task, let us define the following class of functions: $\\mathcal {C}^{\\Gamma ,L}(\\mathcal {Y}):=\\lbrace f:\\mathcal {Y}\\rightarrow \\mathbb {R}, f\\ \\mathrm {is}\\ \\mathrm {convex,} \\ \\Vert f\\Vert _{\\infty ,\\mathcal {Y}}\\le \\Gamma , \\ \\Vert f\\Vert _{L^2(\\nu )}\\le L\\rbrace .$ By setting $\\Gamma :=K^{\\prime }(\\log {m})^{1/\\alpha }$ and $L:=K^{\\prime }$ , (REF ) yields the following conclusion: $&\\;\\;\\;\\limsup _{M\\rightarrow \\infty }\\limsup \\limits _{m,n\\rightarrow \\infty }\\mathbb {P}\\left(\\Bigg |\\int \\Psi _{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n}^*(u)\\,d(\\widetilde{\\nu }_n-\\nu )\\Bigg |\\ge Mr_d^{(n,n)}(\\log {(1+m)})^{t_{d,\\alpha }}\\right)\\\\ & \\le \\limsup _{M\\rightarrow \\infty }\\limsup _{m,n\\rightarrow \\infty }\\mathbb {P}\\left(\\sup _{f\\in \\mathcal {C}^{\\Gamma ,L}(\\mathcal {Y})}\\Bigg |\\int f\\,d(\\widetilde{\\nu }_n-\\nu )\\Bigg |\\ge Mr_d^{(n,n)}(\\log {(1+m)})^{t_{d,\\alpha }}\\right).$ By an application of Markov's inequality, it thus suffices to show that: $\\mathbb {E}\\left[\\sup _{f\\in \\mathcal {C}^{\\Gamma ,L}(\\mathcal {Y})}\\Bigg |\\int f\\,d(\\widetilde{\\nu }_n-\\nu )\\Bigg |\\right]=\\mathcal {O}\\left(r_d^{(n,n)}(\\log {(1+m)})^{t_{d,\\alpha }}\\right).$ In order to bound (REF ), we will use some standard empirical process techniques.","In particular, by using [136], the following bound holds: $&\\;\\;\\;\\mathbb {E}\\left[\\sup _{f\\in \\mathcal {C}^{\\Gamma ,L}(\\mathcal {Y})}\\Bigg |\\int f\\,d(\\widetilde{\\nu }_n-\\nu )\\Bigg |\\right]\\nonumber \\\\ &\\le D\\inf \\left\\lbrace a\\ge \\frac{\\Gamma }{\\sqrt{n}}:a\\ge \\frac{D}{\\sqrt{n}}\\int _a^{\\Gamma }\\sqrt{\\log N_{[]}(\\varepsilon ,\\mathcal {C}^{\\Gamma ,L}(\\mathcal {Y}),L^2(\\nu ))}\\,d\\varepsilon \\right\\rbrace ,$ for some positive constant $D>0$ , where $N_{[]}(\\varepsilon ,\\mathcal {C}^{\\Gamma ,L}(\\mathcal {Y}),L^2(\\nu ))$ is the $\\varepsilon $ -bracketing number of the class of functions $\\mathcal {C}^{\\Gamma ,L}(\\mathcal {Y})$ with respect to the $L^2(\\nu )$ norm.","Note that by [91], we have: $\\log N_{[]}(\\varepsilon ,\\mathcal {C}^{\\Gamma ,L}(\\mathcal {Y}),L^2(\\nu ))\\le \\gamma _d\\left(\\log {\\frac{\\Gamma }{\\varepsilon }}\\right)^{d+1}\\left(\\frac{L}{\\varepsilon }\\right)^{d/2}$ for some $\\gamma _d>0$ depending only on fand the diameter of $\\mathcal {Y}$ .","We will now bound the right hand side of (REF ).","Also we will use $D_d$ to denote changing constants which can depend on $d$ .","When $d=1,2,3$: Choose $a=D_d\\frac{(\\log {n})^{\\frac{1}{\\alpha }\\vee \\frac{2\\alpha +2d\\alpha -d+4}{4\\alpha }}}{\\sqrt{n}}$ .","Observe that: $\\frac{1}{\\sqrt{n}}\\int _a^{\\Gamma }\\sqrt{\\log N_{[]}(\\varepsilon ,\\mathcal {C}^{\\Gamma ,L}(\\mathcal {Y}),L^2(\\nu ))}\\,d\\varepsilon &\\le \\frac{(\\log {n})^{(d+1)/2}}{\\sqrt{n}}\\cdot \\left[\\frac{\\varepsilon ^{1-d/4}}{1-d/4}\\right]_{0}^{\\Gamma }\\\\ &\\lesssim \\frac{(\\log {n})^{(4-d)/(4\\alpha )}\\times (\\log {n})^{(d+1)/2}}{\\sqrt{n}}\\lesssim a.$ When $d=4$: Choose $a=D_d\\frac{(\\log {n})^{\\frac{1}{\\alpha }\\vee \\frac{7}{2}}}{\\sqrt{n}}$ .","Observe that: $\\frac{1}{\\sqrt{n}}\\int _a^{\\Gamma }\\sqrt{\\log N_{[]}(\\varepsilon ,\\mathcal {C}^{\\Gamma ,L}(\\mathcal {Y}),L^2(\\nu ))}\\,d\\varepsilon &\\le \\frac{(\\log {n})^{5/2}}{\\sqrt{n}}\\cdot \\left[\\log {\\varepsilon }\\right]_{D_d\\Gamma /\\sqrt{n}}^{\\Gamma }\\\\ &\\lesssim \\frac{(\\log {n})^{(7/2)}}{\\sqrt{n}}\\lesssim a.$ When $d>4$: Choose $a=D_d\\frac{(\\log {n})^{2(1+d^{-1})}}{n^{2/d}}$ .","Observe that: $\\frac{1}{\\sqrt{n}}\\int _a^{\\Gamma _0}\\sqrt{\\log N_{[]}(\\varepsilon ,\\mathcal {C}^{\\Gamma ,L}(\\mathcal {Y}),L^2(\\nu ))}\\,d\\varepsilon &\\le \\frac{(\\log {n})^{(d+1)/2}}{\\sqrt{n}}\\cdot \\left[\\frac{\\varepsilon ^{1-d/4}}{1-d/4}\\right]_{a}^{\\Gamma }\\\\ &\\lesssim \\frac{a^{1-d/4}(\\log {n})^{(d+1)/2}}{\\sqrt{n}}\\lesssim a.$ This completes the proof after applying the same technique on the other 3 terms on the right hand side of (REF ).","[Proof of cor:fsam] First observe that $\\mathbb {E}\\left[\\int \\varphi _0^*\\,d\\widehat{\\nu }_n\\right]=\\mathbb {E}\\left[\\int \\varphi _0^*\\,d\\overline{\\nu }_m\\right]=\\int \\varphi _0^*\\,d\\nu .$ Using the above observation and the same approach used as in the proof of thm:nsmooth, we will only focus on bounding $\\mathbb {E}\\bigg |\\int \\Psi _{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n}^* \\,d(\\widetilde{\\nu }_n-\\nu )\\bigg |.$ The general strategy to bound the term in (REF ) is derived from some intermediate steps in the proofs of [33].","We still present a sketch here for completeness.","By the same argument as in the proof of thm:nsmooth and using the fact that there exists fixed $R>0$ such that $\\max _{1\\le i\\le m}\\Vert X_i\\Vert \\le R$ , we have $\\Psi _{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n}^*(\\cdot )$ is a convex and $R$ -Lipschitz function on $\\mathcal {Y}$ .","This observation implies: $\\mathbb {E}\\bigg |\\int \\Psi _{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n}^* \\,d(\\widetilde{\\nu }_n-\\overline{\\nu }_m)\\bigg |\\le \\mathbb {E}\\left[\\sup _{\\psi \\in \\mathcal {F}_R(\\mathcal {Y})}\\left|\\int \\psi \\,d(\\widehat{\\nu }_n-\\nu )\\right|\\right]$ where $\\mathcal {F}_R(\\mathcal {Y})$ is the set of convex and $R$ -Lipschitz functions on $\\mathcal {Y}$ .","By [141], we then have: $\\mathbb {E}\\left[\\sup _{\\psi \\in \\mathcal {F}_R(\\mathcal {Y})}\\left|\\int \\psi \\,d(\\widehat{\\nu }_n-\\nu )\\right|\\right]\\lesssim \\inf _{\\delta >0}\\left(\\delta +n^{-1/2}\\int _{\\delta }^{R^2}\\sqrt{\\log {\\mathcal {N}_{\\infty }(\\mathcal {F}_R(\\mathcal {Y}),\\varepsilon )}}\\,d\\varepsilon \\right),$ where $\\mathcal {N}_{\\infty }(\\mathcal {F}_R(\\mathcal {Y}),\\varepsilon )$ is the $\\varepsilon $ -covering number of the set $\\mathcal {F}_R(\\mathcal {Y})$ with respect to the uniform metric.","By using [74] (also see [22]), there exists constants $C_1,C_2>0$ such that whenever $\\varepsilon /R^2\\le C_1$ , then $\\log {\\mathcal {N}_{\\infty }(\\mathcal {F}_R(\\mathcal {Y}),\\varepsilon )}\\le C_2(u/R^2)^{-d/2}$ .","By using this bound in (REF ), we get: $\\mathbb {E}\\left[\\sup _{\\psi \\in \\mathcal {F}_R(\\mathcal {Y})}\\left|\\int \\psi \\,d(\\widehat{\\nu }_n-\\nu )\\right|\\right]\\lesssim \\inf _{\\delta >0}\\left(\\delta +n^{-1/2}\\int _{\\delta }^1 \\varepsilon ^{-d/4}\\,d\\varepsilon \\right).$ Setting $\\delta =0$ for $d<4$ and $\\delta =n^{-2/d}$ for $d\\ge 4$ in (REF ), followed by a direct application of (REF ), we have: $\\mathbb {E}\\bigg |\\int \\Psi _{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n}^* \\,d(\\widetilde{\\nu }_n-\\overline{\\nu }_m)\\bigg |\\lesssim r_d^{(n,n)}.$ This completes the proof.","[Proof of thm:smoothwav] For this proof, we will use an intermediate step in the proof of thm:newubd, which is (REF ), that can alternatively be written as: $\\mathbb {E}\\left[\\int \\Vert \\widetilde{T}_{m,n}^{\\gamma }(x)-T_0(x)\\Vert ^2\\,d\\widetilde{\\mu }_m(x)\\right]\\lesssim \\mathbb {E}|W_2^2(\\widetilde{\\mu }_m,\\widetilde{\\nu }_n)-W_2^2(\\mu ,\\nu )|+\\mathbb {E}\\left|\\int h(y)\\,d(\\widetilde{\\nu }_n-\\overline{\\nu }_m)(y)\\right|$ where $h(y):=\\varphi _0^*(y)-(1/2)\\Vert y\\Vert ^2$ and $C>0$ is some constant.","As $\\mathcal {X}$ and $\\mathcal {Y}$ are compact sets, the function $h(\\cdot )$ is Lipschitz.","Therefore, $\\mathbb {E}\\left|\\int h(y)\\,d(\\widetilde{\\nu }_n-\\nu )(y)\\right|\\lesssim W_1(\\widetilde{\\nu }_n,\\nu )\\le W_2(\\widetilde{\\nu }_n,\\nu ).$ Further, as $T_0(\\cdot )$ is also Lipschitz, we further have: $\\mathbb {E}\\left|\\int h(y)\\,d(\\overline{\\nu }_m-\\nu )(y)\\right|\\lesssim W_1(T_0\\#\\widetilde{\\mu }_m,T_0\\#\\mu )\\lesssim W_1(\\widetilde{\\mu }_m,\\mu )\\le W_2(\\widetilde{\\mu }_m,\\mu ).$ Finally, by the triangle inequality, we also have: $\\mathbb {E}|W_2^2(\\widetilde{\\mu }_m,\\widetilde{\\nu }_n)-W_2^2(\\mu ,\\nu )|&\\lesssim \\mathbb {E}|W_2(\\widetilde{\\mu }_m,\\widetilde{\\nu }_n)-W_2(\\widetilde{\\mu }_m,\\nu )|+\\mathbb {E}|W_2(\\widetilde{\\mu }_m,\\nu )-W_2(\\mu ,\\nu )|\\\\ &\\le \\mathbb {E}W_2(\\widetilde{\\mu }_m,\\mu )+\\mathbb {E}W_2(\\widetilde{\\nu }_n,\\nu ).$ Combining the three displays above and plugging them back in (REF ), we get: $\\mathbb {E}\\left[\\int \\Vert \\widetilde{T}_{m,n}^{\\gamma }(x)-T_0(x)\\Vert ^2\\,d\\widetilde{\\mu }_m(x)\\right]\\lesssim \\mathbb {E}W_2(\\widetilde{\\mu }_m,\\mu )+\\mathbb {E}W_2(\\widetilde{\\nu }_n,\\nu ).$ The conclusion then follows from [143].","[Proof of thm:smooth] Part 1.","By the same arguments (see e.g., (REF )) as used in the proof of thm:nsmooth, it suffices to show that $\\mathbb {E}\\left|\\int \\Psi _{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n}^*(u)(\\widetilde{f}_{\\nu }^{M^{\\prime }}(u)-f_{\\nu }(u)) \\,du\\right|\\lesssim r_{d,s}^{(n,n)}$ for some $M^{\\prime }>0$ .","The general structure of the proof is similar to that of thm:nsmooth.","The crucial observation is that $\\widetilde{f}_{\\mu }^{M^{\\prime }}(\\cdot )$ and $\\widetilde{f}_{\\nu }^{M^{\\prime }}(\\cdot )$ are elements of $C^s(\\mathcal {X};TM)$ and $C^s(\\mathcal {Y};TM)$ respectively, for any $M^{\\prime }>0$ .","Note that, by Caffarelli regularity theory; see [80], there exists $M^{\\prime }>0$ such that $\\Vert \\Psi _{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n}^*(\\cdot )\\Vert _{C^{s+2}(\\mathcal {Y})}\\le M^{\\prime }.$ Next, let us define the following class of functions: $\\mathcal {G}_t^L(\\mathcal {Y}):=\\lbrace g:\\mathcal {Y}\\rightarrow \\mathbb {R},\\ g(\\cdot )\\ \\mbox{is}\\ \\mbox{convex},\\ \\Vert g\\Vert _{C^t(\\mathcal {Y})}\\le L\\rbrace .$ Observe that $\\mathbb {E}\\left|\\int \\Psi _{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n}^*(u)(\\widetilde{f}_{\\nu }^{M^{\\prime }}(u)-f_{\\nu }(u)) \\,du\\right|&\\le \\mathbb {E}\\sup _{g\\in \\mathcal {G}_{s+2}^{M^{\\prime }}(\\mathcal {Y})}\\left|\\int g(u)(\\widetilde{f}_{\\nu }^{M^{\\prime }}(u)-f_{\\nu }(u)) \\,du\\right|\\nonumber \\\\ &\\le 2\\mathbb {E}\\sup _{g\\in \\mathcal {G}_{s+2}^{M^{\\prime }}(\\mathcal {Y})}\\left|\\int g(u)(\\widehat{f}_{\\nu }(u)-f_{\\nu }(u)) \\,du\\right|+r_{d,s}^{(n,n)}$ where the last line follows from (REF ).","Set $K_{d,h_n}(\\cdot ):=h_n^{-d}K_d(\\cdot /h_n)$ .","Following the same decomposition as in [115], we write: $&\\;\\;\\;\\;\\mathbb {E}\\sup _{g\\in \\mathcal {G}_{s+2}^{M^{\\prime }}(\\mathcal {Y})}\\left|\\int g(u)(\\widehat{f}_{\\nu }(u)-f_{\\nu }(u)) \\,du\\right|\\nonumber \\\\ &=\\mathbb {E}\\sup _{g\\in \\mathcal {G}_{s+2}^{M^{\\prime }}(\\mathcal {Y})}\\left|\\int g(u+u^{\\prime })K_{d,h_n}(u^{\\prime })\\,d\\widehat{\\nu }_n(u)\\,du^{\\prime }-\\int g(u)f_{\\nu }(u)\\,du\\right|\\nonumber \\\\ &\\le \\mathbb {E}\\sup _{g\\in \\mathcal {G}_{s+2}^{M^{\\prime }}(\\mathcal {Y})}\\left|\\int g(u+u^{\\prime })K_{d,h_n}(u^{\\prime })\\,d(\\widehat{\\nu }_n-\\nu )(u)\\,du^{\\prime }\\right|\\nonumber \\\\ &\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;+\\sup _{g\\in \\mathcal {G}_{s+2}^{M^{\\prime }}(\\mathcal {Y})}\\left|\\int g(u+u^{\\prime })K_{d,h_n}(u^{\\prime })f_{\\nu }(u)\\,du\\,du^{\\prime }-\\int g(u)f_{\\nu }(u)\\,du\\right|.$ We will now bound the two terms on the right hand side of (REF ).","For the first term, define $\\overline{g}_n(u):=\\int g(u+u^{\\prime })K_{d,h_n}(u)\\,du^{\\prime }.$ If $g\\in \\mathcal {G}_t^L(\\mathcal {Y}^o)$ , then by [53] and using as:kernel, we have $\\overline{g}_n\\in \\mathcal {G}_{s+2}^{cM^{\\prime }}(\\mathcal {Y}^o)$ , for some constant $c>0$ (depending on the constants involved in as:kernel and the diameter of $\\mathcal {Y}$ ).","Combining these observations with (REF ), we get: $&\\;\\;\\;\\;\\mathbb {E}\\sup _{g\\in \\mathcal {G}_{s+2}^{M^{\\prime }}(\\mathcal {Y})}\\left|\\int g(u+u^{\\prime })K_{d,h_n}(u^{\\prime })\\,d(\\widehat{\\nu }_n-\\nu )(u)\\,du^{\\prime }\\right|\\nonumber \\\\ &\\le \\mathbb {E}\\sup _{g\\in \\mathcal {G}_{s+2}^{cM^{\\prime }}(\\mathcal {Y})}\\left|\\int g(u)\\,d(\\widehat{\\nu }_n-\\nu )(u)\\right|\\nonumber \\\\ &\\le D\\inf \\left\\lbrace a\\ge \\frac{cM^{\\prime }}{\\sqrt{n}}:a\\ge \\frac{D}{\\sqrt{n}}\\int _a^{cM^{\\prime }}\\sqrt{\\log N_{[]}(\\varepsilon ,\\mathcal {G}_{s+2}^{cM^{\\prime }}(\\mathcal {Y}),L^2(\\nu ))}\\,d\\varepsilon \\right\\rbrace ,$ for some positive constant $D>0$ , where $N_{[]}(\\varepsilon ,\\mathcal {G}_{s+2}^{cM^{\\prime }}(\\mathcal {Y}),L^2(\\nu ))$ is the $\\varepsilon $ -bracketing entropy of the class of functions $\\mathcal {G}_{s+2}^{cM^{\\prime }}(\\mathcal {Y})$ with respect to the $L^2(\\nu )$ norm.","The last line follows from standard empirical process theory as used in the proof of thm:nsmooth; see (REF ).","Note that by [138], we have: $\\log N_{[]}(\\varepsilon ,\\mathcal {G}_{s+2}^{cL}(\\mathcal {Y}^o),L^2(\\nu ))\\le \\gamma _d\\left(\\frac{1}{\\varepsilon }\\right)^{d/(s+2)}$ for some $\\gamma _d>0$ depending only on dimension and the diameter of $\\mathcal {Y}$ .","We now plug-in the above bound into (REF ).","By using $D_d$ to denote constants that change with $d$ and choosing $a=D_d n^{-1/2}$ for $2(s+2)>d$ , $a=D_d n^{-1/2}\\log {(1+n)}$ for $2(s+2)=d$ and $a=D_d n^{-(s+2)/d}$ for $2(s+2)\\frac{1}{2}$ .","This implies $n^{-\\frac{2s+2}{d+2s}}(\\log {n})^{2s+2}\\lesssim n^{-\\frac{1}{2}}=r_{d,s}^{(n,n)}$ .","When $d=2(s+2)$ : In this case $n^{-\\frac{2s+2}{d+2s}}(\\log {n})^{2s+2}\\lesssim n^{-\\frac{1}{2}}(\\log {n})^{2s+2}\\lesssim r_{d,s}^{(n,n)}$ .","When $d>2(s+2)$ : Note that $\\frac{2s+2}{d+2s}>\\frac{s+2}{d}\\Leftrightarrow 2ds+2d>sd+2s^2+2d+4s\\Leftrightarrow d>2(s+2).$ Therefore, once again $n^{-\\frac{2s+2}{d+2s}}(\\log {n})^{2s+2}\\lesssim n^{-\\frac{s+2}{d}}=r_{d,s}^{(n,n)}$ .","Therefore, combining the above observations, we have: $\\left|\\int g(u+u^{\\prime })K_{d,h_n}(u^{\\prime })f_{\\nu }(u)\\,du\\,du^{\\prime }-\\int g(u)f_{\\nu }(u)\\,du\\right|\\lesssim r_{d,s}^{(n,n)}.$ Combining the above display with (REF ) establishes (REF ).","Part 2.","This proof uses ideas from [76], [112] and [5].","First recall all the notation introduced in def:holder.","Next, we will prove the following sequence of displays: $\\limsup \\limits _{m,n\\rightarrow \\infty }\\max _{k\\le s}\\max _{|\\mathfrak {m}|=k}\\Vert \\partial ^{\\mathfrak {m}}\\mathbb {E}\\widehat{f}_{\\mu }\\Vert _{L^{\\infty }(\\widetilde{\\mathcal {X}})}\\le (T-1)M,$ $\\limsup _{m,n\\rightarrow \\infty }\\ \\Vert \\mathbb {E}\\widehat{f}_{\\mu }-f_{\\mu }\\Vert _{L^{\\infty }(\\widetilde{\\mathcal {X}})}=0$ $\\limsup _{m,n\\rightarrow \\infty }\\mathbb {P}\\left(\\Vert \\widehat{f}_{\\mu }-\\mathbb {E}\\widehat{f}_{\\mu }\\Vert _{C^{s}(\\widetilde{\\mathcal {X}})}\\ge \\varepsilon \\right)=0,$ for any arbitrary $\\varepsilon >0$ and $L^{\\infty }(\\mathcal {X})$ denotes the uniform norm on $\\mathcal {X}$ .","Clearly, (REF ), (REF ), and (REF ) together yield part 1 of the theorem.","Proof of (REF ).","Observe that $\\mathbb {E}\\widehat{f}_{\\mu }(x)=\\frac{1}{h_m^d}\\mathbb {E}K_d\\left(\\frac{x-X_1}{h_m}\\right)=\\frac{1}{h_m^d}\\int K_d\\left(\\frac{x-z}{h_m}\\right)f_{\\mu }(z)\\,dz.$ Since the maximums taken in (REF ) are over finite sets, it suffices to show that for any fixed $\\mathfrak {m}$ with $|\\mathfrak {m}|\\le s$ , we have: $\\sup _{x\\in \\widetilde{\\mathcal {X}}}|\\partial ^{\\mathfrak {m}}\\mathbb {E}\\widehat{f}_{\\mu }(x)|=\\sup _{x\\in \\mathcal {X}}\\bigg |\\frac{1}{h_n^d}\\int K_d\\left(\\frac{z}{h_n}\\right)\\partial ^{\\mathfrak {m}} f(x+z)\\,dz\\Bigg |\\le (T-1)M.$ Here the first equality in the above display follows from (REF ) and Fubini's Theorem.","Here $\\partial ^{\\mathfrak {m}}f_{\\mu }(\\cdot )$ is defined in the weak sense, i.e., it is defined naturally in the interior of the support of $f_{\\mu }(\\cdot )$ , denoted by $\\mathcal {X}$ ; it is set to be 0 outside $\\mathcal {X}$ and defined arbitrarily on the boundary of $\\mathcal {X}$ .","Note that the definition on the boundary doesn't matter as we are integrating with respect to the Lebesgue measure and the boundary of $\\mathcal {X}$ has Lebesgue measure 0.","Next note that, by (REF ), we have: $\\sup _{x\\in \\widetilde{\\mathcal {X}}}|\\partial ^{\\mathfrak {m}}\\mathbb {E}\\widehat{f}_{\\mu }(x)|\\le \\Vert f_{\\mu }\\Vert _{C^s(\\mathcal {X})}h_m^{-d}\\int |K_d(z/h_m)|\\,dz\\le (T-1)\\Vert f_{\\mu }\\Vert _{C^s(\\mathcal {X})}.$ This establishes (REF ).","Proof of (REF ).","First note that, as $\\widetilde{\\mathcal {X}}$ is a compact subset of $\\mathcal {X}^o$ , there exists $\\delta >0$ such that $\\widetilde{\\mathcal {X}}_{\\delta ^{\\prime }}:=\\lbrace x+z:\\ \\Vert z\\Vert \\le \\delta ;,\\ x\\in \\widetilde{\\mathcal {X}}\\rbrace \\subseteq \\mathcal {X}^o\\qquad \\forall \\ 0<\\delta ^{\\prime }\\le \\delta .$ Clearly, $\\widetilde{\\mathcal {X}}_{\\delta ^{\\prime }}$ is compact for all $\\delta ^{\\prime }>0$ .","Fix an arbitrary $\\delta ^{\\prime }\\le \\delta $ .","By using (REF ) and a change of variable formula, we have: $\\Vert \\mathbb {E}\\widehat{f}_{\\mu }-f_{\\mu }\\Vert _{L^{\\infty }(\\widetilde{\\mathcal {X}})}&=\\sup _{x\\in \\widetilde{\\mathcal {X}}}\\bigg |\\frac{1}{h_m^d}\\int K_d\\left(\\frac{z}{h_m}\\right)(f(x+z)-f(x))\\,dz\\bigg | \\\\& \\le (T-1)\\sup _{x\\in \\widetilde{\\mathcal {X}}}\\sup _{\\Vert z\\Vert \\le \\delta ^{\\prime }}|f(x+z)-f(x)|+2M\\int _{\\Vert z\\Vert >\\delta ^{\\prime } h_m^{-1}} |K_d(z)|\\,dz\\\\ & \\le (T-1)M\\delta ^{\\prime }+2M\\left(\\frac{h_m}{\\delta ^{\\prime }}\\right)^{2s+2}\\int \\Vert z\\Vert ^{2s+2}|K_d(z)|\\,dz.$ Observe that as $m,n\\rightarrow \\infty $ , the second term on the right hand side of the above display converges to 0.","This implies $\\limsup \\limits _{m,n\\rightarrow \\infty }\\Vert \\mathbb {E}\\widehat{f}_{\\mu }-f_{\\mu }\\Vert _{L^{\\infty }(\\widetilde{\\mathcal {X}})}\\le (T-1)M\\delta ^{\\prime }.$ As $\\delta ^{\\prime }$ can be chosen arbitrarily small, this completes the proof of (REF ).","Proof of (REF ).","The main technical tool for this part is lem:emproc which we borrow from [5] (also see [99]).","The proof is very similar to [5].","Consider the following class of functions: $\\mathcal {G}=\\left\\lbrace g_x(z,h):\\ g_x(z,h)=\\partial ^{\\mathfrak {m}}K_d\\left(\\frac{(x-z)}{h}\\right),\\ x\\in \\widetilde{\\mathcal {X}},\\ |\\mathfrak {m}|\\le s \\right\\rbrace .$ Observe that $\\sup _{|\\mathfrak {m}|\\le s}\\sup _{x\\in \\widetilde{\\mathcal {X}}}\\sup _{h\\in (0,1)} h^{-d} \\mathbb {E}\\left[\\partial ^{\\mathfrak {m}}K_d\\left(\\frac{(x-z)}{h}\\right)\\right]^2\\le \\Vert K\\Vert _{C^s(\\mathbb {R}^d)}\\Vert f\\Vert _{C^s(\\mathcal {X})}\\sup _{|\\mathfrak {m}|\\le s}\\int |\\partial ^{\\mathfrak {m}} K_d(v)|\\,dv<\\infty .$ Further, by as:kernel, $\\partial ^{\\mathfrak {m}} K_d(\\cdot )$ is differentiable for each $|\\mathfrak {m}|\\le s$ .","Consequently $\\mathcal {G}$ is point wise measurable and of VC-type (see [49]; also see [138] for definitions of point wise differentiability and VC classes).","This verifies the assumptions of lem:emproc.","Observe that $\\frac{1}{n}\\sum _{i=1}^m \\partial ^{\\mathfrak {m}} K\\left(\\frac{x-X_i}{h_m}\\right)=h_m^{d+|\\mathfrak {m}|}\\partial ^{\\mathfrak {m}} \\widehat{f}_{\\mu }(x),\\qquad \\mathbb {E}\\left[\\partial ^{\\mathfrak {m}}K\\left(\\frac{x-X}{h_m}\\right)\\right]=h_m^{d+|\\mathfrak {m}|}\\mathbb {E}\\left[\\partial ^{\\mathfrak {m}}\\widehat{f}_{\\mu }(x)\\right].$ A direct application of lem:emproc for all $|\\mathfrak {m}|\\le s$ , then implies $\\sup _{x\\in \\widetilde{\\mathcal {X}}}\\sqrt{\\frac{m}{h_m^d\\log {m}}}\\cdot h_m^{s+d} \\Vert \\widehat{f}_{\\mu }-\\mathbb {E}\\widehat{f}_{\\mu }\\Vert _{C^s(\\widetilde{\\mathcal {X}})}=O_p(1).$ Using the observation that $mh_m^{d+2s}/\\log {m}\\rightarrow 0$ as $m\\rightarrow \\infty $ then completes the proof.","[Proof of prop:wassrate] As $\\mu \\ne \\nu $ , we have $W_2(\\mu ,\\nu )>0$ .","Therefore, $|W_2(\\widetilde{\\mu }_m,\\widetilde{\\nu }_n)-W_2(\\mu ,\\nu )|=\\frac{W_2^2(\\widetilde{\\mu }_m,\\widetilde{\\nu }_n)-W_2^2(\\mu ,\\nu )|}{W_2(\\widetilde{\\mu }_m,\\widetilde{\\nu }_n)+W_2(\\mu ,\\nu )}\\le \\frac{W_2^2(\\widetilde{\\mu }_m,\\widetilde{\\nu }_n)-W_2^2(\\mu ,\\nu )|}{W_2(\\mu ,\\nu )}.$ The conclusion then follows from thm:smooth.","[Proof of thm:dismooth] Recall that $\\widetilde{\\mu }_m$ and $\\widetilde{\\nu }_n$ are defined as the empirical distributions induced by $M=n^{\\frac{s+2}{2}}$ random samples drawn from $\\widehat{f}_{\\mu }$ and $\\widehat{f}_{\\nu }$ respectively, where $\\widehat{f}_{\\mu }$ , $\\widehat{f}_{\\nu }$ are the kernel density estimates as presented in (REF ).","Let us write $\\mu _{h_n}$ and $\\nu _{h_n}$ for the probability measure induced by the kernel density estimates $\\widehat{f}_{\\mu }$ and $\\widehat{f}_{\\nu }$ respectively.","Once again, by using thm:nsmooth, (REF ), it suffices to prove the following: $\\mathbb {E}\\left|W_2^2(\\widetilde{\\mu }_m,\\widetilde{\\nu }_n)-W_2^2(\\mu ,\\nu )\\right|.$ Next note that by the triangle inequality, (REF ) can be bounded above by: $\\mathbb {E}\\big |W_2^2(\\widetilde{\\mu }_m,\\nu _{h_n})&-W_2^2(\\widetilde{\\mu }_m,\\widetilde{\\nu }_n)\\big |+\\mathbb {E}\\big |W_2^2(\\widetilde{\\mu }_m,\\nu _{h_n})-W_2^2(\\mu _{h_n},\\nu _{h_n})\\big |\\nonumber \\\\&+\\mathbb {E}\\big |W_2^2(\\mu _{h_n},\\nu _{h_n})-W_2^2(\\widetilde{\\mu }_m,\\widetilde{\\nu }_n)\\big |.$ Next note that, by thm:smooth, we have: $\\mathbb {E}\\left|W_2^2(\\mu _{h_n},\\nu _{h_n})-W_2^2(\\widetilde{\\mu }_m,\\widetilde{\\nu }_n)\\right|\\lesssim r_{d,s}^{(n,n)}.$ Next we show that $\\mathbb {E}\\left|W_2^2(\\widetilde{\\mu }_m,\\nu _{h_n})-W_2^2(\\mu _{h_n},\\nu _{h_n})\\right|\\lesssim r_{d,s}^{(n,n)}.$ The other term in (REF ) can be bounded similarly.","Note that, conditioned on $X_1,\\ldots ,X_n,Y_1,\\ldots ,Y_n$ , $\\mu _{h_n}$ and $\\nu _{h_n}$ are non-random measures and $\\widetilde{\\mu }_m$ and $\\widetilde{\\nu }_n$ are the empirical distributions on $M=n^{\\frac{s+2}{2}}$ random samples from the measures $\\mu _{h_n}$ and $\\nu _{h_n}$ , respectively.","Therefore, conditioned on $X_1,\\ldots ,X_n,Y_1,\\ldots ,Y_n$ (which have fixed compact supports), we can invoke cor:fsam to get: $\\mathbb {E}\\left|W_2^2(\\widetilde{\\mu }_m,\\nu _{h_n})-W_2^2(\\mu _{h_n},\\nu _{h_n})\\right| \\lesssim r_{d}^{(M,M)},$ with $M=n^{\\frac{s+2}{2}}$ .","Recall that: $r_d^{(M,M)}={\\left\\lbrace \\begin{array}{ll} n^{-\\frac{s+2}{4}} & \\mbox{if}\\ d\\le 3\\\\ n^{-\\frac{s+2}{4}}\\log {(1+n)} & \\mbox{if}\\ d=4\\\\ n^{-\\frac{s+2}{d}} & \\mbox{if}\\ d>4\\end{array}\\right.","}.$ It therefore only remains to compare $r_d^{(M,M)}$ and $r_{d,s}^{(n,n)}$ .","Case 1: $d\\le 2(s+2)$ .","In this case, if $d=1,2,3$ , then $r_d^{(M,M)}=n^{-\\frac{s+2}{4}}=n^{-\\frac{1}{2}}\\times n^{-\\frac{s}{4}}\\lesssim n^{-\\frac{1}{2}}$ .","If $d=4$ , then $r_d^{(M,M)}= n^{-\\frac{s+2}{4}}\\log {(1+n)}=n^{-\\frac{1}{2}}\\times \\left(n^{-\\frac{s}{4}}\\log {n}\\right)\\lesssim n^{-\\frac{1}{2}}$ .","If $d>4$ , then $r_d^{(M,M)}=n^{-\\frac{s+2}{d}}\\lesssim n^{-\\frac{1}{2}}$ as $\\frac{s+2}{d}\\ge \\frac{1}{2}$ .","Therefore, in all the cses, $r_d^{(M,M)}\\lesssim n^{-\\frac{1}{2}}=r_{d,s}^{(n,n)}$ for $d\\le 2(s+2)$ .","Case 2: $d>2(s+2)$ .","As $s>0$ , then $d>4$ .","In this case, once again $r_d^{(M,M)}=n^{-\\frac{s+2}{d}}=r_{d,s}^{(n,n)}$ .","This establishes (REF ) and completes the proof."],["Proofs from sec:App","[Proof of thm:barate] First define the following measure: $\\rho _0^{\\mathrm {OR}}:=\\left(\\frac{1}{2}\\mbox{Id}+\\frac{1}{2}T_0\\right)\\#\\widetilde{\\mu }_m.$ Fix any $\\gamma \\in \\widetilde{\\Gamma }_{\\mathrm {min}}$ .","By applying the triangle inequality followed by a power mean inequality, we have: $\\sup _{\\gamma \\in \\widetilde{\\Gamma }_{\\mathrm {min}}}W_2^2\\big (\\widehat{\\rho }_0^{\\gamma },\\rho _0\\big )\\lesssim W_2^2\\big (\\rho _0^{\\mathrm {OR}},\\rho _0\\big )+\\sup _{\\gamma \\in \\widetilde{\\Gamma }_{\\mathrm {min}}} W_2^2\\big (\\widehat{\\rho }_0^{\\gamma },\\rho _0^{\\mathrm {OR}}\\big ).$ Next observe that $\\rho _0^{\\mathrm {OR}}$ is the empirical distribution corresponding to $m$ random samples drawn according to $\\rho _0$ .","Therefore, by using [55], we get: $ W_2^2\\big (\\rho _0^{\\mathrm {OR}},\\rho _0\\big )\\lesssim r_d^{(m,m)}.$ Next we will bound the second term on the right hand side of (REF ).","Towards this direction, recall the definition of $\\Pi (\\cdot ,\\cdot )$ from sec:setting.","Consider the following coupling: $\\pi _0^{\\gamma }:=\\left(\\frac{1}{2}\\mbox{Id}+\\frac{1}{2}\\widetilde{T}_{m,n}^{\\gamma },\\frac{1}{2}\\mbox{Id}+\\frac{1}{2}T_0\\right)\\#\\widetilde{\\mu }_m.$ Observe that $\\pi _0^{\\gamma }\\in \\Pi \\big (\\widehat{\\rho }_0^{\\gamma },\\rho _0^{\\mathrm {OR}}\\big )$ .","By plugging the coupling $\\pi _0^{\\gamma }$ into the definition of 2-Wasserstein distance in (REF ), we further get: $\\sup _{\\gamma \\in \\widetilde{\\Gamma }_{\\mathrm {min}}} W_2^2\\big (\\widehat{\\rho }_0^{\\gamma },\\rho _0^{\\mathrm {OR}}\\big )&\\le \\sup _{\\gamma \\in \\widetilde{\\Gamma }_{\\mathrm {min}}} \\int \\Vert x-y\\Vert ^2\\,d\\pi _0^{\\gamma }(x,y)\\nonumber \\\\ &=\\sup _{\\gamma \\in \\widetilde{\\Gamma }_{\\mathrm {min}}} \\int \\Vert \\widetilde{T}_{m,n}^{\\gamma }(x)-T_0(x)\\Vert ^2\\,d\\widetilde{\\mu }_m(x)\\nonumber \\\\ &= O_p\\left(r^{(m,n)}_d\\times (\\log {(1+\\max \\lbrace m,n\\rbrace )})^{t_{d,\\alpha }}\\right)$ where the last inequality follows from thm:nsmooth.","Combining (REF ) and (REF ) with (REF ) completes the proof.","[Proof of thm:dethresh] Let $T_1^{(n)}(\\cdot )$ and $T_2^{(n)}(\\cdot )$ be the optimal transport maps from $\\mu ^{(n)}$ to $\\upsilon _1$ and $\\nu ^{(n)}$ to $\\upsilon _2$ .","Set $\\widehat{x}^{\\mathrm {OR}}_{ij}:=K_1(T_1^{(n)}(X_i),T_1^{(n)}(X_j)),\\qquad \\widehat{y}^{\\mathrm {OR}}_{ij}:=K_2(T_2^{(n)}(Y_i),T_2^{(n)}(Y_j))$ and define the oracle version of $\\widehat{\\mathrm {rHSIC}}$ as follows: $\\widehat{\\mathrm {rHSIC}}^{\\mathrm {OR}}:=\\underbrace{n^{-2}\\sum _{i,j}\\widehat{x}^{\\mathrm {OR}}_{ij}\\widehat{y}^{\\mathrm {OR}}_{ij}}_{\\widehat{A}_{n,1}^{\\mathrm {OR}}}+\\underbrace{n^{-4}\\sum _{i,j,r,s}\\widehat{x}^{\\mathrm {OR}}_{ij}\\widehat{y}^{\\mathrm {OR}}_{rs}}_{\\widehat{A}_{n,2}^{\\mathrm {OR}}}-2\\underbrace{n^{-3}\\sum _{i,j,r} \\widehat{x}^{\\mathrm {OR}}_{ij}\\widehat{y}^{\\mathrm {OR}}_{ir}}_{\\widehat{A}_{n,3}^{\\mathrm {OR}}}.$ The proof of thm:dethresh now proceeds using the following steps: Step I: We show that: $\\mathbb {E}\\big |\\widehat{\\mathrm {rHSIC}}^{\\mathrm {OR}}-\\mathrm {rHSIC}(\\pi ^{(n)}|\\mu ^{(n)}\\times \\nu ^{(n)})\\big |\\lesssim n^{-1/2},$ where $\\widehat{\\mathrm {rHSIC}}^{\\mathrm {OR}}(\\cdot |\\cdot )$ is defined in (REF ).","Step II: We prove that: $ \\mathbb {E}\\big |\\widehat{\\mathrm {rHSIC}}^{\\mathrm {OR}}-\\widehat{\\mathrm {rHSIC}}\\big |\\lesssim \\sqrt{r_d^{(n,n)}}.$ Step III: We combine steps I and II to prove thm:dethresh.","Let us begin with this step first.","Note that by using the triangle inequality, we have: $\\widehat{\\mathrm {rHSIC}}\\ge \\mathrm {rHSIC}(\\pi ^{(n)}|\\mu ^{(n)}\\times \\nu ^{(n)})-\\big |\\widehat{\\mathrm {rHSIC}}^{\\mathrm {OR}}-\\mathrm {rHSIC}\\big |-\\big |\\widehat{\\mathrm {rHSIC}}^{\\mathrm {OR}}-\\widehat{\\mathrm {rHSIC}}\\big |.$ Next observe that by steps I and II, $\\max \\bigg \\lbrace \\big |\\widehat{\\mathrm {rHSIC}}^{\\mathrm {OR}}-\\mathrm {rHSIC}\\big |,\\big |\\widehat{\\mathrm {rHSIC}}^{\\mathrm {OR}}-\\widehat{\\mathrm {rHSIC}}\\big |\\bigg \\rbrace =O_p\\big (\\sqrt{r_{d_1,d_2}^{(n,n)}}\\big ).$ Using the above display with (REF ) and the assumption $(r_{d_1,d_2}^{(n,n)})^{-1/2}\\mathrm {rHSIC}(\\pi ^{(n)}|\\mu ^{(n)}\\times \\nu ^{(n)})\\rightarrow \\infty $ , we have: $\\big (r_{d_1,d_2}^{(n,n)}\\big )^{-1/2}\\widehat{\\mathrm {rHSIC}}\\overset{P}{\\longrightarrow }\\infty .$ Therefore, as $n\\sqrt{r_{d_1,d_2}^{(n,n)}}\\rightarrow \\infty $ and $c_{n,\\alpha }=O(1)$ (see [44]), we have: $\\mathbb {E}\\phi _{n,\\alpha }=\\mathbb {P}(n\\times \\widehat{\\mathrm {rHSIC}}\\ge c_{n,\\alpha })\\rightarrow 1$ under $(r_{d_1,d_2}^{(n,n)})^{-1/2}\\mathrm {rHSIC}(\\pi ^{(n)}|\\mu ^{(n)}\\times \\nu ^{(n)})\\rightarrow \\infty $ .","This completes the proof.","It therefore remains to prove steps I and II.","For step I, let $(X_1^{\\prime },Y_1^{\\prime }),\\ldots ,(X_n^{\\prime },Y_n^{\\prime })\\overset{i.i.d.","}{\\sim }\\pi ^{(n)}$ .","Fix an arbitrary $1\\le j\\le n$ .","Let $\\widehat{A}_{n,1,j}^{\\mathrm {OR},^{\\prime }}$ be the same as $\\widehat{A}_{n,1}^{\\mathrm {OR}}$ except with $(X_j,Y_j)$ replaced by $(X_j^{\\prime },Y_j^{\\prime })$ .","It is easy to check by the compactness of supports of all distributions involved, that: $\\max \\limits _{1\\le j\\le n}\\big |\\widehat{A}_{n,1}^{\\mathrm {OR}}-\\widehat{A}_{n,1,j}^{\\mathrm {OR},^{\\prime }}\\big |\\lesssim n^{-1}.$ Therefore by using Mcdiarmid's inequality (see [20]), we have, for any $t>0$ , $\\mathbb {P}\\left(\\sqrt{n}(\\widehat{A}_{n,1}^{\\mathrm {OR}}-\\mathbb {E}\\widehat{A}_{n,1}^{\\mathrm {OR}})\\ge t\\right)\\le \\exp (-Ct^2)$ for some constant $C>0$ free of $n$ and $t$ .","Similar concentrations can be derived for $\\widehat{A}_{n,2}^{\\mathrm {OR}}$ and $\\widehat{A}_{n,3}^{\\mathrm {OR}}$ .","Combining these concentrations with the observation that $\\mathrm {rHSIC}(\\pi ^{(n)}|\\mu ^{(n)}\\times \\nu ^{(n)})=\\mathbb {E}\\widehat{A}_{n,1}^{\\mathrm {OR}}+\\mathbb {E}\\widehat{A}_{n,2}^{\\mathrm {OR}}-2\\mathbb {E}\\widehat{A}_{n,3}^{\\mathrm {OR}}$ completes the proof of step I.","We now move on to step II.","Recall the definition of $\\widehat{\\mathrm {rHSIC}}$ from (REF ) and write: $\\widehat{\\mathrm {rHSIC}}=\\underbrace{n^{-2}\\sum _{i,j}\\widehat{x}_{ij}\\widehat{y}_{ij}}_{\\widehat{A}_{n,1}}+\\underbrace{n^{-4}\\sum _{i,j,r,s}\\widehat{x}_{ij}\\widehat{y}_{rs}}_{\\widehat{A}_{n,2}}-2\\underbrace{n^{-3}\\sum _{i,j,r} \\widehat{x}_{ij}\\widehat{y}_{ir}}_{\\widehat{A}_{n,3}}.$ By the Lipschitzness of $K_1(\\cdot ,\\cdot )$ and $K_2(\\cdot ,\\cdot )$ , we have: $\\big |\\widehat{x}_{ij}-\\widehat{x}^{\\mathrm {OR}}_{ij}\\big |\\lesssim \\Vert \\widehat{T}_{1,n}(X_i)-T_1^{(n)}(X_i)\\Vert +\\Vert \\widehat{T}_{1,n}(X_j)-T_1^{(n)}(X_j)\\Vert ,$ $\\big |\\widehat{y}_{ij}-\\widehat{y}^{\\mathrm {OR}}_{ij}\\big |\\lesssim \\Vert \\widehat{T}_{2,n}(Y_i)-T_2^{(n)}(Y_i)\\Vert +\\Vert \\widehat{T}_{2,n}(Y_j)-T_2^{(n)}(Y_j)\\Vert .$ Therefore, by using the fact that the probability measures $\\upsilon _1$ and $\\upsilon _2$ are compactly supported, we get: $\\big |\\widehat{A}_{n,1}-\\widehat{A}_{n,1}^{\\mathrm {OR}}\\big |\\lesssim \\frac{1}{n}\\sum _{i=1}^n \\Vert \\widehat{T}_{1,n}(X_i)-T_1^{(n)}(X_i)\\Vert +\\frac{1}{n}\\sum _{j=1}^n \\Vert \\widehat{T}_{2,n}(Y_j)-T_2^{(n)}(Y_j)\\Vert .$ The same bound can similarly be verified for $|\\widehat{A}_{n,2}-\\widehat{A}_{n,2}^{\\mathrm {OR}}|$ and $|\\widehat{A}_{n,3}-\\widehat{A}_{n,3}^{\\mathrm {OR}}|$ .","Combining these observations, we have: $\\big |\\widehat{\\mathrm {rHSIC}}-\\widehat{\\mathrm {rHSIC}}^{\\mathrm {OR}}\\big |&\\lesssim \\frac{1}{n}\\sum _{i=1}^n \\Vert \\widehat{T}_{1,n}(X_i)-T_1^{(n)}(X_i)\\Vert +\\frac{1}{n}\\sum _{j=1}^n \\Vert \\widehat{T}_{2,n}(Y_j)-T_2^{(n)}(Y_j)\\Vert \\nonumber \\\\ &\\le \\sqrt{\\frac{1}{n}\\sum _{i=1}^n \\Vert \\widehat{T}_{1,n}(X_i)-T_1^{(n)}(X_i)\\Vert ^2}+\\sqrt{\\frac{1}{n}\\sum _{j=1}^n \\Vert \\widehat{T}_{2,n}(Y_j)-T_2^{(n)}(Y_j)\\Vert ^2}.$ Step II then follows by invoking cor:fsam."],["Auxiliary definitions and results","[Subdifferential set and subgradient] Given a convex function $f:\\mathbb {R}^d\\rightarrow \\mathbb {R}\\cup \\lbrace \\infty \\rbrace $ , we define the subdifferential set of $f(\\cdot )$ at $x\\in \\mbox{dom}(f):=\\lbrace z\\in \\mathbb {R}^d:f(z)<\\infty \\rbrace $ as follows: $\\partial f(x):=\\lbrace \\xi \\in \\mathbb {R}^d:\\ f(x)+\\langle \\xi , y-x\\rangle \\le f(y),\\quad \\mbox{for}\\ \\mbox{all}\\ y\\in \\mathbb {R}^d\\rbrace .$ Any element in the set $\\partial f(x)$ is called a subgradient of $f(\\cdot )$ at $x$ .","[Strong convexity] A function $f:\\mathbb {R}^d\\rightarrow \\mathbb {R}\\cup \\lbrace \\infty \\rbrace $ is strongly convex with parameter $\\lambda >0$ , if, for all $x,y\\in \\mbox{dom}(f)=\\lbrace z\\in \\mathbb {R}^d:f(z)<\\infty \\rbrace $ , the following holds: $f(y)\\ge f(x) + \\langle \\xi _x,y-x\\rangle +\\frac{\\lambda }{2}\\Vert y-x\\Vert ^2,$ where $\\xi _x\\in \\partial f(x)$ , the subgradient of $f(\\cdot )$ at $x$ as in def:subg.","[Wavelet basis] We present our main assumptions on the wavelet basis discussed in sec:wavelet only for the wavelets on the space $\\mathcal {X}$ .","The same assumptions are also required for the wavelets on $\\mathcal {Y}$ .","These are essentially a subset of the assumptions laid out in [143] as we heavily rely on [143] for proving thm:smoothwav.","(Regularity).","Fix $r>\\max \\lbrace s,1\\rbrace $ .","The functions in $\\Phi $ and $\\Psi _j$ , $j\\ge 0$ have $r$ continuous derivatives, and all polynomials of degree at most $r$ on $\\mathcal {X}$ lie in the span of the functions in $\\Phi $ .","(Tensor construction).","Each $\\psi (\\cdot )\\in \\Psi _j$ can be expressed as $\\psi (x)=\\prod _{i=1}^d \\psi _i(x_i)$ , where $x=(x_1,\\ldots ,x_d)$ , for some univariate functions $\\psi _i(\\cdot )$ 's.","(Locality).","For each $\\psi (\\cdot )\\in \\Psi _j$ there exists a rectangle $I_{\\psi }\\subseteq \\mathcal {X}$ such that $\\mbox{supp}(\\psi )\\subseteq I_{\\psi }$ , $\\mbox{diam}(I_{\\psi })\\le C_1\\cdot 2^{-j}$ , and $\\sup _{x\\in \\mathcal {X}}\\sum _{\\psi (\\cdot )\\in \\Psi _j}\\mathbb {1}(x\\in I_{\\psi })\\le C_2$ for some constants $C_1,C_2>0$ .","(Bernstein estimate).","$\\Vert \\nabla f\\Vert _{L^2(\\mathcal {X})}\\le C_3\\cdot 2^j\\Vert f\\Vert _{L^2(\\mathcal {X})}$ for any $f(\\cdot )$ in the span of the functions in $\\mbox{span}\\left(\\Phi \\cup \\left\\lbrace \\cup _{0\\le k0$ ).\",\n \"Then, $\\\\sup \\\\limits _{\\\\gamma \\\\in \\\\widetilde{\\\\Gamma }_{\\\\mathrm {min}}} \\\\int \\\\Vert \\\\widetilde{T}_{m,n}^{\\\\gamma }(x)-T_0(x)\\\\Vert ^2& \\\\,d\\\\widetilde{\\\\mu }_m(x)\\\\le L \\\\max \\\\left\\\\lbrace \\\\bigg |\\\\int \\\\Psi _{\\\\widetilde{\\\\mu }_m,\\\\widetilde{\\\\nu }_n}^* \\\\,d(\\\\widetilde{\\\\nu }_n-\\\\overline{\\\\nu }_m)\\\\bigg |,\\\\bigg |\\\\int \\\\Psi _{\\\\widetilde{\\\\mu }_m,\\\\overline{\\\\nu }_m}^* \\\\,d(\\\\widetilde{\\\\nu }_n-\\\\overline{\\\\nu }_m)\\\\bigg |\\\\right\\\\rbrace \\\\nonumber \\\\\\\\ &+2L\\\\int \\\\varphi _0^*(y)\\\\,d(\\\\widetilde{\\\\nu }_n-\\\\overline{\\\\nu }_m)(y),$ where $\\\\overline{\\\\nu }_m=T_0\\\\#\\\\widetilde{\\\\mu }_m$ , $\\\\varphi _0^*(\\\\cdot )$ is defined as in (REF ), and with $\\\\mathcal {S}_{\\\\cdot ,\\\\cdot }(\\\\cdot )$ defined as in (), $\\\\Psi _{\\\\widetilde{\\\\mu }_m,\\\\widetilde{\\\\nu }_n}(\\\\cdot ):=\\\\operatornamewithlimits{\\\\arg \\\\!\\\\min }_{f\\\\in \\\\mathcal {F}} \\\\mathcal {S}_{\\\\widetilde{\\\\mu }_m,\\\\widetilde{\\\\nu }_n}(f),\\\\quad \\\\Psi _{\\\\widetilde{\\\\mu }_m,\\\\overline{\\\\nu }_m}(\\\\cdot ):=\\\\operatornamewithlimits{\\\\arg \\\\!\\\\min }_{f\\\\in \\\\mathcal {F}} \\\\mathcal {S}_{\\\\widetilde{\\\\mu }_m,\\\\overline{\\\\nu }_m}(f)$ .\",\n \"The proof of thm:newubd (see sec:mainrespf) starts along the same lines as the proof of the curvature estimate in [62].\",\n \"This is followed by some careful manipulations of $W_2^2(\\\\cdot ,\\\\cdot )$ (as in (REF )) and an application of the conditional version of Jensen's inequality, see (REF ).\",\n \"The final step of the proof uses the dual representation in (REF ) with techniques similar to some intermediate steps in the proof of [101] and [33].\",\n \"Remark 2.1 (Comparison with other stability estimates) thm:newubd provides some important advantages to existing stability estimates in the literature.\",\n \"One of the earliest results in this direction can be found in [62] but their bound involves a push-forward constraint which makes it hard to use for rate of convergence analysis.\",\n \"A bound similar to thm:newubd is presented in [61] but there the authors assume the existence of an OT map from $\\\\widetilde{\\\\mu }_m$ to $\\\\widetilde{\\\\nu }_n$ .\",\n \"Therefore, it does not apply to the discrete-discrete problem where $\\\\widetilde{\\\\mu }_m=\\\\widehat{\\\\mu }_m$ and $\\\\widetilde{\\\\nu }_n=\\\\widehat{\\\\nu }_n$ with $m\\\\ne n$ .\",\n \"Overcoming all these limitations is an important contribution of thm:newubd and allows us to deal with popular plug-in estimators all in one go.\",\n \"The stability estimate in [80] on the other hand requires $\\\\widetilde{\\\\mu }_m$ , $\\\\widetilde{\\\\nu }_n$ to be sufficiently smooth and hence it does not hold for discrete-discrete or semi-discrete plug-in estimators (see ex:baseplug).\",\n \"Further their result requires all the measures involved to be compactly supported unlike the much milder requirements of thm:newubd.\",\n \"However, a shortcoming of thm:newubd is that it is hard to obtain rates faster than $n^{-1/2}$ using it directly, whereas [80] can obtain rates arbitrarily close to $n^{-1}$ .\",\n \"This is a price we pay for analyzing natural and popular plug-in estimators as opposed to the (more intractable) wavelet based estimators in [80].\",\n \"Remark 2.2 (How to use thm:newubd to obtain rates of convergence?)\",\n \"Note that the second term on the right hand side of (REF ), under appropriate moment assumptions, is $O_p(m^{-1/2}+n^{-1/2})$ (free of dimension) by a direct application of Markov's inequality.\",\n \"We therefore focus on the first term.\",\n \"By (), $\\\\Psi _{\\\\widetilde{\\\\mu }_m,\\\\widetilde{\\\\nu }_n}^*(\\\\cdot )$ , $\\\\Psi _{\\\\widetilde{\\\\mu }_m,\\\\overline{\\\\nu }_m}^*(\\\\cdot )\\\\in \\\\mathcal {F}$ .\",\n \"Further, by Caffarelli's regularity theory [23], [24], [25], depending on the “smoothness\\\" of $\\\\widetilde{\\\\mu }_m$ , $\\\\widetilde{\\\\nu }_n$ , it can be shown that there exists a further class of functions $\\\\mathcal {F}_s$ (see Remarks REF and REF ) such that $\\\\Psi _{\\\\widetilde{\\\\mu }_m,\\\\widetilde{\\\\nu }_n}^*(\\\\cdot )$ , $\\\\Psi _{\\\\widetilde{\\\\mu }_m,\\\\overline{\\\\nu }_m}^*(\\\\cdot )\\\\in \\\\mathcal {F}\\\\cap \\\\mathcal {F}_s$ .\",\n \"Thus, we can bound the first term on the right hand side of (REF ) as: $\\\\max \\\\left\\\\lbrace \\\\bigg |\\\\int \\\\Psi _{\\\\widetilde{\\\\mu }_m,\\\\widetilde{\\\\nu }_n}^* \\\\,d(\\\\widetilde{\\\\nu }_n-\\\\overline{\\\\nu }_m)\\\\bigg |,\\\\bigg |\\\\int \\\\Psi _{\\\\widetilde{\\\\mu }_m,\\\\overline{\\\\nu }_m}^* \\\\,d(\\\\widetilde{\\\\nu }_n-\\\\overline{\\\\nu }_m)\\\\bigg |\\\\right\\\\rbrace \\\\le \\\\sup _{f\\\\in \\\\mathcal {F}\\\\cap \\\\mathcal {F}_s}\\\\bigg |\\\\int f\\\\,d(\\\\widetilde{\\\\nu }_n-\\\\overline{\\\\nu }_m)\\\\bigg |.$ The right hand side of (REF ) can now be bounded using the corresponding Dudley's entropy integral bounds using empirical process techniques, see [137].\",\n \"To conclude, the two main steps in our strategy are identifying the family of functions $\\\\mathcal {F}_s$ and computing Dudley's entropy integral.\",\n \"Further, the more the smoothness of $\\\\widetilde{\\\\mu }_m$ , $\\\\widetilde{\\\\nu }_n$ , the smaller is the class of functions $\\\\mathcal {F}_s$ and smaller the supremum on the right hand side of (REF ).\",\n \"This shows why better rates can be expected under smoothness assumptions.\"\n ],\n [\n \"Natural non-smooth plug-in estimator\",\n \"In this case, we discuss the rates of convergence for the discrete-discrete problem and the semi-discrete problem, where no smoothness is available on $\\\\widetilde{\\\\mu }_m$ and $\\\\widetilde{\\\\nu }_n$ .\",\n \"Theorem 2.2 Suppose that $T_0(\\\\cdot )$ is $L$ -Lipschitz, $\\\\nu $ is compactly supported and $\\\\mathbb {E}\\\\exp (t\\\\Vert X_1\\\\Vert ^{\\\\alpha })<\\\\infty $ for some $t>0$ , $\\\\alpha >0$ .\",\n \"(Discrete-discrete): Set $\\\\widetilde{\\\\mu }_m=\\\\widehat{\\\\mu }_m$ and $\\\\widetilde{\\\\nu }_n=\\\\widehat{\\\\nu }_n$ .\",\n \"Then the following holds: $\\\\sup \\\\limits _{\\\\gamma \\\\in \\\\widetilde{\\\\Gamma }_{\\\\mathrm {min}}} \\\\int \\\\Vert \\\\widetilde{T}_{m,n}^{\\\\gamma }(x)-T_0(x)\\\\Vert ^2\\\\,d\\\\widetilde{\\\\mu }_m(x)=O_p\\\\left(r^{(m,n)}_d\\\\times (\\\\log {(1+\\\\max \\\\lbrace m,n\\\\rbrace )})^{t_{d,\\\\alpha }}\\\\right),$ $\\\\mbox{where}\\\\quad r^{(m,n)}_d:={\\\\left\\\\lbrace \\\\begin{array}{ll} m^{-1/2}+n^{-1/2} & \\\\mbox{for}\\\\ d=2,3,\\\\\\\\ m^{-1/2}\\\\log {(1+m)}+n^{-1/2}\\\\log {(1+n)} & \\\\mbox{for}\\\\ d=4,\\\\\\\\ m^{-2/d}+n^{-2/d} & \\\\mbox{for}\\\\ d\\\\ge 5, \\\\end{array}\\\\right.\",\n \"}$ and $t_{d,\\\\alpha }:={\\\\left\\\\lbrace \\\\begin{array}{ll} (4\\\\alpha )^{-1}(4+((2\\\\alpha +2d\\\\alpha -d)\\\\vee 0)) & \\\\mbox{for}\\\\ d<4,\\\\\\\\ (\\\\alpha ^{-1}\\\\vee 7/2)-1 & \\\\mbox{for}\\\\ d=4,\\\\\\\\ 2(1+d^{-1}) & \\\\mbox{for}\\\\ d>4.\\\\end{array}\\\\right.\",\n \"}$ The same bound holds for $|W_2^2(\\\\widetilde{\\\\mu }_m,\\\\widetilde{\\\\nu }_n)-W_2^2(\\\\mu ,\\\\nu )|$ without assuming $T_0(\\\\cdot )$ is Lipschitz.\",\n \"(Semi-discrete): Set $\\\\widetilde{\\\\mu }_m=\\\\mu $ , $\\\\widetilde{\\\\nu }_n=\\\\widehat{\\\\nu }_n$ or $\\\\widetilde{\\\\mu }_m=\\\\widehat{\\\\mu }_m$ , $\\\\widetilde{\\\\nu }_n=\\\\nu $ .\",\n \"Then the left hand side of (REF ) is $O_p(r_d^{(n,n)}\\\\times (\\\\log {(1+n)})^{t_{d,\\\\alpha }})$ or $O_p(r_d^{(m,m)}\\\\times (\\\\log {(1+m)})^{t_{d,\\\\alpha }})$ respectively.\",\n \"A stronger result can be proved if both $\\\\mu $ and $\\\\nu $ are compactly supported.\",\n \"Corollary 2.3 Consider the setting from thm:nsmooth and assume further that $\\\\mu $ is compactly supported.\",\n \"Then, with $r_d^{(m,n)}$ defined as in (REF ), we have: $\\\\mathbb {E}\\\\left[\\\\sup \\\\limits _{\\\\gamma \\\\in \\\\widetilde{\\\\Gamma }_{\\\\mathrm {min}}} \\\\int \\\\Vert \\\\widetilde{T}_{m,n}^{\\\\gamma }(x)-T_0(x)\\\\Vert ^2\\\\,d\\\\widetilde{\\\\mu }_m(x)\\\\right]\\\\le C r^{(m,n)}_d,$ for some constant $C>0$ , in both the discrete-discrete and semi-discrete settings from thm:nsmooth.\",\n \"A brief description of the proof technique of thm:nsmooth using thm:newubd is provided in rem:pfnstech below, and the actual proof is presented in sec:mainrespf.\",\n \"Remark 2.3 (Proof technique) The proof of thm:nsmooth proceeds using the strategy outlined in rem:pftech.\",\n \"We first show that $\\\\mathcal {F}_s$ (see rem:pftech) can be chosen as a certain class of convex functions which are in $L^2(\\\\nu )$ .\",\n \"We then use Dudley's entropy integral type bounds which in turn requires the bracketing entropy [137] of $\\\\mathcal {F}_s$ , recently proved in [91].\",\n \"This strategy is slightly different from that used in the proof of [33], where the authors assume that $\\\\mu $ is compactly supported whereas we only assume the finiteness of $\\\\mathbb {E}\\\\exp (t\\\\Vert X_1\\\\Vert ^{\\\\alpha })$ for some $t>0$ , $\\\\alpha >0$ .\",\n \"The compactness assumption on $\\\\mu $ allows one to further restrict $\\\\mathcal {F}_s$ to the class of Lipschitz functions.\",\n \"This additional restriction does not seem to be immediate without the compactness assumption.\",\n \"As discussed in sec:contrib, the exponents obtained in thm:nsmooth are minimax optimal under bare minimal smoothness assumptions (see [80]).\",\n \"To the best of our knowledge, rates for the discrete-discrete case for $m\\\\ne n$ and those for the semi-discrete case were not known previously in the literature.\",\n \"Our rates are also strictly better than those (for different estimators, based on space tessellations) obtained in [10], [95] and require less stringent assumptions than those in [33].\",\n \"In the next section, we show how smoothness assumptions can be leveraged to mitigate the curse of dimensionality in thm:nsmooth.\"\n ],\n [\n \"Smooth plug-in estimator: mitigating the curse of dimensionality\",\n \"In this section, we focus on two types of plug-in estimators for the densities associated with the probability measures $\\\\mu $ and $\\\\nu $ : (a) wavelet based estimators (see [143], [135], [47], [86], [142]) in sec:wavelet, and (b) kernel based estimators (see [63], [64], [112], [125], [108]) in sec:kernel.\",\n \"In both these cases, we will show, using thm:newubd, that the corresponding estimators of $T_0(\\\\cdot )$ achieve (near) dimension-free rates under sufficient smoothness assumptions.\"\n ],\n [\n \"Wavelet based estimators\",\n \"We begin this subsection by defining the Besov class of functions which will play a pivotal role in the sequel.\",\n \"[Besov class of functions] We describe Besov classes following the notation from [143].\",\n \"Suppose $s>0$ and let $n>s$ be a positive integer.\",\n \"Given $\\\\Omega \\\\subseteq \\\\mathbb {R}^d$ , $h\\\\in \\\\mathbb {R}^d$ and $f(\\\\cdot ):\\\\mathbb {R}^d\\\\rightarrow \\\\mathbb {R}^d$ , set $\\\\Delta _h^1 f(x):=f(x+h)-f(x),$ $\\\\Delta _h^k f(x):= \\\\Delta _h^1\\\\left(\\\\Delta _h^{k-1} f\\\\right)(x), \\\\quad \\\\forall \\\\ 2\\\\le k\\\\le n,$ where these functions are defined on $\\\\Omega _{h,n}:=\\\\lbrace x\\\\in \\\\Omega :x+nh\\\\in \\\\Omega \\\\rbrace $ .\",\n \"For $t>0$ , we then define $\\\\omega _n(f,t):=\\\\sup _{\\\\Vert h\\\\Vert \\\\le t} \\\\Vert \\\\Delta _h^n f\\\\Vert _{L^2(\\\\Omega _{h,n})}.$ Finally, we define the space $\\\\mathcal {B}^s(\\\\Omega )$ to be the set of functions for which the quantity $\\\\Vert f\\\\Vert _{\\\\mathcal {B}^s(\\\\Omega )}:=\\\\Vert f\\\\Vert _{L^2(\\\\Omega )}+\\\\sum _{j\\\\ge 0} 2^{sj}\\\\omega _n(f,2^{-j})$ is finite.\",\n \"The above expression can also be used to define Besov spaces (and norms) for $s<0$ ; see [36].\",\n \"In this subsection, we assume that $\\\\mu $ and $\\\\nu $ admit Besov smooth densities $f_{\\\\mu }(\\\\cdot )$ and $f_{\\\\nu }(\\\\cdot )$ (see [36] and def:besov above for details).\",\n \"Given $\\\\Omega \\\\subseteq \\\\mathbb {R}^d$ and $s>0$ , let $\\\\mathcal {B}^s(\\\\Omega )$ denote the set of Besov smooth functions on $\\\\Omega $ of order $s$ .\",\n \"Assumption (A1) (Regularity of the densities) Suppose that: $f_{\\\\mu }$ and $f_{\\\\nu }$ are supported on compact and convex subsets of $\\\\mathbb {R}^d$ , say $\\\\mathcal {X}$ and $\\\\mathcal {Y}$ respectively.\",\n \"There exists $s,M> 0$ such that $\\\\Vert f_{\\\\mu }\\\\Vert _{\\\\mathcal {B}^{s}(\\\\mathcal {X})}\\\\le M$ , $\\\\Vert f_{\\\\nu }\\\\Vert _{\\\\mathcal {B}^{s}(\\\\mathcal {Y})}\\\\le M$ and $f_{\\\\mu }(x),f_{\\\\nu }(y)\\\\ge M^{-1}$ for all $x\\\\in \\\\mathcal {X}$ , $y\\\\in \\\\mathcal {Y}$ .\",\n \"We now present our wavelet based estimators for $f_{\\\\mu }(\\\\cdot )$ and $f_{\\\\nu }(\\\\cdot )$ .\",\n \"Towards this direction, we begin with sets of functions in $L^2(\\\\mathcal {X})$ (set of square integrable functions on $\\\\mathcal {X}$ ), $\\\\Phi $ and $\\\\lbrace \\\\Psi _j\\\\rbrace _{j\\\\ge 0}$ , which form an orthonormal basis of $L^2(\\\\mathcal {X})$ and satisfy the standard regularity assumptions for a wavelet basis (see [104], [77], [143]).\",\n \"We defer a formal discussion on these assumptions to def:wavelet in the Appendix so as not to impede the flow of the paper.\",\n \"For the moment, it is worth noting that such sets of functions (e.g., Haar wavelets, Daubechies wavelets) are readily available in standard statistical softwares, see e.g., the R package wavelets.\",\n \"Next, fix $J_m\\\\in \\\\mathbb {N}$ (a truncation parameter to be chosen later depending on the sample size $m$ ).\",\n \"Consider the following: $\\\\widehat{f}_{\\\\mu }(x):=\\\\sum _{\\\\phi \\\\in \\\\Phi } a_{\\\\phi }\\\\phi (x)+\\\\sum _{j=0}^{J_m} \\\\sum _{\\\\psi \\\\in \\\\Psi _j} b_{\\\\psi }\\\\psi (x),$ where $a_{\\\\phi }:=\\\\frac{1}{m}\\\\sum _{i=1}^m \\\\phi (X_i), \\\\qquad b_{\\\\psi }:=\\\\frac{1}{m}\\\\sum _{i=1}^m \\\\psi (X_i).$ Unfortunately $\\\\widehat{f}_{\\\\mu }(\\\\cdot )$ as defined in (REF ) may not be a probability density and consequently cannot be used to obtain plug-in estimators for $\\\\widetilde{T}_{m,n}^{\\\\gamma }(\\\\cdot )$ .\",\n \"We therefore take the same route as in [143] to define the following estimator for $f_{\\\\mu }(\\\\cdot )$ : $\\\\widetilde{f}_{\\\\mu }:=\\\\min _{g\\\\in \\\\mathcal {D}(\\\\mathcal {X})}\\\\Vert g-\\\\widehat{f}_{\\\\mu }\\\\Vert _{\\\\mathcal {B}^{-1}(\\\\mathcal {X})},$ where $\\\\mathcal {D}(\\\\mathcal {X})$ is the space of probability density functions on $\\\\mathcal {X}$ and $\\\\mathcal {B}^{-1}(\\\\mathcal {X})$ is the Besov norm on $\\\\mathcal {X}$ of order $-1$ as stated in def:besov.\",\n \"We can define $\\\\widetilde{f}_{\\\\nu }(\\\\cdot )$ similarly.\",\n \"Computing both $\\\\widetilde{f}_{\\\\mu }(\\\\cdot )$ and $\\\\widehat{f}_{\\\\mu }(\\\\cdot )$ (as it involves infinite sums) is challenging and we would refer the interested reader to [143] and the references therein, for details.\",\n \"Further discussion of this aspect is beyond the scope of this paper.\",\n \"We are now in a position to present the main theorem of this subsection.\",\n \"Theorem 2.4 Suppose that $T_0(\\\\cdot )$ is $L$ -Lipschitz, and $\\\\widetilde{\\\\mu }_m$ and $\\\\widetilde{\\\\nu }_n$ are the probability measures corresponding to the probability densities $\\\\widetilde{f}_{\\\\mu }(\\\\cdot )$ and $\\\\widetilde{f}_{\\\\nu }(\\\\cdot )$ with $m^{\\\\frac{1}{d+2s}}\\\\le 2^{J_m}\\\\le m^{\\\\frac{1}{d}}$ and $n^{\\\\frac{1}{d+2s}}\\\\le 2^{J_n}\\\\le n^{\\\\frac{1}{d}}$ , then the following holds for some constant $C>0$ : $\\\\mathbb {E}\\\\left[\\\\sup \\\\limits _{\\\\gamma \\\\in \\\\widetilde{\\\\Gamma }_{\\\\mathrm {min}}} \\\\int \\\\Vert \\\\widetilde{T}_{m,n}^{\\\\gamma }(x)-T_0(x)\\\\Vert ^2\\\\,d\\\\widetilde{\\\\mu }_m(x)\\\\right]\\\\le C \\\\widetilde{r}^{(m,n)}_{d,s},$ $\\\\mbox{where}\\\\quad \\\\widetilde{r}^{(m,n)}_{d,s}:={\\\\left\\\\lbrace \\\\begin{array}{ll} m^{-1/2}\\\\log {(1+m)}+n^{-1/2}\\\\log {(1+n)} & \\\\mbox{for}\\\\ d=2,\\\\\\\\ m^{-\\\\frac{1+s}{d+2s}}+n^{-\\\\frac{1+s}{d+2s}} & \\\\mbox{for}\\\\ d\\\\ge 3, \\\\end{array}\\\\right.\",\n \"}$ The same bound also holds for $\\\\mathbb {E}|W_2^2(\\\\widetilde{\\\\mu }_m,\\\\widetilde{\\\\nu }_n)-W_2^2(\\\\mu ,\\\\nu )|$ .\",\n \"Note that $\\\\frac{1+s}{d+2s}\\\\rightarrow \\\\frac{1}{2}$ as $s\\\\rightarrow \\\\infty $ .\",\n \"Therefore thm:smoothwav shows that, when $m=n$ , the rate of convergence for the wavelet based estimator is “close\\\" to $n^{-1/2}$ provided $s$ is large enough for each fixed $d$ .\",\n \"This shows that $T_0(\\\\cdot )$ obtained using the wavelet estimators for $f_{\\\\mu }(\\\\cdot )$ and $f_{\\\\nu }(\\\\cdot )$ mitigates the curse of dimensionality, contrast this with the estimator in thm:barate.\",\n \"To avoid repetition, we defer further discussions on the rates observed in thm:smoothwav to rem:curdim where a holistic comparison is drawn with two other “smooth” plug-in estimators.\"\n ],\n [\n \"Kernel based estimators\",\n \"We first introduce the Sobolev class of functions which we will exploit in this subsection to construct estimators that achieve rates of convergence which mitigate the curse of dimensionality under sufficient smoothness.\",\n \"[Uniform Sobolev class of functions] Let $\\\\Omega \\\\subseteq \\\\mathbb {R}^d$ and $f(\\\\cdot )$ be uniformly continuous on $\\\\Omega $ and admits uniformly continuous derivatives up to order $s$ on $\\\\Omega $ for some $s\\\\in \\\\mathbb {N}$ .\",\n \"For any $\\\\mathfrak {m}:=(m_1,\\\\ldots ,m_d)\\\\in \\\\mathbb {N}^d$ , let $\\\\partial ^{\\\\mathfrak {m}} f:=\\\\frac{\\\\partial }{\\\\partial _{x_1}^{m_1}}\\\\ldots \\\\frac{\\\\partial }{\\\\partial _{x_d}^{m_d}} f,\\\\quad |\\\\mathfrak {m}|:=\\\\sum _{i=1}^d m_i.$ For any $k\\\\le s$ , we further define, $\\\\Vert f\\\\Vert _{C^k(\\\\Omega )}:=\\\\sum _{|\\\\mathfrak {m}|\\\\le k} \\\\Vert \\\\partial ^{\\\\mathfrak {m}} f\\\\Vert _{L^{\\\\infty }(\\\\Omega )}.$ The space $C^{s}(\\\\Omega )$ is defined as the set of functions $f(\\\\cdot )$ for which $\\\\Vert f\\\\Vert _{C^k(\\\\Omega )}<\\\\infty $ for all $k\\\\le s$ .\",\n \"For this subsection, assume that $\\\\mu $ and $\\\\nu $ admit Sobolev smooth densities $f_{\\\\mu }(\\\\cdot )$ and $f_{\\\\nu }$ in the uniform norm (see def:holder above).\",\n \"Given $\\\\Omega \\\\subseteq \\\\mathbb {R}^d$ and $s\\\\in \\\\mathbb {N}$ , let $C^s(\\\\Omega )$ denote the set of Sobolev smooth functions on $\\\\Omega $ of order $s$ .\",\n \"Assumption (A2) (Regularity of the densities) Suppose that $f_{\\\\mu }$ and $f_{\\\\nu }$ are supported on compact and convex subsets of $\\\\mathbb {R}^d$ , say $\\\\mathcal {X}$ and $\\\\mathcal {Y}$ respectively.\",\n \"There exists $s,M> 0$ such that $f_{\\\\mu }(\\\\cdot )\\\\in C^{s}(\\\\mathcal {X};M)$ and $f_{\\\\nu }(\\\\cdot )\\\\in C^{s}(\\\\mathcal {Y};M)$ where $C^{s}(\\\\mathcal {X};M)$ is the space of real valued functions supported on $\\\\mathcal {X}$ such that for all $f(\\\\cdot )\\\\in C^s(\\\\mathcal {X};M)$ , we have $M^{-1}\\\\le f(x)\\\\le M$ for all $x\\\\in \\\\mathcal {X}$ and $\\\\Vert f\\\\Vert _{C^s(\\\\mathcal {X})}\\\\le M$ .\",\n \"Here $\\\\Vert \\\\cdot \\\\Vert _{C^s(\\\\mathcal {X})}$ is the standard uniform Sobolev norm as defined in def:holder.\",\n \"The space $C^s(\\\\mathcal {Y};M)$ is defined analogously.\",\n \"We now define our estimators for $f_{\\\\mu }(\\\\cdot )$ and $f_{\\\\nu }(\\\\cdot )$ using the standard kernel density estimation technique (see [135]).\",\n \"Set $\\\\widehat{f}_{\\\\mu }(x):=\\\\frac{1}{mh_m^d}\\\\sum _{i=1}^m K_d\\\\left(\\\\frac{X_i-x}{h_m}\\\\right),$ for some bandwidth parameter $h_m>0$ and $d$ -variate kernel $K_d(\\\\cdot )$ .\",\n \"We assume that $K_d(\\\\cdot )$ is the $d$ -fold product of univariate kernels, i.e., there exists a kernel $K(\\\\cdot )$ such that for $u=(u_1,\\\\ldots ,u_d)\\\\in \\\\mathbb {R}^d$ , $K_d(u)=\\\\prod _{i=1}^d K(u_i)$ .\",\n \"We define $\\\\widehat{f}_{\\\\nu }(\\\\cdot )$ similarly with the same univariate kernel and bandwidth.\",\n \"Assumption (A3) (Regularity of the kernel) Assume that $K(\\\\cdot )$ is a symmetric, bounded, $s+1$ times differentiable kernel on $\\\\mathbb {R}^d$ with all $s+1$ derivatives bounded and integrable.\",\n \"Further, suppose that $K(\\\\cdot )$ is of order $2s+2$ , i.e., $\\\\int u^j K(u)\\\\,du=\\\\mathbb {1}(j=0),\\\\qquad \\\\mbox{for }\\\\ j=\\\\lbrace 0,1,2,\\\\ldots ,2s+1\\\\rbrace ,\\\\qquad \\\\mbox{and} \\\\ \\\\int |u|^{2s+2}|K(u)|\\\\,du<\\\\infty .$ The above assumptions on $K(\\\\cdot )$ are standard for estimating smooth densities and their derivatives of different orders in the kernel density estimation literature; see e.g.\",\n \"[76], [5], [63], [135], [64].\",\n \"There are several natural ways to construct kernels satisfying as:kernel, see [135]; an example is also provided in ex:ordker below.\",\n \"[Example of a kernel satisfying as:kernel] Let $\\\\psi _m(\\\\cdot )$ be the $m$ -th Hermite polynomial on $\\\\mathbb {R}$ (see [92]).\",\n \"Then the kernel function defined as $K(u):=\\\\sum _{m=0}^{2s+2} \\\\psi _{m}(0)\\\\psi _m(u)\\\\exp (-u^2/2)$ satisfies as:kernel.\",\n \"It is evident from as:kernel that $K(\\\\cdot )$ may take some negative values, in which case, $\\\\widehat{f}_{\\\\mu }(\\\\cdot )$ (respectively $\\\\widehat{f}_{\\\\nu }(\\\\cdot )$ ) may not be a probability density.\",\n \"Consequently the barycentric projection (see def:bcenterproj) between $\\\\widehat{f}_{\\\\mu }(\\\\cdot )$ and $\\\\widehat{f}_{\\\\nu }(\\\\cdot )$ is not well-defined.\",\n \"We get around this by resorting to the same (approximate) projection technique we used in (REF ) for the wavelet based estimators.\",\n \"In this case however, instead of (approximately) projecting using an appropriate Besov norm as in (REF ), we use a integral probability metric (see def:ipm; also see [127], [107], [114] for examples, computational procedures and applications of such metrics).\",\n \"The corresponding measure is defined below: [Integral probability metric] Given a class $\\\\mathcal {F}$ of bounded functions on $\\\\mathbb {R}^d$ and two probability densities $g_1(\\\\cdot )$ and $g_2(\\\\cdot )$ on $\\\\mathbb {R}^d$ , the integral probability metric/distance between $g_1(\\\\cdot )$ and $g_2(\\\\cdot )$ with respect to $\\\\mathcal {F}$ is defined as $d_{\\\\mathrm {IP}}(g_1,g_2;\\\\mathcal {F}):=\\\\sup _{\\\\psi (\\\\cdot )\\\\in \\\\mathcal {F}} \\\\bigg |\\\\int \\\\psi (x)(g_1(x)-g_2(x))\\\\,dx\\\\bigg |.$ Sufficient conditions on $\\\\mathcal {F}$ for $d_{\\\\mathrm {IP}}(\\\\cdot ,\\\\cdot ;\\\\mathcal {F})$ to be a metric on the space of probability measures (not on the space of probability densities as they can be altered on set of Lebesgue measure 0 without altering the underlying probability measures) on $\\\\mathbb {R}^d$ have been discussed in [107].\",\n \"Observe that the measure $d_{\\\\mathrm {IP}}(g_1,g_2;\\\\mathcal {F})$ is well defined even when $g_1(\\\\cdot )$ and $g_2(\\\\cdot )$ are not probability densities.\",\n \"In thm:smooth below, we use $\\\\mathcal {F}=C^{s+2}(\\\\mathcal {X},M^{\\\\prime })$ .\",\n \"Note that any function in $C^{s+2}(\\\\mathcal {X},M^{\\\\prime })$ can be extended to a function in $C^{s+2}(\\\\mathbb {R}^d;M^{\\\\prime })$ (see [80] and [134]).\",\n \"The fact that this choice of $\\\\mathcal {F}$ results in a metric follows from the argument in [107].\",\n \"We are now in a position to describe the projection estimators for $f_{\\\\mu }(\\\\cdot )$ and $f_{\\\\nu }(\\\\cdot )$ , and the rates achieved by the corresponding plug-in estimator.\",\n \"Theorem 2.5 Assume that $T_0(\\\\cdot )$ is $L$ -Lipschitz and $f_{\\\\mu }$ , $f_{\\\\nu }$ are Lebesgue densities satisfying as:smdensities.\",\n \"Also suppose that $K(\\\\cdot )$ satisfies as:kernel.\",\n \"Define $h_m:=m^{-\\\\frac{1}{d+2s}}\\\\log {m}$ , $h_n:=n^{-\\\\frac{1}{d+2s}}\\\\log {n}$ and $T:=\\\\int |K_d(u)|\\\\,du+1$ .\",\n \"Fix any $M^{\\\\prime }>0$ .\",\n \"Consider any probability density $\\\\widetilde{f}_{\\\\mu }^{M^{\\\\prime }}(\\\\cdot )\\\\in C^s(\\\\mathcal {X};TM)$ (where $M$ is defined as in as:smdensities) which satisfies $d_{\\\\mathrm {IP}}\\\\left(\\\\widetilde{f}_{\\\\mu }^{M^{\\\\prime }},\\\\widehat{f}_{\\\\mu };C^{s+2}(\\\\mathcal {X};M^{\\\\prime })\\\\right)\\\\le \\\\inf _{\\\\begin{array}{c}f(\\\\cdot )\\\\in C^s(\\\\mathcal {X};TM)\\\\\\\\ f\\\\ge 0,\\\\ \\\\int f=1\\\\end{array}} d_{\\\\mathrm {IP}}\\\\left(\\\\widehat{f}_{\\\\mu },f;C^{s+2}(\\\\mathcal {X};M^{\\\\prime })\\\\right)+r_{d,s}^{(m,n)}$ where $r_{d,s}^{(m,n)}$ is defined as in (REF ) and $d_{\\\\mathrm {IP}}(\\\\cdot ,\\\\cdot ;C^{s+2}(\\\\mathcal {X};M^{\\\\prime }))$ is the integral probability metric defined in def:ipm.\",\n \"We define $\\\\widetilde{f}_{\\\\nu }^{M^{\\\\prime }}(\\\\cdot )$ analogously as in (REF ) with $\\\\mathcal {X}$ , $\\\\widehat{f}_{\\\\mu }(\\\\cdot )$ replaced by $\\\\mathcal {Y}$ , $\\\\widehat{f}_{\\\\nu }(\\\\cdot )$ .\",\n \"Then the following conclusions hold.\",\n \"There exists $M^{\\\\prime }>0$ (depending on $M$ ) such that, if, $\\\\widetilde{\\\\mu }_m$ and $\\\\widetilde{\\\\nu }_n$ are the probability measures corresponding to the probability densities $\\\\widetilde{f}_{\\\\mu }^{M^{\\\\prime }}(\\\\cdot )$ and $\\\\widetilde{f}_{\\\\nu }^{M^{\\\\prime }}(\\\\cdot )$ , then the following holds for some constant $C>0$ : $\\\\mathbb {E}\\\\left[\\\\sup \\\\limits _{\\\\gamma \\\\in \\\\widetilde{\\\\Gamma }_{\\\\mathrm {min}}} \\\\int \\\\Vert \\\\widetilde{T}_{m,n}^{\\\\gamma }(x)-T_0(x)\\\\Vert ^2\\\\,d\\\\widetilde{\\\\mu }_m(x)\\\\right]\\\\le C r_{d,s}^{(m,n)},$ $ \\\\mbox{where}\\\\quad r_{d,s}^{(m,n)} := {\\\\left\\\\lbrace \\\\begin{array}{ll} m^{-1/2}+n^{-1/2} & \\\\mbox{for}\\\\ d<2(s+2),\\\\\\\\ m^{-1/2}\\\\left(\\\\log {(1+m)}\\\\right)^d+n^{-1/2}\\\\left(\\\\log {(1+n)}\\\\right)^d & \\\\mbox{for}\\\\ d=2(s+2),\\\\\\\\ m^{-\\\\frac{s+2}{d}}+n^{-\\\\frac{s+2}{d}} & \\\\mbox{for}\\\\ d\\\\ge 2(s+2).\",\n \"\\\\end{array}\\\\right.\",\n \"}$ The same bound also holds for $\\\\mathbb {E}|W_2^2(\\\\widetilde{\\\\mu }_m,\\\\widetilde{\\\\nu }_n)-W_2^2(\\\\mu ,\\\\nu )|$ .\",\n \"$\\\\widehat{f}_{\\\\mu }(\\\\cdot )$ satisfies $\\\\lim \\\\limits _{n\\\\rightarrow \\\\infty } \\\\max \\\\left\\\\lbrace \\\\mathbb {P}\\\\left(\\\\Vert \\\\widehat{f}_{\\\\mu }\\\\Vert _{C^s(\\\\widetilde{\\\\mathcal {X}})}\\\\ge TM\\\\right),\\\\mathbb {P}\\\\left(\\\\sup _{x\\\\in \\\\widetilde{\\\\mathcal {X}}} |\\\\widehat{f}_{\\\\mu }(x)-f_{\\\\mu }(x)|\\\\ge \\\\varepsilon \\\\right)\\\\right\\\\rbrace =0$ for any $\\\\varepsilon >0$ , where $\\\\widetilde{\\\\mathcal {X}}$ is any compact subset of $\\\\mathcal {X}^o$ .\",\n \"The same conclusion holds for $\\\\widehat{f}_{\\\\nu }(\\\\cdot )$ with $\\\\mathcal {X}$ replaced by $\\\\mathcal {Y}$ .\",\n \"In thm:smooth, $\\\\widetilde{f}_{\\\\mu }^{M^{\\\\prime }}(\\\\cdot )$ can be viewed as an approximate minimizer of $d_{\\\\mathrm {IP}}(\\\\widehat{f}_{\\\\mu },\\\\cdot ;C^{s+2}(\\\\mathcal {X},M^{\\\\prime }))$ over an appropriate class of Sobolev smooth probability densities.\",\n \"This is carried out because $\\\\widehat{f}_{\\\\mu }(\\\\cdot )$ by itself may not be a probability density.\",\n \"Further note that $\\\\widetilde{\\\\mu }_m,\\\\widetilde{\\\\nu }_n$ as specified in thm:smooth are both smooth, and consequently $\\\\widetilde{\\\\Gamma }_{\\\\mathrm {min}}$ is a singleton and the supremum in thm:smooth can be dropped.\",\n \"A brief description of the proof technique for thm:smooth is presented in rem:pfstech below and the actual proof is given in sec:mainrespf.\",\n \"Remark 2.4 (Proof technique) The proof of thm:smooth proceeds along the same lines as rem:pfnstech.\",\n \"We first show that $\\\\mathcal {F}_s$ (see rem:pftech) can be chosen as a certain subset of $C^{s+2}(\\\\mathcal {Y}^{\\\\circ })$ .\",\n \"We then use Dudley's entropy integral type bounds which in turn requires the bracketing entropy [137] of the class of compactly supported Sobolev smooth functions which can be found in [138].\",\n \"We now explain the implications of both the parts of thm:smooth in the following two remarks.\",\n \"Remark 2.5 (Mitigating the curse of dimensionality) thm:smooth shows that, under enough smoothness, i.e., when $2(s+2)>d$ , both the upper bounds for (REF ) and (REF ) are $O_p(n^{-1/2})$ .\",\n \"This shows that, for large dimensions, provided $\\\\mu $ and $\\\\nu $ admit smooth enough densities, it is possible to construct plug-in estimators that mitigate the curse of dimensionality.\",\n \"Note that a similar estimator was analyzed in [73] when $m=n$ .\",\n \"However, the rates obtained in thm:smooth are strictly better than those in [73].\",\n \"For $m=n$ , when $d<2(s+2)$ , [73] obtained a rate of $n^{-\\\\frac{s+2}{2(s+2)+d}}$ which is worse than $n^{-1/2}$ obtained in thm:smooth.\",\n \"For the other regimes, [73] obtains rates (up to log factors) of $n^{-1/4}$ and $n^{-\\\\frac{1}{(s+2)(d+2(s+2))}}$ which are both worse than the respective rates of $n^{-1/2}$ and $n^{-\\\\frac{s+2}{d}}$ in thm:smooth.\",\n \"In fact, the rates obtained in thm:smoothwav are also strictly better than the rates obtained in [73] (for every fixed $d$ ) described above, but they are strictly worse than the rates obtained in thm:smooth.\",\n \"When the degree of smoothness $s$ is large, both Theorems REF and REF lead to rates of (approximately) $n^{-1/2}$ .\",\n \"However when $s$ is small, the rate in thm:smooth is much faster than that in thm:smoothwav, e.g., if $s$ is close to 0, the rate in thm:smoothwav is approximately $n^{-\\\\frac{1}{d}}$ whereas that in thm:smooth is the faster rate of $n^{-\\\\frac{2}{d}}$ .\",\n \"It must be noted however that the smoothness assumptions are different in Theorems REF and REF .\",\n \"Remark 2.6 (Computational aspects of thm:smooth) In thm:smooth, we have shown that the plug-in estimator for $T_0(\\\\cdot )$ using $\\\\widetilde{f}_{\\\\mu }^{M^{\\\\prime }}(\\\\cdot )$ and $\\\\widetilde{f}_{\\\\nu }^{M^{\\\\prime }}(\\\\cdot )$ achieve rates that mitigate the curse of dimensionality under sufficient smoothness.\",\n \"However, as is evident, $\\\\widetilde{f}_{\\\\mu }^{M^{\\\\prime }}(\\\\cdot )$ is hard to compute whereas $\\\\widehat{f}_{\\\\mu }(\\\\cdot )$ is computable easily in linear time.\",\n \"Note that if $\\\\widehat{f}_{\\\\mu }(\\\\cdot )$ itself were a probability density in $C^s(\\\\mathcal {X};TM)$ , then we would have $\\\\widehat{f}_{\\\\mu }=\\\\widetilde{f}_{\\\\mu }^{M^{\\\\prime }}$ .\",\n \"While thm:smooth does not establish that, it does come close in part 2, from which we can easily derive the following: $\\\\lim _{n\\\\rightarrow \\\\infty }\\\\mathbb {P}(\\\\widehat{f}_{\\\\mu }(\\\\cdot )\\\\notin C^s(\\\\widetilde{\\\\mathcal {X}};TM))=0.$ The above shows that $\\\\widehat{f}_{\\\\mu }(\\\\cdot )$ is indeed bounded below by $(TM)^{-1}$ on $\\\\widetilde{\\\\mathcal {X}}$ (any compact subset of the interior of $\\\\mathcal {X}$ ), and additionally belongs to $C^s(\\\\widetilde{\\\\mathcal {X}};TM)$ with probability converging to 1.\",\n \"This leads us to conjecture that the natural density version of $\\\\widehat{f}_{\\\\mu }(\\\\cdot )$ , i.e., $\\\\frac{\\\\max \\\\lbrace \\\\widehat{f}_{\\\\mu }(\\\\cdot ),0\\\\rbrace }{\\\\int \\\\max \\\\lbrace \\\\widehat{f}_{\\\\mu }(x),0\\\\rbrace \\\\,dx}$ should serve as a good proxy for $\\\\widetilde{f}_{\\\\mu }^{M^{\\\\prime }}(\\\\cdot )$ and lead to rates of convergence that mitigate the curse of dimensionality.\",\n \"From a computational perspective, the density specified above is easy to simulate from using an accept-reject algorithm without computing the integral in the denominator (see [110]).\",\n \"However, our current proof technique does not provide rates of convergence for the above density estimator based on $\\\\widehat{f}_{\\\\mu }(\\\\cdot )$ .\",\n \"Another important implication of thm:smooth is the bound obtained on $|W_2(\\\\widetilde{\\\\mu }_m,\\\\widetilde{\\\\nu }_n)-W_2(\\\\mu ,\\\\nu )|$ when $\\\\mu \\\\ne \\\\nu $ .\",\n \"We first present the result and then describe the implication.\",\n \"Proposition 2.6 Consider the setting in thm:smooth.\",\n \"Then, provided $\\\\mu \\\\ne \\\\nu $ , the following holds: $|W_2(\\\\widetilde{\\\\mu }_m,\\\\widetilde{\\\\nu }_n)-W_2(\\\\mu ,\\\\nu )|=O_p(r_{d,s}^{(m,n)}).$ prop:wassrate (see sec:mainrespf for a proof) shows an interesting distinction between the $\\\\mu \\\\ne \\\\nu $ case and the $\\\\mu =\\\\nu $ case.\",\n \"For $\\\\mu =\\\\nu $ , the best possible exponent is $n^{-\\\\frac{1+s}{2s+d}}$ for $d\\\\ge 3$ (see [143] where the result was established under more general Besov smoothness assumptions).\",\n \"On the contrary, when $\\\\mu \\\\ne \\\\nu $ , prop:wassrate establishes a rate of $n^{-\\\\frac{s+2}{d}}$ for the Wasserstein distance which is strictly better than the minimax achievable rate mentioned above when $\\\\mu =\\\\nu $ .\",\n \"This observation complements [33] where the authors make a similar observation for the special case of $s=0$ .\"\n ],\n [\n \"Discretized plug-in estimator under smoothness assumptions\",\n \"In sec:nsmooth, we discussed how smoothness can be incorporated into the plug-in procedure to get faster rates of convergence.\",\n \"Such plug-in estimators are popular in the computational OT literature (see [8], [9], [28], [39]).\",\n \"However, even after $\\\\widetilde{f}_{\\\\mu }(\\\\cdot )$ , $\\\\widetilde{f}_{\\\\nu }(\\\\cdot )$ are calculated, $\\\\widetilde{T}_{m,n}^{\\\\gamma }$ as in thm:smooth cannot be computed explicitly from data if $\\\\widetilde{f}_{\\\\mu }(\\\\cdot )$ and $\\\\widetilde{f}_{\\\\nu }(\\\\cdot )$ are continuous densities.\",\n \"This is in contrast to $\\\\widetilde{T}_{m,n}^{\\\\gamma }$ from thm:nsmooth in the discrete-discrete case which is explicitly computable using a standard linear program, but achieves worse rates of convergence.\",\n \"This is not unexpected.\",\n \"Thanks to the no free lunch principle, better statistical accuracy is naturally accompanied by heavier computational challenges.\",\n \"Therefore, our goal here is to construct estimators, under smoothness assumptions as in sec:smooth, which are computable in polynomial time (with complexity increasing with smoothness) provided $\\\\widetilde{f}_{\\\\mu }(\\\\cdot )$ and $\\\\widetilde{f}_{\\\\nu }(\\\\cdot )$ can be sampled from, and also attain rates that mitigate the curse of dimensionality.\",\n \"Construction: We will illustrate the discretized estimator using the kernel based estimator from sec:kernel.\",\n \"Similar results also hold for the wavelet based estimator from sec:wavelet.\",\n \"Recall the kernel density estimators $\\\\widetilde{f}_{\\\\mu }(\\\\cdot )$ and $\\\\widetilde{f}_{\\\\nu }(\\\\cdot )$ (see (REF )).\",\n \"Sample $M\\\\ge 1$ random points from both $\\\\widetilde{f}_{\\\\mu }(\\\\cdot )$ and $\\\\widetilde{f}_{\\\\nu }(\\\\cdot )$ .\",\n \"Let $\\\\widehat{\\\\mu }_{m,M}$ and $\\\\widehat{\\\\nu }_{n,M}$ denote the standard empirical measures on the $M$ points sampled from $\\\\widetilde{f}_{\\\\mu }(\\\\cdot )$ and $\\\\widetilde{f}_{\\\\nu }(\\\\cdot )$ respectively.\",\n \"Finally construct $\\\\widetilde{T}_{m,n}\\\\equiv \\\\widetilde{T}_{m,n}^{\\\\gamma }$ as in def:bcenterproj with $\\\\widetilde{\\\\mu }_m=\\\\widehat{\\\\mu }_{m,M}$ and $\\\\widetilde{\\\\nu }_n=\\\\widehat{\\\\nu }_{n,M}$ .\",\n \"It should be pointed out that a similar construction was also used in [143] for estimating probability densities under the Wasserstein loss.\",\n \"Based on this construction, the main result of this section is as follows: Theorem 2.7 Consider the setting in thm:smooth and the same construction of $\\\\widetilde{T}_{m,n}^{\\\\gamma }$ as above.\",\n \"For simplicity, let's also assume $m=n$ .\",\n \"Accordingly set $M=n^{\\\\frac{s+2}{2}}$ .\",\n \"Then $\\\\widetilde{\\\\Gamma }_{\\\\mathrm {min}}$ is a singleton and consequently the following conclusion holds for some constant $C>0$ : $\\\\mathbb {E}\\\\left[\\\\int \\\\Vert \\\\widetilde{T}_{m,n}(x)-T_0(x)\\\\Vert ^2\\\\,d\\\\widetilde{\\\\mu }_m(x)\\\\right]\\\\le C r_{d,s}^{(n,n)}.$ The same rates also hold for $\\\\mathbb {E}|W_2^2(\\\\widetilde{\\\\mu }_m,\\\\widetilde{\\\\nu }_n)-W_2^2(\\\\mu ,\\\\nu )|$ .\",\n \"The proof of thm:dismooth is given in sec:mainrespf.\",\n \"Once the empirical measures $\\\\widehat{\\\\mu }_{m,M}$ and $\\\\widehat{\\\\nu }_{n,M}$ have been obtained, an explicit computation of $\\\\widetilde{T}_{m,n}$ as described above requires $O(M^3)=O(n^{\\\\frac{3(s+2)}{2}})$ steps using the Hungarian algorithm, see [81].\",\n \"This highlights the statistical versus computational trade off, i.e., in order to mitigate the curse of dimensionality in convergence rates by exploiting smoothness, the computational complexity gets progressively worse by polynomial factors in $n$ .\",\n \"It should be mentioned that (approximate) algorithms faster than the Hungarian algorithm stated above, can be found in [59], [1], [39] to name a few.\",\n \"Due to space constraints, we avoid a detailed discussion on this.\",\n \"In the above construction, sampling from the smoothed kernel densities $\\\\widetilde{f}_{\\\\mu }(\\\\cdot )$ and $\\\\widetilde{f}_{\\\\nu }(\\\\cdot )$ is crucial.\",\n \"If we would simply draw $M$ bootstrap samples from the empirical distributions $\\\\widehat{\\\\mu }_m$ and $\\\\widehat{\\\\nu }_n$ , the rates of convergence wouldn't improve from those observed in thm:nsmooth no matter how large $M$ is chosen.\",\n \"In this section, we will apply our results to two popular problems, namely — estimating the Wasserstein barycenter between two probability distributions (see [2], [40], [15], [27]) in sec:bar, and obtaining detection thresholds in some recent optimal transport based independence testing procedures (see [44], [43], [61], [123], [124]) in sec:nonptest.\"\n ],\n [\n \"Wasserstein barycenter estimation\",\n \"Let $\\\\mu ,\\\\nu \\\\in \\\\mathcal {P}_2(\\\\mathbb {R}^d)$ .\",\n \"The Wasserstein barycenter between $\\\\mu $ and $\\\\nu $ is then given by: $\\\\rho _0:=\\\\min \\\\limits _{\\\\rho \\\\in \\\\mathcal {P}_{\\\\mathrm {ac}}(\\\\mathbb {R}^d)} \\\\left(\\\\frac{1}{2}W_2^2(\\\\mu ,\\\\rho )+\\\\frac{1}{2}W_2^2(\\\\rho ,\\\\nu )\\\\right).$ In fact, by prop:bmopt, there exists an optimal transport map $T_0$ from $\\\\mu $ to $\\\\nu $ and by [2], [15], [18], an alternative characterization of $\\\\rho _0$ is as follows: $\\\\rho _0=\\\\left(\\\\frac{1}{2}\\\\mbox{Id}+\\\\frac{1}{2}T_0\\\\right)\\\\#\\\\mu ,\\\\qquad \\\\mbox{where}\\\\qquad \\\\mbox{Id}(x)=x.$ Estimating $\\\\rho _0$ as in (REF ) has attracted significant attention over the past few years in economics [31], [26], Bayesian learning [130], [129], dynamic formulations [35], [30], algorithmic fairness [68], [34], etc.\",\n \"The most natural strategy employed in estimating $\\\\rho _0$ is to use the empirical plug-in estimator, i.e., replacing $\\\\mu ,\\\\nu $ in (REF ) with $\\\\widehat{\\\\mu }_m,\\\\widehat{\\\\nu }_n$ .\",\n \"This strategy has been used, approximated and analyzed extensively in e.g., [40], [27], [93], [18].\",\n \"Based on (REF ), the natural plug-in estimator of $\\\\rho _0$ would be: $\\\\widehat{\\\\rho }_0^{\\\\gamma }=\\\\left(\\\\frac{1}{2}\\\\mbox{Id}+\\\\frac{1}{2}\\\\widetilde{T}_{m,n}^{\\\\gamma }\\\\right)\\\\#\\\\widetilde{\\\\mu }_m$ where $\\\\widetilde{T}_{m,n}^{\\\\gamma }$ is the plug-in estimator of $T_0$ obtained by solving (REF ), with $\\\\mu $ and $\\\\nu $ replaced by $\\\\widetilde{\\\\mu }_m$ and $\\\\widehat{\\\\nu }_n$ respectively and $\\\\gamma \\\\in \\\\widetilde{\\\\Gamma }_{\\\\mathrm {min}}$ .\",\n \"While the consistency of $\\\\widehat{\\\\rho }_0^{\\\\gamma }$ has been analyzed for $m=n$ in [93] and rates have been obtained for $d=1$ in [14], the more general question of obtaining rates of convergence for $\\\\widehat{\\\\rho }_0^{\\\\gamma }$ for general dimensions $d\\\\ge 1$ is yet unanswered.\",\n \"We address this question in the following result (see sec:Appf for a proof).\",\n \"Theorem 3.1 Suppose that the same assumptions from thm:nsmooth hold.\",\n \"Then, with $\\\\widehat{\\\\rho }_0^{\\\\gamma }$ as defined in (REF ) and $r_d^{(m,n)}$ , $t_{d,\\\\alpha }$ defined in thm:nsmooth, the following holds: $\\\\sup \\\\limits _{\\\\gamma \\\\in \\\\widetilde{\\\\Gamma }_{\\\\mathrm {min}}}W_2^2(\\\\widehat{\\\\rho }_0^{\\\\gamma },\\\\rho _0)=O_p\\\\left(r^{(m,n)}_d\\\\times (\\\\log {(1+\\\\max \\\\lbrace m,n\\\\rbrace )})^{t_{d,\\\\alpha }}\\\\right).$\"\n ],\n [\n \"Nonparametric independence testing: Optimal transport based Hilbert-Schmidt independence criterion\",\n \"Let $(X_1,Y_1),\\\\ldots ,(X_n,Y_n)\\\\overset{i.i.d.\",\n \"}{\\\\sim }\\\\pi $ , a probability measure on $\\\\mathbb {R}^{d_1+d_2}$ , with marginals $\\\\mu \\\\in \\\\mathcal {P}_{\\\\mathrm {ac}}(\\\\mathbb {R}^{d_1})$ and $\\\\nu \\\\in \\\\mathcal {P}_{\\\\mathrm {ac}}(\\\\mathbb {R}^{d_2})$ .\",\n \"Our problem of interest is the following hypothesis testing problem, given as: $\\\\mathrm {H}_0:\\\\pi =\\\\mu \\\\otimes \\\\nu \\\\qquad \\\\mbox{versus} \\\\qquad \\\\mathrm {H}_1:\\\\pi \\\\ne \\\\mu \\\\otimes \\\\nu .$ This is the classical nonparametric independence testing problem which has received a lot of attention in the statistics and machine learning literature (see [132], [71], [78], [12], and [48], [82] for a review).\",\n \"In keeping with the overall theme of this paper, our focus here will be on a large class of OT based independence testing procedures, introduced first in [44] followed by recent developments in [123], [124], [43].\",\n \"These tests bear resemblance to the Hilbert-Schmidt independence criterion (HSIC); see [71], [69], [70] and have attractive properties such as distribution-freeness (see prop:testprop), consistency without moment assumptions and robustness against heavy-tailed distributions and against contamination [44], [123].\",\n \"Below, we describe this class of tests, see (REF ) and (REF ).\",\n \"Our main theoretical contribution of this section will be to provide detection thresholds of these OT based tests.\",\n \"Construction: Suppose $\\\\upsilon _1$ , $\\\\upsilon _2$ be two compactly supported probability distributions on $\\\\mathbb {R}^{d_1}$ and $\\\\mathbb {R}^{d_2}$ respectively (e.g., $\\\\upsilon _1\\\\equiv \\\\mathrm {Unif}[0,1]^{d_1}$ , $\\\\upsilon _2\\\\equiv \\\\mathrm {Unif}[0,1]^{d_2}$ ).\",\n \"Let $U_1,\\\\ldots ,U_n\\\\overset{i.i.d.\",\n \"}{\\\\sim }\\\\upsilon _1$ , $V_1,\\\\ldots ,V_n\\\\overset{i.i.d.\",\n \"}{\\\\sim }\\\\upsilon _2$ , $\\\\widehat{u}_n:=n^{-1}\\\\sum _{i=1}^n \\\\delta _{U_i}$ and $\\\\widehat{v}_n:=n^{-1}\\\\sum _{j=1}^n \\\\delta _{V_j}$ .\",\n \"Recall the definitions of $\\\\widehat{\\\\mu }_n$ (with $m=n$ ) and $\\\\widehat{\\\\nu }_n$ from (REF ).\",\n \"Let $\\\\widehat{T}_{1,n}$ ($\\\\widehat{T}_{2,n}$ ) be obtained by solving (REF ), with $\\\\mu $ and $\\\\nu $ replaced by $\\\\widehat{\\\\mu }_n$ and $\\\\widehat{u}_n$ ($\\\\widehat{\\\\nu }_n$ and $\\\\widehat{v}_n$ ) respectively.\",\n \"Consider two non negative definite, continuous, characteristic kernels (see [58], [128] for definitions) $K_1(\\\\cdot ,\\\\cdot )$ and $K_2(\\\\cdot ,\\\\cdot )$ on $(\\\\mbox{supp}(\\\\upsilon _1))^2$ and $(\\\\mbox{supp}(\\\\upsilon _2))^2$ .\",\n \"Set $\\\\widehat{x}_{ij}:=K_1(\\\\widehat{T}_{1,n}(X_i),\\\\widehat{T}_{1,n}(X_j))$ and $\\\\widehat{y}_{ij}:=K_2(\\\\widehat{T}_{2,n}(Y_i),\\\\widehat{T}_{2,n}(Y_j))$ .\",\n \"Our test statistic is as follows: $\\\\widehat{\\\\mathrm {rHSIC}}:=n^{-2}\\\\sum _{i,j}\\\\widehat{x}_{ij}\\\\widehat{y}_{ij}+n^{-4}\\\\sum _{i,j,r,s}\\\\widehat{x}_{ij}\\\\widehat{y}_{rs}-2n^{-3}\\\\sum _{i,j,r} \\\\widehat{x}_{ij}\\\\widehat{y}_{ir}.$ Proposition 3.2 (See [44], [124]) (Distribution-freeness) When $X_1$ and $Y_1$ are independent, the distribution of $n\\\\times \\\\widehat{\\\\mathrm {rHSIC}}$ is universal, i.e., it does not depend on $\\\\mu $ and $\\\\nu $ for every fixed $n$ .\",\n \"(Consistency against fixed alternatives) Let $c_{n,\\\\alpha }$ be the upper $(1-\\\\alpha )$ -th quantile from the universal distribution in part 1 above.\",\n \"Then $\\\\widehat{\\\\mathrm {rHSIC}}\\\\overset{P}{\\\\longrightarrow }\\\\mathrm {rHSIC}(\\\\pi |\\\\mu \\\\otimes \\\\nu )$ where $&\\\\mathrm {rHSIC}(\\\\pi | \\\\mu \\\\otimes \\\\nu ):=\\\\mathbb {E}[K_1(T_1(X_1),T_1(X_2))K_2(T_2(Y_1),T_2(Y_2))]+\\\\mathbb {E}[K_1(T_1(X_1),T_1(X_2))]\\\\nonumber \\\\\\\\&\\\\times \\\\mathbb {E}[K_2(T_2(Y_1),T_2(Y_2))]-2\\\\mathbb {E}[K_1(T_1(X_1),T_1(X_2))K_2(T_2(Y_1),T_2(Y_3))],$ where $T_{1}(\\\\cdot )$ (respectively $T_2(\\\\cdot )$ ) is the optimal transport map from $\\\\mu $ ($\\\\nu $ ) to $\\\\upsilon _1$ ($\\\\upsilon _2$ ); see def:otmpm.\",\n \"Further $\\\\mathrm {rHSIC}(\\\\pi |\\\\mu \\\\otimes \\\\nu )=0$ if and only if $\\\\pi =\\\\mu \\\\otimes \\\\nu $ .\",\n \"Define the following test function: $\\\\phi _{n,\\\\alpha }:=\\\\mathbb {1}(n\\\\times \\\\widehat{\\\\mathrm {rHSIC}}\\\\ge c_{n,\\\\alpha }).$ Then $\\\\mathbb {E}[\\\\phi _{n,\\\\alpha }]\\\\rightarrow 1$ as $n\\\\rightarrow \\\\infty $ under $\\\\mathrm {H}_1$ , i.e., when $\\\\pi \\\\ne \\\\mu \\\\otimes \\\\nu $ .\",\n \"prop:testprop shows that the test based on $\\\\widehat{\\\\mathrm {rHSIC}}$ (see (REF )), i.e., $\\\\phi _{n,\\\\alpha }$ (see (REF )), can be carried out without resorting to the permutation principle as is necessary for the usual HSIC based test (see [69]).\",\n \"Further, when the sampling distribution is fixed, prop:testprop shows that $\\\\widehat{\\\\mathrm {rHSIC}}$ consistently estimates $\\\\mathrm {rHSIC}(\\\\pi |\\\\mu \\\\otimes \\\\nu )$ , a quantity which equals 0 if and only if $\\\\pi =\\\\mu \\\\otimes \\\\nu $ (this yields the consistency of $\\\\phi _{n,\\\\alpha })$ against fixed alternatives.\",\n \"While consistency against fixed alternatives is an attractive feature of $\\\\phi _{n,\\\\alpha }$ , a more intricate question of statistical interest is to understand the local power of $\\\\phi _{n,\\\\alpha }$ under “changing sequence of alternatives converging to the null\\\" as $n\\\\rightarrow \\\\infty $ .\",\n \"To study the local power of $\\\\phi _{n,\\\\alpha }$ , we need to consider a triangular array setting, where the data distribution changes with $n$ , i.e., $(X_1,Y_1),\\\\ldots ,(X_n,Y_n)\\\\overset{i.i.d.\",\n \"}{\\\\sim }\\\\pi ^{(n)}$ , a probability measure on $\\\\mathbb {R}^{d_1+d_2}$ , with marginals $\\\\mu ^{(n)}\\\\in \\\\mathcal {P}_{\\\\mathrm {ac}}(\\\\mathbb {R}^{d_1})$ and $\\\\nu ^{(n)}\\\\in \\\\mathcal {P}_{\\\\mathrm {ac}}(\\\\mathbb {R}^{d_2})$ .\",\n \"As $\\\\mathrm {rHSIC}(\\\\cdot |\\\\cdot )$ characterizes independence, a mathematical formulation of “alternatives converging to null\\\" would be to say $\\\\mathrm {rHSIC}(\\\\pi ^{(n)}|\\\\mu ^{(n)}\\\\otimes \\\\nu ^{(n)})\\\\rightarrow 0$ as $n\\\\rightarrow \\\\infty $ .\",\n \"Similar questions have attracted a lot of attention in modern statistics, featuring measures (other than $\\\\mathrm {rHSIC}(\\\\cdot |\\\\cdot )$ ) which characterize independence, see e.g., [12], [87], [94], [6].\",\n \"In the following result (see sec:Appf for a proof), we show that if $\\\\mathrm {rHSIC}(\\\\pi ^{(n)}|\\\\mu ^{(n)}\\\\otimes \\\\nu ^{(n)})\\\\rightarrow 0$ slowly enough with $n$ , then $\\\\phi _{n,\\\\alpha }$ yields a consistent sequence of tests for problem (REF ).\",\n \"Theorem 3.3 Consider problem (REF ) with $\\\\pi ^{(n)}$ , $\\\\mu ^{(n)}$ , $\\\\nu ^{(n)}$ (changing with $n$ ) and suppose $T_{1,n}(\\\\cdot )$ and $T_{2,n}(\\\\cdot )$ are both $L$ -Lipschitz ($L$ is free of $n$ ).\",\n \"Also assume $K_1(\\\\cdot )$ , $K_2(\\\\cdot )$ are Lipschitz, $\\\\mu ^{(n)}$ , $\\\\nu ^{(n)}$ are supported on fixed compact sets (supports are free of $n$ ).\",\n \"Set $r_{d_1,d_2}^{(n,n)}:=r_{d_1}^{(n,n)}+r_{d_2}^{(n,n)}$ where $r_{d_1}^{(n,n)},r_{d_2}^{(n,n)}$ is defined via (REF ).\",\n \"Then, $\\\\mathbb {E}[\\\\phi _{n,\\\\alpha }]\\\\rightarrow 1\\\\qquad \\\\mbox{if} \\\\qquad (r_{d_1,d_2}^{(n,n)})^{-1/2}\\\\times \\\\mathrm {rHSIC}(\\\\pi ^{(n)}|\\\\mu ^{(n)}\\\\otimes \\\\nu ^{(n)})\\\\rightarrow \\\\infty ,$\"\n ],\n [\n \"Appendix\",\n \"This section is devoted to proving our main results and is organized as follows: In sec:mainrespf, we present the proofs of results from sec:mainres and in sec:Appf, we present the proofs from sec:App.\",\n \"Throughout this section, we will use the $\\\\lesssim $ sign to hide constants that are free of $m,n$ .\"\n ],\n [\n \"Proofs from sec:mainres\",\n \"[Proof of thm:newubd] We begin the proof by observing that $\\\\varphi _0^*(\\\\cdot )$ is convex and finite on $\\\\mbox{supp}(\\\\nu )$ , and hence differentiable $\\\\nu $ almost everywhere (a.e.).\",\n \"Further by lem:gradual, we also have: $\\\\nabla \\\\varphi _0^*(T_0(x))=x\\\\qquad \\\\mbox{$\\\\mu $-a.e.~$x$.\",\n \"}$ Fix any arbitrary $\\\\gamma \\\\in \\\\widetilde{\\\\Gamma }_{\\\\mathrm {min}}$ and suppose that $\\\\gamma (y|x)$ denotes the conditional distribution of $y$ given $x$ under $\\\\gamma $ .\",\n \"Define, $D_1:= \\\\int \\\\varphi _0^*(y)\\\\,d\\\\widetilde{\\\\nu }_n(y) - \\\\int \\\\varphi _0^*(y) d \\\\overline{\\\\nu }_m(y).$ As $\\\\gamma $ has marginals $\\\\widetilde{\\\\mu }_m$ and $\\\\widetilde{\\\\nu }_n$ , we have: $D_1=\\\\int _{x,y} \\\\varphi _0^*(y)\\\\,d\\\\gamma (y|x)\\\\,d\\\\widetilde{\\\\mu }_m(x)-\\\\int _{x} \\\\varphi _0^*(T_0(x))\\\\,d\\\\widetilde{\\\\mu }_m(x).$ Next, by applying the conditional version of Jensen's inequality, $\\\\int _{x} \\\\left(\\\\int _{y} \\\\varphi _0^*(y)\\\\,d\\\\gamma (y|x)\\\\right)\\\\,d\\\\widetilde{\\\\mu }_m(x)&\\\\ge \\\\int _{x}\\\\varphi _0^*\\\\left(\\\\int _{y} y\\\\,d\\\\gamma (y|x)\\\\right)\\\\,d\\\\widetilde{\\\\mu }_m(x)\\\\nonumber \\\\\\\\&=\\\\int _{x}\\\\varphi _0^*(\\\\widetilde{T}_{m,n}^{\\\\gamma }(x))\\\\,d\\\\widetilde{\\\\mu }_m(x).$ Using (REF ) with (REF ) yields, $D_1&\\\\ge \\\\int [\\\\varphi _0^*(\\\\widetilde{T}_{m,n}^{\\\\gamma }(x)) - \\\\varphi _0^*(T_0(x))] d \\\\widetilde{\\\\mu }_m(x) \\\\nonumber \\\\\\\\& \\\\overset{(a)}{\\\\ge } \\\\int \\\\left\\\\lbrace \\\\nabla \\\\varphi _0^*(T_0(x))^\\\\top (\\\\widetilde{T}_{m,n}^{\\\\gamma }(x) - T_0(x)) + \\\\frac{1}{2L} \\\\Vert \\\\widetilde{T}_{m,n}^{\\\\gamma }(x)-T_0(x)\\\\Vert ^2 \\\\right\\\\rbrace \\\\,d {\\\\widetilde{\\\\mu }_m}(x) \\\\nonumber \\\\\\\\& \\\\overset{(b)}{=} \\\\underbrace{\\\\int x^\\\\top (\\\\widetilde{T}_{m,n}^{\\\\gamma }(x) - T_0(x)) \\\\,d {\\\\widetilde{\\\\mu }_m}(x)}_{D_2} + \\\\frac{1}{2L} \\\\int \\\\Vert \\\\widetilde{T}_{m,n}^{\\\\gamma }(x)-T_0(x)\\\\Vert ^2 \\\\, d {\\\\widetilde{\\\\mu }_m}(x).$ Here (a) follows from the strong convexity of $\\\\varphi _0^*(\\\\cdot )$ with parameter $(1/L)$ (see lem:assm) and (b) follows from (REF ).\",\n \"Next, we will simplify the term $D_2$ .\",\n \"Towards this direction, observe that for every $\\\\gamma \\\\in \\\\widetilde{\\\\Gamma }_{\\\\mathrm {min}}$ , $W_2^2(\\\\widetilde{\\\\mu }_m,\\\\widetilde{\\\\nu }_n)&=\\\\int \\\\Vert x-y\\\\Vert ^2\\\\,d\\\\gamma (x,y)\\\\nonumber \\\\\\\\ &=\\\\int \\\\Vert x\\\\Vert ^2\\\\,d\\\\widetilde{\\\\mu }_m(x)+\\\\int \\\\Vert y\\\\Vert ^2\\\\,d\\\\widetilde{\\\\nu }_n(y)-2\\\\int _{x} \\\\left(x^{\\\\top }\\\\int _{y} y\\\\,d\\\\gamma (y|x)\\\\right)\\\\,d\\\\widetilde{\\\\mu }_m(x)\\\\nonumber \\\\\\\\&=\\\\int \\\\Vert x\\\\Vert ^2\\\\,d\\\\widetilde{\\\\mu }_m(x)+\\\\int \\\\Vert y\\\\Vert ^2\\\\,d\\\\widetilde{\\\\nu }_n(y)-2\\\\int _{x} x^{\\\\top }\\\\widetilde{T}_{m,n}^{\\\\gamma }(x)\\\\,d\\\\widetilde{\\\\mu }_m(x).$ Also, as $T_0$ is the gradient of a convex function, it is also an OT map from $\\\\widetilde{\\\\mu }_m$ to $\\\\overline{\\\\nu }_m$ (see [3]), we have: $W_2^2(\\\\widetilde{\\\\mu }_m,\\\\overline{\\\\nu }_m)&=\\\\int \\\\Vert x-T_0(x)\\\\Vert ^2\\\\,d\\\\widetilde{\\\\mu }_m(x)\\\\nonumber \\\\\\\\ &=\\\\int \\\\Vert x\\\\Vert ^2\\\\,d\\\\widetilde{\\\\mu }_m(x)+\\\\int \\\\Vert y\\\\Vert ^2\\\\,d\\\\overline{\\\\nu }_m(y)-2\\\\int _{x} x^{\\\\top }T_0(x)\\\\,d\\\\widetilde{\\\\mu }_m(x).$ Now (REF ) and (REF ) imply $D_2=\\\\frac{1}{2}\\\\big (W_2^2(\\\\widetilde{\\\\mu }_m,\\\\overline{\\\\nu }_m)-W_2^2(\\\\widetilde{\\\\mu }_m,\\\\widetilde{\\\\nu }_n)\\\\big )+\\\\frac{1}{2}\\\\int \\\\Vert y\\\\Vert ^2\\\\,d(\\\\widetilde{\\\\nu }_n-\\\\overline{\\\\nu }_m)(y).$ Finally by combining (REF ) and (REF ), we get: $&\\\\;\\\\;\\\\;\\\\;\\\\frac{1}{2L}\\\\int \\\\Vert \\\\widetilde{T}_{m,n}^{\\\\gamma }(x)-T_0(x)\\\\Vert ^2\\\\,d\\\\widetilde{\\\\mu }_m(x)\\\\nonumber \\\\\\\\ &\\\\le \\\\frac{1}{2}\\\\big (W_2^2(\\\\widetilde{\\\\mu }_m,\\\\widetilde{\\\\nu }_n)-W_2^2(\\\\widetilde{\\\\mu }_m,\\\\overline{\\\\nu }_m)\\\\big )+\\\\int (\\\\varphi _0^*(y)-(1/2)\\\\Vert y\\\\Vert ^2)\\\\,d(\\\\widetilde{\\\\nu }_n-\\\\overline{\\\\nu }_m)(y).$ Now note that the bound on the right hand side of the above display is free of the particular choice of $\\\\gamma \\\\in \\\\widetilde{\\\\Gamma }_{\\\\mathrm {min}}$ .\",\n \"Therefore, the same bound holds if we take a supremum over $\\\\gamma \\\\in \\\\widetilde{\\\\Gamma }_{\\\\mathrm {min}}$ on the left hand side.\",\n \"We will now provide an upper bound for the right hand side of (REF ).\",\n \"The remainder of the proof proceeds as in the proof of [101].\",\n \"By the dual representation presented in (REF ) and (), and the definitions of $\\\\Psi _{\\\\widetilde{\\\\mu }_m,\\\\widetilde{\\\\nu }_n}(\\\\cdot )$ and $\\\\Psi _{\\\\widetilde{\\\\mu }_m,\\\\overline{\\\\nu }_m}(\\\\cdot )$ in the statement of thm:newubd, we have $\\\\frac{1}{2}W_2^2(\\\\widetilde{\\\\mu }_m,\\\\widetilde{\\\\nu }_n)=\\\\frac{1}{2}\\\\int \\\\Vert x\\\\Vert ^2\\\\,d\\\\widetilde{\\\\mu }_m(x)+\\\\frac{1}{2}\\\\int \\\\Vert y\\\\Vert ^2\\\\,d\\\\widetilde{\\\\nu }_n(y)-\\\\mathcal {S}_{\\\\widetilde{\\\\mu }_m,\\\\widetilde{\\\\nu }_n}(\\\\Psi _{\\\\widetilde{\\\\mu }_m,\\\\widetilde{\\\\nu }_n}),$ $\\\\mbox{and}\\\\qquad \\\\frac{1}{2}W_2^2(\\\\widetilde{\\\\mu }_m,\\\\overline{\\\\nu }_m)=\\\\frac{1}{2}\\\\int \\\\Vert x\\\\Vert ^2\\\\,d\\\\widetilde{\\\\mu }_m(x)+\\\\frac{1}{2}\\\\int \\\\Vert y\\\\Vert ^2\\\\,d\\\\overline{\\\\nu }_m(y)-\\\\mathcal {S}_{\\\\widetilde{\\\\mu }_m,\\\\overline{\\\\nu }_m}(\\\\Psi _{\\\\widetilde{\\\\mu }_m,\\\\overline{\\\\nu }_m}).$ By subtracting the two equations above, we get: $\\\\frac{1}{2}W_2^2(\\\\widetilde{\\\\mu }_m,\\\\widetilde{\\\\nu }_n)-\\\\frac{1}{2}W_2^2(\\\\widetilde{\\\\mu }_m,\\\\overline{\\\\nu }_m)=\\\\frac{1}{2}\\\\int \\\\Vert y\\\\Vert ^2\\\\,d(\\\\widetilde{\\\\nu }_n-\\\\overline{\\\\nu }_m)-\\\\mathcal {S}_{\\\\widetilde{\\\\mu }_m,\\\\widetilde{\\\\nu }_n}(\\\\Psi _{\\\\widetilde{\\\\mu }_m,\\\\widetilde{\\\\nu }_n})+\\\\mathcal {S}_{\\\\widetilde{\\\\mu }_m,\\\\overline{\\\\nu }_m}(\\\\Psi _{\\\\widetilde{\\\\mu }_m,\\\\overline{\\\\nu }_m}).$ Next, we use () to make the following observations: $\\\\mathcal {S}_{\\\\widetilde{\\\\mu }_m,\\\\widetilde{\\\\nu }_n}(\\\\Psi _{\\\\widetilde{\\\\mu }_m,\\\\widetilde{\\\\nu }_n})\\\\le \\\\mathcal {S}_{\\\\widetilde{\\\\mu }_m,\\\\widetilde{\\\\nu }_n}(\\\\Psi _{\\\\widetilde{\\\\mu }_m,\\\\overline{\\\\nu }_m}),\\\\qquad \\\\mathcal {S}_{\\\\widetilde{\\\\mu }_m,\\\\overline{\\\\nu }_m}(\\\\Psi _{\\\\widetilde{\\\\mu }_m,\\\\overline{\\\\nu }_m})\\\\le \\\\mathcal {S}_{\\\\widetilde{\\\\mu }_m,\\\\overline{\\\\nu }_m}(\\\\Psi _{\\\\widetilde{\\\\mu }_m,\\\\widetilde{\\\\nu }_n}).$ Note that (REF ) immediately yields the following conclusions: $\\\\mathcal {S}_{\\\\widetilde{\\\\mu }_m,\\\\overline{\\\\nu }_m}(\\\\Psi _{\\\\widetilde{\\\\mu }_m,\\\\overline{\\\\nu }_m})-\\\\mathcal {S}_{\\\\widetilde{\\\\mu }_m,\\\\widetilde{\\\\nu }_n}(\\\\Psi _{\\\\widetilde{\\\\mu }_m,\\\\overline{\\\\nu }_m})\\\\le \\\\mathcal {S}_{\\\\widetilde{\\\\mu }_m,\\\\overline{\\\\nu }_m}(\\\\Psi _{\\\\widetilde{\\\\mu }_m,\\\\overline{\\\\nu }_m})-\\\\mathcal {S}_{\\\\widetilde{\\\\mu }_m,\\\\widetilde{\\\\nu }_n}(\\\\Psi _{\\\\widetilde{\\\\mu }_m,\\\\widetilde{\\\\nu }_n}),$ and $\\\\mathcal {S}_{\\\\widetilde{\\\\mu }_m,\\\\overline{\\\\nu }_m}(\\\\Psi _{\\\\widetilde{\\\\mu }_m,\\\\overline{\\\\nu }_m})-\\\\mathcal {S}_{\\\\widetilde{\\\\mu }_m,\\\\widetilde{\\\\nu }_n}(\\\\Psi _{\\\\widetilde{\\\\mu }_m,\\\\widetilde{\\\\nu }_n})\\\\le \\\\mathcal {S}_{\\\\widetilde{\\\\mu }_m,\\\\overline{\\\\nu }_m}(\\\\Psi _{\\\\widetilde{\\\\mu }_m,\\\\widetilde{\\\\nu }_n})-\\\\mathcal {S}_{\\\\widetilde{\\\\mu }_m,\\\\widetilde{\\\\nu }_n}(\\\\Psi _{\\\\widetilde{\\\\mu }_m,\\\\widetilde{\\\\nu }_n}).$ By combining the above two displays, we have: $&\\\\;\\\\;\\\\;\\\\;\\\\left|\\\\mathcal {S}_{\\\\widetilde{\\\\mu }_m,\\\\overline{\\\\nu }_m}(\\\\Psi _{\\\\widetilde{\\\\mu }_m,\\\\overline{\\\\nu }_m})-\\\\mathcal {S}_{\\\\widetilde{\\\\mu }_m,\\\\widetilde{\\\\nu }_n}(\\\\Psi _{\\\\widetilde{\\\\mu }_m,\\\\widetilde{\\\\nu }_n})\\\\right|\\\\nonumber \\\\\\\\ &\\\\le \\\\max \\\\left\\\\lbrace \\\\left|\\\\mathcal {S}_{\\\\widetilde{\\\\mu }_m,\\\\overline{\\\\nu }_m}(\\\\Psi _{\\\\widetilde{\\\\mu }_m,\\\\widetilde{\\\\nu }_n})-\\\\mathcal {S}_{\\\\widetilde{\\\\mu }_m,\\\\widetilde{\\\\nu }_n}(\\\\Psi _{\\\\widetilde{\\\\mu }_m,\\\\widetilde{\\\\nu }_n})\\\\right|,\\\\left|\\\\mathcal {S}_{\\\\widetilde{\\\\mu }_m,\\\\overline{\\\\nu }_m}(\\\\Psi _{\\\\widetilde{\\\\mu }_m,\\\\overline{\\\\nu }_m})-\\\\mathcal {S}_{\\\\widetilde{\\\\mu }_m,\\\\widetilde{\\\\nu }_n}(\\\\Psi _{\\\\widetilde{\\\\mu }_m,\\\\overline{\\\\nu }_m})\\\\right|\\\\right\\\\rbrace .$ By () and some simple algebra, the following holds: $ \\\\left|\\\\mathcal {S}_{\\\\widetilde{\\\\mu }_m,\\\\overline{\\\\nu }_m}(\\\\Psi _{\\\\widetilde{\\\\mu }_m,\\\\widetilde{\\\\nu }_n})-\\\\mathcal {S}_{\\\\widetilde{\\\\mu }_m,\\\\widetilde{\\\\nu }_n}(\\\\Psi _{\\\\widetilde{\\\\mu }_m,\\\\widetilde{\\\\nu }_n})\\\\right|=\\\\left|\\\\int \\\\Psi _{\\\\widetilde{\\\\mu }_m,\\\\widetilde{\\\\nu }_n}^*\\\\,d(\\\\overline{\\\\nu }_m-\\\\widetilde{\\\\nu }_n)\\\\right|.$ A similar expression holds for $|\\\\mathcal {S}_{\\\\widetilde{\\\\mu }_m,\\\\overline{\\\\nu }_m}(\\\\Psi _{\\\\widetilde{\\\\mu }_m,\\\\overline{\\\\nu }_m})-\\\\mathcal {S}_{\\\\widetilde{\\\\mu }_m,\\\\widetilde{\\\\nu }_n}(\\\\Psi _{\\\\widetilde{\\\\mu }_m,\\\\overline{\\\\nu }_m})|$ .\",\n \"Using the above observation in (REF ), we get: $\\\\left|\\\\mathcal {S}_{\\\\widetilde{\\\\mu }_m,\\\\overline{\\\\nu }_m}(\\\\Psi _{\\\\widetilde{\\\\mu }_m,\\\\overline{\\\\nu }_m})-\\\\mathcal {S}_{\\\\widetilde{\\\\mu }_m,\\\\widetilde{\\\\nu }_n}(\\\\Psi _{\\\\widetilde{\\\\mu }_m,\\\\widetilde{\\\\nu }_n})\\\\right|\\\\le \\\\max \\\\left\\\\lbrace \\\\bigg |\\\\int \\\\Psi _{\\\\widetilde{\\\\mu }_m,\\\\widetilde{\\\\nu }_n}^* \\\\,d(\\\\widetilde{\\\\nu }_n-\\\\overline{\\\\nu }_m)\\\\bigg |,\\\\bigg |\\\\int \\\\Psi _{\\\\widetilde{\\\\mu }_m,\\\\overline{\\\\nu }_m}^* \\\\,d(\\\\widetilde{\\\\nu }_n-\\\\overline{\\\\nu }_m)\\\\bigg |\\\\right\\\\rbrace .$ Combining the above display with (REF ), we further have: $&\\\\;\\\\;\\\\;\\\\;\\\\left|\\\\frac{1}{2}W_2^2(\\\\widetilde{\\\\mu }_m,\\\\widetilde{\\\\nu }_n)-\\\\frac{1}{2}W_2^2(\\\\widetilde{\\\\mu }_m,\\\\overline{\\\\nu }_m)-\\\\left(\\\\frac{1}{2}\\\\int \\\\Vert y\\\\Vert ^2\\\\,d(\\\\widetilde{\\\\nu }_n-\\\\overline{\\\\nu }_m)\\\\right)\\\\right|\\\\nonumber \\\\\\\\ &\\\\le \\\\max \\\\left\\\\lbrace \\\\bigg |\\\\int \\\\Psi _{\\\\widetilde{\\\\mu }_m,\\\\widetilde{\\\\nu }_n}^* \\\\,d(\\\\widetilde{\\\\nu }_n-\\\\overline{\\\\nu }_m)\\\\bigg |,\\\\bigg |\\\\int \\\\Psi _{\\\\widetilde{\\\\mu }_m,\\\\overline{\\\\nu }_m}^* \\\\,d(\\\\widetilde{\\\\nu }_n-\\\\overline{\\\\nu }_m)\\\\bigg |\\\\right\\\\rbrace .$ Combining (REF ) with (REF ) then completes the proof.\",\n \"[Proof of thm:nsmooth] First observe that $\\\\limsup _{M\\\\rightarrow \\\\infty }\\\\limsup _{m,n\\\\rightarrow \\\\infty } \\\\mathbb {P}\\\\left(\\\\Big |\\\\int \\\\varphi _0^*\\\\,d(\\\\widehat{\\\\nu }_n-\\\\overline{\\\\nu }_m)\\\\big |\\\\ge M \\\\left(r_d^{(m,m)}+r_d^{(n,n)}\\\\right)\\\\right)=0$ by the weak law of large numbers as $(r_{d}^{(n,n)})^{-1}n^{-1/2}=O(1)$ and $(r_{d}^{(m,m)})^{-1}m^{-1/2}=O(1)$ .\",\n \"Combining the above observation with thm:newubd, we have: $&\\\\;\\\\;\\\\;\\\\;\\\\limsup _{M\\\\rightarrow \\\\infty }\\\\limsup _{m,n\\\\rightarrow \\\\infty }\\\\mathbb {P}\\\\left(\\\\sup \\\\limits _{\\\\gamma \\\\in \\\\widetilde{\\\\Gamma }_{\\\\mathrm {min}}} \\\\int \\\\Vert \\\\widetilde{T}_{m,n}^{\\\\gamma }(x)-T_0(x)\\\\Vert ^2 \\\\,d\\\\widetilde{\\\\mu }_m(x)\\\\ge Mr_d^{(m,n)}\\\\right)\\\\nonumber \\\\\\\\ &\\\\le \\\\limsup _{M\\\\rightarrow \\\\infty }\\\\limsup _{m,n\\\\rightarrow \\\\infty }\\\\mathbb {P}\\\\left(\\\\max \\\\left\\\\lbrace \\\\bigg |\\\\int \\\\Psi _{\\\\widetilde{\\\\mu }_m,\\\\widetilde{\\\\nu }_n}^* \\\\,d(\\\\widetilde{\\\\nu }_n-\\\\overline{\\\\nu }_m)\\\\bigg |,\\\\bigg |\\\\int \\\\Psi _{\\\\widetilde{\\\\mu }_m,\\\\overline{\\\\nu }_m}^* \\\\,d(\\\\widetilde{\\\\nu }_n-\\\\overline{\\\\nu }_m)\\\\bigg |\\\\right\\\\rbrace \\\\ge \\\\frac{M}{2} r_d^{(m,n)}\\\\right)\\\\nonumber \\\\\\\\ &\\\\le \\\\limsup _{M\\\\rightarrow \\\\infty }\\\\limsup _{m,n\\\\rightarrow \\\\infty }\\\\Bigg [\\\\mathbb {P}\\\\left(\\\\left|\\\\int \\\\Psi _{\\\\widetilde{\\\\mu }_m,\\\\widetilde{\\\\nu }_n}^* \\\\,d(\\\\widetilde{\\\\nu }_n-\\\\nu )\\\\right|\\\\ge \\\\frac{M}{2}r_d^{(n,n)}\\\\right)+\\\\mathbb {P}\\\\left(\\\\left|\\\\int \\\\Psi _{\\\\widetilde{\\\\mu }_m,\\\\widetilde{\\\\nu }_n}^* \\\\,d(\\\\overline{\\\\nu }_m-\\\\nu )\\\\right|\\\\ge \\\\frac{M}{2}r_d^{(m,m)}\\\\right)\\\\nonumber \\\\\\\\&+\\\\mathbb {P}\\\\left(\\\\left|\\\\int \\\\Psi _{\\\\widetilde{\\\\mu }_m,\\\\overline{\\\\nu }_m}^* \\\\,d(\\\\widetilde{\\\\nu }_n-\\\\nu )\\\\right|\\\\ge \\\\frac{M}{2} r_d^{(n,n)}\\\\right)+\\\\mathbb {P}\\\\left(\\\\left|\\\\int \\\\Psi _{\\\\widetilde{\\\\mu }_m,\\\\overline{\\\\nu }_m}^* \\\\,d(\\\\overline{\\\\nu }_m-\\\\nu )\\\\right|\\\\ge \\\\frac{M}{2} r_d^{(m,m)}\\\\right)\\\\Bigg ].$ In the sequel, we will only discuss how to bound the first term on the right hand side of (REF ).\",\n \"Once that is understood, the other terms can be bounded similarly.\",\n \"Therefore, our focus is on bounding $\\\\limsup _{M\\\\rightarrow \\\\infty }\\\\limsup _{m,n\\\\rightarrow \\\\infty }\\\\mathbb {P}\\\\left(\\\\left|\\\\int \\\\Psi _{\\\\widetilde{\\\\mu }_m,\\\\widetilde{\\\\nu }_n}^* \\\\,d(\\\\widetilde{\\\\nu }_n-\\\\nu )\\\\right|\\\\ge \\\\frac{M}{2}r_d^{(n,n)}(\\\\log {(1+\\\\max \\\\lbrace m,n\\\\rbrace )})^{t_{d,\\\\alpha }}\\\\right).$ For the next part, to simplify notation, let us begin with some notation.\",\n \"Set $\\\\mathcal {Y}:=\\\\mbox{supp}(\\\\nu )$ and $\\\\mathcal {X}_{n,\\\\mu }$ denote the closure of the convex hull of $X_1,\\\\ldots ,X_n$ .\",\n \"Note that if we replace $\\\\Psi _{\\\\widetilde{\\\\mu }_m,\\\\widetilde{\\\\nu }_n}(\\\\cdot )$ by $\\\\Psi _{\\\\widetilde{\\\\mu }_m,\\\\widetilde{\\\\nu }_n}(\\\\cdot )-C$ for some constant $C>0$ , then $\\\\Psi _{\\\\widetilde{\\\\mu }_m,\\\\widetilde{\\\\nu }_n}^*(\\\\cdot )\\\\mapsto \\\\Psi _{\\\\widetilde{\\\\mu }_m,\\\\widetilde{\\\\nu }_n}^*+C$ .\",\n \"However replacing $\\\\Psi _{\\\\widetilde{\\\\mu }_m,\\\\widetilde{\\\\nu }_n}^*(\\\\cdot )$ by $\\\\Psi _{\\\\widetilde{\\\\mu }_m,\\\\widetilde{\\\\nu }_n}^*(\\\\cdot )+C$ in (REF ) doesn't change its value as $\\\\widetilde{\\\\nu }_n$ and $\\\\nu $ are both probability measures.\",\n \"Therefore, without loss of generality, we can assume that $\\\\Psi _{\\\\widetilde{\\\\mu }_m,\\\\widetilde{\\\\nu }_n}(X_1)=0$ for all $m,n$ .\",\n \"We will stick to this convention for the rest of the proof.\",\n \"Also note that $\\\\Psi _{\\\\widetilde{\\\\mu }_m,\\\\widetilde{\\\\nu }_n}(\\\\cdot )$ is only determined at the data points $X_1,\\\\ldots ,X_n$ .\",\n \"Without loss of generality, we extend $\\\\Psi _{\\\\widetilde{\\\\mu }_m,\\\\widetilde{\\\\nu }_n}(\\\\cdot )$ to the whole of $\\\\mathbb {R}^d$ by linear interpolation for any $x\\\\in \\\\mathcal {X}_{n,\\\\mu }$ and setting $\\\\Psi _{\\\\widetilde{\\\\mu }_m,\\\\widetilde{\\\\nu }_n}(x)=\\\\infty $ for $x\\\\in \\\\mathcal {X}_{n,\\\\mu }^c$ .\",\n \"The proof now proceeds using the following steps: Step I: There exists a constant $C_1>0$ and $y_n\\\\in \\\\mbox{supp}(\\\\nu )=\\\\mathcal {Y}$ such that $|\\\\Psi _{\\\\widetilde{\\\\mu }_m,\\\\widetilde{\\\\nu }_n}^*(y_n)|\\\\le \\\\max _{1\\\\le i\\\\le m}\\\\Vert X_i\\\\Vert .$ [Proof of step I] By Kantorovich duality, there exists $y_n$ such that $\\\\Psi _{\\\\widetilde{\\\\mu }_m,\\\\widetilde{\\\\nu }_n}^*(y_n)+\\\\Psi _{\\\\widetilde{\\\\mu }_m,\\\\widetilde{\\\\nu }_n}(X_1)=\\\\langle X_1,y_n\\\\rangle \\\\quad \\\\Rightarrow \\\\quad |\\\\Psi _{\\\\widetilde{\\\\mu }_m,\\\\widetilde{\\\\nu }_n}^*(y_n)|\\\\le C_1\\\\Vert X_1\\\\Vert \\\\le C_1\\\\max _{1\\\\le i\\\\le m}\\\\Vert X_i\\\\Vert ,$ where $C_1:=\\\\sup \\\\lbrace \\\\Vert y\\\\Vert :\\\\ y\\\\in \\\\mathcal {Y}\\\\rbrace $ .\",\n \"Step II: There exists a constant $C_2>0$ such that the following holds: $\\\\Vert \\\\Psi _{\\\\widetilde{\\\\mu }_m,\\\\widetilde{\\\\nu }_n}^*\\\\Vert _{\\\\infty ,\\\\mathcal {Y}}\\\\le C_2\\\\max _{1\\\\le i\\\\le n} \\\\Vert X_i\\\\Vert ,$ where $\\\\Vert \\\\cdot \\\\Vert _{\\\\infty ,\\\\mathcal {Y}}$ is the uniform norm on the support of $\\\\nu $ .\",\n \"[Proof of step II] As $\\\\Psi _{\\\\widetilde{\\\\mu }_m,\\\\widetilde{\\\\nu }_n}(x)=\\\\infty $ for $x\\\\in \\\\mathcal {X}_{n,\\\\mu }^c$ , using (REF ), we can write $\\\\Psi _{\\\\widetilde{\\\\mu }_m,\\\\widetilde{\\\\nu }_n}^*(y)=\\\\max _{x\\\\in \\\\mathcal {X}_{n,\\\\mu }} (\\\\langle x,y\\\\rangle -\\\\Psi _{\\\\widetilde{\\\\mu }_m,\\\\widetilde{\\\\nu }_n}(x))$ for all $y\\\\in \\\\mathcal {Y}$ .\",\n \"For any $y_0\\\\in \\\\mathcal {Y}$ , let $x_0\\\\in \\\\mathcal {X}_{n,\\\\mu }$ be such that $\\\\Psi _{\\\\widetilde{\\\\mu }_m,\\\\widetilde{\\\\nu }_n}^*(y_0)=\\\\langle x_0,y_0\\\\rangle -\\\\Psi _{\\\\widetilde{\\\\mu }_m,\\\\widetilde{\\\\nu }_n}(x_0)$ .\",\n \"Then, for any $y\\\\in \\\\mathcal {X}$ , we have: $&{\\\\left\\\\lbrace \\\\begin{array}{ll} \\\\Psi _{\\\\widetilde{\\\\mu }_m,\\\\widetilde{\\\\nu }_n}^*(y_0)=\\\\langle x_0,y_0\\\\rangle -\\\\Psi _{\\\\widetilde{\\\\mu }_m,\\\\widetilde{\\\\nu }_n}(x_0)\\\\\\\\ \\\\Psi _{\\\\widetilde{\\\\mu }_m,\\\\widetilde{\\\\nu }_n}^*(y)\\\\ge \\\\langle x_0,y\\\\rangle -\\\\Psi _{\\\\widetilde{\\\\mu }_m,\\\\widetilde{\\\\nu }_n}(x_0)\\\\end{array}\\\\right.\",\n \"}\\\\\\\\ \\\\Rightarrow &|\\\\Psi _{\\\\widetilde{\\\\mu }_m,\\\\widetilde{\\\\nu }_n}^*(y_0)-\\\\Psi _{\\\\widetilde{\\\\mu }_m,\\\\widetilde{\\\\nu }_n}^*(y)|\\\\le |\\\\langle x_0,y_0-y\\\\rangle |\\\\le \\\\left(\\\\max _{1\\\\le i\\\\le m}\\\\Vert X_i\\\\Vert \\\\right)\\\\Vert y_0-y\\\\Vert .$ where the last line uses the fact that $y_0,y$ are arbitrary.\",\n \"In particular, by setting $y_0:=y_n$ from step I, we get: $\\\\Vert \\\\Psi _{\\\\widetilde{\\\\mu }_m,\\\\widetilde{\\\\nu }_n}^*\\\\Vert _{\\\\infty ,\\\\mathcal {Y}}\\\\le |\\\\Psi _{\\\\widetilde{\\\\mu }_m,\\\\widetilde{\\\\nu }_n}^*(y_n)|+ \\\\left(\\\\max _{1\\\\le i\\\\le m}\\\\Vert X_i\\\\Vert \\\\right)\\\\sup _{y\\\\in \\\\mathcal {Y}}\\\\Vert y_n-y\\\\Vert \\\\le C_2\\\\left(\\\\max _{1\\\\le i\\\\le m}\\\\Vert X_i\\\\Vert \\\\right),$ where $C_2:=3C_1$ with $C_1$ defined as specified in the proof of step I.\",\n \"The above lemma allows us to bound (with high probability) the $L^{\\\\infty }$ -norm of $\\\\Psi _{\\\\widetilde{\\\\mu }_m,\\\\widetilde{\\\\nu }_n}^*(\\\\cdot )$ on $\\\\mathcal {Y}$ , using the tail assumption $\\\\mathbb {E}\\\\exp (t\\\\Vert X_1\\\\Vert ^{\\\\alpha })<\\\\infty $ for some $t>0$ and $\\\\alpha >0$ .\",\n \"This is the focus of the next step.\",\n \"Step III: For $K>0$ , define the following two sets: $A_{m,n,K}:=\\\\left\\\\lbrace \\\\int (\\\\Psi _{\\\\widetilde{\\\\mu }_m,\\\\widetilde{\\\\nu }_n}^*(u))^2\\\\,d\\\\nu (u)\\\\ge K\\\\right\\\\rbrace ,\\\\quad \\\\mathrm {and},$ $\\\\widetilde{A}_{m,n,K}:=\\\\left\\\\lbrace \\\\Vert \\\\Psi _{\\\\widetilde{\\\\mu }_m,\\\\widetilde{\\\\nu }_n}^*\\\\Vert _{\\\\infty ,\\\\mathcal {Y}}\\\\ge K\\\\big (\\\\log {n}\\\\big )^{1/\\\\alpha }\\\\right\\\\rbrace .$ Then there exists $K_0>0$ such that for any $K\\\\ge K_0$ , we have: $\\\\lim _{m,n\\\\rightarrow \\\\infty }\\\\mathbb {P}(\\\\widetilde{A}_{m,n,K})=0.$ and $\\\\lim _{m,n\\\\rightarrow \\\\infty }\\\\mathbb {P}(A_{m,n,K})=0.$ [Proof of step III] By using the exponential Markov's inequality coupled with the standard union bound, we have: $\\\\mathbb {P}\\\\left(\\\\max _{1\\\\le i\\\\le m} \\\\Vert X_i\\\\Vert \\\\ge K(\\\\log {m})^{1/\\\\alpha }\\\\right)&\\\\le m\\\\mathbb {P}\\\\left(\\\\Vert X_1\\\\Vert \\\\ge K(\\\\log {m})^{1/\\\\alpha }\\\\right)\\\\\\\\ &\\\\le m\\\\exp (-tK^{\\\\alpha }(\\\\log {m}))\\\\mathbb {E}\\\\exp (t\\\\Vert X_1\\\\Vert ^{\\\\alpha })\\\\overset{m\\\\rightarrow \\\\infty }{\\\\longrightarrow }0$ provided $K>t^{-\\\\alpha }$ .\",\n \"Using the above observation coupled with step II, (REF ) follows by choosing $K_0>C_2 t^{-\\\\alpha }$ .\",\n \"For the next part, we define another set: $B_{m,n,\\\\varepsilon }:=\\\\left\\\\lbrace \\\\int \\\\big |\\\\Psi _{\\\\widetilde{\\\\mu }_m,\\\\widetilde{\\\\nu }_n}^*(u)-\\\\Psi _{\\\\mu ,\\\\nu }^*(u)|^2\\\\,d\\\\nu (u)\\\\ge \\\\varepsilon \\\\right\\\\rbrace $ for $\\\\varepsilon >0$ , where, as in (), we have: $W_2^2(\\\\mu ,\\\\nu )=\\\\int \\\\Vert x\\\\Vert ^2\\\\,d\\\\mu (x)+\\\\int \\\\Vert y\\\\Vert ^2\\\\,d\\\\nu (y)-2\\\\left(\\\\int \\\\Psi _{\\\\mu ,\\\\nu }(x)\\\\,d\\\\mu (x)+\\\\int \\\\Psi _{\\\\mu ,\\\\nu }^*(y)\\\\,d\\\\nu (y)\\\\right).$ Now by using [45], we have $\\\\mathbb {P}(B_{m,n,\\\\varepsilon })\\\\rightarrow 0$ as $m,n\\\\rightarrow \\\\infty $ for all $\\\\varepsilon >0$ .\",\n \"As $\\\\int (\\\\Psi _{\\\\widetilde{\\\\mu }_m,\\\\widetilde{\\\\nu }_n}^*(u))^2\\\\,d\\\\nu (u)\\\\le 2\\\\int \\\\big |\\\\Psi _{\\\\widetilde{\\\\mu }_m,\\\\widetilde{\\\\nu }_n}^*(u)-\\\\Psi _{\\\\mu ,\\\\nu }^*(u)|^2\\\\,d\\\\nu (u)+2\\\\int (\\\\Psi _{\\\\mu ,\\\\nu }^*(u))^2\\\\,d\\\\nu (u),$ (REF ) follows with $K_0>2\\\\int (\\\\Psi _{\\\\mu ,\\\\nu }^*(u))^2\\\\,d\\\\nu (u)+1$ if we choose $\\\\epsilon =1/2$ .\",\n \"We are now in a position to complete the proof of thm:nsmooth using steps I-III.\",\n \"Towards this direction, set $K^{\\\\prime }:=2K_0$ where $K_0$ is defined as in the proof of step III and observe that for any $M>0$ , $&\\\\;\\\\;\\\\;\\\\limsup _{M\\\\rightarrow \\\\infty }\\\\limsup _{m,n\\\\rightarrow \\\\infty }\\\\mathbb {P}\\\\left(\\\\Bigg |\\\\int \\\\Psi _{\\\\widetilde{\\\\mu }_m,\\\\widetilde{\\\\nu }_n}^*(u)\\\\,d(\\\\widetilde{\\\\nu }_n-\\\\nu )\\\\Bigg |\\\\ge Mr_d^{(n,n)}(\\\\log {(1+m)})^{t_{d,\\\\alpha }}\\\\right)\\\\nonumber \\\\\\\\&\\\\le \\\\limsup _{M\\\\rightarrow \\\\infty }\\\\limsup _{m,n\\\\rightarrow \\\\infty }\\\\mathbb {P}\\\\left(\\\\Bigg |\\\\int \\\\Psi _{\\\\widetilde{\\\\mu }_m,\\\\widetilde{\\\\nu }_n}^*(u)\\\\,d(\\\\widetilde{\\\\nu }_n-\\\\nu )\\\\Bigg |\\\\ge Mr_d^{(n,n)}(\\\\log {(1+m)})^{t_{d,\\\\alpha }}, A_{m,n,K^{\\\\prime }}^c\\\\cap \\\\widetilde{A}_{m,n,K^{\\\\prime }}^c\\\\right)\\\\nonumber \\\\\\\\ &+\\\\limsup \\\\limits _{n\\\\rightarrow \\\\infty }\\\\mathbb {P}(\\\\widetilde{A}_{m,n,K^{\\\\prime }})+\\\\limsup \\\\limits _{m,n\\\\rightarrow \\\\infty }\\\\mathbb {P}(A_{m,n,K^{\\\\prime }})\\\\nonumber \\\\\\\\&\\\\le \\\\limsup _{M\\\\rightarrow \\\\infty }\\\\limsup \\\\limits _{m,n\\\\rightarrow \\\\infty }\\\\mathbb {P}\\\\left(\\\\Bigg |\\\\int \\\\Psi _{\\\\widetilde{\\\\mu }_m,\\\\widetilde{\\\\nu }_n}^*(u)\\\\,d(\\\\widetilde{\\\\nu }_n-\\\\nu )\\\\Bigg |\\\\ge Mr_d^{(n,n)}(\\\\log {(1+m)})^{t_{d,\\\\alpha }},A_{m,n,K^{\\\\prime }}^c\\\\cap \\\\widetilde{A}_{m,n,K^{\\\\prime }}^c\\\\right),$ where the last step follows from step III.\",\n \"Observe that the left hand side of (REF ) is the same as (REF ).\",\n \"Therefore, it is now enough to bound the right hand side of (REF ).\",\n \"In order to achieve the above task, let us define the following class of functions: $\\\\mathcal {C}^{\\\\Gamma ,L}(\\\\mathcal {Y}):=\\\\lbrace f:\\\\mathcal {Y}\\\\rightarrow \\\\mathbb {R}, f\\\\ \\\\mathrm {is}\\\\ \\\\mathrm {convex,} \\\\ \\\\Vert f\\\\Vert _{\\\\infty ,\\\\mathcal {Y}}\\\\le \\\\Gamma , \\\\ \\\\Vert f\\\\Vert _{L^2(\\\\nu )}\\\\le L\\\\rbrace .$ By setting $\\\\Gamma :=K^{\\\\prime }(\\\\log {m})^{1/\\\\alpha }$ and $L:=K^{\\\\prime }$ , (REF ) yields the following conclusion: $&\\\\;\\\\;\\\\;\\\\limsup _{M\\\\rightarrow \\\\infty }\\\\limsup \\\\limits _{m,n\\\\rightarrow \\\\infty }\\\\mathbb {P}\\\\left(\\\\Bigg |\\\\int \\\\Psi _{\\\\widetilde{\\\\mu }_m,\\\\widetilde{\\\\nu }_n}^*(u)\\\\,d(\\\\widetilde{\\\\nu }_n-\\\\nu )\\\\Bigg |\\\\ge Mr_d^{(n,n)}(\\\\log {(1+m)})^{t_{d,\\\\alpha }}\\\\right)\\\\\\\\ & \\\\le \\\\limsup _{M\\\\rightarrow \\\\infty }\\\\limsup _{m,n\\\\rightarrow \\\\infty }\\\\mathbb {P}\\\\left(\\\\sup _{f\\\\in \\\\mathcal {C}^{\\\\Gamma ,L}(\\\\mathcal {Y})}\\\\Bigg |\\\\int f\\\\,d(\\\\widetilde{\\\\nu }_n-\\\\nu )\\\\Bigg |\\\\ge Mr_d^{(n,n)}(\\\\log {(1+m)})^{t_{d,\\\\alpha }}\\\\right).$ By an application of Markov's inequality, it thus suffices to show that: $\\\\mathbb {E}\\\\left[\\\\sup _{f\\\\in \\\\mathcal {C}^{\\\\Gamma ,L}(\\\\mathcal {Y})}\\\\Bigg |\\\\int f\\\\,d(\\\\widetilde{\\\\nu }_n-\\\\nu )\\\\Bigg |\\\\right]=\\\\mathcal {O}\\\\left(r_d^{(n,n)}(\\\\log {(1+m)})^{t_{d,\\\\alpha }}\\\\right).$ In order to bound (REF ), we will use some standard empirical process techniques.\",\n \"In particular, by using [136], the following bound holds: $&\\\\;\\\\;\\\\;\\\\mathbb {E}\\\\left[\\\\sup _{f\\\\in \\\\mathcal {C}^{\\\\Gamma ,L}(\\\\mathcal {Y})}\\\\Bigg |\\\\int f\\\\,d(\\\\widetilde{\\\\nu }_n-\\\\nu )\\\\Bigg |\\\\right]\\\\nonumber \\\\\\\\ &\\\\le D\\\\inf \\\\left\\\\lbrace a\\\\ge \\\\frac{\\\\Gamma }{\\\\sqrt{n}}:a\\\\ge \\\\frac{D}{\\\\sqrt{n}}\\\\int _a^{\\\\Gamma }\\\\sqrt{\\\\log N_{[]}(\\\\varepsilon ,\\\\mathcal {C}^{\\\\Gamma ,L}(\\\\mathcal {Y}),L^2(\\\\nu ))}\\\\,d\\\\varepsilon \\\\right\\\\rbrace ,$ for some positive constant $D>0$ , where $N_{[]}(\\\\varepsilon ,\\\\mathcal {C}^{\\\\Gamma ,L}(\\\\mathcal {Y}),L^2(\\\\nu ))$ is the $\\\\varepsilon $ -bracketing number of the class of functions $\\\\mathcal {C}^{\\\\Gamma ,L}(\\\\mathcal {Y})$ with respect to the $L^2(\\\\nu )$ norm.\",\n \"Note that by [91], we have: $\\\\log N_{[]}(\\\\varepsilon ,\\\\mathcal {C}^{\\\\Gamma ,L}(\\\\mathcal {Y}),L^2(\\\\nu ))\\\\le \\\\gamma _d\\\\left(\\\\log {\\\\frac{\\\\Gamma }{\\\\varepsilon }}\\\\right)^{d+1}\\\\left(\\\\frac{L}{\\\\varepsilon }\\\\right)^{d/2}$ for some $\\\\gamma _d>0$ depending only on fand the diameter of $\\\\mathcal {Y}$ .\",\n \"We will now bound the right hand side of (REF ).\",\n \"Also we will use $D_d$ to denote changing constants which can depend on $d$ .\",\n \"When $d=1,2,3$: Choose $a=D_d\\\\frac{(\\\\log {n})^{\\\\frac{1}{\\\\alpha }\\\\vee \\\\frac{2\\\\alpha +2d\\\\alpha -d+4}{4\\\\alpha }}}{\\\\sqrt{n}}$ .\",\n \"Observe that: $\\\\frac{1}{\\\\sqrt{n}}\\\\int _a^{\\\\Gamma }\\\\sqrt{\\\\log N_{[]}(\\\\varepsilon ,\\\\mathcal {C}^{\\\\Gamma ,L}(\\\\mathcal {Y}),L^2(\\\\nu ))}\\\\,d\\\\varepsilon &\\\\le \\\\frac{(\\\\log {n})^{(d+1)/2}}{\\\\sqrt{n}}\\\\cdot \\\\left[\\\\frac{\\\\varepsilon ^{1-d/4}}{1-d/4}\\\\right]_{0}^{\\\\Gamma }\\\\\\\\ &\\\\lesssim \\\\frac{(\\\\log {n})^{(4-d)/(4\\\\alpha )}\\\\times (\\\\log {n})^{(d+1)/2}}{\\\\sqrt{n}}\\\\lesssim a.$ When $d=4$: Choose $a=D_d\\\\frac{(\\\\log {n})^{\\\\frac{1}{\\\\alpha }\\\\vee \\\\frac{7}{2}}}{\\\\sqrt{n}}$ .\",\n \"Observe that: $\\\\frac{1}{\\\\sqrt{n}}\\\\int _a^{\\\\Gamma }\\\\sqrt{\\\\log N_{[]}(\\\\varepsilon ,\\\\mathcal {C}^{\\\\Gamma ,L}(\\\\mathcal {Y}),L^2(\\\\nu ))}\\\\,d\\\\varepsilon &\\\\le \\\\frac{(\\\\log {n})^{5/2}}{\\\\sqrt{n}}\\\\cdot \\\\left[\\\\log {\\\\varepsilon }\\\\right]_{D_d\\\\Gamma /\\\\sqrt{n}}^{\\\\Gamma }\\\\\\\\ &\\\\lesssim \\\\frac{(\\\\log {n})^{(7/2)}}{\\\\sqrt{n}}\\\\lesssim a.$ When $d>4$: Choose $a=D_d\\\\frac{(\\\\log {n})^{2(1+d^{-1})}}{n^{2/d}}$ .\",\n \"Observe that: $\\\\frac{1}{\\\\sqrt{n}}\\\\int _a^{\\\\Gamma _0}\\\\sqrt{\\\\log N_{[]}(\\\\varepsilon ,\\\\mathcal {C}^{\\\\Gamma ,L}(\\\\mathcal {Y}),L^2(\\\\nu ))}\\\\,d\\\\varepsilon &\\\\le \\\\frac{(\\\\log {n})^{(d+1)/2}}{\\\\sqrt{n}}\\\\cdot \\\\left[\\\\frac{\\\\varepsilon ^{1-d/4}}{1-d/4}\\\\right]_{a}^{\\\\Gamma }\\\\\\\\ &\\\\lesssim \\\\frac{a^{1-d/4}(\\\\log {n})^{(d+1)/2}}{\\\\sqrt{n}}\\\\lesssim a.$ This completes the proof after applying the same technique on the other 3 terms on the right hand side of (REF ).\",\n \"[Proof of cor:fsam] First observe that $\\\\mathbb {E}\\\\left[\\\\int \\\\varphi _0^*\\\\,d\\\\widehat{\\\\nu }_n\\\\right]=\\\\mathbb {E}\\\\left[\\\\int \\\\varphi _0^*\\\\,d\\\\overline{\\\\nu }_m\\\\right]=\\\\int \\\\varphi _0^*\\\\,d\\\\nu .$ Using the above observation and the same approach used as in the proof of thm:nsmooth, we will only focus on bounding $\\\\mathbb {E}\\\\bigg |\\\\int \\\\Psi _{\\\\widetilde{\\\\mu }_m,\\\\widetilde{\\\\nu }_n}^* \\\\,d(\\\\widetilde{\\\\nu }_n-\\\\nu )\\\\bigg |.$ The general strategy to bound the term in (REF ) is derived from some intermediate steps in the proofs of [33].\",\n \"We still present a sketch here for completeness.\",\n \"By the same argument as in the proof of thm:nsmooth and using the fact that there exists fixed $R>0$ such that $\\\\max _{1\\\\le i\\\\le m}\\\\Vert X_i\\\\Vert \\\\le R$ , we have $\\\\Psi _{\\\\widetilde{\\\\mu }_m,\\\\widetilde{\\\\nu }_n}^*(\\\\cdot )$ is a convex and $R$ -Lipschitz function on $\\\\mathcal {Y}$ .\",\n \"This observation implies: $\\\\mathbb {E}\\\\bigg |\\\\int \\\\Psi _{\\\\widetilde{\\\\mu }_m,\\\\widetilde{\\\\nu }_n}^* \\\\,d(\\\\widetilde{\\\\nu }_n-\\\\overline{\\\\nu }_m)\\\\bigg |\\\\le \\\\mathbb {E}\\\\left[\\\\sup _{\\\\psi \\\\in \\\\mathcal {F}_R(\\\\mathcal {Y})}\\\\left|\\\\int \\\\psi \\\\,d(\\\\widehat{\\\\nu }_n-\\\\nu )\\\\right|\\\\right]$ where $\\\\mathcal {F}_R(\\\\mathcal {Y})$ is the set of convex and $R$ -Lipschitz functions on $\\\\mathcal {Y}$ .\",\n \"By [141], we then have: $\\\\mathbb {E}\\\\left[\\\\sup _{\\\\psi \\\\in \\\\mathcal {F}_R(\\\\mathcal {Y})}\\\\left|\\\\int \\\\psi \\\\,d(\\\\widehat{\\\\nu }_n-\\\\nu )\\\\right|\\\\right]\\\\lesssim \\\\inf _{\\\\delta >0}\\\\left(\\\\delta +n^{-1/2}\\\\int _{\\\\delta }^{R^2}\\\\sqrt{\\\\log {\\\\mathcal {N}_{\\\\infty }(\\\\mathcal {F}_R(\\\\mathcal {Y}),\\\\varepsilon )}}\\\\,d\\\\varepsilon \\\\right),$ where $\\\\mathcal {N}_{\\\\infty }(\\\\mathcal {F}_R(\\\\mathcal {Y}),\\\\varepsilon )$ is the $\\\\varepsilon $ -covering number of the set $\\\\mathcal {F}_R(\\\\mathcal {Y})$ with respect to the uniform metric.\",\n \"By using [74] (also see [22]), there exists constants $C_1,C_2>0$ such that whenever $\\\\varepsilon /R^2\\\\le C_1$ , then $\\\\log {\\\\mathcal {N}_{\\\\infty }(\\\\mathcal {F}_R(\\\\mathcal {Y}),\\\\varepsilon )}\\\\le C_2(u/R^2)^{-d/2}$ .\",\n \"By using this bound in (REF ), we get: $\\\\mathbb {E}\\\\left[\\\\sup _{\\\\psi \\\\in \\\\mathcal {F}_R(\\\\mathcal {Y})}\\\\left|\\\\int \\\\psi \\\\,d(\\\\widehat{\\\\nu }_n-\\\\nu )\\\\right|\\\\right]\\\\lesssim \\\\inf _{\\\\delta >0}\\\\left(\\\\delta +n^{-1/2}\\\\int _{\\\\delta }^1 \\\\varepsilon ^{-d/4}\\\\,d\\\\varepsilon \\\\right).$ Setting $\\\\delta =0$ for $d<4$ and $\\\\delta =n^{-2/d}$ for $d\\\\ge 4$ in (REF ), followed by a direct application of (REF ), we have: $\\\\mathbb {E}\\\\bigg |\\\\int \\\\Psi _{\\\\widetilde{\\\\mu }_m,\\\\widetilde{\\\\nu }_n}^* \\\\,d(\\\\widetilde{\\\\nu }_n-\\\\overline{\\\\nu }_m)\\\\bigg |\\\\lesssim r_d^{(n,n)}.$ This completes the proof.\",\n \"[Proof of thm:smoothwav] For this proof, we will use an intermediate step in the proof of thm:newubd, which is (REF ), that can alternatively be written as: $\\\\mathbb {E}\\\\left[\\\\int \\\\Vert \\\\widetilde{T}_{m,n}^{\\\\gamma }(x)-T_0(x)\\\\Vert ^2\\\\,d\\\\widetilde{\\\\mu }_m(x)\\\\right]\\\\lesssim \\\\mathbb {E}|W_2^2(\\\\widetilde{\\\\mu }_m,\\\\widetilde{\\\\nu }_n)-W_2^2(\\\\mu ,\\\\nu )|+\\\\mathbb {E}\\\\left|\\\\int h(y)\\\\,d(\\\\widetilde{\\\\nu }_n-\\\\overline{\\\\nu }_m)(y)\\\\right|$ where $h(y):=\\\\varphi _0^*(y)-(1/2)\\\\Vert y\\\\Vert ^2$ and $C>0$ is some constant.\",\n \"As $\\\\mathcal {X}$ and $\\\\mathcal {Y}$ are compact sets, the function $h(\\\\cdot )$ is Lipschitz.\",\n \"Therefore, $\\\\mathbb {E}\\\\left|\\\\int h(y)\\\\,d(\\\\widetilde{\\\\nu }_n-\\\\nu )(y)\\\\right|\\\\lesssim W_1(\\\\widetilde{\\\\nu }_n,\\\\nu )\\\\le W_2(\\\\widetilde{\\\\nu }_n,\\\\nu ).$ Further, as $T_0(\\\\cdot )$ is also Lipschitz, we further have: $\\\\mathbb {E}\\\\left|\\\\int h(y)\\\\,d(\\\\overline{\\\\nu }_m-\\\\nu )(y)\\\\right|\\\\lesssim W_1(T_0\\\\#\\\\widetilde{\\\\mu }_m,T_0\\\\#\\\\mu )\\\\lesssim W_1(\\\\widetilde{\\\\mu }_m,\\\\mu )\\\\le W_2(\\\\widetilde{\\\\mu }_m,\\\\mu ).$ Finally, by the triangle inequality, we also have: $\\\\mathbb {E}|W_2^2(\\\\widetilde{\\\\mu }_m,\\\\widetilde{\\\\nu }_n)-W_2^2(\\\\mu ,\\\\nu )|&\\\\lesssim \\\\mathbb {E}|W_2(\\\\widetilde{\\\\mu }_m,\\\\widetilde{\\\\nu }_n)-W_2(\\\\widetilde{\\\\mu }_m,\\\\nu )|+\\\\mathbb {E}|W_2(\\\\widetilde{\\\\mu }_m,\\\\nu )-W_2(\\\\mu ,\\\\nu )|\\\\\\\\ &\\\\le \\\\mathbb {E}W_2(\\\\widetilde{\\\\mu }_m,\\\\mu )+\\\\mathbb {E}W_2(\\\\widetilde{\\\\nu }_n,\\\\nu ).$ Combining the three displays above and plugging them back in (REF ), we get: $\\\\mathbb {E}\\\\left[\\\\int \\\\Vert \\\\widetilde{T}_{m,n}^{\\\\gamma }(x)-T_0(x)\\\\Vert ^2\\\\,d\\\\widetilde{\\\\mu }_m(x)\\\\right]\\\\lesssim \\\\mathbb {E}W_2(\\\\widetilde{\\\\mu }_m,\\\\mu )+\\\\mathbb {E}W_2(\\\\widetilde{\\\\nu }_n,\\\\nu ).$ The conclusion then follows from [143].\",\n \"[Proof of thm:smooth] Part 1.\",\n \"By the same arguments (see e.g., (REF )) as used in the proof of thm:nsmooth, it suffices to show that $\\\\mathbb {E}\\\\left|\\\\int \\\\Psi _{\\\\widetilde{\\\\mu }_m,\\\\widetilde{\\\\nu }_n}^*(u)(\\\\widetilde{f}_{\\\\nu }^{M^{\\\\prime }}(u)-f_{\\\\nu }(u)) \\\\,du\\\\right|\\\\lesssim r_{d,s}^{(n,n)}$ for some $M^{\\\\prime }>0$ .\",\n \"The general structure of the proof is similar to that of thm:nsmooth.\",\n \"The crucial observation is that $\\\\widetilde{f}_{\\\\mu }^{M^{\\\\prime }}(\\\\cdot )$ and $\\\\widetilde{f}_{\\\\nu }^{M^{\\\\prime }}(\\\\cdot )$ are elements of $C^s(\\\\mathcal {X};TM)$ and $C^s(\\\\mathcal {Y};TM)$ respectively, for any $M^{\\\\prime }>0$ .\",\n \"Note that, by Caffarelli regularity theory; see [80], there exists $M^{\\\\prime }>0$ such that $\\\\Vert \\\\Psi _{\\\\widetilde{\\\\mu }_m,\\\\widetilde{\\\\nu }_n}^*(\\\\cdot )\\\\Vert _{C^{s+2}(\\\\mathcal {Y})}\\\\le M^{\\\\prime }.$ Next, let us define the following class of functions: $\\\\mathcal {G}_t^L(\\\\mathcal {Y}):=\\\\lbrace g:\\\\mathcal {Y}\\\\rightarrow \\\\mathbb {R},\\\\ g(\\\\cdot )\\\\ \\\\mbox{is}\\\\ \\\\mbox{convex},\\\\ \\\\Vert g\\\\Vert _{C^t(\\\\mathcal {Y})}\\\\le L\\\\rbrace .$ Observe that $\\\\mathbb {E}\\\\left|\\\\int \\\\Psi _{\\\\widetilde{\\\\mu }_m,\\\\widetilde{\\\\nu }_n}^*(u)(\\\\widetilde{f}_{\\\\nu }^{M^{\\\\prime }}(u)-f_{\\\\nu }(u)) \\\\,du\\\\right|&\\\\le \\\\mathbb {E}\\\\sup _{g\\\\in \\\\mathcal {G}_{s+2}^{M^{\\\\prime }}(\\\\mathcal {Y})}\\\\left|\\\\int g(u)(\\\\widetilde{f}_{\\\\nu }^{M^{\\\\prime }}(u)-f_{\\\\nu }(u)) \\\\,du\\\\right|\\\\nonumber \\\\\\\\ &\\\\le 2\\\\mathbb {E}\\\\sup _{g\\\\in \\\\mathcal {G}_{s+2}^{M^{\\\\prime }}(\\\\mathcal {Y})}\\\\left|\\\\int g(u)(\\\\widehat{f}_{\\\\nu }(u)-f_{\\\\nu }(u)) \\\\,du\\\\right|+r_{d,s}^{(n,n)}$ where the last line follows from (REF ).\",\n \"Set $K_{d,h_n}(\\\\cdot ):=h_n^{-d}K_d(\\\\cdot /h_n)$ .\",\n \"Following the same decomposition as in [115], we write: $&\\\\;\\\\;\\\\;\\\\;\\\\mathbb {E}\\\\sup _{g\\\\in \\\\mathcal {G}_{s+2}^{M^{\\\\prime }}(\\\\mathcal {Y})}\\\\left|\\\\int g(u)(\\\\widehat{f}_{\\\\nu }(u)-f_{\\\\nu }(u)) \\\\,du\\\\right|\\\\nonumber \\\\\\\\ &=\\\\mathbb {E}\\\\sup _{g\\\\in \\\\mathcal {G}_{s+2}^{M^{\\\\prime }}(\\\\mathcal {Y})}\\\\left|\\\\int g(u+u^{\\\\prime })K_{d,h_n}(u^{\\\\prime })\\\\,d\\\\widehat{\\\\nu }_n(u)\\\\,du^{\\\\prime }-\\\\int g(u)f_{\\\\nu }(u)\\\\,du\\\\right|\\\\nonumber \\\\\\\\ &\\\\le \\\\mathbb {E}\\\\sup _{g\\\\in \\\\mathcal {G}_{s+2}^{M^{\\\\prime }}(\\\\mathcal {Y})}\\\\left|\\\\int g(u+u^{\\\\prime })K_{d,h_n}(u^{\\\\prime })\\\\,d(\\\\widehat{\\\\nu }_n-\\\\nu )(u)\\\\,du^{\\\\prime }\\\\right|\\\\nonumber \\\\\\\\ &\\\\;\\\\;\\\\;\\\\;\\\\;\\\\;\\\\;\\\\;\\\\;\\\\;+\\\\sup _{g\\\\in \\\\mathcal {G}_{s+2}^{M^{\\\\prime }}(\\\\mathcal {Y})}\\\\left|\\\\int g(u+u^{\\\\prime })K_{d,h_n}(u^{\\\\prime })f_{\\\\nu }(u)\\\\,du\\\\,du^{\\\\prime }-\\\\int g(u)f_{\\\\nu }(u)\\\\,du\\\\right|.$ We will now bound the two terms on the right hand side of (REF ).\",\n \"For the first term, define $\\\\overline{g}_n(u):=\\\\int g(u+u^{\\\\prime })K_{d,h_n}(u)\\\\,du^{\\\\prime }.$ If $g\\\\in \\\\mathcal {G}_t^L(\\\\mathcal {Y}^o)$ , then by [53] and using as:kernel, we have $\\\\overline{g}_n\\\\in \\\\mathcal {G}_{s+2}^{cM^{\\\\prime }}(\\\\mathcal {Y}^o)$ , for some constant $c>0$ (depending on the constants involved in as:kernel and the diameter of $\\\\mathcal {Y}$ ).\",\n \"Combining these observations with (REF ), we get: $&\\\\;\\\\;\\\\;\\\\;\\\\mathbb {E}\\\\sup _{g\\\\in \\\\mathcal {G}_{s+2}^{M^{\\\\prime }}(\\\\mathcal {Y})}\\\\left|\\\\int g(u+u^{\\\\prime })K_{d,h_n}(u^{\\\\prime })\\\\,d(\\\\widehat{\\\\nu }_n-\\\\nu )(u)\\\\,du^{\\\\prime }\\\\right|\\\\nonumber \\\\\\\\ &\\\\le \\\\mathbb {E}\\\\sup _{g\\\\in \\\\mathcal {G}_{s+2}^{cM^{\\\\prime }}(\\\\mathcal {Y})}\\\\left|\\\\int g(u)\\\\,d(\\\\widehat{\\\\nu }_n-\\\\nu )(u)\\\\right|\\\\nonumber \\\\\\\\ &\\\\le D\\\\inf \\\\left\\\\lbrace a\\\\ge \\\\frac{cM^{\\\\prime }}{\\\\sqrt{n}}:a\\\\ge \\\\frac{D}{\\\\sqrt{n}}\\\\int _a^{cM^{\\\\prime }}\\\\sqrt{\\\\log N_{[]}(\\\\varepsilon ,\\\\mathcal {G}_{s+2}^{cM^{\\\\prime }}(\\\\mathcal {Y}),L^2(\\\\nu ))}\\\\,d\\\\varepsilon \\\\right\\\\rbrace ,$ for some positive constant $D>0$ , where $N_{[]}(\\\\varepsilon ,\\\\mathcal {G}_{s+2}^{cM^{\\\\prime }}(\\\\mathcal {Y}),L^2(\\\\nu ))$ is the $\\\\varepsilon $ -bracketing entropy of the class of functions $\\\\mathcal {G}_{s+2}^{cM^{\\\\prime }}(\\\\mathcal {Y})$ with respect to the $L^2(\\\\nu )$ norm.\",\n \"The last line follows from standard empirical process theory as used in the proof of thm:nsmooth; see (REF ).\",\n \"Note that by [138], we have: $\\\\log N_{[]}(\\\\varepsilon ,\\\\mathcal {G}_{s+2}^{cL}(\\\\mathcal {Y}^o),L^2(\\\\nu ))\\\\le \\\\gamma _d\\\\left(\\\\frac{1}{\\\\varepsilon }\\\\right)^{d/(s+2)}$ for some $\\\\gamma _d>0$ depending only on dimension and the diameter of $\\\\mathcal {Y}$ .\",\n \"We now plug-in the above bound into (REF ).\",\n \"By using $D_d$ to denote constants that change with $d$ and choosing $a=D_d n^{-1/2}$ for $2(s+2)>d$ , $a=D_d n^{-1/2}\\\\log {(1+n)}$ for $2(s+2)=d$ and $a=D_d n^{-(s+2)/d}$ for $2(s+2)\\\\frac{1}{2}$ .\",\n \"This implies $n^{-\\\\frac{2s+2}{d+2s}}(\\\\log {n})^{2s+2}\\\\lesssim n^{-\\\\frac{1}{2}}=r_{d,s}^{(n,n)}$ .\",\n \"When $d=2(s+2)$ : In this case $n^{-\\\\frac{2s+2}{d+2s}}(\\\\log {n})^{2s+2}\\\\lesssim n^{-\\\\frac{1}{2}}(\\\\log {n})^{2s+2}\\\\lesssim r_{d,s}^{(n,n)}$ .\",\n \"When $d>2(s+2)$ : Note that $\\\\frac{2s+2}{d+2s}>\\\\frac{s+2}{d}\\\\Leftrightarrow 2ds+2d>sd+2s^2+2d+4s\\\\Leftrightarrow d>2(s+2).$ Therefore, once again $n^{-\\\\frac{2s+2}{d+2s}}(\\\\log {n})^{2s+2}\\\\lesssim n^{-\\\\frac{s+2}{d}}=r_{d,s}^{(n,n)}$ .\",\n \"Therefore, combining the above observations, we have: $\\\\left|\\\\int g(u+u^{\\\\prime })K_{d,h_n}(u^{\\\\prime })f_{\\\\nu }(u)\\\\,du\\\\,du^{\\\\prime }-\\\\int g(u)f_{\\\\nu }(u)\\\\,du\\\\right|\\\\lesssim r_{d,s}^{(n,n)}.$ Combining the above display with (REF ) establishes (REF ).\",\n \"Part 2.\",\n \"This proof uses ideas from [76], [112] and [5].\",\n \"First recall all the notation introduced in def:holder.\",\n \"Next, we will prove the following sequence of displays: $\\\\limsup \\\\limits _{m,n\\\\rightarrow \\\\infty }\\\\max _{k\\\\le s}\\\\max _{|\\\\mathfrak {m}|=k}\\\\Vert \\\\partial ^{\\\\mathfrak {m}}\\\\mathbb {E}\\\\widehat{f}_{\\\\mu }\\\\Vert _{L^{\\\\infty }(\\\\widetilde{\\\\mathcal {X}})}\\\\le (T-1)M,$ $\\\\limsup _{m,n\\\\rightarrow \\\\infty }\\\\ \\\\Vert \\\\mathbb {E}\\\\widehat{f}_{\\\\mu }-f_{\\\\mu }\\\\Vert _{L^{\\\\infty }(\\\\widetilde{\\\\mathcal {X}})}=0$ $\\\\limsup _{m,n\\\\rightarrow \\\\infty }\\\\mathbb {P}\\\\left(\\\\Vert \\\\widehat{f}_{\\\\mu }-\\\\mathbb {E}\\\\widehat{f}_{\\\\mu }\\\\Vert _{C^{s}(\\\\widetilde{\\\\mathcal {X}})}\\\\ge \\\\varepsilon \\\\right)=0,$ for any arbitrary $\\\\varepsilon >0$ and $L^{\\\\infty }(\\\\mathcal {X})$ denotes the uniform norm on $\\\\mathcal {X}$ .\",\n \"Clearly, (REF ), (REF ), and (REF ) together yield part 1 of the theorem.\",\n \"Proof of (REF ).\",\n \"Observe that $\\\\mathbb {E}\\\\widehat{f}_{\\\\mu }(x)=\\\\frac{1}{h_m^d}\\\\mathbb {E}K_d\\\\left(\\\\frac{x-X_1}{h_m}\\\\right)=\\\\frac{1}{h_m^d}\\\\int K_d\\\\left(\\\\frac{x-z}{h_m}\\\\right)f_{\\\\mu }(z)\\\\,dz.$ Since the maximums taken in (REF ) are over finite sets, it suffices to show that for any fixed $\\\\mathfrak {m}$ with $|\\\\mathfrak {m}|\\\\le s$ , we have: $\\\\sup _{x\\\\in \\\\widetilde{\\\\mathcal {X}}}|\\\\partial ^{\\\\mathfrak {m}}\\\\mathbb {E}\\\\widehat{f}_{\\\\mu }(x)|=\\\\sup _{x\\\\in \\\\mathcal {X}}\\\\bigg |\\\\frac{1}{h_n^d}\\\\int K_d\\\\left(\\\\frac{z}{h_n}\\\\right)\\\\partial ^{\\\\mathfrak {m}} f(x+z)\\\\,dz\\\\Bigg |\\\\le (T-1)M.$ Here the first equality in the above display follows from (REF ) and Fubini's Theorem.\",\n \"Here $\\\\partial ^{\\\\mathfrak {m}}f_{\\\\mu }(\\\\cdot )$ is defined in the weak sense, i.e., it is defined naturally in the interior of the support of $f_{\\\\mu }(\\\\cdot )$ , denoted by $\\\\mathcal {X}$ ; it is set to be 0 outside $\\\\mathcal {X}$ and defined arbitrarily on the boundary of $\\\\mathcal {X}$ .\",\n \"Note that the definition on the boundary doesn't matter as we are integrating with respect to the Lebesgue measure and the boundary of $\\\\mathcal {X}$ has Lebesgue measure 0.\",\n \"Next note that, by (REF ), we have: $\\\\sup _{x\\\\in \\\\widetilde{\\\\mathcal {X}}}|\\\\partial ^{\\\\mathfrak {m}}\\\\mathbb {E}\\\\widehat{f}_{\\\\mu }(x)|\\\\le \\\\Vert f_{\\\\mu }\\\\Vert _{C^s(\\\\mathcal {X})}h_m^{-d}\\\\int |K_d(z/h_m)|\\\\,dz\\\\le (T-1)\\\\Vert f_{\\\\mu }\\\\Vert _{C^s(\\\\mathcal {X})}.$ This establishes (REF ).\",\n \"Proof of (REF ).\",\n \"First note that, as $\\\\widetilde{\\\\mathcal {X}}$ is a compact subset of $\\\\mathcal {X}^o$ , there exists $\\\\delta >0$ such that $\\\\widetilde{\\\\mathcal {X}}_{\\\\delta ^{\\\\prime }}:=\\\\lbrace x+z:\\\\ \\\\Vert z\\\\Vert \\\\le \\\\delta ;,\\\\ x\\\\in \\\\widetilde{\\\\mathcal {X}}\\\\rbrace \\\\subseteq \\\\mathcal {X}^o\\\\qquad \\\\forall \\\\ 0<\\\\delta ^{\\\\prime }\\\\le \\\\delta .$ Clearly, $\\\\widetilde{\\\\mathcal {X}}_{\\\\delta ^{\\\\prime }}$ is compact for all $\\\\delta ^{\\\\prime }>0$ .\",\n \"Fix an arbitrary $\\\\delta ^{\\\\prime }\\\\le \\\\delta $ .\",\n \"By using (REF ) and a change of variable formula, we have: $\\\\Vert \\\\mathbb {E}\\\\widehat{f}_{\\\\mu }-f_{\\\\mu }\\\\Vert _{L^{\\\\infty }(\\\\widetilde{\\\\mathcal {X}})}&=\\\\sup _{x\\\\in \\\\widetilde{\\\\mathcal {X}}}\\\\bigg |\\\\frac{1}{h_m^d}\\\\int K_d\\\\left(\\\\frac{z}{h_m}\\\\right)(f(x+z)-f(x))\\\\,dz\\\\bigg | \\\\\\\\& \\\\le (T-1)\\\\sup _{x\\\\in \\\\widetilde{\\\\mathcal {X}}}\\\\sup _{\\\\Vert z\\\\Vert \\\\le \\\\delta ^{\\\\prime }}|f(x+z)-f(x)|+2M\\\\int _{\\\\Vert z\\\\Vert >\\\\delta ^{\\\\prime } h_m^{-1}} |K_d(z)|\\\\,dz\\\\\\\\ & \\\\le (T-1)M\\\\delta ^{\\\\prime }+2M\\\\left(\\\\frac{h_m}{\\\\delta ^{\\\\prime }}\\\\right)^{2s+2}\\\\int \\\\Vert z\\\\Vert ^{2s+2}|K_d(z)|\\\\,dz.$ Observe that as $m,n\\\\rightarrow \\\\infty $ , the second term on the right hand side of the above display converges to 0.\",\n \"This implies $\\\\limsup \\\\limits _{m,n\\\\rightarrow \\\\infty }\\\\Vert \\\\mathbb {E}\\\\widehat{f}_{\\\\mu }-f_{\\\\mu }\\\\Vert _{L^{\\\\infty }(\\\\widetilde{\\\\mathcal {X}})}\\\\le (T-1)M\\\\delta ^{\\\\prime }.$ As $\\\\delta ^{\\\\prime }$ can be chosen arbitrarily small, this completes the proof of (REF ).\",\n \"Proof of (REF ).\",\n \"The main technical tool for this part is lem:emproc which we borrow from [5] (also see [99]).\",\n \"The proof is very similar to [5].\",\n \"Consider the following class of functions: $\\\\mathcal {G}=\\\\left\\\\lbrace g_x(z,h):\\\\ g_x(z,h)=\\\\partial ^{\\\\mathfrak {m}}K_d\\\\left(\\\\frac{(x-z)}{h}\\\\right),\\\\ x\\\\in \\\\widetilde{\\\\mathcal {X}},\\\\ |\\\\mathfrak {m}|\\\\le s \\\\right\\\\rbrace .$ Observe that $\\\\sup _{|\\\\mathfrak {m}|\\\\le s}\\\\sup _{x\\\\in \\\\widetilde{\\\\mathcal {X}}}\\\\sup _{h\\\\in (0,1)} h^{-d} \\\\mathbb {E}\\\\left[\\\\partial ^{\\\\mathfrak {m}}K_d\\\\left(\\\\frac{(x-z)}{h}\\\\right)\\\\right]^2\\\\le \\\\Vert K\\\\Vert _{C^s(\\\\mathbb {R}^d)}\\\\Vert f\\\\Vert _{C^s(\\\\mathcal {X})}\\\\sup _{|\\\\mathfrak {m}|\\\\le s}\\\\int |\\\\partial ^{\\\\mathfrak {m}} K_d(v)|\\\\,dv<\\\\infty .$ Further, by as:kernel, $\\\\partial ^{\\\\mathfrak {m}} K_d(\\\\cdot )$ is differentiable for each $|\\\\mathfrak {m}|\\\\le s$ .\",\n \"Consequently $\\\\mathcal {G}$ is point wise measurable and of VC-type (see [49]; also see [138] for definitions of point wise differentiability and VC classes).\",\n \"This verifies the assumptions of lem:emproc.\",\n \"Observe that $\\\\frac{1}{n}\\\\sum _{i=1}^m \\\\partial ^{\\\\mathfrak {m}} K\\\\left(\\\\frac{x-X_i}{h_m}\\\\right)=h_m^{d+|\\\\mathfrak {m}|}\\\\partial ^{\\\\mathfrak {m}} \\\\widehat{f}_{\\\\mu }(x),\\\\qquad \\\\mathbb {E}\\\\left[\\\\partial ^{\\\\mathfrak {m}}K\\\\left(\\\\frac{x-X}{h_m}\\\\right)\\\\right]=h_m^{d+|\\\\mathfrak {m}|}\\\\mathbb {E}\\\\left[\\\\partial ^{\\\\mathfrak {m}}\\\\widehat{f}_{\\\\mu }(x)\\\\right].$ A direct application of lem:emproc for all $|\\\\mathfrak {m}|\\\\le s$ , then implies $\\\\sup _{x\\\\in \\\\widetilde{\\\\mathcal {X}}}\\\\sqrt{\\\\frac{m}{h_m^d\\\\log {m}}}\\\\cdot h_m^{s+d} \\\\Vert \\\\widehat{f}_{\\\\mu }-\\\\mathbb {E}\\\\widehat{f}_{\\\\mu }\\\\Vert _{C^s(\\\\widetilde{\\\\mathcal {X}})}=O_p(1).$ Using the observation that $mh_m^{d+2s}/\\\\log {m}\\\\rightarrow 0$ as $m\\\\rightarrow \\\\infty $ then completes the proof.\",\n \"[Proof of prop:wassrate] As $\\\\mu \\\\ne \\\\nu $ , we have $W_2(\\\\mu ,\\\\nu )>0$ .\",\n \"Therefore, $|W_2(\\\\widetilde{\\\\mu }_m,\\\\widetilde{\\\\nu }_n)-W_2(\\\\mu ,\\\\nu )|=\\\\frac{W_2^2(\\\\widetilde{\\\\mu }_m,\\\\widetilde{\\\\nu }_n)-W_2^2(\\\\mu ,\\\\nu )|}{W_2(\\\\widetilde{\\\\mu }_m,\\\\widetilde{\\\\nu }_n)+W_2(\\\\mu ,\\\\nu )}\\\\le \\\\frac{W_2^2(\\\\widetilde{\\\\mu }_m,\\\\widetilde{\\\\nu }_n)-W_2^2(\\\\mu ,\\\\nu )|}{W_2(\\\\mu ,\\\\nu )}.$ The conclusion then follows from thm:smooth.\",\n \"[Proof of thm:dismooth] Recall that $\\\\widetilde{\\\\mu }_m$ and $\\\\widetilde{\\\\nu }_n$ are defined as the empirical distributions induced by $M=n^{\\\\frac{s+2}{2}}$ random samples drawn from $\\\\widehat{f}_{\\\\mu }$ and $\\\\widehat{f}_{\\\\nu }$ respectively, where $\\\\widehat{f}_{\\\\mu }$ , $\\\\widehat{f}_{\\\\nu }$ are the kernel density estimates as presented in (REF ).\",\n \"Let us write $\\\\mu _{h_n}$ and $\\\\nu _{h_n}$ for the probability measure induced by the kernel density estimates $\\\\widehat{f}_{\\\\mu }$ and $\\\\widehat{f}_{\\\\nu }$ respectively.\",\n \"Once again, by using thm:nsmooth, (REF ), it suffices to prove the following: $\\\\mathbb {E}\\\\left|W_2^2(\\\\widetilde{\\\\mu }_m,\\\\widetilde{\\\\nu }_n)-W_2^2(\\\\mu ,\\\\nu )\\\\right|.$ Next note that by the triangle inequality, (REF ) can be bounded above by: $\\\\mathbb {E}\\\\big |W_2^2(\\\\widetilde{\\\\mu }_m,\\\\nu _{h_n})&-W_2^2(\\\\widetilde{\\\\mu }_m,\\\\widetilde{\\\\nu }_n)\\\\big |+\\\\mathbb {E}\\\\big |W_2^2(\\\\widetilde{\\\\mu }_m,\\\\nu _{h_n})-W_2^2(\\\\mu _{h_n},\\\\nu _{h_n})\\\\big |\\\\nonumber \\\\\\\\&+\\\\mathbb {E}\\\\big |W_2^2(\\\\mu _{h_n},\\\\nu _{h_n})-W_2^2(\\\\widetilde{\\\\mu }_m,\\\\widetilde{\\\\nu }_n)\\\\big |.$ Next note that, by thm:smooth, we have: $\\\\mathbb {E}\\\\left|W_2^2(\\\\mu _{h_n},\\\\nu _{h_n})-W_2^2(\\\\widetilde{\\\\mu }_m,\\\\widetilde{\\\\nu }_n)\\\\right|\\\\lesssim r_{d,s}^{(n,n)}.$ Next we show that $\\\\mathbb {E}\\\\left|W_2^2(\\\\widetilde{\\\\mu }_m,\\\\nu _{h_n})-W_2^2(\\\\mu _{h_n},\\\\nu _{h_n})\\\\right|\\\\lesssim r_{d,s}^{(n,n)}.$ The other term in (REF ) can be bounded similarly.\",\n \"Note that, conditioned on $X_1,\\\\ldots ,X_n,Y_1,\\\\ldots ,Y_n$ , $\\\\mu _{h_n}$ and $\\\\nu _{h_n}$ are non-random measures and $\\\\widetilde{\\\\mu }_m$ and $\\\\widetilde{\\\\nu }_n$ are the empirical distributions on $M=n^{\\\\frac{s+2}{2}}$ random samples from the measures $\\\\mu _{h_n}$ and $\\\\nu _{h_n}$ , respectively.\",\n \"Therefore, conditioned on $X_1,\\\\ldots ,X_n,Y_1,\\\\ldots ,Y_n$ (which have fixed compact supports), we can invoke cor:fsam to get: $\\\\mathbb {E}\\\\left|W_2^2(\\\\widetilde{\\\\mu }_m,\\\\nu _{h_n})-W_2^2(\\\\mu _{h_n},\\\\nu _{h_n})\\\\right| \\\\lesssim r_{d}^{(M,M)},$ with $M=n^{\\\\frac{s+2}{2}}$ .\",\n \"Recall that: $r_d^{(M,M)}={\\\\left\\\\lbrace \\\\begin{array}{ll} n^{-\\\\frac{s+2}{4}} & \\\\mbox{if}\\\\ d\\\\le 3\\\\\\\\ n^{-\\\\frac{s+2}{4}}\\\\log {(1+n)} & \\\\mbox{if}\\\\ d=4\\\\\\\\ n^{-\\\\frac{s+2}{d}} & \\\\mbox{if}\\\\ d>4\\\\end{array}\\\\right.\",\n \"}.$ It therefore only remains to compare $r_d^{(M,M)}$ and $r_{d,s}^{(n,n)}$ .\",\n \"Case 1: $d\\\\le 2(s+2)$ .\",\n \"In this case, if $d=1,2,3$ , then $r_d^{(M,M)}=n^{-\\\\frac{s+2}{4}}=n^{-\\\\frac{1}{2}}\\\\times n^{-\\\\frac{s}{4}}\\\\lesssim n^{-\\\\frac{1}{2}}$ .\",\n \"If $d=4$ , then $r_d^{(M,M)}= n^{-\\\\frac{s+2}{4}}\\\\log {(1+n)}=n^{-\\\\frac{1}{2}}\\\\times \\\\left(n^{-\\\\frac{s}{4}}\\\\log {n}\\\\right)\\\\lesssim n^{-\\\\frac{1}{2}}$ .\",\n \"If $d>4$ , then $r_d^{(M,M)}=n^{-\\\\frac{s+2}{d}}\\\\lesssim n^{-\\\\frac{1}{2}}$ as $\\\\frac{s+2}{d}\\\\ge \\\\frac{1}{2}$ .\",\n \"Therefore, in all the cses, $r_d^{(M,M)}\\\\lesssim n^{-\\\\frac{1}{2}}=r_{d,s}^{(n,n)}$ for $d\\\\le 2(s+2)$ .\",\n \"Case 2: $d>2(s+2)$ .\",\n \"As $s>0$ , then $d>4$ .\",\n \"In this case, once again $r_d^{(M,M)}=n^{-\\\\frac{s+2}{d}}=r_{d,s}^{(n,n)}$ .\",\n \"This establishes (REF ) and completes the proof.\"\n ],\n [\n \"Proofs from sec:App\",\n \"[Proof of thm:barate] First define the following measure: $\\\\rho _0^{\\\\mathrm {OR}}:=\\\\left(\\\\frac{1}{2}\\\\mbox{Id}+\\\\frac{1}{2}T_0\\\\right)\\\\#\\\\widetilde{\\\\mu }_m.$ Fix any $\\\\gamma \\\\in \\\\widetilde{\\\\Gamma }_{\\\\mathrm {min}}$ .\",\n \"By applying the triangle inequality followed by a power mean inequality, we have: $\\\\sup _{\\\\gamma \\\\in \\\\widetilde{\\\\Gamma }_{\\\\mathrm {min}}}W_2^2\\\\big (\\\\widehat{\\\\rho }_0^{\\\\gamma },\\\\rho _0\\\\big )\\\\lesssim W_2^2\\\\big (\\\\rho _0^{\\\\mathrm {OR}},\\\\rho _0\\\\big )+\\\\sup _{\\\\gamma \\\\in \\\\widetilde{\\\\Gamma }_{\\\\mathrm {min}}} W_2^2\\\\big (\\\\widehat{\\\\rho }_0^{\\\\gamma },\\\\rho _0^{\\\\mathrm {OR}}\\\\big ).$ Next observe that $\\\\rho _0^{\\\\mathrm {OR}}$ is the empirical distribution corresponding to $m$ random samples drawn according to $\\\\rho _0$ .\",\n \"Therefore, by using [55], we get: $ W_2^2\\\\big (\\\\rho _0^{\\\\mathrm {OR}},\\\\rho _0\\\\big )\\\\lesssim r_d^{(m,m)}.$ Next we will bound the second term on the right hand side of (REF ).\",\n \"Towards this direction, recall the definition of $\\\\Pi (\\\\cdot ,\\\\cdot )$ from sec:setting.\",\n \"Consider the following coupling: $\\\\pi _0^{\\\\gamma }:=\\\\left(\\\\frac{1}{2}\\\\mbox{Id}+\\\\frac{1}{2}\\\\widetilde{T}_{m,n}^{\\\\gamma },\\\\frac{1}{2}\\\\mbox{Id}+\\\\frac{1}{2}T_0\\\\right)\\\\#\\\\widetilde{\\\\mu }_m.$ Observe that $\\\\pi _0^{\\\\gamma }\\\\in \\\\Pi \\\\big (\\\\widehat{\\\\rho }_0^{\\\\gamma },\\\\rho _0^{\\\\mathrm {OR}}\\\\big )$ .\",\n \"By plugging the coupling $\\\\pi _0^{\\\\gamma }$ into the definition of 2-Wasserstein distance in (REF ), we further get: $\\\\sup _{\\\\gamma \\\\in \\\\widetilde{\\\\Gamma }_{\\\\mathrm {min}}} W_2^2\\\\big (\\\\widehat{\\\\rho }_0^{\\\\gamma },\\\\rho _0^{\\\\mathrm {OR}}\\\\big )&\\\\le \\\\sup _{\\\\gamma \\\\in \\\\widetilde{\\\\Gamma }_{\\\\mathrm {min}}} \\\\int \\\\Vert x-y\\\\Vert ^2\\\\,d\\\\pi _0^{\\\\gamma }(x,y)\\\\nonumber \\\\\\\\ &=\\\\sup _{\\\\gamma \\\\in \\\\widetilde{\\\\Gamma }_{\\\\mathrm {min}}} \\\\int \\\\Vert \\\\widetilde{T}_{m,n}^{\\\\gamma }(x)-T_0(x)\\\\Vert ^2\\\\,d\\\\widetilde{\\\\mu }_m(x)\\\\nonumber \\\\\\\\ &= O_p\\\\left(r^{(m,n)}_d\\\\times (\\\\log {(1+\\\\max \\\\lbrace m,n\\\\rbrace )})^{t_{d,\\\\alpha }}\\\\right)$ where the last inequality follows from thm:nsmooth.\",\n \"Combining (REF ) and (REF ) with (REF ) completes the proof.\",\n \"[Proof of thm:dethresh] Let $T_1^{(n)}(\\\\cdot )$ and $T_2^{(n)}(\\\\cdot )$ be the optimal transport maps from $\\\\mu ^{(n)}$ to $\\\\upsilon _1$ and $\\\\nu ^{(n)}$ to $\\\\upsilon _2$ .\",\n \"Set $\\\\widehat{x}^{\\\\mathrm {OR}}_{ij}:=K_1(T_1^{(n)}(X_i),T_1^{(n)}(X_j)),\\\\qquad \\\\widehat{y}^{\\\\mathrm {OR}}_{ij}:=K_2(T_2^{(n)}(Y_i),T_2^{(n)}(Y_j))$ and define the oracle version of $\\\\widehat{\\\\mathrm {rHSIC}}$ as follows: $\\\\widehat{\\\\mathrm {rHSIC}}^{\\\\mathrm {OR}}:=\\\\underbrace{n^{-2}\\\\sum _{i,j}\\\\widehat{x}^{\\\\mathrm {OR}}_{ij}\\\\widehat{y}^{\\\\mathrm {OR}}_{ij}}_{\\\\widehat{A}_{n,1}^{\\\\mathrm {OR}}}+\\\\underbrace{n^{-4}\\\\sum _{i,j,r,s}\\\\widehat{x}^{\\\\mathrm {OR}}_{ij}\\\\widehat{y}^{\\\\mathrm {OR}}_{rs}}_{\\\\widehat{A}_{n,2}^{\\\\mathrm {OR}}}-2\\\\underbrace{n^{-3}\\\\sum _{i,j,r} \\\\widehat{x}^{\\\\mathrm {OR}}_{ij}\\\\widehat{y}^{\\\\mathrm {OR}}_{ir}}_{\\\\widehat{A}_{n,3}^{\\\\mathrm {OR}}}.$ The proof of thm:dethresh now proceeds using the following steps: Step I: We show that: $\\\\mathbb {E}\\\\big |\\\\widehat{\\\\mathrm {rHSIC}}^{\\\\mathrm {OR}}-\\\\mathrm {rHSIC}(\\\\pi ^{(n)}|\\\\mu ^{(n)}\\\\times \\\\nu ^{(n)})\\\\big |\\\\lesssim n^{-1/2},$ where $\\\\widehat{\\\\mathrm {rHSIC}}^{\\\\mathrm {OR}}(\\\\cdot |\\\\cdot )$ is defined in (REF ).\",\n \"Step II: We prove that: $ \\\\mathbb {E}\\\\big |\\\\widehat{\\\\mathrm {rHSIC}}^{\\\\mathrm {OR}}-\\\\widehat{\\\\mathrm {rHSIC}}\\\\big |\\\\lesssim \\\\sqrt{r_d^{(n,n)}}.$ Step III: We combine steps I and II to prove thm:dethresh.\",\n \"Let us begin with this step first.\",\n \"Note that by using the triangle inequality, we have: $\\\\widehat{\\\\mathrm {rHSIC}}\\\\ge \\\\mathrm {rHSIC}(\\\\pi ^{(n)}|\\\\mu ^{(n)}\\\\times \\\\nu ^{(n)})-\\\\big |\\\\widehat{\\\\mathrm {rHSIC}}^{\\\\mathrm {OR}}-\\\\mathrm {rHSIC}\\\\big |-\\\\big |\\\\widehat{\\\\mathrm {rHSIC}}^{\\\\mathrm {OR}}-\\\\widehat{\\\\mathrm {rHSIC}}\\\\big |.$ Next observe that by steps I and II, $\\\\max \\\\bigg \\\\lbrace \\\\big |\\\\widehat{\\\\mathrm {rHSIC}}^{\\\\mathrm {OR}}-\\\\mathrm {rHSIC}\\\\big |,\\\\big |\\\\widehat{\\\\mathrm {rHSIC}}^{\\\\mathrm {OR}}-\\\\widehat{\\\\mathrm {rHSIC}}\\\\big |\\\\bigg \\\\rbrace =O_p\\\\big (\\\\sqrt{r_{d_1,d_2}^{(n,n)}}\\\\big ).$ Using the above display with (REF ) and the assumption $(r_{d_1,d_2}^{(n,n)})^{-1/2}\\\\mathrm {rHSIC}(\\\\pi ^{(n)}|\\\\mu ^{(n)}\\\\times \\\\nu ^{(n)})\\\\rightarrow \\\\infty $ , we have: $\\\\big (r_{d_1,d_2}^{(n,n)}\\\\big )^{-1/2}\\\\widehat{\\\\mathrm {rHSIC}}\\\\overset{P}{\\\\longrightarrow }\\\\infty .$ Therefore, as $n\\\\sqrt{r_{d_1,d_2}^{(n,n)}}\\\\rightarrow \\\\infty $ and $c_{n,\\\\alpha }=O(1)$ (see [44]), we have: $\\\\mathbb {E}\\\\phi _{n,\\\\alpha }=\\\\mathbb {P}(n\\\\times \\\\widehat{\\\\mathrm {rHSIC}}\\\\ge c_{n,\\\\alpha })\\\\rightarrow 1$ under $(r_{d_1,d_2}^{(n,n)})^{-1/2}\\\\mathrm {rHSIC}(\\\\pi ^{(n)}|\\\\mu ^{(n)}\\\\times \\\\nu ^{(n)})\\\\rightarrow \\\\infty $ .\",\n \"This completes the proof.\",\n \"It therefore remains to prove steps I and II.\",\n \"For step I, let $(X_1^{\\\\prime },Y_1^{\\\\prime }),\\\\ldots ,(X_n^{\\\\prime },Y_n^{\\\\prime })\\\\overset{i.i.d.\",\n \"}{\\\\sim }\\\\pi ^{(n)}$ .\",\n \"Fix an arbitrary $1\\\\le j\\\\le n$ .\",\n \"Let $\\\\widehat{A}_{n,1,j}^{\\\\mathrm {OR},^{\\\\prime }}$ be the same as $\\\\widehat{A}_{n,1}^{\\\\mathrm {OR}}$ except with $(X_j,Y_j)$ replaced by $(X_j^{\\\\prime },Y_j^{\\\\prime })$ .\",\n \"It is easy to check by the compactness of supports of all distributions involved, that: $\\\\max \\\\limits _{1\\\\le j\\\\le n}\\\\big |\\\\widehat{A}_{n,1}^{\\\\mathrm {OR}}-\\\\widehat{A}_{n,1,j}^{\\\\mathrm {OR},^{\\\\prime }}\\\\big |\\\\lesssim n^{-1}.$ Therefore by using Mcdiarmid's inequality (see [20]), we have, for any $t>0$ , $\\\\mathbb {P}\\\\left(\\\\sqrt{n}(\\\\widehat{A}_{n,1}^{\\\\mathrm {OR}}-\\\\mathbb {E}\\\\widehat{A}_{n,1}^{\\\\mathrm {OR}})\\\\ge t\\\\right)\\\\le \\\\exp (-Ct^2)$ for some constant $C>0$ free of $n$ and $t$ .\",\n \"Similar concentrations can be derived for $\\\\widehat{A}_{n,2}^{\\\\mathrm {OR}}$ and $\\\\widehat{A}_{n,3}^{\\\\mathrm {OR}}$ .\",\n \"Combining these concentrations with the observation that $\\\\mathrm {rHSIC}(\\\\pi ^{(n)}|\\\\mu ^{(n)}\\\\times \\\\nu ^{(n)})=\\\\mathbb {E}\\\\widehat{A}_{n,1}^{\\\\mathrm {OR}}+\\\\mathbb {E}\\\\widehat{A}_{n,2}^{\\\\mathrm {OR}}-2\\\\mathbb {E}\\\\widehat{A}_{n,3}^{\\\\mathrm {OR}}$ completes the proof of step I.\",\n \"We now move on to step II.\",\n \"Recall the definition of $\\\\widehat{\\\\mathrm {rHSIC}}$ from (REF ) and write: $\\\\widehat{\\\\mathrm {rHSIC}}=\\\\underbrace{n^{-2}\\\\sum _{i,j}\\\\widehat{x}_{ij}\\\\widehat{y}_{ij}}_{\\\\widehat{A}_{n,1}}+\\\\underbrace{n^{-4}\\\\sum _{i,j,r,s}\\\\widehat{x}_{ij}\\\\widehat{y}_{rs}}_{\\\\widehat{A}_{n,2}}-2\\\\underbrace{n^{-3}\\\\sum _{i,j,r} \\\\widehat{x}_{ij}\\\\widehat{y}_{ir}}_{\\\\widehat{A}_{n,3}}.$ By the Lipschitzness of $K_1(\\\\cdot ,\\\\cdot )$ and $K_2(\\\\cdot ,\\\\cdot )$ , we have: $\\\\big |\\\\widehat{x}_{ij}-\\\\widehat{x}^{\\\\mathrm {OR}}_{ij}\\\\big |\\\\lesssim \\\\Vert \\\\widehat{T}_{1,n}(X_i)-T_1^{(n)}(X_i)\\\\Vert +\\\\Vert \\\\widehat{T}_{1,n}(X_j)-T_1^{(n)}(X_j)\\\\Vert ,$ $\\\\big |\\\\widehat{y}_{ij}-\\\\widehat{y}^{\\\\mathrm {OR}}_{ij}\\\\big |\\\\lesssim \\\\Vert \\\\widehat{T}_{2,n}(Y_i)-T_2^{(n)}(Y_i)\\\\Vert +\\\\Vert \\\\widehat{T}_{2,n}(Y_j)-T_2^{(n)}(Y_j)\\\\Vert .$ Therefore, by using the fact that the probability measures $\\\\upsilon _1$ and $\\\\upsilon _2$ are compactly supported, we get: $\\\\big |\\\\widehat{A}_{n,1}-\\\\widehat{A}_{n,1}^{\\\\mathrm {OR}}\\\\big |\\\\lesssim \\\\frac{1}{n}\\\\sum _{i=1}^n \\\\Vert \\\\widehat{T}_{1,n}(X_i)-T_1^{(n)}(X_i)\\\\Vert +\\\\frac{1}{n}\\\\sum _{j=1}^n \\\\Vert \\\\widehat{T}_{2,n}(Y_j)-T_2^{(n)}(Y_j)\\\\Vert .$ The same bound can similarly be verified for $|\\\\widehat{A}_{n,2}-\\\\widehat{A}_{n,2}^{\\\\mathrm {OR}}|$ and $|\\\\widehat{A}_{n,3}-\\\\widehat{A}_{n,3}^{\\\\mathrm {OR}}|$ .\",\n \"Combining these observations, we have: $\\\\big |\\\\widehat{\\\\mathrm {rHSIC}}-\\\\widehat{\\\\mathrm {rHSIC}}^{\\\\mathrm {OR}}\\\\big |&\\\\lesssim \\\\frac{1}{n}\\\\sum _{i=1}^n \\\\Vert \\\\widehat{T}_{1,n}(X_i)-T_1^{(n)}(X_i)\\\\Vert +\\\\frac{1}{n}\\\\sum _{j=1}^n \\\\Vert \\\\widehat{T}_{2,n}(Y_j)-T_2^{(n)}(Y_j)\\\\Vert \\\\nonumber \\\\\\\\ &\\\\le \\\\sqrt{\\\\frac{1}{n}\\\\sum _{i=1}^n \\\\Vert \\\\widehat{T}_{1,n}(X_i)-T_1^{(n)}(X_i)\\\\Vert ^2}+\\\\sqrt{\\\\frac{1}{n}\\\\sum _{j=1}^n \\\\Vert \\\\widehat{T}_{2,n}(Y_j)-T_2^{(n)}(Y_j)\\\\Vert ^2}.$ Step II then follows by invoking cor:fsam.\"\n ],\n [\n \"Auxiliary definitions and results\",\n \"[Subdifferential set and subgradient] Given a convex function $f:\\\\mathbb {R}^d\\\\rightarrow \\\\mathbb {R}\\\\cup \\\\lbrace \\\\infty \\\\rbrace $ , we define the subdifferential set of $f(\\\\cdot )$ at $x\\\\in \\\\mbox{dom}(f):=\\\\lbrace z\\\\in \\\\mathbb {R}^d:f(z)<\\\\infty \\\\rbrace $ as follows: $\\\\partial f(x):=\\\\lbrace \\\\xi \\\\in \\\\mathbb {R}^d:\\\\ f(x)+\\\\langle \\\\xi , y-x\\\\rangle \\\\le f(y),\\\\quad \\\\mbox{for}\\\\ \\\\mbox{all}\\\\ y\\\\in \\\\mathbb {R}^d\\\\rbrace .$ Any element in the set $\\\\partial f(x)$ is called a subgradient of $f(\\\\cdot )$ at $x$ .\",\n \"[Strong convexity] A function $f:\\\\mathbb {R}^d\\\\rightarrow \\\\mathbb {R}\\\\cup \\\\lbrace \\\\infty \\\\rbrace $ is strongly convex with parameter $\\\\lambda >0$ , if, for all $x,y\\\\in \\\\mbox{dom}(f)=\\\\lbrace z\\\\in \\\\mathbb {R}^d:f(z)<\\\\infty \\\\rbrace $ , the following holds: $f(y)\\\\ge f(x) + \\\\langle \\\\xi _x,y-x\\\\rangle +\\\\frac{\\\\lambda }{2}\\\\Vert y-x\\\\Vert ^2,$ where $\\\\xi _x\\\\in \\\\partial f(x)$ , the subgradient of $f(\\\\cdot )$ at $x$ as in def:subg.\",\n \"[Wavelet basis] We present our main assumptions on the wavelet basis discussed in sec:wavelet only for the wavelets on the space $\\\\mathcal {X}$ .\",\n \"The same assumptions are also required for the wavelets on $\\\\mathcal {Y}$ .\",\n \"These are essentially a subset of the assumptions laid out in [143] as we heavily rely on [143] for proving thm:smoothwav.\",\n \"(Regularity).\",\n \"Fix $r>\\\\max \\\\lbrace s,1\\\\rbrace $ .\",\n \"The functions in $\\\\Phi $ and $\\\\Psi _j$ , $j\\\\ge 0$ have $r$ continuous derivatives, and all polynomials of degree at most $r$ on $\\\\mathcal {X}$ lie in the span of the functions in $\\\\Phi $ .\",\n \"(Tensor construction).\",\n \"Each $\\\\psi (\\\\cdot )\\\\in \\\\Psi _j$ can be expressed as $\\\\psi (x)=\\\\prod _{i=1}^d \\\\psi _i(x_i)$ , where $x=(x_1,\\\\ldots ,x_d)$ , for some univariate functions $\\\\psi _i(\\\\cdot )$ 's.\",\n \"(Locality).\",\n \"For each $\\\\psi (\\\\cdot )\\\\in \\\\Psi _j$ there exists a rectangle $I_{\\\\psi }\\\\subseteq \\\\mathcal {X}$ such that $\\\\mbox{supp}(\\\\psi )\\\\subseteq I_{\\\\psi }$ , $\\\\mbox{diam}(I_{\\\\psi })\\\\le C_1\\\\cdot 2^{-j}$ , and $\\\\sup _{x\\\\in \\\\mathcal {X}}\\\\sum _{\\\\psi (\\\\cdot )\\\\in \\\\Psi _j}\\\\mathbb {1}(x\\\\in I_{\\\\psi })\\\\le C_2$ for some constants $C_1,C_2>0$ .\",\n \"(Bernstein estimate).\",\n \"$\\\\Vert \\\\nabla f\\\\Vert _{L^2(\\\\mathcal {X})}\\\\le C_3\\\\cdot 2^j\\\\Vert f\\\\Vert _{L^2(\\\\mathcal {X})}$ for any $f(\\\\cdot )$ in the span of the functions in $\\\\mbox{span}\\\\left(\\\\Phi \\\\cup \\\\left\\\\lbrace \\\\cup _{0\\\\le k4$ , there is no comparable algebraic structure and the Pfaffian could only be given explicitly in terms of the components of the transformed Hamiltonian.","The number of terms in the resulting expression is exponentially large in $N$ .","It is useful to review and compare our results for one-dimensional $A_{2g+}$ pairing and multidimensional $T_{1g+}$ pairing, which illustrate some of our general remarks.","Note that it is hard to envision local degrees of freedom that realize an operator with $A_{2g+}$ symmetry, ultimately because of the high minimum order $l=6$ of basis functions.","Hence, purely local pairing is unlikely to exist and it indeed does not appear in the examples we have considered.","For infinitesimal pairing, the $A_{2g+}$ pairing state for any model has six symmetry-imposed line nodes in the $\\lbrace 110\\rbrace $ mirror planes.","Since they are present in the real and imaginary parts of the gap function, they persist for TRS-broken states.","We find that beyond infinitesimal pairing, the nodes of the TRS-broken $A_{2g+}$ states either persist as line nodes or are inflated into BFSs.","Specifically, the nodes are inflated in the mirror planes for the electrons with effective spin $j=3/2$ but not for the cases of two s-orbitals and of two orbitals of opposite parity.","The origin of this difference is that for the $j=3/2$ case, in the mirror planes, the two amplitudes $\\delta _{Eg+}$ and $\\delta _{T2g+}$ contribute to the Pfaffian whenever there is a phase difference between them.","For the other two cases, there is a single amplitude in the mirror planes and its phase can be gauged away.","Thus there is no inflation.","A general insight here is that if only a single amplitude contributes in some high-symmetry direction or plane the breaking of TRS cannot lead to the formation of a BFS or of a gap since the physics is (gauge) invariant under changes of the phase of this amplitude.","This argument also applies to multidimensional irreps.","TRS-breaking pairing states are more natural for multidimensional irreps.","In our context, $T_{1g+}$ is an interesting pairing symmetry.","Whereas it appears for purely local pairing in the case of two s-orbitals, it only appears for nonlocal pairing for effective-spin-$3/2$ fermions.","Our study suggests that for both cases all point and line nodes appearing for TRS-broken $T_{1g+}$ pairing state with order parameter $(1,i,0)$ are inflated for noninfinitesimal pairing.","The point nodes on the $k_z$ -axis are the only ones which remains attached to the normal-state Fermi surface, while all other point and line nodes are shifted away from the normal-state Fermi surface and thus could annihilate for strong coupling.","To conclude, nonlocal pairing typically permits a much larger number of possible pairing symmetries.","Their nodal structure, including the possibility of BFSs, can be analyzed based on symmetry.","For nodes at infinitesimal pairing, there is a simple yet powerful criterion in terms of a scalar product of form factors.","The known criterion for the appearance of BFSs in terms of a Pfaffian extends to nonlocal pairing and general internal degrees of freedom and shows that BFSs generically exist if the superconducting state breaks TRS.","The authors thank D. F. Agterberg, P. M. R. Brydon, I. F. Herbut, A. Knoll, S. Kobayashi, and J. M. Link for useful discussions.","Financial support by the Deutsche Forschungsgemeinschaft through the Collaborative Research Center SFB 1143, Project A04, the Research Training Group GRK 1621, and the Cluster of Excellence on Complexity and Topology in Quantum Matter ct.qmat (EXC 2147) is gratefully acknowledged."],["Enumeration of basis matrices","In this Appendix, we obtain the number of basis matrices that can occur and thus generically do occur in the normal-state Hamiltonian $H_N(\\mathbf {k})$ .","The same basis matrices can appear in the pairing matrix $D(\\mathbf {k})$ for $s_T=-1$ and even-parity superconductivity as well as for $s_T=+1$ and odd-parity superconductivity, while in the other two cases, only the remaining basis matrices can appear in $D(\\mathbf {k})$ .","The statements can be obtained in a more general framework but it might be useful to present them using representation theory of point groups.","The following theorem is proven: Let $N$ be the dimension of the Hilbert space describing the local degrees of freedom in the normal state.","Let $P$ be the unitary parity operator on this Hilbert space and let $\\mathcal {T}$ be the antiunitary time-reversal operator.","Then the number of Hermitian basis matrices $h_n$ that can appear in a normal-state Hamiltonian that respects $P\\mathcal {T}$ symmetry is $n_h = \\left\\lbrace \\begin{array}{ll}\\displaystyle \\frac{N(N+1)}{2} & \\mbox{for $(P\\mathcal {T})^2 = +1$} , \\\\[1.5ex]\\displaystyle \\frac{N(N-1)}{2} & \\mbox{for $(P\\mathcal {T})^2 = -1$} .\\end{array} \\right.$ As we shall see, the case $(P\\mathcal {T})^2 = -1$ can only occur for even $N$ .","Results for small $N$ are given in Table REF .","Table: Number of basis matrices h n h_n that appear in a normal-state Hamiltonian H N (𝐤)H_N(\\mathbf {k}) with P𝒯P\\mathcal {T} symmetry for the two cases that P𝒯P\\mathcal {T} squares to ±1\\pm 1.","NN is the dimension of the internal Hilbert space.","For (P𝒯) 2 =-1(P\\mathcal {T})^2=-1, which isthe standard case, NN has to be even.To show this, the normal-state Hamiltonian is expanded as $H_N(\\mathbf {k}) = \\sum _{n=1}^{n_h} c_n(\\mathbf {k})\\, h_n .$ Under the magnetic point group $M$ , the functions $c_n(\\mathbf {k})$ must transform like the corresponding $h_n$ to ensure that $H_N(\\mathbf {k})$ is invariant.","The operation $P\\mathcal {T}$ is an element of $M$ .","The irreps of $M$ can be uniquely and exhaustively divided into irreps that are even or odd under $P\\mathcal {T}$ .","Only the $P\\mathcal {T}$ -even irreps have momentum-space basis functions since the momentum $\\mathbf {k}$ is $P\\mathcal {T}$ even.","Hence, all $P\\mathcal {T}$ -even and no $P\\mathcal {T}$ -odd $h_\\nu $ can occur in Eq.","(REF ).","The proposition is thus a statement about the number $n_h$ of $N\\times N$ basis matrices $h_\\nu $ that are even under $P\\mathcal {T}$ .","Since $P\\mathcal {T}$ is antiunitary there exists a unitary $N\\times N$ matrix $U_{PT}$ with $P\\mathcal {T} = U_{PT} \\mathcal {K}$ , where $\\mathcal {K}$ is the complex conjugation.","Then we have $(P\\mathcal {T})^2 = U_{PT} \\mathcal {K} U_{PT} \\mathcal {K} = U_{PT} U_{PT}^* .$ Thus $(P\\mathcal {T})^2 = \\pm 1$ is equivalent to $U_{PT} U_{PT}^* = \\pm 1$ and, since $U_{PT}^*$ is unitary, to $U_{PT} = \\pm (U_{PT}^*)^{-1} = \\pm U_{PT}^T$ .","Case 1: $(P\\mathcal {T})^2 = +1$ and symmetric $U_{PT}$ .","For any unitary symmetric $N\\times N$ matrix $U_{PT}$ , there exists a unitary matrix $Q$ such that $U_{PT} = Q Q^T .$ $h_\\nu $ being even/odd under $P\\mathcal {T}$ means $U_{PT} h_\\nu ^* U_{PT}^\\dagger = \\pm h_\\nu $ , which due to the Hermiticity of $h_\\nu $ is equivalent to $U_{PT} h_\\nu ^T U_{PT}^\\dagger = \\pm h_\\nu .$ Equation (REF ) then gives $Q Q^T h_\\nu ^T Q^* Q^\\dagger = \\pm h_\\nu ,$ which is equivalent to $Q^T h_\\nu ^T Q^* = (Q^\\dagger h_\\nu Q)^T = \\pm Q^\\dagger h_\\nu Q .$ Note that $k_\\nu \\equiv Q^\\dagger h_\\nu Q$ is Hermitian.","The dimension of the vector space over $\\mathbb {R}$ of Hermitian $N\\times N$ matrices that are also symmetric is ${N(N+1)}/{2}$ .","Hence, the dimension of the vector space of Hermitian $P\\mathcal {T}$ -even $N\\times N$ and thus the number of $P\\mathcal {T}$ -even basis elements $h_\\nu $ also equals ${N(N+1)}/{2}$ .","Analogously, the dimension of the space of Hermitian $N\\times N$ matrices that are antisymmetric and thus the number of $P\\mathcal {T}$ -odd basis matrices equals ${N(N-1)}/{2}$ .","Note that the sum of the two numbers is $N^2$ , as expected.","Case 2: $(P\\mathcal {T})^2 = -1$ and antisymmetric $U_{PT}$ .","For any unitary antisymmetric $N\\times N$ matrix $U_{PT}$ , there exists a unitary matrix $Q$ such that $U_{PT} = Q \\Lambda Q^T ,$ with $\\Lambda = i\\sigma _2 \\otimes ,$ which clearly means that $N$ must be even.","We therefore write $N=2M$ .","$h_\\nu $ being even/odd under $P\\mathcal {T}$ means $U_{PT} h_\\nu ^T U_{PT}^\\dagger = \\pm h_\\nu .$ Equation (REF ) gives $Q \\Lambda Q^T h_\\nu ^T Q^* \\Lambda ^\\dagger Q^\\dagger = \\pm h_\\nu ,$ which is equivalent to $\\Lambda Q^T h_\\nu ^T Q^* \\Lambda ^\\dagger = \\Lambda (Q^\\dagger h_\\nu Q)^T\\Lambda ^\\dagger = \\pm Q^\\dagger h_\\nu Q .$ Let $k_\\nu \\equiv Q^\\dagger h_\\nu Q$ (Hermitian).","We write $\\Lambda $ and $k_\\nu $ in block form as $\\Lambda &= \\left(\\begin{array}{cc}0 & \\\\ -& 0\\end{array}\\right) , \\\\k_\\nu &= \\left(\\begin{array}{cc}\\kappa _{11} & \\kappa _{12} \\\\ \\kappa _{12}^\\dagger & \\kappa _{22}\\end{array}\\right) ,$ where $\\kappa _{11}$ and $\\kappa _{22}$ are Hermitian.","Equation (REF ) can then be written as $\\Lambda k_\\nu ^T \\Lambda ^\\dagger = \\begin{pmatrix}\\kappa _{22}^T & -\\kappa _{12}^T \\\\ -\\kappa _{12}^* & \\kappa _{11}^T\\end{pmatrix} = \\begin{pmatrix}\\pm \\kappa _{11} & \\pm \\kappa _{12} \\\\ \\pm \\kappa _{12}^\\dagger & \\pm \\kappa _{22}\\end{pmatrix} = \\pm k_\\nu .$ This yields the relations $\\kappa _{12}^T = \\mp \\kappa _{12}$ and $\\kappa _{22} = \\pm \\kappa _{11}^T$ , and $k_\\nu $ thus assumes the form $k_\\nu = \\begin{pmatrix}\\kappa _{11} & \\kappa _{12} \\\\ \\kappa _{12}^\\dagger & \\pm \\kappa _{11}^T\\end{pmatrix} ,$ with $\\kappa _{11}^\\dagger = \\kappa _{11}$ and $\\kappa _{12}^T = \\mp \\kappa _{12}$ .","The blocks are $M\\times M$ matrices.","The dimension of the vector space spanned by the $k_\\nu $ is $M^2 + M(M-1) = M(2M-1)$ for the upper sign and $M^2 + M(M+1) = M(2M+1)$ for the lower sign.","The sum is $4M^2 = N^2$ , as expected.","Hence, the dimension of the vector space of Hermitian $N\\times N$ matrices that are even under $P\\mathcal {T}$ is ${N(N-1)}/{2}$ , whereas the dimension for $P\\mathcal {T}$ -odd matrices is ${N(N+1)}/{2}$ .","Note that the two numbers are interchanged compared to the case of $(P\\mathcal {T})^2 = +1$ .","This completes the proof.","We are concerned with systems that satisfy TRS and inversion symmetry separately.","Then the $P\\mathcal {T}$ -even (odd) irreps are the $g+$ and $u-$ ($g-$ and $u+$ ) irreps.","Moreover, the two operations commute [19], [29] and $P$ squares to $+1$ .","Hence, $(P\\mathcal {T})^2 = \\mathcal {T}^2$ .","If the internal degrees of freedom include the electron spin we have $\\mathcal {T}^2 = -1$ [19] and even dimension $N$ .","The case $\\mathcal {T}^2 = +1$ can only be realized if the spin does not occur explicitly, for example because one spin state is pushed to high energies by a magnetic field.","Then $\\mathcal {T}$ is not the physical TRS but an effective antiunitary symmetry."],["Infinitesimal-pairing nodes","In this Appendix, we discuss the IP nodes for $s_T=-1$ and odd-parity pairing and for the unconventional sign $s_T=+1$ of time reversal squared.","For $s_T=-1$ and odd-parity pairing, it is still true that infinitesimal pairing can be described in a single-band, pseudospin picture.","However, it is now in the pseudospin-triplet channel.","The pairing matrix in the effective single-band picture thus has the form $D_\\mathrm {eff}(\\mathbf {k}) = \\mathbf {d}(\\mathbf {k}) \\cdot \\mbox{$\\sigma $}$ , where $\\mbox{$\\sigma $}$ is the vector of Pauli matrices representing the pseudospin.","Since the pseudospin is even under inversion and odd under time reversal its components belong to one or more $g-$ irreps.","One can use representation theory to work out which irreps the components of $\\mathbf {d}(\\mathbf {k})$ must belong to in order to obtain a pairing state of a certain symmetry.","The condition $\\mathbf {d}(\\mathbf {k})=0$ then gives the symmetry-imposed IP nodes.","Nodal gaps are thus less likely than for singlet pairing since they must satisfy three scalar conditions.","We now turn to the nonstandard case $s_T=+1$ .","According to Appendix , this allows an effective single-band model with Hilbert-space dimension $N=1$ .","Equation (REF ) then implies that $U_T=1$ and thus $\\Delta (\\mathbf {k})=D(\\mathbf {k})$ .","The only Hermitian basis matrix is $h_0=1$ .","Hence, for a single-band model, the full symmetry information is carried by the form factor $f_0(\\mathbf {k})$ .","$h_0$ belongs to the trivial irrep, which of course is a $g+$ irrep.","Table REF then shows that for $N=1$ only odd-parity pairing with $u-$ form factor $f_0(\\mathbf {k})$ is possible.","The analysis of possible pairing states is analogous to the case with $s_T=-1$ and even parity, except that $g+$ irreps are replaced by $u-$ irreps.","Even-parity pairing states for $s_T=+1$ cannot be described by an effective $N=1$ model but are possible for $N=2$ .","In fact, there are multiple possibilities to implement this case because Eq.","(REF ) now allows $U_T$ to be any symmetric unitary $2\\times 2$ matrix, while $U_P$ can be any unitary $2\\times 2$ matrix that squares to $$ .","The specific $U_T$ and $U_P$ and thus the symmetry properties of the $2\\times 2$ basis matrices $h_1$ , $h_2$ , $h_3$ (which are linear combinations of Pauli matrices) depend on the underlying system.","Universal properties are therefore unlikely and we do not pursue this here."],["Existence and properties of the Pfaffian","In this Appendix, we review the main results for the Pfaffian.","A simpler proof than in [1], [2] is presented.","The BdG Hamiltonian (REF ) satisfies the charge-conjugation symmetry $\\mathcal {U}_C \\mathcal {H}^T(-\\mathbf {k}) \\mathcal {U}_C^\\dagger = - \\mathcal {H}(\\mathbf {k})$ , where $\\mathcal {U}_C = \\sigma _1 \\otimes $ .","For it to also satisfy inversion symmetry, there must exist a unitary matrix $U_P$ such that $\\mathcal {U}_P\\, \\mathcal {H}(-\\mathbf {k})\\, \\mathcal {U}_P^\\dagger = \\mathcal {H}(\\mathbf {k}) ,$ where $\\mathcal {U}_P = \\begin{pmatrix}U_P & 0 \\\\ 0 & U_P^*\\end{pmatrix} .$ This is a special case of Eqs.","(REF ) and (REF ).","The two symmetries imply $\\mathcal {C}P$ symmetry, $\\mathcal {U}_{CP}\\, \\mathcal {H}^T(\\mathbf {k})\\, \\mathcal {U}_{CP}^\\dagger =- \\mathcal {H}(\\mathbf {k}) ,$ with $\\mathcal {U}_{CP} = \\mathcal {U}_C \\mathcal {U}_P^*$ .","We find that $\\mathcal {C}P$ squares to the identity since $(\\mathcal {U}_{CP} \\mathcal {K})^2 = \\mathcal {U}_{CP} \\mathcal {U}_{CP}^*= \\begin{pmatrix}U_P^2 & 0 \\\\ 0 & (U_P^*)^2\\end{pmatrix}= \\begin{pmatrix}& 0 \\\\ 0 & \\end{pmatrix} .$ This implies that $\\mathcal {U}_{CP} = (\\mathcal {U}_{CP}^*)^{-1} = (\\mathcal {U}_{CP}^*)^\\dagger = \\mathcal {U}_{CP}^T$ so that $\\mathcal {U}_{CP}$ is symmetric.","For any (complex) symmetric matrix $\\mathcal {U}_{CP}$ , there exists a unitary matrix $\\Omega $ such that $\\Lambda = \\Omega \\mathcal {U}_{CP} \\Omega ^T$ (note the transpose) is a diagonal matrix with real nonnegative components (Autonne-Takagi factorization [44], [45]).","$\\Lambda $ is evidently unitary.","A diagonal unitary matrix with nonnegative components must be $\\Lambda =$ .","We thus obtain $\\mathcal {U}_{CP} = \\Omega ^\\dagger \\Omega ^*$ and Eq.","(REF ) becomes $\\Omega ^\\dagger \\Omega ^*\\, \\mathcal {H}^T(\\mathbf {k})\\, \\Omega ^T \\Omega =- \\mathcal {H}(\\mathbf {k}) .$ This implies that $\\Omega ^*\\, \\mathcal {H}^T(\\mathbf {k})\\, \\Omega ^T= - \\Omega \\, \\mathcal {H}(\\mathbf {k})\\, \\Omega ^\\dagger .$ Hence, $\\tilde{\\mathcal {H}}(\\mathbf {k}) \\equiv \\Omega \\, \\mathcal {H}(\\mathbf {k})\\, \\Omega ^\\dagger $ is antisymmetric: $\\tilde{\\mathcal {H}}^T(\\mathbf {k}) = - \\tilde{\\mathcal {H}}(\\mathbf {k}) .$ This shows that for systems with inversion symmetry, the BdG Hamiltonian can always be unitarily transformed into antisymmetric form.","For the antisymmetric matrix $\\tilde{\\mathcal {H}}(\\mathbf {k})$ , the Pfaffian $\\mathop {\\textrm {Pf}}\\tilde{\\mathcal {H}}(\\mathbf {k})$ is well defined.","Note that $\\Omega $ is independent of $\\mathbf {k}$ (and is in fact solely determined by $\\mathcal {U}_{CP}$ ) so that the components of $\\tilde{\\mathcal {H}}(\\mathbf {k})$ are smooth functions of $\\mathbf {k}$ .","Hence, the Pfaffian, which is a polynomial of these components, is an smooth function of $\\mathbf {k}$ .","The sign of the Pfaffian is inverted under simultaneous interchange of two rows and the corresponding two columns.","This is a unitary transformation, which can be absorbed into $\\Omega $ .","Hence, the above derivation only determines the Pfaffian up to a sign.","If the dimension $N$ of the local Hilbert space is even the Pfaffian is real: The dimension $2N$ of the BdG Hamiltonian $\\tilde{\\mathcal {H}}(\\mathbf {k})$ is a multiple of four and thus the Pfaffian is a polynomial of even degree of its the components.","Since $\\tilde{\\mathcal {H}}(\\mathbf {k})$ is Hermitian and antisymmetric these components are purely imaginary.","Conversely, for odd $N$ , the Pfaffian is purely imaginary.","We define $P(\\mathbf {k}) \\equiv \\left\\lbrace \\begin{array}{ll}\\mathop {\\textrm {Pf}}\\tilde{\\mathcal {H}}(\\mathbf {k}) & \\mbox{for $N$ even,} \\\\i\\mathop {\\textrm {Pf}}\\tilde{\\mathcal {H}}(\\mathbf {k}) & \\mbox{for $N$ odd}\\end{array} \\right.$ so that $P(\\mathbf {k})$ is always real.","Due to $\\mathcal {C}P$ symmetry, the spectrum of the BdG Hamiltonian is symmetric.","We thus have $\\det \\mathcal {H}(\\mathbf {k}) = (-1)^N \\prod _{j=1}^N E_j^2(\\mathbf {k}) ,$ where $\\pm E_j(\\mathbf {k})$ are the quasiparticle energies.","We assume $E_j(\\mathbf {k})\\ge 0$ without loss of generality.","The determinant equals the square of the Pfaffian.","This implies that $P(\\mathbf {k}) = \\pm \\prod _{j=1}^N E_j(\\mathbf {k}) .$ At any momentum $\\mathbf {k}$ not on a node, $P(\\mathbf {k})$ is strictly positive or negative.","We now choose $\\Omega $ in such a way that in the normal state $P(\\mathbf {k}_\\infty )$ is positive at some momentum $\\mathbf {k}_\\infty $ far from the normal-state Fermi surface.","Then for not too large superconducting energy scale, the energies $E_j(\\mathbf {k}_\\infty )$ do not change sign when superconductivity is switched on and $P(\\mathbf {k}_\\infty )$ remains positive.","Hence, at $\\mathbf {k}_\\infty $ , the sign in Eq.","(REF ) is $+$ .","Since the Pfaffian and thus $P(\\mathbf {k})$ are smooth functions of momentum $P(\\mathbf {k})$ can only change sign at nodes.","Conversely, if $P(\\mathbf {k})$ is negative somewhere in the Brillouin zone there must be a surface of zeros, i.e., a BFS, separating the regions of positive and negative $P(\\mathbf {k})$ .","To determine under what conditions $P(\\mathbf {k})$ can actually become negative, we need to analyze Eq.","(REF ).","The eigenenergies $E_j(\\mathbf {k})$ are continuous functions and are smooth, except at zeros and potentially at crossing points.","If a single $E_j(\\mathbf {k})$ approaches zero linearly the smoothness of $P(\\mathbf {k})$ and the choice $E_j(\\mathbf {k})\\ge 0$ requires the explicit sign in Eq.","(REF ) to flip.","Hence, $P(\\mathbf {k})$ changes sign and there must be a closed BFS.","On the other hand, if two eigenenergies approach zero linearly and simultaneously, for example because they are degenerate, the explicit sign does not flip.","In this case, $P(\\mathbf {k})$ does not change sign at the zero but has a second-order zero there.","The same applies if a single eigenenergy approaches zero quadratically.","If the Pfaffian does not change sign the $\\mathbb {Z}_2$ topological invariant, which is the relative sign of $P(\\mathbf {k})$ [1], [2], exists but is trivial.","This makes BFSs unstable since for any low-symmetry momentum $\\mathbf {k}$ with $P(\\mathbf {k})=0$ , an infinitesimal change of parameters can make $P(\\mathbf {k})$ strictly positive.","For $s_T = -1$ , which implies even $N$ , and preserved TRS, Kramers' theorem [43], [19], [29] shows that the spectrum has twofold degeneracy for all $\\mathbf {k}$ .","Then, the latter case applies, $P(\\mathbf {k})$ does not change sign, and BFSs are not stable.","On the other hand, for $s_T = +1$ or broken TRS, there is no mechanism that leads to twofold degeneracy everywhere, the $P(\\mathbf {k})$ generically changes sign, and BFSs are stable."],["The algebra of basis matrices","As shown in Appendix , for $N=4$ and $(P\\mathcal {T})^2=-1$ , six basis matrices $h_0$ , ..., $h_5$ appear in the normal-state Hamiltonian.","One of them is (proportional to) the identity matrix, as discussed in Sec.",".","We choose $h_0 = $ .","In this Appendix, we show that the basis matrices can always be chosen in such a way that they satisfy the generalized commutation relations $h_0 h_n = h_n h_0$ for $n = 1,2,3,4,5$ and $h_m h_n = -h_n h_m$ for $m, n = 1,2,3,4,5$ and $m\\ne n$ .","It is well known that (several sets of) six $4\\times 4$ matrices with these properties exist [28], [29].","The point here is that the basis matrices always realize this structure.","The matrices $h_n$ can be written as $k_n = Q^\\dagger h_n Q$ , with $Q$ unitary, where the $k_n$ satisfy Eq.","(REF ) with the upper sign and $\\kappa _{11}^\\dagger = \\kappa _{11}$ and $\\kappa _{12}^T = - \\kappa _{12}$ .","For $N=4$ , $\\kappa _{11}$ and $\\kappa _{12}$ are $2\\times 2$ matrices.","Then, $\\kappa _{11}$ has to be a linear combination of $\\sigma _0$ , $\\sigma _1$ , $\\sigma _2$ , $\\sigma _3$ with real coefficients and $\\kappa _{12}$ can be $\\sigma _2$ with an arbitrary complex prefactor.","A maximal set of linearly independent matrices is then $k_0 &= \\left(\\begin{array}{cc}\\sigma _0 & 0 \\\\ 0 & \\sigma _0\\end{array}\\right) = \\sigma _0 \\otimes \\sigma _0 , \\\\k_1 &= \\left(\\begin{array}{cc}\\sigma _1 & 0 \\\\ 0 & \\sigma _1\\end{array}\\right) = \\sigma _0 \\otimes \\sigma _1 , \\\\k_2 &= \\left(\\begin{array}{cc}\\sigma _2 & 0 \\\\ 0 & -\\sigma _2\\end{array}\\right) = \\sigma _3 \\otimes \\sigma _2 , \\\\k_3 &= \\left(\\begin{array}{cc}\\sigma _3 & 0 \\\\ 0 & \\sigma _3\\end{array}\\right) = \\sigma _0 \\otimes \\sigma _3 , \\\\k_4 &= \\left(\\begin{array}{cc}0 & \\sigma _2 \\\\ \\sigma _2 & 0\\end{array}\\right) = \\sigma _1 \\otimes \\sigma _2 , \\\\k_5 &= \\left(\\begin{array}{cc}0 & -i\\sigma _2 \\\\ i\\sigma _2 & 0\\end{array}\\right) = \\sigma _2 \\otimes \\sigma _2 .$ These matrices satisfy $k_0 k_n = k_n k_0$ for $n = 1,2,3,4,5$ and $k_m k_n = -k_n k_m$ for $m, n = 1,2,3,4,5$ and $m\\ne n$ .","Moreover, they are orthonormal with respect to the scalar product $\\mathop {\\textrm {Tr}}k_m k_n$ .","The basis matrices $h_n$ , $n=0,\\ldots ,5$ , are related to the $k_n$ by a unitary transformation.","Since the $k_n$ satisfy the generalized commutation relations so do the $h_n$ .","The upshot is that while the specific form of the basis matrices $h_n$ depends on the model, their algebra does not.","This result extends to general Hilbert-space dimension $N$ but the algebraic properties are more complicated for $N>4$ ."],["Pfaffian for the four-dimensional case","Here, we briefly discuss the analytical expression for the Pfaffian $P(\\mathbf {k})$ of the transformed BdG Hamiltonian $\\tilde{\\mathcal {H}}(\\mathbf {k})$ for the case of $s_T=-1$ , $N=4$ , and even-parity pairing.","The Pfaffian exists and can be chosen to be a smooth function of momentum $\\mathbf {k}$ , as shown in Appendix .","Then the property $P^2(\\mathbf {k}) = \\det \\mathcal {H}(\\mathbf {k})$ fixes $P(\\mathbf {k})$ up to an overall sign.","As discussed in Appendix , we choose this sign so that $P(\\mathbf {k})>0$ far from the normal-state Fermi surface.","If the superconducting energy scale is not too large, the Pfaffian is given in terms of the coefficients in Eq.","(REF ) as $P&(\\mathbf {k}) = \\langle c,c\\rangle ^2 + \\langle f^1,f^1\\rangle ^2+ \\langle f^2,f^2\\rangle ^2 \\nonumber \\\\&{}+ 4\\, \\big ( \\langle c,f^1\\rangle ^2 + \\langle f^1,f^2\\rangle ^2+ \\langle f^2,c\\rangle ^2 \\big ) \\nonumber \\\\&{}- 2\\, \\big ( \\langle c,c\\rangle \\, \\langle f^1,f^1\\rangle + \\langle f^1,f^1\\rangle \\, \\langle f^2,f^2\\rangle + \\langle f^2,f^2\\rangle \\, \\langle c,c\\rangle \\big ) ,$ with the Minkowski-type scalar product $\\langle A,B\\rangle \\equiv A_0 B_0 - \\sum _{n=1}^5 A_n B_n .$ This proposition is proved by evaluating $P^2(\\mathbf {k})$ and showing that it agrees with the determinant of the BdG Hamiltonian.","This cumbersome calculation can be simplified by realizing that the Pfaffian is invariant under simultaneous rotations of the five-vectors $\\vec{c} &= (c_1,c_2,c_3,c_4,c_5) , \\\\\\vec{f}^{\\,1} &= (f^1_1,f^1_2,f^1_3,f^1_4,f^1_5) , \\\\\\vec{f}^{\\,2} &= (f^2_1,f^2_2,f^2_3,f^2_4,f^2_5) .$ The sign of $P(\\mathbf {k})$ is also correct: The assumption of not too large superconducting energy scale means that the $f^1_n$ and $f^2_n$ for $n=0,\\ldots ,5$ are small compared to the $c_n$ far from the normal-state Fermi energy.","Then, at such momenta we get $P(\\mathbf {k}) \\cong \\langle c,c\\rangle ^2 > 0$ .","For large superconducting energy scale, the whole Brillouin zone is affected by superconductivity and we cannot choose the sign of $P(\\mathbf {k})$ by continuity from the normal state.","This simply means that there is no useful distinction between the inside and the outside of the BFS.","The conclusions of this paper remain valid, though."]],"string":"[\n [\n \"Symmetry, nodal structure, and Bogoliubov Fermi surfaces for nonlocal\\n pairing\"\n ],\n [\n \"Abstract Multiband effects can lead to fundamentally different electronic behavior of solids, as exemplified by the possible emergence of Fermi surfaces of Bogoliubov quasiparticles in centrosymmetric superconductors which break time-reversal symmetry.\",\n \"We extend the analysis of possible pairing symmetries, the corresponding nodal structure, and the Bogoliubov Fermi surfaces in two directions: We include nonlocal pairing and we consider internal degrees of freedom other than the effective angular momentum of length $j=3/2$ examined so far.\",\n \"Since our main focus is on the Bogoliubov Fermi surfaces we concentrate on even-parity pairing.\",\n \"The required symmetry analysis is illustrated for several examples, as a guide for the reader.\",\n \"We find that the inclusion of nonlocal pairing leads to a much larger range of possible pairing symmetries.\",\n \"For infinitesimal pairing strength, we find a simple yet powerful criterion for nodes in terms of a scalar product of form factors.\"\n ],\n [\n \"Introduction\",\n \"In condensed matter physics, we are used to study the electronic properties of materials by considering one band at a time.\",\n \"Only when we are interested in excitations at higher energies, e.g., by electromagnetic waves, we include multiple bands.\",\n \"However, it is not always true that the low-energy and equilibrium properties of a solid can be understood based on the single-band paradigm.\",\n \"A case in point is the recent realization that in centrosymmetric multiband superconductors that break time-reversal symmetry (TRS) and have gap nodes, these nodes are generically two dimensional [1], [2].\",\n \"We call these two-dimensional nodes Bogoliubov Fermi surfaces (BFSs).\",\n \"This result has put meat on the bones of the proof [3], [4] that in such systems two-dimensional Fermi surfaces can be protected by a $\\\\mathbb {Z}_2$ topological invariant.\",\n \"This invariant is related to the Pfaffian of the Bogoliubov–de Gennes (BdG) Hamiltonian, unitarily transformed into antisymmetric form [1], [2].\",\n \"Another example is the belief that optical excitations across the gap are forbidden in clean superconductors, which is based on the single-band paradigm but does not hold for multiband superconductors, as recently shown by Ahn and Nagaosa [5].\",\n \"One criterion for when optical excitations across the gap are allowed is the existence of BFSs—this holds both for centrosymmetric and for noncentrosymmetric superconductors.\",\n \"Further experimental signatures of BFSs [6] and their instability against spontaneous breaking of inversion symmetry either electronically [7], [8], [9] or by lattice distortions [10] have been considered by several groups.\",\n \"BFSs can be protected by multiple topological invariants [11], [2], which imposes constraints on how they can merge and gap out for strong coupling.\",\n \"Furthermore, Herbut and Link [9] have recently revealed an analogy between the antisymmetric form of the BdG Hamiltonian—the existence of which is essential for the definition of the Pfaffian [1], [2]—and classical relativity.\",\n \"In this analogy, one can understand the BFSs from a condition of orthogonal fictitious electric and magnetic fields in momentum space [9].\",\n \"The analogy is useful for studying the interaction-induced instability of the BFSs.\",\n \"However, it is restricted to four-dimensional internal Hilbert spaces.\",\n \"Link and Herbut [12] have also presented a complementary study of BFSs in noncentrosymmetric multiband superconductors with broken TRS and gap nodes.\",\n \"Although a $\\\\mathbb {Z}_2$ topological invariant does not exist, BFSs are typically present since the breaking of TRS causes bands shifts that are larger than the induced gaps [12].\",\n \"In the language of tight-binding models, the presence of multiple bands in the vicinity of the Fermi energy results from the existence of multiple (Wannier) orbitals per unit cell that appreciably contribute to the Bloch states at the Fermi energy.\",\n \"These orbitals can be located at the same site or, for structures with a basis, at different sites.\",\n \"We will refer to these orbital and site degrees of freedom, together with the electron spin, as internal degrees of freedom.\",\n \"Superconducting pairing states can be nontrivial with respect to these internal degrees of freedom (internally anisotropic [2]).\",\n \"This allows nontrivial pairing states even for perfectly local pairing, which corresponds to a momentum-independent gap matrix, as we will discuss further below.\",\n \"If the orbital degrees of freedom form a degenerate triplet, for example $p_x$ , $p_y$ , $p_z$ or $d_{yz}$ , $d_{zx}$ , $d_{xy}$ in a cubic crystal field, they can be combined with the spin to form states with effective angular momentum $j=1/2$ or $j=3/2$ [13], [1], [14], [15], [16], [2], [7], [8].\",\n \"The latter leads to a natural description of the fourfold $\\\\Gamma _8$ band-touching points in cubic crystals.\",\n \"While the results concerning the existence of BFSs and their $\\\\mathbb {Z}_2$ topological invariant were general, they were mostly illustrated by the example of the $j=3/2$ model [1], [2], [7], [8], [10].\",\n \"It is important to realized that the possibilities for internal degrees of freedom are much richer.\",\n \"Many superconductors that do not fit the $j=3/2$ description nevertheless have multiple bands close together and close to the Fermi energy.\",\n \"Moreover, the examples studied so far were restricted to local pairing.\",\n \"However, nonlocal pairing is generically present and is often necessary to obtain any superconductivity if local pairing is excluded by a repulsive Hubbard interaction.\",\n \"We will see that nonlocal pairing typically allows for a large range of additional pairing states with symmetries that do not appear for local pairing.\",\n \"In this paper, we extend the analysis of BFSs in centrosymmetric superconductors in two directions: We include nonlocal pairing and we consider internal degrees of freedom other than an effective angular momentum $j=3/2$ .\",\n \"While we mainly discuss the physically most relevant case that the internal degrees of freedom include the spin and the time-reversal transformation squares to minus the identity operation, we also derive results for the opposite case.\",\n \"We are mainly interested in the properties of BFSs in this more general setting and therefore concentrate on even-parity pairing.\",\n \"We provide details on the symmetry analysis to help readers perform such an analysis for specific systems of interest.\",\n \"Everything said here applies to three-dimensional crystals.\",\n \"To be clear, we also specify what we do not consider: We do not address systems with a normal-state Fermi surface that is not topologically equivalent to a sphere enclosing the $\\\\Gamma $ point, for example quasi-two-dimensional systems such as $\\\\mathrm {Sr_2RuO_4}$ [17].\",\n \"In such cases, directions in momentum space with symmetry-imposed nodes for infinitesimal pairing might not intersect with the normal-state Fermi surface so that these symmetry-imposed nodes would not be present.\",\n \"Furthermore, we do not consider accidental gap nodes, pairing states that combine different irreducible representations (irreps) of the point group, and new BFSs that emerge for strong coupling away from the normal-state Fermi surface.\",\n \"The description of such phenomena would only require straightforward extensions of the theory.\",\n \"The remainder of this paper is organized as follows: In Sec.\",\n \", we describe the symmetry analysis that produces all symmetry-allowed contributions to pairing of any symmetry for any crystallographic point group, together with the nodal structure for infinitesimal pairing strength and criteria for the inflation of nodes into BFSs beyond infinitesimal pairing.\",\n \"In Sec.\",\n \", we apply this general framework to a number of model systems of increasing complexity.\",\n \"Finally, in Sec.\",\n \", general insights gained by the preceding sections are discussed and an outlook on open questions is given.\",\n \"Several formal points are presented in Appendices.\"\n ],\n [\n \"General analysis\",\n \"In this section, we describe the general symmetry analysis.\",\n \"We assume the normal state to satisfy inversion symmetry and TRS.\",\n \"The first condition implies that the only possible point groups are $C_i$ , $C_{2h}$ , $D_{2h}$ , $D_{3d}$ , $C_{4h}$ , $D_{4h}$ , $C_{6h}$ , $D_{6h}$ , $S_6 = C_{3i}$ , $T_h$ , and $O_h$ .\",\n \"Of these groups, $C_i$ , $C_{2h}$ , and $D_{2h}$ have only one-dimensional irreps.\",\n \"$C_{4h}$ , $C_{6h}$ , and $S_6$ in addition have two-dimensional real irreps which decompose into one-dimensional complex irreps.\",\n \"The real irreps are relevant for the analysis of Hermitian irreducible tensor operators.\",\n \"Finally, $D_{3d}$ , $D_{4h}$ , $D_{6h}$ , $T_h$ , and $O_h$ also have multidimensional complex irreps.\",\n \"Pairing states described by multidimensional irreps are the most interesting for us since they lead to multicomponent order parameters, which naturally accommodate the breaking of TRS.\",\n \"It is advantageous to consider the magnetic point group of the crystal since this allows us to treat point-group symmetries and TRS on equal footing [19], [20], [21], [22], [23], [18].\",\n \"Since TRS is preserved in the normal state, the antiunitary time-reversal operator $\\\\mathcal {T}$ is an element of the magnetic point group.\",\n \"Hence, we are dealing with a gray group: If $G$ is the structural point group, then the gray point group is $M = G + \\\\mathcal {T}G$ , where $\\\\mathcal {T}G$ contains all elements of $G$ multiplied by time reversal $\\\\mathcal {T}$ (the elements of $G$ commute with $\\\\mathcal {T}$ [23]).\",\n \"The theory of complex corepresentations of magnetic groups [19], [20], [21], [22] has been reviewed for example by Bradley and Davies [23].\",\n \"However, this theory is not the appropriate one for our purposes since it is based on the notions of unitary equivalence of matrices and reducibility of corepresentations by unitary transformations.\",\n \"This leads to the result that two corepresentations that represent $\\\\mathcal {T}$ by $+$ and $-$ , respectively, can be equivalent.\",\n \"Since we need to distinguish operators that are even or odd under time reversal this is not the appropriate equivalence relation.\",\n \"We rather require real corepresentations based on orthogonal equivalence, which leaves the properties under time reversal invariant, and reducibility by orthogonal transformations.\",\n \"By using Wigner's construction of corepresentations [20], [23] but restricting oneself to orthogonal transformations, it is fairly easy to see that for every irrep $\\\\Gamma $ of $G$ , the gray magnetic point group $M$ has two irreps $\\\\Gamma _+$ and $\\\\Gamma _-$ , which are distinguished by a (further) index $\\\\pm $ which indicates the sign under time reversal.\",\n \"Character tables of real corepresentations of the magnetic point groups are given by Erb and Hlinka [24].\",\n \"One more piece of group theory is needed: Since we are studying single-fermion Hamiltonians, we have to consider double groups, i.e., a rotation by $2\\\\pi $ does not give the group identity but only a rotation by $4\\\\pi $ does.\",\n \"This leads to additional “double-valued” or spinor irreps [18].\",\n \"However, the double-valued irreps do not play any role in our analysis for the following reason: We will expand Hamiltonians into sums of basis matrices with momentum-dependent coefficients which are basis functions of irreps.\",\n \"The momentum components $k_x$ , $k_y$ , $k_z$ are basis functions of single-valued irreps, which implies that momentum-dependent functions can only be basis functions of single-valued irreps.\",\n \"Hence, double-valued irreps do not occur in the expansion.\",\n \"We now introduce relevant notations.\",\n \"The superconductor is described by a BdG Hamiltonian of the form $\\\\mathcal {H}(\\\\mathbf {k}) = \\\\begin{pmatrix}H_N(\\\\mathbf {k}) & \\\\Delta (\\\\mathbf {k}) \\\\\\\\\\\\Delta ^\\\\dagger (\\\\mathbf {k}) & -H_N^T(-\\\\mathbf {k})\\\\end{pmatrix} ,$ where $H_N(\\\\mathbf {k})$ is the normal-state Hamiltonian and $\\\\Delta (\\\\mathbf {k})$ is the pairing potential.\",\n \"Both are operators on the $N$ -dimensional Hilbert space of internal degrees of freedom and are given by $N\\\\times N$ matrices for a particular basis.\",\n \"Antisymmetry of fermionic states implies that [25], [26], [27] $\\\\Delta ^T(-\\\\mathbf {k}) = -\\\\Delta (\\\\mathbf {k}) .$ By construction, the BdG Hamiltonian satisfies particle-hole (charge-conjugation) symmetry $\\\\mathcal {C}$ , which is expressed as $\\\\mathcal {U}_C\\\\, \\\\mathcal {H}^T(-\\\\mathbf {k})\\\\, \\\\mathcal {U}_C^\\\\dagger = - \\\\mathcal {H}(\\\\mathbf {k}) ,$ with $\\\\mathcal {U}_C = \\\\sigma _1 \\\\otimes $ .\",\n \"We denote the Pauli matrices by $\\\\sigma _1$ , $\\\\sigma _2$ , $\\\\sigma _3$ and the $2\\\\times 2$ identity matrix by $\\\\sigma _0$ .\",\n \"Identity matrices in any dimension are denoted by $$ .\",\n \"Symmetries in the structural point group $G$ are expressed as $\\\\mathcal {U}\\\\, \\\\mathcal {H}(R^{-1} \\\\mathbf {k})\\\\, \\\\mathcal {U}^\\\\dagger = \\\\mathcal {H}(\\\\mathbf {k}) ,$ where $R$ is an appropriate three-dimensional generalized rotation matrix and $\\\\mathcal {U} = \\\\begin{pmatrix}U & 0 \\\\\\\\ 0 & U^*\\\\end{pmatrix}$ is unitary.\",\n \"This form of $\\\\mathcal {U}$ follows from particle-hole symmetry.\",\n \"The most important case of us is inversion symmetry or parity $P$ , which is implemented by a unitary matrix $U_P$ .\",\n \"We assume inversion symmetry of the normal state, i.e., $U_P\\\\, H_N(-\\\\mathbf {k})\\\\, U_P^\\\\dagger = H_N(\\\\mathbf {k}) .$ Finally, TRS takes the form $\\\\mathcal {U}_T\\\\, \\\\mathcal {H}^T(-\\\\mathbf {k})\\\\, \\\\mathcal {U}_T^\\\\dagger = \\\\mathcal {H}(\\\\mathbf {k}) ,$ where $\\\\mathcal {U}_T = \\\\begin{pmatrix}U_T & 0 \\\\\\\\ 0 & U_T^*\\\\end{pmatrix}$ is (the matrix form of) the unitary part of the antiunitary time-reversal operator.\",\n \"Time reversal $\\\\mathcal {T}$ can square to plus or minus the identity.\",\n \"We denote the sign of $\\\\mathcal {T}^2$ by $s_T = \\\\pm 1$ .\",\n \"Then $U_T U_T^* = s_T\\\\, $ and thus $U_T^T = s_T\\\\, U_T .$ If the internal degrees of freedom include the electron spin we have $s_T = -1$ and $U_T$ is antisymmetric.\",\n \"The matrix $U_T$ is also unitary so that its dimension $N$ must be even since its spectrum consists of pairs $\\\\pm e^{i\\\\phi }$ .\",\n \"The case $s_T = +1$ can only be realized if the spin does not occur explicitly, for example because electrons in one spin state are pushed to high energies by a strong magnetic field.\",\n \"Then $\\\\mathcal {T}$ is not the physical TRS but an effective antiunitary symmetry.\",\n \"For $s_T = +1$ , $U_T$ is symmetric and the dimension $N$ is not restricted.\",\n \"Beyond these considerations, the specific form of $U_T$ as well as the specific form of structural point-group transformations depend on the physical nature of the internal degrees of freedom.\",\n \"It is useful to write the pairing matrix as $\\\\Delta (\\\\mathbf {k}) = D(\\\\mathbf {k})\\\\, U_T .$ Under a general unitary symmetry transformation, Eq.\",\n \"(REF ), the pairing matrix transforms as $\\\\Delta (\\\\mathbf {k}) \\\\mapsto U\\\\, \\\\Delta (R^ {-1}\\\\mathbf {k})\\\\, U^T ,$ i.e., not like a matrix.\",\n \"One easily sees that $D(\\\\mathbf {k})$ transforms as $D(\\\\mathbf {k}) \\\\mapsto U\\\\, D(R^{-1}\\\\mathbf {k})\\\\, U^\\\\dagger ,$ i.e., like a matrix.\",\n \"Analogously, one finds that under time reversal, $D(\\\\mathbf {k})$ transforms as $D(\\\\mathbf {k}) \\\\mapsto U_T\\\\, D^*(-\\\\mathbf {k})\\\\, U_T^\\\\dagger .$ Hence, $D(\\\\mathbf {k})$ transforms like $H_N(\\\\mathbf {k})$ under the magnetic point group, noting that $H_N^T(\\\\mathbf {k}) = H_N^*(\\\\mathbf {k})$ because of Hermiticity.\",\n \"The condition (REF ) from fermionic antisymmetry together with Eqs.\",\n \"(REF ) and (REF ) implies $U_T\\\\, D^T(-\\\\mathbf {k})\\\\, U_T^\\\\dagger = -s_T\\\\, D(\\\\mathbf {k}) .$ For the standard case of $s_T=-1$ , this relation is similar to TRS but differs from it since $D(\\\\mathbf {k})$ is generally not Hermitian so that $D^T(\\\\mathbf {k})$ is not the same as $D^*(\\\\mathbf {k})$ .\",\n \"The general steps of the symmetry analysis are now as follows: (1) Construct a basis $\\\\lbrace h_\\\\nu \\\\rbrace $ of Hermitian matrices on the space of the internal degrees of freedom so that the $h_\\\\nu $ transform as irreducible tensor operators of the magnetic point group $M$ .\",\n \"In the case of point groups with two-dimensional real irreps that decompose into two one-dimensional complex irreps, use the real irreps since this allows one to find Hermitian $h_\\\\nu $ ; the irreducible tensor operators of the corresponding one-dimensional complex irreps are generally not Hermitian.\",\n \"We call the $h_\\\\nu $ basis matrices and normalize them in such a way that $\\\\mathop {\\\\textrm {Tr}}h_\\\\nu ^2 = N$ (the identity matrix is then normalized).\",\n \"It is possible that not all irreps occur.\",\n \"If the dimension of the internal Hilbert space is $N$ there are $N^2$ basis matrices.\",\n \"To find the appropriate basis matrices and their irreps, it is necessary to determine the explicit forms of the symmetry operators, i.e., of the unitary matrices $U_T$ for time reversal and $U_g$ for at least a set of generators $g$ of the structural point group $G$ .\",\n \"(2) Generate a list of all irreps of $M$ that possess basis functions of momentum.\",\n \"These are all irreps that have the same parity under time reversal and inversion since the momentum $\\\\mathbf {k}$ is odd under both.\",\n \"This allows $g+$ and $u-$ irreps but forbids $g-$ and $u+$ irreps.\",\n \"It is also useful to obtain characteristic basis functions for those irreps that have them but it should be kept in mind that these are understood as placeholders for arbitrary sets of functions with the same symmetry under operations from $M$ .\",\n \"In this paper, we will usually represent basis functions by the lowest-order polynomials.\",\n \"(3) Construct the general form of the normal-state Hamiltonian $H_N(\\\\mathbf {k})$ by expanding it into the previously constructed basis, $H_N(\\\\mathbf {k}) = \\\\sum _n c_n(\\\\mathbf {k})\\\\, h_n .$ We enumerate all basis matrices by $\\\\nu $ but the subset that occurs in $H_N(\\\\mathbf {k})$ by $n$ .\",\n \"The Hamiltonian and every term in the expansion must be invariant under $M$ , i.e., it must transform as an irreducible tensor operator belonging to the trivial irrep $A_\\\\mathrm {triv}$ of $M$ .\",\n \"This irrep is even under inversion and under time reversal, i.e., it is $A_{g+}$ or $A_{1g+}$ depending on the group.\",\n \"This requires the form factors $c_n(\\\\mathbf {k})$ to transform as basis functions of the same irrep to which $h_n$ belongs.\",\n \"Moreover, for multidimensional irreps, $c_n(\\\\mathbf {k})$ and $h_n$ must transform as the same component of the irrep and all of them must have the same amplitude if the basis functions and tensor operators are properly normalized.\",\n \"If there is no corresponding basis function for the irrep of some $h_\\\\nu $ , this matrix does not occur in Eq.\",\n \"(REF ).\",\n \"This excludes $h_\\\\nu $ belonging to $g-$ or $u+$ irreps.\",\n \"Note that $h_0 \\\\equiv $ is always an allowed basis matrix since it is a reducible tensor operator of the trivial irrep $A_\\\\mathrm {triv}$ and $c_0(\\\\mathbf {k}) \\\\sim 1$ is always an appropriate basis function.\",\n \"As noted above, for the standard case that the internal degrees of freedom include the electron spin, time reversal squares to $-1$ and the dimension $N$ of the internal Hilbert space must be even.\",\n \"Then the number of allowed basis matrices $h_n$ appearing in $H_N(\\\\mathbf {k})$ is $N(N-1)/2$ , as shown in Appendix .\",\n \"The case of $N=2$ corresponds to spin being the only internal degree of freedom.\",\n \"There is only a single allowed basis matrix, namely $h_0=$ .\",\n \"This means that the normal-state Hamiltonian is independent of spin, which is required by TRS.\",\n \"For $N=4$ , there are 6 basis matrices $h_0=$ , $h_1$ , ..., $h_5$ .\",\n \"These matrices have the special property that $h_1$ , ..., $h_5$ anticommute pairwise [28], [29], while all matrices commute with $h_0$ .\",\n \"Results restricted to this case are discussed in Subsection REF .\",\n \"(4) Construct the allowed pairing states.\",\n \"Here, we have much greater freedom than in constructing $H_N(\\\\mathbf {k})$ since the superconducting state may break symmetries contained in $M$ .\",\n \"We write the pairing matrix as $D(\\\\mathbf {k}) = \\\\sum _j \\\\delta _j D_j(\\\\mathbf {k}) ,$ where the $D_j(\\\\mathbf {k})$ are linearly independent matrix-valued functions transforming like (being irreducible tensor operators belonging to) components of a specific irrep $\\\\Gamma _s$ , $s=\\\\pm $ , of the magnetic point group $M$ and the $\\\\delta _j$ are complex pairing amplitudes.\",\n \"Recall that $D(\\\\mathbf {k})$ transforms like a matrix.\",\n \"$D_j(\\\\mathbf {k})$ can be chosen to be either Hermitian or anti-Hermitian.\",\n \"This can be seen as follows: Note first that Eq.\",\n \"(REF ) has to be satisfied by each component separately since the $D_j(\\\\mathbf {k})$ are independent functions, $U_T\\\\, D_j^T(-\\\\mathbf {k})\\\\, U_T^\\\\dagger = -s_T\\\\, D_j(\\\\mathbf {k}) .$ Depending on the irrep $\\\\Gamma _s$ , $D_j(\\\\mathbf {k})$ is either even ($s=+$ ) or odd ($s=-$ ) under time reversal.\",\n \"Hence, Eq.\",\n \"(REF ) implies that $U_T\\\\, D_j^*(-\\\\mathbf {k})\\\\, U_T^\\\\dagger = s\\\\, D_j(\\\\mathbf {k}) .$ It follows that $D_j^T(\\\\mathbf {k}) = -s_T\\\\, s\\\\, D_j^*(\\\\mathbf {k})$ and thus $D_j^\\\\dagger (\\\\mathbf {k}) = -s_T\\\\, s\\\\, D_j(\\\\mathbf {k}) .$ This equation states that the matrix functions $D_j(\\\\mathbf {k})$ are Hermitian or anti-Hermitian depending on the sign $s_T$ of $\\\\mathcal {T}^2$ and on the pairing state being even or odd under time reversal.\",\n \"Since we consider pure-irrep pairing all appearing $D_j(\\\\mathbf {k})$ have the same sign $-s_T\\\\,s$ under Hermitian conjugation.\",\n \"For the standard case of $s_T = -1$ , $D_j(\\\\mathbf {k})$ is Hermitian (anti-Hermitian) for time-reversal-even (time-reversal-odd) irreps.\",\n \"For a time-reversal-odd irrep $\\\\Gamma _-$ , we can pull a common factor of $i$ out of all $D_j(\\\\mathbf {k})$ and absorb it into the order parameters $\\\\delta _j$ in Eq.\",\n \"(REF ).\",\n \"This makes $D_j(\\\\mathbf {k})$ Hermitian and changes the irrep from $\\\\Gamma _-$ to $\\\\Gamma _+$ since the behavior under spatial transformations is unaffected.\",\n \"Hence, for $s_T = -1$ it is sufficient to consider only the time-reversal-even $g+$ and $u+$ irreps for pairing states.\",\n \"This has the desirable consequence that the breaking of TRS is only encoded in the complex order parameters $\\\\delta _j$ .\",\n \"If and only if all $\\\\delta _j$ can be made real by a global phase rotation the system respects TRS.\",\n \"This is equivalent to the nonvanishing $\\\\delta _j$ having phase differences of 0 or $\\\\pi $ .\",\n \"Conversely, for the nonstandard sign $s_T = +1$ , $D_j(\\\\mathbf {k})$ is anti-Hermitian (Hermitian) for time-reversal-even (time-reversal-odd) irreps.\",\n \"Pulling out a factor of $i$ , we can make sure that $D_j(\\\\mathbf {k})$ is Hermitian and the irrep is odd under time reversal ($g-$ or $u-$ ).\",\n \"Again, the breaking of TRS is only encoded in the order parameters $\\\\delta _j$ .\",\n \"If and only if all $\\\\delta _j$ can be made purely imaginary by a global phase rotation the system respects TRS.\",\n \"This is again equivalent to the nonvanishing $\\\\delta _j$ having phase differences of 0 or $\\\\pi $ .\",\n \"A given model does not necessarily permit all time-reversal-even (or odd) irreps, though.\",\n \"To see this, note that the Hermitian-matrix-valued functions $D_j(\\\\mathbf {k})$ can be expanded into the Hermitian basis matrices $h_\\\\nu $ as $D_j(\\\\mathbf {k}) = \\\\sum _\\\\nu d_{j\\\\nu }(\\\\mathbf {k})\\\\, h_\\\\nu ,$ with real functions $d_{j\\\\nu }(\\\\mathbf {k})$ .\",\n \"Consider all products of momentum basis functions $g_l(\\\\mathbf {k})$ belonging to irreps $\\\\Gamma _l$ and of matrices $h_\\\\nu $ belonging to irreps $\\\\Gamma _\\\\nu $ .\",\n \"Any such product transforms according to the product representation $\\\\Gamma _l \\\\otimes \\\\Gamma _\\\\nu $ , which is generally reducible.\",\n \"A reduction into irreps by standard methods reveals which symmetries of pairing states can occur.\",\n \"Recall that only $g+$ and $u-$ irreps possess momentum basis functions.\",\n \"The possible irreps of basis matrices $h_\\\\nu $ have been obtained in step (1).\",\n \"By reducing all possible products and keeping only those irreps that are even (odd) under time reversal for $s_T=-1$ ($s_T = +1$ ), we obtain the possible irreps.\",\n \"The coefficients $d_{j\\\\nu }(\\\\mathbf {k})$ in Eq.\",\n \"(REF ) can be constructed out of the functions $g_l(\\\\mathbf {k})$ by standard methods.\",\n \"The full pairing matrix then has the form $D(\\\\mathbf {k}) = \\\\sum _\\\\nu \\\\sum _{j=1}^d \\\\delta _j d_{j\\\\nu }(\\\\mathbf {k})\\\\,h_\\\\nu \\\\equiv \\\\sum _\\\\nu f_\\\\nu (\\\\mathbf {k})\\\\, h_\\\\nu .$ The sum in Eq.\",\n \"(REF ) generally contains many terms: While the number of possible basis matrices $h_\\\\nu $ is finite, there are infinitely many smooth functions of momentum that transform according to the same irrep.\",\n \"This implies that TRS can be broken spontaneously for pairing belonging to any irrep, by having amplitudes $\\\\delta _j$ with nontrivial phase differences [30].\",\n \"Consideration of energetics [31], [26], [13], [1], [2] is useful to find plausible modes of TRS breaking.\",\n \"Spontaneous breaking of TRS occurs most naturally for multidimensional irreps, in the form of nontrivial phase factors of contributions belonging to different components of the irrep.\",\n \"Table: Possible combinations of the signs under inversion (parity, gg for even, uu for odd) and under time reversal (++ for even, -- for odd) of irreps of pairing states for time reversal squaring to s T =±1s_T=\\\\pm 1.\",\n \"Recall that s T =-1s_T=-1 is the standard case for electrons.Since the normal state is inversion symmetric the superconducting pairing is either even ($g$ irreps) or odd ($u$ irreps) under inversion (parity).\",\n \"Table REF shows the possible combinations of signs under inversion and time reversal.\",\n \"We note that for $s_T=-1$ and even-parity pairing, only basis matrices belonging to $g+$ and $u-$ irreps occur.\",\n \"These are the same matrices $h_n$ that appear in the normal-state Hamiltonian $H_N(\\\\mathbf {k})$ in Eq.\",\n \"(REF ).\",\n \"On the other hand, for $s_T=-1$ and odd-parity pairing, only those basis matrices $h_\\\\nu $ that do not occur in $H_N(\\\\mathbf {k})$ can appear in $D(\\\\mathbf {k})$ .\",\n \"The numbers of allowed basis matrices are given in Appendix .\",\n \"(5) Analyze the nodal structure for each pairing symmetry (irrep) assuming infinitesimal pairing strength.\",\n \"Since we are interested in the fate of BFSs we concentrate on the case $s_T=-1$ and even-parity pairing, while the other cases are briefly discussed in Appendix .\",\n \"In the limit of infinitesimal pairing amplitudes $\\\\delta _j$ , superconductivity can be described for each band separately.\",\n \"This is because the superconducting gap is of first order in $\\\\delta _j$ , whereas interband effects are of second order.\",\n \"Moreover, there are at most point or line nodes but no BFSs since interband pairing is responsible for the latter [1], [2].\",\n \"The normal-state bands are twofold degenerate because of inversion symmetry and TRS.\",\n \"Hence, in an effective description of a single band, the dimension of the internal Hilbert space is $N=2$ and the internal degree of freedom can be described by a pseudospin of length $1/2$ [2], [10].\",\n \"The superconducting pairing must be in the pseudospin-singlet channel since it is of even parity.\",\n \"The pairing matrix is then $f_0(\\\\mathbf {k})\\\\, \\\\sigma _0$ .\",\n \"Since $\\\\sigma _0$ belongs to a $g+$ irrep, namely the trivial one, $f_0(\\\\mathbf {k})$ must be a basis function of a $g+$ irrep; see Table REF .\",\n \"The symmetry of the pairing state under the magnetic point group can then only be encoded in the function $f_0(\\\\mathbf {k})$ .\",\n \"This function thus generically transforms like the pairing matrix $D(\\\\mathbf {k})$ of the full model.\",\n \"Moreover, the symmetry-imposed zeros of $f_0(\\\\mathbf {k})$ in momentum space correspond to gap nodes for infinitesimal pairing (IP nodes).\",\n \"One of the main messages of Refs.\",\n \"[13], [1], [2] was that a momentum-independent but internally anisotropic pairing matrix can lead to a momentum-dependent function $f_0(\\\\mathbf {k})$ and to gap nodes.\",\n \"A simple but powerful criterion for IP nodes can be obtained as follows: As noted above, for $s_T=-1$ and even-parity pairing, the same basis matrices $h_n$ appear in $H_N(\\\\mathbf {k})$ and $D(\\\\mathbf {k})$ .\",\n \"From Eq.\",\n \"(REF ), we know that the normal-state coefficients $c_n(\\\\mathbf {k})$ transform like the basis matrices $h_n$ under the magnetic point group.\",\n \"Hence, the scalar function $F(\\\\mathbf {k}) \\\\equiv \\\\sum _n c_n(\\\\mathbf {k})\\\\, f_n(\\\\mathbf {k})$ transforms like the pairing matrix $D(\\\\mathbf {k})$ and thus also like the form factor $f_0(\\\\mathbf {k})$ in the single-band picture and, in particular, has the same symmetry-induced nodes.\",\n \"Therefore, we can use $F(\\\\mathbf {k})$ as a proxy for the IP nodal structure.\",\n \"In fact, instead of the normal-state coefficients $c_n(\\\\mathbf {k})$ , we could use any set of basis functions belonging to the same irreps.\",\n \"We will see that an analogous measure emerges naturally for the case of $N=4$ .\",\n \"(6) Check whether the nodes thereby obtained are inflated if the pairing amplitudes are not infinitesimal.\",\n \"The main tool is the Pfaffian $\\\\mathop {\\\\textrm {Pf}}\\\\tilde{\\\\mathcal {H}}(\\\\mathbf {k})$ of an antisymmetric Hamiltonian $\\\\tilde{\\\\mathcal {H}}(\\\\mathbf {k})$ that is unitarily equivalent to the BdG Hamiltonian $\\\\mathcal {H}(\\\\mathbf {k})$ .\",\n \"Such a unitary transformation is guaranteed to exist if the point group contains the inversion, as shown in [1].\",\n \"A simpler version of the proof is presented in Appendix .\",\n \"The square of the Pfaffian equals the determinant of the BdG Hamiltonian and thus the product of the quasiparticle energies.\",\n \"Hence, nodes of any kind correspond to $\\\\mathop {\\\\textrm {Pf}}\\\\tilde{\\\\mathcal {H}}(\\\\mathbf {k}) = 0$ .\",\n \"As shown in Appendix , the Pfaffian is real for even $N$ and imaginary for odd $N$ .\",\n \"We define $P(\\\\mathbf {k}) \\\\equiv \\\\left\\\\lbrace \\\\begin{array}{ll}\\\\mathop {\\\\textrm {Pf}}\\\\tilde{\\\\mathcal {H}}(\\\\mathbf {k}) & \\\\mbox{for $N$ even,} \\\\\\\\[0.5ex]i\\\\mathop {\\\\textrm {Pf}}\\\\tilde{\\\\mathcal {H}}(\\\\mathbf {k}) & \\\\mbox{for $N$ odd}\\\\end{array} \\\\right.$ to obtain a real quantity.\",\n \"We will simply call $P(\\\\mathbf {k})$ the Pfaffian in the following.\",\n \"The sign of $P(\\\\mathbf {k})$ turns out to depend on the choice of unitary transformation which leads to the antisymmetric matrix $\\\\tilde{\\\\mathcal {H}}(\\\\mathbf {k})$ .\",\n \"We choose this transformation in such a way that the Pfaffian is positive at some point far from the normal-state Fermi surface.\",\n \"Since the Pfaffian is a smooth function of momentum this fixes the sign for all $\\\\mathbf {k}$ .\",\n \"For $s_T = -1$ and preserved TRS, $P(\\\\mathbf {k})$ is nonnegative for all $\\\\mathbf {k}$ , the topological $\\\\mathbb {Z}_2$ invariant is thus trivial, and there are no topologically protected BFSs, as shown in [1], [2].\",\n \"Conversely, such BFSs are expected for broken TRS.\",\n \"The argument is reviewed in Appendix .\",\n \"In addition, we there show that the Pfaffian can change sign and BFSs are expected also for $s_T = +1$ , regardless of symmetry.\"\n ],\n [\n \"Four-dimensional internal Hilbert space\",\n \"Even-parity superconductors with a four-dimensional internal Hilbert space (and time reversal squaring to $-1$ ) constitute the simplest case beyond the single-band paradigm.\",\n \"According to Appendix , the normal-state Hamiltonian $H_N(\\\\mathbf {k}) = \\\\sum _n c_n(\\\\mathbf {k})\\\\, h_n$ is a superposition of six basis matrices $h_0$ , ..., $h_5$ .\",\n \"As noted above, the same six basis matrices appear in the pairing matrix $D(\\\\mathbf {k})$ .\",\n \"These matrices realize a nice algebraic structure of $4\\\\times 4$ gamma matrices: One can always choose the $h_n$ in such a way that $h_1$ , ..., $h_5$ anticommute pairwise, while $h_0=$ commutes with any matrix [28], [29]; see Appendix .\",\n \"This implies that for any such model the eigenenergies in the normal state are $E_{N\\\\pm }(\\\\mathbf {k}) = c_0(\\\\mathbf {k}) \\\\pm \\\\sqrt{ c_1^2(\\\\mathbf {k}) + \\\\ldots + c_5^2(\\\\mathbf {k}) } ,$ both twofold degenerate.\",\n \"The algebraic structure also allows to derive analytical results for the quasiparticle energies in the superconducting state and for the Pfaffian $P(\\\\mathbf {k})$ .\",\n \"We obtain universal results when expressing these quantities in terms of coefficients of basis matrices.\",\n \"Since we have found negative values of the Pfaffian and thus BFSs there, the same should be true for any model with $N=4$ in this class.\",\n \"For this conclusion to hold, it is important that the prefactors of the matrices are not constrained by symmetries so that all values of the Pfaffian can actually occur.\",\n \"Table: Commutation relations of the matrices defined in Eqs.\",\n \"()–().\",\n \"++ (--) denotes commutation (anticommutation).The BdG Hamiltonian in Eq.\",\n \"(REF ) can be written as a linear combination of 18 Hermitian $8 \\\\times 8$ matrices, $\\\\mathcal {H}(\\\\mathbf {k}) = \\\\sum _{n=0}^5 c_n(\\\\mathbf {k})\\\\, H_n+ \\\\sum _{n=0}^5 f^1_n(\\\\mathbf {k})\\\\, \\\\Gamma _n+ \\\\sum _{n=0}^5 f^2_n(\\\\mathbf {k})\\\\, \\\\Phi _n .$ The coefficients are all real functions of momentum.\",\n \"The 18 matrices are $H_n &= \\\\left(\\\\begin{array}{cc}h_n & 0 \\\\\\\\0 & - U_P^* h_n^T U_P^T\\\\end{array}\\\\right) , \\\\\\\\\\\\Gamma _n &= \\\\left(\\\\begin{array}{cc}0 & h_n U_T \\\\\\\\U_T^\\\\dagger h_n & 0\\\\end{array}\\\\right) , \\\\\\\\\\\\Phi _n &= \\\\left(\\\\begin{array}{cc}0 & ih_n U_T \\\\\\\\-i U_T^\\\\dagger h_n & 0\\\\end{array}\\\\right) ,$ where $n=0,\\\\ldots ,5$ .\",\n \"The definition of $H_n$ requires some discussion.\",\n \"The matrices $h_n$ are irreducible tensor operators of $g$ or $u$ irreps and $H_n = \\\\left\\\\lbrace \\\\begin{array}{ll}\\\\displaystyle \\\\left(\\\\begin{array}{cc}h_n & 0 \\\\\\\\0 & - h_n^T\\\\end{array}\\\\right) & \\\\mbox{for $g$ irreps,} \\\\\\\\[2.5ex]\\\\displaystyle \\\\left(\\\\begin{array}{cc}h_n & 0 \\\\\\\\0 & h_n^T\\\\end{array}\\\\right) & \\\\mbox{for $u$ irreps.\",\n \"}\\\\end{array}\\\\right.$ For $g$ irreps, the minus sign of $-\\\\mathbf {k}$ in the lower right block of Eq.\",\n \"(REF ) drops out and the first term in Eq.\",\n \"(REF ) is obvious.\",\n \"For $u$ irreps, the form factor $c_n(\\\\mathbf {k})$ is an odd function and the lower right block obtains an additional sign change.\",\n \"Since in Eq.\",\n \"(REF ) the coefficient of $H_n$ is $c_n(\\\\mathbf {k})$ this sign must be incorporated into $H_n$ .\",\n \"The matrices $H_n$ , $\\\\Gamma _n$ , and $\\\\Phi _n$ are all Hermitian, traceless, square to $$ , and either commute or anticommute according to Table REF .\",\n \"Moreover, if $A_\\\\alpha $ , $\\\\alpha =1,\\\\ldots ,18$ denote all 18 matrices, we have $\\\\mathop {\\\\textrm {Tr}}\\\\lbrace A_\\\\alpha , A_\\\\beta \\\\rbrace \\\\equiv \\\\mathop {\\\\textrm {Tr}}(A_\\\\alpha A_\\\\beta + A_\\\\beta A_\\\\alpha ) = 16\\\\, \\\\delta _{\\\\alpha \\\\beta } .$ The Pfaffian can be expressed in closed form, as discussed in more detail in Appendix .\",\n \"We suppress momentum arguments for the rest of this section.\",\n \"It is useful to define the real five-vectors $\\\\vec{c} &\\\\equiv (c_1,c_2,c_3,c_4,c_5) , \\\\\\\\\\\\vec{f}^{\\\\,1} &\\\\equiv (f^1_1,f^1_2,f^1_3,f^1_4,f^1_5) , \\\\\\\\\\\\vec{f}^{\\\\,2} &\\\\equiv (f^2_1,f^2_2,f^2_3,f^2_4,f^2_5)$ and the Minkowski-type scalar product $\\\\langle A,B\\\\rangle \\\\equiv A_0 B_0 - \\\\vec{A}\\\\cdot \\\\vec{B} .$ The Pfaffian can then be written as $P(\\\\mathbf {k}) &= \\\\langle c,c\\\\rangle ^2 + \\\\langle f^1,f^1\\\\rangle ^2+ \\\\langle f^2,f^2\\\\rangle ^2 \\\\nonumber \\\\\\\\&\\\\quad {}+ 4\\\\, \\\\big ( \\\\langle c,f^1\\\\rangle ^2 + \\\\langle f^1,f^2\\\\rangle ^2+ \\\\langle f^2,c\\\\rangle ^2 \\\\big ) \\\\nonumber \\\\\\\\&\\\\quad {}- 2\\\\, \\\\big ( \\\\langle c,c\\\\rangle \\\\, \\\\langle f^1,f^1\\\\rangle + \\\\langle f^1,f^1\\\\rangle \\\\, \\\\langle f^2,f^2\\\\rangle \\\\nonumber \\\\\\\\&\\\\quad {}+ \\\\langle f^2,f^2\\\\rangle \\\\, \\\\langle c,c\\\\rangle \\\\big ) .$ Another form that will prove useful is $P(\\\\mathbf {k}) &= \\\\big ( \\\\langle c,c\\\\rangle - \\\\langle f^1,f^1\\\\rangle - \\\\langle f^2,f^2\\\\rangle \\\\big )^2\\\\nonumber \\\\\\\\&\\\\quad {}+ 4\\\\, \\\\big ( \\\\langle c,f^1\\\\rangle ^2 + \\\\langle c,f^2\\\\rangle ^2 + \\\\langle f^1,f^2\\\\rangle ^2\\\\nonumber \\\\\\\\&\\\\quad {}- \\\\langle f^1,f^1\\\\rangle \\\\langle f^2,f^2\\\\rangle \\\\big ) .$ Nodes are signaled by $P(\\\\mathbf {k})=0$ .\",\n \"If $P(\\\\mathbf {k})$ becomes negative for some momenta $\\\\mathbf {k}$ we obtain two-dimensional BFSs [1], [2].\",\n \"The algebraic structure and the expressions for the Pfaffian are the same for all models with even-parity superconductors, time reversal squaring to $-1$ , and $N=4$ .\",\n \"Hence, the conclusion of Refs.\",\n \"[1], [2] that nodes are inflated into BFSs for TRS-breaking superconducting states applies to all such models.\",\n \"For infinitesimal pairing, we can neglect terms of fourth order in the amplitudes $f^\\\\alpha _n$ compared to terms of second order in Eq.\",\n \"(REF ).\",\n \"The result $P(\\\\mathbf {k}) &\\\\cong \\\\big ( \\\\langle c,c\\\\rangle - \\\\langle f^1,f^1\\\\rangle - \\\\langle f^2,f^2\\\\rangle \\\\big )^2\\\\nonumber \\\\\\\\&\\\\quad {}+ 4\\\\, \\\\big ( \\\\langle c,f^1\\\\rangle ^2 + \\\\langle c,f^2\\\\rangle ^2 \\\\big )$ is nonnegative.\",\n \"Hence, the momentum-space volume of the BFSs shrinks to zero for infinitesimal pairing, leaving only point and line nodes.\",\n \"Since the expression in Eq.\",\n \"(REF ) is a sum of squares IP nodes occur when three conditions hold simultaneously.\",\n \"The first reads as $\\\\langle c,c\\\\rangle - \\\\langle f^1,f^1\\\\rangle - \\\\langle f^2,f^2\\\\rangle = 0 .$ Since $\\\\langle c,c\\\\rangle = E_{N+} E_{N-} = 0$ is a criterion for the normal-state Fermi surface we can say that Eq.\",\n \"(REF ) describes a renormalized Fermi surface.\",\n \"It will be close to the normal-state Fermi surface in the typical case that the pairing energy is small compared to the chemical potential.\",\n \"The second and third condition read as $\\\\langle c,f^1\\\\rangle = \\\\langle c,f^2\\\\rangle = 0$ , which are equivalent to $\\\\langle c,f\\\\rangle = 0 ,$ where $f\\\\equiv f^1+if^2$ .\",\n \"Explicitly, this condition reads as $c_0\\\\, (f^1_0 + i f^2_0) - \\\\sum _{n=1}^5 c_n\\\\, (f^1_n + i f^2_n)\\\\equiv c_0 f_0 - \\\\sum _{n=1}^5 c_n f_n = 0 .$ Except for the signs, which do not matter, this agrees with the function $F(\\\\mathbf {k})$ that we found above to encode the IP nodal structure; see Eq.\",\n \"(REF ).\"\n ],\n [\n \"Applications\",\n \"In the following, we illustrate the general procedure for specific examples.\",\n \"We will mainly consider a familiar setting: the dimension of the internal Hilbert space is $N=4$ , resulting from spin and either orbital or basis site, and the model is described by the cubic point group $O_h$ .\",\n \"This point group has ten irreps, $A_{1g}$ , $A_{2g}$ , $E_g$ , $T_{1g}$ , $T_{2g}$ , $A_{1u}$ , $A_{2u}$ , $E_u$ , $T_{1u}$ , and $T_{2u}$ .\",\n \"For the corresponding gray magnetic point group, the number of irreducible real corepresentations is doubled to $A_{1g+}$ , $A_{1g-}$ , $A_{2g+}$ , etc.\"\n ],\n [\n \"Two \",\n \"We first consider a lattice without basis and with two orbitals per site that are invariant under all point-group transformations.\",\n \"This means that they transform according to the trivial irrep $A_{1g}$ or, in other words, like s-orbitals.\",\n \"The interesting point here is that even such a simple model supports nontrivial multiband superconductivity with BFSs.\",\n \"For the internal Hilbert space, we use the basis $\\\\lbrace |1{\\\\uparrow }\\\\rangle , |1{\\\\downarrow }\\\\rangle , |2{\\\\uparrow }\\\\rangle , |2{\\\\downarrow }\\\\rangle \\\\rbrace $ , where 1, 2 refers to the orbital and $\\\\uparrow $ , $\\\\downarrow $ to the spin.\",\n \"In this section, the first factor in Kronecker products refers to the orbital and the second to the spin.\",\n \"The matrix representation of the inversion or parity operator $P$ has the trivial form $U_P = = \\\\sigma _0 \\\\otimes \\\\sigma _0 .$ The unitary part of the time-reversal operator is $U_T = \\\\sigma _0 \\\\otimes i\\\\sigma _2$ since the orbitals are invariant under time reversal, while in the spin sector we have the standard form $i\\\\sigma _2$ .\",\n \"Table: Basis matrices on the internal Hilbert space for the case of two s-orbitals and point group O h O_h.\",\n \"The basis matrices are irreducible tensor operators of the irreps listed in the second column.\",\n \"For multidimensional irreps, the states transforming into each other under point-group operations are distinguished by the index in the third column.The 16 basis matrices $h_\\\\nu $ of the space of Hermitian $4\\\\times 4$ matrices obtained as Kronecker products are listed in Table REF , together with the corresponding irreps.\",\n \"To understand the table, first consider the structural point group.\",\n \"Since the orbitals transform trivially under all point-group elements the spin alone determines the irrep.\",\n \"Then $\\\\sigma _0$ obviously transforms trivially, i.e., according to $A_{1g}$ , while $\\\\mbox{$\\\\sigma $}=(\\\\sigma _1,\\\\sigma _2,\\\\sigma _3)$ is a pseudovector, which transforms according to $T_{1g}$ .\",\n \"Regarding time reversal, $\\\\sigma _0$ in the spin sector is of course even, whereas $\\\\mbox{$\\\\sigma $}$ is odd.\",\n \"However, the time-reversal operator $\\\\mathcal {T}$ is antilinear so that the imaginary Pauli matrix $\\\\sigma _2$ in the orbital sector gives another sign change.\",\n \"Note that although the orbital degree of freedom appears to be a trivial spectator, it does lead to the appearance of the additional irreps $A_{1g-}$ and $T_{1g+}$ .\",\n \"Next, we consider momentum basis functions.\",\n \"As noted above, they only exist for $g+$ and $u-$ irreps.\",\n \"Low-order polynomial basis functions can be found in tables [18], [33].\",\n \"It is important to note that for our purposes the constant function and the second-order function $k_x^2+k_y^2+k_z^2$ are allowed basis functions of $A_{1g+}$ but are not listed in some tables.\",\n \"The tables usually do not show a basis function for $A_{1u-}$ since the simplest one is of order $l=9$ , specifically $k_xk_yk_z\\\\, [ k_x^4 (k_y^2-k_z^2) + k_y^4 (k_z^2-k_x^2) + k_z^4 (k_x^2-k_y^2) ]$ [34], [18].\",\n \"The possible irreps of pairing states are now obtained by reducing all products of the allowed irreps of the momentum-dependent form factor and of pairing matrices and excluding the ones that are odd under time reversal and thus violate fermionic antisymmetry.\",\n \"The reduction of the remaining combinations is shown in Table REF .\",\n \"The normal-state Hamiltonian $H_N(\\\\mathbf {k})$ can, and generically does, contain all combinations that transform according to $A_{1g+}$ , set in bold face.\",\n \"Only the first row of the table (form-factor irrep $A_{1g+}$ ) is compatible with purely local pairing, which can thus have $A_{1g+}$ or $T_{1g+}$ symmetry.\",\n \"Note that the latter is impossible for a single-orbital system.\",\n \"Table: Reduction of product representations of the allowed irreps of 𝐤\\\\mathbf {k}-dependent form factors (rows) and basis matrices h ν h_\\\\nu (columns) for two s-orbitals.\",\n \"For the form factors, the minimum order of polynomial basis functions is given in the second column.\",\n \"“∘\\\\circ ” indicates products that are forbidden since they violate fermionic antisymmetry.The normal-state Hamiltonian contains two types of terms, generated by $A_{1g+}\\\\otimes A_{1g+}$ and by $T_{1g+}\\\\otimes T_{1g+}$ , respectively.\",\n \"For the first, there are three basis matrices belonging to $A_{1g+}$ according to Table REF , hence we get $H_{N1}(\\\\mathbf {k}) = c_{00}(\\\\mathbf {k})\\\\, \\\\sigma _0 \\\\otimes \\\\sigma _0+ c_{10}(\\\\mathbf {k})\\\\, \\\\sigma _1 \\\\otimes \\\\sigma _0+ c_{30}(\\\\mathbf {k})\\\\, \\\\sigma _3 \\\\otimes \\\\sigma _0 ,$ where $c_{00}$ , $c_{10}$ , and $c_{30}$ are generally distinct basis functions of $A_{1g+}$ .\",\n \"In other words, they are invariant under all elements of the magnetic point group.\",\n \"The leading polynomial terms read a [18], [33] $c_{m0}(\\\\mathbf {k}) &= c_{m0}^{(0)} + c_{m0}^{(2)}\\\\, (k_x^2+k_y^2+k_z^2)+ c_{m0}^{(4)}\\\\, (k_x^4+k_y^4+k_z^4) \\\\nonumber \\\\\\\\&\\\\quad {} + c_{m0}^{(6)}\\\\, k_x^2k_y^2k_z^2 + \\\\ldots $ for $m=0,1,3$ .\",\n \"For the second type, we observe that there is a single triplet of matrices belonging to $T_{1g+}$ , namely $\\\\sigma _2 \\\\otimes \\\\mbox{$\\\\sigma $}$ .\",\n \"To obtain an invariant Hamiltonian, they must each be multiplied by the corresponding momentum basis function, which gives $H_{N2}(\\\\mathbf {k}) &= c_{21}(\\\\mathbf {k})\\\\, \\\\sigma _2 \\\\otimes \\\\sigma _1+ c_{22}(\\\\mathbf {k})\\\\, \\\\sigma _2 \\\\otimes \\\\sigma _2 \\\\nonumber \\\\\\\\&\\\\quad {} + c_{23}(\\\\mathbf {k})\\\\, \\\\sigma _2 \\\\otimes \\\\sigma _3 .$ The functions $c_{2n}$ are not independent but must transform into each other under the magnetic point group.\",\n \"The leading terms are [18], [33] $c_{21}(\\\\mathbf {k}) &= c_{2}^{(4)}\\\\, k_yk_z (k_y^2-k_z^2)+ c_{2}^{(6)}\\\\, k_yk_z (k_y^4-k_z^4) + \\\\ldots , \\\\\\\\c_{22}(\\\\mathbf {k}) &= c_{2}^{(4)}\\\\, k_zk_x (k_z^2-k_x^2)+ c_{2}^{(6)}\\\\, k_zk_x (k_z^4-k_x^4) + \\\\ldots , \\\\\\\\c_{23}(\\\\mathbf {k}) &= c_{2}^{(4)}\\\\, k_xk_y (k_x^2-k_y^2)+ c_{2}^{(6)}\\\\, k_xk_y (k_x^4-k_y^4) + \\\\ldots $ The full normal-state Hamiltonian $H_N(\\\\mathbf {k}) = H_{N1}(\\\\mathbf {k}) + H_{N2}(\\\\mathbf {k})= \\\\sum _{n=0}^5 c_n(\\\\mathbf {k})\\\\, h_n$ is a linear combination of the six basis matrices $h_0 &\\\\equiv \\\\sigma _0 \\\\otimes \\\\sigma _0 & A_{1g+}, \\\\\\\\h_1 &\\\\equiv \\\\sigma _1 \\\\otimes \\\\sigma _0 & A_{1g+}, \\\\\\\\h_2 &\\\\equiv \\\\sigma _3 \\\\otimes \\\\sigma _0 & A_{1g+}, \\\\\\\\h_3 &\\\\equiv \\\\sigma _2 \\\\otimes \\\\sigma _1 & T_{1g+}, \\\\\\\\h_4 &\\\\equiv \\\\sigma _2 \\\\otimes \\\\sigma _2 & T_{1g+}, \\\\\\\\h_5 &\\\\equiv \\\\sigma _2 \\\\otimes \\\\sigma _3 & T_{1g+}, $ where the irreps are also given.\",\n \"$h_1$ , ..., $h_5$ anticommute pairwise; see Appendices and .\",\n \"Table REF also provides useful information on superconductivity: (a) All ten irreps that are even under time reversal appear as symmetries of possible pairing states.\",\n \"In fact, every $g+$ irrep $\\\\Gamma _{g+}$ is possible for any model since such a symmetry can be realized by combining an even momentum-space basis function belonging to $\\\\Gamma _{g+}$ with the $A_{1g+}$ basis matrix $h_0=$ .\",\n \"On the other hand, $u+$ pairing state require $g-$ basis matrices, here belonging to $A_{1g-}$ and $T_{1g-}$ [35].\",\n \"(b) For the even-parity pairing states, only the five $g+$ irreps are relevant.\",\n \"From Table REF , we see that then only the basis matrices transforming according to $A_{1g+}$ or $T_{1g+}$ occur.\",\n \"Further inspection shows that both types of basis matrices contribute to all $g+$ pairing states (all five occur in both columns).\",\n \"We conclude that for pairing states belonging to any single $g+$ irrep, all $A_{1g+}$ and $T_{1g+}$ basis matrices can appear in the pairing matrix $D(\\\\mathbf {k}) = \\\\sum _{n=0}^5 f_n(\\\\mathbf {k})\\\\, h_n .$ What changes between different pairing states are the momentum-dependent form factors $f_n(\\\\mathbf {k})$ .\",\n \"In the following, we analyze the nodal structure for several exemplary pairing symmetries.\",\n \"We start with the simplest pairing symmetry, $A_{1g+}$ .\",\n \"The construction of allowed terms in $D(\\\\mathbf {k})$ is analogous to the construction of $H_N(\\\\mathbf {k})$ .\",\n \"From Table REF , we find two contributions to $A_{1g+}$ pairing: (a) form factors that transform according to $A_{1g+}$ combined with the three $A_{1g+}$ basis matrices and (b) a triplet of $T_{1g+}$ form factors combined with the triplet of $T_{1g+}$ basis matrices.\",\n \"We discuss these two contributions in turn.\",\n \"It will prove useful to separate momentum-independent pairing amplitudes denoted by $\\\\delta _{\\\\cdots }$ from suitably normalized momentum basis functions denoted by $d_{\\\\cdots }(\\\\mathbf {k})$ , as done in Eq.\",\n \"(REF ).\",\n \"(a) This contribution to the pairing matrix reads as $D_1(\\\\mathbf {k}) &= \\\\delta _{00} d_{00}(\\\\mathbf {k})\\\\, \\\\sigma _0 \\\\otimes \\\\sigma _0+ \\\\delta _{10} d_{10}(\\\\mathbf {k})\\\\, \\\\sigma _1 \\\\otimes \\\\sigma _0 \\\\nonumber \\\\\\\\&\\\\quad {}+ \\\\delta _{30} d_{30}(\\\\mathbf {k})\\\\, \\\\sigma _3 \\\\otimes \\\\sigma _0 ,$ where $d_{00}(\\\\mathbf {k})$ , $d_{10}(\\\\mathbf {k})$ , and $d_{30}(\\\\mathbf {k})$ are basis functions of $A_{1g+}$ , and $\\\\delta _{00}$ , $\\\\delta _{10}$ , and $\\\\delta _{30}$ denote the corresponding pairing amplitudes.\",\n \"The leading polynomial forms of the basis functions are $d_{m0}(\\\\mathbf {k}) = d_{m0}^{(0)} + d_{m0}^{(2)}\\\\, (k_x^2+k_y^2+k_z^2) + \\\\ldots ,$ where we can set the three constants $d_{m0}^{(0)}$ to unity as a normalization.\",\n \"The higher-order coefficients are then generally distinct for different $m$ .\",\n \"These contributions can be interpreted as s-wave pairing since the minimum order of the basis functions is $l=0$ .\",\n \"We use the terms s-wave, p-wave, etc.\",\n \"to describe only the momentum dependence, not the symmetry of the full pairing state, for which we always use the irreps.\",\n \"The contribution is evidently spin-singlet pairing because the matrix acting in spin space is $\\\\sigma _0$ .\",\n \"(b) The reducible representation $T_{1g+}\\\\otimes T_{1g+}$ has nine matrix-valued basis functions $d_m(\\\\mathbf {k})\\\\, \\\\sigma _2 \\\\otimes \\\\sigma _n$ , $m,n=1,2,3$ , where $d_m(\\\\mathbf {k})$ are momentum basis functions of $T_{1g+}$ .\",\n \"The reduction $T_{1g+}\\\\otimes T_{1g+} = A_{1g+} \\\\oplus E_{g+} \\\\oplus T_{1g+} \\\\oplus T_{2g+}$ tells us that a basis change to matrix basis functions of the four indicated irreps exists.\",\n \"We here need to find the linear combination of the $d_m(\\\\mathbf {k})\\\\, \\\\sigma _2 \\\\otimes \\\\sigma _n$ that transforms according to $A_{1g+}$ .\",\n \"This is simply the sum over products of corresponding components with identical coefficients, i.e., $D_2(\\\\mathbf {k}) &= \\\\delta _t\\\\, \\\\big [ d_{21}(\\\\mathbf {k})\\\\, \\\\sigma _2 \\\\otimes \\\\sigma _1+ d_{22}(\\\\mathbf {k})\\\\, \\\\sigma _2 \\\\otimes \\\\sigma _2 \\\\nonumber \\\\\\\\&\\\\quad {} + d_{23}(\\\\mathbf {k})\\\\, \\\\sigma _2 \\\\otimes \\\\sigma _3 \\\\big ] ,$ where $d_{21}(\\\\mathbf {k})$ , $d_{22}(\\\\mathbf {k})$ , and $d_{23}(\\\\mathbf {k})$ form a triplet of $T_{1g+}$ basis functions and $\\\\delta _t$ is their common amplitude.\",\n \"The leading polynomials are $d_3(\\\\mathbf {k}) &\\\\equiv d_{21}(\\\\mathbf {k}) = d_t^{(4)}\\\\, k_y k_z (k_y^2-k_z^2) + \\\\ldots , \\\\\\\\d_4(\\\\mathbf {k}) &\\\\equiv d_{22}(\\\\mathbf {k}) = d_t^{(4)}\\\\, k_z k_x (k_z^2-k_x^2) + \\\\ldots , \\\\\\\\d_5(\\\\mathbf {k}) &\\\\equiv d_{23}(\\\\mathbf {k}) = d_t^{(4)}\\\\, k_x k_y (k_x^2-k_y^2) + \\\\ldots ,$ where we can choose $d_t^{(4)}=1$ as a normalization.\",\n \"This is g-wave spin-triplet (hence the subscript “t”) pairing since the minimum order is $l = 4$ and the Pauli matrices $\\\\sigma _1$ , $\\\\sigma _2$ , $\\\\sigma _3$ act on the spin Hilbert space.\",\n \"This combination is made possible by the nontrivial orbital content.\",\n \"The full $A_{1g+}$ pairing matrix has the usual form $D(\\\\mathbf {k}) \\\\equiv D_1(\\\\mathbf {k}) + D_2(\\\\mathbf {k}) = \\\\sum _{n=0}^5f_n(\\\\mathbf {k})\\\\, h_n ,$ where the symmetry properties of the form factors $f_n(\\\\mathbf {k})$ have been obtained above.\",\n \"We first consider pairing that respects TRS.\",\n \"Then, all $f_n(\\\\mathbf {k})$ can be chosen real.\",\n \"Equation (REF ) gives the condition for IP nodes.\",\n \"Each pairing form factor $f_n(\\\\mathbf {k})$ is multiplied by the corresponding normal-state form factor $c_n(\\\\mathbf {k})$ .\",\n \"The contribution (a) give, to leading order, $c_0&(\\\\mathbf {k}) f_0(\\\\mathbf {k}) - c_1(\\\\mathbf {k}) f_1(\\\\mathbf {k}) - c_2(\\\\mathbf {k}) f_2(\\\\mathbf {k})\\\\nonumber \\\\\\\\&= c_{00}^{(0)} \\\\delta _{00} - c_{10}^{(0)} \\\\delta _{10} - c_{30}^{(0)} \\\\delta _{30} + \\\\ldots ,$ which is generically nonzero and nodeless.\",\n \"The expression can of course have accidental nodes from higher-order terms, which we disregard here.\",\n \"For the contribution (b), $(c_3,c_4,c_5)$ and $(f_3,f_4,f_5)$ are corresponding basis functions of $T_{1g+}$ , and we find $- c_3&(\\\\mathbf {k}) f_3(\\\\mathbf {k}) - c_4(\\\\mathbf {k}) f_4(\\\\mathbf {k}) - c_5(\\\\mathbf {k}) f_5(\\\\mathbf {k})\\\\nonumber \\\\\\\\&= - c_2^{(4)} \\\\delta _t\\\\big [ k_y^2 k_z^2 (k_y^2-k_z^2)^2 + k_z^2 k_x^2 (k_z^2-k_x^2)^2 \\\\nonumber \\\\\\\\&\\\\quad {} + k_x^2 k_y^2 (k_x^2-k_y^2)^2 \\\\big ] + \\\\ldots $ This expression vanishes if the conditions $k_y^2k_z^2 (k_y-k_z)^2 (k_y+k_z)^2 &= 0 , \\\\\\\\k_z^2k_x^2 (k_z-k_x)^2 (k_z+k_x)^2 &= 0 , \\\\\\\\k_x^2k_y^2 (k_x-k_y)^2 (k_x+k_y)^2 &= 0$ hold simultaneously.\",\n \"This is the case for the $6+8+12=26$ high-symmetry directions in the $O_h$ Brillouin zone.\",\n \"Hence, there are 26 point nodes on a spheroidal normal-state Fermi surface around the $\\\\Gamma $ point.\",\n \"The higher-order terms in the basis functions do not change this picture since the nodes are imposed by $T_{1g+}$ symmetry.\",\n \"Since the conditions only contain squares, they are second-order (“double Weyl”) point nodes [26], [2], [36].\",\n \"Together with the generically nodeless contribution (a), $\\\\langle c,f\\\\rangle $ can contain first-order line nodes provided that the amplitude $c_2^{(4)} \\\\delta _t$ is sufficiently large and not all terms have the same sign.\",\n \"The location of these line nodes is not fixed by symmetries.\",\n \"In this respect, the situation is similar to the case of mixed singlet-triplet pairing in noncentrosymmetric superconductors [37].\",\n \"However, there is nothing that prevents a full gap, which is typically energetically favorable.\",\n \"Breaking of TRS is possible for one-dimensional irreps, see Sec.\",\n \".\",\n \"However, since the $A_{1g+}$ time-reversal-symmetric state is generically nodeless, the breaking of TRS is not expected to lead to a reduction of the internal energy [26].\",\n \"If TRS does break, then the condition (REF ) for IP nodes splits into two independent conditions for the real and imaginary parts of $\\\\langle c,f\\\\rangle $ .\",\n \"Hence, TRS-breaking $A_{1g+}$ pairing states are even less likely to have nodes than time-reversal-symmetric ones.\"\n ],\n [\n \"$A_{2g+}$ pairing\",\n \"$A_{2g+}$ pairing is potentially interesting since it is governed by a nontrivial one-dimensional irrep.\",\n \"It appears in two places in Table REF : (a) $A_{2g+}\\\\otimes A_{1g+}$ and (b) $T_{2g+}\\\\otimes T_{1g+}$ .\",\n \"Hence, it is incompatible with purely local pairing, for which the first factor must be $A_{1g+}$ .\",\n \"We discuss the two cases in turn.\",\n \"(a) Each of the three $A_{1g+}$ basis matrices is combined with a $A_{2g+}$ form factor, giving $D_1(\\\\mathbf {k}) &= \\\\delta _{00} d_{00}(\\\\mathbf {k})\\\\, \\\\sigma _0 \\\\otimes \\\\sigma _0+ \\\\delta _{10} d_{10}(\\\\mathbf {k})\\\\, \\\\sigma _1 \\\\otimes \\\\sigma _0 \\\\nonumber \\\\\\\\&\\\\quad {}+ \\\\delta _{30} d_{30}(\\\\mathbf {k})\\\\, \\\\sigma _3 \\\\otimes \\\\sigma _0 .$ The leading-order polynomial form is $d_{m0}(\\\\mathbf {k}) &= d_{m0}^{(6)}\\\\, \\\\big [ k_x^4 (k_y^2-k_z^2) + k_y^4 (k_z^2-k_x^2) \\\\nonumber \\\\\\\\&\\\\quad {} + k_z^4 (k_x^2-k_y^2) \\\\big ] + \\\\ldots ,$ where we set $d_{m0}^{(6)}=1$ as a normalization.\",\n \"These are i-wave ($l=6$ ) spin-singlet contributions.\",\n \"Based on the rule of thumb that terms of lower order in $\\\\mathbf {k}$ are energetically favored since they have fewer nodes or nodes of lower order and thus lead to higher condensation energy, we expect this contribution to be weak compared to the following one.\",\n \"(b) The reducible representation $T_{2g+}\\\\otimes T_{1g+}$ has nine matrix-valued basis functions $d_m(\\\\mathbf {k})\\\\, \\\\sigma _2 \\\\otimes \\\\sigma _n$ , $m,n=1,2,3$ , where $d_m(\\\\mathbf {k})$ are basis functions of $T_{2g+}$ .\",\n \"The construction parallels the one for $A_{1g+}$ pairing.\",\n \"The leading polynomial form factors read as $d_1(\\\\mathbf {k}) &\\\\equiv d_{21}(\\\\mathbf {k}) = d_2^{(2)}\\\\, k_y k_z + \\\\ldots , \\\\\\\\d_2(\\\\mathbf {k}) &\\\\equiv d_{22}(\\\\mathbf {k}) = d_2^{(2)}\\\\, k_z k_x + \\\\ldots , \\\\\\\\d_3(\\\\mathbf {k}) &\\\\equiv d_{23}(\\\\mathbf {k}) = d_2^{(2)}\\\\, k_x k_y + \\\\ldots $ We choose $d_2^{(2)}=1$ as normalization.\",\n \"The $A_{2g+}$ part of $T_{2g+}\\\\otimes T_{1g+}$ has the matrix-valued basis function $D_{A_{2g+}}(\\\\mathbf {k}) &= d_1(\\\\mathbf {k})\\\\, h_3 + d_2(\\\\mathbf {k})\\\\, h_4 + d_3(\\\\mathbf {k})\\\\, h_5\\\\nonumber \\\\\\\\&\\\\cong k_y k_z\\\\, \\\\sigma _2 \\\\otimes \\\\sigma _1 + k_z k_x\\\\, \\\\sigma _2 \\\\otimes \\\\sigma _2+ k_x k_y\\\\, \\\\sigma _2 \\\\otimes \\\\sigma _3 ,$ which describes d-wave ($l=2$ ) spin-triplet pairing, allowed due to nontrivial orbital content.\",\n \"Thus the second contribution to the pairing matrix is $D_2(\\\\mathbf {k}) \\\\cong \\\\delta _t ( k_y k_z\\\\, \\\\sigma _2 \\\\otimes \\\\sigma _1+ k_z k_x\\\\, \\\\sigma _2 \\\\otimes \\\\sigma _2 + k_x k_y\\\\, \\\\sigma _2 \\\\otimes \\\\sigma _3 ) .$ The leading order form factors $f_n(\\\\mathbf {k})$ can now be read off.\",\n \"They are summarized in Table REF .\",\n \"Table: Leading-order polynomial forms of the form factors f n (𝐤)f_n(\\\\mathbf {k}) describing A 2g+ A_{2g+} pairing for a model with two s-orbitals.Table: Leading-order polynomial forms of the products c n (𝐤)f n (𝐤)c_n(\\\\mathbf {k}) f_n(\\\\mathbf {k}) of form factors describing A 2g+ A_{2g+} pairing for a model with two s-orbitals.\",\n \"The amplitudes of the leading terms in c n (𝐤)c_n(\\\\mathbf {k}) have been absorbed into new pairing amplitudes marked by a tilde.For time-reversal-symmetric pairing, we can choose all $f_n(\\\\mathbf {k})$ real.\",\n \"For the condition for IP nodes, Eq.\",\n \"(REF ), we require the products $c_n(\\\\mathbf {k}) f_n(\\\\mathbf {k})$ , which are listed in Table REF .\",\n \"The amplitudes appearing the these products are distinguished by a tilde.\",\n \"We obtain, to leading order, $c_0&(\\\\mathbf {k})\\\\, f_0(\\\\mathbf {k}) - \\\\vec{c}(\\\\mathbf {k})\\\\cdot \\\\vec{f}(\\\\mathbf {k}) \\\\nonumber \\\\\\\\&\\\\cong (\\\\tilde{\\\\delta }_{00} - \\\\tilde{\\\\delta }_{10} - \\\\tilde{\\\\delta }_{30} - \\\\tilde{\\\\delta }_t)\\\\,[ k_x^4 (k_y^2-k_z^2) + k_y^4 (k_z^2-k_x^2) \\\\nonumber \\\\\\\\&\\\\quad {}+ k_z^4 (k_x^2-k_y^2) ] \\\\nonumber \\\\\\\\&= -(\\\\tilde{\\\\delta }_{00} - \\\\tilde{\\\\delta }_{10} - \\\\tilde{\\\\delta }_{30} - \\\\tilde{\\\\delta }_t)\\\\,(k_x-k_y)(k_x+k_y) \\\\nonumber \\\\\\\\&\\\\quad {}\\\\times (k_y-k_z)(k_y+k_z)(k_z-k_x)(k_z+k_x) .$ This product clearly vanishes if two of the three components of $\\\\mathbf {k}$ are equal in magnitude.\",\n \"The IP gap thus generically has six line nodes in the $\\\\lbrace 110\\\\rbrace $ planes.\",\n \"They are of first order since $\\\\langle c,f\\\\rangle $ changes sign at the nodes.\",\n \"The most obvious way to break TRS is to have a nontrivial phase difference between at least two of the amplitudes $\\\\tilde{\\\\delta }_{00}$ , $\\\\tilde{\\\\delta }_{10}$ , $\\\\tilde{\\\\delta }_{30}$ , and $\\\\tilde{\\\\delta }_t$ .\",\n \"Then IP nodes exist where both the real part and the imaginary part of $\\\\langle c,f\\\\rangle $ vanish.\",\n \"Equation (REF ) shows that the real and imaginary parts have the same symmetry-imposed line nodes so that the TRS-breaking state also has these line nodes for infinitesimal pairing.\",\n \"To check whether these line nodes are inflated beyond infinitesimal pairing, we consider the Pfaffian.\",\n \"It is useful to keep the full momentum dependence of the normal-state form factors, not just the leading terms.\",\n \"The form factors $c_0(\\\\mathbf {k})$ , $c_1(\\\\mathbf {k})$ , and $c_2(\\\\mathbf {k})$ are independent functions with $A_{1g+}$ symmetry, while the remaining three form factors can be written as $c_3(\\\\mathbf {k}) &= a_T(\\\\mathbf {k})\\\\, k_y k_z (k_y^2-k_z^2) , \\\\\\\\c_4(\\\\mathbf {k}) &= a_T(\\\\mathbf {k})\\\\, k_z k_x (k_z^2-k_x^2) , \\\\\\\\c_5(\\\\mathbf {k}) &= a_T(\\\\mathbf {k})\\\\, k_x k_y (k_x^2-k_y^2) ,$ where $a_T(\\\\mathbf {k})$ is another function with $A_{1g+}$ symmetry.\",\n \"Without loss of generality, we consider the plane $k_x=k_y$ , which is nodal for infinitesimal pairing.\",\n \"In this plane, the generalized scalar products read as $\\\\langle c,c\\\\rangle &= c_0^2(\\\\mathbf {k}) - c_1^2(\\\\mathbf {k}) - c_2^2(\\\\mathbf {k}) \\\\nonumber \\\\\\\\&\\\\quad {}- 2 a_T^2(\\\\mathbf {k})\\\\, k_x^2 k_z^2 (k_x^2-k_z^2)^2 , \\\\\\\\\\\\langle c,f_1\\\\rangle &= \\\\langle c,f_2\\\\rangle = 0 , \\\\\\\\\\\\langle f^1,f^1\\\\rangle &= - (\\\\mathop {\\\\textrm {Re}}\\\\delta _t)^2\\\\, k_x^2 (2 k_z^2 + k_x^2) , \\\\\\\\\\\\langle f^2,f^2\\\\rangle &= - (\\\\mathop {\\\\textrm {Im}}\\\\delta _t)^2\\\\, k_x^2 (2 k_z^2 + k_x^2) , \\\\\\\\\\\\langle f^1,f^2\\\\rangle &= - \\\\mathop {\\\\textrm {Re}}\\\\delta _t \\\\mathop {\\\\textrm {Im}}\\\\delta _t\\\\, k_x^2 (2 k_z^2 + k_x^2) ,$ where $\\\\mathbf {k}=(k_x,k_x,k_z)$ .\",\n \"Equation (REF ) then gives the Pfaffian $P(\\\\mathbf {k})&= \\\\big ( \\\\langle c,c\\\\rangle - \\\\langle f^1,f^1\\\\rangle - \\\\langle f^2,f^2\\\\rangle \\\\big )^2 \\\\nonumber \\\\\\\\&\\\\quad {}+ 4\\\\, \\\\big ( \\\\langle f^1,f^2\\\\rangle ^2 - \\\\langle f^1,f^1\\\\rangle \\\\langle f^2,f^2\\\\rangle \\\\big ) .$ The first term is a complete square and its zeros define the renormalized Fermi surface discussed above.\",\n \"Using Eqs.\",\n \"(REF )–(), the second term obviously vanishes, which can be attributed to the fact that in the plane $k_x=k_y$ only a single pairing channel ($T_{2g+}\\\\otimes T_{1g+}$ ) contributes to the pairing.\",\n \"The phase of the corresponding amplitude $\\\\delta _t$ can always be chosen real so that TRS breaking does not affect the superconducting state.\",\n \"The upshot is that for noninfinitesimal pairing the Pfaffian still has second-order zeros in the $\\\\lbrace 110\\\\rbrace $ planes and thus does not change sign.\",\n \"At least within these planes the line nodes are shifted but neither gapped out nor inflated.\",\n \"The question arises of what happens in the vicinity of these line nodes when we go off the high-symmetry planes.\",\n \"The first term of the general Pfaffian given in Eq.\",\n \"(REF ) has second-order zeros at the renormalized Fermi surface.\",\n \"We expand the second term about a point on the plane $k_x=k_y$ by setting $\\\\mathbf {k} = (k_x+q/\\\\sqrt{2},k_x-q/\\\\sqrt{2},k_z)$ .\",\n \"The leading form in $q$ reads as $4\\\\, &\\\\big ( \\\\langle c,f^1\\\\rangle ^2 + \\\\langle c,f^2\\\\rangle ^2+ \\\\langle f^1,f^2\\\\rangle ^2 - \\\\langle f^1,f^1\\\\rangle \\\\langle f^2,f^2\\\\rangle \\\\big ) \\\\nonumber \\\\\\\\&\\\\cong 32\\\\, k_x^2 (k_x^2-k_z^2)^4\\\\, \\\\Big (\\\\big | c_0 \\\\delta _{00} - c_1 \\\\delta _{10} - c_2 \\\\delta _{30} - a_T \\\\delta _t \\\\big |^2\\\\nonumber \\\\\\\\&\\\\quad {} + k_x^2 (k_x^2 + 2 k_z^2)\\\\, |\\\\delta _t|^2\\\\,\\\\big [ |\\\\delta _{00}|^2 \\\\sin ^2(\\\\phi _{00} - \\\\phi _t) \\\\nonumber \\\\\\\\&\\\\qquad {} - |\\\\delta _{10}|^2 \\\\sin ^2(\\\\phi _{10} - \\\\phi _t)- |\\\\delta _{30}|^2 \\\\sin ^2(\\\\phi _{30} - \\\\phi _t) \\\\big ] \\\\Big )\\\\, q^2 ,$ where $\\\\delta _{00} = |\\\\delta _{00}| e^{i\\\\phi _{00}}$ etc.\",\n \"The expression contains contributions of second and fourth order in the pairing amplitudes.\",\n \"At weak coupling, we can neglect the fourth-order contributions.\",\n \"Then, the leading correction to the Pfaffian away from the $(110)$ plane is nonnegative and generically is strictly positive for $k_x \\\\ne k_z$ .\",\n \"Hence, in this case, there is no BFS in the vicinity of the shifted line node, in any direction.\",\n \"On the other hand, for strong coupling, the coefficient in Eq.\",\n \"(REF ) can become negative.\",\n \"In this case, BFSs can exist on both sides of the $\\\\lbrace 110\\\\rbrace $ planes and touching each other at these planes.\",\n \"For the special case $k_x=k_z$ , the whole $q^2$ term in Eq.\",\n \"(REF ) vanishes.\",\n \"Since we already had assumed $k_x=k_y$ this corresponds to the threefold rotation axis $[111]$ .\",\n \"Here, for infinitesimal pairing three nodal lines intersect.\",\n \"We consider $\\\\mathbf {k} = (k_x+q/\\\\sqrt{2}, k_x-q/\\\\sqrt{2},k_x)$ .\",\n \"The leading form in the expansion of the second term of the Pfaffian here reads as $12&8\\\\, k_x^6\\\\, \\\\Big (\\\\big | c_0 \\\\delta _{00} - c_1 \\\\delta _{10} - c_2 \\\\delta _{30} - a_T \\\\delta _d \\\\big |^2\\\\nonumber \\\\\\\\&{} + 3\\\\, k_x^4\\\\, |\\\\delta _t|^2\\\\, \\\\big [ |\\\\delta _{00}|^2 \\\\sin ^2(\\\\phi _{00} -\\\\phi _d) \\\\nonumber \\\\\\\\&\\\\quad {} - |\\\\delta _{10}|^2 \\\\sin ^2(\\\\phi _{10} - \\\\phi _d)- |\\\\delta _{30}|^2 \\\\sin ^2(\\\\phi _{30} - \\\\phi _d) \\\\big ] \\\\Big )\\\\, q^6 .$ This term is also nonnegative at weak coupling so that there are no BFSs close to the $\\\\langle 111\\\\rangle $ directions.\",\n \"In conclusion, the line nodes for the TRS-breaking $A_{2g+}$ pairing state are not inflated into BFSs.\",\n \"However, they are shifted away from the normal-state Fermi surface everywhere but remain within the high-symmetry (mirror) planes.\",\n \"The lack of inflation within the high-symmetry planes can be understood on the basis that there is only a single relevant pairing amplitude, which can be chosen real.\"\n ],\n [\n \"$E_{g+}$ pairing\",\n \"Pairing conforming to the two-dimensional irrep $E_{g+}$ is of interest since the breaking of TRS occurs naturally for multidimensional irreps.\",\n \"$E_{g+}$ pairing can emerge from the products (a) $E_{g+} \\\\otimes A_{1g+}$ , (b) $T_{1g+} \\\\otimes T_{1g+}$ , and (c) $T_{2g+} \\\\otimes T_{1g+}$ in Table REF .\",\n \"Hence, it is incompatible with purely local pairing.\",\n \"We discuss the contributions in turn.\",\n \"(a) The three $A_{1g+}$ matrices are each combined with a doublet of $E_{g+}$ form factors, giving $D_1(\\\\mathbf {k})&= \\\\left[\\\\delta ^1_{00} d^1_{00}(\\\\mathbf {k}) + \\\\delta ^2_{00} d^2_{00}(\\\\mathbf {k})\\\\right]\\\\sigma _0 \\\\otimes \\\\sigma _0 \\\\nonumber \\\\\\\\&\\\\quad {} + \\\\left[\\\\delta ^1_{10} d^1_{10}(\\\\mathbf {k}) + \\\\delta ^2_{10} d^2_{10}(\\\\mathbf {k})\\\\right]\\\\sigma _1 \\\\otimes \\\\sigma _0 \\\\nonumber \\\\\\\\&\\\\quad {} + \\\\left[\\\\delta ^1_{30} d^1_{30}(\\\\mathbf {k}) + \\\\delta ^2_{30} d^2_{30}(\\\\mathbf {k})\\\\right]\\\\sigma _3 \\\\otimes \\\\sigma _0 .$ The leading polynomial terms are $d^1_{m0}(\\\\mathbf {k}) &= d^{(2)}_{m0}\\\\, (k_x^2-k_y^2) + d^{(4)}_{m0}\\\\, (k_x^4-k_y^4)+ \\\\ldots , \\\\\\\\d^2_{m0}(\\\\mathbf {k}) &= \\\\frac{d^{(2)}_{m0}}{\\\\sqrt{3}}\\\\,(2k_z^2-k_x^2-k_y^2) + \\\\frac{d^{(4)}_{m0}}{\\\\sqrt{3}}\\\\, (2k_z^4-k_x^4-k_y^4) \\\\nonumber \\\\\\\\&\\\\quad {} + \\\\ldots ,$ where we can set the coefficients $d^{(2)}_{m0}$ of the leading terms to unity as a normalization.\",\n \"Recall that the higher-order coefficients are then generally distinct for different $m$ .\",\n \"These contributions can be described as d-wave ($l=2$ ) spin-singlet pairing.\",\n \"(b) The reducible representation $T_{1g+}\\\\otimes T_{1g+}$ has nine matrix-valued basis functions $d_m(\\\\mathbf {k})\\\\, \\\\sigma _2 \\\\otimes \\\\sigma _n$ , $m,n=1,2,3$ , where $d_m(\\\\mathbf {k})$ are momentum basis functions of $T_{1g+}$ .\",\n \"The leading polynomial terms read as $d_1(\\\\mathbf {k}) &= d^{(4)}\\\\, k_y k_z (k_y^2-k_z^2) + \\\\ldots , \\\\\\\\d_2(\\\\mathbf {k}) &= d^{(4)}\\\\, k_z k_x (k_z^2-k_x^2) + \\\\ldots , \\\\\\\\d_3(\\\\mathbf {k}) &= d^{(4)}\\\\, k_x k_y (k_x^2-k_y^2) + \\\\ldots ,$ where $d^{(4)}$ can be chosen to be unity.\",\n \"The reduction $T_{1g+}\\\\otimes T_{1g+} = A_{1g+} \\\\oplus E_{g+} \\\\oplus T_{1g+} \\\\oplus T_{2g+}$ implies that a basis change to matrix basis functions of the four indicated irreps exists.\",\n \"The linear combinations of the functions $d_m(\\\\mathbf {k})\\\\, \\\\sigma _2 \\\\otimes \\\\sigma _n$ that transform according to $E_{g+}$ are $D_{x^2-y^2}(\\\\mathbf {k}) &= d_1(\\\\mathbf {k})\\\\, \\\\sigma _2 \\\\otimes \\\\sigma _1- d_2(\\\\mathbf {k})\\\\, \\\\sigma _2 \\\\otimes \\\\sigma _2 , \\\\\\\\D_{3z^2-r^2}(\\\\mathbf {k}) &= \\\\frac{2 d_3(\\\\mathbf {k})}{\\\\sqrt{3}}\\\\, \\\\sigma _2 \\\\otimes \\\\sigma _3- \\\\frac{d_1(\\\\mathbf {k})}{\\\\sqrt{3}}\\\\, \\\\sigma _2 \\\\otimes \\\\sigma _1 \\\\nonumber \\\\\\\\&\\\\quad {}- \\\\frac{d_2(\\\\mathbf {k})}{\\\\sqrt{3}}\\\\, \\\\sigma _2 \\\\otimes \\\\sigma _2 .$ These matrix basis functions are no longer simply the product of a scalar momentum-dependent form factor and a momentum-independent matrix.\",\n \"Their contribution to the pairing matrix is $D_2(\\\\mathbf {k}) = \\\\delta ^1_{2t}\\\\, D_{x^2-y^2}(\\\\mathbf {k})+ \\\\delta ^2_{2t}\\\\, D_{3z^2-r^2}(\\\\mathbf {k}) .$ This describes g-wave ($l=4$ ) spin-triplet pairing, made possible by nontrivial orbital content.\",\n \"(c) The analysis for $T_{2g+} \\\\otimes T_{1g+} = A_{2g+} \\\\oplus E_{g+} \\\\oplus T_{1g+} \\\\oplus T_{2g+}$ is analogous, except that now the momentum basis functions belong to $T_{2g+}$ , $d^{\\\\prime }_1(\\\\mathbf {k}) &= d^{\\\\prime (2)}\\\\, k_y k_z + \\\\ldots , \\\\\\\\d^{\\\\prime }_2(\\\\mathbf {k}) &= d^{\\\\prime (2)}\\\\, k_z k_x + \\\\ldots , \\\\\\\\d^{\\\\prime }_3(\\\\mathbf {k}) &= d^{\\\\prime (2)}\\\\, k_x k_y + \\\\ldots ,$ where we choose $d^{\\\\prime (2)}=1$ .\",\n \"We have the matrix basis functions $D^{\\\\prime }_{x^2-y^2}(\\\\mathbf {k}) &= \\\\frac{2 d^{\\\\prime }_3(\\\\mathbf {k})}{\\\\sqrt{3}}\\\\, \\\\sigma _2 \\\\otimes \\\\sigma _3- \\\\frac{d^{\\\\prime }_1(\\\\mathbf {k})}{\\\\sqrt{3}}\\\\, \\\\sigma _2 \\\\otimes \\\\sigma _1 \\\\nonumber \\\\\\\\&\\\\quad {} - \\\\frac{d^{\\\\prime }_2(\\\\mathbf {k})}{\\\\sqrt{3}}\\\\, \\\\sigma _2 \\\\otimes \\\\sigma _2 , \\\\\\\\D^{\\\\prime }_{3z^2-r^2}(\\\\mathbf {k}) &= -d^{\\\\prime }_1(\\\\mathbf {k})\\\\, \\\\sigma _2 \\\\otimes \\\\sigma _1+ d^{\\\\prime }_2(\\\\mathbf {k})\\\\, \\\\sigma _2 \\\\otimes \\\\sigma _2 .$ Note that the forms of the expressions for the two components of $E_g$ , i.e., for the $x^2-y^2$ and the $3z^2-r^2$ matrix basis functions, are interchanged compared to case (b).\",\n \"To determine the correct components, their behavior under twofold rotation about the $[110]$ direction has been examined.\",\n \"Furthermore, to find the relative factor, which turns out to be $-1$ , the behavior under threefold rotation about $[111]$ has been considered.\",\n \"The contribution to the pairing matrix is $D_3(\\\\mathbf {k}) = \\\\delta ^1_{2t^{\\\\prime }}\\\\, D^{\\\\prime }_{x^2-y^2}(\\\\mathbf {k})+ \\\\delta ^2_{2t^{\\\\prime }}\\\\, D^{\\\\prime }_{3z^2-r^2}(\\\\mathbf {k}) .$ This is d-wave spin-triplet pairing, again made possible by nontrivial orbital content.\",\n \"The matrix $D(\\\\mathbf {k}) = D_1(\\\\mathbf {k}) + D_2(\\\\mathbf {k}) + D_3(\\\\mathbf {k})$ is evidently of the form of Eq.\",\n \"(REF ).\",\n \"The form factors $f_n(\\\\mathbf {k})$ are complicated functions of $\\\\mathbf {k}$ with nodes in different places.\",\n \"The leading-order polynomial forms are listed in Table REF .\",\n \"Table: Leading-order polynomial forms of the form factors f n (𝐤)f_n(\\\\mathbf {k}) describing E g+ E_{g+} pairing for a model with two s-orbitals.Table: Leading-order polynomial forms of the products c n (𝐤)f n (𝐤)c_n(\\\\mathbf {k}) f_n(\\\\mathbf {k}) of form factors describing E g+ E_{g+} pairing for a model with two s-orbitals.\",\n \"The amplitudes of the leading terms in c n (𝐤)c_n(\\\\mathbf {k}) have been absorbed into new pairing amplitudes marked by a tilde.In the condition for IP nodes, Eq.\",\n \"(REF ), each pairing form factor $f_n(\\\\mathbf {k})$ is multiplied by the normal-state form factor $c_n(\\\\mathbf {k})$ .\",\n \"We list the leading-order polynomial form of these products in Table REF .\",\n \"The contributions for $n=0,1,2$ have the same form and can be grouped together with new amplitudes $\\\\tilde{\\\\delta }^1_0$ and $\\\\tilde{\\\\delta }^2_0$ .\",\n \"With this, we obtain, to leading order, $&c_0(\\\\mathbf {k})\\\\, f_0(\\\\mathbf {k}) - \\\\vec{c}(\\\\mathbf {k}) \\\\cdot \\\\vec{f}(\\\\mathbf {k})\\\\cong \\\\tilde{\\\\delta }^1_0\\\\, (k_x^2 - k_y^2) \\\\nonumber \\\\\\\\&{} + \\\\tilde{\\\\delta }^1_{2t}\\\\, k_z^2 (k_x^2 - k_y^2)(k_x^4 + k_y^4 + k_z^4 + k_x^2 k_y^2 - 2 k_x^2 k_z^2 - 2 k_y^2 k_z^2)\\\\nonumber \\\\\\\\&{} + \\\\frac{\\\\tilde{\\\\delta }^1_{2t^{\\\\prime }}}{\\\\sqrt{3}}\\\\, (k_x^2 - k_y^2)(k_z^4 - 2 k_x^2 k_y^2 - k_x^2 k_z^2 - k_y^2 k_z^2) \\\\nonumber \\\\\\\\&{} + \\\\frac{\\\\tilde{\\\\delta }^2_0}{\\\\sqrt{3}}\\\\, (2k_z^2 - k_x^2 - k_y^2) \\\\nonumber \\\\\\\\&{} + \\\\frac{\\\\tilde{\\\\delta }^2_{2t}}{\\\\sqrt{3}}\\\\, \\\\big ({-}2 k_x^6 k_y^2 + 4 k_x^4 k_y^4 - 2 k_x^2 k_y^6 + k_x^6 k_z^2+ k_y^6 k_z^2 \\\\nonumber \\\\\\\\&\\\\quad {} - 2 k_x^4 k_z^4 - 2 k_y^4 k_z^4 + k_x^2 k_z^6 + k_y^2 k_z^6\\\\big )\\\\nonumber \\\\\\\\&{} + \\\\tilde{\\\\delta }^2_{2t^{\\\\prime }}\\\\, k_z^2 (k_x^4 + k_y^4 - k_x^2 k_z^2 - k_y^2 k_z^2) .$ We first discuss time-reversal-symmetric pairing, for which all amplitudes can be chosen real.\",\n \"Based on the order alone, we expect that typically $\\\\tilde{\\\\delta }^1_0$ and $\\\\tilde{\\\\delta }^2_0$ dominate.\",\n \"Unless both amplitudes vanish these contributions give two first-order line nodes—the function changes sign at these nodes.\",\n \"Note that all contributions belonging to the first component of $E_{g+}$ have nodal planes at $k_y=\\\\pm k_x$ .\",\n \"This is because these nodes are symmetry induced and must lie in the diagonal mirror planes.\",\n \"On the other hand, the contributions belonging to the second component also have two nodal surfaces (two line nodes on the normal-state Fermi surface) but these are not pinned to high-symmetry planes.\",\n \"Hence, higher-order terms can shift them around.\",\n \"Their only symmetry-enforced property is to pass through the $\\\\langle 111\\\\rangle $ directions, where they intersect with the nodes of the first component.\",\n \"If both components have nonzero amplitudes there are still generically two line nodes, which have to pass through the $\\\\langle 111\\\\rangle $ directions [38].\",\n \"If TRS is broken and only $\\\\tilde{\\\\delta }^1_0$ and $\\\\tilde{\\\\delta }^2_0$ are nonzero there must be a nontrivial phase difference between these two amplitudes and we find different line nodes in the real and imaginary parts, resulting in point nodes at their crossings.\",\n \"Since the real and imaginary parts are zero in the $\\\\langle 111\\\\rangle $ directions, we obtain at least eight point nodes in these directions.\",\n \"All higher-order terms are zero there so that they cannot shift or gap out these point nodes.\",\n \"We next turn to the possibility of BFSs.\",\n \"Without loss of generality, we consider the IP point node in the $[111]$ direction.\",\n \"For $\\\\mathbf {k} = k\\\\,(1,1,1)/\\\\sqrt{3} \\\\equiv k\\\\, \\\\hat{\\\\mathbf {n}}_{111}$ , the normal-state form factors $c_0(\\\\mathbf {k}) \\\\equiv c_{00}(\\\\mathbf {k})$ , $c_1(\\\\mathbf {k}) \\\\equiv c_{10}(\\\\mathbf {k})$ , and $c_2(\\\\mathbf {k}) \\\\equiv c_{30}(\\\\mathbf {k})$ are independent even functions of $k$ ; see Eq.\",\n \"(REF ).\",\n \"On the other hand, $c_3(\\\\mathbf {k}) \\\\equiv c_{21}(\\\\mathbf {k})$ , $c_4(\\\\mathbf {k}) \\\\equiv c_{22}(\\\\mathbf {k})$ , and $c_5(\\\\mathbf {k}) \\\\equiv c_{23}(\\\\mathbf {k})$ vanish in this direction; see Eqs.\",\n \"(REF )–().\",\n \"Furthermore, Table REF shows that $f_0(\\\\mathbf {k}) &= f_1(\\\\mathbf {k}) = f_2(\\\\mathbf {k}) = 0 , \\\\\\\\f_3(\\\\mathbf {k}) &= - \\\\frac{\\\\delta ^1_{2t^{\\\\prime }}}{3\\\\sqrt{3}}\\\\, k^2 - \\\\frac{\\\\delta ^2_{2t^{\\\\prime }}}{3}\\\\, k^2 , \\\\\\\\f_4(\\\\mathbf {k}) &= - \\\\frac{\\\\delta ^1_{2t^{\\\\prime }}}{3\\\\sqrt{3}}\\\\, k^2 + \\\\frac{\\\\delta ^2_{2t^{\\\\prime }}}{3}\\\\, k^2 , \\\\\\\\f_5(\\\\mathbf {k}) &= \\\\frac{2\\\\delta ^1_{2t^{\\\\prime }}}{3\\\\sqrt{3}}\\\\, k^2 .$ This implies that $\\\\langle c,c\\\\rangle &= c_0^2(k) - c_1^2(k) - c_2^2(k) , \\\\\\\\\\\\langle c,f^1\\\\rangle &= \\\\langle c,f^2\\\\rangle = 0 , \\\\\\\\\\\\langle f^1,f^1\\\\rangle &= - \\\\frac{2k^2}{9} \\\\big [ (\\\\mathop {\\\\textrm {Re}}\\\\delta ^1_{2t^{\\\\prime }})^2 + (\\\\mathop {\\\\textrm {Re}}\\\\delta ^2_{2t^{\\\\prime }})^2 \\\\big ] , \\\\\\\\\\\\langle f^2,f^2\\\\rangle &= - \\\\frac{2k^2}{9} \\\\big [ (\\\\mathop {\\\\textrm {Im}}\\\\delta ^1_{2t^{\\\\prime }})^2 + (\\\\mathop {\\\\textrm {Im}}\\\\delta ^2_{2t^{\\\\prime }})^2 \\\\big ] , \\\\\\\\\\\\langle f^1,f^2\\\\rangle &= -\\\\frac{2k^2}{9} \\\\big [ \\\\mathop {\\\\textrm {Re}}\\\\delta ^1_{2t^{\\\\prime }} \\\\mathop {\\\\textrm {Im}}\\\\delta ^1_{2t^{\\\\prime }}+ \\\\mathop {\\\\textrm {Re}}\\\\delta ^2_{2t^{\\\\prime }} \\\\mathop {\\\\textrm {Im}}\\\\delta ^2_{2t^{\\\\prime }} \\\\big ] .$ Equation (REF ) then gives $P&(k\\\\, \\\\hat{\\\\mathbf {n}}_{111}) \\\\nonumber \\\\\\\\&= \\\\bigg [ c_0^2(k) - c_1^2(k) - c_2^2(k) + \\\\frac{2}{9}\\\\, k^2\\\\, |\\\\delta ^1_{2t^{\\\\prime }}|^2+ \\\\frac{2}{9}\\\\, k^2\\\\, |\\\\delta ^2_{2t^{\\\\prime }}|^2 \\\\bigg ]^{\\\\!2} \\\\nonumber \\\\\\\\&\\\\quad {}- \\\\frac{16}{81}\\\\, k^4\\\\, |\\\\delta ^1_{2t^{\\\\prime }}|^2 |\\\\delta ^2_{2t^{\\\\prime }}|^2 \\\\sin ^2(\\\\phi _1 - \\\\phi _2) ,$ where the two relevant pairing amplitudes are written as $\\\\delta ^{1,2}_{2t^{\\\\prime }} = |\\\\delta ^{1,2}_{2t^{\\\\prime }}|\\\\, e^{i\\\\phi _{1,2}}$ .\",\n \"The first term has a second-order zero at the renormalized normal-state Fermi surface.\",\n \"The second term is negative whenever the phase difference between $\\\\delta ^1_{2t^{\\\\prime }}$ and $\\\\delta ^2_{2t^{\\\\prime }}$ is not an integer multiple of $\\\\pi $ .\",\n \"This is generically the case for broken TRS.\",\n \"This means that in the vicinity of the renormalized normal-state Fermi surface we find a region with $P(k\\\\, \\\\hat{\\\\mathbf {n}}_{111})<0$ and thus the point node is inflated into a BFS pierced by the $[111]$ axis [39].\",\n \"If the superconducting energy scale becomes comparable to normal-state energies the BFSs are no longer spheroidal pockets close to the IP point nodes.\",\n \"The BFSs might then merge and could move either to the $\\\\Gamma $ point or to the edge of the Brillouin zone and annihilate there [2].\",\n \"We now check whether this can happen.\",\n \"On the unrenormalized normal-state Fermi surface in the $[111]$ direction, $c_0^2 - c_1^2 - c_2^2$ vanishes.\",\n \"The Pfaffian can then be written as $P(k_F\\\\, \\\\hat{\\\\mathbf {n}}_{111}) &= \\\\frac{4}{81}\\\\, k_F^4 \\\\big ( |\\\\delta ^1_{2t^{\\\\prime }}|^2- |\\\\delta ^2_{2t^{\\\\prime }}|^2 \\\\big )^2 \\\\nonumber \\\\\\\\&\\\\quad {} + \\\\frac{16}{81}\\\\, k_F^4\\\\, |\\\\delta ^1_{2t^{\\\\prime }}|^2 |\\\\delta ^2_{2t^{\\\\prime }}|^2\\\\cos ^2(\\\\phi _1 - \\\\phi _2) .$ This means that for the special TRS-breaking state with $|\\\\delta ^1_{2t^{\\\\prime }}| = |\\\\delta ^2_{2t^{\\\\prime }}|$ and phase difference $\\\\pm \\\\pi $ or equivalent, the Pfaffian vanishes at $k=k_F$ so that the BFS must touch the normal-state Fermi surface.\",\n \"In this case, the BFSs cannot annihilate for strong pairing.\",\n \"The special conditions of equal amplitudes and phase difference of $\\\\pm \\\\pi $ are quite natural from the point of view of energetics [31], [26], [13], [1], [2].\",\n \"In conclusion, at infinitesimal pairing, the gap generically has point nodes in the $\\\\langle 111\\\\rangle $ directions if TRS is broken.\",\n \"These nodes are expected to be inflated into BFSs if the amplitudes $\\\\delta ^1_{2t^{\\\\prime }}$ and $\\\\delta ^2_{2t^{\\\\prime }}$ are both nonzero and have a nontrivial phase difference.\",\n \"All other amplitudes do not contribute to the inflation of nodes along the $\\\\langle 111\\\\rangle $ directions since the corresponding form factors $f_n(\\\\mathbf {k})$ vanish there.\",\n \"For the energetically favored $(1,i)$ state, the BFSs stick to the normal-state Fermi surface at the former point nodes.\"\n ],\n [\n \"$T_{1g+}$ pairing\",\n \"The analysis for the three-dimensional irrep $T_{1g+}$ is analogous and we will be brief.\",\n \"All functions of momentum are represented by the lowest-order polynomials of correct symmetry.\",\n \"$T_{1g+}$ pairing appears in the following products in Table REF : (a) $A_{1g+}\\\\otimes T_{1g+}$ , (b) $E_{g+}\\\\otimes T_{1g+}$ , (c) $T_{1g+}\\\\otimes A_{1g+}$ , (d) $T_{1g+}\\\\otimes T_{1g+}$ , (e) $T_{2g+}\\\\otimes T_{1g+}$ .\",\n \"$T_{1g+}$ symmetry is possible even for purely local pairing due to the momentum-independent contribution (a).\",\n \"(a) For the contribution $A_{1g+}\\\\otimes T_{1g+}$ , we find the matrix basis functions $D_{x,A_{1g+}}(\\\\mathbf {k}) &\\\\cong h_3 = \\\\sigma _2 \\\\otimes \\\\sigma _1 , \\\\\\\\D_{y,A_{1g+}}(\\\\mathbf {k}) &\\\\cong h_4 = \\\\sigma _2 \\\\otimes \\\\sigma _2 , \\\\\\\\D_{z,A_{1g+}}(\\\\mathbf {k}) &\\\\cong h_5 = \\\\sigma _2 \\\\otimes \\\\sigma _3 .$ These describe s-wave spin-triplet pairing, made possible by the nontrivial orbital structure.\",\n \"(b) For $E_{g+}\\\\otimes T_{1g+}$ , the basis functions read as $D_{x,E_{g+}}(\\\\mathbf {k}) &\\\\cong \\\\frac{1}{3}\\\\, (2k_x^2-k_y^2-k_z^2)\\\\, h_3 \\\\nonumber \\\\\\\\&= \\\\frac{1}{3}\\\\, (2k_x^2-k_y^2-k_z^2)\\\\, \\\\sigma _2 \\\\otimes \\\\sigma _1 , \\\\\\\\D_{y,E_{g+}}(\\\\mathbf {k}) &\\\\cong \\\\frac{1}{3}\\\\, (2k_y^2-k_z^2-k_x^2)\\\\, h_4 \\\\nonumber \\\\\\\\&= \\\\frac{1}{3}\\\\, (2k_y^2-k_z^2-k_x^2)\\\\, \\\\sigma _2 \\\\otimes \\\\sigma _2 , \\\\\\\\D_{z,E_{g+}}(\\\\mathbf {k}) &\\\\cong \\\\frac{1}{3}\\\\, (2k_z^2-k_x^2-k_y^2)\\\\, h_5 \\\\nonumber \\\\\\\\&= \\\\frac{1}{3}\\\\, (2k_z^2-k_x^2-k_y^2)\\\\, \\\\sigma _2 \\\\otimes \\\\sigma _3 .$ This is d-wave spin-triplet pairing.\",\n \"(c) $T_{1g+}\\\\otimes A_{1g+}$ involves three triplets of basis functions, $D_{x,m0}(\\\\mathbf {k}) &\\\\cong k_yk_z (k_y^2-k_z^2)\\\\, \\\\sigma _m \\\\otimes \\\\sigma _0 , \\\\\\\\D_{y,m0}(\\\\mathbf {k}) &\\\\cong k_zk_x (k_z^2-k_x^2)\\\\, \\\\sigma _m \\\\otimes \\\\sigma _0 , \\\\\\\\D_{z,m0}(\\\\mathbf {k}) &\\\\cong k_xk_y (k_x^2-k_y^2)\\\\, \\\\sigma _m \\\\otimes \\\\sigma _0 ,$ for $m=0,1,3$ .\",\n \"This is g-wave spin-singlet pairing.\",\n \"(d) For $T_{1g+}\\\\otimes T_{1g+}$ , we get the basis functions $&D_{x,T_{1g+}}(\\\\mathbf {k}) \\\\cong k_z k_x (k_z^2-k_x^2)\\\\, h_5 - k_x k_y (k_x^2-k_y^2)\\\\, h_4 \\\\nonumber \\\\\\\\&\\\\quad = k_z k_x (k_z^2-k_x^2)\\\\, \\\\sigma _2 \\\\otimes \\\\sigma _3- k_x k_y (k_x^2-k_y^2)\\\\, \\\\sigma _2 \\\\otimes \\\\sigma _2 , \\\\\\\\&D_{y,T_{1g+}}(\\\\mathbf {k}) \\\\cong k_x k_y (k_x^2-k_y^2)\\\\, h_3 - k_y k_z (k_y^2-k_z^2)\\\\, h_5 \\\\nonumber \\\\\\\\&\\\\quad = k_x k_y (k_x^2-k_y^2)\\\\, \\\\sigma _2 \\\\otimes \\\\sigma _1- k_y k_z (k_y^2-k_z^2)\\\\, \\\\sigma _2 \\\\otimes \\\\sigma _3 , \\\\\\\\&D_{z,T_{1g+}}(\\\\mathbf {k}) \\\\cong k_y k_z (k_y^2-k_z^2)\\\\, h_4 - k_z k_x (k_z^2-k_x^2)\\\\, h_3 \\\\nonumber \\\\\\\\&\\\\quad = k_y k_z (k_y^2-k_z^2)\\\\, \\\\sigma _2 \\\\otimes \\\\sigma _2- k_z k_x (k_z^2-k_x^2)\\\\, \\\\sigma _2 \\\\otimes \\\\sigma _1 .$ This is g-wave spin-triplet pairing.\",\n \"(e) For $T_{2g+}\\\\otimes T_{1g+}$ , we get the basis functions $D_{x,T_{2g+}}(\\\\mathbf {k}) &\\\\cong k_z k_x\\\\, h_5 + k_x k_y\\\\, h_4 \\\\nonumber \\\\\\\\&= k_z k_x\\\\, \\\\sigma _2 \\\\otimes \\\\sigma _3 + k_x k_y\\\\, \\\\sigma _2 \\\\otimes \\\\sigma _2 , \\\\\\\\D_{y,T_{2g+}}(\\\\mathbf {k}) &\\\\cong k_x k_y\\\\, h_3 + k_y k_z\\\\, h_5 \\\\nonumber \\\\\\\\&= k_x k_y\\\\, \\\\sigma _2 \\\\otimes \\\\sigma _1 + k_y k_z\\\\, \\\\sigma _2 \\\\otimes \\\\sigma _3 , \\\\\\\\D_{z,T_{2g+}}(\\\\mathbf {k}) &\\\\cong k_y k_z\\\\, h_4 + k_z k_x\\\\, h_3 \\\\nonumber \\\\\\\\&= k_y k_z\\\\, \\\\sigma _2 \\\\otimes \\\\sigma _2 + k_z k_x\\\\, \\\\sigma _2 \\\\otimes \\\\sigma _1 .$ This is d-wave spin-triplet pairing.\",\n \"Table: Leading-order polynomial forms of the form factors f n (𝐤)f_n(\\\\mathbf {k}) describing T 1g+ T_{1g+} pairing for a model with two s-orbitals.Table: Leading-order polynomial forms of the products c n (𝐤)f n (𝐤)c_n(\\\\mathbf {k}) f_n(\\\\mathbf {k}) of form factors describing T 1g+ T_{1g+} pairing for a model with two s-orbitals.\",\n \"The amplitudes of theleading terms in c n (𝐤)c_n(\\\\mathbf {k}) have been absorbed into new pairing amplitudes marked by a tilde.The resulting form factors are summarized in Table REF .\",\n \"To determine the IP nodes, we require the products $c_n(\\\\mathbf {k})\\\\, f_n(\\\\mathbf {k})$ , which are listed in Table REF .\",\n \"Defining $\\\\tilde{\\\\delta }_{\\\\nu ,0} \\\\equiv \\\\tilde{\\\\delta }_{\\\\nu ,00} - \\\\tilde{\\\\delta }_{\\\\nu ,10} - \\\\tilde{\\\\delta }_{\\\\nu ,30} - \\\\tilde{\\\\delta }_{\\\\nu ,A_{1g+}}$ for $\\\\nu =x,y,z$ , we obtain $c_0&(\\\\mathbf {k})\\\\, f_0(\\\\mathbf {k}) - \\\\vec{c}(\\\\mathbf {k}) \\\\cdot \\\\vec{f}(\\\\mathbf {k}) \\\\nonumber \\\\\\\\&= \\\\bigg [ \\\\tilde{\\\\delta }_{x,0} - \\\\frac{\\\\tilde{\\\\delta }_{x,E_{g+}}}{3}\\\\, (2k_x^2-k_y^2-k_z^2)+ \\\\tilde{\\\\delta }_{x,T_{2g+}}\\\\, k_x^2 \\\\bigg ] \\\\nonumber \\\\\\\\&\\\\quad {}\\\\times k_y k_z (k_y^2-k_z^2) + \\\\ldots ,$ where two terms with cyclically permuted indices $x$ , $y$ , and $z$ have been suppressed.\",\n \"Note that contribution (d) has dropped out.\",\n \"This is an artifact of having used the same leading-order basis functions for $c_n(\\\\mathbf {k})$ and $f_n(\\\\mathbf {k})$ .\",\n \"Using different ones, we see that the terms do not cancel.\",\n \"They do not change the following discussion, though.\",\n \"If only the $\\\\tilde{\\\\delta }_x$ amplitudes are different from zero, i.e., for pairing of $(1,0,0)$ type [26], [13], [2], we expect four first-order line nodes in the planes $k_y=0$ , $k_z=0$ , $k_y=k_z$ , and $k_y=-k_z$ .\",\n \"This is a new example of a state that is necessarily nodal even for purely local pairing, in which case $\\\\tilde{\\\\delta }_{x,0}$ is the only nonvanishing amplitude.\",\n \"Time-reversal-symmetric superpositions of $(1,0,0)$ , $(0,1,0)$ , and $(0,0,1)$ pairing generically also have four line nodes.\",\n \"If TRS is broken, the states with $(1,i,0)$ and $(1,\\\\omega ,\\\\omega ^2)$ where $\\\\omega = e^{2\\\\pi i/3}$ are plausible [31], [26], [13].\",\n \"The $(1,i,0)$ state has IP nodes where both the real and the imaginary part of $c_0 (\\\\mathbf {k})\\\\, f_0(\\\\mathbf {k}) - \\\\vec{c}(\\\\mathbf {k}) \\\\cdot \\\\vec{f}(\\\\mathbf {k})$ vanish.\",\n \"This leads to 18 point nodes in the $\\\\langle 001\\\\rangle $ , $\\\\langle 101\\\\rangle $ , $\\\\langle 111\\\\rangle $ directions outside of the $k_z=0$ plane, and one line node in the $k_z=0$ plane.\",\n \"Compare the $T_{2g+}$ , $(1,i,0)$ pairing state for the $j=3/2$ example [1], [2], where we found two point nodes and one line node in the $k_z=0$ plane.\",\n \"Figure: Zeros of real (blue) and imaginary (red) parts of 〈c,f〉\\\\langle c,f\\\\rangle for the T 1g+ T_{1g+} pairing state with order parameter 1,ω,ω 2 1,\\\\omega ,\\\\omega ^2, with ω=e 2πi/3 \\\\omega =e^{2\\\\pi i/3}, as functions of the spherical polar angles θ\\\\theta , φ\\\\phi of 𝐤\\\\mathbf {k}.\",\n \"Lowest-order polynomial basis functions and the parameter values |𝐤|=1|\\\\mathbf {k}|=1, δ ˜ i,E g+ /δ ˜ i,0 =0.5\\\\tilde{\\\\delta }_{i,E_{g+}}/\\\\tilde{\\\\delta }_{i,0}=0.5, and δ ˜ i,T 2g+ /δ ˜ i,0 =0.3\\\\tilde{\\\\delta }_{i,T_{2g+}}/\\\\tilde{\\\\delta }_{i,0}=0.3 for i=x,y,zi=x,y,z have been used.For the $(1,\\\\omega ,\\\\omega ^2)$ state, there are point nodes where both the real and the imaginary part vanish.\",\n \"Figure REF illustrates the zeros of the real and imaginary parts for a typical parameter set.\",\n \"For generic parameters, there are point nodes in the 26 cubic high-symmetry directions.\",\n \"Six of these are special in that either the zero contours of the real and imaginary part are cotangent or in that the imaginary zero contour has a self crossing.\",\n \"In these cases, the quasiparticle dispersion close to the point node is linear in all directions except along a single axis, where it is quadratic to leading order.\",\n \"The other 20 point nodes show linear dispersion.\",\n \"For noninfinitesimal pairing, we expect the nodes to be inflated.\",\n \"We here only consider the $(1,i,0)$ state.\",\n \"We write the pairing amplitudes as $\\\\delta _{x,00} = \\\\delta _{00}$ , $\\\\delta _{y,00} = i\\\\delta _{00}$ , and $\\\\delta _{z,00} = 0$ , etc.\",\n \"The superconducting form factors then read as $f_0(\\\\mathbf {k}) &= \\\\delta _{00}\\\\, \\\\big [ k_y k_z (k_y^2-k_z^2)+ i\\\\, k_z k_x (k_z^2-k_x^2) \\\\big ] , \\\\\\\\f_1(\\\\mathbf {k}) &= \\\\delta _{10}\\\\, \\\\big [ k_y k_z (k_y^2-k_z^2)+ i\\\\, k_z k_x (k_z^2-k_x^2) \\\\big ] , \\\\\\\\f_2(\\\\mathbf {k}) &= \\\\delta _{30}\\\\, \\\\big [ k_y k_z (k_y^2-k_z^2)+ i\\\\, k_z k_x (k_z^2-k_x^2) \\\\big ] , \\\\\\\\f_3(\\\\mathbf {k}) &= \\\\delta _{A_{1g+}} + \\\\frac{\\\\delta _{E_{g+}}}{3}\\\\, (2k_x^2-k_y^2-k_z^2) \\\\nonumber \\\\\\\\&\\\\quad {}+ i\\\\, \\\\delta _{T_{1g+}}\\\\, k_x k_y (k_x^2-k_y^2)+ i\\\\, \\\\delta _{T_{2g+}}\\\\, k_x k_y , \\\\\\\\f_4(\\\\mathbf {k}) &= i\\\\delta _{A_{1g+}} + \\\\frac{i\\\\,\\\\delta _{E_{g+}}}{3}\\\\, (2k_y^2-k_z^2-k_x^2) \\\\nonumber \\\\\\\\&\\\\quad {}- \\\\delta _{T_{1g+}}\\\\, k_x k_y (k_x^2-k_y^2) + \\\\delta _{T_{2g+}}\\\\, k_x k_y , \\\\\\\\f_5(\\\\mathbf {k}) &= \\\\delta _{T_{1g+}}\\\\, \\\\big [ k_z k_x (k_z^2-k_x^2) - i\\\\,k_y k_z (k_y^2-k_z^2) \\\\big ]\\\\nonumber \\\\\\\\&\\\\quad {}+ \\\\delta _{T_{2g+}}\\\\, \\\\big [ k_z k_x + i\\\\, k_y k_z \\\\big ] .$ Equation (REF ) shows that $\\\\langle c,f^1\\\\rangle = \\\\langle c,f^2\\\\rangle = 0$ remains valid in the radial direction through all point nodes.\",\n \"To go on, we have to distinguish between the inequivalent point nodes.\",\n \"In the $[001]$ direction, we have $f_0(\\\\mathbf {k}) &= f_1(\\\\mathbf {k}) = f_2(\\\\mathbf {k}) = f_5(\\\\mathbf {k}) = 0 , \\\\\\\\f_3(\\\\mathbf {k}) &= \\\\delta _{A_{1g+}} - \\\\frac{\\\\delta _{E_{g+}}}{3}\\\\, k^2 , \\\\\\\\f_4(\\\\mathbf {k}) &= i\\\\delta _{A_{1g+}} - \\\\frac{i\\\\delta _{E_{g+}}}{3}\\\\, k^2 .$ We find $\\\\langle f^1,f^2\\\\rangle = 0$ and $\\\\langle f^1,f^1\\\\rangle = \\\\langle f^2,f^2\\\\rangle = -(\\\\delta _{A_{1g+}}-\\\\delta _{E_{g+}}k^2/3)^2$ and thus for the Pfaffian $P(\\\\mathbf {k}) = \\\\langle c,c\\\\rangle \\\\left[ \\\\langle c,c\\\\rangle + 4 \\\\left( \\\\delta _{A_{1g+}} - \\\\frac{\\\\delta _{E_{g+}}}{3}\\\\, k^2 \\\\right)^{\\\\!2} \\\\right] .$ The first factor changes sign at the normal-state Fermi surface.\",\n \"The second is generically nonzero there and thus the Pfaffian also changes sign at the normal-state Fermi surface.\",\n \"Hence, the point nodes in the $\\\\langle 001\\\\rangle $ directions are inflated and touch the normal-state Fermi surface at the IP point nodes.\",\n \"In the $[101]$ direction, we have $\\\\mathbf {k}=k\\\\,(1,0,1)/\\\\sqrt{2}$ and $f_0(\\\\mathbf {k}) &= f_1(\\\\mathbf {k}) = f_2(\\\\mathbf {k}) = 0 , \\\\\\\\f_3(\\\\mathbf {k}) &= \\\\delta _{A_{1g+}} + \\\\frac{\\\\delta _{E_{g+}}}{6}\\\\, k^2 , \\\\\\\\f_4(\\\\mathbf {k}) &= i\\\\delta _{A_{1g+}} - \\\\frac{i\\\\delta _{E_{g+}}}{3}\\\\, k^2 , \\\\\\\\f_5(\\\\mathbf {k}) &= \\\\frac{\\\\delta _{T_{2g+}}}{2}\\\\, k^2 .$ This implies that $\\\\langle f^1,f^2\\\\rangle = 0$ but $\\\\langle f^1,f^1\\\\rangle $ and $\\\\langle f^2,f^2\\\\rangle $ are generally unequal.\",\n \"The Pfaffian thus reads as $P(\\\\mathbf {k})= \\\\left[ \\\\langle c,c\\\\rangle - \\\\langle f^1,f^1\\\\rangle - \\\\langle f^2,f^2\\\\rangle \\\\right]^2- 4\\\\, \\\\langle f^1,f^1\\\\rangle \\\\langle f^2,f^2\\\\rangle .$ The first term goes to zero on a renormalized Fermi surface.\",\n \"The second term $- 4\\\\, &\\\\langle f^1,f^1\\\\rangle \\\\langle f^2,f^2\\\\rangle = -4 \\\\left( \\\\delta _{A_{1g+}} - \\\\frac{\\\\delta _{E_{g+}}}{3}\\\\, k^2 \\\\right)^{\\\\!2} \\\\nonumber \\\\\\\\&{}\\\\times \\\\left[ \\\\left( \\\\delta _{A_{1g+}} + \\\\frac{\\\\delta _{E_{g+}}}{6}\\\\, k^2 \\\\right)^{\\\\!2}+ \\\\frac{\\\\delta _{T_{2g+}}^2}{4}\\\\, k^4 \\\\right]$ generically is strictly negative.\",\n \"Then, the Pfaffian becomes negative and we find a BFS for a range of $k$ values in the vicinity of but not usually touching the normal-state Fermi surface.\",\n \"The Pfaffian is identical for the directions $[\\\\bar{1}01]$ , $[011]$ , $[0\\\\bar{1}1]$ , and their negatives.\",\n \"Finally, along $[111]$ , we have $\\\\mathbf {k}=k\\\\,(1,1,1)/\\\\sqrt{3}$ and $f_0(\\\\mathbf {k}) &= f_1(\\\\mathbf {k}) = f_2(\\\\mathbf {k}) = 0 , \\\\\\\\f_3(\\\\mathbf {k}) &= \\\\delta _{A_{1g+}} + \\\\frac{i\\\\delta _{T_{2g+}}}{3}\\\\, k^2 , \\\\\\\\f_4(\\\\mathbf {k}) &= i\\\\delta _{A_{1g+}} + \\\\frac{\\\\delta _{T_{2g+}}}{3}\\\\, k^2 , \\\\\\\\f_5(\\\\mathbf {k}) &= \\\\frac{1+i}{3}\\\\, \\\\delta _{T_{2g+}}\\\\, k^2 .$ We thus obtain $\\\\langle f^1,f^2\\\\rangle = -\\\\frac{1}{3}\\\\, \\\\delta _{T_{2g+}} k^2 \\\\left( 2\\\\, \\\\delta _{A_{1g+}}+ \\\\frac{1}{3}\\\\, \\\\delta _{T_{2g+}} k^2 \\\\right)$ and $\\\\langle f^1,f^1\\\\rangle = \\\\langle f^2,f^2\\\\rangle = - \\\\left(\\\\delta _{A_{1g+}}^2 + \\\\frac{2}{9}\\\\, \\\\delta _{T_{2g+}}^2 k^4 \\\\right) .$ The Pfaffian is $P(\\\\mathbf {k}) &= \\\\left[ \\\\langle c,c\\\\rangle - \\\\langle f^1,f^1\\\\rangle - \\\\langle f^2,f^2\\\\rangle \\\\right]^2 \\\\nonumber \\\\\\\\&\\\\quad {} + 4 \\\\left[ \\\\langle f^1,f^2\\\\rangle ^2 - \\\\langle f^1,f^1\\\\rangle \\\\langle f^2,f^2\\\\rangle \\\\right] ,$ wherein the second term evaluates to $4 &\\\\left[ \\\\langle f^1,f^2\\\\rangle ^2 - \\\\langle f^1,f^1\\\\rangle \\\\langle f^2,f^2\\\\rangle \\\\right] \\\\nonumber \\\\\\\\&= -\\\\frac{4}{27} \\\\left( 3\\\\, \\\\delta _{A_{1g+}} - \\\\delta _{T_{2g+}}\\\\, k^2 \\\\right)^2 \\\\nonumber \\\\\\\\&\\\\quad {}\\\\times \\\\left[ 2\\\\, \\\\delta _{A_{1g+}}^2 + ( \\\\delta _{A_{1g+}} + \\\\delta _{T_{2g+}}\\\\, k^2 )^2 \\\\right] .$ Since this is generally negative we also expect BFSs that do not touch the normal-state Fermi surface in the $\\\\langle 111\\\\rangle $ directions.\",\n \"For the equatorial line node, we take $\\\\mathbf {k}=(k_x,k_y,0)$ .\",\n \"The superconducting form factors are $f_0(\\\\mathbf {k}) &= f_1(\\\\mathbf {k}) = f_2(\\\\mathbf {k}) = f_5(\\\\mathbf {k}) = 0 , \\\\\\\\f_3(\\\\mathbf {k}) &= \\\\delta _{A_{1g+}} + \\\\frac{\\\\delta _{E_{g+}}}{3}\\\\, (2k_x^2-k_y^2) \\\\nonumber \\\\\\\\&\\\\quad {}+ i\\\\, \\\\delta _{T_{1g+}}\\\\, k_x k_y (k_x^2-k_y^2)+ i\\\\, \\\\delta _{T_{2g+}}\\\\, k_x k_y , \\\\\\\\f_4(\\\\mathbf {k}) &= i\\\\delta _{A_{1g+}} + \\\\frac{i\\\\,\\\\delta _{E_{g+}}}{3}\\\\, (2k_y^2-k_x^2) \\\\nonumber \\\\\\\\&\\\\quad {}- \\\\delta _{T_{1g+}}\\\\, k_x k_y (k_x^2-k_y^2) + \\\\delta _{T_{2g+}}\\\\, k_x k_y .$ This gives $\\\\langle f^1,&f^2\\\\rangle = -\\\\frac{1}{3}\\\\, k_x k_y \\\\big [ 6\\\\, \\\\delta _{A_{1g+}} \\\\delta _{T_{2g+}} \\\\nonumber \\\\\\\\&{}+ 3\\\\, \\\\delta _{E_{g+}} \\\\delta _{T_{1g+}} (k_x^2-k_y^2)^2+ \\\\delta _{E_{g+}} \\\\delta _{T_{2g+}} (k_x^2+k_y^2) \\\\big ] , \\\\\\\\\\\\langle f^1,&f^1\\\\rangle = -\\\\frac{1}{9} \\\\left[ 3 \\\\delta _{A_{1g+}}+ \\\\delta _{E_{g+}} (2k_x^2 - k_y^2) \\\\right]^2 \\\\nonumber \\\\\\\\&{}- \\\\left[ \\\\delta _{T_{1g+}} (k_x^2-k_y^2) - \\\\delta _{T_{2g+}} \\\\right]^2 k_x^2 k_y^2 , \\\\\\\\\\\\langle f^2,&f^2\\\\rangle = -\\\\frac{1}{9} \\\\left[ 3 \\\\delta _{A_{1g+}}+ \\\\delta _{E_{g+}} (2k_y^2 - k_x^2) \\\\right]^2 \\\\nonumber \\\\\\\\&{}- \\\\left[ \\\\delta _{T_{1g+}} (k_x^2-k_y^2) + \\\\delta _{T_{2g+}} \\\\right]^2 k_x^2 k_y^2 .$ The Pfaffian again has the form of Eq.\",\n \"(REF ), where the second term $4 &\\\\left[ \\\\langle f^1,f^2\\\\rangle ^2 - \\\\langle f^1,f^1\\\\rangle \\\\langle f^2,f^2\\\\rangle \\\\right] \\\\nonumber \\\\\\\\&= -\\\\frac{4}{81}\\\\, \\\\Big [ 9\\\\, \\\\delta _{A_{1g+}}^2+ 2\\\\, \\\\delta _{A_{1g+}} \\\\delta _{E_{g+}} (k_x^2 + k_y^2) \\\\nonumber \\\\\\\\&\\\\quad {}+ \\\\delta _{E_{g+}}^2 (2k_x^2-k_y^2)(2k_y^2-k_x^2) \\\\nonumber \\\\\\\\&\\\\quad {}+ 9\\\\, \\\\delta _{T_{1g+}}^2\\\\, k_x^2 k_y^2 (k_x^2 - k_y^2)^2- 9\\\\, \\\\delta _{T_{2g+}}^2\\\\, k_x^2 k_y^2 \\\\Big ]^2$ is non-positive and generically negative so that the line node is inflated by noninfinitesimal pairing.\",\n \"The resulting BFS is toroidal but may be pinched off, i.e., have self crossings, at some momenta, but these self crossings would be accidental.\",\n \"Since the first term in Eq.\",\n \"(REF ) becomes zero close to but not at the normal-state Fermi surface the BFS generically does not touch the normal-state Fermi surface.\",\n \"In summary, the point and line nodes of the $T_{1g+}$ , $(1,i,0)$ pairing state are all inflated by noninfinitesimal pairing.\",\n \"Only the inflated point nodes on the $k_z$ -axis touch the normal-state Fermi surface, the other pockets are shifted away from it and could thus be annihilated at strong coupling.\"\n ],\n [\n \"Two orbitals of opposite parity\",\n \"To have a single even and a single odd orbital per unit cell for the point group $O_h$ , the first must transform like a one-dimensional $g$ irrep and the second like a one-dimensional $u$ irrep.\",\n \"The most natural possibilities are $A_{1g}$ (s-orbital) and $A_{2u}$ ($f_{xyz}$ -orbital).\",\n \"Now the inversion or parity matrix is nontrivial: $U_P = \\\\sigma _3 \\\\otimes \\\\sigma _0 .$ Moreover, the $f_{xyz}$ -orbital is odd under fourfold rotations but even under threefold rotations.\",\n \"Models of this symmetry have been analyzed in the context of Dirac and Weyl semimetals [32].\",\n \"The unitary part of time reversal is $U_T = \\\\sigma _0 \\\\otimes i\\\\sigma _2$ since the orbitals are invariant.\",\n \"The basis matrices can be written as Kronecker products, which transform according to the irreps as summarized in Table REF .\",\n \"Compared to the example of two s-orbitals, Table REF , the Pauli matrices $\\\\sigma _1$ and $\\\\sigma _2$ for the orbital degree of freedom are now odd under inversion.\",\n \"The new element is that $u$ irreps occur, due to the nontrivial parity operator.\",\n \"This provides additional possibilities for the products of irreps of $\\\\mathbf {k}$ -dependent form factors and basis matrices.\",\n \"Table REF shows all relevant reductions of product representations.\",\n \"Table: Basis matrices on the internal Hilbert space for the case of one s-orbital and one f xyz f_{xyz}-orbital and point group O h O_h.\",\n \"The basis matrices are irreducible tensor operators of the irreps listed in the second column.\",\n \"For multidimensional irreps, the states transforming into each other under point-group operations are distinguished by the index in the third column.Table: Reduction of product representations of the allowed irreps of 𝐤\\\\mathbf {k}-dependent form factors (rows) and basis matrices h ν h_\\\\nu (columns) for one s-orbital and one f xyz f_{xyz}-orbital.\",\n \"For the form factors, the minimum order of polynomial basis functions is given in the second column.\",\n \"“∘\\\\circ ” indicates products that are forbidden since they violate fermionic antisymmetry.The normal-state Hamiltonian is a linear combination of all basis matrices that allow to form products with full $A_{1g+}$ symmetry, marked in bold face in Table REF .\",\n \"These basis matrices, together with their irreps, are $h_0 &\\\\equiv \\\\sigma _0 \\\\otimes \\\\sigma _0 & A_{1g+}, \\\\\\\\h_1 &\\\\equiv \\\\sigma _3 \\\\otimes \\\\sigma _0 & A_{1g+}, \\\\\\\\h_2 &\\\\equiv \\\\sigma _2 \\\\otimes \\\\sigma _0 & A_{2u-}, \\\\\\\\h_3 &\\\\equiv \\\\sigma _1 \\\\otimes \\\\sigma _1 & T_{2u-}, \\\\\\\\h_4 &\\\\equiv \\\\sigma _1 \\\\otimes \\\\sigma _2 & T_{2u-}, \\\\\\\\h_5 &\\\\equiv \\\\sigma _1 \\\\otimes \\\\sigma _3 & T_{2u-} .$ There are thus six matrices that satisfy the same algebra as for the case of two s-orbitals; see Appendices and .\",\n \"The normal-state Hamiltonian reads as $&H_N(\\\\mathbf {k}) \\\\nonumber \\\\\\\\&\\\\:= c_{00}(\\\\mathbf {k})\\\\, \\\\sigma _0 \\\\otimes \\\\sigma _0+ c_{30}(\\\\mathbf {k})\\\\, \\\\sigma _3 \\\\otimes \\\\sigma _0+ c_{20}(\\\\mathbf {k})\\\\, \\\\sigma _2 \\\\otimes \\\\sigma _0 \\\\nonumber \\\\\\\\&\\\\:\\\\quad {} + c_{11}(\\\\mathbf {k})\\\\, \\\\sigma _1 \\\\otimes \\\\sigma _1+ c_{12}(\\\\mathbf {k})\\\\, \\\\sigma _1 \\\\otimes \\\\sigma _2+ c_{13}(\\\\mathbf {k})\\\\, \\\\sigma _1 \\\\otimes \\\\sigma _3 ,$ where the leading polynomial forms are $c_{00}(\\\\mathbf {k}) &= c_{00}^{(0)} + c_{00}^{(2)}\\\\, (k_x^2 + k_y^2 + k_z^2) + \\\\ldots , \\\\\\\\c_{30}(\\\\mathbf {k}) &= c_{30}^{(0)} + c_{30}^{(2)}\\\\, (k_x^2 + k_y^2 + k_z^2) + \\\\ldots , \\\\\\\\c_{20}(\\\\mathbf {k}) &= c_{20}^{(3)}\\\\, k_x k_y k_z + \\\\ldots , \\\\\\\\c_{11}(\\\\mathbf {k}) &= c_1^{(3)}\\\\, k_x (k_y^2-k_z^2) + \\\\ldots , \\\\\\\\c_{12}(\\\\mathbf {k}) &= c_1^{(3)}\\\\, k_y (k_z^2-k_x^2) + \\\\ldots , \\\\\\\\c_{13}(\\\\mathbf {k}) &= c_1^{(3)}\\\\, k_z (k_x^2-k_y^2) + \\\\ldots $ Four of the form factors are odd in momentum.\",\n \"They do not break inversion symmetry since they multiply orbital matrices that are also odd under inversion.\",\n \"Turning to superconducting pairing, it is interesting that local pairing is now either trivial ($A_{1g+}$ ) or has odd parity ($A_{2u+}$ , $T_{2u+}$ ), as seen from the first row of Table REF .\",\n \"Furthermore, for even-parity pairing ($g+$ irreps), only the basis matrices belonging to $A_{1g+}$ , $T_{2u-}$ , and $A_{2u-}$ can occur, i.e., the same matrices $h_0$ , ..., $h_5$ as in $H_N(\\\\mathbf {k})$ .\",\n \"Since $\\\\mathcal {C}P$ squares to $+$ the Hamiltonian can be unitarily transformed into antisymmetric form, guaranteeing the existence of a Pfaffian [1].\",\n \"In the present example, where $U_P = \\\\sigma _3 \\\\otimes \\\\sigma _0$ , the matrix $\\\\Omega $ mediating this transformation reads as $\\\\Omega = \\\\frac{1}{\\\\sqrt{2}} \\\\left(\\\\begin{array}{cc} 1 & 1 \\\\\\\\ i & -i \\\\end{array}\\\\right)\\\\otimes \\\\exp \\\\!\\\\left(-i\\\\,\\\\frac{\\\\pi }{2}\\\\, \\\\frac{\\\\sigma _3}{2}\\\\right) \\\\otimes \\\\sigma _0 .$ This specific form does not affect the eigenvalues, though.\",\n \"Since the algebra of the basis matrices is unchanged, the expressions for the eigenvalues, the Pfaffian, and the condition for IP nodes remain unchanged.\",\n \"In the following, we briefly discuss the $A_{2g+}$ and $E_{g+}$ pairing states and compare them to the case of two s-orbitals.\"\n ],\n [\n \"$A_{2g+}$ pairing\",\n \"$A_{2g+}$ appears in three places in Table REF : (a) $A_{2g+}\\\\otimes A_{1g+}$ , (b) $A_{1u-}\\\\otimes A_{2u-}$ , and (c) $T_{1u-}\\\\otimes T_{2u-}$ .\",\n \"Note that the minimum orders of form factors are (a) 6, (b) 9, and (c) 1 so that one expects that the $T_{1u-}\\\\otimes T_{2u-}$ contribution typically dominates.\",\n \"(a) For $A_{2g+}\\\\otimes A_{1g+}$ : $D_{A_{2g+}}(\\\\mathbf {k})= \\\\delta _{00} d_{00}(\\\\mathbf {k})\\\\, \\\\sigma _{0} \\\\otimes \\\\sigma _{0}+ \\\\delta _{30} d_{30}(\\\\mathbf {k})\\\\, \\\\sigma _{3} \\\\otimes \\\\sigma _{0} .$ To the leading order, $d_{m0}(\\\\mathbf {k})$ takes the form $d_{m0}(\\\\mathbf {k}) \\\\cong d_{m0}^{(6)}\\\\, \\\\big [ k_{x}^4 (k_{y}^2-k_{z}^2)+ k_{y}^4 (k_{z}^2-k_{x}^2) + k_{z}^4 (k_{x}^2-k_{y}^2) \\\\big ] .$ $d_{m0}^{(6)}$ is set to unity.\",\n \"(b) For $A_{1u-}\\\\otimes A_{2u-}$ : $D_{A_{1u-}}(\\\\mathbf {k}) = \\\\delta _{A_{1u-}} d_{A_{1u-}}(\\\\mathbf {k})\\\\, \\\\sigma _{2} \\\\otimes \\\\sigma _{0} ,$ with $d_{A_{1u-}}(\\\\mathbf {k})$ to the leading order given by $d_{A_{1u-}}(\\\\mathbf {k}) &\\\\cong d_{A_{1u-}}^{(9)}\\\\, k_{x} k_{y} k_{z}\\\\, \\\\big [k_{x}^4 (k_{y}^2-k_{z}^2) + k_{y}^4 (k_{z}^2-k_{x}^2) \\\\nonumber \\\\\\\\&\\\\quad {}+ k_{z}^4 (k_{x}^2-k_{y}^2) \\\\big ] .$ We set $d_{A_{1u-}}^{(9)}$ to unity.\",\n \"(c) For $T_{1u-}\\\\otimes T_{2u-}$ : $D_{T_{1u-}}(\\\\mathbf {k}) \\\\cong \\\\delta _{T_{1u-}} ( k_{x}\\\\, \\\\sigma _{1}\\\\otimes \\\\sigma _{1}+ k_{y}\\\\, \\\\sigma _{1}\\\\otimes \\\\sigma _{2} + k_{z}\\\\, \\\\sigma _{1}\\\\otimes \\\\sigma _{3} )$ to leading order.\",\n \"The components are assigned such that the whole term $D_{T_{1u-}}(\\\\mathbf {k})$ changes sign under any four-fold rotation [40].\",\n \"Table: Leading-order polynomial forms of the form factors f n (𝐤)f_n(\\\\mathbf {k}) describing A 2g+ A_{2g+} pairing for a model with one s-orbital and one f xyz f_{xyz}-orbital.Table: Leading-order polynomial forms of the products c n (𝐤)f n (𝐤)c_n(\\\\mathbf {k}) f_n(\\\\mathbf {k}) of form factors describing A 2g+ A_{2g+} pairing for a model with one s-orbital and one f xyz f_{xyz}-orbital.\",\n \"The amplitudes of the leading terms in c n (𝐤)c_n(\\\\mathbf {k}) have been absorbed into new pairing amplitudes marked by a tilde.The resulting superconducting form factors $f_{n}$ and the products $c_{n} f_{n}$ , which are required to determine the IP nodes, are listed in Tables REF and REF , respectively, to the leading order.\",\n \"The condition for IP nodes reads as $&c_0(\\\\mathbf {k})\\\\, f_0(\\\\mathbf {k}) - \\\\vec{c}(\\\\mathbf {k}) \\\\cdot \\\\vec{f}(\\\\mathbf {k})= \\\\big ( \\\\tilde{\\\\delta }_{00} - \\\\tilde{\\\\delta }_{30} - \\\\tilde{\\\\delta }_{A_{1u-}} k_{x}^2 k_{y}^2 k_{z}^2 \\\\big ) \\\\nonumber \\\\\\\\&\\\\qquad {}\\\\times \\\\big [ k_{x}^4 (k_{y}^2-k_{z}^2) + k_{y}^4 (k_{z}^2-k_{x}^2)+ k_{z}^4 (k_{x}^2-k_{y}^2) \\\\big ] = 0 .$ Contribution (c) has dropped out.\",\n \"This is again an artifact of using the same basis functions for $c_n$ and $f_n$ .\",\n \"Going beyond leading order, a contribution remains but does not affect the conclusions.\",\n \"For example, the $T_{1u-}$ basis functions $k_x^3$ , $k_y^3$ , $k_z^3$ generate another term proportional to $k_{x}^4 (k_{y}^2-k_{z}^2) + k_{y}^4 (k_{z}^2-k_{x}^2) + k_{z}^4 (k_{x}^2-k_{y}^2)$ .\",\n \"Equation (REF ) is satisfied whenever any two of the components of $\\\\mathbf {k}$ are equal.\",\n \"Thus there are line nodes in the $\\\\lbrace 110\\\\rbrace $ planes for infinitesimal pairing.\",\n \"Form factors in the $(110)$ plane read as $f_0(\\\\mathbf {k}) &= f_1(\\\\mathbf {k}) = f_2(\\\\mathbf {k}) = 0 , \\\\\\\\f_3(\\\\mathbf {k}) &= f_4(\\\\mathbf {k}) = \\\\delta _{T_{1u-}}k_{x} , \\\\\\\\f_5(\\\\mathbf {k}) &= \\\\delta _{T_{1u-}}k_{z} ,$ which gives $\\\\langle f^1 ,f^1 \\\\rangle &= -(\\\\mathop {\\\\textrm {Re}}\\\\delta _{T_{1u-}})^2 (2 k_{x}^2+k_{z}^2) , \\\\\\\\\\\\langle f^2 ,f^2 \\\\rangle &= -(\\\\mathop {\\\\textrm {Im}}\\\\delta _{T_{1u-}})^2 (2 k_{x}^2+k_{z}^2) , \\\\\\\\\\\\langle f^1 ,f^2 \\\\rangle &= -\\\\mathop {\\\\textrm {Re}}\\\\delta _{T_{1u-}}\\\\, \\\\mathop {\\\\textrm {Im}}\\\\delta _{T_{1u-}} (2 k_{x}^2+k_{z}^2)$ and also $\\\\langle c ,f^1 \\\\rangle =\\\\langle c ,f^2 \\\\rangle =0$ .\",\n \"In this case, the Pfaffian simplifies to the form of Eq.\",\n \"(REF ).\",\n \"The second term $4\\\\,[\\\\langle f^1 ,f^2 \\\\rangle ^2-\\\\langle f^1 ,f^1 \\\\rangle \\\\langle f^2 ,f^2 \\\\rangle ]$ vanishes, which implies that there is no inflation of the line nodes in the mirror plane.\",\n \"The vanishing can be attributed to the fact that in the $\\\\lbrace 110\\\\rbrace $ planes, only a single amplitude $\\\\delta _{T_{1u-}}$ leads to a superconducting gap, the phase of which can always be chosen real so that the TRS breaking is irrelevant.\",\n \"On the other hand, $\\\\langle f^1 ,f^1 \\\\rangle +\\\\langle f^2 ,f^2 \\\\rangle $ is nonzero so that the nodes are shifted.\",\n \"This is analogous to the case of two s-orbitals.\"\n ],\n [\n \"$E_{g+}$ pairing\",\n \"In Table REF , pairing with $E_{g+}$ symmetry occurs in (a) $E_{g+}\\\\otimes A_{1g+}$ , (b) $E_{u-}\\\\otimes A_{2u-}$ , (c) $T_{1u-}\\\\otimes T_{2u-}$ , and (d) $T_{2u-}\\\\otimes T_{2u-}$ .\",\n \"The matrix-valued basis functions are given in the following to leading order only.\",\n \"(a) For $E_{g+}\\\\otimes A_{1g+}$ : $D_{x^2-y^2,00}(\\\\mathbf {k}) &\\\\cong (k_x^2-k_y^2)\\\\, \\\\sigma _0 \\\\otimes \\\\sigma _0 , \\\\\\\\D_{3z^2-r^2,00}(\\\\mathbf {k}) &\\\\cong \\\\frac{1}{\\\\sqrt{3}}\\\\, (2k_z^2-k_x^2-k_y^2)\\\\, \\\\sigma _0 \\\\otimes \\\\sigma _0 , \\\\\\\\D_{x^2-y^2,30}(\\\\mathbf {k}) &\\\\cong (k_x^2-k_y^2)\\\\, \\\\sigma _3 \\\\otimes \\\\sigma _0 , \\\\\\\\D_{3z^2-r^2,30}(\\\\mathbf {k}) &\\\\cong \\\\frac{1}{\\\\sqrt{3}}\\\\, (2k_z^2-k_x^2-k_y^2)\\\\, \\\\sigma _3 \\\\otimes \\\\sigma _0 .$ (b) For $E_{u-}\\\\otimes A_{2u-}$ : $D_{x^2-y^2,20}(\\\\mathbf {k}) &\\\\cong k_xk_yk_z (k_x^2-k_y^2)\\\\, \\\\sigma _2 \\\\otimes \\\\sigma _0 , \\\\\\\\D_{3z^2-r^2,20}(\\\\mathbf {k}) &\\\\cong \\\\frac{1}{\\\\sqrt{3}}\\\\, k_xk_yk_z (2k_z^2-k_x^2-k_y^2)\\\\,\\\\sigma _2 \\\\otimes \\\\sigma _0 .$ Note that $k_xk_yk_z\\\\, \\\\sigma _2\\\\times \\\\sigma _0$ is invariant under $O_h$ .\",\n \"(c) For $T_{1u-}\\\\otimes T_{2u-}$ : $D_{x^2-y^2,T_{1u-}}(\\\\mathbf {k}) &\\\\cong \\\\frac{1}{\\\\sqrt{3}}\\\\, (-k_x\\\\, \\\\sigma _1 \\\\otimes \\\\sigma _1- k_y\\\\, \\\\sigma _1 \\\\otimes \\\\sigma _2 \\\\nonumber \\\\\\\\&\\\\quad {}+ 2\\\\, k_z\\\\, \\\\sigma _1 \\\\otimes \\\\sigma _3 ) , \\\\\\\\D_{3z^2-r^2,T_{1u-}}(\\\\mathbf {k}) &\\\\cong - (k_x\\\\, \\\\sigma _1 \\\\otimes \\\\sigma _1- k_y\\\\, \\\\sigma _1 \\\\otimes \\\\sigma _2) .$ For this and the following contribution, the transformation properties under three- and four-fold rotations have been used to determine the two components.\",\n \"(d) For $T_{2u-}\\\\otimes T_{2u-}$ : $&D_{x^2-y^2,T_{2u-}}(\\\\mathbf {k}) \\\\cong \\\\big [ k_x (k_y^2-k_z^2)\\\\, \\\\sigma _1\\\\otimes \\\\sigma _1 \\\\nonumber \\\\\\\\&\\\\quad {}- k_y (k_z^2-k_x^2)\\\\, \\\\sigma _1 \\\\otimes \\\\sigma _2 \\\\big ] , \\\\\\\\&D_{3z^2-r^2,T_{2u-}}(\\\\mathbf {k}) \\\\cong \\\\frac{1}{\\\\sqrt{3}}\\\\, \\\\big [ {-}k_x(k_y^2-k_z^2)\\\\,\\\\sigma _1 \\\\otimes \\\\sigma _1 \\\\nonumber \\\\\\\\&\\\\quad {}- k_y (k_z^2-k_x^2)\\\\, \\\\sigma _1 \\\\otimes \\\\sigma _2+ 2\\\\, k_z (k_x^2-k_y^2)\\\\, \\\\sigma _1 \\\\otimes \\\\sigma _3 \\\\big ] .$ The resulting superconducting form factors $f_n(\\\\mathbf {k})$ are given in Table REF and the products $c_n(\\\\mathbf {k}) f_n(\\\\mathbf {k})$ appearing in the condition (REF ) for IP nodes are shown in Table REF .\",\n \"The total contribution to $\\\\langle c,f\\\\rangle $ from $x^2-y^2$ basis functions (with amplitudes $\\\\tilde{\\\\delta }^1_{\\\\cdots }$ ) has two symmetry-imposed first-order line nodes at $k_y=\\\\pm k_x$ .\",\n \"In a time-reversal-symmetric state, the inclusion of $3z^2-r^2$ basis functions (with amplitudes $\\\\tilde{\\\\delta }^2_{\\\\cdots }$ ) generically leads to two line nodes elsewhere on the normal-state Fermi surface.\",\n \"The nodes must intersect with the $\\\\langle 111\\\\rangle $ axes, though, since there the full expression vanishes.\",\n \"TRS-breaking states generically lead to point nodes in the $\\\\langle 111\\\\rangle $ directions.\",\n \"This is for example the case for the generalized $(1,i)$ -type state.\",\n \"These point nodes are solely determined by symmetry and therefore agree with the case of two s-orbitals.\",\n \"Table: Leading-order polynomial forms of the form factors f n (𝐤)f_n(\\\\mathbf {k}) describing E g+ E_{g+} pairing for a model with one s-orbital and one f xyz f_{xyz}-orbital.Table: Leading-order polynomial forms of the products c n (𝐤)f n (𝐤)c_n(\\\\mathbf {k}) f_n(\\\\mathbf {k}) of form factors describing E g+ E_{g+} pairing for a model with one s-orbital and one f xyz f_{xyz}-orbital.\",\n \"The amplitudes of the leading terms in c n (𝐤)c_n(\\\\mathbf {k}) have been absorbed into new pairing amplitudes marked by a tilde.For noninfinitesimal pairing that breaks TRS, the point nodes are inflated.\",\n \"This is seen by considering the Pfaffian on the high-symmetry axis $\\\\mathbf {k}=k\\\\,(1,1,1)/\\\\sqrt{3}$ through a IP point node.\",\n \"On this axis, we have, to leading order, $f_0(\\\\mathbf {k}) &= f_1(\\\\mathbf {k}) = f_2(\\\\mathbf {k}) = 0 , \\\\\\\\f_3(\\\\mathbf {k}) &= -\\\\frac{\\\\delta ^1_{T_{1u-}}}{3}\\\\, k- \\\\frac{\\\\delta ^2_{T_{1u-}}}{\\\\sqrt{3}}\\\\, k , \\\\\\\\f_4(\\\\mathbf {k}) &= -\\\\frac{\\\\delta ^1_{T_{1u-}}}{3}\\\\, k+ \\\\frac{\\\\delta ^2_{T_{1u-}}}{\\\\sqrt{3}}\\\\, k , \\\\\\\\f_5(\\\\mathbf {k}) &= \\\\frac{2\\\\, \\\\delta ^1_{T_{1u-}}}{3}\\\\, k .$ For the $(1,i)$ pairing state with $\\\\delta ^2_{T_{1u-}} = i\\\\delta ^1_{T_{1u-}}$ , $\\\\delta ^1_{T_{1u-}}\\\\in \\\\mathbb {R}$ , we find $\\\\langle f^1,f^1\\\\rangle &= \\\\langle f^2,f^2\\\\rangle = -\\\\frac{2k^2}{3}\\\\, \\\\big (\\\\delta ^1_{T_{1u-}}\\\\big )^2 , \\\\\\\\\\\\langle f^1,f^2\\\\rangle &= 0 .$ On the other hand, the normal-state form factors are, to leading order, $c_0(\\\\mathbf {k}) &= c_{00}^{(0)} , \\\\\\\\c_1(\\\\mathbf {k}) &= c_{30}^{(0)} , \\\\\\\\c_2(\\\\mathbf {k}) &= \\\\frac{1}{3\\\\sqrt{3}}\\\\, c_{20}^{(3)}\\\\, k^3 , \\\\\\\\c_3(\\\\mathbf {k}) &= c_4(\\\\mathbf {k}) = c_5(\\\\mathbf {k}) = 0 .$ We thus find $\\\\langle c,f^1\\\\rangle = \\\\langle c,f^2\\\\rangle = 0$ and the analysis is analogous to the one for the $(1,i)$ state for two s-orbitals.\",\n \"Hence, we expect BFSs that touch the normal-state Fermi surface at the IP nodes.\"\n ],\n [\n \"Two-side basis: Diamond structure\",\n \"Another origin of internal degrees of freedom is a nontrivial basis of the crystal.\",\n \"This is a good place to consider an example: the diamond structure with one s-orbital per basis site.\",\n \"The space group is 227, belonging to the point group $O_h$ .\",\n \"We write matrices as Kronecker products of a matrix acting on site space and a matrix on spin space.\",\n \"Table: Basis matrices on the internal Hilbert space for the case of s-orbitals forming a diamond structure.\",\n \"The basis matrices are irreducible tensor operators of the irreps listed in the second column.\",\n \"For multidimensional irreps, the states transforming into each other under point-group operations are distinguished by the index in the third column.The parity matrix $U_P = \\\\sigma _1 \\\\otimes \\\\sigma _0$ is nontrivial since inversion interchanges the basis sites.\",\n \"This case has also been analyzed in the context of semimetals [32].\",\n \"Moreover, the fourfold axes also interchange the basis sites, whereas the threefold axes do not.\",\n \"Time reversal is unchanged, $U_T = \\\\sigma _0 \\\\otimes i\\\\sigma _2$ .\",\n \"The basis matrices are listed in Table REF .\",\n \"This is the same scheme as for two orbitals of opposite parity, see Table REF , except that the Pauli matrices $\\\\sigma _1$ and $\\\\sigma _3$ in the first (orbital/site) factor are interchanged.\",\n \"Thus the results for the pairing can be mapped over from Sec.\",\n \"REF without effort.\"\n ],\n [\n \"Effective spin 3/2\",\n \"Here, we consider electrons with effective angular momentum $j=3/2$ .\",\n \"It is of interest to check whether the results obtained for local pairing in such a model [1], [2] are robust under nonlocal pairing and which additional pairing states are allowed for nonlocal pairing.\",\n \"The Hilbert space for $j=3/2$ is four dimensional.\",\n \"In this case, it is useful to express all matrices as polynomials of the standard angular-momentum-$3/2$ matrices $J_x &= \\\\left(\\\\begin{array}{cccc}0 & \\\\sqrt{3}/2 & 0 & 0 \\\\\\\\\\\\sqrt{3}/2 & 0 & 1 & 0 \\\\\\\\0 & 1 & 0 & \\\\sqrt{3}/2 \\\\\\\\0 & 0 & \\\\sqrt{3}/2 & 0\\\\end{array}\\\\right) , \\\\\\\\J_y &= \\\\left(\\\\begin{array}{cccc}0 & -i\\\\, \\\\sqrt{3}/2 & 0 & 0 \\\\\\\\i\\\\, \\\\sqrt{3}/2 & 0 & -i & 0 \\\\\\\\0 & i & 0 & -i\\\\,\\\\sqrt{3}/2 \\\\\\\\0 & 0 & i\\\\,\\\\sqrt{3}/2 & 0\\\\end{array}\\\\right) , \\\\\\\\J_z &= \\\\left(\\\\begin{array}{cccc}3/2 & 0 & 0 & 0 \\\\\\\\0 & 1/2 & 0 & 0 \\\\\\\\0 & 0 & -1/2 & 0 \\\\\\\\0 & 0 & 0 & -3/2\\\\end{array}\\\\right) ,$ and the $4\\\\times 4$ identity matrix $J_0 \\\\equiv $ .\",\n \"Table: Basis matrices on the internal Hilbert space for the case of electrons with angular momentum j=3/2j=3/2.\",\n \"The basis matrices are irreducible tensor operators of the irreps listed in the second column.\",\n \"For multidimensional irreps, the states transforming into each other under point-group operations are distinguished by the index in the third column.The parity matrix is trivial, $U_P = = J_0$ , since the angular momentum is invariant under inversion.\",\n \"The unitary part of the time-reversal operator now reads as $U_T = e^{i J_y \\\\pi }= \\\\begin{pmatrix}0 & 0 & 0 & 1 \\\\\\\\0 & 0 & -1 & 0 \\\\\\\\0 & 1 & 0 & 0 \\\\\\\\-1 & 0 & 0 & 0\\\\end{pmatrix} .$ The 16 basis matrices $h_\\\\nu $ of the space of Hermitian $4\\\\times 4$ matrices are listed in Table REF , together with the corresponding irreps.\",\n \"We normalize the basis matrices in such a way that $\\\\mathop {\\\\textrm {Tr}}h_\\\\nu ^2 = 4$ .\",\n \"Apart from this, the entries in the table follow from known basis functions [33], taking into account that $\\\\mathbf {J}=(J_x,J_y,J_z)$ is even under inversion and odd under time reversal, and symmetrizing products of angular-momentum matrices so as to generate Hermitian matrices [16].\",\n \"A new feature is the presence of basis matrices belonging to the two-dimensional irrep $E_{g+}$ .\",\n \"Table: Reduction of product representations of the allowed irreps of 𝐤\\\\mathbf {k}-dependent form factors (rows) and basis matrices h ν h_\\\\nu (columns) for electrons with angular momentum j=3/2j=3/2.\",\n \"For the form factors, the minimum order of polynomial basis functions is given in the second column.\",\n \"“∘\\\\circ ” indicates products that are forbidden since they violate fermionic antisymmetry.\",\n \"For brevity, the symbols 𝒜≡A 1u+ ⊕E u+ ⊕T 1u+ ⊕T 2u+ \\\\mathcal {A} \\\\equiv A_{1u+} \\\\oplus E_{u+} \\\\oplus T_{1u+} \\\\oplus T_{2u+} and ℬ≡A 2u+ ⊕E u+ ⊕T 1u+ ⊕T 2u+ \\\\mathcal {B} \\\\equiv A_{2u+} \\\\oplus E_{u+} \\\\oplus T_{1u+} \\\\oplus T_{2u+} are used.Table REF shows all relevant reductions of product representations.\",\n \"The normal-state Hamiltonian can only contain the highlighted $A_{1g+}$ combinations and local pairing is only compatible with the first row of the table—this reproduces the known three irreps $A_{1g+}$ , $T_{2g+}$ , and $E_{g+}$ [1], [2].\",\n \"Again, all ten time-reversal-even pairing symmetries can occur and we restrict ourselves to $g+$ irreps (even parity).\",\n \"All of these occur for any of the three irreps $A_{1g+}$ , $T_{2g+}$ , and $E_{g+}$ of basis matrices.\",\n \"The normal-state Hamiltonian $H_N(\\\\mathbf {k})$ is a linear combination of the basis matrices $h_0 &\\\\equiv J_0 & A_{1g+}, \\\\\\\\h_1 &\\\\equiv \\\\frac{1}{\\\\sqrt{3}}\\\\, (J_yJ_z + J_zJ_y) & T_{2g+}, \\\\\\\\h_2 &\\\\equiv \\\\frac{1}{\\\\sqrt{3}}\\\\, (J_zJ_x + J_xJ_z) & T_{2g+}, \\\\\\\\h_3 &\\\\equiv \\\\frac{1}{\\\\sqrt{3}}\\\\, (J_xJ_y + J_yJ_x) & T_{2g+}, \\\\\\\\h_4 &\\\\equiv \\\\frac{1}{\\\\sqrt{3}}\\\\, (J_x^2 - J_y^2) & E_{g+}, \\\\\\\\h_5 &\\\\equiv \\\\frac{1}{3}\\\\, (2J_z^2 - J_x^2 - J_y^2) & E_{g+} ,$ which again satisfy the universal algebra; see Appendices and .\",\n \"The normal-state Hamiltonian contains these matrices with form factors $c_0(\\\\mathbf {k}),\\\\ldots ,c_5(\\\\mathbf {k})$ , which must transform in the same way as $h_0,\\\\ldots ,h_5$ .\"\n ],\n [\n \"$A_{2g+}$ pairing\",\n \"$A_{2g+}$ pairing appears in three places in Table REF : (a) $A_{2g+}\\\\otimes A_{1g+}$ , (b) $E_{g+}\\\\otimes E_{g+}$ , and (c) $T_{1g+}\\\\otimes T_{2g+}$ .\",\n \"This is a potentially interesting pairing state since it is impossible for purely local pairing and it is an example of a nontrivial one-dimensional irrep.\",\n \"In the following, we give the basis functions to the leading order only.\",\n \"(a) For $A_{2g+}\\\\otimes A_{1g+}$ : $D_{A_{2g+}}(\\\\mathbf {k}) &\\\\cong \\\\big [ k_{x}^4 (k_{y}^2-k_{z}^2) + k_{y}^4 (k_{z}^2-k_{x}^2) \\\\nonumber \\\\\\\\&\\\\quad {} + k_{z}^4 (k_{x}^2-k_{y}^2) \\\\big ]\\\\, h_0 .$ (b) For $E_{g+}\\\\otimes E_{g+}$ : $D_{E_{g+}}(\\\\mathbf {k}) &\\\\cong (k_x^2 - k_y^2)\\\\, h_5 - \\\\frac{1}{\\\\sqrt{3}}\\\\, (2k_z^2 - k_x^2 - k_y^2)\\\\, h_4 .$ (c) For $T_{1g+}\\\\otimes T_{2g+}$ : $D_{T_{1g+}}(\\\\mathbf {k}) &\\\\cong k_y k_z (k_y^2 - k_z^2)\\\\, h_1 + k_z k_x (k_z^2 - k_x^2)\\\\, h_2 \\\\nonumber \\\\\\\\&\\\\quad {} + k_x k_y (k_x^2 - k_y^2)\\\\, h_3 .$ The assignment of the components is done in such a way that the form factor changes sign under any fourfold rotation.\",\n \"Table: Leading-order polynomial forms of the form factors f n (𝐤)f_n(\\\\mathbf {k}) describing A 2g+ A_{2g+} pairing for electrons withangular momentum j=3/2j=3/2.Table: Leading-order polynomial forms of the products c n (𝐤)f n (𝐤)c_n(\\\\mathbf {k}) f_n(\\\\mathbf {k}) of form factors describing A 2g+ A_{2g+} pairing for electrons with angular momentum j=3/2j=3/2.\",\n \"The amplitudes of the leading terms in c n (𝐤)c_n(\\\\mathbf {k}) have been absorbed into new pairing amplitudes marked by a tilde.The resulting superconducting form factors $f_{n}$ and the products $c_{n} f_{n}$ , which are required to determine the IP nodes, are listed in Tables REF and REF , respectively, to the leading order.\",\n \"The condition for IP nodes reads as $c_0&(\\\\mathbf {k})\\\\, f_0(\\\\mathbf {k}) - \\\\vec{c}(\\\\mathbf {k}) \\\\cdot \\\\vec{f}(\\\\mathbf {k}) \\\\nonumber \\\\\\\\&= \\\\tilde{\\\\delta }_{A_{2g+}}\\\\, \\\\big [ k_{x}^4 (k_{y}^2-k_{z}^2) + k_{y}^4 (k_{z}^2-k_{x}^2)+ k_{z}^4 (k_{x}^2-k_{y}^2) \\\\big ] \\\\nonumber \\\\\\\\&\\\\quad {}- \\\\tilde{\\\\delta }_{T_{1g+}}\\\\, \\\\big [ k_y^2k_z^2 (k_y^2-k_z^2)+ k_z^2k_x^2 (k_z^2-k_x^2) \\\\nonumber \\\\\\\\&\\\\quad {}+ k_x^2k_y^2 (k_x^2-k_y^2)\\\\big ] = 0 ,$ where the contributions of type (b) cancel.\",\n \"This is again an artifact of having used the same basis functions for $c_n$ and $f_n$ .\",\n \"For general basis functions, the terms do not cancel, but they do not change the conclusions.\",\n \"The above expression vanishes whenever any two of the three components of $\\\\mathbf {k}$ are equal.\",\n \"There are line nodes in the $\\\\lbrace 110\\\\rbrace $ planes for infinitesimal pairing.\",\n \"Next, we consider the TRS-breaking state where the amplitude from $T_{1g+}\\\\otimes T_{2g+}$ has a phase shift of $\\\\pi /2$ relative to the amplitude from $E_{g+}\\\\otimes E_{g+}$ .\",\n \"The real and imaginary parts of the condition for IP nodes have the same momentum dependence.\",\n \"Hence, the line nodes of the real and imaginary part coincide and the TRS-breaking state retains the six line nodes in the mirror planes.\",\n \"The form factors in the $(110)$ plane read as $f_0(\\\\mathbf {k}) &= f_3(\\\\mathbf {k}) = f_5(\\\\mathbf {k}) = 0 , \\\\\\\\f_1(\\\\mathbf {k}) &=- i \\\\delta _{T_{1g+}} k_zk_x (k_z^2-k_x^2) , \\\\\\\\f_2(\\\\mathbf {k}) &= i \\\\delta _{T_{1g+}} k_zk_x (k_z^2-k_x^2) , \\\\\\\\f_4(\\\\mathbf {k}) &= -\\\\frac{2}{\\\\sqrt{3}}\\\\, \\\\delta _{E_{g+}} (k_z^2-k_x^2) ,$ with $\\\\delta _{T_{1g+}}$ and $\\\\delta _{E_{g+}}$ real, which gives $\\\\langle f^1 ,f^1 \\\\rangle &= -\\\\frac{4}{3}\\\\, \\\\delta _{E_{g+}}^2 (k_z^2-k_x^2)^2 , \\\\\\\\\\\\langle f^2 ,f^2 \\\\rangle &= -2\\\\, \\\\delta _{T_{1g+}}^2 k_z^2 k_x^2 (k_z^2-k_x^2)^2 , \\\\\\\\\\\\langle f^1 ,f^2 \\\\rangle &= 0 .$ and also $\\\\langle c ,f^1 \\\\rangle =\\\\langle c ,f^2 \\\\rangle =0$ .\",\n \"In this case, the Pfaffian simplifies to $P(\\\\textbf {k}) &= (\\\\langle c ,c \\\\rangle -\\\\langle f^1 ,f^1 \\\\rangle -\\\\langle f^2 ,f^2 \\\\rangle )^2- 4\\\\langle f^1 ,f^1 \\\\rangle \\\\langle f^2 ,f^2 \\\\rangle .$ The second term $-4\\\\langle f^1 ,f^1 \\\\rangle \\\\langle f^2 ,f^2 \\\\rangle =-\\\\frac{32}{3}\\\\,\\\\delta _{E_{g+}}^2 \\\\delta _{T_{1g+}}^2 k_z^2 k_x^2 (k_z^2-k_x^2)^4$ is generically negative.\",\n \"Thus we expect all the line nodes to inflate for strong coupling unlike for the two examples of $A_{2g+}$ pairing discussed above.\",\n \"The inflated line nodes are not attached to the normal-state Fermi surface.\",\n \"However, the inflation vanishes in special high-symmetry directions: On the $[111]$ axis, $\\\\langle f^1 ,f^1 \\\\rangle =\\\\langle f^2 ,f^2 \\\\rangle =0$ , thus the nodes are not inflated and stick to the normal-state Fermi surface.\",\n \"For $[001]$ and $[110]$ , nodes are also not inflated because there is only a single amplitude from $E_{g+}\\\\otimes E_{g+}$ and its phase can be gauged away.\",\n \"Moreover, along these directions $\\\\langle f^1 ,f^1 \\\\rangle $ and $\\\\langle f^2 ,f^2 \\\\rangle $ are not both zero.\",\n \"Thus here the nodes are neither inflated nor attached to the normal-state Fermi surface.\",\n \"The vanishing inflation in high-symmetry directions implies that the BFSs have self-touching points there.\",\n \"Interestingly, $[111]$ is the direction in which three weak-coupling line nodes intersect whereas two intersect in the $[001]$ direction and there is no intersection in the $[110]$ direction.\"\n ],\n [\n \"$E_{g+}$ pairing\",\n \"We consider $E_{g+}$ pairing as an example for a symmetry that is also possible for purely local pairing.\",\n \"The question is what changes for nonlocal pairing.\",\n \"$E_{g+}$ appears in six places in Table REF : (a) $A_{1g+}\\\\otimes E_{g+}$ , (b) $A_{2g+}\\\\otimes E_{g+}$ , (c) $E_{g+}\\\\otimes A_{1g+}$ , (d) $E_{g+}\\\\otimes E_{g+}$ , (e) $T_{1g+}\\\\otimes T_{2g+}$ , and (f) $T_{2g+}\\\\otimes T_{2g+}$ .\",\n \"The matrix-valued basis functions are given in the following to leading order only.\",\n \"(a) For $A_{1g+}\\\\otimes E_{g+}$ , we find constants to leading order: $D_{x^2-y^2,A_{1g+}}(\\\\mathbf {k}) &\\\\cong h_4 = J_x^2 - J_y^2 , \\\\\\\\D_{3z^2-r^2,A_{1g+}}(\\\\mathbf {k}) &\\\\cong h_5 = \\\\frac{1}{\\\\sqrt{3}}\\\\, (2J_z^2 - J_x^2 - J_y^2) .$ These are of course the contributions from local pairing [1], [2].\",\n \"(b) For $A_{2g+}\\\\otimes E_{g+}$ : $&D_{x^2-y^2,A_{2g+}}(\\\\mathbf {k}) \\\\nonumber \\\\\\\\&\\\\quad \\\\cong \\\\big [ k_x^4 (k_y^2-k_z^2) + k_y^4 (k_z^2-k_x^2) + k_z^4 (k_x^2-k_y^2) \\\\big ]\\\\, h_5 \\\\\\\\&D_{3z^2-r^2,A_{2g+}}(\\\\mathbf {k}) \\\\nonumber \\\\\\\\&\\\\quad \\\\cong -\\\\big [ k_x^4 (k_y^2-k_z^2) + k_y^4 (k_z^2-k_x^2) + k_z^4 (k_x^2-k_y^2) \\\\big ]\\\\, h_4 .$ (c) For $E_{g+}\\\\otimes A_{1g+}$ : $D_{x^2-y^2,0}(\\\\mathbf {k}) &\\\\cong (k_x^2 - k_y^2)\\\\, h_0 , \\\\\\\\D_{3z^2-r^2,0}(\\\\mathbf {k}) &\\\\cong \\\\frac{1}{\\\\sqrt{3}}\\\\, (2k_z^2 - k_x^2 - k_y^2)\\\\, h_0 .$ (d) For $E_{g+}\\\\otimes E_{g+}$ : $D_{x^2-y^2,E_{g+}}(\\\\mathbf {k}) &\\\\cong \\\\frac{1}{\\\\sqrt{3}}\\\\, (2k_z^2 - k_x^2 - k_y^2)\\\\, h_4+ (k_x^2 - k_y^2)\\\\, h_5 , \\\\\\\\D_{3z^2-r^2,E_{g+}}(\\\\mathbf {k}) &\\\\cong (k_x^2 - k_y^2)\\\\, h_4- \\\\frac{1}{\\\\sqrt{3}}\\\\, (2k_z^2 - k_x^2 - k_y^2)\\\\, h_5 .$ (e) For $T_{1g+}\\\\otimes T_{2g+}$ : $&D_{x^2-y^2,T_{1g+}}(\\\\mathbf {k}) \\\\cong \\\\frac{1}{\\\\sqrt{3}}\\\\, \\\\big [ k_y k_z(k_y^2-k_z^2)\\\\, h_1 \\\\nonumber \\\\\\\\&\\\\quad {}+ k_z k_x (k_z^2-k_x^2)\\\\, h_2 - 2\\\\, k_x k_y (k_x^2-k_y^2)\\\\, h_3 \\\\big ] , \\\\\\\\&D_{3z^2-r^2,T_{1g+}}(\\\\mathbf {k}) \\\\cong k_y k_z (k_y^2-k_z^2)\\\\, h_1 - k_zk_x (k_z^2-k_x^2)\\\\, h_2 .$ (f) For $T_{2g+}\\\\otimes T_{2g+}$ : $D_{x^2-y^2,T_{2g+}}(\\\\mathbf {k}) &\\\\cong - k_y k_z\\\\, h_1 + k_z k_x\\\\, h_2 , \\\\\\\\D_{3z^2-r^2,T_{2g+}}(\\\\mathbf {k}) &\\\\cong \\\\frac{1}{\\\\sqrt{3}}\\\\,( k_y k_z\\\\, h_1 + k_z k_x\\\\, h_2 - 2\\\\, k_x k_y\\\\, h_3 ) .$ The resulting superconducting form factors $f_n(\\\\mathbf {k})$ are given in Table REF and the products $c_n(\\\\mathbf {k}) f_n(\\\\mathbf {k})$ appearing in the condition (REF ) for IP nodes are shown in Table REF .\",\n \"The analysis is analogous to the previous cases of $E_{g+}$ pairing: By inserting $k_y=\\\\pm k_x$ , one can see that the contribution to $\\\\langle c,f\\\\rangle $ from $x^2-y^2$ basis functions (with amplitudes $\\\\tilde{\\\\delta }^1_{\\\\cdots }$ ) has two symmetry-imposed first-order line nodes for $k_y=\\\\pm k_x$ .\",\n \"In a time-reversal-symmetric state, the inclusion of $3z^2-r^2$ basis functions generically leads to two line nodes elsewhere on the normal-state Fermi surface.\",\n \"The nodes must intersect with the $\\\\langle 111\\\\rangle $ axes.\",\n \"TRS-breaking states generically lead to point nodes in the $\\\\langle 111\\\\rangle $ directions, for example for order parameters proportional to $(1,i)$ .\",\n \"These point nodes are solely determined by symmetry.\",\n \"The presence of these eight point nodes was also found for purely local pairing [2].\",\n \"We thus find that the inclusion of nonlocal pairing does not change the nodal structure.\",\n \"Table: Leading-order polynomial forms of the form factors f n (𝐤)f_n(\\\\mathbf {k}) describing E g+ E_{g+} pairing for electrons with angular momentum j=3/2j=3/2.Table: Leading-order polynomial forms of the products c n (𝐤)f n (𝐤)c_n(\\\\mathbf {k}) f_n(\\\\mathbf {k}) of form factors describing E g+ E_{g+} pairing for electrons with angular momentum j=3/2j=3/2.\",\n \"The amplitudes of the leading terms in c n (𝐤)c_n(\\\\mathbf {k}) have been absorbed into new pairing amplitudes marked by a tilde.For purely local pairing, the point nodes are inflated into BFSs for noninfinitesimal pairing [2].\",\n \"We briefly sketch the analysis when nonlocal pairing is included.\",\n \"We consider the Pfaffian on the $[111]$ axis, $\\\\mathbf {k} = k\\\\,(1,1,1)/\\\\sqrt{3}$ .\",\n \"Table REF then shows that $f_0(\\\\mathbf {k}) &= 0 , \\\\\\\\f_1(\\\\mathbf {k}) &= -\\\\frac{\\\\delta ^1_{T_{2g+}}}{3}\\\\, k^2 + \\\\frac{\\\\delta ^2_{T_{2g+}}}{3\\\\sqrt{3}}\\\\, k^2 , \\\\\\\\f_2(\\\\mathbf {k}) &= \\\\frac{\\\\delta ^1_{T_{2g+}}}{3}\\\\, k^2 + \\\\frac{\\\\delta ^2_{T_{2g+}}}{3\\\\sqrt{3}}\\\\, k^2 , \\\\\\\\f_3(\\\\mathbf {k}) &= -\\\\frac{2\\\\, \\\\delta ^2_{T_{2g+}}}{3\\\\sqrt{3}}\\\\, k^2 , \\\\\\\\f_4(\\\\mathbf {k}) &= \\\\delta ^1_{A_{1g+}} , \\\\\\\\f_5(\\\\mathbf {k}) &= \\\\delta ^2_{A_{1g+}} .$ For the generalized $(1,i)$ pairing state with $\\\\delta ^2_{A_{1g+}} = i\\\\delta ^1_{A_{1g+}}$ , $\\\\delta ^2_{T_{2g+}} = i\\\\delta ^1_{T_{2g+}}$ , and $\\\\delta ^1_{A_{1g+}}, \\\\delta ^1_{T_{2g+}} \\\\in \\\\mathbb {R}$ , we find $\\\\langle f^1,f^1\\\\rangle &= \\\\langle f^2,f^2\\\\rangle = -\\\\frac{2k^4}{9}\\\\, \\\\big ( \\\\delta ^1_{T_{2g+}} \\\\big )^2- \\\\big ( \\\\delta ^1_{A_{1g+}} \\\\big )^2 , \\\\\\\\\\\\langle f^1,f^2\\\\rangle &= 0 .$ On the other hand, the normal-state form factors are, to leading order, $c_0(\\\\mathbf {k}) &= c_0^{(0)} , \\\\\\\\c_1(\\\\mathbf {k}) &= c_2(\\\\mathbf {k}) = c_3(\\\\mathbf {k}) = \\\\frac{c_2^{(2)}}{3}\\\\, k^2 , \\\\\\\\c_4(\\\\mathbf {k}) &= c_5(\\\\mathbf {k}) = 0 .$ It follows that $\\\\langle c,c\\\\rangle &= \\\\big (c_0^{(0)}\\\\big )^2 - \\\\frac{\\\\big (c_2^{(2)}\\\\big )^2}{9}\\\\, k^4 , \\\\\\\\\\\\langle c,f^1\\\\rangle &= \\\\langle c,f^2\\\\rangle = 0 .$ The analysis is thus analogous to the previous two examples with $E_{g+}$ pairing.\",\n \"The point nodes are inflated into BFSs, which touch the normal-state Fermi surface.\",\n \"Hence, the inclusion of nonlocal pairing does not affect the phenomenology for this pairing state.\",\n \"Note that only two of the six contributions (a)–(f) lead to inflation in the $[111]$ direction, namely the local $A_{1g+}\\\\otimes E_{g+}$ contribution and the $T_{2g+}\\\\otimes T_{2g+}$ contribution.\",\n \"For this to happen, there must be a pair of nonzero amplitudes with nontrivial phase difference in at least one of these two channels.\"\n ],\n [\n \"$T_{1g+}$ pairing\",\n \"The irrep $T_{1g+}$ provides an example for a pairing state that cannot occur for local pairing in the $j=3/2$ model but unlike $A_{2g+}$ is multidimensional.\",\n \"$T_{1g+}$ pairing emerges in seven places in Table REF : (a) $A_{2g+}\\\\otimes T_{2g+}$ , (b) $E_{g+}\\\\otimes T_{2g+}$ , (c) $T_{1g+}\\\\otimes A_{1g+}$ , (d) $T_{1g+}\\\\otimes T_{2g+}$ , (e) $T_{1g+}\\\\otimes E_{g+}$ , (f) $T_{2g+}\\\\otimes T_{2g+}$ , and (g) $T_{2g+}\\\\otimes E_{g+}$ .\",\n \"We give the the matrix-valued basis functions to the leading order only.\",\n \"(a) For $A_{2g+}\\\\otimes T_{2g+}$ : $& D_{x,A_{2g+}}(\\\\mathbf {k}) \\\\nonumber \\\\\\\\&\\\\quad \\\\cong \\\\big [ k_x^4 (k_y^2-k_z^2) + k_y^4 (k_z^2-k_x^2) + k_z^4 (k_x^2-k_y^2) \\\\big ]\\\\, h_1 , \\\\\\\\& D_{y,A_{2g+}}(\\\\mathbf {k}) \\\\nonumber \\\\\\\\&\\\\quad \\\\cong \\\\big [ k_x^4 (k_y^2-k_z^2) + k_y^4 (k_z^2-k_x^2) + k_z^4 (k_x^2-k_y^2) \\\\big ]\\\\, h_2 , \\\\\\\\& D_{z,A_{2g+}}(\\\\mathbf {k}) \\\\nonumber \\\\\\\\&\\\\quad \\\\cong \\\\big [ k_x^4 (k_y^2-k_z^2) + k_y^4 (k_z^2-k_x^2) + k_z^4 (k_x^2-k_y^2) \\\\big ]\\\\, h_3 .$ (b) For $E_{g+}\\\\otimes T_{2g+}$ : $D_{x,E_{g+}}(\\\\mathbf {k}) &\\\\cong (k_y^2-k_z^2)\\\\, h_1 , \\\\\\\\D_{y,E_{g+}}(\\\\mathbf {k}) &\\\\cong (k_z^2-k_x^2)\\\\, h_2 , \\\\\\\\D_{z,E_{g+}}(\\\\mathbf {k}) &\\\\cong (k_x^2-k_y^2)\\\\, h_3 .$ (c) For $T_{1g+}\\\\otimes A_{1g+}$ : $D_{x,T_{1g+}}(\\\\mathbf {k}) &\\\\cong k_y k_z (k_y^2-k_z^2)\\\\, h_0 , \\\\\\\\D_{y,T_{1g+}}(\\\\mathbf {k}) &\\\\cong k_z k_x (k_z^2-k_x^2)\\\\, h_0 , \\\\\\\\D_{z,T_{1g+}}(\\\\mathbf {k}) &\\\\cong k_x k_y (k_x^2-k_y^2)\\\\, h_0 .$ (d) For $T_{1g+}\\\\otimes T_{2g+}$ : $D^{\\\\prime }_{x,T_{1g+}}(\\\\mathbf {k}) &\\\\cong k_z k_x (k_z^2-k_x^2)\\\\, h_3 + k_x k_y (k_x^2-k_y^2)\\\\, h_2 ,\\\\\\\\D^{\\\\prime }_{y,T_{1g+}}(\\\\mathbf {k}) &\\\\cong k_x k_y (k_x^2-k_y^2)\\\\, h_1 + k_y k_z (k_y^2-k_z^2)\\\\, h_3 ,\\\\\\\\D^{\\\\prime }_{z,T_{1g+}}(\\\\mathbf {k}) &\\\\cong k_y k_z (k_y^2-k_z^2)\\\\, h_2 + k_z k_x (k_z^2-k_x^2)\\\\, h_1 .$ (e) For $T_{1g+}\\\\otimes E_{g+}$ : $D^{\\\\prime \\\\prime }_{x,T_{1g+}}(\\\\mathbf {k}) &\\\\cong k_y k_z (k_y^2-k_z^2)\\\\bigg (\\\\frac{\\\\sqrt{3}}{2}\\\\, h_4- \\\\frac{1}{2} \\\\, h_5\\\\bigg ) , \\\\\\\\D^{\\\\prime \\\\prime }_{y,T_{1g+}}(\\\\mathbf {k}) &\\\\cong -k_z k_x (k_z^2-k_x^2)\\\\bigg (\\\\frac{\\\\sqrt{3}}{2}\\\\, h_4+ \\\\frac{1}{2} \\\\, h_5\\\\bigg ) , \\\\\\\\D^{\\\\prime \\\\prime }_{z,T_{1g+}}(\\\\mathbf {k}) &\\\\cong k_x k_y (k_x^2-k_y^2)\\\\, h_5 .$ (f) For $T_{2g+}\\\\otimes T_{2g+}$ : $D_{x,T_{2g+}}(\\\\mathbf {k}) &\\\\cong k_z k_x \\\\, h_3 - k_x k_y \\\\, h_2 , \\\\\\\\D_{y,T_{2g+}}(\\\\mathbf {k}) &\\\\cong k_x k_y \\\\, h_1 - k_y k_z \\\\, h_3 , \\\\\\\\D_{z,T_{2g+}}(\\\\mathbf {k}) &\\\\cong k_y k_z \\\\, h_2 - k_z k_x \\\\, h_1 .$ (g) For $T_{2g+}\\\\otimes E_{g+}$ : $D^{\\\\prime }_{x,T_{2g+}}(\\\\mathbf {k}) &\\\\cong -k_y k_z\\\\bigg (\\\\frac{1}{2} \\\\, h_4 + \\\\frac{\\\\sqrt{3}}{2}\\\\, h_5\\\\bigg ) , \\\\\\\\D^{\\\\prime }_{y,T_{2g+}}(\\\\mathbf {k}) &\\\\cong k_z k_x\\\\bigg ({-}\\\\frac{1}{2} \\\\, h_4 + \\\\frac{\\\\sqrt{3}}{2}\\\\, h_5\\\\bigg ) , \\\\\\\\D^{\\\\prime }_{z,T_{2g+}}(\\\\mathbf {k}) &\\\\cong k_x k_y \\\\, h_4 .$ Table: Leading-order polynomial forms of the form factors f n (𝐤)f_n(\\\\mathbf {k}) describing T 1g+ T_{1g+} pairing for electrons with angular momentum j=3/2j=3/2.Table: Leading-order polynomial forms of the products c n (𝐤)f n (𝐤)c_n(\\\\mathbf {k}) f_n(\\\\mathbf {k}) of form factors describing T 1g+ T_{1g+} pairing for electrons with angular momentum j=3/2j=3/2.\",\n \"The amplitudes of the leading terms in c n (𝐤)c_n(\\\\mathbf {k}) have been absorbed into new pairing amplitudes marked by a tilde.The analysis is analogous to the one for a system with two s-orbitals; see Sec.\",\n \"REF .\",\n \"The resulting superconducting form factors $f_n(\\\\mathbf {k})$ are given in Table REF and the products $c_n(\\\\mathbf {k}) f_n(\\\\mathbf {k})$ appearing in the condition (REF ) for IP nodes are shown in Table REF .\",\n \"Hence, $c_0&(\\\\mathbf {k})\\\\, f_0(\\\\mathbf {k}) - \\\\vec{c}(\\\\mathbf {k}) \\\\cdot \\\\vec{f}(\\\\mathbf {k}) \\\\nonumber \\\\\\\\&= \\\\bigg [ \\\\tilde{\\\\delta }_{x,T_{1g+}} - \\\\tilde{\\\\delta }_{x,E_{g+}}+ \\\\tilde{\\\\delta }_{x,A_{2g+}} (k_z^2-k_x^2)(k_x^2-k_y^2) \\\\nonumber \\\\\\\\&{}+ \\\\tilde{\\\\delta }_{x,T_{1g+}^{\\\\prime }}\\\\, k_x^2- \\\\frac{\\\\tilde{\\\\delta }_{x,T_{1g+}^{\\\\prime \\\\prime }}}{\\\\sqrt{3}}\\\\, (2 k_x^2-k_y^2-k_z^2) \\\\nonumber \\\\\\\\&{}- \\\\tilde{\\\\delta }_{x,T_{2g+}^{\\\\prime }} \\\\bigg ]\\\\, k_y k_z (k_y^2-k_z^2) + \\\\ldots ,$ where two terms with cyclically permuted indices $x$ , $y$ , and $z$ have been suppressed.\",\n \"For broken TRS, the pairing states $(1,i,0)$ and $(1,\\\\omega ,\\\\omega ^2)$ with $\\\\omega =e^{2\\\\pi i/3}$ , are plausible [31], [26], [13].\",\n \"We here consider the simpler $(1,i,0)$ state, which has 18 point nodes in the $\\\\langle 001\\\\rangle $ , $\\\\langle 101\\\\rangle $ , and $\\\\langle 111\\\\rangle $ directions and one line node in the $k_z=0$ plane.\",\n \"For noninfinitesimal pairing, we expect the nodes to be inflated.\",\n \"For the nodes along the $[001]$ direction, the form factors read as $f_0(\\\\mathbf {k}) &= f_3(\\\\mathbf {k}) = f_4(\\\\mathbf {k}) = f_5(\\\\mathbf {k}) = 0 , \\\\\\\\f_1(\\\\mathbf {k}) &= -\\\\delta _{E_{g+}} k^2 , \\\\\\\\f_2(\\\\mathbf {k}) &= i\\\\delta _{E_{g+}} k^2 .$ We find $\\\\langle f^1, f^1 \\\\rangle =\\\\langle f^2, f^2 \\\\rangle =-\\\\delta ^2_{E_{g+}} k^4$ and $\\\\langle f^1, f^2 \\\\rangle =0$ as well as $\\\\langle c^1, f^1 \\\\rangle =\\\\langle c^2, f^2 \\\\rangle =0$ .\",\n \"In this case, the Pfaffian simplifies to $P(\\\\mathbf {k})=\\\\langle c, c \\\\rangle \\\\, \\\\big (\\\\langle c, c \\\\rangle + 4 \\\\delta ^2_{E_{g+}} k^4 \\\\big ) .$ The first factor changes sign at the normal-state Fermi surface but the second factor does not and thus the Pfaffian changes sign at the normal-state Fermi surface.\",\n \"Hence, the point nodes in the $\\\\langle 001 \\\\rangle $ directions are inflated and remain attached to the normal-state Fermi surfaces for arbitrarily strong coupling.\",\n \"In the $[101]$ direction, we have $\\\\mathbf {k}=k(1,0,1)/\\\\sqrt{2}$ and the form factors read as $f_0(\\\\mathbf {k}) &= f_1(\\\\mathbf {k})=f_2(\\\\mathbf {k})= 0 , \\\\\\\\f_3(\\\\mathbf {k}) &= \\\\frac{\\\\delta _{T_{2g+}}}{2}\\\\, k^2 , \\\\\\\\f_4(\\\\mathbf {k}) &= - i\\\\, \\\\frac{\\\\delta _{T_{2g+}^{\\\\prime }}}{4}\\\\, k^2 , \\\\\\\\f_5(\\\\mathbf {k}) &= i\\\\, \\\\frac{\\\\sqrt{3}\\\\,\\\\delta _{T_{2g+}^{\\\\prime }}}{4}\\\\, k^2 .$ This implies $\\\\langle f^1, f^2 \\\\rangle =0$ but $\\\\langle f^1, f^1 \\\\rangle $ and $\\\\langle f^2, f^2 \\\\rangle $ are generally unequal.\",\n \"The Pfaffian is thus $P(\\\\mathbf {k}) = \\\\left[ \\\\langle c,c\\\\rangle - \\\\langle f^1,f^1\\\\rangle - \\\\langle f^2,f^2\\\\rangle \\\\right]^2 - 4\\\\, \\\\langle f^1,f^1\\\\rangle \\\\langle f^2,f^2\\\\rangle $ .\",\n \"The first term vanishes on a renormalized Fermi surface.\",\n \"The second term $- 4\\\\, \\\\langle f^1,f^1\\\\rangle \\\\langle f^2,f^2\\\\rangle = -\\\\frac{1}{4}\\\\, \\\\delta _{T_{2g+}}^2 \\\\delta _{T_{2g+}^{\\\\prime }}^2 k^8$ is generically negative.\",\n \"Hence, the Pfaffian generically becomes negative in the vicinity of the normal-state Fermi surface but the BFS does not usually touch it.\",\n \"The Pfaffian is identical for all $\\\\langle 101\\\\rangle $ directions that are not in the $k_z=0$ plane.\",\n \"Along $[111]$ , we have $\\\\mathbf {k}=k\\\\,(1,1,1)/\\\\sqrt{3}$ and $f_0(\\\\mathbf {k}) &= 0 , \\\\\\\\f_1(\\\\mathbf {k}) &= i\\\\, \\\\frac{\\\\delta _{T_{2g+}}}{3} \\\\,k^2 , \\\\\\\\f_2(\\\\mathbf {k}) &= - \\\\frac{\\\\delta _{T_{2g+}}}{3} \\\\,k^2 , \\\\\\\\f_3(\\\\mathbf {k}) &= \\\\frac{\\\\delta _{T_{2g+}}}{3} \\\\,k^2- i\\\\frac{\\\\delta _{T_{2g+}}}{3} \\\\,k^2 , \\\\\\\\f_4(\\\\mathbf {k}) &= - \\\\frac{\\\\delta _{T_{2g+}^{\\\\prime }}}{6} \\\\,k^2- i \\\\frac{\\\\delta _{T_{2g+}^{\\\\prime }}}{6} \\\\,k^2 , \\\\\\\\f_5(\\\\mathbf {k}) &= - \\\\frac{\\\\delta _{T_{2g+}^{\\\\prime }}}{2\\\\sqrt{3}} \\\\,k^2+ i \\\\frac{\\\\delta _{T_{2g+}^{\\\\prime }}}{2\\\\sqrt{3}} \\\\,k^2 .$ We thus obtain $\\\\langle f^1,f^2\\\\rangle = \\\\frac{k^4}{18}\\\\, \\\\big ( 2 \\\\delta _{T_{2g+}}^2+ \\\\delta _{T_{2g+}^{\\\\prime }}^2 \\\\big )$ and $\\\\langle f^1,f^1\\\\rangle = \\\\langle f^2,f^2\\\\rangle = -\\\\frac{k^4}{9}\\\\, \\\\big ( 2 \\\\delta _{T_{2g+}}^2 + \\\\delta _{T_{2g+}^{\\\\prime }}^2 \\\\big ) .$ The Pfaffian is $P(\\\\mathbf {k}) &= \\\\left[ \\\\langle c,c\\\\rangle - \\\\langle f^1,f^1\\\\rangle - \\\\langle f^2,f^2\\\\rangle \\\\right]^2\\\\nonumber \\\\\\\\&\\\\quad {} + 4 \\\\left[ \\\\langle f^1,f^2\\\\rangle ^2 - \\\\langle f^1,f^1\\\\rangle \\\\langle f^2,f^2\\\\rangle \\\\right] ,$ wherein the second term evaluates to $-\\\\frac{k^8}{27}\\\\, \\\\big ( 2 \\\\delta _{T_{2g+}}^2 + \\\\delta _{T_{2g+}^{\\\\prime }}^2 \\\\big )^2 .$ Since this is generally negative we also expect the nodes to inflate in the $\\\\langle 111\\\\rangle $ directions but they are not attached to the normal-state Fermi surface.\",\n \"For the equatorial line node, we take $\\\\mathbf {k}=(k_x,k_y,0)$ .\",\n \"The superconducting form factors are $f_0(\\\\mathbf {k}) &= f_3(\\\\mathbf {k}) = f_4(\\\\mathbf {k}) = f_5(\\\\mathbf {k}) = 0 , \\\\\\\\f_1(\\\\mathbf {k}) &= \\\\delta _{A_{2g+}} (k_x^4 k_y^2-k_y^4 k_x^2)+\\\\delta _{E_{g+}}\\\\, k_y^2+ i \\\\delta _{T_{2g+}} k_x k_y \\\\nonumber \\\\\\\\&\\\\quad {} + i \\\\delta _{T_{1g+}^{\\\\prime }} k_x k_y (k_x^2-k_y^2) \\\\\\\\f_2(\\\\mathbf {k}) &= i\\\\delta _{A_{2g+}} (k_x^4 k_y^2-k_y^4 k_x^2)-i \\\\delta _{E_{g+}}\\\\, k_x^2- \\\\delta _{T_{2g+}}\\\\, k_x k_y \\\\nonumber \\\\\\\\&\\\\quad {} + \\\\delta _{T_{1g+}^{\\\\prime }} k_x k_y (k_x^2-k_y^2) .$ We find $\\\\langle f^1,f^2\\\\rangle \\\\ne 0$ and $\\\\langle f^1,f^1\\\\rangle \\\\ne \\\\langle f^2,f^2\\\\rangle $ .\",\n \"The Pfaffian thus again has the form of Eq.\",\n \"(REF ).\",\n \"The second term reads as $& - 4 k_x^2 k_y^2\\\\, \\\\big [ \\\\delta _{T_{2g+}}^2 - \\\\delta _{E_{g+}}^2- \\\\delta _{A_{2g+}} \\\\delta _{E_{g+}} (k_x^2-k_y^2)^2 \\\\nonumber \\\\\\\\&\\\\quad {}+ \\\\big ( \\\\delta _{A_{2g+}}^2 k_x^2 k_y^2- \\\\delta _{T_{1g+}^{\\\\prime }}^2 \\\\big ) (k_x^2 + k_y^2) (k_x^2 - k_y^2) \\\\big ]^2$ and is generically negative.\",\n \"We conclude that the equatorial line node is inflated by noninfinitesimal pairing.\",\n \"The resulting BFS is toroidal but pinched on the $k_x$ and $k_y$ axes since Eq.\",\n \"(REF ) gives zero there.\",\n \"Since the first term in Eq.\",\n \"(REF ) becomes zero close to but not at the normal-state Fermi surface the BFS generically does not touch the normal-state Fermi surface.\",\n \"This also holds on the $k_x$ and $k_y$ axes.\",\n \"The behavior of the nodes is identical to the case of $T_{1g+}$ pairing for two s-orbitals.\"\n ],\n [\n \"Orbital doublet\",\n \"The discussion in Sec.\",\n \"REF suggests how to construct further orbital models: the set of orbitals must be closed under the action of the point group.\",\n \"This implies that the orbitals must transform like basis functions of one-dimensional irreps or as complete sets of basis functions of multidimensional irreps.\",\n \"In the previous examples, we have considered two $A_{1g}$ orbitals and one $A_{1g}$ and one $A_{2u}$ orbital.\",\n \"As the simplest example with a multidimensional irrep we here analyze the case of two orbitals transforming like basis functions of $E_g$ .\",\n \"This is naturally realized by a doublet of $e_g$ orbitals ($d_{x^2-y^2}$ and $d_{3z^2-r^2}$ ) per site.\",\n \"Since they are of even parity we have $U_P = \\\\sigma _0 \\\\otimes \\\\sigma _0$ .\",\n \"Also, $U_T = \\\\sigma _0 \\\\otimes i\\\\sigma _2$ holds.\",\n \"The new aspect here is that the orbital part alone can have higher-dimensional irreps.\",\n \"The irreps of Pauli matrices in orbital space are the following: $\\\\eta _0 \\\\equiv \\\\sigma _0$ belongs to $A_{1g+}$ .\",\n \"The two matrices $\\\\eta _1 \\\\equiv \\\\sigma _1$ and $\\\\eta _2 \\\\equiv -\\\\sigma _3$ form an $E_{g+}$ doublet.\",\n \"It is easy to check that under rotations $\\\\eta _1$ and $\\\\eta _2$ transform like $k_x^2-k_y^2$ and $(2k_z^2-k_x^2-k_y^2)/\\\\sqrt{3}$ , respectively.\",\n \"Finally, $\\\\eta _3 \\\\equiv \\\\sigma _2$ belongs to $A_{2g-}$ .\",\n \"In spin space, $\\\\sigma _0$ of course transforms according to $A_{1g+}$ and $(\\\\sigma _1,\\\\sigma _2,\\\\sigma _3)$ form a $T_{1g-}$ triplet.\",\n \"Combining higher-dimensional irreps from the orbital and spin parts, we obtain reducible product representations.\",\n \"Thus Kronecker products of Pauli matrices in orbital and spin space have to be linearly combined to construct the proper basis matrices.\",\n \"(Such a construction is also implicit in Table REF of basis matrices for the $j=3/2$ case above.)\",\n \"Specifically, we require the nontrivial reduction $E_{g+} \\\\otimes T_{1g-} = T_{1g-} \\\\oplus T_{2g-}$ .\",\n \"The resulting basis matrices are presented in Table REF .\",\n \"Table: Basis matrices on the internal Hilbert spacefor the case of an E g E_g doublet of orbitals with spin and point group O h O_h.\",\n \"The basis matrices are irreducible tensor operators of the irreps listed in the second column.\",\n \"For multidimensional irreps, the states transforming into each other under point-group operations are distinguished by the index in the third column.The basis matrices relevant for the normal-state Hamiltonian and for even-parity pairing are $h_0 &\\\\equiv \\\\eta _0 \\\\otimes \\\\sigma _0 & A_{1g+}, \\\\\\\\h_1 &\\\\equiv \\\\eta _3 \\\\otimes \\\\sigma _1 & T_{2g+}, \\\\\\\\h_2 &\\\\equiv \\\\eta _3 \\\\otimes \\\\sigma _2 & T_{2g+}, \\\\\\\\h_3 &\\\\equiv \\\\eta _3 \\\\otimes \\\\sigma _3 & T_{2g+}, \\\\\\\\h_4 &\\\\equiv \\\\eta _1 \\\\otimes \\\\sigma _0 & E_{g+}, \\\\\\\\h_5 &\\\\equiv \\\\eta _2 \\\\otimes \\\\sigma _0 & E_{g+}.$ The symmetry properties are thus the same as for the $j=3/2$ case.\",\n \"Hence, the analysis of pairing states is completely analogous to Sec.\",\n \"REF , except for the different definition of the basis matrices $h_n$ .\"\n ],\n [\n \"Orbital triplet\",\n \"We briefly consider the case of three $t_{2g}$ orbitals per site of a cubic lattice, i.e., $d_{yz}$ , $d_{zx}$ , and $d_{xy}$ orbitals.\",\n \"In this example, the dimension of the internal Hilbert space is $N=6$ and according to Appendix there are 15 basis matrices for the normal-state Hamiltonian and even-parity superconductivity.\",\n \"Their algebra is much more complicated than for $N=4$ .\",\n \"In particular, they do not anticommute pairwise.\",\n \"We require a basis of $3\\\\times 3$ matrices that act on the orbital space.\",\n \"We take the Gell-Mann matrices [17] $\\\\lambda _0 = &\\\\begin{pmatrix} 1 & 0 & 0 \\\\\\\\ 0 & 1 & 0 \\\\\\\\ 0 & 0 & 1 \\\\end{pmatrix} \\\\!, \\\\\\\\\\\\lambda _1 = \\\\begin{pmatrix} 0 & 1 & 0 \\\\\\\\ 1 & 0 & 0 \\\\\\\\ 0 & 0 & 0 \\\\end{pmatrix} \\\\!,\\\\:\\\\lambda _2 = &\\\\begin{pmatrix} 0 & 0 & 1 \\\\\\\\ 0 & 0 & 0 \\\\\\\\ 1 & 0 & 0 \\\\end{pmatrix} \\\\!,\\\\:\\\\lambda _3 = \\\\begin{pmatrix} 0 & 0 & 0 \\\\\\\\ 0 & 0 & 1 \\\\\\\\ 0 & 1 & 0 \\\\end{pmatrix} \\\\!, \\\\\\\\\\\\lambda _4 = \\\\begin{pmatrix} 0 & -i & 0 \\\\\\\\ i & 0 & 0 \\\\\\\\ 0 & 0 & 0 \\\\end{pmatrix} \\\\!,\\\\:\\\\lambda _5 = &\\\\begin{pmatrix} 0 & 0 & -i \\\\\\\\ 0 & 0 & 0 \\\\\\\\ i & 0 & 0 \\\\end{pmatrix} \\\\!,\\\\:\\\\lambda _6 = \\\\begin{pmatrix} 0 & 0 & 0 \\\\\\\\ 0 & 0 & -i \\\\\\\\ 0 & i & 0 \\\\end{pmatrix} \\\\!, \\\\\\\\\\\\lambda _7 = \\\\begin{pmatrix} 1 & 0 & 0 \\\\\\\\ 0 & -1 & 0 \\\\\\\\ 0 & 0 & 0 \\\\end{pmatrix} \\\\!,\\\\hspace{3.50006pt}&\\\\lambda _8 = \\\\frac{1}{\\\\sqrt{3}} \\\\begin{pmatrix} 1 & 0 & 0 \\\\\\\\ 0 & 1 & 0 \\\\\\\\ 0 & 0 & -2 \\\\end{pmatrix} \\\\!.$ The associated irreps are given in Table REF .\",\n \"The Gell-Mann matrices have to be combined with the Pauli matrices acting on spin space to form basis matrices for the combined internal degrees of freedom.\",\n \"$\\\\sigma _0$ of course transforms according to $A_{1g+}$ and $(\\\\sigma _1,\\\\sigma _2,\\\\sigma _3)$ to $T_{1g-}$ .\",\n \"Possible basis matrices for $H_N(\\\\mathbf {k})$ and even-parity pairing must belong to $g+$ irreps.\",\n \"They can thus be constructed by combining $\\\\lambda _j$ for $j \\\\in \\\\lbrace 0,1,2,3,7,8\\\\rbrace $ with $\\\\sigma _0$ and by combining $\\\\lambda _j$ with $j \\\\in \\\\lbrace 4,5,6\\\\rbrace $ with $\\\\sigma _1$ , $\\\\sigma _2$ , or $\\\\sigma _3$ .\",\n \"The relevant product representations are either trivial or involve the reduction $T_{1g-} \\\\otimes T_{1g-} = A_{1g+} \\\\oplus E_{g+} \\\\oplus T_{1g+} \\\\oplus T_{2g+}$ .\",\n \"The 15 allowed basis matrices and the associated irreps are shown in Table REF .\",\n \"Here, we have not chosen any special normalization, except that the entries belonging to the same multiplets have consistent numerical factors.\",\n \"There are also 21 basis matrices belonging to $g-$ irreps, which are disallowed as pairing matrices and are not listed for simplicity.\",\n \"Table: Basis matrices on the internal Hilbert space for the case of an T 2g T_{2g} triplet of orbitals with spin and point group O h O_h.\",\n \"Unlike for the previous examples, only the basis matrices h n h_n allowed in the normal-state Hamiltonian and for even-parity pairing are listed.\",\n \"The basis matrices are irreducible tensor operators of the irreps listed in the second column.\",\n \"For multidimensional irreps, the states transforming into each other under point-group operations are distinguished by the index in the third column.The normal-state Hamiltonian reads as $H_N(\\\\mathbf {k}) = \\\\sum _{n=0}^{14} c_n(\\\\mathbf {k})\\\\, h_n ,$ where $c_n(\\\\mathbf {k})$ transforms like $h_n$ .\",\n \"For even-parity superconductivity, we can combine the 15 matrices $h_n$ with form factors $f_n(\\\\mathbf {k})$ belonging to all the $g+$ irreps.\",\n \"This obviously generates pairing states for all $g+$ irreps.\",\n \"Any pairing state can be expressed in terms of a pairing matrix of the form $D(\\\\mathbf {k}) = \\\\sum _{n=0}^{14} f_n(\\\\mathbf {k})\\\\, h_n .$ Moreover, all even-parity pairing states other than $A_{2g+}$ can occur for purely local pairing, i.e., with constant form factors, because the basis matrices $h_n$ in Table REF cover all $g+$ irreps except $A_{2g+}$ .\",\n \"Another interesting observation is that purely local pairing with trivial ($A_{1g+}$ ) symmetry now allows for an orbitally nontrivial contribution from $h_6$ .\",\n \"All pairing states including nonlocal contributions can be constructed as described above.\",\n \"For the present case of $N=6$ , we typically find a larger number of contributions than for $N=4$ .\",\n \"For example, $E_{g+}$ pairing can result from the products (form factor times basis matrix) (a) $A_{1g+} \\\\otimes E_{g+}$ , (b) $A_{2g+} \\\\otimes E_{g+}$ , (c) $E_{g+} \\\\otimes A_{1g+}$ , (d) $E_{g+} \\\\otimes E_{g+}$ , (e) $T_{1g+} \\\\otimes T_{1g+}$ , (f) $T_{1g+} \\\\otimes T_{2g+}$ , (g) $T_{2g+} \\\\otimes T_{1g+}$ , and (h) $T_{2g+} \\\\otimes T_{2g+}$ .\",\n \"The leading-order contribution of type (a) to the pairing matrix reads as $D_{x^2-y^2,A_{1g+}}(\\\\mathbf {k}) &\\\\cong \\\\delta _0\\\\, \\\\lambda _7 \\\\otimes \\\\sigma _0+ \\\\delta _1\\\\, ( \\\\lambda _6 \\\\otimes \\\\sigma _1 + \\\\lambda _5 \\\\otimes \\\\sigma _2 ) , \\\\\\\\D_{3z^2-r^2,A_{1g+}}(\\\\mathbf {k}) &\\\\cong -\\\\delta _0\\\\, \\\\lambda _8 \\\\otimes \\\\sigma _0+ \\\\frac{\\\\delta _1}{\\\\sqrt{3}}\\\\, ( 2\\\\lambda _4 \\\\otimes \\\\sigma _3 \\\\nonumber \\\\\\\\&\\\\quad {}- \\\\lambda _6 \\\\otimes \\\\sigma _1 + \\\\lambda _5 \\\\otimes \\\\sigma _2 ) ,$ where one of the constants $\\\\delta _0$ and $\\\\delta _1$ could be set to unity.\",\n \"We omit the construction of the other contributions.\",\n \"The expression $F(\\\\mathbf {k}) = \\\\sum _n c_n(\\\\mathbf {k})\\\\, f_n(\\\\mathbf {k})$ transforms like the pairing matrix $D(\\\\mathbf {k})$ and the condition $F(\\\\mathbf {k})=0$ determines the location of IP nodes, as shown in Sec.\",\n \".\",\n \"In principle, we can now obtain the form factors $c_n(\\\\mathbf {k})$ appearing in Eq.\",\n \"(REF ) and $f_n(\\\\mathbf {k})$ in Eq.\",\n \"(REF ) and thus $F(\\\\mathbf {k})$ and the IP nodal structure.\",\n \"Pairing of not infinitesimal amplitude, in particular the existence of BFSs, could then be analyzed following Sec.\",\n \".\",\n \"This requires the calculation of the Pfaffian $\\\\mathop {\\\\textrm {Pf}}\\\\tilde{\\\\mathcal {H}}(\\\\mathbf {k})$ .\",\n \"Due to the absence of simple algebraic relations between the basis matrices $h_n$ , there is no simple analytical expression in terms of the functions $c_n(\\\\mathbf {k})$ and $f_n(\\\\mathbf {k})$ so that such an analysis would likely require a numerical study of $\\\\mathop {\\\\textrm {Pf}}\\\\tilde{\\\\mathcal {H}}(\\\\mathbf {k})$ .\",\n \"We do not execute this program here.\",\n \"BFSs have been predicted for $E_{g+}$ pairing in an $N=6$ model for $\\\\mathrm {Sr}_2\\\\mathrm {RuO}_4$ [17], which has the point group $D_{4h}$ .\"\n ],\n [\n \"Discussion and conclusions\",\n \"We have analyzed possible superconducting pairing symmetries for materials with local degrees of freedom beyond the electronic spin, such as orbital or basis site.\",\n \"Local degrees of freedom can enable unconventional pairing, in particular with BFSs.\",\n \"We find that this is the case even in the simple (and somewhat artificial) case of two s-orbitals at each lattice site, where the orbital index appears to be a spectator: Such a model permits orbitally nontrivial pairing of $T_{1g}$ symmetry.\",\n \"The main step taken in this paper is to go beyond local pairing.\",\n \"This implies that not only the normal state is characterized by momentum-dependent form factors $c_n(\\\\mathbf {k})$ but also the superconducting pairing is characterized by momentum-dependent form factors $f_n(\\\\mathbf {k})$ .\",\n \"In case of even-parity pairing, the functions $f_n(\\\\mathbf {k})$ have the same symmetry properties as the $c_n(\\\\mathbf {k})$ .\",\n \"There are three distinct types of contributions to pairing with nontrivial symmetry: (a) purely local pairing which is internally anisotropic, (b) internally isotropic pairing with nontrivial momentum-space form factors, and (c) contributions with nontrivial internal and momentum-space structure.\",\n \"Type (a) has been studied in Refs.\",\n \"[1], [2], whereas type (b) is exemplified by $d_{x^2-y^2}$ pairing in cuprates.\",\n \"Types (a) and (c) are internally anisotropic, which means that the pairing matrix $\\\\Delta (\\\\mathbf {k}) = D(\\\\mathbf {k})\\\\, U_T$ acts nontrivially on the internal degrees of freedom.\",\n \"$D(\\\\mathbf {k})$ then contains one or more basis matrices on the internal Hilbert space that are not proportional to the identity matrix.\",\n \"Nonlocal pairing permits symmetries of pairing states that usually cannot all be realized for local pairing.\",\n \"In fact, all even-parity ($g$ ) irreps of the magnetic point group occur for nonlocal pairing since they all have momentum-space basis functions and thus at least permit a contribution of type (b).\",\n \"In particular, there is always at least one basis matrix belonging to the trivial irrep, which is even, namely the identity matrix.\",\n \"Since the constant function of momentum also has full symmetry, local pairing with full, trivial symmetry is always allowed.\",\n \"Of course, it can be energetically suppressed by a local repulsive interaction.\",\n \"The example of two orbitals of opposite parity reveals that if parity acts nontrivially on the internal degrees of freedom, basis matrices become possible that are odd under parity; see Sec.\",\n \"REF .\",\n \"These basis matrices can combine with odd-in-momentum form factors to form even-parity superconducting states [41], [42].\",\n \"Odd-parity superconductivity is characterized by $u+$ irreps (odd under inversion, even under time reversal), which requires $g-$ basis matrices and $u-$ form factors $f_\\\\nu (\\\\mathbf {k})$ .\",\n \"All $u-$ irreps possess momentum-space basis functions and thus allow to write down such form factors.\",\n \"Moreover, inspection of multiplication tables for irreps shows that for any magnetic point group with inversion, the existence of one $g-$ basis matrix is sufficient to generate pairing states belonging to all $u+$ irreps.\",\n \"The IP nodes are determined by both the normal-state and the corresponding superconducting form factors, $c_n(\\\\mathbf {k})$ and $f_n(\\\\mathbf {k})$ , respectively, not by the superconducting factors alone.\",\n \"Specifically, the simple criterion $\\\\sum _n c_n(\\\\mathbf {k})\\\\, f_n(\\\\mathbf {k})=0$ diagnoses the presence of IP nodes.\",\n \"The position of the IP nodes is generically unaffected by nonlocal contributions to the extent that it is determined by symmetry.\",\n \"It is then only determined by the irrep of the pairing state.\",\n \"As a counterexample, for pairing belonging to the second ($3z^2-r^2$ ) component of $E_g$ for the point group $O_h$ , there are always two line nodes but these do not lie in high-symmetry planes and are only constrained to pass through the $\\\\langle 111\\\\rangle $ directions.\",\n \"Pairing states belonging to one-dimensional irreps can break TRS if the pairing has more than one contribution, which allows nontrivial phase factors.\",\n \"In the resulting TRS-breaking state, the symmetry-imposed line nodes of the time-reversal-symmetric state persist for infinitesimal pairing.\",\n \"The same mechanism is also possible for multidimensional irreps, in addition to the more natural case of phase factors between different components.\",\n \"If the pairing strength is not infinitesimal the point or line nodes can be inflated into BFSs.\",\n \"Like for local pairing [1], [2], the BFSs are given by the zeros of the Pfaffian $P(\\\\mathbf {k})$ of the BdG Hamiltonian, unitarily transformed into antisymmetric form.\",\n \"There are three possible cases for the inflated nodes: (1) they are forced to contain the original node and thus remain attached to the normal-state Fermi surface, (2) they do not remain attached to the original node and thus generically shift away from the normal-state Fermi surface with increasing pairing strength, or (3) inflated line nodes remain attached to the original line node only in high-symmetry directions (we have only observed the situation that the inflation also vanishes there).\",\n \"For increasing pairing, the BFSs typically grow and eventually merge.\",\n \"In case (2), the merged pockets can eventually shrink and finally disappear if this does not violate any remaining nonzero topological invariants they possess [11], [2].\",\n \"In real materials, this only seems likely if the BFSs are located inside small normal-state Fermi pocket(s) of a poor metal.\",\n \"It is an intriguing open question whether the resulting fully gapped superconducting state is topologically nontrivial.\",\n \"In principle, new BFSs can emerge at strong coupling, when quasiparticle bands are shifted through the Fermi energy.\",\n \"However, we expect that such BFSs are usually energetically disfavored and that the system can avoid them by developing a suitable momentum-dependent pairing amplitude.\",\n \"The case of an internal Hilbert space of dimension $N=4$ is special [9].\",\n \"We here only discuss the standard case where the internal degrees of freedom include the electron spin so that the transformation $P\\\\mathcal {T}$ squares to $-$ .\",\n \"Then, there are exactly six basis matrices $h_n$ with simple algebraic properties: One of them, $h_0\\\\propto $ , commutes with all others, which anticommute pairwise.\",\n \"This structures allows us to find relatively compact expressions for the Pfaffian $P(\\\\mathbf {k})$ in terms of the form factors $c_n(\\\\mathbf {k})$ and $f_n(\\\\mathbf {k})$ .\",\n \"For $N>4$ , there is no comparable algebraic structure and the Pfaffian could only be given explicitly in terms of the components of the transformed Hamiltonian.\",\n \"The number of terms in the resulting expression is exponentially large in $N$ .\",\n \"It is useful to review and compare our results for one-dimensional $A_{2g+}$ pairing and multidimensional $T_{1g+}$ pairing, which illustrate some of our general remarks.\",\n \"Note that it is hard to envision local degrees of freedom that realize an operator with $A_{2g+}$ symmetry, ultimately because of the high minimum order $l=6$ of basis functions.\",\n \"Hence, purely local pairing is unlikely to exist and it indeed does not appear in the examples we have considered.\",\n \"For infinitesimal pairing, the $A_{2g+}$ pairing state for any model has six symmetry-imposed line nodes in the $\\\\lbrace 110\\\\rbrace $ mirror planes.\",\n \"Since they are present in the real and imaginary parts of the gap function, they persist for TRS-broken states.\",\n \"We find that beyond infinitesimal pairing, the nodes of the TRS-broken $A_{2g+}$ states either persist as line nodes or are inflated into BFSs.\",\n \"Specifically, the nodes are inflated in the mirror planes for the electrons with effective spin $j=3/2$ but not for the cases of two s-orbitals and of two orbitals of opposite parity.\",\n \"The origin of this difference is that for the $j=3/2$ case, in the mirror planes, the two amplitudes $\\\\delta _{Eg+}$ and $\\\\delta _{T2g+}$ contribute to the Pfaffian whenever there is a phase difference between them.\",\n \"For the other two cases, there is a single amplitude in the mirror planes and its phase can be gauged away.\",\n \"Thus there is no inflation.\",\n \"A general insight here is that if only a single amplitude contributes in some high-symmetry direction or plane the breaking of TRS cannot lead to the formation of a BFS or of a gap since the physics is (gauge) invariant under changes of the phase of this amplitude.\",\n \"This argument also applies to multidimensional irreps.\",\n \"TRS-breaking pairing states are more natural for multidimensional irreps.\",\n \"In our context, $T_{1g+}$ is an interesting pairing symmetry.\",\n \"Whereas it appears for purely local pairing in the case of two s-orbitals, it only appears for nonlocal pairing for effective-spin-$3/2$ fermions.\",\n \"Our study suggests that for both cases all point and line nodes appearing for TRS-broken $T_{1g+}$ pairing state with order parameter $(1,i,0)$ are inflated for noninfinitesimal pairing.\",\n \"The point nodes on the $k_z$ -axis are the only ones which remains attached to the normal-state Fermi surface, while all other point and line nodes are shifted away from the normal-state Fermi surface and thus could annihilate for strong coupling.\",\n \"To conclude, nonlocal pairing typically permits a much larger number of possible pairing symmetries.\",\n \"Their nodal structure, including the possibility of BFSs, can be analyzed based on symmetry.\",\n \"For nodes at infinitesimal pairing, there is a simple yet powerful criterion in terms of a scalar product of form factors.\",\n \"The known criterion for the appearance of BFSs in terms of a Pfaffian extends to nonlocal pairing and general internal degrees of freedom and shows that BFSs generically exist if the superconducting state breaks TRS.\",\n \"The authors thank D. F. Agterberg, P. M. R. Brydon, I. F. Herbut, A. Knoll, S. Kobayashi, and J. M. Link for useful discussions.\",\n \"Financial support by the Deutsche Forschungsgemeinschaft through the Collaborative Research Center SFB 1143, Project A04, the Research Training Group GRK 1621, and the Cluster of Excellence on Complexity and Topology in Quantum Matter ct.qmat (EXC 2147) is gratefully acknowledged.\"\n ],\n [\n \"Enumeration of basis matrices\",\n \"In this Appendix, we obtain the number of basis matrices that can occur and thus generically do occur in the normal-state Hamiltonian $H_N(\\\\mathbf {k})$ .\",\n \"The same basis matrices can appear in the pairing matrix $D(\\\\mathbf {k})$ for $s_T=-1$ and even-parity superconductivity as well as for $s_T=+1$ and odd-parity superconductivity, while in the other two cases, only the remaining basis matrices can appear in $D(\\\\mathbf {k})$ .\",\n \"The statements can be obtained in a more general framework but it might be useful to present them using representation theory of point groups.\",\n \"The following theorem is proven: Let $N$ be the dimension of the Hilbert space describing the local degrees of freedom in the normal state.\",\n \"Let $P$ be the unitary parity operator on this Hilbert space and let $\\\\mathcal {T}$ be the antiunitary time-reversal operator.\",\n \"Then the number of Hermitian basis matrices $h_n$ that can appear in a normal-state Hamiltonian that respects $P\\\\mathcal {T}$ symmetry is $n_h = \\\\left\\\\lbrace \\\\begin{array}{ll}\\\\displaystyle \\\\frac{N(N+1)}{2} & \\\\mbox{for $(P\\\\mathcal {T})^2 = +1$} , \\\\\\\\[1.5ex]\\\\displaystyle \\\\frac{N(N-1)}{2} & \\\\mbox{for $(P\\\\mathcal {T})^2 = -1$} .\\\\end{array} \\\\right.$ As we shall see, the case $(P\\\\mathcal {T})^2 = -1$ can only occur for even $N$ .\",\n \"Results for small $N$ are given in Table REF .\",\n \"Table: Number of basis matrices h n h_n that appear in a normal-state Hamiltonian H N (𝐤)H_N(\\\\mathbf {k}) with P𝒯P\\\\mathcal {T} symmetry for the two cases that P𝒯P\\\\mathcal {T} squares to ±1\\\\pm 1.\",\n \"NN is the dimension of the internal Hilbert space.\",\n \"For (P𝒯) 2 =-1(P\\\\mathcal {T})^2=-1, which isthe standard case, NN has to be even.To show this, the normal-state Hamiltonian is expanded as $H_N(\\\\mathbf {k}) = \\\\sum _{n=1}^{n_h} c_n(\\\\mathbf {k})\\\\, h_n .$ Under the magnetic point group $M$ , the functions $c_n(\\\\mathbf {k})$ must transform like the corresponding $h_n$ to ensure that $H_N(\\\\mathbf {k})$ is invariant.\",\n \"The operation $P\\\\mathcal {T}$ is an element of $M$ .\",\n \"The irreps of $M$ can be uniquely and exhaustively divided into irreps that are even or odd under $P\\\\mathcal {T}$ .\",\n \"Only the $P\\\\mathcal {T}$ -even irreps have momentum-space basis functions since the momentum $\\\\mathbf {k}$ is $P\\\\mathcal {T}$ even.\",\n \"Hence, all $P\\\\mathcal {T}$ -even and no $P\\\\mathcal {T}$ -odd $h_\\\\nu $ can occur in Eq.\",\n \"(REF ).\",\n \"The proposition is thus a statement about the number $n_h$ of $N\\\\times N$ basis matrices $h_\\\\nu $ that are even under $P\\\\mathcal {T}$ .\",\n \"Since $P\\\\mathcal {T}$ is antiunitary there exists a unitary $N\\\\times N$ matrix $U_{PT}$ with $P\\\\mathcal {T} = U_{PT} \\\\mathcal {K}$ , where $\\\\mathcal {K}$ is the complex conjugation.\",\n \"Then we have $(P\\\\mathcal {T})^2 = U_{PT} \\\\mathcal {K} U_{PT} \\\\mathcal {K} = U_{PT} U_{PT}^* .$ Thus $(P\\\\mathcal {T})^2 = \\\\pm 1$ is equivalent to $U_{PT} U_{PT}^* = \\\\pm 1$ and, since $U_{PT}^*$ is unitary, to $U_{PT} = \\\\pm (U_{PT}^*)^{-1} = \\\\pm U_{PT}^T$ .\",\n \"Case 1: $(P\\\\mathcal {T})^2 = +1$ and symmetric $U_{PT}$ .\",\n \"For any unitary symmetric $N\\\\times N$ matrix $U_{PT}$ , there exists a unitary matrix $Q$ such that $U_{PT} = Q Q^T .$ $h_\\\\nu $ being even/odd under $P\\\\mathcal {T}$ means $U_{PT} h_\\\\nu ^* U_{PT}^\\\\dagger = \\\\pm h_\\\\nu $ , which due to the Hermiticity of $h_\\\\nu $ is equivalent to $U_{PT} h_\\\\nu ^T U_{PT}^\\\\dagger = \\\\pm h_\\\\nu .$ Equation (REF ) then gives $Q Q^T h_\\\\nu ^T Q^* Q^\\\\dagger = \\\\pm h_\\\\nu ,$ which is equivalent to $Q^T h_\\\\nu ^T Q^* = (Q^\\\\dagger h_\\\\nu Q)^T = \\\\pm Q^\\\\dagger h_\\\\nu Q .$ Note that $k_\\\\nu \\\\equiv Q^\\\\dagger h_\\\\nu Q$ is Hermitian.\",\n \"The dimension of the vector space over $\\\\mathbb {R}$ of Hermitian $N\\\\times N$ matrices that are also symmetric is ${N(N+1)}/{2}$ .\",\n \"Hence, the dimension of the vector space of Hermitian $P\\\\mathcal {T}$ -even $N\\\\times N$ and thus the number of $P\\\\mathcal {T}$ -even basis elements $h_\\\\nu $ also equals ${N(N+1)}/{2}$ .\",\n \"Analogously, the dimension of the space of Hermitian $N\\\\times N$ matrices that are antisymmetric and thus the number of $P\\\\mathcal {T}$ -odd basis matrices equals ${N(N-1)}/{2}$ .\",\n \"Note that the sum of the two numbers is $N^2$ , as expected.\",\n \"Case 2: $(P\\\\mathcal {T})^2 = -1$ and antisymmetric $U_{PT}$ .\",\n \"For any unitary antisymmetric $N\\\\times N$ matrix $U_{PT}$ , there exists a unitary matrix $Q$ such that $U_{PT} = Q \\\\Lambda Q^T ,$ with $\\\\Lambda = i\\\\sigma _2 \\\\otimes ,$ which clearly means that $N$ must be even.\",\n \"We therefore write $N=2M$ .\",\n \"$h_\\\\nu $ being even/odd under $P\\\\mathcal {T}$ means $U_{PT} h_\\\\nu ^T U_{PT}^\\\\dagger = \\\\pm h_\\\\nu .$ Equation (REF ) gives $Q \\\\Lambda Q^T h_\\\\nu ^T Q^* \\\\Lambda ^\\\\dagger Q^\\\\dagger = \\\\pm h_\\\\nu ,$ which is equivalent to $\\\\Lambda Q^T h_\\\\nu ^T Q^* \\\\Lambda ^\\\\dagger = \\\\Lambda (Q^\\\\dagger h_\\\\nu Q)^T\\\\Lambda ^\\\\dagger = \\\\pm Q^\\\\dagger h_\\\\nu Q .$ Let $k_\\\\nu \\\\equiv Q^\\\\dagger h_\\\\nu Q$ (Hermitian).\",\n \"We write $\\\\Lambda $ and $k_\\\\nu $ in block form as $\\\\Lambda &= \\\\left(\\\\begin{array}{cc}0 & \\\\\\\\ -& 0\\\\end{array}\\\\right) , \\\\\\\\k_\\\\nu &= \\\\left(\\\\begin{array}{cc}\\\\kappa _{11} & \\\\kappa _{12} \\\\\\\\ \\\\kappa _{12}^\\\\dagger & \\\\kappa _{22}\\\\end{array}\\\\right) ,$ where $\\\\kappa _{11}$ and $\\\\kappa _{22}$ are Hermitian.\",\n \"Equation (REF ) can then be written as $\\\\Lambda k_\\\\nu ^T \\\\Lambda ^\\\\dagger = \\\\begin{pmatrix}\\\\kappa _{22}^T & -\\\\kappa _{12}^T \\\\\\\\ -\\\\kappa _{12}^* & \\\\kappa _{11}^T\\\\end{pmatrix} = \\\\begin{pmatrix}\\\\pm \\\\kappa _{11} & \\\\pm \\\\kappa _{12} \\\\\\\\ \\\\pm \\\\kappa _{12}^\\\\dagger & \\\\pm \\\\kappa _{22}\\\\end{pmatrix} = \\\\pm k_\\\\nu .$ This yields the relations $\\\\kappa _{12}^T = \\\\mp \\\\kappa _{12}$ and $\\\\kappa _{22} = \\\\pm \\\\kappa _{11}^T$ , and $k_\\\\nu $ thus assumes the form $k_\\\\nu = \\\\begin{pmatrix}\\\\kappa _{11} & \\\\kappa _{12} \\\\\\\\ \\\\kappa _{12}^\\\\dagger & \\\\pm \\\\kappa _{11}^T\\\\end{pmatrix} ,$ with $\\\\kappa _{11}^\\\\dagger = \\\\kappa _{11}$ and $\\\\kappa _{12}^T = \\\\mp \\\\kappa _{12}$ .\",\n \"The blocks are $M\\\\times M$ matrices.\",\n \"The dimension of the vector space spanned by the $k_\\\\nu $ is $M^2 + M(M-1) = M(2M-1)$ for the upper sign and $M^2 + M(M+1) = M(2M+1)$ for the lower sign.\",\n \"The sum is $4M^2 = N^2$ , as expected.\",\n \"Hence, the dimension of the vector space of Hermitian $N\\\\times N$ matrices that are even under $P\\\\mathcal {T}$ is ${N(N-1)}/{2}$ , whereas the dimension for $P\\\\mathcal {T}$ -odd matrices is ${N(N+1)}/{2}$ .\",\n \"Note that the two numbers are interchanged compared to the case of $(P\\\\mathcal {T})^2 = +1$ .\",\n \"This completes the proof.\",\n \"We are concerned with systems that satisfy TRS and inversion symmetry separately.\",\n \"Then the $P\\\\mathcal {T}$ -even (odd) irreps are the $g+$ and $u-$ ($g-$ and $u+$ ) irreps.\",\n \"Moreover, the two operations commute [19], [29] and $P$ squares to $+1$ .\",\n \"Hence, $(P\\\\mathcal {T})^2 = \\\\mathcal {T}^2$ .\",\n \"If the internal degrees of freedom include the electron spin we have $\\\\mathcal {T}^2 = -1$ [19] and even dimension $N$ .\",\n \"The case $\\\\mathcal {T}^2 = +1$ can only be realized if the spin does not occur explicitly, for example because one spin state is pushed to high energies by a magnetic field.\",\n \"Then $\\\\mathcal {T}$ is not the physical TRS but an effective antiunitary symmetry.\"\n ],\n [\n \"Infinitesimal-pairing nodes\",\n \"In this Appendix, we discuss the IP nodes for $s_T=-1$ and odd-parity pairing and for the unconventional sign $s_T=+1$ of time reversal squared.\",\n \"For $s_T=-1$ and odd-parity pairing, it is still true that infinitesimal pairing can be described in a single-band, pseudospin picture.\",\n \"However, it is now in the pseudospin-triplet channel.\",\n \"The pairing matrix in the effective single-band picture thus has the form $D_\\\\mathrm {eff}(\\\\mathbf {k}) = \\\\mathbf {d}(\\\\mathbf {k}) \\\\cdot \\\\mbox{$\\\\sigma $}$ , where $\\\\mbox{$\\\\sigma $}$ is the vector of Pauli matrices representing the pseudospin.\",\n \"Since the pseudospin is even under inversion and odd under time reversal its components belong to one or more $g-$ irreps.\",\n \"One can use representation theory to work out which irreps the components of $\\\\mathbf {d}(\\\\mathbf {k})$ must belong to in order to obtain a pairing state of a certain symmetry.\",\n \"The condition $\\\\mathbf {d}(\\\\mathbf {k})=0$ then gives the symmetry-imposed IP nodes.\",\n \"Nodal gaps are thus less likely than for singlet pairing since they must satisfy three scalar conditions.\",\n \"We now turn to the nonstandard case $s_T=+1$ .\",\n \"According to Appendix , this allows an effective single-band model with Hilbert-space dimension $N=1$ .\",\n \"Equation (REF ) then implies that $U_T=1$ and thus $\\\\Delta (\\\\mathbf {k})=D(\\\\mathbf {k})$ .\",\n \"The only Hermitian basis matrix is $h_0=1$ .\",\n \"Hence, for a single-band model, the full symmetry information is carried by the form factor $f_0(\\\\mathbf {k})$ .\",\n \"$h_0$ belongs to the trivial irrep, which of course is a $g+$ irrep.\",\n \"Table REF then shows that for $N=1$ only odd-parity pairing with $u-$ form factor $f_0(\\\\mathbf {k})$ is possible.\",\n \"The analysis of possible pairing states is analogous to the case with $s_T=-1$ and even parity, except that $g+$ irreps are replaced by $u-$ irreps.\",\n \"Even-parity pairing states for $s_T=+1$ cannot be described by an effective $N=1$ model but are possible for $N=2$ .\",\n \"In fact, there are multiple possibilities to implement this case because Eq.\",\n \"(REF ) now allows $U_T$ to be any symmetric unitary $2\\\\times 2$ matrix, while $U_P$ can be any unitary $2\\\\times 2$ matrix that squares to $$ .\",\n \"The specific $U_T$ and $U_P$ and thus the symmetry properties of the $2\\\\times 2$ basis matrices $h_1$ , $h_2$ , $h_3$ (which are linear combinations of Pauli matrices) depend on the underlying system.\",\n \"Universal properties are therefore unlikely and we do not pursue this here.\"\n ],\n [\n \"Existence and properties of the Pfaffian\",\n \"In this Appendix, we review the main results for the Pfaffian.\",\n \"A simpler proof than in [1], [2] is presented.\",\n \"The BdG Hamiltonian (REF ) satisfies the charge-conjugation symmetry $\\\\mathcal {U}_C \\\\mathcal {H}^T(-\\\\mathbf {k}) \\\\mathcal {U}_C^\\\\dagger = - \\\\mathcal {H}(\\\\mathbf {k})$ , where $\\\\mathcal {U}_C = \\\\sigma _1 \\\\otimes $ .\",\n \"For it to also satisfy inversion symmetry, there must exist a unitary matrix $U_P$ such that $\\\\mathcal {U}_P\\\\, \\\\mathcal {H}(-\\\\mathbf {k})\\\\, \\\\mathcal {U}_P^\\\\dagger = \\\\mathcal {H}(\\\\mathbf {k}) ,$ where $\\\\mathcal {U}_P = \\\\begin{pmatrix}U_P & 0 \\\\\\\\ 0 & U_P^*\\\\end{pmatrix} .$ This is a special case of Eqs.\",\n \"(REF ) and (REF ).\",\n \"The two symmetries imply $\\\\mathcal {C}P$ symmetry, $\\\\mathcal {U}_{CP}\\\\, \\\\mathcal {H}^T(\\\\mathbf {k})\\\\, \\\\mathcal {U}_{CP}^\\\\dagger =- \\\\mathcal {H}(\\\\mathbf {k}) ,$ with $\\\\mathcal {U}_{CP} = \\\\mathcal {U}_C \\\\mathcal {U}_P^*$ .\",\n \"We find that $\\\\mathcal {C}P$ squares to the identity since $(\\\\mathcal {U}_{CP} \\\\mathcal {K})^2 = \\\\mathcal {U}_{CP} \\\\mathcal {U}_{CP}^*= \\\\begin{pmatrix}U_P^2 & 0 \\\\\\\\ 0 & (U_P^*)^2\\\\end{pmatrix}= \\\\begin{pmatrix}& 0 \\\\\\\\ 0 & \\\\end{pmatrix} .$ This implies that $\\\\mathcal {U}_{CP} = (\\\\mathcal {U}_{CP}^*)^{-1} = (\\\\mathcal {U}_{CP}^*)^\\\\dagger = \\\\mathcal {U}_{CP}^T$ so that $\\\\mathcal {U}_{CP}$ is symmetric.\",\n \"For any (complex) symmetric matrix $\\\\mathcal {U}_{CP}$ , there exists a unitary matrix $\\\\Omega $ such that $\\\\Lambda = \\\\Omega \\\\mathcal {U}_{CP} \\\\Omega ^T$ (note the transpose) is a diagonal matrix with real nonnegative components (Autonne-Takagi factorization [44], [45]).\",\n \"$\\\\Lambda $ is evidently unitary.\",\n \"A diagonal unitary matrix with nonnegative components must be $\\\\Lambda =$ .\",\n \"We thus obtain $\\\\mathcal {U}_{CP} = \\\\Omega ^\\\\dagger \\\\Omega ^*$ and Eq.\",\n \"(REF ) becomes $\\\\Omega ^\\\\dagger \\\\Omega ^*\\\\, \\\\mathcal {H}^T(\\\\mathbf {k})\\\\, \\\\Omega ^T \\\\Omega =- \\\\mathcal {H}(\\\\mathbf {k}) .$ This implies that $\\\\Omega ^*\\\\, \\\\mathcal {H}^T(\\\\mathbf {k})\\\\, \\\\Omega ^T= - \\\\Omega \\\\, \\\\mathcal {H}(\\\\mathbf {k})\\\\, \\\\Omega ^\\\\dagger .$ Hence, $\\\\tilde{\\\\mathcal {H}}(\\\\mathbf {k}) \\\\equiv \\\\Omega \\\\, \\\\mathcal {H}(\\\\mathbf {k})\\\\, \\\\Omega ^\\\\dagger $ is antisymmetric: $\\\\tilde{\\\\mathcal {H}}^T(\\\\mathbf {k}) = - \\\\tilde{\\\\mathcal {H}}(\\\\mathbf {k}) .$ This shows that for systems with inversion symmetry, the BdG Hamiltonian can always be unitarily transformed into antisymmetric form.\",\n \"For the antisymmetric matrix $\\\\tilde{\\\\mathcal {H}}(\\\\mathbf {k})$ , the Pfaffian $\\\\mathop {\\\\textrm {Pf}}\\\\tilde{\\\\mathcal {H}}(\\\\mathbf {k})$ is well defined.\",\n \"Note that $\\\\Omega $ is independent of $\\\\mathbf {k}$ (and is in fact solely determined by $\\\\mathcal {U}_{CP}$ ) so that the components of $\\\\tilde{\\\\mathcal {H}}(\\\\mathbf {k})$ are smooth functions of $\\\\mathbf {k}$ .\",\n \"Hence, the Pfaffian, which is a polynomial of these components, is an smooth function of $\\\\mathbf {k}$ .\",\n \"The sign of the Pfaffian is inverted under simultaneous interchange of two rows and the corresponding two columns.\",\n \"This is a unitary transformation, which can be absorbed into $\\\\Omega $ .\",\n \"Hence, the above derivation only determines the Pfaffian up to a sign.\",\n \"If the dimension $N$ of the local Hilbert space is even the Pfaffian is real: The dimension $2N$ of the BdG Hamiltonian $\\\\tilde{\\\\mathcal {H}}(\\\\mathbf {k})$ is a multiple of four and thus the Pfaffian is a polynomial of even degree of its the components.\",\n \"Since $\\\\tilde{\\\\mathcal {H}}(\\\\mathbf {k})$ is Hermitian and antisymmetric these components are purely imaginary.\",\n \"Conversely, for odd $N$ , the Pfaffian is purely imaginary.\",\n \"We define $P(\\\\mathbf {k}) \\\\equiv \\\\left\\\\lbrace \\\\begin{array}{ll}\\\\mathop {\\\\textrm {Pf}}\\\\tilde{\\\\mathcal {H}}(\\\\mathbf {k}) & \\\\mbox{for $N$ even,} \\\\\\\\i\\\\mathop {\\\\textrm {Pf}}\\\\tilde{\\\\mathcal {H}}(\\\\mathbf {k}) & \\\\mbox{for $N$ odd}\\\\end{array} \\\\right.$ so that $P(\\\\mathbf {k})$ is always real.\",\n \"Due to $\\\\mathcal {C}P$ symmetry, the spectrum of the BdG Hamiltonian is symmetric.\",\n \"We thus have $\\\\det \\\\mathcal {H}(\\\\mathbf {k}) = (-1)^N \\\\prod _{j=1}^N E_j^2(\\\\mathbf {k}) ,$ where $\\\\pm E_j(\\\\mathbf {k})$ are the quasiparticle energies.\",\n \"We assume $E_j(\\\\mathbf {k})\\\\ge 0$ without loss of generality.\",\n \"The determinant equals the square of the Pfaffian.\",\n \"This implies that $P(\\\\mathbf {k}) = \\\\pm \\\\prod _{j=1}^N E_j(\\\\mathbf {k}) .$ At any momentum $\\\\mathbf {k}$ not on a node, $P(\\\\mathbf {k})$ is strictly positive or negative.\",\n \"We now choose $\\\\Omega $ in such a way that in the normal state $P(\\\\mathbf {k}_\\\\infty )$ is positive at some momentum $\\\\mathbf {k}_\\\\infty $ far from the normal-state Fermi surface.\",\n \"Then for not too large superconducting energy scale, the energies $E_j(\\\\mathbf {k}_\\\\infty )$ do not change sign when superconductivity is switched on and $P(\\\\mathbf {k}_\\\\infty )$ remains positive.\",\n \"Hence, at $\\\\mathbf {k}_\\\\infty $ , the sign in Eq.\",\n \"(REF ) is $+$ .\",\n \"Since the Pfaffian and thus $P(\\\\mathbf {k})$ are smooth functions of momentum $P(\\\\mathbf {k})$ can only change sign at nodes.\",\n \"Conversely, if $P(\\\\mathbf {k})$ is negative somewhere in the Brillouin zone there must be a surface of zeros, i.e., a BFS, separating the regions of positive and negative $P(\\\\mathbf {k})$ .\",\n \"To determine under what conditions $P(\\\\mathbf {k})$ can actually become negative, we need to analyze Eq.\",\n \"(REF ).\",\n \"The eigenenergies $E_j(\\\\mathbf {k})$ are continuous functions and are smooth, except at zeros and potentially at crossing points.\",\n \"If a single $E_j(\\\\mathbf {k})$ approaches zero linearly the smoothness of $P(\\\\mathbf {k})$ and the choice $E_j(\\\\mathbf {k})\\\\ge 0$ requires the explicit sign in Eq.\",\n \"(REF ) to flip.\",\n \"Hence, $P(\\\\mathbf {k})$ changes sign and there must be a closed BFS.\",\n \"On the other hand, if two eigenenergies approach zero linearly and simultaneously, for example because they are degenerate, the explicit sign does not flip.\",\n \"In this case, $P(\\\\mathbf {k})$ does not change sign at the zero but has a second-order zero there.\",\n \"The same applies if a single eigenenergy approaches zero quadratically.\",\n \"If the Pfaffian does not change sign the $\\\\mathbb {Z}_2$ topological invariant, which is the relative sign of $P(\\\\mathbf {k})$ [1], [2], exists but is trivial.\",\n \"This makes BFSs unstable since for any low-symmetry momentum $\\\\mathbf {k}$ with $P(\\\\mathbf {k})=0$ , an infinitesimal change of parameters can make $P(\\\\mathbf {k})$ strictly positive.\",\n \"For $s_T = -1$ , which implies even $N$ , and preserved TRS, Kramers' theorem [43], [19], [29] shows that the spectrum has twofold degeneracy for all $\\\\mathbf {k}$ .\",\n \"Then, the latter case applies, $P(\\\\mathbf {k})$ does not change sign, and BFSs are not stable.\",\n \"On the other hand, for $s_T = +1$ or broken TRS, there is no mechanism that leads to twofold degeneracy everywhere, the $P(\\\\mathbf {k})$ generically changes sign, and BFSs are stable.\"\n ],\n [\n \"The algebra of basis matrices\",\n \"As shown in Appendix , for $N=4$ and $(P\\\\mathcal {T})^2=-1$ , six basis matrices $h_0$ , ..., $h_5$ appear in the normal-state Hamiltonian.\",\n \"One of them is (proportional to) the identity matrix, as discussed in Sec.\",\n \".\",\n \"We choose $h_0 = $ .\",\n \"In this Appendix, we show that the basis matrices can always be chosen in such a way that they satisfy the generalized commutation relations $h_0 h_n = h_n h_0$ for $n = 1,2,3,4,5$ and $h_m h_n = -h_n h_m$ for $m, n = 1,2,3,4,5$ and $m\\\\ne n$ .\",\n \"It is well known that (several sets of) six $4\\\\times 4$ matrices with these properties exist [28], [29].\",\n \"The point here is that the basis matrices always realize this structure.\",\n \"The matrices $h_n$ can be written as $k_n = Q^\\\\dagger h_n Q$ , with $Q$ unitary, where the $k_n$ satisfy Eq.\",\n \"(REF ) with the upper sign and $\\\\kappa _{11}^\\\\dagger = \\\\kappa _{11}$ and $\\\\kappa _{12}^T = - \\\\kappa _{12}$ .\",\n \"For $N=4$ , $\\\\kappa _{11}$ and $\\\\kappa _{12}$ are $2\\\\times 2$ matrices.\",\n \"Then, $\\\\kappa _{11}$ has to be a linear combination of $\\\\sigma _0$ , $\\\\sigma _1$ , $\\\\sigma _2$ , $\\\\sigma _3$ with real coefficients and $\\\\kappa _{12}$ can be $\\\\sigma _2$ with an arbitrary complex prefactor.\",\n \"A maximal set of linearly independent matrices is then $k_0 &= \\\\left(\\\\begin{array}{cc}\\\\sigma _0 & 0 \\\\\\\\ 0 & \\\\sigma _0\\\\end{array}\\\\right) = \\\\sigma _0 \\\\otimes \\\\sigma _0 , \\\\\\\\k_1 &= \\\\left(\\\\begin{array}{cc}\\\\sigma _1 & 0 \\\\\\\\ 0 & \\\\sigma _1\\\\end{array}\\\\right) = \\\\sigma _0 \\\\otimes \\\\sigma _1 , \\\\\\\\k_2 &= \\\\left(\\\\begin{array}{cc}\\\\sigma _2 & 0 \\\\\\\\ 0 & -\\\\sigma _2\\\\end{array}\\\\right) = \\\\sigma _3 \\\\otimes \\\\sigma _2 , \\\\\\\\k_3 &= \\\\left(\\\\begin{array}{cc}\\\\sigma _3 & 0 \\\\\\\\ 0 & \\\\sigma _3\\\\end{array}\\\\right) = \\\\sigma _0 \\\\otimes \\\\sigma _3 , \\\\\\\\k_4 &= \\\\left(\\\\begin{array}{cc}0 & \\\\sigma _2 \\\\\\\\ \\\\sigma _2 & 0\\\\end{array}\\\\right) = \\\\sigma _1 \\\\otimes \\\\sigma _2 , \\\\\\\\k_5 &= \\\\left(\\\\begin{array}{cc}0 & -i\\\\sigma _2 \\\\\\\\ i\\\\sigma _2 & 0\\\\end{array}\\\\right) = \\\\sigma _2 \\\\otimes \\\\sigma _2 .$ These matrices satisfy $k_0 k_n = k_n k_0$ for $n = 1,2,3,4,5$ and $k_m k_n = -k_n k_m$ for $m, n = 1,2,3,4,5$ and $m\\\\ne n$ .\",\n \"Moreover, they are orthonormal with respect to the scalar product $\\\\mathop {\\\\textrm {Tr}}k_m k_n$ .\",\n \"The basis matrices $h_n$ , $n=0,\\\\ldots ,5$ , are related to the $k_n$ by a unitary transformation.\",\n \"Since the $k_n$ satisfy the generalized commutation relations so do the $h_n$ .\",\n \"The upshot is that while the specific form of the basis matrices $h_n$ depends on the model, their algebra does not.\",\n \"This result extends to general Hilbert-space dimension $N$ but the algebraic properties are more complicated for $N>4$ .\"\n ],\n [\n \"Pfaffian for the four-dimensional case\",\n \"Here, we briefly discuss the analytical expression for the Pfaffian $P(\\\\mathbf {k})$ of the transformed BdG Hamiltonian $\\\\tilde{\\\\mathcal {H}}(\\\\mathbf {k})$ for the case of $s_T=-1$ , $N=4$ , and even-parity pairing.\",\n \"The Pfaffian exists and can be chosen to be a smooth function of momentum $\\\\mathbf {k}$ , as shown in Appendix .\",\n \"Then the property $P^2(\\\\mathbf {k}) = \\\\det \\\\mathcal {H}(\\\\mathbf {k})$ fixes $P(\\\\mathbf {k})$ up to an overall sign.\",\n \"As discussed in Appendix , we choose this sign so that $P(\\\\mathbf {k})>0$ far from the normal-state Fermi surface.\",\n \"If the superconducting energy scale is not too large, the Pfaffian is given in terms of the coefficients in Eq.\",\n \"(REF ) as $P&(\\\\mathbf {k}) = \\\\langle c,c\\\\rangle ^2 + \\\\langle f^1,f^1\\\\rangle ^2+ \\\\langle f^2,f^2\\\\rangle ^2 \\\\nonumber \\\\\\\\&{}+ 4\\\\, \\\\big ( \\\\langle c,f^1\\\\rangle ^2 + \\\\langle f^1,f^2\\\\rangle ^2+ \\\\langle f^2,c\\\\rangle ^2 \\\\big ) \\\\nonumber \\\\\\\\&{}- 2\\\\, \\\\big ( \\\\langle c,c\\\\rangle \\\\, \\\\langle f^1,f^1\\\\rangle + \\\\langle f^1,f^1\\\\rangle \\\\, \\\\langle f^2,f^2\\\\rangle + \\\\langle f^2,f^2\\\\rangle \\\\, \\\\langle c,c\\\\rangle \\\\big ) ,$ with the Minkowski-type scalar product $\\\\langle A,B\\\\rangle \\\\equiv A_0 B_0 - \\\\sum _{n=1}^5 A_n B_n .$ This proposition is proved by evaluating $P^2(\\\\mathbf {k})$ and showing that it agrees with the determinant of the BdG Hamiltonian.\",\n \"This cumbersome calculation can be simplified by realizing that the Pfaffian is invariant under simultaneous rotations of the five-vectors $\\\\vec{c} &= (c_1,c_2,c_3,c_4,c_5) , \\\\\\\\\\\\vec{f}^{\\\\,1} &= (f^1_1,f^1_2,f^1_3,f^1_4,f^1_5) , \\\\\\\\\\\\vec{f}^{\\\\,2} &= (f^2_1,f^2_2,f^2_3,f^2_4,f^2_5) .$ The sign of $P(\\\\mathbf {k})$ is also correct: The assumption of not too large superconducting energy scale means that the $f^1_n$ and $f^2_n$ for $n=0,\\\\ldots ,5$ are small compared to the $c_n$ far from the normal-state Fermi energy.\",\n \"Then, at such momenta we get $P(\\\\mathbf {k}) \\\\cong \\\\langle c,c\\\\rangle ^2 > 0$ .\",\n \"For large superconducting energy scale, the whole Brillouin zone is affected by superconductivity and we cannot choose the sign of $P(\\\\mathbf {k})$ by continuity from the normal state.\",\n \"This simply means that there is no useful distinction between the inside and the outside of the BFS.\",\n \"The conclusions of this paper remain valid, though.\"\n ]\n]"},"id":{"kind":"string","value":"2107.01839"}}},{"rowIdx":1591,"cells":{"text":{"kind":"list like","value":[["Weinhold geometry and thermodynamics of Bardeen AdS black holes"],["Abstract Thermodynamics of Bardeen AdS black holes has attracted a great deal of attentions due to its intrinsic complications and rich phase structures.","However, the entropy and thermodynamic volume are incorrect in some literatures.","In this paper we revisit the thermodynamics of Bardeen AdS black holes and provide the correct entropy and thermodynamic volume.","Furthermore, thermodynamic geometries are a powerful tool to probe the microstructure of black holes.","Based on the Hessian matrix of black hole mass, we introduce a thermodynamic metric and give its scalar curvature in Weinhold's geometry.","The conformal relation between Weinhold's geometry and Ruppeiner's geometry will be changed due to the specific first law of thermodynamics for regular black holes like the Bardeen AdS black hole.","We also investigate the critical behaviour of phase transitions in an extended phase space, and find that the critical behaviour of the Bardeen AdS black hole coincides with that of liquid-gas systems.","In particular, based on the Weinhold geometry we unveil a repulsive interaction in the microstructure of the Bardeen AdS black hole under its small volume state, while it is known that only attractive interaction exists in the microstructure of the van der Waals fluid."],["Introduction","In the past decades, a complete statistical mechanics that describes the microstructure of black holes was still lacking although there was a wealth of work in the thermodynamics of black holes.","A geometric description of thermodynamic systems may provide an alternative attempt to probe the microstructure of black holes.","Weinhold developed [1] the mass (energy) representation by constructing a linear vector space with a full metric structure, where the vector space can be constructed directly from empirical laws of thermodynamics.","For each extensity $X^{\\mu }$ of a thermodynamic system, its conjugate field variable $v_{\\mu }$ is defined by $v_{\\mu }\\equiv \\partial M /\\partial X^{\\mu },$ which constitutes a thermodynamic phase space together with $X^{\\mu }$ that is required to have the properties of a vector space.","Thus, an abstract space having the full properties of inner product on the thermodynamic phase space is given, which implies that the thermodynamic laws can show the underlying structure of a geometric object.","Based on the fluctuation theory of thermodynamics, on the other hand, Ruppeiner introduced [2] the entropy representation, which shows that thermodynamic systems can be described by Riemannian manifolds.","Quite interesting is that there exists [3] a conformal equivalence between Weinhold'geometry and Ruppeiner's geometry.","In information geometry, a parameterized statistical model is considered as a Riemannian manifold in which the corresponding probability distributions can be defined.","In this view, Ruppeiner's geometry is one of information theories with the probability distribution [4], $\\mathcal {P}(X)\\propto \\sqrt{g^R}\\, e^{-\\frac{1}{2}g^R_{\\mu \\nu }X^{\\mu }X^{\\nu }},$ where $g^R_{\\mu \\nu }$ is the Ruppeiner metric and ${g^R}$ is the determinant of the metric.","Thermodynamic geometries have been extensively studied and applied [5], [6], [7], [8], [9], [10], [11], [12] to numerous systems.","In particular, the interpretations of thermodynamic curvatures are linked [13] to the interactions in the microstructure of ideal quantum gases.","In recent years, Ruppeiner's geometry has promoted substantially the connection between microstructure and thermodynamics of black holes, furthermore, such a connection has been shown [14], [15], [16], [17] to be analogous to that in the van der Waals fluid.","These interpretations are new attempts to extract information from the thermodynamic geometry of black holes.","The thermodynamic curvature does not depend [18] on the thermodynamic metric since it is an invariant for a given thermodynamic state, which has recently been verified [19], [20] again in two new phase spaces.","As is known, there is only an attractive interaction in the van der Waals fluid.","However, there exists [21] a weak repulsive interaction in the microstructure of charged AdS black holes.","We may argue whether the existence of a weak repulsive interaction has something to do with the singularity of charged AdS black hole spacetimes.","On the other hand, the current thermodynamic geometries are generally based on the first law of black hole mechanics associated with linear electrodynamics.","They cannot be applied directly to interpret the thermodynamic geometry of regular black holes that have no singularity.","We need to construct the thermodynamic geometry in the context of the nonlinear electrodynamics, which is able to give an interpretation of thermodynamic geometry for regular black holes.","It is challenging to generalize thermodynamic geometry to regular black holes due to the intrinsic complications in the configurations of the first law and the Lagrangian coupled with nonlinear electrodynamics.","In this paper, we study the Weinhold geometry of regular black holes in the context of nonlinear electrodynamics.","Our main task is to construct the Weinhold geometry by using the normalized scalar curvature [21] and reveal the behaviour of thermodynamic curvatures through macroscopic properties.","The outline of this paper is as follows.","In Sec.","we review briefly the properties of thermodynamic geometry and then give the general form of Weinhold scalar curvatures.","Next, in Sec.","we derive the entropy of Bardeen-AdS black holes in terms of the modified integration approach, and demonstrate the critical behaviour of thermodynamics in Bardeen-AdS black holes and the analogous behaviour in liquid-gas systems.","In Sec.","we compute the thermodynamic curvatures of Bardeen-AdS black holes in terms of Weinhold's geometry and Ruppeiner's geometry and make a comparison between our results and the thermodynamic curvature of the van der Waals system.","Finally, we give our conclusions in Sec. .","We start with reviewing some general properties of thermodynamic geometry theories.","The thermodynamic geometry links the statistical mechanics to thermodynamics, in which an appropriate line element is crucial in the equilibrium state space of a thermodynamic system.","In the Ruppeiner theory [2], [22], since the thermodynamic fluctuation theory originates from statistical mechanics, certain properties of thermodynamics and statistical mechanics for a thermodynamic system are encoded into a single thermodynamic manifold equipped with the line element, $ds^2_R=g^R_{\\mu \\nu }dX^\\mu dX^\\nu ,$ where the independent variables $X^\\mu $ with the Greek index, $\\mu =0,1,2,\\cdots $ , denote the extensive quantities of a thermodynamic system and $g^R_{\\mu \\nu }$ is the so-called Ruppeiner metric defined by the Hessian matrix of thermodynamic entropy, $g^R_{\\mu \\nu }=-\\frac{\\partial ^2S(X)}{\\partial X^\\mu \\partial X^ \\nu }.$ Here $S(X)$ is the entropy of the thermodynamic system with Boltzmann's constant, $k_B=1$ , and $g^R_{\\mu \\nu }$ must be a positive definite matrixThermodynamic metrics must be positive definite since the entropy has [18] a maximum value in equilibrium.","This is the condition to ensure thermodynamic stability.","However, the positive definiteness may fail [19] due to the non-independence between entropy and volume.","It is required [23], [24] to impose Sylvetser's criterion for a global thermodynamic stability.","in order to ensure thermodynamic stability.","Furthermore, we take note of a related metric which is defined by the Hessian matrix of black hole mass, known as the Weinhold metric [1], $g^W_{\\mu \\nu }=\\frac{\\partial ^2M(X)}{\\partial X^\\mu \\partial X^ \\nu },$ which can be obtained from the Ruppeiner metric via a transformation of fluctuation coordinates.","Technically, the two metrics that describe Ruppeiner's and Weinhold's geometries, respectively, are conformally related [3] to each other, $ds^2_R=\\frac{1}{T}ds^2_W,$ where $T$ is the temperature of the thermodynamic system under consideration."],["First law of black hole thermodynamics with nonlinear electrodynamics","For the stationary black holes in nonlinear electrodynamics, the first law of black hole thermodynamics has been established [25], [26] via a covariant approach, $dM=TdS+\\Phi dQ_e +\\Psi dQ_m, $ where $Q_e$ and $Q_m$ denote the electric charge and magnetic charge, respectively.","It was claimed [27] that the first law of black hole thermodynamics holds for the black holes with nonlinear electrodynamics, but the Smarr formula seems to be violated.","It is easy to check that Eq.","(REF ) is satisfied for Born-Infeld black holes but not for Bardeen black holes.","The reason why Eq.","(REF ) breaks down for Bardeen black holes has been clarified [28], where the Lagrangian should include the contribution of additional terms besides being a function of electromagnetic invariants when one tries to derive the first law of black hole thermodynamics.","Differing from Rasheed's formula, the first law of Bardeen black hole thermodynamics takes the form, $dM=TdS+\\Psi dq+\\mathcal {K} dq + \\Pi dM, $ which contains the extra terms,The extra terms originate from the partial derivative of action with respect to additional parameters that are introduced to give the correct Smarr's formula (see Ref.","[28] for the specific expression of extra terms).","For Bardeen black holes, $\\Pi $ takes the form of $1-r_h^3(q^2+r_h^2)^{-3/2}$ which is obviously a dimensionless constant with the value less than one when the charge is set to be a constant.","For more details on the first law and Smarr's formula of black hole thermodynamics in the context of nonlinear electrodynamics coupled to Einstein gravity, please refer to Refs.","[25], [26], [28], [29], [30], [31].","$\\mathcal {K} dq$ and $\\Pi dM$ , due to the variation with respect to mass and charge.","In an AdS spacetime, the cosmological constant is interpreted as thermodynamic pressure, $P=-\\Lambda /(8\\pi )$ .","When $PdV$ term is included, the first law can be rewritten to be $dM=\\frac{T}{1-\\Pi }dS+\\sum _{i}\\frac{Y_i}{1-\\Pi }dX^i, $ where the Latin index takes $1, 2, 3\\cdots $ , $X^i=(V,\\cdots )$ and $Y_i=(-P,\\cdots )$ .","When the notations, $X^\\mu =(S,V\\cdots )$ and $Y_\\mu =(T/(1-\\Pi ),Y_i/(1-\\Pi ))$ , are taken, Eq.","(REF ) becomes a compact form, $dM=Y_\\mu dX^\\mu .$ We note that the Einstein notation has been adopted in Eq.","(REF ) and will be used in the following contexts."],["Thermodynamic curvatures","Given the first law of black hole thermodynamics Eq.","(REF ), the thermodynamic line elements of Weinhold's geometry can be written as $ds^2_W=\\frac{1}{1-\\Pi }dTdS+\\frac{1}{1-\\Pi }dY_idX^i.$ Comparing with the line element of Ruppeiner's geometry [15], $ds^2_R=\\frac{1}{T}dTdS+\\frac{1}{T}dY_idX^i,$ we can see that there exists an conformal factor $(1-\\Pi )/T$ between the Weinhold line element Eq.","(REF ) and Ruppeiner line element Eq.","(REF ).","The conformal factor in Eq.","(REF ) is $1/T$ , now it changes to $(1-\\Pi )/T$ due to the introduction of extra terms in the first law of black hole thermodynamics.","When considering the phase space coordinates $(T,X^i)$ , we have $dS=\\left( \\frac{\\partial S}{\\partial T}\\right)dT+\\left( \\frac{\\partial S}{\\partial X^i}\\right)dX^i,$ $dY_j=\\left( \\frac{\\partial Y_j}{\\partial T}\\right)dT+\\left( \\frac{\\partial Y_j}{\\partial X^i}\\right)dX^i.$ Substituting Eqs.","(REF ) and (REF ) into Eq.","(REF ) and using the relation, $\\frac{\\partial S}{\\partial X^i}=-\\frac{\\partial Y_i}{\\partial T},$ we obtain the line element of Weinhold's geometry, $ds^2_W=\\frac{1}{1-\\Pi }\\frac{C_{X^i}}{T}dT^2+\\frac{1}{1-\\Pi }\\left( \\frac{\\partial Y_i}{\\partial X^j}\\right)_TdX^idX^j, $ where $C_{X^i}\\equiv T(\\partial S/\\partial T)_{X^i}$ is the heat capacity at constant $X^i$ .","Since $X^i$ can take anyone of extensive quantities, the metric Eq.","(REF ) is two dimensional.","In the coordinates $(T, X)$ , we obtain the general form of a scalar curvature by using its definition in the Riemannian geometry, $ \\mathbb {R}_W&=&\\frac{1-\\Pi }{2C_{X}^2(\\partial _{X}Y)^2}\\bigg \\lbrace C_{X}\\bigg [(\\partial _{X}C_{X})(\\partial _{X,X}Y)-T(\\partial _{T,X}Y)^2 \\bigg ]+(\\partial _{X}Y)\\bigg [(\\partial _{X}C_{X})^2 \\nonumber \\\\& &-T(\\partial _{T}C_{X})(\\partial _{T,X}Y)+C_{X} \\bigg (-2(\\partial _{X, X}C_{X})+\\partial _{T,X}Y+2T(\\partial _{T,T,X}Y) \\bigg ) \\bigg ] \\bigg \\rbrace ,$ where $\\partial _{T, X}Y\\equiv \\partial ^2 Y/(\\partial T \\partial X)$ as known in General Relativity.","We note that Eq.","(REF ) shares the same divergent points at $C_{X}=0$ or $\\partial _{X}Y=0$ with the scalar curvature of Ruppeiner's geometry given [15] by $ \\mathbb {R}_R&=&\\frac{1}{2C_{X}^2(\\partial _{X}Y)^2}\\bigg \\lbrace T(\\partial _{X}Y)\\bigg [(\\partial _{X}C_{X})^2+(\\partial _{T}C_{X})(\\partial _{X}Y-T(\\partial _{T,X}Y))\\bigg ] +C_{X}\\bigg [(\\partial _{X}Y)^2\\nonumber \\\\& &+T[(\\partial _{X}C_{X})(\\partial _{X,X}Y)-T(\\partial _{T,X}Y)^2]+2T(\\partial _{X}Y)(-(\\partial _{X,X}C_{X})+T(\\partial _{T,T,X}Y)) \\bigg ) \\bigg ] \\bigg \\rbrace .$ Following Refs.","[21], [15] for the treatment to the divergence of Ruppeiner scalar curvature, we introduce the normalized scalar curvatureSince the thermodynamic scalar curvature $\\mathbb {R}$ diverges at $C_{X}=0$ , $C_{X}$ is treated as a constant with the value of infinitely close to zero, namely, $C_{X}\\rightarrow 0^+$ .","Such a normalization just removes the divergence of $\\mathbb {R}$ at $C_{X}=0$ , see Ref.","[15] for the details, but not the divergence at $\\partial _{X}Y=0$ , which will be shown in Fig.","REF and Fig.","REF .","defined by $\\mathcal {R}=\\mathbb {R}\\,C_{X},$ and then reduce the Weinhold scalar curvature Eq.","(REF ) and Ruppeiner scalar curvature Eq.","(REF ) to be $ \\mathcal {R}_W=\\frac{(1-\\Pi )[\\partial _{T,X}Y-T(\\partial _{T,X}Y)^2+2T(\\partial _{T,T,X}Y)]}{2(\\partial _{X}Y)^2},$ $ \\mathcal {R}_R=\\frac{(\\partial _{X}Y)^2-T^2((\\partial _{T,X}Y)^2)+2T^2(\\partial _{X}Y)(\\partial _{T,T,X}Y)}{2(\\partial _{X}Y)^2},$ which are so-called normalized thermodynamic scalar curvatures because their divergent behaviourThere exists [22], [15] a relation between the divergence of correlation length and the divergence of thermodynamic curvature occurring at the critical point of phase transitions.","We shall discuss this issue in detail in section REF .","at the critical point of phase transitions has been removed.","In this section, we start with a brief review on the thermodynamic properties of Bardeen-AdS black holes and investigate the phase transition via $P-V$ criticality in an extended phase space.","The action of Bardeen-AdS black holes with the cosmological constant $\\Lambda $ in the four-dimensional spacetime reads [32], [33] $\\mathcal {S}=\\frac{1}{16\\pi }\\int d^4x\\sqrt{-g}\\left( R-2\\Lambda -4\\mathcal {L}(F) \\right),$ where $g$ is the determinant of the metric tensor, $R$ is the Ricci scalar, $\\Lambda $ is related to the AdS radius $l$ via the relation $\\Lambda =-3/l^2$ , and the function of electromagnetic invariant $F$ , $\\mathcal {L}(F)$ is given by $\\mathcal {L}=\\frac{M}{q^3}\\left( \\frac{\\sqrt{4q^2F}}{1+\\sqrt{4q^2F}}\\right) ^{5/2},$ where $M$ and $q$ denote mass and charge of Bardeen-AdS black holes, respectively.","In four dimensions, the line element of spherically symmetric Bardeen-AdS black holes takes the form, $ds^2=-f(r)dt^2+\\frac{dr^2}{f(r)}+r^2d\\Omega ^2,$ with the shape function, $f(r)=1-\\frac{2Mr^2}{(q^2+r^2)^3/2}+\\frac{r^2}{l^2},$ where $d\\Omega ^2$ is the line element of the unit two sphere.","The Hawking temperature of Bardeen-AdS black holes can be calculated, $T=\\frac{-2q^2+r_h^2+3r_h^4/l^2}{4\\pi r_h(q^2+r_h^2)},$ where $r_h$ stands for the horizon radius which is the solution of the equation $f(r_h)=0$ .","Note that we have to use the modified first law, Eq.","(REF ), to compute the entropy of the Bardeen black hole, which leads to $S=\\int \\frac{1-\\Pi }{T}\\,dM=\\pi r_h^2,$ where the form of $\\Pi $ has been given in footnote 2.","We see that the Bardeen black hole obeys the area law.","Moreover, one can derive the thermodynamic volume conjugated to the thermodynamic pressure, $V=\\left(\\frac{\\partial M}{\\partial P} \\right)_{S,q}= \\frac{4\\pi r_h^3}{3}\\left(1+\\frac{q^2}{r_h^2} \\right)^{3/2}.$ In the above considerations, the black hole mass has been identified with the enthalpy throughout the thermodynamic process in the extended phase space that includes $(P, V)$ .","With the clear representations of the pressure and volume, the heat capacity measuring the thermodynamic stability can be obtained, $C_{_V}&=&T\\left(\\frac{\\partial S}{\\partial T} \\right)_V=0,\\\\C_{_P}&=&T\\left(\\frac{\\partial S}{\\partial T} \\right)_P=\\frac{2\\pi (q^2+r_h^2)^{5/2}(-2q^2+r_h^2+8P\\pi r_h^4)}{2q^4r_h+7q^2r_h^3+(-1+24P\\pi q^2)r_h^5+8P\\pi r_h^7}.$ Unlike the van der Waals fluid, the heat capacity of Bardeen-AdS black holes at constant volume is zero, and the heat capacity at constant pressure is finite.","For Bardeen-AdS black holes, the equation of state $P=(T,V)$ can be derived from the Hawking temperature Eq.","(REF ), $P=\\frac{12q^2+\\left( \\frac{6V}{\\pi }\\right) ^{2/3} \\left( -1+2\\pi T\\sqrt{-4q^2+\\left( \\frac{6V}{\\pi }\\right) ^{2/3}} \\right) }{2\\pi \\left( -4q^2+\\left( \\frac{6V}{\\pi }\\right) ^{2/3}\\right)^{2} }.$ The behaviour of isotherms in $P-V$ diagram is shown in Fig.","REF .","Figure: P-VP-V diagram of Bardeen-AdS black holes.","Isotherms descend from top to bottom when the temperature is decreasing, where the black dashed curve corresponds to T>T c T>T_c, the red curve to the critical temperature T=T c T=T_c, and the two blue curves to T0.16V_c$ .","In general, we point out that the normalized Ruppeiner curvature of Bardeen-AdS black holes has a very similar microstructure to that of van der Waals fluids [15].","Figure: Ruppeiner thermodynamic curvature of Bardeen-AdS black holes.","This picture shows the characteristic behaviour of the Ruppeiner thermodynamic curvature as a function of the reduced temperature and volume."],["Weinhold's geometry","In $(T,V)$ fluctuation coordinates, the line element Eq.","(REF ) of Weinhold's geometry reads $ds^2_W=\\frac{1}{1-\\Pi }\\frac{C_{_V}}{T}dT^2+\\frac{1}{1-\\Pi }\\left( \\frac{\\partial P}{\\partial V}\\right)_TdV^2,$ where $C_{_V}=0$ still leads to a degenerate metric which can be cured by the normalization [15].","We note that $\\Pi $ satisfies $0<\\Pi <1$ and so it changes only the quantity rather than the sign of thermodynamic curvature.","This means that a repulsive interaction cannot be reversed to an attractive interaction by the presence of $\\Pi $ , and vice versa.","Without loss of generality, we set the dimensionless constant $\\Pi =0$ .","Using formula Eq.","(REF ), we obtain the normalized Weinhold thermodynamic scalar curvature straightforwardly, $\\mathcal {R}_W=\\bigg \\lbrace \\bigg ((\\sqrt{273}+17) \\tilde{V}^{2/3}-2\\bigg )^7 \\bigg ((\\sqrt{273}+17) \\tilde{V}^{2/3}+(17 \\sqrt{273}+281) \\tilde{V}^{4/3}-4\\bigg ) \\bigg [\\frac{-57 \\sqrt{\\sqrt{273}+15} \\@root 3 \\of {\\tilde{V}}}{[(\\sqrt{273}+17) \\tilde{V}^{2/3}-2]^{7/2}}\\nonumber \\\\-\\frac{8 (\\sqrt{273}+15) (13 \\sqrt{273}+219) \\tilde{T} [(\\sqrt{273}+17) \\tilde{V}^{2/3}+(17 \\sqrt{273}+281) \\tilde{V}^{4/3}-4]}{[(\\sqrt{273}+17) \\tilde{V}^{2/3}-2]^7} \\bigg ]\\bigg \\rbrace \\bigg /\\bigg \\lbrace 4 (13 \\sqrt{273}+219) \\nonumber \\\\ \\times \\bigg [3 \\sqrt{(\\sqrt{273}+17) \\tilde{V}^{2/3}-2} [(17 \\sqrt{273}+281) \\tilde{V}^{2/3}-5 (\\sqrt{273}+17)]-2 \\sqrt{\\sqrt{273}+15} T [(\\sqrt{273}+17) \\tilde{V}^{2/3}\\nonumber \\\\+(17 \\sqrt{273}+281) \\tilde{V}^{4/3}-4]\\bigg ]^2\\bigg \\rbrace .\\nonumber \\\\ $ We plot the behaviour of normalized Weinhold thermodynamic curvature in Fig.","REF .","Figure: Weinhold thermodynamic curvature of Bardeen-AdS black holes.","This picture shows the characteristic behaviour of the Weinhold thermodynamic curvature as a function of the reduced temperature and volume.From the two expressions in Eqs.","(REF ) and (REF ), it is obvious that the normalized thermodynamic scalar curvatures in the two geometry theories are different.","There is, however, a qualitative similarity between the structures of Ruppeiner's geometry and Weinhold's geometry as displayed in Fig.","REF and Fig.","REF .","The normalized thermodynamic curvatures diverge to negative infinity in very low temperatures.","In other words, for both the Weinhold's geometry and Ruppeiner's geometry there exists a strongly attractive interaction in the microstructure of Bardeen-AdS black holes in this region of temperature, which can be explicitly seen in the two figures.","Furthermore, it is worth noting that the normalized Weinhold curvature is negative and does not change sign, which is due to the existence of conformal relation between Weinhold's and Ruppeiner's geometries.","According to the microscopic interpretation of interactions [13], the positive and negative curvatures imply the repulsive and attractive interactions, respectively.","Hence, for Weinhold's geometry there are only attractive interactions but no repulsive interactions in the microstructure of Bardeen-AdS black holes, which coincides with the interacting behaviour in the van der Waals fluid."],["Critical phenomena","In order to get a deep understanding of the normalized thermodynamic curvatures of Bardeen-AdS black holes, we shall focus on the critical phenomena in which the Bardeen-AdS black hole has a markedly different behaviour from that of the charged AdS black holes with spacetime singularities, such as the Reissner-Nordström AdS and Gauss-Bonnet AdS black holes [16], [43], [14].","The normalized Ruppeiner thermodynamic curvature Eq.","(REF ) diverges and vanishes along the following curves on the $(V/V_c, T/T_c)$ plane, respectively, $\\tilde{T}|_{\\mathcal {R}_R\\rightarrow \\infty }&=& \\frac{3 \\sqrt{\\left(\\sqrt{273}+17\\right) \\tilde{V}^{2/3}-2} \\left(\\left(17 \\sqrt{273}+281\\right) \\tilde{V}^{2/3}-5 \\left(\\sqrt{273}+17\\right)\\right)}{2 \\sqrt{\\sqrt{273}+15} \\left(\\left(\\sqrt{273}+17\\right) \\tilde{V}^{2/3}+\\left(17 \\sqrt{273}+281\\right) \\tilde{V}^{4/3}-4\\right)},\\\\\\tilde{T}|_{\\mathcal {R}_R=0}&=&3 \\bigg [35 (285 \\sqrt{273}+4709) \\tilde{V}^{2/3}-11 (4777 \\sqrt{273}+78929) \\tilde{V}^{4/3}+(80069 \\sqrt{273}+1322957) \\tilde{V}^2\\nonumber \\\\& & -25 (17 \\sqrt{273}+281)\\bigg ]\\bigg /\\bigg [2 \\sqrt{\\sqrt{273}+15} \\sqrt{(\\sqrt{273}+17) \\tilde{V}^{2/3}-2}\\, \\bigg (-7 (17 \\sqrt{273}+281) \\tilde{V}^{2/3}\\nonumber \\\\& &-4 (285 \\sqrt{273}+4709) \\tilde{V}^{4/3}+(4777 \\sqrt{273}+78929) \\tilde{V}^2+10 (\\sqrt{273}+17)\\bigg )\\bigg ],$ which is plotted in Fig.","REF .","Figure: Critical behaviour of thermodynamic curvatures.","The upper (red) curve stands for the diverging case of ℛ R \\mathcal {R}_R and ℛ W \\mathcal {R}_W.","The lower (blue) curve corresponds to the vanishing case of ℛ R \\mathcal {R}_R only.From Eq.","(REF ) we find that the Weinhold curvature shares the same diverging curve as the Ruppeiner curvature, i.e., $\\tilde{T}|_{\\mathcal {R}_W\\rightarrow \\infty }=\\tilde{T}|_{\\mathcal {R}_R\\rightarrow \\infty }$ .","This happens due to the existence of conformal relation between the Weinhold's geometry and Ruppeiner's geometry.","However, the Weinhold curvature does not vanish at any finite positive temperature, which is quite different from the Ruppeiner curvature.","For Weinhold's geometry and Ruppeiner's geometry the normalized thermodynamic curvatures will diverge at the critical point of a phase transition as shown in Fig.","REF .","We can easily see that this divergent point is indeed located at the maximum of the red curve, where the temperature and volume just take their critical values, i.e., $T=T_c$ and $V=V_c$ .","This phenomenon was also noticed [44] for the charged AdS black holes with spacetime singularities, where the divergent points of specific heat match those of thermodynamic curvature.","Furthermore, a positive normalized thermodynamic curvature was obtained [15], [16] in the small volume phase of the charged AdS black holes with spacetime singularities, namely, a weak repulsive interaction appears in the microstructure, while it does not in the van der Waals fluid.","Interestingly, for the Bardeen-AdS black hole that has no spacetime singularities, the temperature converges to zero as the volume decreases, see Fig.","REF .","Such a behaviour has been observed [15] in the van der Waals fluid.","Hence, the Bardeen-AdS black hole has the critical phenomena similar to those of van der Waals fluid."],["Conclusions","In the present work, we revisit the thermodynamics and investigate the thermodynamic geometry for Bardeen black holes in the anti-de Sitter spacetime.","We find that the entropy of Bardeen-AdS black holes still obeys the area law if the modified first law of thermodynamics is considered.","We derive the normalized scalar curvature in the Weinhold's geometry and find that it has the same divergent behaviour as the Ruppeiner scalar curvature at the critical point of a phase transition, but it has no vanishing behaviour at any finite positive temperature while the Ruppeiner scalar curvature has.","Moreover, we notice that the additional parameter $\\Pi $ appeared in the conformal factor of Weinhold's geometry does not reverse the interaction in the microstructure of Bardeen-AdS black holes.","When compared to the charged AdS black holes with spacetime singularities, the Bardeen-AdS black hole as a regular black hole has a rather different behaviour, that is, it has no weak repulsive interaction in the microstructure.","When compared to the van der Waals fluid, the Bardeen-AdS black hole has a critical behaviour resembling the liquid-gas phase transition and both its Weinhold scalar curvature and Ruppeiner scalar curvature have a very similar microstructure to that of the van der Waals fluid.","Our results seem to justify the mathematical analogy of the Bardeen-AdS black hole with the van der Waals fluid.","Whether the vanishing of a weak repulsive interaction in the Bardeen-AdS black hole is universal or not in the other regular black holes, we shall deal with this issue by analyzing their thermodynamic geometries in detail."],["Acknowledgements","The authors would like to thank R.-G Cai and T. Vetsov for their helpful comments and disscussions.","This work was supported in part by the National Natural Science Foundation of China under Grant No.","11675081."]],"string":"[\n [\n \"Weinhold geometry and thermodynamics of Bardeen AdS black holes\"\n ],\n [\n \"Abstract Thermodynamics of Bardeen AdS black holes has attracted a great deal of attentions due to its intrinsic complications and rich phase structures.\",\n \"However, the entropy and thermodynamic volume are incorrect in some literatures.\",\n \"In this paper we revisit the thermodynamics of Bardeen AdS black holes and provide the correct entropy and thermodynamic volume.\",\n \"Furthermore, thermodynamic geometries are a powerful tool to probe the microstructure of black holes.\",\n \"Based on the Hessian matrix of black hole mass, we introduce a thermodynamic metric and give its scalar curvature in Weinhold's geometry.\",\n \"The conformal relation between Weinhold's geometry and Ruppeiner's geometry will be changed due to the specific first law of thermodynamics for regular black holes like the Bardeen AdS black hole.\",\n \"We also investigate the critical behaviour of phase transitions in an extended phase space, and find that the critical behaviour of the Bardeen AdS black hole coincides with that of liquid-gas systems.\",\n \"In particular, based on the Weinhold geometry we unveil a repulsive interaction in the microstructure of the Bardeen AdS black hole under its small volume state, while it is known that only attractive interaction exists in the microstructure of the van der Waals fluid.\"\n ],\n [\n \"Introduction\",\n \"In the past decades, a complete statistical mechanics that describes the microstructure of black holes was still lacking although there was a wealth of work in the thermodynamics of black holes.\",\n \"A geometric description of thermodynamic systems may provide an alternative attempt to probe the microstructure of black holes.\",\n \"Weinhold developed [1] the mass (energy) representation by constructing a linear vector space with a full metric structure, where the vector space can be constructed directly from empirical laws of thermodynamics.\",\n \"For each extensity $X^{\\\\mu }$ of a thermodynamic system, its conjugate field variable $v_{\\\\mu }$ is defined by $v_{\\\\mu }\\\\equiv \\\\partial M /\\\\partial X^{\\\\mu },$ which constitutes a thermodynamic phase space together with $X^{\\\\mu }$ that is required to have the properties of a vector space.\",\n \"Thus, an abstract space having the full properties of inner product on the thermodynamic phase space is given, which implies that the thermodynamic laws can show the underlying structure of a geometric object.\",\n \"Based on the fluctuation theory of thermodynamics, on the other hand, Ruppeiner introduced [2] the entropy representation, which shows that thermodynamic systems can be described by Riemannian manifolds.\",\n \"Quite interesting is that there exists [3] a conformal equivalence between Weinhold'geometry and Ruppeiner's geometry.\",\n \"In information geometry, a parameterized statistical model is considered as a Riemannian manifold in which the corresponding probability distributions can be defined.\",\n \"In this view, Ruppeiner's geometry is one of information theories with the probability distribution [4], $\\\\mathcal {P}(X)\\\\propto \\\\sqrt{g^R}\\\\, e^{-\\\\frac{1}{2}g^R_{\\\\mu \\\\nu }X^{\\\\mu }X^{\\\\nu }},$ where $g^R_{\\\\mu \\\\nu }$ is the Ruppeiner metric and ${g^R}$ is the determinant of the metric.\",\n \"Thermodynamic geometries have been extensively studied and applied [5], [6], [7], [8], [9], [10], [11], [12] to numerous systems.\",\n \"In particular, the interpretations of thermodynamic curvatures are linked [13] to the interactions in the microstructure of ideal quantum gases.\",\n \"In recent years, Ruppeiner's geometry has promoted substantially the connection between microstructure and thermodynamics of black holes, furthermore, such a connection has been shown [14], [15], [16], [17] to be analogous to that in the van der Waals fluid.\",\n \"These interpretations are new attempts to extract information from the thermodynamic geometry of black holes.\",\n \"The thermodynamic curvature does not depend [18] on the thermodynamic metric since it is an invariant for a given thermodynamic state, which has recently been verified [19], [20] again in two new phase spaces.\",\n \"As is known, there is only an attractive interaction in the van der Waals fluid.\",\n \"However, there exists [21] a weak repulsive interaction in the microstructure of charged AdS black holes.\",\n \"We may argue whether the existence of a weak repulsive interaction has something to do with the singularity of charged AdS black hole spacetimes.\",\n \"On the other hand, the current thermodynamic geometries are generally based on the first law of black hole mechanics associated with linear electrodynamics.\",\n \"They cannot be applied directly to interpret the thermodynamic geometry of regular black holes that have no singularity.\",\n \"We need to construct the thermodynamic geometry in the context of the nonlinear electrodynamics, which is able to give an interpretation of thermodynamic geometry for regular black holes.\",\n \"It is challenging to generalize thermodynamic geometry to regular black holes due to the intrinsic complications in the configurations of the first law and the Lagrangian coupled with nonlinear electrodynamics.\",\n \"In this paper, we study the Weinhold geometry of regular black holes in the context of nonlinear electrodynamics.\",\n \"Our main task is to construct the Weinhold geometry by using the normalized scalar curvature [21] and reveal the behaviour of thermodynamic curvatures through macroscopic properties.\",\n \"The outline of this paper is as follows.\",\n \"In Sec.\",\n \"we review briefly the properties of thermodynamic geometry and then give the general form of Weinhold scalar curvatures.\",\n \"Next, in Sec.\",\n \"we derive the entropy of Bardeen-AdS black holes in terms of the modified integration approach, and demonstrate the critical behaviour of thermodynamics in Bardeen-AdS black holes and the analogous behaviour in liquid-gas systems.\",\n \"In Sec.\",\n \"we compute the thermodynamic curvatures of Bardeen-AdS black holes in terms of Weinhold's geometry and Ruppeiner's geometry and make a comparison between our results and the thermodynamic curvature of the van der Waals system.\",\n \"Finally, we give our conclusions in Sec. .\",\n \"We start with reviewing some general properties of thermodynamic geometry theories.\",\n \"The thermodynamic geometry links the statistical mechanics to thermodynamics, in which an appropriate line element is crucial in the equilibrium state space of a thermodynamic system.\",\n \"In the Ruppeiner theory [2], [22], since the thermodynamic fluctuation theory originates from statistical mechanics, certain properties of thermodynamics and statistical mechanics for a thermodynamic system are encoded into a single thermodynamic manifold equipped with the line element, $ds^2_R=g^R_{\\\\mu \\\\nu }dX^\\\\mu dX^\\\\nu ,$ where the independent variables $X^\\\\mu $ with the Greek index, $\\\\mu =0,1,2,\\\\cdots $ , denote the extensive quantities of a thermodynamic system and $g^R_{\\\\mu \\\\nu }$ is the so-called Ruppeiner metric defined by the Hessian matrix of thermodynamic entropy, $g^R_{\\\\mu \\\\nu }=-\\\\frac{\\\\partial ^2S(X)}{\\\\partial X^\\\\mu \\\\partial X^ \\\\nu }.$ Here $S(X)$ is the entropy of the thermodynamic system with Boltzmann's constant, $k_B=1$ , and $g^R_{\\\\mu \\\\nu }$ must be a positive definite matrixThermodynamic metrics must be positive definite since the entropy has [18] a maximum value in equilibrium.\",\n \"This is the condition to ensure thermodynamic stability.\",\n \"However, the positive definiteness may fail [19] due to the non-independence between entropy and volume.\",\n \"It is required [23], [24] to impose Sylvetser's criterion for a global thermodynamic stability.\",\n \"in order to ensure thermodynamic stability.\",\n \"Furthermore, we take note of a related metric which is defined by the Hessian matrix of black hole mass, known as the Weinhold metric [1], $g^W_{\\\\mu \\\\nu }=\\\\frac{\\\\partial ^2M(X)}{\\\\partial X^\\\\mu \\\\partial X^ \\\\nu },$ which can be obtained from the Ruppeiner metric via a transformation of fluctuation coordinates.\",\n \"Technically, the two metrics that describe Ruppeiner's and Weinhold's geometries, respectively, are conformally related [3] to each other, $ds^2_R=\\\\frac{1}{T}ds^2_W,$ where $T$ is the temperature of the thermodynamic system under consideration.\"\n ],\n [\n \"First law of black hole thermodynamics with nonlinear electrodynamics\",\n \"For the stationary black holes in nonlinear electrodynamics, the first law of black hole thermodynamics has been established [25], [26] via a covariant approach, $dM=TdS+\\\\Phi dQ_e +\\\\Psi dQ_m, $ where $Q_e$ and $Q_m$ denote the electric charge and magnetic charge, respectively.\",\n \"It was claimed [27] that the first law of black hole thermodynamics holds for the black holes with nonlinear electrodynamics, but the Smarr formula seems to be violated.\",\n \"It is easy to check that Eq.\",\n \"(REF ) is satisfied for Born-Infeld black holes but not for Bardeen black holes.\",\n \"The reason why Eq.\",\n \"(REF ) breaks down for Bardeen black holes has been clarified [28], where the Lagrangian should include the contribution of additional terms besides being a function of electromagnetic invariants when one tries to derive the first law of black hole thermodynamics.\",\n \"Differing from Rasheed's formula, the first law of Bardeen black hole thermodynamics takes the form, $dM=TdS+\\\\Psi dq+\\\\mathcal {K} dq + \\\\Pi dM, $ which contains the extra terms,The extra terms originate from the partial derivative of action with respect to additional parameters that are introduced to give the correct Smarr's formula (see Ref.\",\n \"[28] for the specific expression of extra terms).\",\n \"For Bardeen black holes, $\\\\Pi $ takes the form of $1-r_h^3(q^2+r_h^2)^{-3/2}$ which is obviously a dimensionless constant with the value less than one when the charge is set to be a constant.\",\n \"For more details on the first law and Smarr's formula of black hole thermodynamics in the context of nonlinear electrodynamics coupled to Einstein gravity, please refer to Refs.\",\n \"[25], [26], [28], [29], [30], [31].\",\n \"$\\\\mathcal {K} dq$ and $\\\\Pi dM$ , due to the variation with respect to mass and charge.\",\n \"In an AdS spacetime, the cosmological constant is interpreted as thermodynamic pressure, $P=-\\\\Lambda /(8\\\\pi )$ .\",\n \"When $PdV$ term is included, the first law can be rewritten to be $dM=\\\\frac{T}{1-\\\\Pi }dS+\\\\sum _{i}\\\\frac{Y_i}{1-\\\\Pi }dX^i, $ where the Latin index takes $1, 2, 3\\\\cdots $ , $X^i=(V,\\\\cdots )$ and $Y_i=(-P,\\\\cdots )$ .\",\n \"When the notations, $X^\\\\mu =(S,V\\\\cdots )$ and $Y_\\\\mu =(T/(1-\\\\Pi ),Y_i/(1-\\\\Pi ))$ , are taken, Eq.\",\n \"(REF ) becomes a compact form, $dM=Y_\\\\mu dX^\\\\mu .$ We note that the Einstein notation has been adopted in Eq.\",\n \"(REF ) and will be used in the following contexts.\"\n ],\n [\n \"Thermodynamic curvatures\",\n \"Given the first law of black hole thermodynamics Eq.\",\n \"(REF ), the thermodynamic line elements of Weinhold's geometry can be written as $ds^2_W=\\\\frac{1}{1-\\\\Pi }dTdS+\\\\frac{1}{1-\\\\Pi }dY_idX^i.$ Comparing with the line element of Ruppeiner's geometry [15], $ds^2_R=\\\\frac{1}{T}dTdS+\\\\frac{1}{T}dY_idX^i,$ we can see that there exists an conformal factor $(1-\\\\Pi )/T$ between the Weinhold line element Eq.\",\n \"(REF ) and Ruppeiner line element Eq.\",\n \"(REF ).\",\n \"The conformal factor in Eq.\",\n \"(REF ) is $1/T$ , now it changes to $(1-\\\\Pi )/T$ due to the introduction of extra terms in the first law of black hole thermodynamics.\",\n \"When considering the phase space coordinates $(T,X^i)$ , we have $dS=\\\\left( \\\\frac{\\\\partial S}{\\\\partial T}\\\\right)dT+\\\\left( \\\\frac{\\\\partial S}{\\\\partial X^i}\\\\right)dX^i,$ $dY_j=\\\\left( \\\\frac{\\\\partial Y_j}{\\\\partial T}\\\\right)dT+\\\\left( \\\\frac{\\\\partial Y_j}{\\\\partial X^i}\\\\right)dX^i.$ Substituting Eqs.\",\n \"(REF ) and (REF ) into Eq.\",\n \"(REF ) and using the relation, $\\\\frac{\\\\partial S}{\\\\partial X^i}=-\\\\frac{\\\\partial Y_i}{\\\\partial T},$ we obtain the line element of Weinhold's geometry, $ds^2_W=\\\\frac{1}{1-\\\\Pi }\\\\frac{C_{X^i}}{T}dT^2+\\\\frac{1}{1-\\\\Pi }\\\\left( \\\\frac{\\\\partial Y_i}{\\\\partial X^j}\\\\right)_TdX^idX^j, $ where $C_{X^i}\\\\equiv T(\\\\partial S/\\\\partial T)_{X^i}$ is the heat capacity at constant $X^i$ .\",\n \"Since $X^i$ can take anyone of extensive quantities, the metric Eq.\",\n \"(REF ) is two dimensional.\",\n \"In the coordinates $(T, X)$ , we obtain the general form of a scalar curvature by using its definition in the Riemannian geometry, $ \\\\mathbb {R}_W&=&\\\\frac{1-\\\\Pi }{2C_{X}^2(\\\\partial _{X}Y)^2}\\\\bigg \\\\lbrace C_{X}\\\\bigg [(\\\\partial _{X}C_{X})(\\\\partial _{X,X}Y)-T(\\\\partial _{T,X}Y)^2 \\\\bigg ]+(\\\\partial _{X}Y)\\\\bigg [(\\\\partial _{X}C_{X})^2 \\\\nonumber \\\\\\\\& &-T(\\\\partial _{T}C_{X})(\\\\partial _{T,X}Y)+C_{X} \\\\bigg (-2(\\\\partial _{X, X}C_{X})+\\\\partial _{T,X}Y+2T(\\\\partial _{T,T,X}Y) \\\\bigg ) \\\\bigg ] \\\\bigg \\\\rbrace ,$ where $\\\\partial _{T, X}Y\\\\equiv \\\\partial ^2 Y/(\\\\partial T \\\\partial X)$ as known in General Relativity.\",\n \"We note that Eq.\",\n \"(REF ) shares the same divergent points at $C_{X}=0$ or $\\\\partial _{X}Y=0$ with the scalar curvature of Ruppeiner's geometry given [15] by $ \\\\mathbb {R}_R&=&\\\\frac{1}{2C_{X}^2(\\\\partial _{X}Y)^2}\\\\bigg \\\\lbrace T(\\\\partial _{X}Y)\\\\bigg [(\\\\partial _{X}C_{X})^2+(\\\\partial _{T}C_{X})(\\\\partial _{X}Y-T(\\\\partial _{T,X}Y))\\\\bigg ] +C_{X}\\\\bigg [(\\\\partial _{X}Y)^2\\\\nonumber \\\\\\\\& &+T[(\\\\partial _{X}C_{X})(\\\\partial _{X,X}Y)-T(\\\\partial _{T,X}Y)^2]+2T(\\\\partial _{X}Y)(-(\\\\partial _{X,X}C_{X})+T(\\\\partial _{T,T,X}Y)) \\\\bigg ) \\\\bigg ] \\\\bigg \\\\rbrace .$ Following Refs.\",\n \"[21], [15] for the treatment to the divergence of Ruppeiner scalar curvature, we introduce the normalized scalar curvatureSince the thermodynamic scalar curvature $\\\\mathbb {R}$ diverges at $C_{X}=0$ , $C_{X}$ is treated as a constant with the value of infinitely close to zero, namely, $C_{X}\\\\rightarrow 0^+$ .\",\n \"Such a normalization just removes the divergence of $\\\\mathbb {R}$ at $C_{X}=0$ , see Ref.\",\n \"[15] for the details, but not the divergence at $\\\\partial _{X}Y=0$ , which will be shown in Fig.\",\n \"REF and Fig.\",\n \"REF .\",\n \"defined by $\\\\mathcal {R}=\\\\mathbb {R}\\\\,C_{X},$ and then reduce the Weinhold scalar curvature Eq.\",\n \"(REF ) and Ruppeiner scalar curvature Eq.\",\n \"(REF ) to be $ \\\\mathcal {R}_W=\\\\frac{(1-\\\\Pi )[\\\\partial _{T,X}Y-T(\\\\partial _{T,X}Y)^2+2T(\\\\partial _{T,T,X}Y)]}{2(\\\\partial _{X}Y)^2},$ $ \\\\mathcal {R}_R=\\\\frac{(\\\\partial _{X}Y)^2-T^2((\\\\partial _{T,X}Y)^2)+2T^2(\\\\partial _{X}Y)(\\\\partial _{T,T,X}Y)}{2(\\\\partial _{X}Y)^2},$ which are so-called normalized thermodynamic scalar curvatures because their divergent behaviourThere exists [22], [15] a relation between the divergence of correlation length and the divergence of thermodynamic curvature occurring at the critical point of phase transitions.\",\n \"We shall discuss this issue in detail in section REF .\",\n \"at the critical point of phase transitions has been removed.\",\n \"In this section, we start with a brief review on the thermodynamic properties of Bardeen-AdS black holes and investigate the phase transition via $P-V$ criticality in an extended phase space.\",\n \"The action of Bardeen-AdS black holes with the cosmological constant $\\\\Lambda $ in the four-dimensional spacetime reads [32], [33] $\\\\mathcal {S}=\\\\frac{1}{16\\\\pi }\\\\int d^4x\\\\sqrt{-g}\\\\left( R-2\\\\Lambda -4\\\\mathcal {L}(F) \\\\right),$ where $g$ is the determinant of the metric tensor, $R$ is the Ricci scalar, $\\\\Lambda $ is related to the AdS radius $l$ via the relation $\\\\Lambda =-3/l^2$ , and the function of electromagnetic invariant $F$ , $\\\\mathcal {L}(F)$ is given by $\\\\mathcal {L}=\\\\frac{M}{q^3}\\\\left( \\\\frac{\\\\sqrt{4q^2F}}{1+\\\\sqrt{4q^2F}}\\\\right) ^{5/2},$ where $M$ and $q$ denote mass and charge of Bardeen-AdS black holes, respectively.\",\n \"In four dimensions, the line element of spherically symmetric Bardeen-AdS black holes takes the form, $ds^2=-f(r)dt^2+\\\\frac{dr^2}{f(r)}+r^2d\\\\Omega ^2,$ with the shape function, $f(r)=1-\\\\frac{2Mr^2}{(q^2+r^2)^3/2}+\\\\frac{r^2}{l^2},$ where $d\\\\Omega ^2$ is the line element of the unit two sphere.\",\n \"The Hawking temperature of Bardeen-AdS black holes can be calculated, $T=\\\\frac{-2q^2+r_h^2+3r_h^4/l^2}{4\\\\pi r_h(q^2+r_h^2)},$ where $r_h$ stands for the horizon radius which is the solution of the equation $f(r_h)=0$ .\",\n \"Note that we have to use the modified first law, Eq.\",\n \"(REF ), to compute the entropy of the Bardeen black hole, which leads to $S=\\\\int \\\\frac{1-\\\\Pi }{T}\\\\,dM=\\\\pi r_h^2,$ where the form of $\\\\Pi $ has been given in footnote 2.\",\n \"We see that the Bardeen black hole obeys the area law.\",\n \"Moreover, one can derive the thermodynamic volume conjugated to the thermodynamic pressure, $V=\\\\left(\\\\frac{\\\\partial M}{\\\\partial P} \\\\right)_{S,q}= \\\\frac{4\\\\pi r_h^3}{3}\\\\left(1+\\\\frac{q^2}{r_h^2} \\\\right)^{3/2}.$ In the above considerations, the black hole mass has been identified with the enthalpy throughout the thermodynamic process in the extended phase space that includes $(P, V)$ .\",\n \"With the clear representations of the pressure and volume, the heat capacity measuring the thermodynamic stability can be obtained, $C_{_V}&=&T\\\\left(\\\\frac{\\\\partial S}{\\\\partial T} \\\\right)_V=0,\\\\\\\\C_{_P}&=&T\\\\left(\\\\frac{\\\\partial S}{\\\\partial T} \\\\right)_P=\\\\frac{2\\\\pi (q^2+r_h^2)^{5/2}(-2q^2+r_h^2+8P\\\\pi r_h^4)}{2q^4r_h+7q^2r_h^3+(-1+24P\\\\pi q^2)r_h^5+8P\\\\pi r_h^7}.$ Unlike the van der Waals fluid, the heat capacity of Bardeen-AdS black holes at constant volume is zero, and the heat capacity at constant pressure is finite.\",\n \"For Bardeen-AdS black holes, the equation of state $P=(T,V)$ can be derived from the Hawking temperature Eq.\",\n \"(REF ), $P=\\\\frac{12q^2+\\\\left( \\\\frac{6V}{\\\\pi }\\\\right) ^{2/3} \\\\left( -1+2\\\\pi T\\\\sqrt{-4q^2+\\\\left( \\\\frac{6V}{\\\\pi }\\\\right) ^{2/3}} \\\\right) }{2\\\\pi \\\\left( -4q^2+\\\\left( \\\\frac{6V}{\\\\pi }\\\\right) ^{2/3}\\\\right)^{2} }.$ The behaviour of isotherms in $P-V$ diagram is shown in Fig.\",\n \"REF .\",\n \"Figure: P-VP-V diagram of Bardeen-AdS black holes.\",\n \"Isotherms descend from top to bottom when the temperature is decreasing, where the black dashed curve corresponds to T>T c T>T_c, the red curve to the critical temperature T=T c T=T_c, and the two blue curves to T0.16V_c$ .\",\n \"In general, we point out that the normalized Ruppeiner curvature of Bardeen-AdS black holes has a very similar microstructure to that of van der Waals fluids [15].\",\n \"Figure: Ruppeiner thermodynamic curvature of Bardeen-AdS black holes.\",\n \"This picture shows the characteristic behaviour of the Ruppeiner thermodynamic curvature as a function of the reduced temperature and volume.\"\n ],\n [\n \"Weinhold's geometry\",\n \"In $(T,V)$ fluctuation coordinates, the line element Eq.\",\n \"(REF ) of Weinhold's geometry reads $ds^2_W=\\\\frac{1}{1-\\\\Pi }\\\\frac{C_{_V}}{T}dT^2+\\\\frac{1}{1-\\\\Pi }\\\\left( \\\\frac{\\\\partial P}{\\\\partial V}\\\\right)_TdV^2,$ where $C_{_V}=0$ still leads to a degenerate metric which can be cured by the normalization [15].\",\n \"We note that $\\\\Pi $ satisfies $0<\\\\Pi <1$ and so it changes only the quantity rather than the sign of thermodynamic curvature.\",\n \"This means that a repulsive interaction cannot be reversed to an attractive interaction by the presence of $\\\\Pi $ , and vice versa.\",\n \"Without loss of generality, we set the dimensionless constant $\\\\Pi =0$ .\",\n \"Using formula Eq.\",\n \"(REF ), we obtain the normalized Weinhold thermodynamic scalar curvature straightforwardly, $\\\\mathcal {R}_W=\\\\bigg \\\\lbrace \\\\bigg ((\\\\sqrt{273}+17) \\\\tilde{V}^{2/3}-2\\\\bigg )^7 \\\\bigg ((\\\\sqrt{273}+17) \\\\tilde{V}^{2/3}+(17 \\\\sqrt{273}+281) \\\\tilde{V}^{4/3}-4\\\\bigg ) \\\\bigg [\\\\frac{-57 \\\\sqrt{\\\\sqrt{273}+15} \\\\@root 3 \\\\of {\\\\tilde{V}}}{[(\\\\sqrt{273}+17) \\\\tilde{V}^{2/3}-2]^{7/2}}\\\\nonumber \\\\\\\\-\\\\frac{8 (\\\\sqrt{273}+15) (13 \\\\sqrt{273}+219) \\\\tilde{T} [(\\\\sqrt{273}+17) \\\\tilde{V}^{2/3}+(17 \\\\sqrt{273}+281) \\\\tilde{V}^{4/3}-4]}{[(\\\\sqrt{273}+17) \\\\tilde{V}^{2/3}-2]^7} \\\\bigg ]\\\\bigg \\\\rbrace \\\\bigg /\\\\bigg \\\\lbrace 4 (13 \\\\sqrt{273}+219) \\\\nonumber \\\\\\\\ \\\\times \\\\bigg [3 \\\\sqrt{(\\\\sqrt{273}+17) \\\\tilde{V}^{2/3}-2} [(17 \\\\sqrt{273}+281) \\\\tilde{V}^{2/3}-5 (\\\\sqrt{273}+17)]-2 \\\\sqrt{\\\\sqrt{273}+15} T [(\\\\sqrt{273}+17) \\\\tilde{V}^{2/3}\\\\nonumber \\\\\\\\+(17 \\\\sqrt{273}+281) \\\\tilde{V}^{4/3}-4]\\\\bigg ]^2\\\\bigg \\\\rbrace .\\\\nonumber \\\\\\\\ $ We plot the behaviour of normalized Weinhold thermodynamic curvature in Fig.\",\n \"REF .\",\n \"Figure: Weinhold thermodynamic curvature of Bardeen-AdS black holes.\",\n \"This picture shows the characteristic behaviour of the Weinhold thermodynamic curvature as a function of the reduced temperature and volume.From the two expressions in Eqs.\",\n \"(REF ) and (REF ), it is obvious that the normalized thermodynamic scalar curvatures in the two geometry theories are different.\",\n \"There is, however, a qualitative similarity between the structures of Ruppeiner's geometry and Weinhold's geometry as displayed in Fig.\",\n \"REF and Fig.\",\n \"REF .\",\n \"The normalized thermodynamic curvatures diverge to negative infinity in very low temperatures.\",\n \"In other words, for both the Weinhold's geometry and Ruppeiner's geometry there exists a strongly attractive interaction in the microstructure of Bardeen-AdS black holes in this region of temperature, which can be explicitly seen in the two figures.\",\n \"Furthermore, it is worth noting that the normalized Weinhold curvature is negative and does not change sign, which is due to the existence of conformal relation between Weinhold's and Ruppeiner's geometries.\",\n \"According to the microscopic interpretation of interactions [13], the positive and negative curvatures imply the repulsive and attractive interactions, respectively.\",\n \"Hence, for Weinhold's geometry there are only attractive interactions but no repulsive interactions in the microstructure of Bardeen-AdS black holes, which coincides with the interacting behaviour in the van der Waals fluid.\"\n ],\n [\n \"Critical phenomena\",\n \"In order to get a deep understanding of the normalized thermodynamic curvatures of Bardeen-AdS black holes, we shall focus on the critical phenomena in which the Bardeen-AdS black hole has a markedly different behaviour from that of the charged AdS black holes with spacetime singularities, such as the Reissner-Nordström AdS and Gauss-Bonnet AdS black holes [16], [43], [14].\",\n \"The normalized Ruppeiner thermodynamic curvature Eq.\",\n \"(REF ) diverges and vanishes along the following curves on the $(V/V_c, T/T_c)$ plane, respectively, $\\\\tilde{T}|_{\\\\mathcal {R}_R\\\\rightarrow \\\\infty }&=& \\\\frac{3 \\\\sqrt{\\\\left(\\\\sqrt{273}+17\\\\right) \\\\tilde{V}^{2/3}-2} \\\\left(\\\\left(17 \\\\sqrt{273}+281\\\\right) \\\\tilde{V}^{2/3}-5 \\\\left(\\\\sqrt{273}+17\\\\right)\\\\right)}{2 \\\\sqrt{\\\\sqrt{273}+15} \\\\left(\\\\left(\\\\sqrt{273}+17\\\\right) \\\\tilde{V}^{2/3}+\\\\left(17 \\\\sqrt{273}+281\\\\right) \\\\tilde{V}^{4/3}-4\\\\right)},\\\\\\\\\\\\tilde{T}|_{\\\\mathcal {R}_R=0}&=&3 \\\\bigg [35 (285 \\\\sqrt{273}+4709) \\\\tilde{V}^{2/3}-11 (4777 \\\\sqrt{273}+78929) \\\\tilde{V}^{4/3}+(80069 \\\\sqrt{273}+1322957) \\\\tilde{V}^2\\\\nonumber \\\\\\\\& & -25 (17 \\\\sqrt{273}+281)\\\\bigg ]\\\\bigg /\\\\bigg [2 \\\\sqrt{\\\\sqrt{273}+15} \\\\sqrt{(\\\\sqrt{273}+17) \\\\tilde{V}^{2/3}-2}\\\\, \\\\bigg (-7 (17 \\\\sqrt{273}+281) \\\\tilde{V}^{2/3}\\\\nonumber \\\\\\\\& &-4 (285 \\\\sqrt{273}+4709) \\\\tilde{V}^{4/3}+(4777 \\\\sqrt{273}+78929) \\\\tilde{V}^2+10 (\\\\sqrt{273}+17)\\\\bigg )\\\\bigg ],$ which is plotted in Fig.\",\n \"REF .\",\n \"Figure: Critical behaviour of thermodynamic curvatures.\",\n \"The upper (red) curve stands for the diverging case of ℛ R \\\\mathcal {R}_R and ℛ W \\\\mathcal {R}_W.\",\n \"The lower (blue) curve corresponds to the vanishing case of ℛ R \\\\mathcal {R}_R only.From Eq.\",\n \"(REF ) we find that the Weinhold curvature shares the same diverging curve as the Ruppeiner curvature, i.e., $\\\\tilde{T}|_{\\\\mathcal {R}_W\\\\rightarrow \\\\infty }=\\\\tilde{T}|_{\\\\mathcal {R}_R\\\\rightarrow \\\\infty }$ .\",\n \"This happens due to the existence of conformal relation between the Weinhold's geometry and Ruppeiner's geometry.\",\n \"However, the Weinhold curvature does not vanish at any finite positive temperature, which is quite different from the Ruppeiner curvature.\",\n \"For Weinhold's geometry and Ruppeiner's geometry the normalized thermodynamic curvatures will diverge at the critical point of a phase transition as shown in Fig.\",\n \"REF .\",\n \"We can easily see that this divergent point is indeed located at the maximum of the red curve, where the temperature and volume just take their critical values, i.e., $T=T_c$ and $V=V_c$ .\",\n \"This phenomenon was also noticed [44] for the charged AdS black holes with spacetime singularities, where the divergent points of specific heat match those of thermodynamic curvature.\",\n \"Furthermore, a positive normalized thermodynamic curvature was obtained [15], [16] in the small volume phase of the charged AdS black holes with spacetime singularities, namely, a weak repulsive interaction appears in the microstructure, while it does not in the van der Waals fluid.\",\n \"Interestingly, for the Bardeen-AdS black hole that has no spacetime singularities, the temperature converges to zero as the volume decreases, see Fig.\",\n \"REF .\",\n \"Such a behaviour has been observed [15] in the van der Waals fluid.\",\n \"Hence, the Bardeen-AdS black hole has the critical phenomena similar to those of van der Waals fluid.\"\n ],\n [\n \"Conclusions\",\n \"In the present work, we revisit the thermodynamics and investigate the thermodynamic geometry for Bardeen black holes in the anti-de Sitter spacetime.\",\n \"We find that the entropy of Bardeen-AdS black holes still obeys the area law if the modified first law of thermodynamics is considered.\",\n \"We derive the normalized scalar curvature in the Weinhold's geometry and find that it has the same divergent behaviour as the Ruppeiner scalar curvature at the critical point of a phase transition, but it has no vanishing behaviour at any finite positive temperature while the Ruppeiner scalar curvature has.\",\n \"Moreover, we notice that the additional parameter $\\\\Pi $ appeared in the conformal factor of Weinhold's geometry does not reverse the interaction in the microstructure of Bardeen-AdS black holes.\",\n \"When compared to the charged AdS black holes with spacetime singularities, the Bardeen-AdS black hole as a regular black hole has a rather different behaviour, that is, it has no weak repulsive interaction in the microstructure.\",\n \"When compared to the van der Waals fluid, the Bardeen-AdS black hole has a critical behaviour resembling the liquid-gas phase transition and both its Weinhold scalar curvature and Ruppeiner scalar curvature have a very similar microstructure to that of the van der Waals fluid.\",\n \"Our results seem to justify the mathematical analogy of the Bardeen-AdS black hole with the van der Waals fluid.\",\n \"Whether the vanishing of a weak repulsive interaction in the Bardeen-AdS black hole is universal or not in the other regular black holes, we shall deal with this issue by analyzing their thermodynamic geometries in detail.\"\n ],\n [\n \"Acknowledgements\",\n \"The authors would like to thank R.-G Cai and T. Vetsov for their helpful comments and disscussions.\",\n \"This work was supported in part by the National Natural Science Foundation of China under Grant No.\",\n \"11675081.\"\n ]\n]"},"id":{"kind":"string","value":"2107.01866"}}},{"rowIdx":1592,"cells":{"text":{"kind":"list like","value":[["A theoretical analysis of one-dimensional discrete generation ensemble\n Kalman particle filters"],["Abstract Despite the widespread usage of discrete generation Ensemble Kalman particle filtering methodology to solve nonlinear and high dimensional filtering and inverse problems, little is known about their mathematical foundations.","As genetic-type particle filters (a.k.a.","sequential Monte Carlo), this ensemble-type methodology can also be interpreted as mean-field particle approximations of the Kalman-Bucy filtering equation.","In contrast with conventional mean-field type interacting particle methods equipped with a globally Lipschitz interacting drift-type function, Ensemble Kalman filters depend on a nonlinear and quadratic-type interaction function defined in terms of the sample covariance of the particles.","Most of the literature in applied mathematics and computer science on these sophisticated interacting particle methods amounts to designing different classes of useable observer-type particle methods.","These methods are based on a variety of inconsistent but judicious ensemble auxiliary transformations or include additional inflation/localisationtype algorithmic innovations, in order to avoid the inherent time-degeneracy of an insufficient particle ensemble size when solving a filtering problem with an unstable signal.","To the best of our knowledge, the first and the only rigorous mathematical analysis of these sophisticated discrete generation particle filters is developed in the pioneering articles by Le Gland-Monbet-Tran and by Mandel-Cobb-Beezley, which were published in the early 2010s.","Nevertheless, besides the fact that these studies prove the asymptotic consistency of the Ensemble Kalman filter, they provide exceedingly pessimistic meanerror estimates that grow exponentially fast with respect to the time horizon, even for linear Gaussian filtering problems with stable one dimensional signals.","In the present article we develop a novel self-contained and complete stochastic perturbation analysis of the fluctuations, the stability, and the long-time performance of these discrete generation ensemble Kalman particle filters, including time-uniform and non-asymptotic mean-error estimates that apply to possibly unstable signals.","To the best of our knowledge, these are the first results of this type in the literature on discrete generation particle filters, including the class of genetic-type particle filters and discrete generation ensemble Kalman filters.","The stochastic Riccati difference equations considered in this work are also of interest in their own right, as a prototype of a new class of stochastic rational difference equation."],["Introduction","The Ensemble Kalman filter (abbreviated EnKF) is a class of interacting particle system methodologies for solving nonlinear filtering and inverse problems.","They were introduced by Evensen in the seminal article [44] published in 1994, see also [45], [46] for a more recent overview.","In the last three decades the EnKF methodology has become one of the most used numerical techniques for solving high dimensional forecasting and data assimilation problems in a variety of applications including ocean and atmosphere sciences [1], [67], [68], [58], [73], fluid mechanics [14], [74], [78], image inverse problems [15], weather forecasting [3], [4], [30], [49], environmental and ecological statistics [43], [52] and oil reservoir simulations [47], [72], [82], [83], [94].","To connect these EnKF techniques with particle filtering methodology for solving high dimensional problems arising in fluid mechanics, we also refer to [59].","In contrast with genetic-type particle filters (a.k.a.","Sequential Monte Carlo, often abbreviated SMC), the EnKF is defined by a system of particles evolving as the signal in some state space with an interaction function that depends on the sample covariance matrices of the system.","For a detailed mathematical description of particle filters and ensemble Kalman filter methodologies in both continuous and discrete time settings we refer, for instance, to the book [39] and the references therein.","See also section REF in the present article dedicated to nonlinear Kalman-Bucy type Markov chains and their mean-field particle interpretations.","There exists a vast literature in data assimilation theory dedicated to the numerical analysis of the EnKF view as a useable observer-type algorithm with a very small sample size and equipped with a variety of judicious ensemble square-root type transformations, as well as inflation/localisation-type algorithmic innovations to control the unstable directions of the signal, see for instance [5], [6], [60], [61], [92], [93], as well as [21], [23] and the references therein.","From this point of view, the EnKF is no longer interpreted as a true approximation of the optimal filter but as an observer.","Here, the terminology “well-posedness\" is not understood as a traditional mathematical consistency-type property of some Monte Carlo statistical estimate but as a lack of divergence of the algorithm with respect to the time horizon.","Despite the widespread use of EnKF techniques to solve nonlinear and high dimensional filtering and inverse problems, little is known about their theoretical performance and their mathematical foundations.","The first rigorous mathematical analysis of discrete generation EnKF appeared in 2011 in the independent pioneering works of Le Gland, Monbet and Tran [48], and Mandel, Cobb and Beezley [70].","These two seminal articles provide mean error estimates for discrete generation EnKF and show that the EnKF converges towards the Kalman filter for linear-Gaussian problems as the number of samples, say $N$ , tends to infinity.","The article [48] also shows that the EnKF doesn't converge to the optimal filter for nonlinear or non-Gaussian filtering problems.","The consistent-type EnKF methodology for linear-Gaussian models can be extended to nonlinear filtering problems using the new class feedback particle filter methodology introduced by Mehta and Meyn and their co-authors in the series of seminal articles [84], [85], [86], [87], [88].","To obtain a consistent EnKF for nonlinear filtering problems one needs to solve a Poisson-type equation at every time step, which often requires an additional level of approximation.","The theoretical analysis of this type of sophisticated nonlinear EnKF is not discussed in the present article but we refer the reader to [88], [89], [90] for the analysis of its performance and the convergence.","Besides these fundamental theoretical advances in the understanding of EnKF filters in linear-Gaussian models, the non-asymptotic analysis developed in the aforementioned articles yields exceedingly pessimistic estimates that grow exponentially fast with respect to the time horizon.","Note that for a time horizon $t=39$ an innocent exponential bound of the form $55\\times e^{5 t}/N\\ge 10^{86}/N$ will require a sample size $N$ that is 10 times larger than the number $10^{86}$ of elementary particles of matter in the visible universe in order to obtain a poor $10\\%$ accuracy after 39 runs.","The mathematical analysis of the continuous-time version of the EnKF methodology has started more recently in [37], [38], followed by the series of articles [18], [20], [22], see also review article [23].","Some extensions to nonlinear filtering problems are also developed in the series of more recent articles [62], [63], [64], [65], [75].","In contrast with the feedback particle filter methodology discussed above, these articles are concerned with a class of EnKF particle filters that are only consistent for linear-Gaussian models.","As expected, for nonlinear or non-Gaussian filtering problems, none of these EnKF converge to the optimal filter as the number of particles tends to infinity.","From a probabilistic viewpoint, in the linear-Gaussian case, continuous time EnKF are represented by a Kalman-Bucy-type diffusion coupled with a stochastic Riccati matrix nonlinear diffusion equation (see for instance Theorem 3.1 in [37]).","In the series of articles [37], [38], [18], [20], [22], the authors present a refined stochastic stability analysis of these rather sophisticated diffusion-type perturbation models, including central limit theorems, as well as uniform bias and mean error estimates with respect to the time horizon for several classes of consistent EnKF stochastic models.","To describe briefly these results, as underlined in the review article [23], we emphasise that the continuous-time EnKF methodology (for linear-Gaussian models) may be broadly divided into three different classes of probabilistic models, according to the level of fluctuation added via sampling noise needed to ensure that the EnKF sample mean and covariance are consistent in the linear-Gaussian setting.","By chronological order of appearance in the literature, these three different classes of continuous-time EnKF are briefly discussed below.","The continuous time version of the EnKF discussed in section REF in the present article coincides with the so-called “Vanilla” discrete time original form of the EnKF [44], [30].","This first class of continuous-time EnKF exhibits the most fluctuations due to sampling both signal and observation noises (see for instance the evolution equation (REF )).","The second class of EnKF is a continuous-time version of the Sakov and Oke square root filter (a.k.a.","“deterministic EnKF”) introduced in [80], [81], see also [76], [13].","Unfortunately, this class of “deterministic EnKF” is only consistent for continuous time models.","In the discrete time situation, it fails to converge to the optimal filter as the number of particles tends to infinity, even in linear-Gaussian settings, see for instance [23] and references therein.","This class of discrete generation EnKF is not discussed in the present article.","Finally, the third class consists of purely deterministic transport-inspired EnKF with randomness coming only from the initial conditions.","By construction, this class of filters can be analysed directly using the evolution semigroup of the optimal filter and that of the Riccati matrix equations.","Further details on this subject can be found in the review article [23] on continuous time EnKF particle filters and in the articles [16], [17] dedicated to the stability of Kalman-Bucy diffusion processes and related Riccati matrix differential equations.","The stochastic analysis of the Riccati matrix diffusion associated with the sample covariance of these three classes of continuous-time EnKF is rather well understood for multivariate models.","The articles [37], [38], [18], [20], [22] present several multivariate fluctuation theorems for Riccati matrix diffusions, as well as a refined non-asymptotic analysis, including several time-uniform mean error estimates.","The extension of these uniform results to the Vanilla EnKF is developed in [37], [38] but only for stable and ergodic signals.","The stability analysis and the long-time performance of Vanilla EnKF with possibly unstable and multivariate transient signals remains partially understood.","To understand the difficulties that arise these problems it is worth mentioning that even for elementary one-dimensional problems, the stochastic Riccati diffusions describing the random evolution of the sample variances of the EnKF may exhibit surprisingly heavy tailed or Gaussian tailed invariant distributions, depending on the class of EnKF one uses as well as the number of samples [22], [23].","In this setting, some raw moments of the sample variances of the Vanilla continuous time EnKF for one-dimensional filtering problems are infinite, for any chosen finite sample size lower than some critical value.","To the best of our knowledge, the question of finding uniform mean error estimates for the Vanilla EnKF sample means remains an important open research question for both discrete generation and continuous-time multivariate models.","The only work in this direction appears to be the recent article [22].","This article is dedicated to the long time performance and the stability properties of the stochastic Riccati diffusion associated with the three different classes of continuous-time EnKF filters discussed above for one-dimensional and linear-Gaussian filtering problems with possibly unstable and transient signals.","The extension of these results to multivariate models remains open.","The main advantage in working with continuous-time models comes from the fact that the sample mean and the sample covariance of the EnKF filters satisfy a coupled nonlinear diffusion equation which can be handled using conventional stochastic analysis.","Note however that in the multivariate case, the sample covariance matrices satisfy a stochastic Riccati equation involving matrix valued martingales which require one to develop an appropriate and more sophisticated stochastic analysis in matrix spaces [18], [20].","Physical systems and most of the filtering problems arising in signal processing and tracking are typically defined by continuous time models.","Nevertheless, for obvious reasons, any numerical integration of these models including all known filtering sensors are defined by discrete-time models.","As a result, the stochastic analysis used for the continuous-time EnKF is no longer applicable without adding an additional level of approximation.","The analysis of discrete generation EnKF is more delicate.","To handle the structure and the statistical difficulty of discrete time models, new stochastic tools need to be developed.","For instance, in contrast with continuous-time models, discrete generation EnKF are not defined by a single coupled diffusion process but in terms of coupled two-step prediction-updating process (a.k.a.","forecasting-analysis steps in EnKF and data assimilation literature).","Furthermore, the Gaussian-nature of the diffusion models arising in the analysis of continuous-time EnKF theory is also lost.","Another inherent difficulty is that discrete generation EnKF involve more sophisticated non-central $\\chi $ -square nonlinear fluctuations (cf.","for instance Theorem REF and Corollary REF ).","We expect the multivariate version of the EnKF evolution to involve a similar two-step prediction-updating transition involving sophisticated non-central Wishart matrix local fluctuations.","Up to some additional technicalities and at the cost of multi-index and tensor theoretical notation, we believe that the analysis of multivariate and discrete generation stochastic Riccati equations can be conducted by extending the theory of discrete generation one-dimensional models developed in the present article in the spirit of the stochastic analysis of continuous time EnKF models developed in [18], [22].","As for continuous time models, we also believe that one cannot expect to directly deduce uniform state estimates from those at the level of the sample covariance matrices.","Due to these reasons, in the present article we have chosen to concentrate our study to one-dimensional models.","We intend to extend these results to more sophisticated multivariate models in a subsequent article.","The main objective of this article is to develop a novel and complete stochastic analysis on the fluctuations and the long time performance of discrete generation EnKF particle filters that applies to filtering problems involving possibly unstable signals.","Our main contributions are listed succinctly below.","For more precise statements of the results, we refer the reader to the series of theorems stated in section .","$\\bullet $ Up to some orthogonal transformations on the sample space, our first main result, Theorem REF , states that the sample variances of the EnKF satisfy an autonomous stochastic Riccati-type evolution equation, which is independent of the sequence of observations, and that the EnKF sample means evolve as a Kalman filter driven by sample variances.","In addition, the local perturbations of the EnKF and the sample variance Riccati equations are independent.","An alternative Markov chain realisation of the stochastic Riccati difference equation is also discussed in Corollary REF .","Theorem REF is an extended version of the stochastic perturbation theorem presented in [37].","The continuous time version of the sample variance equations stated in Theorem REF also coincides with the one-dimensional stochastic Riccati equation discussed in [22], see also [18] for the extended matrix version of these models in the context of multivariate filtering problems.","$\\bullet $ Section REF is concerned with the stability properties of the optimal filter and the stochastic Riccati equations describing the evolution of the sample variances of the EnKF for possibly unstable signals.","In this context, we show that the discrete generation stochastic Riccati equations converge exponentially fast to a single invariant measure.","The contraction theorem, Theorem REF , and the exponential semigroup estimates (REF ) stated in Theorem REF are partial extensions to discrete generation models of Theorem 2.2 in the article [22], dedicated to the stability of one dimensional stochastic Riccati equations.","In Theorem REF we also show that the sample variances and the difference between the sample means and the true (possibly transient) signal forms a Markov chain that converges exponentially fast to a unique invariant measure.","This seems to be the first result of this kind in the field of interacting particle-type filters, including EnKF and genetic-type particle filters.","$\\bullet $ Our third main result is concerned with uniform mean-error and bias estimates with respect to the time horizon for both the sample variances and the sample means delivered by the EnKF.","Theorem REF shows that the sample variances are uniformly close to the true optimal variances for all times, even when the signal and the ensemble are transient.","The EnKF sample mean uniform accuracy for possibly unstable signals is discussed in theorem REF .","This result ensures that the sample means are uniformly close to the true optimal filter at any time horizon.","The continuous time version of these results are discussed in Corollary 2.4 in [22].","$\\bullet $ Last, but not least, we present a novel fluctuation analysis of EnKF filters.","Theorems REF and REF present new multivariate central limit theorems for the sample mean and the sample variance processes.","These fluctuation results, at the level of the processes, are an extended multivariate version of central limit theorem presented in [22] in the context of continuous time models."],["The Kalman filter","Consider a one dimensional, time homogeneous linear-Gaussian filtering model of the following form $\\left\\lbrace \\begin{array}{rcl}Y_n&=&C\\,X_n+D\\,V_n\\\\X_{n+1}&=&A\\,X_{n}+B\\,W_{n+1},\\end{array}\\right.\\qquad n \\ge 0.$ In the above display, $(V_n,W_{n+1})$ is a sequence of 2-dimensional Gaussian centered random variables with unit variance, the initial condition of the signal $X_0$ is a Gaussian random variable with mean and variance denoted by $(\\widehat{X}^-_0,P_0)$ (independent of $(V_n,W_{n+1})$ ), and $(A,B,C,D)$ are some non-zero parameters.","The latter condition can be interpreted as a controllability and observability condition for one-dimensional filtering problems.","Let $ {\\cal Y }_n=\\sigma \\left(Y_k,~k\\le n\\right)$ be the filtration generated by the observation process.","The optimal filter is defined by the conditional distribution $\\widehat{\\eta }_n$ of the signal state $X_n$ given $ {\\cal Y }_n$ .","The distribution $\\widehat{\\eta }_n$ is a Gaussian distribution with conditional mean and variance $\\widehat{X}_n:=\\mathbb {E}(X_n~|~ {\\cal Y }_n)\\quad \\mbox{\\rm and}\\quad \\widehat{P}_n:=\\mathbb {E}\\left(\\left(X_n-\\widehat{X}_n\\right)^{2}\\right).$ The optimal one-step predictor is defined by the conditional distribution $\\eta _{n}$ of the signal state $X_n$ given $ {\\cal Y }_{n-1}$ .","The distribution $\\eta _n$ is also a Gaussian distribution with conditional mean and variance $\\widehat{X}^-_n:=\\mathbb {E}(X_n~|~ {\\cal Y }_{n-1})\\quad \\mbox{\\rm and}\\quad P_n:=\\mathbb {E}\\left(\\left(X_{n}-\\widehat{X}_{n}^{-}\\right)^{2}\\right).$ The updating-prediction steps are described by the following synthetic diagram $(\\widehat{X}_n^-,P_n)\\longrightarrow (\\widehat{X}_{n},\\widehat{P}_{n})\\longrightarrow (\\widehat{X}^-_{n+1},P_{n+1}),$ with the well-known updating-prediction Kalman filter equations: $\\left\\lbrace \\begin{array}{rcl}\\widehat{X}_n&=&\\widehat{X}_n^-+G_n~\\left(Y_n-C\\widehat{X}_n^-\\right) \\\\\\widehat{P}_n&=&(1-G_n C)P_n\\end{array}\\right.\\quad \\mbox{\\rm and} \\quad \\left\\lbrace \\begin{array}{rclcrcl}\\widehat{X}_{n+1}^-&=&A \\widehat{X}_{n}\\\\P_{n+1}&=& A^2\\widehat{P}_{n}+R,\\end{array}\\right.$ where $R = B^2$ .","In the above display, $G_n$ stands for the so-called gain parameter defined by the formula $G_n:=CP_n/(C^2P_n+D^2)\\Longrightarrow 1-G_nC=1/(1+SP_n)\\quad \\mbox{\\rm with}\\quad S:=(C/D)^2.$ The evolution equations of the variance parameters $(P_n,\\widehat{P}_n)$ satisfy the Riccati rational difference equations $P_{n+1}=\\phi \\left(P_n\\right):=\\frac{aP_n+b}{cP_n+d}\\quad \\mbox{\\rm and}\\quad \\widehat{P}_{n+1}=\\widehat{\\phi }\\left(\\widehat{P}_n\\right):=\\frac{\\widehat{a}\\,\\widehat{P}_n+\\widehat{b}}{\\widehat{c}\\,\\widehat{P}_n+\\widehat{d}}$ with the parameters $\\begin{array}{l}(a,b,c,d):=\\left(A^2+RS,R,S,1\\right)\\quad \\mbox{\\rm and}\\quad (\\widehat{a},\\widehat{b},\\widehat{c},\\widehat{d})=(A^2,R,A^2S,1+SR)\\\\\\\\\\Longrightarrow ad-bc=A^2=\\widehat{a}\\,\\widehat{d}-\\widehat{b}\\,\\widehat{c}>0.\\end{array}$ One-dimensional rational difference equations are rather well understood, see for instance [26].","Section REF also provides a refined stability analysis and several comparison properties of rational difference equations including non-asymptotic Taylor expansions any order."],["Ensemble Kalman filters","Let $({X}_0, {W}_n,{V}_n)$ be independent copies of $(X_0, W_n,V_n)$ .","Consider the nonlinear Markov chain starting at ${X}_0$ and defined sequentially for any $n\\ge 0$ by the updating-correction formulae $\\left\\lbrace \\begin{array}{rcl}\\widehat{{X}}_{n}&=&{X}_n+{G}_{\\eta _n}~(Y_n-(C\\,{X}_n+D\\,{V}_n))\\quad \\mbox{\\rm with}\\quad {G}_{\\eta _n}~:=C{Q}_{\\eta _n}/(C^2{Q}_{\\eta _n}+D^2)\\\\&&\\\\{X}_{n+1}&=& A\\,{X}_{n}+B\\,{W}_{n+1}.\\\\\\end{array}\\right.$ In the above display, ${Q}_{\\eta _n}$ stands for the variance parameter ${Q}_{\\eta _n}:=\\int \\left(x- \\int z~ \\eta _n(dz)\\right)^2\\eta _n(dx)\\quad \\mbox{\\rm indexed by}\\quad \\eta _n=\\mbox{\\rm Law}({X}_n~|~ {\\cal Y }_{n-1}).$ Using a simple induction argument, it is straightforward to show that $\\eta _n= {\\cal N }\\left(\\widehat{X}^-_n,P_n\\right)\\quad \\mbox{\\rm and}\\quad \\widehat{\\eta }_n= {\\cal N }\\left(\\widehat{X}_n,\\widehat{P}_n\\right)=\\quad \\mbox{\\rm Law}(\\widehat{{X}}_n~|~ {\\cal Y }_{n}).$ Also observe that the above stochastic model can be interpreted as a Markov chain with a two-step updating-prediction transition described by the following synthetic diagram ${X}_n\\longrightarrow \\widehat{{X}}_{n}\\longrightarrow {X}_{n+1}.$ Note that the updating transition ${X}_n\\longrightarrow \\widehat{{X}}_{n}$ depends on the observation $Y_n$ delivered by the sensor as well as on the conditional law of ${X}_n$ given the information $ {\\cal Y }_{n-1}$ .","The EnKF associated to the filtering problem (REF ) coincides with the mean-field particle interpretation of the nonlinear Markov chain (REF ).","To define these models, we let $\\xi _0=(\\xi _0^i)_{1\\le i\\le N+1}$ be a sequence of $(N+1)$ independent copies of $X_0$ , for some parameter $N\\ge 1$ .","We also denote by $ {\\cal W }_n=( {\\cal W }^i_n)_{i\\ge 1}$ and $ {\\cal V }_n=( {\\cal V }^i_n)_{i\\ge 1}$ a sequence of independent copies of $W_n$ and $V_n$ .","The mean field particle interpretation of the nonlinear Markov chain discussed above is given by the interacting particle system defined sequentially for any $1\\le i\\le N+1$ and $n\\ge 0$ by the formulae $\\left\\lbrace \\begin{array}{rcl}\\widehat{\\xi }^i_{n}&=&\\xi ^i_n+g_n~(Y_n-(C\\xi ^i_n+D {\\cal V }^i_n))\\quad \\mbox{\\rm with}\\quad g_n:=Cp_n/(C^2p_n+D^2)\\\\&&\\\\\\xi ^i_{n+1}&=& A\\,\\widehat{\\xi }^i_{n}+B\\, {\\cal W }^i_{n+1}.\\end{array}\\right.$ In the above display $p_n$ stands for the normalised sample variance $p_n:=\\frac{1}{N}\\sum _{1\\le i\\le N+1}(\\xi ^i_n-m_n)^2=\\left(1+\\frac{1}{N}\\right) {Q}_{\\eta ^N_n}\\quad \\mbox{\\rm with}\\quad \\eta ^N_n:=\\frac{1}{N+1}\\sum _{1\\le i\\le N+1}\\delta _{\\xi ^i_n}.$ We also consider the sample means $m_n:=\\frac{1}{N+1}\\sum _{1\\le i\\le N+1}\\xi ^i_n\\quad \\mbox{\\rm and}\\quad \\widehat{m}_n:=\\frac{1}{N+1}\\sum _{1\\le i\\le N+1}\\widehat{\\xi }^i_n$ and the $\\sigma $ -field filtrations $\\begin{array}{l} {\\cal G }_n:=\\sigma (\\xi _0, {\\cal W }_k, {\\cal V }_k, 0\\le k\\le n)\\supset \\sigma (\\xi _k,\\widehat{\\xi }_k, ~0\\le k\\le n),\\\\\\\\\\widehat{ {\\cal F }}_n:= {\\cal Y }_{n}\\vee {\\cal G }_n\\supset {\\cal F }_n:=\\widehat{ {\\cal F }}_{n-1}\\vee \\sigma ( {\\cal W }_n)\\supset \\widehat{ {\\cal F }}_{n-1}\\vee \\sigma (\\xi _n).\\end{array}$ Note that this implies that $\\widehat{ {\\cal F }}_n\\supset {\\cal Y }_{n}\\quad \\mbox{\\rm and}\\quad {\\cal F }_n\\supset {\\cal Y }_{n-1}.$ In the rest of the article we shall assume that $\\xi _0$ , $ {\\cal V }_n$ and $ {\\cal W }_n$ are column vectors.","We also write $X~\\underline{\\in }~ {\\cal F }$ when a random variable is $ {\\cal F }$ -measurable.","With this notation, at every time $n\\ge 0$ , the updating-prediction steps of the ensemble Kalman filter are described by the following synthetic diagram $(m_n,p_n)~\\underline{\\in }~ {\\cal F }_n\\longrightarrow (\\widehat{m}_{n},\\widehat{p}_{n})~\\underline{\\in }~ \\widehat{ {\\cal F }}_n\\longrightarrow (m_{n+1},p_{n+1})~\\underline{\\in }~ {\\cal F }_{n+1}.$"],["Some basic notation","This section presents some basic notation and preliminary results necessary for the statement of our main results.","Throughout the rest of the article, we write $\\tau $ , $\\tau _k$ as well as $\\tau (l)$ and $\\tau _k(l)$ for some collection of non-universal constants whose values may vary from line to line, but only depend on the parameters $k$ and $l$ in some given parameter spaces.","We use the letters $\\iota $ , $\\iota _k$ , $\\iota (l)$ , $\\iota _k(l)$ as well as $\\epsilon $ and $\\epsilon _k\\in ]0,1]$ when these constants may also depend on the parameters $(A,B,C,D)$ of the filtering model.","Importantly these constants do not depend on the time parameter nor on the number of particles.","We denote by $\\chi ^{2}_{N ,Nx}$ a collection a non-central $\\chi $ -square random variables indexed by the degree of freedom $N\\ge 1$ and non centrality parameter $N x\\ge 0$ .","We recall that for any $N \\ge 1$ , we have $\\begin{array}{l}\\displaystyle \\frac{1}{N}~\\chi ^2_{N,Nx}=\\frac{1}{N}~\\sum _{1\\le i\\le N}~\\left(Z_i+\\sqrt{x}\\right)^2=1+x+\\sqrt{x}~\\frac{2}{N}~\\sum _{1\\le i\\le N}~Z_i+\\frac{\\sqrt{2}}{N}~\\sum _{1\\le i\\le N}~\\frac{Z_i^2-1}{\\sqrt{2}},\\end{array}$ for some given sequence $(Z_i)_{i\\ge 1}$ of independent and centered Gaussian random variables.","Rewritten in a slightly different form, for any $u,v$ with $v\\ne 0$ we have $\\frac{v}{N}~\\chi ^2_{N,N(u/v)^2}&=&\\displaystyle v+\\frac{u^2}{v}+\\frac{2}{\\sqrt{N}}~u~ { {\\cal Z }}+\\sqrt{\\frac{2}{N}}~v~ \\widetilde{ {\\cal Z }},$ where ${ {\\cal Z }}:=\\frac{1}{\\sqrt{N}}~\\sum _{1\\le i\\le N}~Z_i\\quad \\mbox{\\rm and}\\quad \\widetilde{ {\\cal Z }}:=\\frac{1}{\\sqrt{N}}~\\sum _{1\\le i\\le N}~\\frac{Z_i^2-1}{\\sqrt{2}}.$ Note that these definitions imply that ${ {\\cal Z }}$ is a centred Gaussian random variable with unit variance and $\\widetilde{ {\\cal Z }}$ is a centered $\\chi $ -square variable.","We also consider the extended random vector $\\Delta :=\\left(\\begin{array}{c}\\widehat{Z}\\\\{ {\\cal Z }}\\\\\\widetilde{ {\\cal Z }}\\end{array}\\right)\\quad \\mbox{\\rm with}\\quad \\widehat{Z}:=\\frac{1}{\\sqrt{N+1}}~\\sum _{1\\le i\\le N+1}~Z_{N+i}.$ In a similar manner, we define $(\\widehat{\\Delta }_{n},\\Delta _{n+1})$ as the sequence of random vectors $\\widehat{\\Delta }_{n}:=\\left(\\begin{array}{c}\\widehat{\\Delta }^1_n\\\\\\widehat{\\Delta }^2_n\\\\\\widehat{\\Delta }^3_n\\end{array}\\right)=\\left(\\begin{array}{l}\\widehat{V}_{n}\\\\{ {\\cal V }}_{n}\\\\\\displaystyle \\widetilde{ {\\cal V }}_n\\end{array}\\right)\\quad \\mbox{\\rm and}\\quad \\Delta _{n+1}:=\\left(\\begin{array}{c}\\Delta _{n+1}^1\\\\\\Delta _{n+1}^2\\\\\\Delta _{n+1}^3\\end{array}\\right)=\\left(\\begin{array}{l}\\widehat{W}_{n+1}\\\\{ {\\cal W }}_{n+1}\\\\\\displaystyle \\widetilde{ {\\cal W }}_{n+1}\\end{array}\\right)$ indexed by the time parameter $n\\ge 0$ and defined as $\\Delta $ by replacing the sequence $(Z_i)_{i\\ge 1}$ by the sequences of Gaussian variables $ {\\cal V }_n=( {\\cal V }^i_n)_{i\\ge 1}$ and $ {\\cal W }_{n+1}=( {\\cal W }^i_{n+1})_{i\\ge 1}$ introduced in in the definition of the EnKF (REF ).","For $n=0$ we also set $\\Delta _0:=\\left(\\begin{array}{c}\\Delta ^1_0\\\\\\Delta ^2_0\\end{array}\\right)=\\left(\\begin{array}{c}\\widehat{W}_0\\\\\\widetilde{ {\\cal W }}_0\\end{array}\\right)$ with $\\widehat{W}_0$ and $\\widetilde{ {\\cal W }}_0$ defined as $\\widehat{Z}$ and $\\widetilde{ {\\cal Z }}$ by replacing the sequence $Z_i$ with the sequence of Gaussian variables $ {\\cal W }_0^i:=\\frac{\\xi ^i_0-\\widehat{X}^-_0}{\\sqrt{P_0}},$ where we recall that the $\\xi ^i_0$ are independent copies of the Gaussian initial condition $X_0$ of the signal used to initialise the EnKF in (REF ).","In this notation, up to a sequence of Helmert orthogonal transformations on the sequence of Gaussian vectors $ {\\cal W }_0^i$ , the initial condition of the ensemble Kalman filter is given by the stochastic perturbation formulae $m_0=\\widehat{X}^-_0+\\frac{1}{\\sqrt{N+1}}~\\upsilon _0\\quad \\mbox{and}\\quad p_0=P_0+\\frac{1}{\\sqrt{N}}~\\nu _0,$ with the initial local perturbations defined by $\\upsilon _0:=\\sqrt{P_0}~\\Delta ^1_0\\quad \\mbox{and}\\quad \\nu _0:=\\sqrt{2}~P_0~\\Delta ^2_0.$ Observe that the sample means are written in terms of random perturbations $\\upsilon _0$ which are independent of the perturbations $\\nu _0$ of the sample variances.","The independence between the complete sufficient statistic $m_0$ and the ancillary statistic $p_0$ can also be checked using Basu's theorem [11].","An alternative algebraic and direct approach based on Helmert orthogonal matrices can be conducted using the same arguments as those used in the proof of Theorem REF ."],["Local perturbation theorems","Consider the local stochastic perturbations defined for any $n\\ge 0$ by the formulae $\\left\\lbrace \\begin{array}{rcl}\\widehat{\\upsilon }_n&:=&-g_nD~\\widehat{\\Delta }^1_n\\\\&&\\\\\\widehat{\\nu }_n&:=&-2g_nD(1-g_nC)~\\sqrt{p_n}~\\widehat{\\Delta }^2_n+\\sqrt{2}~g_n^2D^2~\\widehat{\\Delta }^3_n\\end{array}\\right.~\\left\\lbrace \\begin{array}{rcl}\\upsilon _{n+1}&:=&B~\\Delta ^1_{n+1}\\\\&&\\\\\\nu _{n+1}&:=&2AB~\\sqrt{\\widehat{p}_n}~\\Delta ^{2}_{n+1}+\\sqrt{2}~R~\\Delta ^3_{n+1}.\\end{array}\\right.$ In the above display, $(\\Delta _n,\\widehat{\\Delta }_n)_{n\\ge 0}$ are the random vectors defined in (REF ), and $g_n$ is the particle gain defined in (REF ).","Theorem 3.1 Up to a sequence of orthogonal transformations on the sequence of Gaussian vectors $ {\\cal W }_n$ and $ {\\cal V }_n$ , the ensemble Kalman filter and the Riccati equation are given by the initial condition (REF ) and the updating-prediction transitions for $n \\ge 0$ $\\left\\lbrace \\begin{array}{rcl}\\displaystyle \\widehat{m}_{n}&=&\\displaystyle m_n+g_n~(Y_n-Cm_n)+\\frac{1}{\\sqrt{N+1}}~\\widehat{\\upsilon }_n\\\\&&\\\\\\widehat{p}_n&=&\\displaystyle (1-g_nC)~p_n+\\frac{1}{\\sqrt{N}}~ \\widehat{\\nu }_n\\end{array}\\right.~\\left\\lbrace \\begin{array}{rcl}\\displaystyle m_{n+1}&=&\\displaystyle A~\\widehat{m}_{n}+\\frac{1}{\\sqrt{N+1}}~\\upsilon _{n+1}\\\\&&\\\\p_{n+1}&=&\\displaystyle A^2~\\widehat{p}_{n}+R+\\frac{1}{\\sqrt{N}}~\\nu _{n+1}.\\end{array}\\right.$ The proof of the above theorem is provided in section REF .","Merging the updating and prediction steps we check the following corollary.","Corollary 3.2 The sample variance $p_n$ of the EnKF satisfies the stochastic Riccati rational difference equation given by the formula $p_{n+1}=\\phi (p_n)+\\frac{1}{\\sqrt{N}}~\\delta _{n+1}\\quad \\mbox{with}\\quad \\delta _{n+1}:=A^2~\\widehat{\\nu }_n+\\nu _{n+1}.$ In the above display, $\\phi (p)$ is the one step mapping of the Riccati equation defined in (REF ), and the random variables $(\\widehat{\\nu }_n,\\nu _{n+1})$ on the right hand side of (REF ) are the local fluctuations defined in (REF ).","Let $ {\\cal R }_n:=\\sigma (p_k,~0\\le k\\le n)\\vee {\\cal Y }_{n}$ be the $\\sigma $ -field generated by the stochastic Riccati equation $p_k$ and the observation sequence up to the time horizon $k\\le n$ and set $\\widehat{m}_{n}^r:=\\mathbb {E}(\\widehat{m}_{n}~|~ {\\cal R }_{n})\\quad \\mbox{\\rm and}\\quad m_{n+1}^r:= \\mathbb {E}(m_{n+1}~|~ {\\cal R }_{n}).$ Using this notation, the next corollary is a direct consequence of Theorem REF .","Corollary 3.3 Given the sample variances and the sequence of observations, for any $n \\ge 0$ , the updating-prediction steps of the sample mean of the EnKF satisfy the quenched Kalman-filter equations $\\widehat{m}_{n}^r =m_n^r+ g_n(Y_n-C\\,m_n^r)\\quad \\mbox{and}\\quad m_{n+1}^r=\\displaystyle A~\\widehat{m}_{n}^r,$ with the initial condition $m_{0}^r=\\widehat{X}^-_0$ .","The next corollary provides an alternative interpretation of the updating-prediction steps in terms of a two-steps Markov chain involving non-central $\\chi $ -square random variables.","Corollary 3.4 The updating-prediction steps of the stochastic Riccati difference equations stated in Theorem REF take the following form $\\widehat{p}_n=\\left(\\frac{p_n}{1+Sp_n}\\right)^2~\\frac{S}{N}~\\widehat{\\chi }^{(n,2)}_{N,N/(Sp_n)}\\quad \\mbox{and}\\quad p_{n+1}=\\frac{R}{N}~\\chi ^{(n+1,2)}_{N,N(A^2/R)\\widehat{p}_{n}},$ with the initial condition $p_0=\\frac{P_0}{N}~\\chi ^{(0,2)}_{N,0}.$ In the above display, $\\chi ^{(n,2)}_{N,x}$ and $\\widehat{\\chi }^{(n,2)}_{N,x}$ stand for a collection of independent non-central $\\chi $ -square random variables with $N$ degrees of freedom and non-centrality parameter $x$ , which are independent of the local perturbation sequence $(\\upsilon _{k+1},\\widehat{\\upsilon }_k)$ of the EnKF filter defined in Theorem REF .","The proof of the above corollary is a direct consequence of the decomposition (REF ), thus it is left as an exercise for the reader."],["Stability theorems","We denote by $\\phi ^n=\\phi ^{n-1}\\circ \\phi $ the composition evolution semigroup of the Riccati difference equation (REF ).","We also let $\\widehat{X}_n(x,p)$ be the solution of the Kalman filter associated with the initial values $(\\widehat{X}_0,P_0)=(x,p)$ .","The next theorem summarises the main stability properties of the Kalman filter and the Riccati difference equations.","Theorem 3.5 The Riccati equation (REF ) has an unique positive fixed point $P_{\\infty }=\\phi (P_{\\infty })>0$ and for any $n\\ge 0$ and any $p=(p_1,p_2)\\in \\mathbb {R}_+^2$ , we have the exponential contraction estimates $\\vert \\phi ^n(p_1)-\\phi ^n(p_2)\\vert \\le \\iota _1~\\left(1-\\epsilon _1\\right)^n~\\vert p_1-p_2\\vert .$ In addition, there exists some integer $k(p_1)\\ge 1$ such that for any $x=(x_1,x_2)\\in \\mathbb {R}^2$ we have $\\mathbb {E}\\left(\\left(\\widehat{X}_n(x_1,p_1)-\\widehat{X}_n(x_2,p_2)\\right)^2\\right)^{1/2}\\le ~\\iota (x,p)~(1-\\epsilon _2)^{n-k(p_1)}~\\left(\\vert p_1-p_2\\vert +\\vert x_1-x_2\\vert \\right).$ The proof of the above theorem is provided in section REF .","Let $ {\\cal P }(p, dq)$ denote the Markov transitions associated with the stochastic Riccati Markov chain $p_n$ discussed in (REF ).","Given some non-negative function $ {\\cal U }$ on $\\mathbb {R}_+ := [0, \\infty [$ we define the $ {\\cal U }$ -norm of some function $f: \\mathbb {R}_+ \\rightarrow \\mathbb {R}$ and some locally finite signed measure $\\mu (dp)$ on $\\mathbb {R}_+$ by $\\Vert f\\Vert _{\\tiny {\\cal U }}:=\\sup _{p\\ge 0}\\left|\\frac{f(p)}{ {\\cal U }(p)+1/2}\\right|\\quad \\mbox{\\rm and}\\quad \\Vert \\mu \\Vert _{\\tiny {\\cal U }}:=\\sup {\\lbrace \\vert \\mu (f)\\vert ~:~f~\\mbox{\\rm s.t.","}~\\Vert f\\Vert _{\\tiny {\\cal U }}\\le 1\\rbrace }.$ The $ {\\cal U }$ -Dobrushin ergodic coefficient of $ {\\cal P }$ is defined by $\\beta _{ {\\cal U }}( {\\cal P }) = \\sup _{(p,q) \\in \\mathbb {R}_+^2}\\frac{\\Vert {\\cal P }(p, \\cdot ) - {\\cal P }(q, \\cdot ) \\Vert _{ {\\cal U }}}{1 + {\\cal U }(p) + {\\cal U }(q)}.$ The next theorem concerns the stability properties of the stochastic Riccati rational difference equation presented in corollary REF .","Theorem 3.6 There exists an unique invariant measure $\\pi $ such that $\\pi = \\pi {\\cal P }$ and some Lyapunov function of the form $ {\\cal U }(p)=u p+v$ such that $\\beta _{ {\\cal U }}( {\\cal P }) < 1$ for some $u,v>0$ .","In addition, for any probability measures $\\mu _1$ and $\\mu _2$ on $\\mathbb {R}_+$ and for any $n\\ge 1$ we have $\\Vert \\mu _1 {\\cal P }^n - \\mu _2 {\\cal P }^n\\Vert _{ {\\cal U }} \\le \\beta _{ {\\cal U }}( {\\cal P })^n\\Vert \\mu _1 - \\mu _2\\Vert _{ {\\cal U }}.$ Moreover, for any function $f$ such that $|f(p)| \\le 1/2 + {\\cal U }(p)$ , for any $p \\in \\mathbb {R}_+$ , we have $| {\\cal P }^n(f)(p) - \\pi (f)| \\le \\beta _{ {\\cal U }}( {\\cal P })^n~(1 + {\\cal U }(p) + \\pi ( {\\cal U })).$ The above theorem comes from the fact that the Riccati map $\\phi (p)$ is a uniformly bounded function (cf.","for instance (REF ) and (REF )).","In addition, using (REF ) or (REF ) we readily check that $ {\\cal P }(p,dq)$ has a continuous density $k(p,q)>0$ with respect to the variables $(p,q)$ and the Lebesgue measure $dq$ on $\\mathbb {R}_+$ .","It is rather well-known that these two properties ensure the existence of a Lyapunov function of the form $ {\\cal U }(u,v)=u p+v$ such that $\\beta _{ {\\cal U }}( {\\cal P }) < 1$ .","For the convenience of the reader a detailed proof of the above theorem is provided in the appendix.","The above theorem is an extension of the stability theorem for stochastic Riccati diffusions presented in the article [22] (see also section 2.2 and Theorem 2.4 in [18] for the multivariate case) to the stochastic Riccati rational equations presented in Corollary REF .","We now turn to quantifying the stability properties of the EnKF sample means.","Since the one-step optimal predictor $\\widehat{X}_n^-$ is a minimal variance estimate of $X_n$ we already know that $M_n:=(m_n-X_n)\\Longrightarrow P_n= \\mathbb {E}((\\widehat{X}_n^--X_n)^2)\\le \\mathbb {E}(M_n^2).$ Thus for small sample sizes we expect $\\mathbb {E}((m_n-X_n)^2)$ to be much larger than $P_n$ .","To find some useful quantitative estimate, observe that $\\displaystyle M_{n+1}=\\displaystyle \\frac{A}{1+Sp_n}~M_n+\\Upsilon _{n+1},$ with the conditionally centered Gaussian random variables $\\Upsilon _{n+1}:=Ag_nDV_n-BW_{n+1}+\\frac{1}{\\sqrt{N+1}}~\\left(A~\\widehat{\\upsilon }_n+\\upsilon _{n+1}\\right).$ The decomposition (REF ) shows that the global stability of the stochastic process $M_n$ depends on the long-time behavior of the random products defined for any $0\\le l\\le n$ by the formula $ {\\cal E }_{l,n}:=\\prod _{l\\le k\\le n}\\frac{A}{1+Sp_k}.$ For stable signal drifts, i.e.","when $\\vert A\\vert <1$ , the exponential decays of these random products is immediate.","In the unstable case, that is when $\\vert A\\vert >1$ , none of the uniform estimates stated later in Theorem REF appear to be useful to quantify directly these random products.","Theorem 3.7 For any $k\\ge 1$ there exist some parameters $N_k\\ge 1$ and $\\epsilon _k\\in ]0,1]$ such that for any $N\\ge N_k$ and any time horizon $n\\ge l\\ge 1$ we have the uniform almost sure exponential decays and the time-uniform estimates $\\mathbb {E}\\left(\\left| {\\cal E }_{l,n}\\right|^k ~|~p_{l-1}\\right)\\le \\iota _k~\\left(1-\\epsilon _k\\right)^{n-l}\\quad \\mbox{and}\\quad \\mathbb {E}\\left(\\vert M_{n}\\vert ^k\\right)\\le \\iota _k.$ The proof of this theorem is provided at the end of section REF .","Let $ {\\cal M }((p,x), d(q,z))$ denote the Markov transitions associated with the coupled Markov chain $(p_n,M_n)$ defined by formulae (REF ) and (REF ).","The $ {\\cal V }$ -norms and the $ {\\cal V }$ -Dobrushin ergodic coefficient $\\beta _{ {\\cal V }}( {\\cal M })$ of $ {\\cal M }$ associated with some non-negative function $ {\\cal V }$ on $\\mathbb {R}_+\\times \\mathbb {R}$ are defined as in (REF ) and (REF ) by replacing the state space $\\mathbb {R}_+$ by $\\mathbb {R}_+\\times \\mathbb {R}$ and the function $ {\\cal U }$ by $ {\\cal V }$ .","For any $\\epsilon \\in ]0,1]$ there exists some time horizon $k\\ge 1$ such that $ {\\cal V }(p,x)= p+\\vert x\\vert \\Longrightarrow {\\cal M }^k( {\\cal V })(p,x)\\le \\epsilon ~ {\\cal V }(p,x)+\\iota .$ A proof of the above Lyapunov inequality is provided in section REF .","Arguing as in the proof of Theorem REF we readily prove the following theorem.","Theorem 3.8 There exists a unique invariant measure $\\mu $ such that $\\mu = \\mu {\\cal M }$ , a Lyapunov function of the form $ {\\cal V }(p,x)=u p+v \\vert x\\vert +w$ for some $u,v,w>0$ , and some integer $k\\ge 1$ such that $\\beta _{ {\\cal V }}( {\\cal M }^k) < 1$ .","In addition, for any probability measures $\\eta _1$ and $\\eta _2$ on $\\mathbb {R}_+\\times \\mathbb {R}$ and any $n\\ge 1$ we have $\\Vert \\eta _1 {\\cal M }^{nk} - \\eta _2 {\\cal M }^{nk}\\Vert _{ {\\cal V }} \\le \\beta _{ {\\cal V }}( {\\cal M }^k)^n\\Vert \\eta _1 - \\eta _2\\Vert _{ {\\cal V }}.$ Moreover, for any function $f$ on $\\mathbb {R}_+\\times \\mathbb {R}$ such that $|f(p,x)| \\le 1/2 + {\\cal V }(p,x)$ , for any $(p,x) \\in \\mathbb {R}_+\\times \\mathbb {R}$ , we have $| {\\cal M }^{nk}(f)(p,x) - \\mu (f)| \\le \\beta _{ {\\cal V }}( {\\cal M }^k)^n~(1 + {\\cal V }(p,x) + \\mu ( {\\cal V })).$"],["Uniform mean-error estimates","The next theorem provides uniform mean-error and bias estimates for the particle gain defined in (REF ) and the stochastic Riccati equations presented in Theorem REF .","Theorem 3.9 For any $k\\ge 1$ and any $N\\ge 1$ and $n\\ge 0$ we have the time uniform estimates $\\mathbb {E}\\left(\\vert p_n-P_n\\vert ^k\\right)^{1/k}\\vee \\mathbb {E}\\left(\\vert \\widehat{p}_n-\\widehat{P}_n\\vert ^k\\right)^{1/k}\\vee \\mathbb {E}\\left(\\vert g_n-G_n\\vert ^k\\right)^{1/k} \\le \\iota _k~(1\\vee P_0) /\\sqrt{N}.$ In addition, we have the uniform bias estimate $0\\le P_n-\\mathbb {E}(p_n)\\le \\frac{\\iota _1}{N}~\\left[1\\vee P_0\\right]^{2}\\quad \\mbox{and}\\quad 0\\le G_n-\\mathbb {E}(g_n)\\le \\frac{\\iota _2}{N}~\\left[1\\vee P_0\\right]^{2}.$ The same formula on the l.h.s.","of the above display holds by replacing $(p_n,P_n)$ by $(\\widehat{p}_n,\\widehat{P}_n)$ .","The proof of the uniform mean error estimates stated in the above theorem is provided in section REF .","The proof of the bias estimates is provided in the end of section REF .","Now we turn to quantifying the fluctuations of the sample means around the optimal filter.","We set $\\widehat{m}_{n}^o:=\\mathbb {E}(\\widehat{m}_{n}~|~ {\\cal Y }_{n})\\quad \\mbox{\\rm and}\\quad m_{n+1}^o:= \\mathbb {E}(m_{n+1}~|~ {\\cal Y }_{n}).$ The next theorem provides uniform mean-error and bias estimates for the sample means of the EnKF filter defined in (REF ).","Theorem 3.10 For any $k\\ge 1$ there exists some parameter $N_k\\ge 1$ such that for any $N\\ge N_k$ and $n\\ge 0$ we have the time-uniform estimates $\\mathbb {E}\\left(\\vert m_{n}-\\widehat{X}^-_{n}\\vert ^k\\right)^{1/k}\\vee \\mathbb {E}\\left(\\vert \\widehat{m}_{n}-\\widehat{X}_{n}\\vert ^k\\right)^{1/k}\\le {\\iota _k}/{\\sqrt{N}}.$ In addition, we have the time-uniform bias estimates $\\mathbb {E}\\left(\\vert m_{n}^o-\\widehat{X}^-_{n}\\vert ^k\\right)^{1/k}\\vee \\mathbb {E}\\left(\\vert \\widehat{m}^o_{n}-\\widehat{X}_{n}\\vert ^k\\right)^{1/k}\\le \\iota _k~(1\\vee P_0)^2/N.$ The proof of the mean error estimates (REF ) is provided in the beginning of section REF .","The uniform bias estimates (REF ) are proved in the end of section REF ."],["Central limit theorems","In the further development of this section, $Z^i_k$ and $\\widehat{Z}^i_k$ stand for a sequence of independent, centered Gaussian random variables with unit variance indexed by $i\\ge 1$ , $k\\ge 0$ .","We also consider the variables $(\\mathbb {U}_0,\\mathbb {V}_0):=(\\sqrt{P_0}~Z^1_0,\\sqrt{2}~P_0~Z^2_0)$ and the sequence of the Gaussian random variables $\\left\\lbrace \\begin{array}{rcl}\\widehat{\\mathbb {U}}_n&:=&\\displaystyle -G_nD~\\widehat{Z}^1_n\\\\&&\\\\\\widehat{\\mathbb {V}}_n&:=&\\displaystyle -2G_nD(1-G_nC)~\\sqrt{P_n}~\\widehat{Z}^2_n+\\sqrt{2}~G_n^2D^2~\\widehat{Z}^3_n\\end{array}\\right.\\left\\lbrace \\begin{array}{rcl}\\mathbb {U}_{n+1}&:=&\\displaystyle B~Z^1_{n+1}\\\\&&\\\\\\mathbb {V}_{n+1}&:=&\\displaystyle 2AB~\\sqrt{\\widehat{P}_n}~Z^{2}_{n+1}+\\sqrt{2}~R~Z^3_{n+1}.\\end{array}\\right.$ Observe that the above sequence of independent and centered Gaussian variables are defined in a similar manner to the local stochastic perturbations (REF ) by replacing the sample variances by their limiting values and the random vectors $(\\widehat{\\Delta }_n,\\Delta _{n+1})$ by the Gaussian random variables $(\\widehat{Z}_n,Z_{n+1})$ .","Theorem 3.11 For any time horizon $n\\ge 0$ we have the weak convergence $(\\upsilon _0,\\nu _0,(\\widehat{\\upsilon }_{k},\\widehat{\\nu }_{k},\\upsilon _{k+1},\\nu _{k+1})_{0\\le k\\le n})\\longrightarrow _{N\\rightarrow \\infty }~(\\mathbb {U}_0,\\mathbb {V}_0,(\\widehat{\\mathbb {U}}_{k},\\widehat{\\mathbb {V}}_{k},\\mathbb {U}_{k+1},\\mathbb {V}_{k+1})_{0\\le k\\le n}).$ The proof of the above theorem is provided in section REF .","Applying the continuous mapping theorem we obtain the following corollary.","Corollary 3.12 For any time horizon $n\\ge 0$ we have the weak convergence $(\\nu _0,(\\delta _k)_{1\\le k\\le n})\\longrightarrow _{N\\rightarrow \\infty }~(\\mathbb {Z}_0,(\\mathbb {Z}_k)_{1\\le k\\le n}),$ with the independent Gaussian random variables $\\mathbb {Z}_n$ defined by $\\mathbb {Z}_{n+1}:=A^2~\\widehat{\\mathbb {V}}_{n}+\\mathbb {V}_{n+1}\\quad \\mbox{and}\\quad \\mathbb {Z}_{0}=\\mathbb {V}_0.$ In the above display, $\\nu _0$ and $\\delta _k$ are the local fluctuations of the stochastic Riccati equation discussed in (REF ).","We now consider the collection of stochastic processes $(\\widehat{\\mathbb {Q}}_n^N,\\mathbb {Q}^N_{n+1})$ indexed by $N\\ge 1$ defined for any $n\\ge 0$ by the updating-prediction synthetic diagram $\\mathbb {Q}^N_{n}:=\\sqrt{N}(p_{n}-P_{n})\\longrightarrow \\widehat{\\mathbb {Q}}_n^N:=\\sqrt{N}(\\widehat{p}_n-\\widehat{P}_n)\\longrightarrow \\mathbb {Q}^N_{n+1}.$ Theorem 3.13 The stochastic processes $(\\widehat{\\mathbb {Q}}_n^N,\\mathbb {Q}^N_{n+1})$ converge in law in the sense of convergence of finite dimensional distributions, and as the number of particles $N\\rightarrow \\infty $ , to a sequence of centered stochastic processes $(\\widehat{\\mathbb {Q}}_n,\\mathbb {Q}_{n+1})$ with initial condition $\\mathbb {Q}_0 = \\mathbb {V}_0$ and updating-prediction transitions given by $\\left\\lbrace \\begin{array}{rcl}\\widehat{\\mathbb {Q}}_n&=&(1-G_nC)^{2}~\\mathbb {Q}_n+\\widehat{\\mathbb {V}}_n\\\\&&\\\\\\mathbb {Q}_{n+1}&=&A^2\\,\\widehat{\\mathbb {Q}}_n+\\mathbb {V}_{n+1}.\\end{array}\\right.$ The proof of the above theorem is provided in section REF .","We consider the collection of stochastic processes $(\\widehat{\\mathbb {X}}_n^N,\\mathbb {X}^N_{n+1})$ indexed by $N\\ge 1$ and defined for any $n\\ge 0$ by the updating-prediction synthetic diagram $\\mathbb {X}^N_{n}:=\\sqrt{N}(m_{n}-\\widehat{X}^-_{n})\\longrightarrow \\widehat{\\mathbb {X}}_n^N:=\\sqrt{N}(\\widehat{m}_n-\\widehat{X}_n)\\longrightarrow \\mathbb {X}^N_{n+1}.$ Theorem 3.14 The stochastic processes $(\\widehat{\\mathbb {X}}_n^N,\\mathbb {X}^N_{n+1})$ converge in law in the sense of convergence of finite dimensional distributions, as the number of particles $N\\rightarrow \\infty $ , to a sequence of centered stochastic processes $(\\widehat{\\mathbb {X}}_n,\\mathbb {X}_{n+1})$ with initial condition $\\mathbb {X}_0 = \\mathbb {U}_0$ and updating-prediction transitions given by $\\left\\lbrace \\begin{array}{rcl}\\widehat{\\mathbb {X}}_{n}&=&(1-G_nC)\\,\\mathbb {X}_n+\\mathbb {G}_n\\,(Y_n-C\\widehat{X}^-_n)+\\widehat{\\mathbb {U}}_n\\\\&&\\\\\\mathbb {X}_{n+1}&=&A\\,\\widehat{\\mathbb {X}}_{n}+\\mathbb {U}_{n+1}.\\end{array}\\right.$"],["Some comments and comparisons","As shown in section REF , the EnKF is a rather simple numerical filtering-type technique defined by an ensemble of particles mimicking the evolution of well-known Kalman filter equations, replacing `the true' covariances by the ensemble sample-covariances.","Besides the fact that the EnKF is only consistent for linear Gaussian filtering problems (cf.","[48]), these popular particle methodologies are applied to complex nonlinear and high dimensional filtering problems arising in fluid mechanics [14], [74], [78], weather forecasting [3], [4], [30], [49], geosciences and reservoir simulation [47], [72], [82], [83], [94], and many other data assimilation disciplines.","To reduce the computational cost, the ensemble sample size is generally chosen to be several orders of magnitude below the very large effective dimension of the underlying signal.","To prevent the ensemble covariance degeneracy in time, small sample EnKF requires one to combine several customisation techniques such as adaptive covariance/gain inflation and related localisation methodologies [12], [13], [21], [93].","The first rigorous analysis of the bias and the consistency of these additional levels of approximations for linear Gaussian models can be found in [21], [23].","As shown in [38], for nonlinear filtering problems, the EnKF can be interpreted as particle-type extended Kalman filter.","As any observer-type algorithm, the well-posedness is not connected to any kind of distance to the optimal filter for any small or large sample sizes, but in terms of non-degeneracy of the observer with respect to time.","As underlined by Majda and Tong in [69] “Why EnKF works well with a small ensemble has remained a complete mystery.","Practitioners often attribute the EnKF success to the existence of a low effective filtering dimension.","But the definition of the effective filtering dimension has remained elusive, as its associated subspace often evolves with the dynamics”.","In the article [69], the authors suggest several theoretical guidelines to choose judiciously the ensemble sample size using covariance inflation and spectral projection techniques on the unstable directions of the signal.","From a purely mathematical perspective, the first important question is clearly to find conditions that ensure that the EnKF is well-founded and consistent, in the sense that it converges to the optimal filter as the ensemble sample size, or equivalently the computational power, tends to infinity (cf.","for instance [48], [70] in discrete time settings and [37], [38] for continuous time EnKF models).","Of course, the consistency property and asymptotic analysis of any statistical Monte Carlo estimate such as the sample means or the sample covariances delivered by EnKF is reassuring but it only gives a partial answer to the performance of these ensemble filters when used with small sample sizes.","The non-asymptotic analysis for finite sample sizes is clearly a more delicate subject as the continuous or the discrete generation EnKF belong to unconventional classes of nonlinear Markov processes and mean-field particle processes interacting through the sample covariance of the system.","Here the interaction covariance function of the EnKF particle filter is not a bounded nor a Lipschitz function as is it is commonly assumed in traditional nonlinear and interacting diffusion theory.","As a result, none of the stochastic tools developed in this field, including the rather elementary variational methodology for nonlinear diffusions developed in [9], [10] nor the more recent powerful differential calculus developed in [31], can be used to analyse this class of continuous-time models equipped with a nonlinear quadratic-type interacting function.","Here, the sample covariance matrices satisfy a rather sophisticated quadratic-type stochastic Riccati matrix diffusion equation, which requires one to develop new stochastic analysis tools [18].","For a more thorough discussion on mean-field type particle systems and a probabilistic description of genetic type particle filters and Ensemble Kalman type particle filters within this framework we refer to the review article [23], the books [35], [37], [39] and references therein.","The analysis of the discrete generation EnKF discussed in the present article is more delicate and requires the development of new stochastic methodologies.","For instance, in discrete time settings, the EnKF particle system evolves as the Kalman filter (REF ) in terms of a two-step prediction-updating interacting Markov chain (REF ).","In contrast with the Gaussian nature of the continuous time EnKF and stochastic Riccati diffusions discussed above, Theorem REF presented in this article shows that these two transitions combine both Gaussian and $\\chi $ -square type perturbations.","To the best of our knowledge, the analysis of stochastic Riccati rational difference equations, such as those discussed in Corollary REF , has not been covered in the literature.","Corollary REF also provides an alternative description of the sample variances in terms of a Markov chain with non-central $\\chi $ -square nonlinear fluctuations.","Again, to the best of our knowledge, these stochastic perturbation theorems and the stability analysis of the analysis of stochastic Riccati rational difference equations are the first of this type in the applied probability literature.","In a study by Tong, Majda and Kelly [93], the authors analyse the long-time behaviour and ergodicity of discrete generation EnKF using Foster-Lyapunov techniques ensuring that the filter is asymptotically stable with respect to any erroneous initial condition as soon as the signal is stable.","These important properties ensure that the EnKF has a single invariant measure and initialisation errors of the EnKF will dissipate with respect to the time parameter.","Nevertheless, the only ergodicity of the signal and thus the ensemble process does not give any useful information on the convergence and the accuracy of the EnKF towards the optimal filter as the number of samples tends to infinity, nor does it allow one to quantify the fluctuations around the optimal filter for a given small or even very large ensemble sample size.","On the other hand, effective unstable directions are connected to unstable-transient signals that may grow exponentially fast.","In this context, for obvious reasons, any well tracking EnKF needs to be an unstable-transient particle process.","As a consequence, we cannot expect to obtain any type of filter ergodicity properties, nor any time-uniform estimates for the raw moments of the EnKF filter.","From our point of view, the terminology \"effective filtering dimensions\" discussed above should be understood as the unstable directions of the signal.","For time varying and multivariate linear-Gaussian filtering problems, these effective dimensions are rarely known as they are partially observed and they may change in time.","For a more detailed discussion on the stability properties of systems of time varying linear stochastic differential equations with random coefficients we refer to [19].","To avoid the time degeneracy of any type of filtering approximation, a crucial question in practice is to check wether or not the convergence is uniform with respect to the time parameter.","Equivalently, we need to obtain estimates of the error between the optimal filter and its approximation that are uniform with respect to any time horizon even if the signal and thus any well tracking approximating filter are both unstable and transient.","We emphasise that in this setting, any good approximating filter will diverge.","In this context, the filtering step is part of a control loop that allows one to track the transient signal at all times.","The main advantage of one-dimensional filtering problems over multivariate problems with unstable signals is that they have a single effective dimension.","In this context, the EnKF is also unstable and transient but the theorems REF and REF presented in this article show that the EnKF tracks and converges to the optimal filter uniformly in time.","These theorems also ensure that the precision error to the optimal filter can be quantified for any given finite sample size, and it will not degrade with respect to the time horizon.","Theorem REF also indicates that the sample variances satisfy an autonomous stochastic Riccati-type evolution equation, independent of the sequence of observations.","From a different angle, theorem REF also shows that the sample variances of the EnKF behaves as a Markov chain that converge exponentially fast as the time horizon tends to infinity to a single invariant measure, even if the signal of the filtering problem is transient.","To the best of our knowledge, this stability theorem and the non-asymptotic theorems discussed above are the first results of this type in the literature on particle filter theory, including the literature of discrete generation EnKF.","Next we provide a more detailed discussion on the stability of particle filters and their particle approximations, with precise reference pointers.","We also suggest an avenue of open research problems.","Most of the stability properties of genetic-type particle filters [34], [36], including continuous time interacting jump Feynman-Kac particle systems [8], are only valid for stable and ergodic signals.","It is clearly out the scope of this article to review in detail all contributions related to the stability of genetic type particle filters, thus we refer to the books [34], [35], [36] and references therein.","The analysis of these continuous-time or discrete generation genetic type processes for unstable signals remains an important and open research question.","Continuous time EnKF filters for stable signals are also discussed in [16], [38], including in the context of the extended Kalman filter.","The analysis of unstable and possibly transient signals is more recent, starting at the end of the 2010s in the context of continuous time filtering problems [20], [18], [22], see also [23] for a comprehensive review of this subject.","The articles [18], [20] analyse the stability and the fluctuation properties of the sample covariance matrices of continuous time EnKF filters for multivariate models under natural and minimal observability and controllability conditions.","To better understand the difficulty in handling unstable effective signal directions, we recall that under natural observability and controllability conditions the Kalman filtering theory developed by Kalman and Bucy in the end of the 1960s ensures that the optimal filter is able to track any possibly unstable signals uniformly with respect to the time horizon.","To answer any stability question related to EnKF particle filters or genetic type particle filters, we first need to extend the theory of Kalman and Bucy to this class of approximating particle filters.","This research project remains open for genetic type filters.","The present article provide a partial answer for discrete generation EnKF particle filters.","To better understand the difficulty in handling and extending the stability theory of Kalman and Bucy to particle filters, it is worth noting a brief of history.","The stability properties of continuous time or discrete generation Kalman filters and their associated Riccati equations for multidimensional filtering problems are usually conducted under some observability/controllability-type conditions using sophisticated and tedious matrix manipulations.","The first stability properties for continuous time models go back to the beginning of the 1960s with the celebrated work of Kalman and Bucy [57], with related prior work by Kalman [53], [54], [56] and later work by Bucy [27].","In [2], the stability of this filter was analysed under a relaxed controllability condition.","The first stability properties for discrete time models go back to to the end of the 1960s with the work of Deyst and Price [41], which was incorporated in the seminal lecture book of Jazwinski [51].","However, as noted in [50], there was in both cases a crucial and commonly made error in the proof which invalidated the results, see [71] and the more recent articles [16], [79] for a more detailed discussion and references on these issues.","This error was repeated in numerous subsequent works.","A first correction [28] was noted in a reply to [50]; see Bucy's reply [29] and a separate reply by Kalman [55].","However, a complete reworking of the result did not appear in entirety until 2001 by Delyon in [40].","The stability of Kalman filter and Riccati equations is nowadays rather well understood.","Nevertheless it is not always simple to find a self-contained, rigorous and easy-to-read study on these subjects.","For a complete and self-contained analysis on the stability and convergence of Kalman-Bucy filtering in continuous time settings, we also refer to [16].","A self-contained analysis of Riccati differential equations and discrete-time Kalman filters for one-dimensional models is provided in section REF and section REF in the present article, see also Theorem REF .","The stability analysis of continuous time stochastic Riccati matrix diffusions is also rather well understood, see for instance [16], [18], [20], [23].","The extension of these multivariate results in the discrete time case remains an important open question.","Theorem REF as well as (REF ) and Theorem REF provide a complete answer to the stability of discrete generation EnKF for one dimensional filtering problems, with possibly unstable signals, yielding what seems to be the first results of this type for this class of particle filters, including both EnKF and genetic type particle filters."],["Some preliminary results","In this section we give several technical results that will be useful in the further development of the article.","We start by considering the inverse moments of the non-central $\\chi $ -square random variables introduced in (REF ).","We then provide a self-contained analysis of the Riccati difference equations defined in (REF )."],["Non-central $\\chi $ -square moments","In this section, we will prove the following technical result that allows one to estimate the inverse raw moments of non-central $\\chi $ -square random variables.","Proposition 4.1 For any $k\\ge 0$ and $n>(k+1)$ we have the uniform estimate $0\\le \\mathbb {E}\\left(\\left(\\frac{2n}{\\chi ^2_{2n,2nx}}\\right)^{k}\\right)-\\frac{1}{\\left(1+x\\right)^{k}}\\le \\frac{k+1}{n}~k~\\left(1+\\frac{k+1}{n-(k+1)}\\right)^{k}.$ In addition, there exists a constant $\\omega _k$ such that, for any $n\\ge 2(k+2)$ we have $\\mathbb {E}\\left(\\left(\\frac{1+x}{\\frac{1}{2n}\\chi ^2_{2n,2nx}}-1\\right)^{k}\\right)\\le \\frac{\\omega _k}{n}~\\left(1+x\\right)^{k}.$ In order to prove this result, let us first consider the following Taylor series expansion, $e^{-x}=\\sum _{0\\le k\\le n}\\frac{(-x)^k}{k!}+\\frac{(-x)^{n+1}}{(n+1)!","}~\\tau _{n+1}(x), n \\ge 0,$ with the collection of integral remainders given by $\\tau _{n+1}(x)=(n+1)~\\int _{0}^1~(1-t)^ne^{-tx}~dt=1-\\frac{x}{n+2}~\\tau _{n+2}(x).$ For any $k\\ge 1$ we denote by $\\tau ^{(k)}_n$ s the $k$ -th differential of the function $\\tau _n$ .","In this notation, the inverse moment of a non-central $\\chi $ -square random variable (cf.","section 3 in [25]) are given, for any $k\\ge 0$ , by the formulae $\\begin{array}{rcl}\\displaystyle \\mathbb {E}\\left(\\left(\\frac{2}{\\chi ^2_{2(n+1),2x}}\\right)^{(k+1)}\\right)&=&\\displaystyle \\frac{(-1)^{k}}{k!","}~\\frac{1}{n-k}~\\tau ^{(k)}_{n-k}(x).\\end{array}$ Setting $k=0$ we find the formulae $\\frac{1}{n}~\\tau _{n}(x)=\\mathbb {E}\\left(\\frac{2}{\\chi ^2_{2(n+1),2x}}\\right)\\ge \\frac{2}{\\mathbb {E}(\\chi ^2_{2(n+1),2x})}=\\frac{1}{x+n+1},$ where we have used Jensen's inequality to obtain the lower bound.","This combined with another application of Jensen's inequality implies that $\\begin{array}{rcl}\\displaystyle \\frac{(-1)^{k}}{k!","}~\\frac{1}{n-k}~\\tau ^{(k)}_{n-k}(x) =\\mathbb {E}\\left(\\left(\\frac{2}{\\chi ^2_{2(n+1),2x}}\\right)^{(k+1)}\\right)&\\ge &\\displaystyle \\left(\\frac{1}{n}~\\tau _{n}(x)\\right)^{k+1}\\ge \\frac{1}{(x+n+1)^{k+1}}.\\end{array}$ On the other hand, note that for any $0\\le z<1$ we have $\\log {(1-z)}\\le -z\\Longrightarrow n\\log {\\left(1-\\frac{s}{m}\\right)}-sx\\le -\\left(x+\\frac{n}{m}\\right)~s.$ This yields the upper bound $m\\tau _{n}(mx) = \\int _0^{m}~\\left(1-\\frac{s}{m}\\right)^n~e^{-sx}~ds\\le \\int _0^{m}~e^{-\\left(x+\\frac{n}{m}\\right)s}~ds=\\frac{1}{x+\\frac{n}{m}}~\\left(1-e^{-\\left(x+\\frac{n}{m}\\right)m}\\right)\\le \\frac{1}{x+\\frac{n}{m}}.$ Thus, for any $m,n\\ge 1$ and any $x\\ge 0$ we have the estimate $\\frac{1}{x+(n+1)/m}\\le \\frac{m}{n}~\\tau _{n}(mx)\\le \\frac{1}{x+{(n-1)}/{m}}.$ More generally, for any $k,m,n\\ge 0$ we have $\\frac{(-1)^k}{n+1}~\\tau _{n+1}^{(k)}(mx)&=&\\int _0^1 t^k~(1-t)^n~e^{-mxt}~dt\\\\&=&\\frac{1}{m}\\int _0^{m}~\\left(\\frac{s}{m}\\right)^k~\\left(1-\\frac{s}{m}\\right)^n~e^{-sx}~ds\\\\&\\le & \\frac{1}{m^{k+1}}\\int _0^{m}~s^k~e^{-\\left(x+\\frac{n}{m}\\right)s}~ds\\le \\frac{1}{m^{k+1}}~\\frac{1}{\\left(x+\\frac{n}{m}\\right)}~\\frac{k!","}{\\left(x+\\frac{n}{m}\\right)^{k}}.$ This discussion can be summarised with the following lemma.","Lemma 4.2 For any $n> k\\ge 0$ , $m\\ge 1$ and any $x\\ge 0$ we have the estimate $\\frac{1}{\\left(x+\\frac{n+1}{m}\\right)^{k+1}}\\le \\frac{(-1)^{k}}{k!","}~\\frac{m^{k+1}}{n-k}~\\tau ^{(k)}_{n-k}(mx)& \\le & ~\\frac{1}{\\left(x+\\frac{n-k-1}{m}\\right)^{k+1}}.$ With this lemma in hand, we are now in a position to prove Proposition REF .","[Proof of Proposition REF ] By Lemma REF we have $\\frac{1}{\\left(x+\\frac{n+1}{m}\\right)^{k+1}}\\le \\mathbb {E}\\left(\\left(\\frac{2m}{\\chi ^2_{2(n+1),2mx}}\\right)^{(k+1)}\\right)& \\le & ~\\frac{1}{\\left(x+\\frac{n-k-1}{m}\\right)^{k+1}}.$ This yields the estimate $\\frac{1}{\\left(1+x\\right)^{k}}\\le \\mathbb {E}\\left(\\left(\\frac{2n}{\\chi ^2_{2n,2nx}}\\right)^{k}\\right)\\le ~\\frac{1}{\\left((1+x)-\\frac{k+1}{n}\\right)^{k}}.$ This implies that for any $n> k$ we have $0\\le \\mathbb {E}\\left(\\left(\\frac{2n}{\\chi ^2_{2n,2nx}}\\right)^{k}\\right)-\\frac{1}{\\left(1+x\\right)^{k}}\\le \\theta _k(x) \\le \\frac{k+1}{n}~\\frac{k}{\\left(x+\\frac{n-(k+1)}{n}\\right)^{k}\\left(1+x\\right)}$ with the function $\\theta _k(x):=~\\frac{1}{\\left((1+x)-\\frac{k+1}{n}\\right)^{k}}-\\frac{1}{\\left(1+x\\right)^{k}}.$ This ends the proof of the first assertion.","This first estimate also yields, for any $n\\ge k+2$ , the rather crude estimate $0\\le \\mathbb {E}\\left(\\left(\\frac{1+x}{\\frac{1}{2n}\\chi ^2_{2n,2nx}}\\right)^{k}\\right)-1\\le \\frac{k}{n}~(k+1)\\left(1+x\\right)^{k}\\left(k+2\\right)^{k}\\le \\frac{1}{n}~\\left(1+x\\right)^{k}\\left(k+2\\right)^{k+2}.$ This implies that $0\\le \\mathbb {E}\\left(\\left(\\frac{1+x}{\\frac{1}{2n}\\chi ^2_{2n,2nx}}\\right)^{k}\\right)-1\\le \\frac{\\omega _k}{n}~\\left(1+x\\right)^{k}\\quad \\mbox{\\rm with}\\quad \\omega _k:=\\left(k+2\\right)^{k+2}.$ Using the decomposition $\\left(y-1\\right)^k=\\sum _{0\\le l\\le k}\\left(\\begin{array}{c}k\\\\l\\end{array}\\right)~(-1)^{l}~y^{k-l}=\\sum _{0\\le l< k}\\left(\\begin{array}{c}k\\\\l\\end{array}\\right)~(-1)^{l}~\\left(y^{k-l}-1\\right),$ we check that $\\mathbb {E}\\left(\\left(\\frac{1+x}{\\frac{1}{2n}\\chi ^2_{2n,2nx}}-1\\right)^{k}\\right)=\\sum _{0\\le l< k}\\left(\\begin{array}{c}k\\\\l\\end{array}\\right)~(-1)^{l}~\\underbrace{\\left(\\mathbb {E}\\left(\\left(\\frac{1+x}{\\frac{1}{2n}\\chi ^2_{2n,2nx}}\\right)^{k-l}\\right)-1\\right)}_{\\ge 0}.$ Thus, for any $n\\ge 2(2k+2)$ we have the estimate $\\mathbb {E}\\left(\\left(\\frac{1+x}{\\frac{1}{2n}\\chi ^2_{2n,2nx}}-1\\right)^{2k}\\right)&\\le & \\sum _{1\\le l\\le k}\\left(\\begin{array}{c}2k\\\\2l\\end{array}\\right)~\\left(\\mathbb {E}\\left(\\left(\\frac{1+x}{\\frac{1}{2n}\\chi ^2_{2n,2nx}}\\right)^{2l}\\right)-1\\right).$ Using (REF ), this implies that $\\mathbb {E}\\left(\\left(\\frac{1+x}{\\frac{1}{2n}\\chi ^2_{2n,2nx}}-1\\right)^{2k}\\right)\\le \\frac{1}{n}~\\left(1+x\\right)^{2k}\\omega ^{\\prime }_{2k}.$ with $\\omega ^{\\prime }_{2k}:=\\sum _{1\\le l\\le k}\\left(\\begin{array}{c}2k\\\\2l\\end{array}\\right)~\\left(2(l+1)\\right)^{2(l+1)}.$ In the same vein, for any $n\\ge 2((2k+1)+2)$ we have $\\mathbb {E}\\left(\\left(\\frac{1+x}{\\frac{1}{2n}\\chi ^2_{2n,2nx}}-1\\right)^{2k+1}\\right)&\\le & \\sum _{0\\le 2l< 2k+1}\\left(\\begin{array}{c}2k+1\\\\2l\\end{array}\\right)~\\underbrace{\\left(\\mathbb {E}\\left(\\left(\\frac{1+x}{\\frac{1}{2n}\\chi ^2_{2n,2nx}}\\right)^{2(k-l)+1}\\right)-1\\right)}_{\\ge 0}\\\\&\\le & \\sum _{0\\le l\\le k}\\left(\\begin{array}{c}2k+1\\\\2(k-l)\\end{array}\\right)~\\underbrace{\\left(\\mathbb {E}\\left(\\left(\\frac{1+x}{\\frac{1}{2n}\\chi ^2_{2n,2nx}}\\right)^{2l+1}\\right)-1\\right)}_{\\le \\frac{\\omega _{2l+1}}{n}~ (1+x)^{2l+1}}.$ This implies that $\\mathbb {E}\\left(\\left(\\frac{1+x}{\\frac{1}{2n}\\chi ^2_{2n,2nx}}-1\\right)^{2k+1}\\right)\\le \\frac{\\omega ^{\\prime }_{2k+1}}{n}~ (1+x)^{2k+1}$ with $\\omega ^{\\prime }_{2k+1}=\\sum _{0\\le l\\le k}\\left(\\begin{array}{c}2k+1\\\\2(k-l)\\end{array}\\right)~(2l+3)^{2l+3}.$ This ends the proof of the proposition."],["Riccati difference equations ","We associate with some parameters $(a,b,c,d)\\in \\mathbb {R}_+^4$ satisfying $c>0$ and $ad>bc$ the Riccati difference equation on $\\mathbb {R}_+:=[0,\\infty [$ given by the recursion $P_n:=\\phi (P_{n-1})\\quad \\mbox{\\rm with}\\quad \\phi (x):=\\frac{a x+b}{cx+d}\\quad \\mbox{\\rm with}\\quad x\\ge 0.$ Observe that $\\partial \\phi (x)=\\frac{\\rho }{(cx+d)^2}> 0\\quad \\mbox{\\rm and}\\quad \\partial ^2 \\phi (x)=-\\frac{2\\rho c}{(cx+d)^3}< 0\\quad \\mbox{\\rm with}\\quad \\rho :=ad-bc>0.\\\\$ This shows that $\\phi $ is an increasing and concave function.","Rational difference equation of the form (REF ) can be solved explicitly.","This section provides a brief review on these equations.","We set $(u,v,w):=(a/c,d/c,b/c)\\in \\mathbb {R}_+^4\\quad \\mbox{\\rm and}\\quad \\lambda =\\lambda _1/\\lambda _2\\in ]0,1[$ with the parameters $\\lambda _2-v:=\\frac{u-v}{2}+\\sqrt{\\left(\\frac{v-u}{2}\\right)^2+w}>0 \\quad \\mbox{\\rm and}\\quad v-\\lambda _1:=\\frac{v-u}{2}+\\sqrt{\\left(\\frac{v-u}{2}\\right)^2+w}> 0.$ Observe that $(v-\\lambda _1) (\\lambda _2-v)=w \\qquad \\mbox{\\rm and}\\qquad r=\\phi (r)>0\\Longleftrightarrow r=\\lambda _2-v.$ Lemma 4.3 For any $x\\ge 0$ and $n\\ge 1$ the evolution semigroup $\\phi ^n=\\phi \\circ \\phi ^{n-1}$ of the Riccati difference equation (REF ) is positive and is given by the formula $\\phi ^n(x)=r+(x-r)~\\frac{\\left(\\lambda _2-\\lambda _1\\right)~\\lambda ^n}{\\left(x+(v-\\lambda _1)\\right)~\\left(1-\\lambda ^{n}\\right)+(\\lambda _2-\\lambda _1)~\\lambda ^{n}}$ with the parameters $\\lambda $ and $(\\lambda _1,\\lambda _2)$ defined in (REF ) and the unique positive fixed point $r$ given in (REF ).","In addition, for any $n\\ge 1$ and any $x\\in \\mathbb {R}_+$ we have the estimates ${b}/{d}\\le \\phi ^n(x)\\le {a}/{c}.$ For any $x,y\\in \\mathbb {R}_+$ and $n\\ge 0$ we have the exponential decay to equilibrium $\\vert \\phi ^n(x)-r\\vert \\le \\frac{\\lambda _2-\\lambda _1}{v-\\lambda _1}~\\lambda ^n~\\vert x-r\\vert \\quad \\mbox{and}\\quad \\vert \\phi ^n(x)-\\phi ^n(y)\\vert \\le \\vert x-y\\vert ~\\left(\\frac{\\lambda _2-\\lambda _1}{v-\\lambda _1}\\right)^2~\\lambda ^n.$ In addition, we have the first order derivative $\\partial \\phi ^n(y):=\\frac{(\\lambda _2-\\lambda _1)^2~\\lambda ^n}{\\big (\\left(y+(v-\\lambda _1)\\right)~\\left(1-\\lambda ^{n}\\right)+(\\lambda _2-\\lambda _1)~\\lambda ^{n}\\big )^2}=\\prod _{0\\le k0.$ This shows that the boundedness properties of Riccati type differential inequalities of the form $r_n\\le \\phi _{\\epsilon }(r_{n-1})$ can be deduced from those of the solution to the evolution equation $s_n=\\phi _{\\epsilon }(s_{n-1})$ .","For instance, since $\\phi _{\\epsilon }$ is increasing we have $\\phi _{\\epsilon }(r_{n-1})\\le \\phi _{\\epsilon }^n(x)\\le x\\vee s^{\\epsilon }\\le x\\vee \\frac{a_{\\epsilon }}{c}= x\\vee (\\epsilon +\\frac{a}{c})\\quad \\mbox{\\rm with the fixed point}\\quad s^{\\epsilon }:= \\phi _{\\epsilon }(s^{\\epsilon })>0.$ Moreover, observe that for any $\\epsilon \\le \\epsilon _1$ we have $0<2c~s^{\\epsilon }&=&a-d+\\epsilon c+\\sqrt{\\left((a-d)+\\epsilon c\\right)^2+4c~(b+\\epsilon d)}\\\\&\\le &a-d+\\sqrt{2(a-d)^2+4bc}+2\\sqrt{\\epsilon _1c~\\left(d+\\frac{\\epsilon _1 c}{2}\\right)}+\\epsilon _1 c.$ In the same vein, for any $k\\ge 1$ we have $\\phi (x)^k=\\left(\\frac{a x+b}{cx+d}\\right)^k\\le \\phi _k(x^k)\\quad \\mbox{\\rm with}\\quad \\phi _k(x):=\\frac{a_{k}x+b_{k}}{c_kx+d_k}$ and where $(a_{k},b_{k},c_k,d_k):=(2^{k-1}a^k,2^{k-1}b,c^k,d^k)\\in \\mathbb {R}_+^4.$ Note that we have $a_{k}d_k-b_{k}c_k=2^{k-1}((ad)^k-(bc)^k)>0\\quad \\mbox{\\rm since}\\quad ad-bc>0.$ As above, this shows that the boundedness properties of Riccati type inequalities of the form $r_n\\le \\phi (r_{n-1})^k$ can be deduced from the solution of the recursion $s_n=\\phi _k(s_{n-1})$ .","Indeed, since $\\phi _k$ is increasing we have $r_n\\le \\phi (r_{n-1})^k\\le \\phi _{k}^n(r_0) \\le x\\vee s^{k}\\quad \\mbox{\\rm with the fixed point}\\quad s^{k}:= \\phi _{k}(s^{k})>0.$"],["Stability of Kalman filters","This section is mainly concerned with the proof of Theorem REF .","Note that Lemma REF yields the existence of the positive fixed point $P_{\\infty }=\\phi (P_{\\infty })>0$ , as well as the estimate (REF ).","The fixed point $P_\\infty $ can be written in closed form in terms of the parameters $(a,b,c)$ (defined in (REF )) by the formula $P_{\\infty }= \\frac{(a-1)+r}{2c}\\quad \\mbox{\\rm with}\\quad r:=\\sqrt{\\left( a-1\\right)^2+4bc} \\,\\frac{\\vert A\\vert -1}{S}.$ This implies that $\\vert A\\vert (1-G_\\infty C)=:1- \\epsilon _{\\infty }<1,$ where $G_\\infty $ is he gain associated with the steady state $P_{\\infty }$ of the Riccati equation and is given by $G_{\\infty }=CP_{\\infty }/(C^2P_{\\infty }+D^2)\\Longleftrightarrow 1-G_{\\infty }C={1}/{(1+SP_{\\infty })}.$ On the other hand, using (REF ), for any $p\\ge 0$ there exists some integer $k(p)\\ge 1$ such that for any $k> k(p)$ , $\\frac{1+SP_{\\infty }}{1+S\\phi ^k(p)}=1+\\frac{S}{1+S\\phi ^k(p)}~(P_{\\infty }-\\phi ^k(p))\\le 1+S~\\kappa _1~\\left(1-\\epsilon _1\\right)^k~\\vert p-P_{\\infty }\\vert \\le 1+\\epsilon _{\\infty }.$ with the parameter $\\epsilon _{\\infty }$ defined in (REF ).","This yields the following technical lemma.","Lemma 4.4 For any $p\\ge 0$ there exists some integer $k(p)\\ge 1$ such that for any $n> k(p)$ we have $\\sqrt{\\vert (\\partial \\phi )(\\phi ^n(p))\\vert }=\\frac{\\vert A\\vert }{1+S\\phi ^{n}(p)} \\le 1-\\epsilon ^2_{\\infty },$ with the parameter $\\epsilon _{\\infty }$ defined in (REF ).","In addition, for any $n\\ge k\\ge k(p)\\ge l$ we have $\\vert E_{l,n}(p)\\vert \\le \\left(\\frac{\\vert A\\vert }{1+SB^2}\\right)^{k(p)-l}~(1-\\epsilon ^2_{\\infty })^{n-k(p)}\\quad \\mbox{\\rm and}\\quad \\vert E_{k,n}(p)\\vert \\le (1-\\epsilon ^2_{\\infty })^{n-k}.$ We are now in position to state and to prove the main result of this section, which also completes the proof of Theorem REF .","Theorem 4.5 For any $p_1\\ge 0$ there exists some integer $k(p_1)\\ge 1$ for any $n> k(p_1)$ and any $x_1,x_2\\in \\mathbb {R}$ we have the almost sure estimate $\\left|\\widehat{X}_n(x_1,p_1)- \\widehat{X}_n(x_2,p_1)\\right|\\le \\left(\\frac{\\vert A\\vert }{1+SB^2}\\right)^{k(p_1)}~(1-\\epsilon ^2_{\\infty })^{n-k(p_1)}~\\vert x_1-x_2\\vert .$ with the parameter $\\epsilon _{\\infty }$ defined in (REF ).","In addition, for any $p_2\\ge 0$ we have $\\mathbb {E}\\left(\\left(\\widehat{X}_n(x_1,p_1)- \\widehat{X}_n(x_2,p_2)\\right)^2\\right)^{1/2}\\le ~\\kappa (p_1,p_2,x_2)~(1-\\epsilon ^2_{\\infty })^{n-k(p_1)}~\\vert x_1-x_2\\vert .$ The estimate (REF ) is a direct consequence of the formula (REF ) and the exponential estimate (REF ).","On the other hand, we have $\\widehat{X}_n(x,p)-X_n&=&\\frac{1}{1+S\\phi ^n(p)} ~(A\\widehat{X}_{n-1}(x,p)-X_n)+\\frac{S\\phi ^n(p)}{1+S\\phi ^n(p)}~(C^{-1}~Y_n-X_n)\\\\&=&\\frac{A}{1+S\\phi ^n(p)} ~(\\widehat{X}_{n-1}(x,p)-X_{n-1})+\\widehat{U}_n(p),$ where $\\widehat{U}_n(p):=\\frac{S\\phi ^n(p)}{1+S\\phi ^n(p)}~\\frac{V_n}{\\sqrt{S}}-\\frac{1}{1+S\\phi ^n(p)}~BW_n\\Longrightarrow \\mathbb {E}(\\widehat{U}_n(p)^2)\\le R+1/S.$ Iterating, this implies that $\\widehat{X}_n(x,p)-X_n= E_{n}(p) (x-X_0)+\\sum _{1\\le k\\le n}~ E_{k,n}(p)~\\widehat{U}_k(p)$ and hence $\\begin{array}{l}\\displaystyle \\mathbb {E}\\left(\\left(\\widehat{X}_n(x,p)-X_n\\right)^2 \\right)=\\vert E_{0,n}(p)\\vert ~\\mathbb {E}((x-X_0)^2)+\\sum _{1\\le l\\le n}~\\vert E_{l,n}(p)\\vert ~ \\mathbb {E}(\\widehat{U}_l(p)^2).\\end{array}$ Using (REF ) we check the uniform estimate $\\mathbb {E}\\left(\\left(\\widehat{X}_n(x,p)-X_n\\right)^2 \\right)\\le \\kappa _1(p)~(1+\\mathbb {E}((x-X_0)^2)).$ In the same vein, for any given $q\\in \\mathbb {R}$ and for any $(p_1,p_2)\\in \\mathbb {R}_+^2$ we check that $\\widehat{X}_n(x,p_1)- \\widehat{X}_n(x,p_2)&=&\\sum _{1\\le l\\le n}~ E_{l,n}(p_1)~\\rho _l(p_1,p_2)~U_l(x,p_2)$ with the function $\\begin{array}{l}\\displaystyle \\rho _n(p_1,p_2):=\\frac{S(\\phi ^n(p_1)-\\phi ^n(p_2))}{(1+S\\phi ^n(p_1))(1+S\\phi ^n(p_2))} \\le S\\kappa _1^2~\\left(1-\\epsilon _1\\right)^n~\\vert p_1-p_2\\vert \\quad \\mbox{\\rm (by (\\ref {ricc-lower-cv-2})})\\end{array}$ and the collection of random variables $\\begin{array}{l}\\displaystyle U_n(x,p):= A(X_{n-1}-\\widehat{X}_{n-1}(x,p))+BW_n+V_n/\\sqrt{S}\\\\\\\\\\Longrightarrow \\mathbb {E}\\left(U_n(x,p)^2 \\right)\\le A^2\\kappa _1(p)~(1+\\mathbb {E}((x-X_0)^2))+R+1/S\\quad \\mbox{\\rm (by (\\ref {X-p-q-X}))}.\\end{array}$ This implies that $\\mathbb {E}\\left(\\left(\\widehat{X}_n(x,p_1)- \\widehat{X}_n(x,p_2)\\right)^2\\right)^{1/2}\\le \\left(\\sum _{1\\le l\\le n}~\\vert E_{l,n}(p_1)\\vert ~\\rho _l(p_1,p_2)~\\right)~\\kappa _2(x,p_2)$ with $\\kappa _2(x,p):=(A^2\\kappa _1(p)+B^2+1/S)~(1+\\mathbb {E}((x-X_0)^2)).$ Using (REF ), for any $n>k(p_1)$ we have $\\sum _{1\\le l\\le n}~\\vert E_{l,n}(p_1)\\vert ~\\rho _l(p_1,p_2)\\le \\kappa _3(p_1)~(1-\\epsilon ^2_{\\infty })^{n-k(p_1)}~\\vert p_1-p_2\\vert .$ This ends the proof of (REF ) and hence the theorem.","Summing the two estimates (REF ) and (REF ) we obtain the estimate (REF ) with $\\epsilon _2=\\epsilon ^2_{\\infty }$ ."],["Stochastic perturbation analysis","This section is mainly concerned with the proof of Theorem REF .","In addition, we also consider the bias and some time-uniform bounds for the sample covariances $p_n$ and $\\widehat{p}_n$ ."],["A local perturbation theorem","We fix some parameter $N\\ge 1$ and we let 1 be the $(N+1)$ column vector with unit entries, $I$ the $(N+1)\\times (N+1)$ identity matrix and $J={1}{1}^{\\prime }$ the $(N+1)\\times (N+1)$ matrix with unit entries, where $(\\cdot )^{\\prime }$ denotes the transpose operator.","We also let $\\epsilon $ be the $(N+1)\\times (N+1)$ matrix given by $\\epsilon = I-{J}/{(N+1)}\\quad \\Longrightarrow \\quad \\epsilon ^2=\\epsilon .$ For any given $(N+1)$ -column $p$ we have $\\overline{p}:=\\frac{1}{N+1}\\sum _{1\\le j\\le N+1} p_j\\quad \\Longrightarrow \\quad p-\\overline{p}\\,{1}:=\\left(\\begin{array}{c}p_1-\\overline{p}\\\\\\vdots \\\\p_{N+1}-\\overline{p}\\end{array}\\right)=\\epsilon p.$ From the mean-field evolution equations (REF ) and the definitions of $m_n$ and $\\widehat{m}_n$ in (REF ), it is straightforward to show that $\\left\\lbrace \\begin{array}{rcl}\\displaystyle (\\xi ^i_n-m_n)&=& A~(\\widehat{\\xi }^i_{n-1}-\\widehat{m}_{n-1})+BW^{i,\\epsilon }_n\\\\\\\\(\\widehat{\\xi }^i_n-\\widehat{m}_n)&=&(1-g_nC)(\\xi ^i_n-m_n)-g_nDV^{i,\\epsilon }_n,\\end{array}\\right.$ with the random variables $W^{i,\\epsilon }_n:=\\sum _{1\\le j\\le N+1}\\epsilon ^i_j~W^j_n\\quad \\mbox{\\rm and}\\quad V^{i,\\epsilon }_n:=\\sum _{1\\le j\\le N+1}\\epsilon ^i_j~V^j_n.$ Taking the square we obtain the formulae $\\left\\lbrace \\begin{array}{rcl}\\displaystyle (\\xi ^i_n-m_n)^2&=& A^2~(\\widehat{\\xi }^i_{n-1}-\\widehat{m}_{n-1})^2+B^2(W^{i,\\epsilon }_n)^2+2AB~(\\widehat{\\xi }^i_{n-1}-\\widehat{m}_{n-1})W^{i,\\epsilon }_n\\\\\\\\(\\widehat{\\xi }^i_n-\\widehat{m}_n)^2&=&(1-g_nC)^2(\\xi ^i_n-m_n)^2+g_n^2D^2(V^{i,\\epsilon }_n)^2-2g_n(1-g_nC)D(\\xi ^i_n-m_n)V^{i,\\epsilon }_n.\\end{array}\\right.$ This implies that $\\left\\lbrace \\begin{array}{rcl}\\displaystyle p_n&=&\\displaystyle A^2~\\widehat{p}_{n-1}+B^2+\\frac{1}{\\sqrt{N}}~\\nu _n\\\\\\\\\\widehat{p}_n&=&\\displaystyle (1-g_nC)^2p_n+g_n^2D^2+\\frac{1}{\\sqrt{N}}~ \\widehat{\\nu }_n=(1-g_nC)p_n+\\frac{1}{\\sqrt{N}}~ \\widehat{\\nu }_n\\end{array}\\right.$ with the orthogonal variables $\\nu _n:=2AB~\\chi _n+B^2~\\varsigma _n\\quad \\mbox{\\rm and}\\quad \\widehat{\\nu }_n:=-2g_n(1-g_nC)D~\\widehat{\\chi }_n+g_n^2D^2~\\widehat{\\varsigma }_n$ and the centered random variables $\\begin{array}{rclcrcl}\\chi _n&:=&\\displaystyle \\frac{1}{\\sqrt{N}}~\\sum _{1\\le i\\le N+1}~(\\widehat{\\xi }^i_{n-1}-\\widehat{m}_{n-1})W^{i,\\epsilon }_n&&\\varsigma _n&:=&\\displaystyle \\sqrt{N}~\\left(\\frac{1}{N}\\sum _{1\\le i\\le N+1}(W^{i,\\epsilon }_n)^2-1\\right)\\\\&&&&&&\\\\\\widehat{\\chi }_n&:=&\\displaystyle \\frac{1}{\\sqrt{N}}~\\sum _{1\\le i\\le N+1}(\\xi ^i_n-m_n)V^{i,\\epsilon }_n&&\\widehat{\\varsigma }_n&:=&\\displaystyle \\sqrt{N}\\left(\\frac{1}{N}\\sum _{1\\le i\\le N+1}(V^{i,\\epsilon }_n)^2-1\\right).\\end{array}$ Further, denote by $\\mathbb {H}$ the $((N+1)\\times (N+1))$ -Helmert matrix whose first row is defined by $\\mathbb {H}_{1,j}=1/\\sqrt{N+1}\\quad \\forall 1\\le j\\le (N+1)\\quad \\Longrightarrow \\quad \\mathbb {H}_{1,\\mbox{\\LARGE .","}}={1}^{\\prime }/\\sqrt{N+1}.$ and whose $i$ -th row for $2\\le i\\le (N+1)$ is given by $\\begin{array}{l}\\mathbb {H}_{i,j}:=\\overline{\\mathbb {H}}_{i,j}:=\\left\\lbrace \\begin{array}{ccl}1/\\sqrt{i(i-1)}&\\mbox{\\rm if}& 1\\le j0.\\end{array}$ Combining the above with Corollary REF and the inequality $|a + b|^k \\le 2^{k-1}(|a|^k + |b|^k)$ , $a, b \\in \\mathbb {R}$ , $k \\ge 1$ , we obtain $p_{n+1}^{2k}\\le \\phi _{2k}(p_n^{2k})+\\frac{2^{2k-1}}{N^k}~\\left(A^{4k}~\\widehat{\\nu }_n^{2k}+\\nu ^{2k}_{n+1}\\right).$ By Jensen's inequality, we check that $q^{2k}_{n+1}\\le \\phi _{2k}(q^{2k}_n)+\\frac{2^{2k-1}}{N^k}~\\left(A^{4k}~\\mathbb {E}(\\widehat{\\nu }_n^{2k})+\\mathbb {E}(\\nu ^{2k}_{n+1})\\right).$ To control the expectations on the right hand side above, note that for any $k\\ge 1$ we have the crude estimates $(g_nD)^{2k}&=&\\left(\\frac{Sp_n}{1+Sp_n}~\\frac{p_n}{1+Sp_n}\\right)^{k}\\le S^{-k},\\\\(g_nD(1-g_nC))^{k}~p_{n}^{k/2}&=&\\left(\\frac{Sp_n}{1+Sp_n}~\\frac{1}{1+Sp_n}~\\left(\\frac{p_n}{1+Sp_n}\\right)^2\\right)^{k/2}\\le S^{-k}.$ Combining these estimates with (REF ) in (REF ), we check that $\\mathbb {E}\\left(\\widehat{\\nu }_n^{k}\\right)\\le \\iota _1(k)\\quad \\mbox{\\rm and}\\quad \\mathbb {E}(\\nu _{n+1}^{k})\\le \\iota _2(k)~\\left(1\\vee q_n^{k/2}\\right) = \\iota _2(k)~\\left(1\\vee q_n\\right)^{k/2}.$ Now, (REF ) and (REF ) imply that $q_n\\le \\iota _0~(P_0\\vee 1).$ Combining this with (REF ) yields the required bounds for $\\nu _n^k$ , $\\widehat{\\nu }_n^k$ and $\\delta _n^k$ .","Returning to the moments of $p_n$ , note that the previous estimates imply that $q^{2k}_{n+1}\\le \\phi _{2k}(q^{2k}_n)+\\frac{\\iota _5(k)}{N^k}~(q^k_n\\vee 1).$ Applying the above estimate to $k=1$ , along with (REF ), we have $q^{2}_{n+1}\\le \\phi _{2}(q^{2}_n)+\\frac{\\iota _1}{N}~( P_0\\vee 1).$ Using the comparison properties (REF ) and (REF ) we conclude that $\\sup _{N\\ge 1}\\sup _{n\\ge 0}q^{2}_{n}<\\iota _2~(P_0\\vee 1)^2.$ Iterating the argument for any $k\\ge 1$ we readily check that $\\sup _{N\\ge 1}\\sup _{n\\ge 0}q^{k}_{n}<\\iota _k~(P_0\\vee 1)^k$ and therefore $\\mathbb {E}\\left(\\vert \\nu _{n+1}\\vert ^{k}\\right)\\le \\iota _6(k)~\\left(1\\vee q_n\\right)^{k/2}\\le \\iota _7(k)~(P_0\\vee 1)^{k/2}\\quad \\mbox{\\rm and}\\quad \\mathbb {E}\\left(\\vert \\delta _{n+1}\\vert ^k\\right)\\le \\iota _k~(P_0\\vee 1)^{k/2}$ Finally, we have $\\widehat{p}_n=\\frac{p_n}{1+Sp_n}+\\frac{1}{\\sqrt{N}}~\\widehat{\\nu }_n\\Longrightarrow \\mathbb {E}(\\widehat{p}_{n}^{k})\\le S^{-k} +\\frac{1}{N^{k/2}}~ \\mathbb {E}(\\vert \\widehat{\\nu }_n\\vert ^{k})\\le \\iota _k,$ which ends the proof of the proposition."],["Stochastic Riccati equations","In this section we focus on the proof of Theorem REF , as well as the central limit theorems presented in section REF ."],["Perturbation analysis","We start by proving the time uniform estimates (REF ).","To this end, first note that we have the telescoping sum formula $p_n-P_n=\\left(\\phi ^n(p_0)-\\phi ^n(P_0)\\right)+\\sum _{1\\le k\\le n} \\left(\\phi ^{n-k}(p_k)-\\phi ^{n-(k-1)}(p_{k-1})\\right).$ To deal with the summands above, observe that $\\begin{array}{rcl}\\phi ^{n-k}(p_k)-\\phi ^{n-(k-1)}(p_{k-1})&=&\\phi ^{n-k}(p_k)-\\phi ^{n-k}(\\phi (p_{k-1}))\\\\&&\\\\&=&\\displaystyle \\phi ^{n-k}\\left(\\phi (p_{k-1})+\\frac{1}{\\sqrt{N}}~ \\delta _k\\right)-\\phi ^{n-k}\\left(\\phi (p_{k-1})\\right).\\end{array}$ Using the Lipschitz estimates (REF ) we check that $\\vert \\phi ^{n-k}(p_k)-\\phi ^{n-k}(\\phi (p_{k-1}))\\vert \\le \\frac{\\kappa _1}{\\sqrt{N}}~(1-\\epsilon _1)^{n-k}~\\left|\\delta _k\\right|,$ with the parameter $\\epsilon _1$ defined in (REF ).","Similarly, we have $\\vert \\phi ^n(p_0)-\\phi ^n(P_0)\\vert \\le \\frac{\\kappa _2}{\\sqrt{N}}~(1-\\epsilon _1)^{n}~\\left|\\nu _0\\right|.$ This yields the almost sure estimate $\\begin{array}{l}\\displaystyle \\sqrt{N}~\\vert p_n-P_n\\vert \\le \\kappa _3~\\left[(1-\\epsilon _1)^{n}\\vert \\nu _0\\vert +\\sum _{1\\le k\\le n} ~(1-\\epsilon _1)^{n-k}~\\left|\\delta _k\\right|\\right].\\end{array}$ Next we note that $\\vert \\widehat{p}_n-\\widehat{P}_n\\vert =\\left|\\frac{p_n-P_n}{(1+SP_n)(1+Sp_n)}\\right|+\\frac{1}{\\sqrt{N}}~\\vert \\widehat{\\nu }_n\\vert \\le \\left|p_n-P_n\\right|+\\frac{1}{\\sqrt{N}}~\\vert \\widehat{\\nu }_n\\vert .$ Finally, we note the following decomposition for the gain parameters $\\sqrt{N}(g_n - G_n)&= \\sqrt{N}C\\left(\\frac{p_n}{C^2p_n + D^2} - \\frac{P_n}{C^2P_n + D^2}\\right) \\\\&= CD^2 \\frac{\\sqrt{N}(p_n - P_n)}{(C^2p_n + D^2)(C^2P_n + D^2)}.$ The above estimates and decompositions allow one to derive several mean error bounds as soon as we control the moments of the local errors.","For instance, using the uniform moments estimates (REF ) we obtain the mean error estimates stated in Theorem REF ."],["Fluctuation analysis","This section is mainly concerned with the proofs of Theorem REF and Theorem REF .","We start with some technical results that will immediately lead to the proof of the former.","Lemma 6.1 The characteristic function of random vector $\\Delta $ defined in (REF ) satisfies for any $w\\in \\mathbb {R}^3$ we have the Gaussian type estimate $\\left|\\mathbb {E}\\left(e^{i w^{\\prime }\\Delta }\\right)-e^{-\\frac{\\Vert w\\Vert ^2}{2}}\\right|\\le \\epsilon (w)~e^{-\\frac{\\Vert w\\Vert ^2}{2}},$ with the function $\\epsilon (w):=\\vert w_3\\vert \\sqrt{\\frac{2}{N}}\\left( \\frac{w_2^2}{2}+w_3^2\\right)~\\exp {\\left(\\vert w_3\\vert \\sqrt{\\frac{2}{N}}\\left( \\frac{w_2^2}{2}+w_3^2\\right)\\right)}.$ The proof of the above lemma follows elementary calculations.","For the convenience of the reader a detailed proof is provided in the appendix.","Lemma 6.2 For any $n\\ge 0$ we have the weak convergence $\\left(\\Delta _0,(\\widehat{\\Delta }^i_k,\\Delta ^i_{k+1})_{0\\le k\\le n,~1\\le i\\le 3}\\right)\\hookrightarrow _{N\\rightarrow \\infty }~\\left((Z^i_0)_{1\\le i\\le 2},(\\widehat{Z}^i_k,Z^i_{k+1})_{0\\le k\\le n,~1\\le i\\le 3}\\right),$ where the $Z^i_k,\\widehat{Z}^i_k$ are independent sequences of independent and centered Gaussian random variables with unit variance.","Applying (REF ) for any $w\\in \\mathbb {R}^3$ we check the rather crude estimates $\\left|\\mathbb {E}\\left(e^{i~w^{\\prime }\\Delta _{n+1}}\\right)-e^{-\\frac{\\Vert w\\Vert ^2}{2}}\\right|\\le \\epsilon (w)~\\quad \\mbox{\\rm and}\\quad \\left|\\mathbb {E}\\left(e^{i~w^{\\prime }\\widehat{\\Delta }_{n}}\\right)-e^{-\\frac{\\Vert w\\Vert ^2}{2}}\\right|\\le \\epsilon (w)~$ with the function $\\epsilon (w)\\longrightarrow _{N\\rightarrow \\infty }0$ defined in (REF ).","For any $u,v\\in \\mathbb {R}^3$ and $n\\ge 0$ we have the decomposition $\\begin{array}{l}\\displaystyle \\mathbb {E}\\left(e^{i~u^{\\prime }\\Delta _{n+1}}~e^{i~v^{\\prime }\\widehat{\\Delta }_{n}}\\right)-e^{-\\frac{\\Vert u\\Vert ^2}{2}}e^{-\\frac{\\Vert v\\Vert ^2}{2}}\\\\\\\\\\displaystyle =\\mathbb {E}\\left(e^{i~v^{\\prime }\\widehat{\\Delta }_{n}}\\right)~\\left(\\mathbb {E}\\left(e^{i~u^{\\prime }\\Delta _{n+1}}\\right)-e^{-\\frac{\\Vert u\\Vert ^2}{2}}\\right)+\\left(\\mathbb {E}\\left(e^{i~v^{\\prime }\\widehat{\\Delta }_{n}}\\right)-e^{-\\frac{\\Vert v\\Vert ^2}{2}}\\right)e^{-\\frac{\\Vert u\\Vert ^2}{2}}.\\end{array}$ This yields the estimate $\\left|\\mathbb {E}\\left(e^{i~u^{\\prime }\\Delta _{n+1}}~e^{i~v^{\\prime }\\widehat{\\Delta }_{n}}\\right)-e^{-\\frac{\\Vert u\\Vert ^2+\\Vert v\\Vert ^2}{2}}\\right|\\le \\epsilon (u,v):=\\epsilon (u)+\\epsilon (v).$ Consider the sequence of random vectors $\\Lambda _n:=\\left(\\begin{array}{c}\\widehat{\\Delta }_{n}\\\\\\Delta _{n+1}\\end{array}\\right)\\in \\mathbb {R}^6.$ In this notation, for any $n\\ge 0$ and $w_n\\in \\mathbb {R}^6$ we have proven the following estimate $\\left|\\mathbb {E}\\left(e^{i~w_n^{\\prime }\\Lambda _n}\\right)-e^{-\\frac{\\Vert w_n\\Vert ^2}{2}}\\right|\\le \\epsilon (w_n).$ Using the decomposition $\\begin{array}{l}\\mathbb {E}\\left(e^{i~\\left(w_{n-1}^{\\prime }\\Lambda _{n-1}+w_n^{\\prime }\\Lambda _n\\right)}\\right)-e^{-\\frac{\\Vert w_{n-1}\\Vert ^2}{2}}e^{-\\frac{\\Vert w_n\\Vert ^2}{2}}\\\\\\\\=\\mathbb {E}\\left(e^{i~w_{n-1}^{\\prime }\\Lambda _{n-1}}\\right)~\\left(\\mathbb {E}\\left(e^{i~w_n^{\\prime }\\Lambda _n}\\right)-e^{-\\frac{\\Vert w_n\\Vert ^2}{2}}\\right)\\\\\\\\\\hspace{85.35826pt}+\\left(\\mathbb {E}\\left(e^{i~w_{n-1}^{\\prime }\\Lambda _{n-1}}\\right)-e^{-\\frac{\\Vert w_{n-1}\\Vert ^2}{2}}~\\right)e^{-\\frac{\\Vert w_n\\Vert ^2}{2}},\\end{array}$ we also check that $\\begin{array}{l}\\left|\\mathbb {E}\\left(e^{i~\\left(w_{n-1}^{\\prime }\\Lambda _{n-1}+w_n^{\\prime }\\Lambda _n\\right)}\\right)-e^{-\\frac{\\Vert w_{n-1}\\Vert ^2}{2}}e^{-\\frac{\\Vert w_n\\Vert ^2}{2}}\\right|\\le \\epsilon (w_{n-1})+\\epsilon (w_n).\\end{array}$ Iterating the argument, we conclude that $\\begin{array}{l}\\left|\\mathbb {E}\\left(e^{i~\\left(u_1\\Delta ^1_0+u_2\\Delta ^2_0+\\sum _{0\\le k\\le n}w_{k}^{\\prime }\\Lambda _k\\right)}\\right)-e^{-\\frac{ u_1^2/2+u_2^2/2+\\sum _{0\\le k\\le n}\\Vert w_{k}\\Vert ^2}{2}}\\right|.\\\\\\\\\\le \\epsilon (u_1,0,u_2)+ \\sum _{0\\le k\\le n}\\epsilon (w_{k})\\longrightarrow _{N\\rightarrow \\infty }0,\\end{array}$ which ends the proof of the lemma.","Combining Lemma REF with Theorem REF and Slutsky's lemma yields Theorem REF .","We are now in position to prove theorem REF and the bias estimates (REF ).","Proof of Theorem REF : By the second order estimate (REF ) we have $\\begin{array}{l}\\displaystyle \\vert \\phi ^{n-k}(p_k)-\\phi ^{n-(k-1)}(p_{k-1})-\\partial \\phi ^{n-k}\\left(\\phi (p_{k-1})\\right)\\frac{1}{\\sqrt{N}}~\\delta _k\\vert \\le \\frac{\\iota _1}{N}~(1-\\epsilon _1)^{n-k}~\\delta _k^2.\\end{array}$ Combining this with the telescoping sum formula (REF ) yields the decomposition $\\mathbb {Q}^N_n:=\\sqrt{N}(p_n-P_n)=\\partial \\phi ^n(P_0)~\\nu _0+\\sum _{1\\le k\\le n} \\partial \\phi ^{n-k}\\left(\\phi (p_{k-1})\\right)~\\delta _k+\\theta _1^N(1),$ with the remainder term $\\vert \\theta ^N_n(1)\\vert \\le \\epsilon _n^N(1):= \\frac{\\iota _2}{\\sqrt{N}}~(1-\\epsilon _1)^n~\\nu _0^2~+\\frac{\\iota _1}{\\sqrt{N}}~\\sum _{1\\le k\\le n} ~(1-\\epsilon _1)^{n-k}~\\delta _k^2\\longrightarrow _{N\\rightarrow \\infty }~0\\quad \\mbox{a.s.}.$ On the other hand, using the Lipschitz estimates (REF ) and (REF ), we have $\\begin{array}{l}\\displaystyle \\left|\\sum _{1\\le k\\le n} \\left[\\partial \\phi ^{n-k}(\\phi (p_{k-1}))-\\partial \\phi ^{n-k}(\\phi (P_{k-1}))\\right]~\\delta _k~\\right|\\\\\\\\\\displaystyle \\le \\epsilon _n^N(2):=\\iota _3 \\sum _{1\\le k\\le n}~(1-\\epsilon )^{n-k}~\\vert \\delta _k\\vert ~ \\vert p_{k-1}-P_{k-1}\\vert \\longrightarrow _{N\\rightarrow \\infty }~0\\quad \\mbox{a.s.}.\\end{array}$ This yields the formula $\\mathbb {Q}^N_n=F_n\\left(\\nu _0,\\delta _1,\\ldots ,\\delta _n\\right)+\\theta _n^N(2)$ with the function $\\displaystyle F_n\\left(\\nu _0,\\delta _1,\\ldots ,\\delta _n\\right):=\\partial \\phi ^n(P_0)~\\nu _0+\\sum _{1\\le k\\le n} \\partial \\phi ^{n-k}\\left(P_k\\right)~\\delta _k,$ and the remainder term $\\theta _n^N(2)$ satisfying $\\vert \\theta _n^N(2)\\vert \\le \\epsilon _n^N(1)+\\epsilon _n^N(2).$ Combining Slutsky's lemma with the continuous mapping theorem and Corollary REF , for any time horizon $n\\ge 0$ we conclude that $(F_k\\left(\\nu _0,\\delta _1,\\ldots ,\\delta _k\\right))_{0\\le k\\le n}\\longrightarrow _{N\\rightarrow \\infty }~(F_k\\left(\\mathbb {Z}_0,\\mathbb {Z}_1,\\ldots ,\\mathbb {Z}_k\\right))_{0\\le k\\le n}$ with the Gaussian random variables $\\mathbb {Z}_k$ defined in Corollary REF .","Thus, we have shown that $\\mathbb {Q}_n^N \\longrightarrow _{N \\rightarrow \\infty } F_n\\left(\\mathbb {Z}_0,\\mathbb {Z}_1,\\ldots ,\\mathbb {Z}_n\\right) =: \\mathbb {Q}_n.$ The recursive formulation (REF ) comes from the decomposition $\\widehat{\\mathbb {Q}}^N_n=\\frac{1}{(1+SP_n)(1+Sp_n)}~\\mathbb {Q}^N_n+\\widehat{\\nu }_n\\longrightarrow _{N\\rightarrow \\infty } \\widehat{\\mathbb {Q}}_n:=\\frac{1}{(1+SP_n)^2}~\\mathbb {Q}_n+\\widehat{\\mathbb {V}}_n.$ In the same vein, we have $\\mathbb {Q}^N_{n+1}=A^2~\\widehat{\\mathbb {Q}}^N_n+\\nu _{n+1}\\Longrightarrow \\mathbb {Q}_{n+1}=A^2\\widehat{\\mathbb {Q}}_n+\\mathbb {V}_{n+1}.$ The proof of the uniform bias estimates (REF ) follows the second order Taylor expansions discussed in the proof of Theorem REF as we will now demonstrate.","Proof of the bias estimate (REF ): Using the second order Taylor expansions discussed in the proof of Theorem REF we have the bias estimates $\\begin{array}{l}\\displaystyle 0\\le P_n-\\mathbb {E}(p_n)\\le \\frac{ \\iota _4}{N}~\\left[\\mathbb {E}(\\nu _0^2)~(1-\\epsilon )^n+~\\sum _{1\\le k\\le n}(1-\\epsilon )^{n-k}~ \\mathbb {E}(\\delta _k^2)\\right]\\\\\\\\\\displaystyle \\hspace{142.26378pt}+\\frac{\\iota _5}{\\sqrt{N}} \\sum _{1\\le k\\le n}~(1-\\epsilon )^{n-k}~\\mathbb {E}(\\vert \\delta _k\\vert ~ \\vert p_{k-1}-P_{k-1}\\vert ).\\end{array}$ The first bias estimte stated in (REF ) now follows from (REF ) and the estimates stated in (REF ).","blackIndeed, using (REF ) we check that $0\\le P_n-\\mathbb {E}(p_n)&\\le & \\frac{ \\iota _6}{N}~\\left[P_0~(1-\\epsilon )^n+~(1\\vee P_0)\\sum _{1\\le k\\le n}(1-\\epsilon )^{n-k}\\right] \\\\&&\\hspace{42.67912pt}+\\frac{\\iota _7}{\\sqrt{N}}~(1\\vee P_0)~ \\sum _{1\\le k\\le n}~(1-\\epsilon )^{n-k}~~\\mathbb {E}( \\vert p_{k-1}-P_{k-1}\\vert ^2)^{1/2}.$ The end of the proof of the l.h.s.","estimate in (REF ) is now a consequence of (REF ).","We also have the second order decomposition $g_n - G_n&=& \\frac{CD^2}{(C^2P_n + D^2)^2}~(p_n - P_n)-\\frac{CD^2}{(C^2P_n + D^2)^2(C^2p_n + D^2)}~(p_n - P_n)^2$ Using the bias estimate stated in the l.h.s.","of (REF ) we readily check that $0\\le G_n-\\mathbb {E}(g_n) \\le \\frac{CD^2}{(C^2P_n + D^2)^2}~\\frac{\\iota _1}{N}~\\left[1\\vee P_0\\right]^{2}+\\frac{C}{(C^2P_n + D^2)^2}~\\mathbb {E}\\left((p_n - P_n)^2\\right)$ Again, using the $\\mathbb {L}_k$ -mean error estimates stated in (REF ) we check the r.h.s.","estimate stated in (REF ).","This ends the proof of the bias estimate."],["Inverse raw moments","We end the section with some bounds on the moments of ratios of the form $\\phi (p_n)/p_{n+1}$ , $n \\ge 1$ .","Using (REF ) we have the formulae, which will be used throughout this section.","$\\frac{1}{\\widehat{p}_n}=\\left(S+\\frac{1}{p_n}\\right)~\\widehat{\\Upsilon }_n(p_n)\\quad \\mbox{\\rm and}\\quad \\frac{1}{p_{n+1}}=\\frac{1}{A^2\\widehat{p}_n+R}~\\Upsilon _{n+1}(\\widehat{p}_n),$ with the inverse non-central $\\chi $ -square random variables $\\widehat{\\Upsilon }_n(p_n):=\\frac{1+1/(Sp_n)}{\\frac{1}{N}~\\widehat{\\chi }^{(n,2)}_{N,N/(Sp_n)}}\\quad \\mbox{\\rm and}\\quad \\Upsilon _{n+1}(\\widehat{p}_n):=\\frac{1+\\left(A^2/R\\right)\\widehat{p}_n}{\\frac{1}{N}~\\chi ^{(n+1,2)}_{N,N\\left(A^2/R\\right)\\widehat{p}_n}}.$ Lemma 6.3 For any $n\\ge 0$ we have $\\begin{array}{l}\\displaystyle \\frac{\\phi (p_n)}{p_{n+1}}-1=\\frac{1}{N}\\frac{A^4}{\\phi (p_n)}~\\frac{\\widehat{\\nu }_n^2}{\\phi (p_n)+\\frac{A^2}{\\sqrt{N}}~\\widehat{\\nu }_n}-\\frac{A^2}{\\sqrt{N}}~ \\frac{\\widehat{\\nu }_n}{\\phi (p_n)}\\\\\\\\\\hspace{142.26378pt}\\displaystyle +\\frac{\\phi (p_n)}{A^2\\widehat{p}_n+R}~\\left(\\Upsilon _{n+1}(\\widehat{p}_n)-1\\right).\\end{array}$ First recall that $\\widehat{p}_n=\\displaystyle (1-g_nC)~p_n+\\frac{1}{\\sqrt{N}}~ \\widehat{\\nu }_n.$ Combining this with the second identity in (REF ), we have $\\begin{array}{l}\\displaystyle \\frac{\\phi (p_n)}{p_{n+1}}=\\frac{1}{1+\\frac{A^2}{\\sqrt{N}}~ \\frac{\\widehat{\\nu }_n}{\\phi (p_n)}}+\\frac{\\phi (p_n)}{A^2\\widehat{p}_n+R}~\\left(\\Upsilon _{n+1}(\\widehat{p}_n)-1\\right).\\end{array}$ The result now follows from the decomposition $\\frac{1}{1+\\frac{A^2}{\\sqrt{N}}~ \\frac{\\widehat{\\nu }_n}{\\phi (p_n)}}=1-\\frac{A^2}{\\sqrt{N}}~ \\frac{\\widehat{\\nu }_n}{\\phi (p_n)}+\\frac{1}{N}\\frac{A^4}{\\phi (p_n)}~\\frac{\\widehat{\\nu }_n^2}{\\phi (p_n)+\\frac{A^2}{\\sqrt{N}}~\\widehat{\\nu }_n}.$ Proposition 6.4 For any $n\\ge 0$ and any even number of particles $N>4$ we have the almost sure uniform drift estimates $1\\le \\mathbb {E}\\left(\\frac{\\phi (p_n)}{p_{n+1}}~|~p_{n}\\right)\\le 1+\\frac{\\iota }{N}.$ First note that from the previous lemma, we have $\\displaystyle \\mathbb {E}\\left(\\frac{\\phi (p_n)}{p_{n+1}}~\\bigg |~p_n,\\widehat{p}_n\\right)\\displaystyle =1-\\frac{A^2}{\\sqrt{N}}~ \\frac{\\widehat{\\nu }_n}{\\phi (p_n)}+\\frac{1}{N}\\frac{A^4}{\\phi (p_n)}~\\frac{\\widehat{\\nu }_n^2}{\\phi (p_n)+\\frac{A^2}{\\sqrt{N}}~\\widehat{\\nu }_n}\\\\+~\\frac{\\phi (p_n)}{A^2\\widehat{p}_n+R}~\\mathbb {E}\\left(\\left(\\Upsilon _{n+1}(\\widehat{p}_n)-1\\right)~|~\\widehat{p}_n\\right).$ Due to (REF ), for any even number of particles $N>4$ we have the estimate $0\\le \\mathbb {E}\\left(\\Upsilon _n(x)\\right)-1\\le \\frac{4}{N}~\\left(1+\\frac{A^2}{R}~x\\right)~\\left(1+\\frac{4}{N-4}\\right).$ Since both $\\widehat{p}_n$ and $\\phi (p_n)$ are bounded, it follows that $\\mathbb {E}\\left(\\frac{\\phi (p_n)}{p_{n+1}}~\\bigg |~p_n,\\widehat{p}_n\\right) \\le 1-\\frac{A^2}{\\sqrt{N}}~ \\frac{\\widehat{\\nu }_n}{\\phi (p_n)}+\\frac{1}{N}\\frac{A^4}{\\phi (p_n)}~\\frac{\\widehat{\\nu }_n^2}{\\phi (p_n)+\\frac{A^2}{\\sqrt{N}}~\\widehat{\\nu }_n}+\\frac{\\iota _1}{N}.$ for some finite constant $\\iota _1$ .","Taking the expectation we check that $\\displaystyle \\mathbb {E}\\left(\\frac{\\phi (p_n)}{p_{n+1}}~|~p_n\\right)\\displaystyle &\\le \\mathbb {E}\\left(1+\\frac{1}{N}\\frac{A^4}{\\phi (p_n)}~\\frac{\\widehat{\\nu }_n^2}{\\phi (p_n)+\\frac{A^2}{\\sqrt{N}}~\\widehat{\\nu }_n}~|~p_n\\right)+\\frac{\\iota _1}{N}\\\\\\ \\\\\\displaystyle &\\le 1+\\frac{1}{N}\\left(\\iota _1+\\frac{A^4}{R^2}~\\mathbb {E}\\left(~\\widehat{\\nu }_n^2~|~p_n\\right) \\right),$ where the last line follows from the estimates $\\phi (p_n)\\ge R\\quad \\mbox{\\rm and}\\quad R\\le A^2\\widehat{p}_n+R=\\phi (p_n)+\\frac{A^2}{\\sqrt{N}}~ \\widehat{\\nu }_n.$ The proposition now follows from the almost sure estimate $\\mathbb {E}\\left(\\widehat{\\nu }_n^2~|~p_n\\right)&=&(2g_n(1-g_nC)D~\\sqrt{p_n})^2+(\\sqrt{2}~g_n^2D^2)^2 \\\\&=&2S\\frac{p_n^3}{(1+Sp_n)^4}~\\left(2+Sp_n\\right)\\le 4S~\\frac{p_n^3}{(1+Sp_n)^3}\\le \\frac{4}{S^2}.$ Theorem 6.5 For any even $k\\ge 2$ there exists some finite constant $\\iota _k$ such that for any $n\\ge 0$ and any even parameter $N>2(k+2)$ we have the almost sure uniform estimates $1\\le \\mathbb {E}\\left(\\left(\\frac{\\phi (p_n)}{p_{n+1}}\\right)^k~|~p_n\\right)\\le 1+\\frac{\\iota _k}{N}.$ We will show that (REF ) holds for the cases $k = 2, 3$ and then outline how the general case may be obtained inductively.","By (REF ), for any $n\\ge 1$ any $k\\ge 0$ and any even $N>2(k+2)$ we have the uniform estimates $\\mathbb {E}\\left(\\left(\\frac{\\Upsilon _{n}(x)-1}{A^2x+R}\\right)^k\\right)\\le \\frac{\\iota _k}{N}.$ Now, recall that $\\frac{\\phi (p_n)}{p_{n+1}}-1=\\widehat{\\alpha }_n(p_n)+\\beta _{n+1}(\\widehat{p}_n),$ with the random variables $\\widehat{\\alpha }_n(p_n)=-\\frac{1}{\\sqrt{N}}~\\frac{A^2}{A^2\\widehat{p}_n+R}~ \\widehat{\\nu }_n\\quad \\mbox{\\rm and}\\quad \\beta _{n+1}(\\widehat{p}_n):=\\phi (p_n)~\\frac{\\Upsilon _{n+1}(\\widehat{p}_n)-1}{A^2\\widehat{p}_n+R}.$ Using the estimates (REF ), (REF ) and (REF ), we check the almost sure uniform estimate $\\begin{array}{l}\\displaystyle \\mathbb {E}\\left(\\left(\\widehat{\\alpha }_n(p_n)+\\beta _{n+1}(\\widehat{p}_n)\\right)~|~p_n\\right)\\\\\\\\\\displaystyle \\le \\frac{1}{N}\\frac{A^4}{R^2}~\\mathbb {E}(\\widehat{\\nu }_n^2~|~p_n)+\\phi (p_n)~\\mathbb {E}\\left(~\\frac{\\Upsilon _{n+1}(\\widehat{p}_n)-1}{A^2\\widehat{p}_n+R}~|~p_n\\right)\\le \\frac{\\iota _1}{N}.\\end{array}$ In the same vein, we have $\\begin{array}{l}\\displaystyle \\mathbb {E}\\left(\\left(\\widehat{\\alpha }_n(p_n)+\\beta _{n+1}(\\widehat{p}_n)\\right)^2~|~p_n,\\widehat{p}_n\\right)\\\\\\\\\\displaystyle =\\widehat{\\alpha }_n(p_n)^2+\\phi (p_n)^2~\\mathbb {E}\\left(\\left(\\frac{\\Upsilon _{n+1}(\\widehat{p}_n)-1}{A^2\\widehat{p}_n+R}\\right)^2~|~p_n,\\widehat{p}_n\\right)\\\\\\\\\\displaystyle \\hspace{85.35826pt}+2~\\phi (p_n)~\\widehat{\\alpha }_n(p_n)~\\mathbb {E}\\left(\\frac{\\Upsilon _{n+1}(\\widehat{p}_n)-1}{A^2\\widehat{p}_n+R}~|~p_n,\\widehat{p}_n\\right).\\end{array}$ On the other hand, we have $\\begin{array}{l}\\displaystyle \\widehat{\\alpha }_n(p_n)~\\mathbb {E}\\left(\\frac{\\Upsilon _{n+1}(\\widehat{p}_n)-1}{A^2\\widehat{p}_n+R}~|~p_n,\\widehat{p}_n\\right)\\\\\\\\\\displaystyle \\le \\widehat{\\alpha }_n(p_n)^2+\\mathbb {E}\\left(\\frac{\\Upsilon _{n+1}(\\widehat{p}_n)-1}{A^2\\widehat{p}_n+R}~|~p_n,\\widehat{p}_n\\right)^2\\le \\widehat{\\alpha }_n(p_n)^2+\\iota _2/N^2.\\end{array}$ Along with the almost sure uniform bound $\\mathbb {E}\\left(\\widehat{\\alpha }_n(p_n)^2~|~p_n\\right)\\le {\\iota _3}/{N},$ which follows from (REF ), we conclude that $\\mathbb {E}\\left(\\left(\\widehat{\\alpha }_n(p_n)+\\beta _{n+1}(\\widehat{p}_n)\\right)^2~|~p_n\\right)\\le {\\iota _4}/{N}.$ Combining this with (REF ), it follows that $1\\le \\mathbb {E}\\left(\\left(\\frac{\\phi (p_n)}{p_{n+1}}\\right)^2~|~p_n=x\\right)\\le 1+{\\iota _5}/{N}.$ For odd powers, for instance we have $\\begin{array}{l}\\displaystyle \\mathbb {E}\\left(\\left(\\widehat{\\alpha }_n(p_n)+\\beta _{n+1}(\\widehat{p}_n)\\right)^3~|~p_n,\\widehat{p}_n\\right)\\\\\\\\\\displaystyle =\\widehat{\\alpha }_n(p_n)^3+3~\\widehat{\\alpha }_n(p_n)^2\\phi (p_n)~\\mathbb {E}\\left(\\frac{\\Upsilon _{n+1}(\\widehat{p}_n)-1}{A^2\\widehat{p}_n+R}~|~\\widehat{p}_n\\right)\\\\\\\\\\displaystyle +3~\\widehat{\\alpha }_n(p_n)\\phi (p_n)^2~\\mathbb {E}\\left(\\left(\\frac{\\Upsilon _{n+1}(\\widehat{p}_n)-1}{A^2\\widehat{p}_n+R}\\right)^2~|~\\widehat{p}_n\\right)+\\phi (p_n)^3~\\mathbb {E}\\left(\\left(~\\frac{\\Upsilon _{n+1}(\\widehat{p}_n)-1}{A^2\\widehat{p}_n+B^2}\\right)^3~|~\\widehat{p}_n\\right).\\end{array}$ Now observe that $\\begin{array}{l}\\displaystyle 6~\\widehat{\\alpha }_n(p_n)\\phi (p_n)^2~\\mathbb {E}\\left(\\left(\\frac{\\Upsilon _{n+1}(\\widehat{p}_n)-1}{A^2\\widehat{p}_n+R}\\right)^2~|~\\widehat{p}_n\\right)\\\\\\\\\\displaystyle \\le \\left(3\\widehat{\\alpha }_n(p_n)\\phi (p_n)^2\\right)^2+\\mathbb {E}\\left(\\left(\\frac{\\Upsilon _{n+1}(\\widehat{p}_n)-1}{A^2\\widehat{p}_n+R}\\right)^4~|~\\widehat{p}_n\\right)\\le \\left(3\\widehat{\\alpha }_n(p_n)\\phi (p_n)^2\\right)^2+\\frac{\\iota _6}{N}.\\end{array}$ Using the estimates (REF ), this implies that $\\begin{array}{l}\\displaystyle \\mathbb {E}\\left(\\left(\\widehat{\\alpha }_n(p_n)+\\beta _{n+1}(\\widehat{p}_n)\\right)^3~|~p_n\\right)\\\\\\\\\\displaystyle \\le \\mathbb {E}\\left(\\widehat{\\alpha }_n(p_n)^3~|~p_n\\right)+\\frac{1}{2}\\left(3\\phi (p_n)^2\\right)^2~\\mathbb {E}\\left(\\widehat{\\alpha }_n(p_n)^2~|~p_n\\right)+\\frac{\\iota _6}{2N}\\\\\\\\\\displaystyle \\hspace{85.35826pt}+3\\,\\phi (p_n)~\\mathbb {E}\\left(\\widehat{\\alpha }_n(p_n)^2~|~p_n\\right)~\\frac{\\iota _1}{N}+\\phi (p_n)^3~\\frac{\\iota _3}{N}.\\end{array}$ Using the fact that $\\mathbb {E}\\left(\\vert \\widehat{\\alpha }_n(p_n)\\vert ^3~|~p_n\\right)\\le \\frac{\\iota _7}{N\\sqrt{N}},$ it follows that $\\begin{array}{l}\\displaystyle \\mathbb {E}\\left(\\left(\\widehat{\\alpha }_n(p_n)+\\beta _{n+1}(\\widehat{p}_n)\\right)^3~|~p_n\\right)\\le \\frac{\\iota _8}{N}.\\end{array}$ Using (REF ) and Jensen's inequality we check that $1\\le \\mathbb {E}\\left(\\left(\\frac{\\phi (p_n)}{p_{n+1}}\\right)^3~|~p_n\\right)\\le 1+\\frac{\\iota _9}{N}.$ More generally, for any $l\\ge 2$ and $k\\ge 1$ we have $\\mathbb {E}\\left(\\vert \\widehat{\\alpha }_n(p_n)\\vert ^l~|~p_n=x\\right)\\le \\frac{\\iota _l(1)}{N}\\quad \\mbox{\\rm and}\\quad \\mathbb {E}\\left(\\beta _{n+1}(\\widehat{p}_n)^{2k}~|~\\widehat{p}_n=x\\right)\\le \\frac{\\iota _2(k)}{N}$ for some finite constants $\\iota _1(k),\\iota _2(k)$ .","Also note that due to the binomial formula, we have $\\begin{array}{l}\\displaystyle \\mathbb {E}\\left(\\left(\\frac{\\phi (p_n)}{p_{n+1}}\\right)^k~|~p_n,\\widehat{p}_n\\right)\\\\\\\\\\displaystyle =1+\\mathbb {E}\\left(\\left(\\widehat{\\alpha }_n(p_n)+\\beta _{n+1}(\\widehat{p}_n)\\right)~|~p_n,\\widehat{p}_n\\right)+\\sum _{2\\le l\\le k}\\left(\\begin{array}{c}k\\\\l\\end{array}\\right)~\\mathbb {E}\\left(\\left(\\widehat{\\alpha }_n(p_n)+\\beta _{n+1}(\\widehat{p}_n)\\right)^{l}~|~p_n,\\widehat{p}_n\\right)\\end{array}.$ We may estimate the above summands, for any $l\\ge 2$ by $\\begin{array}{l}\\displaystyle \\mathbb {E}\\left(\\left(\\widehat{\\alpha }_n(p_n)+\\beta _{n+1}(\\widehat{p}_n)\\right)^{l}~|~p_n,\\widehat{p}_n\\right)\\\\\\\\\\displaystyle \\le \\vert \\widehat{\\alpha }_n(p_n)\\vert ^{l}+\\frac{1}{2}~\\sum _{0\\le i< l}\\left(\\begin{array}{c}l\\\\i\\end{array}\\right)\\left(\\widehat{\\alpha }_n(p_n)^{2l}+\\mathbb {E}\\left( \\left(\\beta _{n+1}(\\widehat{p}_n)\\right)^{2(l-i)}~|~p_n,\\widehat{p}_n\\right)\\right)\\\\\\\\\\displaystyle \\le \\vert \\widehat{\\alpha }_n(p_n)\\vert ^{l}+\\frac{1}{2}~\\sum _{0\\le i< l}\\left(\\begin{array}{c}.l\\\\i\\end{array}\\right)\\left(\\widehat{\\alpha }_n(p_n)^{2l}+\\frac{\\iota _{2}(l-i)}{N}\\right)\\end{array}$ The end of the proof of (REF ) is now a direct consequence of the moment estimates stated above."],["A Feynman-Kac formula","Theorem 7.1 Let $X_n$ be a Markov chain on some measurable state space $E$ with Markov transitions $M(x,dy)$ and starting at some state $X_0=x$ .","Also let $H(x)$ be some positive measurable function on $E$ .","Assume there exists some function $h$ and some parameters $\\epsilon _h\\in ]0,1[$ and $\\kappa _h \\in ]0, \\infty [$ such that $H^h(x):=H(x)~h(x)M(1/h)(x)\\le (1-\\epsilon _h)\\quad \\mbox{and}\\quad M(1/h)(x)\\le \\kappa _h ~M(1/h)(y).$ Then, for any $n\\ge 1$ and any bounded measurable function $F$ on the path space $E^n$ we have the Feynman-Kac change of measure formula $\\begin{array}{l}\\displaystyle \\mathbb {E}\\left(F(X_1,\\ldots ,X_n)~\\left(\\prod _{1\\le k\\le n}H(X_k)\\right)~|~X_0=x\\right)\\\\\\\\\\displaystyle =\\mathbb {E}\\left(F(X^{1/h}_1,\\ldots ,X^{1/h}_n)~~\\frac{M(h^{-1})(X^{1/h}_{0})}{M(h^{-1})(X^{1/h}_{n})}\\left(\\prod _{1\\le k\\le n}H^h(X^{1/h}_{k})\\right)~|~X^{1/h}_0=x\\right).\\end{array}$ In the above display, $X^{1/h}_n$ stands for the Markov chain with Markov transitions $M^{1/h}(x,dy)=M(x,dy)~\\frac{h^{-1}(y)}{M(h^{-1})(x)}.$ In addition, we have the uniform exponential decay estimate $\\sup _{x} \\mathbb {E}\\left(\\prod _{1\\le k\\le n}H(X_k)~|~X_0=x\\right)\\le \\kappa _h~(1-\\epsilon _h)^n.$ We set $h^{-1}(x)=1/h(x)$ .","Let $X^{1/h}_n$ be the Markov chain starting at $X^{1/h}_0=X_0=x_0$ whose Markov transitions $M^{1/h}$ are given by (REF ) so that $M^{1/h}(h)(x)=\\frac{1}{M(h^{-1})(x)}.$ Observe that $\\begin{array}{l}\\mathbb {P}^{1/h}_n(d(x_0,\\ldots ,x_n))\\\\\\\\:=\\mathbb {P}((X^{1/h}_0,\\ldots ,X^{1/h}_n)\\in d(x_0,\\ldots ,x_n))\\\\\\\\\\displaystyle =\\eta _0(dx_0)~M(x_0,dx_1)~\\frac{h^{-1}(x_1)}{M(h^{-1})(x_0)}\\ldots M(x_{n-1},dx_n)~\\frac{h^{-1}(x_n)}{M(h^{-1})(x_{n-1})}.\\end{array}$ Rewritten in terms of expectations we have $\\mathbb {E}\\left(F(X_0,\\ldots ,X_n)~\\prod _{1\\le k\\le n}\\frac{h^{-1}(X_k)}{M(h^{-1})(X_{k-1})}\\right)=\\mathbb {E}\\left(F(X_0^{1/h},\\ldots ,X_n^{1/h})\\right).$ Observe that $\\begin{array}{l}\\mathbb {P}^{1/h}_n(d(x_0,\\ldots ,x_n))\\\\\\\\\\displaystyle =\\mathbb {P}_n(d(x_0,\\ldots ,x_n))~\\left(\\frac{h^{-1}(x_1)}{M(h^{-1})(x_0)}\\ldots ~\\frac{h^{-1}(x_n)}{M(h^{-1})(x_{n-1})} \\right)\\end{array}$ with $\\mathbb {P}_n(d(x_0,\\ldots ,x_n)):=\\eta _0(dx_0)~M(x_0,dx_1)~\\ldots M(x_{n-1},dx_n).$ This clearly implies that $\\begin{array}{l}\\mathbb {P}_n(d(x_0,\\ldots ,x_n))\\\\\\\\\\displaystyle =\\mathbb {P}^{1/h}_n(d(x_0,\\ldots ,x_n))~\\left(\\left(h(x_1)~M(h^{-1})(x_0)\\right)\\ldots ~\\left(h(x_n)~M(h^{-1})(x_{n-1})\\right) \\right)\\\\\\\\\\displaystyle =\\mathbb {P}^{1/h}_n(d(x_0,\\ldots ,x_n))~\\left(\\frac{h(x_1)}{M^{1/h}(h)(x_0)}\\ldots ~\\frac{h(x_n)}{M^{1/h}(h)(x_{n-1})} \\right),\\end{array}$ where we have used (REF ) in the final step.","Rewritten in terms of expectations, the above formula takes the following form $\\mathbb {E}\\left(F(X_0^{1/h},\\ldots ,X_n^{1/h})\\prod _{1\\le k\\le n}\\frac{h(X_k^{1/h})}{M^{1/h}(h)(X^{1/h}_{k-1})}\\right)=\\mathbb {E}\\left(F(X_0,\\ldots ,X_n)\\right).$ Replacing $F(X_0,\\ldots ,X_n)$ by $F(X_0,\\ldots ,X_n)\\prod _{1\\le k\\le n}H(X_k),$ and again recalling (REF ), we obtain the formula $\\begin{array}{l}\\displaystyle \\mathbb {E}\\left(F(X_0,\\ldots ,X_n)\\prod _{1\\le k\\le n}H(X_k)\\right)\\\\\\\\\\displaystyle =\\mathbb {E}\\left(F(X_0^{1/h},\\ldots ,X^{1/h}_n)\\left(\\prod _{1\\le k\\le n}H(X^{1/h}_k)\\right)\\left(\\prod _{1\\le k\\le n}\\frac{h(X_k^{1/h})}{M^{1/h}(h)(X^{1/h}_{k-1})}\\right)\\right)\\\\\\\\\\displaystyle =\\mathbb {E}\\left(F(X^{1/h}_0,\\ldots ,X^{1/h}_n)\\left(\\prod _{1\\le k\\le n}(H h)(X^{1/h}_k)\\right)\\left(\\prod _{0\\le k< n}M(h^{-1})(X^{1/h}_{k})\\right)\\right)\\\\\\\\\\displaystyle =\\mathbb {E}\\left(F(X^{1/h}_0,\\ldots ,X^{1/h}_n)~M(h^{-1})(X^{1/h}_{0})(Hh)(X_{n}^{1/h})\\left(\\prod _{1\\le k< n}(Hh)(X^{1/h}_k)M(h^{-1})(X^{1/h}_{k})\\right)\\right)\\\\\\\\\\displaystyle =\\mathbb {E}\\left(F(X^{1/h}_0,\\ldots ,X^{1/h}_n)~\\frac{M(h^{-1})(X^{1/h}_{0})}{M(h^{-1})(X^{1/h}_n)}\\left(\\prod _{1\\le k\\le n}(Hh)(X^{1/h}_k)M(h^{-1})(X^{1/h}_{k})\\right)\\right).\\end{array}$ This ends the proof of the the Feynman-Kac change of measure formula.","The exponential decay estimate (REF ) is now a direct consequence of the regularity condition (REF ).","We are now in position to prove the uniform almost sure exponential decays stated in Theorem REF .","Proof of first inequality in (REF ) Up to a time change, it suffices to prove the result for $l=1$ and for any given initial condition $p_0=x$ .","Recall the Markov transition, $\\mathcal {P}$ , of the stochastic Riccati process $p_n$ .","We choose in Theorem REF the function $H(x)=\\left(\\frac{\\vert A\\vert }{1+Sx}\\right)^k\\quad \\mbox{\\rm and}\\quad h(x)=x^k.$ Suppose that $\\vert A\\vert <1$ .","In this situation, we have $\\mathbb {E}\\left(\\prod _{1\\le l\\le n}\\left(\\frac{\\vert A\\vert }{1+Sp_l}\\right)^k\\right)\\le (1-\\epsilon _k)^n\\quad \\mbox{\\rm with}\\quad \\epsilon _k:=1-\\vert A\\vert ^k.$ Now assume that $\\vert A\\vert \\ge 1$ .","In this situation, we have $\\vert A\\vert <(A^2+RS)$ .","Observe that $H^h(x)=H(x)~h(x)\\mathcal {P}(h^{-1})(x)=\\left(\\frac{\\vert A\\vert }{1+Sx}\\right)^k~\\mathbb {E}\\left(\\left(\\frac{p_0}{p_1}\\right)^k~|~p_0=x\\right).$ Using (REF ) we check that $H^h(x)=\\left(\\frac{\\vert A\\vert }{1+Sx}~\\frac{x}{\\phi (x)}\\right)^k~\\mathbb {E}\\left(\\left(\\frac{\\phi (p_0)}{p_1}\\right)^k~|~p_0=x\\right)\\le \\left(\\frac{\\vert A\\vert }{1+Sx}~\\frac{x}{\\phi (x)}\\right)^k~\\left(1+\\frac{\\iota _k}{N}\\right).$ This yields the estimate $H^h(x)\\le \\left(\\frac{\\vert A\\vert }{(A^2+RS)+R/x}\\right)^k~\\left(1+\\frac{\\iota _k}{N}\\right)\\le \\left(\\frac{\\vert A\\vert }{A^2+RS}\\right)^k~\\left(1+\\frac{\\iota _k}{N}\\right).$ Choosing $N$ such that $\\displaystyle N\\ge N_{k}:=\\iota _k \\left(1-\\left(\\frac{\\vert A\\vert }{A^2+RS}\\right)^k\\right)^{-1}$ yields $\\left(1+\\frac{\\iota _k}{N}\\right)\\le 1+\\sqrt{\\epsilon _k},\\quad \\mbox{\\rm with}\\quad \\sqrt{\\epsilon _k}:=\\left(1-\\left(\\frac{\\vert A\\vert }{A^2+RS}\\right)^k\\right).$ We readily check that $H^h(x)&\\le & \\left(1-\\sqrt{\\epsilon _k}\\right)\\left(1+\\sqrt{\\epsilon _k}\\right)=1-\\epsilon _k<1.$ This implies that $\\mathbb {E}\\left(\\left(\\prod _{1\\le l\\le n}\\frac{\\vert A\\vert }{1+Sp_l}\\right)^k~|~p_0=x\\right)\\le \\left(1-\\epsilon _k\\right)^{n-1}\\mathbb {E}\\left(\\frac{ {\\cal P }(h^{-1})(x)}{ {\\cal P }(h^{-1})(X^{1/h}_{n-1})}\\right).$ On the other hand, using (REF ) and (REF ), for any even parameter $N>2(k+2)$ we have $\\frac{1}{(A^2/S+R)^k}\\le {\\cal P }(h^{-1})(x)=\\frac{1}{\\phi (p_0)^k}~\\mathbb {E}\\left(\\left(\\frac{\\phi (p_0)}{p_1}\\right)^k~|~p_0=x\\right)\\le \\frac{1}{R^k}~\\left(1+\\frac{\\iota _k}{N}\\right)$ We conclude that $\\mathbb {E}\\left(\\left(\\prod _{1\\le l\\le n}\\frac{\\vert A\\vert }{1+Sp_l}\\right)^k\\right)\\le \\left(1-\\epsilon _k\\right)^{n-1}~\\left(A^2R/S+R^2\\right)^k \\left(1+\\frac{\\iota _k}{N}\\right).$ This ends the proof of the exponential decays estimates stated in the l.h.s.","of (REF )."],["Fluctuation analysis","Lemma 7.2 For any $k\\ge 1$ there exists some parameter $N_k\\ge 1$ such that for any $N\\ge N_k$ and $n\\ge 0$ we have the time uniform estimates $\\mathbb {E}\\left(\\vert m_{n}-X_{n}\\vert ^k\\right)\\vee \\mathbb {E}\\left(\\vert \\widehat{m}_{n}-X_{n}\\vert ^k\\right)\\vee \\mathbb {E}\\left(\\vert Y_n-Cm_{n}\\vert ^k\\right)\\le \\iota _k.$ Using (REF ) we check the recursion $(m_{n+1}-X_{n+1})=\\alpha _n~(X_n-m_n)+\\beta _{n+1},$ with the random variables $\\alpha _n&:=A(1-g_nC)=\\frac{A}{1+Sp_n} \\\\\\beta _{n+1}&:=Ag_nDV_n-BW_{n+1}+\\frac{1}{\\sqrt{N+1}}~\\left(A\\widehat{\\upsilon }_n+\\upsilon _{n+1}\\right).$ This yields the formula $(m_{n+1}-X_{n+1})= {\\cal E }_{0,n}~(X_0-m_0)+\\beta _{n+1}+\\sum _{0\\le k\\le n} {\\cal E }_{k,n}~\\beta _{k}.$ The raw moment estimates of the differences $(m_{n}-X_{n})$ and $(Y_n-Cm_{n})$ stated in (REF ) are now a direct consequence of the exponential decay estimate stated in (REF ).","The raw moment estimates of the difference $(\\widehat{m}_{n}-X_{n})$ is easily checked using the following decomposition $\\widehat{m}_{n}-X_{n}=\\frac{A}{1+Sp_n}~(m_n-X_n)+g_nCDV_n+\\frac{1}{\\sqrt{N+1}}~\\widehat{\\upsilon }_n$ This ends the proof of the lemma.","Note that this completes the proof of Theorem REF .","Now we come to the proof of the Lyapunov estimate (REF ).","Proof of (REF ): Combining (REF ) with the uniform almost sure exponential decays stated in (REF ) we check the uniform estimate $\\mathbb {E}\\left(\\vert M_{n+1}\\vert ~|~(M_0,p_0)\\right)\\le \\mathbb {E}( {\\cal E }_{0,n}~|~p_0)~\\vert M_0\\vert +\\iota _1\\le \\iota _1+\\iota _2 (1-\\epsilon _2)^{n}\\vert M_0\\vert .$ Thus, for any $\\epsilon \\in ]0,1]$ there exists some time horizon $n_{\\epsilon }\\ge 1$ such that for any $n\\ge n_{\\epsilon }$ we have $\\mathbb {E}\\left(\\vert M_{n}\\vert ~|~(M_0,p_0)\\right)\\le \\epsilon ~\\vert M_0\\vert +\\iota .$ The estimate (REF ) now comes from the fact that the one-step Riccati map $\\phi $ is uniformly bounded.","Proof of Theorem REF : First note that $m_{n+1}-\\widehat{X}_{n+1}^-=\\displaystyle A~(\\widehat{m}_{n}-\\widehat{X}_n)+\\frac{1}{\\sqrt{N+1}}~\\upsilon _{n+1}.$ We also have the decomposition $\\begin{array}{l}\\displaystyle (\\widehat{m}_{n}-\\widehat{X}_{n})\\\\\\\\=\\displaystyle (m_n-\\widehat{X}^-_{n})+(g_n-G_n)~(Y_n-Cm_n)-G_n~C(m_n-\\widehat{X}^-_{n})+\\frac{1}{\\sqrt{N+1}}~\\widehat{\\upsilon }_n\\\\\\\\=\\displaystyle (1-G_nC)(m_n-\\widehat{X}^-_{n})+\\frac{1}{\\sqrt{N}}~\\sqrt{N}(g_n-G_n)~(Y_n-Cm_n)+\\frac{1}{\\sqrt{N+1}}~\\widehat{\\upsilon }_n.\\end{array}$ This yields the recursion $(\\widehat{m}_{n}-\\widehat{X}_{n}) =\\alpha _n~(\\widehat{m}_{n-1}-\\widehat{X}_{n-1})+\\frac{1}{\\sqrt{N}}~ \\beta _n,$ with the parameters $\\alpha _n&:=&A(1-G_nC)= \\frac{A}{1+SP_n}\\\\\\beta _n&:=&\\frac{1}{\\sqrt{1+1/N}}~ \\left((1-G_nC) \\upsilon _{n}+\\widehat{\\upsilon }_n\\right)+\\sqrt{N}(g_n-G_n)~(Y_n-Cm_n).$ We conclude that $(\\widehat{m}_{n}-\\widehat{X}_{n}) =E_n(P_0)~(\\widehat{m}_{0}-\\widehat{X}_{0})+\\frac{1}{\\sqrt{N}}~\\sum _{1\\le l\\le n}~ E_{l,n}(P_0)~\\beta _l.$ Finally, observe that $(\\widehat{m}_{0}-\\widehat{X}_{0})=\\displaystyle (1-G_0C)(m_0-\\widehat{X}^-_{0})+\\frac{1}{\\sqrt{N}}~\\left(\\sqrt{N}(g_0-G_0)~(Y_0-Cm_0)+\\frac{1}{\\sqrt{1+1/N}}~\\widehat{\\upsilon }_0\\right).$ The estimate (REF ) is now a consequence of the exponential decays (REF ), the uniform bounds (REF ) and the mean error estimates stated in Theorem REF .","The proof of (REF ) follows similar arguments.","Combining (REF ) and (REF ) we check the decomposition $\\begin{array}{l}\\displaystyle m_{n+1}-\\widehat{X}_{n+1}^-=A~ (1-G_nC)(m_n-\\widehat{X}^-_{n})+A(g_n-G_n)~(Y_n-C\\widehat{X}_n^-)\\\\\\\\\\hspace{56.9055pt}\\displaystyle -\\frac{1}{N}~\\sqrt{N}~A(g_n-G_n)C~~\\sqrt{N}(m_n-\\widehat{X}^-_n)+\\frac{1}{\\sqrt{N+1}}~A\\widehat{\\upsilon }_n+\\frac{1}{\\sqrt{N+1}}~\\upsilon _{n+1}.\\end{array}$ Taking the expectations we obtain the formula $\\begin{array}{l}\\displaystyle (m_{n+1}^o-\\widehat{X}_{n+1}^-)=\\alpha _n~(m_n^o-\\widehat{X}^-_{n})+\\beta _n/N\\end{array}$ with the parameters $(\\alpha _n,\\beta _n)$ now given by $\\alpha _n&:=&A(1-G_nC)= {A}/{(1+SP_n)}\\\\\\beta _n&:=&A~N(\\mathbb {E}(g_n)-G_n)~(Y_n-C\\widehat{X}_n^-)-AC~\\mathbb {E}\\left(\\sqrt{N}~(g_n-G_n)\\sqrt{N}(m_n-\\widehat{X}^-_n)~|~ {\\cal Y }_{n-1}\\right).$ Using the uniform estimates stated in Theorem REF and the gain bias estimates (REF ), we obtain, for $k \\ge 1$ $\\sup _{n\\ge 0}\\mathbb {E}\\left(\\vert \\beta _n\\vert ^k\\right)^{1/k}\\le \\iota _k~(1\\vee P_0)^2.$ The end of the proof of (REF ) now follows the same lines of arguments as the proof of the estimates (REF ), thus we leave the details as an exercise for the reader.","Proof of Theorem REF : First note that from the proof of (REF ), we have the following decomposition $\\sqrt{N}(\\widehat{m}_n - \\widehat{X}_n) = \\mathcal {E}_n(P_0)\\sqrt{N}(\\widehat{m}_0 - \\widehat{X}_0) + \\sum _{1 \\le \\ell \\le n}\\mathcal {E}_{\\ell , n}(P_0)\\beta _\\ell ,$ where we recall that $\\mathcal {E}_{\\ell , n}(p) &= \\prod _{\\ell < k \\le n}\\frac{A}{1 + SP_\\ell (p)},\\\\\\beta _\\ell &= \\frac{1}{\\sqrt{1+1/N}}~ \\left((1-G_nC) \\upsilon _{n}+\\widehat{\\upsilon }_n\\right)+\\sqrt{N}(g_n-G_n)~(Y_n-Cm_n).$ We first deal with $\\sqrt{N}(\\widehat{m}_0 - \\widehat{X}_0)$ on the right hand side of (REF ).","Again, from the proof of (REF ), recall that $\\sqrt{N}(\\widehat{m}_{0}-\\widehat{X}_{0})=\\displaystyle (1-G_0C)\\sqrt{N}(m_0-\\widehat{X}^-_{0})+\\frac{1}{\\sqrt{1+1/N}}~\\widehat{\\upsilon }_0+\\sqrt{N}(g_0-G_0)~(Y_0-Cm_0).$ Further observe that $(Y_0-Cm_0)=(Y_0-C\\widehat{X}^-_0)+C(\\widehat{X}^-_0-m_0)\\quad \\mbox{\\rm and}\\quad (\\widehat{X}^-_0-m_0)\\longrightarrow _{N\\rightarrow \\infty } 0\\quad \\mbox{\\rm a.e.","}.$ Applying Slutsky's lemma, this yields the weak convergence $\\sqrt{N}(g_0-G_0)~C(\\widehat{X}^-_0-m_0)\\longrightarrow _{N\\rightarrow \\infty } 0.$ Using the fluctuation theorems REF and REF we also have $\\sqrt{N}(m_0-\\widehat{X}^-_{0}) &\\longrightarrow _{N\\rightarrow \\infty } \\mathbb {U}_0,\\\\\\sqrt{N}(g_0-G_0)(Y_0-C\\widehat{X}^-_0) &\\longrightarrow _{N\\rightarrow \\infty } \\mathbb {G}_0(Y_0-C\\widehat{X}^-_0),\\\\\\frac{1}{\\sqrt{1+1/N}}~\\widehat{\\upsilon }_0 &\\longrightarrow _{N\\rightarrow \\infty } \\widehat{\\mathbb {U}}_0.$ In the same vein, applying the continuous mapping theorem or the $\\delta $ -method (see for instance Theorem 9.3.3 and Lemma 9.3.1 in [36]) we check that $\\sqrt{N}(\\widehat{m}_{0}-\\widehat{X}_{0})\\longrightarrow _{N\\rightarrow \\infty } (1-G_0C) \\mathbb {U}_0+\\widehat{\\mathbb {U}}_0+\\mathbb {G}_0(Y_0-C\\widehat{X}^-_0).$ To deal with the sum on the right hand side of (REF ), again, thanks to Theorem REF , and arguing as above $\\beta _\\ell $ converges weakly to $(1-G_{\\ell }C)\\mathbb {U}_{\\ell } + \\widehat{\\mathbb {U}}_{\\ell }+\\mathbb {G}_{\\ell } (Y_{\\ell }-C\\widehat{X}_{\\ell }^-).$ In the same vein, we check that $\\widehat{\\mathbb {X}}_n^N &\\longrightarrow _{N\\rightarrow \\infty } \\widehat{\\mathbb {X}}_n:= \\sum _{0 \\le \\ell \\le n}E_{\\ell , n}(P_0)\\left((1-G_{\\ell }C)\\mathbb {U}_{\\ell } + \\widehat{\\mathbb {U}}_{\\ell }+\\mathbb {G}_{\\ell } (Y_{\\ell }-C\\widehat{X}_{\\ell }^-)\\right).$ The proof of the fluctuation theorem at the level of the process in the sense of convergence of finite dimensional distributions is conducted combining the continuous mapping theorem with Slutsky's lemma.","The proof of the recursive formulation (REF ) is also easily checked using the decompositions (REF ) and (REF ).","This ends the proof of the theorem."],["Proof of Theorem ","For the identity function $ {\\cal U }(p):=p$ , due to (REF ), we have $ {\\cal P }( {\\cal U })(p)=\\mathbb {E}( {\\cal U }(p_{n+1})~|~p_n=p)=\\phi (p)\\le \\alpha =A^2/S+R.$ In addition, for any $\\epsilon \\in ]0,1]$ we have $ {\\cal U }_{\\epsilon }:=1+\\frac{\\epsilon }{\\alpha }~ {\\cal U }\\ge 1\\Longrightarrow {\\cal P }( {\\cal U }_{\\epsilon })(p)\\le 1+\\epsilon \\Longrightarrow {\\cal P }( {\\cal U }_{\\epsilon })\\le 1+\\epsilon ~ {\\cal U }_{\\epsilon }.$ The above estimate implies that $ {\\cal U }_{\\epsilon }\\ge 1$ is a Lyapunov function.","On the other hand, $ {\\cal P }(p,dq)=k(p,q)~dq$ has a continuous density $k(p,q)>0$ with respect to the variables $(p,q)$ and the Lebesgue measure $dq$ on $\\mathbb {R}_+$ .","Thus, for any compact set $\\varpi \\subset \\mathbb {R}_+$ there exists some function $\\varphi $ from $[0,\\infty [$ into itself such that $k_{\\varpi }(q):=\\inf _{p\\in \\varpi }k(p,q)=k(\\varphi _{\\varpi }(q),q)>0 \\Longrightarrow 0<\\epsilon _{\\varpi }:=\\int ~k_{\\varpi }(q)~dq<1.$ This implies that $ {\\cal P }(p,dq) \\ge \\epsilon _{\\varpi }~\\nu _{\\varpi }(dq)\\quad \\mbox{\\rm with}\\quad \\nu _{\\varpi }(dq):=\\frac{k_{\\varpi }(q)~dq}{\\int ~k_{\\varpi }(p)~dp},$ from which we prove the Dobrushin local contraction property $\\varpi _r:=[0,r]\\Longrightarrow \\beta _{r}( {\\cal P }):=\\sup _{(p,q)\\in \\varpi _r^2}\\Vert {\\cal K }(p, \\mbox{\\LARGE .","})- {\\cal K }(q, \\mbox{\\LARGE .})","\\Vert _{\\tiny tv}<1-\\epsilon _{\\varpi }<1.$ For any $\\epsilon _1,\\epsilon _2\\in ]0,1]$ we set $ {\\cal U }_{\\epsilon _1,\\epsilon _2}:=\\epsilon _1 {\\cal U }_{\\epsilon _2}\\Longrightarrow {\\cal P }( {\\cal U }_{\\epsilon _1,\\epsilon _2})\\le \\epsilon _1\\left(1+\\epsilon _2 {\\cal U }_{\\epsilon _2}\\right)\\le \\epsilon _1+\\epsilon _2~ {\\cal U }_{\\epsilon _1,\\epsilon _2}.$ Choosing the parameters $(r,\\epsilon _2)$ such that $(1/r+\\epsilon _2)<1$ for any $\\epsilon _1\\in [0,1[$ we can check that $p,q\\ge r\\Longrightarrow \\frac{\\Vert {\\cal P }(p, \\cdot ) - {\\cal P }(q, \\cdot ) \\Vert _{ {\\cal U }_{\\epsilon _1,\\epsilon _2}}}{1 + {\\cal U }_{\\epsilon _1,\\epsilon _2}(p) + {\\cal U }_{\\epsilon _1,\\epsilon _2}(q)}<1.$ The detailed proof of the above assertion can be found in section 8.2.7 in [39].","In addition, for any $x,y\\le r$ we also have $\\epsilon _1<\\frac{1-\\beta _r( {\\cal K })}{4r}\\Longrightarrow \\frac{\\Vert {\\cal P }(x, \\cdot ) - {\\cal P }(y, \\cdot ) \\Vert _{ {\\cal U }_{\\epsilon _1,\\epsilon _2}}}{1 + {\\cal U }_{\\epsilon _1,\\epsilon _2}(x) + {\\cal U }_{\\epsilon _1,\\epsilon _2}(y)}\\le \\beta _r( {\\cal K })+4\\epsilon _1 r<1.$ We conclude that $(1/r+\\epsilon _2)<1\\quad \\mbox{\\rm and}\\quad \\epsilon _1<\\frac{1-\\beta _r( {\\cal P })}{4r}\\quad \\Longrightarrow \\quad \\beta _{ {\\cal U }_{\\epsilon _1,\\epsilon _2}}( {\\cal P })<1,$ which concludes the proof.","Following [26], we have $\\phi ^n(x)+v=\\lambda _2~\\frac{\\left(x+(v-\\lambda _1)\\right)~\\left(1-\\lambda ^{n+1}\\right)+(\\lambda _2-\\lambda _1)~\\lambda ^{n+1}}{\\left(x+(v-\\lambda _1)\\right)~\\left(1-\\lambda ^{n}\\right)+(\\lambda _2-\\lambda _1)~\\lambda ^{n}}.$ Next, we check that the r.h.s.","expression in the above display is non negative.","Notice that $\\phi ^n(x)=\\frac{\\left(x+(v-\\lambda _1)\\right)\\left(\\lambda _2~\\left(1-\\lambda ^{n+1}\\right)-v~\\left(1-\\lambda ^{n}\\right)\\right)-(v-\\lambda _1)(\\lambda _2-\\lambda _1)~\\lambda ^{n}}{\\left(x+(v-\\lambda _1)\\right)~\\left(1-\\lambda ^{n}\\right)+(\\lambda _2-\\lambda _1)~\\lambda ^{n}}.$ Rewritten in a slightly different form (REF ) takes the following form $\\phi ^n(x)=(\\lambda _2-v)+\\frac{\\left(x+(v-\\lambda _1)\\right)~-(\\lambda _2-\\lambda _1)}{\\left(x+(v-\\lambda _1)\\right)~\\left(1-\\lambda ^{n}\\right)+(\\lambda _2-\\lambda _1)~\\lambda ^{n}}~\\left(\\lambda _2-\\lambda _1\\right)~\\lambda ^n.$ This yields the formula $\\phi ^n(x)-r=\\frac{x-r}{\\left(x+(v-\\lambda _1)\\right)~\\left(1-\\lambda ^{n}\\right)+(\\lambda _2-\\lambda _1)~\\lambda ^{n}}~\\left(\\lambda _2-\\lambda _1\\right)~\\lambda ^n,$ which ends the proof of (REF ).","Recalling that $\\phi $ is an increasing function, we check that $\\phi (0)=\\frac{b}{d}\\le \\phi (x)\\le \\lim _{x\\rightarrow \\infty }\\phi (x)=\\frac{a}{c}\\quad \\mbox{\\rm and}\\quad r=\\phi (r)\\le \\frac{a}{c},$ which ends the proof of (REF ).","For any $x,y\\in \\mathbb {R}_+$ we also have $\\begin{array}{l}\\displaystyle \\frac{\\phi ^n(x)-\\phi ^n(y)}{x-y}\\\\\\\\\\displaystyle =\\frac{(\\lambda _2-\\lambda _1)^2~\\lambda ^n}{\\left[\\left(x+(v-\\lambda _1)\\right)~\\left(1-\\lambda ^{n}\\right)+(\\lambda _2-\\lambda _1)~\\lambda ^{n}\\right]\\left[\\left(y+(v-\\lambda _1)\\right)~\\left(1-\\lambda ^{n}\\right)+(\\lambda _2-\\lambda _1)~\\lambda ^{n}\\right]}.\\end{array}$ This ends the proof of the r.h.s.","estimate in (REF ).","The proof of the l.h.s.","estimate in (REF ) is a direct consequence of the easily checked estimate $\\vert \\phi ^n(x)-r\\vert \\le \\vert x-r\\vert ~\\frac{\\left(\\lambda _2-\\lambda _1\\right)}{\\left(x+(v-\\lambda _1)\\right)~\\wedge (\\lambda _2-\\lambda _1)}~~\\lambda ^n.$ We check (REF ) using the second order formula $\\begin{array}{l}\\displaystyle \\frac{\\phi ^n(x)-\\phi ^n(y)}{x-y}\\\\\\\\\\displaystyle =\\frac{(\\lambda _2-\\lambda _1)^2~\\lambda ^n}{\\left[\\left(y+(v-\\lambda _1)\\right)~\\left(1-\\lambda ^{n}\\right)+(\\lambda _2-\\lambda _1)~\\lambda ^{n}\\right]^2}\\\\\\\\\\displaystyle -\\frac{(\\lambda _2-\\lambda _1)^2~\\lambda ^n}{\\left[\\left(y+(v-\\lambda _1)\\right)~\\left(1-\\lambda ^{n}\\right)+(\\lambda _2-\\lambda _1)~\\lambda ^{n}\\right]^2}\\times \\left(\\frac{(x-y)\\left(1-\\lambda ^{n}\\right)}{\\left[\\left(x+(v-\\lambda _1)\\right)~\\left(1-\\lambda ^{n}\\right)+(\\lambda _2-\\lambda _1)~\\lambda ^{n}\\right]}\\right).\\end{array}$ Finally observe that $(\\ref {def-rho})\\Longrightarrow \\partial \\phi ^n(x)=\\partial (\\phi (\\phi ^{n-1}(x)))=(\\partial \\phi )(\\phi ^{n-1}(x))~\\partial \\phi ^{n-1}(x)=\\prod _{0\\le k0.\\\\end{array}$ One-dimensional rational difference equations are rather well understood, see for instance [26].\",\n \"Section REF also provides a refined stability analysis and several comparison properties of rational difference equations including non-asymptotic Taylor expansions any order.\"\n ],\n [\n \"Ensemble Kalman filters\",\n \"Let $({X}_0, {W}_n,{V}_n)$ be independent copies of $(X_0, W_n,V_n)$ .\",\n \"Consider the nonlinear Markov chain starting at ${X}_0$ and defined sequentially for any $n\\\\ge 0$ by the updating-correction formulae $\\\\left\\\\lbrace \\\\begin{array}{rcl}\\\\widehat{{X}}_{n}&=&{X}_n+{G}_{\\\\eta _n}~(Y_n-(C\\\\,{X}_n+D\\\\,{V}_n))\\\\quad \\\\mbox{\\\\rm with}\\\\quad {G}_{\\\\eta _n}~:=C{Q}_{\\\\eta _n}/(C^2{Q}_{\\\\eta _n}+D^2)\\\\\\\\&&\\\\\\\\{X}_{n+1}&=& A\\\\,{X}_{n}+B\\\\,{W}_{n+1}.\\\\\\\\\\\\end{array}\\\\right.$ In the above display, ${Q}_{\\\\eta _n}$ stands for the variance parameter ${Q}_{\\\\eta _n}:=\\\\int \\\\left(x- \\\\int z~ \\\\eta _n(dz)\\\\right)^2\\\\eta _n(dx)\\\\quad \\\\mbox{\\\\rm indexed by}\\\\quad \\\\eta _n=\\\\mbox{\\\\rm Law}({X}_n~|~ {\\\\cal Y }_{n-1}).$ Using a simple induction argument, it is straightforward to show that $\\\\eta _n= {\\\\cal N }\\\\left(\\\\widehat{X}^-_n,P_n\\\\right)\\\\quad \\\\mbox{\\\\rm and}\\\\quad \\\\widehat{\\\\eta }_n= {\\\\cal N }\\\\left(\\\\widehat{X}_n,\\\\widehat{P}_n\\\\right)=\\\\quad \\\\mbox{\\\\rm Law}(\\\\widehat{{X}}_n~|~ {\\\\cal Y }_{n}).$ Also observe that the above stochastic model can be interpreted as a Markov chain with a two-step updating-prediction transition described by the following synthetic diagram ${X}_n\\\\longrightarrow \\\\widehat{{X}}_{n}\\\\longrightarrow {X}_{n+1}.$ Note that the updating transition ${X}_n\\\\longrightarrow \\\\widehat{{X}}_{n}$ depends on the observation $Y_n$ delivered by the sensor as well as on the conditional law of ${X}_n$ given the information $ {\\\\cal Y }_{n-1}$ .\",\n \"The EnKF associated to the filtering problem (REF ) coincides with the mean-field particle interpretation of the nonlinear Markov chain (REF ).\",\n \"To define these models, we let $\\\\xi _0=(\\\\xi _0^i)_{1\\\\le i\\\\le N+1}$ be a sequence of $(N+1)$ independent copies of $X_0$ , for some parameter $N\\\\ge 1$ .\",\n \"We also denote by $ {\\\\cal W }_n=( {\\\\cal W }^i_n)_{i\\\\ge 1}$ and $ {\\\\cal V }_n=( {\\\\cal V }^i_n)_{i\\\\ge 1}$ a sequence of independent copies of $W_n$ and $V_n$ .\",\n \"The mean field particle interpretation of the nonlinear Markov chain discussed above is given by the interacting particle system defined sequentially for any $1\\\\le i\\\\le N+1$ and $n\\\\ge 0$ by the formulae $\\\\left\\\\lbrace \\\\begin{array}{rcl}\\\\widehat{\\\\xi }^i_{n}&=&\\\\xi ^i_n+g_n~(Y_n-(C\\\\xi ^i_n+D {\\\\cal V }^i_n))\\\\quad \\\\mbox{\\\\rm with}\\\\quad g_n:=Cp_n/(C^2p_n+D^2)\\\\\\\\&&\\\\\\\\\\\\xi ^i_{n+1}&=& A\\\\,\\\\widehat{\\\\xi }^i_{n}+B\\\\, {\\\\cal W }^i_{n+1}.\\\\end{array}\\\\right.$ In the above display $p_n$ stands for the normalised sample variance $p_n:=\\\\frac{1}{N}\\\\sum _{1\\\\le i\\\\le N+1}(\\\\xi ^i_n-m_n)^2=\\\\left(1+\\\\frac{1}{N}\\\\right) {Q}_{\\\\eta ^N_n}\\\\quad \\\\mbox{\\\\rm with}\\\\quad \\\\eta ^N_n:=\\\\frac{1}{N+1}\\\\sum _{1\\\\le i\\\\le N+1}\\\\delta _{\\\\xi ^i_n}.$ We also consider the sample means $m_n:=\\\\frac{1}{N+1}\\\\sum _{1\\\\le i\\\\le N+1}\\\\xi ^i_n\\\\quad \\\\mbox{\\\\rm and}\\\\quad \\\\widehat{m}_n:=\\\\frac{1}{N+1}\\\\sum _{1\\\\le i\\\\le N+1}\\\\widehat{\\\\xi }^i_n$ and the $\\\\sigma $ -field filtrations $\\\\begin{array}{l} {\\\\cal G }_n:=\\\\sigma (\\\\xi _0, {\\\\cal W }_k, {\\\\cal V }_k, 0\\\\le k\\\\le n)\\\\supset \\\\sigma (\\\\xi _k,\\\\widehat{\\\\xi }_k, ~0\\\\le k\\\\le n),\\\\\\\\\\\\\\\\\\\\widehat{ {\\\\cal F }}_n:= {\\\\cal Y }_{n}\\\\vee {\\\\cal G }_n\\\\supset {\\\\cal F }_n:=\\\\widehat{ {\\\\cal F }}_{n-1}\\\\vee \\\\sigma ( {\\\\cal W }_n)\\\\supset \\\\widehat{ {\\\\cal F }}_{n-1}\\\\vee \\\\sigma (\\\\xi _n).\\\\end{array}$ Note that this implies that $\\\\widehat{ {\\\\cal F }}_n\\\\supset {\\\\cal Y }_{n}\\\\quad \\\\mbox{\\\\rm and}\\\\quad {\\\\cal F }_n\\\\supset {\\\\cal Y }_{n-1}.$ In the rest of the article we shall assume that $\\\\xi _0$ , $ {\\\\cal V }_n$ and $ {\\\\cal W }_n$ are column vectors.\",\n \"We also write $X~\\\\underline{\\\\in }~ {\\\\cal F }$ when a random variable is $ {\\\\cal F }$ -measurable.\",\n \"With this notation, at every time $n\\\\ge 0$ , the updating-prediction steps of the ensemble Kalman filter are described by the following synthetic diagram $(m_n,p_n)~\\\\underline{\\\\in }~ {\\\\cal F }_n\\\\longrightarrow (\\\\widehat{m}_{n},\\\\widehat{p}_{n})~\\\\underline{\\\\in }~ \\\\widehat{ {\\\\cal F }}_n\\\\longrightarrow (m_{n+1},p_{n+1})~\\\\underline{\\\\in }~ {\\\\cal F }_{n+1}.$\"\n ],\n [\n \"Some basic notation\",\n \"This section presents some basic notation and preliminary results necessary for the statement of our main results.\",\n \"Throughout the rest of the article, we write $\\\\tau $ , $\\\\tau _k$ as well as $\\\\tau (l)$ and $\\\\tau _k(l)$ for some collection of non-universal constants whose values may vary from line to line, but only depend on the parameters $k$ and $l$ in some given parameter spaces.\",\n \"We use the letters $\\\\iota $ , $\\\\iota _k$ , $\\\\iota (l)$ , $\\\\iota _k(l)$ as well as $\\\\epsilon $ and $\\\\epsilon _k\\\\in ]0,1]$ when these constants may also depend on the parameters $(A,B,C,D)$ of the filtering model.\",\n \"Importantly these constants do not depend on the time parameter nor on the number of particles.\",\n \"We denote by $\\\\chi ^{2}_{N ,Nx}$ a collection a non-central $\\\\chi $ -square random variables indexed by the degree of freedom $N\\\\ge 1$ and non centrality parameter $N x\\\\ge 0$ .\",\n \"We recall that for any $N \\\\ge 1$ , we have $\\\\begin{array}{l}\\\\displaystyle \\\\frac{1}{N}~\\\\chi ^2_{N,Nx}=\\\\frac{1}{N}~\\\\sum _{1\\\\le i\\\\le N}~\\\\left(Z_i+\\\\sqrt{x}\\\\right)^2=1+x+\\\\sqrt{x}~\\\\frac{2}{N}~\\\\sum _{1\\\\le i\\\\le N}~Z_i+\\\\frac{\\\\sqrt{2}}{N}~\\\\sum _{1\\\\le i\\\\le N}~\\\\frac{Z_i^2-1}{\\\\sqrt{2}},\\\\end{array}$ for some given sequence $(Z_i)_{i\\\\ge 1}$ of independent and centered Gaussian random variables.\",\n \"Rewritten in a slightly different form, for any $u,v$ with $v\\\\ne 0$ we have $\\\\frac{v}{N}~\\\\chi ^2_{N,N(u/v)^2}&=&\\\\displaystyle v+\\\\frac{u^2}{v}+\\\\frac{2}{\\\\sqrt{N}}~u~ { {\\\\cal Z }}+\\\\sqrt{\\\\frac{2}{N}}~v~ \\\\widetilde{ {\\\\cal Z }},$ where ${ {\\\\cal Z }}:=\\\\frac{1}{\\\\sqrt{N}}~\\\\sum _{1\\\\le i\\\\le N}~Z_i\\\\quad \\\\mbox{\\\\rm and}\\\\quad \\\\widetilde{ {\\\\cal Z }}:=\\\\frac{1}{\\\\sqrt{N}}~\\\\sum _{1\\\\le i\\\\le N}~\\\\frac{Z_i^2-1}{\\\\sqrt{2}}.$ Note that these definitions imply that ${ {\\\\cal Z }}$ is a centred Gaussian random variable with unit variance and $\\\\widetilde{ {\\\\cal Z }}$ is a centered $\\\\chi $ -square variable.\",\n \"We also consider the extended random vector $\\\\Delta :=\\\\left(\\\\begin{array}{c}\\\\widehat{Z}\\\\\\\\{ {\\\\cal Z }}\\\\\\\\\\\\widetilde{ {\\\\cal Z }}\\\\end{array}\\\\right)\\\\quad \\\\mbox{\\\\rm with}\\\\quad \\\\widehat{Z}:=\\\\frac{1}{\\\\sqrt{N+1}}~\\\\sum _{1\\\\le i\\\\le N+1}~Z_{N+i}.$ In a similar manner, we define $(\\\\widehat{\\\\Delta }_{n},\\\\Delta _{n+1})$ as the sequence of random vectors $\\\\widehat{\\\\Delta }_{n}:=\\\\left(\\\\begin{array}{c}\\\\widehat{\\\\Delta }^1_n\\\\\\\\\\\\widehat{\\\\Delta }^2_n\\\\\\\\\\\\widehat{\\\\Delta }^3_n\\\\end{array}\\\\right)=\\\\left(\\\\begin{array}{l}\\\\widehat{V}_{n}\\\\\\\\{ {\\\\cal V }}_{n}\\\\\\\\\\\\displaystyle \\\\widetilde{ {\\\\cal V }}_n\\\\end{array}\\\\right)\\\\quad \\\\mbox{\\\\rm and}\\\\quad \\\\Delta _{n+1}:=\\\\left(\\\\begin{array}{c}\\\\Delta _{n+1}^1\\\\\\\\\\\\Delta _{n+1}^2\\\\\\\\\\\\Delta _{n+1}^3\\\\end{array}\\\\right)=\\\\left(\\\\begin{array}{l}\\\\widehat{W}_{n+1}\\\\\\\\{ {\\\\cal W }}_{n+1}\\\\\\\\\\\\displaystyle \\\\widetilde{ {\\\\cal W }}_{n+1}\\\\end{array}\\\\right)$ indexed by the time parameter $n\\\\ge 0$ and defined as $\\\\Delta $ by replacing the sequence $(Z_i)_{i\\\\ge 1}$ by the sequences of Gaussian variables $ {\\\\cal V }_n=( {\\\\cal V }^i_n)_{i\\\\ge 1}$ and $ {\\\\cal W }_{n+1}=( {\\\\cal W }^i_{n+1})_{i\\\\ge 1}$ introduced in in the definition of the EnKF (REF ).\",\n \"For $n=0$ we also set $\\\\Delta _0:=\\\\left(\\\\begin{array}{c}\\\\Delta ^1_0\\\\\\\\\\\\Delta ^2_0\\\\end{array}\\\\right)=\\\\left(\\\\begin{array}{c}\\\\widehat{W}_0\\\\\\\\\\\\widetilde{ {\\\\cal W }}_0\\\\end{array}\\\\right)$ with $\\\\widehat{W}_0$ and $\\\\widetilde{ {\\\\cal W }}_0$ defined as $\\\\widehat{Z}$ and $\\\\widetilde{ {\\\\cal Z }}$ by replacing the sequence $Z_i$ with the sequence of Gaussian variables $ {\\\\cal W }_0^i:=\\\\frac{\\\\xi ^i_0-\\\\widehat{X}^-_0}{\\\\sqrt{P_0}},$ where we recall that the $\\\\xi ^i_0$ are independent copies of the Gaussian initial condition $X_0$ of the signal used to initialise the EnKF in (REF ).\",\n \"In this notation, up to a sequence of Helmert orthogonal transformations on the sequence of Gaussian vectors $ {\\\\cal W }_0^i$ , the initial condition of the ensemble Kalman filter is given by the stochastic perturbation formulae $m_0=\\\\widehat{X}^-_0+\\\\frac{1}{\\\\sqrt{N+1}}~\\\\upsilon _0\\\\quad \\\\mbox{and}\\\\quad p_0=P_0+\\\\frac{1}{\\\\sqrt{N}}~\\\\nu _0,$ with the initial local perturbations defined by $\\\\upsilon _0:=\\\\sqrt{P_0}~\\\\Delta ^1_0\\\\quad \\\\mbox{and}\\\\quad \\\\nu _0:=\\\\sqrt{2}~P_0~\\\\Delta ^2_0.$ Observe that the sample means are written in terms of random perturbations $\\\\upsilon _0$ which are independent of the perturbations $\\\\nu _0$ of the sample variances.\",\n \"The independence between the complete sufficient statistic $m_0$ and the ancillary statistic $p_0$ can also be checked using Basu's theorem [11].\",\n \"An alternative algebraic and direct approach based on Helmert orthogonal matrices can be conducted using the same arguments as those used in the proof of Theorem REF .\"\n ],\n [\n \"Local perturbation theorems\",\n \"Consider the local stochastic perturbations defined for any $n\\\\ge 0$ by the formulae $\\\\left\\\\lbrace \\\\begin{array}{rcl}\\\\widehat{\\\\upsilon }_n&:=&-g_nD~\\\\widehat{\\\\Delta }^1_n\\\\\\\\&&\\\\\\\\\\\\widehat{\\\\nu }_n&:=&-2g_nD(1-g_nC)~\\\\sqrt{p_n}~\\\\widehat{\\\\Delta }^2_n+\\\\sqrt{2}~g_n^2D^2~\\\\widehat{\\\\Delta }^3_n\\\\end{array}\\\\right.~\\\\left\\\\lbrace \\\\begin{array}{rcl}\\\\upsilon _{n+1}&:=&B~\\\\Delta ^1_{n+1}\\\\\\\\&&\\\\\\\\\\\\nu _{n+1}&:=&2AB~\\\\sqrt{\\\\widehat{p}_n}~\\\\Delta ^{2}_{n+1}+\\\\sqrt{2}~R~\\\\Delta ^3_{n+1}.\\\\end{array}\\\\right.$ In the above display, $(\\\\Delta _n,\\\\widehat{\\\\Delta }_n)_{n\\\\ge 0}$ are the random vectors defined in (REF ), and $g_n$ is the particle gain defined in (REF ).\",\n \"Theorem 3.1 Up to a sequence of orthogonal transformations on the sequence of Gaussian vectors $ {\\\\cal W }_n$ and $ {\\\\cal V }_n$ , the ensemble Kalman filter and the Riccati equation are given by the initial condition (REF ) and the updating-prediction transitions for $n \\\\ge 0$ $\\\\left\\\\lbrace \\\\begin{array}{rcl}\\\\displaystyle \\\\widehat{m}_{n}&=&\\\\displaystyle m_n+g_n~(Y_n-Cm_n)+\\\\frac{1}{\\\\sqrt{N+1}}~\\\\widehat{\\\\upsilon }_n\\\\\\\\&&\\\\\\\\\\\\widehat{p}_n&=&\\\\displaystyle (1-g_nC)~p_n+\\\\frac{1}{\\\\sqrt{N}}~ \\\\widehat{\\\\nu }_n\\\\end{array}\\\\right.~\\\\left\\\\lbrace \\\\begin{array}{rcl}\\\\displaystyle m_{n+1}&=&\\\\displaystyle A~\\\\widehat{m}_{n}+\\\\frac{1}{\\\\sqrt{N+1}}~\\\\upsilon _{n+1}\\\\\\\\&&\\\\\\\\p_{n+1}&=&\\\\displaystyle A^2~\\\\widehat{p}_{n}+R+\\\\frac{1}{\\\\sqrt{N}}~\\\\nu _{n+1}.\\\\end{array}\\\\right.$ The proof of the above theorem is provided in section REF .\",\n \"Merging the updating and prediction steps we check the following corollary.\",\n \"Corollary 3.2 The sample variance $p_n$ of the EnKF satisfies the stochastic Riccati rational difference equation given by the formula $p_{n+1}=\\\\phi (p_n)+\\\\frac{1}{\\\\sqrt{N}}~\\\\delta _{n+1}\\\\quad \\\\mbox{with}\\\\quad \\\\delta _{n+1}:=A^2~\\\\widehat{\\\\nu }_n+\\\\nu _{n+1}.$ In the above display, $\\\\phi (p)$ is the one step mapping of the Riccati equation defined in (REF ), and the random variables $(\\\\widehat{\\\\nu }_n,\\\\nu _{n+1})$ on the right hand side of (REF ) are the local fluctuations defined in (REF ).\",\n \"Let $ {\\\\cal R }_n:=\\\\sigma (p_k,~0\\\\le k\\\\le n)\\\\vee {\\\\cal Y }_{n}$ be the $\\\\sigma $ -field generated by the stochastic Riccati equation $p_k$ and the observation sequence up to the time horizon $k\\\\le n$ and set $\\\\widehat{m}_{n}^r:=\\\\mathbb {E}(\\\\widehat{m}_{n}~|~ {\\\\cal R }_{n})\\\\quad \\\\mbox{\\\\rm and}\\\\quad m_{n+1}^r:= \\\\mathbb {E}(m_{n+1}~|~ {\\\\cal R }_{n}).$ Using this notation, the next corollary is a direct consequence of Theorem REF .\",\n \"Corollary 3.3 Given the sample variances and the sequence of observations, for any $n \\\\ge 0$ , the updating-prediction steps of the sample mean of the EnKF satisfy the quenched Kalman-filter equations $\\\\widehat{m}_{n}^r =m_n^r+ g_n(Y_n-C\\\\,m_n^r)\\\\quad \\\\mbox{and}\\\\quad m_{n+1}^r=\\\\displaystyle A~\\\\widehat{m}_{n}^r,$ with the initial condition $m_{0}^r=\\\\widehat{X}^-_0$ .\",\n \"The next corollary provides an alternative interpretation of the updating-prediction steps in terms of a two-steps Markov chain involving non-central $\\\\chi $ -square random variables.\",\n \"Corollary 3.4 The updating-prediction steps of the stochastic Riccati difference equations stated in Theorem REF take the following form $\\\\widehat{p}_n=\\\\left(\\\\frac{p_n}{1+Sp_n}\\\\right)^2~\\\\frac{S}{N}~\\\\widehat{\\\\chi }^{(n,2)}_{N,N/(Sp_n)}\\\\quad \\\\mbox{and}\\\\quad p_{n+1}=\\\\frac{R}{N}~\\\\chi ^{(n+1,2)}_{N,N(A^2/R)\\\\widehat{p}_{n}},$ with the initial condition $p_0=\\\\frac{P_0}{N}~\\\\chi ^{(0,2)}_{N,0}.$ In the above display, $\\\\chi ^{(n,2)}_{N,x}$ and $\\\\widehat{\\\\chi }^{(n,2)}_{N,x}$ stand for a collection of independent non-central $\\\\chi $ -square random variables with $N$ degrees of freedom and non-centrality parameter $x$ , which are independent of the local perturbation sequence $(\\\\upsilon _{k+1},\\\\widehat{\\\\upsilon }_k)$ of the EnKF filter defined in Theorem REF .\",\n \"The proof of the above corollary is a direct consequence of the decomposition (REF ), thus it is left as an exercise for the reader.\"\n ],\n [\n \"Stability theorems\",\n \"We denote by $\\\\phi ^n=\\\\phi ^{n-1}\\\\circ \\\\phi $ the composition evolution semigroup of the Riccati difference equation (REF ).\",\n \"We also let $\\\\widehat{X}_n(x,p)$ be the solution of the Kalman filter associated with the initial values $(\\\\widehat{X}_0,P_0)=(x,p)$ .\",\n \"The next theorem summarises the main stability properties of the Kalman filter and the Riccati difference equations.\",\n \"Theorem 3.5 The Riccati equation (REF ) has an unique positive fixed point $P_{\\\\infty }=\\\\phi (P_{\\\\infty })>0$ and for any $n\\\\ge 0$ and any $p=(p_1,p_2)\\\\in \\\\mathbb {R}_+^2$ , we have the exponential contraction estimates $\\\\vert \\\\phi ^n(p_1)-\\\\phi ^n(p_2)\\\\vert \\\\le \\\\iota _1~\\\\left(1-\\\\epsilon _1\\\\right)^n~\\\\vert p_1-p_2\\\\vert .$ In addition, there exists some integer $k(p_1)\\\\ge 1$ such that for any $x=(x_1,x_2)\\\\in \\\\mathbb {R}^2$ we have $\\\\mathbb {E}\\\\left(\\\\left(\\\\widehat{X}_n(x_1,p_1)-\\\\widehat{X}_n(x_2,p_2)\\\\right)^2\\\\right)^{1/2}\\\\le ~\\\\iota (x,p)~(1-\\\\epsilon _2)^{n-k(p_1)}~\\\\left(\\\\vert p_1-p_2\\\\vert +\\\\vert x_1-x_2\\\\vert \\\\right).$ The proof of the above theorem is provided in section REF .\",\n \"Let $ {\\\\cal P }(p, dq)$ denote the Markov transitions associated with the stochastic Riccati Markov chain $p_n$ discussed in (REF ).\",\n \"Given some non-negative function $ {\\\\cal U }$ on $\\\\mathbb {R}_+ := [0, \\\\infty [$ we define the $ {\\\\cal U }$ -norm of some function $f: \\\\mathbb {R}_+ \\\\rightarrow \\\\mathbb {R}$ and some locally finite signed measure $\\\\mu (dp)$ on $\\\\mathbb {R}_+$ by $\\\\Vert f\\\\Vert _{\\\\tiny {\\\\cal U }}:=\\\\sup _{p\\\\ge 0}\\\\left|\\\\frac{f(p)}{ {\\\\cal U }(p)+1/2}\\\\right|\\\\quad \\\\mbox{\\\\rm and}\\\\quad \\\\Vert \\\\mu \\\\Vert _{\\\\tiny {\\\\cal U }}:=\\\\sup {\\\\lbrace \\\\vert \\\\mu (f)\\\\vert ~:~f~\\\\mbox{\\\\rm s.t.\",\n \"}~\\\\Vert f\\\\Vert _{\\\\tiny {\\\\cal U }}\\\\le 1\\\\rbrace }.$ The $ {\\\\cal U }$ -Dobrushin ergodic coefficient of $ {\\\\cal P }$ is defined by $\\\\beta _{ {\\\\cal U }}( {\\\\cal P }) = \\\\sup _{(p,q) \\\\in \\\\mathbb {R}_+^2}\\\\frac{\\\\Vert {\\\\cal P }(p, \\\\cdot ) - {\\\\cal P }(q, \\\\cdot ) \\\\Vert _{ {\\\\cal U }}}{1 + {\\\\cal U }(p) + {\\\\cal U }(q)}.$ The next theorem concerns the stability properties of the stochastic Riccati rational difference equation presented in corollary REF .\",\n \"Theorem 3.6 There exists an unique invariant measure $\\\\pi $ such that $\\\\pi = \\\\pi {\\\\cal P }$ and some Lyapunov function of the form $ {\\\\cal U }(p)=u p+v$ such that $\\\\beta _{ {\\\\cal U }}( {\\\\cal P }) < 1$ for some $u,v>0$ .\",\n \"In addition, for any probability measures $\\\\mu _1$ and $\\\\mu _2$ on $\\\\mathbb {R}_+$ and for any $n\\\\ge 1$ we have $\\\\Vert \\\\mu _1 {\\\\cal P }^n - \\\\mu _2 {\\\\cal P }^n\\\\Vert _{ {\\\\cal U }} \\\\le \\\\beta _{ {\\\\cal U }}( {\\\\cal P })^n\\\\Vert \\\\mu _1 - \\\\mu _2\\\\Vert _{ {\\\\cal U }}.$ Moreover, for any function $f$ such that $|f(p)| \\\\le 1/2 + {\\\\cal U }(p)$ , for any $p \\\\in \\\\mathbb {R}_+$ , we have $| {\\\\cal P }^n(f)(p) - \\\\pi (f)| \\\\le \\\\beta _{ {\\\\cal U }}( {\\\\cal P })^n~(1 + {\\\\cal U }(p) + \\\\pi ( {\\\\cal U })).$ The above theorem comes from the fact that the Riccati map $\\\\phi (p)$ is a uniformly bounded function (cf.\",\n \"for instance (REF ) and (REF )).\",\n \"In addition, using (REF ) or (REF ) we readily check that $ {\\\\cal P }(p,dq)$ has a continuous density $k(p,q)>0$ with respect to the variables $(p,q)$ and the Lebesgue measure $dq$ on $\\\\mathbb {R}_+$ .\",\n \"It is rather well-known that these two properties ensure the existence of a Lyapunov function of the form $ {\\\\cal U }(u,v)=u p+v$ such that $\\\\beta _{ {\\\\cal U }}( {\\\\cal P }) < 1$ .\",\n \"For the convenience of the reader a detailed proof of the above theorem is provided in the appendix.\",\n \"The above theorem is an extension of the stability theorem for stochastic Riccati diffusions presented in the article [22] (see also section 2.2 and Theorem 2.4 in [18] for the multivariate case) to the stochastic Riccati rational equations presented in Corollary REF .\",\n \"We now turn to quantifying the stability properties of the EnKF sample means.\",\n \"Since the one-step optimal predictor $\\\\widehat{X}_n^-$ is a minimal variance estimate of $X_n$ we already know that $M_n:=(m_n-X_n)\\\\Longrightarrow P_n= \\\\mathbb {E}((\\\\widehat{X}_n^--X_n)^2)\\\\le \\\\mathbb {E}(M_n^2).$ Thus for small sample sizes we expect $\\\\mathbb {E}((m_n-X_n)^2)$ to be much larger than $P_n$ .\",\n \"To find some useful quantitative estimate, observe that $\\\\displaystyle M_{n+1}=\\\\displaystyle \\\\frac{A}{1+Sp_n}~M_n+\\\\Upsilon _{n+1},$ with the conditionally centered Gaussian random variables $\\\\Upsilon _{n+1}:=Ag_nDV_n-BW_{n+1}+\\\\frac{1}{\\\\sqrt{N+1}}~\\\\left(A~\\\\widehat{\\\\upsilon }_n+\\\\upsilon _{n+1}\\\\right).$ The decomposition (REF ) shows that the global stability of the stochastic process $M_n$ depends on the long-time behavior of the random products defined for any $0\\\\le l\\\\le n$ by the formula $ {\\\\cal E }_{l,n}:=\\\\prod _{l\\\\le k\\\\le n}\\\\frac{A}{1+Sp_k}.$ For stable signal drifts, i.e.\",\n \"when $\\\\vert A\\\\vert <1$ , the exponential decays of these random products is immediate.\",\n \"In the unstable case, that is when $\\\\vert A\\\\vert >1$ , none of the uniform estimates stated later in Theorem REF appear to be useful to quantify directly these random products.\",\n \"Theorem 3.7 For any $k\\\\ge 1$ there exist some parameters $N_k\\\\ge 1$ and $\\\\epsilon _k\\\\in ]0,1]$ such that for any $N\\\\ge N_k$ and any time horizon $n\\\\ge l\\\\ge 1$ we have the uniform almost sure exponential decays and the time-uniform estimates $\\\\mathbb {E}\\\\left(\\\\left| {\\\\cal E }_{l,n}\\\\right|^k ~|~p_{l-1}\\\\right)\\\\le \\\\iota _k~\\\\left(1-\\\\epsilon _k\\\\right)^{n-l}\\\\quad \\\\mbox{and}\\\\quad \\\\mathbb {E}\\\\left(\\\\vert M_{n}\\\\vert ^k\\\\right)\\\\le \\\\iota _k.$ The proof of this theorem is provided at the end of section REF .\",\n \"Let $ {\\\\cal M }((p,x), d(q,z))$ denote the Markov transitions associated with the coupled Markov chain $(p_n,M_n)$ defined by formulae (REF ) and (REF ).\",\n \"The $ {\\\\cal V }$ -norms and the $ {\\\\cal V }$ -Dobrushin ergodic coefficient $\\\\beta _{ {\\\\cal V }}( {\\\\cal M })$ of $ {\\\\cal M }$ associated with some non-negative function $ {\\\\cal V }$ on $\\\\mathbb {R}_+\\\\times \\\\mathbb {R}$ are defined as in (REF ) and (REF ) by replacing the state space $\\\\mathbb {R}_+$ by $\\\\mathbb {R}_+\\\\times \\\\mathbb {R}$ and the function $ {\\\\cal U }$ by $ {\\\\cal V }$ .\",\n \"For any $\\\\epsilon \\\\in ]0,1]$ there exists some time horizon $k\\\\ge 1$ such that $ {\\\\cal V }(p,x)= p+\\\\vert x\\\\vert \\\\Longrightarrow {\\\\cal M }^k( {\\\\cal V })(p,x)\\\\le \\\\epsilon ~ {\\\\cal V }(p,x)+\\\\iota .$ A proof of the above Lyapunov inequality is provided in section REF .\",\n \"Arguing as in the proof of Theorem REF we readily prove the following theorem.\",\n \"Theorem 3.8 There exists a unique invariant measure $\\\\mu $ such that $\\\\mu = \\\\mu {\\\\cal M }$ , a Lyapunov function of the form $ {\\\\cal V }(p,x)=u p+v \\\\vert x\\\\vert +w$ for some $u,v,w>0$ , and some integer $k\\\\ge 1$ such that $\\\\beta _{ {\\\\cal V }}( {\\\\cal M }^k) < 1$ .\",\n \"In addition, for any probability measures $\\\\eta _1$ and $\\\\eta _2$ on $\\\\mathbb {R}_+\\\\times \\\\mathbb {R}$ and any $n\\\\ge 1$ we have $\\\\Vert \\\\eta _1 {\\\\cal M }^{nk} - \\\\eta _2 {\\\\cal M }^{nk}\\\\Vert _{ {\\\\cal V }} \\\\le \\\\beta _{ {\\\\cal V }}( {\\\\cal M }^k)^n\\\\Vert \\\\eta _1 - \\\\eta _2\\\\Vert _{ {\\\\cal V }}.$ Moreover, for any function $f$ on $\\\\mathbb {R}_+\\\\times \\\\mathbb {R}$ such that $|f(p,x)| \\\\le 1/2 + {\\\\cal V }(p,x)$ , for any $(p,x) \\\\in \\\\mathbb {R}_+\\\\times \\\\mathbb {R}$ , we have $| {\\\\cal M }^{nk}(f)(p,x) - \\\\mu (f)| \\\\le \\\\beta _{ {\\\\cal V }}( {\\\\cal M }^k)^n~(1 + {\\\\cal V }(p,x) + \\\\mu ( {\\\\cal V })).$\"\n ],\n [\n \"Uniform mean-error estimates\",\n \"The next theorem provides uniform mean-error and bias estimates for the particle gain defined in (REF ) and the stochastic Riccati equations presented in Theorem REF .\",\n \"Theorem 3.9 For any $k\\\\ge 1$ and any $N\\\\ge 1$ and $n\\\\ge 0$ we have the time uniform estimates $\\\\mathbb {E}\\\\left(\\\\vert p_n-P_n\\\\vert ^k\\\\right)^{1/k}\\\\vee \\\\mathbb {E}\\\\left(\\\\vert \\\\widehat{p}_n-\\\\widehat{P}_n\\\\vert ^k\\\\right)^{1/k}\\\\vee \\\\mathbb {E}\\\\left(\\\\vert g_n-G_n\\\\vert ^k\\\\right)^{1/k} \\\\le \\\\iota _k~(1\\\\vee P_0) /\\\\sqrt{N}.$ In addition, we have the uniform bias estimate $0\\\\le P_n-\\\\mathbb {E}(p_n)\\\\le \\\\frac{\\\\iota _1}{N}~\\\\left[1\\\\vee P_0\\\\right]^{2}\\\\quad \\\\mbox{and}\\\\quad 0\\\\le G_n-\\\\mathbb {E}(g_n)\\\\le \\\\frac{\\\\iota _2}{N}~\\\\left[1\\\\vee P_0\\\\right]^{2}.$ The same formula on the l.h.s.\",\n \"of the above display holds by replacing $(p_n,P_n)$ by $(\\\\widehat{p}_n,\\\\widehat{P}_n)$ .\",\n \"The proof of the uniform mean error estimates stated in the above theorem is provided in section REF .\",\n \"The proof of the bias estimates is provided in the end of section REF .\",\n \"Now we turn to quantifying the fluctuations of the sample means around the optimal filter.\",\n \"We set $\\\\widehat{m}_{n}^o:=\\\\mathbb {E}(\\\\widehat{m}_{n}~|~ {\\\\cal Y }_{n})\\\\quad \\\\mbox{\\\\rm and}\\\\quad m_{n+1}^o:= \\\\mathbb {E}(m_{n+1}~|~ {\\\\cal Y }_{n}).$ The next theorem provides uniform mean-error and bias estimates for the sample means of the EnKF filter defined in (REF ).\",\n \"Theorem 3.10 For any $k\\\\ge 1$ there exists some parameter $N_k\\\\ge 1$ such that for any $N\\\\ge N_k$ and $n\\\\ge 0$ we have the time-uniform estimates $\\\\mathbb {E}\\\\left(\\\\vert m_{n}-\\\\widehat{X}^-_{n}\\\\vert ^k\\\\right)^{1/k}\\\\vee \\\\mathbb {E}\\\\left(\\\\vert \\\\widehat{m}_{n}-\\\\widehat{X}_{n}\\\\vert ^k\\\\right)^{1/k}\\\\le {\\\\iota _k}/{\\\\sqrt{N}}.$ In addition, we have the time-uniform bias estimates $\\\\mathbb {E}\\\\left(\\\\vert m_{n}^o-\\\\widehat{X}^-_{n}\\\\vert ^k\\\\right)^{1/k}\\\\vee \\\\mathbb {E}\\\\left(\\\\vert \\\\widehat{m}^o_{n}-\\\\widehat{X}_{n}\\\\vert ^k\\\\right)^{1/k}\\\\le \\\\iota _k~(1\\\\vee P_0)^2/N.$ The proof of the mean error estimates (REF ) is provided in the beginning of section REF .\",\n \"The uniform bias estimates (REF ) are proved in the end of section REF .\"\n ],\n [\n \"Central limit theorems\",\n \"In the further development of this section, $Z^i_k$ and $\\\\widehat{Z}^i_k$ stand for a sequence of independent, centered Gaussian random variables with unit variance indexed by $i\\\\ge 1$ , $k\\\\ge 0$ .\",\n \"We also consider the variables $(\\\\mathbb {U}_0,\\\\mathbb {V}_0):=(\\\\sqrt{P_0}~Z^1_0,\\\\sqrt{2}~P_0~Z^2_0)$ and the sequence of the Gaussian random variables $\\\\left\\\\lbrace \\\\begin{array}{rcl}\\\\widehat{\\\\mathbb {U}}_n&:=&\\\\displaystyle -G_nD~\\\\widehat{Z}^1_n\\\\\\\\&&\\\\\\\\\\\\widehat{\\\\mathbb {V}}_n&:=&\\\\displaystyle -2G_nD(1-G_nC)~\\\\sqrt{P_n}~\\\\widehat{Z}^2_n+\\\\sqrt{2}~G_n^2D^2~\\\\widehat{Z}^3_n\\\\end{array}\\\\right.\\\\left\\\\lbrace \\\\begin{array}{rcl}\\\\mathbb {U}_{n+1}&:=&\\\\displaystyle B~Z^1_{n+1}\\\\\\\\&&\\\\\\\\\\\\mathbb {V}_{n+1}&:=&\\\\displaystyle 2AB~\\\\sqrt{\\\\widehat{P}_n}~Z^{2}_{n+1}+\\\\sqrt{2}~R~Z^3_{n+1}.\\\\end{array}\\\\right.$ Observe that the above sequence of independent and centered Gaussian variables are defined in a similar manner to the local stochastic perturbations (REF ) by replacing the sample variances by their limiting values and the random vectors $(\\\\widehat{\\\\Delta }_n,\\\\Delta _{n+1})$ by the Gaussian random variables $(\\\\widehat{Z}_n,Z_{n+1})$ .\",\n \"Theorem 3.11 For any time horizon $n\\\\ge 0$ we have the weak convergence $(\\\\upsilon _0,\\\\nu _0,(\\\\widehat{\\\\upsilon }_{k},\\\\widehat{\\\\nu }_{k},\\\\upsilon _{k+1},\\\\nu _{k+1})_{0\\\\le k\\\\le n})\\\\longrightarrow _{N\\\\rightarrow \\\\infty }~(\\\\mathbb {U}_0,\\\\mathbb {V}_0,(\\\\widehat{\\\\mathbb {U}}_{k},\\\\widehat{\\\\mathbb {V}}_{k},\\\\mathbb {U}_{k+1},\\\\mathbb {V}_{k+1})_{0\\\\le k\\\\le n}).$ The proof of the above theorem is provided in section REF .\",\n \"Applying the continuous mapping theorem we obtain the following corollary.\",\n \"Corollary 3.12 For any time horizon $n\\\\ge 0$ we have the weak convergence $(\\\\nu _0,(\\\\delta _k)_{1\\\\le k\\\\le n})\\\\longrightarrow _{N\\\\rightarrow \\\\infty }~(\\\\mathbb {Z}_0,(\\\\mathbb {Z}_k)_{1\\\\le k\\\\le n}),$ with the independent Gaussian random variables $\\\\mathbb {Z}_n$ defined by $\\\\mathbb {Z}_{n+1}:=A^2~\\\\widehat{\\\\mathbb {V}}_{n}+\\\\mathbb {V}_{n+1}\\\\quad \\\\mbox{and}\\\\quad \\\\mathbb {Z}_{0}=\\\\mathbb {V}_0.$ In the above display, $\\\\nu _0$ and $\\\\delta _k$ are the local fluctuations of the stochastic Riccati equation discussed in (REF ).\",\n \"We now consider the collection of stochastic processes $(\\\\widehat{\\\\mathbb {Q}}_n^N,\\\\mathbb {Q}^N_{n+1})$ indexed by $N\\\\ge 1$ defined for any $n\\\\ge 0$ by the updating-prediction synthetic diagram $\\\\mathbb {Q}^N_{n}:=\\\\sqrt{N}(p_{n}-P_{n})\\\\longrightarrow \\\\widehat{\\\\mathbb {Q}}_n^N:=\\\\sqrt{N}(\\\\widehat{p}_n-\\\\widehat{P}_n)\\\\longrightarrow \\\\mathbb {Q}^N_{n+1}.$ Theorem 3.13 The stochastic processes $(\\\\widehat{\\\\mathbb {Q}}_n^N,\\\\mathbb {Q}^N_{n+1})$ converge in law in the sense of convergence of finite dimensional distributions, and as the number of particles $N\\\\rightarrow \\\\infty $ , to a sequence of centered stochastic processes $(\\\\widehat{\\\\mathbb {Q}}_n,\\\\mathbb {Q}_{n+1})$ with initial condition $\\\\mathbb {Q}_0 = \\\\mathbb {V}_0$ and updating-prediction transitions given by $\\\\left\\\\lbrace \\\\begin{array}{rcl}\\\\widehat{\\\\mathbb {Q}}_n&=&(1-G_nC)^{2}~\\\\mathbb {Q}_n+\\\\widehat{\\\\mathbb {V}}_n\\\\\\\\&&\\\\\\\\\\\\mathbb {Q}_{n+1}&=&A^2\\\\,\\\\widehat{\\\\mathbb {Q}}_n+\\\\mathbb {V}_{n+1}.\\\\end{array}\\\\right.$ The proof of the above theorem is provided in section REF .\",\n \"We consider the collection of stochastic processes $(\\\\widehat{\\\\mathbb {X}}_n^N,\\\\mathbb {X}^N_{n+1})$ indexed by $N\\\\ge 1$ and defined for any $n\\\\ge 0$ by the updating-prediction synthetic diagram $\\\\mathbb {X}^N_{n}:=\\\\sqrt{N}(m_{n}-\\\\widehat{X}^-_{n})\\\\longrightarrow \\\\widehat{\\\\mathbb {X}}_n^N:=\\\\sqrt{N}(\\\\widehat{m}_n-\\\\widehat{X}_n)\\\\longrightarrow \\\\mathbb {X}^N_{n+1}.$ Theorem 3.14 The stochastic processes $(\\\\widehat{\\\\mathbb {X}}_n^N,\\\\mathbb {X}^N_{n+1})$ converge in law in the sense of convergence of finite dimensional distributions, as the number of particles $N\\\\rightarrow \\\\infty $ , to a sequence of centered stochastic processes $(\\\\widehat{\\\\mathbb {X}}_n,\\\\mathbb {X}_{n+1})$ with initial condition $\\\\mathbb {X}_0 = \\\\mathbb {U}_0$ and updating-prediction transitions given by $\\\\left\\\\lbrace \\\\begin{array}{rcl}\\\\widehat{\\\\mathbb {X}}_{n}&=&(1-G_nC)\\\\,\\\\mathbb {X}_n+\\\\mathbb {G}_n\\\\,(Y_n-C\\\\widehat{X}^-_n)+\\\\widehat{\\\\mathbb {U}}_n\\\\\\\\&&\\\\\\\\\\\\mathbb {X}_{n+1}&=&A\\\\,\\\\widehat{\\\\mathbb {X}}_{n}+\\\\mathbb {U}_{n+1}.\\\\end{array}\\\\right.$\"\n ],\n [\n \"Some comments and comparisons\",\n \"As shown in section REF , the EnKF is a rather simple numerical filtering-type technique defined by an ensemble of particles mimicking the evolution of well-known Kalman filter equations, replacing `the true' covariances by the ensemble sample-covariances.\",\n \"Besides the fact that the EnKF is only consistent for linear Gaussian filtering problems (cf.\",\n \"[48]), these popular particle methodologies are applied to complex nonlinear and high dimensional filtering problems arising in fluid mechanics [14], [74], [78], weather forecasting [3], [4], [30], [49], geosciences and reservoir simulation [47], [72], [82], [83], [94], and many other data assimilation disciplines.\",\n \"To reduce the computational cost, the ensemble sample size is generally chosen to be several orders of magnitude below the very large effective dimension of the underlying signal.\",\n \"To prevent the ensemble covariance degeneracy in time, small sample EnKF requires one to combine several customisation techniques such as adaptive covariance/gain inflation and related localisation methodologies [12], [13], [21], [93].\",\n \"The first rigorous analysis of the bias and the consistency of these additional levels of approximations for linear Gaussian models can be found in [21], [23].\",\n \"As shown in [38], for nonlinear filtering problems, the EnKF can be interpreted as particle-type extended Kalman filter.\",\n \"As any observer-type algorithm, the well-posedness is not connected to any kind of distance to the optimal filter for any small or large sample sizes, but in terms of non-degeneracy of the observer with respect to time.\",\n \"As underlined by Majda and Tong in [69] “Why EnKF works well with a small ensemble has remained a complete mystery.\",\n \"Practitioners often attribute the EnKF success to the existence of a low effective filtering dimension.\",\n \"But the definition of the effective filtering dimension has remained elusive, as its associated subspace often evolves with the dynamics”.\",\n \"In the article [69], the authors suggest several theoretical guidelines to choose judiciously the ensemble sample size using covariance inflation and spectral projection techniques on the unstable directions of the signal.\",\n \"From a purely mathematical perspective, the first important question is clearly to find conditions that ensure that the EnKF is well-founded and consistent, in the sense that it converges to the optimal filter as the ensemble sample size, or equivalently the computational power, tends to infinity (cf.\",\n \"for instance [48], [70] in discrete time settings and [37], [38] for continuous time EnKF models).\",\n \"Of course, the consistency property and asymptotic analysis of any statistical Monte Carlo estimate such as the sample means or the sample covariances delivered by EnKF is reassuring but it only gives a partial answer to the performance of these ensemble filters when used with small sample sizes.\",\n \"The non-asymptotic analysis for finite sample sizes is clearly a more delicate subject as the continuous or the discrete generation EnKF belong to unconventional classes of nonlinear Markov processes and mean-field particle processes interacting through the sample covariance of the system.\",\n \"Here the interaction covariance function of the EnKF particle filter is not a bounded nor a Lipschitz function as is it is commonly assumed in traditional nonlinear and interacting diffusion theory.\",\n \"As a result, none of the stochastic tools developed in this field, including the rather elementary variational methodology for nonlinear diffusions developed in [9], [10] nor the more recent powerful differential calculus developed in [31], can be used to analyse this class of continuous-time models equipped with a nonlinear quadratic-type interacting function.\",\n \"Here, the sample covariance matrices satisfy a rather sophisticated quadratic-type stochastic Riccati matrix diffusion equation, which requires one to develop new stochastic analysis tools [18].\",\n \"For a more thorough discussion on mean-field type particle systems and a probabilistic description of genetic type particle filters and Ensemble Kalman type particle filters within this framework we refer to the review article [23], the books [35], [37], [39] and references therein.\",\n \"The analysis of the discrete generation EnKF discussed in the present article is more delicate and requires the development of new stochastic methodologies.\",\n \"For instance, in discrete time settings, the EnKF particle system evolves as the Kalman filter (REF ) in terms of a two-step prediction-updating interacting Markov chain (REF ).\",\n \"In contrast with the Gaussian nature of the continuous time EnKF and stochastic Riccati diffusions discussed above, Theorem REF presented in this article shows that these two transitions combine both Gaussian and $\\\\chi $ -square type perturbations.\",\n \"To the best of our knowledge, the analysis of stochastic Riccati rational difference equations, such as those discussed in Corollary REF , has not been covered in the literature.\",\n \"Corollary REF also provides an alternative description of the sample variances in terms of a Markov chain with non-central $\\\\chi $ -square nonlinear fluctuations.\",\n \"Again, to the best of our knowledge, these stochastic perturbation theorems and the stability analysis of the analysis of stochastic Riccati rational difference equations are the first of this type in the applied probability literature.\",\n \"In a study by Tong, Majda and Kelly [93], the authors analyse the long-time behaviour and ergodicity of discrete generation EnKF using Foster-Lyapunov techniques ensuring that the filter is asymptotically stable with respect to any erroneous initial condition as soon as the signal is stable.\",\n \"These important properties ensure that the EnKF has a single invariant measure and initialisation errors of the EnKF will dissipate with respect to the time parameter.\",\n \"Nevertheless, the only ergodicity of the signal and thus the ensemble process does not give any useful information on the convergence and the accuracy of the EnKF towards the optimal filter as the number of samples tends to infinity, nor does it allow one to quantify the fluctuations around the optimal filter for a given small or even very large ensemble sample size.\",\n \"On the other hand, effective unstable directions are connected to unstable-transient signals that may grow exponentially fast.\",\n \"In this context, for obvious reasons, any well tracking EnKF needs to be an unstable-transient particle process.\",\n \"As a consequence, we cannot expect to obtain any type of filter ergodicity properties, nor any time-uniform estimates for the raw moments of the EnKF filter.\",\n \"From our point of view, the terminology \\\"effective filtering dimensions\\\" discussed above should be understood as the unstable directions of the signal.\",\n \"For time varying and multivariate linear-Gaussian filtering problems, these effective dimensions are rarely known as they are partially observed and they may change in time.\",\n \"For a more detailed discussion on the stability properties of systems of time varying linear stochastic differential equations with random coefficients we refer to [19].\",\n \"To avoid the time degeneracy of any type of filtering approximation, a crucial question in practice is to check wether or not the convergence is uniform with respect to the time parameter.\",\n \"Equivalently, we need to obtain estimates of the error between the optimal filter and its approximation that are uniform with respect to any time horizon even if the signal and thus any well tracking approximating filter are both unstable and transient.\",\n \"We emphasise that in this setting, any good approximating filter will diverge.\",\n \"In this context, the filtering step is part of a control loop that allows one to track the transient signal at all times.\",\n \"The main advantage of one-dimensional filtering problems over multivariate problems with unstable signals is that they have a single effective dimension.\",\n \"In this context, the EnKF is also unstable and transient but the theorems REF and REF presented in this article show that the EnKF tracks and converges to the optimal filter uniformly in time.\",\n \"These theorems also ensure that the precision error to the optimal filter can be quantified for any given finite sample size, and it will not degrade with respect to the time horizon.\",\n \"Theorem REF also indicates that the sample variances satisfy an autonomous stochastic Riccati-type evolution equation, independent of the sequence of observations.\",\n \"From a different angle, theorem REF also shows that the sample variances of the EnKF behaves as a Markov chain that converge exponentially fast as the time horizon tends to infinity to a single invariant measure, even if the signal of the filtering problem is transient.\",\n \"To the best of our knowledge, this stability theorem and the non-asymptotic theorems discussed above are the first results of this type in the literature on particle filter theory, including the literature of discrete generation EnKF.\",\n \"Next we provide a more detailed discussion on the stability of particle filters and their particle approximations, with precise reference pointers.\",\n \"We also suggest an avenue of open research problems.\",\n \"Most of the stability properties of genetic-type particle filters [34], [36], including continuous time interacting jump Feynman-Kac particle systems [8], are only valid for stable and ergodic signals.\",\n \"It is clearly out the scope of this article to review in detail all contributions related to the stability of genetic type particle filters, thus we refer to the books [34], [35], [36] and references therein.\",\n \"The analysis of these continuous-time or discrete generation genetic type processes for unstable signals remains an important and open research question.\",\n \"Continuous time EnKF filters for stable signals are also discussed in [16], [38], including in the context of the extended Kalman filter.\",\n \"The analysis of unstable and possibly transient signals is more recent, starting at the end of the 2010s in the context of continuous time filtering problems [20], [18], [22], see also [23] for a comprehensive review of this subject.\",\n \"The articles [18], [20] analyse the stability and the fluctuation properties of the sample covariance matrices of continuous time EnKF filters for multivariate models under natural and minimal observability and controllability conditions.\",\n \"To better understand the difficulty in handling unstable effective signal directions, we recall that under natural observability and controllability conditions the Kalman filtering theory developed by Kalman and Bucy in the end of the 1960s ensures that the optimal filter is able to track any possibly unstable signals uniformly with respect to the time horizon.\",\n \"To answer any stability question related to EnKF particle filters or genetic type particle filters, we first need to extend the theory of Kalman and Bucy to this class of approximating particle filters.\",\n \"This research project remains open for genetic type filters.\",\n \"The present article provide a partial answer for discrete generation EnKF particle filters.\",\n \"To better understand the difficulty in handling and extending the stability theory of Kalman and Bucy to particle filters, it is worth noting a brief of history.\",\n \"The stability properties of continuous time or discrete generation Kalman filters and their associated Riccati equations for multidimensional filtering problems are usually conducted under some observability/controllability-type conditions using sophisticated and tedious matrix manipulations.\",\n \"The first stability properties for continuous time models go back to the beginning of the 1960s with the celebrated work of Kalman and Bucy [57], with related prior work by Kalman [53], [54], [56] and later work by Bucy [27].\",\n \"In [2], the stability of this filter was analysed under a relaxed controllability condition.\",\n \"The first stability properties for discrete time models go back to to the end of the 1960s with the work of Deyst and Price [41], which was incorporated in the seminal lecture book of Jazwinski [51].\",\n \"However, as noted in [50], there was in both cases a crucial and commonly made error in the proof which invalidated the results, see [71] and the more recent articles [16], [79] for a more detailed discussion and references on these issues.\",\n \"This error was repeated in numerous subsequent works.\",\n \"A first correction [28] was noted in a reply to [50]; see Bucy's reply [29] and a separate reply by Kalman [55].\",\n \"However, a complete reworking of the result did not appear in entirety until 2001 by Delyon in [40].\",\n \"The stability of Kalman filter and Riccati equations is nowadays rather well understood.\",\n \"Nevertheless it is not always simple to find a self-contained, rigorous and easy-to-read study on these subjects.\",\n \"For a complete and self-contained analysis on the stability and convergence of Kalman-Bucy filtering in continuous time settings, we also refer to [16].\",\n \"A self-contained analysis of Riccati differential equations and discrete-time Kalman filters for one-dimensional models is provided in section REF and section REF in the present article, see also Theorem REF .\",\n \"The stability analysis of continuous time stochastic Riccati matrix diffusions is also rather well understood, see for instance [16], [18], [20], [23].\",\n \"The extension of these multivariate results in the discrete time case remains an important open question.\",\n \"Theorem REF as well as (REF ) and Theorem REF provide a complete answer to the stability of discrete generation EnKF for one dimensional filtering problems, with possibly unstable signals, yielding what seems to be the first results of this type for this class of particle filters, including both EnKF and genetic type particle filters.\"\n ],\n [\n \"Some preliminary results\",\n \"In this section we give several technical results that will be useful in the further development of the article.\",\n \"We start by considering the inverse moments of the non-central $\\\\chi $ -square random variables introduced in (REF ).\",\n \"We then provide a self-contained analysis of the Riccati difference equations defined in (REF ).\"\n ],\n [\n \"Non-central $\\\\chi $ -square moments\",\n \"In this section, we will prove the following technical result that allows one to estimate the inverse raw moments of non-central $\\\\chi $ -square random variables.\",\n \"Proposition 4.1 For any $k\\\\ge 0$ and $n>(k+1)$ we have the uniform estimate $0\\\\le \\\\mathbb {E}\\\\left(\\\\left(\\\\frac{2n}{\\\\chi ^2_{2n,2nx}}\\\\right)^{k}\\\\right)-\\\\frac{1}{\\\\left(1+x\\\\right)^{k}}\\\\le \\\\frac{k+1}{n}~k~\\\\left(1+\\\\frac{k+1}{n-(k+1)}\\\\right)^{k}.$ In addition, there exists a constant $\\\\omega _k$ such that, for any $n\\\\ge 2(k+2)$ we have $\\\\mathbb {E}\\\\left(\\\\left(\\\\frac{1+x}{\\\\frac{1}{2n}\\\\chi ^2_{2n,2nx}}-1\\\\right)^{k}\\\\right)\\\\le \\\\frac{\\\\omega _k}{n}~\\\\left(1+x\\\\right)^{k}.$ In order to prove this result, let us first consider the following Taylor series expansion, $e^{-x}=\\\\sum _{0\\\\le k\\\\le n}\\\\frac{(-x)^k}{k!}+\\\\frac{(-x)^{n+1}}{(n+1)!\",\n \"}~\\\\tau _{n+1}(x), n \\\\ge 0,$ with the collection of integral remainders given by $\\\\tau _{n+1}(x)=(n+1)~\\\\int _{0}^1~(1-t)^ne^{-tx}~dt=1-\\\\frac{x}{n+2}~\\\\tau _{n+2}(x).$ For any $k\\\\ge 1$ we denote by $\\\\tau ^{(k)}_n$ s the $k$ -th differential of the function $\\\\tau _n$ .\",\n \"In this notation, the inverse moment of a non-central $\\\\chi $ -square random variable (cf.\",\n \"section 3 in [25]) are given, for any $k\\\\ge 0$ , by the formulae $\\\\begin{array}{rcl}\\\\displaystyle \\\\mathbb {E}\\\\left(\\\\left(\\\\frac{2}{\\\\chi ^2_{2(n+1),2x}}\\\\right)^{(k+1)}\\\\right)&=&\\\\displaystyle \\\\frac{(-1)^{k}}{k!\",\n \"}~\\\\frac{1}{n-k}~\\\\tau ^{(k)}_{n-k}(x).\\\\end{array}$ Setting $k=0$ we find the formulae $\\\\frac{1}{n}~\\\\tau _{n}(x)=\\\\mathbb {E}\\\\left(\\\\frac{2}{\\\\chi ^2_{2(n+1),2x}}\\\\right)\\\\ge \\\\frac{2}{\\\\mathbb {E}(\\\\chi ^2_{2(n+1),2x})}=\\\\frac{1}{x+n+1},$ where we have used Jensen's inequality to obtain the lower bound.\",\n \"This combined with another application of Jensen's inequality implies that $\\\\begin{array}{rcl}\\\\displaystyle \\\\frac{(-1)^{k}}{k!\",\n \"}~\\\\frac{1}{n-k}~\\\\tau ^{(k)}_{n-k}(x) =\\\\mathbb {E}\\\\left(\\\\left(\\\\frac{2}{\\\\chi ^2_{2(n+1),2x}}\\\\right)^{(k+1)}\\\\right)&\\\\ge &\\\\displaystyle \\\\left(\\\\frac{1}{n}~\\\\tau _{n}(x)\\\\right)^{k+1}\\\\ge \\\\frac{1}{(x+n+1)^{k+1}}.\\\\end{array}$ On the other hand, note that for any $0\\\\le z<1$ we have $\\\\log {(1-z)}\\\\le -z\\\\Longrightarrow n\\\\log {\\\\left(1-\\\\frac{s}{m}\\\\right)}-sx\\\\le -\\\\left(x+\\\\frac{n}{m}\\\\right)~s.$ This yields the upper bound $m\\\\tau _{n}(mx) = \\\\int _0^{m}~\\\\left(1-\\\\frac{s}{m}\\\\right)^n~e^{-sx}~ds\\\\le \\\\int _0^{m}~e^{-\\\\left(x+\\\\frac{n}{m}\\\\right)s}~ds=\\\\frac{1}{x+\\\\frac{n}{m}}~\\\\left(1-e^{-\\\\left(x+\\\\frac{n}{m}\\\\right)m}\\\\right)\\\\le \\\\frac{1}{x+\\\\frac{n}{m}}.$ Thus, for any $m,n\\\\ge 1$ and any $x\\\\ge 0$ we have the estimate $\\\\frac{1}{x+(n+1)/m}\\\\le \\\\frac{m}{n}~\\\\tau _{n}(mx)\\\\le \\\\frac{1}{x+{(n-1)}/{m}}.$ More generally, for any $k,m,n\\\\ge 0$ we have $\\\\frac{(-1)^k}{n+1}~\\\\tau _{n+1}^{(k)}(mx)&=&\\\\int _0^1 t^k~(1-t)^n~e^{-mxt}~dt\\\\\\\\&=&\\\\frac{1}{m}\\\\int _0^{m}~\\\\left(\\\\frac{s}{m}\\\\right)^k~\\\\left(1-\\\\frac{s}{m}\\\\right)^n~e^{-sx}~ds\\\\\\\\&\\\\le & \\\\frac{1}{m^{k+1}}\\\\int _0^{m}~s^k~e^{-\\\\left(x+\\\\frac{n}{m}\\\\right)s}~ds\\\\le \\\\frac{1}{m^{k+1}}~\\\\frac{1}{\\\\left(x+\\\\frac{n}{m}\\\\right)}~\\\\frac{k!\",\n \"}{\\\\left(x+\\\\frac{n}{m}\\\\right)^{k}}.$ This discussion can be summarised with the following lemma.\",\n \"Lemma 4.2 For any $n> k\\\\ge 0$ , $m\\\\ge 1$ and any $x\\\\ge 0$ we have the estimate $\\\\frac{1}{\\\\left(x+\\\\frac{n+1}{m}\\\\right)^{k+1}}\\\\le \\\\frac{(-1)^{k}}{k!\",\n \"}~\\\\frac{m^{k+1}}{n-k}~\\\\tau ^{(k)}_{n-k}(mx)& \\\\le & ~\\\\frac{1}{\\\\left(x+\\\\frac{n-k-1}{m}\\\\right)^{k+1}}.$ With this lemma in hand, we are now in a position to prove Proposition REF .\",\n \"[Proof of Proposition REF ] By Lemma REF we have $\\\\frac{1}{\\\\left(x+\\\\frac{n+1}{m}\\\\right)^{k+1}}\\\\le \\\\mathbb {E}\\\\left(\\\\left(\\\\frac{2m}{\\\\chi ^2_{2(n+1),2mx}}\\\\right)^{(k+1)}\\\\right)& \\\\le & ~\\\\frac{1}{\\\\left(x+\\\\frac{n-k-1}{m}\\\\right)^{k+1}}.$ This yields the estimate $\\\\frac{1}{\\\\left(1+x\\\\right)^{k}}\\\\le \\\\mathbb {E}\\\\left(\\\\left(\\\\frac{2n}{\\\\chi ^2_{2n,2nx}}\\\\right)^{k}\\\\right)\\\\le ~\\\\frac{1}{\\\\left((1+x)-\\\\frac{k+1}{n}\\\\right)^{k}}.$ This implies that for any $n> k$ we have $0\\\\le \\\\mathbb {E}\\\\left(\\\\left(\\\\frac{2n}{\\\\chi ^2_{2n,2nx}}\\\\right)^{k}\\\\right)-\\\\frac{1}{\\\\left(1+x\\\\right)^{k}}\\\\le \\\\theta _k(x) \\\\le \\\\frac{k+1}{n}~\\\\frac{k}{\\\\left(x+\\\\frac{n-(k+1)}{n}\\\\right)^{k}\\\\left(1+x\\\\right)}$ with the function $\\\\theta _k(x):=~\\\\frac{1}{\\\\left((1+x)-\\\\frac{k+1}{n}\\\\right)^{k}}-\\\\frac{1}{\\\\left(1+x\\\\right)^{k}}.$ This ends the proof of the first assertion.\",\n \"This first estimate also yields, for any $n\\\\ge k+2$ , the rather crude estimate $0\\\\le \\\\mathbb {E}\\\\left(\\\\left(\\\\frac{1+x}{\\\\frac{1}{2n}\\\\chi ^2_{2n,2nx}}\\\\right)^{k}\\\\right)-1\\\\le \\\\frac{k}{n}~(k+1)\\\\left(1+x\\\\right)^{k}\\\\left(k+2\\\\right)^{k}\\\\le \\\\frac{1}{n}~\\\\left(1+x\\\\right)^{k}\\\\left(k+2\\\\right)^{k+2}.$ This implies that $0\\\\le \\\\mathbb {E}\\\\left(\\\\left(\\\\frac{1+x}{\\\\frac{1}{2n}\\\\chi ^2_{2n,2nx}}\\\\right)^{k}\\\\right)-1\\\\le \\\\frac{\\\\omega _k}{n}~\\\\left(1+x\\\\right)^{k}\\\\quad \\\\mbox{\\\\rm with}\\\\quad \\\\omega _k:=\\\\left(k+2\\\\right)^{k+2}.$ Using the decomposition $\\\\left(y-1\\\\right)^k=\\\\sum _{0\\\\le l\\\\le k}\\\\left(\\\\begin{array}{c}k\\\\\\\\l\\\\end{array}\\\\right)~(-1)^{l}~y^{k-l}=\\\\sum _{0\\\\le l< k}\\\\left(\\\\begin{array}{c}k\\\\\\\\l\\\\end{array}\\\\right)~(-1)^{l}~\\\\left(y^{k-l}-1\\\\right),$ we check that $\\\\mathbb {E}\\\\left(\\\\left(\\\\frac{1+x}{\\\\frac{1}{2n}\\\\chi ^2_{2n,2nx}}-1\\\\right)^{k}\\\\right)=\\\\sum _{0\\\\le l< k}\\\\left(\\\\begin{array}{c}k\\\\\\\\l\\\\end{array}\\\\right)~(-1)^{l}~\\\\underbrace{\\\\left(\\\\mathbb {E}\\\\left(\\\\left(\\\\frac{1+x}{\\\\frac{1}{2n}\\\\chi ^2_{2n,2nx}}\\\\right)^{k-l}\\\\right)-1\\\\right)}_{\\\\ge 0}.$ Thus, for any $n\\\\ge 2(2k+2)$ we have the estimate $\\\\mathbb {E}\\\\left(\\\\left(\\\\frac{1+x}{\\\\frac{1}{2n}\\\\chi ^2_{2n,2nx}}-1\\\\right)^{2k}\\\\right)&\\\\le & \\\\sum _{1\\\\le l\\\\le k}\\\\left(\\\\begin{array}{c}2k\\\\\\\\2l\\\\end{array}\\\\right)~\\\\left(\\\\mathbb {E}\\\\left(\\\\left(\\\\frac{1+x}{\\\\frac{1}{2n}\\\\chi ^2_{2n,2nx}}\\\\right)^{2l}\\\\right)-1\\\\right).$ Using (REF ), this implies that $\\\\mathbb {E}\\\\left(\\\\left(\\\\frac{1+x}{\\\\frac{1}{2n}\\\\chi ^2_{2n,2nx}}-1\\\\right)^{2k}\\\\right)\\\\le \\\\frac{1}{n}~\\\\left(1+x\\\\right)^{2k}\\\\omega ^{\\\\prime }_{2k}.$ with $\\\\omega ^{\\\\prime }_{2k}:=\\\\sum _{1\\\\le l\\\\le k}\\\\left(\\\\begin{array}{c}2k\\\\\\\\2l\\\\end{array}\\\\right)~\\\\left(2(l+1)\\\\right)^{2(l+1)}.$ In the same vein, for any $n\\\\ge 2((2k+1)+2)$ we have $\\\\mathbb {E}\\\\left(\\\\left(\\\\frac{1+x}{\\\\frac{1}{2n}\\\\chi ^2_{2n,2nx}}-1\\\\right)^{2k+1}\\\\right)&\\\\le & \\\\sum _{0\\\\le 2l< 2k+1}\\\\left(\\\\begin{array}{c}2k+1\\\\\\\\2l\\\\end{array}\\\\right)~\\\\underbrace{\\\\left(\\\\mathbb {E}\\\\left(\\\\left(\\\\frac{1+x}{\\\\frac{1}{2n}\\\\chi ^2_{2n,2nx}}\\\\right)^{2(k-l)+1}\\\\right)-1\\\\right)}_{\\\\ge 0}\\\\\\\\&\\\\le & \\\\sum _{0\\\\le l\\\\le k}\\\\left(\\\\begin{array}{c}2k+1\\\\\\\\2(k-l)\\\\end{array}\\\\right)~\\\\underbrace{\\\\left(\\\\mathbb {E}\\\\left(\\\\left(\\\\frac{1+x}{\\\\frac{1}{2n}\\\\chi ^2_{2n,2nx}}\\\\right)^{2l+1}\\\\right)-1\\\\right)}_{\\\\le \\\\frac{\\\\omega _{2l+1}}{n}~ (1+x)^{2l+1}}.$ This implies that $\\\\mathbb {E}\\\\left(\\\\left(\\\\frac{1+x}{\\\\frac{1}{2n}\\\\chi ^2_{2n,2nx}}-1\\\\right)^{2k+1}\\\\right)\\\\le \\\\frac{\\\\omega ^{\\\\prime }_{2k+1}}{n}~ (1+x)^{2k+1}$ with $\\\\omega ^{\\\\prime }_{2k+1}=\\\\sum _{0\\\\le l\\\\le k}\\\\left(\\\\begin{array}{c}2k+1\\\\\\\\2(k-l)\\\\end{array}\\\\right)~(2l+3)^{2l+3}.$ This ends the proof of the proposition.\"\n ],\n [\n \"Riccati difference equations \",\n \"We associate with some parameters $(a,b,c,d)\\\\in \\\\mathbb {R}_+^4$ satisfying $c>0$ and $ad>bc$ the Riccati difference equation on $\\\\mathbb {R}_+:=[0,\\\\infty [$ given by the recursion $P_n:=\\\\phi (P_{n-1})\\\\quad \\\\mbox{\\\\rm with}\\\\quad \\\\phi (x):=\\\\frac{a x+b}{cx+d}\\\\quad \\\\mbox{\\\\rm with}\\\\quad x\\\\ge 0.$ Observe that $\\\\partial \\\\phi (x)=\\\\frac{\\\\rho }{(cx+d)^2}> 0\\\\quad \\\\mbox{\\\\rm and}\\\\quad \\\\partial ^2 \\\\phi (x)=-\\\\frac{2\\\\rho c}{(cx+d)^3}< 0\\\\quad \\\\mbox{\\\\rm with}\\\\quad \\\\rho :=ad-bc>0.\\\\\\\\$ This shows that $\\\\phi $ is an increasing and concave function.\",\n \"Rational difference equation of the form (REF ) can be solved explicitly.\",\n \"This section provides a brief review on these equations.\",\n \"We set $(u,v,w):=(a/c,d/c,b/c)\\\\in \\\\mathbb {R}_+^4\\\\quad \\\\mbox{\\\\rm and}\\\\quad \\\\lambda =\\\\lambda _1/\\\\lambda _2\\\\in ]0,1[$ with the parameters $\\\\lambda _2-v:=\\\\frac{u-v}{2}+\\\\sqrt{\\\\left(\\\\frac{v-u}{2}\\\\right)^2+w}>0 \\\\quad \\\\mbox{\\\\rm and}\\\\quad v-\\\\lambda _1:=\\\\frac{v-u}{2}+\\\\sqrt{\\\\left(\\\\frac{v-u}{2}\\\\right)^2+w}> 0.$ Observe that $(v-\\\\lambda _1) (\\\\lambda _2-v)=w \\\\qquad \\\\mbox{\\\\rm and}\\\\qquad r=\\\\phi (r)>0\\\\Longleftrightarrow r=\\\\lambda _2-v.$ Lemma 4.3 For any $x\\\\ge 0$ and $n\\\\ge 1$ the evolution semigroup $\\\\phi ^n=\\\\phi \\\\circ \\\\phi ^{n-1}$ of the Riccati difference equation (REF ) is positive and is given by the formula $\\\\phi ^n(x)=r+(x-r)~\\\\frac{\\\\left(\\\\lambda _2-\\\\lambda _1\\\\right)~\\\\lambda ^n}{\\\\left(x+(v-\\\\lambda _1)\\\\right)~\\\\left(1-\\\\lambda ^{n}\\\\right)+(\\\\lambda _2-\\\\lambda _1)~\\\\lambda ^{n}}$ with the parameters $\\\\lambda $ and $(\\\\lambda _1,\\\\lambda _2)$ defined in (REF ) and the unique positive fixed point $r$ given in (REF ).\",\n \"In addition, for any $n\\\\ge 1$ and any $x\\\\in \\\\mathbb {R}_+$ we have the estimates ${b}/{d}\\\\le \\\\phi ^n(x)\\\\le {a}/{c}.$ For any $x,y\\\\in \\\\mathbb {R}_+$ and $n\\\\ge 0$ we have the exponential decay to equilibrium $\\\\vert \\\\phi ^n(x)-r\\\\vert \\\\le \\\\frac{\\\\lambda _2-\\\\lambda _1}{v-\\\\lambda _1}~\\\\lambda ^n~\\\\vert x-r\\\\vert \\\\quad \\\\mbox{and}\\\\quad \\\\vert \\\\phi ^n(x)-\\\\phi ^n(y)\\\\vert \\\\le \\\\vert x-y\\\\vert ~\\\\left(\\\\frac{\\\\lambda _2-\\\\lambda _1}{v-\\\\lambda _1}\\\\right)^2~\\\\lambda ^n.$ In addition, we have the first order derivative $\\\\partial \\\\phi ^n(y):=\\\\frac{(\\\\lambda _2-\\\\lambda _1)^2~\\\\lambda ^n}{\\\\big (\\\\left(y+(v-\\\\lambda _1)\\\\right)~\\\\left(1-\\\\lambda ^{n}\\\\right)+(\\\\lambda _2-\\\\lambda _1)~\\\\lambda ^{n}\\\\big )^2}=\\\\prod _{0\\\\le k0.$ This shows that the boundedness properties of Riccati type differential inequalities of the form $r_n\\\\le \\\\phi _{\\\\epsilon }(r_{n-1})$ can be deduced from those of the solution to the evolution equation $s_n=\\\\phi _{\\\\epsilon }(s_{n-1})$ .\",\n \"For instance, since $\\\\phi _{\\\\epsilon }$ is increasing we have $\\\\phi _{\\\\epsilon }(r_{n-1})\\\\le \\\\phi _{\\\\epsilon }^n(x)\\\\le x\\\\vee s^{\\\\epsilon }\\\\le x\\\\vee \\\\frac{a_{\\\\epsilon }}{c}= x\\\\vee (\\\\epsilon +\\\\frac{a}{c})\\\\quad \\\\mbox{\\\\rm with the fixed point}\\\\quad s^{\\\\epsilon }:= \\\\phi _{\\\\epsilon }(s^{\\\\epsilon })>0.$ Moreover, observe that for any $\\\\epsilon \\\\le \\\\epsilon _1$ we have $0<2c~s^{\\\\epsilon }&=&a-d+\\\\epsilon c+\\\\sqrt{\\\\left((a-d)+\\\\epsilon c\\\\right)^2+4c~(b+\\\\epsilon d)}\\\\\\\\&\\\\le &a-d+\\\\sqrt{2(a-d)^2+4bc}+2\\\\sqrt{\\\\epsilon _1c~\\\\left(d+\\\\frac{\\\\epsilon _1 c}{2}\\\\right)}+\\\\epsilon _1 c.$ In the same vein, for any $k\\\\ge 1$ we have $\\\\phi (x)^k=\\\\left(\\\\frac{a x+b}{cx+d}\\\\right)^k\\\\le \\\\phi _k(x^k)\\\\quad \\\\mbox{\\\\rm with}\\\\quad \\\\phi _k(x):=\\\\frac{a_{k}x+b_{k}}{c_kx+d_k}$ and where $(a_{k},b_{k},c_k,d_k):=(2^{k-1}a^k,2^{k-1}b,c^k,d^k)\\\\in \\\\mathbb {R}_+^4.$ Note that we have $a_{k}d_k-b_{k}c_k=2^{k-1}((ad)^k-(bc)^k)>0\\\\quad \\\\mbox{\\\\rm since}\\\\quad ad-bc>0.$ As above, this shows that the boundedness properties of Riccati type inequalities of the form $r_n\\\\le \\\\phi (r_{n-1})^k$ can be deduced from the solution of the recursion $s_n=\\\\phi _k(s_{n-1})$ .\",\n \"Indeed, since $\\\\phi _k$ is increasing we have $r_n\\\\le \\\\phi (r_{n-1})^k\\\\le \\\\phi _{k}^n(r_0) \\\\le x\\\\vee s^{k}\\\\quad \\\\mbox{\\\\rm with the fixed point}\\\\quad s^{k}:= \\\\phi _{k}(s^{k})>0.$\"\n ],\n [\n \"Stability of Kalman filters\",\n \"This section is mainly concerned with the proof of Theorem REF .\",\n \"Note that Lemma REF yields the existence of the positive fixed point $P_{\\\\infty }=\\\\phi (P_{\\\\infty })>0$ , as well as the estimate (REF ).\",\n \"The fixed point $P_\\\\infty $ can be written in closed form in terms of the parameters $(a,b,c)$ (defined in (REF )) by the formula $P_{\\\\infty }= \\\\frac{(a-1)+r}{2c}\\\\quad \\\\mbox{\\\\rm with}\\\\quad r:=\\\\sqrt{\\\\left( a-1\\\\right)^2+4bc} \\\\,\\\\frac{\\\\vert A\\\\vert -1}{S}.$ This implies that $\\\\vert A\\\\vert (1-G_\\\\infty C)=:1- \\\\epsilon _{\\\\infty }<1,$ where $G_\\\\infty $ is he gain associated with the steady state $P_{\\\\infty }$ of the Riccati equation and is given by $G_{\\\\infty }=CP_{\\\\infty }/(C^2P_{\\\\infty }+D^2)\\\\Longleftrightarrow 1-G_{\\\\infty }C={1}/{(1+SP_{\\\\infty })}.$ On the other hand, using (REF ), for any $p\\\\ge 0$ there exists some integer $k(p)\\\\ge 1$ such that for any $k> k(p)$ , $\\\\frac{1+SP_{\\\\infty }}{1+S\\\\phi ^k(p)}=1+\\\\frac{S}{1+S\\\\phi ^k(p)}~(P_{\\\\infty }-\\\\phi ^k(p))\\\\le 1+S~\\\\kappa _1~\\\\left(1-\\\\epsilon _1\\\\right)^k~\\\\vert p-P_{\\\\infty }\\\\vert \\\\le 1+\\\\epsilon _{\\\\infty }.$ with the parameter $\\\\epsilon _{\\\\infty }$ defined in (REF ).\",\n \"This yields the following technical lemma.\",\n \"Lemma 4.4 For any $p\\\\ge 0$ there exists some integer $k(p)\\\\ge 1$ such that for any $n> k(p)$ we have $\\\\sqrt{\\\\vert (\\\\partial \\\\phi )(\\\\phi ^n(p))\\\\vert }=\\\\frac{\\\\vert A\\\\vert }{1+S\\\\phi ^{n}(p)} \\\\le 1-\\\\epsilon ^2_{\\\\infty },$ with the parameter $\\\\epsilon _{\\\\infty }$ defined in (REF ).\",\n \"In addition, for any $n\\\\ge k\\\\ge k(p)\\\\ge l$ we have $\\\\vert E_{l,n}(p)\\\\vert \\\\le \\\\left(\\\\frac{\\\\vert A\\\\vert }{1+SB^2}\\\\right)^{k(p)-l}~(1-\\\\epsilon ^2_{\\\\infty })^{n-k(p)}\\\\quad \\\\mbox{\\\\rm and}\\\\quad \\\\vert E_{k,n}(p)\\\\vert \\\\le (1-\\\\epsilon ^2_{\\\\infty })^{n-k}.$ We are now in position to state and to prove the main result of this section, which also completes the proof of Theorem REF .\",\n \"Theorem 4.5 For any $p_1\\\\ge 0$ there exists some integer $k(p_1)\\\\ge 1$ for any $n> k(p_1)$ and any $x_1,x_2\\\\in \\\\mathbb {R}$ we have the almost sure estimate $\\\\left|\\\\widehat{X}_n(x_1,p_1)- \\\\widehat{X}_n(x_2,p_1)\\\\right|\\\\le \\\\left(\\\\frac{\\\\vert A\\\\vert }{1+SB^2}\\\\right)^{k(p_1)}~(1-\\\\epsilon ^2_{\\\\infty })^{n-k(p_1)}~\\\\vert x_1-x_2\\\\vert .$ with the parameter $\\\\epsilon _{\\\\infty }$ defined in (REF ).\",\n \"In addition, for any $p_2\\\\ge 0$ we have $\\\\mathbb {E}\\\\left(\\\\left(\\\\widehat{X}_n(x_1,p_1)- \\\\widehat{X}_n(x_2,p_2)\\\\right)^2\\\\right)^{1/2}\\\\le ~\\\\kappa (p_1,p_2,x_2)~(1-\\\\epsilon ^2_{\\\\infty })^{n-k(p_1)}~\\\\vert x_1-x_2\\\\vert .$ The estimate (REF ) is a direct consequence of the formula (REF ) and the exponential estimate (REF ).\",\n \"On the other hand, we have $\\\\widehat{X}_n(x,p)-X_n&=&\\\\frac{1}{1+S\\\\phi ^n(p)} ~(A\\\\widehat{X}_{n-1}(x,p)-X_n)+\\\\frac{S\\\\phi ^n(p)}{1+S\\\\phi ^n(p)}~(C^{-1}~Y_n-X_n)\\\\\\\\&=&\\\\frac{A}{1+S\\\\phi ^n(p)} ~(\\\\widehat{X}_{n-1}(x,p)-X_{n-1})+\\\\widehat{U}_n(p),$ where $\\\\widehat{U}_n(p):=\\\\frac{S\\\\phi ^n(p)}{1+S\\\\phi ^n(p)}~\\\\frac{V_n}{\\\\sqrt{S}}-\\\\frac{1}{1+S\\\\phi ^n(p)}~BW_n\\\\Longrightarrow \\\\mathbb {E}(\\\\widehat{U}_n(p)^2)\\\\le R+1/S.$ Iterating, this implies that $\\\\widehat{X}_n(x,p)-X_n= E_{n}(p) (x-X_0)+\\\\sum _{1\\\\le k\\\\le n}~ E_{k,n}(p)~\\\\widehat{U}_k(p)$ and hence $\\\\begin{array}{l}\\\\displaystyle \\\\mathbb {E}\\\\left(\\\\left(\\\\widehat{X}_n(x,p)-X_n\\\\right)^2 \\\\right)=\\\\vert E_{0,n}(p)\\\\vert ~\\\\mathbb {E}((x-X_0)^2)+\\\\sum _{1\\\\le l\\\\le n}~\\\\vert E_{l,n}(p)\\\\vert ~ \\\\mathbb {E}(\\\\widehat{U}_l(p)^2).\\\\end{array}$ Using (REF ) we check the uniform estimate $\\\\mathbb {E}\\\\left(\\\\left(\\\\widehat{X}_n(x,p)-X_n\\\\right)^2 \\\\right)\\\\le \\\\kappa _1(p)~(1+\\\\mathbb {E}((x-X_0)^2)).$ In the same vein, for any given $q\\\\in \\\\mathbb {R}$ and for any $(p_1,p_2)\\\\in \\\\mathbb {R}_+^2$ we check that $\\\\widehat{X}_n(x,p_1)- \\\\widehat{X}_n(x,p_2)&=&\\\\sum _{1\\\\le l\\\\le n}~ E_{l,n}(p_1)~\\\\rho _l(p_1,p_2)~U_l(x,p_2)$ with the function $\\\\begin{array}{l}\\\\displaystyle \\\\rho _n(p_1,p_2):=\\\\frac{S(\\\\phi ^n(p_1)-\\\\phi ^n(p_2))}{(1+S\\\\phi ^n(p_1))(1+S\\\\phi ^n(p_2))} \\\\le S\\\\kappa _1^2~\\\\left(1-\\\\epsilon _1\\\\right)^n~\\\\vert p_1-p_2\\\\vert \\\\quad \\\\mbox{\\\\rm (by (\\\\ref {ricc-lower-cv-2})})\\\\end{array}$ and the collection of random variables $\\\\begin{array}{l}\\\\displaystyle U_n(x,p):= A(X_{n-1}-\\\\widehat{X}_{n-1}(x,p))+BW_n+V_n/\\\\sqrt{S}\\\\\\\\\\\\\\\\\\\\Longrightarrow \\\\mathbb {E}\\\\left(U_n(x,p)^2 \\\\right)\\\\le A^2\\\\kappa _1(p)~(1+\\\\mathbb {E}((x-X_0)^2))+R+1/S\\\\quad \\\\mbox{\\\\rm (by (\\\\ref {X-p-q-X}))}.\\\\end{array}$ This implies that $\\\\mathbb {E}\\\\left(\\\\left(\\\\widehat{X}_n(x,p_1)- \\\\widehat{X}_n(x,p_2)\\\\right)^2\\\\right)^{1/2}\\\\le \\\\left(\\\\sum _{1\\\\le l\\\\le n}~\\\\vert E_{l,n}(p_1)\\\\vert ~\\\\rho _l(p_1,p_2)~\\\\right)~\\\\kappa _2(x,p_2)$ with $\\\\kappa _2(x,p):=(A^2\\\\kappa _1(p)+B^2+1/S)~(1+\\\\mathbb {E}((x-X_0)^2)).$ Using (REF ), for any $n>k(p_1)$ we have $\\\\sum _{1\\\\le l\\\\le n}~\\\\vert E_{l,n}(p_1)\\\\vert ~\\\\rho _l(p_1,p_2)\\\\le \\\\kappa _3(p_1)~(1-\\\\epsilon ^2_{\\\\infty })^{n-k(p_1)}~\\\\vert p_1-p_2\\\\vert .$ This ends the proof of (REF ) and hence the theorem.\",\n \"Summing the two estimates (REF ) and (REF ) we obtain the estimate (REF ) with $\\\\epsilon _2=\\\\epsilon ^2_{\\\\infty }$ .\"\n ],\n [\n \"Stochastic perturbation analysis\",\n \"This section is mainly concerned with the proof of Theorem REF .\",\n \"In addition, we also consider the bias and some time-uniform bounds for the sample covariances $p_n$ and $\\\\widehat{p}_n$ .\"\n ],\n [\n \"A local perturbation theorem\",\n \"We fix some parameter $N\\\\ge 1$ and we let 1 be the $(N+1)$ column vector with unit entries, $I$ the $(N+1)\\\\times (N+1)$ identity matrix and $J={1}{1}^{\\\\prime }$ the $(N+1)\\\\times (N+1)$ matrix with unit entries, where $(\\\\cdot )^{\\\\prime }$ denotes the transpose operator.\",\n \"We also let $\\\\epsilon $ be the $(N+1)\\\\times (N+1)$ matrix given by $\\\\epsilon = I-{J}/{(N+1)}\\\\quad \\\\Longrightarrow \\\\quad \\\\epsilon ^2=\\\\epsilon .$ For any given $(N+1)$ -column $p$ we have $\\\\overline{p}:=\\\\frac{1}{N+1}\\\\sum _{1\\\\le j\\\\le N+1} p_j\\\\quad \\\\Longrightarrow \\\\quad p-\\\\overline{p}\\\\,{1}:=\\\\left(\\\\begin{array}{c}p_1-\\\\overline{p}\\\\\\\\\\\\vdots \\\\\\\\p_{N+1}-\\\\overline{p}\\\\end{array}\\\\right)=\\\\epsilon p.$ From the mean-field evolution equations (REF ) and the definitions of $m_n$ and $\\\\widehat{m}_n$ in (REF ), it is straightforward to show that $\\\\left\\\\lbrace \\\\begin{array}{rcl}\\\\displaystyle (\\\\xi ^i_n-m_n)&=& A~(\\\\widehat{\\\\xi }^i_{n-1}-\\\\widehat{m}_{n-1})+BW^{i,\\\\epsilon }_n\\\\\\\\\\\\\\\\(\\\\widehat{\\\\xi }^i_n-\\\\widehat{m}_n)&=&(1-g_nC)(\\\\xi ^i_n-m_n)-g_nDV^{i,\\\\epsilon }_n,\\\\end{array}\\\\right.$ with the random variables $W^{i,\\\\epsilon }_n:=\\\\sum _{1\\\\le j\\\\le N+1}\\\\epsilon ^i_j~W^j_n\\\\quad \\\\mbox{\\\\rm and}\\\\quad V^{i,\\\\epsilon }_n:=\\\\sum _{1\\\\le j\\\\le N+1}\\\\epsilon ^i_j~V^j_n.$ Taking the square we obtain the formulae $\\\\left\\\\lbrace \\\\begin{array}{rcl}\\\\displaystyle (\\\\xi ^i_n-m_n)^2&=& A^2~(\\\\widehat{\\\\xi }^i_{n-1}-\\\\widehat{m}_{n-1})^2+B^2(W^{i,\\\\epsilon }_n)^2+2AB~(\\\\widehat{\\\\xi }^i_{n-1}-\\\\widehat{m}_{n-1})W^{i,\\\\epsilon }_n\\\\\\\\\\\\\\\\(\\\\widehat{\\\\xi }^i_n-\\\\widehat{m}_n)^2&=&(1-g_nC)^2(\\\\xi ^i_n-m_n)^2+g_n^2D^2(V^{i,\\\\epsilon }_n)^2-2g_n(1-g_nC)D(\\\\xi ^i_n-m_n)V^{i,\\\\epsilon }_n.\\\\end{array}\\\\right.$ This implies that $\\\\left\\\\lbrace \\\\begin{array}{rcl}\\\\displaystyle p_n&=&\\\\displaystyle A^2~\\\\widehat{p}_{n-1}+B^2+\\\\frac{1}{\\\\sqrt{N}}~\\\\nu _n\\\\\\\\\\\\\\\\\\\\widehat{p}_n&=&\\\\displaystyle (1-g_nC)^2p_n+g_n^2D^2+\\\\frac{1}{\\\\sqrt{N}}~ \\\\widehat{\\\\nu }_n=(1-g_nC)p_n+\\\\frac{1}{\\\\sqrt{N}}~ \\\\widehat{\\\\nu }_n\\\\end{array}\\\\right.$ with the orthogonal variables $\\\\nu _n:=2AB~\\\\chi _n+B^2~\\\\varsigma _n\\\\quad \\\\mbox{\\\\rm and}\\\\quad \\\\widehat{\\\\nu }_n:=-2g_n(1-g_nC)D~\\\\widehat{\\\\chi }_n+g_n^2D^2~\\\\widehat{\\\\varsigma }_n$ and the centered random variables $\\\\begin{array}{rclcrcl}\\\\chi _n&:=&\\\\displaystyle \\\\frac{1}{\\\\sqrt{N}}~\\\\sum _{1\\\\le i\\\\le N+1}~(\\\\widehat{\\\\xi }^i_{n-1}-\\\\widehat{m}_{n-1})W^{i,\\\\epsilon }_n&&\\\\varsigma _n&:=&\\\\displaystyle \\\\sqrt{N}~\\\\left(\\\\frac{1}{N}\\\\sum _{1\\\\le i\\\\le N+1}(W^{i,\\\\epsilon }_n)^2-1\\\\right)\\\\\\\\&&&&&&\\\\\\\\\\\\widehat{\\\\chi }_n&:=&\\\\displaystyle \\\\frac{1}{\\\\sqrt{N}}~\\\\sum _{1\\\\le i\\\\le N+1}(\\\\xi ^i_n-m_n)V^{i,\\\\epsilon }_n&&\\\\widehat{\\\\varsigma }_n&:=&\\\\displaystyle \\\\sqrt{N}\\\\left(\\\\frac{1}{N}\\\\sum _{1\\\\le i\\\\le N+1}(V^{i,\\\\epsilon }_n)^2-1\\\\right).\\\\end{array}$ Further, denote by $\\\\mathbb {H}$ the $((N+1)\\\\times (N+1))$ -Helmert matrix whose first row is defined by $\\\\mathbb {H}_{1,j}=1/\\\\sqrt{N+1}\\\\quad \\\\forall 1\\\\le j\\\\le (N+1)\\\\quad \\\\Longrightarrow \\\\quad \\\\mathbb {H}_{1,\\\\mbox{\\\\LARGE .\",\n \"}}={1}^{\\\\prime }/\\\\sqrt{N+1}.$ and whose $i$ -th row for $2\\\\le i\\\\le (N+1)$ is given by $\\\\begin{array}{l}\\\\mathbb {H}_{i,j}:=\\\\overline{\\\\mathbb {H}}_{i,j}:=\\\\left\\\\lbrace \\\\begin{array}{ccl}1/\\\\sqrt{i(i-1)}&\\\\mbox{\\\\rm if}& 1\\\\le j0.\\\\end{array}$ Combining the above with Corollary REF and the inequality $|a + b|^k \\\\le 2^{k-1}(|a|^k + |b|^k)$ , $a, b \\\\in \\\\mathbb {R}$ , $k \\\\ge 1$ , we obtain $p_{n+1}^{2k}\\\\le \\\\phi _{2k}(p_n^{2k})+\\\\frac{2^{2k-1}}{N^k}~\\\\left(A^{4k}~\\\\widehat{\\\\nu }_n^{2k}+\\\\nu ^{2k}_{n+1}\\\\right).$ By Jensen's inequality, we check that $q^{2k}_{n+1}\\\\le \\\\phi _{2k}(q^{2k}_n)+\\\\frac{2^{2k-1}}{N^k}~\\\\left(A^{4k}~\\\\mathbb {E}(\\\\widehat{\\\\nu }_n^{2k})+\\\\mathbb {E}(\\\\nu ^{2k}_{n+1})\\\\right).$ To control the expectations on the right hand side above, note that for any $k\\\\ge 1$ we have the crude estimates $(g_nD)^{2k}&=&\\\\left(\\\\frac{Sp_n}{1+Sp_n}~\\\\frac{p_n}{1+Sp_n}\\\\right)^{k}\\\\le S^{-k},\\\\\\\\(g_nD(1-g_nC))^{k}~p_{n}^{k/2}&=&\\\\left(\\\\frac{Sp_n}{1+Sp_n}~\\\\frac{1}{1+Sp_n}~\\\\left(\\\\frac{p_n}{1+Sp_n}\\\\right)^2\\\\right)^{k/2}\\\\le S^{-k}.$ Combining these estimates with (REF ) in (REF ), we check that $\\\\mathbb {E}\\\\left(\\\\widehat{\\\\nu }_n^{k}\\\\right)\\\\le \\\\iota _1(k)\\\\quad \\\\mbox{\\\\rm and}\\\\quad \\\\mathbb {E}(\\\\nu _{n+1}^{k})\\\\le \\\\iota _2(k)~\\\\left(1\\\\vee q_n^{k/2}\\\\right) = \\\\iota _2(k)~\\\\left(1\\\\vee q_n\\\\right)^{k/2}.$ Now, (REF ) and (REF ) imply that $q_n\\\\le \\\\iota _0~(P_0\\\\vee 1).$ Combining this with (REF ) yields the required bounds for $\\\\nu _n^k$ , $\\\\widehat{\\\\nu }_n^k$ and $\\\\delta _n^k$ .\",\n \"Returning to the moments of $p_n$ , note that the previous estimates imply that $q^{2k}_{n+1}\\\\le \\\\phi _{2k}(q^{2k}_n)+\\\\frac{\\\\iota _5(k)}{N^k}~(q^k_n\\\\vee 1).$ Applying the above estimate to $k=1$ , along with (REF ), we have $q^{2}_{n+1}\\\\le \\\\phi _{2}(q^{2}_n)+\\\\frac{\\\\iota _1}{N}~( P_0\\\\vee 1).$ Using the comparison properties (REF ) and (REF ) we conclude that $\\\\sup _{N\\\\ge 1}\\\\sup _{n\\\\ge 0}q^{2}_{n}<\\\\iota _2~(P_0\\\\vee 1)^2.$ Iterating the argument for any $k\\\\ge 1$ we readily check that $\\\\sup _{N\\\\ge 1}\\\\sup _{n\\\\ge 0}q^{k}_{n}<\\\\iota _k~(P_0\\\\vee 1)^k$ and therefore $\\\\mathbb {E}\\\\left(\\\\vert \\\\nu _{n+1}\\\\vert ^{k}\\\\right)\\\\le \\\\iota _6(k)~\\\\left(1\\\\vee q_n\\\\right)^{k/2}\\\\le \\\\iota _7(k)~(P_0\\\\vee 1)^{k/2}\\\\quad \\\\mbox{\\\\rm and}\\\\quad \\\\mathbb {E}\\\\left(\\\\vert \\\\delta _{n+1}\\\\vert ^k\\\\right)\\\\le \\\\iota _k~(P_0\\\\vee 1)^{k/2}$ Finally, we have $\\\\widehat{p}_n=\\\\frac{p_n}{1+Sp_n}+\\\\frac{1}{\\\\sqrt{N}}~\\\\widehat{\\\\nu }_n\\\\Longrightarrow \\\\mathbb {E}(\\\\widehat{p}_{n}^{k})\\\\le S^{-k} +\\\\frac{1}{N^{k/2}}~ \\\\mathbb {E}(\\\\vert \\\\widehat{\\\\nu }_n\\\\vert ^{k})\\\\le \\\\iota _k,$ which ends the proof of the proposition.\"\n ],\n [\n \"Stochastic Riccati equations\",\n \"In this section we focus on the proof of Theorem REF , as well as the central limit theorems presented in section REF .\"\n ],\n [\n \"Perturbation analysis\",\n \"We start by proving the time uniform estimates (REF ).\",\n \"To this end, first note that we have the telescoping sum formula $p_n-P_n=\\\\left(\\\\phi ^n(p_0)-\\\\phi ^n(P_0)\\\\right)+\\\\sum _{1\\\\le k\\\\le n} \\\\left(\\\\phi ^{n-k}(p_k)-\\\\phi ^{n-(k-1)}(p_{k-1})\\\\right).$ To deal with the summands above, observe that $\\\\begin{array}{rcl}\\\\phi ^{n-k}(p_k)-\\\\phi ^{n-(k-1)}(p_{k-1})&=&\\\\phi ^{n-k}(p_k)-\\\\phi ^{n-k}(\\\\phi (p_{k-1}))\\\\\\\\&&\\\\\\\\&=&\\\\displaystyle \\\\phi ^{n-k}\\\\left(\\\\phi (p_{k-1})+\\\\frac{1}{\\\\sqrt{N}}~ \\\\delta _k\\\\right)-\\\\phi ^{n-k}\\\\left(\\\\phi (p_{k-1})\\\\right).\\\\end{array}$ Using the Lipschitz estimates (REF ) we check that $\\\\vert \\\\phi ^{n-k}(p_k)-\\\\phi ^{n-k}(\\\\phi (p_{k-1}))\\\\vert \\\\le \\\\frac{\\\\kappa _1}{\\\\sqrt{N}}~(1-\\\\epsilon _1)^{n-k}~\\\\left|\\\\delta _k\\\\right|,$ with the parameter $\\\\epsilon _1$ defined in (REF ).\",\n \"Similarly, we have $\\\\vert \\\\phi ^n(p_0)-\\\\phi ^n(P_0)\\\\vert \\\\le \\\\frac{\\\\kappa _2}{\\\\sqrt{N}}~(1-\\\\epsilon _1)^{n}~\\\\left|\\\\nu _0\\\\right|.$ This yields the almost sure estimate $\\\\begin{array}{l}\\\\displaystyle \\\\sqrt{N}~\\\\vert p_n-P_n\\\\vert \\\\le \\\\kappa _3~\\\\left[(1-\\\\epsilon _1)^{n}\\\\vert \\\\nu _0\\\\vert +\\\\sum _{1\\\\le k\\\\le n} ~(1-\\\\epsilon _1)^{n-k}~\\\\left|\\\\delta _k\\\\right|\\\\right].\\\\end{array}$ Next we note that $\\\\vert \\\\widehat{p}_n-\\\\widehat{P}_n\\\\vert =\\\\left|\\\\frac{p_n-P_n}{(1+SP_n)(1+Sp_n)}\\\\right|+\\\\frac{1}{\\\\sqrt{N}}~\\\\vert \\\\widehat{\\\\nu }_n\\\\vert \\\\le \\\\left|p_n-P_n\\\\right|+\\\\frac{1}{\\\\sqrt{N}}~\\\\vert \\\\widehat{\\\\nu }_n\\\\vert .$ Finally, we note the following decomposition for the gain parameters $\\\\sqrt{N}(g_n - G_n)&= \\\\sqrt{N}C\\\\left(\\\\frac{p_n}{C^2p_n + D^2} - \\\\frac{P_n}{C^2P_n + D^2}\\\\right) \\\\\\\\&= CD^2 \\\\frac{\\\\sqrt{N}(p_n - P_n)}{(C^2p_n + D^2)(C^2P_n + D^2)}.$ The above estimates and decompositions allow one to derive several mean error bounds as soon as we control the moments of the local errors.\",\n \"For instance, using the uniform moments estimates (REF ) we obtain the mean error estimates stated in Theorem REF .\"\n ],\n [\n \"Fluctuation analysis\",\n \"This section is mainly concerned with the proofs of Theorem REF and Theorem REF .\",\n \"We start with some technical results that will immediately lead to the proof of the former.\",\n \"Lemma 6.1 The characteristic function of random vector $\\\\Delta $ defined in (REF ) satisfies for any $w\\\\in \\\\mathbb {R}^3$ we have the Gaussian type estimate $\\\\left|\\\\mathbb {E}\\\\left(e^{i w^{\\\\prime }\\\\Delta }\\\\right)-e^{-\\\\frac{\\\\Vert w\\\\Vert ^2}{2}}\\\\right|\\\\le \\\\epsilon (w)~e^{-\\\\frac{\\\\Vert w\\\\Vert ^2}{2}},$ with the function $\\\\epsilon (w):=\\\\vert w_3\\\\vert \\\\sqrt{\\\\frac{2}{N}}\\\\left( \\\\frac{w_2^2}{2}+w_3^2\\\\right)~\\\\exp {\\\\left(\\\\vert w_3\\\\vert \\\\sqrt{\\\\frac{2}{N}}\\\\left( \\\\frac{w_2^2}{2}+w_3^2\\\\right)\\\\right)}.$ The proof of the above lemma follows elementary calculations.\",\n \"For the convenience of the reader a detailed proof is provided in the appendix.\",\n \"Lemma 6.2 For any $n\\\\ge 0$ we have the weak convergence $\\\\left(\\\\Delta _0,(\\\\widehat{\\\\Delta }^i_k,\\\\Delta ^i_{k+1})_{0\\\\le k\\\\le n,~1\\\\le i\\\\le 3}\\\\right)\\\\hookrightarrow _{N\\\\rightarrow \\\\infty }~\\\\left((Z^i_0)_{1\\\\le i\\\\le 2},(\\\\widehat{Z}^i_k,Z^i_{k+1})_{0\\\\le k\\\\le n,~1\\\\le i\\\\le 3}\\\\right),$ where the $Z^i_k,\\\\widehat{Z}^i_k$ are independent sequences of independent and centered Gaussian random variables with unit variance.\",\n \"Applying (REF ) for any $w\\\\in \\\\mathbb {R}^3$ we check the rather crude estimates $\\\\left|\\\\mathbb {E}\\\\left(e^{i~w^{\\\\prime }\\\\Delta _{n+1}}\\\\right)-e^{-\\\\frac{\\\\Vert w\\\\Vert ^2}{2}}\\\\right|\\\\le \\\\epsilon (w)~\\\\quad \\\\mbox{\\\\rm and}\\\\quad \\\\left|\\\\mathbb {E}\\\\left(e^{i~w^{\\\\prime }\\\\widehat{\\\\Delta }_{n}}\\\\right)-e^{-\\\\frac{\\\\Vert w\\\\Vert ^2}{2}}\\\\right|\\\\le \\\\epsilon (w)~$ with the function $\\\\epsilon (w)\\\\longrightarrow _{N\\\\rightarrow \\\\infty }0$ defined in (REF ).\",\n \"For any $u,v\\\\in \\\\mathbb {R}^3$ and $n\\\\ge 0$ we have the decomposition $\\\\begin{array}{l}\\\\displaystyle \\\\mathbb {E}\\\\left(e^{i~u^{\\\\prime }\\\\Delta _{n+1}}~e^{i~v^{\\\\prime }\\\\widehat{\\\\Delta }_{n}}\\\\right)-e^{-\\\\frac{\\\\Vert u\\\\Vert ^2}{2}}e^{-\\\\frac{\\\\Vert v\\\\Vert ^2}{2}}\\\\\\\\\\\\\\\\\\\\displaystyle =\\\\mathbb {E}\\\\left(e^{i~v^{\\\\prime }\\\\widehat{\\\\Delta }_{n}}\\\\right)~\\\\left(\\\\mathbb {E}\\\\left(e^{i~u^{\\\\prime }\\\\Delta _{n+1}}\\\\right)-e^{-\\\\frac{\\\\Vert u\\\\Vert ^2}{2}}\\\\right)+\\\\left(\\\\mathbb {E}\\\\left(e^{i~v^{\\\\prime }\\\\widehat{\\\\Delta }_{n}}\\\\right)-e^{-\\\\frac{\\\\Vert v\\\\Vert ^2}{2}}\\\\right)e^{-\\\\frac{\\\\Vert u\\\\Vert ^2}{2}}.\\\\end{array}$ This yields the estimate $\\\\left|\\\\mathbb {E}\\\\left(e^{i~u^{\\\\prime }\\\\Delta _{n+1}}~e^{i~v^{\\\\prime }\\\\widehat{\\\\Delta }_{n}}\\\\right)-e^{-\\\\frac{\\\\Vert u\\\\Vert ^2+\\\\Vert v\\\\Vert ^2}{2}}\\\\right|\\\\le \\\\epsilon (u,v):=\\\\epsilon (u)+\\\\epsilon (v).$ Consider the sequence of random vectors $\\\\Lambda _n:=\\\\left(\\\\begin{array}{c}\\\\widehat{\\\\Delta }_{n}\\\\\\\\\\\\Delta _{n+1}\\\\end{array}\\\\right)\\\\in \\\\mathbb {R}^6.$ In this notation, for any $n\\\\ge 0$ and $w_n\\\\in \\\\mathbb {R}^6$ we have proven the following estimate $\\\\left|\\\\mathbb {E}\\\\left(e^{i~w_n^{\\\\prime }\\\\Lambda _n}\\\\right)-e^{-\\\\frac{\\\\Vert w_n\\\\Vert ^2}{2}}\\\\right|\\\\le \\\\epsilon (w_n).$ Using the decomposition $\\\\begin{array}{l}\\\\mathbb {E}\\\\left(e^{i~\\\\left(w_{n-1}^{\\\\prime }\\\\Lambda _{n-1}+w_n^{\\\\prime }\\\\Lambda _n\\\\right)}\\\\right)-e^{-\\\\frac{\\\\Vert w_{n-1}\\\\Vert ^2}{2}}e^{-\\\\frac{\\\\Vert w_n\\\\Vert ^2}{2}}\\\\\\\\\\\\\\\\=\\\\mathbb {E}\\\\left(e^{i~w_{n-1}^{\\\\prime }\\\\Lambda _{n-1}}\\\\right)~\\\\left(\\\\mathbb {E}\\\\left(e^{i~w_n^{\\\\prime }\\\\Lambda _n}\\\\right)-e^{-\\\\frac{\\\\Vert w_n\\\\Vert ^2}{2}}\\\\right)\\\\\\\\\\\\\\\\\\\\hspace{85.35826pt}+\\\\left(\\\\mathbb {E}\\\\left(e^{i~w_{n-1}^{\\\\prime }\\\\Lambda _{n-1}}\\\\right)-e^{-\\\\frac{\\\\Vert w_{n-1}\\\\Vert ^2}{2}}~\\\\right)e^{-\\\\frac{\\\\Vert w_n\\\\Vert ^2}{2}},\\\\end{array}$ we also check that $\\\\begin{array}{l}\\\\left|\\\\mathbb {E}\\\\left(e^{i~\\\\left(w_{n-1}^{\\\\prime }\\\\Lambda _{n-1}+w_n^{\\\\prime }\\\\Lambda _n\\\\right)}\\\\right)-e^{-\\\\frac{\\\\Vert w_{n-1}\\\\Vert ^2}{2}}e^{-\\\\frac{\\\\Vert w_n\\\\Vert ^2}{2}}\\\\right|\\\\le \\\\epsilon (w_{n-1})+\\\\epsilon (w_n).\\\\end{array}$ Iterating the argument, we conclude that $\\\\begin{array}{l}\\\\left|\\\\mathbb {E}\\\\left(e^{i~\\\\left(u_1\\\\Delta ^1_0+u_2\\\\Delta ^2_0+\\\\sum _{0\\\\le k\\\\le n}w_{k}^{\\\\prime }\\\\Lambda _k\\\\right)}\\\\right)-e^{-\\\\frac{ u_1^2/2+u_2^2/2+\\\\sum _{0\\\\le k\\\\le n}\\\\Vert w_{k}\\\\Vert ^2}{2}}\\\\right|.\\\\\\\\\\\\\\\\\\\\le \\\\epsilon (u_1,0,u_2)+ \\\\sum _{0\\\\le k\\\\le n}\\\\epsilon (w_{k})\\\\longrightarrow _{N\\\\rightarrow \\\\infty }0,\\\\end{array}$ which ends the proof of the lemma.\",\n \"Combining Lemma REF with Theorem REF and Slutsky's lemma yields Theorem REF .\",\n \"We are now in position to prove theorem REF and the bias estimates (REF ).\",\n \"Proof of Theorem REF : By the second order estimate (REF ) we have $\\\\begin{array}{l}\\\\displaystyle \\\\vert \\\\phi ^{n-k}(p_k)-\\\\phi ^{n-(k-1)}(p_{k-1})-\\\\partial \\\\phi ^{n-k}\\\\left(\\\\phi (p_{k-1})\\\\right)\\\\frac{1}{\\\\sqrt{N}}~\\\\delta _k\\\\vert \\\\le \\\\frac{\\\\iota _1}{N}~(1-\\\\epsilon _1)^{n-k}~\\\\delta _k^2.\\\\end{array}$ Combining this with the telescoping sum formula (REF ) yields the decomposition $\\\\mathbb {Q}^N_n:=\\\\sqrt{N}(p_n-P_n)=\\\\partial \\\\phi ^n(P_0)~\\\\nu _0+\\\\sum _{1\\\\le k\\\\le n} \\\\partial \\\\phi ^{n-k}\\\\left(\\\\phi (p_{k-1})\\\\right)~\\\\delta _k+\\\\theta _1^N(1),$ with the remainder term $\\\\vert \\\\theta ^N_n(1)\\\\vert \\\\le \\\\epsilon _n^N(1):= \\\\frac{\\\\iota _2}{\\\\sqrt{N}}~(1-\\\\epsilon _1)^n~\\\\nu _0^2~+\\\\frac{\\\\iota _1}{\\\\sqrt{N}}~\\\\sum _{1\\\\le k\\\\le n} ~(1-\\\\epsilon _1)^{n-k}~\\\\delta _k^2\\\\longrightarrow _{N\\\\rightarrow \\\\infty }~0\\\\quad \\\\mbox{a.s.}.$ On the other hand, using the Lipschitz estimates (REF ) and (REF ), we have $\\\\begin{array}{l}\\\\displaystyle \\\\left|\\\\sum _{1\\\\le k\\\\le n} \\\\left[\\\\partial \\\\phi ^{n-k}(\\\\phi (p_{k-1}))-\\\\partial \\\\phi ^{n-k}(\\\\phi (P_{k-1}))\\\\right]~\\\\delta _k~\\\\right|\\\\\\\\\\\\\\\\\\\\displaystyle \\\\le \\\\epsilon _n^N(2):=\\\\iota _3 \\\\sum _{1\\\\le k\\\\le n}~(1-\\\\epsilon )^{n-k}~\\\\vert \\\\delta _k\\\\vert ~ \\\\vert p_{k-1}-P_{k-1}\\\\vert \\\\longrightarrow _{N\\\\rightarrow \\\\infty }~0\\\\quad \\\\mbox{a.s.}.\\\\end{array}$ This yields the formula $\\\\mathbb {Q}^N_n=F_n\\\\left(\\\\nu _0,\\\\delta _1,\\\\ldots ,\\\\delta _n\\\\right)+\\\\theta _n^N(2)$ with the function $\\\\displaystyle F_n\\\\left(\\\\nu _0,\\\\delta _1,\\\\ldots ,\\\\delta _n\\\\right):=\\\\partial \\\\phi ^n(P_0)~\\\\nu _0+\\\\sum _{1\\\\le k\\\\le n} \\\\partial \\\\phi ^{n-k}\\\\left(P_k\\\\right)~\\\\delta _k,$ and the remainder term $\\\\theta _n^N(2)$ satisfying $\\\\vert \\\\theta _n^N(2)\\\\vert \\\\le \\\\epsilon _n^N(1)+\\\\epsilon _n^N(2).$ Combining Slutsky's lemma with the continuous mapping theorem and Corollary REF , for any time horizon $n\\\\ge 0$ we conclude that $(F_k\\\\left(\\\\nu _0,\\\\delta _1,\\\\ldots ,\\\\delta _k\\\\right))_{0\\\\le k\\\\le n}\\\\longrightarrow _{N\\\\rightarrow \\\\infty }~(F_k\\\\left(\\\\mathbb {Z}_0,\\\\mathbb {Z}_1,\\\\ldots ,\\\\mathbb {Z}_k\\\\right))_{0\\\\le k\\\\le n}$ with the Gaussian random variables $\\\\mathbb {Z}_k$ defined in Corollary REF .\",\n \"Thus, we have shown that $\\\\mathbb {Q}_n^N \\\\longrightarrow _{N \\\\rightarrow \\\\infty } F_n\\\\left(\\\\mathbb {Z}_0,\\\\mathbb {Z}_1,\\\\ldots ,\\\\mathbb {Z}_n\\\\right) =: \\\\mathbb {Q}_n.$ The recursive formulation (REF ) comes from the decomposition $\\\\widehat{\\\\mathbb {Q}}^N_n=\\\\frac{1}{(1+SP_n)(1+Sp_n)}~\\\\mathbb {Q}^N_n+\\\\widehat{\\\\nu }_n\\\\longrightarrow _{N\\\\rightarrow \\\\infty } \\\\widehat{\\\\mathbb {Q}}_n:=\\\\frac{1}{(1+SP_n)^2}~\\\\mathbb {Q}_n+\\\\widehat{\\\\mathbb {V}}_n.$ In the same vein, we have $\\\\mathbb {Q}^N_{n+1}=A^2~\\\\widehat{\\\\mathbb {Q}}^N_n+\\\\nu _{n+1}\\\\Longrightarrow \\\\mathbb {Q}_{n+1}=A^2\\\\widehat{\\\\mathbb {Q}}_n+\\\\mathbb {V}_{n+1}.$ The proof of the uniform bias estimates (REF ) follows the second order Taylor expansions discussed in the proof of Theorem REF as we will now demonstrate.\",\n \"Proof of the bias estimate (REF ): Using the second order Taylor expansions discussed in the proof of Theorem REF we have the bias estimates $\\\\begin{array}{l}\\\\displaystyle 0\\\\le P_n-\\\\mathbb {E}(p_n)\\\\le \\\\frac{ \\\\iota _4}{N}~\\\\left[\\\\mathbb {E}(\\\\nu _0^2)~(1-\\\\epsilon )^n+~\\\\sum _{1\\\\le k\\\\le n}(1-\\\\epsilon )^{n-k}~ \\\\mathbb {E}(\\\\delta _k^2)\\\\right]\\\\\\\\\\\\\\\\\\\\displaystyle \\\\hspace{142.26378pt}+\\\\frac{\\\\iota _5}{\\\\sqrt{N}} \\\\sum _{1\\\\le k\\\\le n}~(1-\\\\epsilon )^{n-k}~\\\\mathbb {E}(\\\\vert \\\\delta _k\\\\vert ~ \\\\vert p_{k-1}-P_{k-1}\\\\vert ).\\\\end{array}$ The first bias estimte stated in (REF ) now follows from (REF ) and the estimates stated in (REF ).\",\n \"blackIndeed, using (REF ) we check that $0\\\\le P_n-\\\\mathbb {E}(p_n)&\\\\le & \\\\frac{ \\\\iota _6}{N}~\\\\left[P_0~(1-\\\\epsilon )^n+~(1\\\\vee P_0)\\\\sum _{1\\\\le k\\\\le n}(1-\\\\epsilon )^{n-k}\\\\right] \\\\\\\\&&\\\\hspace{42.67912pt}+\\\\frac{\\\\iota _7}{\\\\sqrt{N}}~(1\\\\vee P_0)~ \\\\sum _{1\\\\le k\\\\le n}~(1-\\\\epsilon )^{n-k}~~\\\\mathbb {E}( \\\\vert p_{k-1}-P_{k-1}\\\\vert ^2)^{1/2}.$ The end of the proof of the l.h.s.\",\n \"estimate in (REF ) is now a consequence of (REF ).\",\n \"We also have the second order decomposition $g_n - G_n&=& \\\\frac{CD^2}{(C^2P_n + D^2)^2}~(p_n - P_n)-\\\\frac{CD^2}{(C^2P_n + D^2)^2(C^2p_n + D^2)}~(p_n - P_n)^2$ Using the bias estimate stated in the l.h.s.\",\n \"of (REF ) we readily check that $0\\\\le G_n-\\\\mathbb {E}(g_n) \\\\le \\\\frac{CD^2}{(C^2P_n + D^2)^2}~\\\\frac{\\\\iota _1}{N}~\\\\left[1\\\\vee P_0\\\\right]^{2}+\\\\frac{C}{(C^2P_n + D^2)^2}~\\\\mathbb {E}\\\\left((p_n - P_n)^2\\\\right)$ Again, using the $\\\\mathbb {L}_k$ -mean error estimates stated in (REF ) we check the r.h.s.\",\n \"estimate stated in (REF ).\",\n \"This ends the proof of the bias estimate.\"\n ],\n [\n \"Inverse raw moments\",\n \"We end the section with some bounds on the moments of ratios of the form $\\\\phi (p_n)/p_{n+1}$ , $n \\\\ge 1$ .\",\n \"Using (REF ) we have the formulae, which will be used throughout this section.\",\n \"$\\\\frac{1}{\\\\widehat{p}_n}=\\\\left(S+\\\\frac{1}{p_n}\\\\right)~\\\\widehat{\\\\Upsilon }_n(p_n)\\\\quad \\\\mbox{\\\\rm and}\\\\quad \\\\frac{1}{p_{n+1}}=\\\\frac{1}{A^2\\\\widehat{p}_n+R}~\\\\Upsilon _{n+1}(\\\\widehat{p}_n),$ with the inverse non-central $\\\\chi $ -square random variables $\\\\widehat{\\\\Upsilon }_n(p_n):=\\\\frac{1+1/(Sp_n)}{\\\\frac{1}{N}~\\\\widehat{\\\\chi }^{(n,2)}_{N,N/(Sp_n)}}\\\\quad \\\\mbox{\\\\rm and}\\\\quad \\\\Upsilon _{n+1}(\\\\widehat{p}_n):=\\\\frac{1+\\\\left(A^2/R\\\\right)\\\\widehat{p}_n}{\\\\frac{1}{N}~\\\\chi ^{(n+1,2)}_{N,N\\\\left(A^2/R\\\\right)\\\\widehat{p}_n}}.$ Lemma 6.3 For any $n\\\\ge 0$ we have $\\\\begin{array}{l}\\\\displaystyle \\\\frac{\\\\phi (p_n)}{p_{n+1}}-1=\\\\frac{1}{N}\\\\frac{A^4}{\\\\phi (p_n)}~\\\\frac{\\\\widehat{\\\\nu }_n^2}{\\\\phi (p_n)+\\\\frac{A^2}{\\\\sqrt{N}}~\\\\widehat{\\\\nu }_n}-\\\\frac{A^2}{\\\\sqrt{N}}~ \\\\frac{\\\\widehat{\\\\nu }_n}{\\\\phi (p_n)}\\\\\\\\\\\\\\\\\\\\hspace{142.26378pt}\\\\displaystyle +\\\\frac{\\\\phi (p_n)}{A^2\\\\widehat{p}_n+R}~\\\\left(\\\\Upsilon _{n+1}(\\\\widehat{p}_n)-1\\\\right).\\\\end{array}$ First recall that $\\\\widehat{p}_n=\\\\displaystyle (1-g_nC)~p_n+\\\\frac{1}{\\\\sqrt{N}}~ \\\\widehat{\\\\nu }_n.$ Combining this with the second identity in (REF ), we have $\\\\begin{array}{l}\\\\displaystyle \\\\frac{\\\\phi (p_n)}{p_{n+1}}=\\\\frac{1}{1+\\\\frac{A^2}{\\\\sqrt{N}}~ \\\\frac{\\\\widehat{\\\\nu }_n}{\\\\phi (p_n)}}+\\\\frac{\\\\phi (p_n)}{A^2\\\\widehat{p}_n+R}~\\\\left(\\\\Upsilon _{n+1}(\\\\widehat{p}_n)-1\\\\right).\\\\end{array}$ The result now follows from the decomposition $\\\\frac{1}{1+\\\\frac{A^2}{\\\\sqrt{N}}~ \\\\frac{\\\\widehat{\\\\nu }_n}{\\\\phi (p_n)}}=1-\\\\frac{A^2}{\\\\sqrt{N}}~ \\\\frac{\\\\widehat{\\\\nu }_n}{\\\\phi (p_n)}+\\\\frac{1}{N}\\\\frac{A^4}{\\\\phi (p_n)}~\\\\frac{\\\\widehat{\\\\nu }_n^2}{\\\\phi (p_n)+\\\\frac{A^2}{\\\\sqrt{N}}~\\\\widehat{\\\\nu }_n}.$ Proposition 6.4 For any $n\\\\ge 0$ and any even number of particles $N>4$ we have the almost sure uniform drift estimates $1\\\\le \\\\mathbb {E}\\\\left(\\\\frac{\\\\phi (p_n)}{p_{n+1}}~|~p_{n}\\\\right)\\\\le 1+\\\\frac{\\\\iota }{N}.$ First note that from the previous lemma, we have $\\\\displaystyle \\\\mathbb {E}\\\\left(\\\\frac{\\\\phi (p_n)}{p_{n+1}}~\\\\bigg |~p_n,\\\\widehat{p}_n\\\\right)\\\\displaystyle =1-\\\\frac{A^2}{\\\\sqrt{N}}~ \\\\frac{\\\\widehat{\\\\nu }_n}{\\\\phi (p_n)}+\\\\frac{1}{N}\\\\frac{A^4}{\\\\phi (p_n)}~\\\\frac{\\\\widehat{\\\\nu }_n^2}{\\\\phi (p_n)+\\\\frac{A^2}{\\\\sqrt{N}}~\\\\widehat{\\\\nu }_n}\\\\\\\\+~\\\\frac{\\\\phi (p_n)}{A^2\\\\widehat{p}_n+R}~\\\\mathbb {E}\\\\left(\\\\left(\\\\Upsilon _{n+1}(\\\\widehat{p}_n)-1\\\\right)~|~\\\\widehat{p}_n\\\\right).$ Due to (REF ), for any even number of particles $N>4$ we have the estimate $0\\\\le \\\\mathbb {E}\\\\left(\\\\Upsilon _n(x)\\\\right)-1\\\\le \\\\frac{4}{N}~\\\\left(1+\\\\frac{A^2}{R}~x\\\\right)~\\\\left(1+\\\\frac{4}{N-4}\\\\right).$ Since both $\\\\widehat{p}_n$ and $\\\\phi (p_n)$ are bounded, it follows that $\\\\mathbb {E}\\\\left(\\\\frac{\\\\phi (p_n)}{p_{n+1}}~\\\\bigg |~p_n,\\\\widehat{p}_n\\\\right) \\\\le 1-\\\\frac{A^2}{\\\\sqrt{N}}~ \\\\frac{\\\\widehat{\\\\nu }_n}{\\\\phi (p_n)}+\\\\frac{1}{N}\\\\frac{A^4}{\\\\phi (p_n)}~\\\\frac{\\\\widehat{\\\\nu }_n^2}{\\\\phi (p_n)+\\\\frac{A^2}{\\\\sqrt{N}}~\\\\widehat{\\\\nu }_n}+\\\\frac{\\\\iota _1}{N}.$ for some finite constant $\\\\iota _1$ .\",\n \"Taking the expectation we check that $\\\\displaystyle \\\\mathbb {E}\\\\left(\\\\frac{\\\\phi (p_n)}{p_{n+1}}~|~p_n\\\\right)\\\\displaystyle &\\\\le \\\\mathbb {E}\\\\left(1+\\\\frac{1}{N}\\\\frac{A^4}{\\\\phi (p_n)}~\\\\frac{\\\\widehat{\\\\nu }_n^2}{\\\\phi (p_n)+\\\\frac{A^2}{\\\\sqrt{N}}~\\\\widehat{\\\\nu }_n}~|~p_n\\\\right)+\\\\frac{\\\\iota _1}{N}\\\\\\\\\\\\ \\\\\\\\\\\\displaystyle &\\\\le 1+\\\\frac{1}{N}\\\\left(\\\\iota _1+\\\\frac{A^4}{R^2}~\\\\mathbb {E}\\\\left(~\\\\widehat{\\\\nu }_n^2~|~p_n\\\\right) \\\\right),$ where the last line follows from the estimates $\\\\phi (p_n)\\\\ge R\\\\quad \\\\mbox{\\\\rm and}\\\\quad R\\\\le A^2\\\\widehat{p}_n+R=\\\\phi (p_n)+\\\\frac{A^2}{\\\\sqrt{N}}~ \\\\widehat{\\\\nu }_n.$ The proposition now follows from the almost sure estimate $\\\\mathbb {E}\\\\left(\\\\widehat{\\\\nu }_n^2~|~p_n\\\\right)&=&(2g_n(1-g_nC)D~\\\\sqrt{p_n})^2+(\\\\sqrt{2}~g_n^2D^2)^2 \\\\\\\\&=&2S\\\\frac{p_n^3}{(1+Sp_n)^4}~\\\\left(2+Sp_n\\\\right)\\\\le 4S~\\\\frac{p_n^3}{(1+Sp_n)^3}\\\\le \\\\frac{4}{S^2}.$ Theorem 6.5 For any even $k\\\\ge 2$ there exists some finite constant $\\\\iota _k$ such that for any $n\\\\ge 0$ and any even parameter $N>2(k+2)$ we have the almost sure uniform estimates $1\\\\le \\\\mathbb {E}\\\\left(\\\\left(\\\\frac{\\\\phi (p_n)}{p_{n+1}}\\\\right)^k~|~p_n\\\\right)\\\\le 1+\\\\frac{\\\\iota _k}{N}.$ We will show that (REF ) holds for the cases $k = 2, 3$ and then outline how the general case may be obtained inductively.\",\n \"By (REF ), for any $n\\\\ge 1$ any $k\\\\ge 0$ and any even $N>2(k+2)$ we have the uniform estimates $\\\\mathbb {E}\\\\left(\\\\left(\\\\frac{\\\\Upsilon _{n}(x)-1}{A^2x+R}\\\\right)^k\\\\right)\\\\le \\\\frac{\\\\iota _k}{N}.$ Now, recall that $\\\\frac{\\\\phi (p_n)}{p_{n+1}}-1=\\\\widehat{\\\\alpha }_n(p_n)+\\\\beta _{n+1}(\\\\widehat{p}_n),$ with the random variables $\\\\widehat{\\\\alpha }_n(p_n)=-\\\\frac{1}{\\\\sqrt{N}}~\\\\frac{A^2}{A^2\\\\widehat{p}_n+R}~ \\\\widehat{\\\\nu }_n\\\\quad \\\\mbox{\\\\rm and}\\\\quad \\\\beta _{n+1}(\\\\widehat{p}_n):=\\\\phi (p_n)~\\\\frac{\\\\Upsilon _{n+1}(\\\\widehat{p}_n)-1}{A^2\\\\widehat{p}_n+R}.$ Using the estimates (REF ), (REF ) and (REF ), we check the almost sure uniform estimate $\\\\begin{array}{l}\\\\displaystyle \\\\mathbb {E}\\\\left(\\\\left(\\\\widehat{\\\\alpha }_n(p_n)+\\\\beta _{n+1}(\\\\widehat{p}_n)\\\\right)~|~p_n\\\\right)\\\\\\\\\\\\\\\\\\\\displaystyle \\\\le \\\\frac{1}{N}\\\\frac{A^4}{R^2}~\\\\mathbb {E}(\\\\widehat{\\\\nu }_n^2~|~p_n)+\\\\phi (p_n)~\\\\mathbb {E}\\\\left(~\\\\frac{\\\\Upsilon _{n+1}(\\\\widehat{p}_n)-1}{A^2\\\\widehat{p}_n+R}~|~p_n\\\\right)\\\\le \\\\frac{\\\\iota _1}{N}.\\\\end{array}$ In the same vein, we have $\\\\begin{array}{l}\\\\displaystyle \\\\mathbb {E}\\\\left(\\\\left(\\\\widehat{\\\\alpha }_n(p_n)+\\\\beta _{n+1}(\\\\widehat{p}_n)\\\\right)^2~|~p_n,\\\\widehat{p}_n\\\\right)\\\\\\\\\\\\\\\\\\\\displaystyle =\\\\widehat{\\\\alpha }_n(p_n)^2+\\\\phi (p_n)^2~\\\\mathbb {E}\\\\left(\\\\left(\\\\frac{\\\\Upsilon _{n+1}(\\\\widehat{p}_n)-1}{A^2\\\\widehat{p}_n+R}\\\\right)^2~|~p_n,\\\\widehat{p}_n\\\\right)\\\\\\\\\\\\\\\\\\\\displaystyle \\\\hspace{85.35826pt}+2~\\\\phi (p_n)~\\\\widehat{\\\\alpha }_n(p_n)~\\\\mathbb {E}\\\\left(\\\\frac{\\\\Upsilon _{n+1}(\\\\widehat{p}_n)-1}{A^2\\\\widehat{p}_n+R}~|~p_n,\\\\widehat{p}_n\\\\right).\\\\end{array}$ On the other hand, we have $\\\\begin{array}{l}\\\\displaystyle \\\\widehat{\\\\alpha }_n(p_n)~\\\\mathbb {E}\\\\left(\\\\frac{\\\\Upsilon _{n+1}(\\\\widehat{p}_n)-1}{A^2\\\\widehat{p}_n+R}~|~p_n,\\\\widehat{p}_n\\\\right)\\\\\\\\\\\\\\\\\\\\displaystyle \\\\le \\\\widehat{\\\\alpha }_n(p_n)^2+\\\\mathbb {E}\\\\left(\\\\frac{\\\\Upsilon _{n+1}(\\\\widehat{p}_n)-1}{A^2\\\\widehat{p}_n+R}~|~p_n,\\\\widehat{p}_n\\\\right)^2\\\\le \\\\widehat{\\\\alpha }_n(p_n)^2+\\\\iota _2/N^2.\\\\end{array}$ Along with the almost sure uniform bound $\\\\mathbb {E}\\\\left(\\\\widehat{\\\\alpha }_n(p_n)^2~|~p_n\\\\right)\\\\le {\\\\iota _3}/{N},$ which follows from (REF ), we conclude that $\\\\mathbb {E}\\\\left(\\\\left(\\\\widehat{\\\\alpha }_n(p_n)+\\\\beta _{n+1}(\\\\widehat{p}_n)\\\\right)^2~|~p_n\\\\right)\\\\le {\\\\iota _4}/{N}.$ Combining this with (REF ), it follows that $1\\\\le \\\\mathbb {E}\\\\left(\\\\left(\\\\frac{\\\\phi (p_n)}{p_{n+1}}\\\\right)^2~|~p_n=x\\\\right)\\\\le 1+{\\\\iota _5}/{N}.$ For odd powers, for instance we have $\\\\begin{array}{l}\\\\displaystyle \\\\mathbb {E}\\\\left(\\\\left(\\\\widehat{\\\\alpha }_n(p_n)+\\\\beta _{n+1}(\\\\widehat{p}_n)\\\\right)^3~|~p_n,\\\\widehat{p}_n\\\\right)\\\\\\\\\\\\\\\\\\\\displaystyle =\\\\widehat{\\\\alpha }_n(p_n)^3+3~\\\\widehat{\\\\alpha }_n(p_n)^2\\\\phi (p_n)~\\\\mathbb {E}\\\\left(\\\\frac{\\\\Upsilon _{n+1}(\\\\widehat{p}_n)-1}{A^2\\\\widehat{p}_n+R}~|~\\\\widehat{p}_n\\\\right)\\\\\\\\\\\\\\\\\\\\displaystyle +3~\\\\widehat{\\\\alpha }_n(p_n)\\\\phi (p_n)^2~\\\\mathbb {E}\\\\left(\\\\left(\\\\frac{\\\\Upsilon _{n+1}(\\\\widehat{p}_n)-1}{A^2\\\\widehat{p}_n+R}\\\\right)^2~|~\\\\widehat{p}_n\\\\right)+\\\\phi (p_n)^3~\\\\mathbb {E}\\\\left(\\\\left(~\\\\frac{\\\\Upsilon _{n+1}(\\\\widehat{p}_n)-1}{A^2\\\\widehat{p}_n+B^2}\\\\right)^3~|~\\\\widehat{p}_n\\\\right).\\\\end{array}$ Now observe that $\\\\begin{array}{l}\\\\displaystyle 6~\\\\widehat{\\\\alpha }_n(p_n)\\\\phi (p_n)^2~\\\\mathbb {E}\\\\left(\\\\left(\\\\frac{\\\\Upsilon _{n+1}(\\\\widehat{p}_n)-1}{A^2\\\\widehat{p}_n+R}\\\\right)^2~|~\\\\widehat{p}_n\\\\right)\\\\\\\\\\\\\\\\\\\\displaystyle \\\\le \\\\left(3\\\\widehat{\\\\alpha }_n(p_n)\\\\phi (p_n)^2\\\\right)^2+\\\\mathbb {E}\\\\left(\\\\left(\\\\frac{\\\\Upsilon _{n+1}(\\\\widehat{p}_n)-1}{A^2\\\\widehat{p}_n+R}\\\\right)^4~|~\\\\widehat{p}_n\\\\right)\\\\le \\\\left(3\\\\widehat{\\\\alpha }_n(p_n)\\\\phi (p_n)^2\\\\right)^2+\\\\frac{\\\\iota _6}{N}.\\\\end{array}$ Using the estimates (REF ), this implies that $\\\\begin{array}{l}\\\\displaystyle \\\\mathbb {E}\\\\left(\\\\left(\\\\widehat{\\\\alpha }_n(p_n)+\\\\beta _{n+1}(\\\\widehat{p}_n)\\\\right)^3~|~p_n\\\\right)\\\\\\\\\\\\\\\\\\\\displaystyle \\\\le \\\\mathbb {E}\\\\left(\\\\widehat{\\\\alpha }_n(p_n)^3~|~p_n\\\\right)+\\\\frac{1}{2}\\\\left(3\\\\phi (p_n)^2\\\\right)^2~\\\\mathbb {E}\\\\left(\\\\widehat{\\\\alpha }_n(p_n)^2~|~p_n\\\\right)+\\\\frac{\\\\iota _6}{2N}\\\\\\\\\\\\\\\\\\\\displaystyle \\\\hspace{85.35826pt}+3\\\\,\\\\phi (p_n)~\\\\mathbb {E}\\\\left(\\\\widehat{\\\\alpha }_n(p_n)^2~|~p_n\\\\right)~\\\\frac{\\\\iota _1}{N}+\\\\phi (p_n)^3~\\\\frac{\\\\iota _3}{N}.\\\\end{array}$ Using the fact that $\\\\mathbb {E}\\\\left(\\\\vert \\\\widehat{\\\\alpha }_n(p_n)\\\\vert ^3~|~p_n\\\\right)\\\\le \\\\frac{\\\\iota _7}{N\\\\sqrt{N}},$ it follows that $\\\\begin{array}{l}\\\\displaystyle \\\\mathbb {E}\\\\left(\\\\left(\\\\widehat{\\\\alpha }_n(p_n)+\\\\beta _{n+1}(\\\\widehat{p}_n)\\\\right)^3~|~p_n\\\\right)\\\\le \\\\frac{\\\\iota _8}{N}.\\\\end{array}$ Using (REF ) and Jensen's inequality we check that $1\\\\le \\\\mathbb {E}\\\\left(\\\\left(\\\\frac{\\\\phi (p_n)}{p_{n+1}}\\\\right)^3~|~p_n\\\\right)\\\\le 1+\\\\frac{\\\\iota _9}{N}.$ More generally, for any $l\\\\ge 2$ and $k\\\\ge 1$ we have $\\\\mathbb {E}\\\\left(\\\\vert \\\\widehat{\\\\alpha }_n(p_n)\\\\vert ^l~|~p_n=x\\\\right)\\\\le \\\\frac{\\\\iota _l(1)}{N}\\\\quad \\\\mbox{\\\\rm and}\\\\quad \\\\mathbb {E}\\\\left(\\\\beta _{n+1}(\\\\widehat{p}_n)^{2k}~|~\\\\widehat{p}_n=x\\\\right)\\\\le \\\\frac{\\\\iota _2(k)}{N}$ for some finite constants $\\\\iota _1(k),\\\\iota _2(k)$ .\",\n \"Also note that due to the binomial formula, we have $\\\\begin{array}{l}\\\\displaystyle \\\\mathbb {E}\\\\left(\\\\left(\\\\frac{\\\\phi (p_n)}{p_{n+1}}\\\\right)^k~|~p_n,\\\\widehat{p}_n\\\\right)\\\\\\\\\\\\\\\\\\\\displaystyle =1+\\\\mathbb {E}\\\\left(\\\\left(\\\\widehat{\\\\alpha }_n(p_n)+\\\\beta _{n+1}(\\\\widehat{p}_n)\\\\right)~|~p_n,\\\\widehat{p}_n\\\\right)+\\\\sum _{2\\\\le l\\\\le k}\\\\left(\\\\begin{array}{c}k\\\\\\\\l\\\\end{array}\\\\right)~\\\\mathbb {E}\\\\left(\\\\left(\\\\widehat{\\\\alpha }_n(p_n)+\\\\beta _{n+1}(\\\\widehat{p}_n)\\\\right)^{l}~|~p_n,\\\\widehat{p}_n\\\\right)\\\\end{array}.$ We may estimate the above summands, for any $l\\\\ge 2$ by $\\\\begin{array}{l}\\\\displaystyle \\\\mathbb {E}\\\\left(\\\\left(\\\\widehat{\\\\alpha }_n(p_n)+\\\\beta _{n+1}(\\\\widehat{p}_n)\\\\right)^{l}~|~p_n,\\\\widehat{p}_n\\\\right)\\\\\\\\\\\\\\\\\\\\displaystyle \\\\le \\\\vert \\\\widehat{\\\\alpha }_n(p_n)\\\\vert ^{l}+\\\\frac{1}{2}~\\\\sum _{0\\\\le i< l}\\\\left(\\\\begin{array}{c}l\\\\\\\\i\\\\end{array}\\\\right)\\\\left(\\\\widehat{\\\\alpha }_n(p_n)^{2l}+\\\\mathbb {E}\\\\left( \\\\left(\\\\beta _{n+1}(\\\\widehat{p}_n)\\\\right)^{2(l-i)}~|~p_n,\\\\widehat{p}_n\\\\right)\\\\right)\\\\\\\\\\\\\\\\\\\\displaystyle \\\\le \\\\vert \\\\widehat{\\\\alpha }_n(p_n)\\\\vert ^{l}+\\\\frac{1}{2}~\\\\sum _{0\\\\le i< l}\\\\left(\\\\begin{array}{c}.l\\\\\\\\i\\\\end{array}\\\\right)\\\\left(\\\\widehat{\\\\alpha }_n(p_n)^{2l}+\\\\frac{\\\\iota _{2}(l-i)}{N}\\\\right)\\\\end{array}$ The end of the proof of (REF ) is now a direct consequence of the moment estimates stated above.\"\n ],\n [\n \"A Feynman-Kac formula\",\n \"Theorem 7.1 Let $X_n$ be a Markov chain on some measurable state space $E$ with Markov transitions $M(x,dy)$ and starting at some state $X_0=x$ .\",\n \"Also let $H(x)$ be some positive measurable function on $E$ .\",\n \"Assume there exists some function $h$ and some parameters $\\\\epsilon _h\\\\in ]0,1[$ and $\\\\kappa _h \\\\in ]0, \\\\infty [$ such that $H^h(x):=H(x)~h(x)M(1/h)(x)\\\\le (1-\\\\epsilon _h)\\\\quad \\\\mbox{and}\\\\quad M(1/h)(x)\\\\le \\\\kappa _h ~M(1/h)(y).$ Then, for any $n\\\\ge 1$ and any bounded measurable function $F$ on the path space $E^n$ we have the Feynman-Kac change of measure formula $\\\\begin{array}{l}\\\\displaystyle \\\\mathbb {E}\\\\left(F(X_1,\\\\ldots ,X_n)~\\\\left(\\\\prod _{1\\\\le k\\\\le n}H(X_k)\\\\right)~|~X_0=x\\\\right)\\\\\\\\\\\\\\\\\\\\displaystyle =\\\\mathbb {E}\\\\left(F(X^{1/h}_1,\\\\ldots ,X^{1/h}_n)~~\\\\frac{M(h^{-1})(X^{1/h}_{0})}{M(h^{-1})(X^{1/h}_{n})}\\\\left(\\\\prod _{1\\\\le k\\\\le n}H^h(X^{1/h}_{k})\\\\right)~|~X^{1/h}_0=x\\\\right).\\\\end{array}$ In the above display, $X^{1/h}_n$ stands for the Markov chain with Markov transitions $M^{1/h}(x,dy)=M(x,dy)~\\\\frac{h^{-1}(y)}{M(h^{-1})(x)}.$ In addition, we have the uniform exponential decay estimate $\\\\sup _{x} \\\\mathbb {E}\\\\left(\\\\prod _{1\\\\le k\\\\le n}H(X_k)~|~X_0=x\\\\right)\\\\le \\\\kappa _h~(1-\\\\epsilon _h)^n.$ We set $h^{-1}(x)=1/h(x)$ .\",\n \"Let $X^{1/h}_n$ be the Markov chain starting at $X^{1/h}_0=X_0=x_0$ whose Markov transitions $M^{1/h}$ are given by (REF ) so that $M^{1/h}(h)(x)=\\\\frac{1}{M(h^{-1})(x)}.$ Observe that $\\\\begin{array}{l}\\\\mathbb {P}^{1/h}_n(d(x_0,\\\\ldots ,x_n))\\\\\\\\\\\\\\\\:=\\\\mathbb {P}((X^{1/h}_0,\\\\ldots ,X^{1/h}_n)\\\\in d(x_0,\\\\ldots ,x_n))\\\\\\\\\\\\\\\\\\\\displaystyle =\\\\eta _0(dx_0)~M(x_0,dx_1)~\\\\frac{h^{-1}(x_1)}{M(h^{-1})(x_0)}\\\\ldots M(x_{n-1},dx_n)~\\\\frac{h^{-1}(x_n)}{M(h^{-1})(x_{n-1})}.\\\\end{array}$ Rewritten in terms of expectations we have $\\\\mathbb {E}\\\\left(F(X_0,\\\\ldots ,X_n)~\\\\prod _{1\\\\le k\\\\le n}\\\\frac{h^{-1}(X_k)}{M(h^{-1})(X_{k-1})}\\\\right)=\\\\mathbb {E}\\\\left(F(X_0^{1/h},\\\\ldots ,X_n^{1/h})\\\\right).$ Observe that $\\\\begin{array}{l}\\\\mathbb {P}^{1/h}_n(d(x_0,\\\\ldots ,x_n))\\\\\\\\\\\\\\\\\\\\displaystyle =\\\\mathbb {P}_n(d(x_0,\\\\ldots ,x_n))~\\\\left(\\\\frac{h^{-1}(x_1)}{M(h^{-1})(x_0)}\\\\ldots ~\\\\frac{h^{-1}(x_n)}{M(h^{-1})(x_{n-1})} \\\\right)\\\\end{array}$ with $\\\\mathbb {P}_n(d(x_0,\\\\ldots ,x_n)):=\\\\eta _0(dx_0)~M(x_0,dx_1)~\\\\ldots M(x_{n-1},dx_n).$ This clearly implies that $\\\\begin{array}{l}\\\\mathbb {P}_n(d(x_0,\\\\ldots ,x_n))\\\\\\\\\\\\\\\\\\\\displaystyle =\\\\mathbb {P}^{1/h}_n(d(x_0,\\\\ldots ,x_n))~\\\\left(\\\\left(h(x_1)~M(h^{-1})(x_0)\\\\right)\\\\ldots ~\\\\left(h(x_n)~M(h^{-1})(x_{n-1})\\\\right) \\\\right)\\\\\\\\\\\\\\\\\\\\displaystyle =\\\\mathbb {P}^{1/h}_n(d(x_0,\\\\ldots ,x_n))~\\\\left(\\\\frac{h(x_1)}{M^{1/h}(h)(x_0)}\\\\ldots ~\\\\frac{h(x_n)}{M^{1/h}(h)(x_{n-1})} \\\\right),\\\\end{array}$ where we have used (REF ) in the final step.\",\n \"Rewritten in terms of expectations, the above formula takes the following form $\\\\mathbb {E}\\\\left(F(X_0^{1/h},\\\\ldots ,X_n^{1/h})\\\\prod _{1\\\\le k\\\\le n}\\\\frac{h(X_k^{1/h})}{M^{1/h}(h)(X^{1/h}_{k-1})}\\\\right)=\\\\mathbb {E}\\\\left(F(X_0,\\\\ldots ,X_n)\\\\right).$ Replacing $F(X_0,\\\\ldots ,X_n)$ by $F(X_0,\\\\ldots ,X_n)\\\\prod _{1\\\\le k\\\\le n}H(X_k),$ and again recalling (REF ), we obtain the formula $\\\\begin{array}{l}\\\\displaystyle \\\\mathbb {E}\\\\left(F(X_0,\\\\ldots ,X_n)\\\\prod _{1\\\\le k\\\\le n}H(X_k)\\\\right)\\\\\\\\\\\\\\\\\\\\displaystyle =\\\\mathbb {E}\\\\left(F(X_0^{1/h},\\\\ldots ,X^{1/h}_n)\\\\left(\\\\prod _{1\\\\le k\\\\le n}H(X^{1/h}_k)\\\\right)\\\\left(\\\\prod _{1\\\\le k\\\\le n}\\\\frac{h(X_k^{1/h})}{M^{1/h}(h)(X^{1/h}_{k-1})}\\\\right)\\\\right)\\\\\\\\\\\\\\\\\\\\displaystyle =\\\\mathbb {E}\\\\left(F(X^{1/h}_0,\\\\ldots ,X^{1/h}_n)\\\\left(\\\\prod _{1\\\\le k\\\\le n}(H h)(X^{1/h}_k)\\\\right)\\\\left(\\\\prod _{0\\\\le k< n}M(h^{-1})(X^{1/h}_{k})\\\\right)\\\\right)\\\\\\\\\\\\\\\\\\\\displaystyle =\\\\mathbb {E}\\\\left(F(X^{1/h}_0,\\\\ldots ,X^{1/h}_n)~M(h^{-1})(X^{1/h}_{0})(Hh)(X_{n}^{1/h})\\\\left(\\\\prod _{1\\\\le k< n}(Hh)(X^{1/h}_k)M(h^{-1})(X^{1/h}_{k})\\\\right)\\\\right)\\\\\\\\\\\\\\\\\\\\displaystyle =\\\\mathbb {E}\\\\left(F(X^{1/h}_0,\\\\ldots ,X^{1/h}_n)~\\\\frac{M(h^{-1})(X^{1/h}_{0})}{M(h^{-1})(X^{1/h}_n)}\\\\left(\\\\prod _{1\\\\le k\\\\le n}(Hh)(X^{1/h}_k)M(h^{-1})(X^{1/h}_{k})\\\\right)\\\\right).\\\\end{array}$ This ends the proof of the the Feynman-Kac change of measure formula.\",\n \"The exponential decay estimate (REF ) is now a direct consequence of the regularity condition (REF ).\",\n \"We are now in position to prove the uniform almost sure exponential decays stated in Theorem REF .\",\n \"Proof of first inequality in (REF ) Up to a time change, it suffices to prove the result for $l=1$ and for any given initial condition $p_0=x$ .\",\n \"Recall the Markov transition, $\\\\mathcal {P}$ , of the stochastic Riccati process $p_n$ .\",\n \"We choose in Theorem REF the function $H(x)=\\\\left(\\\\frac{\\\\vert A\\\\vert }{1+Sx}\\\\right)^k\\\\quad \\\\mbox{\\\\rm and}\\\\quad h(x)=x^k.$ Suppose that $\\\\vert A\\\\vert <1$ .\",\n \"In this situation, we have $\\\\mathbb {E}\\\\left(\\\\prod _{1\\\\le l\\\\le n}\\\\left(\\\\frac{\\\\vert A\\\\vert }{1+Sp_l}\\\\right)^k\\\\right)\\\\le (1-\\\\epsilon _k)^n\\\\quad \\\\mbox{\\\\rm with}\\\\quad \\\\epsilon _k:=1-\\\\vert A\\\\vert ^k.$ Now assume that $\\\\vert A\\\\vert \\\\ge 1$ .\",\n \"In this situation, we have $\\\\vert A\\\\vert <(A^2+RS)$ .\",\n \"Observe that $H^h(x)=H(x)~h(x)\\\\mathcal {P}(h^{-1})(x)=\\\\left(\\\\frac{\\\\vert A\\\\vert }{1+Sx}\\\\right)^k~\\\\mathbb {E}\\\\left(\\\\left(\\\\frac{p_0}{p_1}\\\\right)^k~|~p_0=x\\\\right).$ Using (REF ) we check that $H^h(x)=\\\\left(\\\\frac{\\\\vert A\\\\vert }{1+Sx}~\\\\frac{x}{\\\\phi (x)}\\\\right)^k~\\\\mathbb {E}\\\\left(\\\\left(\\\\frac{\\\\phi (p_0)}{p_1}\\\\right)^k~|~p_0=x\\\\right)\\\\le \\\\left(\\\\frac{\\\\vert A\\\\vert }{1+Sx}~\\\\frac{x}{\\\\phi (x)}\\\\right)^k~\\\\left(1+\\\\frac{\\\\iota _k}{N}\\\\right).$ This yields the estimate $H^h(x)\\\\le \\\\left(\\\\frac{\\\\vert A\\\\vert }{(A^2+RS)+R/x}\\\\right)^k~\\\\left(1+\\\\frac{\\\\iota _k}{N}\\\\right)\\\\le \\\\left(\\\\frac{\\\\vert A\\\\vert }{A^2+RS}\\\\right)^k~\\\\left(1+\\\\frac{\\\\iota _k}{N}\\\\right).$ Choosing $N$ such that $\\\\displaystyle N\\\\ge N_{k}:=\\\\iota _k \\\\left(1-\\\\left(\\\\frac{\\\\vert A\\\\vert }{A^2+RS}\\\\right)^k\\\\right)^{-1}$ yields $\\\\left(1+\\\\frac{\\\\iota _k}{N}\\\\right)\\\\le 1+\\\\sqrt{\\\\epsilon _k},\\\\quad \\\\mbox{\\\\rm with}\\\\quad \\\\sqrt{\\\\epsilon _k}:=\\\\left(1-\\\\left(\\\\frac{\\\\vert A\\\\vert }{A^2+RS}\\\\right)^k\\\\right).$ We readily check that $H^h(x)&\\\\le & \\\\left(1-\\\\sqrt{\\\\epsilon _k}\\\\right)\\\\left(1+\\\\sqrt{\\\\epsilon _k}\\\\right)=1-\\\\epsilon _k<1.$ This implies that $\\\\mathbb {E}\\\\left(\\\\left(\\\\prod _{1\\\\le l\\\\le n}\\\\frac{\\\\vert A\\\\vert }{1+Sp_l}\\\\right)^k~|~p_0=x\\\\right)\\\\le \\\\left(1-\\\\epsilon _k\\\\right)^{n-1}\\\\mathbb {E}\\\\left(\\\\frac{ {\\\\cal P }(h^{-1})(x)}{ {\\\\cal P }(h^{-1})(X^{1/h}_{n-1})}\\\\right).$ On the other hand, using (REF ) and (REF ), for any even parameter $N>2(k+2)$ we have $\\\\frac{1}{(A^2/S+R)^k}\\\\le {\\\\cal P }(h^{-1})(x)=\\\\frac{1}{\\\\phi (p_0)^k}~\\\\mathbb {E}\\\\left(\\\\left(\\\\frac{\\\\phi (p_0)}{p_1}\\\\right)^k~|~p_0=x\\\\right)\\\\le \\\\frac{1}{R^k}~\\\\left(1+\\\\frac{\\\\iota _k}{N}\\\\right)$ We conclude that $\\\\mathbb {E}\\\\left(\\\\left(\\\\prod _{1\\\\le l\\\\le n}\\\\frac{\\\\vert A\\\\vert }{1+Sp_l}\\\\right)^k\\\\right)\\\\le \\\\left(1-\\\\epsilon _k\\\\right)^{n-1}~\\\\left(A^2R/S+R^2\\\\right)^k \\\\left(1+\\\\frac{\\\\iota _k}{N}\\\\right).$ This ends the proof of the exponential decays estimates stated in the l.h.s.\",\n \"of (REF ).\"\n ],\n [\n \"Fluctuation analysis\",\n \"Lemma 7.2 For any $k\\\\ge 1$ there exists some parameter $N_k\\\\ge 1$ such that for any $N\\\\ge N_k$ and $n\\\\ge 0$ we have the time uniform estimates $\\\\mathbb {E}\\\\left(\\\\vert m_{n}-X_{n}\\\\vert ^k\\\\right)\\\\vee \\\\mathbb {E}\\\\left(\\\\vert \\\\widehat{m}_{n}-X_{n}\\\\vert ^k\\\\right)\\\\vee \\\\mathbb {E}\\\\left(\\\\vert Y_n-Cm_{n}\\\\vert ^k\\\\right)\\\\le \\\\iota _k.$ Using (REF ) we check the recursion $(m_{n+1}-X_{n+1})=\\\\alpha _n~(X_n-m_n)+\\\\beta _{n+1},$ with the random variables $\\\\alpha _n&:=A(1-g_nC)=\\\\frac{A}{1+Sp_n} \\\\\\\\\\\\beta _{n+1}&:=Ag_nDV_n-BW_{n+1}+\\\\frac{1}{\\\\sqrt{N+1}}~\\\\left(A\\\\widehat{\\\\upsilon }_n+\\\\upsilon _{n+1}\\\\right).$ This yields the formula $(m_{n+1}-X_{n+1})= {\\\\cal E }_{0,n}~(X_0-m_0)+\\\\beta _{n+1}+\\\\sum _{0\\\\le k\\\\le n} {\\\\cal E }_{k,n}~\\\\beta _{k}.$ The raw moment estimates of the differences $(m_{n}-X_{n})$ and $(Y_n-Cm_{n})$ stated in (REF ) are now a direct consequence of the exponential decay estimate stated in (REF ).\",\n \"The raw moment estimates of the difference $(\\\\widehat{m}_{n}-X_{n})$ is easily checked using the following decomposition $\\\\widehat{m}_{n}-X_{n}=\\\\frac{A}{1+Sp_n}~(m_n-X_n)+g_nCDV_n+\\\\frac{1}{\\\\sqrt{N+1}}~\\\\widehat{\\\\upsilon }_n$ This ends the proof of the lemma.\",\n \"Note that this completes the proof of Theorem REF .\",\n \"Now we come to the proof of the Lyapunov estimate (REF ).\",\n \"Proof of (REF ): Combining (REF ) with the uniform almost sure exponential decays stated in (REF ) we check the uniform estimate $\\\\mathbb {E}\\\\left(\\\\vert M_{n+1}\\\\vert ~|~(M_0,p_0)\\\\right)\\\\le \\\\mathbb {E}( {\\\\cal E }_{0,n}~|~p_0)~\\\\vert M_0\\\\vert +\\\\iota _1\\\\le \\\\iota _1+\\\\iota _2 (1-\\\\epsilon _2)^{n}\\\\vert M_0\\\\vert .$ Thus, for any $\\\\epsilon \\\\in ]0,1]$ there exists some time horizon $n_{\\\\epsilon }\\\\ge 1$ such that for any $n\\\\ge n_{\\\\epsilon }$ we have $\\\\mathbb {E}\\\\left(\\\\vert M_{n}\\\\vert ~|~(M_0,p_0)\\\\right)\\\\le \\\\epsilon ~\\\\vert M_0\\\\vert +\\\\iota .$ The estimate (REF ) now comes from the fact that the one-step Riccati map $\\\\phi $ is uniformly bounded.\",\n \"Proof of Theorem REF : First note that $m_{n+1}-\\\\widehat{X}_{n+1}^-=\\\\displaystyle A~(\\\\widehat{m}_{n}-\\\\widehat{X}_n)+\\\\frac{1}{\\\\sqrt{N+1}}~\\\\upsilon _{n+1}.$ We also have the decomposition $\\\\begin{array}{l}\\\\displaystyle (\\\\widehat{m}_{n}-\\\\widehat{X}_{n})\\\\\\\\\\\\\\\\=\\\\displaystyle (m_n-\\\\widehat{X}^-_{n})+(g_n-G_n)~(Y_n-Cm_n)-G_n~C(m_n-\\\\widehat{X}^-_{n})+\\\\frac{1}{\\\\sqrt{N+1}}~\\\\widehat{\\\\upsilon }_n\\\\\\\\\\\\\\\\=\\\\displaystyle (1-G_nC)(m_n-\\\\widehat{X}^-_{n})+\\\\frac{1}{\\\\sqrt{N}}~\\\\sqrt{N}(g_n-G_n)~(Y_n-Cm_n)+\\\\frac{1}{\\\\sqrt{N+1}}~\\\\widehat{\\\\upsilon }_n.\\\\end{array}$ This yields the recursion $(\\\\widehat{m}_{n}-\\\\widehat{X}_{n}) =\\\\alpha _n~(\\\\widehat{m}_{n-1}-\\\\widehat{X}_{n-1})+\\\\frac{1}{\\\\sqrt{N}}~ \\\\beta _n,$ with the parameters $\\\\alpha _n&:=&A(1-G_nC)= \\\\frac{A}{1+SP_n}\\\\\\\\\\\\beta _n&:=&\\\\frac{1}{\\\\sqrt{1+1/N}}~ \\\\left((1-G_nC) \\\\upsilon _{n}+\\\\widehat{\\\\upsilon }_n\\\\right)+\\\\sqrt{N}(g_n-G_n)~(Y_n-Cm_n).$ We conclude that $(\\\\widehat{m}_{n}-\\\\widehat{X}_{n}) =E_n(P_0)~(\\\\widehat{m}_{0}-\\\\widehat{X}_{0})+\\\\frac{1}{\\\\sqrt{N}}~\\\\sum _{1\\\\le l\\\\le n}~ E_{l,n}(P_0)~\\\\beta _l.$ Finally, observe that $(\\\\widehat{m}_{0}-\\\\widehat{X}_{0})=\\\\displaystyle (1-G_0C)(m_0-\\\\widehat{X}^-_{0})+\\\\frac{1}{\\\\sqrt{N}}~\\\\left(\\\\sqrt{N}(g_0-G_0)~(Y_0-Cm_0)+\\\\frac{1}{\\\\sqrt{1+1/N}}~\\\\widehat{\\\\upsilon }_0\\\\right).$ The estimate (REF ) is now a consequence of the exponential decays (REF ), the uniform bounds (REF ) and the mean error estimates stated in Theorem REF .\",\n \"The proof of (REF ) follows similar arguments.\",\n \"Combining (REF ) and (REF ) we check the decomposition $\\\\begin{array}{l}\\\\displaystyle m_{n+1}-\\\\widehat{X}_{n+1}^-=A~ (1-G_nC)(m_n-\\\\widehat{X}^-_{n})+A(g_n-G_n)~(Y_n-C\\\\widehat{X}_n^-)\\\\\\\\\\\\\\\\\\\\hspace{56.9055pt}\\\\displaystyle -\\\\frac{1}{N}~\\\\sqrt{N}~A(g_n-G_n)C~~\\\\sqrt{N}(m_n-\\\\widehat{X}^-_n)+\\\\frac{1}{\\\\sqrt{N+1}}~A\\\\widehat{\\\\upsilon }_n+\\\\frac{1}{\\\\sqrt{N+1}}~\\\\upsilon _{n+1}.\\\\end{array}$ Taking the expectations we obtain the formula $\\\\begin{array}{l}\\\\displaystyle (m_{n+1}^o-\\\\widehat{X}_{n+1}^-)=\\\\alpha _n~(m_n^o-\\\\widehat{X}^-_{n})+\\\\beta _n/N\\\\end{array}$ with the parameters $(\\\\alpha _n,\\\\beta _n)$ now given by $\\\\alpha _n&:=&A(1-G_nC)= {A}/{(1+SP_n)}\\\\\\\\\\\\beta _n&:=&A~N(\\\\mathbb {E}(g_n)-G_n)~(Y_n-C\\\\widehat{X}_n^-)-AC~\\\\mathbb {E}\\\\left(\\\\sqrt{N}~(g_n-G_n)\\\\sqrt{N}(m_n-\\\\widehat{X}^-_n)~|~ {\\\\cal Y }_{n-1}\\\\right).$ Using the uniform estimates stated in Theorem REF and the gain bias estimates (REF ), we obtain, for $k \\\\ge 1$ $\\\\sup _{n\\\\ge 0}\\\\mathbb {E}\\\\left(\\\\vert \\\\beta _n\\\\vert ^k\\\\right)^{1/k}\\\\le \\\\iota _k~(1\\\\vee P_0)^2.$ The end of the proof of (REF ) now follows the same lines of arguments as the proof of the estimates (REF ), thus we leave the details as an exercise for the reader.\",\n \"Proof of Theorem REF : First note that from the proof of (REF ), we have the following decomposition $\\\\sqrt{N}(\\\\widehat{m}_n - \\\\widehat{X}_n) = \\\\mathcal {E}_n(P_0)\\\\sqrt{N}(\\\\widehat{m}_0 - \\\\widehat{X}_0) + \\\\sum _{1 \\\\le \\\\ell \\\\le n}\\\\mathcal {E}_{\\\\ell , n}(P_0)\\\\beta _\\\\ell ,$ where we recall that $\\\\mathcal {E}_{\\\\ell , n}(p) &= \\\\prod _{\\\\ell < k \\\\le n}\\\\frac{A}{1 + SP_\\\\ell (p)},\\\\\\\\\\\\beta _\\\\ell &= \\\\frac{1}{\\\\sqrt{1+1/N}}~ \\\\left((1-G_nC) \\\\upsilon _{n}+\\\\widehat{\\\\upsilon }_n\\\\right)+\\\\sqrt{N}(g_n-G_n)~(Y_n-Cm_n).$ We first deal with $\\\\sqrt{N}(\\\\widehat{m}_0 - \\\\widehat{X}_0)$ on the right hand side of (REF ).\",\n \"Again, from the proof of (REF ), recall that $\\\\sqrt{N}(\\\\widehat{m}_{0}-\\\\widehat{X}_{0})=\\\\displaystyle (1-G_0C)\\\\sqrt{N}(m_0-\\\\widehat{X}^-_{0})+\\\\frac{1}{\\\\sqrt{1+1/N}}~\\\\widehat{\\\\upsilon }_0+\\\\sqrt{N}(g_0-G_0)~(Y_0-Cm_0).$ Further observe that $(Y_0-Cm_0)=(Y_0-C\\\\widehat{X}^-_0)+C(\\\\widehat{X}^-_0-m_0)\\\\quad \\\\mbox{\\\\rm and}\\\\quad (\\\\widehat{X}^-_0-m_0)\\\\longrightarrow _{N\\\\rightarrow \\\\infty } 0\\\\quad \\\\mbox{\\\\rm a.e.\",\n \"}.$ Applying Slutsky's lemma, this yields the weak convergence $\\\\sqrt{N}(g_0-G_0)~C(\\\\widehat{X}^-_0-m_0)\\\\longrightarrow _{N\\\\rightarrow \\\\infty } 0.$ Using the fluctuation theorems REF and REF we also have $\\\\sqrt{N}(m_0-\\\\widehat{X}^-_{0}) &\\\\longrightarrow _{N\\\\rightarrow \\\\infty } \\\\mathbb {U}_0,\\\\\\\\\\\\sqrt{N}(g_0-G_0)(Y_0-C\\\\widehat{X}^-_0) &\\\\longrightarrow _{N\\\\rightarrow \\\\infty } \\\\mathbb {G}_0(Y_0-C\\\\widehat{X}^-_0),\\\\\\\\\\\\frac{1}{\\\\sqrt{1+1/N}}~\\\\widehat{\\\\upsilon }_0 &\\\\longrightarrow _{N\\\\rightarrow \\\\infty } \\\\widehat{\\\\mathbb {U}}_0.$ In the same vein, applying the continuous mapping theorem or the $\\\\delta $ -method (see for instance Theorem 9.3.3 and Lemma 9.3.1 in [36]) we check that $\\\\sqrt{N}(\\\\widehat{m}_{0}-\\\\widehat{X}_{0})\\\\longrightarrow _{N\\\\rightarrow \\\\infty } (1-G_0C) \\\\mathbb {U}_0+\\\\widehat{\\\\mathbb {U}}_0+\\\\mathbb {G}_0(Y_0-C\\\\widehat{X}^-_0).$ To deal with the sum on the right hand side of (REF ), again, thanks to Theorem REF , and arguing as above $\\\\beta _\\\\ell $ converges weakly to $(1-G_{\\\\ell }C)\\\\mathbb {U}_{\\\\ell } + \\\\widehat{\\\\mathbb {U}}_{\\\\ell }+\\\\mathbb {G}_{\\\\ell } (Y_{\\\\ell }-C\\\\widehat{X}_{\\\\ell }^-).$ In the same vein, we check that $\\\\widehat{\\\\mathbb {X}}_n^N &\\\\longrightarrow _{N\\\\rightarrow \\\\infty } \\\\widehat{\\\\mathbb {X}}_n:= \\\\sum _{0 \\\\le \\\\ell \\\\le n}E_{\\\\ell , n}(P_0)\\\\left((1-G_{\\\\ell }C)\\\\mathbb {U}_{\\\\ell } + \\\\widehat{\\\\mathbb {U}}_{\\\\ell }+\\\\mathbb {G}_{\\\\ell } (Y_{\\\\ell }-C\\\\widehat{X}_{\\\\ell }^-)\\\\right).$ The proof of the fluctuation theorem at the level of the process in the sense of convergence of finite dimensional distributions is conducted combining the continuous mapping theorem with Slutsky's lemma.\",\n \"The proof of the recursive formulation (REF ) is also easily checked using the decompositions (REF ) and (REF ).\",\n \"This ends the proof of the theorem.\"\n ],\n [\n \"Proof of Theorem \",\n \"For the identity function $ {\\\\cal U }(p):=p$ , due to (REF ), we have $ {\\\\cal P }( {\\\\cal U })(p)=\\\\mathbb {E}( {\\\\cal U }(p_{n+1})~|~p_n=p)=\\\\phi (p)\\\\le \\\\alpha =A^2/S+R.$ In addition, for any $\\\\epsilon \\\\in ]0,1]$ we have $ {\\\\cal U }_{\\\\epsilon }:=1+\\\\frac{\\\\epsilon }{\\\\alpha }~ {\\\\cal U }\\\\ge 1\\\\Longrightarrow {\\\\cal P }( {\\\\cal U }_{\\\\epsilon })(p)\\\\le 1+\\\\epsilon \\\\Longrightarrow {\\\\cal P }( {\\\\cal U }_{\\\\epsilon })\\\\le 1+\\\\epsilon ~ {\\\\cal U }_{\\\\epsilon }.$ The above estimate implies that $ {\\\\cal U }_{\\\\epsilon }\\\\ge 1$ is a Lyapunov function.\",\n \"On the other hand, $ {\\\\cal P }(p,dq)=k(p,q)~dq$ has a continuous density $k(p,q)>0$ with respect to the variables $(p,q)$ and the Lebesgue measure $dq$ on $\\\\mathbb {R}_+$ .\",\n \"Thus, for any compact set $\\\\varpi \\\\subset \\\\mathbb {R}_+$ there exists some function $\\\\varphi $ from $[0,\\\\infty [$ into itself such that $k_{\\\\varpi }(q):=\\\\inf _{p\\\\in \\\\varpi }k(p,q)=k(\\\\varphi _{\\\\varpi }(q),q)>0 \\\\Longrightarrow 0<\\\\epsilon _{\\\\varpi }:=\\\\int ~k_{\\\\varpi }(q)~dq<1.$ This implies that $ {\\\\cal P }(p,dq) \\\\ge \\\\epsilon _{\\\\varpi }~\\\\nu _{\\\\varpi }(dq)\\\\quad \\\\mbox{\\\\rm with}\\\\quad \\\\nu _{\\\\varpi }(dq):=\\\\frac{k_{\\\\varpi }(q)~dq}{\\\\int ~k_{\\\\varpi }(p)~dp},$ from which we prove the Dobrushin local contraction property $\\\\varpi _r:=[0,r]\\\\Longrightarrow \\\\beta _{r}( {\\\\cal P }):=\\\\sup _{(p,q)\\\\in \\\\varpi _r^2}\\\\Vert {\\\\cal K }(p, \\\\mbox{\\\\LARGE .\",\n \"})- {\\\\cal K }(q, \\\\mbox{\\\\LARGE .})\",\n \"\\\\Vert _{\\\\tiny tv}<1-\\\\epsilon _{\\\\varpi }<1.$ For any $\\\\epsilon _1,\\\\epsilon _2\\\\in ]0,1]$ we set $ {\\\\cal U }_{\\\\epsilon _1,\\\\epsilon _2}:=\\\\epsilon _1 {\\\\cal U }_{\\\\epsilon _2}\\\\Longrightarrow {\\\\cal P }( {\\\\cal U }_{\\\\epsilon _1,\\\\epsilon _2})\\\\le \\\\epsilon _1\\\\left(1+\\\\epsilon _2 {\\\\cal U }_{\\\\epsilon _2}\\\\right)\\\\le \\\\epsilon _1+\\\\epsilon _2~ {\\\\cal U }_{\\\\epsilon _1,\\\\epsilon _2}.$ Choosing the parameters $(r,\\\\epsilon _2)$ such that $(1/r+\\\\epsilon _2)<1$ for any $\\\\epsilon _1\\\\in [0,1[$ we can check that $p,q\\\\ge r\\\\Longrightarrow \\\\frac{\\\\Vert {\\\\cal P }(p, \\\\cdot ) - {\\\\cal P }(q, \\\\cdot ) \\\\Vert _{ {\\\\cal U }_{\\\\epsilon _1,\\\\epsilon _2}}}{1 + {\\\\cal U }_{\\\\epsilon _1,\\\\epsilon _2}(p) + {\\\\cal U }_{\\\\epsilon _1,\\\\epsilon _2}(q)}<1.$ The detailed proof of the above assertion can be found in section 8.2.7 in [39].\",\n \"In addition, for any $x,y\\\\le r$ we also have $\\\\epsilon _1<\\\\frac{1-\\\\beta _r( {\\\\cal K })}{4r}\\\\Longrightarrow \\\\frac{\\\\Vert {\\\\cal P }(x, \\\\cdot ) - {\\\\cal P }(y, \\\\cdot ) \\\\Vert _{ {\\\\cal U }_{\\\\epsilon _1,\\\\epsilon _2}}}{1 + {\\\\cal U }_{\\\\epsilon _1,\\\\epsilon _2}(x) + {\\\\cal U }_{\\\\epsilon _1,\\\\epsilon _2}(y)}\\\\le \\\\beta _r( {\\\\cal K })+4\\\\epsilon _1 r<1.$ We conclude that $(1/r+\\\\epsilon _2)<1\\\\quad \\\\mbox{\\\\rm and}\\\\quad \\\\epsilon _1<\\\\frac{1-\\\\beta _r( {\\\\cal P })}{4r}\\\\quad \\\\Longrightarrow \\\\quad \\\\beta _{ {\\\\cal U }_{\\\\epsilon _1,\\\\epsilon _2}}( {\\\\cal P })<1,$ which concludes the proof.\",\n \"Following [26], we have $\\\\phi ^n(x)+v=\\\\lambda _2~\\\\frac{\\\\left(x+(v-\\\\lambda _1)\\\\right)~\\\\left(1-\\\\lambda ^{n+1}\\\\right)+(\\\\lambda _2-\\\\lambda _1)~\\\\lambda ^{n+1}}{\\\\left(x+(v-\\\\lambda _1)\\\\right)~\\\\left(1-\\\\lambda ^{n}\\\\right)+(\\\\lambda _2-\\\\lambda _1)~\\\\lambda ^{n}}.$ Next, we check that the r.h.s.\",\n \"expression in the above display is non negative.\",\n \"Notice that $\\\\phi ^n(x)=\\\\frac{\\\\left(x+(v-\\\\lambda _1)\\\\right)\\\\left(\\\\lambda _2~\\\\left(1-\\\\lambda ^{n+1}\\\\right)-v~\\\\left(1-\\\\lambda ^{n}\\\\right)\\\\right)-(v-\\\\lambda _1)(\\\\lambda _2-\\\\lambda _1)~\\\\lambda ^{n}}{\\\\left(x+(v-\\\\lambda _1)\\\\right)~\\\\left(1-\\\\lambda ^{n}\\\\right)+(\\\\lambda _2-\\\\lambda _1)~\\\\lambda ^{n}}.$ Rewritten in a slightly different form (REF ) takes the following form $\\\\phi ^n(x)=(\\\\lambda _2-v)+\\\\frac{\\\\left(x+(v-\\\\lambda _1)\\\\right)~-(\\\\lambda _2-\\\\lambda _1)}{\\\\left(x+(v-\\\\lambda _1)\\\\right)~\\\\left(1-\\\\lambda ^{n}\\\\right)+(\\\\lambda _2-\\\\lambda _1)~\\\\lambda ^{n}}~\\\\left(\\\\lambda _2-\\\\lambda _1\\\\right)~\\\\lambda ^n.$ This yields the formula $\\\\phi ^n(x)-r=\\\\frac{x-r}{\\\\left(x+(v-\\\\lambda _1)\\\\right)~\\\\left(1-\\\\lambda ^{n}\\\\right)+(\\\\lambda _2-\\\\lambda _1)~\\\\lambda ^{n}}~\\\\left(\\\\lambda _2-\\\\lambda _1\\\\right)~\\\\lambda ^n,$ which ends the proof of (REF ).\",\n \"Recalling that $\\\\phi $ is an increasing function, we check that $\\\\phi (0)=\\\\frac{b}{d}\\\\le \\\\phi (x)\\\\le \\\\lim _{x\\\\rightarrow \\\\infty }\\\\phi (x)=\\\\frac{a}{c}\\\\quad \\\\mbox{\\\\rm and}\\\\quad r=\\\\phi (r)\\\\le \\\\frac{a}{c},$ which ends the proof of (REF ).\",\n \"For any $x,y\\\\in \\\\mathbb {R}_+$ we also have $\\\\begin{array}{l}\\\\displaystyle \\\\frac{\\\\phi ^n(x)-\\\\phi ^n(y)}{x-y}\\\\\\\\\\\\\\\\\\\\displaystyle =\\\\frac{(\\\\lambda _2-\\\\lambda _1)^2~\\\\lambda ^n}{\\\\left[\\\\left(x+(v-\\\\lambda _1)\\\\right)~\\\\left(1-\\\\lambda ^{n}\\\\right)+(\\\\lambda _2-\\\\lambda _1)~\\\\lambda ^{n}\\\\right]\\\\left[\\\\left(y+(v-\\\\lambda _1)\\\\right)~\\\\left(1-\\\\lambda ^{n}\\\\right)+(\\\\lambda _2-\\\\lambda _1)~\\\\lambda ^{n}\\\\right]}.\\\\end{array}$ This ends the proof of the r.h.s.\",\n \"estimate in (REF ).\",\n \"The proof of the l.h.s.\",\n \"estimate in (REF ) is a direct consequence of the easily checked estimate $\\\\vert \\\\phi ^n(x)-r\\\\vert \\\\le \\\\vert x-r\\\\vert ~\\\\frac{\\\\left(\\\\lambda _2-\\\\lambda _1\\\\right)}{\\\\left(x+(v-\\\\lambda _1)\\\\right)~\\\\wedge (\\\\lambda _2-\\\\lambda _1)}~~\\\\lambda ^n.$ We check (REF ) using the second order formula $\\\\begin{array}{l}\\\\displaystyle \\\\frac{\\\\phi ^n(x)-\\\\phi ^n(y)}{x-y}\\\\\\\\\\\\\\\\\\\\displaystyle =\\\\frac{(\\\\lambda _2-\\\\lambda _1)^2~\\\\lambda ^n}{\\\\left[\\\\left(y+(v-\\\\lambda _1)\\\\right)~\\\\left(1-\\\\lambda ^{n}\\\\right)+(\\\\lambda _2-\\\\lambda _1)~\\\\lambda ^{n}\\\\right]^2}\\\\\\\\\\\\\\\\\\\\displaystyle -\\\\frac{(\\\\lambda _2-\\\\lambda _1)^2~\\\\lambda ^n}{\\\\left[\\\\left(y+(v-\\\\lambda _1)\\\\right)~\\\\left(1-\\\\lambda ^{n}\\\\right)+(\\\\lambda _2-\\\\lambda _1)~\\\\lambda ^{n}\\\\right]^2}\\\\times \\\\left(\\\\frac{(x-y)\\\\left(1-\\\\lambda ^{n}\\\\right)}{\\\\left[\\\\left(x+(v-\\\\lambda _1)\\\\right)~\\\\left(1-\\\\lambda ^{n}\\\\right)+(\\\\lambda _2-\\\\lambda _1)~\\\\lambda ^{n}\\\\right]}\\\\right).\\\\end{array}$ Finally observe that $(\\\\ref {def-rho})\\\\Longrightarrow \\\\partial \\\\phi ^n(x)=\\\\partial (\\\\phi (\\\\phi ^{n-1}(x)))=(\\\\partial \\\\phi )(\\\\phi ^{n-1}(x))~\\\\partial \\\\phi ^{n-1}(x)=\\\\prod _{0\\\\le k 0) and sweep ($u$ > 0, $w$ < 0) events.","Notably, the velocity field associated with the representative $\\Lambda $ -eddy packet (figure REF ) is also consistent with this notion and models features associated with both ejection and sweep events.","Hence, we compute the conditionally averaged wall-parallel flow fields for the coherent regions associated with strong ejections and sweeps by following the same methodology as adopted previously by [52] and [20].","This is performed by attaching a frame of reference to the centroid of each flow feature representing strong $u$ and $w$ fluctuations, which are used as the conditioning points.","Here, the strong fluctuations correspond to $u^+$ $\\lesssim $ -1 and $w^+$ $\\gtrsim $ 1 for ejections and vice versa for the sweeps.","The mean flow feature is obtained by averaging multiple such strong features extracted from several flow fields.","Figure: Instantaneous streamwise velocity fluctuations on a wall-normal plane.","Data in (a) corresponds to the dd-AEM developed in the present study, that in (b) is from the AEM of , while that in (c,d) corresponds to the same ZPG TBL DNS flow field , with difference being (c) comprises of solely the wall-coherent motions (represented by subscript wc) and (d) the full flow field (WC+WI).Figure REF presents the averaged velocity signatures conditioned for the strong ejection features, at the lower bound of the log-region, from flow fields obtained from the AEM, dd-AEM and the DNS.","Firstly, it can be noted that the conditionally averaged fields from the DNS (figures REF (c,f,i)) qualitatively resemble the velocity fields around a single $\\Lambda $ -eddy packet (figure REF ), giving further support to the choice of the representative eddy.","On comparing the geometric extents of the velocity features generated from the two AEMs and the DNS, a better match between the dd-AEM and the DNS can be noted, confirming the enhanced performance of the dd-AEM.","To list a few specific improvements, the long streamwise coherence observed particularly in ${\\langle }{w^+_{wc}}{\\rangle }$ (figure REF (i)) is well replicated by ${\\langle }w^+{\\rangle }$ generated from the dd-AEM (figure REF (h)).","Further, the extent of spanwise coherence for all three velocity components, noted in the dd-AEM, compares better with the DNS than that noted for the AEM.","The streamwise extent of the wall-parallel velocity features, however, are still falling short of the estimates from the DNS.","This may be an artefact of the assumption of scale-independent yaw angles, imposed onto the $\\Lambda $ -eddy packets, to replicate `meandering' phenomena in the TBL [20].","It seems plausible that the scales with the longest streamwise extent (i.e.","${\\lambda }_{x}$ ) `meander' with lesser intensity than those with relatively shorter extent, which may favourably alter the comparison being presented in figure REF .","This, however, remains a subject for future work.","It is also worth noting here that the close match of the conditioned fields from the DNS (figures REF (c,f,i)), with the corresponding flow fields from the $\\Lambda $ -eddy packet (figure REF ), is owing to the consideration of solely the WC motions.","Interested readers may refer to figures REF (d,e,f) where full flow fields (WC + WI) have been conditionally averaged based on the same criteria as in figure REF .","The significant mismatch between the velocity features from the WC and full fields, for the DNS, suggests the present representative eddy packet as a suitable choice to model solely the WC subset of the TBL (and not the full flow as a whole)."],["Discussion","The present effort promises enhancement in the spatial representation of an instantaneous wall flow by means of a data-driven coherent structure-based model.","One would hope that such models, which have the capability to predict very high $Re_{\\tau }$ flows in a computationally inexpensive way, will drive future investigations into flow phenomena otherwise difficult to measure experimentally, such as the instantaneous variation of the $w$ -component over the homogeneous wall-parallel plane (as in figure REF (h)) or that of $v$ -component over the wall-normal plane.","While the data driven-AEM developed here seems a promising prospect in this regard, more work needs to be done in terms of also incorporating the WI motions to be able to predict the full inertially-dominated turbulent flow field.","These WI motions not only are as statistically significant as the WC motions in the inertial region [11], [50], but are also the key driver of the flow phenomena in the wake region.","This is evident from the instantaneous $u$ -velocity fluctuations plotted in the wall-normal plane, which are compared for the AEM, dd-AEM and the DNS in figure REF .","As we would expect, the dd-AEM predictions (figure REF (a)) compare well with the instantaneous flow field from the DNS, comprising solely the WC motions (figure REF (c)).","These motions, however, lose their coherence in the wake region where the WI motions predominate, which can be deciphered from the full DNS field in figure REF (d).","Another aspect of coherent structure-based modelling requiring further work is the shape/form of the representative eddy.","The present choice of the $\\Lambda $ -eddy packet for the AEM, although consistent with the statistical picture ($\\Gamma $ ) obtained empirically, does not reflect the typical flow structure observed in an instantaneous flow.","Efficient machine learning-based algorithms [10], which have the capability to analyze big datasets, may be better placed to propose a solution for this.","Besides assisting with the modelling of the WC motions, the empirically determined geometric scalings in (REF ,REF ) can also be used to enhance the active flow control schemes operating based on real-time sensing.","Figure REF depicts a conceptual sketch showcasing the coherent motions in a ZPG TBL, which can be manipulated by cross-flow (wall-normal) jets controlled by a computer, based on data from a spanwise array of skin-friction sensors placed at a sufficiently upstream location [1].","These skin-friction sensors, which are used to detect the incoming high-momentum ($+u$ ) carrying coherent motions, are also capable of detecting the streamwise extent (${\\lambda }_{x}$ ) of these motions.","Utilizing the empirically estimated scalings, the computer controlling the jet actuation can decide how much momentum to inject through them, based on the estimated wall-normal extent ($z$ ) of the incoming motions (indicated as control system 1 in figure REF ).","Alternatively, in scenarios where sufficiently long separations between the sensors and the jets cannot be maintained, the spanwise sensor array can be used to estimate the spanwise coherence (${\\Delta }y$ ) of the incoming motions, through which the wall-normal extent could be predicted (indicated as control system 2 in figure REF ).","Such empirically-driven flow control systems, which also consider the wall-normal extent of the incoming motions, could potentially manipulate the turbulent inertial wall flow more efficiently, as demonstrated previously by [62].","Given the significant contribution of the attached eddy hierarchy to the mean skin friction at high $Re_{\\tau }$ [23], availability of their corresponding 3-D geometric scalings can also be used to decide the spanwise spacing between the cross-flow jets targeting these eddies.","Figure: Conceptualization of an active-flow control system aimed at efficiently manipulating the wall-coherent inertial motions in a ZPG TBL by utilizing the scalings determined from high Re τ Re_{\\tau } datasets.","Concept of the flow control system has been adapted from .","U ∞ U_{\\infty } denotes mean freestream speed."],["Concluding remarks","The present study analyses a unique set of multi-point datasets to reconstruct the 3-D statistical picture of the inertial wall-coherent (WC) turbulence in all three canonical wall-bounded flows.","Previous studies have found these motions to be responsible for both, the $Re_{\\tau }$ -dependence of the skin-friction drag [45], [14], [23], [54] as well as the bulk production and the inter-scale energy transfer in high $Re_{\\tau }$ flows [38], [35], [29].","The aforementioned characteristics make these motions a key target of coherent structure-based models, which the present study attempts to enhance.","Here, the statistical picture is reconstructed by computing the cross-correlation spectra ($\\Gamma $ ) using the streamwise velocity fluctuations mapped across the 3-D space in the wall-bounded shear flow.","The intermediate- and large-scaled inertial WC motions are found to exhibit geometric self-similarity with respect to the distance from the wall $z$ , expressible by simple linear relationships, which are universal across all canonical flows and independent of flow $Re_{\\tau }$ .","The geometry of the very-large-scaled motions, on the other hand, is found to exhibit $\\delta $ -scaling, associating them with the superstructures and VLSMs for external and internal flows, respectively.","The present study also confirms the 3-D geometry of the VLSMs to be much larger than the superstructures even in high $Re_{\\tau }$ flows, highlighting the role of the T/NTI in inhibiting the growth of these very-large-scaled motions in external flows [44], [33].","Alongside testing the universality of the 3-D statistical picture, the geometric scalings brought out from the data (given by REF and REF ) are also proposed to be used as a metric to estimate the spanwise and wall-normal extent corresponding to each WC scale/eddy (${\\lambda }_{x}$ ).","Application of these linear relationships to any coherent structure-based model can provide a data-driven basis to the geometry of the representative structure, which is demonstrated in this study using the attached eddy model (AEM; [46]) for a ZPG TBL.","Here, the choice of AEM (over other models) is also driven by the empirical analysis, given the latter provides evidence of geometric self-similarity of the inertial motions and also supports the $\\Lambda $ -eddy packet as the representative eddy (both of which are built into the AEM framework; [39]).","The present study further extends the scope of the AEM by using its framework to also model the $\\delta $ -scaled superstructures, which correspond to the very-large-scales beyond the self-similar hierarchy.","The data-driven AEM (or dd-AEM), which has the representative eddies defined based on the data, is shown to improve upon recent works on the AEM [20] in replicating instantaneous flow phenomena associated with all three velocity components.","This sets up the platform for future work, which would include both the WC and WI (wall-incoherent) motions modelled via the AEM framework, to replicate the full turbulent wall-bounded flow that can provide realistic inflow conditions for use in future numerical simulations [55], [61].","Figure: Instantaneous (a) streamwise, (b) spanwise and (c) wall-normal velocity fluctuations on a wall-parallel plane at the lower bound of the log-region (zz ≈\\approx 0.05δ\\delta for the present Re τ Re_{\\tau }) extracted from ZPG TBL DNS dataset (𝒯 1 {\\mathcal {T}}_{1}) of .","Unlike figures or , these flow fields comprise contributions from both WC and WI motions, i.e.","they are the raw flow fields directly from the dataset.","(d) Streamwise, (e) spanwise and (f) wall-normal velocity fluctuations conditioned for u + u^+ ≲\\lesssim -1 and w + w^+ ≳\\gtrsim 1 on the same wall-parallel plane as in (a-c)."],["Acknowledgement","The authors wish to acknowledge the Australian Research Council for financial support.","They are also grateful to the authors of [49], [4] and [7] for making their respective data available.","Sandia National Laboratories is a multimission laboratory managed and operated by National Technology and Engineering Solutions of Sandia, LLC., a wholly owned subsidiary of Honeywell International, Inc., for the U.S. Department of Energy's National Nuclear Security Administration under contract DE-NA0003525.","This paper describes objective technical results and analysis.","Any subjective views or opinions that might be expressed in the paper do not necessarily represent the views of the U.S. Department of Energy or the United States Government."],["SLSE-based decomposition of the instantaneous flow fields","As discussed previously in $§$, the flow field in the inertial region comprises contributions from both wall-coherent (WC) and wall-incoherent (WI) eddies.","Here, we demonstrate a methodology, based on the spectral linear stochastic estimate (SLSE) approach, to estimate the WC subset of a velocity fluctuation (say for $u$ -component) at any wall-normal location $z$ in the inertial region.","This procedure has been utilized in previous studies [3], [36], [17] for similar purposes (i.e.","to obtain a subset of the full flow field comprising selected coherent motions), which can be directly referred for an elaborate introduction to this technique.","According to SLSE, a scale-specific unconditional input at any near-wall location, $z_{w}$ (with $z^+_w$ $\\lesssim $ 15) can be used to obtain a scale-specific conditional input at $z$ following: ${\\widetilde{u}}^{E}(z^{+};{\\lambda }^{+}_{x},{\\lambda }^{+}_{y}) = {{H^{u}_{L}}(z^{+},z^{+}_{w};{\\lambda }^{+}_{x},{\\lambda }^{+}_{y})}\\widetilde{u}(z^{+}_{w};{\\lambda }^{+}_{x},{\\lambda }^{+}_{y}),$ where ${\\tilde{u}}({z_{w}};{\\lambda }_{x},{\\lambda }_{y})$ is the 2-D Fourier transform of the instantaneous wall-parallel flow field, $u$ (${z_w};x,y$ ) in space.","Here, the superscript $E$ and $H^{u}_{L}$ respectively represent the estimated quantity and the scale-specific linear transfer kernel (for $u$ -component).","$u^{E}$ in equation (REF ), essentially, corresponds to the energy distribution (at $z$ ) across wavelengths ${\\lambda }_{x}$ and ${\\lambda }_{y}$ , which are coherent across $z$ and $z_w$ .","This is facilitated by first computing $H^{u}_{L}$ from an ensemble of data as per: ${H^{u}_{L}}(z^{+},z^{+}_{w};{\\lambda }^{+}_{x},{\\lambda }^{+}_{y}) = \\frac{ \\lbrace \\widetilde{u}(z^{+};{\\lambda }^{+}_{x},{\\lambda }^{+}_{y}){{\\widetilde{u}}^{\\ast }}(z^{+}_{w};{\\lambda }^{+}_{x},{\\lambda }^{+}_{y}) \\rbrace }{ \\lbrace {\\widetilde{u}(z^{+}_{w};{\\lambda }^{+}_{x},{\\lambda }^{+}_{y})}{{\\widetilde{u}}^{\\ast }(z^{+}_{w};{\\lambda }^{+}_{x},{\\lambda }^{+}_{y})} \\rbrace },$ where the curly brackets ($\\lbrace \\rbrace $ ) and asterisk ($\\ast $ ) denote the ensemble averaging and complex conjugate, respectively.","Given that $z^+_{w}$ $\\lesssim $ 15 and $z^+_{w}$ $\\ll $ $z^+$ , ${\\widetilde{u}}^{E}(z^{+};{\\lambda }^{+}_{x},{\\lambda }^{+}_{y})$ thus represents energy contributions at $z$ from solely WC motions, that are taller than $z$ [36], [17], leading to: ${{{{\\widetilde{u}}^{E}}(z^{+};{\\lambda }^{+}_{x},{\\lambda }^{+}_{y})}{{\\big |}_{{z^{+}_{w}} {\\lesssim } 15}}}\\; {\\rightarrow }\\; {{{\\widetilde{u}}_{wc}}(z^{+};{\\lambda }^{+}_{x},{\\lambda }^{+}_{y})},$ where $u_{wc}$ ($z$ ) represents the subset of $u$ ($z$ ) comprising solely the WC motions.","On obtaining ${{\\widetilde{u}}_{wc}}$ , the corresponding flow field in physical space can be obtained by: ${{{u}_{wc}}(z;x,y)} = {\\mathcal {F}^{-1}}({{{\\widetilde{u}}_{wc}}(z;{\\lambda }_{x},{\\lambda }_{y})}),$ where ${\\mathcal {F}^{-1}}$ represents the inverse Fourier transform.","Here, although the entire procedure has been demonstrated solely for $u$ -fluctuations, the same can be applied to other (lateral) velocity fluctuations by estimating the respective linear transfer kernel ($H^{v}_{L}$ or $H^{w}_{L}$ ).","To give an example, figures REF (a-c) plot the instantaneous wall-parallel flow fields, comprising both WC + WI motions (i.e.","the full flow fields) at the lower bound of the inertial region.","While, figures REF (b,d,f) plot the subset of the corresponding full fields comprising solely the WC motions, estimated using the aforementioned procedure."]],"string":"[\n [\n \"Data-driven enhancement of coherent structure-based models for\\n predicting instantaneous wall turbulence\"\n ],\n [\n \"Abstract Predictions of the spatial representation of instantaneous wall-bounded flows, via coherent structure-based models, are highly sensitive to the geometry of the representative structures employed by them.\",\n \"In this study, we propose a methodology to extract the three-dimensional (3-D) geometry of the statistically significant eddies from multi-point wall-turbulence datasets, for direct implementation into these models to improve their predictions.\",\n \"The methodology is employed here for reconstructing a 3-D statistical picture of the inertial wall coherent turbulence for all canonical wall-bounded flows, across a decade of friction Reynolds number ($Re_{\\\\tau}$).\",\n \"These structures are responsible for the $Re_{\\\\tau}$-dependence of the skin-friction drag and also facilitate the inner-outer interactions, making them key targets of structure-based models.\",\n \"The empirical analysis brings out the geometric self-similarity of the large-scale wall-coherent motions and also suggests the hairpin packet as the representative flow structure for all wall-bounded flows, thereby aligning with the framework on which the attached eddy model (AEM) is based.\",\n \"The same framework is extended here to also model the very-large-scaled motions, with a consideration of their differences in internal versus external flows.\",\n \"Implementation of the empirically-obtained geometric scalings for these large structures into the AEM is shown to enhance the instantaneous flow predictions for all three velocity components.\",\n \"Finally, an active flow control system driven by the same geometric scalings is conceptualized, towards favourably altering the influence of the wall coherent motions on the skin-friction drag.\"\n ],\n [\n \"Introduction\",\n \"Owing to its highly chaotic and random nature, an accurate prediction of the instantaneous velocity in a turbulent flow remains the most challenging demand from any turbulence model.\",\n \"Despite its inherent complexities, turbulence flow modelling has remained an active area of research for over half a century [10], given the numerous incentives on offer.\",\n \"In case of wall-bounded flows, for example, the ability to predict/replicate the instantaneous flow phenomena can greatly facilitate the design of active flow control techniques [12], and also aid in enhancing future high-fidelity numerical simulations, by providing realistic inflow boundary conditions and improving computational efficiency [55], [61].\",\n \"One of the popular approaches of modelling wall-turbulence, amongst many, has been by focusing on the recurring and statistically significant `coherent' motions [48], [12] omnipresent in these flows.\",\n \"These motions play a crucial role in both the kinematics as well as dynamics of wall-bounded flows [48], [30], [35] and have been directly associated with the behaviour of several flow statistics – both averaged [21], [27], [26] and instantaneous [2], [51], [52].\",\n \"Hence, several studies proposing coherent structure-based models can be found in the literature [56], [24], [59], [57], [13], amongst which the attached eddy model (AEM; [46], [39]) based on Townsend's attached eddy hypothesis [57], is one of the most cited.\",\n \"Here, the words `structures', `motions' and `eddies' are used interchangeably and essentially conform to the definition of a coherent motion given by [48].\",\n \"According to Townsend's attached eddy hypothesis, an inviscid asymptotically high friction Reynolds number ($Re_{\\\\tau }$ $=$ $\\\\frac{{U_{\\\\tau }}{\\\\delta }}{\\\\nu }$ ) canonical wall-bounded flow can be modelled via a hierarchy of geometrically self-similar attached eddies, having population densities inversely proportional to their sizes, which vary over ${\\\\mathcal {O}}$ ($z$ ) – ${\\\\mathcal {O}}$ ($\\\\delta $ ).\",\n \"The term `attached' here refers to any coherent motion whose geometric extent scales with its distance from the wall, given by $z$ , while the associated velocity fluctuations scale with the friction velocity $U_{\\\\tau }$ , but doesn't necessarily imply its velocity signatures physically extending down to the wall [39].\",\n \"Further, $\\\\delta $ here corresponds to the outer scale of the respective canonical flow geometry, which is the boundary layer thickness for a zero-pressure gradient turbulent boundary layer (ZPG TBL), while it is the pipe radius and channel half-height for internal flows.\",\n \"Increased access to high $Re_{\\\\tau }$ wall-turbulence datasets, over the past two decades, has confirmed that the kinetic energy production and transfer within these flows is mostly influenced by the inertial (inviscid) motions predominating in the inertial region [38], [35], [29], making AEM ideally suited for modelling these flows.\",\n \"These characteristics also make the inertial motions a key target of active flow control schemes [1].\",\n \"Studies in the past have already demonstrated that the AEM is able to predict ensemble/time-averaged statistics, such as the Reynolds stresses [40], [6], higher order moments [60], [53], two-point correlations [37] and the energy spectra [5], [16], [11] across the inertial layer (100 $\\\\lesssim $ $z^+$ $\\\\lesssim $ 0.15$Re_{\\\\tau }$ ) in high $Re_{\\\\tau }$ canonical flows.\",\n \"While there have been a few studies focused on predicting the instantaneous flow phenomena in both the inertial and the wake region [51], [52], [20], success has mostly been limited to the wake region, predominated solely by the largest hierarchy of eddies ($\\\\sim $ $\\\\mathcal {O}$ ($\\\\delta $ )).\",\n \"The primary challenge associated with accurately predicting the representative instantaneous flow in the inertial region, is its high sensitivity to the geometry of the coexisting motions, given that the region is populated by a host of statistically significant eddies, with sizes varying from ${\\\\mathcal {O}}$ ($z$ ) to ${\\\\mathcal {O}}$ ($\\\\delta $ ) [34], [16], [63].\",\n \"These eddies are highly three-dimensional (3-D) in geometry, and hence vary considerably in terms of their spatial coherence, which also needs to be accounted for while attempting to replicate the instantaneous flow.\",\n \"To this end, there are several studies in recent literature [63], [50], [11] which have envisioned and subsequently characterized the inertial region as a superposition of two types of inertial motions: (i) motions with their velocity signatures (i.e.\",\n \"coherence) extending all the way down to the wall (i.e.\",\n \"wall coherent or WC motions) and (ii) those with velocity signatures not extending to the wall (wall incoherent; WI).\",\n \"Interestingly, both these eddy types (WC and WI) have been shown by the same set of studies to depict characteristics consistent with Townsend’s attached eddies, although with certain caveats; for example, apart from the hierarchy of self-similar eddies, the WC inertial motions also comprise $\\\\delta $ -scaled superstructures or very-large-scale motions (VLSMs), which do not conform to Townsend's attached eddies [16], [63].\",\n \"These results, thus, showcase the prospect of predicting the instantaneous wall-bounded flow by using the AEM framework to individually model the WC and WI motions in the inertial region.\",\n \"Figure: (a) Schematic depicting the representative Λ\\\\Lambda -eddy packet used in the AEM , , with the blue and red iso-contours respectively denoting the negative and positive streamwise velocity induced from the Λ\\\\Lambda -shaped vortex rods (indicated in green).\",\n \"(b) Streamwise, (c) spanwise and (d) wall-normal velocity induced by the Λ\\\\Lambda -eddy packet in the reference wall-parallel plane at zz = 0.01z i z_{i} (highlighted in grey) shown in (a).\",\n \"Labels in (a) are used to refer to the 3-D geometry of the packet, with Δy i {\\\\Delta }y_{i} representing the spanwise half-width, owing to the symmetry of the uu-distribution about the yy = 0 plane.\",\n \"This figure has been adapted from .\",\n \"Mean flow direction is along xx.In the present study, we take the first step towards achieving this by establishing an AEM-based methodology to model the inertial WC motions in all three canonical wall-bounded flows.\",\n \"Specifically, to improve its predictive capability, the geometry of the attached eddies is based on estimates extracted directly from published experimental and numerical datasets.\",\n \"It is worth noting here that although the inertial WC motions are only a subset of the full flow, these motions are responsible for the $Re_{\\\\tau }$ -dependence of the wall shear stress fluctuations (and hence, the skin-friction drag characteristics) in any wall-bounded flow [45], [14], [23], [1], [54].\",\n \"Further to that, the WC motions are also known to superimpose onto and modulate the near-wall cycle [27], [41], which has a significant impact on the skin friction drag.\",\n \"This makes the present modelling effort relevant for both fundamental as well as applied research.\",\n \"We begin by first reviewing the current state of the AEM ($§$REF ) and its limitations ($§$), followed by proposal of the new methodology ($§$), which provides a data-driven basis to improve the AEM predictions.\",\n \"Throughout this paper, we use the coordinate system $x$ , $y$ and $z$ to refer to the streamwise, spanwise and wall-normal directions respectively, with $u$ , $v$ and $w$ denoting the corresponding fluctuating velocity components.\",\n \"${\\\\langle }{\\\\rangle }$ and capitalization indicates averaged quantities while the superscript $+$ refers to normalization in viscous units (eg., $u^+$ $=$ $u$ /$U_{\\\\tau }$ and $z^+$ $=$ $z{U_{\\\\tau }}/{\\\\nu }$ , where $\\\\nu $ is the kinematic viscosity).\",\n \"In the original AEM conceptualized by [46], a single hairpin or a simple arch-shaped ($\\\\Lambda $ ) eddy, inclined forwards with respect to the flow direction ($x$ ) at $\\\\sim $ 45$^{\\\\circ }$ [25], [18], was considered as the representative coherent structure/eddy.\",\n \"This shape was inspired from the seminal flow visualization studies of [25] in a low $Re_{\\\\tau }$ TBL.\",\n \"The simplest version of this eddy is essentially made up of two vortex rods, arranged in a $\\\\Lambda $ -shape, with each rod containing a Gaussian distribution of vorticity about its core.\",\n \"The corresponding velocity field for the eddy can be obtained by performing the Biot-Savart calculations (schematically depicted in figures 2, 6 and 7 of [46]).\",\n \"Over time, the representative eddy shape has evolved as more detailed quantitative measurements, revealing the structure of wall-bounded flows were reported, such as the seminal PIV measurements of [2].\",\n \"While this study highlighted the presence of $\\\\Lambda $ -type eddies in the TBL, it was found that the eddies are organized in the form of a packet in fully turbulent flows.\",\n \"The use of a $\\\\Lambda $ -eddy packet (figure REF (a)), instead of a single eddy, as the representative attached eddy was subsequently incorporated into the AEM by [37], and this change was shown to further improve statistical predictions.\",\n \"Figures REF (b-d) depict the near-wall ($z$ = 0.01$z_{i}$ ) planar flow field of the three velocity components associated with a single packet-like eddy, which is simply a superposition of the velocity fields obtained from multiple individual $\\\\Lambda $ -eddies.\",\n \"In case of the AEM, it should be noted that $w$ = 0 $\\\\ne $ $u$ , $v$ at $z$ = 0 due to the impermeability condition being the only condition imposed at the wall, which is achieved by using packet structures with image packet pairs in the plane of the wall.\",\n \"To model the inertially dominated (i.e.\",\n \"outer) region, the flow is simply represented by the superposition of multiple such $\\\\Lambda $ -eddy packets (referred as hierarchies), of varying sizes and population density, randomly distributed in the flow domain.\",\n \"Following Townsend's hypothesis, the geometry of these hierarchies is considered to vary self-similarly, a claim which has received both empirical [4], [16], [63], [28] and theoretical [47], [42] support in the literature.\",\n \"The recent study by [20] is the latest published AEM flow-field configuration (henceforth referred simply as AEM) used to replicate the instantaneous flow-field for a ZPG TBL at $Re_{\\\\tau }$ $\\\\approx $ 3200.\",\n \"In this study, the major axis of the $\\\\Lambda $ -eddy packet was oriented at various angles (with respect to $x$ ) along the $x$ -$y$ plane, to incorporate the `meandering' characteristics of the eddies noted in experiments [27], [31].\",\n \"To summarize, while the meandering features and the eddy inclination angles have been chosen based on the empirical estimates [18], [31], the aspect ratios governing the representative eddy geometry (marked in figure REF (a)) have never had an empirical basis to date, which forms the motivation for the present study.\",\n \"The eddy geometry plays a major role in the visual appearance of the instantaneous flow field generated by the AEM, as will be highlighted in the next section ($§$).\",\n \"The present study, hence, follows in line with previous studies [53], [20] with the aim of enhancing the AEM framework, in order to improve spatial representation of a wall-bounded flow from the model.\",\n \"We note that the typical shape of the representative eddy (of a $\\\\Lambda $ -eddy packet) has been kept the same with the intention to preserve its low-order complexity, and given its past success in replicating turbulent wall flow statistics [60], [6], [5], [16], [11].\",\n \"Further, the present shape is also well suited to model the wall-coherent subset of the wall-bounded flows, given that all three velocity components generated from the eddy extend down to the near-wall region, (figures REF (b-d)).\",\n \"Figure: Instantaneous (a,b) streamwise, (c,d) spanwise and (e,f) wall-normal velocity fluctuations on a wall-parallel plane at the lower bound of the inertial-region (zz ≈\\\\approx 0.05δ\\\\delta for the present Re τ Re_{\\\\tau }).\",\n \"Data in (a,c,e) corresponds to the AEM of , while that in (b,d,e) corresponds to the ZPG TBL DNS of comprising solely of wall-coherent motions (represented by subscript wc).\",\n \"Black dashed lines are used to highlight the largest spatial features of low momentum for a qualitative comparison.\",\n \"Note the difference in axis limits between (a-d) and (e-f).\"\n ],\n [\n \"Motivation for a data-driven AEM\",\n \"Here, we compare the instantaneous flow fields generated by the AEM [20] with the corresponding flow fields from the direct numerical simulation (DNS) of a ZPG TBL [49] to motivate a data-driven definition of the representative eddy geometry used in the former.\",\n \"Given that the AEM simulates purely the WC portion of the TBL, a logical way to go about this would be by comparing it with the subset of the full DNS fields comprising solely the WC motions.\",\n \"Figure REF presents this comparison between the instantaneous velocity fluctuations for all three components, considered along the wall-parallel plane at the lower bound of the inertial region.\",\n \"It brings out significant differences in the geometry of the coherent motions, with the velocity features in the DNS fields substantially longer than those noted in the AEM fields.\",\n \"A plausible reason behind this difference may be the geometry of a $\\\\Lambda $ -eddy packet not defined based on empirical estimates, which we aim to facilitate in the present study.\",\n \"Here, we propose to obtain the geometric estimates by reconstructing the 3-D statistical picture of the wall-coherent turbulence from published datasets ($§$).\",\n \"Another noteworthy limitation in the AEM is that all the coherent structures considered in the current model correspond to the hierarchy of self-similar eddies, following Townsend's hypothesis.\",\n \"A true WC flow field, however, cannot be replicated without also considering the $\\\\delta $ -scaled superstructures or VLSMs [16], [63], which we also intend to include in the model.\",\n \"Interestingly, these motions can also be modelled via the same representative $\\\\Lambda $ -eddy packet shown in figure REF (a), as demonstrated recently by [11] for modelling the 2-D spectra.\",\n \"Readers may note here that the DNS flow fields in figures REF (b,d,f), which comprise solely the WC motions, have been computed using the publicly available full (WC+WI) fields via a spectral linear stochastic estimation (SLSE) based decomposition technique.\",\n \"Interested readers may refer to appendix for more details on how the WC fields were estimated, and also see figure REF , where the full flow field instants corresponding to the WC fields in figure REF have been plotted.\",\n \"Differences between the full (figure REF ) and the WC (figure REF ) flow fields are substantial, especially for the lateral velocity components, with large spatial coherence observed in case of the latter as compared to the former.\",\n \"Further, the oblique features otherwise apparent in the full spanwise velocity field in the log-region (figure REF (b); [49],[52]), cannot be noted in its WC subset (figure REF (d)).\",\n \"These differences underscore the importance of comparing the AEM fields with solely the WC motions, which the representative eddy models.\"\n ],\n [\n \"Datasets and methodology\",\n \"The present study utilizes five previously published multi-point datasets, spanning all three canonical flow geometries.\",\n \"Key parameters of the datasets are summarised in table REF .\",\n \"Dataset ${\\\\mathcal {T}_{1}}$ corresponds to DNS of ZPG TBL, while ${\\\\mathcal {T}_{2}}$ and ${\\\\mathcal {T}_{3}}$ correspond to higher $Re_{\\\\tau }$ experimental data for the same flow geometry.\",\n \"${\\\\mathcal {C}_{1}}$ and ${\\\\mathcal {P}_{1}}$ are respectively the DNS and experimental datasets for a fully turbulent channel and pipe flow.\",\n \"The selected datasets present a unique combination of synchronously acquired $u$ -fluctuations mapped across 3-D space in the wall-bounded shear flow (refer figures REF (a,b) and REF (a-c)).\",\n \"${\\\\mathcal {T}_{1}}$ , ${\\\\mathcal {C}_{1}}$ and ${\\\\mathcal {P}_{1}}$ each comprise $u$ -fluctuations acquired using multiple near-wall fixed probes (placed at $x_w$ $=$ 0,$y_w$ ,$z_w$ ), distributed in log spacing along the spanwise/azimuthal directions, in conjunction with those acquired by the probe traversing along the wall-normal direction ($z$ ).\",\n \"On the other hand, ${\\\\mathcal {T}_{2}}$ and ${\\\\mathcal {T}_{3}}$ comprise $u$ -fluctuations from a single near-wall fixed probe (placed at $x_w$ $=$ 0,$y_w$ $=$ 0,$z_w$ ), acquired synchronously with those measured farther from the wall by a probe traversing along the $z$ - and $y$ -directions, respectively.\",\n \"Considering the fact that the canonical wall-bounded flows are statistically homogeneous along the span, the cumulative 3-D space mapped by combining the datasets ${\\\\mathcal {T}_{2}}$ and ${\\\\mathcal {T}_{3}}$ would be equivalent to that mapped in each of the individual datasets, ${\\\\mathcal {T}_{1}}$ , ${\\\\mathcal {C}_{1}}$ and ${\\\\mathcal {P}_{1}}$ .\",\n \"In case of ${\\\\mathcal {P}}_{1}$ , it should be noted that the near-wall probe is essentially a hot-film sensor attached to the wall for measuring the instantaneous skin-friction velocity ${\\\\vec{U}}_{\\\\tau }$ (= ${U}_{\\\\tau }$ + ${u_{\\\\tau }}$ ), which is associated with the near-wall instantaneous streamwise velocity (${\\\\vec{U}}$ ) following ${\\\\vec{U}}_{\\\\tau }$ = $\\\\sqrt{{\\\\nu }({{\\\\partial }{\\\\vec{U}}}/{{\\\\partial }z})}$ .\",\n \"Interested readers may refer to the individual references for more specific details regarding the measurements/simulations.\",\n \"Table: Table summarizing the details of five previously published datasets comprising synchronized measurements of uu-fluctuations by various near-wall fixed probes (placed at x w x_{w} = 0,y w y_{w},z w z_{w}) and a traversing probe (xx = 0,yy,zz), in turn separated by relative spanwise (Δy{\\\\Delta }y == ∣y-y w ∣{\\\\mid {y}\\\\;{-}\\\\;{y_w}\\\\mid }) and wall-normal (z-z w {z}\\\\;{-}\\\\;{z_w} ≈\\\\approx zz) offsets.In case of dataset 𝒫 1 {\\\\mathcal {P}}_{1}, the hot-film attached to the wall measures the instantaneous skin-friction velocity, U → τ {\\\\vec{U}}_{\\\\tau } == ν(∂U →/∂z)\\\\sqrt{{\\\\nu }({{\\\\partial }{\\\\vec{U}}}/{{\\\\partial }z})}.The probe arrangements associated with each of these datasets have been schematically depicted in figures and , with probes in blue and red respectively denoting the traversing and near-wall fixed probes.We now move on to proposing the methodology used to reconstruct the 3-D statistical picture of the wall-coherent turbulence, from which the geometric estimates for the representative eddy (of the AEM) would be extracted.\",\n \"There have been several studies in the past [21], [27], [32] which have used multi-point datasets to estimate the geometry of the coherent motions, by computing mostly space-time cross-correlations.\",\n \"These cross-correlations, however, represent cumulative contributions from eddies of various length scales at a specific spatial offset (say streamwise offset, ${\\\\Delta }x$ ).\",\n \"Here, since we are interested in estimating the 3-D geometry of individual hierarchies/length scales to be incorporated in the AEM, we compute the cross-correlations in the spectral domain to get a scale-specific estimate of the coherent structure geometry (i.e.\",\n \"as a function of the streamwise wavelength, ${\\\\lambda }_{x}$ ).\",\n \"Given our focus is on modelling the inertial wall-coherent motions coexisting in the outer region, we consider the cross-correlations specifically between $u$ -signals in the outer region (at $z$ ) and those acquired close to the wall (at $z^{+}_{w}$ $\\\\lesssim $ 15), via the cross-correlation spectra ($\\\\Gamma $ ) defined as [8], [19]: $\\\\begin{aligned}{{\\\\Gamma }}({z},{{\\\\Delta }{y}},{\\\\lambda _{x}}) &= \\\\frac{ {\\\\text{Re}}[\\\\lbrace \\\\widetilde{u}(z,y;\\\\lambda _{x}){{\\\\widetilde{u}}^{\\\\ast }}(z_{w},{y_{w}};\\\\lambda _{x}) \\\\rbrace ]}{ {\\\\sqrt{ \\\\lbrace {\\\\mid {\\\\widetilde{u}(z,y;\\\\lambda _{x})} \\\\mid }^{2} \\\\rbrace }}{\\\\sqrt{ \\\\lbrace {\\\\mid {\\\\widetilde{u}({z_{w}},{y_{w}};\\\\lambda _{x})} \\\\mid }^{2} \\\\rbrace }}}\\\\\\\\ &= \\\\frac{ \\\\text{Re}[ { {{\\\\phi }^{\\\\prime }_{{u}{u_{w}}}}(z,z_{w},{\\\\Delta }y;\\\\lambda _{x}) } ] }{ {\\\\sqrt{\\\\phi _{{u}{u}}({z},y;\\\\lambda _{x})}}{\\\\sqrt{\\\\phi _{{u_{w}}{u_{w}}}({z_{w}},{y_{w}};\\\\lambda _{x})}} },\\\\end{aligned}$ where ${\\\\tilde{u}}$ = ${\\\\mathcal {F}}$ ($u$ ) is the Fourier transform of $u$ in either time or $x$ depending on the dataset with ${\\\\lambda }_{x}$ $=$ 2${\\\\pi }/{k_x}$ , where $k_x$ is the streamwise wavenumber.\",\n \"Further, the curly brackets ($\\\\lbrace \\\\rbrace $ ), asterisk ($\\\\ast $ ) and vertical bars ($\\\\mid \\\\mid $ ) indicate the ensemble averaging, complex conjugate and modulus, respectively while Re denotes the real component.\",\n \"${\\\\phi }^{\\\\prime }_{{u}{u_{w}}}$ is, thus, the complex-valued 1-D cross-spectrum between $u$ -signals recorded at $z$ and $z_{w}$ , which quantifies the scale-specific cross-correlation, while $\\\\phi _{{u}{u}}$ and $\\\\phi _{{u_{w}}{u_{w}}}$ respectively are the conventional 1-D energy spectra at ${z}$ and ${z_{w}}$ used for the scale-specific normalization.\",\n \"Such a definition forces $\\\\Gamma $ to vary between -1 to 1, with the former and latter respectively indicating perfect anti-correlation and correlation for each scale, ${\\\\lambda }_{x}$ .\",\n \"In case of dataset ${\\\\mathcal {P}_{1}}$ , notably, $\\\\Gamma $ is computed by substituting the skin-friction velocity fluctuations ($u_{\\\\tau }$ ) measured by the wall-based hot-films, in place of $u$ ($z_w$ ).\",\n \"As can be noted from its definition in (REF ), $\\\\Gamma $ is a function of the spatial offsets in all three directions, which is facilitated by computing the cross-correlations between $u$ -signals recorded at various relative spanwise offsets (${\\\\Delta }y$ $=$ ${\\\\mid }{y - {y_w}}{\\\\mid }$ ) and wall-normal offsets ($z$ $\\\\approx $ ${z - {z_w}}$ , given $z_{w}$ $\\\\ll $ $z$ ).\",\n \"The offset along $x$ is obtained in the form of a streamwise wavelength, which is estimated by using the Taylor's hypothesis in case of the experimental datasets (where the local convection velocity, $U_{c}$ $=$ $U(z)$ ), or obtained directly from the spatial domain for the DNS datasets.\",\n \"$\\\\Gamma $ ($z$ ,${\\\\Delta }y$ ,${\\\\lambda }_{x}$ ), thus, can be interpreted as a coherence metric which associates each WC scale/eddy (${\\\\lambda }_{x}$ ) with its corresponding spanwise (${\\\\Delta }y$ ) and wall-normal extent ($z$ ).\",\n \"This information can be directly used to define the geometry of the representative eddy, for the respective coherent structure-based model, corresponding to the streamwise scale, ${\\\\lambda }_{x}$ .\",\n \"The benefit in case of models such as the AEM, which incorporate vorticity-based structures, is that once the eddy geometry of the $\\\\Lambda $ -eddy packet is defined, velocity field for all three components can be simulated without needing to estimate $\\\\Gamma $ for the other (lateral) velocity components.\",\n \"Figure: (a,b) Experimental setups corresponding to datasets (a) 𝒯 3 {\\\\mathcal {T}}_{3} and (b) 𝒯 2 {\\\\mathcal {T}}_{2}.\",\n \"Hotwire sensors marked in blue correspond to traversing probes, which move along their respective traversing directions also marked in blue, while sensors marked in red correspond to the fixed near-wall probes.\",\n \"(c) Cross-correlation spectra computed as a function of Δy{\\\\Delta }y and λ x {\\\\lambda }_{x} (i.e.\",\n \"Γ\\\\Gamma (Δy{\\\\Delta }y,λ x {\\\\lambda }_{x})) for zz ≈\\\\approx 0.01δ\\\\delta from the dataset 𝒯 3 {\\\\mathcal {T}}_{3}.\",\n \"(d) Γ\\\\Gamma (zz,λ x {\\\\lambda }_{x}) computed for Δy{\\\\Delta }y ≈\\\\approx 0 from the dataset 𝒯 2 {\\\\mathcal {T}}_{2}.\",\n \"Black contour in (c,d) corresponds to Γ\\\\Gamma = 0.1.\",\n \"(e-g) Reconstruction of the 3-D cross-correlation spectra, Γ\\\\Gamma (zz,Δy{\\\\Delta }y,λ x {\\\\lambda }_{x}) across the inertial region (0.01δ\\\\delta ≲\\\\lesssim zz ≲\\\\lesssim 0.15δ\\\\delta ) by fusing the empirical estimates from datasets 𝒯 2 {\\\\mathcal {T}}_{2} and 𝒯 3 {\\\\mathcal {T}}_{3}.\",\n \"(e) is obtained by simply combining data in (c) and (d).\",\n \"(f) and (g) are similarly obtained by fusing data in (d) and that estimated from dataset 𝒯 3 {\\\\mathcal {T}}_{3} over a range of zz: (f) 0.01δ\\\\delta ≲\\\\lesssim zz ≲\\\\lesssim 0.05δ\\\\delta and (g) 0.01δ\\\\delta ≲\\\\lesssim zz ≲\\\\lesssim 0.15δ\\\\delta .\",\n \"Black iso-contour in (e-g) also corresponds to Γ\\\\Gamma = 0.1.\",\n \"Dash-dotted lines in green, orange, indigo and grey colours represent the scalings: λ x {\\\\lambda }_{x} = 14zz, λ x {\\\\lambda }_{x} = 20Δy{\\\\Delta }y, Δy{\\\\Delta }y = 0.17δ\\\\delta and λ x {\\\\lambda }_{x} = 4δ\\\\delta , respectively.It is worth noting here that $\\\\Gamma $ being used here is different from the linear coherence spectrum (LCS; ${\\\\gamma }^{2}_{L}$ ) computed previously by [4] and [7], since $\\\\Gamma $ retains solely the real part of the cross-spectrum while ${\\\\gamma }^{2}_{L}$ uses the absolute value of the cross-spectrum.\",\n \"Here, we prefer $\\\\Gamma $ over ${\\\\gamma }^2_{L}$ , to define the eddy geometry, given that the anti-correlated regions ($\\\\Gamma $ $<$ 0) are indicative of the relative placement of low-momentum ($-u$ ) regions with respect to the high-momentum ($+u$ ) regions, which otherwise can't be inferred from the LCS (0 $\\\\le $ ${\\\\gamma }^2_{L}$ $\\\\le $ 1).\",\n \"For example, a recent study [19] focusing on the $\\\\Gamma $ -distribution over very large spatial extents has confirmed the periodic distribution of $\\\\delta $ -scaled $+u$ and $-u$ motions along the spanwise direction.\",\n \"Investigating the $\\\\Gamma $ distribution in 3-D space, thus, could provide quantitative basis to the choice of the representative eddy for a coherent structure-based model, as will be discussed ahead in this section.\"\n ],\n [\n \"Application to high $Re_{\\\\tau }$ ZPG TBL datasets\",\n \"We begin the empirical analysis by first reconstructing $\\\\Gamma $ ($z,{{\\\\Delta }y},{{\\\\lambda }_{x}}$ ) for high $Re_{\\\\tau }$ ZPG TBL using the two experimental datasets, ${\\\\mathcal {T}}_{3}$ and ${\\\\mathcal {T}}_{2}$ .\",\n \"In case of ${\\\\mathcal {T}}_{2}$ , the traversing probe is constrained to move vertically above the near-wall fixed probe (i.e.\",\n \"maintaining ${\\\\Delta }y$ $=$ 0), owing to which this dataset yields the statistical picture solely in the streamwise wall-normal plane, given by $\\\\Gamma $ (${{\\\\Delta }y}$ $=$ 0$;z,{{\\\\lambda }_{x}}$ ) in figure REF (d).\",\n \"This plot brings out the range of energetic WC scales/motions coexisting in the outer region, which correspond to the wavelength range, ${\\\\lambda }_{x}$ $\\\\gtrsim $ 0.1$\\\\delta $ .\",\n \"For a specific energetic length scale, say ${\\\\lambda }_{x}$ $\\\\sim $ 4$\\\\delta $ , it indicates the presence of a positively correlated region up to $z$ $\\\\sim $ 0.3$\\\\delta $ , followed by a negatively correlated region between 0.3$\\\\delta $ $<$ $z$ $<$ 0.6$\\\\delta $ .\",\n \"When interpreted physically in terms of the spatial extent of the $u$ -velocity distributions around a $\\\\Lambda $ -eddy packet of length $\\\\sim $ 4$\\\\delta $ (figure REF (a)), the aforementioned $z$ -ranges may be respectively associated with the wall-normal extent of the $-u$ region (represented by +$\\\\Gamma $ ) and of the $+u$ region (represented by -$\\\\Gamma $ ).\",\n \"Consequently, the height of the $\\\\Lambda $ -eddy packet may be nominally defined based on the region where $\\\\Gamma $ $\\\\approx $ 0, which is approximately $z$ $\\\\sim $ 0.3$\\\\delta $ .\",\n \"If we consider this for the entire range ${\\\\lambda }_{x}$ $\\\\gtrsim $ 0.1$\\\\delta $ , figure REF (d) indicates that the height ($z$ ) of the inertially dominated eddies varies self-similarly with respect to their streamwise length scale (${\\\\lambda }_{x}$ ), which is given by the linear relationship ${\\\\lambda }_{x}$ $=$ 14$z$ (indicated by a dashed-dotted green line fitted to $\\\\Gamma $ $\\\\approx $ 0).\",\n \"Interestingly, the same linear relationship between ${\\\\lambda }_{x}$ and $z$ was noted by [4] for a ZPG TBL, although based on ${\\\\gamma }^2_{L}$ as the metric.\",\n \"The same analysis is next conducted along the wall-parallel plane, at $z$ corresponding to the inertial-region, by using the dataset ${\\\\mathcal {T}}_{3}$ .\",\n \"Figure REF (c) plots $\\\\Gamma $ ($z$ $=$ 0.01$\\\\delta $ ;${\\\\Delta }y$ ,${\\\\lambda }_{x}$ ) as an example, which corresponds to the 2-D statistical picture at the lower bound of the inertial region.\",\n \"This picture, however, represents the spanwise coherence only along the $+{\\\\Delta }y$ axis and would be complete by considering a mirror image of the same plot about ${\\\\Delta }y$ = 0, owing to symmetry [27].\",\n \"Given that the $\\\\Gamma $ -distribution in figure REF (c) is of the same nature as in figure REF (d), the physical interpretation discussed for the latter is now extended to the former.\",\n \"By fitting a dashed-dotted orange line along $\\\\Gamma $ $\\\\approx $ 0, we can interpret from figure REF (c) that the spanwise half-width (${\\\\Delta }y$ ) of the $\\\\Lambda $ -eddy packet also varies self-similarly with respect to ${\\\\lambda }_x$ , which is represented by the linear relationship ${\\\\lambda }_{x}$ $=$ 20${\\\\Delta }y$ .\",\n \"This self-similar trend, however, is only valid for streamwise wavelengths up to ${\\\\lambda }_{x}$ $\\\\lesssim $ 4$\\\\delta $ , as all the larger length scales are found to have a constant half-width, ${\\\\Delta }y$ $\\\\sim $ 0.17$\\\\delta $ .\",\n \"These large eddies, hence, do not conform to Townsend's hierarchy of attached eddies but, in fact, correspond to the $\\\\delta $ -scaled very-large-scales or superstructures noted previously in the ZPG TBL [27], [32].\",\n \"The spanwise widths of these structures, which is found from the present analysis to be 2${\\\\Delta }y$ $\\\\sim $ 0.35$\\\\delta $ , is consistent with previous findings based on PIV experiments [15], [22].\",\n \"Following the analysis on the individual 2-D planes, the plots in figures REF (c,d) can now be stitched together to reconstruct the 3-D statistical picture of the WC motions in the ZPG TBL, as shown in figure REF (e).\",\n \"By estimating the cross-correlation spectra at various $z$ : 0.01$\\\\delta $ $\\\\lesssim $ $z$ $\\\\lesssim $ 0.15$\\\\delta $ from dataset ${\\\\mathcal {T}}_{3}$ , $\\\\Gamma $ ($z$ ,${\\\\Delta }y$ ,${\\\\lambda }_{x}$ ) can be reconstructed across the inertial region, as depicted in figures REF (e-g).\",\n \"These figures, thus, describe the scale-specific geometry of the range of energetic motions coexisting in the inertial region: the flow at the beginning of the inertial region ($z$ $\\\\sim $ 0.01$\\\\delta $ ) comprises contributions from the widest hierarchy of self-similar eddies, spanning 0.1$\\\\delta $ $\\\\lesssim $ ${\\\\lambda }_{x}$ $\\\\lesssim $ 4$\\\\delta $ , which reduces to 0.8$\\\\delta $ $\\\\lesssim $ ${\\\\lambda }_{x}$ $\\\\lesssim $ 4$\\\\delta $ at $z$ $\\\\approx $ 0.05$\\\\delta $ , and finally negligible contributions from the self-similar hierarchy at the upper bound ($z$ $\\\\approx $ 0.15$\\\\delta $ ).\",\n \"The contributions from the $\\\\delta $ -scaled superstructures (${\\\\lambda }_{x}$ $\\\\gtrsim $ 4$\\\\delta $ ), on the other hand, is consistently present throughout the inertial region.\",\n \"The corresponding spanwise and wall-normal extents for each eddy (${\\\\lambda }_{x}$ ) can also be inferred based on the scalings shown in figure REF (g), and have been summarized as a function of the flow $Re_{\\\\tau }$ below: ${{\\\\lambda }^{+}_x} {\\\\gtrsim } 4{Re_{\\\\tau }} \\\\; {\\\\left\\\\lbrace \\\\begin{array}{ll}{z^{+}} = {{\\\\lambda }^{+}_x}/14; {\\\\Delta }y^{+} = 0.17{Re_{\\\\tau }} & \\\\text{if true}\\\\\\\\\\\\end{array}{z^{+}} = {{\\\\lambda }^{+}_x}/14; {\\\\Delta }y^{+} = {{\\\\lambda }^{+}_x}/20 & \\\\text{otherwise}.\\\\right.\",\n \"}$ This information can be directly used to facilitate definition of the representative eddy geometries in any coherent structure-based model.\",\n \"It is worth noting here though, $\\\\Gamma $ ($z$ ,${\\\\Delta }y$ ,${\\\\lambda }_{x}$ ) in figure REF (g) looks consistent with the shape of a $\\\\Lambda $ -eddy packet when halved along the span (refer to figure REF or REF ), thus providing quantitative support to the choice of the representative eddy shape used in the AEM and making a strong case in favour of implementing (REF ) into AEM.\"\n ],\n [\n \"Extension to various $Re_{\\\\tau }$ and flow geometries\",\n \"Having described the methodology to reconstruct ${\\\\Gamma }$ ($z$ ,${\\\\Delta }y$ ,${{\\\\lambda }_{x}}$ ) and applied it to high $Re_{\\\\tau }$ ZPG TBL data, we now extend the same to datasets at different $Re_{\\\\tau }$ (${\\\\mathcal {T}}_{1}$ ) and other flow geometries (${\\\\mathcal {C}}_{1}$ and ${\\\\mathcal {P}}_{1}$ ) to check for the universality of the scalings noted in figure REF .\",\n \"As discussed previously, each of these datasets are self-sufficient to reconstruct the 3-D statistical picture across the inertial region.\",\n \"Figure REF (d) plots the same for the low $Re_{\\\\tau }$ ZPG TBL data using similar plotting style as in figure REF (g).\",\n \"Interestingly, all the scalings noted previously from the high $Re_{\\\\tau }$ experimental data (expressed in (REF )) are also observed for the low $Re_{\\\\tau }$ case, confirming the $Re_{\\\\tau }$ -invariance of these empirical estimates and consequently, their direct utilization in data-driven coherent structure-based models.\",\n \"Further, the spanwise half-width (${\\\\Delta }y$ ) is also found to tend towards a constant ($\\\\sim $ 0.17$\\\\delta $ ) at ${\\\\lambda }_{x}$ $\\\\gtrsim $ 4$\\\\delta $ for the DNS dataset, confirming that the wavelength range estimated for the $\\\\delta $ -scaled superstructures is independent of the Taylor's hypothesis assumption.\",\n \"It is worth noting here that the low $Re_{\\\\tau }$ for the dataset ${\\\\mathcal {T}}_{1}$ reduces the thickness of the inertial region (0.05$\\\\delta $ $\\\\lesssim $ ${\\\\lambda }_{x}$ $\\\\lesssim $ 0.15$\\\\delta $ ) and consequently the range of energetic scales (${\\\\lambda }_{x}$ ) corresponding to the self-similar hierarchy, which is most evident at $z$ $\\\\approx $ 0.05$\\\\delta $ .\",\n \"Figure: Reconstruction of the 3-D cross-correlation spectra, Γ\\\\Gamma (zz,Δy{\\\\Delta }y,λ x {\\\\lambda }_{x}) across the inertial region for datasets (d) 𝒯 1 {\\\\mathcal {T}}_{1}, (e) 𝒞 1 {\\\\mathcal {C}}_{1} and (f) 𝒫 1 {\\\\mathcal {P}}_{1} by utilizing the uu-fluctuations associated with the probe placements depicted for the respective flow geometries in (a-c).\",\n \"Probes marked in blue correspond to traversing probe which move along their respective traversing directions also marked in blue, while those marked in red correspond to the fixed near-wall probes.\",\n \"Here, only two near-wall probes are shown for representative purposes, with the actual number given in table .\",\n \"Black iso-contour in (d-f) corresponds to Γ\\\\Gamma = 0.1.\",\n \"Dash-dotted lines in green, orange, indigo and grey colours represent scalings indicated in each figure.Figures REF (e,f) respectively depict ${\\\\Gamma }$ ($z$ ,${\\\\Delta }y$ ,${{\\\\lambda }_{x}}$ ) reconstructed using the high $Re_{\\\\tau }$ channel (${\\\\mathcal {C}}_{1}$ ) and pipe flow (${\\\\mathcal {P}}_{1}$ ) datasets.\",\n \"Remarkably, the same self-similar scalings for the WC motions, noted previously for ZPG TBL, are also noted for the case of internal flows.\",\n \"Similarly, it is found that the contribution from the self-similar hierarchies becomes statistically insignificant beyond the upper bound of the inertial region, in both the internal flows.\",\n \"Although not shown here, ${\\\\Gamma }$ ($z$ ,${\\\\Delta }y$ ,${{\\\\lambda }_{x}}$ ) (similar to figure REF (f)) was also reconstructed for the relatively low $Re_{\\\\tau }$ data ($\\\\approx $ 10000 and 20000) acquired at the same facility by [7], which exhibited the same scalings/behaviour.\",\n \"This universality in the self-similar scaling, across all three canonical flows, is consistent with the observation of [28] based on low $Re_{\\\\tau }$ ($\\\\approx $ 1000) DNS datasets.\",\n \"Given that the estimation of the exact linear scalings at low $Re_{\\\\tau }$ may be limited by the narrower range of the self-similar hierarchy, the present study confirms the existence as well as the $Re_{\\\\tau }$ -invariance of these self-similar scalings across all three canonical flows.\",\n \"In case of the internal flows, however, the ${\\\\lambda }_{x}$ $=$ 20${\\\\Delta }y$ relationship is found to be valid up to a much larger length scale (${\\\\lambda }_{x}$ $\\\\approx $ 6$\\\\delta $ ), beyond which the constant spanwise half-width tends to a constant, ${\\\\Delta }y$ $\\\\sim $ 0.3$\\\\delta $ .\",\n \"The scalings describing the 3-D spatial extent of the inertial motions, in case of the internal flows, can thus be expressed as a function of $Re_{\\\\tau }$ following: ${{\\\\lambda }^{+}_x} {\\\\gtrsim } 6{Re_{\\\\tau }} \\\\; {\\\\left\\\\lbrace \\\\begin{array}{ll}{z^{+}} = {{\\\\lambda }^{+}_x}/14; {\\\\Delta }y^{+} = 0.30{Re_{\\\\tau }} & \\\\text{if true}\\\\\\\\\\\\end{array}{z^{+}} = {{\\\\lambda }^{+}_x}/14; {\\\\Delta }y^{+} = {{\\\\lambda }^{+}_x}/20 & \\\\text{otherwise}.\\\\right.\",\n \"}$ These trends suggest that the $\\\\delta $ -scaled VLSMs (${\\\\lambda }_{x}$ $\\\\gtrsim $ 6$\\\\delta $ ) conforming to the internal flow geometries are relatively wider than the superstructures in ZPG TBL [43].\",\n \"These differences between the internal and external flows have been previously noted by [9], [44] and [33] and were attributed to the `persistent growth' of the energetic motions, beyond the inertial region, in case of internal flows, which is otherwise inhibited by the turbulent/non-turbulent interface (T/NTI) in the ZPG TBL [44], [33].\"\n ],\n [\n \"Incorporating the empirically-obtained scaling laws into the AEM\",\n \"With the 3-D statistical picture of the wall-coherent turbulence now characterized for all three canonical flows, we move onto discussing how the empirically obtained scalings can be used in the data-driven AEM (dd-AEM).\",\n \"In the remainder of this manuscript, we concentrate our efforts on the attached eddy modelling of solely the ZPG TBL flow.\",\n \"The methodology, however, can also be applied to the internal flows by using the appropriate empirical estimates (given in (REF )).\",\n \"Although a real ZPG TBL flow comprises of a continuous distribution of statistically energetic motions over the scale range: ${\\\\lambda }_{x}$ $\\\\gtrsim $ 0.1$\\\\delta $ (figure REF (g)), for the case of the AEM, we consider a discretized distribution of eddies across $n$ hierarchies for convenience in modelling.\",\n \"Here, $n$ depends on the $Re_{\\\\tau }$ of the flow being modelled, with higher $Re_{\\\\tau }$ requiring the inclusion of more hierarchies/scales.\",\n \"We follow the convention adopted by [46], of defining the wall-normal extent of the smallest hierarchy equal to the beginning of the inertial region ($z^+$ $\\\\sim $ 100), with subsequent larger hierarchies doubling in height.\",\n \"Hence, the wall-normal extent of the `$i^{th}$ ' hierarchy (figure REF (b)) may be defined as $z_i$ = ${\\\\delta }$ (2$^{i-n}$ ), with $i$ varying from 1 (for the smallest hierarchy) to $n$ (for the largest hierarchy), with $z_{n}$ $=$ $\\\\delta $ by definition.\",\n \"For the present study, we consider a ZPG TBL to be modelled via $n$ $=$ 6 hierarchies.\",\n \"Hence, the height of the smallest ($i$ $=$ 1) hierarchy becomes $z_1$ $=$ ${\\\\delta }/{2^5}$ or $z^+_1$ $=$ ${Re_{\\\\tau }}/{2^5}$ .\",\n \"Enforcing $z^+_1$ $\\\\approx $ 100 following [46] yields the $Re_{\\\\tau }$ $\\\\approx $ 3200 for the dd-AEM.\",\n \"This $Re_{\\\\tau }$ is equal to that considered for the previously published AEM by [20], and is also close to the $Re_{\\\\tau }$ $\\\\approx $ 2000 of the ZPG TBL of [49].\",\n \"To extract the geometric estimates for the eddy packet (figure REF (b)) corresponding to each of the six hierarchies, figure REF (a) shows the same 3-D statistical picture as in figure REF (g) discretized into six individual blocks.\",\n \"Here, the $x$ -axis location of the $i^{th}$ block represents the streamwise length scale (${\\\\lambda }_{x_{i}}$ ) of the $i^{th}$ hierarchy, while the corresponding extents along the $y-$ and $z-$ axis respectively indicate its spanwise half-width (${\\\\Delta }y_{i}$ ) and the wall-normal height ($z_{i}$ ).\",\n \"It is worth noting here that the eddy packet in figure REF (b), with seven $\\\\Lambda $ -eddies, is simply shown for representative purposes and the actual number of $\\\\Lambda $ -eddies in the packet may vary depending on ${\\\\lambda }_{x_{i}}$ /$z_{i}$ (table REF ), such that the inter-eddy spacing is maintained constant for both AEM and dd-AEM.\",\n \"Table REF records the geometric estimates of all six hierarchies for the dd-AEM, and compares them with those considered in the previous AEM by [20].\",\n \"The size of hierarchies 1 to 4 grows self-similarly in all three directions, in accordance to the empirically obtained scalings, which associates them with Townsend's attached eddies.\",\n \"Hierarchies 5 and 6, on the other hand, have a constant spanwise half-width (${\\\\Delta }y$ $\\\\sim $ 0.17$\\\\delta $ ).\",\n \"The two largest hierarchies, thus, represent the $\\\\delta $ -scaled superstructures, which were found to be the sole energetic WC motions coexisting beyond the upper bound of the inertial region (figure REF (g)).\",\n \"The presence of these $\\\\delta $ -scaled eddies, which do not conform to the self-similar hierarchy, is another improvement of the dd-AEM over the AEM, given that all six hierarchies in the latter conformed to the self-similar hierarchy (table REF ).\",\n \"Table: Comparing the geometry of the various eddy hierarchies in the AEM and dd-AEM.Here, SS and NSS respectively denote self-similar and non-self-similar.\",\n \"Other terminologies have been defined in figure (b).Λ\\\\Lambda -eddy packets corresponding to the AEM and dd-AEM comprise of 7 and 30 Λ\\\\Lambda -eddies respectively, to account for the differing λ x i {\\\\lambda }_{x_{i}}/z i z_{i}.Figure: Cross-correlation spectra computed along (a-c) XZ-plane at Δy{\\\\Delta }y ≈\\\\approx 0, Γ\\\\Gamma (Δy{\\\\Delta }y ≈\\\\approx 0;zz,λ x {\\\\lambda }_{x}), (d-f) XY-plane at the lower bound of the inertial region, Γ\\\\Gamma (zz ≈\\\\approx 0.01δ\\\\delta ;Δy{\\\\Delta }y,λ x {\\\\lambda }_{x}) and (g-i) XY-plane at the upper bound of the inertial region, Γ\\\\Gamma (zz ≈\\\\approx 0.15δ\\\\delta ;Δy{\\\\Delta }y,λ x {\\\\lambda }_{x}) from the (a,d,g) AEM of , (b,e,h) data driven-AEM and (c,f,i) datasets 𝒯 2 {\\\\mathcal {T}}_{2} and 𝒯 3 {\\\\mathcal {T}}_{3}.\",\n \"Dash-dotted lines in green, orange, indigo and grey colours represent the same scalings as described in figure .\",\n \"The plots in (c,f) represent the same data as in figures (d,c) respectively, with the only difference being the change in axis limits to match with the plots from the AEM.The best way to quantify the impact of the changed eddy aspect ratios (table REF ) in the AEM would be to compute the cross-correlation spectra ($\\\\Gamma $ ) using both AEM and dd-AEM fields and comparing it with the empirical estimate.\",\n \"This comparison is showcased in figure REF , where $\\\\Gamma $ is considered along the XZ plane (figures REF (a-c)), and that along two wall-parallel planes at the lower (figures REF (d-f)) and upper bound of the inertial region (figures REF (g-i)).\",\n \"Here, for convenience in interpretation, $\\\\Gamma $ computed from both the AEMs is considered only in the ${\\\\lambda }_{x}$ -range encompassing all six hierarchies (table REF ).\",\n \"We can note a significant difference between $\\\\Gamma $ estimated from the AEM and the dd-AEM fields, with the latter following the scaling trends consistent with those obtained empirically.\",\n \"Based on the comparison along both the wall-parallel planes (figures REF (d-i)), we can expect the dd-AEM to perform much better than the AEM throughout the inertial region.\"\n ],\n [\n \"Spatial representation in the inertial region\",\n \"We now extend the comparison between the dd-AEM and the datasets to the instantaneous velocity fluctuations in the inertial region.\",\n \"Figure REF presents this comparison between the instantaneous fields for all three velocity fluctuations estimated from the dd-AEM, AEM and the ZPG TBL DNS dataset (${\\\\mathcal {T}}_{1}$ ) at the lower bound of the inertial region.\",\n \"Amongst these, the plots associated with the latter two are the same as shown previously in figure REF , with significant differences noted between them ($§$).\",\n \"Now, with the geometry of the representative eddy defined based on data in the dd-AEM, along with the inclusion of the $\\\\delta $ -scaled superstructures, the corresponding instantaneous velocity fields (figures REF (b,e,h)) can be observed to closely match the DNS fields.\",\n \"In particular, the extended coherence of the velocity features ($\\\\gtrsim $ 2$\\\\delta $ ) noted in all three velocity components in the DNS have been well replicated by the dd-AEM, highlighting the impact of basing the eddy geometry on the data.\",\n \"This reaffirms the fact that all three components are essentially inter-dependent, and can be modelled via a representative eddy comprising multiple vortex structures (figure REF ).\",\n \"The meandering very-large-scale motions or superstructures which extend beyond 6$\\\\delta $ in length [27], [32], as observed in the DNS (figure REF (c)), are also well represented by the dd-AEM (figure REF (b)).\",\n \"The qualitative agreement between the dd-AEM and the DNS, showcased in figure REF , can be confirmed quantitatively by generating averaged flow fields conditioned on a statistically dominant feature in the flow.\",\n \"One such statistical flow feature, which is predominant in the inertial region, is the negative value of the instantaneous momentum flux ($uw$ $<$ 0; [58], [17]), which is carried by coherent motions associated with ejection ($u$ < 0, $w$ > 0) and sweep ($u$ > 0, $w$ < 0) events.\",\n \"Notably, the velocity field associated with the representative $\\\\Lambda $ -eddy packet (figure REF ) is also consistent with this notion and models features associated with both ejection and sweep events.\",\n \"Hence, we compute the conditionally averaged wall-parallel flow fields for the coherent regions associated with strong ejections and sweeps by following the same methodology as adopted previously by [52] and [20].\",\n \"This is performed by attaching a frame of reference to the centroid of each flow feature representing strong $u$ and $w$ fluctuations, which are used as the conditioning points.\",\n \"Here, the strong fluctuations correspond to $u^+$ $\\\\lesssim $ -1 and $w^+$ $\\\\gtrsim $ 1 for ejections and vice versa for the sweeps.\",\n \"The mean flow feature is obtained by averaging multiple such strong features extracted from several flow fields.\",\n \"Figure: Instantaneous streamwise velocity fluctuations on a wall-normal plane.\",\n \"Data in (a) corresponds to the dd-AEM developed in the present study, that in (b) is from the AEM of , while that in (c,d) corresponds to the same ZPG TBL DNS flow field , with difference being (c) comprises of solely the wall-coherent motions (represented by subscript wc) and (d) the full flow field (WC+WI).Figure REF presents the averaged velocity signatures conditioned for the strong ejection features, at the lower bound of the log-region, from flow fields obtained from the AEM, dd-AEM and the DNS.\",\n \"Firstly, it can be noted that the conditionally averaged fields from the DNS (figures REF (c,f,i)) qualitatively resemble the velocity fields around a single $\\\\Lambda $ -eddy packet (figure REF ), giving further support to the choice of the representative eddy.\",\n \"On comparing the geometric extents of the velocity features generated from the two AEMs and the DNS, a better match between the dd-AEM and the DNS can be noted, confirming the enhanced performance of the dd-AEM.\",\n \"To list a few specific improvements, the long streamwise coherence observed particularly in ${\\\\langle }{w^+_{wc}}{\\\\rangle }$ (figure REF (i)) is well replicated by ${\\\\langle }w^+{\\\\rangle }$ generated from the dd-AEM (figure REF (h)).\",\n \"Further, the extent of spanwise coherence for all three velocity components, noted in the dd-AEM, compares better with the DNS than that noted for the AEM.\",\n \"The streamwise extent of the wall-parallel velocity features, however, are still falling short of the estimates from the DNS.\",\n \"This may be an artefact of the assumption of scale-independent yaw angles, imposed onto the $\\\\Lambda $ -eddy packets, to replicate `meandering' phenomena in the TBL [20].\",\n \"It seems plausible that the scales with the longest streamwise extent (i.e.\",\n \"${\\\\lambda }_{x}$ ) `meander' with lesser intensity than those with relatively shorter extent, which may favourably alter the comparison being presented in figure REF .\",\n \"This, however, remains a subject for future work.\",\n \"It is also worth noting here that the close match of the conditioned fields from the DNS (figures REF (c,f,i)), with the corresponding flow fields from the $\\\\Lambda $ -eddy packet (figure REF ), is owing to the consideration of solely the WC motions.\",\n \"Interested readers may refer to figures REF (d,e,f) where full flow fields (WC + WI) have been conditionally averaged based on the same criteria as in figure REF .\",\n \"The significant mismatch between the velocity features from the WC and full fields, for the DNS, suggests the present representative eddy packet as a suitable choice to model solely the WC subset of the TBL (and not the full flow as a whole).\"\n ],\n [\n \"Discussion\",\n \"The present effort promises enhancement in the spatial representation of an instantaneous wall flow by means of a data-driven coherent structure-based model.\",\n \"One would hope that such models, which have the capability to predict very high $Re_{\\\\tau }$ flows in a computationally inexpensive way, will drive future investigations into flow phenomena otherwise difficult to measure experimentally, such as the instantaneous variation of the $w$ -component over the homogeneous wall-parallel plane (as in figure REF (h)) or that of $v$ -component over the wall-normal plane.\",\n \"While the data driven-AEM developed here seems a promising prospect in this regard, more work needs to be done in terms of also incorporating the WI motions to be able to predict the full inertially-dominated turbulent flow field.\",\n \"These WI motions not only are as statistically significant as the WC motions in the inertial region [11], [50], but are also the key driver of the flow phenomena in the wake region.\",\n \"This is evident from the instantaneous $u$ -velocity fluctuations plotted in the wall-normal plane, which are compared for the AEM, dd-AEM and the DNS in figure REF .\",\n \"As we would expect, the dd-AEM predictions (figure REF (a)) compare well with the instantaneous flow field from the DNS, comprising solely the WC motions (figure REF (c)).\",\n \"These motions, however, lose their coherence in the wake region where the WI motions predominate, which can be deciphered from the full DNS field in figure REF (d).\",\n \"Another aspect of coherent structure-based modelling requiring further work is the shape/form of the representative eddy.\",\n \"The present choice of the $\\\\Lambda $ -eddy packet for the AEM, although consistent with the statistical picture ($\\\\Gamma $ ) obtained empirically, does not reflect the typical flow structure observed in an instantaneous flow.\",\n \"Efficient machine learning-based algorithms [10], which have the capability to analyze big datasets, may be better placed to propose a solution for this.\",\n \"Besides assisting with the modelling of the WC motions, the empirically determined geometric scalings in (REF ,REF ) can also be used to enhance the active flow control schemes operating based on real-time sensing.\",\n \"Figure REF depicts a conceptual sketch showcasing the coherent motions in a ZPG TBL, which can be manipulated by cross-flow (wall-normal) jets controlled by a computer, based on data from a spanwise array of skin-friction sensors placed at a sufficiently upstream location [1].\",\n \"These skin-friction sensors, which are used to detect the incoming high-momentum ($+u$ ) carrying coherent motions, are also capable of detecting the streamwise extent (${\\\\lambda }_{x}$ ) of these motions.\",\n \"Utilizing the empirically estimated scalings, the computer controlling the jet actuation can decide how much momentum to inject through them, based on the estimated wall-normal extent ($z$ ) of the incoming motions (indicated as control system 1 in figure REF ).\",\n \"Alternatively, in scenarios where sufficiently long separations between the sensors and the jets cannot be maintained, the spanwise sensor array can be used to estimate the spanwise coherence (${\\\\Delta }y$ ) of the incoming motions, through which the wall-normal extent could be predicted (indicated as control system 2 in figure REF ).\",\n \"Such empirically-driven flow control systems, which also consider the wall-normal extent of the incoming motions, could potentially manipulate the turbulent inertial wall flow more efficiently, as demonstrated previously by [62].\",\n \"Given the significant contribution of the attached eddy hierarchy to the mean skin friction at high $Re_{\\\\tau }$ [23], availability of their corresponding 3-D geometric scalings can also be used to decide the spanwise spacing between the cross-flow jets targeting these eddies.\",\n \"Figure: Conceptualization of an active-flow control system aimed at efficiently manipulating the wall-coherent inertial motions in a ZPG TBL by utilizing the scalings determined from high Re τ Re_{\\\\tau } datasets.\",\n \"Concept of the flow control system has been adapted from .\",\n \"U ∞ U_{\\\\infty } denotes mean freestream speed.\"\n ],\n [\n \"Concluding remarks\",\n \"The present study analyses a unique set of multi-point datasets to reconstruct the 3-D statistical picture of the inertial wall-coherent (WC) turbulence in all three canonical wall-bounded flows.\",\n \"Previous studies have found these motions to be responsible for both, the $Re_{\\\\tau }$ -dependence of the skin-friction drag [45], [14], [23], [54] as well as the bulk production and the inter-scale energy transfer in high $Re_{\\\\tau }$ flows [38], [35], [29].\",\n \"The aforementioned characteristics make these motions a key target of coherent structure-based models, which the present study attempts to enhance.\",\n \"Here, the statistical picture is reconstructed by computing the cross-correlation spectra ($\\\\Gamma $ ) using the streamwise velocity fluctuations mapped across the 3-D space in the wall-bounded shear flow.\",\n \"The intermediate- and large-scaled inertial WC motions are found to exhibit geometric self-similarity with respect to the distance from the wall $z$ , expressible by simple linear relationships, which are universal across all canonical flows and independent of flow $Re_{\\\\tau }$ .\",\n \"The geometry of the very-large-scaled motions, on the other hand, is found to exhibit $\\\\delta $ -scaling, associating them with the superstructures and VLSMs for external and internal flows, respectively.\",\n \"The present study also confirms the 3-D geometry of the VLSMs to be much larger than the superstructures even in high $Re_{\\\\tau }$ flows, highlighting the role of the T/NTI in inhibiting the growth of these very-large-scaled motions in external flows [44], [33].\",\n \"Alongside testing the universality of the 3-D statistical picture, the geometric scalings brought out from the data (given by REF and REF ) are also proposed to be used as a metric to estimate the spanwise and wall-normal extent corresponding to each WC scale/eddy (${\\\\lambda }_{x}$ ).\",\n \"Application of these linear relationships to any coherent structure-based model can provide a data-driven basis to the geometry of the representative structure, which is demonstrated in this study using the attached eddy model (AEM; [46]) for a ZPG TBL.\",\n \"Here, the choice of AEM (over other models) is also driven by the empirical analysis, given the latter provides evidence of geometric self-similarity of the inertial motions and also supports the $\\\\Lambda $ -eddy packet as the representative eddy (both of which are built into the AEM framework; [39]).\",\n \"The present study further extends the scope of the AEM by using its framework to also model the $\\\\delta $ -scaled superstructures, which correspond to the very-large-scales beyond the self-similar hierarchy.\",\n \"The data-driven AEM (or dd-AEM), which has the representative eddies defined based on the data, is shown to improve upon recent works on the AEM [20] in replicating instantaneous flow phenomena associated with all three velocity components.\",\n \"This sets up the platform for future work, which would include both the WC and WI (wall-incoherent) motions modelled via the AEM framework, to replicate the full turbulent wall-bounded flow that can provide realistic inflow conditions for use in future numerical simulations [55], [61].\",\n \"Figure: Instantaneous (a) streamwise, (b) spanwise and (c) wall-normal velocity fluctuations on a wall-parallel plane at the lower bound of the log-region (zz ≈\\\\approx 0.05δ\\\\delta for the present Re τ Re_{\\\\tau }) extracted from ZPG TBL DNS dataset (𝒯 1 {\\\\mathcal {T}}_{1}) of .\",\n \"Unlike figures or , these flow fields comprise contributions from both WC and WI motions, i.e.\",\n \"they are the raw flow fields directly from the dataset.\",\n \"(d) Streamwise, (e) spanwise and (f) wall-normal velocity fluctuations conditioned for u + u^+ ≲\\\\lesssim -1 and w + w^+ ≳\\\\gtrsim 1 on the same wall-parallel plane as in (a-c).\"\n ],\n [\n \"Acknowledgement\",\n \"The authors wish to acknowledge the Australian Research Council for financial support.\",\n \"They are also grateful to the authors of [49], [4] and [7] for making their respective data available.\",\n \"Sandia National Laboratories is a multimission laboratory managed and operated by National Technology and Engineering Solutions of Sandia, LLC., a wholly owned subsidiary of Honeywell International, Inc., for the U.S. Department of Energy's National Nuclear Security Administration under contract DE-NA0003525.\",\n \"This paper describes objective technical results and analysis.\",\n \"Any subjective views or opinions that might be expressed in the paper do not necessarily represent the views of the U.S. Department of Energy or the United States Government.\"\n ],\n [\n \"SLSE-based decomposition of the instantaneous flow fields\",\n \"As discussed previously in $§$, the flow field in the inertial region comprises contributions from both wall-coherent (WC) and wall-incoherent (WI) eddies.\",\n \"Here, we demonstrate a methodology, based on the spectral linear stochastic estimate (SLSE) approach, to estimate the WC subset of a velocity fluctuation (say for $u$ -component) at any wall-normal location $z$ in the inertial region.\",\n \"This procedure has been utilized in previous studies [3], [36], [17] for similar purposes (i.e.\",\n \"to obtain a subset of the full flow field comprising selected coherent motions), which can be directly referred for an elaborate introduction to this technique.\",\n \"According to SLSE, a scale-specific unconditional input at any near-wall location, $z_{w}$ (with $z^+_w$ $\\\\lesssim $ 15) can be used to obtain a scale-specific conditional input at $z$ following: ${\\\\widetilde{u}}^{E}(z^{+};{\\\\lambda }^{+}_{x},{\\\\lambda }^{+}_{y}) = {{H^{u}_{L}}(z^{+},z^{+}_{w};{\\\\lambda }^{+}_{x},{\\\\lambda }^{+}_{y})}\\\\widetilde{u}(z^{+}_{w};{\\\\lambda }^{+}_{x},{\\\\lambda }^{+}_{y}),$ where ${\\\\tilde{u}}({z_{w}};{\\\\lambda }_{x},{\\\\lambda }_{y})$ is the 2-D Fourier transform of the instantaneous wall-parallel flow field, $u$ (${z_w};x,y$ ) in space.\",\n \"Here, the superscript $E$ and $H^{u}_{L}$ respectively represent the estimated quantity and the scale-specific linear transfer kernel (for $u$ -component).\",\n \"$u^{E}$ in equation (REF ), essentially, corresponds to the energy distribution (at $z$ ) across wavelengths ${\\\\lambda }_{x}$ and ${\\\\lambda }_{y}$ , which are coherent across $z$ and $z_w$ .\",\n \"This is facilitated by first computing $H^{u}_{L}$ from an ensemble of data as per: ${H^{u}_{L}}(z^{+},z^{+}_{w};{\\\\lambda }^{+}_{x},{\\\\lambda }^{+}_{y}) = \\\\frac{ \\\\lbrace \\\\widetilde{u}(z^{+};{\\\\lambda }^{+}_{x},{\\\\lambda }^{+}_{y}){{\\\\widetilde{u}}^{\\\\ast }}(z^{+}_{w};{\\\\lambda }^{+}_{x},{\\\\lambda }^{+}_{y}) \\\\rbrace }{ \\\\lbrace {\\\\widetilde{u}(z^{+}_{w};{\\\\lambda }^{+}_{x},{\\\\lambda }^{+}_{y})}{{\\\\widetilde{u}}^{\\\\ast }(z^{+}_{w};{\\\\lambda }^{+}_{x},{\\\\lambda }^{+}_{y})} \\\\rbrace },$ where the curly brackets ($\\\\lbrace \\\\rbrace $ ) and asterisk ($\\\\ast $ ) denote the ensemble averaging and complex conjugate, respectively.\",\n \"Given that $z^+_{w}$ $\\\\lesssim $ 15 and $z^+_{w}$ $\\\\ll $ $z^+$ , ${\\\\widetilde{u}}^{E}(z^{+};{\\\\lambda }^{+}_{x},{\\\\lambda }^{+}_{y})$ thus represents energy contributions at $z$ from solely WC motions, that are taller than $z$ [36], [17], leading to: ${{{{\\\\widetilde{u}}^{E}}(z^{+};{\\\\lambda }^{+}_{x},{\\\\lambda }^{+}_{y})}{{\\\\big |}_{{z^{+}_{w}} {\\\\lesssim } 15}}}\\\\; {\\\\rightarrow }\\\\; {{{\\\\widetilde{u}}_{wc}}(z^{+};{\\\\lambda }^{+}_{x},{\\\\lambda }^{+}_{y})},$ where $u_{wc}$ ($z$ ) represents the subset of $u$ ($z$ ) comprising solely the WC motions.\",\n \"On obtaining ${{\\\\widetilde{u}}_{wc}}$ , the corresponding flow field in physical space can be obtained by: ${{{u}_{wc}}(z;x,y)} = {\\\\mathcal {F}^{-1}}({{{\\\\widetilde{u}}_{wc}}(z;{\\\\lambda }_{x},{\\\\lambda }_{y})}),$ where ${\\\\mathcal {F}^{-1}}$ represents the inverse Fourier transform.\",\n \"Here, although the entire procedure has been demonstrated solely for $u$ -fluctuations, the same can be applied to other (lateral) velocity fluctuations by estimating the respective linear transfer kernel ($H^{v}_{L}$ or $H^{w}_{L}$ ).\",\n \"To give an example, figures REF (a-c) plot the instantaneous wall-parallel flow fields, comprising both WC + WI motions (i.e.\",\n \"the full flow fields) at the lower bound of the inertial region.\",\n \"While, figures REF (b,d,f) plot the subset of the corresponding full fields comprising solely the WC motions, estimated using the aforementioned procedure.\"\n ]\n]"},"id":{"kind":"string","value":"2107.01750"}}},{"rowIdx":1595,"cells":{"text":{"kind":"list like","value":[["Supporting decisions by unleashing multiple mindsets using pairwise\n comparisons method"],["Abstract Inconsistency in pairwise comparison judgements is often perceived as an unwanted phenomenon and researchers have proposed a number of techniques to either reduce it or to correct it.","We take a viewpoint that this inconsistency unleashes different mindsets of the decision maker(s) that should be taken into account when generating recommendations as decision support.","With this aim we consider the spanning trees analysis which is a recently emerging idea for use with the pairwise comparison approach that represents the plurality of mindsets (in terms of a plurality of vectors corresponding to different spanning trees).","Until now, the multiplicity of the vectors supplied by the spanning trees approach have been amalgamated into a single preference vector, losing the information about the plurality of mindsets.","To preserve this information, we propose a novel methodology taking an approach similar to Stochastic Multi-criteria Acceptability Analysis.","Considering all the rankings of alternatives corresponding to the different mindsets, our methodology gives the probability that an alternative attains a given ranking position as well as the probability that an alternative is preferred to another one.","Since the exponential number of spanning trees makes their enumeration prohibitive, we propose computing approximate probabilities using statistical sampling of the spanning trees.","Our approach is also appealing because it can be applied also to incomplete sets of pairwise comparisons.","We demonstrate its usefulness with a didactic example as well as with an application to a real-life case of selecting a Telecom backbone infrastructure for rural areas."],["Introduction","In any decision of our life we have to compare a plurality of alternatives with respect to several points of views taking into account our preferences.","For example, if we have to choose a city car, we have to consider the models available on the market and we have to compare them with respect to their salient characteristics such as maximum speed, acceleration, space, price, fuel consumption.","To make a well thought-out decision it is necessary (1) to assign a priority to each alternative with respect to each criterion in order to evaluate its related performance, (2) to assign a priority to each criterion in order to define its importance (3) to aggregate the evaluations of performances on considered criteria obtained in point (1) taking into account the importance of criteria obtained in point (2).","In order to assign priorities to alternatives and criteria, the use of the pairwise comparison approach is quite common due to the fact that it allows decision makers to focus on one comparison at a time.","When prioritising alternatives in point (1), the decision maker focuses on two alternatives at a time, deciding which one is more preferred on the given criterion.","In most cases, the decision maker is also asked about the strength of his/her preference.","These two questions can be combined together as a single question, however, one may argue that establishing the strength of preference can only be established after deciding the direction of preference.","The same process can be repeated for prioritising criteria in point (2).","When there are only two options at hand, obviously there will be only one comparison required and therefore the prioritisation problem becomes trivial.","However, when there are three or more options to prioritise, the number of comparisons increases significantly.","When comparing three options, say A, B and C, we have three possibilities of comparing A to B, then B to C, and finally comparing A to C. One may argue that the last comparison is not required as the comparison of A to C can be mathematically derived from the other two comparisons.","However, the counter-argument is that we should still collect this apparently redundant information as the decision maker might have different opinion about options A and C when compared directly without involving option B.","If the derived judgement and the directly obtained judgement turn out to be same, then we can easily calculate the underlying preferences of the decision maker.","Instead, if these two judgements happen to be different, then we cannot straightforwardly calculate the underlying preferences.","Nevertheless, even if these judgements are different, in any case, from a psychological viewpoint, this gives us an additional piece of information about the problem.","In fact, we expose the inconsistency amongst the direct judgements, and there are a number of techniques for eliciting preferences from inconsistent judgements, for example, the row geometric mean [8], the logarithmic least squares method [9], [18], and the eigenvector method [19].","The basic idea of all these techniques for preference elicitation is to average and to amalgamate the preference information supplied by the decision maker (DM), trying to discover the \"true preferences\" behind the DM's inconsistent judgements.","We believe that with the existing methods a relevant wealth of information is lost.","In fact, we believe that more than merely identifying internal inconsistency, the judgements provided by the DM reveal the plurality of his/her mindsets.","Consequently, unleashing these multiple mindsets is an important area of investigation in an effective decision support process.","In this context, a recent work on the use of the spanning trees approach for analysing pairwise comparison judgements [23], [21] has gained our attention due to the fact that the generation of spanning trees does not involve any aggregation of underlying preferences.","The aim of this paper is to further investigate the use of the spanning trees approach to uncover all possible alternative preference vectors that represent the decision maker in one way or another.","Our aim in this paper is to use stochastic approaches to analyse all the possible preference vectors from inconsistent judgements.","In order to appreciate the use of stochastic approaches, one must realise that even a small number of alternatives and criteria can generate a large number of possible combination of priority vectors for the weights of the criteria (we call them weight vectors) and the evaluation of the alternatives on considered criteria (we call them evaluation vectors).","For example, a decision problem having only four criteria and four alternatives will still end up in generating over a million possible combinations of weight vectors and evaluation vectors.","We discuss this in more detail later in Section .","The stochastic approach proposed in this article can be used as a support in practical decision making situations.","A typical Multiple Criteria Decision Analysis (MCDA) support system aims at recommending a solution and to assess its robustness.","On the contrary, our proposed approach provides insights to the DM for better understanding of his/her preferences, which, in turn, will help the DM to make a well-informed decision.","The paper is organised as follows.","The next section provides background on the pairwise comparisons and the potential of the spanning trees approach; Section 3 then introduces the enumeration of all possible combinations of weight vectors and evaluation vectors, and the use of a stochastic approach for eliciting preferences; then the statistical sample selection of a well distributed family of spanning trees based on a random walk procedure is discussed in Section 4.","Section 5 discusses the Telecom backbone case study to demonstrate the use of random spanning trees, and then finally conclusions and future work are discussed in Section 6."],["Background","Pairwise comparisons are used to obtain DM's preferences using a \"one-at-a-time\" approach.","For example, in a car selection problem, we can either ask the DM to directly assign weights to the three criteria of Price, Speed, and Looks; or we can alternatively ask him/her to compare two criteria at a time and make these comparisons for all three possible pairs, that is, Price versus Speed, Speed versus Looks, and then Price versus Looks.","These comparative judgements can then be used to calculate the relative weights in the form of a weight vector (sometimes referred as preference vector).","Whilst comparing the three criteria, one would expect that if Price is considered more important than Speed and Speed more important than Looks, then Price must be more important than Looks.","This is often referred as the ordinal consistency requirement.","A much stronger requirement would be that if Price is preferred to Speed $p$ times, and Speed is preferred to Looks $q$ times, then Price should be preferred to Looks $pq$ times.","This is often termed as cardinal consistency.","Although these are rational and justified requirements from mathematical perspective, DMs often break these rules.","Obviously, if the DM is cardinally consistent, then the ordinal consistency will already be achieved.","However, the opposite is not true, that is, ordinal consistency does not ensure that the judgements are cardinally consistent.","This is demonstrated through an example shown in Fig.","REF where Price is considered twice as important as Speed, Speed is considered three times as important as Looks - which might suggest that that Price is 6 times as important as Looks.","If this is indeed the case then we can easily calculate the preference vector shown on the right side of this figure.","However, if instead of choosing 6, the DM suggests Price is 3 times as important as Looks, then this makes the three judgements cardinally inconsistent - though note that the order of preference is still preserved and so the judgements are not ordinally inconsistent.","In the case of cardinal inconsistency, eliciting the preference vector is not a straight-forward task.","There are numerous algorithms proposed to estimate the preference vector from an inconsistent set of judgements, however, it remains a widely debated issue as there is no single method that can be justified as the most appropriate method for eliciting preferences from inconsistent judgements.","Figure: The PC matrix acquired for the top-level criteriaSo far, we have assumed that the DMs provide a complete set of pairwise comparison judgements.","However, in practice, it is quite common to confront situations where DMs provide an incomplete set of judgements, and where it is not always possible to ask DMs to provide the missing data.","This leaves us in situations where preferences should be elicited from inconsistent, as well as, incomplete sets of judgements.","The most widely-used technique involving pairwise comparison judgements is the Analytic Hierarchy Process (AHP) [20] where DMs structure their criteria into a hierarchy and then evaluate alternatives with respect to each of these criteria.","Fig.","REF demonstrates this technique with the help of the car selection problem discussed earlier.","In this figure, the DM has evaluated four alternatives with respect to Price, Speed and Looks (therefore, providing three sets of pairwise comparison judgements).","Note that these judgements are shown in the form of a 4-by-4 matrix, which is a standard representation used to show pairwise comparisons.","On the top-right of this figure, the DM has also provided the relative importance of the three criteria, again by using pairwise comparison judgements.","These pairwise comparison judgements are then used to elicit preference vectors with the help of some elicitation technique.","Although there are many techniques for elicitation, the most widely used techniques are Right Eigenvector (REV) [20] and Row Geometric Mean (RGM) [7].","These elicited vectors are then used to construct a decision table as shown at the bottom right of this figure (Fig.","REF ).","At the bottom of this table, note that the criteria weights can also be elicited using the same REV or RGM method.","Finally, all these evaluation vectors and criteria weights can be compiled to generate some aggregated scores and/or to produce some form of recommendations to the decision maker(s).","Figure: The PC matrix acquired for the top-level criteriaHistorically, REV method has been widely used for eliciting preference vectors for both consistent and inconsistent judgements.","The inconsistency is usually measured in terms of the Consistency Ratio (CR) which is an Eigenvalue-based measure.","According to this measure, the PC matrix is usually considered acceptable when the CR value remains below a value of 0.1.","However, the REV method has been criticised due to its left-right eigenvector asymmetry, the use of arbitrary thresholds for inconsistency acceptability, as well as a few other further issues [2], [4].","Due to these shortcomings, several other methods have been proposed in the literature.","[6] analysed and numerically compared a variety of these prioritisation methods and concluded that there is no single best method that outperforms the others in every situation.","Although REV is the most commonly used method, the RGM approach has gained popularity due to its mathematical properties, and due to its ease of implementation [14].","While focusing on this \"single solution\" aspect, it can be argued that an in-depth analysis of the inconsistency gets neglected.","We contend that a prioritisation method must have the capabilities to focus on both aspects of the problem i.e.","production of a \"good quality\" preference vector and also facilitation of an in-depth analysis."],["Spanning tree approach","In this context, a graph-theoretic approach was proposed to generate a set of all possible preference vectors through enumeration ([23], see also [21]).","This approach is briefly summarised in Fig.","REF where the DM provides a set of PC judgements (shown on the top-left) which is then translated into a fully connected graph.","This graph is then analysed using spanning tree analysis to identify all possible combination of judgements, and eventually, generating all possible preference vectors.","Each of these alternative preference vectors essentially represents a mindset of the DM.","Figure: The PC matrix acquired for the top-level criteriaThe proposed method was shown to have a number of desirable properties, however, since the original method used the arithmetic mean to calculate the average of all these vectors, it was again focusing on the \"single solution\" aspect.","Also, one may argue that this computationally expensive enumeration will be overkill when the judgements are (fully) consistent, as all vectors will have identical values.","On the contrary, we argue that this rarely happens in real life; in the majority of cases, human judgements are found to be inconsistent, and therefore, unleashing these multiple mindsets can provide useful insights in practical decision making problems.","More recently, [17] proposed a geometric version of this spanning tree approach and established the mathematical equivalence of this approach with RGM.","They also proposed the possibility of using statistical techniques to gain insights into inconsistency.","In this paper, we develop this idea further and demonstrate its usefulness when we apply this technique at a meta-level, that is, applying it to the whole problem involving multiple criteria, instead of processing just one set of judgements."],["Stochastic approach to spanning trees","We have seen that until now the multiplicity of prioritisation vectors supplied by spanning trees approach have been amalgamated into a single priority vector either using the arithmetic mean or geometric mean.","However, with this approach the information about the plurality of mindsets is lost.","This is the price to pay to get a single overall ranking of alternatives.","A different way of thinking could be to consider the plurality of ranking supplied by the plurality of priority vectors corresponding to each combination of (evaluation and weighting) spanning trees, that in turn corresponds to a mindset."],["Problem formulation","Consider a situation where the DM has $n$ alternative options to choose from, and $m$ criteria to consider whilst making this decision.","These alternatives and criteria can be denoted as: Set of alternatives: $A=\\lbrace a_1,\\ldots ,a_i,\\ldots ,a_n\\rbrace ,$ Set of criteria: $G=\\lbrace g_1,\\ldots ,g_j,\\ldots ,g_m\\rbrace $ In order to assess these alternatives with respect to each criterion, $m$ sets of pairwise comparison judgements will be collected.","Let us denote each of these pairwise comparison by $M^j=[c^j_{i_1,i_2}], g_j \\in G$ with $c^j_{i_1,i_2}$ being the pairwise comparison judgement of alternative $a_{i_1}$ with alternative $a_{i_2}$ with respect to criterion $g_j$ .","The DM also needs to assess the relative importance of the criteria, and therefore another pairwise comparison matrix will be required for elicitation, say: $M^G=[c^G_{j_1,j_2}]$ with $c^G_{j_1,j_2}$ being the pairwise comparison judgement of criterion $g_{j_1}$ with criterion $g_{j_2}$ .","Considering a generic pairwise comparison matrix $M=[c_{rs}]$ , a spanning tree $\\tau _{k}$ consists of $(n-1)$ mutually independent judgements out of the total judgements, that is $\\tau _{k}=\\lbrace c^k_{r_1,s_1},\\ldots ,c^k_{r_{n-1},s_{n-1}}\\rbrace $ with $c^k_{r_1,s_1},\\ldots ,c^k_{r_{n-1},s_{n-1}}$ independent judgements.","Each of these spanning trees can be used to calculate a preference vector, and hence, we will interchangeably use the term \"spanning tree\" and \"spanning tree vector\".","For the matrices $M^j$ assessing alternatives with respect to considered criteria $g_j, j=1,\\ldots ,m$ , the set of spanning trees is denoted as: $\\mathcal {T}^j=\\lbrace \\tau ^j_{k}\\rbrace ,$ For the matrix $M^G$ assessing the importance of criteria, the set of spanning trees is denoted as: $\\mathcal {T}^G=\\lbrace \\tau _{k_r}^G\\rbrace ,$ This gives us a total of $m+1$ sets of spanning trees (spanning tree vectors), and therefore, $m+1$ sets of preference vectors.","As shown in Fig.","REF , we can pick one tree from each of these sets and construct a decision table that is traditionally used to calculate overall preferences.","These combinations of spanning trees can be formulated as $(\\tau _{k_1}^1,\\ldots ,\\tau _{k_m}^m,\\tau _{k_G}^G) \\in \\mathcal {T}^1\\times \\ldots \\mathcal {T}^m \\times \\mathcal {T}^G,$ Figure: Creating a number of decision tables by using different combinations of spanning tree vectors"],["Evaluating the combinations of spanning trees","The evaluation vector of alternatives $a_i \\in A$ corresponding to the spanning tree $\\tau ^j_{k_j} \\in \\mathcal {T}^j$ can thus be represented as: $(u^k_j(a_1),\\ldots ,u^k_j(a_n))$ while the weight vector corresponding to the spanning tree $\\tau ^G_{k} \\in \\mathcal {T}^G$ is denoted as: $(w^k_1,\\ldots ,w^k_m)$ Now the overall priority for alternatives $a_i \\in A$ given by the combination of spanning trees $\\mathbf {\\tau }=(\\tau _{k_1}^1,\\ldots ,\\tau _{k_m}^m,\\tau _{k_G}^G) \\in \\mathcal {T}^1\\times \\ldots \\mathcal {T}^m \\times \\mathcal {T}^G$ will be: $u_{\\mathbf {\\tau }}(a_i)=w^{{k_G}}_1 u^{k_1}_1(a_i)+ \\ldots +w^{k_G}_m u^{k_m}_m(a_i)$ From this formulation, we can deduce the set of combinations of spanning trees for which any $a_{i_1},a_{i_2} \\in A$ $a_{i1}$ is preferred to $a_{i2}$ : $B(a_{i_1}\\succ a_{i_2})=\\lbrace \\mathbf {\\tau } \\in \\mathcal {T}^1\\times \\ldots \\mathcal {T}^m \\times \\mathcal {T}^G:u_{\\mathbf {\\tau }}(a_{i_1})>u_{\\mathbf {\\tau }}(a_{i_2})\\rbrace $ $a_{i1}$ is indifferent with $a_{i2}$ : $B(a_{i_1}\\sim a_{i_2})=\\lbrace \\mathbf {\\tau }\\in \\mathcal {T}^1\\times \\ldots \\mathcal {T}^m \\times \\mathcal {T}^G:u_{\\mathbf {\\tau }}(a_{i_1})=u_{\\mathbf {\\tau }}(a_{i_2})\\rbrace $ The set of combinations of spanning trees $\\mathbf {\\tau } \\in \\mathcal {T}^1\\times \\ldots \\mathcal {T}^m \\times \\mathcal {T}^G$ for which alternative $a_i \\in A$ attains the $p-th$ position with respect to overall priority $u_{\\mathbf {\\tau }}(a_{i})\\rbrace $ can be formulated as: $R(a_i,p)=\\lbrace \\mathbf {\\tau } \\in \\mathcal {T}^1\\times \\ldots \\mathcal {T}^m \\times \\mathcal {T}^G:rank(a_i,\\mathbf {\\tau })=p\\rbrace $ with $rank(a_i,\\mathbf {\\tau })=1+\\sum _{i^\\prime \\ne i}\\rho (u_{\\mathbf {\\tau }}(a_{i^\\prime })>u_{\\mathbf {\\tau }}(a_{i}))$ where $\\rho (false)=0$ and $\\rho (true)=1$ .","Inspired by the Stochastic Multicriteria Acceptability Approach or SMAA [15], [16], the probability that alternative $a_{i_1}$ is preferred to alternative $a_{i_2}$ , with $a_{i1},a_{i2}\\in A$ can be represented as: $P(a_{i1} \\succ a_{i2})=\\frac{card(B(a_{i1} \\succ a_{i2}))}{card(\\mathcal {T}^1\\times \\ldots \\mathcal {T}^m \\times \\mathcal {T}^G)}$ that, remembering that the total number of spanning trees in a graph of $k$ nodes is $k^{k-2}$ [5], and, consequently $card(\\mathcal {T}^1\\times \\ldots \\mathcal {T}^m \\times \\mathcal {T}^G)=m^{m-2}n^{m(n-2)}$ , becomes $P(a_{i1} \\succ a_{i2})=\\frac{card(B(a_{i1} \\succ a_{i2}))}{m^{m-2}n^{m(n-2)}},$ the probability that alternative $a_{i} \\in A$ is attaining the $p-th$ ranking position will be: $P(rank(a_i,\\mathbf {\\tau })=p)=\\frac{R(a_i,p)}{card(\\mathcal {T}^1\\times \\ldots \\mathcal {T}^m \\times \\mathcal {T}^G)}=\\frac{R(a_i,p)}{m^{m-2}n^{m(n-2)}},$ [13] investigated incomplete sets of judgements where the DMs are allowed to respond with \"do not know\" or \"not sure\" to some judgements.","This is an important issue to investigate as the probability of acquiring an incomplete set of PC judgements increases with an increase in the total number of items for comparison [10].","Both the REV and the RGM methods are inappropriate in such cases due to the fact that the PC matrix cannot be constructed without estimating/imputing the missing judgements.","The spanning trees can be applied also to partial comparison matrix with the help of the Kirchoff formula (See details in [21]).","This implies that generating combinations of spanning trees is not restricted to only complete sets of judgements, and therefore, generating preference frequencies and rank-order frequencies are possible from incomplete pairwise comparisons, without any modification.","The implication of having an incomplete set of judgements is that the total number of combinations will be smaller than the number possible from a complete sets of judgements.","This permits the DM to only express values for those PC judgements for which he/she is sufficiently confident, avoiding forcing the DM to provide judgements when he/she is not sufficiently confident - in so doing maintaining the reliability of the preference information considered in the decision support process."],["Didactic example","Let us show how our approach works in practice by using the classic example of school selection proposed in [20].","The parents have to decide the high school for their son.","They consider six criteria being the following $g_1$ : Learning, $g_2$ : Friends, $g_3$ : School life, $g_4$ : Vocational training, $g_5$ : College preparation, $g_6$ : Music classes.","There are three alternatives corresponding to three schools denoted $A$ , $B$ and $C$ .","For each criterion, the parents compared in pairs the schools as shown in the following pairwise comparison matrices.","$M_{g_1}=\\begin{pmatrix}3 & \\frac{1}{3} & \\frac{1}{2} \\\\3 & 1 & 3 \\\\2 & \\frac{1}{3} & 1 \\\\\\end{pmatrix}$ $M_{g_2}=\\begin{pmatrix}1 & 1 & 1 \\\\1 & 1 & 1 \\\\1 & 1 & 1 \\\\\\end{pmatrix}$ $M_{g_3}=\\begin{pmatrix}1 & 5 & 1 \\\\\\frac{1}{5} & 1 & \\frac{1}{5} \\\\1 & 5 & 1 \\\\\\end{pmatrix}$ $M_{g_4}=\\begin{pmatrix}1 & 9 & 7 \\\\\\frac{1}{9} & 1 & \\frac{1}{5} \\\\\\frac{1}{7} & 5 & 1 \\\\\\end{pmatrix}$ $M_{g_5}=\\begin{pmatrix}1 & \\frac{1}{2} & 1 \\\\2 & 1 & 2 \\\\1 & \\frac{1}{2} & 1 \\\\\\end{pmatrix}$ $M_{g_6}=\\begin{pmatrix}1 & 6 & 4 \\\\\\frac{1}{6} & 1 & \\frac{1}{3} \\\\\\frac{1}{4} & 3 & 1 \\\\\\end{pmatrix}$ After comparing the alternatives with respect to these criteria, the parents also compared the criteria in terms of their importance as shown below.","$M_{criteria}=\\begin{pmatrix}1 & 4 & 3 & 1 & 3 & 4 \\\\\\frac{1}{4} & 1 & 7 & 3 & \\frac{1}{5} & 1 \\\\\\frac{1}{3} & \\frac{1}{7} & 1 & \\frac{1}{5} & \\frac{1}{5} & \\frac{1}{6} \\\\1 & \\frac{1}{3} & 5 & 1 & 1 & \\frac{1}{3} \\\\\\frac{1}{3} & 5 & 5 & 1 & 1 & 3 \\\\\\frac{1}{4} & 1 & 6 & 3 & \\frac{1}{3} & 1 \\\\\\end{pmatrix}$ The consistency ratios for the above pairwise comparison matrices are as follows: CR($M_{g_1}$ ) = 0.04, CR($M_{g_2}$ ) = 0, CR($M_{g_3}$ ) = 0, CR($M_{g_4}$ ) = 0.18, CR($M_{g_5}$ ) = 0, CR($M_{g_6}$ ) = 0.04, CR($M_{criteria}$ ) = 0.24 As we can see, there are four matrices which are not completely consistent and two of these, namely $M_{g_4}$ and $M_{criteria}$ have unacceptable levels of inconsistency as measured by the usual threshold of 0.1 for the CR value [20].","The priorities $u_j(X), X=A,B,C, j=1,\\ldots ,6$ , obtained from pairwise comparisons matrices $M_{g_1}-M_{g_6}$ representing the evaluations of schools with respect to the considered criteria are shown in Table REF , while the weights of the considered criteria, according to the priority obtained from pairwise comparison's matrix $M_{criteria}$ , are the following: $w_1=0.32, w_2=0.14,w_3=0.03,w_4=0.13,w_5=0.24,w_6=0.14.$ Table: Scores for alternatives with respect to each criterionUsing priorities $w_j$ and $u_j(X), X=A,B,C, j=1,\\ldots ,6$ , we can compute the overall priority $u(X)$ of each school, with $u(X)=\\sum _{j=1}^6 w_j u_j(X)$ obtaining the following results: $u(A)=0.37, u(B)=0.38, u(C)=0.25.$ Let us now handle the same problem with the spanning tree approach.","With this aim, we have to consider all the combinations of spanning trees from the pairwise comparison matrix of criteria $M_{criteria}$ and from the pairwise comparison matrices of alternatives with respect to the considered criteria $M_{g_j}, j=1,\\ldots ,m$ .","Remembering that with $m$ criteria and $n$ alternatives, we have ${n^{(n-2)}}{m^{(m-2)n}}$ combinations of spanning tress, for the decision problem at hand, we have $6^4\\cdot 3^6 = 944784$ combinations of spanning trees.","We compute weights of criteria $w^k_1,\\ldots ,w^k_6$ and evaluations of alternatives with respect to considered criteria $u^k_j(A),u^k_j(B),u^k_j(C)), j=1,\\ldots ,6,$ for each combination of spanning trees $\\mathbf {\\tau }=(\\tau _{k_1}^1,\\ldots ,\\tau _{k_6}^m,\\tau _{k_G}^G) \\in \\mathcal {T}^1\\times \\ldots \\mathcal {T}^6 \\times \\mathcal {T}^G$ .","Figure: Demonstrating the two different combinations of spanning trees, and the difference in their rankings and scores.Considering all the combinations of spanning trees from $\\mathcal {T}^1\\times \\ldots \\mathcal {T}^6 \\times \\mathcal {T}^G$ we can compute Table REF that shows the probability that each alternative attains a given rank (within parenthesis there is the total number of combinations of spanning trees for which there is the preference of the alternative on the row over the alternative on column), Table REF that for each pair of alternatives shows the probability that the alternative in the row is preferred to the alternative in the column (within parenthesis the total number of combinations of spanning tress for which the alternatives attains the considered ranking position).","Table: Preference frequency over 944784 combinations of spanning treesTable: Rank order frequency over 944784 combinations of spanning trees"],["Random spanning trees","We have seen that, for a problem with $n$ alternatives evaluated across $m$ criteria, the total number of combinations of spanning trees increases exponentially with the problem size - making it impractical to calculate the exact probabilities of $P(rank(a_i,\\mathbf {\\tau })=p)$ and of $P(a_{i1} \\succ a_{i2})$ for large problems.","We therefore propose using an approach which, combined with statistical sampling theory, allows us to provide estimates of the required probabilities of $P(rank(a_i,\\mathbf {\\tau })=p)$ and of $P(a_{i1} \\succ a_{i2})$ to within any user defined degree of accuracy according to any user defined level of confidence.","Of course we cannot physically select a statistical random sample from the population of spanning tree combinations since this would require the generation of the population itself and this is exactly what we aim to avoid.","We can however use the 'random walk' procedure to generate a tree and hence a sample of trees (see [1], [3].","Indeed, the 'random walk' procedure has been proven to generate a true statistical random sample in the sense that the generated sample is equivalent to selection of a statistical random sample from the population, that is, where each tree has the same uniform probability of being selected.","See [3], [1] for more details.","As we discussed the possibility of having incomplete sets of pairwise comparison judgements will arise in practice and so it is important to note that the concept of random walks is equally applicable to these incomplete sets without modification.","Figure: The PC matrix acquired for the top-level criteriaSince each iteration of the procedure generates a new member of the random sample, the total number of iterations used in the procedure is equivalent to sample size and we can use statistical large sample theory to determine the number of iterations required to generate sample parameters (including the probabilities of $P(rank(a_i,\\mathbf {\\tau })=p)$ and of $P(a_{i1} \\succ a_{i2})$ ) to any specified degree of accuracy and with any specified level of confidence.","That is, if we want the estimated probability to have an accuracy within $\\lambda $ and with a $C$ percent level of confidence, then the required number of iterations would be [22]: $It(\\lambda ,C) = \\frac{ {Z_C}^2 }{4\\lambda ^2}$ where $Z_C$ is the z-score calculated from the standardised normal distribution curve.","For example, if we want to achieve an accuracy within 0.01 and with a $99\\%$ level of confidence, then the required number of iterations would be $It(0.01,99\\%) = \\frac{ Z_{99}^2 }{4 \\times 0.01^2}$ As we know that the Z value for 99% confidence interval is equal to 2.58, therefore $It(0.01,99\\%) = \\frac{ {(2.58)}^2 }{4 \\times 0.01^2} = 16,641$ This suggests that we need to perform more than 16,641 iterations if we want to achieve an accuracy within $\\pm 0.01$ with $99\\%$ confidence."],["Applying random walk procedure to the school example","In the example below, we illustrate the use of random walks by repeating an experiment multiple times where each experiment itself uses the required number of iterations $It$ determined by the formula above.","We do this on the previously discussed School example as this is a tractable problem where the whole population of solutions is available.","Later, we will also extend this idea to a bigger problem where generating the whole population is impractical (see Section ).","Table REF shows the preference frequencies obtained through random walks, as well as the preference frequencies obtained by taking a sample from the whole population of solutions.","Recall that the whole population was generated by enumerating all possible combinations of spanning trees that was shown in Table REF earlier.","Here we can see that the estimated values from random walks and from the sample are both lying within 1% of the population values that were previously shown in Table REF .","Table: Preference frequencies of the school exampleTable: Rank order profiles generated for the school exampleWe performed the same analysis for rank-order frequencies as well, as summarised in Table REF .","The table on the left side shows that the rank-order frequencies obtained through random walks are also lying within 1% of the population values shown in Table REF .","The rank-order frequencies obtained by taking a sample are shown on the right side of this table.","This also illustrates that the accuracy and confidence level results for the generated random sample echo the same results had we been able to physically select a statistical random sample from the population of spanning tree combinations."],["Telecom backbone example","In order to demonstrate the use of spanning trees to explore multiple mindsets, we consider the practical data acquired in a recent study: the selection of a backbone infrastructure for telecommunication in rural areas [11].","This study focused on the use of a structured decision making approach towards selecting the telecommunication infrastructure for the rural areas of developing countries.","The lack of adequate telecommunications infrastructure in these parts of the world remains a major obstacle for providing affordable services.","The study considered four options of Fiber-optic cable (G1), Power-line communication (G2), Microwave link (G3) and Satellite communication (G4).","The authors of this study used AHP and Analytic Network Process (ANP) to structure the problem and acquired the necessary data on preferences and assessments from key stakeholders.","The criteria used to compare these alternatives were grouped into six major categories including technical, infrastructural, economic, social, regulatory and environmental factors.","These categories and their constituent criteria are presented in Fig.","REF .","Figure: Criteria to compare the available backbone infrastructuresFigure: The PC matrix acquired for the top-level criteriaThe PC matrix, $A_{top}$ , acquired for prioritising these six categories (top-level criteria) is shown in Fig.","REF .","Although $A_{top}$ is a transitive PC matrix, the estimated vectors produced by the widely used EV method does not preserve the original order of preferences in these judgements.","The final weights calculated using the EV and GM methods are found to be almost identical, as given in Table REF in normalised form.","Satellite communication (G4) is considered the most preferred alternative with a weight of 29.95% (using EV), followed by Microwave (G3) with a weight around 28.34% (using EV).","Table: Estimated weights for the available backbone infrastructure optionsMost criteria lie under the Technical and Infrastructure categories.","The Technical category includes nine criteria whilst the Infrastructure category has eight criteria used to compare the alternatives.","The two PC matrices for the Technical and Infrastructure categories, $A_{tech}$ and $A_{infra}$ , have been found to be intransitive and should be investigated along with $A_{top}$ for their impact on the final result."],["Spanning trees analysis using Random walks","In this case study, there are four alternatives and six top-level criteria, therefore, the number of “spanning-trees” solutions for criteria matrix is $m^{(m-2)}$ = $6^4$ = 1296, and the number of “spanning-trees” solutions for each alternative matrix is $n^{(n-2)}$ = $4^2$ = 16.","As we pick one “spanning-trees” solution from each matrix, there are $m^{(m-2)}n^{m(n-2)}$ = $1296\\times 16^6$ = $21,743,271,936$ combinations possible.","Generating these billions of combinations is computationally expensive, so sampling is the logical way to estimate probabilities.","Therefore we generated a sample of 20,000 solutions with the use of random walks.","We generated 20 such samples in order to check whether all these samples produce similar results.","In practise, only one sample should suffice but we took this approach for investigation only.","The average scores for the four alternatives are given in Table REF .","Each row in this table represents one iteration of generating a sample of 20,000 solutions.","Table: Scores calculated usign random walks; each iteration shows an average of 20k solutionsUnlike the School example where we calculated frequencies by generating all the combinations of spanning trees, here we will calculate the frequencies from the samples generated by random walks.","It can be argued that these frequencies may vary as they are based on randomly generated set of solutions.","However, the aim here is to gain statistical insights, and therefore, these findings should remain useful as long as they produce reliable statistical findings.","We investigated this reliability by repeating the same experiment twenty times and comparing the preference frequencies and rank-order frequencies.","The standard deviations of these values do not exceed 0.005, which supports the argument that these stochastic results are quite reliable.","Table REF shows the probability that one alternative is preferred to the other (for all possible pairs).","Looking at the first row, we can say that Fiber is preferred over Powerline in 63% (see $0.63\\pm 0.004$ under the Powerline column) of the random walk solutions, however, it is seldom preferred over Microwave and Satellite.","Microwave is clearly preferred over Fiber (with a probability of 87%) and Powerline (with a probability of 96%), however, it has been marginally preferred over Satellite (with a probability of 51%).","We can argue that the Fiber and Powerline are the two clearly dominated alternatives, while the Microwave and Satellite are the two dominating alternative.","However, when comparing Microwave and Satellite with each other, Microwave is slightly more preferred but there is no clear winner.","We are providing the standard deviations for each of these values to show that these scores didn't deviate much, as discussed earlier.","Table: Preference frequencies for Telecom backbone selectionTable: Rank-order frequencies for Telecom backbone selectionAnother interesting way to assess these options is to estimate the ranking of each alternative.","Table REF shows the probability that each alternative attains a given rank.","In this table, The probability of Fiber and Powerline taking the first rank is too low (only 4.9% and 2.5% respectively).","However, Microwave and Satellite have high probabilities of taking the first rank.","On the other end, although Fiber is more likely to attain the third rank, both Fiber and Powerline have high probabilities to attain the lowest rank (36.5% and 62.2% respectively).","The two alternatives of Microwave and Satellite are least likely to be placed on the lowest rank, as the percentage of combinations placing them at the 4th position were only 0.6% and 0.7%, respectively."],["Conclusions","The spanning trees analysis can help understand the plurality of mindsets in terms of a plurality of prioritisation vectors originating from a plurality of spanning trees of pairwise comparison matrices.","Considering all the rankings of alternatives corresponding to the different mindsets, we propose to estimate the probability that an alternative attains a given ranking position, and the probability that an alternative is preferred to another one.","Moreover, the proposed approach can be applied to incomplete sets of judgements without any modification.","Since the exponential number of spanning trees makes their enumeration prohibitive, we propose computing approximate probabilities using statistical sampling of spanning trees.","The usefulness of statistical sampling is further demonstrated with the case study of Telecom infrastructure selection for rural areas.","An interesting area of future research is to do performance analysis using the sigma-mu approach proposed by [12].","For each alternative, we compute the mean, $\\mu $ , and the standard deviation, $\\sigma $ , of all the solutions generated from combinations of spanning trees.","The objective would be to maximise the mean while minimising the standard deviation.","The use of interval judgements is also an important area to investigate as we know that decision makers often provide their judgements in intervals (for example, stating that X is about 3 to 5 times better than Y).","Therefore, the idea of interval judgements need to be combined with the use of spanning trees combinations.","Considering the lower and upper limit of intervals, the number of combinations will further increase exponentially, therefore the use of a stochastic approach might be the only practical approach for this purpose.","There is a further potential in the use of spanning trees analysis at group-level decision making where multiple judgements are collected for comparing the same pair of alternatives or weighing the same pair of criteria.","The concept of \"aggregated individual preferences\" and \"aggregated individual judgements\" has been widely debated in the context of group decision making, and we believe that the use of spanning tree analysis can offer better insights into these types of problems."]],"string":"[\n [\n \"Supporting decisions by unleashing multiple mindsets using pairwise\\n comparisons method\"\n ],\n [\n \"Abstract Inconsistency in pairwise comparison judgements is often perceived as an unwanted phenomenon and researchers have proposed a number of techniques to either reduce it or to correct it.\",\n \"We take a viewpoint that this inconsistency unleashes different mindsets of the decision maker(s) that should be taken into account when generating recommendations as decision support.\",\n \"With this aim we consider the spanning trees analysis which is a recently emerging idea for use with the pairwise comparison approach that represents the plurality of mindsets (in terms of a plurality of vectors corresponding to different spanning trees).\",\n \"Until now, the multiplicity of the vectors supplied by the spanning trees approach have been amalgamated into a single preference vector, losing the information about the plurality of mindsets.\",\n \"To preserve this information, we propose a novel methodology taking an approach similar to Stochastic Multi-criteria Acceptability Analysis.\",\n \"Considering all the rankings of alternatives corresponding to the different mindsets, our methodology gives the probability that an alternative attains a given ranking position as well as the probability that an alternative is preferred to another one.\",\n \"Since the exponential number of spanning trees makes their enumeration prohibitive, we propose computing approximate probabilities using statistical sampling of the spanning trees.\",\n \"Our approach is also appealing because it can be applied also to incomplete sets of pairwise comparisons.\",\n \"We demonstrate its usefulness with a didactic example as well as with an application to a real-life case of selecting a Telecom backbone infrastructure for rural areas.\"\n ],\n [\n \"Introduction\",\n \"In any decision of our life we have to compare a plurality of alternatives with respect to several points of views taking into account our preferences.\",\n \"For example, if we have to choose a city car, we have to consider the models available on the market and we have to compare them with respect to their salient characteristics such as maximum speed, acceleration, space, price, fuel consumption.\",\n \"To make a well thought-out decision it is necessary (1) to assign a priority to each alternative with respect to each criterion in order to evaluate its related performance, (2) to assign a priority to each criterion in order to define its importance (3) to aggregate the evaluations of performances on considered criteria obtained in point (1) taking into account the importance of criteria obtained in point (2).\",\n \"In order to assign priorities to alternatives and criteria, the use of the pairwise comparison approach is quite common due to the fact that it allows decision makers to focus on one comparison at a time.\",\n \"When prioritising alternatives in point (1), the decision maker focuses on two alternatives at a time, deciding which one is more preferred on the given criterion.\",\n \"In most cases, the decision maker is also asked about the strength of his/her preference.\",\n \"These two questions can be combined together as a single question, however, one may argue that establishing the strength of preference can only be established after deciding the direction of preference.\",\n \"The same process can be repeated for prioritising criteria in point (2).\",\n \"When there are only two options at hand, obviously there will be only one comparison required and therefore the prioritisation problem becomes trivial.\",\n \"However, when there are three or more options to prioritise, the number of comparisons increases significantly.\",\n \"When comparing three options, say A, B and C, we have three possibilities of comparing A to B, then B to C, and finally comparing A to C. One may argue that the last comparison is not required as the comparison of A to C can be mathematically derived from the other two comparisons.\",\n \"However, the counter-argument is that we should still collect this apparently redundant information as the decision maker might have different opinion about options A and C when compared directly without involving option B.\",\n \"If the derived judgement and the directly obtained judgement turn out to be same, then we can easily calculate the underlying preferences of the decision maker.\",\n \"Instead, if these two judgements happen to be different, then we cannot straightforwardly calculate the underlying preferences.\",\n \"Nevertheless, even if these judgements are different, in any case, from a psychological viewpoint, this gives us an additional piece of information about the problem.\",\n \"In fact, we expose the inconsistency amongst the direct judgements, and there are a number of techniques for eliciting preferences from inconsistent judgements, for example, the row geometric mean [8], the logarithmic least squares method [9], [18], and the eigenvector method [19].\",\n \"The basic idea of all these techniques for preference elicitation is to average and to amalgamate the preference information supplied by the decision maker (DM), trying to discover the \\\"true preferences\\\" behind the DM's inconsistent judgements.\",\n \"We believe that with the existing methods a relevant wealth of information is lost.\",\n \"In fact, we believe that more than merely identifying internal inconsistency, the judgements provided by the DM reveal the plurality of his/her mindsets.\",\n \"Consequently, unleashing these multiple mindsets is an important area of investigation in an effective decision support process.\",\n \"In this context, a recent work on the use of the spanning trees approach for analysing pairwise comparison judgements [23], [21] has gained our attention due to the fact that the generation of spanning trees does not involve any aggregation of underlying preferences.\",\n \"The aim of this paper is to further investigate the use of the spanning trees approach to uncover all possible alternative preference vectors that represent the decision maker in one way or another.\",\n \"Our aim in this paper is to use stochastic approaches to analyse all the possible preference vectors from inconsistent judgements.\",\n \"In order to appreciate the use of stochastic approaches, one must realise that even a small number of alternatives and criteria can generate a large number of possible combination of priority vectors for the weights of the criteria (we call them weight vectors) and the evaluation of the alternatives on considered criteria (we call them evaluation vectors).\",\n \"For example, a decision problem having only four criteria and four alternatives will still end up in generating over a million possible combinations of weight vectors and evaluation vectors.\",\n \"We discuss this in more detail later in Section .\",\n \"The stochastic approach proposed in this article can be used as a support in practical decision making situations.\",\n \"A typical Multiple Criteria Decision Analysis (MCDA) support system aims at recommending a solution and to assess its robustness.\",\n \"On the contrary, our proposed approach provides insights to the DM for better understanding of his/her preferences, which, in turn, will help the DM to make a well-informed decision.\",\n \"The paper is organised as follows.\",\n \"The next section provides background on the pairwise comparisons and the potential of the spanning trees approach; Section 3 then introduces the enumeration of all possible combinations of weight vectors and evaluation vectors, and the use of a stochastic approach for eliciting preferences; then the statistical sample selection of a well distributed family of spanning trees based on a random walk procedure is discussed in Section 4.\",\n \"Section 5 discusses the Telecom backbone case study to demonstrate the use of random spanning trees, and then finally conclusions and future work are discussed in Section 6.\"\n ],\n [\n \"Background\",\n \"Pairwise comparisons are used to obtain DM's preferences using a \\\"one-at-a-time\\\" approach.\",\n \"For example, in a car selection problem, we can either ask the DM to directly assign weights to the three criteria of Price, Speed, and Looks; or we can alternatively ask him/her to compare two criteria at a time and make these comparisons for all three possible pairs, that is, Price versus Speed, Speed versus Looks, and then Price versus Looks.\",\n \"These comparative judgements can then be used to calculate the relative weights in the form of a weight vector (sometimes referred as preference vector).\",\n \"Whilst comparing the three criteria, one would expect that if Price is considered more important than Speed and Speed more important than Looks, then Price must be more important than Looks.\",\n \"This is often referred as the ordinal consistency requirement.\",\n \"A much stronger requirement would be that if Price is preferred to Speed $p$ times, and Speed is preferred to Looks $q$ times, then Price should be preferred to Looks $pq$ times.\",\n \"This is often termed as cardinal consistency.\",\n \"Although these are rational and justified requirements from mathematical perspective, DMs often break these rules.\",\n \"Obviously, if the DM is cardinally consistent, then the ordinal consistency will already be achieved.\",\n \"However, the opposite is not true, that is, ordinal consistency does not ensure that the judgements are cardinally consistent.\",\n \"This is demonstrated through an example shown in Fig.\",\n \"REF where Price is considered twice as important as Speed, Speed is considered three times as important as Looks - which might suggest that that Price is 6 times as important as Looks.\",\n \"If this is indeed the case then we can easily calculate the preference vector shown on the right side of this figure.\",\n \"However, if instead of choosing 6, the DM suggests Price is 3 times as important as Looks, then this makes the three judgements cardinally inconsistent - though note that the order of preference is still preserved and so the judgements are not ordinally inconsistent.\",\n \"In the case of cardinal inconsistency, eliciting the preference vector is not a straight-forward task.\",\n \"There are numerous algorithms proposed to estimate the preference vector from an inconsistent set of judgements, however, it remains a widely debated issue as there is no single method that can be justified as the most appropriate method for eliciting preferences from inconsistent judgements.\",\n \"Figure: The PC matrix acquired for the top-level criteriaSo far, we have assumed that the DMs provide a complete set of pairwise comparison judgements.\",\n \"However, in practice, it is quite common to confront situations where DMs provide an incomplete set of judgements, and where it is not always possible to ask DMs to provide the missing data.\",\n \"This leaves us in situations where preferences should be elicited from inconsistent, as well as, incomplete sets of judgements.\",\n \"The most widely-used technique involving pairwise comparison judgements is the Analytic Hierarchy Process (AHP) [20] where DMs structure their criteria into a hierarchy and then evaluate alternatives with respect to each of these criteria.\",\n \"Fig.\",\n \"REF demonstrates this technique with the help of the car selection problem discussed earlier.\",\n \"In this figure, the DM has evaluated four alternatives with respect to Price, Speed and Looks (therefore, providing three sets of pairwise comparison judgements).\",\n \"Note that these judgements are shown in the form of a 4-by-4 matrix, which is a standard representation used to show pairwise comparisons.\",\n \"On the top-right of this figure, the DM has also provided the relative importance of the three criteria, again by using pairwise comparison judgements.\",\n \"These pairwise comparison judgements are then used to elicit preference vectors with the help of some elicitation technique.\",\n \"Although there are many techniques for elicitation, the most widely used techniques are Right Eigenvector (REV) [20] and Row Geometric Mean (RGM) [7].\",\n \"These elicited vectors are then used to construct a decision table as shown at the bottom right of this figure (Fig.\",\n \"REF ).\",\n \"At the bottom of this table, note that the criteria weights can also be elicited using the same REV or RGM method.\",\n \"Finally, all these evaluation vectors and criteria weights can be compiled to generate some aggregated scores and/or to produce some form of recommendations to the decision maker(s).\",\n \"Figure: The PC matrix acquired for the top-level criteriaHistorically, REV method has been widely used for eliciting preference vectors for both consistent and inconsistent judgements.\",\n \"The inconsistency is usually measured in terms of the Consistency Ratio (CR) which is an Eigenvalue-based measure.\",\n \"According to this measure, the PC matrix is usually considered acceptable when the CR value remains below a value of 0.1.\",\n \"However, the REV method has been criticised due to its left-right eigenvector asymmetry, the use of arbitrary thresholds for inconsistency acceptability, as well as a few other further issues [2], [4].\",\n \"Due to these shortcomings, several other methods have been proposed in the literature.\",\n \"[6] analysed and numerically compared a variety of these prioritisation methods and concluded that there is no single best method that outperforms the others in every situation.\",\n \"Although REV is the most commonly used method, the RGM approach has gained popularity due to its mathematical properties, and due to its ease of implementation [14].\",\n \"While focusing on this \\\"single solution\\\" aspect, it can be argued that an in-depth analysis of the inconsistency gets neglected.\",\n \"We contend that a prioritisation method must have the capabilities to focus on both aspects of the problem i.e.\",\n \"production of a \\\"good quality\\\" preference vector and also facilitation of an in-depth analysis.\"\n ],\n [\n \"Spanning tree approach\",\n \"In this context, a graph-theoretic approach was proposed to generate a set of all possible preference vectors through enumeration ([23], see also [21]).\",\n \"This approach is briefly summarised in Fig.\",\n \"REF where the DM provides a set of PC judgements (shown on the top-left) which is then translated into a fully connected graph.\",\n \"This graph is then analysed using spanning tree analysis to identify all possible combination of judgements, and eventually, generating all possible preference vectors.\",\n \"Each of these alternative preference vectors essentially represents a mindset of the DM.\",\n \"Figure: The PC matrix acquired for the top-level criteriaThe proposed method was shown to have a number of desirable properties, however, since the original method used the arithmetic mean to calculate the average of all these vectors, it was again focusing on the \\\"single solution\\\" aspect.\",\n \"Also, one may argue that this computationally expensive enumeration will be overkill when the judgements are (fully) consistent, as all vectors will have identical values.\",\n \"On the contrary, we argue that this rarely happens in real life; in the majority of cases, human judgements are found to be inconsistent, and therefore, unleashing these multiple mindsets can provide useful insights in practical decision making problems.\",\n \"More recently, [17] proposed a geometric version of this spanning tree approach and established the mathematical equivalence of this approach with RGM.\",\n \"They also proposed the possibility of using statistical techniques to gain insights into inconsistency.\",\n \"In this paper, we develop this idea further and demonstrate its usefulness when we apply this technique at a meta-level, that is, applying it to the whole problem involving multiple criteria, instead of processing just one set of judgements.\"\n ],\n [\n \"Stochastic approach to spanning trees\",\n \"We have seen that until now the multiplicity of prioritisation vectors supplied by spanning trees approach have been amalgamated into a single priority vector either using the arithmetic mean or geometric mean.\",\n \"However, with this approach the information about the plurality of mindsets is lost.\",\n \"This is the price to pay to get a single overall ranking of alternatives.\",\n \"A different way of thinking could be to consider the plurality of ranking supplied by the plurality of priority vectors corresponding to each combination of (evaluation and weighting) spanning trees, that in turn corresponds to a mindset.\"\n ],\n [\n \"Problem formulation\",\n \"Consider a situation where the DM has $n$ alternative options to choose from, and $m$ criteria to consider whilst making this decision.\",\n \"These alternatives and criteria can be denoted as: Set of alternatives: $A=\\\\lbrace a_1,\\\\ldots ,a_i,\\\\ldots ,a_n\\\\rbrace ,$ Set of criteria: $G=\\\\lbrace g_1,\\\\ldots ,g_j,\\\\ldots ,g_m\\\\rbrace $ In order to assess these alternatives with respect to each criterion, $m$ sets of pairwise comparison judgements will be collected.\",\n \"Let us denote each of these pairwise comparison by $M^j=[c^j_{i_1,i_2}], g_j \\\\in G$ with $c^j_{i_1,i_2}$ being the pairwise comparison judgement of alternative $a_{i_1}$ with alternative $a_{i_2}$ with respect to criterion $g_j$ .\",\n \"The DM also needs to assess the relative importance of the criteria, and therefore another pairwise comparison matrix will be required for elicitation, say: $M^G=[c^G_{j_1,j_2}]$ with $c^G_{j_1,j_2}$ being the pairwise comparison judgement of criterion $g_{j_1}$ with criterion $g_{j_2}$ .\",\n \"Considering a generic pairwise comparison matrix $M=[c_{rs}]$ , a spanning tree $\\\\tau _{k}$ consists of $(n-1)$ mutually independent judgements out of the total judgements, that is $\\\\tau _{k}=\\\\lbrace c^k_{r_1,s_1},\\\\ldots ,c^k_{r_{n-1},s_{n-1}}\\\\rbrace $ with $c^k_{r_1,s_1},\\\\ldots ,c^k_{r_{n-1},s_{n-1}}$ independent judgements.\",\n \"Each of these spanning trees can be used to calculate a preference vector, and hence, we will interchangeably use the term \\\"spanning tree\\\" and \\\"spanning tree vector\\\".\",\n \"For the matrices $M^j$ assessing alternatives with respect to considered criteria $g_j, j=1,\\\\ldots ,m$ , the set of spanning trees is denoted as: $\\\\mathcal {T}^j=\\\\lbrace \\\\tau ^j_{k}\\\\rbrace ,$ For the matrix $M^G$ assessing the importance of criteria, the set of spanning trees is denoted as: $\\\\mathcal {T}^G=\\\\lbrace \\\\tau _{k_r}^G\\\\rbrace ,$ This gives us a total of $m+1$ sets of spanning trees (spanning tree vectors), and therefore, $m+1$ sets of preference vectors.\",\n \"As shown in Fig.\",\n \"REF , we can pick one tree from each of these sets and construct a decision table that is traditionally used to calculate overall preferences.\",\n \"These combinations of spanning trees can be formulated as $(\\\\tau _{k_1}^1,\\\\ldots ,\\\\tau _{k_m}^m,\\\\tau _{k_G}^G) \\\\in \\\\mathcal {T}^1\\\\times \\\\ldots \\\\mathcal {T}^m \\\\times \\\\mathcal {T}^G,$ Figure: Creating a number of decision tables by using different combinations of spanning tree vectors\"\n ],\n [\n \"Evaluating the combinations of spanning trees\",\n \"The evaluation vector of alternatives $a_i \\\\in A$ corresponding to the spanning tree $\\\\tau ^j_{k_j} \\\\in \\\\mathcal {T}^j$ can thus be represented as: $(u^k_j(a_1),\\\\ldots ,u^k_j(a_n))$ while the weight vector corresponding to the spanning tree $\\\\tau ^G_{k} \\\\in \\\\mathcal {T}^G$ is denoted as: $(w^k_1,\\\\ldots ,w^k_m)$ Now the overall priority for alternatives $a_i \\\\in A$ given by the combination of spanning trees $\\\\mathbf {\\\\tau }=(\\\\tau _{k_1}^1,\\\\ldots ,\\\\tau _{k_m}^m,\\\\tau _{k_G}^G) \\\\in \\\\mathcal {T}^1\\\\times \\\\ldots \\\\mathcal {T}^m \\\\times \\\\mathcal {T}^G$ will be: $u_{\\\\mathbf {\\\\tau }}(a_i)=w^{{k_G}}_1 u^{k_1}_1(a_i)+ \\\\ldots +w^{k_G}_m u^{k_m}_m(a_i)$ From this formulation, we can deduce the set of combinations of spanning trees for which any $a_{i_1},a_{i_2} \\\\in A$ $a_{i1}$ is preferred to $a_{i2}$ : $B(a_{i_1}\\\\succ a_{i_2})=\\\\lbrace \\\\mathbf {\\\\tau } \\\\in \\\\mathcal {T}^1\\\\times \\\\ldots \\\\mathcal {T}^m \\\\times \\\\mathcal {T}^G:u_{\\\\mathbf {\\\\tau }}(a_{i_1})>u_{\\\\mathbf {\\\\tau }}(a_{i_2})\\\\rbrace $ $a_{i1}$ is indifferent with $a_{i2}$ : $B(a_{i_1}\\\\sim a_{i_2})=\\\\lbrace \\\\mathbf {\\\\tau }\\\\in \\\\mathcal {T}^1\\\\times \\\\ldots \\\\mathcal {T}^m \\\\times \\\\mathcal {T}^G:u_{\\\\mathbf {\\\\tau }}(a_{i_1})=u_{\\\\mathbf {\\\\tau }}(a_{i_2})\\\\rbrace $ The set of combinations of spanning trees $\\\\mathbf {\\\\tau } \\\\in \\\\mathcal {T}^1\\\\times \\\\ldots \\\\mathcal {T}^m \\\\times \\\\mathcal {T}^G$ for which alternative $a_i \\\\in A$ attains the $p-th$ position with respect to overall priority $u_{\\\\mathbf {\\\\tau }}(a_{i})\\\\rbrace $ can be formulated as: $R(a_i,p)=\\\\lbrace \\\\mathbf {\\\\tau } \\\\in \\\\mathcal {T}^1\\\\times \\\\ldots \\\\mathcal {T}^m \\\\times \\\\mathcal {T}^G:rank(a_i,\\\\mathbf {\\\\tau })=p\\\\rbrace $ with $rank(a_i,\\\\mathbf {\\\\tau })=1+\\\\sum _{i^\\\\prime \\\\ne i}\\\\rho (u_{\\\\mathbf {\\\\tau }}(a_{i^\\\\prime })>u_{\\\\mathbf {\\\\tau }}(a_{i}))$ where $\\\\rho (false)=0$ and $\\\\rho (true)=1$ .\",\n \"Inspired by the Stochastic Multicriteria Acceptability Approach or SMAA [15], [16], the probability that alternative $a_{i_1}$ is preferred to alternative $a_{i_2}$ , with $a_{i1},a_{i2}\\\\in A$ can be represented as: $P(a_{i1} \\\\succ a_{i2})=\\\\frac{card(B(a_{i1} \\\\succ a_{i2}))}{card(\\\\mathcal {T}^1\\\\times \\\\ldots \\\\mathcal {T}^m \\\\times \\\\mathcal {T}^G)}$ that, remembering that the total number of spanning trees in a graph of $k$ nodes is $k^{k-2}$ [5], and, consequently $card(\\\\mathcal {T}^1\\\\times \\\\ldots \\\\mathcal {T}^m \\\\times \\\\mathcal {T}^G)=m^{m-2}n^{m(n-2)}$ , becomes $P(a_{i1} \\\\succ a_{i2})=\\\\frac{card(B(a_{i1} \\\\succ a_{i2}))}{m^{m-2}n^{m(n-2)}},$ the probability that alternative $a_{i} \\\\in A$ is attaining the $p-th$ ranking position will be: $P(rank(a_i,\\\\mathbf {\\\\tau })=p)=\\\\frac{R(a_i,p)}{card(\\\\mathcal {T}^1\\\\times \\\\ldots \\\\mathcal {T}^m \\\\times \\\\mathcal {T}^G)}=\\\\frac{R(a_i,p)}{m^{m-2}n^{m(n-2)}},$ [13] investigated incomplete sets of judgements where the DMs are allowed to respond with \\\"do not know\\\" or \\\"not sure\\\" to some judgements.\",\n \"This is an important issue to investigate as the probability of acquiring an incomplete set of PC judgements increases with an increase in the total number of items for comparison [10].\",\n \"Both the REV and the RGM methods are inappropriate in such cases due to the fact that the PC matrix cannot be constructed without estimating/imputing the missing judgements.\",\n \"The spanning trees can be applied also to partial comparison matrix with the help of the Kirchoff formula (See details in [21]).\",\n \"This implies that generating combinations of spanning trees is not restricted to only complete sets of judgements, and therefore, generating preference frequencies and rank-order frequencies are possible from incomplete pairwise comparisons, without any modification.\",\n \"The implication of having an incomplete set of judgements is that the total number of combinations will be smaller than the number possible from a complete sets of judgements.\",\n \"This permits the DM to only express values for those PC judgements for which he/she is sufficiently confident, avoiding forcing the DM to provide judgements when he/she is not sufficiently confident - in so doing maintaining the reliability of the preference information considered in the decision support process.\"\n ],\n [\n \"Didactic example\",\n \"Let us show how our approach works in practice by using the classic example of school selection proposed in [20].\",\n \"The parents have to decide the high school for their son.\",\n \"They consider six criteria being the following $g_1$ : Learning, $g_2$ : Friends, $g_3$ : School life, $g_4$ : Vocational training, $g_5$ : College preparation, $g_6$ : Music classes.\",\n \"There are three alternatives corresponding to three schools denoted $A$ , $B$ and $C$ .\",\n \"For each criterion, the parents compared in pairs the schools as shown in the following pairwise comparison matrices.\",\n \"$M_{g_1}=\\\\begin{pmatrix}3 & \\\\frac{1}{3} & \\\\frac{1}{2} \\\\\\\\3 & 1 & 3 \\\\\\\\2 & \\\\frac{1}{3} & 1 \\\\\\\\\\\\end{pmatrix}$ $M_{g_2}=\\\\begin{pmatrix}1 & 1 & 1 \\\\\\\\1 & 1 & 1 \\\\\\\\1 & 1 & 1 \\\\\\\\\\\\end{pmatrix}$ $M_{g_3}=\\\\begin{pmatrix}1 & 5 & 1 \\\\\\\\\\\\frac{1}{5} & 1 & \\\\frac{1}{5} \\\\\\\\1 & 5 & 1 \\\\\\\\\\\\end{pmatrix}$ $M_{g_4}=\\\\begin{pmatrix}1 & 9 & 7 \\\\\\\\\\\\frac{1}{9} & 1 & \\\\frac{1}{5} \\\\\\\\\\\\frac{1}{7} & 5 & 1 \\\\\\\\\\\\end{pmatrix}$ $M_{g_5}=\\\\begin{pmatrix}1 & \\\\frac{1}{2} & 1 \\\\\\\\2 & 1 & 2 \\\\\\\\1 & \\\\frac{1}{2} & 1 \\\\\\\\\\\\end{pmatrix}$ $M_{g_6}=\\\\begin{pmatrix}1 & 6 & 4 \\\\\\\\\\\\frac{1}{6} & 1 & \\\\frac{1}{3} \\\\\\\\\\\\frac{1}{4} & 3 & 1 \\\\\\\\\\\\end{pmatrix}$ After comparing the alternatives with respect to these criteria, the parents also compared the criteria in terms of their importance as shown below.\",\n \"$M_{criteria}=\\\\begin{pmatrix}1 & 4 & 3 & 1 & 3 & 4 \\\\\\\\\\\\frac{1}{4} & 1 & 7 & 3 & \\\\frac{1}{5} & 1 \\\\\\\\\\\\frac{1}{3} & \\\\frac{1}{7} & 1 & \\\\frac{1}{5} & \\\\frac{1}{5} & \\\\frac{1}{6} \\\\\\\\1 & \\\\frac{1}{3} & 5 & 1 & 1 & \\\\frac{1}{3} \\\\\\\\\\\\frac{1}{3} & 5 & 5 & 1 & 1 & 3 \\\\\\\\\\\\frac{1}{4} & 1 & 6 & 3 & \\\\frac{1}{3} & 1 \\\\\\\\\\\\end{pmatrix}$ The consistency ratios for the above pairwise comparison matrices are as follows: CR($M_{g_1}$ ) = 0.04, CR($M_{g_2}$ ) = 0, CR($M_{g_3}$ ) = 0, CR($M_{g_4}$ ) = 0.18, CR($M_{g_5}$ ) = 0, CR($M_{g_6}$ ) = 0.04, CR($M_{criteria}$ ) = 0.24 As we can see, there are four matrices which are not completely consistent and two of these, namely $M_{g_4}$ and $M_{criteria}$ have unacceptable levels of inconsistency as measured by the usual threshold of 0.1 for the CR value [20].\",\n \"The priorities $u_j(X), X=A,B,C, j=1,\\\\ldots ,6$ , obtained from pairwise comparisons matrices $M_{g_1}-M_{g_6}$ representing the evaluations of schools with respect to the considered criteria are shown in Table REF , while the weights of the considered criteria, according to the priority obtained from pairwise comparison's matrix $M_{criteria}$ , are the following: $w_1=0.32, w_2=0.14,w_3=0.03,w_4=0.13,w_5=0.24,w_6=0.14.$ Table: Scores for alternatives with respect to each criterionUsing priorities $w_j$ and $u_j(X), X=A,B,C, j=1,\\\\ldots ,6$ , we can compute the overall priority $u(X)$ of each school, with $u(X)=\\\\sum _{j=1}^6 w_j u_j(X)$ obtaining the following results: $u(A)=0.37, u(B)=0.38, u(C)=0.25.$ Let us now handle the same problem with the spanning tree approach.\",\n \"With this aim, we have to consider all the combinations of spanning trees from the pairwise comparison matrix of criteria $M_{criteria}$ and from the pairwise comparison matrices of alternatives with respect to the considered criteria $M_{g_j}, j=1,\\\\ldots ,m$ .\",\n \"Remembering that with $m$ criteria and $n$ alternatives, we have ${n^{(n-2)}}{m^{(m-2)n}}$ combinations of spanning tress, for the decision problem at hand, we have $6^4\\\\cdot 3^6 = 944784$ combinations of spanning trees.\",\n \"We compute weights of criteria $w^k_1,\\\\ldots ,w^k_6$ and evaluations of alternatives with respect to considered criteria $u^k_j(A),u^k_j(B),u^k_j(C)), j=1,\\\\ldots ,6,$ for each combination of spanning trees $\\\\mathbf {\\\\tau }=(\\\\tau _{k_1}^1,\\\\ldots ,\\\\tau _{k_6}^m,\\\\tau _{k_G}^G) \\\\in \\\\mathcal {T}^1\\\\times \\\\ldots \\\\mathcal {T}^6 \\\\times \\\\mathcal {T}^G$ .\",\n \"Figure: Demonstrating the two different combinations of spanning trees, and the difference in their rankings and scores.Considering all the combinations of spanning trees from $\\\\mathcal {T}^1\\\\times \\\\ldots \\\\mathcal {T}^6 \\\\times \\\\mathcal {T}^G$ we can compute Table REF that shows the probability that each alternative attains a given rank (within parenthesis there is the total number of combinations of spanning trees for which there is the preference of the alternative on the row over the alternative on column), Table REF that for each pair of alternatives shows the probability that the alternative in the row is preferred to the alternative in the column (within parenthesis the total number of combinations of spanning tress for which the alternatives attains the considered ranking position).\",\n \"Table: Preference frequency over 944784 combinations of spanning treesTable: Rank order frequency over 944784 combinations of spanning trees\"\n ],\n [\n \"Random spanning trees\",\n \"We have seen that, for a problem with $n$ alternatives evaluated across $m$ criteria, the total number of combinations of spanning trees increases exponentially with the problem size - making it impractical to calculate the exact probabilities of $P(rank(a_i,\\\\mathbf {\\\\tau })=p)$ and of $P(a_{i1} \\\\succ a_{i2})$ for large problems.\",\n \"We therefore propose using an approach which, combined with statistical sampling theory, allows us to provide estimates of the required probabilities of $P(rank(a_i,\\\\mathbf {\\\\tau })=p)$ and of $P(a_{i1} \\\\succ a_{i2})$ to within any user defined degree of accuracy according to any user defined level of confidence.\",\n \"Of course we cannot physically select a statistical random sample from the population of spanning tree combinations since this would require the generation of the population itself and this is exactly what we aim to avoid.\",\n \"We can however use the 'random walk' procedure to generate a tree and hence a sample of trees (see [1], [3].\",\n \"Indeed, the 'random walk' procedure has been proven to generate a true statistical random sample in the sense that the generated sample is equivalent to selection of a statistical random sample from the population, that is, where each tree has the same uniform probability of being selected.\",\n \"See [3], [1] for more details.\",\n \"As we discussed the possibility of having incomplete sets of pairwise comparison judgements will arise in practice and so it is important to note that the concept of random walks is equally applicable to these incomplete sets without modification.\",\n \"Figure: The PC matrix acquired for the top-level criteriaSince each iteration of the procedure generates a new member of the random sample, the total number of iterations used in the procedure is equivalent to sample size and we can use statistical large sample theory to determine the number of iterations required to generate sample parameters (including the probabilities of $P(rank(a_i,\\\\mathbf {\\\\tau })=p)$ and of $P(a_{i1} \\\\succ a_{i2})$ ) to any specified degree of accuracy and with any specified level of confidence.\",\n \"That is, if we want the estimated probability to have an accuracy within $\\\\lambda $ and with a $C$ percent level of confidence, then the required number of iterations would be [22]: $It(\\\\lambda ,C) = \\\\frac{ {Z_C}^2 }{4\\\\lambda ^2}$ where $Z_C$ is the z-score calculated from the standardised normal distribution curve.\",\n \"For example, if we want to achieve an accuracy within 0.01 and with a $99\\\\%$ level of confidence, then the required number of iterations would be $It(0.01,99\\\\%) = \\\\frac{ Z_{99}^2 }{4 \\\\times 0.01^2}$ As we know that the Z value for 99% confidence interval is equal to 2.58, therefore $It(0.01,99\\\\%) = \\\\frac{ {(2.58)}^2 }{4 \\\\times 0.01^2} = 16,641$ This suggests that we need to perform more than 16,641 iterations if we want to achieve an accuracy within $\\\\pm 0.01$ with $99\\\\%$ confidence.\"\n ],\n [\n \"Applying random walk procedure to the school example\",\n \"In the example below, we illustrate the use of random walks by repeating an experiment multiple times where each experiment itself uses the required number of iterations $It$ determined by the formula above.\",\n \"We do this on the previously discussed School example as this is a tractable problem where the whole population of solutions is available.\",\n \"Later, we will also extend this idea to a bigger problem where generating the whole population is impractical (see Section ).\",\n \"Table REF shows the preference frequencies obtained through random walks, as well as the preference frequencies obtained by taking a sample from the whole population of solutions.\",\n \"Recall that the whole population was generated by enumerating all possible combinations of spanning trees that was shown in Table REF earlier.\",\n \"Here we can see that the estimated values from random walks and from the sample are both lying within 1% of the population values that were previously shown in Table REF .\",\n \"Table: Preference frequencies of the school exampleTable: Rank order profiles generated for the school exampleWe performed the same analysis for rank-order frequencies as well, as summarised in Table REF .\",\n \"The table on the left side shows that the rank-order frequencies obtained through random walks are also lying within 1% of the population values shown in Table REF .\",\n \"The rank-order frequencies obtained by taking a sample are shown on the right side of this table.\",\n \"This also illustrates that the accuracy and confidence level results for the generated random sample echo the same results had we been able to physically select a statistical random sample from the population of spanning tree combinations.\"\n ],\n [\n \"Telecom backbone example\",\n \"In order to demonstrate the use of spanning trees to explore multiple mindsets, we consider the practical data acquired in a recent study: the selection of a backbone infrastructure for telecommunication in rural areas [11].\",\n \"This study focused on the use of a structured decision making approach towards selecting the telecommunication infrastructure for the rural areas of developing countries.\",\n \"The lack of adequate telecommunications infrastructure in these parts of the world remains a major obstacle for providing affordable services.\",\n \"The study considered four options of Fiber-optic cable (G1), Power-line communication (G2), Microwave link (G3) and Satellite communication (G4).\",\n \"The authors of this study used AHP and Analytic Network Process (ANP) to structure the problem and acquired the necessary data on preferences and assessments from key stakeholders.\",\n \"The criteria used to compare these alternatives were grouped into six major categories including technical, infrastructural, economic, social, regulatory and environmental factors.\",\n \"These categories and their constituent criteria are presented in Fig.\",\n \"REF .\",\n \"Figure: Criteria to compare the available backbone infrastructuresFigure: The PC matrix acquired for the top-level criteriaThe PC matrix, $A_{top}$ , acquired for prioritising these six categories (top-level criteria) is shown in Fig.\",\n \"REF .\",\n \"Although $A_{top}$ is a transitive PC matrix, the estimated vectors produced by the widely used EV method does not preserve the original order of preferences in these judgements.\",\n \"The final weights calculated using the EV and GM methods are found to be almost identical, as given in Table REF in normalised form.\",\n \"Satellite communication (G4) is considered the most preferred alternative with a weight of 29.95% (using EV), followed by Microwave (G3) with a weight around 28.34% (using EV).\",\n \"Table: Estimated weights for the available backbone infrastructure optionsMost criteria lie under the Technical and Infrastructure categories.\",\n \"The Technical category includes nine criteria whilst the Infrastructure category has eight criteria used to compare the alternatives.\",\n \"The two PC matrices for the Technical and Infrastructure categories, $A_{tech}$ and $A_{infra}$ , have been found to be intransitive and should be investigated along with $A_{top}$ for their impact on the final result.\"\n ],\n [\n \"Spanning trees analysis using Random walks\",\n \"In this case study, there are four alternatives and six top-level criteria, therefore, the number of “spanning-trees” solutions for criteria matrix is $m^{(m-2)}$ = $6^4$ = 1296, and the number of “spanning-trees” solutions for each alternative matrix is $n^{(n-2)}$ = $4^2$ = 16.\",\n \"As we pick one “spanning-trees” solution from each matrix, there are $m^{(m-2)}n^{m(n-2)}$ = $1296\\\\times 16^6$ = $21,743,271,936$ combinations possible.\",\n \"Generating these billions of combinations is computationally expensive, so sampling is the logical way to estimate probabilities.\",\n \"Therefore we generated a sample of 20,000 solutions with the use of random walks.\",\n \"We generated 20 such samples in order to check whether all these samples produce similar results.\",\n \"In practise, only one sample should suffice but we took this approach for investigation only.\",\n \"The average scores for the four alternatives are given in Table REF .\",\n \"Each row in this table represents one iteration of generating a sample of 20,000 solutions.\",\n \"Table: Scores calculated usign random walks; each iteration shows an average of 20k solutionsUnlike the School example where we calculated frequencies by generating all the combinations of spanning trees, here we will calculate the frequencies from the samples generated by random walks.\",\n \"It can be argued that these frequencies may vary as they are based on randomly generated set of solutions.\",\n \"However, the aim here is to gain statistical insights, and therefore, these findings should remain useful as long as they produce reliable statistical findings.\",\n \"We investigated this reliability by repeating the same experiment twenty times and comparing the preference frequencies and rank-order frequencies.\",\n \"The standard deviations of these values do not exceed 0.005, which supports the argument that these stochastic results are quite reliable.\",\n \"Table REF shows the probability that one alternative is preferred to the other (for all possible pairs).\",\n \"Looking at the first row, we can say that Fiber is preferred over Powerline in 63% (see $0.63\\\\pm 0.004$ under the Powerline column) of the random walk solutions, however, it is seldom preferred over Microwave and Satellite.\",\n \"Microwave is clearly preferred over Fiber (with a probability of 87%) and Powerline (with a probability of 96%), however, it has been marginally preferred over Satellite (with a probability of 51%).\",\n \"We can argue that the Fiber and Powerline are the two clearly dominated alternatives, while the Microwave and Satellite are the two dominating alternative.\",\n \"However, when comparing Microwave and Satellite with each other, Microwave is slightly more preferred but there is no clear winner.\",\n \"We are providing the standard deviations for each of these values to show that these scores didn't deviate much, as discussed earlier.\",\n \"Table: Preference frequencies for Telecom backbone selectionTable: Rank-order frequencies for Telecom backbone selectionAnother interesting way to assess these options is to estimate the ranking of each alternative.\",\n \"Table REF shows the probability that each alternative attains a given rank.\",\n \"In this table, The probability of Fiber and Powerline taking the first rank is too low (only 4.9% and 2.5% respectively).\",\n \"However, Microwave and Satellite have high probabilities of taking the first rank.\",\n \"On the other end, although Fiber is more likely to attain the third rank, both Fiber and Powerline have high probabilities to attain the lowest rank (36.5% and 62.2% respectively).\",\n \"The two alternatives of Microwave and Satellite are least likely to be placed on the lowest rank, as the percentage of combinations placing them at the 4th position were only 0.6% and 0.7%, respectively.\"\n ],\n [\n \"Conclusions\",\n \"The spanning trees analysis can help understand the plurality of mindsets in terms of a plurality of prioritisation vectors originating from a plurality of spanning trees of pairwise comparison matrices.\",\n \"Considering all the rankings of alternatives corresponding to the different mindsets, we propose to estimate the probability that an alternative attains a given ranking position, and the probability that an alternative is preferred to another one.\",\n \"Moreover, the proposed approach can be applied to incomplete sets of judgements without any modification.\",\n \"Since the exponential number of spanning trees makes their enumeration prohibitive, we propose computing approximate probabilities using statistical sampling of spanning trees.\",\n \"The usefulness of statistical sampling is further demonstrated with the case study of Telecom infrastructure selection for rural areas.\",\n \"An interesting area of future research is to do performance analysis using the sigma-mu approach proposed by [12].\",\n \"For each alternative, we compute the mean, $\\\\mu $ , and the standard deviation, $\\\\sigma $ , of all the solutions generated from combinations of spanning trees.\",\n \"The objective would be to maximise the mean while minimising the standard deviation.\",\n \"The use of interval judgements is also an important area to investigate as we know that decision makers often provide their judgements in intervals (for example, stating that X is about 3 to 5 times better than Y).\",\n \"Therefore, the idea of interval judgements need to be combined with the use of spanning trees combinations.\",\n \"Considering the lower and upper limit of intervals, the number of combinations will further increase exponentially, therefore the use of a stochastic approach might be the only practical approach for this purpose.\",\n \"There is a further potential in the use of spanning trees analysis at group-level decision making where multiple judgements are collected for comparing the same pair of alternatives or weighing the same pair of criteria.\",\n \"The concept of \\\"aggregated individual preferences\\\" and \\\"aggregated individual judgements\\\" has been widely debated in the context of group decision making, and we believe that the use of spanning tree analysis can offer better insights into these types of problems.\"\n ]\n]"},"id":{"kind":"string","value":"2107.01731"}}},{"rowIdx":1596,"cells":{"text":{"kind":"list like","value":[["Age of Information in Relay-Assisted Status Updating Systems"],["Abstract In this paper we consider the age of information (AoI) of a status updating system with a relay, where the updates are delivered to destination either from the direct line or the two-hop link via the relay.","An updating packet generated at source is sent to receiver and the relay simultaneously.","When the direct packet transmission fails, the relay replaces the source and retransmits the packet until it is eventually obtained at the receiver side.","Assume that the propagation delay on each link is one time slot, we determine the stationary distribution of the AoI for three cases: (a) relay has no buffer and the packet delivery from relay cannot be preempted by fresher updates from source; (b) relay has no buffer but the packet substitution is allowable; (c) relay has size 1 buffer and the packet in buffer is refreshed when a newer packet is obtained.","The idea is invoking a multiple-dimensional state vector which contains the AoI as a part and constituting the multiple-dimensional AoI stochastic process.","We find the steady state of each multiple-dimensional AoI process by solving the system of stationary equations.","Once the steady-state distribution of larger-dimensional AoI process is known, the stationary AoI distribution is also obtained as it is one of the marginal distributions of that process's steady-state distribution.","For all the situations, we derive the explicit expression of AoI distribution, and calculate the mean and the variance of the stationary AoI.","All the results are compared numerically, including the AoI performance of the non-relay state updating system.","Numerical results show that adding the relay improves the system's timeliness dramatically, and no-buffer-and-preemption setting in relay achieves both minimal average AoI and AoI's variance.","Thus, for the system model discussed in this paper, to reduce the AoI at receiver there is no need to add the buffer in relay."],["Introduction","Apart from the traditional performance targets such as low propagation delay and large throughput, modern network designs put forward new requirements on information freshness and the timeliness of an information transmission system.","In many remote monitoring system, the source has to transmit the messages or signals via a channel fast to keep the receiver's knowledge about certain physical process in source side fresh enough.","These messages are time sensitive, whose freshness is of great importance in some applications.","For example, the controller of a driverless car must obtain sufficiently new messages to keep the car running safely.","The central node in a sensor network needs the fresh data from its neighbours to perform certain real time computations, such that an optimal resource scheduling policy is determined.","In general, this kind of information transmission system is abstracted into a status updating system, where the updates are generated in source and then sent to the receiver.","The state at the receiver is refreshed after an updating packet is obtained successfully.","To characterize the timeliness of the system or measure the freshness of an updating packets, a timeliness metric called age of information (AoI) was introduced in [1].","Since then, lots of articles has been published focusing on different aspects of AoI, such as the AoI performance analysis and designing various optimal status updating system who has minimal average AoI.","A detailed survey about AoI can be found in paper [2], in which the authors summarize the recent contributions of AoI in the aspect of theory and practice.","For simple queue models such as $M/M/1$ , $M/D/1$ , and $M/M/\\infty $ , the average AoI of state updating system were derived in papers [3], [4], [5].","In particular, the Stochastic Hybrid System (SHS) analysis which is a systematic method to calculate the mean of AoI was introduced in paper [5].","The method has also been used to determine the average AoI of some other status updating systems in [6], [7], [8], [9], [10], [11], [12] recently.","Packet managements problem was considered in [13], where the authors defined another metric called peak age of information (PAoI) and derived the average performance of both AoI and PAoI for three small-size updating systems.","Assume that the packet has a fixed or random deadline, the long term average value of steady-state AoI was calculated in work [14].","Apart from the analytical results of AoI described above, in practical applications the AoI has been considered widely in optimal system designs, either as the performance target or acting as one of constraints [15], [16], [17], [18], [19], [20], [21], [22].","Observing that in the majority of systems considered before, an updating packet takes only one hop from source to the destination.","Meanwhile, often there is only one path between the two nodes.","However, both the packet multihop transmissions and the packet multipath transmissions are existent everywhere in practical communication networks, especially in wireless environment.","The AoI of status updating system having two parallel servers was considered in [23], the authors computed the average AoI of this two-server system by sophisticate random events analysis.","In paper [24] the AoI in multihop networks was analyzed.","Recently, the AoI of a three node status updating system was considered in [25], where a relay was added to cooperate the packet transmission from source to receiver.","The authors showed that although two-hop transmission via the relay takes longer propagation delay, but the transmission success probability is raised, in particular when the direct line from source to destination is bad.","Summarize both aspects, they proved that adding a relay can reduce the average AoI at the receiver, thus improving the system's timeliness.","In this paper, we consider a more general relay-assisted status updating system than that was discussed in [25], and derive more fundamental results about system's AoI.","There, the situation that the relay has buffer was not mentioned and, for the non-relay cases, only the mean of steady-state AoI was calculated.","We investigate both non-relay cases and the situation where the relay has a size 1 buffer.","For each cae, the stationary distribution of AoI was derived explicitly, thus obtaining the complete description of the steady-state AoI.","In [25], the authors derived their results using the most initial formula for the limiting time average AoI, which is used to deal with continuous time AoI.","However, as the time is slotted, the AoI is discretized as well.","Thus, the AoI considered in that paper, and in many other papers where the time was slotted is actually discrete.","Here, we show that the analysis of AoI can be done by the idea and standard tools from queueing theory.","The idea is invoking a multiple-dimensional discrete state vector which contains the AoI as a part, and constituting the multiple-dimensional AoI stochastic process.","As long as the steady state of the constituted process is determined, i.e., all the stationary probabilities are solved, the stationary AoI distribution can be found by merging all the stationary probabilities with identical AoI state component, as it is in fact one of the marginal distributions of the larger-dimensional AoI process's steady-state distribution.","We point out that the idea that invoking a multiple-dimensional state vector to characterize the random transfers of AoI and analyzing the steady state of the constituted AoI process is also the idea of the SHS approach when the continuous AoI is discussed.","Differently, finding the steady-state of a AoI stochastic hybrid system, one has to solve a system of differential equations which is in general very hard.","While when the discrete AoI is analyzed, solving the system of stationary equations of constituted discrete AoI process is easier.","Just recently, in paper [26], [27], [28], [29], the authors proved that following above straightforward thinking, the stationary distribution of continuous AoI of status updating system with a few of simple queue models can also be determined, using AoI's Laplace Stieltjes Transform (LST) form.","Finding the stationary distribution of AoI gets more attention in recent two or three years, both for continuous and discrete form of AoI [30], [31], [7], [32].","The rest of the paper is organized as follows.","We describe the model of a relay-assisted state updating system in Section II and introduce the system parameters.","In Section III, assume that the relay has no buffer, we analyze the system's AoI of two cases where the packet retransmissions from relay can and cannot be preempted by later updates from the source node.","The explicit expressions of steady-state AoI are derived by constituting a two-dimensional stochastic process.","For the situation the relay is equipped with a size 1 buffer, the AoI of relay-assisted system is discussed in Section IV.","There, a three-dimensional state vector is defined and the steady state of a three-dimensional AoI stochastic process is analyzed.","The stationary AoI distribution is obtained as well.","Provided the AoI distributions obtained before, for each case, we calculate the mean and the variance of AoI in Section V. Numerical results are given in Section VI, in which we compare the average AoI and the AoI's variance of all three cases.","Besides, to demonstrate the improvement of adding the relay, we also calculate the AoI performance of an updating system without relay.","The paper is summarized in Section VII, where some possible generalizations of relay-assisted system are discussed as well."],["System Model and Problem Formulation","The system model is depicted in Figure REF .","The source $s$ observes certain physical process and samples the state of the process at random times.","The packet arrival is assumed to be a Bernoulli process, that is a packet arrives in each time slot independently with identical probability $p$ .","An updating packet is delivered to destination $d$ , so that the knowledge of receiver about the process is refreshed.","Since the existence of relay $R$ , the packet can be transmitted either from the direct line or the two-hop link via $R$ .","Specifically, when the source $s$ transmits a packet, the same packet is also sent to $R$ at the same time.","Figure: The model of a status updating system with a relay.Assume that $R$ gets a feedback signal when every time an updating packet is obtained at the receiver.","In this case, the packet stored in $R$ is deleted to avoid redundant transmissions.","In particular, if one packet is transmitted on direct link successfully, then all the packets in $R$ need to be deleted, including the one waiting in buffer when we consider the relay-has-buffer case, since all of these packets are out of date.","Otherwise, the relay will replace the source and retransmit the packet to $d$ .","Since $R$ is closer to $d$ , it is more likely that the transmission is successful.","Notice that the $s-R$ link is also unreliable, it is possible that both $s-R$ and $s-d$ transmissions fail at the same time.","The source $s$ triggers the next transmission when a new packet arrives.","We are interested in the stationary AoI of the relay-assisted system for three cases, i.e., when the relay $R$ has no buffer, we discuss the AoI performance assuming that the packet retransmission can and cannot be preempted by fresher packets from source, and the steady-state AoI of the system when the relay is equipped with a size 1 buffer.","The three parameters $P_1$ , $P_2$ , and $P_3$ denote the transmission success probabilities of $s-d$ , $s-R$ , and $R-d$ links, respectively.","Generally speaking, $P_1$ is less than $P_2$ and $P_3$ .","The propagation delays on all links are suppose to be one time slot.","In this paper, we shall prove that for all of three cases the stationary distribution of AoI can be determined, such that the complete description of the steady-state AoI for the considered relay-assisted status updating system is obtained."],["Analysis of Age of Information: Relay has no buffer","In this Section, assume that the relay has no buffer, we analyze the stationary AoI of the relay-assisted status updating system by constituting a two-dimensional discrete stochastic process.","We determine the explicit expressions of the AoI distribution for both the cases where the packet in relay can and cannot be preempted by the newer ones from thee source.","Observing that if no packet arrives to the destination, the AoI increases by 1.","The value of AoI is reduced to the age of the packet, which is actually equal to the system time of the packet until it is eventually received at receiver $d$ .","Define the two-dimensional discrete state vector $(n,m)$ , where $n$ denotes the AoI at $d$ , and $m$ represents the age of packet which is stored in relay $R$ .","When there is no packet in $R$ , we let $m=0$ .","In the following, for each case, we describe the random state transfers of each state vector, establish the stationary equations of the constituted AoI stochastic process and finally find its steady state, i.e., determining the stationary probabilities of every state vector $(n,m)$ .","Then, the stationary distribution of AoI is obtained by merging all the probabilities of those state vectors that have identical AoI-component $n$ .","Notice that in the following paragraphs we also call the state vector as the age-state of the system, or simply the state."],["Analysis of AoI: packet preemption is allowable in relay","Firstly, we analyze the AoI of the system assuming that the packet in $R$ is substituted when every time a fresher update is obtained from the source.","The random transfers of age-state $(n,m)$ are described in the following.","First of all, assume that the age-state is $(n,m)$ where $n> m\\ge 1$ .","In this case, a packet of age at least 1 is contained in $R$ .","If no packet arrives to source at next time slot, then the state transfers are dependent on the packet transmission on $R-d$ link.","With probability $P_3$ , the packet delivery succeeds and the state vector changes to $(m+1,0)$ .","The second state-component is zero because the packet in $R$ is deleted when it was sent to $d$ successfully.","Otherwise, both $n$ and $m$ will increase 1 at next time slot, i.e., the age-state turns to $(n+1,m+1)$ .","For the case a new packet is generated and sent out from source $s$ , we have to analyze all the packet transmissions on links $s-R$ , $R-d$ , and direct link $s-d$ .","Let $A$ be the random variable of packet arrivals, and we use $T_{s-d}$ , $T_{s-R}$ , $T_{R-d}$ to indicate whether the packet transmission is successful in three links, respectively.","$A=0$ denotes no packet arrives and $T_{s-d}=0$ means that the packet transmission on $s-d$ link fails.","Other variables are defined in similar manner.","We list all the possible state transitions in Table REF .","Table: Description of state vector transfers of AoI stochastic process AoI NB P AoI_{NB}^PTable: All the state transfers and corresponding transition probabilities of stochastic process AoI NB P AoI_{NB}^PIt is necessary to explain all the cases including $A=1$ and $T_{s-d}=1$ in Table REF .","When a new packet is delivered successfully over direct link $s-d$ , the value of AoI must reduce to 1 since a packet of age 1 is obtained in $d$ , although it is possible that an another packet is received on $R-d$ link.","At the same time, according to our assumption, a feedback signal is sent to $R$ and the relay deletes the packet received at current time slot or before.","Thus, $m$ is reduced to zero and the state vector jumps to $(1,0)$ .","Still we have to consider the state transfers when the initial age-state is $(n,0)$ , which means the relay $R$ is currently empty.","If no packet arrives, it is easy to see that the age-state transfers to $(n+1,0)$ at next time slot.","On the contrary, for the case $A=1$ , we need to consider the packet transmissions on $s-d$ and $s-R$ links.","All the possible state transitions from age-state $(n,0)$ are given in the last four lines in Table REF .","Constituting the two-dimensional AoI stochastic process $AoI_{NB}^P=\\lbrace (n_k,m_k):n_k> m_k\\ge 0,k\\in \\mathbb {N}\\rbrace ,$ where the subscript $NB$ denotes that at relay $R$ there is no buffer, and the superscript $P$ indicates that the packet retransmission in $R$ can be preempted.","Define $P_{(n,m),(i,j)}$ be the single step transition probability from $(n,m)$ to age-state $(i,j)$ .","Summarize above discussions, we list all the state transfers and corresponding transition probabilities of the AoI process $AoI_{NB}^P$ in Table REF .","Denote that $\\pi _{(n,m)}$ is the probability of age-state $(n,m)$ , $n> m\\ge 0$ when the AoI process runs in steady state.","In following Theorem 1, we establish the stationary equations of the process $AoI_{NB}^P$ .","Theorem 1 Assume that the stochastic process $AoI_{NB}^P$ reaches the steady state, then the stationary probabilities $\\pi _{(n,m)}$ of age-state $(n,m)$ , $n>m\\ge 0$ satisfy that ${\\left\\lbrace \\begin{array}{ll}\\pi _{(n,m)}=\\pi _{(n-1,m-1)}\\big [(1-p)(1-P_3) + p(1-P_1)(1-P_2)(1-P_3)\\big ] & \\quad (n>m\\ge 2) \\\\\\pi _{(n,1)}=\\pi _{(n-1,0)}p(1-P_1)P_2 + \\sum \\nolimits _{j=1}^{n-2}\\pi _{(n-1,j)}p(1-P_1)P_2(1-P_3) \\\\\\qquad \\qquad \\qquad + \\sum \\nolimits _{k=n}^{\\infty }\\pi _{(k,n-1)}p(1-P_1)P_2P_3 & \\quad (n\\ge 3) \\\\\\pi _{(2,1)}=\\pi _{(1,0)}p(1-P_1)P_2 + \\sum \\nolimits _{k=2}^{\\infty }\\pi _{(k,1)}p(1-P_1)P_2P_3 \\\\\\pi _{(n,0)}=\\pi _{(n-1,0)}\\left[(1-p)+p(1-P_1)(1-P_2)\\right] \\\\\\qquad \\qquad \\qquad + \\sum \\nolimits _{k=n}^{\\infty }\\pi _{(k,n-1)}\\left[(1-p)P_3+p(1-P_1)(1-P_2)P_3\\right] & \\quad (n\\ge 2) \\\\\\pi _{(1,0)}=\\left(\\sum \\nolimits _{n=1}^{\\infty }\\pi _{(n,0)}+ \\sum \\nolimits _{m=1}^{\\infty }\\sum \\nolimits _{n=m+1}^{\\infty }\\pi _{(n,m)} \\right) pP_1\\end{array}\\right.","}$ Equations in (1) are explained as follows.","First of all, begin with $(n-1,m-1)$ , as long as the packet transmission on $R-d$ link fails which occurs with probability $1-P_3$ , along with the condition that either no packet is delivered on $s-R$ and $s-d$ links or both transmissions are unsuccessful, then the age-state transfers to $(n,m)$ at next time slot.","This gives the first line of equations (1).","To obtain $(n,1)$ , more cases need to be considered.","From age-state $(n-1,0)$ which means $R$ is initially empty, at source $s$ a packet arrives and is transmitted to $R$ successfully makes the second parameter jump to 1.","On the other hand, let the packet delivery on $s-d$ link fail, such that the AoI in $d$ increases to $n$ at next time slot.","This state transitions occurs with probability $p(1-P_1)P_2$ .","Next, notice that the packet in $R$ is allowed to be refreshed, the second state-component 1 can also be created by packet replacement.","Assume that the initial state vector is $(n-1,j)$ , $1\\le j\\le n-2$ .","When the new packet is transmitted on $s-R$ link successfully but the packet transmissions on both $s-d$ and $R-d$ links fail, it was observed that the state also transfers to $(n,1)$ .","Thus, we obtain the second term of the RHS of second equation.","Now, suppose that a packet of age $n-1$ goes through $R-d$ link to the receiver.","As a result, the AoI in $d$ is reduced to $n$ at next time slot.","At the same time, the relay $R$ obtains a new packet from $s$ via $s-R$ link, which changes the second parameter to 1.","In this case, we can obtain the age-state $(n,1)$ again.","The initial state can be $(k,n-1)$ where $k$ is an arbitrary number greater than $n-1$ .","For the case $n=2$ , packet substitution in $R$ is nonexistent since $R$ is empty before one time slot.","With all above discussions, we explain how the state vector $(n,1)$ , $n\\ge 2$ is obtained and determine its stationary equation.","Then, the random transitions to $(n,0)$ is analyzed.","The relay $R$ is empty either it is empty initially, or is emptied as the contained packet is sent to $d$ successfully.","Therefore, two cases have to be analyzed.","Firstly, from age-state $(n-1,0)$ , as long as no packet arrives to $R$ and $d$ , obviously the state jumps to $(n,0)$ after one time slot.","On the other hand, let the initial state is $(k,n-1)$ , $k\\ge n$ , a packet of age $n-1$ can be transmitted to $d$ via $R-d$ link and assume that no other packet is obtained from $s$ through the direct link $s-d$ , which ensures that the AoI is set to $n$ at next time slot.","The second parameter maintains 0 if no packet comes to $s$ , or the arriving packet is not transmitted to $R$ successfully on $s-R$ link.","Combined with the transition probabilities in Table REF , we give the stationary equation for the state $(n,0)$ in the end.","The only way to get age-state $(1,0)$ is transmitting a packet via the direct link $s-d$ , no matter what the initial state is.","Actually, this last equation determines $\\pi _{(1,0)}=pP_1$ directly, since the probabilities in bracket add up to 1.","So far, all the state transfers are considered, and the proof of Theorem 1 completes.","We solve the system of equations (1) and determine all the stationary probabilities $\\pi _{(n,m)}$ in Theorem 2, whose proof is postponed to Appendix A. Theorem 2 The probabilities $\\pi _{(n,m)}$ of AoI process $AoI_{NB}^P$ are determined as follows.","Firstly, $\\pi _{(n,0)}=(1-\\delta )\\delta ^{n-1} - p(1-P_1)P_2[(1-P_3)\\delta ]^{n-1} \\qquad (n\\ge 1)$ and for $n>m\\ge 1$ $\\pi _{(n,m)}&=\\pi _{(2,1)}[(1-pP_1)(1-P_3)]^{n-2}\\left(\\frac{\\delta }{1-pP_1}\\right)^{m-1} + \\frac{p(1-P_1)P_2P_3(1-\\delta )\\delta }{\\delta -(1-pP_1)(1-P_3)} \\\\& \\qquad \\quad \\times \\left[ \\delta ^{n-2}(1-P_3)^{m-1} - [(1-pP_1)(1-P_3)]^{n-2}\\left(\\frac{\\delta }{1-pP_1} \\right)^{m-1} \\right]$ where $\\delta =1-p(P_1+P_2-P_1P_2), \\qquad \\pi _{(2,1)}=p^2(1-P_1)P_2[P_1+(1-P_1)P_2P_3] $ Provided all the stationary probabilities, the probability that the steady-state AoI equals $n$ can be obtained by $\\Pr \\lbrace \\Delta =n\\rbrace =\\pi _{(n,0)} + \\sum \\nolimits _{m=1}^{n-1}\\pi _{(n,m)} \\qquad (n\\ge 2)$ The probability that AoI takes value 1 is determined as $pP_1$ , which is known directly from the last line of equations (1).","We determine the explicit expression of AoI distribution in following Theorem 3 and give the calculations in Appendix B. Theorem 3 For the considered relay-assisted status updating system, the stationary distribution of AoI $\\Delta _{NB}^P$ is determined as $\\Pr \\lbrace \\Delta _{NB}^P=n\\rbrace =\\frac{(1-pP_1)P_3(1-\\delta )}{(1-pP_1)P_3-p(1-P_1)P_2}\\delta ^{n-1} +\\xi [(1-pP_1)(1-P_3)]^{n-1} \\quad (n\\ge 1)$ in which $\\xi =\\frac{p[P_1+(1-P_1)P_2P_3]}{1-P_3} - \\frac{P_3(1-\\delta )\\delta }{[(1-pP_1)P_3-p(1-P_1)P_2](1-P_3)} $"],["The stationary AoI: packet retransmission is not preempted in relay","In previous part of this Section, we have determined the stationary AoI distribution for the case where the packet in relay can be replaced.","Intuitively, substituting the large-age packet with fresher ones will result to lower AoI and enhance the timeliness of the system.","We will prove this conclusion by finding the steady-state AoI distribution for the $non-preemption$ case explicitly and do the comparisons.","The AoI stochastic process constituted for this case is denoted as $AoI_{NB}^{NP}$ , where the superscript $NP$ indicates that no packet preemption is allowed in relay $R$ .","When the packet in relay cannot be preempted before the receiver gets an updating packet eventually, which may be obtained from $R-d$ link or the $s-d$ link, the discussions of random state transitions are easier, since the relay will always reject the packet from the source as long as there has been one packet within it.","We first describe the possible state transitions when the initial age-state is $(n,m)$ , where $n>m\\ge 1$ .","If $A=1$ , that is a new packet is sent out from the source in this time slot.","Now, only the packet transmissions on $s-d$ and $R-d$ links need to be discussed, since the packet over $s-R$ link is always rejected.","For the case $A=0$ , we have to consider just the packet delivery over $R-d$ link.","All the possible state transfers are listed in Table REF , in which the state transfers from initial age-state $(n,0)$ are also given.","For each state transfer, we determine the corresponding transition probability in Table REF , from which the stationary equations of the AoI process $AoI_{NB}^{NP}$ can be established, when the process reaches the steady state.","Theorem 4 Assume that the AoI stochastic process $AoI_{NB}^{NP}$ reaches the steady-state, the stationary probabilities $\\pi _{(n,m)}$ satisfy following equations ${\\left\\lbrace \\begin{array}{ll}\\pi _{(n,m)}=\\pi _{(n-1,m-1)}[(1-p)(1-P_3)+p(1-P_1)(1-P_3)] & (n>m\\ge 2) \\\\\\pi _{(n,1)}=\\pi _{(n-1,0)}p(1-P_1)P_2 & (n\\ge 2) \\\\\\pi _{(n,0)}=\\pi _{(n-1,0)}[(1-p)+p(1-P_1)(1-P_2)] \\\\\\qquad \\quad \\quad + \\left(\\sum \\nolimits _{k=n}^{\\infty }\\pi _{(k,n-1)}\\right)[(1-p)P_3+p(1-P_1)P_3] & (n\\ge 2) \\\\\\pi _{(1,0)}= \\left(\\sum \\nolimits _{n=1}^{\\infty }\\pi _{(n,0)} + \\sum \\nolimits _{n=2}^{\\infty }\\sum \\nolimits _{m=1}^{n-1}\\pi _{(n,m)}\\right)pP_1\\end{array}\\right.","}$ Assume that the initial state is $(n-1,m-1)$ , to get age-state $(n,m)$ , it needs that no packet is obtained in both relay and destination, which is satisfied either no packet arrives at source and retransmission from relay fails, or a new packet comes to source but the packet transmissions on $s-d$ and $R-d$ links are unsuccessful.","Since the packet in relay cannot be substituted, the state $(n,1)$ can only be obtained from $(n-1,0)$ by letting a new packet arrive to relay $R$ but fail to transmit to receiver $d$ .","This gives the first two lines of equations (6).","Similarly to the explanations for the second to last equation in (1), both the state $(n-1,0)$ and $(k,n-1)$ can transfer to $(n,0)$ .","Beginning with $(n-1,0)$ , if no packet comes to source or the arriving packet is not sent to $R$ and $d$ successfully, no packet is obtained at $R$ or $d$ .","As a result, at the next time slot the AoI increases 1 while the second parameter remains 0.","On the other hand, a packet of age $n-1$ can be sent to $d$ from $R$ which reduces the AoI to $n$ .","Meanwhile, still we have to ensure that the receiver gets no packet from the direct link $s-d$ .","Combining these two cases, we determine the stationary equation of the state vector $(n,0)$ where $n\\ge 2$ .","Finally, it sees that still only one way can decrease the AoI to 1, i.e., the arriving packet is transmitted to $d$ through the direct link $s-d$ , no matter what the original age-state is.","As before, this equation yields that the stationary probability $\\pi _{(1,0)}$ is $pP_1$ directly.","So far, we have explained all the equations in (6).","This completes the proof of Theorem 4.","Table: Random state transfers and transtion probabilities: non-preemption caseSolving the system of equations (6) is easier than (1), since no-packet-preemption in relay $R$ dramatically simplifies how the age-state $(n,1)$ is obtained.","We give the solutions of all the stationary probabilities $\\pi _{(n,m)}$ in Theorem 5 and put the calculation details in Appendix C. Theorem 5 When the AoI process $AoI_{NB}^{NP}$ reaches the steady-state, the stationary probabilities $\\pi _{(n,m)}$ are solved as $\\pi _{(n,0)}=\\beta _1\\delta ^{n-1} + \\beta _2[(1-pP_1)(1-P_3)]^{n-1} \\quad (n\\ge 1)$ Other probabilities $\\pi _{(n,m)}$ , $n>m\\ge 1$ are determined by $\\pi _{(n,m)}&=p(1-P_1)P_2\\bigg \\lbrace \\beta _1\\delta ^{n-2}\\left(\\frac{(1-pP_1)(1-P_3)^{m-1}}{\\delta }\\right) + \\beta _2\\left[ (1-pP_1)(1-P_3) \\right]^{n-2}\\bigg \\rbrace ,$ in which the coefficients $\\delta $ , $\\beta _1$ are $\\beta _2$ are given by $\\delta &=1-p(P_1 + P_2 - P_1P_2), \\\\\\beta _1&= pP_1-\\beta _2, \\\\\\beta _2&= \\frac{[1-(1-pP_1)(1-P_3)]p(1-P_1)P_2P_3}{[1-(1-pP_1)(1-P_3)+p(1-P_1)P_2]} \\frac{1-pP_1}{(1-pP_1)(1-P_3)-\\delta }.$ Now, the stationary distribution of AoI, which we denote as $\\Delta _{NB}^{NP}$ can be determined.","Theorem 6 Assume that the packet in relay cannot be substituted by later packets from source, then for $n\\ge 1$ , the stationary AoI distribution of relay-assisted status updating system is given as $\\Pr \\lbrace \\Delta _{NB}^{NP}=n\\rbrace &=\\left(\\beta _1 + \\frac{p(1-P_1)P_2}{\\delta -(1-pP_1)(1-P_3)}\\beta _1\\right) \\delta ^{n-1} \\\\& \\quad + \\left(\\beta _2 - \\frac{p(1-P_1)P_2}{\\delta -(1-pP_1)(1-P_3)}\\beta _1\\right) [(1-pP_1)(1-P_3)]^{n-1} \\\\& \\quad + p(1-P_1)P_2\\beta _2(n-1)[(1-pP_1)(1-P_3)]^{n-2}$ As before, for $n\\ge 2$ , we first calculate the sum $&\\sum \\nolimits _{m=1}^{n-1}\\pi _{(n,m)} \\\\={}& \\sum \\nolimits _{m=1}^{n-1} p(1-P_1)P_2\\bigg \\lbrace \\beta _1\\delta ^{n-2}\\left(\\frac{(1-pP_1)(1-P_3)^{m-1}}{\\delta }\\right) + \\beta _2\\left[ (1-pP_1)(1-P_3) \\right]^{n-2}\\bigg \\rbrace \\\\={}& \\frac{p(1-P_1)P_2\\beta _1}{\\delta -(1-pP_1)(1-P_3)}\\left\\lbrace \\delta ^{n-1}-[(1-pP_1)(1-P_3)]^{n-1}\\right\\rbrace \\\\{}& \\qquad + p(1-P_1)P_2\\beta _2(n-1)[(1-pP_1)(1-P_3)]^{n-2}$ Thus, we have that $&\\Pr \\lbrace \\Delta _{NB}^{NP}=n\\rbrace \\\\={}& \\pi _{(n,0)} + \\sum \\nolimits _{m=1}^{n-1}\\pi _{(n,m)} \\\\={}& \\left(\\beta _1 + \\frac{p(1-P_1)P_2}{\\delta -(1-pP_1)(1-P_3)}\\beta _1\\right) \\delta ^{n-1} + \\left(\\beta _2 - \\frac{p(1-P_1)P_2}{\\delta -(1-pP_1)(1-P_3)}\\beta _1\\right) [(1-pP_1)(1-P_3)]^{n-1} \\\\{}& \\qquad + p(1-P_1)P_2\\beta _2(n-1)[(1-pP_1)(1-P_3)]^{n-2}$ where the probability expression (7) is substituted.","Let $n=1$ , equation (13) reduces to $\\beta _1 + \\beta _2=pP_1$ from (9), which equals $\\pi _{(1,0)}$ and is also the probability that AoI takes value 1.","Therefore, (11) is valid for all $n\\ge 1$ .","This completes the proof of Theorem 6."],["Characterizing the Age of Information When relay has size 1 buffer","In the previous Section, we have discussed the relay-assisted system where the relay has no buffer.","The explicit expressions of stationary AoI distribution was derived for two cases in which we assume that the new packet can and cannot preempt the packet retransmissions from relay.","In this Section, we will consider the steady-state AoI when the relay is equipped with a size 1 buffer, which is used to store the arriving packet if there is already one packet in relay.","In addition, the packet in buffer is allowed to be updated by later ones from source $s$ .","The purpose of this Section is two-fold.","On the one hand, we will prove that the AoI distribution can still be determined by constituting a multiple-dimensional stochastic process and solving its steady state.","Hence, not only the mean of AoI, but all the AoI performances can be computed.","On the other hand, finding out the influences of storing-another-packet on the system's AoI is practically meaningful, which directly guides the design of a superior statue updating system.","Table: The age-state transfers of relay-assisted status updating system: relay has size 1 bufferHere, a three-dimensional state vector $(n,m,l)$ is defined, and the three-dimensional stochastic process $AoI_B^{P}$ is constituted accordingly.","The added parameter $l$ denotes the age of packet in relay's buffer.","To describe the age-state transfers, again we list all the cases using a Table REF .","Table: Random state transfers and the corresponding transtion probabilities: relay has size 1 bufferThe state transfers are described assuming that the process $AoI_B^{P}$ has different initial state vector, i.e., $(n,m,l)$ , $(n,m,0)$ and $(n,0,0)$ .","Actually, we list all the state transitions in the order of number of packets in system.","For instance, age-state $(n,m,l)$ means that the current value of AoI is $n$ and there are two packets of age $m$ and $l$ in relay, one is transmitted on $R-d$ link and the other is waiting in buffer.","At the next time slot, if $A=0$ , that is no packet is obtained at source, only the packet transmission over $R-d$ link needs to be considered, since no packet is delivered on both $s-d$ and $s-R$ links.","When $A=1$ , we have to describe whether the new packet is successfully transmitted via $s-d$ and $s-R$ links and in relay $R$ , if the packet having age $m$ is obtained at destination $d$ .","This gives first eight rows of Table REF .","All the state tranfers from other initial age-states $(n,m,0)$ and $(n,0,0)$ are discussed similarly.","Notice that still only one way can reduce the AoI to 1, that is a new updating packet is sent to $d$ via the direct link $s-d$ .","Since we have assumed that when everytime a packet is obtained from $s-d$ link, all the packets in system are deleted, which makes the other two state components $m$ and $l$ jump to 0.","Therefore, we can determine that the first stationary probability $\\pi _{(1,0,0)}$ is equal to $pP_1$ immediately.","According to Table REF , in Table REF we summarize every state transfer and determine the transition probability, from which the stationary equations of AoI stochastic process $AoI_B^P$ can be established when the process reaches the steady state.","For the sake of simplicity, in this Section we denote that $\\delta =(1-p)+p(1-P_1)(1-P_2) =1-p(P_1+P_2-P_1P_2) $ and $\\eta =p(1-P_1)P_2 $ Theorem 7 Assume that the three-dimensional stochastic process $AoI_B^P$ runs in steady state, the stationary probabilities $\\pi _{(n,m,l)}$ of all the age-states satisfy following equations ${\\left\\lbrace \\begin{array}{ll}\\pi _{(n,m,l)}=\\pi _{(n-1,m-1,l-1)}\\delta (1-P_3) & \\qquad (n>m>l\\ge 2) \\\\\\pi _{(n,m,1)}=\\left(\\sum \\nolimits _{j=0}^{m-2}\\pi _{(n-1,m-1,j)}\\right)\\eta (1-P_3) & \\qquad (n>m\\ge 2) \\\\\\pi _{(n,m,0)}=\\pi _{(n-1,m-1,0)}\\delta (1-P_3) +\\left(\\sum \\nolimits _{k=n}^{\\infty }\\pi _{(k,n-1,m-1)}\\right)\\delta P_3 & \\qquad (n>m\\ge 2) \\\\\\pi _{(n,1,0)}=\\pi _{(n-1,0,0)}\\eta +\\left(\\sum \\nolimits _{k=n}^{\\infty }\\sum \\nolimits _{j=0}^{n-2}\\pi _{(k,n-1,j)}\\right)\\eta P_3 & \\qquad (n\\ge 2) \\\\\\pi _{(n,0,0)}=\\pi _{(n-1,0,0)}\\delta +\\left(\\sum \\nolimits _{k=n}^{\\infty }\\pi _{(k,n-1,0)}\\right)\\delta P_3 & \\qquad (n\\ge 2) \\\\\\pi _{(1,0,0)}=\\Big (\\sum \\nolimits _{n=1}^{\\infty }\\pi _{(n,0,0)}+ \\sum \\nolimits _{n=2}^{\\infty }\\sum \\nolimits _{m=1}^{n-1}\\pi _{(n,m,0)}\\\\\\qquad \\qquad \\qquad \\qquad \\qquad + \\sum \\nolimits _{n=3}^{\\infty }\\sum \\nolimits _{m=2}^{n-1}\\sum \\nolimits _{l=1}^{m-1}\\pi _{(n,m,l)}\\Big )pP_1\\end{array}\\right.","}$ We explain each line of equations (14) briefly.","For the cases $n>m>l\\ge 2$ , the state $(n,m,l)$ can only be obtained from $(n-1,m-1,l-1)$ when both $d$ and $R$ obtains no packet in the current time slot.","No packet comes to $d$ ensures that the AoI does not drop, while the age of packet stored in relay's buffer also increases 1 as long as no packet is sent to $R$ from $s$ successfully.","Since the packet contained in relay's buffer can be replaced by later updating packets, apart from $(n-1,m-1,0)$ , state vectors of form $(n-1,m-1,j)$ where $1\\le j\\le m-2$ can also transfer to $(n,m,1)$ .","These state transfers occur if the AoI at $d$ does not decrease, and the packet in relay's buffer is updated by a new packet at the same time.","This gives the second line of (14).","Age-state $(n,m,0)$ means that the buffer in $R$ is empty, which can be transferred to in two ways.","At first, if no packet arrives to $R$ and $d$ , then state vector $(n-1,m-1,0)$ transfers to $(n,m,0)$ after one time slot naturally.","Apart from this, notice that when the packet retransmission is successful via $R-d$ link, the AoI is reduced and this packet is then deleted from $R$ .","Next time the packet in buffer is transmitted.","As a result, any state having form $(k,n-1,m-1)$ , $k\\ge n$ will change to $(n,m,0)$ as well.","Combining both cases, we obtain the third equation in (14).","When considering the state $(n,1,0)$ , remember that the packet in buffer can be replaced, making the last parameter $l$ reduce to 1.","Similar to above discussions, we see that all the age-states of form $(k,n-1,j)$ where $k\\ge n$ and $1\\le j\\le n-2$ can transfer to $(n,1,0)$ , so that in this case we have a double sum in the stationary equation.","The empty state $(n,0,0)$ can be obtained from either $(n-1,0,0)$ when no packet arrives to $d$ and $R$ , or age-state $(k,n-1,0)$ , $k\\ge n$ assuming that the packet of age $n-1$ is sent to $d$ through $R-d$ link and no other packet comes to relay $R$ .","Substituting the transition probabilities in Table REF , the stationary equation corresponding to $(n,0,0)$ can be determined.","At last, the AoI is reduced to 1 from any initial age-state when a new packet is delivered to $d$ through the direct link $s-d$ , which yields the last line of equations (14) and in fact determines that $\\pi _{(1,0,0)}=pP_1$ .","This completes the proof of Theorem 7.","Compared with the no-buffer cases discussed before, here we have to find all the stationary probabilities $\\pi _{(n,m,l)}$ , $n>m>l\\ge 0$ .","Then, the probability that stationary AoI, $\\Delta _B^P$ , takes value $n$ is determined by the formula $\\Pr \\lbrace \\Delta _B^P=n\\rbrace =\\pi _{(n,0,0)} &+\\sum \\nolimits _{m=1}^{n-1}\\pi _{(n,m,0)} +\\sum \\nolimits _{m=2}^{n-1}\\sum \\nolimits _{l=1}^{m-1}\\pi _{(n,m,l)}$ Theorem 8 The solutions of equations (14), i.e., all the stationary probabilities $\\pi _{(n,m,l)}$ of three-dimensional stochastic process $AoI_B^P$ are given as follows.","First of all, $\\pi _{(n,0,0)}&=\\left(pP_1+\\eta \\right)\\delta ^{n-1} -\\left\\lbrace \\eta -(1-S)\\eta P_3\\right\\rbrace [\\delta (1-P_3)]^{n-1} \\\\& \\qquad \\qquad \\qquad \\qquad -(1-S)\\eta P_3 n[\\delta (1-P_3)]^{n-1} \\qquad (n\\ge 1)$ For $n>m\\ge 0$ , $&\\pi _{(n,m,0)} \\\\={}&\\eta \\left(pP_1+\\eta \\right)\\delta ^{n-2}\\left(1-P_3\\right)^{m-1}-\\eta ^2\\left[\\delta (1-P_3)\\right]^{n-2}- (1-S)\\eta ^2P_3(n-m-1)\\left[\\delta (1-P_3)\\right]^{n-2} \\\\{}& +\\widetilde{S}\\eta P_3 \\left[(\\delta +\\eta )(1-P_3)\\right]^{n-2} m \\left(\\frac{\\delta }{\\delta +\\eta }\\right)^{m-1} - (1-S)\\delta \\eta P_3^2 m[\\delta (1-P_3)]^{n-2}$ The probability $\\pi _{(n,m,l)}$ , $n>m>l\\ge 1$ is solved as $\\pi _{(n,m,l)}&= \\eta ^2(1-P_3)\\bigg \\lbrace \\left(pP_1+\\eta \\right)\\delta ^{n-3}\\left(\\frac{(\\delta +\\eta )(1-P_3)}{\\delta }\\right)^{m-2} \\left(\\frac{\\delta }{\\delta +\\eta }\\right)^{l-1} \\\\& \\qquad -\\eta [\\delta (1-P_3)]^{n-3}\\left(\\frac{\\delta +\\eta }{\\delta }\\right)^{m-2} \\left(\\frac{\\delta }{\\delta +\\eta }\\right)^{l-1} \\\\& \\qquad -(1-S)\\eta P_3(n-m-1) [\\delta (1-P_3)]^{n-3}\\left(\\frac{\\delta +\\eta }{\\delta }\\right)^{m-2}\\left(\\frac{\\delta }{\\delta +\\eta }\\right)^{l-1} \\bigg \\rbrace \\\\& \\quad + \\eta P_3\\widetilde{S}[(\\delta +\\eta )(1-P_3)]^{n-2}\\left(\\frac{\\delta }{\\delta +\\eta }\\right)^{l-1} - \\eta P_3\\widetilde{S}[(\\delta +\\eta )(1-P_3)]^{n-2}\\left(\\frac{\\delta }{\\delta +\\eta }\\right)^{m-1} \\\\& \\quad -\\delta \\eta P_3^2(1-S)[\\delta (1-P_3)]^{n-2}\\left(\\frac{\\delta +\\eta }{\\delta }\\right)^{m-1} \\left(\\frac{\\delta }{\\delta +\\eta }\\right)^{l-1} \\\\&\\quad + \\delta \\eta P_3^2(1-S)[\\delta (1-P_3)]^{n-2}$ where the numbers $S$ and $\\widetilde{S}$ are determined by $\\widetilde{S}&=S\\eta +(1-S)(\\delta +\\eta )P_3, \\\\S&=\\frac{pP_1[1-\\delta (1-P_3)]^2+\\delta \\eta P_3^2}{(1-\\delta )[1-\\delta (1-P_3)]^2-\\delta \\eta P_3(1-\\delta )(1-P_3)}$ We will solve the stationary equations (14) in Appendix D. Provided all the stationary probabilities, the distribution of steady-state AoI can be obtained by equation (15).","The calculation details are given in Appendix E. Theorem 9 Assume that the relay has a buffer of size 1, for $n\\ge 1$ , the steady-state AoI $\\Delta _B^P$ of the relay-assisted status updating system is distributed as $\\Pr \\lbrace \\Delta _B^P=n\\rbrace &=\\frac{(\\delta +\\eta )P_3(pP_1+\\eta )}{\\delta -(\\delta +\\eta )(1-P_3)}\\delta ^{n-1}-c_1[(\\delta +\\eta )(1-P_3)]^{n-1} +c_2[\\delta (1-P_3)]^{n-1} \\\\& \\quad +\\frac{(1-S)(\\delta +\\eta )P_3^2}{1-P_3}n[\\delta (1-P_3)]^{n-1} + \\frac{P_3\\widetilde{S}}{1-P_3}n[(\\delta +\\eta )(1-P_3)]^{n-1}$ where the coefficients $c_1$ and $c_2$ are given as $c_1&=\\frac{pP_1\\eta (1-P_3)+\\delta \\eta P_3}{[\\delta -(\\delta +\\eta )(1-P_3)](1-P_3)} + \\frac{\\delta P_3[\\eta +(\\delta +\\eta )P_3](1-S)+(\\delta +\\eta )P_3\\widetilde{S}}{\\eta (1-P_3)} \\\\c_2&=\\frac{P_3[\\eta (\\delta +\\eta )+P_3(1-S)(\\delta +\\eta )(2\\delta -\\eta )]}{\\eta (1-P_3)}$"],["Performance metrics of Age of Information","In Section III, for two cases where the packet retransmission from relay can and cannot be preempted by later packets, we derive the stationary AoI distribution of status updating system with a no-buffer relay.","To investigate the influence of storing another packet in relay on system's AoI, the distribution of steady state AoI is determined in Section IV.","Given the stationary AoI distributions, the following corollary shows the mean and the variance of AoI for all three cases, and the calculation details are provided in Appendix F. Corollary 1 For the relay-assisted status updating system, the average AoI for no-buffer-and-preemption case is equal to $\\mathbb {E}[\\Delta _{NB}^P]&=\\frac{(1-pP_1)P_3}{[(1-pP_1)P_3-p(1-P_1)P_2](1-\\delta )} + \\frac{\\xi }{[1-(1-pP_1)(1-P_3)]^2}$ while for the no-buffer-no-preemption case, it shows that $\\mathbb {E}[\\Delta _{NB}^{NP}]&= \\frac{\\beta _1}{(1-\\delta )^2}+\\frac{\\beta _2}{[1-(1-pP_1)(1-P_3)]^2} \\\\& \\quad + \\frac{p(1-P_1)P_2\\beta _1[2-\\delta -(1-pP_1)(1-P_3)]}{(1-\\delta )^2[1-(1-pP_1)(1-P_3)]^2} +\\frac{2p(1-P_1)P_2\\beta _2}{[1-(1-pP_1)(1-P_3)]^3}$ The average AoI of relay-assisted system when the relay is equipped with a buffer of size 1 is calculated as $\\mathbb {E}[\\Delta _B^P]&=\\frac{(\\delta +\\eta )P_3(pP_1+\\eta )}{[\\delta -(\\delta +\\eta )(1-P_3)](1-\\delta )^2}-\\frac{c_1}{[1-(\\delta +\\eta )(1-P_3)]^2} +\\frac{c_2}{[1-\\delta (1-P_3)]^2} \\\\& \\quad +\\frac{(1-S)(\\delta +\\eta )P_3^2[1+\\delta (1-P_3)]}{(1-P_3)[1-\\delta (1-P_3)]^3} +\\frac{P_3\\widetilde{S}[1+(\\delta +\\eta )(1-P_3)]}{(1-P_3)[1-(\\delta +\\eta )(1-P_3)]^3}$ where in all of three equations $\\delta =1-p(P_1+P_2-P_1P_2) $ Besides, $\\xi $ in (24) is given as $\\xi &=\\frac{p[P_1+(1-P_1)P_2P_3]}{1-P_3} - \\frac{P_3(1-\\delta )\\delta }{[(1-pP_1)P_3-p(1-P_1)P_2](1-P_3)} $ and $\\beta _1$ and $\\beta _2$ in expression (25) are defined in (9) and (10).","In (26), $\\eta =p(1-P_1)P_2$ , $S$ , $\\widetilde{S}$ are given in (19) and (20).","Other two coefficients $c_1$ , $c_2$ are introduced in equations (22) and (23).","For different situations, the second moments of AoI can also be obtained.","Then, we can calculate the variance of stationary AoI by $\\emph {Var}[\\Delta ]=\\mathbb {E}[\\Delta ^2]-\\left(\\mathbb {E}[\\Delta ]\\right)^2$ for all three cases.","The expression of $\\emph {Var}[\\Delta ]$ is lengthy but the calculations are direct.","Here, we provide only $\\emph {Var}[\\Delta _{NB}^{NP}]$ , AoI variances for other two cases can be obtained similarly.","From AoI distribution (11), we have $\\mathbb {E}\\left[\\left(\\Delta _{NB}^{NP}\\right)^2\\right] &= \\left(\\beta _1+\\frac{p(1-P_1)P_2\\beta _1}{\\delta -(1-pP_1)(1-P_3)}\\right)\\sum \\nolimits _{n=1}^{\\infty }n^2\\delta ^{n-1} \\\\& \\quad + \\left(\\beta _2 - \\frac{p(1-P_1)P_2\\beta _1}{\\delta -(1-pP_1)(1-P_3)}\\right) \\sum \\nolimits _{n=1}^{\\infty }n^2[(1-pP_1)(1-P_3)]^{n-1} \\\\& \\quad + p(1-P_1)P_2\\beta _2\\sum \\nolimits _{n=1}^{\\infty }n^2(n-1)[(1-pP_1)(1-P_3)]^{n-2} \\\\&= \\left(\\beta _1+\\frac{p(1-P_1)P_2\\beta _1}{\\delta -(1-pP_1)(1-P_3)}\\right)\\frac{1+\\delta }{(1-\\delta )^3} +\\left(\\beta _2 - \\frac{p(1-P_1)P_2\\beta _1}{\\delta -(1-pP_1)(1-P_3)}\\right) \\\\& \\quad \\times \\frac{1+(1-pP_1)(1-P_3)}{[1-(1-pP_1)(1-P_3)]^3} + p(1-P_1)P_2\\beta _2 \\frac{4+2(1-pP_1)(1-P_3)}{[1-(1-pP_1)(1-P_3)]^4}$ where the last sum $&\\sum \\nolimits _{n=1}^{\\infty }n^2(n-1)[(1-pP_1)(1-P_3)]^{n-2} \\\\={}& \\sum \\nolimits _{n=1}^{\\infty }(n+1)n(n-1)[(1-pP_1)(1-P_3)]^{n-2} -\\sum \\nolimits _{n=1}^{\\infty }n(n-1)[(1-pP_1)(1-P_3)]^{n-2} \\\\={}&\\frac{6}{[1-(1-pP_1)(1-P_3)]^4} - \\frac{2}{[1-(1-pP_1)(1-P_3)]^3} \\frac{4+2(1-pP_1)(1-P_3)}{[1-(1-pP_1)(1-P_3)]^4} $ can be derived using following formula $\\sum \\nolimits _{n=1}^{\\infty }n(n-1)\\cdots (n-k+1)x^{n-k}=\\frac{k!","}{(1-x)^{k+1}} \\text{ for }0m\\ge 0$ .","${\\left\\lbrace \\begin{array}{ll}\\pi _{(n,m)}=\\pi _{(n-1,m-1)}\\delta _1 & \\qquad (n>m\\ge 2) \\\\\\pi _{(n,1)}=\\pi _{(n-1,0)}p(1-P_1)P_2 + \\sum \\nolimits _{j=1}^{n-2}\\pi _{(n-1,j)}p(1-P_1)P_2(1-P_3) \\\\\\qquad \\qquad + \\sum \\nolimits _{k=n}^{\\infty }\\pi _{(k,n-1)}p(1-P_1)P_2P_3 & \\qquad (n\\ge 3) \\\\\\pi _{(2,1)}=\\pi _{(1,0)}p(1-P_1)P_2 + \\sum \\nolimits _{k=2}^{\\infty }\\pi _{(k,1)}p(1-P_1)P_2P_3 \\\\\\pi _{(n,0)}=\\pi _{(n-1,0)}\\delta _2 + \\sum \\nolimits _{k=n}^{\\infty }\\pi _{(k,n-1)}\\delta _3 & \\qquad (n\\ge 2) \\\\\\pi _{(1,0)}=\\left(\\sum \\nolimits _{n=1}^{\\infty }\\pi _{(n,0)} + \\sum \\nolimits _{m=1}^{\\infty }\\sum \\nolimits _{n=m+1}^{\\infty }\\pi _{(n,m)} \\right) pP_1\\end{array}\\right.","}$ where we denote $\\delta _2&=(1-p) + p(1-P_1)(1-P_2) = 1-p(P_1+P_2-P_1P_2) $ and $\\delta _1&=(1-p)(1-P_3)+p(1-P_1)(1-P_2)(1-P_3) =(1-P_3)\\delta _2, \\\\\\delta _3&=(1-p)P_3+p(1-P_1)(1-P_2)P_3=P_3\\delta _2 $ First of all, from the first line of (33) we have $\\pi _{(n,m)}&= \\pi _{(n-1,m-1)}\\delta _1=\\dots =\\pi _{(n-m+1,1)}\\delta _1^{m-1} \\\\&= {\\left\\lbrace \\begin{array}{ll}\\left[\\pi _{(1,0)}p(1-P_1)P_2 \\sum \\nolimits _{k=2}^{\\infty }\\pi _{(k,1)}p(1-P_1)P_2P_3 \\right]\\delta _1^{m-1} & (n-m=1) \\\\\\Big [\\pi _{(n-m,0)}p(1-P_1)P_2 \\sum \\nolimits _{j=1}^{n-m-1}\\pi _{(n-m,j)}p(1-P_1)P_2(1-P_3) \\\\\\qquad \\quad + \\sum \\nolimits _{k=n-m+1}^{\\infty }\\pi _{(k,n-m)}p(1-P_1)P_2P_3 \\Big ]\\delta _1^{m-1} & (n-m\\ge 2)\\end{array}\\right.","}$ In above expressions, we have substituted the second and third lines of (33).","For cases $n\\ge 2$ , the fourth equation of (33) says that $\\pi _{(n,0)}=\\pi _{(n-1,0)}\\delta _2 + \\left(\\sum \\nolimits _{k=n}^{\\infty }\\pi _{(k,n-1)}\\right) \\delta _3$ Using probability expressions obtained in (34), we deal with the sum in equation (35) as follows.","$&\\sum \\nolimits _{k=n}^{\\infty }\\pi _{(k,n-1)}= \\pi _{(n,n-1)}+\\sum \\nolimits _{k=n+1}^{\\infty }\\pi _{(k,n-1)} \\\\={}&\\left[\\pi _{(1,0)}p(1-P_1)P_2 + \\sum \\nolimits _{k=2}^{\\infty }\\pi _{(k,1)}p(1-P_1)P_2P_3 \\right]\\delta _1^{n-2} \\\\{}& \\quad + \\sum \\nolimits _{k=n+1}^{\\infty } \\bigg \\lbrace \\pi _{(k-n+1,0)}p(1-P_1)P_2 + \\sum \\nolimits _{j=1}^{k-n}\\pi _{(k-n+1,j)}p(1-P_1)P_2(1-P_3) \\\\{}& \\quad + \\sum \\nolimits _{y=k-n+2}^{\\infty }\\pi _{(y,k-n+1)}p(1-P_1)P_2P_3 \\bigg \\rbrace \\delta _1^{n-2} \\\\={}&\\left[\\pi _{(1,0)}p(1-P_1)P_2 + \\left(\\sum \\nolimits _{k=2}^{\\infty }\\pi _{(k,1)}\\right) p(1-P_1)P_2P_3 \\right]\\delta _1^{n-2} \\\\{}& \\quad + \\bigg [ \\left(\\sum \\nolimits _{n=2}^{\\infty }\\pi _{(n,0)}\\right) p(1-P_1)P_2 + \\left( \\sum \\nolimits _{k=n+1}^{\\infty } \\sum \\nolimits _{j=1}^{k-n}\\pi _{(k-n+1,j)}\\right) p(1-P_1)P_2(1-P_3) \\\\{}& \\quad + \\left(\\sum \\nolimits _{k=n+1}^{\\infty }\\sum \\nolimits _{y=k-n+2}^{\\infty }\\pi _{(y,k-n+1)}\\right) p(1-P_1)P_2P_3 \\bigg ]\\delta _1^{n-2} \\\\={}&\\left[\\pi _{(1,0)}p(1-P_1)P_2 + \\left(\\sum \\nolimits _{n=2}^{\\infty }\\pi _{(n,1)}\\right) p(1-P_1)P_2P_3 \\right]\\delta _1^{n-2} \\\\{}& \\quad + \\Big [ \\left(\\sum \\nolimits _{n=2}^{\\infty }\\pi _{(n,0)}\\right) p(1-P_1)P_2 + \\left( \\sum \\nolimits _{n=2}^{\\infty } \\sum \\nolimits _{m=1}^{n-1}\\pi _{(n,m)}\\right) p(1-P_1)P_2(1-P_3) \\\\{}& \\quad + \\left(\\sum \\nolimits _{m=2}^{\\infty }\\sum \\nolimits _{n=m+1}^{\\infty }\\pi _{(n,m)}\\right) p(1-P_1)P_2P_3 \\Big ]\\delta _1^{n-2} \\\\={}& \\left[\\left(\\sum \\nolimits _{n=1}^{\\infty }\\pi _{(n,0)}\\right)p(1-P_1)P_2 + \\left( \\sum \\nolimits _{m=1}^{\\infty } \\sum \\nolimits _{n=m+1}^{\\infty }\\pi _{(n,m)}\\right) p(1-P_1)P_2 \\right]\\delta _1^{n-2} \\\\={}& p(1-P_1)P_2\\delta _1^{n-2}$ Let $k-n+1=\\tilde{n}$ , $j=\\tilde{m}$ , then $\\sum \\nolimits _{k=n+1}^{\\infty } \\sum \\nolimits _{j=1}^{k-n}\\pi _{(k-n+1,j)}=\\sum \\nolimits _{\\tilde{n}=2}^{\\infty }\\sum \\nolimits _{\\tilde{m}=1}^{\\tilde{n}-1}\\pi _{(\\tilde{n},\\tilde{m})} $ Similarly, do the substitutions $y=\\tilde{n}$ and $k-n+1=\\tilde{m}$ , it shows that $\\sum \\nolimits _{k=n+1}^{\\infty }\\sum \\nolimits _{y=k-n+2}^{\\infty }\\pi _{(y,k-n+1)}=\\sum \\nolimits _{\\tilde{m}=2}^{\\infty }\\sum \\nolimits _{\\tilde{n}=\\tilde{m}+1}^{\\infty }\\pi _{(\\tilde{n},\\tilde{m})} $ Substituting these sums, we obtain the equation (36).","In addition, observing that $&\\sum \\nolimits _{n=2}^{\\infty }\\pi _{(n,1)} + \\sum \\nolimits _{m=2}^{\\infty } \\sum \\nolimits _{n=m+1}^{\\infty }\\pi _{(n,m)} = \\sum \\nolimits _{m=1}^{\\infty } \\sum \\nolimits _{n=m+1}^{\\infty }\\pi _{(n,m)} $ and $\\sum \\nolimits _{m=1}^{\\infty } \\sum \\nolimits _{n=m+1}^{\\infty }\\pi _{(n,m)}=\\sum \\nolimits _{n=2}^{\\infty } \\sum \\nolimits _{m=1}^{n-1}\\pi _{(n,m)} $ which yields the result (37).","Equation (38) holds because the probabilities contained in bracket of (37) add up to 1.","Finally, we derive the following recursive equation $\\pi _{(n,0)}=\\pi _{(n-1,0)}\\delta _2 + p(1-P_1)P_2\\delta _3\\delta _1^{n-2} \\qquad (n\\ge 2)$ Applying (39) iteratively, the general formula of $\\pi _{(n,0)}$ can be obtained.","We show that $\\pi _{(n,0)}&=\\pi _{(1,0)}\\delta _2^{n-1}+p(1-P_1)P_2\\delta _3 \\sum \\nolimits _{j=0}^{n-2}\\delta _2^j\\delta _1^{n-2-j} \\\\&=\\pi _{(1,0)}\\delta _2^{n-1}+p(1-P_1)P_2\\delta _3 \\frac{\\delta _1^{n-1}-\\delta _2^{n-1}}{\\delta _1-\\delta _2}$ The first probability $\\pi _{(1,0)}$ is equal to $pP_1$ , which can be determined directly from the last equation of (33).","Remember that $\\delta _2=1-p(P_1+P_2-P_1P_2)$ and $\\delta _1=(1-P_3)\\delta _2, \\quad \\delta _3=P_3\\delta _2 $ we have $\\pi _{(n,0)}&=pP_1\\delta _2^{n-1}+p(1-P_1)P_2\\delta _3\\frac{\\delta _1^{n-1}-\\delta _2^{n-1}}{(1-P_3)\\delta _2-\\delta _2} \\\\&=pP_1\\delta _2^{n-1}+p(1-P_1)P_2\\left(\\delta _2^{n-1}-\\delta _1^{n-1} \\right) \\\\&=\\left[pP_1+p(1-P_1)P_2\\right]\\delta _2^{n-1} - p(1-P_1)P_2\\delta _1^{n-1} \\\\&=(1-\\delta _2)\\delta _2^{n-1} - p(1-P_1)P_2[(1-P_3)\\delta _2]^{n-1}$ Let $n=1$ , equation (41) gives $\\pi _{(1,0)}=pP_1$ .","Therefore, we prove that expression (41) is valid for all $n\\ge 1$ .","Next, probabilities $\\pi _{(n,m)}$ , $n>m\\ge 1$ are solved.","We first calculate $\\pi _{(n,1)}$ , then by the relation $\\pi _{(n,m)}=\\pi _{(n-m+1,1)}\\delta _1^{m-1}$ all the probabilities $\\pi _{(n,m)}$ are obtained.","When $n\\ge 3$ , from the second line of equations (33), we show that $\\pi _{(n,1)}&= \\pi _{(n-1,0)}p(1-P_1)P_2 + \\sum \\nolimits _{j=1}^{n-2}\\pi _{(n-1,j)}p(1-P_1)P_2(1-P_3) + \\sum \\nolimits _{k=n}^{\\infty }\\pi _{(k,n-1)}p(1-P_1)P_2P_3 \\\\&=\\left\\lbrace (1-\\delta _2)\\delta _2^{n-2} - p(1-P_1)P_2\\delta _1^{n-2}\\right\\rbrace p(1-P_1)P_2 \\\\& \\qquad + \\sum \\nolimits _{j=1}^{n-2}\\pi _{(n-j,1)}\\delta _1^{j-1} p(1-P_1)P_2(1-P_3) + p(1-P_1)P_2\\delta _1^{n-2}p(1-P_1)P_2P_3 \\\\&= p(1-P_1)P_2(1-\\delta _2)\\delta _2^{n-2} - [p(1-P_1)P_2]^2(1-P_3)\\delta _1^{n-2} \\\\& \\qquad + p(1-P_1)P_2(1-P_3)\\sum \\nolimits _{j=1}^{n-2}\\pi _{(n-j,1)}\\delta _1^{j-1}$ In (43), we have substituted equations (41) and (38).","Using relation (42), we can rewrite $\\pi _{(n-1,j)}$ as $\\pi _{(n-j,1)}\\delta _1^{j-1}$ .","Computing the difference $\\pi _{(n,1)}-\\pi _{(n-1,1)}\\delta _1 = p(1-P_1)P_2(1-\\delta _2)\\delta _2^{n-3}(\\delta _2-\\delta _1) + p(1-P_1)P_2(1-P_3)\\pi _{(n-1,1)} $ which is equivalent to $\\pi _{(n,1)}&=\\pi _{(n-1,1)}[\\delta _1+p(1-P_1)P_2(1-P_3)] + p(1-P_1)P_2(1-\\delta _2)P_3\\delta _2^{n-2} \\\\&= \\pi _{(n-1,1)}(1-P_3)(1-pP_1) + p(1-P_1)P_2(1-\\delta _2)P_3\\delta _2^{n-2}$ where $\\delta _1 + p(1-P_1)P_2(1-P_3) &= (1-P_3)\\delta _2 + p(1-P_1)P_2(1-P_3) \\\\&=(1-P_3)\\left\\lbrace 1-p(P_1+P_2-P_1P_2) + p(1-P_1)P_2 \\right\\rbrace \\\\&=(1-P_3)(1-pP_1) $ Equation (45) gives a recursive relation of the probabilities $\\pi _{(n,1)}$ , $n\\ge 3$ , from which we can derive that $\\pi _{(n,1)}&= \\pi _{(2,1)}[(1-P_3)(1-pP_1)]^{n-2} + p(1-P_1)P_2(1-\\delta _2)P_3 \\sum \\nolimits _{j=0}^{n-3}[(1-P_3)(1-pP_1)]^j \\delta _2^{n-2-j} \\\\&=\\pi _{(2,1)}[(1-P_3)(1-pP_1)]^{n-2} + p(1-P_1)P_2P_3(1-\\delta _2)\\delta _2 \\frac{\\delta _2^{n-2}-[(1-P_3)(1-pP_1)]^{n-2}}{\\delta _2-(1-P_3)(1-pP_1)} \\\\&=\\frac{p(1-P_1)P_2P_3(1-\\delta _2)\\delta _2}{\\delta _2-(1-P_3)(1-pP_1)}\\delta _2^{n-2}+ \\left(\\pi _{(2,1)} - \\frac{p(1-P_1)P_2P_3(1-\\delta _2)\\delta _2}{\\delta _2-(1-P_3)(1-pP_1)} \\right) [(1-P_3)(1-pP_1)]^{n-2}$ Using the basic relation in (33), the probability $\\pi _{(2,1)}$ is calculated as $\\pi _{(2,1)}&=\\pi _{(1,0)}p(1-P_1)P_2 + \\sum \\nolimits _{k=2}^{\\infty }\\pi _{(k,1)}p(1-P_1)P_2P_3 \\\\&=pP_1p(1-P_1)P_2 + [p(1-P_1)P_2]^2P_3 \\\\&=p^2(1-P_1)P_2[P_1+(1-P_1)P_2P_3] $ In equation (46), it is easy to see that for the case $n=2$ , expression (47) reduces to $\\pi _{(2,1)}=\\pi _{(2,1)}$ .","Therefore, we show that (47) is actually valid for all $n\\ge 2$ .","Now, for $n>m\\ge 1$ , the probabilities $\\pi _{(n,m)}$ can be obtained eventually.","It shows that $\\pi _{(n,m)}&=\\pi _{(n-m+1,1)}\\delta _1^{m-1} \\\\&= \\bigg \\lbrace \\frac{p(1-P_1)P_2P_3(1-\\delta _2)\\delta _2}{\\delta _2-(1-P_3)(1-pP_1)}\\delta _2^{n-m-1} \\\\& \\quad + \\left(\\pi _{(2,1)} - \\frac{p(1-P_1)P_2P_3(1-\\delta _2)\\delta _2}{\\delta _2-(1-P_3)(1-pP_1)} \\right) [(1-P_3)(1-pP_1)]^{n-m-1} \\bigg \\rbrace \\delta _1^{m-1} \\\\&= \\frac{p(1-P_1)P_2P_3(1-\\delta _2)\\delta _2}{\\delta _2-(1-P_3)(1-pP_1)}\\delta _2^{n-2}(1-P_3)^{m-1} \\\\& \\quad + \\left(\\pi _{(2,1)} - \\frac{p(1-P_1)P_2P_3(1-\\delta _2)\\delta _2}{\\delta _2-(1-P_3)(1-pP_1)} \\right) [(1-P_3)(1-pP_1)]^{n-2}\\left(\\frac{\\delta _2}{1-pP_1}\\right)^{m-1}$ So far, we have obtained all the stationary probabilities of process $AoI_{NB}^P$ and complete the proof of Theorem 2."],["Proof of Theorem 2","In this part, according to equation (4), we calculation the stationary distribution of AoI.","For $n\\ge 2$ , we first compute $&\\sum \\nolimits _{m=1}^{n-1}\\pi _{(n,m)} \\\\={}&\\sum \\nolimits _{m=1}^{n-1}\\Bigg \\lbrace \\pi _{(2,1)}[(1-pP_1)(1-P_3)]^{n-2}\\left(\\frac{\\delta }{1-pP_1}\\right)^{m-1} + \\frac{p(1-P_1)P_2P_3(1-\\delta )\\delta }{\\delta -(1-pP_1)(1-P_3)} \\\\{}& \\quad \\times \\left[ \\delta ^{n-2}(1-P_3)^{m-1} - [(1-pP_1)(1-P_3)]^{n-2}\\left(\\frac{\\delta }{1-pP_1} \\right)^{m-1} \\right] \\Bigg \\rbrace \\\\={}&\\pi _{(2,1)}\\frac{[(1-pP_1)(1-P_3)]^{n-1}-[(1-P_3)\\delta ]^{n-1}}{(1-pP_1-\\delta )(1-P_3)} + \\frac{p(1-P_1)P_2P_3(1-\\delta )\\delta }{\\delta -(1-pP_1)(1-P_3)} \\\\{}& \\quad \\times \\left\\lbrace \\frac{\\delta ^{n-2}-(1-P_3)[(1-P_3)\\delta ]^{n-2}}{P_3} - \\frac{[(1-pP_1)(1-P_3)]^{n-1}-[(1-P_3)\\delta ]^{n-1}}{(1-pP_1-\\delta )(1-P_3)} \\right\\rbrace \\\\={}& \\frac{p(1-P_1)P_2(1-\\delta )}{\\delta -(1-pP_1)(1-P_3)}\\left\\lbrace \\delta ^{n-1}-[(1-P_3)\\delta ]^{n-1}\\right\\rbrace \\\\{}& \\quad + \\left(\\pi _{(2,1)} - \\frac{p(1-P_1)P_2P_3(1-\\delta )\\delta }{\\delta -(1-pP_1)(1-P_3)} \\right) \\frac{[(1-pP_1)(1-P_3)]^{n-1}-[(1-P_3)\\delta ]^{n-1}}{(1-pP_1-\\delta )(1-P_3)} \\\\={}&\\eta _1\\left\\lbrace \\delta ^{n-1}-[(1-P_3)\\delta ]^{n-1}\\right\\rbrace + \\eta _2\\left\\lbrace [(1-pP_1)(1-P_3)]^{n-1}-[(1-P_3)\\delta ]^{n-1}\\right\\rbrace $ where we denote $\\eta _1&=\\frac{p(1-P_1)P_2(1-\\delta )}{\\delta -(1-pP_1)(1-P_3)}, \\\\\\eta _2&=\\left(\\pi _{(2,1)}-\\frac{p(1-P_1)P_2P_3(1-\\delta )\\delta }{\\delta -(1-pP_1)(1-P_3)}\\right) \\frac{1}{(1-pP_1-\\delta )(1-P_3)} $ Therefore, when $n\\ge 2$ , the probability that AoI equals $n$ is calculated as $\\Pr \\lbrace \\Delta =n\\rbrace &=\\pi _{(n,0)}+\\sum \\nolimits _{m=1}^{n-1}\\pi _{(n,m)} \\\\&= [(1-\\delta )+\\eta _1]\\delta ^{n-1}+\\eta _2[(1-pP_1)(1-P_3)]^{n-1} \\\\&\\qquad \\quad -\\left(\\eta _1+\\eta _2+p(1-P_1)P_2\\right)[(1-P_3)\\delta ]^{n-1}$ To obtain equation (50), we have substituted the probability expression (2).","Since $\\delta -(1-pP_1)(1-P_3) &=1-p(P_1+P_2-P_1P_2)-(1-pP_1)(1-P_3) \\\\&=1-pP_1-p(1-P_1)P_2-(1-pP_1)(1-P_3) \\\\&=(1-pP_1)P_3-p(1-P_1)P_2 $ we have the first coefficient $(1-\\delta )+\\eta _1&=1-\\delta + \\frac{p(1-P_1)P_2(1-\\delta )}{\\delta -(1-pP_1)(1-P_3)} \\\\&=(1-\\delta ) \\left( 1 + \\frac{p(1-P_1)P_2}{(1-pP_1)P_3-p(1-P_1)P_2}\\right) \\\\&=\\frac{(1-pP_1)P_3(1-\\delta )}{(1-pP_1)P_3-p(1-P_1)P_2} $ Notice that $1-pP_1-\\delta =1-pP_1-[1-p(P_1+P_2-P_1P_2)] = p(1-P_1)P_2 $ Substituting $\\pi _{(2,1)}$ , it shows that $\\eta _2&=\\frac{p[P_1+(1-P_1)P_2P_3]}{1-P_3} - \\frac{P_3(1-\\delta )\\delta }{[(1-pP_1)P_3-p(1-P_1)P_2](1-P_3)} $ In following paragraph, we will prove that the coefficient before the last term, i.e., $\\eta _1+\\eta _2+p(1-P_1)P_2$ is zero.","Thus, for $n\\ge 2$ we obtain $\\Pr \\lbrace \\Delta _{NB}^P=n\\rbrace &=\\frac{(1-pP_1)P_3(1-\\delta )}{(1-pP_1)P_3-p(1-P_1)P_2}\\delta ^{n-1} + \\eta _2[(1-pP_1)(1-P_3)]^{n-1}$ From equation (49), it is easy to see that the sum in (4) is zero for the case $n=1$ .","So, the expression (51) is valid for all $n\\ge 1$ .","We now show that $\\eta _1+\\eta _2+p(1-P_1)P_2=0$ .","That is $&\\frac{p(1-P_1)P_2(1-\\delta )}{\\delta -(1-pP_1)(1-P_3)} + \\left(p[P_1+(1-P_1)P_2P_3]-\\frac{P_3(1-\\delta )\\delta }{\\delta -(1-pP_1)(1-P_3)}\\right) \\\\& \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\times \\frac{p(1-P_1)P_2}{(1-pP_1-\\delta )(1-P_3)} + p(1-P_1)P_2 =0 $ which is equivalent to $&\\frac{1-\\delta }{\\delta -(1-pP_1)(1-P_3)} \\\\& + \\left(p[P_1+(1-P_1)P_2P_3]-\\frac{P_3(1-\\delta )\\delta }{\\delta -(1-pP_1)(1-P_3)}\\right) \\frac{1}{(1-pP_1-\\delta )(1-P_3)} + 1 =0 \\\\\\Leftrightarrow {}& \\frac{1-\\delta }{\\delta -(1-pP_1)(1-P_3)} + \\frac{p[P_1+(1-P_1)P_2P_3]}{(1-pP_1-\\delta )(1-P_3)} \\\\& - \\frac{P_3(1-\\delta )\\delta }{[\\delta -(1-pP_1)(1-P_3)](1-pP_1-\\delta )(1-P_3)} +1 =0$ In (52), we have $\\delta -(1-pP_1)(1-P_3)=(1-pP_1)P_3 - p(1-P_1)P_2 $ and $(1-pP_1-\\delta )(1-P_3) =\\left[1-pP_1- \\left(1-p(P_1+P_2-P_1P_2) \\right)\\right](1-P_3) =p(1-P_1)P_2(1-P_3) $ Then, equation (52) can be rewritten as $&\\frac{1-\\delta }{(1-pP_1)P_3-p(1-P_1)P_2}\\left(\\frac{P_3\\delta }{p(1-P_1)P_2(1-P_3)}-1\\right) =\\frac{p[P_1+(1-P_1)P_2P_3]}{p(1-P_1)P_2(1-P_3)}+1$ where the RHS of (53) is equal to $&\\frac{p[P_1+(1-P_1)P_2P_3]+p(1-P_1)P_2(1-P_3)}{p(1-P_1)P_2(1-P_3)} = \\frac{p[P_1+(1-P_1)P_2]}{p(1-P_1)P_2(1-P_3)}= \\frac{1-\\delta }{p(1-P_1)P_2(1-P_3)} $ Therefore, to prove (52), it suffices to show that $&\\frac{1}{(1-pP_1)P_3-p(1-P_1)P_2}\\left(\\frac{P_3\\delta }{p(1-P_1)P_2(1-P_3)}-1\\right) = \\frac{1}{p(1-P_1)P_2(1-P_3)} \\\\\\Leftrightarrow {}&\\frac{P_3\\delta }{p(1-P_1)P_2(1-P_3)}-1 = \\frac{(1-pP_1)P_3-p(1-P_1)P_2}{p(1-P_1)P_2(1-P_3)} = \\frac{(1-pP_1)P_3-p(1-P_1)P_2P_3}{p(1-P_1)P_2(1-P_3)}-1 \\\\\\Leftrightarrow {}&\\delta =(1-pP_1)-p(1-P_1)P_2 =1-p(P_1+P_2-P_1P_2)$ Since equation (54) holds, we verify the original equation (52) and show that the coefficient $\\eta _1+\\eta _2+p(1-P_1)P_2$ is equal to zero.","This completes the proof of Theorem 3."],["Proof of Theorem 5","In this Appendix, we solve the system of stationary equations (6) and find all the stationary probabilities $\\pi _{(n,m)}$ of AoI stochastic process $AoI_{NB}^{NP}$ .","For convenience, we copy these equations in the following.","${\\left\\lbrace \\begin{array}{ll}\\pi _{(n,m)}=\\pi _{(n-1,m-1)}[(1-p)(1-P_3)+p(1-P_1)(1-P_3)] & \\qquad (n>m\\ge 2) \\\\\\pi _{(n,1)}=\\pi _{(n-1,0)}p(1-P_1)P_2 & \\qquad (n\\ge 2) \\\\\\pi _{(n,0)}=\\pi _{(n-1,0)}[(1-p)+p(1-P_1)(1-P_2)] \\\\\\qquad \\qquad + \\left(\\sum \\nolimits _{k=n}^{\\infty }\\pi _{(k,n-1)}\\right)[(1-p)P_3+p(1-P_1)P_3] & \\qquad (n\\ge 2) \\\\\\pi _{(1,0)}= \\left(\\sum \\nolimits _{n=1}^{\\infty }\\pi _{(n,0)} + \\sum \\nolimits _{n=2}^{\\infty }\\sum \\nolimits _{m=1}^{n-1}\\pi _{(n,m)}\\right)pP_1\\end{array}\\right.","}$ First of all, applying the first line of (55) repeatedly, we can write $\\pi _{(n,m)}$ as $\\pi _{(n,m)}&=\\pi _{(n-1,m-1)}[(1-pP_1)(1-P_3)] \\\\&=\\pi _{(n-2,m-2)}[(1-pP_1)(1-P_3)]^2 \\\\&\\qquad \\qquad \\quad \\quad \\vdots \\\\&=\\pi _{(n-m+1,1)}[(1-pP_1)(1-P_3)]^{m-1} \\\\&=\\pi _{(n-m,0)}p(1-P_1)P_2[(1-pP_1)(1-P_3)]^{m-1}$ where to obtain the last step (56), the second equation in (55) is applied.","Substituting (56), we first handle the sum in the third line of (55).","It shows that $\\sum \\nolimits _{k=n}^{\\infty }\\pi _{(k,n-1)}&= \\sum \\nolimits _{k=n}^{\\infty }\\pi _{(k-n+1,0)}p(1-P_1)P_2 [(1-pP_1)(1-P_3)]^{n-2}\\\\&= \\left(\\sum \\nolimits _{k=1}^{\\infty }\\pi _{(k,0)}\\right) p(1-P_1)P_2 [(1-pP_1)(1-P_3)]^{n-2} $ Since all the probabilities $\\pi _{(n,m)}$ have to add up to 1, we have the relation $1&=\\sum \\nolimits _{n=1}^{\\infty }\\pi _{(n,0)} + \\sum \\nolimits _{n=m+1}^{\\infty }\\sum \\nolimits _{m=1}^{\\infty }\\pi _{(n,m)} \\\\&=\\sum \\nolimits _{n=1}^{\\infty }\\pi _{(n,0)} + \\sum \\nolimits _{n=m+1}^{\\infty }\\sum \\nolimits _{m=1}^{\\infty }\\pi _{(n-m,0)} p(1-P_1)P_2[(1-pP_1)(1-P_3)]^{m-1} \\\\&=\\sum \\nolimits _{n=1}^{\\infty }\\pi _{(n,0)} + \\left(\\sum \\nolimits _{t=1}^{\\infty }\\pi _{(t,0)}\\right) p(1-P_1)P_2 \\sum \\nolimits _{m=1}^{\\infty } [(1-pP_1)(1-P_3)]^{m-1} \\\\&=\\left(\\sum \\nolimits _{k=1}^{\\infty }\\pi _{(k,0)}\\right)\\left[1 + \\frac{p(1-P_1)P_2}{1-(1-pP_1)(1-P_3)}\\right]$ To obtain (57), do the substitution $t=n-m$ .","From equation (58) we can derive that $\\sum \\nolimits _{k=1}^{\\infty }\\pi _{(k,0)}=\\frac{1-(1-pP_1)(1-P_3)}{1-(1-pP_1)(1-P_3)+p(1-P_1)P_2} $ Combining above results, for $n\\ge 2$ we can write that $\\pi _{(n,0)}=\\pi _{(n-1,0)}\\delta + \\xi \\left[(1-pP_1)(1-P_3)\\right]^{n-1}$ where $\\delta &=1-p(P_1+P_2-P_1P_2), \\\\\\xi &=\\frac{p(1-P_1)P_2P_3[1-(1-pP_1)(1-P_3)]}{[1-(1-pP_1)(1-P_3)+p(1-P_1)P_2](1-P_3)} $ Equation (59) gives a recursive relation of the probabilities $\\pi _{(n,0)}$ .","Using (59) iteratively yields that $\\pi _{(n,0)}&=\\pi _{(1,0)}\\delta ^{n-1} + \\xi \\sum \\nolimits _{j=0}^{n-2}\\delta ^j[(1-pP_1)(1-P_3)]^{n-1-j} \\\\&=\\pi _{(1,0)}\\delta ^{n-1} + \\frac{\\xi (1-pP_1)(1-P_3)}{(1-pP_1)(1-P_3)-\\delta } \\left\\lbrace [(1-pP_1)(1-P_3)]^{n-1} - \\delta ^{n-1}\\right\\rbrace \\\\&=\\beta _1 \\delta ^{n-1} + \\beta _2[(1-pP_1)(1-P_3)]^{n-1}$ Substituting $\\xi $ , it show that the coefficient $\\beta _2$ is equal to $\\beta _2&=\\frac{[1-(1-pP_1)(1-P_3)]p(1-P_1)P_2P_3}{1-(1-pP_1)(1-P_3)+p(1-P_1)P_2} \\frac{1-pP_1}{(1-pP_1)(1-P_3)-\\delta } $ Since $\\pi _{(1,0)}=pP_1$ , we have $\\beta _1=pP_1-\\beta _2$ .","It is easy to see that when $n=1$ , equation (61) reduces to $\\pi _{(1,0)}=\\pi _{(1,0)}$ , thus probability expression (61) is actually valid for all $n\\ge 1$ .","According to (56), for $n>m\\ge 1$ , the probability $\\pi _{(n,m)}$ is then determined as $\\pi _{(n,m)}&= \\pi _{(n-m,0)}p(1-P_1)P_2[(1-pP_1)(1-P_3)]^{m-1} \\\\={}& p(1-P_1)P_2\\left\\lbrace \\beta _1\\delta ^{n-m-1} + \\beta _2[(1-pP_1)(1-P_3)]^{n-m-1} \\right\\rbrace [(1-pP_1)(1-P_3)]^{m-1} \\\\={}& p(1-P_1)P_2 \\left\\lbrace \\beta _1\\delta ^{n-2}\\left(\\frac{(1-pP_1)(1-P_3)}{\\delta }\\right)^{m-1} + \\beta _2[(1-pP_1)(1-P_3)]^{n-2} \\right\\rbrace $ So far, we have completely solved the system of equations (55) and obtained all the stationary probabilities $\\pi _{(n,m)}$ ."],["Proof of Theorem 8","In this appendix, we determine all the stationary probabilities $\\pi _{(n,m,l)}$ of stochastic process $AoI_B^P$ by solving the following system of equations.","${\\left\\lbrace \\begin{array}{ll}\\pi _{(n,m,l)}=\\pi _{(n-1,m-1,l-1)}\\delta (1-P_3) & \\qquad (n>m>l\\ge 2) \\\\\\pi _{(n,m,1)}=\\left(\\sum \\nolimits _{j=0}^{m-2}\\pi _{(n-1,m-1,j)}\\right)\\eta (1-P_3) & \\qquad (n>m\\ge 2) \\\\\\pi _{(n,m,0)}=\\pi _{(n-1,m-1,0)}\\delta (1-P_3) +\\left(\\sum \\nolimits _{k=n}^{\\infty }\\pi _{(k,n-1,m-1)}\\right)\\delta P_3 & \\qquad (n>m\\ge 2) \\\\\\pi _{(n,1,0)}=\\pi _{(n-1,0,0)}\\eta +\\left(\\sum \\nolimits _{k=n}^{\\infty }\\sum \\nolimits _{j=0}^{n-2}\\pi _{(k,n-1,j)}\\right)\\eta P_3 & \\qquad (n\\ge 2) \\\\\\pi _{(n,0,0)}=\\pi _{(n-1,0,0)}\\delta +\\left(\\sum \\nolimits _{k=n}^{\\infty }\\pi _{(k,n-1,0)}\\right)\\delta P_3 & \\qquad (n\\ge 2) \\\\\\pi _{(1,0,0)}=\\Big (\\sum \\nolimits _{n=1}^{\\infty }\\pi _{(n,0,0)}+ \\sum \\nolimits _{n=2}^{\\infty }\\sum \\nolimits _{m=1}^{n-1}\\pi _{(n,m,0)}\\\\\\qquad \\qquad \\qquad \\qquad \\qquad + \\sum \\nolimits _{n=3}^{\\infty }\\sum \\nolimits _{m=2}^{n-1}\\sum \\nolimits _{l=1}^{m-1}\\pi _{(n,m,l)}\\Big )pP_1\\end{array}\\right.","}$ First of all, for $n>m>l\\ge 1$ , using first two lines of (63) we can obtain that $\\pi _{(n,m,l)}&= \\pi _{(n-1,m-1,l-1)}[\\delta (1-P_3)] \\\\{}& \\qquad \\qquad \\qquad \\vdots \\\\&= \\pi _{(n-l+1,m-l+1,1)}[\\delta (1-P_3)]^{l-1} \\\\&= \\left(\\sum \\nolimits _{j=0}^{m-l-1}\\pi _{(n-l,m-l,j)}\\right)\\eta (1-P_3)[\\delta (1-P_3)]^{l-1}$ Substituting expression (64), the sum in the third equation of (63) can be rewritten as $\\sum \\nolimits _{k=n}^{\\infty }\\pi _{(k,n-1,m-1)}&= \\sum \\nolimits _{k=n}^{\\infty }\\left(\\sum \\nolimits _{j=0}^{n-m-1}\\pi _{(k-m+1,n-m,j)} \\right) \\eta (1-P_3)[\\delta (1-P_3)]^{m-2} \\\\&= \\left( \\sum \\nolimits _{k=n-m+1}^{\\infty }\\sum \\nolimits _{j=0}^{n-m-1}\\pi _{(k,n-m,j)}\\right) \\eta (1-P_3)[\\delta (1-P_3)]^{m-2}\\\\&= t_{n-m}\\eta (1-P_3)[\\delta (1-P_3)]^{m-2}$ where we define $t_n= \\sum \\nolimits _{k=n+1}^{\\infty }\\sum \\nolimits _{j=0}^{n-1}\\pi _{(k,n,j)} \\qquad (n\\ge 1)$ which is actually the probability that the middle parameter equals $n$ .","Therefore, for $n>m\\ge 2$ we have $\\pi _{(n,m,0)}&=\\pi _{(n-1,m-1,0)}\\delta (1-P_3) + t_{n-m}\\eta (1-P_3)[\\delta (1-P_3)]^{m-2} \\delta P_3 \\\\&=\\pi _{(n-1,m-1,0)}\\delta (1-P_3) + t_{n-m}\\eta P_3[\\delta (1-P_3)]^{m-1}$ Equation (67) is a recursive formula of the probabilities $\\pi _{(n,m,0)}$ .","By repeatedly using (67), it shows that $\\pi _{(n,m,0)}&=\\pi _{(n-m+1,1,0)}[\\delta (1-P_3)]^{m-1} + (m-1)t_{n-m}\\eta P_3[\\delta (1-P_3)]^{m-1} \\\\&= \\left\\lbrace \\pi _{(n-m,0,0)}\\eta +t_{n-m}\\eta P_3\\right\\rbrace [\\delta (1-P_3)]^{m-1} + (m-1)t_{n-m}\\eta P_3[\\delta (1-P_3)]^{m-1} \\\\&=\\pi _{(n-m,0,0)}\\eta [\\delta (1-P_3)]^{m-1} + m t_{n-m}\\eta P_3[\\delta (1-P_3)]^{m-1}$ In (68), we have used the fourth line in (63) to represent $\\pi _{(n-m+1,1,0)}$ with $\\pi _{(n-m,0,0)}$ and $t_{n-m}$ .","Notice that equation (69) is valid for all $n>m\\ge 1$ .","Substituting equation (69), the summation within the fifth line of (63) can be transformed as follows.","$\\sum \\nolimits _{k=n}^{\\infty }\\pi _{(k,n-1,0)}&=\\sum \\nolimits _{k=n}^{\\infty }\\Big \\lbrace \\pi _{(k-n+1,0,0)}\\eta [\\delta (1-P_3)]^{n-2} +(n-1)t_{k-n+1}\\eta P_3[\\delta (1-P_3)]^{n-2} \\Big \\rbrace \\\\&= \\left(\\sum \\nolimits _{k=1}^{\\infty }\\pi _{(k,0,0)}\\right)\\eta [\\delta (1-P_3)]^{n-2} + \\left(\\sum \\nolimits _{k=1}^{\\infty }t_{k}\\right) \\eta P_3(n-1)[\\delta (1-P_3)]^{n-2} $ Since $t_n$ denotes the probability that the middle parameter equals $n$ , we have the relation $t_0 + \\sum \\nolimits _{n=1}^{\\infty }t_n =1 $ in which $t_0=\\sum \\nolimits _{k=1}^{\\infty }\\pi _{(k,0,0)}=S $ is the probability that the middle parameter takes value 0, which is represented by $S$ in following paragraphs.","Combining above results, for $n\\ge 2$ we have that $\\pi _{(n,0,0)}&=\\pi _{(n-1,0,0)}\\delta + \\Big \\lbrace S\\eta [\\delta (1-P_3)]^{n-2} + (1-S)\\eta P_3(n-1)[\\delta (1-P_3)]^{n-2}\\Big \\rbrace \\delta P_3$ which gives a recursive formula for $\\pi _{(n,0,0)}$ .","Before the general expression of $\\pi _{(n,0,0)}$ is given, we first determine $S$ .","From $n=2$ to $\\infty $ , adding up both sides of equation (70) yields that $S-\\pi _{(1,0,0)}=S\\delta +\\frac{S\\delta \\eta P_3}{1-\\delta (1-P_3)}+\\frac{(1-S)\\delta \\eta P_3^2}{[1-\\delta (1-P_3)]^2}$ Since we have known that $\\pi _{(1,0,0)}=pP_1$ , from (71) we can derive $S=\\frac{pP_1[1-\\delta (1-P_3)]^2+\\delta \\eta P_3^2}{(1-\\delta )[1-\\delta (1-P_3)]^2-\\delta \\eta P_3(1-\\delta )(1-P_3)}$ Applying equation (70) iteratively, we have $\\pi _{(n,0,0)}&=\\pi _{(1,0,0)}\\delta ^{n-1}+S\\delta \\eta P_3\\sum \\nolimits _{j=0}^{n-2}\\delta ^j[\\delta (1-P_3)]^{n-2-j} \\\\&\\qquad +(1-S)\\delta \\eta P_3^2\\sum \\nolimits _{j=0}^{n-2}\\delta ^j(n-1-j)[\\delta (1-P_3)]^{n-2-j} \\\\&= \\left(pP_1+\\eta \\right)\\delta ^{n-1}-\\eta [\\delta (1-P_3)]^{n-1} - (1-S)\\eta P_3(n-1)[\\delta (1-P_3)]^{n-1}$ in which we have substituted $\\pi _{(1,0,0)}=pP_1$ and omit some intermediate calculation details.","Equation (73) is valid for $n=1$ can be verified directly.","In order to determine probabilities $\\pi _{(n,m,0)}$ by (69), we next derive the general expression of $t_n$ .","For $n\\ge 2$ , it shows that $t_n&=\\sum \\nolimits _{k=n+1}^{\\infty }\\sum \\nolimits _{j=0}^{n-1}\\pi _{(k,n,j)} \\\\&=\\sum \\nolimits _{k=n+1}^{\\infty }\\pi _{(k,n,0)}+\\sum \\nolimits _{k=n+1}^{\\infty }\\sum \\nolimits _{j=1}^{n-1}\\pi _{(k,n,j)} \\\\&=\\sum \\nolimits _{k=n+1}^{\\infty }\\Big \\lbrace \\pi _{(k-n,0,0)}\\eta [\\delta (1-P_3)]^{n-1}+n t_{k-n}\\eta P_3[\\delta (1-P_3)]^{n-1}\\Big \\rbrace \\\\& \\qquad +\\sum \\nolimits _{k=n+1}^{\\infty }\\sum \\nolimits _{j=1}^{n-1}\\left(\\sum \\nolimits _{y=0}^{n-j-1}\\pi _{(k-j,n-j,y)}\\right)\\eta (1-P_3) [\\delta (1-P_3)]^{j-1} \\\\&=S\\eta [\\delta (1-P_3)]^{n-1}+(1-S)\\eta P_3n[\\delta (1-P_3)]^{n-1} \\\\&\\qquad + \\eta (1-P_3)\\sum \\nolimits _{j=1}^{n-1}t_{n-j}[\\delta (1-P_3)]^{j-1}$ where in (74) we have used equations (69) and (64).","To obtain (75), notice that $\\sum \\nolimits _{k=n+1}^{\\infty }\\sum \\nolimits _{y=0}^{n-j-1}\\pi _{(k-j,n-j,y)}=\\sum \\nolimits _{k=n-j+1}^{\\infty }\\sum \\nolimits _{y=0}^{n-j-1}\\pi _{(k,n-j,y)}= t_{n-j} $ Similarly, we also have $t_{n-1}&=S\\eta [\\delta (1-P_3)]^{n-2}+(1-S)\\eta P_3(n-1)[\\delta (1-P_3)]^{n-2} \\\\& \\quad + \\eta (1-P_3)\\sum \\nolimits _{j=1}^{n-2}t_{n-1-j}[\\delta (1-P_3)]^{j-1}$ Compute the following difference $t_n-t_{n-1}\\delta (1-P_3)=(1-S)\\eta P_3[\\delta (1-P_3)]^{n-1}+t_{n-1}\\eta (1-P_3) $ which is equivalent to $t_n=t_{n-1}(\\delta +\\eta )(1-P_3)+(1-S)\\eta P_3[\\delta (1-P_3)]^{n-1}$ Applying equation (77) repeatedly yields that $t_n&=t_1 [(\\delta +\\eta )(1-P_3)]^{n-1} +(1-S)\\eta P_3 \\sum \\nolimits _{j=0}^{n-2}[(\\delta +\\eta )(1-P_3)]^j[\\delta (1-P_3)]^{n-1-j} \\\\&=t_1 [(\\delta +\\eta )(1-P_3)]^{n-1} +(1-S)\\delta P_3 \\left\\lbrace [(\\delta +\\eta )(1-P_3)]^{n-1} - [\\delta (1-P_3)]^{n-1} \\right\\rbrace $ Using the general expression (69), we have $t_1=\\sum \\nolimits _{k=2}^{\\infty }\\pi _{(k,1,0)} =\\sum \\nolimits _{k=2}^{\\infty } \\left\\lbrace \\pi _{(k-1,0,0)}\\eta + t_{k-1}\\eta P_3 \\right\\rbrace =S\\eta +(1-S)\\eta P_3$ Combining (78) and (79), eventually we derive that $t_n&=\\left\\lbrace S\\eta +(1-S)(\\delta +\\eta )P_3\\right\\rbrace [(\\delta +\\eta )(1-P_3)]^{n-1} - (1-S)\\delta P_3[\\delta (1-P_3)]^{n-1} \\\\&=\\widetilde{S}[(\\delta +\\eta )(1-P_3)]^{n-1} - (1-S)\\delta P_3[\\delta (1-P_3)]^{n-1}$ where we denote $\\widetilde{S}=S\\eta +(1-S)(\\delta +\\eta )P_3 $ Provided equations (73) and (80), now the probabilities $\\pi _{(n,m,0)}$ can be determined by (69).","$\\pi _{(n,m,0)}&= \\pi _{(n-m,0,0)}\\eta [\\delta (1-P_3)]^{m-1}+ m t_{n-m}\\eta P_3[\\delta (1-P_3)]^{m-1} \\\\&=\\bigg \\lbrace \\left(pP_1+\\eta \\right)\\delta ^{n-m-1}-\\eta [\\delta (1-P_3)]^{n-m-1} \\\\& \\quad -(1-S)\\eta P_3 (n-m-1)[\\delta (1-P_3)]^{n-m-1}\\bigg \\rbrace \\eta [\\delta (1-P_3)]^{m-1} \\\\& \\quad +\\left\\lbrace \\widetilde{S}[(\\delta +\\eta )(1-P_3)]^{n-m-1} - (1-S)\\delta P_3 [\\delta (1-P_3)]^{n-m-1} \\right\\rbrace \\eta P_3 m[\\delta (1-P_3)]^{m-1} \\\\&= \\eta (pP_1+\\eta )\\delta ^{n-2}(1-P_3)^{m-1}-\\eta ^2[\\delta (1-P_3)]^{n-2} - (1-S)\\eta ^2 P_3(n-m-1)[\\delta (1-P_3)]^{n-2} \\\\& \\quad + \\widetilde{S}\\eta P_3[(\\delta +\\eta )(1-P_3)]^{n-2}m\\left(\\frac{\\delta }{\\delta +\\eta }\\right)^{m-1} - (1-S)\\delta \\eta P_3^2 m[\\delta (1-P_3)]^{n-2}$ At last, the probabilities $\\pi _{(n,m,l)}$ , $n>m>l\\ge 1$ remains to be determined.","Since $\\pi _{(n,m,l)}=\\pi _{(n-l+1,m-l+1,1)}[\\delta (1-P_3)]^{l-1} $ thus it suffices to find all the probabilities $\\pi _{(n,m,1)}$ for all the tuples $(n,m)$ .","From the general formula (64), we have $\\pi _{(n,m,1)}&=\\left(\\sum \\nolimits _{j=0}^{m-2}\\pi _{(n-1,m-1,j)}\\right)\\eta (1-P_3) \\\\&=\\left(\\pi _{(n-1,m-1,0)}+\\sum \\nolimits _{j=1}^{m-2}\\pi _{(n-1,m-1,j)}\\right)\\eta (1-P_3) \\\\&=\\pi _{(n-1,m-1,0)}\\eta (1-P_3) + \\left(\\sum \\nolimits _{j=1}^{m-2}\\pi _{(n-j,m-j,1)}[\\delta (1-P_3)]^{j-1}\\right)\\eta (1-P_3) $ Do once iteration, we can obtain the expansion expression of probability $\\pi _{(n-1,m-1,1)}$ .","The difference $&\\pi _{(n,m,1)}-\\pi _{(n-1,m-1,1)}\\delta (1-P_3) \\\\={}& \\left\\lbrace \\pi _{(n-1,m-1,0)}-\\pi _{(n-2,m-2,0)}\\delta (1-P_3)\\right\\rbrace \\eta (1-P_3) +\\pi _{(n-1,m-1,1)}\\eta (1-P_3) \\\\={}&t_{n-m}\\eta ^2 P_3(1-P_3)[\\delta (1-P_3)]^{m-2}+\\pi _{(n,m,1)}\\eta (1-P_3)$ gives the following recursive relation $\\pi _{(n,m,1)}=\\pi _{(n-1,m-1,1)}(\\delta +\\eta )(1-P_3) + t_{n-m}\\eta ^2 P_3(1-P_3)[\\delta (1-P_3)]^{m-2}$ In (82), we have substituted the probability expression (69).","Then, the general formula of $\\pi _{(n,m,1)}$ can be derived by applying (83) repeatedly.","It shows that $\\pi _{(n,m,1)}&=\\pi _{(n-m+2,2,1)}[(\\delta +\\eta )(1-P_3)]^{m-2} \\\\&\\quad + \\eta ^2P_3(1-P_3)t_{n-m}\\sum \\nolimits _{j=0}^{m-3}[(\\delta +\\eta )(1-P_3)]^j[\\delta (1-P_3)]^{m-2-j} \\\\&=\\pi _{(n-m+2,2,1)}[(\\delta +\\eta )(1-P_3)]^{m-2} \\\\&\\quad + \\delta \\eta P_3(1-P_3)t_{n-m} \\left\\lbrace [(\\delta +\\eta )(1-P_3)]^{m-2}-[\\delta (1-P_3)]^{m-2}\\right\\rbrace $ in which $\\pi _{(n-m+2,2,1)}&= \\pi _{(n-m+1,1,0)}\\eta (1-P_3) \\\\&=\\left\\lbrace \\pi _{(n-m,0,0)}\\eta +t_{n-m}\\eta P_3\\right\\rbrace \\eta (1-P_3) \\\\&=\\pi _{(n-m,0,0)}\\eta ^2(1-P_3) + t_{n-m}\\eta ^2 P_3(1-P_3)$ Combining equations (85) and (84) and merging the same terms gives $\\pi _{(n,m,1)}&=\\pi _{(n-m,0,0)}\\eta ^2(1-P_3)[(\\delta +\\eta )(1-P_3)]^{m-2} \\\\&\\qquad + \\eta P_3 t_{n-m} \\left\\lbrace [(\\delta +\\eta )(1-P_3)]^{m-1}-[\\delta (1-P_3)]^{m-1}\\right\\rbrace $ Since both $\\pi _{(n-m,0,0)}$ and $t_{n-m}$ have been obtained, by substituting (73) and (80), we show that the final result is $\\pi _{(n,m,1)}&= \\eta ^2(1-P_3)\\bigg \\lbrace (pP_1+\\eta )\\delta ^{n-3}\\left(\\frac{(\\delta +\\eta )(1-P_3)}{\\delta }\\right)^{m-2} - \\eta [\\delta (1-P_3)]^{n-3}\\left(\\frac{\\delta +\\eta }{\\delta }\\right)^{m-2} \\\\& \\quad -(1-S)\\eta P_3(n-m-1)[\\delta (1-P_3)]^{n-3}\\left(\\frac{\\delta +\\eta }{\\delta }\\right)^{m-2} \\bigg \\rbrace \\\\& \\quad +\\eta P_3\\widetilde{S}[(\\delta +\\eta )(1-P_3)]^{n-2} - \\eta P_3\\widetilde{S}[(\\delta +\\eta )(1-P_3)]^{n-2}\\left(\\frac{\\delta }{\\delta +\\eta }\\right)^{m-1} \\\\& \\quad - \\delta \\eta P_3^2(1-S)[\\delta (1-P_3)]^{n-2}\\left(\\frac{\\delta +\\eta }{\\delta }\\right)^{m-1} + \\delta \\eta P_3^2(1-S)[\\delta (1-P_3)]^{n-2}$ For $n>m>l\\ge 1$ , the last step is $\\pi _{(n,m,l)}&=\\pi _{(n-l+1,m-l+1,1)}[\\delta (1-P_3)]^{l-1} \\\\&=\\eta ^2(1-P_3)\\bigg \\lbrace \\left(pP_1+\\eta \\right)\\delta ^{n-3}\\left(\\frac{(\\delta +\\eta )(1-P_3)}{\\delta }\\right)^{m-2} \\left(\\frac{\\delta }{\\delta +\\eta }\\right)^{l-1} \\\\& \\qquad -\\eta [\\delta (1-P_3)]^{n-3}\\left(\\frac{\\delta +\\eta }{\\delta }\\right)^{m-2} \\left(\\frac{\\delta }{\\delta +\\eta }\\right)^{l-1} -(1-S)\\eta P_3(n-m-1) \\\\& \\qquad \\times [\\delta (1-P_3)]^{n-3}\\left(\\frac{\\delta +\\eta }{\\delta }\\right)^{m-2}\\left(\\frac{\\delta }{\\delta +\\eta }\\right)^{l-1} \\bigg \\rbrace \\\\& \\quad + \\eta P_3\\widetilde{S}[(\\delta +\\eta )(1-P_3)]^{n-2}\\left(\\frac{\\delta }{\\delta +\\eta }\\right)^{l-1} - \\eta P_3\\widetilde{S}[(\\delta +\\eta )(1-P_3)]^{n-2}\\left(\\frac{\\delta }{\\delta +\\eta }\\right)^{m-1} \\\\&\\quad -\\delta \\eta P_3^2(1-S)[\\delta (1-P_3)]^{n-2}\\left(\\frac{\\delta +\\eta }{\\delta }\\right)^{m-1} \\left(\\frac{\\delta }{\\delta +\\eta }\\right)^{l-1} \\\\&\\quad + \\delta \\eta P_3^2(1-S)[\\delta (1-P_3)]^{n-2}$ Collecting the results in equations (73), (81) and (87), we show that all the stationary probabilities of stochastic process $AoI_B^P$ are determined.","This completes the proof of Theorem 8."],["Proof of Theorem 9","The stationary distribution of AoI, $\\Delta _B^P$ , is calculated in this Appendix.","According to formula (15), for $n\\ge 3$ , we have $\\Pr \\lbrace \\Delta _B^P=n\\rbrace = \\pi _{(n,0,0)}+\\sum \\nolimits _{m=1}^{n-1}\\pi _{(n,m,0)}+\\sum \\nolimits _{m=2}^{n-1}\\sum \\nolimits _{l=1}^{m-1}\\pi _{(n,m,l)} $ where $\\pi _{(n,0,0)}$ is given in (16).","Substituting expression (17), the first sum is computed as $&\\sum \\nolimits _{m=1}^{n-1}\\pi _{(n,m,0)} \\\\={}&\\sum \\nolimits _{m=1}^{n-1}\\Big \\lbrace \\eta \\left(pP_1+\\eta \\right)\\delta ^{n-2}\\left(1-P_3\\right)^{m-1}-\\eta ^2\\left[\\delta (1-P_3)\\right]^{n-2} - (1-S)\\eta ^2P_3(n-m-1) \\\\{}& \\times \\left[\\delta (1-P_3)\\right]^{n-2} +\\widetilde{S}\\eta P_3 \\left[(\\delta +\\eta )(1-P_3)\\right]^{n-2} m \\left(\\frac{\\delta }{\\delta +\\eta }\\right)^{m-1} - (1-S)\\delta \\eta P_3^2 m[\\delta (1-P_3)]^{n-2} \\Big \\rbrace \\\\={}&\\eta \\left(pP_1+\\eta \\right)\\delta ^{n-2} \\frac{1-(1-P_3)^{n-1}}{P_3} - \\eta ^2(n-1)\\left[\\delta (1-P_3)\\right]^{n-2} - (1-S)\\eta ^2P_3 \\\\{}& \\times \\frac{(n-1)(n-2)}{2}\\left[\\delta (1-P_3)\\right]^{n-2} +\\widetilde{S}\\eta P_3 \\left[(\\delta +\\eta )(1-P_3)\\right]^{n-2}\\frac{(\\delta +\\eta )^2}{\\eta ^2} \\\\{}&\\times \\left[1- \\left(\\frac{\\delta }{\\delta +\\eta }\\right)^{n-1}- \\frac{\\eta }{\\delta +\\eta }(n-1) \\left(\\frac{\\delta }{\\delta +\\eta }\\right)^{n-1} \\right] -(1-S)\\delta \\eta P_3^2 \\frac{n(n-1)}{2}[\\delta (1-P_3)]^{n-2}$ We omit the further calculations and directly show that $\\sum \\nolimits _{m=1}^{n-1}\\pi _{(n,m,0)}&=\\frac{\\eta (pP_1+\\eta )}{\\delta P_3}\\delta ^{n-1} + \\frac{\\widetilde{S}(\\delta +\\eta )P_3}{\\eta (1-P_3)}[(\\delta +\\eta )(1-P_3)]^{n-1} \\\\&\\qquad - \\left\\lbrace \\frac{\\eta (pP_1+\\eta )}{\\delta P_3}+\\frac{\\widetilde{S}\\delta P_3}{\\eta (1-P_3)}-\\frac{\\eta ^2-(1-S)\\eta ^2 P_3}{\\delta (1-P_3)}\\right\\rbrace [\\delta (1-P_3)]^{n-1} \\\\&\\qquad - \\left\\lbrace \\frac{\\eta ^2-(1-S)\\eta ^2P_3}{\\delta (1-P_3)}+\\frac{\\widetilde{S}\\eta P_3}{\\eta (1-P_3)} \\right\\rbrace n[\\delta (1-P_3)]^{n-1} \\\\&\\qquad -\\frac{(1-S)\\eta P_3(\\delta P_3+\\eta )}{2\\delta (1-P_3)}n(n-1)[\\delta (1-P_3)]^{n-1}$ At last, $&\\sum \\nolimits _{m=2}^{n-1}\\sum \\nolimits _{l=1}^{m-1}\\pi _{(n,m,l)} \\\\={}&\\sum \\nolimits _{m=2}^{n-1}\\sum \\nolimits _{l=1}^{m-1} \\text{Equation (18)} \\\\={}&\\sum \\nolimits _{m=2}^{n-1}\\Bigg \\lbrace \\eta (pP_1+\\eta )\\delta ^{n-2} \\left[\\left(\\frac{(\\delta +\\eta )(1-P_3)}{\\delta }\\right)^{m-1}-(1-P_3)^{m-1}\\right] \\\\{}&\\qquad - \\eta ^2[\\delta (1-P_3)]^{n-2}\\left[\\left(\\frac{\\delta +\\eta }{\\delta }\\right)^{m-1} -1 \\right] \\\\{}&\\qquad \\quad -(1-S)\\eta ^2 P_3 [\\delta (1-P_3)]^{n-2} (n-m-1)\\left[\\left(\\frac{\\delta +\\eta }{\\delta }\\right)^{m-1} -1\\right] \\Bigg \\rbrace \\\\{}&\\quad +\\widetilde{S}(\\delta +\\eta )P_3[(\\delta +\\eta )(1-P_3)]^{n-2}\\left[1-\\left(\\frac{\\delta }{\\delta +\\eta }\\right)^{m-1} \\right] \\\\{}&\\quad -\\widetilde{S}\\eta P_3[(\\delta +\\eta )(1-P_3)]^{n-2}(m-1)\\left(\\frac{\\delta }{\\delta +\\eta }\\right)^{m-1} \\\\{}&\\quad -(1-S)\\delta (\\delta +\\eta )P_3^2[\\delta (1-P_3)]^{n-2}\\left[\\left(\\frac{\\delta +\\eta }{\\delta }\\right)^{m-1}-1 \\right] \\\\{}&\\quad +(1-S)\\delta \\eta P_3^2(m-1)[\\delta (1-P_3)]^{n-2} \\\\={}&\\eta (pP_1+\\eta )\\delta ^{n-2} \\bigg [ \\frac{\\eta (1-P_3)}{[\\delta -(\\delta +\\eta )(1-P_3)]P_3} \\\\{}& \\quad - \\frac{\\delta }{\\delta -(\\delta +\\eta )(1-P_3)} \\left(\\frac{(\\delta +\\eta )(1-P_3)}{\\delta }\\right)^{n-1} +\\frac{(1-P_3)^{n-1}}{P_3}\\bigg ] \\\\{}&-\\eta ^2[\\delta (1-P_3)]^{n-2}\\left[ \\frac{\\delta }{\\eta }\\left(\\frac{\\delta +\\eta }{\\delta }\\right)^{n-1}-\\frac{\\delta +\\eta }{\\eta }-(n-2) \\right] \\\\{}&-(1-S)\\eta ^2 P_3 [\\delta (1-P_3)]^{n-2}\\bigg [-(n-2)\\frac{\\delta +\\eta }{\\eta } \\\\{}& \\quad + \\frac{\\delta ^2}{\\eta ^2}\\left(\\frac{\\delta +\\eta }{\\delta }\\right)^{n-1}-\\frac{\\delta (\\delta +\\eta )}{\\eta ^2} -\\frac{(n-2)(n-3)}{2}\\bigg ] \\\\{}&+\\widetilde{S}(\\delta +\\eta )P_3[(\\delta +\\eta )(1-P_3)]^{n-2} \\left[(n-2)-\\frac{\\delta }{\\eta }+\\frac{\\delta +\\eta }{\\eta } \\left( \\frac{\\delta }{\\delta +\\eta }\\right)^{n-1} \\right] \\\\{}&-\\widetilde{S}\\eta P_3[(\\delta +\\eta )(1-P_3)]^{n-2}\\bigg [ \\frac{\\delta (\\delta +\\eta )}{\\eta ^2} -\\frac{(\\delta +\\eta )^2}{\\eta ^2} \\left(\\frac{\\delta }{\\delta +\\eta }\\right)^{n-1}-\\frac{\\delta +\\eta }{\\eta }(n-2)\\left(\\frac{\\delta }{\\delta +\\eta }\\right)^{n-1} \\bigg ] \\\\{}&-(1-S)\\delta (\\delta +\\eta )P_3^2[\\delta (1-P_3)]^{n-2} \\left[ \\frac{\\delta }{\\eta }\\left(\\frac{\\delta +\\eta }{\\delta }\\right)^{n-1}-\\frac{\\delta +\\eta }{\\eta }-(n-2) \\right] \\\\{}&+(1-S)\\delta \\eta P_3^2\\frac{(n-1)(n-2)}{2}[\\delta (1-P_3)]^{n-2}$ Collecting all the results obtained in (16), (89) and (90), after careful calculations and numerical verification, we derive that $&\\Pr \\lbrace \\Delta _B^P=n\\rbrace \\\\={}& \\frac{(\\delta +\\eta )P_3(pP_1+\\eta )}{\\delta -(\\delta +\\eta )(1-P_3)}\\delta ^{n-1} +\\frac{P_3[\\eta (\\delta +\\eta )+(1-S)(\\delta +\\eta )P_3(2\\delta -\\eta )]}{\\eta (1-P_3)}[\\delta (1-P_3)]^{n-1} \\\\{}& -\\bigg \\lbrace \\frac{pP_1\\eta (1-P_3)+\\delta \\eta P_3}{[\\delta -(\\delta +\\eta )(1-P_3)](1-P_3)} +\\frac{(1-S)\\delta P_3[\\eta +(\\delta +\\eta )P_3]+\\widetilde{S}(\\delta +\\eta )P_3}{\\eta (1-P_3)} \\bigg \\rbrace [(\\delta +\\eta )(1-P_3)]^{n-1} \\\\{}& +\\frac{(1-S)(\\delta +\\eta )P_3^2}{1-P_3}n[\\delta (1-P_3)]^{n-1} +\\frac{\\widetilde{S}P_3}{1-P_3}n[(\\delta +\\eta )(1-P_3)]^{n-1} \\qquad (n\\ge 3)$ It can be checked directly that equation (89) is zero when $n=1$ , and (90) equals zero for both cases $n=1$ and $n=2$ .","Therefore, the probability distribution (91) is in fact valid for all $n\\ge 1$ .","This completes the proof of Theorem 9."],["Proof of Corollary 1","Notice that for $0 m\\\\ge 1$ .\",\n \"In this case, a packet of age at least 1 is contained in $R$ .\",\n \"If no packet arrives to source at next time slot, then the state transfers are dependent on the packet transmission on $R-d$ link.\",\n \"With probability $P_3$ , the packet delivery succeeds and the state vector changes to $(m+1,0)$ .\",\n \"The second state-component is zero because the packet in $R$ is deleted when it was sent to $d$ successfully.\",\n \"Otherwise, both $n$ and $m$ will increase 1 at next time slot, i.e., the age-state turns to $(n+1,m+1)$ .\",\n \"For the case a new packet is generated and sent out from source $s$ , we have to analyze all the packet transmissions on links $s-R$ , $R-d$ , and direct link $s-d$ .\",\n \"Let $A$ be the random variable of packet arrivals, and we use $T_{s-d}$ , $T_{s-R}$ , $T_{R-d}$ to indicate whether the packet transmission is successful in three links, respectively.\",\n \"$A=0$ denotes no packet arrives and $T_{s-d}=0$ means that the packet transmission on $s-d$ link fails.\",\n \"Other variables are defined in similar manner.\",\n \"We list all the possible state transitions in Table REF .\",\n \"Table: Description of state vector transfers of AoI stochastic process AoI NB P AoI_{NB}^PTable: All the state transfers and corresponding transition probabilities of stochastic process AoI NB P AoI_{NB}^PIt is necessary to explain all the cases including $A=1$ and $T_{s-d}=1$ in Table REF .\",\n \"When a new packet is delivered successfully over direct link $s-d$ , the value of AoI must reduce to 1 since a packet of age 1 is obtained in $d$ , although it is possible that an another packet is received on $R-d$ link.\",\n \"At the same time, according to our assumption, a feedback signal is sent to $R$ and the relay deletes the packet received at current time slot or before.\",\n \"Thus, $m$ is reduced to zero and the state vector jumps to $(1,0)$ .\",\n \"Still we have to consider the state transfers when the initial age-state is $(n,0)$ , which means the relay $R$ is currently empty.\",\n \"If no packet arrives, it is easy to see that the age-state transfers to $(n+1,0)$ at next time slot.\",\n \"On the contrary, for the case $A=1$ , we need to consider the packet transmissions on $s-d$ and $s-R$ links.\",\n \"All the possible state transitions from age-state $(n,0)$ are given in the last four lines in Table REF .\",\n \"Constituting the two-dimensional AoI stochastic process $AoI_{NB}^P=\\\\lbrace (n_k,m_k):n_k> m_k\\\\ge 0,k\\\\in \\\\mathbb {N}\\\\rbrace ,$ where the subscript $NB$ denotes that at relay $R$ there is no buffer, and the superscript $P$ indicates that the packet retransmission in $R$ can be preempted.\",\n \"Define $P_{(n,m),(i,j)}$ be the single step transition probability from $(n,m)$ to age-state $(i,j)$ .\",\n \"Summarize above discussions, we list all the state transfers and corresponding transition probabilities of the AoI process $AoI_{NB}^P$ in Table REF .\",\n \"Denote that $\\\\pi _{(n,m)}$ is the probability of age-state $(n,m)$ , $n> m\\\\ge 0$ when the AoI process runs in steady state.\",\n \"In following Theorem 1, we establish the stationary equations of the process $AoI_{NB}^P$ .\",\n \"Theorem 1 Assume that the stochastic process $AoI_{NB}^P$ reaches the steady state, then the stationary probabilities $\\\\pi _{(n,m)}$ of age-state $(n,m)$ , $n>m\\\\ge 0$ satisfy that ${\\\\left\\\\lbrace \\\\begin{array}{ll}\\\\pi _{(n,m)}=\\\\pi _{(n-1,m-1)}\\\\big [(1-p)(1-P_3) + p(1-P_1)(1-P_2)(1-P_3)\\\\big ] & \\\\quad (n>m\\\\ge 2) \\\\\\\\\\\\pi _{(n,1)}=\\\\pi _{(n-1,0)}p(1-P_1)P_2 + \\\\sum \\\\nolimits _{j=1}^{n-2}\\\\pi _{(n-1,j)}p(1-P_1)P_2(1-P_3) \\\\\\\\\\\\qquad \\\\qquad \\\\qquad + \\\\sum \\\\nolimits _{k=n}^{\\\\infty }\\\\pi _{(k,n-1)}p(1-P_1)P_2P_3 & \\\\quad (n\\\\ge 3) \\\\\\\\\\\\pi _{(2,1)}=\\\\pi _{(1,0)}p(1-P_1)P_2 + \\\\sum \\\\nolimits _{k=2}^{\\\\infty }\\\\pi _{(k,1)}p(1-P_1)P_2P_3 \\\\\\\\\\\\pi _{(n,0)}=\\\\pi _{(n-1,0)}\\\\left[(1-p)+p(1-P_1)(1-P_2)\\\\right] \\\\\\\\\\\\qquad \\\\qquad \\\\qquad + \\\\sum \\\\nolimits _{k=n}^{\\\\infty }\\\\pi _{(k,n-1)}\\\\left[(1-p)P_3+p(1-P_1)(1-P_2)P_3\\\\right] & \\\\quad (n\\\\ge 2) \\\\\\\\\\\\pi _{(1,0)}=\\\\left(\\\\sum \\\\nolimits _{n=1}^{\\\\infty }\\\\pi _{(n,0)}+ \\\\sum \\\\nolimits _{m=1}^{\\\\infty }\\\\sum \\\\nolimits _{n=m+1}^{\\\\infty }\\\\pi _{(n,m)} \\\\right) pP_1\\\\end{array}\\\\right.\",\n \"}$ Equations in (1) are explained as follows.\",\n \"First of all, begin with $(n-1,m-1)$ , as long as the packet transmission on $R-d$ link fails which occurs with probability $1-P_3$ , along with the condition that either no packet is delivered on $s-R$ and $s-d$ links or both transmissions are unsuccessful, then the age-state transfers to $(n,m)$ at next time slot.\",\n \"This gives the first line of equations (1).\",\n \"To obtain $(n,1)$ , more cases need to be considered.\",\n \"From age-state $(n-1,0)$ which means $R$ is initially empty, at source $s$ a packet arrives and is transmitted to $R$ successfully makes the second parameter jump to 1.\",\n \"On the other hand, let the packet delivery on $s-d$ link fail, such that the AoI in $d$ increases to $n$ at next time slot.\",\n \"This state transitions occurs with probability $p(1-P_1)P_2$ .\",\n \"Next, notice that the packet in $R$ is allowed to be refreshed, the second state-component 1 can also be created by packet replacement.\",\n \"Assume that the initial state vector is $(n-1,j)$ , $1\\\\le j\\\\le n-2$ .\",\n \"When the new packet is transmitted on $s-R$ link successfully but the packet transmissions on both $s-d$ and $R-d$ links fail, it was observed that the state also transfers to $(n,1)$ .\",\n \"Thus, we obtain the second term of the RHS of second equation.\",\n \"Now, suppose that a packet of age $n-1$ goes through $R-d$ link to the receiver.\",\n \"As a result, the AoI in $d$ is reduced to $n$ at next time slot.\",\n \"At the same time, the relay $R$ obtains a new packet from $s$ via $s-R$ link, which changes the second parameter to 1.\",\n \"In this case, we can obtain the age-state $(n,1)$ again.\",\n \"The initial state can be $(k,n-1)$ where $k$ is an arbitrary number greater than $n-1$ .\",\n \"For the case $n=2$ , packet substitution in $R$ is nonexistent since $R$ is empty before one time slot.\",\n \"With all above discussions, we explain how the state vector $(n,1)$ , $n\\\\ge 2$ is obtained and determine its stationary equation.\",\n \"Then, the random transitions to $(n,0)$ is analyzed.\",\n \"The relay $R$ is empty either it is empty initially, or is emptied as the contained packet is sent to $d$ successfully.\",\n \"Therefore, two cases have to be analyzed.\",\n \"Firstly, from age-state $(n-1,0)$ , as long as no packet arrives to $R$ and $d$ , obviously the state jumps to $(n,0)$ after one time slot.\",\n \"On the other hand, let the initial state is $(k,n-1)$ , $k\\\\ge n$ , a packet of age $n-1$ can be transmitted to $d$ via $R-d$ link and assume that no other packet is obtained from $s$ through the direct link $s-d$ , which ensures that the AoI is set to $n$ at next time slot.\",\n \"The second parameter maintains 0 if no packet comes to $s$ , or the arriving packet is not transmitted to $R$ successfully on $s-R$ link.\",\n \"Combined with the transition probabilities in Table REF , we give the stationary equation for the state $(n,0)$ in the end.\",\n \"The only way to get age-state $(1,0)$ is transmitting a packet via the direct link $s-d$ , no matter what the initial state is.\",\n \"Actually, this last equation determines $\\\\pi _{(1,0)}=pP_1$ directly, since the probabilities in bracket add up to 1.\",\n \"So far, all the state transfers are considered, and the proof of Theorem 1 completes.\",\n \"We solve the system of equations (1) and determine all the stationary probabilities $\\\\pi _{(n,m)}$ in Theorem 2, whose proof is postponed to Appendix A. Theorem 2 The probabilities $\\\\pi _{(n,m)}$ of AoI process $AoI_{NB}^P$ are determined as follows.\",\n \"Firstly, $\\\\pi _{(n,0)}=(1-\\\\delta )\\\\delta ^{n-1} - p(1-P_1)P_2[(1-P_3)\\\\delta ]^{n-1} \\\\qquad (n\\\\ge 1)$ and for $n>m\\\\ge 1$ $\\\\pi _{(n,m)}&=\\\\pi _{(2,1)}[(1-pP_1)(1-P_3)]^{n-2}\\\\left(\\\\frac{\\\\delta }{1-pP_1}\\\\right)^{m-1} + \\\\frac{p(1-P_1)P_2P_3(1-\\\\delta )\\\\delta }{\\\\delta -(1-pP_1)(1-P_3)} \\\\\\\\& \\\\qquad \\\\quad \\\\times \\\\left[ \\\\delta ^{n-2}(1-P_3)^{m-1} - [(1-pP_1)(1-P_3)]^{n-2}\\\\left(\\\\frac{\\\\delta }{1-pP_1} \\\\right)^{m-1} \\\\right]$ where $\\\\delta =1-p(P_1+P_2-P_1P_2), \\\\qquad \\\\pi _{(2,1)}=p^2(1-P_1)P_2[P_1+(1-P_1)P_2P_3] $ Provided all the stationary probabilities, the probability that the steady-state AoI equals $n$ can be obtained by $\\\\Pr \\\\lbrace \\\\Delta =n\\\\rbrace =\\\\pi _{(n,0)} + \\\\sum \\\\nolimits _{m=1}^{n-1}\\\\pi _{(n,m)} \\\\qquad (n\\\\ge 2)$ The probability that AoI takes value 1 is determined as $pP_1$ , which is known directly from the last line of equations (1).\",\n \"We determine the explicit expression of AoI distribution in following Theorem 3 and give the calculations in Appendix B. Theorem 3 For the considered relay-assisted status updating system, the stationary distribution of AoI $\\\\Delta _{NB}^P$ is determined as $\\\\Pr \\\\lbrace \\\\Delta _{NB}^P=n\\\\rbrace =\\\\frac{(1-pP_1)P_3(1-\\\\delta )}{(1-pP_1)P_3-p(1-P_1)P_2}\\\\delta ^{n-1} +\\\\xi [(1-pP_1)(1-P_3)]^{n-1} \\\\quad (n\\\\ge 1)$ in which $\\\\xi =\\\\frac{p[P_1+(1-P_1)P_2P_3]}{1-P_3} - \\\\frac{P_3(1-\\\\delta )\\\\delta }{[(1-pP_1)P_3-p(1-P_1)P_2](1-P_3)} $\"\n ],\n [\n \"The stationary AoI: packet retransmission is not preempted in relay\",\n \"In previous part of this Section, we have determined the stationary AoI distribution for the case where the packet in relay can be replaced.\",\n \"Intuitively, substituting the large-age packet with fresher ones will result to lower AoI and enhance the timeliness of the system.\",\n \"We will prove this conclusion by finding the steady-state AoI distribution for the $non-preemption$ case explicitly and do the comparisons.\",\n \"The AoI stochastic process constituted for this case is denoted as $AoI_{NB}^{NP}$ , where the superscript $NP$ indicates that no packet preemption is allowed in relay $R$ .\",\n \"When the packet in relay cannot be preempted before the receiver gets an updating packet eventually, which may be obtained from $R-d$ link or the $s-d$ link, the discussions of random state transitions are easier, since the relay will always reject the packet from the source as long as there has been one packet within it.\",\n \"We first describe the possible state transitions when the initial age-state is $(n,m)$ , where $n>m\\\\ge 1$ .\",\n \"If $A=1$ , that is a new packet is sent out from the source in this time slot.\",\n \"Now, only the packet transmissions on $s-d$ and $R-d$ links need to be discussed, since the packet over $s-R$ link is always rejected.\",\n \"For the case $A=0$ , we have to consider just the packet delivery over $R-d$ link.\",\n \"All the possible state transfers are listed in Table REF , in which the state transfers from initial age-state $(n,0)$ are also given.\",\n \"For each state transfer, we determine the corresponding transition probability in Table REF , from which the stationary equations of the AoI process $AoI_{NB}^{NP}$ can be established, when the process reaches the steady state.\",\n \"Theorem 4 Assume that the AoI stochastic process $AoI_{NB}^{NP}$ reaches the steady-state, the stationary probabilities $\\\\pi _{(n,m)}$ satisfy following equations ${\\\\left\\\\lbrace \\\\begin{array}{ll}\\\\pi _{(n,m)}=\\\\pi _{(n-1,m-1)}[(1-p)(1-P_3)+p(1-P_1)(1-P_3)] & (n>m\\\\ge 2) \\\\\\\\\\\\pi _{(n,1)}=\\\\pi _{(n-1,0)}p(1-P_1)P_2 & (n\\\\ge 2) \\\\\\\\\\\\pi _{(n,0)}=\\\\pi _{(n-1,0)}[(1-p)+p(1-P_1)(1-P_2)] \\\\\\\\\\\\qquad \\\\quad \\\\quad + \\\\left(\\\\sum \\\\nolimits _{k=n}^{\\\\infty }\\\\pi _{(k,n-1)}\\\\right)[(1-p)P_3+p(1-P_1)P_3] & (n\\\\ge 2) \\\\\\\\\\\\pi _{(1,0)}= \\\\left(\\\\sum \\\\nolimits _{n=1}^{\\\\infty }\\\\pi _{(n,0)} + \\\\sum \\\\nolimits _{n=2}^{\\\\infty }\\\\sum \\\\nolimits _{m=1}^{n-1}\\\\pi _{(n,m)}\\\\right)pP_1\\\\end{array}\\\\right.\",\n \"}$ Assume that the initial state is $(n-1,m-1)$ , to get age-state $(n,m)$ , it needs that no packet is obtained in both relay and destination, which is satisfied either no packet arrives at source and retransmission from relay fails, or a new packet comes to source but the packet transmissions on $s-d$ and $R-d$ links are unsuccessful.\",\n \"Since the packet in relay cannot be substituted, the state $(n,1)$ can only be obtained from $(n-1,0)$ by letting a new packet arrive to relay $R$ but fail to transmit to receiver $d$ .\",\n \"This gives the first two lines of equations (6).\",\n \"Similarly to the explanations for the second to last equation in (1), both the state $(n-1,0)$ and $(k,n-1)$ can transfer to $(n,0)$ .\",\n \"Beginning with $(n-1,0)$ , if no packet comes to source or the arriving packet is not sent to $R$ and $d$ successfully, no packet is obtained at $R$ or $d$ .\",\n \"As a result, at the next time slot the AoI increases 1 while the second parameter remains 0.\",\n \"On the other hand, a packet of age $n-1$ can be sent to $d$ from $R$ which reduces the AoI to $n$ .\",\n \"Meanwhile, still we have to ensure that the receiver gets no packet from the direct link $s-d$ .\",\n \"Combining these two cases, we determine the stationary equation of the state vector $(n,0)$ where $n\\\\ge 2$ .\",\n \"Finally, it sees that still only one way can decrease the AoI to 1, i.e., the arriving packet is transmitted to $d$ through the direct link $s-d$ , no matter what the original age-state is.\",\n \"As before, this equation yields that the stationary probability $\\\\pi _{(1,0)}$ is $pP_1$ directly.\",\n \"So far, we have explained all the equations in (6).\",\n \"This completes the proof of Theorem 4.\",\n \"Table: Random state transfers and transtion probabilities: non-preemption caseSolving the system of equations (6) is easier than (1), since no-packet-preemption in relay $R$ dramatically simplifies how the age-state $(n,1)$ is obtained.\",\n \"We give the solutions of all the stationary probabilities $\\\\pi _{(n,m)}$ in Theorem 5 and put the calculation details in Appendix C. Theorem 5 When the AoI process $AoI_{NB}^{NP}$ reaches the steady-state, the stationary probabilities $\\\\pi _{(n,m)}$ are solved as $\\\\pi _{(n,0)}=\\\\beta _1\\\\delta ^{n-1} + \\\\beta _2[(1-pP_1)(1-P_3)]^{n-1} \\\\quad (n\\\\ge 1)$ Other probabilities $\\\\pi _{(n,m)}$ , $n>m\\\\ge 1$ are determined by $\\\\pi _{(n,m)}&=p(1-P_1)P_2\\\\bigg \\\\lbrace \\\\beta _1\\\\delta ^{n-2}\\\\left(\\\\frac{(1-pP_1)(1-P_3)^{m-1}}{\\\\delta }\\\\right) + \\\\beta _2\\\\left[ (1-pP_1)(1-P_3) \\\\right]^{n-2}\\\\bigg \\\\rbrace ,$ in which the coefficients $\\\\delta $ , $\\\\beta _1$ are $\\\\beta _2$ are given by $\\\\delta &=1-p(P_1 + P_2 - P_1P_2), \\\\\\\\\\\\beta _1&= pP_1-\\\\beta _2, \\\\\\\\\\\\beta _2&= \\\\frac{[1-(1-pP_1)(1-P_3)]p(1-P_1)P_2P_3}{[1-(1-pP_1)(1-P_3)+p(1-P_1)P_2]} \\\\frac{1-pP_1}{(1-pP_1)(1-P_3)-\\\\delta }.$ Now, the stationary distribution of AoI, which we denote as $\\\\Delta _{NB}^{NP}$ can be determined.\",\n \"Theorem 6 Assume that the packet in relay cannot be substituted by later packets from source, then for $n\\\\ge 1$ , the stationary AoI distribution of relay-assisted status updating system is given as $\\\\Pr \\\\lbrace \\\\Delta _{NB}^{NP}=n\\\\rbrace &=\\\\left(\\\\beta _1 + \\\\frac{p(1-P_1)P_2}{\\\\delta -(1-pP_1)(1-P_3)}\\\\beta _1\\\\right) \\\\delta ^{n-1} \\\\\\\\& \\\\quad + \\\\left(\\\\beta _2 - \\\\frac{p(1-P_1)P_2}{\\\\delta -(1-pP_1)(1-P_3)}\\\\beta _1\\\\right) [(1-pP_1)(1-P_3)]^{n-1} \\\\\\\\& \\\\quad + p(1-P_1)P_2\\\\beta _2(n-1)[(1-pP_1)(1-P_3)]^{n-2}$ As before, for $n\\\\ge 2$ , we first calculate the sum $&\\\\sum \\\\nolimits _{m=1}^{n-1}\\\\pi _{(n,m)} \\\\\\\\={}& \\\\sum \\\\nolimits _{m=1}^{n-1} p(1-P_1)P_2\\\\bigg \\\\lbrace \\\\beta _1\\\\delta ^{n-2}\\\\left(\\\\frac{(1-pP_1)(1-P_3)^{m-1}}{\\\\delta }\\\\right) + \\\\beta _2\\\\left[ (1-pP_1)(1-P_3) \\\\right]^{n-2}\\\\bigg \\\\rbrace \\\\\\\\={}& \\\\frac{p(1-P_1)P_2\\\\beta _1}{\\\\delta -(1-pP_1)(1-P_3)}\\\\left\\\\lbrace \\\\delta ^{n-1}-[(1-pP_1)(1-P_3)]^{n-1}\\\\right\\\\rbrace \\\\\\\\{}& \\\\qquad + p(1-P_1)P_2\\\\beta _2(n-1)[(1-pP_1)(1-P_3)]^{n-2}$ Thus, we have that $&\\\\Pr \\\\lbrace \\\\Delta _{NB}^{NP}=n\\\\rbrace \\\\\\\\={}& \\\\pi _{(n,0)} + \\\\sum \\\\nolimits _{m=1}^{n-1}\\\\pi _{(n,m)} \\\\\\\\={}& \\\\left(\\\\beta _1 + \\\\frac{p(1-P_1)P_2}{\\\\delta -(1-pP_1)(1-P_3)}\\\\beta _1\\\\right) \\\\delta ^{n-1} + \\\\left(\\\\beta _2 - \\\\frac{p(1-P_1)P_2}{\\\\delta -(1-pP_1)(1-P_3)}\\\\beta _1\\\\right) [(1-pP_1)(1-P_3)]^{n-1} \\\\\\\\{}& \\\\qquad + p(1-P_1)P_2\\\\beta _2(n-1)[(1-pP_1)(1-P_3)]^{n-2}$ where the probability expression (7) is substituted.\",\n \"Let $n=1$ , equation (13) reduces to $\\\\beta _1 + \\\\beta _2=pP_1$ from (9), which equals $\\\\pi _{(1,0)}$ and is also the probability that AoI takes value 1.\",\n \"Therefore, (11) is valid for all $n\\\\ge 1$ .\",\n \"This completes the proof of Theorem 6.\"\n ],\n [\n \"Characterizing the Age of Information When relay has size 1 buffer\",\n \"In the previous Section, we have discussed the relay-assisted system where the relay has no buffer.\",\n \"The explicit expressions of stationary AoI distribution was derived for two cases in which we assume that the new packet can and cannot preempt the packet retransmissions from relay.\",\n \"In this Section, we will consider the steady-state AoI when the relay is equipped with a size 1 buffer, which is used to store the arriving packet if there is already one packet in relay.\",\n \"In addition, the packet in buffer is allowed to be updated by later ones from source $s$ .\",\n \"The purpose of this Section is two-fold.\",\n \"On the one hand, we will prove that the AoI distribution can still be determined by constituting a multiple-dimensional stochastic process and solving its steady state.\",\n \"Hence, not only the mean of AoI, but all the AoI performances can be computed.\",\n \"On the other hand, finding out the influences of storing-another-packet on the system's AoI is practically meaningful, which directly guides the design of a superior statue updating system.\",\n \"Table: The age-state transfers of relay-assisted status updating system: relay has size 1 bufferHere, a three-dimensional state vector $(n,m,l)$ is defined, and the three-dimensional stochastic process $AoI_B^{P}$ is constituted accordingly.\",\n \"The added parameter $l$ denotes the age of packet in relay's buffer.\",\n \"To describe the age-state transfers, again we list all the cases using a Table REF .\",\n \"Table: Random state transfers and the corresponding transtion probabilities: relay has size 1 bufferThe state transfers are described assuming that the process $AoI_B^{P}$ has different initial state vector, i.e., $(n,m,l)$ , $(n,m,0)$ and $(n,0,0)$ .\",\n \"Actually, we list all the state transitions in the order of number of packets in system.\",\n \"For instance, age-state $(n,m,l)$ means that the current value of AoI is $n$ and there are two packets of age $m$ and $l$ in relay, one is transmitted on $R-d$ link and the other is waiting in buffer.\",\n \"At the next time slot, if $A=0$ , that is no packet is obtained at source, only the packet transmission over $R-d$ link needs to be considered, since no packet is delivered on both $s-d$ and $s-R$ links.\",\n \"When $A=1$ , we have to describe whether the new packet is successfully transmitted via $s-d$ and $s-R$ links and in relay $R$ , if the packet having age $m$ is obtained at destination $d$ .\",\n \"This gives first eight rows of Table REF .\",\n \"All the state tranfers from other initial age-states $(n,m,0)$ and $(n,0,0)$ are discussed similarly.\",\n \"Notice that still only one way can reduce the AoI to 1, that is a new updating packet is sent to $d$ via the direct link $s-d$ .\",\n \"Since we have assumed that when everytime a packet is obtained from $s-d$ link, all the packets in system are deleted, which makes the other two state components $m$ and $l$ jump to 0.\",\n \"Therefore, we can determine that the first stationary probability $\\\\pi _{(1,0,0)}$ is equal to $pP_1$ immediately.\",\n \"According to Table REF , in Table REF we summarize every state transfer and determine the transition probability, from which the stationary equations of AoI stochastic process $AoI_B^P$ can be established when the process reaches the steady state.\",\n \"For the sake of simplicity, in this Section we denote that $\\\\delta =(1-p)+p(1-P_1)(1-P_2) =1-p(P_1+P_2-P_1P_2) $ and $\\\\eta =p(1-P_1)P_2 $ Theorem 7 Assume that the three-dimensional stochastic process $AoI_B^P$ runs in steady state, the stationary probabilities $\\\\pi _{(n,m,l)}$ of all the age-states satisfy following equations ${\\\\left\\\\lbrace \\\\begin{array}{ll}\\\\pi _{(n,m,l)}=\\\\pi _{(n-1,m-1,l-1)}\\\\delta (1-P_3) & \\\\qquad (n>m>l\\\\ge 2) \\\\\\\\\\\\pi _{(n,m,1)}=\\\\left(\\\\sum \\\\nolimits _{j=0}^{m-2}\\\\pi _{(n-1,m-1,j)}\\\\right)\\\\eta (1-P_3) & \\\\qquad (n>m\\\\ge 2) \\\\\\\\\\\\pi _{(n,m,0)}=\\\\pi _{(n-1,m-1,0)}\\\\delta (1-P_3) +\\\\left(\\\\sum \\\\nolimits _{k=n}^{\\\\infty }\\\\pi _{(k,n-1,m-1)}\\\\right)\\\\delta P_3 & \\\\qquad (n>m\\\\ge 2) \\\\\\\\\\\\pi _{(n,1,0)}=\\\\pi _{(n-1,0,0)}\\\\eta +\\\\left(\\\\sum \\\\nolimits _{k=n}^{\\\\infty }\\\\sum \\\\nolimits _{j=0}^{n-2}\\\\pi _{(k,n-1,j)}\\\\right)\\\\eta P_3 & \\\\qquad (n\\\\ge 2) \\\\\\\\\\\\pi _{(n,0,0)}=\\\\pi _{(n-1,0,0)}\\\\delta +\\\\left(\\\\sum \\\\nolimits _{k=n}^{\\\\infty }\\\\pi _{(k,n-1,0)}\\\\right)\\\\delta P_3 & \\\\qquad (n\\\\ge 2) \\\\\\\\\\\\pi _{(1,0,0)}=\\\\Big (\\\\sum \\\\nolimits _{n=1}^{\\\\infty }\\\\pi _{(n,0,0)}+ \\\\sum \\\\nolimits _{n=2}^{\\\\infty }\\\\sum \\\\nolimits _{m=1}^{n-1}\\\\pi _{(n,m,0)}\\\\\\\\\\\\qquad \\\\qquad \\\\qquad \\\\qquad \\\\qquad + \\\\sum \\\\nolimits _{n=3}^{\\\\infty }\\\\sum \\\\nolimits _{m=2}^{n-1}\\\\sum \\\\nolimits _{l=1}^{m-1}\\\\pi _{(n,m,l)}\\\\Big )pP_1\\\\end{array}\\\\right.\",\n \"}$ We explain each line of equations (14) briefly.\",\n \"For the cases $n>m>l\\\\ge 2$ , the state $(n,m,l)$ can only be obtained from $(n-1,m-1,l-1)$ when both $d$ and $R$ obtains no packet in the current time slot.\",\n \"No packet comes to $d$ ensures that the AoI does not drop, while the age of packet stored in relay's buffer also increases 1 as long as no packet is sent to $R$ from $s$ successfully.\",\n \"Since the packet contained in relay's buffer can be replaced by later updating packets, apart from $(n-1,m-1,0)$ , state vectors of form $(n-1,m-1,j)$ where $1\\\\le j\\\\le m-2$ can also transfer to $(n,m,1)$ .\",\n \"These state transfers occur if the AoI at $d$ does not decrease, and the packet in relay's buffer is updated by a new packet at the same time.\",\n \"This gives the second line of (14).\",\n \"Age-state $(n,m,0)$ means that the buffer in $R$ is empty, which can be transferred to in two ways.\",\n \"At first, if no packet arrives to $R$ and $d$ , then state vector $(n-1,m-1,0)$ transfers to $(n,m,0)$ after one time slot naturally.\",\n \"Apart from this, notice that when the packet retransmission is successful via $R-d$ link, the AoI is reduced and this packet is then deleted from $R$ .\",\n \"Next time the packet in buffer is transmitted.\",\n \"As a result, any state having form $(k,n-1,m-1)$ , $k\\\\ge n$ will change to $(n,m,0)$ as well.\",\n \"Combining both cases, we obtain the third equation in (14).\",\n \"When considering the state $(n,1,0)$ , remember that the packet in buffer can be replaced, making the last parameter $l$ reduce to 1.\",\n \"Similar to above discussions, we see that all the age-states of form $(k,n-1,j)$ where $k\\\\ge n$ and $1\\\\le j\\\\le n-2$ can transfer to $(n,1,0)$ , so that in this case we have a double sum in the stationary equation.\",\n \"The empty state $(n,0,0)$ can be obtained from either $(n-1,0,0)$ when no packet arrives to $d$ and $R$ , or age-state $(k,n-1,0)$ , $k\\\\ge n$ assuming that the packet of age $n-1$ is sent to $d$ through $R-d$ link and no other packet comes to relay $R$ .\",\n \"Substituting the transition probabilities in Table REF , the stationary equation corresponding to $(n,0,0)$ can be determined.\",\n \"At last, the AoI is reduced to 1 from any initial age-state when a new packet is delivered to $d$ through the direct link $s-d$ , which yields the last line of equations (14) and in fact determines that $\\\\pi _{(1,0,0)}=pP_1$ .\",\n \"This completes the proof of Theorem 7.\",\n \"Compared with the no-buffer cases discussed before, here we have to find all the stationary probabilities $\\\\pi _{(n,m,l)}$ , $n>m>l\\\\ge 0$ .\",\n \"Then, the probability that stationary AoI, $\\\\Delta _B^P$ , takes value $n$ is determined by the formula $\\\\Pr \\\\lbrace \\\\Delta _B^P=n\\\\rbrace =\\\\pi _{(n,0,0)} &+\\\\sum \\\\nolimits _{m=1}^{n-1}\\\\pi _{(n,m,0)} +\\\\sum \\\\nolimits _{m=2}^{n-1}\\\\sum \\\\nolimits _{l=1}^{m-1}\\\\pi _{(n,m,l)}$ Theorem 8 The solutions of equations (14), i.e., all the stationary probabilities $\\\\pi _{(n,m,l)}$ of three-dimensional stochastic process $AoI_B^P$ are given as follows.\",\n \"First of all, $\\\\pi _{(n,0,0)}&=\\\\left(pP_1+\\\\eta \\\\right)\\\\delta ^{n-1} -\\\\left\\\\lbrace \\\\eta -(1-S)\\\\eta P_3\\\\right\\\\rbrace [\\\\delta (1-P_3)]^{n-1} \\\\\\\\& \\\\qquad \\\\qquad \\\\qquad \\\\qquad -(1-S)\\\\eta P_3 n[\\\\delta (1-P_3)]^{n-1} \\\\qquad (n\\\\ge 1)$ For $n>m\\\\ge 0$ , $&\\\\pi _{(n,m,0)} \\\\\\\\={}&\\\\eta \\\\left(pP_1+\\\\eta \\\\right)\\\\delta ^{n-2}\\\\left(1-P_3\\\\right)^{m-1}-\\\\eta ^2\\\\left[\\\\delta (1-P_3)\\\\right]^{n-2}- (1-S)\\\\eta ^2P_3(n-m-1)\\\\left[\\\\delta (1-P_3)\\\\right]^{n-2} \\\\\\\\{}& +\\\\widetilde{S}\\\\eta P_3 \\\\left[(\\\\delta +\\\\eta )(1-P_3)\\\\right]^{n-2} m \\\\left(\\\\frac{\\\\delta }{\\\\delta +\\\\eta }\\\\right)^{m-1} - (1-S)\\\\delta \\\\eta P_3^2 m[\\\\delta (1-P_3)]^{n-2}$ The probability $\\\\pi _{(n,m,l)}$ , $n>m>l\\\\ge 1$ is solved as $\\\\pi _{(n,m,l)}&= \\\\eta ^2(1-P_3)\\\\bigg \\\\lbrace \\\\left(pP_1+\\\\eta \\\\right)\\\\delta ^{n-3}\\\\left(\\\\frac{(\\\\delta +\\\\eta )(1-P_3)}{\\\\delta }\\\\right)^{m-2} \\\\left(\\\\frac{\\\\delta }{\\\\delta +\\\\eta }\\\\right)^{l-1} \\\\\\\\& \\\\qquad -\\\\eta [\\\\delta (1-P_3)]^{n-3}\\\\left(\\\\frac{\\\\delta +\\\\eta }{\\\\delta }\\\\right)^{m-2} \\\\left(\\\\frac{\\\\delta }{\\\\delta +\\\\eta }\\\\right)^{l-1} \\\\\\\\& \\\\qquad -(1-S)\\\\eta P_3(n-m-1) [\\\\delta (1-P_3)]^{n-3}\\\\left(\\\\frac{\\\\delta +\\\\eta }{\\\\delta }\\\\right)^{m-2}\\\\left(\\\\frac{\\\\delta }{\\\\delta +\\\\eta }\\\\right)^{l-1} \\\\bigg \\\\rbrace \\\\\\\\& \\\\quad + \\\\eta P_3\\\\widetilde{S}[(\\\\delta +\\\\eta )(1-P_3)]^{n-2}\\\\left(\\\\frac{\\\\delta }{\\\\delta +\\\\eta }\\\\right)^{l-1} - \\\\eta P_3\\\\widetilde{S}[(\\\\delta +\\\\eta )(1-P_3)]^{n-2}\\\\left(\\\\frac{\\\\delta }{\\\\delta +\\\\eta }\\\\right)^{m-1} \\\\\\\\& \\\\quad -\\\\delta \\\\eta P_3^2(1-S)[\\\\delta (1-P_3)]^{n-2}\\\\left(\\\\frac{\\\\delta +\\\\eta }{\\\\delta }\\\\right)^{m-1} \\\\left(\\\\frac{\\\\delta }{\\\\delta +\\\\eta }\\\\right)^{l-1} \\\\\\\\&\\\\quad + \\\\delta \\\\eta P_3^2(1-S)[\\\\delta (1-P_3)]^{n-2}$ where the numbers $S$ and $\\\\widetilde{S}$ are determined by $\\\\widetilde{S}&=S\\\\eta +(1-S)(\\\\delta +\\\\eta )P_3, \\\\\\\\S&=\\\\frac{pP_1[1-\\\\delta (1-P_3)]^2+\\\\delta \\\\eta P_3^2}{(1-\\\\delta )[1-\\\\delta (1-P_3)]^2-\\\\delta \\\\eta P_3(1-\\\\delta )(1-P_3)}$ We will solve the stationary equations (14) in Appendix D. Provided all the stationary probabilities, the distribution of steady-state AoI can be obtained by equation (15).\",\n \"The calculation details are given in Appendix E. Theorem 9 Assume that the relay has a buffer of size 1, for $n\\\\ge 1$ , the steady-state AoI $\\\\Delta _B^P$ of the relay-assisted status updating system is distributed as $\\\\Pr \\\\lbrace \\\\Delta _B^P=n\\\\rbrace &=\\\\frac{(\\\\delta +\\\\eta )P_3(pP_1+\\\\eta )}{\\\\delta -(\\\\delta +\\\\eta )(1-P_3)}\\\\delta ^{n-1}-c_1[(\\\\delta +\\\\eta )(1-P_3)]^{n-1} +c_2[\\\\delta (1-P_3)]^{n-1} \\\\\\\\& \\\\quad +\\\\frac{(1-S)(\\\\delta +\\\\eta )P_3^2}{1-P_3}n[\\\\delta (1-P_3)]^{n-1} + \\\\frac{P_3\\\\widetilde{S}}{1-P_3}n[(\\\\delta +\\\\eta )(1-P_3)]^{n-1}$ where the coefficients $c_1$ and $c_2$ are given as $c_1&=\\\\frac{pP_1\\\\eta (1-P_3)+\\\\delta \\\\eta P_3}{[\\\\delta -(\\\\delta +\\\\eta )(1-P_3)](1-P_3)} + \\\\frac{\\\\delta P_3[\\\\eta +(\\\\delta +\\\\eta )P_3](1-S)+(\\\\delta +\\\\eta )P_3\\\\widetilde{S}}{\\\\eta (1-P_3)} \\\\\\\\c_2&=\\\\frac{P_3[\\\\eta (\\\\delta +\\\\eta )+P_3(1-S)(\\\\delta +\\\\eta )(2\\\\delta -\\\\eta )]}{\\\\eta (1-P_3)}$\"\n ],\n [\n \"Performance metrics of Age of Information\",\n \"In Section III, for two cases where the packet retransmission from relay can and cannot be preempted by later packets, we derive the stationary AoI distribution of status updating system with a no-buffer relay.\",\n \"To investigate the influence of storing another packet in relay on system's AoI, the distribution of steady state AoI is determined in Section IV.\",\n \"Given the stationary AoI distributions, the following corollary shows the mean and the variance of AoI for all three cases, and the calculation details are provided in Appendix F. Corollary 1 For the relay-assisted status updating system, the average AoI for no-buffer-and-preemption case is equal to $\\\\mathbb {E}[\\\\Delta _{NB}^P]&=\\\\frac{(1-pP_1)P_3}{[(1-pP_1)P_3-p(1-P_1)P_2](1-\\\\delta )} + \\\\frac{\\\\xi }{[1-(1-pP_1)(1-P_3)]^2}$ while for the no-buffer-no-preemption case, it shows that $\\\\mathbb {E}[\\\\Delta _{NB}^{NP}]&= \\\\frac{\\\\beta _1}{(1-\\\\delta )^2}+\\\\frac{\\\\beta _2}{[1-(1-pP_1)(1-P_3)]^2} \\\\\\\\& \\\\quad + \\\\frac{p(1-P_1)P_2\\\\beta _1[2-\\\\delta -(1-pP_1)(1-P_3)]}{(1-\\\\delta )^2[1-(1-pP_1)(1-P_3)]^2} +\\\\frac{2p(1-P_1)P_2\\\\beta _2}{[1-(1-pP_1)(1-P_3)]^3}$ The average AoI of relay-assisted system when the relay is equipped with a buffer of size 1 is calculated as $\\\\mathbb {E}[\\\\Delta _B^P]&=\\\\frac{(\\\\delta +\\\\eta )P_3(pP_1+\\\\eta )}{[\\\\delta -(\\\\delta +\\\\eta )(1-P_3)](1-\\\\delta )^2}-\\\\frac{c_1}{[1-(\\\\delta +\\\\eta )(1-P_3)]^2} +\\\\frac{c_2}{[1-\\\\delta (1-P_3)]^2} \\\\\\\\& \\\\quad +\\\\frac{(1-S)(\\\\delta +\\\\eta )P_3^2[1+\\\\delta (1-P_3)]}{(1-P_3)[1-\\\\delta (1-P_3)]^3} +\\\\frac{P_3\\\\widetilde{S}[1+(\\\\delta +\\\\eta )(1-P_3)]}{(1-P_3)[1-(\\\\delta +\\\\eta )(1-P_3)]^3}$ where in all of three equations $\\\\delta =1-p(P_1+P_2-P_1P_2) $ Besides, $\\\\xi $ in (24) is given as $\\\\xi &=\\\\frac{p[P_1+(1-P_1)P_2P_3]}{1-P_3} - \\\\frac{P_3(1-\\\\delta )\\\\delta }{[(1-pP_1)P_3-p(1-P_1)P_2](1-P_3)} $ and $\\\\beta _1$ and $\\\\beta _2$ in expression (25) are defined in (9) and (10).\",\n \"In (26), $\\\\eta =p(1-P_1)P_2$ , $S$ , $\\\\widetilde{S}$ are given in (19) and (20).\",\n \"Other two coefficients $c_1$ , $c_2$ are introduced in equations (22) and (23).\",\n \"For different situations, the second moments of AoI can also be obtained.\",\n \"Then, we can calculate the variance of stationary AoI by $\\\\emph {Var}[\\\\Delta ]=\\\\mathbb {E}[\\\\Delta ^2]-\\\\left(\\\\mathbb {E}[\\\\Delta ]\\\\right)^2$ for all three cases.\",\n \"The expression of $\\\\emph {Var}[\\\\Delta ]$ is lengthy but the calculations are direct.\",\n \"Here, we provide only $\\\\emph {Var}[\\\\Delta _{NB}^{NP}]$ , AoI variances for other two cases can be obtained similarly.\",\n \"From AoI distribution (11), we have $\\\\mathbb {E}\\\\left[\\\\left(\\\\Delta _{NB}^{NP}\\\\right)^2\\\\right] &= \\\\left(\\\\beta _1+\\\\frac{p(1-P_1)P_2\\\\beta _1}{\\\\delta -(1-pP_1)(1-P_3)}\\\\right)\\\\sum \\\\nolimits _{n=1}^{\\\\infty }n^2\\\\delta ^{n-1} \\\\\\\\& \\\\quad + \\\\left(\\\\beta _2 - \\\\frac{p(1-P_1)P_2\\\\beta _1}{\\\\delta -(1-pP_1)(1-P_3)}\\\\right) \\\\sum \\\\nolimits _{n=1}^{\\\\infty }n^2[(1-pP_1)(1-P_3)]^{n-1} \\\\\\\\& \\\\quad + p(1-P_1)P_2\\\\beta _2\\\\sum \\\\nolimits _{n=1}^{\\\\infty }n^2(n-1)[(1-pP_1)(1-P_3)]^{n-2} \\\\\\\\&= \\\\left(\\\\beta _1+\\\\frac{p(1-P_1)P_2\\\\beta _1}{\\\\delta -(1-pP_1)(1-P_3)}\\\\right)\\\\frac{1+\\\\delta }{(1-\\\\delta )^3} +\\\\left(\\\\beta _2 - \\\\frac{p(1-P_1)P_2\\\\beta _1}{\\\\delta -(1-pP_1)(1-P_3)}\\\\right) \\\\\\\\& \\\\quad \\\\times \\\\frac{1+(1-pP_1)(1-P_3)}{[1-(1-pP_1)(1-P_3)]^3} + p(1-P_1)P_2\\\\beta _2 \\\\frac{4+2(1-pP_1)(1-P_3)}{[1-(1-pP_1)(1-P_3)]^4}$ where the last sum $&\\\\sum \\\\nolimits _{n=1}^{\\\\infty }n^2(n-1)[(1-pP_1)(1-P_3)]^{n-2} \\\\\\\\={}& \\\\sum \\\\nolimits _{n=1}^{\\\\infty }(n+1)n(n-1)[(1-pP_1)(1-P_3)]^{n-2} -\\\\sum \\\\nolimits _{n=1}^{\\\\infty }n(n-1)[(1-pP_1)(1-P_3)]^{n-2} \\\\\\\\={}&\\\\frac{6}{[1-(1-pP_1)(1-P_3)]^4} - \\\\frac{2}{[1-(1-pP_1)(1-P_3)]^3} \\\\frac{4+2(1-pP_1)(1-P_3)}{[1-(1-pP_1)(1-P_3)]^4} $ can be derived using following formula $\\\\sum \\\\nolimits _{n=1}^{\\\\infty }n(n-1)\\\\cdots (n-k+1)x^{n-k}=\\\\frac{k!\",\n \"}{(1-x)^{k+1}} \\\\text{ for }0m\\\\ge 0$ .\",\n \"${\\\\left\\\\lbrace \\\\begin{array}{ll}\\\\pi _{(n,m)}=\\\\pi _{(n-1,m-1)}\\\\delta _1 & \\\\qquad (n>m\\\\ge 2) \\\\\\\\\\\\pi _{(n,1)}=\\\\pi _{(n-1,0)}p(1-P_1)P_2 + \\\\sum \\\\nolimits _{j=1}^{n-2}\\\\pi _{(n-1,j)}p(1-P_1)P_2(1-P_3) \\\\\\\\\\\\qquad \\\\qquad + \\\\sum \\\\nolimits _{k=n}^{\\\\infty }\\\\pi _{(k,n-1)}p(1-P_1)P_2P_3 & \\\\qquad (n\\\\ge 3) \\\\\\\\\\\\pi _{(2,1)}=\\\\pi _{(1,0)}p(1-P_1)P_2 + \\\\sum \\\\nolimits _{k=2}^{\\\\infty }\\\\pi _{(k,1)}p(1-P_1)P_2P_3 \\\\\\\\\\\\pi _{(n,0)}=\\\\pi _{(n-1,0)}\\\\delta _2 + \\\\sum \\\\nolimits _{k=n}^{\\\\infty }\\\\pi _{(k,n-1)}\\\\delta _3 & \\\\qquad (n\\\\ge 2) \\\\\\\\\\\\pi _{(1,0)}=\\\\left(\\\\sum \\\\nolimits _{n=1}^{\\\\infty }\\\\pi _{(n,0)} + \\\\sum \\\\nolimits _{m=1}^{\\\\infty }\\\\sum \\\\nolimits _{n=m+1}^{\\\\infty }\\\\pi _{(n,m)} \\\\right) pP_1\\\\end{array}\\\\right.\",\n \"}$ where we denote $\\\\delta _2&=(1-p) + p(1-P_1)(1-P_2) = 1-p(P_1+P_2-P_1P_2) $ and $\\\\delta _1&=(1-p)(1-P_3)+p(1-P_1)(1-P_2)(1-P_3) =(1-P_3)\\\\delta _2, \\\\\\\\\\\\delta _3&=(1-p)P_3+p(1-P_1)(1-P_2)P_3=P_3\\\\delta _2 $ First of all, from the first line of (33) we have $\\\\pi _{(n,m)}&= \\\\pi _{(n-1,m-1)}\\\\delta _1=\\\\dots =\\\\pi _{(n-m+1,1)}\\\\delta _1^{m-1} \\\\\\\\&= {\\\\left\\\\lbrace \\\\begin{array}{ll}\\\\left[\\\\pi _{(1,0)}p(1-P_1)P_2 \\\\sum \\\\nolimits _{k=2}^{\\\\infty }\\\\pi _{(k,1)}p(1-P_1)P_2P_3 \\\\right]\\\\delta _1^{m-1} & (n-m=1) \\\\\\\\\\\\Big [\\\\pi _{(n-m,0)}p(1-P_1)P_2 \\\\sum \\\\nolimits _{j=1}^{n-m-1}\\\\pi _{(n-m,j)}p(1-P_1)P_2(1-P_3) \\\\\\\\\\\\qquad \\\\quad + \\\\sum \\\\nolimits _{k=n-m+1}^{\\\\infty }\\\\pi _{(k,n-m)}p(1-P_1)P_2P_3 \\\\Big ]\\\\delta _1^{m-1} & (n-m\\\\ge 2)\\\\end{array}\\\\right.\",\n \"}$ In above expressions, we have substituted the second and third lines of (33).\",\n \"For cases $n\\\\ge 2$ , the fourth equation of (33) says that $\\\\pi _{(n,0)}=\\\\pi _{(n-1,0)}\\\\delta _2 + \\\\left(\\\\sum \\\\nolimits _{k=n}^{\\\\infty }\\\\pi _{(k,n-1)}\\\\right) \\\\delta _3$ Using probability expressions obtained in (34), we deal with the sum in equation (35) as follows.\",\n \"$&\\\\sum \\\\nolimits _{k=n}^{\\\\infty }\\\\pi _{(k,n-1)}= \\\\pi _{(n,n-1)}+\\\\sum \\\\nolimits _{k=n+1}^{\\\\infty }\\\\pi _{(k,n-1)} \\\\\\\\={}&\\\\left[\\\\pi _{(1,0)}p(1-P_1)P_2 + \\\\sum \\\\nolimits _{k=2}^{\\\\infty }\\\\pi _{(k,1)}p(1-P_1)P_2P_3 \\\\right]\\\\delta _1^{n-2} \\\\\\\\{}& \\\\quad + \\\\sum \\\\nolimits _{k=n+1}^{\\\\infty } \\\\bigg \\\\lbrace \\\\pi _{(k-n+1,0)}p(1-P_1)P_2 + \\\\sum \\\\nolimits _{j=1}^{k-n}\\\\pi _{(k-n+1,j)}p(1-P_1)P_2(1-P_3) \\\\\\\\{}& \\\\quad + \\\\sum \\\\nolimits _{y=k-n+2}^{\\\\infty }\\\\pi _{(y,k-n+1)}p(1-P_1)P_2P_3 \\\\bigg \\\\rbrace \\\\delta _1^{n-2} \\\\\\\\={}&\\\\left[\\\\pi _{(1,0)}p(1-P_1)P_2 + \\\\left(\\\\sum \\\\nolimits _{k=2}^{\\\\infty }\\\\pi _{(k,1)}\\\\right) p(1-P_1)P_2P_3 \\\\right]\\\\delta _1^{n-2} \\\\\\\\{}& \\\\quad + \\\\bigg [ \\\\left(\\\\sum \\\\nolimits _{n=2}^{\\\\infty }\\\\pi _{(n,0)}\\\\right) p(1-P_1)P_2 + \\\\left( \\\\sum \\\\nolimits _{k=n+1}^{\\\\infty } \\\\sum \\\\nolimits _{j=1}^{k-n}\\\\pi _{(k-n+1,j)}\\\\right) p(1-P_1)P_2(1-P_3) \\\\\\\\{}& \\\\quad + \\\\left(\\\\sum \\\\nolimits _{k=n+1}^{\\\\infty }\\\\sum \\\\nolimits _{y=k-n+2}^{\\\\infty }\\\\pi _{(y,k-n+1)}\\\\right) p(1-P_1)P_2P_3 \\\\bigg ]\\\\delta _1^{n-2} \\\\\\\\={}&\\\\left[\\\\pi _{(1,0)}p(1-P_1)P_2 + \\\\left(\\\\sum \\\\nolimits _{n=2}^{\\\\infty }\\\\pi _{(n,1)}\\\\right) p(1-P_1)P_2P_3 \\\\right]\\\\delta _1^{n-2} \\\\\\\\{}& \\\\quad + \\\\Big [ \\\\left(\\\\sum \\\\nolimits _{n=2}^{\\\\infty }\\\\pi _{(n,0)}\\\\right) p(1-P_1)P_2 + \\\\left( \\\\sum \\\\nolimits _{n=2}^{\\\\infty } \\\\sum \\\\nolimits _{m=1}^{n-1}\\\\pi _{(n,m)}\\\\right) p(1-P_1)P_2(1-P_3) \\\\\\\\{}& \\\\quad + \\\\left(\\\\sum \\\\nolimits _{m=2}^{\\\\infty }\\\\sum \\\\nolimits _{n=m+1}^{\\\\infty }\\\\pi _{(n,m)}\\\\right) p(1-P_1)P_2P_3 \\\\Big ]\\\\delta _1^{n-2} \\\\\\\\={}& \\\\left[\\\\left(\\\\sum \\\\nolimits _{n=1}^{\\\\infty }\\\\pi _{(n,0)}\\\\right)p(1-P_1)P_2 + \\\\left( \\\\sum \\\\nolimits _{m=1}^{\\\\infty } \\\\sum \\\\nolimits _{n=m+1}^{\\\\infty }\\\\pi _{(n,m)}\\\\right) p(1-P_1)P_2 \\\\right]\\\\delta _1^{n-2} \\\\\\\\={}& p(1-P_1)P_2\\\\delta _1^{n-2}$ Let $k-n+1=\\\\tilde{n}$ , $j=\\\\tilde{m}$ , then $\\\\sum \\\\nolimits _{k=n+1}^{\\\\infty } \\\\sum \\\\nolimits _{j=1}^{k-n}\\\\pi _{(k-n+1,j)}=\\\\sum \\\\nolimits _{\\\\tilde{n}=2}^{\\\\infty }\\\\sum \\\\nolimits _{\\\\tilde{m}=1}^{\\\\tilde{n}-1}\\\\pi _{(\\\\tilde{n},\\\\tilde{m})} $ Similarly, do the substitutions $y=\\\\tilde{n}$ and $k-n+1=\\\\tilde{m}$ , it shows that $\\\\sum \\\\nolimits _{k=n+1}^{\\\\infty }\\\\sum \\\\nolimits _{y=k-n+2}^{\\\\infty }\\\\pi _{(y,k-n+1)}=\\\\sum \\\\nolimits _{\\\\tilde{m}=2}^{\\\\infty }\\\\sum \\\\nolimits _{\\\\tilde{n}=\\\\tilde{m}+1}^{\\\\infty }\\\\pi _{(\\\\tilde{n},\\\\tilde{m})} $ Substituting these sums, we obtain the equation (36).\",\n \"In addition, observing that $&\\\\sum \\\\nolimits _{n=2}^{\\\\infty }\\\\pi _{(n,1)} + \\\\sum \\\\nolimits _{m=2}^{\\\\infty } \\\\sum \\\\nolimits _{n=m+1}^{\\\\infty }\\\\pi _{(n,m)} = \\\\sum \\\\nolimits _{m=1}^{\\\\infty } \\\\sum \\\\nolimits _{n=m+1}^{\\\\infty }\\\\pi _{(n,m)} $ and $\\\\sum \\\\nolimits _{m=1}^{\\\\infty } \\\\sum \\\\nolimits _{n=m+1}^{\\\\infty }\\\\pi _{(n,m)}=\\\\sum \\\\nolimits _{n=2}^{\\\\infty } \\\\sum \\\\nolimits _{m=1}^{n-1}\\\\pi _{(n,m)} $ which yields the result (37).\",\n \"Equation (38) holds because the probabilities contained in bracket of (37) add up to 1.\",\n \"Finally, we derive the following recursive equation $\\\\pi _{(n,0)}=\\\\pi _{(n-1,0)}\\\\delta _2 + p(1-P_1)P_2\\\\delta _3\\\\delta _1^{n-2} \\\\qquad (n\\\\ge 2)$ Applying (39) iteratively, the general formula of $\\\\pi _{(n,0)}$ can be obtained.\",\n \"We show that $\\\\pi _{(n,0)}&=\\\\pi _{(1,0)}\\\\delta _2^{n-1}+p(1-P_1)P_2\\\\delta _3 \\\\sum \\\\nolimits _{j=0}^{n-2}\\\\delta _2^j\\\\delta _1^{n-2-j} \\\\\\\\&=\\\\pi _{(1,0)}\\\\delta _2^{n-1}+p(1-P_1)P_2\\\\delta _3 \\\\frac{\\\\delta _1^{n-1}-\\\\delta _2^{n-1}}{\\\\delta _1-\\\\delta _2}$ The first probability $\\\\pi _{(1,0)}$ is equal to $pP_1$ , which can be determined directly from the last equation of (33).\",\n \"Remember that $\\\\delta _2=1-p(P_1+P_2-P_1P_2)$ and $\\\\delta _1=(1-P_3)\\\\delta _2, \\\\quad \\\\delta _3=P_3\\\\delta _2 $ we have $\\\\pi _{(n,0)}&=pP_1\\\\delta _2^{n-1}+p(1-P_1)P_2\\\\delta _3\\\\frac{\\\\delta _1^{n-1}-\\\\delta _2^{n-1}}{(1-P_3)\\\\delta _2-\\\\delta _2} \\\\\\\\&=pP_1\\\\delta _2^{n-1}+p(1-P_1)P_2\\\\left(\\\\delta _2^{n-1}-\\\\delta _1^{n-1} \\\\right) \\\\\\\\&=\\\\left[pP_1+p(1-P_1)P_2\\\\right]\\\\delta _2^{n-1} - p(1-P_1)P_2\\\\delta _1^{n-1} \\\\\\\\&=(1-\\\\delta _2)\\\\delta _2^{n-1} - p(1-P_1)P_2[(1-P_3)\\\\delta _2]^{n-1}$ Let $n=1$ , equation (41) gives $\\\\pi _{(1,0)}=pP_1$ .\",\n \"Therefore, we prove that expression (41) is valid for all $n\\\\ge 1$ .\",\n \"Next, probabilities $\\\\pi _{(n,m)}$ , $n>m\\\\ge 1$ are solved.\",\n \"We first calculate $\\\\pi _{(n,1)}$ , then by the relation $\\\\pi _{(n,m)}=\\\\pi _{(n-m+1,1)}\\\\delta _1^{m-1}$ all the probabilities $\\\\pi _{(n,m)}$ are obtained.\",\n \"When $n\\\\ge 3$ , from the second line of equations (33), we show that $\\\\pi _{(n,1)}&= \\\\pi _{(n-1,0)}p(1-P_1)P_2 + \\\\sum \\\\nolimits _{j=1}^{n-2}\\\\pi _{(n-1,j)}p(1-P_1)P_2(1-P_3) + \\\\sum \\\\nolimits _{k=n}^{\\\\infty }\\\\pi _{(k,n-1)}p(1-P_1)P_2P_3 \\\\\\\\&=\\\\left\\\\lbrace (1-\\\\delta _2)\\\\delta _2^{n-2} - p(1-P_1)P_2\\\\delta _1^{n-2}\\\\right\\\\rbrace p(1-P_1)P_2 \\\\\\\\& \\\\qquad + \\\\sum \\\\nolimits _{j=1}^{n-2}\\\\pi _{(n-j,1)}\\\\delta _1^{j-1} p(1-P_1)P_2(1-P_3) + p(1-P_1)P_2\\\\delta _1^{n-2}p(1-P_1)P_2P_3 \\\\\\\\&= p(1-P_1)P_2(1-\\\\delta _2)\\\\delta _2^{n-2} - [p(1-P_1)P_2]^2(1-P_3)\\\\delta _1^{n-2} \\\\\\\\& \\\\qquad + p(1-P_1)P_2(1-P_3)\\\\sum \\\\nolimits _{j=1}^{n-2}\\\\pi _{(n-j,1)}\\\\delta _1^{j-1}$ In (43), we have substituted equations (41) and (38).\",\n \"Using relation (42), we can rewrite $\\\\pi _{(n-1,j)}$ as $\\\\pi _{(n-j,1)}\\\\delta _1^{j-1}$ .\",\n \"Computing the difference $\\\\pi _{(n,1)}-\\\\pi _{(n-1,1)}\\\\delta _1 = p(1-P_1)P_2(1-\\\\delta _2)\\\\delta _2^{n-3}(\\\\delta _2-\\\\delta _1) + p(1-P_1)P_2(1-P_3)\\\\pi _{(n-1,1)} $ which is equivalent to $\\\\pi _{(n,1)}&=\\\\pi _{(n-1,1)}[\\\\delta _1+p(1-P_1)P_2(1-P_3)] + p(1-P_1)P_2(1-\\\\delta _2)P_3\\\\delta _2^{n-2} \\\\\\\\&= \\\\pi _{(n-1,1)}(1-P_3)(1-pP_1) + p(1-P_1)P_2(1-\\\\delta _2)P_3\\\\delta _2^{n-2}$ where $\\\\delta _1 + p(1-P_1)P_2(1-P_3) &= (1-P_3)\\\\delta _2 + p(1-P_1)P_2(1-P_3) \\\\\\\\&=(1-P_3)\\\\left\\\\lbrace 1-p(P_1+P_2-P_1P_2) + p(1-P_1)P_2 \\\\right\\\\rbrace \\\\\\\\&=(1-P_3)(1-pP_1) $ Equation (45) gives a recursive relation of the probabilities $\\\\pi _{(n,1)}$ , $n\\\\ge 3$ , from which we can derive that $\\\\pi _{(n,1)}&= \\\\pi _{(2,1)}[(1-P_3)(1-pP_1)]^{n-2} + p(1-P_1)P_2(1-\\\\delta _2)P_3 \\\\sum \\\\nolimits _{j=0}^{n-3}[(1-P_3)(1-pP_1)]^j \\\\delta _2^{n-2-j} \\\\\\\\&=\\\\pi _{(2,1)}[(1-P_3)(1-pP_1)]^{n-2} + p(1-P_1)P_2P_3(1-\\\\delta _2)\\\\delta _2 \\\\frac{\\\\delta _2^{n-2}-[(1-P_3)(1-pP_1)]^{n-2}}{\\\\delta _2-(1-P_3)(1-pP_1)} \\\\\\\\&=\\\\frac{p(1-P_1)P_2P_3(1-\\\\delta _2)\\\\delta _2}{\\\\delta _2-(1-P_3)(1-pP_1)}\\\\delta _2^{n-2}+ \\\\left(\\\\pi _{(2,1)} - \\\\frac{p(1-P_1)P_2P_3(1-\\\\delta _2)\\\\delta _2}{\\\\delta _2-(1-P_3)(1-pP_1)} \\\\right) [(1-P_3)(1-pP_1)]^{n-2}$ Using the basic relation in (33), the probability $\\\\pi _{(2,1)}$ is calculated as $\\\\pi _{(2,1)}&=\\\\pi _{(1,0)}p(1-P_1)P_2 + \\\\sum \\\\nolimits _{k=2}^{\\\\infty }\\\\pi _{(k,1)}p(1-P_1)P_2P_3 \\\\\\\\&=pP_1p(1-P_1)P_2 + [p(1-P_1)P_2]^2P_3 \\\\\\\\&=p^2(1-P_1)P_2[P_1+(1-P_1)P_2P_3] $ In equation (46), it is easy to see that for the case $n=2$ , expression (47) reduces to $\\\\pi _{(2,1)}=\\\\pi _{(2,1)}$ .\",\n \"Therefore, we show that (47) is actually valid for all $n\\\\ge 2$ .\",\n \"Now, for $n>m\\\\ge 1$ , the probabilities $\\\\pi _{(n,m)}$ can be obtained eventually.\",\n \"It shows that $\\\\pi _{(n,m)}&=\\\\pi _{(n-m+1,1)}\\\\delta _1^{m-1} \\\\\\\\&= \\\\bigg \\\\lbrace \\\\frac{p(1-P_1)P_2P_3(1-\\\\delta _2)\\\\delta _2}{\\\\delta _2-(1-P_3)(1-pP_1)}\\\\delta _2^{n-m-1} \\\\\\\\& \\\\quad + \\\\left(\\\\pi _{(2,1)} - \\\\frac{p(1-P_1)P_2P_3(1-\\\\delta _2)\\\\delta _2}{\\\\delta _2-(1-P_3)(1-pP_1)} \\\\right) [(1-P_3)(1-pP_1)]^{n-m-1} \\\\bigg \\\\rbrace \\\\delta _1^{m-1} \\\\\\\\&= \\\\frac{p(1-P_1)P_2P_3(1-\\\\delta _2)\\\\delta _2}{\\\\delta _2-(1-P_3)(1-pP_1)}\\\\delta _2^{n-2}(1-P_3)^{m-1} \\\\\\\\& \\\\quad + \\\\left(\\\\pi _{(2,1)} - \\\\frac{p(1-P_1)P_2P_3(1-\\\\delta _2)\\\\delta _2}{\\\\delta _2-(1-P_3)(1-pP_1)} \\\\right) [(1-P_3)(1-pP_1)]^{n-2}\\\\left(\\\\frac{\\\\delta _2}{1-pP_1}\\\\right)^{m-1}$ So far, we have obtained all the stationary probabilities of process $AoI_{NB}^P$ and complete the proof of Theorem 2.\"\n ],\n [\n \"Proof of Theorem 2\",\n \"In this part, according to equation (4), we calculation the stationary distribution of AoI.\",\n \"For $n\\\\ge 2$ , we first compute $&\\\\sum \\\\nolimits _{m=1}^{n-1}\\\\pi _{(n,m)} \\\\\\\\={}&\\\\sum \\\\nolimits _{m=1}^{n-1}\\\\Bigg \\\\lbrace \\\\pi _{(2,1)}[(1-pP_1)(1-P_3)]^{n-2}\\\\left(\\\\frac{\\\\delta }{1-pP_1}\\\\right)^{m-1} + \\\\frac{p(1-P_1)P_2P_3(1-\\\\delta )\\\\delta }{\\\\delta -(1-pP_1)(1-P_3)} \\\\\\\\{}& \\\\quad \\\\times \\\\left[ \\\\delta ^{n-2}(1-P_3)^{m-1} - [(1-pP_1)(1-P_3)]^{n-2}\\\\left(\\\\frac{\\\\delta }{1-pP_1} \\\\right)^{m-1} \\\\right] \\\\Bigg \\\\rbrace \\\\\\\\={}&\\\\pi _{(2,1)}\\\\frac{[(1-pP_1)(1-P_3)]^{n-1}-[(1-P_3)\\\\delta ]^{n-1}}{(1-pP_1-\\\\delta )(1-P_3)} + \\\\frac{p(1-P_1)P_2P_3(1-\\\\delta )\\\\delta }{\\\\delta -(1-pP_1)(1-P_3)} \\\\\\\\{}& \\\\quad \\\\times \\\\left\\\\lbrace \\\\frac{\\\\delta ^{n-2}-(1-P_3)[(1-P_3)\\\\delta ]^{n-2}}{P_3} - \\\\frac{[(1-pP_1)(1-P_3)]^{n-1}-[(1-P_3)\\\\delta ]^{n-1}}{(1-pP_1-\\\\delta )(1-P_3)} \\\\right\\\\rbrace \\\\\\\\={}& \\\\frac{p(1-P_1)P_2(1-\\\\delta )}{\\\\delta -(1-pP_1)(1-P_3)}\\\\left\\\\lbrace \\\\delta ^{n-1}-[(1-P_3)\\\\delta ]^{n-1}\\\\right\\\\rbrace \\\\\\\\{}& \\\\quad + \\\\left(\\\\pi _{(2,1)} - \\\\frac{p(1-P_1)P_2P_3(1-\\\\delta )\\\\delta }{\\\\delta -(1-pP_1)(1-P_3)} \\\\right) \\\\frac{[(1-pP_1)(1-P_3)]^{n-1}-[(1-P_3)\\\\delta ]^{n-1}}{(1-pP_1-\\\\delta )(1-P_3)} \\\\\\\\={}&\\\\eta _1\\\\left\\\\lbrace \\\\delta ^{n-1}-[(1-P_3)\\\\delta ]^{n-1}\\\\right\\\\rbrace + \\\\eta _2\\\\left\\\\lbrace [(1-pP_1)(1-P_3)]^{n-1}-[(1-P_3)\\\\delta ]^{n-1}\\\\right\\\\rbrace $ where we denote $\\\\eta _1&=\\\\frac{p(1-P_1)P_2(1-\\\\delta )}{\\\\delta -(1-pP_1)(1-P_3)}, \\\\\\\\\\\\eta _2&=\\\\left(\\\\pi _{(2,1)}-\\\\frac{p(1-P_1)P_2P_3(1-\\\\delta )\\\\delta }{\\\\delta -(1-pP_1)(1-P_3)}\\\\right) \\\\frac{1}{(1-pP_1-\\\\delta )(1-P_3)} $ Therefore, when $n\\\\ge 2$ , the probability that AoI equals $n$ is calculated as $\\\\Pr \\\\lbrace \\\\Delta =n\\\\rbrace &=\\\\pi _{(n,0)}+\\\\sum \\\\nolimits _{m=1}^{n-1}\\\\pi _{(n,m)} \\\\\\\\&= [(1-\\\\delta )+\\\\eta _1]\\\\delta ^{n-1}+\\\\eta _2[(1-pP_1)(1-P_3)]^{n-1} \\\\\\\\&\\\\qquad \\\\quad -\\\\left(\\\\eta _1+\\\\eta _2+p(1-P_1)P_2\\\\right)[(1-P_3)\\\\delta ]^{n-1}$ To obtain equation (50), we have substituted the probability expression (2).\",\n \"Since $\\\\delta -(1-pP_1)(1-P_3) &=1-p(P_1+P_2-P_1P_2)-(1-pP_1)(1-P_3) \\\\\\\\&=1-pP_1-p(1-P_1)P_2-(1-pP_1)(1-P_3) \\\\\\\\&=(1-pP_1)P_3-p(1-P_1)P_2 $ we have the first coefficient $(1-\\\\delta )+\\\\eta _1&=1-\\\\delta + \\\\frac{p(1-P_1)P_2(1-\\\\delta )}{\\\\delta -(1-pP_1)(1-P_3)} \\\\\\\\&=(1-\\\\delta ) \\\\left( 1 + \\\\frac{p(1-P_1)P_2}{(1-pP_1)P_3-p(1-P_1)P_2}\\\\right) \\\\\\\\&=\\\\frac{(1-pP_1)P_3(1-\\\\delta )}{(1-pP_1)P_3-p(1-P_1)P_2} $ Notice that $1-pP_1-\\\\delta =1-pP_1-[1-p(P_1+P_2-P_1P_2)] = p(1-P_1)P_2 $ Substituting $\\\\pi _{(2,1)}$ , it shows that $\\\\eta _2&=\\\\frac{p[P_1+(1-P_1)P_2P_3]}{1-P_3} - \\\\frac{P_3(1-\\\\delta )\\\\delta }{[(1-pP_1)P_3-p(1-P_1)P_2](1-P_3)} $ In following paragraph, we will prove that the coefficient before the last term, i.e., $\\\\eta _1+\\\\eta _2+p(1-P_1)P_2$ is zero.\",\n \"Thus, for $n\\\\ge 2$ we obtain $\\\\Pr \\\\lbrace \\\\Delta _{NB}^P=n\\\\rbrace &=\\\\frac{(1-pP_1)P_3(1-\\\\delta )}{(1-pP_1)P_3-p(1-P_1)P_2}\\\\delta ^{n-1} + \\\\eta _2[(1-pP_1)(1-P_3)]^{n-1}$ From equation (49), it is easy to see that the sum in (4) is zero for the case $n=1$ .\",\n \"So, the expression (51) is valid for all $n\\\\ge 1$ .\",\n \"We now show that $\\\\eta _1+\\\\eta _2+p(1-P_1)P_2=0$ .\",\n \"That is $&\\\\frac{p(1-P_1)P_2(1-\\\\delta )}{\\\\delta -(1-pP_1)(1-P_3)} + \\\\left(p[P_1+(1-P_1)P_2P_3]-\\\\frac{P_3(1-\\\\delta )\\\\delta }{\\\\delta -(1-pP_1)(1-P_3)}\\\\right) \\\\\\\\& \\\\qquad \\\\qquad \\\\qquad \\\\qquad \\\\qquad \\\\qquad \\\\qquad \\\\qquad \\\\times \\\\frac{p(1-P_1)P_2}{(1-pP_1-\\\\delta )(1-P_3)} + p(1-P_1)P_2 =0 $ which is equivalent to $&\\\\frac{1-\\\\delta }{\\\\delta -(1-pP_1)(1-P_3)} \\\\\\\\& + \\\\left(p[P_1+(1-P_1)P_2P_3]-\\\\frac{P_3(1-\\\\delta )\\\\delta }{\\\\delta -(1-pP_1)(1-P_3)}\\\\right) \\\\frac{1}{(1-pP_1-\\\\delta )(1-P_3)} + 1 =0 \\\\\\\\\\\\Leftrightarrow {}& \\\\frac{1-\\\\delta }{\\\\delta -(1-pP_1)(1-P_3)} + \\\\frac{p[P_1+(1-P_1)P_2P_3]}{(1-pP_1-\\\\delta )(1-P_3)} \\\\\\\\& - \\\\frac{P_3(1-\\\\delta )\\\\delta }{[\\\\delta -(1-pP_1)(1-P_3)](1-pP_1-\\\\delta )(1-P_3)} +1 =0$ In (52), we have $\\\\delta -(1-pP_1)(1-P_3)=(1-pP_1)P_3 - p(1-P_1)P_2 $ and $(1-pP_1-\\\\delta )(1-P_3) =\\\\left[1-pP_1- \\\\left(1-p(P_1+P_2-P_1P_2) \\\\right)\\\\right](1-P_3) =p(1-P_1)P_2(1-P_3) $ Then, equation (52) can be rewritten as $&\\\\frac{1-\\\\delta }{(1-pP_1)P_3-p(1-P_1)P_2}\\\\left(\\\\frac{P_3\\\\delta }{p(1-P_1)P_2(1-P_3)}-1\\\\right) =\\\\frac{p[P_1+(1-P_1)P_2P_3]}{p(1-P_1)P_2(1-P_3)}+1$ where the RHS of (53) is equal to $&\\\\frac{p[P_1+(1-P_1)P_2P_3]+p(1-P_1)P_2(1-P_3)}{p(1-P_1)P_2(1-P_3)} = \\\\frac{p[P_1+(1-P_1)P_2]}{p(1-P_1)P_2(1-P_3)}= \\\\frac{1-\\\\delta }{p(1-P_1)P_2(1-P_3)} $ Therefore, to prove (52), it suffices to show that $&\\\\frac{1}{(1-pP_1)P_3-p(1-P_1)P_2}\\\\left(\\\\frac{P_3\\\\delta }{p(1-P_1)P_2(1-P_3)}-1\\\\right) = \\\\frac{1}{p(1-P_1)P_2(1-P_3)} \\\\\\\\\\\\Leftrightarrow {}&\\\\frac{P_3\\\\delta }{p(1-P_1)P_2(1-P_3)}-1 = \\\\frac{(1-pP_1)P_3-p(1-P_1)P_2}{p(1-P_1)P_2(1-P_3)} = \\\\frac{(1-pP_1)P_3-p(1-P_1)P_2P_3}{p(1-P_1)P_2(1-P_3)}-1 \\\\\\\\\\\\Leftrightarrow {}&\\\\delta =(1-pP_1)-p(1-P_1)P_2 =1-p(P_1+P_2-P_1P_2)$ Since equation (54) holds, we verify the original equation (52) and show that the coefficient $\\\\eta _1+\\\\eta _2+p(1-P_1)P_2$ is equal to zero.\",\n \"This completes the proof of Theorem 3.\"\n ],\n [\n \"Proof of Theorem 5\",\n \"In this Appendix, we solve the system of stationary equations (6) and find all the stationary probabilities $\\\\pi _{(n,m)}$ of AoI stochastic process $AoI_{NB}^{NP}$ .\",\n \"For convenience, we copy these equations in the following.\",\n \"${\\\\left\\\\lbrace \\\\begin{array}{ll}\\\\pi _{(n,m)}=\\\\pi _{(n-1,m-1)}[(1-p)(1-P_3)+p(1-P_1)(1-P_3)] & \\\\qquad (n>m\\\\ge 2) \\\\\\\\\\\\pi _{(n,1)}=\\\\pi _{(n-1,0)}p(1-P_1)P_2 & \\\\qquad (n\\\\ge 2) \\\\\\\\\\\\pi _{(n,0)}=\\\\pi _{(n-1,0)}[(1-p)+p(1-P_1)(1-P_2)] \\\\\\\\\\\\qquad \\\\qquad + \\\\left(\\\\sum \\\\nolimits _{k=n}^{\\\\infty }\\\\pi _{(k,n-1)}\\\\right)[(1-p)P_3+p(1-P_1)P_3] & \\\\qquad (n\\\\ge 2) \\\\\\\\\\\\pi _{(1,0)}= \\\\left(\\\\sum \\\\nolimits _{n=1}^{\\\\infty }\\\\pi _{(n,0)} + \\\\sum \\\\nolimits _{n=2}^{\\\\infty }\\\\sum \\\\nolimits _{m=1}^{n-1}\\\\pi _{(n,m)}\\\\right)pP_1\\\\end{array}\\\\right.\",\n \"}$ First of all, applying the first line of (55) repeatedly, we can write $\\\\pi _{(n,m)}$ as $\\\\pi _{(n,m)}&=\\\\pi _{(n-1,m-1)}[(1-pP_1)(1-P_3)] \\\\\\\\&=\\\\pi _{(n-2,m-2)}[(1-pP_1)(1-P_3)]^2 \\\\\\\\&\\\\qquad \\\\qquad \\\\quad \\\\quad \\\\vdots \\\\\\\\&=\\\\pi _{(n-m+1,1)}[(1-pP_1)(1-P_3)]^{m-1} \\\\\\\\&=\\\\pi _{(n-m,0)}p(1-P_1)P_2[(1-pP_1)(1-P_3)]^{m-1}$ where to obtain the last step (56), the second equation in (55) is applied.\",\n \"Substituting (56), we first handle the sum in the third line of (55).\",\n \"It shows that $\\\\sum \\\\nolimits _{k=n}^{\\\\infty }\\\\pi _{(k,n-1)}&= \\\\sum \\\\nolimits _{k=n}^{\\\\infty }\\\\pi _{(k-n+1,0)}p(1-P_1)P_2 [(1-pP_1)(1-P_3)]^{n-2}\\\\\\\\&= \\\\left(\\\\sum \\\\nolimits _{k=1}^{\\\\infty }\\\\pi _{(k,0)}\\\\right) p(1-P_1)P_2 [(1-pP_1)(1-P_3)]^{n-2} $ Since all the probabilities $\\\\pi _{(n,m)}$ have to add up to 1, we have the relation $1&=\\\\sum \\\\nolimits _{n=1}^{\\\\infty }\\\\pi _{(n,0)} + \\\\sum \\\\nolimits _{n=m+1}^{\\\\infty }\\\\sum \\\\nolimits _{m=1}^{\\\\infty }\\\\pi _{(n,m)} \\\\\\\\&=\\\\sum \\\\nolimits _{n=1}^{\\\\infty }\\\\pi _{(n,0)} + \\\\sum \\\\nolimits _{n=m+1}^{\\\\infty }\\\\sum \\\\nolimits _{m=1}^{\\\\infty }\\\\pi _{(n-m,0)} p(1-P_1)P_2[(1-pP_1)(1-P_3)]^{m-1} \\\\\\\\&=\\\\sum \\\\nolimits _{n=1}^{\\\\infty }\\\\pi _{(n,0)} + \\\\left(\\\\sum \\\\nolimits _{t=1}^{\\\\infty }\\\\pi _{(t,0)}\\\\right) p(1-P_1)P_2 \\\\sum \\\\nolimits _{m=1}^{\\\\infty } [(1-pP_1)(1-P_3)]^{m-1} \\\\\\\\&=\\\\left(\\\\sum \\\\nolimits _{k=1}^{\\\\infty }\\\\pi _{(k,0)}\\\\right)\\\\left[1 + \\\\frac{p(1-P_1)P_2}{1-(1-pP_1)(1-P_3)}\\\\right]$ To obtain (57), do the substitution $t=n-m$ .\",\n \"From equation (58) we can derive that $\\\\sum \\\\nolimits _{k=1}^{\\\\infty }\\\\pi _{(k,0)}=\\\\frac{1-(1-pP_1)(1-P_3)}{1-(1-pP_1)(1-P_3)+p(1-P_1)P_2} $ Combining above results, for $n\\\\ge 2$ we can write that $\\\\pi _{(n,0)}=\\\\pi _{(n-1,0)}\\\\delta + \\\\xi \\\\left[(1-pP_1)(1-P_3)\\\\right]^{n-1}$ where $\\\\delta &=1-p(P_1+P_2-P_1P_2), \\\\\\\\\\\\xi &=\\\\frac{p(1-P_1)P_2P_3[1-(1-pP_1)(1-P_3)]}{[1-(1-pP_1)(1-P_3)+p(1-P_1)P_2](1-P_3)} $ Equation (59) gives a recursive relation of the probabilities $\\\\pi _{(n,0)}$ .\",\n \"Using (59) iteratively yields that $\\\\pi _{(n,0)}&=\\\\pi _{(1,0)}\\\\delta ^{n-1} + \\\\xi \\\\sum \\\\nolimits _{j=0}^{n-2}\\\\delta ^j[(1-pP_1)(1-P_3)]^{n-1-j} \\\\\\\\&=\\\\pi _{(1,0)}\\\\delta ^{n-1} + \\\\frac{\\\\xi (1-pP_1)(1-P_3)}{(1-pP_1)(1-P_3)-\\\\delta } \\\\left\\\\lbrace [(1-pP_1)(1-P_3)]^{n-1} - \\\\delta ^{n-1}\\\\right\\\\rbrace \\\\\\\\&=\\\\beta _1 \\\\delta ^{n-1} + \\\\beta _2[(1-pP_1)(1-P_3)]^{n-1}$ Substituting $\\\\xi $ , it show that the coefficient $\\\\beta _2$ is equal to $\\\\beta _2&=\\\\frac{[1-(1-pP_1)(1-P_3)]p(1-P_1)P_2P_3}{1-(1-pP_1)(1-P_3)+p(1-P_1)P_2} \\\\frac{1-pP_1}{(1-pP_1)(1-P_3)-\\\\delta } $ Since $\\\\pi _{(1,0)}=pP_1$ , we have $\\\\beta _1=pP_1-\\\\beta _2$ .\",\n \"It is easy to see that when $n=1$ , equation (61) reduces to $\\\\pi _{(1,0)}=\\\\pi _{(1,0)}$ , thus probability expression (61) is actually valid for all $n\\\\ge 1$ .\",\n \"According to (56), for $n>m\\\\ge 1$ , the probability $\\\\pi _{(n,m)}$ is then determined as $\\\\pi _{(n,m)}&= \\\\pi _{(n-m,0)}p(1-P_1)P_2[(1-pP_1)(1-P_3)]^{m-1} \\\\\\\\={}& p(1-P_1)P_2\\\\left\\\\lbrace \\\\beta _1\\\\delta ^{n-m-1} + \\\\beta _2[(1-pP_1)(1-P_3)]^{n-m-1} \\\\right\\\\rbrace [(1-pP_1)(1-P_3)]^{m-1} \\\\\\\\={}& p(1-P_1)P_2 \\\\left\\\\lbrace \\\\beta _1\\\\delta ^{n-2}\\\\left(\\\\frac{(1-pP_1)(1-P_3)}{\\\\delta }\\\\right)^{m-1} + \\\\beta _2[(1-pP_1)(1-P_3)]^{n-2} \\\\right\\\\rbrace $ So far, we have completely solved the system of equations (55) and obtained all the stationary probabilities $\\\\pi _{(n,m)}$ .\"\n ],\n [\n \"Proof of Theorem 8\",\n \"In this appendix, we determine all the stationary probabilities $\\\\pi _{(n,m,l)}$ of stochastic process $AoI_B^P$ by solving the following system of equations.\",\n \"${\\\\left\\\\lbrace \\\\begin{array}{ll}\\\\pi _{(n,m,l)}=\\\\pi _{(n-1,m-1,l-1)}\\\\delta (1-P_3) & \\\\qquad (n>m>l\\\\ge 2) \\\\\\\\\\\\pi _{(n,m,1)}=\\\\left(\\\\sum \\\\nolimits _{j=0}^{m-2}\\\\pi _{(n-1,m-1,j)}\\\\right)\\\\eta (1-P_3) & \\\\qquad (n>m\\\\ge 2) \\\\\\\\\\\\pi _{(n,m,0)}=\\\\pi _{(n-1,m-1,0)}\\\\delta (1-P_3) +\\\\left(\\\\sum \\\\nolimits _{k=n}^{\\\\infty }\\\\pi _{(k,n-1,m-1)}\\\\right)\\\\delta P_3 & \\\\qquad (n>m\\\\ge 2) \\\\\\\\\\\\pi _{(n,1,0)}=\\\\pi _{(n-1,0,0)}\\\\eta +\\\\left(\\\\sum \\\\nolimits _{k=n}^{\\\\infty }\\\\sum \\\\nolimits _{j=0}^{n-2}\\\\pi _{(k,n-1,j)}\\\\right)\\\\eta P_3 & \\\\qquad (n\\\\ge 2) \\\\\\\\\\\\pi _{(n,0,0)}=\\\\pi _{(n-1,0,0)}\\\\delta +\\\\left(\\\\sum \\\\nolimits _{k=n}^{\\\\infty }\\\\pi _{(k,n-1,0)}\\\\right)\\\\delta P_3 & \\\\qquad (n\\\\ge 2) \\\\\\\\\\\\pi _{(1,0,0)}=\\\\Big (\\\\sum \\\\nolimits _{n=1}^{\\\\infty }\\\\pi _{(n,0,0)}+ \\\\sum \\\\nolimits _{n=2}^{\\\\infty }\\\\sum \\\\nolimits _{m=1}^{n-1}\\\\pi _{(n,m,0)}\\\\\\\\\\\\qquad \\\\qquad \\\\qquad \\\\qquad \\\\qquad + \\\\sum \\\\nolimits _{n=3}^{\\\\infty }\\\\sum \\\\nolimits _{m=2}^{n-1}\\\\sum \\\\nolimits _{l=1}^{m-1}\\\\pi _{(n,m,l)}\\\\Big )pP_1\\\\end{array}\\\\right.\",\n \"}$ First of all, for $n>m>l\\\\ge 1$ , using first two lines of (63) we can obtain that $\\\\pi _{(n,m,l)}&= \\\\pi _{(n-1,m-1,l-1)}[\\\\delta (1-P_3)] \\\\\\\\{}& \\\\qquad \\\\qquad \\\\qquad \\\\vdots \\\\\\\\&= \\\\pi _{(n-l+1,m-l+1,1)}[\\\\delta (1-P_3)]^{l-1} \\\\\\\\&= \\\\left(\\\\sum \\\\nolimits _{j=0}^{m-l-1}\\\\pi _{(n-l,m-l,j)}\\\\right)\\\\eta (1-P_3)[\\\\delta (1-P_3)]^{l-1}$ Substituting expression (64), the sum in the third equation of (63) can be rewritten as $\\\\sum \\\\nolimits _{k=n}^{\\\\infty }\\\\pi _{(k,n-1,m-1)}&= \\\\sum \\\\nolimits _{k=n}^{\\\\infty }\\\\left(\\\\sum \\\\nolimits _{j=0}^{n-m-1}\\\\pi _{(k-m+1,n-m,j)} \\\\right) \\\\eta (1-P_3)[\\\\delta (1-P_3)]^{m-2} \\\\\\\\&= \\\\left( \\\\sum \\\\nolimits _{k=n-m+1}^{\\\\infty }\\\\sum \\\\nolimits _{j=0}^{n-m-1}\\\\pi _{(k,n-m,j)}\\\\right) \\\\eta (1-P_3)[\\\\delta (1-P_3)]^{m-2}\\\\\\\\&= t_{n-m}\\\\eta (1-P_3)[\\\\delta (1-P_3)]^{m-2}$ where we define $t_n= \\\\sum \\\\nolimits _{k=n+1}^{\\\\infty }\\\\sum \\\\nolimits _{j=0}^{n-1}\\\\pi _{(k,n,j)} \\\\qquad (n\\\\ge 1)$ which is actually the probability that the middle parameter equals $n$ .\",\n \"Therefore, for $n>m\\\\ge 2$ we have $\\\\pi _{(n,m,0)}&=\\\\pi _{(n-1,m-1,0)}\\\\delta (1-P_3) + t_{n-m}\\\\eta (1-P_3)[\\\\delta (1-P_3)]^{m-2} \\\\delta P_3 \\\\\\\\&=\\\\pi _{(n-1,m-1,0)}\\\\delta (1-P_3) + t_{n-m}\\\\eta P_3[\\\\delta (1-P_3)]^{m-1}$ Equation (67) is a recursive formula of the probabilities $\\\\pi _{(n,m,0)}$ .\",\n \"By repeatedly using (67), it shows that $\\\\pi _{(n,m,0)}&=\\\\pi _{(n-m+1,1,0)}[\\\\delta (1-P_3)]^{m-1} + (m-1)t_{n-m}\\\\eta P_3[\\\\delta (1-P_3)]^{m-1} \\\\\\\\&= \\\\left\\\\lbrace \\\\pi _{(n-m,0,0)}\\\\eta +t_{n-m}\\\\eta P_3\\\\right\\\\rbrace [\\\\delta (1-P_3)]^{m-1} + (m-1)t_{n-m}\\\\eta P_3[\\\\delta (1-P_3)]^{m-1} \\\\\\\\&=\\\\pi _{(n-m,0,0)}\\\\eta [\\\\delta (1-P_3)]^{m-1} + m t_{n-m}\\\\eta P_3[\\\\delta (1-P_3)]^{m-1}$ In (68), we have used the fourth line in (63) to represent $\\\\pi _{(n-m+1,1,0)}$ with $\\\\pi _{(n-m,0,0)}$ and $t_{n-m}$ .\",\n \"Notice that equation (69) is valid for all $n>m\\\\ge 1$ .\",\n \"Substituting equation (69), the summation within the fifth line of (63) can be transformed as follows.\",\n \"$\\\\sum \\\\nolimits _{k=n}^{\\\\infty }\\\\pi _{(k,n-1,0)}&=\\\\sum \\\\nolimits _{k=n}^{\\\\infty }\\\\Big \\\\lbrace \\\\pi _{(k-n+1,0,0)}\\\\eta [\\\\delta (1-P_3)]^{n-2} +(n-1)t_{k-n+1}\\\\eta P_3[\\\\delta (1-P_3)]^{n-2} \\\\Big \\\\rbrace \\\\\\\\&= \\\\left(\\\\sum \\\\nolimits _{k=1}^{\\\\infty }\\\\pi _{(k,0,0)}\\\\right)\\\\eta [\\\\delta (1-P_3)]^{n-2} + \\\\left(\\\\sum \\\\nolimits _{k=1}^{\\\\infty }t_{k}\\\\right) \\\\eta P_3(n-1)[\\\\delta (1-P_3)]^{n-2} $ Since $t_n$ denotes the probability that the middle parameter equals $n$ , we have the relation $t_0 + \\\\sum \\\\nolimits _{n=1}^{\\\\infty }t_n =1 $ in which $t_0=\\\\sum \\\\nolimits _{k=1}^{\\\\infty }\\\\pi _{(k,0,0)}=S $ is the probability that the middle parameter takes value 0, which is represented by $S$ in following paragraphs.\",\n \"Combining above results, for $n\\\\ge 2$ we have that $\\\\pi _{(n,0,0)}&=\\\\pi _{(n-1,0,0)}\\\\delta + \\\\Big \\\\lbrace S\\\\eta [\\\\delta (1-P_3)]^{n-2} + (1-S)\\\\eta P_3(n-1)[\\\\delta (1-P_3)]^{n-2}\\\\Big \\\\rbrace \\\\delta P_3$ which gives a recursive formula for $\\\\pi _{(n,0,0)}$ .\",\n \"Before the general expression of $\\\\pi _{(n,0,0)}$ is given, we first determine $S$ .\",\n \"From $n=2$ to $\\\\infty $ , adding up both sides of equation (70) yields that $S-\\\\pi _{(1,0,0)}=S\\\\delta +\\\\frac{S\\\\delta \\\\eta P_3}{1-\\\\delta (1-P_3)}+\\\\frac{(1-S)\\\\delta \\\\eta P_3^2}{[1-\\\\delta (1-P_3)]^2}$ Since we have known that $\\\\pi _{(1,0,0)}=pP_1$ , from (71) we can derive $S=\\\\frac{pP_1[1-\\\\delta (1-P_3)]^2+\\\\delta \\\\eta P_3^2}{(1-\\\\delta )[1-\\\\delta (1-P_3)]^2-\\\\delta \\\\eta P_3(1-\\\\delta )(1-P_3)}$ Applying equation (70) iteratively, we have $\\\\pi _{(n,0,0)}&=\\\\pi _{(1,0,0)}\\\\delta ^{n-1}+S\\\\delta \\\\eta P_3\\\\sum \\\\nolimits _{j=0}^{n-2}\\\\delta ^j[\\\\delta (1-P_3)]^{n-2-j} \\\\\\\\&\\\\qquad +(1-S)\\\\delta \\\\eta P_3^2\\\\sum \\\\nolimits _{j=0}^{n-2}\\\\delta ^j(n-1-j)[\\\\delta (1-P_3)]^{n-2-j} \\\\\\\\&= \\\\left(pP_1+\\\\eta \\\\right)\\\\delta ^{n-1}-\\\\eta [\\\\delta (1-P_3)]^{n-1} - (1-S)\\\\eta P_3(n-1)[\\\\delta (1-P_3)]^{n-1}$ in which we have substituted $\\\\pi _{(1,0,0)}=pP_1$ and omit some intermediate calculation details.\",\n \"Equation (73) is valid for $n=1$ can be verified directly.\",\n \"In order to determine probabilities $\\\\pi _{(n,m,0)}$ by (69), we next derive the general expression of $t_n$ .\",\n \"For $n\\\\ge 2$ , it shows that $t_n&=\\\\sum \\\\nolimits _{k=n+1}^{\\\\infty }\\\\sum \\\\nolimits _{j=0}^{n-1}\\\\pi _{(k,n,j)} \\\\\\\\&=\\\\sum \\\\nolimits _{k=n+1}^{\\\\infty }\\\\pi _{(k,n,0)}+\\\\sum \\\\nolimits _{k=n+1}^{\\\\infty }\\\\sum \\\\nolimits _{j=1}^{n-1}\\\\pi _{(k,n,j)} \\\\\\\\&=\\\\sum \\\\nolimits _{k=n+1}^{\\\\infty }\\\\Big \\\\lbrace \\\\pi _{(k-n,0,0)}\\\\eta [\\\\delta (1-P_3)]^{n-1}+n t_{k-n}\\\\eta P_3[\\\\delta (1-P_3)]^{n-1}\\\\Big \\\\rbrace \\\\\\\\& \\\\qquad +\\\\sum \\\\nolimits _{k=n+1}^{\\\\infty }\\\\sum \\\\nolimits _{j=1}^{n-1}\\\\left(\\\\sum \\\\nolimits _{y=0}^{n-j-1}\\\\pi _{(k-j,n-j,y)}\\\\right)\\\\eta (1-P_3) [\\\\delta (1-P_3)]^{j-1} \\\\\\\\&=S\\\\eta [\\\\delta (1-P_3)]^{n-1}+(1-S)\\\\eta P_3n[\\\\delta (1-P_3)]^{n-1} \\\\\\\\&\\\\qquad + \\\\eta (1-P_3)\\\\sum \\\\nolimits _{j=1}^{n-1}t_{n-j}[\\\\delta (1-P_3)]^{j-1}$ where in (74) we have used equations (69) and (64).\",\n \"To obtain (75), notice that $\\\\sum \\\\nolimits _{k=n+1}^{\\\\infty }\\\\sum \\\\nolimits _{y=0}^{n-j-1}\\\\pi _{(k-j,n-j,y)}=\\\\sum \\\\nolimits _{k=n-j+1}^{\\\\infty }\\\\sum \\\\nolimits _{y=0}^{n-j-1}\\\\pi _{(k,n-j,y)}= t_{n-j} $ Similarly, we also have $t_{n-1}&=S\\\\eta [\\\\delta (1-P_3)]^{n-2}+(1-S)\\\\eta P_3(n-1)[\\\\delta (1-P_3)]^{n-2} \\\\\\\\& \\\\quad + \\\\eta (1-P_3)\\\\sum \\\\nolimits _{j=1}^{n-2}t_{n-1-j}[\\\\delta (1-P_3)]^{j-1}$ Compute the following difference $t_n-t_{n-1}\\\\delta (1-P_3)=(1-S)\\\\eta P_3[\\\\delta (1-P_3)]^{n-1}+t_{n-1}\\\\eta (1-P_3) $ which is equivalent to $t_n=t_{n-1}(\\\\delta +\\\\eta )(1-P_3)+(1-S)\\\\eta P_3[\\\\delta (1-P_3)]^{n-1}$ Applying equation (77) repeatedly yields that $t_n&=t_1 [(\\\\delta +\\\\eta )(1-P_3)]^{n-1} +(1-S)\\\\eta P_3 \\\\sum \\\\nolimits _{j=0}^{n-2}[(\\\\delta +\\\\eta )(1-P_3)]^j[\\\\delta (1-P_3)]^{n-1-j} \\\\\\\\&=t_1 [(\\\\delta +\\\\eta )(1-P_3)]^{n-1} +(1-S)\\\\delta P_3 \\\\left\\\\lbrace [(\\\\delta +\\\\eta )(1-P_3)]^{n-1} - [\\\\delta (1-P_3)]^{n-1} \\\\right\\\\rbrace $ Using the general expression (69), we have $t_1=\\\\sum \\\\nolimits _{k=2}^{\\\\infty }\\\\pi _{(k,1,0)} =\\\\sum \\\\nolimits _{k=2}^{\\\\infty } \\\\left\\\\lbrace \\\\pi _{(k-1,0,0)}\\\\eta + t_{k-1}\\\\eta P_3 \\\\right\\\\rbrace =S\\\\eta +(1-S)\\\\eta P_3$ Combining (78) and (79), eventually we derive that $t_n&=\\\\left\\\\lbrace S\\\\eta +(1-S)(\\\\delta +\\\\eta )P_3\\\\right\\\\rbrace [(\\\\delta +\\\\eta )(1-P_3)]^{n-1} - (1-S)\\\\delta P_3[\\\\delta (1-P_3)]^{n-1} \\\\\\\\&=\\\\widetilde{S}[(\\\\delta +\\\\eta )(1-P_3)]^{n-1} - (1-S)\\\\delta P_3[\\\\delta (1-P_3)]^{n-1}$ where we denote $\\\\widetilde{S}=S\\\\eta +(1-S)(\\\\delta +\\\\eta )P_3 $ Provided equations (73) and (80), now the probabilities $\\\\pi _{(n,m,0)}$ can be determined by (69).\",\n \"$\\\\pi _{(n,m,0)}&= \\\\pi _{(n-m,0,0)}\\\\eta [\\\\delta (1-P_3)]^{m-1}+ m t_{n-m}\\\\eta P_3[\\\\delta (1-P_3)]^{m-1} \\\\\\\\&=\\\\bigg \\\\lbrace \\\\left(pP_1+\\\\eta \\\\right)\\\\delta ^{n-m-1}-\\\\eta [\\\\delta (1-P_3)]^{n-m-1} \\\\\\\\& \\\\quad -(1-S)\\\\eta P_3 (n-m-1)[\\\\delta (1-P_3)]^{n-m-1}\\\\bigg \\\\rbrace \\\\eta [\\\\delta (1-P_3)]^{m-1} \\\\\\\\& \\\\quad +\\\\left\\\\lbrace \\\\widetilde{S}[(\\\\delta +\\\\eta )(1-P_3)]^{n-m-1} - (1-S)\\\\delta P_3 [\\\\delta (1-P_3)]^{n-m-1} \\\\right\\\\rbrace \\\\eta P_3 m[\\\\delta (1-P_3)]^{m-1} \\\\\\\\&= \\\\eta (pP_1+\\\\eta )\\\\delta ^{n-2}(1-P_3)^{m-1}-\\\\eta ^2[\\\\delta (1-P_3)]^{n-2} - (1-S)\\\\eta ^2 P_3(n-m-1)[\\\\delta (1-P_3)]^{n-2} \\\\\\\\& \\\\quad + \\\\widetilde{S}\\\\eta P_3[(\\\\delta +\\\\eta )(1-P_3)]^{n-2}m\\\\left(\\\\frac{\\\\delta }{\\\\delta +\\\\eta }\\\\right)^{m-1} - (1-S)\\\\delta \\\\eta P_3^2 m[\\\\delta (1-P_3)]^{n-2}$ At last, the probabilities $\\\\pi _{(n,m,l)}$ , $n>m>l\\\\ge 1$ remains to be determined.\",\n \"Since $\\\\pi _{(n,m,l)}=\\\\pi _{(n-l+1,m-l+1,1)}[\\\\delta (1-P_3)]^{l-1} $ thus it suffices to find all the probabilities $\\\\pi _{(n,m,1)}$ for all the tuples $(n,m)$ .\",\n \"From the general formula (64), we have $\\\\pi _{(n,m,1)}&=\\\\left(\\\\sum \\\\nolimits _{j=0}^{m-2}\\\\pi _{(n-1,m-1,j)}\\\\right)\\\\eta (1-P_3) \\\\\\\\&=\\\\left(\\\\pi _{(n-1,m-1,0)}+\\\\sum \\\\nolimits _{j=1}^{m-2}\\\\pi _{(n-1,m-1,j)}\\\\right)\\\\eta (1-P_3) \\\\\\\\&=\\\\pi _{(n-1,m-1,0)}\\\\eta (1-P_3) + \\\\left(\\\\sum \\\\nolimits _{j=1}^{m-2}\\\\pi _{(n-j,m-j,1)}[\\\\delta (1-P_3)]^{j-1}\\\\right)\\\\eta (1-P_3) $ Do once iteration, we can obtain the expansion expression of probability $\\\\pi _{(n-1,m-1,1)}$ .\",\n \"The difference $&\\\\pi _{(n,m,1)}-\\\\pi _{(n-1,m-1,1)}\\\\delta (1-P_3) \\\\\\\\={}& \\\\left\\\\lbrace \\\\pi _{(n-1,m-1,0)}-\\\\pi _{(n-2,m-2,0)}\\\\delta (1-P_3)\\\\right\\\\rbrace \\\\eta (1-P_3) +\\\\pi _{(n-1,m-1,1)}\\\\eta (1-P_3) \\\\\\\\={}&t_{n-m}\\\\eta ^2 P_3(1-P_3)[\\\\delta (1-P_3)]^{m-2}+\\\\pi _{(n,m,1)}\\\\eta (1-P_3)$ gives the following recursive relation $\\\\pi _{(n,m,1)}=\\\\pi _{(n-1,m-1,1)}(\\\\delta +\\\\eta )(1-P_3) + t_{n-m}\\\\eta ^2 P_3(1-P_3)[\\\\delta (1-P_3)]^{m-2}$ In (82), we have substituted the probability expression (69).\",\n \"Then, the general formula of $\\\\pi _{(n,m,1)}$ can be derived by applying (83) repeatedly.\",\n \"It shows that $\\\\pi _{(n,m,1)}&=\\\\pi _{(n-m+2,2,1)}[(\\\\delta +\\\\eta )(1-P_3)]^{m-2} \\\\\\\\&\\\\quad + \\\\eta ^2P_3(1-P_3)t_{n-m}\\\\sum \\\\nolimits _{j=0}^{m-3}[(\\\\delta +\\\\eta )(1-P_3)]^j[\\\\delta (1-P_3)]^{m-2-j} \\\\\\\\&=\\\\pi _{(n-m+2,2,1)}[(\\\\delta +\\\\eta )(1-P_3)]^{m-2} \\\\\\\\&\\\\quad + \\\\delta \\\\eta P_3(1-P_3)t_{n-m} \\\\left\\\\lbrace [(\\\\delta +\\\\eta )(1-P_3)]^{m-2}-[\\\\delta (1-P_3)]^{m-2}\\\\right\\\\rbrace $ in which $\\\\pi _{(n-m+2,2,1)}&= \\\\pi _{(n-m+1,1,0)}\\\\eta (1-P_3) \\\\\\\\&=\\\\left\\\\lbrace \\\\pi _{(n-m,0,0)}\\\\eta +t_{n-m}\\\\eta P_3\\\\right\\\\rbrace \\\\eta (1-P_3) \\\\\\\\&=\\\\pi _{(n-m,0,0)}\\\\eta ^2(1-P_3) + t_{n-m}\\\\eta ^2 P_3(1-P_3)$ Combining equations (85) and (84) and merging the same terms gives $\\\\pi _{(n,m,1)}&=\\\\pi _{(n-m,0,0)}\\\\eta ^2(1-P_3)[(\\\\delta +\\\\eta )(1-P_3)]^{m-2} \\\\\\\\&\\\\qquad + \\\\eta P_3 t_{n-m} \\\\left\\\\lbrace [(\\\\delta +\\\\eta )(1-P_3)]^{m-1}-[\\\\delta (1-P_3)]^{m-1}\\\\right\\\\rbrace $ Since both $\\\\pi _{(n-m,0,0)}$ and $t_{n-m}$ have been obtained, by substituting (73) and (80), we show that the final result is $\\\\pi _{(n,m,1)}&= \\\\eta ^2(1-P_3)\\\\bigg \\\\lbrace (pP_1+\\\\eta )\\\\delta ^{n-3}\\\\left(\\\\frac{(\\\\delta +\\\\eta )(1-P_3)}{\\\\delta }\\\\right)^{m-2} - \\\\eta [\\\\delta (1-P_3)]^{n-3}\\\\left(\\\\frac{\\\\delta +\\\\eta }{\\\\delta }\\\\right)^{m-2} \\\\\\\\& \\\\quad -(1-S)\\\\eta P_3(n-m-1)[\\\\delta (1-P_3)]^{n-3}\\\\left(\\\\frac{\\\\delta +\\\\eta }{\\\\delta }\\\\right)^{m-2} \\\\bigg \\\\rbrace \\\\\\\\& \\\\quad +\\\\eta P_3\\\\widetilde{S}[(\\\\delta +\\\\eta )(1-P_3)]^{n-2} - \\\\eta P_3\\\\widetilde{S}[(\\\\delta +\\\\eta )(1-P_3)]^{n-2}\\\\left(\\\\frac{\\\\delta }{\\\\delta +\\\\eta }\\\\right)^{m-1} \\\\\\\\& \\\\quad - \\\\delta \\\\eta P_3^2(1-S)[\\\\delta (1-P_3)]^{n-2}\\\\left(\\\\frac{\\\\delta +\\\\eta }{\\\\delta }\\\\right)^{m-1} + \\\\delta \\\\eta P_3^2(1-S)[\\\\delta (1-P_3)]^{n-2}$ For $n>m>l\\\\ge 1$ , the last step is $\\\\pi _{(n,m,l)}&=\\\\pi _{(n-l+1,m-l+1,1)}[\\\\delta (1-P_3)]^{l-1} \\\\\\\\&=\\\\eta ^2(1-P_3)\\\\bigg \\\\lbrace \\\\left(pP_1+\\\\eta \\\\right)\\\\delta ^{n-3}\\\\left(\\\\frac{(\\\\delta +\\\\eta )(1-P_3)}{\\\\delta }\\\\right)^{m-2} \\\\left(\\\\frac{\\\\delta }{\\\\delta +\\\\eta }\\\\right)^{l-1} \\\\\\\\& \\\\qquad -\\\\eta [\\\\delta (1-P_3)]^{n-3}\\\\left(\\\\frac{\\\\delta +\\\\eta }{\\\\delta }\\\\right)^{m-2} \\\\left(\\\\frac{\\\\delta }{\\\\delta +\\\\eta }\\\\right)^{l-1} -(1-S)\\\\eta P_3(n-m-1) \\\\\\\\& \\\\qquad \\\\times [\\\\delta (1-P_3)]^{n-3}\\\\left(\\\\frac{\\\\delta +\\\\eta }{\\\\delta }\\\\right)^{m-2}\\\\left(\\\\frac{\\\\delta }{\\\\delta +\\\\eta }\\\\right)^{l-1} \\\\bigg \\\\rbrace \\\\\\\\& \\\\quad + \\\\eta P_3\\\\widetilde{S}[(\\\\delta +\\\\eta )(1-P_3)]^{n-2}\\\\left(\\\\frac{\\\\delta }{\\\\delta +\\\\eta }\\\\right)^{l-1} - \\\\eta P_3\\\\widetilde{S}[(\\\\delta +\\\\eta )(1-P_3)]^{n-2}\\\\left(\\\\frac{\\\\delta }{\\\\delta +\\\\eta }\\\\right)^{m-1} \\\\\\\\&\\\\quad -\\\\delta \\\\eta P_3^2(1-S)[\\\\delta (1-P_3)]^{n-2}\\\\left(\\\\frac{\\\\delta +\\\\eta }{\\\\delta }\\\\right)^{m-1} \\\\left(\\\\frac{\\\\delta }{\\\\delta +\\\\eta }\\\\right)^{l-1} \\\\\\\\&\\\\quad + \\\\delta \\\\eta P_3^2(1-S)[\\\\delta (1-P_3)]^{n-2}$ Collecting the results in equations (73), (81) and (87), we show that all the stationary probabilities of stochastic process $AoI_B^P$ are determined.\",\n \"This completes the proof of Theorem 8.\"\n ],\n [\n \"Proof of Theorem 9\",\n \"The stationary distribution of AoI, $\\\\Delta _B^P$ , is calculated in this Appendix.\",\n \"According to formula (15), for $n\\\\ge 3$ , we have $\\\\Pr \\\\lbrace \\\\Delta _B^P=n\\\\rbrace = \\\\pi _{(n,0,0)}+\\\\sum \\\\nolimits _{m=1}^{n-1}\\\\pi _{(n,m,0)}+\\\\sum \\\\nolimits _{m=2}^{n-1}\\\\sum \\\\nolimits _{l=1}^{m-1}\\\\pi _{(n,m,l)} $ where $\\\\pi _{(n,0,0)}$ is given in (16).\",\n \"Substituting expression (17), the first sum is computed as $&\\\\sum \\\\nolimits _{m=1}^{n-1}\\\\pi _{(n,m,0)} \\\\\\\\={}&\\\\sum \\\\nolimits _{m=1}^{n-1}\\\\Big \\\\lbrace \\\\eta \\\\left(pP_1+\\\\eta \\\\right)\\\\delta ^{n-2}\\\\left(1-P_3\\\\right)^{m-1}-\\\\eta ^2\\\\left[\\\\delta (1-P_3)\\\\right]^{n-2} - (1-S)\\\\eta ^2P_3(n-m-1) \\\\\\\\{}& \\\\times \\\\left[\\\\delta (1-P_3)\\\\right]^{n-2} +\\\\widetilde{S}\\\\eta P_3 \\\\left[(\\\\delta +\\\\eta )(1-P_3)\\\\right]^{n-2} m \\\\left(\\\\frac{\\\\delta }{\\\\delta +\\\\eta }\\\\right)^{m-1} - (1-S)\\\\delta \\\\eta P_3^2 m[\\\\delta (1-P_3)]^{n-2} \\\\Big \\\\rbrace \\\\\\\\={}&\\\\eta \\\\left(pP_1+\\\\eta \\\\right)\\\\delta ^{n-2} \\\\frac{1-(1-P_3)^{n-1}}{P_3} - \\\\eta ^2(n-1)\\\\left[\\\\delta (1-P_3)\\\\right]^{n-2} - (1-S)\\\\eta ^2P_3 \\\\\\\\{}& \\\\times \\\\frac{(n-1)(n-2)}{2}\\\\left[\\\\delta (1-P_3)\\\\right]^{n-2} +\\\\widetilde{S}\\\\eta P_3 \\\\left[(\\\\delta +\\\\eta )(1-P_3)\\\\right]^{n-2}\\\\frac{(\\\\delta +\\\\eta )^2}{\\\\eta ^2} \\\\\\\\{}&\\\\times \\\\left[1- \\\\left(\\\\frac{\\\\delta }{\\\\delta +\\\\eta }\\\\right)^{n-1}- \\\\frac{\\\\eta }{\\\\delta +\\\\eta }(n-1) \\\\left(\\\\frac{\\\\delta }{\\\\delta +\\\\eta }\\\\right)^{n-1} \\\\right] -(1-S)\\\\delta \\\\eta P_3^2 \\\\frac{n(n-1)}{2}[\\\\delta (1-P_3)]^{n-2}$ We omit the further calculations and directly show that $\\\\sum \\\\nolimits _{m=1}^{n-1}\\\\pi _{(n,m,0)}&=\\\\frac{\\\\eta (pP_1+\\\\eta )}{\\\\delta P_3}\\\\delta ^{n-1} + \\\\frac{\\\\widetilde{S}(\\\\delta +\\\\eta )P_3}{\\\\eta (1-P_3)}[(\\\\delta +\\\\eta )(1-P_3)]^{n-1} \\\\\\\\&\\\\qquad - \\\\left\\\\lbrace \\\\frac{\\\\eta (pP_1+\\\\eta )}{\\\\delta P_3}+\\\\frac{\\\\widetilde{S}\\\\delta P_3}{\\\\eta (1-P_3)}-\\\\frac{\\\\eta ^2-(1-S)\\\\eta ^2 P_3}{\\\\delta (1-P_3)}\\\\right\\\\rbrace [\\\\delta (1-P_3)]^{n-1} \\\\\\\\&\\\\qquad - \\\\left\\\\lbrace \\\\frac{\\\\eta ^2-(1-S)\\\\eta ^2P_3}{\\\\delta (1-P_3)}+\\\\frac{\\\\widetilde{S}\\\\eta P_3}{\\\\eta (1-P_3)} \\\\right\\\\rbrace n[\\\\delta (1-P_3)]^{n-1} \\\\\\\\&\\\\qquad -\\\\frac{(1-S)\\\\eta P_3(\\\\delta P_3+\\\\eta )}{2\\\\delta (1-P_3)}n(n-1)[\\\\delta (1-P_3)]^{n-1}$ At last, $&\\\\sum \\\\nolimits _{m=2}^{n-1}\\\\sum \\\\nolimits _{l=1}^{m-1}\\\\pi _{(n,m,l)} \\\\\\\\={}&\\\\sum \\\\nolimits _{m=2}^{n-1}\\\\sum \\\\nolimits _{l=1}^{m-1} \\\\text{Equation (18)} \\\\\\\\={}&\\\\sum \\\\nolimits _{m=2}^{n-1}\\\\Bigg \\\\lbrace \\\\eta (pP_1+\\\\eta )\\\\delta ^{n-2} \\\\left[\\\\left(\\\\frac{(\\\\delta +\\\\eta )(1-P_3)}{\\\\delta }\\\\right)^{m-1}-(1-P_3)^{m-1}\\\\right] \\\\\\\\{}&\\\\qquad - \\\\eta ^2[\\\\delta (1-P_3)]^{n-2}\\\\left[\\\\left(\\\\frac{\\\\delta +\\\\eta }{\\\\delta }\\\\right)^{m-1} -1 \\\\right] \\\\\\\\{}&\\\\qquad \\\\quad -(1-S)\\\\eta ^2 P_3 [\\\\delta (1-P_3)]^{n-2} (n-m-1)\\\\left[\\\\left(\\\\frac{\\\\delta +\\\\eta }{\\\\delta }\\\\right)^{m-1} -1\\\\right] \\\\Bigg \\\\rbrace \\\\\\\\{}&\\\\quad +\\\\widetilde{S}(\\\\delta +\\\\eta )P_3[(\\\\delta +\\\\eta )(1-P_3)]^{n-2}\\\\left[1-\\\\left(\\\\frac{\\\\delta }{\\\\delta +\\\\eta }\\\\right)^{m-1} \\\\right] \\\\\\\\{}&\\\\quad -\\\\widetilde{S}\\\\eta P_3[(\\\\delta +\\\\eta )(1-P_3)]^{n-2}(m-1)\\\\left(\\\\frac{\\\\delta }{\\\\delta +\\\\eta }\\\\right)^{m-1} \\\\\\\\{}&\\\\quad -(1-S)\\\\delta (\\\\delta +\\\\eta )P_3^2[\\\\delta (1-P_3)]^{n-2}\\\\left[\\\\left(\\\\frac{\\\\delta +\\\\eta }{\\\\delta }\\\\right)^{m-1}-1 \\\\right] \\\\\\\\{}&\\\\quad +(1-S)\\\\delta \\\\eta P_3^2(m-1)[\\\\delta (1-P_3)]^{n-2} \\\\\\\\={}&\\\\eta (pP_1+\\\\eta )\\\\delta ^{n-2} \\\\bigg [ \\\\frac{\\\\eta (1-P_3)}{[\\\\delta -(\\\\delta +\\\\eta )(1-P_3)]P_3} \\\\\\\\{}& \\\\quad - \\\\frac{\\\\delta }{\\\\delta -(\\\\delta +\\\\eta )(1-P_3)} \\\\left(\\\\frac{(\\\\delta +\\\\eta )(1-P_3)}{\\\\delta }\\\\right)^{n-1} +\\\\frac{(1-P_3)^{n-1}}{P_3}\\\\bigg ] \\\\\\\\{}&-\\\\eta ^2[\\\\delta (1-P_3)]^{n-2}\\\\left[ \\\\frac{\\\\delta }{\\\\eta }\\\\left(\\\\frac{\\\\delta +\\\\eta }{\\\\delta }\\\\right)^{n-1}-\\\\frac{\\\\delta +\\\\eta }{\\\\eta }-(n-2) \\\\right] \\\\\\\\{}&-(1-S)\\\\eta ^2 P_3 [\\\\delta (1-P_3)]^{n-2}\\\\bigg [-(n-2)\\\\frac{\\\\delta +\\\\eta }{\\\\eta } \\\\\\\\{}& \\\\quad + \\\\frac{\\\\delta ^2}{\\\\eta ^2}\\\\left(\\\\frac{\\\\delta +\\\\eta }{\\\\delta }\\\\right)^{n-1}-\\\\frac{\\\\delta (\\\\delta +\\\\eta )}{\\\\eta ^2} -\\\\frac{(n-2)(n-3)}{2}\\\\bigg ] \\\\\\\\{}&+\\\\widetilde{S}(\\\\delta +\\\\eta )P_3[(\\\\delta +\\\\eta )(1-P_3)]^{n-2} \\\\left[(n-2)-\\\\frac{\\\\delta }{\\\\eta }+\\\\frac{\\\\delta +\\\\eta }{\\\\eta } \\\\left( \\\\frac{\\\\delta }{\\\\delta +\\\\eta }\\\\right)^{n-1} \\\\right] \\\\\\\\{}&-\\\\widetilde{S}\\\\eta P_3[(\\\\delta +\\\\eta )(1-P_3)]^{n-2}\\\\bigg [ \\\\frac{\\\\delta (\\\\delta +\\\\eta )}{\\\\eta ^2} -\\\\frac{(\\\\delta +\\\\eta )^2}{\\\\eta ^2} \\\\left(\\\\frac{\\\\delta }{\\\\delta +\\\\eta }\\\\right)^{n-1}-\\\\frac{\\\\delta +\\\\eta }{\\\\eta }(n-2)\\\\left(\\\\frac{\\\\delta }{\\\\delta +\\\\eta }\\\\right)^{n-1} \\\\bigg ] \\\\\\\\{}&-(1-S)\\\\delta (\\\\delta +\\\\eta )P_3^2[\\\\delta (1-P_3)]^{n-2} \\\\left[ \\\\frac{\\\\delta }{\\\\eta }\\\\left(\\\\frac{\\\\delta +\\\\eta }{\\\\delta }\\\\right)^{n-1}-\\\\frac{\\\\delta +\\\\eta }{\\\\eta }-(n-2) \\\\right] \\\\\\\\{}&+(1-S)\\\\delta \\\\eta P_3^2\\\\frac{(n-1)(n-2)}{2}[\\\\delta (1-P_3)]^{n-2}$ Collecting all the results obtained in (16), (89) and (90), after careful calculations and numerical verification, we derive that $&\\\\Pr \\\\lbrace \\\\Delta _B^P=n\\\\rbrace \\\\\\\\={}& \\\\frac{(\\\\delta +\\\\eta )P_3(pP_1+\\\\eta )}{\\\\delta -(\\\\delta +\\\\eta )(1-P_3)}\\\\delta ^{n-1} +\\\\frac{P_3[\\\\eta (\\\\delta +\\\\eta )+(1-S)(\\\\delta +\\\\eta )P_3(2\\\\delta -\\\\eta )]}{\\\\eta (1-P_3)}[\\\\delta (1-P_3)]^{n-1} \\\\\\\\{}& -\\\\bigg \\\\lbrace \\\\frac{pP_1\\\\eta (1-P_3)+\\\\delta \\\\eta P_3}{[\\\\delta -(\\\\delta +\\\\eta )(1-P_3)](1-P_3)} +\\\\frac{(1-S)\\\\delta P_3[\\\\eta +(\\\\delta +\\\\eta )P_3]+\\\\widetilde{S}(\\\\delta +\\\\eta )P_3}{\\\\eta (1-P_3)} \\\\bigg \\\\rbrace [(\\\\delta +\\\\eta )(1-P_3)]^{n-1} \\\\\\\\{}& +\\\\frac{(1-S)(\\\\delta +\\\\eta )P_3^2}{1-P_3}n[\\\\delta (1-P_3)]^{n-1} +\\\\frac{\\\\widetilde{S}P_3}{1-P_3}n[(\\\\delta +\\\\eta )(1-P_3)]^{n-1} \\\\qquad (n\\\\ge 3)$ It can be checked directly that equation (89) is zero when $n=1$ , and (90) equals zero for both cases $n=1$ and $n=2$ .\",\n \"Therefore, the probability distribution (91) is in fact valid for all $n\\\\ge 1$ .\",\n \"This completes the proof of Theorem 9.\"\n ],\n [\n \"Proof of Corollary 1\",\n \"Notice that for $0}0$ , or have a anti-clockwise rotation if ${\\phi _0}{<}0$ and have no rotation if ${\\phi _0}{=}0$ .","We consider the clockwise rotating vorticity profile has circular symmetry with parameters $a=b=1$ , amplitude ${\\phi _0}=1$ , sharp cutoff at radial distance $|r|$ =6 units away from the centre of the vortex $(0,0)$ .","This vorticity profile (yellow color) is shown in the following Fig.","REF (a) which has circular interface with surrounding stagnant fluid at $t=0$ .","The simulation region is a square box of length 12$\\pi $ units.","Figure: Initial rotating sharp vorticity profiles at t = 0 to study the KH instability.","The color scale corresponding to a particular vorticity profile has been given in the following respective figure which shows its evolution.","The simulation region is a square box of length 12π\\pi units for all these systems.The panels of Fig.","REF shows time evolution of vorticity profile given in Fig.","REF (a) in the inviscid HD fluid.","The sharpness of the clockwise rotating vorticity profile generates a strong rotational sheared flow.","This sheared flow results in creation of small co-rotating (anti–clockwise) KH vortices (dark blue color vortices) at the vorticity interface $|r|=6$ .","These like-sign vortices start merging as rotation progresses that leads the fluid to evolve into an anisotropic isolated structure.","Figure: An inviscid HD fluid.","The time evolution of a sharp clockwise rotating vorticity which has circular interface with surrounding stagnant fluid.","The sharpness of the vorticity profile generates small KH vortices at the interface that results in a anisotropic isolated structure.In Fig.","REF where ${\\eta }=5$ and ${\\tau _m}=20$ , once the vortex begins to rotate, we observe a pair of ingoing and outgoing cylindrical shear waves from the interface along with these small like-sign KH vortices.","During the evolution, it is observe that both the waves carry the like-sign vortices which interact with themselves in order to merge and have interplay with these waves as well.","The fluid within the inner region ($|r| \\le 6$ ) favors mixing due to the ingoing waves, while the stationary fluid in the outer region ($|r| \\ge 6$ ) gets mixing due to the outgoing waves.","Thus, the TS waves assist the process of fluid mixing by convecting it inside and outside the vortex structure.","Unlike the inviscid case, here, the KH vortices are confined to the radial emitted waves.","Since the wavefronts are cylindrical in shapes, the interplay between waves and vortices leads the evolution of the VE fluid toward an isotropic structure.","Figure: Viscoelastic fluid with coupling parameters η=5{\\eta }=5, and τ m =20{\\tau _m}=20.","The time evolution of a sharp rotating vorticity which has circular interface with surrounding stagnant fluid.","The sharpness of the vorticity profile generates small KH vortices at the interface along with a pair of ingoing and outgoing wavefronts of TS waves that assist in fluid mixing by convecting it inside and outside the vortex structure.In Fig.","REF , we have simulated another case of VE fluid which has less coupling strength ($\\eta /\\tau _m$ =0.125) with ${\\eta }=2.5$ and ${\\tau _m}$ =20.","Unlike Fig.","REF , since the TS waves are not strong enough that they can dominant over the KH instability, less confinement of KH vortices to the waves.","This is evident from Fig.","REF where the evolution of medium towards an isotropic structure and mixing process are much slower in comparison to Fig.","REF .","Figure: The time evolution of a sharp rotating vorticity in VE fluid with coupling parameters η=2.5{\\eta }=2.5, and τ m =20{\\tau _m}=20.","Due to the less coupling strength, the evolution of medium towards an isotropic structure and mixing process are much slower in comparison to Fig.",".In conclusion, as a result of greater the coupling strength or stronger TS wave, the medium evolution attempts to possess radial symmetry and shows better mixing.","The mixing is found minimal in inviscid fluid.","The single-circulation sharp profile carries single types of anti-clockwise rotating KH vortices across its only interface.","While a multi-circulation profile with two or more than two interfaces produces clockwise as well as anti-clockwise KH vortices.","Thus, in multi-circulation case apart from merging between like-sign vortices, the propagation of unlike-sign vortices as a dipolar structure also becomes important which can assist to increasing the spatial domain of mixing fluids."],["Multi-circulation vorticity shell profile","Here, we first consider the simplest case of multiple shells of vorticity having two flows: Inner core flow and a outer shell flow, both flows have reversal circulation.","The velocity profile for this configuration is given below.","${\\vec{v}_{0}} = \\left\\lbrace \\begin{array}{ll}v_{x0}=-{\\phi _0}{(y-y_c)};\\: v_{y0}={\\phi _0}{(x-x_c)} & \\quad |r|\\le 5 \\\\v_{x0}={\\phi _0}{(y-y_c)};\\: v_{y0}=-{\\phi _0}{(x-x_c)} & \\quad 5<|r|\\le 10 \\\\0 & \\quad \\mathrm {otherwise}{.","}\\end{array}\\right.$ The vorticity corresponding to the above velocity profile is given below, ${\\xi _{z0}} = \\left\\lbrace \\begin{array}{ll}2{\\phi _0} & \\quad |r| \\le 5 \\\\-2{\\phi _0} & \\quad 5 < |r| \\le 10 \\\\0 & \\quad \\mathrm {otherwise}{.","}\\end{array}\\right.$ With the parameter $\\phi _0$ =1, we consider both flows rotate with equal rotation rates and in opposite directions (Fig.","REF (b)).","Figure REF shows the evolution of this vorticity profile for an inviscid fluid.","At inner interface, the KH vortices rotate anti-clockwise (blue color vortices) while at outer interface rotate clockwise (red color vortices).","As time goes on, the merging between like-sign vortices take place at both the interfaces, and simultaneously growing closeness between both the interfaces due to the radial gradient in vorticity results in interactions between the counter-rotating (red-blue) vortices as well.","These counter-rotating vortices results in the formation of propagating dipolar structures which help to convect the fluid across the wider domain than the single interface (Fig.","REF ) which only favors the merging process.","Figure: An inviscid HD fluid.","Evolution of of vorticity having two circular sharp flows, inner core and an outer shell flows have reversal circulation, in time generates small KH vortices at each interface that results in a anisotropic isolated structure.In Fig.","REF , we observe that a pair of ingoing and outgoing wavefronts emanates from each of the two sharp interfaces of the vortex structure along with KH vortices (see the first panel).","It is observe that both the wavefronts at inner interface carry the blue color (rotating anti-clockwise) vortices while at outer interface carry red color (rotating clockwise) vortices.","During the emission of these shear waves like-sign vortices interact with themselves in order to get merge.","Figure: Viscoelastic fluid with coupling parameters η=5{\\eta }=5, and τ m =20{\\tau _m}=20.","The time evolution of vorticity having two circular sharp interfaces, inner core and outer shell have reversal circulation, generates small KH vortices at each interface along with a pair of ingoing and outgoing TS waves fronts that results in well fluid mixing by convecting it inside and outside the vortex structure.The stagnant fluid in the outermost region ($|r| \\ge 10$ ) undergoes mixing due to the outgoing wave from the outermost interface at 10, while the innermost vortex region undergoes mixing due to the ingoing wave emanating from the sharp interface located at 5.","Interestingly, the vortex region confined within the two sharp interfaces ($5 <|r|\\le 10$ ) undergoes mixing due to the ingoing wave from the outermost interface and the outgoing wave from the innermost interface.","This region is probably convection dominating region due to the higher possibility of formation of propagating dipolar structures.","As the results of multiple interaction processes the mixing becomes fast and efficient than the cases discussed so far.","Next, we consider the another VE fluid in Fig.","REF having lower coupling strength i.e.","${\\eta }=2.5, {\\tau _m}=20, v_p$ =0.35, it is observed that at each time step in the spatial confinement of this medium is lower than Fig.","REF .","Moreover, the comparison manifests that the mixing and evolution symmetry of a medium are proportional to the coupling strength of that medium as observed for earlier cases.","Figure: Viscoelastic fluid with coupling parameters η=2.5{\\eta }=2.5, and τ m =20{\\tau _m}=20.","The time evolution of vorticity having two circular sharp interfaces, both have reversal circulation.","Due to the less coupling strength the spatial confinement of this medium is lower than Fig.",".Next, in order to make a better comparative numerical analysis about the mixing rate, we have considered a more complex scenario of multiple circulations (each consecutive one having a reversal in its circulation) having the following velocity flow profile: ${\\vec{v}_{0}}= \\left\\lbrace \\begin{array}{ll}v_{x0}=-{\\phi _0}{(y-y_c)};\\: v_{y0}={\\phi _0}{(x-x_c)} & \\quad |r|\\le 2.5 \\\\v_{x0}={\\phi _0}{(y-y_c)};\\: v_{y0}=-{\\phi _0}{(x-x_c)} & \\quad 2.5<|r|\\le 5 \\\\v_{x0}=-{\\phi _0}{(y-y_c)};\\: v_{y0}={\\phi _0}{(x-x_c)} & \\quad 5<|r|\\le 7.5 \\\\v_{x0}={\\phi _0}{(y-y_c)};\\: v_{y0}=-{\\phi _0}{(x-x_c)} & \\quad 7.5<|r|\\le 10 \\\\0 & \\quad \\mathrm {otherwise}{.","}\\end{array}\\right.$ The vorticity corresponding to the above velocity profile is given below, ${\\xi _{z0}} = \\left\\lbrace \\begin{array}{ll}2{\\phi _0} & \\quad |r| \\le 2.5 \\\\-2{\\phi _0} & \\quad 2.5 < |r| \\le 5 \\\\2{\\phi _0} & \\quad 5 < |r| \\le 7.5 \\\\-2{\\phi _0} & \\quad 5 < |r| \\le 10 \\\\0 & \\quad \\mathrm {otherwise}{.","}\\end{array}\\right.$ For the parameter $\\phi _0$ =1, the initial vorticity profile is shown in Fig.","REF (c).","The complexity of this motion of multi-circulation structure is evident from the subplots of Fig.","REF for inviscid fluid.","In initial time period, the vortices of KH instability develop across the interface of each shell.","At intermediate time range, this evolution provides a complete picture of a turbulent flow throughout the entire system which is collection of several small symmetric and non-symmetric vortices.","The transport phenomena like merging between two co-rotating vortices, convection due to the evolution of dipolar (two counter-rotating ) vortices, collision between these dipolar vortices, and also the formation and evolution of triplor structures becomes more frequent than above discussed cases.","Figure: An inviscid HD fluid.","The time evolution of multi-circulation vorticity profile, each consecutive one having a reversal in its circulation, generates small KH vortices at each interface.","This evolution provides a complete picture of a turbulent flow which is collection of various kind of vortices, and exhibits transport properties like convection and merging.Figure REF , represents the evolution of same initial profile (Fig.","REF ) of vorticity for VE fluid with coupling parameters ${\\eta }=5$ , ${\\tau _m}$ =20.","From the comparative observations between Fig.","REF and Fig.","REF , it is interesting to notice that the presence of TS waves leads to the relaxing of the turbulent medium to a single vortex faster than in inviscid fluid.","Figure: The time evolution of multi-circulation vorticity profile in VE fluid with η=5{\\eta }=5, and τ m =20{\\tau _m}=20.","The strong interaction between KH vortices and TS waves results in the relaxing the medium into a single vortex faster than inviscid fluid.In Fig.","REF (${\\eta }=2.5, {\\tau _m}=20, v_p$ =0.35), since the TS waves are weaker than Fig.","REF , the less spatial confinement of the turbulent medium that results in slower merging process, due to which the relaxation time of the medium becomes longer.","Figure: The time evolution of multi-circulation vorticity profile in VE fluid with η=2.5{\\eta }=2.5, and τ m =20{\\tau _m}=20.","Since the TS waves are weaker than Fig.",", the relaxation time of the medium becomes longer.The comparison of Fig.","REF and Fig.","REF clearly displays these observations.","In Section we have stated that tracer particles with very low inertia follow the flow passively, while particles with very high inertia will remain almost unaffected by the medium fluctuations.","In between these two limits particles show the strongest response to the medium fluctuations.","The simulations are performed for all these three inertial particles: very low ($\\tau _s$ =0.05), intermediate ($\\tau _s$ =1), and very high inertia ($\\tau _s$ =50).","To observe the exclusive effect of inertia, we transport these particles through a similar smooth rotating vorticity vortex in an inviscid fluid.","An inviscid fluid has no source term which favors the emission of TS waves or dissipative term like viscosity.","And also, we choose the smooth rotating vorticity vortex which does not satisfy the KH destabilization condition anywhere in the vorticity patch.","The equation of such smooth rotating vorticity is given as ${\\xi _{0}}={\\Omega _0}exp\\left(-\\left({x^2+y^2}\\right)/{a^2_c}\\right){.","}$ The evolution of same structure given by Eq.","REF for ${a_c}$ =1.0, ${\\Omega _0}=5$ in an inviscid fluid is shown in Figs.","(REF ), (REF ), and (REF ).","From these figures it is clear that the rotating vortex keeps rotating without any change.","Figure: An inviscid HD fluid carries tracer particles are shown as red dots.","The low inertial tracers (τ s \\tau _s=0.05) follow the dynamics along the smooth rotating vortex.Now, in order to see the response of tracer particles, initially $(t=0)$ , we have distributed 900 inertial particles (shown by red dots) homogeneously throughout the domain.","Figure: An inviscid HD fluid carries tracer particles are shown as red dots.","The high inertial tracers (τ s \\tau _s=50) show negligible response to the rotating vortex.From Fig.","REF , it is clear that low inertial particles ($\\tau _s$ =0.05) follow the dynamics along the rotating vortex, and the particles with higher inertia $\\tau _s$ = 50 (Fig.","REF ) show negligible response to the vorticity gradient.","Figure: An inviscid HD fluid carries tracer particles are shown by red dots.","Smooth rotating vorticity vortex with tracers having intermediate inertia i.e τ s \\tau _s=1 show high response to the vorticity gradient, the particles are pushed away where the flow is strong enough and get accumulate in strain-dominated region.In comparison to previous cases (Fig.","REF and Fig.","REF ), Fig.","REF shows that the particles with intermediate value of $\\tau _s$ =1 counter a significant outward push.","It is because the particle and fluid time- scales are comparable which results the particles experience a notable centrifugal force due to vorticity gradient.","Since, the inertial particles are pushed away from regions where the flow is strong enough, these particles get accumulate in strain-dominated regions.","Thus, particles tend to leave regions of high vorticity and cluster into regions of high strain [61].","Note, we have simulated a range of intermediate inertial particles, and observed the same effect with varying outward push.","With the identification of intermediate inertial particles and understanding of their evolution, next we compare the evolution of intermediate inertial (typical, $\\tau _s$ =1) with non-inertial particles for the inviscid HD and VE ($\\eta =5$ , $\\tau _m=20$ ) fluids in the following Fig.","REF and Fig.","REF , respectively.","For this, we choose a sharp rotating vorticity profile which is given by Eq.","(REF ).","The middle row in Fig.","REF and Fig.","REF represent the time evolution of this profile for inviscid and VE fluids, respectively.","As the sharp vortex starts to rotate, produces larger strain (deformation) in medium along the interface that results in formation of KH instability.","The evolution of this rotor has already been discussed in the earlier Section REF for inviscid fluid in Fig.","REF and for VE fluid in Fig.","REF .","Initially $(t=0)$ , here, we distribute 3600 tracer particles homogeneously throughout these fluids.","Figure: An inviscid HD fluid advect tracer particles.","First and third row show the temporal and spatial distribution of non-inertial particles (shown by magenta dots) and inertial particles (τ s \\tau _s=1, shown by red dots), respectively, corresponding to the sharp vorticity profile evolution shown in second row.In both Figs.","REF and REF , the first and the third row visualize the distribution of non-inertial particles (shown by magenta dots) and inertial particles (shown by red dots) respectively, advect by their respective fluid flow shown in the middle row.","The accumulation of non-inertial particles is observed in rotation-dominated regions over the vortex structures.","While the inertial particles accumulate in strain-dominated regions along the interfaces.","Figure: Viscoelastic fluid with coupling parameters η=5{\\eta }=5, and τ m =20{\\tau _m}=20 advect tracer particles.","First and third row show the spatiotemporal distribution of non-inertial particles (shown by magenta dots) and inertial particles (τ s \\tau _s=1, shown by red dots), respectively, corresponding to the sharp vorticity profile evolution given in second row.This accumulation process leads to the spatial inhomogeneous distribution of particles.","This inhomogeneous distribution of the particles is known as clustering or preferential concentration.","Clustering is well-studied in the case of inertial particles [62], [31], [26], [27], [63], [64], [65], [66], and several studies are available for non-inertial particles [67].","In real flows the tracer particles always have some inertial value, so we compute the ensemble averaged MSD of intermediate inertial particles to analyze the diffusion of particles in the carrier VE fluid.","This diffusion of particles is associated with the mixing of the fluid.","For this, we advect the inertial particles having $\\tau _s$ =1 using the sharp rotating flows discussed above, inviscid fluid in Fig.","REF /Fig.","REF , VE fluid with $\\eta =5$ , $\\tau _m=20$ in Fig.","REF /Fig.","REF , and another VE fluid with $\\eta =2.5$ , $\\tau _m=20$ in Fig.","REF .","Initially t=0, we disperse 3600 tracer particles homogeneously over these fluids.","Figure: The MSD as function of time for tracing particles.","The sharp rotating vorticity advect the particles with (a) τ s \\tau _s=1, and (b) τ s \\tau _s=0.5.","At later time, the MSD is proportional to the coupling strength.Figure REF (a) compares the ensemble averaged MSD values of inertial particles of $\\tau _s$ =1 for all three types of fluids.","The evolution of MSD occurs in three stages.","Initially (here, $0{\\ge }t{\\approx }1$ ), inertial particles do not respond until their time-scale (here, $\\tau _s=1$ ) is less than fluid time-scale or Stokes number (St) is less than unity.","During the second stage the particles start to respond the flow ($1{\\ge }t{\\le }3.8$ ), the slopes grow at the almost same rate for all fluids.","In the final stage ($t{\\ge }3.8$ ), the slope shows the larger value for the higher coupling strength and it is minimal for inviscid fluid.","The slope of the MSD versus time is proportional to the diffusion coefficient of the tracer particles.","Thus, the larger time-MSD slope means the carrier fluid shows a better mixing performance.","Thus, this result is found consistent with our earlier observations made from the comparative analysis of pictorial evolution of vorticities in Figs.","REF , REF , and REF .","In order to substantiate these observations we further compute the MSD of tracers with $\\tau _s$ =0.5 (Figure REF (b)) for the all three flows under the same flow conditions.","From the plot, again the diffusion of particles increasing with coupling strength.","It should be noted that the MSD has been calculated up to time before hitting the flow to the boundaries.","In the present work, although we have undertaken the simplest tracer model and preliminary investigation of tracers distribution (cluster formation) and their transport property (MSD), but it is worth attempting as a foundation for the future study."],["Summary and conclusions","In this paper, we numerically explore the KH instability for a two-dimensional rotating SCDP under the formalism of GHD fluid model.","This model treats the SCDP as a viscoelastic fluid.","Here, we consider the specific cases of vorticity with abrupt radial changes, in particular: single-circulation, and multi-circulation vorticity shell profiles.","We observe the KH instability in form of small vortices at each interface along with a pair of ingoing and outgoing wavefronts of TS waves.","The interactions between KH vortices, and with TS waves govern the mixing which has been quantified using the passive tracer particles simulation.","The advection of tracers with the flow has been noticed.","Some main observations are as follows.","The interplay between the TS waves and interacting KH vortices results in better mixing of VE fluids than standard hydrodynamic fluids where the only interaction between KH vortices happens.","By tweaking the coupling strength parameter (ratio $\\eta /\\tau _m$ ) which usually represents the strength of viscoelasticity of the medium, one can control the evolution of KH instability that results in control over the transport properties like mixing and diffusion.","The GHD model especially developed for the study of SCDPs is extended to include the transport of passive Lagrangian inertial and non-inertial tracer particles.","This extended model is referred as Generalized Hydrodynamic Tracer Transport (GHTT) model.","We observe that the diffusion of intermediate inertial tracer particles in VE fluids is proportional to the coupling strength.","It is least for an inviscid fluid.","The multi-circulation vortex profiles, we found that the relaxing rate of a turbulent medium increases with the increasing coupling strength.","The particle tracking model appears a suitable diagnostic to understand the associated mixing in a fluid through the diffusion process, and nonlinear dust fluid dynamics through the clustering of a tracers.","In present paper, we just discuss the preliminary simulations for tracers which are based on first order schemes like to advance the tracers we use first order RK scheme, and for the velocity the first order interpolation scheme.","Further improvisations of this model would be to include higher order schemes, finite size tracers, other forces e.g.","gravity, inter-particle interactions, feedback from tracers to the carrier fluid, etc.","These improvisations would be extremely fruitful which are left to future publications.","Although, the rotational dust flows are extensively studied, but to our knowledge, no prior studies have explicitly examined the KH instability for the rotating flows, except [21], [22].","An experimental research effort is required in order to valid this numerical work."]],"string":"[\n [\n \"Kelvin-Helmholtz instability in strongly coupled dusty plasma with\\n rotational shear flows and tracer transport\"\n ],\n [\n \"Abstract Kelvin-Helmholtz (KH) instability plays a significant role in transport and mixing properties of any medium.\",\n \"In this paper, we numerically explore this instability for a two-dimensional strongly coupled dusty plasma with rotational shear flows.\",\n \"We study this medium using generalized hydrodynamic fluid model which treats it as viscoelastic fluid.\",\n \"We consider the specific cases of rotating vorticity with abrupt radial profiles of rotation.\",\n \"In particular: single-circulation, and multi-circulation vorticity shell profiles have been chosen.\",\n \"We observe the KH vortices at each circular interface between two relative rotating flows along with a pair of ingoing and outgoing wavefronts of transverse shear waves.\",\n \"Our studies show that due to the interplay between KH vortices and shear waves in the strongly coupled medium, the mixing and transport behaviour are much better than inviscid hydrodynamic fluids.\",\n \"In interests of substantiating the mixing and transport behaviour, the generalized hydrodynamic fluid model is extended to include the Lagrangian tracer particles.\",\n \"The numerical dispersion of these tracer particles in a flow provides an estimate of the diffusion in such a medium.\",\n \"We present the preliminary observations of tracers distribution (cluster formation) and their diffusion (mean square displacement) across the medium.\"\n ],\n [\n \"Introduction\",\n \"The Kelvin-Helmholtz (KH) instability has been ubiquitously observed in hydrodynamic fluids [1], [2], plasmas [3], geophysical flows [4], and astrophysical situations [5].\",\n \"This instability occurs in form of vortices at the interface between two flows in the presence of a velocity shear.\",\n \"The interactions between these KH vortices govern the transport processes like mixing and diffusion [6].\",\n \"The main objectives of this study are to explore the formation and evolution of KH vortices of a rotating dusty plasma and quantifying the mixing they generate using the tracer particles simulation.\",\n \"The formation and evolution of these vortices depend on the shear and nature of the medium (here the viscoelasticity of the system).\",\n \"The KH instability for dust flows has also been investigated theoretically [7], [8], [9], [10], [11] and numerically [12], [13], [14], [15] as well experimentally [16] for planar sheared flows.\",\n \"The rotating vortex flows have been studied considerably [17], [18], [19], [20], but to our knowledge, no prior studies have explicitly examined the KH instability for such flows, except [21], [22].\",\n \"Dharodi $et~al.$ [21] numerically studied shearless smooth rotating flows to avoid KH destabilization and also considered sharp rotating flows where KH arises in homogeneous medium.\",\n \"Dharodi in [22] has explored these rotating sheared flows in heterogeneous medium.\",\n \"A dusty plasma can exist in strong coupling state quite easily because of high charged dust particles, which is called strongly coupled dusty plasma (SCDP).\",\n \"The SCDP has been modelled under the formalism of generalized hydrodynamic (GHD) fluid model.\",\n \"This model treats the SCDP as a VE fluid and characterizes its VE effects through the coupling strength parameter which is often measured as the ratio ${\\\\eta /\\\\tau _m}$ , the coupling parameters $\\\\eta $ and $\\\\tau _m$ are the shear viscosity and the Maxwell relaxation time, respectively.\",\n \"Thus, here, the effects of viscoelasticity on KH vortices are observed by varying the ratio ${\\\\eta /\\\\tau _m}$.\",\n \"We consider the incompressible limit of GHD model which in addition to the evolution of hydrodynamic KH instability, also supports transverse shear (TS) waves that propagate at phase velocity $\\\\sqrt{\\\\eta /\\\\tau _m}$ [23], [21], [24].\",\n \"These propagating shear waves have the same symmetry as that of their source structure, until there is no boundary effect or no interaction with other waves or obstacle like vortex.\",\n \"Since our interest is in KH instability of rotating SCDPs, we consider the specific cases of sharp vorticity patches: (i) single-circulation, and (ii) multi-circulation vorticity shell profiles.\",\n \"The single-circulation case has already been somewhat discussed in [21], in the present paper we explore it in more detail.\",\n \"We observe the KH instability at each circular interface between two relative rotating flows in form of small vortices along with a pair of ingoing and outgoing wavefronts of TS waves.\",\n \"The interactions between interacting KH vortices and TS waves help the VE fluid in better mixing than standard HD fluids where the only interactions between the KH vortices take place.\",\n \"To substantiate this observation, the passive tracers are dispersed throughout the medium.\",\n \"In the context of fluid mechanics the tracer transport has been studied extensively for flow visualization [25], [26] with the help of theoretical [27], [28], [29], [30], computational [31], [32], [33], [34], [35], [36], [37], [38], [39], [40], [41], [42] and experimental [43], [44], [45], [46], [47] approaches.\",\n \"This technique is also used in complex fluids (polymers, colloids and biological materials) [48], [49].\",\n \"An analysis of the separation of the particle trajectories with the 2D hydrodynamic fluid is also being carried out by Falkovich et al.\",\n \"[50].\",\n \"In dusty plasma, Schwabe et al.\",\n \"[51] observed the vortex movements by adding some micro particles around the void.\",\n \"Here, we consider two kinds of point-like tracer particles, (i) non-inertial, and (ii) inertial tracer particles.\",\n \"The tracer dynamics is simulated using a one-way coupled Lagrangian point-particle approach [52], [53] which means the tracers are affected by the fluid flow, but not vice-versa.\",\n \"In case of multi-circulation vortex profiles, at intermediate time range, a complete picture of a turbulent flow is observed which has a collection of small KH vortices and waves.\",\n \"When the system is left for a very long time, it ultimately settles down to a single vortex faster than in HD fluid.\",\n \"It is observed that the relaxing rate of such turbulent medium increases with the increasing coupling strength.\",\n \"This paper has been organized as follows.\",\n \"In section , the GHD model especially developed for the study of SCDPs is extended to include the transport of passive Lagrangian tracers.\",\n \"This extended model is referred as incompressible Generalized Hydrodynamic Tracer Transport (i-GHTT) model.\",\n \"Section presents the numerical procedure in order to solve the set of equations of GHTT model.\",\n \"In Sec.\",\n \", we numerically explore the evolution of different types of sharp rotating vorticities in VE fluids and quantifying the mixing they generate using the tracer particles simulations.\",\n \"Finally, Section contains the discussion and the conclusions.\"\n ],\n [\n \"Generalized Hydrodynamic Tracer Transport (GHTT) Model\",\n \"A dusty plasma can be prepared or found as a strongly coupled dusty plasma (SCDP) rather easily because of high charge on the micron-sized dust particles.\",\n \"Below the crystallization limit, a SCDP behaves like a viscoelastic fluid which favors both the incompressible transverse shear modes and the compressible longitudinal modes.\",\n \"To study such SCDP the generalized hydrodynamic (GHD) fluid model is found to be quite suitable which takes into account both types of modes.\",\n \"To scrutinize the effect of transverse modes and to abate the possible pairing with the longitudinal mode, we consider the incompressible limit of GHD (i-GHD) model.\",\n \"Thus, i-GHD model represents the incompressible SCDPs which support transverse modes only.\",\n \"The momentum and continuity equations for i-GHD of homogeneous strongly coupled dusty plasma can be written as: $\\\\left[1 + \\\\tau _m \\\\left(\\\\frac{\\\\partial }{\\\\partial t}+\\\\vec{v}_d \\\\cdot \\\\nabla \\\\right)\\\\right] \\\\left[{\\\\frac{\\\\partial \\\\vec{v}_d }{\\\\partial t}+\\\\vec{v}_d \\\\cdot \\\\nabla \\\\vec{v}_d } + \\\\frac{\\\\nabla P}{n_d} - {\\\\nabla \\\\phi _d} \\\\right] = {\\\\eta }{\\\\nabla ^2 }{\\\\vec{v}_d}{,}$ and $\\\\nabla \\\\cdot \\\\vec{v}_d= 0{,}$ respectively.\",\n \"$n_d$ is the number density which is normalized by its respective equilibrium value $n_{d0}$ .\",\n \"The scalar potential $\\\\phi _d$ in the dusty plasma system is normalized by ${{K_B}{T_i}}/{e}$ .\",\n \"The parameters $e$ , $T_i$ and $K_B$ are the electronic charge, ion temperature and Boltzmann constant, respectively.\",\n \"The charge fluctuation over each dust grain has been ignored.\",\n \"The time and length are normalized by inverse of dust plasma frequency $\\\\omega ^{-1}_{pd}=\\\\left({4\\\\pi (Z_de)^{2}n_{d0}}/{m_{d0}}\\\\right)^{-1/2}$ , plasma Debye length $\\\\lambda _{d}=\\\\left({K_B T_i}/{4{\\\\pi } {Z_d}{n_{d0}}{e^2}}\\\\right)^{1/2}$ , respectively.\",\n \"In incompressible limit the Poisson equation has been replaced by the quasineutrality condition.\",\n \"The dust fluid velocity $\\\\vec{v}_d$ is normalized by ${\\\\lambda _d}{\\\\omega _{pd}}$ .\",\n \"The term ${\\\\tau _{m}}({\\\\vec{v}_d}{\\\\cdot }{\\\\nabla })$ in the generalized momentum equation is responsible for introducing the collective behavior in the medium.\",\n \"When this term becomes zero ($\\\\tau _m$ =0), the momentum equation becomes Navier-Stokes equation.\",\n \"In other words, the GHD model turns into a standard hydrodynamic fluid model.\",\n \"Moreover, the presence of this term conserves the Gallilean invariance [54].\",\n \"In our earlier manuscript [21], we proposed an idea of extending the GHD model by including the passive tracer particles in interest to estimate of the diffusion in a medium.\",\n \"In order to accomplish this, we consider two kinds of point-like tracer particles, (i) non-inertial tracers, and (ii) inertial tracers.\",\n \"The non-inertial tracers follow the flow exactly while the velocity of inertial tracers differs from flow velocity due to viscous drag force [55].\",\n \"The tracer particles dynamics is simulated using a one-way coupled Lagrangian point-particle approach.\",\n \"One way coupled particle approach means the tracers do not affect the fluid motion, in other words the tracers are passive.\",\n \"We also neglect any kind of interaction between particles and gravity effect on their dynamics.\",\n \"We further assume that the inertial tracer particles with density $\\\\rho _p$ different than the density $\\\\rho _d$ of the fluid.\",\n \"Under these assumptions, the particles are transported in the flow according to the equations: $\\\\frac{d{\\\\vec{v}_{pi}}}{dt} &= \\\\frac{1}{\\\\tau _s}({\\\\vec{v}_d}({\\\\vec{r}_{pi}})-{\\\\vec{v}_{pi}}){,}$ $\\\\frac{d{\\\\vec{r}_{pi}}}{dt} &= {\\\\vec{v}_{pi}}{,}$ where ${\\\\vec{r}_{pi}}$ and $\\\\vec{v}_{pi}$ are the position and velocity of the $i$ th particle, respectively.\",\n \"${\\\\vec{v}_d}({\\\\vec{r}_{pi}})$ is the dust fluid velocity at the particle position, ${\\\\vec{r}_{pi}}$ which is obtained by solving the set of Eqs.\",\n \"(REF ) and (REF ).\",\n \"These equations are a simplified approximation of the Maxey-Riley equations [56].\",\n \"Although, the neglected effects might have significant impacts in real flows, but these could be incorporated into the future study because even after neglecting them, the Eq.\",\n \"(REF ) describe an enough complex system and it is worth studying to set the foundation for the future research.\",\n \"The particle time-scale $\\\\tau _s={2{r^2_0}{\\\\rho _p}}/{(9{\\\\eta }{\\\\rho _d})}$ denotes the response time of the particles is known as Stokes time [57].\",\n \"$r_0$ is the radius of a particle.\",\n \"Although the particles are assumed as point particles, but they do have finite mass and therefore finite inertia.\",\n \"The ratio of a particle time-scale to a fluid time-scale is known as Stokes number (St).\",\n \"The effect of particle inertia is often given by using the $St$ or $\\\\tau _s$ .\",\n \"On the basis of St or $\\\\tau _s$ or inertia: For low value, the particles are predicted to follow the fluid flow passively like fluid particles, while at very high value almost the particles remain unaffected by the medium fluctuations.\",\n \"In-between these two limits, when the particle and fluid time-scales are comparable, the particles respond in fast and strong manner to the fluctuations.\",\n \"The non-inertial tracers follow the fluid flow exactly and can be considered as attached to fluid surface.\",\n \"They are characterised by their position ${\\\\vec{r}_{pi}}$ , and velocity ${\\\\vec{v}_{pi}} = {\\\\vec{v}_d}({\\\\vec{r}_{pi}})$ that is the dust fluid velocity at their position.\",\n \"Their equation of motion corresponds to the limit $\\\\tau _s\\\\rightarrow $ 0 in the set of Eqs.\",\n \"(REF ) and (REF ) becomes: $\\\\frac{d{\\\\vec{r}_{pi}}}{dt}={\\\\vec{v}_d}({\\\\vec{r}_{pi}}){,}$ The set of Eqs.\",\n \"(REF )-(REF )-(REF )-(REF ) and Eqs.\",\n \"(REF )-(REF )-(REF ) represent the viscoelastic model for inertial and non-inertial tracer particles, respectively, in which the tracers follow the evolution of fluid with time in Lagrangian way.\",\n \"Both the set of equations would be referred as incompressible Generalized Hydrodynamic Tracer Transport (i-GHT2 or i-GHTT) model henceforth in the article.\",\n \"It should be noted that in i-GHTT model, in interest to include both the compressible longitudinal and incompressible transverse modes just replace the i-GHD model ((set of Eqs.\",\n \"(REF )-(REF )) with complete GHD model (set of Eqs.\",\n \"(5)- (6)-(7) in [23]), say GHTT model .\"\n ],\n [\n \"Transport Properties\",\n \"To quantify the average diffusion of tracer particles, the ensemble averaged mean square displacement of tracers is measured which is associated with the mixing of the fluid [58].\",\n \"The MSD is defined as, $MSD(t)= {\\\\frac{1}{N}}{\\\\sum ^N_{j=1}}{(r_j(t)-r_j(0))^2}{,}\\\\nonumber $ Here, $r_j(0)$ is initial position of $j$ th particle at t=0 and $r_j(t)$ is position at time $t$ .\",\n \"N is the total number of tracers in the ensemble.\",\n \"The slope of the MSD versus time is proportional to the diffusion coefficient of tracer particles which in turn is supposed to measure the mixing performance of the carrier fluid.\",\n \"Thus, the mixing performance of the carrier fluid can be quantified through the time-MSD slope, the larger slope means the carrier fluid is a better mixture.\"\n ],\n [\n \"Simulation methodology\",\n \"For the numerical simulations, first we need to express the model Eq.\",\n \"(REF ) as per requirements of simulation software, LCPFCT (Boris $et al.$ [59]).\",\n \"To fulfill these requirements split the Eq.\",\n \"(REF ) in following two coupled equations, ${{\\\\frac{\\\\partial \\\\vec{v}_d }{\\\\partial t}+\\\\vec{v}_d\\\\cdot \\\\nabla \\\\vec{v}_d }+ \\\\frac{\\\\nabla P}{n_d} -\\\\nabla \\\\phi _d}={\\\\vec{\\\\psi }} {,}$ $\\\\frac{\\\\partial {\\\\vec{\\\\psi }}}{\\\\partial t}+\\\\vec{v}_d \\\\cdot \\\\nabla {\\\\vec{\\\\psi }}={\\\\frac{\\\\eta }{\\\\tau _m}}{\\\\nabla ^2}{\\\\vec{v}_d }-{\\\\frac{\\\\vec{\\\\psi }}{\\\\tau _m}}{.\",\n \"}$ For two-dimensional (2D) studies all the variables are functions of $x$ and $y$ only.\",\n \"The new introduced quantity ${\\\\vec{\\\\psi }}(x,y)$ in LHS of Eq.\",\n \"(REF ) represents the strain induced in the elastic medium by the time-varying velocity fields.\",\n \"Next, the gradient terms are eliminated by taking the curl of Eq.\",\n \"(REF ) which yields an equation for the evolution of the vorticity field.\",\n \"So the coupled set of Eqs.\",\n \"(REF )-(REF ) has been recast in the following form: $\\\\frac{\\\\partial {\\\\xi }_z}{\\\\partial t}+\\\\left(\\\\vec{v}_d \\\\cdot \\\\vec{\\\\nabla }\\\\right){{\\\\xi }_z}={\\\\vec{\\\\nabla }}{\\\\times }{\\\\vec{\\\\psi }}{,}$ $\\\\frac{\\\\partial {\\\\vec{\\\\psi }}}{\\\\partial t}+\\\\vec{v}_d \\\\cdot \\\\nabla {\\\\vec{\\\\psi }}={\\\\frac{\\\\eta }{\\\\tau _m}}{\\\\nabla ^2}{\\\\vec{v}_d }-{\\\\frac{\\\\vec{\\\\psi }}{\\\\tau _m}}{.\",\n \"}$ Here, ${\\\\xi _z}={\\\\nabla }{\\\\times }{\\\\vec{v}_d}$ is the vorticity.\",\n \"${\\\\vec{\\\\xi }}$ is normalised with dust plasma frequency.\",\n \"The LCPFCT software [59] is based on a finite difference method has been used to solve the coupled set of Eqs.\",\n \"(REF ) and (REF ).\",\n \"Taking the curl of relation ${\\\\xi _z}={\\\\nabla }{\\\\times }{\\\\vec{v}_d}$ and using incompressible condition given by Eq.\",\n \"(REF ) i.e.\",\n \"$\\\\nabla \\\\cdot \\\\vec{v}_d= 0$ , we get ${\\\\nabla }^2{\\\\vec{v}_d} =-{\\\\nabla }{\\\\times }{\\\\xi _z}$ This velocity-vorticity relation is used to update the fluid velocity at each time step using FISHPACK [60].\",\n \"The validation of our numerical code has been done in our earlier papers [21], [22].\",\n \"Further details on simulation procedure in advancing the tracer particles with flow of a VE fluid.\",\n \"We have the dust fluid velocity for each particle at their respective position at each time step.\",\n \"This dust velocity is going to be used in particle momentum Eq.\",\n \"(REF ).\",\n \"Equations (REF ) and (REF ) are numerically integrated together to find the new position and velocity at the end of each time step.\",\n \"This integration is based on the first order Runge-Kutta method.\",\n \"The particle velocity $\\\\vec{v}_p$ is calculated by interpolating the velocity defined on nearby grid points, based on first-order Lagrangian interpolation scheme.\",\n \"The particles are advanced with fluid time step.\"\n ],\n [\n \"Numerical results and discussion\",\n \"The prime objectives of this section are to numerically explore the evolution of different types of sharp rotating vorticity flows in VE fluids and quantifying the mixing they generate using the tracer particles simulations.\",\n \"For each type of flow, the simulations are performed for the varying coupling strength of VE fluid which is usually measured as the ratio ${\\\\eta /\\\\tau _m}$ .\",\n \"All the simulations are performed with periodic boundary conditions in both the x and y directions on simulation box.\"\n ],\n [\n \"KH instability of rotating SCDPs\",\n \"Since our interest is in KH instability of rotating SCDPs, we consider the specific cases of sharp vorticity patches: (i) single-circulation, and (ii) multi-circulation vorticity shell profiles.\",\n \"Before proceeding with the direct assessment of the evolution of KH instability through the numerical results, it is good to understand the process of formation of these vortices for sharp rotating flows through the schematic picture illustrated in Fig.\",\n \"REF .\",\n \"The multi-circulation vorticity profile can be depicted as core-shell fluid flows.\",\n \"In Fig.\",\n \"REF , one of rotating flows forms the inner core (yellow color regime) and the others make the outer shells (cyan and green color regimes).\",\n \"Each rotating flow is divided by a sharp interface; the fluid on either side rotates in opposite directions.\",\n \"The black circle with arrows in the flow regime indicate its direction of rotation.\",\n \"Figure: Formation of KH vortices at the circular sharp interfaces of core-shell flow regions (without any corresponding scale).\",\n \"One of flows forms the inner core and the others make the outer shells.\",\n \"The black circles with arrows in the flow regions indicates the direction of rotation.\",\n \"At core-shell interface, the KH votices rotate anti-clockwise (blue color vortices with blue curved arrows) while at shell-shell interface rotate clockwise (red color vortices with red curved arrows).At each of the interfaces, the counter-rotating flows create a region of high shear which immediately evolve into small co-rotating (like-sign) KH vortices.\",\n \"The schematic diagram clearly illustrates how the direction of rotation of these vortices depend on the relative motion between the alternative flows.\",\n \"At inner (core-shell) interface, the KH votices rotate anti-clockwise (blue color vortices with blue curved arrows) while at outer (shell-shell) interface rotate clockwise (red color vortices with red curved arrows).\",\n \"Thus far, these appraisals are particularly true for an inviscid HD fluid where no source term exists.\",\n \"Whereas an incompressible VE fluid, besides KH instability, would also support the TS waves that propagate through the medium at phase velocity $v_p= \\\\sqrt{\\\\eta /\\\\tau _m}$ .\",\n \"A medium with higher coupling strength (ratio ${\\\\eta /\\\\tau _m}$ ) favors the faster and stronger TS waves.\",\n \"A stronger wave shows a less fall in amplitude with time in comparison to the lower one (see Figs.\",\n \"2 and 4 in [21], and Fig.\",\n \"4 in [22]).\",\n \"Since, the present simulations have been carried out in the x-y plane, which is the plane of rotation of vorticity profile, the shear waves emitted from each interface should be cylindrical in shape.\",\n \"Thus, a stronger cylindrical wave tries to dominate over KH instability and attempts to keep the cylindrical symmetry during the evolution.\",\n \"In evolution of these KH vortices the transport processes like merging and convection become important.\",\n \"When two co-rotating (like-sign) vortices are brought sufficiently close to each other they start to rotate around one another and eventually merge to form a single vortex while in convection the counter-rotating (unlike-sign) propagate together as single structure (dipole) to convect the fluid.\",\n \"The velocity profile for the single-circulation sharp vorticity vortex is given as follows ${\\\\vec{v}_{0}} = \\\\left\\\\lbrace \\\\begin{array}{ll}v_{x0}=-{\\\\phi _0}{\\\\frac{(y-y_c)}{b}};\\\\:v_{y0}={\\\\phi _0}{\\\\frac{(x-x_c)}{a}} & \\\\quad |r| \\\\le 6 \\\\\\\\0 & \\\\quad \\\\mathrm {otherwise}{.\",\n \"}\\\\end{array}\\\\right.$ The vorticity corresponding to the above velocity profile is given below ${\\\\xi _{z0}} = \\\\left\\\\lbrace \\\\begin{array}{ll}{\\\\phi _0}{\\\\left(\\\\frac{1}{a}+\\\\frac{1}{b}\\\\right)} & \\\\quad |r| \\\\le 6.0\\\\\\\\0 & \\\\quad \\\\mathrm {otherwise}{.\",\n \"}\\\\end{array}\\\\right.$ Here $|r|= {\\\\sqrt{{\\\\left({(x-x_c)/a}\\\\right)}^2+{\\\\left({(y-y_c)/b}\\\\right)}^2}}$ , $a$ and $b$ are the major and minor axes, respectively.\",\n \"$x_c$ and $y_c$ are the $x$ and $y$ coordinates of the center of the vorticity profile.\",\n \"The vorticity will have a clockwise rotation if amplitude ${\\\\phi _0}{>}0$ , or have a anti-clockwise rotation if ${\\\\phi _0}{<}0$ and have no rotation if ${\\\\phi _0}{=}0$ .\",\n \"We consider the clockwise rotating vorticity profile has circular symmetry with parameters $a=b=1$ , amplitude ${\\\\phi _0}=1$ , sharp cutoff at radial distance $|r|$ =6 units away from the centre of the vortex $(0,0)$ .\",\n \"This vorticity profile (yellow color) is shown in the following Fig.\",\n \"REF (a) which has circular interface with surrounding stagnant fluid at $t=0$ .\",\n \"The simulation region is a square box of length 12$\\\\pi $ units.\",\n \"Figure: Initial rotating sharp vorticity profiles at t = 0 to study the KH instability.\",\n \"The color scale corresponding to a particular vorticity profile has been given in the following respective figure which shows its evolution.\",\n \"The simulation region is a square box of length 12π\\\\pi units for all these systems.The panels of Fig.\",\n \"REF shows time evolution of vorticity profile given in Fig.\",\n \"REF (a) in the inviscid HD fluid.\",\n \"The sharpness of the clockwise rotating vorticity profile generates a strong rotational sheared flow.\",\n \"This sheared flow results in creation of small co-rotating (anti–clockwise) KH vortices (dark blue color vortices) at the vorticity interface $|r|=6$ .\",\n \"These like-sign vortices start merging as rotation progresses that leads the fluid to evolve into an anisotropic isolated structure.\",\n \"Figure: An inviscid HD fluid.\",\n \"The time evolution of a sharp clockwise rotating vorticity which has circular interface with surrounding stagnant fluid.\",\n \"The sharpness of the vorticity profile generates small KH vortices at the interface that results in a anisotropic isolated structure.In Fig.\",\n \"REF where ${\\\\eta }=5$ and ${\\\\tau _m}=20$ , once the vortex begins to rotate, we observe a pair of ingoing and outgoing cylindrical shear waves from the interface along with these small like-sign KH vortices.\",\n \"During the evolution, it is observe that both the waves carry the like-sign vortices which interact with themselves in order to merge and have interplay with these waves as well.\",\n \"The fluid within the inner region ($|r| \\\\le 6$ ) favors mixing due to the ingoing waves, while the stationary fluid in the outer region ($|r| \\\\ge 6$ ) gets mixing due to the outgoing waves.\",\n \"Thus, the TS waves assist the process of fluid mixing by convecting it inside and outside the vortex structure.\",\n \"Unlike the inviscid case, here, the KH vortices are confined to the radial emitted waves.\",\n \"Since the wavefronts are cylindrical in shapes, the interplay between waves and vortices leads the evolution of the VE fluid toward an isotropic structure.\",\n \"Figure: Viscoelastic fluid with coupling parameters η=5{\\\\eta }=5, and τ m =20{\\\\tau _m}=20.\",\n \"The time evolution of a sharp rotating vorticity which has circular interface with surrounding stagnant fluid.\",\n \"The sharpness of the vorticity profile generates small KH vortices at the interface along with a pair of ingoing and outgoing wavefronts of TS waves that assist in fluid mixing by convecting it inside and outside the vortex structure.In Fig.\",\n \"REF , we have simulated another case of VE fluid which has less coupling strength ($\\\\eta /\\\\tau _m$ =0.125) with ${\\\\eta }=2.5$ and ${\\\\tau _m}$ =20.\",\n \"Unlike Fig.\",\n \"REF , since the TS waves are not strong enough that they can dominant over the KH instability, less confinement of KH vortices to the waves.\",\n \"This is evident from Fig.\",\n \"REF where the evolution of medium towards an isotropic structure and mixing process are much slower in comparison to Fig.\",\n \"REF .\",\n \"Figure: The time evolution of a sharp rotating vorticity in VE fluid with coupling parameters η=2.5{\\\\eta }=2.5, and τ m =20{\\\\tau _m}=20.\",\n \"Due to the less coupling strength, the evolution of medium towards an isotropic structure and mixing process are much slower in comparison to Fig.\",\n \".In conclusion, as a result of greater the coupling strength or stronger TS wave, the medium evolution attempts to possess radial symmetry and shows better mixing.\",\n \"The mixing is found minimal in inviscid fluid.\",\n \"The single-circulation sharp profile carries single types of anti-clockwise rotating KH vortices across its only interface.\",\n \"While a multi-circulation profile with two or more than two interfaces produces clockwise as well as anti-clockwise KH vortices.\",\n \"Thus, in multi-circulation case apart from merging between like-sign vortices, the propagation of unlike-sign vortices as a dipolar structure also becomes important which can assist to increasing the spatial domain of mixing fluids.\"\n ],\n [\n \"Multi-circulation vorticity shell profile\",\n \"Here, we first consider the simplest case of multiple shells of vorticity having two flows: Inner core flow and a outer shell flow, both flows have reversal circulation.\",\n \"The velocity profile for this configuration is given below.\",\n \"${\\\\vec{v}_{0}} = \\\\left\\\\lbrace \\\\begin{array}{ll}v_{x0}=-{\\\\phi _0}{(y-y_c)};\\\\: v_{y0}={\\\\phi _0}{(x-x_c)} & \\\\quad |r|\\\\le 5 \\\\\\\\v_{x0}={\\\\phi _0}{(y-y_c)};\\\\: v_{y0}=-{\\\\phi _0}{(x-x_c)} & \\\\quad 5<|r|\\\\le 10 \\\\\\\\0 & \\\\quad \\\\mathrm {otherwise}{.\",\n \"}\\\\end{array}\\\\right.$ The vorticity corresponding to the above velocity profile is given below, ${\\\\xi _{z0}} = \\\\left\\\\lbrace \\\\begin{array}{ll}2{\\\\phi _0} & \\\\quad |r| \\\\le 5 \\\\\\\\-2{\\\\phi _0} & \\\\quad 5 < |r| \\\\le 10 \\\\\\\\0 & \\\\quad \\\\mathrm {otherwise}{.\",\n \"}\\\\end{array}\\\\right.$ With the parameter $\\\\phi _0$ =1, we consider both flows rotate with equal rotation rates and in opposite directions (Fig.\",\n \"REF (b)).\",\n \"Figure REF shows the evolution of this vorticity profile for an inviscid fluid.\",\n \"At inner interface, the KH vortices rotate anti-clockwise (blue color vortices) while at outer interface rotate clockwise (red color vortices).\",\n \"As time goes on, the merging between like-sign vortices take place at both the interfaces, and simultaneously growing closeness between both the interfaces due to the radial gradient in vorticity results in interactions between the counter-rotating (red-blue) vortices as well.\",\n \"These counter-rotating vortices results in the formation of propagating dipolar structures which help to convect the fluid across the wider domain than the single interface (Fig.\",\n \"REF ) which only favors the merging process.\",\n \"Figure: An inviscid HD fluid.\",\n \"Evolution of of vorticity having two circular sharp flows, inner core and an outer shell flows have reversal circulation, in time generates small KH vortices at each interface that results in a anisotropic isolated structure.In Fig.\",\n \"REF , we observe that a pair of ingoing and outgoing wavefronts emanates from each of the two sharp interfaces of the vortex structure along with KH vortices (see the first panel).\",\n \"It is observe that both the wavefronts at inner interface carry the blue color (rotating anti-clockwise) vortices while at outer interface carry red color (rotating clockwise) vortices.\",\n \"During the emission of these shear waves like-sign vortices interact with themselves in order to get merge.\",\n \"Figure: Viscoelastic fluid with coupling parameters η=5{\\\\eta }=5, and τ m =20{\\\\tau _m}=20.\",\n \"The time evolution of vorticity having two circular sharp interfaces, inner core and outer shell have reversal circulation, generates small KH vortices at each interface along with a pair of ingoing and outgoing TS waves fronts that results in well fluid mixing by convecting it inside and outside the vortex structure.The stagnant fluid in the outermost region ($|r| \\\\ge 10$ ) undergoes mixing due to the outgoing wave from the outermost interface at 10, while the innermost vortex region undergoes mixing due to the ingoing wave emanating from the sharp interface located at 5.\",\n \"Interestingly, the vortex region confined within the two sharp interfaces ($5 <|r|\\\\le 10$ ) undergoes mixing due to the ingoing wave from the outermost interface and the outgoing wave from the innermost interface.\",\n \"This region is probably convection dominating region due to the higher possibility of formation of propagating dipolar structures.\",\n \"As the results of multiple interaction processes the mixing becomes fast and efficient than the cases discussed so far.\",\n \"Next, we consider the another VE fluid in Fig.\",\n \"REF having lower coupling strength i.e.\",\n \"${\\\\eta }=2.5, {\\\\tau _m}=20, v_p$ =0.35, it is observed that at each time step in the spatial confinement of this medium is lower than Fig.\",\n \"REF .\",\n \"Moreover, the comparison manifests that the mixing and evolution symmetry of a medium are proportional to the coupling strength of that medium as observed for earlier cases.\",\n \"Figure: Viscoelastic fluid with coupling parameters η=2.5{\\\\eta }=2.5, and τ m =20{\\\\tau _m}=20.\",\n \"The time evolution of vorticity having two circular sharp interfaces, both have reversal circulation.\",\n \"Due to the less coupling strength the spatial confinement of this medium is lower than Fig.\",\n \".Next, in order to make a better comparative numerical analysis about the mixing rate, we have considered a more complex scenario of multiple circulations (each consecutive one having a reversal in its circulation) having the following velocity flow profile: ${\\\\vec{v}_{0}}= \\\\left\\\\lbrace \\\\begin{array}{ll}v_{x0}=-{\\\\phi _0}{(y-y_c)};\\\\: v_{y0}={\\\\phi _0}{(x-x_c)} & \\\\quad |r|\\\\le 2.5 \\\\\\\\v_{x0}={\\\\phi _0}{(y-y_c)};\\\\: v_{y0}=-{\\\\phi _0}{(x-x_c)} & \\\\quad 2.5<|r|\\\\le 5 \\\\\\\\v_{x0}=-{\\\\phi _0}{(y-y_c)};\\\\: v_{y0}={\\\\phi _0}{(x-x_c)} & \\\\quad 5<|r|\\\\le 7.5 \\\\\\\\v_{x0}={\\\\phi _0}{(y-y_c)};\\\\: v_{y0}=-{\\\\phi _0}{(x-x_c)} & \\\\quad 7.5<|r|\\\\le 10 \\\\\\\\0 & \\\\quad \\\\mathrm {otherwise}{.\",\n \"}\\\\end{array}\\\\right.$ The vorticity corresponding to the above velocity profile is given below, ${\\\\xi _{z0}} = \\\\left\\\\lbrace \\\\begin{array}{ll}2{\\\\phi _0} & \\\\quad |r| \\\\le 2.5 \\\\\\\\-2{\\\\phi _0} & \\\\quad 2.5 < |r| \\\\le 5 \\\\\\\\2{\\\\phi _0} & \\\\quad 5 < |r| \\\\le 7.5 \\\\\\\\-2{\\\\phi _0} & \\\\quad 5 < |r| \\\\le 10 \\\\\\\\0 & \\\\quad \\\\mathrm {otherwise}{.\",\n \"}\\\\end{array}\\\\right.$ For the parameter $\\\\phi _0$ =1, the initial vorticity profile is shown in Fig.\",\n \"REF (c).\",\n \"The complexity of this motion of multi-circulation structure is evident from the subplots of Fig.\",\n \"REF for inviscid fluid.\",\n \"In initial time period, the vortices of KH instability develop across the interface of each shell.\",\n \"At intermediate time range, this evolution provides a complete picture of a turbulent flow throughout the entire system which is collection of several small symmetric and non-symmetric vortices.\",\n \"The transport phenomena like merging between two co-rotating vortices, convection due to the evolution of dipolar (two counter-rotating ) vortices, collision between these dipolar vortices, and also the formation and evolution of triplor structures becomes more frequent than above discussed cases.\",\n \"Figure: An inviscid HD fluid.\",\n \"The time evolution of multi-circulation vorticity profile, each consecutive one having a reversal in its circulation, generates small KH vortices at each interface.\",\n \"This evolution provides a complete picture of a turbulent flow which is collection of various kind of vortices, and exhibits transport properties like convection and merging.Figure REF , represents the evolution of same initial profile (Fig.\",\n \"REF ) of vorticity for VE fluid with coupling parameters ${\\\\eta }=5$ , ${\\\\tau _m}$ =20.\",\n \"From the comparative observations between Fig.\",\n \"REF and Fig.\",\n \"REF , it is interesting to notice that the presence of TS waves leads to the relaxing of the turbulent medium to a single vortex faster than in inviscid fluid.\",\n \"Figure: The time evolution of multi-circulation vorticity profile in VE fluid with η=5{\\\\eta }=5, and τ m =20{\\\\tau _m}=20.\",\n \"The strong interaction between KH vortices and TS waves results in the relaxing the medium into a single vortex faster than inviscid fluid.In Fig.\",\n \"REF (${\\\\eta }=2.5, {\\\\tau _m}=20, v_p$ =0.35), since the TS waves are weaker than Fig.\",\n \"REF , the less spatial confinement of the turbulent medium that results in slower merging process, due to which the relaxation time of the medium becomes longer.\",\n \"Figure: The time evolution of multi-circulation vorticity profile in VE fluid with η=2.5{\\\\eta }=2.5, and τ m =20{\\\\tau _m}=20.\",\n \"Since the TS waves are weaker than Fig.\",\n \", the relaxation time of the medium becomes longer.The comparison of Fig.\",\n \"REF and Fig.\",\n \"REF clearly displays these observations.\",\n \"In Section we have stated that tracer particles with very low inertia follow the flow passively, while particles with very high inertia will remain almost unaffected by the medium fluctuations.\",\n \"In between these two limits particles show the strongest response to the medium fluctuations.\",\n \"The simulations are performed for all these three inertial particles: very low ($\\\\tau _s$ =0.05), intermediate ($\\\\tau _s$ =1), and very high inertia ($\\\\tau _s$ =50).\",\n \"To observe the exclusive effect of inertia, we transport these particles through a similar smooth rotating vorticity vortex in an inviscid fluid.\",\n \"An inviscid fluid has no source term which favors the emission of TS waves or dissipative term like viscosity.\",\n \"And also, we choose the smooth rotating vorticity vortex which does not satisfy the KH destabilization condition anywhere in the vorticity patch.\",\n \"The equation of such smooth rotating vorticity is given as ${\\\\xi _{0}}={\\\\Omega _0}exp\\\\left(-\\\\left({x^2+y^2}\\\\right)/{a^2_c}\\\\right){.\",\n \"}$ The evolution of same structure given by Eq.\",\n \"REF for ${a_c}$ =1.0, ${\\\\Omega _0}=5$ in an inviscid fluid is shown in Figs.\",\n \"(REF ), (REF ), and (REF ).\",\n \"From these figures it is clear that the rotating vortex keeps rotating without any change.\",\n \"Figure: An inviscid HD fluid carries tracer particles are shown as red dots.\",\n \"The low inertial tracers (τ s \\\\tau _s=0.05) follow the dynamics along the smooth rotating vortex.Now, in order to see the response of tracer particles, initially $(t=0)$ , we have distributed 900 inertial particles (shown by red dots) homogeneously throughout the domain.\",\n \"Figure: An inviscid HD fluid carries tracer particles are shown as red dots.\",\n \"The high inertial tracers (τ s \\\\tau _s=50) show negligible response to the rotating vortex.From Fig.\",\n \"REF , it is clear that low inertial particles ($\\\\tau _s$ =0.05) follow the dynamics along the rotating vortex, and the particles with higher inertia $\\\\tau _s$ = 50 (Fig.\",\n \"REF ) show negligible response to the vorticity gradient.\",\n \"Figure: An inviscid HD fluid carries tracer particles are shown by red dots.\",\n \"Smooth rotating vorticity vortex with tracers having intermediate inertia i.e τ s \\\\tau _s=1 show high response to the vorticity gradient, the particles are pushed away where the flow is strong enough and get accumulate in strain-dominated region.In comparison to previous cases (Fig.\",\n \"REF and Fig.\",\n \"REF ), Fig.\",\n \"REF shows that the particles with intermediate value of $\\\\tau _s$ =1 counter a significant outward push.\",\n \"It is because the particle and fluid time- scales are comparable which results the particles experience a notable centrifugal force due to vorticity gradient.\",\n \"Since, the inertial particles are pushed away from regions where the flow is strong enough, these particles get accumulate in strain-dominated regions.\",\n \"Thus, particles tend to leave regions of high vorticity and cluster into regions of high strain [61].\",\n \"Note, we have simulated a range of intermediate inertial particles, and observed the same effect with varying outward push.\",\n \"With the identification of intermediate inertial particles and understanding of their evolution, next we compare the evolution of intermediate inertial (typical, $\\\\tau _s$ =1) with non-inertial particles for the inviscid HD and VE ($\\\\eta =5$ , $\\\\tau _m=20$ ) fluids in the following Fig.\",\n \"REF and Fig.\",\n \"REF , respectively.\",\n \"For this, we choose a sharp rotating vorticity profile which is given by Eq.\",\n \"(REF ).\",\n \"The middle row in Fig.\",\n \"REF and Fig.\",\n \"REF represent the time evolution of this profile for inviscid and VE fluids, respectively.\",\n \"As the sharp vortex starts to rotate, produces larger strain (deformation) in medium along the interface that results in formation of KH instability.\",\n \"The evolution of this rotor has already been discussed in the earlier Section REF for inviscid fluid in Fig.\",\n \"REF and for VE fluid in Fig.\",\n \"REF .\",\n \"Initially $(t=0)$ , here, we distribute 3600 tracer particles homogeneously throughout these fluids.\",\n \"Figure: An inviscid HD fluid advect tracer particles.\",\n \"First and third row show the temporal and spatial distribution of non-inertial particles (shown by magenta dots) and inertial particles (τ s \\\\tau _s=1, shown by red dots), respectively, corresponding to the sharp vorticity profile evolution shown in second row.In both Figs.\",\n \"REF and REF , the first and the third row visualize the distribution of non-inertial particles (shown by magenta dots) and inertial particles (shown by red dots) respectively, advect by their respective fluid flow shown in the middle row.\",\n \"The accumulation of non-inertial particles is observed in rotation-dominated regions over the vortex structures.\",\n \"While the inertial particles accumulate in strain-dominated regions along the interfaces.\",\n \"Figure: Viscoelastic fluid with coupling parameters η=5{\\\\eta }=5, and τ m =20{\\\\tau _m}=20 advect tracer particles.\",\n \"First and third row show the spatiotemporal distribution of non-inertial particles (shown by magenta dots) and inertial particles (τ s \\\\tau _s=1, shown by red dots), respectively, corresponding to the sharp vorticity profile evolution given in second row.This accumulation process leads to the spatial inhomogeneous distribution of particles.\",\n \"This inhomogeneous distribution of the particles is known as clustering or preferential concentration.\",\n \"Clustering is well-studied in the case of inertial particles [62], [31], [26], [27], [63], [64], [65], [66], and several studies are available for non-inertial particles [67].\",\n \"In real flows the tracer particles always have some inertial value, so we compute the ensemble averaged MSD of intermediate inertial particles to analyze the diffusion of particles in the carrier VE fluid.\",\n \"This diffusion of particles is associated with the mixing of the fluid.\",\n \"For this, we advect the inertial particles having $\\\\tau _s$ =1 using the sharp rotating flows discussed above, inviscid fluid in Fig.\",\n \"REF /Fig.\",\n \"REF , VE fluid with $\\\\eta =5$ , $\\\\tau _m=20$ in Fig.\",\n \"REF /Fig.\",\n \"REF , and another VE fluid with $\\\\eta =2.5$ , $\\\\tau _m=20$ in Fig.\",\n \"REF .\",\n \"Initially t=0, we disperse 3600 tracer particles homogeneously over these fluids.\",\n \"Figure: The MSD as function of time for tracing particles.\",\n \"The sharp rotating vorticity advect the particles with (a) τ s \\\\tau _s=1, and (b) τ s \\\\tau _s=0.5.\",\n \"At later time, the MSD is proportional to the coupling strength.Figure REF (a) compares the ensemble averaged MSD values of inertial particles of $\\\\tau _s$ =1 for all three types of fluids.\",\n \"The evolution of MSD occurs in three stages.\",\n \"Initially (here, $0{\\\\ge }t{\\\\approx }1$ ), inertial particles do not respond until their time-scale (here, $\\\\tau _s=1$ ) is less than fluid time-scale or Stokes number (St) is less than unity.\",\n \"During the second stage the particles start to respond the flow ($1{\\\\ge }t{\\\\le }3.8$ ), the slopes grow at the almost same rate for all fluids.\",\n \"In the final stage ($t{\\\\ge }3.8$ ), the slope shows the larger value for the higher coupling strength and it is minimal for inviscid fluid.\",\n \"The slope of the MSD versus time is proportional to the diffusion coefficient of the tracer particles.\",\n \"Thus, the larger time-MSD slope means the carrier fluid shows a better mixing performance.\",\n \"Thus, this result is found consistent with our earlier observations made from the comparative analysis of pictorial evolution of vorticities in Figs.\",\n \"REF , REF , and REF .\",\n \"In order to substantiate these observations we further compute the MSD of tracers with $\\\\tau _s$ =0.5 (Figure REF (b)) for the all three flows under the same flow conditions.\",\n \"From the plot, again the diffusion of particles increasing with coupling strength.\",\n \"It should be noted that the MSD has been calculated up to time before hitting the flow to the boundaries.\",\n \"In the present work, although we have undertaken the simplest tracer model and preliminary investigation of tracers distribution (cluster formation) and their transport property (MSD), but it is worth attempting as a foundation for the future study.\"\n ],\n [\n \"Summary and conclusions\",\n \"In this paper, we numerically explore the KH instability for a two-dimensional rotating SCDP under the formalism of GHD fluid model.\",\n \"This model treats the SCDP as a viscoelastic fluid.\",\n \"Here, we consider the specific cases of vorticity with abrupt radial changes, in particular: single-circulation, and multi-circulation vorticity shell profiles.\",\n \"We observe the KH instability in form of small vortices at each interface along with a pair of ingoing and outgoing wavefronts of TS waves.\",\n \"The interactions between KH vortices, and with TS waves govern the mixing which has been quantified using the passive tracer particles simulation.\",\n \"The advection of tracers with the flow has been noticed.\",\n \"Some main observations are as follows.\",\n \"The interplay between the TS waves and interacting KH vortices results in better mixing of VE fluids than standard hydrodynamic fluids where the only interaction between KH vortices happens.\",\n \"By tweaking the coupling strength parameter (ratio $\\\\eta /\\\\tau _m$ ) which usually represents the strength of viscoelasticity of the medium, one can control the evolution of KH instability that results in control over the transport properties like mixing and diffusion.\",\n \"The GHD model especially developed for the study of SCDPs is extended to include the transport of passive Lagrangian inertial and non-inertial tracer particles.\",\n \"This extended model is referred as Generalized Hydrodynamic Tracer Transport (GHTT) model.\",\n \"We observe that the diffusion of intermediate inertial tracer particles in VE fluids is proportional to the coupling strength.\",\n \"It is least for an inviscid fluid.\",\n \"The multi-circulation vortex profiles, we found that the relaxing rate of a turbulent medium increases with the increasing coupling strength.\",\n \"The particle tracking model appears a suitable diagnostic to understand the associated mixing in a fluid through the diffusion process, and nonlinear dust fluid dynamics through the clustering of a tracers.\",\n \"In present paper, we just discuss the preliminary simulations for tracers which are based on first order schemes like to advance the tracers we use first order RK scheme, and for the velocity the first order interpolation scheme.\",\n \"Further improvisations of this model would be to include higher order schemes, finite size tracers, other forces e.g.\",\n \"gravity, inter-particle interactions, feedback from tracers to the carrier fluid, etc.\",\n \"These improvisations would be extremely fruitful which are left to future publications.\",\n \"Although, the rotational dust flows are extensively studied, but to our knowledge, no prior studies have explicitly examined the KH instability for the rotating flows, except [21], [22].\",\n \"An experimental research effort is required in order to valid this numerical work.\"\n ]\n]"},"id":{"kind":"string","value":"2107.01831"}}},{"rowIdx":1598,"cells":{"text":{"kind":"list like","value":[["Controllable cardiac synthesis via disentangled anatomy arithmetic"],["Abstract Acquiring annotated data at scale with rare diseases or conditions remains a challenge.","It would be extremely useful to have a method that controllably synthesizes images that can correct such underrepresentation.","Assuming a proper latent representation, the idea of a \"latent vector arithmetic\" could offer the means of achieving such synthesis.","A proper representation must encode the fidelity of the input data, preserve invariance and equivariance, and permit arithmetic operations.","Motivated by the ability to disentangle images into spatial anatomy (tensor) factors and accompanying imaging (vector) representations, we propose a framework termed \"disentangled anatomy arithmetic\", in which a generative model learns to combine anatomical factors of different input images such that when they are re-entangled with the desired imaging modality (e.g.","MRI), plausible new cardiac images are created with the target characteristics.","To encourage a realistic combination of anatomy factors after the arithmetic step, we propose a localized noise injection network that precedes the generator.","Our model is used to generate realistic images, pathology labels, and segmentation masks that are used to augment the existing datasets and subsequently improve post-hoc classification and segmentation tasks.","Code is publicly available at https://github.com/vios-s/DAA-GAN."],["Introduction","Whilst large scale public datasets are available for traditional vision tasks, medical data are difficult to acquire.","Even in a large-scale medical training dataset, examples of rare diseases and anatomies are scarce.","As a result, generalisation to observations that are not seen during training will be reduced.","To increase the diversity of training data and for instance, to increase the incidence of rare characteristics, we would like the ability to mix and match factors that encode these variations [4] in a controllable way i.e.","perform controllable image synthesis.","Figure: Top: overview of the “disentangled anatomy arithmetic\" concept illustrated with 3 factors that represent 3 different anatomical parts of the heart (e.g.","left/right ventricle and myocardium).","Bottom: DAA-GAN generated example.","Given a healthy Subject A, we aim to generate an image A' which exhibits hypertrophic cardiomyopathy (HCM).","We select a Subject B with HCM and remove the anatomical factors from A (i.e.","the ones that encode the myocardium and left ventricular cavity) and add the corresponding factors of B (inner part of the red circle).","Arrows in A' point to local deformations showing the non-linear abilities of our arithmetic.","Arithmetic operations are denoted with ±\\pm .The idea of generating realistic images to augment existing limited data is not new in medical image analysis.","Generative Adversarial Networks (GANs) [16] have been used to generate variants for a given input image based on sampling from a random noise vector.","In fact, more recent GAN architectures pursue controllability by disentangling existing factors of variation, conditioning the generation process using semantic priors (e.g.","segmentation masks) [17], [10], [35], [22], [28], [18], [25] and class labels [20], [13], or learning cross-modality translations [29], [3], [24], [8], [11].","An alternative to relying on sampling from a noise vector for new medical images, is the idea of mixing existing information from different populations (e.g.","patients with different anatomical characteristics) to learn the intermediate latent space and generate more realistic data in a more controllable way.","A concept that approximates this idea is “vector arithmetic\" [33], where existing vector-based latent representations are combined using simple arithmetic operations to produce new images.","However, vector representations do not exploit the spatial equivariance of the image content and the respective task (e.g.","segmentation) and have shown poor reconstruction quality.","An alternative is to use to disentangled representations that use both spatial (tensor) and vector representations to capture factors of variation and permit decomposition of the input in spatially equivariant (and closely to be semantic) and imaging information [7], [9], [38].","Herein using disentangled representations we propose the concept of “disentangled anatomy arithmetic\" (DAA), visualized in Fig.","REF , that enables controllable image synthesis of plausible images with a target pathology, which we show to be useful for augmenting existing medical training data.","We design the DAA-GAN model that learns to combine and transform spatial anatomical factors –we provide a visual example of anatomy factors in Fig.","1 of the supplemental– from input images captured by different vendors or from different populations, and then re-entangle them with the chosen imaging factors (e.g.","MRI) to generate unseen intermediate representations.","Inspired by recent findings regarding the advantages of introducing spatial stochasticity in the generation process [23], [1], [14], [2], we propose a convolutional module for the combined anatomical factor representation transformation, in which structured noise is injected at the spatial locations where the arithmetic operations take place.","Our contributions are to: Introduce the concept of “disentangled anatomy arithmetic\" on spatial representations of the anatomy.","Propose DAA-GAN, a generative model that to the best of our knowledge, is the first to condition image generation using spatial anatomy factors as semantic priors.","Propose the noise injection module that encourages local deformations to realistically blend the new factors after the arithmetic step.","Evaluate the impact of using DAA-GAN for cardiac data augmentation in the context of a classification and a semantic segmentation post-hoc task."],["Generative Model Architecture","Our model assumes disentangled anatomy and imaging representations as inputs.","To obtain them, we use SDNet [7] as this model provides binary spatial factors that correspond to the whole anatomy and can be used as semantic priors.","Model overview.","As depicted in Fig.","REF , DAA-GAN has 4 distinct steps: 1) We combine anatomy factors, using disentangled anatomy arithmetic, to obtain a new mixed anatomical representation $\\mathbf {\\hat{C}}$ .","2) A noise injection network $\\mathcal {J}$ takes $\\mathbf {\\hat{C}}$ and aims to create a plausible and more refined anatomical representation $\\mathbf {\\tilde{C}}$ .","3) A generator $\\mathcal {G}$ reconstructs an image corresponding to $\\mathbf {\\tilde{C}}$ and a given imaging representation.","4) Two critics, namely a discriminator $\\mathcal {D}$ and a pathology classifier $\\mathcal {F}$ ensure good image fidelity, but also that the reconstructed image contains the right target characteristics.","We proceed detailing these steps.","Figure: DAA-GAN overview (from left to right): arithmetic operations are performed between anatomical factors 𝐂 a \\mathbf {C}^{a} and 𝐂 b \\mathbf {C}^{b} to produce a mixed representation 𝐂 ^\\mathbf {\\hat{C}} which is then refined by the noise injection network 𝒥\\mathcal {J}.","The generator 𝒢\\mathcal {G} receives this refined representation 𝐂 ˜\\mathbf {\\tilde{C}} and re-entangles it with the the input imaging factors to generate the new image I ˜\\tilde{I}.","Finally, a discriminator is responsible to judge if I ˜\\tilde{I} is real or fake, whilst a pathology classifier assesses if I ˜\\tilde{I} has the desired cardiac pathology.Disentangled anatomy arithmetic.","As shown in Fig.","REF (top), we consider an example with 3 cardiac anatomy populations $\\mathbf {C}^{a}, \\mathbf {C}^{b}, \\mathbf {C}^{c}$ (e.g.","patients from 3 imaging populations $a$ , $b$ , and $c$ ) with 3 anatomical factors per population, which –when combined– form a nested structure that corresponds to the heart region.","These factors are extracted from $I^{a}, I^{b}, I^{c}$ medical images of dataset $Z_{\\sim \\lbrace I^{a}, I^{b}, I^{c}\\rbrace }$ .","Based on this setup, we define factor arithmetic between populations any swapping operation between the corresponding factors.","Following this example, to create a new $\\mathbf {C}^{a}$ anatomy by mixing factors, we first swap $\\mathbf {C}_{1}^{a}$ with $\\mathbf {C}_{1}^{c}$ , and then swap $\\mathbf {C}_{2}^{a}$ with $\\mathbf {C}_{2}^{b}$ .","The result is an intermediate $\\mathbf {\\hat{C}}$ that is used as input to the next module of DAA-GAN.","Note that before swapping two factors, assuming upright anatomies, we perform a registration step (warping) to align the swapped-in factor with the center of mass location of the swapped-out one.","Noise injection.","Since cardiac anatomy is a nested structure, the output of the arithmetic step might be non-realistic, e.g.","have factors that overlap with each other.","This can lead to generated images with ambiguous pathology.","We tackle this problem with module $\\mathcal {J}$ , which receives $\\mathbf {\\hat{C}}$ and produces a refined representation $\\mathbf {\\tilde{C}}$ .","Inspired by recent work of Karras et al.","[23], we introduce stochastic variation (as Gaussian noise) at specific spatial locations of each convolutional (CONV) layer's activations.","This variation is exploited by the network to cause local deformations around the added (swapped-in) factor(s) in order to preserve the non-overlapping nested structure of the heart (see Fig.","REF (bottom)).","$\\mathcal {J}$ consists of 4 CONV layers, whilst the noise is injected in the form of noise patches (see Fig.","REF ) added to each CONV layer's activation features in an element-wise fashion.","The last CONV layer is followed by a Gumbel-Softmax operator that bounds $\\mathbf {\\tilde{C}}$ to $[0,1]$ .","Figure: Visualization of the localized Gaussian noise patch generation process.","⊗\\otimes and ⊙\\odot denote convolution and element-wise multiplication, respectively.Generator.","The generator is responsible for the re-entanglement of anatomy and imaging information, and by extension for the generation of a new image $\\tilde{I}$ .","$G$ consists of 4 CONV-ReLU layers followed by a hyperbolic tangent activation function.","The first CONV layer receives $\\mathbf {\\tilde{C}}$ as input, while after each CONV-ReLU block, there is an Adaptive Instance Normalization (AdaIN) layer [21] that scales and shifts the activations based on the input imaging factors.","Since $\\mathbf {\\tilde{C}}$ has the same dimensions as $I$ and $\\tilde{I}$ , there is no need for any upsampling.","Critics.","Since for $\\tilde{I}$ we have no ground truth we use a discriminator to guide reconstruction.","We adopt the architecture of the LSGAN discriminator [26] for faster convergence, better stability and ultimately better image quality (compared to a variant with spectral normalization layers adopted from [27]).","Additionally, since we want $\\tilde{I}$ to have desired properties, we use a VGG16 model [36] $\\mathcal {F}$ to classify the pathology represented in $\\tilde{I}$ .","$\\mathcal {F}$ has 7 CONV-BN-ReLU blocks and 3 fully-connected layers followed by a softmax function.","Note that $\\mathcal {F}$ is pre-trained on the original data, and is then used as a pathology predictor during training of the generative model.","Having presented each module in detail, we now proceed to describe the 4 losses used to train our model as a whole: Adversarial loss ($\\mathcal {L}_{adv})$ .","We use an adversarial loss to encourage our model to generate realistic images.","We choose to minimize the LSGAN loss [26] as it is more stable during training and leads to higher quality image generation compared to the traditional GAN loss.","Our adversarial loss is defined as: ${\\begin{array}{c}\\mathcal {L}_{adv} = \\mathcal {L}_{\\mathcal {D}} + \\mathcal {L}_{\\mathcal {G}},\\\\\\mathcal {L}_{\\mathcal {D}} = \\frac{1}{2}\\mathbb {E}_{I^{\\prime }\\sim p(Z)}[(\\mathcal {D}(M\\cdot I^{\\prime }) - 1)^2]\\ +\\frac{1}{2}\\mathbb {E}_{\\mathbf {C}\\sim p(\\mathbf {C}_{Z})}[(\\mathcal {D}(M\\cdot \\mathcal {G}(\\mathbf {\\hat{C}})))^2],\\\\\\mathcal {L}_{\\mathcal {G}} = \\frac{1}{2}\\mathbb {E}_{\\mathbf {C}\\sim p(\\mathbf {C}_{Z})}[(\\mathcal {D}(\\mathcal {G}(\\mathbf {\\hat{C}})) - 1)^2],\\end{array}}$ where $I^{\\prime }$ is the image that contributes only the anatomy factor(s) $\\mathbf {C^{\\prime }}$ which are used to form $\\mathbf {\\hat{C}}$ (e.g.","$I^{b}$ in Fig.","REF ).","$M$ is a binary mask produced by the union of the anatomical factors that contain information about the heart.","Note that $M(j)=0$ for the remaining (i.e.","non-heart) pixels.","Pathology classification ($\\mathcal {L}_{path}$ ).","Since we know that we add the anatomical factor $C_{k}$ to the mixed representation, we expect to be able to recognize the pathology corresponding to $C_{k}$ in the generated image $\\tilde{I}$ .","To achieve this we minimize the cross entropy loss, defined as $\\mathcal {L}_{path} = -\\sum _{i=1}^{\\Omega }y_{i}\\log (p(x_{i}))$ , where $y_{i}$ and $p(x_{i})$ are the ground truth and predicted pathology labels, and $\\Omega $ is the number of pathology classes.","Anatomical and background consistency ($\\mathcal {L}_{cons}$ and $\\mathcal {L}_{bg}$ ).","$\\mathcal {L}_{cons}$ encourages the anatomical factors which are not related with the heart to remain unaltered after the arithmetic and noise injection steps.","To find which pixels should not be changed, we use the blurred mask produced during the noise patch generation, denoted as $\\Phi (j)$ , with $\\Phi (j)=0$ for each pixel location $j$ that is not part of the anatomy mixing.","We define $\\mathcal {L}_{cons} = \\frac{1}{N} \\sum _{j}(1-\\Phi (j)) ||\\mathbf {\\hat{C}}(j) - \\mathbf {\\tilde{C}}(j)||_{1}$ .","$\\mathcal {L}_{bg}$ is the same as $\\mathcal {L}_{cons}$ , but on images.","Namely, $\\mathcal {L}_{bg} = \\frac{1}{N} \\sum _{j}(1-M(j)) ||(I^{a}(j) - \\tilde{I(j)}) ||_{1}$ where $M$ is defined previously, and $N$ is a the total number of pixels.","Total loss ($\\mathcal {L}_{total}$ ).","These losses discussed above, are combined as: $\\mathcal {L}_{total} = \\mathcal {L}_{adv} +\\mathcal {L}_{path} + \\lambda _{1}(\\mathcal {L}_{cons} + \\mathcal {L}_{bg}),$ where $\\lambda _{1}=10$ is a weighting hyperparameter for the consistency losses."],["Experiments","In this section we present key information on datasets and metrics, and discuss the results of our experiments (see training details in Sec.","1 of the supplemental).","Data.","We use the ACDC [5] dataset and data from the M$\\&$ Ms challenge [6].","ACDC consists of cardiac images acquired from MRI scanners across 100 subjects, and provides pathology annotations (5 classes) for all images and pixel-level segmentation masks for end-systole (ES) and end-diastole (ED) per subject.","M$\\&$ Ms has cardiac-MR images acquired from 4 different (known) sites (domains), across 345 subjects.","It provides ED/ES pixel-level segmentation masks annotations for 320 (of 345) subjects and pathology annotations (4 classes) for all images (for more dataset details see Sec.","2 of the supplemental).","Metrics.","We approach this in several levels.","To measure image quality we compute the Fréchet Inception Distance (FID) [19], which quantifies the distance between feature vectors from real and generated images.","To quantify the utility of the generated images we study if they help improving the performance of two post-hoc tasks: a) pathology classification, and b) semantic segmentation.","For the former, we measure the classification accuracy achieved by a VGG16 model, whilst for the latter, we measure the Dice score [12], [37] achieved by a U-Net [34] model.","The hardest is to assess controllability, i.e.","the ability to generate images that faithfully create combinations of anatomical factors.","We approximate this by examining the existence of the added or removed pathologies.","We train two VGG16 classifiers, one 5-class for ACDC data and one 4-class for M$\\&$ Ms data, and measure the pathology classification accuracy on the generated images.","Table: Comparing data augmentation methods in the context of the 4 questions defined in the Results section (see text for details).We report average (standard deviation as subscript) classification accuracy (Acc.)","and segmentation performance as average Dice score.","“n/a\" denotes not applicable experiment.","* and ** denote significant improvement over the 2nd best method with p<0.05p<0.05 and p<0.1p<0.1 (Wilcoxon non-parametric test), respectively.Setup.","We generate data with our DAA-GAN; SPADE [31] a model that conditions synthesis using segmentation masks; and AC-GAN [30] a model that conditions synthesis on class rather than semantic priors.","Then, we train the two posthoc task-specific models (i.e.","VGG16 and U-Net) with the following training sets (applies to both datasets): i) original data (OD), ii) OD augmented with data from DAA-GAN, iii) OD augmented with data from SPADE, and iv) OD augmented with data from AC-GAN.","For (i)-(iv) we further augment the training set using traditional intensity (blur, gamma correction) and geometric (crop, rotate, flip, elastic transformation) augmentations for fairness (validation and test sets remain unaltered).","Note that for each generated image, we extract and binarize only the heart-related $\\mathbf {\\tilde{C}}$ factors (output of $\\mathcal {J}$ ) and use them as near-ground truth masks for retraining U-Net in the context of post-hoc segmentation.","Results.","To demonstrate the effectiveness of our method, we answer several key questions referring to quantitative results presented in Table REF .","Does DAA-GAN augmentation improve learning on a balanced dataset?","For this experiment we use ACDC, which is balanced in terms of subjects per pathology class.","We split the OD into 70%, 15%, 15% subjects for training, validation, and testing, and use DAA-GAN to generate 50 new images for the 5 pathology classes (this corresponds to a 25% samples increase).","These images are picked based on $\\mathcal {F}$ confidence.","As Table REF shows (column “Balanced”) data generated by our model lead to a 3.1% absolute classification accuracy improvement compared to using the OD with traditional augmentations, whilst outperforming both AC-GAN and SPADE.","Regarding segmentation, compared to only using the OD, our model improves the Dice score by an absolute 1.6%.","Since AC-GAN does not condition on masks and SPADE does not have a mechanism to refine the fused segmentation masks, they cannot be used in this experiment.","Does DAA-GAN augmentation improve learning of underrepresented classes?","The M$\\&$ Ms dataset is imbalanced in terms of the Abnormal Right Ventricle (ARV) pathology class, which represents only 5% of the dataset.","We showcase here the impact of our generative model in augmenting underrepresented classes.","We split the data in a 70-15-15 percentage fashion (as with ACDC), and use DAA-GAN to generate 168 –high $\\mathcal {F}$ confidence– new images (matching the highest represented class) with the ARV pathology by mixing the factor that corresponds to the RV with healthy anatomies from other subjects.","We use this new balanced training set that is used to re-train VGG16 and U-Net.","As reported in Table REF , column “Un/ed Class”, augmentations improve both accuracy by 3.3% and Dice by 1.7% Dice outperforming AC-GAN and SPADE where applicable.","Does DAA-GAN augmentation improve learning of underrepresented or new domains?","As stated above, M$\\&$ Ms comprises data captured from 4 different sites (A-D), thus 4 different populations.","However, data from site D represent only 14% in the dataset.","Here we showcase the augmentation of site D. Thus, in this experiment we aim to balance the training set of M$\\&$ Ms by augmenting site D data.","We adopt the same split percentage with the previous experiments, and augment the original training set by mixing pathological factors from subjects of vendors A-C with anatomies of vendor D. Results in Table REF , column “Un/ed Vendor”, show that by augmenting site D with 101 generated images of high $\\mathcal {F}$ confidence value, we improve the two tasks' performance by 3.1% in accuracy and 1.2% Dice, respectively, while outperforming AC-GAN and SPADE.","Figure: Two examples of anatomy factor traversals.","For each step ii increases by 3: a) top row depicts the transformed factor CC, b) middle row depicts the generated images I ˜\\tilde{I}, and c) bottom row shows the factor difference between the current generated image I ˜\\tilde{I} and the input image.","LV, MYO denote the left ventricular cavity and the myocardium.Does DAA-GAN achieve good image quality?","Table REF (rightmost) reports generated image quality for each model.","Our model outperforms SPADE, which is unsurprising since SPADE has no mechanism to mix semantic priors, thus generating images with unusual heart representations due to overlapping anatomical factors (see examples generated using SPADE in Fig.","3 of the supplemental).","AC-GAN, has slightly better FID compared to DAA-GAN in both datasets quality but is the worst model in terms of controllability and utility.","Can we control DAA-GAN synthesis?","We explore this visually by generating images through manipulation of the anatomical factors of the same subject (i.e.","without mixing between subjects).","We erode and dilate only a single factor of each subject and generate images based on the altered anatomy.","From the examples in Fig.","REF , we observe that the generated cardiac parts do not correspond linearly to the generative factors when approaching extreme cases, i.e.","for large kernel size.","Thus, we argue that our model's controllability is constrained only by the dataset bias, i.e.","“unseen\" extreme (possibly unrealistic) combinations.","Ablations.","To evaluate the impact of $\\mathcal {J}$ and $\\mathcal {F}$ on the augmentation results, we replicate each experiment removing one or both modules.","From the results in Table REF , we conclude that the two modules have similar contribution to post-hoc classification improvement, $\\mathcal {F}$ slightly improves segmentation, whilst we also observe that $\\mathcal {J}$ plays an important role in the quality of the generated images."],["Conclusions","In this paper we introduced a novel framework for controllable cardiac image synthesis.","In particular, we presented a generative model that learns to produce unseen images from existing images using a user-selected combination of spatial anatomy factors.","We conducted experiments demonstrating the controllability of the generation process, whilst showcasing the potential of augmenting existing medical data with images generated using the concept of “disentangled anatomy arithmetic\".","Future work will focus on extending the capability of our model beyond simple mixing and morphology to richer anatomy arithmetic operations."],["Acknowledgement","This work was supported by the University of Edinburgh, the Royal Academy of Engineering and Canon Medical Research Europe.","This work was partially supported by the Alan Turing Institute under the EPSRC grant EP/N510129/1.","We thank Nvidia for donating a Titan-X GPU.","S. A. Tsaftaris acknowledges the support of Canon Medical and the Royal Academy of Engineering and the Research Chairs and Senior Research Fellowships scheme (grant RCSRF1819\\8\\25)."],["Experimental Setting","Both $\\mathcal {J}$ and $\\mathcal {D}$ were trained for 90 epochs, using the Adam optimizer with $\\beta _{1}=0$ , $\\beta _{2}=0.999$ , and a learning rate of 0.0001.","Our pathology classification model $\\mathcal {F}$ was pre-trained separately using the Adam optimizer with $\\beta _{1}=0.9$ , $\\beta _{2}=0.999$ , and a learning rate of 0.0001, with classification accuracy in the corresponding validation set as the early stopping criterion.","We use the the pre-trained decoder of SDNet as our generator $\\mathcal {G}$ .","During training of the generative model, the weights of $\\mathcal {F}$ and $\\mathcal {G}$ are frozenFor the ablation study, when we remove $\\mathcal {J}$ , we instead fine-tune $\\mathcal {G}$ during generative model training, with learning rate set to 0.0001..","Finally, the weights of $\\mathcal {D}$ were initialized using the Xavier method [15].","DAA-GAN is implemented using PyTorch [32], while all experiments were conducted on an Nvidia GTX 1080 Ti graphics processor.","For all experiments, when performing anatomy arithmetic we allow only one pathology per subject.","That is, we may add a factor exhibiting a certain pathology into an otherwise healthy subject, or swap one factor for another factor of the same pathology, but we do not combine two factors with different pathologies in the same subject.","During inference, our model uses $\\mathcal {J}$ and $\\mathcal {G}$ to generate an image with the targeted pathology and $\\mathcal {F}$ to generate the pathology label, while $\\mathcal {D}$ is discarded.","Figure: Example of MRI sample from ACDC cardiac dataset (top left), the predicted segmentation masks of the ROI (top right), and the 5 (out of 12) most semantic disentangled anatomical factors.","Blue, yellow, and red show the segmentation prediction for the myocardium (MYO), left ventricle (LV) and right ventricle (RV), respectively."],["Dataset Details","ACDC pathology class annotations: a) normal (NOR), b) myocardial infarction (MINF), c) dilated cardiomyopathy (DCM), d) hypertrophic cardiomyopathy (HCM), and e) abnormal right ventricle (ARV).","ACDC segmentation annotations (3 semantic classes): left ventricular cavity (LV), myocardium (MYO) of the LV, and right ventricle (RV).","All images are resampled to 1.37mm$^2$ /pixel resolution and cropped to $224\\times 224$ pixels.","M$\\&$ Ms (see also https://www.ub.edu/mnms/) comprises class annotations for 19 pathologies.","However, 15 of them represent the $12\\%$ of the subjects that corresponds to 1-3 subjects per class, thus we choose to experiment on the 4 most dominant classes that are: a) NOR, b) DCM, c) HCM, and d) ARV.","Note that ARV in our experiments is the underrepresented class, but with 26 subjects.","M$\\&$ M segmentation annotations are identical with ACDC ones.","All images are resampled to 1.2mm$^2$ /pixel resolution and cropped to $224\\times 224$ pixels.","For both datasets during both model training and evaluation we combine pairs of MR images from the same heart axis slice level.","Figure: Four SPADE-generated images using overlapped anatomical factors as input.","Arrows point to the areas where the performed arithmetic operations lead to such overlaps."]],"string":"[\n [\n \"Controllable cardiac synthesis via disentangled anatomy arithmetic\"\n ],\n [\n \"Abstract Acquiring annotated data at scale with rare diseases or conditions remains a challenge.\",\n \"It would be extremely useful to have a method that controllably synthesizes images that can correct such underrepresentation.\",\n \"Assuming a proper latent representation, the idea of a \\\"latent vector arithmetic\\\" could offer the means of achieving such synthesis.\",\n \"A proper representation must encode the fidelity of the input data, preserve invariance and equivariance, and permit arithmetic operations.\",\n \"Motivated by the ability to disentangle images into spatial anatomy (tensor) factors and accompanying imaging (vector) representations, we propose a framework termed \\\"disentangled anatomy arithmetic\\\", in which a generative model learns to combine anatomical factors of different input images such that when they are re-entangled with the desired imaging modality (e.g.\",\n \"MRI), plausible new cardiac images are created with the target characteristics.\",\n \"To encourage a realistic combination of anatomy factors after the arithmetic step, we propose a localized noise injection network that precedes the generator.\",\n \"Our model is used to generate realistic images, pathology labels, and segmentation masks that are used to augment the existing datasets and subsequently improve post-hoc classification and segmentation tasks.\",\n \"Code is publicly available at https://github.com/vios-s/DAA-GAN.\"\n ],\n [\n \"Introduction\",\n \"Whilst large scale public datasets are available for traditional vision tasks, medical data are difficult to acquire.\",\n \"Even in a large-scale medical training dataset, examples of rare diseases and anatomies are scarce.\",\n \"As a result, generalisation to observations that are not seen during training will be reduced.\",\n \"To increase the diversity of training data and for instance, to increase the incidence of rare characteristics, we would like the ability to mix and match factors that encode these variations [4] in a controllable way i.e.\",\n \"perform controllable image synthesis.\",\n \"Figure: Top: overview of the “disentangled anatomy arithmetic\\\" concept illustrated with 3 factors that represent 3 different anatomical parts of the heart (e.g.\",\n \"left/right ventricle and myocardium).\",\n \"Bottom: DAA-GAN generated example.\",\n \"Given a healthy Subject A, we aim to generate an image A' which exhibits hypertrophic cardiomyopathy (HCM).\",\n \"We select a Subject B with HCM and remove the anatomical factors from A (i.e.\",\n \"the ones that encode the myocardium and left ventricular cavity) and add the corresponding factors of B (inner part of the red circle).\",\n \"Arrows in A' point to local deformations showing the non-linear abilities of our arithmetic.\",\n \"Arithmetic operations are denoted with ±\\\\pm .The idea of generating realistic images to augment existing limited data is not new in medical image analysis.\",\n \"Generative Adversarial Networks (GANs) [16] have been used to generate variants for a given input image based on sampling from a random noise vector.\",\n \"In fact, more recent GAN architectures pursue controllability by disentangling existing factors of variation, conditioning the generation process using semantic priors (e.g.\",\n \"segmentation masks) [17], [10], [35], [22], [28], [18], [25] and class labels [20], [13], or learning cross-modality translations [29], [3], [24], [8], [11].\",\n \"An alternative to relying on sampling from a noise vector for new medical images, is the idea of mixing existing information from different populations (e.g.\",\n \"patients with different anatomical characteristics) to learn the intermediate latent space and generate more realistic data in a more controllable way.\",\n \"A concept that approximates this idea is “vector arithmetic\\\" [33], where existing vector-based latent representations are combined using simple arithmetic operations to produce new images.\",\n \"However, vector representations do not exploit the spatial equivariance of the image content and the respective task (e.g.\",\n \"segmentation) and have shown poor reconstruction quality.\",\n \"An alternative is to use to disentangled representations that use both spatial (tensor) and vector representations to capture factors of variation and permit decomposition of the input in spatially equivariant (and closely to be semantic) and imaging information [7], [9], [38].\",\n \"Herein using disentangled representations we propose the concept of “disentangled anatomy arithmetic\\\" (DAA), visualized in Fig.\",\n \"REF , that enables controllable image synthesis of plausible images with a target pathology, which we show to be useful for augmenting existing medical training data.\",\n \"We design the DAA-GAN model that learns to combine and transform spatial anatomical factors –we provide a visual example of anatomy factors in Fig.\",\n \"1 of the supplemental– from input images captured by different vendors or from different populations, and then re-entangle them with the chosen imaging factors (e.g.\",\n \"MRI) to generate unseen intermediate representations.\",\n \"Inspired by recent findings regarding the advantages of introducing spatial stochasticity in the generation process [23], [1], [14], [2], we propose a convolutional module for the combined anatomical factor representation transformation, in which structured noise is injected at the spatial locations where the arithmetic operations take place.\",\n \"Our contributions are to: Introduce the concept of “disentangled anatomy arithmetic\\\" on spatial representations of the anatomy.\",\n \"Propose DAA-GAN, a generative model that to the best of our knowledge, is the first to condition image generation using spatial anatomy factors as semantic priors.\",\n \"Propose the noise injection module that encourages local deformations to realistically blend the new factors after the arithmetic step.\",\n \"Evaluate the impact of using DAA-GAN for cardiac data augmentation in the context of a classification and a semantic segmentation post-hoc task.\"\n ],\n [\n \"Generative Model Architecture\",\n \"Our model assumes disentangled anatomy and imaging representations as inputs.\",\n \"To obtain them, we use SDNet [7] as this model provides binary spatial factors that correspond to the whole anatomy and can be used as semantic priors.\",\n \"Model overview.\",\n \"As depicted in Fig.\",\n \"REF , DAA-GAN has 4 distinct steps: 1) We combine anatomy factors, using disentangled anatomy arithmetic, to obtain a new mixed anatomical representation $\\\\mathbf {\\\\hat{C}}$ .\",\n \"2) A noise injection network $\\\\mathcal {J}$ takes $\\\\mathbf {\\\\hat{C}}$ and aims to create a plausible and more refined anatomical representation $\\\\mathbf {\\\\tilde{C}}$ .\",\n \"3) A generator $\\\\mathcal {G}$ reconstructs an image corresponding to $\\\\mathbf {\\\\tilde{C}}$ and a given imaging representation.\",\n \"4) Two critics, namely a discriminator $\\\\mathcal {D}$ and a pathology classifier $\\\\mathcal {F}$ ensure good image fidelity, but also that the reconstructed image contains the right target characteristics.\",\n \"We proceed detailing these steps.\",\n \"Figure: DAA-GAN overview (from left to right): arithmetic operations are performed between anatomical factors 𝐂 a \\\\mathbf {C}^{a} and 𝐂 b \\\\mathbf {C}^{b} to produce a mixed representation 𝐂 ^\\\\mathbf {\\\\hat{C}} which is then refined by the noise injection network 𝒥\\\\mathcal {J}.\",\n \"The generator 𝒢\\\\mathcal {G} receives this refined representation 𝐂 ˜\\\\mathbf {\\\\tilde{C}} and re-entangles it with the the input imaging factors to generate the new image I ˜\\\\tilde{I}.\",\n \"Finally, a discriminator is responsible to judge if I ˜\\\\tilde{I} is real or fake, whilst a pathology classifier assesses if I ˜\\\\tilde{I} has the desired cardiac pathology.Disentangled anatomy arithmetic.\",\n \"As shown in Fig.\",\n \"REF (top), we consider an example with 3 cardiac anatomy populations $\\\\mathbf {C}^{a}, \\\\mathbf {C}^{b}, \\\\mathbf {C}^{c}$ (e.g.\",\n \"patients from 3 imaging populations $a$ , $b$ , and $c$ ) with 3 anatomical factors per population, which –when combined– form a nested structure that corresponds to the heart region.\",\n \"These factors are extracted from $I^{a}, I^{b}, I^{c}$ medical images of dataset $Z_{\\\\sim \\\\lbrace I^{a}, I^{b}, I^{c}\\\\rbrace }$ .\",\n \"Based on this setup, we define factor arithmetic between populations any swapping operation between the corresponding factors.\",\n \"Following this example, to create a new $\\\\mathbf {C}^{a}$ anatomy by mixing factors, we first swap $\\\\mathbf {C}_{1}^{a}$ with $\\\\mathbf {C}_{1}^{c}$ , and then swap $\\\\mathbf {C}_{2}^{a}$ with $\\\\mathbf {C}_{2}^{b}$ .\",\n \"The result is an intermediate $\\\\mathbf {\\\\hat{C}}$ that is used as input to the next module of DAA-GAN.\",\n \"Note that before swapping two factors, assuming upright anatomies, we perform a registration step (warping) to align the swapped-in factor with the center of mass location of the swapped-out one.\",\n \"Noise injection.\",\n \"Since cardiac anatomy is a nested structure, the output of the arithmetic step might be non-realistic, e.g.\",\n \"have factors that overlap with each other.\",\n \"This can lead to generated images with ambiguous pathology.\",\n \"We tackle this problem with module $\\\\mathcal {J}$ , which receives $\\\\mathbf {\\\\hat{C}}$ and produces a refined representation $\\\\mathbf {\\\\tilde{C}}$ .\",\n \"Inspired by recent work of Karras et al.\",\n \"[23], we introduce stochastic variation (as Gaussian noise) at specific spatial locations of each convolutional (CONV) layer's activations.\",\n \"This variation is exploited by the network to cause local deformations around the added (swapped-in) factor(s) in order to preserve the non-overlapping nested structure of the heart (see Fig.\",\n \"REF (bottom)).\",\n \"$\\\\mathcal {J}$ consists of 4 CONV layers, whilst the noise is injected in the form of noise patches (see Fig.\",\n \"REF ) added to each CONV layer's activation features in an element-wise fashion.\",\n \"The last CONV layer is followed by a Gumbel-Softmax operator that bounds $\\\\mathbf {\\\\tilde{C}}$ to $[0,1]$ .\",\n \"Figure: Visualization of the localized Gaussian noise patch generation process.\",\n \"⊗\\\\otimes and ⊙\\\\odot denote convolution and element-wise multiplication, respectively.Generator.\",\n \"The generator is responsible for the re-entanglement of anatomy and imaging information, and by extension for the generation of a new image $\\\\tilde{I}$ .\",\n \"$G$ consists of 4 CONV-ReLU layers followed by a hyperbolic tangent activation function.\",\n \"The first CONV layer receives $\\\\mathbf {\\\\tilde{C}}$ as input, while after each CONV-ReLU block, there is an Adaptive Instance Normalization (AdaIN) layer [21] that scales and shifts the activations based on the input imaging factors.\",\n \"Since $\\\\mathbf {\\\\tilde{C}}$ has the same dimensions as $I$ and $\\\\tilde{I}$ , there is no need for any upsampling.\",\n \"Critics.\",\n \"Since for $\\\\tilde{I}$ we have no ground truth we use a discriminator to guide reconstruction.\",\n \"We adopt the architecture of the LSGAN discriminator [26] for faster convergence, better stability and ultimately better image quality (compared to a variant with spectral normalization layers adopted from [27]).\",\n \"Additionally, since we want $\\\\tilde{I}$ to have desired properties, we use a VGG16 model [36] $\\\\mathcal {F}$ to classify the pathology represented in $\\\\tilde{I}$ .\",\n \"$\\\\mathcal {F}$ has 7 CONV-BN-ReLU blocks and 3 fully-connected layers followed by a softmax function.\",\n \"Note that $\\\\mathcal {F}$ is pre-trained on the original data, and is then used as a pathology predictor during training of the generative model.\",\n \"Having presented each module in detail, we now proceed to describe the 4 losses used to train our model as a whole: Adversarial loss ($\\\\mathcal {L}_{adv})$ .\",\n \"We use an adversarial loss to encourage our model to generate realistic images.\",\n \"We choose to minimize the LSGAN loss [26] as it is more stable during training and leads to higher quality image generation compared to the traditional GAN loss.\",\n \"Our adversarial loss is defined as: ${\\\\begin{array}{c}\\\\mathcal {L}_{adv} = \\\\mathcal {L}_{\\\\mathcal {D}} + \\\\mathcal {L}_{\\\\mathcal {G}},\\\\\\\\\\\\mathcal {L}_{\\\\mathcal {D}} = \\\\frac{1}{2}\\\\mathbb {E}_{I^{\\\\prime }\\\\sim p(Z)}[(\\\\mathcal {D}(M\\\\cdot I^{\\\\prime }) - 1)^2]\\\\ +\\\\frac{1}{2}\\\\mathbb {E}_{\\\\mathbf {C}\\\\sim p(\\\\mathbf {C}_{Z})}[(\\\\mathcal {D}(M\\\\cdot \\\\mathcal {G}(\\\\mathbf {\\\\hat{C}})))^2],\\\\\\\\\\\\mathcal {L}_{\\\\mathcal {G}} = \\\\frac{1}{2}\\\\mathbb {E}_{\\\\mathbf {C}\\\\sim p(\\\\mathbf {C}_{Z})}[(\\\\mathcal {D}(\\\\mathcal {G}(\\\\mathbf {\\\\hat{C}})) - 1)^2],\\\\end{array}}$ where $I^{\\\\prime }$ is the image that contributes only the anatomy factor(s) $\\\\mathbf {C^{\\\\prime }}$ which are used to form $\\\\mathbf {\\\\hat{C}}$ (e.g.\",\n \"$I^{b}$ in Fig.\",\n \"REF ).\",\n \"$M$ is a binary mask produced by the union of the anatomical factors that contain information about the heart.\",\n \"Note that $M(j)=0$ for the remaining (i.e.\",\n \"non-heart) pixels.\",\n \"Pathology classification ($\\\\mathcal {L}_{path}$ ).\",\n \"Since we know that we add the anatomical factor $C_{k}$ to the mixed representation, we expect to be able to recognize the pathology corresponding to $C_{k}$ in the generated image $\\\\tilde{I}$ .\",\n \"To achieve this we minimize the cross entropy loss, defined as $\\\\mathcal {L}_{path} = -\\\\sum _{i=1}^{\\\\Omega }y_{i}\\\\log (p(x_{i}))$ , where $y_{i}$ and $p(x_{i})$ are the ground truth and predicted pathology labels, and $\\\\Omega $ is the number of pathology classes.\",\n \"Anatomical and background consistency ($\\\\mathcal {L}_{cons}$ and $\\\\mathcal {L}_{bg}$ ).\",\n \"$\\\\mathcal {L}_{cons}$ encourages the anatomical factors which are not related with the heart to remain unaltered after the arithmetic and noise injection steps.\",\n \"To find which pixels should not be changed, we use the blurred mask produced during the noise patch generation, denoted as $\\\\Phi (j)$ , with $\\\\Phi (j)=0$ for each pixel location $j$ that is not part of the anatomy mixing.\",\n \"We define $\\\\mathcal {L}_{cons} = \\\\frac{1}{N} \\\\sum _{j}(1-\\\\Phi (j)) ||\\\\mathbf {\\\\hat{C}}(j) - \\\\mathbf {\\\\tilde{C}}(j)||_{1}$ .\",\n \"$\\\\mathcal {L}_{bg}$ is the same as $\\\\mathcal {L}_{cons}$ , but on images.\",\n \"Namely, $\\\\mathcal {L}_{bg} = \\\\frac{1}{N} \\\\sum _{j}(1-M(j)) ||(I^{a}(j) - \\\\tilde{I(j)}) ||_{1}$ where $M$ is defined previously, and $N$ is a the total number of pixels.\",\n \"Total loss ($\\\\mathcal {L}_{total}$ ).\",\n \"These losses discussed above, are combined as: $\\\\mathcal {L}_{total} = \\\\mathcal {L}_{adv} +\\\\mathcal {L}_{path} + \\\\lambda _{1}(\\\\mathcal {L}_{cons} + \\\\mathcal {L}_{bg}),$ where $\\\\lambda _{1}=10$ is a weighting hyperparameter for the consistency losses.\"\n ],\n [\n \"Experiments\",\n \"In this section we present key information on datasets and metrics, and discuss the results of our experiments (see training details in Sec.\",\n \"1 of the supplemental).\",\n \"Data.\",\n \"We use the ACDC [5] dataset and data from the M$\\\\&$ Ms challenge [6].\",\n \"ACDC consists of cardiac images acquired from MRI scanners across 100 subjects, and provides pathology annotations (5 classes) for all images and pixel-level segmentation masks for end-systole (ES) and end-diastole (ED) per subject.\",\n \"M$\\\\&$ Ms has cardiac-MR images acquired from 4 different (known) sites (domains), across 345 subjects.\",\n \"It provides ED/ES pixel-level segmentation masks annotations for 320 (of 345) subjects and pathology annotations (4 classes) for all images (for more dataset details see Sec.\",\n \"2 of the supplemental).\",\n \"Metrics.\",\n \"We approach this in several levels.\",\n \"To measure image quality we compute the Fréchet Inception Distance (FID) [19], which quantifies the distance between feature vectors from real and generated images.\",\n \"To quantify the utility of the generated images we study if they help improving the performance of two post-hoc tasks: a) pathology classification, and b) semantic segmentation.\",\n \"For the former, we measure the classification accuracy achieved by a VGG16 model, whilst for the latter, we measure the Dice score [12], [37] achieved by a U-Net [34] model.\",\n \"The hardest is to assess controllability, i.e.\",\n \"the ability to generate images that faithfully create combinations of anatomical factors.\",\n \"We approximate this by examining the existence of the added or removed pathologies.\",\n \"We train two VGG16 classifiers, one 5-class for ACDC data and one 4-class for M$\\\\&$ Ms data, and measure the pathology classification accuracy on the generated images.\",\n \"Table: Comparing data augmentation methods in the context of the 4 questions defined in the Results section (see text for details).We report average (standard deviation as subscript) classification accuracy (Acc.)\",\n \"and segmentation performance as average Dice score.\",\n \"“n/a\\\" denotes not applicable experiment.\",\n \"* and ** denote significant improvement over the 2nd best method with p<0.05p<0.05 and p<0.1p<0.1 (Wilcoxon non-parametric test), respectively.Setup.\",\n \"We generate data with our DAA-GAN; SPADE [31] a model that conditions synthesis using segmentation masks; and AC-GAN [30] a model that conditions synthesis on class rather than semantic priors.\",\n \"Then, we train the two posthoc task-specific models (i.e.\",\n \"VGG16 and U-Net) with the following training sets (applies to both datasets): i) original data (OD), ii) OD augmented with data from DAA-GAN, iii) OD augmented with data from SPADE, and iv) OD augmented with data from AC-GAN.\",\n \"For (i)-(iv) we further augment the training set using traditional intensity (blur, gamma correction) and geometric (crop, rotate, flip, elastic transformation) augmentations for fairness (validation and test sets remain unaltered).\",\n \"Note that for each generated image, we extract and binarize only the heart-related $\\\\mathbf {\\\\tilde{C}}$ factors (output of $\\\\mathcal {J}$ ) and use them as near-ground truth masks for retraining U-Net in the context of post-hoc segmentation.\",\n \"Results.\",\n \"To demonstrate the effectiveness of our method, we answer several key questions referring to quantitative results presented in Table REF .\",\n \"Does DAA-GAN augmentation improve learning on a balanced dataset?\",\n \"For this experiment we use ACDC, which is balanced in terms of subjects per pathology class.\",\n \"We split the OD into 70%, 15%, 15% subjects for training, validation, and testing, and use DAA-GAN to generate 50 new images for the 5 pathology classes (this corresponds to a 25% samples increase).\",\n \"These images are picked based on $\\\\mathcal {F}$ confidence.\",\n \"As Table REF shows (column “Balanced”) data generated by our model lead to a 3.1% absolute classification accuracy improvement compared to using the OD with traditional augmentations, whilst outperforming both AC-GAN and SPADE.\",\n \"Regarding segmentation, compared to only using the OD, our model improves the Dice score by an absolute 1.6%.\",\n \"Since AC-GAN does not condition on masks and SPADE does not have a mechanism to refine the fused segmentation masks, they cannot be used in this experiment.\",\n \"Does DAA-GAN augmentation improve learning of underrepresented classes?\",\n \"The M$\\\\&$ Ms dataset is imbalanced in terms of the Abnormal Right Ventricle (ARV) pathology class, which represents only 5% of the dataset.\",\n \"We showcase here the impact of our generative model in augmenting underrepresented classes.\",\n \"We split the data in a 70-15-15 percentage fashion (as with ACDC), and use DAA-GAN to generate 168 –high $\\\\mathcal {F}$ confidence– new images (matching the highest represented class) with the ARV pathology by mixing the factor that corresponds to the RV with healthy anatomies from other subjects.\",\n \"We use this new balanced training set that is used to re-train VGG16 and U-Net.\",\n \"As reported in Table REF , column “Un/ed Class”, augmentations improve both accuracy by 3.3% and Dice by 1.7% Dice outperforming AC-GAN and SPADE where applicable.\",\n \"Does DAA-GAN augmentation improve learning of underrepresented or new domains?\",\n \"As stated above, M$\\\\&$ Ms comprises data captured from 4 different sites (A-D), thus 4 different populations.\",\n \"However, data from site D represent only 14% in the dataset.\",\n \"Here we showcase the augmentation of site D. Thus, in this experiment we aim to balance the training set of M$\\\\&$ Ms by augmenting site D data.\",\n \"We adopt the same split percentage with the previous experiments, and augment the original training set by mixing pathological factors from subjects of vendors A-C with anatomies of vendor D. Results in Table REF , column “Un/ed Vendor”, show that by augmenting site D with 101 generated images of high $\\\\mathcal {F}$ confidence value, we improve the two tasks' performance by 3.1% in accuracy and 1.2% Dice, respectively, while outperforming AC-GAN and SPADE.\",\n \"Figure: Two examples of anatomy factor traversals.\",\n \"For each step ii increases by 3: a) top row depicts the transformed factor CC, b) middle row depicts the generated images I ˜\\\\tilde{I}, and c) bottom row shows the factor difference between the current generated image I ˜\\\\tilde{I} and the input image.\",\n \"LV, MYO denote the left ventricular cavity and the myocardium.Does DAA-GAN achieve good image quality?\",\n \"Table REF (rightmost) reports generated image quality for each model.\",\n \"Our model outperforms SPADE, which is unsurprising since SPADE has no mechanism to mix semantic priors, thus generating images with unusual heart representations due to overlapping anatomical factors (see examples generated using SPADE in Fig.\",\n \"3 of the supplemental).\",\n \"AC-GAN, has slightly better FID compared to DAA-GAN in both datasets quality but is the worst model in terms of controllability and utility.\",\n \"Can we control DAA-GAN synthesis?\",\n \"We explore this visually by generating images through manipulation of the anatomical factors of the same subject (i.e.\",\n \"without mixing between subjects).\",\n \"We erode and dilate only a single factor of each subject and generate images based on the altered anatomy.\",\n \"From the examples in Fig.\",\n \"REF , we observe that the generated cardiac parts do not correspond linearly to the generative factors when approaching extreme cases, i.e.\",\n \"for large kernel size.\",\n \"Thus, we argue that our model's controllability is constrained only by the dataset bias, i.e.\",\n \"“unseen\\\" extreme (possibly unrealistic) combinations.\",\n \"Ablations.\",\n \"To evaluate the impact of $\\\\mathcal {J}$ and $\\\\mathcal {F}$ on the augmentation results, we replicate each experiment removing one or both modules.\",\n \"From the results in Table REF , we conclude that the two modules have similar contribution to post-hoc classification improvement, $\\\\mathcal {F}$ slightly improves segmentation, whilst we also observe that $\\\\mathcal {J}$ plays an important role in the quality of the generated images.\"\n ],\n [\n \"Conclusions\",\n \"In this paper we introduced a novel framework for controllable cardiac image synthesis.\",\n \"In particular, we presented a generative model that learns to produce unseen images from existing images using a user-selected combination of spatial anatomy factors.\",\n \"We conducted experiments demonstrating the controllability of the generation process, whilst showcasing the potential of augmenting existing medical data with images generated using the concept of “disentangled anatomy arithmetic\\\".\",\n \"Future work will focus on extending the capability of our model beyond simple mixing and morphology to richer anatomy arithmetic operations.\"\n ],\n [\n \"Acknowledgement\",\n \"This work was supported by the University of Edinburgh, the Royal Academy of Engineering and Canon Medical Research Europe.\",\n \"This work was partially supported by the Alan Turing Institute under the EPSRC grant EP/N510129/1.\",\n \"We thank Nvidia for donating a Titan-X GPU.\",\n \"S. A. Tsaftaris acknowledges the support of Canon Medical and the Royal Academy of Engineering and the Research Chairs and Senior Research Fellowships scheme (grant RCSRF1819\\\\8\\\\25).\"\n ],\n [\n \"Experimental Setting\",\n \"Both $\\\\mathcal {J}$ and $\\\\mathcal {D}$ were trained for 90 epochs, using the Adam optimizer with $\\\\beta _{1}=0$ , $\\\\beta _{2}=0.999$ , and a learning rate of 0.0001.\",\n \"Our pathology classification model $\\\\mathcal {F}$ was pre-trained separately using the Adam optimizer with $\\\\beta _{1}=0.9$ , $\\\\beta _{2}=0.999$ , and a learning rate of 0.0001, with classification accuracy in the corresponding validation set as the early stopping criterion.\",\n \"We use the the pre-trained decoder of SDNet as our generator $\\\\mathcal {G}$ .\",\n \"During training of the generative model, the weights of $\\\\mathcal {F}$ and $\\\\mathcal {G}$ are frozenFor the ablation study, when we remove $\\\\mathcal {J}$ , we instead fine-tune $\\\\mathcal {G}$ during generative model training, with learning rate set to 0.0001..\",\n \"Finally, the weights of $\\\\mathcal {D}$ were initialized using the Xavier method [15].\",\n \"DAA-GAN is implemented using PyTorch [32], while all experiments were conducted on an Nvidia GTX 1080 Ti graphics processor.\",\n \"For all experiments, when performing anatomy arithmetic we allow only one pathology per subject.\",\n \"That is, we may add a factor exhibiting a certain pathology into an otherwise healthy subject, or swap one factor for another factor of the same pathology, but we do not combine two factors with different pathologies in the same subject.\",\n \"During inference, our model uses $\\\\mathcal {J}$ and $\\\\mathcal {G}$ to generate an image with the targeted pathology and $\\\\mathcal {F}$ to generate the pathology label, while $\\\\mathcal {D}$ is discarded.\",\n \"Figure: Example of MRI sample from ACDC cardiac dataset (top left), the predicted segmentation masks of the ROI (top right), and the 5 (out of 12) most semantic disentangled anatomical factors.\",\n \"Blue, yellow, and red show the segmentation prediction for the myocardium (MYO), left ventricle (LV) and right ventricle (RV), respectively.\"\n ],\n [\n \"Dataset Details\",\n \"ACDC pathology class annotations: a) normal (NOR), b) myocardial infarction (MINF), c) dilated cardiomyopathy (DCM), d) hypertrophic cardiomyopathy (HCM), and e) abnormal right ventricle (ARV).\",\n \"ACDC segmentation annotations (3 semantic classes): left ventricular cavity (LV), myocardium (MYO) of the LV, and right ventricle (RV).\",\n \"All images are resampled to 1.37mm$^2$ /pixel resolution and cropped to $224\\\\times 224$ pixels.\",\n \"M$\\\\&$ Ms (see also https://www.ub.edu/mnms/) comprises class annotations for 19 pathologies.\",\n \"However, 15 of them represent the $12\\\\%$ of the subjects that corresponds to 1-3 subjects per class, thus we choose to experiment on the 4 most dominant classes that are: a) NOR, b) DCM, c) HCM, and d) ARV.\",\n \"Note that ARV in our experiments is the underrepresented class, but with 26 subjects.\",\n \"M$\\\\&$ M segmentation annotations are identical with ACDC ones.\",\n \"All images are resampled to 1.2mm$^2$ /pixel resolution and cropped to $224\\\\times 224$ pixels.\",\n \"For both datasets during both model training and evaluation we combine pairs of MR images from the same heart axis slice level.\",\n \"Figure: Four SPADE-generated images using overlapped anatomical factors as input.\",\n \"Arrows point to the areas where the performed arithmetic operations lead to such overlaps.\"\n ]\n]"},"id":{"kind":"string","value":"2107.01748"}}},{"rowIdx":1599,"cells":{"text":{"kind":"list like","value":[["Toward Increased Airspace Safety: Quadrotor Guidance for Targeting\n Aerial Objects"],["Abstract As the market for commercially available unmanned aerial vehicles (UAVs) booms, there is an increasing number of small, teleoperated or autonomous aircraft found in protected or sensitive airspace.","Existing solutions for removal of these aircraft are either military-grade and too disruptive for domestic use, or compose of cumbersomely teleoperated counter-UAV vehicles that have proven ineffective in high-profile domestic cases.","In this work, we examine the use of a quadrotor for autonomously targeting semi-stationary and moving aerial objects with little or no prior knowledge of the target's flight characteristics.","Guidance and control commands are generated with information just from an onboard monocular camera.","We draw inspiration from literature in missile guidance, and demonstrate an optimal guidance method implemented on a quadrotor but not usable by missiles.","Results are presented for first-pass hit success and pursuit duration with various methods.","Finally, we cover the CMU Team Tartan entry in the MBZIRC 2020 Challenge 1 competition, demonstrating the effectiveness of simple line-of-sight guidance methods in a structured competition setting."],["Motivation","Micro Aerial Vehicles (MAVs), also referred to as drones, Unmanned Aerial Vehicles (UAVs) and small Unmanned Aerial Systems (sUAS), have seen a huge growth in various market sectors across the globe.","Business Insider projects the sale of drones to surpass $12 billion in 2021, of which consumer drone shipments will comprise 29 million units [5].","Enterprise and governmental sectors generally have strict regulations under which MAVs are operated; however, not only is the consumer sector's operation of these aircraft weakly regulated but there is strong community pushback against any such legislation.","Stronger oversight of private drone use is further motivated by numerous incidents involving small, typically teleoperated drones in public spaces.","In December 2018, dozens of drone sighting reports over Gatwick Airport, near London, affected 1,000 flights and required the help of both the police and military, neither of whom were able to capture the drone over a 24-hour period [6].","Worries over the potential of UAVs above crowds at the 2018 Winter Olympics prompted South Korean authorities to train to disable drones, including developing a teleoperated platform which disables others with a dropped net (Figure REF ) [7].","Beyond these documented examples, there are numerous videos online of recreational drone users losing control of their aircraft due to weather conditions, loss of GPS positioning, or low battery behavior.","Figure: Current possible domestic-use counter-UAV systems in development.While these issues could be mitigated by enforcing strict regulations and oversight on the consumer drone market, this may also drastically curb the independence of hobbyists and researchers.","A potential alternative may be to capture or disable rogue UAVs in a non-disruptive way.","Current anti-UAV technology exists primarily in the military sector, in the form of jammers (used to disrupt the teleoperation signal from a nearby radio controller) or Stinger missiles (meant to disable a UAV by impact).","Neither of these options are suitable for domestic use, where both noise and debris are of issue.","Therefore, we need a solution that minimizes destruction while being agile enough to capture a wide variety of MAVs in local airspaces.","Quadrotors benefit from high, 4-degree of freedom (DOF) maneuverability and can accelerate to high speeds quicker than some single-rotor or fixed-wing counterparts.","This implies a higher capability to stay on an impact trajectory towards a target with an unknown flight plan or characteristics.","Furthermore, recent research in aerial robotics has shown that a suite of obstacle avoidance, detection, planning, and control can run fully onboard on an autonomous quadrotor platform.","Common shortfalls of quadrotors include low battery life, but for this mission type, flights are short but with high accelerations (and therefore, higher energy throughput)."],["Challenges and Approach","The challenges of autonomously impacting an unknown small aerial vehicle with a UAV are numerous, involving fast flight through potentially cluttered environments, as well as the development of a mechanically sound method of capturing the target without damage to either agent.","However, the primary challenge addressed in this thesis surrounds guidance and control of a quadrotor UAV towards a target.","In this thesis, we describe two projects to address this challenge.","In the first study, multiple control and guidance methods derived and inspired from different fields of literature are reviewed and modified for use on a simulated UAV-target scenario.","Here, the perception task is simplified and environmental factors are eliminated to focus on the evaluation of several guidance methods.","The second study evaluates LOS guidance in a robotics competition setting, specifically comprising of the CMU Team Tartan entry in the Mohamed Bin Zayed International Robotics Challenge 2020 Challenge 1.","This effort includes work on (a) planning an adjustable and robust path around a fixed arena based on measured GPS coordinates, (b) control towards semi-stationary targets placed throughout the arena, and (c) detection of a small yellow ball moving at 8m/s against a cluttered background.","A further challenge in this work is the localization of the target in the world relative to the UAV.","Depending on the size and shape of target (e.g.","fixed wing, multirotor, helicopter, blimp) as well as its distance, it cannot be assumed that a 3D sensor, such as LIDAR or stereo vision, can be used to accurately localize the target in space.","For example, because of their sparse and sometimes mesh-structured frames, multirotors in particular can be notoriously difficult to localize with cheap and lightweight scanning LIDARs or stereo cameras at long range.","Therefore, the focus in this thesis is to use monocular vision and adapt guidance methods to use only approximate depth estimates when necessary."],["Contribution and Outline","The main contributions of this thesis are as follows.","An evaluation of various guidance and control methods and how they might be adapted for use on a quadrotor in a simulated environment.","A software system using LOS-guidance for finding and targeting semi-stationary and moving targets within a fixed arena.","Chapter presents a short summary of related work in various fields, including classical visual servoing, missile guidance, and trajectory generation and tracking.","The following two chapters, and , expand on the work done specifically towards the two contributions listed above, respectively.","Chapter describes conclusions drawn from this work, shortcomings of the approach, as well as suggested future directions."],["Classical Visual Servoing","Visual servoing spans a wide range of research focusing on controlling robot links relative to input from visual sensors.","The most common application of visual servoing is in pick-and-place operations done with robotic arms fitted with cameras.","These robots generally either have a eye-in-hand (closed-loop control) or eye-to-hand (open-loop control) setup [9].","Two of the most common approaches in this field are image-based visual servoing (IBVS) and pose-based visual servoing (PBVS), with the difference between the two involving the estimation of the target's relative pose to the robot [10].","IBVS, as described in Hutchinson, et al.","(1996), is only useful within a small region of the task space unless the image Jacobian is computed online with knowledge of the distance-to-target, which is further complicated by a monocular vision-based system.","Unless target velocity is constant, errors or lag in the image plane with a moving target introduces errors in servoing with either method, which would in turn have to be tuned out with a more complex control system.","Chaumette and Santos (1993) [11] tracked a moving target with IBVS but assumed a constant acceleration.","When the target maneuvered abruptly, the Kalman filter-based target motion predictor took some cycles of feedback to recalibrate to the new motion.","In [12], the major pitfall of PBVS is pointed out as the need for 3D information of the target, specifically the depth which may not be readily available."],["Missile Guidance","Homing air missiles are singularly focused on ensuring impact with an aerial target.","Since at least 1956, proportional navigation in some form has been a standard in missile guidance methods [1].","Adler (1956) describes a couple of such methods, including constant-bearing navigation and proportional navigation.","It is noted that constant-bearing navigation, which applies acceleration to maintain a constant bearing-to-target, requires instantaneous corrections to deviations in the line-of-sight (LOS) direction.","This renders it incompatible with the dynamics of missiles, which cannot directly satisfy lateral acceleration commands (similar to fixed-wing aircraft); therefore, Adler proposes using 3D proportional navigation (PN) which applies a turning rate proportional to the LOS direction change.","In later texts, the term proportional navigation is used interchangeably between these two schemes, and also extended to other similar methods.","In this thesis, PN will be used as a general term to refer to any control law using the LOS rotation rate to generate an acceleration command.","As noted in [4], PN, when assuming no autopilot lag, is an optimal control law that assumes very little about the acceleration characteristics of the target.","However, variations on classical PN have also been developed that adapt to different flight profiles, including constant-acceleration and constant-jerk [13].","PN is typically split into two categories, the “true\" variant and the “pure\" variant [2].","Though the naming is largely arbitrary, the primary difference lies in the reference frame in which the lateral acceleration is applied to the pursuing missile.","True PN applies this acceleration orthogonal to the current missile velocity; Pure PN applies the acceleration orthogonal to the current LOS towards the target.","Generalized True PN (as seen in Figure REF from [2]) is not covered in this thesis.","Figure: Comparison of pure and true proportional navigation.","A M A_M and V M V_M refer to the missile's desired acceleration and current velocity, respectively ."],["Trajectory Generation and Tracking","Trajectories provide robots with smooth or kinodynamically feasible paths to follow through its state space.","This is opposed to sending raw differential commands, which may exceed the robot's limitations and lead to controller instability or even to physical breakdown of the robot's actuators.","As such, there has been extensive work in the generation and following of trajectories for use with various types of robots and applications, primarily with robot arms for grasping and self-driving vehicles [14][15][16][17].","This has been extended to MAVs to ensure smooth and efficient flight.","Richter, et al.","(2013) [18] showed that polynomial trajectories eliminated the need for an extensive sample-based search over the state space of the vehicle.","This approach, while not providing asymptotic convergence to the global optimal path, ensured smooth and fast flight of the quadrotor.","In [3], it was shown that with continuous minimum-jerk trajectories and fast re-planning, they achieved higher trajectory smoothness compared to other, more reactive planners.","Figure REF shows the smooth trajectory generated by tracking motion primitive-generated paths.","Gao, et al.","(2018) [19] first finds a time-indexed minimal arrival path that may not be feasible for quadrotor flight, and then forms a surrounding free-space flight corridor in which they generate a feasible trajectory.","They use a Bernstein polynomial basis and represent the trajectory as a piecewise Bézier curve.","Figure: Example of concatenated trajectories in a cluttered environment.","Colored lines represent individual motion primitives; the quadrotor tracks the initial part of every generated trajectory, forming a complete, stitched trajectory represented by the black line.","To follow trajectories, controllers take in a desired trajectory typically composed of position waypoints each with an associated velocity, and issue actuator commands to the robot.","In the MAV case, an autopilot software may accept attitude or attitude-rate commands which come from such a controller.","Hoffman, et al.","(2008) [20] demonstrated a controller that took as input a non-feasible trajectory and outputted feasible attitude commands for a quadrotor that accurately followed the original path.","This was demonstrated outdoors with 50cm accuracy.","A similar approach was taken (but extended to 3D) for the path following controller implemented in [21].","Here, the desired position and desired velocity from the closest point on the trajectory are used to find the position and velocity errors of the robot.","These are used to calculate the desired robot acceleration with PD feedback and a feedforward term.","As described in Section , servoing towards moving targets is challenging with classical methods.","As such, LOS-based guidance principals (Section ) and trajectory following methods () may produce better results when target acceleration is nonzero.","In addition, the quadrotor platform's control limits might be avoided with smooth trajectory-based methods.","This chapter focuses on the development and evaluation of various such guidance methods to achieve impact with a generalized form of an aerial, mobile target.","No information is known about the target other than its color, which is used for segmentation in an RGB image to simplify the detection and tracking problem."],["Derivation of True Proportional Navigation Guidance Law","Line-of-sight (LOS) guidance methods are used to apply acceleration commands to the pursuer that minimize change in the LOS vector towards the target.","In this section, the basic LOS geometry is introduced and used to derive proportional navigation (PN) guidance.","Following subsections show how this is used, with target detections, to calculate quantities used for the applied PN guidance.","Figure: LOS coordinate system .As seen in Figure REF , the fixed world coordinate frame is specified by the unit vectors $\\bar{\\mathbf {1}}_{\\mathbf {x}}$ , $\\bar{\\mathbf {1}}_{\\mathbf {y}}$ , $\\bar{\\mathbf {1}}_{\\mathbf {z}}$ .","The LOS coordinate frame, attached to the moving missile, is specified by the unit vectors $\\bar{\\mathbf {1}}_{\\mathbf {r}}$ , $\\bar{\\mathbf {1}}_{\\mathbf {n}}$ , $\\bar{\\mathbf {1}}_{\\mathbf {\\omega }}$ ; $\\bar{\\mathbf {1}}_{\\mathbf {r}}$ points along the LOS $\\bar{\\mathbf {r}}$ ; $\\bar{\\mathbf {1}}_{\\mathbf {n}}$ is the change in direction (i.e.","a rotation) of the LOS vector; $\\bar{\\mathbf {1}}_{\\mathbf {\\omega }}$ is the cross product of the former two, in that order (forming a right-handed coordinate frame).","In general, the angular velocity of the LOS coordinate frame is given by: $\\bar{\\dot{\\phi }}= \\dot{\\Phi }_{\\mathbf {r}} \\bar{\\mathbf {1}}_{\\mathbf {r}}+ \\dot{\\Phi }_{\\mathbf {n}} \\bar{\\mathbf {1}}_{\\mathbf {n}}+ \\dot{\\Phi }_{\\mathbf {\\omega }} \\bar{\\mathbf {1}}_{\\mathbf {\\omega }}$ Where $\\dot{\\Phi }_{\\mathbf {r}}$ , $\\dot{\\Phi }_{\\mathbf {n}}$ , $\\dot{\\Phi }_{\\mathbf {\\omega }}$ are the magnitudes of the components of the angular velocity defined as: $\\dot{\\Phi }_{\\mathbf {r}} &= \\bar{\\dot{\\phi }} \\mathchoice{\\mathbin {\\hbox{\\scalebox {.5}{$\\m@th \\displaystyle \\bullet $}}}}{}{}{}$ 1r n = 1n = 1 As derived in [4] but not reproduced here, the components of the relative acceleration between the missile and target are: $(\\bar{\\mathbf {a}}_T - \\bar{\\mathbf {a}}_M) \\mathchoice{\\mathbin {\\hbox{\\scalebox {.5}{$\\m@th \\displaystyle \\bullet $}}}}{}{}{}$ 1r = R - R 2 (aT - aM) 1n = 2R + R (aT - aM) 1 = R r Where $\\bar{\\mathbf {a}}_T$ and $\\bar{\\mathbf {a}}_M$ are the target and missile accelerations, respectively.","From this result, specifically using the condition in Equation REF , we can list sufficient conditions to satisfy the equation and ensure intercept: (i) interceptor is able to achieve an acceleration along the LOS greater than that of the target ($(\\bar{\\mathbf {a}}_T - \\bar{\\mathbf {a}}_M) \\mathchoice{\\mathbin {\\hbox{\\scalebox {.5}{$\\m@th \\displaystyle \\bullet $}}}}{}{}{}$ 1r < 0$), (\\textit {ii}) the initial rate of change in the range $ R$ is negative ($ R < 0$), which then ensures $ R < 0$ given the first condition, and (\\textit {iii}) the rate of change in the LOS is $ 0$ ($ = 0$).","Condition (\\textit {i}) depends on the nature of the interceptor and target; condition (\\textit {ii}) implies that PN pursuit must be initialized with a positive closing velocity; condition (\\textit {iii}) implies that the interceptor must satisfy acceleration commands such that the LOS vector remains constant.","Palumbo, et al.","(2010) finds the following true PN (TPN) law that ensures system stability:$ $\\bar{\\mathbf {a}}_M \\mathchoice{\\mathbin {\\hbox{\\scalebox {.5}{$\\m@th \\displaystyle \\bullet $}}}}{}{}{}$ 1n = N Vc , N>2 Where $N$ is a proportional gain and $V_c$ is the closing velocity.","In other words, the interceptor acceleration $\\bar{\\mathbf {a}}_M$ must have a component, orthogonal to the LOS, proportional to the rotation rate of the LOS as specified in Equation REF ."],["True Proportional Navigation","To generate the desired acceleration vector with magnitude specified by Equation REF and direction orthogonal to the LOS, we first calculate both $\\dot{\\Phi }_{\\mathbf {\\omega }}$ (directly represented in Equation REF ) and $\\mathbf {1_{\\mathbf {n}}}$ (acceleration direction).","The target's centroid in image frame coordinates at times $t-1$ and $t$ is represented by $(u_{t-1},v_{t-1})$ and $(u_{t},v_{t})$ , respectively, as shown in Figure REF .","The camera principal point, specified in the calibrated camera's intrinsic matrix (typically denoted $\\mathbf {K}$ ), is represented by $(c_x,c_y)$ .","Figure: Top-down diagram showing intermediate quantities in calculation of desired acceleration command.The LOS vector $\\mathbf {r_t}$ in the camera's frame of reference at time $t$ is given by the following.","$\\mathbf {r_t} = \\begin{bmatrix}[1.75] \\dfrac{u_t-c_x}{f_x} \\\\ \\dfrac{v_t-c_y}{f_y} \\\\ 1 \\end{bmatrix}$ In the special case of the first iteration of the algorithm, at $t=0$ , the LOS vector is calculated according to Equation REF then stored for use as $\\mathbf {r_{t-1}}$ the upcoming iteration.","For the $t=0$ computation cycle the control output is set to $\\mathbf {0}$ .","The angle spanned by the two vectors $\\mathbf {r_{t}}$ and $\\mathbf {r_{t-1}}$ is as follows: $\\Phi _{\\mathbf {\\omega }} = \\arccos {\\dfrac{ <\\mathbf {r_{t}},\\mathbf {r_{t-1}}> }{ ||\\mathbf {r_{t}}||||\\mathbf {r_{t-1}}|| }}$ Therefore, if the difference in time for one cycle is represented by $\\Delta t$ , then the magnitude of the rotation rate of the LOS vector is: $\\dot{\\Phi }_{\\mathbf {\\omega }} = \\frac{\\Phi _{\\mathbf {\\omega }}}{\\Delta t}$ The direction of the acceleration (direction of the LOS rotation) $\\mathbf {1_{\\mathbf {n}}}$ is shown in Figure REF and calculated below.","$\\mathbf {n} &= \\mathbf {r_{t}} - \\texttt {proj}(\\mathbf {r_{t}}, \\mathbf {r_{t-1}})\\\\ &= \\mathbf {r_{t}} - <\\mathbf {r_{t}}, \\frac{\\mathbf {r_{t-1}}}{||\\mathbf {r_{t}}||}> \\dfrac{\\mathbf {r_{t-1}}}{||\\mathbf {r_{t}}||}\\\\\\mathbf {1_{\\mathbf {n}}} &= \\dfrac{\\mathbf {n}}{||\\mathbf {n}||}$ Where the vector projection of $\\mathbf {r_{t}}$ onto $\\mathbf {r_{t-1}}$ is represented by the red vector in Figure REF .","With the scalar $\\dot{\\Phi }_{\\mathbf {\\omega }}$ and the vector $\\mathbf {1_{\\mathbf {n}}}$ , we can compute the desired acceleration as follows.","$\\mathbf {a}_{LOS^{\\prime }} = N V_c \\dot{\\Phi }_{\\mathbf {\\omega }} \\mathbf {1_{\\mathbf {n}}}$ In application on a quadrotor, in this work, the acceleration vector is fed into a velocity controller by integration of the command, which submits a roll, pitch, yawrate, thrust command to the internal autopilot controllers.","Therefore, rather than adjusting heading to satisfy lateral accelerations, the application of TPN in this work relies on roll angle control.","This more direct method of achieving lateral accelerations (that does not require forward velocity) is not possible on a missile or fixed-wing aircraft."],["Proportional Navigation with Heading Control","The TPN algorithm presented above maintains the integrity of the algorithm commonly presented in missile guidance literature, but applies the control command more directly by controlling the roll angle of the UAV.","During a missile's flight, the vehicle fulfills desired acceleration commands by flying in an arc, gradually changing its heading by relying on forward motion and the use of thrust vectoring or control surfaces.","A quadrotor, however, has direct control over its heading by applying yaw-rate control.","In this section, we describe an algorithm that uses PN acceleration in all axes but the lateral axis, and instead controls the heading to achieve lateral acceleration.","Since it does not utilize PN in the lateral axis, we do not assign the label of “true\".","We define an inertial reference frame at the center of the UAV, with $x$ pointing forward, $y$ pointing to the left, and $z$ point upward.","The acceleration along the $x$ and $z$ axes are simply taken from Equation REF as the corresponding components: $a_{x} &= \\mathbf {a}_{LOS^{\\prime }} \\mathchoice{\\mathbin {\\hbox{\\scalebox {.5}{$\\m@th \\displaystyle \\bullet $}}}}{}{}{}$ 1x az = aLOS' 1z The heading control composes of a commanded yaw-rate, which includes the computation of the heading: $\\dot{yaw} &= K_{P,yaw} heading\\\\&= K_{P,yaw} \\arctan { \\dfrac{\\mathbf {r_t} \\mathchoice{\\mathbin {\\hbox{\\scalebox {.5}{$\\m@th \\displaystyle \\bullet $}}}}{}{}{}}{\\mathbin {\\hbox{\\scalebox {.5}{$\\m@th \\textstyle \\bullet $}}}}}{\\mathbin {\\hbox{\\scalebox {.5}{$\\m@th \\scriptstyle \\bullet $}}}}$ 1yrt 1x Where $K_{P,yaw}$ is a tuned proportional gain and $\\mathbf {r_t}$ is the current LOS vector."],["Hybrid TPN-Heading Control","There are potential benefits to both methods presented in Sections REF and REF .","TPN specifically applied to quadrotors via roll angle control might yield quicker reaction time for a moving object.","PN while keeping the target centered in the frame ensures that the target is not lost from frame; otherwise, in a full system, the pursuing UAV would have to return to a search pattern.","The goal of the hybrid algorithm is to capture the advantages of both methods.","This method switches between the two modes, PN and Heading.","The transition between them simply relies on a tuned threshold $k_{heading}$ on the heading towards the target.","If $|heading| < k_{heading}$ , enter state PN: $a_{x} &= \\mathbf {a}_{LOS^{\\prime }} \\mathchoice{\\mathbin {\\hbox{\\scalebox {.5}{$\\m@th \\displaystyle \\bullet $}}}}{}{}{}$ 1x ay = aLOS' 1y az = aLOS' 1z yaw = 0.2 KP,yaw heading If $|heading| \\ge k_{heading}$ , enter state Heading Control: $a_{x} &= \\mathbf {a}_{LOS^{\\prime }} \\mathchoice{\\mathbin {\\hbox{\\scalebox {.5}{$\\m@th \\displaystyle \\bullet $}}}}{}{}{}$ 1x ay = 0.2 aLOS' 1y az = 0 yaw = KP,yaw heading Where $k_{heading}$ may be tuned depending on certain factors of the UAV system, including the camera field-of-view (FOV) or the maximum yaw-rate.","Note that at all times, regardless of the heading, both $a_{y}$ and $\\dot{yaw}$ are nonzero, and are instead suppressed with a factor less than 1.","The factor of 0.2 was found empirically in this study to perform well and yield an appropriate influence of both acceleration and yaw-rate.","Using this hybrid method, the UAV may potentially be able to react to changes in target motion while also keeping it in view."],["Trajectory Following","All trajectory following methods were implemented with some replanning rate at which updated trajectories are published.","Replanning is constantly done from the look ahead point, which is maintained by the Trajectory Controller as described in Section REF ."],["$LOS^{\\prime }$ Acceleration Trajectory Following","These trajectories are formed by taking the desired acceleration command calculated with Equation REF and calculating position waypoints and velocities given the starting position set to the look ahead point.","The calculations for the positions and velocities are done with the following kinematic equations, set in the UAV's inertial reference frame.","$\\mathbf {p}_t &= \\mathbf {v}_0t + \\frac{1}{2} \\mathbf {a}_{LOS^{\\prime }} t^2\\\\\\mathbf {v}_t &= \\mathbf {v}_0 + \\mathbf {a}_{LOS^{\\prime }} t , \\, t=0,0.1,...,T$ Where $T$ is the time length of each trajectory.","As $T$ approaches 0, this method becomes equivalent to commanding the desired $LOS^{\\prime }$ acceleration directly.","The discretization of the timesteps is also a tunable parameter."],["Target Motion Forecasting","In its simplest form, target motion forecasting involves estimating the target velocity in 3D, calculating the future location of the target with a straight-line path, and generating a collision course trajectory towards that point in space at a velocity which completes the trajectory at the specified time.","This method makes three critical assumptions: (i) the target has zero acceleration, (ii) we can approximate time-to-collision by calculating the time along the current LOS vector, and (iii) the forecasted target position will change slowly, so generating straight-path trajectories is sufficient to result in a final, smooth stitched trajectory.","First, the LOS unit vectors at two times are calculated.","These are used along with the depth estimation to find the target's velocity: $\\mathbf {v}_{target} = \\dfrac{d_1(\\mathbf {1_{LOS}}_1)-d_0(\\mathbf {1_{LOS}}_0)}{t_1-t_0}$ $d_1$ and $d_0$ , $\\mathbf {1_{LOS}}_1$ and $\\mathbf {1_{LOS}}_0$ , and $t_1$ and $t_0$ are the estimated depth, calculated LOS unit vector, and time, at two timesteps.","Once we have the velocity, we find the approximate time-to-collision along the current LOS vector by using the UAV's velocity component along the LOS: $t_{collision} = \\dfrac{d_1}{<\\mathbf {v}_{UAV}, \\mathbf {1_{LOS}}_1>}$ Where $\\mathbf {v}_{UAV}$ is the current UAV velocity vector.","Therefore, the approximate point in space to plan the trajectory to is as follows, where $\\mathbf {p}_{collision}$ is in the UAV reference frame.","$\\mathbf {p}_{collision} = \\mathbf {v}_{target}t_{collision} + d_1(\\mathbf {1_{LOS}}_1)$"],["System","Figure REF shows a diagram of the most important parts of the system, including perception, control, and the simulation software.","Each block generally consists of one or two nodes in ROS (Robot Operation System), the framework used in this work."],["Object Segmentation","Since the focus for this chapter was on the comparison of guidance methods, the perception challenge was simplified.","The simulation does not include rich background visuals, there is no dropout in the camera feed, and the object was reduced from a potential UAV shape to a yellow sphere.","The segmentation node composes of color thresholding that creates a binary segmentation image from each camera frame in real-time (30Hz), as seen in Figure REF .","In addition, this node publishes the centroid $(C_x,C_y)$ of the detected object by using the image moments.","$\\begin{bmatrix}C_x \\\\ C_y\\end{bmatrix}=\\begin{bmatrix}[2.0]\\dfrac{M_{10}}{M_{00}} \\\\ \\dfrac{M_{01}}{M_{00}}\\end{bmatrix}$"],["Object Depth Estimation","Depth estimation with monocular camera images relies on prior knowledge of the target size as well as the assumption that the target is spherical (which is correct in this simulation, but extendable to real-life targets with approximate knowledge of target or aircraft type).","The estimate takes into account the camera intrinsic parameters and projects the 2D information (pixels) to 3D (rays starting at the camera center) to extract the depth.","Pixel coordinates at two points in the image need to be located which form a 2D line in the image that projects to a 3D line along the diameter of the object.","The easiest found such line is the line through the image principal point that passes through the centroid, since the target is a sphere.","This line is the dotted line passing through the target in Figure REF .","We must first find the pixel coordinates along this line that correspond to the object edges in the image.","First, we find the rotation angle formed by the centroid coordinates with respect to the image horizontal, labeled $\\theta $ in Figure REF .","$\\theta = \\arctan \\dfrac{C_y-c_y}{C_x-c_x}$ With this angle we can use a rotation matrix to rotate the segmented object onto the $y=c_y$ line.","Note that we use $-\\theta $ to achieve the correct rotation, and that we first compute new pixel coordinates $(u_i,v_i)$ relative to the image center, from the original coordinates $(x_i,y_i)$ .","$\\begin{bmatrix}u_{i} \\\\ v_{i}\\end{bmatrix}&=\\begin{bmatrix}x_i - c_x \\\\ y_i - c_y\\end{bmatrix}, \\; i=1,...,N \\\\\\begin{bmatrix}u_{i,R} \\\\ v_{i,R}\\end{bmatrix}&=\\begin{bmatrix}cos(-\\theta ) && -sin(-\\theta )\\\\sin(-\\theta ) && cos(-\\theta )\\end{bmatrix}\\begin{bmatrix}u_{i} \\\\ v_{i}\\end{bmatrix}, \\; i=1,...,N$ Where N is the total number of segmented pixels in the binary segmentation image.","Once the object is rotated onto the image horizontal, simple min and max operations yield the extreme pixel coordinates on the left and right of the target (or right and left, if $C_x2 Where $N$ is a proportional gain and $V_c$ is the closing velocity.\",\n \"In other words, the interceptor acceleration $\\\\bar{\\\\mathbf {a}}_M$ must have a component, orthogonal to the LOS, proportional to the rotation rate of the LOS as specified in Equation REF .\"\n ],\n [\n \"True Proportional Navigation\",\n \"To generate the desired acceleration vector with magnitude specified by Equation REF and direction orthogonal to the LOS, we first calculate both $\\\\dot{\\\\Phi }_{\\\\mathbf {\\\\omega }}$ (directly represented in Equation REF ) and $\\\\mathbf {1_{\\\\mathbf {n}}}$ (acceleration direction).\",\n \"The target's centroid in image frame coordinates at times $t-1$ and $t$ is represented by $(u_{t-1},v_{t-1})$ and $(u_{t},v_{t})$ , respectively, as shown in Figure REF .\",\n \"The camera principal point, specified in the calibrated camera's intrinsic matrix (typically denoted $\\\\mathbf {K}$ ), is represented by $(c_x,c_y)$ .\",\n \"Figure: Top-down diagram showing intermediate quantities in calculation of desired acceleration command.The LOS vector $\\\\mathbf {r_t}$ in the camera's frame of reference at time $t$ is given by the following.\",\n \"$\\\\mathbf {r_t} = \\\\begin{bmatrix}[1.75] \\\\dfrac{u_t-c_x}{f_x} \\\\\\\\ \\\\dfrac{v_t-c_y}{f_y} \\\\\\\\ 1 \\\\end{bmatrix}$ In the special case of the first iteration of the algorithm, at $t=0$ , the LOS vector is calculated according to Equation REF then stored for use as $\\\\mathbf {r_{t-1}}$ the upcoming iteration.\",\n \"For the $t=0$ computation cycle the control output is set to $\\\\mathbf {0}$ .\",\n \"The angle spanned by the two vectors $\\\\mathbf {r_{t}}$ and $\\\\mathbf {r_{t-1}}$ is as follows: $\\\\Phi _{\\\\mathbf {\\\\omega }} = \\\\arccos {\\\\dfrac{ <\\\\mathbf {r_{t}},\\\\mathbf {r_{t-1}}> }{ ||\\\\mathbf {r_{t}}||||\\\\mathbf {r_{t-1}}|| }}$ Therefore, if the difference in time for one cycle is represented by $\\\\Delta t$ , then the magnitude of the rotation rate of the LOS vector is: $\\\\dot{\\\\Phi }_{\\\\mathbf {\\\\omega }} = \\\\frac{\\\\Phi _{\\\\mathbf {\\\\omega }}}{\\\\Delta t}$ The direction of the acceleration (direction of the LOS rotation) $\\\\mathbf {1_{\\\\mathbf {n}}}$ is shown in Figure REF and calculated below.\",\n \"$\\\\mathbf {n} &= \\\\mathbf {r_{t}} - \\\\texttt {proj}(\\\\mathbf {r_{t}}, \\\\mathbf {r_{t-1}})\\\\\\\\ &= \\\\mathbf {r_{t}} - <\\\\mathbf {r_{t}}, \\\\frac{\\\\mathbf {r_{t-1}}}{||\\\\mathbf {r_{t}}||}> \\\\dfrac{\\\\mathbf {r_{t-1}}}{||\\\\mathbf {r_{t}}||}\\\\\\\\\\\\mathbf {1_{\\\\mathbf {n}}} &= \\\\dfrac{\\\\mathbf {n}}{||\\\\mathbf {n}||}$ Where the vector projection of $\\\\mathbf {r_{t}}$ onto $\\\\mathbf {r_{t-1}}$ is represented by the red vector in Figure REF .\",\n \"With the scalar $\\\\dot{\\\\Phi }_{\\\\mathbf {\\\\omega }}$ and the vector $\\\\mathbf {1_{\\\\mathbf {n}}}$ , we can compute the desired acceleration as follows.\",\n \"$\\\\mathbf {a}_{LOS^{\\\\prime }} = N V_c \\\\dot{\\\\Phi }_{\\\\mathbf {\\\\omega }} \\\\mathbf {1_{\\\\mathbf {n}}}$ In application on a quadrotor, in this work, the acceleration vector is fed into a velocity controller by integration of the command, which submits a roll, pitch, yawrate, thrust command to the internal autopilot controllers.\",\n \"Therefore, rather than adjusting heading to satisfy lateral accelerations, the application of TPN in this work relies on roll angle control.\",\n \"This more direct method of achieving lateral accelerations (that does not require forward velocity) is not possible on a missile or fixed-wing aircraft.\"\n ],\n [\n \"Proportional Navigation with Heading Control\",\n \"The TPN algorithm presented above maintains the integrity of the algorithm commonly presented in missile guidance literature, but applies the control command more directly by controlling the roll angle of the UAV.\",\n \"During a missile's flight, the vehicle fulfills desired acceleration commands by flying in an arc, gradually changing its heading by relying on forward motion and the use of thrust vectoring or control surfaces.\",\n \"A quadrotor, however, has direct control over its heading by applying yaw-rate control.\",\n \"In this section, we describe an algorithm that uses PN acceleration in all axes but the lateral axis, and instead controls the heading to achieve lateral acceleration.\",\n \"Since it does not utilize PN in the lateral axis, we do not assign the label of “true\\\".\",\n \"We define an inertial reference frame at the center of the UAV, with $x$ pointing forward, $y$ pointing to the left, and $z$ point upward.\",\n \"The acceleration along the $x$ and $z$ axes are simply taken from Equation REF as the corresponding components: $a_{x} &= \\\\mathbf {a}_{LOS^{\\\\prime }} \\\\mathchoice{\\\\mathbin {\\\\hbox{\\\\scalebox {.5}{$\\\\m@th \\\\displaystyle \\\\bullet $}}}}{}{}{}$ 1x az = aLOS' 1z The heading control composes of a commanded yaw-rate, which includes the computation of the heading: $\\\\dot{yaw} &= K_{P,yaw} heading\\\\\\\\&= K_{P,yaw} \\\\arctan { \\\\dfrac{\\\\mathbf {r_t} \\\\mathchoice{\\\\mathbin {\\\\hbox{\\\\scalebox {.5}{$\\\\m@th \\\\displaystyle \\\\bullet $}}}}{}{}{}}{\\\\mathbin {\\\\hbox{\\\\scalebox {.5}{$\\\\m@th \\\\textstyle \\\\bullet $}}}}}{\\\\mathbin {\\\\hbox{\\\\scalebox {.5}{$\\\\m@th \\\\scriptstyle \\\\bullet $}}}}$ 1yrt 1x Where $K_{P,yaw}$ is a tuned proportional gain and $\\\\mathbf {r_t}$ is the current LOS vector.\"\n ],\n [\n \"Hybrid TPN-Heading Control\",\n \"There are potential benefits to both methods presented in Sections REF and REF .\",\n \"TPN specifically applied to quadrotors via roll angle control might yield quicker reaction time for a moving object.\",\n \"PN while keeping the target centered in the frame ensures that the target is not lost from frame; otherwise, in a full system, the pursuing UAV would have to return to a search pattern.\",\n \"The goal of the hybrid algorithm is to capture the advantages of both methods.\",\n \"This method switches between the two modes, PN and Heading.\",\n \"The transition between them simply relies on a tuned threshold $k_{heading}$ on the heading towards the target.\",\n \"If $|heading| < k_{heading}$ , enter state PN: $a_{x} &= \\\\mathbf {a}_{LOS^{\\\\prime }} \\\\mathchoice{\\\\mathbin {\\\\hbox{\\\\scalebox {.5}{$\\\\m@th \\\\displaystyle \\\\bullet $}}}}{}{}{}$ 1x ay = aLOS' 1y az = aLOS' 1z yaw = 0.2 KP,yaw heading If $|heading| \\\\ge k_{heading}$ , enter state Heading Control: $a_{x} &= \\\\mathbf {a}_{LOS^{\\\\prime }} \\\\mathchoice{\\\\mathbin {\\\\hbox{\\\\scalebox {.5}{$\\\\m@th \\\\displaystyle \\\\bullet $}}}}{}{}{}$ 1x ay = 0.2 aLOS' 1y az = 0 yaw = KP,yaw heading Where $k_{heading}$ may be tuned depending on certain factors of the UAV system, including the camera field-of-view (FOV) or the maximum yaw-rate.\",\n \"Note that at all times, regardless of the heading, both $a_{y}$ and $\\\\dot{yaw}$ are nonzero, and are instead suppressed with a factor less than 1.\",\n \"The factor of 0.2 was found empirically in this study to perform well and yield an appropriate influence of both acceleration and yaw-rate.\",\n \"Using this hybrid method, the UAV may potentially be able to react to changes in target motion while also keeping it in view.\"\n ],\n [\n \"Trajectory Following\",\n \"All trajectory following methods were implemented with some replanning rate at which updated trajectories are published.\",\n \"Replanning is constantly done from the look ahead point, which is maintained by the Trajectory Controller as described in Section REF .\"\n ],\n [\n \"$LOS^{\\\\prime }$ Acceleration Trajectory Following\",\n \"These trajectories are formed by taking the desired acceleration command calculated with Equation REF and calculating position waypoints and velocities given the starting position set to the look ahead point.\",\n \"The calculations for the positions and velocities are done with the following kinematic equations, set in the UAV's inertial reference frame.\",\n \"$\\\\mathbf {p}_t &= \\\\mathbf {v}_0t + \\\\frac{1}{2} \\\\mathbf {a}_{LOS^{\\\\prime }} t^2\\\\\\\\\\\\mathbf {v}_t &= \\\\mathbf {v}_0 + \\\\mathbf {a}_{LOS^{\\\\prime }} t , \\\\, t=0,0.1,...,T$ Where $T$ is the time length of each trajectory.\",\n \"As $T$ approaches 0, this method becomes equivalent to commanding the desired $LOS^{\\\\prime }$ acceleration directly.\",\n \"The discretization of the timesteps is also a tunable parameter.\"\n ],\n [\n \"Target Motion Forecasting\",\n \"In its simplest form, target motion forecasting involves estimating the target velocity in 3D, calculating the future location of the target with a straight-line path, and generating a collision course trajectory towards that point in space at a velocity which completes the trajectory at the specified time.\",\n \"This method makes three critical assumptions: (i) the target has zero acceleration, (ii) we can approximate time-to-collision by calculating the time along the current LOS vector, and (iii) the forecasted target position will change slowly, so generating straight-path trajectories is sufficient to result in a final, smooth stitched trajectory.\",\n \"First, the LOS unit vectors at two times are calculated.\",\n \"These are used along with the depth estimation to find the target's velocity: $\\\\mathbf {v}_{target} = \\\\dfrac{d_1(\\\\mathbf {1_{LOS}}_1)-d_0(\\\\mathbf {1_{LOS}}_0)}{t_1-t_0}$ $d_1$ and $d_0$ , $\\\\mathbf {1_{LOS}}_1$ and $\\\\mathbf {1_{LOS}}_0$ , and $t_1$ and $t_0$ are the estimated depth, calculated LOS unit vector, and time, at two timesteps.\",\n \"Once we have the velocity, we find the approximate time-to-collision along the current LOS vector by using the UAV's velocity component along the LOS: $t_{collision} = \\\\dfrac{d_1}{<\\\\mathbf {v}_{UAV}, \\\\mathbf {1_{LOS}}_1>}$ Where $\\\\mathbf {v}_{UAV}$ is the current UAV velocity vector.\",\n \"Therefore, the approximate point in space to plan the trajectory to is as follows, where $\\\\mathbf {p}_{collision}$ is in the UAV reference frame.\",\n \"$\\\\mathbf {p}_{collision} = \\\\mathbf {v}_{target}t_{collision} + d_1(\\\\mathbf {1_{LOS}}_1)$\"\n ],\n [\n \"System\",\n \"Figure REF shows a diagram of the most important parts of the system, including perception, control, and the simulation software.\",\n \"Each block generally consists of one or two nodes in ROS (Robot Operation System), the framework used in this work.\"\n ],\n [\n \"Object Segmentation\",\n \"Since the focus for this chapter was on the comparison of guidance methods, the perception challenge was simplified.\",\n \"The simulation does not include rich background visuals, there is no dropout in the camera feed, and the object was reduced from a potential UAV shape to a yellow sphere.\",\n \"The segmentation node composes of color thresholding that creates a binary segmentation image from each camera frame in real-time (30Hz), as seen in Figure REF .\",\n \"In addition, this node publishes the centroid $(C_x,C_y)$ of the detected object by using the image moments.\",\n \"$\\\\begin{bmatrix}C_x \\\\\\\\ C_y\\\\end{bmatrix}=\\\\begin{bmatrix}[2.0]\\\\dfrac{M_{10}}{M_{00}} \\\\\\\\ \\\\dfrac{M_{01}}{M_{00}}\\\\end{bmatrix}$\"\n ],\n [\n \"Object Depth Estimation\",\n \"Depth estimation with monocular camera images relies on prior knowledge of the target size as well as the assumption that the target is spherical (which is correct in this simulation, but extendable to real-life targets with approximate knowledge of target or aircraft type).\",\n \"The estimate takes into account the camera intrinsic parameters and projects the 2D information (pixels) to 3D (rays starting at the camera center) to extract the depth.\",\n \"Pixel coordinates at two points in the image need to be located which form a 2D line in the image that projects to a 3D line along the diameter of the object.\",\n \"The easiest found such line is the line through the image principal point that passes through the centroid, since the target is a sphere.\",\n \"This line is the dotted line passing through the target in Figure REF .\",\n \"We must first find the pixel coordinates along this line that correspond to the object edges in the image.\",\n \"First, we find the rotation angle formed by the centroid coordinates with respect to the image horizontal, labeled $\\\\theta $ in Figure REF .\",\n \"$\\\\theta = \\\\arctan \\\\dfrac{C_y-c_y}{C_x-c_x}$ With this angle we can use a rotation matrix to rotate the segmented object onto the $y=c_y$ line.\",\n \"Note that we use $-\\\\theta $ to achieve the correct rotation, and that we first compute new pixel coordinates $(u_i,v_i)$ relative to the image center, from the original coordinates $(x_i,y_i)$ .\",\n \"$\\\\begin{bmatrix}u_{i} \\\\\\\\ v_{i}\\\\end{bmatrix}&=\\\\begin{bmatrix}x_i - c_x \\\\\\\\ y_i - c_y\\\\end{bmatrix}, \\\\; i=1,...,N \\\\\\\\\\\\begin{bmatrix}u_{i,R} \\\\\\\\ v_{i,R}\\\\end{bmatrix}&=\\\\begin{bmatrix}cos(-\\\\theta ) && -sin(-\\\\theta )\\\\\\\\sin(-\\\\theta ) && cos(-\\\\theta )\\\\end{bmatrix}\\\\begin{bmatrix}u_{i} \\\\\\\\ v_{i}\\\\end{bmatrix}, \\\\; i=1,...,N$ Where N is the total number of segmented pixels in the binary segmentation image.\",\n \"Once the object is rotated onto the image horizontal, simple min and max operations yield the extreme pixel coordinates on the left and right of the target (or right and left, if $C_x
text
sequencelengths 2
2.54k
| id
stringlengths 9
16
|
|---|---|
[
[
"Infinite Ramsey-minimal graphs for star forests"
],
[
"Abstract For graphs $F$, $G$, and $H$, we write $F \\to (G,H)$ if every red-blue coloring of the edges of $F$ produces a red copy of $G$ or a blue copy of $H$.",
"The graph $F$ is said to be $(G,H)$-minimal if it is subgraph-minimal with respect to this property.",
"The characterization problem for Ramsey-minimal graphs is classically done for finite graphs.",
"In 2021, Barrett and the second author generalized this problem to infinite graphs.",
"They asked which pairs $(G,H)$ admit a Ramsey-minimal graph and which ones do not.",
"We show that any pair of star forests such that at least one of them involves an infinite-star component admits no Ramsey-minimal graph.",
"Also, we construct a Ramsey-minimal graph for a finite star forest versus a subdivision graph.",
"This paper builds upon the results of Burr et al.",
"in 1981 on Ramsey-minimal graphs for finite star forests."
],
[
"Introduction",
"All our graphs are simple and undirected.",
"We start by stating basic definitions.",
"For graphs $F$ , $G$ and $H$ , we write $F \\rightarrow (G,H)$ if every red-blue coloring of the edges of $F$ produces a red copy of $G$ or a blue copy of $H$ .",
"A red-blue coloring of $F$ is $(G,H)$ -good if it produces neither a red $G$ nor a blue $H$ .",
"If $F \\rightarrow (G,H)$ and every subgraph $F^{\\prime }$ of $F$ is such that $F^{\\prime } \\lnot \\rightarrow (G,H)$ , then $F$ is $(G,H)$ -minimal.",
"The collection of all $(G,H)$ -minimal graphs is denoted by $\\mathcal {R}(G,H)$ .",
"A pair $(G,H)$ admits a Ramsey-minimal graph if $\\mathcal {R}(G,H)$ is nonempty.",
"If $G$ and $H$ are both finite, then a $(G,H)$ -minimal graph exists.",
"Indeed, we can delete finitely many vertices and/or edges of $K_{r(G,H)}$ until it is $(G,H)$ -minimal.",
"This observation does not necessarily hold when at least one of $G$ and $H$ is infinite, even though there exists a graph $F$ such that $F \\rightarrow (G,H)$ in the countable case by the Infinite Ramsey Theorem [20] and in general by the Erdős–Rado Theorem [14].",
"In fact, a pair of countably infinite graphs almost never admits a Ramsey-minimal graph—see Proposition REF .",
"In 2021, Barrett and the second author [1] introduced a characterization problem for pairs of graphs according to whether or not they admit a minimal graph.",
"Main Problem ([1]) Determine which pairs $(G,H)$ admit a Ramsey-minimal graph and which ones do not.",
"The primary motivation for posing Main Problem is the classic problem of determining whether there are finitely or infinitely many $(G,H)$ -minimal graphs.",
"This problem was first introduced in 1976 [11], [17], and it was studied for finite graphs in general by Nešetřil and Rödl [18], [19] and for various classes of graphs by Burr et al.",
"[5], [6], [7], [9], [12], including for star forests [8].",
"The formulation of Main Problem is also motivated by the more recent work of Stein [23], [24], [25] on extremal infinite graph theory.",
"It is a subfield of extremal graph theory that developed after the notion of end degrees was introduced a few years prior [4], [22].",
"This paper primarily focuses on pairs $(G,H)$ involving a star forest—a union of stars.",
"Our first main result shows that any pair of star forests such that at least one of them involves an infinite-star component admits no Ramsey-minimal graph.",
"Theorem 1.1 Let $G$ and $H$ be star forests.",
"If at least one of $G$ and $H$ contains an infinite-star component, then no $(G,H)$ -minimal graph exists.",
"This theorem is in contrast to a result of Burr et al.",
"[8] on finite star forests, where it is shown that there are infinitely many $(G,H)$ -minimal graphs when both $G$ and $H$ are disconnected finite star forests with no single-edge component.",
"Loosely speaking, the existence of infinitely many finite minimal graphs does not give an indication that a corresponding infinite minimal graph exists.",
"Figure: The graph F d F_d.Similarly, the existence of only finitely many finite minimal graphs does not imply that there are only finitely many corresponding infinite minimal graphs.",
"For $n \\in \\mathbb {N}$ —where $\\mathbb {N}$ is the set of positive integers—we denote the $n$ -edge star by $S_n$ .",
"It is known from [10] that there are only finitely many $(nS_1,H)$ -minimal graphs for $n \\in \\mathbb {N}$ and $H$ a finite graph.",
"On the other hand, if $\\mathbb {Z}$ is the double ray—the two-way infinite path—then $(2S_1,\\mathbb {Z}\\cup S_3)$ admits infinitely many minimal graphs.",
"Indeed, we have $2F_d \\in \\mathcal {R}(2S_1,\\mathbb {Z}\\cup S_3)$ for $d \\ge 3$ , where $F_d$ is the graph illustrated in Figure REF .",
"A graph is leafless if it contains no vertex of degree one, and it is non–self-embeddable if it is not isomorphic to any proper subgraph of itself.",
"Following [16], we denote the subdivision graph of $G$ by $S(G)$ , which is a graph obtained from $G$ by performing a subdivision on each one of its edges.",
"For example, if $P_n$ denotes the $n$ -vertex path, then $S(P_n)=P_{2n-1}$ for $n \\in \\mathbb {N}$ .",
"For our second main result, we construct a Ramsey-minimal graph for a finite star forest versus the subdivision graph of a connected, leafless, non–self-embeddable graph.",
"In 2020, subdivision graphs were used by Wijaya et al.",
"[26] to construct new $(nS_1,P_4)$ -minimal graphs.",
"Theorem 1.2 Let $G$ be a connected, leafless, non–self-embeddable graph.",
"For any finite star forest $H$ , there exists a $(S(G),H)$ -minimal graph.",
"For future investigation, it would be interesting to consider whether every pair of non–self-embeddable graphs admits a minimal graph.",
"If true, this would generalize the observation that a pair of finite graphs always admits a minimal graph, since finite graphs are non–self-embeddable.",
"Is it true that every pair $(G,H)$ of non–self-embeddable graphs admits a Ramsey-minimal graph?",
"We give an outline of this paper.",
"Section discusses self-embeddable graphs and their relevance to the study of Ramsey-minimal graphs.",
"In Section , we briefly discuss the Ramsey-minimal properties of $(G,H)$ when $H$ is a union of graphs.",
"Finally, our two main theorems are proved in Sections and ."
],
[
"Self-embeddable graphs",
"We first provide several preliminary definitions.",
"A graph homomorphism $\\varphi \\colon G \\rightarrow H$ is a map from $V(G)$ to $V(H)$ such that $\\varphi (u)\\varphi (v) \\in E(H)$ whenever $uv \\in E(G)$ .",
"A graph homomorphism is an embedding if it is an injective map of vertices.",
"Following [2], [3], we write $G \\le H$ if $G$ embeds into $H$ ; that is, there exists an embedding $\\varphi \\colon G \\rightarrow H$ .",
"Unlike in [2], [3], however, we do not require that the graph image of $\\varphi $ is an induced subgraph of $H$ .",
"A graph $G$ is self-embeddable if $G \\cong G^{\\prime }$ for some proper subgraph $G^{\\prime }$ of $G$ , and the corresponding isomorphism $\\varphi \\colon G \\rightarrow G^{\\prime }$ is its self-embedding.",
"Examples of self-embeddable graphs include the ray $\\mathbb {N}$ —the one-way infinite path—and a complete graph on infinitely many vertices.",
"On the other hand, finite graphs and the double ray $\\mathbb {Z}$ are non–self-embeddable.",
"Proposition REF provides a necessary and sufficient condition for a graph to be self-embeddable in terms of its components.",
"This proposition is quite similar to [21] for self-contained graphs, the “induced” version of self-embeddable graphs.",
"Proposition 2.1 A graph $G$ is self-embeddable if and only if at least one of the following statements holds: There exists a self-embeddable component of $G$ .",
"There exists a sequence of distinct components $(C_i)_{i \\in \\mathbb {N}}$ of $G$ such that $C_1 \\le C_2 \\le \\cdots $ .",
"The backward direction can be easily proved by defining a suitable self-embedding of $G$ for each of the two cases; it remains to show the forward direction.",
"Suppose that $G$ has a self-embedding $\\varphi $ that embeds $G$ into $G-p$ , where $p$ is either a vertex or an edge of $G$ , and $G$ contains no self-embeddable component.",
"Let $C_0$ be the component of $G$ containing $p$ .",
"We write $v \\simeq w$ if the vertices $v$ and $w$ belong to the same component.",
"Claim 1 If $u \\in V(C_0)$ , then for $0 \\le i<j$ , we have $\\varphi ^i(u) \\lnot \\simeq \\varphi ^j(u)$ .",
"We use induction on $i$ .",
"Let $i=0$ , and suppose to the contrary that $u$ and $\\varphi ^j(u)$ , where $j>0$ , both belong to $C_0$ .",
"If $v \\simeq u$ , we then have $\\varphi ^j(v) \\simeq \\varphi ^j(u) \\simeq u,$ so $\\varphi ^j(v) \\in V(C_0)$ for $v \\in V(C_0)$ .",
"Also, since $\\varphi $ embeds $G$ into $G-p$ , the map $\\varphi ^j$ also embeds $G$ into $G-p$ .",
"Hence $\\varphi ^j$ carries $C_0$ into $C_0-p$ , which contradicts the non–self-embeddability of $C_0$ .",
"Now suppose $i \\ge 1$ , and suppose to the contrary that $\\varphi ^i(u)$ and $\\varphi ^j(u)$ , where $j>i$ , both belong to the same component $C$ .",
"If $v \\simeq \\varphi ^i(u)$ , we then have $\\varphi ^{j-i}(v) \\simeq \\varphi ^j(u) \\simeq \\varphi ^i(u),$ so $\\varphi ^{j-i}(v) \\in V(C)$ for $v \\in V(C)$ .",
"We now prove that $\\varphi ^{j-i}$ carries $C$ into $C-\\varphi ^i(u)$ ; this would contradict the non–self-embeddability of $C$ .",
"Suppose that $\\varphi ^{j-i}(v)=\\varphi ^i(u)$ for some vertex $v$ .",
"We have $\\varphi ^{j-i-1}(v)=\\varphi ^{i-1}(u)$ by injectivity.",
"By the induction hypothesis, we also have $\\varphi ^{i-1}(u) \\lnot \\simeq \\varphi ^{j-1}(u)$ , so $\\varphi ^{j-i-1}(v) \\lnot \\simeq \\varphi ^{j-1}(u)$ .",
"It follows that $v \\lnot \\simeq \\varphi ^i(u)$ , and since $\\varphi ^i(u) \\in V(C)$ , we infer that $v \\notin V(C)$ .",
"Therefore, $\\varphi ^i(u)$ cannot be the image of a vertex of $C$ under $\\varphi ^{j-i}$ , as desired.",
"Let $u \\in V(C_0)$ .",
"Define a sequence $(C_i)_{i \\in \\mathbb {N}}$ such that $C_i$ is the component containing $\\varphi ^i(u)$ .",
"This sequence consists of pairwise distinct components by Claim REF .",
"It is clear that $\\varphi $ carries $C_i$ to $C_{i+1}$ , so $C_i \\le C_{i+1}$ for $i \\in \\mathbb {N}$ , and we are done.",
"Proposition REF implies, as an example, that the union of finite paths is self-embeddable, but the union of finite cycles is not.",
"Also, we obtain the following corollary.",
"Corollary 2.2 A star forest is self-embeddable if and only if it is infinite.",
"For nonempty graphs $G$ , a stronger property than self-embeddability is the property that $G \\le G-e$ for $e \\in E(G)$ .",
"The ray and an infinite complete graph, for example, enjoy this stronger property.",
"On the other hand, the disjoint union $\\mathbb {N}\\cup \\mathbb {Z}$ is self-embeddable, but does not embed into $\\mathbb {N}\\cup (\\mathbb {Z}-e)$ , where $e$ is any edge of $\\mathbb {Z}$ .",
"Thus $\\mathbb {N}\\cup \\mathbb {Z}$ does not possess this stronger property.",
"Proposition 2.3 If $G$ is a nonempty graph such that $G \\le G-e$ for $e \\in E(G)$ , then no $(G,H)$ -minimal graph exists for any graph $H$ .",
"We will prove that for every graph $F$ such that $F \\rightarrow (G,H)$ , we have $F-e \\rightarrow (G,H)$ for some $e \\in E(F)$ .",
"This would show that $(G,H)$ admits no minimal graph.",
"Let $F$ be a graph, and let $e$ be any one of its edges.",
"Set $F^{\\prime }=F-e$ .",
"Suppose that $F^{\\prime } \\lnot \\rightarrow (G,H)$ —there exists a $(G,H)$ -good coloring $c^{\\prime }$ of $F^{\\prime }$ .",
"We show that $F \\lnot \\rightarrow (G,H)$ .",
"Define a coloring $c$ on $F$ such that $c\\upharpoonright _{E(F^{\\prime })}=c^{\\prime }$ and $e$ is colored red.",
"By this definition, no blue $H$ is produced in $F$ .",
"We claim that $c$ does not produce a red $G$ either.",
"Suppose to the contrary that a red copy of $G$ , say $\\widehat{G}$ , is produced in $F$ .",
"Since $\\widehat{G} \\le \\widehat{G}-e$ , we can choose a red copy of $G$ in $F$ that does not contain $e$ ; that is, there exists a red $G$ in $F^{\\prime }$ .",
"This contradicts the $(G,H)$ -goodness of $c^{\\prime }$ .",
"As a consequence, $c$ is a $(G,H)$ -good coloring of $F$ , and thus $F \\lnot \\rightarrow (G,H)$ .",
"We note that Proposition REF does not hold for self-embeddable graphs $G$ in general—see Example .",
"If $R$ is the Rado graph, then $R-e$ is also the Rado graph for $e \\in E(R)$ via [13].",
"As a result, the Rado graph satisfies the hypothesis of Proposition REF .",
"Consequently, by [15], the following proposition holds.",
"Proposition 2.4 For $H$ a fixed graph, almost all countably infinite graphs $G$ produce a pair $(G,H)$ which admits no Ramsey-minimal graph."
],
[
"Graph unions",
"Before we focus on star forests proper, we provide a quick background on graph unions in general.",
"Consider graphs $G$ , $H_1$ , and $H_2$ ; let $F_i \\in \\mathcal {R}(G,H_i)$ for $i \\in \\lbrace 1,2\\rbrace $ .",
"Possible candidates for a $(G,H_1 \\cup H_2)$ -minimal graph include $F_1$ , $F_2$ , and $F_1 \\cup F_2$ .",
"Although $F_1 \\cup F_2 \\rightarrow (G,H_1 \\cup H_2)$ , it not necessarily true that $F_1 \\cup F_2 \\in \\mathcal {R}(G,H_1 \\cup H_2)$ .",
"Indeed, let us take $H_1=H_2=S_1$ .",
"For $G$ connected, we have $2G \\in \\mathcal {R}(G,2S_1)$ provided that $G \\in \\mathcal {R}(G,S_1)$ .",
"This was discussed in [1] but also follows from Proposition REF .",
"On the other hand, if $G$ is disconnected, we have $3\\mathbb {Z}\\in \\mathcal {R}(2\\mathbb {Z},2S_1)$ —not $4\\mathbb {Z}$ —even though $2\\mathbb {Z}\\in \\mathcal {R}(2\\mathbb {Z},S_1)$ .",
"Proposition 3.1 Let $G$ and $H$ be nontrivial, connected graphs, and let $n \\in \\mathbb {N}$ .",
"If $F_i \\in \\mathcal {R}(G,H)$ for $1 \\le i \\le n$ , then $\\bigcup _{i=1}^n F_i \\in \\mathcal {R}(G,nH).$ Consequently, the existence of a $(G,nH)$ -minimal graph is assured provided that a $(G,H)$ -minimal graph exists.",
"The arrowing part is obvious, so we only show the minimality of $\\bigcup _{i=1}^n F_i$ .",
"It is clear that $F_i \\lnot \\rightarrow (G,2H)$ , since otherwise we would have $F_i \\notin \\mathcal {R}(G,H)$ .",
"Let $e$ be an edge of $F_k$ for some $1 \\le k \\le n$ .",
"Color $F_k-e$ by a $(G,H)$ -good coloring and $F_i$ , for $i \\ne k$ , by a $(G,2H)$ -good coloring.",
"This coloring on $\\left(\\bigcup _{i=1}^n F_i\\right)-e$ is easily shown to be $(G,nH)$ -good from the connectivity of $G$ and $H$ .",
"Since $e$ is arbitrary, the proposition is proved.",
"In contrast to Proposition REF , the following proposition considers $F_i$ as a candidate for being in $\\mathcal {R}(G,H_1 \\cup H_2)$ .",
"A sufficient condition is provided for a $(G,H_1)$ -minimal graph to be $(G,H_1 \\cup H_2)$ -minimal.",
"Proposition 3.2 Let $G$ , $H_1$ , and $H_2$ be graphs, and let $F \\in \\mathcal {R}(G,H_1)$ .",
"If $F-V(\\widehat{H}_1) \\rightarrow (G,H_2)$ for every $\\widehat{H}_1$ a copy of $H_1$ in $F$ , then $F \\in \\mathcal {R}(G,H_1 \\cup H_2)$ .",
"We first prove that $F \\rightarrow (G,H_1 \\cup H_2)$ .",
"Suppose $c$ is a coloring on $F$ that produces no red $G$ .",
"It follows from $F \\rightarrow (G,H_1)$ that $c$ produces a blue copy of $H_1$ , say $\\widehat{H}_1$ , in $F$ .",
"Let $F^{\\prime }=F-V(\\widehat{H}_1)$ .",
"Since $F^{\\prime } \\rightarrow (G,H_2)$ and $F^{\\prime }$ contains no red $G$ , there exists a blue copy of $H_2$ , say $\\widehat{H}_2$ , in $F^{\\prime }$ .",
"We observe that $\\widehat{H}_1$ and $\\widehat{H}_2$ are disjoint, so $c$ produces a blue $H_1 \\cup H_2$ .",
"Hence $F \\rightarrow (G,H_1 \\cup H_2)$ .",
"Its minimality follows immediately from the $(G,H_1)$ -minimality of $F$ .",
"Let $\\begin{split}&G=2S_1, \\\\&H_1=\\mathbb {Z}, \\\\&H_2=\\mathbb {N},\\text{ and} \\\\&F=2\\mathbb {Z}.\\end{split}$ The graph $2\\mathbb {Z}$ is $(2S_1,\\mathbb {Z})$ -minimal, and $\\mathbb {Z}\\rightarrow (2S_1,\\mathbb {N})$ , so we can conclude by Proposition REF that $2\\mathbb {Z}\\in \\mathcal {R}(2S_1,\\mathbb {Z}\\cup \\mathbb {N})$ .",
"This serves as an example of a pair $(G,H)$ involving a self-embeddable graph that admits a minimal graph.",
"We note, however, that no $(S_1,\\mathbb {Z}\\cup \\mathbb {N})$ -minimal graph exists since $\\mathbb {Z}\\cup \\mathbb {N}$ is self-embeddable.",
"Thus it is possible that a $(2G,H)$ -minimal graph exists even though no $(G,H)$ -minimal graph exists."
],
[
"Proof of Theorem ",
"We fix star forests $G$ and $H$ such that at least one of them contains a star component on infinitely many vertices.",
"We prove in this section that $(G,H)$ admits no Ramsey-minimal graph.",
"Suppose that $F \\rightarrow (G,H)$ .",
"Since one of $G$ and $H$ contains a vertex of infinite degree, there exists a vertex $v$ of infinite degree in $F$ .",
"We choose an arbitrary edge $e$ at $v$ .",
"We prove that $F^{\\prime } \\rightarrow (G,H)$ , where $F^{\\prime }=F-e$ .",
"Toward a contradiction, suppose that $F^{\\prime }$ admits a $(G,H)$ -good coloring $c^{\\prime }$ .",
"Since $\\deg (v)$ is infinite, there are two possible cases: $v$ is incident to infinitely many red edges or infinitely many blue edges under the coloring $c^{\\prime }$ .",
"Suppose that $v$ is incident to infinitely many red edges.",
"Define a coloring $c$ on $F$ such that $c\\upharpoonright _{E(F^{\\prime })}=c^{\\prime }$ and $e$ is colored red.",
"This coloring produces no blue $H$ , so by $F \\rightarrow (G,H)$ it produces a red copy of $G$ , say $\\widehat{G}$ , in $F$ .",
"There exists a star component $S$ of $\\widehat{G}$ that contains $e$ since otherwise, $\\widehat{G} \\subseteq F^{\\prime }$ , which contradicts the $(G,H)$ -goodness of $c^{\\prime }$ .",
"If $S$ is infinite, then $F^{\\prime }$ clearly contains a red copy of $G$ by removing $e$ from $\\widehat{G}$ .",
"On the other hand, let us suppose that $S$ has $n$ vertices.",
"We can pick a red star $S^{\\prime }$ on $n$ vertices that is centered on $v$ but does not contain $e$ , since $v$ is incident to infinitely many red edges.",
"The graph $F^{\\prime }$ can then be shown to contain a red copy of $G$ by exchanging $S$ from $\\widehat{G}$ for $S^{\\prime }$ .",
"In both cases, we obtain a contradiction.",
"The case when $v$ is incident to infinitely many blue edges can be handled similarly, so our proof of Theorem REF is complete."
],
[
"Bipartite graphs",
"Recall that a graph is bipartite if its vertex set can be partitioned into two parts such that each part is an independent set.",
"Let $K$ be a connected, bipartite graph with bipartition $\\lbrace A,B\\rbrace $ such that $\\deg (u)<\\infty $ for $u \\in A$ .",
"Before we work on subdivision graphs $S(G)$ , we construct for $n \\in \\mathbb {N}$ , a graph $\\Gamma (K,A,n)$ such that $\\Gamma (K,A,n) \\rightarrow (K,S_n)$ .",
"Figure: The construction of Γ(K,A,n)\\Gamma (K,A,n) for K=ℤK=\\mathbb {Z} and n=3n=3.We define $\\Gamma (K,A,n)$ by adding additional vertices and edges to $K$ .",
"For $u \\in A$ , we add vertices $u_1,...,u_{m(n-1)}$ —each not already in $V(K)$ —to $K$ , where $m=\\deg (u)$ .",
"We then insert an edge between $u_i$ and a vertex $v \\in V(K)$ if $uv$ exists in $K$ .",
"We denote the resulting graph by $\\Gamma (K,A,n)$ .",
"Also, for $u \\in A$ , we define $A_u$ as the set $\\lbrace u,u_1,...,u_{m(n-1)}\\rbrace $ .",
"As a result, $\\Gamma (K,A,n)$ admits a bipartition $\\lbrace \\bigcup _{u \\in A} A_u,B\\rbrace $ .",
"Figure REF shows the result of this construction when $K=\\mathbb {Z}$ and $n=3$ .",
"There is a natural projection $\\pi \\colon \\Gamma (K,A,n) \\rightarrow K$ that is also a homomorphism.",
"It is defined as $\\pi (v)={\\left\\lbrace \\begin{array}{ll}u, & v \\in A_u \\text{ for some } u \\in A, \\\\ v, & v \\in B.\\end{array}\\right.",
"}$ Proposition 5.1 Let $K$ be a connected, bipartite graph with bipartition $\\lbrace A,B\\rbrace $ such that $\\deg (u)<\\infty $ for $u \\in A$ .",
"For $n \\in \\mathbb {N}$ , we have $\\Gamma (K,A,n) \\rightarrow (K,S_n)$ .",
"Consequently, $\\bigcup _{i=1}^k \\Gamma (K,A,n_i) \\rightarrow \\left(K,\\bigcup _{i=1}^k S_{n_i}\\right)$ for $n_1,\\dots ,n_k \\in \\mathbb {N}$ .",
"Suppose that $c$ is a coloring on $\\Gamma (K,A,n)$ that produces no blue $S_n$ .",
"We prove that $c$ produces a red $K$ .",
"Claim 2 For $u \\in A$ , there exists $v_u \\in A_u$ such that $v_u$ is incident to only red edges.",
"By construction, the vertices in $A_u$ share the same neighborhood $N$ of $m$ vertices, and $|A_u|=m(n-1)+1$ .",
"If every vertex in $A_u$ is incident to at least one blue edge, then the vertices in $N$ in total are incident to at least $m(n-1)+1$ blue edges.",
"Since $|N|=m$ , there exists a vertex in $N$ that is incident to at least $n$ blue edges by the Pigeonhole Principle.",
"This is impossible since $\\Gamma (K,A,n)$ does not contain a blue $S_n$ .",
"Therefore, $A_u$ must contain a vertex that is incident to only red edges.",
"By Claim REF , we can define an embedding $\\varphi \\colon K \\rightarrow \\Gamma (K,A,n)$ as $\\varphi (u)={\\left\\lbrace \\begin{array}{ll}v_u, & u \\in A, \\\\ u, & u \\in B.\\end{array}\\right.",
"}$ The graph image of $\\varphi $ is a red copy of $K$ in $\\Gamma (K,A,n)$ , as desired.",
"The graph $\\Gamma (K,A,n)$ is not necessarily $(K,S_n)$ -minimal in general.",
"For example, let us take $K=S_k$ and $A$ as the set of leaf vertices of $S_k$ .",
"We have $\\Gamma (S_k,A,n)=S_{kn}$ , which is not $(S_k,S_n)$ -minimal for $k,n \\ge 2$ since $S_{k+n-1} \\in \\mathcal {R}(S_k,S_n)$ .",
"However, we potentially have $\\Gamma (K,A,n) \\in \\mathcal {R}(K,S_n)$ when $K=S(G)$ for some graph $G$ as stated in Theorem REF ."
],
[
"Proof of Theorem ",
"Fix a connected, leafless, non–self-embeddable graph $G$ .",
"Building upon Subsection REF , we prove that for $n_1,\\dots ,n_k \\in \\mathbb {N}$ , we have $\\bigcup _{i=1}^k \\Gamma (S(G),A,n_i) \\in \\mathcal {R}\\left(S(G),\\bigcup _{i=1}^k S_{n_i}\\right),$ where $A$ is taken as the set of vertices of $S(G)$ that subdivide the edges of $G$ .",
"We note that $\\deg (u)=2$ for $u \\in A$ .",
"First, we show that the three properties of $G$ transfer to $S(G)$ , and that $S(G)$ is $C_4$ -free—it contains no 4-cycles.",
"The following lemma can be verified using elementary means.",
"Lemma 5.2 Let $G$ and $H$ be connected, bipartite graph with bipartition $\\lbrace A,B\\rbrace $ and $\\lbrace C,D\\rbrace $ , respectively.",
"For any isomorphism $\\varphi \\colon G \\rightarrow H$ , either $\\varphi (A)=C$ and $\\varphi (B)=D$ , or $\\varphi (A)=D$ and $\\varphi (B)=C$ .",
"Proposition 5.3 If $G$ is a connected, leafless, non–self-embeddable graph, then $S(G)$ is also a connected, leafless, non–self-embeddable graph.",
"In addition, $S(G)$ is $C_4$ -free.",
"The first two properties obviously transfer, and $S(G)$ is $C_4$ -free since $G$ contains no multiple edges.",
"We now prove that $G$ is self-embeddable given that $S(G)$ is self-embeddable.",
"Suppose that $\\varphi $ is a self-embedding of $S(G)$ .",
"Let $A$ be the set of vertices of $S(G)$ that subdivide the edges of $G$ , and let $B=V(G)$ .",
"Since $S(G)$ is connected and bipartite with bipartition $\\lbrace A,B\\rbrace $ , there are by Lemma REF two cases to consider.",
"Case 1: $\\varphi (A) \\subseteq A$ and $\\varphi (B) \\subseteq B$ .",
"We claim that $\\varphi $ , restricted to $V(G)$ , gives rise to a self-embedding $\\widehat{\\varphi }$ of $G$ .",
"It is straightforward to show that $\\widehat{\\varphi }$ is an embedding, so we only prove that there is an edge of $G$ not in the image of $\\widehat{\\varphi }$ .",
"Suppose that $uv$ , where $u \\in A$ and $v \\in B$ , is an edge of $S(G)$ not in the image of $\\varphi $ , and suppose that $u$ subdivides an edge $vw$ of $G$ .",
"We prove that $vw$ is not in the image of $\\widehat{\\varphi }$ .",
"Suppose toward a contradiction that $\\widehat{\\varphi }(a)=v$ and $\\widehat{\\varphi }(b)=w$ for two adjacent vertices $a,b \\in V(G)$ .",
"Let $c$ be the vertex that subdivides $ab$ .",
"It is apparent that $\\lbrace \\varphi (c),v\\rbrace $ and $\\lbrace \\varphi (c),w\\rbrace $ are edges of $S(G)$ .",
"Also, we cannot have $\\varphi (c)=u$ since $uv$ is not in the image of $\\varphi $ .",
"But then the vertices in the set $\\lbrace v,u,w,\\varphi (c)\\rbrace $ induce a 4-cycle on $S(G)$ , which contradicts the fact that $S(G)$ is $C_4$ -free.",
"Case 2: $\\varphi (A) \\subseteq B$ and $\\varphi (B) \\subseteq A$ .",
"The map $\\varphi ^2$ is a self-embedding of $S(G)$ that carries $A$ into $A$ , and $B$ into $B$ .",
"So by appealing to Case 1, we can obtain a self-embedding of $G$ .",
"Armed with Proposition REF , we are ready to prove Theorem REF .",
"But first, let us provide a straightforward application of the membership statement of (REF ) that we will prove later.",
"Choose $G=\\mathbb {Z}$ and $H=S_3$ .",
"Since $\\mathbb {Z}$ is connected, leafless, and non–self-embeddable, and $S(\\mathbb {Z})=\\mathbb {Z}$ , the graph of Figure REF (b) is $(\\mathbb {Z},S_3)$ -minimal by (REF ).",
"[Proof of Theorem REF ] First, suppose $H=\\bigcup _{i=1}^k S_{n_i}, \\text{ where } 1 \\le n_1 \\le \\dots \\le n_k.$ Let $A$ be the set of vertices of $S(G)$ that subdivide the edges of $G$ so that $\\deg (u)=2$ for $u \\in A$ .",
"Define $\\Gamma _i=\\Gamma (S(G),A,n_i)$ and $\\Gamma =\\bigcup _{i=1}^k \\Gamma _i$ .",
"Denote the corresponding set to $A_u$ that belongs to $\\Gamma _i$ by $A_{u,i}$ .",
"We have $|A_{u,i}|=2n_i-1$ .",
"If $B_i=V(\\Gamma _i) \\backslash \\bigcup _{u \\in A} A_{u,i}$ , then $\\Gamma _i$ admits a bipartition $\\lbrace \\bigcup _{u \\in A} A_{u,i},B_i\\rbrace $ .",
"We prove for $e \\in E(\\Gamma )$ that there is a $(S(G),H)$ -good coloring of $\\Gamma -e$ .",
"This, along with Proposition REF , would show that $\\Gamma \\in \\mathcal {R}(S(G),H)$ .",
"Lemma 5.4 For $e \\in E(\\Gamma )$ , there exists a coloring $c$ on $\\Gamma -e$ such that both of the following statements hold: The coloring $c$ produces no blue $H$ .",
"There exists $u \\in A$ such that for $1 \\le i \\le k$ , every vertex in $A_{u,i}$ is incident to exactly one red edge.",
"Suppose that $e$ is an edge of some $\\Gamma _j$ , where $1 \\le j \\le k$ , and that $e$ is at a vertex $v \\in A_{u,j}$ for some $u \\in A$ .",
"We color each edge in every $\\Gamma _i$ , minus the edge $e$ for $i=j$ , by the following rules: Case 1: $i<j$ .",
"Recall that $|A_{u,i}|=2n_i-1$ and that the vertices in $A_{u,i}$ share the same neighborhood of two vertices, say $a$ and $b$ .",
"Arbitrarily partition $A_{u,i}$ into sets $S$ and $T$ such that $|S|=n_i$ and $|T|=n_i-1$ .",
"Color all the edges in $E(S,a) \\cup E(T,b)$ blue; this produces two blue stars of sizes $n_i$ and $n_i-1$ , respectively.",
"Color the rest of $\\Gamma _i$ red.",
"Case 2: $i=j$ .",
"As before, let $a$ and $b$ be the vertices adjacent to each vertex in $A_{u,j}$ .",
"Partition $A_{u,j} \\backslash v$ into sets $S$ and $T$ both of size $n_j-1$ .",
"Similarly to Case 1, we color all the edges in $E(S,a) \\cup E(T,b)$ blue.",
"This produces two blue stars of size $n_j-1$ .",
"Color the rest of $\\Gamma _j$ red.",
"Case 3: $i>j$ .",
"Let $a$ be a vertex adjacent to each vertex in $A_{u,i}$ .",
"Color $E(A_{u,i},a)$ blue; this produces a blue star of size $2n_i-1$ .",
"As previously, we color the rest of $\\Gamma _i$ red.",
"Denote the preceding coloring scheme by $c$ .",
"It is obvious from the preceding construction of $c$ that (ii) holds for our $u \\in A$ , so it remains to prove that (i) holds.",
"Let $j^{\\prime }$ be the least positive integer such that $n_{j^{\\prime }}=n_j$ .",
"Observe that we only produce blue stars of size at least $n_j$ in Case 3 and, if $j^{\\prime }<j$ , in Case 1 also.",
"Every $\\Gamma _i$ such that $j^{\\prime }\\le i \\le k$ and $i \\ne j$ contributes exactly one blue star of size at least $n_j$ , so exactly $k-j^{\\prime }$ such blue stars are produced in $\\Gamma -e$ in total.",
"But $H$ contains $k-j^{\\prime }+1$ stars of size at least $n_j$ , so no blue $H$ can be produced in $\\Gamma -e$ by the coloring $c$ .",
"We take the coloring $c$ of Lemma REF .",
"To prove that $c$ is $(S(G),H)$ -good, we need to show that $c$ does not produce a red $S(G)$ in $\\Gamma -e$ .",
"Suppose to the contrary that there exists an embedding $\\xi \\colon S(G) \\rightarrow \\Gamma _i$ such that its graph image is a red copy of $S(G)$ .",
"Set $\\varphi =\\pi \\circ \\xi $ , where $\\pi \\colon \\Gamma _i \\rightarrow S(G)$ is a projection that sends each vertex in $A_{u,i}$ to $u$ and is defined similarly to Eq.",
"(REF ).",
"We prove that $\\varphi $ is a self-embedding of $S(G)$ , which would contradict the non–self-embeddability of $S(G)$ .",
"For illustration, we provide the following commutative diagram of graph homomorphisms: $\\begin{tikzcd}[row sep=9mm,column sep=8mm]S(G) [hook,r,\"\\xi \"] [dr,\"\\varphi \"^{\\prime }] & \\Gamma _i [two heads,d,\"\\pi \"] \\\\& S(G)\\end{tikzcd}$ Suppose that $\\varphi (a)=b$ for some vertices $a$ and $b$ of $S(G)$ .",
"If $b \\in A$ , then the vertex $\\xi (a)$ belongs in $A_{b,i}$ .",
"Recall that $\\deg (a) \\ge 2$ since $S(G)$ is leafless.",
"Since the graph image of $\\xi $ is red, $\\xi (a)$ needs to be incident to at least two red edges as a result.",
"We infer that $b \\ne u$ , where $u \\in A$ is taken from Lemma REF (ii).",
"This shows that the vertex $u$ of Lemma REF (ii) is not in the image of $\\varphi $ .",
"Since $S(G)$ is $C_4$ -free and $\\xi $ is an embedding, there cannot be a $C_4$ in the graph image of $\\xi $ .",
"We now prove that $\\varphi $ is injective.",
"Let $a$ and $b$ be distinct vertices of $S(G)$ .",
"Since $a$ and $b$ have degree at least two, the vertices $\\xi (a)$ and $\\xi (b)$ also have degree at least two.",
"As a result, $\\xi (a)$ and $\\xi (b)$ cannot both belong in $A_{u,i}$ for some $u \\in A$ , since that would create a $C_4$ in the graph image of $\\xi $ .",
"Therefore, $\\varphi $ is injective.",
"This completes the proof that $\\varphi $ is a self-embedding and finishes our proof of Theorem REF ."
]
] | 2107.01710 |
[
[
"MIMO Operations in Molecular Communications: Theory, Prototypes, and\n Open Challenges"
],
[
"Abstract The Internet of Bio-nano Things is a significant development for next generation communication technologies.",
"Because conventional wireless communication technologies face challenges in realizing new applications (e.g., in-body area networks for health monitoring) and necessitate the substitution of information carriers, researchers have shifted their interest to molecular communications (MC).",
"Although remarkable progress has been made in this field over the last decade, advances have been far from acceptable for the achievement of its application objectives.",
"A crucial problem of MC is the low data rate and high error rate inherent in particle dynamics specifications, in contrast to wave-based conventional communications.",
"Therefore, it is important to investigate the resources by which MC can obtain additional information paths and provide strategies to exploit these resources.",
"This study aims to examine techniques involving resource aggregation and exploitation to provide prospective directions for future progress in MC.",
"In particular, we focus on state-of-the-art studies on multiple-input multiple-output (MIMO) systems.",
"We discuss the possible advantages of applying MIMO to various MC system models.",
"Furthermore, we survey various studies that aimed to achieve MIMO gains for the respective models, from theoretical background to prototypes.",
"Finally, we conclude this study by summarizing the challenges that need to be addressed."
],
[
"Introduction",
"The future of beyond 6G and 7G (B6G/7G) networking is expected to enable novel technologies related to health monitoring, where nanoscale networking has emerged as a promising communication paradigm.",
"Nano-communications primarily constitutes the implementation of nanoscale devices and their utilization for in vivo applications.",
"On the other hand, recent studies show that electromagnetic waves may not be compatible with nano-communication systems.",
"Techniques that use chemical particles as substitute messengers are therefore of interest for communications technologies.",
"A macroscopic view of the trajectory and possible future directions of molecular communications (MC) are presented in [1].",
"System design with analytical formulations [2] and testbed studies [3], [4] are both ongoing with open problems and challenges.",
"The main challenge in this field over the next decade is expected to be technology implementation with higher data rates and lower error rates.",
"This article discusses the tasks and challenges of applying multiple-input multiple-output (MIMO) to MC in theory and prototypes, as well as research directions for contemporary researchers.",
"Particle behaviors are dependent on diffusion and drift; thus, increasing the data rate is a crucial challenge for implementing the MC protocols.",
"In radio frequency (RF) communications, the standard strategy to increase data rate is to obtain independent information paths by exploiting the resources of power, time, frequency, and space.",
"An analogous strategy has also been used in MC; molecular type can be one another resource in MC.",
"It is noteworthy that earlier studies on MC focused more on power, time, and type-based modulation techniques, whereas spatial resources have been studied in recent years [1].",
"In [5], the authors introduced modulation techniques to exploit the variable resources of molecular types and power.",
"They claimed that both type- and power-based modulation techniques could either increase the data rate or reduce errors, and that a type domain ensured performance gain while remaining within the same power constraints.",
"The results were numerically validated.",
"The authors in [6] proposed and compared time-based modulation techniques with power modulation.",
"Owing to the high statistical noise from diffusion, the time domain was shown to yield performance gains in both modulations, and their testbed confirmed the results.",
"It is essential to investigate MIMO configurations to utilize spatial domain resources.",
"In MC with a diffusion environment, it can be argued that the MIMO operation may not work as in RF communication because of the completely different channel characteristics.",
"However, the authors in [7] showed that it is feasible to achieve performance gains in MC by applying MIMO systems in practice.",
"They proposed novel detection algorithms for MC with a 2$\\times $ 2 MIMO configuration and implemented a prototype to test the results.",
"Lee et al.",
"[8] designed learning-based algorithms to cope with unknown channel state information (CSI), in addition to developments in [7].",
"Yu et al.",
"[9] studied an MC system with a 4$\\times $ 4 configuration and demonstrated performance enhancements.",
"The authors in [10], [11] investigated general configurations and considered mobility and asymmetry, respectively.",
"To address the interference, both studies successfully applied a zero forcing detector.",
"Damrath et al.",
"provided analytic discussions on the array gain in MC [12].",
"These studies showed promising results wherein a spatial domain could result in performance gains.",
"Therefore, we are motivated to find open problems in this research area, whereas unknown CSI and interference could pose technical challenges.",
"The remainder of this article is organized as follows.",
"We first provide an overview of the MIMO-embedded MC system, including communication models, technical hurdles, and testbed designs.",
"Next, existing solutions are discussed.",
"Thereafter, we discuss the potential of molecular MIMO and conclude this article by summarizing the open challenges.",
"Figure: Conceptual overview of B6G/7G and molecular MIMO communications."
],
[
"Applying MIMO to MC",
"As shown in Fig.",
"1, we expect the research area of MC to augment the B6G/7G communication paradigm.",
"In addition to the other MC techniques, we believe that the role of MIMO technologies deserves special consideration.",
"We briefly introduce the potential gains of MIMO for RF communications and review the applications of MC in various scenarios."
],
[
"Brief Review of MIMO Gain in RF Communications",
"In RF communications, multiple antennas can provide performance benefits via array, diversity, and multiplexing gains.",
"Array gain is the benefit of having a high signal-to-noise ratio (SNR) at the receiver, which can be achieved by coherently adding received signals at the antennas.",
"A transmitter is required to have CSI for signal coherence.",
"Diversity gain offers a low error rate by providing independent channel paths and reducing the fading effects.",
"In contrast, multiplexing can achieve an increase in the data rate using simultaneous data streams through pairs of transmitter-receiver antennas.",
"This technique is also referred to as spatial multiplexing.",
"Knowledge on channel models, signal coherence, and antenna correlations is required to apply MIMO technologies.",
"In MC, a standard channel model is still in the development stage.",
"Furthermore, a diffusion-dominated environment makes it difficult to achieve coherence.",
"Therefore, channel, interference, detection, and verification are prerequisites for applying MIMO schemes to MC.",
"The term 'antenna' for MC in this article can physically be chemical/biological transceivers (molecular emitters and detectors)."
],
[
"MC System Model",
"We consider three models of interest as depicted in Fig.",
"2: three-dimensional (3D) free-space, bounded space, and unestablished channel models (e.g., underwater).",
"The first column shows the free-space channel model, where molecules propagate through diffusion and drift.",
"A possible target scenario for the use of this model is tactical nano-robots because of its strength against eavesdropping and energy efficiency.",
"The second column presents a vessel-bounded channel, that mimics in vivo applications.",
"Health monitoring by nano-robots with medical sensors and theranostics is a potential application for this model.",
"The third column describes underwater communication, in which RF signal experiences severe degradation.",
"In the bottom row, the corresponding state-of-the-art testbeds are introduced.",
"The first two testbeds are equipped with 2$\\times $ 2 MIMO configurations [8], [3] while the third [4] is equipped with a single-input single-output (SISO) configuration.",
"Mimicking nature is one of the strong motivations for MC technology.",
"For instance, ants in a colony communicate with each other by pheromone molecules, bio-organs in a body organize the system with hormone chemicals, and a shark detects its target from a long distance with a minute amount of odor molecules.",
"From these examples, the following directions can be considered: increasing the number of communication units that a network comprises, expanding the size of a set of chemical compounds that the system employs, and enabling position tracking functions with higher sensitivity.",
"In this article, we base our discussion on a 2$\\times $ 2 symmetric configuration for simplicity.",
"Generalized designs are presented in the following section.",
"The transmitter antennas are regarded as point sources that release messenger particles into the channel medium.",
"The receiver antennas were spherical in shape for analytical tractability.",
"These antennas absorb the contacted molecules and count their numbers over time.",
"In addition to the statistical counting noise, additional noise is considered to account for the unmodeled phenomena.",
"It is assumed that the transceivers are synchronized with one another and are aware of the predefined system parameters, including the required CSI and symbol duration.",
"Figure: Three types of channel model and comparisons in MC MIMO, from theory to prototypes."
],
[
"Channel Model and CSI",
"Free space diffusion dynamics inherently have an indeterminacy of exact movement, and analysis based on statistical characteristics is an alternative approach to address this issue.",
"In this regime, the channel that the information carriers experience is the expected fraction of the number of molecules that the receiver captures with respect to time.",
"A closed-form channel model solution for a case without drift and multiple antennas is presented in [13].",
"Furthermore, Jamali et al.",
"presented solutions for a channel with drift and a non-absorbing receiver [2].",
"In MIMO configurations, molecular absorption at one receiver antenna induces dependency on the others, and finding a closed-form channel model remains an open problem.",
"The studies presented in [7], [8] revealed that it is possible to closely approximate the MIMO channel model by tuning the parameters using a SISO solution.",
"The CSI required for the transmitter in this model comprises the drift velocity, diffusion coefficient, distance, and MIMO parameters.",
"The challenge is to tune MIMO parameters for each topology, where the relational formula has not yet been solved.",
"The requirement of retuning parameters for varying topologies makes the system sensitive to displacements of the communication units.",
"Machine learning-based parameter tuning algorithms were introduced in [8], which manage the adjustment of the parameters given the communication distances in the provided training data.",
"We consider in vivo applications, such as the communication units located in blood vessels, where the dominance of the drift effect is high and boundary restrictions affect the channel.",
"The vessel-bounded channel space is regarded as a straight cylinder [2], [7], [6].",
"Threshold-based detection algorithms were proposed in [7], and learning-based algorithms were presented in [6] for SISO configurations.",
"In vessel-bounded MC, the set of channel state parameters consists of the distance, diffusion coefficient, velocity profile, and vessel radius.",
"One research direction for MC channels is the transition from mathematical analysis to prototype implementations.",
"In [2], the authors considered advective flow and chemical reaction kinetics to provide vessel-bounded MC channel responses.",
"However, this becomes increasingly difficult as the channel becomes more complicated, and another research approach is required.",
"An alternative approach of studying channels begins from a testbed with numerous measurements and then involves determining correlations between sets of input and output, as in [3], [14].",
"The authors in [3] discovered that the velocity profile is related to the radius, and both studies showed communication feasibility in vessel-bounded MC.",
"Challenge in these research directions is that prototypes must be developed beforehand.",
"Underwater environments with severe RF signal degradation are among the prospective fields for the application of MC.",
"The olfactory sense of sharks outperforms any other alternative communication for the subaqueous positioning task; thus, underwater MC is desirable in addition to acoustic/RF communications [4].",
"The aforementioned channel models may not be feasible in these environments.",
"It is highly recommended to use a network with multiple devices and antennas for long-distance communication.",
"Therefore, it is desirable to design communication transceivers with multiple components and limited channel knowledge.",
"Figure: ISI and ILI models for MIMO operations in MC."
],
[
"Interference Sources and Mitigation Strategies",
"Diffusion-based MC suffers from statistical noise as a consequence of Brownian motion.",
"Particles that fail to reach their target within the desired time slot are regarded as interference.",
"Therefore, acquiring a higher data rate comes at the cost of interference.",
"As evident in the first column of Fig.",
"3, the average receiver response to the single-shot transmission has a long tail, which implies that MC suffers from severe interference and noise.",
"In SISO MC, the molecules that cannot arrive within the symbol duration are the only interference sources, and are analogous to the inter-symbol interference (ISI) in RF communications.",
"The second column of Fig.",
"3 shows the effect of a long tail on successive signal responses.",
"Removing remnant molecules with appropriate enzymes can mitigate interference while increasing the system costs.",
"Otherwise, the best available strategy to date is to exploit the statistical channel characteristics and optimize the symbol duration.",
"Kim et al.",
"proposed solutions to achieve a high data rate given the constraints on the SNR level [5].",
"Applying MIMO schemes to MC yields additional spatial resources for the system.",
"Attaining an array or diversity gain by scheduling the transmitter antennas and summing the receiver antenna signals is straightforward.",
"Koo et al.",
"reported that achieving multiplexing gain is possible when inter-link interference (ILI) is considered [7].",
"When the antenna of a device is paired with the closest antenna of the communication counterpart, molecules at the wrong destination are regarded as ILI, as shown in the last column of Fig.",
"3.",
"Note that the curves in Fig.",
"3 are averaged values obtained from numerous observations at the receiver, whereas the actual receptions are random variables.",
"In [2], a Poisson random channel model was introduced to describe the actual responses.",
"Based on an MC feature where the signal degradation as a function of range is greater than that in RF communications, a simple idea for mitigation is to consider ILI noise and apply a SISO threshold.",
"The knowledge of ILI characteristics results in even better performance, as shown in [11] with the equalization method inspired by the zero-forcing method.",
"Koo et al.",
"also proposed zero-forcing detection algorithms using statistical characteristics only at the receiver under the assumption of partial CSI available at the receiver [7].",
"The authors in [10] also exploited the mean value of the channel and interference and proposed zero-forcing, minimum mean square error, and least square detectors.",
"They also showed that having decision feedback at the transmitter can achieve enhanced performance by eliminating successive interference.",
"A scenario with multiple transmitters and receivers also introduces an inter-user interference (IUI).",
"IUI can be regarded as an additional ILI, and additional spatial multiplexing gain can be exploited when pairs of communications are predefined.",
"As the medium drift speed increases, the ratio of ILI to the desired signal decreases.",
"In vessel-bounded channels, where drift dominates over diffusion, the ILI is negligible when the antenna configuration is well aligned in a single stream.",
"Thus, SISO detection algorithms can mitigate ISI for the MIMO setup.",
"Example detection algorithms are provided in [14], where it can be observed that the sensor's chemical instability can also yield heavier tails.",
"We can also assume that communication units spread over multiple vessel streams, which could exacerbate the ILI (or IUI) effect.",
"Lee et al.",
"introduced a vessel testbed that used MC MIMO with intersecting vessels and empirically demonstrated that multiplexing gain is achievable with CSI at both the transmitter and receiver [3].",
"The issue of modeling and solving complex vessel topologies is an open problem."
],
[
"Communication Transceiver Design",
"In this section, MC modulation techniques are presented, their capabilities for MIMO applications are discussed, and we also introduce detection algorithms."
],
[
"Modulations in MC",
"Modulation schemes can exploit, several domains simultaneously such as time, molecular type, power, and space.",
"The power in MC is proportional to the number of molecules that are relevant to the concentrations.",
"Power-based modulation in MC is known as concentration shift keying (CSK), which resembles to power amplitude-shift modulations in classical communications.",
"The binary modulation of CSK (BCSK) maps bit-1 into a fixed number of molecules and bit-0 into zero molecules.",
"Sending zero molecules is not mandatory; however, it is a popular option in terms of power and accuracy.",
"Therefore, BCSK is used interchangeably with on-off keying (OOK) in MC.",
"A high order of modulation is achievable using multiple molecular quantity levels.",
"The next common modulation source is the timing of molecular emission.",
"One such example is pulse position modulation (PPM) [15].",
"The system has a fixed symbol time, which is divided into multiple time slots.",
"A single-shot at one of the slots per symbol time was used, and the slot position represented a data symbol.",
"Another example in [6] employed single-slot timing modulation, which is equivalent to PPM with an infinite symbol time and indefinite modulation order.",
"When a transmitter and a receiver can use multiple types of molecules, additional degrees of freedom can be achieved by the type domain.",
"In [5], the authors proposed molecular type-based modulation.",
"Each molecule represents a different symbol.",
"Assuming that the molecules do not interact with each other, the type domain is independent of the others.",
"Therefore, this domain is typically used in combination with other domains.",
"While the domain has unlimited potential, we need to be cautious when planning to use the type domain because the complexity growth is not yet quantifiable and some of the chemicals may not be eligible for use in practice.",
"A device with multiple antennas yields another independent domain, thereby contributing to the performance gains of the system.",
"In [15], the authors proposed index modulation, where each index denoted one transmitter antenna, and the index represented an individual symbol.",
"Yu et al.",
"[9] also presented space shift keying (SSK) modulation and designed joint modulations of SSK and CSK for a 4$\\times $ 4 MIMO configuration.",
"Gursoy et al.",
"addressed two types of molecules using the index modulation in [15] and combined PPM with space in their subsequent study.",
"High-order modulation is achievable by adopting more antennas per unit; however, this increases the ILI and decreases the antenna separations.",
"The order of the modulation is also relevant to the channel and device functions.",
"We assumed an ideal case with $T$ transmitter and $R$ receiver antennas.",
"The maximum number of combinations that can be used for the modulation in theory is $(R+1)^T$ , where each transmit antenna can either choose one of the $R$ antennas or not.",
"The combinations of transmitter antennas are preferred to be uncorrelated so that the received signals at each antenna are separable by having high directional transmissions to achieve the optimum.",
"Figure: Applying MIMO to MC increasing performance in all terms: array, diversity, and multiplexing gains."
],
[
"Detection Algorithms",
"As stated in [7], the best detection algorithm that can be embedded in the receiver is the maximum a posteriori algorithm with perfect CSI.",
"To achieve optimal performance, the knowledge of data statistics, channel governing equations, channel state parameters, and unlimited receiver memory are required.",
"To obtain a practical detection algorithm, the constrained problems must be solved under relaxed conditions.",
"For example, when we have partial CSI, unknown parameters can be estimated, albeit with uncertainty.",
"Assuming that a dataset is non-biased, and we have a reliable channel model, techniques from RF communications such as zero forcing, coding combining [12], and the Viterbi algorithm properly adapted to the MC environment can be applied.",
"The authors in [7] proposed and compared several detection algorithms for cases in which a limited CSI set was provided.",
"The objective was to obtain a threshold for binary detection at the receiver for each bit slot.",
"Increasing the amount of CSI leads to improved performance, although communication is feasible even with a modest amount of CSI.",
"Farsad et al.",
"[6] proved that learning-based approaches can detect messages from empirical observations without any prior information on CSI.",
"The main difference between MIMO and SISO detection algorithms is the presence of multiple streams and the absence of an exact channel model.",
"For example, in pursuing multiplexing gain, it is assumed that the transceiver antennas are paired and handled the ILI by considering it as additional noise.",
"As mentioned previously, this model is feasible under the condition of antenna pairing in the aligned topology.",
"This problem needs to be addressed to mitigate the ILI when the configuration is not aligned, where the ILI can overwhelm the primary signal.",
"Machine learning algorithms are compatible with MC with an unknown channel model.",
"This approach requires a lengthy measurement period for both prototypes and numerical simulators.",
"Here, the challenge is that a sufficiently large dataset may not be available.",
"To obtain the maximum amount of possible data, device sharing or online data harvesting studies will be required in the future.",
"Furthermore, effective learning algorithms must be developed.",
"It was demonstrated in [3] that knowledge of the channel can reduce the burden of training, making it possible to train on a smaller training dataset.",
"Farsad et al.",
"stated that ensuring adequate memory at the receiver enables recurrent neural networks (RNNs) to perform better.",
"Lee et al.",
"presented a learning algorithm for molecular MIMO that trains MIMO channel tuning parameters in combinations or independently.",
"Therefore, further investigation of the channel considering the developments in training neural networks is required."
],
[
"Exploitation of Spatial Resources",
"The number of paths that can be created using $T$ transmitter antennas and $R$ receiver antennas is $T \\times R$ .",
"However, the antenna correlation limits performance gains; the number of uncorrelated signals at the receiver is bounded between $R$ and one.",
"The MC channel attenuates highly over distance; thus, it is sensitive to physical separation.",
"Misalignment can also affect the effectiveness of receivers.",
"To the best of our knowledge, spatial correlation studies between diffusive signals have not yet been conducted.",
"Antenna correlation studies can provide an upper bound of the gains for general configurations and help decide the order of priority to explore.",
"Therefore, it is expected that characterizing the correlation effects for transmitter and receiver antennas can result in further progress in the theory and prototyping of molecular MIMO systems.",
"The effectiveness of the receiver can be further enhanced by ensuring the transmitter antenna directivity.",
"The number of uncorrelated signals at the transmitter antennas depends on their correlation.",
"It is easier to adjust the transmitter antennas than the receiver antennas, because the presence of a single transmitter antenna does not affect the others.",
"After obtaining the information paths, performance gains can be achieved by applying MIMO strategies as in RF communications.",
"The authors in [12] reported that array gain is achievable by adding coding and receiver combining techniques with the aforementioned detection algorithms, including threshold-finding and learning-based algorithms.",
"The numbers below SISO and MIMO in the top part of Fig.",
"4 indicate the number of molecules required to achieve the target accuracy.",
"They proposed that the Alamouti-type code or repetitive-code benefits the system, and a weighted combination of signals at the receiving antennas introduces array gains of 3 dB with a 2$\\times $ 2 MIMO configuration.",
"Their results also indicate that it is feasible to exploit resources for diversity gains in MC MIMO.",
"We present the results from [9] in the middle part of Fig.",
"4 to show the diversity gain effects, where the error rates of using SISO, 2$\\times $ 2 MIMO, and 4$\\times $ 4 MIMO systems are compared under the same SNR.",
"The numbers below the configurations represent the symbol error rates.",
"The results of [7] also indicate that multiplexing gain is achievable, as shown at the bottom of Fig.",
"4.",
"The numbers below the power indicate the number of molecules used to transmit bit-1 while adopting OOK modulation, and the numbers under SISO and MIMO indicate bps per molecule."
],
[
"Validation by Prototypes",
"Three types of prototypes are presented for the three different representative channel types and propagation modes.",
"The first prototype is based on free-space diffusion and drift.",
"Its state-of-the-art version is equipped with a 2$\\times $ 2 MIMO configuration.",
"We expect its further applications in smart dust and tactical nano-robots.",
"The second prototype is a vessel-bounded testbed.",
"We anticipate that the advanced version will be used in in vivo robots where biocompatibility is vital in the future.",
"The third prototype can be operated underwater.",
"This system is capable of supporting futuristic scenarios, including abyssal communications, where RF signal impairments are severe."
],
[
"Free-Space MIMO Testbed",
"Communication via free-space propagation is achieved with fans, programmable sprays, and chemical sensors, as shown in the bottom part of Fig.",
"2.",
"The fan addresses the drift velocity, and its power can adjust the speed.",
"The programmable spray is controlled by an Arduino system on a chip to check the interactive results in a timely manner, while the preprocessed codes are embedded before use.",
"The transmission power is varied by adjusting the size of the spray outlet or spraying duration per shot.",
"The external processors handle the processing of received responses at the sensors; standalone nano-processors can be investigated in future work.",
"There is wide potential extensibility in the spatial domain, but limited progress has been achieved thus far.",
"Researchers have investigated the variation in performance using 2$\\times $ 2 MIMO and SISO configurations and proved that the results obtained using formulation-based analysis agree well with the prototype results [7].",
"Future upgrade to the free-space MC testbed would focus on increasing the number of adoptable antennas based on suitable theory, while enabling gains from multiple domains."
],
[
"In-Vessel MIMO Testbed",
"Researchers have built in-vessel MC systems at multiple scales.",
"Farsad et al.",
"designed a meso-scale prototype utilizing base and acid molecules.",
"Programmable pumps introduced molecules into the channel, and pH sensors captured the responses.",
"The desired channel topologies can be expressed using adjustable rubber pipes and joints.",
"The authors applied machine learning for detection, and the RNN significantly enhanced the performance by efficiently mitigating the ISI.",
"It is also worth noting that drift domination results in the sharpness of the peak reactions.",
"This implies high extensibility in the time domain.",
"Lee et al.",
"[3] expanded the prototype into a 2$\\times $ 2 MIMO setup.",
"The developed testbed was appropriate for modeling various configurations of tube environments such as the human blood vessel network.",
"A 3D printer was used to print the junction parts.",
"Koo et al.",
"[14] implemented a prototype at the nanoscale.",
"Most of the components were analogous to those in [3].",
"One difference was that the sensor was adaptively designed for MC at the nanoscale.",
"In contrast with current nano-sensors, their sensor was rapidly reactive, sensitive to changes in concentration, and inexpensive.",
"Their study is significant for demonstrating the feasibility at the nanoscale and calling for interdisciplinary research for further advancements."
],
[
"Underwater Testbed",
"Guo et al.",
"[4] introduced a macro-scale underwater MC with a prototype that exploited gravity and buoyancy to convey molecules.",
"The objective of this study was to employ MC in liquid environments.",
"In their implementation, two nodes communicated with each other along the vertical axis.",
"The buoy emitted molecules with an initial velocity from the surface, and the molecules were moved by gravity.",
"In contrast, the submarine-unit emitted molecules could float with the help of buoyancy.",
"Generally, most MC studies in a fluid environment assume laminar flow.",
"However, it is challenging to neglect turbulent flow when the physical size of the application becomes enormous, e.g., tracking the target position beneath the deep ocean.",
"The turbulent flow complicates the analysis of the theoretical channel model, and study on the case is a notable open problem."
],
[
"Open Challenges and Concluding Remarks",
"This study aims to provide insights into the research developments in MC, particularly for exploiting spatial resources with multiple antennas.",
"We surveyed channel models and interference sources in MC and discussed potential technical hurdles that need to be overcome.",
"Next, we summarized the communication strategies considered in MC research and listed open challenges for further study.",
"Because some of the system designs include intractable tradeoffs, it is essential to develop prototypes to test their feasibility and quantify the performances.",
"Therefore, we presented testbed studies for three prospective propagation environments for MC applications.",
"The open challenges identified herein can be summarized as follows: Closed-form channel representations: Analytic characterizations of channels remain open problems in many cases: free-space propagations with multiple absorbing receiver antennas, vessel-bounded complex environments with junctions and leakages, and undersea propagations.",
"Testbed design for feasibility study: Various types of hardware should be developed to enable diverse practical applications.",
"Underwater testbeds that perform sensitive position tracking tasks or testbeds with the capability to compose and sense multiple types of chemicals can be one example.",
"Theoretical studies for generalization: Physical principles that describe the effects of deploying an arbitrary number of antennas are crucial to designing general MIMO communications.",
"For example, knowing the impact of signal correlation between multiple antennas can optimize the MIMO configurations.",
"The current state of MC MIMO research indicates that performance gains from spatial domain exploitation are possible and can be coupled to exploit power, time, and molecular type.",
"In [7], the multiplexing gain results in 1.7 to 1.9 times higher transmission rates with a 2$\\times $ 2 MIMO system when compared with the SISO system.",
"Joint optimization techniques with adaptive coding, modulation, and receiver combining schemes can help achieve further efficiency.",
"Joint modulations were proposed and tested virtually in [9], [15].",
"We suggest two research directions for further studies.",
"The first involves determining a theoretical bound by comparing the results of the signal correlations for the generalized configurations and modulation orders.",
"This will determine the capacity of the system and provide researchers with insights regarding the areas where research must be prioritized.",
"The other direction is a prototype investigation to obtain quantitative information to deploy multiple antennas in practice.",
"We foresee the potential of nano-communications in various scenarios.",
"We also anticipate problems in deploying communication systems in new and unexplored environments.",
"The findings of previous studies suggest that learning-based approaches can address unknown channels, provided that we have sufficient observation data.",
"Machine learning is a widely expanding research field, and synergetic effects can be achieved when the learning structure and hardware are specialized for target applications.",
"Hence, interdisciplinary research can boost nano-communication performance, while an optimized learning network is desired beforehand.",
"Hence, we expect that MC hardware, training networks, and transceiver design will be jointly optimized in the future."
],
[
"Acknowledgment",
"This work was supported in part by the Scientific and Technical Research Council of Turkey (TUBITAK) under Grant 119E190 and part by the Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education (NRF-2020R1A2C4001941).",
"Bon-Hong Koo is currently a Ph.D. student at the School of Integrated Technology, Yonsei University, Korea.",
"His research interests include molecular communications, and machine learning.",
"Changmin Lee is currently a Ph.D. student at the School of Integrated Technology, Yonsei University, Korea.",
"His research interests include testbed implementations.",
"Ali Emre Pusane is a Professor in the Department of Electrical and Electronics Engineering, Bogazici University, Turkey.",
"His research interests include wireless communications, molecular communications, information theory, and coding theory.",
"Tuna Tugcu is a Professor in the Department of Computer Engineering, Bogazici University, Turkey.",
"His research interest includes nanonetworking, molecular communications, wireless networks, and IoT.",
"Chan-Byoung Chae is an Underwood Distinguished Professor at Yonsei University, Korea.",
"His research interest includes emerging technologies for 5G/6G and molecular communications.",
"He is now an EiC of the IEEE Trans.",
"Molecular, Biological, and Multi-scale Communications and an IEEE Fellow."
]
] | 2107.01793 |
[
[
"E-SC4R: Explaining Software Clustering for Remodularisation"
],
[
"Abstract Maintenance of existing software requires a large amount of time for comprehending the source code.",
"The architecture of a software, however, may not be clear to maintainers if up to date documentations are not available.",
"Software clustering is often used as a remodularisation and architecture recovery technique to help recover a semantic representation of the software design.",
"Due to the diverse domains, structure, and behaviour of software systems, the suitability of different clustering algorithms for different software systems are not investigated thoroughly.",
"Research that introduce new clustering techniques usually validate their approaches on a specific domain, which might limit its generalisability.",
"If the chosen test subjects could only represent a narrow perspective of the whole picture, researchers might risk not being able to address the external validity of their findings.",
"This work aims to fill this gap by introducing a new approach, Explaining Software Clustering for Remodularisation, to evaluate the effectiveness of different software clustering approaches.",
"This work focuses on hierarchical clustering and Bunch clustering algorithms and provides information about their suitability according to the features of the software, which as a consequence, enables the selection of the most optimum algorithm and configuration from our existing pool of choices for a particular software system.",
"The proposed framework is tested on 30 open source software systems with varying sizes and domains, and demonstrates that it can characterise both the strengths and weaknesses of the analysed software clustering algorithms using software features extracted from the code.",
"The proposed approach also provides a better understanding of the algorithms behaviour through the application of dimensionality reduction techniques."
],
[
"Introduction",
"Software clustering is one of the software remodularisation and software architecture recovery techniques that has received a substantial amount of attention in recent years [1], [2], [3], [4], [5], [6], [7].",
"There are many goals for software remodularisation, including, but not limited to, representing the high-level architectural view of the analysed software, removing potential technical debt due to suboptimal software structure, and reverse documentation of poorly documented systems.",
"Software remodularisation can help software developers and maintainers better understand the interrelationships between software components.",
"As discussed in the work by Teymourian et al.",
"[8], most of the software clustering algorithms fall into two main categories which are agglomerative hierarchical [2], [3], [9], [10] and search-based algorithms [11], [12], [13].",
"In general, software clustering works by choosing from a collection of software entities (methods, classes, or packages) and then forming multiple groups of entities such that the entities within the same group are similar to each other while being dissimilar from entities in other groups.",
"By dividing and grouping software entities based on their functionality, these groups or clusters can be recognised as functionally similar subsystems, which can be used to represent the software architecture of the system.",
"Ultimately, the high-level architecture view of the software system will aid software maintainers in implementing new functionalities or make changes to existing code through better comprehension of the software design.",
"Due to their capabilities to aid in architecture recovery and software remodularisation, software clustering techniques have been widely investigated and a large number of techniques have been introduced [14], [1].",
"These techniques differ greatly in terms of the chosen common clustering features, similarity measures, clustering algorithm, and evaluation metric [14], [15], [12].",
"There is a vast variety of software systems from different domains with unique structures, characteristics, and behaviour.",
"However, the suitability of different clustering algorithms for different software systems are not investigated thoroughly.",
"Different clustering algorithms tend to produce semantically different clustering results.",
"For instance, if classes are chosen as the basis to perform software clustering, the clustering feature extraction method will only look at class-level interaction between those classes.",
"Subsequently, the clustering results produced by the class-level clustering algorithm will be completely different from a method-level clustering algorithm, although both results might be equally feasible.",
"Furthermore, comparing software clustering algorithms within the same level of granularity is also not straightforward, due to different fitness functions and cluster validity metrics employed by different algorithms [16], [9].",
"Even if we were to compare the effectiveness of the clustering algorithms from the same family (i.e., agglomerative hierarchical clustering), there are still different ways to configure them (i.e.",
"different distance metrics, different linkage algorithms, and different validity indices for hierarchical clustering algorithm).",
"It is then up to the researchers to choose a software clustering evaluation method depending upon if they are able to produce the reference decomposition or model, to evaluate the effectiveness of the clustering results.",
"Almost all of the existing studies in software clustering only emphasised on the advantages and benefits brought upon by the proposed clustering technique, while limited studies suggest the limitation of their approaches [17], [15], [14].",
"Therefore, it raises a question as to how software clustering algorithms are evaluated.",
"Most studies which introduce new clustering algorithms often only evaluate their approach on a specific set of problem instances [14], [16], [17].",
"Different from existing studies, this work aims to provide a better understanding of which software/code features (i.e., lines of code, number of methods, etc.)",
"are related to the performance of clustering algorithms, and whether the software/code features can be used to select the most suitable clustering algorithm.",
"Our work is inspired from similar research efforts in optimisation and search-based software testing [18], [19].",
"This work aims to fill this gap by introducing a new approach that evaluates the effectiveness of software clustering techniques by providing information on their suitability according to the software or code features, which as a consequence, enables the selection of the most optimum algorithm and configuration from our existing pool of choices for a particular profile of a software system.",
"Using the proposed framework, the pool of chosen software clustering algorithms only requires profiling to be done for the first time, in order for the proposed framework to recommend the optimum algorithm and configuration from our existing pool of choices.",
"Software systems that exhibit characteristics that match a profile from the pool of choices (software clustering algorithm) will be recommended with the respective clustering algorithm to improve the overall efficiency and effectiveness.",
"The entire workflow is summarised in Figure REF .",
"Different from traditional software clustering research that uses a trial and error approach to identify the optimum clustering algorithm and its configuration (dashed line in Figure REF ), the proposed framework only requires the developer or researcher to extract the software features of the software to be remodularised, in order for the proposed framework to recommend the most suitable clustering algorithm.",
"Figure: E-SC4R framework design and workflow.Note that in this paper, we only focus on comparing and evaluating the effectiveness of different variants of agglomerative hierarchical clustering algorithms and Bunch clustering algorithm because i.)",
"agglomerative clustering and Bunch (which is a search-based software clustering algorithm) are two of the most popular clustering algorithms as discussed in [8], and ii.)",
"results produce by different family of software clustering algorithms exhibit significantly different structure and behaviour, which are difficult to compare directly.",
"Our proposed technique aims to characterise both the strengths and weaknesses of the analysed software clustering algorithms using existing and newly developed software features extracted from the code.",
"The proposed technique also provides a clear understanding of the algorithm's behaviour by showing a 2D representation of the effectiveness of software clustering techniques on the feature space through the application of dimensionality reduction techniques.",
"Note that to avoid confusion, the term clustering features used in this paper refers to the features extracted from the chosen clustering entities (classes), while software features refers to the characteristic of the software/code such as the lines of code, number of public methods, number of static methods, etc.",
"In essence, the proposed approach can be used to characterise the software/code features that have an impact on the effectiveness of clustering algorithms.",
"It is a known fact that the selection of clustering entities and clustering features will directly influence the final clustering results.",
"If we could understand and correlate the relationships between software/code features and the effectiveness of software clustering algorithms, it is then possible to choose the most optimum clustering algorithm and configuration from our existing pool of choices, based on the profiled software/code features.",
"We show how such software/code features can be measured, and how the footprints of software clustering algorithms (regions where clustering algorithms' strengths are expected) can be visualised across the inspected software components.",
"The proposed approach can be used for performance prediction, enabling the selection of the most suitable clustering algorithm according to the software/code features.",
"The results can also lead to algorithm improvements, considering one of the aims of the proposed approach is to reveal the weaknesses of clustering algorithms.",
"The research questions are: RQ1 How can we identify the strengths and weaknesses of clustering techniques for the remodularisation of software systems?",
"RQ2 How can we select the most suitable clustering technique from a portfolio of hierarchical and Bunch clustering algorithms?"
],
[
"The E-SC4R Framework",
"The E-SC4R framework provides a way for objective assessment of the overall effectiveness of software clustering techniques.",
"Understanding the effectiveness of a software clustering technique is critical in selecting the most optimum technique for a particular software, avoiding trial and error application of software clustering techniques.",
"The purpose of E-SC4R is the ability to identify the most optimum algorithm and configuration from our existing pool of choices for software clustering.",
"The approach involves two main parts: Strengths and Weaknesses of Clustering Techniques, which learns significant software features, such as lines of code, cohesion, coupling, and complexity, that reveal why certain remodularisation problems are hard.",
"The first part visualises footprints of clustering techniques which expose their strengths and weaknesses, and Clustering Technique Selection, which addresses the problem of selecting the most suitable technique for software remodularisation.",
"The approach we propose has two main goals: to help designers of software clustering techniques gain insight into why some techniques might be more or less suited to remodularise certain software systems, thus devising new and better techniques that address any challenging areas, and to help software developers select the most effective clustering technique for their software systems.",
"block = [rectangle, draw, fill=mycolor3!15,text width=14em, text centered, minimum height=4em, thick, rounded corners=4pt] block1 = [rectangle, draw, fill=white,text width=14em, text centered, minimum height=4em, thick, rounded corners=4pt] big = [rectangle, draw, inner sep=0.5cm] line = [draw, -latex',thick] Figure: An overview of E-SC4R.An overview of the E-SC4R framework is presented in Figure REF .",
"The boxes represent the artefacts, while the arrows are the processes/steps for creating the artefacts.",
"In the following subsections, we describe our framework in more detail for each artefact/step."
],
[
"Software Systems",
"The software systems, defined as $s\\in S$ in Figure REF are software systems to be remodularised by researchers using software clustering techniques to recover a high-level abstraction view of the software architecture.",
"The software systems used in this study are chosen in a pseudo-random manner, with a mixture of GitHub projects and Apache projects.",
"The following process is used when selecting the projects from the main GitHub and Apache repository.",
"The search parameters are set to filter out Java-based projects at https://github.com/topics/java, Sort projects by the number of stars, Projects are manually chosen if they meet the following criteria, Have at least 10 commits in the past year, README.MD, Project Title, \"About\" and comments are written in English.",
"In order to make sure that the selected project is still currently active, we only chose projects that have at least 10 commits in 2021 (the year in which we conducted the experiment)."
],
[
"Software Clustering",
"Software clustering techniques are defined as $C(s) \\in Y$ in Figure REF , where a specific clustering technique $C$ , which is a subset of $Y$ , applied on software $s$ .",
"Depending on the different interpretation of a meaningful and effective software clustering result, different clustering algorithms proposed in the existing literature address the software remodularisation and architecture recovery problem in a different manner.",
"While these algorithms usually aim to achieve common goals, i.e., improve the software modularity, quality, and software comprehension, they usually produce significantly different clustering results but at the same time, might also offer equally valid high-level abstraction views of the analysed software [20].",
"Hence, it is necessary to discuss the working principle of some of the widely adopted software clustering algorithms available in the current literature."
],
[
"Search-based Software Remodularisation Algorithms",
"Search-based approaches have been successfully utilised to address the software remodularisation problem.",
"In general, the workflow of search-based clustering algorithms consists of the following steps [12] [3].",
"Generate a random seed solution based on some initial parameters, Explore the neighbourhood structure of the solution.",
"If a neighbour has a better fitness, it becomes the new solution, Repeat step (2) until the neighbourhood of the current candidate solution offers no further improvement to fitness function (local optimal), Restart step (1) with different seed to find better solution and fitness (global optimal).",
"Numerous existing studies that adopt search-based approaches are based on the work by Mitchell and Mancoridis [21], [12], where their approaches have been implemented as part of the Bunch software clustering tool.",
"The authors proposed the notion of module dependency graph (MDG) as the basis of their clustering entities.",
"In their context, module represents source code entities that encapsulate data and functions that operate on the data (e.g., Java/C++ classes, C source code files).",
"Dependencies between the modules, on the other hand, are binary relations between the source code entities, depending upon the programming language used to implement the analysed software systems (e.g., function/procedure invocation, variable access, and inheritance).",
"The Bunch clustering algorithm works by generating a random partition of the MDG which is then re-partition systematically by examining neighbouring structures in order to find a better partition.",
"When an improved partition which yields better intra-cluster cohesion and inter-cluster coupling is found, the process repeats by using the newly found partition as the basis for finding the next improved partition.",
"The algorithm stops when it cannot find a better partition.",
"Based on the work by Mitchell and Mancoridis, Harman et al.",
"propose a search-based optimisation method to search for the best optimum partition [22].",
"In their work, Harman et al.",
"proposed a new objective function for the search-based problem such that for each pair of modules in a partition/cluster, the optimisation score is incremented if there is a dependency between the modules; otherwise, it is decremented.",
"The authors do not consider inter-cluster coupling and only focuses on intra-cluster cohesion.",
"Experiment results show that their proposed approach can tolerate noises better than the Bunch clustering algorithm [21] and reach optimum results faster than Bunch-guided approach.",
"In the work by Beck and Diehl [23], the authors discussed that clustering algorithms based on the Bunch tool often only rely on the structural and static information of the source code in order to measure the similarity and dependencies among software entities.",
"The authors attempted to enrich the structural data with some evolutionary aspects (historical data) of the analysed software such as size of packages, ratio of code to comments, and the number of download.",
"Experiment results show that using evolutionary data alone does not perform better than the traditional clustering algorithms that utilise structural data.",
"It is only when both data are integrated, the clustering results achieve much higher accuracy when compared against the reference model."
],
[
"Hierarchical Clustering Algorithms",
"On the other hand, hierarchical clustering iteratively merges smaller clusters into larger ones or divide large clusters into smaller ones, depending on whether it is a bottom-up or top-down approach.",
"Merging or dividing operations are usually dependent on the clustering algorithm used in the existing studies.",
"In general, hierarchical clustering algorithms can be divided into two main approaches, divisive (top-down) and agglomerative (bottom-up) hierarchical clustering algorithms.",
"Divisive clustering is based on a top-down hierarchical clustering approach where the clustering process starts at the top with all data in one big cluster.",
"The cluster is then split into smaller clusters in a recursive manner until all data resides in a single cluster.",
"For problem domains with a fixed number of top levels, using flat algorithms such as $K$ -mean yield lower computational complexity because divisive clustering is linear in the number of clusters [24].",
"Although the computational complexity of divisive clustering is lower than agglomerative clustering, complete information about the global distribution of the data is needed when making the top-level clustering decisions [25].",
"Most of the time, software maintainers are not involved in the earlier software design phases.",
"If the software documentations are not up-to-date, it is hard for maintainers to identify the ideal number of software packages (or the number of clusters in the context of software clustering) before any attempt to remodularise any software systems.",
"On the other hand, the work by Wiggerts [26] discussed how agglomerative clustering, a bottom-up clustering approach would be helpful to software engineers in remodularising legacy and poorly documented software systems.",
"According to the author, the working principle of agglomerative clustering is actually similar to reverse engineering where the abstractions of software design are recovered in a bottom-up manner.",
"Agglomerative hierarchical clustering starts by placing each cluster entity (usually code or classes in the context of software clustering) in a cluster on its own.",
"At each iteration as we move up the hierarchy, two of the most similar clusters from the lower layer are merged and the number of clusters is reduced by one.",
"The decision to merge which clusters differs depending on the similarity measure and the linkage algorithm used, i.e., nearest neighbour between two clusters, furthest neighbour between two clusters, or average distance between two clusters.",
"Once the two chosen clusters have been merged, the strength of similarity between the newly formed cluster and the rest of the clusters are updated to reflect the changes.",
"The merging process will continue until there is only one cluster left.",
"The results of agglomerative clustering are usually presented in a tree diagram, called dendrogram.",
"A dendrogram shows the taxonomic relationships of clusters produced by the clustering algorithm.",
"Cutting the dendrogram at a certain height produces a set of disjoint clusters.",
"In this paper, we will focus on examining the strength and weakness of different configurations for agglomerative hierarchical clustering algorithm and Bunch algorithm because they are one of the most widely used generic clustering algorithm and search-based algorithms in the existing literature [14], [9], [27], [23], [28].",
"It will be interesting to compare the performance of two different families of software remodularisation techniques and examine their suitability on different behaving datasets."
],
[
"Clustering Technique Performance Indicator",
"The Clustering Technique Performance Indicator, denoted as $I(C(s)) \\in R$ takes as input the clustering results generated by a clustering algorithm $C(s)$ for a particular software system $s\\in S$.",
"There exist various ways to evaluate the performance of software clustering algorithms.",
"Typically, the clustering results generated from a clustering algorithm $C(s)$ are measured against a reference model, which refers to a known good clustering result or a reliable reference that can act as a baseline for comparison.",
"Hence, performance indicator typically measures the similarity of the clustering results against the reference model.",
"One way of accessing the reference model is through feedback from domain experts in the analysed system, for instance, the original designer, system architect, or senior developers directly involved in the development of the software.",
"However, this approach is difficult to realise because the software maintainers are usually not involved in the initial design and development of the maintained software.",
"In existing works, when there are no inputs from domain experts to create a reference model or ground truth, several authors have chosen to use the directory structure or package structure of the analysed software to create an artificial ground truth or reference model [4], [27], [23].",
"This method is less expensive compared to retrieving the reference model from domain experts because it can usually be automated.",
"However, the reliability of using the directory or package structure of the analysed system is strongly dependent on the skills and experience of the software developers because it is assumed that software developers follow the best practices of putting functional relevant and similar classes into the same package directory.",
"If an artificial ground truth or reference model can be retrieved, we can then compare it with the clustering results generated by a clustering algorithm $C(s)$ to measure the extent to which two given decompositions of the software are similar to each other.",
"The work by [29] discussed that one of the most popular performance indicator for software clustering results is the MoJo family of metrics [30]: $\\text{MoJoFM}(A,B) = (1- \\dfrac{mno(A,B)}{\\max (mno(\\forall A, B))}) \\times 100\\%$ where $mno(A,B)$ is the minimum number of Move or Join operations needed to transform the clustering result $(A)$ into ground truth $(B)$ , and $max(mno(\\forall A, B)$ is the maximum number of operations to transform the clustering result to ground truth.",
"MoJoFM return 0 if the clustering result is very different from the ground truth, and return 100 if the clustering result is identical to the ground truth.",
"We will be using MoJoFM as the Clustering Technique Performance Indicator $R$ in this paper."
],
[
"Strengths and Weaknesses of Clustering Techniques",
"One of the important steps in E-SC4R is identifying features of software systems $F(s) \\in F$ that have an impact on the effectiveness of software clustering techniques.",
"Software features such as lines of code, number of public methods, number of static methods, etc., are problem dependent and must be chosen such that depending on the type of target software systems $s \\in S$, any known structural properties of the software systems are captured, and any known advantages and limitations of the different software clustering algorithm are related to the features.",
"For each version of the analysed project, software features are extracted using the Java code metrics calculator, which is publicly available online [31].",
"The tool is capable of calculating simple size metrics such as the numbers of methods, lines of code, and number of private fields, to more complex measures such as depth of inheritance, coupling between object, and other CK suite of metrics.",
"In total, there are 40 metrics that we extracted using the tool.",
"These metrics are calculated at the class level.",
"To aggregate the calculated metrics at the project level, we calculate the max, mean, standard deviation, and sum over all classes of a project, resulting in $40 \\times 4 = 160$ metrics for each project.",
"Table REF shows some of the metrics used in this paper as well as their definitions.",
"The full list of metrics along with its description can be found on the tool's GitHub pagehttps://github.com/mauricioaniche/ck..",
"The set of features listed in Table REF is only a subset of the total number of features used in the paper, where we list some of the more general and widely used software features.",
"As such, the terms software metrics and software features will be used interchangeably in this paper to denote the metrics that represent different features of the analysed software.",
"Table: Description of software features.Using the results gathered from clustering algorithms and software feature extraction, we can create the footprint visualisations of each clustering algorithm in order to identify the most significant software features that have an impact on its effectiveness.",
"E-SC4R identifies software features that have an impact on the effectiveness of software clustering techniques.",
"The clustering results can be affected by the program structure and/or source code, the complexity of dependencies between classes, information about the input/output data space, and information dynamically obtained from program execution.",
"All these aspects may influence the suitability of software clustering techniques for a particular software system.",
"In E-SC4R, a subset of features is considered significant if they result in an instance space – as defined by the 2-dimensional projection of the subset of features – with software systems where a particular clustering technique performs well being clustered together.",
"The software instances are initially projected in the 2D instance space in such a way that if two software systems are similar according to some features, they are closer together, and if they are dissimilar, then they are far apart.",
"Since we focus on arranging the software systems in a space where the instances of a technique is effective are separated from the ineffective ones, we represent a software system as a vector of the most significant features that are likely to correlate with a clustering technique's effectiveness.",
"E-SC4R identifies software features that are able to create a clear separation in instance space, such that we can clearly see the different clusters of software systems where the techniques are effective.",
"We refer to these clusters as clustering technique footprints.",
"The most significant features are determined in 2 steps [32].",
"Firstly, MoJo is determined as the performance metric to measure the effectiveness of the clustering algorithm based on the software features (e.g.",
"lines of code, number of methods, etc.).",
"Next, a genetic algorithm is applied to select the set of features that maximises MoJo as given below: Select a set of software features, Generate an instance space using PCA for dimension reduction, as described below, Fitness of the set of software features is evaluated If the set of software features is not suitable, return to Step 1.",
"The genetic algorithm will search the space for possible subsets and determine the optimal subset with the classification accuracy on an out-of-sample test set used as the fitness function.",
"The instance space is generated in iterations, until an optimal subset of features is found [33].",
"A subset of the features is considered of high quality if they result in a footprint visualisation with distinct clusters.",
"The best subset of features is the one that can best discriminate between high performing and low performing clustering algorithms.",
"After the most significant software features (lines of code, number of methods, etc.)",
"are identified, these features are then used as an input into SVM to capture the relationship between the selected features and the MoJo value of the clustering results.",
"Similar to our previous work [19], [34], we use principal component analysis (PCA) as a method for projecting the software instances to two dimensions, while making sure that we retain as much information as possible.",
"PCA rotates the data to a new coordinate system $\\mathbf {R}^k$ , with axes defined by linear combinations of the selected features.",
"The new axes are the eigenvectors of the covariance matrix.",
"The subset of features that have large coefficients and therefore contribute significantly to the variance of each Principal Component (PC), are identified as the significant features.",
"Retaining the two main PCs, the software instances are then projected on this 2D space.",
"Following a similar approach to previous work on dimensionality reduction [35], we accept the new two dimensional instance space as adequate if most of the variance in the data is explained by the two principal axes.",
"The two principal components are then used to visualise the footprints of the clustering technique.",
"The footprint indicates the area of strength (software instances on which the clustering technique is effective) of each clustering technique.",
"This step provides an answer to the first research question, RQ1 : How can we identify the strengths and weaknesses of clustering techniques for the remodularisation of software systems?"
],
[
"Clustering Technique Selection",
"Finally, E-SC4R is used to predict, based on the most significant software features, the most effective clustering technique for new remodularisation and architecture recovery problems.",
"This step answers the second research question: RQ2: How can we select the most suitable clustering technique from a portfolio of hierarchical and Bunch clustering algorithms?",
"This is achieved by modelling the relationship between software features and clustering technique effectiveness by employing a Support Vector Machine [36] to learn this relationship.",
"E-SC4R uses the two-dimensional space as an input to the Support Vector Machine to learn the relationship between the software features and remodularisation method performance.",
"The SVM is C-classification.",
"The cost C in the regularisation term and the RBF hyper-parameter $\\gamma $ are tuned via grid search in [1,10] and [0,1] respectively.",
"We use 10-fold cross validation to train the model and assess the model generalisation ability.",
"The cross-validated Mean Squared Error and Error Rate are used as estimates of the model generalisation ability in classification.",
"At the end of this process, E-SC4R creates a model that can select the most effective technique for remodularisation based on the features of software programs.",
"This model can be retrained and extended further with new remodularisation techniques and software features."
],
[
"Experimental Design",
"In this chapter, we discuss the way how the experiment is designed to examine the strength and weakness of different configurations for agglomerative hierarchical clustering algorithm and Bunch algorithm."
],
[
"Agglomerative Hierarchical Software Clustering",
"The process of agglomerative hierarchical clustering can be summarised in the following steps, illustrated in Figure REF.",
"Identification of clustering entities, Identification of clustering features, Calculation of similarity measure, Application of clustering algorithm, Evaluation of clustering results.",
"Figure: Agglomerative hierarchical and Bunch clustering processes.In software clustering, the typical choices of entities are in the form of methods, classes, or packages because they represent the basic components and functionalities of a software system.",
"In this paper, we focus on representing the software at the class-level because in object-oriented software systems, classes are the main building blocks that contain the implementation details of the examined components."
],
[
"Identification of clustering features",
"The similarities between entities are determined based on their characteristics or clustering features extracted from the available information.",
"Extracting dependencies between code entities is critical in architecture recovery because it helps to understand the static and dynamic relationships between them.",
"Several existing studies have proposed different methods to extract dependencies from software entities at different levels of granularity, including code, class, and package levels [37], [38], [39].",
"The work by Jin et al., [37] in particular, released their open-source dependency extraction tool, Dependshttps://github.com/multilang-depends/depends, which is capable of gathering syntactical relations among source code entities such as files and methods.",
"In this work, we choose to utilise Depends to extract dependencies between classes in the analysed software in order to improve the replicability of our research findings.",
"Depends extracts the following dependency types: Import, Contain, Parameter, Call, Return, Throw, Implement, Extend, Create, Use, Cast, ImplLink, Annotation, Mixin.",
"We are then able to generate a $NxN$ ($N$ = number of classes) matrix that denotes the relationships between all the classes.",
"Sample data that are extracted from Depends are shown in Table REF , which are then aggregated into relationships between clustering entities shown in Table REF .",
"Table: Examples of relationships extracted from Depends.Table: Examples of relationships between clustering entities aggregated from Table ."
],
[
"Calculation of similarity measure",
"The next step is to ascertain the similarity between entities by referring to the clustering features identified in the previous step.",
"In this paper, we choose to use distance measures because we are able to quantify the strength of dependencies between classes with the aid of Depends.",
"We do not attempt to distinguish between the different type of dependencies identified by Depends, but instead aggregate all the dependencies to represent an aggregated strength of dependency between two classes.",
"In order to generate the distance matrix, the following distance measures were taken into consideration to compute the dissimilarity between each class in the examined software: Euclidean distance : least squares, minimising the sum of the square of the differences between a pair of classes $d(x,y) = {\\sqrt{\\sum _{i=1}^{n} (x_i - y_i)^2}}$ where $n$ = number of classes, $x_i$ and $y_i$ are the classes of vectors $x$ and $y$ respectively in the two-dimensional vector space.",
"Manhattan distance : least absolute deviations, minimising the sum of the absolute differences between a pair of classes $d(x,y) = {\\sum _{i=1}^{n}} |x_i - y_i|$ Cosine distance : the cosine of the angle between a pair of classes $d(x,y) = 1 - {\\frac{\\sum _{i=1}^{n} x_i y_i}{\\sqrt{\\sum _{i=1}^{n} x_i^2}{\\sqrt{\\sum _{i=1}^{n} y_i^2}}}}$ These distance/similarity measures are chosen because they have been proven to be effective in measuring the similarity between software components in some of the related studies [14] [40]."
],
[
"Application of clustering algorithm",
"A clustering algorithm is needed to decide upon how and when to merge two clusters.",
"Depending on the algorithm used, certain algorithms merge the most similar pair first while others merge the most dissimilar first.",
"Once the two chosen clusters have been merged, the strength of similarity or dissimilarity between the newly formed cluster and the rest of the clusters are updated to reflect the changes.",
"It is very common that during hierarchical clustering, there exist more than two entities which are equally similar or dissimilar.",
"In this kind of scenario, the selection of candidate entities to be clustered is arbitrary [14].",
"In this work, we use the following three linkage algorithms [14]: Single Linkage Algorithm - defines the similarity of two chosen clusters as the maximum similarity strength among all pairs of entities (classes) in the two clusters Average Linkage Algorithm - defines the similarity measure between two clusters as the arithmetic average of similarity strengths among all pairs of entities (classes) in the two clusters Complete Linkage Algorithm - defines the similarity of two chosen clusters as the minimum similarity strength among all pairs of entities (classes) in the two clusters There exists many other newer algorithms that are proposed for software architecture recovery.",
"Currently, we are only including these three basic linkage algorithms to demonstrate E-SC4R's ability to identify the most optimum algorithm and configuration from our existing pool of choices.",
"Newer algorithms would be added to E-SC4R in future iterations.",
"Apart from that, we also attempt to determine the most optimum range of number of clusters for each of the chosen hierarchical clustering algorithms.",
"In hierarchical clustering, the final output is represented in a dendrogram, which is a tree diagram that shows the taxonomic relationships of clusters of software entities produced by hierarchical clustering.",
"The distance at which the dendrogram tree is cut determines the number of clusters formed.",
"Cutting the dendrogram tree at a higher distance value always yields a smaller number of clusters.",
"However, this decision involves a trade-off with respect to relaxing the constraint of cohesion in the cluster memberships [9], [16], [41].",
"As such, in this work, we attempt to determine the optimal total number of clusters by dividing the total number of classes with the following divisors : 5, 7, 10, 20, and 25.",
"The numbers were chosen based on the package distribution of the ground truth that we generated and depends on the number of classes of the analysed software.",
"In this paper, we choose to use these divisors instead of an exhaustive approach to save computation time and obtain a range of the optimal number of classes.",
"In practice, E-SC4R would allow the user to specify the number of clusters or divisors.",
"We use different configurations of clustering algorithms which differ between the combination of different distance metrics, linkage algorithms, and the number of clusters.",
"We then record the clustering results of each combination of the clustering algorithm on each version of the software, to be compared with the ground truth.",
"For example: Agglomerative Hierarchical Configuration 1 Linkage = Single Distance Metric = Euclidean Cluster Divisor = 5 Agglomerative Hierarchical Configuration 2 Linkage = Complete Distance Metric = Cosine Cluster Divisor = 7"
],
[
"Evaluation of clustering results",
"As mentioned earlier, creating a reference model to act as the ground truth by engaging domain experts is expensive in terms of time and effort.",
"On the other hand, the reliability of the package structure of the analysed software is strongly dependent on the experience of the software developers, as well as maturity of the analysed project.",
"In this paper, we create the reference model (ground truth) by looking at the most commonly occurring directory structure patterns for the 10 previous releases of the analysed software.",
"A more detailed example is available in Section 3.4.",
"The generated ground truth will then be used to compare against the clustering results that are produced by each of the hierarchical clustering algorithms.",
"A more detailed example is available in Section REF .",
"To evaluate the performance of each hierarchical clustering algorithm against the reference model, we use MoJoFM metric proposed in the work by [42], [30].",
"The MoJo family of metrics were widely used in the domain of software clustering to evaluate the performance of different clustering algorithms [14], [16], [27], [28].",
"Hence, in the remaining of this paper, the term performance of clustering algorithm refers to the MoJoFM value computed when comparing between the produced clustering results and the ground truth."
],
[
"Bunch Clustering Algorithm",
"Bunch supports three main clustering algorithms, namely hill-climbing algorithm, exhaustive clustering algorithm, and genetic algorithm.",
"The authors claimed that exhaustive clustering algorithm only works well for small systems [11], and the hill-climbing algorithm performs well for most software systems [12].",
"Bunch starts off by generating a random partition of the MDG.",
"Then, depending on the chosen clustering algorithm (hill climbing, genetic algorithm, or exhaustive), it will cluster each of the random partitions in the population and select the result with the largest Modularisation Quality (MQ) as the suboptimal solution.",
"MQ measures the quality of an MDG partition by taking into consideration the trade-off between the dependencies between the clustering entities (classes) of two distinct clusters (package/subsystem), and the dependencies between the clustering entities (classes) of the same cluster (package/subsystem).",
"The assumption made is that high quality software systems should be designed with cohesive subsystems that are loosely coupled between each other.",
"As the size of the problem (software systems) increases, the probability of finding a good sub-optimal solution (MQ) also increases.",
"In this paper, we will be using a combination of different algorithms and MQ calculator to evaluate their performance against the chosen datasets.",
"For example, Bunch Configuration 1 Algorithm = HillClimbing Calculator = TurboMQ Bunch Configuration 2 Algorithm = GeneticAlgorithm Calculator = TurboMQIncr"
],
[
"Dataset Collection",
"For each chosen project, we compare the clustering results across 10 releases to ensure the stability of the clustering algorithm.",
"Stability in software clustering is defined as the sensitivity of a particular clustering algorithm toward the changes in the dataset [22].",
"For any good clustering algorithm, small changes in the target software (clustering algorithm applied on multiple small increment releases of the same software) should not alter the clustering results significantly.",
"Due to the way how we create the ground truth, 10 prior releases of the examined software will be needed to identify the common directory structure, as shown in Figure REF .",
"As such, 21 releases of the chosen project are required to form one of our selection criteria.",
"Once we identified the suitable software, 21 versions of the project source code were then downloaded using GitHub CLI to a server based on the project link, release tag, and version name.",
"In total, we had chosen 30 Java-based projects collected from GitHub as we could not find more which suited our search criteria.",
"The selected projects are shown in Table REF and the complete set of datasets can be found on our Github pagehttps://github.com/alvintanjianjia/SoftwareRemodularization.",
"Table: List of chosen projects and their versions.The columns firstRelease and lastRelease in Figure REF indicate the versions of the software that we examined and used for our experiments.",
"Note that we use 21 incremental releases in between the stated firstRelease and lastRelease to ensure the stability of the clustering algorithms, and to generate the ground truth."
],
[
"Generation of Ground Truth",
"In this work, we attempt to improve the existing ground truth generation method by looking into the evolution of the analysed software over multiple releases instead of the latest version package structure.",
"The creation of the ground truth is done via extraction of common directories across 10 previous releases of the software.",
"For each version of the software, the previous 10 releases of the software were analysed and only the common file directory structure across all 10 versions will be extracted.",
"Given the scale of the 30 open source projects over multiple releases, it is challenging to find domain experts for each of the software systems, which is a similar problem encountered in the work by [43].",
"Ground truths that are generated and manually approved by senior developers working on the project would only be practical for a handful of projects.",
"However, this approach will not provide our research with enough data points for footprint visualisations during the evaluation of clustering results.",
"Given that the current factual architecture of the system has been created by the open source developers or administrators themselves, it is reasonable to assume the current package structure is held to a certain standard [44] [29].",
"Additionally, these open source projects are among the highest rated Java projects on GitHub based on stars, which provides additional confidence on the correctness of the generated ground truth based on the evolution of the package structure over multiple releases.",
"Figure: a.)",
"Method used to select project releases and generate ground truth b.)",
"Method used to evaluate clustering results against ground truth.In the example shown in Figure REF a, we use the common directories across apache_spark-1.0 to apache_spark-1.9 to generate the ground truth.",
"Subsequently, this ground truth comprising the common directory from releases 1.0 to 1.9 will be used to evaluate against the clustering results that we produce in the next release, which is apache_spark-2.0, as shown in Figure REF b.",
"To illustrate another simple example, given the following directory structure of a software in 3 incremental releases: V 2.5: src/var/c.java, src/var/d.java, tmp/eg/z.java V 2.6: src/var/c.java, src/var/d.java, temp/util/z.java V 2.7: src/var/c.java, src/var/d.java, temp/eg/z.java Based on our approach, only src/var/c.java and src/var/d.java are extracted from the given 3 versions.",
"A parent-child cluster relationship would be defined based on the extracted directory paths, given by the parent contains child.",
"For example, given that the following directory paths are extracted user/spark/java/a.java user/spark/java/b.java user/spark/main/test.java The following parent-child clusters will be created.",
"java contains a.java java contains b.java main contains test.java spark contains java spark contains main user contains spark The final clustering result obtained is then taken as the ground truth, which will be used as the reference model when compared using the MoJoFM metric."
],
[
"Measuring Dependencies between Classes",
"As mentioned earlier, for hierarchical clustering algorithms, we use Depends [37] to quantify the strength of dependencies between classes in the examined software.",
"The tool can create a $NxN$ matrix to show the types and frequency of dependencies between all the classes in the analysed software ($N$ = number of classes).",
"On the other hand, the Bunch tool uses Module Dependency Graph (MDG) to measure the strength of dependencies between classes[21], [12]."
],
[
"Selection and Permutation of Chosen Clustering Algorithms",
"For each of the chosen software projects, we ran different permutation of hierarchical and Bunch clustering algorithms based on the configuration shown in Table REF and Table REF .",
"Table: Summary of parameters and settings associated with hierarchical clustering.Table: Summary of parameters and settings associated with Bunch clustering.In total, we have 45 unique configurations of hierarchical clustering algorithms and 15 unique configurations of Bunch clustering algorithms to run on each chosen project.",
"Furthermore, for each configuration, we ran it against the 10 prior releases of the target software to ensure the stability of the algorithm."
],
[
"Evaluation of Clustering Results",
"The clustering results are compared with the ground truth using MoJoFM [30].",
"Due to the size of the table, we are unable to show the full set of clustering results from all the chosen projects.",
"The complete set of results can be assessed from our GitHub page.",
"The summarised version of the clustering results, showing the top 10 clustering results for hierarchical clustering and Bunch are shown in Table REF and Table REF respectively.",
"Table: Top 10 hierarchical clustering results based on MoJoFM.Table: Top 10 Bunch clustering results based on MoJoFM."
],
[
"Results and Discussion",
"The proposed E-SC4R framework identifies the most significant software features, which have an impact on the performance of clustering techniques.",
"The resulting SVM predictions are plotted in the reduced instance space as shown in Figure REF .",
"Based on the output of the SVM model, we took a deeper dive into the accuracy, precision, and recall scores of each algorithm, and found out that most of the algorithms with distinct separable clusters are with high recall scores, while the algorithms with indistinguishable clusters are with low recall scores.",
"Distinct separable clusters are manually identified from the footprint visualisations, where for a specific range of software metrics, the algorithm performs undoubtedly the best for clustering these software systems.",
"Drawing an example from Figure REF , in the range where 0.2 < z1 < 0.4, and 0.6 < z2 < 0.8, cosine_average_10 performs the best for these software systems.",
"The values that z1 and z2 represent are explained in more detail under Section REF .",
"Given that, True Positive = Clustering Algorithm is Predicted Correctly True Negative = Clustering Algorithm is Rejected Correctly False Positive = Clustering Algorithm is Predicted Wrongly False Negative = Clustering Algorithm is Rejected Wrongly Accuracy = The number of times E-SC4R is able to accurately predict the most suitable clustering algorithm or reject the wrong clustering algorithm out of the total predictions made: $ \\frac{TP + TN}{TP + TN + FP + FN} $ Precision = The number of times E-SC4R is accurate in predicting the best algorithm out of all the times the algorithm is predicted by E-SC4R: $ \\frac{TP}{TP + FP} $ Recall = The number of times E-SC4R is accurate in predicting the best algorithm out of all the times the best algorithm should have been predicted : $ \\frac{TP}{TP + FN} $ We are able to identify 3 main patterns/clusters of algorithms from the results shown in Table REF .",
"Note that due to the size of the table, we only show some of the examples in the last column of Table REF .",
"The information about the average accuracy, precision, and recall of agglomerative and Bunch algorithm produced from the SVM model (on the 300 releases) are shown in Tables REF , REF , REF , and REF .",
"Table: Categorisation of clustering results based on SVM.Algorithms that fall into $c_1$ which possess high recall are preferred as we would be able to easily identify software features that contribute to determining whether a particular algorithm is the most suitable for the given software systems.",
"Table: Performance of SVM model for agglomerative cosine algorithm.Table: Performance of SVM model for agglomerative Euclidean algorithm.Table: Performance of SVM model for agglomerative Manhattan algorithm.Table: Performance of SVM model for Bunch algorithm."
],
[
"PCA Visualisation",
"To visualise the results in a meaningful way, we apply PCA as a dimensionality reduction technique on the optimal subset of software features.",
"The aim is to plot the performance of the different clustering algorithms across the project space in 2D, which is likely to reveal where the clustering algorithms perform well, and where are their weaknesses.",
"Two new axes were created, which are linear combinations of the selected set of software features.",
"Projecting it using the two principal components holds 85% of the variation in the data.",
"A combined visualisation Figure REF is created to have a general overview on the spread of the algorithm and features.",
"Based on the MoJoFM results, an initial comparison was made on which algorithms are to be prioritised among the 300 projects (30 unique projects with 10 releases each), where some examples of the projects are shown in Table REF .",
"Recall that for each of the chosen 30 projects, we perform different configurations of agglomerative and Bunch clustering algorithms over 10 releases.",
"The best performing algorithm (in terms of MoJoFM values) will be prioritised.",
"Figure: Combined algorithm spread.The coordinate system that defines the new instance space is defined as $\\small \\begin{bmatrix}z_1 \\\\z_2\\end{bmatrix}=\\begin{bmatrix}{-0.1027} & {0.0546} \\\\{-0.0137} & {0.1251} \\\\{0.1056} & {-0.1107} \\\\{-0.0784} & {0.0747} \\\\{0.0708} & {0.0566} \\\\{0.091} & {-0.0484} \\\\{0.1015} & {0.0525} \\\\{-0.0418} & {-0.0474} \\\\{0.1215} & {0.0154} \\\\{0.0578} & {0.1086} \\\\\\end{bmatrix}\\begin{bmatrix}{\\text{staticMethods sum}} \\\\{\\text{modifiers mean}} \\\\{\\text{defaultMethods mean}} \\\\{\\text{maxNestedBlocks mean}} \\\\{\\text{totalMethods mean}} \\\\{\\text{protectedMethods mean}} \\\\{\\text{finalFields mean}} \\\\{\\text{stringLiteralsQty mean}} \\\\{\\text{lambdasQty mean}} \\\\{\\text{returnQty mean}} \\\\\\end{bmatrix}$ As seen from the visualisation in Figure REF , agglomerative algorithms are heavily prioritised over Bunch algorithms, where most of the points of the instance space prioritises the usage of agglomerative algorithms.",
"An interpretation of Figure 6 could be seen as such, where by looking at Equation REF and the visualisation generated, when $z_2$ is within the range of -1 to -1.4, and $z_1$ is within the range of 0 to 0.2, the algorithm that is prioritised is cosine_average_10.",
"Based on Table REF and Equation REF , we can see that agglomerative algorithms are prioritised over bunch algorithms if MoJoFM is used as the main evaluation metric based on the output from the SVM framework as well as the raw data from the first part of the experiments.",
"A higher value for the individual feature in Equation REF would mean that the feature has a higher influence on predicting which algorithm is the best, and lower values would mean a lower feature importance.",
"For example, maxNestedBlocks mean in Equation REF has comparatively low values.",
"This means that the same clustering method may be suitable for programs with vastly different values for this feature.",
"Upon further investigation, we discovered that although agglomerative clustering algorithm appears to be the superior algorithm when measured against MoJoFM, it usually generates many small clusters with few classes (5-10 classes).",
"On the other hand, Bunch tends to generate clustering results with lesser number of clusters and more equal classes inside each cluster, which might make it easier for software maintainers to follow the suggested decomposition.",
"Our findings largely agree with the experiments done by Wu et al.",
"[4] where they discovered that algorithms that give good clustering results according to one criterion (i.e.",
"MoJoFM) often do not give good results according to other criterion (i.e.",
"size and number of clusters).",
"Table: Performance of agglomerative vs Bunch clustering algorithm over 300 projects.As such, we have decided to analyse the strengths and weaknesses of agglomerative and Bunch clustering algorithms separately using PCA, instead of combining the two.",
"Summary of Combined PCA Visualisations: Agglomerative algorithms are prioritised over bunch algorithms if MoJoFM is used as the main evaluation metric.",
"Agglomerative algorithms usually generate many small clusters with few classes (5-10).",
"Bunch algorithms usually generate fewer clusters with a more balanced spread of the number of classes inside each cluster."
],
[
"Agglomerative PCA Visualisation",
"Figure REF illustrates the footprint visualisation generated for agglomerative algorithms.",
"Figure: Agglomerative algorithm selection.The coordinate system that defines the new instance space is defined as: $\\small \\begin{bmatrix}z_1 \\\\z_2\\end{bmatrix}=\\begin{bmatrix}{0.0224} & {0.0178} \\\\{-0.1018} & {0.0626} \\\\{-0.0086} & {0.0677} \\\\{-0.0459} & {-0.005} \\\\{-0.0708} & {0.0803} \\\\{0.0695} & {0.1282} \\\\{0.0908} & {0.003} \\\\\\end{bmatrix}\\begin{bmatrix}{\\text{staticMethods std}} \\\\{\\text{privateMethods mean}} \\\\{\\text{subClassesQty mean}} \\\\{\\text{cbo mean}} \\\\{\\text{modifiers max}} \\\\{\\text{publicMethods mean}} \\\\{\\text{anonymousClassQty mean}} \\\\\\end{bmatrix}$ where the footprint shows 3 main clusters.",
"cosine_average_15 (brown) cosine_average_7 (blue) manhattan_average_25 (pink) The seven software features in Equation REF are identified to be the most impactful on the performance of the different clustering algorithms used for software remodularisation.",
"We noticed that six out of the seven software features that form the coordinate system in Figure REF are size metrics.",
"This shows that for agglomerative clustering algorithms, size metrics have a stronger influence over the performance of the algorithm."
],
[
"Agglomerative PCA Relationship between Features and Clusters",
"Using the new coordinate system for agglomerative clustering algorithm, we visualise the footprints of the different techniques as shown in Figures REF , REF and REF .",
"We show the results for the prioritised clustering methods individually (Figures REF -REF ), by setting the threshold of good performance if the quality of the software clustering is above 70% (MoJoFM).",
"Each data point represents a project, which is labelled as good if the performance of the MoJoFM score is above 70%, and as bad otherwise.",
"To better understand why certain clustering algorithms work better for software projects in the cluster, we did a side-by-side comparison of software features and SVM model performance (Figures REF , REF ) to try and draw correlations.",
"By comparing Figures REF , REF and REF , we are able to draw similarities between the algorithm's footprint patterns and the distribution of the features' values.",
"This means that the following features are the most important when determining the priority of the algorithms.",
"Modifiers_max - Figure REF and Manhattan Average 25 (linkage method; distance metric; cluster divisor) - Figure REF There are distinct distributions between the top left and bottom right clusters which are reflected in both footprints.",
"Representing the software features of the target software $s\\in S$ in the new instance space, when the software features fall in the region of z_2 > -0.2 and z_1 < 0.4, modifiers_max is the most important feature in determining whether Manhattan Average 25 is the most suitable clustering algorithm.",
"Public Methods_mean - Figure REF and Cosine Average 15 - Figure REF There are distinct distributions between the left and right clusters which are reflected in both the footprints.",
"Representing the software features of the target software $s\\in S$ in the new instance space, when the software features fall in the region of 0 > z_2 > -0.1 and z_1 > 0.4, publicMethods_mean is the most important feature in determining whether Cosine Average 15 is the most suitable clustering algorithm.",
"AnonymousClassesQty_mean - Figure REF and Cosine Average 7 - Figure REF There are distinct distributions between the left and right clusters which are reflected in both the footprints.",
"Representing the software features of the target software $s\\in S$ in the new instance space, when the software features fall in the region of z_2 < -0.6 and z_1 > 0.1, AnonymousClassesQty_mean is the most important feature in determining whether Cosine Average 7 is the most suitable clustering algorithm.",
"However, there are certain features such as Static Methods_std - Figure REF and SubClassesQty_mean - Figure REF that do not have a clear distinct distribution among the clusters.",
"These features do not have any similar distribution patterns when compared to the prioritised agglomerative algorithms as well.",
"Figure: Footprints of Manhattan Average 25.Figure: Distribution of private methods_mean.Figure: Distribution of AnonymousClassesQty_mean."
],
[
"Findings from Agglomerative Footprints",
"Softwares with a higher value of staticMethods, publicMethods, privateMethods, and modifiers are more complex which leads to more opportunities for remodularisation [45].",
"While there is no existing literature that identifies any correlation between the number of methods and the performance of software clustering algorithms, the work by [46], [47], [45] discusses how metrics related to the size of the software can be effectively used to measure the quality of object-oriented software systems and for fault prediction.",
"Our approach provides clear evidence of the impact these software features have on the effectiveness of software clustering and remodularisation techniques.",
"For agglomerative clustering, the algorithm carries out clustering based on the provided distance matrix given by the Depends tool, where classes with high functional dependency (i.e., minimal distance) would be clustered together to form a cluster (subsystem).",
"The Depends tool measures dependencies between classes by analysing method invocation, type casting, and variable containment, which are strongly correlated with the seven main software features that we have identified - staticMethods, privateMethods, subClassesQty, CBO (Coupling Between Objects), modifiers, publicMethods, and anonymousClassQty.",
"When there are more methods and assignment operations in a class, the probability for the method operations and variable assignments to involve instances from another class is higher.",
"With the richer dependency information extracted from Depends, we can better illustrate the interrelationships between classes in the analysed software.",
"Based on the results, we are able to observe very distinct clusters from the footprints for the prioritised algorithms.",
"When we investigate the distribution of software features in Figures REF and REF , we see a clear gradient of change from top left to bottom right and from bottom left to top right respectively.",
"This means that we are able to clearly assign the most suitable algorithms to these projects based on these software features - modifier_max and publicMethods_mean respectively.",
"Intuitively, this makes sense, as classes with a high number of public methods and modifiers have more opportunities to be remodularised, by splitting large and complex classes into smaller ones with less methods.",
"SubClassesQty and Coupling Between Objects, on the other hand, increase the complexity of the software remodularisation problem.",
"Intuitively, this makes sense, as classes with the presence of SubClasses and high coupling (CBO) makes it harder for us to separate this classes during the clustering process.",
"Hence, the findings from this section help provide answers for the first research question.",
"Table: Performance of agglomerative linkage distribution.As for the linkage algorithm, our results show that single linkage triumphs over average and complete linkage in more than 50% of the projects, reflected in Table REF .",
"However, single linkage is not prioritised as compared to average linkage based on the SVM model, which is due to the overall SVM accuracy, precision, and recall of each individual algorithm as discussed above.",
"Single linkage algorithm belongs to the $c_3$ cluster (high accuracy, high precision, and low recall), where the SVM model generally prioritises other algorithms due to the low recall of single linkage algorithms.",
"This is interesting because the single linkage algorithm tends to form large and less coupled clusters.",
"Upon further investigation, we found that the ground truth extracted from the analysed projects tends to have a large directory structure as well, which contributes toward the finding.",
"While the work by Maqbool [14] claimed that the complete linkage algorithm is capable of forming the best software clustering results in terms of cluster cohesiveness, they used binary clustering features (identify the presence or absence of similar features) to identify the interrelationships between software entities.",
"On the other hand, our work utilises quantifiable measures to assign a relative weight to indicate the strength of dependencies between classes.",
"Apart from that, when running the same linkage algorithm on 10 previous releases of the examined software, we found that single linkage produces much more stable results as compared to complete linkage and average linkage algorithms, which largely agrees with the observation found in the work by [48].",
"Since single linkage outperforms other linkage algorithms in most of the scenarios, we do not include the illustration of the other footprint visualisation in this paper.",
"The work by Tzerpos et al.",
"[48] stated that single linkage forms the least cohesive cluster.",
"However, we would like to argue that the experiments conducted by the authors are performed on software written in C programming languages.",
"Hence, the same might not be applicable to modern Java-based systems that possess a completely different structure compared to software written in C. We theorise that the feature extraction tool used to capture the relationship between the clustering entities (classes) enables us to extract richer information on the interaction between the classes, thus producing slightly different results as compared to the work by Tzerpos et al.",
"To provide a simple illustration on how E-SC4R can effectively recommend the optimum clustering algorithm from the pool of choices, we have compared the E-SC4R framework against some of the baseline agglomerative hierarchical clustering algorithms.",
"Table REF shows a comparison between the MoJo values for \"Average Euclidean 10\", \"Complete Cosine 10\", \"Single Manhattan 10\", and the configuration recommended by E-SC4R.",
"The lower the MoJo value, the more suited the algorithm configuration is for the specific project.",
"The last column in Table REF is the configuration suggested by E-SC4R for the target software.",
"We will like to note that due to space constraints, we are unable to show all the configurations compared against the ones recommended by E-SC4R.",
"The complete information on all clustering results are available on our GitHub page.",
"By using the proposed framework, developers or researchers can easily identify the optimum clustering algorithm and its configuration instead of adopting an exhaustive or trial-and-error approach which is tedious and error prone.",
"Table: Baseline comparison.Summary of Agglomerative Footprints Visualisation: Agglomerative clustering algorithms are most impacted by staticMethods, privateMethods, subClassesQty, cbo, modifiers, publicMethods and anonymousClassQty.",
"Single linkage outperforms average and complete linkage."
],
[
"Bunch Footprints Visualisation",
"Figure REF illustrates the footprint generated from Bunch algorithm.",
"Figure: Bunch algorithm selection.The coordinate system that defines the new instance space is defined as: $\\small \\begin{bmatrix}z_1 \\\\z_2\\end{bmatrix}=\\begin{bmatrix}{0.1127} & {0.0705} \\\\{0.2152} & {0.049} \\\\{-0.0411} & {0.2699} \\\\\\end{bmatrix}\\begin{bmatrix}{\\text{RFC mean}} \\\\{\\text{staticMethods mean}} \\\\{\\text{stringLiteralsQty mean}}\\end{bmatrix}$ The three software features in Equation REF are identified as the most impactful on the performance of the Bunch clustering algorithms used for software remodularisation.",
"staticMethods and stringLiteralsQty are size related metrics, while RFC is a coupling and complexity related metric.",
"All three metrics are correlated because larger projects tend to have more complex classes with higher number of methods that leads to higher RFC.",
"An interesting finding in Figure REF is the cluster of projects (yellow color) where none of the Bunch clustering algorithm is predicted to perform well, labelled as \"None\".",
"The footprint of this cluster is in the \"grey zone\", where all three software features – RFC, staticMethods, and stringLiteralsQty, have medium scores such that $z_2$ is in the range between 0.2 < $z_1$ < 0.4 and -0.2 < $z_2$ < 0.2.",
"This is an indication of the specialisation of the Bunch clustering algorithm, and provides evidence that these methods are good at solving extreme cases.",
"This finding highlights another important aspect of our methodology; by analysing the strengths and weaknesses of existing software clustering techniques, we could identify areas that require improvement.",
"In this case, it is evident that there is a gap in software clustering techniques which are able to solve problems that have a medium number of RFC, staticMethods, and stringLiteralsQty."
],
[
"Bunch PCA Relationship between Features and Clusters",
"Using the new coordinate system for Bunch, we visualise the footprints of the different techniques as shown in Figures REF , REF and REF .",
"To better understand why certain clustering algorithms or parameters works better for software projects in the cluster, we did a side-by-side comparison of feature and SVM model performance to try and draw correlations.",
"Figure: Footprints of GA TurboMQIncrW.Figure: Footprints of HillClimbing TurboMQIncrWFigure: Distribution of StringLiteralsQty_mean.By comparing Figures REF , REF and REF we are able to draw similarities between the algorithm's footprint patterns and the distribution of the values of the features.",
"This means that these features are the most important when determining priority of the algorithms.",
"StringLiteralsQty_mean - Figure REF and HillClimbing TurboMQ - Figure REF There are distinct distributions between the top left and bottom right clusters which are reflected in both the footprints.",
"Representing the software features of the target software $s\\in S$ in the new instance space, when the software features fall in the region of z_2 > 0.4 and z_1 < 0.4, StringLiteralsQty_mean is the most important feature in determining whether HillClimbing TurboMQ is the most suitable clustering algorithm.",
"RFC_mean - Figure REF and GA TurboMQIncrW - Figure REF There are distinct distributions between the right and left clusters which are reflected in both the footprints.",
"Representing the software features of the target software $s\\in S$ in the new instance space, when the software features fall in the region of z_2 > 0.5 and z_1 > 0.4, RFC_mean is the most important feature in determining whether GA TurboMQIncrW is the most suitable clustering algorithm.",
"StaticMethods_mean - Figure REF and GA BasicMQ - Figure REF There are distinct distributions between the bottom right and top left clusters which are reflected in both the footprints.",
"Representing the software features of the target software $s\\in S$ in the new instance space, when the software features fall in the region of z_2 > 0.4 and z_1 > 0.3, StaticMethods_mean is the most important feature in determining whether GA BasicMQ is the most suitable clustering algorithm."
],
[
"Findings from Bunch Footprints Visualisation",
"For Bunch, the algorithm carries out clustering based on source code analysis.",
"Bunch uses a family of source code analysis tools (supports C, C++ and Java) that is based on an entity relationship model, where the source code is scanned and a relational database is constructed to store the entities and relations [12].",
"Bunch also assumes that all relation types have an equal weight.",
"Hence, when taking into account variable references or global variables, the amount of StringType literals can be a distinguishable feature.",
"This is why StringLiteralsQty turns out to be an important feature within Bunch where the more String variables that are present within the software, there will be more relationships that the source code analysis tool can identify, which subsequently helps to correctly identify the distribution of the correct clusters.",
"At the same time, features such as staticMethods and RFC which contributes to the richness of information within the entity relationship model is also important because they can better illustrate the interrelationships between classes.",
"By looking at the footprint visualisations for the prioritised Bunch algorithm, our results agree to a certain extent with the authors' claim that exhaustive clustering algorithm only works well for small systems [11], and the hill-climbing algorithm performs well for most software systems [12].",
"The footprints of Exhaustive BasicMQ (Figure REF ) and Exhaustive TurboMQIncrW (Figure REF ) shows that there are significantly larger clusters of instances where it is labelled as \"Good\" by the SVM model when the values for $z\\_1$ and $z\\_2$ are small (low RFC, staticMethods, and stringLiteralsQty, shown on the bottom left corner of Figure REF ).",
"This shows that exhaustive clustering works better on software projects classified as small based on the software features.",
"One good example is shown on Figure REF , where only Exhaustive TurboMQIncrW is able to show \"Good\" results when $z\\_1$ is between -0.4 to -0.6, and $z\\_2$ is between -0.8 to -1.0.",
"On the other hand, when the size-related metrics of staticMethods and stringLiteralsQty are high (top right corner of Figure REF and Figure REF ), the footprints of HillClimbing BasicMQ (Figure REF ) and TurboMQIncrW (Figure REF ) shows that instances are labelled as \"Good\".",
"This suggests that HillClimbing algorithms performs well on large sized projects.",
"As such, our E-SC4R framework not only reaffirms that hill climbing approach is well suited for large size projects, we further discover that the value of RFC, staticMethods, and stringLiteralsQty can be a good indicator for researchers to decide whether or not to choose exhaustive Bunch or hill climbing Bunch when performing software remodularisation.",
"Another interesting finding from the footprint visualisations for the prioritised Bunch algorithms is that the same algorithm but with a different calculator is able to cater to a entirely different software type.",
"For example, GA BasicMQ performs well for most software systems, but is unable to cater to extremely large software systems.",
"GA TurboMQIncrW on the other hand, only performs well on extremely large software systems.",
"Similarly, for HillClimbing, HillClimbing BasicMQ and HillClimbing TurboMQIncrW performs well for most software systems, but HillClimbing TurboMQ speficially caters for software with size metrics that are extremely high in the $z\\_1$ range ($z\\_1 > 0$ ) and in the middle of the $z\\_2$ range ($-1 < z\\_2 < 1$ ).",
"Summary of Bunch Footprints Visualisation: Bunch clustering algorithms are most impacted by StaticMethods, StringLiteralQtys, RFC.",
"Exhaustive algorithm works better with small systems.",
"Hill-climbing algorithm performs generally well across all systems.The calculator for the algorithm plays a huge role in determining which software type the configuration of the algorithm is best suited for."
],
[
"Summary of Experiment Results",
"By analysing the strengths and weaknesses of existing software clustering techniques, we could identify areas in current software remodularisation and architecture recovery research that requires improvement.",
"Based on our experiment results and footprints, it is evident that there is a gap in clustering techniques that can address software projects that have more distinct and substantial features that are not captured very well by the existing software clustering algorithms, possibly due to their working principles.",
"For example, based on Figure REF , existing agglomerative clustering techniques are unable to cluster software projects based on Coupling Between Objects (CBO).",
"We are not suggesting that CBO is not a good indicator for software remodularisation problems, but rather, existing software clustering algorithms do not benefit from CBO as a feature during the clustering process.",
"The ability to identify the most optimum algorithm and configuration from our existing pool of choices to configure software clustering algorithms (such as the linkage algorithm, distance metric, etc.)",
"based on the characteristics of different software can help software developers and maintainers to reduce the time and effort needed to perform software remodularisation in a more effective manner, rather than having to resort to intuition or trial and error, both of which have far lower accuracy rates and and it is either resource intensive when evaluating multiple approaches, or they might simply just work with a sub-optimal cluster set.",
"The experiment results also suggest that on a larger scale (analysis of more projects and more distinct identification of domains or metrics to classify a pool of software), it would be possible to, at the minimum, eliminate approaches or parameters that are already known to be sub-optimal.",
"The visualisation has the potential to show the most optimum approach that a software maintainer can adopt, thus improving the accuracy of clustering results as well as saving resources that would otherwise potentially be wasted.",
"RQ1: The strengths and weaknesses of agglomerative hierarchical and Bunch clustering algorithms can be evaluated from the footprints generated.",
"Drawing examples from the Bunch footprint, the strength of the algorithm can be seen in its ability to cluster software accurately through RFC, staticMethods, and stringLiteralsQty metrics, where identifiable and significant clusters can be found from the feature footprints The weakness of the algorithm can be instead seen through clusters that are labelled as \"None\" during the prioritised algorithm footprint visualisation.",
"This shows that the techniques are good at solving extreme cases, but unable to properly cluster software projects with medium-sized metrics, identifying the gap in the aforementioned software clustering techniques.",
"As such, the answers to both research questions are discussed below.",
"RQ2: When using MoJoFM as the evaluation criteria, agglomerative hierarchical clustering proves to be the clear winner when compared to different variations and configuration of Bunch clustering algorithm.",
"It is when the two algorithms are evaluated separately using E-SC4R and using the conclusions drawn from the footprints, we are able to more objectively select the most suitable clustering technique based on the software features of each test subject.",
"Based on our results, modifier_max, publicMethods_mean, and anonymousClassQty_mean are the three most prevalent software features that affect the performance of agglomerative hierarchical clustering algorithm, while RFC_mean, staticMethods_mean, and stringLiteralsQty_mean plays essential role in deciding the type of Bunch algorithm to aid in software remodularisation and architecture recovery.",
"As such, using the identified software features as indicators, they can be used to aid researchers in selecting the most suitable clustering technique, which depends on the characteristic of the software remodularisation and architecture recovery problem."
],
[
"Threats to Validity",
"Based on the classification schema of Runeson et al.",
"[49], Construct Validity in our case refers to whether all relevant parameters for hierarchical clustering and Bunch clustering algorithm have been explored to visualise the footprint for software remodularisation.",
"To mitigate this risk, we considered a plethora of parameters such as number of projects, number of revisions, different linkage algorithms, distance metrics, and search-based fitness function.",
"Besides, we take into consideration the past releases of the examined software in generating the ground truth.",
"Internal Validity is related to the examination of causal relations.",
"Our results pinpoint the particular software features that affect the effectiveness of the agglomerative and Bunch clustering method of the analysed project, which is not inferred from causal relationships.",
"With respect to External Validity, the risk is mitigated by selecting a pool of projects that are well-known and popular in the the open-source community (project selected based on the number of stars on GitHub) forming a representative sample for analysis.",
"In order to provide more information about the quality of the chosen project, Sonarqube [50] has been used to analyse the quality of the chosen projects in terms of the number of bugs, code smells, and code duplication presented in Table REF .",
"This is to demonstrate the application on E-SC4R on a variety of software projects in terms of size and quality.",
"However, a replication of this study in a larger scale that comprises projects written in different languages would be valuable in verifying the current findings.",
"We have created a replication package on our GitHub page.",
"Table: Quality metrics of the chosen projects extracted from SonarQube.Ground truth generation plays a vital role in determining the optimum clustering algorithm and configuration from the existing pool.",
"Getting input from domain expert may help reaffirm the validity of our ground truth.",
"However, due to the scope of the project where we experimented on 30 open source projects, it is challenging to get the developers from the open source community to evaluate the ground truth individually for each version for each project.",
"Based on the state-of-the-art, there is no single well-acknowledged method in creating the ground truth for software clustering.",
"One of the most popular approaches, however, is by leveraging on the package structure of the analysed software.",
"Hence, in this research, we have adopted a similar approach to address the problem.",
"We like to note that the proposed E-SC4R framework can work with any kind of clustering algorithm and ground truth, as long as the ground truth is standardised for comparison across the existing pool of clustering algorithms.",
"Software systems with only a few directories with a large number of files in each might not validate some of our results as well.",
"Upon examining the existing ground truth that we have, we found that the effect of having a few directories with large number of files is negligible.",
"In fact, the majority of the ground truths that we use for the experiments consist of projects with large numbers of directories (packages), with a small number of files in each of them.",
"We have uploaded some of the ground truth that we used for the experiments on the GitHub page https://github.com/alvintanjianjia/SoftwareRemodularization/tree/master/sample_groundtruth.",
"On the other hand, code quality, which includes coding style, readability, level of cohesion, and other indicators are factors that might impact the effectiveness of the presented clustering algorithms.",
"In this research, we have only evaluated the proposed approach on open source systems.",
"While the code quality of open source and real-life industrial systems is very subjective and context dependent (the quality of open source projects can be low when compared to industrial project, and vice versa), we expect that applying the proposed E-SC4R framework on project with different code quality (real-life industrial systems included) will yield different results, mainly because the characteristic of software (i.e.",
"CK metrics of the analysed software) will affect the footprint constructed using our framework.",
"The challenge of running the experiments on low code quality projects is the construction of ground truth to validate the clustering algorithm.",
"In our proposed approach and in existing studies, package structure is the most commonly used method to create the artificial ground truth for software clustering.",
"We construct the ground truth by looking at the package structure of the past 10 releases of the software, and find the overlapping and most common directory structure to be used as the ground truth.",
"If the same approach is to be applied to low code quality projects, the ground truth will be heavily skewed and not reliable, which affects the profiling of these systems.",
"With an accurate ground truth generated manually by an expert, the E-SC4R framework is applicable to both open source and industrial projects for reversing the documentation of poorly documented systems with high technical debts."
],
[
"Related Works on Software Clustering for Architecture Recovery",
"In general, software clustering consists of the following four steps.",
"First, common clustering features are chosen to determine the similarity between entities (methods, classes, or packages depending on the level of granularity).",
"Second, a similarity measure is chosen to determine the similarity strength between two entities (method invocation, passing of parameters, sharing of variables, etc.)",
"[14].",
"Third, a clustering algorithm is chosen to group similar entities together.",
"Finally, a form of validation is required to measure the quality of the clustering results.",
"The results of software clustering can be viewed as a high-level abstraction of the software architecture to aid in software comprehension.",
"All four steps mentioned above play a significant role in determining the quality of clustering results because the selection of different clustering features or metrics will produce substantially different clustering results.",
"Although some clustering algorithms produce a single clustering result for any given dataset, a dataset may have more than one natural and optimum clustering result.",
"For instance, source code can only reveal very limited information about the architectural design of a software system since it is a very low-level software artefact.",
"On the other than, some implementation details might be lost if software packages are being used to represent clustering entities.",
"Hence, identifying the most optimum way to choose the most appropriate clustering algorithm and configure the parameters of the algorithm is a non-trivial task in software remodularisation.",
"The work by Deursen and Kuipers [51] adopted a greedy search method by using mathematical analysis to analyse the structure of cluster entities and identify the clustering features that are shared by them.",
"The proposed approach finds all of the possible combinations of clusters and evaluates the quality of each combination.",
"Agglomerative hierarchical clustering is used in this work.",
"The authors discovered that it is hard to analyse all possible combinations, and useful information might be missing if no attention is given to analyse all the results generated from different dendrogram cutting points.",
"In contrast to the greedy search method proposed by Deursen and Kuipers, the work by Fokaefs et al.",
"[52] proposed an approach that produces multiple clustering results from which software developers and maintainers can choose the best result based on their experiences.",
"The goal is to decompose large classes by identifying ‘Extract Class’ refactoring opportunities.",
"Extract class is defined as classes that contain many methods without clear functionality.",
"The authors adopted the agglomerative clustering algorithm to generate a dendrogram and cut the dendrogram at several places to form multiple sets of results.",
"The authors argued that clustering algorithms that produce a single result are too rigid and not feasible to fit into the context of software development.",
"Work by Anquetil and Lethbridge [53] attempted to perform agglomerative clustering on source files and found out that using source code alone to aid in software remodularisation yields poor results.",
"In their study, clustering entities are represented in the form of source code.",
"The authors found that the quantity of information, such as the number of variables used in the source code, the dependency between routines, and the data passed and shared by functions helps in improving the reliability of clustering.",
"The work by Cui and Chae [54] attempted to analyse the performance, strengths, and weaknesses of different agglomerative hierarchical clustering algorithms using multiple case studies and setups.",
"The authors conducted a series of experiments using 18 clustering strategies.",
"The clustering strategies are the combination of different similarity measures, linkage methods, and weighting schemes.",
"The case studies comprise 11 systems where source codes were used as the input parameters.",
"The authors found that it is difficult to identify a perfect clustering strategy which can fulfil all the evaluation criteria proposed by the authors.",
"As discussed in the systematic literature review conducted by Alsarhan et al.",
"[29], the selection of clustering algorithms remains a challenging problem in the area of software clustering for remodularisation and architectural recovery.",
"While there have been attempts to propose guidelines for selecting or rejecting a clustering algorithm for a given software [55], there is a lack of comprehensive methods for clustering algorithm selection.",
"Our work extends existing research in analysing the effectiveness of software clustering techniques by examining what software features impacts the performance of agglomerative and Bunch clustering algorithms.",
"We characterise a software system using software/code features (e.g., depth of inheritance tree, cohesion, and coupling) and determine the most significant features that have an impact on whether a software clustering technique can generate a good clustering result."
],
[
"Conclusion and Future Work",
"Acknowledging the lack of a universal approach in finding the optimum clustering algorithm for any software remodularisation problem given the numerous algorithms that exist in the literature and the various parameters that may be used to configure software clustering algorithms, we are able to provide empirical evidences that help in identifying key characteristics of software/code features that influence the effectiveness of hierarchical and Bunch-based software clustering algorithms.",
"Given the relatively high cost of running a clustering algorithm on large and complex software systems, the proposed approach optimises the resources spent on software remodularisation.",
"The results in this paper, while promising, are constrained in a number of ways.",
"First, while it is one of the most popular software remodularisation approaches, only agglomerative hierarchical clustering and Bunch clustering methods were assessed in this paper because it is difficult to compare the clustering results produced from different families of clustering algorithms in a fair and unbias manner.",
"The relationships that are extracted from Depends are aggregated in the proposed approach which may lead to the loss of some semantic information.",
"One way to address this problem is by running the experiments multiple times using only one type of the extracted relationship at a time.",
"For example, if we have 14 types of relationships extracted using Depends, we can run it 14 times for each version of the project, and evaluate the effectiveness of different clustering algorithms using each type of relationship (or combination of multiple relationships).",
"The creation of the ground truth relies on the past 10 releases of a particular software, which strongly favours legacy software that have more active developers and contributors.",
"Less stable software with radical changes between versions make it difficult to construct a usable ground truth.",
"For future work, a separate method of defining and creating the ground truth can be further explored.",
"An approach similar to the work by Naseem et al.",
"[43] by taking a deep dive into 1 or 2 of our generated ground truth with a few senior and experienced developers who have been working on the respective projects would be one of the future directions of this research, to affirm the validity of our ground truth and clustering results..",
"Nonetheless, the experiment results show that our findings are a step forward in the area of software remodularisation to reveal the strengths and weaknesses of different hierarchical clustering and Bunch clustering algorithms, and it is hoped that the work discussed in this paper can serve as a framework for further analysis and improvements to be made."
],
[
"Acknowledgement",
"This work was carried out within the framework of the research project FRGS/1/2018/ICT01/MUSM/03/1 under the Fundamental Research Grant Scheme provided by the Ministry of Education, Malaysia."
]
] | 2107.01766 |
[
[
"On the quantization of $C^{\\infty}(\\mathbb R^d)$"
],
[
"Abstract An infinitesimal deformation of $C^{\\infty}(\\mathbb R^d)$ has a vanishing primary obstruction if and only if its skew form is Poisson, and therefore is integrable to a full deformation of $C^{\\infty}(\\mathbb R^d)$ by the work of Kontsevich.",
"This is further equivalent to having its coefficients satisfy a set of $\\binom{d}{3}$ nonlinear partial differential equations.",
"The single such equation in the case $d=3$ suggests that a big bang might be localized to an instant in time and point in space.",
"This note also reexamines the Basic Universal Deformation Formula (UDF) which asserts that if $D_1, D_2$ are commuting derivations of an associative algebra $\\mathcal{A}$ over the rationals then exponentiation of $D_1\\smile D_2$ provides a full deformation of $\\mathcal{A}$.",
"It gives further applications to quantization and to UDFs with non-commuting derivations."
],
[
"Introduction",
"The idea that algebras may deform first appeared in [9] and was developed in [11], [12], [13], [14], but has a deep historical background in the deformation theory of complex analytic structures, beginning with Riemann.",
"He showed that compact Riemann surfaces of genus g depend on $3g-3$ complex parameters for $g \\ge 2$ , one for genus one, while for the sphere there is but one possible complex structure, up to analytic isomorphism.",
"Since the parameters are continuous, the concept of infinitesimal deformations was already inherent in Riemann's work, but these were first formalized by Teichmüller, [31], as quadratic differentials.",
"They are, however, meaningful only in one complex dimension; the breakthrough understanding that infinitesimal deformations of a complex manifold exist in the cohomology of its sheaf of germs of holomorphic tangent vectors is due to Frölicher and Nijenhuis, [8].",
"This opened the way for the monumental works of Kodaira and Spencer, for a comprehensive overview of which see [22].",
"That there could be obstructions to infinitesimal deformations, which do not appear for Riemann surfaces since there can be none in complex dimension one, was described by Kodaira and Spencer as an accidental discovery.",
"While these and jump deformations [4], [14],[15], which are important for quantization, were not fully understood until the development of algebraic deformation theory, it is worth remembering that the present theory rests on over a century and a half of prior work.",
"Deformation quantization, an important advance in algebraic deformation introduced by Bayen, Flato, Frønsdal, Lichnerowitz, and Sternheimer, [1], showed that quantum theory, in particular the spectrum of the hydrogen atom, can be understood, without the use of Schrödinger's equation.",
"For a summary of this and some later developments, see [29] and [6].",
"Subsequent to [1], deformation and quantization, as used in the title to this note, have become almost synonymous for algebras.",
"The question originally asked in [9] is, given an algebra $\\mathcal {A}$ over some commutative unital ring $\\mathbf {k}$ , in what ways can one create a deformation of that algebra with multiplication of the form $a\\star b = ab + \\hbar F_1(a,b) + \\hbar ^2F_2(a,b ) + \\cdots \\quad $ while remaining in the same “equationally defined category\" [12], or in present terms, category of algebras defined over a particular operad.",
"The notation and term “star product” were introduced in [1].",
"For associative algebras, the $F_i$ in (REF ) are Hochschild 2-cochains of $\\mathcal {A}$ with coefficients in $\\mathcal {A}$ itself, tacitly extended to be defined over $\\mathcal {A}[[\\hbar ]]$ ; we may write $F_0$ for the original multiplication in $\\mathcal {A}$ .",
"Here $F_1$ must be a 2-cocycle, often loosely called the infinitesimal of the deformation although, because of gauge equivalence, an infinitesimal is properly a cohomology class.",
"A main problem is to determine which infinitesimals, viewed loosely as 2-cocycles, are those of full deformations, something which depends only on its cohomology class.",
"Given a 2-cocycle $F_1$ , there generally is a sequence of cohomological obstructions to constructing the necessary $F_i, i \\ge 2$ .",
"A full deformation as in (REF ), if it exists, is often called an integral of $F_1$ .",
"A Universal Deformation Formula (UDF) is one which, for some class of integrable 2-cycles, gives an explicit integral.",
"The simplest of these asserts that if $D_1, D_2$ are commuting derivations of an algebra $\\mathcal {A}$ , then a suitably defined exponential of $D_1\\!\\!\\smile \\!\\!D_2$ is an integral.",
"This “Basic\" UDF first appeared in [13] but without proof; we give two proofs here.",
"The first shows that the assertion in [13] is formally correct, i.e., that a certain sequence of formal identities holds.",
"The second can deal with convergence issues when $D_1, D_2$ are derivations of $C^{\\infty }(\\mathbb {R}^d)$ .",
"This addresses the problem that the star product of (REF ) is only a formal power series in which it may not be meaningful, a priori, to specialize the deformation parameter $\\hbar $ to an element of the coefficient ring $\\mathbf {k}$ or of an algebraic extension of it.",
"Nevertheless, the Basic UDF exhibits the the first Weyl algebra, which encapsulates the uncertainty relation between position and momentum, as a deformation of the polynomial ring in two variables.",
"Clifford algebras are, likewise, deformations of graded polynomial rings, when degree is taken into account.",
"We also show that the basic UDF can be shown to underly the first UDF with non-commuting derivations, [5].",
"Kontsevich's proof that Poisson infinitesimal deformations of $C^{\\infty }(\\mathbb {R}^d)$ are integrable, [23], also consists in exhibiting a remarkable UDF in which, however, the coefficients are not rational.",
"His proof uses fundamental work of Stasheff, [28], who introduced the concept of homotopy Lie and associative algebras, something now also understood for algebras over operads.",
"Using different methods, Tamarkin [30], almost simultaneously, gave another proof of the integrability of Poisson structures.",
"The passage from quantizing $C^{\\infty }(\\mathbb {R}^d)$ to quantization of the algebra of smooth functions on a $d$ -dimensional smooth manifold was accomplished by Cattaneo, Felder, and Tomassini, [3].",
"Dolgushev, [7], later showed that if a solution exists, then there is a rational one.",
"Perhaps the most important remaining problem is to determine which infinitesimals are Poisson.",
"An infinitesimal deformation is a cohomology class from which we may choose representatives, in particular a unique one which is antisymmetric or skew, as Poisson 2-cocycles are.",
"It is shown here that the primary obstruction to an infinitesimal deformation vanishes if and only if its unique skew representative is Poisson.",
"Any infinitesimal deformation of $C^{\\infty }(\\mathbb {R}^d)$ with vanishing primary obstruction is therefore integrable.",
"Kontsevich showed, in effect, that the primary obstruction is, in this case, the only one.",
"The vanishing of the primary obstruction to an infinitesimal deformation of $C^{\\infty }(\\mathbb {R}^d)$ can be expressed, in dimension $d$ , by a system of $\\binom{d}{3}$ non-linear partial differential obstruction equations which the coefficients of the representative of the infinitesimal must satisfy.",
"When $d = 2$ there are no obstructions; all infinitesimal deformations give rise to quantizations.",
"For $d=3$ , the single obstruction equation shows that an arbitrarily large deformation can be localized to an arbitrarily small region in space and interval of time; in the limit one has a “big bang\" localized to a point in space and moment in time."
],
[
"Deformation theory",
"Star products $a\\star b = \\sum \\hbar ^iF_i$ and $a\\star ^{\\prime } b = \\sum \\hbar ^iF_i^{\\prime }$ are gauge equivalent if there is a one-parameter family $\\gamma $ of $\\mathbf {k}$ -linear automorphisms of $\\mathcal {A}$ of the form $\\gamma (a) = a+ \\hbar \\gamma _1(a) + \\hbar ^2 \\gamma _2(a_) + \\cdots $ , where the $\\gamma _i$ are linear maps from $\\mathcal {A}$ to itself, again tacitly extended to be defined over $\\mathcal {A}[[\\hbar ]]$ , such that $a\\star ^{\\prime }b = \\gamma ^{-1}(\\gamma (a)\\star \\gamma (b))$ .",
"The algebras that $\\star $ and $\\star ^{\\prime }$ define on the underlying $\\mathbf {k}$ -space of $\\mathcal {A}$ are then isomorphic, and one has $F_1^{\\prime } = F_1 +\\delta \\gamma $ , where $\\delta $ is the Hochschild coboundary operator.",
"If $F,G$ are 2-cochains of $\\mathcal {A}$ then $F\\circ _1G,\\,F\\circ _2G, \\, F\\circ G$ are the 3-cochains defined by setting, respectively, $(F\\circ _1G)(a,b,c) = F(G(a,b),c), \\quad (F\\circ _2G)(a,b,c) = F(a,G(b,c)),$ and $(F\\circ G)(a,b,c) = (F\\circ _1G-F\\circ _2G)(a,b,c) = F(G(a,b),c) - F(a, G(b,c));$ these are special cases of the composition products $\\circ _i$ and $\\circ $ introduced in [9].",
"The requirement that $a\\star (b\\star c) - (a\\star b) \\star c$ = 0 can be expressed by the equation $\\sum _{i= 0}^nF_i\\circ F_{n-i}\\quad = 0\\quad \\text{for all} \\quad n.$ Transposing to the right those terms where either $i=0$ or $i= n$ , (REF ) can be rewritten as $\\sum _{i+j=n, \\, i,j > 0}F_i\\circ F_j = -\\delta F_n.$ When $n = 1$ the left side vanishes, so $F_1$ must be a 2-cocycle, usually called, as we may do here, the infinitesimal of the deformation although, because of gauge equivalence, it is the cohomology class of $F_1$ which should be viewed as the infinitesimal.",
"When a 2-cocycle $F_1$ is given, the question of whether one can construct a formal deformation of $\\mathcal {A}$ as in (REF ) depends only on the cohomology class of $F_1$ .",
"When $F_1$ is a cocycle then so is $F_1\\circ F_1$ .",
"Its cohomology class in $H^3(\\mathcal {A},\\mathcal {A})$ , which depends only on the class of $F_1$ , is its primary obstruction, but $F_1\\circ F_1$ is also commonly called the primary obstruction.",
"When $F_1\\circ F_1$ is a coboundary, one can choose an $F_2$ with $-\\delta F_2 = F_1\\circ F_1$ and one can ask if an $F_3$ exists so that one can continue building the series.",
"However, (REF ) with $n =3$ shows that one may encounter another obstruction in $H^3(\\mathcal {A},\\mathcal {A})$ , and so on indefinitely.",
"If, with a given 2-cocycle $F_1$ , we are able to construct a series such as that in (REF ) with the given $F_1$ , then that series is said to be an integral of the infinitesimal $F_1$ and to quantize $\\mathcal {A}$ .",
"When $\\mathcal {A}$ is the algebra $C^{\\infty }(\\mathbb {R}^d)$ of smooth functions on $\\mathbb {R}^d$ the integral is also said to quantize $\\mathbb {R}^d$ .",
"While in principle one may encounter an infinite sequence of obstructions, Kontsevich [23]'s work, together with what is shown here, implies that an infinitesimal deformation $F_1$ of $C^{\\infty }(\\mathbb {R}^d)$ has only the primary obstruction; if that vanishes then there is a star product with the given $F_1$ .",
"An algebra $\\mathcal {A}$ for which all deformations are gauge equivalent to the trivial deformation, i.e., the one where the star product is just the original multiplication, is called rigid.",
"This will be the case if every infinitesimal deformation is ultimately obstructed and certainly if $H^2(A) = 0$ , in which case the algebra is called absolutely rigid or stable, in the terminology of [8].",
"A tensor product of stable algebras need not be stable, as will be seen.",
"When $H^3(\\mathcal {A},\\mathcal {A}) = 0$ every infinitesimal deformation is integrable.",
"In particular, this is the case for $C^{\\infty }(\\mathbb {R}^2)$ by the Hochschild-Kostant-Rosenberg (HKR) Theorem, [21].",
"That theorem asserts that $H^*(C^{\\infty }(\\mathbb {R}^d))$ is isomorphic, as a module over $C^{\\infty }(\\mathbb {R}^d)$ , to the exterior algebra generated by the partial derivatives, $\\partial _1, \\dots ,\\partial _d$ with respect to the coordinates $x_1,\\dots , x_d$ .",
"In particular, $H^3(C^{\\infty }(\\mathbb {R}^2)) = 0$ , so there are no obstructions.",
"The original formulation of the HKR Theorem was more restrictive.",
"For a proof of the present form, cf.",
"Roger, [26].",
"Note, however, that the module isomorphism does not carry the exterior product on $\\wedge ^* \\operatorname{Der}{\\mathcal {A}}$ to the cup product on $H^*(\\mathcal {A},\\mathcal {A})$ , cf [2].",
"If $D_1,\\dots , D_n$ is a sequence of derivations of $\\mathcal {A}$ (which need commute or be distinct), and $I = (i_1,\\dots ,i_r)$ is a subsequence of $(1,\\dots , d)$ , set $D_I = D_{i_1}D_{i_2}\\cdots D_{i_r}$ and let $I^c$ denote the ordered complement of $I$ .",
"Then $\\delta (D_1D_2\\cdots D_n) = -\\sum _{I^c}D_I\\!\\!\\smile \\!\\!D_{I^c}$ , where the sum is over all non-empty ordered proper, i.e., neither empty nor the whole, subsequences of $(i_1,\\dots , i_r)$ .",
"In particular, for any pair of derivations $D_1,D_2$ of $\\mathcal {A}$ , we have $\\delta (D_1D_2) = - (D_1\\!\\!\\smile \\!\\!D_2 + D_2\\!\\!\\smile \\!\\!D_1)$ .",
"Therefore, $D_1\\!\\!\\smile \\!\\!D_2$ is cohomologous to $-D_2\\!\\!\\smile \\!\\!D_1$ and also to $(1/2)(D_1\\!\\!\\smile \\!\\!D_2 - D_2\\!\\!\\smile \\!\\!D_1)$ , provided that 2 is a unit.",
"It follows from the HKR Theorem that every 2-cocycle of $C^{\\infty }(\\mathbb {R}^d)$ is cohomologous both to one in normal form $\\sum _{1\\le i < j \\le d}a_{ij}\\partial _i\\!\\!\\smile \\!\\!\\partial _j$ and to one in skew form $\\sum _{1\\le i < j \\le d}(a_{ij}/2)(\\partial _i\\!\\!\\smile \\!\\!\\partial _j - \\partial _j\\!\\!\\smile \\!\\!\\partial _i)$ .",
"While an infinitesimal deformation is a cohomology class, the skew and normal representatives of the class are commonly also called “infinitesimal deformations\"."
],
[
"The Basic Universal Deformation Formula",
"The concept of a Universal Deformation Formula (UDF) was introduced (without the present name or proof) in [13], where what we here call the Basic UDF was stated in the context of Composition Complexes.",
"These include the cohomology groups of algebras, coalgebras, and simplicial complexes.",
"A UDF exhibits in closed form an explicit integral for some class of 2-cocycles, considered as infinitesimal deformations.",
"The Basic UDF is used here only for deformations of an associative algebra $\\mathcal {A}$ , in which context we give two proofs.",
"The first remains essentially the same for coalgebras and simplicial complexes.",
"The methods of the second can be used to address convergence questions.",
"All Hochschild cohomology groups of an algebra $\\mathcal {A}$ considered here will have coefficients in $\\mathcal {A}$ itself as a bimodule.",
"That cohomology is frequently called the regular (Hochschild) cohomology and will henceforth be denoted simply $H^*(\\mathcal {A})$ , and similarly for cochains.",
"Note in what follows that if $a$ is a central element of $\\mathcal {A}$ and $F$ a cocycle of any dimension, then $aF$ is again a cocycle, and if $D$ is a derivation, then $Da$ is again central.",
"The product of 1-cochains of an associative algebra $\\mathcal {A}$ is always well defined as their composition.",
"For 2-cochains of the form $f\\!\\!\\smile \\!\\!g$ , where $f,g$ are 1-cochains, one can not generally define a product by setting $(f_1\\!\\!\\smile \\!\\!g_1)(f_2 \\!\\!\\smile \\!\\!g_2) = f_1f_2 \\!\\!\\smile \\!\\!g_1g_2$ , for if $a$ is a central element of $\\mathcal {A}$ , then as 2-cochains one has $a(f\\!\\!\\smile \\!\\!g) = af \\!\\!\\smile \\!\\!g = f\\!\\!\\smile \\!\\!ag$ but such changes in representation will usually change the product.",
"Suppose, however, that we have a set of commuting derivations $\\lbrace D_i\\rbrace $ of $\\mathcal {A}$ .",
"When $I = \\lbrace i_1, \\dots , i_r\\rbrace $ is an unordered set of indices of the $D_i$ , set, as before, $D_I = D_{i_1}\\cdots D_{i_r}$ .",
"(When the set $I$ is empty, interpret $D_I$ as the identity map $\\mathcal {A}$ ; when $I$ and $J$ are both empty then $D_I \\!\\!\\smile \\!\\!D_J$ is the multiplication map.)",
"The foregoing problem then does not arise when multiplication is restricted to 2-cochains of the form $D_I \\!\\!\\smile \\!\\!D_J$ .",
"In particular, if $D_1, D_2$ are commuting derivations of $\\mathcal {A}$ then $(D_1 \\!\\!\\smile \\!\\!D_2)^n = D_1^n \\!\\!\\smile \\!\\!D_2^n$ is well defined, as is $\\exp (D_1\\!\\!\\smile \\!\\!D_2) = \\sum _{n=0}^{\\infty }\\frac{1}{n!",
"}(D_1 \\!\\!\\smile \\!\\!D_2)^n$ when $\\mathcal {A}$ is defined over $\\mathbb {Q}$ .",
"In the following, “formal\" means that no assertion is made that the deformation parameter $\\hbar $ in the series defining $a\\star b, a,b \\in \\mathcal {A}$ can actually be specialized to any value in the coefficient ring.",
"The symbol $\\hbar $ originally denoted the reduced Planck's constant, $h/2\\pi $ , where $h \\approx 6.626176 \\times 10^{-34}$ joule-seconds is Planck's original constant; its use to denote a deformation parameter derives from the fact that Planck's constant may, in fact, be viewed as such, cf [1], [16].Correction to Theorem 8 of [16]: In view of the remark at the end of §2 (not §1) it should read, “...the coherent twist of the path algebra induced by $\\omega $ is trivial only if the class of $\\omega $ is trivial as an element of $H^2(\\mathcal {M}.",
"\\mathbb {R}/\\tau \\mathbb {R})$ .\"",
"Theorem 1 (The Basic UDF) If $D_1, \\,D_2$ are commuting derivations of an associative algebra $\\mathcal {A}$ over a ring $\\mathbf {k}$ containing the rationals, $\\mathbb {Q}$ , then the multiplication defined on $\\mathcal {A}[[\\hbar ]]$ by $a\\star b\\, = \\, \\exp \\hbar (D_1\\!\\!\\smile \\!\\!D_2)(\\,a,b)\\, = \\,\\sum _{n=0}^{\\infty }\\frac{\\hbar ^n}{n!",
"}D_1^na\\cdot D_2^nb$ is associative and defines a formal deformation of $\\mathcal {A}$ .",
"Proof 1 of Theorem REF.",
"Setting $F_i = D_1^i \\!\\!\\smile \\!\\!D_2^i$ in (REF ), what must be shown is that for all $n$ one has the following relation among 3-cochains of $\\mathcal {A}$ : $\\sum _{i=0}^n\\frac{1}{i!(n-i)!",
"}D_1^i\\!\\!\\smile \\!\\!D_2^i(D_1^{n-i} \\!\\!\\smile \\!\\!D_2^{n-i}) =\\sum _{i=0}^n\\frac{1}{i!(n-i)!",
"}D_1^i(D_1^{n-i} \\!\\!\\smile \\!\\!D_2^{n-i})\\!\\!\\smile \\!\\!D_2^i .$ By Leibniz' rule, the left side of (REF ) can be written as $\\sum _{i=0}^n \\frac{1}{i!(n-i)!}",
"\\sum _{j=0}^i \\binom{i}{j} D_1^i \\!\\!\\smile \\!\\!D_2^jD_1^{n-i} \\!\\!\\smile \\!\\!D_2^{i-j}D_2^{n-i}\\\\=\\sum _{i,j=0,\\dots ,n, \\, i\\ge j} \\frac{1}{(n-i)!j!(i-j)!}",
"D_1^i \\!\\!\\smile \\!\\!D_2^jD_1^{n-i} \\!\\!\\smile \\!\\!D_2^{n-j}.$ The right side of (REF ) can be written as $\\sum _{i=0}^n \\frac{1}{i!(n-i)!}",
"\\sum _{j=0}^i \\binom{i}{j} D_1^{i-j}D_1^{n-i} \\!\\!\\smile \\!\\!D_1^jD_2^{n-i} \\!\\!\\smile \\!\\!D_2^i\\\\=\\sum _{i,j=0,\\dots ,n, \\, i\\ge j} \\frac{1}{(n-i)!j!(i-j)!}",
"D_1^{n-j}\\!\\!\\smile \\!\\!D_1^jD_2^{n-i} \\!\\!\\smile \\!\\!D_2^i.$ On the right side of (REF ) we can replace now replace the dummy variable $i$ by $n-j$ and $j$ by $n-i$ .",
"The sum remains formally over the same set of indices since $i\\ge j$ if and only if $n-j \\ge n-i.$ As $D_1$ and $D_2$ commute, the right side of (REF ) then becomes identical to the right side of (REF ), proving the assertion.",
"$\\Box $ As an example, suppose that we have algebras $\\mathcal {A}_1, \\mathcal {A}_2$ defined over a ring $\\mathbf {k}$ containing $\\mathbb {Q}$ with respective derivations $D_1, D_2$ which are not inner.",
"Extend $D_1, D_2$ to derivations of $\\mathcal {A}_1 \\otimes _{\\mathbf {k}} \\mathcal {A}_2$ by setting $D_1(a_1 \\otimes a_2) = D_1a_1\\otimes a_2, \\, D_2(a_1\\otimes a_2) = a_1 \\otimes D_2a_2$ .",
"These extensions commute, so $D_1 \\!\\!\\smile \\!\\!D_2$ can be exponentiated to a full non-trivial deformation of $\\mathcal {A}_1 \\otimes \\mathcal {A}_2$ even when both $\\mathcal {A}_1$ and $\\mathcal {A}_2$ are stable.",
"The simplest example is that where $\\mathcal {A}_1 = \\mathcal {A}_2 = \\mathbf {k}[x]$ , a polynomial ring in one variable, in which case $\\mathcal {A}_1 \\otimes \\mathcal {A}_2 \\cong k[x,y]$ , a polynomial ring in two variables, and $D_1= \\partial _x, \\, D_2 = \\partial _y$ .",
"In this case the deformation parameter can be specialized to any element in the coefficient ring because the series defining $a\\star b$ terminates for any two fixed elements $a,\\,b \\in \\mathbf {k}[x,y]$ .",
"The Basic UDF has the following immediate extension.",
"Theorem 2 If $D_i, \\, i = 1, \\dots ,n$ are commuting derivations (not necessarily all distinct) of an algebra $\\mathcal {A}$ over $\\mathbb {Q}$ and $c_{ij}, i,j = 1,\\dots ,n$ are central elements which are constants for all the $D_i$ , i.e., central elements of $\\mathcal {A}$ with $D_kc_{ij}=0$ for all $i,j,k$ then $\\exp \\sum _{i,j=1}^nc_{ij}( D_i \\smile D_j)$ quantizes $\\mathcal {A}$ .",
"$\\Box $ Proof.",
"Any derivation which commutes with all $D_i, D_j$ remains a derivation after the deformation which they induce.",
"A pair of such therefore induces a further deformation.",
"As the derivations commute, the product of the exponentials involved is the exponential of the sum of the exponents.",
"$\\Box $ In the example above, one has $x\\star y = xy + \\hbar $ while $y\\star x = yx$ , so the star commutator is $[x,y]_{\\star } = \\hbar $ .",
"This exhibits the first Weyl algebra as a deformation of $\\mathbb {C}[x,y]$ .",
"It is also an example of a jump deformation: The algebras defined for all values of $\\hbar $ other than zero are isomorphic.",
"As a consequence, the infinitesimal of the deformation, $\\partial _x \\!\\!\\smile \\!\\!\\partial _y$ , becomes a coboundary in the star multiplication, see e.g., [17].",
"By the HKR Theorem, $\\dim H^1(\\mathbb {C}[x,y]) =1$ and $\\dim H^n(\\mathbb {C}[x,y]) = 0$ for $n > 1$ .",
"It follows that the cohomology of the deformed Weyl algebra vanishes in all positive dimensions, see, e.g., [17].",
"This was first proven in Sridharans's thesis, [27].",
"For a discussion of jump deformations see, e.g., [15], [17].",
"Clifford algebras can similarly be viewed as deformations of exterior algebras.",
"The latter are graded algebras, so one must follow Koszul's rule of signs.",
"In the smallest case, let $\\mathcal {A}$ be the four dimensional exterior algebra over a field $\\mathbf {k}$ generated by a two dimensional vector space spanned by $x$ and $y$ , each of which has degree 1.",
"For simplicity, assume that the characteristic of $\\mathbf {k}$ is not 2.",
"Here $\\partial _x, \\partial _y$ are both derivations of $\\mathcal {A}$ of degree -1 which commute in the sense that $\\partial _x\\partial _y= -\\partial _y\\partial _x$ , and each has square equal to zero.",
"(The square of a derivation of odd degree always must be zero when the characteristic is not 2, since it commutes with itself.)",
"The exponential $\\exp (\\partial _x\\!\\!\\smile \\!\\!\\partial _y)$ reduces to $\\operatorname{id}\\!\\!\\smile \\!\\!\\operatorname{id}+ \\, \\partial _x\\!\\!\\smile \\!\\!\\partial _y$ .",
"With the star product it induces, one still has $x\\star x = y\\star y = 0$ but now $x\\star y = xy +1$ , while $y\\star x = yx$ , so $x\\star y + y \\star x = 1$ .",
"Therefore, $(x+y)^{\\star 2} =1, (x-y)^{\\star 2} = -1$ , so the deformed algebra is the Clifford algebra ${\\text{C}l}_{1,1}(\\mathbf {k})$ .",
"Theorem REF , in more general form, was first explicitly stated in [13], but is already implicit in two important papers on quantum mechanics, Groenewold, [20], 1946, and Moyal, [25], 1949.",
"The algebra being deformed in both is the algebra of observables $\\mathbb {C} [p,\\,q]$ in phase space (for simplicity here in one dimesion); before deformation the position variable $q$ and momentum variable $p$ commute, but after deformation satisfy the uncertainty relation $qp-pq = i\\hbar $ .",
"As remarked before, this is an example of a jump deformation, which always has the effect of reducing the space of cohomology classes of infinitesimal deformations because the infinitesimal of a jump deformation becomes a coboundary after deformation, [17].",
"When that space is reduced to zero the algebra involved has become stable.",
"In our evolving understanding of physical laws they seem in certain respects to reflect stable algebraic structures, which is what makes jump deformations important for quantization.",
"If $\\mathcal {A}_1, \\mathcal {A}_2$ are algebras over the same field $\\mathbf {k}$ with vanishing regular cohomology in all positive dimensions then the same is true of of their tensor product, a special case of [24].",
"It follows that if in $\\mathbb {C}^{\\infty }[q_1,\\dots ,q_d,\\,p_1,\\dots , p_d]$ one takes as infinitesimal deformation $i\\hbar \\sum _{i=1}^d(\\partial q_i \\!\\!\\smile \\!\\!\\partial p_i)$ , then the resulting deformed algebra, in which $q_ip_i -p_iq_i = i\\hbar $ for all $i$ but $p_i$ and $q_j$ still commute for $i \\ne j$ , has no regular cohomology in positive dimensions.",
"In particular, it is stable.",
"In the preceding examples, the power series defining the star product of any two fixed elements of the algebra being deformed actually terminates, which permits specialization of the deformation parameter to an element of the ground ring.",
"This will not be the case in general, so we give a second proof of Theorem REF which may allow this in particular for some cases where $\\mathcal {A}$ is $C^{\\infty }(\\mathbb {R}^d)$ .",
"Proof 2 of Theorem REF .",
"As a model for an algebra $\\mathcal {A}$ with a pair of commuting derivations, take the algebra of smooth functions of two complex variables, $x,y$ with $D_1=\\partial _x, D_2 = \\partial _y$ .",
"There is then a subalgebra of this $\\mathcal {A}$ for which the associativity of the star product is easily seen, namely that generated by functions of the form $e^{\\lambda x}e^{\\mu y}$ , so written because the first factor is a constant for $\\partial _y$ and the second for $\\partial _x$ .",
"Computation shows readily that $(e^{\\lambda _1 x}e^{\\mu _1 y})\\star (e^{\\lambda _2 x}e^{\\mu _2 y}) =e^{\\hbar (\\lambda _1+\\mu _2)}(e^{(\\lambda _1 + \\lambda _2)x}e^{(\\mu _1 + \\mu _2)y}),$ from which the associativity of the star multiplication for these functions is immediate.",
"It follows that the associativity of the star product must hold for all functions which can be written as finite double Fourier series.",
"The nature of the resulting product shows that one can then pass to limits, so the Basic UDF is valid for functions representable by Fourier series.",
"If we now take $x$ and $y$ to be periodic, then any polynomial in $x$ and $y$ can be written as a double Fourier series, so the star product must be associative for these.",
"The associativity of the star product is, however, a consequence of a sequence of rational identities, so this implies that it must hold identically.$\\,\\Box $ From Theorem REF the following is immediate.",
"Theorem 3 Suppose that $f_1,\\dots ,f_n$ are smooth functions on $\\mathbb {R}^d$ where each $f_i$ is a function only of the single variable $x_i$ .",
"If $c_{ij},\\, i,j = 1,\\dots , d$ are elements of $\\mathbb {R}$ then $F_1 = \\sum _{i<j}c_{ij}f_i\\partial _i\\!\\!\\smile \\!\\!f_j\\partial _j$ is an infinitesimal deformation of $C^{\\infty }(\\mathbb {R}^d)$ which is integrable by exponentiation.",
"$\\Box $ Infinitesimal deformations of $C^{\\infty }(R^d)$ of the form $\\sum _{i<j}c_{ij}f_i\\partial _i\\!\\!\\smile \\!\\!f_j\\partial _j$ , where the $c_{ij}$ are constants and each $f_i$ is a function only of the one variable $x_i$ , will be called basic.",
"Its powers suggest a definition for the powers of an arbitrary infinitesimal.",
"For brevity, henceforth we will write $(i|j)$ for $\\partial _i\\!\\!\\smile \\!\\!\\partial _j$ , $(ij|k)$ for $\\partial _i\\partial _j\\!\\!\\smile \\!\\!\\partial _k$ , $(i|j|k)$ for $\\partial _i\\!\\!\\smile \\!\\!\\partial _j\\!\\!\\smile \\!\\!\\partial _k$ , and so forth.",
"In the foregoing, setting $a_{ij} = f_if_j$ , one can verify the following.",
"Theorem 4 For all $1 \\le i < j \\le d, 1 \\le \\, k < l \\le d$ , one has $ a_{ij}(i|j)a_{kl}(k|l) = a_{ij}a_{kl}(ik|jl) \\quad \\text{if}\\\\ \\text{ $i$ is distinct from $k$ and $j$ from $l$, but where we may have $i=l$ or $j=k$} \\\\a_{ij}(i|j)a_{il}(i|l) = a_{ij}\\partial _ia_{il}(i|jl) + a_{ij}a_{il}(ii|jl) \\text{ if $j$ is distinct from $l$,} \\\\a_{i}(i|j)a_{kj}(k|j) = a_{ij}\\partial _ja_{kj}(ik|j) + a_{ij}a_{kj}(ik|jj)\\quad \\text{if $i$ is distinct from $k$,} \\\\a_{ii}(i|i)a_{ii}(i|i) = \\partial _ia_{ii}\\partial _ia_{ii}(i|i) + a_{ii}\\partial _ia_{ii}(ii|i) + a_{ii}\\partial _ia_{ii}(i|ii) + a_{ii}a_{ii}(ii|ii).$ $\\Box $ These relations allow one to write $F_1^2$ , and hence all powers $F_1^n$ as well as $\\exp {F_1}$ , in terms of the $a_{ij}$ and their derivatives without reference to the $f_i$ .",
"The same formulas are meaningful for a general infinitesimal $F_1 = \\sum _{1\\le i < j \\le d} a_{ij}(i|j)$ whether basic or not, but $\\exp {F_1}$ will generally not be an integral of $F_1$ , for if that were the case, then by (REF ) we would have $F_1\\circ F_1= \\delta (-\\frac{1}{2} F_1^2)$ .",
"It is the difference between these that gives rise to the obstruction equations discussed later."
],
[
"A UDF with non-commuting derivations.",
"Theorem REF provides another approach to the following proposition from [5], which was the first UDF with non-commuting derivations.",
"In it, $[x]_n$ denotes the “descending factorial\", $[x]_n = x(x-1)\\cdots (x-n+1)$ .",
"Theorem 5 Suppose that $D_1,D_2$ are derivations of an algebra $\\mathcal {A}$ over $\\mathbb {Q}$ such that $[D_1,D_2] =D_1$ .",
"Then $D_1\\!\\!\\smile \\!\\!D_2$ is an integrable infinitesimal deformation of $\\mathcal {A}$ with integral $e(\\hbar , D_1,D_2): = \\\\id_{\\mathcal {A}}\\!\\!\\smile \\!\\!id_{\\mathcal {A}}+ \\hbar D_1\\!\\!\\smile \\!\\!D_2 +(\\hbar ^2/2!",
")D_1^2\\!\\!\\smile \\!\\!",
"[D_2]_2 + \\cdots + (\\hbar ^n/n!",
")D_1^n\\!\\!\\smile \\!\\!",
"[D_2]_n + \\cdots \\quad .$ Proof.",
"The UDF asserted here is essentially a sequence of formal identities in $D_1,D_2$ within the universal enveloping algebra of the two-dimensional Lie algebra generated by $D_1,D_2$ .",
"If we can exhibit an explicit associative algebra $\\mathcal {A}$ with derivations $D_1,D_2$ having an isomorphic universal enveloping algebra then, as in the second proof of Theorem REF , this will serve as a model in the sense that any proposition about the universal enveloping algebra which holds in this model must be true in general.",
"We can model $D_1$ and $D_2$ as derivations of the algebra of smooth functions of $x$ and $y$ by mapping $D_1$ to $e^{-y}\\partial _x$ and $D_2$ to $\\partial _y$ , but must show that the natural map of the universal enveloping algebra of the Lie algebra generated by $D_1$ and $D_2$ onto the algebra of operators generated by $e^{-y}\\partial _x$ and $\\partial _y$ is an isomorphism.",
"The commutation relation between $D_1$ and $D_2$ implies that the universal enveloping algebra is, as a module over the ground ring, free with generators all monomials of the form $D_1^mD_2^n$ , with $m,n \\ge 0$ .",
"The analogue clearly also holds for the algebra of operators generated by $e^{-y}\\partial _x, \\partial _y$ , so they are isomorphic.",
"Now $D_1\\!\\!\\smile \\!\\!D_2 = e^{-y}\\partial _x \\!\\!\\smile \\!\\!\\partial _y$ , but this can also be written as $\\partial _x \\!\\!\\smile \\!\\!e^{-y}\\partial _y$ , which is a basic infinitesimal with integral $\\operatorname{exp}(\\partial _x \\!\\!\\smile \\!\\!e^{-y}\\partial _y) = \\sum _{n=0}^{\\infty }(\\hbar ^n/n!",
")(\\partial _x)^n \\!\\!\\smile \\!\\!",
"(e^{-y}\\partial _y)^n.$ It is a simple induction to show that for every $\\lambda \\in \\mathbb {C}$ and positive integer $n$ one has $(e^{-\\lambda y}\\partial _y)^n = e^{-ny}\\partial _y(\\partial _y-\\lambda )(\\partial _y-2\\lambda )\\cdots (\\partial _y-(n-1)\\lambda ).$ In particular, for $\\lambda = 1$ , one has $(e^{-y}\\partial _y)^n = e^{-ny}[\\partial _y]_n.$ The right side of (REF ) can therefore be written as $\\sum _{n=0}^{\\infty }(\\hbar ^n/n!",
")(e^{-y}\\partial _x)^n \\!\\!\\smile \\!\\!",
"[\\partial _y]_n$ , proving the theorem.",
"$\\Box $"
],
[
"The differential subcomplex",
"There is a differential subcomplex, $C_{\\text{diff}}^{\\bullet }(C^{\\infty }(\\mathbb {R}^d))$ , of the Hochschild complex $C^{\\bullet }(C^{\\infty }(\\mathbb {R}^d))$ of $C^{\\infty }(\\mathbb {R}^d)$ generated by the derivations $\\partial _1, \\dots \\partial _{d}$ with respect to its coordinates $x_1,\\dots ,x_d$ .",
"Its 0-cochains are the elements of $\\mathcal {A}$ .",
"The module of n-multiderivations (biderivations in the case of cohomological dimension $n=2$ ) is spanned by the $n$ -fold cup products of these.",
"Simple 1-cochains are of the form $a\\partial _{i_1}\\partial _{i_2}\\cdots \\partial _{i_r}$ , where $a\\in C^{\\infty }(\\mathbb {R}^d)$ , $r$ is arbitrary, and there may be duplications amongst the indices.",
"The module of 1-cochains is composed of sums of such cochains of various orders.",
"Simple $n$ -cochains are cup products of simple 1-cochains.",
"The differential order of such a cup product is the sum of the orders of its cup factors.",
"A general $n$ -cochain is a sum of simple ones, possibly of different differential orders; it is homogeneous if the orders are the same.",
"The Hochschild coboundary operator preserves differential order.",
"While the differential subcomplex is closed under the composition product, the composition product of homogeneous cochains is generally no longer homogeneous.",
"The HKR theorem implies that the inclusion of the differential subcomplex into the full Hochschild complex induces an isomorphism of cohomology.",
"Using the notation preceding Theorem REF , the HKR theorem also implies that $H^r(\\mathcal {A})$ can be identified with the $\\mathcal {A}$ module spanned by all forms $(i_1|i_2|\\cdots |i_r)$ with $i_1< i_2 < \\cdots < i_r$ , since the class of any skew form also has a unique representative in that module."
],
[
"Poisson structure and the primary obstruction",
"A Poisson structure on a commutative algebra $\\mathcal {A}$ is a second multiplication $[a,b], \\, a,b \\in \\mathcal {A}$ , which is skew, $[a,b] = -[b,a]$ , a biderivation, $[ab,c] = a[b,c] +b[a,c]$ , and satisfies the Jacobi identity, $[a, \\, [b, \\,c]] + [b, \\, [c, \\, a]] + [c, \\, [a, \\, b]] = 0,$ making $\\mathcal {A}$ with this second multipliation into a Lie algebra.",
"The expression on the left of (REF ) is called the Jacobiator of the bracket product; it will be denoted here as a cyclic sum, $\\sum _\\circlearrowleft [a,[b,c]]$ .",
"Note that a Jacobiator is skew in all three variables.",
"A Poisson manifold $\\mathcal {M}$ is one with a Poisson structure on its algebra of smooth functions.",
"Lemma 1 Let $f$ be a skew 2-cocycle of a commutative algebra $\\mathcal {A}$ over a ring $\\mathbf {k}$ in which 2 is a unit.",
"Then $f$ is a biderivation.",
"Proof.",
"For a skew 2-cochain $f$ of a commutative algebra one has the identity $(\\delta f)(a,b,c) +(\\delta f)(c,a,b) - (\\delta f)(a,c,b) =2[af(b,c) - f(ab,c) + bf(a,c)].$ $\\Box $ Lemma REF suggests that a skew n-cocycle of a commutative algebra should be a multiderivation when the characteristic is 0 or sufficiently large.",
"Recall from the HKR Theorem that every element of $H^2(C^{\\infty }(\\mathbb {R}^d))$ has a unique skew representative 2-cocycle, which by Lemma REF is a biderivation.",
"Theorem 6 Let $\\Pi $ be a skew 2-cocycle of a commutative algebra $\\mathcal {A}$ defined over a ring $\\mathbf {k}$ in which $3!$ is a unit.",
"Then $\\Pi $ is Poisson if and only if its primary obstruction vanishes.",
"Proof.",
"The primary obstruction to $\\Pi $ is the cohomology class of $\\Pi \\circ \\Pi $ .",
"In view of Lemma REF it is sufficient now to show that $\\sum _\\circlearrowleft \\Pi \\circ \\Pi (a, b, c)= 0$ .",
"Since $\\Pi $ is skew, we can also write $\\Pi \\circ \\Pi (a,b,c) $ as $\\Pi (\\Pi (a, b),c) +\\Pi (\\Pi (b,c),a)$ .",
"This is necessarily a 3-cocycle which is furthermore skew in its first and third variables, $a$ and $c$ .",
"One can write the skew part of $\\Pi \\circ \\Pi $ as $(1/3!",
")(\\sum _\\circlearrowleft \\Pi \\circ \\Pi - \\sum _\\circlearrowleft (13)\\Pi \\circ \\Pi $ where on the right, the transposition $(13)$ indicates that the first and third variable should be interchanged.",
"In place of the transposition $(13)$ one could have taken any 2-cycle, but choosing (13) shows immediately that within the parentheses we have $2\\sum _\\circlearrowleft \\Pi \\circ \\Pi $ , so the obstruction for $\\Pi $ is $(1/3!",
")\\sum _\\circlearrowleft \\Pi \\circ \\Pi $ .",
"This 3-cocycle is skew in all three variables and by the HKR Theorem is the unique skew representative of its cohomology class.",
"It therefore vanishes if and only if the obstruction to $\\Pi $ does.",
"$ \\Box $ The proof of Theorem REF implicitly reproduces the case of dimension 3 of the decomposition of commutative algebra cohomology introduced in [18].",
"An immediate corollary is Theorem 7 Poisson structures on $C^{\\infty }(\\mathbb {R}^d)$ are in one-to-one correspondence with those elements of $H^2(C^{\\infty }(\\mathbb {R}^d))$ whose primary obstruction vanishes.",
"$\\Box $ It follows from the work of Kontsevitch [23] that the primary obstruction is the only obstruction; an infinitesimal whose primary obstruction vanishes is integrable to a full deformation.",
"Since the higher order terms in Kontsevich's UDF come from various partial derivatives of the coefficients in the infinitesimal deformation, it follows that the full deformation so constructed will vanish wherever all the coefficients of its infinitesimal do."
],
[
"The obstruction equations",
"If an infinitesimal deformation $F_1$ of $C^{\\infty }(\\mathbb {R}^d)$ were basic then, as remarked at the end of Section , $F_1\\circ F_1$ would be identical to $\\delta (-\\frac{1}{2} F_1^2)$ .",
"The obstruction equations, numbering $\\binom{d}{3}$ , are non-linear partial differential equations in the coefficients of $F_1$ which arise from the difference between $F_1\\circ F_1$ and $\\delta (-\\frac{1}{2} F_1^2)$ ; the primary obstruction to $F_1$ vanishes if and only they are satisfied.",
"The terms in $F_1\\circ F_1$ are in natural one-to-one correspondence with those in $F_1^2$ , with $a_{ij}(ij)\\circ a_{kl}(kl)$ in the former corresponding, in the notation of Section , to $a_{ij}(i|j)a_{kl}(k|l)$ in the latter.",
"The same is therefore true of $F_1\\circ F_1$ and $\\delta (-\\frac{1}{2} F_1^2)$ .",
"The terms of differential order 4 in $F_1\\circ F_1$ and $\\delta (-\\frac{1}{2} F_1^2)$ must coincide since that would be the case if all the coefficients were constants.",
"While $F_1^2$ contains terms of differential order 2, these are cocycles and will not contribute to its coboundary.",
"The obstruction equations arise, therefore, from the difference between the terms of order 3 of $F_1\\circ F_1$ and $\\delta (-\\frac{1}{2} F_1^2)$ .",
"Let $F_1 = \\sum _{1\\le i< j \\le d}a_{ij}(i|j)$ .",
"Analogous to the products in Theorem REF , there are the following four kinds of terms in $F_1\\circ F_1:$ $(1)& \\sum _{1\\le i<j\\le d, 1\\le k<l\\le d}a_{ij}(i|j)\\circ a_{kl}(k|l), \\, \\text{where} \\, i,j,k,l \\, \\text{are all distinct}\\\\(2)& \\sum _{1\\le i<j\\le d, 1\\le i<l\\le d}a_{i|j}(i|j)\\circ a_{il}(i|l), \\, \\text{where} \\, j,l \\, \\text{are distinct}\\\\(3)& \\sum _{1\\le i<j\\le d, 1\\le k<j\\le d}a_{ij}(i|j)\\circ a_{kj}(k|j), \\, \\text{where} \\, i,k \\, \\text{are distinct}\\\\(4)& \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\sum _{1\\le i<j\\le d}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,a_{ij}(i|j)\\circ a_{ij}(i|j).$ Each composition product above gives rise to two terms of the form $b_{ijk}(i|j|k)$ , where there may be a single repetition among $i,j,k$ .",
"A brief examination will show the following.",
"Theorem 8 The sum of all the terms $b_{ijk}(i|j|k)$ in $F_1\\circ F_1$ in which there is a repetition amongst $i,j,k$ is precisely $\\delta (-\\frac{1}{2} F_1^2)$ .",
"$\\Box $ It follows that $F_1\\circ F_1 -\\delta (-\\frac{1}{2} F_1^2)$ is just the sum of those terms without repetitions.",
"The vanishing of the primary obstruction to a general infinitesimal $F_1$ is therefore equivalent to having the sum of those terms without repetitions be a coboundary.",
"When the dimension $d = 2$ , the HKR theorem implies that there can be no obstructions; every infinitesimal deformation is then integrable.",
"For every obstruction must lie in $H^3(C^{\\infty }\\mathbb {R}^2)$ , whose elements are uniquely represented by cocycles of the form $\\sum _{1\\le i <j<k \\le d}a_{ijk}(i|j|k)$ , but there can be no such cocycles when $d=2$ .",
"When $d \\ge 3$ , for every triple $(i,j,k)$ with $1\\le i <j <k \\le d$ a term of the form $b_{ijk}(i|j|k)$ appears amongst the composition products above, e.g., in $a_{ii}(i|i)\\circ a_{jk}(j|k)$ .",
"If $(i^{\\prime },j^{\\prime },k^{\\prime })$ is a permutation of $(i,j,k$ ) then $(i^{\\prime }|j^{\\prime }|k^{\\prime })$ is cohomologous to $+(i|j|k)$ if the permutation is even, and to $-(i|j|k)$ if the permutation is odd.",
"Therefore, the sum of the terms without repetitions is cohomologous to a unique sum of the form $\\sum _{1\\le i <j<k \\le d}a_{ijk}(i|j|k)$ .",
"Here the $a_{ijk}$ are expressions in the coefficients $a_{ij}$ of $F_1 = \\sum _{1\\le i< j \\le d}a_{ij}(i|j)$ and their partial derivatives with respect to the variables $x_1,\\dots ,x_d$ .",
"For the primary obstruction to vanish, each of the $\\binom{d}{3}$ coefficients $a_{ijk}$ must vanish, giving rise to $\\binom{d}{3}$ non-linear partial differential equations in the coefficients of $F_1$ ."
],
[
"The obstruction equation in dimension $d=3$",
"In dimension 3, an infinitesimal deformation has the form $F_1= a_{12}(1|2) + a_{13}(1|3) + a_{23}(2|3)$ , and there is just one obstruction equation.",
"One has $F_1\\circ F_1 = &\\\\{}& a_{12}(1|2)\\circ a_{12}(1|2)+ a_{13}(1|3)\\circ a_{13}(1|3) +a_{23}(2|3)\\circ a_{23}(2|3) + \\\\& a_{12}(1|2)\\circ a_{13}(1|3) + a_{13}(1|3)\\circ a_{12}(1|2) + \\\\& a_{12}(1|2)\\circ a_{23}(2|3) + a_{23}(2|3)\\circ a_{12}(1|2) + \\\\& a_{13}(1|3)\\circ a_{23}(2|3) + a_{23}(2|3)\\circ (a_{13}(1|3).$ After expanding the composition products above, one has $F_1\\circ F_1 - \\delta (-\\frac{1}{2} F_1^2) = &\\\\&\\,\\,\\,\\,\\,\\,\\,\\,(a_{12}\\partial _1a_{13})(1|3|2) + (a_{13}\\partial _1a_{12})(1|2|3) +\\\\&-(a_{12}\\partial _2a_{23})(1|2|3) + (a_{23}\\partial _2 a_{12})(1|2|3)+\\\\&-(a_{13}\\partial _3a_{23})(1|2|3) - (a_{23}\\partial _3 a_{13})(2|1|3).$ As $(1|3|2) = -(1|2|3) + \\delta (1|23)$ and $(2|1|3) = - (1|2|3) - \\delta (12|3),$ we can write the sum of the foregoing terms as $[-a_{12}\\partial _1a_{13}+a_{13}\\partial _1a_{12}-a_{12}\\partial _2a_{23} + a_{23}\\partial _2a_{12} -a_{13}\\partial _3a_{23} + a_{23}\\partial _3a_{13}](1|2|3)\\\\+ \\delta (a_{12}\\partial _1a_{13} + a_{23}\\partial _3a_{13})(1|23)$ The single obstruction equation in dimension 3 is therefore $-a_{12}\\partial _1a_{13}+a_{13}\\partial _1a_{12}-a_{12}\\partial _2a_{23} + a_{23}\\partial _2a_{12} -a_{13}\\partial _3a_{23} + a_{23}\\partial _3a_{13} = 0.$ In (REF ), if $F_1$ is replaced by $\\varphi F_1$ , where $\\varphi $ is a smooth function of $x_1,x_2,x_3$ , then the terms in which $\\varphi $ is differentiated cancel, so the summand on the left side is just multiplied by $\\varphi $ .",
"Therefore, if it is satisfied by a triple of smooth functions $(a_{12}, a_{13}, a_{23})$ of $x_1, x_2, x_3$ , then $(\\varphi a_{12}, \\varphi a_{13}, \\varphi a_{23})$ is also a solution.",
"This implies the following.",
"Theorem 9 If $F_1$ is an infinitesimal deformation of $C^{\\infty }(\\mathbb {R}^3)$ whose primary obstruction vanishes then the same is true of $\\varphi F_1$ for any $\\varphi $ in $C^{\\infty }(\\mathbb {R}^3)$ .",
"In particular, if a skew biderivation $F_1$ of $C^{\\infty }(\\mathbb {R}^3)$ is Poisson, then so is $\\varphi F_1$ .",
"$\\Box $ We do not know if the analogous statement holds in dimensions $d > 3$ .",
"In dimension 3, Theorem REF gives a partial order on integrable infinitesimals of $C^{\\infty }(\\mathbb {R}^3)$ , with $F_1 \\prec F_1^{\\prime } $ whenever $F_1^{\\prime } = \\varphi F_1$ for some $\\varphi $ , and an equivalence relation when also $F_ 1\\prec F_1^{\\prime } \\prec F_1$ .",
"It also implies, for dimension 3, that if $F_1$ is a basic infinitesimal of $C^{\\infty }(\\mathbb {R}^3)$ then $\\varphi F_1$ is integrable for any $\\varphi $ in $C^{\\infty }(\\mathbb {R}^3)$ ; such infinitesimals will be called quasibasic.",
"In any open set in which no coefficient of $F_1 = a_{12}\\partial _1\\!\\!\\smile \\!\\!\\partial _2 + a_{13}\\partial _1\\!\\!\\smile \\!\\!\\partial _3 + a_{23}\\partial _2\\!\\!\\smile \\!\\!\\partial _3$ vanishes we can take their quotients and rewrite the obstruction equation as $a_{13}^2\\partial _1(a_{12}/a_{13}) + a_{23}^2\\partial _3(a_{13}/a_{23}) + a_{12}^2 \\partial _2(a_{23}/a_{12})= 0.$ For a basic infinitesimal $F_1 = f_1f_2\\partial _1\\!\\!\\smile \\!\\!\\partial _2+f_1f_3\\partial _1\\!\\!\\smile \\!\\!\\partial _3 + f_2f_3\\partial _2\\!\\!\\smile \\!\\!\\partial _3$ the stronger equations $\\partial _1(a_{12}/a_{13}) = \\partial _3(a_{13}/a_{23}) = \\partial _2(a_{23}/a_{12})= 0 $ hold.",
"The analog is true in all dimensions.",
"Theorem 10 Let $F_1 = \\sum _{i<j}a_{ij}\\partial _i\\!\\!\\smile \\!\\!\\partial _j$ be an infinitesimal deformation of $C^{\\infty }(\\mathbb {R}^d)$ where $d$ is arbitrary.",
"Suppose that the $a_{ij}$ are all invertible, and that for all $\\lbrace i,j,k\\rbrace $ with $i<j<k$ we have $\\partial _i(a_{ij}/a_{ik}) = \\partial _j(a_{ij}/a_{jk}) = \\partial _k(a_{ik}/a_{jk}) = 0$ .",
"Denote $a_{ij}(x_1,\\dots ,x_{i-1}, 0, x_{i+1},\\dots , x_d)$ by $a_{ij}(\\hat{x_i})$ .",
"Then $a_{ij}/a_{ij}(\\hat{x_i})$ can be written in the form $1+x_iu_i$ , where $u_i$ is not a function of $x_i$ .",
"Further, if the dimension is 3 then $F_1$ is quasibasic; there are single variable functions $f_1,f_2,f_3$ of $x_1,x_2,x_3$ , respectively, and a function $\\varphi $ such that $ a_{12} = \\varphi f_1f_2, \\quad a_{13} = \\varphi f_1f_3\\quad a_{23} = \\varphi f_2f_3.$ Proof.",
"For the first part, we can write $a_{ij}/a_{ij}(\\hat{x_i}) = 1+x_iu_{ij}$ for certain functions $u_{ij}$ .",
"Suppose that $i<j<k$ ; the other cases are similar.",
"By the hypotheses, $[a_{ij}/a_{ij}(\\hat{x_i})]/[a_{ik}/a_{ik}(\\hat{x_i})]$ is not a function of $x_i$ but equals $(1+x_iu_{ij})/(1+x_iu_{ik})$ , so the latter is not a function of $x_i$ .",
"When $x_i = 0$ it is equal to 1, so it is identically equal to 1.",
"It follows that $u_{ij} = u_{ik}$ .",
"Now let $d=3$ , so $F_1 = a_{12}\\partial _1\\!\\!\\smile \\!\\!\\partial _2 + a_{13}\\partial _1\\!\\!\\smile \\!\\!\\partial _3 +a_{23}\\partial _2\\!\\!\\smile \\!\\!\\partial _3.",
"$ We may assume, without loss of generality, that $a_{12} = 1$ .",
"From the hypotheses it then follows that $a_{13}$ is not a function of $x_1$ and $a_{23}$ is not a function of $x_2$ .",
"It follows that $a_{13}(\\hat{x_3})$ is a function only of $x_2$ , which we write as $1/f_2$ , and that $a_{23}(\\hat{x_3})$ is a function only of $x_1$ , which we write as $1/f_1$ .",
"Then $a_{13} =(1+x_3u_3)/f_2$ , where $u_3$ can be a function only of $x_2$ and $x_3$ , and similarly $a_{23} = (1+x_3u_3)/f_2$ , where $u_3$ now can be a function only of $x_1$ and $x_3$ .",
"Therefore, it is a function of $x_3$ alone.",
"Denoting $(1+x_3u_3)$ by $f_3$ , the original $F_1$ is therefore equivalent to an infinitesimal with $a_{12} = 1, a_{13} = f_3/f_2, a_{23} = f_3/f_1$ .",
"Multiplying by $f_1f_2$ gives the desired result.",
"$\\Box $ We do not know if the analogous result holds in higher dimensions."
],
[
"localization",
"Kontsevich's UDF is local in the following sense.",
"If $F_1 = \\sum _{ij}a_{ij}\\partial _i \\!\\!\\smile \\!\\!\\partial _j$ is a Poisson infinitesimal deformation of $C^{\\infty }(\\mathbb {R}^d)$ then, in any open set where all the $a_{ij}$ vanish, the star product of functions which that UDF defines is just their ordinary product.",
"In the special case of dimension 3, it follows from Theorem REF that if $\\varphi $ is an arbitrary smooth function of $x_1,x_2,x_3$ , then $\\varphi F_1$ can be integrated to give a star product which reduces to the ordinary multiplication of smooth functions in any open set where $\\varphi $ vanishes.",
"Taking $\\varphi $ to be arbitrarily large at some point and zero outside an arbitrarily small neighborhood of that point, it follows that we can have a star product which in that small neighborhood makes large changes to the multiplication of functions but which outside its closure makes none.",
"We do not know if this can be extended to higher dimensions in a way in which involves differentiation with respect to all the variables, but it can be extended in a simple way.",
"Suppose that the first three variables are spatial variables and that we have a fourth dimension, say time, $t$ .",
"If $\\psi (t)$ is a smooth function then $\\psi (t)\\varphi (x_1,x_2,x_3)F_1$ will again be integrable as an infinitesimal deformation of $\\mathbb {R}^{\\infty }(x_1,x_2,x_3,t)$ since $F_1$ involved only the space variables.",
"We may now take $\\psi (t)$ to be arbitrarily large at some point $t_0$ but vanishing outside an arbitrarily small neighborhood of $t_0$ .",
"In the limit, one has a “big bang” localized at a point in space and time."
]
] | 2107.01754 |
[
[
"A remark on the paper \"A note on the paper Best proximity point results\n for $p$-proximal contractions\""
],
[
"Abstract Recently, In the year 2020, Altun et al.",
"\\cite{AL} introduced the notion of $p$-proximal contractions and discussed about best proximity point results for this class of mappings.",
"Then in the year 2021, Gabeleh and Markin \\cite{GB} showed that the best proximity point theorem proved by Altun et al.",
"in \\cite{AL} follows from the fixed point theory.",
"In this short note, we show that if the $p$-proximal contraction constant $k<\\frac{1}{3}$ then the existence of best proximity point for $p$-proximal contractions follows from the celebrated Banach contraction principle."
],
[
"Metric fixed point theory is an essential part of Mathematics as it gives sufficient conditions which will ensure the existence of solutions of the equation $F(x)=x$ where $F$ is a self mapping defined on a metric space $(M,d).$ Banach contraction principle for standard metric spaces is one of the important results in metric fixed point theory and it has lot of applications.",
"Let $A,B$ be non-empty subsets of a metric space $(M,d)$ and $Q:A\\rightarrow B$ be a non-self mapping.",
"A necessary condition, to guarantee the existence of solutions of the equation $Qx=x,$ is $Q(A)\\cap A\\ne \\phi .$ If $Q(A)\\cap A= \\phi $ then the mapping $Q$ has no fixed points.",
"In this case, one seek for an element in the domain space whose distance from its image is minimum i.e, one interesting problem is to $\\mbox{minimize}~d(x,Qx)$ such that $x\\in A.$ Since $d(x,Qx)\\ge d(A,B)=\\inf ~\\lbrace d(x,y):x\\in A, y\\in B\\rbrace ,$ so, one search for an element $x\\in A$ such that $d(x,Qx)= d(A,B).$ Best proximity point problems deal with this situation.",
"Authors usually discover best proximity point theorems to generalize fixed point theorems in metric spaces.",
"Recently, In the year 2020, Altun et al.",
"[1] introduced the notion of $p$ -proximal contractions and discussed about best proximity point results for this class of mappings.",
"Then, in the year 2021, Gabeleh and Markin [4] showed that the best proximity point theorem proved by Altun et al.",
"in [1] follows from a result in fixed point theory.",
"In this short note, we show that if the $p$ -proximal contraction constant $k<\\frac{1}{3}$ then the existence of best proximity point for $p$ -proximal contractions follows from the Banach contraction principle."
],
[
"We first recall the following definition of $p$ -proximal contraction from [1].",
"Definition 2.1 [1] Let $(A, B)$ be a pair of nonempty subsets of a metric space $(M,d).$ A mapping $f:A\\rightarrow B$ is said to be a $p$ -proximal contraction if there exists $k\\in (0,1)$ such that $\\begin{rcases}d(u_1, f(x_1))= d(A, B)\\\\d(u_2, f(x_2))= d(A, B)\\end{rcases}{\\Longrightarrow d(u_1,u_2)\\le k \\Big (d(x_1,x_2)+|d(u_1,x_1)-d(u_2,x_2)|\\Big )}$ for all $u_1, u_2, x_1, x_2 \\in A,$ where $d(A,B) = \\inf \\Big \\lbrace d(x, y): x\\in A,\\ y\\in B\\Big \\rbrace .$ In this paper, we call the constant $k$ in the above definition as $p$ -proximal contraction constant.",
"The following notations will be needed.",
"Let $(M,d)$ be a metric space and $A,B$ be nonempty subsets of $M.$ Then $A_0=\\lbrace x\\in A: d(x,y)=d(A,B)~\\mbox{for some}~y\\in B\\rbrace .$ $B_0=\\lbrace y\\in B: d(x,y)=d(A,B)~\\mbox{for some}~x\\in A\\rbrace .$ Definition 2.2 [2] Let $(M,d)$ be a metric space and $A,B$ be two non-empty subsets of $M.$ Then $B$ is said to be approximatively compact with respect to $A$ if for every sequence $\\lbrace y_n\\rbrace $ of $B$ satisfying $d(x,y_n)\\rightarrow d(x,B)$ as $n\\rightarrow \\infty $ for some $x\\in A$ has a convergent subsequence.",
"We need the following lemma from [3].",
"Lemma 2.3 [3] Let $(A,B)$ be a nonempty and closed pair of subsets of a metric space $(X,d)$ such that $B$ is approximatively compact with respect to $A.$ Then $A_0$ is closed.",
"In [1], Altun et al.",
"proved the following best proximity point result.",
"Theorem 2.4 [1] Let $A,B$ be nonempty and closed subsets of a complete metric space $(M,d)$ such that $A_0$ is nonempty and $B$ is approximatively compact with respect to $A.$ Let $T:A\\rightarrow B$ be a $p$ -proximal contraction such that $A_0\\ne \\phi $ and $T(A_0)\\subseteq B_0.$ Then $T$ has an unique best proximity point.",
"In [4], Gabeleh and Markin showed that Theorem REF follows from the following fixed point theorem.",
"Theorem 2.5 [5] Let $(M,d)$ be a complete metric space and $T:M\\rightarrow M$ be a $p$ -contraction mapping.",
"Then $T$ has an unique fixed point and for any $x_0\\in M,$ the Picard iteration sequence $\\lbrace T^{n}(x_0)\\rbrace $ converges to the fixed point of $T.$ Now we like to state our main result.",
"Theorem 2.6 If the $p$ -proximal contraction constant $0<k<\\frac{1}{3}$ then Theorem REF follows from the Banach contraction principle.",
"Let $x\\in A_0.$ As $T(A_0)\\subseteq B_0,$ so, $T(x)\\in B_0.$ This implies there exists $y\\in A_0$ such that $d(y,T(x))=d(A,B).$ Now, we will show that $y\\in A_0$ is unique.",
"Suppose there exists $y_1,y_2\\in A_0$ such that $d(y_1,T(x))=d(A,B)$ and $d(y_2,T(x))=d(A,B).$ Since, $T:A\\rightarrow B$ is a $p$ -proximal contraction so we have, $d(y_1,y_2)\\le k\\Big (d(x,x)+|d(y_1,x)-d(y_2,x)|\\Big )\\le k d(y_1,y_2)$ $\\Longrightarrow y_1=y_2.$ Let $S_1:A_0\\rightarrow A_0$ be defined by $S_1(x)=y.$ Now, we will show that $S$ is a contraction mapping.",
"Let $x_1,x_2\\in A_0.$ As $d(S_1(x_1),T(x_1))=d(A,B)$ and $d(S_1(x_2),T(x_2))=d(A,B)$ and $T$ is a $p$ -proximal contraction so, we have, $d(S_1(x_1),S_1(x_2))\\le k\\Big (d(x_1,x_2)+|d(S_1(x_1),x_1)-d(S_1(x_2),x_2)|\\Big )$ $\\Longrightarrow d(S_1(x_1),S_1(x_2))\\le k\\Big (d(x_1,x_2)+ d(S_1(x_1),S_1(x_2))+d(x_1,x_2)|\\Big )$ $\\Longrightarrow d(S_1(x_1),S_1(x_2))\\le \\frac{2k}{1-k} d(x_1,x_2).$ Since $0<k<\\frac{1}{3},$ so, $0<\\frac{2k}{1-k}<1.$ This shows that $S:A_0\\rightarrow A_0$ is a Banach contraction mapping.",
"Now, from lemma REF , we can say $A_0$ is closed.",
"So, $A_0$ is a complete metric space.",
"Then, by Banach contraction principle the mapping $S_1$ has an unique fixed point $z\\in A_0.$ Now, $d(z,T(z))=d(S(z),T(z))=d(A,B).$ This shows that $z$ is a best proximity point for $T.$ Uniqueness follows from the definition of $p$ -proximal contraction.",
"Also, we can conclude that for any $x_0\\in A_0$ the sequence $\\lbrace S_1^{n}(x_0)\\rbrace $ will converge to the unique best proximity point of $T.$"
],
[
"conclusion",
"The main motivation of the current paper is that if the $p$ -proximal contraction constant $0<k<\\frac{1}{3}$ then the best proximity point theorem by Altun [1] follows from the Banach contraction principle.",
"If $\\frac{1}{3}\\le k<1$ then the best proximity point theorem by Altun [1] follows from Theorem REF which is already shown by Gabeleh and Markin in [4]."
]
] | 2107.01685 |
[
[
"Randomized Neural Networks for Forecasting Time Series with Multiple\n Seasonality"
],
[
"Abstract This work contributes to the development of neural forecasting models with novel randomization-based learning methods.",
"These methods improve the fitting abilities of the neural model, in comparison to the standard method, by generating network parameters in accordance with the data and target function features.",
"A pattern-based representation of time series makes the proposed approach useful for forecasting time series with multiple seasonality.",
"In the simulation study, we evaluate the performance of the proposed models and find that they can compete in terms of forecasting accuracy with fully-trained networks.",
"Extremely fast and easy training, simple architecture, ease of implementation, high accuracy as well as dealing with nonstationarity and multiple seasonality in time series make the proposed model very attractive for a wide range of complex time series forecasting problems."
],
[
"Introduction",
"Time series (TS) expressing different phenomena and processes may include multiple seasonal cycles of different lengths.",
"They can be observed in demand variations for various goods, weather conditions, customer numbers, stock market indicators or results of experimental research.",
"Multiple seasonality in TS as well as nonstationarity, nonlinear trend and random fluctuations place high demands on forecasting models.",
"The model should be flexible enough to capture these features without imposing too much computational burden.",
"Over the years, many sophisticated forecasting models for TS with multiple seasonality have been proposed including statistical and machine learning (ML) ones.",
"One of the most commonly employed classical approaches, the autoregressive moving average model (ARMA), can be extended to multiple seasonal cycles by including additional seasonal factors [1].",
"Another popular statistical model, Holt–Winters exponential smoothing (ETS), was developed for forecasting TS data that exhibits both a trend and a seasonal variation.",
"ETS was extended to incorporate a second and a third seasonal component in [2].",
"Both these models, ARMA and seasonal Holt–Winters model, have a significant weakness.",
"They require the same cyclical behavior for each period.",
"In [3], to cope with changing seasonal patterns, innovations state space models for ETS were proposed.",
"The limitation of the model is that it can only be used for double seasonality where one seasonal length is a multiple of the other.",
"A further extension of ETS was proposed in [4].",
"To deal with multiple seasonal periods, high-frequency, non-integer seasonality, and dual-calendar effects, it combines an ETS state space model with Fourier terms, a Box-Cox transformation and ARMA error correction.",
"As an alternative to statistical models, ML models have the ability to learn relationships between predictors and forecasted variables from historical data.",
"One of the most popular in the well-stocked arsenal of ML methods are neural networks (NNs).",
"A huge number of forecasting models based on different NN architectures have been proposed [5].",
"They deal with multiple seasonality differently, depending on the specific architectural features and the creativity of the authors.",
"For example, the model that won the renowned M4 Makridakis competition combines ETS and recurrent NN (RNN) [6].",
"In this approach, ETS produces two seasonal components for TS deseasonalization and adaptive normalization during on-the-fly preprocessing, while RNN, i.e.",
"long short term memory (LSTM), predicts the preprocessed TS.",
"Another example of using LSTM for forecasting TS with multiple seasonal patterns was proposed recently in [7].",
"To deal with multiple seasonal cycles, the model initially deseasonalizes TS using different strategies including Fourier transformation.",
"RNNs, such as LSTM, gated recurrent units, and DeepAR [8], dominate today as NN architectures for TS forecasting thanks to their powerful ability to process sequential data and capture long-term dependencies.",
"But other deep architectures are also useful for forecasting multiple seasonal TS.",
"For example, N-Beats [9] was designed specifically for TS with multiple seasonality.",
"It is distinguished by a specific architecture including backward and forward residual links and a very deep stack of fully-connected layers.",
"The above presented approaches to forecasting TS with multiple seasonal periods rely on incorporating into the model mechanisms which allow it to deal with seasonal components.",
"This complicates the model and makes it difficult to train and optimize.",
"An alternative approach is to simplify the forecasting problem by TS decomposition or preprocessing.",
"In [10], TS with three seasonal cycles was represented by patterns expressing unified shapes of the basic cycle.",
"This preprocessing simplified the relationship between TS elements, making decomposition unnecessary and removing the need to build a complex model.",
"Instead, simple shallow NNs can be used [10] or nonparametric regression models [11].",
"Experimental research has confirmed that these models can compete in terms of accuracy with state-of-the-art deep learning models, like the winning M4 submission [12].",
"In this study, we use a pattern representation of TS to simplify the forecasting problem with multiple seasonality and propose randomization-based shallow NNs to solve it.",
"Randomized learning was proposed as an alternative to gradient-based learning as the latter is known to be time-consuming, sensitive to the initial parameter values and unable to cope with the local minima of the loss function.",
"In randomized learning, the parameters of the hidden nodes are selected randomly and stay fixed.",
"Only the output weights are learned.",
"This makes the optimization problem convex and allows us to solve it without tedious gradient descent backpropagation, but using a standard linear least-squares method instead [13].",
"This leads to a very fast training.",
"The main problem in randomized learning is how to select the random parameters to ensure the high performance of the NN [14], [15].",
"In this study, to generate the random parameters we use three methods recently proposed in [16], [17].",
"These methods distribute the activation functions (sigmoids) of hidden nodes randomly in the input space and adjust their weights (or a weight interval) to the target function (TF) complexity using different approaches.",
"The main goal of this study is to show that randomization-based NNs can compete in terms of forecasting accuracy with fully-trained NNs.",
"The contribution of this study can be summarized as follows: A new forecasting model for TS with multiple seasonality based on randomized NNs is proposed.",
"To deal with multiple seasonality and nonstationarity, the model applies pattern representation of TS in order to simplify the relationship between input and output data.",
"Three randomization-based methods are used to generate the NN hidden node parameters.",
"They introduce steep fragments of sigmoids in the input space, which improves modeling of highly nonlinear TFs.",
"A randomized approach leads to extremely fast and easy training, simple NN architecture and ease of implementation.",
"Numerical experiments on several real-world datasets demonstrate the efficiency of the proposed randomization-based models when compared to fully-trained NNs.",
"The remainder of this work is structured as follows.",
"Section 2 presents the proposed forecasting model based on randomized NNs, and a TS representation using patterns of seasonal cycles and three methods of generating NN parameters are described.",
"The performance of the proposed approach is evaluated in Section 3.",
"Finally, Section 4 concludes the work."
],
[
"Forecasting model",
"The proposed forecasting model is shown in Fig.",
"REF .",
"It is composed of encoder and decoder modules and a randomized feedforward NN (FNN).",
"The model architecture, its specific features, and components are described below.",
"Figure: Block diagram of the proposed forecasting model."
],
[
"Encoder",
"The task of the encoder is to convert an original TS into unified input and output patterns of its seasonal cycles.",
"To create input patterns, the TS expressing multiple seasonality, $\\lbrace E_k\\rbrace _{k=1}^K$ , is divided into seasonal sequences of the shortest length.",
"Let these sequences be expressed by vectors $ \\mathbf {e}_i = [E_{i,1}, E_{i,2}, …, E_{i,n}]^T $ , where $n$ is the seasonal sequence length and $i=1, 2, ..., K/n$ is the sequence number.",
"These sequences are encoded in input patterns $\\mathbf {x}_i = [x_{i,1}, x_{i,2}, …, x_{i,n}]^T$ as follows: $\\mathbf {x}_i = \\frac{\\mathbf {e}_i-\\overline{{e}}_i}{\\widetilde{e}_i}$ where $\\overline{{e}}_i$ is a mean value of sequence $\\mathbf {e}_i$ , and $\\widetilde{e}_i = \\sqrt{\\sum _{t=1}^{n} (E_{i,t}-\\overline{{e}}_i)^2}$ is a measure of sequence $\\mathbf {e}_i$ dispersion.",
"Note that the x-patterns are normalized versions of centered vectors $\\mathbf {e}_i$ .",
"All x-patterns, representing successive seasonal sequences, have zero mean, the same variance and the same unity length.",
"However, they differ in shape.",
"Thus, the original seasonal sequences, which have a different mean value and dispersion, are unified.",
"This is shown in Fig.",
"REF on the example of the hourly electricity demand TS expressing three seasonalities: daily, weekly, and yearly.",
"Note that the x-patterns representing the daily cycles are all normalized and differ only in shape.",
"The output patterns $\\mathbf {y}_i = [y_{i,1}, y_{i,2}, …, y_{i,n}]^T$ represent the forecasted sequences $ \\mathbf {e}_{i+\\tau } = [E_{i+\\tau ,1}, E_{i+\\tau ,2}, …, E_{i+\\tau ,n}]^T $ , where $\\tau \\ge 1$ is a forecast horizon.",
"The y-patterns are determined as follows: $\\mathbf {y}_i = \\frac{\\mathbf {e}_{i+\\tau }-\\overline{{e}}_i}{\\widetilde{{e}}_i}$ where $\\overline{{e}}_i$ and $\\widetilde{e}_i$ are the same as in (REF ).",
"Note that in (REF ), for the $i$ -th output pattern, we use the same coding variables $\\overline{{e}}_i$ and $\\widetilde{e}_i$ as for the $i$ -th input pattern.",
"This is because the coding variables for the forecasted sequence, $\\overline{{e}}_{i+\\tau }$ and $\\widetilde{e}_{i+\\tau }$ , are unknown for the future period.",
"Using the coding variables determined from the previous period has consequences which are demonstrated in Fig.",
"REF .",
"Note that y-patterns in this figure reveal the weekly seasonality.",
"The y-patterns of Mondays are much higher than the patterns of other days of the week because the Monday sequences are coded with the means of Sunday sequences which are much lower than the means of Monday sequences.",
"For similar reasons, y-patterns for Saturdays and Sundays are lower than y-patterns for the other days of the week.",
"Thus, the y-patterns are not unified globally but are unified in groups composed of the same days of the week.",
"For this reason, we construct forecasting models that learn from data representing the same days of the week.",
"For example, when we train the model to forecast the daily sequence for Monday, the training set for it, $\\Phi =\\lbrace (\\mathbf {x}_i,\\mathbf {y}_i)\\rbrace _{i=1}^N$ , is composed of the y-patterns representing all Mondays from history and corresponding x-patterns representing the previous days (depending on the forecast horizon; Sundays for $\\tau = 1$ ).",
"Figure: Real hourly electricity demand TS and its x- and y-patterns."
],
[
"Decoder",
"The decoder converts a forecasted output pattern into a TS seasonal cycle.",
"The output pattern predicted by randomized FNN is decoded using the coding variables of the input query pattern, $\\mathbf {x}$ , using transformed equation (REF ): $\\widehat{\\mathbf {e}} = \\widehat{\\mathbf {y}}\\widetilde{{e}}+\\overline{{e}}$ where $\\widehat{\\mathbf {e}}$ is the forecasted seasonal sequence, $\\widehat{\\mathbf {y}}$ is the forecasted output pattern, $\\widetilde{e}$ and $\\overline{e}$ are the coding variables determined from the TS sequence encoded in query pattern $\\mathbf {x}$ ."
],
[
"Randomized FNN",
"The randomized FNN is composed of $n$ inputs, one hidden layer with $m$ nonlinear nodes, and $n$ outputs.",
"Logistic sigmoid activation functions are employed for hidden nodes.",
"The training set is $ \\Phi = \\left\\lbrace (\\mathbf {x}_i, \\mathbf {y}_i)\\right\\rbrace _{i=1}^N, \\mathbf {x}_i, \\mathbf {y}_i \\in \\mathbb {R}^n$ .",
"The randomized learning algorithm consists of three steps [18].",
"Randomly generate hidden node parameters, i.e.",
"weights $ \\mathbf {a}_j = [ a_{j,1}, a_{j,2}, \\ldots ,\\\\ a_{j,n}]^T $ and biases $ b_j, j = 1, 2, \\ldots , m $ , according to any continuous sampling distribution.",
"Calculate the hidden layer output matrix: $\\mathbf {H} = \\left[\\begin{array}{c}\\mathbf {h}(\\mathbf {x}_1) \\\\\\vdots \\\\\\mathbf {h}(\\mathbf {x}_N)\\end{array}\\right]$ where $ \\mathbf {h}(\\mathbf {x}) = \\left[h_1(\\mathbf {x}), h_2(\\mathbf {x}), \\ldots , h_m(\\mathbf {x})\\right]$ is a nonlinear feature mapping from $n$ -dimensional input space to $ m $ -dimensional feature space, and $ h_j(\\mathbf {x}) $ is an activation function of the $ j $ -th node (a sigmoid in our case).",
"Calculate the output weights: $\\beta = \\mathbf {H}^+\\mathbf {Y}$ where $ \\beta \\in \\mathbb {R}^{m\\times n}$ is a matrix of output weights, $ \\mathbf {Y} \\in \\mathbb {R}^{N\\times n}$ is a matrix of target output patterns, and $ \\mathbf {H}^+ \\in \\mathbb {R}^{m\\times N} $ is the Moore-Penrose generalized inverse of matrix $ \\mathbf {H} $ .",
"Typically, the hidden node weights and biases are i.i.d random variables both generated from the same symmetrical interval $ a_{j,i}, b_j \\sim U(-u, u) $ .",
"It was pointed out in [18] and [16] that as the weights and biases have different functions they should be selected separately.",
"The weights decide about the sigmoid slopes and should reflect the TF complexity, while the biases decide about the sigmoid shift and should ensure the placement of the most nonlinear sigmoid fragments, i.e.",
"the fragments around the sigmoid inflection points, into the input hypercube.",
"These fragments, unlike saturation fragments, are most useful for modeling TF fluctuations.",
"Recently, to improve the performance of randomized FNNs, several new methods of generating the hidden node parameters have been proposed.",
"Among them is the random $a$ method (R$a$ M) which was proposed in [16].",
"In the first step, this method randomly selects weights from the interval whose bounds $u$ are adjusted to the TF complexity, $ a_{j,i} \\sim U(-u, u) $ .",
"Then, to ensure the introduction of the sigmoid inflection points into the input hypercube, the biases are calculated from: $b_j = -\\mathbf {a}_j^T\\mathbf {x}^*_j$ where $\\mathbf {x}^*_j$ is one of the training x-patterns selected for the $j$ -th hidden node at random.",
"The second method proposed in [16], called the random $\\alpha $ method (R$\\alpha $ M), instead of generating weights, generates the slope angles of sigmoids.",
"This changes the distribution of weights, which typically is a uniform one.",
"This new distribution ensures that the slope angles of sigmoids are uniformly distributed, and so improves results by preventing overfitting, especially for highly nonlinear TFs.",
"This method, in the first step, selects randomly the slope angles of the sigmoids, $ \\alpha _{j,i} \\sim U(\\alpha _{\\min }, \\alpha _{\\max }) $ .",
"Then, the the weights are calculated from: $a_{j,i}=4 \\tan \\alpha _{j,i}$ Finally, the biases are determined from (REF ).",
"To simplify the optimization process, the lower bound for the angles, $\\alpha _{\\min }$ , can be set as $0^\\circ $ .",
"In such a case only one parameter decides about the model flexibility, i.e.",
"$\\alpha _{\\max } \\in (0^\\circ , 90^\\circ )$ .",
"This is what we used in our simulation study.",
"To improve further FNN randomized learning, a data-driven method (DDM) was proposed in [17].",
"This method introduces the sigmoids into randomly selected regions of the input space and adjusts the sigmoid slopes to the TF slopes in these regions.",
"As a result, the sigmoids mimic the TF locally, and their linear combination approximates smoothly the entire TF.",
"In the first step, DDM selects the input space regions by selecting randomly the set of training points, $\\lbrace \\mathbf {x}^*_j\\rbrace _{j=1}^m$ .",
"Then, the hyperplanes are fitted to the TF locally in the neighbourhoods of all points $\\mathbf {x}^*_j$ .",
"The neighborhood of point $\\mathbf {x}^*_j$ , $\\Psi (\\mathbf {x}^*_j)$ , contains this point and its $k$ nearest neighbors in $\\Phi $ .",
"The weights are determined based on the hyperplane coefficients from: $a_{j,i} = 4a_{j,i}^{\\prime }$ where $a_{j,i}^{\\prime }$ are the coefficients of the hyperplane fitted to neighbourhood $\\Psi (\\mathbf {x}^*_j)$ .",
"The hidden node biases are calculated from (REF ).",
"Note that the biases in the above-described approaches are determined based on the weights selected first and the data points.",
"Unlike in the standard approach, they are not chosen randomly from the same interval as the weights.",
"Randomized FNN has two hyperparameters to adjust: number of hidden nodes $m$ , and the smoothing parameter, i.e.",
"$u$ , $\\alpha _{\\max }$ or $k$ , depending on the method of generating parameters chosen.",
"These hyperparameters decide about the fitting performance of the model and its bias-variance tradeoff.",
"Their optimal values should be selected by cross-validation for a given forecasting problem.",
"In this section, we apply the proposed randomization-based neural models to forecasting hourly TS with three seasonalities: yearly, weekly and daily.",
"These TS express electricity demand for four European countries: Poland (PL), Great Britain (GB), France (FR) and Germany (DE).",
"We use real-world data collected from www.entsoe.eu.",
"The data period covers the 4 years from 2012 to 2015.",
"Atypical days such as public holidays were excluded from these data (between 10 and 20 days a year).",
"The forecast horizon $\\tau $ is one day, i.e.",
"24 hours.",
"We forecast the daily load profile for each day of 2015.",
"For each forecasted day, a new training set is created and a new randomized model is optimized and trained.",
"The results presented below are averaged over 100 independent training sessions.",
"The hyperparameters of randomized FNNs were selected using grid search and 5-fold cross-validation.",
"The number of hidden nodes was selected from the set $\\lbrace 5, 10, ..., 50\\rbrace $ .",
"The bounds for weights in R$a$ M were selected from $\\lbrace 0.02, 0.04, ..., \\\\ 0.2, 0.4, ..., 1\\rbrace $ .",
"The $\\alpha _{\\max }$ in R$\\alpha $ M was selected from $\\lbrace 2^\\circ , 4^\\circ , ..., 40^\\circ , 45^\\circ , ..., 90^\\circ \\rbrace $ .",
"The number of nearest neighbors in DDM was selected from $\\lbrace 25, 27, ..., 69\\rbrace $ .",
"For comparison, we applied a multilayer perceptron (MLP) for the same forecasting problems.",
"MLP was composed of a single hidden layer with $m$ sigmoid nodes whose number was selected using 5-fold cross-validation from the set $\\lbrace 2, 4, ..., 24\\rbrace $ .",
"MLP was trained using Levenberg-Marquardt backpropagation with early stopping to avoid overtraining (20% of training samples were used as validation samples).",
"Forecasting quality metrics for the test data are presented in Table REF .",
"They include: mean absolute percentage error ($\\operatorname{\\textsc {mape}}$ ), median of $\\operatorname{\\textsc {ape}}$ , root mean square error ($\\operatorname{\\textsc {rmse}}$ ), mean percentage error ($\\operatorname{\\textsc {mpe}}$ ), and standard deviation of percentage error ($\\operatorname{\\textsc {pe}}$ ) as a measure of the forecast dispersion.",
"Table: Forecasting results.More detailed results, i.e.",
"distributions of $\\operatorname{\\textsc {ape}}$ , are shown in Fig.",
"REF .",
"Based on $\\operatorname{\\textsc {ape}}$ , we performed a Wilcoxon signed-rank test with $\\alpha =0.05$ to indicate the most accurate models.",
"Fig.",
"REF depicts pairwise comparisons of the models.",
"The arrow lying at the intersection of the two models indicates which of them gave the significantly lower error.",
"A lack of an arrow means that both models gave statistically indistinguishable errors.",
"Figure: Boxplots of error\\operatorname{\\textsc {ape}}.Figure: Results of the Wilcoxon signed-rank test for error\\operatorname{\\textsc {ape}}.As can be seen from Table REF and Fig.",
"REF , the randomization-based FNNs gave significantly lower errors than fully-trained MLP for each dataset.",
"According to the Wilcoxon test, R$a$ M outperformed the other approaches.",
"$\\operatorname{\\textsc {mpe}}$ shown in Table REF allows us to asses the bias of the forecasts produced by different models.",
"A positive value of $\\operatorname{\\textsc {mpe}}$ indicates underprediction, while a negative value indicates overprediction.",
"As can be seen from Table REF , for PL and DE data the bias was positive, whilst for GB and FR data it was negative.",
"The forecasts produced by MLP for PL and DE were less biased than the forecast produced by randomized FNNs.",
"Fig.",
"REF presents examples of forecasts of the daily load profiles produced by the examined models.",
"Note that the proposed models generate multi-output response, maintaining the relationships between the output variables (y-pattern components).",
"In the case of single-output models, these relationships are ignored because the variables are predicted independently.",
"This may cause a lack of smoothness in the forecasted curve (zigzag effect; see for example [19]).",
"Figure: Examples of forecasts (shaded regions are 5th and 95th percentiles, measured over 100 trials).Fig.",
"REF shows the optimal numbers of hidden nodes selected in the cross-validation procedure.",
"Obviously, the number of hidden nodes is dependent on TF complexity.",
"The forecasting problem for PL required the greatest number of nodes for randomized FNNs, around 30, regardless of the learning method.",
"MLP for PL needed many fewer nodes, 12 on average.",
"Other forecasting problems were solved by randomized FNNs with fewer hidden nodes, from 20 to 30 on average.",
"For these problems, the difference in the number of nodes between MLP and randomized FNNs was not as large as for PL data.",
"The relatively small number of hidden nodes in randomized FNNs (note that randomized learning usually requires hundreds or even thousands of nodes) results from TS representation by unified patterns and the decomposition of the forecasting problem (a separate model for each forecasting task, i.e.",
"every day in 2015, trained on the selected patterns).",
"The optimal values of smoothing parameters for the randomized learning methods are depicted in Fig.",
"REF .",
"As can be seen from this figure, the optimal value of the bound for weights in R$a$ M varies from 0.2 for FR to 0.7 for GB on average, which correspond to sigmoid slope angles from around $3^\\circ $ to $10^\\circ $ (see [16]).",
"The optimal value of the bound for slope angle in R$\\alpha $ M varies from $12^\\circ $ for FR to $32^\\circ $ for DE on average.",
"Note also the high value of $k$ in DDM (from 49 for PL to 65 for FR on average) in relation to the number of training points, which ranged from 150 to 200.",
"Thus, for our forecasting problems we can expect flat TFs without fluctuations.",
"Such TFs can be modeled using R$a$ M. Its competitors, R$\\alpha $ M and DDM, reveal their strengths in modeling highly nonlinear TFs with fluctuations (see [16], [17]).",
"Figure: Boxplots of the optimal number of hidden nodes.Figure: Boxplots of the optimal smoothing parameters."
],
[
"Conclusion",
"Forecasting TS with multiple seasonality is a challenging problem, which we propose to solve with randomized FNNs.",
"Unlike fully-trained FNNs, randomized FNNs learn extremely fast and are easy to implement.",
"The simulation study showed that their forecasting accuracy is comparable to the accuracy of fully-trained NNs.",
"To deal with nonstationary TS with multiple seasonal periods, the proposed approach employs a pattern representation of the TS.",
"This representation simplifies the relationship between input and output data and makes the problem easier to solve using simple regression models.",
"The effectiveness of the randomized FNNs in modeling nonlinear target functions was achieved due to the application of new methods of generating hidden node parameters.",
"These methods, using different approaches, introduce the steepest fragments of sigmoids, which are most useful for modeling TF fluctuations, into the input hypercube and adjust their slopes to TF complexity.",
"This makes the model more flexible, more data-dependent, and more dependent on the complexity of the solved forecasting problem.",
"In a future study, we plan to introduce an attention mechanism into our randomization-based forecasting models to select training data and develop an ensemble approach for these models."
]
] | 2107.01705 |
[
[
"Faster-LTN: a neuro-symbolic, end-to-end object detection architecture"
],
[
"Abstract The detection of semantic relationships between objects represented in an image is one of the fundamental challenges in image interpretation.",
"Neural-Symbolic techniques, such as Logic Tensor Networks (LTNs), allow the combination of semantic knowledge representation and reasoning with the ability to efficiently learn from examples typical of neural networks.",
"We here propose Faster-LTN, an object detector composed of a convolutional backbone and an LTN.",
"To the best of our knowledge, this is the first attempt to combine both frameworks in an end-to-end training setting.",
"This architecture is trained by optimizing a grounded theory which combines labelled examples with prior knowledge, in the form of logical axioms.",
"Experimental comparisons show competitive performance with respect to the traditional Faster R-CNN architecture."
],
[
"Introduction",
"A long-standing problem in Semantic Image Interpretation (SII) and related tasks is how to combine learning from data with existing background knowledge in the form of relational knowledge or logical axioms [1].",
"Neural-Symbolic (NeSy) integration, which aims at integrating symbolic knowledge representation and learning with machine learning techniques [2], can provide an elegant and principled solution to augment state-of-the-art deep neural networks with these novel capabilities, increasing their performance, robustness and explainability.",
"The present work leverages the Logic Tensor Network (LTN) paradigm that was proposed by Serafini, Donadello and d'Avila Garcez [3], [4].",
"In very simple terms, LTNs operate by interpreting (or grounding) a First-Order Logic (FOL) as functions on real vectors, which parameters can be trained via stochastic gradient descents to maximize the satisfiability of a given theory.",
"LTNs have been successfully applied to the tasks of part-of relationship detection [3] and visual relationship detection [5].",
"Previous works have shown how LTNs can compensate the lack of supervision (e.g., in few-shot learning scenarios) by relying on logical axioms derived from pre-existing knowledge bases.",
"To close the semantic gap between the symbolic (concept) and subsymbolic (pixel) levels, LTNs for SII rely on convolutional neural networks (CNNs) to extract semantic features which form the basis for grounding object instances in a real vector.",
"Previous works [5], [3] relied on pre-trained CNNs, which however suffer from all the limitations traditionally associated with deep learning, namely, the need for a large-scale annotated dataset for training, and lack of interpretability.",
"To fully reap the benefits of NeSy techniques in SII, end-to-end architectures in which the LTN is jointly trained with the feature extraction CNN are needed.",
"In this work, we propose Faster-LTN, an object detector which unifies the Faster R-CNN object detector with a LTN-based classification head.",
"Differently from previous works [3], [5], both modules are jointly trained in an end-to-end fashion.",
"The logical constraints imposed by the LTN can thus shape the training of the convolutional layers, that are no longer purely data-driven.",
"To achieve this objective, we propose several modifications to the original LTN formulation to increase the architecture scalability and deal with data imbalance.",
"Experimental results on the PASCAL VOC and PASCAL PART datasets show that Faster-LTN converges to competitive performance with respect to purely neural architectures, thus proving the feasibility of this approach.",
"The Faster-LTN was implemented in Keras and is available at https://gitlab.com/grains2/Faster-LTN.",
"The rest of the paper is organized as follows.",
"In Section , related work is presented.",
"In Section , different variations of the Faster-LTN architecture are presented, after a brief introduction to the theory behind LTNs.",
"Section presents the experimental setting and results.",
"Finally, conclusions are drawn."
],
[
"Related work",
"A natural image is comprised of scenes, objects and parts, all interconnected by a complex network of spatial and semantic relationships.",
"Thus, developing semantic image interpretation (SII) components requires to recognize a hierarchy of components, and entails both robust visual perception and the ability to encode and (reason about) visual relationships.",
"Several techniques have been proposed to augment Convolutional Neural Networks (CNNs) with relationship representation and reasoning capabilities, including Relational Network [6], Graph Neural Networks [7] and Neural-Symbolic (NeSy) techniques [8], [3], [5].",
"For a more general introduction to NeSy techniques, the reader is referred to recent surveys [9], [10].",
"Many recent approaches extract features from CNNs to a subsequent symbolic or neuro-symbolic module [11], [3], [5], [12].",
"Yuke Zhu et al.",
"[11] use a Markov Logic Network (MLN) to process text information with associated visual features; a knowledge base is used to represent relations between objects using visual, physical, and categorical attributes.",
"Kenneth Marino et al.",
"[13] incorporate a Graph Search Neural Network (GSNN) into a classification network.",
"Donatello et al.",
"[3] and Cewu Lu et al.",
"[12] have demonstrated the use of visual features to train LTNs for visual relationship detection, in form of subject-verb-object triplets or part of relationships.",
"These works demonstrate how NeSy techniques enable the definition of logical axioms that serve as high-level inductive biases, driving the network to find the optimal solution that is compatible with said inductive biases.",
"However, since in the above-mentioned cases the feature extraction and the classification networks are trained separately, the CNN cannot leverage these additional inductive biases during training.",
"There are, however, some practical hurdles associated with the training of NeSy architectures.",
"Scalability, when dealing with large amounts of data, is a known issue associated with symbolic AI [14].",
"For this reason, many NeSy architectures rely on a conventional object detector to provide an initial list of candidate objects [3], thus disregarding the effect of the background and simplifying (i.e., reducing) the scale of the problem.",
"In this work, we compare several strategies that are effectively capable of training a LTN-based object detector from scratch, taking into account the effect of the background and the resulting data imbalance.",
"Another aspect related to scalability is the choice of aggregation function and fuzzy logic operators.",
"Emilie van Krieken et al.",
"[14] and Samy Badreddine [4] found substantial differences between differential fuzzy logic operators in terms of computational efficiency, scalability, gradients, and ability to handle exceptions, which are important characteristics in a learning setting.",
"Their analysis lays the groundwork for the present FasterLTN architecture, which incorporates and extends the log-product aggregator analyzed in [14]."
],
[
"The Faster-LTN architecture",
"This section describes the Faster-LTN architecture and training procedure in detail.",
"An overview of the overall architecture is presented in Figure REF .",
"We first summarize the Faster R-CNN overall architecture (Section REF ).",
"Then, we introduce the main concepts behind LTNs (Section REF ) and their application to object detection (Section REF ), referring the reader to [3], [4] for additional details.",
"Finally, the joint training procedure of Faster-LTN is explained in Section REF , highlighting the main changes introduced to make end-to-end training feasible."
],
[
"Faster R-CNN",
"Faster R-CNN is a two-stage object detector composed of a Region Proposal Network (RPN) and a classification network with a shared backbone [15].",
"For each anchor, the RPN generates a binary classification label (Background vs. foreground), while a regression layer computes the bounding box coordinates.",
"Regions of Interest (ROIs) selected by the RPN are fed to an ROI Pooling layer, which extracts and resizes each proposal bounding box's features from the shared backbone.",
"Feature maps of equal size are passed to the classifier.",
"The classifier comprises two convolutional heads, a classification layer (with softmax activation) that computes the final object classification and a regression layer (with linear activation) that computes the bounding box.",
"Training of the RPN and classifier heads is performed jointly in an alternating fashion.",
"At each forward pass (corresponding to one image), the RPN is trained and updated; then, the RPN output is kept fixed, and the detector head is updated.",
"A fixed number of positive (object) and negative (Background) examples are selected at each step to train the classifier head.",
"The loss is as a combination of regression and classification loss: $L(\\lbrace p_i\\rbrace , \\lbrace b_i\\rbrace ) = \\frac{1}{n_c}\\sum _{i}L_{cls}(p_i, p^{\\prime }_{i}) + \\lambda \\frac{1}{n_r}\\sum _{i}{p_i * L_{reg}(b_i, b^{\\prime }_{i})}$ In the Faster-LTN, we keep the RPN module intact and substitute the classifier head with an LTN.",
"In the LTN framework, it is possible to encode a FOL language $\\mathcal {L}$ by defining its interpretation domain as a subset of $\\mathbb {R}^{n}$ .",
"In the LTN formalism, this process is called grounding.",
"Given the vector space $\\mathbb {R}^{n}$ , a grounding $\\mathcal {G}$ for $\\mathcal {L}$ has the following properties: $\\mathcal {G}(c) \\in \\mathbb {R}^n$ , for every $c \\in \\mathcal {C}$ ; $\\mathcal {G}(P) \\in \\mathbb {R}^{n*k} \\rightarrow [0,1]$ , for every $p \\in \\mathcal {P}$ The grounding of a set of closed terms $t_{1},..,t_{m}$ of $\\mathcal {L}$ in an atomic formula is defined as: $\\mathcal {G} \\left( \\mathcal {P}\\left( t_{1},...t_{m} \\right)\\right) =\\mathcal {G}\\left( P\\right) \\left( \\mathcal {G} \\left( t_{1}\\right),...,\\mathcal {G} \\left( t_{m}\\right)\\right) $ Formulas can be connected with fuzzy logic operators such as conjunctions ($\\wedge $ ), disjunctions ($\\vee $ ), and implications ($\\Rightarrow $ ), including logical quantifiers ($\\forall $ and $\\exists $ ).",
"Several real-valued, differentiable implementations are available in the fuzzy logic domain [14].",
"Our implementation, as in [3], is based on the Łukasiewicz [16] formulation: $& \\mathcal {G} \\left( \\lnot \\phi \\right) = 1 - \\mathcal {G}\\left( \\phi \\right) \\\\& \\mathcal {G} \\left( \\phi \\vee \\psi \\right) = min(1,\\mathcal {G}\\left( \\phi \\right) + \\mathcal {G}\\left( \\psi \\right))$ Predicate symbols are interpreted as functions that map real vectors to the interval $[0, 1]$ , which can be interpreted as the predicate's degree of truth.",
"A typical example is the is-a predicate, which quantifies the existence of a given object.",
"For instance, if $b=\\mathcal {G}(x)$ is the grounding of a dog bounding box, than $\\mathsf {\\mathcal {G}(Dog)(v)}\\simeq 1$ .",
"A logical constraint expressed in FOL allows to define its properties, i.e., $\\forall x \\left( \\mathsf {Dog}(x) \\rightarrow \\mathsf {hasMuzzle} \\left( x\\right) \\right)$ .",
"In LTNs, predicates are typically defined as the generalization of the neural tensor network: $\\mathcal {G}\\left( \\mathcal {P} \\right)( \\mathbf {v}) = \\sigma \\left(\\mathit {u_{P}^{T}}\\tanh \\left( \\mathbf {v_{T}} W_{P}^{[1:k]} \\mathbf {v} + V_{P} \\mathbf {v} + \\mathit {b_{p}} \\right) \\right)$ where $\\sigma $ is the sigmoid function, $W[1:k] \\in \\mathbb {R}^{k \\times mn \\times mn}$ , $V_{p} \\in \\mathbb {R}^{k \\times mn}$ ,$u_{p} \\in \\mathbb {R}^{k}$ and $ b_{p} \\in \\mathbb {R} $ are learnable tensors of parameters.",
"With this formulation, the truth value of a clause can be determined by a neural network which first computes the grounding of the literals (i.e., atomic objects) contained in the clause, and then combines them using fuzzy logical operators, as defined by Eqs.",
"3-4."
],
[
"Grounded theory",
"A Grounded Theory (GT) $\\mathcal {T}$ is defined by a pair $\\langle \\mathcal {K},\\hat{\\mathcal {G}} \\rangle $ , where the knowledge base $\\mathcal {K}$ is a set of closed formulas, and $\\hat{\\mathcal {G}}$ is a partial grounding.",
"$\\mathcal {K}$ is constructed from labelled examples, as well as logical axioms, as defined in Section REF .",
"In practice, a partial grounding is optimized since, qualitatively, our set $\\mathcal {K}$ represents a limited and finite set of examples.",
"A grounding $G$ satisfies a GT $\\langle \\mathcal {K},\\hat{\\mathcal {G}} \\rangle $ if $\\mathcal {G}$ completes $\\hat{\\mathcal {G}}$ and $\\mathcal {G}\\left(\\phi \\right) = 1 \\ \\forall \\ \\phi \\in \\mathcal {K}$ ."
],
[
"Best satisfability problem",
"Given a grounding $\\hat{\\mathcal {G}}_{\\theta }$ , where $\\theta $ is the set of parameters of all predicates, the learning problem in LTNs is framed as a best satisfability problem which consists in determining the values of $\\Theta ^{*}$ that maximize the truth values of the conjunction of all clauses $\\ \\phi \\in \\mathcal {K}$ : $\\Theta ^{*}= argmax_{\\Theta } \\hat{\\mathcal {G}}_{\\theta }\\left(\\bigwedge _{\\phi \\in \\mathcal {K}} \\phi \\right) - \\lambda || \\Theta ||_{2}^{2}$ where $ \\lambda || \\Theta ||_{2}^{2}$ is a regularization term.",
"In practical problems, it is unlikely that a grounded theory can be satisfiable in the classical sense.",
"Hence, we opt instead to find the grounding which achieves the best possible satisfaction, while accounting for the inevitable exception to the rule.",
"Such exceptions can easily arise in the visual domain not only to account to allow the occasional deviation from the norm, but also to account for properties that are not visible.",
"For instance, a cat has (usually) a tail, but a few cats may be tail-less; more frequently, the tail will be occluded or cut from the image."
],
[
"A grounded theory for object detection",
"Let us consider a set of bounding boxes $b \\in \\mathcal {B}$ with known class $c \\in \\mathcal {C}$ .",
"An object with bounding box $b_{n}$ is grounded by the vector: $\\mathbf {v_{b_{n}}}= <\\mathbf {z}_{b_{n}},b_{n} >$ Where $\\mathbf {z}_{b_{n}} = f(I,b_{n})$ is an embedding feature vector, calculated by a convolutional neural network $f$ , given an image $I$ and the bounding box coordinates $b_{n}$ predicted by the RPN layer.",
"This is slightly different from previous works [3], where the grounding of a bounding box was defined by the probability vector predicted by a pre-trained Faster R-CNN, and allows to effectively connect the convolutional layers and the LTN.",
"We set the embedding $f(I,b_{n})$ to the output of the last fully connected layer of the classifier head, without softmax activation.",
"Other choices are possible, e.g., by sum pooling the output of an earlier convolutional layer.",
"The is-a predicate for class $c \\in \\mathcal {C}$ is grounded by a tensor network, defined as in Eq.",
"REF , which implements a one-vs-all classifier.",
"It must be noticed that, differently from [3], the is-a predicate takes as input only the embedding features $\\mathbf {z}_{b_{n}}$ , excluding the bounding box coordinates.",
"This allows to retain one of the basic properties of object detectors, i.e., invariance to translation.",
"The part-of predicate is defined over pairs of bounding boxes [3].",
"A pair of two generic bounding boxes $b_{m}$ and $b_{l}$ is grounded by the vector: $\\mathbf {v_{b_{m, l}}}= <\\mathbf {z}_{b_{m}},b_{m},\\mathbf {z}_{b_{l}},b_{l},ir_{m,l}>$ where $ir_{m,l}$ is the containment ratio defined as: $ir_{m,l}= \\frac{Area \\left( b_{m} \\cap b_{l} \\right)}{Area \\left( b_{m} \\right)}$ The grounding $\\mathcal {G}\\left( \\mathsf {part-of} \\right)( \\mathbf {v_{b_{m, l}}})$ is a neural tensor network as in Eq.",
"REF ."
],
[
"Defining a theory from labelled examples",
"Let us now consider how a GT is constructed to solve the best satisfiability problem defined in Eq.",
"REF for object detection.",
"As in [3], two grounded theories $\\mathcal {T}_{expl}$ and $\\mathcal {T}_{prior}$ are defined.",
"The former, $\\mathcal {T}_{expl}$ , aggregates all the clauses derived from the labelled training set, essentially replicating the classical learning-by-example setting.",
"The theory $\\mathcal {T}_{prior}$ , on the contrary, introduces logical and mereological constraints that represent prior knowledge or, in a more general sense, desirable properties of the final solution.",
"In this work, two types of constraints are defined.",
"First, we enforce mutual exclusion through the clause: $\\forall x( P_1(x) \\Rightarrow ( \\lnot P_2(x) \\wedge ... \\wedge \\lnot P_n(x)))$ Eq.",
"REF is translated into $K(K-1))/2$ clauses, corresponding to all unordered class pairs over $K$ classes, e.g., ${\\mathsf {Cat}(x) \\Rightarrow \\lnot \\mathsf {Person}(x)}$ .",
"Secondly, we impose mereological constraints on the grounding of part-of and is-a predicates derived from an existing ontology (e.g., Wordnet) which includes meronimy (i.e., part-whole) relationships.",
"Axioms are included to specify that a part cannot include another part, that a whole object cannot include another whole object, and that each whole is generally associated with a set of given parts.",
"An example of such axioms is as follows: $\\forall x,y \\left(\\mathsf {Cat}(x) \\wedge \\mathsf {partOf}(y,x) \\rightarrow \\mathsf {Tail}\\left(y\\right) \\vee \\mathsf {Head}\\left(y\\right) ... \\vee \\mathsf {Eye}\\left(y\\right)\\right)$ to indicate that if an object $y$ is classified as part of $x$ and $x$ is a cat, than $y$ can be only an object that we know is a part of the whole cat.",
"Mereological constraints were enforced exploiting the KB developed in [3], to which the reader is referred for further information."
],
[
"Faster-LTN",
"The overall architecture, illustrated in Figure REF , is an end-to-end system connecting a convolutional object detector with an LTN.",
"Specifically, the classifier head is modified, by removing the softmax activation, and feeding the output to the LTN.",
"At training time, a GT is constructed as defined in Section REF .",
"The LTN is implemented by defining three additional layers: Predicate, Literal and Clause layers.",
"For each class $c$ , the corresponding literal computes the truth value of all positive (i.e., belonging to class $c$ ) and negative (i.e., not belonging to class $c$ ) examples.",
"The Clause layer aggregates all literals for a given class, using the selected aggregation function.",
"Additionally, it is possible to define clauses (e.g., for part-of predicates) that take as input multiple literals.",
"For the sake of simplicity, in Figure REF only $\\mathcal {T}_{expl}$ is shown.",
"The final loss of the LTN is given by summing $L_{LTN}$ with the regression loss, as for the RPN layer."
],
[
"Training",
"In order to deal with memory constraints, a partial $\\mathcal {T}_{expl}$ needs to be rebuilt with every batch of examples.",
"In the original implementation [3], the LTN was trained on the predictions of a pre-trained object detector, allowing for a relatively large batch size.",
"In our setting, the LTN is trained on all proposals extracted by the RPN, and a separate batch is constructed for each image, taking into account background as well as foreground examples.",
"It is worth noticing that one-vs-all classification amplifies the data imbalance between positive and negative examples for each class, even when the training batch consists of an equal number of objects and background proposals."
],
[
"Aggregation function",
"The chosen aggregator function is the log-product, which was shown in [14] to scale well with the number of inputs, and which formulation is equivalent to the cross-entropy loss.",
"However, in our case, this choice does not weight adequately the contribution of positive examples, given the high level of class imbalance.",
"Hence, inspired by [17], we introduce the focal log-product aggregation defined as: $L_{LTN} = -\\sum _{j = 0}^{K}\\sum _{i = 0}^{N} \\alpha _c (1 - x_{i,j})^{\\gamma }log\\left(x_{i,j}\\right) $ where $\\alpha _c$ is a class-dependent weight factor, $\\gamma $ enhances the contribution of literals with low truth value (i.e., misclassified examples), $x_i$ is the literal of the $i$ -th ROI in the $j$ -th class, $K$ is the number of classes and $N$ is the batch size.",
"To set the value of $\\alpha _c$ , we simply observe that for each training batch and each class $c$ , the number of negative examples is given by the number of background examples (which is fixed during training), plus the positive examples that belong to other classes.",
"Hence, we set $\\alpha _{c} = \\frac{1 - \\beta }{1 - \\beta ^{pos_c}}$ and $\\alpha _c = \\frac{1 - \\beta }{1 - \\beta ^{neg_c}}$ , for positive and negative examples respectively.",
"Let $p(c)$ be the fraction of bounding boxes in the training set belonging to class $c$ .",
"Then, for a given batch the percentage of positive and negative examples becomes ${pos_c = \\frac{N}{2} p\\left( c\\right)}$ and $ neg_c = \\frac{N}{2} + \\frac{N}{2} \\left(1 - p\\left( c\\right)\\right) $ , respectively."
],
[
"Dataset",
"Experiments were performed on the PASCAL VOC 2010 [18] and PASCAL PART [19] benchmarks.",
"For the latter, we selected 20 classes for whole objects and 39 classes for parts.",
"All experiments are conducted on the trainval partition with 80:20 split.",
"For PASCAL PART (10K images), we further experiment reducing the training set by 50% by random selection: the number of images is thus roughly 8K for PASCAL PART and 4K for PASCAL PART REDUCED.",
"The architecture of the Faster R-CNN follows quite closely the original implementation [15].",
"The backbone architecture was ResNet50 pretrained on ImageNet; the anchor scales were set to $128^{2}$ ,$256^{2}$ , and $512^{2}$ , with aspect ratios of 1:1, 1:2,and 2:1.",
"The number of RPN proposals is set to 300.",
"For training the classifier head, 128 bounding boxes were randomly selected, with a ratio of 32:96 positive and negative examples, for the PASCAL VOC dataset; for PASCAL PART, 32 bounding boxes with 16:16 ratio.",
"The network was trained for 100 epochs with the Adam optimizer; the learning rate was set to $10^{-5}$ for the first 60 epochs, and then reduced to $10^{-6}$ .",
"Regularization techniques included data augmentation (horizontal flip) and weight decay (with rate $5 \\times 10^{-4}$ )."
],
[
"Faster-LTN",
"The architecture of Faster-LTN was the same as Faster R-CNN, except for the classifier head in which the LTN was embedded.",
"Each predicate is defined by Eq.",
"REF , with $k=6$ kernels.",
"Łukasiewicz's t–norm was chosen to encode the literals' disjunction, and the focal log-product, with $\\gamma = 2$ , was selected as the aggregation function.",
"$\\mathcal {T}_{prior}$ included mutual exclusion constraints for PASCAL VOC, and mutual exclusion and mereological constraints for PASCAL PART experiments.",
"In the latter case, the LTN was expanded to include part-of predicates, but for the sake of comparison with Faster R-CNN, only the object detection performance was evaluated.",
"On the PASCAL VOC dataset, different experiments were performed with variations of the focal log-product aggregation function: with and without class weights $\\alpha $ , and with and without adding an additional predicate bg to represent the background class.",
"The experiments are denoted as Faster-LTN, Faster-LTN $\\alpha $ , Faster-LTN bg, and Faster-LTN bg$+\\alpha $ .",
"Experiments on PASCAL-PART were performed with the Faster-LTN bg configuration.",
"All networks were trained for 150 epochs using the Adam optimizer, with weight decay (decay rate $5 \\times 10^{-4}$ ), random horizontal flip and L2 regularization ($\\lambda $ is set to $5 \\times 10^{-4}$ .",
").",
"The learning rate was set to $10^{-5}$ for the first 60 epochs, and then reduced to $10^{-6}$ .",
"All experiments were performed on the HPC@Polito cluster, equipped with V100 NVIDIA GPU.",
"The performance metric was the mean Average Precision (MAP) implemented as in the PASCAL VOC challenge 2010 [20].",
"Table: Results of the Faster R-CNN (FR-CNN), Faster R-CNN with focal loss (FR-CNN FL), and Faster-LTN (F-LTN) on PASCAL VOC."
],
[
"Results",
"Experiments on Pascal VOC, summarized in Table REF , show that Faster LTN achieved competitive and even superior results compared to the original Faster R-CNN architecture, with the mAP increasing from 62.6 to 73.8.",
"In this version of the LTN, the only axiomatic constraint was the one imposing mutual exclusivity (see Eq.",
"REF ).",
"We observed comparable performance when including the background as an additional class (mAP from 73.8 to 73.4); on the other hand, weighting positive and negative samples according to their frequency did not improve results (mAP from 73.8 to 72.1).",
"Qualitatively, we observed that Faster LTN was able to detect more objects than Faster R-CNN.",
"Given that log-product aggregation is mathematically equivalent to the cross-entropy loss, and the backbone is the same, this difference can be attributed to the different classification setting ($K$ one-vs-all classifiers instead of a single multi-class classifier) or the use of the focal loss [17].",
"However, when changing the loss of the Faster R-CNN classifier head to the focal loss, performance dropped from 62.6 to 59.2.",
"Hence, we attribute Faster-LTN performance to the greater flexibility offered by a more complex classifier head, with higher number of parameters.",
"In fairness, Faster LTN took a few more epochs to reach convergence.",
"In the PASCAL PART experiments, shown in Table REF , additional mereological axioms were included in $\\mathcal {T}_{prior}$ .",
"This allowed to increase performance from 35.1 to 41.2; when reducing the training set size by half, the performance gap was maintained (28.5 to 32.8).",
"The comparable quality of the learned features is further supported by the t-SNE embeddings of the extracted features, which are shown in Figure REF .",
"Table: Comparison of Faster R-CNN and Faster-LTN (including mereological constraints) on the PASCAL PART dataset.Figure: Comparison of the t-SNE embeddings of the features extracted for the whole objects classes in the test test.",
"Features extracted from Faster R-CNN (left) and Faster-LTN with axiomatic constraints (right)."
],
[
"Conclusion and future works",
"The availability of large scale, high quality, labelled datasets is one of the major hurdles in the application of deep learning.",
"A tighter integration between perception and reasoning, which is enabled by emerging Neural-Symbolic techniques, allows to complement learning by examples with the integration of axiomatic background knowledge.",
"In this paper, we introduced the Faster-LTN architecture, an end-to-end object detector composed by a convolutional backbone and RPN (based on the Faster R-CNN architecture) and a LTN module.",
"The detector is trained end-to-end by maximizing the satisfiability of a grounded theory combining clauses derived from labelled examples with axiomatic constraints.",
"Our goal was to establish the feasibility of this approach, and indeed the results, albeit preliminary, prove that Faster-LTN is competitive or can even outperform the baseline Faster R-CNN.",
"However, the scalability of this approach to larger training sets and other object detector (e.g., single-stage detectors) should be further investigated.",
"Through the Faster-LTN model, available at https://gitlab.com/grains2/Faster-LTN, we aim to provide a baseline architecture on which new experiments and applications can be built.",
"Future work will investigate how high-level symbolic constraints can shape the learning process, increasing robustness in the presence of noise and dataset bias."
],
[
"Acknolewdgement",
"The authors wish to thank Ivan Donadello for the helpful discussions.",
"Computational resources were in part provided by HPC@POLITO, a project of Academic Computing at Politecnico di Torino (http://www.hpc.polito.it)."
]
] | 2107.01877 |
[
[
"Latent structure blockmodels for Bayesian spectral graph clustering"
],
[
"Abstract Spectral embedding of network adjacency matrices often produces node representations living approximately around low-dimensional submanifold structures.",
"In particular, hidden substructure is expected to arise when the graph is generated from a latent position model.",
"Furthermore, the presence of communities within the network might generate community-specific submanifold structures in the embedding, but this is not explicitly accounted for in most statistical models for networks.",
"In this article, a class of models called latent structure block models (LSBM) is proposed to address such scenarios, allowing for graph clustering when community-specific one dimensional manifold structure is present.",
"LSBMs focus on a specific class of latent space model, the random dot product graph (RDPG), and assign a latent submanifold to the latent positions of each community.",
"A Bayesian model for the embeddings arising from LSBMs is discussed, and shown to have a good performance on simulated and real world network data.",
"The model is able to correctly recover the underlying communities living in a one-dimensional manifold, even when the parametric form of the underlying curves is unknown, achieving remarkable results on a variety of real data."
],
[
"Introduction",
"Network-valued data are commonly observed in many real world applications.",
"They are typically represented by graph adjacency matrices, consisting of binary indicators summarising which nodes are connected.",
"Spectral embedding [19] is often the first preprocessing step in the analysis of graph adjacency matrices: the nodes are embedded onto a low-dimensional space via eigendecompositions or singular value decompositions.",
"This work discusses a network model for clustering the nodes of the graph in the embedding space, when community-specific substructure is present.",
"A popular statistical model for graph adjacency matrices is the latent position model [10].",
"Each node is assumed to have a low-dimensional vector representation $x_i\\in \\mathbb {R}^d$ , such that the probability of connection between two nodes is $\\kappa (x_i,x_j)$ , for a given kernel function $\\kappa :\\mathbb {R}^d\\times \\mathbb {R}^d\\rightarrow [0,1]$ .",
"Examples of commonly used kernel functions are the inner product $\\kappa (x_i,x_j)=x_i^\\intercal x_j$ [39], [2], the logistic distance link $\\kappa (x_i,x_j)=[1+\\exp (-\\Vert x_i-x_j\\Vert )]^{-1}$ [10], [31], where $\\Vert \\cdot \\Vert $ is a norm, or the Bernoulli-Poisson link $\\kappa (x_i,x_j)=1-\\exp (-x_i^\\intercal x_j)$ [35], where the latent positions are constrained to lie in $\\mathbb {R}_+^d$ .",
"[29] notes that spectral embeddings of adjacency matrices generated from LPMs produce node representations living near low-dimensional submanifold structures.",
"Therefore, for subsequent inferential tasks, for example clustering, it is necessary to employ methods which take into account this latent manifold structure.",
"If the kernel function of the LPM is the inner product, spectral embedding provides consistent estimates of the latent positions, up to orthogonal rotations [3], [30].",
"This model is usually referred to as the random dot product graph [39], [2].",
"In this work, RDPGs where the latent positions are constrained to lie on a one-dimensional submanifold of the latent space are considered.",
"Such a model is known in the literature as a latent structure model [4].",
"An example is displayed in Figure REF , which shows the latent positions, and corresponding estimates obtained using adjacency spectral embedding (ASE; see Section for details), of a simulated graph with 1000 nodes, where the latent positions are assumed to be drawn from the Hardy-Weinberg curve $f(\\theta )=\\lbrace \\theta ^2,(1-\\theta )^2,2\\theta (1-\\theta )\\rbrace $ , with $\\theta \\sim \\text{Uniform}[0,1]$ [4], [36].",
"In addition, nodes are commonly divided into inferred groups, or communities, such that it can be plausible to assume that the latent positions live on community-specific submanifolds.",
"The objective of community detection on graphs is to recover such latent groups.",
"For example, in computer networks, it is common to observe community-specific curves in the latent space.",
"Figure REF displays an example, representing the two-dimensional latent positions, estimated using directed adjacency spectral embedding (DASE; see Section ), of 278 computers in a network of HTTP/HTTPS connections from machines hosted in two computer laboratories at Imperial College London (further details are given in Section REF ).",
"The latent positions corresponding to the two communities appear to be distributed around two quadratic curves with different parameters.",
"The best fitting quadratic curves in $\\mathbb {R}^2$ passing through the origin are also displayed, representing an estimate of the underlying community-specific submanifolds.",
"Motivated by the practical application to computer networks, this article develops inferential methods for LSMs allowing for latent community structure.",
"Nodes belonging to different communities are assumed to correspond to community-specific submanifolds.",
"This leads to the definition of the latent structure blockmodel (LSBM), which admits communities with manifold structure in the latent space.",
"The main contribution is a Bayesian probability model for LSBM embeddings, which enables community detection in RDPG models with underlying community-specific manifold structure.",
"Figure: Fitted quadratic curves for the two-dimensional (directed) spectral embedding of a computer network graph with two communities.Naturally, for clustering purposes, procedures that exploit the structure of the submanifold are expected to perform more effectively than standard alternative techniques, such as spectral clustering with $k$ -means [37], [27] or Gaussian mixture models [30].",
"Exploiting the underlying structure is often unpractical, since the submanifold is usually unknown.",
"Within the context of LSMs, [36] propose to use an extension of Isomap in the presence of noise to learn a one-dimensional unknown manifold from the estimated latent positions.",
"In this work, the problem of submanifold estimation is addressed using flexible functional models, namely Gaussian processes, in the embedding space.",
"Applications on simulated and real world computer network data show that the proposed methodology is able to successfully recover the underlying communities even in complex cases with substantial overlap between the groups.",
"Geometries arising from network models have been extensively studied in the literature [1], [20], [34].",
"The approach in this paper is also related to the findings of [24], where a semiparametric Gaussian mixture model with quadratic structure is used to model the embeddings arising from a subset of the neurons in the Drosophila connectome.",
"In particular, the Kenyon Cell neurons are not well represented by a stochastic blockmodel, which is otherwise appropriate for other neuron types.",
"Consequently, those neurons are modelled via a continuous curve in the latent space, estimated via semiparametric MLE [16], [18].",
"However, the real world examples presented in this article in Section require community-specific curves in the latent space for all the communities, not only for a subset of the nodes.",
"Furthermore, estimation can be conducted within the Bayesian paradigm, which conveniently allows marginalisation of the model parameters.",
"This article is organised as follows: Section formally introduces RDPGs and spectral embedding methods.",
"Sections and present the main contributions of this work: the latent structure blockmodel (LSBM), and Bayesian inference for the communities of LSBM embeddings.",
"The method is applied to simulated and real world networks in Section , and the article is then concluded with a discussion in Section ."
],
[
"Random dot product graphs and latent structure models",
"Mathematically, a network $\\mathbb {G}=(V,E)$ is represented by a set $V=\\lbrace 1,2,\\dots ,n\\rbrace $ of nodes, and an edge set $E$ , such that $(i,j)\\in E$ if $i$ forms a link with $j$ .",
"If $(i,j)\\in E \\Rightarrow (j,i)\\in E$ , the network is undirected, otherwise it is directed.",
"If the node set is partitioned into two sets $V_1$ and $V_2$ , $V_1\\cap V_2=\\varnothing $ , such that $E\\subset V_1\\times V_2$ , the graph is bipartite.",
"A network can be summarised by the adjacency matrix $\\mathbf {A}\\in \\lbrace 0,1\\rbrace ^{n\\times n}$ , such that $A_{ij}=1_E\\lbrace (i,j)\\rbrace $ , where $1_\\cdot \\lbrace \\cdot \\rbrace $ is an indicator function.",
"Latent position models [10] for undirected graphs postulate that the edges are sampled independently, with $\\mathbb {P}(A_{ij}=1\\mid x_i,x_j)=\\kappa (x_i,x_j).$ A special case of LPMs is the random dot product graph [39], defined below.",
"Definition 1 (Random dot product graph – RDPG, [39]) For an integer $d>0$ , let $F$ be an inner product distribution on $\\mathcal {X}\\subset \\mathbb {R}^d$ , such that $\\forall \\ x,x^\\prime \\in \\mathcal {X}$ , $0\\le x^\\intercal x^\\prime \\le 1$ .",
"Furthermore, let $\\mathbf {X}=(x_1,\\dots ,x_n)^\\intercal \\in \\mathcal {X}^n$ and $\\mathbf {A}\\in \\lbrace 0,1\\rbrace ^{n\\times n}$ be symmetric.",
"Then $(\\mathbf {A},\\mathbf {X})\\sim \\text{RDPG}_d(F^n)$ if $x_1,\\dots ,x_n &\\overset{iid}{\\sim }F\\\\A_{ij}\\vert \\mathbf {X} &\\sim \\textnormal {Bernoulli}(x_i^\\intercal x_j),\\quad 1\\le i<j\\le n. $ Given a realisation of the adjacency matrix $\\mathbf {A}$ , the first inferential objective is to estimate the latent positions $x_1,\\dots , x_n$ .",
"In RDPGs, the latent positions are inherently unidentifiable; in particular, multiplying the latent positions by any orthogonal matrix $\\mathbf {Q}\\in \\mathbb {O}(d)$ , the orthogonal group with signature $d$ , leaves the inner product (REF ) unchanged: $x^\\intercal x^\\prime =(\\mathbf {Q}x)^\\intercal \\mathbf {Q}x^\\prime $ .",
"Therefore, the latent positions can only be estimated up to orthogonal rotations.",
"Under such a restriction, the latent positions are consistently estimated via spectral embedding methods.",
"In particular, the adjacency spectral embedding (ASE), defined below, has convenient asymptotic properties.",
"Definition 2 (ASE – Adjacency spectral embedding) For a given integer $d\\in \\lbrace 1,\\ldots ,n\\rbrace $ and a binary symmetric adjacency matrix $\\mathbf {A}\\in \\lbrace 0,1\\rbrace ^{n\\times n}$ , the $d$ -dimensional adjacency spectral embedding (ASE) $\\hat{\\mathbf {X}}=[\\hat{x}_{1},\\dots ,\\hat{x}_{n}]^\\intercal $ of $\\mathbf {A}$ is $\\hat{\\mathbf {X}} = \\mathbf {\\Gamma }\\mathbf {\\Lambda }^{1/2}\\in \\mathbb {R}^{n\\times d},$ where $\\mathbf {\\Lambda }$ is a $d\\times d$ diagonal matrix containing the absolute values of the $d$ largest eigenvalues in magnitude, in decreasing order, and $\\mathbf {\\Gamma }$ is a $n\\times d$ matrix containing the corresponding orthonormal eigenvectors.",
"Alternatively, the Laplacian spectral embedding (LSE) is also frequently used, and considers a spectral decomposition of the modified Laplacian $\\mathbf {D}^{-1/2}\\mathbf {A}\\mathbf {D}^{-1/2}$ instead, where $\\mathbf {D}$ is the degree matrix.",
"In this work, the focus will be mainly on ASE.",
"Using ASE, the latent positions are consistently estimated up to orthogonal rotations, and a central limit theorem is available.",
"Theorem 1 (ASE central limit theorem, [3], [30], [2]) Let $(\\mathbf {A}^{(n)}, \\mathbf {X}^{(n)}) \\sim \\text{RDPG}_d(F^n), n=1,2,\\dots $ , be a sequence of adjacency matrices and corresponding latent positions, and let $\\hat{\\mathbf {X}}^{(n)}$ be the $d$ -dimensional ASE of $\\mathbf {A}^{(n)}$ .",
"For an integer $m>0$ , and for the sequences of points $x_1,\\dots ,x_m\\in \\mathcal {X}$ and $u_1,\\dots ,u_m\\in \\mathbb {R}^d$ , there exists a sequence of orthogonal matrices $\\mathbf {Q}_1,\\mathbf {Q}_2,\\ldots \\in \\mathbb {O}(d)$ such that: $\\lim _{n\\rightarrow \\infty }\\mathbb {P}\\left\\lbrace \\bigcap _{i=1}^m \\sqrt{n}\\left(\\mathbf {Q}_n\\hat{x}_i^{(n)} - x_i^{(n)}\\right) \\le u_i\\ \\Bigg \\vert \\ x_1^{(n)}=x_1,\\dots , x_m^{(n)}=x_m \\right\\rbrace = \\prod _{i=1}^m \\Phi \\lbrace u_i,\\mathbf {\\Sigma }(x_i)\\rbrace ,$ where $\\Phi \\lbrace \\cdot \\rbrace $ is the CDF of a $d$ -dimensional normal distribution, and $\\mathbf {\\Sigma }(\\cdot )$ is a covariance matrix which depends on the true value of the latent position.",
"Therefore, informally, it could be assumed that, for $n$ large, $\\mathbf {Q}\\hat{x}_i\\sim \\mathbb {N}_d\\left\\lbrace x_i,\\mathbf {\\Sigma }(x_i)\\right\\rbrace $ for some $\\mathbf {Q}\\in \\mathbb {O}(d)$ , where $\\mathbb {N}_d\\lbrace \\cdot \\rbrace $ is the $d$ -dimensional multivariate normal distribution.",
"This article is mainly concerned with community detection in RDPGs when the latent positions are constrained to lie on community-specific one-dimensional submanifolds $\\mathcal {S}_k\\subset \\mathbb {R}^d$ , $k=1,\\dots ,K$ .",
"The proposed methodology builds up upon latent structure models [4], a special subset of RDPGs.",
"In LSMs, it is assumed that the latent positions of the nodes are determined by a univariate underlying distribution $G$ on $[0,1]$ , inducing the distribution $F$ on a structural support univariate submanifold $\\mathcal {S}\\subset \\mathbb {R}^d$ , such that: $x_i \\overset{iid}{\\sim } F = f(\\theta ),\\ i=1,\\dots ,n,$ where $\\theta \\sim G$ and $f:[0,1]\\rightarrow \\mathcal {S}$ is a function mapping draws from $\\theta $ to $\\mathcal {S}$ .",
"The inverse function $f^{-1}:\\mathcal {S}\\rightarrow [0,1]$ could be interpreted as the arc-length parametrisation of $\\mathcal {S}$ .",
"In simple terms, each node is assigned a draw $\\theta _i$ from the underlying distribution $G$ , representing how far along the submanifold $\\mathcal {S}$ the corresponding latent position lies, such that $x_i=f(\\theta _i)$ .",
"If the graph is directed or bipartite, each node is assigned two latent positions $x_i$ and $x_i^\\prime $ , and the random dot product graph model takes the form $\\mathbb {P}(A_{ij}=1\\mid x_i,x_j^\\prime )=x_i^\\intercal x_j^\\prime $ .",
"In this case, the singular value decomposition of $\\mathbf {A}$ can be used over the eigendecomposition.",
"Definition 3 (DASE – Directed adjacency spectral embedding) For a given integer $d\\in \\lbrace 1,\\ldots ,n\\rbrace $ and a binary adjacency matrix $\\mathbf {A}\\in \\lbrace 0,1\\rbrace ^{n\\times n}$ , the $d$ -dimensional directed adjacency spectral embeddings (DASE) $\\hat{\\mathbf {X}}=[\\hat{x}_{1},\\dots ,\\hat{x}_{n}]^\\intercal $ and $\\hat{\\mathbf {X}}^\\prime =[\\hat{x}_{1}^\\prime ,\\dots ,\\hat{x}_{n}^\\prime ]^\\intercal $ of $\\mathbf {A}$ are $\\hat{\\mathbf {X}} = \\mathbf {U}\\mathbf {S}^{1/2} \\in \\mathbb {R}^{n\\times d},\\ \\hat{\\mathbf {X}}^\\prime = \\mathbf {V}\\mathbf {S}^{1/2}\\in \\mathbb {R}^{n\\times d},$ where $\\mathbf {S}$ is a $d\\times d$ diagonal matrix containing the $d$ largest singular values, in decreasing order, and $\\mathbf {U}$ and $\\mathbf {V}$ are a $n\\times d$ matrix containing the corresponding left- and right-singular vectors.",
"Clearly, DASE could be also used on rectangular adjacency matrices arising from bipartite graphs.",
"Furthermore, [14] prove the equivalent of Theorem REF for DASE, demonstrating that DASE also provides a consistent procedure for estimation of the latent positions in directed or bipartite graphs.",
"Analogous constructions to undirected LSMs could be posited for directed or bipartite models, where the $x_i$ 's and $x_i^\\prime $ 's lie on univariate structural support submanifolds."
],
[
"Latent structure blockmodels",
"This section extends LSM, explicitly allowing for community structure.",
"In particular, it is assumed that each node is assigned a latent community membership $z_i\\in \\lbrace 1,\\dots ,K\\rbrace ,\\ i=1,\\dots ,n$ , and each community is associated with a different one-dimensional structural support submanifold $\\mathcal {S}_k\\subset \\mathbb {R}^d,\\ k=1,\\dots ,K$ .",
"Therefore, it is assumed that $F=\\sum _{k=1}^K \\eta _kF_k$ is a mixture distribution with components $F_1,\\dots ,F_K$ supported on $\\mathcal {S}_1,\\dots ,\\mathcal {S}_K$ , with weights $\\eta _1,\\dots ,\\eta _K$ such that for each $k$ , $\\eta _k\\ge 0$ and $\\sum _{k=1}^K\\eta _k=1$ .",
"Assuming community allocations $z=(z_1,\\dots ,z_n)$ , the latent positions are obtained as $x_i\\mid z_i \\sim F_{z_i} = f_{z_i}(\\theta ),\\ i=1,\\dots ,n,$ where $\\theta $ is a latent random variable with distribution $G$ with support $\\mathcal {G}\\subset \\mathbb {R}$ , and $f_k=(f_{k,1},\\dots ,f_{k,d}):\\mathcal {G}\\rightarrow \\mathcal {S}_k$ are vector-valued functions mapping the latent draw from the distribution $G$ to $\\mathcal {S}_k$ .",
"The resulting model will be referred to as the latent structure blockmodel (LSBM).",
"Note that, for generality, the support of the underlying distribution $G$ is assumed here to be $\\mathcal {G}\\subset \\mathbb {R}$ .",
"Furthermore, $G$ is common to all the nodes, and the pair $(\\theta _i,z_i)$ , where $\\theta _i\\sim G$ , uniquely determines the latent position $x_i$ through $f_{z_i}$ , such that $x_i=f_{z_i}(\\theta _i).$ Note that, under the framework described above, the submanifolds $\\mathcal {S}_1,\\dots ,\\mathcal {S}_K$ are one-dimensional, corresponding to curves, but the LSBM could be extended to higher-dimensional settings of the underlying subspaces, postulating underlying draws from a multivariate distribution $G$ supported on $\\mathcal {G}\\subseteq \\mathbb {R}^p$ with $1\\le p<d$ .",
"Since the latent structure blockmodel is a special case of the random dot product graph, the LSBM latent positions are also estimated consistently via ASE, up to orthogonal rotations, conditional on knowing the functions $f_k(\\cdot )$ .",
"Therefore, approximately: $\\mathbf {Q}\\hat{x}_i\\sim \\mathbb {N}\\lbrace f_{z_i}(\\theta _i),\\mathbf {\\Sigma }(x_i)\\rbrace .$ Special cases of the LSBM include the stochastic blockmodel (SBM) and its degree-corrected extension (DCSBM), and other more complex latent structure models with clustering structure, as demonstrated in the following examples.",
"Example 1 (Stochastic blockmodel, SBM) In an SBM [11], the edge probability is determined by the community allocation of the nodes: $A_{ij}\\sim \\textnormal {Bernoulli}(B_{z_i z_j})$ , where $\\mathbf {B}\\in [0,1]^{K\\times K}$ is a matrix of probabilities for connections between communities.",
"An SBM characterised by a non-negative definite matrix $\\mathbf {B}$ of rank $d$ can be expressed as an LSBM, assigning community-specific latent positions $x_i=\\nu _{z_i}\\in \\mathbb {R}^d$ , such that for each $(k,\\ell )$ , $B_{k\\ell }=\\nu _k^\\intercal \\nu _\\ell $ .",
"Therefore, $f_k(\\theta _i)=\\nu _k$ , with each $\\theta _i=1$ for identifiability.",
"It follows that $f_{k,j}(\\theta _i)=\\nu _{k,j}$ .",
"Example 2 (Degree-corrected stochastic blockmodel, DCSBM) DCSBMs [15] extend SBMs, allowing for heterogeneous degree distributions within communities.",
"The edge probability depends on the community allocation of the nodes, and degree-correction parameters $\\theta \\in \\mathbb {R}^n$ for each node, such that $A_{ij}\\sim \\textnormal {Bernoulli}(\\theta _i\\theta _jB_{z_i z_j})$ .",
"In a latent structure blockmodel interpretation, the latent positions are $x_i=\\theta _i\\nu _{z_i}\\in \\mathbb {R}^d$ .",
"Therefore, $f_k(\\theta _i)=\\theta _i\\nu _k$ , with $f_{k,j}(\\theta _i)=\\theta _i\\nu _{k,j}$ .",
"For identifiability, one could set each $\\nu _{k,1}=1$ .",
"Example 3 (Quadratic latent structure blockmodel) For an LSBM with quadratic $f_k(\\cdot )$ , it can be postulated that, conditional on a community allocation $z_i$ , $x_i=f_{z_i}(\\theta _i)=\\mathbf {\\alpha }_{z_i}\\theta _i^2+\\mathbf {\\beta }_{z_i}\\theta _i+\\mathbf {\\gamma }_{z_i}$ , with $\\mathbf {\\alpha }_k,\\mathbf {\\beta }_k,\\mathbf {\\gamma }_k\\in \\mathbb {R}^d$ .",
"Under this model: $f_{k,j}(\\theta _i)=\\alpha _{k,j}\\theta _i^2+\\beta _{k,j}\\theta +\\gamma _{k,j}$ .",
"Note that the model is not identifiable: for $v\\in \\mathbb {R}$ , then $ (\\mathbf {\\alpha }_{z_i}/v^2)(v\\theta _i)^2+(\\mathbf {\\beta }_{z_i}/v)(v\\theta _i)+\\mathbf {\\gamma }_{z_i}$ is equivalent to $f_{z_i}(\\theta _i)$ .",
"A possible solution is to fix each $\\beta _{k,1}=1$ .",
"Figure REF shows example embeddings arising from the three models described above.",
"From these plots, it is evident that taking into account the underlying structure is essential for successful community detection.",
"Figure: Scatterplots of the two-dimensional ASE of simulated graphs arising from the models in Examples , , , and true underlying latent curves (in black).",
"For each graph, n=1000n=1000 with K=2K=2 communities of equal size, and θ i ∼Uniform(0,1)\\theta _i\\sim \\text{Uniform}(0,1).",
"For (a) and (b), ν 1 =[3/4,1/4]\\mathbf {\\nu }_1=[3/4,1/4] and ν 2 =[1/4,3/4]\\mathbf {\\nu }_2=[1/4,3/4], cf.",
"Examples and .",
"For (c), α k =[0,τ k /2]\\mathbf {\\alpha }_k=[0,\\tau _k/2], β k =[1/2,(1-τ k )/2]\\mathbf {\\beta }_k=[1/2,(1-\\tau _k)/2], γ k =[0,0]\\mathbf {\\gamma }_k=[0,0], with τ 1 =2\\tau _1=2 and τ 2 =-2\\tau _2=-2, cf.",
"Example ."
],
[
"Bayesian modelling of LSBM embeddings",
"Under the LSBM, the inferential objective is to recover the community allocations $z=(z_1,\\dots ,z_n)$ given a realisation of the adjacency matrix $\\mathbf {A}$ .",
"Assuming normality of the rows of ASE for LSBMs (REF ), the inferential problem consists of making joint inference about $z$ and the corresponding latent functions $f_j=(f_{j,1},\\dots ,f_{j,d}):\\mathcal {G}\\rightarrow \\mathbb {R}^d$ .",
"The prior specification for $z$ follows the same structure used for SBMs, $z_i &\\sim \\text{Discrete}(\\mathbf {\\eta }),\\ \\mathbf {\\eta }=(\\eta _1,\\dots ,\\eta _K),\\ i=1,\\dots ,n, \\\\\\mathbf {\\eta }&\\sim \\text{Dirichlet}(\\nu /K,\\dots ,\\nu /K),$ where $\\eta _k\\ge 0\\ \\forall \\ k$ and $\\sum _{k=1}^K\\eta _k=1$ , $\\nu \\in \\mathbb {R}_+$ .",
"Following the ASE-CLT in Theorem REF , the estimated latent positions are assumed to be drawn from Gaussian distributions centred at the underlying function value.",
"Conditional on the pair $(\\theta _i,z_i)$ , the following distribution is postulated for $\\hat{x}_i$ : $\\hat{x}_i\\ \\vert \\ \\theta _i,f_{z_i},\\mathbf {\\sigma }^2_{z_i} \\sim \\mathbb {N}_d\\left\\lbrace f_{z_i}(\\theta _i),\\mathbf {\\sigma }^2_{z_i}\\mathbf {I}_d\\right\\rbrace ,\\ i=1,\\dots ,n,$ where $\\mathbf {\\sigma }^2_k=(\\sigma ^2_{k,1},\\dots ,\\sigma ^2_{k,d})\\in \\mathbb {R}_+^d$ is a vector of community-specific variances.",
"Note that, for simplicity, the components of the estimated latent positions are assumed to be independent.",
"This assumption loosely corresponds to the $k$ -means clustering approach, which has been successfully deployed in spectral graph clustering under the stochastic blockmodel [27].",
"Here, the same idea is extended to a functional setting.",
"Furthermore, for tractability (REF ) assumes the variance of $\\hat{x}_i$ does not depend on $x_i$ , but only on the community allocation $z_i$ .",
"Alternatively, an unknown functional form could be given to the components of the community-specific covariance matrix.",
"For a full Bayesian model specification, prior distributions are required for the latent functions and the unknown variances.",
"The most popular prior for unknown functions is the Gaussian process (GP) described, for example, in [26].",
"Here, for each community $k$ , the $j$ th dimension of the true latent positions are assumed to lie on a one-dimensional manifold described by a function $f_{k,j}$ with a hierarchical GP-IG prior, with an inverse gamma (IG) prior on the variance: $f_{k,j} \\vert \\sigma ^2_{k,j} &\\sim \\text{GP}(0,\\sigma ^2_{k,j}\\xi _{k,j}),\\ k=1,\\dots ,K,\\ j=1,\\dots ,d, \\\\\\sigma ^2_{k,j} &\\sim \\text{IG}(a_0,b_0),\\ k=1,\\dots ,K,\\ j=1,\\dots ,d,$ where $\\xi _{k,j}(\\cdot ,\\cdot )$ is a positive semi-definite kernel function and $a_0,b_0\\in \\mathbb {R}_+$ .",
"Note that the terminology kernel is used in the literature for both the GP covariance function $\\xi _{k,j}(\\cdot ,\\cdot )$ and the function $\\kappa (\\cdot ,\\cdot )$ used in LPMs (cf.",
"Section ), but their meaning is fundamentally different.",
"In particular, $\\kappa :\\mathbb {R}^d\\times \\mathbb {R}^d\\rightarrow [0,1]$ is a component of the graph generating process, and assumed here to be the inner product, corresponding to RDPGs.",
"On the other hand, $\\xi _{k,j}:\\mathbb {R}\\times \\mathbb {R}\\rightarrow \\mathbb {R}$ is the scaled covariance function of the GP prior on the unknown function $f_{k,j}$ , which is used for modelling the observed graph embeddings, or equivalently the embedding generating process.",
"There are no restrictions on the possible forms of $\\xi _{k,j}$ , except positive semi-definiteness.",
"Overall, the approach is similar to the overlapping mixture of Gaussian processes method [17].",
"The class of models that can be expressed in the form (REF ) is vast, and includes, for example, polynomial regression and splines, under the conjugate normal-inverse-gamma prior for the regression coefficients.",
"For example, consider any function that can be expressed in the form $f_{z_i,j}(\\theta _i)= \\mathbf {\\phi }_{z_i,j}(\\theta _i)^\\intercal w_{z_i,j}$ for some community-specific basis functions $\\mathbf {\\phi }_{k,j}:\\mathbb {R}\\rightarrow \\mathbb {R}^{q_{k,j}}, q_{k,j}\\in \\mathbb {Z}_+,$ and corresponding coefficients $w_{k,j}\\in \\mathbb {R}^{q_{k,j}}$ .",
"If the coefficients are given a normal-inverse-gamma prior $(w_{k,j},\\sigma ^2_{k,j})\\sim \\text{NIG}(0,\\mathbf {\\Delta }_{k,j},a_0,b_0)=\\mathbb {N}_{q_{k,j}}(0,\\sigma ^2_{k,j}\\mathbf {\\Delta }_{k,j})\\text{IG}(a_0,b_0),$ where $\\mathbf {\\Delta }_{k,j}\\in \\mathbb {R}^{q_{k,j}\\times q_{k,j}}$ is a positive definite matrix, then $f_{k,j}$ takes the form (REF ), with the kernel function $\\xi _{k,j}(\\theta ,\\theta ^\\prime ) = \\mathbf {\\phi }_{k,j}(\\theta )^\\intercal \\mathbf {\\Delta }_{k,j}\\mathbf {\\phi }_{k,j}(\\theta ^\\prime ).",
"$ Considering the examples in Section , the SBM (Example REF ) corresponds to $\\xi _{k,j}(\\theta ,\\theta ^\\prime )=\\Delta _{k,j},\\ \\Delta _{k,j}\\in \\mathbb {R}_+$ , whereas the DCSBM (Example REF ) corresponds to $\\xi _{k,j}(\\theta ,\\theta ^\\prime )=\\theta \\theta ^\\prime \\Delta _{k,j},\\ \\Delta _{k,j}\\in \\mathbb {R}_+$ .",
"For the quadratic LSBM (Example REF ), $\\xi _{k,j}(\\theta ,\\theta ^\\prime )=(1,\\theta ,\\theta ^2)\\mathbf {\\Delta }_{k,j}(1,\\theta ^\\prime ,\\theta ^{\\prime 2})^\\intercal $ for a positive definite $\\mathbf {\\Delta }_{k,j}\\in \\mathbb {R}^{3\\times 3}$ .",
"The LSBM specification is completed with a prior for each $\\theta _i$ value, which specifies the unobserved location of the latent position $x_i$ along each submanifold curve; for $\\mu _\\theta \\in \\mathbb {R}$ , $\\sigma ^2_\\theta \\in \\mathbb {R}_+$ , $\\theta _i \\sim \\mathbb {N}(\\mu _\\theta ,\\sigma ^2_\\theta ),\\ i=1,\\dots ,n. $"
],
[
"Posterior and marginal distributions",
"The posterior distribution for $(f_{k,j},\\sigma ^2_{k,j})$ has the same GP-IG structure as (REF ), with updated parameters: $f_{k,j} \\vert \\sigma ^2_{k,j}, z, \\mathbf {\\theta }, \\hat{\\mathbf {X}} &\\sim \\text{GP}(\\mu _{k,j}^\\star ,\\sigma ^2_{k,j}\\xi _{k,j}^\\star ),\\ k=1,\\dots ,K,\\ j=1,\\dots ,d, \\\\\\sigma ^2_{k,j} \\vert z, \\mathbf {\\theta }, \\hat{\\mathbf {X}} &\\sim \\text{Inv-Gamma}(a_k,b_{k,j}),\\ k=1,\\dots ,K,\\ j=1,\\dots ,d.$ The parameters are updated as follows: $\\mu _{k,j}^\\star (\\theta ) = \\mathbf {\\xi }_{k,j}(\\theta , \\mathbf {\\theta }_k^\\star )\\lbrace \\mathbf {\\Xi }_{k,j}(\\mathbf {\\theta }_k^\\star , \\mathbf {\\theta }_k^\\star ) + \\mathbf {I}_{n_k}\\rbrace ^{-1}\\hat{\\mathbf {X}}_{k,j}, \\\\\\xi _{k,j}^\\star (\\theta ,\\theta ^\\prime ) = \\xi _{k,j}(\\theta ,\\theta ^\\prime ) - \\mathbf {\\xi }_{k,j}(\\theta ,\\mathbf {\\theta }_k^\\star )\\lbrace \\mathbf {\\Xi }_{k,j}(\\mathbf {\\theta }_k^\\star ,\\mathbf {\\theta }_k^\\star )+\\mathbf {I}_{n_k}\\rbrace ^{-1} \\mathbf {\\xi }_{k,j}(\\mathbf {\\theta }_k^\\star ,\\theta ^\\prime ), \\\\a_k = a_0 + n_k/2,\\ b_{k,j} = b_0+\\hat{\\mathbf {X}}_{k,j}^\\intercal \\lbrace \\mathbf {\\Xi }_{k,j}(\\mathbf {\\theta }_k^\\star ,\\mathbf {\\theta }_k^\\star )+\\mathbf {I}_{n_k}\\rbrace ^{-1}\\hat{\\mathbf {X}}_{k,j}/2,$ where $n_k=\\sum _{i=1}^n1_k\\lbrace z_i\\rbrace $ , $\\hat{\\mathbf {X}}_{k,j}\\in \\mathbb {R}^{n_k}$ is the subset of values of $\\hat{\\mathbf {X}}_j$ for which $z_i=k$ , and $\\mathbf {\\theta }_k^\\star \\in \\mathbb {R}^{n_k}$ is the vector $\\mathbf {\\theta }$ , restricted to the entries such that $z_i=k$ .",
"Furthermore, $\\mathbf {\\xi }_{k,j}$ is a vector-valued extension of $\\xi _{k,j}$ , such that $[\\mathbf {\\xi }_{k,j}(\\theta ,\\mathbf {\\theta }^\\prime )]_\\ell =\\xi _{k,j}(\\theta ,\\theta _\\ell ^\\prime )$ .",
"The structure of the GP-IG yields an analytic expression for the posterior predictive distribution for a new observation $x^\\ast =(x^\\ast _1,\\dots ,x^\\ast _d)$ assigned to community $z^\\ast $ , ${\\hat{x}}^\\ast _j\\vert z^\\ast ,z,\\mathbf {\\theta },\\theta ^\\ast ,\\hat{\\mathbf {X}} \\sim t_{2a_{z^\\ast }}\\left(\\mu _{z^\\ast ,j}^{\\star }(\\theta ^\\ast ),\\frac{b_{z^\\ast ,j}}{a_{z^\\ast }}\\left\\lbrace 1+\\xi _{z^\\ast ,j}^{\\star }(\\theta ^\\ast , \\theta ^\\ast )\\right\\rbrace \\right), $ where $t_\\nu (\\mu ,\\sigma )$ denotes a Student's $t$ distribution with $v$ degrees of freedom, mean $\\mu $ and scale parameter $\\sigma $ .",
"Furthermore, the prior probabilities $\\mathbf {\\eta }$ for the community assignments can be integrated out, obtaining $p(z) = \\frac{\\Gamma (\\nu )\\prod _{k=1}^K \\Gamma (n_k+\\nu /K)}{\\Gamma (\\nu /K)^K\\Gamma (n+\\nu )}, $ where $n_k=\\sum _{i=1}^n 1_k\\lbrace z_i\\rbrace $ .",
"The two distributions (REF ) and (REF ) are key components for the Monte Carlo sampling algorithm for Bayesian inference discussed in the next section."
],
[
"Posterior inference",
"After marginalisation of the pairs $(f_{k,j},\\sigma ^2_{k,j})$ and $\\mathbf {\\eta }$ , inference is limited to the community allocations $z$ and latent parameters $\\mathbf {\\theta }$ .",
"The marginal posterior distribution $p(z,\\mathbf {\\theta }\\mid \\hat{\\mathbf {X}})$ is analytically intractable; therefore, inference is performed using collapsed Metropolis-within-Gibbs Markov Chain Monte Carlo (MCMC) sampling.",
"For the community allocations $z$ , the Gibbs sampling step uses the following decomposition: $p(z_i=k\\mid z^{-i}, \\hat{\\mathbf {X}},\\mathbf {\\theta }) \\propto p(z_i=k\\mid z^{-i}) p(\\hat{x}_i\\mid z_i=k,z^{-i},\\mathbf {\\theta },\\hat{\\mathbf {X}}^{-i}),$ where the superscript $-i$ denotes that the $i$ -th row (or element) is removed from the corresponding matrix (or vector).",
"Using (REF ), the first term is $p(z_i=k\\mid z^{-i})=\\frac{n^{-i}_k+\\nu /K}{n-1+\\nu }.$ For the second term, using (REF ), the posterior predictive distribution for $\\hat{x}_i$ given $z_i=k$ can be written as the product of $d$ independent Student's $t$ distributions, where $\\hat{x}_{i,j}\\vert z_i=k,z^{-i},\\mathbf {\\theta },\\hat{\\mathbf {X}}^{-i} \\sim t_{2a_k^{-i}}\\left(\\mu _{k,j}^{\\star -i}(\\theta _i),\\frac{b_{k,j}^{-i}}{a_k^{-i}}\\left\\lbrace 1+\\xi _{k,j}^{\\star -i}(\\theta _i, \\theta _i)\\right\\rbrace \\right).$ Note that the quantities $\\mu _{k,j}^{\\star -i},\\xi _{k,j}^{\\star -i}, a_k^{-i}$ and $b_{k,j}^{-i}$ are calculated as described in (REF ), excluding the contribution of the $i$ -th node.",
"In order to mitigate identifiability issues, it is necessary to assume that some of the parameters are known a priori.",
"For example, assuming for each community $k$ that $f_{k,1}(\\theta _i)=\\theta _i$ , corresponding to a linear model in $\\theta $ with no intercept and unit slope in the first dimension, gives the predictive distribution: $\\hat{x}_{i,1}\\vert z_i=k,z^{-i},\\mathbf {\\theta },\\hat{\\mathbf {X}}^{-i} \\sim t_{2a_{k,j}^{-i}}\\left(\\theta _i,\\frac{1}{a_k^{-i}} \\left\\lbrace b_0 + \\frac{1}{2}\\sum _{h\\ne i:z_h=k} (\\hat{x}_{h,1} - \\theta _h)^2 \\right\\rbrace \\right).$ Finally, for updates to $\\mathbf {\\theta }_i$ , a standard Metropolis-within-Gibbs step can be used.",
"For a proposed value $\\theta ^\\ast $ sampled from a proposal distribution $q(\\cdot \\mid \\theta _i)$ , the acceptance probability takes the value $\\min \\left\\lbrace 1,\\frac{p(\\hat{x}_i\\mid z_i,z^{-i},\\theta ^\\ast ,\\mathbf {\\theta }^{-i},\\hat{\\mathbf {X}}^{-i})p(\\theta ^\\ast )q(\\theta _i\\mid \\theta ^\\ast )}{p(\\hat{x}_i\\mid z_i,z^{-i},\\theta _i,\\mathbf {\\theta }^{-i},\\hat{\\mathbf {X}}^{-i})p(\\theta _i)q(\\theta ^\\ast \\mid \\theta _i)}\\right\\rbrace .",
"$ The proposal distribution $q(\\theta ^\\ast \\mid \\theta _i)$ in this work is a normal distribution $\\mathbb {N}(\\theta ^\\ast \\mid \\theta _i,\\sigma ^2_\\ast )$ , $\\sigma ^2_\\ast \\in \\mathbb {R}_+$ , implying that the ratio of proposal distributions in (REF ) cancels out by symmetry."
],
[
"Applications and results",
"Inference for the LSBM is tested on synthetic LSBM data and on three real world networks.",
"As discussed in Section REF , the first dimension $\\hat{\\mathbf {X}}_1$ is assumed to be linear in $\\theta _i$ , with no intercept and unit slope.",
"It follows that $\\theta _i$ is initialised to $\\hat{x}_{i,1}+\\varepsilon _i$ , where $\\varepsilon _i\\sim \\mathbb {N}(0,\\sigma ^2_\\varepsilon )$ , for a small $\\sigma ^2_\\varepsilon $ , usually equal to $0.01$ .",
"Note that such an assumption links the proposed Bayesian model for LSBM embeddings to Bayesian errors-in-variables models [6].",
"In the examples in this section, the kernel function is assumed to be of the dot product form (REF ), with Zellner's $g$ -prior such that $\\mathbf {\\Delta }_{k,j} = n^2\\lbrace \\mathbf {\\Phi }_{k,j}(\\mathbf {\\theta })^\\intercal \\mathbf {\\Phi }_{k,j}(\\mathbf {\\theta })\\rbrace ^{-1}$ , where $\\mathbf {\\Phi }_{k,j}(\\mathbf {\\theta })\\in \\mathbb {R}^{n\\times q_{k,j}}$ such that the $i$ -th row corresponds to $\\mathbf {\\phi }_{k,j}(\\theta _i)$ .",
"For the remaining parameters: $a_0=1$ , $b_0=0.001$ , $\\mu _\\theta =\\sum _{i=1}^n \\hat{x}_{i,1}/n$ , $\\sigma ^2_\\theta =10$ , $\\nu =1$ .",
"The community allocations are initialised using $k$ -means with $K$ groups, unless otherwise specified.",
"The final cluster configuration is estimated from the output of the MCMC algorithm described in Section REF , using the estimated posterior similarity between nodes $i$ and $j$ , $\\hat{\\pi }_{ij}=\\hat{\\mathbb {P}}(z_i=z_j\\mid \\hat{\\mathbf {X}})=\\sum _{s=1}^{M} 1_{z^\\star _{i,s}}\\lbrace z^\\star _{j,s}\\rbrace /M$ , where $M$ is the total number of posterior samples and $z^\\star _{i,s}$ is the $s$ -th sample for $z_i$ .",
"The posterior similarity is not affected by the issue of label switching [13].",
"The clusters are subsequently estimated using hierarchical clustering with average linkage, with distance measure $1-\\hat{\\pi }_{ij}$ [21].",
"The quality of the estimated clustering compared to the true partition, when available, is evaluated using the Adjusted Rand Index [12].",
"The results presented in this section are based on $M={10000}$ posterior samples, with 1000 burn-in period."
],
[
"Simulated data",
"First, the performance of the LSBM and related inferential procedures is assessed on simulated data.",
"In [4] and [36], a Hardy-Weinberg latent structure model is used, with $\\mathcal {G}=[0,1]$ and $f(\\theta )=\\lbrace \\theta ^2,2\\theta (1-\\theta ),(1-\\theta )^2\\rbrace $ .",
"In order to introduce community structure, a permutation of the Hardy-Weinberg curve is considered here.",
"A graph with $n={1000}$ nodes is simulated, with $K=2$ communities of equal size.",
"Each node is assigned a latent score $\\theta _i\\sim \\text{Uniform}(0,1)$ , which is used to obtain the latent position $x_i$ through the mapping $f_{z_i}(\\theta _i)$ .",
"In particular: $f_1(\\theta ) =\\lbrace (1-\\theta )^2,\\theta ^2,2\\theta (1-\\theta )\\rbrace , & &f_2(\\theta )=\\lbrace \\theta ^2,2\\theta (1-\\theta ),(1-\\theta )^2\\rbrace .$ Using the latent positions, the graph adjacency matrix is then generated under the random dot product graph kernel $\\mathbb {P}(A_{ij}=1 \\mid x_i,x_j)=x_i^\\intercal x_j$ .",
"The resulting scatterplot of the latent positions estimated via ASE is plotted in Figure REF .",
"For visualisation purposes, the estimated latent positions $\\hat{\\mathbf {X}}$ have been aligned to the true underlying latent positions $\\mathbf {X}$ using a Procrustes transformation [7].",
"Figure: Scatterplots of {𝐗 ^ 1 ,𝐗 ^ 2 ,𝐗 ^ 3 }\\lbrace \\hat{\\mathbf {X}}_1,\\hat{\\mathbf {X}}_2,\\hat{\\mathbf {X}}_3\\rbrace , coloured by community, and true underlying latent positions (in black).The inferential procedure is first run assuming that the parametric form of the underlying latent function is fully known.",
"Therefore, the kernels are set to $\\xi _{k,j}(\\theta ,\\theta ^\\prime )=(1,\\theta ,\\theta ^2)\\mathbf {\\Delta }_{k,j}(1,\\theta ^\\prime ,\\theta ^{\\prime 2})^\\intercal ,\\ k=1,2,\\ j=1,2,3$ .",
"Figure REF shows the best-fitting curves for the two estimated communities after MCMC, which are almost indistinguishable from the true underlying latent curves.",
"The Markov Chain was initialised setting $\\theta _i=\\vert {\\hat{x}_{i,1}}\\vert ^{1/2}$ , and obtaining initial values of the allocations $z$ from $k$ -means.",
"The resulting ARI is $0.7918$ , which is not perfect since some of the nodes at the intersection between the two curves are not classified correctly, but still corresponds to approximately $95\\%$ of nodes correctly classified.",
"If the underlying functional relationship is unknown, a realistic guess could be given by examining the scatterplots of the embedding.",
"The scatterplots in Figure REF show that, assuming linearity in $\\theta _i$ on $\\hat{\\mathbf {X}}_1$ with no intercept and unit slope, a quadratic or cubic polynomial function is required to model $\\hat{\\mathbf {X}}_2$ and $\\hat{\\mathbf {X}}_3$ .",
"Therefore, for the purposes of the MCMC inference algorithm in Section REF , $f_2(\\cdot )$ and $f_3(\\cdot )$ are assumed to be cubic functions, corresponding to the kernel $\\xi _{k,j}(\\theta ,\\theta ^\\prime )=(1,\\theta ,\\theta ^2,\\theta ^3)\\mathbf {\\Delta }_{k,j}(1,\\theta ^\\prime ,\\theta ^{\\prime 2},\\theta ^{\\prime 3})^\\intercal $ , $j\\in \\lbrace 2,3\\rbrace $ , whereas $f_{k,1}(\\theta ) = \\theta \\ \\forall k$ .",
"The curves corresponding to the estimated clustering obtained using MCMC are plotted in Figure REF .",
"Also in this case, the algorithm is able to approximately recover the curves that generated the graph.",
"The imperfect choice of the latent functions makes the ARI decrease to $0.6687$ , which still corresponds to over $90\\%$ of nodes correctly classified.",
"Figure: Scatterplot of 𝐗 ^\\hat{\\mathbf {X}} and estimated generating curves obtained from the estimated clustering, coloured by community, and true underlying latent positions (in black)."
],
[
"Undirected graphs: Harry Potter enmity graph",
"The LSBM is also applied to the Harry Potter enmity graphThe network is publicly available at https://github.com/efekarakus/potter-network., an undirected network with $n=51$ nodes representing characters of J.K. Rowling's series of fantasy novels.",
"In the graph, $A_{ij}=A_{ji}=1$ if the characters $i$ and $j$ are enemies, and 0 otherwise.",
"A degree-corrected stochastic blockmodel represents a reasonable assumption for such a graph [22]: Harry Potter, the main character, and Lord Voldemort, his antagonist, attract many enemies, resulting in a large degree, whereas their followers are expected to have lower degree.",
"The graph might be expected to contain $K=2$ communities, since Harry Potter's friends are usually Lord Voldemort's enemies in the novels.",
"Theorem REF suggests that for a degree corrected blockmodel the embeddings are expected to appear as rays passing through the origin.",
"Therefore, $f_k(\\cdot )$ is assumed to be composed of linear functions such that $f_{k,1}(\\theta )=\\theta $ and $\\xi _{k,2}(\\theta ,\\theta ^\\prime )=\\theta \\theta ^\\prime \\Delta _{k,2}$ .",
"The resulting 2-dimensional ASE embedding of $\\mathbf {A}$ and the estimated clustering obtained using the linear LSBM are pictured in Figure REF .",
"Alternatively, if a polynomial form for $f_k(\\cdot )$ is unknown, a more flexible model is represented by regression splines, which can be also expressed in the form (REF ).",
"A common choice for $\\mathbf {\\phi }_{k,j}(\\cdot )$ is a cubic truncated power basis expansion $\\mathbf {\\phi }_{k,j}=(\\phi _{k,j,1},\\dots ,\\phi _{k,j,6})$ , $\\ell \\in \\mathbb {Z}_+$ , such that: $\\phi _{k,j,1}(\\theta )=\\theta ,\\ \\phi _{k,j,2}(\\theta )=\\theta ^2,\\ \\phi _{k,j,3}(\\theta )=\\theta ^3,\\ \\phi _{k,j,3+\\ell }(\\theta )=(\\theta -\\tau _\\ell )_+^3,\\ \\ell =1,2,3,$ where $(\\tau _1,\\tau _2,\\tau _3)$ are knots, and $(\\cdot )_+=\\max \\lbrace 0,\\cdot \\rbrace $ .",
"In this application, the knots were selected as three equispaced points in the range of $\\hat{\\mathbf {X}}_1$ .",
"The results are plotted in Figure REF .",
"Using either functional form, the algorithm is clearly able to recover the two communities, meaningfully clustering Harry Potter's and Lord Voldemort's followers.",
"Figure: Scatterplots of {𝐗 ^ 2 ,𝐗 ^ 3 }\\lbrace \\hat{\\mathbf {X}}_2,\\hat{\\mathbf {X}}_3\\rbrace vs. 𝐗 ^ 1 \\hat{\\mathbf {X}}_1, coloured by community allocation, and best fitting line and cubic truncated power spline passing through the origin, obtained from the estimated clustering."
],
[
"Directed graphs: ",
"LSBM are also useful to cluster the larval Drosophila mushroom body connectome [9], a directed graph representing connections between $n=213$ neurons in the brain of a species of flyThe data are publicly available at https://github.com/youngser/mbstructure..",
"The right hemisphere mushroom body connectome contains of $K=4$ groups of neurons: Kenyon Cells, Input Neurons, Output Neurons and Projection Neurons.",
"If two neurons are connected, then $A_{ij}=1$ , otherwise $A_{ij}=0$ , forming an asymmetric adjacency matrix $\\mathbf {A}\\in \\lbrace 0,1\\rbrace ^{n\\times n}$ .",
"The network has been extensively analysed in [24] and [2].",
"Following [24] and [2], after applying the DASE (Definition REF ) for $d=3$ , a joint concatenated embedding $\\hat{\\mathbf {Y}}=[\\hat{\\mathbf {X}},\\hat{\\mathbf {X}}^\\prime ]\\in \\mathbb {R}^{n\\times 2d}$ is obtained from $\\mathbf {A}$ .",
"Based on the analysis of [24], it should be assumed that three of the communities (Input Neurons, Output Neurons and Projection Neurons) correspond to a stochastic blockmodel, resulting in Gaussian clusters, whereas the Kenyon Cells form a quadratic curve with respect to the first dimension in the embedding space.",
"Therefore, the kernel functions implied in [24] are $\\xi _{1,j}(\\theta ,\\theta ^\\prime )=(\\theta ,\\theta ^2)\\mathbf {\\Delta }_{1,j}(\\theta ^\\prime ,\\theta ^{\\prime 2})^\\intercal ,\\ j=2,\\dots ,2d$ , with $f_{1,1}(\\theta )=\\theta $ , for the first community (corresponding to the Kenyon Cells), and $\\xi _{k,j}(\\theta ,\\theta ^\\prime )=\\Delta _{k,j},\\ k=2,3,4,\\ j=1,\\dots ,2d$ for the three remaining groups of neurons.",
"Following the discussion in [24], the LSBM inferential procedure is initialised using $k$ -means with $K=6$ , and grouping three of the clusters to obtain $K=4$ initial groups.",
"Note that, since the output of $k$ -means is invariant to permutations of the $K=4$ labels for the community allocations, careful relabelling of the initial values is necessary to ensure that the Kenyon Cells effectively correspond to the first community which assumes a quadratic functional form.",
"The most appropriate relabelling mapping is chosen here by repeatedly initialising the model with all possible permutations of the labels, and choosing the permutation that maximises the marginal likelihood.",
"Note that the marginal likelihood under the Bayesian model (REF ) for LSBMs is analytically available in closed form [26].",
"The results obtained after MCMC sampling are plotted in Figure REF .",
"The estimated clustering has ARI $0.8643$ , corresponding to only 10 misclassified nodes out of 213.",
"Figure: Scatterplots of {𝐘 ^ 2 ,𝐘 ^ 3 ,𝐘 ^ 4 ,𝐘 ^ 5 ,𝐘 ^ 6 }\\lbrace \\hat{\\mathbf {Y}}_2,\\hat{\\mathbf {Y}}_3,\\hat{\\mathbf {Y}}_4,\\hat{\\mathbf {Y}}_5,\\hat{\\mathbf {Y}}_6\\rbrace vs. 𝐘 ^ 1 \\hat{\\mathbf {Y}}_1, coloured by neuron type, and best fitting latent functions obtained from the estimated clustering.The performance of the LSBM is also compared to alternative methods for clustering in Table REF .",
"In particular, Gaussian mixture models with $K=4$ components were fitted on the DASE embedding $\\hat{\\mathbf {X}}$ [30], and its row-normalised version $\\tilde{\\mathbf {X}}$ [23], [25].",
"The ARI is averaged over 500 different initialisations.",
"Furthermore, the LSBM is also compared to the hierarchical Louvain (HLouvain in Table REF ) algorithmImplemented in python in the library scikit-network.",
"for graphs, corresponding to hierarchical clustering by successive instances of the Louvain algorithm [5] adapted to directed graphs [8].",
"Finally, the LSBM is also compared to hierarchical clustering with complete linkage and Euclidean distance (HClust in Table REF ), applied on $\\hat{\\mathbf {X}}$ .",
"The results in Table REF show that the LSBM largely outperform the alternative clustering techniques, which are not able to account for the non-linearity in the community corresponding to the Kenyon Cells.",
"Table: ARI for communities estimated via LSBM and alternative methodologies on the Drosophila connectome.From the scatterplots in Figure REF , it appears that the embedding for the Input Neurons, Output Neurons and Projection Neurons could be also represented using linear functions.",
"Furthermore, a quadratic curve for the Kenyon Cells might be too restrictive.",
"The LSBM framework allows specification of all such choices.",
"Here, assuming $f_{k,1}(\\theta )=\\theta $ for each $k$ , the first community (Kenyon Cells) is given a cubic latent function, implying $\\xi _{1,j}(\\theta ,\\theta ^\\prime )=(\\theta ,\\theta ^2,\\theta ^3)\\mathbf {\\Delta }_{1,j}(\\theta ^\\prime ,\\theta ^{\\prime 2},\\theta ^{\\prime 3})^\\intercal ,$ .",
"For the second community (Input Neurons), a latent linear function is used: $\\xi _{2,j}(\\theta ,\\theta ^\\prime )=(1,\\theta )\\mathbf {\\Delta }_{2,j}(1,\\theta ^\\prime )^\\intercal ,\\ j=2,\\dots ,6$ .",
"Similarly, from observation of the scatterplots in Figure REF , the following kernels, corresponding to linear latent functions, are assigned to the remaining communities: $\\xi _{3,j}(\\theta ,\\theta ^\\prime )=\\theta \\theta ^\\prime \\Delta _{3,j},\\ j=2,\\dots ,6$ , $\\xi _{4,2}(\\theta ,\\theta ^\\prime )=\\theta \\theta ^\\prime \\Delta _{4,2}$ , and $\\xi _{4,j}(\\theta ,\\theta ^\\prime )=(1,\\theta )\\mathbf {\\Delta }_{k,j}(1,\\theta ^\\prime )^\\intercal ,\\ j=3,\\dots ,6$ .",
"The results for these choices of covariance kernels are plotted in Figure REF , resulting in ARI $0.8754$ for the estimated clustering, again corresponding to 10 misclassified nodes.",
"Note that this representation seems to capture more closely the structure of the embeddings.",
"Figure: Scatterplots of {𝐘 ^ 2 ,𝐘 ^ 3 ,𝐘 ^ 4 ,𝐘 ^ 5 ,𝐘 ^ 6 }\\lbrace \\hat{\\mathbf {Y}}_2,\\hat{\\mathbf {Y}}_3,\\hat{\\mathbf {Y}}_4,\\hat{\\mathbf {Y}}_5,\\hat{\\mathbf {Y}}_6\\rbrace vs. 𝐘 ^ 1 \\hat{\\mathbf {Y}}_1, coloured by neuron type, and best fitting latent functions obtained from the estimated clustering, plotted only over the range of nodes assigned to each community."
],
[
"Bipartite graphs: Imperial College London computer laboratories",
"The LSBM methodology is finally applied to a bipartite graph obtained from computer network flow data collected at Imperial College London.",
"The source nodes are $\\vert {V_1}\\vert =n=439$ client machines located in four computer laboratories in different departments at Imperial College London, whereas the destination nodes are $\\vert {V_2}\\vert ={60635}$ internet servers, connected to by HTTP and HTTPS in January 2020 from one or more of the 439 client computers.",
"A total of 717912 edges are observed.",
"The inferential objective is to identify the location of the machines in the network, represented by their department, from the realisation of the rectangular adjacency matrix $\\mathbf {A}$ , where $A_{ij}=1$ if at least one connection is observed between client computer $i\\in V_1$ and the server $j\\in V_2$ , and $A_{ij}=0$ otherwise.",
"It could be assumed that $K=4$ , representing the departments of Chemistry, Civil Engineering, Mathematics, and Medicine.",
"After taking the DASE of $\\mathbf {A}$ , the machines are clustered using the LSBM.",
"The value $d=5$ is selected using the criterion of [40], choosing the second elbow of the scree-plot of singular values.",
"Computer network graphs of this kind have been seen to present quadratic curves in the embedding, as demonstrated, for example, by Figure REF in the introduction, which refers to a different set of machines.",
"Therefore, it seems reasonable to assume that $\\xi _{k,j}(\\theta ,\\theta ^\\prime )=(\\theta ,\\theta ^2)\\mathbf {\\Delta }_{k,j}(\\theta ^\\prime ,\\theta ^{\\prime 2})^\\intercal ,\\ j=2,\\dots ,d$ , which implies quadratic functions passing through the origin, and $f_{k,1}(\\theta )=\\theta $ .",
"The quadratic model with $K=4$ is fitted to the 5-dimensional embedding, obtaining ARI $0.9402$ , corresponding to just 9 misclassified nodes.",
"The results are plotted in Figure REF , with the corresponding best fitting quadratic curves obtained from the estimated clustering.",
"The result is particularly remarkable, considering that the communities are highly overlapping.",
"Figure: Scatterplots of {𝐗 ^ 2 ,𝐗 ^ 3 ,𝐗 ^ 4 ,𝐗 ^ 5 }\\lbrace \\hat{\\mathbf {X}}_2,\\hat{\\mathbf {X}}_3,\\hat{\\mathbf {X}}_4,\\hat{\\mathbf {X}}_5\\rbrace vs. 𝐗 ^ 1 \\hat{\\mathbf {X}}_1, coloured by department, and best fitting quadratic curves passing through the origin.These LSBM results are also compared to alternative methodologies in Table REF .",
"LSBM achieves the best performance in clustering the nodes, followed by Gaussian mixture modelling of the row-normalised adjacency spectral embedding.",
"The GMM on $\\tilde{\\mathbf {X}}$ sometimes converges to competitive solutions, reaching ARI up to $0.94$ , but usually converges to sub-optimal solutions, as demonstrated by the average ARI of $0.7608$ .",
"Table: ARI for communities estimated via LSBM and alternative methodologies on the ICL NetFlow network.As before, if a parametric form for $f_k(\\cdot )$ is unknown, regression splines can be used, for example the truncated power basis (REF ), with three equispaced knots in the range of $\\hat{\\mathbf {X}}_1$ .",
"The results for the initial three dimensions are plotted in Figure REF .",
"The communities are recovered correctly, and the ARI is $0.9360$ , corresponding to 10 misclassified nodes.",
"Figure: Scatterplots of {𝐗 ^ 2 ,𝐗 ^ 3 ,𝐗 ^ 4 ,𝐗 ^ 5 }\\lbrace \\hat{\\mathbf {X}}_2,\\hat{\\mathbf {X}}_3,\\hat{\\mathbf {X}}_4,\\hat{\\mathbf {X}}_5\\rbrace vs. 𝐗 ^ 1 \\hat{\\mathbf {X}}_1, coloured by department, and estimated cubic truncated power splines passing through the origin.A similar performance on this network is achieved by [33]; there, the number of misclassified nodes is 9 with ARI $0.938$ .",
"[33] assume normality on a reparametrisation to spherical coordinates of the embedding, whereas the methodology presented in this paper does not require any transformation."
],
[
"Conclusion",
"An extension of the latent structure model [4] for networks has been introduced, and inferential procedures based on Bayesian modelling of spectrally embedded nodes have been proposed.",
"The model, referred to as the latent structure blockmodel (LSBM), allows for latent positions living on community-specific univariate structural support manifolds.",
"Under the Bayesian paradigm, most model parameters can be integrated out and inference can be performed efficiently.",
"The performance of the model inference has been evaluated on simulated and real world networks.",
"In particular, excellent results have been obtained on complex clustering tasks concerning the Drosophila connectome and the Imperial College NetFlow data, where a substantial overlap between communities is observed.",
"Despite these challenges, the proposed methodology is still able to recover a correct clustering.",
"Overall, this work provides a modelling framework for graph embeddings arising from random dot product graphs where it is suspected that nodes belong to community-specific lower-dimensional subspaces.",
"In particular, this article discusses the case of curves, which are one-dimensional structural support submanifolds.",
"The methodology has been demonstrated to have the potential to recover the correct clustering structure even if the underlying parametric form of the underlying structure is unknown, using a flexible Gaussian process prior on the unknown functions.",
"In particular, regression splines with a truncated power basis have been used, showing good performance in recovering the underlying curves.",
"It has been assumed that the latent dimension $d$ and the number of communities $K$ are fixed and possibly known.",
"As an extension, $K$ could be assumed to be unknown with a geometric prior, and a split-merge sampling scheme similar to [32] could be used for resampling the parameter and community allocations.",
"An alternative approach when $K$ is unknown could also be a nonparametric overlapping mixture of Gaussian processes [28].",
"The techniques in [32], [33], [38] could be used to propose a model when $d$ is also unknown.",
"In that setting, for an $m$ -dimensional embedding with $m>d$ , relevant cluster-specific manifold structure would be observed only in the initial $d$ dimensions, whereas the communities would only differ in the amount of noise around zero in the last $m-d$ components.",
"Furthermore, in the Bayesian model proposed in this work, it has been assumed that the variance only depends on the community allocation.",
"This enables marginalising most of the parameters leading to efficient inference.",
"On the other hand, this is potentially an oversimplification, since the ASE central limit theorem (Theorem REF ) establishes that the covariance structure depends on the latent position.",
"Further work should study efficient algorithms for estimating the parameters when an explicit functional form dependent on $\\theta _i$ is incorporated in the covariance."
],
[
"Code",
"The python code used for this work is available in the GitHub repository fraspass/lsbm.",
"FSP and NAH gratefully acknowledge funding from the Microsoft Security AI research grant \"Understanding the enterprise: Host-based event prediction for automatic defence in cyber-security\"."
]
] | 2107.01734 |
[
[
"The Propagation of Strong Shocks into Planetary and Stellar Atmospheres"
],
[
"Abstract In this work we present a mathematical model for the propagation of the shock waves that occur in graded density profiles.",
"These waves can occur in a wide range of astrophysical events, such as collisions in planetary and stellar atmospheres, common envelope explosions and peculiar type Ia supernovae.",
"The behaviour of the shock wave and its evolution can be modelled using type II self similar solutions.",
"In such solutions the evolution of the shock wave is determined by boundary conditions at the shock front and a singular point in the shocked region.",
"We show how the evolution can be determined for different equations of state and density profiles, and compare these results to numerical simulations.",
"These findings are also applied to a variety of astrophysical phenomena to further test their validity."
],
[
"Introduction",
"The problem of a surface explosion has been considered by many authors in the past, using numerical simulations [33], [26] and lab experiments [16], [23], [15].",
"The primarily incentive for these works was to study cratering events due to planetary impacts.",
"Among the terrestrial impact events considered are the Chicxulub Cretaceous/Tertiary impact [32], [29] and the Tunguska event [3].",
"Related phenomena occurring on other astronomical bodies are being explored as well.",
"Some of these include asteroid impacts on the atmosphere of Venus [21], meteoroid collisions causing particle ejections from the Bennu asteroid [5], and various impacting objects colliding with Jupiter causing flashes of certain light curves [17], the most famous of which is the light curve generated by the Shoemaker Levy 9 collision [46].",
"That being said, subsurface explosions are not restricted to planets, and can also occur in star.",
"For example, it has been suggested that the 1954 precursor to the peculiar supernova iptf2014hls was caused by an explosion in a common envelope [43].",
"In this work, we develop an analytic formalism to model the propagation of strong shocks into an increasing density profiles, such as planetary and stellar atmospheres.",
"For simplicity, we will assume that the explosion is so strong that we can neglect the ambient pressure, and that the radius is much larger than the size of the initial hot spot.",
"We will also neglect gravity and assume an ideal gas equation of state for all materials.",
"Under these assumptions, we can use the method of self similarity to describe the evolution of the shock wave [4].",
"The propagation of such a shock wave will be markedly different from the propagation of a shock wave into a medium with flat or shallow density profile.",
"The latter can be described by the celebrated Sedov Taylor solution [37], [40], where behaviour of the shock is governed by energy conservation.",
"In contrast, in the problem we discuss the energy available to drive the shock decreases with time.",
"This is because shocked fluid elements expand and accelerate away from the shock front, and at some point cross a critical position called the sonic point where information cannot travel back to the shock front.",
"For this reason, the behaviour of this shock is determined by the behaviour near this sonic point.",
"Such solutions to the hydrodynamic equations are termed “type II”, to distinguish them from the “type I” solutions, which are governed by energy conservation.",
"These type II solutions have previously been used to model accelerating shocks propagating into a steeply declining density profiles, [41], [35].",
"In contrast, in this work we consider a decelerating type II shocks propagating into an increasing density profile.",
"Our model is a novel generalisation to the classical impulsive piston problem [1], [44].",
"In the classical problem, a thin wafer collides with a semi infinite space filled with a cold gas with a uniform density.",
"One can consider this problem as the one dimensional analogue of an impact event.",
"Since this problem is one dimensional and slab symmetric, it can be solved using the self similar method.",
"The solution predicts a power law relation between the shock velocity and the swept up mass $v \\propto m^{-\\beta }$ .",
"Interestingly, the same power law relation also holds well in the three dimensional case (i.e.",
"a sphere hitting a flat surface head on).",
"In this study we consider a variation on the classical problem where the density of target can vary as some power law of distance from the impact site $\\rho \\propto x^{\\omega }$ .",
"Not only will this description allow us to consider an atmospheric graded density profile, but it will also help us understand why the one dimensional analogue describes the three dimensional case well.",
"This is because one way to think about the difference between the slab symmetric and three dimensional cases is how the mass increases with the distance.",
"For example, if the density is uniform, then in slab geometry the mass increases linearly with the distance while in three dimensions the mass increases as a cube of the distance.",
"These differences can be accounted for in the slab symmetric analogue by changing the density profile in the target.",
"The plan of the paper is as follows.",
"In section we show how self similarity can be used to solve the generalised impulsive piston problem.",
"In section we demonstrate how the solutions we found can be used to model a variety of astrophysical problems.",
"In section we discuss the results and conclude.",
"Let us consider an explosion close to the surface of a large star or planet, so that the radius of curvature is unimportant and the atmospheric layers can be considered planar.",
"This explosion will generate a shock wave that travels into the atmosphere.",
"While the distance between the centre of the shock and the shock front is much smaller than the atmospheric scale height, the shock front interacts with an ambient medium with the same density, and hence the behaviour can be described by the classical Sedov Taylor explosion [37], [40].",
"When the distance between the shock front and explosion centre becomes comparable to the scale height, the top of the shock encounters a declining density profile and accelerates.",
"A mathematical model describing this part of the shock was developed in [34].",
"In this work we focus on the part of the shock that propagates deeper into the atmosphere, and since the density the shock encounters increases the shock decelerates.",
"When the distance between the centre of the explosion and the deepest point on the shock front is much larger than the atmospheric scale height the shock becomes self similar.",
"Therefore, the effective radius of the shock grows as some power law of time $R \\propto t^{\\alpha }$ , where We define the effective radius of the shock as the distance between the deepest point of the shock front to the centre of the explosion.",
"As was done in a previous paper [44], instead of considering a three dimensional problem, we consider the one-dimensional analogue called the impulsive piston problem [1], [47].",
"In this scenario, a thin wafer hits a much thicker slab of material (with a certain density distribution) and both are perfectly cold prior to the collision.",
"As a result of this collision a shock wave emerges from the contact surface and travels into the target.",
"Once the shock wave has travelled a distance much larger than the width of the wafer, there is no other relevant length scale in this process other than the position of the shock front, and hence we expect the shock wave to evolve in a self similar way.",
"In the original one dimensional impulsive piston problem, the target is considered to have uniform density.",
"However, in this work we consider a graded density profile defined by a power law dependence on the distance from the edge $\\rho _a = k x^{\\omega }$ where $k$ and $\\omega >0$ are constants and $x$ is the distance from the edge.",
"In the next sections we describe how self similarity allows us to solve the impulsive piston problem.",
"It does so by reduce the hydrodynamic equations, which are partial differential equations, to ordinary differential equations.",
"We also describe how these equation can be numerically integrated, and how the value of $\\alpha $ can be obtained, as was done for the case $\\omega =0$ in [44]."
],
[
"Self Similar Equations",
"The mathematical machinery presented in the following subsections is very similar to the methods used to analyse cratering events in planetary collisions in a previous paper [44].",
"The main difference is that the previous work considered a uniform density profile, compared to the graded density profile examined here.",
"The slab symmetric hydrodynamic equations in one dimension are given by [22], which are respectively the continuity, momentum, and entropy conservation, where $\\rho $ is density, $v$ is velocity, $p$ is pressure, and $s$ is entropy: $\\frac{\\partial \\rho }{\\partial t}+\\rho \\frac{\\partial v }{\\partial x} + v \\frac{\\partial \\rho }{\\partial x} = 0 $ $\\frac{\\partial v}{\\partial t} + v\\frac{\\partial v}{\\partial x}+\\frac{1}{\\rho } \\frac{\\partial p}{\\partial x} = 0 $ $\\frac{\\partial s}{\\partial t} + v \\frac{\\partial s}{\\partial x}= 0 \\, .",
"$ We assume an ideal gas equation of state $s = \\ln p - \\gamma \\ln \\rho $ where $\\gamma $ is the adiabatic index.",
"We can replace the pressure by the sound speed using the relation $p = \\rho c^2 / \\gamma $ .",
"After this substitution we have a system of partial differential equations in $v, c$ and $\\rho $ .",
"We make the assumption that the one dimensional shock has a self similar behaviour, so the position of the shock front $X(t)$ evolves as a power law in time $X(t) \\propto t^\\alpha \\, .",
"$ where $\\alpha $ is a constant.",
"One of the main goals of the analysis is determining $\\alpha $ .",
"The partial differential equations can be reduced to ordinary differential equations in the dimensionless position $\\chi = x/X$ .",
"Following [22], we define dimensionless hydrodynamical variables $V, C, D$ in the following way: $v(x,t) = \\frac{d X(t)}{dt}\\chi V\\left(\\chi \\right)$ $c(x,t) = \\frac{d X(t)}{dt} \\chi C\\left(\\chi \\right)$ $\\rho (x,t) = k X^{\\omega }(t) D\\left(\\chi \\right) \\, .",
"$ After this substitution, the equations contain terms involving different time derivatives of $X$ .",
"These time derivatives can be factored out using equation REF .",
"We introduce the following parameter $\\delta = \\frac{\\ddot{X} X}{\\dot{X}^2} = 1-\\frac{1}{\\alpha } \\,.$ This parameter is the power law index of the shock velocity - position relation $d X/dt \\propto X^{\\delta }$ .",
"After making these adjustments and simplifying, the hydrodynamic equations equations REF , REF and REF become the matrix equation $\\overleftrightarrow{M} d\\vec{A}/\\chi = \\vec{B}$ where: $M = \\begin{bmatrix}\\chi \\left(V - 1\\right) & \\chi D & 0\\\\\\chi C^{2} & \\gamma \\chi (V - 1) D & 2 \\chi C D\\\\ C \\chi (\\gamma - 1) \\left(1-V\\right) & 0 & 2 D \\chi \\left(V - 1\\right) \\end{bmatrix}$ $\\vec{A} = \\begin{bmatrix} D\\\\ V\\\\ C\\end{bmatrix}$ $\\vec{B}= \\begin{bmatrix}-\\omega D - D V \\\\- \\delta \\gamma D V - \\gamma (V^{2} - V) D - 2 C^{2} D\\\\ -(-\\gamma \\omega + \\omega ) C D - 2 C D V\\end{bmatrix}$ These equations are linear in the derivatives $dV/d\\chi , dC/d\\chi , dD/d\\chi $ so these can be isolated.",
"Our choice of parameters allows us to go a step further and obtain a single ordinary differential equation involving just $C$ and $V$ .",
"We note that $\\frac{d V}{d\\chi } = \\frac{\\tilde{N}_1}{\\tilde{D}_1} $ and $\\frac{d C}{d \\chi } = \\frac{N_1}{D_1}$ where $\\tilde{N}_1 = - C^{2} V \\gamma - 2 C^{2} \\delta + V^{3} \\gamma + V^{2} \\delta \\gamma - 2 V^{2} \\gamma - V \\delta \\gamma + V \\gamma - \\omega C^2$ $\\tilde{D}_1 = \\chi \\gamma \\left(C^{2} - V^{2} + 2 V - 1\\right)$ $N_1 = C \\left((V + \\omega ) \\gamma \\left(V - 1\\right) \\left(V \\gamma - V - \\gamma + 1\\right) - \\left(C^{2} - \\gamma \\left(V - 1\\right)^{2}\\right)\\times \\right.$ $\\left.",
"\\times \\left(2V + 2\\delta - \\omega \\gamma + \\omega - 2\\right) - \\left(2 C^{2} + V^{2} \\gamma + V \\delta \\gamma - V \\gamma \\right) \\left(V \\gamma - V - \\gamma + 1\\right)\\right)$ $D_1 = 2 \\chi \\left(C^{2} \\left(V - 1\\right) + C^{2} \\left(V \\gamma - V - \\gamma + 1\\right) - \\gamma \\left(V - 1\\right)^{3}\\right)$ Dividing one equation by the other, we get a single ODE $\\frac{dC}{d V} = \\frac{N_2}{D_2} $ where: $N_2 = C \\gamma \\left(- (V + \\omega ) \\gamma \\left(V - 1\\right) \\left(V \\gamma - V - \\gamma + 1\\right) +\\right.$ $\\left.",
"\\left(C^{2} - \\gamma \\left(V - 1\\right)^{2}\\right) \\left(2V + 2\\delta -\\gamma \\omega + \\omega - 2\\right) + \\right.$ $\\left.",
"+ \\left(2 C^{2} + V^{2} \\gamma + V \\delta \\gamma - V \\gamma \\right) \\left(V \\gamma - V - \\gamma + 1\\right)\\right) \\times $ $\\left(C^{2} - V^{2} + 2 V - 1\\right)$ and $D_2 = 2 \\left(C^{2} \\left(V - 1\\right) + C^{2} \\left(V \\gamma - V - \\gamma + 1\\right) - \\gamma \\left(V - 1\\right)^{3}\\right) \\times $ $\\times \\left(C^{2} V \\gamma + 2 C^{2} \\delta + C^2 \\omega - V^{3} \\gamma - V^{2} \\delta \\gamma + 2 V^{2} \\gamma + V \\delta \\gamma - V \\gamma \\right) \\, .$"
],
[
"Boundary Conditions",
"In order to integrate equation REF , boundary conditions and the value of $\\delta $ are needed.",
"The boundary conditions at the shock front are given by the Rankine Hugoniot conditions for a strong shock [47], [22].",
"$V_f = \\frac{2}{\\gamma +1} $ and $C_f = \\frac{\\sqrt{2 \\gamma \\left(\\gamma -1\\right)}}{\\gamma +1} \\, .",
"$ In some self similar problems, like the Sedov Taylor explosion, the parameter $\\delta $ can be be inferred directly from conservation laws.",
"Such problems are known as Type I solutions.",
"In our case, however, $\\delta $ is determined by the condition that the hydrodynamic trajectory passes smoothly through some singularity.",
"These are called Type II solutions [41].",
"The singularity occurs at a point where information cannot propagate back to the shock front, and is therefore also referred to as the sonic point.",
"Energy that flows through the sonic point cannot travel back to the shock front, and so the energy available to sustain the shock is not conserved.",
"Equation REF has a singularity when $C = 1- V $ This curve is known as the sonic line.",
"On the sonic line, the denominator (equation REF ) vanishes.",
"To prevent a divergence, the numerator also has to vanish.",
"This happens on a specific point on the sonic line, which we call the sonic point $V_s = \\frac{2\\delta + \\omega }{\\delta (2-\\gamma )+\\omega } $ $C_s = - \\frac{\\delta \\gamma }{\\delta (2- \\gamma ) + \\omega } $ Knowing the boundary conditions and $\\frac{d C}{d V}$ , we can integrate from the sonic point to the shock front.",
"To avoid a numerical divergence, the starting point is slightly shifted from the sonic point in the positive $V$ direction by $dV_i$ , and in $C$ direction by $d V_i \\frac{d C}{d V}\\Bigr |_{sonic}$ for some small $d V_i \\ll 1$ .",
"The correct value of $\\delta $ is such that the curve $C\\left(V\\right)$ satisfies both boundary conditions at the sonic point and the shock front.",
"We use the shooting method to find the value of $\\delta $ .",
"We guess a value for $\\delta $ , numerically integrate equation REF w.r.t to $V$ from the sonic point to the shock front, and note the value of $C$ at the shock front.",
"We then use the bisection method to refine the value of $\\delta $ to minimise the distance between the value of $C$ obtained from numeric integration and its theoretical value (equation REF ).",
"An example for some hydrodynamic $C\\left(V\\right)$ trajectories for the same value of $\\gamma , \\omega $ but different values of $\\delta $ can be seen in figure REF .",
"In the case $C_s>0$ , numerical integration is straightforward since both the sonic point and the shock front are on the same side of the impact site.",
"An example for several hydrodynamic trajectories for fixed $\\gamma , \\omega $ such that $C_s > 0$ can be seen in figure REF .",
"The case $C_s < 0$ is more complicated because the integration goes through $x=0$ .",
"Because of the way we defined the self similar variables (equation REF ), they diverge at $x=0$ , even though the physical quantities $v$ and $c$ remain finite.",
"We can circumvent this difficulty by noting that the Mach number, i.e.",
"the ratio between the velocity and the speed of sound, remains finite and changes continuously across $x=0$ .",
"The sonic point in the case $C_s < 0$ lies in the fourth quadrant of the of the $C-V$ plane (i.e.",
"positive $V$ and negative $C$ ).",
"Numerical integration proceeds to even higher values of $V$ , moving away from the shock front.",
"Far away from the sonic point, the curve attains some asymptotic slope, and this slope is the Mach number at $x=0$ , which we'll denote by $\\mathcal {M}_0$ .",
"Once we have this piece of information, we can restart the numerical integration in the second quadrant ($C$ is positive and $V$ is negative), beginning from some arbitrarily highly negative $V=V_r$ (subscript r for restart) and $C_r = V_r/\\mathcal {M}_0$ .",
"From this point we can continue the numerical integration all the way to the shock front.",
"An example for several hydrodynamic trajectories for fixed $\\gamma , \\omega $ such that $C_s < 0$ can be seen in figure REF .",
"Figure: First type of integration, for C s >0C_s > 0.",
"Here γ=3.5\\gamma = 3.5 and ω=3\\omega = 3.",
"The numbers in the legend show the values of δ\\delta used in each integration.",
"This integration is straightforward and there are no coordinate singularities.Figure: Second type of integration, for C s <0C_s < 0, and γ=1.47,ω=0.3\\gamma = 1.47, \\omega = 0.3.",
"In this situation, the shock and sonic points are on opposite sides of the boundary separating the fluid and the vacuum, so a mathematical singularity occurs when integrating from the sonic point to the shock front.In this manner we calculated $\\delta $ for different values of $\\gamma $ and $\\omega $ .",
"The results are displayed in the top panel of figure REF .",
"This figure shows that $\\delta $ is mostly dependent on $\\omega $ , and weakly dependent on $\\gamma $ .",
"For completeness, the asymptotic cases of $\\gamma \\rightarrow \\infty $ ; $\\omega \\rightarrow \\infty $ ; and $\\omega \\rightarrow \\infty ,\\gamma \\rightarrow \\infty $ are covered in the appendix.",
"This result prompted us to consider another power law index, $\\beta = d \\ln v / d \\ln m$ , which relates the velocity of the shock to the swept up mass.",
"This parameter can be related to $\\delta $ via $\\beta = \\frac{\\delta }{1+\\omega } $ A contour plot of $\\beta (\\gamma , \\omega )$ is shown in figure REF , in which one can see that $\\beta $ does not vary much as a function of $\\gamma $ and $\\omega $ .",
"Conservation of energy and momentum laws allows us to set upper and lower limits on the power law parameters [47] $-\\frac{1}{2}>\\beta >-1 $ $-\\frac{\\omega + 1}{2} > \\delta > -\\omega -1 $ and $\\frac{2}{3+\\omega } > \\alpha > \\frac{1}{2+\\omega } \\, .$"
],
[
"Comparison with Simulations",
"In order to verify our results, we compare them to the numerical simulations performed in [43].",
"The authors used hydrodynamic simulations to obtain the original power law parameter $\\alpha $ , defined in REF .",
"They obtained $\\alpha \\left(\\gamma = 5/3, \\omega =3\\right) = 0.19$ and $\\alpha \\left(\\gamma =5/3, \\omega =3/2\\right) = 0.25$ .",
"We note that these calculations were performed in spherical geometry, so in order to calculate $\\beta $ we use $\\beta = \\frac{\\delta }{3+\\omega } = \\frac{1-1/\\alpha }{3+\\omega } \\, .",
"$ Using equation REF we calculate the numerical values $\\beta \\left(\\gamma =5/3, \\omega =3\\right) = -0.71$ and $\\beta \\left(\\gamma =5/3, \\omega =3/2\\right) = -0.67$ .",
"Using the numerical integration methods, we obtained $\\beta \\left(\\gamma =5/3, \\omega =3\\right) = -0.671$ and $\\beta \\left(\\gamma =5/3, \\omega =3/2\\right) = -0.663$ .",
"Despite using different geometric setups and approaches to determining the power law scaling parameters, the values for $\\beta $ obtained here are reasonably close to those obtained previously with hydrodynamic simulations.",
"Figure: Contour plots of δ(γ,ω)\\delta (\\gamma , \\omega ) and β(γ,ω)\\beta (\\gamma , \\omega )."
],
[
"Applications",
"A number of previous studies have already considered the role of shock waves from a surface explosions in different astrophysical phenomena.",
"[45] studied atmospheric mass loss from planetary collisions.",
"[43] studied the optical transient that results from a rapid release of energy by a neutron star while inside the envelope of a giant companion star.",
"Finally, [42] showed that the shapes of craters due to collisions with primordial black holes would differ from typical craters, and hence the moon can serve as a dark matter detector.",
"In addition to these cases, in this section we present more possible astrophysical applications to the formalism developed here.",
"In particular, we discuss how the analytical formalism presented above can reproduce results from numerical hydrodynamic simulations and observations."
],
[
"Bolides",
"Even when objects burn up completely in the atmosphere, they produce a pressure wave that can cause damage on the ground.",
"The most recent example is the Chelyabinsk event [9].",
"The damage caused by the explosion (measured by the reports on damage per capita) declined with distance according to $r^{-2.4}$ , similar to the measured decline in over-pressure $r^{-2.6}$ .",
"We can interpret this decline in pressure using the model developed above.",
"After the shock wave reaches the ground, it expands further only in the horizontal direction.",
"For this reason the swept up mass scales with the area $m \\propto r^2$ .",
"The density at ground level is roughly constant, and so the pressure scales as $p \\propto \\rho v^2 \\propto m^{-2 \\beta } \\propto r^{-4 \\beta }$ .",
"Choosing $\\beta = 2/3$ , we find $p \\propto r^{-2.7}$ , which is relatively close to the empirical result.",
"The damage caused by the shock depends on whether the shock is strong by the time it reaches the ground, i.e.",
"whether the shock pressure is much larger than the atmospheric pressure when the shock reaches the ground.",
"As the bolide moves downward, it interacts with the atmosphere, decelerates and heats up.",
"This heat causes it to expand, which increases the deceleration and heating rate and so forth.",
"The air density at the the altitude at which the bolide decelerates considerably is [8] $\\rho _d \\approx \\rho _i \\left(\\frac{R_i}{h}\\right)^2$ where $\\rho _i$ is the density of the impactor, $R_i$ its radius and $h$ is the scale height.",
"The bolide then deposits the energy in the gas and creates a shock wave that expands outward.",
"At early times all parts of the shock interact with roughly the same density, so the shock evolves as a classical Sedov Taylor solution [40], [37].",
"The difference between density at different points on the shock becomes significant when the shock radius is comparable to the scale height.",
"At that point the swept up mass is $M_{ST} \\approx \\rho _d h^3 \\approx \\rho _i R_i^2 h \\, .$ Since energy is conserved, the velocity at the end of the Sedov Taylor phase is $v_s \\approx v_i \\sqrt{\\frac{m_i}{M_{ST}}} \\approx v_i \\sqrt{\\frac{R_i}{h}}$ where $m_i \\approx \\rho _i R_i^3$ is the mass of the impactor.",
"As the shock moves to lower altitudes, it follows the solution found in the previous section, so the shock velocity on the ground is $v_g \\approx v_{ST} \\left(\\frac{M_{ST}}{\\rho _0 h^3}\\right)^{\\beta } \\approx v_i \\sqrt{\\frac{R_i}{h}} \\left(\\frac{R_i^2 \\rho _i}{h^2 \\rho _0}\\right)^{\\beta } \\, .$ where $\\rho _0$ is the air density on the ground so $\\rho _0 h^3$ is the air mass enclosed within a radius $h$ from any point on the ground.",
"The pressure on the ground is $p_g \\approx \\rho _0 v_g^2 \\approx \\rho _0 v_i^2 \\frac{R_i}{h} \\left(\\frac{R_i^2 \\rho _i}{ h^2 \\rho _0}\\right)^{2 \\beta } \\, .$ By comparing this pressure to the atmospheric pressure at ground level we can obtain a critical value for the impactor radius $R_i \\approx h \\left(\\frac{\\rho _0}{\\rho _i}\\right)^{\\frac{2 \\beta }{4 \\beta +1}} \\left(\\frac{p_0}{\\rho _0 v_i^2}\\right)^{\\frac{1}{4 \\beta +1}} \\, .$ Substituting typical values and using $\\beta = 2/3$ yields $R_i \\approx 66 \\tilde{h} \\tilde{\\rho }_0^{1/11} \\tilde{\\rho }_i^{-4/11} \\tilde{p}_0^{3/11} \\tilde{v}_i^{-6/11} \\, \\rm m$ where $\\tilde{h} = h / 8 \\rm \\, km$ , $\\tilde{\\rho }_0 = \\rho _0 / 10^{-3} \\rm \\, \\frac{g}{cm^3}$ , $\\tilde{\\rho }_i = \\rho _i/ 3 \\rm \\, \\frac{g}{cm^3}$ , $\\tilde{p}_0 = p_0 / 10^5 \\, \\rm Pa$ and $\\tilde{v}_i = v_i/10 \\, \\rm \\frac{km}{s}$ .",
"A impactor with a larger radius creates a shock that remains strong when it reaches the ground.",
"In numerical simulations, the critical impactor size is around 50 metres [39].",
"Incidentally, the over-pressure threshold for substantial damage to brick and concrete surfaces is also comparable to the ambient atmospheric pressure [12].",
"Therefore, if the shock wave is no longer strong by the time it reaches the ground, it is unlikely to cause substantial damage to buildings.",
"Weak shocks can still cause damage to more brittle materials such as glass and wood, as was the case with the Chelyabinsk bolide.",
"Finally, it is possible to estimate the atmospheric mass loss from bolides.",
"An impactor moving faster than the escape velocity will create a decelerating shock wave, so that after it has swept up a certain amount of mass the velocity will drop below the escape velocity.",
"Since the velocity does not change considerably after the shock, the amount of mass lost in this way will be comparable to the mass accelerated to velocities exceeding the escape velocity $v_e$ .",
"Hence, we can approximate the amount of atmospheric mass loss $M$ by $M \\approx m_i \\left(\\frac{h}{R_i}\\right)^{\\frac{2 \\beta -1}{\\beta }} \\left(\\frac{v_i}{v_e}\\right)^{1/\\beta }\\, .$ Verification of this relation and discussion of its implication for planet formation is relegated to a future work."
],
[
"Shoemaker Levy 9",
"The impact of the fragments of the comet Shoemaker Levy 9 produced bright visible and infrared flashes that lasted, depending on the band, between tens of seconds and tens of minutes.",
"A direct comparison between the observational data and theory is problematic because of a number of reasons.",
"First, some of the fragments landed on the far side of Jupiter, so that only the spacecraft Galileo was able to observe the early phases of the impact [28], while ground based instruments could only begin to observe after the plumes rose above the horizon (although some did detect a faint precursor due to the passage of the fragments through the tenuous upper layers of Jupiter's atmosphere).",
"Second, some of the ground based telescopes became saturated and thus missed the peak [13].",
"Nevertheless, such a comparison has been performed [46], using a similar model for the shock propagation, but without the analytic justification.",
"We will not repeat that analysis here, but instead focus on the analytic predictions that can be obtained from our formalism.",
"Also, for simplicity in this section we assume that $\\beta = 2/3$ .",
"Most of the light observed after the impact of Shoemaker Levy 9 on Jupiter did not originate from the bolide itself, but from ejecta from the impact site that collided with more distant parts on the surface of Jupiter [46].",
"The range of every fluid shell depends on the velocity with which it emerges from the basin $r \\approx \\frac{v^2}{g} \\approx \\frac{R_{i}^{\\frac{10}{3}} h^{\\frac{2}{3}} \\rho _{i}^{\\frac{4}{3}} v_{i}^{2}}{g m^{\\frac{4}{3}}} \\, .$ where $g$ is the Jovian surface gravity.",
"The time it takes a fluid shell to fall down to the surface is $t_f \\approx \\frac{v}{g} \\approx \\frac{R_{i}^{\\frac{5}{3}} \\@root 3 \\of {h} \\rho _{i}^{\\frac{2}{3}} v_{i}}{g m^{\\frac{2}{3}}} \\, .$ At any moment an observer can only see fluid elements that had time to fall $t>t_f$ .",
"When the ejecta collides with the surface, it is shocked, and its temperature is given by $T \\approx \\mu v^2 / k \\approx \\frac{g^{2} \\mu t^{2}}{k} \\, .$ where $\\mu $ is the atomic mass and $k$ is the Boltzmann constant.",
"The luminosity is given by blackbody emission $L \\approx \\sigma T^4 r^2 \\approx \\frac{g^{10} \\mu ^{4} \\sigma t^{12}}{k^{4}} \\, .$ We find that the luminosity rises very steeply, and so attains its peak value almost instantaneously, in accordance with observations [7].",
"This peak luminosity comes from the fastest shell, which is the Sedov mass.",
"The corresponding peak bolometeric luminosity is $L_p \\approx \\frac{R_{i}^{12} \\mu ^{4} \\sigma v_{i}^{12}}{g^{2} h^{12} k^{4}} \\approx 10^{22} \\tilde{R}_i^{12} \\tilde{\\mu }^4 \\tilde{v}_i^{12} \\tilde{g}^{-2} \\tilde{h}^{-12} \\, \\rm erg/s$ where $\\tilde{R}_i = R_i/24 \\, \\rm km$ , $\\tilde{v}_i = v_i / 60 \\rm \\, km/s$ , $\\tilde{g} = g/60 \\rm m/s^2$ , $\\tilde{h} = h / 24 \\, \\rm km$ , $\\tilde{\\mu } = \\mu / m_p$ and $m_p$ is the proton mass.",
"This result is a factor of a few larger than the observed luminosity [7], however the results are very sensitive to the parameters.",
"The duration of the transient will be comparable to the fallback time $t_p \\approx \\frac{v_i R_i}{g h} \\approx 40 \\tilde{v}_i \\tilde{R}_i \\tilde{g}^{-1} \\tilde{h}^{-1} \\, \\rm s.$ This estimate is also in agreement with the observations.",
"The temperature at the peak is given by $T_p \\approx \\frac{R_{i}^{2} \\mu v_{i}^{2}}{h^{2} k} \\approx 750 \\tilde{R}_i^2 \\tilde{\\mu } \\tilde{v}_i^2 \\tilde{h}^{-2} \\, \\rm K \\, .$ The observed temperature is larger than our result by a factor of three [30]."
],
[
"Asteroid Impact Avoidance",
"The impact of rocky bodies on Earth poses a danger to the life and safety of its inhabitants.",
"For this reason, over the years different strategies for asteroid impact avoidance have been studied [27].",
"Asteroids whose diameter is less than a few hundred meters can be deflected using kinetic impactors (sometimes called mass drivers).",
"The net change in the momentum of the target is larger than just the momentum of the impactor, since upon impact debris from the target fly in the opposite direction.",
"We can use the formalism above to estimate the net change in the momentum of the target.",
"Let us consider an impactor of mass $m$ that collides with a much larger target at a velocity $v_i$ .",
"As a result, a shock wave emerges from the impact site and excavates a crater.",
"Crater excavation stops when the shock velocity drops below some critical velocity $v_e$ , which is comparable to the velocity with which elastic waves propagate.",
"The mass of the excavated material is $M \\approx m \\left(\\frac{v_i}{v_e}\\right)^{1/\\beta } \\, .$ Most of the ejected material moves with a velocity $v_e$ , so the net change in momentum is $M v_e$ .",
"For convenience, we normalise this change in target momentum with the momentum of the incident impactor $\\xi \\approx \\frac{M v_e}{m v_i} \\approx \\left(v_i/v_e\\right)^{\\frac{1-\\beta }{\\beta }} \\, .$ For $\\beta = 2/3$ , we get $\\xi \\propto v_i^{0.5}$ , which is similar to the behaviour found in numerical simulations $\\xi \\propto v_i^{0.44} $ [6].",
"Larger asteroid can only be deflected using nuclear surface explosions [2].",
"We can use our formalism to relate the energy and depth of the explosion to the net momentum change.",
"If the initial depth is $h$ and the ambient density of the asteroid is $\\rho _0$ , then the shock wave conserves energy until it sweeps up a mass $\\rho _0 h^3$ , at which point the shock velocity is $\\sqrt{E/\\rho _0 h^3}$ .",
"At later times the shock evolves according to our new formalism.",
"Thus, the mass of the ejecta is $M \\approx \\rho _0 h^3 \\left(\\sqrt{\\frac{E}{\\rho _0 h^3 v_e^2}}\\right)^{1/\\beta }$ where $v_e$ is the escape velocity from the asteroid.",
"The net change momentum is $I \\approx v_e M \\approx v_e \\rho _0 h^3 \\left(\\sqrt{\\frac{E}{\\rho _0 h^3 v_e^2}}\\right)^{1/\\beta } \\propto h^{3-3/2\\beta } \\, .$ If $\\beta =2/3$ , we have $I \\propto h^{0.75}$ , which is similar to the predictions based on empirical scaling laws [24].",
"If the explosion is too deep, then by the time the explosion makes it to the surface the shock velocity drops below the escape velocity and the net change in momentum is zero.",
"The optimal depth is determined by the condition that the shock is at the escape velocity when it reaches the surface $h_{o} \\approx \\left(\\frac{E}{\\rho _0 v_e^4}\\right)^{1/3} \\, .$"
],
[
"Hypervelocity White Dwarfs",
"Recent observations revealed an intriguing class of celestial objects.",
"These are white dwarfs with velocities that are markedly different than the other stars in their neighbourhoods, and also feature unusual surface composition [20], [38].",
"It has been suggested that these are white dwarfs that underwent some episode of runaway thermonuclear burning that stopped before it consumed the entire white dwarf [11].",
"If the reaction did not stop, then the white dwarf would have explode as a type Ia supernova and leave no remnant behind.",
"For this reason, a partially burned white dwarf is sometimes referred to as a supernova survivor.",
"It has also been suggested that these events give rise to subluminous supernovae [10].",
"Many models have been proposed to explain type Ia supernova [14], and in most of them, the explosion is initiated at a point that is offset from the centre.",
"Numerical simulations have shown that sometimes the reaction will only run through a portion of the white dwarf rather and not destroy it entirely [19].",
"We can use the formalism developed here to calculate the aftermath of such a failed supernova.",
"For simplicity, let us consider the white dwarf to be a sphere of uniform density $\\rho $ .",
"Suppose further that the runaway reaction has burned through a hemisphere of radius $h$ below the surface.",
"The mass of the burnt material is therefore $\\rho h^3$ .",
"We also assume that energy released per unit mass is $\\varepsilon $ , so the burnt material is accelerated to a velocity $\\sqrt{\\varepsilon }$ .",
"From that hot spot a shock wave emerges and travels through the white dwarf, but does not ignite more material.",
"If the escape velocity from the white dwarf is $v_e$ , then the net mass loss is $\\Delta M \\approx \\rho h^3 \\left(\\frac{\\varepsilon }{v_e^2}\\right)^{1/2\\beta } \\, .$ The mass ejected from the explosion moves at the escape velocity, so the velocity kick to the survivor is $\\frac{\\Delta v}{v_e} \\approx \\left(\\frac{h}{R}\\right)^3 \\left(\\frac{\\varepsilon }{v_e^2}\\right)^{1/2\\beta }$ where $R$ is the radius of the white dwarf.",
"For $\\beta = 2/3$ we find $\\frac{\\Delta M}{M_{WD}} \\approx \\left(\\frac{h}{R}\\right)^3 \\left(\\frac{\\varepsilon }{v_e^2}\\right)^{3/4} \\approx 0.05 \\left(\\frac{h/R}{0.3}\\right)^3 \\left(\\varepsilon /\\frac{{\\rm MeV}}{m_p}\\right)^{0.75} \\left(\\frac{v_e}{6 \\cdot 10^3 \\, \\rm \\frac{km}{s}}\\right)^{-1.5}$ where $M_{WD}$ is the mass of the white dwarf and $m_p$ is the proton mass.",
"The kick velocity is $\\Delta v \\approx 340 \\left(\\frac{h/R}{0.3}\\right)^3 \\left(\\varepsilon / \\frac{\\rm MeV}{m_p}\\right)^{0.75} \\left(\\frac{v_e}{6 \\cdot 10^3 \\, \\rm \\frac{km}{s}}\\right)^{-0.5} \\, \\rm \\frac{km}{s} \\, .$ These values are similar to those found in numerical simulations [19].",
"In this paper, we develop a self similar model for shocks propagating down stellar or planetary atmospheres with graded density profiles as a result of a rapid, localised deposition of energy.",
"We achieve this by generalising the impulsive piston problem to account for a graded density profile.",
"In this model, instead of considering the full three dimensional problem, we consider a one dimensional slab symmetric analogue.",
"In the analogue problem, a thin wafer hits a half space target whose density is allowed to vary in the direction normal to the surface.",
"The evolution of the resulting shock is determined by the conditions at the sonic point, a singularity beyond which information cannot travel to the shock front.",
"Under these assumptions, we can apply the self similar method to solve the hydrodynamic equations [22].",
"We do so by reducing the partial differential flow equations to a single ordinary differential equation.",
"We determine the deceleration parameter of the shock $\\beta = d \\ln \\dot{X} / d \\ln m$ (where $\\dot{X}$ is the shock velocity and $m$ is the swept up mass) by imposing boundary conditions at the shock front and at the sonic point, and using the shooting method.",
"We find that the parameter $\\beta $ is weakly dependent on the density profile.",
"For this reason, we can use this scaling relation even for different geometries, since the main difference between slab, cylindrical and spherical shock is the relation between the swept up mass and the distance from the centre of the explosion.",
"We also demonstrate that the method we developed can be used to study a wide range of astrophysical scenarios.",
"These scenarios include the transient resulting from the impact of meteors on Earth and other planets, mass loss due to giant planetary collisions, crater morphology, explosions in a common envelope and failed type Ia supernovae."
],
[
"Acknowledgements",
"We would like to thank Jonathan Fortney for suggesting the title for the paper.",
"AY is supported by the Natural Sciences and Engineering Research Council of Canada (NSERC), funding reference #CITA 490888-16.",
"This work made use of the sympy [25], numpy [31] and matplotlib [18] python packages.",
"The equations and results for Section 2 and the appendix can be found in: ."
],
[
"Asymptotic Behaviour for an Infinite Adiabatic Index",
"In the limit $\\gamma \\rightarrow \\infty $ , the equations used in Section 2.2 break down.",
"This is because the integration interval $V \\in \\left[\\frac{2\\delta + \\omega }{\\delta (2-\\gamma )+\\omega }, \\frac{2}{\\gamma +1}\\right]$ shrinks to zero width in this limit.",
"To overcome this difficulty, we define a new variable $W = \\gamma V$ , as was done in [44].",
"In this new variable, the integration domain becomes $W \\in \\left[-2-\\frac{\\omega }{\\delta },2\\right]$ , and so remains finite in the limit $\\gamma \\rightarrow \\infty $ .",
"The boundary conditions for $C$ simplifies as well, and become: $C = 1$ at the sonic point and $C = \\sqrt{2}$ at the shock front.",
"The expression for the derivative involving the new variable is given by $\\frac{d C}{d W} = \\frac{C (W \\delta + C^{2} \\omega + 2 C^{2} + 2 \\delta - 2)}{2 (W C^{2} + W \\delta - W + 2 C^{2} \\delta + C^{2} \\omega )} \\,$ At the sonic point, the slope is given by $\\left.",
"\\frac{d C}{d W} \\right|_s = - \\frac{\\delta (\\delta - \\omega + \\sqrt{9 \\delta ^{2} + 2 \\delta \\omega - 12 \\delta + \\omega ^{2} + 4} - 2)}{8 \\delta ^{2} + 4 \\delta \\omega - 8 \\delta - 4 \\omega }\\,$ We can determine $\\delta (\\omega )$ and $\\beta (\\omega )$ using the shooting integration method described in the previous section, since we know the boundary conditions and $\\frac{dC}{dW}$ .",
"A set of sample trajectories, for $\\omega = 0.5$ , is shown in figure REF .",
"Applying the shooting integration method with varying $\\omega $ allows one to obtain the dependencies of $\\delta $ and $\\beta $ on $\\omega $ , shown in figure REF .",
"The values of $\\beta $ and $\\delta $ in this asymptotic case fall below their respective energy conservation limits in equations REF , REF , unlike in the opposite limit $\\gamma = 1$ , where the values of these parameters are equal to the momentum conservation limits.",
"Figure: Hydrodynamic trajectories for the case γ→∞,ω=0.5\\gamma \\rightarrow \\infty , \\omega = 0.5.",
"Different coloured lined represent trajectories with different values of δ\\delta .",
"The sonic line is represented by the black line, and the shock front by the black cross.Figure: Using the shooting integration method with a variety of ω\\omega , plots of δ(ω)\\delta (\\omega ) (at the top) and β(ω)\\beta (\\omega ) (at the bottom) for γ→∞\\gamma \\rightarrow \\infty can be obtained.",
"Although δ\\delta appears to be linearly dependent on ω\\omega , this is actually not the case, since dδ dω\\frac{d\\delta }{d\\omega } is not a constant function of ω\\omega .",
"It is noted that β\\beta has a much lower dependence on ω\\omega than δ\\delta ."
],
[
"Asymptotic behaviour for an Infinite Density Slope",
"The limit $\\omega \\rightarrow \\infty $ is equivalent to an exponential density profile.",
"In this scenario, different equations for the density profile, dimensionless hydrodynamic quantities and shock front trajecory have to be used.",
"A similar problem has been considered [36], so many of the expressions are similar.",
"We note however that [36] considered the portion of the shock wave that travels in the opposite direction, towards lower densities, and so their power law index is different.",
"The density profile is defined by $\\rho (x) = \\rho _0 \\exp \\left(\\frac{x}{h}\\right)$ where $\\rho _0$ is the density at a reference point (where the explosion started), x is the distance from the reference point, and h is the atmospheric scale height.",
"In our situation, the shock wave gets slower as time progresses, while in [36] it gets faster and faster and traverses an infinite distance after a finite time.",
"The dimensionless position is $\\chi = \\frac{X(t)-x}{h}$ and the dimensionless hydrodynamic variables are $v(x,t) = \\frac{d X(t)}{dt} V \\quad \\quad c(x,t) = \\frac{d X(t)}{dt} C \\quad \\quad \\rho (x,t) = \\rho _0 e^{X(t)/h} D $ In the new framework, we start again with the same slab symmetric hydrodynamic equations REF , REF and REF and ideal gas equation of state, and then substitute in the new dimensionless parameters defined in REF .",
"Like in section 2.2, after these substitutions, the equations contain terms involving different time derivatives of $X$ .",
"The time derivatives can be factored out using equation, by eliminating the second derivative using $\\frac{d^2 X}{d t^2} = -\\frac{\\delta }{h} \\left(\\frac{d X}{d t}\\right)^2 \\,.$ This equation looks different from equation REF because the trajectory of the shock front is not a power law of time, but instead has a logarithmic dependence on time given by $X(t) = \\frac{h}{\\delta } \\ln (1 + t/t_0$ ) where h is the atmosphere scale height and $t_0$ is a scale time (equation 11 in [36]).",
"In this limit, the hydrodynamic equations equations REF , REF and REF can be represented as a matrix equation $\\overleftrightarrow{M} d\\vec{A}/\\chi = \\vec{B}$ where: $M = \\begin{bmatrix}1 - V & - D & 0\\\\- C^{2} & \\gamma (1 - V) D & - 2 C D\\\\- \\gamma C - (- \\gamma C + C) V + C & 0 & - 2 D V + 2 D\\end{bmatrix}$ $\\vec{A} = \\begin{bmatrix} D\\\\ V\\\\ C\\end{bmatrix}$ $\\vec{B} = \\begin{bmatrix}- D\\\\- \\gamma \\delta D V\\\\\\gamma C D - C D - 2 \\delta C D\\end{bmatrix}$ Once again, these equations are linear in the derivatives $\\frac{dV}{d\\chi }, \\frac{dC}{d\\chi }, \\frac{dD}{d\\chi }$ , so we can isolate them, and obtain a single differential equation of $\\frac{dC}{dV}$ by going through the same process as was done for equations 11 to 14: $\\frac{dC}{dV} = \\frac{C ( \\frac{C^2 \\gamma }{\\delta } - \\frac{C^2}{\\delta } - 2 C^2 - V^2 \\gamma ^2 + 3 V^2 \\gamma + V \\gamma ^2 - 5 V \\gamma + 2 \\gamma )}{- \\frac{2 C^2 V}{\\delta } - 4 C^2 V + \\frac{2 C^2}{\\delta }+ 4 C^2 + 2 V^3 \\gamma - 4 V^2 \\gamma + 2 V \\gamma } $ Like in section 2.3, in order integrate equation REF , boundary conditions and the value of $\\alpha $ are needed.",
"Once again, the boundary conditions at the shock front are: $V_f = \\frac{2}{\\gamma +1} $ and $C_f = \\frac{\\sqrt{2 \\gamma \\left(\\gamma -1\\right)}}{\\gamma +1} \\, .",
"$ The denominator of equation REF vanishes on the sonic line, given by equation REF .",
"At the sonic point, the numerator also vanishes.",
"This allows us to obtain the sonic point boundary conditions for this scenario: $V_s = \\frac{\\frac{1}{\\delta }+2}{\\frac{1}{\\delta }+2-\\gamma } $ $C_s = -\\frac{\\gamma }{\\frac{1}{\\delta }+2-\\gamma } $ Knowing the boundary conditions and $\\frac{d C}{d V}$ , we can integrate from the sonic point to the shock front.",
"The integration process, with the exception of the exact formulas for $\\frac{dC}{dV}$ and boundary conditions, is equivalent to section 2.3 above, so the details will not be repeated here.",
"However, one thing to note is that the $C_{s} < 0$ never arises, because $\\frac{1}{\\delta } + 2 < \\gamma $ for all physically possible ranges of $\\delta , \\gamma $ .",
"This means that only the first integration case, where the sonic point and shock front are on the same side of the impact site, occurs.",
"An example of a hydrodynamic trajectory for this scenario is shown in figure REF .",
"The bisection solving method can be used to obtain $\\delta $ for each possible value of $\\gamma $ , and then taking the reciprocal gets $\\delta $ .",
"This relation $\\delta (\\gamma )$ is plotted in figure REF .",
"In the limit of $\\gamma = 1$ , we get $\\delta = -1$ , which corresponds to momentum conservation.",
"In the limit $\\gamma \\rightarrow \\infty $ , which corresponds to energy conservation, the solution converges to approximately $\\delta =-0.597$ .",
"Obtaining this value requires the same substitution $W = \\gamma V$ present in section A1."
],
[
"Asymptotic behaviour for Infinite Adiabatic Index and Density Slope",
"Once again, when $\\gamma \\rightarrow \\infty $ , the integration along $V \\in \\left[\\frac{\\frac{1}{\\delta }+2}{\\frac{1}{\\delta }+2-\\gamma }, \\frac{2}{\\gamma +1}\\right]$ shrinks to zero width, and using the new variable $W = \\gamma V$ instead of $V$ remedies this.",
"Now the integration range for $W$ is $\\left[-2-\\frac{1}{\\delta },2\\right]$ , and this is not empty, so the integration process can occur.",
"The boundary conditions in $C$ are $C = 1$ at the sonic point and $C = \\sqrt{2}$ at the shock front.",
"The expression for the derivative in this situation is: $\\frac{d C}{d W} = \\frac{C (W + \\frac{C^{2}}{\\delta } + 2)}{2 (W + \\frac{C^{2}}{\\delta } + 2 C^{2})} \\,$ And at the sonic point this slope is: $\\left.",
"\\frac{d C}{d W} \\right|_s = - \\frac{\\frac{3}{\\delta } + \\sqrt{\\frac{9}{\\delta ^{2}} + \\frac{8}{\\delta } +16}}{4( \\frac{1}{\\delta } +2)}\\,$ We can now determine $\\delta (\\gamma )$ using the shooting integration method, since the boundary conditions and $\\frac{dC}{dW}$ are known.",
"Some hydrodynamic trajectories for different values of $\\delta $ close to the solution are shown in figure REF .",
"Using a bisection solver, we find that $\\lim _{\\gamma \\rightarrow \\infty }\\delta \\approx -0.597$ .",
"Figure: Hydrodynamic trajectories for the case ω→∞,γ→∞\\omega \\rightarrow \\infty , \\gamma \\rightarrow \\infty .",
"Different coloured lined represent trajectories with different values of 1 δ\\frac{1}{\\delta }.",
"The sonic line is represented by the black line, and the shock front by the black cross.Figure: Hydrodynamic trajectories for the exponential density profile and γ=3.5\\gamma = 3.5.",
"Fortunately, in this scenario, the integration is always straightforward and there are no mathematical singularities.",
"Once a 1 δ\\frac{1}{\\delta } value is found whose integration curve ends close enough to the shock front boundary condition, its reciprocal is taken to obtain the corresponding δ\\delta value.Figure: A plot of δ(γ)\\delta (\\gamma ) for the exponential density profile scenario (ω→∞\\omega \\rightarrow \\infty ) can be obtained by applying the shooting integration method with a variety of γ\\gamma and taking the reciprocal of the respective 1 δ\\frac{1}{\\delta } values."
]
] | 2107.01701 |
[
[
"A convex optimization approach to online set-membership EIV\n identification of LTV systems"
],
[
"Abstract This paper addresses the problem of recursive set-membership identification for linear time varying (LTV) systems when both input and output measurements are affected by bounded additive noise.",
"First we formulate the problem of online computation of the parameter uncertainty intervals (PUIs) in terms of nonconvex polynomial optimization.",
"Then, we propose a convex relaxation approach based on McCormick envelopes to solve the formulated problem to the global optimum by means of linear programming.",
"The effectiveness of the proposed identification scheme is demonstrated by means of two simulation examples."
],
[
"Introduction",
"Recursive parameter estimation of linear systems has continuously attracted the attention of the automatic control community in the last decades.",
"This topic is of particular interest in the context of linear time-varying (LTV) systems, where the parameter variations need to be tracked online.",
"A relevant number of contributions addressing the problem of recursive identification of LTV systems can be found in the context of classical system identification, where the noise affecting the measurements is assumed to be statistically described.",
"The interested reader can find details in the survey paper [1] and in the book [2], [3].",
"A worthwhile alternative to the stochastic noise description, inspired by the seminal work of Schweppe [4], is the so-called bounded-errors or set-membership (SM) characterization, where uncertainties are assumed to belong to a given set, see, e.g., the book [5] for an introduction to the theory.",
"In the SM framework, all parameter vectors belonging to the feasible parameter set (FPS), i.e., parameters consistent with the measurements, the error bounds and the assumed model structure, are feasible solutions to the identification problem.",
"The objective of any SM algorithm is either to optimally select a single solution in the FPS (pointwise SM estimators) or to compute uncertainty bounds for the parameters (set-valued SM estimators).",
"In this work we focus our attention on this second class.",
"A number of algorithms have been proposed to address the problem of computing parameter bounds for LTV systems.",
"The idea common to all the approaches is to recursively approximate the FPS by means of simply-shaped sets: ellipsoids are considered in [6], polyedrals in [7] and zonotopes in [8], [9], while orthotopic regions have been recently considered in [10].",
"All the aforementioned papers formulate the identification problem with reference to the equation error structure.",
"To the best of the authors' knowledge, the first attempt to address the problem of recursive SM identification for LTV systems in the errors-in-variables (EIV) framework, i.e., when both the input and the output measurements are affected by noise, has been presented in our previous contribution [11].",
"In this paper, we show that the parameter uncertainty intervals (PUIs) can be exactly computed at each time iteration by solving a set of simple linear programming problems, provided that the sign of the parameters are a-priori known.",
"However, if such an information is not available, the problem requires the solution of a number of computationally expensive non-convex polynomial optimization problems.",
"In order to overcome this limitation, in this work we propose a different convex relaxation strategy which does not require any a-priori information on the parameters sign.",
"The proposed approach is based on the concept of McCormick envelopes originally proposed in [12].",
"The paper is organized as follows.",
"Section is devoted to the problem formulation.",
"In Section , we briefly review the results of our previous contribution [11] in order to highlight the mathematical structure of the problem.",
"The novel convex relaxation approach, based on McCormick envelopes, is presented in Section .",
"The effectiveness of the proposed identification scheme is shown in Section by means of two simulation examples.",
"Concluding remarks end the paper."
],
[
"Problem Formulation",
"Let us consider the SISO discrete-time LTV system, depicted in Fig.",
"REF , described in terms of the following linear difference equation $A(t,q^{-1})w(t)=B(t,q^{-1})x(t),$ where $x(t)$ and $w(t)$ are the noise free input and output signals respectively, and $A(t,\\cdot )$ , $B(t,\\cdot )$ are polynomials in the backward shift operator $q^{-1} (i.e., q^{-1}u(t) = u(t-1)$ ) given by $A(t,q^{-1})=1+a_{1}(t)q^{-1}+\\dots +a_{n_a}(t)q^{-n_{a}},$ $B(t,q^{-1})=b_{0}(t)+b_{1}(t)q^{-1}+\\dots +b_{n_b}(t)q^{-n_{b}},$ where $n_a \\ge n_b$ .",
"The unknown parameter vector $\\theta (t) \\in {\\mathbb {R}}^{n_p}$ to be estimated at each time instant $t$ is $\\theta (t)=[a_{1}(t),\\dots ,a_{n_a}(t),b_{0}(t),\\dots ,b_{n_b}(t)]^T,$ where $n_p=n_a+n_b+1$ .",
"Figure: EIV model structureAt each generic time instant $t$ , the value of $k$ -th component $\\theta _k(t)$ of the parameter vector $\\theta (t)$ is described as $\\theta _k(t)=\\theta _k(t-1)+\\delta _{\\theta _k}(t), \\quad \\quad k = 1,\\dots ,n_p,$ where $\\delta _{\\theta _k}(t)$ is the parameter variation between two consecutive generic time instants $t-1$ and $t$ , assumed to be unknown but bounded, i.e., $|\\delta _{\\theta _k}(t)| \\le \\Delta _{\\theta _k}, \\forall t$ where $\\Delta _{\\theta _k}$ , $k=1,\\ldots ,n_p$ , are known variation bounds.",
"Both input and output data are corrupted by additive noise $\\zeta (t)$ and $\\eta (t)$ respectively, $u(t) = x(t)+\\zeta (t),$ $y(t) = w(t)+\\eta (t).$ Each sample of the noise sequences $\\zeta (t)$ and $\\eta (t)$ is bounded by known constants $\\Delta _{\\zeta }$ and $\\Delta _{\\eta }$ , i.e.",
"$|\\zeta (t)| \\le \\Delta _{\\zeta }, \\forall t$ $|\\eta (t)| \\le \\Delta _{\\eta }, \\forall t.$ According to the problem formulation presented in [11], the FPS, at a generic time instant $t$ , is defined as, $\\begin{split}{\\mathcal {D}}_{\\theta }(t) = \\lbrace & \\theta (t) \\in {\\mathbb {R}}^{n_p}:A(q^{-1},t)(y(t)-\\eta (t))=B(q^{-1},t)(u(t)-\\zeta (t)),\\\\&\\theta _k(t)=\\theta _k(t-1)+\\delta _{\\theta _k}(t), \\ |\\delta _{\\theta _k}(t)| \\le \\Delta _{\\theta _k},\\\\ &\\underline{\\theta }_{k}(t-1)\\le \\theta _k(t-1)\\le \\overline{\\theta }_{k}(t-1),\\\\& k = 1,...,n_p, \\ |\\eta (t)|\\le \\Delta _{\\eta }, \\ |\\zeta (t)|\\le \\Delta _{\\zeta }\\rbrace ,\\end{split}$ where $\\underline{\\theta }_{k}(t-1)$ and $\\overline{\\theta }_{k}(t-1)$ are bounds on $\\theta _k(t-1)$ computed at time $t-1$ .",
"In this work we address the problem of online computation of the parameter uncertainty intervals (PUIs) defined as $PUI_{k}(t) = [\\underline{\\theta }_{k}(t) \\ , \\ \\overline{\\theta }_{k}(t)],$ where $\\begin{aligned}\\underline{\\theta }_{k}(t) = \\underset{\\theta (t) \\in {\\mathcal {D}}_{\\theta }(t)}{\\text{min}} \\theta _{k}(t),\\end{aligned}$ $\\begin{aligned}\\overline{\\theta }_{k}(t) = \\underset{\\theta (t) \\in {\\mathcal {D}}_{\\theta }(t)}{\\text{max}} \\theta _{k}(t).\\end{aligned}$ Initial bounds $\\underline{\\theta }_{k}(0)$ and $\\overline{\\theta }_{k}(0)$ are assumed to be a-priori known."
],
[
"Bounding the parameters of the LTV system",
"In this section we briefly review the approach proposed in our previous contribution [11] for the computation of the solution to problems (REF ) and (REF ).",
"The following result provides insight into the mathematical structure of the optimization problem to be solved.",
"Result 1 Computation of $\\underline{\\theta }_{j}(t)$ and $\\overline{\\theta }_{j}(t)$ via polynomial optimization Lower and upper bounds on the $k-$ th component $\\theta _k(t)$ of the parameter vector can be computed solving the following (nonconvex) polynomial optimization problem : $\\begin{aligned}& \\underset{\\theta (t)}{\\text{min}} \\ J_{\\theta _k}(t)\\\\& \\ s.t:\\\\& \\left\\lbrace \\begin{matrix}A(t,q^{-1})(y(t)-\\eta (t)) = B(t,q^{-1})(u(t)-\\zeta (t)), \\vspace{5.69046pt}\\\\\\hspace{-128.0374pt}-\\Delta _\\eta \\le \\eta _t \\le \\Delta _{\\eta _t},\\vspace{5.69046pt}\\\\\\hspace{-128.0374pt}-\\Delta _\\zeta \\le \\zeta _t \\le \\Delta _{\\zeta _t},\\vspace{5.69046pt}\\\\ \\hspace{2.84544pt}\\underline{\\theta }_{k}(t-1)-\\Delta _{\\theta _k}(t)\\le \\theta _k(t)\\le \\overline{\\theta }_{k}(t-1)+\\Delta _{\\theta _k}(t),\\vspace{5.69046pt}\\\\\\hspace{-136.5733pt}k = 1,\\ldots ,n_p\\end{matrix}\\right.\\end{aligned}$ where $\\ J_{\\theta _k}(t)= {\\theta _k}(t)$ for computing the lower bound $\\underline{\\theta }_{k}(t)$ , or $\\ J_{\\theta _k}(t)= -{\\theta _k}(t)$ for the upper bound $\\overline{\\theta }_{k}(t)$ .$\\Box $ Problem (REF ) is directly derived from (REF ) and (REF ) by rewriting the constraints describing the FPS (11) in compact form.",
"Non-convexity of optimization problem (REF ) is due to the presence of bilinear terms (involving unknown variables $\\theta $ , $\\eta $ and $\\zeta $ ) in the equality $A(t,q^{-1})(y(t)-\\eta (t)) = B(t,q^{-1})(u(t)-\\zeta (t))$ .",
"As discussed in our previous contribution [11], problem (REF ) can be solved to global optimum by means of linear programming if a-priori information on the sign of the parameters are available.",
"In fact, under such an assumption, the bilinear model equation can be rewritten as: $(\\varphi (t)-\\Delta _{\\varphi }(t))\\theta \\le y_t+\\Delta _\\eta ,$ $(\\varphi (t)+\\Delta _{\\varphi }(t))\\theta \\ge y_t-\\Delta _\\eta ,$ where $\\varphi (t)$ is the measurement regressor defined as $\\varphi (t) = [-y(t-1),\\ldots ,-y(t-n_a),u(t),\\ldots u(t-n_b)],$ while $\\Delta _{\\varphi }(t)$ is given by: $\\begin{aligned}\\Delta _{\\varphi }(t) = [&\\Delta _\\eta sgn(a_1(t)),\\ldots ,\\Delta _\\eta sgn(a_{n_a}(t))\\\\&\\Delta _\\zeta sgn(b_0(t)),\\ldots ,\\Delta _\\zeta sgn(b_{n_b}(t))].\\end{aligned}$ Remark 1 It is worth noting that, in case no information is available about the sign of the parameters, the approach proposed in paper [11] cannot be applied to convert (REF ) to a linear program.",
"To address this drawback, on the one hand, one can resort to prior estimation of the signs, see, e.g., [13]; nevertheless, this approach may be computationally unfeasible in online identifcation.",
"On the other hand, convex relaxation techniques guaranteed to converge to the global optimum of nonconvex polynomial optimization problems are available in the literature, see, e.g., [14], [15], [16].",
"Nevertheless, such methods require the solution of large-dimensional semidefinite programming problems even when the number of parameters is relatively small; therefore, they cannot be applied in the framework of online estimation of LTV systems, due to their high computational complexity in terms of both computational time and memory resources requirements.",
"In this work, we propose a novel approach which does not require any information about the signs of the parameters."
],
[
"McCormick envelopes based convex relaxation",
"In this section, we propose an approach to reformulate problem (REF ) in terms of convex optimization.",
"The main idea is to exploit the concept of McCormick envelopes [12] to replace the bilinear terms in (REF ) with a set of linear inequalities, without introducing any conservativeness.",
"Let us first rewrite model equations (REF ), (REF ) and (REF ) in the following compact form: $\\begin{aligned}y(t)-\\eta (t) = &-\\sum _{i=1}^{n_a}a_i(t)y(t-i)+\\sum _{j=0}^{n_b}b_j(t)u(t-j)+\\\\&+\\sum _{i=1}^{n_a}a_i(t)\\eta (t-i)-\\sum _{j=0}^{n_b}b_j(t)\\zeta (t-j).\\end{aligned}$ Then, in order to eliminate the bilinear (nonconvex) terms in (REF ), we define the following new variables ${\\mathcal {M}}_{a_i}(t) = a_i(t)\\eta (t-i), \\quad \\quad i = 1,\\ldots ,n_a,$ ${\\mathcal {M}}_{b_j}(t) = b_j(t)\\zeta (t-j), \\quad \\quad j = 0,\\ldots ,n_b$ which allow us to rewrite (REF ) as follows $\\begin{aligned}&y(t)+\\sum _{i=1}^{n_a}a_i(t)y(t-i)-\\sum _{j=0}^{n_b}b_j(t)u(t-j)\\\\&-\\sum _{i=1}^{n_a}{\\mathcal {M}}_{a_i}(t)+\\sum _{j=0}^{n_b}{\\mathcal {M}}_{b_j}(t) = \\eta (t).\\end{aligned}$ Since $\\eta $ is known to be bounded according to (REF ), the following inequality is finally obtained: $ \\begin{aligned}&|y(t)+\\sum _{i=1}^{n_a}a_i(t)y(t-i)-\\sum _{j=0}^{n_b}b_j(t)u(t-j)-\\\\&-\\sum _{i=1}^{n_a}{\\mathcal {M}}_{a_i}(t)+\\sum _{j=0}^{n_b}{\\mathcal {M}}_{b_j}(t)|\\le \\Delta _\\eta .\\end{aligned}$ Upper and lower bounds on ${\\mathcal {M}}_{a_i}(t)$ and ${\\mathcal {M}}_{b_j}(t)$ , can be obtained by relying on the concept of McCormick envelopes.",
"Definition 1 (McCormick envelopes [12]) Given two bounded variables $x, y \\in {\\mathbb {R}}$ , $x^{LB}\\le x \\le x^{UB}$ and $y^{LB}\\le y \\le y^{UB}$ , the product $w = xy$ , satisfies the following inequalities: $w \\ge x^{LB}y+xy^{LB}-x^{LB}y^{LB},$ $w \\ge x^{UB}y+xy^{UB}-x^{UB}y^{UB},$ $w \\le x^{UB}y+xy^{LB}-x^{UB}y^{LB},$ $w \\le xy^{UB}+x^{LB}y-x^{LB}y^{UB}.$ $\\Box $ Direct application of the concept of McCormick envelopes in Definition 1 to equations (REF ) and (REF ) leads to the following result.",
"Result 2 Computation of ${\\mathcal {M}}_{\\theta _k}(t)$ bounds Let ${\\mathcal {M}}_{\\theta _k}(t)$ be the generic term in either equation (REF ) or equation (REF ).",
"${\\mathcal {M}}_{\\theta _k}(t)$ satisfies the following set of inequalities ${\\mathcal {M}}_{\\theta _k}(t) \\ge \\theta _k^{LB}(t)\\epsilon (t-\\lambda )-\\theta _k(t)\\Delta _\\epsilon +\\theta _k^{LB}(t)\\Delta _\\epsilon ,$ ${\\mathcal {M}}_{\\theta _k}(t) \\ge \\theta _k^{UB}(t)\\epsilon (t-\\lambda )+\\theta _k(t)\\Delta _\\epsilon -\\theta _k^{UB}(t)\\Delta _\\epsilon ,$ ${\\mathcal {M}}_{\\theta _k}(t) \\le \\theta _k^{UB}(t)\\epsilon (t-\\lambda )-\\theta _k(t)\\Delta _\\epsilon +\\theta _k^{UB}(t)\\Delta _\\epsilon ,$ ${\\mathcal {M}}_{\\theta _k}(t) \\le \\theta _k(t)\\Delta _\\epsilon +\\theta _k^{LB}(t)\\epsilon (t-\\lambda )-\\theta _k^{LB}(t)\\Delta _\\epsilon ,$ where $\\epsilon (t)=\\eta (t)$ and $\\Delta _\\epsilon =\\Delta _\\eta $ for ${\\mathcal {M}}_{\\theta _k}(t)={\\mathcal {M}}_{a_i}(t)$ in equation (REF ), while $\\epsilon (t)=\\zeta (t)$ and $\\Delta _\\epsilon =\\Delta _\\zeta $ for ${\\mathcal {M}}_{\\theta _k}(t)={\\mathcal {M}}_{b_j}(t)$ in equation (REF ).",
"Bounds $\\theta _k^{LB}(t)$ and $\\theta _k^{UB}(t)$ are given by $\\theta _k^{LB}(t) = \\underline{\\theta }_k(t-1)-\\Delta _{\\theta _k},$ $\\theta _k^{UB}(t) = \\overline{\\theta }_k(t-1)+\\Delta _{\\theta _k}.$ Variable $\\lambda =i$ for $\\theta _k=a_i$ in equation (REF ), while $\\lambda =j$ for $\\theta _k=b_j$ in equation (REF ).",
"$\\Box $ Thanks to Result REF , we are now in the position to state the main result of the paper.",
"Result 3 Computation of PUIs by means of linear programming The global optimal solution to optimization problem (REF ) can be computed solving the following linear program: $\\begin{aligned}& \\underset{\\theta (t)}{\\text{min}} \\ J_{\\theta _k}(t)\\\\& \\ s.t:\\\\& \\left\\lbrace \\begin{matrix}\\begin{aligned}&|y(t)+\\sum _{i=1}^{n_a}a_i(t)y(t-i)-\\sum _{j=0}^{n_b}b_j(t)u(t-j)\\\\&-\\sum _{i=1}^{n_a}{\\mathcal {M}}_{a_i}(t)+\\sum _{j=0}^{n_b}{\\mathcal {M}}_{b_j}(t)|\\le \\Delta _\\eta , \\vspace{5.69046pt}\\\\&{\\mathcal {M}}_{\\theta _k}(t) \\ge \\theta _k^{LB}(t)\\epsilon (t-\\lambda )-\\theta _k(t)\\Delta _\\epsilon +\\theta _k^{LB}(t)\\Delta _\\epsilon ,\\vspace{5.69046pt}\\\\&{\\mathcal {M}}_{\\theta _k}(t) \\ge \\theta _k^{UB}(t)\\epsilon (t-\\lambda )+\\theta _k(t)\\Delta _\\epsilon -\\theta _k^{UB}(t)\\Delta _\\epsilon ,\\vspace{5.69046pt}\\\\&{\\mathcal {M}}_{\\theta _k}(t) \\le \\theta _k^{UB}(t)\\epsilon (t-\\lambda )-\\theta _k(t)\\Delta _\\epsilon +\\theta _k^{UB}(t)\\Delta _\\epsilon ,\\vspace{5.69046pt}\\\\&{\\mathcal {M}}_{\\theta _k}(t) \\le \\theta _k(t)\\Delta _\\epsilon +\\theta _k^{LB}(t)\\epsilon (t-\\lambda )-\\theta _k^{LB}(t)\\Delta _\\epsilon ,\\vspace{5.69046pt}\\\\&\\theta _k^{LB}(t) = \\underline{\\theta }_k(t-1)-\\Delta _{\\theta _k},\\vspace{5.69046pt}\\\\&\\theta _k^{UB}(t) = \\overline{\\theta }_k(t-1)+\\Delta _{\\theta _k},\\vspace{5.69046pt}\\\\&\\theta _k^{LB}(t)\\le \\theta _k(t)\\le \\theta _k^{UB}(t),\\vspace{5.69046pt}\\vspace{5.69046pt}\\\\&k = 1,\\ldots ,n_p\\end{aligned}\\end{matrix}\\right.\\end{aligned}$ $\\Box $ Result REF is proved by replacing the first constraint in (REF ) ($A(t,q^{-1})(y(t)-\\eta (t)) = B(t,q^{-1})(u(t)-\\zeta (t))$ ) with the linear inequality (REF ) and the bounds on ${\\mathcal {M}}_{\\theta _k}$ defined in Result REF ."
],
[
"Simulation examples",
"In order to demonstrate the effectiveness of the proposed approach, two numerical examples are presented in this section.",
"Computations are performed on an Intel Core i7-10510U $@$ 1.80GHz computer with 16 GB RAM, using IBM ILOG CPLEX optimizer under Matlab R2018b."
],
[
"Example 1",
"Let us consider the first order LTV system described by the following input-output equation, $w(t) = -a_1(t)w(t-1)+b_1(t)x(t-1),$ where, $b_{1}(t) & = -2+0.5\\sin {\\left[\\frac{2\\pi t}{750}\\right]},\\\\a_{1}(t) & = 0.2+0.4\\sin {\\left[\\frac{2\\pi t}{500}\\right]},$ and parameter variation bounds are $\\Delta _{b_1}=\\pi /750$ and $\\Delta _{a_1}=0.8\\pi /500$ .",
"The input is a random sequence uniformly distributed between $[-1 \\ , \\ +1]$ .",
"Both input and output sequence are corrupted by random additive noise, uniformly distributed between $[-\\Delta _{\\zeta } \\ , \\ \\Delta _{\\zeta }]$ and $[-\\Delta _{\\eta } \\ , \\ \\Delta _{\\eta }]$ , respectively.",
"The error bounds $\\Delta _{\\zeta }$ and $\\Delta _{\\eta }$ are chosen in such a way as to simulate two different values for both the input ($SNR_x = [47, 27]$ ) and the output ($SNR_w = [46, 26]$ ) signal-to-noise ratios, respectively defined as: $SNR_x = 10\\log \\left\\lbrace \\sum _{t=1}^{N}x_{t}^{2}\\bigg / \\sum _{t=1}^{N}\\zeta _{t}^{2} \\right\\rbrace ,$ $SNR_w = 10\\log \\left\\lbrace \\sum _{t=1}^{N}w_{t}^{2}\\bigg / \\sum _{t=1}^{N}\\eta _{t}^{2} \\right\\rbrace ,.$ In this example we consider a data set of length $N=1500$ .",
"Fig.",
"REF shows the computed bounds $\\underline{\\theta }$ and $\\overline{\\theta }$ , at each sampling instant, alongside the central estimates $\\theta ^c$ given by $\\theta ^{c}_{k}(t) = \\frac{\\overline{\\theta }_{k}(t)+\\underline{\\theta }_{k}(t)}{2}, \\quad \\quad k = 1,\\ldots ,n_p,$ which represent the Chebyshev centers in the $\\ell _\\infty $ -norm of ${\\mathcal {D}}_{\\theta }(t)$ and enjoys peculiar optimality properties (see [17] for details).",
"Average CPU time, at each recursion, is about 1.5 ms. We can clearly observe from these figures that parameter $a_1$ changes sign several times, but this has no effect on the performance of the algorithm, and the true parameter is always included in the interval between $\\underline{\\theta }$ and $\\overline{\\theta }$ ."
],
[
"Example 2",
"This second example is taken from [11].",
"The system to be identified is a second order LTV system described by the following transfer function, $G(q^{-1},t) = \\frac{b_{1}(t)q^{-2}}{1+a_{1}(t)q^{-1}+a_{2}(t)q^{-2}},$ where $a_2 = 0.25$ is a fixed parameters, while $a_1$ and $b_1$ vary according to, $b_{1}(t) & = 0.8+0.3\\sin {\\left[\\frac{2\\pi t}{2000}\\right]},\\\\a_{1}(t) & = \\ \\ 1+0.1\\sin {\\left[\\frac{2\\pi t}{1000}\\right]},$ and parameter variation bounds are $\\Delta _{b_1}=0.6\\pi /2000$ and $\\Delta _{a_1}=0.2\\pi /1000$ .",
"The input is a random sequence uniformly distributed between $[-1 \\ , \\ +1]$ .",
"Both input and output sequence are corrupted by random additive noise, uniformly distributed between $[-\\Delta _{\\zeta } \\ , \\ \\Delta _{\\zeta }]$ and $[-\\Delta _{\\eta } \\ , \\ \\Delta _{\\eta }]$ , respectively.",
"The following values for the input and output signal-to-noise ratios have been considered in this example: $SNR_x = [52, 32]$ dB and $SNR_w = [51, 31]$ dB.",
"The length of the data set is $N=2000$ .",
"In Fig.",
"REF , we show a comparison between $\\underline{\\theta }$ , $\\overline{\\theta }$ and $\\theta ^{c}$ computed through the algorithm proposed in [11], referred to as recursive set-membership with known signs ($RSM$ -$S$ ), and the one presented in this work, recursive set-membership with McCormick relaxation ($RSM$ -$M$ ).",
"Average elapsed CPU time are quite the same for the two algorithms (1.8 ms for the $RSM$ -$S$ , and 2.3 ms for $RSM$ -$M$ ).",
"It can be clearly noticed that the bounds computed through both algorithms are overlapping and are perfectly aligned confirming that the algorithm proposed in this work is able to compute tight PUI (global optimal solution of problem (15)) despite no information on the parameter sign is exploited.",
"Figure: Example 1: Computed PUIs and central estimate through the proposed online identification scheme.Figure: Example 2: Comparison between the PUIs computed through RSMRSM-SS and RSMRSM-MM."
],
[
"Conclusions",
"A novel recursive parameter bounding procedure for SISO discrete-time LTV systems in presence of input and output bounded measurement noise is presented.",
"First the problem is formulated as a nonconvex polynomial optimization problem.",
"Then, based on McCormick envelopes convex relaxation, we show that the parameter uncertainty intervals for the LTV system can be computed by means of linear programming without assuming any a-priori information on the parameter signs.",
"The effectiveness of the proposed identification scheme is demonstrated by means of two simulation examples."
]
] | 2107.01714 |
[
[
"Learning a Model for Inferring a Spatial Road Lane Network Graph using\n Self-Supervision"
],
[
"Abstract Interconnected road lanes are a central concept for navigating urban roads.",
"Currently, most autonomous vehicles rely on preconstructed lane maps as designing an algorithmic model is difficult.",
"However, the generation and maintenance of such maps is costly and hinders large-scale adoption of autonomous vehicle technology.",
"This paper presents the first self-supervised learning method to train a model to infer a spatially grounded lane-level road network graph based on a dense segmented representation of the road scene generated from onboard sensors.",
"A formal road lane network model is presented and proves that any structured road scene can be represented by a directed acyclic graph of at most depth three while retaining the notion of intersection regions, and that this is the most compressed representation.",
"The formal model is implemented by a hybrid neural and search-based model, utilizing a novel barrier function loss formulation for robust learning from partial labels.",
"Experiments are conducted for all common road intersection layouts.",
"Results show that the model can generalize to new road layouts, unlike previous approaches, demonstrating its potential for real-world application as a practical learning-based lane-level map generator."
],
[
"INTRODUCTION",
"The concept of road lanes is central for safe and efficient sharing of the road by multiple road users, and by knowing how lanes are connected, it is possible to navigate the road according to traffic rules and conventions using existing motion planning methods [1], [2].",
"However, inferring the directional connectivity between lanes is a non-trivial task.",
"Most autonomous vehicle (AV) systems rely on pre-constructed, high-definition (HD) maps [3] for tasks such as ego-localization and path planning.",
"These maps use a graphical representation of the road scene to enable sensor-based localization [4], [5], [6], efficient rule-based methods [7], [8] or learned methods [9], [10] for motion planning, as well as being advantageous for the prediction task [11].",
"Mapping is still a challenging aspect of AV deployment in the real world, with limited consensus on what kind of HD map to use and even what data and sensors should be employed in the localization and path planning pipeline [3].",
"The variety of map types, and the fact that HD map construction still usually relies on labor-intensive human annotation, make the mapping process difficult to scale.",
"It remains unclear when, how, and if large regions will be mapped with lane-level information for commercial use, thus tending to limit the adoption of AV technology to small geofenced and predetermined routes.",
"In addition, the dependence on the HD map in the ego-localization and motion planning pipeline renders the vehicle unusable when either an HD map is unavailable for a particular road to be traversed, or when a failure in any of the map loading or localization systems occurs.",
"This can also become apparent when changes in road layout, construction, and environmental changes affect the performance of localization and planning.",
"These problems illustrate the importance of AV technology which allows a vehicle to navigate encountered road scenes based on information from onboard sensors only.",
"Figure: The hybrid neural and search-based graph model infers the spatial connectivity of road lanes (graph output) and directional lane affordance (dense affordance map output) for a road scene represented by drivable regions and road markings.",
"Training samples consist of two input layers coupled with one example trajectory label represented as four dense layers.",
"A self-supervised learning method enables the model to learn to infer the complete road lane network graph from single trajectory examples.",
"The model generalizes to new road layouts and partially observed road scenes.Unlike conventional AVs, humans can infer road lanes and their directional connectivity based on contextual features observed in the road scene, having learned to generalize patterns observed from example trajectories in similar road scenes.",
"However, designing an algorithmic model to infer the lane-level road network graph based on observable features is difficult due to regional differences, local feature variation, and noise, as exemplified by varying or missing road markings, curbs, road geometries, and surface materials.",
"This paper presents the first self-supervised learning method for training a model to infer a spatially grounded directional road lane network graph based on a dense segmented representation of the road scene that can be generated online from onboard sensor data [12].",
"The graph can have many uses; as an online substitute for an HD map, to evaluate the consistency of the onboard map with the observed environment, as well as repairing the map.",
"Training samples consist of single trajectories driven through a road scene, which can be automatically generated from human driving data without human labeling effort, allowing learning from an easily obtainable large and varied real-world dataset.",
"Our novel contributions are threefold: The first self-supervised learning method for training a model to output a spatially grounded road lane network graph based on dense segmentation maps representing the road scene.",
"The method learns how to infer the complete graph from single isolated driving trajectory examples without requiring human labeling effort.",
"A novel barrier function loss formulation for robust learning from partial labels, demonstrating practically perfect true positive road lane output (99.93% test accuracy), while keeping the number of false positives low, resulting in distinctly separated lanes.",
"A formal road lane network model proving that any structured road scene can be represented by a directed acyclic graph (DAG) of at most depth three while retaining the notion of intersection regions, and that this is the most compressed representation.",
"The presented hybrid model implements the formal model.",
"Recent works have presented approaches for reducing the amount of human labor needed for map generation.",
"Iesaki et al.",
"[13] presented a method for generating polynomial curves between intersection road lanes, fitted according to a learned cost function.",
"Zhao et al.",
"[14] presented a SLAM-based method to generate a closed vector map including road lanes.",
"Guo et al.",
"[15] presented a method for generating a lane-level road network graph based on superimposed vehicle trajectories and road markings.",
"Road lane connectivity in intersections is inferred from trajectories, and connecting paths are fitted using a heuristic.",
"Neither of these approaches learns contextual features corresponding to lane connectivity, and thus cannot generalize beyond the road scenes the models have been trained on.",
"Additionally, previous models as represented by [15] tend to use heuristics that depend on a particular feature such as road markings.",
"Our work extends upon these studies by introducing a method that can generalize to new road scenes by learning general contextual features extracted from online sensor data.",
"Homayounfar et al.",
"[16] presented a fully automatic approach for inferring discrete road lanes in highway road scenes as polylines using a recurrent neural network model.",
"An extension of this work [17] also introduces forking and merging road lane topology, resulting in a DAG road lane model.",
"Our work further extends their work by demonstrating applicability to road scenes including complex intersections, meaning our approach is applicable for any general structured road scene.",
"Additional merits of our method are that it is self-supervised, formally proved to infer the most compressed representation of the road lane network graph, as well as better suitability for real-time application by not being a recursive model."
],
[
"HD map free road nagivation",
"Related works include Salzmann et al.",
"[12] who trained a probabilistic CNN model to output dense ego-vehicle path affordance from recorded driving trajectories with a weighted mask loss.",
"Amini et al.",
"[18] demonstrated a mapless driving approach for urban roads using a variational neural network (NN) model to navigate an AV.",
"Ort et al.",
"[19] demonstrated an approach for topologically simple rural roads based on combining driving trajectories with a sparse map containing road-level traffic rules.",
"Pérez-Higueras et al.",
"[20] trained a CNN model to infer a multimodal path conditioned on a start and goal point in a grid map environment.",
"Training labels consist of human example trajectories and the output is used to guide an RRT* planner.",
"Barnes et al.",
"[21] demonstrated a self-supervised learning method to train a CNN model to generate dense multimodal paths for an ego-vehicle.",
"While these methods have been shown successful in relatively unconstrained environments, it is not clear how to apply these methods to complicated intersections requiring navigation constrained by traffic rules.",
"Our model adds to overcoming these limitations by inferring a discrete graphical representation that can be used by conventional motion planning algorithms."
],
[
"Road scene understanding",
"Recent studies include Wang et al.",
"[22] who trained a model to generate top-down semantic representations of the road scene for high-level decision making.",
"However, the representation is not spatially grounded nor does it contain lane connectivity and thus cannot be used directly for local planning.",
"Kunze et al.",
"[23] presented a model for generating a hierarchical lane-level scene graph based on road marking and curb detection.",
"The graph is not spatially grounded and thus has the same limitations as [22].",
"Geiger et al.",
"[24] presented a model which infers a dense semantic directional lane affordance representation of the road scene and object poses, based on image sequences of other vehicles moving in the scene.",
"However, the output fidelity is low and relies on observing human driving trajectories in the operating environment instead of learning contextual features as in our proposed model.",
"One can define many ways to represent the road lane network graph.",
"For example, Homayounfar et al.",
"[17] represents the graph densely as sequentially inferred equidistant points.",
"On the other extreme, one can imagine only connecting road lane entry and exit points to represent the graph.",
"To answer the question \"what is the best representation\", we present a formal road lane network graph model capable of expressing the road lane connectivity of any structured road scene.",
"The formal model is found to correspond to a directed acyclic graph (DAG) of maximum depth 3, and is formally proved to be the most compressed graph able to represent the lane network of any road scene while retaining the notion of intersection regions.",
"The formal model along with proofs are presented in the Appendix.",
"The formal road lane network model is implemented as a hybrid two-stage model depicted as a block diagram in Fig.",
"REF .",
"The flow of the hybrid model is as follows.",
"First, dense input features representing the road scene are generated.",
"The input features are feed to a neural deep learning model (Sec.",
"), which outputs a set of dense road lane affordance maps, a multimodal directional field, and road lane entry and exit points.",
"Finally, a road lane graph describing the global connectivity of the road scene is generated by a search-based graph generation model (Sec. )",
"constrained by the dense affordance map output.",
"Figure: The hybrid model consists of two sequential models.",
"The first neural model (Sec. )",
"takes in a set of input features and outputs a set of dense affordances representing road lanes, multimodal directionality, and road lane entry and exit points.",
"The second graph model (Sec. )",
"generates a road lane network graph based on the dense output.The input road scene representation consists of a two-layered 256x256 segmented drivable region and road marking top-down view tensor, spanning a region large enough to encompass a single intersection as exemplified by Fig.",
"REF .",
"The value of each grid map point $(i,j)$ is a probabilistic measure where positive, unknown, and negative observations equal 1, 0.5, and 0, respectively.",
"Previous work demonstrates that such input representations can be generated in real-time based on onboard camera and/or lidar data [12]."
],
[
"Neural model",
"The first part of the hybrid model depicted in Fig.",
"REF generates dense affordance map output that is used by the graph model (Sec. )",
"to generate the road lane network graph.",
"Input features representing dense semantic information of the road scene are fed to the model, which outputs a dense intermediate representation of the directional road lanes, predicting the local structure of the road scene.",
"The neural model is based on the Directional Soft Lane Affordance (DSLA) model for road scene understanding published by Karlsson et al. [25].",
"The multi-task network architecture consists of a single encoder-decoder network with separate output heads for each output type.",
"Three dense affordance networks heads each output a probability value $y_{i,j}$ for every grid map point $(i,j)$ being part of a road lane, lane entry point, and lane exit point.",
"Directionality is modeled by a probabilistic von Mises mixture density network, predicting the directionality of the road at $(i,j)$ as a multimodal directional distribution $p_{i,j}(\\theta )$ for $\\theta ~\\in ~[0, 2 \\pi ]$ .",
"Three directional components are found to be sufficient for representing road lane directionality.",
"However, the model can be modified to output more modalities or even an infinite number of modalities through a sampling-based implementation [26].",
"Each directional mode is visualized as a small arrow in Fig.",
"REF ."
],
[
"Self-supervised neural training process",
"This section describes the self-supervised method used to train the neural model.",
"Self-supervision is implied in the sense that the training data is generated automatically without requiring human labeling, as similarly demonstrated by Nava et al.",
"[27] and Barns et al. [21].",
"Self-supervised methods possess considerable real-world application advantages over conventional supervised methods, as the cost and human effort of obtaining training data can be orders of magnitude lower compared with human-annotated labels.",
"The partial label tensor consists of a 4x128x128 tensor; a trajectory label encoded by a binary mask, two directional 'x' and 'y' labels encoding trajectory direction as element-wise unit vectors $(\\hat{n}_x, \\hat{n}_y)$ , as well as the trajectory entry and exit locations represented as dense points.",
"Labels can be generated from driving data by pairing a representation of the road scene with a path of a vehicle going through it.",
"A training sample is visualized in Fig.",
"REF .",
"The dense affordance networks that predict road lane, entry point, and exit point affordances $y$ from partial labels $\\hat{y}$ are trained by a novel barrier function loss formulation given in Eq.",
"(REF ).",
"The new loss function $L_{aff}$ displays favorable characteristics for learning categorical predictions from partial labels compared with the masked L2 loss formulation used in [25], such as improved generalization ability and lack of occasional discontinuities in road lane affordance map output.",
"$ L_{aff} = \\frac{1}{N} \\left( \\sum _{i,j} \\vert \\hat{y} - y \\vert - \\alpha \\: CE(y,\\hat{y}) \\right)$ The underlying principle of the barrier loss formulation is that the model prediction elements $y_{i,j}$ should at least always correspond to known positive label elements $\\hat{y}_{i,j}$ , and as such the model is heavily penalized by a theoretically infinite loss when mispredicting elements encompassed by the partial label.",
"This is achieved by multiplying the cross entropy term by a large value $\\alpha = 10^5$ .",
"Gradient clipping is utilized to prevent exploding gradients.",
"The unknown elements of the label $\\hat{y}$ penalizes the model output $y$ through an L1 loss term, which is known to be inherently robust to uncertainty and noisy labels [28].",
"In case both positive and negative partial label elements are available, an additional cross entropy term for the known negative elements can be added.",
"Parallels can be drawn between the process of learning with a barrier function as loss, and the version space algorithm in symbolic machine learning [29] that reduces an a priori set of all possible hypotheses to a subset of hypotheses satisfying all observed examples.",
"For the case of learning road lane affordance with partially known positive labels (i.e.",
"an example trajectory), and lack of any known negative labels, the model initially learns to trivially predict all elements as positive, analogous to starting with the initial set of all possible hypotheses in the version space algorithm.",
"The L1 loss component gradually drives the model output towards a default prediction value which is negative in this work, without mispredicting known positives and thus avoids incurring a large penalty by the barrier function loss, which is analogous to gradual hypothesis reduction in the version space algorithm.",
"Further mathematical analysis of this intuitive similarity is believed to reveal why the proposed barrier function loss formulation is so effective at learning and generalizing from partial labels compared with simply increasing the overall learning rate."
],
[
"Directional affordance learning",
"The second loss formulation involves minimizing the KL divergence loss for every grid map point $(i,j)$ encompassed by the trajectory label following [25].",
"$ L_{DA} = \\frac{1}{N} \\sum _{i,j \\in mask} D_{KL} ( p(\\theta )_{i,j} || \\hat{p} (\\theta )_{i,j} )$ The multimodal directional distribution prediction $p(\\theta )_{i,j}$ is generated by three output network heads.",
"The directional label is obtained from the example trajectory, resulting in a monomodal distribution $\\hat{p}(\\theta )_{i,j}$ , approximating the true multimodal directionality at the point."
],
[
"Graph model",
"The final part of the hybrid model depicted in Fig.",
"REF generates the spatially grounded discrete road lane network graph based on the dense affordance predicted by the neural model.",
"First, a weighted adjacency matrix $A$ is generated, representing the navigational constraints imposed by the dense affordance output, defining the cost function for the A$^*$ search algorithm [30] used to search for valid paths connecting entrance and exit points.",
"Secondly, all found paths are decomposed into a set of entrance, intersection, and exit lanes by unifying paths corresponding to the same road lane.",
"The resulting graph is a DAG of maximum depth three, satisfying the requirements stipulated by the formal model defined in the Appendix.",
"The rest of this section explains each step in more detail.",
"Figure: To ease the unification of paths corresponding to the same road lane, the dense road lane affordance map output is smoothed, so that all optimal paths found by a search algorithm coincide with lane centers.",
"[t] InputInput OutputOutput Initialize empty path sets $P_{entry}$ , $P_{inter}$ , $P_{exit}$ , $P_{con}$ Find entry paths All entry points $q_{entry} \\in Q_{entry}$ $P_{entry\\_tree} = \\lbrace \\rbrace $ All exit points $q_{exit} \\in Q_{exit}$ Path $p$ = A$^{*}$ ($q_{entry}$ , $q_{exit}$ , $A$ ) $p$ exists Add $p$ to $P_{entry\\_tree}$ $p_{entry}$ , $P^{*}_{con}$ = unify($P_{entry\\_tree}$ ) Add $p_{entry} \\rightarrow P_{entry}$ Add $P^{*}_{con} \\rightarrow P_{con}$ Find intersection, exit paths All exit points $q_{exit} \\in Q_{exit}$ $P_{exit\\_tree} \\leftarrow $ Paths in $P_{con}$ ending at exit point $p_{exit}$ , $P^{*}_{inter}$ = reverse_unify($P_{exit\\_tree}$ ) Add $p_{exit} \\rightarrow P_{exit}$ Add $P^{*}_{inter} \\rightarrow P_{inter}$ Construct graph $G$ from $P_{entry}$ , $P_{inter}$ , $P_{exit}$ Road lane network graph $G = (V, E)$ Road lane network graph generation"
],
[
"Weighted adjacency matrix generation",
"The weighted adjacency matrix $A$ defines the cost function for the A$^*$ search algorithm.",
"The generation of $A$ starts with an 8-directional 2D grid map adjacency matrix that is initialized with infinite weight assigned to all edges.",
"The edge weights $e_{A,B}$ are subsequently reset according to the dense affordance output as follows.",
"First, the dense road lane affordance output $y_{i,j}$ of each grid map point $(i,j)$ is thresholded to 0 and 1 and subsequently smoothed by an 8x8 kernel and transformed by the function$ f(\\tilde{y}) = \\tilde{y}^8$ , so that optimal paths found by the search algorithm will coincide with lane centers.",
"The effect of smoothing is displayed in Fig.",
"REF .",
"Next, for each gird map point $A$ at $(i,j)$ , a neigboring point $B$ is reachable, if the direction $\\overrightarrow{A \\: B}$ is within an angle $\\Delta \\theta $ of any dense directional affordance components $\\lbrace \\theta _1, \\theta _2, \\theta _3\\rbrace $ predicted by the neural model at point $(i,j)$ .",
"The edge weight to all reachable neighbors is reset according to Eq.",
"(REF ).",
"$e_{A,B} = \\vert \\overrightarrow{A \\: B} \\vert - \\log {\\tilde{y}_{B}}$"
],
[
"Search-based path algorithm",
"The rest of this section explains the search-based algorithm which finds paths between road lane entry and exit points with $A$ as the cost function, and eventually decomposes the found paths into a road lane network graph $G$ corresponding to the formal model defined in the Appendix.",
"First, a set of discrete entry and exit points $Q_{entry}$ and $Q_{exit}$ are computed from the dense entry and exit affordance map output of the neural model, determined as the center of momentum coordinate $(i,j)$ of the contour constituting each separable cluster of dense output.",
"The road lane network graph $G$ computation is summarized in Algorithm and explained as follows.",
"Three empty path sets $P_{entry}$ , $P_{inter}$ , $P_{exit}$ are initialized (line 1), denoting the set of entry, intersection, and exit paths, each corresponding to the equivalent set of entry, intersection, and exit lanes defined in the formal model.",
"This decomposition is proved to be possible for any structured road scene.",
"For each entry point $q_{entry}$ (line 2), the paths to all reachable exit points $q_{exit}$ are found using the A$^*$ search algorithm with the weighted adjacency matrix $A$ as the cost function.",
"All such paths are rooted at a single entry point and thus constitute a tree $P_{entry\\_tree}$ (lines 3-7).",
"The tree is unified from the root up until the first point where the tree diverges into separate branches, meaning all paths in the tree are modified to consist of the same entry path (line 8).",
"The unification algorithm returns the common entry path $p_{entry}$ and a new set of connecting paths $P^{*}_{con}$ forming a tree rooted at the end of $p_{entry}$ .",
"These paths are added to the maintained sets (lines 9-10).",
"Repeating these steps for all entrance points results in the set $P_{entry}$ constituting the paths of all entry lanes in the road scene.",
"The metric used for tree divergence is the total angle $\\Theta = \\sum ^{N-1}_{i=0}{\\Delta \\theta _{i,i+1}}$ spanned between all $N$ path directions arranged in counterclockwise order.",
"The largest angle $\\Delta \\theta _{N-1, N}$ is excluded as it represents the angle not spanned by the path directions.",
"Path directionality is represented by a vector $\\overrightarrow{p_{t} \\: p_{t+\\Delta t}}$ spanned by a point $p_{t}$ and a future point $p_{t + \\Delta t}$ on the same path.",
"This vector approximates the future directionality given a suitable number of lookahead steps $\\Delta t$ .",
"In this work $\\Delta t = 6$ .",
"The next part of Algorithm applies the same process in the reverse direction.",
"For each exit point $q_{exit}$ (line 11), all paths ending at $q_{exit}$ form a reverse path tree rooted at $q_{exit}$ .",
"After reversing all path directions, the same unification algorithm can be applied to unify the reverse tree from the end to the point where all paths converge into a single path, returning the common exit path $p_{exit}$ and a new set of paths $P^*_{inter}$ forming a reverse tree into the start of $p_{exit}$ (line 13).",
"These paths are added to the maintained sets (lines 14-15).",
"Repeating these steps for all exit points results in the set $P_{exit}$ constituting the paths of all exit lanes in the road scene.",
"Additionally, the remaining set of pruned paths $P_{inter}$ constitute all paths between connected entry and exit paths.",
"The road network graph $G$ is constructed by letting the first and last point of every path $p \\in P_{entry}$ represent an entry and fork vertex sharing a directed edge.",
"Similarly, the first and last point of every path $p \\in P_{exit}$ represents a merge and exit vertex sharing a directed edge.",
"The first and last point of all paths $p \\in P_{inter}$ represents directed edges between a pair of fork and merge vertices, or in other words, intersection connectivity.",
"The formal model is proved to be able to represent any structured road scene, and as $G$ is an arbitrary but particular instance of the formal model, by universal modus ponens, $G$ can represent any structured road scene given the dense affordance output by the neural model correctly represent the local structure of the road scene.",
"Note that the method does not only infer connectivity between road lanes, but also infers intuitive spatial locations for fork and merge points, and consequently, the spatial extent of intersections."
],
[
"Experiments",
"The model is trained and tested on randomly generated samples drawn from two data distributions $p_{train}(x)$ and $p_{test}(x)$ representing the two sets of artificially created road layouts visualized in Fig.",
"REF .",
"Training samples are generated by randomly augmenting road layouts from $p_{train}(x)$ together with one example trajectory as visualized in Fig.",
"REF .",
"$p_{train}(x)$ is intended to cover all fundamental components of intersection layouts [31].",
"Evaluation samples are generated from road layouts in both $p_{train}(x)$ and $p_{test}(x)$ by the same augmentation process, but with all valid trajectories superimposed.",
"Generalization ability is evaluated by how well the model performs on samples drawn from $p_{test}(x)$ , consisting of road layouts with similar components as in $p_{train}(x)$ but with new combinations, varying number of road lanes, and asymetric structure.",
"The neural model is trained on an Nvidia RTX 3090 GPU using stochastic gradient descent (SGD) with mini-batches of 28 samples and polynomial learning rate decay, reducing the initial learning rate 10$^{-3}$ to 10$^{-5}$ over 180,000 iterations with power 0.9."
],
[
"Data augmentation",
"Data augmentation is a critical part of training the neural model to be invariant to particular road geometries as all training examples are unique, and thus improve generalization ability.",
"This work follows the data augmentation approach presented in [25] for generating randomly translated, rotated, and warped road scene samples."
],
[
"Performance metrics",
"This work presents two new metrics $Acc^{eval}_{pos}$ and $L1^{eval}_{neg}$ to evaluate the performance of dense road lane, entry and exit point affordance predictions.",
"Directional affordance $D^{eval}_{KL}$ is evaluated by computing KL divergence between the model output and evaluation label [25].",
"A set of 20 pregenerated samples per road layout is used for evaluation.",
"Desirable dense road lane affordance predictions have two requirements; all true positive elements should be predicted positive, and the less false positives the better.",
"The former requirement can be represented by $Acc^{eval}_{pos}$ as the ratio of all true positive elements actually predicted positive (i.e.",
"$y_{i,j} > 0.5$ ).",
"The later requirement can be quantified by the linear L1 metric $L1^{eval}_{neg} = \\sum ^N_i (\\vert \\hat{y_i} - y_i \\vert ) / N $ for $i \\in $ set of negative evaluation label elements.",
"The topological correctness of a road lane network graph $G$ is measured in terms of the number of missing or erroneous edges compared with the graph generated from the evaluation sample label."
],
[
"Results",
"Output visualizations of testing road layouts sampled from $p_{test}(x)$ are shown in Fig.",
"REF and Fig.",
"REF .",
"The results demonstrate how the neural model has learned a generalized local representation of road scenes, which enables the graph algorithm to infer the global connectivity for new road scenes, including asymmetric intersections and one-way roads.",
"In Fig.",
"REF white heatmap regions represent predicted dense road lane affordance, The red and blue colored blobs represent predicted dense entry and exit points.",
"The small arrows represents predicted local multimodal directionality.",
"The road lane network graph is superimposed over the dense output, with entry edges colored blue, intersection edges colored red, exit edges colored green, and non-intersection lanes reduced to a single purple edge.",
"Results are visualized by a neural model trained for 100,000 iterations, which takes about 26 hours on our hardware.",
"Fig.",
"REF provides a quantitative evaluation of training and generalization performance of the neural model output.",
"The dense affordances learned by the barrier function loss formulation Eq.",
"(REF ) results in comparable performance between train and test samples and continue to improve with further training iterations beyond 140,000 iterations, indicating strong generalization and thus learning of sound abstract representations of road scene structures.",
"On the other hand, directional affordance performance converges relatively early after about 40,000 iterations and there is a larger gap between training and testing performance, both indicating that there is room for further improvements in directional modeling.",
"The result of road lane network graph evaluation is that 337/340 (99.1%) training layouts are error free.",
"Two errors are related to scenes with disconnected merging lanes.",
"For testing layouts, 199/220 (90.5%) samples are error free.",
"14 failure cases are related to inconsistent directional affordance in asymmetric intersection layouts, such as one-way roadways and lanes with ambiguous directionality.",
"The other 7 failures are for the forking lane intersection where the forked road lane is disconnected or otherwise improperly masked out and obstructs the search algorithm.",
"As explained in Sec.",
"REF , our proposed method is the first self-supervised approach for road lane network learning, and extending prior methods to become self-supervised is nontrivial, making direct comparison difficult."
],
[
"Conclusions",
"This work presents the first self-supervised method for learning a hybrid neural and search-based model to infer a spatially grounded graphical representation of the road lane network, based on a dense segmented input representation of the road scene generatable from sensor data as input.",
"The model is formally proved to be the most compressed representation of the road lane network while retaining the notion of intersection regions.",
"Training data can be automatically and cost-effectively generated from driving data.",
"Data augmentation enables the model to learn invariance to particular geometries and instead learn representations of general structures of road layouts.",
"Experiments demonstrate the model successfully generalizing to new intersection layouts not encountered during training, therefore advancing the state of the art in automatic road lane network modeling.",
"Experiment results demonstrate the potential of our approach to enable conventional HD map dependent AV motion planning and prediction systems to operate in the absence of human annotated HD map information, by utilizing artificial intelligence to generate a discrete graphical representation of the road lane network based on onboard sensor data.",
"Future work includes adding a symbolic learning and reasoning step to refine the road lane network graph and resolve ambiguity associated with highly asymmetric road scenes, as well as refine edges to conform with explicit and implicit traffic rules, such as considering traffic signs and generally prohibiting U-turns.",
"The method is to be demonstrated using real-world data.",
"Previous works [12], [18], [20], [21], [22] have demonstrated that CNN-based navigation model approaches can work with noisy and partially occluded real-world input representations of urban road scenes, and thus it is reasonable to expect our method and experimental results to also transfer to real-world inputs."
],
[
"APPENDIX: Formal road lane network model",
"The road lane network model is a formal model to represent the road lane connectivity of any structured road scene.",
"The model is formally proved to be the most compressed representation retaining the notion of intersection regions.",
"Definition 1 (Road scene) The approximate representation of the external world, to the extent relevant for the navigation task, is defined as the road scene [32].",
"This representation includes a description of the spatial configuration and semantics of static and dynamic elements constituting the scene, such as drivable regions and road markings.",
"Definition 2 (Road lane network) The directional connectivity between lanes in a road scenes can be mathematically represented as directed graph $G(V, E)$ .",
"The set of vertices $V = \\lbrace v_1 , \\dots , v_N\\rbrace $ constitute structurally representative points where lanes enter and exit the road scene (i.e.",
"entry and exit vertices), as well as internal points associated with intersection.",
"The set of directed edges $E = \\lbrace e_1, \\dots , e_M \\rbrace $ represents the connectivity between the structural points.",
"Definition 3 (Intersection) An intersection or junction denotes a part of the road scene where incoming road lanes branch and connect with lanes from other entry points.",
"In other words, intersections present a navigational branching choice.",
"All road lanes exiting the intersection at the same point are merged into a single exit lane.",
"Mathematically, an intersection is a subgraph $H$ of the road lane network graph $G$ consisting of at least one fork vertices with outdegree $deg^{-}{(v)} \\ge 2$ (i.e.",
"two or more outgoing edges) or merge vertices with indegree $deg^{+}(v) \\ge 2$ (i.e.",
"two or more incoming edges).",
"If both fork and merge vertices exist, then for every fork vertex there exists at least one edge to some merge vertex and vice versa.",
"A road lane network can contain multiple intersections if and only if the intersections are disjoint subgraphs.",
"Lemma 1 Any road scene can be spatially reduced so that every road lane is part of only one intersection while still being a sufficient representation for the navigation task.",
"A human driver perceiving the road one intersection at a time can successfully navigate any road lane of any road scene.",
"Therefore, by modus ponens, a sufficiently intelligent AV system utilizing a road scene representation encompassing only a single intersection per road lane can also successfully navigate any road scene.",
"Lemma 2 The road lane network for any road scene can be represented by a directed acyclic graph (DAG) of at most depth three.",
"Suppose a directed graph $G = (V, E)$ represents the road lane network graph of a particular but arbitrary road scene.",
"By the definition of a road lane network, all valid road lanes begin at an entry node $v_{entry}$ and end at an exit node $v_{exit}$ .",
"By Lemma REF , all valid road lanes can be a part of only one intersection.",
"By the definition of an intersection, each incoming road lane is associated with one fork node $v_{fork}$ and a merge node $v_{merge}$ , or only one of the nodes.",
"Thus, any valid road lane originating from any road scene entry point can be expressed by at most four vertices, connected by three directional edges.",
"Additionally, as every child vertex is strictly closer to an exit vertex than its parent vertex, every path through the graph decreases the distance to an exit vertex, and the graph is acyclic.",
"Therefore, the directed graph $G$ has a depth of at most three edges and is acyclic.",
"Corollary 1 Any road lane that is part of an intersection can be decomposed into three edges; an entry edge from an entry vertex to a fork vertex, an intersection edge from a fork vertex to a merge vertex, and an exit edge from a merge vertex to an exit vertex.",
"Corollary 2 A road lane not part of an intersection can be represented by a single directional edge.",
"Corollary 3 A special case of intersections is the branching of a single road lane, or merging of two road lanes, constituting a point intersection represented by a DAG of depth two, which is the most compact representation.",
"Corollary 4 Road scenes with an intersection consisting of two or more incoming and outgoing road lanes can be represented by a DAG of depth three, which is the most compact representation.",
"This research was supported by Program on Open Innovation Platform with Enterprises, Research Institute and Academia, Japan Science and Technology Agency (JST, OPERA, JPMJOP1612)."
]
] | 2107.01784 |
[
[
"Real-time Detection and Adaptive Mitigation of Power-based Side-Channel\n Leakage in SoC"
],
[
"Abstract Power-based side-channel is a serious security threat to the System on Chip (SoC).",
"The secret information is leaked from the power profile of the system while a cryptographic algorithm is running.",
"The mitigation requires efforts from both the software level and hardware level.",
"Currently, there is no comprehensive solution that can guarantee the whole complex system is free of leakage and can generically protect all cryptographic algorithms.",
"In this paper, we propose a real-time leakage detection and mitigation system which enables the system to monitor the side-channel leakage effects of the hardware.",
"Our proposed system has extensions that provide a real-time monitor of power consumption, detection of side-channel leakage, and real-time adaptive mitigation of detected side-channel leakage.",
"Our proposed system is generic and can protect any algorithm running on it."
],
[
"Introduction",
"Side-channel leakage (SCL) discloses secret information through power consumption, electromagnetic dissipation, execution time, and other sources.",
"In this work, our focus is on power-based SCL.",
"Power side-channel analysis (SCA) has made the hardware attacker a particularly powerful adversary, relevant to embedded applications of secure System-on-Chips (SoC).",
"When an SoC handles secret values, such as secret keys used in a symmetric cryptographic algorithm, the physical side-effects of these computations may be exploited as SCL.",
"SCL is a critical vulnerability of secure SoCs as sophisticated SCA enables attackers to learn about the secret information even when the cryptographic algorithm is computationally secure.",
"When a program is running on a processor, unexpected SCL can happen.",
"A software program is optimized and transformed into an executable file by a compiler.",
"Generally, compilers are designed to optimize the speed or memory footprint of code, however, that is not always aligned with critical software security concerns.",
"For example, in a masked implementation, compiler tasks such as register allocation and strength reduction (leading to shift operations), might cause unintended unmasking.",
"Furthermore, even the existing algorithms for secure software design, like probing secure multiplication (ISW) [1] or bounded-moment secure multiplication [2], have been shown to have leakage caused by unintended hardware effects [3], [4].",
"While the software implementation causes the side-channel leakage, it is the processor hardware that creates the physical effects of SCL.",
"Therefore, even if the software includes countermeasures against SCL, it is very hard to guarantee that the underlying hardware will be leakage free while running the software program.",
"A processor consists of many elements including memory units and pipeline stages.",
"Power SCL can arise from several parts of the micro-architecture simultaneously at times and individually at others.",
"Therefore it is challenging to pin-point the exact part of the micro-architecture causing this leakage.",
"Figure: Proposed mechanism.",
"Power sensors monitor the power.",
"Leakage detection module predicts if a power SCL is happening based on the monitored power and alarms the adaptive controller.",
"Adaptive countermeasure controller enables local countermeasures through ACCs.Current research efforts attempt to mitigate side-channel leakage from two ends.",
"From the software side, researchers attempt to develop side-channel resistant software.",
"This includes secure programming mechanisms [2], [5], and leakage-mitigating code-generation techniques [6].",
"From the hardware side, the hardware designers focus on building side-channel resistant processor designs and instruction set extensions to eliminate the dependencies of power consumption on the secret values and prevent accidental unmasking in masked software programs [7], [8], or building up design-time (pre-silicon) side-channel evaluation [9], [10].",
"Software-only and hardware-only approaches are sub-optimal.",
"The software designer will have to apply countermeasure to a substantial part of the code, if not the entire code, including the leaking section while being oblivious to the detailed design of the hardware.",
"The processor hardware designer will have to build a completely new processor with corresponding countermeasures applied [11].",
"These approaches cause a large overhead in software execution time and code size, or in hardware area overhead and performance.",
"We propose to enable the processor system to monitor the side-channel leakage.",
"We introduce a sensor that detects leakage in real-time and that triggers adaptive countermeasures.",
"We present a proactive security mechanism based on hardware extensions.The extensions provide a real-time monitor of the power consumption (using sensor cells in Fig.",
"REF ), detection of side-channel leakage (using leakage detection cells in Fig.",
"REF ), and real-time adaptive mitigation of detected side-channel leakage (using adaptive countermeasure cells (ACC) in Fig.",
"REF ).",
"When the system is running, the proposed security mechanism (implemented as sensors) continuously monitors the leakage status of the system.",
"Once a leakage is detected by the implemented sensors, the alarm signal will be set.",
"This alarm signal has two functions: the alarm points out the precise leakage source in the hardware, and it triggers the adaptive countermeasure to mitigate the leakage."
],
[
"System Design",
"We propose to build an insight into tracking, identifying, and mitigating side-channel leakage, while maintaining minimal implementation overhead.",
"Using this methodology, the underlying hardware (processor) provides protection against power SCA for any program running on it.",
"In our solution, then, the software programmer can be unaware of the power consumption of the hardware; the programmer may write code and generate the assembly using off-the-shelf compilers.",
"The design time is therefore dramatically reduced, and the existing compiler optimizations are applied to provide a performance boost while ensuring resistance to power analysis attacks.",
"The challenges facing the implementation of such a system are three-fold: detecting leakage, finding the source of leakage, and dynamically eliminating leakage.",
"To address these challenges, our proposed real-time leakage detection system contains the following modules:"
],
[
"In situ power measurement",
"Traditionally, power measurements were applied to the prototype of the chip.",
"To achieve real-time leakage detection, we will implement an on-chip in situ power measurement based on power modeling.",
"We also intend to study the possibility of utilizing existing on-chip sensors for the same purpose.",
"Figure: Each ACC turns on as soon as the leakage metric corresponding to its nearby power sensor passes the high threshold (Th high Th_{high}).",
"Once the leakage decrements to lower than Th low Th_{low}, ACC will turn off."
],
[
"Leakage detection",
"After getting the real-time in situ power measurement, we will come up with the leakage evaluation mechanism and implement the corresponding hardware extension within the processor to achieve leakage detection.",
"The key challenge in leakage detection is building a suitable leakage evaluation metric that fits well within the needs of real-time leakage detection and can be used for general-purpose programs.",
"TVLA [12] is currently the most popular leakage assessment technique, however, it requires categorization of inputs into fixed and random sets and therefore cannot be directly applied to our real-time leakage detection.",
"Possible leakage metrics include power distribution-based characterization or machine learning-based metrics."
],
[
"Adaptive countermeasure",
"Our goal is to keep the leakage probability low; start the protection once the probability of leakage is higher than a threshold and stop it when the probability is lower than another (as shown in Fig.",
"REF ).",
"This has the advantage that while the design is kept secure, only the necessary parts of the design are protected.",
"As the first step to build up the whole real-time system, we implemented the power sensors as hardware extensions to an SoC with a RISC-V processor and fabricated the chip with the CMOS 180nm technology for further testing.",
"We chose digital Ring-Oscillators (RO) as our power sensors to be able to mirror the changes in the power consumption of the chip in through their oscillating frequency.",
"The power sensors are added as co-processors in the SoC to communicate with the processor.",
"In the chip layout, as shown in Fig.REF , the power sensors are evenly distributed throughout the design to monitor the local power consumption in real-time while the chip is running a program.",
"Figure: Layout of the chip, integrating the power sensors scattered throughout the layout.",
"The yellow dashed squares highlight the power sensors."
],
[
"Conclusion and Future Work",
"We believe that our work is the first to build a mechanism that provides security to any algorithm and dynamically applies corresponding countermeasures.",
"The presented system is expected to be very generic; once we have an implementation of such a system, we can securely run any algorithm on it.",
"This is also the case for a protected processor implementation.",
"Whereas, in a software protection realm, any new algorithm would need its own security implementation from the ground up, hence having a low genericness.",
"All these advantages of our proposed method come at the cost of silicon area overhead which is still expected to be lower than that of a protected processor implementation."
]
] | 2107.01725 |
[
[
"Improve Agents without Retraining: Parallel Tree Search with Off-Policy\n Correction"
],
[
"Abstract Tree Search (TS) is crucial to some of the most influential successes in reinforcement learning.",
"Here, we tackle two major challenges with TS that limit its usability: \\textit{distribution shift} and \\textit{scalability}.",
"We first discover and analyze a counter-intuitive phenomenon: action selection through TS and a pre-trained value function often leads to lower performance compared to the original pre-trained agent, even when having access to the exact state and reward in future steps.",
"We show this is due to a distribution shift to areas where value estimates are highly inaccurate and analyze this effect using Extreme Value theory.",
"To overcome this problem, we introduce a novel off-policy correction term that accounts for the mismatch between the pre-trained value and its corresponding TS policy by penalizing under-sampled trajectories.",
"We prove that our correction eliminates the above mismatch and bound the probability of sub-optimal action selection.",
"Our correction significantly improves pre-trained Rainbow agents without any further training, often more than doubling their scores on Atari games.",
"Next, we address the scalability issue given by the computational complexity of exhaustive TS that scales exponentially with the tree depth.",
"We introduce Batch-BFS: a GPU breadth-first search that advances all nodes in each depth of the tree simultaneously.",
"Batch-BFS reduces runtime by two orders of magnitude and, beyond inference, enables also training with TS of depths that were not feasible before.",
"We train DQN agents from scratch using TS and show improvement in several Atari games compared to both the original DQN and the more advanced Rainbow."
],
[
"Introduction",
"Tree search (TS) is a fundamental component of Reinforcement Learning (RL) [41] used in some of the most successful RL systems [37], [39].",
"One popular TS method, Monte-Carlo Tree Search (MCTS) [10], achieved superhuman performance in board games like Go [39], Chess [40], and Bridge [5].",
"MCTS gradually unfolds the tree by adding nodes and visitation counts online and storing them in memory for future traversals.",
"This paradigm is suitable for discrete state-spaces where counts are aggregated across multiple iterations as the tree is built node-by-node, but less suitable for continuous state-spaces or image-based domains like robotics and autonomous driving.",
"For the same reason, MCTS cannot be applied for improving pre-trained agents without collecting their visitation statistics in training iterations.",
"Instead, we explore the possibility of conducting the TS “on-demand” by expanding the tree up to a given depth at each state.",
"Our approach handles continuous and large state-spaces like images without requiring any memorization.",
"This on-demand TS can be performed both at training or inference time.",
"Here, we focus our attention on the second case, which leads to score improvement without any re-training.",
"This allows one to better utilize existing pre-trained agents even without having the ability or resources to train them.",
"For example, a single AlphaGo training run is estimated to cost 35 million USD [1].",
"In other cases, even when compute budget is not a limitation, the setup itself makes training inaccessible.",
"For example, when models are distributed to end-clients with too few computational resources to train agents in their local custom environments.",
"We run TS for inference as follows.",
"For action selection, we feed the states at the leaves of the spanned tree to the pre-trained value function.",
"We then choose the action at the root according to the branch with the highest discounted sum of rewards and value at the leaves.",
"Our approach instantly improves the scores of agents that were already trained for long periods (see Sec.",
"REF ).",
"Often, such improvement is possible because the value function is not fully realizable with a function approximator, e.g., a deep neural network; TS can then overcome the limitation of the model.",
"In practice, TS requires access to a forward model that is fed with actions to advance states and produce rewards.",
"Here, we build on the recently published CuLE [11] – an Atari emulator that runs on GPU.",
"This allows us to isolate the fundamental properties of TS without the added noise of learned models such as those described in Sec..",
"Performing TS on-demand has many benefits, but it also faces limitations.",
"We identify and analyze two major obstacles: distribution shift and scalability.",
"First, we report a counter-intuitive phenomenon when applying TS to pre-trained agents.",
"As TS looks into the future, thus utilizing more information from the environment, one might expect that searching deeper should yield better scores.",
"Surprisingly, we find that in many cases, the opposite happens: action selection based on vanilla TS can drastically impair performance.",
"We show that performance deteriorates due to a distribution shift from the original pre-trained policy to its corresponding tree-based policy.",
"We analyze this phenomenon by quantifying the probability of choosing a sub-optimal action when the value function error is high.",
"This occurs because for values of out-of-distribution states, larger variance translates to a larger bias of the maximum.",
"Our analysis leads to a simple, computationally effective off-policy correction term based on the Bellman error.",
"We refer to the resulting TS as the Bellman-Corrected Tree-Search (BCTS) algorithm.",
"BCTS yields monotonically improving scores as the tree depth increases.",
"In several Atari games, BCTS even more than doubles the scores of pre-trained Rainbow agents [20].",
"The second limitation is scalability: the tree grows exponentially with its depth, making the search process computationally intensive and limiting the horizon of forward-simulation steps.",
"To overcome this limitation, we propose Batch-BFS: a parallel GPU adaptation of Breadth-First Search (BFS), which brings the runtime down to a practical regime.",
"We measured orders-of-magnitude speed-up compared to alternative approaches.",
"Thus, in addition to improving inference, it also enables training tree-based agents in the same order of training time without a tree.",
"By combining Batch-BFS with DQN [28] and training it with multiple depths, we achieve performance comparable or superior to Rainbow – one of the highest-scoring variants of the DQN-based algorithm family.",
"Our Contributions.",
"(1) We identify and analyze a distribution-shift that impairs post-training TS.",
"(2) We introduce a correction mechanism and use it to devise BCTS: an efficient algorithm that improves pre-trained agents, often doubling their scores or more.",
"(3) We create Batch-BFS, an efficient TS on GPU.",
"(4) We use Batch-BFS to train tree-based DQN agents and obtain higher scores than DQN and Rainbow."
],
[
"Preliminaries",
"Our framework is an infinite-horizon discounted Markov Decision Process (MDP) [34].",
"An MDP is defined as the 5-tuple $(\\mathcal {S}, {\\mathcal {A}},P,r,\\gamma )$ , where ${\\mathcal {S}}$ is a state space, ${\\mathcal {A}}$ is a finite action space, $P(s^{\\prime }|s,a)$ is a transition kernel, $r(s,a)$ is a reward function, $\\gamma \\in (0,1)$ is a discount factor.",
"At each step $t=0,1,\\dots ,$ the agent observes the last state $s_t$ , performs an action $a_t$ and receives a reward $r_t$ .",
"The next state is then sampled by $s_{t+1} \\sim P(\\cdot | s_t, a_t)$ .",
"For brevity, we denote $A:=|{\\mathcal {A}}|.$ Let $\\pi : \\mathcal {S}\\rightarrow {\\mathcal {A}}$ be a stationary policy.",
"Let $Q^\\pi : {\\mathcal {S}}\\times {\\mathcal {A}}\\rightarrow \\mathbb {R}$ be the state-action value of a policy $\\pi ,$ defined in state $s$ as $Q^\\pi (s, a) \\equiv \\mathbb {E}^\\pi \\left[ \\sum _{t=0}^\\infty \\gamma ^tr(s_t,\\pi (s_t)) \\big | s_0=s, a_0=a \\right]$ , where $\\mathbb {E}^\\pi $ denotes expectation w.r.t.",
"the distribution induced by $\\pi $ .",
"Our goal is to find a policy $\\pi ^*$ yielding the optimal value $Q^*$ such that $Q^*(s,a) = \\max _\\pi r(s,a) + \\gamma \\mathbb {E}_{s^{\\prime }\\sim P(\\cdot | s, a)} \\max _{a^{\\prime }} Q^\\pi (s^{\\prime }, a^{\\prime })$ .",
"It is well known that $Q^*(s, a) = r(s,a) + \\gamma \\mathbb {E}_{s^{\\prime }\\sim P(\\cdot | s, a)} \\max _{a^{\\prime }} Q^* (s^{\\prime }, a^{\\prime }), \\quad \\quad \\quad \\pi ^*(s) = \\operatornamewithlimits{arg\\,max}_a Q^*(s,a).$ Vanilla tree search.",
"To ease notations and make the results concise, we limit the analysis to deterministic transitionsThe Atari environments we experiment on here are indeed close to deterministic., i.e., an action sequence $(a_0,\\dots ,a_{d-1})$ , starting at $s_0$ leads to a corresponding trajectory $(s_0,\\dots ,s_d).$ Nonetheless, the results can be extended to a stochastic setup by working with the marginal probability over the trajectory.",
"Then, for a policy $\\pi _o,$ let the $d$ -step Q-function ${Q^{\\pi _o}_d}(s, a) = \\Bigg [ \\max _{(a_k)_{k=1}^d \\in {\\mathcal {A}}} \\Bigg [ \\sum _{t=0}^{d-1} \\gamma ^t r(s_t, a_t) \\bigg ] + \\gamma ^d {Q^{\\pi _o}}(s_d, a_d) \\Bigg ]_{s_0=s, a_0=a},$ and similarly let $\\hat{Q}^{\\pi _o}_d(s,a)$ be the d-step Q-function estimator that uses an estimated Q-function $\\hat{Q}^{\\pi _o}$ instead of ${Q^{\\pi _o}}.$ Finally, denote by $\\pi _d$ the $d$ -step greedy policy $\\pi _d(s) := \\arg \\max _{a \\in {\\mathcal {A}}} \\hat{Q}^{\\pi _o}_d(s, a).$"
],
[
"Solving the Tree Search Distribution Shift",
"In this section, we show how to leverage TS to improve a pre-trained policy.",
"We start by demonstrating an intriguing phenomenon: The quality of agents degrades when using vanilla TS.",
"We then analyze the problem and devise a solution."
],
[
"Performance degradation with vanilla tree search",
"We focus on applying TS at inference time, without learning.",
"We begin with a simple experiment that quantifies the benefit of using TS given a pre-trained policy.",
"A TS policy has access to future states and rewards and, by definition, is optimal when the tree depth goes to infinity.",
"Hence, intuitively, one may expect a TS policy to improve upon $\\pi _o$ for finite depths as well.",
"To test this, we load Rainbow agents $\\hat{Q}^{\\pi _o}$ , pre-trained on 50M frames; they are publicly available in [22] and achieve superhuman scores as in [20].",
"We use them to test TS policies $\\pi _d$ with multiple depths $d$ on several Atari benchmarks.",
"Surprisingly, the results (red curves in Fig.",
"REF , Sec.",
"REF ) show that TS reduces the total reward, sometimes to scores of random policies, in various games and depths.",
"The drop is particularly severe for TS policies with $d=1$ — a fact later explained by our analysis in Thm.",
"REF .",
"Figure: A failure of vanilla tree search.",
"Left: An Atari Breakout frame.",
"Right: Q-values of TS for the frame on the left.",
"Rows correspond to the action taken at the considered depth, which is d=0d=0 for the first column and d=1d=1 for the four others.",
"The action at the root of the tree is color-coded: Red for `Right', and blue for `Left'.We find the reason for this performance drop to be the poor generalization of the value function to states outside the stationary $\\pi _o$ 's distribution.",
"Fig.",
"REF shows a typical, bad action selection in Atari Breakout by a depth-1 TS.",
"The table on the right reports the estimated Q-values of the root state (first column) and of every state at depth 1 (last four columns).",
"Since the ball is dropping, 'Left' is the optimal action.",
"Indeed, this corresponds with the Q-values at the root.",
"However, at depth 1, the Q-values of the future state that corresponds to choosing 'Left' (second column) are the lowest among all depth-1 future states.",
"Subsequently, the depth-1 TS policy selects 'Right'.",
"During training, towards convergence, the trained policy mostly selects 'Left' while other actions are rarely sampled.",
"Therefore, expanding the tree at inference time generates states that have been hardly observed during training and are consequently characterized by inaccurate Q-value estimates.",
"In the case of Fig.",
"REF , the Q-value for 'Left' should indeed be low because the agent is about to lose the game.",
"As for the other, poorly sampled states, regression towards a higher mean leads to over-estimated Q-values.",
"Similar poor generalization has been observed in previous studies [16] and is interpreted as an off-policy distribution shift [30].",
"Beyond the anecdotal example in Fig.",
"REF , additional evidence support our interpretation regarding the distribution shift.",
"We first consider the error in the value estimate captured by the Bellman error minimized in training.",
"We compare the average Bellman error of the action chosen by $\\pi _o$ to all other actions at the tree root.",
"When averaging over 200 episodes, we find that the error for actions chosen by $\\pi _o$ is consistently lower than for other actions: $\\times 1.5$ lower for Breakout, and $\\times 2$ for Frostbite.",
"We also measure the off-policy distribution shift between $\\pi _o$ and the TS policy by counting disagreements on actions between the two policies.",
"In Breakout, $\\pi _o$ and $\\pi _1$ agreed only in $18\\%$ of the states; in Frostbite, the agreement is only $1.96\\%$ .",
"Such a level of disagreement between a well-trained agent and its one-step look-ahead extension is surprising.",
"On the other hand, it accounts for the drastic drop in performance when applying TS, especially in Frostbite (Fig.",
"REF )."
],
[
"Analysis of the degradation",
"We analyze the decision process of a policy $\\pi _d$ , given a pre-trained value function estimator, in a probabilistic setting.",
"Our analysis leads to a simple correction term given at the end of the section.",
"We also show how to compute this correction from the TS.",
"Formally, we are given a policy represented as a value function $\\hat{Q}^{\\pi _o},$ which we feed to the $d$ -step greedy policy (REF ) for each action selection.",
"Policy training is a stochastic process due to random start states, exploration, replay buffer sampling, etc.",
"The value estimator $\\hat{Q}^{\\pi _o}$ can thus be regarded as a random variable, with an expectation that is a deterministic function ${Q^{\\pi _o}}$ with a corresponding policy $\\pi _o,$ where `$o$ ' stands for `original'.",
"In short, we bound the probability that a sub-optimal action falsely appears more attractive than the optimal one.",
"We thus wish to conclude with high probability whether $a_0=\\operatornamewithlimits{arg\\,max}_a\\hat{Q}^{\\pi _o}_d(s,a)$ is indeed optimal, i.e., ${Q^{\\pi _o}_d}(s,a_0)\\ge {Q^{\\pi _o}_d}(s,a)~\\forall a\\in {\\mathcal {A}}$ .",
"As we have shown in Sec.",
"REF that states belonging to trajectories that follow $\\pi _o$ have lower value estimation noise, we model this effect via the following two assumptions.",
"Here, we denote by $t=0$ the time when an agent acts and not as the time step of the episode.",
"Assumption 1 Let $\\sigma _o, \\sigma _e \\in \\mathbb {R}^+$ s.t.",
"$0 < \\sigma _o < \\sigma _e$ .",
"For action sequence $(a_0, \\dots , a_{d-1})$ and corresponding state trajectory $(s_0,\\dots , s_d),$ $\\hat{Q}^{\\pi _o}(s_d, a_d) \\sim {\\left\\lbrace \\begin{array}{ll}\\mathcal {N}({Q^{\\pi _o}}(s_d, a_d), \\sigma ^2_o) &\\mbox{ if } a_0=\\pi _o(s_0) \\\\\\mathcal {N}({Q^{\\pi _o}}(s_d, a_d), \\sigma ^2_e) &\\mbox{ otherwise}.\\end{array}\\right.",
"}$ Assumption REF presumes that a different choice of the first action $a_0$ yields a different last state $s_d$ .",
"This is commonly the case for environments with large state spaces, especially when stacking observations as done in Atari.",
"While assuming a normal distribution is simplistic, it still captures the essence of the search process.",
"Regarding the expectation, recall that $\\pi _o$ is originally obtained from the previous training stage via gradient descent with a symmetric loss function ${\\mathcal {L}}.$ Due to the symmetric loss, the estimate $\\hat{Q}^{\\pi _o}_\\theta $ is unbiased, i.e., ${\\mathbb {E}}[\\hat{Q}^{\\pi _o}_\\theta ] = {Q^{\\pi _o}}.$ Regarding the variance, towards convergence, the loss is computed on replay buffer samples generated according to the stationary distribution of $\\pi _o$ .",
"The estimate of the value function for states outside the stationary distribution is consequently characterized by a higher variance, i.e.",
"$\\sigma _o <\\sigma _e$ .",
"We also show that this separation of variance occurs in the data, as detailed in the last paragraph of Sec.",
"REF .",
"After conditioning the variance on whether $\\pi _o$ was followed, we similarly split the respective sub-trees.",
"To split, we assume the cumulative reward along the tree and values at the leaves depend only on (i) the root state and (ii) whether $\\pi _o$ was selected at that state.",
"Assumption 2 For action sequence $(a_0, \\dots a_{d-1})$ and corresponding trajectory $(s_0, \\dots , s_d),$ $\\sum _{t=0}^{d-1} \\gamma ^t r(s_t, a_t) ={\\left\\lbrace \\begin{array}{ll}R_o(s_0) \\mbox{ if } a_0=\\pi _o(s_0) \\\\R_e(s_0) \\mbox{ otherwise},\\end{array}\\right.",
"}{Q^{\\pi _o}}(s_d, a_d) ={\\left\\lbrace \\begin{array}{ll}\\mu _o(s_0) \\mbox{ if } a_0=\\pi _o(s_0) \\\\\\mu _e(s_0) \\mbox{ otherwise},\\end{array}\\right.",
"}$ with $R_o, R_e, \\mu _o, \\mu _e$ being functions of $s$ .",
"Assumption REF could be replaced with a more detailed consideration of each trajectory, but we make it for simplicity.",
"This assumption considers the worst-case scenario: the rewards are unhelpful in determining the optimal policy, and all leaves are equally likely to mislead the $d$ -step greedy policy.",
"Assuming from now on that Assumptions REF and REF hold, we can now explicitly express the distribution of the maximal value among the leaves using Generalized Extreme Value (GEV) theory [9].",
"Lemma 3.1 It holds that $ \\hat{Q}^{\\pi _o}_d(s, \\pi _o(s)) = R_o(s) + \\gamma ^d G_o(s),\\quad \\max _{a\\ne \\pi _o(s)}\\hat{Q}^{\\pi _o}_d(s, a) = R_e(s) + \\gamma ^d G_e(s),$ with $G_o(s) \\sim \\text{GEV}(\\mu ^{\\text{GEV}}_o(s), \\sigma ^{\\text{GEV}}_o, 0), \\quad G_e(s) \\sim \\text{GEV}(\\mu ^{\\text{GEV}}_e(s), \\sigma ^{\\text{GEV}}_e, 0),$ where GEV is the Generalized Extreme Value distribution and $\\mu ^{\\text{GEV}}_o(s), \\sigma ^{\\text{GEV}}_o, \\mu ^{\\text{GEV}}_e(s), \\sigma ^{\\text{GEV}}_e$ are given in Appendix REF .",
"All proofs are deferred to Appendix .",
"Using the GEV distribution, we can now quantify the bias stemming from the maximization in each of two sub-trees corresponding $\\pi _o$ vs. all other actions.",
"Lemma 3.2 It holds that ${\\mathbb {E}}\\left[\\hat{Q}^{\\pi _o}_d(s, \\pi _o(s))\\right] &= {Q^{\\pi _o}_d}(s, \\pi _o(s)) + \\gamma ^d B_o(\\sigma _o, A, d) \\\\{\\mathbb {E}}\\left[\\max _{a\\ne \\pi _o(s)}\\hat{Q}^{\\pi _o}_d(s, a)\\right] &= \\max _{a\\ne \\pi _o(s)}{Q^{\\pi _o}_d}(s, a) + \\gamma ^d B_e(\\sigma _e, A, d),$ where the biases $B_o,~ B_e$ are given in Appendix REF , and satisfy $0 \\le B_o(\\sigma _o, A, d)< B_e(\\sigma _e, A, d)$ .",
"Lemma REF conveys the main message of our analysis: the variance of terms being maximized translates to a positive shift in the expectation of the maximum.",
"Hence, even if $\\mu _o(s) > \\mu _e(s)$ for a certain $s,$ a different action than $\\pi _o(s)$ can be chosen with non-negligible probability as the bias in $\\mu _e(s)$ is greater than the one in $\\mu _o(s)$ .",
"To compensate for this bias that gives an unfair advantage to the noisier nodes of the tree, we introduce a penalty term that precisely cancels it."
],
[
"BCTS: Bellman Corrected Tree Search",
"Instead of selecting actions via (REF ), in BCTS we replace $\\hat{Q}^{\\pi _o}_d$ with the corrected $Q^{\\text{BCTS}}_d$ defined by $\\hat{Q}^{\\text{BCTS},\\pi _o}_d(s,a) :={\\left\\lbrace \\begin{array}{ll}\\hat{Q}^{\\pi _o}_d(s,a) & \\mbox{ if } a_0=\\pi _o(s_0), \\\\\\hat{Q}^{\\pi _o}_d(s,a) - \\gamma ^d \\left( B_e(\\sigma _e, A, d) - B_o(\\sigma _o, A, d) \\right) & \\mbox{ otherwise},\\end{array}\\right.",
"}$ and we denote the BCTS policy by $\\pi ^{\\text{BCTS}}_d:=\\operatornamewithlimits{arg\\,max}_a \\hat{Q}^{\\text{BCTS},\\pi _o}_d(s,a).$ In the following result, we prove that BCTS indeed eliminates undesirable bias.",
"Theorem 3.3 The relation ${\\mathbb {E}}\\Bigg [ \\hat{Q}^{\\text{BCTS},\\pi _o}_d(s,\\pi _o(s)) \\bigg ] > {\\mathbb {E}}\\Bigg [ \\max _{a\\ne \\pi _o(s)}\\hat{Q}^{\\text{BCTS},\\pi _o}_d(s,a) \\bigg ]$ holds if and only if ${Q^{\\pi _o}_d}(s, \\pi _o(s)) > \\max _{a \\ne \\pi _o(s)}{Q^{\\pi _o}_d}(s, a).$ The biases $B_o,~B_e$ include the inverse of the cumulative standard normal distribution $\\Phi ^{-1}$ (Appendix REF ).",
"We now approximate them with simple closed-form expressions that are highly accurate for $d \\ge 2$ (Appendix REF ).",
"These approximations help revealing how the problem parameters dictate prominent quantities such as the correction term in (REF ) and the probability in Thm.",
"REF below.",
"Lemma 3.4 When $A^{d-1} \\gg 1$ , the correction term in (REF ) can be approximated with $B_e(\\sigma _e, A, d) - B_o(\\sigma _o, A, d) \\approx \\sqrt{2\\log A}\\left(\\sigma _e\\sqrt{d} - \\sigma _o\\sqrt{d-1}\\right) - (\\sigma _e - \\sigma _o) / 2.$ The bias gap in (REF ) depends on the ratio between $\\sigma _e$ and $\\sigma _o;$ this suggests that TS in different environments will be affected differently.",
"As $\\sigma _e > \\sigma _o,$ the bias gap is positive.",
"This is indeed expected, since the maximum over the sub-trees of $a_0 \\ne \\pi _o(s)$ includes noisier elements than those in the sub-tree of $a_0 = \\pi _o(s).$ Also, in (REF ), $\\sigma _e\\sqrt{d}$ dominates $\\sigma _o\\sqrt{d-1},$ making the bias gap grow asymptotically with $\\sqrt{d\\log A}.$ This rate reflects how the number of elements being maximized over affects the bias of their maximum.",
"Next, we bound the probability of choosing a sub-optimal action when using BCTS.",
"In Appendix , Thm.",
"REF , we give an exact bound to that probability without assuming $A^{d-1} \\gg 1.$ Here, we apply Lemma.",
"REF to give the result in terms of $\\sigma _o,~\\sigma _e.$ Theorem 3.5 When $A^{d-1} \\gg 1$ , the policy $\\pi ^{\\text{BCTS}}_d(s)$ (see (REF )) chooses a sub-optimal action with probability bounded by: $\\Pr \\big (\\pi ^{\\text{BCTS}}_d(s) &\\notin \\arg \\max _a {Q^{\\pi _o}_d}(s,a)\\big )\\\\ & \\le \\left(1 + \\frac{6d\\log A \\left({Q^{\\pi _o}_d}(s, \\pi _o(s)) - \\max _{a \\ne \\pi _o(s)}{Q^{\\pi _o}_d}(s, a)\\right)^2}{\\gamma ^{2d}\\pi ^2\\left(\\sigma _o^2+\\sigma _e^2\\right)}\\right)^{-1}.",
"$ The fraction in the above bound is a signal-to-noise ratio: The expected value difference in the numerator represents the signal, while the variances in the denominator represent the noise.",
"In addition, the signal is “amplified” by $d\\log A$ because, after applying the correction from (REF ), a larger number of tree nodes amount to a more accurate maximum estimator.",
"A similar amplification also occurs due to nodes being deeper and is captured by $\\gamma ^d$ in the denominator.",
"The factor $\\gamma ^d$ also appears in the correction term from (REF ).",
"This correction is the bias gap from (REF ), scaled by $\\gamma ^d.$ Hence, asymptotically, while the bias gap grows with $\\sqrt{d},$ the exponential term is more dominant; so, overall, this product decays with $d.$ This decay reduces the difference between the vanilla and BCTS policies for deeper trees by lowering the portion of the estimated value compared to the exact reward.",
"As we show later, this phenomenon is consistent with our experimental observations.",
"Figure: NO_CAPTIONBCTS Algorithm.",
"Exploring the full tree reveals out-of-distribution states (red).",
"These were not visited during training and tend to have highly variable scores, leading to high overestimation error.",
"The penalty term $B(\\hat{\\delta }_e,\\hat{\\delta }_o,A,d)$ (see (REF )) cancels the excess bias.",
"The Bellman errors $\\hat{\\delta }_e,\\hat{\\delta }_o$ are extracted from the tree at depth $1.$ Computing the correction term requires estimates of $\\sigma _o$ and $\\sigma _e$ .",
"As a surrogate for the error, we use the Bellman error.",
"To justify it, let us treat the values of $\\hat{Q}^{\\pi _o}_d$ at different depths as samples of $\\hat{Q}^{\\pi _o}.$ Then, the following result holds for its variance estimator.",
"Note that we do not need Assumptions REF and REF to prove the following result.",
"Proposition 3.6 Let $\\hat{{\\text{var}}}_n[X]$ be the variance estimator based on $n$ samples of X.",
"Then, $\\hat{\\text{var}}_{n=2}[\\hat{Q}^{\\pi _o}(s, a)] = {\\left(\\hat{Q}^{\\pi _o}_1(s, a) - \\hat{Q}^{\\pi _o}_0(s, a)\\right)^2} / {2} = {\\delta ^2(s,a)} / {2},$ where $\\delta (s,a)$ is the Bellman error.",
"Note that during a TS, at depth 1 we have access to $\\delta (s_0,a)$ of all $a \\in {\\mathcal {A}}$ without additional computation.",
"For depths 2 and above, the Bellman error is defined only for actions chosen by $\\pi _o,$ corresponding to a single trajectory down the tree.",
"For these reasons, we base the above result on samples from depths 0 and $1.$ Thanks to Prop.",
"REF , we can estimate the bias correction term in Lemma REF directly from the TS operation.",
"The resulting correction term is $B(\\hat{\\delta }_e, \\hat{\\delta }_o, A, d) = \\sqrt{\\log A}\\left(\\hat{\\delta }_e\\sqrt{d} - \\hat{\\delta }_o\\sqrt{d-1}\\right),$ where $\\hat{\\delta }_o$ is the Bellman error corresponding to $a=\\pi _o(s)$ at the root, and $\\hat{\\delta }_e$ and is the average Bellman error of all other actions.",
"A visualization of the resulting BCTS algorithm is in Fig.",
"REF ."
],
[
"Solving Scalability via Batch-BFS",
"The second drawback of TS is scalability: exhaustive TS is impractical for non-trivial tree depths because of the exponential growth of the tree dimension.",
"As TS requires generating $|A|^d$ leaves at depth $d$ , it has been rarely considered as a viable solution.",
"To solve this issue, we propose Batch-BFS, an efficient, parallel, TS scheme based on BFS, built upon the ability to advance multiple environments simultaneously; see Alg.",
".",
"It achieves a significant runtime speed-up when a forward model is implemented on a GPU.",
"Such GPU-based environments are becoming common nowadays because of their advantages (including parallelization and higher throughput) over their CPU counterparts.",
"E.g., Isaac-Gym [32] provides a GPU implementation robotic manipulation tasks [41], whereas Atari-CuLE [11] is a CUDA-based version of the AtariGym benchmarks [4].",
"Batch-BFS is not limited to exact simulators and can be applied to learned deep forward models, like those in [31], [21].",
"Since Batch-BFS simultaneously advances the entire tree, it enables exhaustive tree expansion to previously infeasible depths.",
"It also allows access to the cumulative reward and estimated value over all the tree nodes.",
"Such access to all future values paves a path to new types of algorithms for, e.g., risk reduction and early tree-pruning.",
"We leave such directions to future work.",
"In addition to Alg.",
", we also provide a visualization of Batch-BFS in Appendix , Fig.",
"REF .",
"[tb] Batch-BFS Input: GPU environment ${\\mathcal {G}}$ , value network $Q_\\theta $ , depth $d$ Init tensors: state $\\bar{S} = [s_0],$ action $\\bar{A} = \\left[0, 1, 2, .., A-1\\right]$ , reward $\\bar{R}=[0]$ $i_d=0$ to $d-1$ $\\bar{S} \\leftarrow \\bar{S} \\times A,~~\\bar{R} \\leftarrow \\bar{R} \\times A$ Replicate state and reward tensors $A$ times $\\bar{r},\\bar{S}^{\\prime } = {\\mathcal {G}}([\\bar{S},\\bar{A}])$ Feed $[\\bar{S},\\bar{A}]$ to simulator and advance $\\bar{R} \\leftarrow \\bar{R} + \\gamma ^{i_d} \\bar{r}, ~~\\bar{S} \\leftarrow \\bar{S}^{\\prime }$ Accumulate discounted reward $\\bar{A} \\leftarrow \\bar{A} \\times A$ Replicate action tensor $A$ times $\\bar{R} \\leftarrow \\bar{R} + \\gamma ^d \\max _a Q_\\theta (\\bar{S},a) $ Accumulate discounted value of states at depth $d$ Return $\\lfloor (\\operatornamewithlimits{arg\\,max}\\bar{R}) / A^{d-1} \\rfloor $ Return optimal action at the root Figure: Average tree-search time per action selection.",
"Left: Atari-CuLE Breakout, Pong, and VideoPinball.",
"Right: A randomly generated neural network to mimic a learned forward-model with A∈{2,10}.A \\in \\lbrace 2, 10\\rbrace .",
"Note that xx and yy axes are in log-scale."
],
[
"Runtime experiments",
"To showcase the efficacy of Batch-BFS, we compare it to a CPU-based BFS and to non-parallel TS, i.e., Depth-First-Search (DFS).",
"We measure the duration of an average single TS operation on two environments: Atari-CuLE [11] and a Deep NN (DNN) mimicking a learned forward model.",
"The DNN is implemented in pytorch and uses cudnn [8].",
"It consists of three randomly-initialized hidden layers of width 100 with input size 100 for the state and 2 or 10 for the actions.",
"The results are given in Fig.",
"REF .",
"We run our experiments on a 8 core Intel(R) Xeon(R) CPU E5-2698 v4 @ 2.20GHz equipped with one NVIDIA Tesla V100 16GB.",
"Although we used a single GPU for our experiments, we expect a larger parallelization (and thus computational efficiency) to be potentially achieved in the future through a multi-GPU implementation.",
"As expected, DFS scales exponentially in depth and is slower than BFS.",
"When comparing BFS on CPU vs. GPU, we see that CPU is more efficient in low depths.",
"This is indeed expected, as performance of GPUs without parallelization is inferior to that of CPUs.",
"This issue is often addressed by distributing the simulators across a massive number of CPU cores [15].",
"We leverage this phenomenon in Batch-BFS by finding the optimal “cross-over depth” per game and swap the compute device in the middle of the search from CPU to GPU."
],
[
"Experiments",
"In this section, we report our results on two sets of experiments: the first deals solely with TS for inference, without learning, whereas the second includes the case of TS used in training.",
"Figure: Inference only: Vanilla TS vs. BCTS.",
"Median scores with lower (0.250.25) and upper (0.750.75) quantiles over 200 episodes, as a function of the tree depth.",
"Surprisingly, a deeper tree often degrades the performance of vanilla TS.",
"BCTS (blue) improves upon vanilla TS (red) for all depths, except for Asteroids.",
"The improvement grows monotonically with the tree depth."
],
[
"Inference with tree search",
"Using the pre-trained Rainbow agents in [22] (see Sec.",
"), we test vanilla TS and show (Fig.",
"REF , red plots) that it leads to a lower score than the baseline $\\pi _o$ .",
"The largest drop is for $d=1$ as supported by our analysis.",
"The game that suffered the most is Frostbite.",
"This can be explained from it having the largest number of actions ($A=18$ ), which increases the bias in (REF ).",
"As for BCTS, we found that multiplying its correction term (see (REF )) by a constant that we sweep over can improve performance; we applied this method for the experiments here.",
"Recall that BCTS is a TS applied on the pre-trained Rainbow baseline; i.e., the case of $d=0$ is Rainbow itself.",
"The results in Fig.",
"REF show that BCTS significantly improves the scores, monotonically in depth, in all games.",
"It improves the Rainbow baseline already for $d=1$ , while for $d=4,$ the score more than doubles.",
"For BeamRider, BCTS with $d=4$ achieves roughly $\\times 5$ improvement.",
"Notice that without the computationally efficient implementation of Batch-BFS, the results for deeper trees would not have been obtainable in a practical time.",
"We provide timing measurements per game and depth in Appendix REF .",
"Finally, notice that the advantage provided by BCTS is game-specific.",
"Different games benefit from it by a different amount.",
"In one of the games tested, Asteroids, vanilla TS was as useful as BCTS.",
"Our findings reported in the last paragraph of Sec.",
"REF hint why certain games benefit from BCTS more than others.",
"Nonetheless, a more thorough study on how the dynamics and state distribution in different games affect TS constitutes an interesting topic for future research."
],
[
"Training with tree search",
"To further demonstrate the potential benefits of TS once a computationally efficient implementation (Section ) is available, we show how it affects training agents from scratch on CuLE-Atari environments.",
"We extend classic DQN [28] with TS using Batch-BFS for each action selection.",
"Notice that training with TS does not suffer from the distribution shift studied in Sec. .",
"Hence, the experiments below use vanilla TS and not BCTS.",
"Our experiment is even more significant considering that Efroni et al.",
"[14] recently proved that the Bellman update should be replaced so that contraction is guaranteed for tree-based policies only when the value at the leaves is backed.",
"However, this theory was not supported by empirical evidence beyond a toy maze.",
"As far as we know, our work is the first to adopt this corrected update to obtain favorable results in state-of-the-art domains, thanks to the computationally efficient TS implementation achieved by Batch-BFS.",
"We find this method to be beneficial in several of the games we tested.",
"In the experiments below, we treat the correction from [14] as a hyper-parameter and include ablation studies of it in Appendix REF .",
"We show the training scores in Table REF and convergence plots in Appendix REF .",
"For a fair comparison of different TS depths, we stop every run after two days on the same hardware (see Appendix ), not considering the total iteration count.",
"To compare our results against classic DQN and Rainbow, we measure the number of iterations completed in two days by DQN with TS, $d=0$ .",
"In Table REF we report the corresponding intermediate results for DQN and Rainbow reported by the original authors in [35].",
"In most games, it amounts to roughly 30 million iterations.",
"As already shown in the case of inference with no training, the achieved score increases monotonically with the tree depth.",
"In four of the five games, DQN with TS even surpasses the more advanced Rainbow.",
"Since all results were obtained for identical runtime, improvement per unit of time is higher for higher depths.",
"This essentially translates to better efficiency of compute resources.",
"Convergence plots as a function of wall-clock time are shown in Appendix .",
"We also tested TS on two additional games not included in Table REF : Boxing and Pong.",
"Interestingly, TS with $d=4$ immediately obtained the highest possible scores in both these games already in the first training iteration.",
"Table: Atari scores after two days of training.",
"We follow the evaluation method in : Average of 200 testing episodes, from the agent snapshot that obtained the highest score during training."
],
[
"Related work",
" The idea of searching forward in time has been employed extensively in control theory via methods such as A* [19], RRT [26], and MPC [2].",
"The latter is quite popular in the context of efficient planning in RL [31], [42], [43].",
"MPC-based controllers rely on recourse planning by solving an optimization program given the continuous structure of the dynamics.",
"In our setup, the controls are discrete and the forward model is a black-box that cannot be directly used in an optimization scheme.",
"We leverage an efficient GPU simulator to conduct the look-ahead search.",
"When such a simulator is out of reach, it can be learned from data.",
"There is vast literature regarding learned deep forward models without optimizing a policy [24], [33], [25].",
"Other works [23], [21], [18], [36] used the learned model for planning but not with a TS.",
"Few considered roll-outs of learned dynamics [6], [17] but only for evaluation purposes.",
"Additional relevant works are MuZero [37] and “MuZero Unplugged” [38] which utilized a learned forward-model for prediction in an MCTS-based policy.",
"In [29], the trade-off between learning and planning using a TS was empirically tested.",
"Finally, look-ahead policies for RL were also studied theoretically; bounds on the suboptimality of the learned policy were given in [12], [13], [14].",
"There, the focus was on the effect of planning on the learning process."
],
[
"Discussion",
" Our study of the degradation with vanilla TS implies that the learned value function is not representative of the actual states that can be visited.",
"This conclusion can be helpful in debugging RL systems.",
"It can also be used to better tune the approximation architecture, or guide exploration.",
"Our solution to the above performance degradation is an off-policy correction that penalizes high-error trajectories.",
"It can be further improved with different notions of uncertainty for the value function, e.g., Bayesian or bootstrapped models.",
"Also, while some simulators are available on GPU, such as Atari [11] and robotic manipulation [27], in other cases, a learned model can be used.",
"In these cases, one could include the simulator quality in the off-policy penalty term.",
"Finally, a limitation of our work is that we focus on problems with a discrete action space.",
"Handling problems with continuous action tasks is a challenging direction for future work.",
"Broader Impact.",
"The paper proposes a method to improve existing RL algorithms.",
"As such, its main impact is to make RL more easily and widely deployable.",
"Since our method can be applied to policies trained with any algorithm, it can be viewed as a generic “policy booster”, and find applications with access to the environment model or its approximation."
],
[
"Proof of Lemma ",
"Lemma It holds that $ \\hat{Q}^{\\pi _o}_d(s, \\pi _o(s)) = R_o(s) + \\gamma ^d G_o(s),\\quad \\max _{a\\ne \\pi _o(s)}\\hat{Q}^{\\pi _o}_d(s, a) = R_e(s) + \\gamma ^d G_e(s),$ with $G_o(s) \\sim \\text{GEV}(\\mu ^{\\text{GEV}}_o(s), \\sigma ^{\\text{GEV}}_o, 0), \\quad G_e(s) \\sim \\text{GEV}(\\mu ^{\\text{GEV}}_e(s), \\sigma ^{\\text{GEV}}_e, 0),$ where GEV is the Generalized Extreme Value distribution and $\\mu ^{\\text{GEV}}_o(s) &= \\mu _o(s) + \\sigma _o \\Phi ^{-1}\\left( 1 - \\frac{1}{A^{d-1}}\\right), \\nonumber \\\\\\sigma ^{\\text{GEV}}_o &=\\sigma _o \\left[\\Phi ^{-1}\\left(1 - \\frac{1}{eA^{d-1}}\\right) - \\Phi ^{-1}\\left(1 - \\frac{1}{A^{d-1}}\\right) \\right], \\nonumber \\\\\\mu ^{\\text{GEV}}_e(s) &= \\mu _e(s) + \\sigma _e \\Phi ^{-1}\\left( 1 - \\frac{1}{A^d - A^{d-1}}\\right), \\nonumber \\\\\\sigma ^{\\text{GEV}}_e &= \\sigma _e \\left[\\Phi ^{-1}\\left(1 - \\frac{1}{e(A^d - A^{d-1})}\\right) - \\Phi ^{-1}\\left(1 - \\frac{1}{A^d - A^{d-1}}\\right) \\right].",
"$ The function $\\Phi ^{-1}$ is the inverse of the CDF of the standard normal distribution.",
"For $1 \\le i \\le A^{d-1},$ let $ N_o^{(i)}$ be independent random variables distributed ${N_o^{(i)}\\sim \\mathcal {N}(\\mu _o(s), \\sigma _o^2)}.$ According to (REF ), we have that $\\hat{Q}^{\\pi _o}_d(s, a) = \\Bigg [ \\max _{(a_k)_{k=1}^d \\in {\\mathcal {A}}} \\Bigg [ \\sum _{t=0}^{d-1} \\gamma ^t r(s_t, a_t) \\bigg ] + \\gamma ^d \\hat{Q}^{\\pi _o}(s_d, a_d) \\Bigg ]_{s_0=s, a_0=a}.$ For the case of $a=\\pi _o(s),$ using Assumption REF , we can replace the cumulative reward above with $R_o(s)$ and be left with $\\hat{Q}^{\\pi _o}_d(s, \\pi _o(s)) & = R_o(s) + \\gamma ^d \\max _{(a_k)_{k=1}^d \\in {\\mathcal {A}}} \\hat{Q}^{\\pi _o}(s_d, a_d) \\big |_{s_0=s, a_0=\\pi _o(s)} \\\\& = R_o(s) + \\gamma ^d \\max _{1 \\le i \\le A^{d-1}} N_o^{(i)} \\\\& = R_o(s) + \\gamma ^d G_o(s),$ where the second relation is due to Assumptions REF together with REF , and the third follows from the maximum of $\\lbrace N_o^{(i)}\\rbrace _{i=1}^{A^{d-1}}$ being GEV-distributed with the parameters in the statement, derived as in [9].",
"Similarly, for $a_0\\ne \\pi _0(s),$ we will have $A^d - A^{d-1}$ iid variables ${N_e^{(i)} \\sim \\mathcal {N}(\\mu _e(s), \\sigma _e^2)}$ and, consequently, $G_e(s)$ as given in the statement."
],
[
"Proof of Lemma ",
"Lemma It holds that ${\\mathbb {E}}\\left[\\hat{Q}^{\\pi _o}_d(s, \\pi _o(s))\\right] &= {Q^{\\pi _o}_d}(s, \\pi _o(s)) + \\gamma ^d B_o(\\sigma _o, A, d), \\\\{\\mathbb {E}}\\left[\\max _{a\\ne \\pi _o(s)}\\hat{Q}^{\\pi _o}_d(s, a)\\right] &= {Q^{\\pi _o}_d}(s, a\\ne \\pi _o(s)) + \\gamma ^d B_e(\\sigma _e, A, d),$ where the biases $B_o$ and $B_e,$ satisfying $0 \\le B_o(\\sigma _o, A, d) < B_e(\\sigma _e, A, d),$ are given by $B_o(\\sigma _o,A,d) &= {\\left\\lbrace \\begin{array}{ll}0 & \\mbox{if } d=1, \\\\\\sigma _o \\Phi ^{-1}\\left( 1 - \\frac{1}{A^{d-1}}\\right) + \\gamma _{\\text{EM}} \\sigma ^{\\text{GEV}}_o & \\mbox{otherwise},\\end{array}\\right.}",
"\\nonumber \\\\B_e(\\sigma _e,A,d) &= {\\left\\lbrace \\begin{array}{ll}0, & \\mbox{if } d=1 \\mbox{ and } A=2 \\\\\\sigma _e\\Phi ^{-1}\\left( 1 - \\frac{1}{A^d - A^{d-1}}\\right) + \\gamma _{\\text{EM}} \\sigma ^{\\text{GEV}}_e, & \\mbox{otherwise}.\\end{array}\\right.}",
"$ The constant $\\gamma _{\\text{EM}} \\approx 0.58$ is the Euler–Mascheroni constant.",
"First, obviously, the maximum over a set containing a single random variable has the distribution of that single element.",
"Hence, there is no overestimation bias in the single-element case; i.e., the bias is 0 for the sub-tree of $a=\\pi _o(s)$ with $d=1,$ and the sub-tree of $a\\ne \\pi _o(s)$ with $d=1$ and $A=2.$ Next, let $X$ be s.t.",
"$X\\sim \\text{GEV}(\\mu ^{\\text{GEV}}, \\sigma ^{\\text{GEV}}, 0).$ From [9], we have that ${\\mathbb {E}}\\left[ X \\right] = \\mu ^{\\text{GEV}} + \\gamma _{\\text{EM}} \\sigma ^{\\text{GEV}}.$ Applying Lemma REF , we have that ${\\mathbb {E}}\\left[\\hat{Q}^{\\pi _o}_d(s, \\pi _o(s))\\right] &= {\\mathbb {E}}\\left[R_o(s) + \\gamma ^d G_o(s)\\right] \\nonumber \\\\&= R_o(s) + \\gamma ^d{\\mathbb {E}}\\left[G_o(s)\\right] \\\\&= R_o(s) + \\gamma ^d\\left[\\mu _o(s) + \\sigma _o \\Phi ^{-1}\\left(1 - \\frac{1}{A^{d-1}}\\right) + \\gamma _{\\text{EM}}\\sigma ^{\\text{GEV}}_o\\right] \\\\&= {Q^{\\pi _o}_d}(s, \\pi _o(s)) + \\gamma ^d B_o(\\sigma _o, A, d), $ where relation (REF ) is due to the rewards being deterministic (Assumption REF ), relation () follows from Lemma REF together with (REF ), and relation () is due to the definition of ${Q^{\\pi _o}_d}(s, \\pi _o(s))$ in (REF ), together with Assumptions REF and REF .",
"The calculation for the expectation ${\\mathbb {E}}\\left[\\max _{a\\ne \\pi _o(s)}\\hat{Q}^{\\pi _o}_d(s, a)\\right]$ follows the same steps.",
"Next, we show that $0 \\le B_o(\\sigma _o, A, d)< B_e(\\sigma _e, A, d).$ For this, let us define $B(n) = {\\left\\lbrace \\begin{array}{ll}0 & n = 1 \\\\\\gamma _{\\text{EM}}\\Phi ^{-1}\\left(1 - \\frac{1}{en}\\right) + (1 - \\gamma _{\\text{EM}})\\Phi ^{-1}\\left( 1 - \\frac{1}{n}\\right), & n\\ne 1.\\end{array}\\right.",
"}$ Notice we now have $B_o(\\sigma _o, A, d)=\\sigma _o B(A^{d-1})$ and $B_e(\\sigma _e, A, d)=\\sigma _e B(A^d - A^{d-1}).$ Since by Assumption REF we have that $\\sigma _e > \\sigma _o > 0,$ to prove (REF ) it is sufficient to show that ${0\\le {B(A^{d-1}) < B(A^d - A^{d-1})}}.$ Since $B(n)$ is composed of two positive monotonically increasing functions, it is a positive monotonically increasing function.",
"So, whenever $A^{d-1} < A^d - A^{d-1},$ we also have that $0\\le {B(A^{d-1}) < B(A^d - A^{d-1})}$ and, consequently, (REF ).",
"Taking a log, we see it is indeed the case for $A>2.$ For $A=2,$ we have equality and ${B(A^{d-1}) = B(A^d - A^{d-1})}.$ But then, (REF ) holds again, since $\\sigma _e > \\sigma _o > 0.$"
],
[
"Proof of Theorem ",
"Theorem The relation ${\\mathbb {E}}\\Bigg [ \\hat{Q}^{\\text{BCTS},\\pi _o}_d(s,\\pi _o(s)) \\bigg ] > {\\mathbb {E}}\\Bigg [\\max _{a\\ne \\pi _o(s)} \\hat{Q}^{\\text{BCTS},\\pi _o}_d(s,a) \\bigg ],$ holds if and only if ${Q^{\\pi _o}_d}(s, \\pi _o(s)) > \\max _{a \\ne \\pi _o(s)}{Q^{\\pi _o}_d}(s, a).$ For the case of $a=\\pi _o(s),$ we have ${\\mathbb {E}}\\Bigg [ \\hat{Q}^{\\text{BCTS},\\pi _o}_d(s,\\pi _o(s)) \\bigg ] = {\\mathbb {E}}\\Bigg [ \\hat{Q}^{\\pi _o}_d(s,\\pi _o(s)) \\bigg ] = {Q^{\\pi _o}_d}(s,\\pi _o(s)) + \\gamma ^d B_o(\\sigma _o, A, d),$ where the first relation holds by the definition in (REF ) and the second relation follows from Lemma REF .",
"Using the same arguments, for $a\\ne \\pi _o(s),$ $&{\\mathbb {E}}\\Bigg [ \\max _{a\\ne \\pi _o(s)}\\hat{Q}^{\\text{BCTS},\\pi _o}_d(s,a) \\bigg ] \\\\= {} & {\\mathbb {E}}\\Bigg [ \\max _{a\\ne \\pi _o(s)}\\hat{Q}^{\\pi _o}_d(s,a) \\bigg ] - \\gamma ^d \\left[ B_e(\\sigma _e, A, d) - B_o(\\sigma _o, A, d) \\right] + \\gamma ^d B_e(\\sigma _e, A, d) \\\\= {} & \\max _{a\\ne \\pi _o(s)}{Q^{\\pi _o}_d}(s,a) + \\gamma ^d B_o(\\sigma _o, A, d),$ and the result immediately follows."
],
[
"Proof of Lemma ",
"Lemma When $A^{d-1} \\gg 1$ , the correction term in (REF ) can be approximated with $B_e(\\sigma _e, A, d) - B_o(\\sigma _o, A, d) \\approx \\sqrt{2\\log A}\\left(\\sigma _e\\sqrt{d} - \\sigma _o\\sqrt{d-1}\\right) - (\\sigma _e - \\sigma _o) / 2.$ In the following, we apply the approximation $\\Phi ^{-1}\\left(1 - \\frac{1}{n}\\right) \\approx \\sqrt{2\\log n} - 0.5$ from [3], which we empirically show to be highly accurate in Appendix REF .",
"By definition (REF ), $\\sigma ^{\\text{GEV}}_e &= \\sigma _e\\left[\\Phi ^{-1}\\left(1 - \\frac{1}{e(A^d - A^{d-1})}\\right) - \\Phi ^{-1}\\left(1 - \\frac{1}{A^d - A^{d-1}}\\right) \\right] \\nonumber \\\\&\\approx \\sigma _e \\left[\\Phi ^{-1}\\left(1 - \\frac{1}{eA^d}\\right) - \\Phi ^{-1}\\left(1 - \\frac{1}{A^d}\\right) \\right] \\\\&\\approx \\sigma _e \\left[\\sqrt{2\\log \\left(eA^d\\right)} - \\sqrt{2\\log \\left(A^d\\right)} \\right] \\\\&= \\sigma _e \\frac{2\\log (eA^d) - 2\\log A^d}{\\sqrt{2\\log \\left(eA^d)\\right)} + \\sqrt{2\\log A^d}} \\nonumber \\\\&\\approx \\frac{\\sigma _e}{\\sqrt{2d\\log A}}, $ where in (REF ) we applied (REF ); in relation () we use that $A^d - A^{d-1} \\approx A^d$ since $A^d \\gg 1;$ and in relation () we approximate $1 + \\log (A^d) \\approx \\log (A^d),$ again because $A^d \\gg 1.$ When $A^d \\gg 1,$ the second case of (REF ) holds, and thus $B_e(\\sigma _e, A, d) &= \\sigma _e \\Phi ^{-1}\\left( 1 - \\frac{1}{A^d - A^{d-1}}\\right) + \\gamma _{\\text{EM}} \\sigma ^{\\text{GEV}}_e\\\\&\\approx \\sigma _e \\left(\\sqrt{2d\\log A} - 0.5\\right) + \\gamma _{\\text{EM}} \\frac{\\sigma _e}{\\sqrt{2d\\log A}} \\\\&\\approx \\sigma _e \\left(\\sqrt{2d\\log A} - 0.5\\right),$ where the second relation follows from (REF ) and (), while for the third relation we approximate $\\sqrt{2\\log (A^d)} (1 + \\frac{\\gamma _{\\text{EM}}}{{2\\log (A^d)}}) \\approx \\sqrt{2\\log (A^d)}$ since $A^d \\gg 1$ (recall that $\\gamma _{\\text{EM}} \\approx 0.58$ ).",
"Applying the same derivation for $B_o$ gives that $B_o(\\sigma _o, A, d) \\approx \\sigma _o \\left(\\sqrt{2(d-1)\\log A} - 0.5\\right),$ and the result follows directly."
],
[
"Proof of Theorem ",
"To prove Theorem REF , we first obtain the following non-approximate result.",
"Theorem A.1 The policy $\\pi ^{\\text{BCTS}}_d(s)$ (see (REF )) chooses a sub-optimal action with probability bounded by: $\\Pr \\left(\\pi ^{\\text{BCTS}}_d(s) \\notin \\arg \\max _a {Q^{\\pi _o}_d}(s,a)\\right) \\le \\left(1 + \\frac{6\\left({Q^{\\pi _o}_d}(s, \\pi _o(s)) - \\max _{a \\ne \\pi _o(s)}{Q^{\\pi _o}_d}(s, a)\\right)^2}{\\gamma ^{2d}\\pi ^2\\left({(\\sigma _o^{\\text{GEV}})}^2+ {(\\sigma _e^{\\text{GEV}})}^2\\right)}\\right)^{-1}.$ We first recall Cantelli's inequality [7]: Let $X$ be a real-valued random variable.",
"Then, for some $\\lambda > 0,$ $\\Pr \\left(X - {\\mathbb {E}}\\left[ X \\right] \\ge \\lambda \\right) \\le \\left(1 + \\frac{\\lambda ^2}{\\text{Var}[X]} \\right)^{-1}.$ We choose $X := \\max _{a \\ne \\pi _o(s)} \\hat{Q}^{BCTS, \\pi _o}_d(s, a) - \\hat{Q}^{BCTS, \\pi _o}_d(s, \\pi _o(s)).$ We can now split the event of sub-optimal action choice as follows: $& \\lbrace \\pi ^{\\text{BCTS}}_d(s) \\notin \\arg \\max _a {Q^{\\pi _o}_d}(s,a)\\rbrace \\nonumber \\\\={} & \\lbrace {\\pi _o(s) \\in \\arg \\max _a {Q^{\\pi _o}_d}(s,a)} \\cap X > 0\\rbrace \\bigcup \\lbrace {\\pi _o(s) \\notin \\arg \\max _a {Q^{\\pi _o}_d}(s,a)} \\cap X < 0 \\rbrace .",
"$ Note that this division is possible because according to Assumptions REF and REF , all actions different than $\\pi _o(s)$ have equal cumulative reward and value in expectation.",
"Next, we consider the two (deterministic) following possible cases.",
"Case I: ${\\pi _o(s) \\in \\arg \\max _a {Q^{\\pi _o}_d}(s,a).",
"}$ Then, the second event in (REF ) is an empty set and we have that ${\\lbrace \\pi ^{\\text{BCTS}}_d(s) \\notin \\arg \\max _a {Q^{\\pi _o}_d}(s,a)\\rbrace = \\lbrace X > 0\\rbrace }.$ Case II: ${\\pi _o(s) \\notin \\arg \\max _a {Q^{\\pi _o}_d}(s,a)}.$ Then, from symmetry, we will get ${\\lbrace \\pi ^{\\text{BCTS}}_d(s) \\notin \\arg \\max _a {Q^{\\pi _o}_d}(s,a)\\rbrace = \\lbrace X < 0\\rbrace }.$ In the rest of the proof, we shall apply Cantelli's inequality to upper bound $P(X>0)$ in Case I, i.e., to bound the sub-optimal action selection event.",
"Afterward, we will explain how Case II yields the same bound as in Case I.",
"Let us set $\\lambda = - {\\mathbb {E}}\\left[ X \\right] = {Q^{\\pi _o}_d}(s, \\pi _o(s)) - \\max _{a \\ne \\pi _o(s)}{Q^{\\pi _o}_d}(s, a),$ where the second relation follows from the proof of Theorem REF .",
"Next, we calculate the variance of $X$ using GEV theory [9].",
"For $G\\sim \\text{GEV}(\\mu , \\sigma , 0),$ we have that $\\text{Var}\\left[G\\right] = \\sigma ^2\\frac{\\pi ^2}{6}.$ Then, $\\text{Var}\\left[X\\right] &= \\text{Var}\\left[ \\max _{a \\ne \\pi (s)} \\hat{Q}^{BCTS, \\pi _o}_d(s, a) - \\hat{Q}^{BCTS, \\pi _o}_d(s, \\pi _o(s)) \\right] \\nonumber \\\\&= \\text{Var}\\left[ \\max _{a \\ne \\pi (s)} \\hat{Q}^{BCTS, \\pi _o}_d(s, a)\\right] + \\text{Var}\\left[ \\hat{Q}^{BCTS, \\pi _o}_d(s, \\pi _o(s)) \\right] \\nonumber \\\\&= \\text{Var}\\left[ \\max _{a \\ne \\pi (s)} \\hat{Q}^{\\pi _o}_d(s, a)\\right] + \\text{Var}\\left[ \\hat{Q}^{\\pi _o}_d(s, \\pi _o(s)) \\right] \\nonumber \\\\&= \\frac{\\gamma ^{2d}\\left(\\sigma _e^{\\text{GEV}}\\right)^2\\pi ^2}{6} + \\frac{\\gamma ^{2d}\\left(\\sigma _o^{\\text{GEV}}\\right)^2\\pi ^2}{6}, $ where the third relation is because $\\text{Var} \\left[\\hat{Q}^{BCTS, \\pi _o}_d(s, a)\\right] = \\text{Var}\\left[\\hat{Q}^{\\pi _o}_d(s,a)\\right]$ following (REF ); the last relation follows from $\\hat{Q}^{\\pi _o}_d$ having a GEV distribution as given in Lemma REF .",
"Plugging (REF ), (REF ), and (REF ) into (REF ) gives the desired result for Case I.",
"Finally, for Case II, we define $Y=-X$ and repeat exactly the same process to upper bound $P(Y>0).$ Since $({\\mathbb {E}}[Y])^2=({\\mathbb {E}}[X])^2,$ and $\\text{Var}[Y]=\\text{Var}[X],$ we obtain exactly the same bound as in Case I.",
"This concludes the proof.",
"Theorem REF now follows from Theorem REF after plugging the approximation () from the proof of Lemma REF in (REF ), and upper bounding the resulting expressions: $\\sigma ^{\\text{GEV}}_e \\approx \\frac{\\sigma _e}{\\sqrt{2d\\log A}} \\le \\frac{\\sigma _e}{\\sqrt{d\\log A}}, \\quad \\quad \\quad \\sigma ^{\\text{GEV}}_o \\approx \\frac{\\sigma _o}{\\sqrt{2(d-1)\\log A}} \\le \\frac{\\sigma _o}{\\sqrt{d\\log A}}.$"
],
[
"Proof of Proposition ",
"Proposition Let $\\hat{{\\text{var}}}_n[X]$ be the variance estimator based on $n$ samples of X.",
"Then, $\\hat{\\text{var}}_{n=2}[\\hat{Q}^{\\pi _o}(s, a)] = {\\left(\\hat{Q}^{\\pi _o}_1(s, a) - \\hat{Q}^{\\pi _o}_0(s, a)\\right)^2} / {2} = {\\delta ^2(s,a)} / {2},$ where $\\delta (s,a)$ is the Bellman error.",
"We shall employ the known unbiased variance estimator: $& \\hat{\\text{var}}_{n=2}[\\hat{Q}^{\\pi _o}(s, a)] \\\\= & {} \\frac{1}{n-1}\\sum _{d=0}^n\\left(\\hat{Q}^{\\pi _o}_d(s, a) - \\frac{1}{n}\\sum _{d=0}^n\\hat{Q}^{\\pi _o}_d(s, a) \\right)^2 \\\\\\overset{(\\mathrm {n=2})}{=} & {} \\left(\\hat{Q}^{\\pi _o}_0(s, a) - \\frac{\\hat{Q}^{\\pi _o}_0(s, a) + \\hat{Q}^{\\pi _o}_1(s, a)}{2}\\right)^2 + \\left(\\hat{Q}^{\\pi _o}_1(s, a) - \\frac{\\hat{Q}^{\\pi _o}_0(s, a) + \\hat{Q}^{\\pi _o}_1(s, a)}{2}\\right)^2 \\\\= & {} \\frac{1}{4}\\left(\\hat{Q}^{\\pi _o}_0(s, a) - \\hat{Q}^{\\pi _o}_1(s, a)\\right)^2 + \\frac{1}{4}\\left(\\hat{Q}^{\\pi _o}_1(s, a) - \\hat{Q}^{\\pi _o}_0(s, a)\\right)^2 \\\\= & {} \\frac{\\delta ^2(s,a)}{2}$"
],
[
"Approximate bounds bias",
"To show the validity of our approximation, we numerically evaluate the two sides of (REF ), i.e.",
"the LHS $B_e(\\sigma _e, A, d) - B_o(\\sigma _o, A, d)$ as given in (REF ), vs. the RHS $\\sqrt{2\\log A}\\left(\\sigma _e\\sqrt{d} - \\sigma _o\\sqrt{d-1}\\right) - (\\sigma _e - \\sigma _o) / 2.$ To compute the values, we arbitrarily choose $A=3,\\sigma _o=1,\\sigma _e=4,$ and increase $d.$ We plot the results in Fig.",
"REF .",
"As seen, the approximation is highly accurate for all values of $A^d.$ Figure: Quality of approximation in Lemma ."
],
[
"Batch-BFS algorithm",
"In continuation of the discussion in Section , Fig.",
"REF illustrates the Batch-BFS implementation.",
"The tree expansion is visualized on the left, with the corresponding batch GPU operations on the right.",
"In every tree expansion, the state $S_t$ is duplicated and concatenated with all possible actions.",
"The resulting tensor is fed into the GPU forward model to generate the tensor of next states $(S^0_{t+1},\\dots ,S^{A-1}_{t+1})$ .",
"The next-state tensor is then duplicated and concatenated again with all possible actions, fed into the forward model, etc.",
"This procedure is performed until the final depth is reached, in which case the Q-function is applied per state.",
"Tensors of the cumulative rewards and whether the episode has finished in the simulated trajectory are also duplicated and updated on every expansion to calculate the total value per leaf.",
"Figure: Visualization of Batch-BFS.",
"The tree expansion is illustrated on the left, with the corresponding batch GPU operations on the right."
],
[
"Inference timing",
"We measure the average complete TS time per action selection.",
"We provide the results together with the respective scores per game and depth in Fig.",
"REF .",
"The scores are obtained via BCTS with a Batch-BFS implementation, as reported in Section REF .",
"As depth increases, the number of explored nodes in the tree grows exponentially, with runtime increasing accordingly.",
"The plots depict the tradeoff between improved scores and the corresponding price in terms of runtime per action selection.",
"Figure: Score vs. inference time of each TS operation as a function of TS depth."
],
[
"Training",
"In our training experiments we use the same training hyper-parameters as the original DQN paper [28]."
],
[
"Ablation study: Training with propagated value from the tree nodes",
"Here, we present an ablation study for the correction to the Bellman update proposed in [14] in the case of a TS policy.",
"This correction modifies the training target: Instead of bootstrapping the value from the transition sampled from the replay buffer, we use the cumulative reward and value computed during the TS.",
"In Fig.",
"REF , we present training plots for Atari Space-Invaders for DQN with TS of depths 2,3, and 4.",
"Note that for depth 1, the correction is vacuous since it coincides with the classic Bellman update.",
"As seen, for Space-Invaders, the correction improves convergence in all tested depths.",
"Figure: Propagated value from the tree nodes: Ablation study for Space-Invaders convergence.",
"Episodic training cumulative reward of DQN with TS based on 5 seeds.",
"We compare the standard update method with the update based on the propagated value from the tree nodes, as proposed in .Lastly, we summarize the results for all tested games in Table REF .",
"The table reveals that the correction often improves training, though not always.",
"Therefore, we treat it as a hyper-parameter that we sweep over.",
"Table: Ablation study: Propagated value (PV) from the tree nodes: Ablation study for scores of all tested games.All agents of all depths were trained using DQN with TS, with similar train time that amounts to 30 million frames of DQN.",
"The reported scores were obtained as in : Average of 200 testing episodes, from the agent snapshot that obtained the highest score during training."
],
[
"Training: Wall-clock time",
"We provide in Fig.",
"REF convergence plots of DQN with TS for all tested Atari games, for depths 0 to 4.",
"To showcase the time efficiency of using a tree-based policy, we give the scores with respect to the wall-clock time.",
"We run each training experiment for two days, which amount to roughly 30 million frames for depth 0 and 1 million frames for depth $4.$ For each run, we display the average score together with std using 5 seeds.",
"Fig.",
"REF reveals the trade-off between deeper TS inference (action selection) time and the improvement thanks to the deeper TS.",
"For regular DQN (depth=0), each inference is the fastest, but generally leads to lower scores than deeper trees.",
"On the other hand, the deepest tree is not necessarily the most time-efficient.",
"In most cases here, there is a sweet-spot in depth 2 or 3 that gives the best score for the same training time.",
"Figure: Training convergence plots for tree search with DQN of depths up to 4."
],
[
"Hardware",
"We run our training experiments on a 10 core Intel(R) CPU i9-10900X @ 3.70GHz equipped with NVIDIA Quadro GV100 32GB."
]
] | 2107.01715 |
[
[
"Optimal Binary Classification Beyond Accuracy"
],
[
"Abstract The vast majority of statistical theory on binary classification characterizes performance in terms of accuracy.",
"However, accuracy is known in many cases to poorly reflect the practical consequences of classification error, most famously in imbalanced binary classification, where data are dominated by samples from one of two classes.",
"The first part of this paper derives a novel generalization of the Bayes-optimal classifier from accuracy to any performance metric computed from the confusion matrix.",
"Specifically, this result (a) demonstrates that stochastic classifiers sometimes outperform the best possible deterministic classifier and (b) removes an empirically unverifiable absolute continuity assumption that is poorly understood but pervades existing results.",
"We then demonstrate how to use this generalized Bayes classifier to obtain regret bounds in terms of the error of estimating regression functions under uniform loss.",
"Finally, we use these results to develop some of the first finite-sample statistical guarantees specific to imbalanced binary classification.",
"Specifically, we demonstrate that optimal classification performance depends on properties of class imbalance, such as a novel notion called Uniform Class Imbalance, that have not previously been formalized.",
"We further illustrate these contributions numerically in the case of $k$-nearest neighbor classification"
],
[
"Introduction",
"Many binary classification problems exhibit class imbalance, in which one of the two classes vastly outnumbers the other.",
"Classifiers that perform well with balanced classes routinely fail for imbalanced classes, and developing reliable techniques for classification in the presence of severe class imbalance remains a challenging area of research [26], [30], [21].",
"Many practical approaches have been proposed to improve performance under class imbalance, including reweighting plug-in estimates of class probabilities [32], resampling data to improve class imbalance [13], or reformulating classification algorithms to optimize different performance metrics [16], [20], [28].",
"Extensive discussion of practical methods for handling class imbalance are surveyed in the books of [26] and [21].",
"Despite the pervasive challenge of class imbalance in practical problems, our theoretical understanding of class imbalance is quite limited.",
"The vast majority of theoretical performance guarantees for classification characterize classification accuracy (or, equivalently, misclassification risk) [42], which is typically an uninformative measure of performance for imbalanced classes.",
"Under measures that are used with imbalanced classes in practice, such as precision, recall, $F_\\beta $ scores, and class-weighted scores [54], [55], existing theoretical guarantees are limited to statistical consistency, in that the algorithm under consideration asymptotically optimizes the metric of choice [29], [40], [43]; specifically, there is no finite-sample theory that would allow comparison of an algorithm's performance to that of other algorithms or to theoretically optimal performance levels.",
"Additionally, existing theory for classification does not explicitly model the effects of class imbalance, especially severe imbalance (i.e., as the proportion of samples from the rare class vanishes), and hence sheds little light on how severe imbalance should influence the behavior of a classifier.",
"This paper provides two main contributions.",
"First, in Section , we provide a novel characterization of classifiers optimizing general performance metrics that are functions of a classifier's confusion matrix.",
"This characterization generalizes a classical result – that the Bayes classifier optimizes classification accuracy – to a much larger class of performance measures, including those commonly used in imbalanced classification.",
"While this generalization may be of independent interest, we use it here to show how to provide relative performance guarantees under these more general performance measures.",
"We show, in particular, that performance guarantees can be derived in terms of the error of estimating the class probability function under uniform ($\\mathcal {L}_\\infty $ ) loss.",
"This motivates our second main contribution, which is to provide an analysis of $k$ -nearest neighbor ($k$ NN) classification under uniform loss.",
"In doing so, we also propose an explicit model of a sub-type of class imbalance, which we call Uniform Class Imbalance, and we show that the $k$ NN classifier behaves quite differently under Uniform Class Imbalance than under other sub-types of class imbalance.",
"To the best of our knowledge, such sub-types of class imbalance have not previously been distinguished in either the theoretical or practical literature, and we hope that identifying such relevant features of imbalanced datasets may facilitate the development of classifiers that perform well on specific imbalanced classification problems of practical importance.",
"Collectively, these contributions allow us to provide the first finite-sample performance guarantees for nonparametric binary classification in terms of performance metrics that are appropriate for imbalanced data and to show how these guarantees depend on the nature of imbalance in the data."
],
[
"Related Work",
"Here, we discuss how our results relate to existing theoretical guarantees for imbalanced binary classification and prior analyses of $k$ NN methods."
],
[
"Theoretical Guarantees for Imbalanced Binary Classification",
"To the best of our knowledge, there are no existing statistical guarantees in terms of the functions of the confusion matrix in the generality that we propose.",
"Extensive statistical learning theory for classification in terms of accuracy can be found in [42].",
"However, in the presence of severe class imbalance, accuracy ceases to be an informative measure of performance [14].",
"A straightforward alternative is cost-weighting.",
"This is intuitive in applications where costs can be explicitly assigned, and statistically, weighting is equivalent to threshold selection in the case of binary classification.",
"Interestingly, cost-weighting has been studied under the motivation of addressing class imbalance before [48], but in the context of calibrated losses, i.e., the guarantee that minimizing a surrogate loss for the zero-one classification loss leads to a Bayes-optimal estimator in the classical sense (see Eq.",
"(REF ); [35], [50]).",
"In addition to class-weighting, other methods commonly used for imbalanced binary classification include resampling, margin adjustment, and Neyman-Pearson classification.",
"Various forms of resampling appear most common in practice [26].",
"Undersampling the dominant class is straightforward and provides computational benefits, often with little loss in statistical performance [22], while interest in oversampling rare classes, sometimes referred to as data augmentation, has grown with the advent of sophisticated generative models (such as generative adversarial networks (GANs)) to produce additional data [38].",
"However, the theoretical ramifications of oversampling techniques used for imbalanced classification, most commonly variants of SMOTE [13], are poorly understood.",
"Margin-adjustment involves adjusting the margins of support vector machines (SVMs) to appropriately handle class-weighting [33], and [48] allows for class-based modifications to the margin as well.",
"More recent work adjusts the margins for deep neural network classifiers [11].",
"A family of techniques related to our work comes from the perspective of Neyman-Pearson classification, which attempts to minimize misclassification error on one class subject to a constraint on the maximum misclassification error on a second class, following the structure of statistical hypothesis testing.",
"Our first main result (Theorem REF ) implicitly involves re-framing optimization of general classification performance measures in the Neyman-Pearson framework, and some of our results for classification have analogues in the classical hypothesis testing literature [31].",
"Unlike many methods in imbalanced classification, substantial theoretical guarantees do exist for Neyman-Pearson classification [45], [51], [52], but these results focus on performance within the Neyman-Pearson framework, rather than in terms of general classification performance measures, which are the focus of our work.",
"Finally, the most similar existing results to our Theorem REF are due to [59] and [58]; we discuss these results in detail after presenting Theorem REF in Section ."
],
[
"$k$ NN Classification and Regression",
"The $k$ NN classifier, first published by [23], is one of the oldest and most well-studied nonparametric classifiers.",
"Early theoretical results include those of [15], who showed that the misclassification risk of the $k$ NN classifier with $k = 1$ is at most twice that of the Bayes-optimal classifier, and [49], who showed that the $k$ NN classifier is Bayes-consistent if $k \\rightarrow \\infty $ and $k/n \\rightarrow 0$ .",
"An extensive literature on the accuracy/misclassification risk of $k$ NN classification has since developed [18], [25], [46], [12], [5], [24], [19], [9].",
"Rather than accuracy bounds for $k$ NN classification, the bounds on uniform error we present in Section are most closely related to risk bounds for $k$ NN regression, of which the results of [6] are representative.",
"[6] gives convergence rates for $k$ NN regression in $\\mathcal {L}_2$ risk, weighted by the covariate distribution, in terms of noise variance and covering numbers of the covariate space.",
"While closely related to our bounds on uniform ($\\mathcal {L}_\\infty $ ) risk, their results differ in at least three main ways.",
"First, minimax rates under $\\mathcal {L}_\\infty $ risk are necessarily worse than under $\\mathcal {L}_2$ risk by a logarithmic factor (as implied by our lower bounds).",
"Second, that fact that [6] use a risk that is weighted by the covariate distribution allows them to avoid our assumption that the covariate density is lower bounded away from 0, whereas, the lower boundedness assumption is unavoidable under $\\mathcal {L}_\\infty $ risk and is ultimately necessary for analyzing general classification performance measures.",
"Finally, instead of Bernoulli noise, [6] assume additive noise with finite variance; using Bernoulli noise is crucial for us to accurately model the effect of severe class imbalance.",
"Research on improving $k$ NN for imbalanced classification has focused on algorithmic modifications, which are surveyed by [21].",
"Examples include prototype selection, in which representative points are selected from the training set to use with the $k$ NN classifier [34], [36], [57], and gravitational methods, in which the distance function is modified to resemble the gravitational force [10], [61].",
"However, to the best of our knowledge, no statistical guarantees exist for such methods."
],
[
"Setup and Notation",
"Let $(\\mathcal {X}, \\rho )$ be a separable metric space, and let $\\mathcal {Y}= \\lbrace 0, 1\\rbrace $ denote the set of classes.",
"Consider a dataset of $n$ independent samples $(X_{1}, Y_{1}), \\ldots , (X_{n}, Y_{n})$ drawn from a distribution $P_{X, Y}$ on $\\mathcal {X}\\times \\mathcal {Y}$ with marginals $P_X$ and $P_Y$ .",
"To construct classifiers that optimize general performance metrics, it will be necessary to consider stochastic classifiers, which may assign inputs to classes nondeterministically.",
"Formally, letting $\\mathcal {B}:= \\lbrace Y \\sim \\text{Bernoulli}(p) : p \\in [0, 1]\\rbrace $ denote the set of binary random variables, a stochastic classifier can be modeled as a mapping $\\widehat{Y} : \\mathcal {X}\\rightarrow \\mathcal {B}$ , where, for any $x \\in \\mathcal {X}$ , $\\operatornamewithlimits{\\mathbb {E}}[\\widehat{Y}(x)]$ is the probability that the classifier assigns $x$ to class 1.",
"We will use $\\mathcal {SC}$ to denote the class of all stochastic classifiers.",
"The true regression function $\\eta : \\mathcal {X}\\rightarrow [0, 1]$ is defined by $\\eta ^*(x) = \\operatornamewithlimits{\\mathbb {P}}\\left[Y = 1| X = x \\right] = \\operatornamewithlimits{\\mathbb {E}}\\left[ Y | X = x \\right];$ that is, given an instance $X_i$ , the label $Y_i$ has a Bernoulli distribution with mean $\\eta (X_i)$ .",
"As we show in the next section, an optimal classifier can always be written in terms of the true regression function $\\eta $ , motivating estimates $\\widehat{\\eta }: \\mathcal {X}\\rightarrow [0, 1]$ of $\\eta $ .",
"Such estimates $\\widehat{\\eta }$ are called “regressors”."
],
[
"Optimal Classification beyond Accuracy",
"A famous result states that classification accuracy is maximized by the “Bayes” classifier $\\widehat{Y}(x) \\sim \\text{Bernoulli}\\left( 1\\lbrace \\eta ^*(x) > 0.5\\rbrace \\right).$ This result is of limited practical value when $\\eta ^*$ is unknown, but it is a cornerstone of the statistical theory of binary classification because it provides an optimal performance benchmark against which a classifier of interest can be evaluated in terms of classification accuracy [18], [41], [27].",
"As discussed previously, classification accuracy can be a poor measure of performance in the presence of class imbalance.",
"Therefore, the main contribution of this section, provided in Theorem REF below, is to generalize this result to a broad class of classification performance measures, including those commonly used in imbalanced classification.",
"First, we specify the performance measures for which our results apply."
],
[
"Confusion Matrix Measures (CMMs)",
"Nearly all measures of classification performance, including accuracy, precision, recall, $F_\\beta $ scores, and others, can be computed from the confusion matrix, which counts the number of test samples in each $(\\text{true class, estimated class})$ pair.",
"More formally, let $= \\lbrace C \\in [0, 1]^{2 \\times 2} : C_{1,1} + C_{1, 2} + C_{2, 1} + C_{2,2} = 1\\rbrace $ denote the set of all possible binary confusion matrices.",
"Given a classifier $\\widehat{Y}$ , the confusion matrix $C_{\\widehat{Y}}$ and empirical confusion matrix $\\widehat{C}_{\\widehat{Y}}$ are given respectively by $C_{\\widehat{Y}} =\\begin{bmatrix}\\text{TN}_{\\widehat{Y}} & \\text{FP}_{\\widehat{Y}} \\\\\\text{FN}_{\\widehat{Y}} & \\text{TP}_{\\widehat{Y}}\\end{bmatrix} \\in \\text{ and } \\quad \\widehat{C}_{\\widehat{Y}} =\\begin{bmatrix}\\widehat{\\text{TN}}_{\\widehat{Y}} & \\widehat{\\text{FP}}_{\\widehat{Y}} \\\\\\widehat{\\text{FN}}_{\\widehat{Y}} & \\widehat{\\text{TP}}_{\\widehat{Y}}\\end{bmatrix} \\in $ wherein the true positive rate $\\text{TP}_{\\widehat{Y}}$ and the empirical true positive rate $\\widehat{\\text{TP}}_{\\widehat{Y}}$ are given respectively by $\\text{TP}_{\\widehat{Y}} = \\operatornamewithlimits{\\mathbb {E}}\\left[ \\eta ^*(X) \\widehat{Y}(X) \\right]\\quad \\text{ and } \\quad \\widehat{\\text{TP}}_{\\widehat{Y}} = \\frac{1}{n} \\sum _{i = 1}^{n} Y_i \\widehat{Y}(X_i),$ and the true and empirical false positive ($\\text{FP}_{\\widehat{Y}}$ and $\\widehat{\\text{FP}}_{\\widehat{Y}}$ ), false negative ($\\text{FN}_{\\widehat{Y}}$ and $\\widehat{\\text{FN}}_{\\widehat{Y}}$ ), and true positive ($\\text{TP}_{\\widehat{Y}}$ and $\\widehat{\\text{TP}}_{\\widehat{Y}}$ ) rates are defined similarly.",
"Note that the expectation in Eq.",
"(REF ) is taken over both stochasticity in the data and stochasticity in the classifier.",
"Intuitively, any measure of a classifier's performance should improve as $\\text{TN}$ and $\\text{TP}$ increase and $\\text{FN}$ and $\\text{FP}$ decrease.",
"We therefore define the class of Confusion Matrix Measures (CMMs) as follows: Definition 1 (Confusion Matrix Measure (CMM)) A function $M : \\mathbb {R}$ is called a confusion matrix measure (CMM) if, for any confusion matrix $C =\\begin{bmatrix}\\text{TN}& \\text{FP}\\\\\\text{FN}& \\text{TP}\\end{bmatrix} \\in \\text{ $\\epsilon _1 \\in [0, \\text{FP}]$, and $\\epsilon _2 \\in [0, \\text{FN}]$, we have }M(C) \\le M \\left(\\begin{bmatrix}\\text{TN}+ \\epsilon _1 & \\text{FP}- \\epsilon _1 \\\\\\text{FN}- \\epsilon _2 & \\text{TP}+ \\epsilon _2\\end{bmatrix}\\right).$ Under this definition, correcting any incorrect classification should not reduce a CMM.",
"This mild condition is satisfied by all performance measures commonly used in practice, including weighted accuracies, precision, recall, $F_\\beta $ scores, and MCC.",
"Our main result, presented below, will generalize the Bayes classifier (REF ) to arbitrary CMMs."
],
[
"Generalizing the Bayes Classifier",
"The Bayes classifier given in Eq.",
"(REF ) thresholds the regression function deterministically at the value $0.5$ .",
"The following definition generalizes this to a stochastic threshold at an arbitrary value: Definition 2 (Regression-Thresholding Classifier) A classifier $\\widehat{Y} : \\mathcal {X}\\rightarrow \\mathcal {B}$ is called a regression-thresholding classifier if, for some $p, t \\in [0, 1]$ and some $\\eta : \\mathcal {X}\\rightarrow [0, 1]$ , $\\widehat{Y}(x) \\sim \\text{Bernoulli}(p1\\lbrace \\eta (x) = t\\rbrace + 1\\lbrace \\eta (x) > t\\rbrace ), \\quad \\text{ for all } x \\in \\mathcal {X}.$ In the sequel, we will denote such classifiers $\\widehat{Y}_{p,t,\\eta }$ , and refer to the pair $(p, t)$ as the threshold.",
"Given Definitions REF and REF , we can state the main result of this section: Theorem 3 If $\\operatornamewithlimits{\\arg \\!\\max }_{\\widehat{Y} \\in \\mathcal {SC}} M(C_{\\widehat{Y}}) \\ne \\varnothing $ , then there exists a regression-thresholding classifier $\\widehat{Y}_{p,t,\\eta } \\in \\operatornamewithlimits{\\arg \\!\\max }_{\\widehat{Y} \\in \\mathcal {SC}} M(C_{\\widehat{Y}}).$ Theorem REF states that, if a CMM $M$ is maximized by any stochastic classifier, then $M$ can be maximized by some regression-thresholding classifier.",
"As a special case, the optimality of the classical Bayes classifier corresponds to $M(C) = \\text{TN}+ \\text{TP}$ , $p = 0$ , and $t = 0.5$ .",
"Since classifiers of the form (REF ) generalize both the regression-thresholding structure and optimality properties of the Bayes classifier, we will refer to them, in the sequel, as generalized Bayes classifiers.",
"Note that the existence of any maximizer $\\widehat{Y}$ of $M(C_{\\widehat{Y}})$ depends on specific properties, such as (semi)continuity or convexity of $M$ ; we do not investigate the question of existence here and focus only on characterizing a particular maximizer when it exists.",
"Moreover, maximizers $\\widehat{Y}$ of $M(C_{\\widehat{Y}})$ are typically not unique, since, for example, changing $\\widehat{Y}$ on sets of $P_X$ -measure 0 does not change $C_{\\widehat{Y}}$ ; however, among maximizers, generalized Bayes classifiers are an important, mathematically well-behaved subclass.",
"To the best of our knowledge, no previous results provide such a precise and general characterization of an optimal classifier.",
"The most similar results of which we are aware are those of [59] and [58].",
"Theorem 3.1 of [59] shows that, under a “karmic” assumption slightly stronger than our monotonicity assumption in Definition REF , if the random variable $\\eta (X)$ is absolutely continuous (i.e., has a density with respect to Lebesgue measure on $[0, 1]$ ), then an optimal classifier can be obtained by (deterministically) thresholding the regression function $\\eta $ .",
"Our Theorem REF is strictly more general, since, if $\\eta (X)$ is absolutely continuous, then one can set $p = 0$ without changing $C_{\\widehat{Y}_{p,t,\\eta }}$ .",
"More recently, [58] showed (in their Corollary 2), under absolute continuity of $\\eta (X)$ and a monotonicity assumption comparable to our Definition REF , that the optimal classifier can always be written as a mixture of two deterministic classifiers.",
"Although [58] consider more general multiclass and multilabel settings, in the binary setting, our result is more precise than theirs, since our regression-thresholding classifiers are deterministic except for a single value of $\\eta $ .",
"We leave it to future work to develop a similarly precise generalization of this result to multiclass and multilabel settings.",
"Interestingly, [58] claim that regularity assumptions on $\\eta (X)$ such as absolute continuity “seem to be unavoidable”, and our Theorem REF appears to be the first result to omit such assumptions; specifically, we show that this comes at the cost of the optimal classifier possibly being non-deterministic for a single atom of $\\eta (X)$ .",
"Finally, we note that there exist a number of relevant results that are specific to certain CMMs, such as Lemma 12 of [60], which provides a similar characterization of the (deterministic) classifier that maximizes the $F_1$ score.",
"We prove Theorem REF in Appendix using a series of variational arguments.",
"Roughly speaking, given a classifier $\\widehat{Y}$ is not of the form (REF ), we construct a perturbation $\\widehat{Y}^{\\prime }$ of $\\widehat{Y}$ such that either $M(C_{\\widehat{Y}}) < M(C_{\\widehat{Y}^{\\prime }})$ or $\\widehat{Y}^{\\prime }$ is of the form (REF ) and $M(C_{\\widehat{Y}}) \\le M(C_{\\widehat{Y}^{\\prime }})$ .",
"Since, in general, the classifier $\\widehat{Y}$ might be quite poorly behaved (e.g., its behavior on sets of $P_X$ -measure 0 could be arbitrary), the main technical complexity lies in constructing admissible perturbations (i.e., those that are well-defined classifiers).",
"For this reason, the proof of Theorem REF involves a series of constructions of increasingly well-behaved classifiers.",
"Theorem REF shows that the generalized Bayes classifier can be written in terms of the regression function $\\eta $ and at most 2 scalar parameters $p$ and $t$ depending on the distribution of $\\eta (X)$ and the CMM $M$ .",
"We now provide a simple example showing that the generalized Bayes classifier cannot be further simplified without additional assumptions: Example 4 Suppose that $\\mathcal {X}= \\lbrace 0\\rbrace $ is a singleton, that $\\eta (0) \\in (0, 1)$ , and, for some $\\theta > 0$ , $M(C) = (\\text{TP})^\\theta \\text{TN}$ .",
"One can easily check that $M$ is a valid CMM.",
"Suppose $\\widehat{Y}$ is a classifier of the form (REF ), with generalized threshold $(p, t)$ .",
"It is straightforward to compute that $M(C_{\\widehat{Y}}) = (p \\eta (0))^\\theta (1 - p) (1 - \\eta (0)) 1\\lbrace t = \\eta (0)\\rbrace $ , and that $M(C_{\\widehat{Y}})$ is uniquely maximized by $p = \\frac{\\theta }{\\theta + 1} \\in (0, 1)$ and $t = \\eta (0) \\in (0, 1)$ .",
"This shows that both generalized threshold parameters $p$ and $t$ in Eq.",
"(REF ) are necessary in the absence of further assumptions on $M$ or $\\eta $ .",
"This example also illustrates the need for stochasticity in classifiers optimizing general CMMs.",
"Specifically, for any deterministic classifier $\\widehat{Y}$ , either $\\widehat{Y}(0) = 0$ (so that $\\text{TP}= 0$ ) or $\\widehat{Y}(0) = 1$ (so that $\\text{TN}= 0$ ); in either case, $M(C_{\\widehat{Y}}) = 0$ ."
],
[
"Relative Performance Guarantees in terms of the Generalized Bayes Classifier",
"Theorem REF motivates a two-step approach to imbalanced classification in which one first estimates the regression function $\\eta $ and then selects a generalized threshold $(p, t)$ that optimizes empirical performance $M(\\widehat{C}_{\\widehat{Y}})$ .",
"Such an approach can have a number of practical advantages; for example, one can address covariate shift (specifically, a change in $P_X$ without a corresponding change in $P_{Y|X}$ ), or retune a classifier trained under one CMM to perform well under another CMM, by simply re-selecting the two scalar parameters $p$ and $t$ .",
"This may be statistically and computationally much easier than retraining a classifier from scratch.",
"In this section, we focus on an advantage for theoretical analysis, namely that the error of such a classifier can be decomposed into errors in selecting $(p, t)$ and errors in estimating $\\eta $ , facilitating the derivation of performance guarantees relative to the generalized Bayes classifier (REF ).",
"All results in this section are proven in Appendix .",
"Our first lemma bounds the performance difference of thresholding two different regressors in terms of their uniform distance.",
"This will allow us to bound the error of using a regressor $\\widehat{\\eta }$ instead of the true regression function $\\eta $ .",
"Lemma 5 Let $p,t \\in [0, 1]$ and let $\\eta , \\eta ^{\\prime } : \\mathcal {X}\\rightarrow [0, 1]$ .",
"Then, $\\left\\Vert C_{\\widehat{Y}_{p,t,\\eta }} - C_{\\widehat{Y}_{p,t,\\eta ^{\\prime }}} \\right\\Vert _\\infty \\le \\operatornamewithlimits{\\mathbb {P}}\\left[ |\\eta (X) - t| \\le \\left\\Vert \\eta - \\eta ^{\\prime }\\right\\Vert _\\infty \\right].$ Intuitively, Lemma REF bounds the largest difference in the confusion matrices of $\\widehat{Y}_{p,t,\\eta }$ and $\\widehat{Y}_{p,t,\\eta ^{\\prime }}$ by the probability that the threshold $t$ lies between $\\eta $ and $\\eta ^{\\prime }$ .",
"As we will show later, under standard margin assumptions, this can be bounded by the $\\mathcal {L}_\\infty $ distance $\\left\\Vert \\eta - \\eta ^{\\prime }\\right\\Vert _\\infty $ between $\\eta $ and $\\eta ^{\\prime }$ .",
"Our second lemma bounds the worst-case error (over all thresholds $(p, t) \\in [0, 1]$ ) of our estimated confusion matrix.",
"This will allow us to bound the error due to using an empirically selected threshold $(\\widehat{p}, \\widehat{t})$ instead of the threshold $(p^*, t^*)$ that is optimal for the true regression function.",
"Lemma 6 Let $\\eta : \\mathcal {X}\\rightarrow [0, 1]$ be any regression function.",
"Then, with probability at least $1 - \\delta $ , $\\sup _{p, t \\in [0, 1]} \\left\\Vert \\widehat{C}_{\\widehat{Y}_{p,t,\\eta }} - C_{\\widehat{Y}_{p,t,\\eta }} \\right\\Vert _\\infty \\le \\sqrt{\\frac{8}{n} \\log \\left( \\frac{32(2n + 1)}{\\delta } \\right)}.$ This result follows from a standard Vapnik-Chervonenkis-type bound on the complexity of the set $\\lbrace \\widehat{Y}_{p,t,\\eta } : p, t \\in [0, 1]\\rbrace $ of possible regression-thresholding classifiers with fixed regression function $\\eta $ .",
"In fact, in the Appendix, we prove a generalization of Lemma REF that bounds the error between the empirical and true confusion matrices uniformly over any family $\\mathcal {F}$ of (potentially stochastic) classifiers in terms of the growth function of $\\mathcal {F}$ .",
"As a consequence, for any class $\\mathcal {F}$ with finite VC dimension, we obtain uniform convergence at the fairly fast rate $\\sqrt{\\frac{\\log (n/\\delta )}{n}}$ .",
"As we will formalize later in terms of Lipschitz constants, this suggests that the difficulty in tuning an imbalanced classifier to optimize a particular CMM $M$ comes not from difficulty in estimating the confusion matrix but rather from the instability of many commonly used CMMs.",
"Because Theorem REF shows that any CMM can be optimized by a regression-thresholding classifier, to reduce notational complexity, we state here only the specific result for regression-thresholding classifiers.",
"Before combining Lemmas REF and REF to give the main result of this section, we present a widely-used margin condition, which characterizes the separation between the two classes.",
"Definition 7 (Tsybakov Margin Condition) Let $C, \\beta \\ge 0$ , $t \\in (0, 1)$ .",
"A classification problem specified by covariate distribution $P_X$ and regression function $\\eta $ is said so satisfy a $(C, \\beta )$ -margin condition around $t$ if, for any $\\delta > 0$ , $\\operatornamewithlimits{\\mathbb {P}}\\left[ |\\eta (X) - t| \\le \\epsilon \\right] \\le C \\epsilon ^\\beta .$ The Tsybakov margin condition, introduced by [37] in the special case $t = 0.5$ , has been widely used to establish fast convergence rates for classification in terms of accuracy [2], [1], [12].",
"Together with the margin condition and a Lipschitz smoothness assumption on the CMM $M$ , Lemmas REF and REF give the following bound on the sub-optimality of any regression-thresholding classifier, where the threshold is selected by maximizing the CMM $M$ on the empirical confusion matrix.",
"Corollary 8 Let $\\eta : \\mathcal {X}\\rightarrow [0, 1]$ denote the true regression function, and let $\\widehat{\\eta }: \\mathcal {X}\\rightarrow [0, 1]$ denote any regressor.",
"Let $\\left( \\widehat{p}, \\widehat{t} \\right) := \\operatornamewithlimits{\\arg \\!\\max }_{(p, t) \\in [0, 1]^2} M \\left( \\widehat{C}_{\\widehat{Y}_{p,t,\\widehat{\\eta }}} \\right)\\quad \\text{ and } \\quad \\left( p^*, t^* \\right) := \\operatornamewithlimits{\\arg \\!\\max }_{(p, t) \\in [0, 1]^2} M \\left(C_{\\widehat{Y}_{p,t,\\eta }} \\right)$ denote the empirically selected and true optimal thresholds, respectively.",
"Suppose that $M$ is Lipschitz continuous with constant $L_M$ with respect to the uniform ($\\mathcal {L}_\\infty $ ) metric on $.",
"Finally, suppose that $ PX$ and $$ satisfies a $ (C, )$-margin condition around $ t*$.",
"Then, with probability at least $ 1 - $,{\\begin{@align}{1}{-1}M\\left(C_\\eta \\left(p^*, t^*\\right)\\right) - M\\left(C_{\\widehat{\\eta }}\\left(\\widehat{p}, \\widehat{t}\\right)\\right)& \\le L_M \\left( C\\left\\Vert \\eta - \\widehat{\\eta }\\right\\Vert _\\infty ^\\beta + 2 \\sqrt{\\frac{8}{n} \\log \\frac{32(2n + 1)}{\\delta }} \\right).\\end{@align}}$ The computation of a Lipschitz constant $L_M$ is straightforward for some CMMs, such as weighted accuracy, but is more involved for many other CMMs such as precision, recall, and $F_\\beta $ scores, which which the exact value of $L_M$ depends on the true distribution of class labels in the data.",
"In Appendix REF , we therefore demonstrate how to compute $L_M$ for a few of these standard CMMs."
],
[
"Uniform Error of the $k$ -Nearest Neighbor Regressor",
"In the previous section, for classifiers that threshold an estimate (a “regressor”) of the regression function, we bounded relative performance, as measured by arbitrary CMMs, in terms of the uniform ($\\mathcal {L}_\\infty $ ) loss of the regression function estimate.",
"In this section, we bound the uniform loss of one such regressor, the widely used $k$ -nearest neighbor ($k$ NN) regressor.",
"Our analyses include a parameter $r$ , introduced in Section REF , that characterizes a novel sub-type of class imbalance, which we call Uniform Class Imbalance.",
"As we discuss later, this leads to insights about how the behavior of the $k$ NN classifier depends not only on the degree, but also on the structure, of class imbalance in a given dataset."
],
[
"$k$ -Nearest Neighbor Regressor",
"Given a point $x \\in \\mathcal {X}$ , order the training data $X_{\\sigma _{1}(x)}, \\ldots , X_{\\sigma _{n}(x)}$ such that $\\rho \\left(X_{\\sigma _{1}(x)}, \\; x\\right)\\le \\ldots \\le \\rho \\left(X_{\\sigma _{n}(x)}, \\; x\\right);$ i.e., $X_{\\sigma _i(x)}$ is the $i^{th}$ -nearest neighbor of $x$ among $X_1,...,X_n$ .",
"For an integer $k > 0$ , the $k$ NN regressor $\\widehat{\\eta }_k : \\mathcal {X}\\rightarrow [0, 1]$ , is defined by the proportion $\\widehat{\\eta }_k(x) = \\frac{1}{k} \\sum _{i = 1}^k Y_{\\sigma _i(x)},\\quad \\text{ for all } x \\in \\mathcal {X},$ of $x$ 's $k$ -nearest neighbors in class 1."
],
[
"Uniform Class Imbalance",
"In this paper, we formalize a sub-type of class imbalance, which we refer to as Uniform Class Imbalance.",
"We decompose the regression function as $\\eta = r \\zeta $ , where $r \\in (0, 1]$ and $\\zeta : \\mathcal {X}\\rightarrow [0, 1]$ is a regression function with $\\sup _{x \\in \\mathcal {X}} \\zeta (x) = 1$ .",
"Note that this decomposition loses no generality, as any regression function $\\eta $ can be written in this form.",
"In Uniform Class Imbalance, $r \\approx 0$ , so that the class $Y = 1$ is rare regardless of the covariate $X$ (hence the name “uniform”).",
"Uniform Class Imbalance tends to occur in “challenging” classification problems in which the covariate $X$ provides only partial information about the class $Y$ .",
"Practical examples include rare disease diagnosis [47], credit card fraud detection [3], or predicting whether an applicant will be offered a job when there are many more qualified applicants than openings.",
"In practice, in such problems, the classifier's role is often not so much to make a final class determination as to identify “high-risk” samples $X$ such that $\\eta (X)$ is relatively elevated, for follow-up investigation.",
"Uniform Class Imbalance can be distinguished from “easier” classification problems in which, for some values $x \\in \\mathcal {X}$ , $\\eta (X) \\approx 1$ and so a good classifier can confidently assign the label $Y = 1$ .",
"These include well-separated classes, or the extreme case where $Y$ is a deterministic function of $X$ , such as in certain protein structure prediction problems [44]."
],
[
"Upper Bounds",
"In this section, we present bounds on the uniform error $U(\\widehat{\\eta }) := \\left\\Vert \\eta - \\widehat{\\eta }\\right\\Vert _\\infty $ of the $k$ NN regressor $\\widehat{\\eta }_k$ , where, for a function $f : \\mathcal {X}\\rightarrow \\mathbb {R}$ , $\\Vert f\\Vert _\\infty := \\sup _{x \\in \\mathcal {X}} |f(x)|$ denotes the $\\sup $ -norm of $f$ .",
"Before presenting our bounds, we define two standard quantities, covering numbers and shattering coefficients, by which we measure the complexity of the feature space.",
"Definition 9 (Covering Number) Suppose $(\\mathcal {X}, \\rho )$ is a totally bounded metric space.",
"Then, for any $\\epsilon > 0$ , the $\\epsilon $ -covering number $N(\\epsilon )$ of $(\\mathcal {X}, \\rho )$ is the smallest integer such that there exist $N(\\epsilon )$ points $x_1,...,x_{N(\\epsilon )} \\in X$ satisfying $\\mathcal {X}\\subseteq \\bigcup _{i = 1}^{N(\\epsilon )} B(x_i, \\epsilon )$ .",
"Definition 10 (Shattering Coefficient of Balls) For positive integers $n$ , $S(n):= \\sup _{x_1,...,x_n \\in \\mathcal {X}} \\left| \\left\\lbrace \\lbrace x_1,...,x_n\\rbrace \\cap B(x, \\epsilon ) : x \\in \\mathcal {X}, \\epsilon \\ge 0 \\right\\rbrace \\right|$ denotes the shattering coefficient of open balls in $(\\mathcal {X}, \\rho )$ .",
"We now state two assumptions we make on the joint distribution $P_{X, Y}$ of the data Assumption 11 (Dense Covariates Assumption) The marginal distribution $P_X$ of the covariates is lower bounded in the sense that, for some constants $p_*,\\epsilon ^*,d > 0$ , for any point $x$ in $\\mathcal {X}$ and radius $\\epsilon $ in $(0,\\epsilon ^*]$ , we have the inequality $P_X(B_\\epsilon (x)) \\ge p_* \\epsilon ^d$ .",
"Assumption REF ensures that each query point's nearest neighbor is sufficiently near.",
"We will also need to assume that the regression function $\\zeta $ is sufficiently smooth: Assumption 12 (Hölder Continuity) $\\zeta $ is $(\\alpha , L)$ -Hölder continuous; that is, for all $x, x^{\\prime } \\in \\mathcal {X}$ , $\\left| \\zeta (x) - \\zeta (x^{\\prime }) \\right| \\le L \\rho ^\\alpha (x, x^{\\prime })$ .",
"We now provide our upper bound on the uniform error, proven in Appendix REF .",
"Theorem 13 Under Assumptions REF and REF , whenever $k / n \\le p_*(\\epsilon ^*)^d / 2$ , for any $\\delta > 0$ , with probability at least $1 - N\\left( \\left( 2k / (p_* n) \\right)^{1/d} \\right) e^{-k/4} - \\delta $ , we have the uniform error bound $\\left\\Vert \\eta - \\widehat{\\eta }\\right\\Vert _\\infty \\le 2^\\alpha Lr\\left( \\frac{2k}{p_* n} \\right)^{\\alpha /d}+ \\frac{2}{3k} \\log \\frac{2 S(n)}{\\delta } + \\sqrt{\\frac{2r}{k} \\log \\frac{2 S(n)}{\\delta }}.$ If $r \\in O \\left( \\frac{\\log S(n)}{n} \\right)$ , this bound is minimized by $k \\asymp n$ , giving $\\left\\Vert \\eta - \\widehat{\\eta }\\right\\Vert _\\infty \\in O_P \\left( \\frac{\\log S(n)}{n} \\right)$ .",
"Otherwise, under a mild simplifying assumption that $N(\\epsilon )$ increases at most polynomially with $1/\\epsilon $ , this bound is minimized by $k \\asymp n^{\\frac{2\\alpha }{2\\alpha +d}} (\\log S(n))^{\\frac{d}{2\\alpha +d}} r^{-\\frac{d}{2\\alpha + d}}$ , giving $\\left\\Vert \\eta - \\widehat{\\eta }\\right\\Vert _\\infty \\in O_P \\left( \\left( \\frac{\\log S(n)}{n} \\right)^\\frac{\\alpha }{2\\alpha + d} r^\\frac{\\alpha + d}{2\\alpha + d} \\right).$ Of the three terms in (REF ), the first term, of order $r(k/n)^{\\alpha /d}$ , comes from smoothing bias of the $k$ NN classifier.",
"The second and third terms are due to label noise, with the second term dominating under extreme class imbalance ($r \\in O \\left( \\frac{\\log S(n)}{n} \\right)$ ) and the third term dominating otherwise.",
"Theorem REF shows that the optimal choice of the tuning parameter $k$ is much larger under Uniform Class Imbalance than in the case of balanced classes; indeed, one can check that setting $k \\asymp n^{\\frac{2\\alpha }{2\\alpha +d}} (\\log S(n))^{\\frac{d}{2\\alpha +d}}$ , which is optimal in the balanced case, gives a rate that slower by a factor of $r^{-\\frac{d}{4\\alpha + d}}$ .",
"One interpretation is that a larger number of neighbors is needed to obtain enough samples from the rare class to make a reliable prediction at any given point.",
"The following two examples demonstrate how to apply Theorem REF in specific settings of interest: Corollary 14 (Euclidean, Absolutely Continuous Case) Suppose $(\\mathcal {X},\\rho ) = ([0,1]^d,\\Vert \\cdot \\Vert _2)$ is the unit cube in $\\mathbb {R}^d$ , equipped with the Euclidean metric, and $P_X$ has a density that is lower bounded away from 0 on $\\mathcal {X}$ .",
"Then, $N(\\epsilon ) \\le (2/\\epsilon )^d$ and $S(n) \\le 2n^{d + 1} + 2$ , and so, for $k \\asymp n^{\\frac{2\\alpha }{2\\alpha +d}} (\\log n)^{\\frac{d}{2\\alpha +d}} r^{-\\frac{d}{2\\alpha + d}}$ , by Theorem REF , $\\left\\Vert \\eta - \\widehat{\\eta }\\right\\Vert _\\infty \\in O_P \\left( \\left( \\frac{\\log n}{n} \\right)^{\\frac{\\alpha }{2\\alpha +d}} r^\\frac{\\alpha + d}{2\\alpha + d} \\right).$ The most problematic term in this bound is the exponential dependence on the dimension $d$ of the covariates.",
"Fortunately, since Theorem REF utilizes covering numbers, it improves if the covariates exhibit structure, such as that of a low-dimensional manifold.",
"The next example formalizes this.",
"Corollary 15 (Implicit Manifold Case) Suppose $Z$ is a $[0,1]^d$ -valued random variable with a density lower bounded away from 0, and suppose that, for some Lipschitz map $T : [0,1]^d \\rightarrow \\mathbb {R}^D$ , $X = T(Z)$ .",
"Then, $N(\\epsilon ) \\le (2/\\epsilon )^d$ , and $S(n) \\le 2n^{D + 1} + 2$ , and so, by Theorem REF , $k \\asymp n^{\\frac{2\\alpha }{2\\alpha +d}} (\\log n)^{\\frac{d}{2\\alpha +d}} r^{-\\frac{d}{2\\alpha + d}}$ , $\\left\\Vert \\eta - \\widehat{\\eta }\\right\\Vert _\\infty \\in O_P \\left( \\left( \\frac{\\log n}{n} \\right)^{\\frac{\\alpha }{2\\alpha +d}} r^\\frac{\\alpha + d}{2\\alpha + d} \\right).$ This shows that, if the $D$ covariates lie implicitly on a $d$ -dimensional manifold (e.g., if the covariates are strongly correlated), convergence rates depend on $d$ , which may be much smaller than $D$ .",
"We close this section with a lower bound, proven in Appendix REF , on the minimax uniform error, showing that the rate provided in Theorem REF is minimax optimal over $(\\alpha , L)$ -Hölder regression functions, up to a polylogarithmic factor in $r$ : Theorem 16 Suppose $\\mathcal {X}= [0,1]^d$ is the $d$ -dimensional unit cube and the marginal distribution of $X$ is uniform on $\\mathcal {X}$ .",
"Let $\\Sigma ^\\alpha (L)$ denote the family of $(\\alpha , L)$ -Hölder continuous regression function.",
"Then, for any $\\alpha , L > 0$ , there exist constants $n_0$ and $c > 0$ (depending only on $\\alpha $ , $L$ , and $d$ ) such that, for all $n \\ge n_0$ and any estimator $\\widehat{\\eta }$ , $\\sup _{\\zeta \\in \\Sigma ^\\alpha (L)} \\mathop {\\operatornamewithlimits{\\mathbb {P}}}\\left[ \\left\\Vert \\eta - \\widehat{\\eta }\\right\\Vert _\\infty \\ge c \\left( \\frac{\\log (nr)}{n} \\right)^{\\frac{\\alpha }{2\\alpha + d}} r^\\frac{\\alpha + d}{2\\alpha + d} \\right] \\ge \\frac{1}{8}.$"
],
[
"Discussion",
"The upper bounds on $\\left\\Vert \\eta - \\widehat{\\eta }\\right\\Vert _\\infty $ given in this section can be plugged directly into Corollary to provide error bound under arbitrary CMMs, in terms of the sample size $n$ , hyperparameter $k$ , degree $r$ of Uniform Class Imbalance, and complexity parameters (margin $\\beta $ , smoothness $\\alpha $ , intrinsic dimension $d$ , etc.)",
"of $\\mathcal {X}$ and $P_{X,Y}$ .",
"Thus, these results collectively give some of the first finite-sample guarantees under general performance metrics used for imbalanced classification.",
"As noted previously, our analysis shows that, under severe Uniform Class Imbalance, the optimal choice of the hyperparameter $k$ is much larger than in balanced classification.",
"Importantly, this larger choice of $k$ , leads to sub-optimal, or even inconsistent, estimates of the regression function under other (nonuniform) forms of class imbalance.",
"The following example illustrates this: Example 17 Suppose $\\mathcal {X}= [0, 1]$ , $X \\sim \\operatorname{Uniform}([0, 1])$ , and $r \\in (0, 1)$ .",
"Consider two regression functions $\\eta _1(x) = r(1 - x)$ and $\\eta _2(x) = \\max \\lbrace 0, 1 - x/r\\rbrace $ .",
"Both $\\eta _1$ and $\\eta _2$ exhibit the same degree of overall class imbalance, with the proportion of samples from class 1 being $r/2$ .",
"The regression function $\\eta _1$ satisfies Uniform Class Imbalance of degree $r$ , whereas $\\eta _2$ does not satisfy a nontrivial degree of Uniform Class Imbalance, since $\\eta _2(0) = 1$ .",
"For sufficiently small $r \\in (0, 1)$ , specifically $r \\in o \\left( n^{\\frac{-d}{2(\\alpha + d)}} \\right)$ , Theorem REF gives that the optimal choice of $k$ under $\\eta _1$ satisfies $k \\in \\omega (rn)$ .",
"On the other hand, if $k \\in \\omega (rn)$ , then, under $\\eta _2$ , $\\operatornamewithlimits{\\mathbb {E}}\\left[ \\widehat{\\eta }_k(0) \\right] \\rightarrow 0$ , so that $\\widehat{\\eta }_k(0)$ is an inconsistent estimate of $\\eta _2(0) = 1$ .",
"This example demonstrates that constructing classifiers that perform well under severe class imbalance may require distinguishing different sub-types of class imbalance, such as Uniform Class Imbalance."
],
[
"Conclusions",
"The main conclusions of this paper are as follows.",
"First, the Bayes classifier, which optimizes classification performance in terms of accuracy, can be generalized to many other measures of classification performance using a simple thresholding procedure with only two additional scalar parameters.",
"This Generalized Bayes Classifier provides an optimal performance benchmark, relative to which one can evaluate many classifiers of interest in terms of their ability to estimate the regression function in uniform loss.",
"This includes the widely-used $k$ NN classifier, for which we provided a number of guarantees, showing that it performs minimax optimally under uniform loss in a number of settings, including that of severe Uniform Class Imbalance.",
"On the other hand, we showed that the optimal tuning of $k$ can differ significantly between different sub-types of imbalanced classification, suggesting that developing reliable classifiers for severely imbalanced classification may require a more nuanced understanding of the nature of class imbalance intrinsic to the problem at hand.",
"We hope that some of these ideas, especially the generalized Bayes classifier and the distinction of sub-types of class imbalance, will play a role in developing a coherent and insightful statistical theory of imbalanced classification, and that this theory will inform the construction of more reliable classifiers for challenging real-world imbalanced classification problems.",
"Additionally, these ideas should be generalized to the multi-class case, in which severe class imbalance emerges naturally when the number of classes is large and existing statistical theory is quite limited."
],
[
"Derivation of the Generalized Bayes Classifier (Theorem ",
"In this Appendix, we prove Theorem REF , in which we characterize a generalization of the Bayes classifier to arbitrary CMMs.",
"We reiterate the theorem for the reader here: thm:generalizedbayesTheorem REF If $\\operatornamewithlimits{\\arg \\!\\max }_{\\widehat{Y} \\in \\mathcal {SC}} M(C_{\\widehat{Y}}) \\ne \\varnothing $ , then there exists a regression-thresholding classifier $\\widehat{Y}_{p,t,\\eta } \\in \\operatornamewithlimits{\\arg \\!\\max }_{\\widehat{Y} \\in \\mathcal {SC}} M(C_{\\widehat{Y}}).$ As described in the main paper, the proof of Theorem REF is given in a sequence of steps constructing optimal classifiers in forms progressively closer to that of the generalized Bayes classifier described in Theorem REF .",
"Specifically, we first show, in Lemma REF , that there exists an optimal classifier that is a (stochastic) function of the regression function $\\eta $ .",
"We then construct an optimal classifier in which this function of $\\eta $ is non-decreasing.",
"Finally, we construct an optimal classifier in which this function of $\\eta $ is a threshold function, as in Theorem REF .",
"Lemma 18 For any stochastic classifier $\\widehat{Y} : \\mathcal {X}\\rightarrow \\mathcal {P}(\\lbrace 0, 1\\rbrace )$ , there is a stochastic classifier $\\widehat{Y}^{\\prime } : \\mathcal {X}\\rightarrow \\mathcal {P}(\\lbrace 0, 1\\rbrace )$ of the form $\\widehat{Y}^{\\prime }(x) \\sim \\text{Bernoulli}(f(\\eta (x))),$ for some $f : [0, 1] \\rightarrow [0, 1]$ , such that $C_{\\widehat{Y}^{\\prime }} = C_{\\widehat{Y}}$ .",
"Define $\\widehat{Y}^{\\prime } : \\mathcal {X}\\rightarrow \\mathcal {P}(\\lbrace 0, 1\\rbrace )$ by $\\widehat{Y}^{\\prime }(x) \\sim \\text{Bernoulli} \\left( \\operatornamewithlimits{\\mathbb {E}}_{Z \\sim P_X} \\left[ \\widehat{Y}(Z) | \\eta (Z) = \\eta (x) \\right] \\right),$ and note that $\\widehat{Y}^{\\prime }$ has the desired form.",
"Note that both $\\widehat{Y}(X)$ and $\\widehat{Y}^{\\prime }(X)$ are conditionally independent of the true label $Y$ given $\\eta (X)$ .",
"Thus, $\\operatornamewithlimits{\\mathbb {E}}\\left[ Y \\widehat{Y}^{\\prime }(X) \\right]& = \\operatornamewithlimits{\\mathbb {E}}\\left[ \\operatornamewithlimits{\\mathbb {E}}\\left[ Y \\widehat{Y}^{\\prime }(X) | \\eta (X) \\right] \\right] \\\\& = \\operatornamewithlimits{\\mathbb {E}}\\left[ \\operatornamewithlimits{\\mathbb {E}}\\left[ Y | \\eta (X) \\right] \\operatornamewithlimits{\\mathbb {E}}\\left[ \\widehat{Y}^{\\prime }(X) | \\eta (X) \\right] \\right] \\\\& = \\operatornamewithlimits{\\mathbb {E}}\\left[ \\eta (X) \\operatornamewithlimits{\\mathbb {E}}\\left[ \\widehat{Y}^{\\prime }(X) | \\eta (X) \\right] \\right]$ and, similarly, $\\operatornamewithlimits{\\mathbb {E}}\\left[ Y \\widehat{Y}(X) \\right]= \\operatornamewithlimits{\\mathbb {E}}\\left[ \\eta (X) \\operatornamewithlimits{\\mathbb {E}}\\left[ \\widehat{Y}(X) | \\eta (X) \\right] \\right].$ Additionally, by construction of $\\widehat{Y}^{\\prime }$ , $\\operatornamewithlimits{\\mathbb {E}}\\left[ \\widehat{Y}^{\\prime }(X) | \\eta (X) \\right]= \\operatornamewithlimits{\\mathbb {E}}\\left[ \\operatornamewithlimits{\\mathbb {E}}_{Z \\sim P_X} \\left[ \\widehat{Y}(Z) | \\eta (Z) = \\eta (X) \\right] | \\eta (X) \\right]= \\operatornamewithlimits{\\mathbb {E}}\\left[ \\widehat{Y}(X) | \\eta (X) \\right].$ Putting these together, we have $\\text{TP}_{\\widehat{Y}^{\\prime }}= \\operatornamewithlimits{\\mathbb {E}}\\left[ Y \\widehat{Y}^{\\prime }(X) \\right]& = \\operatornamewithlimits{\\mathbb {E}}\\left[ \\eta (X) \\operatornamewithlimits{\\mathbb {E}}\\left[ \\widehat{Y}^{\\prime }(X) | \\eta (X) \\right] \\right] \\\\& = \\operatornamewithlimits{\\mathbb {E}}\\left[ \\eta (X) \\operatornamewithlimits{\\mathbb {E}}\\left[ \\widehat{Y}(X) | \\eta (X) \\right] \\right]= \\operatornamewithlimits{\\mathbb {E}}\\left[ Y \\widehat{Y}(X) \\right]= \\text{TP}_{\\widehat{Y}}.$ Similarly, one can check that $C_{\\widehat{Y}^{\\prime }} = C_{\\widehat{Y}}$ .",
"It follows from Lemma REF that, if $M(C_{\\widehat{Y}})$ is maximized by any classifier, then it is maximized by a classifier $\\widehat{Y}$ of the form in Eq.",
"(REF ).",
"It remains to show that $f$ in Eq.",
"(REF ) can be of the form $z \\mapsto p 1\\lbrace z = t\\rbrace + 1\\lbrace z > t\\rbrace $ for some threshold $(p, t) \\in [0, 1]^2$ .",
"Before proving this, we give a simplifying lemma showing that the problem of maximizing a CMM can be equivalently framed as a particular functional optimization problem.",
"This will allow us to to significantly simplify the notation in the subsequent proofs.",
"Lemma 19 Let $M$ be a CMM, and suppose that $M(C_{\\widehat{Y}})$ is maximized (over $\\mathcal {SC}$ ) by a classifier $\\widehat{Y}$ of the form $\\widehat{Y}(x) \\sim \\operatorname{Bernoulli}(f^*(\\eta (x))),$ for some $f^* : [0, 1] \\rightarrow [0, 1]$ .",
"Let $f$ be a solution to the optimization problem $\\max _{f : [0, 1] \\rightarrow [0, 1]} \\operatornamewithlimits{\\mathbb {E}}[\\eta (X) f(\\eta (X))]\\quad \\text{ subject to } \\quad \\operatornamewithlimits{\\mathbb {E}}[(1 - \\eta (X)) f(\\eta (X))] \\le \\operatornamewithlimits{\\mathbb {E}}[(1 - \\eta (X)) f^*(\\eta (X))].$ Then, the classifier $\\widehat{Y}^{\\prime }(x) \\sim \\operatorname{Bernoulli}(f(\\eta (x))),$ also maximizes $M(C_{\\widehat{Y}^{\\prime }})$ (over $\\mathcal {SC}$ ).",
"This result follows from the definition (Definition REF ) of a CMM.",
"Specifically, by construction of $\\widehat{Y}^{\\prime }$ , $\\text{TP}_{\\widehat{Y}^{\\prime }} = \\operatornamewithlimits{\\mathbb {E}}[\\eta (X) f(\\eta (X))] \\ge \\operatornamewithlimits{\\mathbb {E}}[\\eta (X) f^*(\\eta (X))] = \\text{TP}_{\\widehat{Y}}$ and $\\text{FP}_{\\widehat{Y}^{\\prime }} = \\operatornamewithlimits{\\mathbb {E}}[(1 - \\eta (X)) f(\\eta (X))] \\le \\operatornamewithlimits{\\mathbb {E}}[(1 - \\eta (X)) f^*(\\eta (X))] = \\text{FP}_{\\widehat{Y}}.$ Moreover, since, since the actual proportions of positives and negatives are fixed (i.e., $\\text{TP}_{\\widehat{Y}^{\\prime }} + \\text{FN}_{\\widehat{Y}^{\\prime }} = \\text{TP}_{\\widehat{Y}} + \\text{FN}_{\\widehat{Y}}$ and $\\text{FP}_{\\widehat{Y}^{\\prime }} + \\text{TN}_{\\widehat{Y}^{\\prime }} = \\text{FP}_{\\widehat{Y}} + \\text{TN}_{\\widehat{Y}}$ ), we have $C_{\\widehat{Y}^{\\prime }} =\\begin{bmatrix}\\text{TN}_{\\widehat{Y}} + \\epsilon _1 & \\text{FP}_{\\widehat{Y}} - \\epsilon _1 \\\\\\text{FN}_{\\widehat{Y}} - \\epsilon _2 & \\text{TP}_{\\widehat{Y}} + \\epsilon _2,\\end{bmatrix},$ where $\\epsilon _1 := \\text{FP}_{\\widehat{Y}} - \\text{FP}_{\\widehat{Y}^{\\prime }} \\in [0, \\text{FP}_{\\widehat{Y}}]$ and $\\epsilon _2 := \\text{TP}_{\\widehat{Y}^{\\prime }} - \\text{TP}_{\\widehat{Y}} \\in [0, \\text{FN}]$ .",
"Thus, by the definition (Definition REF ) of a CMM, $M(C_{\\widehat{Y}^{\\prime }}) \\ge M(C_{\\widehat{Y}})$ .",
"Lemma REF essentially shows that maximizing any CMM $M$ is equivalent to performing Neyman-Pearson classification, at a some particular false positive level $\\alpha $ depending on $M$ (through $f^*$ ) and on the distribution of $\\eta (X)$ .",
"For our purposes, this simplifies the remaining steps in proving Theorem REF by allowing us to ignore the details of the particular CMM $M$ and regression function $\\eta $ and focus on characterizing solutions to an optimization problem of the form (REF ) (see, specifically, (REF ) below).",
"To characterize solutions to this optimization problem, we will repeatedly utilize the following measure-theoretic technical lemma: Lemma 20 Let $\\mu $ be a measure on $[0, 1]$ with $\\mu ([0, 1]) > 0$ .",
"Then, there exists $z \\in \\mathbb {R}$ such that, for all $\\epsilon > 0$ , $\\mu ([0, 1] \\cap (z - \\epsilon , z)) > 0$ .",
"We prove the contrapositive.",
"Suppose that, for every $z \\in [0, 1]$ , there exists $\\epsilon _z > 0$ such that $\\mu ([0, 1] \\cap (z - \\epsilon _z, z)) = 0$ .",
"The family $\\mathcal {S} := \\lbrace [0, 1] \\cap (z - \\epsilon _z, z) : z \\in \\mathbb {R}\\rbrace $ is an open cover of $[0, 1]$ .",
"Since $[0, 1]$ is compact, there exists a finite sub-cover $\\mathcal {S}^{\\prime } \\subseteq \\mathcal {S}$ of $[0, 1]$ .",
"Thus, by countable subaddivity of measures, $\\mu ([0, 1]) \\le \\sum _{S \\in \\mathcal {S}^{\\prime }} \\mu (\\mathcal {S}) = 0.$ We are now ready for the main remaining step in the proof of Theorem REF , namely characterizing solutions of (a generalization of) the optimization problem (REF ): Lemma 21 Let $Z$ be a $[0, 1]$ -valued random variable, and let $c \\in [0, 1]$ .",
"Suppose that the optimization problem $\\max _{f : [0, 1] \\rightarrow [0, 1]} \\operatornamewithlimits{\\mathbb {E}}[Z f(Z)]\\quad \\text{ subject to } \\quad \\operatornamewithlimits{\\mathbb {E}}[(1 - Z) f(Z)] \\le c$ has a solution.",
"Then, there is a solution to (REF ) that is a generalized threshold function.",
"Suppose that there exists a solution $f$ to (REF ).",
"We will construct a generalized threshold function that solves (REF ) in two main steps.",
"First, we will construct a monotone solution to (REF ).",
"Second, we will show that this monotone solution is equal to a generalized threshold function except perhaps on a set of probability 0 with respect to $Z$ .",
"This generalized threshold function is therefore a solution to (REF ).",
"Construction of Monotone Solution to (REF ): Define $g(z) := \\operatornamewithlimits{\\text{ess sup}}_{P_Z} f([0, z])\\quad \\text{ and } \\quad h(z) := \\operatornamewithlimits{\\text{ess inf}}_{P_Z} f((z, 1]),$ where the essential supremum and infimum are taken with respect to the measure $P_Z$ of $Z$ , with the conventions $g(z) = 0$ whenever $P_Z([0, z]) = 0$ and $h(z) = 1$ whenever $P_Z((z, 1]) = 0$ .",
"We first show that, for all $z \\in [0, 1]$ , $g(z) \\le h(z)$ .",
"We will then use this to show that $g = f$ except on a set of $P_Z$ measure 0 (i.e., $P_Z(\\lbrace z \\in [0, 1] : g(z) \\ne f(z)\\rbrace ) = 0$ ).",
"Therefore, both $\\operatornamewithlimits{\\mathbb {E}}[Z g(Z)] = \\operatornamewithlimits{\\mathbb {E}}[Z f(Z)]$ and $\\operatornamewithlimits{\\mathbb {E}}[(1 - Z) g(Z)] = \\operatornamewithlimits{\\mathbb {E}}[(1 - Z) f(Z)]$ .",
"Since $g : [0, 1] \\rightarrow [0, 1]$ is clearly monotone non-decreasing, the result follows.",
"Suppose, for sake of contradiction, that, for some $z \\in [0, 1]$ , $g(z) > h(z)$ .",
"Then there exist $A \\subseteq [0, z]$ and $B \\subseteq (z, 1]$ such that $\\inf _{z \\in A} f(z) > \\sup _{z \\in B} f(z)$ and $P_Z(A), P_Z(B) > 0$ .",
"Define $z_A := \\operatornamewithlimits{\\mathbb {E}}[Z|Z \\in A]$ and $z_B := \\operatornamewithlimits{\\mathbb {E}}[Z|Z \\in B]$ , and note that, since $A \\subseteq [0, z]$ and $B \\subseteq (z, 1]$ , $z_A < z_B$ .",
"Define, $\\epsilon := \\min \\left\\lbrace \\frac{P_Z(A) (1 - z_A)}{P_Z(B) (1 - z_B)} \\inf _{z \\in A} f(z),\\quad \\sup _{z \\in B} f(z) \\right\\rbrace > 0$ and define $\\phi : [0, 1] \\rightarrow [0, 1]$ by $\\phi (z):= \\left\\lbrace \\begin{array}{cc}f(z) - \\epsilon \\frac{P_Z(B) (1 - z_B)}{P_Z(A) (1 - z_A)} & \\text{ if } z \\in A \\\\f(z) + \\epsilon & \\text{ if } z \\in B \\\\f(z) & \\text{ otherwise,}\\end{array}\\right.$ noting that, by construction of $\\epsilon $ , $\\phi (z) \\in [0, 1]$ for all $z \\in [0, 1]$ .",
"Then, by construction of $\\phi $ , $\\operatornamewithlimits{\\mathbb {E}}[(1 - Z) \\phi (Z)] - \\operatornamewithlimits{\\mathbb {E}}[(1 - Z) f(Z)]& = - (1 - z_A) \\epsilon \\frac{P_Z(B) (1 - z_B)}{P_Z(A) (1 - z_A)} P_Z(A) + (1 - z_B) \\epsilon P_Z(B) \\\\& = 0,$ while $\\operatornamewithlimits{\\mathbb {E}}[Z \\phi (Z)] - \\operatornamewithlimits{\\mathbb {E}}[Z f(Z)]& = - z_A \\epsilon \\frac{P_Z(B) (1 - z_B)}{P_Z(A) (1 - z_A)} P_Z(A) + z_B \\epsilon P_Z(B) \\\\& = \\left( - \\frac{z_A}{1 - z_A}(1 - z_B) + z_B \\right) \\epsilon P_Z(B)> 0,$ since the function $z \\mapsto \\frac{z}{1 - z}$ is strictly increasing.",
"This contradicts the assumption that $f$ optimizes (REF ), implying $g \\le h$ .",
"We now show that $g = f$ except on a set of $P_Z$ measure 0.",
"First, note that, if $g(z) \\ne f(z)$ , then $g(z) = \\operatornamewithlimits{\\text{ess sup}}f([0, z]) = \\operatornamewithlimits{\\text{ess sup}}f([0, z))$ , and so $g$ is left-continuous at $z$ .",
"For any $\\delta > 0$ , define $A_\\delta := \\left\\lbrace z \\in [0, 1] : g(z) < f(z) - \\delta \\right\\rbrace \\quad \\text{ and } \\quad B_\\delta := \\left\\lbrace z \\in [0, 1] : g(z) > f(z) + \\delta \\right\\rbrace .$ Since $\\lbrace z \\in [0, 1] : g(z) < f(z)\\rbrace = \\bigcup _{j = 1}^\\infty \\left\\lbrace z \\in [0, 1] : g(z) < f(z) - \\frac{1}{j} \\right\\rbrace $ and $\\lbrace z \\in [0, 1] : g(z) > f(z)\\rbrace = \\bigcup _{j = 1}^\\infty \\left\\lbrace z \\in [0, 1] : g(z) > f(z) + \\frac{1}{j} \\right\\rbrace ,$ by countable subadditivity, it suffices to show that $P_Z(A_\\delta ) = P_Z(B_\\delta ) = 0$ for all $\\delta > 0$ .",
"Suppose, for sake of contradiction, that $P_Z(A_\\delta ) > 0$ .",
"Applying Lemma REF to the measure $E \\mapsto P_Z(A_\\delta \\cap E)$ , there exists $z \\in \\mathbb {R}$ such that, for any $\\epsilon > 0$ , $P_Z(A_\\delta \\cap (z - \\epsilon , z)) > 0$ .",
"Since $g$ is continuous at $z$ , there exists $\\epsilon > 0$ such that $g(z - \\epsilon ) \\ge g(z) - \\delta $ , so that, for all $z \\in A_\\delta \\cap (z - \\epsilon , z)$ , $f(z) > g(z) + \\delta $ .",
"Then, since $P_Z(A_\\delta \\cap (z - \\epsilon , z)) > 0$ , we have the contradiction $g(z) \\ge \\operatornamewithlimits{\\text{ess sup}}f(A_\\delta \\cap (z - \\epsilon , z)) > g(z).$ On the other hand, suppose, for sake of contradiction, that $P_Z(B_\\delta ) > 0$ .",
"Applying Lemma REF to the measure $E \\mapsto P_Z(B_\\delta \\cap E)$ , there exists $z \\in \\mathbb {R}$ such that, for any $\\epsilon > 0$ , $P_Z(B_\\delta \\cap (z - \\epsilon , z)) > 0$ .",
"Since $g$ is continuous at $z$ , there exists $\\epsilon > 0$ such that $g(z - \\epsilon ) \\ge g(z) - \\delta $ .",
"At the same time, since $g$ is non-decreasing, for $t \\in B_\\delta \\cap (z - \\epsilon , z)$ , $f(t) < g(t) - \\delta \\le g(z) - \\delta $ .",
"Thus, since $P_Z(B_\\delta \\cap (z - \\epsilon , z)) > 0$ , we have $h(z - \\epsilon ) < g(z) - \\delta < g(z - \\epsilon )$ , contradicting the previously shown fact that $g \\le h$ .",
"To conclude, we have shown that $P_Z(\\lbrace z \\in [0, 1] : g(z) \\ne f(z)\\rbrace ) = 0$ .",
"Construction of a Generalized Threshold Solution: We now construct a solution to (REF ) that is equal to a generalized threshold function (i.e., a function that has the form $p 1\\lbrace z = t\\rbrace + 1\\lbrace z > t\\rbrace $ ) except on a set of $P_Z$ -measure 0.",
"To show this, it suffices to construct a function $f : [0, 1] \\rightarrow [0, 1]$ such that (a) $f$ is monotone non-decreasing and (b) the set $f^{-1}((0, 1))$ is the union of the singleton $\\lbrace t\\rbrace $ and a set of $P_Z$ -measure 0.",
"From the previous step of this proof, we may assume that we have a solution $f$ to (REF ) that is monotone non-decreasing.",
"It suffices therefore to show that $A := f^{-1}((0, 1))$ is the union of a singleton and a set of $P_Z$ -measure 0.",
"Define $t_0 := \\inf \\lbrace z \\in [0, 1] : P_Z(A \\cap [0, z]) > 0\\rbrace \\quad \\text{ and } \\quad t_1 := \\sup \\lbrace z \\in [0, 1] : P_Z(A \\cap [z, 1]) > 0\\rbrace .$ Then, for all $\\epsilon > 0$ , $P_Z(A \\cap [0, t_0 - \\epsilon ]) = P_Z(A \\cap [t_1 + \\epsilon , 1]) = 0$ .",
"If $t_0 = t_1$ , then, since $A \\backslash \\lbrace t_0\\rbrace = \\bigcup _{j = 1}^\\infty A \\cap \\left( [0, t_0 - 1/j] \\cup [t_0 + 1/j, 1] \\right)$ by countable subadditivity, $P_Z(A \\backslash \\lbrace t_0\\rbrace ) = 0$ , which implies that $A = \\lbrace t_0\\rbrace \\cup (A \\backslash \\lbrace t_0\\rbrace )$ is the union of a singleton and a set of measure 0.",
"It suffices therefore to prove that $t_0 = t_1$ .",
"Suppose, for sake of contradiction, that $t_0 < t_1$ .",
"Then, there exists $t \\in (t_0, t_1)$ , and, by definition of $t_0$ and $t_1$ , both $P_Z(A \\cap [0, t)) > 0$ and $P_Z(A \\cap (t, 1]) > 0$ .",
"For any $\\delta \\ge 0$ , define $B_\\delta := \\lbrace z \\in [0, t): \\delta < f(z) < 1 - \\delta \\rbrace \\quad \\text{ and } \\quad C_\\delta := \\lbrace z \\in (t, 1]: \\delta < f(z) < 1 - \\delta \\rbrace ,$ so that $P_Z(B_0) > 0$ and $P_Z(C_0) > 0$ .",
"By countable subadditivity, there exists $\\delta > 0$ such that $P_Z(B_\\delta ) > 0$ and $P_Z(C_\\delta ) > 0$ .",
"Define $\\epsilon := \\delta \\cdot \\min \\lbrace P_Z(B_\\delta ), P_Z(C_\\delta )\\rbrace \\rbrace > 0$ .",
"Define $g : [0, 1] \\rightarrow \\mathbb {R}$ for all $z \\in [0, 1]$ by $g(z)= \\left\\lbrace \\begin{array}{cc}f(z) - \\frac{\\epsilon }{P_Z(B_\\delta )} & \\text{ if } z \\in B_\\delta \\\\f(z) + \\frac{\\epsilon }{P_Z(C_\\delta )} & \\text{ if } z \\in C_\\delta \\\\f(z) & \\text{ otherwise}.\\end{array}\\right.,$ and note that, by definition of $\\epsilon $ , $B_\\delta $ , and $C_\\delta $ , $g : [0, 1] \\rightarrow [0, 1]$ .",
"Then, $\\operatornamewithlimits{\\mathbb {E}}[g(Z)] - \\operatornamewithlimits{\\mathbb {E}}[f(Z)]= -\\frac{\\epsilon }{P_Z(B_\\delta )} P_Z(B_\\delta ) + \\frac{\\epsilon }{P_Z(C_\\delta )} P_Z(C_\\delta )= 0,$ while $\\operatornamewithlimits{\\mathbb {E}}[Z g(Z)] - \\operatornamewithlimits{\\mathbb {E}}[Z f(Z)]& = -\\operatornamewithlimits{\\mathbb {E}}[Z | Z \\in B_\\delta ] \\frac{\\epsilon }{P_Z(B_\\delta )} P_Z(B_\\delta )+ \\operatornamewithlimits{\\mathbb {E}}[Z | Z \\in C_\\delta ] \\frac{\\epsilon }{P_Z(C_\\delta )} P_Z(C_\\delta ) \\\\& = \\epsilon \\left( \\operatornamewithlimits{\\mathbb {E}}[Z | Z \\in C_\\delta ] - \\operatornamewithlimits{\\mathbb {E}}[Z | Z \\in B_\\delta ] \\right).$ Since $B_\\delta \\subseteq [0, t)$ and $C_\\delta \\subseteq (t, 1]$ , this difference is strictly positive, contradicting the assumption that $f$ optimizes (REF ).",
"Combining Lemma REF with Lemma REF completes the proof of our main result, Theorem REF ."
],
[
"Relative Performance Guarantees in terms of the Generalized Bayes Classifier",
"In this Appendix, we prove Lemmas REF and REF , as well as their consequence, Corollary .",
"Also, in Section REF , we demonstrate, in a few key examples, how to compute the Lipschitz constant used in Corollary .",
"We begin with the proof of Lemma REF , which, at a given threshold $(p, t)$ , bounds the difference between the confusion matrices of the true regression function $\\eta $ and an estimate $\\eta ^{\\prime }$ of $\\eta $ .",
"We restate the result for the reader's convenience: lemma:approximationerrorLemma REF Let $p,t \\in [0, 1]$ and let $\\eta , \\eta ^{\\prime } : \\mathcal {X}\\rightarrow [0, 1]$ .",
"Then, $\\left\\Vert C_{\\widehat{Y}_{p,t,\\eta }} - C_{\\widehat{Y}_{p,t,\\eta ^{\\prime }}} \\right\\Vert _\\infty \\le \\operatornamewithlimits{\\mathbb {P}}\\left[ |\\eta (X) - t| \\le \\left\\Vert \\eta - \\eta ^{\\prime }\\right\\Vert _\\infty \\right].$ For the true negative rate, we have $\\left| \\text{TN}_{\\widehat{Y}_{p,t,\\eta }} - \\text{TN}_{\\widehat{Y}_{p,t,\\eta ^{\\prime }}} \\right|& = \\left| \\operatornamewithlimits{\\mathbb {P}}\\left[ Y = 0, \\eta ^{\\prime }(X) \\le t < \\eta (X) \\right]- \\operatornamewithlimits{\\mathbb {P}}[ Y = 0, \\eta (X) \\le t < \\eta ^{\\prime }(X)] \\right| \\\\& \\le \\operatornamewithlimits{\\mathbb {P}}\\left[ |\\eta (X) - t| \\le \\Vert \\eta - \\eta ^{\\prime }\\Vert _\\infty \\right].$ This type of inequality is standard and follows from the fact that, if $t$ lies between $\\eta $ and $\\eta ^{\\prime }$ , then the difference of $\\eta $ and $t$ is necessarily less than $\\eta $ and $\\eta ^{\\prime }$ .",
"Repeating this calculation for the true positive, false positive, and false negative rates gives (REF ).",
"Note that, in the presence of degree $r$ Uniform Class Imbalance (see Section REF ), one can obtain a tighter error bound $r\\operatornamewithlimits{\\mathbb {P}}\\left[ |\\eta (X) - t| \\le \\Vert \\eta - \\eta ^{\\prime }\\Vert _\\infty \\right]$ for the true positive and false negative rates because, for all $x \\in \\mathcal {X}$ , $\\operatornamewithlimits{\\mathbb {P}}[Y = 1|X = x] \\le r$ .",
"However, the weaker bound (REF ) simplifies the exposition.",
"We now turn to proving Lemma REF , which we use to bound the maximum difference between the empirical and true confusion matrices of a regression-thresholding classifier over thresholds $(p, t)$ .",
"Specifically, we will use this result to bound the difference in confusion matrices between the optimal threshold $(p^*, t^*)$ and the threshold $(\\widehat{p}, \\widehat{t})$ selected by maximizing the empirical CMM.",
"We actually prove a more general version of Lemma REF , for arbitrary classifiers, based on the following definition: Definition 22 (Stochastic Growth Function) Let $\\mathcal {F}$ be a family of $[0, 1]$ -valued functions on $\\mathcal {X}$ .",
"The stochastic growth function $\\Pi _\\mathcal {F}: \\mathbb {N} \\rightarrow \\mathbb {N}$, defined by $\\Pi _\\mathcal {F}(n) := \\max _{\\begin{array}{c}x_1,...,x_n \\in \\mathcal {X},\\\\z_1,...,z_n \\in [0, 1]\\end{array}} \\left| \\left\\lbrace \\left( 1\\lbrace f(x_i) > z_i\\rbrace \\right)_{i = 1}^n : f \\in \\mathcal {F}\\right\\rbrace \\right|\\quad \\text{ for all } \\quad n \\in \\mathbb {N},$ is the maximum number of distinct classifications of $n$ points $x_1,...,x_n$ by a stochastic classifier $\\widehat{Y}$ with $(x \\mapsto \\operatornamewithlimits{\\mathbb {E}}[\\widehat{Y}(x)]) \\in \\mathcal {F}$ and randomness given by $z_1,...,z_n$ .",
"Definition REF generalizes the growth function [42], a classical measure of the complexity of a hypothesis class originally due to [56], to non-deterministic classifiers.",
"Importantly for our purposes, one can easily bound the stochastic growth function of regression-thresholding classifiers: Example 23 (Stochastic Growth Function of Regression-Thresholding Classifiers) Suppose $\\mathcal {F}= \\left\\lbrace f : \\mathcal {X}\\rightarrow [0, 1] | \\text{ for some } p, t \\in [0, 1], f(x) = p \\cdot 1\\lbrace \\eta (x) = t\\rbrace + 1\\lbrace \\eta (x) > t\\rbrace \\text{ for all } x \\in \\mathcal {X}\\right\\rbrace ,$ so that $\\lbrace \\widehat{Y}_{f, \\eta } : f \\in \\mathcal {F}\\rbrace $ is the class of regression-thresholding classifiers.",
"Any set of points $(x_1,z_1),...,(x_n,z_n)$ , can be sorted in increasing order by $\\eta (x)$ 's, breaking ties in decreasing order by $z$ 's.",
"Having sorted the points in this way, $\\lbrace f(x) > z\\rbrace = 0$ for the first $j$ points and $\\lbrace f(x) > z\\rbrace = 1$ for the remaining $n - j$ points, for some $j \\in [n] \\cup \\lbrace 0\\rbrace $ .",
"Thus, $\\Pi _\\mathcal {F}(n) = n + 1$ .",
"We will now prove the following result, from which, together with Example REF , Lemma REF follows immediately: lemma:estimationerrorLemma REF (Generalized Version) Let $\\mathcal {F}$ be a family of $[0, 1]$ -valued functions on $\\mathcal {X}$ .",
"Then, with probability at least $1 - \\delta $ , $\\sup _{f \\in \\mathcal {F}} \\left\\Vert \\widehat{C}_{\\widehat{Y}_f} - C_{\\widehat{Y}_f} \\right\\Vert _\\infty \\le \\sqrt{\\frac{8}{n} \\log \\frac{32\\Pi _\\mathcal {F}(2n)}{\\delta }}.$ Before proving Lemma REF , we note a standard symmetrization lemma, which allows us to replace the expectation of $\\widehat{\\text{TN}}_{\\widehat{Y}_{p,t,\\eta }}$ with its value on an independent, identically distributed “ghost sample”.",
"Lemma 24 (Symmetrization; Lemma 2 of [8]) Let $X$ and $X^{\\prime }$ be independent realizations of a random variable with respect to which $\\mathcal {F}$ is a family of integrable functions.",
"Then, for any $\\epsilon > 0$ , $\\operatornamewithlimits{\\mathbb {P}}\\left[ \\sup _{f \\in \\mathcal {F}} f(X) - \\operatornamewithlimits{\\mathbb {E}}f(X) > \\epsilon \\right]\\le 2\\operatornamewithlimits{\\mathbb {P}}\\left[ \\sup _{f \\in \\mathcal {F}} f(X) - f(X^{\\prime }) > \\frac{\\epsilon }{2} \\right].$ We now use this lemma to prove Lemma REF .",
"To facilitate analyzing the stochastic aspect of the classifier $\\widehat{Y}_{f,\\eta }$ , let $Z_1,...,Z_n \\stackrel{IID}{\\sim } \\operatorname{Uniform}([0, 1])$ , such that $\\widehat{Y}_{f,\\eta }(X_i) = 1\\lbrace Z_i < f(\\eta ((X_i))\\rbrace $ .",
"Now suppose that we have a ghost sample $(X_1^{\\prime },Y_1^{\\prime },Z_1^{\\prime }),...,(X_n^{\\prime },Y_n^{\\prime },Z_n^{\\prime })$ .",
"Let $\\widehat{\\text{TN}}^{\\prime }_{\\widehat{Y}_{f,\\eta }}$ denote the empirical true negative rate computed on this ghost sample, and let $\\widehat{\\text{TN}}^{(i)}_{\\widehat{Y}_{f,\\eta }}$ denote the empirical true negative rate computed on $(X_1,Y_1,Z_1),...,(X_{i-1},Y_{i-1}Z_{i-1}),(X_i^{\\prime },Y_i^{\\prime },Z_i^{\\prime }),(X_{i+1},Y_{i+1},Z_{i+1}),...(X_n,Y_n,Z_n)$ (i.e., replacing only the $i^{th}$ sample with its ghost).",
"By the Symmetrization Lemma, $\\operatornamewithlimits{\\mathbb {P}}\\left[ \\sup _{f \\in \\mathcal {F}} \\widehat{\\text{TN}}_{\\widehat{Y}_{f,\\eta }} - \\operatornamewithlimits{\\mathbb {E}}\\widehat{\\text{TN}}_{\\widehat{Y}_{f,\\eta }} > \\epsilon \\right]& \\le 2\\operatornamewithlimits{\\mathbb {P}}\\left[ \\sup _{f \\in \\mathcal {F}} \\widehat{\\text{TN}}_{\\widehat{Y}_{f,\\eta }} - \\widehat{\\text{TN}}^{\\prime }_{\\widehat{Y}_{f,\\eta }} > \\epsilon /2 \\right] \\\\& \\le 2\\Pi _\\mathcal {F}(2n) \\sup _{f \\in \\mathcal {F}} \\operatornamewithlimits{\\mathbb {P}}\\left[ \\widehat{\\text{TN}}_{\\widehat{Y}_{f,\\eta }} - \\widehat{\\text{TN}}^{\\prime }_{\\widehat{Y}_{f,\\eta }} > \\epsilon /2 \\right] \\\\& \\le 4\\Pi _\\mathcal {F}(2n) \\sup _{f \\in \\mathcal {F}} \\operatornamewithlimits{\\mathbb {P}}\\left[ \\widehat{\\text{TN}}_{\\widehat{Y}_{f,\\eta }} - \\operatornamewithlimits{\\mathbb {E}}\\widehat{\\text{TN}}_{\\widehat{Y}_{f,\\eta }} > \\epsilon /4 \\right],$ where the second inequality is a union bound over the $\\Pi _\\mathcal {F}(2n)$ distinct classifications of $2n$ points that can be assigned by $\\widehat{Y}_{f,\\eta }$ with $f \\in \\mathcal {F}$ , and the last inequality is from the fact that $\\widehat{\\text{TN}}_{\\widehat{Y}_{f,\\eta }}$ and $\\widehat{\\text{TN}}^{\\prime }_{\\widehat{Y}_{f,\\eta }}$ are identically distributed and the algebraic fact that, if $a - b > \\epsilon $ , then either $a - c > \\epsilon /2$ or $b - c > \\epsilon /2$ .",
"For any particular $f \\in \\mathcal {F}$ , by McDiarmid's inequality [39], $\\operatornamewithlimits{\\mathbb {P}}\\left[ \\widehat{\\text{TN}}_{\\widehat{Y}_{f,\\eta }} - \\operatornamewithlimits{\\mathbb {E}}\\widehat{\\text{TN}}_{\\widehat{Y}_{f,\\eta }} > \\epsilon /4 \\right]\\le e^{-n\\epsilon ^2/8},$ since, for any $i \\in [n]$ , $\\left| \\widehat{\\text{TN}}_{\\widehat{Y}_{f,\\eta }} - \\widehat{\\text{TN}}^{(i)}_{\\widehat{Y}_{f,\\eta }} \\right|= \\frac{1}{n} \\left| 1\\left\\lbrace Y_i = \\widehat{Y}_{f,\\eta }(X_i) = 0 \\right\\rbrace - 1\\left\\lbrace Y_i^{\\prime } = \\widehat{Y}_{f,\\eta }(X_i^{\\prime }) = 0 \\right\\rbrace \\right|\\le \\frac{1}{n}.$ Plugging Inequality (REF ) into Inequality (REF ) gives $\\operatornamewithlimits{\\mathbb {P}}\\left[ \\sup _{f \\in \\mathcal {F}} \\widehat{\\text{TN}}_{\\widehat{Y}_{f,\\eta }} - \\operatornamewithlimits{\\mathbb {E}}\\widehat{\\text{TN}}_{\\widehat{Y}_{f,\\eta }} > \\epsilon \\right]\\le 4 \\Pi _\\mathcal {F}(2n) e^{-n\\epsilon ^2/8}.$ Repeating this argument with $-\\widehat{\\text{TN}}$ instead of $\\widehat{\\text{TN}}$ , as well as with $\\widehat{\\text{TP}}$ , $\\widehat{\\text{FN}}$ , $\\widehat{\\text{FP}}$ and their negatives, and taking a union bound over these 8 cases, gives the desired result.",
"Finally, we will use these two lemmas, together with the margin and Lipschitz assumptions, to prove Corollary , which bounds the sub-optimality of the trained classifier, relative to the generalized Bayes classifier, in terms of the desired CMM.",
"corr:CMMerrordecompositionCorollary Let $\\eta : \\mathcal {X}\\rightarrow [0, 1]$ denote the true regression function, and let $\\widehat{\\eta }: \\mathcal {X}\\rightarrow [0, 1]$ denote any empirical regressor.",
"Let $\\left( \\widehat{p}, \\widehat{t} \\right) := \\operatornamewithlimits{\\arg \\!\\max }_{(p, t) \\in [0, 1]^2} M \\left( \\widehat{C}_{\\widehat{Y}_{p,t,\\widehat{\\eta }}} \\right)\\quad \\text{ and } \\quad \\left( p^*, t^* \\right) := \\operatornamewithlimits{\\arg \\!\\max }_{(p, t) \\in [0, 1]^2} M \\left(C_{\\widehat{Y}_{p,t,\\eta }} \\right)$ denote the empirically selected and true optimal thresholds, respectively.",
"Suppose that $M$ is Lipschitz continuous with constant $L_M$ with respect to the uniform ($\\mathcal {L}_\\infty $ ) metric on $.",
"Finally, suppose that $ PX$ and $$ satisfies a $ (C, )$-margin condition around $ t*$.",
"Then, with probability at least $ 1 - $,{\\begin{@align}{1}{-1}M\\left(C_{\\widehat{Y}_{p,t,\\eta }}\\left(p^*, t^*\\right)\\right) - M\\left(C_{\\widehat{Y}_{p,t,\\widehat{\\eta }}}\\left(\\widehat{p}, \\widehat{t}\\right)\\right)& \\le L_M \\left( C\\left\\Vert \\eta - \\widehat{\\eta }\\right\\Vert _\\infty ^\\beta + 2 \\sqrt{\\frac{8}{n} \\log \\frac{32(2n + 1)}{\\delta }} \\right).\\end{@align}}$ First, note that $M\\left(C_{\\widehat{Y}_{p^*, t^*,\\eta }}\\right) - M\\left(C_{\\widehat{Y}_{\\widehat{p}, \\widehat{t},\\widehat{\\eta }}}\\right)& \\le M\\left(C_{\\widehat{Y}_{p^*, t^*,\\eta }}\\right) - M\\left(C_{\\widehat{Y}_{p^*, t^*,\\widehat{\\eta }}}\\right) \\\\& + M\\left(C_{\\widehat{Y}_{p^*, t^*,\\widehat{\\eta }}}\\right) - M\\left(\\widehat{C}_{\\widehat{Y}_{p^*, t^*,\\widehat{\\eta }}}\\right) \\\\& + M\\left(\\widehat{C}_{\\widehat{Y}_{\\widehat{p}, \\widehat{t},\\widehat{\\eta }}}\\right) - M\\left(C_{\\widehat{Y}_{\\widehat{p}, \\widehat{t},\\widehat{\\eta }}}\\right),$ since, by definition of $(\\widehat{p}, \\widehat{t})$ , $M\\left(\\widehat{C}_{\\widehat{Y}_{p^*,t^*,\\widehat{\\eta }}}\\right) - M\\left(\\widehat{C}_{\\widehat{Y}_{\\widehat{p},\\widehat{t},\\widehat{\\eta }}} \\right) \\le 0;$ this term sits between the second and third lines above.",
"By the Lipschitz assumption, $& M\\left(C_{\\widehat{Y}_{p^*, t^*,\\eta }}\\right) - M\\left(C_{\\widehat{Y}_{\\widehat{p}, \\widehat{t},\\widehat{\\eta }}}\\right)\\\\&\\le L_M \\bigg ( \\left\\Vert C_{\\widehat{Y}_{p^*, t^*,\\eta }} - C_{\\widehat{Y}_{p^*, t^*,\\widehat{\\eta }}} \\right\\Vert _\\infty \\\\ & \\hspace{26.19995pt}+ \\left\\Vert C_{\\widehat{Y}_{p^*, t^*,\\widehat{\\eta }}} - \\widehat{C}_{\\widehat{Y}_{p^*, t^*,\\widehat{\\eta }}} \\right\\Vert _\\infty \\\\& \\hspace{26.19995pt}+ \\left\\Vert \\widehat{C}_{\\widehat{Y}_{\\widehat{p}, \\widehat{t},\\widehat{\\eta }}} - C_{\\widehat{Y}_{\\widehat{p}, \\widehat{t},\\widehat{\\eta }}} \\right\\Vert _\\infty \\bigg ).$ Corollary follows by applying Lemma REF and the $(C,\\beta )$ -margin condition to (REF ) and applying Lemma REF to both terms () and ()."
],
[
"Lipschitz constants for some common CMMs",
"Corollary assumed that the CMM $M$ was Lipschitz continuous with respect to the $\\sup $ -norm on confusion matrices.",
"In this section, we show how to compute appropriate Lipschitz constants for several simple example CMMs.",
"We begin with a simple example: Example 25 (Weighted Accuracy) For a fixed $w \\in (0, 1)$ , the $w$ -weighted accuracy is given by $M(C) = (1 - w) \\text{TP}+ w \\text{TN}$ .",
"In this case, $M$ clearly has Lipschitz constant $L_M = \\max \\lbrace w, 1 - w\\rbrace $ .",
"For the remainder of this section (only), we will use $P := \\operatornamewithlimits{\\mathbb {E}}[Y]$ to denote the positive rate of the true labels and $\\widehat{P} := \\frac{1}{n} \\sum _{i = 1}^n Y_i$ to denote the empirical positive rate of the true labels.",
"Many CMMs of interest, such as Recall and $F_\\beta $ scores, are not Lipschitz continuous over all of $.",
"Fortunately, inspecting the proof of Corollary~\\ref {corr:CMM_error_decomposition}, it suffices for the CMM $ M$ to be Lipschitz continuous on the line segments between three specific pairs of confusion matrices, given in Eqs.~(\\ref {term:approximation_error}), (\\ref {term:estimation_error_term1}), and (\\ref {term:estimation_error_term2}).",
"Deriving the appropriate Lipschitz constants is a bit more complex, and we demonstrate here how to derive them for the specific CMMs of Recall and $ F$ scores.$ Of the six confusion matrices in Eqs.",
"(REF ), (), and (), four are true confusion matrices.",
"These four matrices have the same positive rate $\\text{TP}+ \\text{FN}= P$ , which is a function of the true distribution of labels.",
"The remaining two matrices are empirical confusion matrices, and hence have the positive rate $\\widehat{\\text{TP}}+ \\widehat{\\text{FN}}= \\widehat{P}$ , which is a function of the data.",
"By a multiplicative Chernoff bound, with probability at least $1 - e^{-nP/8}$ , $\\widehat{P} \\ge P/2$ .",
"Thus, with high probability, it suffices for the CMM $M$ to be Lipschitz continuous over confusion matrices with positive rate at least $P/2$ .",
"For Recall and $F_\\beta $ scores, this gives the following Lipschitz constants: Example 26 (Recall) Recall is given by $M(C) = \\frac{\\text{TP}}{\\text{TP}+ \\text{FN}} = \\frac{\\text{TP}}{P}$ .",
"Thus, $M$ is Lipschitz continuous with constant $L_M = \\frac{2}{P}$ over the confusion matrices in Eqs.",
"(REF ), (), and ().",
"Example 27 ($F_\\beta $ Score) For $\\beta \\in (0, \\infty )$ , the $F_\\beta $ score is given by $M(C)= \\frac{(1 + \\beta ^2) \\text{TP}}{(1 + \\beta ^2) \\text{TP}+ \\text{FP}+ \\beta ^2 \\text{FN}}= \\frac{(1 + \\beta ^2) \\text{TP}}{\\text{TP}+ \\text{FP}+ \\beta ^2 P}.$ Hence, $\\left| \\frac{\\partial }{\\partial \\text{TP}} M(C) \\right|= (1 + \\beta ^2) \\frac{\\text{FP}+ \\beta ^2 P}{\\left( \\text{TP}+ \\text{FP}+ \\beta ^2 P \\right)^2}\\le \\frac{1 + \\beta ^2}{\\beta ^2 P},$ while, since $\\text{TP}\\le P$ , $\\left| \\frac{\\partial }{\\partial \\text{FP}} M(C) \\right|= (1 + \\beta ^2) \\frac{\\text{TP}}{\\left( \\text{TP}+ \\text{FP}+ \\beta ^2 P \\right)^2}\\le \\frac{1 + \\beta ^2}{\\beta ^4 P}.$ Hence, $M$ is Lipschitz continuous with constant $\\frac{2(1 + \\beta ^2)}{P} \\max \\left\\lbrace \\beta ^{-2}, \\beta ^{-4} \\right\\rbrace $ over the confusion matrices in Eqs.",
"(REF ), (), and ().",
"As Examples REF and REF demonstrate, the Lipschitz constants of many CMMs can become large when the proportion $P$ is positive samples is small.",
"In particular, when $P \\in O \\left( \\sqrt{\\frac{\\log n}{n}} \\right)$ , the $\\asymp L_M \\sqrt{\\frac{\\log (n/\\delta )}{n}}$ term of Corollary fails to vanish as $n \\rightarrow \\infty $ .",
"We believe that some loss of convergence rate is inevitable if $P \\rightarrow 0$ as $n \\rightarrow \\infty $ , due to the inherent instability of such metrics, but further work is needed to understand if the rates given by Corollary are optimal under these metrics.",
"See also [17] for detailed discussion of Lipschitz constants of many common CMMs."
],
[
"Bounds on Uniform Error of the Nearest Neighbor Regressor",
"In this appendix, we prove our upper bound on the uniform risk of the $k$ NN regressor (Theorem REF ), as well as the corresponding minimax lower bound (Theorem REF )."
],
[
"Upper Bounds",
"In this section, we prove Theorem REF , our upper bound on the uniform error of the $k$ -NN regressor.",
"The main result is restated below: thm:unifconvergenceTheorem REF Under Assumptions REF and REF , whenever $k / n \\le p_*(\\epsilon ^*)^d / 2$ , for any $\\delta > 0$ , with probability at least $1 - N\\left( \\left( 2k / (p_* n) \\right)^{1/d} \\right) e^{-k/4} - \\delta $ , we have the uniform error bound $\\left\\Vert \\eta - \\widehat{\\eta }\\right\\Vert _\\infty \\le 2^\\alpha Lr\\left( \\frac{2k}{p_* n} \\right)^{\\alpha /d}+ \\frac{2}{3k} \\log \\frac{2 S(n)}{\\delta } + \\sqrt{\\frac{2r}{k} \\log \\frac{2 S(n)}{\\delta }}.$ For any $x \\in \\mathcal {X}$ , let $\\widetilde{\\eta }_k(x) := \\frac{1}{k} \\sum _{j = 1}^k \\eta (X_{\\sigma _j(x)})$ denote the mean of the true regression function over the $k$ nearest neighbors of $x$ .",
"By the triangle inequality, $\\Vert \\eta - \\widehat{\\eta }\\Vert _\\infty \\le \\Vert \\eta - \\widetilde{\\eta }_k\\Vert _\\infty + \\Vert \\widetilde{\\eta }_k - \\widehat{\\eta }\\Vert _\\infty ,$ wherein $\\Vert \\eta - \\widetilde{\\eta }_k\\Vert _\\infty $ captures bias due to smoothing and $\\Vert \\widetilde{\\eta }_k - \\widehat{\\eta }\\Vert _\\infty $ captures variance due to label noise.",
"We separately show that, with probability at least $1 - N \\left( \\left( \\frac{2k}{p_* n} \\right)^{1/d} \\right) e^{-k/4}$ , $\\left\\Vert \\eta - \\widetilde{\\eta }_k \\right\\Vert _\\infty \\le 2^\\alpha Lr \\left( \\frac{2k}{p_* n} \\right)^{\\alpha /d},$ and that, with probability at least $1 - \\delta $ , $\\Vert \\widetilde{\\eta }_k - \\widehat{\\eta }\\Vert _\\infty \\le \\frac{2}{3k} \\log \\frac{2 S(n)}{\\delta }+ \\sqrt{\\frac{2r}{k} \\log \\frac{2 S(n)}{\\delta }}.$"
],
[
"Bounding the smoothing bias",
"Fix some $r > 0$ to be determined, and let $\\lbrace B_r(z_1),...,B_r(z_{N(r)})\\rbrace $ be a covering of $(\\mathcal {X}, \\rho )$ by $N(r)$ balls of radius $r$ , with centers $z_1,...,z_{N(r)} \\in \\mathcal {X}$ .",
"By the lower bound assumption on $P_X$ , each $P_X(B_r(z_j)) \\ge p_* r^d$ .",
"Therefore, by a multiplicative Chernoff bound, with probability at least $1 - N(r) e^{-p_* n r^d/8}$ , each $B_r(z_j)$ contains at least $p_* n r^d/2$ samples.",
"In particular, if $r \\ge \\left( \\frac{2k}{p_* n} \\right)^{1/d}$ , then each $B_k$ contains at least $k$ samples, and it follows that, for every $x \\in \\mathcal {X}$ , $\\rho (x, X_{\\sigma _k(x)}) \\le 2r$ .",
"Thus, by Hölder continuity of $\\eta $ , $\\left| \\eta (x) - \\widetilde{\\eta }_k(x) \\right|= \\left| \\eta (x) - \\frac{1}{k} \\sum _{j = 1}^k \\eta (X_{\\sigma _j(x)}) \\right|\\le \\frac{1}{k} \\sum _{j = 1}^k \\left| \\eta (x) - \\eta (X_{\\sigma _j(x)}) \\right|\\le L(2r)^\\alpha .$ Finally, if $\\frac{k}{n} \\le \\frac{p_*}{2} (r^*)^d$ , then we can let $r = \\left( \\frac{2k}{p_* n} \\right)^{1/d}$ ."
],
[
"Bounding variance due to label noise",
"Let $\\Sigma := \\lbrace \\sigma (x) \\in [n]^k : x \\in \\mathcal {X}\\rbrace $ denote the set of possible $k$ -nearest neighbor index sets.",
"One can check from the definition of the shattering coefficient that $|\\Sigma | \\le S(n)$ .",
"For any $\\sigma \\in [n]^k$ , let $Z_\\sigma := \\sum _{j = 1}^k Y_{\\sigma _j}$ and let $\\mu _\\sigma := \\operatornamewithlimits{\\mathbb {E}}\\left[ Z_\\sigma \\right]$ .",
"Note that the conditional random variables $Y_{\\sigma _j}|X_1,...,X_n$ have conditionally independent Bernoulli distributions with means $\\operatornamewithlimits{\\mathbb {E}}[Y_{\\sigma _j}|X_1,...,X_n] = \\eta (X_{\\sigma _j})$ and variances $\\operatornamewithlimits{\\mathbb {E}}\\left[ \\left( Y_{\\sigma _j} - \\eta (X_{\\sigma _j}) \\right)^2 |X_1,...,X_n \\right] = \\eta (X_{\\sigma _j}) (1 - \\eta (X_{\\sigma _j})) \\le r$ .",
"Therefore, by Bernstein's inequality (Eq.",
"(2.10) of [7]), for any $\\epsilon > 0$ , $\\operatornamewithlimits{\\mathbb {P}}\\left[ |Z_\\sigma /k - \\mu _\\sigma | \\ge \\epsilon \\right] \\le 2 \\exp \\left( -\\frac{k\\epsilon ^2}{2(r + \\epsilon /3)} \\right).$ Moreover, for any $x \\in \\mathcal {X}$ , $\\mu _{\\sigma (x)} = \\widetilde{\\eta }_k(x)$ and $Z_{\\sigma (x)}/k = \\widehat{\\eta }(x)$ .",
"Hence, by a union bound over $\\sigma $ in $\\Sigma $ , $\\operatornamewithlimits{\\mathbb {P}}\\left( \\sup _{x \\in \\mathcal {X}} \\left| \\widetilde{\\eta }_k(x) - \\widehat{\\eta }(x) \\right| > \\epsilon | X_1,...,X_n \\right)& = \\operatornamewithlimits{\\mathbb {P}}\\left( \\sup _{x \\in \\mathcal {X}} \\left| \\mu _{\\sigma (x)} - Z_{\\sigma (x)}/k\\right| > \\epsilon | X_1,...,X_n \\right) \\\\& \\le \\operatornamewithlimits{\\mathbb {P}}\\left( \\sup _{\\sigma \\in \\Sigma } \\left| \\mu _\\sigma - Z_\\sigma /k\\right| > \\epsilon | X_1,...,X_n \\right) \\\\& \\le |\\Sigma | \\sup _{\\sigma \\in \\Sigma } \\operatornamewithlimits{\\mathbb {P}}\\left( \\left| \\mu _\\sigma - Z_\\sigma /k\\right| > \\epsilon | X_1,...,X_n \\right) \\\\& \\le 2S(n) \\exp \\left( -\\frac{k\\epsilon ^2}{2(r + \\epsilon /3)} \\right).$ Since the right-hand side is independent of $X_1,...,X_n$ , the unconditional bound $\\operatornamewithlimits{\\mathbb {P}}\\left( \\sup _{x \\in \\mathcal {X}} \\left\\Vert \\widetilde{\\eta }_k(x) - \\widehat{\\eta }(x) \\right\\Vert _\\infty > \\epsilon \\right)\\le 2S(n) \\exp \\left( -\\frac{k\\epsilon ^2}{2(r + \\epsilon /3)} \\right)$ follows.",
"Plugging in $\\epsilon = \\frac{1}{3k} \\log \\frac{2 S(n)}{\\delta } + \\sqrt{\\left( \\frac{1}{3k} \\log \\frac{2 S(n)}{\\delta } \\right)^2 + \\frac{2r}{k} \\log \\frac{2 S(n)}{\\delta }}\\le \\frac{2}{3k} \\log \\frac{2 S(n)}{\\delta } + \\sqrt{\\frac{2r}{k} \\log \\frac{2 S(n)}{\\delta }}$ and simplifying gives the final result.",
"Recall that there is a small (polylogarithmic in $r$ ) gap between our upper and lower bounds.",
"We believe that the upper bound may be slightly loose, and that this might be tightened by using a stronger concentration inequality, such as Bennett's inequality [4], instead of Bernstein's inequality in Inequality (REF )."
],
[
"Lower Bounds",
"In this section, we prove Theorem REF , our lower bound on the minimax uniform error of estimating a Hölder continuous regression function.",
"We use a standard approach based on the following version of Fano's lemma: Lemma 28 (Fano's Lemma; Simplified Form of Theorem 2.5 of [53]) Fix a family $\\mathcal {P}$ of distributions over a sample space $\\mathcal {X}$ and fix a pseudo-metric $\\rho : \\mathcal {P}\\times \\mathcal {P}\\rightarrow [0,\\infty ]$ over $\\mathcal {P}$ .",
"Suppose there exist $P_0 \\in \\mathcal {P}$ and a set $T \\subseteq \\mathcal {P}$ such that $\\sup _{P \\in T} D_{KL}(P,P_0)\\le \\frac{\\log |T|}{16},$ where $D_{KL} : \\mathcal {P}\\times \\mathcal {P}\\rightarrow [0,\\infty ]$ denotes Kullback-Leibler divergence.",
"Then, $\\inf _{\\widehat{P}} \\sup _{P \\in \\mathcal {P}}\\operatornamewithlimits{\\mathbb {P}}\\left( \\rho (P,\\widehat{P})\\ge \\frac{1}{2} \\inf _{P \\in T} \\rho (P,P_0) \\right) \\ge 1/8,$ where the first $\\inf $ is taken over all estimators $\\widehat{P}$ .",
"Now, we proceed with the proof.",
"We now proceed to construct an appropriate $P_0 \\in \\mathcal {P}$ and $T \\subseteq \\mathcal {P}$ .",
"Let $g : [-1,1]^d \\rightarrow [0,1]$ defined by $g(x) = \\left\\lbrace \\begin{array}{cc}\\exp \\left( 1 - \\frac{1}{1 - \\Vert x\\Vert _2^2} \\right) & \\text{ if } \\Vert x\\Vert _2 < 1 \\\\0 & \\text{ else }\\end{array} \\right.$ denote the standard bump function supported on $[-1,1]^d$ , scaled to have $\\Vert g\\Vert _{\\mathcal {X},\\infty } = 1$ .",
"Since $g$ is infinitely differentiable and compactly supported, it has a finite $\\alpha $ -Hölder semi-norm: $\\Vert g\\Vert _{\\Sigma ^\\alpha }:= \\sup _{\\ell \\in \\mathbb {N}^d : \\Vert \\ell \\Vert _1 \\le \\alpha } \\quad \\sup _{x \\ne y \\in \\mathcal {X}} \\quad \\frac{|g^\\ell (x) - g^\\ell (y)|}{\\Vert x - y\\Vert ^{\\alpha - \\Vert \\ell \\Vert _1}}< \\infty ,$ where $\\ell $ is any $\\lfloor \\beta \\rfloor $ -order multi-index and $g^\\ell $ is the corresponding mixed derivative of $g$ .",
"Define $M := \\left( \\frac{64(2\\alpha + d)nr}{d \\log (nr)} \\right)^{\\frac{1}{2\\alpha + d}} \\ge 1$ , since $r \\ge 1/n$ .",
"For each $m \\in [M]^d$ , define $g_m : \\mathcal {X}\\rightarrow [0,1]$ by $g_m (x) := g\\left( Mx - \\frac{2m - 1_d}{2} \\right),$ so that $\\lbrace g_m : m \\in [M]^d\\rbrace $ is a grid of $M^d$ bump functions with disjoint supports.",
"Let $\\zeta _0 \\equiv \\frac{1}{4}$ denote the constant-$\\frac{1}{4}$ function on $\\mathcal {X}$ .",
"Finally, for each $m \\in [M]^d$ , define $\\zeta _m : \\mathcal {X}\\rightarrow [0,1]$ by $\\zeta _m := \\zeta _0 + \\min \\left\\lbrace \\frac{1}{2}, \\frac{L}{\\Vert g\\Vert _{\\Sigma ^\\alpha }} \\right\\rbrace M^{-\\alpha } g_m.$ Note that, for any $m \\in [M]^d$ , $\\Vert \\zeta _m\\Vert _{\\Sigma ^\\alpha }\\le L M^{-\\alpha } \\frac{\\Vert g_m\\Vert _{\\Sigma ^\\alpha }}{\\Vert g\\Vert _{\\Sigma ^\\alpha }}= L,$ so that $\\zeta _m$ satisfies the Hölder smoothness condition.",
"For any particular $\\eta $ , let $P_\\eta $ denote the joint distribution of $(X, Y)$ .",
"Note that $P_\\zeta (x, 1) = \\zeta (x) \\ge 1/4$ .",
"Moreover, one can check that, for all $x \\ge -2/3$ , $-\\log (1 + x) \\le x^2 - x$ .",
"Hence, for any $x \\in \\mathcal {X}$ , $P_{\\eta _m}(x, 1) \\log \\frac{P_{\\eta _m}(x, 1)}{P_{\\eta }(x, 1)}& = rP_{\\zeta _m}(x, 1) \\log \\frac{P_{\\zeta _m}(x, 1)}{P_{\\zeta }(x, 1)} \\\\& = r\\zeta _m(x) \\log \\frac{\\zeta _m(x)}{\\zeta (x)} \\\\& = -r\\zeta _m(x) \\log \\left( 1 + \\frac{\\zeta (x) - \\zeta _m(x)}{\\zeta _m(x)} \\right) \\\\& \\le r\\zeta _m(x) \\left( \\left( \\frac{\\zeta (x) - \\zeta _m(x)}{\\zeta _m(x)} \\right)^2 - \\frac{\\zeta (x) - \\zeta _m(x)}{\\zeta _m(x)} \\right) \\\\& = r \\left( \\frac{\\left( \\zeta (x) - \\zeta _m(x)\\right)^2 }{\\zeta _m(x)} - \\zeta (x) + \\zeta _m(x) \\right) \\\\& \\le r \\left( 4\\left( \\zeta (x) - \\zeta _m(x)\\right)^2 - \\zeta (x) + \\zeta _m(x) \\right),$ and, similarly, since $P_\\zeta (x,0) = 1 - \\zeta (x) \\ge 1/4$ , $P_{r\\eta _m}(x, 0) \\log \\frac{P_{r\\eta _m}(x, 0)}{P_{r\\eta }(x, 0)}\\le r\\left( 4\\left( \\zeta (x) - \\zeta _m(x)\\right)^2 + \\zeta (x) - \\zeta _m(x) \\right).$ Adding these two terms gives $D_{\\text{KL}}\\left( P_{r\\eta }^n, P_{r\\eta _m}^n \\right)& = n \\left( \\int _\\mathcal {X}P_{r\\eta _m}(x, 0) \\log \\frac{P_{r\\eta }(x, 0)}{P_{r\\eta _m}(x, 0)} \\, dx + \\int _\\mathcal {X}P_{r\\eta _m}(x, 1) \\log \\frac{P_{r\\eta }(x, 1)}{P_{r\\eta _m}(x, 1)} \\, dx \\right) \\\\& \\le 8nr\\int _\\mathcal {X}\\left( \\zeta (x) - \\zeta _m(x)\\right)^2 \\\\& = 8nr\\Vert \\zeta - \\zeta _m\\Vert _2^2 \\\\& \\le 2nr M^{-2\\alpha } \\Vert g_m\\Vert _2^2 \\\\& = 2nr M^{-(2\\alpha + d)} \\Vert g\\Vert _2^2 \\\\& = 2nr \\left( \\left( \\frac{64 (2\\alpha + d) nr}{d \\log (nr)} \\right)^{\\frac{1}{2\\alpha + d}}\\right)^{-(2\\alpha + d)} \\Vert g\\Vert _2^2 \\\\& = \\frac{1}{32} \\frac{d}{2\\alpha + d} \\Vert g\\Vert _2^2 \\log (nr) \\\\& \\le \\frac{1}{16} \\frac{d}{2\\alpha + d} \\left( \\log (nr) - \\log \\log (nr) + \\log \\frac{64 (2\\alpha + d)}{d} \\right)\\\\ &= \\frac{\\log |[M]^d|}{16},$ where the second inequality comes from the definition of $\\zeta _m$ (Eq.",
"REF ) and the third inequality comes from the facts that $\\Vert g\\Vert _2^2 \\le 1$ and $\\log \\log x \\le \\frac{1}{2} \\log x$ for all $x > 1$ .",
"Fano's lemma therefore implies the lower bound $\\inf _{\\widehat{\\eta }} \\sup _{r \\in (0, 1], \\zeta \\in \\Sigma ^\\alpha (L)}\\operatornamewithlimits{\\mathbb {P}}_{\\lbrace (X_i,Y_i)\\rbrace _{i = 1}^n \\sim P_\\eta ^n} \\left( \\left\\Vert r\\zeta - r\\widehat{\\zeta }\\right\\Vert _\\infty \\ge \\frac{1}{2} \\min \\left\\lbrace \\frac{1}{2}, \\frac{L}{\\Vert g\\Vert _{\\Sigma ^\\alpha }} \\right\\rbrace \\left( \\frac{d \\log (nr)}{64(2\\alpha + d) n} \\right)^{\\frac{\\alpha }{2\\alpha + d}} r^\\frac{\\alpha + d}{2\\alpha + d} \\right)\\ge \\frac{1}{8},$ which completes the proof."
],
[
"Uniform Error of the $k$ -Nearest Neighbor Regressor",
"In the previous section, for classifiers that threshold an estimate (a “regressor”) of the regression function, we bounded relative performance, as measured by arbitrary CMMs, in terms of the uniform ($\\mathcal {L}_\\infty $ ) loss of the regression function estimate.",
"In this section, we bound the uniform loss of one such regressor, the widely used $k$ -nearest neighbor ($k$ NN) regressor.",
"Our analyses include a parameter $r$ , introduced in Section REF , that characterizes a novel sub-type of class imbalance, which we call Uniform Class Imbalance.",
"As we discuss later, this leads to insights about how the behavior of the $k$ NN classifier depends not only on the degree, but also on the structure, of class imbalance in a given dataset."
],
[
"$k$ -Nearest Neighbor Regressor",
"Given a point $x \\in \\mathcal {X}$ , order the training data $X_{\\sigma _{1}(x)}, \\ldots , X_{\\sigma _{n}(x)}$ such that $\\rho \\left(X_{\\sigma _{1}(x)}, \\; x\\right)\\le \\ldots \\le \\rho \\left(X_{\\sigma _{n}(x)}, \\; x\\right);$ i.e., $X_{\\sigma _i(x)}$ is the $i^{th}$ -nearest neighbor of $x$ among $X_1,...,X_n$ .",
"For an integer $k > 0$ , the $k$ NN regressor $\\widehat{\\eta }_k : \\mathcal {X}\\rightarrow [0, 1]$ , is defined by the proportion $\\widehat{\\eta }_k(x) = \\frac{1}{k} \\sum _{i = 1}^k Y_{\\sigma _i(x)},\\quad \\text{ for all } x \\in \\mathcal {X},$ of $x$ 's $k$ -nearest neighbors in class 1."
],
[
"Uniform Class Imbalance",
"In this paper, we formalize a sub-type of class imbalance, which we refer to as Uniform Class Imbalance.",
"We decompose the regression function as $\\eta = r \\zeta $ , where $r \\in (0, 1]$ and $\\zeta : \\mathcal {X}\\rightarrow [0, 1]$ is a regression function with $\\sup _{x \\in \\mathcal {X}} \\zeta (x) = 1$ .",
"Note that this decomposition loses no generality, as any regression function $\\eta $ can be written in this form.",
"In Uniform Class Imbalance, $r \\approx 0$ , so that the class $Y = 1$ is rare regardless of the covariate $X$ (hence the name “uniform”).",
"Uniform Class Imbalance tends to occur in “challenging” classification problems in which the covariate $X$ provides only partial information about the class $Y$ .",
"Practical examples include rare disease diagnosis [47], credit card fraud detection [3], or predicting whether an applicant will be offered a job when there are many more qualified applicants than openings.",
"In practice, in such problems, the classifier's role is often not so much to make a final class determination as to identify “high-risk” samples $X$ such that $\\eta (X)$ is relatively elevated, for follow-up investigation.",
"Uniform Class Imbalance can be distinguished from “easier” classification problems in which, for some values $x \\in \\mathcal {X}$ , $\\eta (X) \\approx 1$ and so a good classifier can confidently assign the label $Y = 1$ .",
"These include well-separated classes, or the extreme case where $Y$ is a deterministic function of $X$ , such as in certain protein structure prediction problems [44]."
],
[
"Upper Bounds",
"In this section, we present bounds on the uniform error $U(\\widehat{\\eta }) := \\left\\Vert \\eta - \\widehat{\\eta }\\right\\Vert _\\infty $ of the $k$ NN regressor $\\widehat{\\eta }_k$ , where, for a function $f : \\mathcal {X}\\rightarrow \\mathbb {R}$ , $\\Vert f\\Vert _\\infty := \\sup _{x \\in \\mathcal {X}} |f(x)|$ denotes the $\\sup $ -norm of $f$ .",
"Before presenting our bounds, we define two standard quantities, covering numbers and shattering coefficients, by which we measure the complexity of the feature space.",
"Definition 9 (Covering Number) Suppose $(\\mathcal {X}, \\rho )$ is a totally bounded metric space.",
"Then, for any $\\epsilon > 0$ , the $\\epsilon $ -covering number $N(\\epsilon )$ of $(\\mathcal {X}, \\rho )$ is the smallest integer such that there exist $N(\\epsilon )$ points $x_1,...,x_{N(\\epsilon )} \\in X$ satisfying $\\mathcal {X}\\subseteq \\bigcup _{i = 1}^{N(\\epsilon )} B(x_i, \\epsilon )$ .",
"Definition 10 (Shattering Coefficient of Balls) For positive integers $n$ , $S(n):= \\sup _{x_1,...,x_n \\in \\mathcal {X}} \\left| \\left\\lbrace \\lbrace x_1,...,x_n\\rbrace \\cap B(x, \\epsilon ) : x \\in \\mathcal {X}, \\epsilon \\ge 0 \\right\\rbrace \\right|$ denotes the shattering coefficient of open balls in $(\\mathcal {X}, \\rho )$ .",
"We now state two assumptions we make on the joint distribution $P_{X, Y}$ of the data Assumption 11 (Dense Covariates Assumption) The marginal distribution $P_X$ of the covariates is lower bounded in the sense that, for some constants $p_*,\\epsilon ^*,d > 0$ , for any point $x$ in $\\mathcal {X}$ and radius $\\epsilon $ in $(0,\\epsilon ^*]$ , we have the inequality $P_X(B_\\epsilon (x)) \\ge p_* \\epsilon ^d$ .",
"Assumption REF ensures that each query point's nearest neighbor is sufficiently near.",
"We will also need to assume that the regression function $\\zeta $ is sufficiently smooth: Assumption 12 (Hölder Continuity) $\\zeta $ is $(\\alpha , L)$ -Hölder continuous; that is, for all $x, x^{\\prime } \\in \\mathcal {X}$ , $\\left| \\zeta (x) - \\zeta (x^{\\prime }) \\right| \\le L \\rho ^\\alpha (x, x^{\\prime })$ .",
"We now provide our upper bound on the uniform error, proven in Appendix REF .",
"Theorem 13 Under Assumptions REF and REF , whenever $k / n \\le p_*(\\epsilon ^*)^d / 2$ , for any $\\delta > 0$ , with probability at least $1 - N\\left( \\left( 2k / (p_* n) \\right)^{1/d} \\right) e^{-k/4} - \\delta $ , we have the uniform error bound $\\left\\Vert \\eta - \\widehat{\\eta }\\right\\Vert _\\infty \\le 2^\\alpha Lr\\left( \\frac{2k}{p_* n} \\right)^{\\alpha /d}+ \\frac{2}{3k} \\log \\frac{2 S(n)}{\\delta } + \\sqrt{\\frac{2r}{k} \\log \\frac{2 S(n)}{\\delta }}.$ If $r \\in O \\left( \\frac{\\log S(n)}{n} \\right)$ , this bound is minimized by $k \\asymp n$ , giving $\\left\\Vert \\eta - \\widehat{\\eta }\\right\\Vert _\\infty \\in O_P \\left( \\frac{\\log S(n)}{n} \\right)$ .",
"Otherwise, under a mild simplifying assumption that $N(\\epsilon )$ increases at most polynomially with $1/\\epsilon $ , this bound is minimized by $k \\asymp n^{\\frac{2\\alpha }{2\\alpha +d}} (\\log S(n))^{\\frac{d}{2\\alpha +d}} r^{-\\frac{d}{2\\alpha + d}}$ , giving $\\left\\Vert \\eta - \\widehat{\\eta }\\right\\Vert _\\infty \\in O_P \\left( \\left( \\frac{\\log S(n)}{n} \\right)^\\frac{\\alpha }{2\\alpha + d} r^\\frac{\\alpha + d}{2\\alpha + d} \\right).$ Of the three terms in (REF ), the first term, of order $r(k/n)^{\\alpha /d}$ , comes from smoothing bias of the $k$ NN classifier.",
"The second and third terms are due to label noise, with the second term dominating under extreme class imbalance ($r \\in O \\left( \\frac{\\log S(n)}{n} \\right)$ ) and the third term dominating otherwise.",
"Theorem REF shows that the optimal choice of the tuning parameter $k$ is much larger under Uniform Class Imbalance than in the case of balanced classes; indeed, one can check that setting $k \\asymp n^{\\frac{2\\alpha }{2\\alpha +d}} (\\log S(n))^{\\frac{d}{2\\alpha +d}}$ , which is optimal in the balanced case, gives a rate that slower by a factor of $r^{-\\frac{d}{4\\alpha + d}}$ .",
"One interpretation is that a larger number of neighbors is needed to obtain enough samples from the rare class to make a reliable prediction at any given point.",
"The following two examples demonstrate how to apply Theorem REF in specific settings of interest: Corollary 14 (Euclidean, Absolutely Continuous Case) Suppose $(\\mathcal {X},\\rho ) = ([0,1]^d,\\Vert \\cdot \\Vert _2)$ is the unit cube in $\\mathbb {R}^d$ , equipped with the Euclidean metric, and $P_X$ has a density that is lower bounded away from 0 on $\\mathcal {X}$ .",
"Then, $N(\\epsilon ) \\le (2/\\epsilon )^d$ and $S(n) \\le 2n^{d + 1} + 2$ , and so, for $k \\asymp n^{\\frac{2\\alpha }{2\\alpha +d}} (\\log n)^{\\frac{d}{2\\alpha +d}} r^{-\\frac{d}{2\\alpha + d}}$ , by Theorem REF , $\\left\\Vert \\eta - \\widehat{\\eta }\\right\\Vert _\\infty \\in O_P \\left( \\left( \\frac{\\log n}{n} \\right)^{\\frac{\\alpha }{2\\alpha +d}} r^\\frac{\\alpha + d}{2\\alpha + d} \\right).$ The most problematic term in this bound is the exponential dependence on the dimension $d$ of the covariates.",
"Fortunately, since Theorem REF utilizes covering numbers, it improves if the covariates exhibit structure, such as that of a low-dimensional manifold.",
"The next example formalizes this.",
"Corollary 15 (Implicit Manifold Case) Suppose $Z$ is a $[0,1]^d$ -valued random variable with a density lower bounded away from 0, and suppose that, for some Lipschitz map $T : [0,1]^d \\rightarrow \\mathbb {R}^D$ , $X = T(Z)$ .",
"Then, $N(\\epsilon ) \\le (2/\\epsilon )^d$ , and $S(n) \\le 2n^{D + 1} + 2$ , and so, by Theorem REF , $k \\asymp n^{\\frac{2\\alpha }{2\\alpha +d}} (\\log n)^{\\frac{d}{2\\alpha +d}} r^{-\\frac{d}{2\\alpha + d}}$ , $\\left\\Vert \\eta - \\widehat{\\eta }\\right\\Vert _\\infty \\in O_P \\left( \\left( \\frac{\\log n}{n} \\right)^{\\frac{\\alpha }{2\\alpha +d}} r^\\frac{\\alpha + d}{2\\alpha + d} \\right).$ This shows that, if the $D$ covariates lie implicitly on a $d$ -dimensional manifold (e.g., if the covariates are strongly correlated), convergence rates depend on $d$ , which may be much smaller than $D$ .",
"We close this section with a lower bound, proven in Appendix REF , on the minimax uniform error, showing that the rate provided in Theorem REF is minimax optimal over $(\\alpha , L)$ -Hölder regression functions, up to a polylogarithmic factor in $r$ : Theorem 16 Suppose $\\mathcal {X}= [0,1]^d$ is the $d$ -dimensional unit cube and the marginal distribution of $X$ is uniform on $\\mathcal {X}$ .",
"Let $\\Sigma ^\\alpha (L)$ denote the family of $(\\alpha , L)$ -Hölder continuous regression function.",
"Then, for any $\\alpha , L > 0$ , there exist constants $n_0$ and $c > 0$ (depending only on $\\alpha $ , $L$ , and $d$ ) such that, for all $n \\ge n_0$ and any estimator $\\widehat{\\eta }$ , $\\sup _{\\zeta \\in \\Sigma ^\\alpha (L)} \\mathop {\\operatornamewithlimits{\\mathbb {P}}}\\left[ \\left\\Vert \\eta - \\widehat{\\eta }\\right\\Vert _\\infty \\ge c \\left( \\frac{\\log (nr)}{n} \\right)^{\\frac{\\alpha }{2\\alpha + d}} r^\\frac{\\alpha + d}{2\\alpha + d} \\right] \\ge \\frac{1}{8}.$"
],
[
"Discussion",
"The upper bounds on $\\left\\Vert \\eta - \\widehat{\\eta }\\right\\Vert _\\infty $ given in this section can be plugged directly into Corollary to provide error bound under arbitrary CMMs, in terms of the sample size $n$ , hyperparameter $k$ , degree $r$ of Uniform Class Imbalance, and complexity parameters (margin $\\beta $ , smoothness $\\alpha $ , intrinsic dimension $d$ , etc.)",
"of $\\mathcal {X}$ and $P_{X,Y}$ .",
"Thus, these results collectively give some of the first finite-sample guarantees under general performance metrics used for imbalanced classification.",
"As noted previously, our analysis shows that, under severe Uniform Class Imbalance, the optimal choice of the hyperparameter $k$ is much larger than in balanced classification.",
"Importantly, this larger choice of $k$ , leads to sub-optimal, or even inconsistent, estimates of the regression function under other (nonuniform) forms of class imbalance.",
"The following example illustrates this: Example 17 Suppose $\\mathcal {X}= [0, 1]$ , $X \\sim \\operatorname{Uniform}([0, 1])$ , and $r \\in (0, 1)$ .",
"Consider two regression functions $\\eta _1(x) = r(1 - x)$ and $\\eta _2(x) = \\max \\lbrace 0, 1 - x/r\\rbrace $ .",
"Both $\\eta _1$ and $\\eta _2$ exhibit the same degree of overall class imbalance, with the proportion of samples from class 1 being $r/2$ .",
"The regression function $\\eta _1$ satisfies Uniform Class Imbalance of degree $r$ , whereas $\\eta _2$ does not satisfy a nontrivial degree of Uniform Class Imbalance, since $\\eta _2(0) = 1$ .",
"For sufficiently small $r \\in (0, 1)$ , specifically $r \\in o \\left( n^{\\frac{-d}{2(\\alpha + d)}} \\right)$ , Theorem REF gives that the optimal choice of $k$ under $\\eta _1$ satisfies $k \\in \\omega (rn)$ .",
"On the other hand, if $k \\in \\omega (rn)$ , then, under $\\eta _2$ , $\\operatornamewithlimits{\\mathbb {E}}\\left[ \\widehat{\\eta }_k(0) \\right] \\rightarrow 0$ , so that $\\widehat{\\eta }_k(0)$ is an inconsistent estimate of $\\eta _2(0) = 1$ .",
"This example demonstrates that constructing classifiers that perform well under severe class imbalance may require distinguishing different sub-types of class imbalance, such as Uniform Class Imbalance."
],
[
"Conclusions",
"The main conclusions of this paper are as follows.",
"First, the Bayes classifier, which optimizes classification performance in terms of accuracy, can be generalized to many other measures of classification performance using a simple thresholding procedure with only two additional scalar parameters.",
"This Generalized Bayes Classifier provides an optimal performance benchmark, relative to which one can evaluate many classifiers of interest in terms of their ability to estimate the regression function in uniform loss.",
"This includes the widely-used $k$ NN classifier, for which we provided a number of guarantees, showing that it performs minimax optimally under uniform loss in a number of settings, including that of severe Uniform Class Imbalance.",
"On the other hand, we showed that the optimal tuning of $k$ can differ significantly between different sub-types of imbalanced classification, suggesting that developing reliable classifiers for severely imbalanced classification may require a more nuanced understanding of the nature of class imbalance intrinsic to the problem at hand.",
"We hope that some of these ideas, especially the generalized Bayes classifier and the distinction of sub-types of class imbalance, will play a role in developing a coherent and insightful statistical theory of imbalanced classification, and that this theory will inform the construction of more reliable classifiers for challenging real-world imbalanced classification problems.",
"Additionally, these ideas should be generalized to the multi-class case, in which severe class imbalance emerges naturally when the number of classes is large and existing statistical theory is quite limited."
],
[
"Derivation of the Generalized Bayes Classifier (Theorem ",
"In this Appendix, we prove Theorem REF , in which we characterize a generalization of the Bayes classifier to arbitrary CMMs.",
"We reiterate the theorem for the reader here: thm:generalizedbayesTheorem REF If $\\operatornamewithlimits{\\arg \\!\\max }_{\\widehat{Y} \\in \\mathcal {SC}} M(C_{\\widehat{Y}}) \\ne \\varnothing $ , then there exists a regression-thresholding classifier $\\widehat{Y}_{p,t,\\eta } \\in \\operatornamewithlimits{\\arg \\!\\max }_{\\widehat{Y} \\in \\mathcal {SC}} M(C_{\\widehat{Y}}).$ As described in the main paper, the proof of Theorem REF is given in a sequence of steps constructing optimal classifiers in forms progressively closer to that of the generalized Bayes classifier described in Theorem REF .",
"Specifically, we first show, in Lemma REF , that there exists an optimal classifier that is a (stochastic) function of the regression function $\\eta $ .",
"We then construct an optimal classifier in which this function of $\\eta $ is non-decreasing.",
"Finally, we construct an optimal classifier in which this function of $\\eta $ is a threshold function, as in Theorem REF .",
"Lemma 18 For any stochastic classifier $\\widehat{Y} : \\mathcal {X}\\rightarrow \\mathcal {P}(\\lbrace 0, 1\\rbrace )$ , there is a stochastic classifier $\\widehat{Y}^{\\prime } : \\mathcal {X}\\rightarrow \\mathcal {P}(\\lbrace 0, 1\\rbrace )$ of the form $\\widehat{Y}^{\\prime }(x) \\sim \\text{Bernoulli}(f(\\eta (x))),$ for some $f : [0, 1] \\rightarrow [0, 1]$ , such that $C_{\\widehat{Y}^{\\prime }} = C_{\\widehat{Y}}$ .",
"Define $\\widehat{Y}^{\\prime } : \\mathcal {X}\\rightarrow \\mathcal {P}(\\lbrace 0, 1\\rbrace )$ by $\\widehat{Y}^{\\prime }(x) \\sim \\text{Bernoulli} \\left( \\operatornamewithlimits{\\mathbb {E}}_{Z \\sim P_X} \\left[ \\widehat{Y}(Z) | \\eta (Z) = \\eta (x) \\right] \\right),$ and note that $\\widehat{Y}^{\\prime }$ has the desired form.",
"Note that both $\\widehat{Y}(X)$ and $\\widehat{Y}^{\\prime }(X)$ are conditionally independent of the true label $Y$ given $\\eta (X)$ .",
"Thus, $\\operatornamewithlimits{\\mathbb {E}}\\left[ Y \\widehat{Y}^{\\prime }(X) \\right]& = \\operatornamewithlimits{\\mathbb {E}}\\left[ \\operatornamewithlimits{\\mathbb {E}}\\left[ Y \\widehat{Y}^{\\prime }(X) | \\eta (X) \\right] \\right] \\\\& = \\operatornamewithlimits{\\mathbb {E}}\\left[ \\operatornamewithlimits{\\mathbb {E}}\\left[ Y | \\eta (X) \\right] \\operatornamewithlimits{\\mathbb {E}}\\left[ \\widehat{Y}^{\\prime }(X) | \\eta (X) \\right] \\right] \\\\& = \\operatornamewithlimits{\\mathbb {E}}\\left[ \\eta (X) \\operatornamewithlimits{\\mathbb {E}}\\left[ \\widehat{Y}^{\\prime }(X) | \\eta (X) \\right] \\right]$ and, similarly, $\\operatornamewithlimits{\\mathbb {E}}\\left[ Y \\widehat{Y}(X) \\right]= \\operatornamewithlimits{\\mathbb {E}}\\left[ \\eta (X) \\operatornamewithlimits{\\mathbb {E}}\\left[ \\widehat{Y}(X) | \\eta (X) \\right] \\right].$ Additionally, by construction of $\\widehat{Y}^{\\prime }$ , $\\operatornamewithlimits{\\mathbb {E}}\\left[ \\widehat{Y}^{\\prime }(X) | \\eta (X) \\right]= \\operatornamewithlimits{\\mathbb {E}}\\left[ \\operatornamewithlimits{\\mathbb {E}}_{Z \\sim P_X} \\left[ \\widehat{Y}(Z) | \\eta (Z) = \\eta (X) \\right] | \\eta (X) \\right]= \\operatornamewithlimits{\\mathbb {E}}\\left[ \\widehat{Y}(X) | \\eta (X) \\right].$ Putting these together, we have $\\text{TP}_{\\widehat{Y}^{\\prime }}= \\operatornamewithlimits{\\mathbb {E}}\\left[ Y \\widehat{Y}^{\\prime }(X) \\right]& = \\operatornamewithlimits{\\mathbb {E}}\\left[ \\eta (X) \\operatornamewithlimits{\\mathbb {E}}\\left[ \\widehat{Y}^{\\prime }(X) | \\eta (X) \\right] \\right] \\\\& = \\operatornamewithlimits{\\mathbb {E}}\\left[ \\eta (X) \\operatornamewithlimits{\\mathbb {E}}\\left[ \\widehat{Y}(X) | \\eta (X) \\right] \\right]= \\operatornamewithlimits{\\mathbb {E}}\\left[ Y \\widehat{Y}(X) \\right]= \\text{TP}_{\\widehat{Y}}.$ Similarly, one can check that $C_{\\widehat{Y}^{\\prime }} = C_{\\widehat{Y}}$ .",
"It follows from Lemma REF that, if $M(C_{\\widehat{Y}})$ is maximized by any classifier, then it is maximized by a classifier $\\widehat{Y}$ of the form in Eq.",
"(REF ).",
"It remains to show that $f$ in Eq.",
"(REF ) can be of the form $z \\mapsto p 1\\lbrace z = t\\rbrace + 1\\lbrace z > t\\rbrace $ for some threshold $(p, t) \\in [0, 1]^2$ .",
"Before proving this, we give a simplifying lemma showing that the problem of maximizing a CMM can be equivalently framed as a particular functional optimization problem.",
"This will allow us to to significantly simplify the notation in the subsequent proofs.",
"Lemma 19 Let $M$ be a CMM, and suppose that $M(C_{\\widehat{Y}})$ is maximized (over $\\mathcal {SC}$ ) by a classifier $\\widehat{Y}$ of the form $\\widehat{Y}(x) \\sim \\operatorname{Bernoulli}(f^*(\\eta (x))),$ for some $f^* : [0, 1] \\rightarrow [0, 1]$ .",
"Let $f$ be a solution to the optimization problem $\\max _{f : [0, 1] \\rightarrow [0, 1]} \\operatornamewithlimits{\\mathbb {E}}[\\eta (X) f(\\eta (X))]\\quad \\text{ subject to } \\quad \\operatornamewithlimits{\\mathbb {E}}[(1 - \\eta (X)) f(\\eta (X))] \\le \\operatornamewithlimits{\\mathbb {E}}[(1 - \\eta (X)) f^*(\\eta (X))].$ Then, the classifier $\\widehat{Y}^{\\prime }(x) \\sim \\operatorname{Bernoulli}(f(\\eta (x))),$ also maximizes $M(C_{\\widehat{Y}^{\\prime }})$ (over $\\mathcal {SC}$ ).",
"This result follows from the definition (Definition REF ) of a CMM.",
"Specifically, by construction of $\\widehat{Y}^{\\prime }$ , $\\text{TP}_{\\widehat{Y}^{\\prime }} = \\operatornamewithlimits{\\mathbb {E}}[\\eta (X) f(\\eta (X))] \\ge \\operatornamewithlimits{\\mathbb {E}}[\\eta (X) f^*(\\eta (X))] = \\text{TP}_{\\widehat{Y}}$ and $\\text{FP}_{\\widehat{Y}^{\\prime }} = \\operatornamewithlimits{\\mathbb {E}}[(1 - \\eta (X)) f(\\eta (X))] \\le \\operatornamewithlimits{\\mathbb {E}}[(1 - \\eta (X)) f^*(\\eta (X))] = \\text{FP}_{\\widehat{Y}}.$ Moreover, since, since the actual proportions of positives and negatives are fixed (i.e., $\\text{TP}_{\\widehat{Y}^{\\prime }} + \\text{FN}_{\\widehat{Y}^{\\prime }} = \\text{TP}_{\\widehat{Y}} + \\text{FN}_{\\widehat{Y}}$ and $\\text{FP}_{\\widehat{Y}^{\\prime }} + \\text{TN}_{\\widehat{Y}^{\\prime }} = \\text{FP}_{\\widehat{Y}} + \\text{TN}_{\\widehat{Y}}$ ), we have $C_{\\widehat{Y}^{\\prime }} =\\begin{bmatrix}\\text{TN}_{\\widehat{Y}} + \\epsilon _1 & \\text{FP}_{\\widehat{Y}} - \\epsilon _1 \\\\\\text{FN}_{\\widehat{Y}} - \\epsilon _2 & \\text{TP}_{\\widehat{Y}} + \\epsilon _2,\\end{bmatrix},$ where $\\epsilon _1 := \\text{FP}_{\\widehat{Y}} - \\text{FP}_{\\widehat{Y}^{\\prime }} \\in [0, \\text{FP}_{\\widehat{Y}}]$ and $\\epsilon _2 := \\text{TP}_{\\widehat{Y}^{\\prime }} - \\text{TP}_{\\widehat{Y}} \\in [0, \\text{FN}]$ .",
"Thus, by the definition (Definition REF ) of a CMM, $M(C_{\\widehat{Y}^{\\prime }}) \\ge M(C_{\\widehat{Y}})$ .",
"Lemma REF essentially shows that maximizing any CMM $M$ is equivalent to performing Neyman-Pearson classification, at a some particular false positive level $\\alpha $ depending on $M$ (through $f^*$ ) and on the distribution of $\\eta (X)$ .",
"For our purposes, this simplifies the remaining steps in proving Theorem REF by allowing us to ignore the details of the particular CMM $M$ and regression function $\\eta $ and focus on characterizing solutions to an optimization problem of the form (REF ) (see, specifically, (REF ) below).",
"To characterize solutions to this optimization problem, we will repeatedly utilize the following measure-theoretic technical lemma: Lemma 20 Let $\\mu $ be a measure on $[0, 1]$ with $\\mu ([0, 1]) > 0$ .",
"Then, there exists $z \\in \\mathbb {R}$ such that, for all $\\epsilon > 0$ , $\\mu ([0, 1] \\cap (z - \\epsilon , z)) > 0$ .",
"We prove the contrapositive.",
"Suppose that, for every $z \\in [0, 1]$ , there exists $\\epsilon _z > 0$ such that $\\mu ([0, 1] \\cap (z - \\epsilon _z, z)) = 0$ .",
"The family $\\mathcal {S} := \\lbrace [0, 1] \\cap (z - \\epsilon _z, z) : z \\in \\mathbb {R}\\rbrace $ is an open cover of $[0, 1]$ .",
"Since $[0, 1]$ is compact, there exists a finite sub-cover $\\mathcal {S}^{\\prime } \\subseteq \\mathcal {S}$ of $[0, 1]$ .",
"Thus, by countable subaddivity of measures, $\\mu ([0, 1]) \\le \\sum _{S \\in \\mathcal {S}^{\\prime }} \\mu (\\mathcal {S}) = 0.$ We are now ready for the main remaining step in the proof of Theorem REF , namely characterizing solutions of (a generalization of) the optimization problem (REF ): Lemma 21 Let $Z$ be a $[0, 1]$ -valued random variable, and let $c \\in [0, 1]$ .",
"Suppose that the optimization problem $\\max _{f : [0, 1] \\rightarrow [0, 1]} \\operatornamewithlimits{\\mathbb {E}}[Z f(Z)]\\quad \\text{ subject to } \\quad \\operatornamewithlimits{\\mathbb {E}}[(1 - Z) f(Z)] \\le c$ has a solution.",
"Then, there is a solution to (REF ) that is a generalized threshold function.",
"Suppose that there exists a solution $f$ to (REF ).",
"We will construct a generalized threshold function that solves (REF ) in two main steps.",
"First, we will construct a monotone solution to (REF ).",
"Second, we will show that this monotone solution is equal to a generalized threshold function except perhaps on a set of probability 0 with respect to $Z$ .",
"This generalized threshold function is therefore a solution to (REF ).",
"Construction of Monotone Solution to (REF ): Define $g(z) := \\operatornamewithlimits{\\text{ess sup}}_{P_Z} f([0, z])\\quad \\text{ and } \\quad h(z) := \\operatornamewithlimits{\\text{ess inf}}_{P_Z} f((z, 1]),$ where the essential supremum and infimum are taken with respect to the measure $P_Z$ of $Z$ , with the conventions $g(z) = 0$ whenever $P_Z([0, z]) = 0$ and $h(z) = 1$ whenever $P_Z((z, 1]) = 0$ .",
"We first show that, for all $z \\in [0, 1]$ , $g(z) \\le h(z)$ .",
"We will then use this to show that $g = f$ except on a set of $P_Z$ measure 0 (i.e., $P_Z(\\lbrace z \\in [0, 1] : g(z) \\ne f(z)\\rbrace ) = 0$ ).",
"Therefore, both $\\operatornamewithlimits{\\mathbb {E}}[Z g(Z)] = \\operatornamewithlimits{\\mathbb {E}}[Z f(Z)]$ and $\\operatornamewithlimits{\\mathbb {E}}[(1 - Z) g(Z)] = \\operatornamewithlimits{\\mathbb {E}}[(1 - Z) f(Z)]$ .",
"Since $g : [0, 1] \\rightarrow [0, 1]$ is clearly monotone non-decreasing, the result follows.",
"Suppose, for sake of contradiction, that, for some $z \\in [0, 1]$ , $g(z) > h(z)$ .",
"Then there exist $A \\subseteq [0, z]$ and $B \\subseteq (z, 1]$ such that $\\inf _{z \\in A} f(z) > \\sup _{z \\in B} f(z)$ and $P_Z(A), P_Z(B) > 0$ .",
"Define $z_A := \\operatornamewithlimits{\\mathbb {E}}[Z|Z \\in A]$ and $z_B := \\operatornamewithlimits{\\mathbb {E}}[Z|Z \\in B]$ , and note that, since $A \\subseteq [0, z]$ and $B \\subseteq (z, 1]$ , $z_A < z_B$ .",
"Define, $\\epsilon := \\min \\left\\lbrace \\frac{P_Z(A) (1 - z_A)}{P_Z(B) (1 - z_B)} \\inf _{z \\in A} f(z),\\quad \\sup _{z \\in B} f(z) \\right\\rbrace > 0$ and define $\\phi : [0, 1] \\rightarrow [0, 1]$ by $\\phi (z):= \\left\\lbrace \\begin{array}{cc}f(z) - \\epsilon \\frac{P_Z(B) (1 - z_B)}{P_Z(A) (1 - z_A)} & \\text{ if } z \\in A \\\\f(z) + \\epsilon & \\text{ if } z \\in B \\\\f(z) & \\text{ otherwise,}\\end{array}\\right.$ noting that, by construction of $\\epsilon $ , $\\phi (z) \\in [0, 1]$ for all $z \\in [0, 1]$ .",
"Then, by construction of $\\phi $ , $\\operatornamewithlimits{\\mathbb {E}}[(1 - Z) \\phi (Z)] - \\operatornamewithlimits{\\mathbb {E}}[(1 - Z) f(Z)]& = - (1 - z_A) \\epsilon \\frac{P_Z(B) (1 - z_B)}{P_Z(A) (1 - z_A)} P_Z(A) + (1 - z_B) \\epsilon P_Z(B) \\\\& = 0,$ while $\\operatornamewithlimits{\\mathbb {E}}[Z \\phi (Z)] - \\operatornamewithlimits{\\mathbb {E}}[Z f(Z)]& = - z_A \\epsilon \\frac{P_Z(B) (1 - z_B)}{P_Z(A) (1 - z_A)} P_Z(A) + z_B \\epsilon P_Z(B) \\\\& = \\left( - \\frac{z_A}{1 - z_A}(1 - z_B) + z_B \\right) \\epsilon P_Z(B)> 0,$ since the function $z \\mapsto \\frac{z}{1 - z}$ is strictly increasing.",
"This contradicts the assumption that $f$ optimizes (REF ), implying $g \\le h$ .",
"We now show that $g = f$ except on a set of $P_Z$ measure 0.",
"First, note that, if $g(z) \\ne f(z)$ , then $g(z) = \\operatornamewithlimits{\\text{ess sup}}f([0, z]) = \\operatornamewithlimits{\\text{ess sup}}f([0, z))$ , and so $g$ is left-continuous at $z$ .",
"For any $\\delta > 0$ , define $A_\\delta := \\left\\lbrace z \\in [0, 1] : g(z) < f(z) - \\delta \\right\\rbrace \\quad \\text{ and } \\quad B_\\delta := \\left\\lbrace z \\in [0, 1] : g(z) > f(z) + \\delta \\right\\rbrace .$ Since $\\lbrace z \\in [0, 1] : g(z) < f(z)\\rbrace = \\bigcup _{j = 1}^\\infty \\left\\lbrace z \\in [0, 1] : g(z) < f(z) - \\frac{1}{j} \\right\\rbrace $ and $\\lbrace z \\in [0, 1] : g(z) > f(z)\\rbrace = \\bigcup _{j = 1}^\\infty \\left\\lbrace z \\in [0, 1] : g(z) > f(z) + \\frac{1}{j} \\right\\rbrace ,$ by countable subadditivity, it suffices to show that $P_Z(A_\\delta ) = P_Z(B_\\delta ) = 0$ for all $\\delta > 0$ .",
"Suppose, for sake of contradiction, that $P_Z(A_\\delta ) > 0$ .",
"Applying Lemma REF to the measure $E \\mapsto P_Z(A_\\delta \\cap E)$ , there exists $z \\in \\mathbb {R}$ such that, for any $\\epsilon > 0$ , $P_Z(A_\\delta \\cap (z - \\epsilon , z)) > 0$ .",
"Since $g$ is continuous at $z$ , there exists $\\epsilon > 0$ such that $g(z - \\epsilon ) \\ge g(z) - \\delta $ , so that, for all $z \\in A_\\delta \\cap (z - \\epsilon , z)$ , $f(z) > g(z) + \\delta $ .",
"Then, since $P_Z(A_\\delta \\cap (z - \\epsilon , z)) > 0$ , we have the contradiction $g(z) \\ge \\operatornamewithlimits{\\text{ess sup}}f(A_\\delta \\cap (z - \\epsilon , z)) > g(z).$ On the other hand, suppose, for sake of contradiction, that $P_Z(B_\\delta ) > 0$ .",
"Applying Lemma REF to the measure $E \\mapsto P_Z(B_\\delta \\cap E)$ , there exists $z \\in \\mathbb {R}$ such that, for any $\\epsilon > 0$ , $P_Z(B_\\delta \\cap (z - \\epsilon , z)) > 0$ .",
"Since $g$ is continuous at $z$ , there exists $\\epsilon > 0$ such that $g(z - \\epsilon ) \\ge g(z) - \\delta $ .",
"At the same time, since $g$ is non-decreasing, for $t \\in B_\\delta \\cap (z - \\epsilon , z)$ , $f(t) < g(t) - \\delta \\le g(z) - \\delta $ .",
"Thus, since $P_Z(B_\\delta \\cap (z - \\epsilon , z)) > 0$ , we have $h(z - \\epsilon ) < g(z) - \\delta < g(z - \\epsilon )$ , contradicting the previously shown fact that $g \\le h$ .",
"To conclude, we have shown that $P_Z(\\lbrace z \\in [0, 1] : g(z) \\ne f(z)\\rbrace ) = 0$ .",
"Construction of a Generalized Threshold Solution: We now construct a solution to (REF ) that is equal to a generalized threshold function (i.e., a function that has the form $p 1\\lbrace z = t\\rbrace + 1\\lbrace z > t\\rbrace $ ) except on a set of $P_Z$ -measure 0.",
"To show this, it suffices to construct a function $f : [0, 1] \\rightarrow [0, 1]$ such that (a) $f$ is monotone non-decreasing and (b) the set $f^{-1}((0, 1))$ is the union of the singleton $\\lbrace t\\rbrace $ and a set of $P_Z$ -measure 0.",
"From the previous step of this proof, we may assume that we have a solution $f$ to (REF ) that is monotone non-decreasing.",
"It suffices therefore to show that $A := f^{-1}((0, 1))$ is the union of a singleton and a set of $P_Z$ -measure 0.",
"Define $t_0 := \\inf \\lbrace z \\in [0, 1] : P_Z(A \\cap [0, z]) > 0\\rbrace \\quad \\text{ and } \\quad t_1 := \\sup \\lbrace z \\in [0, 1] : P_Z(A \\cap [z, 1]) > 0\\rbrace .$ Then, for all $\\epsilon > 0$ , $P_Z(A \\cap [0, t_0 - \\epsilon ]) = P_Z(A \\cap [t_1 + \\epsilon , 1]) = 0$ .",
"If $t_0 = t_1$ , then, since $A \\backslash \\lbrace t_0\\rbrace = \\bigcup _{j = 1}^\\infty A \\cap \\left( [0, t_0 - 1/j] \\cup [t_0 + 1/j, 1] \\right)$ by countable subadditivity, $P_Z(A \\backslash \\lbrace t_0\\rbrace ) = 0$ , which implies that $A = \\lbrace t_0\\rbrace \\cup (A \\backslash \\lbrace t_0\\rbrace )$ is the union of a singleton and a set of measure 0.",
"It suffices therefore to prove that $t_0 = t_1$ .",
"Suppose, for sake of contradiction, that $t_0 < t_1$ .",
"Then, there exists $t \\in (t_0, t_1)$ , and, by definition of $t_0$ and $t_1$ , both $P_Z(A \\cap [0, t)) > 0$ and $P_Z(A \\cap (t, 1]) > 0$ .",
"For any $\\delta \\ge 0$ , define $B_\\delta := \\lbrace z \\in [0, t): \\delta < f(z) < 1 - \\delta \\rbrace \\quad \\text{ and } \\quad C_\\delta := \\lbrace z \\in (t, 1]: \\delta < f(z) < 1 - \\delta \\rbrace ,$ so that $P_Z(B_0) > 0$ and $P_Z(C_0) > 0$ .",
"By countable subadditivity, there exists $\\delta > 0$ such that $P_Z(B_\\delta ) > 0$ and $P_Z(C_\\delta ) > 0$ .",
"Define $\\epsilon := \\delta \\cdot \\min \\lbrace P_Z(B_\\delta ), P_Z(C_\\delta )\\rbrace \\rbrace > 0$ .",
"Define $g : [0, 1] \\rightarrow \\mathbb {R}$ for all $z \\in [0, 1]$ by $g(z)= \\left\\lbrace \\begin{array}{cc}f(z) - \\frac{\\epsilon }{P_Z(B_\\delta )} & \\text{ if } z \\in B_\\delta \\\\f(z) + \\frac{\\epsilon }{P_Z(C_\\delta )} & \\text{ if } z \\in C_\\delta \\\\f(z) & \\text{ otherwise}.\\end{array}\\right.,$ and note that, by definition of $\\epsilon $ , $B_\\delta $ , and $C_\\delta $ , $g : [0, 1] \\rightarrow [0, 1]$ .",
"Then, $\\operatornamewithlimits{\\mathbb {E}}[g(Z)] - \\operatornamewithlimits{\\mathbb {E}}[f(Z)]= -\\frac{\\epsilon }{P_Z(B_\\delta )} P_Z(B_\\delta ) + \\frac{\\epsilon }{P_Z(C_\\delta )} P_Z(C_\\delta )= 0,$ while $\\operatornamewithlimits{\\mathbb {E}}[Z g(Z)] - \\operatornamewithlimits{\\mathbb {E}}[Z f(Z)]& = -\\operatornamewithlimits{\\mathbb {E}}[Z | Z \\in B_\\delta ] \\frac{\\epsilon }{P_Z(B_\\delta )} P_Z(B_\\delta )+ \\operatornamewithlimits{\\mathbb {E}}[Z | Z \\in C_\\delta ] \\frac{\\epsilon }{P_Z(C_\\delta )} P_Z(C_\\delta ) \\\\& = \\epsilon \\left( \\operatornamewithlimits{\\mathbb {E}}[Z | Z \\in C_\\delta ] - \\operatornamewithlimits{\\mathbb {E}}[Z | Z \\in B_\\delta ] \\right).$ Since $B_\\delta \\subseteq [0, t)$ and $C_\\delta \\subseteq (t, 1]$ , this difference is strictly positive, contradicting the assumption that $f$ optimizes (REF ).",
"Combining Lemma REF with Lemma REF completes the proof of our main result, Theorem REF ."
],
[
"Relative Performance Guarantees in terms of the Generalized Bayes Classifier",
"In this Appendix, we prove Lemmas REF and REF , as well as their consequence, Corollary .",
"Also, in Section REF , we demonstrate, in a few key examples, how to compute the Lipschitz constant used in Corollary .",
"We begin with the proof of Lemma REF , which, at a given threshold $(p, t)$ , bounds the difference between the confusion matrices of the true regression function $\\eta $ and an estimate $\\eta ^{\\prime }$ of $\\eta $ .",
"We restate the result for the reader's convenience: lemma:approximationerrorLemma REF Let $p,t \\in [0, 1]$ and let $\\eta , \\eta ^{\\prime } : \\mathcal {X}\\rightarrow [0, 1]$ .",
"Then, $\\left\\Vert C_{\\widehat{Y}_{p,t,\\eta }} - C_{\\widehat{Y}_{p,t,\\eta ^{\\prime }}} \\right\\Vert _\\infty \\le \\operatornamewithlimits{\\mathbb {P}}\\left[ |\\eta (X) - t| \\le \\left\\Vert \\eta - \\eta ^{\\prime }\\right\\Vert _\\infty \\right].$ For the true negative rate, we have $\\left| \\text{TN}_{\\widehat{Y}_{p,t,\\eta }} - \\text{TN}_{\\widehat{Y}_{p,t,\\eta ^{\\prime }}} \\right|& = \\left| \\operatornamewithlimits{\\mathbb {P}}\\left[ Y = 0, \\eta ^{\\prime }(X) \\le t < \\eta (X) \\right]- \\operatornamewithlimits{\\mathbb {P}}[ Y = 0, \\eta (X) \\le t < \\eta ^{\\prime }(X)] \\right| \\\\& \\le \\operatornamewithlimits{\\mathbb {P}}\\left[ |\\eta (X) - t| \\le \\Vert \\eta - \\eta ^{\\prime }\\Vert _\\infty \\right].$ This type of inequality is standard and follows from the fact that, if $t$ lies between $\\eta $ and $\\eta ^{\\prime }$ , then the difference of $\\eta $ and $t$ is necessarily less than $\\eta $ and $\\eta ^{\\prime }$ .",
"Repeating this calculation for the true positive, false positive, and false negative rates gives (REF ).",
"Note that, in the presence of degree $r$ Uniform Class Imbalance (see Section REF ), one can obtain a tighter error bound $r\\operatornamewithlimits{\\mathbb {P}}\\left[ |\\eta (X) - t| \\le \\Vert \\eta - \\eta ^{\\prime }\\Vert _\\infty \\right]$ for the true positive and false negative rates because, for all $x \\in \\mathcal {X}$ , $\\operatornamewithlimits{\\mathbb {P}}[Y = 1|X = x] \\le r$ .",
"However, the weaker bound (REF ) simplifies the exposition.",
"We now turn to proving Lemma REF , which we use to bound the maximum difference between the empirical and true confusion matrices of a regression-thresholding classifier over thresholds $(p, t)$ .",
"Specifically, we will use this result to bound the difference in confusion matrices between the optimal threshold $(p^*, t^*)$ and the threshold $(\\widehat{p}, \\widehat{t})$ selected by maximizing the empirical CMM.",
"We actually prove a more general version of Lemma REF , for arbitrary classifiers, based on the following definition: Definition 22 (Stochastic Growth Function) Let $\\mathcal {F}$ be a family of $[0, 1]$ -valued functions on $\\mathcal {X}$ .",
"The stochastic growth function $\\Pi _\\mathcal {F}: \\mathbb {N} \\rightarrow \\mathbb {N}$, defined by $\\Pi _\\mathcal {F}(n) := \\max _{\\begin{array}{c}x_1,...,x_n \\in \\mathcal {X},\\\\z_1,...,z_n \\in [0, 1]\\end{array}} \\left| \\left\\lbrace \\left( 1\\lbrace f(x_i) > z_i\\rbrace \\right)_{i = 1}^n : f \\in \\mathcal {F}\\right\\rbrace \\right|\\quad \\text{ for all } \\quad n \\in \\mathbb {N},$ is the maximum number of distinct classifications of $n$ points $x_1,...,x_n$ by a stochastic classifier $\\widehat{Y}$ with $(x \\mapsto \\operatornamewithlimits{\\mathbb {E}}[\\widehat{Y}(x)]) \\in \\mathcal {F}$ and randomness given by $z_1,...,z_n$ .",
"Definition REF generalizes the growth function [42], a classical measure of the complexity of a hypothesis class originally due to [56], to non-deterministic classifiers.",
"Importantly for our purposes, one can easily bound the stochastic growth function of regression-thresholding classifiers: Example 23 (Stochastic Growth Function of Regression-Thresholding Classifiers) Suppose $\\mathcal {F}= \\left\\lbrace f : \\mathcal {X}\\rightarrow [0, 1] | \\text{ for some } p, t \\in [0, 1], f(x) = p \\cdot 1\\lbrace \\eta (x) = t\\rbrace + 1\\lbrace \\eta (x) > t\\rbrace \\text{ for all } x \\in \\mathcal {X}\\right\\rbrace ,$ so that $\\lbrace \\widehat{Y}_{f, \\eta } : f \\in \\mathcal {F}\\rbrace $ is the class of regression-thresholding classifiers.",
"Any set of points $(x_1,z_1),...,(x_n,z_n)$ , can be sorted in increasing order by $\\eta (x)$ 's, breaking ties in decreasing order by $z$ 's.",
"Having sorted the points in this way, $\\lbrace f(x) > z\\rbrace = 0$ for the first $j$ points and $\\lbrace f(x) > z\\rbrace = 1$ for the remaining $n - j$ points, for some $j \\in [n] \\cup \\lbrace 0\\rbrace $ .",
"Thus, $\\Pi _\\mathcal {F}(n) = n + 1$ .",
"We will now prove the following result, from which, together with Example REF , Lemma REF follows immediately: lemma:estimationerrorLemma REF (Generalized Version) Let $\\mathcal {F}$ be a family of $[0, 1]$ -valued functions on $\\mathcal {X}$ .",
"Then, with probability at least $1 - \\delta $ , $\\sup _{f \\in \\mathcal {F}} \\left\\Vert \\widehat{C}_{\\widehat{Y}_f} - C_{\\widehat{Y}_f} \\right\\Vert _\\infty \\le \\sqrt{\\frac{8}{n} \\log \\frac{32\\Pi _\\mathcal {F}(2n)}{\\delta }}.$ Before proving Lemma REF , we note a standard symmetrization lemma, which allows us to replace the expectation of $\\widehat{\\text{TN}}_{\\widehat{Y}_{p,t,\\eta }}$ with its value on an independent, identically distributed “ghost sample”.",
"Lemma 24 (Symmetrization; Lemma 2 of [8]) Let $X$ and $X^{\\prime }$ be independent realizations of a random variable with respect to which $\\mathcal {F}$ is a family of integrable functions.",
"Then, for any $\\epsilon > 0$ , $\\operatornamewithlimits{\\mathbb {P}}\\left[ \\sup _{f \\in \\mathcal {F}} f(X) - \\operatornamewithlimits{\\mathbb {E}}f(X) > \\epsilon \\right]\\le 2\\operatornamewithlimits{\\mathbb {P}}\\left[ \\sup _{f \\in \\mathcal {F}} f(X) - f(X^{\\prime }) > \\frac{\\epsilon }{2} \\right].$ We now use this lemma to prove Lemma REF .",
"To facilitate analyzing the stochastic aspect of the classifier $\\widehat{Y}_{f,\\eta }$ , let $Z_1,...,Z_n \\stackrel{IID}{\\sim } \\operatorname{Uniform}([0, 1])$ , such that $\\widehat{Y}_{f,\\eta }(X_i) = 1\\lbrace Z_i < f(\\eta ((X_i))\\rbrace $ .",
"Now suppose that we have a ghost sample $(X_1^{\\prime },Y_1^{\\prime },Z_1^{\\prime }),...,(X_n^{\\prime },Y_n^{\\prime },Z_n^{\\prime })$ .",
"Let $\\widehat{\\text{TN}}^{\\prime }_{\\widehat{Y}_{f,\\eta }}$ denote the empirical true negative rate computed on this ghost sample, and let $\\widehat{\\text{TN}}^{(i)}_{\\widehat{Y}_{f,\\eta }}$ denote the empirical true negative rate computed on $(X_1,Y_1,Z_1),...,(X_{i-1},Y_{i-1}Z_{i-1}),(X_i^{\\prime },Y_i^{\\prime },Z_i^{\\prime }),(X_{i+1},Y_{i+1},Z_{i+1}),...(X_n,Y_n,Z_n)$ (i.e., replacing only the $i^{th}$ sample with its ghost).",
"By the Symmetrization Lemma, $\\operatornamewithlimits{\\mathbb {P}}\\left[ \\sup _{f \\in \\mathcal {F}} \\widehat{\\text{TN}}_{\\widehat{Y}_{f,\\eta }} - \\operatornamewithlimits{\\mathbb {E}}\\widehat{\\text{TN}}_{\\widehat{Y}_{f,\\eta }} > \\epsilon \\right]& \\le 2\\operatornamewithlimits{\\mathbb {P}}\\left[ \\sup _{f \\in \\mathcal {F}} \\widehat{\\text{TN}}_{\\widehat{Y}_{f,\\eta }} - \\widehat{\\text{TN}}^{\\prime }_{\\widehat{Y}_{f,\\eta }} > \\epsilon /2 \\right] \\\\& \\le 2\\Pi _\\mathcal {F}(2n) \\sup _{f \\in \\mathcal {F}} \\operatornamewithlimits{\\mathbb {P}}\\left[ \\widehat{\\text{TN}}_{\\widehat{Y}_{f,\\eta }} - \\widehat{\\text{TN}}^{\\prime }_{\\widehat{Y}_{f,\\eta }} > \\epsilon /2 \\right] \\\\& \\le 4\\Pi _\\mathcal {F}(2n) \\sup _{f \\in \\mathcal {F}} \\operatornamewithlimits{\\mathbb {P}}\\left[ \\widehat{\\text{TN}}_{\\widehat{Y}_{f,\\eta }} - \\operatornamewithlimits{\\mathbb {E}}\\widehat{\\text{TN}}_{\\widehat{Y}_{f,\\eta }} > \\epsilon /4 \\right],$ where the second inequality is a union bound over the $\\Pi _\\mathcal {F}(2n)$ distinct classifications of $2n$ points that can be assigned by $\\widehat{Y}_{f,\\eta }$ with $f \\in \\mathcal {F}$ , and the last inequality is from the fact that $\\widehat{\\text{TN}}_{\\widehat{Y}_{f,\\eta }}$ and $\\widehat{\\text{TN}}^{\\prime }_{\\widehat{Y}_{f,\\eta }}$ are identically distributed and the algebraic fact that, if $a - b > \\epsilon $ , then either $a - c > \\epsilon /2$ or $b - c > \\epsilon /2$ .",
"For any particular $f \\in \\mathcal {F}$ , by McDiarmid's inequality [39], $\\operatornamewithlimits{\\mathbb {P}}\\left[ \\widehat{\\text{TN}}_{\\widehat{Y}_{f,\\eta }} - \\operatornamewithlimits{\\mathbb {E}}\\widehat{\\text{TN}}_{\\widehat{Y}_{f,\\eta }} > \\epsilon /4 \\right]\\le e^{-n\\epsilon ^2/8},$ since, for any $i \\in [n]$ , $\\left| \\widehat{\\text{TN}}_{\\widehat{Y}_{f,\\eta }} - \\widehat{\\text{TN}}^{(i)}_{\\widehat{Y}_{f,\\eta }} \\right|= \\frac{1}{n} \\left| 1\\left\\lbrace Y_i = \\widehat{Y}_{f,\\eta }(X_i) = 0 \\right\\rbrace - 1\\left\\lbrace Y_i^{\\prime } = \\widehat{Y}_{f,\\eta }(X_i^{\\prime }) = 0 \\right\\rbrace \\right|\\le \\frac{1}{n}.$ Plugging Inequality (REF ) into Inequality (REF ) gives $\\operatornamewithlimits{\\mathbb {P}}\\left[ \\sup _{f \\in \\mathcal {F}} \\widehat{\\text{TN}}_{\\widehat{Y}_{f,\\eta }} - \\operatornamewithlimits{\\mathbb {E}}\\widehat{\\text{TN}}_{\\widehat{Y}_{f,\\eta }} > \\epsilon \\right]\\le 4 \\Pi _\\mathcal {F}(2n) e^{-n\\epsilon ^2/8}.$ Repeating this argument with $-\\widehat{\\text{TN}}$ instead of $\\widehat{\\text{TN}}$ , as well as with $\\widehat{\\text{TP}}$ , $\\widehat{\\text{FN}}$ , $\\widehat{\\text{FP}}$ and their negatives, and taking a union bound over these 8 cases, gives the desired result.",
"Finally, we will use these two lemmas, together with the margin and Lipschitz assumptions, to prove Corollary , which bounds the sub-optimality of the trained classifier, relative to the generalized Bayes classifier, in terms of the desired CMM.",
"corr:CMMerrordecompositionCorollary Let $\\eta : \\mathcal {X}\\rightarrow [0, 1]$ denote the true regression function, and let $\\widehat{\\eta }: \\mathcal {X}\\rightarrow [0, 1]$ denote any empirical regressor.",
"Let $\\left( \\widehat{p}, \\widehat{t} \\right) := \\operatornamewithlimits{\\arg \\!\\max }_{(p, t) \\in [0, 1]^2} M \\left( \\widehat{C}_{\\widehat{Y}_{p,t,\\widehat{\\eta }}} \\right)\\quad \\text{ and } \\quad \\left( p^*, t^* \\right) := \\operatornamewithlimits{\\arg \\!\\max }_{(p, t) \\in [0, 1]^2} M \\left(C_{\\widehat{Y}_{p,t,\\eta }} \\right)$ denote the empirically selected and true optimal thresholds, respectively.",
"Suppose that $M$ is Lipschitz continuous with constant $L_M$ with respect to the uniform ($\\mathcal {L}_\\infty $ ) metric on $.",
"Finally, suppose that $ PX$ and $$ satisfies a $ (C, )$-margin condition around $ t*$.",
"Then, with probability at least $ 1 - $,{\\begin{@align}{1}{-1}M\\left(C_{\\widehat{Y}_{p,t,\\eta }}\\left(p^*, t^*\\right)\\right) - M\\left(C_{\\widehat{Y}_{p,t,\\widehat{\\eta }}}\\left(\\widehat{p}, \\widehat{t}\\right)\\right)& \\le L_M \\left( C\\left\\Vert \\eta - \\widehat{\\eta }\\right\\Vert _\\infty ^\\beta + 2 \\sqrt{\\frac{8}{n} \\log \\frac{32(2n + 1)}{\\delta }} \\right).\\end{@align}}$ First, note that $M\\left(C_{\\widehat{Y}_{p^*, t^*,\\eta }}\\right) - M\\left(C_{\\widehat{Y}_{\\widehat{p}, \\widehat{t},\\widehat{\\eta }}}\\right)& \\le M\\left(C_{\\widehat{Y}_{p^*, t^*,\\eta }}\\right) - M\\left(C_{\\widehat{Y}_{p^*, t^*,\\widehat{\\eta }}}\\right) \\\\& + M\\left(C_{\\widehat{Y}_{p^*, t^*,\\widehat{\\eta }}}\\right) - M\\left(\\widehat{C}_{\\widehat{Y}_{p^*, t^*,\\widehat{\\eta }}}\\right) \\\\& + M\\left(\\widehat{C}_{\\widehat{Y}_{\\widehat{p}, \\widehat{t},\\widehat{\\eta }}}\\right) - M\\left(C_{\\widehat{Y}_{\\widehat{p}, \\widehat{t},\\widehat{\\eta }}}\\right),$ since, by definition of $(\\widehat{p}, \\widehat{t})$ , $M\\left(\\widehat{C}_{\\widehat{Y}_{p^*,t^*,\\widehat{\\eta }}}\\right) - M\\left(\\widehat{C}_{\\widehat{Y}_{\\widehat{p},\\widehat{t},\\widehat{\\eta }}} \\right) \\le 0;$ this term sits between the second and third lines above.",
"By the Lipschitz assumption, $& M\\left(C_{\\widehat{Y}_{p^*, t^*,\\eta }}\\right) - M\\left(C_{\\widehat{Y}_{\\widehat{p}, \\widehat{t},\\widehat{\\eta }}}\\right)\\\\&\\le L_M \\bigg ( \\left\\Vert C_{\\widehat{Y}_{p^*, t^*,\\eta }} - C_{\\widehat{Y}_{p^*, t^*,\\widehat{\\eta }}} \\right\\Vert _\\infty \\\\ & \\hspace{26.19995pt}+ \\left\\Vert C_{\\widehat{Y}_{p^*, t^*,\\widehat{\\eta }}} - \\widehat{C}_{\\widehat{Y}_{p^*, t^*,\\widehat{\\eta }}} \\right\\Vert _\\infty \\\\& \\hspace{26.19995pt}+ \\left\\Vert \\widehat{C}_{\\widehat{Y}_{\\widehat{p}, \\widehat{t},\\widehat{\\eta }}} - C_{\\widehat{Y}_{\\widehat{p}, \\widehat{t},\\widehat{\\eta }}} \\right\\Vert _\\infty \\bigg ).$ Corollary follows by applying Lemma REF and the $(C,\\beta )$ -margin condition to (REF ) and applying Lemma REF to both terms () and ()."
],
[
"Lipschitz constants for some common CMMs",
"Corollary assumed that the CMM $M$ was Lipschitz continuous with respect to the $\\sup $ -norm on confusion matrices.",
"In this section, we show how to compute appropriate Lipschitz constants for several simple example CMMs.",
"We begin with a simple example: Example 25 (Weighted Accuracy) For a fixed $w \\in (0, 1)$ , the $w$ -weighted accuracy is given by $M(C) = (1 - w) \\text{TP}+ w \\text{TN}$ .",
"In this case, $M$ clearly has Lipschitz constant $L_M = \\max \\lbrace w, 1 - w\\rbrace $ .",
"For the remainder of this section (only), we will use $P := \\operatornamewithlimits{\\mathbb {E}}[Y]$ to denote the positive rate of the true labels and $\\widehat{P} := \\frac{1}{n} \\sum _{i = 1}^n Y_i$ to denote the empirical positive rate of the true labels.",
"Many CMMs of interest, such as Recall and $F_\\beta $ scores, are not Lipschitz continuous over all of $.",
"Fortunately, inspecting the proof of Corollary~\\ref {corr:CMM_error_decomposition}, it suffices for the CMM $ M$ to be Lipschitz continuous on the line segments between three specific pairs of confusion matrices, given in Eqs.~(\\ref {term:approximation_error}), (\\ref {term:estimation_error_term1}), and (\\ref {term:estimation_error_term2}).",
"Deriving the appropriate Lipschitz constants is a bit more complex, and we demonstrate here how to derive them for the specific CMMs of Recall and $ F$ scores.$ Of the six confusion matrices in Eqs.",
"(REF ), (), and (), four are true confusion matrices.",
"These four matrices have the same positive rate $\\text{TP}+ \\text{FN}= P$ , which is a function of the true distribution of labels.",
"The remaining two matrices are empirical confusion matrices, and hence have the positive rate $\\widehat{\\text{TP}}+ \\widehat{\\text{FN}}= \\widehat{P}$ , which is a function of the data.",
"By a multiplicative Chernoff bound, with probability at least $1 - e^{-nP/8}$ , $\\widehat{P} \\ge P/2$ .",
"Thus, with high probability, it suffices for the CMM $M$ to be Lipschitz continuous over confusion matrices with positive rate at least $P/2$ .",
"For Recall and $F_\\beta $ scores, this gives the following Lipschitz constants: Example 26 (Recall) Recall is given by $M(C) = \\frac{\\text{TP}}{\\text{TP}+ \\text{FN}} = \\frac{\\text{TP}}{P}$ .",
"Thus, $M$ is Lipschitz continuous with constant $L_M = \\frac{2}{P}$ over the confusion matrices in Eqs.",
"(REF ), (), and ().",
"Example 27 ($F_\\beta $ Score) For $\\beta \\in (0, \\infty )$ , the $F_\\beta $ score is given by $M(C)= \\frac{(1 + \\beta ^2) \\text{TP}}{(1 + \\beta ^2) \\text{TP}+ \\text{FP}+ \\beta ^2 \\text{FN}}= \\frac{(1 + \\beta ^2) \\text{TP}}{\\text{TP}+ \\text{FP}+ \\beta ^2 P}.$ Hence, $\\left| \\frac{\\partial }{\\partial \\text{TP}} M(C) \\right|= (1 + \\beta ^2) \\frac{\\text{FP}+ \\beta ^2 P}{\\left( \\text{TP}+ \\text{FP}+ \\beta ^2 P \\right)^2}\\le \\frac{1 + \\beta ^2}{\\beta ^2 P},$ while, since $\\text{TP}\\le P$ , $\\left| \\frac{\\partial }{\\partial \\text{FP}} M(C) \\right|= (1 + \\beta ^2) \\frac{\\text{TP}}{\\left( \\text{TP}+ \\text{FP}+ \\beta ^2 P \\right)^2}\\le \\frac{1 + \\beta ^2}{\\beta ^4 P}.$ Hence, $M$ is Lipschitz continuous with constant $\\frac{2(1 + \\beta ^2)}{P} \\max \\left\\lbrace \\beta ^{-2}, \\beta ^{-4} \\right\\rbrace $ over the confusion matrices in Eqs.",
"(REF ), (), and ().",
"As Examples REF and REF demonstrate, the Lipschitz constants of many CMMs can become large when the proportion $P$ is positive samples is small.",
"In particular, when $P \\in O \\left( \\sqrt{\\frac{\\log n}{n}} \\right)$ , the $\\asymp L_M \\sqrt{\\frac{\\log (n/\\delta )}{n}}$ term of Corollary fails to vanish as $n \\rightarrow \\infty $ .",
"We believe that some loss of convergence rate is inevitable if $P \\rightarrow 0$ as $n \\rightarrow \\infty $ , due to the inherent instability of such metrics, but further work is needed to understand if the rates given by Corollary are optimal under these metrics.",
"See also [17] for detailed discussion of Lipschitz constants of many common CMMs."
],
[
"Bounds on Uniform Error of the Nearest Neighbor Regressor",
"In this appendix, we prove our upper bound on the uniform risk of the $k$ NN regressor (Theorem REF ), as well as the corresponding minimax lower bound (Theorem REF )."
],
[
"Upper Bounds",
"In this section, we prove Theorem REF , our upper bound on the uniform error of the $k$ -NN regressor.",
"The main result is restated below: thm:unifconvergenceTheorem REF Under Assumptions REF and REF , whenever $k / n \\le p_*(\\epsilon ^*)^d / 2$ , for any $\\delta > 0$ , with probability at least $1 - N\\left( \\left( 2k / (p_* n) \\right)^{1/d} \\right) e^{-k/4} - \\delta $ , we have the uniform error bound $\\left\\Vert \\eta - \\widehat{\\eta }\\right\\Vert _\\infty \\le 2^\\alpha Lr\\left( \\frac{2k}{p_* n} \\right)^{\\alpha /d}+ \\frac{2}{3k} \\log \\frac{2 S(n)}{\\delta } + \\sqrt{\\frac{2r}{k} \\log \\frac{2 S(n)}{\\delta }}.$ For any $x \\in \\mathcal {X}$ , let $\\widetilde{\\eta }_k(x) := \\frac{1}{k} \\sum _{j = 1}^k \\eta (X_{\\sigma _j(x)})$ denote the mean of the true regression function over the $k$ nearest neighbors of $x$ .",
"By the triangle inequality, $\\Vert \\eta - \\widehat{\\eta }\\Vert _\\infty \\le \\Vert \\eta - \\widetilde{\\eta }_k\\Vert _\\infty + \\Vert \\widetilde{\\eta }_k - \\widehat{\\eta }\\Vert _\\infty ,$ wherein $\\Vert \\eta - \\widetilde{\\eta }_k\\Vert _\\infty $ captures bias due to smoothing and $\\Vert \\widetilde{\\eta }_k - \\widehat{\\eta }\\Vert _\\infty $ captures variance due to label noise.",
"We separately show that, with probability at least $1 - N \\left( \\left( \\frac{2k}{p_* n} \\right)^{1/d} \\right) e^{-k/4}$ , $\\left\\Vert \\eta - \\widetilde{\\eta }_k \\right\\Vert _\\infty \\le 2^\\alpha Lr \\left( \\frac{2k}{p_* n} \\right)^{\\alpha /d},$ and that, with probability at least $1 - \\delta $ , $\\Vert \\widetilde{\\eta }_k - \\widehat{\\eta }\\Vert _\\infty \\le \\frac{2}{3k} \\log \\frac{2 S(n)}{\\delta }+ \\sqrt{\\frac{2r}{k} \\log \\frac{2 S(n)}{\\delta }}.$"
],
[
"Bounding the smoothing bias",
"Fix some $r > 0$ to be determined, and let $\\lbrace B_r(z_1),...,B_r(z_{N(r)})\\rbrace $ be a covering of $(\\mathcal {X}, \\rho )$ by $N(r)$ balls of radius $r$ , with centers $z_1,...,z_{N(r)} \\in \\mathcal {X}$ .",
"By the lower bound assumption on $P_X$ , each $P_X(B_r(z_j)) \\ge p_* r^d$ .",
"Therefore, by a multiplicative Chernoff bound, with probability at least $1 - N(r) e^{-p_* n r^d/8}$ , each $B_r(z_j)$ contains at least $p_* n r^d/2$ samples.",
"In particular, if $r \\ge \\left( \\frac{2k}{p_* n} \\right)^{1/d}$ , then each $B_k$ contains at least $k$ samples, and it follows that, for every $x \\in \\mathcal {X}$ , $\\rho (x, X_{\\sigma _k(x)}) \\le 2r$ .",
"Thus, by Hölder continuity of $\\eta $ , $\\left| \\eta (x) - \\widetilde{\\eta }_k(x) \\right|= \\left| \\eta (x) - \\frac{1}{k} \\sum _{j = 1}^k \\eta (X_{\\sigma _j(x)}) \\right|\\le \\frac{1}{k} \\sum _{j = 1}^k \\left| \\eta (x) - \\eta (X_{\\sigma _j(x)}) \\right|\\le L(2r)^\\alpha .$ Finally, if $\\frac{k}{n} \\le \\frac{p_*}{2} (r^*)^d$ , then we can let $r = \\left( \\frac{2k}{p_* n} \\right)^{1/d}$ ."
],
[
"Bounding variance due to label noise",
"Let $\\Sigma := \\lbrace \\sigma (x) \\in [n]^k : x \\in \\mathcal {X}\\rbrace $ denote the set of possible $k$ -nearest neighbor index sets.",
"One can check from the definition of the shattering coefficient that $|\\Sigma | \\le S(n)$ .",
"For any $\\sigma \\in [n]^k$ , let $Z_\\sigma := \\sum _{j = 1}^k Y_{\\sigma _j}$ and let $\\mu _\\sigma := \\operatornamewithlimits{\\mathbb {E}}\\left[ Z_\\sigma \\right]$ .",
"Note that the conditional random variables $Y_{\\sigma _j}|X_1,...,X_n$ have conditionally independent Bernoulli distributions with means $\\operatornamewithlimits{\\mathbb {E}}[Y_{\\sigma _j}|X_1,...,X_n] = \\eta (X_{\\sigma _j})$ and variances $\\operatornamewithlimits{\\mathbb {E}}\\left[ \\left( Y_{\\sigma _j} - \\eta (X_{\\sigma _j}) \\right)^2 |X_1,...,X_n \\right] = \\eta (X_{\\sigma _j}) (1 - \\eta (X_{\\sigma _j})) \\le r$ .",
"Therefore, by Bernstein's inequality (Eq.",
"(2.10) of [7]), for any $\\epsilon > 0$ , $\\operatornamewithlimits{\\mathbb {P}}\\left[ |Z_\\sigma /k - \\mu _\\sigma | \\ge \\epsilon \\right] \\le 2 \\exp \\left( -\\frac{k\\epsilon ^2}{2(r + \\epsilon /3)} \\right).$ Moreover, for any $x \\in \\mathcal {X}$ , $\\mu _{\\sigma (x)} = \\widetilde{\\eta }_k(x)$ and $Z_{\\sigma (x)}/k = \\widehat{\\eta }(x)$ .",
"Hence, by a union bound over $\\sigma $ in $\\Sigma $ , $\\operatornamewithlimits{\\mathbb {P}}\\left( \\sup _{x \\in \\mathcal {X}} \\left| \\widetilde{\\eta }_k(x) - \\widehat{\\eta }(x) \\right| > \\epsilon | X_1,...,X_n \\right)& = \\operatornamewithlimits{\\mathbb {P}}\\left( \\sup _{x \\in \\mathcal {X}} \\left| \\mu _{\\sigma (x)} - Z_{\\sigma (x)}/k\\right| > \\epsilon | X_1,...,X_n \\right) \\\\& \\le \\operatornamewithlimits{\\mathbb {P}}\\left( \\sup _{\\sigma \\in \\Sigma } \\left| \\mu _\\sigma - Z_\\sigma /k\\right| > \\epsilon | X_1,...,X_n \\right) \\\\& \\le |\\Sigma | \\sup _{\\sigma \\in \\Sigma } \\operatornamewithlimits{\\mathbb {P}}\\left( \\left| \\mu _\\sigma - Z_\\sigma /k\\right| > \\epsilon | X_1,...,X_n \\right) \\\\& \\le 2S(n) \\exp \\left( -\\frac{k\\epsilon ^2}{2(r + \\epsilon /3)} \\right).$ Since the right-hand side is independent of $X_1,...,X_n$ , the unconditional bound $\\operatornamewithlimits{\\mathbb {P}}\\left( \\sup _{x \\in \\mathcal {X}} \\left\\Vert \\widetilde{\\eta }_k(x) - \\widehat{\\eta }(x) \\right\\Vert _\\infty > \\epsilon \\right)\\le 2S(n) \\exp \\left( -\\frac{k\\epsilon ^2}{2(r + \\epsilon /3)} \\right)$ follows.",
"Plugging in $\\epsilon = \\frac{1}{3k} \\log \\frac{2 S(n)}{\\delta } + \\sqrt{\\left( \\frac{1}{3k} \\log \\frac{2 S(n)}{\\delta } \\right)^2 + \\frac{2r}{k} \\log \\frac{2 S(n)}{\\delta }}\\le \\frac{2}{3k} \\log \\frac{2 S(n)}{\\delta } + \\sqrt{\\frac{2r}{k} \\log \\frac{2 S(n)}{\\delta }}$ and simplifying gives the final result.",
"Recall that there is a small (polylogarithmic in $r$ ) gap between our upper and lower bounds.",
"We believe that the upper bound may be slightly loose, and that this might be tightened by using a stronger concentration inequality, such as Bennett's inequality [4], instead of Bernstein's inequality in Inequality (REF )."
],
[
"Lower Bounds",
"In this section, we prove Theorem REF , our lower bound on the minimax uniform error of estimating a Hölder continuous regression function.",
"We use a standard approach based on the following version of Fano's lemma: Lemma 28 (Fano's Lemma; Simplified Form of Theorem 2.5 of [53]) Fix a family $\\mathcal {P}$ of distributions over a sample space $\\mathcal {X}$ and fix a pseudo-metric $\\rho : \\mathcal {P}\\times \\mathcal {P}\\rightarrow [0,\\infty ]$ over $\\mathcal {P}$ .",
"Suppose there exist $P_0 \\in \\mathcal {P}$ and a set $T \\subseteq \\mathcal {P}$ such that $\\sup _{P \\in T} D_{KL}(P,P_0)\\le \\frac{\\log |T|}{16},$ where $D_{KL} : \\mathcal {P}\\times \\mathcal {P}\\rightarrow [0,\\infty ]$ denotes Kullback-Leibler divergence.",
"Then, $\\inf _{\\widehat{P}} \\sup _{P \\in \\mathcal {P}}\\operatornamewithlimits{\\mathbb {P}}\\left( \\rho (P,\\widehat{P})\\ge \\frac{1}{2} \\inf _{P \\in T} \\rho (P,P_0) \\right) \\ge 1/8,$ where the first $\\inf $ is taken over all estimators $\\widehat{P}$ .",
"Now, we proceed with the proof.",
"We now proceed to construct an appropriate $P_0 \\in \\mathcal {P}$ and $T \\subseteq \\mathcal {P}$ .",
"Let $g : [-1,1]^d \\rightarrow [0,1]$ defined by $g(x) = \\left\\lbrace \\begin{array}{cc}\\exp \\left( 1 - \\frac{1}{1 - \\Vert x\\Vert _2^2} \\right) & \\text{ if } \\Vert x\\Vert _2 < 1 \\\\0 & \\text{ else }\\end{array} \\right.$ denote the standard bump function supported on $[-1,1]^d$ , scaled to have $\\Vert g\\Vert _{\\mathcal {X},\\infty } = 1$ .",
"Since $g$ is infinitely differentiable and compactly supported, it has a finite $\\alpha $ -Hölder semi-norm: $\\Vert g\\Vert _{\\Sigma ^\\alpha }:= \\sup _{\\ell \\in \\mathbb {N}^d : \\Vert \\ell \\Vert _1 \\le \\alpha } \\quad \\sup _{x \\ne y \\in \\mathcal {X}} \\quad \\frac{|g^\\ell (x) - g^\\ell (y)|}{\\Vert x - y\\Vert ^{\\alpha - \\Vert \\ell \\Vert _1}}< \\infty ,$ where $\\ell $ is any $\\lfloor \\beta \\rfloor $ -order multi-index and $g^\\ell $ is the corresponding mixed derivative of $g$ .",
"Define $M := \\left( \\frac{64(2\\alpha + d)nr}{d \\log (nr)} \\right)^{\\frac{1}{2\\alpha + d}} \\ge 1$ , since $r \\ge 1/n$ .",
"For each $m \\in [M]^d$ , define $g_m : \\mathcal {X}\\rightarrow [0,1]$ by $g_m (x) := g\\left( Mx - \\frac{2m - 1_d}{2} \\right),$ so that $\\lbrace g_m : m \\in [M]^d\\rbrace $ is a grid of $M^d$ bump functions with disjoint supports.",
"Let $\\zeta _0 \\equiv \\frac{1}{4}$ denote the constant-$\\frac{1}{4}$ function on $\\mathcal {X}$ .",
"Finally, for each $m \\in [M]^d$ , define $\\zeta _m : \\mathcal {X}\\rightarrow [0,1]$ by $\\zeta _m := \\zeta _0 + \\min \\left\\lbrace \\frac{1}{2}, \\frac{L}{\\Vert g\\Vert _{\\Sigma ^\\alpha }} \\right\\rbrace M^{-\\alpha } g_m.$ Note that, for any $m \\in [M]^d$ , $\\Vert \\zeta _m\\Vert _{\\Sigma ^\\alpha }\\le L M^{-\\alpha } \\frac{\\Vert g_m\\Vert _{\\Sigma ^\\alpha }}{\\Vert g\\Vert _{\\Sigma ^\\alpha }}= L,$ so that $\\zeta _m$ satisfies the Hölder smoothness condition.",
"For any particular $\\eta $ , let $P_\\eta $ denote the joint distribution of $(X, Y)$ .",
"Note that $P_\\zeta (x, 1) = \\zeta (x) \\ge 1/4$ .",
"Moreover, one can check that, for all $x \\ge -2/3$ , $-\\log (1 + x) \\le x^2 - x$ .",
"Hence, for any $x \\in \\mathcal {X}$ , $P_{\\eta _m}(x, 1) \\log \\frac{P_{\\eta _m}(x, 1)}{P_{\\eta }(x, 1)}& = rP_{\\zeta _m}(x, 1) \\log \\frac{P_{\\zeta _m}(x, 1)}{P_{\\zeta }(x, 1)} \\\\& = r\\zeta _m(x) \\log \\frac{\\zeta _m(x)}{\\zeta (x)} \\\\& = -r\\zeta _m(x) \\log \\left( 1 + \\frac{\\zeta (x) - \\zeta _m(x)}{\\zeta _m(x)} \\right) \\\\& \\le r\\zeta _m(x) \\left( \\left( \\frac{\\zeta (x) - \\zeta _m(x)}{\\zeta _m(x)} \\right)^2 - \\frac{\\zeta (x) - \\zeta _m(x)}{\\zeta _m(x)} \\right) \\\\& = r \\left( \\frac{\\left( \\zeta (x) - \\zeta _m(x)\\right)^2 }{\\zeta _m(x)} - \\zeta (x) + \\zeta _m(x) \\right) \\\\& \\le r \\left( 4\\left( \\zeta (x) - \\zeta _m(x)\\right)^2 - \\zeta (x) + \\zeta _m(x) \\right),$ and, similarly, since $P_\\zeta (x,0) = 1 - \\zeta (x) \\ge 1/4$ , $P_{r\\eta _m}(x, 0) \\log \\frac{P_{r\\eta _m}(x, 0)}{P_{r\\eta }(x, 0)}\\le r\\left( 4\\left( \\zeta (x) - \\zeta _m(x)\\right)^2 + \\zeta (x) - \\zeta _m(x) \\right).$ Adding these two terms gives $D_{\\text{KL}}\\left( P_{r\\eta }^n, P_{r\\eta _m}^n \\right)& = n \\left( \\int _\\mathcal {X}P_{r\\eta _m}(x, 0) \\log \\frac{P_{r\\eta }(x, 0)}{P_{r\\eta _m}(x, 0)} \\, dx + \\int _\\mathcal {X}P_{r\\eta _m}(x, 1) \\log \\frac{P_{r\\eta }(x, 1)}{P_{r\\eta _m}(x, 1)} \\, dx \\right) \\\\& \\le 8nr\\int _\\mathcal {X}\\left( \\zeta (x) - \\zeta _m(x)\\right)^2 \\\\& = 8nr\\Vert \\zeta - \\zeta _m\\Vert _2^2 \\\\& \\le 2nr M^{-2\\alpha } \\Vert g_m\\Vert _2^2 \\\\& = 2nr M^{-(2\\alpha + d)} \\Vert g\\Vert _2^2 \\\\& = 2nr \\left( \\left( \\frac{64 (2\\alpha + d) nr}{d \\log (nr)} \\right)^{\\frac{1}{2\\alpha + d}}\\right)^{-(2\\alpha + d)} \\Vert g\\Vert _2^2 \\\\& = \\frac{1}{32} \\frac{d}{2\\alpha + d} \\Vert g\\Vert _2^2 \\log (nr) \\\\& \\le \\frac{1}{16} \\frac{d}{2\\alpha + d} \\left( \\log (nr) - \\log \\log (nr) + \\log \\frac{64 (2\\alpha + d)}{d} \\right)\\\\ &= \\frac{\\log |[M]^d|}{16},$ where the second inequality comes from the definition of $\\zeta _m$ (Eq.",
"REF ) and the third inequality comes from the facts that $\\Vert g\\Vert _2^2 \\le 1$ and $\\log \\log x \\le \\frac{1}{2} \\log x$ for all $x > 1$ .",
"Fano's lemma therefore implies the lower bound $\\inf _{\\widehat{\\eta }} \\sup _{r \\in (0, 1], \\zeta \\in \\Sigma ^\\alpha (L)}\\operatornamewithlimits{\\mathbb {P}}_{\\lbrace (X_i,Y_i)\\rbrace _{i = 1}^n \\sim P_\\eta ^n} \\left( \\left\\Vert r\\zeta - r\\widehat{\\zeta }\\right\\Vert _\\infty \\ge \\frac{1}{2} \\min \\left\\lbrace \\frac{1}{2}, \\frac{L}{\\Vert g\\Vert _{\\Sigma ^\\alpha }} \\right\\rbrace \\left( \\frac{d \\log (nr)}{64(2\\alpha + d) n} \\right)^{\\frac{\\alpha }{2\\alpha + d}} r^\\frac{\\alpha + d}{2\\alpha + d} \\right)\\ge \\frac{1}{8},$ which completes the proof."
],
[
"Bounds on Uniform Error of the Nearest Neighbor Regressor",
"In this appendix, we prove our upper bound on the uniform risk of the $k$ NN regressor (Theorem REF ), as well as the corresponding minimax lower bound (Theorem REF )."
],
[
"Upper Bounds",
"In this section, we prove Theorem REF , our upper bound on the uniform error of the $k$ -NN regressor.",
"The main result is restated below: thm:unifconvergenceTheorem REF Under Assumptions REF and REF , whenever $k / n \\le p_*(\\epsilon ^*)^d / 2$ , for any $\\delta > 0$ , with probability at least $1 - N\\left( \\left( 2k / (p_* n) \\right)^{1/d} \\right) e^{-k/4} - \\delta $ , we have the uniform error bound $\\left\\Vert \\eta - \\widehat{\\eta }\\right\\Vert _\\infty \\le 2^\\alpha Lr\\left( \\frac{2k}{p_* n} \\right)^{\\alpha /d}+ \\frac{2}{3k} \\log \\frac{2 S(n)}{\\delta } + \\sqrt{\\frac{2r}{k} \\log \\frac{2 S(n)}{\\delta }}.$ For any $x \\in \\mathcal {X}$ , let $\\widetilde{\\eta }_k(x) := \\frac{1}{k} \\sum _{j = 1}^k \\eta (X_{\\sigma _j(x)})$ denote the mean of the true regression function over the $k$ nearest neighbors of $x$ .",
"By the triangle inequality, $\\Vert \\eta - \\widehat{\\eta }\\Vert _\\infty \\le \\Vert \\eta - \\widetilde{\\eta }_k\\Vert _\\infty + \\Vert \\widetilde{\\eta }_k - \\widehat{\\eta }\\Vert _\\infty ,$ wherein $\\Vert \\eta - \\widetilde{\\eta }_k\\Vert _\\infty $ captures bias due to smoothing and $\\Vert \\widetilde{\\eta }_k - \\widehat{\\eta }\\Vert _\\infty $ captures variance due to label noise.",
"We separately show that, with probability at least $1 - N \\left( \\left( \\frac{2k}{p_* n} \\right)^{1/d} \\right) e^{-k/4}$ , $\\left\\Vert \\eta - \\widetilde{\\eta }_k \\right\\Vert _\\infty \\le 2^\\alpha Lr \\left( \\frac{2k}{p_* n} \\right)^{\\alpha /d},$ and that, with probability at least $1 - \\delta $ , $\\Vert \\widetilde{\\eta }_k - \\widehat{\\eta }\\Vert _\\infty \\le \\frac{2}{3k} \\log \\frac{2 S(n)}{\\delta }+ \\sqrt{\\frac{2r}{k} \\log \\frac{2 S(n)}{\\delta }}.$"
],
[
"Bounding the smoothing bias",
"Fix some $r > 0$ to be determined, and let $\\lbrace B_r(z_1),...,B_r(z_{N(r)})\\rbrace $ be a covering of $(\\mathcal {X}, \\rho )$ by $N(r)$ balls of radius $r$ , with centers $z_1,...,z_{N(r)} \\in \\mathcal {X}$ .",
"By the lower bound assumption on $P_X$ , each $P_X(B_r(z_j)) \\ge p_* r^d$ .",
"Therefore, by a multiplicative Chernoff bound, with probability at least $1 - N(r) e^{-p_* n r^d/8}$ , each $B_r(z_j)$ contains at least $p_* n r^d/2$ samples.",
"In particular, if $r \\ge \\left( \\frac{2k}{p_* n} \\right)^{1/d}$ , then each $B_k$ contains at least $k$ samples, and it follows that, for every $x \\in \\mathcal {X}$ , $\\rho (x, X_{\\sigma _k(x)}) \\le 2r$ .",
"Thus, by Hölder continuity of $\\eta $ , $\\left| \\eta (x) - \\widetilde{\\eta }_k(x) \\right|= \\left| \\eta (x) - \\frac{1}{k} \\sum _{j = 1}^k \\eta (X_{\\sigma _j(x)}) \\right|\\le \\frac{1}{k} \\sum _{j = 1}^k \\left| \\eta (x) - \\eta (X_{\\sigma _j(x)}) \\right|\\le L(2r)^\\alpha .$ Finally, if $\\frac{k}{n} \\le \\frac{p_*}{2} (r^*)^d$ , then we can let $r = \\left( \\frac{2k}{p_* n} \\right)^{1/d}$ ."
],
[
"Bounding variance due to label noise",
"Let $\\Sigma := \\lbrace \\sigma (x) \\in [n]^k : x \\in \\mathcal {X}\\rbrace $ denote the set of possible $k$ -nearest neighbor index sets.",
"One can check from the definition of the shattering coefficient that $|\\Sigma | \\le S(n)$ .",
"For any $\\sigma \\in [n]^k$ , let $Z_\\sigma := \\sum _{j = 1}^k Y_{\\sigma _j}$ and let $\\mu _\\sigma := \\operatornamewithlimits{\\mathbb {E}}\\left[ Z_\\sigma \\right]$ .",
"Note that the conditional random variables $Y_{\\sigma _j}|X_1,...,X_n$ have conditionally independent Bernoulli distributions with means $\\operatornamewithlimits{\\mathbb {E}}[Y_{\\sigma _j}|X_1,...,X_n] = \\eta (X_{\\sigma _j})$ and variances $\\operatornamewithlimits{\\mathbb {E}}\\left[ \\left( Y_{\\sigma _j} - \\eta (X_{\\sigma _j}) \\right)^2 |X_1,...,X_n \\right] = \\eta (X_{\\sigma _j}) (1 - \\eta (X_{\\sigma _j})) \\le r$ .",
"Therefore, by Bernstein's inequality (Eq.",
"(2.10) of [7]), for any $\\epsilon > 0$ , $\\operatornamewithlimits{\\mathbb {P}}\\left[ |Z_\\sigma /k - \\mu _\\sigma | \\ge \\epsilon \\right] \\le 2 \\exp \\left( -\\frac{k\\epsilon ^2}{2(r + \\epsilon /3)} \\right).$ Moreover, for any $x \\in \\mathcal {X}$ , $\\mu _{\\sigma (x)} = \\widetilde{\\eta }_k(x)$ and $Z_{\\sigma (x)}/k = \\widehat{\\eta }(x)$ .",
"Hence, by a union bound over $\\sigma $ in $\\Sigma $ , $\\operatornamewithlimits{\\mathbb {P}}\\left( \\sup _{x \\in \\mathcal {X}} \\left| \\widetilde{\\eta }_k(x) - \\widehat{\\eta }(x) \\right| > \\epsilon | X_1,...,X_n \\right)& = \\operatornamewithlimits{\\mathbb {P}}\\left( \\sup _{x \\in \\mathcal {X}} \\left| \\mu _{\\sigma (x)} - Z_{\\sigma (x)}/k\\right| > \\epsilon | X_1,...,X_n \\right) \\\\& \\le \\operatornamewithlimits{\\mathbb {P}}\\left( \\sup _{\\sigma \\in \\Sigma } \\left| \\mu _\\sigma - Z_\\sigma /k\\right| > \\epsilon | X_1,...,X_n \\right) \\\\& \\le |\\Sigma | \\sup _{\\sigma \\in \\Sigma } \\operatornamewithlimits{\\mathbb {P}}\\left( \\left| \\mu _\\sigma - Z_\\sigma /k\\right| > \\epsilon | X_1,...,X_n \\right) \\\\& \\le 2S(n) \\exp \\left( -\\frac{k\\epsilon ^2}{2(r + \\epsilon /3)} \\right).$ Since the right-hand side is independent of $X_1,...,X_n$ , the unconditional bound $\\operatornamewithlimits{\\mathbb {P}}\\left( \\sup _{x \\in \\mathcal {X}} \\left\\Vert \\widetilde{\\eta }_k(x) - \\widehat{\\eta }(x) \\right\\Vert _\\infty > \\epsilon \\right)\\le 2S(n) \\exp \\left( -\\frac{k\\epsilon ^2}{2(r + \\epsilon /3)} \\right)$ follows.",
"Plugging in $\\epsilon = \\frac{1}{3k} \\log \\frac{2 S(n)}{\\delta } + \\sqrt{\\left( \\frac{1}{3k} \\log \\frac{2 S(n)}{\\delta } \\right)^2 + \\frac{2r}{k} \\log \\frac{2 S(n)}{\\delta }}\\le \\frac{2}{3k} \\log \\frac{2 S(n)}{\\delta } + \\sqrt{\\frac{2r}{k} \\log \\frac{2 S(n)}{\\delta }}$ and simplifying gives the final result.",
"Recall that there is a small (polylogarithmic in $r$ ) gap between our upper and lower bounds.",
"We believe that the upper bound may be slightly loose, and that this might be tightened by using a stronger concentration inequality, such as Bennett's inequality [4], instead of Bernstein's inequality in Inequality (REF )."
],
[
"Lower Bounds",
"In this section, we prove Theorem REF , our lower bound on the minimax uniform error of estimating a Hölder continuous regression function.",
"We use a standard approach based on the following version of Fano's lemma: Lemma 28 (Fano's Lemma; Simplified Form of Theorem 2.5 of [53]) Fix a family $\\mathcal {P}$ of distributions over a sample space $\\mathcal {X}$ and fix a pseudo-metric $\\rho : \\mathcal {P}\\times \\mathcal {P}\\rightarrow [0,\\infty ]$ over $\\mathcal {P}$ .",
"Suppose there exist $P_0 \\in \\mathcal {P}$ and a set $T \\subseteq \\mathcal {P}$ such that $\\sup _{P \\in T} D_{KL}(P,P_0)\\le \\frac{\\log |T|}{16},$ where $D_{KL} : \\mathcal {P}\\times \\mathcal {P}\\rightarrow [0,\\infty ]$ denotes Kullback-Leibler divergence.",
"Then, $\\inf _{\\widehat{P}} \\sup _{P \\in \\mathcal {P}}\\operatornamewithlimits{\\mathbb {P}}\\left( \\rho (P,\\widehat{P})\\ge \\frac{1}{2} \\inf _{P \\in T} \\rho (P,P_0) \\right) \\ge 1/8,$ where the first $\\inf $ is taken over all estimators $\\widehat{P}$ .",
"Now, we proceed with the proof.",
"We now proceed to construct an appropriate $P_0 \\in \\mathcal {P}$ and $T \\subseteq \\mathcal {P}$ .",
"Let $g : [-1,1]^d \\rightarrow [0,1]$ defined by $g(x) = \\left\\lbrace \\begin{array}{cc}\\exp \\left( 1 - \\frac{1}{1 - \\Vert x\\Vert _2^2} \\right) & \\text{ if } \\Vert x\\Vert _2 < 1 \\\\0 & \\text{ else }\\end{array} \\right.$ denote the standard bump function supported on $[-1,1]^d$ , scaled to have $\\Vert g\\Vert _{\\mathcal {X},\\infty } = 1$ .",
"Since $g$ is infinitely differentiable and compactly supported, it has a finite $\\alpha $ -Hölder semi-norm: $\\Vert g\\Vert _{\\Sigma ^\\alpha }:= \\sup _{\\ell \\in \\mathbb {N}^d : \\Vert \\ell \\Vert _1 \\le \\alpha } \\quad \\sup _{x \\ne y \\in \\mathcal {X}} \\quad \\frac{|g^\\ell (x) - g^\\ell (y)|}{\\Vert x - y\\Vert ^{\\alpha - \\Vert \\ell \\Vert _1}}< \\infty ,$ where $\\ell $ is any $\\lfloor \\beta \\rfloor $ -order multi-index and $g^\\ell $ is the corresponding mixed derivative of $g$ .",
"Define $M := \\left( \\frac{64(2\\alpha + d)nr}{d \\log (nr)} \\right)^{\\frac{1}{2\\alpha + d}} \\ge 1$ , since $r \\ge 1/n$ .",
"For each $m \\in [M]^d$ , define $g_m : \\mathcal {X}\\rightarrow [0,1]$ by $g_m (x) := g\\left( Mx - \\frac{2m - 1_d}{2} \\right),$ so that $\\lbrace g_m : m \\in [M]^d\\rbrace $ is a grid of $M^d$ bump functions with disjoint supports.",
"Let $\\zeta _0 \\equiv \\frac{1}{4}$ denote the constant-$\\frac{1}{4}$ function on $\\mathcal {X}$ .",
"Finally, for each $m \\in [M]^d$ , define $\\zeta _m : \\mathcal {X}\\rightarrow [0,1]$ by $\\zeta _m := \\zeta _0 + \\min \\left\\lbrace \\frac{1}{2}, \\frac{L}{\\Vert g\\Vert _{\\Sigma ^\\alpha }} \\right\\rbrace M^{-\\alpha } g_m.$ Note that, for any $m \\in [M]^d$ , $\\Vert \\zeta _m\\Vert _{\\Sigma ^\\alpha }\\le L M^{-\\alpha } \\frac{\\Vert g_m\\Vert _{\\Sigma ^\\alpha }}{\\Vert g\\Vert _{\\Sigma ^\\alpha }}= L,$ so that $\\zeta _m$ satisfies the Hölder smoothness condition.",
"For any particular $\\eta $ , let $P_\\eta $ denote the joint distribution of $(X, Y)$ .",
"Note that $P_\\zeta (x, 1) = \\zeta (x) \\ge 1/4$ .",
"Moreover, one can check that, for all $x \\ge -2/3$ , $-\\log (1 + x) \\le x^2 - x$ .",
"Hence, for any $x \\in \\mathcal {X}$ , $P_{\\eta _m}(x, 1) \\log \\frac{P_{\\eta _m}(x, 1)}{P_{\\eta }(x, 1)}& = rP_{\\zeta _m}(x, 1) \\log \\frac{P_{\\zeta _m}(x, 1)}{P_{\\zeta }(x, 1)} \\\\& = r\\zeta _m(x) \\log \\frac{\\zeta _m(x)}{\\zeta (x)} \\\\& = -r\\zeta _m(x) \\log \\left( 1 + \\frac{\\zeta (x) - \\zeta _m(x)}{\\zeta _m(x)} \\right) \\\\& \\le r\\zeta _m(x) \\left( \\left( \\frac{\\zeta (x) - \\zeta _m(x)}{\\zeta _m(x)} \\right)^2 - \\frac{\\zeta (x) - \\zeta _m(x)}{\\zeta _m(x)} \\right) \\\\& = r \\left( \\frac{\\left( \\zeta (x) - \\zeta _m(x)\\right)^2 }{\\zeta _m(x)} - \\zeta (x) + \\zeta _m(x) \\right) \\\\& \\le r \\left( 4\\left( \\zeta (x) - \\zeta _m(x)\\right)^2 - \\zeta (x) + \\zeta _m(x) \\right),$ and, similarly, since $P_\\zeta (x,0) = 1 - \\zeta (x) \\ge 1/4$ , $P_{r\\eta _m}(x, 0) \\log \\frac{P_{r\\eta _m}(x, 0)}{P_{r\\eta }(x, 0)}\\le r\\left( 4\\left( \\zeta (x) - \\zeta _m(x)\\right)^2 + \\zeta (x) - \\zeta _m(x) \\right).$ Adding these two terms gives $D_{\\text{KL}}\\left( P_{r\\eta }^n, P_{r\\eta _m}^n \\right)& = n \\left( \\int _\\mathcal {X}P_{r\\eta _m}(x, 0) \\log \\frac{P_{r\\eta }(x, 0)}{P_{r\\eta _m}(x, 0)} \\, dx + \\int _\\mathcal {X}P_{r\\eta _m}(x, 1) \\log \\frac{P_{r\\eta }(x, 1)}{P_{r\\eta _m}(x, 1)} \\, dx \\right) \\\\& \\le 8nr\\int _\\mathcal {X}\\left( \\zeta (x) - \\zeta _m(x)\\right)^2 \\\\& = 8nr\\Vert \\zeta - \\zeta _m\\Vert _2^2 \\\\& \\le 2nr M^{-2\\alpha } \\Vert g_m\\Vert _2^2 \\\\& = 2nr M^{-(2\\alpha + d)} \\Vert g\\Vert _2^2 \\\\& = 2nr \\left( \\left( \\frac{64 (2\\alpha + d) nr}{d \\log (nr)} \\right)^{\\frac{1}{2\\alpha + d}}\\right)^{-(2\\alpha + d)} \\Vert g\\Vert _2^2 \\\\& = \\frac{1}{32} \\frac{d}{2\\alpha + d} \\Vert g\\Vert _2^2 \\log (nr) \\\\& \\le \\frac{1}{16} \\frac{d}{2\\alpha + d} \\left( \\log (nr) - \\log \\log (nr) + \\log \\frac{64 (2\\alpha + d)}{d} \\right)\\\\ &= \\frac{\\log |[M]^d|}{16},$ where the second inequality comes from the definition of $\\zeta _m$ (Eq.",
"REF ) and the third inequality comes from the facts that $\\Vert g\\Vert _2^2 \\le 1$ and $\\log \\log x \\le \\frac{1}{2} \\log x$ for all $x > 1$ .",
"Fano's lemma therefore implies the lower bound $\\inf _{\\widehat{\\eta }} \\sup _{r \\in (0, 1], \\zeta \\in \\Sigma ^\\alpha (L)}\\operatornamewithlimits{\\mathbb {P}}_{\\lbrace (X_i,Y_i)\\rbrace _{i = 1}^n \\sim P_\\eta ^n} \\left( \\left\\Vert r\\zeta - r\\widehat{\\zeta }\\right\\Vert _\\infty \\ge \\frac{1}{2} \\min \\left\\lbrace \\frac{1}{2}, \\frac{L}{\\Vert g\\Vert _{\\Sigma ^\\alpha }} \\right\\rbrace \\left( \\frac{d \\log (nr)}{64(2\\alpha + d) n} \\right)^{\\frac{\\alpha }{2\\alpha + d}} r^\\frac{\\alpha + d}{2\\alpha + d} \\right)\\ge \\frac{1}{8},$ which completes the proof."
]
] | 2107.01777 |
[
[
"Kapitsa pendulum effects in Josephson junction + nanomagnet under\n external periodic drive"
],
[
"Abstract We investigate reorientation effects under external periodic drive in the nanomagnet dynamics coupled to a Josephson junction.",
"The Kapitsa pendulum is introduced as a mechanical analog to this system and we demonstrate the reorientation of the easy axis of the nanomagnet.",
"The magnetic field generated by the Josephson junction and external drive plays the role of the oscillating force of the suspension point in the Kapitsa pendulum.",
"The high frequency oscillations change the orientation of the magnetic moment.",
"The magnetic field of the quasiparticle current determines the frequency dependence of the magnetic moment's orientation.",
"We obtain simple analytical formulas for the stable position of the magnetic moment, both under the external periodic drive and without it.",
"The influence of external periodic drive on the voltage of complete reorientation have been demonstrated."
],
[
"Kapitsa pendulum effects in Josephson junction + nanomagnet under external periodic drive K. V. Kulikov$^{1}$ , D. V. Anghel$^{2}$ , A. T. Preda$^{2,3}$ , M. Nashaat$^{1,4}$ , M. Sameh$^{4}$ and Yu.",
"M. Shukrinov$^{1,5}$ $^1$ BLTP, JINR, Dubna, Moscow region, 141980, Russia $^2$ Horia Hulubei National Institute for R& D in Physics and Nuclear Engineering, Măgurele, Romania $^3$ University of Bucharest, Faculty of Physics, Bucharest, Romania $^4$ Department of Physics, Faculty of Science, Cairo University, 12613, Giza, Egypt $^5$ Dubna State University, Dubna, Russia We investigate Kapitsa pendulum-like effects in the magnetic moment dynamics of a nanomagnet coupled to a Josephson junction under external periodic drive.",
"Generated by the Josephson junction and external drive, the magnetic field plays the role of the oscillating force of the suspension point in the Kapitsa pendulum.",
"The high frequency oscillations change the orientation of magnetic moment.",
"The magnetic field of the quasiparticle current of the Josephson junction determines the frequency dependence of the magnetic moment's orientation.",
"We obtain simple analytical formulas for the stable position of magnetic moment, both under the external periodic drive and without it.",
"The influence of external periodic drive on the voltage of complete reorientation have been demonstrated.",
"05.70.Ln, 05.30.Rt, 71.10.Pm Kapitsa's pioneering work [1] initiated the field of vibrational mechanics, and his method is used to describe periodic processes in a variety of different physical systems, like atomic physics [2], [3], [4], [5], plasma physics, optics [6], condensed matter physics, biophysics [7] and cybernetical physics (see [8], [9], [10], [11], [12], [13] and references therein).",
"In particular, imposing vibrational quantum coherence into topological states of matter may become a universal light control principle for reinforcing the symmetry-protected helical transport [14].",
"Coherent lattice vibrations can have direct and profound effects on surface transport of Dirac fermions, via periodic modulation of electronic states.",
"Ultrafast phononics has been explored as a new avenue to manipulate properties of superconductors, oxides, semimetal and photovoltaic semiconductors.",
"In a chain of spins with long-range ferromagnetic interactions, a magnetic field with periodic modulation can lead to stability regions of ferromagnetic spins around unstable paramagnetic configuration [15].",
"In nonlinear control theory, the Kapitsa pendulum is used as an example of a parametric oscillator that demonstrates the concept of “dynamic stabilization”.",
"In Ref.",
"[16], the authors realized experimentally the Kapitsa pendulum at the micrometre scale using a colloidal particle suspended in water and trapped by optical tweezers.",
"Moreover, it was analytically and experimentally demonstrated that if the oscillation direction of the pendulum suspension point change over time, so does the pendulum equilibrium point and active damping control can take place.",
"The Kapitsa quantum pendulum can be stabilized in the form of quantum states near a local minimum of the effective potential energy [17].",
"One way to probe magnetization dynamics involves the interaction of a nanomagnet with a Josephson junction (JJ).",
"The physics of such JJ-nanomagnet systems has received significant attention recently, both theoretically and experimentally.",
"In Ref.",
"[18], the authors introduce the Kapitsa pendulum as a mechanical analog to the $\\varphi _{0}$ -junction and demonstrated the reorientation of the easy axis of the magnetic moment of the ferromagnetic layer.",
"In this case, the Josephson to magnetic energy ratio $G$ corresponds to the amplitude of the variable force of the Kapitsa pendulum, the Josephson frequency $\\Omega _J$ corresponds to the oscillation frequency of the suspension point, and the averaged magnetic moment components specify the stable position.",
"However, the results show an opposite frequency dependence from the characteristic Kapitsa pendulum behavior, where the reorientation value is proportional to the frequency of the force applied to the suspension point.",
"In [18], the increase in $\\Omega _J$ leads to a smaller reorientation value at the same value of $G$ .",
"On the other hand, an effect corresponding to the Kapitsa pendulum has been observed in Ref.",
"[19], where for the JJ-nanomagnet system the increase in $\\Omega _J$ led to the larger reorientation value.",
"Figure: (a) Schematic diagram of the considered system with the system geometry.",
"(b) Average magnetic moment component m z m_z of the nanomagnet as a function of the voltage across the JJ demonstrating magnetization reorientation in voltage bias regime at α=0.1\\alpha = 0.1, G=3πG = 3\\pi , k=0.1k = 0.1, Ω F =0.5\\Omega _F = 0.5 and Ω=0.8\\Omega = 0.8.",
"The dashed arrow indicates the voltage of complete reorientation.In this Letter we demonstrate the dynamics of a nanomagnet coupled to a Josephson junction (see Fig.REF a) under external periodic drive.",
"We show an important role of the quasiparticle current in the effective field of the system: it can change the frequency dependence of the Kapitsa pendulum characteristics.",
"We also investigate the effect of external drive on the reorientation of the nanomagnet easy axis.",
"We consider voltage biased Josephson junction with length $l$ coupled to a nonomagnet with magnetic moment $\\textbf {M}=(M_x,M_y,M_z)$ located at distance $\\textbf {r}_{M}=a \\textbf {e}_x$ from the center of the junction as shown in Fig.REF a.",
"The dynamics of magnetic moment can be described by Landau-Lifshiz-Gilbert (LLG) equation $\\dfrac{d\\textbf {M}}{dt} = \\gamma \\textbf {H}_{eff} \\times \\textbf {M}+\\dfrac{\\alpha }{M_0}\\left(\\textbf {M}\\times \\dfrac{d\\textbf {M}}{d\\tau }\\right)$ where $\\alpha $ is the Gilbert damping parameter, $M_0=\\mid \\textbf {M}\\mid $ , $\\textbf {H}_{eff}$ is the effective field with components $(H_x, H_y, H_z)$ , $\\gamma $ is the gyromagnetic ratio.",
"We use normalized units, $m_{i}=M_{i}/M_{s}$ ($i=1,2,3$ ), $t=\\tau \\omega _c$ and $h_{i}=H_{i}/H_{F}$ , where $M_{s}$ is the saturation magnetization, $H_{F}=\\Omega _{F}/\\gamma $ , $\\Omega _{F}=\\omega _F/\\omega _c$ is frequency of the ferromagnetic resonance, $\\omega _{c}$ =$2eRI_{c}/\\hbar $ is characteristic Josephson frequency, $I_{c}$ is the critical current of the JJ, and $R$ is the resistance of the JJ in normal state.",
"The effective field is generated by the total current flowing through the JJ (determined by its superconducting and quasiparticle components [20], [19]) and external periodic drive.",
"The components of the total effective field are given by $h_{x}&=&0, h_{y} = m_{y}, \\nonumber \\\\ h_{z}& =& \\epsilon [\\sin (V t - k m_{z}+\\frac{A}{\\Omega }\\sin (\\Omega t))\\nonumber \\\\&+& \\delta ( V+A \\cos (\\Omega t)) - \\delta k \\dot{m}_{z}]$ where $V$ is the dc voltage bias normalized to $V_c=\\hbar \\omega _c/ 2e$ , $A=V_{ac}/V_c$ is the amplitude of external drive, $\\Omega $ is the frequency of the external drive normalized to $\\omega _c$ , $\\epsilon =Gk$ , $G=\\epsilon _J / K_{an} v$ is the Josephson to magnetic energy ratio, $\\epsilon _{J}=\\Phi _{0}I_{c}/2\\pi $ is the Josephson energy, $\\Phi _{0}$ is the flux quantum, $v$ is the volume of the nanomagnet, $K_{an}$ is the magnetic anisotropy constant, $k=(2\\pi /\\Phi _{0})\\mu _{0} M_{s}l /a\\sqrt{l^{2}+a^{2}}$ is the coupling constant between the JJ and the nanomagnet, whereas $\\delta = 1, 0$ is the parameter which switches on and off the quasiparticle current in the effective field.",
"Notice also that in our normalization the dc voltage bias $V$ is equal to the Josephson frequency $\\Omega _J$ .",
"We use Eq.",
"(REF ) with the effective field (REF ) to numerically calculate the dynamics of the magnetic moment projections on the coordinate axes, with the initial conditions $m_{x}=0$ , $m_{y}=1$ , and $m_{z}=0$ .",
"As in a Kaptitsa pendulum [21], [1], the applied voltage across the Josephson junction in our system generates a high frequency magnetic field that reorients the magnetic moment of the nanomagnet.",
"Figure REF b shows the reorientation of the magnetic moment as a function of the DC bias voltage, i.e., a manifestation of the Kapitsa pendulum feature in the “Josephson junction–nanomagnet” system.",
"The stabilization of the magnetic moment components dynamics occurs at $M = (0,0,1)$ , when $V$ exceeds a certain reorientation value $V_r$ , which indicates a complete reorientation of the magnetic moment.",
"We study analytically the magnetic moment dynamics of the nanomagnet in the approximation $V, \\Omega \\gg \\Omega _F$ , that is, when the frequencies of the JJ and of the external drive are much higher than the eigen-frequency of the nanomagnet.",
"These will produce small and fast oscillations of the magnetic moment, similar to the oscillations in the Kapitsa pendulum [21].",
"We use the spherical coordinates $\\theta , \\phi $ to write $(m_x,m_y,m_z) \\equiv (\\sin \\theta \\cos \\varphi ,\\sin \\theta \\sin \\varphi , \\cos \\theta )$ and separate them into fast and slow variables by introducing the notations $\\theta \\equiv \\Theta + \\xi $ and $\\phi \\equiv \\Phi + \\zeta $ .",
"Here, $\\Theta $ and $\\Phi $ describe the “slower” motion, relevant on longer time scales (comparable to the period of the oscillations of the system in the absence of the external drive and the Josephson oscillations), whereas the variables $\\xi $ and $\\zeta $ describe the “fast” oscillations of the system, which take place on shorter time scales (comparable to $1/V$ and $1/\\Omega $ ).",
"Writing Eq.",
"(REF ) in the variables $(\\theta ,\\phi )$ [22] and expanding it in a Taylor series to the first order in $(\\xi ,\\zeta )$ around $(0,0)$ , we obtain $&& \\dot{\\theta }= \\dot{\\Theta }+ \\dot{\\xi }\\approx M_0(\\Theta ) F(\\Theta , \\Phi , \\xi , \\zeta , t)\\\\&& \\dot{\\phi }= \\dot{\\Phi }+ \\dot{\\zeta }\\approx M_0(\\Theta ) Q(\\Theta , \\Phi , \\xi , \\zeta , t)$ with $F(\\Theta , \\Phi , \\xi , \\zeta , t)= F_0(\\Theta , \\Phi ) + F_{\\xi }(\\Theta , \\Phi , t) \\xi + F_{\\zeta }(\\Theta , \\Phi , t) \\zeta + F_t(\\Theta ,t)$ and $Q(\\Theta , \\Phi , \\xi , \\zeta , t)=Q_0(\\Theta , \\Phi ) + Q_{\\xi }(\\Theta , \\Phi , t) \\xi + Q_{\\zeta }(\\Theta , \\Phi , t) \\zeta + Q_t(\\Theta ,t)$ , where $&&M_0(\\Theta ) \\equiv \\frac{ \\Omega _F }{ 1 + \\alpha ^2 + \\delta \\alpha \\epsilon k \\sin ^2\\Theta \\, \\Omega _F }, \\quad F_0(\\Theta , \\Phi ) = ( \\alpha \\sin \\Phi \\cos \\Theta + \\cos \\Phi ) \\sin \\Theta \\sin \\Phi - \\alpha \\epsilon \\delta V \\sin \\Theta , \\\\&&F_{\\xi }(\\Theta , \\Phi , t) = \\Big [- \\cos \\Phi \\cos \\Theta \\Big (\\sin ^2\\Theta \\alpha \\delta \\epsilon k \\Omega _F - \\alpha ^2 - 1 \\Big ) + 2 \\alpha ( \\alpha ^2 + 1) \\sin \\Phi \\cos ^2\\Theta - \\alpha (\\alpha \\delta \\epsilon k \\Omega _F \\sin ^2\\Theta + \\alpha ^2 + 1) \\sin \\Phi \\Big ] \\nonumber \\\\&& \\times \\frac{\\sin \\Phi }{ 1 + \\alpha ^2 + \\delta \\alpha \\epsilon k \\sin ^2\\Theta \\, \\Omega _F }- \\alpha \\epsilon k \\sin ^2\\Theta \\cos \\left[ V t - k \\cos \\Theta + \\frac{A}{\\Omega } \\sin (\\Omega t) \\right] , \\\\&&F_{\\zeta }(\\Theta , \\Phi , t) = (2 \\cos \\Phi \\sin \\Phi \\cos \\Theta \\alpha + 2 \\cos ^2\\Phi - 1) \\sin \\Theta , \\\\&&F_t(\\Theta ,t) = - \\alpha \\sin \\Theta \\epsilon \\delta A \\cos (\\Omega t)- \\alpha \\epsilon \\sin \\Theta \\sin \\left[ V t - k \\cos \\Theta + \\frac{A}{\\Omega } \\sin (\\Omega t) \\right] , \\\\&&Q_0(\\Theta , \\Phi ) = \\Big [\\epsilon \\delta V + (\\delta \\epsilon k \\Omega _F \\sin ^2\\Theta \\cos \\Phi - \\cos \\Theta \\sin \\Phi + \\alpha \\cos \\Phi ) \\sin \\Phi \\Big ] , \\\\&&Q_{\\xi }(\\Theta , \\Phi , t) = \\left\\lbrace \\Bigg ( \\frac{ \\Big [\\Omega _F \\alpha \\delta \\epsilon k (\\cos ^2\\Theta + 1) + \\alpha ^2 + 1\\Big ] \\sin \\Phi + 2 \\cos \\Phi \\cos \\Theta \\delta \\epsilon k \\Omega _F}{ 1 + \\alpha ^2 + \\delta \\alpha \\epsilon k \\sin ^2\\Theta \\, \\Omega _F } \\Bigg )\\sin \\Phi \\right.",
"\\nonumber \\\\&& \\left.",
"+ \\epsilon k \\cos \\left[ V t - k \\cos \\Theta + \\frac{A}{\\Omega } \\sin (\\Omega t) \\right] \\right\\rbrace \\sin \\Theta , \\\\&&Q_{\\zeta }(\\Theta , \\Phi , t) = \\Big [ (2\\cos ^2\\Phi - 1) (k \\delta \\epsilon \\Omega _F \\sin ^2\\Theta + \\alpha )- 2 \\cos \\Phi \\cos \\Theta \\sin \\Phi \\Big ] , \\\\&&Q_t(\\Theta , t) = \\epsilon \\delta A \\cos (\\Omega t)+ \\epsilon \\sin \\left[ V t - k \\cos \\Theta + \\frac{A}{\\Omega } \\sin (\\Omega t) \\right] ,$ with $\\sin [ V t - k \\cos \\Theta + \\frac{A}{\\Omega } \\sin ( \\Omega t )] &=& \\sum _{m=-\\infty }^{\\infty } \\text{sign}^m(m) J_{|m|}\\left(\\frac{A}{\\Omega }\\right)\\sin [ (V + m \\Omega ) t - k \\cos \\Theta ] , \\\\\\cos [ V t - k \\cos \\Theta + \\frac{A}{\\Omega } \\sin ( \\Omega t ) ] &=& \\sum _{m=-\\infty }^{\\infty } \\text{sign}^m(m) J_{|m|}\\left(\\frac{A}{\\Omega }\\right)\\cos [ ( V + m \\Omega ) t - k \\cos \\Theta ] .$ In the definitions (REF ), $M_0$ , $F_0$ , and $Q_0$ do not depend explicitly on time.",
"Plugging Eqs.",
"(REF ) into (REF ), we see that $F_t$ and $Q_t$ may be written as infinite sums of terms that oscillate with the frequencies $\\Omega $ or $V+m\\Omega $ .",
"We denote $m_0\\equiv -V/\\Omega $ .",
"If $m_0$ is an integer, then $Q_{m_0}(\\Theta ) \\equiv -\\epsilon \\text{sign}^{m_0}(m_0) J_{|m_0|}\\left(\\frac{A}{\\Omega }\\right) \\sin [k\\cos \\Theta ]$ and $F_{m_0}(\\Theta ) \\equiv - \\alpha \\sin \\Theta \\, Q_{m_0}(\\Theta )$ , otherwise, $Q_{m_0}(\\Theta )=F_{m_0}(\\Theta )=0$ .",
"Then, in the zeroth order of approximation we have $&& \\dot{\\Theta }_0 \\equiv M_0(\\Theta ) \\left[ F_0(\\Theta ,\\Phi ) + F_{m_0}(\\Theta ) \\right] , \\nonumber \\\\&& \\dot{\\Phi }_0 \\equiv M_0(\\Theta ) \\left[ Q_0(\\Theta ,\\Phi ) + Q_{m_0}(\\Theta ) \\right] .$ Since $\\Omega ,V \\gg \\Omega _{F}$ , and if $|V+m\\Omega |\\gg \\Omega _{F}$ for any $m\\ne m_0$ , the terms that depend explicitly on time in $F_t$ and $Q_t$ oscillate fast as compared to the period of oscillation in the absence of perturbation, so we have $&& \\dot{\\zeta }_0(\\Theta ,t) \\equiv M_0(\\Theta ) [Q_t(\\Theta ,t) - Q_{m_0}(\\Theta )] ,\\nonumber \\\\&& \\dot{\\xi }_0(\\Theta ,t) = -\\alpha \\sin \\Theta \\, \\dot{\\zeta _0},\\nonumber \\\\&& \\zeta _0(\\Theta ,t) = \\int _{0}^{t}\\dot{\\zeta }_0(\\Theta ,t) \\, dt , \\ \\xi _0(\\Theta ,t) = \\int _{0}^{t}\\dot{\\xi }_0(\\Theta ,t) dt.$ In the order $n\\ge 1$ of approximation we have $\\dot{\\xi }_n = \\dot{\\xi }_{n-1} + M_0(\\Theta ) [F_{\\xi }(\\Theta , \\Phi , t) \\xi _{n-1} + F_{\\zeta }(\\Theta , \\Phi , t) \\zeta _{n-1}],$ $\\dot{\\zeta }_n = \\dot{\\zeta }_{n-1} + M_0(\\Theta ) [Q_{\\xi }(\\Theta , \\Phi , t) \\xi _{n-1} + Q_{\\zeta }(\\Theta , \\Phi , t) \\zeta _{n-1}] .",
"$ If $(\\dot{\\xi }_{n-1}, \\dot{\\zeta }_{n-1})$ contain terms oscillating with frequencies $nV+m\\Omega $ , then, from $F_{\\xi }(\\Theta , \\Phi , t) \\xi + F_{\\zeta }(\\Theta , \\Phi , t) \\zeta $ , $Q_{\\xi }(\\Theta , \\Phi , t) \\xi + Q_{\\zeta }(\\Theta , \\Phi , t) \\zeta $ , and Eqs.",
"(REF ), $(\\dot{\\xi }_{n},\\dot{\\zeta }_{n})$ contain terms oscillating with frequencies $(n+1)V+m\\Omega $ , for any $m$ , such that $(n+1)V+m\\Omega \\ne 0$ [22].",
"If exists an integer $m^{(n)}_{0}$ , such that $(n+1)V+m^{(n)}_0\\Omega = 0$ , this would give a term which does not explicitly depend on time and therefore will be incorporated into the $n^{\\rm th}$ order contribution to the slow motion $(\\dot{\\Theta }_n, \\dot{\\Phi }_n)$ .",
"This contribution will contain a factor $M_0^n(\\Theta )$ , as compared to $(\\dot{\\Theta }_0, \\dot{\\Phi }_0)$ , which decreases to zero with $n$ if $M_0(\\Theta )<1$ (we have in general $\\Omega _{F}\\ll 1$ ).",
"The presence of the terms that oscillate in time may be emphasized by the fast Fourier transform (FFT) of $m_{z}(t)$ .",
"The oscillating frequencies should be $|nV+m\\Omega |$ , where $n, m$ are integers and $n \\ge 0$ .",
"This can be clearly seen in Fig.",
"REF (a), where we present the results of $m_{z}(t)$ 's precession frequency $\\Omega _{p}$ for $V=0.75$ and $\\Omega =0.65$ , the other parameters being $A=0$ or 2, $\\Omega _{F}=10^{-2}$ , $G=2\\pi $ , $k=0.1$ and $\\alpha =0.1$ .",
"Figure: (Color online) Results of FFT analysis of m z (t)m_{z}(t) at V=0.75V=0.75 without and with external drive of amplitude A=2A=2 and Ω=0.65\\Omega =0.65.Due to the coupling between JJ and nonomagnet through the current phase relation, at $A=0$ , the magnetization $m_{z}$ oscillates only with the Josephson frequency $V$ , hence we see only one frequency line (blue line) at $\\Omega _{p}=V$ .",
"If $A\\ne 0$ we have two frequencies, $V$ and $\\Omega $ , which affect the precession of $m_{z}(t)$ .",
"In this case, we observe two harmonic frequency lines at $\\Omega _{p}=V$ and $\\Omega _{p}=\\Omega $ .",
"In addition, the FFT analysis gives several frequency lines correspond to the subharmonics with $\\Omega _{p} =n V \\pm m \\Omega $ , where $n$ and $m$ are integers.",
"When $A = 0$ , eqs.",
"(REF ) and (REF ) simplify considerably and the equilibrium conditions $\\dot{\\Theta }= \\dot{\\Phi }= 0$ give [22] $\\Phi = \\pi /2 \\qquad {\\rm or} \\qquad \\Phi = 3\\pi /2 $ and an equation for $\\Theta $ $\\langle m_z \\rangle =\\cos \\Theta = \\epsilon \\delta V + \\frac{ \\alpha \\epsilon ^2 k \\sin ^4\\Theta \\Omega _F }{ 2V(1 + \\alpha ^2 + \\delta \\alpha \\epsilon k \\sin ^2\\Theta \\, \\Omega _F)^2 } $ Equation (REF ) is valid for $-1 \\le \\epsilon V \\le 1$ ; if $|\\epsilon V| > 1$ , then $m_z = {\\rm sign}(V)$ .",
"Results of numerical calculations of the averaged $m_z$ as a function of $G$ at two frequencies, presented in Fig.",
"REF (a), demonstrate the changes of stability position.",
"The analytical dependence calculated by (REF ) shows an excellent agreement with numerical data.",
"Our results explain the unusual frequency dependence of the reorientation in the $\\varphi _0$ Josephson junction [18], where only the superconducting current in the effective field of the LLG equation was taken into account.",
"This discrepancy with the usual Kapitsa pendulum is due to the omission of the quasiparticle current in the effective field.",
"It can be clearly seen in (REF ) that at $\\delta =0$ (there is no quasiparticle current in the effective field [18]) only the second term will contribute, which is proportional to $1/V$ , but at $\\delta =1$ the first term in the equation, which is proportional to $V$ , will rise much faster.",
"In our considerations, both currents are included in the effective field, therefore, the frequency dependence shown in Fig.REF (b) coincide with the usual Kapitsa pendulum features.",
"Note also that the value of $G$ for the complete reorientation, which indicates the stabilization of the magnetic moment dynamics at $\\delta =1$ is much smaller then at $\\delta =0$ (see [18]).",
"Figure: (color online) (a) The average values of <m z ><m_z> versus the Josephson-to-magnetic energy ratio GG for the two indicated voltages.",
"The symbols show the values obtained by the numerical calculation of eqs.",
"().",
"The lines show analytical results obtained from eqs.().",
"(b) The average value of the magnetic moment component m z m_z as a function of amplitude of external drive AA at α=0.1\\alpha = 0.1, G=3πG = 3\\pi , k=0.1k = 0.1, Ω F =0.001\\Omega _F = 0.001, V=0.75V=0.75, Ω=0.75\\Omega = 0.75, m 0 =-V/Ω=-1m_0=-V/\\Omega =-1.",
"The symbols indicate the average values obtained by numerical calculation of Eqs.",
"().",
"The lines indicate the analytical results obtained from eqs.",
"() and ().",
"(c) Variation of the reorientation voltage with the external drive amplitude.",
"The green line represents the fitting with Bessel function.If we have a zeroth order resonance $m_0$ , then $V+m_0\\Omega = 0$ .",
"From Eqs.",
"(REF ) and $\\dot{\\Theta }= \\dot{\\Phi }= 0$ we obtain $\\Phi = \\pi /2$ or $3\\pi /2$ and an equation for $\\Theta $ : $ \\cos \\Theta = \\epsilon \\delta V - \\epsilon \\text{sign}^{m_0}(m_0) J_{m_0}\\left(\\frac{A}{\\Omega }\\right)\\sin (k \\cos \\Theta ) .$ Based on it, we calculate the average value of magnetic moment component $m_z$ as a function of amplitude of external drive $A$ .",
"The results are shown in Fig.",
"REF (b) and we see a very good agreement with the direct numerical calculations.",
"Note that the average $m_z$ as a function of $A$ demonstrates Bessel behavior.",
"The applied external periodic drive also affects the voltage of complete reorientation $V_{r}$ , which indicates the stabilization of the magnetic moment dynamics (see Fig.",
"REF b).",
"This effect is demonstrated in Fig.",
"REF (c), where the results of numerical calculations of $V_{r}$ as a function of external drive amplitude $A$ are presented (symbols).",
"$V_{r}$ as a function of $\\Omega $ is the same not shown here.",
"The numerical data are well fitted by the Bessel function.",
"At chosen parameters $G=3\\pi , ~\\alpha =0.1, ~k=0.1, ~\\Omega _F=0.5$ and $\\Omega =0.8$ the data presents a good agreement with Bessel dependence $0.5J(A/\\Omega )+0.54$ , shown by green solid line.",
"Note, that this effect can also be seen from (REF ) at $\\cos \\Theta =1 $ .",
"We obtained an analytical expressions to describe the movement of the stability position in the $yz$ plane and had a very good agreement with the direct numerical simulations.",
"It was shown that the magnetic field of the quasiparticle current determines the frequency dependence of magnetic moment's stable position.",
"It also decreases the value of Josephson to magnetic energy ratio $G$ necessary for the complete reorientation.",
"Therefore, the quasiparticle current magnetic field plays an important role for the reorientation problem.",
"The influence of external periodic drive on the Kapitsa pendulum-like effects has been demonstrated.",
"It was shown that the dependence of the average value of the magnetic moment component $m_z$ on the amplitude of external drive $A$ is described by the Bessel function.",
"This result significantly expand the possibilities of controlling the dynamics of magnetization.",
"We emphasize that this influence is orders of magnitude more pronounced when the Josephson frequency $V$ is equal to an integer number of external drive frequencies $\\Omega $ (i.e.",
"$V+m_0\\Omega = 0$ , where $m_0$ is negative integer).",
"Otherwise, the influence of $A$ is very small and the Kapitsa pendulum-like effects come mainly from the Josephson oscillations.",
"We also showed that the voltage of complete reorientation $V_{r}$ depends as the Bessel function on the external drive amplitude.",
"We consider that the obtained results open a wide field of research and applications related to the possibility of reorientation of nanomagnet's easy axis.",
"Such a realization might play a crucial role in quantum information processing and spintronics.",
"The experimental verification of our work would involve preparing a voltage-biased JJ-nanomagnet system with sufficiently small values of $\\epsilon $ .",
"We expect that a 2D thin-film niobium superconducting junction in the x-y plane coupled to the nanomagnet could be a potential candidate for experimental realization.",
"Note that the value of the Josephson energy in such a junction is $\\epsilon _J \\sim 2 \\times 10^{-18} \\; J$ , while the resistance is $\\sim 3 \\; m \\Omega $ and $\\omega _c = 50 \\; GHz$ .",
"So, the Josephson frequency is in the range of $GHz$ and the voltage in $\\mu V$ .",
"The nanomagnet is assumed to have a radius of $7 - 30 \\; nm$ in thickness with magnetic anisotropy constant of $K_{an} \\sim 20\\; kJ/m^3$ and a saturation magnetization of $1950\\; kA/m$ .",
"The Josephson junction induces an electromagnetic radiation of frequency around $1 GHz$ .",
"We also note that such experiments should also be possible with 1D junctions using nanowires with spin-orbit coupling [23], [24].",
"The coupling of such JJs with Majorana bound states to nanomagnets in the presence of a magnetic field may lead to new experimental signatures of such states [25]."
]
] | 2107.01882 |
[
[
"Spread balanced Wannier functions: Robust and automatable orbital\n localization"
],
[
"Abstract We introduce a new type of Wannier functions (WFs) obtained by minimizing the conventional spread functional with a penalty term proportional to the variance of the spread distribution.",
"This modified Wannierisation scheme is less prone to produce ineffective solutions featuring one or several poorly localized orbitals, making it well suited for complex systems or high-throughput applications.",
"Furthermore, we propose an automatable protocol for selecting the initial guess and determine the optimal number of bands (or equivalently WFs) for the localization algorithm.",
"The improved performance and robustness of the approach is demonstrated for a diverse set of test systems including the NV center in diamond, metal slabs with atomic adsorbates, spontaneous polarization of ferroelectrics and 30 inorganic monolayer materials comprising both metals and semiconductors.",
"The methods are implemented in Python as part of the Atomic Simulation Environment (ASE)."
],
[
"Introduction",
"As computational codes are being increasingly automated it becomes possible to perform complex, high-throughput investigations with minimal human efforts, creating new advantages and opportunities.",
"Within materials science, these developments significantly expand the range of materials/properties that can be examined by a single researcher.",
"Moreover, it increases data quality by reducing the risk of human errors, and enables researchers to address materials phenomena or properties outside his/her domain expertise by lowering barriers related to the technical aspects of the calculation.",
"While some computational tasks are straightforward to automate, others are more challenging.",
"An important example of the latter is the generation of localized representations of the delocalized Bloch states of a crystal, i.e.",
"Wannier functions [1] (WFs).",
"In a seminal paper, Marzari and Vanderbilt [2] introduced a practical scheme for calculating maximally localized Wannier functions (MLWFs) that overcomes the problem of the non-uniqueness (or \"gauge dependence\").",
"For simple systems, i.e.",
"when the bands of interest form an isolated group and/or there are few atoms in a unit cell, the standard algorithms typically yield well localized WFs.",
"In general, however, the construction of useful WFs requires some hand-holding rendering automation highly non-trivial.",
"When the bands of interest (from hereon referred to as the \"target bands\") are isolated from all higher and lower lying bands by energy gaps, the MLWFs are obtained by minimizing the sum of the quadratic spread of all the WFs.",
"In the general case, i.e.",
"when the target bands do not form an isolated group, the problem of finding a proper localized representation becomes significantly harder.",
"In this case, extra degrees of freedom (EDF) in the form of states orthogonal to the target bands, must be included to aid the localization.",
"Since the target bands typically contain the occupied manifold, the problem of identifying the optimum EDF can be seen as the process of augmenting the target bands by their anti-bonding states [3].",
"The so-called disentanglement procedure [4] identifies the EDF by minimizing the dispersion of the $\\mathbf {k}$ -subspaces (the subspace spanned by the target bands and the EDF at a given $\\mathbf {k}$ ) across the Brillouin zone (BZ).",
"Having identified the optimal $\\mathbf {k}$ -subspaces the MLWFs are obtained following the usual localization procedure.",
"For perfect crystals, this approach is well suited.",
"On the other hand, for systems where crystal momentum is not a good quantum number, e.g.",
"molecules, amorphous solids, crystals with defects, etc.",
"the idea of minimizing the $\\mathbf {k}$ -dispersion does not seem a natural strategy.",
"As an alternative, one can determine the EDF by direct minimization of the spread functional.",
"With this strategy, the selection of the EDF and the localization into WFs is cast as one global optimization problem rather the two-step strategy applied by the disentanglement procedure.",
"This idea leads to the partly occupied Wannier functions (POWF) developed in Refs.",
"[3], [5].",
"Being the result of a one- rather than two-step optimization, the POWFs have smaller spread than the MLWFs.",
"Moreover, the scheme avoids reference to the $\\mathbf {k}$ -dispersion, which seems natural for non-periodic systems.",
"We note that the POWFs were rediscovered in a different but equivalent form in Ref.",
"[6].",
"Regardless of how the EDF are selected the standard localization procedure, i.e.",
"the minimization of the sum of the quadratic spreads, is not always straightforward.",
"One manifestation of this problem is its starting-point dependence, i.e.",
"that different WFs are obtained depending on the initial guess for the orbitals, e.g.",
"orbital type ($s$ ,$p$ ,$d$ ) and position (atom- or bond centered).",
"This indicates that the conventional spread functional exhibits several local minima.",
"From a practical point of view, the lack of robustness/reproducibility arising from the starting point dependence is not a problem in itself as long as a decent set of WFs is obtained.",
"Unfortunately, even that can be challenging and sometimes requires tuning of the initial guess, the target bands, and number of EDF.",
"One specific problem sometimes encountered is that all WFs become well localized except for one or a few which remain delocalized; this renders the entire set of WFs useless for many purposes and significantly complicates automatization.",
"In this paper, we introduce a new class of spread functionals that explicitly penalize delocalization of individual WFs.",
"The minimization of these functionals generally produces WFs with a more balanced spread distribution.",
"In particular, the problem of \"sacrificing\" one WF to improve the total spread does not occur.",
"We also introduce a specific protocol for automatically initializing the WFs based on the valence configuration of the involved atoms.",
"Leveraging the properties of the POWFs, we device a simple trial-and-error, yet easy to automate, procedure for selecting the optimal number of EDF.",
"Putting it all together we arrive at a highly robust and fully automatic scheme for constructing spread balanced WFs for general types of materials.",
"We demonstrate the method for a number of challenging systems, including atoms adsorbed on metal slabs, the NV defect in diamond, and a set of 30 two-dimensional (2D) materials arbitrarily selected from the Computational 2D Materials Database (C2DB) [7].",
"The WFs are used to obtain electronic band structures and spontaneous polarisations within the framework of the modern theory of polarization [8].",
"All methods are implemented in the open source Atomic Simulation Environment (ASE) [9]."
],
[
"Theory",
"In this section we review the theory and construction of partly occupied WFs.",
"We then introduce our new spread functionals designed to produce WFs with narrow size distributions.",
"While we have investigated several different functionals, we focus on the best performing one, the minimal variance spread functional, throughout this paper.",
"Finally, we describe our protocols for initializing WFs and selecting the optimal number of WFs, respectively."
],
[
"Partly occupied Wannier functions",
"The partly occupied Wannier functions were introduced in 2005 [3], [5] and were recently demonstrated [6] to represent the global minimum of the quadratic spread functional.",
"The POWFs are related to the maximally localized Wannier functions [2], [4] but avoid explicit reference to the wave vector in the band disentanglement procedure, and are directly applicable to non-periodic systems.",
"Instead of maximizing the reciprocal space smoothness the POWF method is entirely based on the minimization of the real space spread of the WFs.",
"For systems with periodic boundary conditions and a sufficiently large supercell, the minimization of the conventional Marzari-Vanderbilt spread functional [2] for a set of WFs $\\lbrace w_n({r})\\rbrace _{n=1}^{N_w}$ is equivalent [10] to the maximization of $\\Omega = \\sum _{n=1}^{N_w} \\sum _{\\alpha =1}^{N_G} W_{\\alpha } |Z_{\\alpha , nn}|^2$ where the matrix $Z_{\\alpha }$ is defined as $Z_{\\alpha , nm} = \\langle w_n | e^{-i {G}_{\\alpha } \\cdot {r}} | w_m \\rangle .$ The $\\lbrace {G}_{\\alpha }\\rbrace $ is a set of $N_G$ reciprocal lattice vectors that connect each $k$ -point to its neighbors and $W_{\\alpha }$ are corresponding weights accounting for the shape of the unit cell.",
"The value of $N_G$ can range from 3 to 6 depending on the symmetry of the unit cell.",
"For a discussion about these vectors and weights we refer to Ref.",
"[11], [12].",
"The definition of localization we impose with the functional $\\Omega $ , as in the case of the Marzari-Vanderbilt spread functional, is equivalent to the Foster-Boys method [13].",
"We emphasize that the assumption of a large supercell with periodic boundary conditions does not represent a limitation.",
"For example, it applies to a pristine periodic crystal (with the primitive cell repeated a number of times in all directions), isolated entities like molecules or clusters surrounded by a sufficiently large vacuum region, a surface slab (possibly with the primitive cell repeated in the in-plane directions), or a solid with periodically repeated disorder, e.g.",
"an impurity or point defect.",
"The goal of this approach is to obtain a set of $N_w$ localized WFs that can reproduce any eigenstate below an energy threshold, $E_0$ , exactly.",
"Given $N_b$ available eigenstates, a localization subspace is defined as the space spanned by the $M$ eigenstates with energy below $E_0$ and additional $L$ extra degrees of freedom (EDF), where $M + L = N_w \\le N_b$ .",
"Each WF is then defined as $w_n = \\sum _{m=1}^{M} U_{mn} \\psi _m + \\sum _{l=1}^L U_{M+l,n} \\phi _l$ where the EDFs $\\phi _l$ are defined as $\\phi _l = \\sum _{m=1}^{N_b - M} c_{ml} \\psi _{M+m}.$ The matrix $c$ has orthonormal columns while the matrix $U$ is unitary.",
"All the expressions in this section refer to the simple case where the eigenstates have been obtained in a large supercell for which a $\\Gamma $ -point sampling of the Brillouin zone (BZ) is a good approximation.",
"We stress, however, that for systems exhibiting periodicity on a smaller scale than the supercell dimensions, e.g.",
"for a perfect crystal, it is possible to formulate the theory in terms of the eigenstates of the primitive unit cell sampled on a uniform grid of $k$ points.",
"[5] In this case, the number of fixed states, $M^{{k}}$ , and EDF, $L^{{k}}$ , become ${k}$ -dependent.",
"The localization functional $\\Omega $ can be maximized with respect to $U$ and $c$ using any gradient-dependent algorithm under the constraint of orthonormality of the EDFs, $\\phi _l$ , implemented e.g.",
"via the method of Lagrange multipliers.",
"This step is referred to as the iterative localization procedure.",
"The POWF method was originally implemented in the open source ASE [9] Python package.",
"All the methods described in the following were implemented as extensions/improvements to the existing POWF-ASE code."
],
[
"Variance reducing spread functionals",
"The maximization of $\\Omega $ in Eq.",
"(REF ) is equivalent to the minimization of a cost functional given by the sum of the quadratic spreads (second moments) of the individual WFs.",
"In our experience, this approach is not robust and can produce delocalized WFs, in particular, for large numbers of WFs ($> 50$ ).",
"We hypothetize that this happens because the cost of delocalizing a single function can be compensated by a small improvement in the localization of a number of other functions.",
"To circumvent this problem, we have explored different types of cost functionals designed to share the spread more evenly across the entire set of WFs.",
"One approach is to apply a function on top of the $Z$ -matrix elements of the original functional (these are related to the inverse spread of the corresponding WF) $\\Omega _f = \\sum _{n=1}^{N_w} \\sum _{\\alpha =1}^{N_G} W_{\\alpha } f \\left( |Z_{\\alpha , nn}|^2 \\right).$ We have tested different functions $f(x)$ : square root ($\\sqrt{x}$ ), scaled error function ($\\frac{2}{\\sqrt{\\pi }} \\int _0^{2x} e^{-t^2} dt$ ), scaled and translated sigmoid function ($1 / \\left[1 + e^{-10(x-0.5)} \\right]$ ), see Figure REF .",
"They span the same range as the original matrix elements, but introduce flat plateaus for well localized and a steeper slope for delocalized WFs.",
"The effect of the steeper slope increases the gain of localizing a delocalized WF relative to the cost of delocalizing an already localized WF.",
"In particular, the sigmoid function has an additional penalty for delocalized functions ($|Z_{\\alpha , nn}|^2 < 0.5$ ; the threshold can be tuned if needed).",
"We mention that the modification function, $f$ , may alternatively be applied to the $\\alpha $ -sum instead of the individual $Z$ -matrix elements, but this was not pursued in the current study.",
"Figure: Different functions applied to the |Z| 2 |Z|^2 matrix elements entering the localization functional Eq.",
"().The original localization functional is corresponds to f(x)=xf(x)=x (blue dots).Figure: Effect of changing the weight parameter, w var w_{\\mathrm {var}}, of the variance penalty term in Ω var \\Omega _{\\mathrm {var}} for Wannier functions of bulk silicon.",
"For larger values of w var w_{\\mathrm {var}}, the spread of the most delocalised WF (maximal spread) decreases while the average spread of the WFs increases slightly.",
"The symbols represent the mean values over 10 independent Wannierisations with different initial guess and the bars indicate the standard deviations.The definition of the minimal N w N_w is given in Eq.",
".Our second approach adds a penalty term to the original functional proportional to the variance of the spread distribution $\\mathrm {\\Omega _{var}} = \\Omega - w_{\\mathrm {var}} \\mathrm {Var} \\left[ \\sum _{\\alpha =1}^{N_G} W_{\\alpha } | Z_{\\alpha ,nn} |^2 \\right]$ where $w_{\\mathrm {var}}$ is a parameter setting the weight for the variance term.",
"In all our calculations we have set $w_{\\mathrm {var}} = N_w$ (the weight of the penalty term should grow linearly with $N_w$ as the same holds for $\\Omega $ ).",
"Of all the functionals tested in this study, $\\mathrm {\\Omega _{var}}$ has the best overall performance, and we therefore focus on this functional in the rest of the paper.",
"In particular, the functional based on the square root only lead to minor and inconsistent improvements in the localization compared to $\\Omega $ .",
"The functionals based on the error function and the sigmoid greatly increased the average spread $\\bar{s}$ in order to minimize the maximum spread $s_{\\mathrm {max}}$ of the set of WFs, see App.",
"for the definition of $\\bar{s}$ and $s_{\\mathrm {max}}$ .",
"We stress that $\\mathrm {\\Omega _{var}}$ properly converges to real-valued WFs, as expected when reaching the global maximum [14].",
"We now return to $\\Omega _{\\mathrm {var}}$ and the role of the weight parameter of the penalty term, $w_{\\mathrm {var}}$ .",
"Fig.",
"REF shows the average and maximum spread of the set of WFs of bulk silicon obtained by maximizing $\\Omega _{\\mathrm {var}}$ with different prefactors included in $w_{\\mathrm {var}}$ .",
"As expected, increasing the weight of the penalty term leads to a more narrow spread distribution and thus a smaller spread of the least localised WF (maximum spread), but at the same time leads to an increase of the average spread of the WFs.",
"Therefore, this parameter can be tuned as needed and may even be optimized for specific applications.",
"The initial value of $w_{\\mathrm {var}} = N_w$ worked consistently across the set of materials we tested, hence we did not perform any further optimization.",
"Finally, we note that the use of different objective functions may in general lead to a different characters of the localized orbitals, as described in Sec.",
"IIIA of Ref.",
"[14].",
"In particular, the $\\Omega _{\\mathrm {var}}$ functional may produce WFs that differ somewhat in shape from those of the original $\\Omega $ functional.",
"A detailed investigation of this aspect is, however, beyond the scope of the current study where we focus solely on the capability of $\\Omega _{\\mathrm {var}}$ to produce WFs with narrow spread distributions."
],
[
"Selecting the number of Wannier functions",
"For a given set of target bands, defined by the energy threshold $E_0$ , the number of WFs, $N_w = M^{{k}} + L^{{k}}$ , is a most important parameter for successful Wannerisation.",
"Depending on the application, different criteria may be used to quantify the quality of a set of WFs.",
"We have found that the spread (see App. )",
"of the most delocalized WF, $s_{\\mathrm {max}}$ , is generally a good quality-indicator, and we will use this measure along with the maximum and average band interpolation errors (see App. )",
"throughout the paper.",
"In simple cases, a natural value of $N_w$ may be guessed by analysing the band structure, considering symmetries, or using chemical intuition.",
"In the general case, however, the optimal $N_w$ cannot be guessed and a more systematic approach is desired, see Sec.",
"REF .",
"We note that the minimum possible value for $N_w$ is given by the largest number of bands lying below $E_0$ at any $\\mathbf {k}$ , $N_w^{\\mathrm {min}}(E_0) = \\max _{{k}} {\\sum _n H(E_0 - \\varepsilon _{n{k}}) }$ where $H$ is the Heaviside step function.",
"At such $\\mathbf {k}$ -points we have $M^{{k}} = N_w^{\\mathrm {min}}$ and thus $L^{{k}}=0$ , i.e.",
"no EDF.",
"In the POWF formalism there is no upper limit to $N_w$ (apart from the total number of bands available, $N_b$ ).",
"To illustrate how the $N_w$ parameter may influence the Wannierisation, we consider the case of monolayer MoS$_2$ .",
"In Fig.",
"REF (left) we show the interpolated band structure obtained from the POWFs obtained by maximising the $\\Omega $ spread functional.",
"Results are shown with an energy threshold ($E_0$ ) of 1 eV above the conduction band minimum (CBM) for $N_w=16$ and $N_w=17$ , respectively.",
"(The choice $N_w=16$ corresponds to the minimum number of WFs because exactly 16 bands fall below $E_0$ at approximately 1/3 along the $\\Gamma $ -M path.)",
"It is clear that the choice $N_w=17$ improves both the band structure interpolation and $s_{\\mathrm {max}}$ .",
"In the present case, this result could have been anticipated by an analysis of the band structure, which present an energy gap separating the lowest 17 bands from all higher lying bands.",
"Fig.",
"REF (right) shows resolution of the resulting set of WF over the eigenstates (at the $\\Gamma $ point).",
"From this analysis it is clear, that the target states, consisting of states below $E_0$ , can be perfectly completed by including the lowest 17 bands in the Wannierization.",
"In contrast, a frustrated solution is obtained for $N_w=16$ where one of the EDF becomes a mix of eigenstate 15 and 22.",
"In general, the effect of varying $N_w$ can be difficult to predict.",
"A prototype example where this happens is a non-elemental, low symmetry material with no natural band gaps above $E_0$ .",
"In such cases, the POWF localization algorithm, which may be seen as a bonding-antibonding completion procedure, might select EDF corresponding to high-energy eigenstates well separated from the target bands, or even mixtures of such [5]."
],
[
"Initial guess",
"The iterative localization procedure requires an initial guess for the rotation matrix $U$ and the EDF coefficient matrix $c$ .",
"The quality of the initial guess is essential.",
"This is particularly true for systems with many WFs where the iterative optimization algorithm is more likely to get trapped in a local minimum if the initial guess is far from the global minimum.",
"A natural choice is to start from a set of $N_{\\mathrm {AO}}$ atomic orbitals $\\lbrace g_i\\rbrace $ , and then project these onto the available eigenstates, producing the $N_b \\times N_{\\mathrm {AO}}$ matrix $P_{ni} = \\langle \\psi _n | g_i \\rangle $ .",
"A prescription for extracting $U$ and $c$ from $P$ can be found in Ref.",
"[5].",
"We not that $N_{\\mathrm {AO}} = N_w$ in this procedure."
],
[
"Protocol for automated Wannierization",
"In this section we present a protocol for automated construction of POWF suitable for high-throughput calculations.",
"We stress that the protocol can be used with any spread functional, e.g.",
"$\\Omega $ or $\\mathrm {\\Omega _{var}}$ ."
],
[
"Initial guess",
"It is possible, and sometimes useful, to rely on chemical intuition when selecting a set of atomic orbitals as initial guess for the Wannierisation.",
"On the other hand, chemical intuition is not easy to schematize in a form valid for general materials.",
"With automation a key motivation, we therefore propose a simple, generally applicable protocol for an initial guess, which does not require any parameters and which proved to be highly effective: For each atom with d-states in its valence electron configuration, we include a group of 5 atom-centered d orbitals, one for each value of the magnetic quantum number.",
"This is motivated by our observation that in such systems one typically finds WFs that closely mimick atom-centered $d$ -orbitals.",
"The total set of $d$ -orbitals ($N_d$ equals 5 times the number of transition metal atoms in a unit cell) then sets a lower limit for $N_w$ , in addition to the one set by the threshold energy, $E_0$ .",
"The set of $d$ -orbitals is complemented by $N_s=N_w-N_d$ $s$ -orbitals placed at random positions, but always within a radius of 1.5 of an atom.",
"These $s$ -orbitals can act as “nucleation centers” for atom- as well as bond-centered $s, p$ or $sp$ -like WFs.",
"All the aforementioned atomic orbitals are always set with a Gaussian of half-width 1 as radial dependence.",
"We did not perform any tests on materials with $f$ -electrons.",
"However, due to the highly localised nature of f-orbitals in general, we propose to treat such states similarly to $d$ -states, i.e.",
"include a group of 7 atom-centered f orbitals in the initial guess.",
"We have found that this choice of initial guess is both effective and robust in the sense that leads to fairly rapid convergence toward solutions representing either the global minimum or a local minimum close to the global one.",
"The variations induced by the randomness in the initial guess generally produce small variations in the resulting WFs.",
"Nevertheless, to reduce the influence of the randomness we always perform five independent localizations and pick the best solution (see caption of Figure REF as example).",
"Figure: Spread distribution of several calculations with different random seed for the initial guess.Each data point represents the result of a single calculation, for a total of 100 for each functional.We mention that an alternative method for initializing the Wannierisation is the SCDM proposed by Damle et al.",
"[15], that also does not require human intervention.",
"The SCDM did, however, not lead to improved performance over the initial guess protocol described above when applied to a set of materials with isolated groups of bands.",
"It is not straightforward to combine the SCDM with the disentanglement method of our POWF scheme and thus we were not able to test the SCDM for more complex situations.",
"The element of randomness in our initialization scheme may seem to be a weakness.",
"However, we see it as a strength because it allows us to repeat the Wannierisation with slightly different starting points, which often leads to slightly different outcomes of which the \"best\" solution can be selected.",
"The results in Fig.",
"REF show the presence of multiple nearby local maxima of both $\\Omega $ and $\\Omega _{\\mathrm {var}}$ for bulk silicon.",
"In this case, all the solutions are valid in the sense that the WFs are all well localized and real valued.",
"Nonetheless, in a given situation one may prefer a specific solution satisfying certain problem specific requirements.",
"The figure also clearly reveals the consistent reduction of the maximum spread $s_{\\mathrm {max}}$ when using the $\\mathrm {\\Omega _{var}}$ functional, in particular when used with a non-optimal $N_w$ , such as the minimum value.",
"The improvement becomes less pronounced when used with the optimal $N_w$ , but is still visible.",
"A more detailed discussion of these aspects are provided in Sec.",
"REF ."
],
[
"Optimal number of Wannier functions",
"When the optimal number of WFs cannot be guessed, c.f.",
"Sec.",
"REF , it must be computed.",
"We do this by constructing the POWFs for a range of $N_w \\ge N_w^{\\mathrm {min}}$ and selecting the solution presenting the smallest $s_{\\mathrm {max}}$ .",
"In other words, we add EDF to the Wannierization space as long as it reduces the spread of the least localized WF.",
"For the calculations presented in this work we have varied $N_w$ from $N_w^{\\mathrm {min}}$ to $N_w^{\\mathrm {min}}+5$ .",
"Depending on the size of the system (number of bands) and the available computational resources, a higher upper limit may be chosen.",
"Due to the randomness in the initial guess, we run 5 optimizations for each value of $N_w$ and select the best solution.",
"In total we thus perform 25 Wannierisations for each material.",
"Based on our experience, the resulting optimal $N_w$ is the same for $\\Omega $ and $\\mathrm {\\Omega _{var}}$ .",
"As an example, Fig.",
"REF shows $s_{\\mathrm {max}}$ for bulk GaAs.",
"The vertical lines represent the variations due to the randomness in the initial guess.",
"The minimal $\\langle s_{\\mathrm {max}}\\rangle $ appears for $N_w = 8$ .",
"As was found for MoS$_2$ , the optimal $N_w$ also produces a better interpolation of the band structure, in particular around the energy threshold ($E_0$ =CBM+2 eV), see Fig.",
"REF .",
"We do stress, however, that the size of $s_{\\mathrm {max}}$ is not always directly correlated with the band interpolation error[16], [6]."
],
[
"Results",
"In this section we present the results we obtained with our methods.",
"We start presenting the differences we observed with the new localization functional on few specific systems, then we move to a verification of the automated procedure on a set of 30 two-dimensional materials."
],
[
"An illustrative example: WMo$_3$ Te{{formula:c6ed5deb-ed28-46af-afad-c4db997ebddb}}",
"In order to demonstrate the effect of the variance term in our newly defined localization functional $\\mathrm {\\Omega _{var}}$ , we construct the Wannier functions of monolayer WMo$_3$ Te$_8$ .",
"This material is a 2D semiconductor with 12 atoms per unit cell, 52 occupied bands, and a band gap of 0.9 obtained from a DFT calculation with the PBE xc-functional [17], [7].",
"In particular, we set an energy threshold $E_0 = \\mathrm {CBM} + 2$ and computed 64 WFs, that is the minimal number of WFs needed to describe the states up to $E_0$ .",
"The large number of WFs helps in proving our thesis, the optimization algorithm is in fact more likely to choose a solution with few delocalized WFs, if the relative contribution to the total localization functional is lower.",
"For monolayer WMo$_3$ Te$_8$ we obtain an average spread (see App. )",
"of $\\bar{s} = 2.7$ 2 and a maximum spread of $s_{\\mathrm {max}} = 21.5$ 2 with the standard functional $\\Omega $ , while the variance minimizing functional $\\mathrm {\\Omega _{var}}$ converges to $\\bar{s} = 2.8$ 2 and $s_{\\mathrm {max}} = 5.1$ 2.",
"The most delocalized WFs for both functionals are shown in Figure REF .",
"For the band structure interpolation error (see App. )",
"we obtain $\\eta = 21$ ($\\eta _{\\mathrm {max}} = 420$ ) for the standard MLWFs produced by $\\Omega $ and $\\eta = 9$ ($\\eta _{\\mathrm {max}} = 143$ ) for the spread balanced WFs generated with $\\mathrm {\\Omega _{var}}$ .",
"As expected, the average localization of the WFs generated with the two different spread functionals is almost identical while the spread of the most delocalized WFs is greatly improved by $\\mathrm {\\Omega _{var}}$ .",
"Moreover, the variance reducing functional leads to a significant improvement in the tight-binding interpolation of the band structure.",
"We stress that the significant improvements found with $\\mathrm {\\Omega _{var}}$ for this specific material may not be fully representative.",
"Although we do generally find a significant improvement, i.e.",
"reduction, of $s_{\\mathrm {max}}$ , the band interpolation error is often similar to that obtained with $\\Omega $ (see Sec.",
"REF ).",
"The effect of the variance penalty term on the optimization of $\\mathrm {\\Omega _{var}}$ is plotted in Figure REF (c).",
"Comparing the value of the variance term between the two iterative optimizations, with $\\Omega $ and $\\mathrm {\\Omega _{var}}$ , it is clear that the penalty term leads to a different convergence path.",
"For the vast majority of the materials we tested, the number of steps required for the optimization of $\\mathrm {\\Omega _{var}}$ was comparable to or slightly larger than required for $\\Omega $ .",
"However, due to the additional terms in the gradient of $\\mathrm {\\Omega _{var}}$ , each step is roughly twice as expensive to evaluate in terms computational time.",
"Figure: Isosurface plots of the most delocalized WF of monolayer WMo 3 _3Te 8 _8 obtained with (a) the standard localization functional Ω\\Omega yielding a spread of s max =21.5s_{\\mathrm {max}} = 21.5 2 and (b) the variance reducing functional Ω var \\mathrm {\\Omega _{var}} yielding a spread of s max =5.1s_{\\mathrm {max}} = 5.1 2.",
"The Wannierisation has been performed for the minimal number of WFs consistent with an energy threshold of E 0 = CBM +2E_0 = \\mathrm {CBM} + 2 .",
"The isosurface level is 0.7-3/2.",
"(c) The value of the variance penalty term in Ω var \\mathrm {\\Omega _{var}} (see Eq.",
"()) during the iterative optimization.",
"Additional optimization steps with Ω var \\mathrm {\\Omega _{var}} can further decrease the variance term by a factor of 2."
],
[
"Spontaneous polarization",
"The spontaneous polarization of ferrolectrics comprises a prominent example of a physical quantity that is easily accessible from Wannier functions.",
"As shown by King-Smith and Vanderbilt [8] the change in polarization under an adiabatic deformation can be calculated by a Berry-phase type formula.",
"Typically the polarization of ferroelectrics is measured with respect to a centrosymmetric phase which is known to have vanishing polarization and the spontaneous polarization can then be computed by a single calculation in the polar phase.",
"To unravel the relation to Wannier functions is straightforward to show that the Wannier charge centers can be written as [14] rnwn|r|wn=V(2)3dkunk|ik|unk, where $V$ is the unit cell volume and $u_{n\\mathbf {k}}$ is the periodic part of a Bloch function.",
"Except for the factor of $V$ this expression is exactly the Berry phase formula for the electric contribution to the polarization for a single band and the full polarization can thus be written as P=1VaZara-1Vnoccrn, where $\\mathbf {r}_a$ is the position of nucleus $a$ with charge $Z_a$ .",
"Eq.",
"(REF ) is formally equivalent to the Berry phase expression for the polarization and the $2\\pi $ ambiguity in the Berry phase is reflected by the fact that the nuclei positions as well as the Wannier functions can be chosen in an arbitrary unit cell.",
"This expression for the polarization has the advantage of providing a clear physical interpretation of the polarization as the dipole resulting from a diplacement of Wannier charge centers from the nuclei positions.",
"Moreover, it strongly facilitates a microscopic analysis of how the polarization is affected by impurities or interfaces since one may monitor the shift in Wannier charge centers under an applied perturbation.",
"We have compared the calculated spontaneous polarization of tetragonal BaTiO$_3$ obtained with a direct implementation of the Berry phase method [18] with that obtained from Eq.",
"(REF ) using the $\\mathrm {\\Omega _{var}}$ spread functional Eq.",
"(REF ) to construct Wannier functions from the occupied states.",
"In the present case we have performed a full relaxation with the PBE functional using a $8\\times 8\\times 8$ $k$ -point mesh and 800 eV plane wave cutoff.",
"The result from both calculations is 45.4 $\\mu \\textrm {C}/\\textrm {cm}^2$ , which is in agreement with previous calculations [19], [20].",
"As expected, the new type of spread balanced Wannier functions thus reproduce the result obtained with standard maximally localized Wannier functions."
],
[
"Complex systems",
"While the generation of well localized Wannier functions is usually relatively straightforward for simple systems (small number of atoms, isolated group of bands, etc.",
"), it can be significantly more challenging in the general case.",
"To test the variance reducing localization functional on more complex systems we compare its performance to the standard $\\Omega $ functional for a nitrogen-vacancy (NV) defect center in a diamond crystal and a Ru(111) surface slab with adsorbed H, N, and O atoms, respectively.",
"The results are summarized in Table REF and confirm the previous conclusions.",
"In particular, the spread of the most delocalized WF is significantly reduced when using the $\\mathrm {\\Omega _{var}}$ functional.",
"Additional computational details are provided in App.",
"."
],
[
"Towards high-throughput applications",
"In the previous sections we have demonstrated our new Wannierisation scheme for several different types of systems.",
"In this section, we apply the scheme to a larger set of materials comprising 30 atomically thin two-dimensional (2D) materials randomly selected from the Computational 2D Materials Database [7].",
"The set includes 22 materials with finite band gaps and 8 metals, and covers several different crystal lattices and a large set of chemical elements.",
"The complete list of materials is provided in App.",
"and the computational details in App.",
".",
"Although we focus on 2D materials we expect our results to be representative for general material types.",
"We applied our Wannierisation scheme to each material and determined the optimal $N_w$ as the one yielding the lowest maximum spread, $s_{\\mathrm {max}}$ , see Sec.",
"REF .",
"For each value of $N_w$ in the range from $N_w^{\\mathrm {min}}$ to $N_w^{\\mathrm {min}}+5$ , we performed 5 independent Wannierisations using different initializations (differing due to the arbitrariness in the position of the $s$ -orbitals).",
"Following the procedure visualised in Fig.",
"REF (top), we used the average $s_{\\mathrm {max}}$ over the 5 sets of WFs to determine the optimal $N_w$ .",
"In all cases, we include eigenstates up to 2 above the conduction band minimum (Fermi level) for insulating (metallic) materials.",
"The results of the Wannierisation procedure for the 30 materials are summarised in Fig.",
"REF .",
"The left column shows the average and maximal spreads of the WFs generated with the $\\Omega $ and $\\Omega _{\\mathrm {var}}$ functionals for the minimal and optimal number of WFs, respectively.",
"Note the different scales on the axes.",
"The symbols indicate the mean values of $\\bar{s}$ and $s_{\\mathrm {max}}$ over the 5 initializations, and the lines indicate the standard deviation.",
"The green and red dashed lines indicate $\\langle \\bar{s} \\rangle = \\langle s_{\\mathrm {max}}\\rangle $ and $\\langle \\bar{s} \\rangle = 2\\langle s_{\\mathrm {max}}\\rangle $ , respectively.",
"For generating well-localized WFs, the number of WFs, i.e.",
"using $N_w^{opt}$ rather than $N_w^{min}$ , was found to be more critical than the type spread functional, i.e.",
"$\\Omega $ versus $\\Omega _{\\mathrm {var}}$ .",
"However, the use of $\\Omega _{\\mathrm {var}}$ instead of $\\Omega $ does lead to a significant improvement in the localisation for a fixed $N_w$ .",
"This improvement is most pronounced for non-optimal values of $N_w$ , e.g.",
"the minimal $N_w$ .",
"In particular, for the two materials WMo$_3$ Te$_8$ and ZrTi$_3$ Te$_8$ (both with $N_w>60$ and indicated by red circles), we were not able to localize all the WFs when using the $\\Omega $ -functional and the minimal $N_w$ .",
"This issue did not occur with the $\\Omega _{\\mathrm {var}}$ -functional.",
"Even when using the optimal $N_w$ , the improvement by the $\\Omega _{\\mathrm {var}}$ -functional is significant.",
"Not only do we obtain better localisation of the least localised WF ($s_{\\mathrm {max}}$ ) without sacrificing the average localisation ($\\bar{s}$ ), the standard deviations on both $\\bar{s}$ and $s_{\\mathrm {max}}$ is also lowered.",
"The band interpolation errors (again averaged over the 5 initializations) are shown in the right column of Fig.",
"REF .",
"As previously found and discussed in Section REF , the errors decrease significantly when using the optimized $N_w$ as compared to the minimal $N_w$ .",
"With the optimal $N_w$ the majority of the materials show a maximum error below 20.",
"A few materials show higher band errors, which is related to significant band crossings (band entanglement) with higher energy bands close to the $E_0$ energy cutoff.",
"We note in passing that if accurate band interpolation in this region is required, one may simply increase $E_0$ , which will push the inaccuracies to higher band energies.",
"While the $\\Omega _{\\mathrm {var}}$ spread functional improves the localisation properties of WFs, it does not represent a significant improvement over $\\Omega $ in terms of the band interpolation error.",
"This shows that the band interpolation error is not directly correlated with the localisation properties of the WFs.",
"More precisely, it is not directly correlated with the spread of the most delocalised WF, $s_{\\mathrm {max}}$ , which is always significantly and consistently reduced by using $\\Omega _{\\mathrm {var}}$ .",
"We stress, however, that there are many other applications of WFs where robust localisation of all WFs, is of key importance.",
"These include the interpolation of electron-phonon matrix elements[21], [22], calculation of Berry curvatures and conductivities[23], and basis sets for electron transport calculations based on non-equilibrium Green's functions[24], [25].",
"Table: Comparison of the spread distribution over the entire set of 2D materials.The definitions of s ¯ tot \\bar{s}_{\\mathrm {tot}}, σ ¯ tot \\bar{\\sigma }_{\\mathrm {tot}}, s ¯ max \\bar{s}_{\\mathrm {max}} and σ ¯ max \\bar{\\sigma }_{\\mathrm {max}} are given in Eqs.",
", and the following text.For each value we also report the standard deviation of the mean (i.e.",
"the standard error).The effect of penalizing the spread variance in $\\Omega _{\\mathrm {var}}$ can be illustrated by considering the localisation properties of the resulting sets of WFs averaged over all 30 materials for a fixed $N_w$ .",
"Specifically, we fix $N_w$ at either its minimal or optimal value, and consider the mean of the average spread $\\langle \\langle \\bar{s} \\rangle _{\\mathrm {init}} \\rangle _{\\mathrm {mat}} = \\frac{1}{N_{\\mathrm {mat}}} \\sum ^{N_{\\mathrm {mat}}}_{m=1} \\left( \\frac{1}{N_{\\mathrm {init}}} \\sum ^{N_{\\mathrm {init}}}_{i=1} \\bar{s}_i^m \\right)$ and the mean standard deviation of the average spread over 5 initialisations $\\langle \\sigma _{\\mathrm {init}}(\\bar{s}) \\rangle _{\\mathrm {mat}} = \\frac{1}{N_{\\mathrm {mat}}} \\sum ^{N_{\\mathrm {mat}}}_{m=1} \\sqrt{\\mathrm {Var}_{\\textrm {init}}[\\bar{s}_i^m]},$ where $N_{\\mathrm {mat}}$ is the total number of materials (here $N_{\\mathrm {mat}}=30$ ), $N_{\\mathrm {init}}$ is the number of independent optimizations with different initial guess (here $N_{\\mathrm {init}}=5$ ), $\\bar{s}_i^m$ is the average spread of the WFs of material $m$ with initialization $i$ , and $\\mathrm {Var}_{\\mathrm {init}}$ is the variance of the $N_{\\mathrm {init}}$ independent runs.",
"In the same way, we can define $\\langle \\langle s_{\\mathrm {max}} \\rangle _{\\mathrm {init}} \\rangle _{\\mathrm {mat}}$ and $\\langle \\sigma _{\\mathrm {init}}(s_{\\mathrm {max}}) \\rangle _{\\mathrm {mat}}$ by replacing $\\bar{s}$ with $s_{\\mathrm {max}}$ in the above equations.",
"Note, that $\\langle \\sigma _{\\mathrm {init}}(\\bar{s}) \\rangle _{\\mathrm {mat}}$ and $\\langle \\sigma _{\\mathrm {init}}(s_{\\mathrm {max}}) \\rangle _{\\mathrm {mat}}$ measure the variation in the average and maximal spread in dependence of the initial guess, i.e.",
"the robustness of the Wannierisation.",
"In particular, it does not reflect the variation of the spreads within a given set of WFs.",
"As demonstrated on several places in the paper, the latter is always significantly and consistently reduced when using $\\Omega _{\\mathrm {var}}$ .",
"Table REF shows the results for the four quantities $\\langle \\langle \\bar{s} \\rangle _{\\mathrm {init}} \\rangle _{\\mathrm {mat}}$ , $\\langle \\langle s_{\\mathrm {max}} \\rangle _{\\mathrm {init}} \\rangle _{\\mathrm {mat}}$ , $\\langle \\sigma _{\\mathrm {init}}(\\bar{s}) \\rangle _{\\mathrm {mat}}$ , and $\\langle \\sigma _{\\mathrm {init}}(s_{\\mathrm {max}}) \\rangle _{\\mathrm {mat}}$ for each of the spread functionals $\\Omega $ and $\\Omega _{\\mathrm {var}}$ .",
"These numbers summarise the information in Fig.",
"REF .",
"On this basis we conclude that the penalization of the spread variance as done in the $\\mathrm {\\Omega _{var}}$ -functional, does not affect the average spread of the WFs, but consistently decreases the maximum spread and improves the robustness with respect to the choice of initial orbitals."
],
[
"Data availability",
"The data presented in this article together with information required for its reproduction, are available in a Git repository [26].",
"For more details see also App.",
".",
"We have introduced a new localization functional for generating Wannier functions (WFs) with balanced spread distributions.",
"By penalizing the variance of the spread distribution, the algorithm becomes less prone to produce individual WFs with large spreads.",
"It thereby resolves a well known problem of standard Wannierisation schemes, which is particularly important when applied to complex systems.",
"Furthermore, we have proposed a general and fully automatic algorithm for selecting the optimal number of WFs and the initial set of orbitals for the Wannierisation procedure.",
"Application to an extensive test suite comprising both bulk and monolayer materials, point defects and atoms on metal slabs, consistently show that these algorithms comprise a highly robust approach for generating WFs, which should be particularly useful for applications to complex systems and/or high-throughput studies.",
"The methods are implemented in Python and are available as part of the open-source Atomic Simulation Environment (ASE)."
],
[
"Acknowledgments",
"The Center for Nanostructured Graphene (CNG) is sponsored by the Danish National Research Foundation, Project DNRF103.",
"This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program grant agreement No 773122 (LIMA)."
],
[
"Wannier function spread",
"We define the spread of the single WF $s_n = - \\frac{1}{(2 \\pi )^2} \\sum _{\\alpha =1}^{N_G} W_{\\alpha } \\log \\left(|Z_{\\alpha , nn}|^2 \\right).$ This quantity is an approximation for the spread of a single WF in 2 [11], if the weights $W_{\\alpha }$ retain the physical dimensions.",
"This definition has the feature of exponentially increasing in case of completely delocalized WFs, which can help with the identification of localization issues but at the same time it does not retain any physical meaning for delocalized WFs.",
"Although, the latter applies for every approximation of the WF spread, to some extent [11].",
"In addition, most definitions of spread or localization functional have a convergence with the $k$ -point grid density, as observed and studied in Ref.",
"[27].",
"In the present work we often refer to $\\bar{s}$ , the average of $s_n$ over the set of WFs, and $s_{max}$ , the maximum value of $s_n$ over the set of WFs."
],
[
"Band interpolation error",
"In order to quantitatively estimate the quality of the band interpolation we introduce a quantity called band distance or band interpolation error, that measures the difference between the Kohn-Sham and the Wannier-interpolated band structures.",
"The definition is the same that appears in published papers [16], [6] and allows to easily compare the results.",
"The band distance is computed up to the energy threshold $E_0$ , introduced in Section REF .",
"If the band structure has a gap at the energy threshold, then the distance is computed only on the energy bands below the gap.",
"We define the average band distance $\\eta $ and the maximum contribution to the band distance $\\eta _{\\mathrm {max}}$ as $\\eta = \\sqrt{\\sum _{n{k}}\\frac{(\\varepsilon ^{\\mathrm {KS}}_{n{k}} - \\varepsilon ^{\\mathrm {Wan}}_{n{k}})^2}{N_b N_{{k}}}}$ $\\eta _{\\mathrm {max}} = \\text{max}_{n{k}}(|\\varepsilon ^{\\mathrm {KS}}_{n{k}} - \\varepsilon ^{\\mathrm {Wan}}_{n{k}}|).$ where $\\varepsilon ^{KS}$ and $\\varepsilon ^{Wan}$ are the Kohn-Sham eigenvalues and their Wannier interpolation respectively, $N_b$ is the number of bands and $N_{{k}}$ the number of $k$ points.",
"In the case of entangled bands we introduce a different definition which uses a smearing function to weight the contributions from the bands around the energy threshold $\\eta = \\sqrt{\\frac{\\sum _{n{k}}(\\varepsilon ^{\\mathrm {KS}}_{n{k}} - \\varepsilon ^{\\mathrm {Wan}}_{n{k}})^2 \\tilde{f}_{n{k}}}{\\sum _{n{k}} \\tilde{f}_{n{k}}}}$ $\\eta _{\\mathrm {max}} = \\text{max}_{n{k}}(\\tilde{f}_{n{k}} |\\varepsilon ^{\\mathrm {KS}}_{n{k}} - \\varepsilon ^{\\mathrm {Wan}}_{n{k}}|)$ where $\\tilde{f}_{n{k}} = \\sqrt{f^{\\mathrm {KS}}_{n{k}}(\\nu , \\tau ) f^{\\mathrm {Wan}}_{n{k}}(\\nu , \\tau )}$ and $f_{n{k}}(\\nu , \\tau )$ is the Fermi-Dirac distribution for the state at energy $\\varepsilon _{n{k}}$ , $\\nu $ is a fictitious chemical potential fixed at $E_0$ and $\\tau $ is a smearing width fixed to 0.1."
],
[
"Computational details",
"For each material in this work we performed a self-consistent PBE calculation with the GPAW code[28] using a Monkhorst-Pack grid [29] with a minimum density of 5 $k$ -points per and a real-space grid with a spacing of 0.2.",
"Several unoccupied states were included in the calculation.",
"The band structure calculation, used to evaluate the band interpolation accuracy, was performed along a path with minimum density of 50 $k$ -points per .",
"For all materials in Sec.",
"REF we used the structure available in the public C2DB database [7]."
],
[
"Methods for complex systems",
"For the NV center in diamond we started from a pristine diamond bulk, in a cubic unit cell of 64 atoms we substituted one carbon atom with a nitrogen atom.",
"We then proceeded in the structure relaxation using GPAW [28], with a 2x2x2 Monkhorst-Pack grid of $k$ -points in the Brillouin zone (BZ), plane waves basis set with an energy cutoff of 400 and an initial charge of -1 elementary charges.",
"The optimization algorithm was LBFGS [30] and the force threshold was at 0.05 a.u.. After the structure was relaxed we ran a self-consistent and a non self-consistent calculation, with a BZ sampling density of 2 and 5 $k$ -points per respectively.",
"The self-consistent calculation was converged for every state up to 4 from the conduction band minimum (CBM).",
"The rest of the methods were in line with Section REF .",
"With the adsorption systems we followed a similar workflow with ASE [9].",
"We created a 2x2x4 slab of ruthenium with HCP(0001) surface and a vacuum layer of 7 along the $z$ direction.",
"Then the adsorbate was placed on the HCP site at 1, 1.108, 1.257 for H, N, O respectively.",
"The structure was then relaxed, fixing the first two layers of Ru, using the FD method and the PBE [17] xc-functional in GPAW, again using LBFGS as optimization algorithm and 0.05 a.u.",
"as forces threshold.",
"After the relaxation, a self-consistent and a non self-consistent calculations were ran on each system, following the workflow for the NV center in diamond."
],
[
"List of 2D materials",
"The 2D structures in Table REF are picked from the C2DB database [7].",
"The values for the maximum spread are the mean and the standard deviation over 5 calculations with different random seeds for the initial guess.",
"The standard deviation for the maximum band interpolation error is computed with the same method.",
"For the maximum band interpolation and the number of WFs (the latter varies only in the case of optimal $N_w$ ) we use the value from the most localized set (lowest maximum spread) over 5 calculations, instead of the mean value.",
"In the last row we show the average value and the average standard deviation for each column.",
"Table: List of the 2D materials considered in Sec.",
".",
"The \"Crystal type\" in the second column follows the convention of the C2DB database and stands for stoichiometry-space group-occupied Wyckoff positions."
],
[
"Software availability",
"Most of the software used and developed in this project is open-source and available for free.",
"The density functional theory code used is GPAW 20.10 [28] with version 0.9.2 of the atomic setups, together with ASE 3.20 [9].",
"The latter also includes the Wannier module in which the new spread functional has been implemented.",
"The Wannier functions are represented with Vesta [31], which is available for free.",
"The plots are produced with Matplotlib [32] and the independent calculations were run in parallel with the help of GNU parallel [33], both of them are open-source software."
]
] | 2107.01722 |
[
[
"Exploring Data Pipelines through the Process Lens: a Reference Model\n forComputer Vision"
],
[
"Abstract Researchers have identified datasets used for training computer vision (CV) models as an important source of hazardous outcomes, and continue to examine popular CV datasets to expose their harms.",
"These works tend to treat datasets as objects, or focus on particular steps in data production pipelines.",
"We argue here that we could further systematize our analysis of harms by examining CV data pipelines through a process-oriented lens that captures the creation, the evolution and use of these datasets.",
"As a step towards cultivating a process-oriented lens, we embarked on an empirical study of CV data pipelines informed by the field of method engineering.",
"We present here a preliminary result: a reference model of CV data pipelines.",
"Besides exploring the questions that this endeavor raises, we discuss how the process lens could support researchers in discovering understudied issues, and could help practitioners in making their processes more transparent."
],
[
"Introduction",
"Training data can have a considerable impact on the outputs of a machine learning system, by affecting its accuracy, or causing undesirable or even harmful effects when the system is deployed in applications that act in the world.",
"In response to these harms, machine-learning and interdisciplinary researchers who tackle the societal impact of computer vision typically a) explore characteristics of datasets with respect to their potentially harmful impacts on society [8], [28], [22], e.g., taxonomies with offensive labels, often resulting from uncritical practices [2]; b) identify and mitigate harms by adapting single, discrete processes in the data pipeline, e.g.",
"filtering out the offensive and non-visual labels of the taxonomy, [33], [2], [17]; c) propose techniques to make datasets more transparent [12], to encourage analysts to be more accountable and reflective of potential harms.",
"These forms of enquiry presuppose considering datasets as objects with configurable and conceivable properties.",
"Recent exceptions [23], [26], [7], [10], [16] hint at another direction, that explores not only the datasets themselves but considers the data pipelines and actors behind their creation and use more broadly.",
"In this paper, we take inspiration from these works and explore a process-oriented lens on data pipelines.",
"Such a lens may encourage researchers and practitioners to be more rigorous in the production and the analysis of datasets vis a vis their potential societal impact.",
"Most importantly, a process lens will enable to recognize and capture the following: Datasets as complex, living objects.",
"Datasets are not inert but living objects.",
"They are regularly updated in production to improve models, while they are often re-purposed and adapted for new applications.",
"Moreover, datasets interact with each other, particularly through models, e.g.",
"they are composed using transfer learning [19] such as for improving facial recognition systems meant to be applied in a wide variety of contexts [25].",
"However, while datasets can create harms at each phase of their life, academic works on datasets typically are ambiguous about these phases and seem to be based only on their first release.",
"The pipeline as a vector of harms.",
"Not only the processes of the pipeline that follow the first release might raise issues, but also the interplay between these processes and the ones leading to the first release, as it defines the potential errors of a model.",
"For instance, in Detroit, following harmful mistakes of a facial recognition system in deployment, the police has decided to only apply it to still images, as it is closer to the training data collected in a static setting in development, which shows how some harms can occur due to problematic mismatches between processes in deployment and development phases [1].",
"Some existing works have already investigated the impact of pipeline processes such as rotation, cropping, and other pre-processing on models' accuracy [11], [15], [34], although not necessarily on societal harms [8].",
"Nonetheless, little attention was given to studying the impact of sequences of processes.",
"Reasoning behind data pipeline design.",
"The lack of transparency about the design of a dataset's pipeline (see examples in sec:sketches) makes it challenging for researchers to analyse the procedural harms or downstream harms of models training on a dataset.",
"Such transparency would reveal not only the processes themselves, but also the reasoning behind them.",
"For instance, while it is valuable to study datasets in-depth, it can be intractable to analyse each label and image due to the scale.",
"Documentation of the reasoning behind the choices of label ontologies, and the process of collecting corresponding images, would improve the analysis.",
"E.g., knowing that WordNet was used to select the labels of ImageNet [9] enables us to directly reflect on harms from the point of view of the taxonomy [8], and knowing that images were collected from country-specific search engines enables us to foresee the cultural skews in label representations [28].",
"Such matters are not consistently included in academic works, and they are not always explicitly asked for in documentation methodologies [12].",
"Our contributions.",
"In this paper, we devise a systematic approach to study the data pipelines in computer vision.",
"For this, we combine techniques from the development of reference models, method engineering, and process engineering.",
"These are well-established, intertwined fields [31] that propose methods to capture complex sets of processes.",
"Using these techniques, we develop an empirically informed reference model of data pipelines in computer vision.",
"We then reflect on its potential usefulness, as well as the possible limitations of this approach for creating transparency and accountability for practitioners and researchers."
],
[
"Methodology",
"Our objective is to build a reference model of the data pipeline in machine learning-based computer vision tasks.",
"The literature on method, process and reference-model engineering identifies the following requirements towards reference models: representativity of the pipeline used in practice at an abstraction level high enough for including comprehensively each of their processes [21], [13], [4], modularity to easily add new processes [14], [4], clarity for individuals interested in different parts of the pipeline [21], [4], and sufficient level of detail to enable actionable reporting, and reflection on potential ethical issues.",
"Following these requirements, we build: a high-level reference model that serves as an ontology of our main concepts; a set of low-level models of each identifiable process within existing pipelines; a set of process maps reflecting the pipelines of the main computer vision practices.",
"Populating the models and maps.",
"The methods from each of the mentioned fields have some advantages and limitations.",
"Process patterns in process engineering allow for a fine-grained description of the processes, but generally impose an order on these processes, which does not correspond to the computer vision pipelines we analysed.",
"Reference models do not necessarily impose an order, but can be too generic for our desired level of detail.",
"However, the method engineering formalisms can alleviate these issues as they support processes of different granularities, and the interactions between the processes can be represented in a map without necessarily enforcing order.",
"Thus, we employed a mix of methods from the different disciplines.",
"The details can be found in sec:method.",
"In short: 1) We did a systematic literature review that presents computer vision datasets (51 publications) or specific processes to build datasets (16 publications) selected out of 220 relevant papers found using different search methods.",
"2) We identified and noted the main processes within each publication.",
"3) By comparing each description, we iteratively identified common processes, refined their granularity, and aggregated them into meaningful method processes.",
"4) We described each of them with a relevant formalism, and drew the maps of each dataset pipeline.",
"5) We searched for grey areas and are in the process of conducting expert interviews to further validate and refine our reference model, and to model processes that are not objects of scientific publications (e.g.",
"processes for serving data in deployment)."
],
[
"Results: Reference model of data processes",
"In this section, we present our formalism of the components of the reference model summarized in fig:UML, and provide examples of its application.",
"Figure: UML diagram describing the main components of our reference model formalism, and activity diagram for the process map.",
"Relationships: association, ◊\\lozenge aggregation (i.e.",
"“part-of”), ⧫\\blacklozenge composition, ⊳\\vartriangleright inheritance (i.e.",
"“is-a”)."
],
[
"Products",
"The products are the essential components of the datasets.",
"Each product is defined by an object (e.g.",
"an image, a set of images, an image-annotation pair, etc.)",
"and a set of properties.",
"Based on the literature review, we have identified the following types of products: Label-related products.",
"A dataset comprises a set of target labels, often organized into a hierarchical taxonomy, with higher levels of the taxonomy representing more abstract concepts (e.g.",
"animal) and lower levels representing more fine-grained concepts (e.g.",
"types of animals).",
"The main properties considered in the reviewed literature are the size (number of labels) and the number of levels in the hierarchy.",
"Many properties of labels are currently implicit, whereas other properties, like where the taxonomy originates from, can be relevant for capturing potential harms.",
"Image-related products.",
"A dataset typically contains a set of images.",
"The size of the set, the content of the images and especially its diversity are put forward as main properties in the reviewed literature.",
"Additional properties can be considered, especially the metadata attached to the images (e.g.",
"author of the images, etc.",
"), physical characteristics of the images such as their quality, resolution, etc., and the distribution of these characteristics across the set.",
"Annotation-related products.",
"These products can be thought as a list of tuples, each composed of an image and one or multiple annotations.",
"The properties put forward are often the number and quality of annotations per image."
],
[
"Chunks",
"Borrowing a standard term from method engineering, we refer to the processes of the data pipeline as chunks.",
"They can be composed: a chunk can consist of multiple lower-level chunks.",
"We define 3 major granularity levels.",
"1) The pipeline level—the main sections of the data pipeline.",
"These are the development of training data, the processes of improving and updating the training data in deployment (feedback loops), and the processing of data at inference time.",
"2) The process level—the individual activities taking place in the chunks of the pipeline level (e.g., data collection).",
"3) The task level—the sub-activities required to conduct the activities of the prior level (e.g.",
"query preparation and image crawling for data collection).",
"These chunks can be further divided into sub-tasks when necessary.",
"Each chunk has an associated guideline that defines the activity it represents in relation to the products; and a descriptor that uniquely identifies it.",
"A guideline is composed of: i) an intention description, i.e.",
"the goal of the chunk, ii) a situation description, i.e.",
"the input and output products of the chunk, iii) a strategy description that describes the activity textually (i.e.",
"on the finest granularity), or indicates the composition of a sequence of lower level chunks to perform the activity, iv) a set of strategy hyperparameters allowing to be precise about the design choices in the strategy.",
"Based on the literature review, we have identified the following process-level chunks: label definition, data collection, data annotation, data filtering, data processing, data augmentation, data splitting, and product refinement.",
"We provide a short description of these, and some task-level chunks that we have identified in sec:activity."
],
[
"Process maps",
"Each dataset is created in a pipeline that uses different chunks, in different orders, at different granularities.",
"Composing chunks in various ways impacts the final dataset products and the outputs of a resulting model.",
"We formalise the overall process, i.e.",
"the connections between chunks, through intention-strategy maps and their sections, as inspired from method engineering that assembles chunks to compose individual situated methods.",
"We develop these maps to materialize the process lens mentioned in the sec:intro, and to evaluate its potential and limitations.",
"Intuitively, an intention-strategy map represents a sequence of chunks.",
"Formally, the intention-strategy map is an ordered sequence of intentions connected by the strategies to realize them (see fig:UML), each intention-strategy pair referring to a chunk guideline.",
"Attached to the sequence are sections.",
"Each section consists of the initial and subsequent intention of each of the chunks, and the strategy to realize the subsequent intention, i.e.",
"the description of how the chunk fulfills its intention.",
"Additionally, the section also contains an intention selection guideline specifying the reason to go from the first intention to the second, and a strategy selection guideline specifying the reason for choosing this particular strategy to perform this intention.",
"As an illustration, fig:imagenet provides an example of a segment of the process map for the ImageNet creation process.",
"Figure: Example segments of the process map associated to the development of ImageNet .We identified multiple chunks and process maps in our literature review, which shows that a plurality of sequences exists.",
"This suggests that there is no standardized data pipeline in computer vision.",
"The process-level chunks are not always the same, and the order might differ widely or may not be reported at all.",
"The data-processing chunks, for example, happen either after collection, after annotation, or after filtering.",
"The task-level chunks and their parameters may also vary between the training and deployment phases.",
"While both may impact the outputs of the model, they are often not clearly specified in the papers."
],
[
"Discussion",
"Introducing processes via the formalism presents advantages for researchers and practitioners of computer vision."
],
[
"Insights for researchers",
"Identification of knowledge gaps.",
"Our formalism enables to systematically analyze typical processes employed in computer vision across tasks.",
"This unifying lens uncovers the grey areas that escape scrutiny.",
"In fact, we believe that only a fraction of existing process chunks have been analyzed on the subject of ethical considerations and downstream negative consequences [24].",
"For example, our framework highlighted an understudied issue: the papers we reviewed rarely report on data processing chunks—yet the interaction of these chunks with other pipeline chunks can potentially have a significant impact.",
"For instance, cropping images after labeling might exclude the visual information relevant to the label, making the model learn wrong associations, a matter that seems to not have been studied previously.",
"Our formalism allows to surface this issue as it makes apparent the related chunks, and the current lack of reporting on them.",
"Process details.",
"Applying our formalism also uncovers what chunks, metrics, and intentions the researchers deem worth documenting in the publications.",
"This can guide researchers towards identifying the chunks that should merit more detailed descriptions.",
"For example, while a few interdisciplinary publications mention issues with the label-related products, less than 10% of the reviewed papers were found to refer to some of the properties of these products, e.g.",
"only the SUN database [32] and MS-COCO [20] outline their reasoning around the completeness of the set of labels employed.",
"As for data collection and filtering, only the authors of Tiny Images [30] make transparent the potential biases in images retrieved from search engines, and include a study of the dataset noise with and without filtering these images.",
"A more rigorous documentation could support a more systematic analysis of potential issues."
],
[
"Insights for practitioners",
"Supporting developers or auditors in making processes surrounding data more transparent and structured with our formalism could help with capturing or fostering reflection around potential issues.",
"Compared to previous frameworks [12], our formalism enables to report on the entire pipeline surrounding the dataset instead of solely the dataset or a subset of processes executed to develop it.",
"For instance, it outlines both the development and deployment pipelines, which makes it easier to identify potential differences in terms of chunk, chunk order, and chunk hyperparameters between them.",
"These differences can lead to distribution shift, resulting in misclassifications, unfairness in outputs, or non-robustness, as the Detroit police example showed.",
"The body of documented chunks of an organization can grow over time.",
"The identical format and reusable nature of chunks may help practitioners to easily identify the chunks of a new pipeline, and to build on existing documentation.",
"It may also encourage them to share ethical considerations corresponding to chunks across pipelines."
],
[
"Reflections and limitations",
"Despite potential advantages, the feasibility of building a satisfying reference model for systematizing the analysis of data pipelines for harms requires further discussion.",
"Most papers we studied, and our preliminary interviews with practitioners, have shown a divergence in data pipeline design, including lack of common chunks or order of processes.",
"This has parallels in software engineering.",
"Prior work in method engineering recognizes that all projects are different and cannot be supported by a single method, but instead adapted methodological guidance should be proposed [4].",
"However, the divergences in CV data pipelines pose challenges in fulfilling our requirements towards representativity, comprehensiveness, and clarity.",
"This raises a number of questions.",
"For example: Can we develop a reference model that would capture all these messy or idiosyncratic processes?",
"Would its use in production be tedious and easily out of date?",
"Could modularity help in incorporating further processes and details?",
"Would such a model give sufficient structure to the reporting of the data pipelines for purposes of increased transparency and accountability?",
"Finally, our primary interest lies in exploring the use of such formalisms in systematizing the identification of harms that may stem from data pipelines and surrounding processes.",
"Whether such use of a reference model would aid in more effective and efficient identification and action on harms in data pipelines requires further study.",
"Other fields such as information retrieval have developed a common language and benchmarks to study their systems of interest and foster communication between industry, academia and governments, enabling progress on their respective problems [27].",
"Whether this would also apply to computer vision and analysis of associated harms is yet to be seen.",
"Our empirical analysis of data pipelines in this paper has illustrated some of the envisioned benefits of a process-oriented lens.",
"The sole fact that the analysis has also raised the diversity of questions above hints at the richness of this lens, that we plan to investigate in the future."
],
[
"The Art of Sketching Data Pipelines",
"It is difficult to even find sketches of data pipelines in publications fig:impubli.",
"They are easier to find in grey materials of companies, such as for Amazon fig:amazon, Google fig:google, and DynamAI fig:dynamai.",
"We show examples of these sketches below.",
"The main observations one can make out of these sketches is that they are very diverse, without any uniformity.",
"They do not all use the same level of granularity to talk about the data pipelines, they do not all use the same vocabulary, and they refer to various different processes, often mentioning the training data development process but not the deployment processes.",
"This variability in the sketches does not allow to reflect on the potential harms of the pipelines in a global way, that can be re-used to talk about different pipelines.",
"Figure: Dataset creation pipeline from .Figure: Dataset creation pipeline from a Google course on machine learning https://developers.google.com/machine-learning/data-prep/process?hl=frFigure: Dataset creation pipeline from Dynam.AI https://www.dynam.ai/computer-vision-projects-management-part-1/, B2B company that implements computer vision solutions.Figure: Dataset creation pipeline from Amazon https://aws.amazon.com/blogs/iot/sagemaker-object-detection-greengrass-part-1-of-3/"
],
[
"Systematic review of the literature",
"We systematically surveyed the computer science publications that present computer vision datasets, or that propose methods pertaining to the creation of datasets.",
"For that, we went through the list of publications of the following top conferences CVPR, ECCV, ICCV, NeurIPS, ICLR and associated workshops in the general field of machine learning and in the field of computer vision more specifically –the assumption here is that in the more general conferences, some computer vision publications might appear, and some papers might mention general data pipelines with example in computer vision–, for the years 2015 to June 2020.",
"Then we proceeded to snowball sampling from the references of the papers initially collected.",
"We only retained publications related to images (we excluded video dataset papers and 3D based works), that touch upon classification or segmentation tasks, as the data pipelines for other types of data and tasks vary even more than just for these image tasks.",
"It amounted to around 270 papers.",
"We filtered out other types of tasks such as referring expression comprehension [6], image captioning and visual question answering because the data pipelines of many of these datasets are not described in details in the papers, and are not always published in the field of computer vision but also natural language processing.",
"The set of applications of this pool of publications (around 220 papers) is large.",
"We further scoped the study by selecting datasets and research publications that have as object of study objects (things), scenes and stuffs, or persons (mainly faces and emotions), as they are one of the main current interests in computer vision research.",
"We filtered out datasets for autonomous driving and aerial views as they consist mostly of road and field views, which is a more restricted set of classification labels and hence present mostly a subset of processes retrieved from the publications we selected (and there are –too– many of them), and a subset of potential harms.",
"Although works relying on these types of data and / or application are not included and we assume that many points we learn from our survey will apply to them, it would surely be beneficial to repeat our analysis on these datasets in the future, as even more harms might arise from them.",
"In the end, we selected 51 dataset papers to study thoroughly.",
"We also analysed shorter datasets of natural topics such as animals (e.g.",
"[18]) and flora [29], as we noticed that although they serve for the same type of classification tasks, their collection process differs from the other works.",
"Concerning computer vision papers that relate to processes of the pipelines, we solely found papers that studied and proposed ways to optimize the crowdsourcing components of the pipelines amounting to the number of 16.",
"Only one paper [5] seems to study the impact of the image signal processing pipeline on the accuracy and energy consumption of the following learning pipeline."
],
[
"Interviews",
"As for the interviews, we plan to interview two groups of individuals: researchers working in the field of computer vision, and practitioners who develop computer vision pipelines.",
"In both cases, our goal is to identify limitations of the initial reference model and its population that we drew from the literature review.",
"For that, we first ask the participants to talk about the pipelines they know of, and to describe them (this way, we can also identify the processes they deem worth reporting).",
"We then present to the interview participants a drawing of the reference model, and we ask them to discuss it critically, i.e.",
"to identify chunks that could be missing or hyperparameters that could be different, as well as to re-order the chunks based on their experience.",
"This way, we can iteratively improve our reference model and validate it.",
"Finally, we interrogate them around harms they know of in relation to the different processes."
],
[
"Reference model -further details",
"[leftmargin=*,topsep=0pt, noitemsep] Label definition.",
"This chunk consists in defining the label-related products.",
"The method employed to define such objects (which maps to various sub-chunks depending on the dataset), their source, and the individuals who define them are the main hyperparameters.",
"Data collection.",
"It consists in collecting a set of potential image-related products for the dataset.",
"A multitude of hyperparameters lay within this activity, stemming again from the source of images, the individuals collecting them, and the general approach to gather them.",
"Data annotation.",
"It consists in collecting the set of image-annotation tuples of the dataset.",
"It is not needed when images are photographed by the researchers for pre-defined labels, but it is when they are collected from the Internet.",
"This chunk strategy is instantiated in different task-chunks, with an automatic process (pre-trained machine learning models, or meta data of the images), a human process or a hybrid one.",
"The human process chunks can then be defined into sub-chunks.",
"Data filtering.",
"It consists in fine-tuning the set of image products by removing images that do not fit the requirements fixed for the dataset.",
"It impacts the properties of the image-related and possibly annotation-related products.",
"Data processing.",
"It consists in transforming the individual images in the image-related products according to certain processing criteria for the images to verify certain properties usually in terms of size (downsampling or upsampling), and possibly color hue, lighting, etc.",
"This is done to support their use within machine learning classifiers, and in some cases to simplify the learning tasks.",
"Data augmentation.",
"Although not an activity mentioned in dataset papers, it is a standard practice in industry, and re-uses task-chunks of data processing for modifying image-related products (especially the size of the set).",
"Data splitting.",
"The dataset is divided in a training(, validation) and test sets, that create new versions of all the dataset products.",
"Product refinement.",
"Although not considered in any computer vision paper, there is an additional set of chunks related to dataset transformations generally happening after deployment.",
"Datasets might be reduced in size, augmented with new labels and images, or deprived from some, as ethical or performance issues are identified or distribution shifts happen.",
"This impacts the properties of each dataset product.",
"Surfacing these chunks might provide insights to develop best practices or critical views on the handling of harms."
]
] | 2107.01824 |
[
[
"Automated Recovery of Issue-Commit Links Leveraging Both Textual and\n Non-textual Data"
],
[
"Abstract An issue documents discussions around required changes in issue-tracking systems, while a commit contains the change itself in the version control systems.",
"Recovering links between issues and commits can facilitate many software evolution tasks such as bug localization, and software documentation.",
"A previous study on over half a million issues from GitHub reports only about 42.2% of issues are manually linked by developers to their pertinent commits.",
"Automating the linking of commit-issue pairs can contribute to the improvement of the said tasks.",
"By far, current state-of-the-art approaches for automated commit-issue linking suffer from low precision, leading to unreliable results, sometimes to the point that imposes human supervision on the predicted links.",
"The low performance gets even more severe when there is a lack of textual information in either commits or issues.",
"Current approaches are also proven computationally expensive.",
"We propose Hybrid-Linker to overcome such limitations by exploiting two information channels; (1) a non-textual-based component that operates on non-textual, automatically recorded information of the commit-issue pairs to predict a link, and (2) a textual-based one which does the same using textual information of the commit-issue pairs.",
"Then, combining the results from the two classifiers, Hybrid-Linker makes the final prediction.",
"Thus, every time one component falls short in predicting a link, the other component fills the gap and improves the results.",
"We evaluate Hybrid-Linker against competing approaches, namely FRLink and DeepLink on a dataset of 12 projects.",
"Hybrid-Linker achieves 90.1%, 87.8%, and 88.9% based on recall, precision, and F-measure, respectively.",
"It also outperforms FRLink and DeepLink by 31.3%, and 41.3%, regarding the F-measure.",
"Moreover, Hybrid-Linker exhibits extensive improvements in terms of performance as well."
],
[
"Introduction",
"Issues and commits are two software artifacts commonly used for various tasks in software hosting platforms such as GitHub, Jira, and Bugzilla.",
"Issue reports encapsulate user discussions around different aspects of a software, as a sort of documentation.",
"Commits contain source code changes required to fix bugs, add features, improvements, etc discussed in the issues.",
"Issues are usually reported in bug-tracking systems such as Bugzilla or Jira, on the other hand, corresponding commits are stored in version control systems such as GitHub [1].",
"There are also cases that they are both maintained in one system.",
"When a developer commits a change in a project, it is a good practice to mention the issue in the commit to document the relationship between the two.",
"However, it is seldom the case due to the deadline's pressure, lack of motivation, etc. [1].",
"To quantify the prevalence of missing issue-commit links, Ruan et al.",
"[2] analyzed over half a million issues from GitHub.",
"They report only 42.2% of issues were linked to corresponding commits.",
"Recovering issue-commit links is deemed important for improving bug prediction solutions [3], [2], bug assignment [4], feature location techniques [5], and other software maintenance tasks.",
"It can also be used to evaluate software maintenance efforts and quality [6].",
"Thus, an automated method for recovering links between issues and their corresponding commits can be of high value.",
"The first challenge for such an approach is to use a proper dataset of True and False Links between issues and commits.",
"True Links are the correct links between issues and their related commits.",
"All the other combinations of links can be considered False Links.",
"Current approaches build these links manually.",
"This affects the reliability of results.",
"Moreover, some issues have more than one related commit.",
"An automatic solution to recovering True Links should be able to handle these relationships.",
"Another important aspect is the performance of proposed approaches.",
"Current studies mostly focus on the precision and recall scores of the predictions.",
"However, the prediction time and complexity of the models are also important.",
"In this work, we introduce a novel approach, named Hybrid-Linker to address the above-mentioned problems.",
"Hybrid-Linker exploits both textual and non-textual data to achieve higher performance.",
"Textual information includes the issue title, description, code difference, and commit messages.",
"Non-textual information consists of various characteristics of an issue and commits, such as the author of an issue, the committer, commit time, type of an issue (bug, feature, task), and state of a issue (open, closed, or resolved).",
"We first identify all the relevant information and then perform feature engineering to extract the most important ones.",
"The reason for incorporating non-textual data is to enable Hybrid-Linker to exploit this knowledge when there is little textual information available (e.g., there is no commit messages), or there are few similarities between the description of an issue and textual information of a commit.",
"We train a hybrid model consisting of two classifiers and a module to achieve the best linear composition of these classifiers.",
"The non-textual component is an ensemble of two classifiers.",
"The textual component is created using TF-IDF word embeddings and a single classifier.",
"We evaluated Hybrid-Linker against two baseline methods, FRLink and DeepLink for 12 projects with different characteristics.",
"In summary, our contributions are as follows: Proposing an automatic approach, called Hybrid-Linker, for recovering the links between issues and commits using a hybrid model of classical classifiers.",
"Our results show that Hybrid-Linker outperforms the competing approaches, FRLink and DeepLink, by 31.3%, and 41.3% respectively, regarding the F-measure.",
"Moreover, our proposed approach shows extensive improvements in terms of required training time.",
"Finally, we release our source code and data publicly.https://github.com/MalihehIzadi/hybrid-linker"
],
[
"Motivating Example",
"Here, we illustrate an example as the motivation for enhancing automatic link recovering approaches between issues and commits.",
"Figure REF is an example of an issuehttps://issues.apache.org/jira/browse/FLINK-17012.",
"Figure REF shows an example of a commithttps://bit.ly/2PCsQu6 related to the above-mentioned issue.",
"The issue and the commit are selected from Flink project.",
"Apache Flink is an open-source, unified stream-processing and batch-processing framework developed by the Apache Software Foundation.",
"An issue has different fields like type, status, release note, description, created date, updated date, and resolved data.",
"A commit contains commit message, committer ID, author ID, name of changed files, and Diff of changed files.",
"Note that other information such as comments and code snippets attached to some issues do not always exist.",
"As shown, there is no compelling similarity between the text of issue description, its release note and the respective commit message.",
"Due to lack of similarity in textual information of this issue and commit, FRLink approach fails to discover the True Link between them [6], Moreover, DeepLink [2] approach also struggles to identify this link as there is no code snippet in the description section of the issue.",
"Thus, DeepLink will find little semantic relation between the issue and the source code in this commit.",
"To address these problems, we propose to extract knowledge from both textual and non-textual channels of issues and commits.",
"Then combine this information in a hybrid model to train stronger link recovery models.",
"Figure: Example of a True Link between an issue and a commit"
],
[
"Proposed Approach",
"In this section, we present the main steps of our approach, namely: (1) data crawling, (2) data preparation, (3) feature engineering, (4) model training, and (5) linear accumulator hyper-tuning.",
"figure:approachoverview illustrates an overview of the approach and the following provides a detailed description of each of the five aforementioned steps.",
"Figure: Overview of the proposed approach"
],
[
"Data Crawling",
"While we utilize a dataset from Claes et al.",
"[7], this dataset does not satisfy our needs for the approach.",
"More specifically, we aim to incorporate the code diff data of the commits into the solution, which is missing from the shared dataset due to the large volume of such data.",
"To be able to crawl the excessively large code diff data, we first reduce the number of projects to a sample of 12 projects.",
"Then, for the sampled projects, we crawl the missing data from the projects' code bases to complement the dataset.",
"Moreover, the dataset is provided in a segmented state (separated commits and issues).",
"To uniformize it, for each project, we concatenated segments of data."
],
[
"Data Preparation",
"In this step, we prepare the dataset from two aspects.",
"In the link generation process, we generate the data points, i.e., the issue-commit pairs.",
"These data points are assigned with True Link labels when the link between the issue and its paired commit is in place and False Link otherwise, i.e., the issue and commit are irrelevant.",
"We also perform textual data preprocessing techniques on the textual data of issues and commits separately to prepare them for feature engineering.",
"The following elaborates on these two data preparation steps."
],
[
"Link Generation",
"In the dataset, there are instances of issues and commits that are already linked by the developers, which means they are established and validated as True Links.",
"We take such pairs of issue-commits as data points with the True Link label.",
"To train the classifiers, we need to provide the model with data points labeled as False Link as well.",
"However, such data points are not explicitly included in the dataset.",
"Thus, we need to generate them such that their label, False Link, is ensured.",
"To do so, we pair the commits that are already linked to an issue by the developers with any issue other than the ones they are already linked to.",
"Since the commits only make a True Link with the issues selected by the developers, pairing them with any other issue makes a False Link.",
"However, taking all the generated links as False Link data points makes the dataset extremely imbalanced.",
"To perceive the severity of this problem in the context of a project, consider a project with $c$ number of commits that are already linked to an issue by the developers.",
"If the same project contains $i$ number of issues, $c$ of which are already linked to the aforementioned commits.",
"For each commit already linked to an issue, there are $i - 1$ issues each posing as a potential False Link pair for the commit.",
"Hence, the number of False Link data points adds up to $c*(i-1)$ issue-commit pairs.",
"To address this issue, we use the criteria used by previous work [6], [2] to generate the False Links.",
"Thus we compare the two relevant submission dates of a commit with the three date attributes in an issue report and construct a new False link if the commit is submitted seven days before or after any of these three issue attributes.",
"Table REF presents the information of the resultant dataset.",
"As shown in this table, even after employing the above-mentioned criteria, the number of False Links is much larger than those of True Links for all projects.",
"To further alleviate the imbalanced nature of this dataset, we apply a common data balancing technique.",
"More specifically, we randomly select the same number of False Links as the True Links in each project, to provide our classifier with completely balanced datasets.",
"Table: Selected projects' information"
],
[
"Textual Data Preprocessing",
"The resultant dataset contains textual and non-textual data on issues and commits from the sampled projects.",
"The textual data contains both natural language text such as issue title, issue description and commit message, and the code diff.",
"We first clean and preprocess the input textual data.",
"We perform the three commonly-used strategies of tokenizing, removing stop words, and stemming on the natural language text data as the preprocessing step.",
"These preprocessing actions not only reduce the vocabulary size, which in turn makes the feature set a compact one, but also they integrate different forms of words by replacing them with their roots.",
"As for the diff data, while they do include multiple lines of code per sample, only the identifiers, i.e., method and variable names, carry valuable information about the changes in a commit.",
"That is because many of the keywords and commonly used method calls in the diff appear all over the code without indicating the purpose of the code snippet, while identifiers, if named according to the software development guidelines, refer to their purpose, role, and/or task.",
"Hence, we aim to extract only the identifiers through the use of code term patterns.",
"The code term patterns we employ are the ones previously used by Sun et al.",
"[6] and Ruan et al.",
"[2] (defined in table:code patterns).",
"Table: Code term patterns introduced in FRLink"
],
[
"Feature Engineering",
"We leverage both textual and non-textual data to improve the results of the True Link prediction task.",
"However, the features in the textual and non-textual feature vectors are not equally valuable in terms of being determinative of a True Link.",
"In the textual data context, there might be distinct words throughout the dataset that appear in a significant number of the data points.",
"This signifies that they are simply common tokens throughout the project and can not be considered as the indicator of the subject of a commit.",
"In the non-textual context, this problem manifests itself in highly correlated columns of data or even almost identical ones.",
"There is also the case of almost empty columns in which the data is null-valued more often than not.",
"This makes the feature vector unnecessarily extensive, which makes it harder for the classifier models to converge due to the multitude of parameters they are to optimize.",
"Even if the classifier does converge and yield better results with such data included, the improvement is negligible and unjustifiable when evaluated against the computational costs.",
"For these reasons, we perform a feature engineering process on both the textual and non-textual data to reduce the size of feature vectors and keep both the performance of the solution and the computational costs of the model optimal.",
"The feature engineering processes performed are detailed in the following."
],
[
"Textual Feature Engineering",
"We employ the widely used data modeling technique, TF-IDF, which captures the importance of the tokens based on probabilistic measures over the dataset.",
"This data modeling technique computes the term frequency of each term (token) in each document and document frequency of each term over the dataset and combines the former with the inverse of the latter to calculate a measure of importance for each term in the dataset.",
"The higher the value of the TF-IDF measure for a term, the more probable the term is to contribute to the label prediction.",
"We apply the TF-IDF technique on natural language textual data of commits and issues and the code diff textual data separately.",
"This generates three vectors of TF-IDF features for each data point.",
"Then, we concatenate the resultant vectors and construct one textual feature vector per data point.",
"The reason for such an approach is that the information and vocabulary in issues and commits inherently differ.",
"Moreover, according to our experiments, this approach leads to higher accuracy compared to when we combine the three input data first and then apply the TF-IDF technique on the concatenated text.",
"Our textual features are a commit's Message and Diff, and an issue's Summary and Description.",
"Note that we have also applied Word2Vec and Doc2Vec techniques, however, TF-IDF embeddings produced the best result."
],
[
"Non-textual Feature Engineering",
"In the case of highly correlated columns, reducing them to a single column can improve the computational costs of the model training process by limiting the number of optimization parameters, while preserving the performance of the classifier.",
"We extensively inspect the dataset for such strongly correlated columns among commit features an issue features by calculating the similarity and correlation among the columns of similar types.",
"We discover that among the issue data columns, over $99\\%$ of the data points have the same value as the reporter and the creator.",
"This makes these two columns practically duplicate.",
"Hence, we drop one and keep the other.",
"We also detect that over $65\\%$ of the data points have the same author and committer in their commit data columns.",
"As we believe a similarity of $65\\%$ is not high enough to justify the omission of one of the columns, so we keep both columns in the dataset.",
"Since the categorical data will be converted to a one-hot model, each distinct value in the categorical data column will serve as a Boolean feature.",
"Thus, the multitude of distinct values in a categorical column results in an over-complicated feature vector with too many features but very few true points, also known as a sparse matrix.",
"To avoid such an occurrence, we study the histograms of the categorical data and discovered that due to differences in labeling style across projects, the distinct values of the commit_status and issue_type columns can be mapped to two reduced sets of values.",
"For commits, status values was reduced from a set of 11 statuses to three main categories of open, closed, and resolved.",
"We also reduced the set of 15 distinct values of issue types to three main categories of task, new feature, and bug.",
"While there are two columns of highly correlated dates for issues, namely the create_date and the update_date, these dates prove as important features for the prediction of True Links.",
"The same goes for the author_time_date and the commit_time_date among the data of the commits.",
"We keep these columns intact to the dataset.",
"Finally, we drop the columns which have a significant number of null values.",
"After one-hot transformation of the categorical data, we calculate the correlations among all the columns, including the label column, for issues and commits separately.",
"This is to verify that there are no correlations among the features and target column.",
"After it is verified that the dataset is not biased, the resultant commit and issue feature vectors are concatenated to compose a single feature vector for each data point.",
"Our non-textual features are commit time, authoring time, author hash, and commit hash of commits.",
"We also include updated date, created date, status (closed, open, resolved), issue type (bug, new feature, task) and creator hash from issue reports."
],
[
"Model Training",
"We aim to keep the classifier model simple to lower the computational costs of training and prediction.",
"We believe one can improve the prediction accuracy of these models by augmenting the input data.",
"To do so, we leverage both textual and non-textual data on the commits and issues and construct a hybrid model by training two classifiers, one that operates on textual data and calculates the probability of labels, and another one that does the same using non-textual data."
],
[
"Textual Classifier Model",
"As the textual classifier component, we train multiple classification models, namely a Decision Tree (DT), a Gradient Boosting (GB), a Logistic Regression (LR), and a Stochastic Gradient Descent (SGD) model to choose the model with the best performance among them.",
"We feed these models the resultant feature vectors from Section REF and train them.",
"The trained models take as input the processed vector of textual data and predict a label, either True or False Link for an issue-commit pair."
],
[
"Non-textual Classifier Model",
"Here, we also use single and ensemble models to achieve the best results [8].",
"As simple classifier models, we train a Gradient Boosting [9], a Naive Bayes (NB) [10], a Generalized Linear (GL), a Random Forest (RF) [11], and a XGBoost [12] model.",
"To construct the ensemble models, following the overview illustrated in figure:non-textualapproachoverview, we combine the models and make four ensemble models accordingly.",
"The ensemble models are a $RF+GB$ , a $GB+XGBoost$ , a $RF+XGBoost$ , and a $RF+GB+XGBoost$ combinations.",
"Figure: Overview of ensemble models for non-textual classifier component"
],
[
"Linear Accumulator Hyper-tuning",
"The last step in our approach is to combine the predictions of the two classifier components and generate the final prediction on the label of the data points.",
"To do so, we combine the predicted probability of a data point being a True Link from the two components and with a linear accumulator function, defined as the following: $P_f = \\alpha \\times P_{nt} + (1-\\alpha ) \\times P_t$ in which $P_f$ is the final calculated probability of a commit-issue pair being a true-link, $P_{nt}$ is the probability of the same pair being a true-link according to the non-textual classifier component, and $P_t$ is the same probability according to the textual classifier component.",
"In eq:alpha, $\\alpha $ is the hyper-parameter by which we tune the model to produce the best results tailored to the characteristics of each project.",
"To do so, we vary the value of $\\alpha $ from $0.00$ to $1.00$ in $0.05$ steps.",
"The value $\\alpha $ by which best results regarding the F1-score is yielded is taken as the optimal $\\alpha $ value."
],
[
"Research Questions",
"We define three Research Questions (RQ) to measure the effectiveness of our proposed approach.",
"We review these questions in the following.",
"RQ1: Compared to the state-of-the-art approaches, how effective is our approach in recovering the missing links between issues and commits?",
"To answer this question, we evaluate our method's performance using the 20MAD dataset (reviewed in the next section) [7].",
"We use 12 projects from this dataset for training and testing our model.",
"There are different approaches for commit-issue recovery.",
"We use two of the state-of-the-art models, namely FRLink [6] and DeepLink [2], to compare with the proposed approach.",
"RQ2: How to combine the two components of the model to achieve the best outcome?",
"There are different ways to combine our two models (textual and non-textual).",
"With this question, we aim to identify the best method to build a hybrid model.",
"RQ3: What is the effect of each component of the model on the outcome?",
"As our model is constructed of different components, we assess the benefits of adding each through an ablation study.",
"That is, we evaluate the model using each of the two components separately and then compare the results by those of the Hybrid-Linker."
],
[
"Data Selection",
"As the data of previous work were not publicly shared, we utilized the dataset presented by Claes and Mantyla [7] in the MSR conference, 2020.",
"From the Apache projects, we chose 12 based on two criteria; (1) having a repository with more than 500 stars (to have good input data for training the models), and (2) having a diverse number of issues for different projects (to be fair).",
"As of September 2020, the number of stars for the selected projects was in the range of 580 to 18800.",
"The second criterion let us choose projects with different number of issues from small to large software projects.",
"The number of issues among our projects range from a couple hundred to more than 25K issues.",
"To prepare this data for feeding our Machine-Learning-based models, we complement and transform the selected 12 projects from the 20MAD dataset as explained in Sections REF and REF ."
],
[
"Evaluation Metrics",
"As previously used in the related work, we use three metrics of Precision, Recall, and F-measure to evaluate the performance of the approach [2], [6].",
"These metrics are calculated using the following equations.",
"$Precision=\\frac{TP}{TP + FP}$ $Recall=\\frac{TP}{TP + FN}$ $F1=\\frac{2 \\times Precision \\times Recall}{Precision + Recall}$"
],
[
"Experiment Setting",
"For preprocessing, we use Pandas [13] library.",
"For training the classifiers in the non-textual component of our approach, we use H2O.ai [14] library.",
"H2O benefits from distributed, in-memory processing which results in faster models.",
"It is also able to manage hash data better than the Sci-Kit Learn [15] library.",
"This library can be used in different programming languages such as Python and R. We use the Python version and added a Java Runtime Environment for the backend.",
"For the textual component, we use the Sci-Kit Learn library.",
"It has great I/O which lets us use different types of data like Parquet and Pickle files simultaneously.",
"We use five-fold cross-validation to evaluate the models more thoroughly.",
"That is, we break the data into five parts randomly, choose one as the test set and use the others for training.",
"After repeating this process five times, iterating over the parts as the test set, we report the average as the result of evaluations.",
"This also helps with the generalizability of the approach and avoiding the overfitting problem.",
"To find the best parameters for the ensemble model in the non-textual component, we perform a Random Grid search for each project of the dataset.",
"The result of the search indicate that (n_trees=60, max_depth=15, min_rows=2, learn_rate=0.1, learn_rate_annealing=1) for the Gradient Boosting model and (n_trees=60, max_depth=15, min_rows=2, learn_rate=0.1) for the XGBoost model are the best choices.",
"To find the best parameters for the Gradient Boosting in the textual component, we perform another Random Grid search for each project of the dataset.",
"The result of the search indicate that (n_estimators=300, max_features=None, max_depth=50 and learning_rate=0.1) are the best choices.",
"To build the TF-IDF embedding vectors, we experiment with unigram, bigram, and union of unigram, bigram, and trigram word embedding.",
"The best case is the union of unigram, bigram, and trigram as it finds all the important individuals and combinations of words.",
"For TF-IDF embeddings, we set a maximum number of features to 10K.",
"For all of our experiments, we used the same machine with 32GB memory and a 4-core Intel i7-7700k 4.2G processor.",
"The baselines here are FRLink [6] and DeepLink [2] as they achieve the state-of-the-art results in the problem of automatically recovering links between issues and their corresponding commits.",
"FRLink uses a set of features and complementary documents such as non-source documents to learn from relevant data for recovering links [6].",
"They analyze and filter out irrelevant source code files to reduce data noise.",
"On the other hand, DeepLink uses a semantically-enhanced link recovery method based on deep neural networks [2].",
"The authors apply a recurrent neural network on the textual information of issues and commits for training their model.",
"They also disregard issue comments due to their length and noise.",
"While DeepLink outperforms FRLink in terms of F-measure, it achieves lower recall scores [2].",
"Thus, we use both these techniques here as the baselines to compare our approach with.",
"We use the replication packages provided by Ruan et al.",
"[2] for these two modelshttps://github.com/ruanhang1993/DeepLink.",
"We slightly modified their input reader function to be able to read our data.",
"Moreover, we set all the parameters as specified in the original papers."
],
[
"Results",
"In this section, we answer the research questions by providing the results of the experiments.",
"We first, compare the performance of our proposed approach with the state-of-the-art ones.",
"Next, we review the results of our investigations on how to build a hybrid model.",
"Finally, we present the results of the ablation study to show the effectiveness and impact of each component of the proposed approach."
],
[
"RQ1",
"Compared to the state-of-the-art approaches, how effective is our approach in recovering the missing links between issues and commits?",
"To answer this RQ, we built our approach with two classifier components, a textual classifier and a non-textual one that each predict the probability of a issue-commit pair being a true-link.",
"We plugged multiple classifier models into each of the said components and chose the models with best performances as our proposed ones.",
"For the textual classifier component, we fed the concatenated TF-IDF vectors to four classifier models widely used for text classification purposes and study the results to determine the best performance among the models.",
"table:textualresults shows the outcome of the trained models.",
"The results indicate the best algorithm for classifying issue and commits linkage based on their textual data is the Gradient Boosting model.",
"For training a classifier on non-textual information, we experimented with well-known classical classifiers to identify the best classifier for our case.",
"As seen in Table REF Gradient Boosting, Random Forest, and XGBoost have higher overall metrics results.",
"Moreover, ensemble methods have been shown to outperform simple models.",
"Thus, we also opt for an ensemble model of the above algorithms to identify the best combination here.",
"Based on results in table:nontextualresults, the ensemble of Gradient Boosting and XGBoost produce the best result for our non-textual data.",
"Table REF reports the average score of precision, recall and F-measure for each model.",
"Table: The average performance of models on textual data.Table: The average performance of models on non-textual data.The effectiveness of our proposed method is evaluated based on three metrics, namely Precision, Recall, and F-measure.",
"Table REF presents a summary of the average performance of our approach across projects.",
"According to the results, our approach achieves an average of $90.14\\%$ on Recall, $87.78\\%$ on Precision, and $88.88\\%$ of F-measure.",
"Respectively, the lowest Recall is $84.41\\%$ for Arrow and the highest is $100\\%$ for Cassandra project.",
"On the other hand, the lowest precision is $81.81\\%$ for Cassandra and the highest is $96.04\\%$ for Ambari.",
"Table: Performance of the modelsWe compare our approach with two of the competing models, namely FRLink and DeepLink.",
"On average, our approach has $34.17\\%$ higher precision and $21.21\\%$ higher F-measure scores than FRLink.",
"Although FRLink achieves higher recall than our proposed approach, its precision score is much lower compared to our model.",
"Hence, Hybrid-Linker ultimately outperforms FRLink based on F-measure which is the harmonic mean of recall and precision.",
"Moreover, obtaining high recall but low precision calls for manual assessment of the predictions.",
"That is, a developer needs to check the predicted links and remove the incorrect ones.",
"This adversely affects the automated feature of the approach.",
"Hybrid-Linker outperforms DeepLink by $50.40\\%$ , $26.99\\%$ , and $41.34\\%$ regarding the average recall, precision, and F-measure.",
"Previous studies have shown deep learning-based models tend to outperform classical machine learning models.",
"However, as shown in a study by Hellendoorn et al.",
"[16], it is possible to achieve better results using simple and well-engineered approaches compared with vanilla deep neural networks.",
"According to our results, we are also able to surpass DeepLink as we carefully inspect the domain of the problem, identify and incorporate more relevant information from the non-textual channel in addition to the textual information of issues and commits.",
"Evidently, these types of information can help boost the performance of automatic link recovery models.",
"Our results are also compatible with those reported by Ruan et al.",
"[2] where the overall recall score of FRlink is higher than DeepLink.",
"However, Ruan et al.",
"[2] originally evaluated using six projects with almost identical number of true/false links, while in this study we have included 12 projects with various number of true/false links and sizes to improve diversity of our dataset.",
"This may cause the drop in individual scores reported in this work and Ruan et al.",
"'s [2] study (regarding comparison with FRlink).",
"Furthermore, our approach uses fewer computational resources and time while training the models.",
"For instance, pertaining the Airflow project, the required time to train Hybrid-Linker is 25 minutes, while it takes about 7 hours to train DeepLink.",
"able REF provides execution times per project.",
"DeepLink has also reported a 5X overhead comparing their approach to FRLink.",
"The large overhead of DeepLink is probably due to the complex nature of deep models.",
"Our results indicate we can train simpler models which incorporate more relevant information, thus, achieving higher accuracy and less overhead.",
"We also calculated the standard deviation of the F-measure for the 12 projects in the dataset.",
"Taking all projects into account, the standard deviations of the F-measure are $3.01$ , $3.92$ , and $4.68$ for Hybrid-Linker, DeepLink, and FRLink, respectively.",
"That is, our approach is more stable than the other two, hence proving to be a more generalizable approach.",
"Table: execution time for each project on our hardware"
],
[
"RQ2",
"How to combine the two components of the model to achieve the best outcome?",
"To incorporate as much information as possible and consequently boost the performance, we propose a hybrid model of our two distinct classifiers.",
"To combine the predictions of the two components, we create a linear composition of their outputs.",
"figure:project base alphas presents the results of using different values of alpha ranging from 0 to 1 for the 12 projects under study.",
"As can be seen, each project requires a different value of alpha.",
"Thus, selecting a constant alpha for all projects will result in weaker results.",
"table:best alpha project base lists the best $\\alpha $ values for each project.",
"In most cases, $\\alpha $ is above $0.5$ , with the average $\\alpha $ being $0.66$ for all the projects.",
"This means, interestingly, in most cases the non-textual component plays a more important role in the final decision making of Hybrid-linker.",
"This highlights the importance of incorporating these types of information while recovering links.",
"The only exception to this finding occurs in the Ambari project with $\\alpha $ of $0.45$ .",
"This implies an approximately equal contribution of the two components of our proposed approach for this project.",
"On the other hand, for Calcite, the best results are achieved with an $\\alpha $ of $0.95$ .",
"This can be a indicator that this project lacks adequate textual information useful for recovering links.",
"Figure: Tuning Alpha per projectTable: Best value of Alpha per project"
],
[
"RQ3",
"What is the effect of each component of the model on the outcome?",
"In this section, we present the results of our ablation study on assessing the effect of each component of the proposed model.",
"table:textual, non-textual and hybrid comparison presents a summary of our model's performance based on each project.",
"The results indicate that on average, the performance of the textual model is lower than both the non-textual and hybrid models.",
"The textual model also marks the highest standard deviation among the models with $5.83$ .",
"Interestingly, the non-textual model outperforms the hybrid model regarding precision by $4.38\\%$ .",
"On the other hand, the standard deviation of the non-textual model is $3.10$ which is slightly higher than the standard deviation of the hybrid model, $3.00$ .",
"As natural language is more complex, text-based approaches may require more complex techniques to perform fairly good.",
"The higher performance of the non-textual component is probably due to (1) having more explicit data, and (2) the advantage of ensemble models.",
"To conclude, the hybrid model has higher recall and F-measure scores.",
"It also obtains the lowest standard deviation regarding its performance on all the projects.",
"This means, by employing both of the textual and non-textual components, the hybrid model achieves higher results, while preserving the stability of the proposed approach.",
"Table: Results of the ablation study"
],
[
"Discussion",
"Here, we present an example where our model successfully recovers the True Link between an issue and its corresponding commit.",
"Table REF summarizes the information of these two artifacts.",
"Although there are few similarities in textual information, the baselines and our textual component are unable to recognize this connection.",
"However, our non-textual component compensates for this shortcoming and predicts the correct connection.",
"As it is shown, our model is capable of correct predictions both (1) when there is little textual information available or (2) when there is no explicit relation among the text of the two artifacts.",
"Note that non-textual data are often available as they are automatically recorded.",
"Table: An example of a True Link prediction"
],
[
"Threats to Validity",
"Here we discuss the threats to the validity of our work, organized into internal, external, and construct validity."
],
[
"Internal Validity",
"Internal validity is the extent to which a piece of evidence supports a claim about cause and effect, within the context of a particular study [17].",
"The first threat to the internal validity of our study is the True Link trustworthiness and False Link trustworthiness in our dataset.",
"In the case of building True Links, we have used the links provided by Claes and Mantyla in [7].",
"Although this dataset is validated by the authors [7], incorrect links may still be present due to human error.",
"Any combination other than a True Link can be considered a False Link.",
"However, due to the diversity and multitude of choices for creating False Links, we had to employ several constraints as explained in Section REF .",
"These constraints affect our results.",
"According to previous studies, if an issue is related to a commit, there is a higher chance it will be answered/solved by a commit within seven days.",
"Thus, by creating different combinations of False Links within seven days, we aim to create a more relevant and appropriate False Link dataset for training the models.",
"Lastly, data balancing is an important issue to keep in mind.",
"Although one can easily create a large number of False Links, lack of enough True Links adversely affects the performance of classifiers.",
"To tackle this problem, we balanced the dataset by selecting a random subset of the over-presented class before training.",
"Other balancing techniques are also viable."
],
[
"External Validity",
"External validity is concerned with the generalizability of the approach and results [17].",
"In that regard, the dataset used in this study affects the outcome of the models.",
"The size and quality of the data play an important role in having a good issue and commit link predictor.",
"We addressed this threat by evaluating our approach against data from multiple projects and studying the results.",
"As discussed in Section , the lower standard deviation achieved by Hybrid-Linker indicates that results from this approach are more stable across projects.",
"That is, the approach is more generalizable than the state-of-the-art baselines and produces results in an expected range when applied on data from different projects."
],
[
"Construct Validity",
"Construct validity is concerned with the evaluation of the models [17].",
"Similar to previous work [2], [6], [1], [18], [3], we use precision, recall, and F-measure to evaluate the performance of our approach.",
"To evaluate our proposed model more fairly, we also use five-fold cross-validation in all model evaluation steps of the study and report the average of the metrics.",
"By breaking data into five smaller chunks and re-evaluating the model, we ensure that all of the data has been used for training and testing.",
"In this section, we review the related studies with the purpose of linking issues to their corresponding commits.",
"We categorize these approaches into three major groups of heuristic-based, Machine-Learning-based, and Deep-learning-based studies."
],
[
"Heuristic-based approaches",
"These studies simply define a set of heuristics to find the links between issues and commits.",
"ReLink [19], MLink [20], and PaLiMod [18] fall into this category.",
"Wu et al.",
"[19], introduced ReLink, an approach that builds on top of traditional heuristics for creating True Links.",
"The traditional heuristics used in this work mostly rely on hints or links developers leave about bug fixes in changelogs.",
"For instance, they search for keywords such as `fixed' and `bug', or bug ID references in changelogs.",
"Moreover, they would try to find the link by using features extracted from linked issues and commits.",
"They obtained $89\\%$ precision and $78\\%$ recall on average.",
"Nguyen et al.",
"[20] presented MLink, a layered approach that exploits both textual and code-related features.",
"They outperform ReLink by $13\\%$ to $17\\%$ on recall and $8\\%$ to $17\\%$ on precision [20].",
"However, they used only three projects when evaluating their work.",
"Moreover, their results showed that some individual layer's precision or recall are very low.",
"Finally, Schermann et al.",
"[18] introduced PaLiMod to enable the analysis of interlinking characteristics in commit and issue data.",
"They used this analysis to define their heuristics.",
"PaLiMod achieves a precision of $96\\%$ and recall of $92\\%$ in the case of the Loner heuristic which are single commits with no link to the addressed issues.",
"Also, their method reach overall precision of $73\\%$ with a recall of $53\\%$ in the case of the Phantom heuristic which are commits without a link in a series of commits that address a certain issue.",
"Although the idea of the Phantom case was novel, the results were not significant compared to former heuristic methods such as MLink.",
"One of the drawbacks of these studies is using a manually-created dataset by the authors themselves [21].",
"Most of these cases used manually labeled data which reduces the confidence in the results."
],
[
"Machine-Learning-based approaches",
"The second approach to recovering links is to use traditional binary classifications, including RCLinker [3], FRLink [6] and PULink [1].",
"RcLinker employed ChangeScribe, a tool for creating a commit message and used a set of features to recover the links.",
"They outperformed MLink in terms of F-measure by $138.66\\%$ [3].",
"ChangeScribe creates highly detailed commits which are not very suitable for feature extraction in this context.",
"Recently, FRLink was introduced which uses its own set of features [6].",
"The authors also use complementary documents such as non-source documents to learn from relevant data They analyze and filter out irrelevant source code files to reduce data noise.",
"FRLink outperforms RCLinker in F-measure by $40.75\\%$ when achieving the highest recalls.",
"However, their approach encounter problems when (1) a dataset has a low percentage of non-source documents in commits, or (2) it has few or no similar code terms in the issue report and corresponding fixing commits.",
"Also, text and code features were equally weighted in this approach.",
"A close study to FRLink is PULink [1], where authors labeled their data as a True Link/unlabeled instead of True/False Links.",
"They claim they can obtain the same value of precision and recall with almost $70\\%$ of the number of True Links in other approaches.",
"However, they too had a problem when a dataset has a low percentage of True Links.",
"Generally, the main problem of these studies is the low performance based on metrics like F1, precision, and recall.",
"Although FRLink achieves higher recall scores, its precision and F1 are very low."
],
[
"Deep-Learning-based approaches",
"Xie et al.",
"[22] proposed DeepLink [22], which incorporates a knowledge based graph and deep learning to solve this problem.",
"Using class embeddings in commit codes, the authors created this graph.",
"Authors also use CBOW and Word2Vec embedding for commit and issue documentation.",
"As we did not have access to the knowledge graph and replication package, we were not able to replicate this approach.",
"Another publication also named DeepLink [2] uses a semantically-enhanced link recovery method based on deep neural networks to tackle this problem.",
"The authors use recurrent neural networks on the textual information of issues and commits to train their model.",
"They disregard comments because of their length and noise.",
"They have added semantic to their model to have a better prediction.",
"DeepLink outperforms FRLink in terms of F-measure by $10\\%$ [2].",
"The challenge with deep learning algorithms lies in the need for a large amount of data and high computational resources.",
"Moreover, training these models takes a lot of time.",
"We propose a model that outperforms the baselines by exploiting information from both textual and non-textual channels.",
"We use fewer resources and our training and prediction time are much lower.",
"We also train with projects where fewer issues and commits are available.",
"Thus our model will not fail when there is little historical data available for a project."
],
[
"Conclusion and Future Work",
"The importance of recovering true connections between issues and their corresponding commits greatly affects various software maintenance tasks.",
"Previous studies mostly focused on exploiting textual information to train their models to identify the links.",
"However, we introduced a hybrid method, called Hybrid-Linker based on classical ML-based classifiers, that employs both textual and non-textual information to recover the links.",
"For each project, we tune alpha, as an indication of the importance of each information channel.",
"The results suggest that the non-textual information indeed help the predictions.",
"This is highlighted in cases that there is little textual information available.",
"Moreover, our approach requires shorter training time and outperforms both the competing methods, namely DeepLink [2] and FRLink [6] by $41.3\\%$ and $31.3\\%$ on F-measure, respectively.",
"In the future, we plan to boost our proposed classifier by identifying new features from different bug tracking and version control systems.",
"We will also investigate other classifier architectures."
],
[
"Acknowledgement",
"We would like to acknowledge Mahtab Nejati for her valuable comments and help on this work."
]
] | 2107.01894 |
[
[
"Massively parallelizable proximal algorithms for large-scale stochastic\n optimal control problems"
],
[
"Abstract Scenario-based stochastic optimal control problems suffer from the curse of dimensionality as they can easily grow to six and seven figure sizes.",
"First-order methods are suitable as they can deal with such large-scale problems, but may fail to achieve accurate solutions within a reasonable number of iterations.",
"To achieve solutions of higher accuracy and high speed, in this paper we propose two proximal quasi-Newtonian limited-memory algorithms - MinFBE applied to the dual problem and the Newton-type alternating minimization algorithm (NAMA) - which can be massively parallelized on lockstep hardware such as graphics processing units (GPUs).",
"We demonstrate the performance of these methods, in terms of convergence speed and parallelizability, on large-scale problems involving millions of variables."
],
[
"Background",
"Stochastic optimal control is the backbone of stochastic mpc, which is known for its appealing stability and constraint satisfaction properties [1], [2] and has found several applications [3], [4], [5].",
"More specifically, scenario-based stochastic mpc is gaining great popularity [6], [7], [8], [9] due to its applicability to virtually any stochastic model of uncertainty that can be reasonably approximated by a discrete distribution.",
"However, the limiting factor towards its industrial uptake is the computational time required to solve numerically the resulting large-scale optimisation problem.",
"Indeed, multistage scenario-based stochastic optimal control problems suffer from the curse of dimensionality and can lead to problems with millions of decision variables [7].",
"gpu have been used for their massive parallelization capabilities in applications as diverse as cryptocurrency mining [10], cosmology [11], medical image processing [12], simulations of molecular dynamics [13], machine learning [14], and a lot more.",
"gpu are suitable for lockstep parallelization, where the same elementary operations are applied to different memory positions using dedicated functions known as kernels.",
"Programming gpu for general-purpose data-parallel computations is facilitated by programming languages and frameworks such as CUDA [15], [16] (for NVIDIA GPUs, used by well-known software such as Tensorflow [17] and Caffe [18]), OpenCL, OpenACC, OpenGL and more.",
"In recent years, a number of papers have proposed parallelizable variants of numerical optimization methods such as the interior point method [19], parallel quadratic programming [20], admm [21], [22], [23] and other proximal algorithms [24], [25].",
"In these approaches, gpu are used to parallelize the involved algebraic operations and the solution of linear systems: the primal-dual optimalily conditions in interior point algorithms and equality-constrained QPs in admm.",
"Given the lockstep data parallelization paradigm of gpu, numerical methods that aim at splitting the problem into smaller optimization problems that are to be executed in parallel (such as [26] and [27]) do not lend themselves to gpu implementations.",
"Scenario-based problems possess a certain structure that can be exploited to design very efficient ad hoc GPU-enabled implementations leading to a higher acceleration as discussed in [7].",
"It has been shown that first-order algorithms such as the accelerated proximal gradient method can be used to achieve significant speed-ups [28], [7], [29].",
"However, first-order methods tend to be prone to ill-conditioning as they disregard curvature information.",
"This motivates the development of numerical methods that can exploit the underlying problem structure of scenario-based optimal control problems, come with good convergence characteristics, and are amenable to lockstep parallelisation on gpu.",
"In this paper we propose two massively parallelizable numerical methods that exploit the structure of scenario-based stochastic optimal control problems, building up on (i) the minfbe method [30] applied to the dual problem, (ii) the Newton-type alternating minimization algorithm (nama) [31] algorithms, as well as (iii) on our previous work on GPU-accelerated optimization [32].",
"All methods lend themselves to highly parallelizable implementations and lead to similar convergence speeds.",
"However, we will show that nama allows a significantly higher parallelizability and lower computation times.",
"minfbe and nama involve only simple algebraic operations, use limited-memory BFGS directions and can achieve better accuracy and significantly faster convergence than the accelerated proximal gradient method of [7] (linear convergence rate instead of $\\mathcal {O}(1/k^2)$ )."
],
[
"Notation",
"Let $$ , $$ , $^n$ and $^{m{}\\times {}n}$ denote the sets of nonnegative integers, real numbers, $n$ -dimensional vectors and $m$ -by-$n$ matrices respectively.",
"Let $_{[k_1, k_2]} \\lbrace n\\in {}:{} k_1\\le {}n{}\\le {}k_2\\rbrace $ .",
"Let $={}\\cup {}\\lbrace +\\infty \\rbrace $ be the set of extended-real numbers.",
"Given a set $X{}\\subseteq {}^n$ and $x\\in ^n$ we define the indicator of $X$ as the extended-real-valued function $\\delta ({}\\cdot {}\\mid {}X):^n\\rightarrow $ with $\\delta (x{}\\mid {}X) = 0$ for $x{}\\in {}X$ and $\\delta (x {}\\mid {} X)=\\infty $ otherwise.",
"For $A\\in ^{m{}\\times {}n}$ , $A^\\top $ denotes the transpose of $A$ .",
"For $A,B\\in ^{m\\times n}$ , we write $A\\succ {}B$ ($A \\succcurlyeq B$ ) if $A-B$ is positive (semi)definite.",
"For a convex function $f:^n\\rightarrow $ , its convex conjugate function $f^*$ is defined as $f^*(y) = \\sup _{x}\\lbrace x^\\top y-f(x)\\rbrace .$ Lastly, given a nonempty, closed, convex set $X{}\\subseteq {}^n$ , we define the projection operator onto $X$ as $_X(x) = _{y{}\\in {}X}\\Vert y-x\\Vert $ .",
"We start by stating the stochastic optimal control problem we will study in this paper."
],
[
"Stochastic dynamics on scenario trees",
"Consider a discrete-time stochastic dynamical system of the form $x_{t+1} = A_{w_t}x_t + B_{w_t}u_t + c_{w_t},$ with state $x_t\\in ^{n_x}$ and input $u_t\\in ^{n_u}$ , which is driven by the stochastic process $w_t$ .",
"For example, Markov jump affine systems fall into this category [33].",
"The evolution of this system over a finite sequence of time instants, $t\\in _{[0, N]}$ , can be described using a scenario tree: a directed graph of the form shown in fig:scenariotree.",
"The scenario tree structure is essentially the representation of a discrete multistage probability distribution.",
"A scenario tree represents the evolution of the system states as more information becomes available: at every stage $t$ , we assume that the state, $x_t$ , can be measured and a control action $u_t$ can be decided based on that measurement, thus modeling an entire feedback policy.",
"Figure: Scenario tree structure with three stages and the system dynamicson its nodes.",
"The cost associated with node i=1i=1 is ℓ 1 (x 0 ,u 0 ,w 1 )\\ell _1(x^0, u^0, w^1).The nodes of the scenario tree are organised in stages, $t\\in _{[0, N]}$ , and indexed by a unique integer $i$ .",
"At stage $t=0$ we assume that the state — which is the current state in an mpc setting — is known; this corresponds to the root node of the tree, which is indexed by $i=0$ .",
"The nodes at a stage $t$ are denoted by $(t)$ and the nodes at stage $t=N$ are called the leaf nodes of the tree.",
"For notational convenience, we will denote the nodes at stages $t\\in _{[t_1, t_2]}$ , with $0 \\le t_1 \\le t_2 \\le N$ , by $(t_1, t_2) = \\bigcup _{t=t_1}^{t_2}(t)$ .",
"The set $(t)$ is a probability space: every node $i\\in (t)$ is assigned a nonzero probability value $\\pi ^i$ .",
"Naturally $\\pi ^0 = 1$ and $\\sum _{i\\in (t)}\\pi ^i = 1$ for all $t\\in _{[0, N]}$ .",
"Every node $i$ at a stage $t\\in _{[1, N]}$ has an ancestor, $(i) \\in (t-1)$ , and all nodes at a stage $t\\in _{[0, N-1]}$ have a set of children, $(i) \\subseteq (t+1)$ .",
"The set $(i)$ is a probability space with probability vector $\\pi ^{[i]} \\in ^{|(i)|}$ .",
"This is a vector whose $i_{\\scriptscriptstyle +}$ -th element is equal to ${\\pi ^{i_{\\scriptscriptstyle +}}}/{\\pi ^i}$ — for short $\\pi ^{[i]}{}={}\\tfrac{1}{\\pi ^i}(\\pi ^{i_{{\\scriptscriptstyle +}}})_{i_{{\\scriptscriptstyle +}}\\in (i)}.$ The system dynamics, (REF ), across the nodes of the scenario tree can be stated as $x^{i_{\\scriptscriptstyle +}}{}={}A^{i_{\\scriptscriptstyle +}}x^{i}{}+{}B^{i_{\\scriptscriptstyle +}} u^{i}{}+{}c^{i_{\\scriptscriptstyle +}},$ for $i\\in (0, N-1)$ , $i_{\\scriptscriptstyle +}\\in (i)$ .",
"Note that the total number of scenarios coincides with the number of leaf nodes, and the number of non-leaf nodes with the number of free input variables (see fig:scenariotree)."
],
[
"Stochastic optimal control problem",
"A multistage stochastic optimal control problem for (REF ) with horizon $N$ can be formulated as $\\mathbb {P}(p){}:{}_{\\lbrace u_t\\rbrace _{t=0}^{N-1},\\lbrace x_t\\rbrace _{t=0}^{N}}\\ \\mathbb {E}\\left[V_f(x_N) + \\sum _{t=0}^{N-1}\\ell _t(x_t, u_t,w_t)\\right],$ subject to (REF ) and the condition $x_0=p$ .",
"Note that in this formulation, $\\lbrace u_t\\rbrace _{t=0}^{N-1}$ and $\\lbrace x_t\\rbrace _{t=0}^{N}$ are random variables.",
"The stage cost at stage $t\\in _{[1, N]}$ is a random variable which admits the values $\\ell ^{i}(x^{(i)}, u^{(i)}) \\ell _t(x^{(i)}, u^{(i)}, w^{i})$ , for $i\\in (t)$ , with probability $\\pi ^{i}$ .",
"The terminal cost function is also a random variable which admits the values $V_f(x^i)$ for $i\\in (N)$ with probability $\\pi ^i$ .",
"That said, the optimal control problem can be written as $\\mathbb {P}(p){}:{} _{\\begin{array}{c}\\lbrace u^i\\rbrace _{i\\in (0, N-1)}\\\\\\lbrace x^i\\rbrace _{i\\in (0, N)}\\end{array}\\hspace{6.99997pt}}\\sum _{i\\in (1, N)}\\hspace{-15.0pt}\\pi ^i\\ell ^i(x^{(i)}, u^{(i)})\\\\+\\sum _{i\\in (N)}\\hspace{-10.0pt}\\pi ^i V_f^i(x^i),$ subject to the system dynamics (REF ) and the condition $x_0=p$ .",
"The stage cost function, $\\ell ^{i}:^{n_x}\\times ^{n_u}\\rightarrow $ , at node $i\\in (1, N)$ , is an extended-real-valued function which can be decomposed as follows $\\ell ^i(x, u) = \\phi ^i(x, u) + \\bar{\\phi }^i(F^i x + G^i u),$ where $\\phi ^i:^{n_x}\\times ^{n_u}\\rightarrow $ is a smooth convex function and $\\bar{\\phi }^i:^{m_i}\\rightarrow $ is a proper, extended-real-valued, possibly nonsmooth, convex, lower semicontinuous function and $F^i \\in ^{m_i\\times n_x}$ , $G^i\\in ^{m_i\\times n_u}$ .",
"Functions $\\bar{\\phi }^i$ can be taken to be indicator functions so as to model constraints on inputs and states.",
"We can also decompose the terminal cost function, $V_f^i:^{n_x}\\rightarrow $ , as follows $V_f^i(x) = \\phi _N^i(x) + \\bar{\\phi }_N^i(F_N^i x),$ where $F_N^i{}\\in {}^{m_{N,i}{}\\times {}n}$ , $\\phi _N^i:^{n_x}\\rightarrow $ is real valued, smooth, convex function and $\\bar{\\phi }_N^i:^{m_{N,i}}\\rightarrow $ is a proper extended-real-valued, convex, lower semicontinuous function.",
"Functions $\\bar{\\phi }^{i}$ need not be smooth.",
"They can be used to describe hard joint state-input constraints of the form $F^{i} x + G^{i} u \\in Y^{i}$ by taking $\\bar{\\phi }^i(\\cdot )=\\delta ({}\\cdot {}|{}Y^i)$ .",
"Similarly, $\\bar{\\phi }^{i}$ can describe soft constraints simply by replacing the indicator function $\\delta ({}\\cdot {}|{}Y^i)$ by a distance-to-set function.",
"On the other hand, functions $\\phi ^i$ and $\\phi _N^i$ are typically taken to be convex quadratic (and $\\phi ^i$ are assumed to be strongly convex with respect to $u$ and jointly convex in $(x, u)$ ).",
"Hereafter, we consider the quadratic cost functions $\\phi ^i(x,u)= \\begin{bmatrix}x\\\\u\\end{bmatrix}^\\top \\begin{bmatrix}Q_i & S_i^\\top \\\\ S_i & R_i\\end{bmatrix}\\begin{bmatrix}x\\\\u\\end{bmatrix} +q_i^\\top x +r_i^\\top u,$ for $x\\in ^{n_x}$ and $u\\in ^{n_u}$ , with $Q_i = Q_i^\\top \\succcurlyeq 0$ , and $R_i=R_i^\\top \\succ 0$ , and $\\begin{bmatrix}Q_i & S_i^\\top \\\\ S_i & R_i\\end{bmatrix}{}\\succcurlyeq {}0,$ for all $i\\in (0, N-1)$ .",
"Lastly, $V_f(x) = x^\\top P_N x + p_N^\\top x$ for $x\\in ^{n_x}$ with $P_N = P_N^\\top \\succ 0$ ."
],
[
"Formulation of optimization problem",
"The decision variable of $\\mathbb {P}(p)$ is the vector $x=\\left((u^i)_{i\\in (0, N-1)}, (x^{i})_{i\\in (1, N)}\\right)\\in ^n$ , where $n=|(0, N-1)|n_u + |(1, N)|n_x$ .",
"Let us define the affine space $\\mathcal {Z}(p) = \\left\\lbrace x{}\\left|\\begin{array}{l}x^0 = p, x^{i_{\\scriptscriptstyle +}} = A^{i^{\\scriptscriptstyle +}} x^i + B^{i_{\\scriptscriptstyle +}}u^{i} + c^{i_{\\scriptscriptstyle +}},\\\\i\\in (0, N-1), i_{\\scriptscriptstyle +}\\in (i).\\end{array}\\right.\\right\\rbrace ,$ which describes the system dynamics.",
"Let us also define the functions $f(\\cdot ;p):^n\\rightarrow $ and $g{}:{}^m\\rightarrow $ that maps $z=((z^i)_{i\\in (0,N)},(z_{N}^{i})_{i\\in (N)})$ with $z^i\\in ^{m_i}$ and $z_{N}^{i}\\in ^{m_{N,i}}$ and is given by $f(x) & = \\sum _{i\\in (1, N)}\\hspace{-10.0pt}\\pi ^i\\phi ^i(x^{(i)},u^{(i)})\\\\& \\qquad {}+{}\\hspace{-10.0pt}\\sum _{i\\in (N)}\\hspace{-10.0pt}\\pi ^i \\phi _N(x^i){}+{}\\delta (x| \\mathcal {Z}(p)),\\\\g(z) & {}={}\\hspace{-8.00003pt}\\sum _{i\\in (1, N)}\\hspace{-10.0pt}\\pi ^i\\bar{\\phi }^i(z^i){}+{}\\hspace{-3.99994pt}\\sum _{i\\in (N)}\\hspace{-10.0pt}\\hspace{-3.99994pt}\\pi ^i\\bar{\\phi }_N(z_N^i),$ where $z=((z^i)_{i\\in (0, N)}, (z_N^i)_{i\\in (N)})$ and define $H:^n\\rightarrow ^m$ as a linear operator that maps $x$ to a vector $z\\in ^m$ as above with $z^i = F^i x^{(i)} + G^i u^{(i)}$ for $i\\in (0, N)$ and $z_N^i = F_N^i x^i$ for $i\\in (N)$ .",
"Given that functions $\\phi ^{i}$ are quadratic as described in the previous section, function $f$ is strongly convex (as it follows from [34]), therefore the convex conjugate of $f$ , $f^*$ , is differentiable with $L$ -Lipschitz gradient because of [35].",
"Problem $\\mathbb {P}(p)$ can be written as $\\mathbb {P}(p){}:{} _{x\\in ^n} f(x; p) + g(Hx).$ Hereafter, we assume that $\\mathbb {P}(p)$ is feasible.",
"The Fenchel dual of Problem $\\mathbb {P}(p)$ in eq:p-star is $\\mathbb {D}(p){}:{} _{y\\in ^m} f^*(-H^\\top y; p) + g^*(y).$ Let us define the function $\\hat{f}:^m\\rightarrow $ as $\\hat{f}(y; p)f^*(-H^\\top y; p).$ Then, Problem $\\mathbb {D}(p)$ can be written as $\\mathbb {D}(p){}:{} _{y\\in ^m} \\hat{f}(y; p) + g^*(y).$ For given $p$ , strong duality holds if there is an $x\\in \\mathcal {Z}(p)$ such that $Hx \\in \\operatorname{relint} g$ [36] — we will hereafter assume that this assumption is satisfied."
],
[
"Optimality conditions",
"The proximal operator of a proper, closed, convex function $g$ plays a major role in modern optimization theory and is defined as $_{\\lambda g}(v) = _{z}\\lbrace g(z) + \\tfrac{1}{2\\lambda }\\Vert v-z\\Vert ^2\\rbrace ,$ with $\\lambda {}>{}0$ .",
"Proximal operators of a great variety of functions including indicators of sets, distance-to-set functions and norms can be easily evaluated analytically and at a very low computational cost [37].",
"For example, the proximal operator of the indicator of a set $Y$ is the projection on $Y$ , that is $_{\\lambda \\delta (\\cdot \\mid Y)}(v)=(v\\mid Y)$ .",
"A simple optimality condition for (REF ) is $y-_{\\lambda g^*}(y-\\lambda \\nabla \\hat{f}(y)) = 0,$ for some $\\lambda >0$ [38].",
"By virtue of the Moreau decomposition formula, (REF ) is equivalently written as $\\nabla \\hat{f}(y) + _{\\lambda ^{-1}g}(\\lambda ^{-1} y - \\nabla \\hat{f}(y)) = 0.$ We define the forward-backward mapping $T_\\lambda (y) & _{\\lambda g^*}(y-\\lambda \\nabla \\hat{f}(y)),$ which, using the Moreau decomposition property, becomes $T_\\lambda (y) = y - \\lambda \\nabla \\hat{f}(y) - \\lambda _{\\lambda ^{-1}g}(\\lambda ^{-1}y {-} \\nabla \\hat{f}(y)),$ and we also define the fixed-point residual mapping $R_\\lambda (y) & \\lambda ^{-1}(y-T_\\lambda (y)) \\\\& = z_{\\lambda }(y)-Hx(y),$ where $x(y)$ and $z_\\lambda (y)$ are defined as $x(y) & \\nabla f^*(-H^\\top y), \\\\z_{\\lambda }(y) & _{\\lambda ^{-1}g}\\left(\\lambda ^{-1}y+Hx(y)\\right),$ therefore, $x(y)=_{z}\\lbrace z,H^\\top y\\: + f(z)\\rbrace .$ Note also that $T_\\lambda (y)$ can be computed from eq:tlambdaprox as $T_\\lambda (y)=y - \\lambda \\left(\\nabla \\hat{f}(y) - z_\\lambda (y)\\right).$ The aforementioned optimality condition in eq:optimality-condition is equivalently written as $R_\\lambda (y)=0$ , that is, solving the dual optimization problem (REF ) becomes equivalent to finding a zero of the operator $R_\\lambda $ ."
],
[
"Forward-backward envelope",
"The forward-backward envelope (FBE) of (REF ) is a real-valued function $\\varphi _\\lambda $ given by [39], [40] $\\varphi _\\lambda (y) = & \\hat{f}(y)+ g^*(T_\\lambda (y))\\\\&- \\lambda \\nabla \\hat{f}( y), R_\\lambda (y)\\:+ \\tfrac{\\lambda }{2}\\Vert R_\\lambda (y)\\Vert ^2.$ If $\\hat{f}$ is twice continuously differentiable — conditions under which this is the case can be found in [41] — then $\\varphi _\\lambda $ is continuously differentiable with $\\nabla \\varphi _\\lambda (y) = (I-\\lambda \\nabla ^2 \\hat{f}(y))R_\\lambda (y).$ Note that in practice it is not necessary to compute or store the Hessian matrix $\\nabla ^2 \\hat{f}(y)$ .",
"Instead, it suffices to implement an algorithm that returns Hessian-vector products of the form $\\nabla ^2 \\hat{f}(y){}\\cdot {}z$ .",
"The most important property of the FBE is that for $\\lambda \\in (0,{1}{L})$ , the set of minimizers of (REF ) coincides with $\\varphi _\\lambda & \\equiv \\nabla \\varphi _{\\lambda }\\lbrace y: \\nabla \\varphi _\\lambda (y) = 0 \\rbrace \\\\& = \\hat{f}(y) + g^*(y)=R_\\lambda .$ Essentially, the problem of solving the dual optimization problem (REF ) is equivalent to the unconstrained minimization of the continuously differentiable function $\\varphi _\\lambda $ , that is $\\inf \\hat{f}(y) + g^*(y) {}={} & \\inf \\varphi _\\lambda , \\\\\\hat{f}(y) + g^*(y) {}={} & \\varphi _\\lambda .$ Moreover, the above is equivalent to finding a zero of the fixed-point residual operator.",
"In the common case where $\\hat{f}$ is strongly convex quadratic, $\\phi _\\lambda $ is both continuously differentiable and convex."
],
[
"Dual ",
"If $\\hat{f}$ is twice differentiable, according to (REF ) the original (dual) optimization problem can be cast as an unconstrained optimization problem with a smooth cost function.",
"As a result we can use an appropriate unconstrained optimization method to solve such problems, such as limited-memory BFGS [32], however, convergence is only guaranteed under restrictive requirements (such as twice differentiability and uniform convexity of the FBE [42]).",
"Instead, minfbe is a method that can be applied to problems with nonsmooth cost functions using the forward-backward envelope as a merit function using a simple line search [30].",
"minfbe involves simple and computationally inexpensive iterations, and exhibits superior global convergence properties.",
"The application of minfbe to the dual optimization problem, $\\mathbb {D}(p)$ , leads to alg:FB-LBFGS.",
"[H] Dual minfbe with L-BFGS directions [1] $\\lambda \\in (0,{1}{L})$ , $y^0$ , $m$ (memory), $\\epsilon $ (tolerance) Primal-dual solution triple $(x, z, y)$ Initialize an L-BFGS buffer with memory $m$ $\\Vert R_{\\lambda }(y^k)\\Vert _{\\infty } {}>{} \\epsilon $ $d^k = -B^k \\nabla \\varphi _{\\lambda }(y^k)$ (Compute an L-BFGS direction using the L-BFGS buffer) Choose the smallest $\\tau _k \\in \\lbrace 2^{-\\nu }\\rbrace _{\\nu \\in }$ so that $\\varphi _{\\lambda }(w^{k})\\le \\varphi _{\\lambda }(y^k),$ where $w^{k} = y^k + \\tau _k d^k$ $x^{k+1} = T_\\lambda (w^k)$ Compute the L-BFGS-related quantities $s^k = y^{k+1} - y^k$ , $q^k = \\nabla \\varphi _{\\lambda }(y^{k+1}) - \\nabla \\varphi _{\\lambda }(y^{k})$ and $\\rho _k = \\langle s^k, q^k \\rangle $ $s^k, q^k\\: > \\epsilon ^{\\prime } \\Vert s^k\\Vert ^2 \\Vert \\nabla \\varphi _\\lambda (y^k)\\Vert ^2$ Push $(s^k,q^k,\\rho ^k)$ into the LBFGS buffer $k = k + 1$ return $(x, z, y) = (x(y^k), z(y^k), y^k)$ minfbe consists in applying the forward-backward mapping on the extrapolated vector $w^k = y^k + \\tau _k d^k$ which satisfies the decrease condition (REF ).",
"The L-BFGS buffer is updated with the vectors $s^k$ , $q^k$ , and their inner product $\\rho _k$ , provided that the minimum-curvature condition in line 7 is satisfied for a small tolerance $\\epsilon ^{\\prime }{}>{}0$ , following [43].",
"The algorithm iterates on the dual vectors $y^k$ and returns a triple $(x, z, y)$ which satisfies the termination condition $\\Vert R_{\\lambda }(y^k)\\Vert _{\\infty } \\le \\epsilon $ , which, in light of eq:fpr2 means that $\\Vert z - Hx\\Vert _{\\infty } \\le \\epsilon ,\\\\-H^\\top y \\in \\partial f(x),\\\\_{\\Vert \\cdot \\Vert _\\infty }(y, \\partial g(z)) \\le \\lambda \\epsilon ,$ where $_{\\Vert \\cdot \\Vert _\\infty }$ denotes the point-to-set distance with repsect to the $\\infty $ -norm.",
"We should highlight that the line search in line 4 of alg:FB-LBFGS is a simple descent condition on the FBE, which is simpler than the Wolfe conditions used in [32].",
"Moreover, although in alg:FB-LBFGS we use L-BFGS directions, the method works with any direction of descent $d^k$ with respect to the FBE, that is, if $d^k, \\nabla \\varphi _\\lambda (y^k)\\: \\le 0$ .",
"If $f$ is quadratic plus the indicator of an affine subspace, $x(y)$ turns out to be linear, that is $x(w) = x(y{}+{}\\tau d) = x(y) + \\tau x(d),$ and $\\hat{f}$ is a quadratic function, that is $\\nabla \\hat{f}$ is linear and $\\hat{f}(y) = y, \\nabla \\hat{f}(y)\\:,$ from which we can see that $\\hat{f}(y+\\tau d) {}={} & y + \\tau d, \\nabla \\hat{f}(y+ \\tau d)\\:\\\\{}={} & \\hat{f}(y) + \\tau ^2 \\hat{f}(d) + 2\\tau y, \\nabla \\hat{f}(d)\\:.$ By virtue of the last two properties and after some algebraic manipulations, we find that the line search condition $\\varphi _{\\lambda }(w^{k}){}-{}\\varphi _{\\lambda }(y^k) {}\\le {}0$ is equivalent to $\\alpha _2(y, d) \\tau ^2 + \\alpha _1(d) \\tau + \\alpha _0(\\tau ; y, d) \\le 0,$ where $\\alpha _0(\\tau ; y, d) {}={} & g^*(T_\\lambda (y+\\tau d)) - g^*(T_\\lambda (y)) \\\\& + \\tfrac{\\lambda }{2}[\\Vert z_\\lambda (y+\\tau d)\\Vert ^2 - \\Vert z_\\lambda (y)\\Vert ^2],\\\\\\alpha _1(y,d) {}={} & Hx(d), 2y - \\lambda Hx(y)\\:,\\\\\\alpha _2(d) {}={} & \\hat{f}(d) - \\tfrac{\\lambda }{2}\\Vert Hx(d)\\Vert ^2,$ and $\\hat{f}(d)$ can be computed by invoking [44], from which $\\hat{f}(d) = - Hx(d), d\\: - f(x(d)).$ Note that $\\alpha _1$ and $\\alpha _2$ do not depend on $\\tau $ , therefore, can be computed once per iteration.",
"This leads to a significant reduction of the involved floating point operations per iteration.",
"The most computationally demanding parts of minfbe are (i) the computation of $x(y)$ and $x(d)$ , and (ii) the computation of the Hessian-vector product $\\nabla \\hat{f}(y^k)R_\\lambda (y^k)$ that is required to determine $\\nabla \\varphi _\\lambda (y^k)$ in line 3 of alg:FB-LBFGS.",
"The involved operations can be parallelized on a gpu as we shall discuss in Section REF , but the computations of $x(y)$ , $x(d)$ and $\\nabla \\hat{f}(y^k)R_\\lambda (y^k)$ cannot be parallelized.",
"Often, the Lipshcitz constant of the gradient of $\\hat{f}$ is not known and needs to be estimated with a backtracking procedure.",
"The original backtracking proposed in [30] halves the value of $\\lambda $ after the line search in line 4 if the following condition is satisfied $\\hat{f}(T_\\lambda (w^k)) > \\hat{f}(y^k) - \\lambda \\nabla \\hat{f}(y^k), R_\\lambda (y^k)\\: \\\\+ \\tfrac{(1-\\beta )\\lambda }{2}\\Vert R_\\lambda (y^k)\\Vert ^2,$ for some $\\beta \\in [0, 1)$ .",
"The values $\\hat{f}(y^k)$ , $\\nabla \\hat{f}(y^k), R_\\lambda (y^k)\\:$ and $\\Vert R_\\lambda (y)\\Vert ^2$ are known from the preceding line search, so the cost of the backtracking is that of computing $\\hat{f}(T_\\lambda (w^k))$ .",
"Alternatively, we may use the backtracking method proposed in [45] which halves $\\lambda $ if $\\lambda \\Vert \\nabla \\hat{f}(T_\\lambda (y^k)) - \\nabla \\hat{f}(y^k)\\Vert {}>{}\\epsilon ^{\\prime \\prime } \\Vert T_\\lambda (y^k)-y^y\\Vert ,$ where $\\epsilon ^{\\prime \\prime }{}\\in {}(0, {1}{2})$ .",
"This backtracking procedure has a lower computational cost compared to eq:original-backtracking.",
"In both cases, the L-BFGS buffer is emptied when the value of $\\lambda $ is updated."
],
[
"Parallelizable Newton-type Alternating Minimization Algorithm",
"The Newton-type alternating minimization algorithm (nama) can be used to solve the dual optimization problem $\\mathbb {D}(p)$ in eq:dual-problem without the need to compute the gradient of the FBE [31].",
"nama, applied to the dual optimization problem is given in alg:nama.",
"nama involves a simple line search which consists in determining a $\\tau _k$ so that the dual vector defined as $w^{k} = y^k + \\tau _k d^k + (1-\\tau _k)r^k$ satisfies the descent condition $\\varphi _\\lambda (w^{k})\\le \\varphi _\\lambda (y^k)$ .",
"Again, if $\\hat{f}$ is a quadratic function, we can precompute certain quantities in a fashion akin to eq:linesearchtrick.",
"In particular, before the line search in line 7 of alg:nama we need to compute $x(r)$ and $x(d)$ .",
"[htb] nama method for the dual optimization problem [1] $\\lambda \\in (0,\\mu _f/\\Vert H\\Vert ^2)$ , $y^0$ , $\\epsilon >0$ (tolerance) Primal-dual solution triple $(x, z, y)$ $k {}={} 0$ $\\Vert R_\\lambda (y^k)\\Vert >\\epsilon $ $x^k {}={} x(y^k)$ , $z^k {}={} z_\\lambda (y^k)$ $r^k {}={} z^k - Hx^k$ $d^k {}={} -B^k r^k$ (Compute an L-BFGS direction using the L-BFGS buffer) Choose the smallest $\\tau _k \\in \\lbrace 2^{-\\nu }\\rbrace _{\\nu \\in }$ so that $\\varphi _{\\lambda }(w^{k})\\le \\varphi _{\\lambda }(y^k),$ where $w^{k} = y^k + \\tau _k d^k + (1-\\tau _k)r^k$ $\\tilde{x}^k {}={} x(w^k)$ , $\\tilde{z}^k {}={} z_\\lambda (w^k)$ $y^{k+1} {}={} y^k + \\lambda (H \\tilde{x}^k - \\tilde{z}^k)$ Compute the L-BFGS-related quantities $s^k = y^{k+1} - y^k$ , $q^k = R_{\\lambda }(y^{k+1}) - r^{k}$ and $\\rho _k = \\langle s^k, q^k \\rangle $ $s^k, q^k\\: > \\epsilon ^{\\prime } \\Vert s^k\\Vert ^2 \\Vert r^{k}\\Vert ^2$ Push $(s^k,q^k,\\rho ^k)$ into the L-BFGS buffer $k = k + 1$ return $(x, z, y)=(x(y^k), z(y^k), y^k)$ The main computational cost involved in alg:nama comes from the evaluation of $x(y)$ , $x(r)$ , and $x(d)$ .",
"Note that if $x$ is linear, $x(w)$ can be computed at a very low computational cost.",
"In particular, the extrapolated vector $w^k$ can be written as $w^k = \\tilde{y}^k + \\tau _k \\tilde{d}^k$ , where $\\tilde{y}^k = y^k + r^k$ and $\\tilde{d}^k = d^k - r^k$ , therefore the decrease condition of nama in eq:decreaseconditionnama is equivalent to eq:linesearchtrick with $\\tilde{y}^k$ and $\\tilde{d}^k$ in lieu of $y^k$ and $d^k$ respectively, that is, $\\alpha _2(\\tilde{y}, \\tilde{d}) \\tau ^2+\\alpha _1(\\tilde{d}) \\tau +\\alpha _0(\\tau ; \\tilde{y}, \\tilde{d}) \\le 0.$ Overall, given that the computation of Hessian-vector products in minfbe comes at approximately the same cost as computing the dual gradient, and given that the computation of $x(r)$ and $x(d)$ can be carried out in parallel, nama has a lower per-iteration computation cost.",
"Although minfbe and nama exhibit similar convergence properties, with nama we can afford a greater parallelizability that leads to superior performance in practice as we shall show in sec:numericalsimulations."
],
[
"Efficient parallel computations",
"gpu have a hardware architecture that allows the execution of the same set of instructions on different memory positions.",
"gpu are equipped with a set of SIMD stream processors, each having its own computing resources, that execute “compute kernels,” that is, functions that are executed simultaneously on different data.",
"NVIDIA's gpu use the CUDA programming interface where kernels are executed in parallel threads, which are organised in blocks which can share memory and which are in turn organised in grids.",
"At a hardware level, threads are executed in parallel in warps of 32 threads.",
"Threads in the same block have asynchronous read/write access to a local shared memory and can synchronize.",
"Each thread has its own local memory, and all threads have access to the device's global memory.",
"Modern gpu count several streaming multiprocessors with hundreds of cores, possess a computing throughput of several Tera-FLOPs, and have a significant memory capacity of several GBs.",
"The hardware architecture and programming model of gpu necessitates a fresh look at parallelization approaches for numerical optimization.",
"Kernels are best suited for the parallel execution of simple numerical operations.",
"The efficient computation of the dual gradient is of crucial importance for the performance of the algorithm we are about to describe.",
"By virtue of the Conjugate Subgradient Theorem [44], we have that $x(y){}={}_{z\\in \\mathcal {Z}(p)} \\bigg \\lbrace \\sum _{i\\in (1, N)}\\hspace{-11.99997pt}\\pi ^i\\hat{\\phi }^i(x^{(i)},u^{(i)})\\\\[-0.5em]{}+{}\\sum _{i\\in (N)}\\hspace{-8.99994pt}\\pi ^i \\hat{\\phi }_N(x^i)\\bigg \\rbrace ,$ where $\\hat{\\phi }^i(x^{(i)},u^{(i)}) = \\phi ^i(x^{(i)},u^{(i)})+ y^i, F^i x^{(i)}+ G^{i}u^{(i)}\\:$ for $i\\in (1, N)$ , and $\\hat{\\phi }_N(x^i)=\\phi _N(x^i)+y^i, F_N^i x^i\\:$ , for $i\\in (1, N)$ .",
"The solution of this problem can be determined via a dynamic programming in a way akin to [29] leading to alg:solve wherein $\\Phi _k^i$ , $\\Theta _k^i$ , $D_k^i$ , $\\Lambda _k^i$ , $K_k^i$ $\\sigma _k^i$ , $c_k^i$ are computed once offline following a Riccati-type recursion.",
"In cases where the data of the optimal control problem need to be updated (e.g., if the dynamical system is time varying, or the parameters of the cost must be updated in real time), the computation of these matrices can be carried out on a gpu and in fact the time for their computation is negligible compared to that of solving the problem.",
"[htbp] Computation of the dual gradient, $x(y)$ [1] Dual vector $y\\in ^m$ $x(y)$ $\\hat{q}^i\\leftarrow y^i,$ for all $i\\in (N)$ [in parallel]$i\\in (0, N-1)$ $\\phantom{\\hat{q}^i}{u^i}\\leftarrow \\Phi ^i y^i +\\sigma ^i$ $\\hat{q}^i\\leftarrow D^{i\\top } y^i + \\hat{c}^i$ $k = N - 1, \\ldots , 0$ [in parallel]$i\\in (k)$ $\\phantom{\\hat{q}^i}{u^i}\\leftarrow \\sum _{i_{\\scriptscriptstyle +}\\in (i)}\\Theta ^{i_{\\scriptscriptstyle +}} \\hat{q}^{i_{\\scriptscriptstyle +}}$ $\\hat{q}^i\\leftarrow \\sum _{i_{\\scriptscriptstyle +}\\in (i)} \\Lambda ^{i_{\\scriptscriptstyle +}\\top } \\hat{q}^{i_{\\scriptscriptstyle +}}$ $x^0=p$ $i\\in (0, N-1)$ $u^i\\leftarrow K^ix^i+u^i$ $i_{\\scriptscriptstyle +}\\in (i)$ $x^{i_{\\scriptscriptstyle +}}\\leftarrow A^{i_{\\scriptscriptstyle +}} x^i+B^{i_{\\scriptscriptstyle +}} u^i +c^{i_{\\scriptscriptstyle +}}$ $x(y) = (\\lbrace x^i\\rbrace ,\\lbrace u^i\\rbrace )$ .",
"The computation of Hessian-vector products of the form $\\nabla ^2 \\hat{f}(y){}\\cdot {}r$ that is required for the computation of the gradient of the FBE is given in alg:hessian-vec-products.",
"alg:solve and alg:hessian-vec-products incur roughly the same computation cost.",
"[ht] Computation of Hessian-vector products required for the computation of $\\nabla \\varphi _\\lambda $ [1] Vector $r$ Hessian-vector product, $\\nabla ^2 \\hat{f}(y){}\\cdot {}r$ $\\hat{q}^i\\leftarrow r^i,$ for all $i\\in (N)$ $k = N - 1, \\ldots , 0$ [in parallel]$i\\in (k)$ $\\phantom{\\hat{q}^i}{\\hat{u}^i}\\leftarrow \\Phi ^i r^i + \\sum _{i_{\\scriptscriptstyle +}\\in (i)}\\Theta ^{i_{\\scriptscriptstyle +}} \\hat{q}^{i_{\\scriptscriptstyle +}}$ $\\hat{q}^i\\leftarrow D^{i\\top } r^i + \\sum _{i_{\\scriptscriptstyle +}\\in (i)} \\Lambda ^{i_{\\scriptscriptstyle +}\\top } \\hat{q}^{i_{\\scriptscriptstyle +}}$ $\\hat{x}^0=0$ $i\\in (0, N-1)$ $\\hat{u}^i\\leftarrow K^i\\hat{x}^i+\\hat{u}^i$ $i_{\\scriptscriptstyle +}\\in (i)$ $\\hat{x}^{i_{\\scriptscriptstyle +}} {}\\leftarrow {} A^{i_{\\scriptscriptstyle +}} \\hat{x}^i+B^{i_{\\scriptscriptstyle +}} \\hat{u}^i$ $\\nabla ^2 \\hat{f}(y){}\\cdot {}r {}={} (\\lbrace \\hat{x}^i\\rbrace ,\\lbrace \\hat{u}^i\\rbrace )$ .",
"Lastly, most proximal operations can be massively parallelized.",
"For example, if $\\bar{\\phi }^{i}(z) = \\delta (z {}\\mid {} Y^i)$ and $Y^i=\\lbrace z{}:{}z^{i}_{\\min } \\le z \\le z^i_{\\max }\\rbrace $ , then the computation of $_{\\lambda \\bar{\\phi }^i} = _{Y^i}$ is element-wise independent and can be easily parallelized.",
"Likewise, a great many proximal operators, such as those of the indicators of rectangles and common norm-balls, and functions such as $\\Vert {}\\cdot {}\\Vert _1$ , the Huber loss function and more, lend themselves to high parallelizability [38].",
"In general, the total memory that needs to be allocated on the gpu grows linearly with the length of the L-BFGS buffer, linearly with the prediction horizon, and linearly with the number of nodes of the tree, and quadratically with the system states and inputs.",
"The additional parallelisation in nama requires the allocation of additional memory on the gpu, but leads to a higher throughput and occupancy of the device."
],
[
"Preconditioning",
"Stochastic optimal control problems tend to be ill conditioned because of the presence of generally small probability values.",
"As first-order methods are known to be affected by the problem being ill conditioned, here we make use of a simple diagonal preconditioning heuristic where we scale the original dual variables $y=((y^i)_{i\\in (0, N)},(y_{N}^{i})_{i\\in (N)})$ by introducing the scaled dual variables $\\bar{y}=((\\bar{y}^i)_{i\\in (0, N)}, (\\bar{y}_N^i)_{i\\in (N)})$ with $\\bar{y}^i = \\frac{y^i}{\\sqrt{\\pi ^i}},$ for $i\\in (0, N)$ and $\\bar{y}_N^i = \\frac{y_N^i}{\\sqrt{\\pi ^i}},$ for $i\\in (N)$ .",
"This scaling is a heuristic similar to the Jacobi preconditioning discussed in [46]."
],
[
"Warm start",
"Generally, the accelerated projected gradient method converges at a rate $\\mathcal {O}(1/k^2)$ and although it may exhibit slow convergence, its iterations are computationally cheap, so it can be used to warm start minfbe and nama.",
"We have observed that running as few as five iterations of gpad [47], [7] can provide a good warm starting point for minfbe and nama."
],
[
"Numerical Simulations",
"This section is organised in two parts: in sec:spring-mass we compare minfbe and nama with the accelerated projected gradient method and discuss the convergence rate of each method.",
"In particular, we demonstrate that a serial implementation of nama and minfbe leads to superior performance compared to the accelerated proximal gradient method.",
"The two methods exhibit comparable convergence speed.",
"Next, in sec:barcelona-dwn we apply minfbe and nama to solve a large-scale stochastic optimal control problem for the operating management of the drinking water network of Barcelona taken from [7].",
"We show that nama affords a higher parallelisation leading to a significant performance improvement."
],
[
"Spring-mass-damper array",
"Consider an array of $M$ consecutive point particles of mass $m$ connected to each other through elastic springs of stiffness $k_{\\rm s}$ and linear dampers with viscous damping coefficients $b_{\\rm d}$ illustrated in fig:springmass.",
"Figure: Array of MM consecutive interconnected masses, M+1M+1 elastic springs and dampersand M-1M-1 actuators.In between the successive masses $j$ and $j+1$ , for $j=1,\\ldots ,M-1$ , there is an actuator that can apply a force $u_j\\in [u_{\\rm min}, u_{\\rm max}]$ .",
"The state variable of this sytem comprises of the positions $p_j$ of the masses and their velocities $v_j$ , which are constrained in $[p_{\\rm min}, p_{\\rm max}]$ and $[v_{\\rm min}, v_{\\rm max}]$ , respectively.",
"The system is described by a set of linear differential equations which can be obtained by the application of Newton's second law of motion, which, after discretisation with sampling time $T_s$ and a zero-order hold, yields a discrete-time linear time invariant system.",
"Furthermore, we assume that there is an external additive disturbance $c_{w_k}$ , as in eq:affinestochasticdynamics, which is driven by a discrete Markov process, $w_k$ , with two modes.",
"In this example, we consider a stochastic optimal control problem with prediction horizon $N$ , quadratic stage cost functions $\\phi ^i(x, u) = x^\\top Q x + u^\\top R u$ , and quadratic terminal costs $\\phi _N^i(x)=x^\\top Q_N x$ .",
"Moreover, we have $M=5$ masses with $m=[5]{kg}$ , $k_{\\rm s}=[1]{N}{m}$ , $b_{\\rm d}=[0.1]{Ns}{m}$ , $u_{\\rm max} = -u_{\\rm min} = [2]{N}$ and the maximum allowed velocity is $[5]{m}{s}$ .",
"The prediction horizon is $N=11$ and the external disturbance $c_k$ is driven by a Markov chain with two modes with initial probability distribution $p_c=(0.5, 0.5)$ and probability transition matrix $P_c={0.1 & 0.9\\\\0.9 & 0.1}$ ; at mode 1 the value of $c$ is zero and at mode 2, $c$ takes the value $0.1$ .",
"The sampling time is $T_s=[0.5]{s}$ .",
"Lastly, the weights of the stage and terminal cost functions are $Q=5I_{10}$ , $R=2I_{4}$ and $Q_N = 100I_{10}$ .",
"No warm starting is used in any of the algorithms.",
"Figure: Comparison of the convergence of nama, minfbe, and the accelerated projected gradient (gpad) methodapplied to the dual problem.Figure: Distribution of the number of oracle calls required for the computation of the dual gradient (alg:solve)and Hessian-vector products (alg:hessian-vec-products) for gpad, minfbe and nama.We ran the stochastic optimal control problem for 300 initial states $x_0=p$ , sampled uniformly from the problem's domain.",
"These problems were solved with nama, minfbe and the accelerated projected gradient method applied to the dual problem (gpad) following [7].",
"In nama and minfbe we used L-BFGS directions with a memory of 5.",
"We used the same termination condition in all methods with $\\epsilon =5\\cdot 10^{-4}$ .",
"gpad is known to converge at a rate of $\\mathcal {O}(1/k^2)$ , which can be observed in fig:Convergencesamplepoint; clearly, gpad can only achieve low to medium accuracy solutions within a few hundred iterations.",
"On the other hand, minfbe and nama exhibit a significantly faster convergence rate and require fewer iterations to achieve solutions of higher accuracy.",
"In fig:histogramhessiancall we show the number of calls of alg:solve and alg:hessian-vec-products required to solve the aforementioned collection of 300 random problems up to the desired accuracy.",
"We may observe that in the majority of cases ($84\\%$ ), minfbe and nama can solve the problems with no more than 50 calls, whereas the median of the number of calls corresponding to gpad is 188.",
"Note that nama and minfbe appear to perform on a par.",
"However, in the next section we will demonstrate that nama allows for greater parallelizability leading to superior performance on a gpu."
],
[
"Large-scale drinking water network",
"In this section we apply the proposed numerical optimization methods for the solution of a model predictive control problem for a drinking water network, whose transportation dynamics is described by $x_{t+1} {}={} & Ax_t + Bu_t + G d_{t},\\\\0 {}={} & E_u u_t + E_d d_{t},$ where $x_t$ is the vector of the volume of water in the reservoirs of the network, $u_t$ is the vector of pumping set points and $d_t$ is the vector of water demands from the various distribution nodes.",
"The value of $d_t$ is measured at time $t$ and future demand values are predicted by a model that returns estimates $\\hat{d}_{t+t^{\\prime }{}\\mid {}t}$ , for $t^{\\prime } \\ge t$ , while $d_{t+t^{\\prime }} = \\hat{d}_{t+t^{\\prime }{}\\mid {}t} + \\epsilon _{t^{\\prime }}$ , where $\\epsilon _{t^{\\prime }}$ is a random process that can be described by a scenario tree [7].",
"The water network model (REF ) comprises 63 states corresponding to water level in the tanks, 114 inputs corresponding to flow control devices (pumps and valves), 88 disturbance variables corresponding to the demand sectors and input-disturbance relationship corresponding to the 17 mixing nodes.",
"The detailed stochastic optimal control problem and the formulation of the optimisation problem is discussed in [7].",
"The operation of the water network is subject to uncertainty in water demand and electricity prices.",
"Figure: Box plots of the computational time with different scenario tree sizesfor the gpu implementation of the algorithms:(parallel) nama, minfbe and gpad.The nama and minfbe algorithms are implemented in the RapidNethttps://github.com/GPUEngineering/RapidNet software package that is developed for the operational control of water network problems.",
"All simulations presented in this section were carried out on an NVIDIA Tesla C2075 gpu which counts 448 CUDA cores running at $[1.15]{GHz}$ and $[6]{GB}$ of dedicated memory.",
"In order to demonstrate the effect of the additional parallelisation in nama that we discussed in sec:parallelizable-nama, we provide results for the method with that additional parallelization in the computation of the line search (p-nama) and nama without that additional parallelization.",
"The parallel computations involved in Algorithms REF and REF are carried out using cuBLAS's cublasSgemmBatched and cublasSgemm.",
"In this example, matrices $A$ , $B$ , $G_d$ , $E_u$ and $E_d$ are sparse, and this has been used to tailor the implementations of Algorithms REF and REF to be more efficient.",
"The L-BFGS memory is set to 15.",
"In the case with 577 scenarios, the problem involves 2.1 million primal and 3.8 million dual variables and nama and minfbe algorithms require an excess of $2.9\\%$ ($[172]{MB}$ ) of memory and p-nama requires an excess of $4.1\\%$ ($[242]{MB}$ ) of memory than dual accelerated proximal gradient (gpad) algorithm.",
"The solve times of p-nama, nama, minfbe and gpad are shown in fig:Computationtime, where note that the horizontal axis is logarithmic.",
"It can be observed that p-nama is noticeably faster compared to nama, minfbe and gpad."
],
[
"Conclusions",
"In this paper we proposed the use of minfbe and nama for solving large-scale scenario-based convex stochastic optimal control problems.",
"Both methods use limited-memory quasi-Newtonian, L-BFGS, directions and exhibit a very fast convergence rate.",
"They are both suitable for parallelization on GPUs, but nama lends itself to a significantly higher parallelization.",
"We presented compelling results on two stochastic optimal control problems, namely a spring-mass-damper array and the drinking water network of Barcelona, demonstrating that the two methods significantly outperform gpad, whose parallelizable implementation on a gpu has been previously shown to outperform Gurobi's interior point solver [7].",
"Future work will focus on the development of parallelizable methods for large-scale scenario-based risk-averse optimal control problems [48]."
]
] | 2107.01745 |
[
[
"Many-body coherence and entanglement from randomized correlation\n measurements"
],
[
"Abstract We show that $k$-point correlation measurements on output of a non-interacting, multimode, random unitary allow to quantify the $k$-particle coherence of $N\\geq k$ identical (bosonic or fermionic) particles.",
"For separable many-particle input states, we establish an on average strictly monotonic relationship between such $k$-particle coherences, the interference contrast in the experimentally accessible counting statistics and the degree of the particles' mutual distinguishability, as controlled by their internal degrees of freedom.",
"Non-separability on input can be unveiled by comparison of correlation measurements of different orders."
],
[
"Relation of the mean coherence to quantities defined in the literature",
"For Fock states (i.e.",
"eigenstates of all number operators $\\hat{N}_{p\\alpha }=\\hat{a}^\\dagger _{p\\alpha } \\hat{a}_{p\\alpha }$ ), the degree of indistinguishability $\\mathcal {I} =\\sum _{m \\ne n\\in \\mathcal {B}{}_\\mathrm {ext}}\\sum _{\\alpha \\in \\mathcal {B}{}_\\mathrm {ext}} N_{m\\alpha }N_{n\\alpha }\\ \\Big / \\!",
"\\sum _{m \\ne n\\in \\mathcal {B}{}_\\mathrm {ext}} N_{m}N_{n}$ was introduced in [54], [49] to quantify partial distinguishability in multi-component bosonic systems.",
"In particular, $\\mathcal {I}$ was shown to correlate with the time-average of the density variances $\\mathinner {\\langle {N_m^2(t)}\\rangle }-\\mathinner {\\langle {N_m(t)}\\rangle }^2$ , which probe the $2P$ reduced state evolving from the initial Fock state.",
"The degree of indistinguishability is related to the 2P mean coherence for arbitrary definite external mode occupations $N_p \\in \\mathbb {N}$ , by $\\mathcal {I} =N (N-1) (\\mathcal {W}^{(2)}-1)\\ \\Big / \\!",
"\\sum _{m \\ne n\\in \\mathcal {B}{}_\\mathrm {ext}} N_{m}N_{n}~.$ For an $N$ P state $\\rho $ , the reduced external state $\\rho {}_\\mathrm {ext}=\\rho {}_\\mathrm {ext}^{(N)}$ coincides with $1/N!$ times the $J$ matrix introduced in [24], [55], [15] if ideal detectors are assumed.",
"Actually, the author of [15] writes “Note that quantum coherence of photon paths is reflected in the $J$ matrix in a way very similar as in the usual density matrix of a quantum system” but does not push the connection further.",
"One can measure the bosonic character of the external reduced state $\\rho {}_\\mathrm {ext}$ by its projection onto the symmetric subspace [55], [27] $p_s=\\mathrm {tr}(\\rho {}_\\mathrm {ext}P_S) \\,,$ where $P_S=\\frac{1}{N!",
"}\\sum _{\\pi \\in S_N} \\pi $ is the symmetrizer and $\\pi $ acts on $\\mathbf {m}\\in \\mathcal {H}{}_\\mathrm {ext}^{(N)}$ as $\\pi \\mathinner {|{\\mathbf {m}}\\rangle }=\\mathinner {|{ m_{\\pi ^{-1}(1)} ,\\dots , m_{\\pi ^{-1}(N)} }\\rangle }$ .",
"This quantity is proportional to the $N$ P mean coherence, with $p_s=\\mathcal {W}^{(N)}\\prod _{m\\in \\mathcal {B}{}_\\mathrm {ext}} N_p!/N!",
"\\,.$ For particles with individual pure internal states $\\mathinner {|{\\phi _i}\\rangle }$ , this is also equal to $1/N!$ times the permanent of the distinguishability matrix $\\mathcal {S}=(\\mathinner {\\langle {\\phi _i|\\phi _j}\\rangle })_{i,j}$ introduced in [14]."
],
[
"Sampling of internal states",
"To map out the full transition from indistinguishable fermions to bosons via the intermediate case of distinguishable particles, in terms of the $k$ P mean coherences $\\mathcal {W}^{(k)}$ as uniformly as possible, we use the following two-step sampling procedure of pure internal states for each of the particles: The dimension of the internal Hilbert space has to be taken larger than the particle number.",
"To map out the neighborhood of indistinguishable particles, we start from a unit vector $\\mathinner {|{e}\\rangle }\\in \\mathcal {H}{}_\\mathrm {int}$ and add a perturbation $\\mathinner {|{f_i}\\rangle }$ , with the real and imaginary parts of the components of $\\mathinner {|{f_i}\\rangle }$ sampled from a normal distribution with zero mean and variance $\\epsilon $ .",
"By choosing $\\epsilon $ sufficiently small, the resulting internal states $\\mathinner {|{\\phi _i}\\rangle }= \\mathinner {|{e}\\rangle } + \\mathinner {|{f_i}\\rangle }$ , after normalization, are almost parallel.",
"The larger $\\epsilon $ gets, the smaller the relative contribution of the constant vector $\\mathinner {|{e}\\rangle }$ becomes, after renormalization, and we sample the unit sphere in $\\mathcal {H}{}_\\mathrm {int}$ almost uniformly.",
"As a second step, we sample the neighborhood of perfectly distinguishable particles by choosing orthogonal unit vectors $\\mathinner {|{e_i}\\rangle }\\in \\mathcal {H}{}_\\mathrm {int}$ for each particle, perturbed by vectors $\\mathinner {|{f_i}\\rangle }$ sampled as before with normally distributed components in $\\mathbb {C}$ , followed by renormalization.",
"As before, for large $\\epsilon $ the contributions from the constant vectors $\\mathinner {|{e_i}\\rangle }$ in $\\mathinner {|{\\phi _i}\\rangle }= \\mathinner {|{e_i}\\rangle } + \\mathinner {|{f_i}\\rangle }$ vanish after renormalization, and we approach uniform sampling of the unit sphere in $\\mathcal {H}{}_\\mathrm {int}$ .",
"For sufficiently small $\\epsilon $ , we generate states $\\mathinner {|{\\phi _i}\\rangle }$ in the vicinity of perfect distinguishability."
]
] | 2107.01686 |
[
[
"Integrating Expert Knowledge with Domain Adaptation for Unsupervised\n Fault Diagnosis"
],
[
"Abstract Data-driven fault diagnosis methods often require abundant labeled examples for each fault type.",
"On the contrary, real-world data is often unlabeled and consists of mostly healthy observations and only few samples of faulty conditions.",
"The lack of labels and fault samples imposes a significant challenge for existing data-driven fault diagnosis methods.",
"In this paper, we aim to overcome this limitation by integrating expert knowledge with domain adaptation in a synthetic-to-real framework for unsupervised fault diagnosis.",
"Motivated by the fact that domain experts often have a relatively good understanding on how different fault types affect healthy signals, in the first step of the proposed framework, a synthetic fault dataset is generated by augmenting real vibration samples of healthy bearings.",
"This synthetic dataset integrates expert knowledge and encodes class information about the fault types.",
"However, models trained solely based on the synthetic data often do not perform well because of the distinct distribution difference between the synthetically generated and real faults.",
"To overcome this domain gap between the synthetic and real data, in the second step of the proposed framework, an imbalance-robust domain adaptation~(DA) approach is proposed to adapt the model from synthetic faults~(source) to the unlabeled real faults~(target) which suffer from severe class imbalance.",
"The framework is evaluated on two unsupervised fault diagnosis cases for bearings, the CWRU laboratory dataset and a real-world wind-turbine dataset.",
"Experimental results demonstrate that the generated faults are effective for encoding fault type information and the domain adaptation is robust against the different levels of class imbalance between faults."
],
[
"Introduction",
"Data-driven fault diagnosis methods often require a large number of labeled data to generalize well.",
"Faults are, however, rare in real-world complex and safety critical systems.",
"Therefore, a sufficient number of representative samples of faulty conditions is often impossible to be collected in real world applications.",
"Recordings from industry assets often consist of a majority of healthy states and only few faults.",
"In addition, not all fault types may have been captured by the different assets.",
"Moreover, these recordings are often unlabeled because precisely identifying when and which fault is emerging can be difficult even for experienced domain experts.",
"These real-world restrictions make learning fault patterns from unlabeled real-world data a challenging task.",
"One potential solution to overcome these challenges in this unsupervised fault diagnosis setup is to use synthetic faults as the supervision for the data-driven diagnosis models.",
"For example, for bearing fault diagnosis, given operating conditions and bearing characteristics as input, synthetic vibration signals can be generated by highly accurate physical models, e.g, [1].",
"By generating a large number of synthetic faults, a data-driven model can be trained solely based on the synthetic faults, and then evaluated on the real target data [2], [3], [4], [5].",
"However, this way of using synthetic faults has several drawbacks.",
"Firstly, detailed operating conditions can be unknown in order to achieve a realistic physical model.",
"For example, in bearing vibration modeling we typically see that only the bearing is modeled ignoring surrounding rotating equipment, which can have a strong impact on the diagnosis performance.",
"Secondly, even advanced simulations are not perfect, there will always be a distribution gap between the synthetic data and experimental measurements of mechanical systems [6].",
"This domain gap between synthetic source and real target often leads to a significant performance degradation [7] if the model is solely trained on the synthetic data.",
"Thirdly, this pure synthetic data approach fails to make use of the available unlabeled real data, which can be potentially useful for providing additional information on the real faults.",
"To tackle the challenges of synthetic fault generation outlined above, we propose a novel framework for unsupervised fault diagnosis which relaxes the need of highly accurate physical models, while performing well on the real target data.",
"This is achieved by integrating expert knowledge in synthetic data with imbalance-robust domain adaptation for unsupervised fault diagnosis.",
"Unlike previous works [5], [6] which use faults generated by highly accurate physical simulation models as the supervision, the proposed framework uses a relatively simple process for fault generation by augmenting healthy samples.",
"To compensate on the potentially large domain gap between synthetic and real faults, the idea of domain adaptation (DA) is adopted.",
"We propose to align the conditional distributions between synthetic and real features.",
"The proposed DA approach relies less on the quality of the synthetic fault simulator and is robust to the class imbalance in the target domain.",
"Specifically, the proposed framework consists of two complementary parts.",
"In the first stage, expert knowledge is used to generate synthetic data.",
"In the second stage, DA is applied on the synthetic source and real target data to alleviate the domain gap under severe class imbalance.",
"In this work, we apply our framework to bearing fault diagnosis.",
"More specifically, we rely on a relative simple approach where we synthesize vibration signals with fault-initiated pulse-trains corresponding to certain surface defects [1], [8].",
"We include more realistic vibration disturbances from surrounding rotating equipment, by mixing synthetic signals with healthy conditions.",
"As a consequence, a reasonable number of samples can be generated for each fault, forming a balanced synthetic dataset.",
"This synthetic dataset, thus makes use of expert knowledge, and can then facilitate the training of a data-driven fault diagnosis model.",
"This proposed generation process relaxes the need of highly accurate physical models.",
"Although mixing healthy with synthetic defect signals will result in more realistic samples, a domain gap is still inevitable.",
"To overcome this, the idea of unsupervised domain adaptation [9], [7], [10], [11] offers a potential solution.",
"DA aims at learning a representative model from the labeled synthetic source domain, at the same time leveraging unlabeled data from the target domain to improve the model's generalization ability on the target data.",
"Popular adversarial DA approaches [7], [12] use a domain discriminator to directly align the unconditional source and target distributions and minimize the discrepancy between them.",
"They were originally designed and validated for image classification which often consists of balanced classes.",
"Figure: The synthetic faults often suffer a distribution shift from the real faults, which leads to misclassified examples when models are trained only on the synthetic data.",
"We propose to alleviate this problem by using domain adaptation to align the labeled synthetic domain with unlabeled target domain.",
"The rare faults in the unlabeled target domain have very few samples, which makes the alignment harder.",
"The proposed alignment method is robust to this class imbalance by aligning the conditional distributions.",
"The model performance on the target domain can largely be improved by our approach.However, these standard adversarial DA methods make the implicit assumption that source and target domain share similar class distributions, and directly align the source and target unconditional data without using class information.",
"While this assumption is often realistic for the original image datasets, this can be largely violated in the unsupervised fault diagnosis setups because of the nature of rare faults.",
"Given an unlabeled dataset from real-world (target) domain, its class distribution is most likely largely skewed towards the healthy class.",
"Moreover, no more information can be inferred about the target class distribution or its level of class imbalance because the dataset is unlabeled.",
"On the synthetic (source) side, samples are usually generated for each fault class based on the healthy samples, leading to a class-balanced synthetic dataset.",
"If we directly adopt the standard adversarial DA methods, the alignment is then performed between a class-balanced synthetic source domain and a highly imbalanced real-world target domain consisting of a majority of healthy data.",
"This mismatch can further lead to a performance drop on the target data.",
"On the one hand, the different faults in the source domain would mostly be aligned to the healthy state which dominates the target data.",
"On the other hand, rare faults in the target data would be poorly adapted.",
"To address the challenge of imbalanced datasets in unsupervised fault diagnostics setup, we propose a novel imbalance-robust DA approach which overcomes the imbalance problem by making better use of available class information for the alignment.",
"Specifically, we design the discriminator to align the distributions conditioned on classes instead of the unconditional ones.",
"This is achieved by feeding the class information encoded by pseudo-labels to the discriminator.",
"In addition, to make the training of the discriminator more stable and provide more training examples from rare faults, we propose to use a mixup [13]-inspired augmentation to provide more support for the conditioned distributions on these rare classes.",
"We illustrate the effect of the proposed DA approach in Figure REF .",
"We demonstrate the effectiveness of our proposed framework on two different case studies.",
"The first case study is based on the publicly available CWRU laboratory dataset.",
"The second case study is based on the data collected from bearings of real wind turbine generators in the field.",
"The proposed framework uses a relatively simple method for synthesizing the faults.",
"However, it is still able to achieve a good performance with the help of the proposed DA approach.",
"While the evaluation of the proposed framework was performed on two bearing datasets, the proposed synthetic-to-real framework can be easily implemented for most industry assets.",
"Moreover, the proposed imbalance-robust domain adaptation method can be generalized to other applications because it is in theory useful for any adaptation task which is facing the imbalanced data challenge between source and target datasets.",
"Our contributions can be summarized as follows: We propose an unsupervised fault diagnosis framework which builds on simple fault generation process but performs well with the help of domain adaptation.",
"The generation-adaptation framework makes use the power of both expert knowledge and domain adaptation to relieve the need of fault labels.",
"A novel imbalance-robust domain adaptation approach is proposed for unsupervised fault diagnosis which is robust against different imbalance levels between different health conditions.",
"In the field of fault diagnosis, synthetic faults have been used as a reasonable substitute when data on real faults is not available [2], [3], [14].",
"For example, synthetic faults can be induced from healthy samples using an analytical model [2].",
"Faults can also be simulated by highly accurate physical simulation such as finite element method (FEM) models [5], [6].",
"In most works, the learning of the models is purely based on the synthetic faults, while the rare fault data collected from the real applications is preserved as a test set to evaluate the model performance [2], [3], [4], [5].",
"Some recent research studies, such as [15], [16], [17] make the assumption that they have access to a very small set of labeled real faults.",
"A large number of synthetic faults is then generated to mimic the real faults.",
"The corresponding data-driven models can be trained based on these imitated faults.",
"This approach is different from previous works which learn solely from the synthetic data, because the model now also have an implicit or explicit access to information of the small real set of labeled faults.",
"However, in these works, the unavoidable domain shift between the synthetic and real faults is still overlooked.",
"In addition, the requirement of access to the labeled real faults imposes a real limitation on the generalization ability.",
"If the target operating condition is different from the observed one, the learned method may fail because of the existence of a further gap between operating conditions.",
"To the best of our knowledge, these domain gaps between synthetic and real data in the field of fault diagnosis has been long overlooked.",
"Very recently, [18] used both synthetic data and real data to learn a model for remaining useful life prediction in the related field of prognostics.",
"Our work contributes to this by considering a mix of both the healthy and synthetic data which will result in a more realistic source domain."
],
[
"Domain Adaptation",
"Unsupervised domain adaptation [19], [20] is a powerful tool to alleviate the domain shift between synthetic and real data.",
"DA approaches often consider the case where there is a labeled source domain (synthetic in our case) and an unlabeled target (real) domain.",
"The methods improve the performance on the target by making use of both labeled source data and additional unlabeled target data.",
"DA has been widely studied in fields such as computer vision [21], [22], [23], [24] and natural language processing [25], [26].",
"The alleviation of the distribution difference between source and target is often the motivation for applying DA approaches.",
"By aligning the distributions, models can effectively benefit from both the source and target data.",
"Following this motivation, a series of approaches have explored different ways of alignments.",
"Discrepancies such as Maximum Mean Discrepancy (MMD) were used [10] as a guide to align the distributions.",
"Domain Adversarial Neural Networks (DANN) [7] use a domain discriminator to adversarially align the features from source and target domains.",
"The method usually have a classifier branch and a discriminator branch.",
"Given a feature extractor $f$ parameterized by $\\theta _f$ , a classifier $g$ parameterized by $\\theta _g$ , and a discriminator $h$ parameterized by $\\theta _h$ , the DANN method is essentially solving the following equations: $\\mathcal {L}(\\theta _f, \\theta _g, \\theta _h) &= \\mathcal {L}_{clf}(g(f(x))) - \\lambda _d \\mathcal {L}_d(h(f(x))),\\\\\\hat{\\theta }_f, \\hat{\\theta }_g &= \\arg \\min _{\\theta _f, \\theta _g} \\mathcal {L}(\\theta _f, \\theta _g, \\hat{\\theta }_h),\\\\\\hat{\\theta }_h &= \\arg \\max _{\\theta _h} \\mathcal {L}(\\hat{\\theta }_f, \\hat{\\theta }_g, \\theta _h),$ where $\\mathcal {L}_{clf}$ is the cross entropy loss function for the main classifier.",
"$\\mathcal {L}_{d}$ is the cross entropy loss for the domain classification.",
"The classifier branch is trained to minimize the classification loss $\\mathcal {L}_{clf}$ on source data.",
"The discriminator is trained to generate unbiased features for both source and target data.",
"One the one hand, the optimization is updating the discriminator's weight by minimizing discriminator loss $\\mathcal {L}_{d}$ to make it distinguish the source and target features as well as possible.",
"On the other hand, the equation is forcing the feature extractor $f$ to generate unbiased features such that the discriminator loss is large.",
"This minimax game effectively aligns the distributions between source and target.",
"CDAN [27] takes this concept one step further and conditions the adaptation models on discriminative information conveyed in the classifier predictions."
],
[
"Domain Adaptation for Fault Diagnosis",
"In recent years, domain adaptation methods have raised a strong interest in the fault diagnosis community [28], [29], [30], [31], [32], [33].",
"Most existing works focus on the adaptation between operating conditions, and have found classic DA methods beneficial [34], [35], [36], [37], [38].",
"While many works make the assumption that both domains have a balanced number of samples for each class, [34], [39] explore the scenario where the target data has missing fault classes.",
"Another related line of work is to adapt models between different units of a fleet, particularly focusing on the complementary operating conditions [40].",
"Synthetic-to-real adaptation is relatively new and an unexplored research direction in the field of fault diagnostics.",
"To the best of our knowledge, there has been no prior work which explores it for fault diagnosis where the imbalanced faults impose a great challenge to the domain adaptation task.",
"To exploit the possibility of effectively learning and adapting from synthetic data in real industrial scenarios, we consider a challenging and realistic setup where we only have access to segments of condition monitoring vibration recordings.",
"Given unlabeled training data from the target real domain: $\\mathcal {D}_{t}=\\lbrace x_{t}^1, ...,x_{t}^m\\rbrace , $ where $x_{t}^i$ is a vector of a real vibration recording.",
"The target in this unsupervised fault diagnosis task is to train a model which performs well on this target domain."
],
[
"Overview of the Proposed Framework",
"The proposed framework integrates expert knowledge with imbalance-robust domain adaptation for unsupervised fault diagnosis tasks.",
"In the first part, the fault generation module makes use of expert knowledge on the fault types and generates synthetic faults based on the unlabeled real recordings of the bearings.",
"Unlike previous works which utilize highly accurate physical models to synthesize the faults, our generation stage is deliberately designed to be simple such that the overall framework does not rely on highly accurate physical simulators.",
"In the second part, a novel imbalance-robust domain adaption approach is proposed to alleviate the distribution gap between the synthetic and real features, specifically for fault diagnosis.",
"Unlike classic DA methods which typically deal with balanced data, the proposed method is able to align the features when the class distribution is different between the synthetic source and real target.",
"This imbalance-robust framework is proposed for realisitic scenarios as real datasets are often very imbalanced because of the rare faults.",
"We would like to highlight that the proposed domain adaptation approach does not rely on specific generation process in the previous step.",
"Thus, it can be easily generalized to other fault generation process or even to other classification problems that suffer from a class-imbalance between source and target datasets.",
"Integrating both parts together leads to our proposed framework which makes effective use of both expert knowledge and unlabeled target data.",
"The An overview of our proposed framework is shown in Figure REF ."
],
[
"Bearing Fault Generation",
"The challenge of fault generation in our proposed framework is how to make the synthetic faults adaptable to the given target (real) domain without using highly accurate physical models.",
"It is important to take the limited information provided in the unlabeled target samples into account.",
"As a prerequisite, we assume that we have access to some real healthy samples.",
"This is usually achievable by using the early recordings of each asset where we can safely assume that the faults have not yet emerged.",
"Thus, this is a realistic assumption also in real application conditions.",
"The healthy real samples are then used as the base signal for the synthetic process and inject the fault patterns using expert knowledge.",
"Since the base signal encodes information about the operating and environmental conditions, the generated signals can be more adaptable to the target domain.",
"Following traditional vibration modeling of bearings [1], a surface defect typically generates a pulse-train, where the pulse frequency relates to the location of the defect.",
"The amplitude of the pulse is modulated depending on the load during the moment of contact, where its transfer path to the accelerometer is typically described as a linear time-invariant process.",
"Let $w[n]$ denote a pulse-train with sampling-index $n$ , where $\\omega $ and $\\omega _a$ describe its repetition frequency and amplitude modulation frequency, respectively.",
"If $h$ describes the transfer function, we can model a defect initiated vibration signal $\\epsilon $ as, $\\epsilon [n] = h[n]*\\left( w[\\omega n] + w[\\omega n]\\sum _{k=1}^{4} \\alpha _k \\cos {\\omega _a k n} \\right)$ where $*$ denotes the convolution.",
"For the amplitude modulation term we use a sum of harmonics to introduce multiple side-bands with amplitudes $\\alpha =[0.76, 0.38, 0.11, 0.05]$ normalized such that the modulation term has unit amplitude.",
"These numbers are found such to have a number of side-bands close to a typical radially loaded bearing.",
"An individual pulse in $w$ is generated by means of a Hann-window with a duration of 5% of the pulse train period.",
"To introduce natural randomness, the pulses have normally distributed amplitudes with unit mean and a standard deviation of 0.1.",
"We assume that a defect signal can be described as a superposition of the healthy signal $x_h$ , and the defect-initiated pulse-train signal.",
"By mixing with the healthy signals, our method also has to compensate for cross-talk of other rotating components, reducing the signal-to-noise-ratio (SNR).",
"We, thus, synthesize fault signals based on healthy signal with $x_h[n] + \\beta \\epsilon [n]$ , where $\\beta $ is uniformly distributed between 0.25 and 2 to make our method robust for various SNRs."
],
[
"Augmented Conditional Adversarial Alignment",
"The proposed generation approach provide us with the source dataset $\\mathcal {D}_s$ which contains the synthetic faults.",
"To make efficient use of the unlabeled data from the real target domain, we propose to re-formulated the unsupervised fault diagnosis task as a unsupervised domain adaptation problem.",
"Synthetic source domain data with balanced samples across all classes $\\mathcal {D}_s=\\lbrace (x_{s}^1, y_{s}^1), ...,(x_{s}^n, y_{s}^n)\\rbrace , y_{s}^i \\in Y.$ Unlabeled real data from target domain with an imbalance across classes $\\mathcal {D}_{t}=\\lbrace x_{t}^1, ...,x_{t}^m\\rbrace ,$ where $Y$ is the set of discrete health states.",
"Our aim is to improve the model performance in the target domain $\\mathcal {D}_{t}$ .",
"The setup is now similar to transductive DA problems [41], [42]."
],
[
"Imbalance Issue for Direct Alignment",
"There is one essential difference between the standard DA commonly seen in image classification and our setup which could potentially harm the alignment quality.",
"In most DA setups, the source and target domains are assumed to have the same class distributions, meaning that either both source and target are class balanced, or both follow a similar class distribution.",
"However, for our synthetic-to-real setup, the scenario is quite different.",
"On the synthetic data side, since faults are all generated based on the healthy samples, each health condition has the exact same number of samples, leading to a balanced source dataset.",
"On the target real data side, the class distribution contains a majority of healthy states and few faults, leading to an imbalanced dataset.",
"Figure: Standard adversarial domain adaptation performance decreases more rapidly when the target domain becomes more imbalanced.",
"A synthetic-to-real CWRU experiment with different levels of imbalance on the rolling element fault class.",
"1% on the x-axis means that only 1% samples are available for rare fault class, compared to the balanced case.",
"The orange dash-dot line indicates the performance when the dataset is fully balanced (100%).",
"Blue dash line indicates the source-only baseline.This mismatch between the source and target domain can in fact lead to a significant decline of the adaptation performance.",
"We show empirically in Figure REF the performance of a naive synthetic-to-real adversarial adaptation based on the CWRU dataset using DANN [7].",
"In this setup, the source synthetic domain is balanced, while for the target domain, one of the faults (rolling element fault) has a smaller number of samples.",
"The experiment shows that, compared to a fully balanced case, the stronger class imbalance in the target leads to a steeper performance decline.",
"To answer why such a performance decline occurs, we need to look back into the assumptions that the classical DA methods make.",
"Classical DANN methods [7], [43], [35] are based on the following domain adaptation theory: the error function of the domain discriminator corresponds well to the discrepancy between unconditional feature distributions $P$ and $Q$ .",
"Thus, minimizing the error rate of the discriminator can lead to aligned feature distributions.",
"However, when the joint distributions of feature and class, i.e.",
"$P(x_s, y_s)$ and $Q(x_t, y_t)$ , are non-identical across domains, adapting only the feature representation may be insufficient [27].",
"This can be an especially large problem when the target class distributions in fault diagnosis are heavily skewed towards the healthy class.",
"In this case, simply aligning the features, in fact, gives no theoretical guarantee that two different distributions are identical even if the discriminator is fully confused [44]."
],
[
"Class Information Taken into Account",
"The performance degradation stems from the fact that the class distributions are mismatched and the discriminator fails to take any class information into account.",
"We, thus, propose to use the class information of the representations as additional input to the discriminator in order to counteract the class mismatch from rare faults.",
"By providing the generator hints on the class information, the discriminator can better align the distributions.",
"We use the multi-linear map to fuse the class information with features to improve the domain adaptation performance under severe class imbalance.",
"As shown in Fig REF , given a feature extractor $f$ parameterized by $\\theta _f$ , a classifier $g$ parameterized by $\\theta _g$ , and a discriminator $h$ parameterized by $\\theta _h$ , the loss function then becomes: $\\mathcal {L}(\\theta _f, \\theta _g, \\theta _h) = \\mathcal {L}_{clf}(g(f(x))) - \\lambda _d \\mathcal {L}_d(h(f(x)\\otimes g(f(x)))),$ where $\\otimes $ is a multi-linear map, and $\\hat{y} = g(f(x))$ represents the predicted pseudo-labels.",
"Comparing to equation REF of the standard adversarial training, the main difference is that instead of using features $f(x)$ alone as input to the discriminator, the information of features and classes (provided by pseudo-labels) are combined together via a multi-linear map.",
"By applying the map, the discriminator can gain the information from both the feature distribution and class distribution and better align the class conditioned distributions.",
"The proposed method is inspired by conditional adversarial methods in other application fields.",
"For example, for image generation tasks, Conditional Generative Adversarial Networks [45] concatenate the class vector with the feature vector to generate images conditioned on the single classes.",
"In domain adaptation for image classification, CDAN [27] uses multi-linear conditioning to align multi-modal distributions.",
"These methods often deal with balanced datasets and does not focus on severe class-imbalanced scenarios, but the idea of aligning the conditional distribution is especially suitable for fault diagnosis."
],
[
"Augmented Distributions",
"One potential issue with the above solution is that even though the important class information is provided for the discriminator, the fact that the rare fault classes have so few samples can potentially make the optimization of the discriminator unstable.",
"This can also lead to a decline of the alignment performance.",
"To reinforce the alignment, it is essential to provide better distribution support for the conditional distributions on the rare classes.",
"We propose to augment the features and pseudo-labels used as input to the discriminator.",
"In particular, for a batch of $x$ from the target real dataset, we can get its corresponding feature embedding $e=f(x)$ , and pseudo-label $\\hat{y}=g(f(x))$ vectors.",
"Whereby, $\\hat{y}$ is generated by using the prediction from the model trained in the previous iteration.",
"Then, the augmented interpolated sample can be represented as, $\\tilde{e} &=& \\lambda e + (1-\\lambda ) e[\\text{idx}], \\\\\\tilde{y} &=& \\lambda \\hat{y} + (1-\\lambda ) \\hat{y}[\\text{idx}],\\\\\\tilde{z} &=& \\tilde{e} \\otimes \\tilde{y},$ where $\\text{idx}$ is the shuffled index of a batch, $\\lambda $ is generated from an prior $\\beta $ distribution, i.e.",
"$ \\lambda \\sim \\beta (\\alpha , \\alpha )$ with $\\alpha $ controls the shape of the $\\beta $ distribution, $\\tilde{e}$ is the interpolated features, $\\tilde{y}$ is its class label, and $\\tilde{z}$ is the multi-linear input to the discriminator.",
"We use $\\alpha =1$ for all our experiments.",
"This idea is inspired by MixUp [13] in image classification, where the authors mix input images and labels to provide more training samples in a fully supervised training setup.",
"Instead of directly mixing the input images in MixUp, our proposed distribution augmentation is conducted on the feature space and used for our unsupervised domain adaptation setup.",
"The motivation behind this is to augment the conditional feature distributions of rare classes for fault diagnosis.",
"By mixing up the features within a batch, the fault information becomes present in more samples within a batch.",
"The interpolated samples enlarge the target training dataset for the rare fault classes, making the learning process for the discriminator more stable.",
"We highlight the difference between DANN and our alignment method in blue on the right side of Figure REF ."
],
[
"Datasets",
"To facilitate our adaptation experiments, as described in Section REF , we generate the synthetic faults for two datasets.",
"Figure: CWRU normalized spectral envelope of an inner ring defect example with corresponding defect ball pass frequency (BPFI) denoted by the dashed vertical lines.",
"The difference between the real and synthetic fault in BPFI frequencies and sidebands (showed by vertical dotted lines) motivates the use of domain adaptation.Figure: Wind turbine generator dataset: normalized spectral envelope of an outer ring defect example.",
"In this example, the unknown interference is realistically transferred from healthy to synthetic.",
"However, the synthetic fault does not match perfectly with the real fault, and the difference motivates the use of domain adaptation.",
"Notice the frequency range between 6-12 orders such that only two and three times the ball-pass frequency of the outer ring (BPFO) is visible."
],
[
"CWRU Bearing Data",
"The Case Western Reserve University (CWRU) [46] bearing dataset is used.",
"It is a benchmark dataset for DA in bearing fault diagnosis [34], [33], [29], [35].",
"Drive-end accelerometer data of 12 kHz sampling rate is used.",
"We consider four health states (classes) in this paper: healthy, inner race fault (IF), rolling element fault (REF), and outer race fault (OF).",
"We group the sub-fault-types of different spall sizes together.",
"For each health state, we sample 1200 segments.",
"Each signal segment contains 4096 points.",
"This results in 4800 samples for the real dataset.",
"Subsequently, our generation method is applied to the real healthy signals to synthesize the real defects.",
"To avoid data leakage during evaluation, we split half of the real healthy samples to the synthetic source data and up-sample the number of samples in healthy class in both domains back to 1200.",
"Figure REF shows the spectral envelop of one synthetic and one real example of the inner race fault.",
"Defect frequencies are indicated by the dashed lines, where a dotted line shows the first three side-bands of an inner ring defect around its ball-pass frequency (BPFI).",
"The BPFI and its first harmonic in the synthetic and real examples are roughly aligned.",
"However, differences can still be observed: 1) the real defect has more side-bands around the BPFI 2) the 2nd harmonic is not present in the real defect and 3) the defect frequencies have a slightly higher frequency compared to the analytic defect frequencies.",
"These differences motivate the use of domain adaptation to bridge the gap between the real and synthetic signals."
],
[
"Wind turbine bearing data",
"A real-world dataset from generator bearings of multiple wind turbines is collected and used to evaluate the method.",
"This data origins from a condition monitoring service, where bearings are monitored by human experts aided by analysis software.",
"For each turbine, historic data is available varying between 3-5 years, where on a daily basis two recordings are made.",
"Each vibration recording has a length of 1.28 seconds and a sample-rate of 12.8 kHz.",
"Rotational speeds vary between 900 and 1700 RPM.",
"Speed recordings are available via a tachometer.",
"There are three health states (classes), healthy, outer-ring defects, and inner-ring defects.",
"Based on a combination of manual analysis of vibration and process data, knowledge of bearing replacements and anonymized customer feedback on bearing defects, the data was carefully labeled.",
"In total, we collected 1643 samples for the healthy state, 2990 samples for the outer-ring defects, and 192 samples for the inner-ring defects from the generator drive-end bearings each on a different wind turbine.",
"Note that during the labeling of the data we had access to additional process data such as electrical power and wind-speed but were also able to inspect trends and high SNR averaged spectra of many vibration recordings.",
"As a consequence, a single vibration recording, as used by our proposed domain adaptation method, is only a fraction of the information used in the labeling procedure.",
"In many situations, all these additional information sources are unavailable, which motivates the use of single vibration recording predictions as in this research."
],
[
"Model Architecture",
"The network consists of a feature extractor, classifier, and a discriminator.",
"The architecture of the feature extractor and classifier are taken from [34].",
"The details are summarized in Table REF ."
],
[
"Data Preprocessing",
"A preprocessing step is applied to all data.",
"First, the time-domain waveforms are normalized to unit standard deviation and subsequently converted to a spectral envelope representation.",
"It is obtained by taking the Fourier transform magnitudes of the full-wave rectified and band-pass filtered signals.",
"A band-pass filter with a pass-band between 500-4000 Hz is used.",
"The spectral representation is then interpolated to a speed-normalized axis of 1000 values between 0-30 repetition orders."
],
[
"Evaluation Metric",
"Since the classes are imbalanced, we report a balanced version of the accuracy.",
"This metric was used by imbalanced classification tasks such as [47].",
"For a dataset with $K$ classes, the reported accuracy is defined as: $\\frac{1}{K} \\sum _{k=1}^{K}{\\frac{P_k}{M_k}} $ where $M_k$ is the number of test samples with class $k$ as ground truth, and $P_k$ is the number of correct predictions for the given $M_k$ samples in class $k$ .",
"This metric provides a fair comparison also in highly imbalanced datasets.",
"All reported results are based on the average of ten runs using this metric."
],
[
"CWRU Synthetic-to-Real Experiments with Different Levels of Imbalance on One Fault. Results are shown in balanced accuracy.",
"We evaluate the proposed method against a source-only baseline and the standard DANN alignment.",
"In this ablation study we reduce the complexity by assuming that all classes have the same number of samples except the rolling element fault.",
"More realistic setups are evaluated in later sections.",
"To show the effectiveness of the proposed methodology against different levels of class imbalance, we change the number of samples with the rolling element fault.",
"For example, when the balance level is 1%, the unlabeled target training data contains $1\\% * 1200 = 12$ samples for the rolling element fault class, while 1200 samples are available for every other class.",
"In the $10\\%$ setup, the target set contains $10\\% * 1200 = 120$ samples for the rolling element fault class.",
"A smaller value of level of balance (1%) indicates a larger level of class imbalance as the number of faults is considerably smaller compared to the number of healthy class samples.",
"We use a batch size of 128 and a learning rate of 0.001 for the CWRU experiments.",
"As shown in Table REF , without any alignment, the source-only accuracy is 59.55% in this 4-class-classification problem.",
"This is already higher than the random guess performance of 25% and proves that our synthetic faults encode the fault information in a meaningful way.",
"On the other hand, DANN can perform quite well when all the classes are balanced, but degrades dramatically when the level of balance decreases.",
"By taking the class information into account, conditional alignment alone shows resistance to the degradation from the class imbalance.",
"However, the performance gap still exists, and the performance becomes less stable when the balance level decreases.",
"By further augmenting the conditional distributions, the proposed method can significantly strengthen the quality of the alignment and lead to a performance that is at a similar level as for the balanced dataset, even when there are only $1\\%$ rolling element fault samples in the target dataset.",
"This demonstrates the effectiveness of our proposed method.",
"We also report the improvement of the proposed method over the standard DANN.",
"An interesting observation is that with an increasing level of imbalance, our method provides an increasing amount of improvement over DANN.",
"This suggests that the method can be especially beneficial when imbalance in the target dataset is severe.",
"Furthermore, the proposed method appears to be robust to the degree of imbalance.",
"The results for all the degrees of imbalance demonstrate a similar accuracy.",
"Even with a relatively strong imbalance ($1\\%$ balance level) , the performance is similar to the the fully balanced setup of the standard DANN.",
"Therefore, the methodology is suitable for cases where the degree of class imbalance in the target dataset is not known which is a typical setup in real applications.",
"Thus, the method is quite suitable for real fault diagnosis tasks."
],
[
"CWRU Synthetic-to-Real Experiment with a More Realistic Class Imbalance among Faults",
"In reality, the imbalance does not only exist for one single fault, but also exists between different faults.",
"To mimic also the imbalance between several faults in the target domain, we consider the following setup.",
"The unlabeled target dataset consists of $N=1200$ healthy samples, 10% $\\times $ N samples for the OF, 5% $\\times $ N samples for the IF and 1% $\\times $ N sample for the REF.",
"This constructs a highly imbalanced target dataset.",
"All labels are again removed during the training.",
"This setup makes it a more challenging task as the imbalance levels of the three faults are different.",
"Results in Table REF show that when class imbalance is severe among different faults, the proposed method can still provide an improved alignment for the different classes.",
"Compared to the simple AdaBN [22] and DANN [43] alignment method which does not consider the imbalance between different classes, both the conditional alignment and our proposed mixup augmentation provide a significant improvement.",
"Conditional alignment alone provides a 2.81% absolute improvement over DANN by simply providing additional class information.",
"By combining the proposed augmentation with the class conditional alignment, we achieve a much stronger accuracy of 82.29% on the target data.",
"This is a 9.72% absolute improvement.",
"Again, the performance of the proposed methodology is similar to that of DANN in the ideal fully balanced case.",
"This makes the proposed framework applicable for unknown imbalance levels achieving same level of performance as in a balanced dataset.",
"This finding is particularly encouraging for real applications where the imbalance level is unknown apriori.",
"Table: Experiment results reported in balanced accuracy on the Wind Turbine dataset."
],
[
"Wind Turbine Generator Synthetic-to-Real Experiment",
"To evaluate the effectiveness of the proposed methodology in a more realistic scenario, we additionally conduct an experiment on the real-world wind turbine bearings.",
"As described in the dataset section, the outer race fault has a significantly larger number of samples than the inner race fault.",
"In this setup, the source-only baseline achieves an accuracy of 60.85%, showing that the synthetic data we generated for the wind turbine bearing is meaningful.",
"The DANN method can improve the performance and achieve 64.47%.",
"By providing class information to the discriminator, we can achieve 68.28%.",
"If we additionally use our augmentation to enhance the distribution support, the final proposed method can achieve an accuracy of 70.76%, yielding an almost 10% absolute improvement on this real dataset, compared to the source-only baseline.",
"The proposed method also outperform the popular AdaBN [22] approach by a large margin.",
"The AdaBN method does not improve over the baseline most likely because of the severe imbalance between the classes.",
"We would like to emphasize that in all experiments, we only use knowledge of the healthy labels from the target domain.",
"All class information on the faults is learned from the synthetic data where the expert knowledge is encoded.",
"This result shows that given an unlabeled target bearing dataset, it is possible to make use of expert knowledge and train a data-driven fault diagnosis model for it.",
"This is achieved by our proposed framework that combines a fault generation process and our proposed synthetic-to-real adaptation approach which is specifically designed for the imbalanced target data."
],
[
"Conclusions",
"We proposed a novel fault diagnosis framework which can learn effective models from unlabeled real bearing data.",
"In particular, we showed that by generating synthetic faults using expert knowledge and conducting imbalance-robust domain adaptation, a fault diagnosis model can be learned without any supervision from the real faults.",
"We showed that a good approach to compensate the imbalance from rare target faults is key to the synthetic-to-real adaptation performance.",
"A class conditioned adversarial adaptation method is, thus, proposed to address this issue.",
"An additional augmentation based on the mixup approach was further proposed to deal with the limited number of fault samples and bridge the class distribution gap.",
"The two components for domain adaptation can be easily applied in combination with other fault generation frameworks.",
"The proposed methodology does not require any assumption on the degree of the underlying class imbalance and achieves a similar performance in the imbalanced setup as the standard DANN on the fully balanced setup, demonstrating its robustness to different imbalance levels.",
"Experiments on the benchmark CWRU bearing dataset and a wind turbine generator bearing dataset have validated the effectiveness of our approach also in real applications and under realistic assumptions.",
"The framework can be implemented easily, and have the potential to be applied on other industry assets."
]
] | 2107.01849 |
[
[
"The Siebeck-Marden-Northshield Theorem and the Real Roots of the\n Symbolic Cubic Equation"
],
[
"Abstract The isolation intervals of the real roots of the symbolic monic cubic polynomial $x^3 + a x^2 + b x + c$ are determined, in terms of the coefficients of the polynomial, by solving the Siebeck-Marden-Northshield triangle - the equilateral triangle that projects onto the three real roots of the cubic polynomial and whose inscribed circle projects onto an interval with endpoints equal to stationary points of the polynomial"
],
[
"straight[ 0.1.0.01em * 0pt3.25ex plus 1ex minus .2ex1.5ex plus .2ex 4 The Siebeck–Marden–Northshield Theorem and the Real Roots of the Symbolic Cubic Equation Emil M. Prodanov School of Mathematical Sciences, Technological University Dublin, Park House, Grangegorman, 191 North Circular Road, Dublin D07 EWV4, Ireland, e-mail: [email protected] The isolation intervals of the real roots of the symbolic monic cubic polynomial $x^3 + a x^2 + b x + c$ are determined, in terms of the coefficients of the polynomial, by solving the Siebeck–Marden–Northshield triangle — the equilateral triangle that projects onto the three real roots of the cubic polynomial and whose inscribed circle projects onto an interval with endpoints equal to stationary points of the polynomial.",
"Mathematics Subject Classification Codes (2020): 26C10, 12D10, 11D25.",
"Keywords: Polynomials; Cubic equation; Siebeck–Marden–Northshield theorem; Roots; Isolation intervals; Root bounds.",
"1 Introduction The elegant theorem of Siebeck and Marden (often referred to as Marden's theorem) [1]–[5] relates geometrically the complex non-collinear roots of a cubic polynomial with complex coefficients to a triangle whose vertices project onto them, on one hand, and, on the other, the critical points of the polynomial to the projections of the foci of the inellipse of this triangle.",
"This ellipse is unique and is called Steiner inellipse [6].",
"It is inscribed in the triangle in such way that it is tangent to the sides of the triangle at their midpoints.",
"The real version of the Siebeck–Marden Theorem, as given by Northshield [7], states that the three real roots (not all of which are equal) of a cubic polynomial are projections of the vertices of some equilateral triangle in the plane.",
"However, it is the inscribed circle of the equilateral triangle that projects onto an interval the endpoints of which are the stationary points of the polynomial.",
"The goal of this work is to consider a cubic equation with real coefficients and, using the Siebeck–Marden–Northshield theorem [7], solve the equilateral triangle and find the isolation intervals of the real roots of the symbolic monic cubic polynomial $x^3 + a x^2 + b x + c$ .",
"2 Analysis The vertices of the equilateral triangle that projects onto the three real roots of the cubic polynomial $x^3 + a x^2 + b x + c$ are points $P$ , $Q$ , and $R$ with coordinates $(x_1, (x_2 - x_3)/\\sqrt{3})$ , $(x_2, (x_3 - x_1)/\\sqrt{3})$ , and $(x_3, (x_1 - x_2)/\\sqrt{3})$ , respectively [7].",
"Lemma 1 The monic cubic polynomial $p(x) = x^3 + a x^2 + b x + c$ with $b > a^2/3$ has only one real root.",
"The discriminant of the monic cubic polynomial $x^3 + a x^2 + b x + c$ is $\\Delta _3 = -27 c^2 + (18 a b - 4 a^3) c + a^2 b^2 - 4b^3.$ It is quadratic in $c$ and the discriminant of this quadratic is $\\Delta _2 = 16 (a^2 - 3b)^3$ As $b > a^2/3$ , one has $\\Delta _2 < 0$ for all $a$ and thus $\\Delta _3 < 0$ for all $a$ and $c$ .",
"Hence, the cubic polynomial $p(x) = x^3 + a x^2 + b x + c$ with $b > a^2/3$ has only one real root (and a pair of complex conjugate roots).",
"Lemma 2 The monic cubic polynomial $p(x) = x^3 + a x^2 + b x + c$ with $b \\le a^2/3$ has three real roots, provided that $c \\in [c_2, c_1]$ , where $c_{1,2}$ are the roots of the quadratic equation $x^2 + \\left( \\frac{4}{27} \\, a^3 - \\frac{2}{3}\\, a b \\right) x - \\frac{1}{27} \\, a^2 b^2 + \\frac{4}{27}\\, b^3 = 0,$ namely: $c_{1,2}(a,b) = c_0 \\, \\pm \\, \\frac{2}{27} \\, \\sqrt{( a^2 - 3 b)^3},$ where $c_0(a,b) = -\\frac{2}{27} \\, a^3 + \\frac{1}{3} \\, a b.$ The discriminant $\\Delta _3 = -27 c^2 + (18 a b - 4 a^3) c + a^2 b^2 - 4b^3$ of the monic cubic polynomial $x^3 + a x^2 + b x + c$ is positive between the roots of the equation $\\Delta _3 = 0$ , which is quadratic in $c$ .",
"This is exactly equation (REF ) and its roots are the ones given in (REF ) and (REF ).",
"Lemma 3 (Solving the Siebeck–Marden–Northshield Triangle) The centre of the inscribed circle of the equilateral triangle that projects onto the three real roots of the monic cubic polynomial $p(x) = x^3 + a x^2 + b x + c$ is point $(-a/3, 0)$ , the projection onto the abscissa of the inflection point of $p(x)$ , and the radius of the inscribed circle is $r = (1/3) \\sqrt{a^2 - 3b}$ .",
"The radius of the circumscribed circle is $2r = (2/3) \\sqrt{a^2 - 3b}$ .",
"The inflection point of the graph of the monic cubic polynomial $p(x) = x^3 + a x^2 + b x + c$ occurs at the root of $p^{\\prime \\prime }(x) = 6x + 2a$ , namely at $x = \\phi = - a/3$ .",
"Given that the vertices $P$ , $Q$ , and $R$ of the triangle are points of coordinates $(x_1, (x_2 - x_3)/\\sqrt{3})$ , $(x_2, (x_3 - x_1)/\\sqrt{3})$ , and $(x_3, (x_1 - x_2)/\\sqrt{3})$ , respectively, the centroid of the triangle is point of coordinates $(-a/3, 0)$ — the first coordinate projection of the inflection point.",
"Each side of the triangle is equal to $\\alpha = (\\sqrt{12}/3) \\sqrt{a^2 - 3b}$ .",
"The radius of a circle inscribed in equilateral triangle with side $\\alpha $ is $r = \\alpha /\\sqrt{12} = (1/3) \\sqrt{a^2 - 3b}$ .",
"The radius of the circumscribed circle of an equilateral triangle with side $\\alpha $ is $2r = (2/3) \\sqrt{a^2 - 3b}$ — see Figure 1.",
"Lemma 4 The maximum distance between the three real roots of the monic cubic polynomial $p(x) = x^3 + a x^2 + b x + c$ is $\\sqrt{12} r = (\\sqrt{12}/3) \\sqrt{a^2 - 3b}$ .",
"In this case, one side of the equilateral triangle that projects onto the roots of the monic cubic polynomial $p(x) = x^3 + a x^2 + b x + c$ is parallel to the abscissa.",
"Figure: NO_CAPTIONGiven that the root $x_2 = \\nu _2 = \\phi = -a/3$ of the “balanced\" cubic equation $x^3 + a x^2 + b x + c_0 = 0$ , where $c_0 = - 2a^3/27 + a b/3$ , is the midpoint between its other two roots $x_{1,3} = \\nu _{1,3} = - a/3 \\pm \\sqrt{a^2/3 - b}$ , one has $x_1 - x_2$ (the second coordinate of point $R$ ) being equal to $x_2 - x_3$ (the second coordinate of point $R$ ).",
"Hence $P$ and $R$ are both above the abscissa and are equidistant from it.",
"Thus $PR$ is parallel to the abscissa.",
"Hence, the distance between $x_3$ and $x_1$ is exactly equal to the length $\\alpha = (\\sqrt{12}/3) \\sqrt{a^2 - 3b}$ of the side $PR$ .",
"In any other case of three real roots ($c \\in [c_2, c_1]$ and $c \\ne c_0$ ), the side $PR$ will not be parallel to the abscissa and hence the projection of $PR$ onto the abscissa will be shorter than the length of $PR$ , that is, the three real roots of the cubic polynomial will lie in an interval of length smaller than $\\alpha = (\\sqrt{12}/3) \\sqrt{a^2 - 3b}$ .",
"Theorem 1 The monic cubic polynomial $p(x) = x^3 + a x^2 + b x + c$ , for which $b < a^2/3$ and $c \\in [c_2, c_1]$ , has three real roots $x_3 \\le x_2 \\le x_1$ , at least two of which are different and any two of which are not farther apart than $(\\sqrt{12}/3) \\sqrt{a^2 - 3b}$ , with the following isolation intervals: (I) For $c_2 \\le c \\le c_0$ : $x_3 \\in [\\nu _3, \\mu _2], \\,\\, x_2 \\in [\\mu _2, \\phi ]$ , and $x_1 \\in [\\nu _1, \\xi _2]$ .",
"(II) For $c_0 \\le c \\le c_1$ : $x_3 \\in [\\xi _1, \\nu _3], \\,\\, x_2 \\in [\\phi ,\\mu _1]$ , and $x_1 \\in [\\mu _1, \\nu _1]$ , For $c_2 \\le c \\le c_0$ : $x_3 \\in [\\nu _3, \\mu _2], \\,\\, x_2 \\in [\\mu _2, \\phi ]$ , and $x_1 \\in [\\nu _1, \\xi _2]$ .",
"For $c_0 \\le c \\le c_1$ : $x_3 \\in [\\xi _1, \\nu _3], \\,\\, x_2 \\in [\\phi ,\\mu _1]$ , and $x_1 \\in [\\mu _1, \\nu _1]$ , where: (i) $\\mu _{1,2}$ is the double root and $\\xi _{1,2}$ is the simple root of $p_{1,2}(x) = x^3 + a x^2 + b x + c_{1,2}$ , that is, $\\mu _{1,2}$ are the roots of $3 x^2 + 2 a x + b = 0$ , namely: $\\mu _{1,2} = -a/3 \\pm r = -a/3 \\pm (1/3) \\sqrt{a^2 - 3 b}$ and $\\xi _{1,2} = - a - 2 \\mu _{1,2} = - a/3 \\mp 2r = -a/3 \\mp (2/3) \\sqrt{a^2 - 3 b}$ .",
"(ii) $\\nu _{1,2,3}$ are the roots of the “balanced\" cubic equation $p_0(x) = x^3 + a x^2 + b x + c_0$ , namely: $\\nu _{1,3} = -a/3 \\pm \\alpha /2 = -a/3 \\pm (\\sqrt{3}/3) \\sqrt{a^2 - 3 b}$ and $\\nu _2 = \\phi = - a/3$ .",
"Figure: NO_CAPTION$\\mu _{1,2}$ is the double root and $\\xi _{1,2}$ is the simple root of $p_{1,2}(x) = x^3 + a x^2 + b x + c_{1,2}$ , that is, $\\mu _{1,2}$ are the roots of $3 x^2 + 2 a x + b = 0$ , namely: $\\mu _{1,2} = -a/3 \\pm r = -a/3 \\pm (1/3) \\sqrt{a^2 - 3 b}$ and $\\xi _{1,2} = - a - 2 \\mu _{1,2} = - a/3 \\mp 2r = -a/3 \\mp (2/3) \\sqrt{a^2 - 3 b}$ .",
"$\\nu _{1,2,3}$ are the roots of the “balanced\" cubic equation $p_0(x) = x^3 + a x^2 + b x + c_0$ , namely: $\\nu _{1,3} = -a/3 \\pm \\alpha /2 = -a/3 \\pm (\\sqrt{3}/3) \\sqrt{a^2 - 3 b}$ and $\\nu _2 = \\phi = - a/3$ .",
"Due to Lemma 2, the discriminant $\\Delta _3 = -27 c^2 + (18 a b - 4 a^3) c + a^2 b^2 - 4b^3$ of the monic cubic polynomial $x^3 + a x^2 + b x + c$ is non-negative for all $a$ and $b \\le a^2/3$ , if $c$ is between the roots $c_{1,2} = c_0 \\pm (2/27) \\sqrt{( a^2 - 3 b)^3}$ (with $c_0= - 2 a^3/27 + a b/3$ ) of the quadratic equation $x^2 + (4 a^3/27 - 2 a b/3 ) x - a^2 b^2/27 + 4 b^3/27 = 0$ .",
"Then $x^3 + a x^2 + b x + c$ will have three real roots.",
"The two “extreme\" cases, the cubics $x^3 + a x^2 + b x + c_1$ and $x^3 + a x^2 + b x + c_2$ , will each have a double root (as $\\Delta _3$ vanishes for $c = c_{1,2}$ ) and a simple root.",
"Otherwise, for $c_2 < c < c_1$ , the cubic polynomial will have three distinct roots.",
"Let $\\mu _{1,2}$ denote the double root of the “extreme\" cubic $x^3 + a x^2 + b x + c_{1,2}$ and $\\xi _{1,2}$ — the corresponding simple root.",
"When $c = c_{1,2}$ , one has, due to Viète formulæ: $2 \\mu _i + \\xi _i = - a, \\,\\, \\mu _i^2 + 2 \\mu _i \\xi _i = b,$ and $\\mu _i^2 \\xi _i = -c$ (for $i = 1, 2$ ).",
"Expressing from the first $\\xi _i = - a - 2 \\mu _i$ and substituting into the second yields $-3 \\mu _i^2 - 2 a \\mu _i - b = 0$ , that is, the double roots $\\mu _{1,2}$ of each of the “extreme\" cubics $x^3 + a x^2 + b x + c_{1,2}$ are the roots of the quadratic equation $3x^2 + 2 a x + b = 0$ , that is $\\mu _{1,2} = -a/3 \\pm r = -a/3 \\pm (1/3) \\sqrt{a^2 - 3 b}$ .",
"Hence one finds: $\\xi _{1,2} = - a - 2 \\mu _{1,2} = - a/3 \\mp 2r = -a/3 \\mp (2/3) \\sqrt{a^2 - 3 b}$ .",
"Due to Lemma 4, the biggest distance between the roots of the cubic will be $\\alpha = (\\sqrt{12}/3) \\sqrt{a^2 - 3b}$ .",
"The roots of the “balanced\" cubic equation $x^3 + a x^2 + b x - 2a^3/27 + a b/3 = 0$ (see the proof of Lemma 4) are symmetric with respect to the centre of the inscribed circle: $\\nu _3 = -a/3 - \\sqrt{a^2/3 - b}$ , $\\nu _2 = \\phi = -a/3$ , and $\\nu _1 = - a/3 + \\sqrt{a^2/3 - b}$ .",
"The “balanced\" equation has triangle $P_0 Q_0 R_0$ and the side $P_0 R_0$ is parallel to the abscissa (Figure 2).",
"When $c = c_1 > c_0$ , the Siebeck–Marden–Northshield triangle is $P_1 Q_1 R_1$ and its side $P_1 Q_1$ is perpendicular to the abscissa.",
"Hence the roots $x_2$ and $x_1$ coalesce into the double root $\\mu _1$ .",
"The vertex $R_1$ is on the abscissa at the smallest root $\\xi _1$ (Figure 2).",
"When $c = c_2 < c_0$ , the Siebeck–Marden–Northshield triangle is $P_2 Q_2 R_2$ and its side $R_2 Q_2$ is perpendicular to the abscissa.",
"The roots $x_3$ and $x_2$ coalesce into the double root $\\mu _2$ , while the biggest root $x_1$ is equal to $\\xi _2$ , as the vertex $P_2$ is on the abscissa at $\\xi _2$ (Figure 2).",
"Increasing $c$ rotates the Siebeck–Marden–Northshield triangle counterclockwise about its centroid.",
"Decreasing $c$ results in its clockwise rotation.",
"The isolation intervals of the roots of the cubic polynomial are then easily read geometrically — see Figure 2.",
"Theorem 2 The monic cubic polynomial $p(x) = x^3 + a x^2 + b x + c$ , for which $b < a^2/3$ and (I) $c < c_2$ , has only one real root: $x_1 > \\xi _2 = - a - 2 \\mu _2 = - a/3 + 2r = - a/3 + (2/3) \\sqrt{a^2 - 3 b}$ (it can be bounded from above by a polynomial root bound).",
"(II) $c > c_1$ , has only one real root: $x_1 < \\xi _1 = - a - 2 \\mu _1 = - a/3 - 2r = - a/3 - (2/3) \\sqrt{a^2 - 3 b}$ (it can be bounded from below by a polynomial root bound).",
"$c < c_2$ , has only one real root: $x_1 > \\xi _2 = - a - 2 \\mu _2 = - a/3 + 2r = - a/3 + (2/3) \\sqrt{a^2 - 3 b}$ (it can be bounded from above by a polynomial root bound).",
"$c > c_1$ , has only one real root: $x_1 < \\xi _1 = - a - 2 \\mu _1 = - a/3 - 2r = - a/3 - (2/3) \\sqrt{a^2 - 3 b}$ (it can be bounded from below by a polynomial root bound).",
"See the caption of Figure 3.",
"As polynomial upper root bound, one can take one of the many existing root bounds.",
"For example, it could be the bigger of 1 and the sum of the absolute values of all negative coefficients [8].",
"Or one can consider the bound [9]: $1 + \\@root k \\of {H}$ , where $k = 1$ if $a < 0, \\,\\, k = 2$ if $a > 0$ and $b <0$ , and $k = 3$ if $a > 0$ and $b > 0,$ and $c < 0$ (if $a$ , $b$ , and $c$ are all positive, the upper root bound is zero).",
"$H$ is the biggest absolute value of all negative coefficients in $x^3 + a x^2 + b x + c$ .",
"The lower root bound is the negative of the upper root bound of $-x^3 + a x^2 - b x + c$ .",
"Theorem 3 The monic cubic polynomial $p(x) = x^3 + a x^2 + b x + c$ , for which $b = a^2/3$ and (I) $c < (1/27) a^3$ , has only one real root: $x_1 = -a/3 + \\@root 3 \\of {a^3/27 - c} \\, > \\, -a/3$ .",
"(II) $c = (1/27) a^3$ , has a triple real root: $x_1 = x_2 = x_3 = -a/3$ .",
"(III) $c > (1/27) a^3$ , has only one real root: $x_1 = -a/3 + \\@root 3 \\of {a^3/27 - c} \\, < \\, -a/3$ .",
"$c < (1/27) a^3$ , has only one real root: $x_1 = -a/3 + \\@root 3 \\of {a^3/27 - c} \\, > \\, -a/3$ .",
"$c = (1/27) a^3$ , has a triple real root: $x_1 = x_2 = x_3 = -a/3$ .",
"$c > (1/27) a^3$ , has only one real root: $x_1 = -a/3 + \\@root 3 \\of {a^3/27 - c} \\, < \\, -a/3$ .",
"See the caption of Figure 4.",
"Table: NO_CAPTION Theorem 4 The only real root $x_1$ of monic cubic polynomial $p(x) = x^3 + a x^2 + b x + c$ with $b > a^2/3$ (due to Lemma 1) has the following isolation interval: (I) If $a \\ge 0$ and $c \\le 0: \\,\\, 0 \\le x_1 \\le -c/b$ .",
"(II) If $a \\ge 0$ and $c > 0: \\,\\,$ min$\\lbrace -a, -c/b\\rbrace \\le x_1 \\le $ max$\\lbrace -a, -c/b\\rbrace $ .",
"(III) If $a < 0$ and $c < 0: \\,\\,$ min$\\lbrace -a, -c/b\\rbrace \\le x_1 \\le $ max$\\lbrace -a, -c/b\\rbrace $ .",
"(IV) If $a < 0$ and $c \\ge 0: \\,\\, -c/b \\le x_1 \\le 0$ .",
"If $a \\ge 0$ and $c \\le 0: \\,\\, 0 \\le x_1 \\le -c/b$ .",
"If $a \\ge 0$ and $c > 0: \\,\\,$ min$\\lbrace -a, -c/b\\rbrace \\le x_1 \\le $ max$\\lbrace -a, -c/b\\rbrace $ .",
"If $a < 0$ and $c < 0: \\,\\,$ min$\\lbrace -a, -c/b\\rbrace \\le x_1 \\le $ max$\\lbrace -a, -c/b\\rbrace $ .",
"If $a < 0$ and $c \\ge 0: \\,\\, -c/b \\le x_1 \\le 0$ .",
"Re-write the cubic equation $x^3 + a x^2 + b x + c = 0$ as $x^3 + a x^2 = - b x - c$ .",
"Such “split\" of polynomial equations of different degrees has been proposed and studied in [10], [11], [12] The rest of the proof is graphic — see the captions of Figures 5–8 for the four cases (I)–(IV) respectively.",
"Table: NO_CAPTION Table: NO_CAPTION 3 Roles of the Coefficients of the Symbolic Cubic Equation $\\Large \\hspace{-8.5359pt}{\\,\\, x^3 + a x^2 + b x + c \\,\\,}$ and Isolation Intervals of its Real Roots — Summary and Application of the Analysis (a) The coefficient $a$ of the quadratic term of $x^3 + a x^2 + b x + c$ selects the centre $\\phi = - a/3$ of the inscribed circle of the equilateral triangle that projects onto the roots of $x^3 + a x^2 + b x + c$ , in the case of three real roots.",
"The centre of this circle is also the projection of the inflection point of the graph of $x^3 + a x^2 + b x + c$ onto the abscissa.",
"The inscribed circle projects to an interval on the abscissa with endpoints equal to the projections of the stationary points of $x^3 + a x^2 + b x + c$ (Figure 1).",
"(b) For any given $a$ , the coefficients $b$ of the linear term of $x^3 + a x^2 + b x + c$ determines the radius $r = (1/3) \\sqrt{a^2 - 3b}$ of the inscribed circle.",
"The circumscribed circle of the equilateral triangle has radius $2 r = (2/3) \\sqrt{a^2 - 3b}$ .",
"If a cubic polynomial has two stationary points, the distance between them is always $2 r = (2/3) \\sqrt{a^2 - 3b}$ .",
"The inflection point of the graph of $x^3 + a x^2 + b x + c$ is always the midpoint ($-a/3$ ) between the stationary points of the cubic polynomial.",
"Hence, the analysis of the cubic polynomial $x^3 + a x^2 + b x + c$ should start with what the value of $b$ , relative to $a^2/3$ , is.",
"(I) If ${b < a^2/3}$ and if: (i) ${c_2 \\le c \\le c_0}$ , then the polynomial $x^3 + a x^2 + b x + c$ has three real roots with the following isolation intervals: $x_3 \\in [\\nu _3, \\mu _2], \\,\\, x_2 \\in [\\mu _2, \\phi ]$ , and $x_1 \\in [\\nu _1, \\xi _2]$ (Figure 2).",
"(ii) ${c_0 \\le c \\le c_1}$ , then the polynomial $x^3 + a x^2 + b x + c$ has three real roots with the following isolation intervals: $x_3 \\in [\\xi _1, \\nu _3], \\,\\, x_2 \\in [\\phi ,\\mu _1]$ , and $x_1 \\in [\\mu _1, \\nu _1]$ (Figure 2).",
"In the above, $c_{1,2} = c_0 \\pm (2/27) \\sqrt{( a^2 - 3 b)^3}$ , with $c_0 = - 2a^3/27 + a b/3$ , are the values of $c$ for which, for any $a$ and $b < a^2/3$ , the discriminant $\\Delta _3$ of the cubic polynomial $x^3 + a x^2 + b x + c$ is zero ($\\Delta _3$ positive for $c$ between $c_2$ and $c_1$ ).",
"Namely, these are the roots of the quadratic equation (REF ): $x^2 + (4 a^3/27 - 2 a b/3 ) x - a^2 b^2/27 + 4 b^3/27 = 0$ .",
"Also in the above, $\\nu _3 = -a/3 - \\sqrt{a^2/3 - b}$ , $\\nu _2 = \\phi = -a/3$ , and $\\nu _1 = - a/3 + \\sqrt{a^2/3 - b}$ are three real roots of the “balanced\" cubic polynomial $x^3 + a x^2 + b x + c_0$ (Figure 2).",
"The roots of the “extreme\" cubic $x^3 + a x^2 + b x + c_1$ are the double root $\\mu _1 = -a/3 + (\\sqrt{3}/3) \\, \\sqrt{a^2/3-b}$ and the simple root $\\xi _1 = - a - 2 \\mu _1 = - a/3 - 2r = - a/3 - (2/3) \\sqrt{a^2 - 3 b}$ .",
"Likewise, the roots of the “extreme\" cubic $x^3 + a x^2 + b x + c_1$ are the double root $\\mu _2 = -a/3 - (\\sqrt{3}/3) \\, \\sqrt{a^2/3-b}$ and the simple root $\\xi _2 = - a - 2 \\mu _2 = - a/3 + 2r = - a/3 + (2/3) \\sqrt{a^2 - 3 b}$ (Figure 2 and Figure 3).",
"The biggest distance between any two of the three real roots of the cubic equation $x^3 + a x^2 + b x + c = 0$ is $\\alpha = \\sqrt{12} r = (\\sqrt{12}/3) \\sqrt{a^2 - 3b}$ — achieved for the roots of the “balanced\" cubic equation $x^3 + a x^2 + b x + c_0$ (Figure 2).",
"For any other cubic equation with $c_2 \\le c \\le c_1$ , the three real roots are within an interval of length $3 r = \\sqrt{a^2 - 3b} < \\alpha $ (Figure 2).",
"(iii) ${c < c_2}$ , then the polynomial $x^3 + a x^2 + b x + c$ has only one real root: $x_1 > \\xi _2 = - a - 2 \\mu _2 = - a/3 + 2r = - a/3 + (2/3) \\sqrt{a^2 - 3 b}$ (Figure 3).",
"The root $x_1$ can be bounded from above by a polynomial root bound.",
"(iv) ${c > c_1}$ , then the polynomial $x^3 + a x^2 + b x + c$ has only one real root: $x_1 < \\xi _1 = - a - 2 \\mu _1 = - a/3 - 2r = - a/3 - (2/3) \\sqrt{a^2 - 3 b}$ (Figure 3).",
"The root $x_1$ can be bounded from below by a polynomial root bound.",
"(II) If ${b = a^2/3}$ and if: (i) ${c < (1/27) a^3}$ , then the polynomial $x^3 + a x^2 + b x + c$ has only one real root: $x_1 = -a/3 + \\@root 3 \\of {a^3/27 - c} \\, > \\, -a/3$ (Figure 4).",
"(ii) ${c = (1/27) a^3}$ , then the polynomial $x^3 + a x^2 + b x + c$ has a triple real root: $x_1 = x_2 = x_3 = -a/3$ (Figure 4).",
"(iii) ${c > (1/27) a^3}$ , then the polynomial $x^3 + a x^2 + b x + c$ has only one real root: $x_1 = -a/3 + \\@root 3 \\of {a^3/27 - c} \\, < \\, -a/3$ (Figure 4).",
"(III) If ${b > a^2/3}$ , the discriminant of the cubic polynomial is negative and thus $x^3 + a x^2 + b x + c$ has one real root $x_1$ and a pair of complex conjugate roots.",
"The isolation interval of $x_1$ depends on the signs of $a$ and $c$ and is as follows: (i) If ${a \\ge 0}$ and ${c \\le 0}\\!",
": \\,\\, 0 \\le x_1 \\le -c/b$ (Figure 5).",
"(ii) If ${a \\ge 0}$ and ${c > 0}\\!",
": \\,\\,$ min$\\lbrace -a, -c/b\\rbrace \\le x_1 \\le $ max$\\lbrace -a, -c/b\\rbrace $ (Figure 6).",
"(iii) If ${a < 0}$ and ${c < 0}\\!",
": \\,\\,$ min$\\lbrace -a, -c/b\\rbrace \\le x_1 \\le $ max$\\lbrace -a, -c/b\\rbrace $ (Figure 7).",
"(iv) If ${a < 0}$ and ${c \\ge 0}\\!",
": \\,\\, -c/b \\le x_1 \\le 0$ (Figure 8).",
"(c) The coefficient $c$ of $x^3 + a x^2 + b x + c$ rotates the equilateral triangle (which exists if $b < a^2/3$ ) that projects onto the roots $x_3 \\le x_2 \\le x_1$ (at least two of which are different) of the cubic polynomial.",
"The vertices $P$ , $Q$ , and $R$ of the triangle are points of coordinates $(x_1, (x_2 - x_3)/\\sqrt{3})$ , $(x_2, (x_3 - x_1)/\\sqrt{3})$ , and $(x_3, (x_1 - x_2)/\\sqrt{3})$ , respectively.",
"Point $Q$ is always below the abscissa and points $P$ and $R$ — always above it.",
"When $c = c_0 = - 2a^3/27 + a b/3$ , the side $PR$ is parallel to the abscissa.",
"This corresponds to the “balanced\" cubic equation $x^3 + a x^2 + b x - 2a^3/27 + a b/3 = 0$ , the roots of which are symmetric with respect to the centre of the inscribed circle: $\\nu _3 = -a/3 - \\sqrt{a^2/3 - b}$ , $\\nu _2 = \\phi = -a/3$ , and $\\nu _1 = - a/3 + \\sqrt{a^2/3 - b}$ .",
"The “balanced\" equation has triangle $P_0 Q_0 R_0$ (Figure 2).",
"When $c$ increases from $c_0$ towards $c_1 > c_0$ , the equilateral triangle $PQR$ rotates counterclockwise around its centre from the position of triangle $P_0 Q_0 R_0$ of the “balanced\" equation.",
"When $c = c_1$ , the roots $x_2$ and $x_1$ coalesce into the double root $\\mu _1$ , while the smallest root $x_3$ becomes equal to $\\xi _1 = - a - 2 \\mu _1 = - a/3 - 2r = - a/3 - (2/3) \\sqrt{a^2 - 3 b}$ .",
"The triangle in this case is $P_1 Q_1 R_1$ and its side $P_1 Q_1$ is perpendicular to the abscissa.",
"The vertex $R_1$ is on the abscissa.",
"The triangle cannot be rotated further counterclockwise as, when $c > c_1$ , the polynomial $x^3 + a x^2 + b x + c$ has only one real root (Figure 2).",
"When $c$ decreases from $c_0$ towards $c_2 < c_0$ , the equilateral triangle $PQR$ rotates clockwise around its centre from the position of triangle $P_0 Q_0 R_0$ of the “balanced\" equation.",
"When $c = c_2$ , the roots $x_3$ and $x_2$ coalesce into the double root $\\mu _2$ , while the biggest root $x_1$ becomes equal to $\\xi _2 = - a - 2 \\mu _2 = - a/3 + 2r = - a/3 + (2/3) \\sqrt{a^2 - 3 b}$ .",
"The triangle in this case is $P_2 Q_2 R_2$ and its side $R_2 Q_2$ is perpendicular to the abscissa.",
"The vertex $P_2$ is on the abscissa.",
"The triangle cannot be rotated further clockwise as, when $c < c_2$ , the polynomial $x^3 + a x^2 + b x + c$ has only one real root (Figure 2).",
"The coefficient $a$ of the quadratic term of $x^3 + a x^2 + b x + c$ selects the centre $\\phi = - a/3$ of the inscribed circle of the equilateral triangle that projects onto the roots of $x^3 + a x^2 + b x + c$ , in the case of three real roots.",
"The centre of this circle is also the projection of the inflection point of the graph of $x^3 + a x^2 + b x + c$ onto the abscissa.",
"The inscribed circle projects to an interval on the abscissa with endpoints equal to the projections of the stationary points of $x^3 + a x^2 + b x + c$ (Figure 1).",
"For any given $a$ , the coefficients $b$ of the linear term of $x^3 + a x^2 + b x + c$ determines the radius $r = (1/3) \\sqrt{a^2 - 3b}$ of the inscribed circle.",
"The circumscribed circle of the equilateral triangle has radius $2 r = (2/3) \\sqrt{a^2 - 3b}$ .",
"If a cubic polynomial has two stationary points, the distance between them is always $2 r = (2/3) \\sqrt{a^2 - 3b}$ .",
"The inflection point of the graph of $x^3 + a x^2 + b x + c$ is always the midpoint ($-a/3$ ) between the stationary points of the cubic polynomial.",
"Hence, the analysis of the cubic polynomial $x^3 + a x^2 + b x + c$ should start with what the value of $b$ , relative to $a^2/3$ , is.",
"(I) If ${b < a^2/3}$ and if: (i) ${c_2 \\le c \\le c_0}$ , then the polynomial $x^3 + a x^2 + b x + c$ has three real roots with the following isolation intervals: $x_3 \\in [\\nu _3, \\mu _2], \\,\\, x_2 \\in [\\mu _2, \\phi ]$ , and $x_1 \\in [\\nu _1, \\xi _2]$ (Figure 2).",
"(ii) ${c_0 \\le c \\le c_1}$ , then the polynomial $x^3 + a x^2 + b x + c$ has three real roots with the following isolation intervals: $x_3 \\in [\\xi _1, \\nu _3], \\,\\, x_2 \\in [\\phi ,\\mu _1]$ , and $x_1 \\in [\\mu _1, \\nu _1]$ (Figure 2).",
"In the above, $c_{1,2} = c_0 \\pm (2/27) \\sqrt{( a^2 - 3 b)^3}$ , with $c_0 = - 2a^3/27 + a b/3$ , are the values of $c$ for which, for any $a$ and $b < a^2/3$ , the discriminant $\\Delta _3$ of the cubic polynomial $x^3 + a x^2 + b x + c$ is zero ($\\Delta _3$ positive for $c$ between $c_2$ and $c_1$ ).",
"Namely, these are the roots of the quadratic equation (REF ): $x^2 + (4 a^3/27 - 2 a b/3 ) x - a^2 b^2/27 + 4 b^3/27 = 0$ .",
"Also in the above, $\\nu _3 = -a/3 - \\sqrt{a^2/3 - b}$ , $\\nu _2 = \\phi = -a/3$ , and $\\nu _1 = - a/3 + \\sqrt{a^2/3 - b}$ are three real roots of the “balanced\" cubic polynomial $x^3 + a x^2 + b x + c_0$ (Figure 2).",
"The roots of the “extreme\" cubic $x^3 + a x^2 + b x + c_1$ are the double root $\\mu _1 = -a/3 + (\\sqrt{3}/3) \\, \\sqrt{a^2/3-b}$ and the simple root $\\xi _1 = - a - 2 \\mu _1 = - a/3 - 2r = - a/3 - (2/3) \\sqrt{a^2 - 3 b}$ .",
"Likewise, the roots of the “extreme\" cubic $x^3 + a x^2 + b x + c_1$ are the double root $\\mu _2 = -a/3 - (\\sqrt{3}/3) \\, \\sqrt{a^2/3-b}$ and the simple root $\\xi _2 = - a - 2 \\mu _2 = - a/3 + 2r = - a/3 + (2/3) \\sqrt{a^2 - 3 b}$ (Figure 2 and Figure 3).",
"The biggest distance between any two of the three real roots of the cubic equation $x^3 + a x^2 + b x + c = 0$ is $\\alpha = \\sqrt{12} r = (\\sqrt{12}/3) \\sqrt{a^2 - 3b}$ — achieved for the roots of the “balanced\" cubic equation $x^3 + a x^2 + b x + c_0$ (Figure 2).",
"For any other cubic equation with $c_2 \\le c \\le c_1$ , the three real roots are within an interval of length $3 r = \\sqrt{a^2 - 3b} < \\alpha $ (Figure 2).",
"(iii) ${c < c_2}$ , then the polynomial $x^3 + a x^2 + b x + c$ has only one real root: $x_1 > \\xi _2 = - a - 2 \\mu _2 = - a/3 + 2r = - a/3 + (2/3) \\sqrt{a^2 - 3 b}$ (Figure 3).",
"The root $x_1$ can be bounded from above by a polynomial root bound.",
"(iv) ${c > c_1}$ , then the polynomial $x^3 + a x^2 + b x + c$ has only one real root: $x_1 < \\xi _1 = - a - 2 \\mu _1 = - a/3 - 2r = - a/3 - (2/3) \\sqrt{a^2 - 3 b}$ (Figure 3).",
"The root $x_1$ can be bounded from below by a polynomial root bound.",
"(II) If ${b = a^2/3}$ and if: (i) ${c < (1/27) a^3}$ , then the polynomial $x^3 + a x^2 + b x + c$ has only one real root: $x_1 = -a/3 + \\@root 3 \\of {a^3/27 - c} \\, > \\, -a/3$ (Figure 4).",
"(ii) ${c = (1/27) a^3}$ , then the polynomial $x^3 + a x^2 + b x + c$ has a triple real root: $x_1 = x_2 = x_3 = -a/3$ (Figure 4).",
"(iii) ${c > (1/27) a^3}$ , then the polynomial $x^3 + a x^2 + b x + c$ has only one real root: $x_1 = -a/3 + \\@root 3 \\of {a^3/27 - c} \\, < \\, -a/3$ (Figure 4).",
"(III) If ${b > a^2/3}$ , the discriminant of the cubic polynomial is negative and thus $x^3 + a x^2 + b x + c$ has one real root $x_1$ and a pair of complex conjugate roots.",
"The isolation interval of $x_1$ depends on the signs of $a$ and $c$ and is as follows: (i) If ${a \\ge 0}$ and ${c \\le 0}\\!",
": \\,\\, 0 \\le x_1 \\le -c/b$ (Figure 5).",
"(ii) If ${a \\ge 0}$ and ${c > 0}\\!",
": \\,\\,$ min$\\lbrace -a, -c/b\\rbrace \\le x_1 \\le $ max$\\lbrace -a, -c/b\\rbrace $ (Figure 6).",
"(iii) If ${a < 0}$ and ${c < 0}\\!",
": \\,\\,$ min$\\lbrace -a, -c/b\\rbrace \\le x_1 \\le $ max$\\lbrace -a, -c/b\\rbrace $ (Figure 7).",
"(iv) If ${a < 0}$ and ${c \\ge 0}\\!",
": \\,\\, -c/b \\le x_1 \\le 0$ (Figure 8).",
"If ${b < a^2/3}$ and if: (i) ${c_2 \\le c \\le c_0}$ , then the polynomial $x^3 + a x^2 + b x + c$ has three real roots with the following isolation intervals: $x_3 \\in [\\nu _3, \\mu _2], \\,\\, x_2 \\in [\\mu _2, \\phi ]$ , and $x_1 \\in [\\nu _1, \\xi _2]$ (Figure 2).",
"(ii) ${c_0 \\le c \\le c_1}$ , then the polynomial $x^3 + a x^2 + b x + c$ has three real roots with the following isolation intervals: $x_3 \\in [\\xi _1, \\nu _3], \\,\\, x_2 \\in [\\phi ,\\mu _1]$ , and $x_1 \\in [\\mu _1, \\nu _1]$ (Figure 2).",
"${c_2 \\le c \\le c_0}$ , then the polynomial $x^3 + a x^2 + b x + c$ has three real roots with the following isolation intervals: $x_3 \\in [\\nu _3, \\mu _2], \\,\\, x_2 \\in [\\mu _2, \\phi ]$ , and $x_1 \\in [\\nu _1, \\xi _2]$ (Figure 2).",
"${c_0 \\le c \\le c_1}$ , then the polynomial $x^3 + a x^2 + b x + c$ has three real roots with the following isolation intervals: $x_3 \\in [\\xi _1, \\nu _3], \\,\\, x_2 \\in [\\phi ,\\mu _1]$ , and $x_1 \\in [\\mu _1, \\nu _1]$ (Figure 2).",
"In the above, $c_{1,2} = c_0 \\pm (2/27) \\sqrt{( a^2 - 3 b)^3}$ , with $c_0 = - 2a^3/27 + a b/3$ , are the values of $c$ for which, for any $a$ and $b < a^2/3$ , the discriminant $\\Delta _3$ of the cubic polynomial $x^3 + a x^2 + b x + c$ is zero ($\\Delta _3$ positive for $c$ between $c_2$ and $c_1$ ).",
"Namely, these are the roots of the quadratic equation (REF ): $x^2 + (4 a^3/27 - 2 a b/3 ) x - a^2 b^2/27 + 4 b^3/27 = 0$ .",
"Also in the above, $\\nu _3 = -a/3 - \\sqrt{a^2/3 - b}$ , $\\nu _2 = \\phi = -a/3$ , and $\\nu _1 = - a/3 + \\sqrt{a^2/3 - b}$ are three real roots of the “balanced\" cubic polynomial $x^3 + a x^2 + b x + c_0$ (Figure 2).",
"The roots of the “extreme\" cubic $x^3 + a x^2 + b x + c_1$ are the double root $\\mu _1 = -a/3 + (\\sqrt{3}/3) \\, \\sqrt{a^2/3-b}$ and the simple root $\\xi _1 = - a - 2 \\mu _1 = - a/3 - 2r = - a/3 - (2/3) \\sqrt{a^2 - 3 b}$ .",
"Likewise, the roots of the “extreme\" cubic $x^3 + a x^2 + b x + c_1$ are the double root $\\mu _2 = -a/3 - (\\sqrt{3}/3) \\, \\sqrt{a^2/3-b}$ and the simple root $\\xi _2 = - a - 2 \\mu _2 = - a/3 + 2r = - a/3 + (2/3) \\sqrt{a^2 - 3 b}$ (Figure 2 and Figure 3).",
"The biggest distance between any two of the three real roots of the cubic equation $x^3 + a x^2 + b x + c = 0$ is $\\alpha = \\sqrt{12} r = (\\sqrt{12}/3) \\sqrt{a^2 - 3b}$ — achieved for the roots of the “balanced\" cubic equation $x^3 + a x^2 + b x + c_0$ (Figure 2).",
"For any other cubic equation with $c_2 \\le c \\le c_1$ , the three real roots are within an interval of length $3 r = \\sqrt{a^2 - 3b} < \\alpha $ (Figure 2).",
"(iii) ${c < c_2}$ , then the polynomial $x^3 + a x^2 + b x + c$ has only one real root: $x_1 > \\xi _2 = - a - 2 \\mu _2 = - a/3 + 2r = - a/3 + (2/3) \\sqrt{a^2 - 3 b}$ (Figure 3).",
"The root $x_1$ can be bounded from above by a polynomial root bound.",
"(iv) ${c > c_1}$ , then the polynomial $x^3 + a x^2 + b x + c$ has only one real root: $x_1 < \\xi _1 = - a - 2 \\mu _1 = - a/3 - 2r = - a/3 - (2/3) \\sqrt{a^2 - 3 b}$ (Figure 3).",
"The root $x_1$ can be bounded from below by a polynomial root bound.",
"${c < c_2}$ , then the polynomial $x^3 + a x^2 + b x + c$ has only one real root: $x_1 > \\xi _2 = - a - 2 \\mu _2 = - a/3 + 2r = - a/3 + (2/3) \\sqrt{a^2 - 3 b}$ (Figure 3).",
"The root $x_1$ can be bounded from above by a polynomial root bound.",
"${c > c_1}$ , then the polynomial $x^3 + a x^2 + b x + c$ has only one real root: $x_1 < \\xi _1 = - a - 2 \\mu _1 = - a/3 - 2r = - a/3 - (2/3) \\sqrt{a^2 - 3 b}$ (Figure 3).",
"The root $x_1$ can be bounded from below by a polynomial root bound.",
"If ${b = a^2/3}$ and if: (i) ${c < (1/27) a^3}$ , then the polynomial $x^3 + a x^2 + b x + c$ has only one real root: $x_1 = -a/3 + \\@root 3 \\of {a^3/27 - c} \\, > \\, -a/3$ (Figure 4).",
"(ii) ${c = (1/27) a^3}$ , then the polynomial $x^3 + a x^2 + b x + c$ has a triple real root: $x_1 = x_2 = x_3 = -a/3$ (Figure 4).",
"(iii) ${c > (1/27) a^3}$ , then the polynomial $x^3 + a x^2 + b x + c$ has only one real root: $x_1 = -a/3 + \\@root 3 \\of {a^3/27 - c} \\, < \\, -a/3$ (Figure 4).",
"${c < (1/27) a^3}$ , then the polynomial $x^3 + a x^2 + b x + c$ has only one real root: $x_1 = -a/3 + \\@root 3 \\of {a^3/27 - c} \\, > \\, -a/3$ (Figure 4).",
"${c = (1/27) a^3}$ , then the polynomial $x^3 + a x^2 + b x + c$ has a triple real root: $x_1 = x_2 = x_3 = -a/3$ (Figure 4).",
"${c > (1/27) a^3}$ , then the polynomial $x^3 + a x^2 + b x + c$ has only one real root: $x_1 = -a/3 + \\@root 3 \\of {a^3/27 - c} \\, < \\, -a/3$ (Figure 4).",
"If ${b > a^2/3}$ , the discriminant of the cubic polynomial is negative and thus $x^3 + a x^2 + b x + c$ has one real root $x_1$ and a pair of complex conjugate roots.",
"The isolation interval of $x_1$ depends on the signs of $a$ and $c$ and is as follows: (i) If ${a \\ge 0}$ and ${c \\le 0}\\!",
": \\,\\, 0 \\le x_1 \\le -c/b$ (Figure 5).",
"(ii) If ${a \\ge 0}$ and ${c > 0}\\!",
": \\,\\,$ min$\\lbrace -a, -c/b\\rbrace \\le x_1 \\le $ max$\\lbrace -a, -c/b\\rbrace $ (Figure 6).",
"(iii) If ${a < 0}$ and ${c < 0}\\!",
": \\,\\,$ min$\\lbrace -a, -c/b\\rbrace \\le x_1 \\le $ max$\\lbrace -a, -c/b\\rbrace $ (Figure 7).",
"(iv) If ${a < 0}$ and ${c \\ge 0}\\!",
": \\,\\, -c/b \\le x_1 \\le 0$ (Figure 8).",
"If ${a \\ge 0}$ and ${c \\le 0}\\!",
": \\,\\, 0 \\le x_1 \\le -c/b$ (Figure 5).",
"If ${a \\ge 0}$ and ${c > 0}\\!",
": \\,\\,$ min$\\lbrace -a, -c/b\\rbrace \\le x_1 \\le $ max$\\lbrace -a, -c/b\\rbrace $ (Figure 6).",
"If ${a < 0}$ and ${c < 0}\\!",
": \\,\\,$ min$\\lbrace -a, -c/b\\rbrace \\le x_1 \\le $ max$\\lbrace -a, -c/b\\rbrace $ (Figure 7).",
"If ${a < 0}$ and ${c \\ge 0}\\!",
": \\,\\, -c/b \\le x_1 \\le 0$ (Figure 8).",
"The coefficient $c$ of $x^3 + a x^2 + b x + c$ rotates the equilateral triangle (which exists if $b < a^2/3$ ) that projects onto the roots $x_3 \\le x_2 \\le x_1$ (at least two of which are different) of the cubic polynomial.",
"The vertices $P$ , $Q$ , and $R$ of the triangle are points of coordinates $(x_1, (x_2 - x_3)/\\sqrt{3})$ , $(x_2, (x_3 - x_1)/\\sqrt{3})$ , and $(x_3, (x_1 - x_2)/\\sqrt{3})$ , respectively.",
"Point $Q$ is always below the abscissa and points $P$ and $R$ — always above it.",
"When $c = c_0 = - 2a^3/27 + a b/3$ , the side $PR$ is parallel to the abscissa.",
"This corresponds to the “balanced\" cubic equation $x^3 + a x^2 + b x - 2a^3/27 + a b/3 = 0$ , the roots of which are symmetric with respect to the centre of the inscribed circle: $\\nu _3 = -a/3 - \\sqrt{a^2/3 - b}$ , $\\nu _2 = \\phi = -a/3$ , and $\\nu _1 = - a/3 + \\sqrt{a^2/3 - b}$ .",
"The “balanced\" equation has triangle $P_0 Q_0 R_0$ (Figure 2).",
"When $c$ increases from $c_0$ towards $c_1 > c_0$ , the equilateral triangle $PQR$ rotates counterclockwise around its centre from the position of triangle $P_0 Q_0 R_0$ of the “balanced\" equation.",
"When $c = c_1$ , the roots $x_2$ and $x_1$ coalesce into the double root $\\mu _1$ , while the smallest root $x_3$ becomes equal to $\\xi _1 = - a - 2 \\mu _1 = - a/3 - 2r = - a/3 - (2/3) \\sqrt{a^2 - 3 b}$ .",
"The triangle in this case is $P_1 Q_1 R_1$ and its side $P_1 Q_1$ is perpendicular to the abscissa.",
"The vertex $R_1$ is on the abscissa.",
"The triangle cannot be rotated further counterclockwise as, when $c > c_1$ , the polynomial $x^3 + a x^2 + b x + c$ has only one real root (Figure 2).",
"When $c$ decreases from $c_0$ towards $c_2 < c_0$ , the equilateral triangle $PQR$ rotates clockwise around its centre from the position of triangle $P_0 Q_0 R_0$ of the “balanced\" equation.",
"When $c = c_2$ , the roots $x_3$ and $x_2$ coalesce into the double root $\\mu _2$ , while the biggest root $x_1$ becomes equal to $\\xi _2 = - a - 2 \\mu _2 = - a/3 + 2r = - a/3 + (2/3) \\sqrt{a^2 - 3 b}$ .",
"The triangle in this case is $P_2 Q_2 R_2$ and its side $R_2 Q_2$ is perpendicular to the abscissa.",
"The vertex $P_2$ is on the abscissa.",
"The triangle cannot be rotated further clockwise as, when $c < c_2$ , the polynomial $x^3 + a x^2 + b x + c$ has only one real root (Figure 2)."
]
] | 2107.01847 |
[
[
"Positivity vs. Lorentz-violation: an explicit example"
],
[
"Abstract We show how a class of multi-field scalar-field theories in a Lorentz-breaking background imposes consistency conditions on its effective theory of a single field and provides an example of order-unity violation of a naively applied positivity bound, assuming a large hierarchy between the masses of the lightest field and the others."
],
[
"Introduction",
"Underlying assumptions on ultraviolet (UV) completion can impose constraints on its low-energy effective field theories (EFTs), meaning that not all EFTs may be consistent with the assumed UV physics even if they are “consistent” from low-energy perspective.",
"One of the most well-established constraints is called positivity bounds [1], provided that EFTs admit a unitary, Poincaré-invariant, analytic, and bounded UV completion.",
"The last two assumptions are inferred from causality and locality.",
"Although whether the UV theory in nature indeed satisfies these assumptions is unknown, they are well-defined and self-consistent, and the positivity bounds can be used to test whether our working assumptions about UV physics are compatible with low-energy experiments/observations.",
"The positivity bounds are often derived by means of scattering amplitudes where asymptotic states have to be well-defined.",
"As for EFTs around a Poincaré invariant background, there would be no subtleties about the states and the bounds provide remarkably strong constraints on higher derivative operators of EFTs [1], [2], [3], [4], [5].",
"Furthermore, the positivity bounds provide a cutoff scale of a renormalizable theory when it couples to gravity [6], [7], [8], [9]: for example, the Standard Model of particle physics coupled to general relativity violates the positivity bound when it is extrapolated up to $10^{16}$ GeV, suggesting that quantum gravity should be needed around or below $10^{16}$ GeV [8].",
"The framework of EFTs is robust even around a non-trivial background, which is a typical situation in realistic setups including (but not limited to) cosmology.",
"As is well-known, the notion of “particle” is ambiguous in field theories in curved spacetimes and, in particular, there would be no definite notion of particles in the infrared (IR) limit.",
"The Poincaré-invariant positivity bounds may not be directly applicable to EFTs around such non-trivial backgrounds.",
"We study EFTs without the Lorentz symmetry, which naturally arise in cosmology, for instance.",
"The states may be well-defined as long as the temporal and spatial translation symmetries are preserved.",
"Assuming dispersion relations (linear equations of motion) of particles, one may discuss positivity bounds by considering scatterings of the particles even when the Lorentz invariance is absent.",
"This is indeed analysed in [10], [11], and the positivity bounds are discussed.",
"However, when the Lorentz invariance is spontaneously broken by a background configuration of a field, the background serves as a source of gravity and then the temporal and/or spatial translation symmetry should be generically broken as well.",
"The existence of gravity should impose a strong constraint on the consistency of the arguments.Positivity bounds on gravitational EFTs are already non-trivial due to the pole associated with the graviton $t$ -channel exchange even around the Minkowski background.",
"Gravitational positivity bounds can be derived by assuming the Regge behaviour of the amplitude, which is a consistent satisfaction of the Froissart bound [12], [13] in gravitational theories, to cancel the pole of the graviton exchange [14], [15], [16] (see also [17], [6], [7], [18] for related discussions).",
"In fact, the paper [19] showed that the existence of graviton enforces three-particle amplitudes to be Lorentz invariant from unitarity, spacetime translation symmetries, spatial rotational symmetry, relativistic dispersion relations, and analyticity, implying that the Lorentz symmetry is an emergent symmetry from some of the assumptions.",
"A similar observation can be found in [20].",
"One should carefully examine working assumptions in the bottom-up approaches, as the input may already include the Lorentz symmetry, implicitly.",
"It is therefore important to consider different approaches that can be applied to systems in which gravity is essential.",
"Some of such approaches are called swampland conjectures, ranging from the weak gravity conjecture [21] to the de Sitter conjecture [22], [23].",
"Those swampland conjectures are supposed to tell which EFTs are consistent with quantum gravity and which ones are not (see [24] for a review).",
"However, none of the swampland conjectures enjoys a rigorous proof based on a fundamental theory, and indeed there can be exceptions and/or counterexamples (e.g.",
"the KKLT scenario [25] and Large Volume Scenario [26] against the de Sitter conjecture).",
"Therefore, the statements of the swampland conjectures should at best be considered as some properties that most (but not necessarily all) of consistent EFTs tend to possess.",
"Another related approach is based on the generalized second law of black holes [27], [28], aiming to test the consistency of a particular Lorentz violating theory called ghost condensate [29], [30].",
"It was later shown that in this theory the generalized second law is actually protected in a rather non-trivial way because of the accretion of the Nambu-Goldstone mode [31], [32].",
"Yet another consistency test was proposed in the context of an accelerated expansion of the universe, leading to the so-called de Sitter entropy bound [33].",
"Again, the bound was recently shown to hold in the ghost condensate [34].",
"These examples clearly illustrate the importance of model-independent consistency conditions that are applicable to systems with gravity.",
"On the other hand, model-dependent approaches are also useful, as, at the very least, they provide smoking guns for the corresponding UV physics that can be tested by observations/experiments at low energy.",
"They also serve as examples upon which less model-dependent bounds and/or conjectures can be built.",
"In the present paper, we therefore take a complementary top-down approach: we assume a particular but sufficiently wide class of (partial) UV completion and discuss constraints on its low-energy EFTs.",
"While it does not apply to other types of (partial) UV completion such as the idea in [35], an advantage of this approach is that, once we admit such a particular class of (partial) UV completion, it can robustly be applied to the situation where Lorentz symmetry is broken and/or gravity cannot be ignored.",
"Also, if one finds an example with a special feature then it serves as an existence proof of an EFT with the feature.",
"The low-energy EFT that we are interested in is given by the action of the form $\\int \\mathrm {d}^d x \\sqrt{-g}P(\\varphi , X)$ , called k-essence, where $\\varphi $ is a scalar field, $X=-g^{\\mu \\nu }\\partial _{\\mu }\\varphi \\partial _{\\nu }\\varphi /2$ its kinetic term, and $d$ the spacetime dimension.",
"This action can be regarded as the leading operators in the context of EFT of single-field inflation/dark energy models [36], [37], [38], and its (partial) UV completion has recently been developed in [39], [40], [41], [42], [43].",
"The single-field k-essence theory can be obtained from a non-linear sigma model with one light direction in the field space of an arbitrary geometry by integrating out all the massive modes while keeping the light one.",
"We mainly study the k-essence theory in the Einstein frame, that is, gravity is minimally coupled to $\\varphi $ .",
"We will briefly discuss a (partial) UV completion of a subclass of degenerate higher-order scalar-tensor (DHOST) theories [44], [45], [46] by considering non-minimal couplings.",
"The single-field low-energy EFT action that we obtain from the multi-field UV model is Lorentz-invariant, and thus the Lorentz violation is spontaneous.",
"For a trivial background $\\partial _{\\mu }\\varphi =0$ , the positivity bound would conclude that the coefficient of the four-point interactions has to be positive [1], namely $P_{XX}>0$ , where a subscript $X$ denotes a derivative with respect to $X$ .",
"On the other hand, our target is the case where $\\varphi $ has a non-vanishing gradient $C_{\\mu }=\\partial _{\\mu }\\varphi \\ne 0$ at the background level.",
"Then, the vector $C_{\\mu }$ determines a preferred direction, and the Lorentz symmetry is spontaneously broken.",
"The standard positivity bounds may not be applied in such situations.",
"In particular, we show by an explicit example that $P_{XX}$ is allowed to be negative in Lorentz-violating backgrounds, without any inconsistency in UV.",
"The rest of the paper is organized as follows.",
"In Sec.",
"we consider a few simple toy examples to illustrate how the reduction to a low-energy effective theory works.",
"In Sec.",
"we describe the procedure of EFT reductions for two-field models and general multi-field nonlinear sigma models.",
"Here our analysis is done at fully nonlinear levels with minimally coupled gravity.",
"Then in Sec.",
", we perform perturbative analyses concentrating on nonlinear sigma models composed of two scalars without gravity.",
"We derive the consistency conditions with clear validity ranges; when the background $C_{\\mu }=\\partial _{\\mu }\\varphi $ is spacelike, namely $X<0$ , the ghost-free condition and the no-tachyon condition immediately conclude $P_X>0$ and $P_{XX}>0$ , while precautions are needed in the timelike background $X>0$ .",
"In particular we show an explicit example where $P_{XX}$ can be negative in the timelike background.",
"We also conclude that the same result holds in a general multi-field case.",
"In Sec.",
", extensive discussions on several aspects of our EFT reduction are provided.",
"We discuss higher-derivative corrections to the k-essence and also comment on the implications for screening effects and the preservation of the null energy condition.",
"We further revisit the UV consistency of the k-essence by using the Feynman-like diagrams.",
"We then consider effects of non-minimal matter coupling and implications to a subset of DHOST theories.",
"Section is devoted to the summary and conclusions.",
"In Appendix , we perform the analysis for models with the DBI-type kinetic terms in an arbitrary field space and derive essentially the same results as in the case of the non-linear sigma models.",
"We demonstrate the EFT reduction from the $U(1)$ scalar theory as a concrete example of our general argument in Appendix , and finally in Appendix we make a one-to-one comparison of our EFT to the models of ghost condensate."
],
[
"Preparation for EFT reduction",
"The purpose of this section is to demonstrate the procedure of EFT reduction using a few explicit examples, before exhibiting more general multi-field cases in Sec. .",
"We first consider a simple toy example of a classical-mechanical system to illustrate how the reduction to a low-energy effective theory works in Sec.",
"REF and then study two-field-scalar systems and its reduction to a single-field EFT."
],
[
"Warm-up: low-energy effective theory of coupled oscillators",
"As a first concrete toy model of EFT reduction, we consider a classical system of coupled oscillators $x(t)$ and $y(t)$ described by the Lagrangian, $L = \\frac{1}{2}\\left(\\dot{x}^2 + \\dot{y}^2 + 2\\alpha \\dot{x}\\dot{y}\\right) - \\frac{1}{2}\\left(\\mu ^2x^2 + \\eta M^2 y^2 + 2\\beta \\mu Mxy\\right) - \\gamma M y\\dot{x}\\,, $ where a dot represents derivative with respect to the time $t$ , and $\\alpha ,\\beta ,\\gamma ,\\eta , \\mu $ and $M$ are all constants.",
"We do not perform a diagonalization to eliminate the kinetic mixing $\\alpha \\dot{x} \\dot{y}$ at this stage, in order to make the analogy to the cases in later sections easier.",
"We suppose the hierarchy between the two mass scales $\\mu $ and $M$ as $\\epsilon \\equiv \\frac{\\mu }{M} \\ll 1\\,, $ and ($\\alpha $ , $\\beta $ , $\\gamma $ , $\\eta $ ) are dimensionless constants of at most order unity.",
"The avoidance of a ghost mode is achieved by the condition $\\vert \\alpha \\vert <1$ .",
"The general solution for $x$ and $y$ is a linear combination of solutions of the form, $x = x_0 e^{-i\\omega \\tau } \\,, \\quad y = y_0 e^{-i\\omega \\tau } \\,, \\quad \\tau \\equiv \\mu t\\,,$ where $x_0$ and $y_0$ are constants.",
"The equations of motion are reduced to $(1-\\alpha ^2)\\epsilon ^2 \\omega ^4 - (\\eta +\\gamma ^2-2\\alpha \\beta \\epsilon + \\epsilon ^2)\\omega ^2 + (\\eta -\\beta ^2) = 0\\,, $ and $y_0 (\\eta - \\epsilon ^2 \\omega ^2)=-\\epsilon x_0 [\\beta - \\omega (i \\gamma +\\epsilon \\alpha \\omega )]\\,.$ Eq.",
"(REF ) admits a couple of “fast” solutions, $\\omega ^2 = \\frac{\\eta +\\gamma ^2}{1-\\alpha ^2}\\epsilon ^{-2} + \\mathcal {O}(\\epsilon ^{-1})\\,,$ and a couple of “slow” solutions, $\\omega ^2 = \\frac{\\eta - \\beta ^2}{\\eta + \\gamma ^2} + 2\\alpha \\beta \\frac{\\eta -\\beta ^2}{(\\eta +\\gamma ^2)^2} \\, \\epsilon + \\frac{\\eta -\\beta ^2}{(\\eta +\\gamma ^2)^3} \\left(5\\alpha ^2\\beta ^2 -\\eta \\alpha ^2 - \\beta ^2 - \\gamma ^2 \\right) \\epsilon ^2 + \\mathcal {O}(\\epsilon ^3)\\,.",
"$ The “slow” and “fast” solutions are characterized by $\\omega ^2=\\mathcal {O}(\\epsilon ^0)$ and $\\omega ^2=\\mathcal {O}(\\epsilon ^{-2})$ , respectively, where the time scale is normalized by using the mass of the “light” oscillator.",
"The “slow” solutions (REF ) describe the low-energy physics of the “light” oscillator $x(t)$ under the influence of the “heavy” oscillator $y(t)$ .",
"On the other hand, the “fast” solutions describe the high-energy physics of the “heavy” oscillator $y(t)$ under the influence of the “light” oscillator $x(t)$ .",
"The “fast” solutions (REF ) are stable as long as $\\frac{\\eta +\\gamma ^2}{1-\\alpha ^2} M^2 > 0\\,.",
"$ This condition can be satisfied even for $\\eta \\le 0$ .",
"Although the general solution is a linear combination of the “fast” solutions and the “slow” solutions, $x=x_{\\rm fast}+x_{\\rm slow}\\,, \\quad y=y_{\\rm fast}+y_{\\rm slow}\\,,$ we may extract the “slow” solutions by restricting our consideration to low energy phenomena since it costs energies of order $M$ ($\\gg \\mu $ ) to excite the “fast” solution.",
"Since (REF ) is a linear system, the “fast” solutions do not affect the “slow” solutions.",
"Therefore, restricting our consideration to low energy phenomenon simply means ignoring the “fast” solutions, $x \\simeq x_{\\rm slow}\\,, \\quad y \\simeq y_{\\rm slow}\\; .$ This procedure is justified as long as the “fast” solutions do not develop instabilities, i.e.",
"the condition (REF ) is respected.",
"We would like to find a simple effective theory that describes the low-energy physics of the “light” oscillator $x(t)$ under the influence of the “heavy” oscillator $y(t)$ .",
"For this purpose, we first rewrite the Lagrangian (REF ) as $L = \\mu ^2 \\tilde{L}\\,, \\quad \\tilde{L} = \\frac{1}{2}\\left[(\\partial _{\\tau } x)^2 + \\epsilon ^2 (\\partial _{\\tau }Y)^2 + 2 \\alpha \\epsilon \\partial _{\\tau }x \\partial _{\\tau }Y\\right] - \\frac{1}{2}( x^2 + \\eta Y^2 + 2\\beta xY) - \\gamma Y\\partial _{\\tau }x\\,, \\quad Y \\equiv \\epsilon ^{-1}y\\,.$ where we have introduced the variable $Y$ of which amplitude scales as $\\mathcal {O}(\\epsilon ^0)$ when the “slow” solutions are considered.",
"Since we are interested in the “slow” dynamics, the frequency in the unit of $\\mu $ is of the order of $\\epsilon ^0$ , meaning $\\partial _{\\tau }=\\mathcal {O}(\\epsilon ^0)$ .",
"Therefore, the scaling of $\\epsilon $ is explicit in each term of the Lagrangian (REF ).",
"The Euler-Lagrange equation for $Y(t)$ is $\\epsilon ^2\\partial _{\\tau }^2Y + \\alpha \\epsilon \\partial _{\\tau }^2x + \\eta Y + \\beta x + \\gamma \\partial _{\\tau }x = 0\\,.",
"$ Considering the hierarchy (REF ), we expand $Y$ with respect to $\\epsilon $ and obtain the “slow” solution of $Y$ as $Y = - \\frac{\\beta }{\\eta } x - \\frac{\\gamma }{\\eta }\\partial _{\\tau }x - \\frac{\\alpha }{\\eta } \\epsilon \\partial _{\\tau }^2 x + \\frac{\\beta }{\\eta ^2}\\epsilon ^2\\partial _{\\tau }^2 x + \\frac{\\gamma }{\\eta ^2} \\epsilon ^2\\partial _{\\tau }^3x + \\mathcal {O}(\\epsilon ^3)\\,, $ where we have assumed that $\\eta $ is non-vanishing and of order unity.",
"By substituting this to the Lagrangian and dropping total derivative, one obtains $\\tilde{L} = \\frac{1}{2}\\left(1 + \\frac{\\gamma ^2}{\\eta } - \\frac{2\\alpha \\beta }{\\eta }\\epsilon + \\frac{\\beta ^2}{\\eta ^2}\\epsilon ^2\\right)(\\partial _{\\tau }x)^2 - \\frac{1}{2}\\left(1 - \\frac{\\beta ^2}{\\eta }\\right)x^2 + \\frac{1}{2}\\left(\\frac{\\alpha ^2}{\\eta }+\\frac{\\gamma ^2}{\\eta ^2}\\right)\\epsilon ^2(\\partial _{\\tau }^2x)^2 + \\mathcal {O}(\\epsilon ^3)\\,.",
"$ This is the low-energy effective theory describing the “slow” solutions.",
"Indeed, it is easy to show that the Euler-Lagrange equation from this effective action admits a couple of “fast” solutions and a couple of “slow” solutions.",
"As expected, while the former do not agree with the “fast” solutions (REF ) from the original Lagrangian (REF ) (since they are outside the regime of validity of the low-energy effective theory (REF )), the latter correctly reproduce (REF ) up to $\\mathcal {O}(\\epsilon ^2)$ .",
"If one wants, one can easily increase the precision of the “slow” solutions by systematically expanding $Y$ up to any order in $\\epsilon $ .",
"This procedure is justified under the stability condition (REF ), which may be satisfied even when $\\eta <0$ (negative mass square of $y$ in the absence of $x$ ), as far as the properties of the “slow” solutions are concerned."
],
[
"General procedure",
"In the rest of the present paper we shall perform essentially the same analysis for several multi-field scalar systems in order to derive a single-field effective field theory that describes the low-energy/momentum physics of a “light” degree of freedom under the influence of “heavy” degrees of freedom.",
"Before starting a concrete analysis, we find it convenient to make a general argument of the EFT reduction, extending the mechanical example in the previous subsection to field-theoretical ones.",
"An EFT is derived from a theory by integrating out modes of which dynamics we are not interested in.",
"In many situations, we are interested in low-energy/momentum dynamics, so we integrate out high-energy/momentum degrees of freedom while keeping low-energy/momentum degrees of freedom.",
"In general, the term “integrating out” refers to performing integrations of uninterested degrees of freedom in path integral.",
"In the present paper, we will restrict our attention to the case when the dynamics is well-approximated by tree-level calculations, i.e.",
"classical dynamics.",
"In this case, integrating out is performed by solving classical equations of motion for uninterested modes and then by substituting the solutions into the action where the initial conditions of the uninterested modes have to be uniquely determined by the modes which are kept in the EFT.",
"Let us denote the original action by $S_{\\rm UV}[\\chi ,\\varphi ]$ and the EFT action by $S_{\\rm IR}[\\varphi ]$ where $\\chi $ and $\\varphi $ are the modes which are to be integrated out and to be kept in the EFT reduction, respectively.",
"Note that $\\chi $ and $\\varphi $ here are collective notations and not necessarily different fields: for instance, a field $\\phi $ may be split into the high-energy/momentum, namely UV, modes $\\phi _{\\rm UV} \\in \\chi $ and the low-energy/momentum (IR) modes $\\phi _{\\rm IR} \\in \\varphi $ to derive a low-energy EFT.",
"The original equations of motion are $\\frac{\\delta S_{\\rm UV}}{\\delta \\chi (x)}=0\\,, \\quad \\frac{\\delta S_{\\rm UV}}{\\delta \\varphi (x)}=0\\,,$ while the EFT equation of motion from $S_{\\rm IR}[\\varphi ]=S_{\\rm UV}|_{\\chi =\\chi (\\varphi )}$ is $\\frac{\\delta S_{\\rm IR}}{\\delta \\varphi (x)} = \\left.",
"\\frac{\\delta S_{\\rm UV}}{\\delta \\varphi (x)} \\right|_{\\chi =\\chi [\\varphi ]} + \\int \\mathrm {d}^d y \\, \\left.",
"\\frac{\\delta \\chi [\\varphi ](y)}{\\delta \\varphi (x)} \\, \\frac{\\delta S_{\\rm UV}}{\\delta \\chi (y)} \\right|_{\\chi =\\chi (\\varphi )}\\,,$ where $\\chi =\\chi [\\varphi ]$ is the solution to the first equation in (REF ).",
"Note that we have assumed that the solution $\\chi =\\chi [\\varphi ]$ is unique at least locally so that $\\delta \\chi [\\varphi ](y)/\\delta \\varphi (x)$ is well-defined.",
"Then, the second term in (REF ) vanishes since $\\chi [\\varphi ]$ is a solution to the original equation of motion, and thus (REF ) correctly reproduces (REF ) under the solution $\\chi =\\chi [\\varphi ]$ .",
"Hence, a necessary step for the EFT reduction is just to solve the equations of motion for $\\chi $ under an appropriate initial and boundary conditions.",
"This is not an easy task, in general; thus, an EFT derivation is usually accompanied with derivative expansions.",
"The derivative expansion is essentially the same as what we did in the previous subsection as the series expansion in terms of $\\epsilon $ .",
"As we have seen in (REF ) and (REF ), the solution of $Y$ and the low-energy effective Lagrangian can be systematically obtained as a series of $\\epsilon $ .",
"The parameter $\\epsilon $ in Sec.",
"REF was defined as the ratio between $\\mu $ and $M$ , which are the only dimensional parameters characterizing the dynamics of the “light” and the “heavy” oscillators, respectively.",
"However, this definition cannot be directly applied to field theories since frequencies depend on momenta.",
"Also, there is no mass parameter when $\\varphi $ is massless.",
"Instead, one can introduce $\\epsilon \\equiv E/M$ as an expansion parameter where $E$ is a reference scale satisfying $\\partial = \\mathcal {O}(E)$ , where $\\partial $ is a temporal and/or spatial derivative.",
"We thus have $\\partial /M = \\mathcal {O}(\\epsilon )$ and then the series expansion in terms of the derivative $\\partial /M$ is nothing but the series expansion in terms of $\\epsilon $ .",
"We will give concrete examples to explain how the derivative expansion works in the EFT reduction in the following subsubsections.",
"We would like to investigate situations where the Lorentz symmetry is spontaneously broken by a background, and care is needed to accommodate this nature.",
"We for now focus on a massless field $\\varphi $ for simplicity and will generalize the argument to the massive case later.",
"Let us split the massless field $\\varphi $ into a background $\\bar{\\varphi }$ and perturbation $\\pi $ , $\\varphi =\\bar{\\varphi }+\\pi \\,.$ For a concrete discussion, we here restrict our interest to the solutions of the form $\\varphi (t,x^i) \\simeq C_0 \\Delta t + \\pi (t,x^i) \\,, \\quad \\pi (t,x^i) = \\int _{ p \\ll M} \\frac{\\mathrm {d}^d p}{(2\\pi )^d} \\pi (p) e^{ip_{\\mu }x^{\\mu }} + {\\rm h.c}\\,,\\quad \\Delta t = t-t_0 \\, ,$ where $t_0$ is some (arbitrary) reference time and $C_0$ is a constant.The constancy of $C_0$ can be an approximate one.",
"The following arguments hold as long as $\\varphi $ can be expanded in the form (REF ) locally in spacetime.",
"The background $\\bar{\\varphi }=C_0 \\Delta t$ has a non-vanishing gradient $C_{\\mu }\\equiv \\partial _{\\mu }\\bar{\\varphi }=(C_0, \\mathbf {0})$ .",
"As a result, the dynamics of $\\pi $ does not respect the Lorentz invariance due to the preferred direction $C_{\\mu }$ .",
"Let us set the mass dimension of $\\varphi $ to be $[\\varphi ]=-1$ so that the vector $C_{\\mu }$ is dimensionless.",
"Then, the amount of the symmetry breaking is of order unity if $\\partial _{\\mu }\\bar{\\varphi }=\\mathcal {O}(1)\\,,$ and the Lorentz symmetry is recovered in the limit $\\partial _{\\mu }\\bar{\\varphi } \\rightarrow 0$ .",
"The case (REF ) may appear to spoil the convergence of the derivative expansion, but this is not the case.",
"Using $p_{\\mu }=\\mathcal {O}(E)$ , we find the following scaling $\\partial ^n \\varphi = \\mathcal {O}(E^{n-1})\\,,$ that is, all the higher derivatives, $\\partial ^n$ with $n\\ge 2$ , are tiny compared with the UV physics scale, $M^{n-1} $ , provided $\\epsilon \\ll 1$ .",
"Therefore, we can still use the derivative expansion treating the second- and higher- than second-order derivatives as perturbations while keeping first order derivatives non-perturbatively.",
"We will indeed confirm that the scaling (REF ) does not prevent the usage of the derivative expansion in a concrete example.",
"When the field $\\varphi $ is massive, the background $\\varphi \\propto t$ cannot be an exact solution.",
"Nonetheless, (REF ) can be regarded as an approximate solution.",
"Let $m$ be the mass scale of the light field $\\varphi $ .",
"As long as we focus on the scales satisfying $m \\lesssim E \\ll M$ , $C_{\\mu }=\\mathcal {O}(1)$ survives and we can use (REF ) as an approximate solution during some finite time interval $\\Delta t$ with $m^{-1} \\gtrsim \\Delta t \\gg M^{-1}$ .",
"In the inflationary cosmology for instance, the timescale $m^{-1}$ is typically supposed to be the inflationary Hubble timescale or longer so that $\\Delta t$ is long enough to describe inflationary observables.",
"In general, the background can be a function of space as well.",
"All in all, (REF ) can be used as an approximate solution for the physics within temporal/spatial intervals $\\Delta t$ and $\\Delta x$ satisfying $m^{-1} \\gtrsim \\Delta t \\gg M^{-1}$ and $m^{-1} \\gtrsim \\Delta x \\gg M^{-1}$ , respectively, even with the spontaneous broken Lorentz symmetry by the background, $C_{\\mu }=\\mathcal {O}(1)$ .",
"Before closing this subsection, let us discuss a consistency condition for the EFT reduction.",
"The low-energy EFT is dedicated to studying long-time/large-scale dynamics of the original theory.",
"The EFT can make robust predictions at low-energies/momenta as far as the UV modes are stable.",
"However, when there exists an unstable UV mode, there is no proper justification to describe long-time/large-scale dynamics by ignoring short-time dynamics.",
"Such an unstable UV mode develops the instability during a short period which the low-energy EFT cannot resolve.",
"As a result, such an EFT is inconsistent, or at best UV sensitive, and cannot make robust predictions.",
"Therefore, a consistency condition for the EFT reduction is the stability condition of the modes which are integrated out.",
"We emphasize that there is no inconsistency for the IR modes to be unstable, as long as the stability of the UV modes is respected.",
"The predictions of the low-energy EFT are trustable even if the IR degree of freedom is unstable, similarly to situations like the Jeans instability.",
"The purpose of the present paper is to clarify the UV stability conditions of multi-field models and to show how such stability conditions of UV modes impose constraints on the resulting single-field EFT action especially in Lorentz-violating backgrounds."
],
[
"Example 1",
"As a concrete example of the EFT reduction, we consider a theory $S_{\\rm UV}=\\int \\mathrm {d}^4 x \\mathcal {L}_{\\rm UV}\\,, \\quad \\mathcal {L}_{\\rm UV} = -\\frac{1}{2}\\partial _{\\mu } \\chi \\, \\partial ^{\\mu } \\chi -\\frac{1}{2}\\partial _{\\mu } \\varphi \\, \\partial ^{\\mu } \\varphi -\\frac{M^2}{2}\\chi ^2-\\frac{m^2}{2}\\varphi ^2 - g M \\chi \\varphi ^2\\,,$ where $M$ and $m$ are mass parameters satisfying $m\\ll M$ and $g$ is a coupling constant.",
"Here, we canonically normalize the kinetic terms so that the mass dimension of $\\varphi $ is $[\\varphi ]=1$ .",
"Although we consider a four-dimensional flat spacetime, extensions to other dimensions and to curved spacetimes are straightforward.",
"The equation of motion for $\\chi $ is $(\\Box -M^2) \\chi = gM \\varphi ^2\\,,$ of which solution is formally expressed as $\\chi = \\frac{ gM}{\\Box -M^2 } \\varphi ^2\\,,$ provided that initial and boundary conditions are properly specified.",
"In field theories, $\\chi $ and $\\varphi $ consist of a collection of modes characterized by their momenta, and we call the ones with high momenta $\\chi _{\\rm UV}$ and $\\varphi _{\\rm UV}$ and those with low momenta $\\chi _{\\rm IR}$ and $\\varphi _{\\rm IR}$ , respectively.",
"As long as IR-UV mixing can be ignored, the UV modes do not affect the IR dynamics (decoupling theorem), and we can simply set $\\chi =\\chi _{\\rm IR}$ and $\\varphi =\\varphi _{\\rm IR}$ to analyze the low-energy behavior, similarly to (REF ) and (REF ).",
"Then, the solution of $\\chi $ is $\\chi = \\chi _{\\rm IR} = \\frac{ gM}{\\Box -M^2 } \\varphi ^2_{\\rm IR} = -\\frac{g}{M} \\sum _{n^{\\prime }=0}^{\\infty } \\left( \\frac{\\Box }{M^2} \\right)^{n^{\\prime }} \\varphi _{\\rm IR}^2\\,,$ where $\\chi $ is uniquely determined by $\\varphi _{\\rm IR}$ when the infinite series converges.",
"In this context, “IR modes” are defined as the modes that respect this convergence.",
"For later convenience, we put a prime to the index of the sum here.",
"We can truncate the series when $\\varphi _{\\rm IR}$ is so low-energy/momentum that the solution $\\chi =\\chi _{\\rm IR}$ in (REF ) can be well-approximated by the first few terms of the infinite series.",
"For instance, using the approximate solution $\\chi _{\\rm IR} \\approx -\\frac{g}{M}\\varphi _{\\rm IR}^2-\\frac{g}{M^3}\\Box \\varphi _{\\rm IR}^2$ , the EFT action up to this order is $S_{\\rm IR}=\\int \\mathrm {d}^4 x \\left[ -\\frac{1}{2}\\partial _{\\mu } \\varphi _{\\rm IR} \\partial ^{\\mu } \\varphi _{\\rm IR}-\\frac{m^2}{2} \\varphi _{\\rm IR} + g^2 \\varphi ^4_{\\rm IR} + \\frac{g^2}{M^2} \\varphi _{\\rm IR}^2 \\Box \\varphi _{\\rm IR}^2 + \\cdots \\right].$ The higher order terms can be added to improve the accuracy of EFT so long as the infinite series (REF ) converge.",
"The convergence condition is formally given by $|\\Box /M^2| < 1$ and thus $M$ is the cutoff of the derivative expansion.",
"As we have explained, the expansion in terms of $\\Box /M^2$ is essentially the same as the series expansion of $\\epsilon =E/M$ because we have $\\Box /M^2=\\mathcal {O}(\\epsilon ^2)$ ."
],
[
"Example 2",
"In the previous example, we did not need to care the scaling of $\\partial _\\mu \\varphi $ since the UV Lagrangian (REF ) only has one interaction that involves no derivatives.",
"On the other hand, we should take it into account when there are other interactions, especially interactions arising from the field-space metric.",
"We consider the UV theory containing the interactions $\\mathcal {L}_{\\rm UV} = &-\\frac{1}{2}\\partial _{\\mu } \\chi \\, \\partial ^{\\mu } \\chi -\\frac{1}{2}\\partial _{\\mu } \\varphi \\, \\partial ^{\\mu } \\varphi -\\frac{M^2}{2} \\, \\chi ^2-\\frac{m^2}{2} \\, \\varphi ^2 - g M \\chi \\varphi ^2\\nonumber \\\\ &-h \\, \\partial _{\\mu } \\varphi \\, \\partial ^{\\mu } \\chi - h_{\\varphi } \\, \\varphi \\, \\partial _{\\mu } \\varphi \\, \\partial ^{\\mu } \\chi - \\frac{f_{\\chi }}{2} \\, \\chi \\, \\partial _{\\mu } \\varphi \\, \\partial ^{\\mu } \\varphi \\,,$ where the first line is the same as (REF ), and the second line is added with $h,h_{\\varphi }$ and $f_{\\chi }$ taken to be constants.",
"Variation with respect to $\\chi $ leads to $(\\Box -M^2)\\chi +h\\Box \\varphi + \\frac{1}{2}(2h_{\\varphi }-f_{\\chi }) \\partial _{\\mu }\\varphi \\partial ^{\\mu } \\varphi + h_{\\varphi } \\varphi \\Box \\varphi -gM\\varphi ^2=0,$ which we now solve via the derivative expansion.",
"Note that we only consider IR solutions and suppress the subscripts, IR and UV, hereinafter.",
"The solution can be found as $\\chi &=\\frac{1}{M^2}\\sum _{n^{\\prime }=0}^{\\infty } \\left( \\frac{\\Box }{M^2} \\right)^{n^{\\prime }} \\left[ h\\Box \\varphi + \\frac{1}{2}(2h_{\\varphi }-f_{\\chi }) \\partial _{\\mu }\\varphi \\partial ^{\\mu } \\varphi + h_{\\varphi } \\varphi \\Box \\varphi \\right]-\\frac{g}{M} \\sum _{n^{\\prime }=0}^{\\infty } \\left( \\frac{\\Box }{M^2} \\right)^{n^{\\prime }} \\varphi ^2\\nonumber \\\\&= \\frac{2h_{\\varphi }-f_{\\chi }}{2M^2} \\, \\partial _{\\mu }\\varphi \\partial ^{\\mu } \\varphi +\\frac{1}{M^2}(h \\Box \\varphi + h_{\\varphi } \\varphi \\Box \\varphi ) + \\frac{2h_{\\varphi }-f_{\\chi }}{2M^4} \\Box (\\partial _{\\mu }\\varphi \\partial ^{\\mu } \\varphi )-\\frac{g}{M}\\,\\varphi ^2-\\frac{g}{M^3}\\,\\Box \\varphi ^2+\\cdots \\,.$ If we simply counted the number of derivatives, we would need to equally treat the interactions, the second line of (REF ), in the derivative expansion, as all the terms inside the square brackets in the first line of (REF ) have two derivatives.",
"However, this is not what we are interested in; rather, we aim to investigate the cases where the Lorentz symmetry is significantly violated by the gradient of $\\varphi $ as in (REF ).",
"As one can see from (REF ), the convergence condition of the derivative expansion requires that higher derivatives of $\\varphi $ , $\\partial ^n$ with $n\\ge 2$ , must be tiny compared with the mass scale $M$ while it does not require the smallness of the first derivatives.",
"The maximal amount of the Lorentz symmetry breaking that is allowed in the EFT below the large mass scale $M$ can be characterized by $\\partial _{\\mu }\\varphi =\\mathcal {O}(M^2)\\,.$ with the scaling $\\partial ^n \\varphi = \\mathcal {O}(M^2 E^{n-1})\\,.$ Note that (REF ) leads to the variation $\\Delta \\varphi =\\mathcal {O}(M^2 E^{-1}) \\gg M$ in the interval $\\Delta t \\sim E^{-1}$ , implying that, in order to achieve this scaling compatible with the equations of motion, an approximate shift symmetry of $\\varphi $ should be introduced.",
"To this end, let us suppose the hierarchy of the couplings: $|f_{\\chi }|=\\mathcal {O}(M^{-1})\\,, \\quad |h|=\\mathcal {O}(1)\\,, \\quad |h_{\\varphi }| = \\mathcal {O}(m M^{-2})\\,, \\quad |g| = \\mathcal {O}(m^2 M^{-2})\\,,$ where the shift symmetry becomes exact in the limit $m\\rightarrow 0$ .",
"As a result, the solution $\\chi $ is organized into $\\chi &=\\sum _{n=0}^{\\infty } \\chi _n =\\chi _0+\\chi _1+\\chi _2+\\cdots \\,,\\\\\\chi _0&=-\\frac{f_{\\chi }}{2M^2} \\, \\partial _{\\mu }\\varphi \\partial ^{\\mu } \\varphi -\\frac{g}{M} \\, \\varphi ^2\\,, \\\\\\chi _1&=\\frac{h}{M^2}\\,\\Box \\varphi + \\frac{h_{\\varphi }}{M^2}(\\varphi \\Box \\varphi + \\partial _{\\mu }\\varphi \\partial ^{\\mu } \\varphi )\\,, \\\\\\chi _2&=-\\frac{f_{\\chi }}{2M^4}\\,\\Box (\\partial _{\\mu }\\varphi \\partial ^{\\mu } \\varphi ) - \\frac{g}{M^3}\\,\\Box \\varphi ^2\\,,$ with $\\chi _n=\\mathcal {O}(\\epsilon ^n M)\\,,$ where $\\chi _0$ and $\\chi _1$ are originated from $n^{\\prime }=0$ terms of (REF ) and $\\chi _n~(n \\ge 2)$ are from $n^{\\prime }\\ge 1$ terms.",
"The subscript $n$ denotes the order of the solution where the scaling (REF ) is taken into account.The equation of motion (REF ) may be solved by using the derivative expansion even if we do not assume the approximate shift symmetry; however, the series (REF ) is not a systematic series in this case.",
"For instance, we have $\\frac{g}{M^3}\\Box \\varphi ^2 = g \\times \\mathcal {O}(M)$ which is comparable to the term $\\frac{f_{\\chi }}{M^2}(\\partial _{\\mu }\\varphi \\partial ^{\\mu }\\varphi )=f_{\\chi } \\times \\mathcal {O}(M^2)$ if $|f_{\\chi }|=\\mathcal {O}(M^{-1}),~ |g|=\\mathcal {O}(1)$ and (REF ) are assumed.",
"Since the solution of $\\chi $ is found as a series of $\\epsilon $ , the effective Lagrangian is easily computed accordingly."
],
[
"Bookkeeping parameter $\\tilde{\\epsilon }$",
"In this subsubsection, we introduce a simple, systematic way to perform truncation of the series expansion that is employed in the previous subsubsections.",
"It practically deduce the same effective description but can make it more transparent to solve the equations of motion via the derivative expansion.",
"We again consider the example described by the action (REF ).",
"Let us replace $\\partial \\chi $ with $\\tilde{\\epsilon } \\partial \\chi $ to rewrite the action as $S_{\\rm UV}=\\int d^4 x \\mathcal {L}_{\\rm UV}\\,, \\quad \\mathcal {L}_{\\rm UV} = -\\frac{1}{2}\\,\\tilde{\\epsilon }^2\\partial _{\\mu } \\chi \\partial ^{\\mu } \\chi -\\frac{1}{2}\\,\\partial _{\\mu } \\varphi \\partial ^{\\mu } \\varphi -\\frac{M^2}{2}\\,\\chi ^2-\\frac{m^2}{2}\\,\\varphi ^2 - g M \\chi \\varphi ^2\\,.$ This action shares the same concept with (REF ).",
"The equation of motion of $\\chi $ is $(\\tilde{\\epsilon }^2 \\Box -M^2) \\chi = gM \\varphi ^2\\,.$ We solve this equation of motion order by order by using the ansatz $\\chi =\\sum _{n=0}^{\\infty } \\tilde{\\epsilon }^n \\chi _n = \\chi _0+\\tilde{\\epsilon }\\chi _1+\\tilde{\\epsilon }^2 \\chi _2 +\\cdots \\,.$ We can easily find the solution $\\chi _0=-\\frac{g}{M}\\varphi ^2\\,,\\quad \\chi _1=0\\,, \\quad \\chi _2=-\\frac{g}{M} \\frac{\\Box }{M^2}\\varphi ^2 \\,, \\cdots \\; .$ Plugging these back into the action, we obtain the IR action $S_{\\rm IR}=\\int d^4 x \\left[ -\\frac{1}{2}\\partial _{\\mu } \\varphi \\partial ^{\\mu } \\varphi -\\frac{m^2}{2}\\varphi ^2 + g^2 \\varphi ^4 + \\tilde{\\epsilon }^2 \\frac{g^2}{M^2} \\varphi ^2 \\Box \\varphi ^2 + \\mathcal {O}(\\tilde{\\epsilon }^4) \\right],$ which recovers the result (REF ) by setting $\\tilde{\\epsilon }=1$ .",
"In this prescription, $\\tilde{\\epsilon }$ denotes the order of the derivative expansion, and the expansion parameter should be understood as $\\tilde{\\epsilon }\\partial /M = \\mathcal {O}(\\epsilon )$ rather than $\\tilde{\\epsilon }$ itself.",
"The same procedure can be applied into general UV Lagrangians, e.g.",
"the theory including the interaction (REF ).",
"We emphasize that the replacement rule $\\partial \\chi \\rightarrow \\tilde{\\epsilon } \\partial \\chi $ should be applied when the Lagrangian contains at most first-order derivatives of the fields.",
"We consider the Lagrangian $\\mathcal {L}_{\\rm UV}=-\\frac{1}{2}\\tilde{\\epsilon }^2\\partial _{\\mu } \\chi \\partial ^{\\mu } \\chi - \\tilde{\\epsilon } (h+h_{\\varphi } \\varphi ) \\partial _{\\mu } \\varphi \\partial ^{\\mu } \\chi - \\frac{1}{2}(1+f_{\\chi }\\chi ) \\partial _{\\mu } \\varphi \\partial ^{\\mu } \\varphi -\\frac{M^2}{2}\\chi ^2-\\frac{m}{2}\\varphi ^2- g M \\chi \\varphi ^2\\,,$ and find the solution order by order as $\\chi =\\sum _{n=0}^{\\infty }\\tilde{\\epsilon }^n \\chi _n=\\chi _0+ \\tilde{\\epsilon }\\chi _1 + \\tilde{\\epsilon }^2 \\chi _2 +\\cdots \\,.$ One can easily confirm that the solutions $\\chi _i$ agree with eqs.",
"()-() at each order.",
"The original result is recovered by setting $\\tilde{\\epsilon }=1$ at the end of calculations.",
"We treat the operators $\\partial _{\\mu } \\chi \\partial ^{\\mu } \\chi $ and $ \\partial _{\\mu } \\varphi \\partial ^{\\mu } \\chi $ as perturbations while we keep non-linearity of $\\partial _{\\mu } \\varphi \\partial ^{\\mu } \\varphi $ because we are interested in the Lorentz-violating background with a sufficiently large gradient $\\partial _{\\mu } \\varphi \\lesssim \\mathcal {O}(M^2)$ .",
"The order of the bookkeeping parameter $\\tilde{\\epsilon }$ represents the order of the derivative expansion with taking the scaling (REF ) into account.",
"Needless to say, the scaling of $E$ is important for the derivative expansion while the overall normalization is irrelevant.",
"Although we have used the canonical normalization with $[\\varphi ]=1$ to follow the convention in the above examples, it is straightforward to conduct the same analysis with other normalizations of $\\varphi $ ."
],
[
"Setup",
"Our aim is to deduce the consistency conditions in deriving the low-energy single-field effective field theory (EFT) from classes of UV models.",
"The gist of the EFT reduction procedure has been demonstrated in Sec.",
", and we now proceed to more general setups.",
"A key ingredient in our study is a kinetic coupling from a field space in UV.",
"We consider a general field-space metric $\\gamma _{AB}(\\Phi ) \\, \\mathrm {d}\\Phi ^A \\mathrm {d}\\Phi ^B \\; ,$ where the upper-case Latin alphabets $A,B, \\dots $ denote the field indices, and the metric $\\gamma _{AB}$ is a function of the fields $\\Phi ^A$ .",
"The field space metric is supposed to be positive definite not to have ghost states.",
"We can define the covariant derivative, the curvature tensors, and the scalar associated with $\\gamma _{AB}$ in the same manner as for the spacetime metric and characterize the geometrical structure of the field space in terms of their properties.",
"Nevertheless, our UV consistency conditions can and will be derived without specifying the structure of the field space explicitly.",
"The works [47], [43] have shown that the only two classes of kinetic terms that are free from the formation of caustics singularity in a planar configuration in the Minkowski spacetime are linear and DBI kinetic terms.",
"Hence, the DBI-type kinetic term can be also regarded as a partial UV model of the single-field EFT, which we will investigate in Appendix .",
"In this section, however, we focus on a class of models that have linear kinetic terms with the curved field space metric (REF ), characterized by the action of a non-linear sigma model $S_{\\rm UV} = \\int d^dx\\sqrt{-g}\\,\\mathcal {L}_{\\rm UV}\\,, \\quad \\mathcal {L}_{\\rm UV}=-\\frac{1}{2} \\, \\gamma _{AB}(\\Phi ) \\, g^{\\mu \\nu }\\partial _\\mu \\Phi ^A \\partial _\\nu \\Phi ^B - V(\\Phi )\\,,$ as a (partial) UV completion of a single-field EFT.",
"Here the Greek alphabets $\\mu , \\nu , \\dots $ denote the spacetime indices, $g^{\\mu \\nu }$ and $g$ are the inverse and determinant of the spacetime metric $g_{\\mu \\nu }$ .",
"As mentioned in Introduction, we consider the theory in the Einstein frame, and the Einstein-Hilbert action is implicit in this section since the Einstein-Hilbert action is irrelevant to the EFT reduction here.",
"In this section, we perform the EFT reduction by using the prescription introduced in Sec.",
"REF without assuming any explicit form of either the field space metric $\\gamma _{AB}(\\Phi )$ or the potential $V(\\Phi )$ .",
"In order to integrate out $N-1$ heavy fields to reduce the non-linear sigma model to a single-field EFT of one light degree of freedom, our only assumption is a large hierarchy between the “mass” of the lightest field $\\varphi $ and those of other heavy fields $\\chi ^a$ , namely the approximate shift symmetry to the direction of $\\varphi $ , where the index $a$ runs through the $(N-1)$ -dimensional subspace that excludes the $\\varphi $ direction.",
"Note that, due to the presence of the non-linear interactions through $\\gamma _{AB}$ , the “mass” is not simply the second derivatives of $V$ around a trivial background, and moreover, the notion of invariant mass is no longer available around a Lorentz-violating background in general.",
"Nonetheless, this does not prevent a self-consistent EFT reduction, which we perform below.",
"Let $m$ be the mass scale of the lightest field $\\varphi $ .",
"We suppose that the theory enjoys the exact shift symmetry $\\varphi \\rightarrow \\varphi + c$ in the limit $m\\rightarrow 0$ , similarly to the scaling (REF ) discussed in Sec.",
"REF .",
"More precisely, the approximate shift symmetry is introduced as follows.",
"By setting the mass dimension of $\\varphi $ to be $-1$ so that $\\partial _{\\mu }\\varphi $ is dimensionless, we would like to study situations where the Lorentz symmetry is spontaneously broken by the background $\\partial _{\\mu }\\varphi = \\mathcal {O}(1)$ .",
"We assume that $\\varphi $ respects an approximate shift symmetry, which is reflected by the $\\varphi $ -dependence of the potential and the field space metric that is suppressed by the small mass scale $m$ , compared to other dimensional quantities that are normalized by the scale of the heavy physic, $M$ .",
"In this case, the change of the Lagrangian under the change $\\Delta \\varphi = \\mathcal {O}(\\Delta t)$ (or $\\Delta \\varphi = \\mathcal {O}(\\Delta x)$ ) is at most of order unity for the intervals $\\Delta t, \\Delta x=\\mathcal {O}(E^{-1})\\lesssim \\mathcal {O}(m^{-1})$ , and, in particular, the change is negligible during the period $\\Delta t, \\Delta x\\ll \\mathcal {O}(m^{-1})$ .",
"Hence, in this case the effect due to he violation of the shift symmetry would not be significant within the temporal/spatial intervals $\\Delta t = \\mathcal {O}(E^{-1})$ and $\\Delta x = \\mathcal {O}(E^{-1})$ that we are interested in, with $m\\lesssim E \\ll M$ , i.e.",
"one can safely assume that the shift symmetry holds approximately.",
"If preferred, one can canonically normalize $\\varphi $ (or use other normalizations) after introducing the shift symmetry in this way.",
"Note that the approximate shift symmetry does not need to exist globally in the field space.",
"We only assume that the theory enjoys the approximate shift symmetry at least during the intervals $\\Delta t, \\Delta x=\\mathcal {O}(E^{-1})\\lesssim \\mathcal {O}(m^{-1})$ and consider the effective description at the scale $m \\lesssim E \\ll M$ during this phase.",
"This observation concludes that the resultant EFT does not necessarily have a Lorentz-invariant vacuum in the regime of validity; some EFTs are well-defined only in the Lorentz-violating phase, $\\partial _{\\mu }\\varphi = \\mathcal {O}(1)$ (see Appendix for a concrete example).",
"We also note that the regime of our consideration, $m \\lesssim E$ , is demanded due to our interest in the physics around the scale $E^{-1}$ for which $\\varphi $ is well approximated by the solution (REF ) with $C_0 = \\mathcal {O}(1)$ .",
"Of course our EFT itself can be also used to describe the low-energy physics $E\\ll m$ if $\\partial _{\\mu }\\varphi \\ll 1 $ is well-defined for the intervals $\\Delta t = \\mathcal {O}(E^{-1})$ and $\\Delta x = \\mathcal {O}(E^{-1})$ .",
"Our formulation in this section applies at fully nonlinear orders, without relying on explicit configurations of the fields as long as the derivative expansion converges, i.e.",
"$E\\ll M$ .",
"We do not need to explicitly split the field into the background and perturbations since the analysis is nonlinear.",
"This section is dedicated to providing the general relations between the UV action and the EFT action within our setup."
],
[
"Two-field UV models",
"We first concentrate on two-field models $\\Phi ^A=\\lbrace \\chi ,\\varphi \\rbrace $ with $N=2$ and will discuss the generic multi-field models with an arbitrary $N$ in Sec.",
"REF .",
"The components of the field space metric are $\\gamma _{AB} \\, \\mathrm {d}\\Phi ^A \\mathrm {d}\\Phi ^B= \\gamma (\\chi ,\\varphi ) \\, \\mathrm {d}\\chi ^2 + 2 h(\\chi ,\\varphi ) \\, \\mathrm {d}\\chi \\mathrm {d}\\varphi + f(\\chi ,\\varphi ) \\, \\mathrm {d}\\varphi ^2\\,,$ where $\\gamma ,h$ and $f$ are in general functions of $\\chi $ and $\\varphi $ .",
"As we explained in Sec.",
"REF , the equation of motion for $\\chi $ can be systematically solved by using the derivative expansion when the parameter $\\tilde{\\epsilon }$ is introduced.",
"We first write the UV action as $S_{\\rm UV}=\\int d^dx \\sqrt{-g}\\left[ -\\frac{1}{2}\\tilde{\\epsilon }^2 \\gamma (\\chi ,\\varphi ) (\\nabla \\chi )^2 - \\tilde{\\epsilon } h(\\chi ,\\varphi ) \\nabla _{\\mu } \\chi \\nabla ^{\\mu } \\varphi + f(\\chi ,\\varphi ) X-V\\right],$ where $X\\equiv -(\\nabla \\varphi )^2/2$ .",
"The equation of motion for $\\chi $ is $\\tilde{\\epsilon }^2 \\left( \\gamma \\Box \\chi + \\frac{1}{2} \\gamma _{\\chi } (\\nabla \\chi )^2 +\\gamma _{\\varphi } \\nabla _{\\mu }\\chi \\nabla ^{\\mu }\\varphi \\right)+\\tilde{\\epsilon } \\left(h\\Box \\varphi -2X h_{\\varphi } \\right) +Xf_{\\chi }-V_{\\chi }=0\\,,$ where the subscripts $\\chi $ and $\\varphi $ represent derivatives with respect to the specified variables.",
"We can then find a solution for the heavy field $\\chi $ order by order as a series in terms of $\\tilde{\\epsilon }$ as $\\chi = \\chi [\\varphi ] = \\sum _{n=0}^{\\infty }\\tilde{\\epsilon }^n\\chi _n[\\varphi ]\\,.",
"$ For the expansion (REF ), the equation of motion (REF ) reads $f_{\\chi }(\\chi _0,\\varphi ) X-V_{\\chi }(\\chi _0, \\varphi ) &=0\\,,&{\\rm at}~\\mathcal {O}(\\tilde{\\epsilon }^0)\\,, \\\\h(\\chi _0,\\varphi ) \\Box \\varphi -2h_{\\varphi }(\\chi _0,\\varphi ) X &=M^2(\\chi _0, \\varphi ) \\chi _1 \\,,&{\\rm at}~\\mathcal {O}(\\tilde{\\epsilon }^1)\\,, \\\\\\gamma (\\chi _0,\\varphi ) \\Box \\chi _0 + \\frac{1}{2} \\gamma _{\\chi }(\\chi _0,\\varphi ) (\\nabla \\chi _0 )^2 +\\gamma _{\\varphi }(\\chi _0,\\varphi ) \\nabla _{\\mu }\\chi _0 \\nabla ^{\\mu }\\varphi & \\nonumber \\\\+\\chi _1 \\left( h_{\\chi }(\\chi _0,\\varphi ) \\Box \\varphi -2X h_{\\varphi \\chi }(\\chi _0,\\varphi ) \\right)-\\frac{1}{2}\\chi _1^2 \\left( V_{\\chi \\chi \\chi }(\\chi _0,\\varphi ) - X f_{\\chi \\chi \\chi }(\\chi _0,\\varphi ) \\right)&=M^2(\\chi _0,\\varphi ) \\chi _2 \\,,&{\\rm at}~\\mathcal {O}(\\tilde{\\epsilon }^2)\\,,$ and so on.",
"Here, we define the function $M^2& \\equiv V_{\\chi \\chi }-X f_{\\chi \\chi }\\,,$ which corresponds to a squared mass scale related to the cutoff of the derivative expansion.",
"In fact, as in () and (), the equation of motion at $\\mathcal {O}(\\tilde{\\epsilon }^n)$ ($n=1,2,\\cdots $ ) generically takes the form, $\\mathcal {F}_n[\\varphi ; \\chi _0, \\cdots , \\chi _{n-1}]=M^2 \\chi _n\\,,$ because $\\chi _n$ appears only from the last two terms of (REF ).",
"Here $\\mathcal {F}_n$ are those functionals of $\\varphi $ and lower orders of $\\chi $ which correspond to the $n$ -th order of equation of motion for $\\chi $ .",
"The leading-order equation of motion (REF ) is a “constraint” equation of $\\chi _0$ .",
"The implicit function theorem guarantees that at least locally there exists a function $\\chi _0=\\chi _0(\\varphi ,X)$ that satisfies the leading-order equation of motion (REF ) if $M^2\\ne 0$ .",
"Using this leading-order solution, the solutions higher-order in $\\tilde{\\epsilon }$ are uniquely determined as long as $M^2\\ne 0$ ; schematically, $\\chi _0=\\chi _0[\\varphi ]\\,, \\quad \\chi _1=\\chi _1[\\varphi ,\\chi _0[\\varphi ] ]=\\chi _1[\\varphi ]\\,, \\quad \\chi _2=\\chi _2[\\varphi ,\\chi _0[\\varphi ],\\chi _1[\\varphi ] ]=\\chi _2[\\varphi ]\\,, ~\\cdots \\,.$ This solution describes the response of the heavy field $\\chi $ to the low-energy physics of the light field $\\varphi $ and thus does not allow for independent initial conditions for $\\chi $ and $\\dot{\\chi }$ on the initial Cauchy hypersurface.",
"It takes high energies or/and momenta of order $M$ for $\\chi $ to deviate from this particular solution.",
"Since we are interested in physics at low energies and momenta sufficiently below $M$ , we employ the solution (REF ) with (REF ).",
"The effective Lagrangian for $\\varphi $ is then obtained by substituting the solution $\\chi =\\chi [\\varphi ]$ given by the expansion (REF ) with (REF ) into the Lagrangian (REF ).",
"The solution up to $\\mathcal {O}(\\tilde{\\epsilon }^1)$ is needed to obtain the effective Lagrangian up to $\\mathcal {O}(\\tilde{\\epsilon }^2)$ .",
"The EFT Lagrangian is $\\mathcal {L}_{\\rm IR}&=fX-V- \\tilde{\\epsilon } h \\nabla _{\\mu }\\chi _0 \\nabla ^{\\mu }\\varphi \\nonumber \\\\&+\\tilde{\\epsilon }^2\\left(-\\frac{1}{2}\\gamma (\\nabla \\chi _0)^2 + \\frac{h^2}{2M^2}(\\Box \\varphi )^2-\\frac{2h h_{\\varphi }}{M^2} X \\Box \\varphi +\\frac{2h_{\\varphi }^2 }{M^2}X^2 \\right)+\\mathcal {O}(\\tilde{\\epsilon }^3)\\,,$ after integration by parts and using the equations (REF ) and (), where $\\chi _0=\\chi _0(\\varphi ,X)$ is understood as the solution to (REF ).",
"Here the arguments of each function are evaluated at $\\chi = \\chi _0(\\varphi ,X)$ .",
"One can confirm that the effective Lagrangian (REF ) correctly reproduces the original equation of motion for $\\varphi $ under the solution (REF ) up to $\\mathcal {O}(\\tilde{\\epsilon }^2)$ .The agreement is confirmed by using e.g.",
"the Mathematica package xTras [48].",
"Also, if needed, one can systematically increase the accuracy of EFT by using a higher-order solution.",
"The k-essence theory is obtained as the leading-order EFT of (REF ), $\\mathcal {L}_{\\rm IR} &\\, =P(\\varphi ,X)+\\mathcal {O}(\\tilde{\\epsilon }) \\,,$ where $P(\\varphi ,X)& \\equiv f(\\chi _0,\\varphi ) X-V(\\chi _0,\\varphi ) \\, \\big \\vert _{\\chi _0 = \\chi _0(\\varphi ,X)}\\,.$ We compute the first and second derivatives of $P$ with respect to $X$ by using the chain rule and the implicit function theorem: $P_X&=\\frac{\\partial P}{\\partial X}=\\left.",
"f+(f_{\\chi }X-V_\\chi )\\frac{\\partial \\chi _0}{\\partial X} \\right|_{\\chi _0 = \\chi _0(\\varphi ,X)}= f \\, \\big \\vert _{\\chi _0 = \\chi _0(\\varphi ,X)}\\,, \\\\P_{XX}&=\\frac{\\partial ^2 P}{\\partial X^2}=\\left.",
"f_{\\chi } \\frac{\\partial \\chi _0}{\\partial X} \\right|_{\\chi _0 = \\chi _0(\\varphi ,X)} = \\left.",
"\\frac{f_{\\chi }^2}{M^2} \\right|_{\\chi _0 = \\chi _0(\\varphi ,X)}\\,, $ and other derivatives are also computed accordingly.",
"Note that, as was shown in [43], the EFT derivation at the leading order in $\\tilde{\\epsilon }$ is the Legendre transformation especially in the shift symmetric case and then the relations (REF ) and () are simply the properties of the Legendre transformation.",
"We have not specified the sign of $M^2$ so far.",
"The necessary condition for the existence of the solution $\\chi $ required $M^2\\ne 0$ while it said nothing about the sign of $M^2$ .",
"On the other hand, the positivity bounds conclude $P_{XX}>0$ around a Lorentz-invariant background.",
"In order to see how this constraint arises in the present setup restricted to a Lorentz-invariant background, let us consider the Lorentz-invariant background, $\\chi , \\varphi =$ constant, where the background values of $\\chi ,\\varphi $ are determined by $V_{\\chi }=0$ and $V_{\\varphi }=0$ .",
"One can easily find that the no-tachyon condition of $\\chi $ requires $V_{\\chi \\chi }|_{\\chi , \\varphi = {\\rm constant} }>0\\,,$ as a necessary condition around the Lorentz-invariant background.",
"Since the background $\\varphi $ is constant, we have $M^2=V_{\\chi \\chi }>0$ evaluated at this background, concluding $P_{XX}(X=0)>0$ as a consistency condition for the Lorentz-invariant EFT reduction.",
"The stability conditions around generic Lorentz-violating backgrounds are not straightforwardly obtained, on the other hand.",
"As we have mentioned, the “mass” of the field is not simply evaluated by the second derivative of the potential.",
"Furthermore, the friction term may allow to have a stable UV state even for the convex shape of the potential as we have seen in Sec.",
"REF .",
"This issue in fact contains a rich content, and we thus leave the analysis on the stability conditions around Lorentz-violating backgrounds to Sec.",
", while in this section we only state that $M^2>0$ is not an immediate consequence of the UV stability if the Lorentz symmetry is spontaneously broken.",
"Before moving to the next subsection, let us discuss a freedom of field redefinitions.",
"The description of the theory is not unique and the EFT operators can be changed via field redefinitions.",
"We focus on the shift symmetric theory and consider transformations which preserve the shift symmetry manifestly.",
"First of all, transformations of $\\chi $ are irrelevant for the EFT because the field $\\chi $ is integrated out.",
"A field transformation that preserves the shift symmetry is the change according to $\\varphi \\rightarrow \\varphi + \\tilde{\\epsilon } g(\\chi )\\,,$ where $g$ is an arbitrary function of $\\chi $ .",
"The change must be of the order of $\\tilde{\\epsilon }$ because this transformation leads to $\\mathrm {d}\\varphi \\rightarrow \\mathrm {d}\\varphi + \\tilde{\\epsilon } g_{\\chi }(\\chi ) \\mathrm {d}\\chi \\; ,$ and $g_{\\chi }$ contributes to $\\tilde{\\epsilon }^2 \\mathrm {d}\\chi ^2$ and $\\tilde{\\epsilon } \\mathrm {d}\\chi \\mathrm {d}\\varphi $ .",
"Therefore, the field redefinition (REF ) does not contribute to the leading operator $P(X)$ and only changes the subleading operators.",
"One can also consider a perturbative field redefinition including derivatives after (or before) the EFT is derived, say $\\varphi \\rightarrow \\varphi ^{\\prime }=\\varphi - \\tilde{\\epsilon }^2 \\, \\frac{\\Box \\varphi }{M^2}\\,,$ which generates the $(\\Box \\varphi )^2$ operator from the leading operator $P(X)$ , $P \\rightarrow P + \\tilde{\\epsilon }^2 \\frac{P_X}{M^2} \\nabla _{\\mu }\\varphi \\nabla ^{\\mu } \\Box \\varphi +\\cdots = P - \\tilde{\\epsilon }^2 \\frac{P_X}{M^2} (\\Box \\varphi )^2 +\\cdots $ where $\\cdots $ are terms irrelevant for our consideration here, and we take integration by parts to obtain the last expression.",
"The inverse of the transformation is given as a series in $\\tilde{\\epsilon }$ .",
"We can change the coefficient of $(\\Box \\varphi )^2$ in (REF ) by using this transformation if $P_X\\ne 0$ , for instance.",
"Nonetheless, the derivative terms are higher orders in $\\tilde{\\epsilon }$ and change the subleading operators only.",
"All in all, the field redefinitions can change the subleading operators whereas the leading operators described by $P(X)$ are invariant.",
"We can use the freedom of the field redefinitions to discuss the UV consistency conditions in Sec.",
"when the bounds on the shift symmetric parts of the leading operators $P(X)$ are concerned.On the other hand, non-shift symmetric parts of $P(\\varphi ,X)$ can be changed by field redefinitions.",
"For instance, one can consider the transformation according to $\\varphi \\rightarrow \\varphi +m\\varphi ^2/M^2$ ."
],
[
"General multi-field UV models",
"In this subsection, we continue to perform the EFT reduction by extending the two-field model to a generic multi-field model with a field space geometry of any (finite) dimension $N$ .",
"We introduce a mass scale $m$ as a controlling parameter so that the theory enjoys the exact shift symmetry in the limit $m\\rightarrow 0$ .",
"In this limit, the existence of the exact shift symmetry, $\\varphi \\rightarrow \\varphi + c\\,,$ implies that the field space metric admits a Killing vector $\\xi = \\frac{\\partial }{\\partial \\varphi } \\equiv \\partial _\\varphi \\; ,$ or $\\xi ^A = \\delta ^A_\\varphi $ in the component notation, and that the potential is independent of $\\varphi $ .",
"Using $\\varphi $ as a coordinate This way, $\\varphi $ is prefixed to become the actual dynamical degree of freedom in the corresponding low-energy EFT.",
"Our setup is in this sense modified and/or illustrative as compared to other scenarios that share some common philosophy, such as the completely generic multi-field setup [49], the gelaton scenario [50], field space with sharp turns [51], [52], and the geometrical destabilization [53]., $\\Phi ^N=\\varphi $ , and denoting other coordinates by $\\Phi ^a=\\chi ^a$ ($a=1,\\cdots ,N-1$ ), so that $\\Phi ^A=(\\chi ^a, \\varphi )$ , the general shift-symmetric field-space metric is written in the form $\\gamma _{AB}\\mathrm {d}\\Phi ^A \\mathrm {d}\\Phi ^B=\\gamma _{ab}(\\chi ) \\mathrm {d}\\chi ^a \\mathrm {d}\\chi ^b + 2 h_a(\\chi ) \\mathrm {d}\\chi ^a \\mathrm {d}\\varphi + f(\\chi ) \\mathrm {d}\\varphi ^2\\,.$ According to Frobenius's theorem, the Killing vector is hypersurface orthogonal if and only if $\\xi _{[A} \\mathcal {D}_B \\xi _{C]}=0 \\,,$ is satisfied where $\\mathcal {D}_A$ is the covariant derivative compatible with the field space metric $\\gamma _{AB}$ (see e.g. [54]).",
"When (REF ) holds, the variable $\\varphi $ can be chosen to respect the reflection symmetry, $\\varphi \\rightarrow -\\varphi \\,,$ and we can set $h_{a}(\\chi )=0$ without loss of generality.",
"For $N=2$ , the existence of the Killing vector (the shift symmetry) immediately concludes the hypersurface orthogonality (REF ) while it does not necessarily hold for general $N$ .",
"Note, however, that the leading shift-symmetric operators $P(X)$ enjoys the accidental reflection symmetry whether or not the full theory does.",
"The field space metric and the potential can depend on $\\varphi $ in the way that the $\\varphi $ -dependence is suppressed by the small mass scale $m$ associated with $\\varphi $ , when the shift symmetric is not exact.",
"We now recover a small but non-vanishing $m$ .",
"We keep generality and do not add any extra ingredient such as the reflection and exact shift symmetries.",
"We only assume that the non-linear sigma model has one light direction described by the approximate shift symmetry at least locally.",
"The general Lagrangian is given by $\\mathcal {L}_{\\rm UV}=-\\frac{1}{2}\\tilde{\\epsilon }^2 \\gamma _{ab}(\\chi ,\\varphi ) \\nabla _{\\mu } \\chi ^a \\nabla ^{\\mu } \\chi ^b - \\tilde{\\epsilon } h_a(\\chi ,\\varphi ) \\nabla _{\\mu } \\chi ^a \\nabla ^{\\mu } \\varphi + f(\\chi ,\\varphi ) X- V(\\chi ,\\varphi )\\,,$ where the parameter $\\tilde{\\epsilon }$ is introduced.",
"The solutions for the heavy fields $\\chi ^a$ can be found as a series in terms of $\\tilde{\\epsilon }$ : $\\chi ^a=\\sum _{n=0}^{\\infty }\\tilde{\\epsilon }^n \\chi ^a_n[\\varphi ]$ where the field space indices $a,b,\\cdots $ and the order of the $\\tilde{\\epsilon }$ expansion, $n$ , should not be confused.",
"The zeroth-order equation of motion reads $\\partial _a f(\\chi _0,\\varphi ) X-\\partial _a V(\\chi _0,\\varphi )=0\\,,$ where $\\partial _a$ is the derivative with respect to the heavy fields $\\chi ^a$ .",
"Provided that the matrix $M^2_{ab}\\equiv \\partial _a \\partial _b V - \\partial _a \\partial _b f \\, X\\,,$ has a non-zero determinant, we can solve (REF ) for $\\chi _0^a$ in favor of $\\varphi $ and $X$ according to the implicit function theorem.",
"The equation of motion at $\\mathcal {O}( \\tilde{\\epsilon }^n )$ with $n\\ge 1$ takes the form, with $\\mathcal {F}_{a,n}$ abbreviating the $n$ -th order equation of motion for $\\chi $ , $\\mathcal {F}_{a,n}[\\varphi ; \\chi _0,\\cdots ,\\chi _{n-1}] = M^2_{ab}\\chi ^b_n\\,,$ implying that the solution at $\\mathcal {O}(\\tilde{\\epsilon }^n)$ is uniquely determined by the solutions at $\\mathcal {O}(\\tilde{\\epsilon }^m)$ with $m<n$ as far as ${\\rm det}M^2_{ab}\\ne 0$ is respected.",
"Solving the equations of motion recurrently, all $n$ -th order solutions are uniquely obtained as a function of $\\varphi $ and its derivatives.",
"Let us focus on the leading-order EFT Lagrangian, ${\\cal L}_{\\rm IR} = f(\\chi _0,\\varphi ) \\, X - V(\\chi _0,\\varphi ) \\, \\vert _{\\chi _0^a = \\chi _0^a(\\varphi , X)} +\\mathcal {O}(\\tilde{\\epsilon }^1)= P(\\varphi , X) +\\mathcal {O}(\\tilde{\\epsilon }^1)\\; ,$ where $\\chi _0^a(\\varphi , X)$ is the solution to (REF ) and $P(\\varphi , X)$ is a function of $\\varphi $ and $X$ .",
"Now we observe $P_X & = \\frac{\\partial P}{\\partial X} = f + \\left( \\partial _a f \\, X - \\partial _a V \\right) \\frac{\\partial \\chi ^a}{\\partial X} \\, \\bigg \\vert _{\\chi _0^a = \\chi ^a(\\varphi , X)} = f \\, \\big \\vert _{\\chi _0^a = \\chi ^a(\\varphi , X)} \\; , \\\\P_{XX} & = \\frac{\\partial ^2 P}{\\partial X^2}= \\partial _a f \\, \\frac{\\partial \\chi ^a}{\\partial X} \\, \\bigg \\vert _{\\chi _0^a = \\chi _0^a(\\varphi , X)}= \\partial _a f \\left( M^{-2} \\right)^{ab} \\partial _b f \\, \\Big \\vert _{\\chi _0^a = \\chi _0^a(\\varphi , X)} \\; ,$ where $\\left( M^{-2} \\right)^{ab} $ is the inverse of $M^2_{ab}$ .",
"In order to obtain the above relations, we have used (REF ) and its variation for a fixed $\\varphi $ , $M^2_{ab} \\, \\delta \\chi ^b = \\partial _a f \\, \\delta X\\Rightarrow \\frac{\\partial \\chi ^a}{\\partial X} \\, \\bigg \\vert _{\\chi _0^a = \\chi ^a(\\varphi , X)} =(M^{-2})^{ab} \\partial _b f \\, \\Big \\vert _{\\chi _0^a = \\chi _0^a(\\varphi , X)}\\,,$ which is a direct consequence of the implicit function theorem.",
"It is important to stress that $P_{XX}$ is given by a quadratic form; thus, the sign of $P_{XX}$ is fixed when $M^2_{ab}$ is either the positive definite or the negative definite.",
"We discuss the signature of $M^2_{ab}$ in Sec.",
"."
],
[
"UV consistency of single-field EFT",
"The EFT reduction performed in Sec.",
"applies at fully nonlinear orders and is valid as far as the second and higher derivatives of $\\varphi $ are sufficiently small.",
"The typical energy/momentum scale of our interest is denoted by $E$ and the derivative expansion converges in $E \\ll |M|$ where $M$ is defined via (REF ) ($|M|$ is understood as the square root of the smallest absolute value of eigenvalues of the matrix $M^2_{ab}$ in the multi-field UV models).",
"Here, the sign of $M^2$ is crucial and $M^2$ is precisely defined by (REF ) (the multi-field extension $M^2_{ab}$ is defined by (REF )).",
"We also note that we have introduced the approximate shift symmetry and the scale of the light field, $m$ , is supposed to satisfy $m \\ll |M|$ where $m$ is still a schematic symbol denoting the scale of the light field.",
"In this section, we investigate the spectra of the full theory (REF ) to derive the UV consistency conditions of the resultant EFT.",
"The relevant modes of our interest in this section is the modes satisfying $E_{\\rm UV} \\gtrsim |M|$ .",
"Therefore, the scale $m$ is irrelevant to the analysis.",
"We thus concentrate on the shift symmetric theories, $m\\rightarrow 0$ , in this section.",
"We also ignore the curvature of the spacetime since we are interested in high-energy/momentum modes.",
"Note, however, that the presence of gravity puts a consistency of the argument in the way of energy conditions even if we ignore the curvature of the spacetime, which will be discussed in Sec.",
"REF .",
"We give detailed analysis about the two-field model in Sec.REF -REF and then extend the results into the multi-field (partial) UV completion in Sec.",
"REF ."
],
[
"Quadratic action around Lorentz-violating backgrounds",
"We focus on the shift symmetric two-field model for this and following two subsections.",
"The field-space metric is $\\gamma _{AB} \\, \\mathrm {d}\\Phi ^A \\mathrm {d}\\Phi ^B= \\tilde{\\epsilon }^2\\gamma (\\chi ) \\, \\mathrm {d}\\chi ^2 + 2 \\tilde{\\epsilon } h(\\chi ) \\, \\mathrm {d}\\chi \\mathrm {d}\\varphi + f(\\chi ) \\, \\mathrm {d}\\varphi ^2\\,.$ As we explained, the Killing vector for $N=2$ is hypersurface orthogonal and $\\varphi $ can be chosen to respect the reflection symmetry.",
"Concretely, we can use a freedom to change $\\varphi $ according to $\\varphi \\rightarrow \\varphi ^{\\prime }=\\varphi +\\tilde{\\epsilon } g(\\chi )$ where the Lagrangian is invariant under the shift of the new $\\varphi $ .",
"Using this freedom, we can eliminate the kinetic mixing between the light direction $\\varphi $ and the heavy field $\\chi $ , i.e.",
"$h=0$ , without loss of generality.",
"Then, we can conduct a transformation $\\chi \\rightarrow \\chi ^{\\prime }=\\mathcal {X}(\\chi )$ to set $\\gamma =1$ , as long as the kinetic term of $\\chi $ is healthy, $\\gamma >0$ .",
"All these transformations keep the leading shift-symmetric operator $P(X)$ invariant.",
"As a result, the most general Lagrangian of the two-field model under the shift symmetry is given by $\\mathcal {L}_{\\rm UV}&=-\\frac{1}{2}\\tilde{\\epsilon }^2 (\\nabla \\chi )^2+f(\\chi )X-V(\\chi )\\,.$ We have the same relations as before: $P=fX-V|_{\\chi _0=\\chi _0(X)},~P_X=f|_{\\chi _0=\\chi _0(X)},~P_{XX}=f_{\\chi }^2/M^2|_{\\chi _0=\\chi _0(X)}$ where $M^2\\equiv V_{\\chi \\chi }-f_{\\chi \\chi }X \\ne 0$ .",
"The Lorentz symmetry is spontaneously broken by the gradient of $\\varphi $ .",
"We split the fields into the background part and perturbations, $\\chi =\\bar{\\chi }+\\delta \\chi \\,, \\quad \\varphi = \\bar{\\varphi }+ \\pi \\,,$ where the background is supposed to provide the large gradient, $C_{\\mu }\\equiv \\partial _{\\mu }\\bar{\\varphi }=\\mathcal {O}(1)$ .",
"In general, the background $C_{\\mu }$ can be a function of time and/or space but these dependency must be tiny compared with $|M|$ so that the same configuration of the fields can be described by the EFT (recall that the second and higher than second order derivatives of $\\varphi $ must be tiny).",
"Such a change of $C_{\\mu }$ is negligible for the UV modes $E_{\\rm UV} \\gtrsim |M|$ .",
"Therefore, for simplicity, we consider the constant backgrounds, $\\bar{\\chi }={\\rm constant}\\,, \\quad C_{\\mu }={\\rm constant}\\,,$ to realize the spontaneous Lorentz symmetry breaking.In a more general sense, the solutions (REF ) should be regarded as approximate background solutions in the adiabatic limit.",
"In particular, $C_{\\mu }$ and $\\bar{\\chi }$ should be functions of time and/or space either when the shift symmetry is not exact or gravity is turned on.",
"The background equation of motion for $\\bar{\\chi }$ reads $V_{\\chi }-f_{\\chi }X=0\\,,$ while the equation of motion for $\\bar{\\varphi }$ is trivially satisfied.",
"From here on in this section, functions such as $X$ , $f$ and $V$ and their derivatives are understood to be evaluated at the background values $\\chi = \\bar{\\chi }$ and $\\varphi = \\bar{\\varphi }$ .",
"The background equation (REF ) is identical to the leading-order equation of the derivative expansion (REF ) because the background should be the low-energy/momentum part of the fields.",
"The equation (REF ) determines $\\bar{\\chi }$ in terms of $X=-\\frac{1}{2}C_{\\mu }C^{\\mu }$ as far as $M^2 \\ne 0$ .",
"On the other hand, the constant vector $C_{\\mu }$ is undetermined by the equation of motion and the value and direction of $C_{\\mu }$ determines how the Lorentz symmetry is broken by the background (REF ).",
"The quadratic action for the perturbations is $\\mathcal {L}^{(2)}_{\\rm UV}=-\\frac{1}{2}\\tilde{\\epsilon }^2 (\\partial \\delta \\chi )^2 -\\frac{1}{2}M^2 \\delta \\chi ^2 -\\frac{1}{2}f (\\partial \\pi )^2 - f_{\\chi } \\delta \\chi (C \\cdot \\partial \\pi )\\,,$ where all the coefficients are evaluated at the background (REF ) and thus become functions of $X=-\\frac{1}{2}C_{\\mu }C^{\\mu }$ .",
"Here, we use the notation $C\\cdot \\partial \\pi =C^{\\mu }\\partial _{\\mu }\\pi $ .",
"The first three terms are nothing but the Lorentz-invariant kinetic terms and the mass term, preserving the Lorentz symmetry of the perturbations.",
"The last term, $f_{\\chi } \\delta \\chi (C \\cdot \\partial \\pi )$ , is precisely the origin of the Lorentz-violation.",
"The dispersion relation of the perturbations are obtained by solving the equation ${\\rm det}\\begin{pmatrix}\\tilde{\\epsilon }^2 p^2+M^2 & f_{\\chi } (ip\\cdot C) \\\\-f_{\\chi } (ip\\cdot C) & p^2 f\\end{pmatrix}=p^2f(\\tilde{\\epsilon }^2 p^2+M^2)-f_{\\chi }^2(p\\cdot C)^2=0\\,,$ where $p^{\\mu }=(\\omega ,k^i)$ .",
"The relativistic dispersion relations, $p^2=0$ and $p^2+M^2=0$ , are obtained when $f_{\\chi }p\\cdot C=0$ while the dispersion relations are in general nonlinear due to the presence of the last term, $f_{\\chi } \\delta \\chi (C \\cdot \\partial \\pi )$ .",
"Let us clarify the stability conditions of the perturbations.",
"The ghost-free condition is the positive definiteness of the field space metric, leading to $f>0$ in the present parametrization.",
"We also need to identify the mass of the heavy mode.",
"The mass of the heavy mode may be evaluated by identifying the energy at the rest frame, $p^{\\mu }=(\\omega ,{\\mathbf {0}})$ .",
"In the Lorentz-invariant background, we can always perform the Lorentz transformation to go the rest frame of the massive particle.",
"However, we are interested in situations $X\\ne 0$ where $X<0$ may be relevant to e.g.",
"astrophysical environments [55], [56] and $X>0$ has been discussed in cosmology.",
"There is a background vector field $C_{\\mu }$ that spontaneously breaks the Lorentz symmetry.",
"Nonetheless, since both $\\bar{\\chi }$ and $C_\\mu $ are constant on the background (REF ), we can change frames globally by performing Lorentz transformations but cannot in general go to the preferred frame of the background $C_\\mu $ and the rest frame of $p^\\mu $ simultaneously.",
"We should first define the preferred frame in connection with $C_\\mu $ and then the mass of the particle.",
"In the following, we shall discuss the spacelike $C_{\\mu }$ case and the timelike one in order.",
"As for the spacelike background $X=-\\frac{1}{2}C\\cdot C<0$ , we can perform the Lorentz transformation so that $C^{\\mu }=(0,C^i)$ .",
"We define particles at rest in this frame.",
"This choice would be most useful because (REF ) contains $\\omega ^2$ and $\\omega ^4$ only, meaning that positive and negative frequency modes obey the same dispersion relation.",
"The mass of the heavy particle can be identified in the limit of vanishing spatial momentum in this frame, i.e.",
"by substituting $p^{\\mu }=(\\omega ,{\\mathbf {0}})$ and $C^{\\mu }=(0,C^i)$ into (REF ) and solving the equation in terms of $\\omega $ .",
"The last term in (REF ) does not contribute to the equation.",
"The mass of the heavy mode is then simply given by $M$ .",
"The dispersion relations for generic spatial momentum $k^i$ are $\\omega ^2_{\\rm heavy}&=k^2+\\frac{M^2}{2\\tilde{\\epsilon }^2 }\\left( 1 +\\sqrt{1+ \\tilde{\\epsilon }^2\\frac{4 f_{\\chi }^2 (C^i k_i)^2}{f M^4}} \\right)\\,,\\\\\\omega ^2_{\\rm light}&=k^2+\\frac{M^2}{2\\tilde{\\epsilon }^2 }\\left( 1 - \\sqrt{1+ \\tilde{\\epsilon }^2 \\frac{4 f_{\\chi }^2 (C^i k_i)^2}{f M^4}} \\right)\\nonumber \\\\&=k^2-\\frac{f_{\\chi }^2}{f M^2} (C^i k_i)^2 +\\tilde{\\epsilon }^2 \\frac{f_{\\chi }^4}{f^2M^6}(C^i k_i)^4+ \\mathcal {O}(\\tilde{\\epsilon }^4)\\,, $ where $k \\equiv \\vert k^i \\vert $ , and $f>0$ and $M^2>0$ have been used.",
"The conditions $f>0$ and $M^2>0$ are the stability conditions for the UV modes, and we assume them throughout the analysis around the spacelike background.",
"We introduce $k_{\\perp }^i$ and $k_{\\parallel }$ which are the perpendicular and the parallel components of $k^i$ to $C_i$ , that is, $k_{\\perp }^i C_i =0$ and $C_i k^i =\\sqrt{|2X|} \\, k_{\\parallel }$ , respectively.",
"Since the background configuration breaks the spatial rotational symmetry, the dispersion relation is anisotropic.",
"For the modes $k^i=k^i_{\\perp }, k_{\\parallel }=0$ , the standard dispersion relations $\\omega _{\\rm heavy}^2=M^2+k^2_{\\perp }, \\omega _{\\rm light}^2=k^2_{\\perp }$ are recovered while the dispersion relations take non-linear forms for $k_{\\parallel }\\ne 0$ .",
"For an illustrative purpose, Fig.",
"REF shows some concrete forms of the dispersion relations of both heavy and light modes with $k^i_{\\perp }=0$ .",
"Figure: M 2 <k * 2 M^2<k_*^2Let us recall that all the UV modes satisfying $\\omega ,k>M$ are integrated out to derive the single-field EFT via the derivative expansion.",
"The UV modes include not only the heavy mode $\\delta \\chi $ but also the light mode $\\pi $ .",
"The conditions $f>0$ and $M^2>0$ are the necessary conditions for the UV consistency, but we should discuss whether any other consistency conditions arise by checking the full spectra of the theory.",
"The dispersion relations show qualitatively different features depending on the background value of $|X|$ .",
"We consider the modes $k^i_{\\perp }=0$ for which the dispersion relations deviate from the standard form most significantly.",
"We set $\\tilde{\\epsilon }=1$ in the rest of this subsection (see subsubsection REF ).",
"The roots of $\\omega _{\\rm light}^2=0$ are, without expanding for small $\\tilde{\\epsilon }$ , $k_{\\parallel }^2=0\\,,~k_*^2\\,,$ where $k_*^2 \\equiv 2|X|\\frac{f_{\\chi }^2}{f}-M^2 \\; .$ Therefore, all the light modes are stable if $ k_*^2 <0$ whereas there exist unstable modes at IR satisfying $k_{\\parallel }^2< k_*^2$ if $k_*^2>0$ .",
"Since the modes $k_{\\parallel }^2>M^2$ are integrated out, the EFT reduction would be inconsistent if the instability existed at $k_{\\parallel }>M$ .",
"This requires $k_*^2<M^2$ as a consistency of the EFT reduction.",
"The typical behaviours are shown in Fig.",
"REF -(b) and REF -(c): the EFT reduction is consistent in (b) while is inconsistent in (c).Nevertheless, there can be a consistent single-field EFT even for Fig.",
"REF -(c) because the instability timescale is still longer than the dynamical timescale of the heavy mode $M^{-1}$ .",
"This requires to extend the validity of the EFT into the momentum domain $k_{\\parallel }^2>M^2$ while satisfying $|\\omega ^2|<M^2$ .",
"We elaborate on such an extension around the timelike background in the following section.",
"Note that the stability of the heavy mode is guaranteed by $M^2>0$ and the instability exists only in the IR part of the light field.",
"We investigate the timescale of the instability in the case $M^2>k_*^2>0$ .",
"The momentum giving the minimum of $\\omega ^2_{\\rm light}$ is determined by the condition $\\frac{\\partial \\omega _{\\rm light}^2(k^i)}{\\partial k_{\\parallel }^2}=0\\,,$ and then the minimum value is $\\omega _{\\rm min}^2=k^2_{\\perp }-\\frac{1}{4} k_*^2 \\left( 1 - \\frac{M^2f}{2|X|f_{\\chi }^2} \\right) > -\\frac{1}{4}k_*^2\\,.$ The instability timescale is thus $\\Delta t_{\\rm ins}\\equiv \\frac{1}{|\\omega _{\\rm min}|} > \\frac{2}{k_*}\\,.$ The timescale is long enough to be resolved by the EFT if $k_*$ is sufficiently small.",
"In particular, $M>k_*>0$ results in $\\Delta t_{\\rm ins}>2/M$ , that is, the instability timescale is longer than (twice) the cutoff scale $M^{-1}$ .",
"Therefore, such IR instabilities do not render the EFT reduction inconsistent."
],
[
"UV consistency around the spacelike background",
"We are ready to derive the UV consistency conditions on the EFT around the spacelike background, $X<0$ .",
"As observed in the previous subsubsection, the ghost-free and stability conditions are, respectively, $f>0\\,, \\quad M^2>0\\,,$ and $M^2 > k_*^2\\,,$ where $k_*^2$ can be either positive or negative.",
"The conditions (REF ) arise from the stability conditions of the UV modes while the condition (REF ) is the condition that the IR instability is under control if exists.",
"The EFT Lagrangian up to $\\mathcal {O}(\\tilde{\\epsilon }^2)$ is given by $\\mathcal {L}_{\\rm IR}=P(X)-\\tilde{\\epsilon }^2 \\frac{P_{XX}}{2M^2} (\\partial X)^2 + \\mathcal {O}\\left(\\tilde{\\epsilon }^2 \\right)\\,,$ where the Lagrangian at subleading orders is simplified from (REF ) because of the field redefinition.",
"Using the relations $P_X=f|_{\\chi _0=\\chi _0(X)},~P_{XX}=f_{\\chi }^2/M^2|_{\\chi _0=\\chi _0(X)}$ , the conditions (REF ) immediately conclude $P_X(X<0)>0\\,, \\quad P_{XX}(X<0)>0$ as the UV consistency of the EFT.Here, we implicitly assume $f_{\\chi }\\ne 0$ to have the Lorentz-violation effect in the quadratic Lagrangian.",
"We also have the relation $k_*^2 = 2|X| \\frac{f_{\\chi }^2}{f}-M^2=\\left(2|X|\\frac{P_{XX}}{P_X}-1 \\right) M^2\\,.$ Therefore, the condition (REF ) leads to $\\left| \\frac{P_X+2XP_{XX}}{P_X} \\right| < 1\\quad {\\rm if}\\quad \\frac{P_X+2XP_{XX}}{P_X} <0\\,,$ where $X<0$ is taken into account.",
"Note that the condition (REF ) is trivially satisfied if $(P_X+2XP_{XX})/P_X$ is positive."
],
[
"Matching IR dispersion relation",
"In order to verify that the procedure in the previous subsubsection correctly reproduces the IR physics, let us consider the quadratic Lagrangian of the EFT $\\mathcal {L}_{\\rm IR}=P(X)$ around the constant background, $\\mathcal {L}_{\\rm IR}^{(2)}=-\\frac{1}{2}\\left[P_X(\\partial \\pi )^2-P_{XX}(C\\cdot \\partial \\pi )^2+\\tilde{\\epsilon }^2 \\frac{P_{XX}}{M^2} \\partial _{\\mu }(C\\cdot \\partial \\pi ) \\partial ^{\\mu }(C\\cdot \\partial \\pi ) +\\mathcal {O}(\\tilde{\\epsilon }^4) \\right].",
"$ The EFT dispersion relation is a root of $&p^2P_X-P_{XX} \\left( C\\cdot p \\right)^2+\\tilde{\\epsilon }^2 \\frac{P_{XX}}{M^2} \\, p^2 \\left( C\\cdot p \\right)^2 + \\mathcal {O}(\\tilde{\\epsilon }^4)=0$ which indeed reproduces the original dispersion relation of the light mode up to $\\mathcal {O}(\\tilde{\\epsilon }^2)$ .",
"Using $C_{\\mu }=(0,C_i)$ , the EFT dispersion relation of the light mode is explicitly given by $\\omega ^2_{\\rm EFT}&=k_{\\perp }^2 + c_{\\parallel }^2 k_{\\parallel }^2 + \\tilde{\\epsilon }^2 \\frac{(2XP_{XX})^2}{P_X^2 M^2} k_{\\parallel }^4+\\mathcal {O}(\\tilde{\\epsilon }^4)\\,,\\\\c_{\\parallel }^2&\\equiv \\frac{P_X+2XP_{XX}}{P_X}\\,.$ One can easily confirm the agreement with (REF ) up to $\\mathcal {O}(\\tilde{\\epsilon }^2)$ .",
"Also, the conditions (REF ) and (REF ) lead to the bounds on the transverse part of the sound speed, $-1<c_{\\parallel }^2<1\\,.$"
],
[
"Spectra in full theory",
"Let us now turn to the timelike case, $X>0$ .",
"We first set $C_{\\mu }=(C_0, {\\mathbf {0}})$ by the use of the Lorentz transformation.",
"The dispersion relations are then $\\omega _{\\rm heavy}^2&=k^2+\\frac{\\mathcal {M} ^2}{2\\tilde{\\epsilon }^2 }\\left( 1+\\sqrt{1+ \\tilde{\\epsilon }^2\\frac{4 \\, \\delta M^2 k^2}{\\mathcal {M} ^4}} \\right)\\,, \\\\\\omega _{\\rm light}^2&=k^2+\\frac{\\mathcal {M} ^2}{2\\tilde{\\epsilon }^2 }\\left( 1-\\sqrt{1+ \\tilde{\\epsilon }^2 \\frac{4 \\, \\delta M^2 k^2}{\\mathcal {M} ^4}} \\right)\\nonumber \\\\&=c_s^2 k^2 + \\tilde{\\epsilon }^2 \\frac{\\delta M^4}{\\mathcal {M} ^6} k^4 + \\mathcal {O}(\\tilde{\\epsilon }^4)\\,,$ where $\\mathcal {M} ^2 =M^2 + \\delta M^2 = \\frac{M^2}{c_s^2}\\,, \\qquad \\delta M^2 = \\frac{C_0^2 f_{\\chi }^2}{f}= 2X \\frac{f_{\\chi }^2}{f}\\,, \\qquad c_s^2 =1-\\frac{\\delta M^2}{\\mathcal {M} ^2}=\\frac{M^2}{\\mathcal {M} ^2} \\,,$ and the conditions $f>0$ and $\\mathcal {M} ^2>0$ have been used.",
"The mass of the heavy mode is now given by $\\mathcal {M} ^2$ rather than $M^2$ where the correction $\\delta M^2$ comes from the friction term, $f_{\\chi }\\delta \\chi C\\cdot \\partial \\pi =f_{\\chi }C_0 \\delta \\chi \\dot{\\pi }$ .",
"Let us emphasize that $M^2$ can take either a positive or a negative value on the timelike background, and the heavy mode is stable so long as $\\mathcal {M} ^2>0$ regardless of the sign of $M^2$ .",
"The same observation was found in Sec.",
"REF (see (REF ) that can be satisfied even for $\\eta \\le 0$ .",
").",
"Hence $M^2$ can be either positive or negative for the timelike $C_{\\mu }=\\partial _{\\mu }\\bar{\\varphi }$ .We do not consider $M^2=0$ because the background $\\bar{\\chi }$ would be undetermined by $\\bar{\\varphi }$ .",
"On the other hand, we can discuss the limit $M^2 \\rightarrow 0$ .",
"Typical behaviours of the dispersion relations are shown in Fig.",
"REF .",
"Figure: -2(2-1)δM 2 >M 2 -2(\\sqrt{2}-1)\\delta M^2>M^2At first glance, the dispersion relations appear to behave similarly to those around the spacelike background.",
"IR instabilities can appear depending on the parameters.",
"The roots of $\\omega _{\\rm light}^2=0$ are, absorbing $\\tilde{\\epsilon }$ into the definition of $\\chi $ (equivalently setting $\\tilde{\\epsilon }= 1$ ), $k^2=0\\,, ~-M^2\\,,$ around the timelike background.",
"Therefore, the light modes are always stable for $M^2>0$ while the modes $k^2<|M^2|$ are unstable for $M^2<0$ .",
"The timescale of the instability is estimated by the same way as the spacelike background.",
"The momentum corresponding to the minimum of $\\omega _{\\rm light}^2$ is determined by $\\frac{\\mathrm {d}\\omega _{\\rm light}^2(k^2)}{\\mathrm {d}k^2}=0\\,,$ yielding $\\omega _{\\rm min}^2 = - \\frac{M^4}{4\\delta M^2}\\,.$ The timescale is $t_{\\rm uns}\\equiv \\frac{1}{|\\omega _{\\rm min}|} = \\frac{2\\delta M}{|M^2|}\\,,$ where $\\delta M^2>0$ is used.",
"Nonetheless, there is a crucial difference from the spacelike background.",
"In the case of the spacelike background, the boundary of the instability is determined by $k_*^2$ which can be chosen to satisfy $k_*^2< M^2$ .",
"Here, recall that the absolute value of the parameter $M$ determines the convergence radius of the derivative expansion with respect to spacetime derivatives, see (REF ).",
"On the other hand, around the timelike background with $M^2<0$ , the parameter $|M^2|$ serves as not only the boundary of the convergence but also that of the instability in the EFT.",
"In fact all the modes with $k^2 \\lesssim \\vert M^2 \\vert $ would be unstable if $M^2<0$ .",
"Nevertheless, this would not invoke any instabilities in the UV, and therefore the UV modes should be safely integrated out, keeping the validity of the effective description.",
"The time and length scales of applicability of such an unstable EFT might be rather limited in realistic setups, as the instability of the modes, especially the ones near $k^2 \\sim \\vert M^2 \\vert $ , could drive the system out of the EFT's validity range and even excite high-energy/momentum modes via nonlinear interactions.",
"However, the effective mass of the heavy modes is given by $\\mathcal {M} ^2$ rather than $M^2$ , and therefore the practical cutoff scale of the single-field EFT can be raised accordingly.",
"Indeed, we can extend the validity of the EFT into the domain $|\\omega ^2| < \\mathcal {M} ^2 \\,, \\quad k^2 < \\Lambda ^2 \\equiv \\mathcal {M} ^2+\\delta M^2\\,,$ where $k^2= \\Lambda ^2$ corresponds to $\\omega _{\\rm light}^2=\\mathcal {M} ^2$ .",
"In this case, the single-field EFT can accommodate the stable modes above $\\vert M \\vert $ as well, and one can introduce hierarchy between the threshold of the instability, $|M^2|$ , and the actual cutoff $\\Lambda ^2$ .",
"For consistency, we need to assume the condition that the timescale of the instability is sufficiently longer than the cutoff timescale, which is given by $|\\omega _{\\rm min}^2|< \\mathcal {M} ^2 \\Rightarrow |M^2| < 2(\\sqrt{2}-1) \\delta M^2 \\quad {\\rm if} \\quad M^2<0\\,,$ As a result, the theory corresponding to Fig.REF -(b) can be described by a single-field EFT while Fig.REF -(c) could not as instabilities would develop before the heavy modes could be stabilized.",
"In the following subsubsections, we separately discuss the EFTs that are applicable to the two different (but not necessarily exclusive) domains.",
"We first consider the case in which the stability of deep IR modes is assured by the condition $M^2 >0$ , and later we study the general EFT by extending the validity of the EFT into (REF )."
],
[
"UV consistency around the timelike background without IR instability",
"Let us first consider the UV consistency of the EFT (REF ) under the additional assumption of the absence of IR instabilities, i.e.",
"$M^2>0$ .",
"In this case we have $f>0\\,, \\quad M^2>0\\,,$ where the first condition is due to the UV consistency while the second one is to avoid IR instabilities.",
"The remaining UV consistency, i.e.",
"the no-tachyon condition $\\mathcal {M} ^2>0$ is automatically satisfied by these conditions.",
"We then find $P_X(X>0)>0\\,, \\quad P_{XX}(X>0)>0\\,,$ as the UV consistency conditions of (REF ) and the above-mentioned additional assumption.",
"The square of the sound speed of the perturbations, $c_s^2=\\frac{P_X}{P_X+2XP_{XX}}\\,,$ is positive and bounded by the speed of light, $0<c_s^2<1$ as a consequence of the UV consistency (REF ).",
"One can confirm that the EFT dispersion relation (REF ) correctly reproduces the original relation (REF ) up to $\\mathcal {O}(\\tilde{\\epsilon }^2)$ under $C_{\\mu }=(C_0,\\mathbf {0})$ .The dispersion relation (REF ) generically contains a ghost mode due to the truncation of the higher derivative operators.",
"One has to only consider the light (physical) mode of the solution.",
"As we mentioned in the spacelike case, our consistency conditions (REF ) hold even in the largely broken Lorentz symmetry, $C_{\\mu }=\\mathcal {O}(1)$ , as far as we assume the class of (partial) UV completion and the absence of IR instabilities.",
"Although we have assumed the constant $C_{\\mu }$ , the background can depend on time (and/or space) as long as its change is adiabatic.",
"Furthermore, the bounds (REF ) (and (REF )) are applicable to the EFT that has no consistent Lorentz-invariant background (see Appendix )."
],
[
"UV consistency around the timelike background with IR instability: apparent violation of positivity",
"We now discuss how we can extend the validity of EFT to go beyond the threshold of the IR instability around the timelike background, even in the case with $M^2<0$ .",
"Let us rediscuss the quadratic UV Lagrangian (REF ), $S^{(2)}_{\\rm UV}=\\int \\mathrm {d}^dx \\left[-\\frac{1}{2}(\\partial \\delta \\chi )^2 -\\frac{1}{2}M^2 \\delta \\chi ^2 -\\frac{1}{2}f (\\partial \\pi )^2 - f_{\\chi } \\delta \\chi (C \\cdot \\partial \\pi ) \\right]\\,,$ where we set $\\tilde{\\epsilon }=1$ .",
"The parameter $\\tilde{\\epsilon }$ is no longer useful because we cannot use the derivative expansion for the present purpose.",
"In the momentum space, the action is $S^{(2)}_{\\rm UV}=\\int \\mathrm {d}t \\int \\frac{\\mathrm {d}^{d-1} k}{(2\\pi )^{d-1}} \\left[ \\frac{1}{2} \\left( \\left|\\delta \\dot{\\chi } \\right|^2- \\left( k^2+M^2 \\right) \\left|\\delta \\chi \\right|^2 \\right) +\\frac{f}{2} \\left( \\left|\\dot{\\pi } \\right|^2-k^2 \\left|\\pi \\right|^2 \\right) + \\frac{C_0 f^{\\prime }}{2} \\left( \\delta \\chi ^\\dagger \\dot{\\pi } + {\\rm h.c.} \\right)\\right]\\,.$ The complication arises from the fact that the variables $\\delta \\chi $ and $\\pi $ are not eigenstates of propagations.",
"The mass of the heavy mode is corrected by the friction term, $\\mathcal {M} ^2=M^2+\\delta M^2$ .",
"To overcome this point, we perform field redefinition to diagonalize the fields.",
"As for the Lorentz-invariant terms (the kinetic terms and the mass term), we can take a general linear transformation of $\\delta \\chi $ and $\\pi $ to diagonalize them which is indeed what we did at the beginning of this section, see (REF ).",
"On the other hand, the friction term cannot be diagonalized in this way; instead, it is more convenient to perform a canonical transformation (see e.g.",
"[57], [58]).",
"We thus consider the Hamiltonian rather than the Lagrangian.",
"From (REF ), defining the conjugate momenta via $p&=\\delta \\dot{\\chi } \\,, \\\\p_{\\pi }&= f\\dot{\\pi }+f_{\\chi } C_0 \\delta \\chi \\,,$ the Hamiltonian around the timelike background is $H_{\\rm UV}^{(2)}&=\\int \\frac{\\mathrm {d}^{d-1}k}{(2\\pi )^{d-1}} \\mathcal {H}_{\\rm UV}^{(2)}\\,, \\\\\\mathcal {H}^{(2)}_{\\rm UV}&=\\frac{1}{2} \\left|\\delta \\dot{\\chi } \\right|^2+\\frac{1}{2} \\left( k^2+M^2 \\right) \\left|\\delta \\chi \\right|^2 + \\frac{f}{2} \\left|\\dot{\\pi } \\right|^2 + \\frac{f}{2} \\, k^2 \\left|\\pi \\right|^2\\nonumber \\\\&=\\frac{1}{2} \\left|p \\right|^2+\\frac{1}{2} \\left( k^2+M^2 \\right) \\left|\\delta \\chi \\right|^2 + \\frac{1}{2f} \\left|p_{\\pi }-f_{\\chi } C_0 \\delta \\chi \\right|^2 + \\frac{f}{2} \\, k^2 \\left|\\pi \\right|^2$ in the momentum space.",
"Here, on the first line of (REF ), $\\delta \\dot{\\chi }$ and $\\dot{\\pi }$ are understood as functions of canonical variables.",
"We then take a canonical transformation to remove the mixing term $p_{\\pi }\\delta \\chi $ from the Hamiltonian by considering an appropriate generating function.",
"One example of such a transformation is to take linear combinations to define a set of new conjugate variables by $& \\tilde{\\pi }\\equiv c_1 \\, \\delta \\chi + d_1 \\, p_\\pi \\; , \\qquad \\tilde{p}_\\pi \\equiv c_2 \\, p + d_2 \\, \\pi \\; ,\\\\ &\\delta \\tilde{\\chi }\\equiv c_3 \\, \\pi + d_3 \\, p \\; , \\qquad \\tilde{p} \\equiv c_4 \\, p_\\pi + d_4 \\, \\delta \\chi \\; ,$ where $c_i$ and $d_i$ are constants.",
"The exact forms of these coefficients are rather lengthy and not illuminating, so we only write the first two terms of each in the small $k$ expansion, reading $c_1 & = \\frac{\\delta M}{\\mathcal {M} ^2} \\, k - \\frac{3 \\, \\delta M^3}{2 \\, \\mathcal {M} ^6} \\, k^3+ {\\cal O}(k^5) \\; , \\qquad d_1 = \\frac{1}{\\sqrt{f} \\, k} - \\frac{\\delta M^2}{2 \\sqrt{f} \\, \\mathcal {M} ^4} \\, k+ {\\cal O}(k^3) \\; ,\\\\c_2 & = \\frac{\\delta M}{\\mathcal {M} ^2} \\, k - \\frac{3 \\, \\delta M^3}{2 \\, \\mathcal {M} ^6} \\, k^3+ {\\cal O}(k^5) \\; , \\qquad d_2 = - \\sqrt{f} \\, k + \\frac{\\sqrt{f} \\, \\delta M^2}{2 \\, \\mathcal {M} ^4} \\, k^3+ {\\cal O}(k^5) \\; ,\\\\c_3 & = \\frac{\\sqrt{f} \\, \\delta M}{\\mathcal {M} ^3} \\, k^2 - \\frac{\\sqrt{f} \\, \\delta M \\left( \\mathcal {M} ^2 + 4 \\, \\delta M^2 \\right)}{2 \\, \\mathcal {M} ^7} \\, k^4+ {\\cal O}(k^6) \\; , \\qquad d_3 = \\frac{1}{\\mathcal {M} } - \\frac{\\mathcal {M} ^2 + 2 \\, \\delta M^2}{2 \\, \\mathcal {M} ^5} \\, k^2+ {\\cal O}(k^4) \\; ,\\\\c_4 & = \\frac{\\delta M}{\\sqrt{f} \\, \\mathcal {M} } + \\frac{\\delta M \\left( \\mathcal {M} ^2 - 2\\, \\delta M^2 \\right)}{2 \\sqrt{f} \\, \\mathcal {M} ^5} \\, k^2+ {\\cal O}(k^4) \\; , \\qquad d_4 = - \\frac{k^2 + 2 \\, \\mathcal {M} ^2}{2 \\, \\mathcal {M} }+ {\\cal O}(k^4) \\; .$ Then $\\lbrace \\tilde{\\pi }, \\tilde{p}_\\pi \\rbrace $ and $\\lbrace \\delta \\tilde{\\chi }, \\tilde{p} \\rbrace $ are conjugate pairs, and the Hamiltonian in terms of the new conjugate variables is diagonalized, ${\\cal H}^{(2)}_{\\rm UV} = \\frac{1}{2} \\left( \\left|\\tilde{p} \\right|^2 + \\Omega _\\chi ^2 \\left|\\delta \\tilde{\\chi }\\right|^2+ \\left|\\tilde{p}_\\pi \\right|^2 + \\Omega _\\pi ^2 \\left|\\tilde{\\pi }\\right|^2 \\right) \\; ,$ where $\\Omega _\\chi ^2 & = \\mathcal {M} ^2 + k^2 \\left( 1 + \\frac{\\delta M^2}{\\mathcal {M} ^2} \\right)+ {\\cal O}(k^4) \\; ,\\qquad \\Omega _\\pi ^2 = c_s^2 \\, k^2 + \\frac{\\delta M^4}{\\mathcal {M} ^6} \\, k^4 + {\\cal O}(k^6) \\; ,$ which perfectly coincide with (REF ) up to these orders.",
"The heavy mode variables $\\lbrace \\delta \\tilde{\\chi }, \\tilde{p} \\rbrace $ are now decoupled.",
"Integrating the UV modes out, we find the quadratic Hamiltonian of the EFT, $H^{(2)}_{\\rm IR}&=\\int ^{\\Lambda }\\!\\!",
"\\frac{\\mathrm {d}^{d-1}k}{(2\\pi )^{d-1}} \\mathcal {H}_{\\rm UV}^{(2)}\\,, \\quad {\\cal H}^{(2)}_{\\rm IR} = \\frac{1}{2} \\left( \\left|\\tilde{p}_\\pi \\right|^2 + \\Omega _\\pi ^2 \\left|\\tilde{\\pi }\\right|^2 \\right) \\; ,$ where $\\Lambda $ is added to the integral symbol in order to represent that the domain of integration is limited to $k^2<\\Lambda ^2 = \\mathcal {M} ^2+\\delta M^2$ so that frequencies of $\\lbrace \\tilde{p}_{\\pi }, \\tilde{\\pi }\\rbrace $ do not exceed the mass of the heavy mode.",
"After the Legendre transformation, the quadratic action of the EFT is $S^{(2)}_{\\Lambda }[\\tilde{\\pi }]=\\int \\mathrm {d}t \\int ^{\\Lambda }\\!\\!",
"\\frac{\\mathrm {d}^{d-1}k}{(2\\pi )^{d-1}} \\!\\!\\left( \\frac{1}{2} |\\dot{\\tilde{\\pi }}|^2 + \\frac{1}{2} \\Omega _\\pi ^2 \\left|\\tilde{\\pi }\\right|^2 \\right),$ which is valid even for $M^2<0$ .",
"We also note the the Lagrangian is well-behaved even in the limit $M^2 \\rightarrow 0$ .",
"Although we only compute the quadratic action, the non-linear interactions of the EFT may be computed accordingly.",
"It deserves care to compare the previous EFT (REF ) with the present EFT (REF ) because of the field transformation.",
"Let us call the previous one the $M$ -EFT and the present one the $\\Lambda $ -EFT, respectively, since the cutoffs of these EFTs are determined by $M^2$ and $\\Lambda ^2=\\mathcal {M} ^2+\\delta M^2$ .",
"The on-shell relation between the variables is $\\tilde{\\pi }=\\frac{f(d_1d_4-c_1c_4)}{d_4+c_4 \\sqrt{f} \\delta M} \\, \\dot{\\pi }\\simeq \\frac{\\mathcal {M} ^2}{\\sqrt{f}k(k^2+M^2)} \\, \\dot{\\pi }\\,,$ where we have assumed $k^2 \\ll \\mathcal {M} ^2$ and $|M^2|\\ll \\mathcal {M} ^2$ to get the last expression.",
"This implies that $\\pi $ and $\\tilde{\\pi }$ are related in a non-local way in both time and space.",
"The action of $\\Lambda $ -EFT in terms of the variable $\\pi $ can be obtained by taking the canonical transformation.",
"Here, we perform the canonical transformation in the Lagrangian level by following the technique [59].",
"For simplicity, we consider the case $|M^2| \\ll \\mathcal {M} ^2$ where $\\Omega _{\\pi }^2$ is approximated as $\\Omega _{\\pi }^2 \\simeq c_s^2 k^2 + \\frac{1}{\\mathcal {M} ^2}k^4$ up to the subleading order in $k^2$ .",
"We integrate in the variable $\\pi $ and write the action (REF ) as $S^{(2)}_{\\Lambda }[\\tilde{\\pi },\\pi ]=\\int \\mathrm {d}t \\int ^{\\Lambda }\\!\\!",
"\\frac{\\mathrm {d}^{d-1}k}{(2\\pi )^{d-1}}\\left[ -\\frac{1}{2}f k^2|\\pi |^2 - \\frac{k^2(k^2+M^2)}{2\\mathcal {M} ^2} |\\tilde{\\pi }|^2 - \\frac{\\sqrt{f} k}{2}(\\pi ^\\dagger \\dot{\\tilde{\\pi }} + {\\rm h.c.} ) \\right],$ which recovers (REF ) when $\\pi $ is integrated out under the approximation (REF ).",
"Instead, we integrate $\\tilde{\\pi }$ out.",
"The equation of motion for $\\tilde{\\pi }$ yields the relation (REF ).",
"Substituting (REF ) into $S^{(2)}_{\\Lambda }[\\tilde{\\pi },\\pi ]$ , we obtain $S^{(2)}_{\\Lambda }[\\pi ]&=\\int \\mathrm {d}t \\int ^{\\Lambda }\\!\\!",
"\\frac{\\mathrm {d}^{d-1}k}{(2\\pi )^{d-1}}\\frac{f}{2} \\left[ \\frac{\\mathcal {M} ^2}{k^2+M^2}|\\dot{\\pi }|^2 - k^2 |\\pi |^2 \\right].",
"$ Note that the action (REF ) can be directly derived from the UV action (REF ) by integrating out $\\delta \\chi $ if we keep the nonlinearity of $k^2$ .",
"The case $|M^2| \\ll \\mathcal {M} ^2$ corresponds to $|c_s^2| \\ll 1 \\Rightarrow |\\omega ^2| \\ll k^2$ , meaning that there is no need to equally treat the time and the space.",
"We can perturbatively treat the time derivative $\\delta \\dot{\\chi }$ while keeping the spacial derivative $\\partial _i \\delta \\chi =i k_i\\delta \\chi $ non-perturbatively.",
"The action (REF ) is obtained by dropping $|\\delta \\dot{\\chi }|^2$ from (REF ) and then by integrating out $\\delta \\chi $ .",
"In the position space, the solution of $\\delta \\chi $ is given by a spatial integral, yielding the non-local EFT action as one can see from the $k$ -dependent denominator in (REF ).",
"Nonetheless, such a non-locality does not provide any pathology since $\\delta \\chi $ can be uniquely determined by $\\pi $ under an appropriate boundary condition.",
"The action (REF ) is non-local in space but local in time.",
"The quadratic action of the $M$ -EFT around the background $C_{\\mu }=(C_0,\\mathbf {0})$ is $S_{\\rm IR}^{(2)}[\\pi ]=\\int \\mathrm {d}t \\int ^{M}\\!\\!\\!",
"\\frac{\\mathrm {d}^{d-1}k}{(2\\pi )^{d-1}}\\frac{1}{2}\\left[ (P_X+2XP_{XX})|\\dot{\\pi }|^2 - P_X k^2 |\\pi |^2 + \\frac{2XP_{XX}}{M^2}(|\\ddot{\\pi }|^2 - k^2 |\\dot{\\pi }|^2) +\\cdots \\right],$ where we have the relations $\\delta M^2 = \\frac{2XP_{XX}}{P_X} M^2 \\,, \\quad \\mathcal {M} ^2 = \\frac{P_X+2XP_{XX}}{P_X} M^2\\,, \\quad c_s^2= \\frac{P_X}{P_X+2XP_{XX}}\\,.$ We find $S_{\\rm IR}^{(2)}[\\pi ]=\\int \\mathrm {d}t \\int ^{M}\\!\\!\\!",
"\\frac{\\mathrm {d}^{d-1}k}{(2\\pi )^{d-1}}\\frac{1}{2}\\left[ (P_X+2XP_{XX})|\\dot{\\pi }|^2 - P_X k^2 |\\pi |^2 - \\frac{2XP_{XX}}{M^2} k^2 |\\dot{\\pi }|^2 +\\cdots \\right],$ when $|c_s^2| \\ll 1~\\Rightarrow |\\ddot{\\pi }|^2 \\ll k^2|\\dot{\\pi }|^2$ .",
"This indeed agrees with the low-momentum part of the $\\Lambda $ -EFT, $S^{(2)}_{\\Lambda }[\\pi ]&=\\int \\mathrm {d}t \\int ^{\\Lambda }\\!\\!",
"\\frac{\\mathrm {d}^{d-1}k}{(2\\pi )^{d-1}}\\frac{1}{2} \\left[ \\frac{P_X+2XP_{XX}}{k^2+M^2}|\\dot{\\pi }|^2 - P_Xk^2 |\\pi |^2 \\right]\\nonumber \\\\&\\simeq \\int \\mathrm {d}t \\int ^{\\Lambda }\\!\\!",
"\\frac{\\mathrm {d}^{d-1}k}{(2\\pi )^{d-1}}\\frac{1}{2} \\left[ (P_X+2XP_{XX})|\\dot{\\pi }|^2 - P_Xk^2 |\\pi |^2 - \\frac{2XP_{XX}}{M^2}k^2 |\\dot{\\pi }|^2 +\\cdots \\right],$ where $|M^2|\\ll \\mathcal {M} ^2 \\Rightarrow P_X+2XP_{XX} \\simeq 2XP_{XX}$ is used to make the $k^2 |\\dot{\\pi }|^2$ term equal to that of (REF ).",
"One should, however, notice that the integration domains are different.",
"In the $\\Lambda $ -EFT, we need a resummation of the infinite number of spatial derivatives to make the EFT valid even at $|M^2|<k^2 (<\\Lambda ^2)$ .",
"As we mentioned, the $\\Lambda $ -EFT is thus spatially non-local in terms of the variable $\\pi $ .",
"We have constructed the single-field EFT that is valid in the extended domain $|M^2|<k^2<\\Lambda ^2$ , hence the name $\\Lambda $ -EFT, and that accommodates the IR instability caused by $M^2<0$ .",
"One of the advantages of the $\\Lambda $ -EFT over the $M$ -EFT is that, due to the wider regime of validity, it can describe the truncation of the IR instabilities at $k=|M|$ without need to go back to the multi-field completion.",
"The UV consistency conditions, combined with the condition for the existence of the $\\Lambda $ -EFT with this advantage, are (i) the stability conditions of the UV modes, $f>0 \\,, \\quad \\mathcal {M} ^2>0\\,,$ and (ii) the condition that the IR instability, if exists $(M^2<0)$ , is under control within the EFT, $|M^2| < 2(\\sqrt{2}-1) \\delta M^2\\,,$ where the condition (ii) does not exist if $M^2$ is positive.",
"Using $P_X$ and $P_{XX}$ , these conditions are rewritten as $P_X>0\\,,$ and ${\\left\\lbrace \\begin{array}{ll}P_{XX}>0 &{\\rm for}~ M^2>0\\,, \\\\|P_{XX}| > \\left(\\frac{1}{2}+\\frac{1}{\\sqrt{2}} \\right)\\frac{P_X}{2X} &{\\rm for }~M^2<0~\\Leftrightarrow P_{XX}<0\\,,\\end{array}\\right.",
"}$ around the timelike background, $X>0$ .",
"It is interesting that two consistent regions of $P_{XX}$ are disconnected, that is, $P_{XX}$ is either positive or large negative.",
"The region $-\\left(\\frac{1}{2}+\\frac{1}{\\sqrt{2}} \\right)\\frac{P_X}{2X}< P_{XX}<0$ is inconsistent.",
"We provide an illustrative figure in Fig.",
"REF .",
"The conditions (REF ) are summarized in a simple form in terms of the sound speed, given by $-\\left(\\frac{1}{\\sqrt{2}}-\\frac{1}{2}\\right)\\simeq -0.2 < c_s^2 < 1\\,.$ Albeit its appearance, the imaginary sound speed is in fact consistent with the UV stability, and its absolute value is bounded by a finite value of at most order unity.",
"Figure: An example of P X P_X and P XX P_{XX}.",
"The UV consistency conditions () and () conclude that the single-field EFT can only have a partial UV completion by means of a two-field model either in the red domain (P XX >0)(P_{XX}>0) or the blue domain (P XX <0)(P_{XX}<0).",
"Note that the two consistent domains are disconnected.",
"If ϕ\\varphi tends to go outside the consistent domains by considering an adiabatic motion of XX, one should return to the UV theory before violating either () or ().This figure is solely for an illustrative purpose, and our choice of the shapes of P X P_X and P XX P_{XX} has no physical consequence.The negative sign of $P_{XX}$ apparently contradicts the conventional positivity bound [1].",
"One should recall that our setup is completely different from the argument of the conventional positivity bounds.",
"First of all, the positivity bounds are the bounds on the EFT around the Lorentz-invariant background and are not applicable to the EFT around the Lorentz-violating background.",
"In fact, the branch $|P_{XX}| > \\left(\\frac{1}{2}+\\frac{1}{\\sqrt{2}} \\right)\\frac{P_X}{2X}$ cannot have a continuous limit to the Lorentz-invariant background due to the denominator $X$ .",
"The flat limit $X\\rightarrow +0$ yields $P_{XX}<-\\infty $ since $P_X$ is related to the kinetic term of the UV theory and thus has to be strictly positive.",
"This means that there is no consistent single-field EFT that can accommodate both the Lorentz-invariant background and the Lorentz-violating background with negative $P_{XX}$ .",
"(This does not exclude the possibility that a UV theory may reduce to a single-field EFT with positive $P_{XX}$ around a Lorentz-invariant background and to a different single-field EFT with negative $P_{XX}$ around a Lorentz-violating background.",
"In order to connect the two single-field EFTs one needs to go back to the UV theory.)",
"Furthermore, the negative $P_{XX}$ exhibits the IR instability, meaning that the scattering amplitudes may not be well-defined.",
"Nonetheless, such an EFT is consistent and can describe transient IR instabilities (see e.g.",
"[60])."
],
[
"UV consistency for multi-field UV models",
"We generalize the previous analysis to the multi-field UV models.",
"Assuming the shift symmetry, the UV Lagrangian is given by $\\mathcal {L}_{\\rm UV}^{(2)}&=- \\frac{1}{2}\\tilde{\\epsilon }^2 \\gamma _{ab}(\\chi ) (\\partial \\chi ^a \\cdot \\partial \\chi ^b)- \\tilde{\\epsilon } h_a(\\chi ) (\\partial \\chi ^a \\cdot \\partial \\varphi ) - \\frac{1}{2}f(\\chi ) (\\partial \\varphi )^2- V(\\chi )\\; ,$ where lower-case alphabets $a,b,\\dots $ run for the heavy fields $\\chi ^a$ .",
"The metric $\\gamma _{AB}$ is supposed to be positive definite which is equivalent to $\\gamma _{ab}-h_a h_b/f>0$ and $f>0$ .",
"The Killing vector is $\\xi ^A=\\lbrace 0, 1 \\rbrace $ which is hypersurface orthogonal if and only if $f\\partial _{[a}h_{b]} + h_{[a} \\partial _{b]} f=0\\,,$ according to Frobenius's theorem.",
"We consider the constant background, $\\bar{\\chi }^a={\\rm constant}\\,, \\quad C_{\\mu }=\\partial _{\\mu }\\bar{\\varphi }={\\rm constant}\\,,$ being subject to the equations $f_a(\\bar{\\chi }) X -V_a(\\bar{\\chi })=0\\,,$ where the subscript indices $a,b,\\cdots $ is the derivative with respect to $\\chi ^a$ , e.g.",
"$f_a=\\frac{\\partial f}{\\partial \\chi ^a}$ .",
"All $\\chi ^a$ are determined by $X$ as long as the determinant of the matrix $M^2_{ab}= -f_{ab }X+V_{ab}\\,,$ is non-zero.",
"The quadratic action for the perturbations, $\\chi ^a=\\bar{\\chi }^a+\\delta \\chi ^a \\,, \\quad \\varphi =\\bar{\\varphi }+\\pi \\,,$ is $\\mathcal {L}_{\\rm UV}^{(2)}= -\\frac{1}{2}\\tilde{\\epsilon }^2 \\gamma _{ab} (\\partial \\delta \\chi ^a \\cdot \\partial \\delta \\chi ^b)-\\frac{1}{2}M^2_{ab} \\delta \\chi ^a \\delta \\chi ^b - \\tilde{\\epsilon }h_{[ab]} (C \\cdot \\partial \\delta \\chi ^a) \\delta \\chi ^b -\\frac{1}{2} f (\\partial \\pi )^2 -\\tilde{\\epsilon } h_a (\\partial \\pi \\cdot \\partial \\delta \\chi ^a) - f_a \\delta \\chi ^a (C\\cdot \\partial \\pi )\\,,$ where the coefficients are evaluated at the background.",
"We can use the freedom of the field redefinition $\\varphi \\rightarrow \\varphi ^{\\prime } = \\varphi + \\tilde{\\epsilon } g(\\chi ^a) \\Rightarrow \\pi \\rightarrow \\pi ^{\\prime }=\\pi + \\tilde{\\epsilon } g_a(\\bar{\\chi }) \\delta \\chi ^a$ so that $h_a(\\bar{\\chi })=0$ evaluated at the background (REF ).",
"The quadratic Lagrangian is simplified to be $\\mathcal {L}_{\\rm UV}^{(2)}= -\\frac{1}{2} \\tilde{\\epsilon }^2 \\gamma _{ab} (\\partial \\delta \\chi ^a \\cdot \\partial \\delta \\chi ^b)-\\frac{1}{2}M^2_{ab} \\delta \\chi ^a \\delta \\chi ^b - \\tilde{\\epsilon } h_{[ab]} (C \\cdot \\partial \\delta \\chi ^a) \\delta \\chi ^b -\\frac{1}{2} f (\\partial \\pi )^2 - f_a \\delta \\chi ^a (C\\cdot \\partial \\pi )\\; ,$ Note that $h_{[ab]}=\\frac{1}{2}(\\partial _b h_a - \\partial _a h_b)$ does not vanish, in general, since we can only set $h_a(\\bar{\\chi })=0$ via a field redefinition while we cannot set $h_{[ab]}(\\bar{\\chi })=0$ (i.e.",
"we cannot set $h_a(\\chi )=0$ unless the Killing vector is hypersurface orthogonal).",
"The third term vanishes only when (REF ) holds.",
"We have an additional freedom to change $\\chi ^a$ according to $\\chi ^a \\rightarrow \\mathcal {X}^a{}=\\mathcal {X}^a{}(\\chi ) \\Rightarrow \\delta \\chi \\rightarrow \\delta \\chi ^{\\prime }=G^a{}_b \\delta \\chi ^b\\,,$ which will be used later.",
"The dispersion relations of (REF ) are determined by the equation $D&={\\rm det}\\begin{pmatrix}D_{ab} & f_a (i C \\cdot p) \\\\- f_b (i C \\cdot p) & p^2 f\\end{pmatrix}=0\\,,$ where $D_{ab}&\\equiv \\tilde{\\epsilon }^2 p^2\\gamma _{ab} +2\\tilde{\\epsilon } h_{[ab]} (iC\\cdot p) + M^2_{ab}\\,.$ The equation (REF ) is transformed to $D={\\rm det}D_{ab} \\times \\left[ p^2f -f_a (D^{-1})^{ab} f_b (C\\cdot p)^2 \\right]=0,\\quad ({\\rm det}D_{ab}\\ne 0)\\,,$ and then the dispersion relation of the light mode is given by $p^2f -f_a (D^{-1})^{ab} f_b (C\\cdot p)^2=0\\,.$ The EFT coefficients $P_X$ and $P_{XX}$ are related to the UV coefficients via $P_X=f\\,, \\quad P_{XX}= \\left.",
"f_a (D^{-1})^{ab} f_b \\right|_{p=0} = f_a (M^{-2})^{ab} f_b\\,.$ Hereinafter, we set $\\tilde{\\epsilon }=1$ since it is irrelevant for the following discussions.",
"As in the two-field case in Sec.",
"REF , when $C_{\\mu }$ is spacelike, the terms proportional to $C\\cdot p$ do not contribute to the masses of the heavy modes.",
"The masses are determined by the eigenvalues of $M_{ab}^2$ and the no-tachyon condition is the positive definiteness of $M_{ab}^2$ .",
"We immediately obtain the same consistency conditions $P_X(X<0)>0\\,, \\quad P_{XX}(X<0)>0$ from the UV stability conditions $f>0$ and $M_{ab}^2>0$ around the spacelike background.",
"We then discuss the timelike background.",
"It is convenient to use the Hamiltonian to find the stability conditions around the timelike background $C_\\mu = (C_0 , \\mathbf {0})$ .",
"Before discussing the multi-field system, let us revisit the Hamiltonian of the two-field system, $\\mathcal {H}^{(2)}_{\\rm UV}&=\\frac{1}{2} \\left|p \\right|^2+\\frac{1}{2} M^2 \\left|\\delta \\chi \\right|^2 + \\frac{1}{2f} \\left|p_{\\pi }-f_{\\chi } C_0 \\delta \\chi \\right|^2$ where we have taken the limit $k \\rightarrow 0$ since we are interested in the stability of the zero momentum modes.",
"The naive boundedness of the Hamiltonian, i.e.",
"$M^2>0$ , does not tell us the stability condition of the heavy mode since we have seen that $M^2<0$ is allowed.",
"What we need to see is the boundedness of the Hamiltonian incorporating its dynamics.",
"The conjugate $p_{\\pi }$ is a constant of motion in $k=0$ .",
"The solution describing the coherent oscillation of the heavy mode around the constant background can always be taken to be $p_{\\pi }=0$ by absorbing any non-zero constant into a shift of the background $\\bar{\\chi }$ .",
"Therefore, the stability of the heavy mode follows the boundedness of the Hamiltonian after substituting $p_{\\pi }=0$ , $\\left.",
"\\mathcal {H}^{(2)}_{\\rm UV} \\right|_{p_{\\pi }=0}&=\\frac{1}{2} \\left|p \\right|^2+\\frac{1}{2} \\left(M^2 + \\frac{f_{\\chi }^2}{f}C_0^2 \\right) \\left|\\delta \\chi \\right|^2\\Rightarrow M^2 + \\frac{f_{\\chi }^2}{f}C_0^2 >0\\,,$ which correctly reproduces the condition $\\mathcal {M} ^2>0$ .",
"Returning to the multi-field models, we study the Hamiltonian of this system and impose the boundedness of the Hamiltonian (no-ghost and no-tachyon conditions) in the UV.",
"The conjugate momenta are $p_a&=\\gamma _{ab}\\delta \\dot{\\chi }^b+h_{[ab]}C_0 \\delta \\chi ^b\\,, \\\\p_{\\pi }&=f\\dot{\\pi }+f_a C_0 \\delta \\chi ^a\\,,$ and the Hamiltonian is $\\mathcal {H}^{(2)}_{\\rm UV} &=\\frac{1}{2}\\gamma _{ab} \\delta \\dot{\\chi }^a{}^{\\dagger } \\delta \\dot{\\chi }^b+\\frac{1}{2}f |\\dot{\\pi }|^2+ \\frac{1}{2}(k^2 \\gamma _{ab}+M^2_{ab})\\delta \\chi ^a{}^{\\dagger } \\delta \\chi ^b+ \\frac{1}{2}k^2 f |\\pi |^2\\,,$ where $\\delta \\dot{\\chi }^a$ and $\\dot{\\pi }$ are understood as the functions of the canonical variables.",
"The boundedness of the Hamiltonian for the mode $k=0$ with $p_{\\pi }=0$ requires $\\mathcal {M} ^2_{ab}&>0\\,, \\quad \\mathcal {M} ^2_{ab}\\equiv M^2_{ab}+\\frac{f_af_b}{f}C_0^2\\; ,$ at least under the situation we are considering.",
"The condition $M^2_{ab}>0$ is sufficient but not necessary, just like the two-field model.",
"On the other hand, the Hamiltonian is unbounded at low $k$ if $M^2_{ab}$ is indefinite, leading to an instability at IR, whose threshold is determined by ${\\rm det} (k^2 \\gamma _{ab}+M^2_{ab})=0\\,.$ Since the heavy modes are stable at $k=0$ , such unstable modes must be the IR modes of the light degree of freedom $\\pi $ .",
"This fact still renders the EFT reduction consistent, as the UV sector suffers no pathological behavior.",
"We finally show that $P_{XX}$ is negative if $M_{ab}$ contains a negative eigenvalue.",
"For this purpose, we use the freedom of the change of $\\chi ^a$ in the following sequence.",
"Performing a general linear transformation of $\\delta \\chi ^a$ , we can diagonalize the matrix $M^2_{ab}$ and set $f_a=(f_1,f_{a^{\\prime }})=(f_1,0)$ simultaneously where the indices with a prime run over $a^{\\prime }=2,3,\\cdots $ .",
"Denoting the eigenvalues of $M_{ab}^2$ by $M_1^2$ and $M^2_{a^{\\prime }}~(a^{\\prime }=2,3,\\cdots )$ , the stability condition $\\mathcal {M} ^2_{ab}>0$ is reduced to $\\mathcal {M} ^2_1 = M_{1}^2+\\frac{f_1^2}{f} C_0^2>0 \\,, \\quad M^2_{a^{\\prime }}>0\\,.$ Hence, the number of negative eigenvalues of $M_{ab}^2$ is at most one, which is $M_1^2$ in this basis.",
"With this choice of field-space coordinates, we simply have $P_{XX}=f_a (M^{-2})^{ab} f_b = \\frac{f_1^2}{M_1^2}\\,,$ that is, the sign of $M_1^2$ is that of $P_{XX}$ .",
"The condition $\\mathcal {M} _1^2>0$ yields $|P_{XX}|>P_X/2X$ as a consistency in the case of $P_{XX}<0$ .",
"All in all, the same consistency conditions are obtained as in the two-field UV model.",
"The condition $P_X>0$ is universal in both spacelike and timelike cases.",
"In addition, the consistency condition on $P_{XX}$ is $P_{XX}>0$ around the spacelike background while the negative $P_{XX}$ is allowed around the timelike background.",
"The positive region $P_{XX}>0$ and the negative region $P_{XX}<0$ are disconnected since the consistency condition is either $P_{XX}>0$ or $P_{XX}<-P_X/2X<0$ .",
"Note that these conditions are necessary conditions and may be sharpened by studying detailed spectra of the UV theory as we did in the case of the two-field UV model."
],
[
"Role of higher derivative operators",
"We have seen that the IR instabilities around the spacelike and timelike backgrounds are qualitatively different: the threshold of the former instability can be chosen so that $k_*^2\\approx M^2$ , where a finite number of higher derivatives well approximate the dynamics, while the latter one requires a resummation of higher derivative operators to cure it.",
"In this subsection, we discuss how this distinction can be understood.",
"We consider the k-essence part of the quadratic Lagrangian $\\mathcal {L}_{\\rm IR}^{(2)}=-\\frac{1}{2}\\left[P_X(\\partial \\pi )^2-P_{XX}(C\\cdot \\partial \\pi )^2+\\cdots \\right] = -\\frac{1}{2}g^{\\mu \\nu }_{\\rm eff}\\partial _{\\mu }\\pi \\partial _{\\nu } \\pi +\\cdots \\,,$ where $g^{\\mu \\nu }_{\\rm eff}\\equiv P_X \\eta ^{\\mu \\nu } - P_{XX} C^{\\mu }C^{\\nu }\\; ,$ and we remind $C_\\mu \\equiv \\partial _\\mu \\bar{\\varphi }$ and $\\eta ^{\\mu \\nu }$ is the Minkowski metric.",
"We study a $1+1$ dimensional spacetime for simplicity of the explanation, but essentially the same discussions follow in general dimensions.",
"In both spacelike and timelike cases, the IR instability exists if $P_X + 2XP_{XX}<0$ is satisfied in terms of the k-essence coefficient.",
"This parameter region leads to the gradient instability around the spacelike background, $C_{\\mu }=(0,\\sqrt{-2X})$ , because the effective metric is given by $g^{\\mu \\nu }_{\\rm eff}&=\\begin{pmatrix}-P_X & 0 \\\\0 & P_X+2XP_{XX}\\end{pmatrix}=\\begin{pmatrix}- & 0 \\\\0 & -\\end{pmatrix}\\; , \\qquad \\text{(spacelike background)}\\,.$ On the other hand, the unstable parameter corresponds to the ghost instability around the timelike background, $C_{\\mu }=(\\sqrt{2X},0)$ , $g^{\\mu \\nu }_{\\rm eff}&=\\begin{pmatrix}-(P_X+2XP_{XX}) & 0 \\\\0 & P_X\\end{pmatrix}=\\begin{pmatrix}+ & 0 \\\\0 & +\\end{pmatrix}\\; , \\qquad \\text{(timelike background)}\\,.$ Therefore, since the no ghost condition in the UV ensures $P_X > 0$ , the IR instabilities can be distinguished by whether the instability is caused by flipping the sign of the spatial derivative (spacelike background) or the sign of the time derivative (timelike background).",
"The gradient instability in EFT can be cured by adding a local higher derivative operator, à la ghost condensate [29], [30].",
"In the present case, such a higher derivative operator is $\\partial _{\\mu }X\\partial ^{\\mu }X$ rather than $(\\Box \\varphi )^2$ , see (REF ).",
"Considering the perturbations $\\pi $ around the spacelike background $C_{\\mu }=(0,C_i)$ , the higher derivative operator $(\\partial X)^2$ yields $-C^i \\partial _i \\dot{\\pi } C^j \\partial _j \\dot{\\pi } + C^i \\partial _k \\partial _i \\pi C^j \\partial ^k \\partial _j \\pi $ at the quadratic order.",
"In general, higher derivative operators should be treated as perturbations since the higher derivative terms yield Ostrogradsky ghost state(s), which in turn indicate that the regime of validity of the EFT is restricted below the ghost state(s).",
"On the other hand, no higher time derivative term is generated by $(\\partial X)^2$ at the quadratic order around the purely spacelike background $C_0=0$ (while the higher time derivatives appear at nonlinear level of perturbations).",
"The contribution from $(\\partial X)^2$ can consistently dominate over the “leading” lower derivative contribution $g^{\\mu \\nu }_{\\rm eff}\\partial _{\\mu }\\pi \\partial _{\\nu } \\pi $ and can change the sign of the gradient term.",
"After adding the higher derivative operator $(\\partial X)^2$ , the quadratic Lagrangian (REF ) around the spacelike background is explicitly given by $\\mathcal {L}^{(2)}_{\\rm IR}=\\frac{1}{2}\\left[ \\left( P_X + \\frac{2|X|P_{XX}}{M^2}k^2 \\right) |\\dot{\\pi }|^2 - \\left( (P_X+2XP_{XX}) k^2 + \\frac{2|X|P_{XX}}{M^2} k^4 \\right) |\\pi |^2 +\\cdots \\right]$ in the momentum space where we have $k^2=k_{\\parallel }^2$ since we are considering $1+1$ dimensions, for simplicity.",
"Even if there is the IR instability, $P_{X}+2XP_{XX}<0$ , the presence of $k^4$ term cures the instability within the regime of validity of the EFT.",
"On the other hand, the IR ghost instability can be resolved by using the idea of Jeans ghost [61].",
"Even though the Lagrangian in terms of $\\pi $ has the ghost instability below some value of $k$ , we can perform the canonical transformation to write the action in terms of $\\tilde{\\pi }$ as in the form of (REF ).",
"The kinetic term of (REF ) is positive definite, and thus there is no ghost instability.",
"The gradient term is proportional to $\\Omega _{\\pi }^2=c_s^2 k^2 + \\frac{\\delta M^4}{\\mathcal {M} ^6}k^4 +\\cdots $ where $P_X+2XP_{XX}<0$ corresponds to $c_s^2<0$ .",
"The IR ghost instability is translated into the IR gradient instability via the canonical transformation which is as harmless as the IR instability around the spacelike background.",
"There is, however, a clear distinction between the remedies of the IR gradient and ghost instabilities.",
"Let us discuss the role of the higher derivative operator $(\\partial X)^2$ around the timelike background.",
"In contrast to the spacelike background, $(\\partial X)^2$ in the timelike background yields higher time derivative term $|\\ddot{\\pi }|^2$ in addition to $k^2|\\dot{\\pi }|^2$ .",
"Nonetheless, this is not a crucial difference since we may treat $|\\ddot{\\pi }|^2$ as perturbations when the Lorentz symmetry is largely broken, $|c_s^2| \\ll 1~\\Rightarrow |\\ddot{\\pi }|^2 \\ll k^2|\\dot{\\pi }|^2$ , as used to derive (REF ).",
"The crucial point is the strong coupling.",
"In the example (REF ), the sign of the time derivative is negative even if $k^2|\\dot{\\pi }|^2$ is taken into account.",
"Hence, we try to consider the EFT with a “wrong” sign of the $(\\partial X)^2$ operator to change the sign of the kinetic term: $\\mathcal {L}^{(2)}_{\\rm IR} &= \\frac{1}{2}\\left[ (P_X+2XP_{XX})|\\dot{\\pi }|^2 + \\frac{2XP_{XX}}{M^2}k^2|\\dot{\\pi }|^2 +\\cdots \\right]\\nonumber \\\\&\\simeq \\frac{1}{2} \\frac{2XP_{XX}}{M^2}\\left( -|M^2| + k^2 \\right) |\\dot{\\pi }|^2 +\\cdots $ where $k^2|\\dot{\\pi }|^2$ is taken to be opposite in sign to (REF ) and we have used $P_X+2XP_{XX}\\simeq 2XP_{XX}$ and $M^2<0$ .",
"Then, one realizes that the sign of the kinetic tern cannot be flipped since the light mode $\\pi $ is infinitely strongly coupled at the critical point $k^2-|M^2|=0$ .",
"The EFT description breaks down before approaching $k^2-|M^2|=0$ : the ghost cannot be cured by adding a local higher derivative operator, contrary to the gradient instability.",
"On the other hand, one may consider a resummation of spatial derivative operators to construct the non-local kinetic term $\\mathcal {L}^{(2)}_{\\rm IR} \\simeq \\frac{1}{2} \\frac{2XP_{XX} M^2}{k^2-|M^2|} |\\dot{\\pi }|^2 + \\cdots $ just like (REF ).",
"The critical point $k^2-|M^2|=0$ now corresponds to the infinitely weakly coupling, implying that the IR ghost can be resolved be the use of a non-local operator within the EFT.",
"In fact, this is what we have observed in the EFT from our concrete UV models (see also the theories in [61]).",
"Alternatively, one may consider a canonical transformation to translate the IR ghost instability into the IR gradient instability.",
"Nevertheless, such a canonical transformation should be a non-local map of the on-shell variables as we have seen in (REF ) and the non-locality again plays an important role in this procedure."
],
[
"UV obstruction to screening",
"Nonlinear kinetic terms have gained attention phenomenologically since it can provide a screening mechanism, in which the fifth force carried by a light scalar can be screened at a sufficiently small scale.",
"It has been known, however, that the requirement from the positivity bounds is incompatible with a successful screening at least in some particular theories [56], [62].",
"Here, we discuss whether or not this obstruction is a general feature under our UV consistency conditions.",
"We study a four-dimensional spacetime to make discussions concrete and assume that the light field $\\varphi $ is canonically normalized.",
"We consider a situation when the light field $\\varphi $ couples with a localized source via the interaction $\\frac{1}{M_{\\rm Pl}} \\varphi T$ where $T$ is the trace of the energy-momentum tensor of the source.",
"Then, the field $\\varphi $ gives rise to a force in addition to the standard gravitational force.",
"In the asymptotic region of the spacetime outside the source, the equation of motion of $\\varphi $ may be approximated by the linear equation, $\\Box \\varphi = 0 $ , whose solution should behave as $\\varphi \\propto r^{-1}$ for large $r$ , where $r$ is the distance from the localized source.",
"Both the fifth force and the gravitational force obey the Newtonian law $r^{-2}$ and are comparable in the asymptotic region.",
"As we get closer to the source, the gradient of $\\varphi $ becomes larger and larger, and at some point nonlinear kinetic terms can no longer be ignored, modifying the scaling law of $\\varphi $ .",
"The expectation is that nonlinear kinetic terms may modify $\\varphi $ by making its distant behavior milder than $r^{-1}$ , so that the force carried by $\\varphi $ is smaller than the gravitational force in the small scales.",
"For instance, the model $P(X)=X-X^2/M^4$ leads to $\\varphi \\propto r^{1/3}$ in the nonlinear region, $\\partial \\varphi \\gtrsim M^2$ , which is a successful example of the screening by the nonlinear kinetic term $-X^2/M^4$ [56].",
"The EFT can be valid even in the nonlinear region as far as higher derivatives are small, $\\partial ^n \\varphi \\ll M^{n+1}~(n\\ge 2)$ .",
"The existence of the screening can be understood by the weak coupling (see e.g. [63]).",
"We may split the field and the source into backgrounds $\\lbrace \\bar{\\varphi }, \\bar{T} \\rbrace $ and perturbations $\\lbrace \\pi , \\delta T \\rbrace $ .",
"Due to the presence of the nonlinear kinetic terms, the fluctuations of the scalar field propagate on the effective metric, not the spacetime metric, leading to $\\mathcal {L}^{(2)}_{\\rm IR}=-\\frac{1}{2}g_{\\rm eff}^{\\mu \\nu }(\\bar{\\varphi })\\partial _{\\mu } \\pi \\partial _{\\nu } \\pi + \\frac{1}{M_{\\rm Pl}} \\pi \\delta T\\,,$ where we ignore the higher derivative terms.",
"As we have seen, the k-essence theory yields $g_{\\rm eff}^{00} = -P_X$ around the spacelike background.",
"The model $P(X)=X-X^2/M^4$ reads $P_X \\rightarrow 1 $ as $X\\rightarrow 0$ (the asymptotic region) while $P_X \\gg 1$ as $|X| \\gg M^4$ (the nonlinear region).",
"We then introduce the canonically normalized field via $\\hat{\\pi } \\equiv \\sqrt{P_X} \\pi \\,,$ by which the Lagrangian is written as $\\mathcal {L}^{(2)}_{\\rm IR}=-\\frac{1}{2}\\hat{g}_{\\rm eff}^{\\mu \\nu }(\\bar{\\varphi })\\partial _{\\mu } \\hat{\\pi } \\partial _{\\nu } \\hat{\\pi } + \\frac{1}{ M_{\\rm Pl}\\sqrt{P_X}} \\hat{\\pi } \\delta T+\\cdots \\,.$ with $\\hat{g}^{00}_{\\rm eff}=-1$ where $\\cdots $ are lower derivative terms of $\\hat{\\pi }$ .",
"One can see that the effective coupling constant is given by $(M_{\\rm Pl}\\sqrt{P_X})^{-1}$ which is much smaller than the gravitational one $M_{\\rm Pl}^{-1}$ in the nonlinear region, $P_X \\gg 1$ .",
"Indeed, $P_X$ is nothing but the coefficient in front of the kinetic term of $\\varphi $ before integrating out the heavy modes in our partial UV models, cf.",
"(REF ).",
"This means that $P_X=f \\gg 1$ is the weak coupling limit of $\\varphi $ .",
"We then use the UV consistency condition $P_{XX}(X<0)>0$ which should hold for a spacelike configuration of $\\partial _{\\mu } \\varphi $ (recall that $P_{XX}<0$ is allowed only around timelike backgrounds, $X>0$ ).",
"As we explained, the scalar field should obey the Newtonian law $\\varphi \\propto r^{-1}$ in the asymptotic region, implying $-X|_{ \\text{at a finite } r} > -X|_{r\\rightarrow \\infty }=0\\,.",
"$ Since the condition $P_{XX}>0$ implies that $P_X$ is a monotonically increasing function of $X$ , the inequality (REF ) concludes $P_X|_{\\text{at a finite } r} < P_X|_{r\\rightarrow \\infty }\\,.$ As a result, the UV consistency condition $P_{XX}>0$ obstructs the requirement for the screening, $P_X|_{\\text{at a finite } r} > P_X|_{r\\rightarrow \\infty }$ .",
"The fifth force at a finite $r$ ($X<0$ ) should be stronger than the force in the asymptotic region $X\\rightarrow 0$ , which is a requirement from admitting a UV completion at least by means of multi-field models.",
"When this manuscript was being completed, the paper [64] appeared in arXiv which would contradict our discussion about the screening.",
"The paper [64] uses the result of [11], which claims that, under the dispersion relation $\\omega ^2=c_s^2 k^2$ , there exist positivity constraints on four-point amplitudes even around the timelike background, and applies the result into spacelike configurations of $\\partial _{\\mu }\\varphi $ .",
"A claim of [64] is that the positivity bounds can be compatible with the screening in models $P(X)=X+c_n X^n$ with an odd number $n$ (note that our definition of $X$ is opposite in sign to their definition of $X$ ).",
"However, as we have clarified in this paper, the dispersion relation has a rich structure which is inevitably nonlinear around Lorentz-violating backgrounds.",
"Careful analysis should be required to obtain generic consistency conditions which we will further discuss in Sec.",
"REF .",
"Furthermore, the constraints obtained around the timelike background cannot be naively applied for the spacelike configurations.",
"For instance, as for the models $P(X)=X+c_n X^n$ with an odd number $n$ , our bound for a stable EFT, $P_{XX}>0$ , reads $c_n>0$ around the timelike background while reads $c_n<0$ around the spacelike background.",
"Recall that the EFT may be consistent only around particular backgrounds, see e.g. Fig.",
"REF and Appendix .",
"If one chooses the bound obtained around the timelike background, $c_n>0$ , and naively extrapolates the theory into the spacelike background, the model looks compatible with the requirement from the screening, $P_{XX}<0$ with $X<0$ .",
"However, the model with $c_n>0$ is inconsistent around the spacelike background at least in our consistency conditions, and cannot be used for screening phenomena.",
"The consistent sign around the spacelike background is $c_n<0$ which is incompatible with the screening as we have discussed with a more general argument.",
"Nonetheless, our analysis relies on a particular class of the partial UV completion and it must be interesting to investigate a theory with the screening mechanism to be consistent with the UV requirements."
],
[
"Null energy condition",
"Gravity can play an important role implicitly through the energy conditions.",
"In this subsection we thus consider the implication of the null energy condition (NEC) in our framework.",
"Since the class of (partial) UV completion considered in the present paper does not involve any sources of NEC violation, the corresponding single-field EFT at low energy should also respect NEC.",
"The stress energy tensor in the UV theory and that in the single-field EFT are defined respectively by $T_{\\rm UV}^{\\mu \\nu } = \\frac{2}{\\sqrt{-g}}\\frac{\\delta S_{\\rm UV}}{\\delta g_{\\mu \\nu }}\\,, \\quad T_{\\rm IR}^{\\mu \\nu } = \\frac{2}{\\sqrt{-g}}\\frac{\\delta S_{\\rm IR}}{\\delta g_{\\mu \\nu }}\\,.$ They should agree with each other in the regime of validity of the EFT: $T_{\\rm UV}^{\\mu \\nu }\\simeq T_{\\rm IR}^{\\mu \\nu }$ .",
"For the UV theory (REF ) that we have assumed throughout the present paper, we have $T_{\\rm UV}^{\\mu \\nu } = \\gamma _{AB}\\nabla ^{\\mu }\\Phi ^A\\nabla ^{\\nu }\\Phi ^B + \\mathcal {L}_{UV}g^{\\mu \\nu }\\,,$ and thus NEC is respected: $T_{\\rm UV}^{\\mu \\nu } \\ell _{\\mu } \\ell _{\\nu } = \\gamma _{AB} (\\ell \\cdot \\partial \\Phi ^A) (\\ell \\cdot \\partial \\Phi ^B) \\ge 0\\,,$ for an arbitrary null vector $\\ell ^{\\mu }$ , provided that the matrix $\\gamma _{AB}$ is positive definite.",
"Therefore, all single-field EFTs that result from this UV theory inevitably satisfy the null energy condition.",
"Indeed, for the EFT Lagrangian $\\mathcal {L}_{\\rm IR} = P(X)$ and the background $\\partial _{\\mu }\\varphi = C_{\\mu }$ , the energy-momentum tensor is $T_{\\rm IR}^{\\mu \\nu } = P_X C^{\\mu }C^{\\nu }+g^{\\mu \\nu } P\\,,$ and thus $T_{\\rm IR}^{\\mu \\nu }\\ell _{\\mu }\\ell _{\\nu }=P_X (C \\cdot \\ell )^2 \\ge 0\\,,$ for an arbitrary null vector $\\ell ^{\\mu }$ .",
"Here, we have used $P_X>0$ .",
"To be more concrete, let us consider the local description of the $\\Lambda $ -EFT (REF ).",
"Considering the case $|M^2|\\ll \\mathcal {M} ^2$ , the quadratic Lagrangian looks similar to that of ghost condensate, $\\mathcal {L}_{\\rm GC}=\\tilde{P}(\\tilde{X}) - \\frac{1}{2\\tilde{\\mathcal {M} }^2} (\\Box \\tilde{\\varphi })^2 +\\cdots \\,, $ where $\\cdots $ are even higher derivative terms.",
"The quadratic Lagrangian of the ghost condensate for the perturbation $\\tilde{\\varphi }=\\bar{\\varphi }+\\tilde{\\pi }$ is $\\mathcal {L}_{\\rm GC}^{(2)}=\\frac{1}{2}\\left[ (\\tilde{P}_X+2 \\tilde{X}\\tilde{P}_{XX}) \\left|\\dot{\\tilde{\\pi }} \\right|^2- \\left( \\tilde{P}_X k^2 + \\frac{1}{\\tilde{\\mathcal {M} }^2} k^4 \\right) \\left|\\tilde{\\pi } \\right|^2 +\\cdots \\right] \\; ,$ where dots include higher orders in perturbations and in time derivatives.",
"It might look plausible to identify the $\\Lambda $ -EFT (REF ) with the ghost condensate.",
"However, this is not possible, as we shall see now.",
"Around the timelike background, the sound speed of perturbations is given by $c_s^2\\equiv P_X/(P_X+2XP_{XX})$ , and from a low-energy perspective there are in principle two possible realizations of the imaginary sound speed, $P_X<0$ or $P_X+2XP_{XX}<0$ .",
"The first case, which violates NEC, is known in the context of ghost condensate [29], [30] with the soft breaking of the shift symmetry [36] If the shift symmetry is exact then the averaged NEC holds [32]., and the latter one, which possesses the IR ghost, is discussed in [60].",
"These EFTs can be distinguished by the way how the imaginary sound speed is realized: $\\tilde{P}_X>0$ and $\\tilde{P}_X+2 \\tilde{X}\\tilde{P}_{XX}<0$ in the case of the $\\Lambda $ -EFT; and $\\tilde{P}_X<0$ and $\\tilde{P}_X+2 \\tilde{X}\\tilde{P}_{XX}>0$ in the case of the ghost condensate.",
"This means that the overall sign of the quadratic action is opposite in these two EFTs.",
"As a result, physical degrees of freedom in these two EFTs gravitate differently.",
"(We provide a further comparison in Appendix .)",
"Therefore, our UV models cannot deduce the ghost condensate in the low-energy limit since NEC is preserved.",
"On the other hand, the latter one is compatible with the assumed (partial) UV completion, even though it apparently contradicts the naively applied positivity bound $P_{XX}>0$ .",
"While our results are obtained from a particular class of partial UV completion, still they generically hold for nonlinear sigma models independently from the number of the fields, the field space metric, and the potential (see also Appendix for the DBI-type generalization in which the same consistency conditions are obtained).",
"It would be interesting to investigate whether our UV consistency conditions have to hold even for other types of UV completion.",
"It is also important to stress that, even if the same conditions are obtained from generic assumptions, one should not a priori exclude other possibilities like the EFT of ghost condensate [29], [30], the scordatura theory [65], [66], a stable violation of NEC [36], and theories exhibiting the screening mechanism [55], [56], as long as they are consistent on their own.",
"Rather, any observational signatures of these theories should be understood as a smoking gun for a richer structure in the UV, for example, sources of NEC violation such as objects with negative tension motivated by orientifold constructions in string theory [67]"
],
[
"Diagrammatic understanding",
"We revisit our UV consistency conditions from a different perspective in this subsection.",
"So far, we have avoided the use of amplitudes because we are interested in non-trivial backgrounds where there could in general be subtleties to define the S-matrix in Lorentz-violating systems, and we have mainly considered the linear order of perturbations.",
"In this section, we re-discuss the two-field model and the UV consistency of the k-essence by using the Feynman-like diagrams, putting aside the subtleties about the rigorous definitions.",
"This provides simple and intuitive explanations to the results of the previous sections and explain why we have discussed the quadratic Lagrangian rather than the four-point interaction.",
"The essential difference from Lorentz-symmetric cases is the existence of the mixing term $-f_{\\chi } (C\\cdot \\partial \\pi ) \\delta \\chi $ in (REF ), where $C_{\\mu }=\\partial _{\\mu }\\bar{\\varphi }\\ne 0$ is the origin of the spontaneous Lorentz-symmetry breaking.",
"The dispersion relation of the two-field model around a background $C_{\\mu }\\ne 0$ is given by a root of ${\\rm det}\\begin{pmatrix}p^2+M^2 & f_{\\chi } (ip\\cdot C) \\\\-f_{\\chi } (ip \\cdot C) & p^2 f\\end{pmatrix}&=(p^2+M^2)\\left[ p^2f - \\Pi (p) \\right] &(p^2+M^2 \\ne 0)\\nonumber \\\\&=0\\,,$ where $i\\Pi (p)&=i\\frac{f_{\\chi }^2 (p\\cdot C)^2}{p^2+M^2}\\,.$ This result is diagrammatically reproduced by $\\frac{i\\Pi (p)}{|i p\\cdot C|^2} &=\\begin{tikzpicture}[baseline=0]\\begin{feynhand}[particle] (i1) at (0,0) {};[particle] (o1) at (3,0) {};[crossdot] (a1) at (1,0) {} ;[crossdot] (a2) at (2,0) {} ;[plain] (i1) to (a1);[scalar] (a1) to (a2);[plain] (a2) to (o1);\\end{feynhand}\\end{tikzpicture}\\,, \\\\$ with the rules $\\begin{tikzpicture}[baseline=0]\\begin{feynhand}[dot] (a1) at (1,0) ;[dot] (a2) at (2,0) ;[scalar] (a1) to (a2);\\end{feynhand}\\end{tikzpicture}= \\frac{-i}{p^2+M^2}\\,, \\qquad \\otimes = -if_{\\chi }\\,,$ where the left-hand side is divided by $|i p\\cdot C|^2$ so that the vertex $\\otimes $ is normalized not to include the factor $ip\\cdot C$ .",
"The dashed line is interpreted as the propagator of $\\delta \\chi $ while the solid line is the external leg of $\\pi $ .",
"We use this diagrammatic notation to compute the leading-order EFT coefficients, $P^{(n)}\\equiv \\mathrm {d}^n P/d X^n$ .",
"We consider the leading-order EFT, namely the k-essence theory $\\mathcal {L}_{\\rm IR}=P(X)$ .",
"The coefficients $P^{(n)}~(n\\ge 2)$ are the coefficients of the $n$ -point interactions, $\\mathcal {L}_{\\rm IR} \\supset (-1)^n \\frac{P^{(n)}}{n!}",
"(C\\cdot \\partial \\pi )^n$ .",
"Since the EFT is obtained by integrating out the heavy modes, $P^{(n)}$ can be computed as the Feynman diagrams of which all $n$ external legs are connected to $C\\cdot \\partial \\pi $ , while the internal lines are the UV mode $\\delta \\chi $ .",
"We define $\\mathcal {V}(\\chi ,X)\\equiv V(\\chi )-f(\\chi )X$ and its derivatives, $\\mathcal {V}_{\\chi \\chi }=\\left.",
"\\frac{\\partial ^2 \\mathcal {V}}{\\partial \\chi ^2} \\right|_{\\chi =\\chi _0(X)}\\,, \\quad \\mathcal {V}_{\\chi \\chi \\chi }=\\left.",
"\\frac{\\partial ^3 \\mathcal {V}}{\\partial \\chi ^3} \\right|_{\\chi =\\chi _0(X)}\\,, \\cdots \\,,\\mathcal {V}^{(n)}=\\left.",
"\\frac{\\partial ^n \\mathcal {V}}{\\partial \\chi ^n} \\right|_{\\chi =\\chi _0(X)}\\,.$ We put the superscript $(n)$ to denote the general $n$ -th derivative of $f$ and $\\mathcal {V}$ with respect to $\\chi $ evaluated at $\\chi =\\chi _0(X)$ , instead of writing $n$ subscripts of $\\chi $ .",
"Note the relation $\\mathcal {V}_{\\chi \\chi }=M^2|_{\\chi =\\chi _0(X)}$ .",
"In the two-field (partial) UV models, all the interactions of perturbations are $-\\frac{f^{(n)}}{n!",
"}(C\\cdot \\partial \\pi ) \\delta \\chi ^n \\,, \\quad -\\frac{f^{(n)}}{2\\cdot n!",
"}(\\partial \\pi )^2 \\delta \\chi ^n\\,, \\quad -\\frac{\\mathcal {V}^{(n)}}{n!",
"}\\delta \\chi ^n$ around the constant background where the coefficients of $(C\\cdot \\partial \\pi )\\delta \\chi ^n$ and $(\\partial \\pi )^2 \\delta \\chi ^n$ are not independent due to the original implementation of the Lorentz invariance at UV.",
"The UV coefficients, $f^{(n)}$ and $\\mathcal {V}^{(n)}$ , can be arbitrary in the IR perspective unless additional assumptions on UV are imposed.",
"For instance, $P^{(2)}=P_{XX}$ is diagrammatically computed as $iP_{XX}(X\\ne 0) &=\\begin{tikzpicture}[baseline=0]\\begin{feynhand}[particle] (i1) at (0,0) { C \\!\\cdot \\!",
"\\partial \\pi };[particle] (o1) at (3,0) { C \\!\\cdot \\!",
"\\partial \\pi };[crossdot] (a1) at (1,0) {} ;[crossdot] (a2) at (2,0) {} ;[plain] (i1) to (a1);[scalar] (a1) to (a2);[plain] (a2) to (o1);\\end{feynhand}\\end{tikzpicture}= \\left|\\begin{tikzpicture} [baseline=-0.1cm]\\begin{feynhand}[particle] (i1) at (0,0) {C \\!\\cdot \\!",
"\\partial \\pi };[crossdot] (a1) at (1,0) {} ;[particle] (a2) at (2,0) {\\delta \\chi };[plain] (i1) to (a1);[scalar] (a1) to (a2);\\end{feynhand}\\end{tikzpicture}\\right|^2,$ where the propagator of $\\delta \\chi $ is approximated by $-i/M^2$ since we are focusing on the leading-order EFT coefficients.",
"We put $C\\cdot \\partial \\pi $ to the external legs to represent that the diagram corresponds to the EFT coefficient of $(C\\cdot \\partial \\pi )^2$ .",
"The $|{\\rm diagram}|^2$ means that the diagram is factorized by the same diagrams connected by a propagator which is positive, provided the condition $M^2>0$ .",
"The positivity of $P_{XX}$ is guaranteed by the factorization property of the diagram in addition to the positivity of $M^2$ .",
"This is the same as the diagrammatic explanation of the optical theorem.",
"Let us investigate third and fourth derivatives of $P(X)$ with respect to $X$ , namely $P_{XXX}$ and $P_{XXXX}$ .",
"From the relation $P=fX-V$ , by the use of the chain rule and the implicit function theorem, the third derivative of $P(X)$ is computed as $P_{XXX} &=\\frac{\\mathrm {d}}{\\mathrm {d}X}P_{XX}=\\left[ \\frac{\\partial }{\\partial X} \\left( \\frac{f_{\\chi } ^2}{M^2} \\right) + \\frac{\\mathrm {d}\\chi }{\\mathrm {d}X} \\frac{\\partial }{\\partial \\chi } \\left( \\frac{f_{\\chi }^2}{M^2} \\right) \\right]_{\\chi =\\chi _0(X)}\\nonumber \\\\&=\\left[3 \\, \\frac{f_{\\chi }^2f_{\\chi \\chi }}{M^4}-\\frac{\\mathcal {V}_{\\chi \\chi \\chi } f_{\\chi }^3}{M^6} \\right]_{\\chi =\\chi _0(X)}\\,,$ and then $P_{XXXX}&=\\left[ 4 \\, \\frac{f_{\\chi }^3 f_{\\chi \\chi \\chi }}{M^6} - \\frac{f_{\\chi }^4 \\mathcal {V}_{\\chi \\chi \\chi \\chi }}{M^8} + \\frac{3}{M^2} \\left( 2\\, \\frac{f_{\\chi } f_{\\chi \\chi }}{M^2} -\\frac{f_{\\chi }^2 \\mathcal {V}_{\\chi \\chi \\chi }}{M^4}\\right)^2 \\right]_{\\chi =\\chi _0(X)}\\,.$ The same results can be also computed by the use of Feynman-like diagrams as follows: $-iP_{XXX}(X\\ne 0)&=\\begin{tikzpicture}[baseline=0]\\begin{feynhand}[particle] (i1) at (-1,1) {C \\!\\cdot \\!",
"\\partial \\pi };[particle] (i2) at (-1,-1) {C \\!\\cdot \\!",
"\\partial \\pi };[particle] (i3) at (1.3,0) {C \\!\\cdot \\!",
"\\partial \\pi };[crossdot] (a1) at (-1/2,1/2) {};[crossdot] (a2) at (-1/2,-1/2) {};[dot] (b1) at (0,0) {};[plain] (i1) to (a1);[plain] (i2) to (a2);[plain] (i3) to (b1);[scalar] (a1) to (b1);[scalar] (a2) to (b1);\\node at (0.2,-0.3) {f_{\\chi \\chi }};\\end{feynhand}\\end{tikzpicture}+\\begin{tikzpicture}[baseline=0]\\begin{feynhand}[particle] (i1) at (-1,1) {C \\!\\cdot \\!",
"\\partial \\pi };[particle] (i2) at (-1,-1) {C \\!\\cdot \\!",
"\\partial \\pi };[particle] (i3) at (1.6,0) {C \\!\\cdot \\!",
"\\partial \\pi };[crossdot] (a1) at (-1/2,1/2) {};[crossdot] (a2) at (-1/2,-1/2) {};[crossdot] (a3) at (1.3/2, 0) {};[dot] (b1) at (0,0) {};[plain] (i1) to (a1);[plain] (i2) to (a2);[plain] (i3) to (a3);[scalar] (a1) to (b1);[scalar] (a2) to (b1);[scalar] (a3) to (b1);\\node at (0.3,-0.3) {\\mathcal {V}_{\\chi \\chi \\chi }};\\end{feynhand}\\end{tikzpicture}, \\\\iP_{XXXX}(X\\ne 0) &=\\begin{tikzpicture}[baseline=0]\\begin{feynhand}[particle] (i1) at (-1,1) {C \\!\\cdot \\!",
"\\partial \\pi };[particle] (i2) at (-1,-1) {C \\!\\cdot \\!",
"\\partial \\pi };[particle] (i3) at (1,-1) {C \\!\\cdot \\!",
"\\partial \\pi };[particle] (i4) at (1,1) {C \\!\\cdot \\!",
"\\partial \\pi };[crossdot] (a1) at (-1/2,1/2) {};[crossdot] (a2) at (-1/2,-1/2) {};[crossdot] (a3) at (1/2, -1/2) {};[dot] (b1) at (0,0) {};[plain] (i1) to (a1);[plain] (i2) to (a2);[plain] (i3) to (a3);[plain] (i4) to (b1);[scalar] (a1) to (b1);[scalar] (a2) to (b1);[scalar] (a3) to (b1);\\node at (0.6,0) {f_{\\chi \\chi \\chi }};\\end{feynhand}\\end{tikzpicture}+\\begin{tikzpicture}[baseline=0]\\begin{feynhand}[particle] (i1) at (-1,1) {C \\!\\cdot \\!",
"\\partial \\pi };[particle] (i2) at (-1,-1) {C \\!\\cdot \\!",
"\\partial \\pi };[particle] (i3) at (1,-1) {C \\!\\cdot \\!",
"\\partial \\pi };[particle] (i4) at (1,1) {C \\!\\cdot \\!",
"\\partial \\pi };[crossdot] (a1) at (-1/2,1/2) {};[crossdot] (a2) at (-1/2,-1/2) {};[crossdot] (a3) at (1/2, -1/2) {};[crossdot] (a4) at (1/2, 1/2) {};[dot] (b1) at (0,0) {};[plain] (i1) to (a1);[plain] (i2) to (a2);[plain] (i3) to (a3);[plain] (i4) to (a4);[scalar] (a1) to (b1);[scalar] (a2) to (b1);[scalar] (a3) to (b1);[scalar] (a4) to (b1);\\node at (0.7,0) {\\mathcal {V}_{\\chi \\chi \\chi \\chi }};\\end{feynhand}\\end{tikzpicture}+\\left|\\begin{tikzpicture}[baseline=0]\\begin{feynhand}[particle] (i1) at (-1,1) {C \\!\\cdot \\!",
"\\partial \\pi };[particle] (i2) at (-1,-1) {C \\!\\cdot \\!",
"\\partial \\pi };[particle] (i3) at (1.2,0) {\\delta \\chi };[crossdot] (a1) at (-1/2,1/2) {};[dot] (b1) at (0,0) {};[plain] (i1) to (a1);[plain] (i2) to (b1);[scalar] (i3) to (b1);[scalar] (a1) to (b1);\\node at (0.2,-0.3) {f_{\\chi \\chi }};\\end{feynhand}\\end{tikzpicture}+\\begin{tikzpicture}[baseline=0]\\begin{feynhand}[particle] (i1) at (-1,1) {C \\!\\cdot \\!",
"\\partial \\pi };[particle] (i2) at (-1,-1) {C \\!\\cdot \\!",
"\\partial \\pi };[particle] (i3) at (1.2,0) {\\delta \\chi };[crossdot] (a1) at (-1/2,1/2) {};[crossdot] (a2) at (-1/2,-1/2) {};[dot] (b1) at (0,0) {};[plain] (i1) to (a1);[plain] (i2) to (a2);[scalar] (a1) to (b1);[scalar] (a2) to (b1);[scalar] (b1) to (i3);\\node at (0.3,-0.3) {\\mathcal {V}_{\\chi \\chi \\chi }};\\end{feynhand}\\end{tikzpicture}\\right|^2,$ where each vertex has a $-i$ factor in addition to the specified factors.",
"We have not shown the crossed versions of the diagrams, e.g.",
"there are two other geometrically equivalent diagrams corresponding to the first diagram of (REF ).",
"Note that we can compute $P^{(n)}$ by using other diagrams.",
"For example, $P_{XXX}$ is the coefficient of $-\\frac{1}{2}(C\\cdot \\partial \\pi )^2 (\\partial \\pi )^2$ , and thus, $-iP_{XXX}(X\\ne 0)&=\\begin{tikzpicture}[baseline=0]\\begin{feynhand}[particle] (i1) at (-1.1,1.1) {C \\!\\cdot \\!",
"\\partial \\pi };[particle] (i2) at (-1.1,-1.1) { \\partial \\pi };[particle] (i3) at (1.1,-1.1) {\\partial \\pi };[particle] (i4) at (1.1,1.1) {C \\!\\cdot \\!",
"\\partial \\pi };[crossdot] (a1) at (-1/2,1/2) {};[crossdot] (a4) at (1/2,1/2) {};[dot] (b1) at (0,0) {};[plain] (i1) to (a1);[plain] (i4) to (a4);[scalar] (a1) to (b1);[plain] (i2) to (b1);[plain] (i3) to (b1);[scalar] (a4) to (b1);\\node at (0.5,0) {f_{\\chi \\chi }};\\end{feynhand}\\end{tikzpicture}+\\begin{tikzpicture}[baseline=0]\\begin{feynhand}[particle] (i1) at (-1.2,1.1) {C \\!\\cdot \\!",
"\\partial \\pi };[particle] (i2) at (-1.2,-1.1) { \\partial \\pi };[particle] (i3) at (1.2,-1.1) {\\partial \\pi };[particle] (i4) at (1.2,1.1) {C \\!\\cdot \\!",
"\\partial \\pi };[crossdot] (a1) at (-1.2*2/5, 0.3+0.8*2/5) {};[dot] (b1) at (0,0.3) {};[dot] (b2) at (0,-0.3) {};[plain] (i1) to (a1);[plain] (i4) to (b1);[plain] (i2) to (b2);[plain] (i3) to (b2);[scalar] (b1) to (b2);[scalar] (a1) to (b1);\\node at (0.1,0.7) {f_{\\chi \\chi }};\\node at (0,-0.6) {f_{\\chi }};\\end{feynhand}\\end{tikzpicture}+\\begin{tikzpicture}[baseline=0]\\begin{feynhand}[particle] (i1) at (-1.2,1.1) {C \\!\\cdot \\!",
"\\partial \\pi };[particle] (i2) at (-1.2,-1.1) { \\partial \\pi };[particle] (i3) at (1.2,-1.1) {\\partial \\pi };[particle] (i4) at (1.2,1.1) {C \\!\\cdot \\!",
"\\partial \\pi };[crossdot] (a1) at (-1.2*2/5, 0.3+0.8*2/5) {};[crossdot] (a4) at (1.2*2/5, 0.3+0.8*2/5) {};[dot] (b1) at (0,0.3) {};[dot] (b2) at (0,-0.3) {};[plain] (i1) to (a1);[plain] (i4) to (a4);[plain] (i2) to (b2);[plain] (i3) to (b2);[scalar] (b1) to (b2);[scalar] (a1) to (b1);[scalar] (a4) to (b1);\\node at (0.6,0.2) {\\mathcal {V}_{\\chi \\chi \\chi }};\\node at (0,-0.6) {f_{\\chi }};\\end{feynhand}\\end{tikzpicture}\\,.$ All the diagrammatic computations agree with the direct calculations (REF ) and (REF ) as they should be.",
"The essential point is the presence of the mixing term, $\\otimes $ .",
"The coefficients $P_{XXX},P_{XXXX}$ are generated by not only factorized exchanging diagrams but also contact diagrams since $C\\cdot \\partial \\pi $ can be transformed into $\\delta \\chi $ via the mixing.",
"The value and sign of $P^{(n)}$ with $n\\ge 3$ could be arbitrary unless there are additional constraints on both field space metric and potential.",
"It would be interesting to compare the $(\\partial \\pi )^4$ diagram, $iP_{XX}=\\left|\\begin{tikzpicture}[baseline=0]\\begin{feynhand}[particle] (i1) at (-0.7,0.7) {\\partial \\pi };[particle] (i2) at (-0.7,-0.7) {\\partial \\pi };[particle] (i3) at (1,0) {\\delta \\chi };[dot] (b1) at (0,0) {};[plain] (i1) to (b1);[plain] (i2) to (b1);[scalar] (b1) to (i3);\\node at (0.2,-0.3) {f_{\\chi }};\\end{feynhand}\\end{tikzpicture}\\right|^2\\,.$ The derivative interaction $(\\partial \\pi )^4$ is generated only through the $\\delta \\chi $ exchange which is factorized and guaranteed to be positive provided the positive mass square of $\\delta \\chi $ , $M^2>0$ .",
"In the Lorentz-invariant background, (REF ) is the only diagram of the four-point function of $\\pi $ because there is no mixing (and no potential here).",
"On the other hand, when $\\varphi $ has a non-vanishing gradient at the background, other diagrams contribute to the four-point function.",
"The only factorized diagram is the two-point diagram (REF ), which indeed yields the bound $P_{XX}>0$ provided $M^2>0$ .",
"The same is true for the general multi-field UV models: the difference is that each vertex has field space indices while the geometrical structure of the diagrams is the same.",
"This consideration suggests that the generic UV consistency conditions around Lorentz-violating backgrounds may be obtained by two-point functions, not four-point functions.",
"This should be regarded physically reasonable, since the linear perturbations can be scattered by the background field.",
"We will further discuss this point in the next subsubsection.",
"Note that, as shown in Sec.",
"REF , the EFT may be well defined even in the case of $M^2 < 0$ with a timelike Lorentz violation, and the domain $P_{XX} < 0$ does not necessarily conflict with the UV physics, as seen in (REF ).",
"We have the relation $P_{XX}=\\frac{f_{\\chi }^2}{M^2}$ , and the factorization property guarantees the positivity of the numerator only.",
"Our UV theories are just partial ones, in the sense that they still admit a large freedom for model parameters.",
"When one considers the fundamental theory, say quantum gravity, there could be non-trivial constraints on the structure of the field space and the form of the potential.",
"For instance, the $U(1)$ scalar $\\Phi =\\chi e^{i\\varphi }$ , which corresponds to a flat field space, with the Higgs-type potential $V=-\\frac{M_{\\Phi }^2}{2}|\\Phi |^2+\\frac{\\lambda }{4} |\\Phi |^4$ yields $P^{(n)}=0$ for $n>2$ .",
"In general, the multi-field theory should be allowed to have more complicated field space and potential, and the swampland conjectures aim to find general constraints on them.",
"We may thus be able to obtain constraints on higher derivative coefficients, $P^{(n)}$ , thanks to the consistency with UV, at least in principle.",
"Conversely, bottom-up constraints on $P^{(n)}$ with $n>2$ (either theoretically or observationally) may be translated into constraints on the field space metric and the potential of the UV theory."
],
[
"Toward bottom-up derivation",
"Let us make a similar argument to the derivation of the Lorentz-invariant positivity bounds to clarify similarities and differences in the Lorentz-violating case.",
"As we explained, we consider the two-point function rather than the four-point function.One might consider the four-point function by setting special configurations of momenta and/or taking appropriate subtractions so that the non-factorized diagrams do not contribute.",
"In general, the dispersion relation of the light mode around Lorentz-violating backgrounds may be represented by $p^2-\\Pi (p^{\\mu })=0\\,,$ in the adiabatic limit of the background where we have set $f=1$ by canonically normalizing the field $\\pi $ .",
"We assume that $\\Pi $ is a scalar function of $p^{\\mu }$ and $C_{\\mu }$ the latter of which determines the preferred direction.",
"Therefore, the “self-energy” $\\Pi $ must be a function of two variables defined by $s\\equiv -p_{\\mu }p^{\\mu }\\,, \\quad q \\equiv p^{\\mu }C_{\\mu }\\,.$ Then, $P_{XX}$ can be defined as $P_{XX}\\equiv \\left.",
"\\frac{1}{2} \\frac{\\partial ^2}{\\partial q^2}\\Pi (s,q) \\right|_{s=0,q=0}\\,,$ since $P_{XX}$ is the coefficient of $q^2$ in the dispersion relation (REF ).",
"Let us see properties of $\\Pi (s,q)$ in our partial UV models.",
"In the case of the general multi-field UV completion, $\\Pi $ is given by $\\Pi (s,q)=q^2 f_a (D^{-1})^{ab}f_{b}$ where $(D^{-1})^{ab}$ is the inverse of $D_{ab}(s,q)=-s\\gamma _{ab}+2i q h_{[ab]}+M^2_{ab}\\,.$ We can perform the field redefinition so that $\\gamma _{ab}=\\delta _{ab}={\\rm diag}[1,1,\\cdots ]\\,, \\quad M^2_{ab}={\\rm diag}[M_1^2,M_2^2,\\cdots ]\\,,$ that is, the fields $\\delta \\chi ^a$ are canonically normalized and diagonalized when $q=0$ .",
"We would like to clarify the poles and the residues of $\\Pi $ as a function of $s$ for a fixed $q$ .If $q$ is also regarded as a (complex) variable, the analytic structure of $\\Pi (s,q)$ would be more complicated.",
"See, e.g.",
"(REF ) below.",
"With the diagonalized $\\gamma $ and $M^2$ as in (REF ), the matrix $D_{ab}$ is diagonalizable by using a unitary matrix $U$ since it is hermitian (as far as $q$ is real), to have $\\tilde{D}_{ab}(s,q)=(U^{\\dagger }DU)_{ab}=-s\\delta _{ab}+\\tilde{M}_{ab}(q)\\,, \\quad \\tilde{M}^2_{ab}(q)={\\rm diag}[\\tilde{M}_1^2(q),\\tilde{M}_2^2(q),\\cdots ]$ where the components $\\tilde{M}^2_a~(a=1,2,\\cdots )$ are generically functions of $q$ when $h_{[ab]}\\ne 0$ , namely in the absence of the reflection symmetry on $\\varphi $ .",
"The values of $\\tilde{M}^2_a$ are real for a real $q$ since they are the eigenvalues of the hermitian matrix.",
"Note also that the unitary matrix $U(q)$ becomes the identity matrix when $q=0$ because $D_{ab}$ is already diagonal at $q=0$ .",
"Defining $\\tilde{f}_a=(Uf)_a$ , $\\Pi (s,q)$ is expressed as $\\Pi (s,q)= \\sum _{a}\\frac{|q\\tilde{f}_a(q)|^2}{\\tilde{M}_a^2(q) - s }\\,.$ This implies ${\\rm Im}\\, \\Pi (s,q) = \\sum _{a} \\pi \\delta (\\tilde{M}_a^2(q) - s) |q\\tilde{f}_a(q)|^2 \\ge 0$ when we add $+i\\epsilon $ to the denominator as usual.",
"These properties are indeed expected properties: the singularities of $\\Pi $ should arise from the “particle” exchanges and the imaginary part of singularity should be positive.",
"However, we recall again that ${\\rm det}D_{ab}=0\\Leftrightarrow s-\\tilde{M}_a^2=0$ is not an on-shell state of the heavy mode because of the coupling $\\delta \\chi ^a (C\\cdot \\partial \\pi )$ , that is, ${\\rm det}D_{ab}=0$ is not a solution to the original dispersion relation (REF ), in general.",
"On the other hand, $s-M_a^2=0$ with $q=0$ is a solution to the original dispersion relation (REF ) of the heavy mode.",
"In practice, the necessary properties for our purpose are $\\Pi ^{(2)}(s) &= \\sum _{a}\\frac{|f_a|^2}{M_a^2 - s }\\,, \\\\{\\rm Im}\\, \\Pi ^{(2)} (s) & =\\sum _{a} \\pi \\delta ( M_a^2 - s) |f_a|^2 \\ge 0\\,,$ where $\\Pi ^{(2)}(s)\\equiv \\left.",
"\\frac{1}{2}\\frac{\\partial ^2}{\\partial q^2} \\Pi (s,q) \\right|_{q=0}\\,.$ Here, we have used $\\tilde{f}_a|_{q=0} =f_a$ and $\\tilde{M}_a^2|_{q=0} =M^2_a$ .",
"In this case, the poles of $\\Pi ^{(2)}(s)$ correspond to $s-M_a^2=0$ with $q=0$ , and the imaginary part of $\\Pi ^{(2)}$ is still non-negative.",
"We also see the asymptotic behaviour $\\lim _{|s|\\rightarrow \\infty } \\Pi ^{(2)} \\rightarrow 0$ regarding $s$ as a complex variable.",
"This asymptotic behaviour could be expected because the Lorentz-violating correction to the dispersion relation (REF ) may vanish in the high-energy limit $|s|\\rightarrow \\infty $ while keeping $q$ finite.",
"As a result, we find $\\Pi ^{(2)}(s) &= \\oint _C \\frac{ds^{\\prime }}{2\\pi i} \\frac{ \\Pi ^{(2)}(s^{\\prime })}{s^{\\prime }-s} \\nonumber \\\\&= \\int _{-\\infty }^{\\infty } \\frac{ds^{\\prime }}{2\\pi i} \\frac{\\Pi ^{(2)}(s^{\\prime }+i\\epsilon )}{s^{\\prime }-s+i\\epsilon } + \\int _{\\infty }^{-\\infty } \\frac{ds^{\\prime }}{2\\pi i} \\frac{\\Pi ^{(2)}(s^{\\prime }-i\\epsilon )}{s^{\\prime }-s-i\\epsilon }+\\int _{\\mathcal {C}^{\\pm }_{\\infty }} \\frac{ds^{\\prime }}{2\\pi i} \\frac{ \\Pi ^{(2)}(s^{\\prime })}{s^{\\prime }-s}\\nonumber \\\\&= \\int _{-\\infty }^{\\infty } \\frac{ds^{\\prime }}{\\pi } \\frac{{\\rm Im}\\, \\Pi ^{(2)}(s^{\\prime })}{s^{\\prime }-s}$ where we use Cauchy's integral formula in the first equality and then we deform the integration contour to the contour along the real axis and the infinitely large semi-circles $\\mathcal {C}^{\\pm }_{\\infty }$ by using the fact that the poles $s=M_a^2$ exist on the real axis.",
"We finally use Schwarz reflection principle $\\Pi ^{(2)}(s^{\\prime }-i\\epsilon )={\\Pi ^{(2)}}^*(s^{\\prime }+i\\epsilon )$ and (REF ) to get the last expression.",
"This looks a deliberately complicated derivation of the relation (REF ) from () and (REF ).",
"However, this kind of deformation of the equation is the basis of the derivation of the positivity bounds in the Lorentz-invariant system.",
"In fact, the following three properties are sufficient to conclude $P_{XX}>0$ : (i) the singularities of $\\Pi ^{(2)}(s)$ are on the positive real axis; (ii) the imaginary part is non-negative; and (iii) the asymptotic behaviour $\\lim _{|s|\\rightarrow \\infty } \\Pi ^{(2)} \\rightarrow 0$ .",
"We then find $\\Pi ^{(2)}(s)=\\int ^{\\infty }_0 \\frac{ds^{\\prime }}{\\pi } \\frac{{\\rm Im}\\, \\Pi ^{(2)}(s^{\\prime })}{s^{\\prime }-s}\\Rightarrow P_{XX}=\\Pi ^{(2)}|_{s=0}= \\int ^{\\infty }_0 \\frac{ds^{\\prime }}{\\pi } \\frac{{\\rm Im}\\, \\Pi ^{(2)}(s^{\\prime })}{s^{\\prime }} > 0$ where ${\\rm Im}\\, \\Pi ^{(2)}(s)$ is assumed to take a non-zero value at some $s>0$ .",
"Note that the domain of the integration is $s>0$ since we have assumed that all the singularities exist on the positive real axis here.",
"These three assumptions are similar to the assumptions in the Lorentz-invariant positivity bounds.",
"However, as we have seen, an eigenvalue of $M^2_a$ can be negative around the timelike background, which means that there can exist a pole on the negative real axis.",
"Then, the positivity of $P_{XX}$ does not necessarily hold.",
"The fact that such a “tachyonic” pole is physically allowed is one of the major differences from the standard argument of the positivity bounds.",
"In this subsection, we have been identifying the poles with the on-shell heavy states by taking the limit $q\\rightarrow 0$ .",
"Part of our consistency conditions for the EFT reduction arise from the stability conditions of the heavy modes.",
"Therefore, if the limit $q \\rightarrow 0$ could be properly taken, tachyonic poles could not be identified as physical states, since they would have to represent unstable heavy state(s).",
"The limit would make sense around the spacelike background $C_{\\mu }=(0,C_i)$ , since $q=0$ only corresponds to a particular configuration of the momentum, $p^{\\mu }=(\\omega , k^i_{\\perp })$ .",
"Indeed, we have found that the positivity of $P_{XX}$ has to be satisfied in the spacelike case.",
"On the other hand, $q \\rightarrow 0$ would result in $p^{\\mu }=(0,k^i)$ around the timelike background $C_{\\mu }=(C_0,\\mathbf {0})$ , implying that the poles in the limit $q \\rightarrow 0$ could not be identified with on-shell heavy states.",
"Hence, there would be no reason to forbid a “tachyonic” pole in the timelike case.",
"In fact, we have found that non-zero $C_0$ stabilizes the heavy modes as seen in (REF ) and that the “tachyonic” pole is allowed around the timelike background at least in our models.",
"The purpose of this subsection is not to derive the positivity $P_{XX}>0$ (and the negativity $-P_{XX} > P_X/2X>0$ ) from general properties such as unitarity.",
"We have just illustrated similarities and differences from the standard argument of the positivity bounds around the Lorentz-invariant background based on our concrete UV setup.",
"In addition, we should carefully think of particles either when gravity is turned on or the shift symmetry is not exact, since the background should depend on time and/or space, i.e.",
"the translation invariance does not hold."
],
[
"Non-minimal coupling",
"Throughout the present work, we have assumed that scalar fields $\\Phi ^A$ minimally couple to gravity.",
"However, one can introduce a non-minimal coupling, say $\\chi ^2 R$ which is not forbidden by any symmetries and should be added in the spirit of EFT.",
"Nonetheless, we can perform a field redefinition to eliminate such a non-minimal coupling.",
"Then, the effects of the non-minimal coupling are embedded in the way of the matter coupling, $S_{\\rm m}=S_{\\rm m}[\\psi , g^J_{\\mu \\nu } ]\\,,$ where $g^J_{\\mu \\nu }$ is the Jordan frame metric which is not the same as $g_{\\mu \\nu }$ , in general.",
"When the Jordan frame metric depends on $\\chi $ , e.g.",
"$g^J_{\\mu \\nu }=\\Omega ^2(\\chi )g_{\\mu \\nu }$ , the matter fields act as a source of the equation of motion of $\\chi $ .",
"Under the condition that the backreaction from the matter fields onto the equation for the heavy field $\\chi $ is negligible, then the solution of $\\chi $ is given by the same solution $\\chi =\\chi (\\varphi ,X)$ as in the cases we have considered in the main body of this paper.",
"The matter action is thus $S_{\\rm m}=S_{\\rm m}[\\psi , \\Omega ^2(\\varphi ,X)g_{\\mu \\nu }]\\,,$ where $\\Omega (\\varphi ,X)=\\Omega (\\chi )|_{\\chi =\\chi (\\varphi ,X)}$ .",
"This type of matter coupling yields a subclass of DHOST theories [68], [69], [45].",
"A class of the DHOST theory is obtained as an EFT of the two-field model with the non-minimal coupling under the assumption of small backreaction of matter fields to the heavy field.",
"Our results hold in the DHOST theory in the Einstein frame as long as effects from the matter field are subdominant.",
"The theory deviates from the DHOST theory when the backreaction of the matter to the heavy field is included.",
"It would be intriguing to see how two theories, the EFT of two-field model and the DHOST theory, could differ due to the backreaction of matter."
],
[
"Conclusion",
"In the present paper, we have clarified UV consistency conditions of the single-field effective field theory (EFT) with spontaneously broken Lorentz symmetry in the Einstein frame by studying explicit EFT reductions from multi-scalar theories.",
"All the classifications of the EFT reduction we consider are summarized in Table REF , and the conclusions of the present paper are listed as follows: The EFT preserves the null energy condition (NEC); The sound speed of perturbations can be either subluminal or imaginary; The case with imaginary sound speed can accommodate a violation of the naively applied positivity bounds, $P_{XX}>0$ ; Both ghost and gradient instabilities at IR can be as harmless as Jeans instability; The effective coupling of the fifth force at a finite distance should be larger than that at infinity; and These properties are inferred from the two-point function, not four-point functions, in Lorentz-violating EFTs.",
"A few comments are in order.",
"First of all, UV consistency conditions arise from stability conditions of the UV modes which are integrated out.",
"The NEC preservation in IR is a direct consequence of the lack of the source of NEC violation (ghost-free condition at UV) in the particular class of our partial UV completion, and the subluminal propagation is anticipated by the standard positivity bounds [1].",
"We emphasize, however, that stability of IR degrees of freedom is not mandatory in considering the UV consistency, since our EFTs do not necessarily describe the true vacuum.",
"This is one of the differences from the existing arguments about the Lorentz-invariant positivity bounds, and, indeed, the order-unity violation of the naively applied positivity bound is found when a system exhibits the IR instability.",
"Some EFTs are defined only around a Lorentz-violating background in which there is no need to respect the Lorentz-invariant positivity bounds.",
"In Appendix , we provide concrete examples which explain how EFT properties are related to properties of the UV theory.",
"Physically, the IR instability present in the EFT is analogous to the Jeans instability and is automatically cured at a relatively low scale either by higher-derivative terms that are also present in the EFT or by the multi-field dynamics of the assumed (partial) UV completion.",
"It is interesting that the naively applied positivity bound $P_{XX}>0$ does not necessarily hold around a timelike background, whereas it does hold around a spacelike background.",
"The consistency condition $P_{XX}>0$ for the spacelike ($X<0$ ) case concludes that there should be no screening mechanism by the nonlinear kinetic terms, if we assume the particular partial UV completion that we studied in the present paper.",
"Finally, we emphasize that these conclusions are obtained by studying the quadratic action, rather than four-point functions, which are typically used to obtain the positivity bounds on Lorentz-invariant EFTs."
],
[
"Acknowledgments",
"K.A.",
"would like to thank Toshifumi Noumi for insightful discussions, and R.N.",
"is grateful to Masahito Yamazaki for inspiring comments on related topics during the period of “Workshop on Gravity and Cosmology by Young Researchers 2021” (YITP-W-20-11) at the Yukawa Institute for Theoretical Physics, Kyoto University.",
"The work of K.A.",
"was supported in part by Grants-in-Aid from the Scientific Research Fund of the Japan Society for the Promotion of Science, No.",
"19J00895 and No. 20K14468.",
"The work of S.M was supported in part by Japan Society for the Promotion of Science Grants-in-Aid for Scientific Research No.",
"17H02890, No.",
"17H06359, and by World Premier International Research Center Initiative, MEXT, Japan."
],
[
"General multi-field UV models with DBI-type kinetic terms",
"Ref.",
"[47] has shown that DBI-type kinetic terms lead to another class of models without caustic singularities.",
"In this sense, this class can also be regarded as a candidate of partially UV-complete models.",
"While ref.",
"[47] focuses on the case of a single light field, we can extend it to a general multi-field case with a curved field space.",
"The Lagrangian of this class thus reads ${\\cal L}_{\\rm DBI} = - \\sqrt{1 + \\gamma _{AB}(\\Phi ) \\, \\mathrm {d}\\Phi ^A \\mathrm {d}\\Phi ^B} - V(\\Phi ) \\; ,$ where the $N$ fields $\\Phi ^A$ contains one light field degree $\\varphi $ that respects a (approximate) shift symmetry.",
"Note that the caustic free condition does not exclude the DBI model with the opposite sign, $\\mathcal {L}=+ \\sqrt{1 - \\gamma _{AB}(\\Phi ) \\, \\mathrm {d}\\Phi ^A \\mathrm {d}\\Phi ^B} - V(\\Phi )$ ; nonetheless, this model admits the Minkowski vacuum and then the standard positivity bounds exclude this model.",
"In addition, although the cuscuton theory [70], [71], [72] is free from caustic singularities, the cuscuton is non-local in the sense that there exists an instantaneous/shadowy mode [73] and will not be discussed here in a sprite of the positivity bounds.",
"We only consider (REF ) as a partial UV completion of the single-field EFT.",
"In this appendix, we only focus on the EFT at the leading order in $\\tilde{\\epsilon }$ , ${\\cal L}_{\\rm DBI}&=Q\\left( \\chi , \\varphi , X \\right) +\\mathcal {O}(\\tilde{\\epsilon }) \\; ,\\\\Q\\left( \\chi , \\varphi , X \\right) &\\equiv - \\sqrt{1 + f(\\chi ,\\varphi ) \\left( \\partial \\varphi \\right)^2} - V(\\chi ,\\varphi )$ where $Q$ is defined as a function of $\\chi ^a,\\varphi $ and $X = - \\left( \\partial \\varphi \\right)^2 / 2$ for a compact notation.",
"The kinetic terms for $\\chi ^a$ are absent in (REF ) at the leading-order expansion.",
"The “constraint” equations, which are the same as the equations of motion for $\\chi ^a$ , are obtained by variations of (REF ) with respect to $\\chi ^a$ , i.e.",
"$\\partial _a Q = 0 \\; .$ Then the resultant single-field EFT must obey the following relations: ${\\cal L}_{\\rm IR} = P(\\varphi , X)+\\mathcal {O}(\\tilde{\\epsilon })&=Q \\, \\big |_{\\chi ^a=\\chi _0^a(\\varphi , X)} +\\mathcal {O}(\\tilde{\\epsilon }) \\; ,$ with $P_X&=\\left.",
"Q_X+ \\partial _a Q \\, \\frac{\\partial \\chi ^a}{\\partial X} \\right|_{\\chi ^a=\\chi ^a_0(\\varphi , X)}= Q_X \\, \\big \\vert _{\\chi ^a=\\chi ^a_0(\\varphi , X)}\\,, \\\\P_{XX}&=\\left.",
"Q_{XX}+ \\partial _a Q_X \\, \\frac{\\partial \\chi ^a}{\\partial X} \\right|_{\\chi ^a=\\chi ^a_0(\\varphi , X)}= \\left.",
"Q_{XX} + \\partial _a Q_X \\left( M^{-2} \\right)^{ab} \\partial _b Q_X \\, \\right|_{\\chi ^a = \\chi _0^a(\\varphi , X)} \\; ,$ where $\\chi _0^a(X)$ are the solutions to (REF ) and $M^2_{ab} \\equiv - \\partial _a \\partial _b Q = \\partial _a \\partial _b V + \\cdots $ .",
"Note that the DBI-type kinetic term yields $Q_X&=\\frac{f}{\\sqrt{1+ \\gamma _{AB}\\, \\mathrm {d}\\Phi ^A \\mathrm {d}\\Phi ^B} } >0\\,, \\\\Q_{XX}&=\\frac{f^2}{ ( 1+ \\gamma _{AB}\\, \\mathrm {d}\\Phi ^A \\mathrm {d}\\Phi ^B)^{3/2} } >0$ where the components of the field space metric is written as (REF ).",
"In particular, the positive definiteness of the field space metric implies $P_X>0$ .",
"We then investigate the stability conditions of the UV modes to discuss the sign of $P_{XX}$ .",
"Following the main text, we assume the exact shift symmetry and consider the perturbations around the constant background, $\\bar{\\chi }^a={\\rm constant}\\,, \\quad C_{\\mu }\\equiv \\partial _{\\mu }\\bar{\\varphi }={\\rm constant}\\,.$ We perform a field redefinition to eliminate the kinetic mixing between $\\varphi $ and $\\chi ^a$ , as we did in Sec.",
"REF .",
"Then, the quadratic Lagrangian of the DBI-type partial UV model is $\\mathcal {L}_{\\rm DBI}^{(2)} &= -\\frac{1}{2}\\Gamma _{ab} (\\partial \\delta \\chi ^a \\cdot \\partial \\delta \\chi ^b)-\\frac{1}{2}M^2_{ab} \\delta \\chi ^a \\delta \\chi ^b - H_{[ab]} (C \\cdot \\partial \\delta \\chi ^a) \\delta \\chi ^b-\\frac{1}{2} F (\\partial \\pi )^2 - F_a \\delta \\chi ^a (C\\cdot \\partial \\pi )+\\frac{1}{2}Q_{XX}(C\\cdot \\partial \\pi )^2\\,,$ where the coefficients are evaluated at the background and $\\Gamma _{ab} &\\equiv \\frac{\\gamma _{ab}}{\\sqrt{1-2f X}}\\,, \\quad F \\equiv \\frac{f}{\\sqrt{1-2fX }}\\,, \\quad F_a \\equiv \\partial _a F\\,,\\quad H_{[ab]}\\equiv \\frac{h_{[ab]} }{\\sqrt{1-2fX }}\\,.$ Note that $F_a=\\partial _a Q_X$ holds when evaluated at the background.",
"The no-tachyon condition of the heavy mode is immediately found as $M^2_{ab}>0$ around the spacelike background.",
"Since the DBI theory satisfies $Q_{XX}>0$ , the positive definiteness of $M^2_{ab}$ concludes $P_{XX}>0$ .",
"We study the Hamiltonian to obtain the stability conditions around the timelike background $C_\\mu = (C_0 , \\mathbf {0})$ .",
"The conjugate momenta are $p_a&=\\Gamma _{ab}\\delta \\dot{\\chi }^b+H_{[ab]}C_0\\delta \\chi ^b\\,,\\\\p_{\\pi }&=(F+2X Q_{XX})\\dot{\\pi }+F_a C_0 \\delta \\chi ^a\\,.$ and the Hamiltonian is $\\mathcal {H}^{(2)}_{\\rm DBI}&=\\frac{1}{2}\\Gamma _{ab} \\delta \\dot{\\chi }^a{}^{\\dagger } \\delta \\dot{\\chi }^b+\\frac{1}{2}(F+2X Q_{XX}) |\\dot{\\pi }|^2+ \\frac{1}{2}(k^2 \\gamma _{ab}+M^2_{ab})\\delta \\chi ^a{}^{\\dagger } \\delta \\chi ^b+ \\frac{1}{2}k^2 F |\\pi |^2\\,,$ in the momentum space where $\\delta \\dot{\\chi }^a$ and $\\dot{\\pi }$ are understood as the functions of the conjugate variables.",
"We can repeat the same analysis as in Sec.",
"REF .",
"We impose the boundedness of the Hamiltonian under $k=0$ and $p_{\\pi }=0$ , leading to $M^2_{ab}+\\frac{2X F_a F_b}{F+2X Q_{XX}}>0\\; ,$ at least under the situation we are considering.",
"We then use the freedom of field redefinitions to diagonalize $M^2_{ab}={\\rm diag}[M^2_1,M^2_2,\\cdots ]$ and set $F_a=(F_1,0,0,\\cdots )$ .",
"In this coordinate choice, the conditions (REF ) are $M_1^2+ \\frac{2X F_1^2}{F+2XQ_{XX}}>0 \\,, \\quad M^2_{a^{\\prime }}>0 \\; ,$ where the primed index $a^{\\prime }$ denotes the fields other than $\\delta \\chi ^1$ in this field basis.",
"Hence the relation to the k-essence function is $P_{XX}=Q_{XX} + \\frac{F_1^2}{M_1^2}\\,.$ The case $M_1^2>0$ , i.e.",
"$M^2_{ab}>0$ , concludes $P_{XX}>0$ .",
"On the other hand, the case $M_1^2<0$ does not immediately suggest the negative sign of $P_{XX}$ .",
"Nonetheless, the former one of the condition (REF ) is given by $F+2X\\left( Q_{XX}+\\frac{F_1^2}{M_1^2} \\right)<0\\Rightarrow P_X+2XP_{XX}<0\\,,$ where $M_1^2<0$ and $F+2XQ_{XX}>0$ are used, the latter is required by the no ghost for $\\pi $ .",
"Since we have $P_X>0$ and $X>0$ , $P_{XX}$ has to be negative if there is a negative eigenvalue of $M^2_{ab}$ .",
"As a result, the UV consistency conditions for the DBI-type partial UV completion are the same as those in the general multi-field models shown in Sec.",
"REF ."
],
[
"EFT from $U(1)$ scalar field",
"As a concrete example, we consider a $U(1)$ scalar field described by the Lagrangian $\\mathcal {L}_{U(1)}=-\\frac{1}{2}\\partial _{\\mu }\\Phi ^\\dagger \\partial ^{\\mu } \\Phi - \\sigma \\frac{M_{\\Phi }^2}{2}|\\Phi |^2 - \\frac{\\lambda }{4}|\\Phi |^4\\,, $ with $\\sigma = \\pm 1$ .",
"We introduce the real variables $\\chi $ and $\\varphi $ via $\\Phi = \\cos \\theta \\chi ^{1+i \\tan \\theta } e^{i \\varphi / \\cos \\theta }\\,,$ with $\\chi >0$ where $\\theta $ is a constant parameter which determines the kinetic mixing between $\\chi $ and $\\varphi $ .",
"In this appendix, the variable $\\varphi $ is dimensionless since it is essentially the angular coordinate of the field space.",
"The $U(1)$ symmetry of $\\Phi $ recasts the shift symmetry of $\\varphi $ .",
"The Lagrangian is written as $\\mathcal {L}_{U(1)}=-\\frac{1}{2}\\tilde{\\epsilon }^2 (\\partial \\chi )^2 +\\tilde{\\epsilon } \\sin \\theta \\chi \\nabla _{\\mu } \\chi \\nabla ^{\\mu } \\varphi - \\frac{1}{2} \\chi ^2 (\\partial \\varphi )^2 - \\sigma \\frac{M_{\\Phi }^2}{2}\\chi ^2 - \\frac{\\lambda }{4}\\chi ^4 \\,,$ reading $f=\\chi ^2\\,, \\quad V=\\sigma \\frac{M_{\\Phi }^2}{2}\\chi ^2 + \\frac{\\lambda }{4}\\chi ^4 \\,,$ where $\\tilde{\\epsilon }$ is introduced which will be set to be $\\tilde{\\epsilon }=1$ at the end of the calculation (see Sec.",
"REF ).",
"In this example, the constant background, $\\partial _{\\mu }\\bar{\\varphi },\\bar{\\chi }={\\rm constant}~(\\bar{\\chi }\\ne 0)$ , is determined by $-(\\partial \\bar{\\varphi })^2 = \\sigma M^2_{\\Phi }+\\lambda \\bar{\\chi }^2\\,.$ In particular, as for the timelike background, $\\bar{\\chi }=$ constant and $\\bar{\\varphi }\\propto t$ , the background motion is a uniform circular rotation in the field space which is realized by balancing the potential force $V_{\\chi }$ and the centrifugal force $-f_{\\chi } X$ .",
"We consider the solution up to $\\mathcal {O}(\\tilde{\\epsilon }^2)$ .",
"Writing $\\chi =\\chi _0+\\tilde{\\epsilon } \\chi _1 + \\tilde{\\epsilon }^2 \\chi _2 + \\mathcal {O}(\\tilde{\\epsilon }^3)$ , we find $\\chi _0&= \\sqrt{ \\frac{2X-\\sigma M_{\\Phi }^2}{\\lambda } }\\,, \\\\\\chi _1&=-\\frac{\\sin \\theta }{2\\sqrt{\\lambda ( 2X-\\sigma M_{\\Phi }^2) } } \\Box \\varphi \\,, \\\\\\chi _2&=\\frac{1}{2\\sqrt{ \\lambda (2X -\\sigma M_{\\Phi }^2)^3}}\\left( \\Box X - \\frac{1}{4}\\sin ^2 \\theta (\\Box \\varphi )^2 - \\frac{\\nabla _{\\mu } X \\nabla ^{\\mu }X }{2X-\\sigma M_{\\Phi }^2} \\right)\\,, $ where $(2X-\\sigma M_{\\Phi }^2)/\\lambda >0$ has to be satisfied so that $\\chi $ is real.",
"In this example, the positivity of $M^2$ is related to the boundedness of the potential $\\lambda >0$ because $M^2 \\equiv V_{\\chi \\chi }-f_{\\chi \\chi } X=-2\\sigma M_{\\chi }^2+ 4X =2 \\lambda \\chi _0^2(X)\\,,$ while $\\mathcal {M} ^2$ , which is the effective mass around the timelike background, is given by $\\mathcal {M} ^2\\equiv M^2+ 2X \\frac{f_{\\chi }^2}{f}=-2\\sigma M_{\\chi }^2+12 X\\,.$ Following the analysis in Sec.",
"REF , the EFT Lagrangian is $\\mathcal {L}_{\\rm IR}=\\frac{1}{\\lambda } \\left[ (X-\\sigma M_{\\Phi }^2/2)^2 - \\tilde{\\epsilon } \\sin \\theta X \\Box \\varphi + \\tilde{\\epsilon }^2 \\left( -\\frac{(\\nabla X)^2}{M^2} +\\frac{1}{4}\\sin ^2 \\theta (\\Box \\varphi )^2 \\right) + \\mathcal {O}(\\tilde{\\epsilon }^3) \\right]\\,.$ The k-essence part is $P(X)=\\frac{1}{\\lambda } (X-\\sigma M_{\\Phi }^2/2)^2\\,,$ which yields $P_X=\\frac{2X-\\sigma M_{\\Phi }^2}{\\lambda } \\,, \\quad P_{XX}=\\frac{2}{\\lambda }\\,.$ One can explicitly confirm the general relations $P_{X}=f$ and $P_{XX}=f_{\\chi }^2/M^2$ .",
"We first discuss the positive $\\lambda $ where the potential is bounded from below.",
"The sign of $\\sigma $ determines the sign of the linear kinetic term $P(X)=-\\sigma M_{\\Phi }^2 X/\\lambda + \\cdots $ .",
"From the UV point of view, $\\sigma $ determines whether $\\chi =0$ is the true vacuum or the false one.",
"Let us set $\\theta =0$ to have an intuition.",
"Then, the variables $\\chi $ and $\\varphi $ are nothing but the radial and the angular coordinates in the field space.",
"The motion $\\chi =$ constant and $\\varphi \\propto t$ is a uniform circular motion.",
"When $\\sigma = -1$ , the fields settle down the true vacuum $\\chi =M_{\\Phi }/\\sqrt{\\lambda }$ when $\\varphi $ stops where there exists the massless direction $\\varphi $ .",
"Hence, the EFT is still valid at $X=0$ which can be seen as the positive sign of the linear kinetic term of $P(X)$ .",
"On the other hand, the true vacuum is at $\\chi =0$ for $\\sigma =+1$ where the EFT reduction is no longer consistent since there is no massless mode.",
"From the EFT perspective, the invalidity of EFT reduction can be seen as the opposite sign of the linear kinetic term $P(X)=-M_{\\Phi }^2 X /\\lambda +\\cdots $ .",
"However, the massless direction appears when the field $\\chi $ goes away from the origin $\\chi =0$ thanks to the non-zero angular momentum $\\varphi \\propto t$ .",
"As a result, the EFT reduction becomes consistent around such a Lorentz-violating background.",
"This consistency condition is captured by our conditions $P_X>0$ and $P_{XX}>0$ , which conclude $X>M_{\\Phi }^2/2$ for $\\sigma =+1$ .",
"This EFT is valid if and only if $\\varphi $ has a non-vanishing timelike gradient.",
"When we have a kinetic mixing at UV $(\\Leftrightarrow \\theta \\ne 0)$ , the EFT has the operator $(\\Box \\varphi )^2$ .",
"It seems that the operator $(\\Box \\varphi )^2$ pushes up the validity of the EFT under the limit $P_X \\rightarrow 0$ like the ghost condensate.",
"However, this is not the case.",
"The positive definiteness of the kinetic matrix at UV requires that the coefficient of $(\\Box \\varphi )^2$ is not arbitrary large, i.e.",
"$|\\sin \\theta | < 1$ .",
"The operator $(\\nabla X)^2$ eventually dominates over $(\\Box \\varphi )^2$ in the limit $P_X \\propto M^2 \\rightarrow 0$ which spoils the scenario of the ghost condensate.",
"In fact, changing $\\theta $ is just a change of the variable at UV which does not change physics.",
"We then consider the non-relativistic limit of the $U(1)$ scalar which corresponds to the limit $M^2 \\rightarrow 0$ .",
"As discussed in REF , the validity of EFT can be extended around the timelike background.",
"Then $M^2$ is no longer the cutoff of the single-field EFT.",
"Note that the EFT breaks down when we take the limit $\\chi _0 \\rightarrow 0$ .",
"Therefore, we should take the limit $\\lambda \\rightarrow 0$ while keeping $\\chi _0$ finite to consistently take the limit $M^2 \\rightarrow 0$ .",
"In terms of the k-essence $P(X)$ , this limit yields $P_X \\rightarrow $ finite and $P_{XX}\\rightarrow \\infty $ , implying that $\\varphi $ is not a good variable.",
"We instead use the following parametrization $\\Phi =\\psi e^{-iM_{\\Phi }t}\\,,$ where $\\psi $ is a complex variable.",
"We suppose that the variable $\\psi $ slowly varies with respect to $M_{\\Phi }$ under $\\sigma =+1$ and $\\lambda \\rightarrow 0$ .",
"The choice (REF ) with $\\sigma =+1$ and $\\lambda \\rightarrow 0$ is a free massive $U(1)$ scalar, meaning that the general solution is given by a superposition of the negative frequency mode solutions $\\Phi \\propto e^{i(\\omega t - k_i x^i)}$ as well as the positive frequency mode solutions $\\Phi \\propto e^{-i(\\omega t - k_i x^i)}$ with $\\omega = \\sqrt{M_{\\Phi }^2+k^2}$ .",
"The assumption of the slow variation of $\\psi $ is thus to ignore, namely to integrate out, the negative frequency modes as well as the positive frequency modes with $k\\gtrsim M_{\\Phi }$ .",
"We only consider the dynamics of the positive frequency modes with $k \\ll M_{\\Phi }$ .",
"The Lagrangian of the free massive $U(1)$ scalar in terms of $\\psi $ is $\\mathcal {L}_{\\lambda \\rightarrow 0}=\\frac{1}{2}|\\dot{\\psi }|^2 + \\frac{1}{2}iM_{\\Phi }(\\psi ^\\dagger \\dot{\\psi }-\\dot{\\psi }^\\dagger \\psi ) -\\frac{1}{2}\\partial _i \\psi ^\\dagger \\partial ^i \\psi $ where the dot is the time derivative and $\\partial _i$ is the spatial derivative, respectively.",
"Since we have assumed the slowly varying $\\psi $ , the first term can be ignored compared with the second term.",
"We then obtain the Lagrangian for the free Schrödinger field, $\\mathcal {L}_{\\rm Sch}= \\frac{1}{2}iM_{\\Phi }(\\psi ^\\dagger \\dot{\\psi }-\\dot{\\psi }^\\dagger \\psi ) -\\frac{1}{2}\\partial _i \\psi ^\\dagger \\partial ^i \\psi \\,,$ of which equation of motion is the well-known Schrödinger equation, $i\\dot{\\psi }=-\\frac{1}{2M_{\\Phi }}\\partial ^i \\partial _i \\psi \\,.$ This confirms that $\\lambda \\rightarrow 0 ~(\\Rightarrow M^2 \\rightarrow 0)$ consistently provides the effective theory with the non-relativistic dispersion relation.",
"The physical meaning of the effective mass, $\\mathcal {M} ^2 = (2M_{\\Phi })^2 \\quad {\\rm as}\\quad \\lambda \\rightarrow 0\\,,$ is the energy gap between the positive and negative frequency modes.",
"The relation to the polar coordinates $\\Phi =\\chi e^{i\\varphi }$ with $\\chi =\\bar{\\chi }+\\delta \\chi $ and $\\varphi =-M_{\\Phi } t + \\pi $ is ${\\rm Re}\\, \\psi = \\bar{\\chi }+\\delta \\chi + \\cdots \\,, \\quad {\\rm Im}\\, \\psi = \\bar{\\chi } \\pi + \\cdots \\,,$ up to linear in the perturbations.",
"One can consider the quadratic action in terms of $\\delta \\chi $ and $\\pi $ instead of the complex variable $\\psi $ .",
"Integrating out the imaginary part $\\delta \\chi $ , one then finds a non-local action $S_{\\Lambda }[\\pi ]$ which agrees with (REF ) in the limit $M^2 \\rightarrow 0$ .",
"Finally, we discuss the unstable theory $\\lambda <0$ .",
"Since one can add a higher-order polynomial of $|\\Phi |$ to bound the potential at a sufficiently large $\\chi $ , this theory can be still regarded as a partially UV complete theory.",
"The inequalities $M^2<0$ and $\\mathcal {M} ^2>0$ are satisfied when $2X<M_{\\Phi }^2<6X$ with $\\sigma =+1$ .",
"The simplest example of $\\lambda <0$ is found by taking $\\lambda \\rightarrow -0$ and $2X\\rightarrow M^2_{\\Phi }$ .",
"We again use the parametrization (REF ) and include the leading order effect of $\\lambda $ .",
"Now, $\\psi $ contains not only inhomogeneous oscillations but also a homogeneous oscillation since the background equation (REF ) implies $\\Phi \\propto e^{iC_0 t}$ with $C_0=-M_{\\Phi }+\\mathcal {O}(\\lambda )$ .",
"Nonetheless, we have $|\\dot{\\psi }|^2=\\mathcal {O}(\\lambda ^2)$ from the homogeneous oscillation which can be ignored at the leading order in $\\lambda $ .",
"As a result, the effective Lagrangian is given by the Gross–Pitaevskii form (see e.g.",
"[74]), $\\mathcal {L}_{\\rm GP}= \\frac{1}{2}iM_{\\Phi }(\\psi ^\\dagger \\dot{\\psi }-\\dot{\\psi }^\\dagger \\psi ) -\\frac{1}{2}\\partial _i \\psi ^\\dagger \\partial ^i \\psi -\\frac{\\lambda }{4}|\\psi |^4\\,,$ for a small $\\lambda $ .",
"The negative sign $\\lambda <0$ corresponds to the attractive interaction, leading to the simple physical interpretation of the IR instability discussed in Sec.",
"REF : the IR instability is the consequence of the attractive force, similarly to the Jeans instability."
],
[
"Comparison with ghost condensate",
"In order to examine the claim after (REF ), that is the non-equivalence between the $\\Lambda $ -EFT and the ghost condensate in the presence of gravity, we now introduce the gravity sector into the two-field model we have been discussing.",
"In fact, this is rather a natural consideration, since Lorentz-violating backgrounds are most relevant for gravitating systems.",
"We consider coupling the two-field model (REF ) to gravity in the Einstein frame as a minimal setup.",
"The (partial) UV Lagrangian is $\\mathcal {L}_{\\rm UV}=\\frac{M_{\\rm Pl}^2}{2} \\, R-\\frac{1}{2} \\left( \\nabla \\chi \\right)^2-\\frac{1}{2} \\, f(\\chi ) \\left( \\nabla \\varphi \\right)^2-V(\\chi )\\; ,$ where $M_{\\rm Pl}$ is the reduced Planck mass, and $R$ and $\\nabla _\\mu $ the Ricci scalar and the covariant derivative, respectively, associated with the spacetime metric $g_{\\mu \\nu }$ .",
"When the gravitational interactions are turned on, the background of $\\chi $ and $\\varphi $ gravitates, and then the constant solution (REF ) is no longer a solution of the system.",
"For instance, one can consider the Friedmann-Lemaître-Robertson-Walker (FLRW) metric for the background, $\\mathrm {d}s^2=g_{\\mu \\nu } \\, \\mathrm {d}x^{\\mu } \\mathrm {d}x^{\\nu }=-\\bar{N}^2(t) \\, \\mathrm {d}t^2 + a^2(t) \\, \\delta _{ij} \\, \\mathrm {d}x^i \\mathrm {d}x^j\\,,$ where $\\bar{N}$ and $a$ are the background lapse and the scale factor, respectively.",
"We can then search for a time-dependent homogeneous solution $\\bar{\\chi }= \\bar{\\chi }(t)$ and $\\bar{\\varphi }= C_0 t$ , where $C_0$ is a constant and $\\bar{\\chi }$ is now promoted to a time-dependent function instead of a constant.In fact, this form of the solution can always be achieved, at the cost of the choice of $\\bar{N}$ .",
"Since the general homogeneous solution for $\\bar{\\varphi }$ is $\\partial _t \\bar{\\varphi }= C_0 / ( \\bar{N} a^3 f)$ with an integration constant $C_0$ , we can choose the (background) gauge by $\\bar{N} = (a^3 f t)^{-1}$ to achieve the aforementioned form of the background solution.",
"This way all the complication of time dependence is taken care of by $\\bar{\\chi }$ .",
"A tiny variation of the background is allowed to integrate out the UV modes, and the backreaction of gravity does not change the main part of the previous analysis as long as the effect of gravity is small compared to the mass scale of the decoupling.",
"In order to make comparison, we now prepare the candidate effective Lagrangian of ghost condensate, $\\mathcal {L}_{\\rm GC}=\\frac{M_{\\rm Pl}^2}{2} \\, R+P(\\tilde{X})- \\frac{1}{2\\tilde{\\mathcal {M}}^2} (\\Box \\tilde{\\varphi })^2 +\\cdots \\; ,$ with a scalar field $\\tilde{\\varphi }=\\bar{\\varphi }+\\tilde{\\pi }$ .",
"As we have discussed in the main text, the linear dynamics of the ghost condensate is the same as the EFT of the two-field model if gravity is absent.",
"We study linear perturbations around the homogeneous and isotropic background to clarify the non-equivalence in the presence of gravity.",
"We should compare two theories in the same gauge choice.",
"Here, we adopt the unitary gauge in which the quadratic Lagrangian is given by a simple form.",
"Looking into the sector of scalar modes, the lapse $N$ , the shift $N^i$ , and the spatial metric $\\gamma _{ij}$ are given by $N=\\bar{N}(t)(1+\\alpha )\\,,\\quad N^i=\\bar{N}(t)\\delta ^{ij}\\partial _j \\beta \\,, \\quad \\gamma _{ij}=a^2 e^{2\\zeta } \\delta _{ij}\\,.$ The scalar fields are perturbed as $\\chi =\\bar{\\chi }(t)+\\delta \\chi \\,, \\quad \\varphi =C_0 t\\,,$ as for the UV two-field model (REF ), whereas the scalar field for the ghost condensate (REF ) is $\\tilde{\\varphi }=\\tilde{C}_0 t\\,.$ As we mentioned after (REF ), the $\\Lambda $ -EFT can be derived by treating the time derivative of $\\chi $ as perturbations.",
"At the leading order, the kinetic term of $\\chi $ in (REF ) is approximated by $(\\nabla \\chi )^2\\simeq \\gamma ^{ij}\\partial _i \\chi \\partial _j \\chi \\,.$ On the other hand, we only keep the highest spatial derivative term for $\\tilde{\\varphi }$ in (REF ), i.e., $(\\Box \\tilde{\\varphi })^2 \\simeq 2\\tilde{X} (\\partial _i \\partial ^i \\beta )^2\\, .$ The quadratic Lagrangian for the curvature perturbation $\\zeta $ of (REF ) after integrating out $\\delta \\chi $ and that of (REF ) are $\\mathcal {L}^{(2)}_{\\Lambda } & \\simeq \\left( \\frac{f X}{H^2} \\right)\\left[ \\frac{\\mathcal {M} ^2}{k^2/a^2+M^2} \\, \\frac{\\vert \\dot{\\zeta } \\vert ^2}{\\bar{N}^2}- \\frac{k^2}{a^2} \\left|\\zeta \\right|^2 \\right]\\nonumber \\\\&=\\frac{M^2}{k^2/a^2 +M^2} \\left( \\frac{X}{H^2} \\right)\\left[ (P_X+2XP_{XX}) \\frac{\\vert \\dot{\\zeta } \\vert ^2}{\\bar{N}^2}- \\left(P_X + P_X \\, \\frac{k^2}{a^2 M^2} \\right) \\frac{k^2}{a^2} \\, \\vert \\zeta \\vert ^2 \\right],\\\\\\mathcal {L}^{(2)}_{\\rm GC} & \\simeq \\left( \\frac{ \\tilde{X} }{H^2} \\right)\\left[ ( \\tilde{P}_X+2\\tilde{X} \\tilde{P}_{XX})\\frac{\\vert \\dot{\\zeta } \\vert ^2}{\\bar{N}^2}-\\left( \\tilde{P}_X + \\frac{k^2}{a^2 \\tilde{\\mathcal {M}}^2} \\right) \\frac{k^2}{a^2} \\, \\vert \\zeta \\vert ^2 \\right],$ in the momentum space, respectively, where we have used the relations $f=P_X$ and $f_{\\chi }^2/M^2=P_{XX}$ .",
"It is clear that two theories (REF ) and () describe the same dynamics only for the modes $k^2 / a^2 \\ll |M^2|$ due to the prefactor $\\frac{M^2}{k^2/a^2+M^2}$ .",
"Although two theories describe the same dynamics when the perturbations are decoupled from the background, the background motion of $\\frac{M^2}{k^2/a^2+M^2}$ provides the difference.",
"The EFT of (REF ) and the ghost condensate (REF ) are distinguished by how they gravitate."
]
] | 2107.01755 |
[
[
"Entanglement entropy and out-of-time-order correlator in the long-range\n Aubry-Andr\\'e-Harper model"
],
[
"Abstract We investigate the nonequilbrium dynamics of entanglement entropy and out-of-time-order correlator (OTOC) of noninteracting fermions at half-filling starting from a product state to distinguish the delocalized, multifractal (in the limit of nearest neighbor hopping), localized and mixed phases hosted by the quasiperiodic Aubry-Andr\\'e-Harper (AAH) model in the presence of long-range hopping.",
"For sufficiently long-range hopping strength a secondary logarithmic behavior in the entanglement entropy is found in the mixed phases whereas the primary behavior is a power-law the exponent of which is different in different phases.",
"The saturation value of entanglement entropy in the delocalized, multifractal and mixed phases depends linearly on system size whereas in the localized phase (in the short-range regime) it is independent of system size.",
"The early-time growth of OTOC shows very different power-law behaviors in the presence of nearest neighbor hopping and long-range hopping.",
"The late time decay of OTOC leads to noticeably different power-law exponents in different phases.",
"The spatial profile of OTOC and its system-size dependence also provide distinct features to distinguish phases.",
"In the mixed phases the spatial profile of OTOC shows two different dependences on space for small and large distances respectively.",
"Interestingly the spatial profile contains large fluctuations at the special locations related to the quasiperiodicity parameter in the presence of multifractal states."
],
[
"Introduction",
"The nature of correlations between different parts of a system is of fundamental interest in physics.",
"Entanglement entropy has been a popular measure of quantum correlations in many-body systems [1].",
"The study of entanglement in stationary, equilibrium and nonequilibrium states has proven to be insightful in a wide variety of contexts [2], [3], [4], [5], [6].",
"In recent years, out-of-time-order correlators or OTOC, which have emerged as a useful probe of quantum chaos [7], have gained importance in a diverse set of fields ranging from high energy physics [8], [9], [10], [11] to condensed matter physics [12], [13], [14], [15], [16], [17] to quantum information [18], [19].",
"Devised originally as a theoretical measure [20], [21], considerable excitement has been generated from the recent experimental measurement of OTOC using nuclear magnetic resonance [22], [23], [24] and trapped ions [25], [26].",
"The OTOC is generically defined as: $C(x,t)=\\langle [ \\hat{W}(x,t), \\hat{V}(0,0) ]^\\dagger [\\hat{W}(x,t), \\hat{V}(0,0)] \\rangle ,$ where $\\hat{W}$ and $\\hat{V}$ are arbitrary local operators separated by a displacement $x$ and commute at $t=0$ .",
"Here $\\langle .\\rangle $ typically refers to a thermal average, although the expectation value in specific states may also be of interest.",
"Choosing both $W$ and $V$ to be both Hermitian and unitary is particularly advantageous as Eq.",
"REF reduces to the compact expression: $C(x,t)= 2(1-Re[F(x,t)]),$ where $F(x,t)=\\langle \\hat{W}(x,t) \\hat{V}(0) \\hat{W}(x,t)\\hat{V}(0)\\rangle $ .",
"At $t=0$ $C(x,t)$ is zero.",
"Then it increases for $t>0$ due to non-commutativity of $\\hat{W}(x,t)$ and $\\hat{V}(0)$ .",
"Models that exhibit localization are a natural setup for investigation of OTOC, in condensed matter systems.",
"A particularly important class of such models is the family of models with quasi-periodic disorder, that have sustained interest over several decades [27], [28], [29], [30].",
"Unlike with Anderson localization where even an infinitesimal random disorder results in localization, a quasiperiodic disorder of finite strength is essential for localization of a single particle even in one dimension [31], [32].",
"There has been a revival of interest in quasiperiodic systems since their experimental realization using ultra-cold atoms [33], [34], [35], [36].",
"Furthermore the possibility of many-body localization in such models has triggered a lot of interest both from a theoretical [37], [38], [39] and an experimental [40] perspective.",
"Apart from the delocalized and localized phases, quasiperiodic systems can also host other nonergodic phases [41], [42] with their characteristic properties.",
"In this study, we numerically probe the different phases using quantum dynamics of out-of-time-order correlators.",
"We also study the quantum dynamics of entanglement entropy to complement and contrast against OTOC.",
"If $C(x,t)$ remains non-zero for an extended period of time one says that the system has `scrambled'.",
"For early time approach to scrambling one expects $C(x,t)\\sim e^{\\lambda _{quant}(t-x/v_B)}$ where $\\lambda _{quant}$ is the `quantum Lyapunov exponent' which is bounded by $\\lambda _{quant}\\le 2\\pi k_BT/\\hbar $ as conjectured in maldacena2016bound.",
"$v_B$ is called the `Butterfly velocity' which is also bounded by the Lieb-Robinson bound [43].",
"Quantum systems in which $\\lambda _{quant}$ approaches its bound are called fast scramblers [44], [45].",
"However, many condensed matter systems exhibit a much slower growth and hence are called slow scramblers.",
"This includes the many-body localized systems showing a power law growth [13], [15], [46], [16] which itself may be contrasted with Anderson localized systems where $C(x,t)$ is expected to be a constant [16].",
"It should be noted that $\\lambda _{quant}$ , although inspired by classical chaos is quite different from its classical counterpart $\\lambda _L$ that characterizes chaotic motion in classical systems [14], [47], [48].",
"The OTOC corresponding to classical chaos was found to grow as $C(t)=\\langle [q(t),p]^2\\rangle \\sim e^{2\\lambda _Lt}$ , where $\\lambda _L$ may become arbitrarily large.",
"Also the late time dynamics of $C(x,t)$ has turned out to be quite interesting.",
"An inverse power-law behavior has been seen in integrable quantum spin chains [49], [50] and many-body localized systems [13].",
"Recently late time behavior of $C(x,t)$ has been proposed as a diagnostic to distinguish regular and chaotic quantum systems [51], [52].",
"Although OTOC has been studied extensively in quantum systems, not many disordered integrable models have been addressed [53], [54] in the context of the delocalization-localization transition.",
"In addition to studies that look at the evolution of an initial thermal state, studies involving an initial product state in a nonequilibrium setting have also been carried out [15], [55], [56], [53].",
"Here we study OTOC starting from a CDW-type initial product state.",
"We also study entanglement entropy which has been one of the most popular tools to characterize different many-body phases, especially in disordered quantum systems [5].",
"This paper is organized as follows.",
"In Section we introduce the model and briefly discuss the various single particle phases shown by it [41], [42].",
"In Section we describe the results obtained from the nonequilbrium dynamics of the entanglement entropy.",
"In Section we study the nonequlibrium dynamics of OTOC.",
"This section consists of two subsections: Subsection REF where we briefly describe the formalism for noninteracting fermions and Subsection REF where we discuss the results for our model.",
"Finally we conclude in Section ."
],
[
"The model",
"The one dimensional long-range Harper (LRH) model is given by the Hamiltonian: $H = -\\sum \\limits _{i<j}^{N} \\bigg (\\frac{J}{r_{ij}^\\sigma } \\hat{c}_i^\\dagger \\hat{c}_j + H.c.\\bigg )\\nonumber + \\lambda \\sum \\limits _{i=1}^{N} \\cos (2\\pi \\alpha i + \\theta _p)\\hat{n}_i,\\nonumber \\\\$ where $\\hat{c}_i^\\dagger $ $(\\hat{c}_i)$ represents the single particle creation (destruction) operator at site $i$ and $\\hat{n}_i=\\hat{c}_i^\\dagger \\hat{c}_i$ , the number operator acting at site $i$ .",
"We consider a lattice of total number of sites $N$ , where $r_{ij}$ is the geometric distance between the sites $i$ and $j$ in an open chain.",
"Here $\\lambda $ is the strength of the quasi-periodic potential with the quasiperiodicity parameter $\\alpha $ which is a Diophantine irrational number [57] e.g.",
"$\\alpha _g=(\\sqrt{5}-1)/2$ , $\\alpha _s=(\\sqrt{2}-1)$ , $\\alpha _b=(\\sqrt{13}-3)/2$ etc [58], [59], also known as the `golden mean', `silver mean', `bronze mean' etc.",
"$\\theta _p$ is an arbitrary global phase.",
"The strength of the long range hopping is controlled by $J$ and the long range parameter in the hopping $\\sigma $ .",
"We set our units such that $J=1$ throughout this article.",
"In the $\\sigma \\rightarrow \\infty $ limit, this model is the well-known Aubry-André-Harper(AAH) model [31], [32].",
"The AAH model has a self-dual point $\\lambda =2J$ , where the model in position space maps to itself in momentum space.",
"As a consequence, all the eigenstates are delocalized in position space for $\\lambda <2J$ and localized for $\\lambda >2J$ [57].",
"Some filling-fraction dependent properties of the AAH model have also been reported [60], [42].",
"The single particle phase diagram of the LRH model has been chalked out recently [41], [42].",
"Along with the delocalized and localized phases the phase digram contains mixed phases where a certain fraction of delocalized eigenstates coexists with multifractal or localized eigenstates.",
"For the `golden mean' $\\alpha _g$ the mixed phases can be denoted as $P_q$ $(q=1,2,3...)$ where $\\alpha _g^q$ fraction of eigenstates are delocalized and $(1-\\alpha _g^q)$ fraction of eigenstates are multifractal or localized depending on whether $\\sigma <1$ or $\\sigma >1$ .",
"Hence $P_q$ phases for $\\sigma <1$ contain the delocalized-multifractal (DM) edges.",
"$P_q$ phases for $\\sigma >1$ contain the delocalized-localized (DL) edges, also known as mobility edges.",
"For the present numerical study we have chosen some specific ($\\lambda ,\\sigma $ ) values.",
"For $\\sigma =0.5$ , we consider $\\lambda =0.1,0.5,1.0,2.0$ which correspond to the delocalized, $P_1$ , $P_2$ and $P_3$ phases (with DM edge) respectively.",
"For $\\sigma =1.5$ , we look at $\\lambda =0.1,1.3,2.0,3.0$ which correspond to the delocalized, $P_1$ , $P_2$ and $P_3$ phases (with DL edge) respectively.",
"For $\\sigma =3.0$ , we look at $\\lambda =0.1,1.7,2.1,2.5,5.0$ which correspond to the delocalized, $P_1$ , $P_2$ , $P_3$ phases (with DL edge) and localized phases respectively with $\\sigma =3.0$ being essentially the short-range limit.",
"Next we discuss the nonequilibrium dynamics of free fermions in the AAH and LRH models."
],
[
"Entanglement entropy",
"The study of out-of-equilibrium properties of disordered quantum systems has been proved to be a very efficient tool to detect delocalized and localized phases.",
"The system is initially prepared in a suitable state, and the properties of the time-evolved state are tracked.",
"Since a charge density wave (CDW) type of state (for fermions at half-filling) is easily prepared in experiments involving ultra-cold atoms, we consider a CDW state as the initial state in our study.",
"The initial state can be written as: $\\mathinner {|{\\Psi _{in}}\\rangle }=\\prod \\limits _{i=1}^{N/2} \\hat{c}_{2i}^\\dagger \\mathinner {|{0}\\rangle }.$ We are mainly interested in the dynamics of entanglement entropy and the out-of-time-order correlator which are of current interest for integrable disordered quantum systems [61].",
"In this section we discuss the dynamics of entanglement entropy.",
"OTOC will be discussed in the following section.",
"We will stick to the quasiperiodicity parameter $\\alpha _g=(\\sqrt{5}-1)/2$ unless otherwise mentioned.",
"Figure: Entanglement entropy in the AAH model.",
"(a) The dynamics ofthe half-chain entanglement entropy S A S_A with increasing values ofλ\\lambda for free fermions at half-filling.",
"Here system sizeN=512N=512.",
"(b) The system size NN dependence of the saturation valueof the half-chain entanglement entropy S A ∞ S_A^\\infty of free fermionsat half-filling for increasing values of λ\\lambda .",
"For all theplots, total number of θ p \\theta _p realizations is 100 withquasi-periodicity fixed to be α g \\alpha _g.Figure: (a-c) The dynamics of the half-chain entanglemententropy S A S_A with increasing values of λ\\lambda for freefermions at half-filling and for σ=0.5,1.5\\sigma =0.5,1.5 and 3.03.0respectively.",
"For all the plots system size N=1024N=1024.When the overall state of the system is pure, entanglement entropy is simply given by $S_A=-Tr(\\rho _A \\ln \\rho _A)$ where $\\rho _A$ is the reduced density matrix of the subsystem A.",
"We calculate the dynamics of the half-chain entanglement entropy using free fermionic techniques [62], [63] that allow for the study of significantly large system sizes.",
"In the AAH model, the growth of $S_A$ is ballistic in time in the delocalized phase $(\\lambda =1)$ and (almost) diffusive at the critical point $(\\lambda =2)$ whereas there is essentially no growth in the localized phase $(\\lambda =3)$ as shown in Fig.",
"REF (a).",
"These results are in agreement with those of an earlier study of quench dynamics in the AAH model [64].",
"Fig.",
"REF (b) shows that the saturation value $S_A^\\infty $ scales linearly with system sizes $(S_A^\\infty \\propto N)$ at $\\lambda =1$ and $\\lambda =2$ , while $S_A^\\infty \\propto N^0$ for $\\lambda =3$ .",
"Also we have checked that these results remain independent of the choice of the quasiperiodicity parameter $\\alpha $ .",
"The plots of $S_A$ as a function of time for the LRH model are shown in Fig.",
"REF (a-c) for increasing values of $\\lambda $ and $\\sigma =0.5,1.5$ and $3.0$ respectively.",
"In the plots for $\\sigma =0.5$ and $\\sigma =1.5$ each, $S_A$ shows two different behaviors with time which can be noticed both in Fig.",
"REF (a) and Fig.",
"REF (b).",
"In Fig.",
"REF (a) after the initial transient a power-law growth is found followed by a secondary logarithmic growth (see Fig.",
"REF (a)).",
"The secondary growth appears presumably due to the presence of the DM edge.",
"It is to be noted that the secondary growth is absent for $\\lambda =0.1$ for which all the eigenstates are delocalized.",
"The primary growth in the dynamics of $S_A$ can be fitted with a function $S_A(t)=c_1t^\\beta +c_2$ to extract the values of the power-law exponent $\\beta $ .",
"For $\\lambda =0.1$ , $\\beta $ turns out to be $0.53$ .",
"For other values of $\\lambda =0.5,1.0,2.0$ which correspond to mixed phases with DM edges, $\\beta =0.45,0.38$ and $0.31$ respectively.",
"In Fig.",
"REF (b) for $\\sigma =1.5$ a primary power-law growth and a subsequent secondary logarithmic (see Fig.",
"REF (b)) growth is observed.",
"For $\\sigma =1.5$ , $\\lambda =0.1$ corresponds to the delocalized phase whereas $\\lambda =1.3,2.0,3.0$ here correspond to mixed phases with DL edges.",
"For $\\lambda =0.1,1.3,2.0$ and $3.0$ , the power-law exponent $\\beta =0.89,0.82,0.80$ and $0.76$ respectively.",
"The secondary growth is again absent for $\\lambda =0.1$ for which there is no DL edge.",
"For $\\sigma =3.0$ the secondary growth is absent as seen from Fig.",
"REF (c) since the LRH model approaches the short-range AAH limit at this point.",
"For $\\lambda =0.1$ the growth of $S_A$ happens ballistically as $\\beta =1.0$ as in the delocalized phase of the short-range AAH model.",
"For $\\lambda =1.7,2.1,2.6$ the system is in the mixed phases with the DL edges.",
"In the mixed phases the growth of $S_A$ is initially less sensitive to the delocalized eigenstates due to the short-rangeness of the system.",
"After some time the delocalized eigenstates start to dominate as indicated by the increasing change of rate of $S_A$ in Fig.",
"REF (c).",
"Right before reaching saturation the power-law fit provides $\\beta =0.84,0.82,0.79$ for $\\lambda =1.7,2.1,2.6$ respectively.",
"The secondary logarithmic growth for $\\sigma =0.5, 1.5$ are depicted in Fig.",
"REF (a,b) respectively where the plots are fitted with the function $S_A(t)=a_1\\ln t + b_1$ .",
"Lots of intrinsic fluctuations can be seen in the plots due to the quasiperiodicity in the system.",
"The secondary logarithmic growth tends to vanish in the short-range limit of hopping as these are barely seen for $\\sigma =3.0$ (see Fig.",
"REF (c)).",
"Logarithmic growth of entanglement entropy has been seen recently in a few noninteracting randomly disordered systems [61], [65].",
"Logarithmic growth of entanglement in longrange interacting systems has also recently been addressed [66].",
"The logarithmic behavior in quasiperiodically disordered long-ranged LRH model is attributed to the presence of mixed phases in the longrange regime since this feature is not found in the short-range regime or in absence of mixed phases.",
"Figure: (a) The secondary logarithmic growth of the half-chainentanglement entropy S A S_A in the LRH model for σ=0.5\\sigma =0.5 andλ=0.5,1.0,2.0\\lambda =0.5,1.0,2.0 for which the best fits are 18.98lnt+15.6418.98\\ln t +15.64, 12.18lnt+32.3812.18\\ln t + 32.38 and 15.21lnt-15.5415.21\\ln t - 15.54respectively.",
"(b) Similar plots for σ=1.5\\sigma =1.5 andλ=1.3,2.0,3.0\\lambda =1.3,2.0,3.0 for which best fits are 17.61lnt+24.1517.61\\ln t + 24.15,13.39lnt+17.9413.39\\ln t + 17.94 and 3.89lnt+41.933.89\\ln t + 41.93 respectively.",
"The solidlines are best fits whereas the scattered points represent thecorresponding data-points.",
"The xx-axis is in the log scale.",
"For allthe plots system size N=1024N=1024 and fermionic filling fraction is1/21/2.We notice that the power-law exponent $\\beta $ is larger for $\\sigma >1$ as compared to $\\sigma <1$ .",
"The counter-intuitive behavior of power-law exponent in the entanglement growth has been addressed earlier in a clean free fermionic long-range model [67].",
"It is noteworthy that the exponent $\\beta $ changes very little with $\\lambda $ for $\\sigma =1.5$ and $3.0$ for each of which $(\\lambda ,\\sigma )$ combinations correspond to the same $P_1, P_2$ and $P_3$ phases with DL edges.",
"This happens possibly because the properties of the localized states barely vary in the different $P_q$ phases.",
"On the other hand $\\beta $ changes rapidly with $\\lambda $ for $\\sigma =0.5$ in the presence of multifractal states the properties of which may change significantly as one moves from $P_1$ to $P_2$ to $P_3$ and so on.",
"Another observation is that the late time dynamics of $S_A$ slows down for $\\sigma =1.5$ whereas it speeds up for $\\sigma =3.0$ .",
"This happens due to varying degrees of effectiveness of the delocalized eigenstates in the presence of long-range and short-range hopping.",
"In a particular $P_q$ phase (with DM or DL edges) the values of all the exponents discussed here barely change with $\\lambda $ for a fixed value of $\\sigma $ .",
"Similar results have been discussed in a recent work [68].",
"Also we have checked that the qualitative behaviors of all the $S_A$ plots and the values of the exponent $\\beta $ change very little if, instead of $\\alpha _g$ , one uses $\\alpha _s$ or $\\alpha _b$ for an initial half filled CDW state.",
"However, the exponents associated with the secondary $S_A$ growth may change significantly as this part of the dynamics is dominated by the multifractal or the localized single particle eigenstates, the fraction of which depends on the choice of the quasiperiodicity parameter in a particular $P_q$ $(q=1,2,3...)$ phase.",
"Figure: (a) The saturation value of the half-system S A S_A asa function of λ\\lambda for σ=0.5,1.5\\sigma =0.5,1.5 and 3.03.0respectively for fermions at half-filling.",
"(b) The systemsize NN dependence of the saturation value of thehalf-chain entanglement entropy S A ∞ S_A^\\infty of freefermions at half-filling for different combinations ofλ\\lambda and σ\\sigma .The saturation value of entanglement entropy $S_A^\\infty $ turns out to be a useful quantity.",
"Fig.",
"REF (a) shows $S_A^\\infty $ as a function of $\\lambda $ for $\\sigma =0.5,1.5,3.0$ .",
"The steps appearing in the plots denote the transitions from the delocalized-to $P_1$ -to-$P_2$ -to-$P_3$ etc.",
"phases.",
"The $P_q$ phases have a fraction of eigenstates that are multifractal for $\\sigma =0.5$ and a fraction of eigenstates that are localized for $\\sigma =1.5,3.0$ .",
"Hence $S_A^\\infty $ is much lower for $\\sigma =1.5,3.0$ than for $\\sigma =0.5$ in these phases.",
"Also we have looked at the system size $N$ dependence of $S_A^\\infty $ in these phases as shown in Fig.",
"REF (b).",
"The combinations of ($\\lambda $ ,$\\sigma $ ) are chosen in such a way that the system is in the delocalized phase for $(0.1,0.5)$ ; $P_2$ phase with DM edge for $(1.0,0.5)$ ; $P_2$ phase with DL edge for $(2.0,1.5)$ , and $(2.1,3.0)$ ; and localized phase for $(5.0,3.0)$ .",
"For the delocalized and mixed phases with DM or DL edges $S_A^\\infty \\propto N$ .",
"In the localized phase $S_A^\\infty \\propto N^0$ , which is obtained effectively in the short-range AAH limit.",
"Typically in a sufficiently long-ranged regime one can obtain algebraic localization such as seen in the random long-range hopping model [63].",
"In the random long-range hopping model an algebraic localization dominated phase is found for $1<\\sigma <2$ for which $S_A^\\infty $ varies sub-linearly with $N$ [63]."
],
[
"Out-of-time-order correlator",
"Out-of-time-order correlators (OTOC) are good observables to capture chaos or information scrambling in quantum systems.",
"The majority of studies looking at OTOC have been in the context of localization transitions in interacting systems [15], [16], [17], [13], [46].",
"However, OTOC has been barely [53], [54] addressed in the literature in relation to the localization transition in disordered noninteracting (quadratic) Hamiltonians.",
"Our goal here is to investigate OTOC as a distinguisher for the various phases found in the AAH and LRH models.",
"In this work we choose the two unitary-and-Hermitian operators $\\hat{\\sigma }_i^z$ and $\\hat{\\sigma }_j^z$ at a distance $x=|i-j|$ .",
"The function $F(x,t)$ in Eq.",
"REF is then given by $F(x,t)=\\langle \\hat{\\sigma }_i^z(t) \\hat{\\sigma }_j^z(0) \\hat{\\sigma }_i^z(t) \\hat{\\sigma }_j^z(0) \\rangle .$ We keep the position of the time evolved operator fixed at $i=N/2$ .",
"By varying $j$ we study the scrambling of quantum information over the lattice as a function of time.",
"The initial state is fixed as the product state of half-filled fermions defined in Eq.",
"REF .",
"For free fermions one can use the Jordan-Wigner transformation $\\hat{\\sigma }_j^z=2\\hat{n}_j-1$ to simplify the expression of $F(x,t)$ [53].",
"We elaborate on this ahead."
],
[
"Formalism",
"Here we provide a brief description of the formalism in relation to OTOC which is used in this work.",
"Let us consider a generic quadratic Hamiltonian: $\\hat{H}_{free} = \\sum \\limits _{i,j} H_{ij} \\hat{c}_i^{\\dagger } \\hat{c}_j,$ where $H_{ij}$ 's are the elements of a Hermitian matrix $H$ and $\\hat{c}_i^{\\dagger }$ 's ($\\hat{c}_i$ 's) are fermionic creation (annihilation) operators obeying the following anti-commutation relations : $\\lbrace \\mathit {\\hat{c}_i^{\\dagger }}\\textsf {,}\\hspace{2.84526pt}\\mathit {\\hat{c}_j} \\rbrace = \\delta _{ij} \\textsf {;}\\hspace{11.38109pt} \\lbrace \\mathit {\\hat{c}_i^{\\dagger }}\\textsf {,}\\hspace{2.84526pt}\\mathit {\\hat{c}_j^{\\dagger }} \\rbrace \\hspace{5.69054pt} = \\hspace{5.69054pt} \\lbrace \\mathit {\\hat{c}_i}\\textsf {,}\\hspace{2.84526pt}\\mathit {\\hat{c}_j} \\rbrace \\hspace{5.69054pt} = \\hspace{5.69054pt}0.$ Using the eigenvectors of the coupling matrix $H$ , we can define new fermionic operators that diagonalize the Hamiltonian.",
"If $A_{jk}$ represent the coefficients of the eigenvectors of the matrix $H$ , we introduce the fermionic operators: $\\mathit {\\hat{d}_k^{\\dagger }} = \\sum \\limits _j A_{jk}^\\ast \\mathit {\\hat{c}_j^{\\dagger }},\\hspace{11.38109pt}\\mathit {\\hat{d}_k} = \\sum \\limits _j A_{jk} \\mathit {\\hat{c}_j}$ that transform the Hamiltonian into a diagonal form: $\\hat{H}_{free} = \\sum \\limits _{k} \\epsilon _k \\mathit {\\hat{d}_k^{\\dagger }}\\mathit {\\hat{d}_k}.$ Here $\\mathit {\\hat{d}_k^{\\dagger }}$ ($\\mathit {\\hat{d}_k}$ ) creates (annihilates) a particle with energy $\\epsilon _k$ and obeys similar anti-commutation relations as $\\hat{c_i}$ 's: $\\lbrace \\mathit {\\hat{d}_k^{\\dagger }}\\textsf {,}\\mathit {\\hat{d}_l}\\rbrace = \\delta _{kl}, \\hspace{5.69054pt}\\lbrace \\mathit {\\hat{d}_k}\\textsf {,}\\mathit {\\hat{d}_l}\\rbrace = 0, \\hspace{5.69054pt}\\lbrace \\mathit {\\hat{d}_k^{\\dagger }}\\textsf {,}\\mathit {\\hat{d}_l^{\\dagger }}\\rbrace = 0.$ Using the Heisenberg equation for operators, the time-evolved operators $\\mathit {\\hat{d}_k^{\\dagger }}(t)$ and $\\mathit {\\hat{d}_k}(t)$ can be found.",
"$\\frac{d}{dt}\\mathit {\\hat{d}_k} = \\imath \\left[\\hat{H}_{free}\\textsf {,} \\mathit {\\hat{d}_k} \\right]=-\\imath \\epsilon _k \\mathit {\\hat{d}}_k,$ which leads to $\\mathit {\\hat{d}_k} (t) = e^{-\\imath \\epsilon _k t} \\mathit {\\hat{d}_k}(t=0)$ and hence $\\mathit {\\hat{d}_k}^{\\dagger } (t) = e^{\\imath \\epsilon _k t} \\mathit {\\hat{d}_k}^{\\dagger }(t=0)$ .",
"Using the relations: $\\mathit {\\hat{c}_j^{\\dagger }}(t) = \\sum \\limits _k A_{jk}\\mathit {\\hat{d}_k^{\\dagger }}(t), \\hspace{5.69054pt}\\mathit {\\hat{c}_j}(t) = \\sum \\limits _k A^*_{jk}\\mathit {\\hat{d}_k}(t)$ one finds the following anti-commutation relations between creation and annihilation operators at different times in position space.",
"$&&\\lbrace \\hat{c}_i^{\\dagger } (t), \\hat{c}_j\\rbrace = \\sum \\limits _{k} e^{\\imath \\epsilon _kt} A_{ik}^\\ast A_{jk} = a_{ij}(t)\\nonumber \\\\&&\\lbrace \\hat{c}_i(t), \\hat{c}_j^{\\dagger }\\rbrace = \\sum \\limits _{k} e^{-\\imath \\epsilon _kt} A_{ik}A_{jk}^\\ast = {a}_{ij}^\\ast (t)$ along with $\\lbrace \\hat{c}_i^{\\dagger } (t), \\hat{c}_j^{\\dagger }\\rbrace = \\lbrace \\hat{c}_i (t), \\hat{c}_j\\rbrace = 0$ , which are trivially satisfied.",
"Here the parantheses used to denote time are dropped from the operators for $t=0$ .",
"This convention is used further in the paper.",
"In this work we consider an initial product state of the form $\\mathinner {|{\\Psi }\\rangle } = \\prod \\limits _{j\\in S}\\hspace{2.84526pt}\\mathit {\\hat{c}_j^{\\dagger }} \\mathinner {|{0}\\rangle }$ where $j$ refers to the index of the site which is occupied.",
"Let $S$ be the set consisting of site indices of sites which are occupied.",
"The initial occupation matrix in position space is then given by $<\\mathit {\\hat{c}_i^{\\dagger }}\\mathit {\\hat{c}_j}> = \\bigg \\lbrace {1 \\hspace{11.38109pt} \\textsf {if} \\hspace{2.84526pt}i=j\\hspace{5.69054pt} \\forall \\hspace{5.69054pt} i\\in S \\atop 0 \\hspace{34.1433pt}\\textsf {otherwise} }.$ Using the Jordan-Wigner transformation $\\hat{\\sigma }_i^z=2\\hat{n}_i -1$ with $\\hat{n}_i=\\hat{c}_i^\\dagger \\hat{c}_i$ in Eq.",
"REF we have: $F(x\\textsf {,} t) &=& 16\\langle \\hat{n}_i(t)\\hat{n}_j\\hat{n}_i(t)\\hat{n}_j\\rangle + 4\\langle \\hat{n}_j\\hat{n}_i(t)\\rangle -4\\langle \\hat{n}_i(t)\\hat{n}_j\\rangle \\nonumber \\\\ &&-8\\langle \\hat{n}_i(t)\\hat{n}_j\\hat{n}_i(t)\\rangle -8\\langle \\hat{n}_j\\hat{n}_i(t)\\hat{n}_j\\rangle + 1 .$ In this work we have kept $i=N/2$ where $N$ is the number of sites in the lattice and calculated $F(x,t)$ by varying $j$ .",
"For the case $j \\in S$ such that $\\mathit {\\hat{c}_j^{\\dagger }}\\mathinner {|{\\Psi }\\rangle } = 0$ , Eq.",
"REF can be written as [53] $F(x\\textsf {,} t) = 8\\vert a_{ij}\\vert ^2\\left<\\hat{n}_i(t)\\right> - 8\\vert a_{ij}\\vert ^2 + 1 .$ For $j\\notin S$ , $\\mathit {\\hat{c}_j}\\mathinner {|{\\Psi }\\rangle } = 0$ which leads to $F(x\\textsf {,} t) = 1 - 8\\vert a_{ij}\\vert ^2\\left<\\hat{n}_i(t)\\right>.$"
],
[
"Results",
"We now discuss the OTOC-related results for the AAH and LRH models.",
"AAH model: First we calculate $C(x,t)$ in the AAH model.",
"The profiles of $C(x,t)$ in position space for increasing instants of time are shown in Fig.",
"REF (a-c) for $\\lambda =1.0,2.0$ and $3.0$ respectively.",
"At $t=0$ $C(x)$ is zero for all $x$ because $F(x,0)$ reduces to the squares of Pauli matrices yielding unity in Eq.",
"REF .",
"Then $C(x)$ starts developing for small values of the distance $x$ due to the non-commutation of the matrices $\\hat{\\sigma }_i^z(t)$ and $\\hat{\\sigma }_j^z(0)$ for small $x$ at early times.",
"During this period of time $C(x)$ attains high values for small $x$ while the maximum value of $C(x)$ happens to be at $x=1$ .",
"This is shown in Fig.",
"REF (a) for $\\lambda =1$ .",
"Then $C(x,t)$ starts decreasing for small $x$ whereas it keeps growing for large values of $x$ due to the spreading of non-commutativity among Pauli matrices.",
"In the long run $C(x,t)$ shows a uniform dependence on $x$ for $\\lambda =1$ (see Fig.",
"REF (a)) when $S_A$ also reaches saturation.",
"Figure: OTOC in the AAH model.",
"(a-c) OTOC C(x,t)C(x,t) as a function ofdistance xx at different instants tt for λ=1.0,2.0\\lambda =1.0,2.0 and3.03.0 respectively.",
"System size N=256N=256.",
"The plot legend shown in(a) also applies to (b) and (c).",
"(d-f) Saturation value C ∞ (x)C^\\infty (x)as a function of distance xx for increasing system sizes NN andfor λ=1.0,2.0\\lambda =1.0,2.0 and 3.03.0 respectively.",
"The plot legendshown in (d) also applies to (e) and (f).",
"For all the plots, totalnumber of θ p \\theta _p realizations is 500.For the critical point $\\lambda =2$ the initial dynamics of $C(x,t)$ shown in Fig.",
"REF (b) is similar to that for $\\lambda =1$ .",
"But in the long-time limit $C(x,t)$ shows a non-uniform dependence on $x$ with occasionally large fluctuations especially at $x=21,34,55$ etc.",
"which are terms in the Fibonacci sequence of the `golden mean' [69].",
"These large fluctuations appear possibly due to the multifractal nature of the eigenstates.",
"In Fig.",
"REF (c) for the localized phase at $\\lambda =3$ , $C(x,t)$ grows for small $x$ at early times while the subsequent decay is absent in the dynamics.",
"Eventually in the long-time limit $C(x)$ drops exponentially with $x$ i.e.",
"$C(x)\\sim e^{-x/{\\xi _{OTOC}}}$ [53] such that $C(x)\\ne 0$ for $x<\\xi _{OTOC}$ but is zero for large $x$ .",
"$\\xi _{OTOC}$ decreases with $\\lambda $ in the localized phase.",
"Also we analyze the system size $N$ -dependence of the spatial profile of $C^\\infty (x)$ in the long-time limit as shown in Fig.",
"REF (d-f) for $\\lambda =1,2,3$ respectively.",
"For $\\lambda =1$ , $C^\\infty \\propto 1/N$ .",
"This can be explained by looking at the long-time behavior of $|a_{ij}(t)|^2$ defined in Section REF .",
"$\\lim \\limits _{T\\rightarrow \\infty }\\frac{1}{T}\\int \\limits _{0}^{T} dt |a_{ij}(t)|^2=\\sum \\limits _{k}|A_{ik}|^2 |A_{jk}|^2$ , which scales with $1/N$ as $A_{ik}\\propto 1/\\sqrt{N}$ in the delocalized phase.",
"At the critical point $\\lambda =2$ , $C^\\infty $ depends on $x$ and shows a sub-linearly decreasing dependence with $N$ except on the points where large fluctuations are observed due to the multifractal nature of the eigenstates.",
"At these special points the $N$ -dependence is not regular.",
"The number of these large fluctuations increases with $N$ .",
"However, in the localized phase for $\\lambda =3$ , $C^\\infty \\propto N^0$ for $x<\\xi _{OTOC}$ and is in any case zero for large $x$ .",
"Figure: Time dynamics of OTOC in the AAH model for increasingvalues of xx.",
"(a-c) C(x,t)C(x,t) vs tt plots for early times forλ=1.0,2.0\\lambda =1.0,2.0 and 3.03.0 respectively.",
"The plot legend shown in(a) also applies to (b) and (c).",
"(d-f) C(x,t)C(x,t) vs ttplots for late times for λ=1.0,2.0\\lambda =1.0,2.0 and 3.03.0 respectively.The plot legend shown in (d) also applies to (e) and (f).For all the plots, system size N=1024N=1024 and total number ofθ p \\theta _p realizations is 500.The early-time growth of OTOC in the AAH model is shown in Fig.",
"REF (a-c) for small values of $x$ and for $\\lambda =1,2,3$ respectively.",
"For all values of $\\lambda $ we notice that $C(x,t)\\sim t^{2x}$ $\\forall $ odd $x$ and $C(x,t)\\sim t^{2(x+1)}$ $\\forall $ even $x$ , which is also found in translationally invariant models [49].",
"This can be understood by writing the Heisenberg time evolution of $\\hat{W}(t)$ using Hausdorff-Baker-Campbel (HBC) formula $e^{it\\hat{H}} \\hat{W} e^{-it\\hat{H}} = \\sum \\limits _{m=0}^{\\infty } \\frac{(it)^m}{m!}",
"\\hat{L}^m(\\hat{W}) ,$ where $\\hat{L}(\\hat{W})=[\\hat{H},\\hat{W}]$ and $\\hat{W}=\\hat{\\sigma }^z_{L/2}$ .",
"The power-law growth obtained in the early-time dynamics is controlled by the term with smallest $m$ such that $[\\hat{L}^m,\\hat{\\sigma }^z_{L/2+x}])]\\ne 0$ .",
"For short-range AAH Hamiltonian it is clear that this happens when $m=x$ leading to $C(x,t)\\sim t^{2x}$ [49].",
"For $x=2,4,6,..$ one includes the next leading term which gives $C(x,t)\\sim t^{2(x+1)}$ [53].",
"This shows that the quasiperiodic disorder does not play any important role in the initial dynamics.",
"However, in the long-time limit OTOC is found to decay as $1/t^\\gamma $ with time and the power-law exponent $\\gamma $ depends on $\\lambda $ as shown in Fig.",
"REF (d-f) for $\\lambda =1,2,3$ respectively.",
"We find that the values of $\\gamma \\approx 1.0,0.3,0.0$ for $\\lambda =1,2,3$ respectively which correspond to the delocalized, critical and localized phases respectively.",
"The $t^{-1}$ decay in the delocalized phase is also seen in a clean system [49].",
"The extended (ergodic or nonergodic) states are responsible for the correlation wavefront to reach a particular distant site in the lattice (leading to OTOC growth) and then proceed further (leading to OTOC decay) until OTOC reaches a saturation.",
"Although the decay rate is expectedly less in presence of (nonergodic) multifractal phase in comparison to (ergodic) delocalized phase.",
"A lot of intrinsic fluctuations are found in these plots due to the presence of quasiperiodic disorder.",
"We also note that in the late time dynamics for a fixed value of $\\lambda $ the value of $\\gamma $ does not depend on $x$ unlike the early-time growth.",
"Figure: OTOC in the LRH model.",
"(a-d) OTOC C(x,t)C(x,t) as a function ofdistance xx at different instants tt for(λ=1.0,σ=0.5)(\\lambda =1.0,\\sigma =0.5), (λ=2.0,σ=1.5)(\\lambda =2.0,\\sigma =1.5),(λ=2.1,σ=3.0)(\\lambda =2.1,\\sigma =3.0) and (λ=5.0,σ=3.0)(\\lambda =5.0,\\sigma =3.0)respectively.",
"System size N=1024N=1024 The plot legend shown in(a) also applies to (b), (c) and (d).",
"(e-h) Saturation valueC ∞ (x)C^\\infty (x) as a function of distance xx for increasing system sizesNN and for (λ=1.0,σ=0.5)(\\lambda =1.0,\\sigma =0.5), (λ=2.0,σ=1.5)(\\lambda =2.0,\\sigma =1.5),(λ=2.1,σ=3.0)(\\lambda =2.1,\\sigma =3.0) and (λ=5.0,σ=3.0)(\\lambda =5.0,\\sigma =3.0)respectively.",
"The plot legend shown in figure (e) also applies tofigures (f), (g) and (h).",
"For all the plots, totalnumber of θ p \\theta _p realizations is 500.LRH model: The spatial distribution of OTOC for the LRH model of fermions at half-filling is shown in Fig.",
"REF .",
"We have chosen the combination of parameters $(\\lambda ,\\sigma )$ in such a way that the system is in four different types of phases: (i) $P_2$ phase with DM edge ($\\sigma =0.5$ ), (ii) $P_2$ phase with DL edge where the hopping is relatively long-range ($\\sigma =1.5$ ), (iii) $P_2$ phase with DL edge where the hopping is short-range ($\\sigma =3.0$ ) and (iv) the localized phase.",
"The spatial profiles of $C(x,t)$ for each of the above kinds of parameter combinations are shown in Fig.",
"REF (a-d) respectively.",
"For early times $C(x,t)$ shows $1/x^{2\\sigma }$ dependence for all the choices of parameters.",
"Figure: Time dynamics of OTOC in the LRH model for increasing valuesof xx.",
"(a-d) C(x,t)C(x,t) vs tt plots for early times for(λ=1.0,σ=0.5)(\\lambda =1.0,\\sigma =0.5), (λ=2.0,σ=1.5)(\\lambda =2.0,\\sigma =1.5),(λ=2.1,σ=3.0)(\\lambda =2.1,\\sigma =3.0) and (λ=5.0,σ=3.0)(\\lambda =5.0,\\sigma =3.0)respectively.",
"The plot legend shown in (a) also applies to(b), (c) and (d).",
"(e-h) C(x,t)C(x,t) vs tt plots for late times for(λ=1.0,σ=0.5)(\\lambda =1.0,\\sigma =0.5), (λ=2.0,σ=1.5)(\\lambda =2.0,\\sigma =1.5),(λ=2.1,σ=3.0)(\\lambda =2.1,\\sigma =3.0) and (λ=5.0,σ=3.0)(\\lambda =5.0,\\sigma =3.0)respectively.",
"The plot legend shown in figure (e) also applies tofigures (f), (g) and (h).For all the plots, system size N=1024N=1024 and totalnumber of θ p \\theta _p realizations is 500.In the mixed phases we see that in the long time limit $C(x,t)$ follows $1/x^{\\delta }$ (power-law) behavior for small $x$ and almost $x$ -independent behavior for large $x$ .",
"In Fig.",
"REF (a) $\\delta \\approx 1.0$ for small $x$ whereas occasional large fluctuations can be seen for large values of $x$ which are terms in the Fibonacci sequence of the `golden mean'.",
"The occasional large fluctuations are signatures of the multifractal states similar to the AAH model at the transition point.",
"In Fig.",
"REF (b) for intermediate times $(t\\sim 10)$ $C(x,t)$ shows $1/x^{1.5}$ dependence for small $x$ and $1/x^{2\\sigma }$ dependence for large $x$ .",
"However, in the long time limit $x$ -independent behavior of $C(x,t)$ is seen for large $x$ along with the $1/x^{1.5}$ dependence for small $x$ .",
"Here $\\delta \\approx 1.5$ .",
"In Fig.",
"REF (c) for intermediate times $C(x,t)$ shows $1/x^{1.5}$ dependence for small $x$ and $1/x^{2\\sigma }$ dependence for large $x$ .",
"A sharp boundary can be seen between these two behaviors, which is a characteristic signature of the short-range regime [43].",
"In the long time limit $C(x,t)$ does not depend on $x$ for large $x$ whereas it continues to show the $1/x^{1.5}$ dependence for small $x$ corresponding to $\\delta \\approx 1.5$ once again.",
"In Fig.",
"REF (d) corresponding to the localized phase the spatial profile $C(x,t)$ continues to be $1/x^{2\\sigma }$ $\\forall x$ .",
"We do not a see a `mixed' behavior in this case as the system is unambiguously in the localized phase.",
"We notice that the value of $\\delta \\approx 1.0$ for all the mixed phases with DM edges ($\\sigma <1$ ) while $\\delta \\approx 1.5$ for the mixed phases with DL edges ($\\sigma >1$ ).",
"It is noticeable that the value of $\\delta $ is larger in the presence of localized states than that in the presence of multifractal states.",
"In Fig.",
"REF (e-h) we show the system size dependence of the spatial profile of OTOC in the long time limit $C^\\infty (x)$ corresponding to phases described in Fig.",
"REF (a-d) respectively.",
"Since we have already mentioned earlier that the calculation of $C^\\infty (x)$ involves each eigenstate in the spectrum, in the mixed phases ($P_2$ phase with DL/DM edge), in Fig.",
"REF (e-g) $C^\\infty (x)$ for large $x$ decreases with the system size $N$ although its functional dependence on $N$ is not very clear due to fluctuations.",
"In Fig.",
"REF (e) we see the occasional large fluctuations due to the presence of multifractal states increase and become more prominent with $N$ .",
"In Fig.",
"REF (f) and Fig.",
"REF (g) the large fluctuations are not seen due to the absence of multifractal states.",
"In the presence of DM edge $C(x)$ depends on $N$ for small $x$ whereas in the presence of DL edge $C(x)$ remains invariant with the change in $N$ for small $x$ .",
"In Fig.",
"REF (h) the spatial profile of $C^\\infty (x)$ becomes system size independent which is characteristic of a localized phase.",
"We note that in the presence of localized states the peak of the profile of $C^\\infty (x)$ has a higher value (Fig.",
"REF (f-h)) than in the absence of localized states (Fig.",
"REF (e)).",
"The early-time growth of OTOC in the LRH model is shown for small $x$ in Fig.",
"REF for the $P_2$ phase with DM edge in Fig.",
"REF (a), $P_2$ phase with DL edge in Fig.",
"REF (b-c) and the localized phase in Fig.",
"REF (d).",
"Independent of the values of $\\lambda $ and $\\sigma $ , we find that $C(x,t)\\sim t^2$ $\\forall $ odd $x$ and $C(x,t)\\sim t^4$ $\\forall $ even $x$ as also found for the translationally invariant long-range hopping model (we have checked).",
"Unlike the short-range AAH model here OTOC does not have a power-law behaviour with $x$ and the growth here is in fact largely $x$ -independent.",
"This can be again understood from Eq.",
"REF .",
"Since LRH Hamiltonian is long-ranged $\\hat{\\sigma }^z_{L/2}$ and $\\hat{\\sigma }^z_{L/2+x}$ immediately gets connected for smallest $m=1$ which gives $C(x,t)\\sim t^2$ [70].",
"For even $x$ $C(x,t)\\sim t^4$ by including next term as the leading order.",
"For the same phases of the LRH model, the late-time decay of OTOC is shown in Fig.",
"REF (e-h).",
"From Fig.",
"REF (e) we see that in the $P_2$ phase with DM edge the power-law decay exponent $\\gamma =0.15$ $\\forall $ $x$ .",
"In Fig.",
"REF (f-g) we find that in the $P_2$ phase with DL edge the decay exponent gets smaller, and is difficult to determine.",
"A power-law decay is found due to the presence of the delocalized states in the phase.",
"However it is smaller as compared to that in Fig.",
"REF (e) due to the presence of localized states instead of (extended nonergodic) multifractal states.",
"In Fig.",
"REF (h) we do not see any decay of OTOC after the early-time growth due to the absence of delocalized or multifractal states.",
"All the dynamical behaviors shown by both the entanglement entropy and OTOC can be seen more clearly as one increases the system sizes.",
"We would also like to mention that we have checked that for a clean (undisordered) system in the presence of long-range hopping, in the long-time limit $C(x,t)$ decays as $1/t$ independent of $x$ and the long-range parameter $\\sigma $ .",
"The values of $C^\\infty $ are also independent of $x$ and $\\sigma $ as the phases are delocalized for all $\\sigma $ .",
"On the other hand in the LRH model, the power-law decay exponent $\\gamma $ is much lower than that in the clean system.",
"In the LRH model $\\gamma $ depends on the values of $\\sigma $ .",
"The values of $C^\\infty $ depend on $x$ as the phases are (nonergodic) mixed or localized.",
"However, the values of $\\gamma $ and $C^\\infty $ change very little with the fraction of delocalized states present in the mixed phases, especially in the presence of the DM edge."
],
[
"Conclusion",
"To conclude we study the nonequilibrium dynamics of entanglement entropy and out-of-time-order correlator of noninteracting fermions at half-filling starting from a product state to distinguish different phases hosted by the quasiperiodic Aubry-André-Harper model with long-range hopping.",
"Apart from the delocalized and localized phases, the model also shows mixed phases which consist of delocalized and multifractal or localized states.",
"In the nearest neighbor hopping limit due to the restoration of self-duality the model hosts delocalized, multifractal and localized phases.",
"When the hopping is sufficiently long-ranged a secondary logarithmic behavior in the entanglement entropy is seen in the mixed phases whereas the primary behavior is a power-law growth which can be different in different phases.",
"The saturation value of entanglement entropy in the delocalized, multifractal and mixed phases depends linearly on system size whereas in the localized phase (in the short-range regime) it is independent of system size.",
"The secondary growth is a unique feature that we expect to see in the long-ranged mixed phases of other models as this feature seems to be absent in the short-range regime.",
"Although the logarithmic behavior in our case is surprising and it may not necessarily be logarithmic in nature for other cases.",
"In early-time dynamics OTOC shows very different behavior in the presence of nearest neighbor hopping and long-range hopping, like is seen also in clean systems.",
"The late-time decay rate of OTOC is different in the delocalized and multifractal phases of the nearest neighbor AAH model whereas the localized phase of the same model shows no such decay.",
"In the long-time limit the spatial profile of OTOC is independent, dependent (with large fluctuations) and exponentially dependent on space in the delocalized, multifractal and localized phases respectively.",
"Also the profile decreases linearly and sub-linearly with system size in the delocalized and multifractal phases respectively whereas it is independent of system size in the localized phase.",
"In the multifractal phase, large fluctuations are observed at the special points which are related to the Fibonacci sequence of the quasiperiodicity parameter.",
"In the long-range Harper model, the late-time power-law decay is present in the mixed phases due to the presence of extended states although the power-law decay exponent is smaller compared to the inverse power-law behavior found in the delocalized phase of the (clean) system.",
"The power-law exponent barely changes with the change in the fraction of delocalized states in the mixed phases showing the dominace of the nonergodic states in the dynamics.",
"Among the mixed phases the presence of localized states supresses the late-time decay even more than that of multifractal states.",
"The localized phase of this model does not show any such decay due to absence of extended states.",
"The dynamics of the spatial profile of OTOC in the mixed phase in the short-range limit reveals a sharp boundary which is typical of longrange models [67].",
"The spatial profile of OTOC in the long-time limit in the mixed phases shows a mixed behavior: power-law dependence for small distance (nonergodic behavior) and no dependence for large distance (ergodic behavior).",
"In the mixed phases containing multifractal states the profile shows large fluctuations at special points for large distance similar to the critical point of the AAH model.",
"In the localized phase the spatial dependence of OTOC is a power-law one for all distances and is also independent of system size.",
"Also in the mixed phases the spatial profile shows different system-size dependences for small and large distances which is expected.",
"One may expect to see these behaviors in the mixed phases of other long-range (Harper-like) models.",
"Entanglement entropy and OTOC are two quantities that are of great interest in dynamical studies of quantum systems with the second one being easier to be implemented in experiments.",
"Very recently, a surge of interest in the community [71], [72], [73], [74] has been seen in the experimental detection of quantum phase transitions using OTOC.",
"At this point our work provides the temporal and spatial features of OTOC to detect a host of different quantum phases which can potentially be implemented in the ongoing experiments.",
"Also there are possibilities of studying the temperature dependence of OTOC in the longrange Harper model using a thermal state which one can address in the future."
],
[
"Acknowledgements",
"NR would like to acknowledge University Grants Commision (UGC), India for providing a PhD fellowship and thanks Kamanpreet Singh Manoor for fruitful discussions on OTOC.",
"A.S acknowledges financial support from SERB via the grant (File Number: CRG/2019/003447), and from DST via the DST-INSPIRE Faculty Award [DST/INSPIRE/04/2014/002461]."
]
] | 2107.01712 |
[
[
"Fast Rate Learning in Stochastic First Price Bidding"
],
[
"Abstract First-price auctions have largely replaced traditional bidding approaches based on Vickrey auctions in programmatic advertising.",
"As far as learning is concerned, first-price auctions are more challenging because the optimal bidding strategy does not only depend on the value of the item but also requires some knowledge of the other bids.",
"They have already given rise to several works in sequential learning, many of which consider models for which the value of the buyer or the opponents' maximal bid is chosen in an adversarial manner.",
"Even in the simplest settings, this gives rise to algorithms whose regret grows as $\\sqrt{T}$ with respect to the time horizon $T$.",
"Focusing on the case where the buyer plays against a stationary stochastic environment, we show how to achieve significantly lower regret: when the opponents' maximal bid distribution is known we provide an algorithm whose regret can be as low as $\\log^2(T)$; in the case where the distribution must be learnt sequentially, a generalization of this algorithm can achieve $T^{1/3+ \\epsilon}$ regret, for any $\\epsilon>0$.",
"To obtain these results, we introduce two novel ideas that can be of interest in their own right.",
"First, by transposing results obtained in the posted price setting, we provide conditions under which the first-price biding utility is locally quadratic around its optimum.",
"Second, we leverage the observation that, on small sub-intervals, the concentration of the variations of the empirical distribution function may be controlled more accurately than by using the classical Dvoretzky-Kiefer-Wolfowitz inequality.",
"Numerical simulations confirm that our algorithms converge much faster than alternatives proposed in the literature for various bid distributions, including for bids collected on an actual programmatic advertising platform."
],
[
"Introduction",
"We consider the problem of setting a bid in repeated first-price auctions.",
"First-price auctions are widely used in practice, partly because they constitute the most natural and simple type of auctions.",
"In particular, they have been largely adopted in the field of programmatic advertising, where they have progressively replaced second-price auctions [25], [24].",
"This recent transition took place for various reasons.",
"First, whereas second-price auctions have the advantage of being dominant-strategy incentive-compatible and hence allow for simple bidding strategies [28], they were made obsolete by the widespread use of header bidding, a technology that puts different ad-exchange platforms in competition.",
"With this technology, every participating ad-exchange has to provide the winning bid of the auction organized on its platform; a second-level auction is then organized between all the winners to determine which bidder earns the right of displaying its banner.",
"Second price auctions would hence jeopardize the fairness of the attribution of the placement at sale with header bidding.",
"Second, sellers have benefited from the transition, since many bidders continued to bid as in second-price auctions and despite the automated implementation of so-called bid shading by demand-side platforms, meant to adjust their bids to this new situation [26].",
"The transition to first price auctions raises questions for advertisers who need new bidding strategies.",
"In general, bidders participating in auctions in the context of programmatic advertising do not know the bidding strategies of the other contestants in advance, or anything about the valuations that other bidders attribute to the advertisement slot.",
"Not only do they have to learn other bidders' behavior on the go, but they also need to understand how valuable the placement is for their own use (how many clicks or actions the display of their ad on this placement will lead to), which is usually not the same for all bidders.",
"In this work, we model the problem faced by a single bidder in repeated stochastic first-price auctions, that is, when the contestants' bids are drawn from a stationary distribution.",
"We consider that the learner's bids will not influence the others' bidding strategies.",
"This approximation is sensible in contexts where the major part of the stakeholders do not have an elaborate bidding strategy.",
"More precisely, many stakeholders never modify their bids or do so at a very low frequency.",
"Moreover, the poll of bidders is very large and each bidder only participates in a fraction of the auctions, which argues in favor of the assumption that the influence of one bidder on the rest of the participants can be neglected."
],
[
"Model",
"We consider that similar items are sold in $T$ sequential first price auctions.",
"For $t=1,\\ldots ,T$ , the auction mechanism unfolds in the following way.",
"First, the bidder submits her bid $B_t$ for the item that is of unknown value $V_t$ .",
"The other players submit their bids, the maximum of which is called $M_t$ .",
"If $M_t\\le B_t$ (which includes the case of ties), the bidder observes and receives $V_t$ and pays $B_t$ .",
"If $B_t< M_t$ , the bidder loses the auction and does not observe $V_t$ .",
"We make the following additional assumptions: $\\lbrace V_t\\rbrace _{t\\ge 1}$ are independent and identically distributed random variables in the unit interval $[0,1]$ ; their expectation is denoted by $v:=\\mathbb {E}(V_t)$ .",
"The $\\lbrace M_t\\rbrace _{t\\ge 1}$ are independent and identically distributed random variables in the unit interval $[0,1]$ with a cumulative distribution function (CDF) $F$ , independent from the $\\lbrace V_t\\rbrace _{t\\ge 1}$ .",
"When applicable, we denote by $f=F^{\\prime }$ the associated probability density function.",
"The first-price utility of the bidder is $U_{v,F}(b) := \\mathbb {E}\\big [(V_t- b)\\mathbb {1}\\lbrace M_t\\le b\\rbrace \\big ] = (v-b)F(b)$ .",
"The regret is defined as $R_T^{v,F} &= T \\max _{b \\in [0,1]} U_{v,F}(b)- \\sum _{t=1}^T \\mathbb {E}[U_{v,F}(B_t)]\\;.$ We denote by $b^*_{v,F} = \\max \\big \\lbrace \\operatornamewithlimits{arg\\,max}_{b\\in [0,1]}U_{v,f}(b)\\big \\rbrace $ the (highest) optimal bid.",
"Note that the outer max is required as the utility may have multiple maxima (see Section below): in that case, we define the optimal bid as the one that has the largest winning rate.",
"In the sequel, we exclude the particular case where $F(b^*_{v,F})=0$ , since in this hopeless situation the contestants always bid above the value of the item and the best strategy is not to bid at all ($B_t\\equiv 0$ ): we thus assume that $F(b^*_{v,F})>0.$ In Section , we will first assume that $F$ is known to the learner.",
"This setting bears some similarities with the case of second-price auctions considered by [29], [1]: the truthfulness of second-price auctions makes it sufficient for the bidder to learn the value of $v$ and the valuation of the item is the only parameter to estimate in that case.",
"However, an important feature of the second-price auction mechanism is that the utility of the bidder is quadratic in $v$ under very mild assumptions on the bidding distribution $F$ .",
"In the case of first-price auctions, the utility is no longer guaranteed to be unimodal, neither is the optimal bid $b^*_{v,F}$ a regular function of $v$ .",
"We treat the case, in Section , where the CDF $F$ of the opponents' maximal bid is initially unknown to the learner, assuming that the maximal bid $M_t$ is observed for each auction.",
"Note that in this more realistic setting, the bidder could not infer the optimal bid $b^*_{v,F}$ even if she had perfect knowledge of the item value $v$ .",
"The bidder consequently needs to estimate $F$ and $v$ simultaneously, which makes it a clearly harder task.",
"This second setting bears some similarities with the task of fixing a price in the posted price problem [16], [17], [4], [7], in which a seller needs to estimate the distribution of the valuations of buyers, in order to set the optimal price in terms of her revenue.",
"However, in contrast to the posted-price setting, there is an additional unknown parameter $v$ that also impacts the utility function.",
"In both of these settings, the learner is faced with a structured continuously-armed bandit problem with censored feedback.",
"Indeed, the bidder only observes the reward associated with the chosen bid, but she observes the value only when she wins.",
"This introduces a specific exploitation/exploration dilemma, where exploitation is achieved by bidding close to one of the optimal bids but exploration requires that the bids are not set too low.",
"This structure seems to call for algorithms that bid above the optimal bid with high probability, as in [29], [1] for the second-price case, but we will see in the following that it is not necessarily true."
],
[
"Related Works",
"A major line of research in the field of online learning in repeated auctions is devoted to fixing a reserve price for second-price auctions or a selling price in posted price auctions, see [22] for a general survey.",
"In the posted price setting, arbitrarily bad distributions of bids give rise to very hard optimization problems [23].",
"That is why regularity assumptions are often used, like e.g.",
"the monotonic hazard rate (MHR) condition.",
"Most notably, [16], [8], [10] use this assumption to bound the sample complexity of finding the monopoly price.",
"Regarding online learning in the posted price setting, [17] and [7] introduce algorithms for the stochastic case, respectively in the cases where the distribution of the prices are continuous and discrete.",
"[4] study the adversarial counterpart.",
"[2], [6] study online strategies that aim at setting the optimal reserve price in second-price auctions while learning the distribution of the buyer's bids.",
"[6] assume that bidders are symmetric, but that the bids distribution is not necessarily MHR.",
"They introduce an optimistic algorithm based on two ideas.",
"Firstly they observe that exploitation is achieved by submitting a price smaller than the optimal reserve price, and secondly they use the fact that the utility can be bounded in infinite norm, thanks to the Dvoretzky-Kiefer-Wolfowitz (DKW) inequality [20].",
"The problem of learning in repeated auctions from the point of view of the buyer was originally addressed in the setting of second-price auctions.",
"For the stochastic setting, [29] propose an algorithm that overbids with high probability, and that is shown to have a regret of the order of $\\log ^2{T}$ under mild assumptions on the distribution of the bids.",
"They also provide algorithms for the adversarial case, that have a regret scaling in $\\sqrt{T}$ .",
"[1] extend their work by proposing tighter optimistic strategies that show better worst case performances.",
"They also analyze non-overbidding strategies, proving that such strategies can perform well on a large class of second-price auctions instances.",
"[13] consider the contextual set-up where the value associated to an item is linear with respect to a context vector associated to the item, and revealed before each action.",
"Learning in repeated stochastic first price auctions is a difficult problem that has given rise to a number of very different though equally interesting modelizations.",
"[12] consider auctions in which the values of all the bidders are revealed as a context before each turn, proving that the bids of bidders who use no regret contextual learning strategies in first price auctions converge to Bayes Nash equilibria.",
"[15] also consider the case where the values are assumed to be revealed as an element of context before each auction takes place and the highest bid among others' bids is only shown to the learner when she loses.",
"This setting interestingly introduces a censoring structure that is opposed to the one we consider: in this context, exploitation is achieved by not bidding too high.",
"[15] provide new algorithms for this setting which have a regret of the order of $\\sqrt{T}$ .",
"A setting somewhat closer to ours is studied by [11].",
"This work deals with the setting of a bid in an adversarial fashion, when the other bids are revealed at each time step and the value is revealed only upon winning an auction.",
"However the proposed algorithm is based on a discretization of the bidding space which relies on the prior knowledge of the smallest gap between two distinct bids.",
"With this knowledge, the proposed algorithm achieves an adversarial regret of the order of $\\sqrt{T}$ ."
],
[
"Contributions",
"The highlights of Sections – are the following.",
"In Section we stress the hardness of the first-price bid optimization task, showing that in general it necessarily leads to high minimax regret rates.",
"We however transplant ideas introduced in the case of posted prices to exhibit natural assumptions ensuring that the first-price utility is smooth, paving the way for faster learning.",
"In Section , we consider the case where the learner can assume knowledge of $F$ and propose a new UCB-type algorithm called UCBid1 for learning the optimal bid with low regret.",
"UCBid1 is adaptive to the difficulty of the problem in the sense that its regret is $O(\\sqrt{T})$ in difficult cases, but comes down to $O(\\log ^2 T)$ when the first-price utility is smooth.",
"We also provide lower-bound results suggesting that these rates are nearly optimal.",
"In Section , we consider the more general setting where $F$ is initially unknown to the learner.",
"By leveraging the structure of the first-price bidding problem, we are able to propose an algorithm, termed UCBid1+, which is a direct generalization of UCBid1.",
"Interestingly, this algorithm is not optimistic anymore: it does not submit bids which are with high probability above the (unknown) optimal bid.",
"However, it can still be proved to achieve a regret rate of $O(\\sqrt{T})$ in the most general case and, more importantly, a regret rate upper bounded by $O(T^{1/3+ \\epsilon })$ for every $\\epsilon >0$ when the first-price utility satisfies the regularity assumptions mentioned in Section .",
"The latter result relies on an original proof notably based on the use of a local concentration inequality on the empirical CDF.",
"All the proofs corresponding to these three sections are presented in appendix.",
"Section closes the paper with numerical simulations where we compare the proposed algorithms with continuously-armed bandit strategies and tailored strategies from the literature, both using simulated and real-world data."
],
[
"Properties of Stochastic First-Price auctions",
"There are two important difficulties with first price auctions.",
"The first one lies in the fact that the utility can have multiple maximizers (or multiple modes with arbitrarily close values) and thus lead to arbitrarily hard optimization problems.",
"To illustrate this, we provide in Figure REF an example of value $v$ and discrete distribution, supported on two values $m_0,m_1$ , that leads to a utility having two global maximizers.",
"Note that the utility $U_{v,F}(b)$ is the area of the rectangle with vertices $(b,F(b)), (b,0),(v,F(b)), (v,0)$ .",
"This observation makes it easy to build examples with multiple maxima.",
"Discrete examples like the one in Figure REF are intuitive because the utility is decreasing between two successive points of the support, but there also exist similar cases with continuous distributions (see for example Appendix REF ).",
"This example also shows that there exist combinations of bids distributions and values for which the utility is not regular around its maximum.",
"The second difficulty comes from the fact that the mapping from $v$ to the largest maximizer, $\\psi _F: v \\mapsto b^*_{v,F}$ may also lack regularity.",
"Indeed, keeping the distribution in Figure REF but setting the value to $v^{\\prime } = v + \\Delta $ , with a positive $\\Delta $ (resp.",
"to $v^{\\prime }=-\\Delta $ ) yields that the set of maximizers is $\\lbrace m_1\\rbrace $ (resp.",
"$\\lbrace m_0\\rbrace $ ).",
"Even though $\\psi _F$ can not be proved to be regular in all generality, it always holds that $\\psi _F$ is increasing.",
"This is intuitive: the optimal bid grows with the private valuation.",
"Lemma 1 For any cumulative distribution $F$ , $\\psi _F: v \\mapsto b^*_{v,F}$ is non decreasing.",
"The two aforementioned difficulties contribute to making the problem at hand particularly hard.",
"In the following theorem, we show that any algorithm is bound to have a worst case regret growing at least like $\\sqrt{T}$ .",
"Theorem 1 Let $\\mathcal {C}$ denote the class of cumulative distribution functions on $[0,1]$ .",
"Any strategy, whether it assumes knowledge of $F$ or not, must satisfy $\\liminf _{T \\rightarrow \\infty } \\frac{\\max _{v \\in [0,1], F \\in \\mathcal {C}}R_T^{v,F}}{\\sqrt{T}}&\\ge \\frac{1}{64},$ Theorem REF corresponds to Theorem 6 in [15].",
"For completeness, we prove it in Appendix .",
"The proof relies on specifically hard instances of CDF that are perturbations of the example of Figure REF .",
"It illustrates the complexity of bidding in first-price auctions, when $F$ and $v$ are arbitrary.",
"This complexity stems from specifically hard instances of $F$ and $v$ .",
"We present a natural assumption that avoids these pathological cases.",
"Assumption 1 $F$ is continuously differentiable and is strictly log-concave.",
"This assumption is reminiscent of the monotonic hazard rate (MHR) condition (see e.g.",
"[8]), that appears in the analysis of the posted price problem.",
"While MHR requires ${f}/{(1-F)}$ to be increasing, Assumption REF requires ${f}/{F}$ to be decreasing.",
"In particular, this condition is satisfied by truncated exponentials and Beta distributions with $f$ of the form $C x^{\\alpha -1}$ where $\\alpha >1$ or $C (1-x)^{\\beta -1}$ where $\\beta >1$ , or Beta distributions in which $\\alpha + \\beta < \\alpha \\beta $ (see Lemma REF in Appendix ).",
"Assumption REF plays roughly the same role for first price auctions than MHR for the posted price setting.",
"It guarantees in particular that there is a unique optimal bid.",
"Note that if $F$ satisfies Assumption REF , $F$ is increasing, and admits an inverse which we denote by $F^{-1}$ .",
"Lemma 2 Under Assumption REF , for any $v\\in [0,1]$ the mapping $b\\mapsto U_{v,F}(b)$ has a unique maximizer.",
"As does the MHR assumption for the posted-prices setting, Assumption REF ensures that the utility is strictly concave when expressed as a function of the quantile $q=F(b)$ associated with the bid $b$ .",
"Another important consequence of Assumption REF is that the mapping from $v$ to the optimal bid $b^*_{v,F}$ is guaranteed to be regular.",
"Lemma 3 If Assumption REF is satisfied and $f$ is continuously differentiable, then $\\psi _F: v \\mapsto b^*_{v, F}$ is Lipschitz continuous with a Lipschitz constant 1.",
"Indeed, if $f$ is continuously differentiable and if $f$ does not vanish on $[0,1[$ (which is implied by Assumption REF ), $\\psi _F$ is invertible and it inverse $\\phi _F$ writes $\\phi _F : b \\mapsto b+ {F(b)}/{f(b)}$ .",
"Assumption REF ensures that $\\phi _F$ admits a derivative that is lower-bounded by $\\phi _F^{\\prime }(b) > 1$ .",
"Assumption REF also implies the important property that the probability of winning the auction at the optimal bid $F(b^*_{v,F})$ cannot be arbitrarily small when compared to $F(v)$ .",
"Lemma 4 If Assumption REF is satisfied, then $F(b^*_{v,F})\\ge \\frac{F(v)}{e}\\;.$ We conclude this section by additional properties that are essential for obtaining low regret rates: the utility is second-order regular, when expressed as a function of the quantiles.",
"Let $W_{v,F}$ denote the utility expressed as a function of the quantile, $W_{v,F}: q\\mapsto U_{v,F}(F^{-1}(q))$ , and let $q^*_{v,F}:= F(b^*_{v,F})$ be its maximizer.",
"Under Assumption REF , the deviations of $W_{v,F}$ from its maximum are lower-bounded by a quadratic function.",
"Lemma 5 Under Assumption REF , for any $q\\in [0, 1]$ , $W_{v,F}(q^*_{v,F}) - W_{v,F}(q) \\ge \\frac{1}{4}(q^*_{v,F} - q)^2 W_{v,F}(q^*_{v,F}).$ This property relies, among other arguments, on the observation that $W^{\\prime }_{v,F}(q) =v- \\phi _F(F^{-1}(q))=\\phi _F(F^{-1}(q^*_{v,F}))- \\phi _F(F^{-1}(q))$ and that $\\phi _F^{\\prime }$ is lower-bounded by 1 under Assumption REF (see discussion of Lemma REF above).",
"Similarly, in order to obtain a quadratic lower bound on $W_{v,F}(q)$ , one needs to show that $\\phi _F^{\\prime }$ may be upper bounded.",
"However, Assumption REF is not sufficient for this purpose and we require some additional regularity assumptions on $F$ .",
"Assumption 2 $F$ admits a density $f$ such that $c_f<f(b)<C_f, \\forall b \\in [b^*_{v,F} - \\Delta ,b^*_{v,F} + \\Delta ]$ and $\\phi _F: b \\mapsto b+ {F(b)}/{f(b)}$ admits a derivative that is upper-bounded by a constant $\\lambda \\in \\mathbb {R}^+$ on $[b^*_{v,F},b^*_{v,F} + \\Delta ]$ .",
"Assumption REF holds, in particular, when $F$ is twice differentiable, $f$ is lower-bounded by a positive constant and $f^{\\prime }$ is upper-bounded by a positive constant on a neighborhood of $b^*_{v,F}$ .",
"This assumption implies the following lower bound for the utility expressed as a function of the quantiles.",
"Lemma 6 Under Assumption REF , for any $q \\in [q^*_{v,F}, q^*_{v,F}+ C_f \\Delta ]$ , $W_{v,F}(q^*_{v,F}) - W_{v,F}(q) \\le \\frac{1}{c_f}\\lambda (q^*_{v,F} - q)^2 .$"
],
[
"Known Bid Distribution",
"In this section we address the online learning task in the setting where the bid distribution $F$ is known to the learner from the start.",
"In order to set the bid $B_t$ at time $t$ , the available information consists in $N_{t}:= \\sum _{s=1}^{t-1} \\mathbb {1} \\lbrace M_s \\le B_s\\rbrace $ , the number of observed values before time $t$ , and $\\hat{V}_t:= \\frac{1}{N_{t}}\\sum _{s=1}^{t-1} V_s \\mathbb {1}(M_s\\le B_s)$ the average of those values.",
"Let $\\epsilon _t := \\sqrt{{\\gamma \\log (t-1)}/{2 N_t}}$ denote a confidence bonus depending on a parameter $\\gamma >0$ to be specified below.",
"Algorithm 1 (UCBid1) Initially set $B_1 = 1$ and, for $t \\ge 2$ , bid according to $ B_t = \\max \\Big \\lbrace \\operatornamewithlimits{arg\\,max}_{b \\in [0,1]}(\\hat{V}_t + \\epsilon _t -b)F(b)\\Big \\rbrace .$ This algorithm, strongly inspired by UCB-like methods designed for second-price auctions by [29], [1], is a natural approach to first-price auctions.",
"The idea behind this kind of method is that one should rather overestimate the optimal bid, so as to guarantee a sufficient rate of observation.",
"As an UCB-like algorithm, UCBid1 submits an (high probability) upper bound $\\psi _F(\\hat{V}_t + \\epsilon _t)$ of $b^*_{v,F}$ , thanks to Lemma REF and since $\\psi _F$ is non decreasing.",
"In practice, the algorithm requires a line search at each step as the utility maximization task is usually non-trivial, as discussed in Section .",
"In the most general case, the regret of UCBid1 admits an upper bound of the order of $\\sqrt{T\\log (T)}$ .",
"Theorem 2 When $\\gamma >1$ , the regret of UCBid1 is upper-bounded as $R_T^{v,F} \\le \\frac{ \\sqrt{2 \\gamma }}{F(b^*_{v,F})} \\sqrt{ \\log T} \\sqrt{T} + O(\\log T)\\;.$ Note that $\\sqrt{T}$ is the order of the regret of UCB strategies designed for second-price auctions in the absence of regularity assumptions on $F$ [29].",
"However, under the regularity assumptions introduced in Section , it is possible to achieve faster learning rates.",
"Theorem 3 If $F$ satisfies Assumption REF and REF , then, for any $\\gamma >1$ , $R_T^{v,F}\\le \\frac{2\\gamma \\lambda C_f^2}{F(b^*_{v,F}) c_f}\\log ^2(T) + O(\\log T) .$ The $\\log ^2(T)$ rate of the regret comes from the Lipschitz nature of $\\psi _F$ , that makes it possible to bound the gap $B_t-b^*_{v,F}$ , and from the obervation that the utility is quadratic around its optimum.",
"This explains the similarity with the order of the regret of UCBID in [29], when the distribution of the bids admits a bounded density.",
"Indeed, in second-price-auctions, when the distribution of the bids admits a bounded density, the utility is locally quadratic around its maximum and the equivalent of $\\psi _F$ is the identity, meaning that the optimal bid is just the value $v$ of the item.",
"The presence of the multiplicative constant $1/F(b^*_{v,F})$ is also expected: it is the average time between two successive observations under the optimal policy.",
"This similarity between the structures of second and first price auctions under Assumptions REF and REF also suggest that the constants in the regret may be further improved by using a tighter confidence interval for $v$ based on Kullback-Leibler divergence, proceeding as in [1].",
"Under Assumption REF , the regret of any optimistic strategy can be shown to satisfy the following lower bound.",
"Theorem 4 Consider all environments where $V_t$ follows a Bernoulli distribution with expectation $v$ and $F$ satisfies Assumption REF and is such that $\\phi ^{\\prime } \\le \\lambda $ , and there exists $c_f$ and $C_f$ such that $0<c_f<f(b)<C_f, ~ \\forall b \\in [0,1]$ .",
"If a strategy is such that, for all such environments, $R_T^{v,F}\\le O(T^{a})$ , for all $a>0$ , and there exists $\\gamma >0$ such that $\\mathbb {P}(B_t< b^*)<t^{-\\gamma }$ , then this strategy must satisfy: $& \\liminf _{T \\rightarrow \\infty } \\frac{R_T^{v,F}}{\\log T}\\ge c_f^2 \\lambda ^2\\left(\\frac{v(1-v)(v- b^*_{v,F})}{32} \\right).$ The first assumption, $R_T^{v,F}\\le O(T^{a})$ , is a common consistency constraint that is used when proving the lower bound of [19] in the well-established theory of multi-armed bandits.",
"The second assumption, $\\mathbb {P}(B_t< v)<t^{-\\gamma }$ , restricts the validity of the lower bound to the class of strategies that overbid with high probability.",
"By construction, this assumption is satisfied for UCBid1.",
"Note that there is a gap between the rates $\\log T$ in the lower bound (Theorem REF ) and $\\log ^2T$ in the performance bound of UCBid1 (Theorem REF ), which we believe is mostly due to the mathematical difficulty of the analysis.",
"The $v(1-v)$ factor may be interpreted as an upper bound on the variance of the value distribution with expectation $v$ .",
"Theorem REF displays a dependence on $v$ of the order of $v^2$ when $v$ tends to 0.",
"However this has to be put in perspective with the fact that the value of the optimal utility $U_{v,F}(b^*_{v,F})$ is also quadratic in $v$ , when $v$ tends to zero under the assumptions of Theorem REF (from Lemma REF )."
],
[
"Unknown Bid Distribution",
"We now turn to the more realistic, but harder, setting where both the parameter $v$ and and the function $F$ need to be estimated simultaneously.",
"For this setting, we propose the following algorithm, which is a natural adaptation of UCBid1, simply plugging in the empirical CDF in place of the unknown $F$ .",
"It may come as a surprise that we do not add any optimistic bonus to the estimate $\\hat{F}_t$ : it is not necessary to be optimistic about $F$ since the observation $M_t$ drawn according to $F$ is observed at each time step whatever the bid submitted.",
"Algorithm 2 (UCBid1+) Submit a bid equal to 1 in the first round, then bid: $ B_t = \\max \\Big \\lbrace \\operatornamewithlimits{arg\\,max}_{b \\in [0,1]}(\\hat{V}_t + \\epsilon _t -b)\\hat{F}_t(b)\\Big \\rbrace ,$ where $\\hat{F}_t(b) := \\frac{1}{t-1}\\sum _{s=1}^{t-1} \\mathbb {1}\\lbrace M_s<b\\rbrace $ and $\\epsilon _t := \\sqrt{{\\gamma \\log (t-1)}/{2 N_t}}.$ Although $B_t$ produced by Algorithm REF could, in principle, be arbitrarily small, it is possible to show that there is no extinction of the observation process.",
"Indeed, after a time that only depends on $v$ and $F$ , $F(B_t)$ is guaranteed to be higher than a strictly positive fraction of $F(b^*_{v,F})$ with high probability (see Lemma REF in Appendix ).",
"This result implies that the number of successful auctions $N_t$ asymptotically grows at a linear rate (with high probability), making it possible to bound the expected difference between $\\hat{V}_t + \\epsilon _t$ and $v$ .",
"Combined with the DKW inequality [20], this allows to bound the difference between the utility and $(\\hat{V}_t + \\epsilon _t -b)\\hat{F}_t(b)$ in infinite norm and hence the difference between $B_t$ and $b^*_{v,F}$ .",
"Putting all the pieces together (see the complete proof in Appendix ) yields the following upper bound on the regret of UCBid1+.",
"Theorem 5 UCBid1+ incurs a regret bounded by $R_T^{v,F}\\le 12 \\sqrt{\\frac{\\gamma v}{U_{v,F}(b^*_{v,F})}} \\sqrt{\\log T} \\sqrt{T} + O(\\log T),$ provided that $\\gamma >2$ .",
"Note that computing the bid $B_t$ for UCBid1+ is easy, as $(\\hat{V}_t + \\epsilon _t -b)\\hat{F}_t(b)$ necessarily lies among the observed bids because this function is linearly decreasing between observed bids.",
"More precisely, $(\\hat{V}_t + \\epsilon _t -b)\\hat{F}_t(b)) =\\hat{F}_t(M^{(i)})(\\hat{V}_t + \\epsilon _t-b),$ for $b \\in [M^{(i)}, M^{(i+1)}[$ , where $M^{(i)}$ is the i-th order statistic of the observed bids (obtained by sorting the bids in ascending order).",
"However, as there is no obvious way to update $B_t$ sequentially, this results in a complexity of UCBid1+ that grows quadratically with the time horizon $T$ .",
"The proof of Theorem REF relies on the DKW inequality to bound the difference between $B_t$ and $b^*$ .",
"This happens to be very conservative and a little misleading in practice.",
"Indeed, what really matters is the local behavior of the empirical utility, and hence, of $\\hat{F}_t$ around $b^*$ .",
"As illustrated by Figure REF , locally, $\\hat{F}_t$ is roughly a translation of $F$ plus a negligible perturbation which can be bounded in infinite norm.",
"This intuition is formalized in Lemma REF , a localized version of the DKW inequality.",
"The fact that $\\hat{F}_t$ is locally almost parallel to $F$ imposes a constraint on $B_t$ that may be used to bound its distance from $b^*$ , yielding an improved regret rate under Assumptions REF and REF , as shown by Theorem REF .",
"Lemma 7 For any $a,b\\in [0,1]$ , if $F$ is increasing, $\\sup _{a\\le x \\le b}|\\hat{F}_t(x) - F(x) - (\\hat{F}_t(a)- F(a))| \\\\\\le \\sqrt{\\frac{2(F(b)- F(a))\\log \\left( \\frac{e \\sqrt{t}}{ \\eta \\sqrt{2(F(b)- F(a))}}\\right)}{t}} + \\frac{\\log (\\frac{t}{2(F(b)- F(a) \\eta ^2 })}{6 t},$ with probability $1-\\eta $ .",
"Figure: Local behavior of the empirical CDFTheorem 6 If $F$ satisfies Assumptions REF and REF , UCBid1+ incurs a regret bounded by $R_T^{v,F}\\le O\\big (T^{1/3+\\epsilon }\\big ),$ for any $\\epsilon >0$ , provided that $\\gamma >2$ .",
"UCBid1+ thus retains the adaptivity of UCBid1.",
"In general, its regret is of the order of $\\sqrt{T}$ (omitting logarithmic terms), matching the lower bound of Theorem REF .",
"But it is reduced to $T^{1/3+\\epsilon }$ , for any $\\epsilon >0$ , in the smooth case defined by Assumptions REF and REF .",
"In practice, the improvement over other $\\sqrt{T}$ -regret algorithms is huge, as shown in the next section."
],
[
"Methods pertaining to black box optimization.",
"Sequential black box optimization algorithms, also known as continuously-armed bandits [18], [3], [21], [27], are algorithms designed to find the optimum of an unknown function by receiving noisy evaluations of that function at points that are chosen sequentially by the learner.",
"They rely on prior assumptions on the smoothness of the unknown function.",
"For first-price bidding, we may consider that the reward $(v-B_t) \\mathbb {1}(M_t\\le B_t)$ is a noisy observation of the utility $U_{v,F}(B_t)$ , with a noise bounded by 1.",
"Moreover, when $F$ admits a density $f$ and $f(b)<C_f$ , then $-1<U_{v,F}^{\\prime }(b)= (v-x) f(b) -F(b)< C_f$ , which implies that $U_{v, F}$ is Lipschitz with constant $\\max (1, C_f)$ .",
"As a consequence, all black-box optimization algorithms that consider an objective function with Lispchitz regularity may be used for learning in stochastic first price auctions.",
"HOO [3] has a parameter $\\rho $ related to the level of smoothness of the objective function which we can set to $1/2$ , corresponding to the observation that the first-price utility is Lipschitz under the assumptions discussed above.",
"This immediately leads to a first baseline approach with $O(\\sqrt{T \\log T})$ regret rate.",
"Setting the parameter related to the Lipschitz constant of HOO so that it is larger than $C_f$ is not possible in practice without prior knowledge on $F$ .",
"More generally, knowing the smoothness is considered a challenge most of the time in black-box optimization, so that several methods have been introduced that are adaptive to the smoothness, e.g.",
"stoSOO [27]."
],
[
"UCB on a smartly chosen discretization.",
"[9] prove that when the reward function is unimodal, a discretization based on the smoothness level of this function suffices to achieve a regret of the order of $\\sqrt{T}$ .",
"If $F$ satisfies Assumption REF , $U_{v,F}$ is unimodal, as shown by the proof of Lemma REF , or is maximized on an interval.",
"Hence, using the right discretization while applying UCB, one can achieve a $O(\\sqrt{T})$ regret.",
"In particular if the utility is quadratic, the advised discretization is a grid of $O(T^{1/4})$ values."
],
[
"O-UCBID1.",
"We also implement the following algorithm, that is reminiscent of the method used by [6] to learn reserve prices.",
"Algorithm 3 (O-UCBid1) Submit a bid equal to 1 in the first round, then bid: $B_t= \\max \\lbrace b \\in [0,\\hat{V}_t+ \\epsilon _t], \\hat{U}_t(b) \\ge \\max _{b \\in [0,1]}\\hat{U}_t(b) - 2 \\epsilon _t\\rbrace ,$ where $\\hat{U}_t(b)=(\\hat{V}_t-b)\\hat{F}_t(b)$ .",
"This algorithm overbids with high probability, by construction.",
"Thanks to the DKW inequality, one can control the difference between the true bid distribution $F$ and its empirical version $\\hat{F}_t$ in infinite norm.",
"Because we observe $M_t$ at each round, $\\Vert F-\\hat{F}_t\\Vert _{\\infty }$ is at most $\\epsilon _t$ with high probability.",
"It is easy to show that $\\Vert U_{v,F}-\\hat{U}_t\\Vert _{\\infty }$ is bounded by a multiple of $\\epsilon _t$ showing that $B_t$ is (again with high probability) larger than the unknown optimal bid $b^*_{v,F}$ .",
"O-UCBid1 is very close to the method used by [6] to set a reserve price in second-price auctions.",
"While in first-price auctions, a bidder needs to overbid in order to favor exploration, sellers in second-price auctions are encouraged to offer a lower price than the optimal one, as they can only observe the second highest bid if their reserve price is set lower than the latter.",
"The approach of [6] requires successive stages as sellers in second-price auctions can only observe the second-price and need to estimate the distribution of all bids based on this information.",
"In our setting, we have direct access to the opponents' highest bid and successive stages are not required any longer.",
"We prove that the regret incurred by O-UCBid1 is of the order of $\\log T \\sqrt{T}$ when $\\gamma >1$ , which makes it an interesting baseline algorithm, that has guarantees similar to those of black box optimization algorithm, without the need of knowing the smoothness or the horizon.",
"We refer to Theorem REF in Appendix for further details."
],
[
"Methods for discrete distributions",
"We run UCBid1+ on discrete examples.",
"In this case, we compare it to UCB on a discretization of $[0,1]$ and to WinExp, a generalization of Exp3 for the problem of learning to bid [11]."
],
[
"Experiments On Simulated Data",
"In this section we focus on two particular instances of the first price auction learning problem.",
"The first instance is characterized by a value distribution set to a Bernoulli distribution of average $0.5$ , and a distribution of the highest contestants' bids set to a Beta(1,6).",
"The second instance only differs by the distribution of the highest contestants' bids, which is set to a mixture of two Beta distributions: $0.55 \\times Beta(500, 2500)+ 0.45 \\times Beta(1000, 2000)$ .",
"This distribution is very close to that used in the proof of Theorem REF , but is continuous.",
"The cumulative distribution and the matching utility of each instance are plotted on Figure REF .",
"Both distributions are smooth but the first one satisfies Assumption REF , while it is not clear that the second one does.",
"Figures REF and REF show the regret of various strategies when $F$ is known.",
"The first (respectively second) figure represents the regrets of these strategies under the first (respectively second) instance of the problem described above.",
"The horizon is set to 10000 and the results of 720 Monte Carlo trials are aggregated.",
"The plots represent the average regret over time (shaded areas correspond to the interquartile range).",
"The strategy termed Greedy is a naive strategy that bids $\\max \\operatornamewithlimits{arg\\,max}\\hat{U}_t(b)$ , whenever it has made more than three observations.",
"It shows a linear regret, which comes from the fact that when it only observes value samples equal to zero during the first three observations, it bids 0 indefinitely, and thus incurs the regret $U_{v,F}(b^*_{v,F})-U_{v,F}(0)$ at each time step.",
"Observing only 0 three times in a row is not very likely: the third quartile is very small, but the consequences are so terrible that the average is many orders of magnitude higher.",
"The strategy termed Balanced consists in bidding the median of the highest contestants' bids.",
"It guarantees that the learner is able to win half of the rounds.",
"As expected, this strategy, which does not adapt to the instance at hand, shows poor performances in both cases.",
"However, it is a better solution than bidding 0 or 1.",
"Finally, we also plot the regret of UCBid1.",
"Note that in order to implement UCBid1 we would have to compute $\\operatornamewithlimits{arg\\,max}_{b \\in [0,1]}(\\hat{V}_t + \\epsilon _t -b)F(b)$ at each round; instead we only use an approximation of this quantity by computing the argmax of the function over a grid of 10000 values.",
"UCBid1 outperforms the naive baseline strategies in both cases.",
"Under the more complex second instance of the problem, it shows a larger regret than under the first one.",
"However, even in this more complex case, the rate of growth of the regret stays very low.",
"In Figure REF , we analyze the regrets of different algorithms when $F$ is unknown.",
"In this setting, we compare UCB on a discretization of $[0,1]$ with 10 arms, HOO [3] with various parameters, O-UCBid1 and UCBid1+ with $\\gamma =1$ and stoSOO [27] with the parameters recommended in the latter paper.",
"For efficiency reasons, we also do not allow the tree built by HOO and stoSOO to have a depth larger than $\\log _2{T}$ .",
"The various versions of HOO, UCB, as well as stoSOO show regret plots that could correspond to a $\\sqrt{T}$ behavior.",
"UCBid1+ shows a dramatically improved regret plot compared to the black box optimization strategies.",
"Figure: Regret plotsFigure REF shows a different example where the distribution of bids is discrete with a probability mass of $0.51$ on $0.1$ and equal probability masses on $i/50, \\forall i \\in [1\\ldots 4, 6, \\ldots , 50]$ .",
"We compare UCBid1+ with UCB, having operated a discretization into 10 arms and with Winexp with a discretization into 50 arms.",
"UCBid1+ again yields a regret at least 5 times smaller than the other algorithms.",
"In addition, it is important to stress that UCBid1+ and O-UCBid1 are anytime algorithms, while all the alternatives shown on Figures REF and REF require, at least, the knowledge of the time horizon.",
"Figure: Regret plots with bidding data"
],
[
"Experiments On a Real Bidding Dataset",
"We also experiment on a real-world bidding dataset representing the highest bids from the contestants of one advertiser on a certain campaign.",
"We have at our disposal a set of 56607 bids that were made on a specific placement and that correspond to the bids made on auctions that a specific advertiser participated to, for a specific campaign.",
"We keep only the bids smaller than the 90% quantile and we normalize them to get data between 0 and 1 (see Figure REF in Appendix for a histogram).",
"The regret plots are represented in Figure REF .",
"As earlier, with discrete simulated data, we compare UCBid1+ with UCB, having operated a discretization into 10 arms and with Winexp with a discretization into 100 arms.",
"Unsurprisingly, the regret plots are similar to those with simulated data, since the distributions at hand are similar.",
"UCBid1+ still largely outperforms the baseline algorithms.",
"Supplementary Material"
],
[
"Outline.",
"We prove in Appendix all the results pertaining to Section apart from Theorem REF , which is proved separately in Appendix .",
"In Appendix , we introduce preliminary results necessary to analyze the regrets of the algorithms presented in main body of the paper.",
"Appendix contains all the proofs of the results of Section , while the theorems of Section are proved in Appendix .",
"A figure related to Section is presented in Appendix ."
],
[
"Notation.",
" In the following we write $U$ instead of $U_{v,F}$ (respectively $W$ instead of $W_{v,F}$ ; $b^*$ instead of $b^*_{v,F}$ ; $q^*$ instead of $q^*_{v,F}$ and $R_T$ instead of $R_T^{v,F}$ ) when there is no ambiguity.",
"$b(q)$ denotes $F^{-1}(q)$ .",
"$\\hat{V}(n):=1/n\\sum _{s=1}^n V(s)$ is the mean of the $n$ first observed values.",
"We set $V^{\\prime }_s = V_s$ if $ M_s\\le B_s,$ and $V^{\\prime }_s = \\emptyset $ otherwise.",
"We set $\\mathcal {F}_t = \\sigma ((M_s,V^{\\prime }_s)_{s\\le t})$ be the $\\sigma $ -algebra generated by the the bid maxima and the values observed up to time $t$ .",
"$S_t :=(V_t-b^*)\\mathbb {1}(M_t < b^*) - (V_t-B_t)\\mathbb {1}(M_t < B_t)$ represents the instantaneous regret."
],
[
"General properties",
"lem:psiFLemma REF For any cumulative distribution function $F$ , $\\psi _F$ is non decreasing.",
"Let $0<v_1<v_2<1$ .",
"We have $U_{v_2,F}(b_{v_2,F}^*)-U_{v_2,F}(b_{v_1,F}^*)\\ge 0$ and $U_{v_1,F}(b_{v_1,F}^*)-U_{v_1,F}(b_{v_2,F}^*)\\ge 0$ , by definition of $b_{v_1,F}^*$ and $b_{v_2,F}^*$ .",
"By summing these two inequalities, $U_{v_2,F}(b_{v_2,F}^*) -U_{v_1,F}(b_{v_2,F}^*)-(U_{v_2,F}(b_{v_1,F}^*)- U_{v_1,F}(b_{v_1,F}^*))\\ge 0.$ Hence $(v_2-v_1)(F(b^*_{v_2,F})-F(b^*_{v_1,F}))\\ge 0.$ We then prove the result by contradiction, by assuming that $b_{v_1,F}^*>b_{v_2,F}^*$ .",
"Then $F(b_{v_1,F}^*)=F( b_{v_2,F}^*)$ , since $F$ is non decreasing.",
"In this case, $U_{v_1, F}(b_{v_1,F}^*)= (v_1- b_{v_1}^*)F(b_{v_1,F}^*)< (v_1- b_{v_2}^*)F(b_{v_2,F}^*)= U_{v_1, F}(b_{v_2,F}^*).$ This is impossible, since $b^*_{v_1,F}$ is an optimizer of $U_{v_1, F}$ .",
"In conclusion, $b_{v_1,F}^*\\le b_{v_2,F}^*$"
],
[
"Properties under regularity assumptions",
"lem:uniquemaxLemma REF If Assumption REF is satisfied, then for any $v\\in [0,1]$ , $U_{v,F}$ has a unique maximizer.",
"If $F$ satisfies Assumption REF then $\\frac{f}{F}$ is decreasing and $\\phi _F: b \\mapsto b + \\frac{F(b)}{f(b)}$ is increasing and $f$ does not vanish on $]0,1[$ .",
"The derivative of $U$ is $U^{\\prime }(b) = \\left(v-b- \\frac{F(b)}{f(b)}\\right)f(b)$ .",
"So $U^{\\prime }(b)=0$ if and only if $v= b + \\frac{F(b)}{f(b)}$ .",
"Since $\\phi _F$ is increasing, this can only be satisfied by a single $b \\in [0,1]$ .",
"Also, since $f$ does not vanish, $U$ is unimodal (increasing then decreasing).",
"Lemma 8 If Assumption REF is satisfied, then $W_{v,F}$ is strongly concave.",
"If $F$ satisfies Assumption REF then $\\frac{f}{F}$ is decreasing and $\\phi _F: b \\mapsto b + \\frac{F(b)}{f(b)}$ is increasing and $f$ does not vanish on $]0,1[$ .",
"The derivative of $U$ is $U^{\\prime }(b) = \\left(v-b- \\frac{F(b)}{f(b)}\\right)f(b)$ .",
"The derivative of $W$ is $W^{\\prime }(q) = \\left(v-b- \\frac{F(F^{-1}(q))}{f(F^{-1}(q))}\\right)= v- \\phi _F^{\\prime }(F^{-1}(q))$ , since $\\phi _F$ is increasing.",
"Consequently, $U^{\\prime }$ is decreasing, and $U^{\\prime }$ is strongly concave.",
"lem:lipschitzLemma REF If Assumption REF is satisfied and $f$ is differentiable, then $\\psi _F: v \\mapsto b^*(v, F)$ is Lipschitz continuous with a Lipschitz constant 1.",
"If $b^*$ is the optimum of the utility $U$ , then it satisfies $(v-b^*)f(b^*)-F(b^*)=0$ .",
"It satisfies $\\phi _F(b^*) :=b^* +\\frac{F(b^*)}{f(b^*)}=v.$ Since $\\phi _F^{\\prime }(b^*) >1$ thanks to Assumption REF , $\\phi _F$ is invertible and $(\\phi _F)^{-1}= \\psi _F$ is Lipschitzian with constant 1 .",
"lem:boundFb*Lemma REF If Assumption REF is satisfied, then $F(b^*)\\ge e^{-1}F(v)$ We know that $b^*<v$ and $\\log \\left(\\frac{F(v)}{F(b^*)}\\right) = \\int _{b^*}^v \\frac{f(u)}{F(u)}du.$ Hence $\\frac{F(v)}{F(b)} = \\exp \\left(\\int _{b^*}^v \\frac{f(u)}{F(u)} du\\right).$ Since $\\frac{f(u)}{F(u)}$ is decreasing, thanks to Assumption REF , $\\frac{F(v)}{F(b)} \\le \\exp \\left((v-b^*)\\frac{f(b^*)}{F(b^*)} \\right).$ We have $v-b^* = \\frac{F(b^*)}{f(b^*)}$ , by definition of $b^*$ .",
"Hence $ \\exp \\left(v-b^*)\\frac{f(b^*)}{F(b^*)}\\right)=\\exp (1)$ and $F(b^*)\\ge \\exp (-1)F(v).$ lem:quadraticLemma REF If Assumption REF is satisfied, for any $0 \\le q^{\\prime } \\le 1$ , $W(q^*) - W(q^{\\prime }) \\le \\frac{1}{4}(q^* - q^{\\prime })^2 W(q^*)$ Note that this proof is an adaptation of the proof of Lemma 3.2 in [16].",
"In this proof, we denote by $b(q)$ $ F^{-1}(q)$ .",
"First of all, let us observe that $U^{\\prime }(b) = (v-\\phi _F(b)) f(b)$ .",
"We have $W^{\\prime }(q)= v- \\phi _F(F^{-1}(q)).$ Assumption REF implies that $\\phi _F^{\\prime }(b) >1, ~ \\forall b \\in [0,1]$ .",
"To prove Lemma REF , we will apply case-based reasoning.",
"There are three cases depending on the relation between $q^{\\prime }$ and $q^*$ : $q^{\\prime } > q^*$ , $q^{\\prime } = q^*$ , and $q^{\\prime } < q^*$ .",
"The second case, i.e., $q^{\\prime } = q^*$ , is trivial.",
"First, consider the case when $q^{\\prime } > q^*$ .",
"It holds $W(q^*) - W(q^{\\prime }) = \\int ^{q^{\\prime }}_{q^*} - W^{\\prime }(q) dq = \\int ^{q^{\\prime }}_{q^*} \\Big (\\phi _F(b(q)) -v\\Big ) dq.$ We therefore need to bound $\\phi _F(b(q), \\forall q \\in [q^*, q^{\\prime }].$ By definition of $q^*$ , for any $q$ s.t.",
"$q^* \\le q \\le q^{\\prime }$ , we have $q (v- b(q)) \\le q^* (v-b(q^*)).$ By rewriting this equation, $ b(q)\\ge \\frac{q v - q^* v + q^* b(q^*)}{q}= v\\left(\\frac{q-q^*}{q}\\right) + \\frac{q^*}{q}b(q^*)$ Secondly, by the intermediate value theorem, there exists $b \\in [b(q^*), b(q)]$ , such that $\\phi _F(b(q)) - \\phi _F(b(q^*)) = \\phi _F^{\\prime }(b)\\Big ( b(q) - b(q^*)\\Big ) \\ge b(q) - b(q^*),$ for any $q^* \\le q \\le q^{\\prime }$ , where the second inequality follows from Assumption REF that $\\tfrac{d \\phi _F(b)}{db} \\ge 1$ and $F$ being increasing thanks to Assumption REF .",
"This in turn yields $\\phi _F(b(q)) \\ge v + b(q) - b(q^*),$ since by definition, $W^{\\prime }(q^*) = \\phi _F(b(q^*))=v$ .",
"Combining with Inequality REF , we get that $\\phi _F(b(q)) -v \\ge v(\\frac{q-q^*}{q}) + \\frac{q^* }{q}b(q^*) - b(q^*)\\ge (v-b(q^*))(\\frac{q-q^*}{q})= \\frac{W(q^*)}{q^*} (\\frac{q-q^*}{q})$ Therefore, we get that $W(q^*) - W(q^{\\prime }) &= \\int ^{q^{\\prime }}_{q^*} - W^{\\prime }(q) dq = \\int ^{q^{\\prime }}_{q^*} \\Big (\\phi _F(b(q)) -v\\Big ) dq\\ge \\frac{W(q^*)}{q^*} \\int ^{q^{\\prime }}_{q^*} \\frac{q - q^*}{q} dq \\\\&\\ge \\frac{W(q^*)}{q^*}\\int _{\\frac{q^{\\prime }+q^*}{2}}^{q^{\\prime }} \\frac{q - q^*}{q} dq,$ since $\\frac{q - q^*}{q} \\ge 0$ for any $q^{\\prime } \\le q \\le q^*$ .",
"Moreover, for any $q \\ge \\tfrac{q^{\\prime } + q^*}{2}$ , we have $\\frac{q - q^*}{q} = 1 - \\frac{q^*}{q}\\ge 1 - \\frac{2 q^*}{q^{\\prime } + q^*} \\ge \\frac{q^{\\prime } - q^*}{q^{\\prime } + q^*}$ .",
"Hence, we can derive the following inequality $W(q^*) - W(q^{\\prime }) \\ge \\int ^{q^{\\prime }}_{\\frac{q^{\\prime } + q^*}{2}} \\frac{q^{\\prime } - q^*}{q^{\\prime } + q^*} \\frac{W(q^*)}{q^*} dq = \\frac{(q^{\\prime } - q^*)^2}{2(q^{\\prime } + q^*)} \\frac{W(q^*)}{q^*} = \\frac{(q^{\\prime } - q^*)^2}{2 q^*(q^{\\prime } + q^*)} W(q^*) \\hspace{5.0pt}.$ The lemma then follows from the fact that $0 \\le q^{\\prime }, q^* \\le 1$ .",
"The second case, $q^{\\prime } > q^*$ has to be treated a little differently than the first, partly because we now need to upper bound $b(q)$ instead of lower-bounding it.",
"We achieve this by using the concavity of $W$ (proved in Lemma REF ).",
"By concavity of the revenue curve, for any $q^{\\prime } \\le q \\le q^*$ , we have $W(q) \\ge \\frac{q - q^{\\prime }}{q^* - q^{\\prime }} W(q^*) + \\frac{q^* - q}{q^* - q^{\\prime }} W(q^{\\prime }) \\hspace{5.0pt},$ because $W$ lies above the segment that connects $(q^{\\prime }, W(q^{\\prime }))$ and $(q^*, W(q^*))$ , between $q^{\\prime }$ and $q^*$ .",
"Hence $(v-b(q))q \\ge \\frac{q - q^{\\prime }}{q^* - q^{\\prime }} (v-b(q^*))q^* + \\frac{q^* - q}{q^* - q^{\\prime }} (v-b(q^{\\prime }))q^{\\prime } \\ge qv - b(q^*) q^* \\frac{q - q^{\\prime }}{q^* - q^{\\prime }} -b(q^{\\prime }) q^{\\prime } \\frac{q^* - q}{q^* - q^{\\prime }} ,$ And $-qb(q) \\ge \\frac{q^* q^{\\prime }}{(q^* - q^{\\prime })}\\Big (b(q^*)-b(q^{\\prime }) \\Big ) + q\\frac{q^{\\prime } b(q^{\\prime }) - q^*b(q^*)}{q^* - q^{\\prime }} ,$ which yields $qb(q) \\le \\frac{q^* q^{\\prime }}{(q^*- q^{\\prime } )}\\Big (b(q^{\\prime })-b(q^*) \\Big ) + q\\frac{ q^*b(q^*) -q^{\\prime } b(q^{\\prime }) }{q^* - q^{\\prime }} ,$ Dividing both sides by $q$ , we have $b(q) \\le \\frac{q^* q^{\\prime }}{q(q^*- q^{\\prime } )}\\Big (b(q^{\\prime })-b(q^*) \\Big ) + \\frac{ q^*b(q^*) -q^{\\prime } b(q^{\\prime }) }{q^* - q^{\\prime }} ,$ Further, by the intermediate value theorem, there exists $b \\in [b(q^*), b(q)]$ , such that $\\phi _F(b(q)) - \\phi _F(b(q^*)) = \\phi _F^{\\prime }(b)\\Big ( b(q) - b(q^*)\\Big ) ,$ for any $q^* \\le q \\le q^{\\prime }$ .",
"Further, by Assumption REF that $\\tfrac{d \\phi _F(b)}{db} \\ge 1$ , and because $b$ is increasing thanks to Assumption REF , for any $q^{\\prime } \\le q \\le q^*$ , $\\phi _F(b(q)) - \\phi _F(b(q^*))\\le b(q) - b(q^*)$ and $\\phi _F(b(q)) \\le v + b(q) - b(q^*) ~=~ v+ b(q) - b(q^*),$ Combining with Inequality REF , we get that $\\phi _F(b(q)) & \\le & v + \\frac{q^* q^{\\prime }}{q(q^*- q^{\\prime } )}\\Big (b(q^{\\prime })-b(q^*) \\Big ) + \\frac{ q^*b(q^*) -q^{\\prime } b(q^{\\prime }) }{q^* - q^{\\prime }} -b(q*) \\\\& = & v+ \\frac{q^{\\prime } (q^* - q)}{q (q^* - q^{\\prime })} \\big ( b(q^{\\prime }) - b(q^*) \\big ) ~\\le ~ v+ \\frac{q^{\\prime } (q^* - q)}{q^* (q^* - q^{\\prime })} \\big ( b(q^{\\prime }) - b(q^*) \\big ) \\hspace{5.0pt},$ where the last inequality is due to $q \\le q^*$ and $b(q^{\\prime }) - b(q^*)<0$ .",
"Hence, we have $W(q^*) - W(q^{\\prime }) & = \\int ^{q^*}_{q^{\\prime }} W^{\\prime }(q) dq \\\\& = ~ \\int ^{q^*}_{q^{\\prime }}v - \\phi _F(b(q)) dq \\\\& \\ge ~ \\int ^{q^*}_{q^{\\prime }} \\frac{q^{\\prime } (q^* - q)}{q^* (q^* - q^{\\prime })} \\big ( b(q^*) - b(q^{\\prime }) \\big ) dq \\\\& = \\frac{q^{\\prime }}{2 q^*} (q^* - q^{\\prime }) \\big ( b(q^*) -b(q^{\\prime })\\big ).",
"$ On the other hand, we have $W(q^*) - W(q^{\\prime }) = (q^*-q^{\\prime })v +q^{\\prime }b(q^{\\prime })-q^* b(q^*).$ Taking the linear combination $\\frac{2 q^*}{3 q^* - q^{\\prime }} \\cdot \\ref {eq:manyprepeak1} + \\frac{q^* - q^{\\prime }}{3 q^* - q^{\\prime }} \\cdot \\ref {eq:manyprepeak2}$ , we have $W(q^*) - W(q^{\\prime }) &\\ge v \\frac{(q^* -q^{\\prime })^2}{3q^* - q^{\\prime })} - \\frac{(q^* - q^{\\prime })^2}{3 q^* - q^{\\prime }} b(q^*)\\\\& =\\frac{1}{q^* (3 q^* - q^{\\prime })} (q^* - q^{\\prime })^2 W(q^*)\\\\&\\ge \\frac{1}{3} (q^* -q^{\\prime })^2 W(q^*) \\hspace{5.0pt},$ where the last inequality holds because $0 \\le q^*, q^{\\prime } \\le 1$ .",
"lem:subquadraticLemma REF If Assumption REF is satisfied, for any $F^{-1}(b^*)\\le q^{\\prime } \\le F^{-1}(b^*+ \\Delta ) \\le b^*+ C_f \\Delta )$ , $W(q^*) - W(q^{\\prime }) \\le \\frac{1}{c_f}\\lambda (q^* - q^{\\prime })^2 ,$ $W(q^*) - W(q^{\\prime }) = \\int ^{q^{\\prime }}_{q^*} - W^{\\prime }(q) dq = \\int ^{q^{\\prime }}_{q^*} \\Big (\\phi _F(b(q)) -v\\Big ) dq.$ by the intermediate value theorem, there exists $b\\in [b(q^*), b(q)]$ , such that $\\phi _F(b(q)) - \\phi _F(b(q^*)) = \\phi _F^{\\prime }(b)\\Big ( b(q) - b(q^*)\\Big ) \\ge \\lambda ( b(q) - b(q^*)),$ so that $\\phi _F(b(q)) -v \\le \\lambda ( b(q)-b(q^*))$ when $q^* \\le q \\le q^{\\prime }$ and $\\phi _F(b(q)) -v \\ge \\lambda (b(q)-b(q^*))$ when $q^{\\prime }\\le q \\le q^*$ .",
"Since $f$ is bounded from below by $c_f$ , and since by the intermediate value theorem $\\exists u \\in [q, q^*], ~ b(q)-b(q^*) = b^{\\prime }(u)(q-q^*)\\ge \\frac{1}{f(u)}(q-q^*)$ , this yields $W(q^*) - W(q^{\\prime }) \\le \\lambda \\frac{1}{c_f}(q^{\\prime }-q^*)^2$ in both cases.",
"Lemma 9 Beta distributions such that $\\alpha + \\beta < \\alpha \\beta $ satisfy Assumption REF .",
"The density of a Beta distribution satisfies $f(x) = \\frac{x^{\\alpha -1}(1-x)^{\\beta -1}}{B(\\alpha , \\beta )}$ And $f^{\\prime }(x) = \\frac{(\\alpha -1) x^{\\alpha -2}(1-x)^{\\beta -1} - (\\beta -1) x^{\\alpha -1}(1-x)^{\\beta -2}}{B(\\alpha , \\beta )},$ where $B(\\alpha , \\beta ) = \\frac{\\Gamma (\\alpha +\\beta )}{\\Gamma (\\alpha )\\Gamma (\\beta )}$ when $\\Gamma $ denotes the Gamma function.",
"$F$ satisfies assumption REF if and only if $\\left(\\frac{f}{F}\\right)^{\\prime }(x) =\\frac{F(x)f^{\\prime }(x)-f^2(x)}{F^2(x)}<0$ , $\\forall x \\in ]0,1[$ , which is equivalent to: $f^{\\prime }(x) F(x) - f^2(x)<0 ,~\\forall x \\in ]0,1[ &\\iff \\frac{f^{\\prime }(x)}{f(x)} F(x) < f(x) ,~\\forall x \\in ]0,1[)\\\\& \\iff F(x) B(\\alpha , \\beta ) \\left[(\\alpha -1)(1-x) - (\\beta -1)x \\right]< x^{\\alpha }(1-x)^{\\beta }, \\\\& ~~~~\\forall x \\in ]0,1[.$ Therefore we study the function $G:x \\mapsto F(x) B(\\alpha , \\beta ) \\left[(\\alpha -1)(1-x) - (\\beta -1)x \\right]- x^{\\alpha }(1-x)^{\\beta }.$ First of all, we observe that $G(0) = 0$ .",
"Next, we note that $G^{\\prime }(x) = &- F(x) ( \\alpha + \\beta -2)B(\\alpha , \\beta ) + ((\\alpha -1)- (\\alpha + \\beta -2)x)x^{\\alpha -1}(1-x)^{\\beta -1}\\\\&- \\left(\\left( \\alpha (1-x) - \\beta x \\right) x^{\\alpha -1}(1-x)^{\\beta -1}\\right)$ and $G^{\\prime }(0) = 0$ .",
"Now, we compute the second derivative of $G$ : $G^{\\prime \\prime }(x) =& -( \\alpha + \\beta - 2 ) x^{\\alpha -1}(1-x)^{\\beta -1} + \\left( (\\alpha -1) - (\\alpha + \\beta -2) x)\\right)^2x^{\\alpha -2}(1-x)^{\\beta -2} \\\\&- (\\alpha + \\beta -2) x^{\\alpha -1}(1-x)^{\\beta -1} - \\left( \\alpha - (\\alpha + \\beta )x\\right) ((\\alpha -1)-\\\\& (\\alpha + \\beta -2)x)x^{\\alpha -2}(1-x)^{\\beta -2} + ( \\alpha + \\beta )x^{\\alpha -1}(1-x)^{\\beta -1}$ The sign of $G^{\\prime \\prime }(x)$ is the same as that of $P(x)=-\\left( \\left( \\alpha + \\beta \\right) -4\\right) (x(1-x)) + (-1+2x) \\left( (\\alpha -1) - (\\alpha + \\beta -2)x\\right)$ .",
"By simplifying, we get $P(x) = - (\\alpha + \\beta )x^2+ 2 \\alpha x -(\\alpha -1)$ .",
"This polynomial is always negative because its maximum is $P(\\frac{\\alpha }{\\alpha + \\beta }) = - \\frac{\\alpha ^2}{\\alpha + \\beta } + 2 \\frac{\\alpha ^2}{\\alpha + \\beta } - \\alpha +1 = \\alpha ^2(\\frac{2}{\\alpha + \\beta }-1) - \\alpha +1 = \\frac{\\alpha ^2}{\\alpha +\\beta } - \\alpha + 1 =\\frac{\\alpha + \\beta - \\alpha \\beta }{\\alpha + \\beta }$ .",
"Since $G^{\\prime \\prime }(x)<0, \\forall x \\in [0,1]$ and $G^{\\prime }(0)=0$ , then $G^{\\prime }(x)<0, ~ \\forall x \\in [0,1]$ .",
"Similarly, $G^{\\prime }(x)<0, \\forall x \\in [0,1]$ and $G(0)=0$ , implies $G^{\\prime }(x)<0, ~ \\forall x \\in [0,1]$ , which in turn implies that $F$ satisfies Assumption REF ."
],
[
"Continuous distribution\nleading to a utility with two global maximizers ",
"Consider a distribution which cumulative distribution function $F$ is piece-wise linear on $[0,v]$ at least.",
"We consider that it changes slope at $a_1 v<v$ , and that it is constant on $[a_2 v,v]$ , as in Figure REF .",
"We denote by $b_1= F(a_1 v)$ and $b_2 = F(a_2 v)$ .",
"For simplicity we assume that $F$ is constant on $[a_2 v, a_3 v]$ it is linear and does not change slope on $[a_3 v,1]$ with $a_3>1$ .",
"We make the following assumptions ${\\left\\lbrace \\begin{array}{ll}a_2 v>v/2, \\\\a_2 v \\le \\frac{v+ a_1 v}{2} - \\frac{a_2 v-a_1 v}{b_2-b_1} \\frac{b_1}{2}.\\end{array}\\right.",
"}$ Figure: Example of FFThen On $[0, a_1 v]$ $U_v(x)=\\frac{b_1}{a_1 v}x$ , and the optimum on this interval is $v/2$ .",
"The optimal value on this interval is $U_v(v/2)= \\frac{b_1}{a_1 v}\\frac{v^2}{4}$ on this interval.",
"On $[a_1 v, a_2 v]$ , $U_v(x)= \\left( \\frac{b_2-b_1}{a_2 v-a_1 v}(x-a_1 v) + b_1 \\right) (v-x)$ , and on this interval, $U_v^{\\prime }(x) = \\frac{b_2-b_1}{a_2 v-a_1 v}(v-2x+a_1 v) -b_1$ and $U^{\\prime }_v(x) = 0 \\iff x= \\frac{v+ a_1 v}{2} - \\frac{a_2 v-a_1 v}{b_2-b_1} \\frac{b_1}{2}.$ The optimizer on this interval is hence $a_2 v$ , if $\\frac{v+ a_1 v}{2} - \\frac{a_2 v-a_1 v}{b_2-b_1} \\frac{b_1}{2}>a_2 v$ .",
"Under this condition, the optimal value is $U_v(a_2 v)= b_2(v-a_1 v)$ on this interval.",
"This can also be extended to the whole interval $[a_1 v, v]$ , since U is decreasing after $a_2 v$ .",
"Setting $\\frac{b_1}{a_1 }\\frac{v}{4} = b_2 $ leads to the utility having two global maximizers, $v/2$ and $a_2 v$ .",
"To summarize, the utility's argmax is $\\lbrace v/2,a_2 v\\rbrace $ if the set of Equations REF holds.",
"We can for example choose : $v = 1/2; ~ a_2 =\\frac{15}{16};~ a_1 = \\frac{29}{32};~b_2 = \\frac{128}{29} b_1; b_1= 0.5$ This choice of parameters satisfies Condition REF and Condition REF .",
"Figure REF shows the corresponding utility on $[0,v]$ .",
"Figure: Associated Utility with two maximizers"
],
[
"Lower Bound",
"th:lowerboundgenTheorem REF Let $\\mathcal {C}$ denote the class of cumulative distribution functions on $[0,1]$ .",
"Any strategy, whether it assumes knowledge of $F$ or not, must satisfy $\\liminf _{T \\rightarrow \\infty } \\frac{\\max _{v \\in [0,1], F \\in \\mathcal {C}}R_T^{v,F}}{\\sqrt{T}}&\\ge \\frac{1}{64},$ We exhibit a choice of $F$ , and two alternative Bernoulli value distributions $Ber(v)$ and $Ber(v^{\\prime })$ that are difficult to distinguish but whose difference is large enough so that mistaking one for the other necessarily leads to a regret of the order of $\\sqrt{T}$ when the cumulative distribution function is $F$ .",
"Let $v<1$ and consider a discrete distribution with support $\\big \\lbrace \\frac{v}{3}, \\frac{2v}{3}, 1\\big \\rbrace $ such that $F(\\frac{v}{3}) = A$ and $F(\\frac{2v}{3})= 2A + 3\\frac{\\Delta _T}{v}$ , where $\\Delta _T$ and $A$ are positive constants, that we will fix later on.",
"A maximizer of the utility can only be a point of the support, since $U_{v,F}$ decreases in the intervals where $F$ is constant.",
"It can not be 1, because $v<1$ .",
"We have $U_{v,F}(\\frac{v}{3}) = \\frac{2vA}{3}$ and $U_{v,F}(\\frac{2v}{3}) = \\frac{2vA}{3} + \\Delta _T$ , while $U_{v,F}(1)\\le 0$ .",
"Consequently, when the value is $v$ , the optimum is achieved by bidding $\\frac{2v}{3}$ and bidding less than $\\frac{2v}{3}$ yields a regret of at least $\\Delta _T$ .",
"Now let us consider the alternative situation in which the value is $v^{\\prime } = v - \\delta _T$ , with $\\delta _T>0.$ We get $U_{v^{\\prime },F}(\\frac{v}{3}) = \\frac{2Av}{3} - \\delta _T A$ and $U_{v^{\\prime },F}(\\frac{2v}{3}) = \\frac{2Av}{3} + \\Delta _T - \\delta _T(2A + \\frac{3 \\Delta _T}{v} )$ .",
"When $\\Delta _T < \\delta _T(2A + \\frac{3 \\Delta _T}{v})$ , the optimal bid is $\\frac{v}{3}$ and the regret incurred by bidding more than $\\frac{2v}{3}$ is at least $\\delta _T(A + \\frac{3 \\Delta _T}{v} ) - \\Delta _T$ .",
"By setting $\\Delta _T = \\frac{A \\delta _T}{2 - 3 \\delta _T /v}$ , we ensure that the regret incurred by bidding on the wrong side of $\\frac{2v}{3}$ is larger than $\\Delta _T$ , whether the value is $v$ or $v^{\\prime }$ .",
"Further, by setting $\\delta _T = \\sqrt{{v(1-v)}/{T}}$ , we force the error $\\Delta _T$ to be of the order of ${1}/{\\sqrt{T}}$ .",
"We also set $A =\\frac{1}{4}$ , and $v=1/2$ .",
"We can prove that $\\forall T>16$ , $2A +3 \\frac{\\Delta _T}{v} < 1$ ; Indeed, if $T>16>(11/3)^2$ , $\\frac{4}{3}<2\\sqrt{T}-6$ hence $\\frac{4}{3\\sqrt{T}}<2 - \\frac{6}{\\sqrt{T}}$ which implies $\\frac{2}{3}\\frac{\\frac{1}{\\sqrt{T}}}{2 - \\frac{6}{\\sqrt{T}}}= 6 \\Delta _T<\\frac{1}{2}= 1 -2A$ .",
"We denote by $\\mathbb {P}_{v,F}(\\cdot )$ the probability of an event under the first configuration (respectively $\\mathbb {E}_{v,F}(\\cdot )$ the expectation of a random variable under the first configuration), and by $\\mathbb {P}_{v^{\\prime },F}(\\cdot )$ the probability of an event under the second configuration (respectively $\\mathbb {E}_{v- \\delta -T, F}(\\cdot )$ the expectation of a random variable under the first configuration).",
"We denote by $I_t$ the information collected up to time $t+1$ : $(M_t, V^{\\prime }_t, \\ldots M_1, V^{\\prime }_1)$ .",
"$\\mathbb {P}_{v,F}^{I_t}$ (respectively $\\mathbb {P}_{v^{\\prime }}^{I_t}$ ) denotes the law of $I_t$ in the first (respectively second) configuration.",
"We consider the Kullback Leibler divergence between $\\mathbb {P}_{v,F}^{I_t}$ and $\\mathbb {P}_{v^{\\prime },F}^{I_t}$ .",
"We prove that it is equal to $ KL(\\mathbb {P}_v^{I_t},\\mathbb {P}_{v^{\\prime },F}^{I_t})= kl(v,v^{\\prime }) \\mathbb {E}[N_t], $ where $kl(\\cdot , \\cdot )$ denotes the Kullback Leibler divergence between two Bernoulli distributions.",
"Indeed, thanks to the chain rule for conditional KL, $KL(\\mathbb {P}_{v,F}^{I_t},\\mathbb {P}_{v^{\\prime },F}^{I_t})= KL(\\mathbb {P}_{v,F}^{I_t},\\mathbb {P}_{v^{\\prime },F}^{I_{t}})\\\\ + KL(\\mathbb {P}_{v,F}^{(M_t,V^{\\prime }_t)|I_{t}},\\mathbb {P}_{v^{\\prime },F}^{(M_t,V^{\\prime }_t)|I_{t}}),$ and $KL(\\mathbb {P}_{v,F}^{(M_t,V^{\\prime }_t)|I_{t}},\\mathbb {P}_{v^{\\prime },F}^{(M_t,V^{\\prime }_t)|I_{t}})&= \\mathbb {E}[\\mathbb {E}[KL(\\nu _{I_t}\\otimes \\mathcal {D}_F, \\nu ^{\\prime }_{I_t}\\otimes \\mathcal {D}_F)|I_{t}]]\\\\&= \\mathbb {E}[kl(v, v^{\\prime })\\mathbb {1}(B_t>M_t)].\\\\$ where $\\nu _{I_t}$ (respectively $\\nu ^{\\prime }_{I_t}$ ) denotes the law of $V^{\\prime }_t$ knowing $I_t$ in the first configuration (respectively the second), and $\\mathcal {D}_F$ the law of $M_t$ .",
"By induction, we obtain $ KL(\\mathbb {P}_{v,F}^{I_t},\\mathbb {P}_{v^{\\prime },F}^{I_t})= kl(v,v^{\\prime }) \\mathbb {E}_{v,F}[N_t].$ We stress that in either of the former configurations (under $(v,F)$ or $(v^{\\prime },F)$ ), playing on the wrong side of $\\frac{2}{3} v$ yields a regret larger than $\\Delta _T$ .",
"Using this, we get that $\\forall T> 16$ , $\\max (R_T^{v,F}, R^{v^{\\prime },F}_T) & \\ge \\frac{1}{2}(R_T^{v,F}+ R^{v-\\delta ,F}_T)\\\\& \\ge \\frac{1}{2}\\sum _{t=1}^T\\left( \\Delta _T \\mathbb {P}_{v,F}\\left(B_t<\\frac{2}{3} v\\right) + \\Delta _T \\mathbb {P}_{v^{\\prime }, F}\\left(B_t>\\frac{2}{3} v\\right)\\right)\\\\& \\ge \\frac{1}{2}\\sum _{t=1}^T\\left( \\Delta _T \\mathbb {P}_{v,F} \\left(B_t<\\frac{2}{3} v\\right) + \\Delta _T \\left(1- \\mathbb {P}_{v^{\\prime },F}(B_t>\\frac{2}{3} v)\\right)\\right)\\\\& \\ge \\frac{1}{2}\\sum _{t=1}^T \\Delta _T \\left(1- TV(\\mathbb {P}_{v,F}^{I_t}, \\mathbb {P}_{v^{\\prime },F}^{I_t})\\right)\\\\& \\ge \\frac{1}{2}\\sum _{t=1}^T \\Delta _T \\left(1- \\sqrt{\\frac{1}{2}KL(\\mathbb {P}_{v,F}^{I_t}, \\mathbb {P}_{v^{\\prime },F}^{I_t})}\\right)\\\\&\\ge \\frac{1}{2}\\sum _{t=1}^T \\Delta _T \\left(1- \\sqrt{\\frac{1}{2} \\mathbb {E}_{v,F}[N_t]kl(v, v^{\\prime })}\\right)\\\\&\\ge \\frac{1}{2}\\sum _{t=2}^T \\Delta _T \\left(1- \\sqrt{\\frac{1}{2} T kl(v, v^{\\prime })}\\right)$ where we used Pinsker's inequality in the fifth inequality and where $TV(\\cdot , \\cdot )$ denotes the total variation.",
"Yet, since $kl(v,v^{\\prime })= \\frac{(v^{\\prime }-v)^2}{2} \\int _0^1 g^{\\prime \\prime }(v^{\\prime } + s(v^{\\prime } + s(v-v^{\\prime }))2(1-s)ds$ , where $g(x) = kl(x,v^{\\prime })$ thanks to Taylor's inequality, $kl(v,v^{\\prime })&\\le \\frac{(v^{\\prime }-v)^2}{2} \\int _0^1 2 \\max _{u\\in [v,v^{\\prime }]} g^{\\prime \\prime }(u)ds\\\\&\\le (v^{\\prime }-v)^2 \\frac{1}{\\min _{u\\in [v,v^{\\prime }]}u(1-u)}\\\\&\\le \\frac{(v^{\\prime }-v)^2}{v^{\\prime }(1-v^{\\prime })},$ since $v= \\frac{1}{2}$ .",
"Therefore, $\\max (R_T^{v,F}, R^{v^{\\prime },F}_T) &\\ge \\frac{1}{2}\\sum _{t=1}^T \\Delta _T \\left(1- \\sqrt{\\frac{1}{2} T kl(v, v^{\\prime })}\\right)\\\\&\\ge \\frac{1}{2}\\sum _{t=1}^T \\Delta _T \\left(1- \\sqrt{\\frac{1}{8} \\frac{1}{(1/2 -\\frac{1}{2\\sqrt{T}})(1/2 +\\frac{1}{2\\sqrt{T}})}}\\right)\\\\&\\ge \\frac{1}{2}\\times \\frac{A \\delta _T}{2 - 3/2 \\delta _T } T\\left(1- \\sqrt{\\frac{1}{8} \\frac{1}{(1/2 -\\frac{1}{2\\sqrt{T}})(1/2 +\\frac{1}{2\\sqrt{T}})}}\\right)\\\\&\\ge \\frac{1}{16 - 12 /\\sqrt{T}} \\sqrt{T} \\left(1- \\sqrt{\\frac{1}{8} \\frac{1}{(1/2 -\\frac{1}{2\\sqrt{T}})(1/2 +\\frac{1}{2\\sqrt{T}})}}\\right)$ Finally $\\liminf _{T \\rightarrow \\infty } \\frac{\\max (R_T^{v,F}, R^{v^{\\prime },F}_T)}{\\sqrt{T}}&\\ge \\frac{1}{16} \\left(1- \\sqrt{\\frac{1}{2}}\\right) \\ge \\frac{1}{64}$"
],
[
" On the value $V_t$",
"Lemma 10 The following concentration inequality on the values holds $&\\sum _{t=2}^T \\mathbb {P}\\left((\\hat{V}_t-v)^2 \\ge \\frac{\\gamma \\log (t-1)}{2 N_t}\\right) \\le \\sum _{t=1}^T 2 e \\sqrt{\\gamma }( \\log (t))t^{-\\gamma }.$ We have, for all $\\eta _{t-1}$ , $\\sum _{t=2}^T \\mathbb {P}\\left((\\hat{V}(N_t)-v)^2 \\ge \\frac{\\eta _{t-1}}{2 N_t}\\right)&\\le \\sum _{t=2}^T \\mathbb {P}\\left(\\exists m : ~1\\le m \\le t, 2m (\\hat{V}(m)-v)^2 \\ge \\eta _{t-1} \\right)\\\\&\\le \\sum _{t=1}^T 2 e \\sqrt{\\eta _{t-1} \\log (t-1) } \\exp (-\\eta _{t-1}) := l_1(T)$ where the second inequality comes from Lemma 11 in [5], and from the fact that $V_t$ is a positive random variable bounded by 1, so $1/2-$ sub-Gaussian.",
"Therefore, if $\\eta _t := \\gamma \\log t$ , $l_1(T) = \\sum _{t=2}^T 2 e \\sqrt{\\gamma }( \\log (t-1) )(t-1)^{-\\gamma }$ which tends to a finite limit as soon as $\\gamma > 1$ ."
],
[
"On the cumulative distribution function of $M_t$",
"Lemma 11 The following concentration inequality holds on the empirical cumulative distribution $\\hat{F}_t$ .",
"$\\sum _{t=2}^T \\mathbb {P}\\left(\\Vert \\hat{F}_t-F\\Vert _{\\infty }\\ge \\frac{\\gamma \\log (t-1)}{2 (t-1)}\\right) \\\\\\le 2 \\sum _{t=1}^T t^{-\\gamma } .$ It holds $&\\sum _{t=2}^T \\mathbb {P}\\left((\\max _{b\\in [0,1]} |F_t(b)-F(b)|)^2 \\ge \\frac{\\gamma \\log (t-1)}{2(t-1)}\\right)\\\\&\\le \\sum _{t=2}^T \\mathbb {P}\\left(\\Vert \\hat{F}_t-F\\Vert _{\\infty }^2\\ge \\frac{\\gamma \\log (t-1)}{2 (t-1)}\\right)\\\\&\\le \\sum _{t=1}^{T-1} 2 e^{-\\frac{2\\gamma \\log (t}{2t}}\\\\&\\le \\sum _{t=1}^T 2 t^{-\\gamma },$ according to the Dvoretzky–Kiefer–Wolfowitz inequality (see [20]).",
"Note that this also yields $\\sum _{t=2}^T \\mathbb {P}\\left(\\Vert \\hat{F}_t-F\\Vert _{\\infty }\\ge \\frac{\\gamma \\log (t-1)}{2 N_t}\\right)&\\le \\sum _{t=2}^T\\mathbb {P}\\left(\\Vert \\hat{F}_t-F\\Vert _{\\infty }\\ge \\frac{\\gamma \\log (t-1)}{2 (t-1)}\\right) \\\\&\\le 2 \\sum _{t=1}^T t^{-\\gamma } .$"
],
[
"Local concentration inequality",
"This lemma is key for the proof of the upper bound of the regret of UCBid1+.",
"It quantifies the variation of $\\hat{F}_t$ on a small interval.",
"lem:localconcentrationinequalityLemma REF For any $a,b\\in [0,1]$ , if $F$ is continuous and increasing, then $\\sup _{a\\le x \\le b}|\\hat{F}_t(x) - F(x) - (\\hat{F}_t(a)- F(a))| \\\\\\le \\sqrt{\\frac{2(F(b)- F(a))\\log \\left( \\frac{e \\sqrt{t}}{\\sqrt{2(F(b)- F(a))}\\eta }\\right)}{t}} + \\frac{\\log (\\frac{t}{2(F(b)- F(a) \\eta ^2 })}{6 t},$ with probability $1-\\eta $ Remark : it follows from the lemma that the the maximal gap between $\\hat{F}_t(x) - F(x)$ and $\\hat{F}_t(\\frac{a+b}{2})- F(\\frac{a+b}{2})$ can easily be bounded by : $\\sup _{a\\le x \\le b}|\\hat{F}_t(x) - F(x) - (\\hat{F}_t(\\frac{a+b}{2})- F(\\frac{a+b}{2}))|\\\\ \\le 2 \\sqrt{\\frac{2(F(b)- F(a))\\log \\left( \\frac{e \\sqrt{t}}{\\sqrt{2\\eta (F(b)- F(a))}}\\right)}{t}} + 2 \\frac{\\log (\\frac{t}{2(F(b)- F(a) \\eta ^2 })}{6 t}$ with probability $1-\\eta $ .",
"Proof: Let $X_1,\\dots ,X_n\\stackrel{iid}{\\sim }dF$ .",
"Let $m>2$ For every $1\\le i\\le m$ , let $x_i$ be such that $F(x_i) = F(a) + \\frac{i}{m}\\big (F(b)-F(a)\\big )\\;.",
"$ By Bernstein's inequality, since $t\\big (\\hat{F}_t(x_i)-\\hat{F}_t(a)\\big ) \\sim \\mathcal {B}(n, F(x_i)-F(a))$ has a variance bounded by $t\\big (F(b)-F(a))$ , there is an event $A$ of probability at least $1- m e^{-z}$ on which $\\max _{0\\le i\\le m} \\big | \\hat{F}_t(x_i)-\\hat{F}_t(a) -(F(x_i)-F(a)) \\big | \\le \\sqrt{\\frac{2\\big (F(b)-F(a)\\big )z}{t}} + \\frac{z}{3t} := \\delta ,$ by a union bound.",
"Besides, for $i=0$ , $\\hat{F}_t(x_i)-\\hat{F}_t(a) -(F(x_i)-F(a))=0$ .",
"On this event, for every $x_{i-1}\\le x \\le x_i$ : $\\hat{F}_t(x)-\\hat{F}_t(a) -(F(x)-F(a)) & \\le \\hat{F}_t(x_i)-\\hat{F}_t(a)-(F(x_i)-F(a)) + F(x_i) - F(x) \\le \\delta + \\frac{1}{m},\\\\\\hat{F}_t(x)-\\hat{F}_t(a) -(F(x)-F(a)) &\\ge \\hat{F}_t(x_{i-1})-\\hat{F}_t(a)-(F(x_{i-1})-F(a)) + F(x_{i-1}) - F(x) \\\\&\\ge -\\delta - \\frac{1}{m}\\;.$ and hence $\\sup _{a\\le t \\le b} \\big | \\hat{F}_t(x)-\\hat{F}_t(a) -(F(x)-F(a)) \\big | \\le \\sqrt{\\frac{2\\big (F(b)-F(a)\\big )z}{t}} + \\frac{z}{3t} + \\frac{1}{m} \\;.$ Now, take $m= \\Big \\lceil \\sqrt{\\frac{t}{2\\big (F(b)-F(a)\\big )}} \\Big \\rceil $ and $z = \\log (m/\\eta ) $ : one gets that with probability at least $1-\\eta $ , $\\sup _{a\\le t \\le b}& \\big | \\hat{F}_t(x)-\\hat{F}_t(a) -(F(x)-F(a)) \\big | \\\\& \\le \\sqrt{\\frac{2\\big (F(b)-F(a)\\big )\\log \\left( \\frac{\\sqrt{\\frac{t}{2(F(b)-F(a))}}}{\\eta }\\right)}{t}} + \\frac{\\log \\left( \\frac{\\sqrt{\\frac{t}{2(F(b)-F(a))}}}{\\eta }\\right)}{3t} + \\sqrt{\\frac{2\\big (F(b)-F(a)\\big )}{t}}\\\\&\\le \\sqrt{\\frac{2\\big (F(b)-F(a)\\big )\\log \\left(\\frac{e\\sqrt{t}}{\\sqrt{2(F(b)-F(a))}\\eta }\\right)}{t}} + \\frac{\\log \\left( \\frac{t}{2(F(b)-F(a))\\eta ^2}\\right)}{6t}\\;.$"
],
[
"General bound on the instantaneous regret",
"In the following, we will repeatedly use the following general bound on the instantaneous regret conditioned on the past and on a current victory.",
"Lemma 12 Let $A$ be an $\\mathcal {F}_{t-1}$ -measurable event.",
"Let $S_t$ denote $(V_t-b^*)\\mathbb {1}(M_t < b^*) - (V_t-B_t)\\mathbb {1}(M_t < B_t) $ .",
"The following inequality holds: $\\mathbb {E}\\left[S_t \\mathbb {1}(B_t>b^*)\\mathbb {1}(A)| \\mathcal {F}_{t-1} \\vee \\sigma (\\mathbb {1}(B_t>M_t))\\right]\\le \\frac{U(b^*)- U(B_t)}{F(b^*)} \\mathbb {1}(M_t\\le B_t)\\mathbb {1}(A).$ When $B_t>b^*$ , the instantaneous regret can be decomposed as follows $S_t \\mathbb {1}(B_t>b^*) =(B_t-v)\\mathbb {1}(M_t\\le b^*) \\mathbb {1}(B_t>b^*)+ (B_t - b^*)\\mathbb {1} \\left\\lbrace (M_t \\le b^* \\le B_t)\\right\\rbrace .$ Note that in particular, there is no instantaneous regret when $M_t> B_t$ .",
"Therefore $&\\mathbb {E}\\left[S_t \\mathbb {1}(B_t>b^*) \\mathbb {1}(A)| \\mathcal {F}_{t-1} \\vee \\mathbb {1}(B_t>M_t)\\right]\\\\& \\le \\frac{(B_t-b^*)F(b^*) + (B_t-v)(F(B_t)-F(b^*))}{F(B_t)}\\mathbb {1}(M_t\\le B_t) \\mathbb {1}(B_t>b^*) \\mathbb {1}(A)\\\\&\\le \\frac{U(b^*)- U(B_t)}{F(b^*)} \\mathbb {1}(M_t\\le B_t) \\mathbb {1}(A),$ since $U(b^*)-U(B_t)= (v-b^*)F(b^*)- (v-B_t)F(B_t)$ , which also equals $(B_t-b^*)F(b^*) + (B_t-v)(F(B_t)-F(b^*))$ ."
],
[
"Other lemmas",
"Lemma 13 The expectations $\\mathbb {E}\\left[\\sum _{t=2}^T\\frac{1}{N_t} \\mathbb {1}\\lbrace M_t\\le B_t\\rbrace \\right]$ and $ \\mathbb {E}\\left[\\sum _{t=2}^T \\sqrt{\\frac{1}{N_t}} \\mathbb {1}\\lbrace M_t\\le B_t\\rbrace \\right]$ can always be bounded as follows ${\\left\\lbrace \\begin{array}{ll}\\mathbb {E}\\left[\\sum _{t=2}^T \\frac{1}{N_t} \\mathbb {1}\\lbrace M_t\\le B_t\\rbrace \\right]\\le 1+\\log T,\\\\\\mathbb {E}\\left[\\sum _{t=2}^T \\sqrt{\\frac{1}{N_t}} \\mathbb {1}\\lbrace M_t\\le B_t\\rbrace \\right] \\le 1+ \\sqrt{T}.\\end{array}\\right.",
"}$ Since winning an auction increments the number of observations $N_t$ by 1, $\\sum _{t=2}^T \\mathbb {E}\\Big [ \\sqrt{\\frac{1}{N_{t}}} \\mathbb {1}(M_t \\le B_t)\\Big ]&\\le \\sum _{t=2}^T \\sum _{n=1}^{T-1} \\sqrt{\\frac{1}{n}} \\mathbb {1}\\lbrace N_t= n,~ N_{t+1} =n+1\\rbrace \\\\&\\le \\sum _{n=1}^{T-1} \\sqrt{\\frac{1}{ n}} \\sum _{t=2}^T \\mathbb {1}\\lbrace N_{t}= n,~ N_{t} =n+1\\rbrace \\\\&\\le \\sum _{n=1}^{T-1} \\sqrt{\\frac{1}{ n}}\\\\&\\le 1 + \\sum _{n=2}^{T-1} \\int _{n-1}^n \\sqrt{\\frac{1}{ u}}du\\\\&\\le 1+\\sqrt{T}.$ Similarly, we get $\\sum _{t=2}^T \\mathbb {E}\\Big [ \\frac{1}{N_{t}} \\mathbb {1}(M_t \\le B_t)\\Big ]&\\le \\sum _{t=2}^T \\sum _{n=1}^{T-1} \\frac{1}{n} \\mathbb {1}\\lbrace N_t= n,~ N_{t+1} =n+1\\rbrace \\\\&\\le \\sum _{n=1}^{T-1} \\frac{1}{ n} \\sum _{t=2}^T \\mathbb {1}\\lbrace N_{t}= n,~ N_{t} =n+1\\rbrace \\\\&\\le \\sum _{n=1}^{T-1} \\frac{1}{ n}\\\\&\\le 1 + \\sum _{n=2}^{T-1} \\int _{n-1}^n \\frac{1}{ u}du\\\\&\\le 1+\\log {T}.$ Lemma 14 If $g_1$ and $g_2$ are two functions such that $\\Vert g_1 - g_2\\Vert _{\\infty } \\le \\delta $ , then $g_1(b_1^*) - g_1(b^*_2) \\le 2 \\delta $ where $b_1^* = \\max (\\operatornamewithlimits{arg\\,max}_{b \\in [0,1]} g_1(b))$ and $b_2^* = \\max (\\operatornamewithlimits{arg\\,max}_{b \\in [0,1]} g_2(b))$ .",
"Indeed, $0\\le g_1(b^*_1) - g_1(b^*_2) &\\le g_1(b_1^*) - g_2(b^*_2) + g_2(b^*_2) - g_1(b^*_2) \\\\& \\le 2 \\delta .$ Lemma 15 For any $a>0$ , $t\\ge 2 a \\log (a)$ implies $t\\ge a \\log t$ .",
"$a \\log t &\\ge a \\left(\\frac{t}{2a} + \\log (2a)\\right)\\\\&\\ge t/2 + a \\log (a),$ where the first inequality follows from the fact that $\\log (x/y)\\le x/y$ for any positive $x$ and $y$ .",
"Hence when $t>2 a \\log (a)$ , $t\\ge t/2+ a \\log t \\ge a \\log t.$"
],
[
"Upper Bounds of the Regret of UCBid1",
"We prove the somewhat more precise form of Theorem REF .",
"th:FPUCBIDgeneralTheorem REF UCBid1 incurs a regret bounded as follows $R_T \\le \\frac{1 }{ F(b^*)} \\sqrt{\\gamma \\log T }(\\sqrt{T}+1) + O(1).$ We denote by $U^{UCBid1}_t$ the function $b \\mapsto (\\hat{V}_t + \\epsilon _t -b)F(b)$ .",
"The regret can be decomposed as follows.",
"$R_T&\\le 1 + \\sum _{t=2}^T\\mathbb {P}\\left(|\\hat{V}_{t}-v|\\ge \\epsilon _{t} \\right) + \\sum _{t=2}^T\\mathbb {E}\\left[S_t \\mathbb {1}\\left\\lbrace |\\hat{V}_{t}-v|\\le \\epsilon _{t} \\right\\rbrace \\right],$ Lemma REF yields the following bound on the probability of over-estimating $\\hat{V}_t$ : $\\sum _{t=2}^T\\mathbb {P}(|\\hat{V}_{t}-v|\\ge \\epsilon _{t}) \\le \\sum _{t=1}^{t}2 e \\sqrt{\\gamma }( \\log t )t^{-\\gamma }.$ Since $F(x)\\le 1, \\forall x\\in [0,1]$ , and $\\Vert U^{UCBid1}_{t}-U\\Vert _{\\infty } = \\Vert (\\hat{V}_{t}-v+ \\epsilon _{t})F(x)\\Vert _{\\infty }\\le |\\hat{V}_{t}-v+ \\epsilon _{t}|$ , we can bound the difference between the utility function and its (upper confidence) estimate with high probability: $\\sum _{t=2}^T\\mathbb {P}(\\Vert U^{UCBid1}_{t}- U\\Vert _{\\infty } \\ge 2\\epsilon _{t}) \\le \\sum _{t=1}^{T}2 e \\sqrt{\\gamma }( \\log t )t^{-\\gamma }.$ When $\\Vert U^{UCBid1}_{t}- U\\Vert _{\\infty } \\le 2 \\epsilon _{t}$ , then $|U(b^*) - U(B_t)| \\le 4 \\epsilon _{t},$ thanks to Lemma REF .",
"Additionally, using Lemma REF , if $\\hat{V}_{t}+\\epsilon _{t}-v\\ge 0$ , then $B_t\\ge b^*$ Therefore, $&\\sum _{t=2}^T\\frac{1}{F(b^*)}\\mathbb {E}\\left[S_t \\mathbb {1}\\left\\lbrace M_t\\le B_t\\right\\rbrace \\mathbb {1}\\left\\lbrace b^*\\le B_t\\right\\rbrace \\mathbb {1}\\left\\lbrace |\\hat{V}_{t}-v|\\le \\epsilon _{t}\\right\\rbrace \\right]\\\\&\\le \\sum _{t=2}^T\\mathbb {E}\\left[ \\frac{U(b^*) - U(B_t)}{F(b^*)} \\mathbb {1}\\left\\lbrace b^*\\le B_t\\right\\rbrace \\mathbb {1}\\left\\lbrace M_t\\le B_t\\right\\rbrace \\mathbb {1}\\left\\lbrace |\\hat{V}_{t}-v|\\le \\epsilon _{t}\\right\\rbrace \\right]\\\\&\\le \\sum _{t=2}^T\\mathbb {E}\\left[ \\frac{U(b^*) - U(B_t)}{F(b^*)} \\mathbb {1}\\left\\lbrace b^*\\le B_t\\right\\rbrace \\mathbb {1}\\left\\lbrace M_t\\le B_t\\right\\rbrace \\mathbb {1}\\left\\lbrace U(b^*)-U(B_t)\\le 4 \\epsilon _{t} \\right\\rbrace \\right]\\\\&\\le \\sum _{t=2}^T\\frac{1}{F(b^*)}\\mathbb {E}\\left[4 \\epsilon _{t} \\mathbb {1}\\left\\lbrace M_t\\le B_t\\right\\rbrace \\mathbb {1}\\left\\lbrace (U(b^*)-U(B_t)\\le 4 \\epsilon _{t}\\right\\rbrace \\right]\\\\&\\le \\sum _{t=2}^T \\frac{1}{F(b^*)}\\sqrt{ 2\\frac{\\gamma \\log T}{ N_{t}}}\\\\&\\le \\frac{1}{F(b^*)} \\sqrt{2 \\gamma \\log T} (1+\\sqrt{T}),$ where the second inequality comes from Lemma REF (in fact $\\left\\lbrace |\\hat{V}_{t}-v|\\le \\epsilon _{t}\\right\\rbrace $ is $\\mathcal {F}_{t-1}$ -measurable) and the last inequality comes from Lemma REF .",
"Using Lemma REF yields $\\sum _{t=2}^T\\mathbb {P}(|\\hat{V}_{t}-v|\\ge \\epsilon _{t}) \\le \\sum _{t=1}^T2 e \\sqrt{\\gamma }( \\log t )t^{-\\gamma } .$ Combining this with the above decomposition of the regret yields $R_T\\le 1 + \\sum _{t=1}^T2 e \\sqrt{\\gamma }( \\log t )t^{-\\gamma } +\\frac{1}{F(b^*)} \\sqrt{2\\log T} (1+\\sqrt{T}),$ When $\\gamma >1$ , $\\sum _{t=1}^T2 e \\sqrt{\\gamma }( \\log t )t^{-\\gamma }$ tends to a constant, and $R_T \\le \\frac{1}{F(b^*)} \\sqrt{2 \\gamma \\log T} (1+\\sqrt{T}) + O(1),$ which concludes the proof.",
"th:FPUCBIDpseudomhrTheorem REF If $F$ satisfies Assumption REF and REF , then $R_T\\le \\frac{2\\gamma \\lambda C_f^2}{F(b^*) c_f}\\log ^2(T) + O(\\log T), $ when $\\gamma >1.$ Thanks to Lemma REF , if $\\hat{V}_{t}+\\epsilon _{t}-v\\ge 0$ , then $B_t\\ge b^*$ .",
"Additionally, $B_t-b^*\\le (\\hat{V}_{t}+ \\epsilon _{t} - v),$ thanks to Lemma REF .",
"In particular, if $ \\hat{V}_{t}+\\epsilon _{t} -v< 2 \\epsilon _{t}$ , $B_t-b^*\\le 2 \\epsilon _{t}.$ The regret can therefore be decomposed as follows : $ R_T \\le 1 +\\sum _{t=2}^T \\mathbb {P}(\\hat{V}_{t}+\\epsilon _{t} -v \\le 0) +\\sum _{t=2}^T \\mathbb {P}(\\hat{V}_{t}-\\epsilon _{t} -v \\ge 0)\\\\ +\\mathbb {E}\\left[\\sum _{t=2}^T S_t \\mathbb {1}(B_t\\in \\left[b^*, b^*+ \\min (2 \\epsilon _{t}, \\Delta )\\right]\\right] + \\sum _{t=2}^T \\mathbb {E}\\left[S_t \\mathbb {1}(B_t\\in \\left[b^*+ \\min (2 \\epsilon _{t}, \\Delta ), b^* + \\Delta \\right])\\right]$ Let us bound the third term of this inequality.",
"Thanks to Lemma REF , $\\mathbb {E}\\left[S_t \\mathbb {1}(B_t\\in \\left[b^*, b^*+\\epsilon _{t} \\right])|\\mathcal {F}_{t-1}\\vee \\sigma ( \\mathbb {1}\\left\\lbrace M_t\\le B_t\\right\\rbrace ) \\right]\\\\\\le \\frac{U(b^*) - U(B_t )}{F(b^*)}\\times \\mathbb {1}\\left\\lbrace M_t\\le B_t\\right\\rbrace \\mathbb {1}\\left\\lbrace b^*\\le B_t \\le b^* + 2 \\epsilon _{t}\\right\\rbrace ,$ because $(B_t\\in \\left[b^*, b^*+\\epsilon _{t}\\right])$ is $\\mathcal {F}_{t-1}$ - measurable.",
"This is why $&\\sum _{t=2}^T\\mathbb {E}\\left[ \\mathbb {E}\\left[ S_t \\mathbb {1}(B_t\\in [b^*, b^*+ \\min (2 \\epsilon _{t}, \\Delta ])|\\mathcal {F}_{t-1}\\vee \\sigma (\\left\\lbrace \\mathbb {1}\\left\\lbrace M_t\\le B_t\\right\\rbrace )\\right\\rbrace \\right]\\right]\\\\&\\le \\sum _{t=2}^T \\mathbb {E}\\left[\\frac{U(b^*) - U(B_t )}{F(b^*)}\\times \\mathbb {1}\\left\\lbrace M_t\\le B_t\\right\\rbrace \\mathbb {1}\\left\\lbrace b^*\\le B_t \\le b^* + \\min (2 \\epsilon _{t}, \\Delta \\right\\rbrace \\right]\\\\&\\le \\sum _{t=2}^T \\mathbb {E}\\left[\\frac{W(q^*) - W(Q_t )}{F(b^*)}\\times \\mathbb {1}\\left\\lbrace M_t\\le B_t\\right\\rbrace \\mathbb {1}\\left\\lbrace q^*\\le Q_t \\le b^* + 2 C_f \\epsilon _{t}\\right\\rbrace \\right]\\\\&\\le \\sum _{t=2}^T \\mathbb {E}\\left[\\frac{\\lambda (q^* - Q_t )^2}{c_fF(b^*)}\\times \\mathbb {1}\\left\\lbrace M_t\\le B_t\\right\\rbrace \\mathbb {1}\\left\\lbrace q^*\\le Q_t \\le b^* + 2 C_f \\epsilon _{t}\\right\\rbrace \\right]\\\\&\\le \\mathbb {E}\\left[ \\frac{\\lambda (2 C_f)^2}{ c_fF(b^*)} \\sum _{t=2}^T \\left(\\frac{\\gamma \\log T }{2N_t}\\right) \\mathbb {1}\\left\\lbrace M_t \\le B_t\\right\\rbrace \\right]\\\\&\\le \\frac{ 2 \\lambda \\gamma \\bar{C_f}}{ c_f F(b^*)} \\log T(\\log T +1),$ where the third inequality comes from Lemma REF and the last one follows from Lemma REF .",
"Thanks to Lemma REF , the sum of the first term and the second term of Equation (REF ) can be bounded by $\\sum _{t=2}^T \\mathbb {P}(\\hat{V}_{t} -v< \\epsilon _{t})+ \\sum _{t=2}^T \\mathbb {P}(\\hat{V}_{t}-\\epsilon _{t} -v \\ge 0) \\le \\sum _{t=1}^T e \\sqrt{\\gamma } \\frac{\\log t}{t^{\\gamma }}$ which is bounded by a constant when $\\gamma >1$ .",
"The last term of Equation (REF ) can be bounded as follows: $\\sum _{t=2}^T \\mathbb {E}\\left[S_t\\mathbb {1}(B_t\\in \\left[b^*+ \\min (2 \\epsilon _{t}, \\Delta ), b^* + \\Delta \\right])\\right]&\\le \\sum _{t=2}^T \\mathbb {P}\\left[(\\Delta > 4 \\epsilon _{t}, M_t \\le B_t,B_t>b^* \\right]\\\\&\\le \\sum _{t=2}^T \\mathbb {P}\\left[\\Delta ^2 > 4 \\frac{\\gamma \\log T}{2 N_{t}}, M_t \\le B_t, B_t>b^* \\right]\\\\&\\le \\sum _{t=2}^T \\sum _{n=1}^{T-1} \\mathbb {P}\\left[\\Delta ^2 > 2 \\frac{\\gamma \\log T}{2 N_{t}}\\right]\\mathbb {1}\\left[N_{t}= n,~ N_{t+1} =n+1 \\right]\\\\&\\le \\sum _{n=1}^{T-1} \\mathbb {1}\\left[n<4\\frac{\\gamma \\log T}{2 \\Delta ^2}\\right] \\sum _{t=2}^T\\mathbb {1}\\left\\lbrace N_{t}= n,~ N_{t+1} =n+1 \\right\\rbrace \\\\&\\le \\sum _{n=1}^{T-1} \\mathbb {1}\\left\\lbrace n<4\\frac{\\gamma \\log T}{2 \\Delta ^2}\\right\\rbrace \\\\& \\le 4\\frac{\\gamma \\log T}{2 \\Delta ^2}$ where the first inequality comes from the fact that when $B_t>b^*$ , a positive instantaneous regret can only occur if $M_t \\le B_t$ .",
"By summing all components of the regret, $R_T\\le 1 + 4\\frac{\\gamma \\log T}{2 \\Delta ^2} + \\frac{2\\gamma \\lambda C_f^2}{F(b^*) c_f}(\\log ^2(T)+ \\log T).$ In conclusion, $ R_T\\le \\frac{2\\gamma \\lambda C_f^2}{F(b^*) c_f}\\log ^2(T) + O(\\log T) \\\\$ when $\\gamma >1.$"
],
[
"Lower bound of the regret of optimistic strategies",
"th:parametriclowerboundLemma REF Consider all environments where $V_t$ follows a Bernoulli distribution with expectation $v$ and $F$ satisfies Assumption REF and is such that $\\phi ^{\\prime } \\le \\lambda $ , and there exists $c_f$ and $C_f$ such that $0<c_f<f(b)<C_f, ~ \\forall b \\in [0,1]$ .",
"If a strategy is such that, for all such environments, $R_T^{v,F}\\le O(T^{a})$ , for all $a>0$ , and there exists $\\gamma >0$ such that $\\mathbb {P}(B_t< b^*)<t^{-\\gamma }$ , then this strategy must satisfy: $& \\liminf _{T \\rightarrow \\infty } \\frac{R_T^{v,F}}{\\log T}\\ge c_f^2 \\lambda ^2\\left(\\frac{v(1-v)(v- b^*_{v,F})}{32} \\right).$ Note that this proof is an adaptation of the proof of the parametric lower bound of [1].",
"Lemma 16 If $~ R_T \\le O(T^{a}), ~ \\forall a>0,$ and $F$ admits a density which is lower bounded by a positive constant and upper bounded.",
"Then, $ \\lim _{t \\rightarrow \\infty } \\mathbb {E}\\left[\\frac{N_t}{t}\\right] = F(b^*).$ The fraction of won auctions is $\\mathbb {E}\\left[\\frac{N_t}{t}\\right]= \\mathbb {E}[\\frac{1}{t}\\sum _{s=1}^t F(B_s]$ , by the tower rule.",
"Since $F$ admits a density $f$ , upper bounded by a constant $C_f$ , $\\mathbb {E}[(F(B_t) - F(b^*))^2]]\\le C_f^2 \\mathbb {E}[(B_t-b^*)^2] .$ The consistency assumption implies $\\sum _{t=1}^{T} \\mathbb {E}[(B_t- b^*)^2]\\le O(T^{a}), ~ \\forall a>0,$ because of Lemma REF .",
"In particular $\\lim _{t \\rightarrow \\infty }\\mathbb {E}[(B_t- b^*)^2] = 0 $ .",
"Combining the two previous arguments yields $\\lim _{t \\rightarrow \\infty }\\mathbb {E}[(F(B_t) - F(b^*))^2]=0$ .",
"Then, because $L_2$ -convergence implies $L_1$ -convergence, $\\lim _{t \\rightarrow \\infty }\\mathbb {E}[F(B_t)] = F(b^*)$ .",
"Together with the equality $\\mathbb {E}\\left[\\frac{N_t}{t}\\right]= \\mathbb {E}[\\frac{1}{t}\\sum _{s=1}^t F(B_s)]$ , and with the Cesaro theorem, this result proves suffices to prove the lemma.",
"We set a time step $t\\in [1,T]$ .",
"We consider two alternative configurations with identical distributions for $M_t$ but that differ by the distribution of $V_t$ .",
"The value $V_t$ is distributed according to a Bernoulli distribution of expectation $v$ in the first configuration, respectively $v^{\\prime }_t = v + \\sqrt{\\frac{v(1-v)}{F(b^*)t}}$ , in the second configuration."
],
[
"Notation.",
"We let $\\mathbb {P}_v(\\cdot )$ denote the probability of an event under the first configuration (respectively $\\mathbb {E}_v(\\cdot )$ the expectation of a random variable under the first configuration), whereas $\\mathbb {P}_{v^{\\prime }_t}(\\cdot )$ denotes the probability of an event under the second configuration (respectively $\\mathbb {E}_{v^{\\prime }_t}(\\cdot )$ the expectation of a random variable under the first configuration).",
"The information collected up to time $t+1$ is denoted $I_t$ : $(M_t, V^{\\prime }_t, \\ldots M_1, V^{\\prime }_1)$ .",
"Finally, $\\mathbb {P}_v^{I_t}$ (respectively $\\mathbb {P}_{v^{\\prime }_t}^{I_t}$ ) is the law of $I_t$ in the first (respectively second) configuration.",
"The Kullback Leibler divergence between $\\mathbb {P}_v^{I_t}$ and $\\mathbb {P}_{v^{\\prime }_t}^{I_t}$ can be proved to satisfy $ KL(\\mathbb {P}_v^{I_t},\\mathbb {P}_{v^{\\prime }_t}^{I_t})= kl(v,v^{\\prime }_t) \\mathbb {E}[N_t],$ exactly like in Equation REF .",
"Using Lemma REF , $\\forall \\epsilon >0, \\exists t_1(\\epsilon ), \\forall t\\ge t_1(\\epsilon ) $ , $ KL(\\mathbb {P}_v^{I_t},\\mathbb {P}_{v^{\\prime }_t}^{I_t}) \\le kl(v,v^{\\prime }_t)(1+\\epsilon )F(b^*).$ Using the data processing inequality (see for example [14]), we get $KL(\\mathbb {P}_v^{I_t},\\mathbb {P}_{v^{\\prime }_t}^{I_t})& \\ge kl \\left(\\mathbb {P}_v \\left( B_t > \\frac{b^*_{v,F} + b^{*}_{v^{\\prime }_t,F}}{2}\\right), \\mathbb {P}_{v^{\\prime }_t} \\left( B_t > \\frac{v+ b^{*}_{v^{\\prime }_t,F}}{2}\\right)\\right)\\\\& \\ge 2 \\left(\\mathbb {P}_v \\left( B_t > \\frac{b^*_{v,F} + b^{*}_{v^{\\prime }_t,F}}{2}\\right)- \\mathbb {P}_{v^{\\prime }_t} \\left( B_t > \\frac{b^*_{v,F} + b^{*}_{v^{\\prime }_t,F}}{2}\\right)\\right)^2\\\\& \\ge 2 \\left(\\mathbb {P}_v \\left( B_t > \\frac{b^*_{v,F} + b^{*}_{v^{\\prime }_t,F}}{2}\\right)+ \\mathbb {P}_{v^{\\prime }_t} \\left( B_t \\le \\frac{b^*_{v,F} + b^{*}_{v^{\\prime }_t,F}}{2}\\right)-1\\right)^2,$ where the second inequality comes from Pinsker inequality.",
"Consequently, we get $\\mathbb {P}_v \\left( B_t > \\frac{b^*_{v,F}+ b^{*}_{v^{\\prime }_t,F}}{2}\\right)+ \\mathbb {P}_{v^{\\prime }_t} \\left( B_t \\le \\frac{b^*_{v,F} + b^{*}_{v^{\\prime }_t,F}}{2}\\right)\\ge 1 - \\sqrt{\\frac{1}{2} KL(\\mathbb {P}_v^{I_t},\\mathbb {P}_{v^{\\prime }_t}^{I_t})}.$ Specifically, $\\forall t>t_0(\\epsilon )$ , $\\mathbb {P}_v \\left( B_t > \\frac{b^*_{v,F} + b^{*}_{v^{\\prime }_t,F}}{2}\\right)+ \\mathbb {P}_{v^{\\prime }_t} \\left( B_t \\le \\frac{b^*_{v,F} + b^{*}_{v^{\\prime }_t,F}}{2}\\right)\\ge 1 - \\sqrt{\\frac{1}{2}kl(v,v^{\\prime }_t)(1+\\epsilon )F(b^*_{v,F})t}.$ Using the fact that $\\mathbb {E}_v[(B_t-b^*_{v,F})^2]\\ge \\left(b^*_{v,F}-\\frac{b^*_{v,F}+ b^{*}_{v^{\\prime }_t,F}}{2}\\right)^2\\mathbb {P}_v \\left( B_t > \\frac{b^*_{v,F}+ b^{*}_{v^{\\prime }_t,F}}{2}\\right)$ yields $\\mathbb {E}_v[(B_t-b^*_{v,F})^2]&\\ge \\left(\\frac{b^*_{v,F}- b^{*}_{v^{\\prime }_t,F}}{2}\\right)^2 \\mathbb {P}_v \\left( B_t > \\frac{b^*_{v,F}+ b^{*}_{v^{\\prime }_t,F}}{2}\\right)\\\\&\\ge \\left(\\lambda \\frac{v- v^{\\prime }_t}{2}\\right)^2 \\mathbb {P}_v \\left( B_t > \\frac{b^*_{v,F}+ b^{*}_{v^{\\prime }_t,F}}{2}\\right)\\\\& \\ge \\lambda ^2 \\frac{v(1-v)}{4F(b^*_{v,F})t} \\left( 1- \\sqrt{ \\frac{1}{2}(1 + \\epsilon ) kl(v, v^{\\prime }_t)F(b^*_{v,F})t} - 1/ {t^{\\gamma }}\\right),$ where the second inequality comes from the fact that $v = \\phi _F(b^{*}_{v,F})$ (resp.",
"$v^{\\prime }_t = \\phi _F(b^{*}_{v^{\\prime }_t,F})$ ) and that $\\phi _F^{\\prime }\\le \\lambda $ and the the second inequality stems from the assumption that the algorithm outputs a bid that does not underestimate $b^{*}_{v^{\\prime }_t,F}$ with high probability: $\\mathbb {P}_{v^{\\prime }_t}(B_t< b^{*}_{v^{\\prime }_t,F})<\\frac{1}{t^{\\gamma }}$ .",
"We use the fact that $\\forall \\epsilon >0, ~\\exists t_2(v, \\epsilon ), ~\\forall t\\ge t_2(v,\\epsilon ),~ kl\\left(v, v + \\sqrt{\\frac{v(1-v)}{F(b^*_{v,F})t}}\\right) \\le \\frac{1 + \\epsilon }{2F(b^*_{v,F})t} $ which is proved by observing that $kl(v,v^{\\prime })= \\frac{(v^{\\prime }-v)^2}{2} \\int _0^1 g^{\\prime \\prime }(v^{\\prime } + s(v^{\\prime } + s(v-v^{\\prime }))2(1-s)ds$ , where $g(x) = kl(x,v^{\\prime })$ ; and that thanks to Taylor's inequality, $kl(v,v^{\\prime })&\\le \\frac{(v^{\\prime }-v)^2}{2} \\int _0^1 2 \\max _{u\\in [v,v^{\\prime }]} g^{\\prime \\prime }(u)ds\\\\&\\le (v^{\\prime }-v)^2 \\frac{1}{\\min _{u\\in [v,v^{\\prime }]}u(1-u)}$ and that $\\forall \\epsilon >0, ~\\exists t_2(v, \\epsilon )$ , such that $\\min _{u\\in [v,v^{\\prime }]}u(1-u)<\\frac{1+\\epsilon }{v(1-v)}$ .",
"Putting all the pieces together yields $\\forall t\\ge \\max ( t_1(\\epsilon ),t_2(v,\\epsilon )), $ $\\mathbb {E}_v[(B_t-b^*_{v,F})^2] \\ge \\frac{v(1-v)}{4F(b^*_{v,F})t} \\left( 1- \\sqrt{ \\frac{1}{4}(1 + \\epsilon )^2 } - 1/ t^{\\gamma }\\right).$ Let $t_0(v,\\epsilon ) =\\max ( t_1(\\epsilon ),t_2(v,\\epsilon )).$ We obtain $\\sum _{t= 1}^T \\mathbb {E}_v[(B_t-b^*_{v,F})^2] \\ge \\sum _{t= t_0(v,\\epsilon )}^T \\lambda ^2 \\frac{v(1-v)}{4F(b^*_{v,F})t} \\left( 1- \\frac{1}{2}(1 + \\epsilon ) - 1/ t^{\\gamma }\\right).$ Recall that, according to Lemma REF , $R_T(v) = \\sum _{t=1}^T \\mathbb {E}\\left[ U(b^*_{v,F}) - U(B_t) \\right]\\ge \\frac{U(b^*_{v,F})}{4} \\sum _{t= 1}^T \\mathbb {E}_v[(Q_t-q^*)^2] \\ge \\frac{c_f^2 U(b^*_{v,F})}{4} \\sum _{t= 1}^T \\mathbb {E}_v[(B_t-b^*_{v,F})^2] .$ Hence, $\\forall \\epsilon >0,$ $R_T(v)\\ge \\lambda ^2 \\frac{c_f^2 U(b^*_{v,F})}{4} \\left(\\frac{v(1-v)}{4} \\left( 1- \\frac{1}{2}(1 + \\epsilon )\\right)\\right) \\log \\frac{T}{t_0(v,\\epsilon )} - O(1).$ And $\\forall \\epsilon >0,$ $& \\liminf _{T \\rightarrow \\infty } \\frac{R_T(v)}{\\log T}\\ge \\frac{c_f^2 \\lambda ^2 U(b^*_{v,F})}{4} \\left(\\frac{v(1-v)}{4F(b^*_{v,F})} \\left( 1- \\frac{1}{2}(1 + \\epsilon )\\right)\\right) .$ Since this holds for all $\\epsilon $ , $\\liminf _{T \\rightarrow \\infty } \\frac{R_T(v)}{\\log T}\\ge \\lambda ^2 c_f^2 \\left(\\frac{v(1-v)(v- b^*_{v,F})}{32} \\right).$"
],
[
"Upper Bound of the Regret of O-UCBid1",
"Theorem 7 O-UCBid1 incurs a regret bounded by $R_T \\le \\frac{4 \\sqrt{2}}{F(b^*)} \\sqrt{ \\gamma \\log T} (\\sqrt{T}+1) + O(1).$ We first observe that the algorithm overbids ($B_t>b^*$ ) when $F$ and $v$ belong to their confidence regions $\\mathbb {F}_t = \\lbrace \\tilde{F}, \\Vert F-\\hat{F}_t\\Vert \\le \\epsilon _t \\rbrace $ and $\\mathbb {V}_t = [v- \\epsilon _t, v+\\epsilon _t]$ .",
"Lemma 17 The bid submitted by O-UCBid1 is an upper bound of $b^*$ when $\\Vert \\hat{U}_{t} - U\\Vert _\\infty \\le 2 \\epsilon _{t}$ .",
"$\\left\\lbrace ~\\Vert \\hat{U}_{t} - U\\Vert _\\infty \\le 2 \\epsilon _{t}~\\right\\rbrace \\text{ implies } b^* \\le B_t.$ Let us pick $\\underline{b}\\in \\operatornamewithlimits{arg\\,max}\\hat{U}_{t}$ .",
"$\\hat{U}_{t}(\\underline{b}) -\\hat{U}_{t}(b^*)= \\hat{U}_{t}(\\underline{b})-U(b^*) +U(b^*)-\\hat{U}_{t}(b^*) \\le 4 \\epsilon _{t}.$ We deduce that $\\hat{U}_{t}(b^*)\\ge \\hat{U}_{t}(\\underline{b}) - 4 \\epsilon _{t} \\ge \\max \\hat{U}_{t} - 4 \\epsilon _{t}$ .",
"Hence, $b^*\\in \\left\\lbrace b\\in [0,1], \\hat{U}_{t}(b)\\ge \\max \\hat{U}_{t} - 2 \\epsilon _{t} \\right\\rbrace .$ By definition of $B_t$ , this yields $B_t\\ge b^*$ .",
"Next we observe that if $F$ and $v$ lie in their confidence regions $\\mathbb {F}_{t}$ and $\\mathbb {V}_{t}$ , then $\\Vert \\hat{U}_{t} - U\\Vert _{\\infty }\\le 2 \\epsilon _t $ .",
"(Recall that $\\hat{U}_t(b)=(\\hat{V}_t-b)\\hat{F}_t(b)$ .)",
"Indeed, we have $\\hat{U}_{t}(b) - U(b) &= (\\hat{V}_{t}-b)\\hat{F}_{t}(b) -(v-b) F(b)\\\\&= (\\hat{V}_{t}-v)F(b) + \\hat{V}_t(\\hat{F}_{t}(b) -F(b)) + b(F(b)-\\hat{F}_b) \\\\&= (\\hat{V}_{t}-v)F(b) + (\\hat{V}_{t}-b)(\\hat{F}_{t}(b) -F(b))$ which yields $|\\hat{U}_{t}(b) - U(b)| \\le |\\hat{V}_{t}-v| + \\Vert F(b) -\\hat{F}_{t}(b)\\Vert _\\infty .$ We then decompose the regret into $&E(R_T ) = \\sum _{t=1}^T \\mathbb {E}(U(b^*)-U(B_t))\\\\&\\le 1+ \\sum _{t=}^T \\mathbb {P}( F \\notin \\mathbb {F}_{t} \\text{ or } v \\notin \\mathbb {V}_{t}) + \\sum _{t=2}^T \\mathbb {E}\\left( S_t \\mathbb {1}(B_t>b^*) \\mathbb {1}(\\Vert \\hat{U}_{t}-U\\Vert _\\infty \\le 2 \\epsilon _{t},~ F \\in \\mathbb {F}_{t} ,~v \\in \\mathbb {V}_{t}\\right) .$ The second term of the second hand side of Equation REF is easily bounded thanks to the concentration inequalities in Lemmas REF and REF .",
"In fact, combining these latter lemmas yields the following bound.",
"Lemma 18 $ \\sum _{t=2}^T \\mathbb {P}(F \\notin \\mathbb {F}_{t} \\text{ or } v \\notin \\mathbb {V}_{t}) \\le 2 \\sum _{t=1}^T2 e \\sqrt{\\gamma }( \\log t )t^{-\\gamma }$ We apply Lemma REF to bound the third term of the second hand side of Equation REF as follows: $\\mathbb {E}\\Big [] S_t \\mathbb {1}(B_t>b^*) \\mathbb {1}(\\Vert \\hat{U}_{t}-U\\Vert _\\infty \\le 2 \\epsilon _{t}, ~ F \\in \\mathbb {F}_{t} ,~v \\in \\mathbb {V}_{t})\\Big ] \\\\\\le \\frac{1}{F(b^*)}\\mathbb {E}\\Big [U(b^*)-U(B_t))\\times \\mathbb {1}(M_t \\le B_t)\\mathbb {1}(\\Vert U-\\hat{U}_{t}\\Vert _{\\infty }\\le 2 \\epsilon _{t}, ~ F \\in \\mathbb {F}_{t} ,~v \\in \\mathbb {V}_{t})\\mathbb {1}(B_t>b^*)\\Big ],$ because $\\mathbb {1}(B_t>b^*) \\mathbb {1}(\\Vert \\hat{U}_{t}-U\\Vert _\\infty \\le 2 \\epsilon _{t}, ~ F \\in \\mathbb {F}_{t} ,~v \\in \\mathbb {V}_{t})$ is $\\mathcal {F}_{t-1}$ -measurable.",
"We then bound the deviation $(U(b^*)-U(B_t))\\mathbb {1}(M_t\\le B_t)$ by $8\\epsilon _t$ by using Lemma REF .",
"Lemma 19 When applying the O-UCBid1 strategy, if $\\Vert U - \\hat{U}_{t}\\Vert _{\\infty }\\le 2 \\epsilon _{t} $ , then $|U(B_t)-U(b^*)|\\le 8 \\epsilon _{t}.$ Assume $\\Vert U - \\hat{U}_{t}\\Vert _{\\infty }\\le 2 \\epsilon _{t} $ .",
"Note that $\\hat{U}_{t}(B_t) - \\hat{U}_{t}(b^*)= \\hat{U}_{t}(B_t) - \\hat{U}_{t}(\\hat{b})+ \\hat{U}_{t}(\\hat{b}) - \\hat{U}_{t}(b^*) $ , where $\\hat{b}= \\max \\operatornamewithlimits{arg\\,max}_{b \\in [0,1]} (\\hat{V}_{t}-b)\\hat{F}_{t}(b)$ .",
"By design , we have $\\hat{U}_{t}(B_t) - \\hat{U}_{t}(\\hat{b})= - 2 \\epsilon _{t} $ .",
"Thanks to Lemma REF , and because $\\Vert U - \\hat{U}_{t}\\Vert _{\\infty }\\le 2 \\epsilon _{t}$ we know that $0 \\le \\hat{U}_{t}(\\hat{b}) - \\hat{U}_{t}(b^*) \\le 4 \\epsilon _{t}$ .",
"This yields $|\\hat{U}_{t}(B_t) - \\hat{U}_{t}(b^*)|\\le 4 \\epsilon _{t}$ .",
"Finally $|U(B_t)- U(b^*)|\\le 8 \\epsilon _{t}.$ Then, by summing, we get $&\\sum _{t=2}^T\\mathbb {E}\\Big [] S_t \\mathbb {1}(B_t>b^*) \\mathbb {1}(\\Vert \\hat{U}_{t}-U\\Vert _\\infty \\le 2 \\epsilon _{t}, ~ F \\in \\mathbb {F}_{t} ,~v \\in \\mathbb {V}_{t})\\Big ] \\\\&\\le \\sum _{t=2}^T \\frac{1}{F(b^*)}\\mathbb {E}\\Big (U(b^*)-U(B_t))\\times \\mathbb {1}(M_t \\le B_t)\\mathbb {1}(B_t>b^*) \\mathbb {1}(\\Vert U-\\hat{U}_{t}\\Vert _{\\infty }\\le 2 \\epsilon _{t}, ~ F \\in \\mathbb {F}_{t} ,~v \\in \\mathbb {V}_{t})\\Big )\\\\&\\le \\sum _{t=2}^T \\frac{1}{F(b^*)}\\mathbb {E}\\Big [8 \\epsilon _{t} \\times \\mathbb {1}(M_t \\le B_t)\\mathbb {1}(\\Vert U-\\hat{U}_{t}\\Vert _{\\infty }\\le 2 \\epsilon _{t})\\mathbb {1}(B_t>b^*)\\Big ]\\\\&\\le \\sum _{t=2}^T \\frac{1}{F(b^*)}\\mathbb {E}\\Big [8 \\sqrt{\\frac{\\log T}{2 N_{t}}} \\mathbb {1}(M_t \\le B_t)\\Big ]\\\\&\\le \\frac{1}{F(b^*)}4 \\sqrt{2\\log T}(\\sqrt{T}+1),$ where the last inequality comes from Lemma REF .",
"Using Equation REF and Lemma REF yields $R_T \\le \\frac{1}{F(b^*)}4 \\sqrt{2\\log T}(\\sqrt{T}+1) + \\sum _{t=2}^T2 e \\sqrt{\\gamma }( \\log t )t^{-\\gamma }.$ Consequently, when $\\gamma >1$ , $R_T \\le \\frac{1}{F(b^*)}4 \\sqrt{2\\log T}(\\sqrt{T}+1) + O(1).$"
],
[
"General Upper Bound of the Regret of UCBid1+",
"We prove a slightly different version of Theorem REF than that of the main paper.",
"th:V-optTheorem REF UCBid1+ incurs a regret bounded by $R_T&\\le 12 \\sqrt{\\frac{\\gamma \\alpha }{F(b^*)}} \\sqrt{\\log T} \\sqrt{T} + O(\\log T)\\\\&\\le 12 \\frac{1}{U(b^*)}\\sqrt{v \\gamma } \\sqrt{\\log T} \\sqrt{T} + O(\\log T),$ where $\\alpha := \\frac{v}{v-b^*} $ , provided that $\\gamma >2$ .",
"We denote by $\\mathcal {E}$ the event $\\lbrace \\forall t_0<t<T, \\;|\\hat{V}_{t}-v|\\le \\epsilon _{t}, \\Vert F- \\hat{F}_{t}\\Vert _{\\infty } \\le \\sqrt{\\frac{\\gamma \\log (t-1)}{2(t-1)}} \\rbrace $ , where $t_0:= \\min (3,1+ 8\\frac{ \\gamma (\\alpha +1)^2}{\\alpha (F(b^*))^2} \\log \\left( 4\\frac{ \\gamma (\\alpha +1)^2}{\\alpha (F(b^*))^2}\\right)).$ Using Lemmas REF and REF , this event happens with high probability, when $\\gamma >2$ .",
"Lemma 20 The probability of the complementary of $\\mathcal {E}$ is bounded as follows $\\mathbb {P}\\left(\\mathcal {E}^C\\right)\\le 4 e(\\gamma -1) (\\log T ) (T)^{1-\\gamma }.$ provided that $\\gamma >2.$ We have $ \\mathbb {P}\\left(\\exists t\\in [t_0,T], ~(\\hat{V}(N_{t})-v)^2 \\ge \\frac{\\gamma \\log (t-1)}{2 N_{t}}\\right) &\\le \\mathbb {P}\\left(\\exists t\\in [2,T], ~(\\hat{V}(N_{t})-v)^2 \\ge \\frac{\\gamma \\log (t-1)}{2 N_{t}}\\right) \\\\& \\le \\sum _{t=2}^T\\mathbb {P}\\left((\\hat{V}(N_{t})-v)^2 \\ge \\frac{\\gamma \\log (t-1)}{2 N_{t}},\\right) \\\\&\\le \\sum _{t=1}^{T} \\ 2 e \\log (t) t^{-\\gamma } \\\\& \\le \\int _{u=1}^{T} 2 e \\log (t) u^{-\\gamma } du\\\\& \\le 2 e(\\gamma -1) \\log (T) (T)^{1-\\gamma },$ thanks to Lemma REF .",
"Similarly, $ \\mathbb {P}\\left(\\exists t\\in [t_0,T], ~\\Vert F- \\hat{F}\\Vert _{\\infty } \\ge \\sqrt{\\frac{\\gamma \\log (t-1)}{2 N_{t}}}\\right)&\\le \\sum _{t=t_0}^T \\mathbb {P}\\left(~\\Vert F- \\hat{F}\\Vert _{\\infty } \\ge \\sqrt{\\frac{\\gamma \\log (t-1)}{2 N_{t}}}\\right)\\\\&\\le 2 \\sum _{t=t_0}^T t^{-\\gamma }\\\\&\\le \\int _{u=2}^{T} 2 u^{-\\gamma } du\\\\&\\le 2 (\\gamma -1) (T)^{1-\\gamma }$ thanks to Lemma REF .",
"When $\\mathcal {E}$ occurs, it is possible to prove that $F(B_t)$ is lower-bounded by a positive constant as soon as $t$ is large enough.",
"Lemma 21 On $\\mathcal {E}$ , provided that $t>t_0:= \\min \\left(3,1+ 8\\frac{ \\gamma (\\alpha +1)^2}{\\alpha (F(b^*))^2} \\log \\left( 4\\frac{ \\gamma (\\alpha +1)^2}{\\alpha (F(b^*))^2}\\right)\\right) $ , $F(B_t)$ is lower bounded by $F(B_t)>\\frac{F(b^*)}{2 \\alpha },$ where $\\alpha = \\frac{v}{v-b^*}$ .",
"$b^*= \\frac{\\alpha -1 }{\\alpha } v$ .",
"Since we are on $\\mathcal {E}$ , $b^*\\le \\frac{\\alpha -1}{\\alpha }(\\hat{V}_{t} + \\epsilon _{t}).$ Hence $\\hat{V}_{t} + \\epsilon _{t} \\le \\alpha (\\hat{V}_{t} + \\epsilon _{t} - b^*).$ Since $B_t >0$ , $\\hat{V}_{t} + \\epsilon _{t} - B_t \\le \\alpha (\\hat{V}_{t} + \\epsilon _{t} - b^*).$ And $\\frac{\\hat{V}_{t} + \\epsilon _{t} - B_t}{\\hat{V}_{t} + \\epsilon _{t} - b^*}\\le \\alpha .$ By definition of $B_t$ , $(\\hat{V}_{t} + \\epsilon _{t} - B_t )\\hat{F}_t(B_t)\\ge (\\hat{V}_{t} + \\epsilon _{t} - b^*) \\hat{F}_t(b^*) $ which implies $ \\hat{F}_t(B_t)\\ge \\frac{\\hat{V}_{t} + \\epsilon _{t} - b^*}{\\hat{V}_{t} + \\epsilon _{t} - B_t}\\hat{F}_t(b^*) \\ge \\frac{1}{\\alpha } \\hat{F}_t(b^*) $ Now, $F(B_t)&\\ge \\hat{F}_t(B_t) - \\sqrt{\\frac{\\gamma \\log (t-1)}{2 (t-1)}} \\\\&\\ge \\frac{1}{\\alpha } \\hat{F}_t(b^*) - \\sqrt{\\frac{\\gamma \\log (t-1)}{2(t-1)}} \\\\&\\ge \\frac{1}{\\alpha } F(b^*) -\\left( \\frac{1}{\\alpha } +1 \\right) \\sqrt{\\frac{\\gamma \\log (t-1)}{2 (t-1)}},$ because we assume that we are on $\\mathcal {E}$ .",
"Note that if $t>t_0$ , then $\\frac{4\\gamma (\\alpha +1)^2}{F(b^*)^2} <\\frac{(t-1)}{\\log (t-1)},$ thanks to Lemma REF , and $\\left( \\frac{1}{\\alpha } +1\\right) \\sqrt{\\frac{\\gamma \\log (t-1)}{2 (t-1)}}< \\frac{1}{2\\alpha } F(b^*),$ so that $F(B_t) \\ge \\frac{ F(b^*)}{2\\alpha },$ which concludes the proof.",
"Lemma 22 $\\forall t>t_0,$ $\\mathbb {P}\\left(N_{t}< \\frac{1}{4\\alpha }F(b^*) (t- t_0), \\mathcal {E} \\right) \\le \\exp \\left(-\\frac{2((\\frac{1}{2\\alpha }F(b^*))^2}{4}(t-t_0)\\right).$ Indeed if $t \\ge t_0$ , then $N_{t}$ is larger than the sum $N^{\\prime }_{t}$ of $t-t_0$ samples from a Bernoulli distribution with average $\\frac{1}{2\\alpha }F(b^*)$ , hence the probability that $N_{t}< \\frac{1}{4\\alpha }F(b^*) (t- t_0)$ intersected with $\\mathcal {E} $ can be bounded as follows.",
"$ &\\mathbb {P}\\left(N_{t}< \\frac{1}{4\\alpha }F(b^*) (t- t_0), \\mathcal {E} \\right)\\\\&\\le \\mathbb {P}\\left(N^{\\prime }_{t}<+ \\frac{1}{4\\alpha }F(b^*) (t- t_0) \\right)\\\\& \\le \\mathbb {P}\\left( \\frac{1}{2\\alpha }F(b^*) (t- t_0) -(N^{\\prime }_{t} - t_0) > \\frac{1}{4\\alpha }F(b^*) (t- t_0)\\right) \\\\& \\le \\exp \\left(-\\frac{2((\\frac{1}{2\\alpha }F(b^*))^2}{4}(t- t_0)\\right)\\\\& \\le \\exp \\left(-\\frac{2((\\frac{1}{2\\alpha }F(b^*))^2}{4}(t-t_0)\\right),$ where we used Hoeffding's inequality for the third inequality.",
"Finally, we can prove that the expected instantaneous regret conditioned on $B_t$ is bounded by a multiple of $\\epsilon _{t}$ .",
"Lemma 23 $U(B_t)-U(b^*)\\le 6 \\epsilon _{t}$ Thanks to Equation REF , we have $\\Vert \\hat{U}_{t}-U\\Vert _{\\infty }\\le \\ 2 \\epsilon _{t}.$ Very similarly we have $\\Vert U^{UCBid1+}_{t} - \\hat{U}\\Vert _{\\infty } = \\max _{b \\in [0,1]}|\\epsilon _{t}\\hat{F}_{t}(b)|\\\\\\le \\epsilon _{t},$ where $U^{UCBid1+}: b \\mapsto (\\hat{V}_{t} + \\epsilon _{t} - b) \\hat{F}_{t}(b)$ .",
"Hence, $\\Vert U^{UCBid1+}_{t} - U\\Vert _{\\infty } \\le 3 \\epsilon _{t}.$ By Lemma REF , this yields $U(B_t)-U(b^*)\\le 6 \\epsilon _{t}$"
],
[
"Proof of the Theorem",
"We use the following decomposition $R_T &\\le T \\times \\mathbb {P}(\\mathcal {E}^c) +\\sum _{t=1}^T \\mathbb {E}[S_t \\mathbb {1}\\lbrace \\mathcal {E}\\rbrace ]\\\\&\\le T \\times \\mathbb {P}(\\mathcal {E}^c) + t_0 + \\sum _{t=t_0}^T \\mathbb {E}[S_t \\mathbb {1}\\lbrace \\mathcal {E}\\rbrace ]$ Thanks to Lemma REF , and when $t>t_0$ , $F(B_t)\\ge \\frac{1}{2\\alpha }F(b^*)$ .",
"Using this, we get $N_{t}> \\frac{1}{4\\alpha }F(b^*)(t-t_0) , \\forall t>t_0$ with high probability.",
"Thanks to Lemma REF , $\\mathbb {E}[S_t \\mathbb {1}\\lbrace \\mathcal {E}\\rbrace ] &\\le \\exp \\left(-\\frac{2((\\frac{1}{2\\alpha }F(b^*))^2}{4}(t-t_0)\\right) + \\mathbb {E}\\left[S_t \\mathbb {1}\\lbrace N_{t} \\ge \\frac{1}{4\\alpha }F(b^*) (t- t_0) \\rbrace \\right]\\\\&\\le \\exp \\left(-\\frac{2((\\frac{1}{2\\alpha }F(b^*))^2}{4}(t-t_0)\\right) + \\mathbb {E}\\left[6 \\sqrt{\\frac{ 4 \\alpha \\gamma \\log T}{ F(b^*) (t- t_0)}}\\mathbb {1}\\lbrace N_{t} \\ge \\frac{1}{4\\alpha }F(b^*) (t- t_0) \\rbrace \\right];$ By summing, $\\sum _{t=t_0}^T\\mathbb {E}[S_{t} \\mathbb {1}\\lbrace \\mathcal {E}\\rbrace ]&\\le \\sum _{t=t_0}^T \\exp \\left(-\\frac{2((\\frac{1}{2\\alpha }F(b^*))^2}{4}(t-t_0)\\right) + \\sum _{t=t_0}^T 6 \\sqrt{\\frac{4 \\alpha \\gamma \\log T}{ F(b^*) (t- t_0)}}\\\\&\\le \\frac{1}{1- \\exp (-\\frac{2(\\frac{1}{2\\alpha }F(b^*))^2}{4})}+ 6 \\sqrt{\\frac{4 \\alpha \\gamma }{ F(b^*)}} \\sqrt{\\log T} \\sqrt{T} \\\\&\\le \\frac{4}{\\frac{1}{2\\alpha }F(b^*)} + 6 \\sqrt{\\frac{4 \\alpha \\gamma }{ F(b^*)}} \\sqrt{\\log T} (\\sqrt{T}) ,$ where the last inequality comes from $1-\\exp (-u)\\ge 2/u$ , for any positive $u$ .",
"Using the decomposition of the regret yields $&R_T\\le t_0 + T \\mathbb {P}(\\mathcal {E}^C) +\\frac{4}{\\frac{1}{2\\alpha }F(b^*)}+ 6 \\sqrt{\\frac{4 \\alpha }{ F(b^*)}} \\sqrt{\\log T } \\sqrt{T}\\\\&\\le 4+ 8\\frac{ \\gamma (\\alpha +1)^2}{\\alpha (F(b^*))^2} \\log \\left( 4\\frac{ \\gamma (\\alpha +1)^2}{\\alpha (F(b^*))^2}\\right) + 4 e(\\gamma -1) \\log T (T)^{2-\\gamma } +\\frac{8 \\alpha }{ F(b^*)} + 12 \\sqrt{\\frac{\\alpha \\gamma }{ F(b^*)}} \\sqrt{\\log T} \\sqrt{T} \\\\&\\le 4+ \\frac{8\\alpha }{ F(b^*)}+ 8\\frac{ \\gamma (\\alpha +1)^2}{\\alpha (F(b^*))^2} \\log \\left( 4\\frac{ \\gamma (\\alpha +1)^2}{\\alpha (F(b^*))^2}\\right) + 4 e(\\gamma -1) \\log T + 12 \\sqrt{\\frac{\\alpha \\gamma }{F(b^*)}} \\sqrt{\\log T} (\\sqrt{T}),$ which concludes the proof."
],
[
"Proof of an Intermediary Regret Rate under Assumptions ",
"In this section, we prove an easier version of Theorem REF .",
"We will use lemmas of the previous subsection for this version as well as for the more complex version.",
"In particular we have already proven that $\\mathcal {E} \\cap \\lbrace N_t\\ge \\frac{1}{4 \\alpha }F(b^*)t\\rbrace $ , occurs with high probability.",
"Under Assumptions REF and REF and on this event, we prove the following result.",
"Lemma 24 Under Assumptions REF and REF and if $t>\\max (t_0, t_1)$ , $\\Vert F- \\hat{F}_t\\Vert _{\\infty } \\le \\epsilon _t^+$ and $|v- \\hat{V}_t| \\le \\epsilon _t^+$ , $ |U(b^*) - U(B_t)| \\le 6 \\epsilon _t^+$ $|b^* - B_t| \\le \\Delta $ , $|b^* - B_t| \\le 1/\\sqrt{c_U} \\sqrt{ 6 \\epsilon _t^+}$ .",
"$ |U(b^*) - U(B_t)| \\le C_U(b^*- B_t)^2$ on $\\mathcal {E} \\cap \\lbrace N_t\\ge \\frac{1}{4 \\alpha }F(b^*)t\\rbrace $ , where ${\\left\\lbrace \\begin{array}{ll}t_0 = \\min \\left(3,1+ 8\\frac{ \\gamma (\\alpha +1)^2}{\\alpha (F(b^*))^2} \\log \\left( 4\\frac{ \\gamma (\\alpha +1)^2}{\\alpha (F(b^*))^2}\\right)\\right)\\\\t_1 = 2 \\sqrt{C_u} \\Delta ^{1/4} \\frac{\\gamma \\alpha }{F(b^*)} \\log T,\\\\\\epsilon _t^+ = \\sqrt{\\frac{2 \\alpha \\gamma \\log t}{ F(b^*)t}},\\\\c_U = c_f \\frac{1}{4} U(b^*),\\\\C_U = \\frac{C_f}{c_f} \\lambda .\\end{array}\\right.",
"}$ On the event $\\mathcal {E} \\cap \\lbrace N_t\\ge \\frac{1}{4 \\alpha }F(b^*)t\\rbrace $ , $\\Vert F- \\hat{F}_t\\Vert _{\\infty } \\le \\epsilon _t^+$ and $|v- \\hat{V}_t| \\le \\epsilon _t^+$ where $\\epsilon _t^+ = \\sqrt{\\frac{2 \\alpha \\gamma \\log t}{ F(b^*)t}}$ from Lemmas, REF ,REF REF and $ |U(b^*) - U(B_t)| \\le 6 \\epsilon _t \\le 6 \\epsilon _t^+$ from Lemmas REF and REF .",
"Under Assumptions REF and REF , we prove that after $t_1$ , we have $|B_t - b^*| \\le \\Delta $ on $\\mathcal {E} \\cap \\lbrace N_t\\ge \\frac{1}{4 \\alpha }F(b^*)t\\rbrace $ , so that we will be able to use the boundedness of the density after this time step.",
"When $F$ satisfies assumption REF , $U$ is unimodal, as shown in the proof of Lemma REF and so if $U(b^*) - U(b) \\le \\min (U(b^*)- U(b^*- \\Delta ),U(b^*)- U(b^*+ \\Delta )) ,$ then $b \\in [b^*- \\Delta , b^* + \\Delta ].$ It follows that if $6 \\epsilon _t^+ \\le \\min (U(b^*)- U(b^* - \\Delta ),U(b^*)- U(b^* + \\Delta )) $ and therefore $6 \\epsilon _t^+ \\le C_u \\Delta $ where $C_u := \\lambda C_f/ c_f$ (see Lemma 7), then $|b^* - B_t| \\le \\Delta $ on $\\mathcal {E} \\cap \\lbrace N_t\\ge \\frac{1}{4 \\alpha }F(b^*)t\\rbrace $ .",
"Then, for all $t> 2 \\sqrt{c_u} \\Delta ^{1/4} \\frac{\\gamma \\alpha }{F(b^*)} \\log T := t_1$ , we have $|B_t - b^*| \\le \\Delta $ on $\\mathcal {E} \\cap \\lbrace N_t\\ge \\frac{1}{4 \\alpha }F(b^*)t\\rbrace $ .",
"Under Assumption REF , for any $q \\in [0,1]$ , $W_{v,F}(q^*_{v,F}) - W_{v,F}(q) \\ge \\frac{1}{4}(q^*_{v,F} - q)^2 W_{v,F}(q^*_{v,F}).$ We have $U = W \\circ F$ , so that if $t>t_1$ , then $B_t \\in [b^*- \\Delta , b^* +\\Delta ]$ and $U(b^*) -U(B_t) \\ge c_f \\frac{1}{4}(b^* -B_t)^2 U(b*) := c_U (b^* -B_t)^2$ .",
"In this case, we can also prove that $|b^* - B_t| \\le 1/\\sqrt{c_U} \\sqrt{ 6\\epsilon _t^+}$ , under $\\mathcal {E} \\cap \\lbrace N_t\\ge \\frac{1}{4 \\alpha }F(b^*)t\\rbrace $ .",
"Proposition 1 Under Assumptions REF and REF and if $t>\\max (t_0, t_1)$ , $\\delta _t< \\Delta $ , $|B_t - b^*|\\le \\delta _t$ , and $\\epsilon _t^+\\le M \\delta _t$ , Then $|B_t - b^*|^2 \\le \\frac{6}{c_U} \\sqrt{\\frac{ C_f \\delta _t \\log \\left( \\frac{M e^2 t\\sqrt{2t}}{2 c_f \\eta ^2}\\right)}{t}} + \\frac{2\\log (\\frac{M t\\sqrt{2t}}{2c_f \\eta ^2 })}{c_U t} + \\frac{2}{c_U} (2 C_f +1) \\delta _t \\sqrt{\\frac{2 \\alpha \\gamma \\log T}{F(b^*)t}},$ with probability $1- \\eta $ on $\\mathcal {E} \\cap \\lbrace N_t\\ge \\frac{1}{4 \\alpha }F(b^*)t\\rbrace $ .",
"It is clear from Lemma REF that $\\sup _{b^* - \\delta _t \\le b \\le b^* + \\delta _t}|\\hat{F}_t(b) - F(b) - (\\hat{F}_t(b^*)- F(b^*))| \\le 2 \\sqrt{\\frac{2 C_f \\delta _t \\log \\left( \\frac{e \\sqrt{t}}{\\sqrt{2c_f \\delta _t}\\eta }\\right)}{t}} + 2 \\frac{\\log (\\frac{t}{2c_f \\delta _t \\eta ^2 })}{6 t} := \\beta _t,$ with probability $1- \\eta $ .",
"We can also decompose $U(b)-U^{UCBid1+}_t(b) - (U^{UCBid1+}_t(b^*)- U(b^*))$ into $&U(b)-U^{UCBid1+}_t(b) - (U^{UCBid1+}_t(b^*)- U(b^*))\\\\&=(v-b)F(b) - (\\hat{V_t}+ \\epsilon _t-b)\\hat{F}_t(b) - \\left((v-b^*)F(b^*) - (\\hat{V_t}+ \\epsilon _t-b^*)\\hat{F^*}_t(b)\\right)\\\\&= (v-b)F(b) - (v-b)\\hat{F}_t(b) - \\left((v-b^*)F(b^*) - (v-b^*)\\hat{F^*}_t(b)\\right) - (\\hat{V_t}+ \\epsilon _t - v) \\left(\\hat{F}_t(b) - \\hat{F}_t(b^*)\\right)\\\\&= (v-b^*)\\left(F(b) - \\hat{F}_t(b) - \\left(F(b^*) -\\hat{F^*}_t(b)\\right)\\right) - (\\hat{V_t}+ \\epsilon _t - v) \\left(\\hat{F}_t(b) - \\hat{F}_t(b^*)\\right)\\\\&~~ + (b^* - b) (\\hat{F}(b) - \\hat{F}_t(b))$ which in turn proves that $|U(b)-U^{UCBid1+}_t(b) - (U^{UCBid1+}_t(b^*)- U(b^*))| &\\le \\beta _t+ 2 \\epsilon _t |\\hat{F}_t(b) - \\hat{F}_t(b^*)| + \\delta _t |\\hat{F}_t(b) - \\hat{F}_t(b)| \\\\&\\le \\beta _t+ 2 \\epsilon _t^+ (C_f \\delta _t + \\beta _t) + \\delta _t \\epsilon _t^+ \\\\& \\le \\beta _t + 2 \\epsilon _t^+ \\beta _t + (2 C_f +1) \\delta _t \\epsilon _t^+\\\\& \\le 3\\beta _t + (2 C_f +1) \\delta _t \\epsilon _t^+ := \\gamma _t,$ for all $b$ in $[b^* - \\delta _t, b^* + \\delta _t]$ .",
"Now, we know that $U(b^*)- U(b)$ is lower bounded by $c_U(b^*-b)^2$ , on this interval and $\\Vert U^{UCBid1+}_t(b)- U(b) + U^{UCBid1+}_t(b^*)- U(b^*) \\Vert _{\\infty } \\le \\gamma _t $ on $[b^* - \\delta _t, b^* + \\delta _t]$ .",
"We call $G$ the shifted version of $U$ defined by $G(b) = U(b) + U^{UCBid1+}_t(b^*)- U(b^*)$ .",
"Its argmax is $b^*$ and $G(b^*)- G(b)$ is lower bounded by $c_U(b^*-b)^2$ then $c_U(B_t- b^*)^2\\le G(b^*)- G(B_t) \\le 2 \\gamma _t $ (see Lemma REF ).",
"Then , by definition of $\\gamma _t$ and $\\beta _t$ : $ (B_t- b^*)^2 &\\le \\frac{6}{c_U} \\sqrt{\\frac{ C_f \\delta _t \\log \\left( \\frac{e^2 t}{2c_f \\delta _t \\eta ^2}\\right)}{t}} + \\frac{2\\log (\\frac{t}{2c_f \\delta _t \\eta ^2 })}{ c_U t} + \\frac{2}{c_U}(2 C_f +1) \\delta _t \\epsilon _t^+\\\\&\\le \\frac{6}{c_U} \\sqrt{\\frac{ C_f \\delta _t \\log \\left(M \\frac{e^2 t}{2c_f \\epsilon _t^+\\eta ^2}\\right)}{t}} + \\frac{2\\log (\\frac{Mt}{2c_f \\epsilon _t^+ \\eta ^2 })}{ c_U t} +\\frac{2}{c_U} (2 C_f +1) \\delta _t \\epsilon _t^+\\\\&\\le \\frac{6}{c_U} \\sqrt{\\frac{ C_f \\delta _t \\log \\left( \\frac{M e^2 t\\sqrt{t}}{2 c_f \\eta ^2}\\right)}{t}} + \\frac{2\\log (\\frac{M t\\sqrt{t}}{2c_f \\eta ^2 })}{ c_U t} + \\frac{2}{c_U} (2 C_f +1) \\delta _t \\sqrt{\\frac{2 \\alpha \\gamma \\log T}{F(b^*)t}}.$ where the last inequality stems from that fact that $1/\\epsilon _t^+ = \\sqrt{\\frac{F(b^*) t}{2\\alpha \\gamma \\log t}}\\le \\sqrt{t}$ since $\\alpha ,\\gamma \\ge 1$ .",
"Theorem 8 Under Assumptions REF and REF , $R_T \\le O(T^{3/8} \\log T).$ From Lemma REF , we have that $|b^* - B_t| \\le 1/\\sqrt{c_U} \\sqrt{ 6 \\epsilon _t^+}$ , on $\\mathcal {E}\\cap \\lbrace N_t\\ge \\frac{1}{4 \\alpha }F(b^*)t\\rbrace \\rbrace $ .",
"Therefore, we can apply Proposition REF with $\\delta _t = \\frac{1}{\\sqrt{c_U}} \\sqrt{ 6\\epsilon _t^+}$ with $M= \\frac{\\sqrt{c_U}}{\\sqrt{6}}$ , and $\\eta = \\frac{1}{t}$ .",
"We use the general fact that $\\log (At^{\\alpha })\\le 2 \\alpha \\log t$ as soon as $t^{\\alpha }>A$ , for all $A,a>0$ , to derive the following two inequalities : $\\forall t\\ge \\left(\\frac{M e^2}{2c_f}\\right)^{\\frac{1}{4}}$ , $ \\frac{6}{c_U} \\sqrt{\\frac{ C_f \\delta _t \\log \\left( \\frac{M e^2 t\\sqrt{t}}{2 c_f \\eta ^2}\\right)}{t}} \\le \\frac{6 \\sqrt{8} \\sqrt{C_f}}{c_U^{\\frac{5}{4}}} \\sqrt{\\frac{\\delta _t\\log t}{t}}= \\frac{24 (72\\alpha \\gamma )^{\\frac{1}{8}} \\sqrt{C_f}}{c_U^{\\frac{5}{4}} {F(b^*)}^{\\frac{1}{8}}} \\sqrt{\\frac{\\log ^2 t }{t^{\\frac{5}{4}}}}.$ $\\forall t\\ge (\\frac{M}{2 c_f})^{\\frac{1}{4}}$ , $ \\frac{2\\log (\\frac{Mt^2 t\\sqrt{t}}{2c_f })}{ c_U t}\\le \\frac{16}{c_U} \\frac{\\log t}{t} .$ We also have, for all t, $\\frac{2}{c_U} (2 C_f +1) \\delta _t \\sqrt{\\frac{ 2 \\gamma \\alpha \\log T}{F(b^*)t}} &\\le \\frac{2}{c_U}(2 C_f + 1)\\sqrt{\\frac{2\\gamma \\alpha }{F(b^*)}} \\delta _t \\sqrt{\\frac{\\log t}{t}}\\\\& = \\frac{2(72\\alpha \\gamma )^{\\frac{1}{4}} }{c_U^{\\frac{3}{2}} {F(b^*)}^{\\frac{1}{4}}}(2 C_f + 1)\\sqrt{\\frac{2\\gamma \\alpha }{F(b^*)}} \\frac{(\\log t)^{\\frac{1}{4}}}{t^\\frac{1}{4}} \\sqrt{\\frac{\\log t}{t}}$ Therefore $|B_t - b^*|^2 \\le \\left( \\frac{24 (72\\alpha \\gamma )^{\\frac{1}{8}} \\sqrt{C_f}}{c_U^{\\frac{5}{4}} {F(b^*)^{\\frac{1}{8}}}}+ \\frac{16}{c_U} +\\frac{2(72\\alpha \\gamma )^{\\frac{1}{4}} }{c_U^{\\frac{3}{2}} {F(b^*)}^{\\frac{1}{4}}}(2 C_f + 1)\\sqrt{\\frac{2\\gamma \\alpha }{F(b^*)^{\\frac{1}{8}}}} \\right)\\frac{\\log t }{t^{\\frac{5}{8}}} $ with probability $1- \\frac{1}{t}$ , for $t \\ge \\max (\\left(\\frac{M e^2}{2c_f}\\right)^{\\frac{1}{4}}, (\\frac{M}{2c_f})^{\\frac{1}{4}}):= t_2$ on $\\mathcal {E} \\cap \\lbrace N_t\\ge \\frac{1}{4 \\alpha }F(b^*)t\\rbrace $ .",
"On this event, $U(b^*) - U(B_t) \\le C_U (b^* - B_t)^2$ We use the following decomposition $R_T &\\le T \\times \\mathbb {P}(\\mathcal {E}^c) +\\sum _{t=1}^T \\mathbb {E}[S_t \\mathbb {1}\\lbrace \\mathcal {E}\\rbrace ]\\\\&\\le T \\times \\mathbb {P}(\\mathcal {E}^c) + \\max (t_0,t_1, t_2) + \\sum _{t=\\max (t_0,t_1, t_2)}^T \\mathbb {E}[S_t \\mathbb {1}\\lbrace \\mathcal {E}\\rbrace ]\\\\&\\le T \\times \\mathbb {P}(\\mathcal {E}^c) + \\max (t_0,t_1, t_2) + \\sum _{t=\\max (t_0,t_1,t_2 )}^T \\mathbb {P}(\\mathcal {E} \\cap \\lbrace N_t< \\frac{1}{4 \\alpha }F(b^*)t\\rbrace ) \\\\&~~~ +\\sum _{t=\\max (t_0,t_1, t_2 )}^T \\mathbb {E}[S_t \\mathbb {1}\\lbrace \\mathcal {E}\\cap \\lbrace N_t\\ge \\frac{1}{4 \\alpha }F(b^*)t\\rbrace \\rbrace ]\\\\&\\le T \\times \\mathbb {P}(\\mathcal {E}^c) + \\max (t_0,t_1,t_2 ) + \\sum _{t=\\max (t_0,t_1, t_2 )}^T \\mathbb {P}(\\mathcal {E} \\cap \\lbrace N_t< \\frac{1}{4 \\alpha }F(b^*)t\\rbrace ) \\\\&~~~ + \\sum _{t=\\max (t_0,t_1, t_2 )}^T C_U \\mathbb {E}[(b^* - B_t)^2]\\\\&\\le T \\times \\mathbb {P}(\\mathcal {E}^c) + \\max (t_0,t_1, t_2 ) + \\sum _{t=\\max (t_0,t_1,t_2 )}^T \\mathbb {P}(\\mathcal {E} \\cap \\lbrace N_t< \\frac{1}{4 \\alpha }F(b^*)t\\rbrace ) \\\\&~~~ + \\sum _{t=\\max (t_0,t_1, t_2 )}^T C_0 \\frac{\\log t }{t^{\\frac{5}{8}}} + \\sum _{t=\\max (t_0,t_1, t_2 )}^T \\frac{1}{t}\\\\&\\le T \\times \\mathbb {P}(\\mathcal {E}^c) + \\max (t_0,t_1, t_2 ) + \\sum _{t=\\max (t_0,t_1, t_2)}^T \\mathbb {P}(\\mathcal {E} \\cap \\lbrace N_t< \\frac{1}{4 \\alpha }F(b^*)t\\rbrace ) \\\\&~~~ +C_0 \\frac{8}{3} T^{\\frac{3}{8}} \\log T + \\log T \\\\&\\le \\log T+ 4 e(\\gamma -1) \\log T (T)^{2-\\gamma } + \\max (t_0,t_1, t_2 )+ \\frac{8 \\alpha }{ F(b^*)} + \\frac{8}{3}C_0 T^{\\frac{3}{8}} \\log T.$ where ${\\left\\lbrace \\begin{array}{ll}t_0 = \\min \\left(3,1+ 8\\frac{ \\gamma (\\alpha +1)^2}{\\alpha (F(b^*))^2} \\log \\left( 4\\frac{ \\gamma (\\alpha +1)^2}{\\alpha (F(b^*))^2}\\right)\\right)\\\\t_1 = 2 \\sqrt{C_u} \\Delta ^{1/4} \\frac{\\gamma \\alpha }{F(b^*)} \\log T,\\\\t_2 = \\max (\\left(\\frac{\\sqrt{c_U} e^2}{2c_f\\sqrt{6}}\\right)^{\\frac{1}{4}}, (\\frac{\\sqrt{c_U}}{2c_f\\sqrt{6}})^{\\frac{1}{4}}) = (\\frac{\\sqrt{c_U} e^2}{2c_f\\sqrt{6}})^{\\frac{1}{4}},\\\\C_0 =\\left( \\frac{24 (72\\alpha \\gamma )^{\\frac{1}{8}} \\sqrt{C_f}}{c_U^{\\frac{5}{4}} {F(b^*)^{\\frac{1}{8}}}}+ \\frac{16}{c_U} +\\frac{2(72\\alpha \\gamma )^{\\frac{1}{4}} }{c_U^{\\frac{3}{2}} {F(b^*)}^{\\frac{1}{4}}}(2 C_f + 1)\\sqrt{\\frac{2\\gamma \\alpha }{F(b^*)}} \\right)C_U.\\end{array}\\right.",
"}$ Therefore $R_T \\le \\frac{8}{3}C_0 T^{\\frac{3}{8}}\\log T + o(T^{\\frac{3}{8}} \\log T).$"
],
[
"Proof of Theorem ",
"Theorem REF is proved by applying Proposition REF once.",
"By iterating the argument, we can actually achieve a regret of the order of $T^a$ , for any $a>\\frac{1}{3}$ .",
"The proof involves an induction argument.",
"The following lemma is the main element of the proof of the induction.",
"Lemma 25 Assume that $t$ and $F$ satisfy the assumptions of Proposition REF .",
"Assume that $|B_t - b^*|$ is bounded by $\\delta _t^{(k)}$ such that $\\delta _t^{(k)}= \\min (1,C^{(k)}\\log (t) t^{- u_k})$ with probability $1-\\eta ^{(k)}$ , and $u_k <2/3$ , $C{(k)} \\ge 1$ .",
"Then $|B_t - b^*|$ is bounded by $\\delta _t^{(k+1)}$ such that $\\delta _t^{(k+1)}=\\min (1, C^{(k+1)}\\log (t) t^{-\\frac{1}{4}(1+u_{k})})$ with probability $1-\\eta ^{(k)}- \\frac{1}{Kt}$ , where $C^{(k+1)} = C \\left(C^{(k)}\\right)^{\\frac{1}{4}}$ and where $C=\\max \\left(1, \\frac{12 \\sqrt{2 C_f}}{c_U}+ \\frac{16}{c_U}+\\frac{2}{c_U}(2 C_f + 1)\\sqrt{\\frac{2\\gamma \\alpha }{F(b^*}}\\right).$ We use Proposition REF , and the fact that $\\epsilon _t^+ \\le \\sqrt{2 \\alpha \\gamma / F(b^*) }\\frac{\\log t}{t^{-u_k}} \\le \\sqrt{2 \\alpha \\gamma / F(b^*) }\\delta ^{(k)}_t$ to prove that $|B_t - b^*|^2\\le \\frac{6}{c_U} \\sqrt{\\frac{ C_f \\delta _t^{(k)} \\log \\left( \\frac{M e^2 t\\sqrt{2t} K^2 t^2}{2 c_f }\\right)}{t}}+ \\frac{2\\log (\\frac{MK^2 t^2 t\\sqrt{2t}}{2c_f })}{ c_U t} + \\frac{2}{c_U} (2 C_f +1) \\delta _t^{(k)} \\sqrt{\\frac{2\\alpha \\gamma \\log t}{F(b^*)t}} ,$ with probability $(1-\\eta ^{(k)})(1- \\frac{1}{Kt})$ and with $M = \\sqrt{2 \\alpha \\gamma / F(b^*) }$ We use the general fact that $\\log (At^{\\alpha })\\le 2 \\alpha \\log t$ as soon as $t^{\\alpha }>A$ , for all $A,a>0$ , to derive the following two inequalities : $\\forall t\\ge \\left(\\frac{M e^2 K^2}{2c_f}\\right)^{\\frac{1}{4}}$ , $ \\frac{6}{c_U} \\sqrt{\\frac{ C_f \\delta _t^{(k)} \\log \\left( \\frac{M e^2 t\\sqrt{2t} K^2 t^2}{2 c_f }\\right)}{t}} \\le \\frac{6 \\sqrt{8 C_f}}{c_U} \\sqrt{\\frac{\\delta _t^{(k)}\\log t}{t}}:= C_1\\sqrt{\\frac{\\delta _t^{(k)}\\log t}{t}}:= C_1 \\beta _{1,t}.$ $\\forall t\\ge (\\frac{MK^2}{2c_f})^{\\frac{1}{4}}$ , $ \\frac{2\\log (\\frac{MK^2 t^2 t\\sqrt{2t}}{2c_f })}{ c_U t}\\le \\frac{16}{ c_U} \\frac{\\log t}{t} := C_2 \\frac{\\log t}{t} := C_2 \\beta _{2,t} .$ We also have, for all t, $ \\frac{2}{c_U} (2 C_f +1) \\delta _t^{(k)} \\sqrt{\\frac{2\\alpha \\gamma \\log T}{F(b^*)t}} \\le \\frac{2}{c_U}(2 C_f + 1)\\sqrt{\\frac{2\\gamma \\alpha }{F(b^*}} \\delta _t^{(k)}\\sqrt{\\frac{\\log t}{t}} := C_3 \\delta _t^{(k)}\\sqrt{\\frac{\\log t}{t}}:= C_3 \\beta _{3,t} $ We can derive the following bounds $\\beta _{3,t} \\le \\beta _{1,t}$ since $\\delta _t^{(k)}\\le 1$ .",
"$\\beta _{2,t} \\le \\beta _{1,t}$ since $\\delta _t^{(k)}= \\min (1,C^{(k)}\\log (t) t^{- u_k})\\ge \\frac{\\log t}{t}$ .",
"Hence $|B_t - b^*|^2\\le (C_1+ C_2+C_3) \\beta _{1,t}= (C_1+ C_2+C_3) \\sqrt{\\frac{\\delta _t^{(k)}\\log t}{t}},$ with probability $1-\\eta ^{(k)} \\frac{1}{Kt}.$ This yields $|B_t - b^*|&\\le \\sqrt{(C_1+ C_2+C_3)} \\left(\\frac{\\delta _t^{(k)}\\log t}{t}\\right)^{\\frac{1}{4}}\\\\&\\le \\sqrt{(C_1+ C_2+C_3)} \\left(\\frac{\\min (1,C^{(k)}\\log ^2(t) t^{- u_k})}{t}\\right)^{\\frac{1}{4}}\\\\&\\le \\sqrt{(C_1+ C_2+C_3)} (C^{(k)})^{1/4} t^{-\\frac{1}{4}(1+u_k)} \\log t\\\\&\\le C \\left(C^{(k)}\\right)^{1/4} t^{-\\frac{1}{4}(1+u_k)} \\log t,$ Proposition 2 Assume that $t$ and $F$ satisfy the assumptions of Proposition REF .",
"If $t>t_3= max \\left((\\frac{\\sqrt{2 \\alpha \\gamma / F(b^*) }K^2}{2c_f})^{\\frac{1}{4}} ,(\\frac{\\sqrt{2 \\alpha \\gamma / F(b^*) } e^2 K^2}{2c_f})^{\\frac{1}{4}} \\right)$ , then on $\\mathcal {E} \\cap \\lbrace N_t\\ge \\frac{1}{4 \\alpha }F(b^*)t\\rbrace $ , $|B_t - b^*|\\le C^{(0)} C^{\\frac{1}{3}} \\log (t) t^{-\\frac{1}{3}+ \\frac{1}{3 \\times 4^K} + \\frac{1}{4^{K+1}}},$ with probability $1-\\frac{1}{t}$ where $C=\\max \\left(1, \\frac{12 \\sqrt{2 C_f}}{c_U}+ \\frac{16}{c_U}+\\frac{2}{c_U}(2 C_f + 1)\\sqrt{\\frac{2 \\gamma \\alpha }{F(b^*)}}\\right)$ , and $C^{(0)} =\\max \\left(1,\\sqrt{\\frac{1}{c_U}}\\left(\\frac{72\\gamma \\alpha }{F(b^*) }\\right)^{\\frac{1}{4}} \\right)$ The proposition follows from using an induction argument based on Lemma REF .",
"We can initiate an induction argument with $\\delta _t^{(0)}$ such that $\\delta _t^{(0)}= \\min (1,C^{(0)}\\log (t) t^{- u_k}),$ writing $u_0= \\frac{1}{4}$ and $C^{(0)} = \\max (1,\\sqrt{\\frac{1}{c_U}}\\left(\\frac{72 \\alpha \\gamma }{ F(b^*)}\\right)^{1/4} )$ , thanks to Lemma REF .",
"The fact that $u_k$ and $C^{(k)}$ as defined as in Lemma REF satisfy $u_{k+1} = \\frac{1}{4}(1+u_k)$ which yields $u_K = \\left(\\frac{1}{4}\\right)^K u_0 + \\sum _{i=1}^K \\frac{1}{4^i} = \\frac{1}{4}^K u_0 + 4\\frac{1/4- (1/4)^{K+1}}{3}$ and $C^{(k+1)} = C \\times (C^{(k)})^{\\frac{1}{4}}$ which yields $C^{(K)}= \\left(C^{(0)}\\right)^{\\frac{1}{4^K}} C^{\\sum _{i=1}^K \\frac{1}{4^i}} \\le C^{\\frac{1}{3}}, $ suffices to complete the induction.",
"We recall Theorem REF .",
"th:fastrateTheorem REF Under Assumptions REF and REF , $R_T \\le O(T^{1/3 + \\epsilon } ),$ for any $\\epsilon >0$ as long as $\\gamma >2$ .",
"We choose $K$ such that $\\frac{1}{3}+ \\frac{2}{3 \\times 4^K} + \\frac{2}{4^{K+1}} < \\frac{1}{3} + \\epsilon $ .",
"(We can choose $K = \\big \\lceil \\log _4 \\left(\\frac{3}{14} \\frac{1}{\\epsilon }\\right) \\big \\rceil ) +1$ for example).",
"Then, thanks to proposition REF , for all $t> t_3$ , on $\\mathcal {E} \\cap \\lbrace N_t\\ge \\frac{1}{4 \\alpha }F(b^*)t\\rbrace $ , $|B_t - b^*|\\le C^{(0)} C^{\\frac{1}{3}} \\log (t) t^{-\\frac{1}{3}+ \\frac{1}{3 \\times 4^K} + \\frac{1}{4^{K+1}}},$ with probability $1-\\frac{1}{t}$ .",
"We can therefore do the same decomposition as in the proof of Theorem REF .",
"$R_T &\\le T \\times \\mathbb {P}(\\mathcal {E}^c) + \\max (t_0,t_1, t_3) + \\sum _{t=\\max (t_0,t_1,t_3 )}^T \\mathbb {P}(\\mathcal {E} \\cap \\lbrace N_t< \\frac{1}{4 \\alpha }F(b^*)t\\rbrace ) \\\\&~~~ +\\sum _{t=\\max (t_0,t_1, t_3 )}^T \\mathbb {E}[S_t \\mathbb {1}\\lbrace \\mathcal {E}\\cap \\lbrace N_t\\ge \\frac{1}{4 \\alpha }F(b^*)t\\rbrace \\rbrace ]\\\\&\\le T \\times \\mathbb {P}(\\mathcal {E}^c) + \\max (t_0,t_1,t_3 ) + \\sum _{t=\\max (t_0,t_1, t_3 )}^T \\mathbb {P}(\\mathcal {E} \\cap \\lbrace N_t< \\frac{1}{4 \\alpha }F(b^*)t\\rbrace ) \\\\&~~~ + \\sum _{t=\\max (t_0,t_1, t_3 )}^T C_U \\mathbb {E}[(b^* - B_t)^2]\\\\&\\le T \\times \\mathbb {P}(\\mathcal {E}^c) + \\max (t_0,t_1, t_3) + \\sum _{t=\\max (t_0,t_1,t_3 )}^T \\mathbb {P}(\\mathcal {E} \\cap \\lbrace N_t< \\frac{1}{4 \\alpha }F(b^*)t\\rbrace ) \\\\&~~~ + \\sum _{t=\\max (t_0,t_1, t_3 )}^T C^{(0)} C_U C^{\\frac{1}{3}}(\\log t) t^{-\\frac{2}{3}+ \\frac{2}{3 \\times 4^K} + \\frac{2}{4^{K+1}}} \\\\& + \\sum _{t=\\max (t_0,t_1, t_3 )}^T \\frac{1}{t}\\\\&\\le T \\times \\mathbb {P}(\\mathcal {E}^c) + \\max (t_0,t_1, t_3 ) + \\sum _{t=\\max (t_0,t_1, t_3)}^T \\mathbb {P}(\\mathcal {E} \\cap \\lbrace N_t< \\frac{1}{4 \\alpha }F(b^*)t\\rbrace ) \\\\&~~~ + C^{(0)}C_U C^{\\frac{1}{3}} \\frac{1}{\\frac{1}{3}+ \\frac{2}{3 \\times 4^K} + \\frac{2}{4^{K+1}}} T^{\\frac{1}{3}+ \\frac{2}{3 \\times 4^K} + \\frac{2}{4^{K+1}}} \\log T + \\log T \\\\&\\le \\log T + 4 e(\\gamma -1) \\log T (T)^{2-\\gamma } + \\max (t_0,t_1, t_3 )\\\\&~~~+ \\frac{8 \\alpha }{ F(b^*)} + 3 C^{(0)} C_U C^{\\frac{1}{3}} T^{\\frac{1}{3}+ \\epsilon }.$ where ${\\left\\lbrace \\begin{array}{ll}t_0 = \\min \\left(3,1+ 8\\frac{ \\gamma (\\alpha +1)^2}{\\alpha (F(b^*))^2} \\log \\left( 4\\frac{ \\gamma (\\alpha +1)^2}{\\alpha (F(b^*))^2}\\right)\\right)\\\\t_1 = 2 \\sqrt{c_u} \\Delta ^{1/4} \\frac{\\gamma \\alpha }{F(b^*)} \\log T,\\\\t_3= (\\frac{\\sqrt{2 \\alpha \\gamma / F(b^*) }e^2 K^2}{2c_f})^{\\frac{1}{4}} ,\\\\C^{(0)}= \\max \\left(1,\\sqrt{\\frac{1}{c_U}}\\left(\\frac{72\\gamma \\alpha }{F(b^*) }\\right)^{\\frac{1}{4}} \\right)\\\\C=\\max \\left(1, \\frac{12 \\sqrt{ 2C_f}}{c_U}+ \\frac{16}{c_U}+\\frac{2}{c_U}(2 C_f + 1)\\sqrt{\\frac{2\\gamma \\alpha }{F(b^*)}}\\right)\\end{array}\\right.",
"}$ .",
"Hence $R_T \\le O(T^{1/3 + \\epsilon }).$"
],
[
"Further figures",
"We present in Figure REF the histogram of the normalized data used to simulate the real-world experiment.",
"Figure: Bidding Data histogram"
]
] | 2107.01835 |
[
[
"Web-Scale Generic Object Detection at Microsoft Bing"
],
[
"Abstract In this paper, we present Generic Object Detection (GenOD), one of the largest object detection systems deployed to a web-scale general visual search engine that can detect over 900 categories for all Microsoft Bing Visual Search queries in near real-time.",
"It acts as a fundamental visual query understanding service that provides object-centric information and shows gains in multiple production scenarios, improving upon domain-specific models.",
"We discuss the challenges of collecting data, training, deploying and updating such a large-scale object detection model with multiple dependencies.",
"We discuss a data collection pipeline that reduces per-bounding box labeling cost by 81.5% and latency by 61.2% while improving on annotation quality.",
"We show that GenOD can improve weighted average precision by over 20% compared to multiple domain-specific models.",
"We also improve the model update agility by nearly 2 times with the proposed disjoint detector training compared to joint fine-tuning.",
"Finally we demonstrate how GenOD benefits visual search applications by significantly improving object-level search relevance by 54.9% and user engagement by 59.9%."
],
[
"Introduction",
"Visual search, in which an image constitutes the user query, is an emerging modality of search that allows users to provide a new class of queries beyond text-based search.",
"This search solution requires us to intelligently identify visual concepts, retrieve visually and semantically similar images, search for product information, or get inspiration from other images.",
"Understanding and representing the query image is the critical first step of visual search.",
"Many commercial visual search systems represent query images with image-level embeddings.",
"However, this assumes that the query image is focused on a single object with a clean and simple background which often does not hold true in real world scenarios with mobile captured images.",
"Figure: Examples of user interfaces with interactive hotspots detected by the Generic Object Detection (GenOD) in the Bing Visual Search experiences.",
"Left: The experience in Bing Image Details Page on desktop, allows users to click on hotspots to search for related products.",
"Right: The Bing Mobile Camera experience detects objects in real-time to allow the user to quickly choose which object is of interest to them.Object detection has been introduced to several visual search engines [11], [9], [1], [37] to better parse user intent.",
"Given an image, object detection aims to locate and recognize objects from a predefined set of categories.",
"Given their business scenarios, these systems tend to use object detection to display hotspots or remove background noise of objects in scoped segments like shopping.",
"For a web-scale, general-purpose visual search engine like Microsoft Bing, there are numerous search query segments and application scenarios and it is imperative to have a comprehensive and scalable object-based understanding of images at a generic level.",
"In this paper, we present how we built Generic Object Detection (GenOD) as a platform service at Microsoft Bing.",
"Figure REF depicts the overview of GenOD in the Bing Visual Search stack.",
"Starting from domain-specific detection models, object detection in Bing has evolved to the generic object level with a taxonomy of over 900 categories, making it one of the largest deployed object detection models in production.",
"With the ability to detect a wide range of objects, GenOD fuels multiple applications including visual product search, image content-based triggering and post-processing, fine-grained entity recognition, fine-grained attribute classification, and real-time object detection in camera search experiences https://bing.com/camera.",
"Figure REF showcases how users interact with detected object hotspots in Bing Images Details Page and Bing Mobile camera based search experience.",
"The challenges of building such a versatile system can be broken down into three main aspects: Data collection and processing for a large vocabulary Collecting object detection annotations for hundreds of categories at the scale required for deep models is much slower and more costly than getting image class labels [32] and can be prohibitively expensive when using expert judges.",
"The task is also fairly complex for crowd platforms, especially because it quickly becomes infeasible for judges to keep track of hundreds of categories.",
"Even if one can leverage recently released large-scale object detection datasets such as OpenImages [16], LVIS [6] and VisualGenome [14] with hundreds to thousands of categories, determining the best way to combine these resources remains an open issue.",
"Compared to conventional object detection models trained on a single dataset with a small vocabulary [22], [2], training a unified large-scale detection model by combining several diverse datasets faces new challenges including: (1) long-tailed category distribution: this is especially the case in natural images when the number of categories grows 10 times larger.",
"The rare classes often perform poorly with few training samples.",
"(2) hierarchical labels: as the taxonomy grows, each object instance naturally has multiple valid labels as part of a hierarchy.",
"For example, an apple can be labeled as \"Apple\" and \"Fruit\" because both categories are in the taxonomy.",
"This will introduce missing and noisy label issues because not all object instances can be exhaustively annotated either by humans or oracle models, so it poses a serious barrier in both model training and evaluation.",
"(3) imbalance between datasets: Some of the datasets are much larger than others in size with specific distributions, which would be likely dominate model training and cause poorer generalized performance.",
"Agility of model development Continuously iterating machine learning models deployed in online systems remains difficult due to: (1) heavy model training: conventional way of model training is an all-or-nothing change, reducing update agility.",
"(2) production non-regression requirements: when a new model is deployed to production, it is important not to regress in performance for any downstream task dependent on the model.",
"However, with the increasing number of categories and dependencies, improving the model for a certain task or subset of categories may lead to a decline in the performance of others, which would block model deployment.",
"Therefore it is imperative to have a novel architecture design to meet such strict requirements.",
"Latency-accuracy tradeoff The visual search stack in Microsoft Bing has strict near real-time inference requirements, especially for applications like Bing mobile camera.",
"Since GenOD is required to run for all requests, latency of the model is a key criterion in model training and selection.",
"The key contribution of this paper is a detailed description of how we overcome the challenges mentioned above to design and deploy a large-scale generic object detection system in an industry setting that is adaptable to rapidly changing business needs.",
"Specifically, our contributions are as follows: We discuss the design of a low-cost, high-throughput data collection pipeline that can easily scale to new categories.",
"We discuss how we handle the imbalance in category and dataset distributions while combining multiple datasets in training a large-scale unified generic object detection model.",
"We evaluate on the various academic and internal benchmarks to demonstrate the efficacy of the model with good speed-accuracy trade-offs and show that a generic large-scale model is able to beat domain-specific models.",
"We propose an architecture design of disjoint detectors on a shared backbone pretrained for general purpose object detection, in order to tackle the challenge of agile model updates in a production context.",
"We describe how we serve GenOD at web scale with low latency, and demonstrate its impact as a fundamental service in Bing Visual Search to improve user engagement and relevance for a wide range of deployed applications in Microsoft through offline and online A/B tests.",
"The rest of the paper is organized as follows: We first review related literature in Section then introduce our data collection pipeline in Section .",
"Our model design, training and deployment is described in Section and we include corresponding experiments in Section .",
"Finally we demonstrate the applications of GenOD in Bing Visual Search in Section .",
"Major companies [5], [15], [11], [1], [35], [9], [36], [31], [37] have been developing visual search systems to satisfy an increasing demand for content-based retrieval.",
"Facebook [1] and Alibaba [37] perform class-agnostic object localization to remove background noise and retrieve images at object level to improve product search relevance.",
"Pinterest [11], [36], [31] displays hotspots on objects in a few shopping segments including fashion, home decor and vehicles.",
"[9] leveraged object detection to improve engagement and relevance in the web-scale responsive visual search system in Microsoft Bing.",
"However, most of these systems target a limited set of shopping domains and only cover a small set of categories.",
"Google Lens [5] was one of the first attempts to apply generic object detection for visual search, but a detailed analysis of their system has not been published yet.",
"To the best of our knowledge, this paper is the first work to comprehensively discuss the challenges and solutions for developing a web-scale generic object detection service in a production system."
],
[
"Large scale generic object detection",
"With the advance of deep neural networks (DNN), the research community is moving towards the challenging goal of building a generic object detection system that can detect a broad or even open-ended range of objects like humans [23].",
"Numerous object detection architectures have emerged during the last decade.",
"Two-stage detectors [29], [7] were first proposed to apply DNNs end-to-end to a region proposal network and a detection stage; one-stage detectors [28], [21], [33] and anchor-free approaches [18], [34] were proposed later with attempts to predict objects without region proposals and anchor boxes, respectively, to further improve speed-accuracy trade-offs [10].",
"For a more comprehensive survey in the area please refer to [23].",
"However, most of these architectures in academic settings seldom consider the agility to add or update categories without regressing others, making them less adaptive in an industry product setting with rapidly changing business needs.",
"Prevalent works on general purpose object detection are mostly performed on a predefined small set of categories with relatively adequate and balanced training samples (e.g.100$\\sim $ 1000+) for each category [22], [2].",
"Generic object detection with large vocabulary, in contrast, poses new challenges including long-tail distribution for data collection and model training.",
"Some large-scale datasets [16], [6], [30] have been collected to facilitate further research in this scenario, in which challenges and solutions to data collection have been discussed.",
"Recent studies to address the challenges of long-tail distribution include data distribution re-balancing [25], [4], class-balanced losses [21], decoupling representation and classifier learning [13], [19].",
"This paper mainly experiments with the data distribution re-balancing approach as a simple but robust baseline, but other directions of research in long-tail object detection could be applied in the future."
],
[
"Data collection",
"In this section, we describe the methodology used to collect data at scale to power the GenOD service.",
"Given the large nature of the vocabulary, it is imperative from a production standpoint to have a robust pipeline that is high quality, cost-efficient, high throughput and agile to taxonomy changes and business needs.",
"Previous iterations [9] which relied on ad-hoc collection of data through 3rd party vendors or managed human judges were slow and expensive.",
"We also found that a unified large vocabulary necessitated careful data collection design since it was infeasible for human judges to label images while keeping hundreds of categories in mind.",
"We leveraged crowd platforms to access a large pool of judges for high throughput and cost-efficient labeling.",
"Since crowd platforms are generally not suited for complex annotation tasks, we adapted the orchestration in [6] for object detection.",
"An overview of the pipeline can be seen in Figure REFImage from https://cutetropolis.com/2016/08/31/links-thats-the-way-they-became-the-brady-bunch by Mike Brailer/ CC BY-SA 4.0.",
"The key stages in the pipeline are described below: Figure: The GenOD data collection pipeline is designed as a chain of micro-tasks suited for judging on crowd platforms.",
"It has 3 main stages: category discovery, instance marking and bounding box drawing.",
"Micro-tasks with complex annotations which cannot be easily aggregated are followed by verification micro-tasks with high overlap to ensure quality.",
"The pipeline is orchestrated so crowd judges only have to annotate a single category or marker at a time."
],
[
"Category Discovery",
"The goal of this stage is to discover all the salient object categories in the image.",
"This is challenging given that there are hundreds of categories and it may be exhausting to label every single object instance in an image.",
"To solve this issue, we ask judges to only discover a single new category by placing a marker on an instance or skip if there are no salient object categories to be added (previously marked categories are shown to the judge).",
"This is repeated until 3 consecutive judges skip, at which point we consider all salient object categories have been discovered.",
"We also employ careful user-interface design so the judge can navigate a hierarchy of categories or directly start typing the name of the category to search for the appropriate category.",
"Unlike [6], the user interface replaces the cursor with a circle with a size corresponding to the minimum area in an image for salient objects.",
"This ensures judges are not spending time marking insignificant objects that are not important from a user scenario standpoint.",
"We also have a simpler variant of this application where a judge only has to spot a specified category rather than provide the names of new categories.",
"The simpler variant is used when we want to quickly collect data for a single new category for business needs.",
"With these two variants, we are able to quickly discover concepts for our vocabulary while also being agile about adding annotations if the taxonomy expands.",
"After category discovery, we run a marker verification micro-task to ensure that all the marked categories are correct.",
"The goal of this stage is to mark all the instances of the categories discovered in the previous stage.",
"We ask a judge to mark every instance of a given category and follow it up with two quality control micro-tasks: (1) Verify that all instances have been marked (2) Verify all the markers are correct.",
"At the end of this stage, we have markers for all the salient object instances in the given set of images.",
"The goal of this stage is to draw a bounding box for a given category marker.",
"This is followed up by a bounding box verification micro-task to ensure quality.",
"By decoupling the drawing of the bounding box from the marker, the data collection pipeline is flexible to accommodate future needs such as segmentation.",
"The goal of this stage is to collect a set of images for a category such that no instance of that category exist in the images.",
"This stage is not necessary while collecting training data, but is useful for the federated measurement design described in [6]."
],
[
"Annotation evaluation",
"We evaluate the proposed pipeline against the baseline data collection approach which used managed vendor judges.",
"To capture the statistics of camera and web-style images appropriately, we randomly sampled 500 images from each distribution for a total evaluation set of 1k images.",
"When comparing the proposed pipeline's results to the existing baseline annotations, we find that 85.75% of baseline instances (93.5% if we exclude objects with smaller dimension < 55 pixels) are correctly localized, and 97% of the markers are verified as correctly categorized.",
"For a more rigorous comparison that is not biased to the baseline or any particular vocabulary as groundtruth, we sent a subsample of 100 images to expert judges to annotate all salient objects as groundtruth and also verify correctness of bounding boxes provided by each data pipeline.",
"We measure precision for each candidate pipeline's provided bounding boxes and recall against the expert-provided salient bounding boxes.",
"We can see the metrics for quality, cost and latency in Table REF .",
"While the throughput at the image level is slightly worse than our baseline approach, this is mainly because our pipeline is more successful at finding more instances to be labeled per image.",
"We ran 2 labeling tasks with 100 and 1000 samples respectively and found that the time taken to get a fully annotated image decreased from 9.3 mins to 4.85 mins.",
"As demonstrated in [17], task interruption and context switching decreases the efficiency of workers while performing micro-tasks on a crowdsourcing platform.",
"Judges are more likely to work on a task when a lot of data is available to be judged.",
"This suggests that even at the image level, throughput can be increased further by sending larger batches which optimize for the capacity of the crowd platform.",
"Table: Comparison of the GenOD data collection pipeline to the previous method.",
"We find that the new pipeline can discover more object instances, annotate faster, and more cost-effectively per bounding box compared to the baseline."
],
[
"Approach",
"In this section, we describe how we developed the GenOD service including data processing to mitigate the serious imbalance in category distribution and dataset sizes, model architecture selection in pursuit of a good speed-accuracy trade-off, and training protocols to achieve a balance between high-performing deep models and model update agility.",
"Our strategy is to first train a single unified base model with a large amount of data to get a generic backbone and have a default detector head for all categories which is easy to maintain and updated less frequently.",
"We improve on this design with the concept of disjoint detectors on the shared backbone, which allows for agile, targeted updates while not disrupting downstream dependencies.",
"Finally we discuss how we deal with the system challenges in scalable serving with low latency."
],
[
"Training Data",
"We combine several large-scale object detection datasets such as Bing internal datasets and open source datasets in our training data.",
"These datasets vary from each other greatly in domain (eg.",
"fine-grained commerce versus generic concepts), the number of images, number and hierarchy of categories as well as the coverage and density of objects per image.",
"Therefore combining these heterogeneous datasets for training with a unified taxonomy is a non-trivial task.",
"Another challenge with such a large vocabulary is the long-tailed, extremely imbalanced category distribution, as shown in the red curve in Figure REF .",
"Directly training on such imbalanced data would lead to poor performance on the tail categories.",
"There is also an imbalance in the number of images from different datasets, ranging from several millions to a few 100 thousands.",
"Therefore, training would be dominated by the distribution in larger datasets while smaller datasets would be under-represented.",
"To combine these diverse and imbalanced datasets, we first built a unified GenOD taxonomy with both top-down human design and bottom-up mapping of the categories from all the datasets; organized in a hierarchical manner.",
"To alleviate the poor performance of rare categories with few training samples, we employ a simple yet effective approach of offline class-aware upsampling [4] on the tail classes, during which all the categories will be upsampled to at least $N_{min}$ instances in the training set.",
"In our experiments we use $N_{min}=2000$ as we found it works well empirically.",
"Figure REF shows the class-wise distribution in our training set before and after the class-aware sampling.",
"With class-aware sampling, we obtained a total of $3.4$ million training images and $29.3$ million objects.",
"We denote this training set as $\\mathcal {D}_{large}$ .",
"Figure: Dataset label distribution before and after applying class-aware sampling.",
"We can see the originally skewed distribution becomes more balanced.",
"The class IDs are sorted by the numberof annotated bounding boxes in the original distribution.To address the imbalance between different datasets, we also downsample the larger datasets offline.",
"This gives us a training set with $1.4$ million images and $10.8$ million objects, which we denote as $\\mathcal {D}_{base}$ ."
],
[
"Base model architectures",
"We have experimented with model architectures including Faster-RCNN [29] and SSD [24] for shopping-segment object detection models in our previous work [9].",
"However, with an order of magnitude more categories, in this work we consider the speed-accuracy trade-off as our first priority, and focus on the evaluation of singe-stage detectors which demonstrate better speed-accuracy trade-offs since their inception [27], [24].",
"We evaluate two variants of single-stage detection models: RetinaNet [21] and FCOS [34], state-of-the-art single-stage detectors for anchor-based and anchor-free models respectively at the time of development of GenOD.",
"Both models have achieved good speed-accuracy trade-off at relatively small tasks like COCO.",
"As the number of categories increases, latency for RetinaNet increases dramatically since it has a large number of per-anchor operations and the last few convolution layers to output the class-wise confidence and bounding box regressions become proportionally larger.",
"On the other hand, since FCOS is anchor-free, it reduces the per anchor operations 9 times compared to RetinaNet.",
"With a few nearly cost-free improvements, FCOS can achieve better results than previous anchor-based detectors with less floating point operations per second (FLOPs) and latency.",
"The experiments in Section provide a comprehensive, quantitative comparison and analysis of these two models."
],
[
"Disjoint detector architecture",
"In a production setting, it is common to have an urgent business need to support a new category or improve a specific category quickly while also not degrading performance on other categories that may have downstream dependencies.",
"With smaller vocabularies, it can be sufficient to retrain the entire model with a new category or more data, but when scaling to a large vocabulary, it becomes very time-consuming to update the entire model and also guarantee no regression in any of the categories.",
"The base model described in the previous section cannot easily accommodate ad-hoc requests or agile updates.",
"To address this, we designed the GenOD model as a federation of disjoint detector heads that share a fixed common backbone.",
"New detector heads, which include classification and bounding box regression networks, can be trained and added on top of the backbone without disrupting the other detectors.",
"When there is a need to quickly add or update a category, the data collection process described in Section allows us to quickly collect data for that category and then the disjoint principle allows us to update GenOD service with much less data and without disrupting any production dependencies.",
"We explore this through a prototypical experiment in Section REF ."
],
[
"Deployment and Serving",
"Service latency is an important factor for a core service like GenOD, therefore we deploy the GenOD models to Microsoft Azure GPU clusters.",
"To serve the GenOD models on GPUs, we first convert them to ONNX models and use ONNX Runtime as the backend for inference, which provides an $18\\%$ speed-up.",
"We built a wrapper of ONNX Runtime on Windows and used the Domain Specific Language (DSL) in [9], which utilizes a graph execution engine to perform thread-safe inference.",
"To address global scalability issues, we leverage a micro-service system built on Microsoft Service Fabric [12] to serve multiple GPU models as micro-services on several global clusters that can scale elastically based on traffic in different regions.",
"Building a cache of detected objects further reduces end-to-end latency.",
"In the end we have built an elastic, scalable GPU serving system for GenOD which can handle hundreds of queries per second across different regions."
],
[
"Evaluation metrics and datasets",
"Unless specified, in the following sections, we use mean average precision defined in [2], [22] as our primary metric, where the average precision (AP) is calculated as the integral of the area under the precision-recall curve in which detections are considered true positives if their intersection-over-union (IOU) with the groudtruths are over 50%.",
"We denote the metric as AP50.",
"AP50 weighs each category equally, however to account for the true distribution of categories seen in production traffic, we also use weighted mean average precision@IOU50, denoted as wAP50: $\\begin{aligned}[rl]wAP = \\frac{\\sum _{c\\in {\\mathcal {C}}} w_c AP_c}{\\sum _{c\\in {\\mathcal {C}}}w_c},\\\\\\end{aligned}$ where $w_c$ and $AP_c$ are the weight and the AP50 for class $c$ in a validation set of $\\mathcal {C}$ classes, respectively.",
"In our setting, we typically set $w_c$ to the number of instances of $c$ in the validation set.",
"Table: Experiments of GenOD models on the 4 validation sets (OpenImagesV5, COCO 2017, Bing internal fashion and home furniture detection datasets), comparing the RetinaNet and FCOS architectures, and two FCOS variants ℳ base \\mathcal {M}_{base} and ℳ large \\mathcal {M}_{large}.",
"Overall GenOD ℳ base \\mathcal {M}_{base} is selected as the model candidate with the best average AP50 and wAP50 metrics.We evaluate candidates on the validation splits of two public datasets (OpenImagesV5[16], COCO 2017[22]) and two of Bing's internal validation sets in fashion and home furnishing, denoted as Bing-Fashion Val and Bing-HF Val, respectively.",
"We use the average of AP50 and wAP50 metrics over the 4 validation sets as the criteria to select the final model candidate.",
"We then measure final performance on 3 internal test sets: Bing-Generic Test, Bing-Fashion Test and Bing-HF Test which are collected by a weighted sampling of Bing traffic.",
"Note that Bing traffic is much more challenging than the validation data due to a higher proportion of noisy, cluttered scenes in real-world data.",
"For the COCO dataset, we follow the evaluation protocol in [22] and also report the AP@IOU[0.5:0.95], which is simply denoted as AP.",
"For the OpenImages dataset, we follow the same federated evaluation protocol in the OpenImages challenges [16]."
],
[
"Base model training",
"We implemented both the RetinaNet and FCOS models based on maskrcnn-benchmark [3].",
"Both models are trained with Feature Pyramid Network (FPN) [20] and Focal Loss [21], using ResNet-101 [8] as backbone.",
"For FCOS we employ Modulated Deformable Convolution (DCNv2) [38] at stage 2$\\sim $ 4 and trained the model with the proposed improvements in [34] to further boost the accuracy.",
"Both variants are trained for 24 epochs on the dataset in Section REF using 8 V100 GPUs, with a batch size of 64 and learning rate of 0.03.",
"To best optimize for online production latency, we use an input image resolution of 400$\\times $ 667.",
"We use multi-scale training with the shorter side ranging from 300 to 500 while keeping aspect ratios to adapt to different scales of inputs."
],
[
"Candidate selection",
"Table REF shows the results of the GenOD models on the four validation sets described in Section REF .",
"During inference, we map the results from the GenOD taxonomy to the corresponding categories in benchmark datasets for a fair comparison.",
"From the table we can see the trained models with FCOS architecture consistently outperform the RetinaNet one.",
"We denote the FCOS models trained with $\\mathcal {D}_{base}$ and $\\mathcal {D}_{large}$ as $\\mathcal {M}_{base}$ and $\\mathcal {M}_{large}$ , respectively.",
"Overall $\\mathcal {M}_{base}$ achieves the best performance in the aggregated metrics, so we select this model as our candidate for further evaluation."
],
[
"Label propagation in the taxonomy hierarchy",
"We also experiment with leveraging the hierarchical information in the GenOD taxonomy to propagate the bounding boxes and scores of the fine-grained categories to their ancestors in the taxonomy at inference time.",
"For example, if a \"blue jay\" is detected, it would also be propagated to generate \"bird\" and \"animal\" labels.",
"We select OpenImages as the benchmark because it has a meaningful generic hierarchy.",
"We evaluate label propagation on the two FCOS model candidates trained on GenOD $\\mathcal {D}_{base}$ and $\\mathcal {D}_{large}$ respectively on the OpenImagesV5 validation sets.",
"Significant improvements have been observed over the original predictions without propagation.",
"Specifically, label propagation improves wAP50 of the $\\mathcal {M}_{base}$ model from $51.34$ to $61.65$ , and improving the performance of the $\\mathcal {M}_{large}$ model from $51.02$ to $63.16$ .",
"Moreover, the AP50 of the $\\mathcal {M}_{large}$ model is competitive among the best single models in the OpenImages Detection Challenge 2019https://storage.googleapis.com/openimages/web/challenge2019.html that are trained with similar backbones on larger resolutions (800$\\times $ 1333), showing the effectiveness of our model training and post-processing approach.",
"Table: Evaluating label propagation for FCOS model candidates trained on GenOD training data on the OpenImagesV5 validation set."
],
[
"Comparison with segment models",
"Table REF shows the test set results of GenOD $\\mathcal {M}_{base}$ using the wAP50 metric.",
"We compare GenOD $\\mathcal {M}_{base}$ to two production segment models trained separately for fashion and home furnishing detection using Bing-Fashion Train and Bing-HF Train datasets respectively.",
"It should be noted that each of those sets is contained within $\\mathcal {D}_{base}$ .",
"We find that GenOD $\\mathcal {M}_{base}$ improves performance on the key product verticals over domain-specific models while significantly reducing the capacity and maintenance cost.",
"Table: Evaluation of weighted AP50 on the Bing object detection measurement set."
],
[
"Latency",
"In Table REF , we benchmark the latency between the RetinaNet and FCOS variants in the single-GPU and batch-1 setting on the COCO validation sets on V100 GPUs with CUDA 11.0 by averaging the five runs.",
"From the table we can see FCOS is $3.9\\times $ faster than RetinaNet.",
"More interestingly, we observed when scaling up from the 80-class COCO taxonomy to the 900-class GenOD taxonomy, RetinaNet becomes nearly 3 times slower while the latency of FCOS remains stable, which further increases the latency gap between the two models from $1.4\\times $ to $3.9\\times $ .",
"This shows that the last few class-wise convolution layers in an anchor-based model generate significant overhead as the number of categories grows and demonstrating the anchor-free approach is robust in latency against the scaling of vocabulary, making it better suited to large-vocabulary object detection.",
"Table: Single GPU batch-1 latency of RetinaNet and FCOS variants of GenOD models on V100 GPU.",
"With the number of categories scaling from 80 classes of COCO to the generic object taxonomy, the speedup of the FCOS architecture grows from 1.4×1.4\\times to 3.9×3.9\\times ."
],
[
"Experiments on COCO benchmark",
"In Table REF we compare the GenOD model to the models trained on COCO with the same architectures.",
"We can see GenOD consistently outperforms the COCO-trained models especially on small and mid-size objects, even though they target a much broader vocabulary and are not specifically trained for COCO objects.",
"Table: Comparison of GenOD ℳ base \\mathcal {M}_{base} model with the FCOS model trained on COCO on the COCO 2017 validation set."
],
[
"Disjoint detector training",
"As described in Section REF , here we compare the conventional joint training approach with our disjoint approach with a prototypical experiment.",
"The baseline is the GenOD model with a jointly trained head for all categories using $\\mathcal {D}_{base}$ , i.e, GenOD $\\mathcal {M}_{base}$ .",
"Given the GenOD $\\mathcal {M}_{base}$ model, suppose our goal is to improve the sofa category in response to user feedback, without performance degradation of other categories within a short development cycle.",
"For the update, we consider an additional set of labeled data: $\\mathcal {D}_{update}=\\mathcal {D}_{large}-\\mathcal {D}_{base}$ .",
"Given this additional data, we conduct three experiments and report the results in Table REF : Joint detector retraining : We train the single-head joint model with all the available data ($\\mathcal {D}_{large}$ ) using the same training scheme as GenOD $\\mathcal {M}_{base}$ as described in Section REF .",
"Joint detector finetuning : We randomly sample 50k images from $\\mathcal {D}_{update}$ and finetune the joint detector starting from the GenOD $\\mathcal {M}_{base}$ model.",
"For this finetuning stage, we use a smaller learning rate of 0.0001 and train on the data for 12 epochs.",
"Disjoint detector finetuning: We split the GenOD $\\mathcal {M}_{base}$ head to create a disjoint detector head for just the sofa category.",
"We finetune this model on the same dataset (50k randomly sampled images from $\\mathcal {D}_{update}$ ) as described in REF above.",
"We use a learning rate of 0.00003 and train the disjoint detector head of the model for 12 epochs.",
"As seen from the experimental results in Table REF , disjoint detector finetuning on just a small amount of data is far more agile and allows us to train $\\sim $300x faster than joint retraining in REF and $\\sim $2x faster than joint finetuning in REF , while also improving on the category AP.",
"This is achieved without disrupting the existing model for any of the other categories which allows for stable updates in the production stack compared to the conventional model retraining process.",
"Table: Evaluation of model update agility for the sofa category.",
"Disjoint training of the targeted category is much faster, while also increasing its AP and keeping other categories stable."
],
[
"Object Detection for Visual Product Search",
"One of the primary applications for GenOD is to help users better formulate visual search queries and improve the relevance of search results.",
"Figure REF showcases the hotspot interactions in the Bing Image Details Page.",
"GenOD assists the user in formulating a query.",
"Instead of the user having to manually crop to region of interest in the image, detected hotspots are shown over the image for users to start their search with just a tap/click.",
"GenOD's detected categories can be passed to downstream tasks like similar image search ranker to filter out semantically irrelevant results and improve relevance.",
"We trained and deployed a lightweight category classifier to the index images.",
"The detected category of the query will be sent to match with the categories of the index images, and filter out those images that do not match.",
"We conducted comprehensive offline and online experiments on the efficacy of GenOD improving visual search experience.",
"We measure the defect rate of the top-5 visual search results from the hotspot clicks on the fashion segment, where defect rate is defined as the average error rate in the categories of the retrieved images.",
"In table REF we can see that after applying the GenOD categories for filtering, the hotspot click-through defect rate has dropped by $54.9\\%$ , substantially improving the relevance and user experience.",
"Table: Defect rates of top-5 visual search results from object detection hotspot clicks (lower is better).",
"With the ranking results filtered semantically by object detection categories, the product search defect rate decreases significantly by 54.9%.Table: Aggregated online user engagement metrics after deploying GenOD to Bing Visual Search, which shows significant gains over baseline.We also set up a series of online A/B tests to measure the user engagement before and after deploying GenOD to all Bing Visual Search requests, as shown in Table REF .",
"After aggregating the online user engagement metrics including the percentage of user entries to visual search, overall entries to visual search and hotspot clicks per unique users, GenOD shows significant gains in bringing in more engaging users in visual search; demonstrating the advantage of expanding object detection to generic objects on all images."
],
[
"Object-Level Triggering for Fine-grained Visual Recognition",
"Bing Visual Search runs multiple fine-grained recognition models including animals, plants, landmarks and more.",
"Since these models usually have high latency, it is necessary to perform lightweight triggering before running them.",
"Previous image-level visual triggering models such as the one described in [9] often fail to trigger on small objects or when multiple different objects are in the scene.",
"For a fair comparison with the previous approach, we compare triggering performance at the image level by aggregating the outputs of the GenOD model.",
"In Table REF we evaluate the image-level triggering precision and recall on the food recognition measurement set.",
"For triggering, we prefer models with higher recall performance.",
"Compared to the image-level triggering model, GenOD improves the triggering recall by detecting and recognizing smaller objects.",
"Table: Comparison of triggering precision and recall between image-level and object-level triggering on the food recognition measurement set."
],
[
"Bing Mobile Camera",
"Bing mobile camera is a real-time mobile web experience allowing users to search without having to manually capture a picture.",
"The mobile interface is shown in Figure REF (right).",
"When a user opens the camera experience and phone is stable, a frame will be captured and sent to the GenOD service to perform real-time object detection.",
"The detected objects will be sent back to the phone to render as hotspots.",
"An in-app object tracking model keeps track of the objects when they are in view, so hotspots can stay on the objects without the need to make additional GenOD requests.",
"Clicking on the detected hotspots provides relevant results to the selected object.",
"GenOD enhances the user experience and simplifies the formulation of visual search queries.",
"Online A/B tests show a reduction of 31.8% in responses with no visual search results compared to our control in which we depend on users to formulate a visual query."
],
[
"Conclusions",
"We presented GenOD, a web-scale generic object detection service that is fundamental in image understanding for Bing Visual Search and fueling multiple downstream applications.",
"We described an efficient data collection workflow for training the models used in the system.",
"We demonstrated with experiments that the GenOD model has achieved competitive object detection results across production measurement sets and academic benchmarks, with good speed-accuracy tradeoff, and can be updated with agility while being stable to dependencies.",
"Specifically, we have shown that by moving to a large-scale single unified generic detector, GenOD can achieve better results than multiple domain-specific models in each vertical, reducing the cost of maintaining several segment models.",
"Finally we also showed how GenOD benefits visual search applications by significantly improving user engagement and search relevance.",
"We would like to thank Arun Sacheti, Meenaz Merchant, Surendra Ulabala, Mikhail Panfilov, Andre Alves, Kiril Moskaev, Vishal Thakkar, Avinash Vemuluru, Souvick Sarkar, Li Zhang, Anil Akurathi, Vladimir Vakhrin, Houdong Hu, Rui Xia, Xiaotian Han, Dongfei Yu, Ye Wu, Vincent Chen, Kelly Huang, Nik Srivastava, Yokesh Kumar, Mark Bolin, Mahdi Hajiaghayi, Pengchuan Zhang, Xiyang Dai, Lu Yuan, Lei Zhang and Jianfeng Gao for their support, collaboration and many interesting discussions."
]
] | 2107.01814 |
[
[
"Matching a Desired Causal State via Shift Interventions"
],
[
"Abstract Transforming a causal system from a given initial state to a desired target state is an important task permeating multiple fields including control theory, biology, and materials science.",
"In causal models, such transformations can be achieved by performing a set of interventions.",
"In this paper, we consider the problem of identifying a shift intervention that matches the desired mean of a system through active learning.",
"We define the Markov equivalence class that is identifiable from shift interventions and propose two active learning strategies that are guaranteed to exactly match a desired mean.",
"We then derive a worst-case lower bound for the number of interventions required and show that these strategies are optimal for certain classes of graphs.",
"In particular, we show that our strategies may require exponentially fewer interventions than the previously considered approaches, which optimize for structure learning in the underlying causal graph.",
"In line with our theoretical results, we also demonstrate experimentally that our proposed active learning strategies require fewer interventions compared to several baselines."
],
[
"Introduction",
"Consider an experimental biologist attempting to turn cells from one type into another, e.g., from fibroblasts to neurons [37], by altering gene expression.",
"This is known as cellular reprogramming and has shown great promise in recent years for regenerative medicine [28].",
"A common approach is to model gene expression of a cell, which is governed by an underlying gene regulatory network, using a structural causal model [12], [4].",
"Through a set of interventions, such as gene knockouts or over-expression of transcription factors [9], a biologist can infer the structure of the underlying regulatory network.",
"After inferring enough about this structure, a biologist can identify the intervention needed to successfully reprogram a cell.",
"More generally, transforming a causal system from an initial state to a desired state through interventions is an important task pervading multiple applications.",
"Other examples include closed-loop control [34] and pathway design of microstructures [38].",
"With little prior knowledge of the underlying causal model, this task is intrinsically difficult.",
"Previous works have addressed the problem of intervention design to achieve full identifiability of the causal model [16], [14], [32].",
"However, since interventional experiments tend to be expensive in practice, one wishes to minimize the number of trials and learn just enough information about the causal model to be able to identify the intervention that will transform it into the desired state.",
"Furthermore, in many realistic cases, the set of interventions which can be performed is constrained.",
"For instance, in CRISPR experiments, only a limited number of genes can be knocked out to keep the cell alive; or in robotics, a robot can only manipulate a certain number of arms at once.",
"Contributions.",
"We take the first step towards the task of causal matching (formalized in sec:problem-step), where an experimenter can perform a series of interventions in order to identify a matching intervention which transforms the system to a desired state.",
"In particular, we consider the case where the goal is to match the mean of a distribution.",
"We focus on a subclass of interventions called shift interventions, which can for example be used to model gene over-expression experiments [35].",
"These interventions directly increase or decrease the values of their perturbation targets, with their effect being propagated to variables which are downstream (in the underlying causal graph) of these targets.",
"We show that there always exists a unique shift intervention (which may have multiple perturbation targets) that exactly transforms the mean of the variables into the desired mean (Lemma REF ).",
"We call this shift intervention the matching intervention.",
"To find the matching intervention, in sec:identifiability we characterize the Markov equivalence class of a causal graph induced by shift interventions, i.e., the edges in the causal graph that are identifiable from shift interventions; in particular, we show that the resulting Markov equivalence classes can be more refined than previous notions of interventional Markov equivalence classes.",
"We then propose two active learning strategies in sec:algorithms based on this characterization, which are guaranteed to identify the matching intervention.",
"These active strategies proceed in an adaptive manner, where each intervention is chosen based on all the information gathered so far.",
"In sec:theory, we derive a worst-case lower bound on the number of interventions required to identify the matching intervention and show that the proposed strategies are optimal up to a logarithmic factor.",
"Notably, the proposed strategies may use exponentially fewer interventions than previous active strategies for structure learning.",
"Finally, in sec:experiments, we demonstrate also empirically that our proposed strategies outperform previous methods as well as other baselines in various settings."
],
[
"Related Works",
" Experimental Design.",
"Previous work on experimental design in causality has considered two closely related goals: learning the most structural information about the underlying DAG given a fixed budget of interventions [13], and fully identifying the underlying DAG while minimizing the total number or cost [30], [21] of interventions.",
"These works can also be classified according to whether they consider a passive setting, i.e., the interventions are picked at a single point in time [19], [30], [21], or an active setting, i.e., interventions are decided based on the results of previous interventions [17], [1], [14], [32].",
"The setting addressed in the current work is closest to the active, full-identification setting.",
"The primary difference is that in order to match a desired mean, one does not require full identification; in fact, as we show in this work, we may require significantly less interventions.",
"Causal Bandits.",
"Another related setting is the bandit problem in sequential decision making, where an agent aims to maximize the cumulative reward by selecting an arm at each time step.",
"Previous works considered the setting where there are causal relations between regrets and arms [25], [26], [39].",
"Using a known causal structure, these works were able to improve the dependence on the total number of arms compared to previous regret lower-bounds [6], [25].",
"These results were further extended to the case when the causal structure is unknown a priori [8].",
"In all these works the variables are discrete, with arms given by do-interventions (i.e., setting variables to a given value), so that there are only a finite number of arms.",
"In our work, we are concerned with the continuous setting and shift interventions, which corresponds to an infinite (continuous) set of arms.",
"Correlation-based Approaches.",
"There are also various correlation-based approaches for this task that do not make use of any causal information.",
"For example, previous works have proposed score-based [7], entropy-based [10] and distance-based approaches [28] for cellular reprogramming.",
"However, as shown in bandit settings [25], when the system follows a causal structure, this structure can be exploited to learn the optimal intervention more efficiently.",
"Therefore, we here focus on developing a causal approach."
],
[
"Problem Setup",
"We now formally introduce the causal matching problem of identifying an intervention to match the desired state in a causal system under a given metric.",
"Following [22], a causal structural model is given by a directed acyclic graph (DAG) $\\mathcal {G}$ with nodes $[p]=\\lbrace 1,\\dots , p\\rbrace $ , and a set of random variables $X=\\lbrace X_1,...,X_p\\rbrace $ whose joint distribution ${\\mathrm {P}}$ factorizes according to $\\mathcal {G}$ .",
"Denote by $\\operatorname{pa}_{\\mathcal {G}}(i)=\\lbrace j\\in [p]\\mid j\\rightarrow i\\rbrace $ the parents of node $i$ in $\\mathcal {G}$ .",
"An intervention $I\\subset [p]$ with multiple perturbation targets $i\\in I$ either removes all incoming edges to $X_i$ (hard intervention) or modifies the conditional probability ${\\mathrm {P}}(X_i|X_{\\operatorname{pa}_{\\mathcal {G}}(i)})$ (soft intervention) for all $i\\in I$ .",
"This results in an interventional distribution ${\\mathrm {P}}^{I}$ .",
"Given a desired joint distribution ${\\mathrm {Q}}$ over $X$ , the goal of causal matching is to find an optimal matching intervention $I$ such that ${\\mathrm {P}}^{I}$ best matches ${\\mathrm {Q}}$ under some metric.",
"In this paper, we address a special case of the causal matching problem, which we call causal mean matching, where the distance metric between ${\\mathrm {P}}^I$ and ${\\mathrm {Q}}$ depends only on their expectations.",
"We focus on causal mean matching for a class of soft interventions, called shift interventions [29].",
"Formally, a shift intervention with perturbation targets $I\\subset [p]$ and shift values $\\lbrace a_i\\rbrace _{i\\in I}$ modifies the conditional distribution as ${\\mathrm {P}}^{I}(X_i=x+a_i|X_{\\operatorname{pa}_{\\mathcal {G}}(i)})={\\mathrm {P}}(X_i=x|X_{\\operatorname{pa}_{\\mathcal {G}}(i)})$ .",
"Here, the shift values $\\lbrace a_i\\rbrace _{i\\in I}$ are assumed to be deterministic.",
"We aim to find $I\\subset [p]$ and $\\lbrace a_i\\rbrace _{i\\in I}\\in \\mathbb {R}^{|I|}$ such that the mean of ${\\mathrm {P}}^I$ is closest to that of ${\\mathrm {Q}}$ , i.e., minimizes $ d(\\mathbb {E}_{{\\mathrm {P}}^{I}}(X), \\mathbb {E}_{{\\mathrm {Q}}}(X))$ for some metric $d$ .",
"In fact, as we show in the following lemma, there always exists a unique shift intervention, which we call the matching intervention, that achieves exact mean matching.To lighten notation, we use $I$ to denote both the perturbation targets and the shift values of this intervention.",
"Lemma 1 For any causal structural model and desired mean $\\mathbb {E}_{\\mathrm {Q}}(X)$ , there exists a unique shift intervention $I^*$ such that $\\mathbb {E}_{{\\mathrm {P}}^{I^*}}(X) = \\mathbb {E}_{{\\mathrm {Q}}}(X)$ .",
"We assume throughout that the underlying causal DAG $\\mathcal {G}$ is unknown.",
"But we assume causal sufficiency [31], which excludes the existence of latent confounders, as well as access to enough observational data to determine the joint distribution ${\\mathrm {P}}$ and thus the Markov equivalence class of $\\mathcal {G}$ [3].",
"It is well-known that with enough interventions, the causal DAG $\\mathcal {G}$ becomes fully identifiable [40].",
"Thus one strategy for causal mean matching is to first use interventions to fully identify the structure of $\\mathcal {G}$ , and then solve for the matching intervention given full knowledge of the graph.",
"However, in general this strategy requires more interventions than needed.",
"In fact, the number of interventions required by such a strategy can be exponentially larger than the number of interventions required by a strategy that directly attempts to identify the matching intervention, as illustrated in Figure REF and proven in Theorem REF .",
"Figure: Completely identifying a DAG can require exponentially more interventions than identifying the matching intervention.Consider a graph constructed by joining rr size-4 cliques, where the matching intervention has the source node as the only perturbation target, as pictured in (a) with r=2r=2 and the source node in purple;(b) shows the minimum size set intervention (in purple) that completely identifies the DAG, which grows as O(r)O(r) .In Theorem , we show that the matching intervention can be identified in O(logr)O(\\log r) single-node interventions.In this work, we consider active intervention designs, where a series of interventions are chosen adaptively to learn the matching intervention.",
"This means that the information obtained after performing each intervention is taken into account for future choices of interventions.",
"We here focus on the noiseless setting, where for each intervention enough data is obtained to decide the effect of each intervention.",
"Direct implications for the noisy setting are discussed in appendix:noisy.",
"To incorporate realistic cases in which the system cannot withstand an intervention with too many target variables, as is the case in CRISPR experiments, where knocking out too many genes at once often kills the cell, we consider the setting where there is a sparsity constraint $S$ on the maximum number of perturbation targets in each intervention, i.e., we only allow $I$ where $|I| \\le S$ ."
],
[
"Identifiability",
"In this section, we characterize and provide a graphical representation of the shift interventional Markov equivalence class (shift-$\\mathcal {I}$ -MEC), i.e., the equivalence class of DAGs that is identifiable by shift interventions $\\mathcal {I}$ .",
"We also introduce mean interventional faithfulness, an assumption that guarantees identifiability of the underlying DAG up to its shift-$\\mathcal {I}$ -MEC.",
"Proofs are given in appendix:identify."
],
[
"Shift-interventional Markov Equivalence Class",
"For any DAG $\\mathcal {G}$ with nodes $[p]$ , a distribution $f$ is Markov with respect to $\\mathcal {G}$ if it factorizes according to $f(X)=\\prod _{i\\in [p]} f(X_i|X_{\\operatorname{pa}_{\\mathcal {G}}(i)})$ .",
"Two DAGs are Markov equivalent or in the same Markov equivalence class (MEC) if any positive distribution $f$ which is Markov with respect to (w.r.t.)",
"one DAG is also Markov w.r.t.",
"the other DAG.",
"With observational data, a DAG is only identifiable up to its MEC [3].",
"However, the identifiability improves to a smaller class of DAGs with interventions.",
"For a set of interventions $\\mathcal {I}$ (not necessarily shift interventions), the pair $(f, \\lbrace f^I\\rbrace _{I\\in \\mathcal {I}})$ is $\\mathcal {I}$ -Markov w.r.t.",
"$\\mathcal {G}$ if $f$ is Markov w.r.t.",
"$\\mathcal {G}$ and $f^I$ factorizes according to $f^I(X) = \\prod _{i\\notin I}f(X_i|X_{\\operatorname{pa}_{\\mathcal {G}}(i)}) \\prod _{i\\in I} f^{I}(X_i|X_{\\operatorname{pa}_{\\mathcal {G}}(i)}),\\quad \\forall I \\in \\mathcal {I}.$ Similarly, the interventional Markov equivalence class ($\\mathcal {I}$ -MEC) of a DAG can be defined, and [40] provided a structural characterization of the $\\mathcal {I}$ -MEC for general interventions $\\mathcal {I}$ (not necessarily shift interventions).",
"Following, we show that if $\\mathcal {I}$ consists of shift interventions, then the $\\mathcal {I}$ -MEC becomes smaller, i.e., identifiability of the causal DAG is improved.",
"The proof utilizes Lemma REF on the relationship between conditional probabilities.",
"For this, denote by $\\operatorname{an}_{\\mathcal {G}}(i)$ the ancestors of node $i$ , i.e., all nodes $j$ for which there is a directed path from $j$ to $i$ in $\\mathcal {G}$ .",
"For a subset of nodes $I$ , we say that $i\\in I$ is a source w.r.t.",
"$I$ if $\\operatorname{an}_{\\mathcal {G}}(i)\\cap I=\\varnothing $ .",
"A subset $I^{\\prime }\\subset I$ is a source w.r.t.",
"$I$ if every node in $I^{\\prime }$ is a source w.r.t.",
"$I$ .",
"Lemma 2 For any distribution $f$ that factorizes according to $\\mathcal {G}$ , the interventional distribution $f^I$ for a shift intervention $I\\subset [p]$ with shift values $\\lbrace a_i\\rbrace _{i\\in I}$ satisfies $\\mathbb {E}_{f^I}(X_i) = \\mathbb {E}_{f}(X_i) + a_i, $ for any source $i\\in I$ .",
"Furthermore, if $i\\in I$ is not a source w.r.t.",
"$I$ , then there exists a positive distribution $f$ such that $\\mathbb {E}_{f^I}(X_i) \\ne \\mathbb {E}_{f}(X_i) + a_i$ .",
"Hence, we can define the shift-$\\mathcal {I}$ -Markov property and shift-interventional Markov equivalence class (shift-$\\mathcal {I}$ -MEC) as follows.",
"Definition 1 For a set of shift interventions $\\mathcal {I}$ , the pair $(f, \\lbrace f^I\\rbrace _{I\\in \\mathcal {I}})$ is shift-$\\mathcal {I}$ -Markov w.r.t.",
"$\\mathcal {G}$ if $(f, \\lbrace f^I\\rbrace _{I\\in \\mathcal {I}})$ is $\\mathcal {I}$ -Markov w.r.t.",
"$\\mathcal {G}$ and $\\mathbb {E}_{f^I}(X_i) = \\mathbb {E}_{f}(X_i) + a_i, \\quad \\forall ~i\\in I\\in \\mathcal {I}~s.t.~\\operatorname{an}_{\\mathcal {G}}(i)\\cap I = \\varnothing .$ Two DAGs are in the same shift-$\\mathcal {I}$ -MEC if any positive distribution that is shift-$\\mathcal {I}$ -Markov w.r.t.",
"one DAG is shift-$\\mathcal {I}$ -Markov also w.r.t.",
"the other DAG.",
"The following graphical characterizations are known: Two DAGs are in the same MEC if and only if they share the same skeleton (adjacencies) and v-structures (induced subgraphs $i\\rightarrow j \\leftarrow k$ ), see [36].",
"For general interventions $\\mathcal {I}$ , two DAGs are in the same $\\mathcal {I}$ -MEC, if they are in the same MEC and they have the same directed edges $\\lbrace i\\rightarrow j|i\\in I, j\\notin I, I\\in \\mathcal {I}, i-j\\rbrace $ , where $i-j$ means that either $i \\rightarrow j$ or $j \\rightarrow i$ [15], [40].",
"In the following theorem, we provide a graphical criterion for two DAGs to be in the same shift-$\\mathcal {I}$ -MEC.",
"Theorem 1 Let $\\mathcal {I}$ be a set of shift interventions.",
"Then two DAGs $\\mathcal {G}_1$ and $\\mathcal {G}_2$ belong to the same shift-$\\mathcal {I}$ -MEC if and only if they have the same skeleton, v-structures, directed edges $\\lbrace i\\rightarrow j|i\\in I, j\\notin I, I\\in \\mathcal {I}, i-j\\rbrace $ , as well as source nodes of $I$ for every $I \\in \\mathcal {I}$ .",
"In other words, two DAGs are in the same shift-$\\mathcal {I}$ -MEC if and only if they are in the same $\\mathcal {I}$ -MEC and they have the same source perturbation targets.",
"Figure REF shows an example; in particular, to represent an MEC, we use the essential graph (EG), which has the same skeleton as any DAG in this class and directed edges $i\\rightarrow j$ if $i\\rightarrow j$ for every DAG in this class.",
"The essential graphs corresponding to the MEC, $\\mathcal {I}$ -MEC and shift-$\\mathcal {I}$ -MEC of a DAG $\\mathcal {G}$ are referred to as EG, $\\mathcal {I}$ -EG and shift-$\\mathcal {I}$ -EG of $\\mathcal {G}$ , respectively.",
"They can be obtained from the aforementioned graphical criteria (along with a set of logical rules known as the Meek rules [27]; see details in appendix:pre).",
"Figure REF shows an example of EG, $\\mathcal {I}$ -EG and shift-$\\mathcal {I}$ -EG of a four-node DAG.",
"Figure: Three types of essential graphs.",
"(a).",
"DAG 𝒢\\mathcal {G}; (b).",
"EG of 𝒢\\mathcal {G}; (c).",
"ℐ\\mathcal {I}-EG of 𝒢\\mathcal {G} where ℐ\\mathcal {I} contains one intervention with perturbation targets X 1 ,X 2 X_1,X_2 (purple); (d).",
"shift-ℐ\\mathcal {I}-EG of 𝒢\\mathcal {G}, which can identify an additional edge compared to ℐ\\mathcal {I}-EG (red)."
],
[
"Mean Interventional Faithfulness",
"For the causal mean matching problem, the underlying $\\mathcal {G}$ can be identified from shift interventions $\\mathcal {I}$ up to its shift-$\\mathcal {I}$ -MEC.",
"However, we may not need to identify the entire DAG to find the matching intervention $I^*$ .",
"Lemma REF implies that if $i$ is neither in nor downstream of $I^*$ , then the mean of $X_i$ already matches the desired state, i.e., $\\mathbb {E}_{{\\mathrm {P}}}(X_i)= \\mathbb {E}_{{\\mathrm {Q}}}(X_i)$ ; this suggest that these variables may be negligible when learning $I^*$ .",
"Unfortunately, the reverse is not true; one may design “degenerate\" settings where a variable is in (or downstream of) $I^*$ , but its marginal mean is also unchanged: Example 1 Let $X_3 = X_1 + 2 X_2$ , with $\\mathbb {E}_{\\mathrm {P}}(X_1) = 1$ and $\\mathbb {E}_{\\mathrm {P}}(X_2) = 1$ , so that $\\mathbb {E}_{\\mathrm {P}}(X_3) = 3$ .",
"Suppose $I^*$ is a shift intervention with perturbation targets $\\lbrace X_1, X_2, X_3 \\rbrace $ , with $a_1 = 1$ , $a_2 = -1$ , and $a_3 = 1$ .",
"Then $\\mathbb {E}_{{\\mathrm {P}}^I} (X_3) = 3$ , i.e., the marginal mean of $X_3$ is unchanged under the intervention.",
"Such degenerate cases arise when the shift on a node $X_j$ (deemed 0 if not shifted) exactly cancels out the contributions of shifts on its ancestors.",
"Formally, the following assumption rules out these cases.",
"Assumption 1 (Mean Interventional Faithfulness) If $i\\in [p]$ satisfies $\\mathbb {E}_{{\\mathrm {P}}}(X_i)= \\mathbb {E}_{{\\mathrm {Q}}}(X_i)$ , then $i$ is neither a nor downstream of any perturbation target, i.e., $i\\notin I^*, \\operatorname{an}_{\\mathcal {G}}(i)\\cap I^*=\\varnothing $ .",
"This is a particularly weak form of faithfulness, which is implied by interventional faithfulness assumptions in prior work [40], [33], [20].",
"Let $T$ be the collection of nodes $i\\in [p]$ for which $\\mathbb {E}_{{\\mathrm {P}}}(X_i)\\ne \\mathbb {E}_{{\\mathrm {Q}}}(X_i)$ .",
"The following lemma shows that under the mean interventional faithfulness assumption we can focus on the subgraph $\\mathcal {G}_{T}$ induced by $T$ , since $I^*\\subset T$ and interventions on $X_{T}$ do not affect $X_{[p]\\setminus T}$ .",
"Lemma 3 If assumption:1 holds, then any edge $i-j$ with $j\\in T$ and $i \\notin T$ has orientation $j\\leftarrow i$ .",
"Conversely, if assumption:1 does not hold, then there exists some $i - j$ , $j \\in T$ , $i \\notin T$ such that $j \\rightarrow i$ ."
],
[
"Algorithms",
"Having shown that shift interventions allow the identification of source perturbation targets and that the mean interventional faithfulness assumption allows reducing the problem to an induced subgraph, we now propose two algorithms to learn the matching intervention.",
"The algorithms actively pick a shift intervention $I_t$ at time $t$ based on the current shift-interventional essential graph (shift-$\\mathcal {I}_{t}$ -EG).",
"Without loss of generality and for ease of discussion, we assume that the mean interventional faithfulness assumption holds and we therefore only need to consider $\\mathcal {G}_{T}$ .",
"In appendix:alg, we show that the faithfulness violations can be identified and thus assumption:1 is not necessary for identifying the matching intervention, but additional interventions may be required.",
"Figure: Learning I * I^* when the structure is known.",
"Undimmed parts represent the current subgraph with source nodes (in purple).",
"I * ={1,2,4,5}I^*=\\lbrace 1,2,4,5\\rbrace is solved in three steps.",
"Shift values are omitted.",
"(a).",
"𝒢 T \\mathcal {G}_{T} and U T U_{T}; (b).",
"𝒢 T 1 \\mathcal {G}_{T_1} and U T 1 U_{T_1}; (c).",
"𝒢 T 2 \\mathcal {G}_{T_2} and U T 2 U_{T_2}.Warm-up: Upstream Search.",
"Consider solving for the matching intervention $I^*$ when the structure of $\\mathcal {G}_{T}$ is known, i.e., the current shift-$\\mathcal {I}_{t}$ -EG is fully directed.",
"Let $U_{T}=\\lbrace i|i\\in T, \\operatorname{an}_{\\mathcal {G}_{T}}(i)\\cap T = \\varnothing \\rbrace $ be the non-empty set of source nodes in $T$ .",
"We make the following key observation.",
"Observation 1 $U_{T}\\subset I^*$ , and the shift values are $a_{i}= \\mathbb {E}_{{\\mathrm {Q}}}(X_i) - \\mathbb {E}_{{\\mathrm {P}}}(X_i)$ for each $i \\in U_T$ .",
"This follows since shifting other variables in $T$ cannot change the mean of nodes in $U_T$ .",
"Further, the shifted means of variables in $U_T$ match the desired mean (Lemma REF ).",
"Given the resulting intervention $U_T$ , we obtain a new distribution ${\\mathrm {P}}^{U_T}$ .",
"Assuming mean interventional faithfulness on this distribution, we may now remove those variables whose means in ${\\mathrm {P}}^{U_T}$ already match ${\\mathrm {Q}}$ .",
"We then repeat this process on the new set of unmatched source nodes, $T_1$ , to compute the corresponding shift intervention $U_{T_1}$ .",
"Repeating until we have matched the desired mean for all variables yields $I^*$ .",
"We illustrate this procedure in Figure REF .",
"The idea of upstream search extends to shift-$\\mathcal {I}_{t}$ -EG with partially directed or undirected $\\mathcal {G}_T$ .",
"In this case, if a node or nodes of $\\mathcal {G}_T$ are identified as source, Observation REF still holds.",
"Hence, we solve a part of $I^*$ with these source nodes and then intervene on them to reduce the unsolved graph size.",
"Decomposition of Shift Interventional Essential Graphs: In order to find the source nodes, we decompose the current shift-$\\mathcal {I}_t$ -EG into undirected components.",
"[16] showed that every interventional essential graph is a chain graph with chordal chain components, where the orientations in one chain component do not affect the orientations in other components.The chain components of a chain graph are the undirected connected components after removing all its directed edges, and an undirected graph is chordal if all cycles of length greater than 3 contain a chord.",
"By a similar argument, we can obtain an analogous decomposition for shift interventional essential graphs, and show that there is at least one chain component with no incoming edges.",
"Let us separate out all of the chain components of shift-$\\mathcal {I}_t$ -EG with no incoming edges.",
"The following lemma proves that all sources are contained within these components.",
"Lemma 4 For any shift-$\\mathcal {I}$ -EG of $\\mathcal {G}$ , each chain component has exactly one source node w.r.t.",
"this component.",
"This node is a source w.r.t.",
"$\\mathcal {G}$ if and only if there are no incoming edges to this component.",
"These results hold when replacing $\\mathcal {G}$ with any induced subgraph of it.",
"Thus, we can find the source nodes in $T$ by finding the source nodes in each of its chain components with no incoming edges."
],
[
"Two Approximate Strategies",
"Following the chain graph decomposition, we now focus on how to find the source node of an undirected connected chordal graph $\\mathcal {C}$ .",
"If there is no sparsity constraint on the number of perturbation targets in each shift intervention, then directly intervening on all of the variables in $\\mathcal {C}$ gives the source node, since by Theorem REF , all DAGs in the shift-$\\mathcal {I}$ -MEC share the same source node.",
"However, when the maximum number of perturbation targets in an intervention is restricted to $S <|\\mathcal {C}|$ , multiple interventions may be necessary to find the source node.",
"After intervening on $S$ nodes, the remaining unoriented part can be decomposed into connected components.",
"In the worst case, the source node of $\\mathcal {C}$ is in the largest of these connected components.",
"Therefore we seek the set of nodes, within the sparsity constraint, that minimizes the largest connected component size after being removed.",
"This is known as the MinMaxC problem [24], which we show is NP-complete on chordal graphs (appendix:alg).",
"We propose two approximate strategies to solve this problem, one based on the clique tree representation of chordal graphs and the other based on robust supermodular optimization.",
"The overall algorithm with these subroutines is summarized in Algorithm REF .",
"We outline the subroutines here, and give further details in appendix:alg.",
"[t] Joint distribution ${\\mathrm {P}}$ , desired joint distribution ${\\mathrm {Q}}$ , sparsity constraint $S$ .",
"Initialize $I^*=\\varnothing $ and $\\mathcal {I}=\\lbrace \\varnothing \\rbrace $ $\\mathbb {E}_{{\\mathrm {P}}^{I^*}}(X)\\ne \\mathbb {E}_{{\\mathrm {Q}}}(X)$ let $T=\\lbrace i|i\\in [p], \\mathbb {E}_{{\\mathrm {P}}^{I^*}}(X_i)\\ne \\mathbb {E}_{{\\mathrm {Q}}}(X_i)\\rbrace $ let $\\mathcal {G}$ be the subgraph of shift-$\\mathcal {I}$ -EG induced by $T$ let $U_T$ be the identified source nodes in $T$ $U_T=\\varnothing $ let $\\mathcal {C}$ be a chain component of $\\mathcal {G}$ with no incoming edges select shift intervention $I$ by running $\\texttt {CliqueTree}(\\mathcal {C},S)$ or $\\texttt {Supermodular}(\\mathcal {C},S)$ perform $I$ and append it to $\\mathcal {I}$ update $\\mathcal {G}$ and $U_T$ as the outer loop set $a_i=\\mathbb {E}_{{\\mathrm {Q}}}(X_i)-\\mathbb {E}_{{\\mathrm {P}}^{I^*}}(X_i)$ for $i$ in $U_T$ include perturbation targets $U_T$ and shift values $\\lbrace a_i\\rbrace _{i\\in U_T}$ in $I^*$ and perform $I^*$ Matching Intervention $I^*$ Active Learning for Causal Mean Matching Clique Tree Strategy.",
"Let $C(\\mathcal {C})$ be the set of maximal cliques in the chordal graph $\\mathcal {C}$ .",
"There exists a clique tree $\\mathcal {T}(\\mathcal {C})$ with nodes in $C(\\mathcal {C})$ and edges satisfying that $\\forall C_1,C_2\\in C(\\mathcal {C})$ , their intersection $C_1\\cap C_2$ is a subset of any clique on the unique path between $C_1,C_2$ in $\\mathcal {T}(\\mathcal {C})$ [5].",
"Thus, deleting a clique which is not a leaf node in the clique tree will break $\\mathcal {C}$ into at least two connected components, each corresponding to a subtree in the clique tree.",
"Inspired by the central node algorithm [14], [32], we find the $S$ -constrained central clique of $\\mathcal {T}(\\mathcal {C})$ by iterating through $C(\\mathcal {C})$ and returning the clique with no more than $S$ nodes that separates the graph most, i.e., solving MinMaxC when interventions are constrained to be maximal cliques.",
"We denote this approach as CliqueTree.",
"Figure: Picking 2 nodes in an undirected connected chordal graph 𝒞\\mathcal {C}.",
"CliqueTree picks {X 4 ,X 5 }\\lbrace X_4,X_5\\rbrace , while Supermodular picks the better {X 3 ,X 6 }\\lbrace X_3,X_6\\rbrace .",
"(a).",
"𝒞\\mathcal {C}; (b).",
"Clique tree 𝒯(𝒞)\\mathcal {T}(\\mathcal {C}).Supermodular Strategy.",
"Our second approach, denoted Supermodular, optimizes a lower bound of the objective of MinMaxC.",
"Consider the following equivalent formulation of MinMaxC $\\min _{A \\subset V_{\\mathcal {C}}} \\max _{i\\in V_{\\mathcal {C}}} f_i(A),\\quad |A|\\le S,$ where $V_{\\mathcal {C}}$ represents the nodes of $\\mathcal {C}$ and $\\forall ~i\\in V_{\\mathcal {C}}$ , $f_i(A) = \\sum _{j\\in V_\\mathcal {C}} g_{i,j}(A)$ with $g_{i,j}(A) = 1$ if $i$ and $j$ are the same or connected after removing nodes in $A$ from $\\mathcal {C}$ and $g_{i,j}(A) = 0$ otherwise.",
"MinMaxC (REF ) resembles the problem of robust supermodular optimization [23].",
"Unfortunately, $f_i$ is not supermodular for chordal graphs (appendix:alg).",
"Therefore, we propose to optimize for a surrogate of $f_i$ defined as $\\hat{f}_i(A) = \\sum _{j\\in \\mathcal {C}} \\hat{g}_{i,j}(A)$ , where $\\hat{g}_{i,j}(A) = {\\left\\lbrace \\begin{array}{ll}\\frac{m_{i,j}(V_{\\mathcal {C}}-A)}{m_{i,j}(V_{\\mathcal {C}})},&\\quad {i--j}~\\mathrm {in}~{\\mathcal {C}}, \\\\0,&\\quad \\mathrm {otherwise}.\\end{array}\\right.",
"}$ Here $m_{i,j}(V_{\\mathcal {C}^{\\prime }})$ is the number of paths without cycles between $i$ and $j$ in $\\mathcal {C}^{\\prime }$ (deemed 0 if $i$ or $j$ does not belong to $\\mathcal {C}^{\\prime }$ and 1 if $i=j\\in \\mathcal {C}^{\\prime }$ ) and $i--j$ means $i$ is either connected or equal to $j$ .",
"Comparing $\\hat{g}_{i,j}$ with $g_{i,j}$ , we see that $\\hat{f}_i(A)$ is a lower bound of $f_i(A)$ for MinMaxC, which is tight when $\\mathcal {C}$ is a tree.",
"We show that $\\hat{f}_i$ is monotonic supermodular for all $i$ (appendix:alg).",
"Therefore, we consider (REF ) with $f_i$ replaced by $\\hat{f}_i$ , which can be solved using the SATURATE algorithm [23].",
"Notably, the results returned by Supermodular can be quite different to those returned by CliqueTree since Supermodular is not constrained to pick a maximal clique; see Figure REF ."
],
[
"Theoretical Results",
" In this section we derive a worst-case lower bound on the number of interventions for any algorithm to identify the source node in a chordal graph.",
"Then we use this lower bound to show that our strategies are optimal up to a logarithmic factor.",
"This contrasts with the structure learning strategy, which may require exponentially more interventions than our strategy (Figure REF ).",
"The worst case is with respect to all feasible orientations of an essential graph [16], [30], i.e., orientations corresponding to DAGs in the equivalence class.",
"Given a chordal chain component $\\mathcal {C}$ of $\\mathcal {G}$ , let $r_{\\mathcal {C}}$ be the number of maximal cliques in $\\mathcal {C}$ , and $m_{\\mathcal {C}}$ be the size of the largest maximal clique in $\\mathcal {C}$ .",
"The following lemma provides a lower bound depending only on $m_{\\mathcal {C}}$ .",
"Lemma 5 In the worst case over feasible orientations of $\\mathcal {C}$ , any algorithm requires at least $\\lceil \\frac{m_{\\mathcal {C}}-1}{S}\\rceil $ shift interventions to identify the source node, under the sparsity constraint $S$ .",
"To give some intuition for this result, consider the case where the largest maximal clique is upstream of all other maximal cliques.",
"Given such an ordering, in the worst case, each intervention rules out only $S$ nodes in this clique (namely, the most downstream ones).",
"Now, we show that our two strategies need at most $ \\lceil \\log _2(r_{\\mathcal {C}}+1)\\rceil \\cdot \\lceil \\frac{m_{\\mathcal {C}}-1}{S}\\rceil $ shift interventions for the same task.",
"Lemma 6 In the worst case over feasible orientations of $\\mathcal {C}$ , both CliqueTree and Supermodular require at most $ \\lceil \\log _2(r_\\mathcal {C}+1)\\rceil \\cdot \\lceil \\frac{m_\\mathcal {C}-1}{S}\\rceil $ shift interventions to identify the source node, under the sparsity constraint $S$ .",
"By combining lemma:oracle-lb and lemma:alg-lb, which consider subproblems of the causal mean matching problem, we obtain a bound on the number of shift interventions needed for solving the full causal mean matching problem.",
"Let $r$ be the largest $r_{\\mathcal {C}}$ for all chain components $\\mathcal {C}$ of $\\mathcal {G}$ : Theorem 2 Algorithm REF requires at most $\\lceil \\log _2(r+1)\\rceil $ times more shift interventions, compared to that required by the optimal strategy, in the worst case over feasible orientations of $\\mathcal {G}$ .",
"A direct application of this theorem is that, in terms of the number of interventions required to solve the causal mean matching problem, our algorithm is optimal in the worst case when $r=1$ , i.e., when every chain component is a clique.",
"All proofs are provided in appendix:bound."
],
[
"Experiments",
"We now evaluate our algorithms in several synthetic settings.",
"Each setting considers a particular graph type, number of nodes $p$ in the graph and number of perturbation targets $|I^*|\\le p$ in the matching intervention.",
"We generate 100 problem instances in each setting.",
"Every problem instance contains a DAG with $p$ nodes generated according to the graph type and a randomly sampled subset of $|I^*|$ nodes denoting the perturbation targets in the matching intervention.",
"We consider both, random graphs including Erdös-Rényi graphs [11] and Barabási–Albert graphs [2], as well as structured chordal graphs, in particular, rooted tree graphs and moralized Erdös-Rényi graphs [30].",
"The graph size $p$ in our simulations ranges from 10 to 1000, while the number of perturbation targets ranges from 1 to $\\min \\lbrace p,100\\rbrace $ .",
"We compare our two subroutines for Algorithm REF , CliqueTree and Supermodular, against three carefully constructed baselines.",
"The UpstreamRand baseline follows Algorithm REF where line 8 is changed to selecting $I$ randomly from $\\mathcal {C}$ without exceeding $S$ , i.e., when there is no identified source node it randomly samples from the chain component with no incoming edge.",
"This strategy highlights how much benefit is obtained from CliqueTree and Supermodular on top of upstream search.",
"The Coloring baseline is modified from the coloring-based policy for structure learning [30], previously shown to perform competitively on large graphs [32].",
"It first performs structure learning with the coloring-based policy, and then uses upstream search with known DAG.",
"We also include an Oracle baseline, which does upstream search with known DAG.",
"In Figure REF we present a subset of our results on Barabási–Albert graphs with 100 nodes; similar behaviors are observed in all other settings and shown in appendix:append-exp.",
"In Figure REF , we consider problem instances with varying size of $|I^*|$ .",
"Each algorithm is run with sparsity constraint $S=1$ .",
"We plot the number of extra interventions compared to Oracle, averaged across the 100 problem instances.",
"As expected, Coloring requires the largest number of extra interventions.",
"This finding is consistent among different numbers of perturbation targets, since the same amount of interventions are used to learn the structure regardless of $I^*$ .",
"As $|I^*|$ increases, CliqueTree and Supermodular outperform UpstreamRand.",
"To further investigate this trend, we plot the rate of extra interventionsThe rate is calculated by (#Strategy-#UpstreamRand)/#UpstreamRand where # denotes the number of extra interventions compared to Oracle and Strategy can be CliqueTree, Supermodular or UpstreamRand.",
"used by CliqueTree and Supermodular relative to UpstreamRand in Figure REF .",
"This figure shows that CliqueTree and Supermodular improve upon upstream search by up to $25\\%$ as the number of perturbation targets increases.",
"Finally, we consider the effect of the sparsity constraint $S$ in Figure REF with $|I^*|=50$ .",
"In line with the discussion in sec:4.3, as $S$ increases, the task becomes easier for plain upstream search.",
"However, when the number of perturbation targets is restricted, CliqueTree and Supermodular are superior, with Supermodular performing best in most cases.",
"Figure: Barabási–Albert graphs with 100 nodes.",
"(a).",
"Averaged (100 instances) numbers of extra interventions each algorithm (with sparsity constraint S=1S=1) requires compared to Oracle, plotted against number of perturbation targets in I * I^*; (b).",
"Rates of extra interventions CliqueTree and Supermodular (S=1S=1) required relative to UpstreamRand, plotted against number of perturbation targets in I * I^*; (c).",
"Relative extra rate (|I * |=50|I^*|=50), plotted against sparsity constraint SS."
],
[
"Discussion",
"In this work, we introduced the causal mean matching problem, which has important applications in medicine and engineering.",
"We aimed to develop active learning approaches for identifying the matching intervention using shift interventions.",
"Towards this end, we characterized the shift interventional Markov equivalence class and showed that it is in general more refined than previously defined equivalence classes.",
"We proposed two strategies for learning the matching intervention based on this characterization, and showed that they are optimal up to a logarithmic factor.",
"We reported experimental results on a range of settings to support these theoretical findings.",
"Limitations and Future Work.",
"This work has various limitations that may be interesting to address in future work.",
"First, we focus on the task of matching a desired mean, rather than an entire distribution.",
"This is an inherent limitation of deterministic shift interventions: as noted by [18], in the linear Gaussian setting, these interventions can only modify the mean of the initial distribution.",
"Thus, matching the entire distribution, or other relevant statistics, will require broader classes of interventions.",
"Assumptions on the desired distribution are also required to rule out possibly non-realizable cases.",
"Second, we have focused on causal DAG models, which assume acyclicity and the absence of latent confounders.",
"In many realistic applications, this could be an overly optimistic assumption, requiring extensions of our results to the cyclic and/or causally insufficient setting.",
"Finally, throughout the main text, we have focused on the noiseless setting; we briefly discuss the noisy setting in appendix:noisy, but there is much room for more extensive investigations."
],
[
"Acknowledgements ",
"C. Squires was partially supported by an NSF Graduate Fellowship.",
"All authors were partially supported by NSF (DMS-1651995), ONR (N00014-17- 1-2147 and N00014-18-1-2765), the MIT-IBM Watson AI Lab, and a Simons Investigator Award to C. Uhler.",
"toc"
],
[
"Meek Rules",
"Given any Markov equivalence class of DAGs with shared directed and undirected edges, the corresponding essential graph $\\mathcal {E}$ can be obtained using a set of logical relations known as Meek rules meek2013causal+.",
"The Meek rules are stated in the following proposition.",
"Proposition 1 (Meek Rules meek2013causal+) We can infer all directed edges in $\\mathcal {E}$ using the following four rules: If $i\\rightarrow j - k$ and $i$ is not adjacent to $k$ , then $j\\rightarrow k$ .",
"If $i\\rightarrow j \\rightarrow k$ and $i-k$ , then $i\\rightarrow k$ .",
"If $i-j, i-k, i-l, j\\rightarrow k, l\\rightarrow k$ and $j$ is not adjacent to $l$ , then $i\\rightarrow k$ .",
"If $i-j, i-k, i-l, j\\leftarrow k, l\\rightarrow k$ and $j$ is not adjacent to $l$ , then $i\\rightarrow j$ .",
"Figure REF illustrates these four rules.",
"Figure: Meek Rules."
],
[
"Proof of Exact Matching",
"[Proof of Lemma REF ] Without loss of generality, assume $1,2,...,p$ is the topological order of the underlying DAG $\\mathcal {G}$ , i.e., $j\\in \\operatorname{pa}_{\\mathcal {G}}(i)$ implies $j<i$ .",
"We will first construct $I^*$ such that $\\mathbb {E}_{{\\mathrm {P}}^{I^*}}(X)=\\mathbb {E}_{{\\mathrm {Q}}}(X)$ , and then show that $I^*$ is unique.",
"Existence: Denote $i_1$ as the smallest $i\\in [p]$ such that $\\mathbb {E}_{{\\mathrm {P}}}(X_{i})\\ne \\mathbb {E}_{{\\mathrm {Q}}}(X_i)$ .",
"Witout loss of generality we assume that $i_1$ exists (if $i_1$ does not exists, then $I^*=\\varnothing $ suffices since $\\mathbb {E}_{{\\mathrm {P}}}(X)=\\mathbb {E}_{{\\mathrm {Q}}}(X)$ ).",
"Let $I_1$ be the shift intervention with perturbation target $i_1$ and shift values $a_{i_1}= \\mathbb {E}_{{\\mathrm {Q}}}(X_{i_1})-\\mathbb {E}_{{\\mathrm {P}}}(X_{i_1})$ .",
"Since ${\\mathrm {P}}^{I_1}(X_{i_1}=x+a_{i_1}|X_{\\operatorname{pa}_{\\mathcal {G}}(i_1)})={\\mathrm {P}}(X_{i_1}=x|X_{\\operatorname{pa}_{\\mathcal {G}}(i_1)})$ and ${\\mathrm {P}}^{I_1}(X_{\\operatorname{pa}_{\\mathcal {G}}(i_1)})={\\mathrm {P}}(X_{\\operatorname{pa}_{\\mathcal {G}}(i_1)})$ by definition, we have ${\\mathrm {P}}^{I_1}(X_{i_1}=x+a_{i_1})={\\mathrm {P}}(X_{i_1}=x).$ Thus $\\mathbb {E}_{{\\mathrm {P}}^{I_1}}(X_{i_1}) = \\mathbb {E}_{{\\mathrm {P}}}(X_{i_1})+a_{i_1} = \\mathbb {E}_{{\\mathrm {Q}}}(X_{i_1})$ .",
"Also $\\mathbb {E}_{{\\mathrm {P}}^{I_1}}(X_{i}) = \\mathbb {E}_{{\\mathrm {Q}}}(X_{i})$ for $i<i_1$ .",
"Denote $i_2$ as the smallest $i\\in [p]$ such that $\\mathbb {E}_{{\\mathrm {P}}^{I_1}}(X_i)\\ne \\mathbb {E}_{{\\mathrm {Q}}}(X_i)$ .",
"If $i_2$ does not exists, then $I^*=I_1$ suffices.",
"Otherwise $i_2>i_1$ .",
"Let $I_2$ be the shift intervention with perturbation target $i_1,i_2$ and corresponding shift values $a_{i_1}$ and $a_{i_2} = \\mathbb {E}_{{\\mathrm {Q}}}(X_{i_2})-\\mathbb {E}_{{\\mathrm {P}}^{I_1}}(X_{i_2})$ .",
"We have ${\\mathrm {P}}^{I_2}(X_{i_2}=x+a_{i_2}|X_{\\operatorname{pa}_{\\mathcal {G}}(i_2)})={\\mathrm {P}}(X_{i_2}=x|X_{\\operatorname{pa}_{\\mathcal {G}}(i_2)})={\\mathrm {P}}^{I_1}(X_{i_2}=x|X_{\\operatorname{pa}_{\\mathcal {G}}(i_2)})$ and ${\\mathrm {P}}^{I_2}(X_{\\operatorname{pa}_{\\mathcal {G}}(i_2)})={\\mathrm {P}}^{I_1}(X_{\\operatorname{pa}_{\\mathcal {G}}(i_2)})$ by definition, the topological order, and $i_2>i_1$ .",
"Then ${\\mathrm {P}}^{I_2}(X_{i_2}=x+a_{i_2}) = {\\mathrm {P}}^{I_1}(X_{i_2}=x).$ Thus $\\mathbb {E}_{{\\mathrm {P}}^{I_2}}(X_{i_2}) = \\mathbb {E}_{{\\mathrm {P}}^{I_1}}(X_{i_2})+a_{i_2} = \\mathbb {E}_{{\\mathrm {Q}}}(X_{i_2})$ .",
"Also $\\mathbb {E}_{{\\mathrm {P}}^{I_2}}(X_{i})=\\mathbb {E}_{{\\mathrm {P}}^{I_1}}(X_{i})=\\mathbb {E}_{{\\mathrm {Q}}}(X_{i})$ for $i<i_2$ .",
"By iterating this process, we will reach $I_k$ for some $k\\le p$ such that there is no $i$ with $\\mathbb {E}_{{\\mathrm {P}}^{I_k}}(X_i)\\ne \\mathbb {E}_{{\\mathrm {Q}}}(X_k)$ .",
"Taking $I^*=I_k$ suffices.",
"Uniqueness: If there exists $I_1^*\\ne I_2^*$ such that $\\mathbb {E}_{{\\mathrm {P}}^{I_1^*}}(X)=\\mathbb {E}_{{\\mathrm {P}}^{I_2^*}}(X)=\\mathbb {E}_{{\\mathrm {Q}}}(X)$ , let $i\\in [p]$ be the smallest index such that either $i$ has different shift values in $I_1^*$ and $I_2^*$ , or $i$ is only in one intervention's perturbation targets.",
"In either case, we have ${\\mathrm {P}}^{I_1^*}(X_{\\operatorname{pa}_{\\mathcal {G}}(i)}) = {\\mathrm {P}}^{I_2^*}(X_{\\operatorname{pa}_{\\mathcal {G}}(i)})$ by the topological order and ${\\mathrm {P}}^{I_1^*}(X_{i}=x|X_{\\operatorname{pa}_{\\mathcal {G}}(i)})= {\\mathrm {P}}^{I_2^*}(X_{i}=x+a|X_{\\operatorname{pa}_{\\mathcal {G}}(i)})$ for some $a\\ne 0$ .",
"Thus ${\\mathrm {P}}^{I_1^*}(X_{i}=x)= {\\mathrm {P}}^{I_2^*}(X_{i}=x+a)$ contradicting $\\mathbb {E}_{{\\mathrm {P}}^{I_1^*}}(X_i)=\\mathbb {E}_{{\\mathrm {P}}^{I_2^*}}(X_i)$ ."
],
[
"Shift Interventional MEC",
"[Proof of Lemma REF ] For any distribution $f$ that factorizes according to $\\mathcal {G}$ and shift intervention $I$ , let $i\\in I$ be any source w.r.t.",
"$I$ .",
"By definition, $\\operatorname{an}_{\\mathcal {G}}(i)\\cap I=\\varnothing $ .",
"Thus $\\operatorname{pa}_{\\mathcal {G}}(i)$ contains neither a member nor a descendant of $I$ , i.e., there does not exists $j\\in \\operatorname{pa}_{\\mathcal {G}}(i)$ and $k\\in I$ such that there is a direct path from $k$ to $j$ or $k=j$ .",
"Hence we have $f^I(X_{\\operatorname{pa}_{\\mathcal {G}}(i)}) = f(X_{\\operatorname{pa}_{\\mathcal {G}}(i)})$ , which gives $f^I(X_{i}=x+a_i) = f(X_{i}=x).$ Therefore $\\mathbb {E}_{f^I}(X_i) = \\mathbb {E}_{f}(X_i)+a_i$ .",
"On the other hand, if $i\\in I$ is not a source w.r.t.",
"$I$ , consider the following linear Gaussian model, $X_j = \\sum _{k\\in \\operatorname{pa}_{\\mathcal {G}}(j)} \\beta _{kj} X_k + \\epsilon _j, \\quad \\forall j\\in [p],$ where $\\beta _{kj}$ are deterministic scalars and $\\epsilon _j\\sim \\mathcal {N}(0,1)$ are i.i.d.",
"random variables.",
"Since $i$ is not a source in $I$ , there exists a source $i^{\\prime }$ in $I$ such that there is a directed path $i^{\\prime } = i_0 \\rightarrow i_1 \\rightarrow \\dots \\rightarrow i_\\ell $ .",
"From above, $\\mathbb {E}_{f^I}(X_{i^{\\prime }})= \\mathbb {E}_{f}(X_{i^{\\prime }})+a_{i^{\\prime }}$ for $a_{i^{\\prime }}\\ne 0$ .",
"Consider setting $\\beta _{i_0,i_1} = 2|a_i|/a_{i^{\\prime }}$ , $\\beta _{i_k,i_{k+1}} = 1$ for $k = 1,\\ldots ,\\ell -1$ , and the remaining edge weights to $\\epsilon > 0$ .",
"For $\\epsilon $ sufficiently small, we have that $\\mathbb {E}_{f^I} (X_i) \\ge \\mathbb {E}_f (X_i) + 1.5|a_i|$ , i.e., we cannot have that $\\mathbb {E}_{f^I}(X_i) = \\mathbb {E}_f (X_i) + a_i$ .",
"[Proof of Theorem REF ] Denote $\\mathcal {I}=\\lbrace I_1,...,I_m\\rbrace $ .",
"For $k\\in [m]=\\lbrace 1,...,m\\rbrace $ , let $\\hat{I}_k$ and $\\hat{I}_k^{\\prime }$ be the collection of source nodes in $I_k$ in $\\mathcal {G}_1$ and $\\mathcal {G}_2$ , respectively.",
"From def:1, we know that $\\mathcal {G}_1$ and $\\mathcal {G}_2$ are in the same shift-$\\mathcal {I}$ -MEC if and only if they are in the same $\\mathcal {I}$ -MEC and, for any pair $(f,\\lbrace f^{I_k}\\rbrace _{k\\in [m]})$ that is $\\mathcal {I}$ -Markov w.r.t.",
"both $\\mathcal {G}_1$ and $\\mathcal {G}_2$ , it satisfies $\\mathbb {E}_{f^{I_k}}(X_i)=\\mathbb {E}_{f}(X_i)+a_{i}, \\quad \\forall i \\in \\hat{I}_k, \\forall k \\in [m],$ if and only if it also satisfies $\\mathbb {E}_{f^{I_k}}(X_i)=\\mathbb {E}_{f}(X_i)+a_{i}, \\quad \\forall i \\in \\hat{I}_k^{\\prime }, \\forall k \\in [m].$ By Lemma REF , we know that $\\hat{I}_k^{\\prime }\\subset \\hat{I}_k$ for all $k\\in [m]$ .",
"Otherwise we can find a pair $(f,\\lbrace f^{I_k}\\rbrace _{k\\in [m]})$ that violates (REF ) for $i\\in \\hat{I}_k^{\\prime }\\setminus \\hat{I}_k$ .",
"Similarly, we have $\\hat{I}_k\\subset \\hat{I}_k^{\\prime }$ .",
"Therefore $\\hat{I}_k=\\hat{I}_k^{\\prime }$ .",
"In this case, (REF ) is equivalent to (REF ).",
"Hence, $\\mathcal {G}_1$ and $\\mathcal {G}_2$ are in the same shift-$\\mathcal {I}$ -MEC if and only if they are in the same $\\mathcal {I}$ -MEC and they have the same source nodes of $I$ for every $I\\in \\mathcal {I}$ .",
"From Theorem 3.9 in yang2018characterizing+, we know that $\\mathcal {G}_1$ and $\\mathcal {G}_2$ are in the same $\\mathcal {I}$ -MEC if and only if they share the same skeleton, $v$ -structures and directed edges $\\lbrace i\\rightarrow j|i\\in I,j\\notin I, I\\in \\mathcal {I}, i-j\\rbrace $ .",
"Therefore, $\\mathcal {G}_1$ and $\\mathcal {G}_2$ are in the same shift-$\\mathcal {I}$ -MEC if and only if they have the same skeleton, $v$ -structures, directed edges $\\lbrace i\\rightarrow j|i\\in I,j\\notin I, I\\in \\mathcal {I}, i-j\\rbrace $ , as well as source nodes of $I$ for every $I\\in \\mathcal {I}$ .",
"Let $\\mathcal {D}$ be any DAG, suppose that $\\mathcal {I}=\\lbrace I_1,...,I_m\\rbrace $ and $\\hat{I}_k$ is the collection of source nodes in $I_k$ in $\\mathcal {D}$ for $k\\in [m]$ .",
"Then as a direct corollary of thm:1, we can represent a shift interventional Markov equivalence class with a (general) interventional Markov equivalence class.",
"Corollary 1 Let $\\hat{\\mathcal {I}}=\\mathcal {I}\\cup \\lbrace \\hat{I}_k|k\\in [m]\\rbrace $ ; a DAG $\\mathcal {D}^{\\prime }$ is shift-$\\mathcal {I}$ -Markov equivalent to $\\mathcal {D}$ if and only if $\\mathcal {D}^{\\prime }$ is $\\hat{\\mathcal {I}}$ -Markov equivalent to $\\mathcal {D}$ .",
"The proof follws as a direct application of thm:1, Theorem 3.9 in yang2018characterizing+, and the fact that there are no edges between nodes in $\\hat{I}_k$ ."
],
[
"Mean Interventional Faithfulness",
"[Proof of Lemma REF ] If Assumption REF holds, then for any $i\\notin T$ , since $\\mathbb {E}_{{\\mathrm {P}}}(X_i)= \\mathbb {E}_{{\\mathrm {Q}}}(X_i)$ , then $i\\notin I^*$ and $\\operatorname{an}_{\\mathcal {G}}(i)\\cap I^*=\\varnothing $ .",
"Let $j\\in T$ such that there is an edge $i-j$ between $i$ and $j$ .",
"Since $\\mathbb {E}_{{\\mathrm {P}}}(X_j)\\ne \\mathbb {E}_{{\\mathrm {Q}}}(X_j)$ , there is either $j\\in I^*$ or $\\operatorname{an}_{\\mathcal {G}}(j)\\cap I^*\\ne \\varnothing $ .",
"Therefore if $j\\rightarrow i$ , then $\\operatorname{an}_{\\mathcal {G}}(i)\\cap I^*\\ne \\varnothing $ , a contradiction.",
"Thus $j\\leftarrow i$ .",
"Conversely, if Assumption REF does not hold, then there exists $i\\notin T$ (i.e., $\\mathbb {E}_{{\\mathrm {P}}}(X_i)= \\mathbb {E}_{{\\mathrm {Q}}}(X_i)$ ) such that either $i\\in I^*$ or $\\operatorname{an}_{\\mathcal {G}}(i)\\cap I^*\\ne \\varnothing $ .",
"If $i\\in I^*$ , then since $\\mathbb {E}_{{\\mathrm {P}}}(X_i)= \\mathbb {E}_{{\\mathrm {Q}}}(X_i)$ and Lemma REF , $i$ must not be a source in $I^*$ .",
"Therefore we only need to discuss the case where $i\\notin T$ and $\\operatorname{an}_{\\mathcal {G}}(i)\\cap I^*\\ne \\varnothing $ .",
"Let $k$ be a source of $\\operatorname{an}_{\\mathcal {G}}(i)\\cap I^*$ , then $k$ must also be a source of $I^*$ .",
"Otherwise there is a directed path from $k^{\\prime }$ to $k$ where $k^{\\prime }\\ne k$ and $k^{\\prime }\\in I^*$ .",
"By definition of ancestors, we know from $k\\in \\operatorname{an}_{\\mathcal {G}}(i)$ that there is also $k^{\\prime }\\in \\operatorname{an}_{\\mathcal {G}}(i)$ .",
"Therefore $k^{\\prime }\\in \\operatorname{an}_{\\mathcal {G}}(i)\\cap I^*$ , which violates $k$ being a source of $\\operatorname{an}_{\\mathcal {G}}(i)\\cap I^*$ .",
"Since $k$ is a source of $I^*$ , by Lemma REF and REF , we know that $\\mathbb {E}_{{\\mathrm {P}}}(X_k)\\ne \\mathbb {E}_{{\\mathrm {Q}}}(X_k)$ , i.e., $k\\in T$ .",
"Notice that $k\\in \\operatorname{an}_{\\mathcal {G}}(i)$ , and thus we must have a directed path from $k\\in T$ to $i\\notin T$ .",
"Thus, there exists some $i-j,j\\in T,i\\notin T$ such that $j\\rightarrow i$ .",
"Using Lemma REF , we know that we can check the authenticity of Assumption REF by looking at the orientation of edges between $T$ and $[p]\\setminus T$ , which is achievable by any (general) intervention on $X_{T}$ (or $X_{[p]\\setminus T}$ ).",
"Corollary 2 Assumption REF holds if and only if the $\\lbrace T\\rbrace $ -essential graph (or $\\lbrace [p]\\setminus T\\rbrace $ -essential graph) of $\\mathcal {G}$ has edges $j\\leftarrow i$ for all $i-j,j\\in T, i\\notin T$ .",
"The proof follows as a direct application of the graphical characterization of interventional equivalence class in sec:id-1 and the results in Lemma REF ."
],
[
"Decomposition of Shift Interventional Essential Graphs",
"Chain Graph Decomposition: hauser2014two+ showed that every interventional essential graph is a chain graph with undirected connected chordal chain components, where the orientations in one component do not affect any other components.",
"This decomposition also holds for shift interventional essential graphs, since every shift interventional essential graph is also an interventional essential graph (Corollary REF ).",
"Below, we show an example of this decomposition (Figure REF ).",
"Figure: Chain graph decomposition of the essential graph in (a).",
"[Proof of Lemma REF ] Suppose an undirected connected chain component $\\mathcal {C}$ of the essential graph has two source nodes $i$ and $j$ w.r.t.",
"$\\mathcal {C}$ .",
"Since $\\mathcal {C}$ is connected, there is a path between $i$ and $j$ in $\\mathcal {C}$ ; let $i-k_1-...-k_r-j$ be the shortest among all these paths.",
"Because $i$ and $j$ are sources of $\\mathcal {C}$ , there must be $i\\rightarrow k_1$ and $k_r\\leftarrow j$ .",
"Therefore, $\\exists l\\in \\lbrace 1,...,r\\rbrace $ such that $k_{l-1}\\rightarrow k_l\\leftarrow k_{l+1}$ (let $k_0=i$ and $k_{r+1}=j$ ).",
"By the shortest path definition, there is no edge between $k_{l-1}$ and $k_{l+1}$ .",
"Therefore there is a v-structure in $\\mathcal {C}$ induced by $k_{l-1}\\rightarrow k_l\\leftarrow k_{l+1}$ .",
"Since all DAGs in the same shift interventional equivalence class share the same v-structures, $k_{l-1}\\rightarrow k_l\\leftarrow k_{l+1}$ must be oriented in the essential graph.",
"This violates $k_{l-1},k_l, k_{l+1}$ belonging to the same undirected chain component $\\mathcal {C}$ .",
"Thus, combining this with the fact that $\\mathcal {C}$ must have one source node, we obtain that $\\mathcal {C}$ has exactly one source node w.r.t.",
"$\\mathcal {C}$ .",
"Next we show that the source node of a chain component is also the source of $\\mathcal {G}$ if and only if there are no incoming edges to this component.",
"Let $i$ be the source of the chain component $\\mathcal {C}$ .",
"On one hand, $i$ must be the source of $\\mathcal {G}$ if there is no incoming edges to $\\mathcal {C}$ .",
"On the other hand, if there is an incoming edge $j\\rightarrow k$ for some $j\\notin \\mathcal {C}$ and $k\\in \\mathcal {C}$ , then since the essential graph is closed under Meek R1 and R2 (Proposition REF ), we know that there must be an edge $j\\rightarrow l$ for all neighbors $l$ of $k$ .",
"Following the same deduction and the fact that $\\mathcal {C}$ is connected, we obtain that $j\\rightarrow l$ for all $l\\in \\mathcal {C}$ (Figure REF ).",
"This means that $j\\rightarrow i$ as well.",
"Therefore $i$ cannot be a source of $\\mathcal {G}$ .",
"Figure: j→lj\\rightarrow l for all l∈𝒞l\\in \\mathcal {C}."
],
[
"NP-completeness of MinMaxC",
"It was shown separately in shen2012exact and lalou2018critical+ that the MinMaxC problem is NP-complete for general graphs and split graphs.",
"Split graphs are a subclass of chordal graphs, where the vertices can be separated into a clique and an independent set (isolated nodes after removing the clique).",
"Thus, MinMaxC is also NP-complete for chordal graphs."
],
[
"Clique Tree Strategy",
"The clique tree strategy takes inputs of an undirected connected chordal graph $\\mathcal {C}$ and the sparsity constraint $S$ , and outputs a shift intervention with no more than $S$ perturbation targets.",
"If $\\mathcal {C}$ contains no more than $S$ nodes, then it returns any shift intervention with perturbation targets in $\\mathcal {C}$ .",
"If $\\mathcal {C}$ contains more than $S$ nodes, it first constructs a clique tree $\\mathcal {T}(\\mathcal {C})$ of $\\mathcal {C}$ by the maximum-weight spanning tree algorithm koller2009probabilistic+.",
"Then it iterates through the nodes in $\\mathcal {T}(\\mathcal {C})$ (which are maximal cliques in $\\mathcal {C}$ ) to find a maximal clique $\\mathcal {K}$ that breaks $\\mathcal {T}(\\mathcal {C})$ into subtrees with sizes no more than half of the size of $\\mathcal {T}(\\mathcal {C})$ .",
"If $\\mathcal {K}$ has no more than $S$ nodes, then it returns any shift intervention with perturbation targets in $\\mathcal {K}$ .",
"Otherwise, it samples $S$ nodes from $\\mathcal {K}$ and returns any shift intervention with these $S$ nodes as perturbation targets.",
"The following subroutine summarizes this procedure.",
"[ht] Chordal chain component $\\mathcal {C}$ , sparsity constraint $S$ .",
"$\\mathcal {C}$ has no more than $S$ nodes set $I$ as any shift intervention on $\\mathcal {C}$ with non-zero shift values let $C(\\mathcal {C})$ be the maximal cliques of the chordal graph $\\mathcal {C}$ let $\\mathcal {T}(\\mathcal {C})$ be a maximum-weight spanning tree of $\\mathcal {C}$ with $C(\\mathcal {C})$ as nodes set $\\mathcal {K}=\\varnothing $ $K$ in $C(\\mathcal {C})$ get the subtrees of $\\mathcal {T}(\\mathcal {C})$ after deleting node $C$ all subtrees has size $\\le \\lceil (|C(\\mathcal {C})|-1)/2\\rceil $ set $\\mathcal {K}=K$ break $|\\mathcal {K}|>S$ set $\\mathcal {K}$ as a random $S$ -subset of $\\mathcal {K}$ set $I$ as any shift intervention on $\\mathcal {K}$ with non-zero shift values Shift Intervention $I$ $\\texttt {CliqueTree}(\\mathcal {C},S)$ Complexity: Let $N$ represent the number of nodes in $\\mathcal {C}$ , i.e., $N=|\\mathcal {C}|$ .",
"All the maximal cliques of the chordal graph $\\mathcal {C}$ can be found in $O(N^2)$ time galinier1995chordal.",
"We use Kruskal's algorithm for computing the maximum-weight spanning tree, which can be done in $O(N^2\\log (N))$ kruskal1956shortest.",
"The remaining procedure of iterating through $C(\\mathcal {C})$ takes no more than $O(N^2)$ since chordal graphs with $N$ nodes have no more than $N$ maximal cliques galinier1995chordal and all subtree sizes can be obtained in $O(N)$ .",
"Therefore this subroutine can be computed in $O(N^2\\log (N))$ time."
],
[
"Supermodular Strategy",
"The supermodular procedure takes as input an undirected connected chordal graph $\\mathcal {C}$ as well as the sparsity constraint $S$ , and outputs a shift intervention with perturbation targets by solving $\\min _{A\\subset V_{\\mathcal {C}}}\\max _{i\\in V_{\\mathcal {C}}} \\hat{f}_i(A),\\quad |A|\\le S,$ with the SATURATE algorithm krause2008robust+.",
"Here $V_{\\mathcal {C}}$ represents nodes of $\\mathcal {C}$ and $\\hat{f}_i(A) =\\sum _{j\\in V_{\\mathcal {C}}} \\hat{g}_{i,j}(A)$ with $\\hat{g}_{i,j}$ defined in (REF ).",
"Algorithm REF summarizes this subroutine.",
"[ht] Chordal chain component $\\mathcal {C}$ , sparsity constraint $S$ .",
"$\\mathcal {C}$ has no more than $S$ nodes set $I$ as any shift intervention on $\\mathcal {C}$ with non-zero shift values let $A$ be the solution of (REF ) returned by SATURATE krause2008robust+ set $I$ as any shift intervention on $A$ with non-zero shift values Shift Intervention $I$ $\\texttt {Supermodular}(\\mathcal {C},S)$ Supermodularity: First we give an example showing that $f_i$ defined in (REF ) is not supermodular for chordal graphs, although it is clearly monotonic decreasing.",
"Example 2 Consider the chordal graph in Figure REF ; we have $f_1(\\lbrace 2\\rbrace )-f_1(\\varnothing )=3-4=-1 > -2 =1-3=f_1(\\lbrace 2,3\\rbrace )-f_1(\\lbrace 3\\rbrace )$ .",
"Therefore $f_1$ is not supermodular for this graph.",
"Figure: f i f_i is not supermodular.Next we prove that $\\hat{f}_i$ is supermodular and monotonic decreasing.",
"Since $\\hat{f}_i(A)=\\sum _{j\\in V_{\\mathcal {C}}} \\hat{g}_{i,j}(A)$ , we only need to show that every $\\hat{g}_{i,j}$ is supermodular and monotonic decreasing.",
"In the following, we refer to a path without cycles as a simple path.",
"For any $A\\subset B\\subset V_{\\mathcal {C}}$ , since $V_{\\mathcal {C}}-B$ is a subgraph of $V_{\\mathcal {C}}-A$ , then any simple path between $i$ and $j$ in $V_{\\mathcal {C}}-B$ must also be in $V_{\\mathcal {C}}-A$ .",
"Hence $m_{i,j}(V_{\\mathcal {C}}-B)\\le m_{i,j}(V_{\\mathcal {C}}-A)$ , which means that $\\hat{g}_{i,j}(A)\\ge \\hat{g}_{i,j}(B),$ i.e., $\\hat{g}_{i,j}$ is monotonic decreasing.",
"For any $x\\in V_{\\mathcal {C}}\\setminus B$ , the difference $m_{i,j}(V_{\\mathcal {C}}-B)-m_{i,j}(V_{\\mathcal {C}}-B\\cup \\lbrace x\\rbrace )$ is the number of simple paths in $V_{\\mathcal {C}}-B$ between $i$ and $j$ that pass through $x$ .",
"Each of these paths must also be in $V_{\\mathcal {C}}-A$ , since $V_{\\mathcal {C}}-B$ is a subgraph of $V_{\\mathcal {C}}-A$ .",
"Therefore, $m_{i,j}(V_{\\mathcal {C}}-B)-m_{i,j}(V_{\\mathcal {C}}-B\\cup \\lbrace x\\rbrace ) \\le m_{i,j}(V_{\\mathcal {C}}-A)-m_{i,j}(V_{\\mathcal {C}}-A\\cup \\lbrace x\\rbrace ),$ which means that $\\hat{g}_{i,j}(A\\cup \\lbrace x\\rbrace )-\\hat{g}_{i,j}(A) \\le \\hat{g}_{i,j}(B\\cup \\lbrace x\\rbrace )-\\hat{g}_{i,j}(B),$ i.e., $\\hat{g}_{i,j}$ is supermodular.",
"SATURATE algorithm krause2008robust+: Having shown that $\\hat{f}_i$ is monotonic supermodular, we solve the robust supermodular optimization problem in (REF ) with the SATURATE algorithm in krause2008robust+.",
"SATURATE performs a binary search for potential objective values and uses a greedy partial cover algorithm to check the feasibility of these objective values; for a detailed description of the algorithm, see krause2008robust+.",
"Complexity: Let $N$ represent the number of nodes in $\\mathcal {C}$ , i.e., $N=|\\mathcal {C}|$ .",
"SATURATE uses at most $O(N^2S\\log (N))$ evaluations of supermodular functions $\\hat{f}_i$ krause2008robust+.",
"Each $\\hat{f}_i$ computes all the simple paths between $i$ and all other $j$ in $\\mathcal {C}$ .",
"A modified depth-first search is used to calculated these paths sedgewick2001algorithms, which results in $\\mathcal {F}(N)$ complexity.",
"For general graphs, this problem is #P-complete valiant1979complexity.",
"However, this might be significantly reduced for chordal graphs.",
"We are unaware of particular complexity results for chordal graphs, which would be an interesting direction for future work.",
"The total runtime of this subroutine is thus bounded by $O(N^2\\mathcal {F}(N)S\\log (N))$ .For a more efficient implementation, one could replace the undirected graph with a DAG in its MEC (which can be found in linear time using L-BFS).",
"All statements hold except that $\\hat{f}_i$ is no longer necessarily tight for tree graphs.",
"This replacement results in a total complexity of $O(N^4 S\\log (N))$ for the subroutine, since directed simple paths can be counted in $O(N^2)$ ."
],
[
"Violation of Faithfulness",
"From Corollary REF , we know that we can check whether Assumption REF holds or not by any intervention on $X_{T}$ (or $X_{[p]\\setminus T}$ ).",
"However, we can run Algorithm REF to obtain $I^*$ without Assumption REF because lines 2-14 in Algorithm REF always return the correct $I^*$ .",
"Let $I\\subset I^*$ be the resolved part of $I^*$ in line 2, i.e., it is a shift intervention constructed by taking a subset of perturbation targets of $I^*$ and their corresponding shift values.",
"Let $I^*-I$ be the remaining shift intervention constructed by deleting $I$ in $I^*$ .",
"Denote $T_{I}=\\lbrace i|i\\in [p], \\mathbb {E}_{{\\mathrm {P}}^{I}}(X_i)\\ne \\mathbb {E}_{{\\mathrm {Q}}}(X_i)\\rbrace $ , which is returned by line 3.",
"If $T_I\\ne \\varnothing $ , then we have solved $I^*$ .",
"Otherwise we have: Lemma 7 The source nodes w.r.t.",
"$T_I$ must be perturbation targets of $I^*-I$ and their corresponding shift values are $\\mathbb {E}_{{\\mathrm {Q}}}(X_i)-\\mathbb {E}_{{\\mathrm {P}}^{I}}(X_i)$ (for source node $i$ ).",
"Let $i$ be a source node w.r.t.",
"$I^*-I$ and $a_i$ be its corresponding shift value.",
"Since intervening on other nodes in $I^*-I$ does not change the marginal distribution of $i$ , we must have that $a_i =\\mathbb {E}_{({\\mathrm {P}}^{I})^{I^*-I}}(X_i)-\\mathbb {E}_{{\\mathrm {P}}^{I}}(X_i)$ .",
"And because $ ({\\mathrm {P}}^{I})^{I^*-I}={\\mathrm {P}}^{I^*} = {\\mathrm {Q}}$ , we know that $a_i = \\mathbb {E}_{{\\mathrm {Q}}}(X_i)-\\mathbb {E}_{{\\mathrm {P}}^{I}}(X_i).$ From this, we also have that $\\mathbb {E}_{{\\mathrm {P}}^{I}}(X_i)\\ne \\mathbb {E}_{{\\mathrm {Q}}}(X_i)$ since $a_i\\ne 0$ .",
"Therefore, all source nodes $i$ w.r.t.",
"$I^*-I$ are in $T_I$ and their corresponding shift values are $\\mathbb {E}_{{\\mathrm {Q}}}(X_i)-\\mathbb {E}_{{\\mathrm {P}}^{I}}(X_i)$ .",
"Let $i$ be a source w.r.t.",
"$T_I$ , then $i$ must also be a source node w.r.t.",
"$I^*-I$ .",
"Since $\\mathbb {E}_{{\\mathrm {P}}^{I}}(X_i)\\ne \\mathbb {E}_{{\\mathrm {Q}}}(X_i)$ , $i$ must be a source node in $I^*-I$ or has a source node in $I^*-I$ as its ancestor.",
"If it is the latter case, then since all source nodes in $I^*-I$ must be in $T_I$ , $i$ cannot be a source node w.r.t.",
"$T_I$ , a contradiction.",
"Therefore the source w.r.t.",
"$T_I$ must also be the source w.r.t.",
"$I^*-I$ .",
"Combined with the result in the previous paragraph, we have that all source nodes $i$ w.r.t.",
"$T_I$ are perturbation targets of $I^*-I$ and their corresponding shift values are $\\mathbb {E}_{{\\mathrm {Q}}}(X_i)-\\mathbb {E}_{{\\mathrm {P}}^{I}}(X_i)$ .",
"This lemma shows that $U_T$ obtained in lines 5-11 of Algorithm REF must be the perturbation targets of $I^*-I$ and line 12 gives the correct shift values.",
"Therefore Algorithm REF must return the correct $I^*$ .",
"However, to be able to obtain the shift-$\\mathcal {I}$ -EG of $\\mathcal {G}$ , we need mean interventional faithfulness to be satisfied by $I\\in \\mathcal {I}$ (replacing $I^*$ with $I$ and ${\\mathrm {Q}}$ with ${\\mathrm {P}}^{I}$ in Assumption REF ) as well as $\\mathcal {I}$ -faithfulness squires2020permutation+ to be satisfied by $({\\mathrm {P}},\\lbrace {\\mathrm {P}}^I\\rbrace _{I\\in \\mathcal {I}})$ with respect to $\\mathcal {G}$ ."
],
[
"Proof of Lemma ",
"To show Lemma REF , we need the following proposition, which states that we can orient any maximal clique of a chordal graph to be most-upstream without creating cycles and v-structures, and the orientation in this clique can be made arbitrary.",
"It was pointed out in vandenberghe2015chordal using similar arguments that any clique of a chordal graph can be most-upstream.",
"Here, we provide the complete proof.",
"Proposition 2 Let $\\mathcal {D}$ be any undirected chordal graph and $K$ be any maximal clique of $\\mathcal {D}$ , for any permutation $\\pi _K$ of the nodes in $K$ , there exists a topological order $\\pi $ of the nodes in $\\mathcal {D}$ such that $\\pi $ starts with $\\pi _K$ and orienting $\\mathcal {D}$ according to $\\pi $ does not create any v-structures.",
"A topological order $\\pi $ of a chordal graph $\\mathcal {D}$ , orienting according to which does not create v-structures, corresponds to the reverse of a perfect elimination order hauser2014two+.",
"A perfect elimination order is an order of nodes in $\\mathcal {D}$ , such that all neighbors of $i$ in $\\mathcal {D}$ that appear after $i$ in this order must constitute a clique in $\\mathcal {D}$ .",
"Any chordal graph has at least one perfect elimination order andersson1997characterization+.",
"In the following, we will use the reverse of a perfect elimination order to refer to a topological order that does not create v-structures.",
"To prove Proposition REF , we first prove the following statement: if $K\\ne \\mathcal {D}$ , then there exists a perfect elimination order of nodes in $\\mathcal {D}$ that starts with a node not in $K$ .",
"To show this, by Proposition 6 in hauser2014two+, we only need to prove that if $K\\ne \\mathcal {D}$ , then there is a node not in $K$ , whose neighbors in $\\mathcal {D}$ constitute a clique.",
"We use induction on the number of nodes in $\\mathcal {D}$ : Consider $|\\mathcal {D}|=1$ .",
"Since $K$ is a maximal clique, $K=\\mathcal {D}$ .",
"This statement holds trivially.",
"Suppose the statement is true for chordal graphs with size $n-1$ .",
"Consider $|\\mathcal {D}|=n$ .",
"Since $\\mathcal {D}$ is a chordal graph, it must have a perfect elimination order.",
"If this perfect elimination order starts with $i\\in K$ , then there is no edge between $i$ and any node $j\\notin K$ .",
"Otherwise, since it is a perfect elimination order starting with $i$ and $K\\ni i$ is a clique, there must be edges $j-k$ for all $k\\in K$ .",
"This is a contradiction to $K$ being a maximal clique.",
"Figure: NO_CAPTIONConsider the chordal graph $\\mathcal {D}^{\\prime }$ by deleting $i$ from $\\mathcal {D}$ , $|\\mathcal {D}^{\\prime }|=n-1$ .",
"Let $K^{\\prime }$ be the maximal clique in $\\mathcal {D}^{\\prime }$ containing $K\\setminus \\lbrace i\\rbrace $ .",
"If $K^{\\prime }=\\mathcal {D}^{\\prime }$ , let $j$ be any node in $\\mathcal {D}\\setminus K$ .",
"Since there is no edge $j-i$ , and $\\mathcal {D}^{\\prime }\\ni j$ is a clique.",
"$j$ 's neighbors in $\\mathcal {D}$ must also constitute a clique.",
"If $K^{\\prime }\\ne \\mathcal {D}^{\\prime }$ , then by induction, we know that there exists $j\\in \\mathcal {D}^{\\prime }\\setminus K^{\\prime }$ such that $j$ 's neighbors in $\\mathcal {D}^{\\prime }$ constitute a clique.",
"Since there is no edge $j-i$ , $j$ 's neighbors in $\\mathcal {D}$ must also constitute a clique.",
"Thus the statement holds for chordal graphs of size $n$ .",
"Therefore the statement holds.",
"Figure: NO_CAPTIONNow, we prove Proposition REF by induction on the number of nodes in $\\mathcal {D}$ : Consider $|\\mathcal {D}|=1$ .",
"Since $K$ is a maximal clique, $K=\\mathcal {D}$ .",
"Thus Proposition REF holds trivially.",
"Suppose Proposition REF holds for chordal graphs of size $n-1$ .",
"Consider $|\\mathcal {D}|=n$ .",
"If $K=\\mathcal {D}$ , then Proposition REF holds.",
"If $K\\ne \\mathcal {D}$ , then by the above statement, there exists $j\\in \\mathcal {D}\\setminus K$ , such that there exists a perfect elimination order of $\\mathcal {D}$ starting with $j$ .",
"Let $\\mathcal {D}^{\\prime }$ be the chordal graph obtained by deleting $j$ from $\\mathcal {D}$ .",
"By induction, there exists $\\pi ^{\\prime }$ , a reverse of perfect elimination order, that starts with $\\pi _K$ .",
"Let $\\pi =(\\pi ^{\\prime },j)$ ; we must have that the reverse of $\\pi $ is a perfect elimination order, since all neighbors of $j$ in $\\mathcal {D}$ constitute a clique.",
"Therefore $\\pi $ gives the wanted topological order and Proposition REF holds for chordal graphs of size $n$ .",
"This completes the proof of Proposition REF .",
"[Proof of Lemma REF ] Given any algorithm $\\mathcal {A}$ , let $S_1,...,S_k$ be the first $k$ shift interventions given by $\\mathcal {A}$ .",
"By Proposition REF , we know that there exists a feasible orientation of $\\mathcal {C}$ such that the largest maximal clique $K$ of $\\mathcal {C}$ is most-upstream and that, for $k^{\\prime }=1,...,k$ , $S_{k^{\\prime }}\\cap K$ is most-downstream of $K-\\cup _{l<k^{\\prime }} S_{l}$ .",
"For example, in the figure below, suppose algorithm $\\mathcal {A}$ chooses $S_1=\\lbrace 3\\rbrace $ based on (a) and $S_2=\\lbrace 2\\rbrace $ based on (b).",
"There is a feasible orientation in (d) such that the largest clique $K=\\lbrace 1,2,3\\rbrace $ is most-upstream and $S_{k^{\\prime }}\\cap K$ is most-downstream of $K$ , for $k^{\\prime }=1,2$ .",
"Figure: Orientation of 𝒞\\mathcal {C}Since $|S_{k^{\\prime }}|\\le S$ and $|K|=m_{\\mathcal {C}}$ , in this worst case, it needs at least $\\lceil \\frac{m_{\\mathcal {C}}-1}{S}\\rceil $ interventions to identify the source of $K$ , i.e., the source of $\\mathcal {C}$ (minus 1 because in this case, if there is only one node left, then it must be the source)."
],
[
"Proof of Lemma ",
"Let $K$ be the clique obtained by lines 7-13 in Algorithm REF ; when $\\mathcal {C}$ has more than $S$ nodes, we refer to $K$ as the central clique.",
"To prove Lemma REF , we need the following proposition.",
"This proposition shows that by looking at the undirected graph $\\mathcal {C}$ , we can find a node in the central clique $K$ satisfying certain properties, which will become useful in the proof of Lemma REF .",
"Proposition 3 Let $\\lbrace \\mathcal {T}_{a}\\rbrace _{a\\in A}$ be the connected subtrees of $\\mathcal {T}(\\mathcal {C})$ after removing $K$ .",
"For a node $k \\in K$ , let $A_k \\subset A$ be the set of indices $a \\in A$ such that the tree $\\mathcal {T}_a$ is connected to $K$ only through the node $k$ .",
"Let $\\mathcal {T}_{A_k}=\\lbrace \\mathcal {T}_{a}\\rbrace _{a\\in A_k}$ be the collection of all such subtrees.",
"If there exists $a \\in A \\setminus A_k$ such that there is an edge between $\\mathcal {T}_a$ and $k$ , let $\\mathcal {T}^*_{k}$ be the one with the largest number of maximal cliques; otherwise let $\\mathcal {T}^*_{k}=\\varnothing $ .",
"Then there exists a node $k$ such that the number of maximal cliques in the subgraph induced by the subtrees $\\mathcal {T}_{A_k} \\cup \\lbrace \\mathcal {T}^*_{k}\\rbrace $ and $k$ itself does not exceed $\\lceil \\frac{r-1}{2}\\rceil $ .",
"Example 3 As an example, the following figure shows the subtrees that are connected to $K$ only through node 1, indexed by $A_1$ (blue).",
"The largest subtree in $A\\setminus A_1$ that is adjacent to node 1 is denoted by $\\mathcal {T}_1^*$ (undimmed in green).",
"Figure: An example of 𝒯 A k \\mathcal {T}_{A_k} and 𝒯 k * \\mathcal {T}_{k}^* for k=1k=1.",
"[Proof of Proposition REF ] Notice the following facts.",
"Fact 1: Let $\\mathcal {T}$ be any subtree in $\\lbrace \\mathcal {T}_{a}\\rbrace _{a\\in A}$ ; then there must exist a node $i\\in K$ such that there is no edge between $i$ and $\\mathcal {T}$ .",
"Proof of Fact 1: For any two nodes $i,i^{\\prime }\\in K$ , because $\\mathcal {C}$ is chordal and $\\mathcal {T}$ is connected, either the neighbors of $i$ in $\\mathcal {T}$ subset that of $i^{\\prime }$ , or the the neighbors of $i^{\\prime }$ in $\\mathcal {T}$ subset that of $i$ .",
"Therefore we can order all nodes $K$ , where all neighbors of $i$ in $\\mathcal {T}$ subset that of $i^{\\prime }$ that appear after $i$ .",
"Then if the first node in this order has some neighbor $t\\in \\mathcal {T}$ , all nodes in $K$ have $t$ as neighbor, contradicting $K$ being a maximal clique.",
"Figure: Contradicting maximal clique KKFact 2: Let $\\bar{\\mathcal {T}}$ be the collection of the subtrees where all edges connecting to $K$ are through a single node $k \\in K$ .",
"We have that $\\bar{\\mathcal {T}}$ is the union of disjoint sets $\\mathcal {T}_{A_k}, k\\in K$ .",
"Proof of Fact 2: This follows directly from the definition of $A_k$ .",
"Fact 3: Let $\\mathcal {T}^*$ be the collection of non-empty $\\mathcal {T}_{k}^*, k \\in K$ .",
"Then $\\mathcal {T}^* \\cap \\bar{\\mathcal {T}}=\\varnothing $ .",
"Furthermore, for any subtree in $\\mathcal {T}^*$ , there is a node $i\\in K$ such that there is no edge between $i$ and this subtree.",
"Proof of Fact 3: This follows directly from the definition of $\\mathcal {T}_{k}^*$ and Fact 1.",
"Now we prove Proposition REF .",
"If $\\mathcal {T}^*=\\varnothing $ , then since $K$ contains at least two nodes (otherwise $A=\\varnothing $ and the proposition holds trivially) and the number of maximal cliques in $\\bar{\\mathcal {T}}$ does not exceed $r-1$ , using Fact 2, we have at least one $k\\in K$ such that the number of maximal cliques in the subgraph induced by $\\mathcal {T}_{A_k}\\cup \\lbrace \\mathcal {T}_{k}^*\\rbrace =\\mathcal {T}_{A_k}$ and $k$ itself does not exceed $\\lceil \\frac{r-1}{2}\\rceil $ .",
"If $\\mathcal {T}^* \\ne \\varnothing $ .",
"Let $\\mathcal {T}^*_{k^{\\prime }}$ be the subtree with the largest number of maximal cliques in $\\mathcal {T}^*$ .",
"Let $k\\in K$ be the node such that there is no edge between $k$ and the subtree $\\mathcal {T}^*_{k^{\\prime }}$ ($k$ exists because of Fact 3).",
"Now suppose that the proposition does not hold.",
"Then the number of maximal cliques in the subgraph induced by $\\mathcal {T}_{A_k}\\cup \\lbrace \\mathcal {T}_{k}^*\\rbrace $ and $k$ itself must exceed $\\lceil \\frac{r-1}{2}\\rceil $ .",
"Also, the number of maximal cliques in the subgraph induced by $\\mathcal {T}_{A_{k^{\\prime }}}\\cup \\lbrace \\mathcal {T}^*_{k^{\\prime }}\\rbrace $ and $k^{\\prime }$ itself exceeds $\\lceil \\frac{r-1}{2}\\rceil $ .",
"Notice that $(\\mathcal {T}_{A_k}\\cup \\lbrace \\mathcal {T}_{k}^*\\rbrace )\\cap (\\mathcal {T}_{A_{k^{\\prime }}}\\cup \\lbrace \\mathcal {T}_{k^{\\prime }}^*\\rbrace )=\\varnothing $ .",
"Therefore $\\mathcal {T}_{A_k}\\cup \\lbrace \\mathcal {T}_{k}\\rbrace $ is connected to $\\mathcal {T}_{A_{k^{\\prime }}}\\cup \\lbrace \\mathcal {T}^*_{k^{\\prime }}\\rbrace $ only through $K$ .",
"Hence the sum of numbers of maximal cliques in $\\mathcal {T}_{A_k}\\cup \\lbrace \\mathcal {T}^*_{k}\\rbrace \\cup \\lbrace \\lbrace k\\rbrace \\rbrace $ and $\\mathcal {T}_{A_{k^{\\prime }}}\\cup \\lbrace \\mathcal {T}^*_{k^{\\prime }}\\rbrace \\cup \\lbrace \\lbrace k^{\\prime }\\rbrace \\rbrace $ does not exceed $r$ .",
"We cannot have both $\\mathcal {T}_{A_k}\\cup \\lbrace \\mathcal {T}^*_{k}\\rbrace \\cup \\lbrace \\lbrace k\\rbrace \\rbrace $ and $\\mathcal {T}_{A_{k^{\\prime }}}\\cup \\lbrace \\mathcal {T}^*_{k^{\\prime }}\\rbrace \\cup \\lbrace \\lbrace k^{\\prime }\\rbrace \\rbrace $ having more than $\\lceil \\frac{r-1}{2}\\rceil $ maximal cliques.",
"Therefore the proposition must hold.",
"[Proof of Lemma REF ] For CliqueTree, we prove this lemma for a “less-adaptive” version for the sake of clearer discussions.",
"In this “less-adaptive” version, instead of output 1 intervention with $S$ perturbation targets sampled from the central clique $K$ (when it has more than $S$ nodes) in Algorithm REF , we directly output $\\lceil \\frac{|K|-1}{S}\\rceil $ interventions with non-overlapping perturbation targets in $K$ .",
"Each of these interventions has no more than $S$ perturbation targets and they contain at least $|K|-1$ nodes in $K$ altogether.",
"Furthermore, we pick these interventions such that if they contain exactly $|K|-1$ nodes, then the remaining node satisfies Proposition REF .",
"After these $\\lceil \\frac{|K|-1}{S}\\rceil $ interventions, we obtain a partially directed $\\mathcal {C}$ , which is a chain graph, with one of its chain components without incoming edges as input to CliqueTree in the next iteration of the inner-loop in Algorithm REF .",
"Denote this chain component as $\\mathcal {C}^{\\prime }$ .",
"We show that $\\mathcal {C}^{\\prime }$ has no more than $\\left\\lceil \\frac{r-1}{2} \\right\\rceil $ maximal cliques each with no more than $m_{\\mathcal {C}}$ nodes.",
"If $\\lceil \\frac{r-1}{2}\\rceil =0$ , then $r=1$ and this trivially holds since the source of $\\mathcal {C}$ must be identified.",
"In the following, we assume $\\lceil \\frac{r-1}{2}\\rceil >0$ .",
"Size of maximal cliques: The maximal clique in $\\mathcal {C}^{\\prime }$ must belong to a maximal clique in $\\mathcal {C}$ , and thus has no more than $m_{\\mathcal {C}}$ nodes.",
"Number of maximal cliques: If the source node is identified, then $\\mathcal {C}^{\\prime }$ only has one node.",
"This trivially holds.",
"Now consider when the source node is not identified.",
"We proceed in two cases.",
"Case I: if these $\\lceil \\frac{|K|-1}{S}\\rceil $ interventions contain all nodes in $K$ , then they break the clique tree $\\mathcal {T}(\\mathcal {C})$ into subtrees each with no more than $\\lceil \\frac{r-1}{2}\\rceil $ maximal cliques.",
"$\\mathcal {C}^{\\prime }$ must belong to one of these subtrees.",
"Therefore it must have no more than $\\lceil \\frac{r-1}{2}\\rceil $ maximal cliques.",
"Case II: if these $\\lceil \\frac{|K|-1}{S}\\rceil $ interventions do not contain all nodes in $K$ , then there is exactly one node left in $K$ that is not a perturbation target, which satisfies Proposition REF .",
"Denote this node as $k$ and the source node w.r.t.",
"the intervened $|K|-1$ nodes as $i$ .",
"From Theorem REF , we have that $i$ is identified and $\\forall j\\in K, j\\ne k$ , the orientation of edge $k-j$ is identified.",
"If $i\\rightarrow k$ , then $i$ is the source w.r.t.",
"$K$ : if $i$ is the source w.r.t.",
"$\\mathcal {C}$ , then $\\mathcal {C}^{\\prime }=\\lbrace i\\rbrace $ has no more than $\\lceil \\frac{r-1}{2}\\rceil $ maximal cliques; otherwise, there is a unique subtree of $\\mathcal {T}(\\mathcal {C})$ after removing $K$ that has an edge pointing to $i$ in $\\mathcal {C}$ (it exists because $i$ is the source of $K$ but not the source of $\\mathcal {C}$ ; it is unique because there is no edge between subtrees and there is no v-structure at $i$ ), and therefore $\\mathcal {C}^{\\prime }$ must belong to this subtree which has no more than $\\lceil \\frac{r-1}{2}\\rceil $ maximal cliques.",
"Figure: If i←ki\\leftarrow k, Fact 1If $i\\leftarrow k$ , then $k$ is the source w.r.t.",
"$K$ : consider all the subtrees of $\\mathcal {T}(\\mathcal {C})$ after removing $K$ .",
"We have the following two facts: Fact 1: Let $\\mathcal {T}^{\\prime }$ be a subtree such that there is an edge between $\\mathcal {T}^{\\prime }$ and $K-\\lbrace k\\rbrace $ and all these edges are pointing towards $\\mathcal {T}^{\\prime }$ .",
"Then all edges between $k$ and $t\\in \\mathcal {T}^{\\prime }$ must be oriented as $k\\rightarrow t$ .",
"Thus $\\mathcal {C}^{\\prime }\\cap \\mathcal {T}^{\\prime }=\\varnothing $ .",
"Proof of Fact 1: Otherwise, suppose $t\\in \\mathcal {T}^{\\prime }$ and $t\\rightarrow k$ .",
"Let $j\\in K-\\lbrace k\\rbrace $ such that there is an edge between $j$ and $\\mathcal {T}^{\\prime }$ .",
"Since $\\mathcal {T}^{\\prime }$ is connected, there must be a path from $j$ to $t$ in $\\mathcal {T}^{\\prime }$ .",
"Let $j=t_0-t_1-...-t_l-t_{l+1}=t$ be the shortest of these path.",
"Since $t_0-t_1-...-t_l-t_{l+1}$ is shortest, there cannot be an edge between $t_{l^{\\prime }}$ and $t_{l^{\\prime \\prime }}$ with $l^{\\prime \\prime }-l^{\\prime }>1$ .",
"And since all edges between $\\mathcal {T}^{\\prime }$ and $K-\\lbrace k\\rbrace $ are pointing towards $\\mathcal {T}^{\\prime }$ , there is an edge $j=t_0\\rightarrow t_1$ .",
"Therefore to avoid v-structures, it must be $j=t_0\\rightarrow t_1\\rightarrow ...\\rightarrow t_l\\rightarrow t_{l+1}=t$ .",
"This creates a directed cycle $k\\rightarrow j \\rightarrow ... \\rightarrow t \\rightarrow k$ , a contradiction.",
"Fact 2: There can be at most one subtree $\\mathcal {T}^{\\prime }$ such that there is an edge pointing from $\\mathcal {T}^{\\prime }$ to $K-\\lbrace k\\rbrace $ and also some $t\\in \\mathcal {T}^{\\prime }$ such that $t\\rightarrow k$ or $t-k$ is unidentified.",
"Therefore at most one subtree $\\mathcal {T}^{\\prime }$ of this type can have $\\mathcal {C}^{\\prime }\\cap \\mathcal {T}^{\\prime }\\ne \\varnothing $ .",
"Proof of Fact 2: Otherwise suppose there are two different subtrees $\\mathcal {T}^{\\prime }_1,\\mathcal {T}^{\\prime }_2$ such that $K-\\lbrace k\\rbrace \\ni j_1\\leftarrow t_1\\in \\mathcal {T}^{\\prime }_1, K-\\lbrace k\\rbrace \\ni j_2\\leftarrow t_2\\in \\mathcal {T}^{\\prime }_2$ .",
"Since there is no edge $t_1-t_2$ , we have $j_1\\ne j_2$ .",
"Without loss of generality, suppose $j_1\\rightarrow j_2$ .",
"Let $t$ be any node in $\\mathcal {T}_2^{\\prime }$ with an edge $t-k$ , since $\\mathcal {T}_2^{\\prime }$ is connected, let $t=t_0^{\\prime }-t_1^{\\prime }-...-t_l^{\\prime }-t_{l+1}^{\\prime }=t_2$ be the shortest path between $t$ and $t_2$ in $\\mathcal {T}_2^{\\prime }$ .",
"Let $l^{\\prime }$ be the maximum in $0,1,...,l$ such that $t^{\\prime }_{l^{\\prime }}\\leftarrow t^{\\prime }_{l^{\\prime }+1}$ .",
"If such $l^{\\prime }$ does not exist, then $t=t_0^{\\prime }\\rightarrow t_1^{\\prime }\\rightarrow ...\\rightarrow t_{l+1}^{\\prime }=t_2$ .",
"Since $j_1\\rightarrow j_2$ and there is no v-structure at $j_2$ , there must be an identified edge $j_1-t_{l+1}^{\\prime }=t_2$ .",
"Notice that there is no edge between $t_2$ and $t_1$ and $t_1\\rightarrow j_1$ , to avoid v-structure, it must be $j_1\\rightarrow t_2$ .",
"The same deduction leads to identified edges $j_1\\rightarrow t_{0}^{\\prime }=t$ .",
"Since $k\\rightarrow j_1$ and there are no cycles, the edge $k\\rightarrow t$ must be identified.",
"If $l^{\\prime }$ exists, since $t=t_0^{\\prime }-t_1^{\\prime }-...-t_l^{\\prime }-t_{l+1}^{\\prime }=t_2$ is the shortest path and there is no v-structure, we must have $t=t_0^{\\prime }\\leftarrow ... \\leftarrow t_{l^{\\prime }+1}^{\\prime }$ .",
"Furthermore, since $l^{\\prime }$ is the largest, $t_{l^{\\prime }+1}^{\\prime }\\rightarrow ... \\rightarrow t_{l+1}^{\\prime } =t_2$ .",
"By a similar deduction as in the case where $l^{\\prime }$ does not exist, we must have an identified edge $j_1\\rightarrow t_{l^{\\prime }+1}^{\\prime }$ .",
"Therefore $k\\rightarrow j_1\\rightarrow t_{l^{\\prime }+1}^{\\prime } \\rightarrow ... t_0^{\\prime }=t$ .",
"To avoid directed cycles, $k\\rightarrow t$ must be identified.",
"Therefore all edges between $k$ and $\\mathcal {T}_2^{\\prime }$ are identified as pointing to $\\mathcal {T}_2^{\\prime }$ .",
"Figure: l ' l^{\\prime } existsUsing the above two facts, let $\\mathcal {T}^{\\prime }$ be the unique subtree in Fact 2 (if it exists); if there is no edge between $\\mathcal {T}^{\\prime }$ and $k$ , then $\\mathcal {C}^{\\prime }$ must be in the subgraph induced by $k$ itself and $\\mathcal {T}_{A_k}$ in Proposition REF , which has no more than $\\lceil \\frac{r-1}{2}\\rceil $ maximal cliques.",
"If there is an edge between $\\mathcal {T}^{\\prime }$ and $k$ , we know that $\\mathcal {C}^{\\prime }$ must be in the joint set of $k$ , $\\mathcal {T}^{\\prime }$ and $\\mathcal {T}_{A_k}$ .",
"Since the number of maximal cliques in $\\mathcal {T}^{\\prime }$ must be no more than that of $\\mathcal {T}_{k}^*$ in Proposition REF , we know that $\\mathcal {C}^{\\prime }$ has no more than $\\lceil \\frac{r-1}{2}\\rceil $ maximal cliques.",
"Therefore, after $\\lceil \\frac{|K|-1}{S}\\rceil \\le \\lceil \\frac{m_{\\mathcal {C}}-1}{S}\\rceil $ interventions, we reduce the number of maximal cliques to at most $\\lceil \\frac{r-1}{2}\\rceil $ while maintaining the size of the largest maximal clique $\\le m_{\\mathcal {C}}$ .",
"Using this iteratively, we obtain that CliqueTree identifies the source node with at most $\\lceil \\log _2(r_\\mathcal {C}+1)\\rceil \\cdot \\lceil \\frac{m_{\\mathcal {C}}-1}{S}\\rceil $ interventions.",
"For Supermodular, we do not discuss the gap between $\\hat{g}_{i,j}$ and $g_{i,j}$ and how well SATURATE solves (REF ).",
"In this case, it is always no worse than the CliqueTree in the worst case over the feasible orientations of $\\mathcal {C}$ , since it solves MinMaxC optimally without constraining to maximal cliques.",
"Therefore, it also takes no more than $\\lceil \\log _2(r_\\mathcal {C}+1)\\rceil \\cdot \\lceil \\frac{m_\\mathcal {C}-1}{S}\\rceil $ to identify the source node."
],
[
"Proof of Theorem ",
"[Proof of Theorem REF ] This result follows from Lemma REF and REF .",
"Divide $I^*$ into $I_1,...,I_k$ such that $I_{k^{\\prime }}$ is the source node of $I^*-\\cup _{l<k^{\\prime }}I_{l}$ .",
"Since shifting $I_{k^{\\prime }}$ affects the marginal of subsequent $I_{k^{\\prime \\prime }}$ with $k^{\\prime \\prime }>k^{\\prime }$ , any algorithm needs to identify $I_1,...,I_k$ sequentially in order to identify the exact shift values.",
"Suppose $I_1,...,I_{k^{\\prime }-1}$ are learned.",
"For $I_{k^{\\prime }}$ , consider the chain components of the subgraph of the shift-$\\lbrace \\cup _{l< k^{\\prime }} I_l\\rbrace $ -EG induced by $T=\\lbrace i|i\\in [p], \\mathbb {E}_{({\\mathrm {P}}^{\\cup _{l<k^{\\prime }} I_l})}(X_i)\\ne \\mathbb {E}_{{\\mathrm {Q}}}(X_i)\\rbrace $ with no incoming edge.",
"Applying Lemma REF for $\\mathcal {I}=\\lbrace \\cup _{l< k^{\\prime }} I_l$ } and Observation REF for this subgraph and $I_{k^{\\prime }}$ , we deduce that there are exactly $|I_{k^{\\prime }}|$ such chain components and $I_{k^{\\prime }}$ has exactly one member in each of these chain components.",
"Let $m_{k^{\\prime },1},...,m_{k^{\\prime },|I_{k^{\\prime }}|}$ be the sizes of the largest maximal cliques in these $|I_{k^{\\prime }}|$ chain components.",
"By Lemma REF , we know that any algorithm needs at least $\\sum _{i=1}^{|I_{k^{\\prime }}|} \\lceil \\frac{m_{k^{\\prime },i}-1}{S}\\rceil $ number of interventions to identify $I_{k^{\\prime }}$ in the worst case.",
"However, since all these chain components contain no more than $r$ maximal cliques, by Lemma REF , we know that our strategies need at most $\\lceil \\log _2(r+1)\\rceil \\cdot \\sum _{i=1}^{|I_{k^{\\prime }}|} \\lceil \\frac{m_{k^{\\prime },i}-1}{S}\\rceil $ to identify $I_{k^{\\prime }}$ .",
"Applying this result for $k^{\\prime }=1,...,k$ , we obtain that our strategies for solving the causal mean matching problem require at most $\\lceil \\log _2(r+1)\\rceil $ times more interventions, compared to the optimal strategy, in the worse case over all feasible orientations."
],
[
"Experimental Setup",
"Graph Generation: We consider two random graph models: Erdös-Rényi graphs [11] and Barabási–Albert graphs [2].",
"The probability of edge creation in Erdös-Rényi graphs is set to $0.2$ ; the number of edges to attach from a new node to existing nodes in Barabási–Albert graphs is set to 2.",
"We then tested on two types of structured chordal graphs: rooted tree with root randomly sampled from all the nodes in this tree, and moralized Erdös-Rényi graphs [30] with the probability of edge creation set to $0.2$ .",
"Multiple Runs: For each instance in the settings of Barabási–Albert graphs with 100 nodes and $S=1$ in Figure REF , we ran the three non-deterministic strategies (UpstreamRand,CliqueTree, Supermodular) for five times and observed little differences across all instances.",
"Therefore, we excluded the error bars when plotting the results as they are visually negligible and the strategies are robust in these settings.",
"Implementation: We implemented our algorithms using the NetworkX package hagberg2008exploring and the CausalDAG package https://github.com/uhlerlab/causaldag.",
"All code is written in Python and run on AMD 2990wx CPU."
],
[
"More Empirical Results",
"In the following, we present additional empirical result.",
"The evaluations are the same as in sec:experiments.",
"The following figures show that We observe similar behaviors as in Figure REF across different settings.",
"Random graphs of size $\\lbrace 10, 50, 100\\rbrace $ : Barabási–Albert and Erdös-Rényi graphs with number of nodes in $\\lbrace 10, 50, 100\\rbrace $ .",
"Figure: Barabási–Albert graphs with 50 nodes.",
"(a).",
"and (b).",
"S=1S=1; (c).",
"|I * |=25|I^*|=25.Figure: Barabási–Albert graphs with 10 nodes.",
"(a).",
"and (b).",
"S=1S=1; (c).",
"|I * |=5|I^*|=5.Figure: Erdös-Rényi graphs with 100 nodes.",
"(a).",
"and (b).",
"S=1S=1; (c).",
"|I * |=50|I^*|=50.Larger Barabási–Albert graphs of size 1000: Figure: Larger Barabási–Albert graphs with 1000 nodes (excluding coloring which takes more than 80 extra interventions).",
"(a).",
"and (b).",
"S=1S=1; (c).",
"|I * |=100|I^*|=100.Two types of structured chordal graphs: Figure: Structured chordal graphs.",
"(a).",
"and (b).",
"rooted tree graphs with 50 nodes and S=1S=1; (c).",
"and (d).",
"moralized Erdös-Rényi graphs with 10 nodes and S=1S=1."
],
[
"Discussion of the Noisy Setting",
"In the noisy setting, an intervention can be repeated many times to obtain an estimated essential graph.",
"Each intervention results in a posterior update of the true DAG $\\mathcal {G}$ over all DAGs in the observational Markov equivalence class.",
"For a tree graph $\\mathcal {G}$ , this corresponds to a probability over all possible roots.",
"To be able to learn the edges, greenewald2019sample+ proposed a bounded edge strength condition on the noise for binary variables.",
"Under this condition, they showed that the root node of a tree graph can be learned in finite steps in expectation with high probability.",
"In our setting, to ensure that the source node w.r.t.",
"an intervention can be learned, we need to repeat this intervention for enough times such that the expectation of each variable $X_i$ can be estimated.",
"Furthermore, to ensure that the edges in the (general) interventional essential graph can be learned, we need a similar condition as in greenewald2019sample+ for general chordal graphs and continuous variables.",
"apalike main"
]
] | 2107.01850 |
[
[
"Reconstruction of multiple Compton scattering events in MeV gamma-ray\n Compton telescopes towards GRAMS: the physics-based probabilistic model"
],
[
"Abstract Aimed at progress in mega-electron volt (MeV) gamma-ray astronomy, which has not yet been well-explored, Compton telescope missions with a variety of detector concepts have been proposed so far.",
"One of the key techniques for these future missions is an event reconstruction algorithm that is able to determine the scattering orders of multiple Compton scattering events and to identify events in which gamma rays escape from the detectors before they deposit all of their energies.",
"We revisit previous event reconstruction methods and propose a modified algorithm based on a probabilistic method.",
"First, we present a general formalism of the probabilistic model of Compton scattering describing physical interactions inside the detector and measurement processes.",
"Then, we also introduce several approximations in the calculation of the probability functions for efficient computation.",
"For validation, the developed algorithm has been applied to simulation data of a Compton telescope using a liquid argon time projection chamber, which is a new type of Compton telescope proposed for the GRAMS project.",
"We have confirmed that it works successfully for up to 8-hit events, including correction of incoming gamma-ray energies for escape events.",
"The proposed algorithm can be used for next-generation MeV gamma-ray missions featured by large-volume detectors, e.g., GRAMS."
],
[
"Introduction",
"Astrophysical observations of gamma rays from a few 100 keV to a few 10 MeV remain to be explored in modern astronomy.",
"COMPTEL aboard the Compton Gamma-Ray Observatory pioneered this MeV gamma-ray astronomy in 1991-2000 [1] and found more than 30 sources in the energy band from 0.75 to 30 MeV [2].",
"After this mission, however, few space satellite missions have been succeeded so far.",
"These days, stimulated by the dawn of multi-messenger astronomy including gravitational wave [3] and neutrino observations [4], this curtained window of electromagnetic waves is drawing increasing attention.",
"Towards high-sensitivity observations in the 2020s or 2030s, several MeV gamma-ray missions have been proposed at this moment (GRAMS [5], AMEGO [6], e-ASTROGAM [7], COSI [8], SMILE [9] e.t.c.).",
"All of the missions above utilize Compton telescopes which are one of the most promising techniques for imaging gamma-ray sources in the sub-MeV/MeV bands [10], [11], [12].",
"A Compton telescope measures the position and deposited energy at each interaction, and calculates the scattering angle by the kinematics of Compton scattering.",
"Then, the incoming gamma-ray direction is constrained on a circle in the sky.",
"If recoiled electron trajectories can be measured additionally, then the gamma-ray direction is constrained on an arc-shaped region [13], [14], [15].",
"After an accumulation of many events, the incoming gamma-ray direction can be identified as intersections of the constrained circles or arcs.",
"This is a basic principle of Compton imaging, which is called a back-projection method.",
"Statistical approaches of the image reconstruction methods have been also proposed [16], [17].",
"To calculate the scattering angle of a gamma-ray event, it is required to identify the scattering order of the detected signals and estimate the incident gamma-ray energy accurately.",
"The scattering order determination becomes more complicated in a higher energy band because gamma rays can easily be scattered multiple times in a detector and escape from it before they are absorbed inside the detector.",
"These multiple scattering events are considered to be dominant in large-volume detectors such as proposed in GRAMS [5] and AMEGO [6].",
"Since in these projects it is difficult to measure time-of-flight between signals, the scattering order cannot be determined directly, and thus, one has to determine it based on the detected energies and positions.",
"In this paper, we focus on this order determination problem of the multiple scattering events.",
"Several simple approaches have been proposed [18], [19], [20], [21].",
"For events with $n~(\\ge 3)$ interactions, the scattering angle at $i$ -th $(2 \\le i \\le n-1)$ interaction site can be calculated in two ways, i.e., from kinematics or geometrical information.",
"Then, by comparing the calculated angles at each site, the most plausible scattering angle is estimated based on statistical quantities.",
"This is a simple method, and thus other information is not included in the calculation, e.g., the anisotropy of Compton scattering or the length between interactions.",
"More sophisticated approaches such as the Bayesian method [22] or the neural network method [23] can consider this additional information and outperform the classical approach in principle.",
"However, these approaches require large simulation data sets and a lot of computer resources.",
"As another issue, the above approaches put weights on events that deposit all of their gamma-ray energies in the detector (full-deposit events), and usually reject events such that gamma rays escape from the detector (escape events) as background.",
"However, if the number of interactions is three or more, the gamma-ray energy of escape events can be estimated by combining the position and energy information (see, e.g., [20]).",
"It suggests that the gamma-ray detection efficiency can be increased if escape events can also be identified accurately and their escape energies are correctly estimated.",
"This consideration would be more essential in a higher energy band since gamma rays become harder to be photo-absorbed in the detector.",
"In this work, we revisit the classical approach and propose a new reconstruction algorithm of multiple Compton scattering events for MeV gamma rays, based on the maximum likelihood method.",
"It is one of the first algorithms that can distinguish the full-deposit and escape events explicitly, and correct energies of gamma rays that escape from the detectors.",
"In Sections and , we formulate probability functions related to physical processes and measurements in Compton telescopes.",
"In Section , we implement the algorithm, introducing several approximations to calculate the likelihood function efficiently.",
"Here we consider a Compton telescope that measures the interaction positions and deposited energies not including electron trajectories.",
"A summary of the developed algorithm is given in Section .",
"As a numerical test, we apply it to simulation data sets in Section .",
"Finally, we summarize our results and discuss further improvements in Section ."
],
[
"Basic Concept",
"In a Compton telescope, gamma rays deposit their energies at interaction sites via Compton scattering or photoabsorption.",
"In general, measured values at each site are a tuple of several physical quantities, which is denoted by $\\mbox{$D$}_I$ .",
"The index $I$ is the label of each tuple in an event, and here the scattering order is unknown.",
"When a Compton telescope measures deposit energies and interaction positions, $\\mbox{$D$}_I$ is a pair of them: $\\mbox{$D$}_I = \\left( \\mbox{$r$}_{I}, \\varepsilon _{I} \\right)~,$ where $\\mbox{$r$}_{I}$ and $\\varepsilon _{I}$ are the measured position and energy, respectively.",
"Since a gamma ray is scattered multiple times or absorbed by a detector, in experiments we obtain a list of $\\mbox{$D$}_I$ : $(\\mbox{$D$}_I) = (\\mbox{$D$}_1, \\mbox{$D$}_2, ...)~.$ We define a hit as a measured interaction with $\\mbox{$D$}_I$ .",
"When the number of hits in an event is $n$ , we refer to the event as an $n$ -hit event.",
"In the reconstruction of Compton scattering events, the task is divided into the following: to determine the event type, i.e., a full-deposit event or an escape event for a given event.",
"to determine the scattering order of detected hits: $(\\mbox{$D$}_I)_{\\mathrm {ordered}} = (\\mbox{$D$}_{\\tau (1)}, \\mbox{$D$}_{\\tau (2)}, ..., \\mbox{$D$}_{\\tau (n)})_{\\mathrm {ordered}}~,$ where $(\\mbox{$D$}_I)_{\\mathrm {ordered}}$ is a re-ordered list of $(\\mbox{$D$}_I)$ by determining or assuming the scattering order.",
"Here we define the function $\\tau (\\cdot )$ which maps the scattering order to the data label ($I$ ), i.e., $i$ -th interaction corresponds to $\\mbox{$D$}_{\\tau (i)}$ .",
"Figure REF shows the scattering order candidates and their corresponding map functions $\\tau (\\cdot )$ for a 3-hit event.",
"to estimate the incident gamma-ray energy.",
"to estimate the incoming gamma-ray direction.",
"Figure: Scattering order candidates of a 3-hit event and their corresponding map functions τ(·)\\tau (\\cdot ).In the classical approach [18], [19], [20], [21], the scattering angles are calculated redundantly from kinematics and from geometrical information, and they are compared by the following quantity: $\\chi ^2_c = \\frac{1}{N-2} \\sum _{i=2}^{N-1} \\frac{(\\cos \\vartheta _{i}^{kin} - \\cos \\vartheta _{i}^{geo} )^2}{\\Delta \\cos ^2 \\vartheta _{i}^{kin} + \\Delta \\cos ^2 \\vartheta _{i}^{geo} }~(N \\ge 3)~,$ where $\\vartheta _{i}^{geo}$ and $\\vartheta _{i}^{kin}$ are $i$ -th scattering angles calculated by kinematics and geometrically, respectively; $\\Delta \\cos ^2 \\vartheta _{i}$ is the measurement uncertainty, and $N$ is the number of hits.",
"This quantity is interpreted as a generalized chi-squared value.",
"For all $N!$ scattering order candidates, $\\chi ^2_c$ are calculated and the best scattering order is determined as one that yields the smallest $\\chi ^2_c$ .",
"In this method, the incoming gamma-ray energy is assumed to be the sum of the detected energies, and only the full-deposit events are treated.",
"Moreover, other physical factors, e.g., the Klein-Nishina differential cross section and the length between hits are not considered as already mentioned in Section .",
"To solve these problems, we consider the probabilistic processes related to Compton telescopes, and construct the likelihood function for both full-deposit and escape events instead of $\\chi ^2_c$ .",
"Then, the most probable event type and scattering order are determined as those that yield the maximum likelihood value."
],
[
"Formalism",
"In this section, we formulate the likelihood function.",
"Here we assume that an incoming gamma ray is scattered in the detector $n-1$ times and photo-absorbed at last, or is scattered $n$ times and escapes from the detector.",
"In both cases, the number of interactions is $n$ .",
"Also, we assume that all interactions are measured and a list of the measurement values $(\\mbox{$D$}_I) = (\\mbox{$D$}_1, \\mbox{$D$}_2, ..., \\mbox{$D$}_n )$ are obtained.",
"Note that in reality there is a case that some interactions are not detected due to interactions in passive materials, the detector threshold, and multiple scattering in the same spatial resolution element of the detector.",
"The probability of these events strongly depends on the actual detector configuration, and we ignore these possibilities in the likelihood formulation presented in this section.",
"As shown in Figure REF , the gamma ray changes its energy and its direction of travel every time it interacts with the detector.",
"Then, the gamma ray after $i$ -th interaction in the detector can be described with the quantities $\\hat{\\mbox{$q$}}_{i}$ defined as $\\begin{aligned}\\hat{\\mbox{$q$}}_{i} &= \\left(\\hat{\\mbox{$r$}}_{i}, \\hat{E}_{i}, \\hat{\\theta }_i, \\hat{\\phi }_i \\right)~,\\\\\\hat{\\mbox{$r$}}_{i} &= \\left(\\begin{array}{c}x_i \\\\y_i \\\\z_i\\end{array}\\right)~,\\\\\\hat{\\mbox{$p$}}_{i} &= \\frac{\\hat{E}_{i}}{c} \\left(\\begin{array}{c}\\sin \\hat{\\theta }_i \\cos \\hat{\\phi }_i \\\\\\sin \\hat{\\theta }_i \\sin \\hat{\\phi }_i \\\\\\cos \\hat{\\theta }_i\\end{array}\\right)~,\\end{aligned}$ where $\\hat{\\mbox{$r$}}_{i}$ represents the $i$ -th interaction position; $\\hat{E}_{i}$ and $\\hat{\\mbox{$p$}}_{i}$ represent the energy and momentum vector, respectively, of the gamma ray after the $i$ -th interaction (see Figure REF ); $\\hat{\\theta }_i$ and $\\hat{\\phi }_i$ describe the direction of travel of the gamma ray.",
"We refer to the quantity $\\hat{\\mbox{$q$}}_{i}$ as gamma-ray state in this work.",
"To distinguish explicitly the parameters in the gamma-ray state and those measured by experiments, we put hats on the former ones.",
"Note that $\\hat{\\mbox{$q$}}_{0}$ represents the initial gamma-ray state.",
"In astrophysical observations, $\\hat{\\mbox{$q$}}_{0}$ is described by just three parameters: $\\hat{\\mbox{$q$}}_{0} &= \\left(\\hat{E}_{0}, \\hat{\\theta }_{0}, \\hat{\\phi }_{0} \\right)~,$ since incoming gamma rays originate from distant sources.",
"The gamma-ray state $\\hat{\\mbox{$q$}}_{i}$ ($i \\ge 1$ ) can be determined when we assume the initial position and energy of the gamma ray ($\\hat{\\mbox{$r$}}_{0}, \\hat{E}_{0}$ ) and the position of each interaction ($\\hat{\\mbox{$r$}}_{i(\\ge 1)}$ ).",
"Except for $i=n$ , $\\hat{\\theta }_i$ and $\\hat{\\phi }_i$ are determined from the interaction positions: $\\hat{\\mbox{$p$}}_{i} \\parallel \\frac{\\hat{\\mbox{$r$}}_{i+1} - \\hat{\\mbox{$r$}}_{i}}{|\\hat{\\mbox{$r$}}_{i+1} - \\hat{\\mbox{$r$}}_{i}|}~.$ The gamma-ray energy $\\hat{E}_{i}$ is determined by the kinematics of Compton scattering: $\\hat{E}_{i} = \\frac{\\hat{E}_{i-1}}{1 + \\dfrac{\\hat{E}_{i-1}}{m_e c^2}\\left(1 - \\cos \\hat{\\vartheta }^\\mathrm {scat}_i \\right)}~,$ where $\\hat{\\vartheta }^\\mathrm {scat}_i$ is the $i$ -th scattering angle which is calculated as $\\cos \\hat{\\vartheta }^\\mathrm {scat}_i = \\frac{ \\hat{\\mbox{$p$}}_{i-1} \\cdot \\hat{\\mbox{$p$}}_{i} }{ |\\hat{\\mbox{$p$}}_{i-1}| |\\hat{\\mbox{$p$}}_{i}| }~.$ In the escape events, $\\hat{\\theta }_n$ and $\\hat{\\phi }_n$ should be also assumed to determine the direction of escape.",
"Note that Eq.",
"REF does not take account of the uncertainty of $\\hat{E}_{i}$ due to the finite momentum fluctuation of the target electrons in the detector material, which is also known as the Doppler broadening effect [24].",
"We will discuss a possible treatment to include this effect in Section .",
"Figure: A schematic of a multiple Compton scattering event in a detector.",
"The red arrows represent the path of the gamma ray.To construct the likelihood function, it is useful to describe the graphical representation of a Compton scattering event as shown in Figure REF .",
"The change of the state $\\mbox{$\\hat{q}$}_i$ is a probabilistic process determined by the physics of Compton scattering.",
"Then, the parameters of the state are related to the measured values $(\\mbox{$D$}_I)$ through measurement processes.",
"For example, $\\hat{E}_{i-1} - \\hat{E}_{i}$ is measured as the deposit energy at $i$ -th interaction.",
"Figure: A graphical representation of Compton scattering in a detector and measurement in Compton telescopes.",
"The red arrows correspond to probabilistic processes that take place only in escape events.As a general expression, the likelihood function for full-deposit events is described as $\\begin{split}L_\\mathrm {fulldep} & \\left((\\hat{\\mbox{$q$}}_{i}), \\tau (\\cdot );(\\mbox{$D$}_I)\\right) \\\\& = P_\\mathrm {abs} \\left( \\hat{\\mbox{$q$}}_{n}, \\mbox{$D$}_{\\tau (n)} \\mid \\hat{\\mbox{$q$}}_{n-1}\\right) \\\\&\\times \\prod _{i=1}^{n-1} P_\\mathrm {scat} \\left( \\hat{\\mbox{$q$}}_{i}, \\mbox{$D$}_{\\tau (i)} \\mid \\hat{\\mbox{$q$}}_{i-1}\\right)~(n \\ge 2)~.\\end{split}$ It is the probability density function (PDF) that a gamma ray with the initial state of $\\hat{\\mbox{$q$}}_{0}$ is scattered at $\\hat{\\mbox{$r$}}_i$ ($1 \\le i \\le n-1)$ and is absorbed at $\\hat{\\mbox{$r$}}_{n}$ , and the deposit energy and position at $i$ -th interaction are measured as the obtained data sample $\\mbox{$D$}_{\\tau (i)}$ .",
"In the likelihood calculation, model parameters are $\\hat{\\mbox{$q$}}_{0}$ , $(\\hat{\\mbox{$r$}}_{i(\\ge 1)})$ and $\\tau (\\cdot )$ .",
"The gamma-ray energy $\\hat{E}_{i}$ and the direction of travel $(\\hat{\\theta }_i$ , $\\hat{\\phi }_i)$ after $i(\\ge 1)$ -th interaction are determined from the model parameters (see Eqs.",
"REF and REF ).",
"Here $P_\\mathrm {scat} \\left( \\hat{\\mbox{$q$}}_{i}, \\mbox{$D$}_{\\tau (i)} \\mid \\hat{\\mbox{$q$}}_{i-1}\\right)$ is the PDF that a gamma ray with $\\hat{\\mbox{$q$}}_{i-1}$ is scattered with changing its state to $\\hat{\\mbox{$q$}}_{i}$ , and the interaction is measured as $\\mbox{$D$}_{\\tau (i)}$ .",
"It is described as the product of three functions: $\\begin{aligned}P_\\mathrm {scat} & \\left( \\hat{\\mbox{$q$}}_{i}, \\mbox{$D$}_{\\tau (i)} \\mid \\hat{\\mbox{$q$}}_{i-1}\\right) = P_\\mathrm {path, scat} \\left( \\hat{\\mbox{$r$}}_{i} \\mid \\hat{\\mbox{$q$}}_{i-1}\\right) \\\\& \\times P_\\mathrm {KN} \\left(\\hat{\\theta }_{i}, \\hat{\\phi }_{i} \\mid \\hat{\\mbox{$q$}}_{i-1}\\right)P_\\mathrm {det} \\left( \\mbox{$D$}_{\\tau (i)} \\mid \\hat{\\mbox{$q$}}_{i-1}, \\hat{\\mbox{$q$}}_{i} \\right)~,\\end{aligned}$ where $P_\\mathrm {path, scat}(\\cdot )$ is the PDF that a gamma ray of $\\hat{\\mbox{$q$}}_{i-1}$ is scattered at $\\hat{\\mbox{$r$}}_{i}$ ; $P_\\mathrm {KN}(\\cdot )$ is the PDF that the scattering direction is $(\\hat{\\theta }_{i}, \\hat{\\phi }_{i})$ and it is determined by Klein-Nishina formula [25]; $P_\\mathrm {det}(\\cdot )$ corresponds to the PDF related to the detector response.",
"Also, $P_\\mathrm {abs} \\left( \\hat{\\mbox{$q$}}_{n}, \\mbox{$D$}_{\\tau (n)} \\mid \\hat{\\mbox{$q$}}_{n-1} \\right)$ is the PDF that a gamma ray of $\\hat{\\mbox{$q$}}_{n-1}$ is absorbed at $\\hat{\\mbox{$r$}}_{n}$ , and the interaction is measured as $\\mbox{$D$}_{\\tau (n)}$ : $\\begin{aligned}P_\\mathrm {abs} & \\left( \\hat{\\mbox{$q$}}_{n}, \\mbox{$D$}_{\\tau (n)} \\mid \\hat{\\mbox{$q$}}_{n-1}\\right) \\\\= P&_\\mathrm {path, abs} \\left( \\hat{\\mbox{$r$}}_{n} \\mid \\hat{\\mbox{$q$}}_{n-1}\\right) P_\\mathrm {det} \\left(\\mbox{$D$}_{\\tau (n)} \\mid \\hat{\\mbox{$q$}}_{n-1}, \\hat{\\mbox{$q$}}_{n} \\right)~,\\end{aligned}$ where $P_\\mathrm {path, abs}(\\cdot )$ is the PDF that a gamma ray with $\\hat{\\mbox{$q$}}_{i-1}$ is absorbed at $\\hat{\\mbox{$r$}}_{i}$ .",
"In the following subsections, we explain these PDFs in detail.",
"Besides, the likelihood function for escape events is described as $\\begin{split}L_\\mathrm {escape} &\\left((\\hat{\\mbox{$q$}}_{i}), \\tau (\\cdot );(\\mbox{$D$}_I)\\right) \\\\&= P_\\mathrm {esc} \\left( \\hat{\\mbox{$q$}}_{n}, \\mbox{$D$}_{\\tau (n)} \\mid \\hat{\\mbox{$q$}}_{n-1}\\right) \\\\&\\times \\prod _{i=1}^{n-1} P_\\mathrm {scat} \\left( \\hat{\\mbox{$q$}}_{i}, \\mbox{$D$}_{\\tau (i)} \\mid \\hat{\\mbox{$q$}}_{i-1}\\right)~(n \\ge 2)~.\\end{split}$ The only difference between Eq.",
"REF and Eq.",
"REF is the treatment of the last interaction.",
"Here we introduce $P_\\mathrm {esc} \\left( \\hat{\\mbox{$q$}}_{n}, \\mbox{$D$}_{\\tau (n)} \\mid \\hat{\\mbox{$q$}}_{n-1}\\right)$ , which is the PDF that a gamma ray of $\\hat{\\mbox{$q$}}_{n-1}$ is scattered with changing its state to $\\hat{\\mbox{$q$}}_{n}$ and escape from the detector, and the interaction is measured as $\\mbox{$D$}_{\\tau (n)}$ .",
"It is described as $\\begin{aligned}P_\\mathrm {esc} \\left( \\hat{\\mbox{$q$}}_{n}, \\mbox{$D$}_{\\tau (n)} \\mid \\hat{\\mbox{$q$}}_{n-1} \\right) =P_\\mathrm {path,scat} \\left( \\hat{\\mbox{$r$}}_{n} \\mid \\hat{\\mbox{$q$}}_{n-1}\\right) &\\\\\\times \\int \\mathrm {d}(\\cos \\hat{\\theta }_n) \\mathrm {d}\\hat{\\phi }_n P_\\mathrm {KN} \\left( \\hat{\\theta }_n, \\hat{\\phi }_n \\mid \\hat{\\mbox{$q$}}_{n-1}\\right) &\\\\\\times P_\\mathrm {det} \\left( \\mbox{$D$}_{\\tau (n)} \\mid \\hat{\\mbox{$q$}}_{n-1}, \\hat{\\mbox{$q$}}_{n}\\right) P_\\mathrm {path, esc} \\left( \\hat{\\mbox{$q$}}_{n} \\right)&~,\\end{aligned}$ where $P_\\mathrm {path, esc}(\\hat{\\mbox{$q$}}_{n})$ is the probability that the gamma ray with a state of $\\hat{\\mbox{$q$}}_{n}$ escapes from the detector.",
"Since we cannot know the momentum direction of the escape gamma ray, we integrate the functions over $\\hat{\\phi }_n$ and $\\cos \\hat{\\theta }_n$ ."
],
[
"Probability functions related to physical processes",
"In the above equations, $P_\\mathrm {path,scat}( \\hat{\\mbox{$r$}}_{i} \\mid \\hat{\\mbox{$q$}}_{i-1} )$ and $P_\\mathrm {path,abs}(\\hat{\\mbox{$r$}}_{i} \\mid \\hat{\\mbox{$q$}}_{i-1})$ describe the PDFs that a gamma ray of $\\hat{\\mbox{$q$}}_{i-1}$ is scattered or absorbed at $\\hat{\\mbox{$r$}}_{i}$ .",
"They are defined as $&\\begin{aligned}&P_\\mathrm {path,scat} \\left( \\hat{\\mbox{$r$}}_{i} \\mid \\hat{\\mbox{$q$}}_{i-1}\\right) \\mathrm {d}^3 \\hat{\\mbox{$r$}}_{i} = \\\\&~~~~~\\frac{\\rho \\sigma _\\mathrm {scat}}{|\\hat{\\mbox{$r$}}_{i} - \\hat{\\mbox{$r$}}_{i-1}|^2} \\exp \\left(-\\rho \\sigma _\\mathrm {all} |\\hat{\\mbox{$r$}}_{i} - \\hat{\\mbox{$r$}}_{i-1}|\\right) \\mathrm {d}^3 \\hat{\\mbox{$r$}}_{i}~,\\end{aligned}\\\\&\\begin{aligned}&P_\\mathrm {path,abs} \\left( \\hat{\\mbox{$r$}}_{i} \\mid \\hat{\\mbox{$q$}}_{i-1} \\right) \\mathrm {d}^3 \\hat{\\mbox{$r$}}_{i} = \\\\&~~~~~\\frac{\\rho \\sigma _\\mathrm {abs}}{|\\hat{\\mbox{$r$}}_{i} - \\hat{\\mbox{$r$}}_{i-1}|^2} \\exp \\left(-\\rho \\sigma _\\mathrm {all} |\\hat{\\mbox{$r$}}_{i} - \\hat{\\mbox{$r$}}_{i-1}|\\right) \\mathrm {d}^3 \\hat{\\mbox{$r$}}_{i}~,\\end{aligned}$ where $\\sigma _\\mathrm {abs}$ , $\\sigma _\\mathrm {scat}$ are the cross sections of photoabsorption, Compton scattering at an energy of $\\hat{E}_{i-1}$ , respectively; $\\rho $ is the number density of the detector material; $\\sigma _\\mathrm {all}$ is the sum of the cross sections of physical processes.",
"When only the pair creation ($\\sigma _\\mathrm {pair}$ ) is considered additionally, it is equal to $\\sigma _\\mathrm {abs} + \\sigma _\\mathrm {scat} + \\sigma _\\mathrm {pair}$ .",
"The term $\\frac{1}{|\\hat{\\mbox{$r$}}_{i} - \\hat{\\mbox{$r$}}_{i-1}|^2}$ corresponds to the solid angle of the volume element at $\\hat{\\mbox{$r$}}_{i}$ seen from $\\hat{\\mbox{$r$}}_{i-1}$ .",
"When incoming gamma rays are considered to be parallel light as in astrophysical observations, this term with $i = 1$ can be removed and Eq.",
"REF is described as $P_\\mathrm {path,scat} \\left( \\hat{\\mbox{$r$}}_{1} \\mid \\hat{\\mbox{$q$}}_{0} \\right) \\mathrm {d}^3 \\hat{\\mbox{$r$}}_{1} \\propto \\rho \\sigma _\\mathrm {scat} \\exp \\left(-\\rho \\sigma _\\mathrm {all} \\hat{l}_{\\mathrm {first}}\\right) \\mathrm {d}^3 \\hat{\\mbox{$r$}}_{1}~,$ where $\\hat{l}_{\\mathrm {first}}$ is the length of the gamma-ray path inside the detector before it arrives at $\\hat{\\mbox{$r$}}_{1}$ .",
"Note that here it is assumed that the detector consists of a single material.",
"If a Compton telescope consists of several detectors with different materials, e.g., semiconductor detectors and scintillators, then $\\sigma _\\mathrm {abs/scat/pair}$ and $\\rho $ also depend on the position.",
"In this case, Eq.",
"REF is modified as $\\begin{aligned}&P_\\mathrm {path,scat} \\left( \\hat{\\mbox{$r$}}_{i} \\mid \\hat{\\mbox{$q$}}_{i-1} \\right) \\mathrm {d}^3 \\hat{\\mbox{$r$}}_{i} = \\\\&\\frac{\\rho (\\hat{\\mbox{$r$}}_{i}) \\sigma _\\mathrm {scat} (\\hat{\\mbox{$r$}}_{i}) }{|\\hat{\\mbox{$r$}}_{i} - \\hat{\\mbox{$r$}}_{i-1}|^2} \\exp \\left(-\\int _{C} \\rho (\\hat{\\mbox{$r$}}) \\sigma _\\mathrm {all} (\\hat{\\mbox{$r$}}) \\mathrm {d}|\\hat{\\mbox{$r$}}| \\right) \\mathrm {d}^3 \\hat{\\mbox{$r$}}_{i}~,\\end{aligned}$ where $C$ is the straight line from $\\hat{\\mbox{$r$}}_{i-1}$ to $\\hat{\\mbox{$r$}}_{i}$ .",
"The same applies to Eqs.",
"and REF .",
"The function $P_\\mathrm {KN} \\left(\\hat{\\theta }_{i}, \\hat{\\phi }_{i} \\mid \\hat{\\mbox{$q$}}_{i-1}\\right)$ is the PDF that the gamma ray of $\\hat{\\mbox{$q$}}_{i-1}$ is scattered to the direction described by $(\\hat{\\theta }_{i}, \\hat{\\phi }_{i})$ .",
"Namely it is the normalized differential cross section of Compton scattering.",
"Following Klein-Nishina's formula [25], it is defined as $\\begin{aligned}P_\\mathrm {KN} & \\left( \\hat{\\theta }_i, \\hat{\\phi }_i \\mid \\hat{\\mbox{$q$}}_{i-1} \\right) \\mathrm {d}\\hat{\\Omega }_i = \\frac{1}{\\sigma _\\mathrm {scat}} \\frac{\\mathrm {d}\\sigma _\\mathrm {scat}}{\\mathrm {d}\\hat{\\Omega }_i} \\mathrm {d}\\hat{\\Omega }_i = \\frac{1}{\\sigma _\\mathrm {scat}} \\frac{r_e^2}{2} \\\\\\times & \\left(\\frac{\\hat{E}_{i}}{\\hat{E}_{i-1}}\\right)^2 \\left( \\frac{\\hat{E}_{i}}{\\hat{E}_{i-1}} + \\frac{\\hat{E}_{i-1}}{\\hat{E}_{i}} - \\sin ^2\\hat{\\vartheta }^\\mathrm {scat}_i\\right) \\mathrm {d}\\hat{\\Omega }_i~,\\end{aligned}$ where $r_e$ is classical electron radius and $\\hat{E}_{i}$ is the energy of the scattered gamma ray calculated by Eq.",
"REF and $\\hat{\\vartheta }^\\mathrm {scat}_i$ is the $i$ -th scattering angle defined in Eq.",
"REF and $\\mathrm {d}\\hat{\\Omega }_i$ is equal to $\\mathrm {d}(\\cos \\hat{\\theta }_i) \\mathrm {d}\\hat{\\phi }_i$ .",
"Finally we define $P_\\mathrm {path,esc} \\left( \\cdot \\right)$ as the probability that a gamma ray escapes from the detector without any interaction.",
"It is described as $P_\\mathrm {path,esc} \\left( \\hat{\\mbox{$q$}}_{n} \\right) = \\exp \\left(-\\rho \\sigma _\\mathrm {all} \\hat{l}_n \\right)~,$ where $\\hat{l}_n$ is the length between $\\mbox{$\\hat{r}$}_n$ and the boundary of the detector from $\\mbox{$\\hat{r}$}_n$ along the direction of escape."
],
[
"Probability functions related to measurements",
"The function $P_\\mathrm {det}(\\cdot )$ corresponds to the measurement process, and then it is determined by the detector response and what kind of quantities a Compton telescope can measure.",
"When a Compton telescope measures the deposit energy and position of each interaction independently, then $P_\\mathrm {det}(\\cdot )$ is usually expressed as the product of two detector response functions defined as $\\begin{aligned}P_\\mathrm {det} \\left( \\mbox{$D$}_{\\tau (i)} \\mid \\hat{\\mbox{$q$}}_{i-1}, \\hat{\\mbox{$q$}}_{i}\\right) = & P_\\mathrm {ene}(\\varepsilon _{\\tau (i)} \\mid \\hat{\\varepsilon }_{i}) \\\\&\\times P_\\mathrm {pos}(\\mbox{$r$}_{\\tau (i)} \\mid \\hat{\\mbox{$r$}}_i)~,\\end{aligned}$ where $\\hat{\\varepsilon }_{i}$ is the true deposited energy at $i$ -th interaction, which is equal to $\\hat{E}_{i-1} - \\hat{E}_{i}$ .",
"Here $P_\\mathrm {ene}(\\varepsilon _{\\tau (i)} \\mid \\hat{\\varepsilon }_{i})$ is the PDF that the true deposit energy of $\\hat{\\varepsilon }_{i}$ is detected as $\\varepsilon _{\\tau (i)}$ , and $P_\\mathrm {pos}(\\mbox{$r$}_{\\tau (i)} \\mid \\hat{\\mbox{$r$}}_i)$ is one that the interaction position of $\\hat{\\mbox{$r$}}_i$ is detected as $\\mbox{$r$}_{\\tau (i)}$ .",
"In general, these two can have any function forms and one can determine them so that they explain experimental data adequately."
],
[
"Implementation",
"In the maximum likelihood estimation, the estimation of parameters is obtained as those that yield the maximum likelihood value.",
"For an event reconstruction in a Compton telescope, one needs to estimate the incident energy ($\\hat{E}_0^{\\mathrm {ML}}$ ), the incoming direction ($\\hat{\\theta }_0^{\\mathrm {ML}}, \\hat{\\phi }_0^{\\mathrm {ML}}$ ), and the scattering order.",
"The estimated parameter set is obtained as one that maximizes Eq.",
"REF or Eq.",
"REF .",
"Ideally, this task is achieved by sweeping the parameter space of the incoming gamma-ray energy, direction, and interaction positions.",
"However, it is not practical in terms of computational time because the parameter space becomes of high dimensions, i.e., $(3n+3)$ dimensions when the number of interactions is $n$ .",
"In this section, we implement the order determination algorithm by focusing on a Compton telescope that measures the interaction positions and deposited energies and does not obtain the trajectories of recoiled electrons.",
"First, we introduce specific forms for the detector response terms.",
"Then, we introduce several approximations to eliminate integral calculations and parameter space search in the likelihood calculation."
],
[
"Detector response",
"To describe the detector response terms $P_\\mathrm {ene}(\\varepsilon \\mid \\hat{\\varepsilon })$ and $P_\\mathrm {pos}(\\mbox{$r$} \\mid \\hat{\\mbox{$r$}})$ , we adopt the Gaussian function, which is sufficient in most cases.",
"Namely, these functions are formulated as $P_\\mathrm {ene}(\\varepsilon \\mid \\hat{\\varepsilon }) \\mathrm {d}\\varepsilon = \\frac{1}{\\sqrt{2 \\pi \\sigma ^2_{\\hat{\\varepsilon }}}} \\exp \\left( - \\frac{(\\hat{\\varepsilon } - \\varepsilon )^2}{2 \\sigma ^2_{\\hat{\\varepsilon }}} \\right) \\mathrm {d}\\varepsilon ~,$ $\\begin{aligned}P_\\mathrm {pos}(\\mbox{$r$} \\mid \\hat{\\mbox{$r$}}) \\mathrm {d}^3 \\mbox{$r$}&= \\frac{1}{\\sqrt{2 \\pi \\sigma ^2_{\\hat{x}}}}\\exp \\left( - \\frac{(\\hat{x} - x)^2}{2 \\sigma ^2_{\\hat{x}}} \\right) \\\\&\\times \\frac{1}{\\sqrt{2 \\pi \\sigma ^2_{\\hat{y}}}}\\exp \\left( - \\frac{(\\hat{y} - y)^2}{2 \\sigma ^2_{\\hat{y}}} \\right) \\\\&\\times \\frac{1}{\\sqrt{2 \\pi \\sigma ^2_{\\hat{z}}}}\\exp \\left( - \\frac{(\\hat{z} - z)^2}{2 \\sigma ^2_{\\hat{z}}} \\right) \\mathrm {d}^3 \\mbox{$r$}~,\\end{aligned}$ where $\\sigma _{\\hat{\\varepsilon }}$ , $\\sigma _{\\hat{x}}$ , $\\sigma _{\\hat{y}}$ , and $\\sigma _{\\hat{z}}$ are the energy and positional resolutions of the detector, respectively.",
"For the full-deposit events, $\\hat{E}_0^{\\mathrm {ML}}$ and the interaction positions are estimated by a good approximation as the sum of the measured deposited energies and the measured positions, respectively: $&\\hat{E}_0^\\mathrm {ML} \\simeq \\sum _{i=1}^{n} \\varepsilon _i~,\\\\&\\hat{\\mbox{$r$}}_i^{\\mathrm {ML}} \\simeq \\mbox{$r$}_{\\tau (i)}~(i \\ge 1)~.$ In this approximation, we ignore the positional errors of the interaction sites.",
"To compensate for this, we modify the energy resolution in Eq.",
"REF .",
"At $i$ -th interaction ($2 \\le i \\le n-1$ ), the scattering angle can be calculated geometrically, and the positional errors produce the uncertainty on it, to which we refer as $\\Delta (\\cos \\vartheta ^\\mathrm {scat}_{i})_{\\mathrm {pos}}$ .",
"Considering the propagation of the positional errors as discussed in the classical approach [19], $\\Delta (\\cos \\vartheta ^\\mathrm {scat}_{i})_{\\mathrm {pos}}$ can be calculated.",
"Then, the partial derivative of Eq.",
"REF with respect to $\\cos \\vartheta ^\\mathrm {scat}_{i}$ yields $\\Delta \\hat{\\varepsilon }_{i} = \\frac{\\hat{E}^2_{i}}{m_e c^2} \\Delta (\\cos \\vartheta ^\\mathrm {scat}_{i})~,$ which means that the positional errors produce uncertainty on the estimation of the gamma-ray energy through the scattering angle calculation.",
"In our algorithm, to include this effectively, for $2 \\le i \\le n-1$ , we modify the energy resolution as $\\sigma ^2_{\\hat{\\varepsilon }} \\rightarrow \\sigma ^2_{\\hat{\\varepsilon }} + \\left(\\frac{\\hat{E}^2_{i}}{m_e c^2} \\Delta (\\cos \\vartheta ^\\mathrm {scat}_{i})_{\\mathrm {pos}}\\right)^2~.$ The incoming direction ($\\hat{\\theta }_0^{\\mathrm {ML}}, \\hat{\\phi }_0^{\\mathrm {ML}}$ ) should be also determined.",
"In Compton telescopes, the incoming gamma-ray direction is constrained on a circle in the sky.",
"We calculate the first scattering angle under assumption of the incident energy ($\\hat{E}_{0}^\\mathrm {ML}$ ) and the deposit energy ($\\varepsilon _{\\tau (1)}$ ) and the first and second interaction positions ($\\mbox{$r$}_{\\tau (1)}, \\mbox{$r$}_{\\tau (2)}$ ), and then the circle can be obtained as the solutions of ($\\hat{\\theta }_0^{\\mathrm {ML}}, \\hat{\\phi }_0^{\\mathrm {ML}}$ ) that satisfies the Compton kinematics: $\\hat{E}_{0}^\\mathrm {ML} = \\hat{E}_{1} (\\hat{E}_{0}^\\mathrm {ML}, \\hat{\\theta }_{0}^\\mathrm {ML}, \\hat{\\phi }_{0}^\\mathrm {ML}, \\mbox{$r$}_{\\tau (1)}, \\mbox{$r$}_{\\tau (2)}) + \\varepsilon _{\\tau (1)}~.$ Here $\\hat{E}_{1}$ is calculated by Eqs.",
"REF and REF .",
"The function $P_\\mathrm {path,scat} \\left( \\hat{\\mbox{$r$}}_{1} \\mid \\hat{\\mbox{$q$}}_{0} \\right)$ also depends on $\\hat{\\theta }_0^{\\mathrm {ML}}$ and $\\hat{\\phi }_0^{\\mathrm {ML}}$ .",
"Here we assume a constant value for the path length in Eq.",
"REF : $\\hat{l}_\\mathrm {first} = L_0~.$ This should be comparable to the size scale of the detector.",
"Then, the likelihood values are the same for the parameters ($\\hat{\\theta }_0^{\\mathrm {ML}}, \\hat{\\phi }_0^{\\mathrm {ML}}$ ) that satisfy Eq.",
"REF , i.e., on a Compton circle, and they are considered to be the approximation of the likelihood value for the assumed scattering order."
],
[
"Escape events",
"For escape events, the incident gamma-ray energy and escape energy can be estimated using the scattering angle measured from geometrical information [18].",
"Here we calculate the incident gamma-ray energy at each $i$ -th interaction site ($2 \\le i \\le n-1$ ), and use the averaged value as the estimation.",
"Based on [20], $\\hat{E}_0^\\mathrm {ML}$ is estimated as $\\hat{E}_0^\\mathrm {ML} = \\frac{1}{n-2} \\sum _{m=2}^{n-1} \\hat{E}_0^\\mathrm {ML, m}~(n \\ge 3)~,$ where, $& \\hat{E}_0^\\mathrm {ML, m} = \\sum _{i=1}^{m} \\varepsilon _{\\tau (i)} + E^{\\prime }_{m}~, \\\\& E^{\\prime }_m = -\\frac{\\varepsilon _{\\tau (m)}}{2} + \\sqrt{\\frac{\\varepsilon _{\\tau (m)}^2}{4} + \\frac{\\varepsilon _{\\tau (m)} m_e c^2}{1 - \\cos \\vartheta ^{\\mathrm {scat},G}_{m}}}~, \\\\& \\cos \\vartheta ^{\\mathrm {scat},G}_{m} = \\frac{(\\mbox{$r$}_{\\tau (m)} - \\mbox{$r$}_{\\tau (m-1)})\\cdot (\\mbox{$r$}_{\\tau (m+1)} - \\mbox{$r$}_{\\tau (m)})}{|\\mbox{$r$}_{\\tau (m)} - \\mbox{$r$}_{\\tau (m-1)}| |\\mbox{$r$}_{\\tau (m+1)} - \\mbox{$r$}_{\\tau (m)}|}~.$ In the calculation of the likelihood function for escape events, the term $P_\\mathrm {esc} \\left( \\hat{\\mbox{$q$}}_{n}, \\mbox{$D$}_{\\tau (n)} \\mid \\hat{\\mbox{$q$}}_{n-1} \\right)$ needs the integration over the direction of escape (see Eq.",
"REF ).",
"To calculate it approximately without integral computation, here we assume that $&P_\\mathrm {ene} (\\varepsilon _{\\tau (n)} \\mid \\hat{\\varepsilon }_{n})\\simeq \\delta \\left( \\varepsilon _{\\tau (n)} - \\hat{\\varepsilon }_{n}\\right)~.$ We also approximate $\\hat{l}_n$ in Eq.",
"REF as a constant value: $\\hat{l}_\\mathrm {esc} = L_\\mathrm {esc}~,$ where $L_\\mathrm {esc}$ should be also comparable to the detector size scale like $L_0$ .",
"Then these approximations yield $\\begin{aligned}&P_\\mathrm {esc} \\left( \\hat{\\mbox{$q$}}_{n}, \\mbox{$D$}_{\\tau (n)} \\mid \\hat{\\mbox{$q$}}_{n-1}\\right) \\\\&= P_\\mathrm {path,scat} \\left( \\mbox{$r$}_{\\tau (n)} \\mid \\hat{\\mbox{$q$}}_{n-1}\\right) \\\\&~~~~\\times \\int \\mathrm {d}(\\cos \\hat{\\vartheta }^\\mathrm {scat}_n) \\mathrm {d}\\hat{\\varphi }^{\\mathrm {scat}}_n P_\\mathrm {KN} \\left(\\hat{\\theta }_{n}, \\hat{\\phi }_{n} \\mid \\hat{\\mbox{$q$}}_{n-1} \\right) \\\\&~~~~~~~~\\times P_\\mathrm {det} \\left( \\mbox{$D$}_{\\tau (n)} \\mid \\hat{\\mbox{$q$}}_{n-1}, \\hat{\\mbox{$q$}}_{n}\\right) P_\\mathrm {path,esc} \\left( \\hat{\\mbox{$q$}}_{n} \\right) \\\\&= P_\\mathrm {path,scat} \\left( \\mbox{$r$}_{\\tau (n)} \\mid \\hat{\\mbox{$q$}}_{n-1}\\right)\\frac{P_{\\mathrm {pos}} (\\mbox{$r$}_{\\tau (n)}, \\mbox{$r$}_{\\tau (n)}) }{\\sigma _\\mathrm {scat}} \\\\&~~~~\\times \\int \\mathrm {d}(\\cos \\hat{\\vartheta }^\\mathrm {scat}_n) \\mathrm {d}\\hat{\\varphi }^{\\mathrm {scat}}_n \\frac{\\mathrm {d}\\sigma _\\mathrm {scat}}{\\mathrm {d}\\Omega _n}\\\\&~~~~~~~~\\times \\delta \\left( \\varepsilon _{\\tau (n)} - \\hat{\\varepsilon }_{n} \\right)\\exp \\left(-\\rho \\sigma _\\mathrm {all} L_\\mathrm {esc} \\right)\\\\&= P_\\mathrm {path,scat} \\left( \\mbox{$r$}_{\\tau (n)} \\mid \\hat{\\mbox{$q$}}_{n-1}\\right)\\frac{P_{\\mathrm {pos}} (\\mbox{$r$}_{\\tau (n)}, \\mbox{$r$}_{\\tau (n)}) }{\\sigma _\\mathrm {scat}} \\\\&~~~~\\times \\int \\mathrm {d}\\hat{\\varepsilon }_{n} \\mathrm {d}\\varphi ^{\\mathrm {scat}}_n \\frac{\\mathrm {d}\\cos \\vartheta ^\\mathrm {scat}_n}{\\mathrm {d}\\hat{\\varepsilon }_{n}} \\frac{\\mathrm {d}\\sigma _\\mathrm {scat}}{\\mathrm {d}\\Omega _n}\\\\&~~~~~~~~\\times \\delta \\left( \\varepsilon _{\\tau (n)} - \\hat{\\varepsilon }_{n} \\right)\\exp \\left(-\\rho \\sigma _\\mathrm {all} L_\\mathrm {esc}\\right) \\\\&= P_\\mathrm {path,scat} \\left( \\mbox{$r$}_{\\tau (n)} \\mid \\hat{\\mbox{$q$}}_{n-1}\\right)\\frac{P_{\\mathrm {pos}} (\\mbox{$r$}_{\\tau (n)}, \\mbox{$r$}_{\\tau (n)}) }{\\sigma _\\mathrm {scat}} \\\\&~~~~\\times \\frac{2 \\pi m_e c^2}{(\\hat{E}_{n-1} - \\varepsilon _{\\tau (n)})^2}\\frac{\\mathrm {d}\\sigma _\\mathrm {scat}}{\\mathrm {d}\\Omega }\\exp \\left(-\\rho \\sigma _\\mathrm {all} L_\\mathrm {esc}\\right)~.\\end{aligned}$ Note that here the direction of escape is described with the scattering angle $\\hat{\\vartheta }^\\mathrm {scat}_n$ at the last interaction and the azimuth angle $\\hat{\\varphi }^{\\mathrm {scat}}_n$ along $\\mbox{$r$}_{\\tau (n)} - \\mbox{$r$}_{\\tau (n-1)}$ , not $\\hat{\\theta }_n$ and $\\hat{\\phi }_n$ , because it makes the calculation simpler, e.g., $\\mathrm {d}\\cos \\vartheta ^\\mathrm {scat}_n / \\mathrm {d}\\hat{\\varepsilon }_{n}$ ."
],
[
"Algorithm Summary",
"By calculating the likelihood as described in the previous section, the scattering order and the event type (full-deposit or escape) can be determined as follows.",
"One selects a scattering order from all candidates.",
"Here the selected order is labeled with $k$ : $( \\mbox{$D$}_I)_{\\mathrm {ordered}}^{k} = (\\mbox{$D$}_{\\tau _{k} (1)}, \\mbox{$D$}_{\\tau _{k} (2)}, ..., \\mbox{$D$}_{\\tau _{k} (n)} )_{\\mathrm {ordered}}~.$ When the number of the hits is $n$ , the number of the candidates is $n!$ , i.e., $1 \\le k \\le n!$ .",
"One calculates the likelihood value using the approximated formula introduced in Section REF .",
"Then, for each scattering candidate, one obtains the two likelihood values $L_\\mathrm {fulldep}^{k}$ and $L_\\mathrm {escape}^{k}$ , which correspond to the full-deposit and escape events, respectively.",
"The calculation procedure of $L_\\mathrm {fulldep}^{k}$ and $L_\\mathrm {escape}^{k}$ is described in the following subsections.",
"One determines the scattering order and the event type as a set of them that yield the maximum value in all of the calculated $L_\\mathrm {fulldep}^{k}$ and $L_\\mathrm {escape}^{k}$ ."
],
[
"Calculation of $L_\\mathrm {fulldep}^{k}$",
" The incoming gamma-ray energy is calculated as the sum of the detected deposit energy (Eq.",
"REF ), and the interaction positions are assumed to be the same as the detected ones (Eq. ).",
"The first scattering angle is calculated by Eq.",
"REF and the incoming gamma-ray direction is constrained on a Compton circle in the sky.",
"The likelihood value is calculated using Eq.",
"REF by applying the approximations described in Eq.",
"REF , REF"
],
[
"Calculation of $L_\\mathrm {escape}^{k}$",
" The incoming gamma-ray energy is calculated by Eq.",
"REF , and the interaction positions are assumed to be the same as the detected ones (Eq. ).",
"The first scattering angle is calculated by Eq.",
"REF and incoming gamma-ray direction is constrained on a Compton circle in the sky.",
"The likelihood value is calculated using Eq.",
"REF by applying the approximations described in Eq.",
"REF , REF , REF .",
"When the number of hits is two or fewer, the escape gamma-ray energy cannot be estimated in Eq.",
"REF .",
"Thus, this algorithm can work for 3 or more hit events.",
"Note that if the incident gamma-ray energy is known a priori, then this algorithm can also work for 2-hit events with minor modification (see Section REF )."
],
[
"Numerical Experiments",
"In order to test the reconstruction algorithm developed in this work, we apply it to a simulation data set generated by ComptonSoft, a software package of Geant4-based simulation and data analysis for Compton telescopes [26], [27], [28].",
"The algorithm has a great advantage in the event reconstruction of Compton telescopes with a large sensitive volume.",
"The GRAMS experiment utilizes such a large volume detector using a liquid argon time projection chamber [5].",
"We choose this type of Compton telescope for a numerical demonstration of the reconstruction algorithm, and evaluate the performance of the algorithm, i.e., the event classification (full-deposit or escape events), the incident gamma-ray energy, the scattering order and the angular resolution."
],
[
"Simulation Setup",
"We assume a simple cubic detector with a size of $30\\times 30\\times 30$ cm$^3$ filled with liquid argon as shown in Figure REF .",
"The energy resolution $\\sigma _{\\hat{\\varepsilon }}$ of the detector is set to as follows [5]: $\\sigma _{\\hat{\\varepsilon }}^2 = (5~\\mathrm {keV})^2 + 0.25 \\times (\\hat{\\varepsilon }/\\mathrm {keV})~.$ This equation is also adopted in the likelihood calculation in Eq.",
"REF .",
"The X-Y positions of signals are pixelized with a size of 3 mm, and we set $\\sigma _{\\hat{x}}$ and $\\sigma _{\\hat{y}}$ to be $3/\\sqrt{12}$ mm in Eq.",
"REF .",
"It is the standard deviation of X or Y positions of events distributed uniformly in a single pixel.",
"The resolution of the Z position is assumed to $\\sigma _{\\hat{z}} = $ 1 mm.",
"In this simulation, the signals in adjacent pixels are merged into a single signal, and the position of the merged signal is set to the center of gravity weighting with the detected energies in the pixels.",
"The energy threshold of each pixel is set to 25 keV.",
"In this demonstration, we set $L_\\mathrm {first}$ in Eq.",
"REF and $L_\\mathrm {esc}$ in Eq.",
"REF to be 15 cm, one-half of the detector size.",
"Figure: The geometry of Geant4 simulation."
],
[
"Results of Event Classification and Energy Reconstruction",
"Here we simulated $10^{7}$ events of 1 MeV gamma-ray beam incoming from the top of the detector, i.e., $\\hat{\\theta }_0 = \\pi $ .",
"The beam has a radius of 10 cm and is co-aligned at the detector center.",
"Figure REF shows the count of the detected events with the number of hits.",
"The counts of the full-deposit event reach the maximum at 4 hits.",
"Here we focus on the events with the number of hits from 3 to 8.",
"The ratio of the number of the events with more than 8 hits to the total detected events is just 0.07% and they are negligible.",
"Figure: The count of the detected events with the number of hits.",
"Here 10 7 10^{7} events of 1 MeV gamma rays are simulated.Note that 0 hit represents the event that gamma ray escapes from the detector without any interaction or all the produced signals are lower than the threshold.First, we examine the algorithm performance against full-deposit events qualitatively.",
"Figure REF shows the energy spectra of all detected events (blue, dotted) and those classified as full-deposit events (red, solid).",
"We confirmed that the events peaking around 1 MeV is correctly classified as full-deposit events and the continuum component below 1 MeV is reduced successfully after the reconstruction algorithm was applied.",
"As the number of hits is increased, the events around 1 MeV are classified as full-deposit more accurately.",
"This is because as the number of hits gets larger, the Compton scattering angle can be calculated redundantly at more sites.",
"Next, the algorithm performance against escape events is checked.",
"Figure REF shows energy spectra of events classified as escape events.",
"The blue solid lines are spectra of the sum of deposit energies of events classified as escape events.",
"We confirmed that most of the continuum components are successfully reconstructed as escape events.",
"Moreover, the spectra of incident gamma-ray energy corrected by Eq.",
"REF (red, solid) shows a clear peak at 1 MeV.",
"It confirms that the algorithm estimates the escape energy correctly.",
"The shape of the reconstructed energy spectrum of escape events strongly depends on the number of hits; when the number of hits is small, the spectrum has relatively long tails.",
"Figure REF shows the standard deviation of the reconstructed energy spectra in 900–1100 keV, with different number of hits.",
"The energy resolution of full-deposit events does not depend on the number of hits so much.",
"On the other hand, the energy resolution of escape events becomes better as the number of hits gets larger, due to the different method of the energy estimation.",
"Though the absolute value of the energy resolution varies by the assumed detector response, in general the energy resolutions of the full-deposit and escape events are different from each other, and they have different dependence on the number of hits.",
"This feature should be treated accurately when one performs spectral analysis or image reconstruction quantitatively.",
"Figure: The energy spectra of 1 MeV gamma-ray events classified as full-deposit events.The blue dashed and red solid lines represent the spectra of the total energy deposit of all events and those of the events classified as full-deposit events by the algorithm, respectively.Figure: The energy spectra of 1 MeV gamma-ray events classified as escape events.The blue solid lines represent the spectra of the total energy deposit of events classified as escape events by the algorithm, and the red solid ones represent the spectra of the incident energy of those events, estimated by the algorithm (see Eq.",
").The blue dashed lines are the same as in Figure .Figure: The energy resolution of the reconstructed spectra for 1 MeV gamma-ray events."
],
[
"Accuracy of the Event Reconstruction",
"In order to quantify the performance of the developed algorithm, we define the reconstruction accuracy $A_{\\mathrm {full}}$ and $A_{\\mathrm {esc}}$ as $A_{\\mathrm {full}} &= \\frac{N_{\\mathrm {full, acc}}}{N_{\\mathrm {full}}}~,\\\\A_{\\mathrm {esc}} &= \\frac{N_{\\mathrm {esc, acc}}}{N_{\\mathrm {esc}}}~,$ where $N_{\\mathrm {full}}$ is the total number of detected full-deposit events and $N_{\\mathrm {full, acc}}$ is the number of full events that are reconstructed as full-deposit events with the accurate scattering order; $N_{\\mathrm {esc}}$ is the total number of detected escape events and $N_{\\mathrm {esc, acc}}$ is the number of escape events that are reconstructed as escape events with the accurate scattering order.",
"We show the reconstruction accuracy for 1 MeV gamma-ray events in Figure REF .",
"In the calculation of $A_{\\mathrm {esc}}$ , we used only events in which the sum of the actual energy deposits is less than 900 keV because the escape energy is too small to detect due to the detector energy resolution when it is close to 1 MeV.",
"At 6–7 hit and 4–5 hit events, $A_{\\mathrm {full}}$ and $A_{\\mathrm {esc}}$ get their maximum, respectively.",
"This result can be interpreted qualitatively as follows.",
"As the number of hits is increased, we obtain more information about Compton scattering which occurred in the detector, which helps to determine the scattering order accurately.",
"On the other hand, the number of the scattering order candidates is also increased as the factorial of the number of hits, which would make the event reconstruction more complicating.",
"The trade-off between these two factors is considered to determine the number of hits that yields the highest accuracy.",
"We also calculated $A_{\\mathrm {full}}$ and $A_{\\mathrm {esc}}$ for gamma rays with different incoming energies from 500 keV to 10 MeV.",
"The results are shown in Figure REF .",
"Below $\\sim $ 1 MeV, 40–50% of 3-hit events and 50–70% of 4 or more hit events are reconstructed accurately.",
"As the incoming energy is increased, $A_{\\mathrm {full}}$ and $A_{\\mathrm {esc}}$ are decreased, especially above $\\sim $ 3 MeV.",
"In the argon detector, at $\\sim 5$ MeV the cross section of pair creation becomes comparable to that of Compton scattering.",
"The contamination of these pair creation events makes the accuracy worse.",
"Moreover, electrons with above few MeV produce gamma rays via bremsstrahlung emission, and these gamma rays are not considered in the algorithm.",
"It also reduces the performance of the event reconstruction.",
"We will discuss it in Section .",
"Figure: The reconstruction accuracy for 1 MeV gamma-ray events and its dependence on the number of hits.Figure: The energy dependence of the accuracy for both full-deposit (left) and escape (right) events.For the escape events,we used only events in which the actual escape energy is more than 100 keV."
],
[
"Angular Resolution and its dependence on the number of hits",
"We also investigated the angular resolution of the reconstructed events and its dependence on the number of hits.",
"The angular resolution is evaluated by the angular resolution measure (ARM), which is defined as $\\mathrm {ARM} = \\theta _K - \\theta _G~,$ where $\\theta _K$ is the estimated first scattering angle: $\\theta _K = \\arccos \\left( 1 + m_e c^2 \\left( \\frac{1}{\\hat{E}_0^\\mathrm {ML}} - \\frac{1}{\\hat{E}_0^\\mathrm {ML} - E_1} \\right) \\right)~,$ and $\\theta _G$ is the first scattering angle calculated from the detected positions of the first and second hits.",
"Figure REF shows distributions of ARM for both full-deposit and escape events up to 6 hits.",
"Here the incoming gamma-ray energy is 1 MeV, and the incoming direction is the same as that in Sec.",
"REF .",
"In both cases, as the number of hits increases, the tail components and a sub-peak at $\\sim 90$ degrees are reduced.",
"The angular resolution depends on the number of hits.",
"Figure REF shows FWHMs of the obtained ARM distributions for different number of hits.",
"While the FWHMs for the escape events are nearly constant, those of the full-deposit events depend on the number of hits significantly.",
"This is because full-deposit events with a small number of hits have large scattering angles at the first interaction.",
"Then, the angular resolution becomes worse by the Doppler broadening effect.",
"For example, in the full-deposit case, the ratio of the events with the first scattering angle larger than 60 degrees is $\\sim $ 70% for 3-hit events and $\\sim $ 10 % for 8-hit events.",
"This can be interpreted as that if the scattering angle is small the scattered gamma rays still have large energy and can be easily scattered in the detector many times.",
"On the other hand, in the escape events, the ratio of the forward scattering events does not depend on the number of hits so much, and the angular resolution is almost constant over the number of hits.",
"This dependence of the angular resolution on the event type and the number of hits should be also treated accurately in the quantitative imaging analysis.",
"Finally, we show a simple back-projection image of the 1 MeV gamma-rays events using both full-deposit and escape events in Figure REF .",
"Here the number of hits is from 3 to 8.",
"It demonstrates that the gamma-ray source is reconstructed successfully by using the multiple Compton scattering events.",
"Figure: ARM distributions of 1 MeV gamma rays with different number of hits.The left and right panels correspond to full-deposit and escape events, respectively.Figure: The angular resolutions of 1 MeV gamma rays for full-deposit and escape events.The FWHMs of the ARM distributions in Figure are shown for different number of hits.The filled round and open square markers correspond to the full-deposit and escape events, respectively.Figure: The simple back-projection image of 1 MeV gamma rays for full-deposit and escape events with the number of hits from 3 to 8."
],
[
"Discussion and Conclusion",
"In this work, we have formulated probability functions of physical processes in Compton telescopes and developed the reconstruction algorithm for the multiple Compton scattering events based on the maximum likelihood method.",
"It can treat both full-deposit and escape events simultaneously.",
"The developed algorithm can be used in large volume Compton telescopes aiming at up to $\\sim 10$ MeV, e.g., GRAMS, AMEGO/e-ASTOGAM, and so on.",
"We also verified its performance using simulation data sets of a $30\\times 30\\times 30$ cm$^3$ liquid argon detector, and confirmed that the algorithm works well for up to 8-hit events for 1 MeV gamma rays.",
"The reconstruction accuracy defined in Eq.",
"REF gets its maximum of $A_\\mathrm {full} \\sim 0.5$ at the number of hits of 6–7 for the full-deposit events.",
"In the case of escape events, it was maximized at 4-5 hit events with $A_\\mathrm {esc} \\sim 0.7$ .",
"The information about the Compton scattering in the detector increases as the number of hits increases, but also the number of the scattering order candidates increases as the factorial of the number of hits.",
"The trade-off between these two factors determines the reconstruction accuracy and its corresponding number of hits.",
"This result suggests that optimizing the number of hits in the detector for a target gamma-ray energy band is an essential factor to achieve good performance of Compton telescopes."
],
[
"The use of the likelihood for background reduction and measurement of gamma rays with known energy",
"The maximum likelihood value obtained from the algorithm is considered to use for background reduction, e.g., random coincidence events.",
"The blue line in Figure REF shows the distribution of the likelihood value of 3-hit full-deposit events using the 1 MeV gamma-ray simulation data set in Section REF .",
"To simply make the random coincidence events, we randomized the positions of the hits in this data set, and applied the reconstruction algorithm to them.",
"The red line in the figure corresponds to the distribution of the likelihood value of these randomized events.",
"As clearly seen, the randomized events have much smaller likelihood values than the gamma-ray events.",
"Thus, such background events can be removed by setting a threshold in the obtained likelihood value.",
"Figure: The distribution of the maximum likelihood value for 3-hit full-deposit events (blue) and the random coincidence events (red).",
"The incoming gamma-ray energy is 1 MeV.Our algorithm can be used in other fields, e.g., imaging in nuclear medicine therapy or monitoring of high-intensity radiation fields.",
"In these cases, incoming gamma-ray energy is often known a priori, and the reconstruction algorithm can be modified by fixing the incoming energy instead of estimating it by Eq.",
"REF or REF .",
"This additional information can improve the reconstruction performance, and allow us to analyze 2-hit events.",
"Figure REF shows the accuracy of 1 MeV gamma-ray event when fixing the incoming energy.",
"The reconstruction accuracy is improved, especially for full-deposit events with a small number of hits and escape events with any number of hits.",
"Moreover, this method could be also useful to estimate the in-orbit instrumental background of a Compton telescope or to search for gamma-ray emission lines from celestial objects because the energy of the target gamma rays is usually known in these cases.",
"Figure: The reconstruction accuracy when the incident gamma-ray energy is known a priori.",
"The incoming gamma rays are 1 MeV."
],
[
"Possible extension of the algorithm for electron-tracking Compton telescopes",
"Though this paper focuses on the case that only the deposit energies and positions are measured, our approach can be extensible even when a Compton telescope can also measure the trajectories of recoiled electrons and estimate their initial momentum directions [13], [14], [15].",
"In this case, $\\mbox{$D$}_I$ is needed to be redefined as $\\mbox{$D$}_I = \\left( \\mbox{$r$}_{I}, \\varepsilon _{I}, \\mbox{$p$}_{\\mathrm {e},I} \\right)~,$ where $\\mbox{$p$}_{\\mathrm {e},I}$ is the measured momentum direction of a recoiled electron.",
"Then, we modify the detector response function $P_\\mathrm {det} \\left( \\mbox{$D$}_{I} \\mid \\hat{\\mbox{$q$}}_{i-1}, \\hat{\\mbox{$q$}}_{i} \\right)$ by including the electron trajectory information.",
"Here we introduce a function $P_{\\mathrm {track}} (\\cdot )$ that compares the expected direction of a recoiled electron $\\left(\\hat{\\mbox{$p$}}_{i} - \\hat{\\mbox{$p$}}_{i-1}\\right)$ and measured one $\\left(\\mbox{$p$}_{\\mathrm {e},I}\\right)$ .",
"The definition of $P_{\\mathrm {track}} (\\cdot )$ would depend on the configuration of Compton telescopes because the obtained electron images vary by the detectors.",
"As an example, if the 3-dimensional trajectories can be obtained and the difference from the expected recoil direction obeys the Gaussian function, then $P_{\\mathrm {track}} (\\cdot )$ is determined as $&\\begin{aligned}P_\\mathrm {track}&( \\mbox{$p$}_{\\mathrm {e},I} \\mid \\hat{\\mbox{$p$}}_{i} - \\hat{\\mbox{$p$}}_{i-1} ) \\mathrm {d}\\Delta \\theta _{\\mathrm {e}} \\\\&= \\frac{1}{\\sqrt{2 \\pi \\sigma ^2_{\\Delta \\theta _{\\mathrm {e}}}}}\\exp \\left( - \\frac{\\Delta \\theta _{\\mathrm {e}}^2}{2 \\sigma ^2_{\\Delta \\theta _{\\mathrm {e}}}} \\right) \\mathrm {d}\\Delta \\theta _{\\mathrm {e}}~,\\end{aligned}\\\\&\\Delta \\theta _{\\mathrm {e}} = \\arccos \\left(\\frac{ \\left(\\hat{\\mbox{$p$}}_{i} - \\hat{\\mbox{$p$}}_{i-1}\\right) \\cdot \\hat{\\mbox{$p$}}_{\\mathrm {e},I} }{ |\\hat{\\mbox{$p$}}_{i} - \\hat{\\mbox{$p$}}_{i-1}| |\\hat{\\mbox{$p$}}_{\\mathrm {e},I}| }\\right)~,$ and $P_\\mathrm {det}(\\cdot )$ is reformulated as $\\begin{aligned}P_\\mathrm {det} \\left( \\mbox{$D$}_{I} \\mid \\hat{\\mbox{$q$}}_{i-1}, \\hat{\\mbox{$q$}}_{i}\\right) = & P_\\mathrm {ene}( \\varepsilon _I \\mid \\hat{\\varepsilon }_{i})\\times P_\\mathrm {pos}( \\mbox{$r$}_I \\mid \\hat{\\mbox{$r$}}_i) \\\\& \\times P_\\mathrm {track}( \\mbox{$p$}_{\\mathrm {e},I} \\mid \\hat{\\mbox{$p$}}_{i} - \\hat{\\mbox{$p$}}_{i-1})~.\\end{aligned}$ After modifying $P_\\mathrm {det}(\\cdot )$ , the scattering order can be determined in the same way as in Section but with one exception.",
"In this case, even if the incoming gamma-ray direction is constrained on a Compton circle, the likelihood value becomes to depend on the location in the circle because the direction of electron recoil varies depending on it.",
"Thus, one needs to specify the gamma-ray incoming direction in a Compton circle, where the likelihood is maximized."
],
[
"For further improvement",
"The developed algorithm successfully reconstructs multiple Compton scattering events with high accuracy.",
"However, two essential factors are not considered.",
"One is the momentum distribution of electrons in the detector materials.",
"When the initial electron momentum is not zero, the scattered gamma-ray energy differs from Eq.",
"REF .",
"This effect is known as the Doppler broadening effect [24], and limits the angular resolution.",
"Effectively, this effect could be considered by adding a function $\\sigma _{\\mathrm {Doppler}}$ in Eq.",
"REF , which is an uncertainty in the energy determination at each interaction site due to the Doppler broadening effect.",
"Since it depends on the gamma-ray energy, the scattering angle, and detector materials, the function $\\sigma _{\\mathrm {Doppler}}$ should be carefully modeled.",
"When the Doppler broadening effect is comparable to or dominates over the position and energy resolutions, then such a modification would improve the reconstruction accuracy.",
"The other factor is bremsstrahlung from scattered electrons in the detectors.",
"Bremsstrahlung becomes the dominant energy loss in high energy band, for example, in argon detector it dominates over the ionization losses at a few MeV.",
"Gamma rays emitted by this process make additional signals, which makes the event reconstruction more complex.",
"Decreasing the accuracy above a few MeV is considered to be partially due to bremsstrahlung (see Figure REF ).",
"To treat this process is not straightforward because it is a stochastic process without any strong restriction like Eq.",
"REF .",
"A possible way is to calculate the probability that a given signal is produced by a bremsstrahlung photon, considering energies and lengths to other interaction sites, and then merge it to nearby signals if the calculated probability is large.",
"This calculation is expected to be too complicated to be described by analytic expressions.",
"In a high energy band, it is also needed to distinguish pair creation events.",
"To consider these factors (the Doppler broadening, bremsstrahlung, and pair creation), other statistical methods using large data sets might be effective, e.g., deep neural network technique.",
"We expect that the combination of the analytical method like this work and simulation-based statistical methods can be one of the promising ways to achieve even better reconstruction performance."
],
[
"Code availability",
"The code for the reconstruction algorithm is available at https://github.com/odakahirokazu/ComptonSoft.",
"We acknowledge support from JSPS KAKENHI grant numbers 20K22355 and 20H00153."
]
] | 2107.01846 |
[
[
"Upper bound inequality for calculation time in simulated annealing\n analogous to adiabatic theorem in quantum systems"
],
[
"Abstract It has been recently reported that classical systems have speed limit for state evolution, although such a concept of speed limit had been considered to be unique to quantum systems.",
"Owing to the speed limit for classical system, the lower bound for calculation time of simulated annealing with desired calculation accuracy can be derived.",
"However, such a lower bound does not work as a criterion for completion of calculation in a practical time.",
"In this paper, we derive an inequality for classical system analogous to the quantum adiabatic theorem that gives calculation time for an accuracy-guaranteed fluctuation-exploiting computation.",
"The trade-off relation between calculation time and accuracy is given in the form tolerable in practical use."
],
[
"Introduction",
"Exploiting fluctuation has become an indispensable technique to solve optimization problems both in classical and quantum computations, i.e., simulated [1], [2], [3] and quantum annealing [4], [5], [6], [7], [8], [9].",
"The original optimization problem is appropriately mapped to a spin model [10].",
"Then the optimum solution for the original problem is translated as a ground state of the spin system that is realized through its natural relaxation process.",
"In such a general calculation scheme, relaxation time of the system is regarded as the calculation time to solve the optimization problem.",
"The relaxation of the quantum system is governed by the so-called quantum speed limit (QSL) [11], [12], [13], [14], [15], [16], [17], [18], [19], [20].",
"QSL gives the lower bound of the transition time of quantum systems from given initial state to the provided final state.",
"QSL yields an uncertainty relation between the time of transition and the energy gap between two states.",
"It has been reported that classical systems also have such a concept of speed limit, namely classical speed limit (CSL) [21], [22], although such an uncertainty relation had been considered to be unique in quantum systems.",
"Then the lower bound of the relaxation time in classical system is roughly evaluated by CSL.",
"The CSL is important for evaluating the limit of the calculation speed in principle.",
"Unlike simulated annealing where Hamiltonian is driven externally, CSL provides the limit on the relaxation speed of the system under a fixed Hamiltonian.",
"However, CSL allows us to estimate the speed limit for simulated annealing to approach a solution.",
"The lower limit of the calculation time given by CSL is roughly determined only by the desired calculation accuracy and initial state, and does not depend on the details of the system Hamiltonian.",
"Therefore, CSL imposes a computational speed limit that cannot be achieved by any arrangement of the Hamiltonian.",
"On the other hand, in order to judge whether the relaxation of the classical system is tolerable for practical use to solve optimization problems, it is required to evaluate the upper bound of the relaxation time rather than the lower bound, which is given by the CSL.",
"In this paper, we will obtain such an upper bound in an inequality form.",
"The derived inequality is regarded as the classical version of adiabatic theorem in quantum systems [23], [24].",
"In the context of fluctuation-exploiting computation, the adiabatic theorem implies that the calculation time is inversely proportional to the calculation accuracy.",
"The coefficient in such a trade-off relation depends on the energy gap between the ground and the first exited states, and the transition amplitude between these two states.",
"Because of the difficulty to concretely evaluate the transition amplitude, it is difficult to evaluate the relaxation time exactly from adiabatic theorem.",
"On the other hand, our inequality has a simple form to easily evaluate the upper bound depending only on the initial state and the energy gap.",
"Therefore, the calculation time in the worst case can be discussed focusing only on the energy gap without being bothered by estimating the transition amplitude in our framework."
],
[
"Classical speed limit",
"CSL was given independently in [21] and [22].",
"Shanahan, Chenu, Margolus, and del Campo derived the CSL as an uncertainty relation by phase-space approach in [21].",
"On the other hand, Okuyama and Ohzeki gave the CSL directly from stochastic dynamics in [22].",
"They consider the Fokker-Planck equation $\\dfrac{\\partial }{\\partial t}\\rho (x, t) = \\dfrac{\\partial }{\\partial x}\\left[2\\dfrac{\\partial W(x)}{\\partial x} + \\dfrac{\\partial }{\\partial x}\\right]\\rho (x, t)\\,,$ which has a steady state solution $\\pi (x) = \\exp \\left[-2W(x)\\right]$ .",
"Rewriting $\\rho (x, t)$ as $\\rho (x, t) = \\exp \\left[-W(x)\\right]\\psi (x, t)$ , we have the imaginary-time Schrödinger equation as $-\\dfrac{\\partial }{\\partial t}\\psi (x, t) &=& \\left[-\\dfrac{\\partial ^2}{\\partial x^2} + \\left(\\dfrac{\\partial W}{\\partial x}\\right)^2 - \\dfrac{\\partial ^2 W}{\\partial x^2}\\right]\\psi (x, t)\\nonumber \\\\&:=& \\hat{H}_{F}\\psi (x, t)\\,.$ The ground state for $\\hat{H}_F$ is given by $\\psi _0(x) = \\exp \\left[-W(x)\\right]$ .",
"Then the CSL corresponding to the QSL known as the Margolus-Levitin bound [12] is given as $\\tau \\ge \\dfrac{\\ln \\left\\langle \\psi (0)|\\psi (0)\\right\\rangle - \\ln \\left\\langle \\psi (0)|\\psi (\\tau )\\right\\rangle }{\\dfrac{\\left\\langle \\psi (0)|\\hat{H}_{F}|\\psi (0)\\right\\rangle }{\\left\\langle \\psi (0)|\\psi (0)\\right\\rangle }} := \\tau _{\\rm min}\\,.$ Note that the overlap between the initial and final states, $\\left\\langle \\psi (0)|\\psi (\\tau )\\right\\rangle $ , characterizes the deviation of the final state $\\psi (\\tau )$ from the ground state $\\psi _0$ , which is regarded as the calculation error in the context of simulated annealing.",
"Note also that the $\\tau _{\\rm min}$ gives the lower limit of relaxation time for the time-independent Fokker-Planck operator.",
"Thus $\\tau _{\\rm min}$ should be used as a guideline for the calculation speed limit, not as the calculation speed limit itself, in the use of simulated annealing where the Fokker-Planck operator varies temporally.",
"Eq.",
"(REF ) implies that, in the context of simulated annealing, the minimum required time to solve the optimization problem is roughly bounded by $\\tau _{\\rm min}$ that is evaluated only by desired calculation accuracy and initial condition.",
"However, such a lower bound of calculation time cannot be a criterion of completion of simulated annealing in a practical time, since $\\tau _{\\rm min}$ underestimates the calculation time.",
"In section , on the contrary, we will derive the upper bound of the calculation time that gives a criterion for a practical use of simulated annealing."
],
[
"Adiabatic theorem",
"In this section, we will overview the quantum adiabatic theorem, which provides the trade-off relation between calculation time and accuracy in quantum annealing.",
"Consider the temporally modified system Hamiltonian $\\hat{H}(t) = \\dfrac{t}{\\tau }\\hat{H}_0 + \\left(1-\\dfrac{t}{\\tau }\\right)\\hat{H}_1\\,,$ where $\\hat{H}_0$ is the target Hamiltonian corresponding to the considered optimization problem and $\\hat{H}_1$ is an initial Hamiltonian whose ground state can be easily prepared by some operations.",
"In our system settings, the temporal evolution of the system state, which is regarded as the calculation process to solve the optimization problem, is assumed to be stopped when the system Hamiltonian $\\hat{H}(t)$ becomes the target Hamiltonian $\\hat{H}_0$ .",
"The calculation time $\\tau $ , which can be an arbitrary positive value, is given by the protocol of the system Hamiltonian.",
"The calculation error $\\delta $ is characterized by the overlap between the ground state $|\\psi _0(\\tau )\\rangle $ of the target Hamiltonian $\\hat{H}_0$ and the system state $|\\psi (\\tau )\\rangle $ at the final time as $\\left| \\left\\langle \\psi (\\tau )|\\psi _0(\\tau )\\right\\rangle \\right|^2 = 1-\\delta ^2\\,.$ We assume here that the error is sufficiently small, i.e., $\\delta ^2 \\ll 1$ .",
"The calculation time $\\tau $ is related to $\\delta $ as $\\dfrac{\\max _t \\left\\langle \\psi _1(t)|\\hat{H}_0-\\hat{H}_1|\\psi _0(t)\\right\\rangle }{\\tau \\min _t \\Delta _t^2} = \\delta \\,,$ where $|\\psi _0(t)\\rangle $ and $|\\psi _1(t)\\rangle $ denote the ground and the first excited states for the Hamiltonian $\\hat{H}(t)$ at time $t$ , respectively, and $\\Delta _t$ is the energy gap between these two states [25].",
"The maximization and minimization are carried out over time $0 \\le t \\le \\tau $ .",
"Thus the trade-off between the calculation time and accuracy is roughly given as $\\tau \\propto \\dfrac{1}{\\delta \\min _t\\Delta _t^2}\\,.$ This is the quantum adiabatic theorem, which implies that longer calculation time leads less error, and greater energy gap is preferable both for less error and shorter calculation time.",
"However, it is difficult to concretely evaluate the calculation time with required accuracy from Eq.",
"(REF ) because of the transition amplitude from the ground to the first excited state in Eq.",
"(REF ).",
"In the next section, we will derive the simple inequality that can be regarded as the relaxed version of Eq.",
"(REF ) for classical systems."
],
[
"upper bound inequality",
"In this section, we will investigate the upper bound of calculation time for simulated annealing, which is realized in classical systems, in contrast to the lower bound given by CSL.",
"Consider a Fokker-Planck equation $\\dfrac{\\partial \\rho (t)}{\\partial t} = -\\hat{L}_{\\gamma (t), u(t)} \\rho (t)\\,,$ where the Fokker-Planck operator $\\hat{L}_{\\gamma (t), u(t)}$ depends on time via parameters $\\gamma $ and $u$ which are controlled during the calculation process.",
"The parameter $\\gamma $ controls the quantities such as noise strength, and it does not violate the detailed balance condition.",
"On the contrary, the parameter $u$ yields a probability flow, which characterizes the violation of detailed balance condition [26], [27].",
"Hereinafter, we assume that the Fokker-Planck operator $\\hat{L}_{\\gamma (t), u(t)}$ has a unique steady state solution for each $t$ .",
"The Fokker-Planck operator $\\hat{L}_{\\gamma , u}$ is assumed to be driven externally by the temporal changes of $\\gamma $ and $u$ .",
"The initial state is set to be the steady state for the operator $\\hat{L}_{\\gamma (0), u(0)}$ , which is easily prepared.",
"The steady state for the final operator $\\hat{L}_{\\gamma (\\tau ), u(\\tau )}$ is set to correspond to the optimum solution for the considered problem.",
"The Fokker-Planck operator $\\hat{L}_{\\gamma (t), u(t)}$ is characterized by its eigenvalues and eigenfunctions as follows: $\\hat{L}_{\\gamma (t), u(t)}\\phi _k^{\\gamma (t), u(t)} &=& \\lambda _k^{\\gamma (t), u(t)} \\phi _k^{\\gamma (t), u(t)}\\,,\\\\\\hat{L}_{\\gamma (t), u(t)}^\\dagger (t)\\psi _k^{\\gamma (t), u(t)} &=& \\lambda _k^{\\gamma (t), u(t)} \\psi _k^{\\gamma (t), u(t)}\\,,$ where $\\hat{L}_{\\gamma , u}^\\dagger $ indicates an adjoint operator of the Fokker-Planck operator.",
"The eigenvalues are ordered as $0 = \\lambda _0^{\\gamma (t), u(t)} \\le {\\rm Re}\\lambda _1^{\\gamma (t), u(t)}\\le {\\rm Re}\\lambda _2^{\\gamma (t), u(t)}\\le \\cdots $ .",
"The eigenfunctions satisfy the orthonormal relations: $\\int dx\\, \\psi _k^{\\gamma (t), u(t)}(x)\\phi _l^{\\gamma (t), u(t)}(x) = \\delta _{kl}\\,,$ where $\\delta _{kl}$ is a Kronecker delta and $\\displaystyle \\sum _n \\psi _n^{\\gamma (t), u(t)}(x) \\phi _n^{\\gamma (t), u(t)}(y) = \\delta (x - y)\\,.$ When the detailed balance condition holds, i.e., $u = 0$ , the eigenvalues are all real and the eigenfunctions satisfies $\\phi _k^{\\gamma (t), 0} = \\psi _k^{\\gamma (t), 0} \\pi ^{\\gamma (t)}\\,,$ where $\\pi ^{\\gamma (t)}$ is the steady state solution $\\hat{L}_{\\gamma (t), 0} \\pi ^{\\gamma (t)} = 0$ , and $\\pi ^{\\gamma (t)} = \\phi _0^{\\gamma (t), 0}$ by definition of eigenfunctions.",
"Eq.",
"(REF ) is straightforwardly shown by substituting it into the characteristic equation (REF ) for the adjoint operator $\\hat{L}_{\\gamma (t), u(t)}^\\dagger $ with $u(t) = 0$ .",
"Note that the steady state $\\pi ^{\\gamma (t)}$ is independent of the value of $u$ which controls a steady state probability flow, i.e., $\\hat{L}_{\\gamma (t), u(t)}\\pi ^{\\gamma (t)} = 0$ for arbitrary $u(t)$ [26].",
"For convenience, the steady state solution is normalized as $\\int dx\\,\\pi ^{\\gamma (t)} = 1$ for arbitrary $t$ , where the integration is carried out over all system degrees of freedom.",
"In the absence of detailed balance condition where $u(t)\\ne 0$ , eigenvalues are complex and the simple relation (REF ) does not hold in general.",
"Consider the expansion of the probability density $\\rho (t)$ , which follows the Fokker-Planck equation (REF ), in terms of the eigenfunctions at the final time in the presence of detailed balance condition, i.e., the eigenfunctions for $\\hat{L}_{\\gamma (\\tau ), 0}$ as $\\rho (t) = \\displaystyle \\sum _n c_n(t)\\phi _n^{\\gamma (\\tau ), 0}\\,.$ By our assumption that $\\hat{L}_{\\gamma (t), u(t)}$ has a unique stationary solution for fixed $t$ , an arbitrary state $\\rho (t)$ converges to the stationary solution $\\pi ^{\\gamma (t)}$ under the dynamics with fixed Fokker-Planck operator $\\hat{L}_{\\gamma (t), u(t)}$ after a long time.",
"Then in addition to Eq.",
"(REF ), we consider the expansion of $\\rho (t)$ by the eigenfunctions for $\\hat{L}_{\\gamma (t), u(t)}$ , which give orthonormal basis varying temporally: $\\rho (t) = \\displaystyle \\sum _n d_n(t)\\phi _n^{\\gamma (t), u(t)}\\,.$ Since the steady state solution $\\phi _0^{\\gamma (t), u(t)} = \\pi ^{\\gamma (t)}$ and $\\rho (t)$ both are normalized as $\\int dx\\, \\pi ^{\\gamma (t)} = 1$ and $\\int dx\\,\\rho (t) = 1$ , we find $d_n(t) = 0$ and $\\int dx\\,\\phi _n^{\\gamma (t), u(t)} = 0$ for $n\\ge 1$ .",
"The relation $c_0(t)=1$ is also easily derived from Eq.",
"(REF ).",
"Comparing the two expression for an arbitrary distribution Eqs.",
"(REF ) and (REF ), we find the coefficient $d_m(t)$ as $d_m(t) = \\displaystyle \\sum _n c_n(t) A_{nm}(t)\\,,$ where the matrix $A(t)$ is defined as $A_{nm}(t) = \\int dx\\,\\psi _m^{\\gamma (t), u(t)}\\phi _n^{\\gamma (\\tau ), 0}\\,.$ Furthermore, it is straightforwardly shown that the inverse of the matrix $A(t)$ is given as $\\left(A^{-1}(t)\\right)_{nm} = \\int dx\\psi _m^{\\gamma (\\tau ), 0}\\phi _n^{\\gamma (t), u(t)}$ by using the orthonormal relations Eqs.",
"(REF ) and (REF ).",
"Substituting the expression (REF ) into the left-hand-side and Eq.",
"(REF ) into the right-hand-side of the Fokker-Planck equation (REF ) respectively, and using Eqs.",
"(REF )–(REF ), we obtain the dynamics for $c_n(t)$ as $\\dot{c}_n(t) = -\\displaystyle \\sum _{m, k}c_m(t) A_{mk}(t)\\lambda _k^{\\gamma (t), u(t)}A_{kn}^{-1}(t)\\,.$ In order to evaluate the calculation error quantitatively, we consider the following overlap between probability densities: $D(\\tau ) = \\int \\dfrac{dx}{\\pi ^{\\gamma (\\tau )}}\\rho (0)\\rho (\\tau )\\,.$ Since $\\phi _k^{\\gamma (\\tau ), 0} = \\psi _k^{\\gamma (\\tau ), 0}\\pi ^{\\gamma (\\tau )}$ , this overlap is essentially same as the overlap $\\left\\langle \\psi (0)|\\psi (\\tau )\\right\\rangle $ appearing in Eq.",
"(REF ).",
"Using the expression of the probability density Eq.",
"(REF ) and the fact that $c_0(t) = 1$ for arbitrary $t$ , the overlap is rewritten as $D(\\tau ) = 1 + \\vec{c}(0)^{\\rm T}\\vec{c}(\\tau )\\,,$ where $\\vec{c}(t) = \\left(c_1(t), c_2(t), \\cdots \\right)^{\\rm T}$ .",
"Here, we have used the relation (REF ) for $\\phi _n^{\\gamma (\\tau ), 0}$ and the orthonormal condition (REF ).",
"According to Schwartz inequality, we obtain $D(\\tau ) \\le 1 + \\left|\\vec{c}(0)\\right|\\left|\\vec{c}(\\tau )\\right|\\,.$ Then $\\left|\\vec{c}(\\tau )\\right|$ is evaluated as follows: in the right-hand-side of Eq.",
"(REF ), $A\\Lambda A^{-1}$ gives a similarity transformation of $\\Lambda = {\\rm diag}(\\lambda _0^{\\gamma , u}, \\lambda _1^{\\gamma , u},\\cdots )$ .",
"Thus the eigenvalues for $A\\Lambda A^{-1}$ are same as those for $\\Lambda $ .",
"Furthermore, considering the order of the eigenvalues $0=\\lambda _0^{\\gamma (t), u(t)}\\le {\\rm Re}\\lambda _1^{\\gamma (t), u(t)}\\le {\\rm Re}\\lambda _2^{\\gamma (t), u(t)}\\le \\cdots $ and the fact that $\\lambda _n^{\\gamma (t), u(t)}$ changes its value temporally, we have $\\left|\\vec{c}(\\tau )\\right| \\le \\exp \\left[-\\tau \\displaystyle \\min _t {\\rm Re}\\lambda _1^{\\gamma (t), u(t)}\\right]\\left| \\vec{c}(0)\\right|\\,.$ Using Eqs.",
"(REF ) and (REF ), we conclude $D(\\tau ) \\le 1 + \\left[D(0)-1\\right]\\exp \\left[-\\tau \\displaystyle \\min _t{\\rm Re}\\lambda _1^{\\gamma (t), u(t)}\\right]\\,.$ This is equivalent to the expression for the calculation time $\\tau $ as $\\tau \\le \\dfrac{1}{\\min _t {\\rm Re}\\lambda _1^{\\gamma (t), u(t)}}\\ln \\left[\\dfrac{D(0)-1}{D(\\tau )-1}\\right] := \\tau _{\\rm max}^{\\gamma , u}\\,.$ Note that $D(\\tau ) = 1$ when $\\rho (\\tau ) = \\pi ^{\\gamma (\\tau )}$ , and $\\pi ^{\\gamma (\\tau )}$ is the steady state corresponding to the true solution of the considered optimization problem.",
"Thus $D(\\tau ) - 1$ in the right-hand-side of Eq.",
"(REF ) is regarded as a calculation error after $\\tau $ .",
"Moreover, $D(0) - 1$ is determined only by the initial state, and ${\\rm Re}\\lambda _1^{\\gamma (t), u(t)}$ plays the role of energy gap between the ground and the first excited states in the case of quantum annealing.",
"Thus the inequality (REF ) is regarded as the classical version of adiabatic theorem (REF ).",
"The inequality (REF ) indicates that the calculation time of $\\tau _{\\rm max}^{\\gamma , u}$ is sufficient to guarantee the calculation error less than $D(\\tau )-1$ .",
"In contrast to the quantum adiabatic theorem (REF ), our bound is loose, but avoid the difficulty of the estimation of transition amplitudes."
],
[
"Summary and Discussion",
"The inequality we have obtained gives the worst evaluation, while the CSL gives the best evaluation for the calculation time and accuracy.",
"In practice, the worst evaluation for the calculation time and accuracy should be used as a criterion for completion of the calculation in practical time.",
"The lower limit of calculation time $\\tau _{\\rm min}$ given by the CSL cannot be used for guaranteeing the calculation accuracy.",
"For example, even if $\\tau _{\\rm min}$ is one second, the actual calculation time to achieve the desired calculation accuracy may be one month.",
"On the other hand, the $\\tau _{\\rm max}^{\\gamma , u}$ obtained by our result guarantees the calculation accuracy.",
"If $\\tau _{\\rm max}^{\\gamma , u}$ is ten second, the desired calculation accuracy is always achieved within ten second.",
"Inequality (REF ) gives the upper bound of the calculation time for simulated annealing in a simple form.",
"Particularly, if the calculation error $D(\\tau ) - 1$ is preset as a target value, $\\ln \\left[\\left(D(0)-1\\right)/\\left(D(\\tau )-1\\right)\\right]$ in the right-hand-side does not depend on the relaxation dynamics.",
"${\\rm Re}\\lambda _1^{\\gamma (t), u(t)}$ only depends on the dynamics of the system.",
"Thus, in order to shorten the calculation time, it is required to focus on a good design of annealing dynamics for greater ${\\rm Re}\\lambda _1^{\\gamma , u}$ .",
"In the context of fasten convergence of Markov chain Monte Carlo algorithms [28], [29], [30], [31], [32], [33], [34], it is known that the violation of detailed balance yields the shift of the real part of the eigenvalues for the Fokker-Planck operator [26], [35], [36]: ${\\rm Re}\\lambda _1^{\\gamma , u} \\ge \\lambda _1^{\\gamma , 0}\\,,$ where $\\lambda _1^{\\gamma , 0}$ is real since the detailed balance condition holds.",
"Thus the annealing schedule without satisfying the detailed balance condition may yield shorter calculation time with same calculation accuracy as one satisfying detailed balance condition: $\\tau \\le \\tau _{\\rm max}^{\\gamma , u}\\le \\tau _{\\rm max}^{\\gamma , 0}\\,.$ The realization of such dynamics remains to be a future work.",
"A. Ichiki was supported by JSPS KAKENHI Grants No.",
"JP17H06469."
]
] | 2107.01792 |
[
[
"Optimum GSSK Transmission in Massive MIMO Systems Using the Box-LASSO\n Decoder"
],
[
"Abstract We propose in this work to employ the Box-LASSO, a variation of the popular LASSO method, as a low-complexity decoder in a massive multiple-input multiple-output (MIMO) wireless communication system.",
"The Box-LASSO is mainly useful for detecting simultaneously structured signals such as signals that are known to be sparse and bounded.",
"One modulation technique that generates essentially sparse and bounded constellation points is the so-called generalized space-shift keying (GSSK) modulation.",
"In this direction, we derive high dimensional sharp characterizations of various performance measures of the Box-LASSO such as the mean square error, probability of support recovery, and the element error rate, under independent and identically distributed (i.i.d.)",
"Gaussian channels that are not perfectly known.",
"In particular, the analytical characterizations can be used to demonstrate performance improvements of the Box-LASSO as compared to the widely used standard LASSO.",
"Then, we can use these measures to optimally tune the involved hyper-parameters of Box-LASSO such as the regularization parameter.",
"In addition, we derive optimum power allocation and training duration schemes in a training-based massive MIMO system.",
"Monte Carlo simulations are used to validate these premises and to show the sharpness of the derived analytical results."
],
[
"Introduction",
"The least absolute selection and shrinkage operator (LASSO) [1] is a widely used method to recover an unknown sparse signal ${\\bf s}_0$ from noisy linear measurements ${\\bf r}= {\\bf A}{\\bf s}_{0} + {\\bf v},$ by solving the following optimization problem: $\\widehat{{\\bf s}}= { \\rm {arg}} \\ \\underset{{{\\bf s}\\in \\mathbb {R}^{n}}}{\\operatorname{\\min }} \\ \\Vert {\\bf A}{\\bf s}- {\\bf r}\\Vert _2^2 + \\gamma \\Vert {\\bf s}\\Vert _1,$ where $\\Vert \\cdot \\Vert _2$ , and $ \\Vert \\cdot \\Vert _1$ represent the $\\ell _2$ -norm and $\\ell _1$ -norm of a vector, respectively.",
"Furthermore, $ {\\bf A}$ is the measurement matrix, ${\\bf v}$ is the noise vector, and $\\gamma >0$ is a regularization parameter that balances between the fidelity of the solution as controlled by the $\\ell _2$ -norm on one side, and the sparsity of the solution as enforced by the $\\ell _1$ -norm on the other hand.",
"It allows for learning a sparse model where few of the entries are non-zero.",
"The LASSO reduces to the linear regression as $\\gamma \\rightarrow 0$ .",
"The LASSO has been widely used in modern data science and signal processing applications such as in [2], [3], [4].",
"The LASSO is a special instance of a class of problems called non-smooth regularized convex optimization problems [5].",
"In recent years, various forms of sharp analysis of the asymptotic performance of such optimization problems have been studied under the assumption of noisy independent and identically distributed (i.i.d.)",
"Gaussian measurements.",
"They mostly take one of the following approaches.",
"The first is the approximate message passing (AMP) framework, which was utilized in [6], [7], [8] to conduct a sharp asymptotic study of the performance of compressed sensing problems under the assumptions of i.i.d.",
"Gaussian sensing matrix.",
"The authors in [9], [10], [11] undertook a different approach that uses the replica method from statistical physics, which is a powerful high-dimensional analysis tool.",
"However, it lacks rigorous mathematical justifications in some steps.",
"In addition, the high-dimensional error performance of different regularized estimators has been previously considered using some heuristic arguments and numerical simulations in [12] and [13].",
"Another approach based on random matrix theory (RMT) [14] was taken in [15], [16] to analyze the high-dimensional squared error performance of ridge regression.",
"One major drawback of RMT is that it requires the involved optimization problems to admit a closed-form solution which is not true for the general non-smooth convex optimization problems.",
"The most recent approach is based on a newly developed framework that uses the convex Gaussian min-max theorem (CGMT) initiated by Stojnic [17] and further extended by Thrampoulidis et al.",
"in [5].",
"It provides the analysis in a more natural and elementary way when compared to the previously discussed methods.",
"The CGMT has been applied to the performance analyses of various optimization problems.",
"The asymptotic mean square error (MSE) have been analyzed for various regularized estimators in [18], [19], [20], [21].",
"Precise analysis of general performance measures such as the probability of support recovery and $\\ell _1$ -reconstruction error of the LASSO and Elastic Net was obtained in [22], [23], [24].",
"The asymptotic symbol error rate (SER) of the box relaxation optimization has been derived for various modulation schemes in [25], [26], [27], [28].",
"CGMT has also been used for the analysis of nonlinear measurements models (e.g., quantized measurements) in [19] and [29].",
"In this paper, instead of the standard LASSO in (REF ), we will use the so-called Box-LASSO [20], which is the same as the LASSO but with an additional box-constraint.",
"We will formally define it in (REF ).",
"We propose using the Box-LASSO as a low-complexity decoder in massive MIMO communication systems with modern modulation methods such as the generalized space-shift keying (GSSK) [30] modulation and the generalized spatial modulation (GSM) [31].",
"In such systems, the transmitted signal vector ${\\bf s}_0$ is inherently sparse and have elements belonging to a finite alphabet, which is an excellent setting for using the Box-LASSO.",
"Using the CGMT framework, this paper derives sharp asymptotic characterizations of the mean square error, support recovery probability, and element error rate of the Box-LASSO under the presence of uncertainties in the channel matrix that has i.i.d.",
"Gaussian entries.",
"In addition, the analysis demonstrate that the Box-LASSO outperforms the standard one in all of the considered metrics.",
"The derived expressions can be used to optimally tune the involved hyper-parameters of the algorithm.",
"Furthermore, we study the application of the Box-LASSO to GSSK modulated massive MIMO systems and optimize their power allocation and training duration.",
"The additional contributions of this paper against the most related works such as [20], [26], [23], [24], [32] are summarized as follows: $\\bullet $ This paper considers the more practical and challenging scenario of imperfect channel state information (CSI), whereas [20], [26], [32] only derived the analysis for the ideal case of perfect CSI.",
"$\\bullet $ Even when the imperfect CSI case was studied in the previous works on the LASSO and Box-Elastic Net in [23], [24], only a theoretical imperfect CSI model (the so-called Gauss-Markov model) was considered.",
"On the other hand, this work presents the analysis under a more practical model of the imperfect CSI, which is the linear minimum mean square error (LMMSE) channel estimate in (REF ) for a massive MIMO application.",
"$\\bullet $ With this massive MIMO application in mind, we derive the asymptotically optimal power allocation and training duration schemes for GSSK signal recovery.",
"$\\bullet $ We show that the derived power allocation optimization is nothing but the well-known scheme that maximizes the effective signal-to-noise ratio (SNR) in [33]."
],
[
"Organization",
"The rest of this paper is organized as follows.",
"The system model, channel estimation and the proposed Box-LASSO decoder are discussed in Section .",
"Section provides the main asymptotic results of the work.",
"The application of Box-LASSO to a massive MIMO system and its power allocation and training duration optimization is presented in Section .",
"Finally, concluding remarks and future research directions are stated in Section .",
"The proof of the main results is deferred to the Appendix."
],
[
"Notations",
"Here, we gather the basic notations that are used throughout the paper.",
"Bold face lower case letters (e.g., ${\\bf x}$ ) represent a column vector while $x_j$ is its $j^{th}$ entry.",
"For ${\\bf x}\\in \\mathbb {R}^n$ , let $\\Vert {\\bf x}\\Vert _2 = \\sqrt{\\sum _{j=1}^n x_j^2}$ , and $\\Vert {\\bf x}\\Vert _1= \\sum _{j=1}^n |x_j|$ .",
"Matrices are denoted by upper case letters such as ${\\bf X}$ , with ${\\bf I}_n$ being the $n \\times n$ identity matrix.",
"$(\\cdot )^T$ and $(\\cdot )^{-1}$ are the transpose and inverse operators respectively.",
"We use the standard notations $\\mathbb {E}[\\cdot ],$ and $\\mathbb {P}[\\cdot ]$ to denote the expectation of a random variable and probability of an event respectively.",
"We write $X \\sim p_x$ to denote that a random variable $X$ has a probability density/mass function $p_x$ .",
"In particular, the notation ${\\bf q}\\sim \\mathcal {N}(\\mathbf {0},{\\bf R}_{{\\bf q}})$ is used to denote that the random vector ${\\bf q}$ is normally distributed with $\\mathbf {0}$ mean and covariance matrix ${\\bf R}_{{\\bf q}} = \\mathbb {E}[{\\bf q}{\\bf q}^T]$ , where $\\mathbf {0}$ represent the all-zeros vector of the appropriate size.",
"The notation $\\delta _\\xi $ is used to represent a point-mass distribution at $\\xi $ .",
"We write $``\\overset{P}{\\longrightarrow }\"$ to denote convergence in probability as $n \\rightarrow \\infty $ .",
"We also use standard notation ${\\rm {plim}}_{n\\rightarrow \\infty } \\Theta _n = \\Theta $ to denote that a sequence of random variables $\\Theta _{n},[n=1,2,...]$ , converges in probability towards a constant $\\Theta $ .",
"When writing $x^\\star = \\mathrm {arg} \\min _x f(x)$ , the operator $\\mathrm {arg} \\min $ returns any one of the possible minimizers of $f$ .",
"The function $Q(x) = \\frac{1}{\\sqrt{2 \\pi }} \\int _{x}^{\\infty } e^{-u^2/2} {\\rm d}u$ is the $Q$ -function associated with the standard normal density.",
"Finally, for $a,\\gamma ,l,u \\in \\mathbb {R}$ , such that $\\gamma ,u \\ge 0, l \\le 0, $ we define the following functions: $\\bullet $ The saturated shrinkage function $\\mathcal {H}(a ; \\gamma , l, u) :=\\operatornamewithlimits{arg\\,min}_{l \\le s \\le u}$ $\\frac{1}{2}(s-a)^2 + \\gamma |s|$ , which is given as: $\\mathcal {H}(a ; \\gamma , l, u) ={\\left\\lbrace \\begin{array}{ll}u & ,\\text{if} \\ a \\ge u + \\gamma \\\\\\end{array}a - \\gamma & ,\\text{if} \\ \\gamma < a < u + \\gamma \\\\\\right.0 & ,\\text{if} \\ |a| \\le \\gamma \\\\}a + \\gamma & ,\\text{if} \\ l - \\gamma < a < - \\gamma \\\\$ l ,if a l - .",
"A plot of this function is depicted in Fig.",
"REF .",
"Figure: Saturated shrinkage function.$\\bullet $ Also, let $\\mathcal {J}(a;\\gamma ,l,u) :=\\min _{l \\le s \\le u} \\frac{1}{2}(s-a)^2 + \\gamma |s|$ , which can be rewritten as: $\\mathcal {J}(a ; \\gamma , l, u) ={\\left\\lbrace \\begin{array}{ll}\\frac{1}{2} (u - a)^2 + \\gamma u & ,\\text{if} \\ a \\ge u + \\gamma \\\\\\end{array}\\gamma a - \\frac{1}{2} \\gamma ^2 & ,\\text{if} \\ \\gamma < a < u + \\gamma \\\\\\right.\\frac{1}{2} a^2 & ,\\text{if} \\ |a| \\le \\gamma \\\\}-\\gamma a - \\frac{1}{2} \\gamma ^2 & ,\\text{if} \\ l - \\gamma < a < -\\gamma \\\\$ 12 (l - a)2 - l ,if a l - ."
],
[
"System Model",
"A massive MIMO system with $n$ transmit (Tx) antennas and $m$ receive (Rx) antennas is considered in this paper.",
"We herein consider a training-based transmission that consists of a coherence interval with $T = T_t + T_d$ symbols in which the channel realization is assumed to be constant.",
"During the first part of this coherence interval, $T_t$ symbol intervals are used as known pilot symbols, with average power, ${P}_t $ .",
"These pilot symbols are employed for channel estimation purposes.",
"The remaining $T_d$ symbols are dedicated to transmitting data symbols with average power, $P_d$ .",
"Conservation of energy implies that $P_t T_t + P_d T_d = {P} \\ T,$ where $P$ is the average total transmission power.",
"Letting $\\nu $ denote the ratio of the total transmission energy allocated to the data, we may write $P_d T_d= \\nu T P, \\ \\ \\quad P_t T_t = (1- \\nu ) T {P}, \\ \\quad \\nu \\in (0,1).$ This system model is illustrated in Fig.",
"REF .",
"The received signal for the data transmission phase can be modeled as ${\\bf r}= \\sqrt{\\frac{P_d}{n}} {\\bf H}{\\bf s}_0 +{\\bf v}\\in \\mathbb {R}^m,$ where the following model-assumptions hold, except if otherwise stated: ${\\bf H}\\in \\mathbb {R}^{m \\times n}$ is the MIMO channel matrix which has i.i.d.",
"standard Gaussian entries (i.e., $\\mathcal {N}(0,1)$ ).",
"${\\bf v}\\in \\mathbb {R}^{m}$ is the noise vector with i.i.d.",
"standard Gaussian entries, i.e., ${\\bf v}\\sim \\mathcal {N}(\\mathbf {0},{\\bf I}_m)$ .",
"The unknown signal vector ${\\bf s}_{0}$ is assumed to be $k$ -sparse, i.e., only $k$ of its entries are sampled i.i.d.",
"from a distribution $ p_{s_0}$ which has zero-mean and unit-variance (i.e., $\\mathbb {E}[S_0^2] = \\sigma _s^2=1$ ), and the remaining entries are zeros.",
"Figure: A training-assisted massive MIMO system."
],
[
"Estimation of the Channel Matrix",
"As indicated in the preceding subsection, a training phase in which the transmitter sends $T_{{t}}$ pilot symbols is used to estimate the channel matrix ${\\bf H}$ , which is unknown to the receiver.In communications literature, this is known as the “imperfect CSI” case.",
"In this training phase, the received signal is represented as ${\\bf R}_{t} = \\sqrt{\\frac{{P}_{t}}{n}} {\\bf H}{\\bf S}_{t} + {\\bf V}_{t},$ where ${\\bf R}_{t}\\in \\mathbb {R}^{m \\times T_{t}}$ is the received signal matrix, ${\\bf S}_{t}\\in \\mathbb {R}^{n \\times T_{t}}$ is the matrix of transmitted pilot symbols, and ${\\bf V}_{t}\\in \\mathbb {R}^{m \\times T_{t}}$ is a zero-mean additive white Gaussian noise (AWGN) matrix with covariance $\\mathbb {E}[{\\bf V}_{{t}} {\\bf V}_{{t}}^T] =T_{{t}} {\\bf I}_m$ .",
"In this paper, we consider the linear minimum mean square error (LMMSE) estimate of the channel matrix, which can be derived based on the knowledge of ${\\bf R}_{t}$ from (REF ) as [34] $\\widehat{{\\bf H}} &= \\sqrt{\\frac{n}{{P}_{t}}}{\\bf R}_{t} {\\bf S}_{{t}}^T \\left( {\\bf S}_{{t}} {\\bf S}_{{t}}^T + \\frac{n}{{P}_{t}} {\\bf I}_{n} \\right)^{-1},\\nonumber \\\\& = {\\bf H}+ \\hbox{$\\Omega $},$ where $\\hbox{$\\Omega $}$ is the channel estimation error matrix, which is uncorrelated with $\\widehat{{\\bf H}}$ , as per the orthogonality principle of the LMMSE estimation [34], [33].",
"For i.i.d.",
"MIMO channels, it has been proven that the optimal ${\\bf S}_{t}$ that minimizes the estimation MSE satisfies [33] ${\\bf S}_{t} {\\bf S}_{{t}}^{T}= T_{t}{\\bf I}_{n}.$ For the above condition to hold, it is required that $T_t \\ge n.$ Moreover, under (REF ), it can be shown that the channel estimate $\\widehat{{\\bf H}}$ has i.i.d.",
"zero-mean Gaussian entries with variance $\\sigma _{{\\widehat{H}}}^2 = 1-\\sigma _{{\\omega }}^2$ [33], where $\\sigma _{{\\omega }}^2 = \\frac{1}{ 1+ \\frac{{P}_{t}}{n} T_{t} }$ is the variance of each element in $\\hbox{$\\Omega $}$ .",
"Furthermore, it can be shown that the entries of $\\hbox{$\\Omega $}$ are i.i.d.",
"$\\mathcal {N}(0,\\sigma _\\omega ^2)$ distributed.",
"Note that the training-phase energy $ {P}_{t} T_{t}$ controls the quality of the estimation as it appears from (REF ).",
"In fact, as ${P}_{t} T_{t} \\rightarrow \\infty $ , $\\sigma _\\omega ^2 \\rightarrow 0$ , and $\\widehat{{\\bf H}} \\rightarrow {\\bf H}$ , which represents the case of perfect CSI."
],
[
"Data Detection via the Box-LASSO",
"In this work, the problem in (REF ) is referred to as the standard LASSO, and we instead introduce the following revised formulation of it termed the Box-LASSO: $\\widehat{{\\bf s}}= {\\rm {arg}} \\ \\underset{{{\\bf s}\\in \\mathbb {B}^n}}{\\operatorname{\\min }} \\bigg \\Vert \\sqrt{ \\frac{P_d}{n}} \\widehat{{\\bf H}} {\\bf s}- {\\bf r}\\bigg \\Vert _2^2 + \\gamma P_d \\Vert {\\bf s}\\Vert _1,$ $ \\text{where}, \\mathbb {B} = [\\ell , \\mu ], \\ \\text{and} \\ \\ell \\le 0, \\mu \\ge 0 \\in \\mathbb {R}.$ When compared to (REF ), we use ${\\bf A}= \\sqrt{ \\frac{P_d}{n}} \\widehat{{\\bf H}} $ here.",
"This is due to the fact that ${\\bf H}$ is not perfectly known and we only have its estimate $\\widehat{{\\bf H}}$ that was obtained by training.",
"In addition, note that the regularization parameter $\\gamma $ is scaled by a factor of $P_d $ .",
"This is made such that the two terms grow with the same pace.",
"The only difference between (REF ) and (REF ), is that (REF ) now has a “box-constraint”.",
"However, as we will show later, in cases where the elements of ${\\bf s}_0$ are bounded or approximately so, this minor modification ensures a considerable gain in performance.",
"Therefore, the Box-LASSO can be used to recover simultaneously structured signals [35], for example, signals that are both bounded and sparse.",
"Such signals appear in various applications including machine learning [4], wireless communications [36], image processing [2], and so on.",
"Although the Box-LASSO is not as well-known as the standard LASSO, there are a few references where it has been applied [37], [38], [39].",
"Of particular interest is the application of the Box-LASSO in spatially modulated MIMO systems such as GSSK modulated signals which we will discuss in Section ."
],
[
"Technical Assumptions",
"In this work, we require the following technical assumptions to hold.",
"Assumption 1 The analysis requires that the system dimensions ($m$ , $n$ and $k$ ) grow simultaneously large (i.e., $m,n,k \\rightarrow \\infty $ ) at fixed ratios: $\\frac{m}{n} \\rightarrow \\eta \\in (0,\\infty ),$ and $\\frac{k}{n} \\rightarrow \\rho \\in (0,1).$ Assumption 2 We assume that the normalized coherence interval, normalized number of pilot symbols and normalized number data symbols are fixed and given as $\\frac{T}{n} \\rightarrow \\tau \\in (1,\\infty ),$ $ \\frac{T_{{t}}}{n} \\rightarrow \\tau _{{t}} \\in [1,\\infty ),$ and $\\frac{T_{{d}}}{n} \\rightarrow \\tau _{{d}},$ respectively.",
"Under these assumptions, the energy conservation in (REF ) becomes $P_t \\tau _t + P_d \\tau _d = {P} \\ \\tau ,$ and the channel estimation error variance in (REF ) reads $\\sigma _{{\\omega }}^2 = \\frac{1}{ 1+ P_t\\tau _{t} }.$"
],
[
"Figures of Merit",
"We measure the performance of the Box-LASSO using following figures of merit: Mean Square Error: A widely used figure of merit is the estimation mean square error (MSE), that measures the divergence of the estimate $\\widehat{{\\bf s}}$ from the original signal ${\\bf s}_0$ .",
"Formally, it is defined as ${\\rm {MSE}} := \\frac{1}{n}\\Vert \\widehat{{\\bf s}} - {\\bf s}_0 \\Vert _2^2.$ Support Recovery: In sparse recovery problems, a natural performance measure that is employed in numerous applications is support recovery, that can be defined as determining whether an element of ${\\bf s}_0$ is non-zero (i.e., on the support), or if it is zero (i.e., off the support).",
"The decision, based on the Box-LASSO solution $\\widehat{{\\bf s}}$ , proceeds as follows: if $| \\widehat{s}_{j}| \\ge $ , then, $\\widehat{s}_j$ is on the support, where $> 0$ is a user-defined hard threshold on the elements of $\\widehat{{\\bf s}}$ .",
"Otherwise, $\\widehat{s}_j$ is off the support.",
"Essentially, we have two measures: the probability of successful on-support recovery denoted by $_{}^{\\text{on}}(\\widehat{{\\bf s}})$ , and the probability of successful off-support recovery, i.e., $_{}^{\\text{off}}(\\widehat{{\\bf s}})$ .",
"Formally, these quantities are defined, respectively, as $_{}^{\\text{on}}(\\widehat{{\\bf s}}) := \\frac{1}{k} \\sum _{j \\in \\mathcal {T}({\\bf s}_0)} \\mathbb {1}_{\\lbrace | \\widehat{s}_{j}| \\ge \\rbrace },\\\\_{}^{\\text{off}}(\\widehat{{\\bf s}}) := \\frac{1}{n-k} \\sum _{j \\notin \\mathcal {T}({\\bf s}_0)} \\mathbb {1}_{\\lbrace | \\widehat{s}_{j}| \\le \\rbrace },$ where $\\mathbb {1}_{\\lbrace .",
"\\rbrace }$ is the indicator function, and $\\mathcal {T}({\\bf s}_0)$ is the support of ${\\bf s}_0$ , i.e., the set of all non-zero elements of ${\\bf s}_0$ ."
],
[
"Performance Characterization",
"In this subsection, we summarize the main theoretical results regarding the asymptotic performance of the Box-LASSO.",
"The first theorem gives the sharp performance analysis of the MSE of the Box-LASSO.",
"Theorem 1 Let $\\widehat{{\\bf s}}$ be a minimizer of the Box-LASSO problem in (REF ), where ${\\bf H}, {\\bf v}$ and ${\\bf s}_0$ satisfy the model assumptions in Section REF .",
"In addition, assume that the optimization problem: $\\max _{\\beta > 0} \\min _{\\lambda >0} \\ \\mathcal {G} (\\beta , \\lambda )$ has a unique optimizer $(\\beta _\\star , \\lambda _\\star )$ , where $\\mathcal {G} (\\beta , \\lambda )&:= \\frac{\\beta \\sqrt{\\eta }}{2 \\lambda }+ \\frac{\\beta \\lambda \\sqrt{\\eta }}{2 } \\left( 1+ \\rho {P_d \\sigma _{\\omega }^2} \\right) -\\frac{\\beta ^2}{4} - \\frac{\\beta }{2 \\lambda \\sqrt{\\eta }} \\nonumber \\\\& +\\beta \\lambda \\sqrt{\\eta } P_d \\sigma _{\\widehat{H}}^2 \\ \\mathbb {E} \\left[\\mathcal {J} \\left(S_{0}+\\frac{Z}{\\lambda \\sqrt{\\eta P_d \\sigma _{\\widehat{H}}^2 }}; \\frac{\\gamma }{\\beta \\lambda \\sqrt{\\eta } \\sigma _{\\widehat{H}}^2 }, \\ell ,\\mu \\right) \\right],$ and the expectation is taken over $S_0 \\sim p_{s_0}$ and $Z \\sim \\mathcal {N}(0,1)$ .",
"Then, under Assumption 1 and Assumption 2, and for a fixed $\\gamma > 0$ , it holds: $&\\underset{n\\rightarrow \\infty }{\\rm {plim}} \\ {\\rm {MSE}}= \\frac{1}{P_d \\sigma _{\\widehat{H}}^2} \\left( \\frac{1}{\\lambda _\\star ^2} -1 - \\rho P_d \\sigma _{\\omega }^2 \\right).$ The proof is relegated to the appendix.",
"Remark 1 (Finding optimal scalars) The scalars $\\beta _\\star $ and $\\lambda _\\star $ can be numerically evaluated by solving the first-order optimality conditions, i.e., $\\nabla _{(\\beta , \\lambda )} \\mathcal {G}(\\beta , \\lambda )=\\bf {0}.$ Remark 2 (Roles of $\\lambda _\\star $ and $\\beta _\\star $ ) From Theorem REF above, we can see that the optimal scalar $\\lambda _\\star $ is related to the asymptotic MSE.",
"However, the role of $\\beta _\\star $ is not evident from the above theorem.",
"Based on our derivations, it turns out that $\\beta _\\star $ is related to another performance metric called the residual [40] between ${\\bf r}$ and the estimate $\\widehat{{\\bf r}}$ which is also called the prediction error.",
"Formally, it is defined as ${\\rm {\\mathcal {R}}} : =\\frac{1}{n} \\bigg \\Vert \\underbrace{ \\sqrt{\\frac{P_d }{n}} \\widehat{{\\bf H}}\\widehat{{\\bf s}}}_{:=\\widehat{{\\bf r}}} - {\\bf r}\\bigg \\Vert _2^2.$ Then, as we will prove in the Appendix, under the same assumptions in Theorem REF , it holds $&\\underset{n\\rightarrow \\infty }{\\rm {plim}} \\ \\mathcal {R}= \\frac{1}{4} \\beta _\\star ^2.$ The above expression clearly shows the role $\\beta _\\star $ in predicting the asymptotic value of the residual, and Fig.",
"REF illustrates its high accuracy.",
"The residual metric is less relevant to the MIMO application considered in this paper, but it is of great interest in other practical data science problems, where you only have an access to the measurement vector ${\\bf r}$ , and not the true vector ${\\bf s}_0$ .",
"Remark 3 (Optimal MSE regularizer) Theorem 1 can be used to find the optimal regularizer $\\gamma _\\star ^{\\mathrm {MSE}}$ that minimizes the MSE.",
"See for example Fig.",
"REF a.",
"Particularly, $\\gamma _\\star ^{\\mathrm {MSE}}$ can be found as follows $\\gamma _\\star ^{\\mathrm {MSE}} = &\\operatornamewithlimits{arg\\,min}_{\\gamma >0} \\frac{1}{\\lambda _\\star },\\nonumber \\\\= &\\operatornamewithlimits{arg\\,max}_{\\gamma >0} {\\lambda _\\star }.$ The above expression can be easily proven, by noting that $\\gamma $ appears in the MSE expression of (REF ) only implicitly through $\\lambda _\\star $ .",
"In the next theorem, we sharply characterizes the support recovery metrics introduced earlier in (REF ).",
"Theorem 2 Under the same settings of Theorem REF , for any fixed $>0$ , and under Assumption 1 and Assumption 2, the on-support and off-support probabilities converge as $\\underset{n\\rightarrow \\infty }{\\rm {plim}}\\ _{}^{\\rm {on}}(\\widehat{{\\bf s}}) = \\mathbb {P} \\biggl [\\biggl | \\mathcal {H}\\biggl ( S_{0}+\\frac{Z}{\\lambda _\\star \\sqrt{\\eta P_d \\sigma _{\\widehat{H}}^2 }}; \\frac{\\gamma }{\\beta _\\star \\lambda _\\star \\sqrt{\\eta } \\sigma _{\\widehat{H}}^2 }, \\ell ,\\mu \\biggr ) \\biggr | \\ge \\biggr ],$ and $\\underset{n\\rightarrow \\infty }{\\rm {plim}} \\ _{}^{\\rm {off}}(\\widehat{{\\bf s}}) = \\mathbb {P} \\biggl [ \\biggl | \\mathcal {H}\\biggl ( \\frac{Z}{\\lambda _\\star \\sqrt{\\eta P_d \\sigma _{\\widehat{H}}^2 }}; \\frac{\\gamma }{\\beta _\\star \\lambda _\\star \\sqrt{\\eta } \\sigma _{\\widehat{H}}^2 }, \\ell ,\\mu \\biggr ) \\biggr | \\le \\biggr ],$ respectively.",
"The proof of Theorem REF to a great extent follows the proof of Theorem REF , but is omitted for briefness of the presentation.",
"See [27], [41] for similar proof techniques.",
"Remark 4 (Regularizer's strength) It is easy to see from Theorem REF that when $\\gamma $ grows larger, $_{}^{\\rm {off}}(\\widehat{{\\bf s}})$ converges to 1 whereas $ _{}^{\\rm {on}}(\\widehat{{\\bf s}})$ converges to 0.",
"When $\\gamma $ approaches 0, opposite behavior is exhibited.",
"This is expected since large values of $\\gamma $ emphasize the $\\ell _1$ -norm term, resulting in a sparser solution.",
"This is illustrated clearly in Fig.",
"REF .",
"Remark 5 (Optimal regularizer) A sensible measure of performance to trade-off between the on-support and off-support recovery probability is $_{}(\\widehat{{\\bf s}}) :=\\theta \\ _{}^{\\rm {on}}(\\widehat{{\\bf s}})+ (1-\\theta ) \\ _{}^{\\rm {off}}(\\widehat{{\\bf s}}), \\ \\text{for} \\ \\theta \\in [0, 1].$ The behavior of this metric as a function of $\\gamma $ is sharply characterized by Theorem REF .",
"As a result, Theorem REF can also be utilized to determine the optimal value of $\\gamma $ which optimizes $_{}(\\widehat{{\\bf s}})$ ."
],
[
"Numerical Illustrations",
"For the sake of illustration, we will simply look at the instance when ${\\bf s}_0$ has elements that are only allowed to take one of two possible values: 0, or $E > 0$ .",
"For a normalized sparsity level $\\rho \\in (0, 1)$ , such prior knowledge is typically modeled using a sparse-Bernoulli distribution on the elements of ${\\bf s}_0$ , i.e., $S_0 \\sim (1-\\rho )\\delta _0 + \\rho \\delta _E $ .",
"This model is frequently seen in MIMO communication systems using generalized space-shift keying (GSSK) modulation [30]; we go over the role of the Box-LASSO in such systems in Section .",
"In this case, setting $\\ell =0$ , and $\\mu =E$ as the box-constraint values is a natural choice.",
"Therefore, in our numerical simulations, we assume that ${\\bf s}_0$ has elements that are sampled i.i.d.",
"from a sparse-Bernoulli distribution with $\\mathbb {P}[S_0 = 0] =0.8$ , $\\mathbb {P}[S_0 = 1] =0.2$ (i.e., $\\rho =0.2$ ) and $E =1$ ; to satisfy the unit-variance assumption.",
"Fig.",
"REF shows the close match between Theorem REF asymptotic prediction of the MSE and residual of the Box-LASSO and the Monte Carlo (MC) simulations.",
"For the simulations, we used $\\eta = 1.5,n = 100$ , $T = 500,T_t =n, \\nu = 0.5,$ and $P= 15 \\ {\\rm {dB}}$ .",
"These results are averaged over 100 independent realizations of ${\\bf s}_0, {\\bf H}$ and ${\\bf v}$ .",
"From this figure, it can be seen that the Box-LASSO outperforms the standard LASSO.",
"We can also see in Fig.",
"REF that as the regularization parameter $\\gamma $ is varied, a pronounced minimum for a certain $\\gamma > 0$ is observed.",
"Figure: MSE/Residual performance of the Box-LASSO and the standard LASSO vs. the regularizer.The analytical prediction is based on Theorem with p s 0 =(1-ρ)δ 0 +ρδ E p_{s_0} = (1-\\rho ) \\ \\delta _0 + \\rho \\ \\delta _E.",
"We used ρ=0.2,η=1.5,n=128\\rho =0.2,\\eta = 1.5,n = 128, T=500,T t =n,ν=0.5,E=1,T = 500,T_t =n, \\nu = 0.5, E=1, and P=15 dB P= 15 \\ {\\rm {dB}}.The analytical expressions of Theorem REF for the support recovery probabilities are compared to the MC simulations and displayed in Fig.",
"REF with the same simulation settings as in the preceding figure.",
"Once again, this figure demonstrates the great accuracy of the provided theoretical expressions.",
"Figure: Probability of successful support recovery of the Box-LASSO.The analytical prediction is based on Theorem with p s 0 =(1-ρ)δ 0 +ρδ E p_{s_0} = (1-\\rho ) \\ \\delta _0 + \\rho \\ \\delta _E.",
"We used ρ=0.2,η=1.5,n=128\\rho =0.2,\\eta = 1.5,n = 128, T=500,T t =n,ν=0.5,E=1,T = 500,T_t =n, \\nu = 0.5, E=1, and P=15 dB P= 15 \\ {\\rm {dB}}.Remark 6 For the previously mentioned sparse-Bernoulli distributed signal, the support recovery probabilities in (REF ), and (REF ) simplify to $\\underset{{n\\rightarrow \\infty } }{\\rm {plim}} \\ _{}^{\\text{on}}(\\widehat{{\\bf s}}) = \\ &Q \\left( (- E) \\lambda _\\star \\sqrt{\\eta P_d \\sigma _{\\widehat{H}}^2} + \\frac{\\gamma \\sqrt{P_d}}{\\beta _\\star \\sigma _{\\widehat{H}}} \\right),$ and $\\underset{{n\\rightarrow \\infty } }{\\rm {plim}} \\ _{}^{\\text{off}}(\\widehat{{\\bf s}}) = \\ 1 - Q \\left( \\lambda _\\star \\sqrt{\\eta P_d \\sigma _{\\widehat{H}}^2} + \\frac{\\gamma \\sqrt{P_d}}{\\beta _\\star \\sigma _{\\widehat{H}}} \\right).$ These expressions have been used in the numerical simulations above with $E =1$ therein.",
"Remark 7 (Unbounded elements) In instances when the elements of the original signal are unbounded but take values in a specific range with high probability, the Box-LASSO can be a valuable decoder as well.",
"To demonstrate this, let us take the example in which the elements of ${\\bf s}_0$ are i.i.d.",
"sparse-Gaussian distributed, i.e., $S_0 \\sim (1-\\rho )\\delta _0 +\\rho \\ \\mathcal {N} (0, \\sigma _s^2)$ .",
"Fig.",
"REF illustrates a case in which the Box-LASSO outperforms the standard LASSO.",
"We used $\\ell = - \\sigma _s$ and $\\mu = \\sigma _s$ as the box-constraints in this example.",
"Figure: MSE of the Box-LASSO and standard LASSO.",
"The analytical prediction is based on Theorem with a sparse-Gaussian 𝐬 0 {\\bf s}_0 signal.",
"We used ρ=0.1,η=1.2,n=400\\rho =0.1,\\eta = 1.2,n = 400, T=1000,T t =456,ν=0.5,σ s 2 =1,T = 1000,T_t =456, \\nu =0.5,\\sigma _s^2=1, and P=10 dB P= 10 \\ {\\rm {dB}}.Remark 8 (Universality) Even without the Gaussianity assumption on the elements of the channel matrix ${\\bf H}$ , our extensive simulations strongly indicate that the statements of Theorem REF and Theorem REF are still valid.",
"This is especially helpful in MIMO applications where the channel matrix elements can be represented beyond the typical fading model (Gaussian), such as in the more involved fading models, e.g., Weibull and Nakagami [42].",
"The same asymptotic statements appear to hold regardless of whether the distribution of ${\\bf H}$ is Gaussian, Binary, or Laplacian (as illustrated in Fig.",
"REF ).",
"Rigorous proofs, known as universality results, in [43], [44], [45] support such a claim.",
"Figure: MSE of the Box-LASSO for different measurement matrices.",
"The theoretical prediction is based on Theorem with a sparse-Bernoulli 𝐬 0 {\\bf s}_0 signal.",
"We used ρ=0.1,η=0.8,n=200\\rho =0.1,\\eta = 0.8,n = 200, T=700,T t =256,ν=0.6,P=5 dB T = 700,T_t =256, \\nu =0.6, P= 5 \\ {\\mathrm {dB}}.Remark 9 (Efficient implementation) It is worth noting that the Box-LASSO can be efficiently implemented via quadratic programming (QP) as in [37], where it was used to implement an efficient algorithm of the constrained LASSO.",
"We applied the same algorithm to the Box-LASSO decoder in the above numerical simulations utilizing MATLAB built-in function $\\mathsf {quadprog}$ ."
],
[
"GSSK Modulated Massive MIMO Systems",
"Traditional linear modulation schemes become expensive in modern massive MIMO systems.",
"This is due to the large required number of radio frequency (RF) chains needed for the massive number of antennas.",
"One promising modulation technique is the so-called spatial modulation (SM), where only the antenna's location relays information and only a small subset of the antennas is active at each time [46].",
"This significantly reduces the system complexity since the required number of RF chains is less now.",
"This modulation scheme saves energy; since using fewer RF chains, we have less power dissipation through the power amplifiers, etc.",
"Modern SM techniques include generalized space-shift keying (GSSK) modulation [30], [36], and the generalized spatial modulation (GSM) [31]."
],
[
"Box-LASSO for Detecting GSSK Modulated Signals",
"As mentioned above, recently developed modulation techniques such as GSM and GSSK modulation, generate signals that are essentially sparse and have elements belonging to a finite alphabet (i.e., bounded).",
"Hence, when such modulations are employed, we will use the Box-LASSO as low-complexity decoding method, instead of the previously proposed standard LASSO decoders [47], [48].",
"For the sake of simplicity, we will focus on GSSK.",
"Considering a modulation setup, where a fixed-size set $\\mathcal {I} \\subset \\lbrace 1, \\cdots , n\\rbrace $ of active antennas transmit $s_{0,j} = 1, j \\in \\mathcal {I}$ at each transmission, while the remaining antennas stay inactive, i.e., $s_{0,j} = 0, j \\notin \\mathcal {I}$ .",
"Hence, only active antennas positions convey information.",
"To decode ${\\bf s}_0$ , firstly, get a solution $\\widehat{{\\bf s}}$ of the Box-LASSO in (REF ), with $\\ell =0$ and $\\mu =1$ .",
"Then, map $\\widehat{{\\bf s}}$ into a vector ${{\\bf s}}^\\star $ which has elements either 0 or 1.",
"In the GSSK context, this typically entails sorting $\\widehat{{\\bf s}}$ and setting its largest $k$ entries to 1 and the remaining to 0 [47].",
"In order to evaluate the performance of the Box-LASSO in this application, we will use the so-called element error rate (EER) [20] as a performance measure.",
"Similar to the support recovery metric, we first hard-thresholding $\\widehat{{\\bf s}}$ by a constant $\\in (0, 1)$ , in order to map each of its element to either 0 or 1.",
"Then, the EER can be defined as ${\\rm {EER}}_:= \\frac{1}{| \\mathcal {I}|} \\sum _{j \\in \\mathcal {I}} \\mathbb {1}_{\\lbrace | \\widehat{s}_{j}| \\le \\rbrace } + \\frac{1}{n-| \\mathcal {I}|} \\sum _{j \\notin \\mathcal {I}} \\mathbb {1}_{\\lbrace | \\widehat{s}_{j}| \\ge \\rbrace }.$ The next proposition gives a sharp asymptotic prediction of the EER in the GSSK modulated MIMO systems application.",
"Proposition 1 Let ${\\rm {EER}}_$ be as defined in the above equation with $|\\mathcal {I}| = \\rho n$ , for $\\rho \\in (0, 1)$ .",
"Also, let $\\beta _\\star $ , and $\\lambda _\\star $ be solutions to the minimax optimization in (REF ), with $p_{s_0} = (1-\\rho ) \\delta _0 +\\rho \\delta _{1} $ therein.",
"Then, for a fixed $\\in (0,1)$ , it holds $\\underset{{n \\rightarrow \\infty }}{\\rm {plim}} \\ {\\rm {EER}}_{} = \\ & Q \\left( (1 -) \\lambda _\\star \\sqrt{\\eta P_d \\sigma _{\\widehat{H}}^2} - \\frac{\\gamma \\sqrt{P_d}}{\\beta _\\star \\sigma _{\\widehat{H}}} \\right)+ Q \\left( \\lambda _\\star \\sqrt{\\eta P_d \\sigma _{\\widehat{H}}^2} + \\frac{\\gamma \\sqrt{P_d}}{\\beta _\\star \\sigma _{\\widehat{H}}} \\right).$ The proof follows from Theorem REF by observing that the EER in (REF ) may be rewritten as ${\\rm {EER}}_= 2 -_{}^{\\text{on}}(\\widehat{{\\bf s}}) -_{}^{\\text{off}}(\\widehat{{\\bf s}}),$ with the on/off support probability expressions of the sparse-Bernoulli distribution derived earlier in (REF ) and (REF ) for $E=1$ .",
"On the receiving side of some MIMO systems, there may not be sufficient number of antennas.",
"This is owing to the receiver's small size, limited cost or weight, and low power consumption.",
"Such MIMO systems, where the number of receive antennas $m$ is less than that of the transmitters $n$ (i.e., $m < n$ ), are known as overloaded (or underdetermined) MIMO systems [49].",
"Fig.",
"REF illustrates the accuracy of the derived EER expression for a case of an overloaded system, with $\\eta =0.8$ .",
"This figure shows that the Box-LASSO outperforms the standard one in the EER sense as well.",
"Figure: Element Error Rate of the Box-LASSO and the standard LASSO for GSSK signal recovery.",
"Here, we used T=500,m=120,n=T t =150,k=15,=0.1,ν=0.5T =500, m=120, n=T_t =150, k =15, = 0.1, \\nu = 0.5, and P=10 dB P = \\mathrm { 10 \\ dB}.",
"The data are averaged over 100 independent iterations."
],
[
"Power Allocation and Training Duration Optimization",
"The overall massive MIMO system's performance can be improved by optimizing the power allocation between the transmitted pilot and data symbols as oppose to equal power distribution [33].",
"Power optimization problems in MIMO systems have been proposed based on different performance metrics.",
"In [50], [51], the authors derived a power allocation scheme based on minimizing the MSE, while minimizing the the bit error rate (BER) and SER was considered in [52], [25], [53].",
"Training optimization based on maximizing the channel capacity was addressed in [54], [55], [33].",
"In addition, the authors in [56], [57], [58] provided power allocation strategies based on maximizing the sum rates.",
"The above list of references is not comprehensive, since power allocation optimization research has very rich literature.",
"However, we cited the most related works to this paper."
],
[
"Optimal Power Allocation",
"In this subsection, we will use the previously derived asymptotic results for the MSE and EER to find an optimal power allocation scheme, in a GSSK modulated system, that minimizes these error measures.",
"For fixed $\\tau _t $ and $\\tau $ , the power allocation optimization problem can be caste as $&\\min _{P_t, P_d} \\rm {MSE} \\\\& \\text{subject to:} \\ P_t \\tau _t + P_d (\\tau -\\tau _t ) = {P} \\tau , \\\\& P_t = (1-\\nu ) {P} \\tau , \\ P_d = \\nu {P} \\tau , \\ \\ 0<\\nu <1.$ It can be shown that the above optimization problem boils down to only optimizing the data energy ratio $\\nu $ .",
"The results are summarized next.",
"For fixed $\\tau _t $ and $\\tau $ , and using Box-LASSO with an optimal regularizer $\\gamma _\\star ^{\\mathrm {MSE}}$ as in (REF ), the optimal power allocation that minimizes the MSE is given by $\\nu _\\star ^{\\rm {MSE}} = \\operatornamewithlimits{arg\\,min}_{0<\\nu <1} \\rm {MSE},$ where $\\rm {MSE}$ is the asymptotic MSE expression in (REF ).",
"Similarly, when using the EER as a performance metric, we have $\\nu _\\star ^{\\rm {EER}} = \\operatornamewithlimits{arg\\,min}_{0<\\nu <1} \\rm {EER}_(\\gamma _\\star ),$ where $\\rm {EER}_(\\gamma _\\star )$ is the asymptotic EER expression in (REF ) with the optimal regularizer $\\gamma _\\star $ that minimizes the EER.",
"For the Box-LASSO decoder, finding $\\nu _\\star ^{\\rm {MSE}}$ or $\\nu _\\star ^{\\rm {EER}}$ in closed-form expressions seems to be a difficult task, but by using a bisection method we can numerically find the optimal power allocation scheme.",
"In Fig.",
"REF , we plotted the MSE and EER of the Box-LASSO and standard LASSO as functions of the data energy ratio $\\nu $ .",
"This figure shows that optimizing the MSE and EER are equivalent with $\\nu _\\star \\approx 0.5373$ .",
"Furthermore, it shows that the optimal power allocation is nothing but the well-known scheme $\\bar{\\nu }_\\star $ which was shown in [25], [33] to maximize the effective SNR, where $\\bar{\\nu }_\\star $ is given as ([25]): $\\bar{\\nu }_\\star ={\\left\\lbrace \\begin{array}{ll}\\vartheta - \\sqrt{\\vartheta (\\vartheta -1)} & \\text{if $\\tau _{d} > 1$,} \\\\\\frac{1}{2} & \\text{if $\\tau _{d} =1$,} \\\\\\vartheta + \\sqrt{\\vartheta (\\vartheta -1)} & \\text{if $\\tau _d< 1$,}\\end{array}\\right.",
"}$ where $\\vartheta = \\frac{1 + P\\cdot \\tau }{P\\cdot \\tau (1 - \\frac{1}{\\tau _d})}.$ This result is significant, since it showed again that the optimal power allocation scheme is nothing but the celebrated one that maximizes the effective SNR of the MIMO system, i.e., $\\bar{\\nu }_\\star $ .",
"The power allocation actually does not depend on the type of the modulation constellation used, the used detector, or the other problem parameters such as $\\eta $ and $\\rho $ .Provided that we use the LMMSE estimator for the channel estimation.",
"For example, in [25], the same power allocation scheme was obtained for a massive MIMO system with $M$ -PAM signals and a Box-regularized least squares (Box-RLS) detector, while this work employs the Box-LASSO decoder for GSSK signal recovery.",
"Figure: This figure shows the MSE/EER of Box-LASSO/LASSO as functions of ν\\nu .",
"We used T=1000,n=400,T t =456,P=12 dB ,δ=1.5,ρ=0.1,T =1000, n=400, T_t =456, P= 12 \\mathrm {dB}, \\delta =1.5,\\rho =0.1, and =0.01 =0.01."
],
[
"Optimal Training Duration",
"In order to optimize the training duration, we introduce the following performance metric, which is called the goodput.",
"The goodput is calculated by dividing the amount of useful transmitted data by the time it takes to send it successfully [59], [25].",
"Formally, it can be defined as ${\\rm {Goodput}}(\\tau _t, \\tau ) = \\bigg (1-\\frac{\\tau _t}{\\tau }\\bigg ) (1-\\rm {EER}).$ The optimal value $T_t^\\star $ that maximizes the goodput is determined in the following Corollary.",
"For a fixed power allocation, the goal is to identify the optimal number of training symbols out of the total coherence interval symbols.",
"From (REF ), we must have $T_t^\\star \\ge n$ (or, $\\tau _t^\\star \\ge 1$ ), and obviously, $T_t^\\star <T$ (or, $\\tau _t^\\star < \\tau $ ).",
"Therefore, $\\tau _t^\\star $ is a solution to the maximization problem: $\\tau _t^\\star = \\mathrm {arg} \\max _{1 \\le \\tau _t < \\tau } \\overline{\\rm {Goodput}}(\\tau _t, \\tau ),$ where $\\overline{\\rm {Goodput}}(\\tau _t, \\tau ): =\\big (1-\\frac{\\tau _t}{\\tau }\\big ) [1-{\\rm {plim}}_{n \\rightarrow \\infty } \\ \\rm {EER}_(\\gamma _\\star )]$ is the asymptotic value of the goodput.",
"Corollary 1 (Optimal Training Duration) Under imperfect CSI, the optimal training duration that maximizes the goodput in (REF ) is given by: $\\tau _t^\\star = 1 \\ (\\text{or} \\ T_t^\\star = n),$ for all ${P}$ and $\\tau $ (or $T$ ).",
"This result can be proven in a similar manner to [25], details are thus omitted.",
"Fig.",
"REF shows the goodput performance of Box-LASSO simulated versus the training duration $T_t$ which confirms the result of Corollary REF .",
"It shows that at $T_t = n=200$ , the goodput is maximized.",
"Figure: Goodput performance vs. T t T_t.",
"We used T=1000,P=12 dB ,n=200,ν=0.5,δ=1.5,ρ=0.1,T = 1000, P=12 \\ \\mathrm {dB},n =200,\\nu =0.5,\\delta =1.5,\\rho =0.1, and =0.01=0.01."
],
[
"Conclusion and Future Work",
"In this work, we derived sharp asymptotic characterizations of the mean square error, probability of support recovery and element error rate of the Box-LASSO under the presence of uncertainties in the channel matrix in the form of channel estimation errors.",
"The analytical tool used in the analysis is the recently developed CGMT framework.",
"The derived expressions can be used to optimally tune the involved hyper-parameters of the algorithm such as the regularizer.",
"Then, we used the Box-LASSO as a low complexity decoder in an application of massive MIMO detection using GSSK modulation, and optimize the power allocation between training and data symbols to minimize the MSE or EER of the system.",
"Furthermore, we derived the optimal training duration that maximizes the system's goodput.",
"Numerical simulations show very close agreement to the derived theoretical predictions.",
"Moreover, we showed that the Box-LASSO outperforms standard one in all of the considered performance metrics.",
"Finally, we highlight that the generalized spacial modulation (GSM) is more involved than the considered GSSK modulation since it uses the positions of active antenna in addition to a constellation symbol (e.g., $M$ -QAM, $M$ -PSK, etc.)",
"to encode information [31].",
"However, we focused in this paper on GSSK systems since the analysis framework, namely the CGMT, requires real-valued quantities (signals and channels).",
"An interesting possible future work is to extend the results of this paper to systems involving complex-valued data such as GSM and investigate their power allocation optimization.",
"Moreover, this paper assumes Gaussian channels matrices with i.i.d.",
"entries, but in numerous wireless communication applications, there are usually spatial correlations between the antennas.",
"Therefore, another possible extension is to study correlated massive MIMO systems, where the matrix entries are no longer i.i.d."
],
[
"Sketch of the Proof",
"In this appendix, we derive the asymptotic analysis of the considered Box-LASSO problem's performance metrics.",
"Our analysis is based on the CGMT, which is discussed more below."
],
[
"Technical Tool: CGMT",
"Firstly, we summarize the CGMT framework [5] before the proof of our main results.",
"For a comprehensive list of technical requirements, please see [5], [27].",
"Consider the following two optimization problems, which we call the Primal Optimization (PO) and Auxiliary Optimization (AO) problems, respectively.",
"$&({\\bf C}) := \\underset{{\\bf x}\\in \\mathcal {S}_{{\\bf x}}}{\\operatorname{\\min }} \\ \\underset{{\\bf y}\\in \\mathcal {S}_{{\\bf y}}}{\\operatorname{\\max }} \\ {\\bf y}^{T} {\\bf C}{\\bf x}+ \\xi ( {\\bf x}, {\\bf y}), \\\\&({\\bf g}_1, {\\bf g}_2) := \\underset{{\\bf x}\\in \\mathcal {S}_{{\\bf x}}}{\\operatorname{\\min }} \\ \\underset{{\\bf y}\\in \\mathcal {S}_{{\\bf y}}}{\\operatorname{\\max }} \\ \\Vert {\\bf x}\\Vert {\\bf g}_1^{T} {\\bf y}+ \\Vert {\\bf y}\\Vert {\\bf g}_2^{T} {\\bf x}+ \\xi ( {\\bf x}, {\\bf y}), $ where ${\\bf C}\\in \\mathbb {R}^{\\tilde{m} \\times \\tilde{n}}, {\\bf g}_1 \\in \\mathbb {R}^{\\tilde{m}}, {\\bf g}_2 \\in \\mathbb {R}^{\\tilde{n}}, \\mathcal {S}_{\\bf x}\\subset \\mathbb {R}^{\\tilde{n}}, \\mathcal {S}_{\\bf y}\\subset \\mathbb {R}^{\\tilde{m}}$ and $\\xi : \\mathbb {R}^{\\tilde{n}} \\times \\mathbb {R}^{\\tilde{m}} \\mapsto \\mathbb {R}$ .",
"Moreover, the function $\\xi $ is assumed to be independent of the matrix ${\\bf C}$ .",
"Denote by ${\\bf x}_{} := {\\bf x}_{}({\\bf C}) $ , and ${\\bf x}_{} := {\\bf x}_{}( {\\bf g}_1, {\\bf g}_2)$ any optimal minimizers of (REF ) and (), respectively.",
"Further let $\\mathcal {S}_{\\bf x},$ and $\\mathcal {S}_{\\bf y}$ be convex and compact sets, $\\xi ({\\bf x},{\\bf y})$ is convex-concave continuous on $\\mathcal {S}_{\\bf x}\\times \\mathcal {S}_{\\bf y}$ , and let ${\\bf C}, {\\bf g}_1$ and ${\\bf g}_2 $ all have i.i.d.",
"standard normal entries.",
"The PO-AO equivalence is formally stated in the next theorem, the proof of which can be found in [5].",
"Theorem 3 (CGMT [5]) Under the above assumptions, the CGMT [5] shows that the following statements are true: (a) For all $c \\in \\mathbb {R}$ and $t>0$ , it holds $\\mathbb {P}\\left[\\left| ({\\bf C})-c \\right|> t\\right] \\le 2 \\mathbb {P}\\left[\\left| ({\\bf g}_1, {\\bf g}_2)-c \\right|> t\\right].$ In words, concentration of the optimal cost of the AO problem around $c$ implies concentration of the optimal cost of the corresponding PO problem around the same value $c$ .",
"According the CGMT, this will finally imply that the original optimal cost (e.g., the Box-LASSO of (REF ) in this paper) will concentrate around $c$ as well.",
"Please see Fig.",
"REF for a numerical illustration.",
"(b) Let $\\mathcal {S}$ be any arbitrary open subset of $\\mathcal {S}_{\\bf x}$ , and $\\mathcal {S}^c = \\mathcal {S}_{\\bf x}\\setminus \\mathcal {S}$ .",
"Denote $_{\\mathcal {S}^c}({\\bf g}_1,{\\bf g}_2)$ the optimal cost of the optimization in (), when the minimization over ${\\bf x}$ is constrained over ${\\bf x}\\in \\mathcal {S}^c$ .",
"Consider the regime $\\tilde{m}, \\tilde{n} \\rightarrow \\infty $ such that $\\frac{\\tilde{m}}{ \\tilde{n}}\\rightarrow \\eta $ .",
"To keep notation short, this regime is denoted by $\\tilde{n}\\rightarrow \\infty $ .",
"Suppose that there exist constants $\\overline{\\phi }$ and $\\overline{\\phi }_{\\mathcal {S}^c}$ such that (i) $\\overline{\\phi } < \\overline{\\phi }_{\\mathcal {S}^c}$ , (ii) $({\\bf g}_1,{\\bf g}_2) \\overset{P}{\\longrightarrow } \\overline{\\phi }$ , and (iii) $_{\\mathcal {S}^c}({\\bf g}_1,{\\bf g}_2) \\overset{P}{\\longrightarrow } \\overline{\\phi }_{\\mathcal {S}^c}$ .",
"Then, if in addition, $\\lim _{\\tilde{n} \\rightarrow \\infty } \\mathbb {P}[{\\bf x}_{} \\in \\mathcal {S}] = 1,$ it also holds that $\\lim _{\\tilde{n} \\rightarrow \\infty } \\mathbb {P}[{\\bf x}_{} \\in \\mathcal {S}] = 1.$ When the assumptions of Theorem REF are met, the CGMT-based proof proceeds in general as follows: $\\bullet $ Identifying the PO and the associated AO: This step involves transforming the original optimization problem into the desired minimax PO form, and then identify its corresponding AO problem.",
"$\\bullet $ Simplifying the AO: In this step, the AO is reduced into a scalar optimization problem.",
"$\\bullet $ Probabilistic analysis of the AO: In this step, we prove that the AO converges to a (deterministic) asymptotic optimization problem which involves only scalar variables.",
"$\\bullet $ Choice of $\\mathcal {S}$: The set $\\mathcal {S}$ should be selected properly based on the measure of interest.",
"For instance, if we want to analyze the MSE or the EER, $\\mathcal {S}$ will be the set in which the MSE or the EER concentrates, respectively.",
"After introducing the CGMT framework, we prove Theorem REF next.",
"For clarity, the steps of the proof are divided into different subsections."
],
[
"PO and AO Identification",
"To obtain the result of the main theorems using CGMT, we need first to rewrite the Box-LASSO optimization problem (REF ) as a PO problem.",
"For convenience, we consider the error vector ${\\bf e}:= {\\bf s}- {\\bf s}_0,$ and the modified Box-set for all $j\\in \\lbrace 1,2, \\cdots , n\\rbrace $ : $\\mathbb {D} =\\bigg \\lbrace e_j \\in \\mathbb {R} : \\ell - s_{0,j} \\le e_j \\le \\mu - s_{0,j} \\bigg \\rbrace ,$ then the problem in (REF ) can be reformulated as $\\widehat{{\\bf e}} = {\\rm {arg}} \\ \\underset{{\\bf e}\\in \\mathbb {D}^n}{\\operatorname{\\min }} \\ & \\bigg \\Vert \\sqrt{ \\frac{P_d}{n}} \\widehat{{\\bf H}} {\\bf e}+ \\sqrt{ \\frac{P_d}{n}} \\hbox{$\\Omega $}{\\bf s}_0 -{\\bf v}\\bigg \\Vert _2^2 + \\gamma P_d \\Vert {\\bf e}+ {\\bf s}_0 \\Vert _1.$ This minimization is not in the PO form as it is missing the max part.",
"So to fix this, let us express the loss function using its Legendre-Fenchel transformationFor any convex function $f$ , we can write: $f({\\bf x}) = \\max _{{\\bf y}} {\\bf y}^T {\\bf x}- f^\\star ({\\bf y})$ , where $f^\\star $ is the Fenchel (convex) conjugate of $f$ .",
": $\\Vert {\\bf x}\\Vert _2^2 = \\max _{{\\bf y}\\in \\mathbb {R}^m} {\\bf y}^T {\\bf x}- \\frac{1}{4} \\Vert {\\bf y}\\Vert _2^2.$ Hence, the problem above is equivalent to the following $\\underset{{\\bf e}\\in \\mathbb {D}^n}{\\operatorname{\\min }} \\ \\underset{{\\bf y}\\in \\mathbb {R}^m}{\\operatorname{\\max }} & \\ \\sqrt{ \\frac{P_d}{n}} {\\bf y}^T \\widehat{{\\bf H}} {\\bf e}+ \\sqrt{ \\frac{P_d}{n}} {\\bf y}^T \\hbox{$\\Omega $}{\\bf s}_0 - {\\bf y}^T {\\bf v}- \\frac{1}{4} \\Vert {\\bf y}\\Vert _2^2+ \\gamma P_d \\Vert {\\bf e}+ {\\bf s}_0 \\Vert _1.$ One technical requirement of the CGMT is the compactness of the feasibility set over ${\\bf y}$ .",
"This can be handled following the approach in [5], by introducing a sufficiently large artificial constraint set $\\mathcal {S}_{\\bf y}= \\biggl \\lbrace {\\bf y}\\in \\mathbb {R}^m: \\Vert {\\bf y}\\Vert _2 \\le C_y \\biggr \\rbrace ,$ for some sufficiently large constant (independent of $n$ ) $C_y>0$ .",
"This will not asymptotically affect the optimization problem.",
"Then, we obtain $\\underset{{\\bf e}\\in \\mathbb {D}^n}{\\operatorname{\\min }} \\ \\underset{{\\bf y}\\in \\mathcal {S}_{\\bf y}}{\\operatorname{\\max }}& \\ \\sqrt{ \\frac{P_d \\sigma _{\\widehat{H}}^2}{n}} {\\bf y}^T \\widetilde{{\\bf H}} {\\bf e}+ \\sqrt{ \\frac{P_d \\sigma _{\\omega }^2}{n}} {\\bf y}^T \\widetilde{\\hbox{$\\Omega $}}{\\bf s}_0 - {\\bf y}^T {\\bf v}- \\frac{1}{4} \\Vert {\\bf y}\\Vert _2^2 + \\gamma P_d \\Vert {\\bf e}+ {\\bf s}_0 \\Vert _1,$ where $\\widetilde{{\\bf H}}$ and $\\widetilde{\\hbox{$\\Omega $}}$ are independent matrices with i.i.d.",
"$\\mathcal {N}(0,1)$ entries each.",
"The above problem is now in a PO form with $\\xi ({\\bf e},{\\bf y}) = \\ &\\sqrt{ \\frac{P_d \\sigma _{\\omega }^2}{n}} {\\bf y}^T \\widetilde{\\hbox{$\\Omega $}}{\\bf s}_0- {\\bf y}^T {\\bf v}- \\frac{1}{4} \\Vert {\\bf y}\\Vert _2^2+ \\gamma P_d \\Vert {\\bf e}+ {\\bf s}_0 \\Vert _1.$ Thus, its associated AO is given as $\\underset{{\\bf e}\\in \\mathbb {D}^n}{\\operatorname{\\min }} \\ \\underset{{\\bf y}\\in \\mathcal {S}_{\\bf y}}{\\operatorname{\\max }} & \\ \\sqrt{ \\frac{P_d \\sigma _{\\widehat{H}}^2}{n}} \\Vert {\\bf e}\\Vert _2 {\\bf g}^T{\\bf y}+ \\sqrt{ \\frac{P_d \\sigma _{\\widehat{H}}^2}{n}} \\Vert {\\bf y}\\Vert _2 {\\bf z}^T{\\bf e}+ \\xi ({\\bf e},{\\bf y}),$ where ${\\bf g}\\sim \\mathcal {N}(\\mathbf {0},{\\bf I}_m)$ and ${\\bf z}\\sim \\mathcal {N}(\\mathbf {0},{\\bf I}_n)$ are independent random vectors."
],
[
"AO Simplification",
"In order to simplify the AO, we first let $\\widetilde{{\\bf g}}:= \\sqrt{ \\frac{P_d \\sigma _{\\widehat{H}}^2}{n}} \\Vert {\\bf e}\\Vert _2 {\\bf g}- {\\bf v}+ \\sqrt{ \\frac{P_d \\sigma _{\\omega }^2}{n}} \\widetilde{\\hbox{$\\Omega $}}{\\bf s}_0.$ It can be shown that $\\widetilde{{\\bf g}} \\sim \\mathcal {N}(\\mathbf {0}, \\hbox{$\\Sigma $}_{\\widetilde{{\\bf g}}}),$ where $\\hbox{$\\Sigma $}_{\\widetilde{{\\bf g}}} = \\left( \\frac{P_d \\sigma _{\\widehat{H}}^2}{n} \\Vert {\\bf e}\\Vert _2^2 +1 +\\frac{P_d \\sigma _{\\omega }^2}{n} \\Vert {\\bf s}_0 \\Vert _2^2 \\right) {\\bf I}_m.$ Then, the AO can be rewritten as $\\underset{{\\bf e}\\in \\mathbb {D}^n}{\\operatorname{\\min }} \\ \\underset{{\\bf y}\\in \\mathcal {S}_{\\bf y}}{\\operatorname{\\max }} \\ & \\widetilde{{\\bf g}}^T{\\bf y}+ \\sqrt{ \\frac{P_d \\sigma _{\\widehat{H}}^2}{n}} \\Vert {\\bf y}\\Vert _2 {\\bf z}^T{\\bf e}-\\frac{\\Vert {\\bf y}\\Vert _2^2}{4} + \\gamma P_d \\Vert {\\bf e}+ {\\bf s}_0 \\Vert _1.$ Fixing the norm of ${\\bf y}$ to $\\alpha : =\\Vert {\\bf y}\\Vert _2$ , we can easily optimize over its direction.",
"This simplifies the AO to $\\underset{{\\bf e}\\in \\mathbb {D}^n}{\\operatorname{\\min }} \\ \\max _{\\alpha \\ge 0} & \\ \\alpha \\Vert \\widetilde{{\\bf g}}\\Vert _2 + \\sqrt{ \\frac{P_d \\sigma _{\\widehat{H}}^2}{n}} \\ \\alpha \\ {\\bf z}^T{\\bf e}-\\frac{\\alpha ^2}{4} + \\gamma P_d \\Vert {\\bf e}+ {\\bf s}_0 \\Vert _1.$"
],
[
"Asymptotic Analysis of the AO",
"Next, we need to normalize the above optimization problem by $\\frac{1}{n}$ , to have all of its terms of the same order, $\\mathcal {O}(1)$ , and also define $\\beta : = \\frac{\\alpha }{\\sqrt{n}}.$ Then, after a change of the order of the $\\min $ -$\\max $[5] shows that flipping the $\\min $ -$\\max $ order is possible for large dimensions., we obtain: $\\max _{\\beta \\ge 0} \\underset{{\\bf e}\\in \\mathbb {D}^n}{\\operatorname{\\min }} & \\ \\beta \\sqrt{\\frac{P_d \\sigma _{\\widehat{H}}^2}{n} \\Vert {\\bf e}\\Vert _2^2 +1 +\\frac{P_d \\sigma _{\\omega }^2}{n} \\Vert {\\bf s}_0 \\Vert _2^2} \\ \\ \\frac{\\Vert {\\bf g}\\Vert _2}{\\sqrt{n}} \\nonumber \\\\&+ \\sqrt{ {P_d \\sigma _{\\widehat{H}}^2}} \\beta \\frac{1}{n} {\\bf z}^T{\\bf e}-\\frac{\\beta ^2}{4} + \\frac{\\gamma P_d}{n} \\Vert {\\bf e}+ {\\bf s}_0 \\Vert _1.$ Note the abuse of notation for ${\\bf g}$ to represent another standard normal vector.",
"To have a separable optimization problem, we use the following variational identity: $\\sqrt{x} = \\min _{\\lambda >0} \\frac{1}{2 \\lambda } + \\frac{ \\lambda x}{2 },\\ \\text{for} \\ x>0,$ with optimum solution $\\hat{\\lambda }= \\frac{1}{\\sqrt{x}}$ .",
"Using this trick, with $x = \\frac{P_d \\sigma _{\\widehat{H}}^2}{n} \\Vert {\\bf e}\\Vert _2^2 +1 +\\frac{P_d \\sigma _{\\omega }^2}{n} \\Vert {\\bf s}_0 \\Vert _2^2,$ the optimization in (REF ) becomes $\\max _{\\beta \\ge 0} \\min _{\\lambda >0} & \\ \\frac{\\beta \\Vert {\\bf g}\\Vert _2}{2 \\lambda \\sqrt{n}}+ \\frac{\\beta \\lambda \\Vert {\\bf g}\\Vert _2}{2 \\sqrt{n}} \\left( 1+ \\frac{P_d \\sigma _{\\omega }^2}{n} \\Vert {\\bf s}_0 \\Vert _2^2 \\right) -\\frac{\\beta ^2}{4} \\nonumber \\\\&+ \\min _{{\\bf e}\\in \\mathbb {D}^n} \\biggl \\lbrace \\frac{\\beta \\lambda \\Vert {\\bf g}\\Vert _2}{2 \\sqrt{n}} \\frac{P_d \\sigma _{\\widehat{H}}^2}{n} \\Vert {\\bf e}\\Vert _2^2 + \\sqrt{ {P_d \\sigma _{\\widehat{H}}^2}} \\beta \\frac{{\\bf z}^T{\\bf e}}{n}+ \\frac{\\gamma P_d}{n} \\Vert {\\bf e}+ {\\bf s}_0 \\Vert _1 \\biggr \\rbrace .$ Using the weak law of large numbers (WLLN), $\\frac{\\Vert {\\bf g}\\Vert _2}{\\sqrt{n}} \\overset{P}{\\longrightarrow }\\sqrt{\\eta },$ and $\\frac{1}{n} \\Vert {\\bf s}_0 \\Vert _2^2 \\overset{P}{\\longrightarrow }\\rho \\sigma _s^2=\\rho .$ Next, using the above convergence results, and working with the original optimization variable ${\\bf s}$ instead of ${\\bf e}$ , we get $\\max _{\\beta \\ge 0} \\min _{\\lambda >0} & \\ \\frac{\\beta \\sqrt{\\eta }}{2 \\lambda }+ \\frac{\\beta \\lambda \\sqrt{\\eta }}{2 } \\left( 1+ {P_d \\sigma _{\\omega }^2} \\rho \\right) -\\frac{\\beta ^2}{4} \\nonumber \\\\&+\\frac{1}{n}\\sum _{j=1}^n \\min _{\\ell \\le s_j \\le \\mu } \\biggl \\lbrace \\frac{\\beta \\lambda \\sqrt{\\eta }}{2 } {P_d \\sigma _{\\widehat{H}}^2} (s_j - s_{0,j})^2+ \\sqrt{ {P_d \\sigma _{\\widehat{H}}^2}} \\beta z_j (s_j -s_{0,j}) + {\\gamma P_d} | s_j | \\biggr \\rbrace .$ After a completion of squares in $s_j$ , and noting that $\\frac{1}{n} {\\bf z}^T {\\bf s}_0 \\overset{P}{\\longrightarrow }0$ , the above problem becomes $\\max _{\\beta \\ge 0} \\min _{\\lambda >0} & \\ \\frac{\\beta \\sqrt{\\eta }}{2 \\lambda }+ \\frac{\\beta \\lambda \\sqrt{\\eta }}{2 } \\left( 1+ {P_d \\sigma _{\\omega }^2} \\rho \\right) -\\frac{\\beta ^2}{4} - \\frac{1}{n} \\sum _{j=1}^n \\frac{\\beta }{2 \\lambda \\sqrt{\\eta }} z_j^2 \\nonumber \\\\&+\\beta \\lambda \\sqrt{\\eta } P_d \\sigma _{\\widehat{H}}^2 \\ \\frac{1}{n}\\sum _{j=1}^n \\min _{\\ell \\le s_j \\le \\mu } \\Biggl \\lbrace \\frac{1}{2} \\Biggl ( s_j- \\biggl ( s_{0,j} + \\frac{z_j}{\\lambda \\sqrt{\\eta P_d \\sigma _{\\widehat{H}}^2 }}\\biggr ) \\Biggr )^2 + \\frac{\\gamma }{\\beta \\lambda \\sqrt{\\eta } \\sigma _{\\widehat{H}}^2} | s_j | \\Biggr \\rbrace \\nonumber \\\\= & \\max _{\\beta \\ge 0} \\min _{\\lambda >0} G(\\beta ,\\lambda ,{\\bf z},{\\bf s}_0).$ The optimization over $s_j$ could be obtained in closed-form using the saturated shrinkage function $\\mathcal {H}(a ; \\gamma , l, u)$ which is defined in (REF ).",
"Also, let its optimal objective be $\\mathcal {J}(a;\\gamma ,l,u) =\\min _{l \\le s \\le u} \\frac{1}{2}(s-a)^2 + \\gamma |s|$ as defined in (REF ).",
"Now, again, using the WLLN, we have $\\frac{1}{n} \\sum _{j=1}^n z_j^2 \\overset{P}{\\longrightarrow }1,$ and for all $\\beta >0$ , and $\\lambda >0$ : $&\\frac{1}{n} \\sum _{j =1}^n \\mathcal {J} \\left(s_{0,j}+\\frac{z_j}{\\lambda \\sqrt{\\eta P_d \\sigma _{\\widehat{H}}^2 }}; \\frac{\\gamma }{\\beta \\lambda \\sqrt{\\eta } \\sigma _{\\widehat{H}}^2 }, \\ell ,\\mu \\right)\\overset{P}{\\longrightarrow }\\mathbb {E} \\left[\\mathcal {J} \\left(S_{0}+\\frac{Z}{\\lambda \\sqrt{\\eta P_d \\sigma _{\\widehat{H}}^2 }}; \\frac{\\gamma }{\\beta \\lambda \\sqrt{\\eta } \\sigma _{\\widehat{H}}^2 }, \\ell ,\\mu \\right) \\right],$ where the expectation is taken over $S_0 \\sim p_{s_0}$ and $Z \\sim \\mathcal {N}(0,1)$ .",
"Consequently, the objective function in (REF ), i.e., $G(\\beta ,\\lambda ,{\\bf z},{\\bf s}_0)$ , converges point-wise to the quantity $\\mathcal {G}(\\beta ,\\lambda )$ , defined in (REF ), in the limit of $n \\rightarrow \\infty $ .",
"Afterwards, observe that $G(\\beta ,\\lambda ,{\\bf z},{\\bf s}_0)$ is convex in $\\lambda $ and concave in $\\beta $ .",
"With these, and using Theorem 2.7 in [60], it follows that $\\max _{\\beta \\ge 0} \\min _{\\lambda >0} G(\\beta ,\\lambda ,{\\bf z},{\\bf s}_0) \\overset{P}{\\longrightarrow }\\max _{\\beta \\ge 0} \\min _{\\lambda >0} \\mathcal {G}(\\beta ,\\lambda ).$ Finally, the optimization problem in (REF ) simplifies to the following Scalar Optimization (SO) problem: $\\max _{\\beta \\ge 0} \\min _{\\lambda >0} & \\ \\frac{\\beta \\sqrt{\\eta }}{2 \\lambda }+ \\frac{\\beta \\lambda \\sqrt{\\eta }}{2 } \\left( 1+ {P_d \\sigma _{\\omega }^2} \\rho \\right) -\\frac{\\beta ^2}{4} - \\frac{\\beta }{2 \\lambda \\sqrt{\\eta }} \\nonumber \\\\&+\\beta \\lambda \\sqrt{\\eta } P_d \\sigma _{\\widehat{H}}^2 \\ \\mathbb {E} \\Biggl [\\mathcal {J} \\biggl (S_{0}+ \\frac{Z}{\\lambda \\sqrt{\\eta P_d \\sigma _{\\widehat{H}}^2 }}; \\frac{\\gamma }{\\beta \\lambda \\sqrt{\\eta } \\sigma _{\\widehat{H}}^2 }, \\ell ,\\mu \\biggr ) \\Biggr ] \\nonumber \\\\&= \\max _{\\beta \\ge 0} \\min _{\\lambda >0} \\mathcal {G}(\\beta ,\\lambda ).$ It worth mentioning that in the above equation, $ \\mathcal {G}(\\beta _\\star ,\\lambda _\\star )$ , where $(\\beta _\\star ,\\lambda _\\star )$ is the unique solution of (REF ), represents the the asymptotic value of the objective function in (REF ) for a minimizer $\\widehat{{\\bf s}}$ , i.e., $\\underset{n\\rightarrow \\infty }{\\rm {plim}} \\ \\frac{1}{n}\\left( \\bigg \\Vert \\sqrt{ \\frac{P_d}{n}} \\widehat{{\\bf H}} \\widehat{{\\bf s}} - {\\bf r}\\bigg \\Vert _2^2 + \\gamma P_d \\Vert \\widehat{{\\bf s}} \\Vert _1 \\right) = \\mathcal {G}(\\beta _\\star ,\\lambda _\\star ).$ Fig.",
"REF shows the great accuracy of the above result.",
"Figure: Optimal objective function value of the Box-LASSO vs. the regularizer for a sparse-Bernoulli vector.",
"We used ρ=0.2,η=1.5,n=128\\rho =0.2,\\eta = 1.5,n = 128, T=500,T t =n,ν=0.5,E=1,T = 500,T_t =n, \\nu = 0.5, E=1, and P=15 dB P= 15 \\ {\\rm {dB}}.After deriving the SO problem, we are now in a position to study the asymptotic convergence of the MSE.",
"The analysis is given in the next subsection."
],
[
"Error Analysis via CGMT (Proof of Theorem 1)",
"In this part, we study the asymptotic convergence of the MSE of the Box-LASSO.",
"First, using the fact that $\\hat{\\lambda }= \\frac{1}{\\sqrt{x}}$ and recalling from (REF ) that $\\frac{1}{n} \\Vert \\widetilde{{\\bf e}}\\Vert _2^2= \\frac{1}{P_d \\sigma _{\\widehat{H}}^2} \\left( \\frac{1}{\\hat{\\lambda }_n^2}-1 -\\frac{P_d \\sigma _{\\omega }^2}{n} \\Vert {\\bf s}_0 \\Vert _2^2 \\right),$ where $\\widetilde{{\\bf e}}$ is the AO solution in (REF ), and $\\hat{\\lambda }_n$ is the solution to (REF ) in $\\lambda $ .",
"Using [5], it can be shown that $\\hat{\\lambda }_n \\overset{P}{\\longrightarrow }\\lambda _\\star $ , where $\\lambda _\\star $ is the solution to (REF ).",
"Then, by the WLLN, $\\frac{1}{n} \\Vert {\\bf s}_0\\Vert _2^2 \\overset{P}{\\longrightarrow }\\rho $ , and hence $\\frac{1}{n} \\Vert \\widetilde{{\\bf e}} \\Vert _2^2&\\overset{P}{\\longrightarrow }\\frac{1}{P_d \\sigma _{\\widehat{H}}^2} \\left( \\frac{1}{\\lambda _\\star ^2}-1 -{P_d \\sigma _{\\omega }^2} \\rho \\right).$ Recall that $\\widetilde{{\\bf e}} =\\widetilde{{\\bf s}} - {\\bf s}_0$ , so the last step is to use the CGMT to prove that the quantities $\\widehat{{\\bf s}} - {\\bf s}_0$ and $\\widetilde{{\\bf s}} - {\\bf s}_0$ are concentrated in the same set with high probability.",
"Formally, for any fixed $\\varepsilon > 0$ , we define the set: $\\mathcal {S}_\\varepsilon = \\left\\lbrace \\mathbf {q} \\in \\mathbb {R}^n: \\left| \\frac{\\Vert {\\bf q}\\Vert _2^2}{n} - \\frac{ (\\frac{1}{\\lambda _\\star ^2}-1 -{P_d \\sigma _{\\omega }^2} \\rho ) }{P_d \\sigma _{\\widehat{H}}^2} \\right| < \\varepsilon \\right\\rbrace .$ Equation (REF ) proves that for any $\\varepsilon >0$ , $\\widetilde{{\\bf s}} - {\\bf s}_0 \\in \\mathcal {S}_\\varepsilon $ with probability approaching one.",
"Then, we conclude by applying the CGMT that $\\widehat{{\\bf s}} - {\\bf s}_0 \\in \\mathcal {S}_\\varepsilon $ with probability approaching one.",
"This proves the asymptotic prediction of the MSE as summarized in Theorem REF .",
"The residual prediction in Remark REF , eq.",
"(REF ), can be proven in a similar way, by first noting that $\\Vert \\hat{{\\bf y}}\\Vert _2^2 = 4n \\mathcal {R},$ where $\\hat{{\\bf y}}$ is the PO solution of (REF ).",
"Using the definition: $\\beta _n^2 = \\frac{\\Vert \\widetilde{{\\bf y}}\\Vert _2^2}{n}$ , where $\\beta _n, $ and $\\widetilde{{\\bf y}}$ are the solutions of (REF ), and (REF ) respectively, and following the same steps as in the MSE proof above, one can show that $\\hat{{\\bf y}}$ and $\\widetilde{{\\bf y}}$ concentrate in the same set with probability approaching one, and then apply the CGMT to reach the proof of the residual result in (REF ).",
"Details are thus omitted."
]
] | 2107.01870 |
[
[
"Anisotropic flow and correlations between azimuthal anisotropy Fourier\n harmonics in Xe-Xe collisions at $\\sqrt{s_{NN}}$ = 5.44 TeV under HYDJET++\n framework"
],
[
"Abstract The study of anisotropic harmonic flow coefficients $ v_{n}$(n=2,3,4) is performed in Xe-Xe collisions at $\\sqrt{s_{NN}}$ = 5.44 TeV under Monte Carlo HYDJET++ model (HYDrodynamics plus JETs) framework.",
"Anisotropic flow of identified particles and correlation between the azimuthal harmonic flow amplitudes is presented.",
"Here, we have considered body-body and tip-tip type of geometrical configurations for Xe-Xe collision systems.",
"The kinematic ranges $|\\eta|<0.8$, $0<p_{T}<5.0$ GeV/c, and $|\\delta / \\eta|> 2$ are considered.",
"The results have been shown for seven classes of centrality and compared with the ALICE experimental data.",
"The anisotropic flow of identified charged particles show a strong centrality dependence.",
"Mass ordering is observed for $v_{2},v_{3}$ and $v_{4}$.",
"Mass ordering is different for different ranges of transverse momentum $p_{T}$.",
"Strong correlation is observed between $v_{3}-v_{2}$, $v_{4}-v_{2}$, and $v_{4}-v_{3}$.",
"Such correlation is centrality dependent and is different in different centrality windows.",
"The anisotropic flow coefficients show a clear dependence on the total charged particle multiplicity.",
"HYDJET++ model justifies experimental data well enough."
],
[
"Introduction",
"Several fascinating and impeccable phenomenon, yet not studied systematically, have been observed in the Relativistic Heavy Ion Collider (RHIC) and Large Hadron Collider (LHC) heavy-ion program.",
"Experiments involved in the investigation of ultra-relativistic collisions aim to explore a deconfined state of quarks and gluons called as the Quark Gluon Plasma (QGP) [1].",
"Quark Gluon Plasma is a new state of nuclear matter existing at high temperatures and densities, formed when the composite states of matter (hadrons) lose their identity and dissolve into a soup of quarks and gluons [2].",
"This Quark Gluon Plasma (QGP) created in high energy heavy ion collisions expands rapidly.",
"Relativistic viscous hydrodynamic models [3], [4], [5] elegantly describe the space-time dynamics of this created Quark Gluon Plasma.",
"During expansion, large pressure gradients of the generated QGP convert the spatial anisotropies in the initial-state geometry to the momentum anisotropies of the final state particles.",
"These momemtum anisotropies are characterized by the Fourier expansion of particle density in the azimuthal angle $\\phi $ [6], [7], $\\dfrac{dN}{d\\phi } \\propto 1+ 2\\sum \\limits _{n=1}^{\\infty } v_{n}\\cos [n(\\phi -\\psi _{n})] \\quad $ where $\\phi $ = azimuthal angle with respect to the reaction plane $\\psi _{n}$ of the produced particle, n = harmonic value, $\\psi _{n}$ = reaction plane, and $v_{n}$ = fourier coefficient of order n representing the flow harmonics given by- $v_{n} = \\langle \\langle \\cos [n(\\phi -\\psi _{n})] \\rangle \\rangle \\quad .$ The second harmonic, n=2 is called as elliptic flow $v_{2}$ that reveals the lenticular shape of the collision overlap region.",
"The elliptic flow relates the anisotropic shape of the overlapped region of the colliding nuclei to the corresponding anisotropy of the outgoing momentum distribution.",
"The higher harmonics $v_{n}$ (n$>$ 2) are produced lesser than $v_{2}$ .",
"These coefficients also carry essential information on the dynamics of the created medium and provide a more clear and complete picture of its bulk properties along with $v_{2}$ .",
"The third harmonic, n=3 is called as triangular flow $v_{3}$ and the fourth harmonic, n=4 is called as quadrangular flow $v_{4}$ .",
"Triangular flow is caused due to the initial-state fluctuations in the nucleon positions at the moment of impact [8] while the higher harmonics are affected by the dynamics of the expanding system.",
"These harmonics have been closely studied so far [9], [10], [8].",
"The pentagonal and hexagonal flows $v_{5}$ and $v_{6}$ respectively are studied to a lesser extent may be due to lack of experimental evidences or so, some predictions from hydrodynamics on them also have been made [11].",
"At relatively low $p_{T}$ , pressure driven anisotropic expansion of the created matter results in the azimuthal anisotropy, emitting more particles in the direction of large pressure gradients [12].",
"At higher transverse momentum $p_{T}$ , the anisotropy is explained using the path-length dependent energy loss of partonic jets as they traverse the matter, emitting more jet particles in the direction of shortest path-length [13].",
"Anisotropic flow develops in the system because the spatial anisotropies $\\epsilon _{n}$ of the overlapping region are transformed into the momentum anisotropies $v_{n}$ of final hadronic distribution.",
"Due to the initial state fluctuations, these spatial anisotropies exist even in very central collisions.",
"Momentum anisotropy also arises from the non-isotropic azimuthal dependence of the transverse velocity of the expanding fireball resulting in collective flow gradients in various directions.",
"These two different sources of the particle momentum anisotropy are called as geometric and dynamical anisotropy, respectively.",
"It might be possible to disentengle these anisotropies by the simultaneous analysis of the flow harmonics, which we aim to perform in our present work.",
"Recently, we performed our study on Xe-Xe collisions, where we emphasized on the motto of performing our analysis on xenon-xenon collision systems [14].",
"The similarities observed between smaller systems, such as p + p and p + Pb and larger Pb-ion systems are debatable as to whether they arise from same physics mechanism.",
"Here, again we will deal with body-body and tip-tip collision configurations [15].",
"The charged particle multiplicity density in the transverse phase space is higher in deformed collision systems than spherical or non-deformed nucleus collisions [16], [17].",
"Lately, HYDJET++ model was modified to study U-U collisions at 193 GeV center-of-mass energy in body-body and tip-tip geometrical configurations [18].",
"Anisotropic flow harmonics and the correlations between them have been studied intensively at both RHIC and LHC energies.",
"Recently, [19], [20], predictions were made for Xe-Xe collisions using event-by-event hydrodynamic simulations discussing anisotropic flow coefficients as a function of centrality and comparing the results with the ATLAS, CMS, ALICE experimental measurements.",
"From the results of anisotropic flow coefficients for Xe-Xe and Pb-Pb collisions, it was confirmed that xenon has a non-spherical shape.",
"In references [21], [22], [23], anisotropic flow coefficients studies were performed in O-O, Al-Al, and Cu-Cu collisions under Color Gluon Condensate (CGC) framework or under AMPT model (a multiphase transport model) approach or in the fusing color string model.",
"Here, a thorough description of bulk observables and multi-particle correlations in several collision systems such as Au+Au, U+U, Ru+Ru, Zr+Zr, and O+O collisions at top RHIC energies and Pb+Pb, Xe+Xe, and O+O collisions at LHC energies are performed.",
"In reference [24], measurements of elliptic and triangular azimuthal anisotropy of charged particles in Au+Au collisions were presented using the multiparticle cumulant technique to study the centrality dependence of $v_{2}$ and $v_{3}$ along with the significances of initial geometrical fluctuations and their translation into the final state momentum distributions.",
"An investigation was performed in Xe-Xe collision systems on the system-size dependence of the longitudinal decorrelations of $v_{2},v_{3}$ and $v_{4}$ and comparing the results with Pb-Pb collisions at 5.02 TeV [25].",
"Recently, measurements of anisotropic flow harmonic coefficients ($v_{n}$ ) for inclusive charged particles and identified hadrons were performed in Cu+Au (asymmetric) collisions at 200 GeV [26] where mass ordering in hydrodynamic flow and the particle azimuthal distributions were studied as a function of $p_{T}$ over various centrality classes at RHIC energies.",
"In another work, mass dependency of hadrons (pions, kaons and protons) for elliptic flow $v_{2}$ ($p_{T}$ ) is observed similar to the observations from Pb-Pb collisions [27].",
"Such study has not been performed for the higher flow harmonics.",
"Correlation between anisotropic flow coefficients $v_{n}$ and average transverse momentum $\\langle p_{T} \\rangle $ of outgoing particles in Pb + Pb collisions is studied at LHC energy tracing back to the initial density profile, i.e., to the early stages of the collision [28].",
"A lot of study in the above mentioned aspect has been done to understand higher flow harmonics in spherical collision systems at RHIC as well as LHC energies but deformed systems have not been touched much till now, especially visualisation in various geometrical configurations.",
"Azimuthal correlations provide valuable information about the relativistic hydrodynamic nature of the medium, about its transport coefficients, and also about the fluctuations in the initial state from which the medium is formed [29].",
"They are extensively studied as a function of centrality of collision and transverse momentum $p_{T}$ [30], [31], produced particle type, rapidity, and expected event-by-event geometrical fluctuations of the nuclei [32].",
"It is to be noted that [9], [33] in the absence of event-by-event fluctuation (involving hydrodynamics with smooth initial condition), the even flow harmonic coefficients are found to be correlated.",
"The reason being that despite the fluid velocity profile is elliptically deformed, complete set of the even flow harmonic coefficients is generated in general as fluid velocity enters as the exponent of the (flow-boosted) thermal distribution on the freeze-out surface.",
"When these event-by-event fluctuations in the initial state are present, the resulting flow fluctuations of different harmonic orders are generally correlated by the geometric constraints on the shapes and positions of these fluctuations within the overlapping spatially deformed region.",
"The availability of experimental data from various collision programmes gives a motivation to perform a study whether the experimentally measured anisotropic flow correlations and their dependence on centrality of collision can be understood and described well through a successful hydrodynamical model approach.",
"In our study of disentangling both geometrical and dynamical anisotropies and performing analysis of the higher anisotropic flow coefficients we choose HYDJET++ Model framework [34].",
"This model allows to switch on/off both the anisotropy parameters independently.",
"The production of higher flow harmonics within HYDJET++ framework is advantageous in the sense that the interplay of ideal hydrodynamics with jets reveal the role of hard processes in the production of secondary hadrons.",
"Also, the existence of $v_{2}$ and $v_{3}$ allows us to understand the contribution of these to all other higher odd and even anisotropic flow coefficients [35].",
"In reference [36], HYDJET++ model was used to study the LHC data on multiplicity, charged hadron spectra, elliptic flow and femtoscopic correlations in Pb-Pb collisions.",
"Considering both soft as well as hard components along with the tuning of the parameters, we can reproduce the experimental data under HYDJET++.",
"In this paper, we have studied the centrality, transverse momentum and total charged particle multiplicity dependence of anisotropic flow coefficients in Xe-Xe collisions at 5.44 TeV center-of-mass energy.",
"$m_{T}$ dependence of anisotropic flow harmonics is visualized here.",
"The correlation between these coefficients is an interesting part of this work.",
"The analysis of our results have been performed in body-body and tip-tip geometrical configurations using the HYDJET++ model.",
"In Sec.",
"II, we have briefly discussed formulation of the HYDJET++ model and the incorporation of deformation in the body of the model.",
"Also, we discuss how the model incorporates the higher flow harmonics in the body of the model.",
"In Sec.",
"III, we present the results and discussions part for the elliptic flow $v_{2}$ , triangular flow $v_{3}$ and quadrangular flow $v_{4}$ distributions.",
"Lastly, we have summarized our results in Sec.",
"IV."
],
[
"Model Formalism",
" HYDJET++ (HYDrodynamics plus JETs) is a Monte Carlo model of relativistic heavy ion collisions which includes the simultaneous superposition of two independent components: the soft hydro-type state and the hard state resulting from the medium-modified multiparton fragmentation.",
"The details of the model and the corresponding simulation procedure can be found in the paper [34], [37] and the references there within.",
"The model parameters have been tuned to reproduce the experimental LHC data on various physical observables measured in Xe-Xe collisions at 5.44 TeV of center-of-mass energy per nucleon pair.",
"A concise view about the physics of the model valuable for our study has been presented in our previous article [14] where we have discussed about both hard as well as soft part of the model.",
"In there, we performed elliptic flow studies showing the transverse momentum ($p_{T}$ ) and centrality dependence of elliptic flow $v_{2}$ but did not work on higher anisotropic fourier harmonics.",
"To simulate higher azimuthal anisotropy harmonics, various alterations were required and have been made in HYDJET++.",
"Basically, the model does not involve the evolution of fireball from the initial state to the final state freeze-out stage.",
"It utilizes simple and often used parameterization of the freeze-out hypersurface rather than using computational relativistic hydrodynamics (time consuming).",
"The anisotropic elliptic shape of the initial overlap of the colliding nuclei results in a corresponding anisotropy of the outgoing momentum distribution.",
"The second harmonic $v_{2}$ is described using the coefficients $\\epsilon _{2}$ (b) and $\\delta _{2}$ (b) known as the spatial anisotropy and momentum anisotropy, respectively.",
"$\\epsilon _{2}$ (b) exemplifies the elliptic modulation of the final freeze-out hypersurface at a given impact parameter b, whereas $\\delta _{2}$ (b) deals with the alteration of flow velocity profile.",
"These two parameters can be treated independently for each centrality or can be made interdependent via the dependence on the initial ellipticity $\\epsilon _{0}(b)=b/2R_{A}$ where $R_{A}$ is the nucleus radius.",
"Here, we are treating them independent of each other.",
"The transverse radius of the fireball is given as: $R_{ell}(b,\\phi )=R_{f}(b)\\frac{\\sqrt{1-\\epsilon _{2}^{2}(b)}}{1+\\epsilon _{2}(b)\\cos 2\\phi },$ where, $R_{f}(b)=R_{0}\\sqrt{1-\\epsilon _{2}(b)}.$ Here $R_{0}$ denotes is the freeze-out transverse radius in absolute central collision with b=0.",
"Then, the spatial anisotropy gets transformed into momentum anisotropy at freeze-out, because each of the fluid cells is carrying some momentum.",
"The term dynamical anisotropy arises here implying that the azimuthal angle of the fluid cell velocity, $\\phi _{cell}$ does not coincide with the azimuthal angle $\\phi $ , instead correlates with it [38] through the non-linear function involving the anisotropy parameter $\\delta _{2}(b)$ $\\frac{\\tan \\phi _{cell}}{\\tan \\phi }=\\sqrt{\\frac{1-\\delta _{2}(b)}{1+\\delta _{2}(b)}}.$ In case where $\\delta \\ne 0$ even the spherically symmetric source can mirror the spatially contracted one.",
"The elliptic flow coefficient $v_{2}(\\epsilon , \\delta _{2})$ in the hydrodynamical approach [39] is given as: $v_{2}(\\epsilon _{2},\\delta _{2}) \\propto \\frac{2(\\delta _{2}-\\epsilon _{2})}{(1-\\delta _{2}^{2})(1-\\epsilon _{2}^{2})}.$ For triangular flow $v_{3}$ in HYDJET++, the model has another parameter $\\epsilon _{3}$ (b),for spatial triangularity of the fireball.",
"Thus the modified radius of the freeze-out hypersurface in azimuthal plane reads: $R(b,\\phi )=R_{ell}(b)\\lbrace 1+\\epsilon _{3}(b)\\cos [3(\\phi -\\psi _{3}^{RP})]+...\\rbrace .$ where, $\\phi $ = spatial azimuthal angle of the fluid element relatively to the direction of the impact parameter.",
"The phase $\\psi _{3}^{RP}$ gives us the advantage to introduce a third harmonic having its own reaction plane, distributed randomly with respect to the direction of the impact parameter($\\psi _{2}^{RP}=0$ ).",
"This new anisotropy parameter, $\\epsilon _{3}(b)$ again can be handled in two ways: independently for each centrality and dependent using $\\epsilon _{0}(b)=b/2R_{A}$ where $R_{A}$ has its meaning unchanged.",
"Such modifications do not affect the elliptic flow (controlled by $\\epsilon (b)$ and $\\delta (b)$ ).",
"Hence, the triangular dynamical anisotropy can be incorporated by the parameterization of the maximal transverse flow rapidity [23], $\\rho _{u}^{max}(b)=\\rho _{u}^{max}(0)\\lbrace 1+ \\rho _{3u}(b)\\cos [3(\\phi -\\psi _{3}^{RP})] +...\\rbrace .$ As a result, the maximal transverse flow rapidity [34] after the parameterization of the four-velocity $u$ upto the fourth order harmonics is given as, $\\rho _{u}^{max}(b)=\\rho _{u}^{max}(0)\\lbrace 1+ \\rho _{3u}(b)\\cos 3\\phi + \\rho _{4u}(b)\\cos 4\\phi +...\\rbrace .$ Hence, we can calculate higher harmonics with respect to the direction of the impact parameter $b \\psi _{2}^{RP}=0$ .",
"Again, these new anisotropy determiners $\\rho _{3u}(b)$ and $\\rho _{4u}(b)$ can be treated both independently and dependent via initial ellipticity $\\epsilon _{0}(b)=b/2R_{A}$ .",
"Now, here we opted the former case and treated the parameters independently and varied them with centrality.",
"The next important part of the HYDJET++ model is the incorporation of the intrinsic deformation in Xe nucleus.",
"This has been already done in our previous work [14], where we performed our study in both tip-tip and body-body geometrical configurations making our modified HYDJET++ model work at both RHIC as well as LHC energies.",
"After implementing higher fourier harmonics the simulation and the optimization of the parameters is verified.",
"We obtain results similar to the figures 2 and 3 in our previous work [14] thereby certifying our HYDJET++ model simulations in both tip-tip and body-body type of geometrical configurations."
],
[
"Results and Discussions",
"We have generated $5 \\times 10^5$ events using the modified HYDJET++ model in different centrality classes for both tip-tip and body-body configurations at 5.44 TeV center of mass energy.",
"We have performed our simulations for n$\\le $ 4 as the model has been designed upto that only.",
"Only those events have been considered for the results which fall in the kinematic range $|\\eta |<0.8$ and $0<p_{T}<5$ GeV/c.",
"In our previous work on Xe-Xe collision systems [14], it was demonstrated that tuned HYDJET++ model can reproduce the LHC data on centrality and transverse momentum dependence of charged particle multiplicity density, transverse momentum $p_{T}$ spectra and elliptic flow coefficient $v_{2}$ up to $p_{T}\\sim $ 2.0 GeV/c and 60% centrality range).",
"However, the reasonable treatment of higher Fourier harmonics of particle azimuthal distribution $v_{n}$ (n$>$ 2) needs additional modifications in the model, which does not affect our previous results.",
"This is evident from figure REF .",
"We have compared the results of HYDJET++ simulations with the LHC (ALICE) experimental data [40] on $v_{n} \\lbrace 2\\rbrace $ second order cumulant for inclusive as well as for identified charged hadrons for our analysis.",
"Figure REF presents the variation of azimuthal anisotropy fourier harmonics ($v_{2},v_{3}$ and $v_{4}$ ) in Xe-Xe collision systems at 5.44 TeV with centrality.",
"The model results in minimum bias have been compared with the ALICE experimental data [40].",
"A strong centrality dependence of elliptic, triangular and quadrangular flows is observed here.",
"There is a fair agreement of the HYDJET++ model results with the ALICE experimental data both qualitatively as well as quantitatively.",
"As we move from most central to most peripheral collisions, elliptic flow increases and then decreases in most peripheral collisions.",
"Similar behaviour is shown by triangular flow but the fall is seen here early (centrality $>$ 40%).",
"However, quadrangular flow shows a gradual increase as we move from most central to most peripheral class of collisions.",
"The elliptic flow $v_{2}$ results from HYDJET++ match very well with the experimental data in all centrality windows except in most peripheral collisions where the model overpredicts the data.",
"The triangular flow $v_{3}$ shows a good agreement with the experimental data in central collisions.",
"As we move from most central, towards semi-peripheral collisions (centrality $<40\\%$ ), the deviation from the experimental results is observed (model underpredicts the experimental data).",
"However, the deviation decreases as we move towards most peripheral collisions (centrality $>40\\%$ ).",
"Lastly, as we move from most central to most peripheral collisions the quandrangular flow $v_{4}$ results from HYDJET++ show a suitable match with the ALICE experimental result at all centrality classes of Xe-Xe collisions.",
"In figure REF , we have compared our model results in body-body and tip-tip collisions with the ALICE experimental data for $v_{2},v_{3}$ and $v_{4}$ .",
"The anisotropic harmonic coefficients obtained from our HYDJET++ model in both the geometrical configurations show strong centrality dependence.",
"The qualitative behaviour of the two geometrical configurations is similar to the the one presented in figure REF .",
"In case of elliptic flow $v_{2}$ , we find that our results agree with experimental results both quantitatively as well as qualitatively.",
"The body-body collision results are higher than tip-tip collision results.",
"However, in most central collisions, there is hardly any difference between the two geometrical configurations.",
"But this difference is clearly visible as we move towards peripheral collisions.",
"In most peripheral collisions, model results in the two geometrical configurations are higher than data thereby overpredicting the experimental result.",
"Moving to triangular flow $v_{3}$ results, our model results for the two geometrical configuartions match ALICE results qualitatively.",
"However, in a closer view, we find that quantitatively HYDJET++ results for the two geometrical configurations underpredict the experimental data.",
"Body-body collision results are higher than tip-tip collision results.",
"The two geometrical configurations cannot be disentangled in most central and most peripheral collisions but the difference can be seen very clearly in semi-peripheral collisions.",
"The HYDJET++ results for quadrangular flow $v_{4}$ in body-body and tip-tip collisions show a suitable match with the experimental data qualitatively.",
"In central collisions (centrality $<20\\%$ ), it is difficult to differentiate between the two geometrical configurations.",
"This can be accomplished as we move towards peripheral collisions.",
"A bump appears in between (20%-30%) class of collision centrality in the ALICE experimental data.",
"This bump can be seen in our HYDJET++ model results, although not so prominent.",
"Again, body-body collision results are higher than tip-tip colision results.",
"Figure REF presents the elliptic flow $v_{2}$ of identified charged particles with respect to transverse momentum in four classes of centrality.",
"As a function of centrality, elliptic flow $v_{2}$ increases as we move from most central collisions to semi-peripheral collisions and then starts to decrease as we enter region of most peripheral ((50-60)%) class of collision.",
"Mass ordering is observed here in each class of collision.",
"At low $p_{T}$ , ($p_{T}<$ 1.6 GeV/c) $v_{2}$ for the lower mass particle (pions) is more than the higher mass particles.",
"In simple words, $v_{2}^{\\Pi }>v_{2}^{K}>v_{2}^{p}$ where $m^{\\Pi }<m^{K}<m^{p}$ .",
"However, for $p_{T}\\ge $ 1.6 GeV/c the situation changes.",
"As we move from most central to peripheral collisions, mass ordering reverses.",
"Higher mass particles are produced more.",
"$v_{2}^{\\Pi }<v_{2}^{K}<v_{2}^{p}$ where $m^{\\Pi }<m^{K}<m^{p}$ .",
"In most peripheral collisions, the scenario appears completely different.",
"At $p_{T}>$ 1.6 GeV/c $v_{2}^{K}>v_{2}^{\\Pi }>v_{2}^{p}$ where $m^{\\Pi }<m^{K}<m^{p}$ .",
"This cut value of transverse momentum may vary from centrality to centrality.",
"At much higher $p_{T}\\ge 2.8\\pm 0.2$ , $v_{2}^{p}>v_{2}^{K}>v_{2}^{\\Pi }$ in most central and semi-peripheral class of collisions.",
"This is absent in peripheral collisions and $v_{2}^{p}$ is found to be lesser.",
"Figure REF shows transverse momentum dependence of triangular flow $v_{3}$ for identified charged particles in four classes of centrality.",
"Triangular flow $v_{3}$ increases as we move from most central collisions and decreases as we move towards most peripheral class of collisions.",
"Mass ordering is observed here in each class of collision.",
"At low $p_{T}$ , ($p_{T}<$ 1.5 GeV/c) $v_{3}$ for the lower mass particle (pions) is more than $v_{3}$ for higher mass particles.",
"In simple words, $v_{3}^{\\Pi }>v_{3}^{K}>v_{3}^{p}$ where $m^{\\Pi }<m^{K}<m^{p}$ .",
"However, for $p_{T}\\ge $ 1.5 GeV/c the situation changes.",
"As we move from most central to peripheral collisions, mass ordering reverses.",
"Higher mass particles are produced more.",
"$v_{3}^{\\Pi }<v_{3}^{K}<v_{3}^{p}$ where $m^{\\Pi }<m^{K}<m^{p}$ .",
"In most peripheral collisions, the scenario appears different.",
"At $p_{T}>$ 1.7 GeV/c $v_{3}^{K}>v_{3}^{\\Pi }>v_{3}^{p}$ where $m^{\\Pi }<m^{K}<m^{p}$ .",
"At much higher values of $p_{T}\\ge 2.8\\pm 0.2$ , again $v_{2}^{p}>v_{2}^{K}>v_{2}^{\\Pi }$ in most central and semi-peripheral class of collisions, and in peripheral collisions $v_{2}^{p}$ is found to be lesser.",
"Figure REF presents the quadrangular flow $v_{4}$ of identified charged particles with respect to transverse momentum in various centrality windows.",
"Quadrangular flow $v_{4}$ increases as we move from most central collisions and decreases as we move towards most peripheral class of collisions.",
"Mass ordering is observed here in each class of collision.",
"At low $p_{T}$ , ($p_{T}<$ 1.7 GeV/c) $v_{4}$ for the lower mass particle (pions) is more than $v_{4}$ for higher mass particles.",
"In simple words, $v_{4}^{\\Pi }>v_{4}^{K}>v_{4}^{p}$ where $m^{\\Pi }<m^{K}<m^{p}$ .",
"However, for $p_{T}>$ 1.7 GeV/c the situation changes.",
"As we move from most central to peripheral collisions, mass ordering reverses.",
"Higher mass particles are produced more.",
"$v_{4}^{\\Pi }<v_{4}^{K}<v_{4}^{p}$ where $m^{\\Pi }<m^{K}<m^{p}$ .",
"In most peripheral collisions, the scenario appears different.",
"At $p_{T}>$ 1.7 GeV/c $v_{4}^{K}>v_{4}^{\\Pi }>v_{4}^{p}$ where $m^{\\Pi }<m^{K}<m^{p}$ .",
"Similar to the observations in figures REF and REF , here too we have $v_{2}^{p}>v_{2}^{K}>v_{2}^{\\Pi }$ in most central and semi-peripheral collisions at $p_{T}\\ge 2.8\\pm 0.2$ , this being absent in peripheral class of collisions.",
"Thus, from the above anaysis of the results in figures REF , REF , and REF we conclude that the $p_{T}$ cut value is different for different flow coefficients but the qualitative behaviour of the identified particles and mass ordering of $v_{n}$ for n=2,3,4 is similar.",
"In figure REF , we present the variation of minimum bias $v_{n}$ with total charged particle multiplicity.",
"Here we have compared our HYDJET++ model results with the ALICE experimental data [40].",
"The qualitative behaviour for different flow coefficients, $v_{2},v_{3}$ and $v_{4}$ is similar to ALICE experimental results.",
"The elliptic flow $v_{2}$ decreases gradually as the total charged particle multiplicity increases.",
"In most peripheral class of collisions, the model results are in very close agreement with the experimental data.",
"As we move towards most central class of collisions, the observed deviation from the experiment results increases.",
"The triangular flow $v_{3}$ shows a linear increase and then falls gently as we move to higher values of total charged particle multiplicity.",
"A soft peak is observed at $n_{ch}\\approx 400$ .",
"Quantitatively, our HYDJET++ results underestimate the ALICE experimental data in all classes of collisions except in most central collisions where our HYDJET++ model results overestimate the experimental results.",
"The plot of quadrangular flow $v_{4}$ with respect to the total charged particle multiplicity shows that the $v_{4}$ decreases as we move from most peripheral to most central class of collisions.",
"However, due to lack of results in some more centrality classes, we do not obtain the exactly similar behaviour.",
"But we can say that our HYDJET++ results match the experimental results quantitatively with least errors.",
"Figure REF presents the correlation between $v_{3}$ and $v_{2}$ for seven centrality windows from our HYDJET++ model and from experimental data [40] for Xe-Xe collisions at the LHC.",
"Here, we observe a qualitative agreement of our model results with the experimental data.",
"The upper panel of the figure shows the comparison of minimum bias results with the experimental data.",
"In central collisions, a linear positive correlation is observed between $v_{3}$ and $v_{2}$ .",
"Also, our model results show suitable match with the experimental data quantitatively.",
"However, in mid-central or semi-peripheral collisions, HYDJET++ results underpredict the data quantitatively.",
"The correlation between $v_{3}$ and $v_{2}$ is not very much positive due to the reason that our model fails to predict $v_{3}$ in these collision centralities.",
"As we move towards peripheral collisions, again positive correlation is seen between $v_{3}$ and $v_{2}$ .",
"In most peripheral collisions, the situation changes, a sharp negative correlation is predicted which is in good agreement with the ALICE experimental results qualitatively.",
"Quantitatively, there exists a clear deviation of our model results from the experimental results in most peripheral collisions.",
"This is because HYDJET++ model fails to handle such collision centralities.",
"Hence, the correlation structure between $v_{3}$ and $v_{2}$ as a function of centrality might be attributed to the fact that $v_{3}$ has weaker centrality dependence as compared to $v_{2}$ .",
"The lower panel of figure REF shows the correlation between $v_{3}$ and $v_{2}$ for seven classes of centrality from our HYDJET++ model in body-body and tip-tip geometrical configurations.",
"These results have been compared with the ALICE experimental data [40] where a complete agreement is observed between model and experiment qualitatively.",
"Quantitatively, body-body results are higher than tip-tip results.",
"In central collisions, the correlation between $v_{3}$ and $v_{2}$ in both the cases is similar to the above described for the upper panel results.",
"However, in mid-central or semi-peripheral collisions, body-body and tip-tip collision results show a positive correlation between $v_{3}$ and $v_{2}$ , tip-tip being weaker than body-body collisions.",
"This correlation indicates a similar correlation between the initial eccentricities $\\epsilon _{3}$ and $\\epsilon _{2}$ [41], [42].",
"Such inference is expected because the hydrodynamic response of $v_{3}$ and $v_{2}$ to $\\epsilon _{3}$ and $\\epsilon _{2}$ , respectively, is linear approximately, especially at small eccentricities.",
"A small dip is observed in the ALICE experimental result as the correlation changes from positive to negative (as we move from peripheral to most-peripheral collisions).",
"This dip is not so prominent in minimum bias and body-body collision results but can be seen in tip-tip collisions.",
"Figure REF depicts the correlation between $v_{4}$ and $v_{2}$ over seven centrality classes from Xe-Xe collisions at $\\sqrt{s_{NN}}$ =5.44TeV.",
"We have compared our model results with ALICE experimental data [40].",
"We observe a qualitative agreement of our model results with the experimental data.",
"Quadrangular flow $v_{4}$ strongly increases as elliptic flow $v_{2}$ increases.",
"This inference is supported well by the work done in article [41].",
"The upper panel of the figure compares minimum bias results with the ALICE experimental data.",
"Our HYDJET++ model predicts somewhat a non-linear positive correlation between $v_{4}$ and $v_{2}$ throughout centrality which is in fair agreement with ALICE experimental data.",
"HYDJET++ successfully produces minimum bias results qualitatively but overestimates quantitatively in central collisions.",
"Moving further, we see that $v_{4}$ decreases and then increases leading to a peak in semi-peripheral collisions.",
"This peak is not so prominent in HYDJET++ results as compared to experiment.",
"Further, as we move from semi-peripheral to most peripheral class of collisions experimental results appear a bit complex, showing a positive correlation followed by a fall (negative correlation) and then a sudden sharp rise (positive correlation).",
"Thus, a bump and a dip is observed here.",
"This bump is visible in our results while the dip is not observed here.",
"HYDJET++ model results again produce a positive correlation between $v_{4}$ and $v_{2}$ and underestimates the experiment in this region thereby failing to explain the ALICE experimental result in such centrality region.",
"The lower panel of figure REF shows the correlation between $v_{4}$ and $v_{2}$ over seven classes of centrality from our HYDJET++ model in body-body and tip-tip geometrical configurations along with ALICE experimental data [40] for comparison.",
"Again, a complete agreement is observed between model and experiment qualitatively.",
"In central collisions, body-body and tip-tip collision results are indistinguishable and overestimate the ALICE experimental data.",
"As we move towards semi-peripheral collisions, the two geometrical configurations can be distinguished.",
"Quantitatively, body-body results are higher than tip-tip results in this region.",
"As we move from central to most peripheral class of collisions, results from body-body and tip-tip collisions fail to explain the experiment (underestimate the ALICE results).",
"The bump is more clear in body-body results than in tip-tip collisions appearing early in tip-tip results.",
"Figure REF presents the correlation between $v_{4}$ and $v_{3}$ for seven centrality windows from our HYDJET++ model and from experimental data [40] for Xe-Xe collisions at the LHC energy.",
"Here, we observe a qualitative agreement of our model results with the experimental data.",
"The upper panel of the figure depicts the comparison of minimum bias results with the experimental data.",
"HYDJET++ model results overpredict the data quantitatively in these centralities.",
"In central collisions, a positive linear correlation is observed between $v_{4}$ and $v_{3}$ .",
"Moving towards higher centralities, we observe the correlation between $v_{4}$ and $v_{3}$ to be a boomerang like.",
"Such inference is in strong agreement with the ALICE experimental data.",
"However, model results strongly deviate from the experiment quantitatively.",
"This is because of the fact that HYDJET++ model for the triangular flow underestimates experimental results in this region or fails to handle such collision centralities.",
"Similar behaviour is observed for Xe-Xe collisions in body-body and tip-tip geometrical configuration, shown in lower panel of figure REF .",
"These results have been compared with the ALICE experimental data [40] where a complete agreement is observed between model and experiment qualitatively.",
"Quantitatively, model results overestimate the experimental data.",
"Body-body results are higher than tip-tip results except in central collisions where the two overlap and thus it is not possible to disentangle body-body configuration from tip-tip.",
"In central collisions, the correlation between $v_{4}$ and $v_{3}$ in both the cases is similar to the above described for the upper panel results.",
"The HYDJET++ model results show a qualitative deviation from experimental results because the triangular flow $v_{3}$ underestimates the experimental data quantitatively in the region between mid-central to peripheral collisions."
],
[
"Summary and Outlook",
"In a brief summary of our work, we have made a scrupulous study of azimuthal anisotropic fourier harmonic coefficients in xenon-xenon collision systems at $\\sqrt{s_{NN}}$ = 5.44-TeV LHC energies performed under the framework of the modified HYDJET++ model, providing the possibility to study the collisions in various geometrical configurations which being cognizant of the initial conditions.",
"Here, we have used tip-tip and body-body configurations for our analysis considering only those events which fall in the kinematic range $|\\eta |<0.8$ and $0<p_{T}<2$ GeV/c and the results have been compared with ALICE experimental data.",
"Our model results show a suitable match with the ALICE experimental data both quantitatively and qualitatively thereby enlighting both geometrical and dynamical anisotropies of the system, respectively.",
"We observe a strong centrality dependence of the azimuthal anisotropic harmonic coefficients $v_{2}$ and $v_{3}$ but a weak dependence of $v_{4}$ on collision centrality.",
"In a recent article [23], O-O, Al-Al, and Cu-Cu collisions at 200 GeV/c as a function of centrality were studied where we observed that $v_{2}$ has a weak centrality dependence whereas both $v_{3}$ and $v_{4}$ fall and rise with centrality.",
"Also, the quadrangular flow $v_{4}$ is observed to be larger than the triangular flow $v_{3}$ , as a function of centrality.",
"But our results are contrary to these where $v_{2}>v_{3}>v_{4}$ as a function of centrality.",
"The minimum bias HYDJET++ model results are consistent with the ALICE experimental data.",
"Elliptic flow results underpredict the experimental data in most peripheral class of collisions whereas triangular flow underpredicts the experimental results except in most central and most peripheral class of collisions.",
"However, quadrangular flow results for the HYDJET++ model are consistent with the data throughout collision centrality.",
"Our model results in body-body and tip-tip collisions show strong dependence on collision centrality.",
"Body-body collision results are higher than tip-tip collision results.",
"It is possible to disentangle the two geometrical configurations in various classes of collisions.",
"For elliptic flow, the two geometrical configuarations can be observed deviating from each other as we move from most central to most peripheral class of collisions.",
"In case of triangular flow, the two geometrical configurations can be disentangled in all classes of collisions except in most central and most peripheral collisions.",
"Similar is the case of quadrangular flow where body-body and tip-tip collision results overlap in most central (0%-5%) and in most peripheral (50%-60%) class of collisions.",
"Anisotropic flow of identified charged particles with respect to transverse momentum is studied.",
"Again centrality dependence of elliptic, triangular and quadrangular flows is observed.",
"Mass ordering is observed in case $v_{2},v_{3}$ and $v_{4}$ .",
"At low $p_{T}$ , ($p_{T}<p_{T}^{cut}$ GeV/c) $v_{n}$ for the lower mass particle (pions) is more than the higher mass particles.",
"However, for $p_{T}>p_{T}^{cut}$ , as we move from most central to peripheral collisions higher mass particles are produced more.",
"In most peripheral collisions, the situation is different.",
"At $p_{T}>p_{T}^{cut}$ $v_{2}^{K}>v_{2}^{\\Pi }>v_{2}^{p}$ where $m^{\\Pi }<m^{K}<m^{p}$ .",
"The $p_{T}^{cut}$ values are different for $v_{2},v_{3}$ and $v_{4}$ being 1.6$\\pm 0.2$ GeV/c, 1.5$\\pm 0.4$ GeV/c, and 1.7$\\pm 0.2$ GeV/c, respectively.",
"At much higher transverse momenta, $p_{T}>2.8\\pm 0.2$ , $v_{2}^{p}$ is quite higher, the order of flow being $v_{2}^{p}>v_{2}^{K}>v_{2}^{\\Pi }$ .",
"The variation of minimum bias $v_{n}$ with respect to the total charged particle multiplicity is presented.",
"Qualitatively, the behaviour for different flow coefficients, $v_{2},v_{3}$ and $v_{4}$ is similar to ALICE experimental results, showing a strong dependence on the total charged particle multiplicity.",
"Quantitatively, the model almost underpredicts the experimental data.",
"The correlation between the different azimuthal anisotropic coefficients is also studied.",
"Qualitative agreement of our model results with the experimental data is observed.",
"Positive linear correlation is observed between $v_{3}$ and $v_{2}$ in central collisions.",
"However, in mid-central or semi-peripheral collisions, correlation between $v_{3}$ and $v_{2}$ is not very much positive whereas in most peripheral collisions, a sharp negative correlation is observed having good agreement with the ALICE experimental results qualitatively.",
"Results for both body-body and tip-tip collisions has also been presented.",
"Quantitatively, body-body results are higher than tip-tip results.",
"The correlation between $v_{3}$ and $v_{2}$ in both the geometrical configurations is similar to the experimental results in central collisions.",
"In central collisions, the two geometrical configurations are indistinguishable whereas as we move towards peripheral collisions it is very much possible to disentangle body-body collisions from tip-tip collisions.",
"A qualitative agreement of HYDJET++ model results with the ALICE experimental data is observed for the correlation between $v_{4}$ and $v_{2}$ .",
"Positive linear correlation between $v_{4}$ and $v_{2}$ is seen throughout centrality.",
"HYDJET++ results overestimate experimental measurements quantitatively as we move from most central to semi-peripheral collisions.",
"As we move from semi-peripheral to most peripheral collisions ALICE experimental results show a positive correlation followed by a fall (negative correlation) and then a sudden sharp rise (positive correlation).",
"However, our model results show a positive correlation between $v_{4}$ and $v_{2}$ and underestimate the experimental results.",
"Hence, failing to explain such ALICE experimental result in this region.",
"In case of the two geometrical configurations, a suitable agreement is observed between model and experiment qualitatively.",
"Quantitatively, body-body results are higher than tip-tip results in semi-peripheral collision region where the two configurations can be distinguished.",
"As we move towards most peripheral collisions, body-body results being higher than tip-tip collision fail to explain the experimental data (underpredict the data).",
"The correlation between $v_{4}$ and $v_{3}$ for Xe-Xe collisions show a qualitative agreement of our model results with the ALICE experimental data.",
"A positive linear correlation is observed between $v_{4}$ and $v_{3}$ in central collisions.",
"As we move towards higher centralities, the correlation between $v_{4}$ and $v_{3}$ is observed to be a boomerang like.",
"This is in strong agreement with the ALICE experimental data.",
"Quantitatively, HYDJET++ model results strongly deviate from the experiment.",
"Similar behaviour is observed in case of body-body and tip-tip collisions.",
"Body-body collision results are higher than tip-tip results.",
"The HYDJET++ model results show a qualitative deviation from ALICE experiment due to the reason that triangular flow $v_{3}$ underestimates the experimental data quantitatively in the region between mid-central to peripheral collisions.",
"the two geometrical configurations are inseperable in central collisions but can be easily differentiated as we move towards most peripheral collisions.",
"Thus, we disintegratd our $v_{n}$ -$v_{m}$ correlations into linear and non-linear contributions having strong dependence on centrality and showing strong agreement with ALICE experiment qualitatively and in some regions quantitatively.",
"Also, at such stages we are quite successful in disentangling the geometrical configurations.",
"Our analysis also showers some light on the geometrical and dynamical anisotropies of the system.",
"This non-linear correlation contribution between the anisotropic coefficients may be attributed to the elliptic geometric deformation of the nuclear overlap region in non-central Xe-Xe collision systems and is visualized here since the nuclear overlap region is elliptically deformed even in most central colisions (at b=0).",
"Further higher harmonic coefficients ($n\\ge 5$ ) can also be studied under HYDJET++ framework in Xe-Xe collision systems but due to lack of experimental evidences, the problem is a bit tacky.",
"The study can be performed in a way by modifying the model for such higher azimuthal anisotropic harmonics and then comparing the results with inferences predicted from various thermodynamical models [43], [44].",
"We leave this part for our future work."
],
[
"ACKNOWLEDGEMENTS",
" We sincerely acknowledge financial support from the Institutions of Eminence (IoE) BHU grant.",
"SP acknowledges the financial support obtained from UGC under research fellowship scheme during the work.",
"Figure: From article Figure: Centrality dependence of v n v_{n} along with ALICE experimental data .Figure: Centrality dependence of v n v_{n} for body-body and tip-tip geometrical configurations along with ALICE experimental data for comparison.Figure: Transverse momentum dependence of v 2 v_{2} for identified particles in different centrality windows.Figure: Transverse momentum dependence of v 3 v_{3} for identified particles in different centrality classes.Figure: Transverse momentum dependence of v 4 v_{4} for identified particles in different centrality windows.Figure: Variation of minimum bias v n v_{n} with total charged particle multiplicity along with ALICE experimental data for comparison.Figure: Correlation between v 3 v_{3} and v 2 v_{2} for p T <p_{T}<2 GeV/c over seven centrality classes in Xe-Xe collisions at 5.44 TeV.",
"The results have been compared with ALICE experimental data .Figure: Correlation between v 4 v_{4} and v 2 v_{2} for p T <p_{T}<2 GeV/c over seven centrality classes in Xe-Xe collisions at 5.44 TeV.",
"The results have been compared with ALICE experimental data .Figure: Correlation between v 4 v_{4} and v 3 v_{3} for p T <p_{T}<2 GeV/c over seven centrality classes in Xe-Xe collisions at 5.44 TeV.",
"The results have been compared with ALICE experimental data ."
]
] | 2107.01880 |
[
[
"Topological pseudo entropy"
],
[
"Abstract We introduce a pseudo entropy extension of topological entanglement entropy called topological pseudo entropy.",
"Various examples of the topological pseudo entropies are examined in three-dimensional Chern-Simons gauge theory with Wilson loop insertions.",
"Partition functions with knotted Wilson loops are directly related to topological pseudo (R\\'enyi) entropies.",
"We also show that the pseudo entropy in a certain setup is equivalent to the interface entropy in two-dimensional conformal field theories (CFTs), and leverage the equivalence to calculate the pseudo entropies in particular examples.",
"Furthermore, we define a pseudo entropy extension of the left-right entanglement entropy in two-dimensional boundary CFTs and derive a universal formula for a pair of arbitrary boundary states.",
"As a byproduct, we find that the topological interface entropy for rational CFTs has a contribution identical to the topological entanglement entropy on a torus."
],
[
"Introduction",
"Entanglement entropy has played an important role as a useful quantum order parameter in various quantum many-body systems [1], [2], [3], [4], [5].",
"In particular, the topological entanglement entropy [4], [5] can characterize topologically ordered phases.",
"A prominent example of topological field theory is a three-dimensional Chern-Simons gauge theory, where the topological entanglement entropy can be computed by the famous surgery method [6] as first shown in [7].",
"Refer to [8], [9], [10], [11], [12], [13], [14], [15], [16] for further developments.",
"Recently, a quantity called the pseudo entropy was introduced in [17], mainly motivated by finding a counterpart to a generalization of holographic entanglement entropy [18], [19], [20], [21], [22] to Euclidean time-dependent backgrounds.",
"The pseudo entropy itself is a generalization of entanglement entropy that depends on both the initial state $|\\psi \\rangle $ and the final state $|\\varphi \\rangle $ , defined as follows.",
"Let $\\mathinner {|{\\psi }\\rangle },\\mathinner {|{\\varphi }\\rangle }\\in \\mathcal {H}_A\\otimes \\mathcal {H}_B$ be unnormalized states satisfying $\\mathinner {\\langle {\\varphi |\\psi }\\rangle }\\ne 0$ .",
"Define the transition matrix as $\\tau ^{\\psi |\\varphi }\\equiv \\frac{\\mathinner {|{\\psi }\\rangle }\\mathinner {\\langle {\\varphi }|}}{\\mathinner {\\langle {\\varphi |\\psi }\\rangle }}\\ ,$ and its reduced version as $\\tau _A^{\\psi |\\varphi }\\equiv {\\rm Tr}_B\\left[\\tau ^{\\psi |\\varphi }\\right]\\ .$ The pseudo Rényi entropy is $S^{(n)}\\left(\\tau _A^{\\psi |\\varphi }\\right)\\equiv \\frac{1}{1-n}\\log \\,{\\rm Tr}_A\\left[\\left(\\tau _A^{\\psi |\\varphi }\\right)^n\\right]\\ ,$ and we define the pseudo entropy by taking a limit $n\\rightarrow 1$ : $S\\left(\\tau _A^{\\psi |\\varphi }\\right)\\equiv \\lim _{n\\rightarrow 1}S^{(n)}\\left(\\tau _A^{\\psi |\\varphi }\\right)=-{\\rm Tr}_A\\left[\\tau _A^{\\psi |\\varphi }\\log \\tau _A^{\\psi |\\varphi }\\right]\\ .$ Since the transition matrix is not Hermitian in general, the pseudo entropy can take complex values.",
"This guides us to define the following real-valued quantity $\\Delta S^{(n)}\\left(\\tau _A^{\\psi |\\varphi }\\right)\\equiv \\frac{1}{2}\\left[S^{(n)}\\left(\\tau _A^{\\psi |\\varphi }\\right)+ S^{(n)}\\left(\\tau _A^{\\varphi |\\psi }\\right)-S^{(n)}\\left(\\tau _A^{\\psi |\\psi }\\right)- S^{(n)}\\left(\\tau _A^{\\varphi |\\varphi }\\right)\\right]\\ ,$ where note the relation $S^{(n)}\\left(\\tau _A^{\\varphi |\\psi }\\right)=S^{(n)}\\left(\\tau _A^{\\psi |\\varphi }\\right)^*$ and the fact that the latter two terms are the standard entanglement Rényi entropy for $|\\psi \\rangle $ and $|\\varphi \\rangle $ , respectively.",
"In other words, $\\Delta S\\left(\\tau _A^{\\psi |\\varphi }\\right)\\equiv \\lim _{n\\rightarrow 1}\\Delta S^{(n)}\\left(\\tau _A^{\\psi |\\varphi }\\right)$ is the difference between the real part of the pseudo entropy and the averaged entanglement entropy.",
"In [23], [24], the pseudo entropy was numerically evaluated for the Lifshitz free scalar field and for Ising and XY spin models.",
"These calculations showed that the difference $\\Delta S\\left(\\tau _A^{\\psi |\\varphi }\\right)$ always takes non-positive values when $|\\psi \\rangle $ and $|\\varphi \\rangle $ belong to the same phase.",
"However, it turns out that when the two states are in different quantum phases, the difference typically takes positive values.",
"This implies that the pseudo entropy can distinguish two different quantum phases.",
"A heuristic explanation of this interesting behavior was given in [24] based on holography, where an anti-de Sitter space emerges in the gravity dual along the interface between two quantum phases, which enhances the pseudo entropy.",
"Motivated by these, the purpose of the present paper is to introduce a pseudo entropy extension of topological entanglement entropy, which we call topological pseudo entropy.",
"We will explicitly evaluate the topological pseudo entropy in various examples in three-dimensional Chern-Simons gauge theory.",
"We will also point out that the pseudo entropy in a class of specific setups is equivalent to the interface entropy [25], [26], [27], [28], [29], [30], [31], [32], [33] in conformal field theories (CFTs).",
"We will also provide and evaluate a pseudo entropy extension of the left-right entanglement entropy [34], [35], [16] in CFTs.",
"This paper is organized as follows.",
"In section we calculate the topological pseudo entropy in various setups of a three-dimensional Chern-Simons gauge theory and provide its interpretations in the light of quantum entanglement and geometry.",
"In section , we explain how to calculate the pseudo entropy in CFTs via conformal transformations and show that the pseudo entropy in a special case of CFTs is equivalent to the interface entropy.",
"In section , we introduce the pseudo entropy extension of the left-right entanglement entropy.",
"In section , we summarize our conclusions.",
"In the appendix , we provide explicit values for the $\\mathrm {SU}(2)$ Chern-Simons gauge theory.",
"In the appendix , we evaluate the pseudo entropy for multi-boundary states in Chern-Simons gauge theory."
],
[
"Topological pseudo entropy in Chern-Simons gauge theory",
"Consider the three-dimensional Chern-Simons gauge theory with the gauge group $\\mathrm {SU}(N)$ at level $k$ .",
"The partition functions of the Chern-Simons theory with Wilson lines can be calculated from the knowledge of two-dimensional (2d) conformal field theory of $\\widehat{\\mathrm {SU}(N)}_k$ Wess-Zumino-Witten (WZW) model [6] as quantum states in the Chern-Simons theory correspond to the conformal blocks of the 2d CFT.",
"First we explain how to calculate pseudo entropy in Chern-Simons theory from section REF to section REF .",
"Next, we calculate the entanglement entropy or the pseudo entropy for states on $\\mathbb {S}^2$ with two excitations in section REF and four excitations in section REF , and states on $\\mathbb {T}^2$ in section REF .",
"In section REF , we consider the geometric interpretation for pseudo entropy from the above calculations.",
"Finally in section REF , we consider the definition of boundary states in Chern-Simons theory by analogy with boundary conformal field theory (BCFT) for comparison with the results in later sections.",
"We investigate another example of multi-boundary states in Chern-Simons theory in appendix ."
],
[
"Replica trick",
"Before considering the Chern-Simons theory, we review how to compute the pseudo entropy in quantum field theory.",
"We can compute the pseudo entropy on a spatial region $\\Sigma =A\\cup B$ as well as the entanglement entropy by using the replica trick.",
"We consider a Euclidean field theory with an action $I[\\Phi ]$ , where $\\Phi $ is the collection of fields.",
"We prepare the two states $\\mathinner {|{\\psi }\\rangle }$ and $\\mathinner {|{\\varphi }\\rangle }$ by inserting operators $\\mathcal {O}_\\psi $ and $\\mathcal {O}_\\varphi $ respectively to the path integral on the past of $\\Sigma $ : $\\mathinner {\\langle {\\Phi _0|\\psi }\\rangle }&=\\int _{\\Phi |_{\\Sigma }=\\Phi _0}{\\cal D}\\Phi \\ \\mathcal {O}_\\psi [\\Phi ]\\,e^{-I[\\Phi ]}=\\begin{tikzpicture}[thick,scale=1.5,baseline={([yshift=-.5ex]current bounding box.center)}][fill=lightgray!20!white] (0.9,0) arc (0:-180:0.9 and 0.8);[dotted] (0.9,0) -- (-0.9,0);\\node at (0,-0.4) {\\mathcal {O}_\\psi };\\node at (-0.5,0.15) {\\small \\Phi _0};\\node at (1.05,0) {\\Sigma };\\end{tikzpicture}\\ , \\\\\\mathinner {\\langle {\\Phi _0|\\varphi }\\rangle }&=\\int _{\\Phi |_{\\Sigma }=\\Phi _0}{\\cal D}\\Phi \\ \\mathcal {O}_\\varphi [\\Phi ]\\,e^{-I[\\Phi ]}=\\begin{tikzpicture}[thick,scale=1.5,baseline={([yshift=-.5ex]current bounding box.center)}][fill=lightgray!20!white] (0.9,0) arc (0:-180:0.9 and 0.8);[dotted] (0.9,0) -- (-0.9,0);\\node at (0,-0.4) {\\mathcal {O}_\\varphi };\\node at (-0.5,0.15) {\\small \\Phi _0};\\node at (1.05,0) {\\Sigma };\\end{tikzpicture}\\ ,$ where $\\Phi _0$ is a boundary condition of $\\Phi $ on $\\Sigma $ and $\\mathinner {|{\\Phi _0}\\rangle }$ is a state on $\\Sigma $ defined by $\\hat{\\Phi }|_\\Sigma \\mathinner {|{\\Phi _0}\\rangle }=\\Phi _0\\mathinner {|{\\Phi _0}\\rangle }$ .",
"The vertical direction in the figure is the imaginary time.",
"The inserted operators $\\mathcal {O}_\\psi $ and $\\mathcal {O}_\\varphi $ may be collections of line operators like Wilson loops as well as local operators.",
"The inner product is given by gluing the manifolds for $\\mathinner {|{\\psi }\\rangle }$ and $\\mathinner {|{\\varphi }\\rangle }$ along $\\Sigma $ and integrating over the boundary condition: $\\mathinner {\\langle {\\varphi |\\psi }\\rangle }=\\int {\\cal D}\\Phi _0\\mathinner {\\langle {\\varphi |\\Phi _0}\\rangle }\\mathinner {\\langle {\\Phi _0|\\psi }\\rangle }=\\begin{tikzpicture}[thick,scale=1.5,baseline={([yshift=-.5ex]current bounding box.center)}][fill=lightgray!20!white] (0,0) ellipse (0.9 and 0.8);[dotted] (0.9,0) -- (-0.9,0);\\node at (0,-0.4) {\\mathcal {O}_\\psi };\\node at (0,0.4) {\\mathcal {O}^\\dagger _\\varphi };\\end{tikzpicture}\\ .$ We call the resulting manifold $\\mathcal {M}_1$ .",
"Then we can interpret $\\mathinner {\\langle {\\varphi |\\psi }\\rangle }$ as a partition function on $\\mathcal {M}_1$ in the presence of $\\mathcal {O}_\\psi $ and $\\mathcal {O}_\\varphi ^{\\dagger }$ , so we denote it by $Z\\left[\\mathcal {M}_1;\\mathcal {O}_\\psi ,\\mathcal {O}_\\varphi ^{\\dagger }\\right]$ .",
"Next, we evaluate ${\\rm Tr}_A\\left[({\\rm Tr}_B\\mathinner {|{\\psi }\\rangle }\\mathinner {\\langle {\\varphi }|})^n\\right]$ .",
"A partial trace over $B$ corresponds to the gluing only over $B$ , thus the unnormalized version of the reduced transition matrix is $\\tilde{\\tau }^{\\psi |\\varphi }_A\\equiv {\\rm Tr}_B\\left[\\mathinner {|{\\psi }\\rangle }\\mathinner {\\langle {\\varphi }|}\\right]=\\int {\\cal D}[\\Phi _0|_B]\\mathinner {\\langle {\\varphi |\\Phi _0}\\rangle }\\mathinner {\\langle {\\Phi _0|\\psi }\\rangle }=\\begin{tikzpicture}[thick,scale=1.5,baseline={([yshift=-.5ex]current bounding box.center)}][fill=lightgray!20!white] (0,0) ellipse (0.9 and 0.8);[dotted,fill=white] (0,0) arc (180:0:0.3 and 0.05);[dotted,fill=white] (0,0) arc (180:360:0.3 and 0.05);[dotted] (0,0) -- (-0.9,0) ;[dotted] (0.6,0) -- (0.9,0) ;\\node at (0,-0.4) {\\mathcal {O}_\\psi };\\node at (0,0.4) {\\mathcal {O}^\\dagger _\\varphi };\\node at (0.35,0.18) {\\small A};\\node at (-0.45,0.1) {\\small B};\\node at (0.75,0.1) {\\small B} ;\\end{tikzpicture}\\ .$ To compute the $n^{\\text{th}}$ power of $\\tilde{\\tau }^{\\psi |\\varphi }_A$ , we prepare $n$ copies of the manifold in (REF ) and glue them along the subregion $A$ cyclically: ${\\rm Tr}_A\\left[\\left(\\tilde{\\tau }^{\\psi |\\varphi }_A\\right)^n\\right]=\\begin{tikzpicture}[thick,scale=1.5,baseline={([yshift=-.5ex]current bounding box.center)}]\\begin{scope}[fill=lightgray!20!white] (0,0) ellipse (0.9 and 0.8);[dotted,fill=white] (0,0) arc (180:0:0.3 and 0.05);[dotted,fill=white] (0,0) arc (180:360:0.3 and 0.05);[dotted] (0,0) -- (-0.9,0) ;[dotted] (0.6,0) -- (0.9,0) ;\\node at (0,-0.4) {\\mathcal {O}_\\psi };\\node at (0,0.4) {\\mathcal {O}^\\dagger _\\varphi };\\end{scope}\\begin{scope}[shift={(2,0)}][fill=lightgray!20!white] (0,0) ellipse (0.9 and 0.8);[dotted,fill=white] (0,0) arc (180:0:0.3 and 0.05);[dotted,fill=white] (0,0) arc (180:360:0.3 and 0.05);[dotted] (0,0) -- (-0.9,0) ;[dotted] (0.6,0) -- (0.9,0) ;\\node at (0,-0.4) {\\mathcal {O}_\\psi };\\end{scope}\\node at (3.4,0) {\\Large \\cdots };\\begin{scope}[shift={(4.8,0)}][fill=lightgray!20!white] (0,0) ellipse (0.9 and 0.8);[dotted,fill=white] (0,0) arc (180:0:0.3 and 0.05);[dotted,fill=white] (0,0) arc (180:360:0.3 and 0.05);[dotted] (0,0) -- (-0.9,0) ;[dotted] (0.6,0) -- (0.9,0) ;\\node at (0,-0.4) {\\mathcal {O}_\\psi };\\end{scope}\\begin{scope}\\begin{knot}[background color=lightgray!20!white][->,OliveGreen] (0.4,0.05) .. controls (0.75,0.3) ..(1.3,0) .. controls (1.75,-0.3) .. (2.2,-0.05) ;[OliveGreen] (2.4,0.05) .. controls (2.75,0.3)..(3.1,0.1);[->,OliveGreen] (3.8,0.1).. controls (4.5,-0.3) .. (5,-0.05);[->,OliveGreen] (5.1,0.05) .. controls (4,1) and (2.5,0.7) .. (1.5,0) .. controls (0.7,-0.5) .. (0.2,-0.05);\\end{knot}\\end{scope}\\node [fill=lightgray!20!white] at (2,0.4) {\\mathcal {O}^\\dagger _\\varphi };\\node [fill=lightgray!20!white] at (4.8,0.4) {\\mathcal {O}^\\dagger _\\varphi };\\end{tikzpicture}\\ .$ We denote the glued manifold in (REF ) by $\\mathcal {M}_n$ and the partition function on $\\mathcal {M}_n$ by $Z \\left[\\mathcal {M}_n;\\mathcal {O}_\\psi ,\\mathcal {O}_\\varphi ^{\\dagger }\\right]$ .",
"Finally we obtain the pseudo entropy $\\begin{aligned}S\\left(\\tau ^{\\psi |\\varphi }_A\\right)&=\\lim _{n\\rightarrow 1}\\frac{1}{1-n}\\log {\\rm Tr}_A\\left[\\left(\\frac{\\tilde{\\tau }^{\\psi |\\varphi }_A}{{\\rm Tr}_A\\left[\\tilde{\\tau }^{\\psi |\\varphi }_A\\right]}\\right)^n\\right] \\\\&=-\\left.\\frac{\\partial }{\\partial n}\\log \\frac{Z \\left[\\mathcal {M}_n;\\mathcal {O}_\\psi ,\\mathcal {O}_\\varphi ^{\\dagger }\\right]}{Z\\left[\\mathcal {M}_1;\\mathcal {O}_\\psi ,\\mathcal {O}_\\varphi ^{\\dagger }\\right]^n}\\right|_{n=1}\\ .\\end{aligned}$"
],
[
"Chern-Simons theory and modular $\\mathcal {S}$ -matrix",
"The Chern-Simons theory on a 3d manifold $\\mathcal {M}$ with gauge group $\\mathrm {SU}(N)$ is defined by the action $I_\\text{CS}[A]= -{\\rm i}\\,\\frac{k}{4\\pi }\\int _{\\mathcal {M}}{\\rm tr}\\left[A\\wedge \\mathrm {d}A+\\frac{2}{3}\\, A\\wedge A\\wedge A\\right]\\ ,$ where $A$ is a connection one-form and the trace is taken over the Lie algebra associated with $\\mathrm {SU}(N)$ .",
"A prefactor $k$ , which has to take an integer value for gauge invariance, is called the level of the Chern-Simons theory.",
"Since the action does not depend on the metric, Chern-Simons theory is a topological field theory.",
"Topological invariance is such a strong property that we can obtain a lot of information from the invariance.",
"We will focus on observables that are also topologically invariant, i.e., Wilson loops, defined by $W_R[A]={\\rm tr}_R\\,\\mathcal {P}\\exp \\left(\\int _CA\\right)\\ ,$ where the trace is taken over the representation space of a representation $R$ of $\\mathrm {SU}(N)$ , $\\mathcal {P}$ means the path ordered integral along a closed loop $C$ .",
"We can evaluate the partition function of a Chern-Simons theory by using the fact that there is a duality between Chern-Simons theories and WZW models [6].",
"Before describing the duality, we recapitulate several notions about WZW models.",
"Let $\\chi _i(\\tau )$ be a character of a WZW model on a torus with a complex structure $\\tau $ , where $i$ denotes an integrable representation of an affine Lie algebra $\\widehat{\\mathrm {SU}(N)}_k$ .",
"The modular invariance of the theory amounts to the transformation law for the character: $\\chi _i(-1/\\tau ) = \\sum _{j}{\\mathcal {S}_{i}}^{j}\\,\\chi _j(\\tau )\\ ,$ where ${\\mathcal {S}_{i}}^{j}$ is called modular $\\mathcal {S}$ -matrix, which is a unitary and symmetric matrix, $ \\sum _{l}{\\mathcal {S}_{i}}^{l}\\,{\\left(\\mathcal {S}^{\\dagger }\\right)_l}^j= {\\delta _{i}}^j\\ , \\qquad {\\mathcal {S}_{i}}^{j} = {\\mathcal {S}_{j}}^{i}\\ .$ Moreover, the square of the modular $\\mathcal {S}$ -matrix is identical to the charge conjugation $\\mathcal {C}$ : $\\sum _{l}{\\mathcal {S}_{i}}^{l}\\,{\\mathcal {S}_{l}}^{j}={\\mathcal {C}_i}^j={\\delta _i}^{\\bar{j}}\\ ,$ where $\\bar{j}$ denotes the charge conjugate representation of $j$ .",
"This leads to the identity ${\\mathcal {S}_{i}}^{j}=\\left({\\mathcal {S}_{i}}^{\\bar{j}}\\right)^*$ .",
"In particular, we find that the matrix element ${\\mathcal {S}_{0}}^{i}={\\mathcal {S}_{i}}^{0}$ is real valued for any $i$ .",
"For an example of $\\widehat{\\mathrm {SU}(2)}_k$ WZW theory, the modular $\\mathcal {S}$ -matrix can be written as ${\\mathcal {S}_{i}}^{j}=\\sqrt{\\frac{2}{k+2}}\\,\\sin \\left[\\frac{\\pi (2i+1)(2j+1)}{k+2}\\right]\\ ,$ where the subscripts $i,j=0,\\ldots ,k/2$ label the integrable representations of $\\widehat{\\mathrm {SU}(2)}_k$ and 0 denotes the identity representation.",
"Note that ${\\cal S}$ -matrix for $\\widehat{\\mathrm {SU}(2)}_k$ is real.",
"We summarize the properties and several explicit values of $\\mathrm {SU}(2)$ ${\\cal S}$ -matrix in appendix .",
"There is another important relation between the modular $\\mathcal {S}$ -matrix and the fusion coefficients ${N_{ij}}^k$ , known as the Verlinde formula [36]: ${N_{ij}}^k=\\sum _l\\frac{{\\mathcal {S}_{i}}^{l}\\,{\\mathcal {S}_{j}}^{l}\\,{\\left({\\cal S}^\\dagger \\right)_l}^k}{{\\mathcal {S}_{0}}^{l}}\\ ,$ or equivalently $\\sum _k {N_{ij}}^k\\,\\frac{{\\mathcal {S}_{k}}^{m}}{{\\mathcal {S}_{0}}^{m}}=\\frac{{\\mathcal {S}_{i}}^{m}}{{\\mathcal {S}_{0}}^{m}}\\,\\frac{{\\mathcal {S}_{j}}^{m}}{{\\mathcal {S}_{0}}^{m}}\\ .$ Regarding ${N_{ij}}^k$ as the $(j,k)$ -component of the matrix $\\mathbf {\\mathrm {N}}_i$ , ${\\mathcal {S}_{i}}^{m}/{\\mathcal {S}_{0}}^{m}$ in (REF ) is an eigenvalue of $\\mathbf {\\mathrm {N}}_i$ .",
"In particular, $m=0$ yields the largest eigenvalue $d_i=\\frac{{\\mathcal {S}_{i}}^{0}}{{\\mathcal {S}_{0}}^{0}}\\ ,$ called quantum dimension for the representation $i$ .",
"Note that ${\\mathcal {S}_{0}}^{0}$ and ${\\mathcal {S}_{i}}^{0}$ are real, so that the quantum dimensions are also real.",
"The total quantum dimension is defined by ${\\cal D}=\\sqrt{\\sum _{i}|d_i|^2}=\\frac{1}{{\\mathcal {S}_{0}}^{0}}\\ .$ The second equality in (REF ) follows from the unitarity condition (REF ).",
"Finally let us describe the duality between Chern-Simons and WZW theories.",
"Consider a Chern-Simons theory with Wilson loops and take a spatial submanifold $\\Sigma \\simeq \\mathbb {S}^2$ .",
"When $\\Sigma $ has some intersections with Wilson loops $W_{R_i}[A]$ , the Hilbert space on $\\Sigma $ is given by $\\mathcal {H}_\\Sigma =\\mathrm {Inv}\\left(\\bigotimes _i{R}_i\\right)\\ ,$ where ${R}_i$ denotes the representation space of an integrable representation $R_i$ , and “$\\mathrm {Inv}$ ” means that it takes only the invariant subspace.",
"The subscript $i$ in (REF ) runs over all the intersections of Wilson lines and $\\Sigma $ .",
"In particular, if there are no intersections, then the Hilbert space is one-dimensional."
],
[
"Computation of partition functions in Chern-Simons gauge theory",
"With the input of the modular properties of 2d CFTs, we can evaluate the partition functions in Chern-Simons theory by Witten's method [6].",
"We cut a manifold ${\\cal M}$ along a submanifold $\\Sigma \\simeq \\mathbb {S}^2$ into two parts ${\\cal M}^{\\prime }$ and ${\\cal M}^{\\prime \\prime }$ .",
"When $\\Sigma $ has no intersections with any Wilson loops, the Hilbert space on $\\Sigma $ is one-dimensional by (REF ).",
"Therefore we can attach a hemisphere to each of the cross-sections, then we have $Z\\left[{\\cal M}\\right]=\\frac{Z\\left[{\\cal M}^{\\prime }\\right]\\, Z\\left[{\\cal M}^{\\prime \\prime }\\right]}{Z\\left[\\mathbb {S}^3\\right]}\\ .$ Figure: A manifold can be decomposed into two by cutting it a half and attaching hemispheres to each of them.Figure REF shows this relation graphically.",
"We can apply this method also to the case where $\\Sigma $ has two punctures $R_i$ and $\\overline{R}_i$ because the Hilbert space is one-dimensional.",
"We consider the case ${\\cal M}=\\mathbb {S}^2\\times \\mathbb {S}^1$ including two Wilson loops wrapping along $\\mathbb {S}^1$ .",
"Applying the above method, we obtain $Z\\left[\\mathbb {S}^2\\times \\mathbb {S}^1;R_i,\\overline{R}_j\\right]={\\delta _{i}}^{j}\\ .$ Next, we would like to evaluate a partition function on a sphere $\\mathbb {S}^3$ .",
"This can be obtained by gluing two solid tori along their common boundary $\\mathbb {T}^2$ .",
"When we glue the two, we perform the modular transformation for one of the tori as depicted in figure REF .",
"Thus the partition function on $\\mathbb {S}^3$ without any Wilson loops becomes $Z\\left[\\mathbb {S}^3\\right]=\\sum _i{\\mathcal {S}_{0}}^{i}\\,Z\\left[\\mathbb {S}^2\\times \\mathbb {S}^1;R_i\\right]={\\mathcal {S}_{0}}^{0}\\ .$ Moreover, the $\\mathbb {S}^3$ partition function with a single Wilson loop in a representation $R_i$ and that with two linked Wilson loops in representations $R_i$ and $R_j$ are given by $ \\begin{aligned}Z\\left[\\mathbb {S}^3; R_i\\right]&={\\mathcal {S}_{0}}^{i}\\ ,\\\\Z\\left[\\mathbb {S}^3; L(R_i,R_j)\\right]&={\\mathcal {S}_{i}}^{j}\\ .\\end{aligned}$ Figure REF shows the calculations for these results.",
"Figure: The modular transformation in Chern-Simons gauge theory and the evaluations of the partition functions with Wilson loops.",
"The horizontal solid tori have a complex structure τ\\tau while the vertical ones have -1/τ-1/\\tau .",
"The dot means the gluing along the torus on the boundaries of two solid tori.By using these results and (REF ), we can calculate the partition functions with multiple disconnected Wilson loops.",
"For example, the $\\mathbb {S}^3$ partition function with two disconnected Wilson loops in representations $R_i$ and $R_j$ (see figure REF ) is computed as $Z\\left[\\mathbb {S}^3; R_i,R_j\\right] = \\frac{{\\mathcal {S}_{0}}^{i}\\, {\\mathcal {S}_{0}}^{j}}{{\\mathcal {S}_{0}}^{0}}\\ .$ Figure: We can calculate Z𝕊 3 ;R i ,R j Z\\left[\\mathbb {S}^3;R_i,R_j\\right] by applying () and ()."
],
[
"Topological entanglement entropy on $\\mathbb {S}^2$ with two excitations",
"Before we go to our main target of topological pseudo entropy, we would like to start with the calculation of topological entanglement entropy in a simple setup.",
"Refer to [7] for more extensive analysis.",
"We consider a setup where a state $|\\psi \\rangle $ is defined by a path integral on a hemisphere $\\mathbb {B}^3$ such that on its boundary $\\mathbb {S}^2$ , there are two quasi-particle (i.e.",
"anyon) excitations one of which is in a representation $R_i$ and the other is in $\\overline{R}_i$ of $\\widehat{\\mathrm {SU}(N)}_k$ , so that they form a singlet.",
"We choose the subsystem $A$ on the sphere $\\mathbb {S}^2$ , such that $A$ includes the excitation in $R_i$ and its complement $B$ does that in $\\overline{R}_i$ .",
"The entanglement entropy $S(\\rho _A)=-{\\rm Tr}_A\\left[\\rho _A\\log \\rho _A\\right]\\ ,$ of the reduced density matrix $\\rho _A=\\mbox{Tr}_B\\left[\\frac{|\\psi \\rangle \\langle \\psi |}{\\mathinner {\\langle {\\psi |\\psi }\\rangle }}\\right]$ can be computed via the replica trick we reviewed in section REF .",
"We can construct the state $\\mathinner {|{\\psi }\\rangle }$ by a path integral over $\\mathbb {B}^3$ inserting a Wilson line operator ending on the excitations ($\\mathcal {O}_\\psi =W_{R_i}$ in (REF )).",
"The partial trace over $B$ can be performed by gluing only the subregion $B$ of $\\mathbb {S}^2$ and the product of two $\\rho _A$ 's can be done by gluing the subregion $A$ , then ${\\rm Tr}_A\\left[({\\rm Tr}_B\\mathinner {|{\\psi }\\rangle }\\mathinner {\\langle {\\psi }|})^n\\right]$ becomes a partition function on $\\mathbb {S}^3$ with a Wilson loop.",
"Figure REF shows the calculation of $n=2$ case.",
"Divided by the normalization factor, we obtain Figure: We can calculate Tr A ( Tr B |ψ〉〈ψ|) 2 {\\rm Tr}_A\\left[({\\rm Tr}_B\\mathinner {|{\\psi }\\rangle }\\mathinner {\\langle {\\psi }|})^2\\right] by gluing BB with the neighboring B ¯\\bar{B}, corresponding to taking the partial trace over BB, and AA with the neighboring A ¯\\bar{A}, corresponding to the product of ρ A \\rho _A.",
"The last A ¯\\bar{A} is glued to the first AA, corresponding to the trace over AA.${\\rm Tr}_A\\left[\\rho _A^n\\right]=\\frac{Z\\left[\\mathbb {S}^3; R_i\\right]}{Z\\left[\\mathbb {S}^3; R_i\\right]^n}\\ .$ Thus, the topological entanglement entropy is given by $\\begin{aligned}S(\\rho _A)&=\\log Z\\left[\\mathbb {S}^3; R_i\\right]\\\\&=\\log {\\mathcal {S}_{0}}^{i}\\\\&=-\\log \\, {\\cal D}+\\log d_i\\ .\\end{aligned}$ If we do not insert any excitation, we have by simply setting $d_j=0$ ,In $d=3$ dimensions, the pseudo entropy can have an area law UV divergent term.",
"In the Chern-Simons theory calculation, however, the partition function is a topological invariant, i.e., independent of any scale, after renormalizing the UV divergence in an appropriate scheme [6].",
"Hence in this case the pseudo entropy is free from the area law term and becomes scale independent.",
"$\\begin{aligned}S(\\rho _A)&=\\log Z\\left[\\mathbb {S}^3\\right] \\\\&=\\log {\\mathcal {S}_{0}}^{0}\\\\&=-\\log {\\cal D}\\ .\\end{aligned}$ This vacuum topological entanglement entropy is related to the total quantum dimension [4], [5] and is expected to measure the degrees of freedom of edge modes, which is analogous to the area term in the holographic entanglement entropy.",
"When we add an anyon, the topological entanglement entropy increases by the amount of log of the quantum dimension as in (REF )."
],
[
"Topological pseudo entropy on $\\mathbb {S}^2$ with four excitations",
"We consider the case that the spatial region is $\\mathbb {S}^2$ , which is divided to two subregions $A$ and $B$ as the figures show in (REF ) and there are four excitations.",
"For simplicity, we only consider fundamental (called $j$ ) or anti-fundamental (called $\\bar{j}$ ) excitations.",
"For the total charge to vanish, two of the four excitations must be fundamental and the others must be anti-fundamental.",
"There are then two possible cases: 1) a pair of $j$ and $\\bar{j}$ in $A$ and the other pair in $B$ , and 2) two $j$ 's in $A$ and two $\\bar{j}$ 's in $B$ .",
"We prepare these states by the path integral.",
"The excitations will be the edges of Wilson lines.",
"There are many ways to connect the excitations so that the Wilson lines make some knots.",
"In what follows we will show they give rise to nontrivial contributions to the pseudo or entanglement entropies."
],
[
"Case 1: $j$ and {{formula:bc161cff-17f8-4ac6-ae5e-3161a78fdfc6}} in {{formula:55eb034b-6d2e-49f0-932c-3a6ad4bcabd0}} , the others in {{formula:05a6e862-d65e-4148-8e48-a4d7789915d8}}",
"In this case, there are two configurations of Wilson lines which end on one $j$ and one $\\bar{j}$ .",
"We set $\\mathinner {|{\\psi }\\rangle }$ and $\\mathinner {|{\\varphi }\\rangle }$ as $\\mathinner {|{\\psi }\\rangle }=\\begin{tikzpicture}[thick,scale=1.5,baseline={([yshift=-.5ex]current bounding box.center)}]\\begin{scope}[decoration={markings, mark=at position 0.5 with {{>}}}][fill=lightgray!20!white] (0,0) circle (1);[dotted] (0,1) arc (90:-90:0.2 and 1);\\node at (-0.9,0.9) {A};\\node at (0.9,0.9) {B};\\begin{scope}[shift={(-0.7,0.4)}] [BrickRed] (-0.05,-0.05)--(0.05,0.05);[BrickRed] (-0.05,0.05)--(0.05,-0.05);\\end{scope}\\begin{scope}[shift={(-0.7,-0.4)}] [BrickRed] (-0.05,-0.05)--(0.05,0.05);[BrickRed] (-0.05,0.05)--(0.05,-0.05);\\end{scope}\\begin{scope}[shift={(0.7,0.4)}] [BrickRed] (-0.05,-0.05)--(0.05,0.05);[BrickRed] (-0.05,0.05)--(0.05,-0.05);\\end{scope}\\begin{scope}[shift={(0.7,-0.4)}] [BrickRed] (-0.05,-0.05)--(0.05,0.05);[BrickRed] (-0.05,0.05)--(0.05,-0.05);\\end{scope}[BrickRed, postaction={decorate}] (-0.7,-0.4) to [out=60,in=300] (-0.7,0.4);[BrickRed,postaction={decorate}] (0.7,0.4) to [out=240,in=120] (0.7,-0.4);\\node [BrickRed] at (-0.55,0.6) {\\footnotesize j} ;\\node [BrickRed] at (-0.55,-0.6) {\\footnotesize \\bar{j}} ;\\node [BrickRed] at (0.55,0.6) {\\footnotesize \\bar{j}} ;\\node [BrickRed] at (0.55,-0.6) {\\footnotesize j} ;\\end{scope}(0,1) arc (90:270:0.2 and 1);\\end{tikzpicture}$ , |= [thick,scale=1.5,baseline=([yshift=-.5ex]current bounding box.center)] [decoration=markings, mark=at position 0.5 with >] [fill=lightgray!20!white] (0,0) circle (1); [dotted] (0,1) arc (90:-90:0.2 and 1); t (-0.9,0.9) $A$ ; t (0.9,0.9) $B$ ; [shift=(-0.7,0.4)] [BrickRed] (-0.05,-0.05)–(0.05,0.05); [BrickRed] (-0.05,0.05)–(0.05,-0.05); [shift=(-0.7,-0.4)] [BrickRed] (-0.05,-0.05)–(0.05,0.05); [BrickRed] (-0.05,0.05)–(0.05,-0.05); [shift=(0.7,0.4)] [BrickRed] (-0.05,-0.05)–(0.05,0.05); [BrickRed] (-0.05,0.05)–(0.05,-0.05); [shift=(0.7,-0.4)] [BrickRed] (-0.05,-0.05)–(0.05,0.05); [BrickRed] (-0.05,0.05)–(0.05,-0.05); [BrickRed, postaction=decorate] (0.7,0.4) to [out=210,in=330] (-0.7,0.4); [BrickRed,postaction=decorate] (-0.7,-0.4) to [out=30,in=150] (0.7,-0.4); BrickRed] at (-0.55,0.6) $j$ ; BrickRed] at (-0.55,-0.6) $\\bar{j}$ ; BrickRed] at (0.55,0.6) $\\bar{j}$ ; BrickRed] at (0.55,-0.6) $j$ ; (0,1) arc (90:270:0.2 and 1); We first calculate the entanglement entropies of $\\mathinner {|{\\psi }\\rangle }$ and $\\mathinner {|{\\varphi }\\rangle }$ .",
"For $\\mathinner {|{\\psi }\\rangle }$ , ${\\rm Tr}_A\\left[\\left(\\tilde{\\rho }^{\\psi }_A\\right)^n\\right]$ equals to the partition function on $\\mathbb {S}^3$ that includes $2n$ Wilson loops in the representation $R_j$ .",
"Thus $\\begin{aligned}{\\rm Tr}_A\\left[\\left(\\rho ^{\\psi }_A\\right)^n\\right]&=\\frac{Z\\left[\\mathbb {S}^3;R_j\\right]^{2n}/Z\\left[\\mathbb {S}^3\\right]^{2n-1}}{\\left(Z\\left[\\mathbb {S}^3;R_j\\right]^2/Z\\left[\\mathbb {S}^3\\right]\\right)^n} \\\\&=Z\\left[\\mathbb {S}^3\\right]^{1-n}\\\\&=\\left({\\mathcal {S}_{0}}^{0}\\right)^{1-n}\\ .\\end{aligned}$ Since ${\\mathcal {S}_{0}}^{0}={\\cal D}^{-1}$ , we have $S\\left(\\rho ^{\\psi }_A\\right) = -\\log {\\cal D}\\ .$ For $\\mathinner {|{\\varphi }\\rangle }$ , ${\\rm Tr}_A\\left[\\left(\\tilde{\\rho }^{\\varphi }_A\\right)^n\\right]$ equals to the partition function on $\\mathbb {S}^3$ that includes two Wilson loops: $\\begin{aligned}{\\rm Tr}_A\\left[\\left(\\rho ^{\\varphi }_A\\right)^n\\right]&=\\frac{Z\\left[\\mathbb {S}^3;R_j\\right]^2/Z\\left[\\mathbb {S}^3\\right]}{\\left(Z\\left[\\mathbb {S}^3;R_j\\right]^2/Z\\left[\\mathbb {S}^3\\right]\\right)^{n}} \\\\&=\\left[\\frac{({\\mathcal {S}_{0}}^{j})^2}{{\\mathcal {S}_{0}}^{0}}\\right]^{1-n}\\ .\\end{aligned}$ Therefore, we have $S\\left(\\rho ^{\\varphi }_A\\right)=-\\log {\\cal D}+2\\log d_j\\ .$ Next, we calculate the pseudo entropy of the reduced transition matrix: $\\tau _A^{\\psi |\\varphi }={\\rm Tr}_B\\left[\\frac{\\mathinner {|{\\psi }\\rangle }\\mathinner {\\langle {\\varphi }|}}{\\mathinner {\\langle {\\varphi |\\psi }\\rangle }}\\right]\\ .$ ${\\rm Tr}_A\\left[\\left(\\tilde{\\tau }_A^{\\psi |\\varphi }\\right)^n\\right]$ equals to the partition function on $\\mathbb {S}^3$ with $n$ Wilson loop, so $\\begin{aligned}{\\rm Tr}_A\\left[\\left(\\tau _A^{\\psi |\\varphi }\\right)^n\\right]&=\\frac{Z\\left[\\mathbb {S}^3;R_j\\right]^n/Z\\left[\\mathbb {S}^3\\right]^{n-1}}{Z\\left[\\mathbb {S}^3;R_j\\right]^n} \\\\&=\\left({\\mathcal {S}_{0}}^{0}\\right)^{1-n}\\ .\\end{aligned}$ Therefore, the pseudo entropy is $S\\left(\\tau _A^{\\psi |\\varphi }\\right)=-\\log {\\cal D}\\ .$ In this case the difference of the pseudo entropy from the entanglement entropy is negative: $\\Delta S=-\\log d_j<0\\ .$ The results (REF ), (REF ), and (REF ) are easily interpreted as follows.",
"$\\mathinner {|{\\psi }\\rangle }$ is not entangled since no Wilson lines connect $A$ and $B$ , so that $S\\left(\\rho ^{\\psi }_A\\right)$ has no non-topological contributions.",
"On the other hand, $S\\left(\\rho ^{\\varphi }_A\\right)$ has the term $2\\log d_j$ because $\\mathinner {|{\\varphi }\\rangle }$ is entangled due to the Wilson lines connecting the two points $A$ and $B$ .",
"As shown in [17], the pseudo entropy is zero when either state has no entanglement.",
"Now $\\mathinner {|{\\psi }\\rangle }$ has no entanglement, so $S\\left(\\tau ^{\\psi |\\varphi }_A\\right)$ has no terms other than the topological term."
],
[
"Case 2: two $j$ 's in {{formula:fc814085-43c2-42db-8216-844fbd6f3558}} , the others in {{formula:9d022ad6-5fd3-4cce-84c8-70af9619e312}}",
"We define states $\\mathinner {|{\\psi _a}\\rangle } (a\\in \\mathbb {Z})$ as follows.",
"First we define at $a=0$ $\\mathinner {|{\\psi _0}\\rangle }=\\begin{tikzpicture}[thick,scale=1.5,baseline={([yshift=-.5ex]current bounding box.center)}]\\begin{scope}[decoration={markings, mark=at position 0.5 with {{>}}}][fill=lightgray!20!white] (0,0) circle (1);[dotted] (0,1) arc (90:-90:0.2 and 1);\\node at (-0.9,0.9) {A};\\node at (0.9,0.9) {B};\\begin{scope}[shift={(-0.7,0.4)}] [BrickRed] (-0.05,-0.05)--(0.05,0.05);[BrickRed] (-0.05,0.05)--(0.05,-0.05);\\end{scope}\\begin{scope}[shift={(-0.7,-0.4)}] [BrickRed] (-0.05,-0.05)--(0.05,0.05);[BrickRed] (-0.05,0.05)--(0.05,-0.05);\\end{scope}\\begin{scope}[shift={(0.7,0.4)}] [BrickRed] (-0.05,-0.05)--(0.05,0.05);[BrickRed] (-0.05,0.05)--(0.05,-0.05);\\end{scope}\\begin{scope}[shift={(0.7,-0.4)}] [BrickRed] (-0.05,-0.05)--(0.05,0.05);[BrickRed] (-0.05,0.05)--(0.05,-0.05);\\end{scope}[BrickRed, postaction={decorate}] (0.7,0.4) to [out=210,in=330] (-0.7,0.4);[BrickRed,postaction={decorate}] (0.7,-0.4) to [out=150,in=30] (-0.7,-0.4);\\node [BrickRed] at (-0.55,0.6) {\\footnotesize j} ;\\node [BrickRed] at (-0.55,-0.6) {\\footnotesize j} ;\\node [BrickRed] at (0.55,0.6) {\\footnotesize \\bar{j}} ;\\node [BrickRed] at (0.55,-0.6) {\\footnotesize \\bar{j}} ;\\end{scope}(0,1) arc (90:270:0.2 and 1);\\end{tikzpicture}$ .",
"Then we define $\\mathinner {|{\\psi _a}\\rangle }(a\\in \\mathbb {Z}_+)$ by twisting the region $B$ $a$ times: $\\mathinner {|{\\psi _1}\\rangle }=\\begin{tikzpicture}[thick,scale=1.5,baseline={([yshift=-.5ex]current bounding box.center)}]\\begin{scope}[decoration={markings, mark=at position 0.3 with {{>}}}][fill=lightgray!20!white] (0,0) circle (1);[dotted] (0,1) arc (90:-90:0.2 and 1);\\node at (-0.9,0.9) {A};\\node at (0.9,0.9) {B};\\begin{scope}[shift={(-0.7,0.4)}] [BrickRed] (-0.05,-0.05)--(0.05,0.05);[BrickRed] (-0.05,0.05)--(0.05,-0.05);\\end{scope}\\begin{scope}[shift={(-0.7,-0.4)}] [BrickRed] (-0.05,-0.05)--(0.05,0.05);[BrickRed] (-0.05,0.05)--(0.05,-0.05);\\end{scope}\\begin{scope}[shift={(0.7,0.4)}] [BrickRed] (-0.05,-0.05)--(0.05,0.05);[BrickRed] (-0.05,0.05)--(0.05,-0.05);\\end{scope}\\begin{scope}[shift={(0.7,-0.4)}] [BrickRed] (-0.05,-0.05)--(0.05,0.05);[BrickRed] (-0.05,0.05)--(0.05,-0.05);\\end{scope}\\begin{knot}[background color=lightgray!20!white][BrickRed, postaction={decorate}] (0.7,0.4) -- (-0.7,-0.4);[BrickRed, postaction={decorate}] (0.7,-0.4) -- (-0.7,0.4);\\end{knot}\\node [BrickRed] at (-0.55,0.6) {\\footnotesize j} ;\\node [BrickRed] at (-0.55,-0.6) {\\footnotesize j} ;\\node [BrickRed] at (0.55,0.6) {\\footnotesize \\bar{j}} ;\\node [BrickRed] at (0.55,-0.6) {\\footnotesize \\bar{j}} ;\\end{scope}(0,1) arc (90:270:0.2 and 1);\\end{tikzpicture}$ , |2= [thick,scale=1.5,baseline=([yshift=-.5ex]current bounding box.center)] [decoration=markings, mark=at position 0.5 with >] [fill=lightgray!20!white] (0,0) circle (1); [dotted] (0,1) arc (90:-90:0.2 and 1); t (-0.9,0.9) $A$ ; t (0.9,0.9) $B$ ; [shift=(-0.7,0.4)] [BrickRed] (-0.05,-0.05)–(0.05,0.05); [BrickRed] (-0.05,0.05)–(0.05,-0.05); [shift=(-0.7,-0.4)] [BrickRed] (-0.05,-0.05)–(0.05,0.05); [BrickRed] (-0.05,0.05)–(0.05,-0.05); [shift=(0.7,0.4)] [BrickRed] (-0.05,-0.05)–(0.05,0.05); [BrickRed] (-0.05,0.05)–(0.05,-0.05); [shift=(0.7,-0.4)] [BrickRed] (-0.05,-0.05)–(0.05,0.05); [BrickRed] (-0.05,0.05)–(0.05,-0.05); [background color=lightgray!20!white,flip crossing=2] [BrickRed, postaction=decorate] (0.7,0.4) .. controls (0,-0.5) .. (-0.7,0.4); [BrickRed, postaction=decorate] (0.7,-0.4) .. controls (0,0.5) .. (-0.7,-0.4); BrickRed] at (-0.55,0.6) $j$ ; BrickRed] at (-0.55,-0.6) $j$ ; BrickRed] at (0.55,0.6) $\\bar{j}$ ; BrickRed] at (0.55,-0.6) $\\bar{j}$ ; (0,1) arc (90:270:0.2 and 1); , |3= On the other hand, we define $\\mathinner {|{\\psi _a}\\rangle }(a\\in \\mathbb {Z}_-)$ by twisting the region $B$ $|a|$ times in the opposite direction: $\\mathinner {|{\\psi _{-1}}\\rangle }=\\begin{tikzpicture}[thick,scale=1.5,baseline={([yshift=-.5ex]current bounding box.center)}]\\begin{scope}[decoration={markings, mark=at position 0.3 with {{>}}}][fill=lightgray!20!white] (0,0) circle (1);[dotted] (0,1) arc (90:-90:0.2 and 1);\\node at (-0.9,0.9) {A};\\node at (0.9,0.9) {B};\\begin{scope}[shift={(-0.7,0.4)}] [BrickRed] (-0.05,-0.05)--(0.05,0.05);[BrickRed] (-0.05,0.05)--(0.05,-0.05);\\end{scope}\\begin{scope}[shift={(-0.7,-0.4)}] [BrickRed] (-0.05,-0.05)--(0.05,0.05);[BrickRed] (-0.05,0.05)--(0.05,-0.05);\\end{scope}\\begin{scope}[shift={(0.7,0.4)}] [BrickRed] (-0.05,-0.05)--(0.05,0.05);[BrickRed] (-0.05,0.05)--(0.05,-0.05);\\end{scope}\\begin{scope}[shift={(0.7,-0.4)}] [BrickRed] (-0.05,-0.05)--(0.05,0.05);[BrickRed] (-0.05,0.05)--(0.05,-0.05);\\end{scope}\\begin{knot}[background color=lightgray!20!white,flip crossing=1][BrickRed, postaction={decorate}] (0.7,0.4) -- (-0.7,-0.4);[BrickRed, postaction={decorate}] (0.7,-0.4) -- (-0.7,0.4);\\end{knot}\\node [BrickRed] at (-0.55,0.6) {\\footnotesize j} ;\\node [BrickRed] at (-0.55,-0.6) {\\footnotesize j} ;\\node [BrickRed] at (0.55,0.6) {\\footnotesize \\bar{j}} ;\\node [BrickRed] at (0.55,-0.6) {\\footnotesize \\bar{j}} ;\\end{scope}(0,1) arc (90:270:0.2 and 1);\\end{tikzpicture}$ , |-2= [thick,scale=1.5,baseline=([yshift=-.5ex]current bounding box.center)] [decoration=markings, mark=at position 0.5 with >] [fill=lightgray!20!white] (0,0) circle (1); [dotted] (0,1) arc (90:-90:0.2 and 1); t (-0.9,0.9) $A$ ; t (0.9,0.9) $B$ ; [shift=(-0.7,0.4)] [BrickRed] (-0.05,-0.05)–(0.05,0.05); [BrickRed] (-0.05,0.05)–(0.05,-0.05); [shift=(-0.7,-0.4)] [BrickRed] (-0.05,-0.05)–(0.05,0.05); [BrickRed] (-0.05,0.05)–(0.05,-0.05); [shift=(0.7,0.4)] [BrickRed] (-0.05,-0.05)–(0.05,0.05); [BrickRed] (-0.05,0.05)–(0.05,-0.05); [shift=(0.7,-0.4)] [BrickRed] (-0.05,-0.05)–(0.05,0.05); [BrickRed] (-0.05,0.05)–(0.05,-0.05); [background color=lightgray!20!white,flip crossing=1] [BrickRed, postaction=decorate] (0.7,0.4) .. controls (0,-0.5) .. (-0.7,0.4); [BrickRed, postaction=decorate] (0.7,-0.4) .. controls (0,0.5) .. (-0.7,-0.4); BrickRed] at (-0.55,0.6) $j$ ; BrickRed] at (-0.55,-0.6) $j$ ; BrickRed] at (0.55,0.6) $\\bar{j}$ ; BrickRed] at (0.55,-0.6) $\\bar{j}$ ; (0,1) arc (90:270:0.2 and 1); , |-3= In other words, $\\mathinner {|{\\psi _a}\\rangle }$ is a state which has $|a|$ crossings.",
"We would like to calculate the pseudo entropy of the transition matrix: $\\tau ^{a|b}\\equiv \\frac{\\mathinner {|{\\psi _a}\\rangle }\\mathinner {\\langle {\\psi _b}|}}{\\mathinner {\\langle {\\psi _b|\\psi _a}\\rangle }}.$ The unnormalized reduced transition matrix $\\tilde{\\tau }^{a|b}_A\\equiv {\\rm Tr}_B[\\mathinner {|{\\psi _a}\\rangle }\\mathinner {\\langle {\\psi _b}|}]$ is $\\tilde{\\tau }^{a|b}_A=\\begin{tikzpicture}[thick,scale=1.5,baseline={([yshift=-.5ex]current bounding box.center)}]\\begin{scope}[decoration={markings, mark=at position 0.7 with {{>}}}][fill=lightgray!20!white] (0,0) ellipse (1.5 and 1);[dotted] (0,1) arc (90:-90:0.2 and 1);\\node at (-1.2,0.9) {A};\\node at (1.2,0.9) {\\bar{A}};\\begin{scope}[shift={(-1.1,0.4)}] [BrickRed] (-0.05,-0.05)--(0.05,0.05);[BrickRed] (-0.05,0.05)--(0.05,-0.05);\\end{scope}\\begin{scope}[shift={(-1.1,-0.4)}] [BrickRed] (-0.05,-0.05)--(0.05,0.05);[BrickRed] (-0.05,0.05)--(0.05,-0.05);\\end{scope}\\begin{scope}[shift={(1.1,0.4)}] [BrickRed] (-0.05,-0.05)--(0.05,0.05);[BrickRed] (-0.05,0.05)--(0.05,-0.05);\\end{scope}\\begin{scope}[shift={(1.1,-0.4)}] [BrickRed] (-0.05,-0.05)--(0.05,0.05);[BrickRed] (-0.05,0.05)--(0.05,-0.05);\\end{scope}\\begin{knot}[background color=lightgray!20!white][BrickRed] (1.1,0.4) .. controls (0.9,-0.45) and (0.7,-0.45) .. (0.52,-0.1);[BrickRed] (1.1,-0.4) .. controls (0.9,0.45) and (0.7,0.45) .. (0.52,0.1);\\end{knot}\\begin{knot}[background color=lightgray!20!white,flip crossing=1][BrickRed] (-1.1,0.4) .. controls (-0.8,-0.5) and (-0.6,-0.5) .. (-0.38,0) .. controls (-0.2,0.42) and (0,0.42) ..(0.15,0.1);[BrickRed] (-1.1,-0.4) .. controls (-0.8,0.5) and (-0.6,0.5) .. (-0.38,0) .. controls (-0.2,-0.42) and (0,-0.42) ..(0.15,-0.1);\\end{knot}\\node [BrickRed] at (-0.95,0.6) {\\footnotesize j} ;\\node [BrickRed] at (-0.95,-0.6) {\\footnotesize j} ;\\node [BrickRed] at (0.95,0.6) {\\footnotesize \\bar{j}} ;\\node [BrickRed] at (0.95,-0.6) {\\footnotesize \\bar{j}} ;\\node [BrickRed] at (0.4, 0) {\\large \\dots };(0,1) arc (90:270:0.2 and 1);\\node [fill=lightgray!20!white] at (0,-0.6) {\\footnotesize |a-b| crossings};\\end{scope}\\end{tikzpicture}\\ ,$ which has the crossing number $|a-b|$ .",
"In the figure, $\\bar{A}$ means the conjugation of $A$ .",
"Therefore ${\\rm Tr}_A\\left[\\left(\\tilde{\\tau }^{a|b}_A\\right)^n\\right]$ has one or two Wilson loops with $n|a-b|$ crossings.When $n|a-b|$ is even, there are two Wilson loops while there is one Wilson loop when $n|a-b|$ is odd.",
"Here let us pause to compute the partition function on $\\mathbb {S}^3$ with a crossing number $m$ .",
"We call such a manifold as $X_m$ .",
"In this case, we also use a technique introduced in [6].",
"We cut along a two-dimensional submanifold that intersects with Wilson lines for four times, and we perform a twisting transformation on the cross section.",
"Then we obtain three states with different links.",
"Since the Hilbert space on the cross section is two-dimensional due to (REF ), these three states are linearly dependent, giving the skein relation: $\\alpha \\, Z\\left[X_m\\right] + \\beta \\, Z\\left[X_{m-1}\\right] + \\gamma \\, Z\\left[X_{m-2}\\right] = 0\\ ,$ where we call $\\mathbb {S}^3$ including $m$ -crossing Wilson lines $X_m$ .",
"Since now our gauge group is $\\mathrm {SU}(N)$ and Wilson loops are in the fundamental representation, the coefficients areIn fact, those coefficients depend on the choice of the “framing\".",
"The framing of Wilson lines in $\\mathbb {S}^3$ can be chosen to be canonical in the sense that the self-interaction numbers of the links are zero.",
"The result $Z\\left[X_m\\right]/Z\\left[X_1\\right]^n$ does not depend on the choice of the framing.",
"$\\alpha =-q^{\\frac{N}{2}}\\ ,\\qquad \\beta =q^\\frac{1}{2}-q^{-\\frac{1}{2}}\\ ,\\qquad \\gamma =q^{-\\frac{N}{2}}\\ ,$ where we define $q=e^{2\\pi {\\rm i}/(N+k)}$ .",
"Then we obtain the recursion relation $Z\\left[X_m\\right] + q^{-\\frac{N+1}{2}}\\,Z\\left[X_{m-1}\\right]=q^{-\\frac{N-1}{2}}\\,\\left(Z\\left[X_{m-1}\\right]+q^{-\\frac{N+1}{2}}\\,Z\\left[X_{m-2}\\right]\\right)\\ .$ Solving this relation with the initial conditions $Z\\left[X_0\\right] = {\\cal S}_0^{~0}\\,d_j^2$ and $Z\\left[X_1\\right] = {\\cal S}_0^{~0}\\,d_j$ , we have $\\frac{Z\\left[X_m\\right]}{{\\mathcal {S}_{0}}^{0}}=\\left(q^{-\\frac{N-1}{2}}\\right)^m\\frac{[N+1]\\,[N]}{[2]}+\\left(-q^{-\\frac{N+1}{2}}\\right)^m\\frac{[N]\\,[N-1]}{[2]}\\ ,$ where $[x]\\equiv \\frac{q^\\frac{x}{2}-q^{-\\frac{x}{2}}}{q^{\\frac{1}{2}}-q^{-\\frac{1}{2}}} \\ ,$ and the quantum dimension is $d_j=[N]$ .",
"This is what we have wanted to obtain.",
"It follows from (REF ) $\\begin{aligned}{\\rm Tr}_A\\left[\\left(\\tau _A^{a|b}\\right)^n\\right]&=\\frac{Z\\left[X_{|(a-b)n|}\\right]}{Z\\left[X_{|a-b|}\\right]^n} \\\\&=\\left({\\mathcal {S}_{0}}^{0}\\,[N]\\right)^{(1-n)}\\,\\frac{\\left(q^{\\frac{1}{2}}\\right)^{|a-b|\\,n}\\frac{[N+1]}{[2]}+\\left(-q^{-\\frac{1}{2}}\\right)^{|a-b|\\,n}\\frac{[N-1]}{[2]}}{\\left[\\left(q^{\\frac{1}{2}}\\right)^{|a-b|}\\frac{[N+1]}{[2]}+\\left(-q^{-\\frac{1}{2}}\\right)^{|a-b|}\\frac{[N-1]}{[2]}\\right]^n}\\ .\\end{aligned}$ When $a=b$ , ${\\rm Tr}_A\\left[\\left(\\rho ^{a}_A\\right)^n\\right] = \\left({\\mathcal {S}_{0}}^{0}\\,[N]^2\\right)^{1-n}\\ ,$ where we have defined $\\rho ^a_A\\equiv \\tau ^{a|a}_A$ and used the relation $[N]=\\frac{[N+1]}{[2]}+\\frac{[N-1]}{[2]}$ .",
"Then the entanglement entropy becomes independent of $a$ : $S\\left(\\rho ^a_A\\right) = -\\log {\\cal D}+ 2\\log \\,[N]\\ .$ We are now ready to calculate the difference $\\Delta S$ of the pseudo entropy from the averaged entanglement entropy, defined by $\\Delta S=-\\frac{1}{2}\\left.\\frac{\\partial }{\\partial n}\\log \\frac{{\\rm Tr}_A\\left[\\left(\\tau _A^{a|b}\\right)^n\\right]\\,{\\rm Tr}_A\\left[\\left(\\tau _A^{a|b}\\right)^n\\right]^*}{{\\rm Tr}_A\\left[\\left(\\rho _A^{a}\\right)^n\\right]\\,{\\rm Tr}_A\\left[\\left(\\rho _A^{b}\\right)^n\\right]}\\right|_{n=1}\\ .$ Here the argument of the logarithm is $\\begin{aligned}&\\frac{{\\rm Tr}_A\\left[\\left(\\tau _A^{a|b}\\right)^n\\right]\\,{\\rm Tr}_A\\left[\\left(\\tau _A^{a|b}\\right)^n\\right]^*}{{\\rm Tr}_A\\left[\\left(\\rho _A^{a}\\right)^n\\right]\\,{\\rm Tr}_A\\left[\\left(\\rho _A^{b}\\right)^n\\right]}\\\\&\\quad =\\left([N]\\,[2]\\right)^{2(n-1)}\\,\\frac{[N+1]^2+[N-1]^2 + 2\\,(-1)^{|a-b|\\,n}\\,\\cos \\left(\\frac{2\\pi \\, |a-b|\\,n}{N+k}\\right) [N+1]\\,[N-1]}{\\left[ [N+1]^2+[N-1]^2+2\\,(-1)^{|a-b|}\\,\\cos \\left(\\frac{2\\pi \\, |a-b|}{N+k}\\right) [N+1]\\,[N-1]\\right]^n}\\ .\\end{aligned}$ Now we analytically continue $n$ in (REF ) or (REF ) to real numbers.",
"However we have to be careful because the phase factor $(-1)^{|a-b|n}$ depends on the way of analytic continuation.",
"In the followings we compute $S\\left(\\tau ^{\\psi |\\varphi }_A\\right)$ and $\\Delta S$ in two different prescriptions of analytic continuations: (1) a naive prescription by deforming $(-1)^{|a-b|n}=e^{{\\rm i}\\pi |a-b|n}$ and (2) restricting $n$ to odd numbers and then analytically continuing to real numbers, which is similar to the replica method for the logarithmic negativity [37]."
],
[
"(1) A naive prescription",
"When $|a-b|$ is even, $(-1)^{|a-b|\\,n}=1$ for any integer $n$ .",
"Therefore there is no ambiguity due to the choice of the prescriptions.",
"Thus the pseudo entropy takes the form: $\\begin{aligned}S\\left(\\tau ^{a|b}_A\\right)&=-\\log \\mathcal {D}+\\log \\left[\\frac{[N]}{[2]}\\right]+\\log \\left[q^{\\frac{|a-b|}{2}}\\,[N+1]+q^{-\\frac{|a-b|}{2}}\\,[N-1]\\right] \\\\&\\qquad \\qquad - {\\rm i}\\,\\frac{\\pi \\,|a-b|}{N+k}\\, \\frac{q^{\\frac{|a-b|}{2}}\\,[N+1]-q^{-\\frac{|a-b|}{2}}\\,[N-1]}{q^{\\frac{|a-b|}{2}}\\,[N+1]+q^{-\\frac{|a-b|}{2}}\\,[N-1]}\\ ,\\end{aligned}$ and $\\Delta S$ becomes $\\begin{aligned}\\Delta S&=-\\log \\left( [N]\\,[2] \\right)+ \\frac{1}{2}\\,\\log \\left[[N+1]^2+[N-1]^2 + 2\\,\\cos \\left(\\frac{2\\pi \\, |a-b|}{N+k}\\right)[N+1]\\,[N-1]\\right]\\\\&\\qquad + \\frac{2\\pi \\,|a-b|}{N+k}\\,\\frac{\\sin \\left(\\frac{2\\pi \\,|a-b|}{N+k}\\right)[N+1]\\,[N-1]}{[N+1]^2+[N-1]^2 + 2\\,\\cos \\left(\\frac{2\\pi \\, |a-b|}{N+k}\\right)[N+1]\\,[N-1] }\\ .\\end{aligned}$ When $|a-b|$ is odd, the factor $(-1)^{|a-b|\\,n}$ remains.",
"Deforming it to $e^{{\\rm i}\\pi \\,|a-b|\\,n}$ , the pseudo entropy results in $\\begin{aligned}S\\left(\\tau ^{a|b}_A\\right)&=-\\log \\mathcal {D}+\\log \\left[\\frac{[N]}{[2]}\\right]+\\log \\left[q^{\\frac{|a-b|}{2}}\\,[N+1]-q^{-\\frac{|a-b|}{2}}\\,[N-1]\\right] \\\\&\\qquad - {\\rm i}\\,\\frac{\\pi \\,|a-b|}{N+k}\\,\\frac{q^{\\frac{|a-b|}{2}}\\,[N+1]+(1-N-k)\\,q^{-\\frac{|a-b|}{2}}\\,[N-1]}{q^{\\frac{|a-b|}{2}}\\,[N+1]-q^{-\\frac{|a-b|}{2}}\\,[N-1]}\\ ,\\end{aligned}$ and $\\Delta S$ becomes $\\begin{aligned}\\Delta S&=-\\log \\left( [N]\\,[2] \\right)+ \\frac{1}{2}\\,\\log \\left[[N+1]^2+[N-1]^2 - 2\\,\\cos \\left(\\frac{2\\pi \\, |a-b|}{N+k}\\right)[N+1]\\,[N-1]\\right]\\\\&\\qquad + \\frac{(2-N-k)\\,\\pi \\,|a-b|}{N+k}\\,\\frac{\\sin \\left(\\frac{2\\pi \\,|a-b|}{N+k}\\right)[N+1]\\,[N-1]}{[N+1]^2+[N-1]^2 - 2\\,\\cos \\left(\\frac{2\\pi \\, |a-b|}{N+k}\\right)[N+1]\\,[N-1] }\\ .\\end{aligned}$ In this calculation, we used the relation $-1=e^{{\\rm i}\\pi }$ .",
"However, more generally it satisfies $-1=e^{{\\rm i}(2m+1)\\pi }\\ (m\\in \\mathbb {Z})$ , which corresponds to choosing a branch of logarithm such that $(2m-1)\\pi <\\mathrm {Im}\\,[\\log z]\\le (2m+1)\\pi $ .",
"The pseudo entropy and $\\Delta S$ depend on which branch we choose because of differentiating $(-1)^{|a-b|\\,n}$ with respect to $n$ .",
"While the usual entanglement entropy also depends on the branch, it does not affect the real part.",
"Therefore, it seems to be unnatural that the real part of the pseudo entropy, and $\\Delta S$ , depends on the branch.",
"To avoid this obstruction, we have to use a prescription that does not include the derivative of $(-1)^{|a-b|\\,n}$ ."
],
[
"(2) Restricting $\\mathbf {n}$ to odd numbers",
"In the previous calculation, the obstruction is the existence of $(-1)^{|a-b|\\, n}$ .",
"Here we restrict $n$ to odd so that $(-1)^{|a-b|\\,n}$ reduces to $(-1)^{|a-b|}$ and after that analytically continue $n$ to real numbers.A similar method was used for the calculation of the entanglement entropy for Dirac fields [38].",
"Also the logarithmic negativity calculation [37] employs the analytic continuation of even $n$ .",
"Here we simply assume odd $n$ in continuing to $n=1$ as the (pseudo) Rényi entropy goes to the (pseudo) entanglement entropy in the limit.",
"In this case, $\\begin{aligned}S\\left(\\tau ^{a|b}_A\\right)&=-\\log \\mathcal {D}+\\log \\left[\\frac{[N]}{[2]}\\right]+\\log \\left[q^{\\frac{|a-b|}{2}}\\,[N+1]+(-1)^{|a-b|}\\,q^{-\\frac{|a-b|}{2}}\\,[N-1]\\right] \\\\&\\qquad \\qquad - {\\rm i}\\,\\frac{\\pi |a-b|}{N+k}\\, \\frac{q^{\\frac{|a-b|}{2}}\\,[N+1]-(-1)^{|a-b|}\\,q^{-\\frac{|a-b|}{2}}\\,[N-1]}{q^{\\frac{|a-b|}{2}}\\,[N+1]+(-1)^{|a-b|}\\,q^{-\\frac{|a-b|}{2}}\\,[N-1]}\\ ,\\end{aligned}$ and $\\begin{aligned}\\Delta S&=-\\log \\left( [N]\\,[2] \\right)+ \\frac{1}{2}\\,\\log \\left[[N+1]^2+[N-1]^2 + 2\\,(-1)^{|a-b|}\\,\\cos \\left(\\frac{2\\pi |a-b|}{N+k}\\right)[N+1]\\,[N-1]\\right]\\\\&\\qquad + (-1)^{|a-b|}\\,\\frac{2\\pi \\,|a-b|}{N+k}\\,\\frac{\\sin \\left(\\frac{2\\pi \\,|a-b|}{N+k}\\right)[N+1]\\,[N-1]}{[N+1]^2+[N-1]^2 + 2\\,(-1)^{|a-b|}\\,\\cos \\left(\\frac{2\\pi |a-b|}{N+k}\\right)[N+1]\\,[N-1] }\\ .\\end{aligned}$ Figure: The difference ΔS\\Delta S of the pseudo entropy from the averaged entanglement entropy as a function of the levels kk when N=5N=5.",
"The left panel shows ΔS\\Delta S of the form () by the second prescription (2) of analytic continuation.",
"The blue, orange, green and red curves represent the cases with |a-b|=1,2,3,4|a-b|=1,2,3,4 respectively.",
"For comparison, the right panel shows ΔS\\Delta S of the form () by a naive prescription (1) when |a-b||a-b| is odd.",
"For even |a-b||a-b|, ΔS\\Delta S takes the same values as the left panel.$\\Delta S$ depends on $a$ and $b$ only through the difference $|a-b|$ and highly depends on whether $|a-b|$ is even or odd through the sign factor $(-1)^{|a-b|}$ .",
"The left panel of figure REF shows the difference $\\Delta S$ for several choices of the level $k$ when $N=5$ (For comparison the right panel shows $\\Delta S$ calculated by the previous prescription only for odd $|a-b|$ in the right panel).",
"The four curves represent the cases of $|a-b|=1,\\ldots ,4$ .",
"The figure shows that $\\Delta S$ can be positive only when $|a-b|$ is even.",
"In the classical limit $k\\rightarrow \\infty $ , $[x]$ reduces to $x$ , so $\\Delta S\\rightarrow {\\left\\lbrace \\begin{array}{ll}0 & |a-b|:\\text{even} \\\\-\\log N & |a-b|:\\text{odd}\\end{array}\\right.",
"}$ Refer also to appendix B for the SU$(2)$ case.",
"This can also be seen in figure REF .",
"We can interpret this behavior as follows.",
"Whether $a$ is even or odd determines the pairs of the excitations connected by Wilson lines in $\\mathinner {|{\\psi _a}\\rangle }$ (see figures in (REF )-(REF )).",
"Therefore if $|a-b|$ is even, the pairs of excitations connected in $\\mathinner {|{\\psi _a}\\rangle }$ and those in $\\mathinner {|{\\psi _b}\\rangle }$ are same, but those are different if $|a-b|$ is odd.",
"(REF ) shows that the links of Wilson lines do not contribute to the pseudo entropy in the classical limit.",
"When $|a-b|$ is even, $\\Delta S$ goes to zero because we can regard $\\mathinner {|{\\psi _a}\\rangle }$ and $\\mathinner {|{\\psi _b}\\rangle }$ as the same states in the classical limit.",
"When $|a-b|$ is odd, $\\Delta S$ has a contribution from the difference of the pairs of excitations.",
"In [17], it was shown in multi-qubit systems that if $\\mathinner {|{\\psi }\\rangle }$ and $\\mathinner {|{\\varphi }\\rangle }$ are related by an entanglement swapping, then $\\Delta S<0$ .",
"Moreover in the case of odd $|a-b|$ , the result (REF ) can be understood as a consequence of entanglement swapping (see figure REF ).",
"Furthermore, it is also important that $\\Delta S$ is non-positive in the classical limit.",
"We can see that $\\Delta S$ can be positive (for $a-b$ even) only when the quantum effect from the links of Wilson loops give a huge contribution to the pseudo entropy.",
"This is also consistent with the results in the transverse Ising model [23] and the XY model [24].",
"In such situations, $\\Delta S$ plays a role of the order parameter diagnosing whether the two states $\\mathinner {|{\\psi }\\rangle }$ and $\\mathinner {|{\\varphi }\\rangle }$ , used in the definition of the transition matrix, are in the same phase or not.",
"The transverse Ising model, for example, has the paramagnetic and ferromagnetic phase, which are called quantum phases because those phases are emergent only in quantum systems.",
"Therefore, we may conclude that $\\Delta S$ captures the quantum-theoretic difference between the two states $\\mathinner {|{\\psi }\\rangle }$ and $\\mathinner {|{\\varphi }\\rangle }$ .",
"Figure: The partition function for the pseudo entropy with nn odd and even and their values.",
"In the odd case, we can interpret that the two states are related by the entanglement swapping.",
"In the even case, it can be regarded as two copies of entangled pairs."
],
[
"Geometrical interpretation",
"Motivated by the geometric formula of holographic entanglement entropy [18], [19], [20], [22], we explore a possible geometric interpretation of topological pseudo entropy in the Chern-Simons gauge theory.",
"Consider Wilson loops on $\\mathbb {S}^3$ and divide the sphere into two hemispheres.",
"The surface of each hemisphere is $\\mathbb {S}^2$ and we separate $\\mathbb {S}^2$ into two regions $A$ and $B$ along a curve $\\Gamma (=\\partial A=\\partial B)$ .",
"When there are no Wilson loops, it is clear that the topological entanglement entropy is simply given by $S\\left(\\rho _A\\right)=-n({\\Gamma })\\, \\log {\\cal D}\\ ,$ where $n({\\Gamma })$ is the number of connected components of $\\Gamma $ .",
"If $\\Gamma $ is connected, i.e., $n(\\Gamma )=1$ and the $\\Gamma $ intersects with only one Wilson line in the fundamental representation (see the left of figure REF ), it is easy to evaluate the topological pseudo entropy: $S\\left(\\tau _A\\right)=\\log d_j-\\log {\\cal D}\\ .$ Figure: The intersections between the entangling surface Γ\\Gamma and the Wilson loops.",
"The left panel describes the setup with n(Γ∪W)=1n(\\Gamma \\cup W)=1.",
"In the right panel, we count it as n(Γ∪W)=2n(\\Gamma \\cup W)=2.However, it is not straightforward to find a simple formula in more general cases.",
"Thus, we focus on the semi-classical limit $k\\rightarrow \\infty $ .",
"In this limit, if $\\Gamma $ is connected, we can find the following simple result: $S\\left(\\tau _A\\right)=\\sum _{j}\\,n_i({\\Gamma }\\cap W)\\, \\log d_j-\\log {\\cal D}\\ .$ We defined $n_j({\\Gamma }\\cap W)$ to be the number of the Wilson loops $W$ in the representation $R_j$ which wrap on $\\Gamma $ , as illustrated in figure REF .",
"We may regard $n({\\Gamma }\\cap W)$ as the number of entangled pairs given by the Wilson lines.",
"This is qualitatively similar to the holographic entanglement entropy, where the entanglement entropy is proportional to the area of codimension-two surface like $\\Gamma $ .",
"The holographic entanglement entropy suggests a heuristic picture of emergent spacetime from quantum entanglement in that a Bell pair per Planck unit area is expected to be penetrated on the codimension-two surface.",
"Indeed in our topological entropy, the Wilson loop is linked with $\\Gamma $ , which gives the contribution proportional to $\\log d_j$ .",
"On the other hand, the term proportional to $-\\log {\\cal D}$ is analogous to the gravity edge mode contribution."
],
[
"Topological pseudo entropy on $\\mathbb {T}^2$ with Wilson loops",
"We move onto the case where the subsystem $A$ is a cylinder on a torus $\\mathbb {T}^2$ and where there is a Wilson loop in the interior of $\\mathbb {T}^2$ winding handle.",
"This is depicted as the vertical subsystem in the upper figure REF .",
"Figure: Topological entanglement entropy of states on a solid torus with a Wilson loop insertion.",
"There are two ways to choose the subsystem AA on the surface; the vertical subsystem [Above] and horizontal subsystem [Below].By the same calculations as [7], we have $\\begin{aligned}{\\rm Tr}_A\\left[\\left({\\rm Tr}_B\\left[\\,\\mathinner {|{R_i}\\rangle }\\mathinner {\\langle {R_j}|}\\,\\right]\\right)^n\\right]&=Z\\left[\\mathbb {S}^3;R_i\\right]^{2(1-n)}\\,\\delta _{ij}\\\\&=\\left({\\mathcal {S}_{0}}^{i}\\right)^{2(1-n)}\\,\\delta _{ij}\\ ,\\end{aligned}$ where $\\mathinner {|{R_j}\\rangle }$ is a state including a Wilson loop in the representation denoted by $j$ .",
"For general unnormalized states $\\mathinner {|{\\psi }\\rangle }=\\sum _i\\psi _i\\mathinner {|{R_i}\\rangle }\\ ,\\qquad \\mathinner {|{\\varphi }\\rangle }=\\sum _i\\varphi _i\\mathinner {|{R_i}\\rangle }\\ ,$ the trace of the $n^{\\text{th}}$ power of the reduced transition matrix is $\\begin{aligned}{\\rm Tr}_A\\left[\\left({\\rm Tr}_B\\left[\\,\\mathinner {|{\\psi }\\rangle }\\mathinner {\\langle {\\varphi }|}\\,\\right]\\right)^n\\right]&=\\sum _i\\varphi _i^*\\psi _i\\cdots \\varphi _i^*\\psi _i\\,{\\rm Tr}_A\\left[\\left({\\rm Tr}_B\\left[\\,\\mathinner {|{R_i}\\rangle }\\mathinner {\\langle {R_i}|}\\,\\right]\\right)^n\\right] \\\\&=\\sum _i(\\varphi _i^*\\psi _i)^n\\left({\\mathcal {S}_{0}}^{i}\\right)^{2(1-n)}\\ .\\end{aligned}$ Thus ${\\rm Tr}_A\\left[\\left(\\tau _A^{\\psi |\\varphi }\\right)^n\\right]=\\frac{\\sum _i(\\varphi _i^*\\psi _i)^n\\left({\\mathcal {S}_{0}}^{i}\\right)^{2(1-n)}}{\\left(\\sum _i\\varphi _i^*\\psi _i\\right)^n}\\ .$ This leads to the pseudo entropy given by $S\\left(\\tau _A^{\\psi |\\varphi }\\right)=\\log \\left[\\sum _{i}\\varphi ^*_i\\psi _i\\right]-\\sum _{i}\\varphi ^*_i\\psi _i \\log \\frac{\\varphi ^*_i\\psi _i}{\\left({\\mathcal {S}_{0}}^{i}\\right)^2} \\ .$ When we consider the topological entanglement entropy for the Wilson line $R_i$ we have $\\begin{aligned}S\\left(\\rho _A\\right)&=2\\log {\\mathcal {S}_{0}}^{i}\\\\&=-2\\log {\\cal D}+ 2\\log d_i\\ .\\end{aligned}$ The difference $\\Delta S$ of the pseudo entropy from the average of entanglement entropy is calculated by $\\begin{aligned}\\Delta S&=\\log \\frac{|\\mathinner {\\langle {\\varphi |\\psi }\\rangle }|^2}{\\sqrt{\\mathinner {\\langle {\\psi |\\psi }\\rangle }\\mathinner {\\langle {\\varphi |\\varphi }\\rangle }}}- \\frac{1}{\\sum _k \\varphi _k^*\\psi _k} \\sum _i\\varphi _i^*\\psi _i\\log \\frac{\\varphi ^*_i\\psi _i}{\\left({\\mathcal {S}_{0}}^{i}\\right)^2} \\\\&\\qquad + \\frac{1}{2}\\left[ \\frac{1}{\\sum _k|\\psi _k|^2}\\sum _i|\\psi _i|^2\\log \\frac{|\\psi _i|^2}{\\left({\\mathcal {S}_{0}}^{i}\\right)^2} + \\frac{1}{\\sum _k|\\varphi _k|^2}\\sum _i|\\varphi _i|^2\\log \\frac{|\\varphi _i|^2}{\\left({\\mathcal {S}_{0}}^{i}\\right)^2}\\right] \\ .\\end{aligned}$ On the other hand, if we consider the horizontal subsystem in figure REF , the topological entanglement entropy is found as follows.",
"First, the replica method gives $\\begin{aligned}{\\rm Tr}_A\\left[\\left(\\rho _A\\right)^n\\right]&=\\frac{Z_{2n}}{(Z_2)^n}\\\\&=\\sum _{j_1,\\ldots ,j_{2n-3}}{N_{\\bar{i}\\bar{i}}}^{j_1}{N_{ij_1}}^{j_2}{N_{\\bar{i}j_2}}^{j_3}\\cdot \\cdot \\cdot {N_{\\bar{i}j_{2n-4}}}^{j_{2n-3}} {N_{ij_{2n-3}}}^{\\bar{i}} \\\\&=\\sum _j\\frac{\\left|{\\mathcal {S}_{i}}^{j}\\right|^{2n}}{\\left|{\\mathcal {S}_{0}}^{j}\\right|^{2n-2}}\\ ,\\end{aligned}$ where $Z_{2n}$ is the partition function on $\\mathbb {S}^1\\times \\mathbb {S}^2$ with $n$ Wilson lines $R_i$ and $n$ Wilson lines $R^*_i$ winding around $\\mathbb {S}^1$ .",
"Finally, we find the entanglement entropy $S_A=-\\sum _{j} \\left|{\\mathcal {S}_{i}}^{j}\\right|^2\\log \\frac{\\left|{\\mathcal {S}_{i}}^{j}\\right|^2}{\\left|{\\mathcal {S}_{0}}^{j}\\right|^2}\\ .$ We will see later that this coincides with a finite term of the topological interface entropy in (REF ).",
"We can also get (REF ) by setting $\\psi _j=\\varphi _j={\\mathcal {S}_{i}}^{j}$ ."
],
[
"Possible definition of boundary states in Chern-Simons theory",
"Consider a path integral on a three-dimensional hemisphere or a ball in Chern-Simons theory.",
"We divide its boundary given by $\\mathbb {S}^2$ into $A$ and $B$ , such that they are two dimensional hemispheres.",
"Now we can define the Ishibashi-type state $|I_i\\savebox {}{\\m@th {\\rangle }}\\mathclose {\\copy \\hspace{0.0pt}\\usebox {}}$ as the path integral on the three-dimensional hemisphere $\\mathbb {B}^3$ with an open Wilson line with the representation $R_i$ such that one of its end points is on $A$ and the other is on $B$ (see figure REF ).The Ishibashi-like state $|I_i\\savebox {}{\\m@th {\\rangle }}\\mathclose {\\copy \\hspace{0.0pt}\\usebox {}}$ we define in 3d Chern-Simons theory is different from the Ishibashi state $|i\\savebox {}{\\m@th {\\rangle }}\\mathclose {\\copy \\hspace{0.0pt}\\usebox {}}$ in 2d BCFT used in section .",
"Obviously, they satisfy the same relation as the Ishibashi-type states in boundary CFT$_2$ : $\\savebox {}{\\m@th {\\langle }}\\mathopen {\\copy \\hspace{0.0pt}\\usebox {}}I_i| I_j\\savebox {}{\\m@th {\\rangle }}\\mathclose {\\copy \\hspace{0.0pt}\\usebox {}}=\\delta _{ij}\\,{\\mathcal {S}_{0}}^{i}\\ .$ It is also straightforward to calculate the entanglement entropy $S_A$ of $|I_i\\savebox {}{\\m@th {\\rangle }}\\mathclose {\\copy \\hspace{0.0pt}\\usebox {}}$ via the replica trick and this leads to $S_A=\\log {\\mathcal {S}_{0}}^{i}= -\\log {\\cal D}+ \\log d_i \\ .$ If we consider the linear combination state $\\begin{aligned}|\\psi \\rangle &=\\sum _{i}\\psi _i\\, |I_i\\savebox {}{\\m@th {\\rangle }}\\mathclose {\\copy \\hspace{0.0pt}\\usebox {}}\\ ,\\\\|\\varphi \\rangle &=\\sum _{i}\\varphi _i\\, |I_i\\savebox {}{\\m@th {\\rangle }}\\mathclose {\\copy \\hspace{0.0pt}\\usebox {}}\\ ,\\end{aligned}$ then the transition matrix looks like $\\tau _A^{\\psi |\\varphi }=\\frac{\\mbox{Tr}_B\\left[\\,|\\psi \\rangle \\langle \\varphi |\\,\\right]}{\\langle \\varphi |\\phi \\rangle }=\\frac{\\sum _i \\varphi ^*_i\\psi _i\\, \\mbox{Tr}_B\\left[\\,|I_i\\savebox {}{\\m@th {\\rangle }}\\mathclose {\\copy \\hspace{0.0pt}\\usebox {}}\\savebox {}{\\m@th {\\langle }}\\mathopen {\\copy \\hspace{0.0pt}\\usebox {}}I_i|\\,\\right]}{\\sum _i \\varphi ^*_i\\psi _i\\, {\\mathcal {S}_{i}}^{0}}\\ .$ We can calculate the pseudo entropy ${\\rm Tr}_A\\left[\\left(\\tau _A^{\\psi |\\varphi }\\right)^n\\right]=\\frac{\\sum _i (\\varphi ^*_i\\psi _i)^n\\, {\\mathcal {S}_{i}}^{0}}{\\left(\\sum _{i} \\left(\\varphi ^*_i\\psi _i\\right)^n\\, {\\mathcal {S}_{i}}^{0}\\right)^n}\\ ,$ leading to the expression $S\\left(\\tau ^{\\psi |\\varphi }_A\\right)=\\log \\left[\\sum _{i}\\varphi ^*_i\\psi _i\\, {\\mathcal {S}_{i}}^{0}\\right]-\\frac{\\sum _i \\varphi ^*_i\\psi _i\\, {\\mathcal {S}_{i}}^{0}\\,\\log (\\varphi ^*_i\\psi _i)}{\\sum _i \\varphi ^*_i \\psi _i\\, {\\mathcal {S}_{i}}^{0}}\\ .$ Figure: Analogues of boundary states in Chern-Simons theory.",
"The Ishibashi-like state is defined as a state on the surface (𝕊 2 \\mathbb {S}^2) of a ball where a pair of excitations is located across the common boundary of the two regions AA and BB [Left].",
"On the other hand the Cardy-like state is defined as a state on the surface of a ball with the inner boundary surface with a specific boundary condition corresponding to () [Right].Next, we introduce the Cardy-type state byWhile the Cardy-like state $\\mathinner {|{B_a}\\rangle }$ satisfies the same relation (REF ) as the Cardy state $\\mathinner {|{a}\\rangle }$ in 2d BCFT they are different states.",
"(refer to figure REF ) $|B_a \\rangle =\\sum _{i}\\frac{{\\mathcal {S}_{a}}^{i}}{\\sqrt{{\\mathcal {S}_{0}}^{i}}}\\,|I_i\\savebox {}{\\m@th {\\rangle }}\\mathclose {\\copy \\hspace{0.0pt}\\usebox {}}\\ .$ It is easy to show that the Cardy-type states are orthogonal to each other $\\langle B_a| B_b\\rangle =\\delta _{ab}\\ .$ Even though in physical two-dimensional CFTs, the Cardy-type state satisfied the open-closed duality, the above result corresponds to the truncation to the lowest energy mode of open string.",
"If we calculate the topological entanglement entropy for $|B_a\\rangle $ we get from (REF ) $S_A=-\\sum _{i} \\left({\\mathcal {S}_{a}}^{i}\\right)^2\\,\\log \\frac{\\left({\\mathcal {S}_{a}}^{i}\\right)^2}{{\\mathcal {S}_{0}}^{i}}\\ .$ This is the same as the finite part of the left-right entanglement entropy (REF ) of the Cardy state characterized by the boundary condition $a$ in 2d boundary CFT examined in section .",
"We can also evaluate the partition function on the hemisphere $\\mathbb {B}^3$ with the boundary condition of $|B_a\\rangle $ : $Z\\left[\\mathbb {B}^3; B_a\\right]=\\savebox {}{\\m@th {\\langle }}\\mathopen {\\copy \\hspace{0.0pt}\\usebox {}}I_0|B_a \\rangle ={\\mathcal {S}_{a}}^{0}\\sqrt{{\\mathcal {S}_{0}}^{0}}\\ .$ We define the $g$ -function by $g_a=\\savebox {}{\\m@th {\\langle }}\\mathopen {\\copy \\hspace{0.0pt}\\usebox {}}\\tilde{I}_0| B_a \\rangle ={\\mathcal {S}_{a}}^{0}\\ ,$ where $|\\tilde{I}_0\\savebox {}{\\m@th {\\rangle }}\\mathclose {\\copy \\hspace{0.0pt}\\usebox {}}=\\frac{1}{\\sqrt{{\\mathcal {S}_{0}}^{0}}}\\,|I_0\\savebox {}{\\m@th {\\rangle }}\\mathclose {\\copy \\hspace{0.0pt}\\usebox {}}$ is the normalized vacuum state.",
"Then we have $Z\\left[\\mathbb {B}^3; B_a\\right]=\\sqrt{{\\mathcal {S}_{0}}^{0}}\\ g_a\\ .$ Note that the partition functions we have obtained above satisfy $Z\\left[\\mathbb {S}^3\\right]=\\sum _a Z\\left[\\mathbb {B}^3; B_a\\right]^*Z\\left[\\mathbb {B}^3; B_a\\right]\\ ,$ which can be regarded as a completeness relation for $\\mathinner {|{B_a}\\rangle }$ .",
"Finally, we calculate the entanglement entropy of the vacuum state on a disk $\\mathbb {D}^2$ with the Cardy-type boundary condition.",
"By (REF ), the resulting entanglement entropy takes the form $\\begin{aligned}\\log Z\\left[\\mathbb {B}^3; B_a\\right]=\\frac{1}{2}\\log Z\\left[\\mathbb {S}^3\\right]+\\log g_a\\ .\\end{aligned}$ The second term is analogous to the boundary entropy in BCFT.",
"Indeed, the form $\\log Z\\left[\\mathbb {B}^d; B_a\\right]-\\frac{1}{2}\\log Z\\left[\\mathbb {S}^d\\right]$ is proposed to be a candidate for a $C$ -function in 3d [39] and 4d [40], and it was shown in $d$ -dimensional BCFT that the boundary entropy is defined by $S_{\\text{bdy}}=S^{\\text{(BCFT)}}-S^{\\text{(CFT)}}/2$ equals to (REF ) up to a UV divergence [41]."
],
[
"Pseudo entropy in CFT",
"We switch gears and move to examining the pseudo entropy for a simple choice of the entangling region in CFT.",
"In section REF we review the Casini-Huerta-Myers (CHM) map for a spherical entangling surface on $\\mathbb {R}^d$ , and describe the pseudo entropy as the path integral on $\\mathbb {S}^1\\times \\mathbb {H}^{d-1}$ .",
"In section REF we illustrate the application of the CHM map by showing the calculation of the pseudo entropy in the three-dimensional Chern-Simons theory, reproducing the results in section REF from a slightly different viewpoint.",
"We then expand on the relation between the pseudo entropy and interface entropy in CFT$_2$ , which allows us to read off the pseudo entropies of non-topological theories from their interface entropies in section REF ."
],
[
"Conformal map",
"We begin with reviewing the CHM map [42] that equates the entanglement entropy across a sphere in CFT$_d$ to the calculation of the partition function on $\\mathbb {S}^d$ or $\\mathbb {S}^1\\times \\mathbb {H}^{d-1}$ .",
"Figure: The spherical entangling surface Σ=∂A\\Sigma =\\partial A at a time slice (t=0t=0) in flat space ℝ d \\mathbb {R}^{d}.In what follows, we bipartite a constant time slice into two regions $A$ and its complement $B$ in flat space $\\mathbb {R}^d$ with the metric: $\\mathrm {d}s^2_\\text{Flat} = \\mathrm {d}t^2 + \\mathrm {d}r^2 + r^2\\,\\mathrm {d}\\Omega _{d-2}^2 \\ ,$ and let the entangling surface $\\Gamma = \\partial A$ be spherical (see figure REF ): $\\Gamma = \\lbrace t=0,\\, r = R \\rbrace \\ .$ The flat space is conformally equivalent to $\\mathbb {S}^1 \\times \\mathbb {H}^{d-1}$ with the metric $\\mathrm {d}s_\\text{Hyp}^2 = \\mathrm {d}\\tau ^2 + \\mathrm {d}u^2 + \\sinh ^2 u\\,\\mathrm {d}\\Omega _{d-2}^2 \\ , \\qquad (0\\le \\tau < 2\\pi , \\, 0\\le u < \\infty ) \\ ,$ by the CHM map [42] $t = R\\, \\frac{\\sin \\tau }{\\cosh u + \\cos \\tau } \\ , \\qquad r = R\\, \\frac{\\sinh u}{\\cosh u + \\cos \\tau } \\ .$ Indeed, the two spaces are related by $\\mathrm {d}s^2_\\text{Flat} = \\Omega _\\text{Hyp}^2\\,\\mathrm {d}s^2_\\text{Hyp} \\ , \\qquad \\Omega _\\text{Hyp} = \\frac{R}{\\cosh u + \\cos \\tau } \\ .$ In the latter geometry, the replica geometry can be given simply by scaling the periodicity of $\\tau $ by $n$ .",
"The entangling region $A$ (and its complement $B$ ) is mapped to the time slice at $\\tau = 0$ (and at $\\tau = \\pi $ ) and $\\Gamma $ is pushed to the infinity of the hyperbolic space (see figure REF ): $\\Gamma = \\lbrace \\tau = 0, \\, u = \\infty \\rbrace \\ .$ Figure: The Euclidean configuration for the pseudo entropy across the spherical entangling surface after the CHM map [Left] and the path integral representation of the transition matrix τ A ψ|ϕ \\tau ^{\\psi |\\varphi }_A [Right].Now let us turn to the pseudo entropy between two states $|\\psi \\rangle $ and $|\\varphi \\rangle $ .",
"To prepare the transition matrix $\\tau _A^{\\psi |\\varphi }$ we use the Euclidean path integral where the ket state $|\\psi \\rangle $ is represented as a path integral from $t=-\\infty $ to $t=0$ while the bra state $\\langle \\varphi |$ represented as a path integral from $t=\\infty $ to $t=0$ in the flat space.",
"After the CHM map, each state covers half of the cylinder as in figure REF : $\\langle \\varphi | :~ \\tau \\in [0, \\pi ] \\ , \\qquad |\\psi \\rangle :~ \\tau \\in [\\pi , 2\\pi ] \\ .$ With this in mind the $n^{\\text{th}}$ pseudo Rényi entropy defined by (REF ) is calculable from the path integral representation of the replica partition function: ${\\rm Tr}_A\\left[ \\left(\\tau _A^{\\psi |\\varphi }\\right)^n\\right] \\equiv \\frac{Z(n)}{(Z(1))^n}\\ , \\qquad Z(1) \\equiv \\langle \\varphi | \\psi \\rangle \\ .$"
],
[
"Chern-Simons calculation revisited",
"To illustrate the application of the CHM map, we revisit the pseudo entropy in the three-dimensional Chern-Simons theory considered in section REF .",
"Hence, we focus on states with two excitations in the entangling region $A$ and the other two excitations in the complement in the Chern-Simons theory.",
"Corresponding to the two cases studied in section REF there are two configurations depending on whether the two excitations in $A$ are in the same representation or not as shown in figure REF .",
"Figure: The spherical entangling region with two excitations inside and the other two outside.",
"There are two configurations: (1) two excitations in AA are in the same representation [Left], (2) two excitations in AA are in the conjugate representation to each other [Right]."
],
[
"Case 1",
"We begin with the case with two excitations in the fundamental representation $R_j$ inside $A$ and two excitations in the anti-fundamental representation $R_{\\bar{j}}$ .",
"The two states $|\\psi \\rangle $ , $|\\varphi \\rangle $ given by (REF ) are conformally equivalent to the configuration in figure REF .",
"Figure: The path integral representation of the transition matrix τ A ψ|ϕ \\tau _A^{\\psi |\\varphi } for the case with two excitations in the fundamental representation inside AA.",
"We draw only the two-dimensional space parametrized by the coordinates τ\\tau and uu.The replica partition function can be given by gluing $n$ copies of the transition matrix cyclically along $\\tau $ , resulting in a cylinder of circumference $2\\pi n$ (times $\\mathbb {H}^2$ ) with $n$ disjoint Wilson loops inserted.",
"The replica manifold is topologically $\\mathbb {S}^3$ , so we find $Z(n) = Z\\left[\\mathbb {S}^3; R_j^{\\otimes n}\\right] \\ .$ One can simplify the right hand side to a product of the partition functions on $\\mathbb {S}^3$ with one Wilson loop insertion using the relation (REF ): $Z(n) = \\frac{Z\\left[\\mathbb {S}^3; R_j\\right]^n}{Z\\left[\\mathbb {S}^3\\right]^{n-1}} \\ .$ Hence we find the pseudo entropy of this configuration: $\\begin{aligned}S\\left( \\tau _A^{\\psi |\\varphi }\\right)&=\\log Z\\left[\\mathbb {S}^3\\right] \\\\&=- \\log {\\cal D}\\ ,\\end{aligned}$ which reproduces the previous result (REF )."
],
[
"Case 2",
"Next we move to the second case with two excitations, one in the fundamental representation and the other in the anti-fundamental representation inside $A$ , and take the two states $|\\psi \\rangle , |\\varphi \\rangle $ as in figure REF .",
"This configuration corresponds to the choice of the states $|\\psi \\rangle = |\\psi _0\\rangle $ and $|\\phi \\rangle = |\\psi _1\\rangle $ in section REF .",
"Figure: The path integral representation of the transition matrix τ A ψ|ϕ \\tau _A^{\\psi |\\varphi } for the case with two excitations, one in the fundamental representation and the other in the anti-fundamental representation inside AA.In this case, the replica partition function falls into two classes depending on whether $n$ is even or not."
],
[
"$n$ : even",
"When $n$ is even the replica manifold is topologically equivalent to $\\mathbb {S}^3$ with two Wilson loops inserted with linking number $n/2$ : $\\begin{aligned}Z(n)&=Z\\left[\\mathbb {S}^3; R_j^{\\otimes 2}\\big |_{n/2\\, \\text{link}}\\right] \\\\&=\\frac{[N]}{{\\cal D}\\,[2]}\\left[ q^{-\\frac{(N-1)n}{2}}\\,[N+1] + q^{-\\frac{(N+1)n}{2}}\\,[N-1] \\right] \\ ,\\end{aligned}$ where we use (REF ) in the last line.",
"If we analytically continue even $n$ to a real number and calculate the pseudo entropy from the above partition function, we find $\\begin{aligned}S\\left( \\tau _A^{\\psi |\\varphi }\\right)&=- \\log {\\cal D}+ \\log \\left[ \\frac{[N]}{[2]}\\right]+ \\log \\left[ q^\\frac{1}{2}\\,[N+1] + q^{-\\frac{1}{2}}\\,[N-1]\\right] \\\\&\\qquad -{\\rm i}\\,\\frac{\\pi }{N+k}\\, \\frac{q^\\frac{1}{2}\\,[N+1] - q^{-\\frac{1}{2}}\\,[N-1]}{q^\\frac{1}{2}\\,[N+1] + q^{-\\frac{1}{2}}\\,[N-1]} \\ .\\end{aligned}$"
],
[
"$n$ : odd",
"When $n$ is odd the replica manifold is topologically equivalent to $\\mathbb {S}^3$ with one Wilson loop inserted with $n$ crossings (self-intersection at $n$ points): $\\begin{aligned}Z(n)&=Z\\left[\\mathbb {S}^3; R_j\\big |_{n\\, \\text{crossing}}\\right] \\\\&=\\frac{[N]}{{\\cal D}\\,[2]}\\left[ q^{-\\frac{(N-1)n}{2}}\\,[N+1] - q^{-\\frac{(N+1)n}{2}}\\,[N-1] \\right] \\ ,\\end{aligned}$ where we use again (REF ) in the last line.",
"By analytically continuing $n$ to a real number, the pseudo entropy is calculated to be $\\begin{aligned}S\\left( \\tau _A^{\\psi |\\varphi }\\right)&=- \\log {\\cal D}+ \\log \\left[ \\frac{[N]}{[2]}\\right]+ \\log \\left[ q^\\frac{1}{2}\\,[N+1] - q^{-\\frac{1}{2}}\\,[N-1]\\right] \\\\&\\qquad -{\\rm i}\\,\\frac{\\pi }{N+k}\\, \\frac{q^\\frac{1}{2}\\,[N+1] + q^{-\\frac{1}{2}}\\,[N-1]}{q^\\frac{1}{2}\\,[N+1] - q^{-\\frac{1}{2}}\\,[N-1]} \\ ,\\end{aligned}$ which reproduces (REF ) for $|a-b|=1$ derived with the odd $n$ analytic continuation."
],
[
"Relation to interface entropy in two dimensions",
"The argument for the CHM map slightly differs in two dimensions from the one given in the previous section.",
"To be concrete, we consider an interval $A=[u,v]$ on $\\mathbb {R}$ at time slice $t=0$ and prepare two different states $\\langle \\varphi |$ and $|\\psi \\rangle $ in the Euclidean path integral.",
"Using a transformation $e^w = \\frac{z-u}{v-z}\\ ,$ the original space (parametrized by the complex coordinates $z$ ) can be mapped to a cylinder of circumference $\\tau \\sim \\tau + 2\\pi $ with the coordinates $w\\equiv \\sigma + {\\rm i}\\,\\tau $ as in figure REF .",
"This is indeed a conformal map as seen from the transformation of the metric: $\\mathrm {d}z\\,\\mathrm {d}\\bar{z} = \\left( \\frac{v-u}{4\\,\\sinh (w/2)\\sinh (\\bar{w}/2)}\\right)^2\\,\\mathrm {d}w\\, \\mathrm {d}\\bar{w}\\ .$ Note that the left and right boundaries correspond to the boundaries of the disks around the endpoints of $A$ which play a role of the UV cutoff in calculation of the partition function.",
"Then $Z(n)$ is given by the partition function on the replica manifold which can be constructed straightforwardly by gluing $n$ copies of the strip as in figure REF .",
"Figure: The pseudo entropy for an interval in CFT 2 _2 [Left] and the conformal transformation to the cylinder [Right].Figure: The replica partition function Z(n)Z(n) for the pseudo entropy.It is sometimes useful to make a further map from the cylinder to a sphere by the following coordinate transformation: $\\mathrm {d}w\\, \\mathrm {d}\\bar{w} = \\mathrm {d}\\sigma ^2 + \\mathrm {d}\\tau ^2 = \\frac{1}{\\sin ^2\\phi } \\left[ \\mathrm {d}\\phi ^2 + \\sin ^2\\phi \\,\\mathrm {d}\\tau ^2\\right] \\ ,$ where $\\sigma = \\log \\tan (\\phi /2)$ with $\\phi \\in [0,\\pi ]$ .",
"Combining the two transformations, we find the map from the original space to the sphere: $\\mathrm {d}z\\,\\mathrm {d}\\bar{z} = \\Omega (\\phi ,\\tau )^2 \\left[ \\mathrm {d}\\phi ^2 + \\sin ^2\\phi \\,\\mathrm {d}\\tau ^2\\right] \\ , \\qquad \\Omega \\equiv \\frac{v-u}{2(1 - \\cos \\tau \\sin \\phi )} \\ .$ A closely related measure to the pseudo entropy is the entanglement entropy across a conformal interface, also known as interface entropy.",
"We here consider a restricted case where two different states $|\\psi \\rangle $ and $|\\varphi \\rangle $ in CFT$_2$ are glued along an interface ${\\cal I}$ at the origin of a time slice as in figure REF .",
"The interface entropy is the entanglement entropy for the entangling region $A$ extending from the origin to the right, which quantifies the difference between the two states [25].",
"Figure: The entanglement entropy across a conformal interface ℐ{\\cal I} (interface entropy) in two dimensions [Left].The entangling region AA is taken to be a half line right to the interface ℐ{\\cal I}.The configuration can be mapped to the cylinder by a conformal transformation [Right].Using a canonical conformal map from flat space to a cylinder and gluing $n$ copies along the entangling surface one obtains the replica manifold as a cylinder of circumference $\\tau ~\\tau + 2\\pi n$ with $n$ interfaces inserted at specific locations (see figure REF ): ${\\cal I}: \\left\\lbrace \\tau = \\frac{(2i -1)\\pi }{2} \\ , \\quad (i=1,2,\\cdots , 2n)\\right\\rbrace \\ .$ Compared with figure REF this is the same replica manifold as the pseudo entropy in the previous subsection with $\\tau $ shifted by $\\pi /2$ .",
"Hence, we establish the equality between the pseudo entropy and interface entropy in CFT$_2$ : $S\\left( \\tau ^{\\psi |\\varphi }_{A:\\, \\text{interval}}\\right) = S^{\\cal I}_{A:\\, \\text{half-line}} \\qquad \\text{in CFT$_2$} \\ .$ Figure: The replica partition function for the interface entropy.More generally, taking the entangling surface to be a hyperplane in flat space the pseudo entropy equals to the interface entropy in any QFT in $d\\ge 2$ dimensions: $S\\left( \\tau ^{\\psi |\\varphi }_{A}\\right) = S^{\\cal I}_{A} \\qquad \\text{for} \\quad \\Gamma = \\partial A : \\lbrace x^0=x^1 =0 \\rbrace \\ .$ This is clear from figure REF where one sees the replica manifold of the former is obtained by rotating that of the latter by $\\pi /2$ degree.",
"Figure: The pseudo entropy and interface entropy across a hyperplane.",
"Their replica manifolds are the same up to the τ\\tau shift by π/2\\pi /2."
],
[
"Compact scalar theory",
"We calculate the pseudo entropy between ground states of massless compact bosons of different radii $R_1$ and $R_2$ in two dimensions.",
"When the entangling region is an interval it amounts to the entanglement entropy across an interface between the two compact boson theories as we saw in section REF .",
"There are four types of conformal interfaces ${\\cal I}^\\pm _{k_1k_2}$ labeled by $\\pm $ and their conjugates, which act as intertwiners from one side of a free boson theory to the other.",
"The pair of relatively prime numbers $(k_1, k_2)$ can be interpreted as the winding numbers of D-brane along the two cycles of the torus parametrized by two compact bosons.",
"We focus on the case with ${\\cal I}^+_{k_1k_2}$ and read off the pseudo entropy from the result of [25] for the interface entropy by translating the parameters appropriately: $S^{{\\cal I}^+_{k_1k_2}}\\left( \\tau _A^{\\psi |\\varphi }\\right) = \\sigma _\\text{s} \\left(|\\sin (2\\theta _+)|\\right)\\, \\log \\left( \\frac{L}{\\epsilon }\\right) - \\log |k_1 k_2| \\ ,$ where $L\\equiv v-u$ , $\\epsilon $ are the lengths of the interval and the UV cutoff for the pseudo entropy, respectively.",
"The parameter $\\theta _+$ defined through the relation $\\tan \\theta _+ \\equiv \\frac{k_2 R_2}{k_1 R_1} \\ ,$ controls the transmittance of the interface.",
"The function $\\sigma _\\text{s} (x)$ defined by $\\sigma _\\text{s} (x ) \\equiv \\frac{1}{6} + \\frac{x}{3} + \\frac{1}{\\pi ^2}\\left[ (x+1)\\log (x+1)\\,\\log x + (x-1)\\,\\text{Li}_2( 1-x) + (x+1)\\,\\text{Li}_2 (-x)\\right] \\ ,$ interpolates between $\\sigma _s(0) = 0$ and $\\sigma _s(1)=1/3$ monotonically."
],
[
"Free fermion",
"The entanglement entropy across a conformal interface in the Ising model in two dimensions was investigated in [28].",
"Interfaces in the Ising model can be described as boundary conditions in either the $\\mathbb {Z}_2$ -orbifold theory of a free boson or the real Majorana fermion theory.",
"In the latter description, there is an independent set of interfaces: NS, R, and neutral interfaces, labeled by a parameter $\\phi $ controlling their transmittance.",
"It follows from the result of [28] the pseudo entropy for the NS interface becomes $S^\\text{NS}\\left( \\tau _A^{\\psi |\\varphi }\\right) = \\sigma _\\text{f} \\left(|\\sin (2\\phi )|\\right)\\, \\log \\left( \\frac{L}{\\epsilon }\\right) \\ ,$ where the function $\\sigma _\\text{f} (x)$ defined by $\\sigma _\\text{f} (x ) \\equiv \\frac{x-1}{6} - \\frac{1}{\\pi ^2}\\left[ (x+1)\\log (x+1)\\,\\log x + (x-1)\\,\\text{Li}_2( 1-x) + (x+1)\\,\\text{Li}_2 (-x)\\right] \\ ,$ interpolates between $\\sigma _f(0) = 0$ and $\\sigma _f(1)=1/6$ monotonically.",
"The pseudo entropies for the R and neutral interfaces are also given by $\\begin{aligned}S^\\text{R}\\left( \\tau _A^{\\psi |\\varphi }\\right)&=S^\\text{NS}\\left( \\tau _A^{\\psi |\\varphi }\\right) \\ , \\\\S^\\text{neutral}\\left( \\tau _A^{\\psi |\\varphi }\\right)&=S^\\text{NS}\\left( \\tau _A^{\\psi |\\varphi }\\right) - \\log 2 \\ .\\end{aligned}$"
],
[
"Topological interface",
"Suppose we are given two CFTs glued along a straight line ${\\cal C}$ .",
"To preserve a part of the conformal symmetry, the energy flow perpendicular to ${\\cal C}$ has to be continuous: $T^{(1)} - \\bar{T}^{(1)}|_{{\\cal C}} = T^{(2)} - \\bar{T}^{(2)}|_{{\\cal C}} \\ .$ The gluing line ${\\cal C}$ may be seen as an defect operator ${\\cal I}$ intertwining the Hilbert space ${\\cal H}^{(1)}$ of one theory with the other ${\\cal H}^{(2)}$ .",
"The condition (REF ) implies that such an operator satisfies the commutation relations: $\\left(L^{(1)}_n - \\bar{L}^{(1)}_n\\right)\\,{\\cal I}= {\\cal I}\\,\\left(L^{(2)}_n - \\bar{L}^{(2)}_n\\right) \\ .$ Finding solutions to these relations is a hard problem, but it simplifies if the defects satisfy stronger conditions: $L^{(1)}_n\\,{\\cal I}= {\\cal I}\\,L^{(2)}_n \\ , \\qquad \\bar{L}^{(1)}_n\\,{\\cal I}= {\\cal I}\\,\\bar{L}^{(2)}_n \\ .$ Since they commute with all the Virasoro generators, their locations can be moved freely, and hence ${\\cal I}$ becomes topological.",
"Topological defects have been extensively studied in a rational CFT (RCFT) whose Hilbert space takes the form: ${\\cal H}= \\bigoplus _{i, \\bar{j}}\\,M_{i \\bar{j}}\\, {\\cal V}_i \\otimes \\bar{{\\cal V}}_{\\bar{j}} \\ ,$ where $i$ and $\\bar{j}$ label (a finite number of) irreducible representations ${\\cal V}_i$ and $\\bar{{\\cal V}}_{\\bar{j}}$ of the Virasoro algebra in chiral and anti-chiral sectors respectively and $M_{i\\bar{j}}$ is the multiplicity of the pair representation $(i, \\bar{j})$ appearing in the spectrum of the theory.",
"In this case, the topological defects between two RCFTs with multiplicity matrices $M_{i\\bar{j}}^{(1)}$ and $M_{i\\bar{j}}^{(2)}$ can be written as [43], [29] ${\\cal I}_K = \\sum _{\\bf i}\\,d_{K\\bf i}\\,P^{\\bf i} \\ ,$ where $K$ labels types of topological interface and ${\\bf i} \\equiv (i, \\bar{j}; \\alpha ,\\beta )$ is the index for the projector $P^{\\bf i}$ intertwining between a representation $\\left({\\cal V}_i\\otimes \\bar{{\\cal V}}_{\\bar{j}}\\right)^{(\\alpha )}$ with $\\alpha = 1, \\cdots , M_{i\\bar{j}}^{(1)}$ of one RCFT to another representation $\\left({\\cal V}_i\\otimes \\bar{{\\cal V}}_{\\bar{j}}\\right)^{(\\beta )}$ with $\\beta = 1, \\cdots , M_{i\\bar{j}}^{(2)}$ of the other RCFT.",
"When the two CFTs are isomorphic and hence $M_{i\\bar{j}}^{(1)} = M_{i\\bar{j}}^{(2)}$ , hence for conformal defects between the same CFT, the projector $P^{\\bf i}$ is given a realization as $P^{(i, \\bar{j}; \\alpha , \\beta )} \\equiv \\sum _{ {\\bf n}, \\bar{\\bf n}}\\, \\left(|i, {\\bf n}\\rangle \\otimes |\\bar{j}, \\bar{\\bf n}\\rangle \\right)^{(\\alpha )}\\, \\left(\\langle i, {\\bf n}|\\otimes \\langle \\bar{j}, \\bar{\\bf n}|\\right)^{(\\beta )} \\ ,$ where $|i, {\\bf n}\\rangle \\otimes |\\bar{j}, \\bar{\\bf n}\\rangle $ is an orthogonal basis for the representation ${\\cal V}_i\\otimes \\bar{{\\cal V}}_{\\bar{j}}$ .",
"Clearly ${\\cal I}_K$ satisfies the conditions (REF ) which now reduce to the commutation relations with the Virasoro generators $[L_n, {\\cal I}_K] = [\\bar{L}_n, {\\cal I}_K] = 0 \\ ,$ and hence correlation functions depend only on the homotopy class of the contour of ${\\cal I}_K$ [43].",
"Defects intertwining between the same theory are called interfaces, so topological interfaces are the solutions to the conditions (REF ).",
"The replica partition function for calculating the entanglement entropy across a topological interface is given by a torus partition function with $2n$ insertion of interface operators: $\\begin{aligned}Z_{K}(n)&= {\\rm tr}\\left[ \\left( {\\cal I}_K\\, e^{-t H}\\, {\\cal I}_K^\\dagger \\,e^{-t H}\\right)^n\\right]\\\\&= {\\rm tr}\\left[ \\left({\\cal I}_K\\,{\\cal I}_K^\\dagger \\right)^n\\,e^{-2t H}\\right] \\\\&= \\sum _{(i,\\bar{j})}\\, {\\rm Tr}\\left[\\left(d_{K {\\bf i}}\\,d_{K^\\ast {\\bf i}}\\right)^{n}\\right]\\, \\chi _i \\left(e^{-2nt}\\right)\\, \\chi _{\\bar{j}} \\left(e^{-2nt}\\right) \\ ,\\end{aligned}$ where $H$ is the Hamiltonian on a cylinder $H = L_0 + \\bar{L}_0 - \\frac{c}{12} \\ ,$ and we used the commutation relation (REF ) in the second equality.",
"$\\chi _i$ is the character in the ${\\cal V}_i$ representation and the parameter $t$ is related to the UV and IR cutoffs $\\epsilon $ , $L$ as $t = \\frac{2\\pi ^2}{\\log ( L/\\epsilon )} \\ .$ We took a trace $\\text{Tr}$ with respect to the multiplicity indices $\\alpha , \\beta $ by regarding $d_{K(i,\\bar{j};\\alpha , \\beta )}$ as a matrix with the notation $d_{K^\\ast (i,\\bar{j};\\alpha , \\beta )} \\equiv d_{K(i,\\bar{j};\\beta , \\alpha )}^\\ast $ .",
"Using the modular property of the character, the entanglement entropy becomes [29] $S_{{\\cal I}_K}=\\frac{c}{6}\\log \\left(\\frac{L}{\\epsilon }\\right) -\\sum _{(i, \\bar{j})}\\,{\\rm Tr}\\left[ p^K_{\\bf i}\\,\\log \\frac{p^K_{\\bf i}}{p^\\text{Id}_{\\bf i}}\\right] \\ ,$ where $p^K_{\\bf i}$ is a probability distribution characterized by the modular ${\\cal S}$ -matrix as $p^K_{\\bf i} \\equiv \\frac{{\\mathcal {S}_{0}}^{i}\\,\\left({\\mathcal {S}_{0}}^{\\bar{j}}\\right)^\\ast }{\\sum _{(i,\\bar{j})}\\,{\\mathcal {S}_{0}}^{i}\\,\\left({\\mathcal {S}_{0}}^{\\bar{j}}\\right)^\\ast \\,{\\rm Tr}\\left[ d_{K {\\bf i}}\\,d_{K^\\ast {\\bf i}}\\right]}\\,d_{K {\\bf i}}\\,d_{K^\\ast {\\bf i}} \\ , \\qquad p^\\text{Id}_{\\bf i} \\equiv {\\mathcal {S}_{0}}^{i}\\,\\left({\\mathcal {S}_{0}}^{\\bar{j}}\\right)^\\ast \\,\\delta _{\\alpha \\beta } \\ .$ For diagonal theories with multiplicity $M_{i\\bar{j}} = \\delta _{i\\bar{j}}$ , the CFTs on both sides are the same theory and topological interfaces are one-to-one correspondence to the primary operators labeled by $a$ : ${\\cal I}_a = \\sum _i\\,\\frac{{\\mathcal {S}_{a}}^{i}}{{\\mathcal {S}_{0}}^{i}}\\,P^{i} \\ ,$ where $P^{i}$ is the projector acting on the representation ${\\cal V}_i\\otimes \\bar{{\\cal V}}_{i}$ : $P^i \\equiv \\sum _{{\\bf n}, \\bar{\\bf n}}\\, |i, {\\bf n}\\rangle \\otimes |i, \\bar{\\bf n}\\rangle \\, \\langle i, {\\bf n}|\\otimes \\langle i, \\bar{\\bf n}| \\ .$ Then the probability distribution simplifies to $p^a_i = \\left|{\\mathcal {S}_{a}}^{i}\\right|^2 \\ ,\\qquad p^\\text{Id}_i = \\left|{\\mathcal {S}_{0}}^{i}\\right|^2 \\ ,$ and we find [26], [29] $S_{{\\cal I}_a} = \\frac{c}{6}\\log \\left(\\frac{L}{\\epsilon }\\right) -2\\sum _i\\,\\left|{\\mathcal {S}_{a}}^{i}\\right|^2\\,\\log \\left| \\frac{{\\mathcal {S}_{a}}^{i}}{{\\mathcal {S}_{0}}^{i}}\\right| \\ .$ It would be worthwhile to note that the constant term contributed from the interface takes the same form as the topological entanglement entropy for any state $|\\psi \\rangle $ in the Chern-Simons theory on a torus we considered in section REF (see also section 2.5.2 in [7]): $S_\\psi = -2 \\sum _{i}\\, |\\psi _i|^2\\, \\log \\frac{|\\psi _i|}{{\\mathcal {S}_{0}}^{i}} \\ ,$ where $\\psi _i$ are coefficients for the state $|\\psi \\rangle $ which is the superposition of a single Wilson loop in the irreducible representation $R_i$ : $|\\psi \\rangle = \\sum _i\\,\\psi _i\\,|R_i\\rangle \\ .$ Indeed one can reproduce the finite part of the interface entropy (REF ) by setting $\\psi _i$ to a specific value $\\psi _i = {\\mathcal {S}_{a}}^{i}$ (see (REF )).",
"This coincidence may not be so surprising given the well-known correspondence between the WZW model and the Chern-Simons theory, but we are not aware of any direct link between them."
],
[
"Left-right pseudo entanglement entropy",
"A closely related measure to the interface entropy is the left-right entanglement entropy (LREE) in BCFT$_2$ [34], [35].",
"For a boundary state $|B\\rangle $ subject to the gluing condition $\\left( L_n - \\bar{L}_{-n}\\right)\\,|B\\rangle = 0 \\ ,$ there exists an orthogonal basis spanned by the Ishibashi states $|i\\savebox {}{\\m@th {\\rangle }}\\mathclose {\\copy \\hspace{0.0pt}\\usebox {}}$ [44]: $|i\\savebox {}{\\m@th {\\rangle }}\\mathclose {\\copy \\hspace{0.0pt}\\usebox {}}= \\sum _{{\\bf n}} |i, {\\bf n}\\rangle \\otimes \\overline{|i, {\\bf n}\\rangle } \\ .$ Note that they are non-normalizable states, but their inner product can be regularized asWe can equally normalize the Ishibashi state as $\\savebox {}{\\m@th {\\langle }}\\mathopen {\\copy \\hspace{0.0pt}\\usebox {}}i | j\\savebox {}{\\m@th {\\rangle }}\\mathclose {\\copy \\hspace{0.0pt}\\usebox {}}= \\delta _{ij}$ .",
"This is the normalization employed in [11], which is equivalent to the rescaling of the coefficient such that $\\psi _i\\rightarrow \\psi _i/\\sqrt{{\\mathcal {S}_{0}}^{i}}$ .",
"(see e.g.",
"[45]) $\\savebox {}{\\m@th {\\langle }}\\mathopen {\\copy \\hspace{0.0pt}\\usebox {}}i | j\\savebox {}{\\m@th {\\rangle }}\\mathclose {\\copy \\hspace{0.0pt}\\usebox {}}= \\delta _{ij}\\,{\\mathcal {S}_{0}}^{i} \\ .$ Hence one can expand any boundary state $|\\psi \\rangle $ by the Ishibashi state as follows: $|\\psi \\rangle = \\sum _{i}\\, \\psi _i\\,|i\\savebox {}{\\m@th {\\rangle }}\\mathclose {\\copy \\hspace{0.0pt}\\usebox {}}\\ .$ The LREE is the von Neumann entropy of the reduced density matrix for the left (holomorphic) sector: $\\rho _L^{\\psi } \\equiv \\frac{1}{\\langle \\psi | \\psi \\rangle }\\,{\\rm Tr}_R\\left[ |\\psi \\rangle \\,\\langle \\psi | \\right] \\ .$ Now we introduce the transition matrix for two boundary states $|\\psi \\rangle $ and $|\\varphi \\rangle $ by $\\tau _L^{\\psi |\\varphi } \\equiv \\frac{1}{\\langle \\varphi | \\psi \\rangle }\\,{\\rm Tr}_R\\left[ |\\psi \\rangle \\,\\langle \\varphi | \\right] \\ ,$ and define the left-right pseudo entropy (LRPE) as the von Neumann entropy of the transition matrix.",
"Since the boundary states are non-normalizable we regularize them by slightly evolving them in the imaginary time direction: $|\\psi \\rangle \\rightarrow e^{-\\epsilon H}\\,|\\psi \\rangle \\ .$ For a theory on a cylinder of circumference $\\ell $ the Hamiltonian becomes $H = \\frac{2\\pi }{\\ell } \\left( L_0 + \\bar{L}_0 - \\frac{c}{12}\\right) \\ ,$ which yields the following expressions: $\\begin{aligned}\\langle \\varphi |\\, e^{-2\\epsilon H} \\,| \\psi \\rangle &= \\sum _i\\,\\psi _i\\,\\varphi _i^\\ast \\,\\chi _i \\left( e^{- \\frac{8\\pi \\epsilon }{\\ell }}\\right) \\ , \\\\{\\rm Tr}_R\\left[ e^{-\\epsilon H}|\\psi \\rangle \\,\\langle \\varphi |\\,e^{-\\epsilon H} \\right]&=\\sum _{i, {\\bf n}}\\,\\psi _i\\,\\varphi _i^\\ast \\,\\,e^{- \\frac{8\\pi \\epsilon }{\\ell }\\left( h_i + N_{\\bf n} - \\frac{c}{24} \\right)}\\,|i, {\\bf n}\\rangle \\, \\langle i, {\\bf n}| \\ ,\\\\{\\rm Tr}_L \\left[ \\left(\\tau _L^{\\psi |\\varphi }\\right)^n \\right]&=\\frac{1}{\\left[ \\sum _i\\,\\psi _i\\,\\varphi _i^\\ast \\,\\chi _i \\left( e^{- \\frac{8\\pi \\epsilon }{\\ell }}\\right) \\right]^n} \\, \\sum _i\\,\\left(\\psi _i\\,\\varphi _i^\\ast \\right)^n\\,\\chi _i \\left( e^{- \\frac{8\\pi \\epsilon n}{\\ell }}\\right)\\ .\\end{aligned}$ Here $h_i$ and $N_{\\bf n}$ are the conformal dimension and the level of the descendant state $|i, {\\bf n}\\rangle $ respectively, and $\\chi _i(q)\\equiv {\\rm tr}_{{\\cal V}_i}\\,q^{L_0 - \\frac{c}{24}}$ is the character for the representation $i$ .",
"Using the modular transformation $\\chi _i \\left( e^{-\\frac{8\\pi \\epsilon n}{\\ell }}\\right) = \\sum _{j}{\\mathcal {S}_{i}}^{j}\\,\\chi _j\\left( e^{-\\frac{\\pi \\ell }{2\\epsilon n}}\\right) \\ ,$ and taking the $\\epsilon \\rightarrow 0$ limit we find the LRPE: $\\begin{aligned}S\\left(\\tau _L^{\\psi |\\varphi }\\right)&=- \\partial _n\\,{\\rm Tr}_L \\left[ \\left(\\tau _L^{\\psi |\\varphi }\\right)^n \\right]\\big |_{n=1} \\\\&=\\frac{\\pi c\\ell }{24\\,\\epsilon } - \\frac{\\sum _i\\,{\\mathcal {S}_{i}}^{0}\\,\\psi _i\\,\\varphi _i^\\ast \\,\\log (\\psi _i\\,\\varphi _i^\\ast \\,)}{\\sum _i\\,{\\mathcal {S}_{i}}^{0}\\,\\psi _i\\,\\varphi _i^\\ast }+\\log \\left[\\sum _i\\,{\\mathcal {S}_{i}}^{0}\\,\\psi _i\\,\\varphi _i^\\ast \\right] \\ .\\end{aligned}$ This expression is not necessarily real, but when $\\psi = \\varphi $ it reduces to the LREE derived in [35]: $S^{(\\text{LR})}\\left( |\\psi \\rangle \\right)=\\frac{\\pi c\\ell }{24\\,\\epsilon } - \\frac{\\sum _i\\,{\\mathcal {S}_{i}}^{0}\\,|\\psi _i|^2\\,\\log |\\psi _i|^2}{\\sum _i\\,{\\mathcal {S}_{i}}^{0}\\,|\\psi _i|^2}+\\log \\left[\\sum _i\\,{\\mathcal {S}_{i}}^{0}\\,|\\psi _i|^2\\right] \\ .$ which is real as expected.",
"In a diagonal theory, the Cardy states $|a\\rangle $ can be written as a superposition of the Ishibashi states:Note that this is the same relation as the expression of the interface operator (REF ) if one replaces ${\\mathcal {S}_{0}}^{i}$ with $\\left({\\mathcal {S}_{0}}^{i}\\right)^2$ .",
"$|a\\rangle = \\sum _i\\,\\frac{{\\mathcal {S}_{a}}^{i}}{\\sqrt{{\\mathcal {S}_{0}}^{i}}}\\,|i\\savebox {}{\\m@th {\\rangle }}\\mathclose {\\copy \\hspace{0.0pt}\\usebox {}}\\ .$ As a simple example let us take $|\\psi \\rangle = |a\\rangle \\ , \\qquad |\\varphi \\rangle = |i\\savebox {}{\\m@th {\\rangle }}\\mathclose {\\copy \\hspace{0.0pt}\\usebox {}}\\ ,$ then the LRPE becomes $S\\left(\\tau _L^{\\psi |\\varphi }\\right)=\\frac{\\pi c\\ell }{24\\,\\epsilon } + \\log \\,{\\mathcal {S}_{i}}^{0} \\ .$ Interestingly, the LRPE does not depend on the choice of the Cardy state $|a\\rangle $ as long as it overlaps with the Ishibashi state $|i\\savebox {}{\\m@th {\\rangle }}\\mathclose {\\copy \\hspace{0.0pt}\\usebox {}}$ Next, consider the LREE for the Ishibashi state $|\\psi \\rangle = |i\\savebox {}{\\m@th {\\rangle }}\\mathclose {\\copy \\hspace{0.0pt}\\usebox {}}$ .",
"It follows from (REF ) with $\\psi _i=\\varphi _i =1$ , $\\psi _{k\\ne i} = \\varphi _{k\\ne i} = 0$ that $S^\\text{(LR)}\\left(|i\\savebox {}{\\m@th {\\rangle }}\\mathclose {\\copy \\hspace{0.0pt}\\usebox {}}\\right)=\\frac{\\pi c\\ell }{24\\,\\epsilon } + \\log \\,{\\mathcal {S}_{i}}^{0} \\ .$ This is the same as the LRPE considered above.",
"Another example is the LREE for the Cardy state $|\\psi \\rangle = |a\\rangle $ , which is given by substituting $\\psi _i = \\varphi _i = {\\mathcal {S}_{a}}^{i}/\\sqrt{{\\mathcal {S}_{0}}^{i}}$ to (REF ) [35], [29] $S^\\text{(LR)}\\left(|a\\rangle \\right) = \\frac{\\pi c\\ell }{24\\,\\epsilon } - \\sum _i\\,\\left({\\mathcal {S}_{a}}^{i}\\right)^2\\,\\log \\left[\\frac{({\\mathcal {S}_{a}}^{i})^2}{{\\mathcal {S}_{0}}^{i}}\\right] \\ ,$ which agrees with the topological entanglement entropy (REF ) of the Chern-Simons theory for the Cardy state.",
"Note that (REF ) is close to but differs from (REF ) by the denominator in the logarithm.",
"This difference can be accounted for by the fact that both holomorphic and anti-holomorphic sectors contribute to the interface entropy while there is only the holomorphic sector in the unfolded theory of BCFT in the LREE.",
"In the latter case the correspondence between the topological entanglement entropy in the Chern-Simons theory and the LREE of a boundary state is clear as the left and right moving CFTs appear as the edge modes of the Chern-Simons theory on either side of the entangling surface [35].",
"It would be interesting to understand the above coincidence between the interface entropy in 2d and the topological entropy in 3d along the same lines of thought."
],
[
"Conclusions",
"In this paper, we studied pseudo entropy in quantum field theory, mainly concentrating on its topological properties.",
"In section , we focused on the excited states in Chern-Simons theory.",
"This provides a class of important examples where pseudo entropy can be analytically computed in quantum field theory.",
"We found non-trivial behavior of the pseudo entropy in the presence of four excitations on $\\mathbb {S}^2$ .",
"Such excited states are prepared by path integrals with inserting appropriate Wilson lines.",
"In contrast to topological entanglement entropy, we have seen that topological (Rényi) pseudo entropies are directly related to partition functions with knotted Wilson loops.",
"In other words, generic partition functions with Wilson loops should be interpreted as topological pseudo entropy rather than topological entropy because the initial state and final state are different.",
"Since the dependence on the crossings vanishes in the classical limit, we can interpret that the crossings give purely quantum contributions to pseudo entropy.",
"In particular, it is remarkable that the pseudo entropy may be larger than the entanglement entropy, i.e., the difference $\\Delta S$ defined in (REF ) can be positive.",
"This contrasts with a standard quantum many-body system or quantum field theory within the same quantum phase, where $\\Delta S$ is always non-positive [23], [24].",
"This is consistent with the known fact that the anyonic states created by Wilson loops in Chern-Simons theory belong to different quantum phases in general.",
"Note that this is the first example in dimensions higher than two, where pseudo entropy was explicitly evaluated in non-trivial topological phases.",
"We also explored a geometric interpretation of topological pseudo entropy in Chern-Simons theory.",
"We found a universal result when a single Wilson loop is linked with the surface $\\Gamma =\\partial A$ once.",
"Although universal results are not available in more general cases, we noted that in the semiclassical limit $k\\rightarrow \\infty $ , the topological pseudo entropy captures the number of Wilson loops which link with the surface $\\Gamma =\\partial A$ .",
"This is analogous to the holographic entanglement entropy in the sense that it also measures entanglement or the number of Bell pairs around an extremal surface $\\Gamma $ .",
"The geometrical interpretation including full quantum effects remains as a future problem.",
"In section , we investigated the properties of the pseudo entropy in CFT.",
"In particular, we found the close relation between the pseudo entropy and the interface entropy in 2d CFT, which can be generalized to any QFT in $d\\ge 2$ with the restriction to the case where the subsystem $A$ is a half space, i.e., $\\partial A:\\lbrace x^0=x^1=0\\rbrace $ .",
"The extension of the relation to a more general $A$ would be challenging and is left as a future problem.",
"We used this relation to calculate the pseudo entropies in several interface CFTs.",
"The finite term in the resulting interface entropy coincides with that in Chern-Simons theory on a torus.",
"The CHM map in section REF can be concatenated by a further conformal map $\\tanh \\frac{u}{2} = \\tan \\frac{\\theta }{2}$ to $\\mathbb {S}^d$ with the metric: $\\mathrm {d}s_\\text{Sph}^2 = \\mathrm {d}\\theta ^2 + \\cos ^2 \\theta \\, \\mathrm {d}\\tau ^2 + \\sin ^2\\theta \\,\\mathrm {d}\\Omega _{d-2}^2 \\ , \\qquad \\left(0\\le \\theta \\le \\frac{\\pi }{2}\\right) \\ .$ The resulting map may open the way to evaluate the pseudo entropy in CFT through the sphere partition function with two states glued alternately along the $\\tau $ coordinate.",
"The same map was applied to the interface entropy to derive a universal relation between the entropy and sphere free energy [41], [46], but there is a crucial difference between the pseudo entropy and interface entropy as the number of the interfaces between the two states depends on the replica parameter in the former while it is independent in the latter.",
"Thus, the calculation of the replica partition function is a highly daunting task.",
"While the exact results of such a partition function are far from our reach in general, it would become tractable for supersymmetric field theories.",
"Supersymmetries are broken on the replica manifold due to the conical singularity, but may be maintained by introducing a sort of chemical potential to the pseudo entropy in a similar manner to the supersymmetric Rényi entropy [47], resulting in being calculable due to the supersymmetric localization (see also [48], [49], [50], [51], [52], [53], [54], [55], [56], [57], [58], [59], [60] for the generalizations in various dimensions).",
"In particular, it would be worthwhile to see if the supersymmetric extended pseudo entropy could probe two difference phases related by duality such as the $S$ -duality wall in ${\\cal N}=4$ supersymmetric Yang-Mills theory in four dimensions [61], [62], [63].",
"We are grateful to Yasuaki Hikida, Ali Mollabashi, Kotaro Tamaoka, and Zixia Wei for valuable discussions.",
"The work of T. N. was supported in part by the JSPS Grant-in-Aid for Scientific Research (C) No.19K03863.",
"The work of T. N. and T. T. is supported by the JSPS Grant-in-Aid for Scientific Research (A) No.21H04469.",
"T. T. is supported by the Simons Foundation through the “It from Qubit” collaboration, Inamori Research Institute for Science and World Premier International Research Center Initiative (WPI Initiative) from the Japan Ministry of Education, Culture, Sports, Science and Technology (MEXT).",
"T. T. is also supported by JSPS Grant-in-Aid for Challenging Research (Exploratory) 18K18766."
],
[
"Modular properties in $\\mathrm {SU}(2)$ case",
"Here we summarize the explicit partition functions for the $\\mathrm {SU}(2)$ i.e.",
"$N=2$ case.",
"The ${\\cal S}$ -matrices are ${\\mathcal {S}_{i}}^{j}=\\sqrt{\\frac{2}{k+2}}\\sin \\left[\\frac{\\pi \\,(2i+1)(2j+1)}{k+2}\\right]\\ .$ In particular we have $\\begin{aligned}{\\mathcal {S}_{0}}^{0}&=\\sqrt{\\frac{2}{k+2}}\\,\\sin \\left[\\frac{\\pi }{k+2}\\right]\\ , \\\\{\\mathcal {S}_{\\frac{1}{2}}}^{0}&=\\sqrt{\\frac{2}{k+2}}\\,\\sin \\left[\\frac{2\\pi }{k+2}\\right]\\ , \\\\{\\mathcal {S}_{\\frac{1}{2}}}^{\\frac{1}{2}}&=\\sqrt{\\frac{2}{k+2}}\\,\\sin \\left[\\frac{4\\pi }{k+2}\\right]\\ .\\end{aligned}$ The quantum dimension reads $d_j=[2j+1]=\\frac{\\sin \\left[\\frac{\\pi (2j+1)}{k+2}\\right]}{\\sin \\left[\\frac{\\pi }{k+2}\\right]}\\ .$ The partition function $Z\\left[ X_n\\right]$ for Wilson loop with $n$ crossing is in general given by $\\frac{Z\\left[X_n\\right]}{{\\mathcal {S}_{0}}^{0}}=\\left(q^{-\\frac{N-1}{2}}\\right)^n\\frac{[N+1][N]}{[2]}+\\left(-q^{-\\frac{N+1}{2}}\\right)^n\\frac{[N][N-1]}{[2]}\\ ,$ where $q=e^{2\\pi {\\rm i}/(k+N)}$ .",
"For $N=2$ we get explicitly $Z\\left[ X_n\\right]=\\sqrt{\\frac{2}{k+2}}\\cdot \\left(\\frac{-i}{2}\\right)\\cdot \\left[e^{\\frac{\\pi {\\rm i}(3-n)}{k+2}}-e^{-\\frac{\\pi {\\rm i}(3+n)}{k+2}}+(-1)^n\\, e^{\\frac{\\pi {\\rm i}(1-3n)}{k+2}}-(-1)^n\\, e^{-\\frac{\\pi {\\rm i}(1+3n)}{k+2}}\\right]\\ .$ In particular, we find $\\begin{aligned}Z\\left[X_0\\right]&=\\frac{\\left({\\mathcal {S}_{0}}^{\\frac{1}{2}}\\right)^2}{{\\mathcal {S}_{0}}^{0}}\\ ,\\\\Z\\left[X_1\\right]&={\\mathcal {S}_{0}}^{\\frac{1}{2}}\\ ,\\\\Z\\left[X_2\\right]&={\\mathcal {S}_{\\frac{1}{2}}}^{\\frac{1}{2}}\\ .\\end{aligned}$ In the $k\\rightarrow \\infty $ limit we obtain the simple result: $Z\\left[ X_n\\right]~ \\longrightarrow ~ \\frac{\\pi \\,\\sqrt{2}}{k^\\frac{3}{2}}\\,\\left[3+(-1)^n\\right]\\ .$"
],
[
"Multi-boundary states in Chern-Simons theory",
"In this section we consider the multi-boundary states of spatial regions $\\Sigma =\\bigsqcup \\mathbb {T}^2$ consisting of several tori without any Wilson loops.",
"The calculation of the entanglement entropies of these states is performed in [10], which is easily generalized to pseudo entropy.",
"These states can be prepared by drilling out the internal region of a subregion $\\Sigma =\\bigsqcup \\mathbb {T}^2$ from $\\mathbb {S}^3$ .",
"The resulting state can be expanded by the states $\\mathinner {|{R_j}\\rangle }$ in figure REF : $\\mathinner {|{\\psi }\\rangle }=\\sum _{j_1,\\ldots j_n}c_{j_1,\\ldots ,j_n}\\mathinner {|{R_{j_1},\\ldots ,R_{j_n}}\\rangle },$ where $n$ is the number of tori and $\\mathinner {|{R_{j_1},\\ldots ,R_{j_n}}\\rangle }\\equiv \\mathinner {|{R_{j_1}}\\rangle }\\otimes \\cdots \\otimes \\mathinner {|{R_{j_n}}\\rangle }$ .",
"The coefficients $c_{j_1,\\ldots ,j_n}&=\\mathinner {\\langle {R_{j_1},\\ldots ,R_{j_n}|\\psi }\\rangle }$ are partition functions on $\\mathbb {S}^3$ with $n$ Wilson loops of representations $R_{j_1},\\ldots ,{R_{j_n}}$ .",
"Here we only consider a simple example of the two tori states $\\Sigma =\\mathbb {T}^2\\sqcup \\mathbb {T}^2$ in $\\mathrm {U}(1)$ Chern-Simons theory.",
"In this case, the coefficients turn out to be $c_{j_1,j_2}=Z\\left[\\mathbb {S}^3;R_{j_1},R_{j_2}\\right]=\\exp \\left(\\frac{2\\pi {\\rm i}}{k}\\,q_1\\,q_2\\,\\ell _{12}\\right),$ where $q_1,q_2$ are the $\\mathrm {U}(1)$ charges of the two loops and $\\ell _{12}$ is the linking number between the two loops.",
"We define two states $\\begin{aligned}\\mathinner {|{\\psi }\\rangle }&=\\frac{1}{k}\\sum _{q_1,q_2}\\exp \\left(\\frac{2\\pi {\\rm i}}{k}\\,q_1\\,q_2\\,\\ell _{12}\\right)\\mathinner {|{R_{j_1}}\\rangle }\\otimes \\mathinner {|{R_{j_2}}\\rangle }, \\\\\\mathinner {|{\\varphi }\\rangle }&=\\frac{1}{k}\\sum _{q_1,q_2}\\exp \\left(\\frac{2\\pi {\\rm i}}{k}\\,q_1\\,q_2\\,\\ell _{12}^{\\prime }\\right)\\mathinner {|{R_{j_1}}\\rangle }\\otimes \\mathinner {|{R_{j_2}}\\rangle } \\ ,\\end{aligned}$ with different linking numbers.",
"We call one of the two tori $A$ , which corresponds to the region including a Wilson loop with $R_{j_1}$ , and the other $B$ .",
"Let us calculate the entanglement entropies between $A$ and $B$ following [10].",
"The reduced density matrix for $\\mathinner {|{\\psi }\\rangle }$ is $\\begin{aligned}\\rho _A^{\\psi }\\equiv {\\rm Tr}_B\\left[\\mathinner {|{\\psi }\\rangle }\\mathinner {\\langle {\\psi }|}\\right]&=\\frac{1}{k^2}\\sum _{q_1,q_1^{\\prime },q_2}e^{\\frac{2\\pi {\\rm i}}{k}(q_1-q_1^{\\prime })\\,\\ell _{12}\\,q_2}\\mathinner {|{R_{j_1}}\\rangle }\\mathinner {\\langle {R_{j_1^{\\prime }}}|} \\\\&=\\frac{1}{k}\\sum _{q_1,q_1^{\\prime }}\\eta _{q_1,q_2}(k,\\ell _{12})\\mathinner {|{R_{j_1}}\\rangle }\\mathinner {\\langle {R_{j_1^{\\prime }}}|}\\ ,\\end{aligned}$ where $\\eta _{q_1,q_2}(k,\\ell _{12})\\equiv {\\left\\lbrace \\begin{array}{ll}1 \\qquad \\ell _{12}(q_1-q_2)=0\\ {\\rm mod}\\ {k}\\\\0 \\qquad \\ell _{12}(q_1-q_2)\\ne 0\\ {\\rm mod}\\ {k}\\end{array}\\right.",
"}.$ The Rényi entropy for $\\mathinner {|{\\psi }\\rangle }$ is $S^{(n)}\\left(\\rho _A^{\\psi }\\right)&=\\frac{1}{1-n}\\log \\left[\\frac{1}{k^n}\\sum _{q_1,\\ldots ,q_n}\\eta _{q_1,q_2}(k,\\ell _{12})\\cdots \\eta _{q_{n},q_1}(k,\\ell _{12})\\right]\\ .$ When we fix $0\\le q_1\\le k-1$ , there are $\\mathrm {gcd}(k,\\ell _{12})$ values of $0\\le q_2\\le k-1$ satisfying the relation $\\ell _{12}(q_1-q_2)=0\\ {\\rm mod}\\ {k}$ .",
"Similarly, $q_3,\\ldots ,q_n$ also take $\\mathrm {gcd}(k,\\ell _{12})$ values.",
"Therefore the summation in the logarithm in (REF ) will be $k\\,(\\mathrm {gcd}(k,\\ell _{12}))^{n-1}$ , so $S^{(n)}\\left(\\rho _A^{\\psi }\\right)=\\log \\left[\\frac{k}{\\mathrm {gcd}(k,\\ell _{12})}\\right]\\ ,$ and the entanglement entropy is clearly $S\\left(\\rho _A^{\\psi }\\right)=\\log \\left[\\frac{k}{\\mathrm {gcd}(k,\\ell _{12})}\\right]\\ .$ Similarly, the entanglement entropy for $\\mathinner {|{\\varphi }\\rangle }$ is $S\\left(\\rho _{A}^{\\varphi }\\right)=\\log \\left[\\frac{k}{\\mathrm {gcd}(k,\\ell _{12}^{\\prime })}\\right]\\ .$ Figure: The difference ΔS\\Delta S of the pseudo entropy from the average of the entanglement entropies varying the level kk.",
"We set ℓ 12 =2,ℓ 12 ' =1\\ell _{12}=2,\\ell _{12}^{\\prime }=1.Next, let us calculate the pseudo entropy.",
"The inner product is $\\begin{aligned}\\mathinner {\\langle {\\varphi |\\psi }\\rangle }&=\\frac{1}{k^2}\\sum _{q_1,q_2}\\exp \\left(\\frac{2\\pi {\\rm i}}{k}\\,q_1\\,q_2\\,(\\ell _{12}-\\ell ^{\\prime }_{12})\\right) \\\\&=\\frac{1}{k}\\sum _{q_1}\\eta _{\\ell _{12},\\ell _{12}^{\\prime }}(q_1,k) \\\\&=\\frac{1}{k}\\,\\mathrm {gcd}(\\ell _{12}-\\ell _{12}^{\\prime },k)\\ .\\end{aligned}$ To obtain the third line, we have used the fact that there are $\\mathrm {gcd}(\\ell _{12}-\\ell _{12}^{\\prime },k)$ values of $0\\le q_1\\le k-1$ satisfying $q_1(\\ell _{12}-\\ell _{12}^{\\prime })=0\\ {\\rm mod}\\ {k}$ .",
"When $\\ell _{12}=\\ell ^{\\prime }_{12}$ , i.e.",
"$\\mathinner {|{\\psi }\\rangle }=\\mathinner {|{\\varphi }\\rangle }$ , the inner product is 1 since $\\mathrm {gcd}(0,k)=k$ .",
"The reduced transition matrix is $\\begin{aligned}\\tau _A^{\\psi |\\varphi }&=\\frac{1}{k^2\\mathinner {\\langle {\\varphi |\\psi }\\rangle }}\\sum _{q_1,q_1^{\\prime },q_2}e^{\\frac{2\\pi {\\rm i}}{k}(q_1\\,\\ell _{12}-q_1^{\\prime }\\,\\ell _{12}^{\\prime })\\,q_2}\\mathinner {|{R_{j_1}}\\rangle }\\mathinner {\\langle {R_{j_1}^{\\prime }}|} \\\\&=\\frac{1}{\\mathrm {gcd}(\\ell _{12}-\\ell _{12}^{\\prime },k)}\\sum _{q_1,q_2}\\tilde{\\eta }_{q_1,q_1^{\\prime }}(\\ell _{12},\\ell _{12}^{\\prime },k)\\mathinner {|{R_{j_1}}\\rangle }\\mathinner {\\langle {R_{j_1}^{\\prime }}|}\\ ,\\end{aligned}$ where $\\tilde{\\eta }_{q_1,q_2}(\\ell _{12},\\ell _{12}^{\\prime }, k)\\equiv {\\left\\lbrace \\begin{array}{ll}1 \\qquad q_1\\,\\ell _{12}-q_1^{\\prime }\\,\\ell _{12}^{\\prime }=0\\ {\\rm mod}\\ {k}\\\\0 \\qquad q_1\\,\\ell _{12}-q_1^{\\prime }\\,\\ell _{12}^{\\prime }\\ne 0\\ {\\rm mod}\\ {k}\\end{array}\\right.",
"}.$ Let $N(\\ell _{12},\\ell ^{\\prime }_{12},k)$ be the number of $0\\le q_1^{\\prime }\\le k-1$ satisfying $q_1\\,\\ell _{12}-q_1^{\\prime }\\,\\ell _{12}^{\\prime }=0\\ {\\rm mod}\\ {k}$ when fixing $q_1$ , which in fact does not depend on $q_1$ .",
"Then the pseudo entropy is $\\begin{aligned}S\\left(\\tau _A^{\\psi |\\varphi }\\right)&=\\lim _{n\\rightarrow 1}\\frac{1}{1-n}\\log \\left[\\frac{N(\\ell _{12},\\ell ^{\\prime }_{12},k)}{\\gcd (\\ell _{12}-\\ell _{12}^{\\prime },k)}\\right]^{n-1} \\\\&=\\log \\left[\\frac{\\gcd (\\ell _{12}-\\ell _{12}^{\\prime },k)}{N(\\ell _{12},\\ell ^{\\prime }_{12},k)}\\right]\\ .\\end{aligned}$ Figure REF shows the difference $\\Delta S$ when the linking numbers are $\\ell _{12}=2,\\ell _{12}^{\\prime }=1$ .",
"We can see that $\\Delta S<0$ similar to the other typical examples."
]
] | 2107.01797 |
[
[
"Popcorn: Paillier Meets Compression For Efficient Oblivious Neural\n Network Inference"
],
[
"Abstract Oblivious inference enables the cloud to provide neural network inference-as-a-service (NN-IaaS), whilst neither disclosing the client data nor revealing the server's model.",
"However, the privacy guarantee under oblivious inference usually comes with a heavy cost of efficiency and accuracy.",
"We propose Popcorn, a concise oblivious inference framework entirely built on the Paillier homomorphic encryption scheme.",
"We design a suite of novel protocols to compute non-linear activation and max-pooling layers.",
"We leverage neural network compression techniques (i.e., neural weights pruning and quantization) to accelerate the inference computation.",
"To implement the Popcorn framework, we only need to replace algebraic operations of existing networks with their corresponding Paillier homomorphic operations, which is extremely friendly for engineering development.",
"We first conduct the performance evaluation and comparison based on the MNIST and CIFAR-10 classification tasks.",
"Compared with existing solutions, Popcorn brings a significant communication overhead deduction, with a moderate runtime increase.",
"Then, we benchmark the performance of oblivious inference on ImageNet.",
"To our best knowledge, this is the first report based on a commercial-level dataset, taking a step towards the deployment to production."
],
[
"Introduction",
"Deep convolutional neural networks have achieved tremendous success in various domains such as facial recognition [33], medical diagnosis [9], and image classification [13], whilst demonstrating beyond-experts performance.",
"The massive (labeled) training data and extensive computational resources are the fuel for breakthroughs in accuracy.",
"However, it also becomes a notorious challenge for individuals and non-specialised institutions to train and deploy state-of-the-art models.",
"Thanks to the advances in cloud computing, companies with sufficient computing power and expertise (e.g., Google and Amazon) can provide machine learning services to the public.",
"NN-IaaS is an important business paradigm, in which a server provides machine learning prediction APIs, built upon its pre-trained machine learning model, to the clients.",
"The latter uploads her data to the server and receives predictions by calling the APIs.",
"Usually, business and privacy protection requirements may prevent the server from providing any information other than prediction results.",
"Similarly, the clients are protective of their private data and cannot disclose it to the server.",
"This dilemma significantly limits the use of cloud-based services, leading to an urgent need for privacy-preserving NN-IaaS.",
"Homomorphic encryption (HE) [10], Garbled Circuits (GC) [3] and secret sharing (SS) [2] are the workhorse driving many exciting recent advances in oblivious neural network inference [11], [21], [30], [16].",
"CryptoNets [11] and its variant [6] adopted HE to support privacy-preserving neural network predictions.",
"Since HE operations are constrained to addition-and-multiplications, the non-linear activation function (e.g., relu$(x,0)$ ) is substituted with a low-degree polynomial; and the max-pooling function is replaced by the mean-pooing function.",
"This approach requires modifying the original model architecture, significantly impacting accuracy.",
"XONN [30] exploited the fact that the XNOR operation is free in the GC protocol [17] to efficiently evaluate binarized neural networks.",
"This approach, however, requires both the weights and activation values to be binarized, which harms accuracy performance.",
"Moreover, completely compiling a neural network into circuits increases communication overhead.",
"MiniONN [21] and Gazelle [16] combined different primitives to keep neural networks unchanged.",
"In their methods, the GC is often adopted to compute non-linear layers.",
"It is worth highlighting that existing methods always leak some information beyond the predictions.",
"For example, in CryptoNets, the client can make inferences about the model, as it would have to generate parameters for the encryption according to the model architecture.",
"In XONN and MiniONN, the client can learn the exact network architecture.",
"Gazelle only reveals the number of neurons of each neural layer.",
"Though there is always some form of information leakage, it is believed that less leakage is more preferable [16], [30].",
"Engineering complexity is also an important problem to be addressed.",
"For instance, Gazelle relies on sophisticated packing schemes.",
"Its implementation depends on specific network parameters.",
"Sometimes, it also requires performing trade-offs between efficiency and privacy.",
"XONN needs to compile a whole network into circuits, the effort is non-trivial; the computational cost is also completely transferred from the server to the client.",
"In this paper, we introduce Popcorn, a concise oblivious inference framework that is entirely built upon the Paillier homomorphic encryption scheme [27].",
"Popcorn is a non-invasive solution which does not require modifications to network architectures (e.g., approximating non-linear activation functions with polynomials).",
"The security model of Popcorn is consistent with Gazelle [16], i.e., this protocol hides the network weights and architecture except for the number of neurons of each layer.",
"The main contribution of this paper consists of three aspects, We introduce a suite of Popcorn protocols for efficiently computing non-linear neural network layers (e.g., $relu$ activation layer and max-pooling layer).",
"Under the Popcorn framework, we leverage network compression (e.g., weight pruning and quantization) to accelerate the computation of linear layers (e.g., convolutional layer and fully-connected layer).",
"We benchmark the oblivious inference performance on the ImageNet dataset, using state-of-the-art models.",
"To our best knowledge, this is the first report on privacy-preserving ImageNet-scale classification tasks.",
"Compared with existing solutions, an important contribution of Popcorn is that its engineering is extremely simple.",
"The framework completely relies on the Paillier HE scheme.",
"In the implementation, we only need to substitute the algebraic operations in plaintext inference to the corresponding HE operations, making (machine learning) engineers agnostic to the obscure cryptography knowledge.",
"Although the framework is built on the Paillier scheme in this paper, we can directly adopt other HE schemes.",
"For example, in the case where a client has a number of images (e.g., $>$ 1000) to classify, we can also adopt a lattice-based additive HE scheme (e.g., [16]) to pack multiple images into one ciphertext, amortizing the computational and communicational overhead."
],
[
"Convolutional Neural Network",
"A typical convolutional neural network (CNN) consists of a sequence of linear and non-linear layers and computes classifications in the last layer.",
"In this section, we describe the functionality of neural network layers."
],
[
"Linear Layers",
"Linear operations in networks are often carried in fully-connected layers ($fc$ ) and convolutional layers ($conv$ ).",
"The $conv$ layer is composed of a 3D tensor input (in the form of $\\mathbb {R}^{(w_i, h_i, c_i)}$ ), a set of 3D tensor filters (in the form of $\\mathbb {R}^{(f_w, f_h, f_c)}$ , s.t.",
"$f_w<w_i, f_h<h_i, f_c=c_i$ ) to extract local features from the input and a 3D tensor output (in the form of $\\mathbb {R}^{(w_o, h_o, c_o)}$ , $c_o$ is the number of channels of the output, which is also the number of filters).",
"Each channel of the output is obtained by a filter that convolves the input along the direction of $w_i$ and $h_i$ (i.e., the width and height of the input).",
"Specifically, each element of a channel is calculated through filter point-wisely multiplicating its perceptive field of the input and accumulating.",
"The $fc$ layer is a matrix-vector multiplication as follows, $\\textbf {y}=\\textbf {W}\\cdot \\textbf {x}+\\textbf {b}$ where $\\textbf {x} \\in \\mathbb {R}^{m\\times 1}$ is the input vector, $\\textbf {y}\\in \\mathbb {R}^{n\\times 1}$ is the output,$\\textbf {W}\\in \\mathbb {R}^{n\\times m}$ is the weight matrix and $\\textbf {b}\\in \\mathbb {R}^{n\\times 1}$ is the bias vector.",
"In fact, a $conv$ layer can be also written in a form of matrix-vector multiplication, as defined in Eq.",
"(REF ).",
"In addition to $conv$ and $fc$ layers, the batch-normalization ($bn$ ) [15] layer can be seen as a linear layer.",
"It is often adopted to normalize the output of a linear layer by re-centering and re-scaling as follows, $\\begin{split}y & =\\gamma \\cdot \\frac{x-\\mu _B}{\\sqrt{\\delta ^2_B+\\epsilon }}+\\beta \\\\& = \\frac{\\gamma }{\\sqrt{\\delta ^2_B+\\epsilon }}\\cdot x - (\\frac{\\gamma \\cdot \\mu _B}{\\sqrt{\\delta ^2_B+\\epsilon }}-\\beta )\\end{split}$ where $\\mu _B$ and $\\delta ^2_B$ are the per-dimension mean and variance of a mini-batch, respectively.",
"$\\epsilon $ is a is an arbitrarily small constant added in the denominator for numerical stability.",
"The $\\gamma $ and $\\beta $ are the re-scaling and re-centering parameters subsequently learned in the optimization process.",
"In the inference phase, all the parameters above are constant.",
"Therefore, the $bn$ computation is a linear computation in the inference and can be easily absorbed in its previous layer and next layer [14]."
],
[
"Non-linear Layers",
"The activation layer and pooling layer are two common non-linear layers.",
"The activation layer performs non-linear transformation for its inputs element-wise.",
"It does not change the input dimension.",
"The $relu$ (i.e., $\\max (x,0)$ ) activation function is widely adopted as the non-linear transformation function.",
"Different from the activation layer, the pooling layer is for feature dimension reduction.",
"The max-pooling ($mp$ ) is a popular pooling method, it outputs the maximal value of each local perception area of its inputs.",
"Suppose the local perception area has $m$ inputs, the max-pooling outputs the result of $\\max (\\lbrace x_0, x_1,\\cdots ,x_m\\rbrace )$ .",
"There is also another pooling operation, i.e., mean-pooling, it outputs the average value of each local perception area.",
"The mean pooling layer can be treated as a special simplified convolution layer with a filter in which the weights share the same value.",
"In practice, the mean pooling can be further simplified and replaced with sum-pooling, which outputs the sum of the perception area."
],
[
"Neural Network Compression",
"To accelerate neural network predictions and minimize network size, the compression technique is developed to discover and remove unimportant weights (i.e., network pruning), and to present weights with fewer bits (i.e., weights quantization), without noticeably decreasing accuracy performance [36], [12].",
"Network Pruning.",
"Weight pruning and filter pruning (a.k.a channel pruning) are two main network pruning methods [36].",
"The former investigates the removal of each individual weight (fine-grained pruning).",
"Normally, the weights that have a small magnitude or contribute less to the network loss function are removed first [12], [20].",
"The latter investigates removing entire filters (coarse-grained pruning).",
"Usually, filters that frequently generate zero outputs after the $relu$ layer are removed first [36].",
"Generally, the weight pruning approach can remove more weights than the filter pruning approach does, but the filter pruning methods can remove more neurons (we call each element of any output layer a neuron).",
"Though both the pruning approaches can benefit the Popcorn framework, we focus on the weight pruning methods in this paper.",
"Weight Quantization.",
"The quantization methods aim to reduce the number of bits to represent neural weights, which can be roughly categorized into two approaches: floating-point preserving [12] and integer-based [18].",
"In the former, the weights are quantized into a small number of bins (e.g., 128), all the weights in the same bin share the same value.",
"Thus, for each weight, we need to store only a small index into a table of shared weights.",
"In the latter, the weights are first quantized into integer representations and then recovered into a more expressive form (e.g., floating-point or more levels) through a de-quantization process, during the inference phase [18].",
"Binarized quantization is an extreme case, where the weights or activation values are represented in 1-bit [29], with a cost of accuracy loss.",
"In this work, we leverage the floating-point preserved approach and binarized neural networks to accelerate oblivious neural network inferences."
],
[
"Paillier Homomorphic Encryption",
"Homomorphic encryption (HE) is a form of encryption that allows computations to be carried over ciphertexts without decryption.",
"The result, after decryption, is the same as if the operations had been performed on the plaintexts.",
"The Paillier Cryptosystem [27] is a well-developed additive HE scheme.",
"We describe the Paillier HE scheme in the form of $\\lbrace $ KeyGen, HEnc, HAdd, HMul, HDec $\\rbrace $ .",
"$\\bullet $ KeyGen (Generate keys: $(pk, sk)$ ).",
"Choose two large prime numbers $p$ and $q$ randomly and independently, such that $\\gcd (pq,(p-1)(q-1))=1$ .",
"Compute $n=pq$ and $\\lambda =lcm(p-1,q-1)$ , where $lcm$ means least common multiple.",
"Obtain public key $pk = n, g$ ; private key $sk = p, q, \\lambda $ .",
"$g\\in \\mathbb {Z}^{*}_{n^2}$ is a multiple of $n$ .",
"$\\bullet $ HEnc (Encryption: $\\llbracket m \\rrbracket $ := HEnc$(m, pk)$ ).",
"Let $m$ be a message to be encrypted, where $0\\le m < n$ .",
"Select a random $r\\in \\mathbb {Z} _{n}^{*}$ , s.t.",
"$\\gcd (r,n)=1$ .",
"Compute the ciphertext of $m$ : $\\llbracket m \\rrbracket \\leftarrow g^mr^n \\mod {n}^2$ .",
"$\\bullet $ HDec (Decryption: $m :=$ HDec$(\\llbracket m \\rrbracket , pk)$ ).",
"Let $\\llbracket m \\rrbracket $ be a cipher to be decrypted, where $ \\llbracket m \\rrbracket < n^2$ .",
"Compute the plaintext of $\\llbracket m \\rrbracket $ : $m \\leftarrow \\frac{L(\\llbracket m \\rrbracket ^{\\lambda } \\mod {n}^2)}{L(g^{\\lambda } \\mod {n}^2)} \\mod {n}$ , where $L(x)=\\frac{x-1}{n}$ .",
"$\\bullet $ HAdd (The HE addition) $m_1+m_2$ = HDec$(\\llbracket m_1 \\rrbracket \\oplus \\llbracket m_2 \\rrbracket , sk )$ .",
"$\\oplus $ is the HE addition operator.",
"$\\bullet $ HAdd (The HE multiplication) $m_1\\times m_2$ = HDec$(\\llbracket m_1 \\rrbracket \\otimes m_2, sk )$ .",
"$\\otimes $ is the HE multiplication operator."
],
[
"Popcorn: Fast Non-linear Computation",
"In this section, we present our fast methods for non-linear layer computations.",
"The computation protocols are built upon the Paillier HE scheme [27]."
],
[
"Fast Secure $relu$ Computation Protocol",
"Given a vector $\\llbracket \\textbf {x} \\rrbracket ^{m \\times 1}$ which is element-wisely encrypted under the Paillier HE scheme (where $x_i\\in \\mathbb {Z}_n^*$ ), the server computes $relu(\\llbracket \\textbf {x} \\rrbracket ^{m \\times 1})$ , without leaking the information of any element $x_{i}$ .",
"In this setting, the client holds the secret key $sk$ , the server holds the public key $pk$ but cannot access to the $sk$ .",
"We describe the secure computation of $relu$ activation layer in Problem REF , presenting the inputs in form of vector.",
"The $relu$ layer performs non-linear transformation on each element of its inputs, i.e., $\\lbrace max(x_i,0) \\rbrace _{i=0}^{n-1}$ .",
"To solve the Problem REF , we start from a simple multiplicative-obfuscation protocol for each $x_i$ (named Version one), which is outlined in Fig.",
"REF .",
"For denotation succinctness, we omit the subscript of $x_{i}$ and we assume the implementation of Paillier HE supports encoding of of negative integers.",
"Figure: Secure relurelu computation (Version one).",
"τ -1 \\tau ^{-1} is the multiplicative inverse of τ\\tau .",
"nn is a large modulo.In the Version one, the server first samples a random positive integer $ \\tau \\in \\mathbb {Z}_n^+$ , then it blinds the $\\llbracket x \\rrbracket $ with $\\tau $ (i.e., $\\llbracket y \\rrbracket _s := \\llbracket x \\rrbracket \\odot \\tau $ ), and sends it to the client.",
"After receiving the $\\llbracket y \\rrbracket _s$ , the client decrypts $\\llbracket y \\rrbracket _s$ , and returns the server with $\\llbracket max(y_s,0) \\rrbracket $ .",
"The server removes the $\\tau $ by multiplying $\\tau ^{-1}$ , where $\\tau ^{-1}$ the inverse of $\\tau $ .",
"We emphasize that $\\tau $ is independently and randomly sampled for each element $x$ .",
"If $y_c=0$ (line 5), the correctness proof is exactly the same of the decryption of Paillier HE scheme.",
"Therefore, we focus the correctness proof on $\\llbracket y \\rrbracket _s := \\llbracket y \\rrbracket _c \\odot \\tau ^{-1}$ (line 8), where $y_c \\ne 0$ , i.e., the decryption of $\\llbracket y \\rrbracket _s$ is $x$ .",
"Clearly, if we can eliminate the random factor $\\tau $ when performing decryption, it is a normal Paillier decryption process(see Section REF , HDec) and the correctness is guaranteed.",
"[Correctness Proof of Version one] To decrypt $\\llbracket y \\rrbracket _s$ , we need to compute $(\\llbracket y \\rrbracket _s^\\lambda \\mod {n}^2)$ as follows $\\nonumber \\begin{split}\\llbracket y \\rrbracket _s^{\\lambda } &= (g^yr^n)^{\\tau ^{-1}{\\lambda }} \\mod {n}^2 \\\\&=g^{\\lambda x\\tau \\tau ^{-1}}r^{\\lambda n\\tau ^{-1}} \\mod {n}^2 \\\\&\\equiv g^{\\lambda x\\tau \\tau ^{-1}} \\mod {n}^2 \\\\&=(1+n)^{\\lambda x\\tau \\tau ^{-1}} \\mod {n}^2\\\\&=1+nx\\lambda \\tau \\tau ^{-1} \\mod {n}^2 \\quad (\\triangleright \\tau \\tau ^{-1} \\equiv 1 \\mod {n})\\\\&=1+nx\\lambda \\mod {n}^2\\end{split}$ As above, the random factor $\\tau $ has been eliminated by its multiplicative inverse $\\tau ^{-1}$ , the rest is exactly the same with the decryption of the Paillier HE scheme.",
"The decryption of $\\llbracket y \\rrbracket _s$ is $x$ , the correctness is established.",
"As the proof above, with protocol Version one , the server correctly gets $max(\\llbracket x \\rrbracket ,0)$ and learns nothing about $x$ .",
"However, since $\\tau \\in \\mathbb {Z}_n^+$ , the client learns two pieces of information about $x$ , (1) the sign of $x$ and (2) whether $x=0$ or not (denoted as $x \\stackrel{?",
"}{=} 0$ ).",
"In the rest of this subsection, we first introduce the method to hide the sign of $x$ then to disguise $x \\stackrel{?",
"}{=} 0$ ."
],
[
"Hide the Sign of $x$",
"The root cause for revealing the sign information of $x$ is that the blind factor $\\tau $ is restricted to be positive (i.e., $\\tau \\in \\mathbb {Z}_n^+$ ).",
"To hide the sign information of $x$ , we revise version one of the protocol (see Fig.",
"REF ) to allow the blind factor $\\tau $ to be a nonzero integer (e.g., $\\tau \\in \\mathbb {Z}_{n}^*, s.t.\\ \\gcd (\\tau , n)=1$ ).",
"We outline the new version, named Version two, in Fig.",
"REF .",
"Figure: Version two: hide the sign information of xx.Compared to the protocol Version one (Fig.",
"REF ), the operations at the client side remain the same.",
"The key difference is that when the server receives the response (i.e., $\\llbracket y \\rrbracket _c$ ) from the client, the server computes the $relu$ activation depending on the blind factor $\\tau $ .",
"If $\\tau >0$ , the $relu$ calculation is the same with version one of the protocol; if $r<0$ , it means the client always provides the server the opposite information (e.g., when $x>0$ , the client returns $\\llbracket 0 \\rrbracket $ .",
").",
"Therefore, the server computes the $relu$ in an opposite way, $\\llbracket y \\rrbracket _s := \\llbracket x \\rrbracket - \\llbracket y \\rrbracket _c \\odot \\tau ^{-1}$ .",
"If $\\tau >0$ , the proof is the same as Version one.",
"Here, we focus the proof of Version two on $\\tau < 0$ .",
"[Correctness Proof of Version two] $\\newline $ If $x>0$ , the client returns $\\llbracket y \\rrbracket _c = \\llbracket 0 \\rrbracket $ , the server computes $\\llbracket y \\rrbracket _s := \\llbracket x \\rrbracket - \\llbracket 0 \\rrbracket \\odot \\tau ^{-1}$ , obtaining $\\llbracket y \\rrbracket _s = \\llbracket x \\rrbracket $ as the expectation (i.e., $\\max (x,0)$ ).",
"$\\newline $ If $x\\le 0$ , the client returns $\\llbracket y \\rrbracket _c = \\llbracket x\\cdot \\tau \\rrbracket $ (line 5, $y_s=x\\cdot \\tau $ ), the server computes $\\llbracket y \\rrbracket _s := \\llbracket x \\rrbracket - \\llbracket x\\cdot \\tau \\rrbracket \\odot \\tau ^{-1}$ , getting $\\llbracket y \\rrbracket _s = \\llbracket 0 \\rrbracket $ as the expectation.",
"The correctness is established."
],
[
"Hide $x_i \\stackrel{?}{=} 0$",
"Straightforwardly applying the protocol Version two (see REF ) to Problem REF leaks $x_i \\stackrel{?",
"}{=} 0$ .",
"We address this issue by randomly shuffling the input elements feeding to the $relu$ layer.",
"After the shuffling, the client cannot trace where a value was originally placed (i.e., un-traceable), thus to hide whether the true value of a specific slot is zero or not.",
"By this, the client only learns the number of zero values, which can be hidden by adding dummy elements.",
"Formally, we describe the un-traceability of $x_i\\in \\textbf {x}^{m\\times 1} $ in the Definition REF .",
"The location of $x_i \\in \\textbf {x}^{m\\times 1}$ is un-traceable if an observer is unable to distinguish $y_i \\in \\textbf {y}^{m\\times 1}$ from a random, where $y_i = x_i^{\\prime }\\cdot \\tau _i$ and $x_i^{\\prime }$ is the value at slot $i$ of $\\textbf {x}^{m\\times 1}$ , not $x_i$ itself (through randomly shuffling).",
"To achieve the un-traceability, the server needs a pair of uniform-random shuffling functions ($\\Pi , \\Pi ^{-1}$ ) as follows, $\\begin{split}& \\llbracket \\textbf {x}^{\\prime } \\rrbracket ^{m\\times 1} \\leftarrow \\Pi (\\llbracket \\textbf {x} \\rrbracket ^{m\\times 1}, \\gamma )\\\\& \\llbracket \\textbf {x} \\rrbracket ^{m\\times 1} \\leftarrow \\Pi ^{-1} (\\llbracket \\textbf {x}^{\\prime } \\rrbracket ^{m\\times 1}, \\gamma )\\end{split}$ where the $\\gamma $ is a private random seed.",
"Specifically, given an input vector $\\llbracket \\textbf {x} \\rrbracket ^{m\\times 1}$ , the server first adds $t$ dummy elements (of which a random portion are set to 0) to the input vector, and gets $\\llbracket \\textbf {x} \\rrbracket ^{(m+t)\\times 1}$ .",
"This step is to hide the number of elements with value of 0.",
"For denotation succinctness, we let $m=m+t$ , i.e., we continue to denote the new input vector as $\\llbracket \\textbf {x} \\rrbracket ^{m\\times 1}$ .",
"The second, the server samples a private random seed $\\gamma $ and applies the random shuffling function $\\Pi $ to the input vector $\\llbracket \\textbf {x} \\rrbracket ^{m\\times 1}$ , obtaining $\\llbracket \\textbf {x}^{\\prime } \\rrbracket ^{n\\times 1}$ .",
"The third, the server blinds each $\\llbracket x_i^{\\prime } \\rrbracket \\in \\llbracket \\textbf {x}^{\\prime } \\rrbracket $ with a independently and randomly sampled integer $\\tau \\in \\mathbb {Z}_{n}^*\\ s.t.",
"\\ \\tau \\ne 0$ ( $\\llbracket y_i \\rrbracket _s = \\llbracket x_i^{\\prime } \\rrbracket \\cdot \\tau _i$ , see Fig.",
"REF ).",
"With the random shuffling and one-time random mask (i.e., $\\tau _i$ ), it is clearly that $y_i$ is indistinguishable from a random value (i.e., $x_i\\in \\textbf {x}^{n\\times 1} $ is un-traceable), thus to hide the sign information of $x_i$ .",
"We emphasize that, after applying the shuffling function $\\Pi $ to the input array $\\textbf {x}^{m\\times 1}$ , $x_i^{\\prime }$ is the value at slot $i$ ($y_i = x_i^{\\prime }\\cdot \\tau _i $ ), not $x_i$ itself.",
"Now, we have introduced the completed version of our secure $relu$ computation protocol, outlined in Fig.",
"REF .",
"Figure: Secure relurelu computation (Complete version).",
"We assume the 〚𝐱〛 m×1 \\llbracket \\textbf {x} \\rrbracket ^{m \\times 1} already contains dummies."
],
[
"Efficient Max-Pooling ($mp$ )",
"Given a matrix $\\llbracket \\textbf {X} \\rrbracket ^{m \\times m}$ which is element-wisely encrypted under the Paillier HE scheme ( $x_{ij}\\in \\mathbb {Z}_n^*$ ), the server computes $mp(\\llbracket \\textbf {X} \\rrbracket ^{m \\times m}, t, s)$ , without leaking the information of any element $x_{ij}$ .",
"$t$ is the pooling-window dimension and $s$ is the stride, where $s\\le t< m$ (usually, $t=s=2$ ).",
"The client has $sk$ , the server has $pk$ but cannot access to $sk$ .",
"The max-pooling ($mp$ ) operation outputs the maximum value of each pooling window (in size of $t \\times t$ ).",
"For succinctness, we denote the pooling window in form of vector, i.e., $max(\\lbrace \\llbracket x_0 \\rrbracket ,\\llbracket x_1 \\rrbracket ,\\cdots ,\\llbracket x_m \\rrbracket \\rbrace )$ , $m=t\\times t$ .",
"In existing solutions, e.g.,[16], [30], [21], the $mp$ operation is notoriously inefficient.",
"In this section, we will show how to efficiently compute the $mp$ layer.",
"We first introduce a new protocol to compute $\\max (\\llbracket x_i \\rrbracket , \\llbracket x_j \\rrbracket )$ , which is the fundamental building block for computing the $mp$ layer.",
"Then, we leverage the $\\llbracket x_i \\rrbracket \\ominus \\llbracket x_j \\rrbracket $ inherent in the $\\max (\\llbracket x_i \\rrbracket , \\llbracket x_j \\rrbracket )$ protocol to reduce homomorphic computations.",
"Lastly, we present a method to absorb the $relu$ computation into the $mp$ layer."
],
[
"Compute $\\max (\\llbracket x_i \\rrbracket , \\llbracket x_j \\rrbracket )$",
"The $\\max (\\llbracket x_i \\rrbracket , \\llbracket x_j \\rrbracket )$ is the fundamental computation for the max-pooling operation.",
"We construct a lightweight protocol to compute $\\max (\\llbracket x_i \\rrbracket , \\llbracket x_j \\rrbracket )$ .",
"The basic idea is first to covert it to a comparison between a ciphertext and 0 (i.e., $\\max (\\llbracket x\\rrbracket ,0 )$ ), then recover the result from the client's response, which is similar to the secure $relu$ computation protocol.",
"We summarize the $\\max (\\llbracket x_i \\rrbracket , \\llbracket x_j \\rrbracket )$ computation protocol in Fig.",
"REF .",
"Figure: Secure max(〚x i 〛,〚x j 〛)\\max (\\llbracket x_i \\rrbracket , \\llbracket x_j \\rrbracket ) computation protocol.",
"〚x i 〛,〚x j 〛\\llbracket x_i \\rrbracket , \\llbracket x_j \\rrbracket are randomly selected from a pooling window, such that we can sample the binder factor τ\\tau from ℤ + \\mathbb {Z}^+.For each pooling window, the max-pooling operation needs $m-1$ comparisons and calls the $\\max (\\llbracket x_i \\rrbracket , \\llbracket x_j \\rrbracket )$ protocol $\\left\\lceil log_2(n) \\right\\rceil $ times.",
"The same with the $relu$ protocol (see Fig.",
"REF ), we adopt the random shuffling method to avoid leaking $x_i - x_j \\stackrel{?",
"}{=} 0$ .",
"Firstly, the server randomly maps the elements of each pooling window into pairs; then, the server shuffles all the pairs from all the pooling windows (each pair as an unit); the third, the server executes the secure $\\max (\\llbracket x_i \\rrbracket , \\llbracket x_j \\rrbracket )$ computation protocol to get the maximum of each pair."
],
[
"$\\llbracket x_i \\rrbracket \\ominus \\llbracket x_j \\rrbracket $ Benefits Computation",
"The secure $\\max (\\llbracket x_i \\rrbracket , \\llbracket x_j \\rrbracket )$ computation protocol contains a subtraction between two ciphertexts (i.e., $\\llbracket x_i \\rrbracket \\ominus \\llbracket x_j \\rrbracket $ ), we can exploit this fact to reduce computations on encrypted data.",
"Figure: An illustration of two adjacent convconv operations, with stride s=1s=1 and convconv window size w=3w=3.",
"x i =conv i ,x j =conv j x_i = conv_i, x_j=conv_j.",
"x * in x_*^{in} indicates the element of the convconv inputs.As shown in Fig.",
"REF , $\\llbracket x_i \\rrbracket =conv_i= \\sum _{u=1}^3\\sum _{v=1}^3a_{ij}\\llbracket x_{ij}^{in} \\rrbracket $ and $\\llbracket x_j \\rrbracket =conv_j=\\sum _{u=1}^3\\sum _{v=2}^4a_{ij}\\llbracket x_{ij}^{in} \\rrbracket $ are the results of two adjacent $conv$ operations.",
"Each $conv$ costs $w^2$ homomorphic multiplication-and-additions (here, $w=3$ ).",
"Instead of independently computing $conv_i$ and $conv_j$ , we leverage the computation of $\\llbracket x_i \\rrbracket \\ominus \\llbracket x_j \\rrbracket $ to reduce HE operations as follows, $\\begin{split}& \\llbracket x_i \\rrbracket \\ominus \\llbracket x_j \\rrbracket = \\sum _{u=1}^3\\sum _{v=1}^3a_{ij} \\llbracket x_{ij}^{in} \\rrbracket \\ominus \\sum _{u=1}^3\\sum _{v=2}^4a_{ij}\\llbracket x_{ij}^{in} \\rrbracket \\\\&= \\sum _{u=1}^3\\sum _{v=2}^3(a_{uv}-b_{uv})\\llbracket x_{uv}^{in}\\rrbracket \\oplus \\sum _u^3x_{u1}^{in} \\ominus \\sum _u^3 \\llbracket x_{u4}^{in} \\rrbracket \\end{split}$ where $a_{uv}$ and $b_{uv}$ denote the weights of the two $conv$ filters, respectively.",
"As shown in Equation (REF ), for two adjacent $conv$ operations, we only need to apply one homomohphic multiplication-and-addition for each element of the overlapped region (i.e., by $(a_{uv}-b_{uv})\\llbracket x_{uv}^{in}\\rrbracket $ ).",
"In general, suppose the $conv$ window size is $w$ and the stride is $s$ .",
"Thanks to the $\\llbracket x_i \\rrbracket \\ominus \\llbracket x_j \\rrbracket $ , we can reduce the number of homomorphic multiplication-and-additions (of two $conv$ operations) from $2w^2$ to $w(w-s)+2ws=w^2+ws$ .",
"The ratio of computation reduction is $1-\\frac{w^2+ws}{2w^2}=0.5-\\frac{s}{2w}$ .",
"In most CNNs, $s\\in \\lbrace 1,2\\rbrace $ and $w\\in \\lbrace 3,5,7,11\\rbrace $ .",
"Therefore, nearly 50% of homomorphic multiplication-and-additions can be reduced when meeting a $mp$ layer."
],
[
"Compute $relu\\rightarrow mp$",
"Usually, a $mp$ layer often directly follows a $relu$ layer, i.e., $relu \\rightarrow mp$ [13], [19].",
"For each pooling window, we can write the computation of $relu \\rightarrow mp$ as follows, $\\max (\\max (x_1,0),\\max (x_2,0),\\cdots ,\\max (x_m,0))$ Computing the Equation (REF ) costs $2n-1$ comparisons.",
"Clearly, we can transform this equation into following form, $\\begin{split}&\\max (\\max (x_1,0),\\max (x_2,0),\\cdots ,\\max (x_m,0)) \\\\=&\\max (max(x_1,x_2,\\cdots ,x_m),0 )\\end{split}$ Compared with Equation (REF ), Equation (REF ) reduces the number of comparisons from $2m-1$ to $m$ .",
"From the perspective of communication overhead, by this transformation, we can compute the max-pooling layer for free."
],
[
"Summary",
"Computing the $mp$ layer often leads to expensive communication overhead in existing methods [16], [21].",
"A commonly-seen trick is to reduce the use of max-pooling layers such as using mean-pooling layers instead.",
"However, this approach may results in a risk of accuracy degradation, as it breaches the original design.",
"In this section, we first present a fast method for computing the max-pooling layer, of which the communicational overhead is equivalent to computing a $relu$ layer.",
"Then, we exploit the stable design pattern in CNN, i.e., $conv \\rightarrow relu \\rightarrow mp$ , to further reduce the computational cost, and to absorb the computation of $relu$ into $mp$ (looks like we can compute the $mp$ layer for free).",
"It means that, in the Popcorn framework, the $mp$ layer can be a factor to improve efficiency, instead of becoming a heavy burden as usual."
],
[
"Popcorn: Fast Linear Computation",
"In the Popcorn framework, the input of each layer is element-wisely encrypted (the client data is the first layer input).",
"Based on this fact, we can exploit neural network compression technologies to reduce HE computations in linear layers (i.e., the $conv$ and $fc$ layers), speeding up the oblivious inference.",
"In this section, we introduce fast linear computation methods, which rely on pruned-and-quantized networks and binarized neural networks, respectively.",
"Usually, the former preserves accuracy well; the latter leads to higher efficiency."
],
[
"Pruning-Then-Quantization",
"Suppose there are two filters (represented in a form of vector) $\\textbf {a}= [a_1, a_2, \\cdots , a_{n-1}, a_{n}]$ , $\\textbf {b}= [b_1, b_2, \\cdots , b_{n-1}, b_{n}]$ , and an encrypted input $\\llbracket \\textbf {x} \\rrbracket = [\\llbracket x_0 \\rrbracket , \\llbracket x_1 \\rrbracket ,\\cdots , \\llbracket x_{n-1} \\rrbracket , \\llbracket x_n \\rrbracket ]$ .",
"Before performing the $conv$ operations, we can adopt network pruning techniques to discover and remove un-important weights of $\\textbf {a},\\textbf {b}$ (e.g., let $a_2=0,\\ a_{n-1}=0,\\ b_1=0$ ), and employ network quantization methods to force multiple weights of $\\textbf {a},\\textbf {b}$ to share the same value (e.g., $a_1=b_1$ ).",
"We illustrate how to use the prunned-and-quantized $conv$ filters to reduce homomorphic computations as follows, $\\begin{split}& \\begin{vmatrix}0 & a_1 & a_2 & \\cdots & 0 &a_{n} \\\\b_0 & b_1 & 0 & \\cdots & b_{n-1} &b_{n}\\end{vmatrix}\\begin{vmatrix}\\llbracket x_0 \\rrbracket \\\\\\llbracket x_1 \\rrbracket \\\\\\llbracket x_2 \\rrbracket \\\\\\cdots \\\\\\llbracket x_{n-1} \\rrbracket \\\\\\llbracket x_{n} \\rrbracket \\end{vmatrix} \\\\=&\\begin{vmatrix}0+a_1\\llbracket x_1 \\rrbracket +a_2\\llbracket x_2 \\rrbracket +\\cdots +0+a_n\\llbracket x_n \\rrbracket \\\\b_0\\llbracket x_0\\rrbracket +0+0+\\cdots +b_{n-1}\\llbracket x_{n-1} \\rrbracket +a_n\\llbracket x_n \\rrbracket \\end{vmatrix}\\end{split}$ Firstly, we can simply skip any computations related to the removed weights (which are permanently set to 0).",
"Then, we can find out which weights connect to the same ciphertext and share the same value, thus reusing the intermediate results.",
"For example, if $a_i=b_i$ (where $0\\le i \\le n$ ) we can reuse the result of $a_i\\llbracket x_i \\rrbracket $ when computing $b_i\\llbracket x_i \\rrbracket $ .",
"Obviously, the inference acceleration relies on the number of removed weights (by pruning) and reused intermediate results (by quantization).",
"This observation straightforwardly applies to the $fc$ layer."
],
[
"Network Compression For Popcorn",
"A number of network compression methods have been proposed for different purposes such as minimizing model size [12], reducing energy consumption [36].",
"However, all the exiting methods are designed for plaintext inputs.",
"There is a lack of investigation for network compression for ciphertext inputs.",
"In this section, we analyze the weight pruning and quantization methods that are fit for the Popcorn framework.",
"Weight Pruning.",
"The aim of weight pruning is to further reduce the number of homomorphic computations, by removing more weights.",
"For a $conv$ layer, each filter weight connects to $\\frac{w^2}{s^2}$ ciphertexts, where $w$ is the input dimension of the layer, $s$ is the stride-size of the $conv$ operations.",
"So a single weight in different $conv$ layers may connect to a different number of ciphertexts.",
"This means that removing the same number of weights from different $conv$ layers can result in a different reduction of homomorphic operations, depending on the input dimension $w$ of each $conv$ layer.",
"The $fc$ layer is a vector-matrix multiplication, each weight connects to one ciphertext.",
"Therefore, for $conv$ layers, it is suggested to first remove weights in a layer that has a larger input dimension.",
"Weight Quantization.",
"The purpose of weight quantization is to reuse the intermediate results computed between weights and encrypted inputs, as much as possible.",
"Hence, the quantization priority of each layer depends on the number of weights that a ciphertext connects to, i.e., the larger the number, the higher the priority of the layer is.",
"The weights of a high priority layer should first be quantized into lower bit representation.",
"For a $conv$ layer, each ciphertext connects to $\\frac{c_of_w^2}{s^2}$ weights, where $c_o$ is the number of filters and $f_w$ is the filter dimension.",
"Assume the weights are represented in $t$ bits, the reuse ratio is $\\ge \\frac{c_ot^2}{s^2\\cdot 2^m}$ .",
"For a $fc$ layer, each ciphertext connects to $c_o$ weights, here we re-define $c_o$ as the $fc$ layer output dimension.",
"The reuse ratio is $\\ge \\frac{c_o}{2^m}$ .",
"With a quantized network, we can limit the number of homomorphic computations to $O(n\\cdot 2^m)$ , where we abuse $n$ to denote the number of ciphertexts.",
"When the $c_0$ and $f_w$ are large while the $m$ and $s$ are small, the reduced homomorphic computations are significant."
],
[
"Summary",
"Based on the analysis above, we suggest a layer-by-layer pruning-then-quantization approach to speed up the linear layer computations in the Popcorn framework.",
"This approach starts from pruning layers with the largest input dimensions or with the most number of multiplication-and-accumulation operations.",
"It can be implemented through the algorithm introduced by [36].",
"After the pruning process, we can quantify the remaining non-zero weights into low bits, beginning with the layers in which each ciphertext connects to the most non-zero weights.",
"For a specific layer, we can adopt the codebook-based method introduced in [12].",
"Compared with other methods, this method can obtain a lower-bit representation while ensuring the same accuracy."
],
[
"Binarized Neural Network",
"The binarized neural network is an extreme case of network quantization, of which the weights are binarized, i.e., $\\lbrace -1,+1 \\rbrace $ .",
"In this section, we introduce how to efficiently evaluate a binarized network in the Popcorn framework.",
"Assume there are two binarized filters (in the form of vector) $\\textbf {a}= [+1, -1, \\cdots , -1, +1]$ , $\\textbf {b}= [-1, -1, \\cdots , +1, -1]$ , and an encrypted input $\\llbracket \\textbf {x} \\rrbracket = [\\llbracket x_0 \\rrbracket , \\llbracket x_1 \\rrbracket ,\\cdots , \\llbracket x_{n-1} \\rrbracket , \\llbracket x_n \\rrbracket ]$ .",
"We describe the $conv$ (as well as the $fc$ ) computation as follows, $\\begin{split}& \\begin{vmatrix}+1 & -1 & \\cdots & -1 &+1 \\\\-1 & +1 & \\cdots & -1 &-1\\end{vmatrix}\\begin{vmatrix}\\llbracket x_0 \\rrbracket \\\\\\llbracket x_1 \\rrbracket \\\\\\cdots \\\\\\llbracket x_{n-1} \\rrbracket \\\\\\llbracket x_{n} \\rrbracket \\end{vmatrix} \\\\=&\\begin{vmatrix}+\\llbracket x_0 \\rrbracket +\\llbracket x_1 \\rrbracket +\\cdots +\\llbracket x_{n-1} \\rrbracket -\\llbracket x_n \\rrbracket \\\\-\\llbracket x_0 \\rrbracket -\\llbracket x_1 \\rrbracket +\\cdots +\\llbracket x_{n-1} \\rrbracket -\\llbracket x_n \\rrbracket \\end{vmatrix}\\end{split}$ It is clear that the computations only rely on efficient homomorphic additions, avoiding expensive multiplications.",
"Therefore, the execution of $conv$ and $fc$ layers can be very efficient.",
"In the sub-section REF , we introduce the \"$+1$ Trick\" to further improve efficiency."
],
[
"$+1$ Trick",
"We exploit the fact that the weights are binarized, $\\textbf {a} \\in \\lbrace +1,-1\\rbrace ^m$ , to roughly halve the computation cost, through a $+1$ trick as follows, $\\begin{split}\\textbf {a}\\textbf {x} & = (\\textbf {1}+\\textbf {a})\\textbf {x}-\\textbf {1}\\textbf {x}\\\\& =\\frac{(\\textbf {1}+\\textbf {a})}{2}\\textbf {x}+\\frac{(\\textbf {1}+\\textbf {a})}{2}\\textbf {x} -\\textbf {1}\\textbf {x}\\end{split}$ where $\\textbf {1}$ indicates a vector in which each element is 1.",
"Two avoid multiplications, we split $(\\textbf {1}+\\textbf {a}) \\in \\lbrace 0, +2 \\rbrace ^m$ into two pieces of $\\frac{(\\textbf {1}+\\textbf {a})}{2} \\in \\lbrace 0, +1 \\rbrace ^m$ .",
"For the same layer, we only need to compute $\\textbf {1}\\textbf {x}$ once, and the cost can be amortized by all the $conv$ filters.",
"For the case that there are more $+1$ than $-1$ , we can adapt the Equation (REF ) to $\\textbf {a}\\textbf {x}=-\\frac{(\\textbf {1}-\\textbf {a})}{2}\\textbf {x}-\\frac{(\\textbf {1}-\\textbf {a})}{2}\\textbf {x} +\\textbf {1}\\textbf {x}$ .",
"Therefore, we can always halve the homomorphic computations by the $+1$ trick.",
"Different from [32], we don't need to binarize the input $\\textbf {x}$ , facilitating accuracy preservation."
],
[
"Comparison and New Benchmark",
"In this section, we first conduct a general comparison between the Popcorn and prior arts regarding the privacy guarantee and utility (Section REF ); then we test Popcorn and compare it with previous arts in term of efficiency; Lastly, we report the benchmarks of oblivious inference on the ImageNet dataset, based on start-of-the-art networks."
],
[
"Evaluation Settings",
"We implement the Popcorn framework based on OPHELib [25], which provides an implementation of the Paillier encryption scheme, written in C++.",
"The code is compiled using GCC with the '-O3' optimization, and the OpenMP for parallel acceleration is activated.",
"The test is performed on (Ubuntu 18.04 LTS) machines with Intel(R) Xeon(R) CPU E5 and 32GB of RAM.",
"The Paillier key size is always set to 2048 bits.",
"We execute the comparison and benchmarks on the MNIST, CIFAR-10, and the ILSVRC2012 ImageNet dataset.",
"To the best of our knowledge, this is the first report for benchmarking oblivious inference on the ImageNet dataset."
],
[
"General Comparison of Prior Arts",
"Existing frameworks are usually efficiency-oriented, with different compromises in terms of privacy and computational guarantees.",
"For clarification, we describe prior arts and our Popcorn framework according to the following guarantees, P1 (data privacy).",
"The framework hides the client data from the server, except for data dimensions.",
"P2 (network privacy).",
"The framework hides the server's network weights (as well the output values of each hidden layer) from the client.",
"P3 (network privacy).",
"The framework hides the server's network (including network weights, architecture and the output values of each hidden layer) from the client, except for (1) the number of layers; (2) the number of activations of each layer; (3) classification results.",
"P4 (network privacy).",
"The framework hides the server's network from the client, except for classification results.",
"V1.",
"The framework supports any type of CNNs.",
"V2.",
"The framework does not rely on any external party.",
"Table: The comparison of different frameworks.As shown in Table REF , all the frameworks can meet the $P1$ criteria, i.e., the client data privacy is well preserved.",
"However, the network information is leaked to different extents within different frameworks.",
"CryptoNets [11] preserves the most network privacy ($P2,P3,P4$ , assume the encryption parameters are large enough), while SecureML [23], MiniONN [21] and XONN [30] leak the most network information (i.e., $P3, P4$ are leaked).",
"Gazelle [16] and Popcorn achieve a compromise ($P2, P3$ are protected).",
"Note that, to improve efficiency, Gazelle introduced a padded patch, but pay the price of privacy, i.e., the $conv$ filter size is disclosed to the client.",
"So far, it is very challenging to completely protect network privacy.",
"For example, in CryptoNets, the activation method and network size can be inferred by the encryption parameters; in Gazelle and Popcorn, the size of each network layer can be deduced by the number of neurons.",
"Though multiparty-computation-based frameworks do not completely protect network privacy, we believe that it is important to hide more information, rather than ignore the privacy breach or directly disclosing the network information to the client.",
"The Versatility, i.e., the support for various CNNs ($V1$ ) and the requirements of server settings ($V2$ ), directly impact the deployments in reality.",
"CryptoNets and SecureML only support linearized CNNs, which require substituting non-linear activation functions with polynomials.",
"This approach may significantly reduce accuracy performance, especially for large neural networks.",
"The SecureML needs two non-colluding servers, which may narrow the applicable scenarios.",
"XONN is applicable only when the weights and activations of a network are binarized.",
"Gazelle, MiniONN, and Popcorn satisfy both the $V1$ and $V2$ criteria, as they don't need to adjust the original design of a network.",
"It is worth mentioning that, using XONN, the client will undertake the most computational overhead, as it is responsible for executing the complied network circuits to obtain classification results.",
"It may become a heavy burden to the client when the network becomes large."
],
[
"Efficiency Comparison with Prior Arts",
"To compare with prior arts, we report runtime (RT), communication bandwidth (COM) and accuracy (acc%) on MNIST and CIFAR-10 classification tasks.",
"Since the Popcorn can leverage compressed and binarized networks to accelerate the oblivious inference.",
"We implement two versions of the Popcorn framework, Popcorn-b and Popcorn-c.",
"The former supports binarized neural networks (section REF ) and the latter supports compressed neural networks ( section REF ).",
"For network binarization, we follow the XNOR-Net method [29] that we only binarize the network weights and leave the activation values as the original.",
"By this, we can preserve the information carried on activations, benefiting accuracy performance [28].",
"For network compression, we adopt the layer-by-layer pruning-then-quantization approach suggested in Section REF .",
"Firstly, we adopt the layer-by-layer weight pruning method introduced by [36], to remove more weights in lower layers.",
"Then, we employ the deep-compression method [12] to quantize the rest non-zero weights, to use lower bits to represent weights in $fc$ layers, and $conv$ layers which contain more filters.",
"As mentioned above, in the implementation, we build the network binarization and the pruning-then-quantization approaches with existing arts (i.e., [29], [36], [12]), thus avoiding tedious accuracy performance evaluation and focusing on the efficiency testing and comparison."
],
[
"Evaluation on MNIST",
"The MNIST is an entry-level image classification dataset.",
"It consists of a set of grayscale images of handwritten digits (i.e., [0,9]), and the dimension of each image is $28\\times 28 \\times 1$ .",
"We perform the experiments with three classical neural networks which were frequently adopted by previous arts, as shown in Table REF .",
"Table: Network architectures for the MNIST dataset.",
"NM stands for Network on MNIST.",
"NM1 is a MLP network.",
"NM2 and NM3 are two small CNNs.",
"In CryptoNets, the relurelu activation layer is replaced by the square (f(x)=x 2 f(x)=x^2) activation layer.",
"For detailed architecture information, please refer to the papers listed in the \"Source\" column.The \"P/Q\" presents the pruning ratio (i.e., the number of removed weights divided by the number of total weights) and the average number of bits to represent the overall weights.",
"As shown in Table REF , we can remove more than 90% weights and quantize the rest weights to $\\le 6$ bits without losing accuracy.",
"Sometimes, the Popcorn-c even results in higher accuracy.",
"Table: Comparison on MNIST.",
"RT means the runtime in second.",
"COM presents the communication bandwidth in megabyte.For the binarized model evaluation, we did not adopt the scaling-factor method introduced by XONN [30].",
"Instead, for all the networks, first we double the output size of the first layer, the others remain the same; then, we follow the XNOR-Net training method [29] to obtain binarized models.",
"There are two main reasons leading us to the current implementation.",
"First, we don't quantize the activation values, more information can be carried to preserve accuracy performance.",
"Second, the networks for the MNIST classification tasks are very small (e.g., small input dimensions, $28\\times 28 \\times 1$ ).",
"Instead of using a complicated method to discover a thin network architecture, we can empirically and efficiently try different settings of the neural networks.",
"Regarding the runtime performance, XONN and Gazelle are in the leading position, the Popcorn follows.",
"For the communication overhead, the Popcorn framework shows a significant advantage.",
"According to the results, all the frameworks achieve equivalent accuracy (from 97.6% to 99.0%, see Table REF ).",
"XONN and CryptoNets require modifying the original network design to fit the constraints of the adopted crypto primitives.",
"The CryptoNets shows that by substituting the non-linear activation method (e.g., $relu$ ) with a square function, it can still get decent accuracy; XONN demonstrates that it is possible to improve accuracy by scaling up the network architecture size, remedying the accuracy loss caused by the binarizing network weights and activations.",
"However, a natural question arises, can such tricks be applied to larger datasets (e.g., larger input dimensions and network size)?"
],
[
"Evaluation on CIFAR-10",
"The CIFAR-10 is another popular image classification dataset, which consists of a number of colorful images and categorized into 10 classes such as bird, truck, cats.",
"The dimension of each image is $32\\times 32 \\times 3$ .",
"Unlike the MNIST, the CIFAR-10 classification tasks require sophisticated design on neural networks.",
"The tricks applied to the MNIST for preserving accuracy may not work on the CIFAR-10.",
"For example, Li et at.",
"[6] investigated different approximation methods to replace non-linear activation functions, but none of them could obtain decent accuracy.",
"Straightforwardly increasing architecture size also becomes struggling to improve accuracy (see Table REF ).",
"We conduct the experiments with two networks adopted by prior arts and a VGG variant for CIFAR-10 [34], [37], summarized in Table REF .",
"Table: Networks on CIFAR-10.",
"NC stands for Nnetworks on CIFAR-10.",
"VGG-c is a customized network for CIFAR-10 .",
"For detailed architecture information, please refer to the papers listed in the \"Source\" column.For network binarization, we also use the XNOR-Net training method.",
"For the NC1 and NC2, we double the number of $conv$ filters of the first 3 layers to obtain equivalent accuracy with their full precision version.",
"For VGG-c, we retain the original network architecture.",
"Note that VGG-c has many more filters than NC1 and NC2.",
"In Popcorn-b, we don't binarize activations.",
"As shown in Table REF , the network NC1 leads to a 10% decrease in accuracy.",
"Therefore, simply scaling up the size of a fully binarized network may not obtain decent accuracy as expected, instead, it is suggested to use state-of-the-art networks for classification tasks.",
"Table: Comparison on CIFAR-10.",
"RT means the runtime in second.",
"COM presents the communication bandwidth in megabyte.As shown in Table REF , at the same accuracy level, Popcorn requires much less communication overhead.",
"For example, to reach the accuracy of 88.0%, the communication overhead of XONN is around 41 GB, while Popcorn-b only needs 704.7 MB, which is $60 \\times $ smaller.",
"In addition, we use the state-of-the-art binarized VGG-c, getting an accuracy of 91%, and communication bandwidth is only 450 MB.",
"This observation shows again the importance of using start-of-the-art networks."
],
[
"Benchmarks on ImageNet",
"With the promising results on the datasets MNIST and CIFAR-10 (especially the communication overhead), we look at a commercial-level dataset, the ImangeNet ILSVRC2012 dataset.",
"Neural networks for classification on this dataset often have large input dimensions (i.e., $224 \\times 224 \\times 3$ ), which is much larger than the input dimension adopted for MNIST ( $28 \\times 28 \\times 1$ ) and CIFAR-10 ( $32 \\times 32 \\times 3$ ).",
"To the best of our knowledge, this is the first report for benchmarking the oblivious inference on ImageNet classification tasks.",
"To evaluate the Popcorn framework, we adopt AlexNet [19] and VGG [34], which are two milestone networks for the ImageNet classification tasks, to benchmark the oblivious inference.",
"Compared with the networks used for MNIST and CIFAR-10 classifications, the most significant difference is that the dimensions of input and each hidden layer are much larger.",
"For ease of future comparison, we describe the network architectures in Table REF and Table REF , respectively.",
"Table: AlexNet : pw stands for the dimension of the max-pooling window; s is the stride.",
"P/Q records the pruning ratio and the number of bits for weights representation of each layer.Table: VGG .",
"For fcfc layers, the output dimension is recorded in the \"Kernel\" column.For network compression and binarization, we use the same method applied to the MNIST and CIFAR-10 classification tasks.",
"Thanks to [36], a pruned AlexNet model already exists, and it was pruned starting from lower layers.",
"So we directly use it as the basis, and quantize the remaining non-zero weights through the deep compression method [12].",
"The compression results of each layer of AlexNet and VGG are summarized in the Table REF and Table REF .",
"Table: benchmarks on ImageNet: RT (m) means the runtime in minute; COM presents the communication bandwidth in megabyte.",
"(redNote: this version corrects a naive but significant typo in our previous version.",
"Previously, we mistakenly indicated the communication cost in COM(g), i.e., communication bandwidth in gigabyte.",
"In fact, it should be COM as in this version, and COM presents the communication bandwidth in megabyte.",
"It is megabyte, not gigabyte.",
")We report the runtime and communication bandwidth in Table REF .",
"Compared with MNIST and CIFAR-10, the runtime is significantly increased, as the network size is orders of magnitude larger than that for MNIST and CIFAR-10.",
"The communication bandwidth is still in a reasonable range, even lower than the bandwidth required by the prior arts to run the CIFAR-10 classification tasks.",
"It is worth stressing that the bandwidth complexity of Popcorn is $O(n)$ , where $n$ is the number of activations.",
"The bandwidth complexity of XONN and Gazelle relies on the network architecture.",
"For example, using XONN, we need to compile the whole network into circuits; using Gazelle, large input dimensions require high degree polynomials (the efficiency is also obviously affected).",
"When we try to evaluate AlexNet using Gazelle, on the same machine that the Popcorn runs, the out-of-memory error always occurs (using the implementation provided by [16]).",
"An estimation of executing AlexNet with Gazelle, the communication overhead is at least 50 GB, the XONN is even worse.",
"Therefore, the Popcorn can have a significant advantage for evaluating state-of-the-art networks (e.g., AlexNet and VGG)."
],
[
"Related Work",
"Barni et al.",
"[1] initiated one of the earliest attempts for oblivious inference, using homomorphic encryption (HE).",
"Since HE is not compatible with non-linear algebraic operations (e.g., comparison, division), they introduced a multiplicative obfuscation to hide intermediate computation results.",
"However, this method leaks information about neural network weights [26].",
"Gilad-Bachrach et al.",
"[11] replaced the non-linear activation function (e.g., $relu(0,x)$ [24]) with a low-degree polynomial ($f(x)=x^2$ ), making the neural network fully compatible with HE.",
"Several works [6], [5], [4] developed different methods to improve the CryptoNets paradigm, in terms of efficiency and accuracy.",
"However, compared with other approaches, the results are still not promising.",
"Rouhani et al.",
"[31] proposed garbled circuits based framework for the oblivious inference, where the server compiles a pre-trained network into circuits and sends the circuits to a client, the client gets the prediction results by evaluating the circuits.",
"However, performing multiplication in GC has quadratic computation and communication complexity with respect to the bit-length of the input operands.",
"This fact rises up a serious efficiency problem for a precise inference (which often needs a high bit-length of the input operands).",
"Riazi et al.",
"[30] leveraged fully binarized neural networks, of which the weights and activations are binary (i.e., $\\lbrace +1,-1\\rbrace $ ), to accelerate the GC-based oblivious inference.",
"However, fully binarized neural networks are not stable in accuracy performance, especially, when used for classification tasks on large-scale datasets (e.g., ImageNet [19]).",
"Liu et al.",
"[21] combined HE, GC and SPDZ [8] to speedup the oblivious inference.",
"It showed a hybrid approach can be promising in both efficiency and accuracy.",
"To balance the communication overhead and accuracy, Mishra et al.",
"[22] proposed to replace partial non-linear activations with low-degree polynomials, the inference computation is similar to [21].",
"However, the SPDZ-based methods directly reveal the network architecture to the client.",
"Juvekar et al.",
"[16] leveraged SIMD to accelerate the linear computation in the inference by packing multiple messages into one ciphertext, and they used GC to compute the $relu$ activation and max-pooling layers.",
"Zhang et al.",
"[38] improved this solution by reducing the permutation operations when performing dot-product based on packed ciphertexts.",
"The trusted execution environment technology (TEE, e.g., Intel SGX [7]) is also an interesting approach to build oblivious inference frameworks (e.g., [35]).",
"Generally, most TEE-based frameworks are more efficient than cryptography-based solutions [35].",
"However, this approach requires trust in hardware vendors and the implementation of the enclave.",
"We leave this discussion of TEE-based solutions out of the scope of this paper."
],
[
"Conclusion",
"This work presented a concise oblivious neural network inference framework, the Popcorn.",
"This framework was completely built on the Paillier HE scheme.",
"It is easy to implement, one only needs to replace the algebraic operations of existing networks with their corresponding homomorphic operations.",
"We conducted experiments on different datasets (MNIST, CIFAR-10 and ImageNet), and showed its significant advantage in communication bandwidth.",
"To our best knowledge, this work is the first report for oblivious inference benchmark on ImageNet-scale classification tasks."
]
] | 2107.01786 |
[
[
"Zero-modified Count Time Series with Markovian Intensities"
],
[
"Abstract This paper proposes a method for analyzing count time series with inflation or deflation of zeros.",
"In particular, zero-modified Poisson and zero-modified negative binomial series with intensities generated by non-negative Markov sequences are studied in detail.",
"Parameters of the model are estimated by the method of estimating equations which is facilitated by expressing the model in a generalized state space form.",
"The latent intensities required for estimation are extracted using generalized Kalman filter.",
"The applications of proposed model and its estimation methods are illustrated using simulated and real data sets."
],
[
"Introduction",
"Time series data in certain areas such as public health and environment are available often in the form of counts.",
"For instance, weekly or monthly occurrence of certain disease in a city over time form a time series of counts.",
"In air pollution studies, modelling annoyance caused by particulate matter such as dust and smoke (or by odor and noise), the observations often are counts and are dependent over time leading to the possible use of count time series models.",
"In such situations it is natural to adopt a Poisson process or some other discrete time counting process to model the data.",
"Some of these time series such as monthly counts of workplace injuries or crimes in a region may contain large number of zeros which requires the use of what is known as zero inflation models or hurdle models.",
"Another special case found in practice is the case of smaller number of occurrences of zeros known as zero deflated data.",
"Moreover, some of the count data may not have zeros.",
"Such data may not be modelled using usual Poisson models.",
"In such situations, zero modified models provide a more general frame work to take care of the count data with inflation or deflation or truncation at zero, when no information about the nature of this situation is known.",
"A recent Handbook of discrete-valued time series edited by Davis et al.",
"(2016) contains several theoretical developments and application avenues of count time series models.",
"Following Cox (1981) one can classify count time series models as (i) observation driven models or (ii) parameter driven models.",
"In these count models, the mean and variance may depend on previous measurements and so it is natural to consider generalized autoregressive conditional heteroscedastic (GARCH) like models and refer to them as observation driven.",
"On the other hand we can think of the mean and variance to have latent models as in Stochastic Volatility (SV) models and such models are called parameter driven.",
"Also, in modelling the number of transactions in financial markets, count series models with dependent intervals between transactions is a natural choice.",
"In this context, Engle and Russell (1998) introduced the Autoregressive Conditional Duration (ACD) model.",
"A simple ACD model can be described as follows: Suppose $X_t$ is a random variable (rv) denoting the duration between the ‘arrival times’ or times of transactions, then $E(X_t|x_{t-1},x_{t-2}, ...)$ = $\\lambda _{t}(x_{t-1},...,\\theta )$ or $X_t = \\lambda _t\\varepsilon _t,$ where $\\lbrace \\varepsilon _t\\rbrace $ is independent and identically distributed (iid) with density function $f(\\varepsilon _t,\\beta ),$ $E(\\varepsilon _t) = 1,$ $\\lambda _{t}$ is a function of the previous durations and parameters $\\theta ,$ where as $\\beta $ is a vector of parameters of the distribution with probability density function (pdf) $f$ .",
"This model is an example of an ‘observation driven model’ and can have many variations by specifying $\\lambda _t$ and $f$ differently.",
"It can also be generalised to address many different situations.",
"A recent survey on these models may be found in Bhogal and Variyam (2019).",
"The method of construction of such models was extended to incorporate the count series $\\lbrace Y_t\\rbrace $ with observation driven intensities.",
"One such model is the observation driven model introduced in Ferland et al.",
"(2006).",
"A very simple case of this is given by the following.",
"Conditional on the past, the counts, $Y_t,$ at time $t$ , has Poisson distribution, $Y_t|{\\cal F}_{t-1} \\sim P(\\lambda _t)$ , where $P$ is a Poisson distribution with intensity parameter $\\lambda _t$ and ${\\cal F}_{t-1}$ contains all the information up to time $t-1.$ In addition, $\\lambda _{t}$ depends on the previous observations through the model, $E(Y_t| y_{t-1},…)=\\lambda _{t}=\\gamma +\\alpha y_{t-1}+ \\beta \\lambda _{t-1}$ .",
"One can see that this is a special case of the ACD model.",
"Tjostheim (2016) discussed the details of count time series under observation driven setup.",
"This model was generalised to the case of random coefficient models (see for instance Fokianos (2016)), where $Y_t|{\\cal F}_{t-1} \\sim P(Z_t\\lambda _t)$ , where $\\lbrace Z_t\\rbrace $ is an iid sequence of positive random variables with mean 1 and independent of $Y_i, i<t$ .",
"This is a case of count process with parameter driven intensity processes.",
"Recently, Yang et al.",
"(2015), proposed a flexible class of dynamic models for zero inflated count time series in the state-space framework and performed a sequential Monte Carlo analysis.",
"A key property of the class of parameter driven count series is that, although the observations are correlated marginally, they are independent, conditional on the latent process.",
"In this paper we study the properties of the count processes with modified frequency of zeroes when their intensities are generated by certain latent parameter driven models.",
"Naturally the proposed model is more general than zero inflated models.",
"In the context of regression, Lambart (1992) proposed zero-inflated Poisson regression, to model the defects in manufacturing.",
"Dietz and Bohning (2000) introduced the zero modified Poisson regression.",
"Barreto-Souza (2015) studied the zero modified geometric INAR(1) model to analyze count time series with deflation or inflation of zeros.",
"Sharafi et al.",
"(2021) constructed a first order integer valued autoregressive process with zero modified Poisson-Lindley distributed innovations to model zero modified count time series.",
"To the best of our knowledge, the case of zero modified dynamic count time series is not discussed anywhere in the literature.",
"We try to fill this gap.",
"In particular we consider the zero modified count processes when the intensities are generated by stationary Markov sequences.",
"We list some specific examples of such Markov sequences in Section REF .",
"Statistical inference for observation-driven count processes are more easier to handle via likelihood based methods.",
"The involvement of unobserved latent intensities make the inference difficult for the parameter driven models.",
"Estimating functions (EFs) are widely used in situations where the explicit form of the likelihood function is not available or intractable (Godambe, 1985).",
"Naik-Nimbalkar and Rajarshi (1995) have used this method in the context of state-space (SS) models, whereas Thavaneswaran et al.",
"(2015) and Thekke et al.",
"(2016) have applied this method in the context of stochastic conditional duration models.",
"We propose methods based on EF to estimate parameters, which are more straight forward compared to the MCMC methods.",
"However, the resulting estimating equations depend on the latent intensities, which are not observable.",
"To circumvent this problem, we propose to filter the intensities from the observed count data.",
"In order to achieve that, we represent the model in a generalized state space (GSS) form and then adopt the generalized Kalman filter (GKF) algorithm proposed by Zenwirth (1988).",
"Rest of the paper is organised as follows.",
"In the next section, we introduce the zero modified count processes induced by latent Markov sequences referred to as zero modified stochastic conditional duration (ZMSCD) models.",
"The basic properties of zero modified Poisson (ZMP) and zero modified negative binomial (ZMNB) processes are described.",
"This section also introduces some Markov sequences suitable for generating the intensities in our ZMSCD models.",
"In Section 3 we formulate the model in GSS form and then write down the GKF algorithm.",
"Section 4 discusses the details of EF method for parameter estimation in our model.",
"Simulation results to illustrate the computation methods are summarized in Section 5.",
"Real data sets are also analyzed in Section 6 and some concluding remarks are provided in Section 7."
],
[
"ZMSCD models generated by Markovian intensities",
"Let $\\lbrace Y_t, t=0, \\pm 1, \\pm 2, ...\\rbrace $ be a discrete time count process on the state space $ \\lbrace 0,1,2,...\\rbrace $ and ${\\cal F}_{t-1}$ be the sigma field generated by $\\lbrace Y_{t-1}, Y_{t-2},...\\rbrace $ .",
"Suppose that, conditional on ${\\cal F}_{t-1},$ the rv $Y_t$ follows a zero-modified distribution (ZMD) with stochastic intensity function $\\lambda _t$ .",
"We assume that the stochastic intensity, $\\lbrace \\lambda _t\\rbrace $ is a stationary Markov sequence of non-negative rvs.",
"Let us now describe two forms of ZMD which were introduced to study the count regression models.",
"The first one initially studied by Dietz and Bohning (2000) for a zero modified Poisson model and then generalized by Bertoli et al (2019) is defined by $P^{M}(Y_t = k|{\\cal F}_{t-1},p_t,\\lambda _t) ={\\left\\lbrace \\begin{array}{ll}(1 - p_t) + p_tP(Y_t = 0 |{\\cal F}_{t-1},\\lambda _t), & k = 0 \\\\p_tP(Y_t = k|{\\cal F}_{t - 1},\\lambda _t), & k > 0, \\quad \\quad \\quad \\end{array}\\right.",
"}$ where $p_t$ is the zero modification parameter such that $0 \\le {p_t} \\le P^{-1}(Y_t = 0|{\\cal F}_{t-1},\\lambda _t).$ The model is quite general in the sense it includes several special cases.",
"For example, if $p_t=0$ , then $P^M(Y_t=0|p_t,\\lambda _t)=1$ , which gives the PMF of a rv degenerate at 0.",
"$p_t=1$ , then $P^M(Y_t=0|p_t,\\lambda _t)=P(Y_t=0|\\lambda _t)$ which is the usual model without any zero modification.",
"$0\\le p_t \\le 1$ , then $(1-p_t)P(Y_t=0|\\lambda _t)>0,$ which implies that the modified distribution has more zeros than the zeros in the base line distribution and hence this is a case of zero inflation.",
"${p_t} \\in \\left[ {1,{P^{ - 1}}\\left( {\\left.",
"{{Y_t} = 0} \\right|{{\\cal F}_{t-1}},{\\lambda _t}} \\right)} \\right]$ , then $(1-p_t)P(Y_t=0|\\lambda _t)<0$ which is a case of zero deflation.",
"${p_t} = {P^{ - 1}}\\left( {\\left.",
"{{Y_t} = 0} \\right|{{\\cal F}_{t-1}},{\\lambda _t}} \\right)$ , then $P(Y_t=0|\\lambda _t)=0$ which implies the zero truncated case with pmf: $\\hspace{-14.45377pt} P_T( Y_t = k|{\\cal F}_{t-1},p_t,\\lambda _t )&=& \\frac{P(Y_t = k |{{\\cal F}_{t-1}},\\lambda _t)}{P(Y_t > 0 |{{\\cal F}_{t-1},\\lambda _t } )}( {1 - {\\delta _Y}}),\\ \\ \\text{ where} \\ \\ \\ \\delta _Y ={\\left\\lbrace \\begin{array}{ll}1\\quad & if\\quad {Y_t} = 0\\\\0\\quad & \\text{otherwise}.\\end{array}\\right.",
"}$ In short, one can easily see that (REF ) is not a kind of mixture distribution typically chosen to handle the zero inflated data.",
"Also, it is straight forward to see that $P^M(Y_t|p_t,\\lambda _t) \\ge 0$ and is a proper PMF.",
"To see the effect of zero modification parameter $p_t$ , the proportion of additional or missing zeros can be computed as ${P^M}\\left( {\\left.",
"{{Y_t} = 0} \\right|{{\\cal F}_{t-1}},{p_t},{\\lambda _t}} \\right) - P\\left( {\\left.",
"{{Y_t} = 0} \\right|{{\\cal F}_{t-1}},{\\lambda _t}} \\right) = \\left( {1 - {p_t}} \\right)P\\left( {\\left.",
"{{Y_t} > 0} \\right|{{\\cal F}_{t-1}},{\\lambda _t}} \\right).$ So for a specified value of $p_t$ , one can identify the nature of zero modifications with respect to the base distribution.",
"An alternative parameterization of the ZMD proposed by Barreto-Souza (2015) is, $P^{M}(Y_t = k|{\\cal F}_{t-1},p_t,\\lambda _t) ={\\left\\lbrace \\begin{array}{ll}p_t +(1- p_t)P(Y_t = 0 |{\\cal F}_{t-1},\\lambda _t), & k = 0 \\\\(1-p_t)P(Y_t = k|{\\cal F}_{t-1},\\lambda _t), & k > 0, \\quad \\quad \\quad \\end{array}\\right.",
"}$ In this parameterziation, the range of zero modification parameter $p_t$ is specified by $- \\frac{{P\\left( {\\left.",
"{{Y_t} = 0} \\right|{{\\cal F}_{t-1}},{\\lambda _t}} \\right)}}{{1 - P\\left( {\\left.",
"{{Y_t} = 0} \\right|{{\\cal F}_{t-1}},{\\lambda _t}} \\right)}} \\le {p_t} \\le 1.$ This representation also covers some of the interesting special cases.",
"For example, if $p_t=0$ , then $P^M(Y_t=0|p_t,\\lambda _t)=P(Y_t=0|\\lambda _t)$ which gives usual model witout any zero modification.",
"$p_t=1$ , then $P^M(Y_t=0|p_t,\\lambda _t)=1$ , which is the case of degeneracy at 0.",
"$0\\le p_t \\le 1$ , then $p_tP(Y_t=0|\\lambda _t)>0$ implies the modified distribution has an excess proportion of zeros larger than the base line distribution and hence there is zero inflation.",
"${p_t} \\in \\left( { - \\frac{{P\\left( {\\left.",
"{{Y_t} = 0} \\right|{{\\cal F}_{t-1}},{\\lambda _t}} \\right)}}{{1 - P\\left( {\\left.",
"{{Y_t} = 0} \\right|{{\\cal F}_{t-1}},{\\lambda _t}} \\right)}},0} \\right)$ , then $p_tP(Y_t=0|\\lambda _t)<0,$ which implies that the modified distribution has lesser proportion of zeros than the base line distribution and hence it is a case of zero deflation.",
"${p_t} = - \\frac{{P\\left( {\\left.",
"{{Y_t} = 0} \\right|{{\\cal F}_{t-1}},{\\lambda _t}} \\right)}}{{1 - P\\left( {\\left.",
"{{Y_t} = 0} \\right|{{\\cal F}_{t-1}},{\\lambda _t}} \\right)}}$ , then $P(Y_t=0|\\lambda _t)=0$ which implies the zero truncated case.",
"Though both these forms (REF and REF ) of ZMD are used in the literature, we consider the second form (REF ) in this paper.",
"Our objective here is to study the properties of ZMSCD models when their intensity functions are generated by stationary non-negative Markov sequences.",
"Further we assume that the zero modification parameter $p_t=p$ as constant.",
"By the basic property of parameter driven models for counting processes, $Cov(Y_t|{\\cal F}_{t},Y_s|{\\cal F}_{s}) =0$ for $s\\ne t$ .",
"The following notations are introduced for a stationary Markov sequence $\\lbrace \\lambda _t\\rbrace $ .",
"Let $\\mu _{\\lambda } = E(\\lambda _t), \\sigma _{\\lambda }^2= Var(\\lambda _t)$ and the $lag$ -$k$ autocovariance and autocorrelation (ACF) of $\\lbrace \\lambda _t\\rbrace $ are respectively denoted by $ \\gamma _{\\lambda }(k) = Cov(\\lambda _t,\\lambda _{t+k})$ and $\\rho _{\\lambda }(k) =\\gamma _{\\lambda }(k)/\\sigma _{\\lambda }^2 .$ We focus on two of the commonly used zero modified counting processes namely zero modified Poisson (ZMP) and zero modified negative binomial (ZMNB).",
"Other zero modified models are also possible based on Generalized Poisson distribution, Poisson Lindley distribution, Poisson Inverse Gaussian distribution, geometric distribution, etc.",
"Some of the elementary properties of the resulting ZMSCD models are described below.",
"Zhu(2012) studied the details of zero inflated observation driven models."
],
[
"Zero Modified Poisson SCD Model",
"We say that $\\lbrace Y_t\\rbrace $ is a zero modified Poison stochastic conditional duration (ZMPSCD) process if the conditional distribution of $Y_t$ given the past information follows a zero modified Poisson distribution defined by $P(Y_t=k|{\\cal F}_{t-1}) ={\\left\\lbrace \\begin{array}{ll}\\omega _t + (1-\\omega _t) e^{-\\lambda _t} \\ \\ \\ \\ \\ \\text{if} \\ \\ \\ \\ \\ \\ k=0, \\\\(1-\\omega _t)e^{-\\lambda _t} \\lambda _t^k /k!",
"\\ \\ \\ \\ \\text{if} \\ \\ \\ \\ \\ \\ k=1,2,...,\\end{array}\\right.",
"}$ where $\\lbrace \\lambda _t\\rbrace $ is a stationary Markov sequence of non-negative rvs.",
"This model is obtained from the ZMSCD model (REF ) by taking $p_t=\\omega _t, 0 < 1-{p _t} < \\frac{{{e^{{\\lambda _t}}}}}{{{e^{{\\lambda _t}}} - 1}}$ and the base line distribution as the Poisson with mean $\\lambda _t$ .",
"When $\\omega _t=\\omega $ and the range of $\\omega $ is restricted to the unit interval, the model reduces to Zero Inflated Poisson(ZIP) model.",
"The conditional moments of any order $Y_t$ given ${\\cal F}_{t-1}$ can be obtained using its probabilty generating function (PGF): $G_y(s)=\\omega +(1-\\omega )e^{(s-1)\\lambda _t}$ , for $0<s<1$ .",
"Thus we have the conditional mean and variance respectively: $E(Y_t|{\\cal F}_{t-1}) = (1-\\omega )\\lambda _t \\ {\\text{and}}\\ \\ \\mbox{Var}(Y_t|{\\cal F}_{t-1}) = (1-\\omega )(1+\\omega \\lambda _t)\\lambda _t.$ Consequently the unconditional mean and variance of $Y_t$ respectively become $E(Y_t) = E(E(Y_t|{\\cal F}_{t-1})) = E((1-\\omega )\\lambda _t) = (1-\\omega ) \\mu _{\\lambda }\\\\V(Y_t) =V(E(Y_t|{\\cal F}_{t-1}))+E(V(Y_t|{\\cal F}_{t-1}))= (1-\\omega )[\\mu _{\\lambda }+\\sigma _{\\lambda }^2 + \\omega \\mu _{\\lambda }^2].$ To compute autocovarance function of $\\lbrace Y_t\\rbrace $ consider $E(Y_tY_{t+k})=E(E(Y_tY_{t+k}|{\\cal F}_{t-1},{\\cal F}_{t+k-1} ))= (1-\\omega )^2 E(\\lambda _t.\\lambda _{t+k}),$ where we have used the conditional independence of $Y_t$ given the past.",
"Thus, we have the autocovarance function of $\\lbrace Y_t\\rbrace $ : $\\gamma _y(k) = Cov(Y_t, Y_{t+k}) = (1-\\omega )^2 \\gamma _{\\lambda }(k)$ and its autocorrelation function (ACF) : $\\rho _y(k) = \\frac{(1-\\omega )^2 \\gamma _{\\lambda }(k)}{(1-\\omega )[\\mu _{\\lambda }+\\sigma _{\\lambda }^2 + \\omega \\mu _{\\lambda }^2]} = \\frac{(1-\\omega )\\sigma _{\\lambda }^2\\rho _{\\lambda }(k)}{\\mu _{\\lambda }+\\sigma _{\\lambda }^2 + \\omega \\mu _{\\lambda }^2} \\le \\rho _{\\lambda }(k).$ The geometrically decreasing behaviour of the ACF of a stationary Markov sequence $\\lbrace \\lambda _t\\rbrace $ is preserved by the corresponding ZMSCD sequence.",
"However, $\\rho _y(k) \\le \\rho _{\\lambda }(k)$ for every $k$ over the whole parameter space.",
"Fig.1a is a sample plot to show this behaviour."
],
[
"Zero modified negative binomial SCD Models",
"In addition to zero modification, overdispersion can also be present in many count time series.",
"The zero modified negative binomial (ZMNB) model can be used in such situations.",
"The ZMNBSCD model is obtained from the general ZMSCD model, (REF ) by taking $p_t=\\omega _t$ such that $0 < 1-{p_t} < {\\left[ {1 - {{\\left( {\\frac{1}{{1 + a\\lambda _t^c}}} \\right)}^{\\frac{{\\lambda _t^{1 - c}}}{a}}}} \\right]^{ - 1}}$ and the base line distribution as the negative binomial with parameters $\\lambda _t>0$ and $a>0$ .",
"In terms of notations of Zhu(2012), the probability mass function of ZMNB distribution with intensity function $\\lambda _t$ is given by $P(Y_t=k|{\\cal F}_{t-1}) ={\\left\\lbrace \\begin{array}{ll}\\omega _t+(1-\\omega _t)\\Big (\\frac{1}{1+a\\lambda _t^c}\\Big )^{\\lambda _t^{1-c}/a}, & \\text{if}\\ \\ k=0,\\\\(1-\\omega _t)\\frac{\\Gamma (k+\\lambda _t^{1-c}/a)}{k!\\Gamma (\\lambda _t^{1-c}/a)} \\Big (\\frac{1}{1+a\\lambda _t^c}\\Big )^{\\lambda _t^{1-c}/a} \\Big (\\frac{a\\lambda _t^c}{1+a\\lambda _t^c}\\Big )^k,& \\text{if} \\ \\ k=1,2,... , \\end{array}\\right.",
"}$ where $\\lambda _t>0, a > 0$ and we denote it by $(Y_t=k|{\\cal F}_{t-1}) \\sim ZMNB(\\lambda _t, a, \\omega _t).$ The index $c=0,1$ identifies the particular form of the negative binomial distribution.",
"When $\\omega _t=\\omega $ and $0<\\omega <1$ , the model becomes zero inflated negative binomial model.",
"The conditional and unconditional moments of such model are similar to those in the case of ZMPSCD model, which are listed below: $E(Y_t|{\\cal F}_{t-1}) = (1-\\omega )\\lambda _t \\ \\ \\mbox{Var}(Y_t|{\\cal F}_{t-1}) = (1-\\omega )(1+\\omega \\lambda _t+a\\lambda _t^c)\\lambda _t, \\ \\ E(Y_t) = (1-\\omega )\\mu _{\\lambda },$ $V(Y_t) ={\\left\\lbrace \\begin{array}{ll}(1-\\omega )[(1+a)\\mu _{\\lambda }+\\sigma _{\\lambda }^2 + \\omega \\mu _{\\lambda }^2] \\ \\ \\ \\ \\text{if} \\ \\ \\ \\ \\ \\ c=0,\\\\(1-\\omega )[\\mu _{\\lambda }+(a+1)\\sigma _{\\lambda }^2 + (\\omega +a)\\mu _{\\lambda }^2] \\ \\ \\ \\ \\text{if} \\ \\ \\ \\ \\ \\ c=1.\\end{array}\\right.",
"}$ Further the ACF may be expressed as $\\rho _y(k) ={\\left\\lbrace \\begin{array}{ll}\\frac{(1-\\omega )\\sigma _{\\lambda }^2\\rho _{\\lambda }(k)}{[(1+a)\\mu _{\\lambda }+\\sigma _{\\lambda }^2 + \\omega \\mu _{\\lambda }^2] } \\ \\ & \\text{if} \\ \\ \\ \\ \\ \\ c=0,\\\\\\frac{(1-\\omega )\\sigma _{\\lambda }^2\\rho _{\\lambda }(k)}{[\\mu _{\\lambda }+(a+1)\\sigma _{\\lambda }^2 + (\\omega +a)\\mu _{\\lambda }^2]} \\ \\ \\ \\ & \\text{if} \\ \\ \\ \\ \\ \\ c=1.\\end{array}\\right.",
"}$ Figure 1 about here.",
"Note 1: As $a\\rightarrow 0$ , the $ZMNB(\\lambda _t, a, \\omega )$ reduces to $ZMP(\\lambda _t, \\omega )$ .",
"Consequently, the mean, variance and ACF of ZMNB count process reduce to those of ZMP count process as can be easily verified from the corresponding expressions.",
"The following fourth moment for the ZMNB process is useful for estimating the parameters.",
"$E(Y_t -(1-\\omega )\\lambda _t)^4|{\\cal F}_{t-1})&=& (1-\\omega )\\lambda _t\\Big [\\lambda _t^3(3\\omega ^3-3\\omega ^2+\\omega ) + \\nonumber 6\\lambda _t^2\\omega ^2+4\\lambda _t\\omega +3\\lambda _t+1 + 6a^3\\lambda _t^{3c}\\\\&& + (12a^2+(3+8\\omega )a^2\\lambda _t)\\lambda _t^{2c} + (6a\\lambda _t^2\\omega ^2+6(1+2\\omega )a\\lambda _t)\\lambda _t^{c}\\Big ].$"
],
[
"Examples for Markov sequences of intensities.",
"1.",
"First order autoregressive (AR(1)) model for non-negative rvs: Let $\\lbrace \\eta _t\\rbrace $ be a sequence of iid rvs with $E(\\eta _t) = \\mu _{\\eta }$ and $V(\\eta _t) = \\sigma ^2_{\\eta }$ .",
"Define $\\lambda _t =\\rho \\lambda _{t-1} +\\eta _t, \\ \\ \\ \\ 0\\le \\rho <1.$ The distribution of $\\eta _t$ is chosen in such a way that $\\lbrace \\lambda _t\\rbrace $ defines a stationary sequence of specified marginal distribution.",
"In particular this model includes the exponential AR(1) (EAR(1)) model and gamma AR(1) (GAR(1)) model defined by Gaver and Lewis (1980).",
"In the case of EAR(1) model, the marginal distribution of $\\lbrace \\lambda _t\\rbrace $ is exponential with pdf: $f(x;\\beta ) = \\beta e^{-\\beta x}, x\\ge 0, \\beta >0$ if and only if the distribution function of the innovation rv, $\\eta _t$ in (REF ) is given by $F_{\\eta }(x) = \\rho + (1-\\rho )(1-e^{-\\beta x}), x\\ge 0, \\beta >0.$ The intensity sequence, $\\lbrace \\lambda _t\\rbrace $ is a GAR(1) sequence if each $\\lambda _t$ follows a gamma marginal ($G(\\beta ,p)$ ) distribution with pdf: $f(x|\\beta ,p) = \\frac{e^{-\\beta x}\\beta ^p x^{p-1}}{\\Gamma (p)}, \\ \\ \\ x\\ge 0, \\beta >0, p>0.$ The marginal distribution of $\\lbrace \\lambda _t\\rbrace $ in (REF ) is $G(\\beta ,p)$ if and only if the distribution of the innovation rv, $\\eta _t$ is given by (cf, Lawrance (1982)) that of $ \\eta _t = \\sum _{i=1}^{N}\\rho ^{U_i}E_i,$ where $\\lbrace U_i\\rbrace $ and $\\lbrace E_i\\rbrace $ are mutually independent iid $Unif(0,1)$ and $Exp(\\beta )$ rvs and $N$ follows a Poisson distribution with mean $plog(1/\\rho ).$ 2.",
"Random Coefficient AR(1) (RCAR(1)) models: Let $\\lbrace J_i\\rbrace $ be a sequence of iid rvs distributed over $(0,1)$ and difine $\\lambda _t =J_{t} \\lambda _{t-1} +\\eta _t, \\ \\ \\ \\ 0\\le E(J_t)<1.$ The gamma Markov sequences defined by Sim (1990) and Beta Gamma AR(1) model difined by Lewis et al (1989) are included in the above RCAR(1) model.",
"The NEAR(1) and TEAR(1) models of Lawrance and Lewis (1981) are also special cases of (REF ).",
"3.",
"Product AR(1) models: Let $\\lbrace V_t\\rbrace $ be a sequence of iid non-negative rvs with $E(V_t) = \\mu _{v}$ and $V(V_t) = \\sigma ^2_{v}$ .",
"Assume that $\\lambda _0$ is independent of $V_1$ and define $\\lambda _t = \\lambda ^{\\rho }_{t-1} V_t, \\ \\ \\ \\ 0\\le \\rho <1, t=1,2,....$ Mckenzie (1982) introduced this model for defining a stationary gamma Markov sequence and compared it with the GAR(1) model.",
"Abraham and Balakrishna (2012) obtained an explicit form of the distribution of the innovation rv, $V_t$ .",
"Muhammed, Balakrishna and Abraham (2019) obtained the innovation distribution of generalized gamma PAR(1) model and proposed methods of estimation for the stochastc volatility models generated by it.",
"4.",
"Pitt-Walker models: Pitt and Walker (2005) proposed a method of constructing stationary AR(1) sequences $\\lbrace \\lambda _t\\rbrace ,$ by choosing marginal and conditional distributions which satisfy a particular equation.",
"They used this idea to construct stationary Markov sequences with stationary marginal distributions such as gamma, inverse gamma, etc.",
"The resulting sequence satisfies the relation: $E(\\lambda _t|\\lambda _{t-1}) = \\rho \\lambda _{t-1} + (1-\\rho )\\mu _{\\lambda },\\ \\ \\ 0\\le \\rho <1.$"
],
[
"State Space representation of ZMSCD models ",
"As stated in Section 1, we propose EF method for parameter estimation, which requires the filtering of the latent intensities.",
"This can be facilitated by expressing the model in a generalized state space (GSS) form and then using generalized Kalman filtering (GKF).",
"To achieve this goal we adapt the method proposed by Zehnwirth (1988), which is summarised below for our reference.",
"The GSS model contains an observation equation in terms of the data and a state equation.",
"$Y_t=F_t\\Lambda _t + \\varepsilon _t \\ \\ \\ \\ \\text{: observation equation}$ $\\Lambda _t = G_t \\Lambda _{t-1} + W_t, \\ \\ \\ \\text{ : state equation,}$ where (a) $\\Lambda _t$ is a q-dimensional state vector; (b) $F_t$ , and $G_t$ are known matrices of dimensions $p \\times q$ and $q \\times q$ , respectively; (c) $\\varepsilon _t$ , is a p-dimensional observation error such that $E[\\varepsilon _t|\\Lambda _t] = 0$ , $Cov[\\varepsilon _t|\\Lambda _t] = V_t(\\Lambda _t),$ where $V_t(.",
")$ is a known function of unknown $\\Lambda _t$ and $Cov[\\varepsilon _t,\\varepsilon _s|\\Lambda _t,\\Lambda _s] = 0$ for all $s \\ne t$ .",
"(d) The q-dimensional random vectors $W_t$ , form an uncorrelated sequence with $E[W_t] = 0$ and $Cov[W_t] = w^*_t$ with ${w^*_t}$ known, and $C[W_t,\\Lambda _s] = 0$ for $t > s$ ; and (e) $Cov(\\varepsilon _t,W_s|\\Lambda _t) = 0$ for all $s$ and $t$ .",
"Consequently it follows that $Cov[Y_t,Y_s|\\Lambda _t] = 0$ for all $t > s.$ Under this set up Zehnwirth (1988), established the following GKF algorithm for the GSS model, where $\\hat{\\Lambda }_t$ denotes the filtered value of $\\Lambda _t$ conditional on the past observations: $\\hat{\\Lambda }_{t|t-1} = G_t\\hat{\\Lambda }_{t-1}, \\\\\\hat{\\Lambda }_{t} = \\hat{\\Lambda }_{t|t-1} + K_t(Y_t - F_t\\hat{\\Lambda }_{t|t-1}), \\\\K_t = C_{t|t-1}F_t^{\\prime }[F_tC_{t|t-1}F_t^{\\prime } + \\bar{V}_t]^{-1}, \\\\C_t = [I - K_tF_t]C_{t|t-1},$ where $\\hat{\\Lambda }_{t|s}$ is the minimum mean squared error linear estimator of $\\hat{\\Lambda }_t$ based on $Y_s, (s\\le t)$ , $C_{t|s}$ is the unconditional error covariance matrix of $\\hat{\\Lambda }_{t|s}$ and $\\bar{V}_t = E(V_t(\\Lambda _t))$ .",
"In particular, one writes $\\hat{\\Lambda }_{t} = \\hat{\\Lambda }_{t|t}$ and $C_{t} =C_{t|t}.$ The major difference between GSS and the usual SS set up is that $V_t(.",
")$ is a known constant in the latter case.",
"But in GSS model, $V_t(\\Lambda _t)$ is a known function of unknown $\\Lambda _t$ .",
"In order to implement the above GKF we replace $V_t(\\Lambda _t)$ by its expectation $\\bar{V}_t$ , computed using the state equation.",
"Next we simplify the algorithm when $\\lbrace Y_t\\rbrace $ follows a zero-modified SCD model with intensities generated by certain non-negative Markov sequences described in Section 2, where $\\lambda _t$ denotes the one-dimensional intensity function.",
"The filtering algorithms also lead to suitable estimating equations to estimate the parameters inolved, which we explore in the next section.",
"See also Thavaneswaran et al.",
"(2015) and Thekke et al.",
"(2016)."
],
[
"Estimation for Zero modified SCD models",
"As discussed in Section 2, conditional on the past, $Y_t$ follows a zero modified distribution with intensity function $\\lambda _t$ .",
"Further we assume that $\\lbrace \\lambda _t\\rbrace $ follows a stationary Markov sequence.",
"Let $\\theta =(\\theta _1,\\theta _2,...,\\theta _p)^{\\prime },$ (where prime denotes the transpose of a vector) be the vector of parameters indexing the finite dimensional distribution of $\\lbrace \\lambda _t\\rbrace $ .",
"There are no parameters other than $\\theta $ and $\\omega $ in the whole model.",
"So the components of $\\theta $ are the parameters present in the stationary marginal distribution of $\\lbrace \\lambda _t\\rbrace $ and its one-step transition distribution.",
"Based on the theory of linear estimating equations for stochastic processes, one may estimate the parameters of interest using the optimal EF based on the following martingale EFs: $g_{1t} = Y_t - E(Y_t|{\\cal F}_{t-1}) = Y_t - (1-\\omega )\\lambda _t \\\\g_{2t} = \\lambda _t - E(\\lambda _t|{\\cal F}_{t-1}) = \\lambda _t - E(\\lambda _t |\\lambda _{t-1}).$ Note that in the models listed above, $E(\\lambda _t|{\\cal F}_{t-1})$ is a (linear or nonlinear) function of $\\lambda _{t-1}$ and the parameters.",
"After establishing the optimal properties of the estimating functions, we can find the estimates by solving the optimal estimating equations based on $g_{1t}, g_{2t}$ .",
"But, $g_{it}, i=1,2 $ contain the latent variables $\\lambda _t$ , which are not observable.",
"So we implement the methods discussed in Section 3 to filter these latent variables and use them for estimation.",
"In fact the GKF alogorithm leads to a useful estimating equation, which may be used for estimation.",
"We illustrate the proposed methods under two special cases, namely ZMPSCD and ZMNBSCD which are discussed in the following subsections."
],
[
"Estimation for ZMPSCD model",
"Under a ZMPSCD model, we have $E(Y_t|{\\cal F}_{t-1}) = (1-\\omega )\\lambda _t$ and $V(Y_t|{\\cal F}_{t-1}) = (1-\\omega )(1+\\omega \\lambda _t)\\lambda _t$ .",
"Accordingly the GKF algorithm described in (3.3) to (3.6), may be written as: $\\hat{\\lambda }_{t} = \\hat{\\lambda }_{t|t-1} + \\frac{(1-\\omega )C_{t|t-1}}{(1-\\omega )^2C_{t|t-1} +(1-\\omega ) v}(Y_t - (1-\\omega )\\hat{\\lambda }_{t|t-1})$ with $v=\\mu _{\\lambda }+\\omega (\\sigma _{\\lambda }^2+\\mu _{\\lambda }^2).$ That is, ${\\hat{\\lambda }_{t|t} = \\rho \\hat{\\lambda }_{t-1|t-1} + (1-\\rho )\\mu _{\\lambda }} \\nonumber \\\\&& \\hspace{14.45377pt}+ \\frac{(1-\\omega )[\\rho ^2C_{t-1|t-1}+(1-\\rho ^2)\\sigma _{\\lambda }^2]}{(1-\\omega )^2[\\rho ^2C_{t-1|t-1}+(1-\\rho ^2)\\sigma _{\\lambda }^2] +(1-\\omega ) v}(Y_t - (1-\\omega )[\\rho \\hat{\\lambda }_{t-1|t-1} + (1-\\rho )\\mu _{\\lambda }]).$ This equation helps in getting filtered values of $\\lambda _t$ provided the parameters and the initial values are known.",
"The suggested values for the initial values above are also in terms of parameters.",
"Next we will identify suitable optimal estimating equations in terms of the filtered values to estimate the unknown parameters.",
"Intoducing notations: $L_t=\\hat{\\lambda }_{t|t},h_t= Y_t - (1-\\omega )[\\rho \\hat{\\lambda }_{t-1|t-1} + (1-\\rho )\\mu _{\\lambda }],P_t=(1-\\omega )[\\rho ^2C_{t-1|t-1}+(1-\\rho ^2)\\sigma _{\\lambda }^2], \\\\J_t=(1-\\omega ) (P_t+v), K_t=\\frac{P_t}{J_t} \\ \\ \\text{and}\\ \\ D_t = L_t - \\rho L_{t-1} + (1-\\rho )\\mu _{\\lambda }$ we can express (REF ) as $ h_t = D_t\\frac{J_t}{P_t}.$ Clearly $E(h_t|{\\cal F}_{t-1}) = 0\\ \\ \\text{and}\\ \\ E(h_t^2|{\\cal F}_{t-1}) = V(h_t|{\\cal F}_{t-1}) = \\frac{J_t^2}{P_t^2}V(D_t|{\\cal F}_{t-1}) = \\frac{J_t^2}{P_t^2}\\frac{P_t}{1-\\omega } = \\frac{J_t^2}{(1-\\omega ) P_t}.$ To illustrate the method of computation, let us consider the ZMPSCD model in which $\\lbrace \\lambda _t\\rbrace $ is a stationary AR(1) sequence of non-negative rvs.",
"In this case the parameter vector to be estimated is $\\theta = (\\omega ,\\mu _{\\lambda },\\rho , \\sigma _{\\lambda }^2)^{\\prime }= (\\theta _1,\\theta _2,\\theta _3,\\theta _4 )^{\\prime } \\ \\ \\text{say}.$ We propose to estimate the first three components, namely $\\omega ,\\mu _{\\lambda },\\rho $ using an optimum unbiased estimating function based on a martingale sequence: $h_t= Y_t - (1-\\omega )[\\rho \\hat{\\lambda }_{t-1|t-1} + (1-\\rho )\\mu _{\\lambda }],$ defined above.",
"The fourth component, $\\sigma _{\\lambda }^2$ will be estimated based on the resulting residuals.",
"To begin with let ${\\cal G}= \\Big \\lbrace g=(g^{(1)}, g^{(2)}, g^{(3)})^{\\prime }= \\sum _{t=1}^na_{t-1}h_t\\Big \\rbrace $ be the class of linear unbiased estimating functions.",
"Then from the theory of martingale estimating functions (cf, Godambe (1985)), the optimal EF is given by $ g^*=(g^{*(1)}, g^{*(2)}, g^{*(3)})^{\\prime }$ with $g^{*(i)} = \\sum _{t=1}^na^{*(i)}_{t-1}h_t \\ \\ \\text{and} \\ \\ a^{*(i)}_{t-1} = \\frac{E\\Big (\\frac{\\partial h_t}{\\partial \\theta _i}|{\\cal F}_{t-1}\\Big )}{Var(h_t|{\\cal F}_{t-1})}\\ \\ \\text{for}\\ \\ i=1,2,3.$ For the above martingale sequence $\\lbrace h_t\\rbrace $ we have, $ \\frac{\\partial h_t}{\\partial \\theta _1} = \\rho \\hat{\\lambda }_{t-1} + (1-\\rho )\\mu _{\\lambda },\\ \\ \\frac{\\partial h_t}{\\partial \\theta _2} = -(1-\\omega )(1-\\rho ),\\ \\ \\frac{\\partial h_t}{\\partial \\theta _3} = -(1-\\omega )( \\hat{\\lambda }_{t-1} -\\mu _{\\lambda }).$ Since $ \\hat{\\lambda }_{t-1}$ is measurable with respect to ${\\cal F}_{t-1}$ , it follows that $E\\Big (\\frac{\\partial h_t}{\\partial \\theta _i}|{\\cal F}_{t-1}\\Big ) = \\frac{\\partial h_t}{\\partial \\theta _i} \\ \\ \\text{and} \\ \\ a^{*(i)}_{t-1} = \\frac{(1-\\omega )P_t}{J_t^2}\\frac{\\partial h_t}{\\partial \\theta _i}, i= 1,2,3.$ Hence the optimal EF is given by $ g^*=(g^{*(1)}, g^{*(2)}, g^{*(3)})^{\\prime }$ where $g^{*(1)} = \\sum _{t=1}^n\\frac{(1-\\omega )P_t}{J_t^2}[\\rho \\hat{\\lambda }_{t-1} + (1-\\rho )\\mu _{\\lambda }](Y_t - (1-\\omega )[\\rho \\hat{\\lambda }_{t-1} + (1-\\rho )\\mu _{\\lambda }])\\\\g^{*(2)} = \\sum _{t=1}^n\\frac{(1-\\omega )P_t}{J_t^2}[-(1-\\omega )(1-\\rho )](Y_t - (1-\\omega )[\\rho \\hat{\\lambda }_{t-1} + (1-\\rho )\\mu _{\\lambda }])\\\\g^{*(3)} = \\sum _{t=1}^n\\frac{(1-\\omega )P_t}{J_t^2}[-(1-\\omega )( \\hat{\\lambda }_{t-1} -\\mu _{\\lambda })](Y_t - (1-\\omega )[\\rho \\hat{\\lambda }_{t-1} + (1-\\rho )\\mu _{\\lambda }]).$ The estimates are obtained by solving the equation $g^*=\\bf {0},$ which requires a suitable iterative method.",
"We suggest ordinary moment estimates for the initial values to implement the iterative procedure.",
"In order to estimate $\\sigma _{\\lambda }^2$ , we propose the EF based on $h_{2t} =\\Big [\\big (\\lambda _t - \\rho \\lambda _{t-1} - (1-\\rho )\\mu _{\\lambda }\\big )^2/(1-\\rho ^2) -\\sigma _{\\lambda }^2 \\Big ].$ Clearly, $E(h_{2t}|{\\cal F}_{t-1}) = 0$ and $Var(h_{2t}|{\\cal F}_{t-1}) =E\\Big (\\lambda _t - \\rho \\lambda _{t-1} - (1-\\rho )\\mu _{\\lambda }\\Big )^4/((1-\\rho ^2))^2 -\\sigma _{\\lambda }^4.$ Now the optimal estimating function is chosen from the class ${\\cal G}_1 = \\Big \\lbrace g= \\sum _{t=1}^nb_{t-1}h_{2t}\\Big \\rbrace $ with optimum coefficients $b^*_{t-1}=\\frac{E\\Big (\\frac{\\partial h_{2t}}{\\partial \\theta _4}|{\\cal F}_{t-1}\\Big )}{Var(h_{2t}|{\\cal F}_{t-1})} = \\frac{-1}{Var(h_{2t}|{\\cal F}_{t-1})}.$ For evaluating the estimate, we can replace $\\lambda _t $ by its filtered value and $\\rho $ and $\\mu _{\\lambda }$ by the respective estimates.",
"If $\\lbrace \\lambda _t\\rbrace $ is a stationary homoscedastic Markov sequence like the $GAR(1)$ then $b^*_{t-1}$ will be a constant and hence the estimate of $\\sigma ^2_{\\lambda }$ will be $\\hat{\\sigma }^2_{\\lambda } = \\frac{1}{n} \\sum _{t=1}^n\\Big (\\lambda _t - \\rho \\lambda _{t-1} - (1-\\rho )\\mu _{\\lambda }\\Big )^2/(1-\\rho ^2).$ In particular if $\\lbrace \\lambda _t\\rbrace $ a GAR(1) process, its acf is $\\rho _{\\lambda }(k)=\\rho ^k$ .",
"Let $\\bar{Y}$ , $s^2$ and $r_1$ be the sample mean, sample variance and the first order sample acf of $\\lbrace Y_t\\rbrace $ .",
"Equating them to the corresponding moments of $\\lbrace Y_t\\rbrace $ in Section 2.1, we can write $\\begin{array}{l}\\bar{Y} = (1 - \\omega )\\mu _\\lambda ,\\quad s^2 = (1 - \\omega )\\left[ {\\mu _\\lambda + \\sigma _\\lambda ^2 +\\omega \\mu _{\\lambda } ^2 } \\right], \\quad r_1 = \\frac{{(1 - \\omega )\\sigma _\\lambda ^2 \\rho }}{{\\mu _\\lambda + \\sigma _\\lambda ^2 +\\omega \\mu _{\\lambda } ^2 }}.",
"\\\\\\end{array}$ Simultaneous solution of these equations will provide a set of moment estimates of the model parameters to initialize the computations.",
"The details are dscribed in the following algorithm.",
"If $p=1$ , we get the intensities generated by an Exponential AR(1) (EAR(1)) model and the corresponding expressions reduce to the following.",
"$\\bar{Y} = (1-\\omega )\\mu _{\\lambda }, s^2 = (1-\\omega )\\mu _{\\lambda }[1 + (1+\\omega ) \\mu _{\\lambda }],r_1=\\frac{(1-\\omega )\\mu _{\\lambda }\\rho }{1+\\mu _{\\lambda }(1 + \\omega )}.$ Superfixing (0) to denote the initial values and then simplifying, we get $\\omega ^{(0)} = \\theta ^{(0)}_1 = \\frac{z-1}{z+1}, {\\text{with}}\\ \\ z=\\frac{1}{\\bar{Y}}(\\frac{s^2}{\\bar{Y}}-1)\\\\\\mu ^{(0)}_{\\lambda } = \\theta ^{(0)}_2 = \\frac{\\bar{Y}}{(1-\\omega )},\\ \\ \\ \\rho ^{(0)} = \\theta ^{(0)}_3 = r_1\\frac{1 + (1+\\omega ) \\mu _{\\lambda }}{(1-\\omega ) \\mu _{\\lambda }}.$ The following algorithm may be used for filtering the stochastic intensity and estimate the parameters iteratively for a ZMP model with GAR(1) intensities.",
"Filtering and estimation algorithm: Initialize $\\theta ^{(0)}_i, i=1,2,3,$ and $C_{1|0}= (1-\\rho ^2)\\sigma _{\\lambda }^2 $ and $\\hat{\\lambda }_{1|0} = \\rho \\lambda _0 + (1-\\rho )\\mu _{\\lambda }, {\\lambda _0\\sim G(\\beta , p)} $ .",
"Use (REF ) and (REF ) for updating $\\lambda $ - values $\\hat{\\lambda }_{t} = \\hat{\\lambda }_{t|t-1} + \\frac{(1-\\omega )C_{t|t-1}}{(1-\\omega )^2C_{t|t-1} +(1-\\omega ) v}(Y_t - (1-\\omega )\\hat{\\lambda }_{t|t-1}).$ Compute $\\theta ^{(k)}_i, i=1,2,3; k=1,2,...$ by solving the estimating equations $g^{*}=\\bf {0}$ .",
"Store the filtered values of $\\lambda _t$ to obtain $\\hat{\\sigma }^2_{\\lambda }$ using (REF ) when $\\lambda _t$ are generated by the GAR(1) model.",
"The parameters $\\beta $ and $p$ may be estimated by $\\hat{\\beta }= \\hat{\\mu }_{\\lambda } /\\hat{\\sigma }^2_{\\lambda } $ and $\\hat{p}=\\hat{\\mu }_{\\lambda } \\hat{\\beta }$ .",
"Repeat the above steps with $k=k+1$ until convergence."
],
[
"Estimation for ZMNBSCD model",
"Suppose that $Y_t$ given the past, follows a ZMNB distribution.",
"From Section 2.2, we have $E(Y_t|{\\cal F}_{t-1}) = (1-\\omega )\\lambda _t \\ \\ {\\text{and}} \\ \\ \\mbox{Var}(Y_t|{\\cal F}_{t-1}) = (1-\\omega )(1+\\omega \\lambda _t + a\\lambda _t^c)\\lambda _t.$ If the sequence of intensities $\\lbrace \\lambda _t\\rbrace $ is generated by a stationary non-negative AR(1) model then the GKF system described in Section 3 will remain same except for the expression of $\\bar{V}_t,$ which is given by $V_t(\\lambda _t)$ .",
"In terms of notations of Section 3, we have $\\bar{V}_t&=& E(Var(Y_t|{\\cal F}_{t-1})) \\nonumber \\\\&=&{\\left\\lbrace \\begin{array}{ll}(1-\\omega )[(1+a)\\mu _{\\lambda } + \\omega (\\sigma _{\\lambda }^2+\\mu _{\\lambda }^2)], & \\ \\ \\ {\\text{if}} \\ \\ c=0 \\\\(1-\\omega )[\\mu _{\\lambda } + (\\omega +a)(\\sigma _{\\lambda }^2+\\mu _{\\lambda }^2)], & \\ \\ \\ {\\text{if}} \\ \\ c=1.\\end{array}\\right.}",
"= (1-\\omega )v_b^{(c)}, {\\text{say}}.$ This $\\bar{V}_t$ reduces to that of ZIPSCD model if $a=0$ .",
"The value of $c$ is taken as either 0 or 1 according to the selected class of ZMNB distribution.",
"So the equation for filtering $\\lambda _t$ is same as (REF ) with $v$ replaced by $v_b^{(c)}$ given in (REF ) .",
"The parameter vector to be estimated here is $\\theta = (\\omega ,\\mu _{\\lambda },\\rho ,a, \\sigma _{\\lambda }^2)^{\\prime }= (\\theta _1,\\theta _2,\\theta _3,\\theta _4, \\theta _5)^{\\prime }$ say.",
"The first three components, namely $\\omega ,\\mu _{\\lambda },\\rho $ can be estimated using the EF resulting from equation (REF ) by repeating the method described in Section 4.1.",
"For estimating the dispersion parameter $\\theta _4 = a$ , we use the following quadratic EF: $h_t^Q&=& (Y_t-E(Y_t|{\\cal F}_{t-1}))^2 - V(Y_t|{\\cal F}_{t-1})\\\\&=& (Y_t-(1-\\omega )\\lambda _t)^2 - (1-\\omega )(1+\\omega \\lambda _t + a\\lambda _t^c)\\lambda _t.$ Clearly $E(h_t^Q|{\\cal F}_{t-1}) = 0$ and $Var(h_t^Q|{\\cal F}_{t-1}) &=& E[(Y_t-(1-\\omega )\\lambda _t)^4|{\\cal F}_{t-1}] - (1-\\omega )^2(1+\\omega \\lambda _t + a\\lambda _t^c)^2\\lambda _t^2 \\nonumber \\\\&=& (1-\\omega )\\lambda _t\\Big [\\lambda _t^3(4\\omega ^3-4\\omega ^2+\\omega ) +\\lambda _t^2(8\\omega ^2-2\\omega )+\\lambda _t(5\\omega +2)+1+6a^3\\lambda _t^{3c}\\\\ \\nonumber &&+\\lambda _t^{2c}(12a^2+2a^2\\lambda _t+9\\omega a^2\\lambda _t)+\\lambda _t^{c}(8\\omega ^2 a\\lambda _t^2 +4a\\lambda _t+14\\omega a\\lambda _t - 2\\omega a\\lambda _t^2) \\Big ].$ Now the optimal estimating function $g^*_Q $ is chosen from the class of EFs ${\\cal G}_Q = \\Big \\lbrace g_Q= \\sum _{t=1}^na^Q_{t-1}h^Q_{t}\\Big \\rbrace \\ \\ \\text{as} \\ \\ g^*_Q= \\sum _{t=1}^na^{Q*}_{t-1}h^Q_{t}\\\\\\text{with}\\ \\ \\ \\ \\ a^{Q*}_{t-1} = \\frac{E\\Big (\\frac{\\partial h^Q_{t}}{\\partial a}|{\\cal F}_{t-1}\\Big )}{Var(h^Q_{t}|{\\cal F}_{t-1})} = \\frac{-(1-\\omega )\\lambda _t\\lambda ^c_t}{Var(h^Q_{t}|{\\cal F}_{t-1})}.$ Then obtain the estimate of $a$ as a solution of the equation $g^*_Q= 0.$ Finally for the parameter $\\theta _5=\\sigma _{\\lambda }^2$ , we propose the same EF based on $h_{2t}$ defined by (REF ) used for estimating $\\sigma _{\\lambda }^2$ in Section 4.1.",
"The computation algorithm developed in Section 4.1 for ZMPSCD model can be used for ZMNBSCD model by replacing $v$ in (REF ) by $v^{(c)}_b$ given in (REF )."
],
[
"Simulation Studies",
"A simulation study was conducted to evaluate the finite sample performance of the proposed estimators.",
"First we consider the simulation studies for ZIP model followed by ZINB model when intensities are generated by GAR(1) and EAR(1) sequences respectively.",
"That is, we choose the values of zero modification parameter $w$ in the interval $(0,1)$ so that the resulting model becomes suitable to analyze zero inflated count data.",
"The ZMSCD models with intensities generated by other models listed in Section 2.3 may also be considered similarly.",
"Inspired by the applications of estimating functions in filtering and smoothing of general state space models (cf; Naik-Nimbalkar and Rajarshi, 1995), we have extended their ideas to the case of zero modified time series models.",
"The computational ease of this method facilitates a near optimal solution to a highly non linear and high dimensional numerical integration problem.",
"The goal of this simulation study is to analyze the sampling behavior of estimators obtained as solution to the appropriate optimal estimating functions rather than to compare with other methods applicable to ZIP or ZINB models.",
"In what follows, we simulate a count series of length $n$ from the models, (REF ), (REF ), (REF ) for different values of parameters and carry out estimation using the algorithm described in Section 4.",
"The moment estimate denoted by $\\tilde{\\theta }$ is used as initial value to start the algorithm.",
"As there are no closed form expressions for the moment estimates, we applied a general grid search for initialization.",
"In some iterations, it is observed that the moment estimates $\\tilde{w}$ and $\\tilde{\\rho }$ become infeasible in the sense that they lie outside the parameter space, especially when the experiment is conducted with values of $\\rho $ close to 1.",
"In such cases, either we took the initial value from the $\\epsilon $ neighborhood of the true value or discard the iteration.",
"The proportion of infeasible solutions was relatively small.",
"This iterative process of estimation was repeated 1000 times for each combination of parameters and obtained the estimates and filtered values of latent intensities.",
"Table 1 summarizes the simulation results for a ZIPSCD model with GAR(1) intensities.",
"This Table gives the average of these 1000 estimates along with the corresponding mean squared error(MSE) in the parenthesis.",
"Based on our simulation results, we can see that the estimates are close to the true values with small MSEs.",
"The MSE of the estimate of shape parameter $p$ is relatively larger but, when the value of $\\rho $ decreases, it also decreases.",
"It is seen that the MSEs decrease when sample size increases.",
"We observe similar behavior when the intensities are generated by EAR(1) model and hence omit the details.",
"Table 1 about here.",
"The details of simulation study for ZINB models when intensities are generated by GAR(1) and EAR(1) sequences are discussed below.",
"We consider the case of $c=1$ in the ZINB model given by (REF ).",
"In this scenario, the moment estimators do not possess closed form expressions for both EAR(1) and GAR(1) based ZINB models.",
"Thus, to solve the estimating equations discussed in Section 4.2, we started with simple grid search.",
"In Table 2 we summarize the simulation results.",
"As in the case of ZIPSCD with GAR(1) intensities, here also the estimates of $\\rho $ , $\\omega $ , and $\\beta $ performs well in all cases as far as the bias and MSEs are concerned.",
"Meanwhile, the estimates of shape parameter $p$ and dispersion parameter $a$ , behave little differently, for instance, when $\\rho =0.9$ , $\\omega =0.2$ , $\\beta =2$ , the bias of both $\\hat{p}$ and $\\hat{a}$ become relatively high.",
"This pattern repeats over the other parameter combinations.",
"Though, the MSE of $\\hat{a}$ decreases when $\\rho $ increases and vice versa for $\\hat{p}$ .",
"The above results show that the performance of EF based estimators is satisfactory for all combinations of parameters considered for ZIPSCD with GAR(1) intensities whereas the same is not fully warranted in ZINBSCD models.",
"This observation was also made by Yang et al (2015).",
"They pointed out that, due to the possibility of estimation problems caused by weak identifiability in ZINB models, the dynamic ZINB model is not a good candidate when sample information is limited.",
"In our case also, the complexity of the ZINBSCD specification may be a reason for comparatively less efficient estimation results.",
"Table 2 about here.",
"A simulation study is also carried out to see the sampling behavior of the proposed method when a zero deflation is present.",
"That is, we allow the zero modification parameter $w$ to take negative values so that the resulting model can be used to analyze the zero deflated data.",
"As in the case of inflated models, we have generated pseudo random numbers of different sizes ($n=200,500, 1000$ ) from the zero deflated model and applied the estimation and filtering algorithm to obtain the parameter estimates.",
"This procedure is repeated 1000 times and the resulting estimates were saved.",
"Using these values, we have computed the mean and MSE of the estimators.",
"Table 3 summarizes the simulation results of ZDPSCD model when a GAR(1) process is used to generate latent intensities.",
"We skip the results for the sample size $n=500$ to save space.",
"It is straightforward to see that the bias and MSE decreases as sample size increases.",
"Table 3 about here.",
"Similar pattern was also observed in the case of ZDNBSCD with GAR(1) intensities, presented in Table 4 .",
"However, a weaker performance (in terms of bias and MSE) of the estimates for ZDNBSCD model is observed compared to that of the ZDPSCD model.",
"This behavior is same as the one observed for the zero inflated models discussed earlier.",
"Table 4 about here."
],
[
"Data Analysis",
"In this Section we analyze two sets of data to illustrate the applications of the proposed models.",
"The first data set consists of weekly number of syphilis cases reported in the state of Maryland in United States from 2007 to 2010.",
"This data set is available in the R package ZIM (see Yang et al.",
"2015).",
"Figure 2 shows the basic structure of the data such as time series, ACF, PACF plots.",
"The histogram clearly indicates that the data contains several zeros.",
"The basic statistics of the data are given in Table 5.",
"Stationarity of the data is confirmed by performing an augmented Dicky-Fuller test at lag 5, yielding $p$ -value 0.01.",
"Figure 2 about here.",
"Table 5 about here.",
"Observe that the variance is larger than mean implying the presence of over-dispersion.",
"Also, the sample auto-correlation function and partial auto-correlation functions of the data exhibit temporal correlations.",
"In fact, the Ljung-Box test for autocorrelation at lag 1 rejects the null hypothesis of zero autocorrelation with $p$ -value 0.04104.",
"To capture all these features, we fit a ZIPSCD model with GAR(1) intensity processes by using the filtering and estimation algorithm described in Section 4.",
"To find the initial values to start the algorithm, we used the following factorial moment equations, namely $\\bar{y}^{(1)} = E\\left( {Y_t } \\right) = (1 - \\omega )\\frac{p}{\\beta },\\ \\ \\ \\ \\bar{y}^{(2)} = E\\left( {Y_t (Y_t - 1)} \\right) = (1 - \\omega )\\frac{{p(p + 1)}}{{\\beta ^2 }} $ and $\\bar{y}^{(3)} = E\\left( {Y_t (Y_t - 1)(Y_t - 2)} \\right) = (1 - \\omega )\\frac{{p(p + 1)(p + 2)}}{{\\beta ^3 }}.$ Once the initial estimates are obtained, we run the filtering algorithm and then use these filtered intensities to compute the estimates.",
"This process is repeated until convergence.",
"The final estimates of model parameters are given in Table 6.",
"Table 6 about here.",
"Following Zeger(1988), we have calculated the standard errors of the estimators by means of simulations.",
"That is, we have simulated a sample of size $n=209$ from a ZIP-GAR(1) model with the final estimates for the data generating process and computed the new estimates.",
"Then we repeated the procedure 1000 times and the empirical standard deviation of these 1000 estimates is taken as the standard error of the estimates.",
"The histogram along with a superimposed gamma density and the plot of filtered intensities are exhibited in Figure 3.",
"Figure 3 about here.",
"To check the model adequacy, we found the Pearson residuals defined by ${e_t} = \\frac{{{y_t} - (1 - w){\\lambda _t}}}{{\\sqrt{\\left( {1 - w} \\right)\\left( {1 + w{\\lambda _t}} \\right){\\lambda _t}} }};t = 1,2,...,n$ with filtered values of $\\lambda _t$ and the estimate of $\\omega $ .",
"The time series plot, acf and pacf of the residulas are given in Figure 4.",
"The residulas show no significant autocorrelation at all lags considered.",
"Further, the Ljung-Box test applied to the residual series at lag 20 confirms this fact with a $p$ -value 0.4852.",
"Finally we computed the theorotical probabilities $P\\left( {Y_t = k} \\right) = \\left\\lbrace {\\begin{array}{*{20}c}{\\omega + (1 - \\omega )\\left( {\\frac{\\beta }{{\\beta + 1}}} \\right)^p ;\\,\\;\\quad k = 0} \\\\{(1 - \\omega )\\frac{{\\beta ^p \\,\\Gamma \\left( {p + k} \\right)}}{{k!\\left( {\\beta + 1} \\right)^{p + k} \\Gamma \\left( p \\right)}}\\,;k \\ge 1} \\\\\\end{array}} \\right.$ and compared it with empirical probabilities.",
"Figure 5 depicts the strength of agreement between the fitted and empirical probabilities.",
"For instance, the sample proportion of zeros is 0.2823 whereas the estimated probability using the ZIPSCD-GAR(1) model is 0.2882.",
"This shows a reasonably good fit.",
"Figure 4 about here.",
"Figure 5 about here.",
"As an application of the model to a zero deflated situation, we consider the monthly count of aggravated assaults reported in the 34th police car beat in Pittsburgh which was analyzed previously by Barreto-Souza (2015) and Sharafi et al (2020).",
"The data is collected during the period January 1990 to December 2001.",
"Figure 6 exhibits the time series plot, acf and histogram of the data.",
"The data is available in Appendix C of the supporting information provided by Barreto-Souza (2015).",
"Figure 6 about here.",
"When a Poisson conditional distribution is fitted to this data we get a deflated number of zeros compared with the empirical count.",
"This fact leads us to fit a zero deflated model using the ZMPSCD model with GAR(1) intensities.",
"The resulting estimators are $\\hat{w} =-0.1161, \\hat{\\rho }=0.4311, \\hat{p}=1.8314$ and $\\hat{\\beta }=2.2575$ .",
"The negative sign of the estimate of $w,$ the zero modification parameter, clearly indicates the presence of zero deflation compared to the base line Poisson model.",
"The fitted probabilities are displayed in the Figure 7.",
"Figure 7 about here.",
"This shows a close agreement of correct zero probability with the proposed model.",
"The Figure 8 shows the filtered values of latent intensities $\\lambda _t$ along with actual data and the acf of Pearson residuals obtained as in (REF ).",
"The acf plot shows no remaining autocorrelations in the residuals.",
"The Ljung-Box test for randomness applied to the residuals confirms the absence of significant autocorrelation upto lag order 20 at $5\\%$ level of significance.",
"Figure 8 about here."
],
[
"Concluding Remarks",
"Count time series with excess or deficient zeros occur in some practical situations and we proposed zero modified count time series with Markov dependent intensities to analyze such situations.",
"It is demonstrated that the method of estimating function performs well in filtering the unobserved intensities and then estimating model parameters.",
"Yang et al (2015) introduced a state space model for zero inflated count series and used a Monte Carlo expectation maximization algorithm for analysis.",
"As an alternative, we recommend the EF based filtering and estimation procedure.",
"Once suitable initial estimates were obtained, the EF method provides feasible estimates for both Poisson and negative binomial models.",
"We plan to develop, in the near future, coherent forecasting of zero modified count data with Markovian latent intensities.",
"Acknowledgement: N. Balakrishna acknowledges the financial support by Science and Engineering Research Board (SERB) of India under MATRICS scheme MTR/2018/000195.",
"The research of Bovas Abraham was supported by a grant from the Natural Sciences and Engineering Research Council (NSERC) of Canada.",
"This work was initiated while Balakrishna spent a part of his sabbatical at University of Waterloo during September to November, 2019.",
"References Abraham, B. and Balakrishna, N. (2012).",
"Product autoregressive models for non-negative variables.",
"Statistics and Probability Letters, 82, 1530 - 1537.",
"Barreto-Souza, W. (2015).",
"Zero-modified geometric INAR(1) process for modelling count time series with deflation or inflation of zeros.",
"Journal of Time Series Analysis.",
"36(6), 839-52.",
"Bertoli, W., Katiane S. Conceição, Marinho G. Andrade and Francisco Louzada (2019).",
"Bayesian approach for the zero-modified Poisson–Lindley regression model.",
"Brazilian Journal of Probability and Statistics, 33, No.",
"4, 826–860.",
"Bhogal SK, Thekke Variyam R.(2019).",
"Conditional duration models for high-frequency data: a review on recent developments.",
"Journal of Economic Survey, 33(1):252-273.",
"Cox, D. R. (1981).",
"Statistical analysis of time series: Some recent developments.",
"Scandinavian Journal of Statistics, 8:93-115.",
"Davis, R. A., Holan, S. H., Lund, R. and Ravishanker, N. (2016).",
"Handbook of discrete-valued time series.",
"Chapman and Hall/ CRC, Boca Raton, FL.",
"Engle, RF & Russell, JR (1998).",
"Autoregressive conditional duration: a new approach for irregularly spaced transaction data.",
"Journal of Econometrics, 119, 381-482.",
"Dietz, E., Bohning, D. (2000).",
"On estimation of the Poisson parameter in zero-modified Poisson models.",
"Computational Statistics and Data Analysis, 34:441-459.",
"Ferland, R., Latour, A., and Oraichi, D. (2006).",
"Integer-valued GARCH model.",
"Journal of Time Series Analysis, 27:923-942.",
"Fokianos, K. (2016).",
"Statistical analysis of count time series models: A GLM perspective.",
"Handbook of discrete-valued time series.",
"Chapman and Hall/ CRC, Boca Raton, FL.",
"Gaver, D. P., Lewis, P. A. W. (1980).",
"First order autoregressive gamma sequences and point processes.",
"Advances in Applied Probability, 12: 727-745.",
"Godambe, V. P. (1985).",
"The Foundations of Finite Sample Estimation in Stochastic Processes.",
"Biometrika, 72: 419-428.",
"Lambert, D. (1992).",
"Zero-inflated Poisson Regression, With an Application to Defects in Manufacturing.",
"Technometrics, 34, 1–14.",
"Lawrance A. J.",
"(1982).",
"The innovation distribution of a gamma distributed autoregressive process.",
"Scandinavian Journal of Statistics , 9, 224-236.",
"Lawrance, A. J. and Lewis, P. A. W. (1981).",
"A new autoregressive time series model in exponential variables (NEAR(1)).",
"Advances in Applied Probability 13, 826 - 845.",
"Lewis, P. A .W., McKenzie, E. & Hugus, D. K. (1989).",
"Gamma processes, Stochastic Models, 5:1, 1-30.",
"Muhammed Anvar, P., Balakrishna, N. and Bovas Abraham (2019).",
"Stochastic volatility generated by product autoregressive models.",
"Stat.,DOI: 10.1002/sta4.232 Naik-Nimbalkar, U. V. and Rajarshi, M. B.",
"(1995).",
"Filtering and Smoothing Via Estimating Functions, Journal of American Statistical Association, 90, 301-306 .",
"Pitt, M.K., Walker, S.G. (2005).",
"Constructing stationary time series models using auxiliary variables with applications.",
"Journal of American statistical association, 100, 554-564.",
"Thekke, R., Mishra, A., and Abraham, B.",
"(2016).",
"Estimation, filtering and smoothing in the stochastic conditional duration model: an estimating function approach.",
"Stat, 5(1), 11-21.",
"Sharafi, M., Sajjadnia, Z. and Zamani, A.",
"(2021): A first-order integer-valued autoregressive process with zero-modified Poisson-Lindley distributed innovations, Communications in Statistics - Simulation and Computation, DOI:10.1080/03610918.2020.1864644 Sim, C. H. (1990).",
"First order autoregressive models for gamma and exponential processes, Journal of Applied Probability, 27, 325-332.",
"Tjostheim, D. (2016).",
"Count time series with observation-driven autoregressive parameter dynamics.",
"Handbook of discrete-valued time series.",
"Chapman and Hall/ CRC.pp 77-100.",
"Thavaneswaran, A., Ravishanker, N., Liang, Y.",
"(2015).",
"Generalized duration models and optimal estimation using estimating functions.",
"Annals of the Institute of Statistical Mathematics, 67(1):129-156.",
"Yang, M., Cavanaugh, J. E., Zamba, G. K.D.",
"(2015).",
"State space models for count time series with excess zeros.",
"Statistical Modelling, 15(1), 70-90.",
"Zehnwirth, B (1988).",
"A generalization of the Kalman filter model for models with state-dependent observation variance.",
"Journal of the American Statistical Association, 83, 164-167.",
"Zeger, S. L. (1988).",
"A regression model for time series of counts.",
"Biometrika, 75(4), 621-629.",
"Zhu, F. (2012).",
"Zero-inflated Poisson and negative binomial integer-valued GARCH models.",
"Journal of Statistical Planning and Inference, 142:826-839."
]
] | 2107.01813 |
[
[
"High-frequency instabilities of the Ostrovsky equation"
],
[
"Abstract We study spectral stability of small amplitude periodic traveling waves of the Ostrovsky equation.",
"We prove that these waves exhibit spectral instabilities arising from a collision of pair of non-zero eigenvalues on the imaginary axis when subjected to square integrable perturbations on the whole real line.",
"We also list all such collisions between pair of eigenvalues on the imaginary axis and do a Krein signature analysis."
],
[
"Introduction",
"The Ostrovsky equation $(u_t - \\beta u_{xxx}+(u^2)_x)_x = \\gamma u ,\\quad x\\in \\mathbb {R}$ was derived by Ostrovsky (see [13]) as a model for the unidirectional propagation of weakly nonlinear long surface and internal waves of small amplitude in rotating liquid.",
"The liquid is assumed to be incompressible and inviscid.",
"Here, $u(x,t)$ represents the free surface of the liquid.",
"The constant $\\gamma >0$ measures the effect of rotation and is rather small for the real conditions of the Earth rotation [4].",
"The parameter $\\beta $ determines the type of dispersion, namely $\\beta <0$ (negative dispersion) for surface and internal waves in the ocean and surface waves in a shallow channel with an uneven bottom and $\\beta >0$ (positive dispersion) for capillary waves on the surface of liquid or for oblique magneto-acoustic waves in plasma [3].",
"Setting $\\gamma = 0$ in (REF ) and integrating with respect to $x \\in \\mathbb {R}$ and assuming the solution $u(x,t)$ and all the derivatives are vanishing at infinity, one obtains the well-known Korteweg-de Vries (KdV) equation $u_t -\\beta u_{xxx}+(u^2)_x = 0.$ The Ostrovsky equation is non-local and dispersive with linear dispersion as $\\omega (k) =\\frac{\\gamma }{k}+\\beta k^3.$ It is also Hamiltonian $u_t = \\frac{\\partial \\mathcal {H}}{\\partial u}$ where $\\mathcal {H} = \\int _{\\mathbb {R}} \\left(\\frac{\\beta }{2}|u_x|^2 +\\frac{\\gamma }{2} |D_x^{-1}u|^2+\\frac{1}{3}u^3 \\right)dx,$ and for $k \\in \\mathbb {N}$ , the operator $D_x^{-k}$ is defined by $\\widehat{(D_x^{-k}f})(\\xi ) = (i\\xi )^{-k}\\hat{f}(\\xi ).$ The Ostrovsky equation, unlike KdV, is nonintegrable by the method of the inverse scattering transform.",
"The local and global-well posedness of the Ostrovsky equation are known in some weighted Sobolev spaces [11].",
"The stability or instability of different type of solutions of the Ostrovsky and related models have been investigated by several authors.",
"The orbital stability of solitary-wave solutions of the Ostrovsky equation has been established in [10].",
"In [9], periodic traveling waves of (REF ) with general nonlinearity has been constructed for small values of $\\gamma $ and shown to be spectrally stable to periodic perturbations of the same period as the wave.",
"In this article, we investigate spectral stability of small amplitude periodic traveling waves of the Ostrovsky equation.",
"We use a standard argument based on implicit function theorem and Lyapunov-Schmidt reduction to establish the existence of a family of periodic traveling waves.",
"As a consequence, we obtain small amplitude expansion of these periodic traveling waves.",
"We linearize (REF ) about obtained periodic traveling wave and examine the $L^2(\\mathbb {R})$ -spectrum of the linearized operator.",
"In case of periodic perturbations, one needs to restrict on mean zero space because of the presence of $\\partial _z^{-1}$ in the linearized operator.",
"But for square integrable perturbations on the whole real line, we use Floquet-Bloch theory which transforms $\\partial _z^{-1}$ to $(\\partial _z+i\\xi )^{-1}$ , where $\\xi $ is the Floquet exponent.",
"As a result, for $\\xi \\ne 0$ , we need not restrict to mean zero space.",
"In terms of perturbations, $\\xi \\ne 0$ corresponds to non-modulational perturbations and resulting spectral instability is termed as high-frequency instability [2].",
"We show that obtained small amplitude periodic traveling waves of (REF ) exhibit high-frequency instability.",
"In Section , we obtain periodic traveling waves of the Ostrovsky equation bifurcating from the trivial solution.",
"We set up the spectral stability problem in Section and prove the existence of high-frequency instabilities in Section ."
],
[
"Notations",
"The following notations are going to be used throughout the article.",
"Here, $L^2(\\mathbb {R})$ denotes the set of real or complex valued, Lebesgue measurable functions $f(x)$ over $\\mathbb {R}$ such that $\\Vert f\\Vert _{L^2(\\mathbb {R})}=\\Big (\\frac{1}{2\\pi }\\int _\\mathbb {R}|f|^2~dx\\Big )^{1/2}<+\\infty \\quad $ and $L^2(\\mathbb {T})$ denote the space of $2\\pi $ -periodic, measurable, real or complex valued functions over $\\mathbb {R}$ such that $\\Vert f\\Vert _{L^2(\\mathbb {T})}=\\Big (\\frac{1}{2\\pi }\\int ^{2\\pi }_0 |f|^2~dx\\Big )^{1/2}<+\\infty .$ For $f \\in L^1(\\mathbb {R})$ , the Fourier transform of $f$ is written as $\\hat{f}$ and defined by $\\hat{f}(t)=\\frac{1}{\\sqrt{2\\pi }}\\int _{\\mathbb {R}} f(x)e^{-itx}dx$ It follows from Parseval Theorem that if $f\\in L^2(\\mathbb {R})$ then $\\Vert \\hat{f}\\Vert _{L^2(\\mathbb {R})} = \\Vert {f}\\Vert _{L^2(\\mathbb {R})}$ .",
"Moreover, for any $s\\in \\mathbb {R}$ , let $H^s(\\mathbb {R})$ consist of tempered distributions such that $\\Vert f\\Vert _{H^s(\\mathbb {R})} = \\left(\\int _{\\mathbb {R}}(1+|t|^2)^s|\\hat{f}(t)|^2dt\\right)^{\\frac{1}{2}} < +\\infty $ Furthermore, $L^2(\\mathbb {T})$ -inner product is defined as $\\langle f,g\\rangle =\\frac{1}{2\\pi }\\int ^{2\\pi }_{0} f(z)\\overline{g}(z)~dz=\\sum _{n\\in \\mathbb {Z}} \\widehat{f}_n\\overline{\\widehat{g}_n}.$ For any $k\\in \\mathbb {N}$ , let $H^k(\\mathbb {T})$ be the space of $L^2(\\mathbb {T})$ functions whose derivatives up to $k$ th order are all in $L^2(\\mathbb {T})$ .",
"Let $H^\\infty (\\mathbb {T})=\\bigcap _{k=1}^\\infty H^k(\\mathbb {T})$ ."
],
[
"Sufficiently small and periodic traveling waves",
"A traveling wave of (REF ) is a solution which propagates at a constant velocity without change of form.",
"That is, $u(x,t)=U(x-ct)$ for some $c \\in \\mathbb {R}$ .",
"Substituting this in (REF ) leads to $cU^{\\prime \\prime }+\\beta U^{\\prime \\prime \\prime \\prime }-(U^2)^{\\prime \\prime }+\\gamma U=0$ We seek a $\\it {periodic\\hspace{3.0pt}pxtraveling\\hspace{3.0pt}pxwave}$ of (REF ).",
"That is, $U$ is a $2\\pi /k$ -periodic function of its argument where $k>0$ is the wave number.",
"Taking $z:=kx$ , the function $w(z):=U(kx)$ is $2\\pi $ -periodic in $z$ and satisfy $ck^2 w^{\\prime \\prime }+\\beta k^4 w^{\\prime \\prime \\prime \\prime }-k^2(w^2)^{\\prime \\prime }+\\gamma w=0.$ Note that (REF ) is invariant under $z\\mapsto z+z_0$ and $z\\mapsto -z$ and therefore, we may assume that $w$ is even.",
"Also, note that (REF ) does not possess scaling invariance.",
"Hence, we may not a priori assume that $k=1$ .",
"In fact, the stability result reported in Theorem REF depends on $k$ .",
"To compare, the KdV equation red(REF ) for periodic traveling waves possesses scaling invariance and stability results are independent of the carrier wave number, see [1], for instance.",
"In what follows, we seek a non-trivial $2\\pi $ -periodic solution $w$ of (REF ).",
"For fixed $\\beta $ and $\\gamma $ , let $F:H^4(\\mathbb {T})\\times \\mathbb {R}\\times \\mathbb {R}^+ \\rightarrow L^2(\\mathbb {T})$ be defined as $F(w,c;k)=ck^2 w^{\\prime \\prime }+\\beta k^4 w^{\\prime \\prime \\prime \\prime }-k^2(w^2)^{\\prime \\prime }+\\gamma w.$ It is well defined by a Sobolev inequality.",
"We seek a solution $w\\in H^4(\\mathbb {T})$ , $c \\in \\mathbb {R}$ and $k>0$ of $F(w,c;k)=0.$ Note that if $w \\in H^4(\\mathbb {T})$ , then from (REF ), $w^{\\prime \\prime \\prime \\prime } \\in H^2(\\mathbb {T})$ by a Sobolev inequality.",
"Therefore, $w \\in H^6(\\mathbb {T})$ .",
"By a bootstrap argument, we obtain that $w \\in H^\\infty (\\mathbb {T})$ .",
"The operator $F$ in (REF ) is a polynomial in parameters $c$ and $k$ .",
"Its Fréchet derivatives with respect to $w$ are all continuous from $H^4(\\mathbb {T})$ to $L^2(\\mathbb {T})$ .",
"Therefore, $F$ is a real analytic operator.",
"Clearly, $F(0,c;k)=0$ for all $c \\in \\mathbb {R}$ and $k>0$ .",
"If non-trivial solutions of $F(w,c;k)=0$ bifurcates from $w \\equiv 0$ for some $c=c_0$ then $L_0 := \\partial _wF(0,c_0;k) = c_0k^2\\partial _z^2 + \\beta k^4 \\partial _z^4 + \\gamma $ from $H^4(\\mathbb {T})$ to $L^2(\\mathbb {T})$ , is not an isomorphism.",
"From a straightforward calculation, $L_0 e^{inz} = (-c_0k^2n^2 + \\beta k^4 n^4 + \\gamma ) e^{inz} = 0, \\quad n \\in \\mathbb {Z}$ if and only if $c_0=\\frac{\\gamma }{k^2 n^2}+\\beta k^2 n^2, \\quad n\\in \\mathbb {Z}.$ Without loss of generality, we take $n=1$ .",
"Note that for $\\beta >0$ , wavenumbers, $k=\\left(\\frac{\\gamma }{\\beta n^2}\\right)^{1/4}$ , $2\\le n\\in \\mathbb {N}$ , satisfy resonance condition $\\frac{\\gamma }{k^2}+\\beta k^2 =\\frac{\\gamma }{k^2 n^2}+\\beta k^2 n^2$ of fundamental mode and $n$ th harmonic then the kernel of $L_0$ is four-dimensional.",
"For all other values of $k$ , $L_0$ is a Fredholm operator of index zero with both kernal and co-kernal spanned by $e^{\\pm iz}$ .",
"Next, we employ a Lyapunov-Schmidt procedure to establish the existence of a one-parameter family of non-trivial solutions of $F(w,c;k)=0$ bifurcating from $w \\equiv 0$ and $c = c_0$ .",
"The proof follows along the same lines as the arguments in [6], [8] and we do not include it here.",
"We summarize the existence result for periodic traveling waves of (REF ) and their small amplitude expansion below.",
"Theorem 2.1 For any $k>0$ if $\\beta <0$ and $k\\ne \\left(\\frac{\\gamma }{\\beta n^2}\\right)^{1/4}$ , $2\\le n\\in \\mathbb {N}$ if $\\beta >0$ , a one parameter family of solutions of (REF ) exists, given by $u(x,t)=w(a;k)(k(x-c(a;k)t))$ for $a \\in \\mathbb {R}$ and $|a|$ sufficiently small; $w(a;k)(\\cdot )$ is $2\\pi $ -periodic, even and smooth in its argument, and $c(a;k)$ is even in $a$ ; $w(a;k)$ and $c(a;k)$ depend analytically on $a$ and $k$ .",
"Moreover, $w(a;k)(z)=a\\cos (z) + a^2A_2\\cos 2z + a^3A_3\\cos 3z +a^4(A_{42}\\cos 2z+A_{44}\\cos 4z)+ O(a^5),$ and $c(a;k)=c_0+a^2c_2+a^4c_4+O(a^6)$ as $a \\rightarrow 0$ , where $c_0$ is in (REF ), $A_2 = \\dfrac{2k^2}{3\\gamma -12\\beta k^4}, \\quad A_3 = \\dfrac{9k^2A_2}{8\\gamma -72\\beta k^4}, \\quad A_{42}=2A_2A_3-2A_2^3, \\quad A_{44}=\\dfrac{8k^2(A_2^2+2A_3)}{15\\gamma -240\\beta k^4},$ $c_2=A_2, \\quad \\text{and} \\quad c_4=3A_2A_3-2A_2^3.$"
],
[
"Linearization and the spectral problem",
"We linearize (REF ) about the solution $w$ in Theorem REF in the coordinate frame moving at the speed $c$ .",
"The result becomes $k(v_t - ck v_z -\\beta k^3 v_{zzz}+2k(w v)_z)_z=\\gamma v.$ We seek a solution of the form $v(z,t) = e^{\\frac{\\lambda }{k} t} \\tilde{v}(z)$ , $\\lambda \\in \\mathbb {C}$ , to arrive at $\\mathcal {T}^\\lambda _{k,a} \\tilde{v} := (\\lambda \\partial _z -k^2\\partial ^2_z(c + \\beta k^2 \\partial ^2_z - 2w )- \\gamma )\\tilde{v} = 0$ The operator $\\mathcal {T}^\\lambda _{k,a}$ is defined on $L^2(\\mathbb {R})$ with dense domain $H^4(\\mathbb {R})$ .",
"We define the spectral stability of the periodic traveling wave solution $w$ with respect to square integrable perturbations as follows: it is spectrally stable if $\\mathcal {T}^\\lambda _{k,a}$ is invertible for any $\\lambda \\in \\mathbb {C}$ with $\\Re (\\lambda )>0$ , otherwise, it is deemed to be spectrally unstable.",
"The operator $\\mathcal {T}^\\lambda _{k,a}$ has continuous spectrum in $L^2(\\mathbb {R})$ .",
"By Floquet theory, since coefficients of $\\mathcal {T}^\\lambda _{k,a}$ are periodic functions, all solutions of (REF ) in $L^2(\\mathbb {R})$ are of the form $\\tilde{v}(z)=e^{i\\xi z}V(z)$ where $\\xi \\in (-1/2,1/2]$ is the Floquet exponent and $V$ is a $2\\pi $ -periodic function, see [5] for a similar situation.",
"This helps to break the invertibility problem of $\\mathcal {T}^\\lambda _{k,a}$ in $L^2(\\mathbb {R})$ into a family of invertibility problems in $L^2(\\mathbb {T})$ .",
"Lemma 3.1 The linear operator $\\mathcal {T}^\\lambda _{k,a}$ is invertible in $L^2(\\mathbb {R})$ if and only if linear operators $\\mathcal {T}^\\lambda _{k,a,\\xi } = \\lambda (\\partial _z+i\\xi ) - k^2(\\partial _z+i\\xi )^2(c + \\beta k^2 (\\partial _z+i\\xi )^2 - 2w )- \\gamma $ acting in $L^2(\\mathbb {T})$ with dense domain $H^4(\\mathbb {T})$ are invertible, for any $\\xi \\in (-1/2,1/2]$ .",
"We refer to [5] for a detailed proof in a similar situation.",
"The $L^2(\\mathbb {T})$ -spectra of operators $T^\\lambda _{k,a,\\xi }$ consist of eigenvalues of finite multiplicity.",
"Therefore, $T^\\lambda _{k,a,\\xi }$ is invertible in $L^2(\\mathbb {T})$ if zero is not an eigenvalue of $T^\\lambda _{k,a,\\xi }$ .",
"Using this, we have the following result.",
"Lemma 3.2 The operator $\\mathcal {T}^{\\lambda }_{k,a,\\xi }$ is not invertible in $L^2(\\mathbb {T})$ for some $\\lambda \\in and $ 0$ if and only if $ (Ak,a,)$, $ L2(T)$-spectrum of the operator,{\\begin{@align*}{1}{-1}\\mathcal {A}_{k,a,\\xi } := k^2(\\partial _z+i\\xi )(c + \\beta k^2 (\\partial _z+i\\xi )^2 - 2w ) + \\gamma (\\partial _z+i\\xi )^{-1}.\\end{@align*}}$ The operator $\\mathcal {T}^{\\lambda }_{k,a,\\xi }$ is not invertible in $L^2(\\mathbb {T})$ for some $\\lambda \\in and $ 0$ if and only if zero is an eigenvalue of $ Tk,a,$.",
"Moreover, for a $ VL2(T)$, $ Tk,a,V=0$ if and only if $ Ak,a,V=V$.",
"The proof follows trivially.$ Note that $\\xi \\ne 0$ is important in Lemma REF .",
"For $\\xi =0$ , $\\mathcal {A}_{k,a,0}$ is not well-defined on $L^2(\\mathbb {T})$ since $\\partial _z^{-1}$ is not well-defined on $L^2(\\mathbb {T})$ .",
"In what follows, we restrict $\\xi $ to be non-zero and examine the $L^2(\\mathbb {T})$ -spectrum of $\\mathcal {A}_{k,a,\\xi }$ .",
"To ease the notation, we will drop $k$ from subscript in $\\mathcal {A}_{k,a,\\xi }$ .",
"We observe that if $\\lambda \\in \\sigma (\\mathcal {A}_{a,\\xi })$ then $\\bar{\\lambda }\\in \\sigma (\\mathcal {A}_{a,-\\xi })$ , therefore, it is enough to consider $\\xi \\in \\left(0,1/2\\right]$ .",
"Also, since $w(z)$ is even in $z$ , we have $\\sigma (\\mathcal {A}_{a,\\xi }) = \\sigma (-\\mathcal {A}_{a,-\\xi }).$ Consequently, we obtain spectral instability of $w$ if $\\sigma (\\mathcal {A}_{a,\\xi })$ is not contained in the imaginary axis for some $\\xi \\in \\left(0,1/2\\right]$ .",
"A straightforward calculation shows that $\\mathcal {A}_{0,\\xi }e^{inz}=i\\omega _{n,\\xi }e^{inz}, \\quad n\\in \\mathbb {Z},$ where $\\omega _{n,\\xi } = k^2(n+\\xi )(c_0 - \\beta k^2(n+\\xi )^2)-\\frac{\\gamma }{n+\\xi }.$ We have $\\sigma (\\mathcal {A}_{0,\\xi })\\subset i\\mathbb {R}$ which should be the case since $a=0$ corresponds to the zero solution which is trivially stable.",
"As $|a|$ increases, the eigenvalues in (REF ) move around and may leave imaginary axis to give spectral instability.",
"Because of the symmetry of the spectrum around real and imaginary axes, spectral instability takes place only if a pair of imaginary eigenvalues collide on the imaginary axis.",
"If the spectral instability arises from a collision away from the origin on imaginary axis, it is termed as High-frequency instability [2].",
"Let $n\\ne m\\in \\mathbb {Z}$ , and $\\xi _{n,m}\\in (0,1/2]$ be such that $\\omega _{n,\\xi _{n,m}}=\\omega _{m,\\xi _{n,m}}.$ A quick calculation reveals that for a fixed value of $\\gamma $ , collisions at the origin take place only for $\\beta >0$ , all $n\\in \\mathbb {Z}$ , $m=-n-1$ , $\\xi _{n,m}=1/2$ , and $k=\\left(\\frac{\\gamma }{\\beta (n+1/2)^2}\\right)^{1/4}$ .",
"There is no collision at the origin if $\\beta <0$ .",
"The collision at the origin for non-zero Floquet exponent is an interesting characteristics of the Ostrovsky equation.",
"In similar studies on other various water wave models such collisions have not been observed, see [6], [7], [8], for example.",
"Here, we seek to find high-frequency instabilities and therefore, lists all collisions away from the origin below.",
"Lemma 3.3 The collision condition in (REF ) is satisfied away from the origin by: all pairs $\\lbrace n,m\\rbrace $ except $\\lbrace -1,1\\rbrace $ , and $\\lbrace -\\Delta n,0\\rbrace $ , $\\Delta n\\geqslant 2$ if $\\beta >0$ , and pairs $\\lbrace n,0\\rbrace $ , $n\\leqslant -2$ , and $\\lbrace -1,1\\rbrace $ if $\\beta < 0$ .",
"Moreover, if one of the colliding indices $n$ and $m$ is zero then the collision occurs for wavenumbers $k\\in (k_{n,m}^{\\min },\\infty )$ otherwise the collision occurs in a finite interval $k\\in (k_{n,m}^{\\min },k_{n,m}^{\\max })$ for some $k_{n,m}^{\\min },k_{n,m}^{\\max } > 0$ .",
"Without loss of generality, we can assume that $n<m$ and $m=n+\\Delta n$ with $\\Delta n\\in \\mathbb {N}$ .",
"Wavenumbers $k$ that satisfy (REF ) for $\\lbrace n,n+\\Delta n\\rbrace $ are given by $k^4=\\frac{\\gamma \\Delta n}{\\beta }K(x,\\Delta n) := \\frac{\\gamma \\Delta n}{\\beta } \\frac{ 1+x(x+\\Delta n) }{ x (x+\\Delta n) ( (x+\\Delta n)^3 - x^3 - \\Delta n ) }$ where $x=n+\\xi $ .",
"In terms of the function $K(x,\\Delta n)$ , since $\\gamma >0$ , collision between $n$ and $n+\\Delta n$ takes place for $\\beta >0$ if $K(n+\\xi , \\Delta n)>0$ for some $\\xi \\in (0,1/2]$ , and for $\\beta <0$ if $K(n+\\xi , \\Delta n)<0$ for some $\\xi \\in (0,1/2]$ .",
"Hence, it is important to know the sign of $K(x,\\Delta n)$ for a fixed $\\Delta n$ and $x\\in \\mathbb {R}$ .",
"We examine this case by case.",
"Case 1 ($\\Delta n=1$ ): The function $K(x,1)$ , see Figure REF , is always positive except at singularities $-1$ and 0.",
"Therefore, there is a collision between $n$ and $n+1$ for all $n\\in \\mathbb {Z}$ when $\\beta >0$ while there is no collision between $n$ and $n+1$ for any $n\\in \\mathbb {Z}$ when $\\beta <0$ .",
"Case 2 ($\\Delta n=2$ ): The function $K(x,2)$ , see Figure REF , is positive for $x\\in (-\\infty ,-2)\\cup (0,\\infty )$ and negative for $x\\in (-2,0)$ .",
"Therefore, there is a collision between $n$ and $n+2$ for all $n\\in \\mathbb {Z}\\backslash \\lbrace -1,-2\\rbrace $ when $\\beta >0$ while there is a collision between $n$ and $n+2$ only for $n=-2,$ and $-1$ when $\\beta <0$ .",
"Case 3 ($\\Delta n\\geqslant 3$ ): The function $K(x,\\Delta n)$ , $\\Delta n\\geqslant 3$ , see Figures REF and REF for example, is positive in $(-\\infty ,-\\Delta n)\\cup \\left(-\\frac{\\Delta n+\\sqrt{\\Delta n^2-4}}{2},-\\frac{\\Delta n-\\sqrt{\\Delta n^2-4}}{2}\\right)\\cup (0,\\infty ),$ and negative in $\\left(-\\Delta n,-\\frac{\\Delta n+\\sqrt{\\Delta n^2-4}}{2}\\right)\\cup \\left(-\\frac{\\Delta n-\\sqrt{\\Delta n^2-4}}{2},0\\right).$ For $\\Delta n\\geqslant 3$ , we have $-\\Delta n<-\\frac{\\Delta n+\\sqrt{\\Delta n^2-4}}{2}<-\\Delta n+\\frac{1}{2}, \\text{ and } -\\frac{1}{2}<-\\frac{\\Delta n-\\sqrt{\\Delta n^2-4}}{2}<0.$ Therefore, there is a collision between $n$ and $n+\\Delta n$ for all $n\\in \\mathbb {Z}\\backslash \\lbrace -\\Delta n\\rbrace $ when $\\beta >0$ while there is a collision between $n$ and $n+\\Delta n$ only for $n=-\\Delta n$ when $\\beta <0$ .",
"This proves the existence of all pairs satisfying collision condition (REF ) away from the origin.",
"Now, for a fixed $\\Delta n$ , if $K(n,\\Delta n)>0$ for some $n\\ne -\\Delta n,0$ then it continue to be positive in $[n,n+1/2]$ and therefore, collision takes place between $n$ and $n+\\Delta n$ for all $\\xi \\in (0,1/2]$ .",
"Since $K(x,\\Delta n)$ restricted to $x\\in [n,n+1/2]$ is a continuous function, it attains maximum and minimum in $[n,n+1/2]$ and therefore, collision takes place in a finite interval of wavenumbers $k\\in (k_{n,m}^{\\min },k_{n,m}^{\\max })\\subset (0,\\infty )$ , see Figure REF for an example.",
"For $n=-\\Delta n$ or 0, $K(x,\\Delta n)$ is positive and unbounded either in $(n,n+1/2]$ and therefore $K(x,\\Delta n)$ is bounded below but unbounded above.",
"In these cases, collision takes place in an interval of wavenumbers $k\\in (k_{n,m}^{\\min },\\infty )\\subset (0,\\infty )$ , see Figure REF for an example.",
"This completes the proof.",
"Figure: Graph of function K(x,Δn)K(x,\\Delta n) vs. xx for Δn=1,2,3\\Delta n=1,2,3, and 4.Figure: Graph of wavenumbers vs. n+ξn+\\xi for two collisions.",
"The range of wavenumbers for which collision is taking place is approximately (0.5,0.73)(0.5,0.73) for the left plot and (0,∞)(0, \\infty ) for the right plot.A necessary condition for collisions in Lemma REF to provide high-frequency instability is that their Krein signatures at collision should be opposite [12].",
"Since the Ostrovsky equation possesses a Hamiltonian structure, the linear operator $\\mathcal {A}_{a,\\xi }$ can be decomposed as $\\mathcal {A}_{a,\\xi } = J_\\xi \\mathcal {L}_{a,\\xi }$ where $J_{\\xi } = \\partial _z + i\\xi $ is skew-adjoint and $\\mathcal {L}_{a,\\xi } = k^2(c + \\beta k^2 (\\partial _z+i\\xi )^2 - 2w) + \\gamma (\\partial _z+i\\xi )^{-2}$ is self-adjoint.",
"With this decomposition, the Krein signature $\\kappa _{n,\\xi }$ of eigenvalues $i\\omega _{n,\\xi }$ in (REF ) of $\\mathcal {A}_{0,\\xi }$ is given by $\\kappa _{n,\\xi } = \\operatorname{sgn}(\\left<\\mathcal {L}_{0,\\xi } e^{inz}, e^{inz}\\right>)=\\operatorname{sgn}\\left( \\frac{1}{n+\\xi }\\omega _{n,\\xi }\\right)$ where $\\operatorname{sgn}$ is the signum function which determines the sign of a real number.",
"If the collision condition (REF ) is satisfied for some $n,m\\in \\mathbb {Z}$ and $\\xi _{n,m}\\in (0,1/2]$ then (REF ) provides that eigenvalues $i\\omega _{n,\\xi }$ and $i\\omega _{m,\\xi }$ have opposite Krein signatures at the collision if $(n+\\xi _{n,m})(m+\\xi _{n,m})<0$ otherwise they have same Krein signatures at the collision.",
"Using (REF ), we can rule out some collisions in Lemma REF which will not lead to high-frequency instability.",
"Lemma 3.4 For $\\beta >0$ , out of all collisions mentioned in Lemma REF , $\\lbrace n,m\\rbrace $ with $n\\leqslant -1$ and $m\\geqslant 1$ , and $\\lbrace -1,0\\rbrace $ have opposite Krein signatures.",
"For $\\beta <0$ , all collisions mentioned in Lemma REF have opposite Krein signatures.",
"If $n\\ne 0$ and $m\\ne 0$ then from (REF ), $n$ and $m$ must be of opposite signs for (REF ) to hold.",
"If one of $n$ or $m$ is zero then the other needs to be negative in order for (REF ) to hold.",
"Then the proof follows.",
"High-frequency instabilities Table: Collisions with opposite Krein signatures for a given Δn\\Delta n for β>0\\beta >0 and β<0\\beta <0.Table REF summarizes all the collisions with opposite Krein signatures based on Lemma REF for a given $\\Delta n$ for both $\\beta >0$ and $\\beta < 0$ .",
"In what follows, we do further analysis to check if collisions in Table REF corresponding to $\\Delta n=1$ , and 2, lead to high-frequency instability.",
"$\\Delta n=1$ calculation and conclusion For a fixed $n\\in \\mathbb {Z}$ , let $\\xi _{0}\\in (0,1/2]$ be such that $0 \\ne \\omega _{n,\\xi _0} = \\omega _{n+1,\\xi _0} =: \\omega .$ Therefore, $i\\omega $ is an eigenvalue of $\\mathcal {A}_{0,\\xi _0}$ of multiplicity two with an orthonormal basis of eigenfunctions $\\lbrace e^{inz},e^{i(n+1)z}\\rbrace $ .",
"For $|a|$ small, let $\\lambda _{n,a,\\xi _0}$ and $\\lambda _{n+1,a,\\xi _0}$ be eigenvalues of $\\mathcal {A}_{a,\\xi _0}$ bifurcating from $i\\omega $ with an orthonormal basis of eigenfunctions $\\lbrace \\phi _{n,a,\\xi _0}(z),\\phi _{n+1,a,\\xi _0}(z)\\rbrace $ .",
"Note that $\\lambda _{n,0,\\xi _0}=\\lambda _{n+1,0,\\xi _0}=i\\omega $ with $\\phi _{n,0,\\xi _0}(z)=e^{inz}$ and $\\phi _{n+1,0,\\xi _0}(z)=e^{i(n+1)z}$ .",
"Let $\\lambda _{n,a,\\xi _0} = i \\omega + i \\mu _{n,a,\\xi _0}\\quad \\quad \\text{and} \\quad \\quad \\lambda _{n+1,a,\\xi _0} = i \\omega + i \\mu _{n+1,a,\\xi _0}.$ We are interested in the location of $\\mu _{n,a,\\xi }$ and $\\mu _{n+1,a,\\xi }$ for $|a|$ small as if they have non-zero imaginary parts then we obtain high-frequency instability.",
"We start with the following expansions of eigenfunctions $\\phi _{n,a,\\xi _0} =& e^{inz}+a\\phi _{n,1}+a^2\\phi _{n,2}+O(a^3), \\\\\\phi _{n+1,a,\\xi _0} =& e^{i(n+1)z}+a\\phi _{n+1,1}+a^2\\phi _{n+1,2}+O(a^3).$ We use orthonormality of $\\phi _{n,a,\\xi _0}$ and $\\phi _{n+1,a,\\xi _0}$ to find that $\\phi _{n,1}=\\phi _{n,2}=\\phi _{n+1,1}=\\phi _{n+1,2}=0.$ To trace the bifurcation of the eigenvalues from the point of the collision on the imaginary axis for $|a|$ sufficiently small, we compute the actions of $\\mathcal {A}_{a,\\xi _0}$ and identity operators on the extended eigenspace $\\lbrace \\phi _{n,a,\\xi _0}(z), \\phi _{n+1,a,\\xi _0}(z)\\rbrace $ viz.",
"$\\mathcal {B}_{a,\\xi _0} = \\left[ \\frac{\\langle \\mathcal {A}_a(\\xi _0)\\phi _{i,a,\\xi _0}(z),\\phi _{j,a,\\xi _0}(z)\\rangle }{\\langle \\phi _{i,a,\\xi _0}(z),\\phi _{i,a,\\xi _0}(z)\\rangle } \\right]_{i,j=n,n+1}\\text{ and }\\mathcal {I}_{a} = \\left[ \\frac{\\langle \\phi _{i,a,\\xi _0}(z),\\phi _{j,a,\\xi _0}(z)\\rangle }{\\langle \\phi _{i,a,\\xi _0}(z),\\phi _{i,a,\\xi _0}(z)\\rangle } \\right]_{i,j=n,n+1}.$ Here $\\langle \\hspace{2.0pt}px\\cdot \\hspace{2.0pt}px,\\hspace{2.0pt}px\\cdot \\hspace{2.0pt}px\\rangle $ denotes the $L^2(\\mathbb {T})$ - inner product as defined in (REF ).",
"Using expansions of $w$ and $c$ in Theorem REF , we expand $\\mathcal {A}_{a,\\xi _0}$ in $a$ as $\\mathcal {A}_{a,\\xi _0}=\\mathcal {A}_{0,\\xi _0}-2ak^2(\\partial _z+i\\xi _0)\\cos z+a^2k^2(\\partial _z+i\\xi _0)(c_2-2A_2\\cos 2z)+O(a^3)$ and use the expansion of eigenfunctions in (REF )-() to find the matrices in (REF ) as $\\mathcal {B}_{a,\\xi _0} =\\begin{bmatrix}i \\omega +ik^2a^2(n+\\xi _0)c_2 & -ik^2a (n+1+\\xi _0) \\\\& \\\\-ik^2a (n+\\xi _0) & i \\omega +ik^2a^2(n+1+\\xi _0)c_2\\end{bmatrix}+O(a^3)$ and $\\mathcal {I}_{a} =\\begin{bmatrix}1 & 0 \\\\0 & 1\\end{bmatrix}+O(a^3).$ Note that $\\mathcal {B}_{0,\\xi _0}=\\operatorname{diag}(i\\omega ,i\\omega )$ which should be the case as $i\\omega $ is an eigenvalue of $\\mathcal {A}_{0,\\xi _0}$ of multiplicity two.",
"The two values of $\\mu $ solving the equation $\\det (\\mathcal {B}_{a,\\xi _0}-(i \\omega + i \\mu ) \\mathcal {I}_{a}) = 0,$ would coincide with $\\mu _{n,a,\\xi _0}$ and $\\mu _{n+1,a,\\xi _0}$ in (REF ) in leading order of $a$ .",
"Plugging the values in (REF ) and calculating the discriminant of the quadratic in $\\mu $ , we arrive at $\\mathbb {D}_{a,\\xi _0} = 4k^4a^2(n+\\xi _0)(n+1+\\xi _0)+O(a^3).$ Therefore, for sufficiently small $|a|$ , if $(n+\\xi _0)(n+1+\\xi _0)$ is negative then we would obtain high-frequency instability.",
"From Table REF , the only collision for $\\Delta n=1$ is when $\\beta >0$ and $n=-1$ .",
"This collision takes place for all values of $\\xi \\in (0,1/2]$ , see Figure REF .",
"Figure: Collision contour describing collision between eigenvalues iω -1,ξ i\\omega _{-1,\\xi } and iω 0,ξ i\\omega _{0,\\xi } for different values of kk and ξ\\xi for γ=6\\gamma =6 and β=1\\beta =1.Analyzing the function $K(x,\\Delta n)$ for $x=\\xi -1$ and $\\Delta n=1$ , we easily deduce that this collision takes place for wavenumbers $k\\in ((4\\gamma /\\beta )^{1/4},\\infty )$ .",
"We summarize the result in the following theorem.",
"Theorem 4.1 For a fixed $\\gamma >0$ and $\\beta >0$ , a $2\\pi /k$ -periodic traveling wave of (REF ) given by $u(x,t)=w(k(x-ct))$ where $w$ and $c$ are given in Theorem REF suffers high-frequency instability if $k>\\@root 4 \\of {\\dfrac{4\\gamma }{\\beta }}.$ $\\Delta n=2$ calculation and conclusion We proceed as in the previous section.",
"For a fixed $n\\in \\mathbb {Z}$ , let $\\xi _{0}\\in (0,1/2]$ be such that $0 \\ne \\omega _{n,\\xi _0} = \\omega _{n+2,\\xi _0} =: \\omega .$ That is, $i\\omega $ is an eigenvalue of $\\mathcal {A}_{0,\\xi _0}$ of multiplicity two with an orthonormal basis of eigenfunctions $\\lbrace e^{inz},e^{i(n+2)z}\\rbrace $ .",
"As before, for $|a|$ small, let $\\lambda _{n,a,\\xi _0}$ and $\\lambda _{n+2,a,\\xi _0}$ be eigenvalues of $\\mathcal {A}_{a,\\xi _0}$ bifurcating from $i\\omega $ with an orthonormal basis of eigenfunctions $\\lbrace \\phi _{n,a,\\xi _0}(z),\\phi _{n+2,a,\\xi _0}(z)\\rbrace $ .",
"Let $\\lambda _{n,a,\\xi _0} = i \\omega + i \\mu _{n,a,\\xi _0}\\quad \\quad \\text{and} \\quad \\quad \\lambda _{n+2,a,\\xi _0} = i \\omega + i \\mu _{n+2,a,\\xi _0}$ and we are interested in the location of $\\mu _{n,a,\\xi }$ and $\\mu _{n+2,a,\\xi }$ for $|a|$ small.",
"Again, using orthonormality of $\\phi _{n,a,\\xi _0}$ and $\\phi _{n+2,a,\\xi _0}$ we find that $\\phi _{n,a,\\xi _0} = e^{inz}+O(a^5)\\quad \\text{ and }\\quad \\phi _{n+2,a,\\xi _0} = e^{i(n+2)z}+O(a^5).$ As before, we compute the action matrices of $\\mathcal {A}_{a,\\xi _0}$ and identity operators on the extended eigenspace $\\lbrace \\phi _{n,a,\\xi _0}(z), \\phi _{n+2,a,\\xi _0}(z)\\rbrace $ .",
"We use expansions of $w$ and $c$ in Theorem REF to expand $\\mathcal {A}_{a,\\xi _0}$ in $a$ as $\\mathcal {A}_{a,\\xi _0}=&\\mathcal {A}_{0,\\xi _0}-2ak^2(\\partial _z+i\\xi _0)\\cos z+a^2k^2(\\partial _z+i\\xi _0)(c_2-2A_2\\cos 2z)-2a^3k^2A_3(\\partial _z+i\\xi _0)\\cos 3z\\\\&+a^4k^2(\\partial _z+i\\xi _0)(c_4-2(A_{42}\\cos 2z+A_{44}\\cos 4z))+O(a^5).$ Using the expansion of eigenfunctions in (REF ), the matrices in (REF ) turn out to be $\\mathcal {B}_{a,\\xi _0} =\\begin{bmatrix}i \\omega + ik^2(a^2A_2+a^4c_4)(n+\\xi _0) & -ik^2(a^2A_2+a^4A_{42}) (n+2+\\xi _0) \\\\& \\\\-ik^2(a^2A_2+a^4A_{42})(n+\\xi _0) & i \\omega + ik^2(a^2A_2+a^4c_4)(n+2+\\xi _0)\\end{bmatrix}+O(a^5)$ and $\\mathcal {I}_{a} =\\begin{bmatrix}1 & 0 \\\\0 & 1\\end{bmatrix}+O(a^5).$ Again, we solve the equation $\\det (\\mathcal {B}_{a,\\xi _0}-(i \\omega + i \\mu ) \\mathcal {I}_{a}) = 0,$ to obtain a quadratic in $\\mu $ whose discriminant is given by $\\mathbb {D}_{a,\\xi _0} = 4k^4a^4A_2^2(n+\\xi _0+1)^2 + O(a^5).$ Note that, irrespective of the values of $n$ and $\\xi _0$ , the leading term in the discriminant is always positive.",
"Therefore, we do not observe any high-frequency instability for $\\Delta n=2$ case by performing the perturbation calculation up to fourth power of the amplitude parameter $a$ .",
"Acknowledgement Bhavna and AKP are supported by the Science and Engineering Research Board (SERB), Department of Science and Technology (DST), Government of India under grant SRG/2019/000741.",
"AK is supported by Junior Research Fellowships (JRF) by Council of Scientific and Industrial Research (CSIR), Government of India.",
"Conflict of interest statement On behalf of all authors, the corresponding author states that there is no conflict of interest.",
"Data availability statement Data sharing not applicable to this article as no datasets were generated or analysed during the current study."
],
[
"High-frequency instabilities",
"Table REF summarizes all the collisions with opposite Krein signatures based on Lemma REF for a given $\\Delta n$ for both $\\beta >0$ and $\\beta < 0$ .",
"In what follows, we do further analysis to check if collisions in Table REF corresponding to $\\Delta n=1$ , and 2, lead to high-frequency instability."
],
[
"$\\Delta n=1$ calculation and conclusion",
"For a fixed $n\\in \\mathbb {Z}$ , let $\\xi _{0}\\in (0,1/2]$ be such that $0 \\ne \\omega _{n,\\xi _0} = \\omega _{n+1,\\xi _0} =: \\omega .$ Therefore, $i\\omega $ is an eigenvalue of $\\mathcal {A}_{0,\\xi _0}$ of multiplicity two with an orthonormal basis of eigenfunctions $\\lbrace e^{inz},e^{i(n+1)z}\\rbrace $ .",
"For $|a|$ small, let $\\lambda _{n,a,\\xi _0}$ and $\\lambda _{n+1,a,\\xi _0}$ be eigenvalues of $\\mathcal {A}_{a,\\xi _0}$ bifurcating from $i\\omega $ with an orthonormal basis of eigenfunctions $\\lbrace \\phi _{n,a,\\xi _0}(z),\\phi _{n+1,a,\\xi _0}(z)\\rbrace $ .",
"Note that $\\lambda _{n,0,\\xi _0}=\\lambda _{n+1,0,\\xi _0}=i\\omega $ with $\\phi _{n,0,\\xi _0}(z)=e^{inz}$ and $\\phi _{n+1,0,\\xi _0}(z)=e^{i(n+1)z}$ .",
"Let $\\lambda _{n,a,\\xi _0} = i \\omega + i \\mu _{n,a,\\xi _0}\\quad \\quad \\text{and} \\quad \\quad \\lambda _{n+1,a,\\xi _0} = i \\omega + i \\mu _{n+1,a,\\xi _0}.$ We are interested in the location of $\\mu _{n,a,\\xi }$ and $\\mu _{n+1,a,\\xi }$ for $|a|$ small as if they have non-zero imaginary parts then we obtain high-frequency instability.",
"We start with the following expansions of eigenfunctions $\\phi _{n,a,\\xi _0} =& e^{inz}+a\\phi _{n,1}+a^2\\phi _{n,2}+O(a^3), \\\\\\phi _{n+1,a,\\xi _0} =& e^{i(n+1)z}+a\\phi _{n+1,1}+a^2\\phi _{n+1,2}+O(a^3).$ We use orthonormality of $\\phi _{n,a,\\xi _0}$ and $\\phi _{n+1,a,\\xi _0}$ to find that $\\phi _{n,1}=\\phi _{n,2}=\\phi _{n+1,1}=\\phi _{n+1,2}=0.$ To trace the bifurcation of the eigenvalues from the point of the collision on the imaginary axis for $|a|$ sufficiently small, we compute the actions of $\\mathcal {A}_{a,\\xi _0}$ and identity operators on the extended eigenspace $\\lbrace \\phi _{n,a,\\xi _0}(z), \\phi _{n+1,a,\\xi _0}(z)\\rbrace $ viz.",
"$\\mathcal {B}_{a,\\xi _0} = \\left[ \\frac{\\langle \\mathcal {A}_a(\\xi _0)\\phi _{i,a,\\xi _0}(z),\\phi _{j,a,\\xi _0}(z)\\rangle }{\\langle \\phi _{i,a,\\xi _0}(z),\\phi _{i,a,\\xi _0}(z)\\rangle } \\right]_{i,j=n,n+1}\\text{ and }\\mathcal {I}_{a} = \\left[ \\frac{\\langle \\phi _{i,a,\\xi _0}(z),\\phi _{j,a,\\xi _0}(z)\\rangle }{\\langle \\phi _{i,a,\\xi _0}(z),\\phi _{i,a,\\xi _0}(z)\\rangle } \\right]_{i,j=n,n+1}.$ Here $\\langle \\hspace{2.0pt}px\\cdot \\hspace{2.0pt}px,\\hspace{2.0pt}px\\cdot \\hspace{2.0pt}px\\rangle $ denotes the $L^2(\\mathbb {T})$ - inner product as defined in (REF ).",
"Using expansions of $w$ and $c$ in Theorem REF , we expand $\\mathcal {A}_{a,\\xi _0}$ in $a$ as $\\mathcal {A}_{a,\\xi _0}=\\mathcal {A}_{0,\\xi _0}-2ak^2(\\partial _z+i\\xi _0)\\cos z+a^2k^2(\\partial _z+i\\xi _0)(c_2-2A_2\\cos 2z)+O(a^3)$ and use the expansion of eigenfunctions in (REF )-() to find the matrices in (REF ) as $\\mathcal {B}_{a,\\xi _0} =\\begin{bmatrix}i \\omega +ik^2a^2(n+\\xi _0)c_2 & -ik^2a (n+1+\\xi _0) \\\\& \\\\-ik^2a (n+\\xi _0) & i \\omega +ik^2a^2(n+1+\\xi _0)c_2\\end{bmatrix}+O(a^3)$ and $\\mathcal {I}_{a} =\\begin{bmatrix}1 & 0 \\\\0 & 1\\end{bmatrix}+O(a^3).$ Note that $\\mathcal {B}_{0,\\xi _0}=\\operatorname{diag}(i\\omega ,i\\omega )$ which should be the case as $i\\omega $ is an eigenvalue of $\\mathcal {A}_{0,\\xi _0}$ of multiplicity two.",
"The two values of $\\mu $ solving the equation $\\det (\\mathcal {B}_{a,\\xi _0}-(i \\omega + i \\mu ) \\mathcal {I}_{a}) = 0,$ would coincide with $\\mu _{n,a,\\xi _0}$ and $\\mu _{n+1,a,\\xi _0}$ in (REF ) in leading order of $a$ .",
"Plugging the values in (REF ) and calculating the discriminant of the quadratic in $\\mu $ , we arrive at $\\mathbb {D}_{a,\\xi _0} = 4k^4a^2(n+\\xi _0)(n+1+\\xi _0)+O(a^3).$ Therefore, for sufficiently small $|a|$ , if $(n+\\xi _0)(n+1+\\xi _0)$ is negative then we would obtain high-frequency instability.",
"From Table REF , the only collision for $\\Delta n=1$ is when $\\beta >0$ and $n=-1$ .",
"This collision takes place for all values of $\\xi \\in (0,1/2]$ , see Figure REF .",
"Figure: Collision contour describing collision between eigenvalues iω -1,ξ i\\omega _{-1,\\xi } and iω 0,ξ i\\omega _{0,\\xi } for different values of kk and ξ\\xi for γ=6\\gamma =6 and β=1\\beta =1.Analyzing the function $K(x,\\Delta n)$ for $x=\\xi -1$ and $\\Delta n=1$ , we easily deduce that this collision takes place for wavenumbers $k\\in ((4\\gamma /\\beta )^{1/4},\\infty )$ .",
"We summarize the result in the following theorem.",
"Theorem 4.1 For a fixed $\\gamma >0$ and $\\beta >0$ , a $2\\pi /k$ -periodic traveling wave of (REF ) given by $u(x,t)=w(k(x-ct))$ where $w$ and $c$ are given in Theorem REF suffers high-frequency instability if $k>\\@root 4 \\of {\\dfrac{4\\gamma }{\\beta }}.$"
],
[
"$\\Delta n=2$ calculation and conclusion",
"We proceed as in the previous section.",
"For a fixed $n\\in \\mathbb {Z}$ , let $\\xi _{0}\\in (0,1/2]$ be such that $0 \\ne \\omega _{n,\\xi _0} = \\omega _{n+2,\\xi _0} =: \\omega .$ That is, $i\\omega $ is an eigenvalue of $\\mathcal {A}_{0,\\xi _0}$ of multiplicity two with an orthonormal basis of eigenfunctions $\\lbrace e^{inz},e^{i(n+2)z}\\rbrace $ .",
"As before, for $|a|$ small, let $\\lambda _{n,a,\\xi _0}$ and $\\lambda _{n+2,a,\\xi _0}$ be eigenvalues of $\\mathcal {A}_{a,\\xi _0}$ bifurcating from $i\\omega $ with an orthonormal basis of eigenfunctions $\\lbrace \\phi _{n,a,\\xi _0}(z),\\phi _{n+2,a,\\xi _0}(z)\\rbrace $ .",
"Let $\\lambda _{n,a,\\xi _0} = i \\omega + i \\mu _{n,a,\\xi _0}\\quad \\quad \\text{and} \\quad \\quad \\lambda _{n+2,a,\\xi _0} = i \\omega + i \\mu _{n+2,a,\\xi _0}$ and we are interested in the location of $\\mu _{n,a,\\xi }$ and $\\mu _{n+2,a,\\xi }$ for $|a|$ small.",
"Again, using orthonormality of $\\phi _{n,a,\\xi _0}$ and $\\phi _{n+2,a,\\xi _0}$ we find that $\\phi _{n,a,\\xi _0} = e^{inz}+O(a^5)\\quad \\text{ and }\\quad \\phi _{n+2,a,\\xi _0} = e^{i(n+2)z}+O(a^5).$ As before, we compute the action matrices of $\\mathcal {A}_{a,\\xi _0}$ and identity operators on the extended eigenspace $\\lbrace \\phi _{n,a,\\xi _0}(z), \\phi _{n+2,a,\\xi _0}(z)\\rbrace $ .",
"We use expansions of $w$ and $c$ in Theorem REF to expand $\\mathcal {A}_{a,\\xi _0}$ in $a$ as $\\mathcal {A}_{a,\\xi _0}=&\\mathcal {A}_{0,\\xi _0}-2ak^2(\\partial _z+i\\xi _0)\\cos z+a^2k^2(\\partial _z+i\\xi _0)(c_2-2A_2\\cos 2z)-2a^3k^2A_3(\\partial _z+i\\xi _0)\\cos 3z\\\\&+a^4k^2(\\partial _z+i\\xi _0)(c_4-2(A_{42}\\cos 2z+A_{44}\\cos 4z))+O(a^5).$ Using the expansion of eigenfunctions in (REF ), the matrices in (REF ) turn out to be $\\mathcal {B}_{a,\\xi _0} =\\begin{bmatrix}i \\omega + ik^2(a^2A_2+a^4c_4)(n+\\xi _0) & -ik^2(a^2A_2+a^4A_{42}) (n+2+\\xi _0) \\\\& \\\\-ik^2(a^2A_2+a^4A_{42})(n+\\xi _0) & i \\omega + ik^2(a^2A_2+a^4c_4)(n+2+\\xi _0)\\end{bmatrix}+O(a^5)$ and $\\mathcal {I}_{a} =\\begin{bmatrix}1 & 0 \\\\0 & 1\\end{bmatrix}+O(a^5).$ Again, we solve the equation $\\det (\\mathcal {B}_{a,\\xi _0}-(i \\omega + i \\mu ) \\mathcal {I}_{a}) = 0,$ to obtain a quadratic in $\\mu $ whose discriminant is given by $\\mathbb {D}_{a,\\xi _0} = 4k^4a^4A_2^2(n+\\xi _0+1)^2 + O(a^5).$ Note that, irrespective of the values of $n$ and $\\xi _0$ , the leading term in the discriminant is always positive.",
"Therefore, we do not observe any high-frequency instability for $\\Delta n=2$ case by performing the perturbation calculation up to fourth power of the amplitude parameter $a$ ."
],
[
"Acknowledgement",
"Bhavna and AKP are supported by the Science and Engineering Research Board (SERB), Department of Science and Technology (DST), Government of India under grant SRG/2019/000741.",
"AK is supported by Junior Research Fellowships (JRF) by Council of Scientific and Industrial Research (CSIR), Government of India.",
"On behalf of all authors, the corresponding author states that there is no conflict of interest.",
"Data sharing not applicable to this article as no datasets were generated or analysed during the current study."
]
] | 2107.01794 |
[
[
"Testing Binomiality of Chemical Reaction Networks Using Comprehensive\n Gr\\\"obner Systems"
],
[
"Abstract We consider the problem of binomiality of the steady state ideals of biochemical reaction networks.",
"We are interested in finding polynomial conditions on the parameters such that the steady state ideal of a chemical reaction network is binomial under every specialisation of the parameters if the conditions on the parameters hold.",
"We approach the binomiality problem using Comprehensive Gr\\\"obner systems.",
"Considering rate constants as parameters, we compute comprehensive Gr\\\"obner systems for various reactions.",
"In particular, we make automatic computations on n-site phosphorylations and biomodels from the Biomodels repository using the grobcov library of the computer algebra system Singular."
],
[
"Introduction",
"A chemical reaction is a transformation between two sets of chemical objects called chemical complexes.",
"The objects that form a chemical complex are chemical species.",
"In fact, complexes are formal sums of chemical species representing the left and the right hand sides of chemical reactions.",
"A chemical reaction network is a set of chemical reactions.",
"For example ${E + S <=>[k_1][k_{-1}] ES ->[k_2] E + P}$ is a chemical reaction network with one reversible reactions and one non-reversible reaction.",
"This reaction network is a well-known network, called the Michaelis–Menton reaction network.",
"A kinetics of a chemical reaction network is an assignment of a rate function to each reaction in the network.",
"The rate function depends on the concentrations of the species.",
"A kinetics for a chemical reaction network is called mass-action if for each reaction in the network, the rate function is a monomial in terms of the concentrations of the species, such that the exponents are given by the numbers of molecules of the species consumed in the reaction, multiplied by a constant called rate constant.",
"In the Michaelis–Menton reaction, $k_{1}$ , $k_{-1}$ , $k_{2}$ are the rate constants.",
"In this article, we assume mass-action kinetics.",
"A system of autonomous ordinary differential equations can be used to describe the change in the concentration of each species over time in a reaction.",
"For example, in the Michaelis–Menton reaction, let the variables $s, p, c, e$ represent the concentrations of the species $S, P, ES, E$ respectively.",
"The ordinary differential equations (ODEs) describing change of the concentrations of the species for this reaction network are the following: $\\dot{s} &= f_s = -k_{1} se + k_{-1} c, \\\\\\dot{p} &= f_p = k_{2} c, \\\\\\dot{c} &= f_c = k_{1} s e - (k_{-1}+k_{2}) c, \\\\\\dot{e} &= -f_c .$ Solutions of the polynomials $f_s$ , $f_p$ , $f_c$ and $-f_c$ give us the concentrations of the species in which the system is in equilibrium.",
"In fact, the solutions of $f_s$ , $f_p$ , $f_c$ and $-f_c$ are called the steady states of the chemical reaction network.",
"Accordingly, the ideal generated by $f_s$ , $f_p$ , $f_c$ and $-f_c$ , i.e., $I=\\langle f_s, f_p, f_c,-f_c \\rangle \\subseteq \\mathbb {K}[k_1,k_{-1},k_2][s,p,c,e] $ , where $\\mathbb {K}$ is a field, is called the steady state ideal of the Michaelis–Menton network.",
"For a thorough introduction on chemical reaction network theory, refer to Feinberg's Book [22] and his lecture notes [21].",
"We follow the notation of Feinberg's book in this article.",
"A binomial ideal is an ideal that is generated by a set of binomials.",
"In this article, we consider the problem of binomiality of steady state ideals when the rate constants are specialised over a field extension of $\\mathbb {K}$ , that is, when the rate constants have been assigned values from an extension of $\\mathbb {K}$ , typically the closure of $\\mathbb {K}$ .",
"More precisely, we are interested in conditions over the rate constants (typically given by polynomial equations on rate constants), such that for every values of the rate constants in the extension field, the steady state ideal is binomial under those conditions.",
"In this article, we often use parameters instead of rate constants, an indication that they can be specialised.",
"Therefore, we consider the parametric binomiality problem.",
"Let us consider the steady state ideal of the Michaelis–Menton reaction: $I = I=\\langle f_s, f_p, f_c\\rangle \\subseteq \\mathbb {K}[k_1,k_{-1},k_2][s,p,c,e],$ given by Equations (REF )–().",
"One can observe that $f_c=-f_s+f_p$ .",
"Hence, $I = \\langle f_s,f_p\\rangle $ .",
"Having fixed the term ordering induced by $c>s>e$ , one may consider further reducing $f_s$ by $f_p$ , i.e., $f_s-f_p = (k_{-1}-k_1)c -k_1 se$ .",
"As the rate constants in a chemical reaction take values, $k_{-1}-k_1$ may vanish.",
"In this case, if the leading term of $f_s-f_p$ vanishes, then it will be a monomial, and therefore, the reduced Gröbner basis of $I$ will be the monomial ideal generated by $\\lbrace k_2c,-k_1 se\\rbrace $ , given that $k_2 \\ne 0$ and $k_{-1} \\ne 0$ .",
"This example shows that the Gröbner basis of the steady state ideal (and the steady states of the reaction) can change depending on the values of the rate constants.",
"Therefore, we must consider distinct cases for the parameters when analysing a reaction network.",
"Thinking purely in terms of computer algebra, this example illustrates the idea behind Comprehensive Gröbner bases.",
"In this article, we investigate the conditions on the parameters of a steady state ideal (or equivalently on the rate constants of a reaction) such that the steady state ideal is binomial when those conditions on the parameters hold.",
"In the literature, a slightly different notions of binomiality has been considered.",
"Eisenbud and Sturmfels in [16] call an ideal binomial if it is generated by polynomials with at most two terms.",
"Some authors, e.g., Pérez-Milán et al.",
"[40], have studied the binomiality of steady state ideals according to the definition in [16].",
"However, in this article, our definition does not include those ideals that include monomials.",
"This difference in the definition, obviously, affects the steady state variety of binomial chemical reaction networks in practice.",
"Binomial ideals and toric varieties have rich history in chemical reaction networks theory.",
"Binomiality corresponds to detailed balance, which is a very important concept in thermodynamics.",
"Detailed balance means that at thermodynamic equilibrium, the forward and backward rates should be equal for all reactions.",
"Detailed balance has been historically used by Einstein [15], Wegscheider [48] and by Onsager [38].",
"Some of the subsystems of molecular devices can satisfy binomiality conditions.",
"Another interesting point to study binomiality is because the analysis of properties such as multi-stationarity and stability are easier to establish for binomial systems.",
"Toricity, also known as complex, or cyclic, or semi-detailed balance is also known since Boltzmann that has used it as a sufficient condition for deriving his famous H-theorem [2].",
"Toricity implies binomiality, but the converse is not true.",
"A toric variety is indeed irreducible, however a binomial steady state ideal may have an irreducible variety, which would not be toric.",
"However, every variety of a binomial ideal includes a toric variety as its irreducible component.",
"A toric system must obey constraints on the rates constants, such as the well known Weigscheider—Kolmogorov condition, which implies the equality of the products of forward and backward rates constants in cycles of reversible reactions.",
"Mathematicians have considered binomiality and toricity and investigated their properties thoroughly, among them existing literature are the work by Fulton [23], Sturmfels [45] and Eisenbud et al. [16].",
"Binomiality implies detailed balancing of reversible chemical reactions, which has been studied by Gorban et al.",
"[24], [25] and Grigoriev and Weber [28].",
"Toric dynamical systems have been studied by Feinberg [20] and Horn and Jackson [30].",
"Over the real numbers Craciun et al.",
"have studied the toricity problem in [9].",
"In the latter work, it hs been shown that complex balanced systems are the same as toric dynamical systems, although toric steady states are different from that.",
"Binomiality implies much simpler criteria for multistationarity [14], [44].",
"Pérez-Milán, et al.",
"presented a sufficient linear algebra conditions with inequalities for binomiality of the steady state ideals [40].",
"The idea in the latter has been developed in [39], where MESSI reactions have been introduced.",
"Conradi and Kahle have proved in [8] that for homogenous ideals the latter sufficient condition is necessary as well, and introduced an algorithm for that case.",
"Their algorithm has been implemented in Maple and Macaulay II in [32], [31].",
"A geometric view towards toricity of chemical reaction networks has been given by Grigoriev et al.",
"in [27], where shifted toricity has been introduced, algorithms presented for testing shifted toricity and complexity bounds and experimental results are discussed.",
"In [27], the two main tools from computer algebra,, quantifier elimination [12], [26], [49] and Gröbner bases [5], [6], [18], [19] are used.",
"Also recently, the authors introduced a first order logic test for toricity [43].",
"An efficient linear algebra method for testing unconditional binomiality has been presented in [42] and a graph-theoretical equivalent of the method is given in [41].",
"Testing binomiality of an ideal is a difficult problem, both from a theoretical and a practical point of view.",
"A typical method to test binomiality is via computing a Gröbner basis.",
"It has been shown that computing a Göbner basis is EXPSPACE-complete [36], which shows the difficulty of the binomiality problem from the computational point of view.",
"The approach proposed for testing binomiality of steady state ideals in [40], [8] relies on linear algebra.",
"In this approach the computations are done without considering the values of the parameters.",
"Also large matrices are constructed in this approach.",
"Existing work on binomiality of chemical reaction networks typically ignores specialisation of the parameters, often treating them as variables and carrying on the computations.",
"For instance, fixing an ordering in which the parameters are smaller than the variables, e.g., lexicographic ordering, one may consider computing a Gröbner basis of the steady state ideal and then eliminating the variables.",
"Then the elimination ideal will be in the ring of parameters and may result in conditions on the parameters such that the original ideal is binomial.",
"However, this approach does not consider the fact that in the process of computations, some terms can be vanished, if parameters are specialised.",
"In contrast, our approach is to use comprehensive Gröbner bases, which considers specialisations of the parameters.",
"A comprehensive Gröbner basis of an ideal is a finite set of polynomials on the parameters and the variables, such that it is a Gröbner basis under every value assignment in the parameters.",
"Therefore, a steady state ideal is binomial if its comprehensive Gröbner basis is binomial.",
"This observation reduces testing binomiality of a steady state ideal under specialisation into testing binomiality of a comprehensive Gröbner basis.",
"Computing a comprehensive Gröbner basis results in a partitioning of the ambient space into certain varieties and computations of certain set of polynomials associated to each of those varieties, such that if the parameters are specialised from the variety, the associated polynomial set is a Gröbner basis.",
"Such a partition with its associated polynomial sets is called a Gröbner system.",
"Computing comprehensive Gröbner bases is at least as difficult as computing Gröbner bases.",
"Hence, testing binomiality via comprehensive Gröbner bases is a hard problem.",
"The concept of comprehensive Gröbner bases has been introduced by Weispfenning in his seminal work [50].",
"He later introduced canonical comprehensive Gröbner bases in [51].",
"A source of introduction to comprehensive Gröbner basis is Becker and Weispfenning's book [1].",
"Weispfenning also worked on the relation between comprehensive Gröbner bases and regular rings [52].",
"Later, several authors worked on the topic and introduced more efficient algorithms and polished the theory of comprehensive Gröbner bases.",
"Suzuki-Sati's approach to Gröbner bases is presented in [46].",
"Montes has worked extensively on comprehensive Gröbner bases, introduced several algorithms and developed the theory [37], [11].",
"In particular, Montes' book, the Gröbner Cover [35] is a great source for computations, among other interesting aspects, that can be used as a guide to the Singular library grobcov.lib [13] for computing comprehensive Gröbner bases.",
"Among the most efficient algorithms for computing comprehensive Gröbner bases are the algorithms given by Kapur et al.",
"[34], [33].",
"Dehghani and Hashemi studied Gröbner walk and FGLM for comprehensive Gröbner bases [10], [29] and implemented several algorithms for computing comprehensive Gröbner bases and related topics in Maple [29].https://amirhashemi.iut.ac.ir/sites/amirhashemi.iut.ac.ir/files//file_basepage/pggw_0.txt To the best of our knowledge, to this date, comprehensive Gröbner bases have not been used in chemical reaction networks theory.",
"Previous studies on binomiality of steady state ideals have considered Gröbner bases, linear algebra on stoichiometric matrices, etc., however, never have considered the change in the polynomials during computations when the values are assigned to the parameters.",
"For instance, it is known that detailed balancing holds under some polynomial conditions on the parameters.",
"However, the fact that specialisation of the rate constants may affect the computations has not beed considered.",
"The authors' previous work on toricity [27], [43] considers the toricity problem when the parameters have already been assigned real positive values.",
"Other articles of the authors have considered unconditional binomiality, that is, when the rate constants are considered variables [42], [41].",
"The present article is the original work that consideres specialisation of the parameters and uses comprehensive Gröbner bases to study the binomiality under specialisations.",
"The plan of the article is as follows.",
"Section gives an introduction to the necessary concepts of chemical reaction network theory, reviews the literature and presents the idea of the present article.",
"Section explains the preliminaries required on comprehensive Gröbner systems, explains the main concepts and sketches the idea behind computing comprehensive Gröbner bases.",
"Section includes the main computations, where we show our computations on $n-$ phosphorylations and biochemical reactions and present the benchmarks.",
"We furthermore compare our computations using comprehensive Gröbner bases with some earlier work on the binomiality problem that does not take into account the specialisation of the rate constants.",
"In Section we summarise our results and draw some conclusions."
],
[
"Preliminaries on Comprehensive Gröbner\nSystem",
"We review the required definitions, theorems and an algorithm on comprehensive Gröbner systems, mainly from the original work of Weispfenning [50] and Kapur, et al.",
"'s work [34].",
"Let $\\mathbb {K}$ be a field, $R =\\mathbb {K}[U] = \\mathbb {K}[u_1,\\dots ,u_m]$ be the ring of polynomials over $\\mathbb {K}$ in the indeterminates $u_1,\\dots ,u_m$ and let $S=\\mathbb {K}[U][X]=\\mathbb {K}[u_1,\\dots ,u_m][x_1,\\dots ,x_n]$ be the ring of polynomials over $\\mathbb {K}[U]$ with the indeterminantes $x_1,\\dots ,x_n$ .",
"Assume that $X \\cap U = \\emptyset $ .",
"We call $u_1,\\dots ,u_m$ the parameters of the ring $S$ and $x_1,\\dots ,x_n$ the variables of $S$ .",
"In fact, the coefficients of every polynomial in $S$ are themselves polynomials in parameters.",
"For every $\\alpha =(\\alpha _1,\\dots ,\\alpha _n) \\in \\mathbb {N}^n$ , by $X^\\alpha $ we denote $x_1^{\\alpha _1}\\dots x_n^{\\alpha _n}$ and by $U^\\alpha $ we denote $u_1^{\\alpha _1}\\dots u_n^{\\alpha _n}$ .",
"In this paper, $\\mathbb {K}$ is either $\\mathbb {R}$ or $\\mathbb {C}$ .",
"By the variety of an ideal $I$ (or a set of polynomials $F$ ), we mean the set of solutions of the ideal $I$ (or the set of polynomials $F$ ) and we denote it by $V(I)$ (or $V(F)$ ).",
"Let $<_1$ and $<_2$ be term orders on $\\mathbb {K}[U]$ and $\\mathbb {K}[X]$ , respectively.",
"We define a block order $<$ produced by the latter on $\\mathbb {K}[U][X]$ .",
"Firstly, define $u_i < x_j$ for all $1\\le i \\le m, 1 \\le j \\le n$ .",
"Secondly, define $X^{\\alpha _1}U^{\\beta _1} < X^{\\alpha _2}U^{\\beta _2}$ if either $X^{\\alpha _1} < X^{\\alpha _2}$ or $(X^{\\alpha _1} = X^{\\alpha _2} \\wedge U^{\\alpha _1} < U^{\\alpha _2})$ .",
"A polynomial of the form $c_\\alpha p(U)X^\\alpha $ , where $\\alpha \\in \\mathbb {N}^n$ , $c_\\alpha \\in \\mathbb {K}$ and $p(U) \\in R$ , is called a term in $\\mathbb {K}[U][X]$ .",
"A monomial is a term of the form $X^\\alpha $ .",
"Leading monomial, leading term and leading coefficient of the polynomials in $\\mathbb {K}[U][X]$ are defined with respect to the block ordering $<$ .",
"A specialisation of $S$ is a ring-homomorphism from the ring of parameters $R=\\mathbb {K}[U]$ into some field $\\mathbb {L}$ , i.e., $\\sigma :R \\rightarrow \\mathbb {L}$ .",
"Obviously $\\mathbb {K}$ is embedded in $\\mathbb {L}$ .",
"We consider $\\mathbb {L}$ to be an algebraically closed field in this paper.",
"Every specialisation is uniquely determined by its restriction to $\\mathbb {K}$ and its images on the parameters $u_1,\\dots ,u_m$ and vice versa.",
"A specialisation $\\sigma : R \\rightarrow \\mathbb {L}$ has a canonical extension to a ring-homomorphism $\\bar{\\sigma }:S \\rightarrow \\mathbb {L}[x_1,\\dots ,x_n]$ , i.e., for every $f=\\sum _{i \\in I}a_i(U)X^{\\alpha _i}, \\bar{\\sigma }(f)=\\sum _{i \\in I}\\sigma (a_i(U)) X^{\\alpha _i}$ , where $a_i(U)\\in R$ and $X^{\\alpha _i}$ is a monomial in $\\mathbb {K}[X]$ .",
"Following Weispfenning's notation, we denote $\\bar{\\sigma }$ by $\\sigma $ as well.",
"Specialisation of a set of polynomials $F$ by $\\sigma $ , denoted by $\\sigma (F)$ , is defined to be the set of specialisations of the polynomials in $F$ .",
"Accordingly, a specialisation of an ideal $I$ by $\\sigma $ is defined, and is denoted by $\\sigma (I)$ .",
"Following Kapur, et al.",
"[34], in this paper we only consider specialisations induced by the elements $a \\in \\mathbb {L}^m$ , that is, $\\sigma _a:f \\rightarrow f(a)$ , where $f \\in R$ .",
"Below we mention the definition of comprehensive Gröbner system and comprehensive Gröbner basis, which are due to Weispfenning.",
"We follow Kapur et al's notation in [34].",
"[Comprehensive Gröbner System] Let $I$ be an ideal in $S$ generated by a finite set $F\\subseteq S$ and $\\mathbb {L}$ be a an algebraically closed field containing $\\mathbb {K}$ .",
"Assume that $V_1, W_1,\\dots ,V_r,W_r$ are varieties in $\\mathbb {L}^n$ , and $G_1$ , $\\dots $ , $G_r$ are finite sets of polynomials in $S$ .",
"A set of tripiles $\\mathcal {G}=\\lbrace (V_1,W_1,G_1),\\dots , (V_r,W_r,G_r)\\rbrace $ is called a comprehensive Gröbner system of $I$ on $V = \\bigcup _{i=1^r} V_i \\setminus W_i$ , if for every $a \\in V$ and every specialisation $\\sigma _a$ of $S$ , $\\sigma _a(G_i)$ is a Gröbner basis of $\\sigma _a(I)$ in $\\mathbb {L}[X]$ when $a$ is in $V(V_i)\\setminus V(W_i)$ , for $i=1,\\dots ,r$ .",
"If $V=\\mathbb {L}^m$ , we simply call $\\mathcal {G}$ a comprehensive Gröbner system of $I$ .",
"Each $(V_i,W_i,G_i)$ is called a branch of $\\mathcal {G}$ .",
"A comprehensive Gröbner system $\\mathcal {G}$ of $I$ is called faithful, if every element of $G_i$ is in $I$ .",
"[Comprehensive Gröbner Basis] Let $I$ be an ideal in $S$ and $\\mathbb {L}$ be an algebraically closed field containing $\\mathbb {K}$ .",
"Assume that $V$ is a subset of $\\mathbb {L}^m$ .",
"A finite subset $G$ of $I$ is called a comprehensive Gröbner basis of $I$ on $V$ , if for all specialisations $\\sigma _a :R \\rightarrow \\mathbb {L}$ of $S$ , where $a \\in V$ , the set $\\sigma _a(G)$ is a Gröbner basis of the ideal generated by $\\sigma _a(I)$ in $\\mathbb {L}[X]$ .",
"If $V=\\mathbb {L}^m$ , we simply call $G$ a comprehensive Gröbner basis of $I$ .",
"A comprehensive Gröbner basis $G$ of $I$ is called faithful, if every element of $G$ is in $I$ .",
"Having defined comprehensive Gröbner bases, Weispfenning proved the existence of a comprehensive Gröbner basis for every ideal in $S$ [50].",
"In the latter reference, he gave a non-constructive proof first, and an algorithm later.",
"Following the first algorithm proposed by Weispfenning, algorithms for computing a comprehensive Gröbner basis essentially construct a faithful comprehensive Gröbner system $\\mathcal {G}=\\lbrace (V_1,W_1,G_1),\\dots , (V_r,W_r,G_r)\\rbrace $ .",
"Then the union $G=\\cup _{i=1}^r G_i$ will be a comprehensive Gröbner basis.",
"Roughly speaking, the varieties $V_i$ and $W_i$ are typically obtained by considering the monomials that are vanished by specialisations, and simultaneously, using a Gröbner basis computation algorithm, a Gröbner basis under the conditions imposed by the specialisations is computed.",
"Below we present a modified version of Kapur, et al.",
"'s algorithm by Dehghani and Hashemi from [29].",
"“Other cases” in line 16 of the algorithm refers to those cases that the Gröbner basis is 1.",
"Dehghani and Hashemi group all those cases together with the aim of speeding the computations up.",
"In line 13, $\\operatorname{MDBasis}$ computes a minimal Dickson basis for a given set of polynomials in $S$ .",
"For more details, refer to [29].",
"$\\operatorname{PGBMAIN}$ [1] 1.",
"$N, W \\subseteq \\mathbb {K}[U]$ finite; 2.",
"$F\\subseteq \\mathbb {K}[U][X]$ finite $PGB$ a Gröbner system of $F$ on $V(N)\\setminus V(W)$ $PGB:=\\emptyset $ $V(N)\\setminus V(W) = \\emptyset $ $\\emptyset $ $G:=\\operatorname{ReducedGroebnerBasis}(F \\cup N,<)$ $1 \\in G$ $\\lbrace (N,W,\\lbrace 1\\rbrace )\\rbrace $ $G_r:=G\\cap \\mathbb {K}[U]$ $V(G_r) \\setminus V(W) = \\emptyset $ $PGB$ $G_m:=\\operatorname{MDBasis}(G\\setminus G_r)$ $h=lcm(h_1,\\dots ,h_k)$ with $h_i=LC_{<_1}(g_i)$ for each $g_i \\in G_m$ $V(G_r) \\setminus V(W\\times \\lbrace h\\rbrace ) \\ne \\emptyset $ $PGB:=PGB \\cup \\lbrace G_r, W\\times \\lbrace h\\rbrace ,G_m\\rbrace $ $PGB \\cup \\bigcup _{h_i \\in \\lbrace h_1,\\dots ,h_k\\rbrace } \\operatorname{PGBMAIN}(G_r\\cup \\lbrace h_i\\rbrace , W \\times \\lbrace h_1h_2\\dots h_{i-1}\\rbrace , G\\setminus G_r)\\cup \\lbrace (\\text{Other Cases}, \\lbrace 1\\rbrace )\\rbrace $"
],
[
"Testing Binomiality of Chemical Reaction Networks Using\nComprehensive Gröbner Systems",
"In this section we present computations on biochemical networks, using comprehensive Gröbner bases, in order to test binomiality of the corresponding steady state ideals.",
"In [16], [9], [39], the authors call an ideal binomial if there exists a basis for the ideal whose polynomials have at most two terms.",
"In particular, as it is discussed in the latter references, one can see that an ideal is binomial if and only if its reduced Gröbner bases with respect to every term order is binomial.",
"Our definition of binomiality is as in [41], [42], which is slightly different from [16], [9], [39].",
"We call an ideal binomial if there exists a basis for the ideal whose polynomials have exactly two terms.",
"That is, we do not consider monomials in the basis.",
"Similar to the definition of binomiality in [16], [9], [39], one can easily observe that, for the case of our definition, an ideal is binomial if and only if its reduced Gröbner bases with respect to every term order is binomial.",
"In terms of parametric polynomial rings, i.e., $\\mathbb {K}[U][X]$ , we discuss the binomiality using a comprehensive Gröbner system.",
"That is in particular the case for the steady state ideals of chemical reaction networks.",
"As computing a comprehensive Gröbner basis is doen via computing the branches of a comprehensive Gröbner system, we basically compute the latter and check the binomiality of the Gröbner basis at each branch.",
"Then a comprehensive Gröbner basis of a steady state ideal will be binomial if and only if the Gröbner basis at each branch of a comprehensive system is binomial.",
"One can consider the generic comprehensive Gröbner bases, introduced in [50], however as it is mentioned in the latter reference, computing a generic comprehensive Gröbner basis is not feasible in practice.",
"In this paper, for our computations on the steady state ideals of the chemical reaction networks, we consider $\\mathbb {L}=\\bar{\\mathbb {K}}$ , the algebraic closure of $\\mathbb {K}$ .",
"In practice, for the computation purpose, the coefficient field is considered to be $\\mathbb {Q}$ , extended by the parameters, i.e., $\\mathbb {Q}(k_1,\\dots ,k_m)$ ; hence the comprehensive Gröbner system computations are carried out over $\\mathbb {Q}(k_1,\\dots ,k_m)[x_1,\\dots ,x_n]$ .",
"Our computations are carried out via version 4.2.0 of the computer algebra system Singular [13]http://www.singular.uni-kl.de, the grobcov package (whose latest version is available at A. Montes' website)https://mat.upc.edu/en/people/antonio.montes.",
"For instructions on the grobcov package we refer the reader to the book [35] and examples by A. Montes.",
"We have done fully automated computations on sets of examples, in particular on biochemical models from the BioModels' repository [7] https://www.ebi.ac.uk/biomodels.",
"Our computations have been done on a 2.48 MHz AMD EPYC 7702 64-Core Processor in a Debian GNU/Linux 10 machine with 211 gB memory."
],
[
"$n$ -Site Phosphorylation",
"Multisite phosphorylation–dephosphorylation cycles or $n$ -site phosphorylations (for $n \\in \\mathbb {N}$ ) are studied by Wang and Sontag in [47] in terms of multi-stationarity.",
"Pérez-Milán et al.",
"in [40] have shown that for every $n \\in \\mathbb {N}$ , $n$ -site phosphorylation has a binomial steady state.",
"As mentioned earlier, in the latter reference, the authors did not take into account the specialisations of the constant rates.",
"In this subsection, we first do some reductions on a basis of the steady state ideal of $n-$ phosphorylations and prove its binomiality.",
"This essentially gives us the unconditional binomiality of $n-$ phosphorylation, defined and investigated in [42], [41].",
"Our algebraic maniplations below are simple and avoid the criterion presented by Pérez-Milán et al.",
"in [40].",
"Using Wang and Sontag's notation in [47] for the variables and parameters, for a fixed positive integer $n$ , the $n$ -site phosphorylation reaction network is the following.",
"${S_0 + E <=>[k_{\\text{on}_0}][k_{\\text{off}_0}] ES_0->[k_{\\text{cat}_0}] S_1 + E} \\\\\\vdots \\\\{S_{n-1} + E <=>[k_{\\text{on}_{n-1}}][k_{\\text{off}_{n-1}}] ES_{n-1}->[k_{\\text{cat}_{n-1}}] S_n + E} \\\\{S_1 + F <=>[l_{\\text{on}_0}][l_{\\text{off}_0}] FS_1->[l_{\\text{cat}_0}] S_0 + F} \\\\\\vdots \\\\{S_n + F <=>[l_{\\text{on}_{n-1}}][l_{\\text{off}_{n-1}}] FS_n->[l_{\\text{cat}_{n-1}}] S_{n-1} + F}$ The parameters of the reaction network are $k_{\\text{on}_0},\\dots , k_{\\text{on}_{n-1}}$ , $k_{\\text{off}_0},\\dots ,k_{\\text{off}_{n-1}}$ , $k_{\\text{cat}_0},\\dots , k_{\\text{cat}_{n-1}}$ , $l_{\\text{on}_0},\\dots , l_{\\text{on}_{n-1}}$ , $l_{\\text{off}_0},\\dots ,l_{\\text{off}_{n-1}}$ , $l_{\\text{cat}_0},\\dots , l_{\\text{cat}_{n-1}}$ .",
"Let the variables $s_0,\\dots ,s_n, c_0,\\dots , c_{n-1}, d_1,\\dots , d_n, e, f$ represent the concentrations of the species ${S_0},\\dots , {S_n}, {ES_0},\\dots , {ES_{n-1}},{FS_1},\\dots , {FS_n}, {E}, {F}$ respectively.",
"The ODEs describing change of the concentrations of the species for this reaction network are the following: $\\dot{s_0} = P_0 = & -k_{\\text{on}_0} s_0e + k_{\\text{off}_0} c_0 +l_{\\text{cat}_0} d_1,\\\\\\dot{s_i} = P_i = & -k_{\\text{on}_i} s_i e + k_{\\text{off}_i} c_i +k_{\\text{cat}_{i-1}}c_{i-1} -l_{\\text{on}_{i-1}} s_i f +l_{\\text{off}_{i-1}} d_i + l_{\\text{cat}_i} d_{i+1}, \\\\& i=1,\\dots ,n-1,\\\\\\dot{c_j} = Q_j = & k_{\\text{on}_j} s_j e -(k_{\\text{off}_j}+k_{\\text{cat}_j}) c_j, \\quad j=0,...,n-1,\\\\\\dot{d_k} = R_k= & l_{\\text{on}_{k-1}} s_kf-(l_{\\text{off}_{k-1}}+l_{\\text{cat}_{k-1}}) d_k, \\quad k=1,...,n.$ The ODEs for $s_n, e$ and $f$ are linear combinations of the above ODEs, hence they are redundant and we skip them in this article.",
"In order to show unconditional binomiality of the steady state ideal of $n$ -phosphorylation, we perform reductions on the the generators of the steady state ideal so that a binomial basis is obtained.",
"First of all, note that polynomials $Q_j$ and $R_k$ are already binomial.",
"Reducing $P_0$ with respect to $Q_0$ , we obtain $P_0^{\\prime }= & P_0+Q_0 \\\\= & -k_{\\text{on}_0} s_0e + k_{\\text{off}_0} c_0 +l_{\\text{cat}_0} d_1 \\\\& + k_{\\text{on}_0} s_0 e -(k_{\\text{off}_0}+k_{\\text{cat}_0}) c_0\\\\= & l_{\\text{cat}_0} d_1 + k_{\\text{cat}_0} c_0,$ which is a binomial.",
"Now we reduce $P_i$ with respect to $P_0^{\\prime }$ , $Q_j$ and $R_k$ as follows.",
"First we reduce $P_i$ with respect to $R_I$ $P_i+R_i =& \\\\& -k_{\\text{on}_i} s_i e + k_{\\text{off}_i} c_i + k_{\\text{cat}_{i-1}}c_{i-1} -l_{\\text{on}_{i-1}} s_i f + l_{\\text{off}_{i-1}} d_i + l_{\\text{cat}_i}d_{i+1}\\\\&+ l_{\\text{on}_{i-1}} s_if-(l_{\\text{off}_{i-1}}+l_{\\text{cat}_{i-1}}) d_i\\\\=& -k_{\\text{on}_i} s_i e + k_{\\text{off}_i} c_i + k_{\\text{cat}_{i-1}}c_{i-1} + l_{\\text{cat}_i}d_{i+1} -l_{\\text{cat}_{i-1}} d_i.$ Then we reduce the result with respect to $Q_i$ $P_i+R_i+Q_i =& \\\\& -k_{\\text{on}_i} s_i e + k_{\\text{off}_i} c_i + k_{\\text{cat}_{i-1}}c_{i-1} + l_{\\text{cat}_i}d_{i+1}-l_{\\text{cat}_{i-1}} d_i\\\\& + k_{\\text{on}_i} s_i e - (k_{\\text{off}_i}+k_{\\text{cat}_i}) c_i \\\\=& k_{\\text{cat}_{i-1}}c_{i-1} + l_{\\text{cat}_i}d_{i+1}-l_{\\text{cat}_{i-1}} d_i +k_{\\text{cat}_i} c_i.$ For $i=1$ , the above can be reduced with respect to $P_0^{\\prime }$ $P_1^{\\prime } = P_1+R_1+Q_1-P_0^{\\prime }= &k_{\\text{cat}_{0}} c_{0} + l_{\\text{cat}_1}d_{1}-l_{\\text{cat}_{0}} d_1 +k_{\\text{cat}_1} c_1 \\\\& -\\left( l_{\\text{cat}_0} d_1 + k_{\\text{cat}_0} c_0 \\right)\\\\=& l_{\\text{cat}_1}d_{1} + k_{\\text{cat}_1} c_1,$ which is a binomial.",
"Similarly, for $i=2,\\dots ,n$ , $P_i$ can be reduced to a binomial with respect to $R_i$ , $Q_I$ and $P_{i-1}^{\\prime }$ .",
"Therefore, a binomial basis can be obtained this way for the steady state ideal.",
"As the algebraic manipulations above do not consider into account the specialisations of the parameters, we computed comprehensive Gröbner system of the steady state ideals for the cases $n=1,2$ to test the binomiality under specialisations.",
"1-site phosphorylation and 2-site phosphorylations have been studied in [40] using the criteria presented in that article as well.",
"Example 1 (1-site phosphorylation, [40], Example 2.1) ${S_0 + E <=>[k_{\\text{on}_0}][k_{\\text{off}_0}] ES_0->[k_{\\text{cat}_0}] S_1 + E} \\\\{S_1 + F <=>[l_{\\text{on}_0}][l_{\\text{off}_0}] FS_1->[l_{\\text{cat}_0}] S_0 + F}.$ Let the variables representing the change of the concentrations of the species ${S_0}$ , ${S_1}$ , ${ES_0}$ , ${FS_1}$ , ${E}$ , ${F}$ be $s_0$ , $s_1$ , $c_0$ , $d_1$ , $e$ , $f$ respectively, and let the parameters be $k_{\\text{on}_0}, k_{\\text{off}_0}, k_{\\text{cat}_0},l_{\\text{on}_0}, l_{\\text{off}_0}, l_{\\text{cat}_0}$ .",
"The steady state ideal for 1-site phosphorylation reaction is generated by $\\dot{s_0} = & -k_{\\text{on}_0} s_0 e + k_{\\text{off}_0} c_0 +l_{\\text{cat}_0} d_1, \\\\\\dot{s_1}= & -k_{\\text{on}_1} s_1 e + k_{\\text{off}_1} c_1 +k_{\\text{cat}_0} c_0 - l_{\\text{on}_0}s_1f +l_{\\text{off}_0}d_1,\\\\\\dot{c_0}=& k_{\\text{on}_0} s_0 e -( k_{\\text{off}_0} +k_{\\text{cat}_0}) c_0,\\\\\\dot{d_1}=& l_{\\text{on}_0} s_1 f -( l_{\\text{off}_0} +l_{\\text{cat}_0}) d_1.$ We skip the ODEs for $e$ and $f$ as they are linear combination of the other ODEs.",
"Renaming the variables as $ s_0 = x_1,s_1 = x_2,c_0 = x_3,d_1 = x_4,e = x_5,f = x_6, $ we computed the comprehensive Gröbner system for the steady state ideal using Singular.",
"It contains 25 branches, out of which 6 are binomial.",
"We recall that in this article, a binomial ideal is an ideal that is generated by a set of binomials (not including monomials).",
"For the last branch, $V_{25}$ and $W_{25}$ are the zero sets of the following sets of polynomials in $\\mathbb {Q}[k_{\\text{on}_0}, k_{\\text{off}_0}, k_{\\text{cat}_0},l_{\\text{on}_0}, l_{\\text{off}_0},l_{\\text{cat}_0}][x_1,\\dots ,x_6]$ respectively: $& \\lbrace l_{\\text{cat}_0}, k_{\\text{on}_0}\\rbrace ,\\\\& \\lbrace k_{\\text{off}_0} k_{\\text{cat}_0}l_{\\text{on}_0}+k_{\\text{cat}_0}^2l_{\\text{on}_0} \\rbrace .$ The corresponding Gröbner basis is $\\lbrace f_1=& k_{\\text{cat}_0} x_3, \\\\f_2=&l_{\\text{on}_0} x_2x_6-l_{\\text{off}_0} x_4 \\rbrace ,$ which obviously is not binomial.",
"An example of a branch with binomial Gröbner basis is branch 24, for which $V_{24}$ and $W_{24}$ are the zero sets of the following sets, respectively: $& \\lbrace k_{\\text{off}_0}+k_{\\text{cat}_0}, k_{\\text{on}_0}\\rbrace ,\\\\& \\lbrace l_{\\text{on}_0} k_{\\text{cat}_0} \\rbrace .$ The corresponding Gröbner basis is $\\lbrace f_1=& k_{\\text{cat}_0} x_3 +l_{\\text{cat}_0} x_4, \\\\f_2=&l_{\\text{on}_0} x_2x_6+(-l_{\\text{off}_0}-l_{\\text{cat}_0}) x_4 \\rbrace .$ Example 2 (2-site phosphorylation, [40], Example 3.13) The steady state ideal for the 2-site phosphorylation reaction is generated by $\\dot{s_0} = P_0 = & -k_{\\text{on}_0} s_0e + k_{\\text{off}_0}c_0 + l_{\\text{cat}_0} d_1,\\\\\\dot{s_1} = P_1 = & -k_{\\text{on}_1} s_1 e + k_{\\text{off}_1} c_1+ k_{\\text{cat}_{0}}c_{0} -l_{\\text{on}_{0}} s_1 f +l_{\\text{off}_{0}} d_1 + l_{\\text{cat}_1} d_{2}, \\\\\\dot{c_0} = Q_0 = & k_{\\text{on}_0} s_0 e -(k_{\\text{off}_0}+k_{\\text{cat}_0}) c_0,\\\\\\dot{c_0} = Q_1 = & k_{\\text{on}_1} s_1 e -(k_{\\text{off}_1}+k_{\\text{cat}_1}) c_1,\\\\\\dot{d_1} = R_1= & l_{\\text{on}_{0}}s_1f-(l_{\\text{off}_{0}}+l_{\\text{cat}_{0}}) d_1, \\\\\\dot{d_2} = R_2= & l_{\\text{on}_{1}}s_2f-(l_{\\text{off}_{1}}+l_{\\text{cat}_{1}} ) d_2 ,$ where the variables are $ s_0, s_1, s_2, c_0, c_1, d_1, d_2, e, f $ and the parameters are $k_{\\text{on}_0},k_{\\text{on}_1},k_{\\text{off}_0},k_{\\text{off}_1},k_{\\text{cat}_0},k_{\\text{cat}_1},l_{\\text{on}_0},l_{\\text{on}_1},l_{\\text{off}_0},l_{\\text{off}_1},l_{\\text{cat}_0},l_{\\text{cat}_1}.$ We have computed a comprehensive Gröbner system for this system using Singular.",
"It has 1187 branches, out of which 36 are binomial.",
"The last branch of the comprehensive Gröbner system is as follows.",
"$V_{1187}$ is the zero set of $l_{\\text{off}_1}+l_{\\text{cat}_1}$ and $W_{1187}$ is the zero set of the following polynomial: $&k_{\\text{on}_0}k_{\\text{on}_1}k_{\\text{off}_0}k_{\\text{off}_1}k_{\\text{cat}_0}k_{\\text{cat}_1}l_{\\text{on}_0}l_{\\text{on}_1}l_{\\text{off}_0}l_{\\text{cat}_0}l_{\\text{cat}_1} \\\\& +k_{\\text{on}_0}k_{\\text{on}_1}k_{\\text{off}_0}k_{\\text{off}_1}k_{\\text{cat}_0}k_{\\text{cat}_1}l_{\\text{on}_0}l_{\\text{on}_1}l_{\\text{cat}_0}^2l_{\\text{cat}_1} \\\\&+k_{\\text{on}_0}k_{\\text{on}_1}k_{\\text{off}_0}k_{\\text{cat}_0}k_{\\text{cat}_1}^2l_{\\text{on}_0}l_{\\text{on}_1}l_{\\text{off}_0}l_{\\text{cat}_0}l_{\\text{cat}_1} \\\\&+k_{\\text{on}_0}k_{\\text{on}_1}k_{\\text{off}_0}k_{\\text{cat}_0}k_{\\text{cat}_1}^2l_{\\text{on}_0}l_{\\text{on}_1}l_{\\text{cat}_0}^2l_{\\text{cat}_1} \\\\&+k_{\\text{on}_0}k_{\\text{on}_1}k_{\\text{off}_1}k_{\\text{cat}_0}^2k_{\\text{cat}_1}l_{\\text{on}_0}l_{\\text{on}_1}l_{\\text{off}_0}l_{\\text{cat}_0}l_{\\text{cat}_1}\\\\&+k_{\\text{on}_0}k_{\\text{on}_1}k_{\\text{off}_1}k_{\\text{cat}_0}^2k_{\\text{cat}_1}l_{\\text{on}_0}l_{\\text{on}_1}l_{\\text{cat}_0}^2l_{\\text{cat}_1} \\\\&+k_{\\text{on}_0}k_{\\text{on}_1}k_{\\text{cat}_0}^2k_{\\text{cat}_1}^2l_{\\text{on}_0}l_{\\text{on}_1}l_{\\text{off}_0}l_{\\text{cat}_0}l_{\\text{cat}_1}\\\\&+k_{\\text{on}_0}k_{\\text{on}_1}k_{\\text{cat}_0}^2k_{\\text{cat}_1}^2l_{\\text{on}_0}l_{\\text{on}_1}l_{\\text{cat}_0}^2l_{\\text{cat}_1}.$ Renaming the variables as $ s_0 = x_1,s_1 = x_2,s_2 = x_3,c_0 = x_4,c_1 = x_5,d_1 = x_6,d_2= x_7,e = x_8,f = x_9,$ the Gröbner basis for every specialisation of the parameters in $V_{1187}\\setminus W_{1187}$ is the following.",
"$f_1=&k_{\\text{cat}_1}x_5-l_{\\text{cat}_1}x_7, \\\\f_2= &k_{\\text{cat}_0}x_4-l_{\\text{cat}_0}x_6,\\\\f_3= &l_{\\text{on}_1}x_3x_9,\\\\f_4= &l_{\\text{on}_0}x_2x_9+(-l_{\\text{off}_0}-l_{\\text{cat}_0})x_6,\\\\f_5= &(k_{\\text{on}_1}l_{\\text{off}_0}+k_{\\text{on}_1}l_{\\text{cat}_0})x_6x_8+(-k_{\\text{off}_1}l_{\\text{on}_0})x_5x_9+(-l_{\\text{on}_0}l_{\\text{cat}_1})x_7x_9,\\\\f_6=&(k_{\\text{on}_1})x_2x_8+(-k_{\\text{off}_1})x_5+(-l_{\\text{cat}_1})x_7,\\\\f_7= &(k_{\\text{on}_0})x_1x_8+(-k_{\\text{off}_0})x_4+(-l_{\\text{cat}_0})x_6,\\\\f_8= &(l_{\\text{on}_1}l_{\\text{off}_0}+l_{\\text{on}_1}l_{\\text{cat}_0})x_3x_6,\\\\f_9=&(k_{\\text{on}_1}k_{\\text{off}_0}k_{\\text{cat}_1}l_{\\text{cat}_0}+k_{\\text{on}_1}k_{\\text{cat}_0}k_{\\text{cat}_1}l_{\\text{cat}_0})x_2x_6\\\\&+(-k_{\\text{on}_0}k_{\\text{off}_1}k_{\\text{cat}_0}l_{\\text{cat}_1}-k_{\\text{on}_0}k_{\\text{cat}_0}k_{\\text{cat}_1}l_{\\text{cat}_1})x_1x_7,\\\\f_{10}=&(k_{\\text{on}_0}k_{\\text{off}_1}l_{\\text{on}_0}l_{\\text{cat}_1}+k_{\\text{on}_0}k_{\\text{cat}_1}l_{\\text{on}_0}l_{\\text{cat}_1})x_1x_7x_9\\\\&+(-k_{\\text{on}_1}k_{\\text{off}_0}k_{\\text{cat}_1}l_{\\text{off}_0}-k_{\\text{on}_1}k_{\\text{off}_0}k_{\\text{cat}_1}l_{\\text{cat}_0})x_4x_6 \\\\& +(-k_{\\text{on}_1}k_{\\text{cat}_1}l_{\\text{off}_0}l_{\\text{cat}_0}-k_{\\text{on}_1}k_{\\text{cat}_1}l_{\\text{cat}_0}^2)x_6^2,\\\\f_{11}=&(k_{\\text{on}_0}k_{\\text{off}_1}k_{\\text{cat}_0}l_{\\text{on}_1}l_{\\text{off}_0}l_{\\text{cat}_1}+k_{\\text{on}_0}k_{\\text{off}_1}k_{\\text{cat}_0}l_{\\text{on}_1}l_{\\text{cat}_0}l_{\\text{cat}_1} \\\\& +k_{\\text{on}_0}k_{\\text{cat}_0}k_{\\text{cat}_1}l_{\\text{on}_1}l_{\\text{off}_0}l_{\\text{cat}_1}+k_{\\text{on}_0}k_{\\text{cat}_0}k_{\\text{cat}_1}l_{\\text{on}_1}l_{\\text{cat}_0}l_{\\text{cat}_1})x_1x_3x_7.$ One can observe that the above branch of the comprehensive Gröbner system is not binomial.",
"We carried on the computations for comprehensive Gröbner system of the steady state ideal of $n-$ phosphorylation for $n=2, 3, 4, 5$ in Singular with the time limit of six hours.",
"The results of the computations are summarised in Table REF .",
"In this table, DNF refers to did not finish.",
"0pt 0pt @|c|c|c|c|c| Comprehensive Gröbner System of n-Phosphorylations $\\#$ branches $\\#$ binomial branches $\\%$ of binomial branches time(s) $2-$ phosph.",
"1187 36 3.03 24 $3-$ phosph.",
"57857 216 0.37 2231 $4-$ phosph.",
"- - - DNF $5-$ phosph.",
"- - - DNF As the number of variables and parameters grow drastically when $n$ increases, comprehensive Gröbner system computations did not finish in a reasonable time period for $n\\ge 4$ .",
"We also computed a comprehensige Groöbner system of 2-phosphorylation in Maple, using Dehghani and Hashemi's PWWG packagehttps://amirhashemi.iut.ac.ir/sites/amirhashemi.iut.ac.ir/files//file_basepage/pggw_0.txt, which uses a modification of Kapur et al.",
"'s algorithm so that the branches with Gröbner basis $\\lbrace 1\\rbrace $ are ignored [29].",
"According to the authors' experiments in [29], this modification results in speed-up of the computations.",
"However, even for 2-phosphorylation the computations did not finish in six hours in Maple.",
"As we see from the computations in this subsection, there are several branches of teh $n-$ phosphorylations that are not binomial.",
"This means that for certain values of the rate constants, $n-$ phosphorylation is not binomial, while the computations without taking into account the specialisations of the rate constants leads to the binomiality."
],
[
"BioModels",
"Our main benchmark for computing comprehensive Gröbner system of steady state ideals, are the biochemical models from the BioModels repository [7], which is typically used for such computations.",
"As a first example, we present biomodel 629 and the corresponding computations in the following example.",
"Example 3 (BIOMD0000000629, [7]) The corresponding ODEs for biomodel 629 are the following; $\\dot{x_1}=& -k_2x_1x_3 + k_3x_2, \\\\\\dot{x_2}=&k_2x_1x_3 - k_3x_2 - k_4x_2x_4 + k_5x_5, \\\\\\dot{x_3}=& -k_2x_1x_3 + k_3x_2, \\\\\\dot{x_4}=& -k_4x_2x_4 + k_5x_5,\\\\\\dot{x_5}=&k_4x_2x_4 - k_5x_5,$ where $k_1,\\dots ,k_5$ are the parameters and $x_1,\\dots ,x_5$ are the variables.",
"Comprehensive Göbner system computation over the ring $\\mathbb {Q}[k_1,\\dots ,k_5][x_1,\\dots ,x_5]$ in Singular results in 10 branches with the following conditions and Gröbner bases.",
"0pt 0pt @|c|c|c|c| Comprehensive Gröbner System of BIOMD0000000629 branch $V$ $W$ GB 1 0 $k_2k_4$ $k_4x_2x_4-k_5x_5, k_2x_1x_3-k_3x_2$ 2 $k_4$ $k_2k_5$ $k_5x_5, k_2x_1x_3-k_3x_2$ 3 $k_5,k_2$ $k_2$ $k_2x_1x_3-k3x_2$ 4 $k_5,k_4,k_2$ $k_3$ $k_3x_2$ 5 $k_5,k_4,k_3,k_2$ 1 0 6 $k_4,k_2$ $k_3k_5$ $k_5x_5,k_3x_2$ 7 $k_4,k_3,k_2$ $k_5$ $k_5x_5$ 8 $k_2$ $k_5,k_4,k_3$ $k_3k_5x_5, k_3x_2$ 9 $k_3, k_2$ $k_4$ $k_4x_2x_4-k_5x_5$ 10 $k_5,k_2$ $k_3k_4$ $k_3x_2$ There are three branches with binomial Gröbner basis for biomodel 629.",
"All the branches have either monomial or binomial Gröbner basis.",
"In Table REF , we present the results of our computations for some biomodels from the Biomodels repository [7].",
"As computing comprehensive Gröbner system of systems with large number of variables is very expensive, we have considered those biomodels that have relatively small number of species (correspondingly, relatively small number of variables), so that the computations took less than ten minutes for those biomodels.",
"In Table REF , one can find the number of branches of the corresponding comprehensive Gröbner systems, the number of branches that are binomial, and their percentage.",
"Except for biomodels 271 and 519 that have no binomial branch, all other biomodels have at least one binomial branch.",
"For two biomodels (283 and 486), at least half of their branches are binomial.",
"The largest biomodel we have considered is the model 26.",
"We note that this model is a MAPK reaction network.",
"It has been studied in [17], where the authors associated a graph to the CRN and used a trick based on vertex cover in order to reduce the number of the polynomials in the steady state ideal into 2 polynomials.",
"0pt 0pt @|c|c|c|c| Branches of Comprehensive Gröbner Systems of Biomodels model $\\#$ branches $\\#$ binomial branches $\\%$ of binomial branches 26 46870 164 0.35 40 35 6 17.00 92 10 4 40.00 101 81 11 13.40 104 4 1 25.00 156 25 5 20.00 159 36 6 16.66 178 24 2 8.33 194 19 5 26.31 233 18 5 27.78 267 12 2 16.67 271 92 0 0.00 272 44 7 15.91 282 18 4 22.22 283 2 1 50.00 289 351 43 12.25 321 26 5 19.23 363 15 2 13.33 459 40 9 22.50 486 3 2 66.67 519 128 0 0.00 546 15 1 6.67 629 10 4 40.00"
],
[
"Conclusion",
"We address the problem of binomiality of the steady state ideal of a chemical reaction network.",
"The binomiality problem has been widely considered in the literature of mathematics and chemical reaction network theory and is still an active research area.",
"Finding binomiality and toricity is a hard problem from both a theoretical and a practical point of view.",
"The computational methods typically rely on Gröbner bases..",
"The authors have recently investigated binomiality and toricity in several papers.",
"We have given efficient algorithms for testing toricity in [27].",
"We also have considered the binomiality from a first-order logic point of view and gave efficient computational results and studied biomodels systematically via quantifier elimination [43], [27].",
"Other than those, we have considered the concept of unconditional binomiality, which considers rate constants as variables, and gave polynomial time linear algebra and graph theoretical approaches for detecting binomiality [41], [42].",
"The existing work on binomiality of steady state ideals do not take into account the effect of assigning values to the rate constats during the computations.",
"In the present work, we consider the problem of binomiality when the parameters can be specialised.",
"Our approach to this parametric binomiality problem is naturally based on comprehensive Gröbner bases.",
"We make systematic computations on $n-$ phosphorylations and biomodels and detect the branches of the Gröbner systems that are binomial.",
"Our computations via comprehensive Gröbner systems show that in several cases, the comprehensive Gröbner bases for steady state ideals are not binomial, while using other methods, e.g., considering rate constants as variables or doing computations without considering the effect of specialisation, one may consider those steady state ideal as binomial ideals.",
"As in this paper the concept of comprehensive Gröbner bases is used for the first time on chemical reaction network theory, we propose using this approach for studying further properties of chemical reaction networks."
],
[
"Acknowledgments",
"This work has been supported by the interdisciplinary bilateral project ANR-17-CE40-0036/DFG-391322026 SYMBIONT [3], [4].",
"We would like to thank A. Hashemi and M. Dehghani for the discussions on comprehensive Gröbner bases and providing us with their Maple package."
]
] | 2107.01706 |
[
[
"Temporally and Spatially Extended Star Formation in the Brightest\n Cluster Galaxy of MACS\\,J0329.7$-$0211 at $z=0.45$: Implications for Stellar\n Growth"
],
[
"Abstract Brightest cluster galaxies (BCGs), particularly those at the centers of cool-core clusters, can exhibit star formation over spatial extents of up to $\\gtrsim$100\\,kpc at inferred rates of up to $\\gtrsim100\\rm\\,M_\\sun\\,yr^{-1}$.",
"Is their star formation also extended over time, as might be expected if fuelled by cooling of the surrounding hot intracluster gas -- a residual cooling flow -- as demonstrated hitherto only for the BCG in the Perseus cluster?",
"Here, to infer the formation history of relatively young stars in the BCG of MACS\\,J0329.7$-$0211, we fit model single-stellar-populations to the spectral energy distributions (spanning near-UV to near-IR) measured along different sightlines towards its young stellar population.",
"Employing a Markov Chain Monte Carlo method, we show that star formation in this BCG has persisted at a relatively constant rate of $\\sim2{\\rm\\,M_\\sun\\,yr^{-1}}$ (factors of 10--40 below the rates previously inferred using simpler methods and/or ad hoc assumptions) over the past $\\sim$400\\,Myr, beyond which any star formation falls below the observational detection threshold.",
"Such persistent star formation from a residual cooling flow can contribute up to $\\sim$10\\% of the original stellar mass of this BCG if its progenitor was among the most massive red nuggets known at $z\\sim$2 having masses of $\\sim1\\times10^{11}\\rm\\,M_\\sun$, but only a few percent of its overall growth in stellar mass to $\\sim8\\times10^{11}\\rm\\,M_\\sun$ at $z=0.45$.",
"Although constituting only a minor pathway for the stellar growth of this BCG, persistent star formation from a residual cooling flow can nevertheless contribute significantly to the enormous number of globular clusters found around BCGs in the local Universe."
],
[
"Introduction",
"Elliptical galaxies are generally red (dominated by an old stellar population that formed in the early Universe) and dead (no detectable ongoing or recent star formation) – apart from a small minority that, in most cases, have been temporarily revived by the accretion of cool gas through interactions or mergers with other galaxies thus fueling a brief episode of star formation.",
"One might therefore expect giant elliptical galaxies at the centers of massive galaxy clusters – brightest cluster galaxies, henceforth BCGs – to be especially lifeless, as ram-pressure stripping by the hot intracluster medium removes much of the gas in cluster member galaxies that move at high speeds on predominantly radial orbits [33] by the time dynamical friction slows these galaxies down sufficiently to be captured by the BCG.",
"Indeed, it has been argued since [72] and [51] that BCGs grew over time in stellar mass and, especially, stellar size primarily through dry mergers with cluster member galaxies (i.e., those lacking cool gas).",
"Such mergers are especially effective in growing galaxies in stellar size, as stars from the more strongly disrupted companion (or, if they have comparable masses and sizes, both disrupted galaxies) are deposited primarily at the outskirts of the BCG (or merger remnant).",
"In wet mergers, by comparison, cool gas accreted from the disrupted companion sinks dissipatively into the center of the BCG (or merger remnant) to fuel star formation; although this cool gas is eventually dispersed by stellar winds, radiation pressure, and supernova explosions associated with the newly-formed stars, along possibly by actions associated with an Active Galactic Nucleus (AGN) fueled by the cool gas, the contraction of the BCG (or merger remnant) in response to its central change in gravitational potential (owing to the newly-formed stars) partially mitigates its overall increase in stellar size owing to the accretion of stars from its disrupted companion.",
"Further underpinning this picture, BCGs have since been postulated to descend from red nuggets, the oldest and among the most massive galaxies discovered (by [14] and [69]) at high redshifts ($z \\sim 2$ ).",
"These galaxies have large stellar masses of $\\sim $$10^{10}$ –$10^{11} {\\rm \\, M_}$ , but remarkably small sizes with effective radii of just $\\lesssim 1$ kpc.",
"They must therefore grow perhaps just modestly in mass (by a factor of a few) but spectacularly in size (by about an order of magnitude) to resemble present-day BCGs, which in the local Universe have stellar masses spanning $\\sim $$10^{11}$ –$10^{12} {\\rm \\, M_}$ (typically several $10^{11} \\, M_{}$ ) and effective radii $\\gtrsim 5$ kpc.",
"The manner by which this growth actually occurs continues to pose an especially big challenge to our understanding of galaxy assembly, and constitutes a key test of galaxy formation models.",
"In contrast to the expectations laid out above, an increasing number of BCGs at relatively low redshifts – mostly $z \\lesssim 0.1$ where searches have been concentrated [21], [61], but recently up to $z \\sim 0.6$ ([10], and references therein) and now even up to $z \\sim 1.2$ [19] – have been found to contain large quantities of molecular gas as traced in CO, the reservoir for star formation.",
"The inferred masses span $\\sim $$10^9$ –$10^{11} {\\rm \\, M_}$ ; to place such large gas masses into context, the upper end of this range in gas mass closely approaches the typical present-day stellar masses of BCGs.",
"In addition, by contrast with theoretical models [16], [37] that invoke very little if any star formation following an initial gas-rich dissipational collapse onto massive dark matter halos fuelling star formation at $z \\gtrsim 2$ [50] thereby giving rise (presumably) to red nuggets, a significant fraction of BCGs at $z \\lesssim 1$ have been found to exhibit ongoing or recent star formation with inferred rates as high as $\\gtrsim 100 {\\rm \\, M_\\ yr^{-1}}$ (e.g., [18], [47]).",
"Appreciable star formation in BCGs is far from exceptional: between one-third and one-half of BCGs in massive X-ray-selected clusters at $z \\lesssim 0.1$ exhibit luminous optical emission-line nebulae, for which their line strengths in H$\\alpha $ are correlated (albeit with substantial scatter) with UV emission from newly-formed stars ([17], and references therein).",
"Even if star formation at rates of $\\sim $$100 {\\rm \\, M_\\ yr^{-1}}$ persist for just 100 Myr (or lower rates over commensurately longer time intervals) to consume $\\sim $$10^{10} {\\rm \\, M_}$ of molecular gas, the newly-formed stars would add $\\sim $ 10%–100% to the original stellar masses of red nuggets.",
"As a fiducial comparison, theoretical models [37] postulate, and observations ([39]; see also discussion and references in [40]) find, that the stellar masses of BCGs have nearly doubled (i.e., increased by nearly 100%) since $z \\sim 1$ (over the past $\\sim $ 8 Gyr).",
"Does in-situ star formation since $z \\lesssim 2$ therefore play a significant role in the stellar growth of BCGs since they were red nuggets?",
"Can such star formation contribute to, or at least not too severely mitigate by deepening the central gravitational potential of the burgeoning galaxy, the overall growth in their stellar sizes so as to become present-day BCGs?",
"To address these questions, we need to better understand the star-formation history of BCGs – both temporally and spatially – since $z \\lesssim 2$ .",
"A first step towards addressing the nature of star formation in BCGs is to address the origin of their molecular gas, the reservoir for star formation.",
"A strong relationship has long been established between BCGs that contain cool gas as traced in optical emission lines and the physical properties of their surrounding hot intracluster gas that emits in X-rays.",
"At low to moderate redshifts ($z \\lesssim 0.5$ ), such BCGs are found exclusively in cool-core clustersA designation introduced by [49] for galaxy clusters that exhibit a temperature decrement in their intracluster X-ray emitting gas at the cluster core.",
"The X-ray gas permeating such clusters exhibits a surface brightness that is strongly centrally peaked, indicating prodigious radiative loss at the cluster core.",
"The resulting loss in pressure support was predicted to result in an inflow of cooling intracluster gas, referred to as an X-ray cooling flow [13], [26].",
"[49] found, as demonstrated more robustly by [15] and [52], that the mass-deposition rate from any such flow is much lower than had previously been inferred.",
"Today, it is widely recognized that X-ray cooling flows are strongly quenched by AGNs in BCGs [25].",
"At low redshifts, cool-core clusters constitute the majority ($\\gtrsim 70\\%$ ) of X-ray selected samples of galaxy clusters (e.g., [23]; [22], [32]), and $\\sim $ 30% of essentially unbiased samples of galaxy clusters [20], [46], specifically those in which the entropy of the hot intracluster gas at the cluster core lies below a well-defined threshold [11]; the same threshold separates BCGs that exhibit star formation from those that do not [56].",
"This relationship implies that the cool gas in BCGs must originate from cooling of the hot intracluster gas [54], albeit counteracted in large part by re-heating owing to the actions of their AGN jets [25].",
"The manner by which a residual cooling nevertheless occurs is not fully understood, although likely to involve a complex interplay between AGN jets and the surrounding intracluster gas [55] rather than a simple inflow of cooling quiescent gas.",
"Despite these uncertainties, the common entropy threshold found for the onset of cool gas and star formation implies that the residual – but still large – amounts of cooled gas (nearly) always fuels star formation.",
"The deposition of cool gas in BCGs owing to a residual cooling of their surrounding intracluster gas – hereafter a residual cooling flow, no matter how this cooling actually occurs – differs from any cool gas accreted through wet mergers with cluster member galaxies in several important ways: (i) the mass of cool gas in BCGs can be, as appears to be observed, substantially higher than what would be expected from mergers with cluster member galaxies that have had much of their gas removed by ram-pressure stripping; (ii) the spatial distribution of cool gas can be highly extended owing to an interplay with the AGN jets, rather than dissipatively accumulating at the centers of BCGs as would be expected in wet mergers; and (iii) the continuous deposition (albeit perhaps at a time-varying rate) and therefore replenishment of cool gas can sustain star formation at a high rate over an indefinite period, by contrast with wet mergers whereby gas is accreted in a single episode to spark a brief period of star formation.",
"Consistent with these expectations, the optical emission-line nebulae of BCGs can, and indeed quite often, extend over several 10 kpc if not over 100 kpc [44], [12], [68], as do their molecular gas as traced in CO [62], [48], [57], [58], [70], [59], [60] and star formation as traced in UV emission [68], [18].",
"In the best studied example, NGC 1275 at the center of the Perseus cluster, its filamentary optical emission-line nebula spans $\\sim $ 140 kpc [44], [12] compared with an effective (optical) radius for this galaxy of $\\sim $ 25 kpc [66].",
"The nebula is multi-phase, containing not just atomic/ionic gas seen in optical emission lines, but also ionized gas seen in X-rays, and a counterpart in molecular gas seen in the rotational-vibrational line of H$_2$ in the IR [42] and as traced in CO [64], [41], [63], [31], [62].",
"NGC 1275 is the only star-forming BCG sufficiently close for individual star clusters to be observable (and even marginally resolved): this galaxy has formed thousands of star clusters having masses ranging from $\\sim $$10^4 {\\rm \\, M_}$ , imposed by the observational detection threshold, up to $\\sim $$10^6 {\\rm \\, M_}$ , thus spanning the same mass range as globular clusters, at an essentially constant rate of $\\sim 0.1 {\\rm \\, M_\\ yr^{-1}}$ over at least the past 1 Gyr, beyond which the relatively young star clusters cannot be distinguished from the even more numerous globular clusters in this galaxy [43].",
"Although the bulk of the molecular gas is located at a projected radius of $\\lesssim 10$ kpc from the center of NGC 1275, the majority of the newly-formed star clusters are located much further out in close association with the outer gas filaments.",
"Beginning their free-fall from far beyond the strong tidal fields at the inner region of the BCG, the more massive star clusters ($\\gtrsim 10^5 {\\rm \\, M_}$ ) are likely to survive a Hubble time, whereas the more numerous star clusters having lower masses are more easily tidally disrupted to help grow the galaxy in, especially, stellar size (see a more detailed discussion in [43], as well as earlier arguments by [12]).",
"Despite the numerous examples of spatially extended star formation as cited above, NGC 1275 was – until the work described here – the only BCG definitively demonstrated to show not just spatially but also temporally extended star formation: implying that star formation in BCGs can be both spatially widespread and sustained indefinitely by residual cooling flows.",
"Nonetheless, there remain concerns that NGC 1275 may be exceptional and therefore its gas properties and star formation extraordinary.",
"This perception stems in large part to the spectacular appearance of its optical emission-line nebula, which in the past has made NGC 1275 unusual, but is owed in part to no more than its relative proximity (at $z=0.01756$ ) making its parent cluster the X-ray brightest cluster in the sky.",
"Here, we make use of the Cluster Lensing And Supernova survey with Hubble (CLASH) program [53], a Hubble Space Telescope (HST) treasury program in which 25 massive galaxy clusters were imaged in 16 filters spanning UV to IR wavelengths, to study the star formation of the BCG in MACS J0329.7$-$ 0211.",
"At $z = 0.45$ , this BCG is seen at an epoch dating back to about two-thirds the present age of the Universe.",
"From the CLASH images of this galaxy, [18] found that its UV emission has a complex morphology, as shown in Figure REF , with a maximal projected spatial extent of $\\sim $ 30 kpc.",
"[10] report a mass in molecular hydrogen gas for this BCG of $3.4 \\pm 1.0 \\times 10^{10} \\rm \\, M_$ as traced in CO.",
"In our work, we extract the spectral energy distributions (SEDs) of the young stellar population in this galaxy along all spatially-separated sightlines to reconstruct its recent star-formation history.",
"The smaller panels in Figure REF provide a preview of our results, showing the ages of the young stellar population and their total stellar masses at birth along different sightlines (upper row), as well as the maximal possible H$\\alpha +$ [NII] emission from HII regions and the minimal H$\\alpha +$ [NII] emission from a nebula not associated with HII regions (lower row).",
"Figure: (a) RGB image of the central region of the cluster MACS J0329.7--0211 composed from images in F435W (for B), F775W (for G) and F105W (for R).",
"A young and spatially complex (blue), as well as an old and spheroidal (red), stellar population is apparent in the BCG.",
"The dashed box encompasses the region over which we determined the physical properties of the young stellar population in the BCG, and includes cluster members as a sanity check of our methodology.",
"The remaining panels have the same size as this box.",
"Nominal (b) ages and (c) total stellar masses at birth inferred using a MCMC approach (Section ) for the young stellar population in the BCG by adopting Z=0.4Z ⊙ Z = 0.4\\,\\rm Z_{\\odot }, the approximate metallicity of the intracluster gas.",
"Our model predicted (d) maximal Hα\\alpha +[NII] emission from HII regions associated with the young stellar population, and (e) minimal Hα\\alpha +[NII] emission from a line-emitting nebula not excited by the young stellar population (Section ).In Section , we explain how we first removed, from the images in each filter, the old stellar population in the BCG so as to isolate the young stellar population along with the emission-line nebula in this galaxy.",
"We present images of the young stellar population and the emission-line nebula combined in Section , and explain how we extracted their SEDs along different sightlines for model fitting.",
"Then, in Section , we describe how we fit model SEDs generated from a stellar population synthesis code to the measured SEDs of the young stellar population along each sightline, taking care to avoid wavelengths in the measured SEDs containing a significant contribution from the emission-line nebula (which, as we show, is largely not associated with HII regions) in this galaxy.",
"We present the results for the ages and masses of the young stellar population along each sightline, and compute the star-formation history – star-formation rate as a function of time – of this population in Section .",
"In this section, we also present the morphology of the line-emission nebula, and examine the extent to which its emission may correspond to HII regions.",
"In Section , we examine and correct for selection biases – to the degree possible – in the inferred star-formation history.",
"We then consider the implications of the temporally and spatially extended star formation on the growth of this BCG.",
"Finally, in Section , we provide a summary of the most important points of this paper.",
"We adopt a distance to MACS J0329.7$-$ 0211 of 2497.6 Mpc based on a standard flat $\\Lambda CDM$ cosmology with $\\Omega _m=0.3$ , $\\Omega _{\\Lambda }=0.7$ , and $H_0 = 70 \\rm \\, km \\, s^{-1} \\, Mpc^{-1}$ , so that $1= 5.76\\,\\rm kpc$ ."
],
[
"Removing old stellar population",
"We downloaded images having a pixel size of 65 mas (those having a smaller pixel size of 30 mas sometime show artefacts owing to the manner in which they were reconstructed from the dithered images) from the CLASH archive.",
"Figure REF (first row) shows cropped images centered on the BCG in three representative filters – rest-frame UV (F435W), optical (F775W), and near-IR (F105W) – revealing: (i) a spatially complex feature extending from the south-east to the north-west with a projected longer dimension of $\\sim $ 6($\\sim $ 30 kpc) that dominates the light at ultraviolet to short optical wavelengths (F225W to F475W filters); and (ii) a spatially simple and nearly circularly-symmetric feature that increasingly contributes and eventually dominates the light at longer wavelengths (e.g., F105W filter).",
"As might be supposed and as we shall demonstrate, the former is produced by a young stellar population and the latter by an old stellar population.",
"Their different spatial distributions permits model fits to the projected 2-dimensional (2-D) light distribution of the old stellar population after masking out the light from the young stellar population (and other features as described below).",
"Subtracting the fitted light profiles of the old stellar population leaves only light from the young stellar population (subject to any internal dust extinction, along with a contribution from an emission-line nebula), thus permitting the spectral energy distributions (SEDs) of just this population to be isolated and modelled.",
"In the remainder of this section, we describe the procedure used to carefully derive the 2-D light distribution of the old stellar population.",
"Figure: Images in or involving the F435W (first column), F775W (middle column), and F105W (right column) filters.",
"First row are the original CLASH images extracted from the archive.",
"Second row are our best-fit 2-dimensional models for the projected light distribution of old stellar population.",
"Third row are color images corresponding to the ratio in intensity between the corresponding filter and the longest wavelength filter, F160W in the near-IR, such that blue corresponds to a higher intensity ratio (i.e., bluer colors) than red.",
"Fourth row are images derived by subtracting images in the second row from those in the first row, thereby removing the old stellar population.",
"Green arrows point to Hα\\alpha +[NII] emitting gas not spatially coincident with any detectable young stellar population in F435W or F775W (nor in the remaining filters not encompassing the Hα\\alpha +[NII] lines)."
],
[
"Masking",
"Owing to their simple morphologies, analytical functions can provide an excellent description of the two-dimensional (2-D) light distribution of the old stellar population in elliptical galaxies.",
"Before we are able to fit analytical models to the 2-D light distribution of the old stellar population in the BCG of MACS J0329.7$-$ 0211, however, we first have to mask out sources unrelated to this population.",
"Such sources include the young stellar population, dust visible in silhouette, a gaseous emission-line nebula, along with neighbouring galaxies.",
"To aid in identifying and/or clarifying the spatial extents of the aforementioned sources, we constructed color images by dividing the image in a given filter with that in the longest-wavelength filter (F160W).",
"Example color images involving a filter in the rest-frame UV (F435W), optical (F775W), and near-IR (F105W) are shown in Figure REF (third row); note that the F105W filter spans the H$\\alpha $ +[NII] line at the redshift of the BCG.",
"As can be seen, the light extending from the south-east to the north-west across the center of the BCG is relatively blue, characteristic of a young stellar population, whereas the nearly circularly-symmetric light in the BCG is relatively red, characteristic of an old stellar population.",
"Silhouette dust is apparent as a reddish patch and curved filament just south of the BCG center, especially in the (F775W$-$ F160W) image.",
"Finally, a prominent blue filament north-north-west of the BCG center (indicated by an arrow) can be seen in the (F105W$-$ F160W) color image but not the (F435W$-$ F160W) nor (F775W$-$ F160W) color images; this feature corresponds to line emission from a gaseous nebula, which is commonly found in BCGs at the centers of cool-core clusters.",
"Such emission-line nebulae are not always spatially coincident with young stars [7], [8], [43], if any are indeed present.",
"Because features such as the emission-line nebula appear only in a restricted number of filters, we did not impose an intensity threshold for computing the color images; as a consequence, regions with neighbouring pixels that very randomly between blue and red correspond to noise.",
"The young stellar population dominates if not produces all the light of the BCG in the UV filters (F225W to F390W).",
"Thus, to mask out the young stellar population in these filters, we simply masked all pixels with intensities above 3$\\sigma _{\\rm noise}$ , where $\\sigma _{\\rm noise}$ is the root-mean-square (rms) noise fluctuation of the image in a given filter (Section REF ).",
"At longer wavelengths, the old stellar population increasingly contributes and eventually dominates the BCG light.",
"In these filters, we manually masked the young stellar population guided by their spatial distribution both in the UV images and in the color images (i.e., relatively blue regions) constructed at optical and near-IR wavelengths (e.g., third row of Fig.",
"REF ).",
"This approach ensures that the entire young stellar population, even if detectable only in the optical and/or near-IR filters, is masked out."
],
[
"Neighbouring Galaxies",
"We used the source catalog generated by [53] using SExtractor to mask all sources (galaxies and stars) in the field apart from the BCG.",
"Although the detectable BCG light is confined to a relatively small area (first row of Fig.",
"REF ) where the only cataloged sources are galaxies, extensive masking of all cataloged sources is necessary for determining the rms noise of the image in each filter as explained in Section REF .",
"In the vicinity of the BCG, we carefully tuned the sizes of the circular or elliptical masks for the cataloged sources in each image to best match their detectable sizes, leaving as much of the old stellar population in the BCG as possible for fitting models to its 2-D light distribution."
],
[
"Silhouette dust",
"Silhouette dust is visible as a dark filamentary patch in the images shown in Figure REF (top row), although its full spatial extent is best revealed as the reddest regions in color images such as those shown in Figure REF (third row).",
"By selecting pixels with an intensity ratio below $1.1$ within a central radius of $\\sim $ 15 kpc in the (F775W-F160W) color image (Fig.",
"REF , third row, middle column), where we find the silhouette dust to have the greatest contrast, we created a common mask for this feature in all the images."
],
[
"Emission-line nebula",
"As mentioned earlier, an emission-line nebula is clearly detected as relatively blue regions away from the young stellar population in color images involving filters that span the H$\\alpha +$ [NII] lines, such as the (F105W$-$ F160W) image shown in Figure REF (third row, right column).",
"To mask the nebula in these filters, we selected relatively blue pixels in the color images having intensity ratios above 1.5 within a radius of $\\sim $ 15 kpc centered on the BCG."
],
[
"Model 2-D light distribution",
"An example of the original image and the same image after masking, both in the longest-wavelength filter (F160W), is shown in Figure REF (upper panels) over an area of size $180 \\times 220$ kpc centered on the BCG.",
"To model the remaining light in the masked image produced solely by an old stellar population in the BCG, we fitted 2-D S$\\rm \\acute{e}$ rsic functions to the masked images over a generous area of size $750 \\times 750$ kpc.",
"We found that a combination of three S$\\rm \\acute{e}$ rsic functions having different power-law slopes ($n$ ), ellipticities ($e$ ), position angles ($PA$ ), and effective radii ($R_{\\rm e}$ ) are required to fully capture the 2-D light distribution of the old stellar population.",
"The different $e$ and $PA$ of the three S$\\rm \\acute{e}$ rsic functions reflect the triaxial shapes of elliptical galaxies projected onto the plane of the sky, resulting in twists in the major axis and changes in the ellipticity of their light isophotes with radius.",
"We found that the parameters of the fitted S$\\rm \\acute{e}$ rsic components vary little among the images in the different filters, providing a measure of their robustness.",
"Because the old stellar population is brightest in the near-IR images, we took the mean of the model parameters for each S$\\rm \\acute{e}$ rsic component in the three longest-wavelength filters (F125W, F140W, and F160W), as listed in Table REF , to represent the optimal model parameters for the 2-D light distribution of the old stellar population.",
"Thus fixing the values of these model parameters, we then fitted for the light intensity at $R_{\\rm e}$ for each of the three S$\\rm \\acute{e}$ rsic components in the individual images.",
"Reassuringly, the relative intensities of the three fitted S$\\rm \\acute{e}$ rsic components at their respective $R_{\\rm e}$ was similar among all the filters.",
"Examples of the model 2-D light distribution for the old stellar population are shown in Figure REF (second row).",
"Figure: Upper row: Original CLASH image of the central region of MACS J0329.7--0211 in F160W (left panel) and the same image after masking (right panel) to enable a fit for the light distribution of the old stellar population in the BCG.",
"Only an area of size 280×220280 \\times 220 kpc is shown centered on the BCG.",
"Lower row: Radial light profile of the original image as indicated by the blue dashed curve, and of the best-fit model for the old stellar population as indicated by the red solid curve, at a position angle of 150 ∘ 150^\\circ chosen to best avoid sources unrelated to the old stellar population.",
"The model is in good agreement with the data except within the masked region, shaded gray, owing to light from a young stellar population and extinction from dust.c c c c[ht] S$\\rm \\acute{e}$ rsic Parameters Outer S$\\rm \\acute{e}$ rsic Middle S$\\rm \\acute{e}$ rsic Inner S$\\rm \\acute{e}$ rsic $n$ 0.788 1.816 1.004 $e$ 0.26 0.13 0.22 $PA$ ($\\deg $ ) 179.1 149.9 80.3 $R_e$ (kpc) 55.0 11.7 2.3 Figure REF (lower panel) compares the 1-D radial light profile of the BCG in the F160W filter (in which the old stellar population most dominates the light) along a position angle of $150^{\\circ }$ (to best avoid, where possible, sources unrelated to the old stellar population), against that extracted from the model image in the same filter and along the same position angle.",
"Except at the inner regions where a young stellar population is evident, the radial light profile of the model image closely follows that of the observed image, providing a measure of the accuracy of our model fit.",
"Figure REF compares the SED measured over a region in the BCG that avoids all other sources apart from its old stellar population, and the SED extracted over the same region from our model images for the old stellar population.",
"Once again there is an excellent match, providing another measure of the accuracy of our model fit.",
"Finally, the measured SEDs of the BCGs in, respectively, MACSJ0416.1$-$ 2403 ($z=0.395$ ) and MACSJ1311.0$-$ 0311 ($z=0.494$ ) are shown also in Figure REF , scaled so as to be superposed against the measured SED of the BCG in MACS J0329.7$-$ 0211 for easy comparison.",
"These galaxy clusters, which were also observed in the CLASH program, have redshifts that straddle the redshift of the galaxy cluster studied here.",
"The BCGs in these clusters show no detectable evidence for a young stellar population or a gaseous nebula (Levitsky et al., in preparation); nonetheless, we masked the respective centers of these BCGs before extracting their SEDs.",
"As is apparent, the measured SED of the old stellar population in the BCG of MACS J0329.7$-$ 0211 closely matches the measured SEDs of the BCGs in MACSJ0416.1$-$ 2403 and MACSJ1311.0$-$ 0311, indicating that we are genuinely tracing the old stellar population in the target BCG.",
"Figure: Measured SED indicated by filled circles with ±1σ noise \\pm 1\\sigma _{\\rm noise} error bars (Section ) extracted from a region of the BCG in MACS J0329.7--0211 that avoids all sources (the young stellar population, silhouette dust, and line-emitting gas) apart from the old stellar population.",
"SED extracted over the same region from our best-fit triple Se ´\\rm \\acute{e}rsic model (Table ) for the old stellar population in this BCG is indicated by the blue curve.",
"For comparison, measured SED of the BCG in MACSJ0416.1--2403 at z=0.395z=0.395 indicated by the dotted green curve, and that of the BCG in MACSJ1311.0--0311 at z=0.494z=0.494 indicated by the dashed orange curve.",
"These BCGs, which do not exhibit any detectable young stellar populations, have redshifts straddling that of MACS J0329.7--0211."
],
[
"Subtracted Images",
"Figure REF (last row) shows examples of the resulting images after subtracting our model images of the old stellar population in the BCG from the observed images.",
"In the subtracted images, the young stellar population now stands out in clear relief, as does the silhouette dust, the emission-line nebula (in the F105W filter), and of course projected neighbouring galaxies.",
"A final stage of image processing is necessary before extracting the SEDs of the young stellar population for model fitting, as is explained next."
],
[
"PSF homogenization",
"The images in the different filters shown in Figure REF all have different point spread functions (PSFs).",
"Before extracting the SEDs of the young stellar population, we therefore convolved the subtracted images to a common angular resolution set by the broadest PSF among the different filters.",
"From measurements of the 2-D radial profiles of field stars visible in each image, we found that the PSFs in all the images are well represented in their cores by a circularly-symmetric Gaussian profile, with the broadest at full-width half-maximum (FWHM) belonging to that of the image in the F110W filter.",
"We therefore convolved every other image to a FWHM of $\\sim $ 023, corresponding to that in the F110W filter.",
"The final subtracted images after PSF homogenization are shown in Figure REF .",
"Also shown in this figure are the passbands of the individual filters used in the CLASH program, along with where the H$\\alpha $ , H$\\beta $ , and [OIII] doublet (which, along with the [NII] doublet closely straddling H$\\alpha $ , are the brightest emission lines from HII regions over the wavelength range encompassed by these filters) would appear given the redshift of the BCG.",
"We note that the feature indicated by an arrow in Figure REF appears in all filters containing the H$\\alpha $ line but not in any of those containing the [OIII] doublet, further indicating that it is not related to HII regions (see Section REF ).",
"Figure: Upper panels: subtracted images in all 16 filters of the CLASH program after PSF homogenization.",
"Filter names are indicated in each panel, along with selected lines in brackets that might be encompassed in each filter.",
"Dashed box in the F160W panel encloses the region over which we extracted SEDs for modelling.",
"Bottom panel is the throughput of all the CLASH filters plotted with python package stsynphot , with wavelengths of the same selected lines at the redshift of the BCG in MACS J0329.7--0211 indicated by dashed vertical lines.",
"Note some filters are translated up for clarity."
],
[
"Noise level",
"The rms noise fluctuation cannot be directly measured from the individual images as each image is constructed from multiple dithered images in a given filter.",
"In the process, the images were sub-sampled so that their pixel sizes (65 mas) are smaller than the pixel sizes of the camera (either the Wide-field Camera 3, WFC3, or the Advanced Camera for Surveys, ACS) used in the observations.",
"As a consequence, neighbouring pixels in these images are not statistically independent.",
"The rms noise fluctuation in each pixel, $\\sigma _{\\rm noise}$ , can be separated into Poisson noise from light sources, $\\sigma _{\\rm source}$ , and that contributed by the CCD (read and dark noise) along with the diffuse astronomical sky both lumped together (as they cannot be distinguished in the archival images) into $\\sigma _{\\rm sky}$ .",
"The latter (i.e., $\\sigma _{\\rm sky}$ ) can be measured from apparently blank regions of the archival images in the following manner.",
"First, we masked out all the cataloged sources (including the BCG) in the original images so as to leave only apparently blank regions of sky, and then convolved the images to the same (common) FWHM for their PSFs like in the final subtracted images.",
"After that, we binned the images over $N$ pixels on a side so that each rebinned pixel has an area of $N^2$ , and for each such rebinned image computed $\\sigma _{\\rm sky}$ as a function of $N^2$ as shown in Figure REF for the image in the F226W filter.",
"As can be seen, for $N^2 \\ge 256$ , the rms noise decreases with increasing number of pixels binned as $1/\\sqrt{N^2}$ , as is expected for a Gaussian noise distribution.",
"At smaller $N^2$ , however, the rms noise departs from this dependence, reflecting the fact that neighbouring pixels (at the number of pixels binned) are not statistically independent.",
"Thus, to estimate $\\sigma _{\\rm sky}$ , we extrapolated the rms noise measured from images binned to $N^2 \\ge 256$ pixels to smaller $N^2$ using a $1/N$ dependence, as is shown by the dashed line in Figure REF .",
"The extrapolated value at $N^2 = 4 \\times 4$ , the area over which we extracted measured SEDs as described next, is taken as $\\sigma _{\\rm sky}$ .",
"By contrast with the calculation for $\\sigma _{\\rm sky}$ , the value of $\\sigma _{\\rm source}$ in a given pixel can simply be directly computed from the count rate in that pixel.",
"In the SED plots shown below, the measurement uncertainty associated with each filter is therefore $\\sigma _{\\rm noise} = \\sqrt{\\sigma _{\\rm source}^2+\\sigma _{\\rm sky}^2}$ .",
"Figure: Measured sky noise, σ sky \\sigma _{\\rm sky}, in the F225W image indicated by triangles, computed from images masked of all sources and binned over NN pixels on a side.",
"An extrapolation of σ sky \\sigma _{\\rm sky} from N 2 ≥256N^2 \\ge 256 to smaller N 2 N^2 based on a 1/N 2 1/\\sqrt{N^2} relation is indicated by a dashed line, demonstrating that the sky noise would be underestimated at N 2 ≲100N^2 \\lesssim 100 if directly computed from the archival image owing to dithering.",
"We determine the true sky noise for the measured SEDs, extracted over 4×44 \\times 4 pixels, from the extrapolation to N 2 =4×4N^2 = 4 \\times 4 for the image in each filter."
],
[
"Extracted SEDs",
"From the subtracted images after PSF homogeneization as shown in Figure REF , we extracted SEDs summed over $4 \\times 4$ pixels corresponding therefore to $026 \\times 026$ ($1.5 \\times 1.5$ kpc); i.e., with sides just slightly larger than the common FWHM of the images.",
"The area over which we extracted SEDs spans $128 \\times 128$ pixels, corresponding to $83 \\times 83$ ($48 \\times 48$ kpc), centered on the BCG as indicated by the dashed box in the last panel of Figure REF .",
"This area encloses a number of neighbouring galaxies, which proved useful for providing a sanity check on our model SED fits.",
"All the SEDs were corrected for Galactic dust extinction by adopting $A_V=0.165$ [65] and $R_V=3.1$ in the dust extinction curve of [9]."
],
[
"Model Single Stellar Populations",
"We started with the hypothesis that the measured SEDs of the young stellar population contained within individual apertures of size $4 \\times 4$ pixels can each be characterised by a coeval stellar population sharing the same metallicity; i.e., a single stellar population (SSP).",
"The task then is to determine which model SED generated for SSPs having different ages and metallicities, and if justified also extinctions, best matches a particular measured SED.",
"As the measured SEDs are each extracted over a region of size $1.5 \\times 1.5$ kpc, it would not be surprising if a mixture of different SSPs (i.e., separate star clusters), including those that overlap along the sightline, contribute to the individual measured SEDs.",
"In such cases, the best-fit model SED will either reflect the SSP deemed to dominate a particular measured SED or still poorly reproduce the measurements, or both.",
"To generate the model SEDs, we used the publicly available software code YGGDRASILhttps://www.astro.uu.se/~ez/yggdrasil/yggdrasil.html [73], [74], which employs model SSPs from Starburst99 [38], [71] based on a Kroupa initial mass function [35], [36] over the stellar mass interval $0.1$ –$100 \\, \\rm M_{\\odot }$ along with Padova stellar evolutionary tracks.",
"At sufficiently young ages for the model SSPs ($\\lesssim $ 10 Myr), surrounding gas that is photoionized to form a HII region can contribute line along with continuum emission, the latter from recombination, free-free, and 2-photon emission.",
"The model brightnesses of such HII regions are parameterised by the fraction of Lyman continuum photons from internal (hot and massive) stars that is absorbed by the surrounding gas, $f_{\\rm cov} = 1 - f_{\\rm esc}$ , where $f_{\\rm esc}$ is the fraction that escapes; i.e., $f_{\\rm cov} = 0$ corresponding to none, and $f_{\\rm cov} = 1$ corresponding to a maximal amount, of the surrounding gas being photoionized.",
"Because stars that produce the required Lyman continuum photons have largely vanished by an age of $\\sim $ 10 Myr, we found that the selected value of $f_{\\rm cov}$ only strongly affects the model spectra at younger ages.",
"Specifically, after convolving over the bandpasses of the individual filters used in the CLASH program so as to provide a direct comparison with the measured SEDs, we found that emission lines do not contribute appreciably to the (convolved) model SEDs beyond an age of 5 Myr.",
"Nonetheless, up to ages of $\\sim $ 100 Myr, the intensities in all filters are slightly elevated for $f_{\\rm cov} = 1$ compared with $f_{\\rm cov} = 0$ .",
"We do not know whether this excess is produced by leftover photoionized gas or, as we suspect, small numerical errors as the reverse can happen at ages beyond 100 Myr.",
"We considered two different metallicities, $Z$ , for the model SSPs of $Z = \\rm Z_{}$ , the solar metallicity, and $Z = 0.4 \\, \\rm Z_{}$ , approximately that of the intracluster gas around the BCG [45].",
"For reasons we shall explain, we also subjected the model spectra to varying amounts of “internal\" dust extinction – that within the BCG to the foreground of the SSP (recall that the measured SEDs have been corrected for Galactic dust extinction) – based on the dust extinction curve of [9], for which we adopted $R_V=3.1$ .",
"Finally, the template spectra were Doppler shifted to the redshift of the BCG and then convolved with the spectral responses of the filters used in the CLASH program, thus generating model SEDs that can be directly compared with the measured SEDs.",
"In this way, we generated model SEDs at logarithmically space intervals beginning at an age of 1 Myr, the youngest permitted by the code, up to an age of 3 Gyr (far exceeding the oldest age found for the young stellar population; see below), except for a denser sampling between 10 Myr and 15 Myr in steps of 1 Myr."
],
[
"Color-Color Diagrams",
"Before evaluating which model SEDs best fit the measured SEDs, we found it instructive to construct color-color diagrams from the measured SEDs to compare against evolutionary tracks generated from the model SEDs.",
"As we shall see, this comparison yielded crucial insights on the overall range of physical parameters – in particular, ages and (internal) dust extinctions – spanned by the young stellar population, thus constraining the parameter space over which we needed to explore to find best-fit model SEDs.",
"Just as importantly, this comparison revealed bright line emission in filters encompassing H$\\alpha $ +[NII] on sightlines towards the young stellar population where they have inferred ages $\\gg 5$ Myr, by which time line emission from HII regions is predicted to be undetectable.",
"An emission-line nebula unrelated to HII regions is therefore present along most if not all sightlines towards the young stellar population, in addition to sighliness away from this population as pointed out earlier in Figure REF .",
"Our model SEDs are not able to account for the contribution from such an emission-line nebula, making it necessary to exclude filters encompassing H$\\alpha $ +[NII] when fitting model SEDs to the measured SEDs as described later.",
"To construct the color-color diagrams presented in Figure REF , we selected sightlines along which the signal-to-noise ratio (S/N) of the young stellar population exceeds $1\\sigma _{\\rm noise}$ in each of the 16 filters.",
"Through trial-and-error, we found this criterion to select a representative range of ages exhibited by the young stellar population while preserving an adequate S/N for their colors.",
"In each panel of Figure REF , evolutionary tracks derived from the model SSPs are plotted for metallicities of $Z = 0.4 \\, \\rm Z_{}$ (left column) and $Z = \\rm Z_{}$ (right column), for which we chose the extreme limits of $f_{\\rm cov} = 0$ (no HII region; red track) and $f_{\\rm cov} = 1$ (maximal emission from a HII region; blue track) to illustrate the maximal possible differences in colors for the evolutionary tracks at a given metallicity.",
"Figure: Color-color diagrams selected to highlight a relatively broad range of ages (top row), Hα\\alpha +[NII] emission in excess of that predicted from HII regions (middle row), and little if any appreciable Hβ\\beta and [OIII] emission (bottom row).",
"Measured SEDs along individual sightlines are indicated by filled circles, with colors corresponding to their measurement uncertainties as indicated in the color bar accompanying each panel.",
"The median ±1σ noise \\pm 1\\sigma _{\\rm noise} uncertainty of the measured SEDs is indicated at the lower left corner of each panel.",
"Evolutionary tracks are plotted in each panel, with the blue tracks corresponding to f cov =0f_{\\rm cov} = 0 (i.e., no HII region) and the red tracks to f cov =1f_{\\rm cov} = 1 (i.e., maximal predicted emission from a HII region) for adopted metallicities of 0.4Z0.4 \\rm Z_ in the left column and Z\\rm Z_ in the right column.",
"The red arrow in each panel indicates the expected color change owing to dust extinction as large as A V =0.5A_V=0.5.Figure REF (top row) shows color-color diagrams selected to best differentiate between age and (internal) dust extinction.",
"As can be seen, the evolutionary tracks (spanning ages from 1 Myr to 400 Myr) are nearly vertical in this color-color diagram, reflecting a rapid change in colors between (rest-frame) UV (F336W) and near-IR (F160W) along the ordinate, but a nearly constant color in UV (F336W and F435W) along the abscissa, as age increases.",
"By contrast, internal dust extinction displaces the colors of the model SSPs along a more tilted direction as indicated by extinction vectors having a length of $A_V = 0.5$ , thus permitting a distinction between age and appreciable dust extinction for a color spread among the young stellar population.",
"The measured colors of the young stellar population straddle the model evolutionary tracks, with no apparent systematic shift to the right of these tracks as might be expected if extinction is appreciable along many sightlines.",
"Their overall distribution in color-color space suggests a collection of un-extincted SSPs having a range of ages spanning a few Myr to a few 100 Myr.",
"The lack of apparent dust extinction is consistent with that observed for young star clusters over the same age range in NGC 1275 [43], where the vast majority of star clusters lie next to rather than within the emission-line nebula associated with that galaxy.",
"Note the weak dependence on the colors of the evolutionary tracks with metallicity: as a consequence, our best-fit model SEDs to the measured SEDs do not permit a strong constraint on metallicity (over, at least, the range 0.4–1.0 $\\rm Z_{}$ ).",
"Figure REF (middle row) shows color-color diagrams selected to highlight, along the abscissa, any line emission in the F850LP filter, which encompasses the H$\\alpha $ line, [NII] doublet, and [SII] doublet at the redshift of the BCG (see bottom panel of Fig.",
"REF ).",
"The other filter involved in the abscissa, F125W in the near-IR, does not encompass any bright emission lines that we are aware of.",
"The ordinate involves the same two filters as before in Figure REF (first row).",
"The extinction vector is nearly parallel to the evolutionary tracks in this color-color diagram, and so the spread in measured colors along the ordinate can in principle reflect either an age spread or spatially-variable extinction (or both); as just demonstrated in Figure REF (top row), however, any extinction is too low to produce a detectable systematic shift in the color-color diagram, and so the color spread along the ordinate must primarily reflect an age spread.",
"A large systematic shift to the left leaving relatively few points to the right of the evolutionary tracks is clearly apparent in Figure REF (middle row), indicating bright emission lines in the F850LP filter even along sightlines where the young stellar population have inferred ages of $\\gg 5$ Myr.",
"Along many sightlines, the total line emission in the F850LP filter is much stronger than the maximal predicted line emission from HII regions (i.e., for $f_{\\rm cov} = 1$ ).",
"At this point, we note that such a systematic shift is observed not only for colors involving the F850LP and F125W filters, but also for colors involving either the F814W, F105W, or F110W – which also encompass the H$\\alpha $ line+[NII] lines – and F125W filters.",
"To further emphasize that emission lines in the F850LP filter are predominantly unrelated to HII regions, Figure REF (bottom row) shows color-color diagrams selected to highlight, along the abscissa, any line emission in the F775W filter, which encompasses the [OIII] doublet and H$\\beta $ at the redshift of the BCG (see bottom panel of Fig.",
"REF ; notice that H$\\beta $ lies at the edge of the passband, where the throughput is only $\\sim $ 50% that of the [OIII] doublet).",
"The ordinate involves the same two filters as before.",
"By contrast with Figure REF (middle row), there is, at best, a weak systematic shift to the left of the evolutionary tracks.",
"The contribution by emission lines to the F775W filter is therefore much weaker if at all appreciable by comparison with that to the F850LP filter.",
"This behavior is inconsistent with expectations for HII regions, whereby emission in the individual [OIII] doublets is typically about as bright as that in H$\\alpha $ , but consistent with expectations for emission-line nebulae in BCGs, whereby [OIII] is, where at all detectable, much dimmer than H$\\alpha $ [30]."
],
[
"Model SED Fitting",
"The color-color diagrams presented in Figure REF provide encouragement that each of the measured SEDs, despite encompassing an area of size $1.5 \\times 1.5$ kpc, can be closely reproduced by model SEDs for SSPs.",
"To assess how well these model SEDs actually fit the measured SEDs, Figure REF shows six representative examples of measured SEDs fitted by model SEDs having different SSP ages – corresponding to the approximate ages inferred from the color-color diagram of Figure REF (top row) – as indicated at the top of each panel, except for the model SEDs indicated by dashed lines in Figure REF $d$ –$e$ that have ages of only 1 Myr.",
"Model SEDs having $Z = 0.4 \\, \\rm Z_{}$ are plotted in green, and those having $Z = \\rm Z_{}$ are plotted in orange.",
"The solid and dashed lines are for model SEDs having, respectively, $f_{\\rm cov} = 0$ and $f_{\\rm cov} = 1$ .",
"The SEDs are plotted at the rest frame of the BCG, with each data point centered at the effective wavelength (i.e., weighted by the spectral response) of the corresponding filter after correcting for the Doppler shift of the BCG.",
"The data points colored red correspond to the four filters (F814W, F850LP, F105W, and F110W) that encompass the (redshifted) H$\\alpha $ +[NII] lines.",
"Figure: Measured SEDs indicated by filled circles selected to highlight a broad range of ages along different sightlines to the young stellar population.",
"All have ±1σ noise \\pm 1\\sigma _{\\rm noise} error bars indicated by vertical bars; those colored red encompass the Hα\\alpha +[NII] lines.",
"Model SEDs having ages as indicated above each panel or for an age of 1 Myr as indicated are plotted as solid curves for f cov =0f_{\\rm cov} = 0 (no HII region) and dashed curves for f cov =1f_{\\rm cov} = 1, and in green for 0.4Z0.4 \\, \\rm Z_ and orange for Z\\rm Z_.",
"The dashed curves in panel (a) underpredict the red data points, suggesting that only a portion of the Hα\\alpha +[NII] emission along this sightline may be generated by HII regions, wheres none of the same line emission in panels (d) and (e) can be generated by HII regions.Apart for the model SEDs plotted as dashed lines in Figure REF $d$ –$e$ , the remaining model SEDs plotted fit the measured SEDs over all filters (panels $b$ , $c$ and $f$ ), or over all filters except those encompassing the H$\\alpha $ +[NII] lines as indicated by the red data points (panels $a$ , $d$ and $e$ ), remarkably well – even though these model SEDs need not necessarily represent the very best fits to the measured SEDs (see Section REF and Section REF ) as their SSP ages were inferred solely and only approximately from the color-color diagram of Figure REF (top row).",
"Over the SSP age range 5–100 Myr encompassed by this figure, both the measured and model SEDs change with age in two important ways: (i) their slope, especially from UV to optical wavelengths, becomes shallower with age; (ii) a discontinuity in the slope near 4000 Å, owing predominantly to the Balmer break at 3646 Å (rather than absorption by ionized metals that create the 4000 Å break, a feature that dominates at ages older than the range relevant here), becomes increasingly prominent with age.",
"The young stellar population must therefore encompass a relatively broad range of ages, confirming this aspect of the color-color diagrams shown earlier (Fig.",
"REF , top row).",
"At a given age, the model SEDs exhibit little change with metallicity, except at an age of 10 Myr whereby the slope of the SED at near-IR wavelengths is different for the two different metallicities plotted – owing to the different timescales over which massive stars evolve to become red supergiants as a function of metallicity.",
"As a consequence, the measured SEDs do not provide strong constraints on metallicity, at least over the range 0.4–$1.0 \\rm \\, Z_$ .",
"In Figure REF $a$ , the model SEDs at an age of 5 Myr formally under-predict the measurements in filters containing H$\\alpha +$ [NII] (red data points) even for $f_{\\rm cov}= 1$ (dashed lines), albeit well reproducing the measurements in the other filters.",
"Even more glaringly, in Figure REF $d$ –$e$ , the model SEDs (solid lines) far under-predict the measurements in filters containing H$\\alpha +$ [NII] albeit well reproducing the measurements in the other filters.",
"The measured SEDs shown in Figure REF $d$ –$e$ exhibit a prominent Balmer break, and must therefore have ages far exceeding 10 Myr by which time any HII regions would have vanished.",
"The dashed lines in the same panels show model SEDs having $f_{\\rm cov}= 1$ and an age of just 1 Myr (the youngest allowed by the SED code used), thereby giving the maximal possible H$\\alpha +$ [NII] emission from an associated HII region.",
"Although these model SEDs can reproduce the measurements in filters containing H$\\alpha +$ [NII] as well as those at longer wavelengths, they far over-predict the measured brightnesses in filters at short optical and UV wavelengths.",
"These examples illustrate the contribution by an emission-line nebula unrelated to HII regions to filters encompassing the H$\\alpha +$ [NII] lines, confirming this aspect of the color-color diagram shown earlier (Fig.",
"REF , middle row).",
"As a consequence, the inclusion of these filters complicates if not renders impossible satisfactory fits to the measured SEDs in instances where the emission-line nebula makes a detectable contribution.",
"Because the emission-line nebula contributes detectably along many sightlines as demonstrated in Figure REF (middle row), we omitted the four filters encompassing H$\\alpha +$ [NII] from the model fits as described next."
],
[
"$\\chi ^2$ minimization",
"As a first step towards evaluating quantitatively how well the model SEDs match the measured SEDs as well as the age range over which individual measured SEDs can be constrained, we computed the reduced $\\chi ^2$ (henceforth, $\\chi _{\\rm red}^2$ ) of the model SEDs fitted to the individual measured SEDs.",
"As explained earlier, model SEDs were generated over the age range of 1 Myr to 3 Gyr for metallicities of either $Z = 0.4 \\, \\rm Z_$ or $Z = \\rm Z_$ .",
"Because we omitted the four filters containing H$\\alpha +$ [NII] from consideration owing to contamination by an emission-line nebula, we henceforth consider only model SEDs having $f_{\\rm cov} = 0$ .",
"Finally, as the color-color diagram of Figure REF (top row) indicates negligible internal dust extinction, we consider for now only model SEDs having $A_V = 0$ .",
"For each model SED, we computed the optimal normalization in brightness – thus yielding the total stellar mass at birth (henceforth, birth mass) for the corresponding model SSP – that provides the best fit to the measured SED as judged by the lowest $\\chi _{\\rm red}^2$ .",
"The lowest two rows of Figure REF shows example plots of the $\\chi _{\\rm red}^2$ versus model SSP age, $t_{\\rm age}$ , of the model SEDs fitted to the measured SEDs shown in the upper two rows of Figure REF .",
"The measured SEDs are the same as those shown in Figure REF .",
"Like before, the green lines are for model SEDs having $Z = 0.4 \\, \\rm Z_$ and the orange lines for those having $Z = \\rm Z_$ .",
"Satisfactory fits having $\\chi _{\\rm red}^2 \\le 1$ can be found for all the examples shown except Figure REF $a$ , where the lowest $\\chi _{\\rm red}^2$ is between 1 and 3 (depending on the metallicity).",
"As can be seen, the $\\chi _{\\rm red}^2$ versus $t_{\\rm age}$ can exhibit either: (i) a single narrow minimum, indicating a narrow (in logarithmic scale) range of best-fit ages (e.g., orange in panels $g$ and $l$ ); (ii) two narrow local minima whereby one is significantly deeper than the other, indicating a relatively narrow range of best-fit ages (e.g., green in panel $a$ ); (iii) two narrow and comparably deep local minima, indicating two relatively narrow ranges of best-fit ages (e.g., both orange and green in panel $h$ ); and most commonly (iv) a broad minimum, indicating a relatively broad range of best-fit ages (e.g., green in panel $i$ , both orange and green in panels $j$ and $k$ ).",
"The model SEDs overlaid on the measured SEDs in the upper two rows of Figure REF correspond to the $t_{\\rm age}$ at the lowest $\\chi _{\\rm red}^2$ as indicated by the dotted vertical green and orange lines (which overlap in panels $g$ and $l$ ).",
"Despite the sometimes broad range of ages spanned by the model SEDs that provide satisfactory fits to the measured SEDs, the measured SEDs shown in Figure REF clearly span about two decades in ages.",
"Figure: Measured SEDs in first and second rows are the same as those in Fig. .",
"Model SEDs having the same color coding as Fig.",
"are plotted at SSP ages, t age t_{\\rm age}, indicated at the top of each panel, corresponding to the lowest χ red 2 \\chi _{\\rm red}^2 among the model SEDs fitted to the individual measured SEDs as indicated by the vertical dotted lines in the χ red 2 \\chi _{\\rm red}^2 versus t age t_{\\rm age} plots in the third and fourth rows.",
"The measurements in red, which encompass the Hα\\alpha +[NII] lines, were excluded when making model SED fits to the measured SEDs for reasons described in the text."
],
[
"Markov Chain Monte Carlo",
"To fully explore and quantify the likelihood of a given model SED in matching an individual measured SED, we used a Markov Chain Monte Carlo (MCMC) method.",
"For this purpose, we developed an algorithm that yields the likelihood surface of individual model SSP parameters used to generate the suite of model SEDs fitted to an individual measured SED, as described in full in Appendix .",
"The free parameters explored are SSP age, $t_{\\rm age}$ , over the range 1 Myr to 1 Gyr (although model SEDs were generated from 1 Myr to 3 Gyr as mentioned in Section REF , from the work described in Section REF we found that all sightlines towards the young stellar population have very large $\\chi _{\\rm red}^2$ for ages beyond several hundreds of Myr), and birth mass $M_*$ (corresponding to the sum over the stellar mass interval 0.1–100 $\\rm M_$ with a Kroupa initial mass function).",
"We set the metallicity at either $Z = 0.4 \\rm \\, Z_$ or $Z = \\rm Z_$ , and set $f_{\\rm cov} = 0$ as all four filters encompassing H$\\alpha +$ [NII] are omitted from the fit.",
"We also set $A_V = 0$ as the color-color diagram indicates negligible (internal) dust extinction, although we later also explored how our results would change if we allow $A_V \\le 0.5$ mag.",
"We assumed flat priors (i.e., uniform initial likelihood) for each free parameter, and extracted the maximum a posteriori estimate of each parameter from the peak of the inferred multi-dimensional likelihood surface – thus deriving parameter values for the most probable model SED (among those considered) that matches a given measured SED.",
"Owing to discrete sampling of the multi-dimensional likelihood surface, the maximum a posteriori estimate may not coincide exactly with – but is always close to – the set of parameter values that give the lowest $\\chi _{\\rm red}^2$ , except possibly in relatively rare cases where the multi-dimensional likelihood surface exhibits two well-separated peaks having comparable heights such that the parameter value having the lowest $\\chi _{\\rm red}^2$ would be found close to the somewhat lower peak if the sampling was continuous.",
"We henceforth refer to the maximum a posteriori estimates derived from the smoothed multi-dimensional likelihood surface as the nominal parameter values.",
"For each free parameter, we compute its one-dimensional likelihood function by marginalizing over all the other free parameters.",
"The uncertainty, $\\widetilde{\\sigma }$ , of a given free parameter is defined, by analogy with the variance in a Gaussian distribution, as the area under its posterior distribution that encompasses $68.2\\%$ (i.e., corresponding to 2$\\widetilde{\\sigma }$ ) of the total area over which this distribution has the highest valuesWe use $\\widetilde{\\sigma }$ rather than $\\sigma $ as a reminder to the reader that the uncertainty inferred for a given parameter from our MCMC method should not be regarded in the same way as the root-mean-square deviation of a Gaussian distribution.. For those not familiar with the MCMC approach, we emphasize here caution in making too close an analogy between the uncertainty as computed using the MCMC approach with the variance as computed for a Gaussian distribution: even if the posterior distribution has a single high and narrow peak but broad wings, the uncertainty can be much broader than the visual width of the peak so as to encompass a broad portion of the wings.",
"Furthermore, because the posterior distribution need neither be unimodal nor symmetric, the absolute values of $\\widetilde{\\sigma }$ need not be the same on opposite sides of the peak in the posterior distribution.",
"If the posterior distribution is multimodal, the range of parameter values encompassed within the uncertainties may include those having very low if not near zero probabilities; e.g., those examples in Figure REF where the $\\chi _{\\rm red}^2$ as a function of age exhibits two local minima having $\\chi _{\\rm red}^2 \\le 1$ separated by a plateau having $\\chi _{\\rm red}^2 > 1$ , such that the parameter values encompassed within the uncertainties span the entire range between two local minima.",
"Finally, we note that even the most probable SED does not necessarily guarantee a satisfactory fit to a given measured SED, so that it is important to check the $\\chi _{\\rm red}^2$ of any fit.",
"Figure: Left panel: Same measured SED as in Fig.",
"(a) and Fig. (a).",
"Red curve is best-fit model SED having Z=ZZ = \\rm Z_ and nominal parameters in t age t_{\\rm age} and M * M_* as listed at the upper right corner, and red shaded region the range of model SEDs having Z=ZZ = \\rm Z_ encompassed within ±1σ ˜\\pm 1 \\widetilde{\\sigma } (68% confidence levels; see text) of the nominal values, inferred based on our MCMC method.",
"The posterior distribution of the individual parameters, and combined posterior distribution of both parameters (shaded dark blue within ±1σ ˜\\pm 1 \\widetilde{\\sigma } and light blue within ±2σ ˜\\pm 2 \\widetilde{\\sigma }), are shown in the panels to the right.To help visualize the output from our MCMC algorithm, in the following we show exemplar results for four of the measured SEDs shown previously in Figures REF –REF .",
"Figure REF shows the same measured SED as in Figure REF $a$ and Figure REF $a$ .",
"The model SEDs with $Z = \\rm Z_$ provide a reasonable fit ($\\chi _{\\rm red}^2 \\lesssim 2$ ) to this measured SED over only a very narrow age range of $\\sim $ 2 Myr centered at $t_{\\rm age} \\sim 6$ Myr as shown earlier in Figure REF $g$ .",
"In Figure REF , we also plot histograms of the posterior distributions in both $t_{\\rm age}$ and $M_*$ as inferred by our MCMC algorithm.",
"As can be seen, the two posterior distributions are unimodal, although not symmetric.",
"The contour plots show the range of parameter values bounded at the 68% (dark blue) and 95% (light blue) confidence levels.",
"The red curve superposed on the measured SED is the model SED having the nominal parameter values; i.e., as extracted at the peak of the multi-dimensional likelihood surface.",
"The nominal parameter values along with their uncertainty bounds thus derived, as well as the $\\chi _{\\rm red}^2$ of the fit, is listed in the left-most panel.",
"The red band bounding the measured SEDs indicates the boundary spanned by all the model SEDs combined having ages within the 68% confidence level of the nominal age (note that different $t_{\\rm age}$ have different corresponding $M_*$ ).",
"Neither the upper nor lower boundaries of the red band necessarily correspond to a specific model SED, as different SEDs can define either boundaries in different filters.",
"Notice that the brightnesses in three of the four filters spanning H$\\alpha +$ [NII] (indicated by red symbols) are in excess of those predicted by the most probable model SED having $t_{\\rm age} = 5$ Myr (with $f_{\\rm cov} = 0$ ); while it is tempting to attribute this excess to line emission from a HII region, as demonstrated above (see Figs.",
"REF –REF ) excess emission in these filters also is seen towards regions that are far too old to exhibit HII regions.",
"Figure: Same as Fig.",
", but for the measured SED in Fig.",
"(b) and Fig.",
"(b).Figure REF shows the MCMC results for the same measured SED as in Figure REF $b$ and Figure REF $b$ .",
"The model SEDs with $Z = \\rm Z_$ provide a good fit ($\\chi _{\\rm red}^2 \\le 1$ ) to this measured SED over the age range $\\sim $ 6-40 Myr as shown earlier in Figure REF $h$ .",
"The relatively broad age range of satisfactory fits reflects the relatively slow evolution in the model SEDs so long as red supergiants, which have lifetimes of up to a few 10s Myr, contribute significantly if not dominate the light of the SSP.",
"As can be seen in Figure REF , the posterior distribution spans a similar decade in ages, with $t_{\\rm age} = 12^{+48}_{-6}$ Myr .",
"Figure REF shows the MCMC results for the same measured SED as in Figure REF $c$ and Figure REF $c$ .",
"The model SEDs with $Z = \\rm Z_$ provide a good fit ($\\chi _{\\rm red}^2 \\le 1$ ) to this measured SEDs over two distinct age ranges of $\\sim $ 5–8 Myr and $\\sim $ 30–70 Myr, separated by a plateau over which $1 < \\chi _{\\rm red}^2 < 2$ , as shown earlier in Figure REF $i$ .",
"Similarly, the posterior distribution in age is clearly bimodal with comparable likelihoods for the younger and older age brackets, such that $t_{\\rm age} = 40^{+40}_{-33}$ Myr.",
"Figure: Same as Fig.",
", but for the measured SED in Fig.",
"(c) and Fig.",
"(c).In the final example, Figure REF shows the MCMC results for the same measured SED as in Figure REF $f$ and Figure REF $f$ .",
"The model SEDs with $Z = \\rm Z_$ provide good fits ($\\chi _{\\rm red}^2 \\le 1$ ) to the measured SED over the age range $\\sim $ 60–150 Myr as shown earlier in Figure REF $l$ .",
"Similarly, the posterior distribution in age is strongly peaked (in a logarithmic plot) at $\\sim $ 100 Myr, such that $t_{\\rm age} = 80^{+120}_{-50}$ Myr.",
"To see how permitting internal dust extinction might change the results, we also fit model SEDs having $A_V \\le 0.5$ as guided by Figure REF (top row).",
"We found that all sightlines with acceptable model SED fits (see Section REF ) – which exclude those along which silhouette dust is clearly visible (see Fig.",
"REF and Fig.",
"REF ) – have nominal $A_V$ either considerably lower than 0.5 or consistent with 0 within the uncertainties.",
"As would be expected, allowing for extinction can expand the age range of acceptable fits owing to an age-extinction degeneracy at optical wavelengths; i.e., increasing either the age (at a fixed dust extinction) or the dust extinction (at a fixed age) both flattens the SED slope at optical wavelengths.",
"Given the very low if not null extinction found for all sightlines with acceptable model SED fits, in agreement with Figure REF (top row), we henceforth present results only for fits by model SEDs having no (internal) dust extinction.",
"Figure: Same as Fig.",
", but for the measured SED in Fig.",
"(f) and Fig.",
"(f)."
],
[
"Results",
"We present here the results from our MCMC approach.",
"First, we explain how we select satisfactory ($\\chi _{\\rm red}$ of order unity) as well as sensible ($t_{\\rm age}$ well constrained) – the two conditions that must be met for selecting acceptable – model SED fits to the measured SEDs of the young stellar population.",
"As will become clear, acceptable fits are found for nearly all sightlines towards the young stellar population except where silhouette dust is visible; on these sightlines, our subtraction of the old stellar population is badly compromised and therefore also the measured SEDs of the young stellar population.",
"This ubiquity of acceptable model SED fits indicates that, oftentimes, a SSP likely dominates the brightness along a given sightline towards the young stellar population.",
"Of course, where the S/N ratio is poor, satisfactory fits can always be found, as is the case on sightlines towards the outer bounds of the young stellar population where ages cannot be sensibly constrained.",
"Furthermore, satisfactory but not sensible fits can always be found towards blank sky where model SEDs are fitted to random noise fluctuations.",
"Based on the model SEDs that yield acceptable fits to the measured SEDs, we construct maps showing the inferred nominal age and birth mass of the young stellar population along individual sightlines.",
"We then compute the star formation history – star formation rate as a function of time – of this population based on their nominal parameters as well as from the full posterior distribution of the relevant parameters."
],
[
"Acceptable Model SED Fits",
"As we shall demonstrate, the key to selecting acceptable model SED fits to the measured SEDs lies in the posterior distribution in $t_{\\rm age}$ given by our MCMC method.",
"Specifically, unacceptable fits are characterised by a relatively flat and broad posterior distribution in $t_{\\rm age}$ , indicating instances whereby model SEDs spanning a broad age range provide either: (i) comparably poor (unsatisfactory) fits to the measured SEDs; or (ii) equally satisfactory but not sensible fits, as in instances where the measured SEDs have poor S/N or correspond to blank sky.",
"Figure REF (first row) shows the standard deviation of the posterior distribution in $t_{\\rm age}$ , $\\delta _{\\rm age}$ , versus the nominal birth mass, $M_*$ , for $Z = 0.4 \\, \\rm Z_$ (first column) and $Z = \\rm Z_$ (second column), in both cases for $A_V = 0$ .",
"The standard deviation of the posterior distribution is defined (following convention) as $\\delta _x^2 = \\int _{x_{\\rm lower}}^{x_{\\rm upper}} \\, (x - \\mu )^2 \\, f(x) \\, dx$ , where $f(x)$ is the probability distribution function in $x$ (in this instance, $t_{\\rm age}$ ) and $\\mu = \\int x f(x) \\, dx$ the weighted average value in $x$ .",
"Three trends are immediately apparent from the distribution of $\\delta _{\\rm age}$ versus $M_*$ : (i) a pileup in $\\delta _{\\rm age}$ over the range $\\sim $ 200–300 Myr (black data points); (ii) a grouping of data points having $\\delta _{\\rm age}$ significantly below this pileup over the range $10^5 \\, \\rm M_$$\\lesssim M_* \\lesssim 10^7$$\\,\\rm M_$ (blue data points); and (iii) a systematic decrease in $\\delta _{\\rm age}$ with increasing $M_*$ at $M_* \\gtrsim 10^7 \\, \\rm M_$ (red data points).",
"The corresponding spatial locations of these three groups of data points are shown in Figure REF (third row).",
"As can be seen, the blue data points coincide with the young stellar population, whereas the red data points coincide almost exclusively if not entirely with neighbouring cluster members.",
"The black data points coincide primarily with regions in between the young stellar population and neighbouring cluster members (i.e., blank sky), although a few coincide with the inner region of the BCG where silhouette dust is clearly visible and where the fits are poor (for the reasons explained above).",
"Figure: First column: Standard deviation in nominal t age t_{\\rm age}, δ age \\delta _{\\rm age} (see text), versus nominal M * M_* for model SEDs having Z=0.4Z ⊙ Z=0.4\\,\\rm Z_{\\odot } (top row) and Z=Z ⊙ Z = \\rm Z_{\\odot } (mid row) along all sightlines encompassed by the square box indicated in Figures and .",
"Second column: nominal M * M_* versus nominal t age t_{\\rm age} for sightlines colored blue.",
"Dotted diagonal line is the approximate detection threshold determined in the manner described in Section and plotted in Fig.",
", and is used to select acceptable fits among these sightlines.",
"Third column: spatial locations of the data points colored black (blank sky), blue (young stellar population), and red (all except one sightline corresponding to cluster members).",
"The RGB image in the last row is the same as in Figure , zoomed in to the dashed box.The pileup at $\\delta _{\\rm age} \\approx 200$ –300 Myr reflects the relatively flat and broad posterior distribution in $t_{\\rm age}$ when model SEDs are fitted to random noise fluctuations (the situation in most cases) or where the fits are poorFor example, in the case of a log-uniform posterior distribution, $\\delta _x^2 = \\left[ \\frac{ x_{\\rm upper}^2 - x_{\\rm lower}^2}{2 \\, {\\rm ln} (x_{\\rm upper}/x_{\\rm lower}) } \\right] - \\left[ \\frac{x_{\\rm upper} - x_{\\rm lower}}{{\\rm ln} (x_{\\rm upper}/x_{\\rm lower}) } \\right]^2$ , so that for $x_{\\rm lower} = 1 \\, {\\rm Myr}$ and $x_{\\rm upper} = 1 \\, {\\rm Gyr}$ (the lower and upper bounds in age range of the model SEDs used in our MCMC computations), $\\delta _{\\rm age} = 227$ Myr..",
"The blue data points (young stellar population) are formally separated from the black data points (blank sky or unacceptable fits) by a demarkation line that we define at $\\delta _{\\rm age} = 190$ Myr, lying just below the pileup in $\\delta _{\\rm age}$ .",
"Figure REF (second row) shows $M_*$ plotted as a function of $t_{\\rm age}$ for the blue data points.",
"As is apparent, the data points are strongly concentrated at the upper left corner, with a trend of an increasing lower bound in $M_*$ with increasing $t_{\\rm age}$ suggestive of an observational detection threshold.",
"Indeed, the approximate detection threshold computed in the manner described in Section REF and indicated by a diagonal line closely defines the lower bound of this main group of data points.",
"The blue data points plotted as open circles mostly lie well below the detection threshold, and have relatively large $\\delta _{\\rm age}$ just below the demarkation line at 190 Myr; they coincide with the outer regions of the young stellar population where the S/N is poor.",
"The remaining blue data points plotted as crosses ($+$ ) are therefore selected as satisfactory and sensible (i.e., acceptable) model SEDs fits to the measured SEDs.",
"Although not plotted in this figure, the red data points are concentrated at the upper bound in age considered in our model SED fits of 1 Gyr, implying that their stellar populations have older ages (as would be expected for early-type galaxies)."
],
[
"Nominal Model Parameters",
"In Figure REF , we present maps of the nominal parameters for the young stellar population in ages (first row) and birth masses (second row) for $Z = 0.4 \\, \\rm Z_$ (left column) and $Z = \\rm Z_$ (right column).",
"As can be seen, the nominal ages for $Z = \\rm Z_$ (second column) tend to be somewhat younger than for $Z = 0.4\\, \\rm Z_$ (first column) owing to an age-metallicity degeneracy.",
"Given the generally small differences in ages between the two metallicities, however, the birth masses are very similar irrespective of whether $Z = 0.4\\, \\rm Z_$ or $Z = \\rm Z_$ .",
"The nominal ages span $\\sim $ 1–100 Myr, and the nominal birth masses (which, along a given sightline, encompasses an area of $1.5 \\times 1.5$ kpc) span $\\sim $$10^5$ –$10^7 {\\rm \\, M_}$ .",
"Figure: Nominal t age t_{\\rm age} (top row) and nominal M * M_* (bottom row) for Z=0.4Z ⊙ Z=0.4\\,\\rm Z_{\\odot } (left column) and Z=Z ⊙ Z=\\rm Z_{\\odot } (right column) inferred from our MCMC method for the young stellar population."
],
[
"Emission-line Nebula",
"In observations utilizing filters, emission-line maps are usually constructed by subtracting an image comprising just stellar continuum from an image comprising both line(s) and stellar continuum together.",
"Before subtraction, the image comprising just stellar continuum is scaled to match the intensity of the stellar continuum in the image comprising both line and stellar continuum.",
"This procedure was used by [27] to produce $\\rm H\\alpha + [NII]$ images for a number of the BCGs in the CLASH program, including that studied here.",
"When scaling for the stellar continuum, [27] adopted the same color throughout the galaxy corresponding to that of its old stellar population.",
"The situation for BCGs hosting a young stellar population, as is the case here, is more complicated as the stellar continuum along some sightlines includes a contribution from this population in addition to that from an old stellar population – in the case of the BCG here, many of the same sightlines that exhibit line emission.",
"To further complicate matters, the young stellar population has a spatially-varying SED owing to their different ages and birth masses along different sightlines, and therefore contribute different amounts of continuum light at different spatial locations.",
"In such instances, the key to producing an accurate emission-line map is to accurately model or infer the stellar continuum of both the young and old stellar population.",
"To estimate the stellar continuum of the young stellar population along different sightlines, we computed its model stellar continuum based on its nominal $t_{\\rm age}$ and $M_*$ along individual sightlines by setting $f_{\\rm cov} = 0$ (i.e., so as not to include any non-stellar contributions from HII regions).",
"We then subtracted this component from the image in the F850LP filter – where H$\\alpha $ +[NII] contribute the largest fractional intensity – from which the continuum light of the old stellar population had already been accurately removed (Section ) as shown in Figure REF .",
"To isolate regions coincident only with either the young stellar population or emission-line gas (e.g., the filament indicated by an arrow in Fig.",
"REF ), as well as to help discriminate against noise peaks, we select relatively blue regions in the color maps having intensity ratios $f_{\\rm 850LP}/f_{\\rm F160W} > 3.0$ and $f_{\\rm 814W}/f_{\\rm F140W} > 2.0$ .",
"This procedure produces a pure line image subject only to uncertainties in the inferred parameters ($t_{\\rm age}$ and $M_*$ ) of the young stellar population (and hence its contribution to the stellar continuum) along each sightline (and, in principle, the degree to which the model SEDs actually match the measured SEDs).",
"The results after subtracting the model continuum light of the young stellar population are shown in Figure REF (a) for $Z = 0.4 \\rm \\, Z_$ (left column) and $Z = \\rm Z_$ (right column).",
"As can be seen, H$\\alpha $ +[NII] emission is apparent along sightlines even where: (i) the nominal ages of the young stellar population, as shown in Figure REF (d), is older than 5 Myr, at which time the model SEDs show no appreciable contribution from H$\\alpha $ +[NII] after convolving over the spectral response of the F850LP filter (see Section REF ); and (ii) there is no detectable young stellar population, most prominently to the north-west of the BCG center (at the location indicated by an arrow in Fig.",
"REF ).",
"These results conform with expectations based on the color-color diagrams presented in Figure REF (middle row) showing H$\\alpha $ +[NII] emission at levels in excess of those predicted for $f_{\\rm cov} = 1$ along many sightlines to the young stellar population, and the color map shown in Figure REF (third row, third column) where pure line emission as indicated by an arrow is visible north-west of the BCG center.",
"Notice that the pure line image sometimes shows no emission along sightlines where the nominal ages of the young stellar population is $\\lesssim 5$ Myr, indicating no detectable HII regions along these sightlines.",
"In this way, we find an integrated line intensity of $3.8 \\times 10^{-15} \\, \\rm erg \\, s^{-1} \\, cm^{-2}$ (i.e., $L_{\\rm H\\alpha +\\rm [NII]}=4.1 \\times 10^{42}\\, \\rm erg \\, s^{-1}$ ) if the young stellar population has $Z = 0.4 \\rm \\, Z_$ , and an integrated line intensity of $3.7 \\times 10^{-15} \\, \\rm erg \\, s^{-1} \\, cm^{-2}$ (i.e., $L_{\\rm H\\alpha +\\rm [NII]}=4.0 \\times 10^{42}\\rm \\, erg \\, s^{-1}$ ) if this population has $Z = \\rm Z_$ .",
"By comparison, [27] report $L_{\\rm H\\alpha +\\rm [NII]} = (25.1 \\pm 2.4) \\times 10^{42}\\, \\rm erg \\, s^{-1}$ (after correction for internal dust extinction; see Section REF ), about six times higher than what we measure.",
"Figure: (a): pure Hα\\alpha +[NII] image after subtracting, where necessary, model stellar continuua for the young stellar population from the F850LP image of Fig.",
", where the stellar continuum from the old stellar population has already been accurately subtracted.",
"(b): Maximal predicted Hα\\alpha +[NII] or relatively weak continuum from HII regions, derived by subtracting model SEDs having f cov =0f_{\\rm cov} = 0 from those having f cov =1f_{\\rm cov} = 1 based on the nominal t age t_{\\rm age} and M * M_* along each sightline.",
"(c): Minimal predicted Hα\\alpha +[NII] from an emission-line nebula not associated with HII regions, derived by subtracting the images in second row from those in the first row.",
"(d): sightlines coded blue where nominal t age ≤5t_{\\rm age} \\le 5 Myr and therefore Hα\\alpha +[NII] from HII regions potentially detectable, and sightlines coded red where nominal t age >5t_{\\rm age} > 5 Myr and any Hα\\alpha +[NII] from HII regions too weak to be detectable, in the F850LP filter.Left column is for Z=0.4Z ⊙ Z=0.4\\, \\rm Z_{\\odot } and right column for Z=Z ⊙ Z= \\rm Z_{\\odot }.To estimate the maximal possible line emission from HII regions, we subtracted model images for $f_{\\rm cov} = 0$ from those for $f_{\\rm cov} = 1$ generated at the same nominal $t_{\\rm age}$ and $M_*$ in the F850LP filter.",
"The results are shown in Figure REF (b) for the two respective metallicities.",
"As would be expected, relatively bright line emission from HII regions is predicted where the nominal $t_{\\rm age} \\le 5$ Myr (note that intensities are color coded on a logarithmic scale, so that the range spanned by light green to yellow is an order of magnitude brighter than the range spanned by black to dark green).",
"We truncate the scale bar at $0.05 \\times 10^{-20} {\\rm erg \\, s^{-1} Å^{-1} \\, cm^{-2}}$ as emission below this level mostly if not entirely arises, we suspect, from small numerical errors in the YGGDRASIL algorithm used to generate the model SEDs for $f_{\\rm cov} = 1$ (see earlier discussion in Section REF ), rather than from genuine HII regions.",
"By subtracting the maximal model HII-region emission in the second row from the pure line image in the first row of Figure REF for the respective metallicities, we generated images showing the minimal line emission from gas not associated with HII regions – hereafter, emission-line nebula – as shown in the Figure REF (c).",
"As is apparent, the emission-line nebula overlaps in large part with the young stellar population, consistent with the color-color diagram shown earlier in Figure REF (b) whereby the contribution from H$\\alpha $ +[NII] along many sightlines to the young stellar population far exceeds that predicted by the model SEDs even for $f_{\\rm cov} = 1$ .",
"Furthermore, the emission-line nebula extends beyond the bounds of the young stellar population (in particular to the north-west), consistent also with the color image shown in Figure REF (third column, third row).",
"From the ratio between the total flux density of the images in the third row and first row of Figure REF , we estimate that the emission-line nebula accounts for at least $\\sim $ 60% (for $Z=\\rm Z_$ ) to $\\sim $ 80% (for $Z = \\rm 0.4 \\, Z_$ ) of the H$\\alpha $ +[NII] line emission from the BCG.",
"Attributing the H$\\alpha $ +[NII] emission predominantly or entirely to star formation would therefore result in an over-estimate of the star-formation rate as computed from the total line luminosity."
],
[
"Star Formation History",
"Based on the nominal $t_{\\rm age}$ and $M_*$ along each sightline towards the young stellar population, we computed the ensemble star formation rate as a function of time, $SFR(t) = \\Delta M_* / \\Delta t_{\\rm age}$ , whereby $\\Delta M_*$ is the sum of the birth masses along all sightlines spanning a selected range in nominal $t_{\\rm age}$ of $\\Delta t_{\\rm age}$ .",
"The results are shown in Figure REF (left panel) for $Z = 0.4 \\, \\rm Z_$ by the green histogram and $Z = \\rm Z_$ by the orange histogram.",
"In this plot, we selected different $\\Delta t_{\\rm age}$ centered at different $t_{\\rm age}$ so as to give a comparable number of samples within each bin.",
"The difference in $SFR(t)$ between the two metallicities are small except at $t_{\\rm age} \\sim 10$ –100 Myr, whereby the model SEDs are dominated by red supergiants having evolutionary timescales that depend on metallicity.",
"Figure: Left panel: star-formation rate, SFR(t)SFR(t), as a function of time, tt, based on the nominal t age t_{\\rm age} and M * M_* inferred from our MCMC method.",
"Right panel: same as left panel, but now utilising the full posterior distribution for t age t_{\\rm age} and M * M_* (see text).",
"The bin widths, corresponding to time intervals, were selected such that there are roughly equal number of samples in each bin.",
"Green color is for Z=0.4Z ⊙ Z=0.4\\,\\rm Z_{\\odot } and orange color is for Z=Z ⊙ Z=\\rm Z_{\\odot }.",
"Solid curves are for SFR(t)=2M yr -1 SFR(t) = 2 \\rm M_\\, {\\rm \\, yr^{-1}}, dotted curves for SFR(t)=5M yr -1 SFR(t) = 5 \\rm M_\\, {\\rm \\, yr^{-1}}, and dashed curves for a starburst that began ∼\\sim 400 Myr ago and has since decayed exponentially with an e-folding timescale of 100 Myr, in all cases after applying selection effect I (an increasing detection threshold in M * M_* with increasing t age t_{\\rm age}) as discussed in the text.Taking into consideration the uncertainties in $t_{\\rm age}$ and $M_*$ along each sightline provided by our MCMC method, Figure REF (right panel) shows the star-formation history computed using the full posterior distribution of these parameters along individual sightlines.",
"The mathematical treatment is described in full in Appendix .",
"In essence, for all sightlines towards the young stellar population, we summed the products between $M_*$ at a particular $t_{\\rm age}$ and the probability given by the posterior distribution for that age bin.",
"We then computed $SFR(t)$ by choosing an appropriate $\\Delta t_{\\rm age}$ centered at a given $t_{\\rm age}$ so as to give roughly equal number of sightlines in each $\\Delta t_{\\rm age}$ bin.",
"At this stage, we caution that the $SFR(t)$ thus derived should not be taken to reflect the intrinsic $SFR(t)$ , as the measurements shown in Figure REF do not take into account selection effects as we shall describe next."
],
[
"Selection Effects",
"Over a given wavelength interval, stellar populations generally fade over time as progressively less massive stars reach the terminal phases of their evolution.",
"Correspondingly, between ages of $\\sim $ 10 Myr and $\\sim $ 100 Myr, our model SSPs fade in brightness by: (i) just over an order of magnitude (factor of $\\sim $ 20) in the rest-frame UV; and (ii) just under to about an order of magnitude in the rest-frame optical to near-IR.",
"Between ages of $\\sim $ 1 Myr and $\\sim $ 10 Myr, the fading at all these wavelengths is significantly less, corresponding to a factor of $\\lesssim 4$ .",
"This fading over time leads to the following selection effects depending on the composition of SSPs (i.e., ensemble of star clusters possibly having different birth masses and/or ages) along individual sightlines: (i) if composed of a singular SSP or multiple SSPs having similar ages, then a limitation on the oldest detectable SSP for a given (total) birth mass (selection effect I); and (ii) if composed of multiple SSPs having a broad range of ages but otherwise similar birth masses, then shifting the inferred age from the mean age towards the age of the youngest (and hence brightest) SSP, as well as overestimating its birth mass as part of the light is actually contributed by SSPs having different ages (selection effect II).",
"Both these selection effects lead to the same bias in the star-formation history as inferred from the measurements: inevitably underestimating at progressively greater severity the star-formation rate in the more distant past.",
"To understand the consequences of selection effect I on our results, we generated from our model SEDs for either $Z = 0.4 \\, \\rm Z_$ or $Z = \\rm Z_$ , and for $M_*$ spanning the range $10^4-10^8 \\, \\rm M_$ at uniform logarithmic steps (every one-tenth of a decade), a corresponding suite of mock model SEDs perturbed in each filter by the maximal noise measured for the image in that filter to produce 30 independent realizations of each model SED.",
"We then fit, using our MCMC approach, these mock noise-perturbed model SEDs by the original noiseless model SEDs having the same corresponding metallicities.",
"In Figure REF , we show the probability of the noise-perturbed suite of model SEDs at a given $t_{\\rm age}$ and $M_*$ passing our selection criterion of ${\\delta }_{\\rm age} < 190 \\, \\rm Myr$ , shaded from blue to red in order of increasing probability such that white corresponds to a probability of $\\sim $ 50%.",
"The methodology of deriving this probability is explained in full in Appendix REF .",
"As can be seen, the boundary between a high (red shaded region) and low (blue shaded region) likelihood of detection, indicated by a black dotted line in each panel, is sharp, increasing approximately linearly in $M_*$ with $t_{\\rm age}$ in a logarithmic plot.",
"Notably: (i) at $t_{\\rm age} \\lesssim 5$ Myr, SSPs having $M_* \\gtrsim 1 \\times 10^5 \\, \\rm M_$ are all detectable; (ii) by $t_{\\rm age} = 10$ Myr, those having only slightly higher $M_* \\gtrsim 3 \\times 10^5 \\, \\rm M_$ remain detectable; (iii) at $t_{\\rm age} = 50$ Myr, only SSPs having $M_* \\gtrsim 10^6 \\, \\rm M_$ are detectable; and (iv) by $t_{\\rm age} = 100$ Myr, only those having $M_* \\gtrsim 4 \\times 10^6 \\, \\rm M_$ remain detectable.",
"Because we perturbed the mock model SEDs at each wavelength by the highest noise measured in the image at that corresponding filter, the actual detection threshold in $M_*$ can be somewhat lower at a given $t_{\\rm age}$ .",
"Figure: Probability of detecting M * M_* as a function of t age t_{\\rm age}, with blue colors indicating a probability less than 50% and red colors indicating a probability above 50%, for Z=0.4Z ⊙ Z=0.4\\,\\rm Z_{\\odot } (left panel) and Z=Z ⊙ Z=\\rm Z_{\\odot } (right panel).",
"The dividing line between a probability of less than 100% and a probability of 100% is quite sharp, as indicated approximately by the black diagonal dotted lines.",
"The same diagonal lines are plotted in Fig.",
".As shown earlier in Figure REF (second column), the nominal $M_*$ inferred along the vast majority of sightlines to the young stellar population lie comfortably above the detection threshold as indicated by the black diagonal dotted lines in this figure (corresponding to the same black diagonal dotted lines in Fig.",
"REF ).",
"Furthermore, the lower bound in the nominal $M_*$ increases with increasing nominal $t_{\\rm age}$ , as would be expected given the increasing detection threshold in $M_*$ with $t_{\\rm age}$ .",
"A few slightlines to the young stellar population, however, have nominal $M_*$ mostly lying far below the detection threshold, corresponding to the blue data points plotted as open circles in Figure REF (first column).",
"These sightlines were excluded from further consideration owing to their poorly constrained fits – all have ${\\delta }_{\\rm age}$ close to our selection threshold of 190 Myr – as a result of the poor S/N of their measured SEDs.",
"The consequences of selection effect II are more difficult to quantify.",
"As explained above, the light from SSPs spanning a range of ages along a given sightline can boost the $M_*$ computed for the singular SSP deemed to have the best-fit age for that sightline.",
"This effect can explain why the inferred $M_*$ along all sightlines to the young stellar population lie comfortably above the detection threshold as indicated by the black diagonal dotted lines in Figure REF ."
],
[
"Bias in Star-Formation History",
"The inevitable rise in the detection threshold for $M_*$ with increasing $t_{\\rm age}$ (selection effect I) biases the measured $SFR(t)$ in two distinct ways: (i) generating an apparent decrease in the measured $SFR(t)$ towards the more distant past even if the intrinsic $SFR(t)$ is constant over time if not higher in the past; and also (ii) producing an apparent cutoff in the measured $SFR(t)$ at a distinct epoch as $M_*$ falls below the detection threshold.",
"As we shall show, both these effects can be seen in the measured $SFR(t)$ of the BCG in MACS J0329.7$-$ 0211.",
"To study the effect of an increasing detection threshold for $M_*$ with increasing $t_{\\rm age}$ (selection effect I), we assume that the bulk of stars are formed in star clusters having a mass function of $dN/dM \\sim M^{-2.1}$ .",
"This functional dependence is the apparent universal mass function of star clusters (see [43]), whether it be open star clusters in our Galaxy, newborn star clusters in interacting or merging galaxies, relatively young and massive star clusters in the BCG of the Perseus cluster, or globular clusters having masses $\\gtrsim 10^5 {\\rm \\, M_}$ (above the peak in their mass function).",
"Adopting this mass function for different model $SFR(t)$ , we then computed how such model $SFR(t)$ would manifest themselves in our measurements following the methodology laid out in Appendix REF .",
"The results are shown in Figure REF for two different underlying model $SFR(t)$ : (i) a starburst that began $\\sim $ 400 Myr ago followed by an exponential decay with an e-folding timescale of $\\sim $$10^8 {\\rm \\, yr}$ (dashed curves), the typical decay timescale of starbursts inferred for local starburst galaxies [3]; and (ii) $SFR(t) = \\rm constant$ at either $2 {\\rm \\, M_\\, yr^{-1}}$ (solid curves) or $5 {\\rm \\, M_\\, yr^{-1}}$ (dotted curves).",
"As can be seen, the model starburst provides a poor description of the measured $SFR(t)$ .",
"Instead, apart for a relatively brief period of elevated $SFR(t) \\approx 10 \\rm \\, M_\\, yr^{-1}$ around $10^7$ yr ago, the measurements are better represented by an approximately constant $SFR(t) = 2 \\rm \\, M_\\, yr^{-1}$ (for either $0.4\\,\\rm Z_$ or $\\rm Z_$ ) over, at least, the past $\\sim $ 400 Myr, beyond which any star formation drops below the detection threshold.",
"We note here that the brief elevation in the $SFR$ about $10^7$ yr ago makes little difference to the time-averaged $SFR$ over the past 100 Myr or longer.",
"As a sanity check, Figure REF shows the SED integrated over all the selected sightlines towards the young stellar population as indicated by the black circles.",
"The red dashed curve corresponds to the spatial integration of model SEDs having $Z = 0.4 \\, \\rm Z_$ and the nominal $t_{\\rm age}$ and $M_*$ inferred along individual sightlines as used for plotting Figure REF (left panel), and the gray dashed curve to the spatial integration of model SEDs having also $Z = 0.4 \\, \\rm Z_$ but now for values of $t_{\\rm age}$ and $M_*$ that take into account the full posterior distribution of these parameters as plotted in Figure REF (right panel).",
"Both these curves are in reasonable agreement with the spatially-integrated SED, providing confidence in our methodology of fitting model SEDs to the measured SEDs along individual sightlines.",
"The solid green curve correspond to $SFR(t) = 2 \\rm \\, M_\\, yr^{-1}$ and the dotted green curve to $SFR(t) =5 \\rm \\, M_\\, yr^{-1}$ , both after correcting for selection effect I using Eq.",
"REF .",
"As can be seen, a constant $SFR(t) = 5 \\rm \\, M_\\, yr^{-1}$ provides a good representation of the spatially-integrated SED at the seven shortest wavelength filters, whereas a constant $SFR(t) = 2 \\rm \\, M_\\, yr^{-1}$ presents a better representation of the spatially-integrated SED at the three longest wavelength filters.",
"Light in the shorter-wavelength filters is dominated by SSPs having relatively young ages, and a constant $SFR(t) = 5 \\rm \\, M_\\, yr^{-1}$ is close to the mean $SFR(t)$ over the past $\\sim $$10^7 {\\rm \\, yr}$ .",
"Light in the longer wavelength filters is less subject to an age bias (selection effect II), and suggests a constant $SFR(t) \\approx 2 \\rm \\, M_\\, yr^{-1}$ .",
"The spatially-integrated SED is therefore compatible with the measured $SFR(t)$ as shown in Figure REF .",
"Figure: Measured SED integrated over all the selected sightlines towards the young stellar population, where as before the data points colored red correspond to filters spanning the Hα\\alpha +[NII] lines.",
"Overlaid on the measured SEDs are: (i) red dashed curve corresponding to the spatial integration of model SEDs having Z=0.4ZZ = 0.4 \\, \\rm Z_ and the nominal t age t_{\\rm age} and M * M_* inferred from our MCMC method along individual sightlines; (ii) gray dashed curve corresponding to the binned star-formation rate over time as in plotted in Figure (right panel) for Z=0.4ZZ = 0.4 \\, \\rm Z_; (iii) green solid curve corresponding to SFR(t)=2M yr -1 SFR(t) = 2 \\rm \\, M_\\, yr^{-1} and green dotted curve corresponding to SFR(t)=5M yr -1 SFR(t) = 5 \\rm \\, M_\\, yr^{-1}, both corrected for the increasing threshold in M * M_* with t age t_{\\rm age}."
],
[
"Comparison with previously inferred Star-Formation Rates",
"[18], [27], and [28] have previously inferred star formation rates for the BCGs in the CLASH program.",
"Using the [34] relationship between UV luminosity and $SFR$ , a relationship that implicitly assumes that $SFR(t) = \\rm constant$ over the past $\\sim $$10^8 \\rm \\, yr$ (the overall lifespan of stars that emit significantly in the UV), [18] derive $SFR = (25.1 \\pm 2.4) \\, \\rm M_\\, yr^{-1}$ for the BCG in MACS J0329.7$-$ 0211, the third highest amongst all the BCGs in the CLASH program.",
"To estimate the UV luminosity of the young stellar population in the F390W filter, [18] corrected for an estimated contribution from evolved stars by adopting a mean color of $\\rm (UV-IR) = 5.5$ for these stars, and also corrected for Galactic dust extinction (but not for any dust extinction intrinsic to the BCG).",
"As we have shown, the $SFR$ for this BCG has not been constant over the past $10^8 \\rm \\, yr$ , such that the $SFR$ inferred by [18] is a factor of $\\sim $ 2.5 higher than that even during a brief period of elevated star formation $\\sim $$10^7$ yr ago and over an order of magnitude higher than the time-averaged $SFR$ over the past $10^8 \\rm \\, yr$ .",
"Repeating the procedure described by [18] to determine the UV luminosity of the young stellar population in the F390W filter, we find a luminosity that is nearly exactly a factor of $1 + z = 1.45$ lower than that inferred by [18] for the BCG in MACS J0329.7$-$ 0211.",
"Even then, this UV luminosity remains a factor of $\\sim $ 2.5 higher than that measured from the subtracted image in the F390W filter shown in Figure REF , indicating that the contribution from the old stellar population is not completely removed using the method adopted by [18].",
"Using the UV luminosity we cleanly measure for the young stellar population from the subtracted image in the F390W filter, we derive $SFR \\approx 7 \\rm \\, M_\\, yr^{-1}$ based on the [34] relationship.",
"This $SFR$ is in much better agreement with, albeit still a factor of $\\sim $ 3.5 higher than, the time-averaged $SFR$ over the past $\\sim $$10^8 {\\rm \\, yr}$ that we infer in our work.",
"Using also the [34] relationship between UV luminosity and $SFR$ , [27] derived $SFR = 42 \\pm 2 \\, \\rm M_\\, yr^{-1}$ for the BCG in MACS J0329.7$-$ 0211.",
"They derived the UV luminosity of the young stellar population in this BCG from images in the four shortest wavelength filters employed in the CLASH program.",
"Using the same method as [18], they corrected for a contribution to the UV emission from the old stellar population, and corrected for Galactic dust extinction.",
"Unlike [18], however, [27] inferred and corrected for dust extinction by assuming an intrinsically flat SED – if flux density is expressed in $f_\\nu $ rather than $f_\\lambda $ – over the rest wavelength range 1500–2800 Å along individual sightlines to the young stellar population, as would be the case if $SFR(t) = \\rm constant$ along all such sightlines.",
"In this way, [27] derived the color excess based on the UV slope of the measured SEDs along individual sightlines, and then used the dust law of [6] to infer spatially-variable dust extinction spanning the range $A_V \\sim 0.3$ –1.8 mag.",
"As we have shown, however, the assumption that $SFR(t) = \\rm constant$ along individual sightlines is not justified, nor do we find significant dust extinction along any sightline to the young stellar population (except over a small area, omitted from our work, exhibiting silhouette dust).",
"As mentioned in Section REF , by subtracting only the estimated stellar continuum of an old stellar population but not that of the young stellar population, [27] derived H$\\alpha $ +[NII] images for a number of the BCGs in the CLASH program.",
"Using the [34] relationship between H$\\alpha $ luminosity and $SFR$ , a nearly instantaneous measure of the $SFR$ owing to the short lifetimes of massive stars capable of ionizing the surrounding leftover gas from star formation to produce HII regions, [27] derive $SFR = 80 \\pm 21 \\, \\rm M_\\, \\rm yr^{-1}$ for the BCG in MACS J0329.7$-$ 0211 after correcting for dust extinction inferred in the manner described above.",
"As we emphasized in Section REF , however, much of the H$\\alpha $ +[NII] emission associated with this BCG is not related to HII regions, resulting in a overestimate of the $SFR$ (which, in the case of [27], is further exacerbated by the large extinctions inferred) if the line emission is attributed entirely to HII regions.",
"Assuming an exponentially decaying star-formation rate to approximate a recent starburst, along also with an exponentially decaying star-formation rate in the distant past for forming an old stellar population, [28] fit model SEDs to the spatially-integrated SED for the BCG in MACS J0329.7$-$ 0211.",
"They infer $SFR(0) = 39.8^{+20.5}_{-14.8} \\, \\rm M_\\, yr^{-1}$ and therefore a progressively higher $SFR(t)$ into the past, and a burst duration – defined as the timescale required to form the inferred mass of stars in the burst – of $1.0^{+1.4}_{-0.6}$ Gyr.",
"The $SFR(0)$ they derive is corrected for an inferred dust extinction of $A_V = 0.56^{+0.18}_{-0.19}$ .",
"As shown in Figure REF , however, there is no evidence to support the argument for an exponential decay in the star-formation rate of this BCG in the recent past."
],
[
"Implications for BCG Growth",
"As explained in Section , in-situ star formation can pose a double-edged sword for the stellar growth of BCGs.",
"While increasing their stellar content, star formation that is concentrated in the central regions of BCGs – as might be expected if cool gas sinks dissipatively into their centers before fuelling vigorous star formation – causes the galaxy to shrink in response to a central deepening in its gravitational potential well.",
"By contrast with this picture, however, star formation in the vast majority of BCGs has been found to be spatially extended (see Section ): in the case of the BCG in MACS J0329.7$-$ 0211, the star formation is spatially extended over $\\sim $ 30 kpc.",
"For the BCG in the Perseus cluster, newly-formed star clusters are found preferentially towards the outskirts of this BCG [43] despite the molecular gas as traced in CO being concentrated at its inner regions.",
"Obviously, star formation at the outskirts rather than the inner regions of a BCG has a different impact on its growth in stellar size.",
"The manner by which the newly-formed stars contribute to the growth in stellar size of BCGs also depends on whether they form in star clusters, as is the case in nearby star-forming galaxies (whether spiral galaxies or in interacting or merging galaxies) as well as in the BCG of the Perseus cluster.",
"In such a case, the less massive (and also more numerous) star clusters may be disrupted by strong tidal forces as they free fall into the inner regions of their host galaxies (see discussion in [43]), thus disgorging stars along their orbits to promote the growth in stellar sizes of their host galaxies.",
"The star-formation rates of BCGs as reported in the literature can reach values $\\gtrsim 100 \\rm \\, M_\\, yr^{-1}$ .",
"Although the veracity of any reported star-formation rate needs to be scrutinized when derived, especially, from line-emitting gas (which, in BCGs, may not primarily constitute HII regions), star-formation rates derived from UV emission – which, using the conversion of [34], implicitly assumes a constant star-formation rate over the past $\\sim $$10^8$ yr – may be on somewhat safer grounds, provided that light from the old stellar population is properly subtracted and any dust extinction inferred with utmost care.",
"High star-formation rates in galaxies have hitherto been regarded as being unsustainable, as the existing gas reservoir is rapidly consumed and, in part, dispersed.",
"In the case of BCGs, however, the continual replenishment of the gas reservoir from a residual cooling flow can sustain star formation over an indefinite period, as has been found for the BCG in the Perseus cluster [43] and now demonstrated also for the BCG in MACS J0329.7$-$ 0211.",
"In apparent agreement with this picture, [19] find that the upper mass bound of molecular gas in BCGs as traced in CO remains constant at $\\sim $$10^{11} \\, \\rm M_$ up to $z \\sim 1.2$ ; they argue that star-forming BCGs “process any accreted molecular gas into stars through means that are agnostic to both their redshift and their cluster mass.\"",
"The role of persistent star formation from a residual cooling flow in contributing to the growth in stellar mass and size of some BCGs – thus helping explain their broad range of physical properties, as there are multiple pathways that may operate together for their growth – therefore deserves due consideration.",
"That said, does sustained star formation fuelled by a residual cooling flow contribute significantly to the stellar growth of BCGs?",
"The BCG in the Perseus cluster has formed numerous star clusters (having masses of $\\sim $$10^4-10^6 \\rm \\, M_$ ) at a relatively steady mass-formation rate of $\\sim 0.1 {\\rm \\, M_\\, yr^{-1}}$ over the past 1 Gyr.",
"Even if sustained since $z \\sim 1$ –2 (over the past $\\sim $ 8–10 Gyr), star formation at this rate would have added just $\\sim $$10^9 \\, \\rm M_$ in stars to this galaxy (of course, we cannot rule out that $SFR(t)$ was higher in the more distant past).",
"Not all these stars may add to the stellar mass or size of the BCG, as a fraction of the stellar mass formed is bound in massive star clusters that may long survive disruption by the tidal field of their host galaxy [43].",
"By comparison with the BCG in the Perseus cluster, the BCG in MACS J0329.7$-$ 0211 has a persistent star-formation rate of $\\sim $$2 {\\rm \\, M_\\, yr^{-1}}$ (albeit with brief excursions to much higher values) that is about 20 times higher.",
"Over the past $\\sim $ 400 Myr alone, this BCG has formed $\\sim $$8 \\times 10^8 {\\rm \\, M_}$ of new stars, accounting for $\\sim $ 1% of the original stellar mass of the most massive red nuggets ($\\sim $$10^{11} {\\rm \\, M_}$ ); if sustained over a $\\sim $ 3 Gyr (or $\\sim $ 5.6 Gyr) interval between $z = 0.45$ and $z \\sim 1$ (or $z \\sim 2$ ), star formation at this rate could have contributed $\\sim $$6 \\times 10^{9} \\, \\rm M_$ (or $\\sim $$1.1 \\times 10^{10} \\, \\rm M_$ ) in stellar mass, and therefore potentially added $\\sim $ 10% to the original stellar mass of the most massive red nuggets.",
"A comparison of the mass of stars potentially formed in a residual cooling flow against the mass of relatively old stars in BCGs provides a gauge of the relative importance of different pathways for the stellar growth of their progenitor red nuggets.",
"By fitting a model SED having an age of 5.3 Gyr (i.e., formation at $z \\sim 2$ ) and $Z = \\rm Z_$ to the measured SED of the old stellar population in the BCG of MACS J0329.7$-$ 0211, we derive a mass of $\\sim $$8 \\times 10^{11} \\rm \\, M_$ for this population.",
"For comparison, [5] estimate a stellar mass of $\\sim $$4 \\times 10^{11} \\rm \\, M_$ for the same BCG using also images from the CLASH program, but extracting and modelling the measured SEDs in a different way.",
"We determine an effective radius for the BCG in MACS J0329.7$-$ 0211 based on its old stellar population of $\\sim $ 20 kpc, at least 20 times the effective radii of red nuggets.",
"Assuming an order of magnitude growth in stellar mass from $\\sim $$8 \\times 10^{10} \\rm \\, M_$ for its progenitor red nugget to that of $\\sim $$8 \\times 10^{11} \\rm \\, M_$ at $z = 0.45$ , a persistent $SFR(t) \\simeq 2 {\\rm \\, M_\\, yr^{-1}}$ from a residual cooling flow since $z \\lesssim 2$ could have contributed at most $\\sim $ 1.5% to the stellar growth of this BCG compared to other pathways such as dry mergers.",
"Unless $SFR(t)$ from a residual cooling flow was highly elevated in the past, then this pathway could have played only a minor role in the stellar growth of the BCG in MACS J0329.7$-$ 0211.",
"We contrast our results with those of [27] and [28], from which one might reach a diametrically opposite conclusion.",
"As mentioned in Section REF , [28] adopted an exponentially decaying starburst to find a current $SFR = 39.8^{+20.5}_{-14.8} \\, \\rm M_\\, yr^{-1}$ (and therefore a higher $SFR(t)$ into the past).",
"[28] do not report the stellar mass formed in the presumed starburst; nonetheless, based on the parameters provided, the stellar mass formed since the starburst began $\\sim $ 1 Gyr is well in excess of $4 \\times 10^{10} {\\rm \\, M_}$ (i.e., given a nominal $SFR(0) = 40 \\rm \\ M_\\, yr^{-1}$ and a $SFR(t)$ that increases into the past until the beginning of the starburst).",
"The stellar mass inferred to have formed over the past 1 Gyr alone is well over half the original stellar mass of even the most massive red nuggets.",
"The conclusions one would reach from the results of [27] and [28] for the BCG in RX J1532.9+3021 ($z=0.363$ ), having a factor of just over 2 higher UV luminosity than the BCG in MACS J0329.7$-$ 0211, provide an even greater contrast.",
"For this galaxy, [27] and [28] fit model SEDs to the spatially-integrated as well as the spatially-resolved SEDs (i.e., over different sightlines) adopting a $SFR(t)$ that has decayed exponentially over time.",
"In both methods, they derived a stellar mass of $\\sim $$1 \\times 10^{11} \\rm \\, M_$ formed by this galaxy during the present starburst of duration $\\sim $ 0.7 Gyr thus far.",
"The inferred mass in new stars is comparable with the stellar mass of the most massive red nuggets known at $z \\gtrsim 2$ .",
"An even more instructive comparison is provided by contrasting the mass of new stars formed with the mass of old stars in this galaxy of $(5.33 \\pm 0.61) \\times 10^{11} \\rm \\, M_$ [5].",
"The present starburst alone, which has lasted $\\sim $ 0.7 Gyr to date, already constitutes $\\sim $ 20% by mass of the entire old stellar population in this galaxy!",
"In brief, at least for the BCGs in the Perseus cluster and MACS J0329.7$-$ 0211 that differ by over an order of magnitude in their star-formation rates, persistent star formation from a residual cooling flow has only contributed in a relatively minor if not negligible manner to their overall stellar growth since $z \\sim 1$ –2 – unless their persistent star-formation rates at their respective epochs are much lower than those in their past and, for the BCG in MACS J0329.7$-$ 0211, possibly also future.",
"Rather than contributing in a major way to the stellar growth of most if not all BCGs, star formation from a residual cooling flowOur work does not address stellar growth from star formation owing to wet mergers, which [47] argue is the major cause of star formation observed in BCGs at $z \\gtrsim $ 0.6. may play a more significant role in contributing to the enormous numbers of globular clusters around BCGs.",
"For example, the BCG in the Perseus cluster has formed, on average, one globular cluster with a mass of $\\sim 10^5 \\rm \\, M_$ (at the peak of the mass distribution in globular clusters) every 1 Myr for the past, at least, 1 Gyr [43].",
"Extrapolated to the BCG in MACS J0329.7$-$ 0211, it has already formed about ten times as many progenitor globular clusters over a timescale that is a factor of $\\sim $ 2 shorter.",
"A sustained formation over time does not fit easily into current narratives for the formation of globular clusters (e.g., see a brief review in [29]), but deserves due consideration in the case of BCGs."
],
[
"Summary and Conclusions",
"We have determined, to the highest degree of accuracy that we believe is possible using data from the CLASH program, the formation history of the young stellar population in the BCG of MACS J0329.7$-$ 0211 at $z=0.45$ , observed at a time when the Universe was two-thirds of its present age.",
"By fitting for and then subtracting the 2-dimensional light distribution of the old stellar population in this BCG, we are able to isolate just the spectral energy distribution (SED) of its young stellar population along individual spatially-resolved sightlines (Section ).",
"The subtracted images, which were convolved to a common angular resolution corresponding to the point spread function of the lowest angular resolution image at F110W, reveal that the young stellar population is detectable from the near-UV to the near-IR, as well as the presence of line (H$\\alpha $ +[NII]) emitting gas not spatially coincident with the young stellar population and a relatively compact dust feature observed in silhouette near the center of the BCG (Section ).",
"We fit model single stellar populations (SSPs; i.e., stars all sharing a common age and metallicity in a given population, and with a Kroupa initial mass function) to the measured SEDs along individual sightlines, each having a cross-sectional area of $1.5 \\times 1.5$ kpc that is comparable to the full-width half-maximum of the common point spread function of the images (Section ).",
"We find, apart from the region where silhouette dust is visible and hence our subtraction of the old stellar population is badly compromised, no significant dust extinction towards the young stellar population.",
"A simple comparison between the model and measured SEDs reveal that the young stellar population span ages from a few Myr to a few hundreds of Myr.",
"To quantify the star-formation history (i.e., star-formation rate as a function of time) of the young stellar population, we employed a Markov Chain Monte Carlo (MCMC) method to fit model SEDs (generated for SSPs) to the measured SEDs along individual sightlines, yielding the full probability distribution in both age and birth mass along each sightline for a given selected metallicity, $Z$ , of either $Z = 0.4 \\rm Z_$ (approximately that of the intracluster gas) or $Z = \\rm Z_$ .",
"We find that H$\\alpha $ +[NII] line emission not associated with HII regions is detectable along many sightlines to the young stellar population as shown in Figure REF , even (and especially) at ages $> 5$ Myr when any line emission from HII regions are predicted to no longer be detectable over the bandpasses of the filters employed (Section ).",
"This line-emitting gas accounts for at least $\\sim $ 60% (for $Z = \\rm Z_$ ) to at least $\\sim $ 80% (for $Z = 0.4 \\rm Z_$ ) of the H$\\alpha $ +[NII] emission from the BCG.",
"Attributing all this line emission to that from HII regions would result in a vast over-estimate of the instantaneous star-formation rate.",
"The result for the star-formation history of the young stellar population is shown in Figure REF (Section ), which presents the central finding of our work.",
"To properly interpret this result, we need to take into account observational selection effects: in doing so, we find that the formation rate of the young stellar population has, apart from a brief elevation to $\\sim $$10 \\rm \\, M_\\, \\rm yr^{-1}$ about $10^7$ yr ago, been approximately constant at $\\sim $$2 \\rm \\, M_\\, \\rm yr^{-1}$ over the past $\\sim $ 400 Myr, beyond which any star formation drops below the observational detection threshold (Section ).",
"Star formation that is extended not only over space (spanning a projected linear dimension of $\\sim $ 30 kpc in this BCG) but also through time provides further support for the argument that young stellar populations in (some) BCGs are produced by a residual cooling flow that sustains a significant rate of star formation over an indefinite period.",
"Such persistent star formation from a residual cooling flow can contribute up to $\\sim $ 10% of the original stellar mass of the BCG in MACS J0329.7$-$ 0211 if its progenitor was among the most massive red nuggets known at $z \\sim 2$ having masses $\\sim $$1 \\times 10^{11} \\rm \\, M_$ , but only a few percent of its overall growth in stellar mass to $\\sim $$8 \\times 10^{11} \\rm \\, M_$ at $z=0.45$ .",
"Instead, the prodigious number of star clusters formed from a residual cooling flow may play a more important role in contributing to the enormous numbers of globular clusters around BCGs.",
"J. Lim acknowledges support from the Research Grant Council of Hong Kong through the grant 17304817, which also supported the MPhil studentship of J. Li.",
"Y.O.",
"acknowledges the support by the Ministry of Science and Technology (MOST) of Taiwan through grants, MOST 109-2112-M-001-021-.",
"HST (ACS and WFC3) Astropy [1], [2], Stsynphot [67], Imfit [24]"
],
[
"MCMC algorithm",
"We adopt a fully Bayesian calculation for extracting physical parameters from the measured SEDs using a Markov Chain Monte Carlo (MCMC) approach.",
"The similarity between the measured SED and the model SED having a given parameter set $\\mathbf {\\Theta } \\equiv \\lbrace Z, f_{cov}, A_v, \\log t_{\\rm age}, \\log M_* \\rbrace $ , where $Z$ is the adopted metallicity (either $0.4\\,\\rm Z_$ or 1.0$\\,\\rm Z_$ ), $f_{cov}$ the adopted covering factor (either 0 or 1), $A_v$ the dust extinction (either set to 0 or, for the purpose of investigating any dust extinction, restricted to $\\le 0.5$ mag), $t_{\\rm age}$ the age of the stellar population (up to 1 Gyr), and $M_*$ the birth mass derived by integrating stars having a Kroupa initial mass function over the stellar mass range 0.1–$100 \\rm M_$ , is evaluated using the standard $\\chi ^2$ statistic: $\\chi ^2(\\mathbf {\\Theta }) \\equiv \\sum _i \\frac{\\bar{f}_i - f_i(\\mathbf {\\Theta })}{\\sigma _{noise, i}^2} \\, \\, ,$ where $i$ runs over all the filters used for constructing the SED, $\\bar{f_i}$ is the flux density of the measured SED in the $i$ 'th filter, $f_i(\\mathbf {\\Theta })$ the flux density of the model SED in the $i$ 'th filter, and $\\sigma _{noise, i}$ the measurement uncertainty in flux density at the $i$ 'th filter.",
"The probability that a given model SED can reproduce the measured SED is given by $p(\\bar{f}_i | \\mathbf {\\Theta }) \\propto \\exp (-\\chi ^2/2)$ , which describes a Gaussian centered at $f(\\mathbf {\\Theta })$ .",
"The posterior distribution, $p(\\mathbf {\\Theta } | \\bar{f})$ , describing the likelihood that the different model SEDs considered fit the measured SED, is simply obtained from the Bayes Theorem: $ p(\\mathbf {\\Theta } | \\bar{f}) \\propto p(\\bar{f}_i | \\mathbf {\\Theta }) p(\\mathbf {\\Theta }) \\, \\, ,$ where $p(\\mathbf {\\Theta })$ is the adopted prior (i.e., initial guess in $\\mathbf {\\Theta }$ , which can be a functional form).",
"To reflect our ignorance of which model SED can best fit the measured SED, we adopt an uniform prior.",
"Equation (REF ) possess an intuitive interpretation: the peak in the posterior distribution corresponds to the model SED having parameters $\\mathbf {\\Theta }$ at which $\\chi ^2$ is minimized, and hence that which best agrees with the measured SED.",
"Equation (REF ) is often numerically intractable because the normalization for the posterior requires expensive numerical integration.",
"Instead, we generated samples from the posterior distribution $p(\\Theta | \\bar{f})$ using a MCMC approach.",
"The posterior distribution can then be approximated by a kernel density estimation of the samples.",
"In this work, we use the PyMultiNest algorithm developed by [4], which is a modification of the standard MCMC implementation so as to provide a faster convergence on multi-modal and asymmetric posteriors.",
"Like for the age parameter $t_{\\rm age}$ , we used a logarithmic instead of a linear scale for $M_*$ to restrict the number of values explored so as to not far outnumber those explored in $A_V$ (where relevant) and $t_{\\rm age}$ .",
"Otherwise, the algorithm that we wrote for performing the MCMC computation would have been much slower or not converge in practise."
],
[
"Computing Star Formation History",
"Our goal is to compute the posterior of $SFR(t)$ , namely $p(SFR(t)| \\lbrace s \\rbrace )$ , given the posteriors of individual parameters in the set $\\Theta _s \\equiv \\lbrace \\log t_{\\rm age,s}, \\log M_{*,s}\\rbrace $ along each sightline, $s$ , for a particular adopted metallicity.",
"To begin, we outline the calculation of the probabilistic average instead of the full probability distribution so as to provide more insight on the full calculation.",
"In practise, when computing the $SFR$ at a given $t = t_{\\rm age}$ having discrete values selected as mentioned in Section REF , we choose a time interval, $\\Delta t$ , equal to the difference between the corresponding adjacent age steps.",
"Probabilistically, if a particular sightline has the probability $p(t_{\\rm age,s}, M_{*,s}|s)$ of having an age $t_{\\rm age,s}$ and a birth mass $M_{*,s}$ , the expected value in birth mass that this sightline contributes to the age bin $\\Delta t^{(i)} \\equiv t^{(i+1)}-t^{(i)}$ is: $ \\langle M | \\Delta t^{(i)} \\rangle _s = \\int ^{t^{(t+1)}}_{t^{(i)}} \\int ^\\infty _0 M_{*,s}\\cdot p(t_{\\rm age,s}, M_{*,s}|s)) \\, dM_{*,s} \\, dt_{\\rm age,s}.$ Summing the expectation along all sightlines, $ \\langle SFR(t\\in \\Delta t^{(i)}) \\rangle = \\frac{1}{\\Delta t^{(i)}}\\sum _s \\langle M | \\Delta t^{(i)} \\rangle _s = \\sum _s \\langle \\frac{ M}{\\Delta t^{(i)}} | \\Delta t^{(i)} \\rangle _s \\,\\, ,$ where the bin width $\\Delta t^{(i)}$ is a fixed value at a given $t = t_{\\rm age}$ as mentioned above.",
"While the double integral in Eq.",
"REF cannot be evaluated analytically, we can estimate the expected value by utilizing the MCMC samples obtained.",
"The above formalism on the expectation $\\langle SFR(t\\in \\Delta t^{(i)}) \\rangle $ can be extended to calculate the full posterior $p(SFR(t)| \\lbrace s \\rbrace )$ .",
"With MCMC, we get sample ages, $\\tilde{t}_{\\rm age, s}$ , and masses, $\\tilde{M}_{*,s}$ , from the probability distribution along each sightline, $\\tilde{M}_{*,s}, \\tilde{t}_{\\rm age, s} \\sim p(t_{\\rm age,s}, M_{*,s}|s)$ .",
"As a consequence, Eq.",
"REF can be replaced as a Monte Carlo sum, and be promoted to a probability by removing the integration: $p( M_s| \\Delta t^{(i)}, s ) = \\frac{1}{N_{i,s}} {1}(M_s = \\tilde{M}_{*,s}) \\,\\, ,$ where $N_{i,s}$ is the count of the number of samples belonging to the age bin $\\Delta t^{(i)}$ for a particular sightline $s$ .",
"The indicator function 1 simply counts the number of mass samples $\\tilde{M}_{*,s}$ (right-hand side) that matches the argument $M_s$ (left-hand side).",
"We can promote Eq.",
"REF to a full posterior in a similar way.",
"As $SFR(t\\in \\Delta ^{(i)}) = \\sum _s M_{s}/\\Delta t^{(i)}$ , we therefore have: $p( SFR(t\\in \\Delta t^{(i)} | \\lbrace s \\rbrace ) = p( \\sum _s M_{s}/\\Delta t^{(i)} | \\lbrace s \\rbrace ) = \\frac{1}{N_{i,s}} {1}( SFR=\\sum _s \\frac{\\tilde{M}_{*,s}}{\\Delta t^{(i)}} ) \\,\\, .$ Again, the indicator function 1 is there to match both sides.",
"In practice, as the indicator function only occupies a point in the entire posterior space, we need to use a kernel density estimation to `smear out' the indicator function according to the density of samples in its neighbourhood.",
"This procedure is exactly the standard procedure used whenever MCMC samples are converted to probability contours plot.",
"One can check, upon taking the expectation (i.e., integrating over the probability), that we can get back to Eq.",
"REF , as is required.",
"We have checked that the resulting posteriors obtained from above procedure are, to a good approximation, a log-normal distribution, despite the individual posteriors for each sightlines being much more complex.",
"Therefore we report in Figure REF (lower panel) just the mean and standard deviation for $SFR(t)$ calculated using this probabilistic approach.",
"The age bins over which we plot the SFH as shown in Figure REF (lower panel) has been designed so as to have approximately equal number of samples in each bin.",
"Conceptually, we generate many different realizations for $t_{\\rm age}$ and $M_*$ over different sightlines, and then selected time intervals over which the number of samples are approximately equal in each time bin."
],
[
"Completeness Limits",
"In Section REF , we explained how we selected acceptable – satisfactory and sensible – model SEDs fits to the measured SEDs of the young stellar population.",
"In brief, we used two criteria: (i) a standard deviation in the posterior distribution for age of $\\delta _{\\rm age} < 190$ Myr (see Section REF for how $\\delta _{\\rm age}$ is computed); together with (ii) a nominal birth mass, $M_*$ , above a mass threshold that increases with age.",
"Below, we describe how the adopted threshold of $\\delta _{\\rm age} < 190$ Myr sets a completeness limit – in a probabilistic sense – for the birth mass at a given age when fitting model SEDs to the measured SEDs using our MCMC approach.",
"This completeness limit defines the threshold in birth mass as a function of age for acceptable fits.",
"As mentioned in Section REF and in Appendix , for the purpose of the MCMC analyses, we generated model SEDs, $\\mathcal {F}$ , at logarithmic intervals in age (except for additional steps of 1 Myr from 10–15 Myr ), $t_{\\rm age}$ , and birth mass, $M_*$ , so as to produce an approximately uniform logarithmic grid in these model parameters, $\\lbrace \\Theta ^{(i,j)} \\rbrace \\equiv \\lbrace t_{\\rm age}^{(i)}, M_{*}^{(j)} \\rbrace $ .",
"The set of model SEDs, $\\mathcal {F}(\\Theta ^{(i,j)})$ , in these parameters alone therefore number $N_{t_{\\rm age}} \\times N_{M_*} = \\sum i \\times \\sum j$ .",
"To simulate the effect of noise, we perturbed each of these model SEDs by a Gaussian distribution, $\\mathcal {G}(0,\\sigma _{noise})$ , centered at 0 and with a standard derivation of $\\sigma _{noise}$ ; the latter is chosen to be the maximal noise over the image in a given filter (Section REF ), thus corresponding to the upper limit for the detection threshold in that filter.",
"In this way, we generated a set of $N_k = 30$ noise-perturbed realizations for each model SED, ${\\mathcal {F}}_k(\\Theta ^{(i,j)})$ .",
"We treat these noise-perturbed model SEDs as mock data to be fitted by the original noiseless model SEDs using the same MCMC approach as for the measured SEDs (Appendix ), thus yielding the posteriors $p \\left( \\Theta | {\\mathcal {F}}_k(\\Theta ^{(i,j)}) \\right)$ .",
"At each grid point $(i,j)$ , we therefore have $N_k$ posteriors, $p \\left(\\Theta ^{(i,j)} | {\\mathcal {F}}_k(\\Theta ^{(i,j)})\\right)$ , and $N_k$ standard deviations for the posterior distribution in ages, $\\delta _{t_{\\rm age}}^{(i,j,k)} \\equiv \\left(\\left[ p(\\tau \\in \\Theta ^{(i,j)} | {\\mathcal {F}}_k(\\Theta ^{(i,j)}) \\right]\\right)^{1/2}$ .",
"The probability for a particular parameter combination $\\Theta ^{(i,j)} = (t_{\\rm age}^{(i)}, M_{*}^{(j)})$ to pass the selection criteria $p\\left(\\delta _{t_{\\rm age}}^{(i,j)} < 190 {\\rm \\, Myr} | \\Theta ^{ij} = ({t_{\\rm age}}^{(i)}, M_{*}^{(j)})\\right)$ is given by: $p\\left({\\sigma }_{t_{\\rm age}}^{(i,j)} < 190 {\\rm \\, Myr} | \\Theta ^{(i,j)} = {t_{\\rm age}^{(i)}}, M{_{*}^{(j)}} \\right) \\approx \\frac{\\sum _k {1}\\left({\\sigma }_{t_{\\rm age}}^{(i,j,k)} < 190 {\\rm \\, Myr} \\right)}{N_k} \\, \\, ,$ whereby $\\sum _k {1} \\left({\\sigma }_{t_{\\rm age}}^{(i,j,k)}\\right)$ is simply the sum of the number of realizations among the noise-perturbed model SEDs that individually yield an age posterior with a standard deviation of ${\\sigma }_{t_{\\rm age}}^{(i,j,k)} < 190$ Myr (and so, divided by $N_k$ , is simply the fraction of realizations for a particular parameter combination $\\Theta ^{ij} = ({t_{\\rm age}^{(i)}}, M_{*}^{(j)})$ that passes the selection criterion).",
"Because the noise-perturbed model SEDs are generated using the maximal noise uncertainty in each filter image, the fraction that passes the selection criterion is a lower limit.",
"The results are plotted in Figure REF for $0.4 \\rm \\, Z_$ (left panel) and $1.0 \\rm \\, Z_$ (right panel), where we show the fraction of realizations that satisfy ${\\sigma }_{t_{\\rm age}}^{(i,j,k)} < 190$ Myr as a function of $\\log M_*$ versus $\\log t_{\\rm age}$ .",
"At a given metallicity, the boundary between $p\\left(\\sigma _{t_{\\rm age}}^{(i,j)}\\right) = 1$ (red region) and $p\\left(\\sigma _{t_{\\rm age}}^{(i,j)}\\right) < 1$ (blue region) increases approximately linearly in $\\log M_*$ with $\\log t_{\\rm age}$ as indicated by a dashed diagonal line, which therefore approximately defines the completeness limit in $M_*$ at a given $t_{\\rm age}$ .",
"The same diagonal lines are drawn in Figure REF (middle column) as one of the two criteria for selecting acceptable model SED fits to the young stellar population in the BCG."
],
[
"Correcting for Completeness Limits in Star Formation History",
"The increasing lower bound in the birth mass, $M_*$ , with age, $t_{\\rm age}$ , found in our model SED fits to the measured SEDs of the young stellar population as shown in Figure REF (middle column) is a selection effect as demonstrated in Appendix REF .",
"This resulting detection threshold is indicated approimately by the dotted diagonal lines in Figure REF (middle column), which imposes a bias on the star formation history as computed from $M_*$ and $t_{\\rm age}$ for each sightline and integrated over all sightlines.",
"For example, an intrinsically constant star-formation rate over time ($SFR(t) \\propto t^0$ ) would be inferred from the measurements to be a decreasing star-formation rate into the past; even an increasing star-formation rate into the past could potentially show the opposite behaviour in the inferred star-formation history.",
"The completeness limit, $M_{*({\\rm min})}(t_{\\rm age})$ , indicated by the dotted lines in Figure REF can be approximately described by the functional form: $M_{*({\\rm min})}(t_{\\rm age}) = M_0 \\left( \\frac{t_{\\rm age}}{\\text{Myr}} \\right)^\\alpha ={\\left\\lbrace \\begin{array}{ll}1.1\\times 10^4 \\left( \\frac{t_{\\rm age}}{\\text{Myr}}\\right)^{1.1} \\, \\text{M}_\\odot ,\\, Z=0.4\\,\\rm Z_{\\odot } \\\\1.5\\times 10^4 \\left( \\frac{t_{\\rm age}}{\\text{Myr}}\\right)^{1.1} \\, \\text{M}_\\odot ,\\, Z=\\rm Z_{\\odot }\\end{array}\\right.",
"}\\, \\,$ to within a scatter of $\\lesssim 0.1$ dex irrespective of the adopted metallicities and internal dust extinction of $0 \\le A_V \\le 0.5$ .",
"The number of sightlines that would be selected, $N_\\text{obs}(M_*,t_{\\rm age})$ , owing to the completeness limit is therefore: $N_\\text{obs}(M_*,t_{\\rm age}) ={\\left\\lbrace \\begin{array}{ll}N(M_*,t_{\\rm age}), & M_* \\ge M_{*({\\rm min})}(t_{\\rm age}) \\\\0, & M_* < M_{*({\\rm min})}(t_{\\rm age})\\end{array}\\right.",
"}\\, \\, ,$ where $N(M_*,t_{\\rm age})$ is the total number of sightlines having $M_*$ and $t_{\\rm age}$ .",
"At a given $t_{\\rm age}$ , the detectable total mass, ${\\mathbb {M}}_{\\rm obs} (t_{\\rm age})$ , having $M_* \\ge M_{*({\\rm min})}(t_{\\rm age})$ along individual relevant sightlines compared with the actual total mass, ${\\mathbb {M}}_{\\rm total} (t_{\\rm age})$ , above an intrinsic low-mass cutoff, $M_{*(\\rm low)}$ , is given by: $ \\frac{{\\mathbb {M}}_{\\rm obs} (t_{\\rm age})}{{\\mathbb {M}}_{\\rm total} (t_{\\rm age})} = \\frac{\\int ^{M_{* \\rm (high)}}_{M_{*({\\rm min})}(t_{\\rm age})} M_* \\cdot dN(M_*, t_{\\rm age})}{\\int ^{M_{* \\rm (high)}}_{M_{*(\\rm low)}} M_* \\cdot dN(M_*, t_{\\rm age})} \\, \\, ,$ where $M_{*(\\rm high)}$ is the intrinsic high-mass cutoff.",
"Evaluating Eq.",
"REF requires knowledge of ${\\partial N(M_*)}/{\\partial M_*}$ , the number of sightlines as a function of $M_*$ at a given $t_{\\rm age}$ .",
"Figure REF shows ${\\partial N(M_*)}/{\\partial M_*}$ versus $M_*$ for all sightlines having $t_{\\rm age} \\le 10\\,{\\rm Myr}$ for both $Z = 0.4\\,\\rm Z_$ (left panel) and $Z =\\rm Z_$ (right panel).",
"The best linear fits to these log-log plots are indicated by the solid lines, and have a power-law index, $\\beta $ (whereby ${\\partial N(M_*)}/{\\partial M_*} \\propto M_*^\\beta $ ), of between $-1.2$ and $-1.5$ .",
"As the adopted model SSPs fade by a factor of 4 between ages of 1 Myr and 10 Myr, the slopes of the best-fit lines represent upper limits; i.e., the actually slope should be steeper after correcting for the completeness limit as a function of age.",
"Dashed lines having a power-law index of $-2.1$ are shown also in Figure REF , corresponding to an apparently universal number dependence of star clusters with mass across all mass scales – whether it be open star clusters in our Galaxy, massive star clusters in interacting or merging galaxies, massive star clusters in the BCG in the Perseus cluster, or globular clusters above the peak in their mass function – thereby implicating a common underlying mechanism for the formation of star clusters over all mass scales (see [43]).",
"If each sightline to the young stellar population in the BCG of MACS J0329.7$-$ 0211 encompasses multiple star clusters having the same number dependence, then statistically we would expect ${\\partial N(M_*)}/{\\partial M_*}$ versus $M_*$ to also exhibit a power-law index of $-2.1$ , as we shall henceforth adopt.",
"Figure: Histograms of inferred masses (M * M_*) with inferred ages (t age ≤10t_{\\rm age} \\le 10\\,Myr), assuming A V =0A_V=0, Z=0.4Z ⊙ Z=0.4\\,\\rm Z_{\\odot } (left) and Z=Z ⊙ Z=\\rm Z_{\\odot } (right).",
"The bestfit power-law functions are plotted as orange lines with the parameters indicated in legends.",
"Power-law with β=-2.1\\beta =-2.1 is also plotted manually as grey dashed line to make visual comparison.Adopting also $M_{*(\\rm low)} = 10^5 {\\rm \\, M_}$ , approximately the lowest $M_*$ detected, and $M_{*(\\rm high)} = 10^7 {\\rm \\, M_}$ , approximately the highest $M_*$ detected, Eq.",
"REF can then be solved to give: $\\frac{{\\mathbb {M}}_{\\rm obs} (t_{\\rm age})}{{\\mathbb {M}}_{\\rm total} (t_{\\rm age})} = \\frac{M_0^{\\beta +2}t_{\\rm age}^{\\alpha (\\beta +2)} - M_{*(\\rm high)}^{\\beta +2}}{M_{*(\\rm low)}^{\\beta +2} -M_{*(\\rm high)}^{\\beta +2} } \\, \\, .$ This scaling relation allows a simple correction to be made to the measured $SFR(t)$ so as to derive the actual $SFR(t)$ ; in practise, owing to the uncertainties associated with the measured $SFR$ at a given $t$ , we compare the latter to that we would infer for different star-formation histories given the selection bias introduced by Equation REF .",
"The results are shown in Fig REF for a constant or, to mimic a short-duration starburst in the local Universe, an exponentially decaying star-formation rate over time.",
"Although we adopt $\\beta = -2.1$ to correct for completeness, motivated by the expectation that the young stellar population in the BCG is composed of an ensemble of star clusters, the correction is only weak sensitive to the exact value of $\\beta $ at least between $-1.0$ and $-2.1$ ."
]
] | 2107.01771 |
[
[
"Towards Real-World Applications of ServiceX, an Analysis Data\n Transformation System"
],
[
"Abstract One of the biggest challenges in the High-Luminosity LHC (HL- LHC) era will be the significantly increased data size to be recorded and analyzed from the collisions at the ATLAS and CMS experiments.",
"ServiceX is a software R&D project in the area of Data Organization, Management and Access of the IRIS- HEP to investigate new computational models for the HL- LHC era.",
"ServiceX is an experiment-agnostic service to enable on-demand data delivery specifically tailored for nearly-interactive vectorized analyses.",
"It is capable of retrieving data from grid sites, on-the-fly data transformation, and delivering user-selected data in a variety of different formats.",
"New features will be presented that make the service ready for public use.",
"An ongoing effort to integrate ServiceX with a popular statistical analysis framework in ATLAS will be described with an emphasis of a practical implementation of ServiceX into the physics analysis pipeline."
],
[
"Introduction",
"ServiceX is a scalable HEP event data location, extraction, filtering, and transformation system that has been developed as part of the Institute for Research and Innovation in Software for High Energy Physics (IRIS-HEP).",
"It runs on any Kubernetes cluster and can be offered as a public service or hosted on an institution's private cluster.",
"ServiceX accepts requests via a REST interface.",
"The requests include a dataset identifier (DID) that resolves to a number of input data files along with a columnar event data selection statement expressed in an elemental expression language called Query AST Language Expressions (Qastle) [1].",
"The service relies on Rucio [2] to lookup file replicas for the requested DID.",
"It will attempt to select the file replicas that are most efficiently accessible by the host Kubernetes cluster.",
"The Qastle syntax is designed to be translated into code that describes operations on input data, with the transformer providing the data handling libraries.",
"There is currently support for C++ code that is run in a transformer based on the ATLAS Event-Loop framework.",
"Additionally, there is a python code generator that produces a script to drive the python Uproot [3] library.",
"It is suited for reading flat ntuples such as CMS NanoAOD [4] files and analysis group generated files.",
"The results from a ServiceX transformation are either flat ROOT files or Parquet columnar data files persisted to an object store which can be accessed via HTTP protocols or with the Amazon S3 API.",
"Figure REF shows how each of these steps is assembled to produce the desired set of transformed files.",
"This paper introduces the latest developments in ServiceX that allow the system to be readily implemented into real-world applications.",
"Analysis pipelines that employ ServiceX for the currently available transformers are outlined.",
"In this paper, an analysis pipeline refers to a chain from the reconstructed object to the final result.",
"The primary goal of this paper follows to establish a practical implementation of ServiceX into the analysis pipeline that can be utilized in physics analysis."
],
[
"New features in ServiceX",
"ServiceX was first presented at CHEP 2019 [7].",
"Since that conference, the system has evolved to make it useful as a production system to solve real world problems.",
"The main enhancements were:"
],
[
"Public access",
"While ServiceX can be deployed in an experiment group's private Kubernetes cluster, users will still wish to connect to it from remote locations.",
"As soon as the service is exposed to the internet, careful consideration must be given to securing it.",
"In version 1.0, ServiceX uses Globus Auth [8] to authenticate users.",
"Administrators are notified of user signups in a private Slack channel and can approve new accounts from there.",
"Approved users are given an API token for making requests to the ServiceX deployment."
],
[
"Auto-scaling of Transformer Pods",
"Initial versions of ServiceX required the user to specify the number of workers to launch to process the transformation job.",
"This was simple to implement, but potentially wasteful in its use of resources, since CPUs could be sitting around idly.",
"The service now makes use of Kubernetes auto-scaling capabilities to only launch new pods if files becoming available from Rucio exceed the existing set."
],
[
"Support for Reading ATLAS xAOD Files",
"Reading of flat ntuples using Uproot libraries and Awkward arrays lends itself quite easily to the columnar style of analysis advanced by ServiceX.",
"The ATLAS xAOD [9] files are another matter all together as xAOD files use custom ROOT objects and require a C++ framework to fully utilize.",
"Research into the analysis description languages, func-adl, provided ServiceX with the ability to translate high level event selection queries into C++ code that is executed by an experiment approved framework inside the transformer pods."
],
[
"Support for High-Level Expressions",
"ServiceX uses an elemental LISP-inspired expression language called Qastle to represent event selection and transformation request.",
"It is not intended for end-users to author selections in this representation.",
"Instead, it is hoped that researchers of analysis description languages will create transpilers to generate Qastle queries from a higher level language.",
"The ServiceX backend will immediately allow them experiment with their language using the full scale resources of the server.",
"Initial work was done using the func-adl language from the Watts lab.",
"As described in more detail in section REF , it is equally important to work with popular event selection languages to encourage analysts to move their research over to this new environment with minimal disruption."
],
[
"ServiceX in analysis pipeline",
"Figure REF shows a schema of the current ATLAS analysis pipeline starting from the AOD (Analysis Object Data) to the final analysis formats in black lines.",
"Many individual and group-based derivations, which reduce the size of AOD by removing unnecessary information, exist and even smaller ROOT ntuples for the final analysis.",
"ServiceX, on the other hand, can access directly to the upstream of the analysis pipeline as it supports different types of transformers for different input file formats.",
"The transformers available today are developed for those file formats that are primary in the analysis pipeline at present: xAOD transformer for the ATLAS xAOD or Derived AOD format (red lines), and Uproot transformer for flat ROOT ntuples (orange line).",
"The latter is also compatible with the CMS NanoAOD format, which also features a flat ROOT TTree structure.",
"A transformer for the CMS MiniAOD format, which also requires dedicated libraries similar to the ATLAS xAOD/DAOD format, is currently being developed.",
"The strong point of ServiceX is the flexibility of the transformer.",
"The architecture implemented in ServiceX relies on a number of containerized microservices which include a transformer.",
"Thus ServiceX is adaptable to future data formats with new transformers, and it is also feasible to have dedicated transformers for specific purposes."
],
[
"ServiceX for statistical analysis framework",
"Given that ServiceX is relatively new to the community and the public release of the service became available recently, there are not many practical implementations of ServiceX into the analysis pipeline up to the present time.",
"It is also because of the fact that most analyzers are accustomed to the traditional analysis pipeline based on ROOT and grid jobs.",
"Therefore, it is helpful to lower the barrier to allow them to experience the new analysis ecosystem in Python.",
"We have been developed a tool that can be implemented into a practical physics analysis with a minimum effort from analyzers.",
"Figure: Current workflow of TRExFitter (top) and alternative ServiceX workflow (bottom) to generate histograms from ROOT ntuples that are produced on the grid."
],
[
"TRExFitter",
"TRExFitter [10] is a framework used by many ATLAS physics analyses for statistical inference via profile likelihood fits.",
"It is designed to handle everything that analysers need for the statistical part of their analysis.",
"It produces RooFit workspaces, perform fits on them, and interfaces with RooStats macros for limit and significance.",
"It also generates publication-level pre-fit and post-fit plots and tables.",
"Many more additional features are also supported to understand the fit behavior.",
"TRExFitter takes ROOT ntuples or histograms as input format, and a configuration file to steer the framework.",
"A configuration file includes high-level physics choices: specification of signal/validation/control regions (the channels in HistFactory schema), observables to be used for the statistical analysis, Monte Carlo samples for signal and background and data samples, sources of systematic uncertainties, details of fit model, parameter of interest, and other general settings.",
"TRExFitter provides the feature to generate histograms for statistical analysis from ROOT ntuples based on the provided configuration file.",
"Hence, it is a typical workflow to download whole ROOT ntuples from the grid to a local cluster or directly accessible machines to generate histograms as shown in the top of Figure REF .",
"On the other hand, ServiceX can deliver only necessary branches or columns with event filtering as shown in the bottom of Figure REF .",
"The advantages of the ServiceX workflow are as follows: Local storage: It takes up less storage space as ServiceX delivers minimal information to generate histograms.",
"Download time: Data transferred over WAN is smaller for the ServiceX workflow.",
"The network speed between a grid site and Kubernetes cluster is sufficiently fast as they are usually co-located."
],
[
"TCut translator for ServiceX",
"ServiceX utilizes func-adl, a python-based declarative analysis description language, to filter events and request branches from the input data file.",
"It is an intuitive language to extract data directly from ROOT ntuples, but TRExFitter relies on the ROOT TTree::Draw method, which uses TCut syntax for TTree selections.",
"Since TCut syntax is not directly readable by the func-adl, a python package for TCut to func-adl translation is developed.",
"The package further converts func-adl to Qastle language, which is the language that ServiceX transformers understand.",
"The package supports arithmetic operators (+, -, $\\ast $ , /), logical operators (!, $\\&\\&$ , ||), and relational and comparison operators (==, !=, >, <, >=, <=).",
"Listing shows an example of translating a ROOT-based query into a ServiceX query.",
"The first argument is the name of the TTree object in an input file.",
"The second is the list of selected branches for delivery.",
"The last argument is the TCut object for TTree selection.",
"Only events that pass the selection will be delivered for the requested branches.",
"Thus, the second argument effectively removes branches from the input ROOT tree, and the third drops events.",
"The package, tcut-to-qastle [11], is published at PyPI for a convenient access.",
"import tcut_to_qastle as tq # Get ServiceX query query = tq.translate(\"nominal\", \"A,B,D\", \"(A && !B) || (C > 0.1)\")"
],
[
"$\\texttt {servicex-for-trexfitter}$",
"The $\\texttt {servicex-for-trexfitter}$ is a python package, which has been developed to provide seamless integration of ServiceX into the TRExFitter framework.",
"It makes use of the building blocks that are described above: Uproot ServiceX to read input ROOT ntuples from the grid and perform transformations; ServiceX frontend library to access ServiceX backend and manage ServiceX delivery requests; the tcut-to-qastle package to translate ROOT-based query into the ServiceX query.",
"Each of the following steps runs within the $\\texttt {servicex-for-trexfitter}$ package: prepares ServiceX requests by analyzing a TRExFitter configuration file to deliver a minimum amount of data from the grid, makes ServiceX requests simultaneously, downloads output of finished transformations asynchronously from the object store of Kubernetes cluster, and converts downloaded output files into a ROOT file for each sample.",
"It takes a TRExFitter configuration file, which has an identical structure with the traditional ROOT ntuple input.",
"The only addition is a new field for grid dataset ID for each sample since ServiceX reads input ROOT ntuples from the grid.",
"The caching feature of the ServiceX frontend library allows only modified or added part of the TRExFitter configuration creates new ServiceX requests."
],
[
"Example",
"The following prerequisites are needed to run the servicex-for-trexfitter package.",
"It is written for Python version equal to or higher than 3.6.",
"Access to an Uproot ServiceX endpoint is also required.",
"Any running Uproot ServiceX instance should work, or access to the centrally-managed ServiceX instance can be granted as described in the ServiceX documentation [12].",
"To convert the outputs from ServiceX into ROOT TTree, PyROOT [13] has to be installed.",
"Lastly, the input ROOT ntuples need to be organized in a way that each sample in the TRExFitter configuration file corresponds to a single Rucio dataset ID.",
"Listing shows an example of how to use the $\\texttt {servicex-for-trexfitter}$.",
"An instance can be created with an argument of TRExFitter configuration file.",
"Data transformation and delivery status can be interactively monitored just after the method get$\\_$ ntuples() is called, and the path to the slimmed/skimmed output ROOT ntuples will be printed once the delivery is completed.",
"from servicex_for_trexfitter import ServiceXTRExFitter sx_trex = ServiceXTRExFitter(\"example.config\") sx_trex.get_ntuples()"
],
[
"Benchmark results",
"The performance of $\\texttt {servicex-for-trexfitter}$ is measured and compared with the current workflow using the same TRExFitter configuration file.",
"A practical TRExFitter configuration file which contains 17 Samples and 34 Systematics is used for the benchmark.",
"The total size of ROOT ntuples is 650 Gigabytes stored at the MidWest Tier-2 Center.",
"The benchmark for ServiceX workflow utilizes the Uproot ServiceX deployed at the University of Chicago SSL-River Kubernetes cluster.",
"Table: Benchmark results from the current TRExFitter workflow and the workflow using 𝚜𝚎𝚛𝚟𝚒𝚌𝚎𝚡-𝚏𝚘𝚛-𝚝𝚛𝚎𝚡𝚏𝚒𝚝𝚝𝚎𝚛\\texttt {servicex-for-trexfitter}.The results are shown in Table REF .",
"The current workflow takes up the same amount of local disk space with the total size of ROOT ntuples on the grid, whereas the workflow using $\\texttt {servicex-for-trexfitter}$ takes up a lot smaller disk space as it delivers only information that is needed to generate histograms defined in the TRExFitter configuration file.",
"The wall time to download ROOT ntuples for the subsequent step, generating histograms from downloaded ROOT ntuples, shows a much shorter time for the workflow using $\\texttt {servicex-for-trexfitter}$.",
"This is due to the fact that ServiceX scales the workers to parallelize transformations and delivers only a subset of ROOT ntuples over WAN.",
"In addition, the time spent on processing downloaded ROOT ntuples is significantly shorter for the workflow using $\\texttt {servicex-for-trexfitter}$ as it needs to process only 0.4 GB than 650 GB.",
"The recent developments in ServiceX put the service in a state of a production system that allows practical implementations into a traditional analysis workflow.",
"The authentication system which enables public access over the internet is particularly beneficial as it opens the door to more users.",
"The auto-scaling of transformer pods makes the system more robust, and the new C++ transformer for the ATLAS xAOD improves the performance significantly.",
"The support of TCut expressions to the func-adl language lowers the threshold to try the service.",
"The real-world application which employs the new features of ServiceX has been developed to provide a novel data delivery method to the popular statistical analysis framework in ATLAS.",
"Accessing input ROOT ntuples directly from the grid saves significant amount of local disk space.",
"Scaling up the transformer pods in a Kubernetes cluster can remarkably reduce a turnaround time by extracting only necessary information from input ROOT ntuples.",
"The primary goal of the $\\texttt {servicex-for-trexfitter}$ package is also fulfilled by requiring a minimal addition to the traditional approach.",
"This paper describes the implementation of ServiceX to interface with the existing ROOT-based statistical analysis framework.",
"The HEP tools in Python are rapidly evolving, and ServiceX can nicely align with those tools to achieve a complete analysis workflow within the Python ecosystem."
]
] | 2107.01789 |
[
[
"Poisoning Attack against Estimating from Pairwise Comparisons"
],
[
"Abstract As pairwise ranking becomes broadly employed for elections, sports competitions, recommendations, and so on, attackers have strong motivation and incentives to manipulate the ranking list.",
"They could inject malicious comparisons into the training data to fool the victim.",
"Such a technique is called poisoning attack in regression and classification tasks.",
"In this paper, to the best of our knowledge, we initiate the first systematic investigation of data poisoning attacks on pairwise ranking algorithms, which can be formalized as the dynamic and static games between the ranker and the attacker and can be modeled as certain kinds of integer programming problems.",
"To break the computational hurdle of the underlying integer programming problems, we reformulate them into the distributionally robust optimization (DRO) problems, which are computationally tractable.",
"Based on such DRO formulations, we propose two efficient poisoning attack algorithms and establish the associated theoretical guarantees.",
"The effectiveness of the suggested poisoning attack strategies is demonstrated by a series of toy simulations and several real data experiments.",
"These experimental results show that the proposed methods can significantly reduce the performance of the ranker in the sense that the correlation between the true ranking list and the aggregated results can be decreased dramatically."
],
[
"Introduction",
"Rank aggregation, in particular estimating a ranking based on comparisons between pairs of objects, arises in a variety of disciplines, including the social choice theory[3], psychology[16], statistics[34], machine learning[39], bioinformatics[37] and others.",
"The convenience of these rank aggregation methods relies on their utilization of the ordinal data.",
"Without features, the comparisons only contain the partial ranking lists generated by human beings.",
"For instance, the voters who participated in an election choose one over the other candidates, which generate pairwise comparisons between the candidates.",
"As another example workers in a crowdsourcing platform are often asked to identify the better advertisement of two possible visualization modes.",
"Competitive sports such as tennis or chess also involve a serious of competitions between two players.",
"From a modeling perspective, the rank aggregation approach treats pairwise comparisons as an access to estimate the underlying “scores” or “qualities” of the items being compared (e.g., preference of candidates, skill levels of tennis players, and advertisement performance).",
"A vast body of prior work has made the significant progress in studying both statistical and computational aspects[61], [64], [63], [58], [59], [76], [65], [57].",
"However, the existing work ignores the security issue.",
"Beyond statistical property and computational complexity, situations become complicated when the pairwise ranking algorithms are utilized in high-stakes applications, e.g.",
"elections, sports competitions, and recommendation.",
"In pursuit of huge economic benefits, the potential attackers have strong motivations and incentives to manipulate or disrupt the aggregated results.",
"When the victims are ranking algorithms, a profit-oriented adversary could try his/her best to manipulate or disrupt the ranking list which will favor his/her demands-say, the attacker could place the special object at the top of the recommendation list, help the particular candidate to win an election or just defeat the candidate who should have won the election.",
"If the attackers compromise the integrity of ranking results, the fairness and rationality will be lost in these high-stakes applications.",
"Unfortunately, the security risk and serious threat of pairwise ranking problem have not been comprehensively examined yet.",
"Can rank aggregation algorithms with pairwise comparisons be easily manipulated or disrupted?",
"How reliable are their results in the high-stakes applications?",
"To the best of our knowledge, the adversarial arsenal for pairwise ranking methods has never been serious studied.",
"On one hand, the pairwise comparisons are the most simple data in the literature as just binary variables can represent them.",
"Due to the absence of features, modifying these binary data is an easy job.",
"On the other hand, any single comparison does not dictate the aggregated result.",
"Even manipulating a small quantity of binary data could not affect the final global ranking.",
"Such a contradiction inspires us to initiate an adversarial investigation of pairwise ranking problem.",
"To execute the attack strategy in the scenario, the adversary must analyze the characteristics of pairwise ranking problems.",
"Unlike the supervised learning tasks (e.g.",
"regression, classification, multi-arm bandit and reinforcement learning), the rank aggregation does not need the test protocol.",
"This means that the evasion attacks (a.k.a adversarial examples[24]) are not realistic.",
"Evasion attack causes the fixed model to misbehave by well-crafted test data.",
"But there is no test phrase to implement such a kind of attack.",
"To archive his/her goal, the adversary needs to inject the manipulated data into the training data.",
"Thus, rank aggregation in an adversarial setting is inherently related to the challenging poisoning attacks[10], [31].",
"Next, the adversary should consider the discrete property of the pairwise comparisons.",
"Unlike the data consisting of features in continuous space, the input of pairwise ranking only consists of binary data.",
"The adversary could only add, delete or flip pairwise comparisons to execute the poisoning attacks.",
"Such limitations make the substantial attack operations on pairwise ranking even harder.",
"How to design efficient algorithms that are able to inject toxic data in a discrete domain?",
"It is the distinguishable characteristic of our work which is different with the existing poisoning attack approaches[47], [43], [42], [49], [48], [33], [31], [79], [15].",
"Given these challenges, we propose a principle framework for adversarial perturbations of pairwise comparisons that aims to break the integrity of rank aggregation result.",
"In particular, we focus on the parametric model solved by maximum likelihood estimation[34].",
"We make the following contributions: We propose two game-theoretic frameworks specifically designed for adversary with the full or limited knowledge of the victim algorithm.",
"By introducing the uncertainty set around the original data, the adversary aims to find a toxic distribution which will maximize the risk of estimating the ranking parameters.",
"The dynamic threat model assumes that the adversary is aware of the original pairwise comparisons, the ranking algorithm and the ranking parameter learned from the original data.",
"This model relates to a dynamic distributionally robust game.",
"Besides, we propose a weaker threat model which assumes that the adversary only predominates the original data and the ranking algorithm.",
"It induces a static distributionally robust game where the adversary can only execute the attacks in the “black-box” attack style.",
"Different statistical attacks corresponding to the dynamic and static threat models are formulated into the bi-level optimization problem and distributionally robust optimization problem.",
"In the bi-level optimization problem, we adopt $\\chi ^2$ divergence to describe the uncertainty set around the original data.",
"The optimal attack strategy can be obtained by the projection onto a simplex.",
"In the distributionally robust optimization problem, the uncertainty set is a Wasserstein ball.",
"Based on the strong duality, the optimal attack behavior is obtained by a least square problem with a special regularization.",
"We prove the existence of robust optimization equilibrium and establish a minimax framework for pairwise ranking under adversarial setting.",
"To the best of our knowledge, this is the first systematic study of attacking rank aggregation under different adversarial models.",
"The extensively evaluations are conducted on several datasets from different high-stake domains, including election, crowdsourcing, and recommendation.",
"Our experiments demonstrate that the proposed poisoning attack could significantly decrease the correlation between the true ranking list and the aggregated result.",
"Notations Let $V$ be a finite set.",
"We will adopt the following notation from combinatorics: $\\binom{V}{k}:=\\text{set of all}\\ k\\ \\text{element subset of}\\ V.$ In particular $\\binom{V}{2}$ would be the set of all unordered pairs of elements of $V$ .",
"The sets of ordered pair will be denoted $V\\times V$ .",
"Ordered and unordered pairs will be delimited by parentheses $(i,\\ j)$ and braces $[i,\\ j]$ respectively.",
"We will use positive integers to indicate alternatives and voters.",
"Henceforth, $V$ will always be the set $[n]=\\lbrace 1,\\dots , n\\rbrace $ and will denote a set of alternatives to be ranked.",
"$\\mathcal {U}=\\lbrace 1,\\dots , m\\rbrace $ will denote a set of voters.",
"For $i,\\ j\\in V$ , we write $i\\succ j$ to mean that alternative $i$ is preferred over alternative $j$ .",
"If we wish to emphasize the preference judgment of a particular voter $u\\in \\mathcal {U}$ , we will write $i\\succ _u j$ .",
"Suppose that $\\Omega \\subset \\mathbb {R}^n$ is the data space, we denote $(\\Omega , d(\\cdot ,\\ \\cdot ))$ as a metric space equipped with some metric $d:\\Omega \\times \\Omega \\rightarrow \\mathbb {R}$ ."
],
[
"Ranking with Pairwise Comparisons",
"Given a collection $V$ of $n$ alternatives, we suppose that each $i\\in V$ has a certain numeric quality score $\\theta ^*_i$ .",
"We represent the quality scores of $V$ as a vector $\\theta ^*\\in \\mathbb {R}^n$ .",
"Suppose that a comparison of any pair $[i,\\ j]\\in \\binom{V}{k}$ is generated via the comparison of the corresponding scores $\\theta ^*_i,\\ \\theta ^*_j$ in the presence of noise.",
"Let $y^*_{ij}$ be the true direction of a pair $[i,\\ j]$ as $y^*_{ij} ={\\left\\lbrace \\begin{array}{ll}\\ \\ \\ 1, & \\theta ^*_i>\\theta ^*_j,\\\\-1, & \\theta ^*_i<\\theta ^*_j.\\end{array}\\right.",
"}$ Let $\\mathcal {C}$ be a collection of $N$ pairwise comparisons $\\mathcal {C} = \\lbrace c=[i,\\ j]\\ |\\ y_{ij}=1,\\ i,\\ j\\in V,\\ i\\ne j\\rbrace ,$ and $y_{ij}$ is the label of pair $[i,\\ j]$ which could not be consist with $y^*_{ij}$ .",
"It is worth noting that $\\mathcal {C}$ is always a multi-set.",
"For any pair $[i,\\ j]$ , it could be labeled by multiple users.",
"Given a set of voter $\\mathcal {U}=\\lbrace u_1,\\dots ,u_m\\rbrace $ , let $y^u_{ij}$ be the judgment of pair $[i,\\ j]$ given by voter $u\\in \\mathcal {U}$ .",
"We can aggregate $y^{u_1}_{ij},\\dots ,y^{u_m}_{ij}$ into a weight $w^0_{ij}$ .",
"Define $w(i,\\ j,\\ u)$ as the indicator of $y^{u}_{ij}$ : $w(i,\\ j,\\ u)={\\left\\lbrace \\begin{array}{ll}\\ 1, & \\text{if}\\ y^{u}_{ij}=1,\\ u\\in \\mathcal {U}\\\\\\ 0, & \\text{otherwise}\\end{array}\\right.",
"}$ and the weight $w^0_{ij}$ of $y_{ij}$ is $w^0_{ij} = \\underset{u\\ \\in \\ \\mathcal {U}}{\\sum }\\ w(i,\\ j,\\ u).$ Moreover, we introduce the comparison matrix $A$ .",
"If there exists a comparison $c\\in \\mathcal {C}$ , it can be described by its label $y_{ij}$ and a row of $A$ as $a^{c}=\\lbrace a^{c}_1,\\dots ,a^{c}_{|\\mathcal {C}|}\\rbrace $ : $a^{c}_k ={\\left\\lbrace \\begin{array}{ll}\\ \\ \\ 1, & k = i,\\\\-1, & k = j,\\\\\\ \\ \\ 0, & \\text{otherwise.}\\end{array}\\right.",
"}$ Then the data of pairwise ranking problem can be represented by $\\mathcal {C}_{\\mathcal {U}}=\\lbrace A,\\ y,\\ w_0\\rbrace $ where $w_0=\\lbrace w^0_{ij}\\rbrace $ , $y=\\lbrace y_{ij}\\rbrace $ is a $n(n-1)/2$ -d binary vector.",
"In statistical ranking or estimation from pairwise comparison, our goal is to obtain a score vector $\\hat{\\theta }$ to minimize a loss of a global ranking on the given data $\\mathcal {C}_{\\mathcal {U}}$ .",
"$\\hat{\\theta }\\in \\underset{\\theta \\ \\in \\ \\mathbb {R}^n}{\\textbf {\\textit {arg min}}}\\ \\ \\ell (\\theta ;\\ \\mathcal {C}_{\\mathcal {U}}).$ In particular, let the estimation of $y_{ij}$ be $\\hat{y}_{ij} = \\textbf {\\textit {sgn}}\\Big (\\langle a^{c},\\ \\theta \\rangle +\\varepsilon _{c}\\Big ),\\ \\forall \\ c\\in \\mathcal {C},$ where $\\textbf {\\textit {sgn}}(\\cdot )$ is the sign function, $\\varepsilon _c$ is the independent and identically distributed (i.i.d) noise variable and has a cumulative distribution function (c.d.f) $F$ .",
"Actually, (REF ) minimizes the derivation between the observed label $y$ and its estimation $\\hat{y}=\\lbrace \\hat{y}_{ij}\\rbrace $ based on the observing data $\\mathcal {C}_{\\mathcal {U}}$ .",
"In addition, the random variable $\\varepsilon _c$ plays the role of a noise parameter, with a higher magnitude of $\\varepsilon _c$ leading to more uncertainty in the comparisons and the higher probability of sign inconsistency occurred between $y_{ij}$ and $\\theta _i-\\theta _j$ .",
"The event that object $i$ dominating object $j$ ($y_{ij}=1$ ) is generally independent of the order of the two items being compared, thus, the following holds: $\\Pr (y_{ij}=1)= 1-\\Pr (y_{ij}=-1)$ and $F$ is a symmetric c.d.f whose continuous inverse is well-defined.",
"Some typical examples of (REF ) are the uniform model [63], the Bradley-Terry- Luce (BTL) model [14], [46], and the Thurstone model with Gaussian noise (Case V) [67], which have been extensively studied in literature (e.g., [17], [78]).",
"In this paper, we focus on the Uniform Model: one can adopt the symmetric c.d.f $F(t) = \\frac{t+1}{2}$ , and the general set-up (REF ) turns to be a linear model.",
"Furthermore, the loss function in (REF ) can be specialized as the weighted sum-of-squares function: $\\begin{aligned}& \\ell \\ (\\theta ;\\ \\mathcal {C}_{\\mathcal {U}})&=&\\ \\ \\frac{1}{2|\\mathcal {C}_{\\mathcal {U}}|}\\ \\Vert \\ y - A\\theta \\ \\Vert ^2_{2,\\ w_0} \\\\& &=&\\ \\ \\frac{1}{2|\\mathcal {C}_{\\mathcal {U}}|}\\ \\underset{(i,\\ j)}{\\sum }w^0_{ij}\\ (\\ y_{ij}-\\theta _i+\\theta _j\\ )^2.\\end{aligned}$"
],
[
"Methodology",
"In this section, we systematically introduce the methodology for poisoning attacks on pairwise ranking.",
"Specifically, we first start by introducing two game-theoretic threat models including the full knowledge and the limited knowledge adversaries.",
"Then we present the corresponding algorithms to generate the optimal strategies of these threat models at different uncertainty budgets.",
"Finally, the existence of equilibrium and the results of generalization analysis are discussed in the end of this section."
],
[
"Poisoning Attack on Pairwise Ranking",
"We provide here a detailed adversarial framework for poisoning attacks against pairwise ranking algorithms.",
"The framework consists of defining the adversary’s goal, knowledge of the attacked method, and capability of manipulating the pairwise data, to eventually define the optimal poisoning attack strategies.",
"The Goal of Adversary.",
"If an adversary executes the poisoning attack, he/she will provide the ranker with the toxic data.",
"This action will mislead its opponent into picking parameters to generate a different ranking result from $\\hat{\\theta }$ obtained by the original data $\\mathcal {C}_{\\mathcal {U}}$ in (REF ).",
"Let $\\bar{\\theta }$ be the solution of (REF ) with the toxic data, it satisfies $d(\\pi _{\\theta ^*},\\ \\pi _{\\hat{\\theta }}) \\le d(\\pi _{\\theta ^*},\\ \\pi _{\\bar{\\theta }}),$ where $\\theta ^*$ is the true quality scores of the objects, $\\pi _{\\theta }$ is the ranking list decided by $\\theta $ and $d(\\pi _1,\\ \\pi _2)$ measures the similarity of two ordered lists $\\pi _1$ and $\\pi _2$ .",
"The Knowledge of Adversary.",
"We assume two distinct attack scenarios which are distinguished by the knowledge of adversary, referred to as dynamic and static attacks in the following.",
"The adversaries in the two scenarios have different knowledge of the victims.",
"In dynamic attacks, the attacker is assumed to know the observed data $\\mathcal {C}_{\\mathcal {U}}$ , the ranking algorithm, and even the ranking parameters $\\hat{\\theta }$ obtained by the original data $\\mathcal {C}_{\\mathcal {U}}$ in (REF ).",
"If a dictator wants to sabotage the election which will subvert his/her predominant, he/she would not need to manipulate the results of the election.",
"Making the most competitive opponent lose the advantage in the key districts will achieve the purpose.",
"The dictator could execute the dynamic strategies as the aggregation process is a “white-box” to him/her.",
"This adversarial mechanism can be implemented by establishing the hierarchical relationship between the ranker and the attacker.",
"The attacker is assumed to anticipate the reactions of the ranker; this allows him/her to choose the best—or optimal—strategy accordingly.",
"Such a hierarchical interaction results in the fact that the mathematical program related to the ranking process is part of the adversary's constraints.",
"It is also known as the dynamic or Stackelberg (leader-follower) game[7] in the literature: the two agents take their actions in a sequential (or repeated) manner.",
"Moreover, the hierarchical relationship is the major feature of bi-level optimization.",
"The bi-level program includes two mathematical programs within a single instance, one of these problems being part of the constraints of the other one.",
"In static attacks, the attacker could not grasp $\\hat{\\theta }$ but is still aware of the observed data $\\mathcal {C}_{\\mathcal {U}}$ and the ranking algorithm.",
"This scenario comes from the fact that the ranking aggregation problem does not need the test protocol.",
"Once the adversary provides the modified data, the victim would generate the ranking list immediately.",
"There is no chance to monitor the ranker's behavior.",
"In most cases, the adversary can not obtain $\\hat{\\theta }$ .",
"There is no feedback for the adversary to update his/her strategies.",
"A competitor of the e-commerce platform, who wants to disrupt the recommendation results and destroy the user experience, would execute the static strategies.",
"Promoting the rank of specific goods is challenging.",
"Disrupting the normal ranking result is sufficient to archive his/her purpose.",
"The competitor could only execute the static strategies as the aggregation process is a “gray-box”.",
"The leading e-commerce platform is the only one who could access the ranking parameters.",
"The objective function and the pairwise comparisons for recommendation can be perceivable to the adversary.",
"This adversarial mechanism should be modeled as a static game.",
"A static game is one in which a single decision is made by each player, and each player has no knowledge of the decision made by the other players before making their own decision.",
"In other words, decisions or actions are made simultaneously (or the order is irrelevant).",
"The Capability of Adversary.",
"To modify the original data $\\mathcal {C}_{\\mathcal {U}}$ in poisoning attacks, the adversary will inject an arbitrary pair $[i,\\ j]\\in \\binom{V}{k}$ with any directions into $\\mathcal {C}_{\\mathcal {U}}$ , delete the existing comparison $c=(i,\\ j)$ in $\\mathcal {C}_{\\mathcal {U}}$ or just flip the label of $c$ .",
"The three kinds of operations require some new representations of the observed set.",
"We augment the observed data $\\mathcal {C}_{\\mathcal {U}}$ with the comparisons which are not labeled by users in $\\mathcal {U}$ .",
"Let $\\mathcal {D} = V\\times V$ be the set of all ordered pairs, and $|\\mathcal {D}|= N = n(n-1)$ .",
"The weights of all possible comparisons are $w^{\\prime }_0$ and there exist 0 entries in $w^{\\prime }_0$ .",
"As $\\mathcal {D}$ is the complete comparison set, the comparison matrix $B$ will be fixed and we can adopt a $n(n-1)$ -d single-value vector to represent the labels, saying that $y^{\\prime }$ is a vector with all entries are 1.",
"Now all attack operations (adding, deleting and flipping) can be executed by increasing or decreasing the corresponding weight $w^{\\prime }_0$ .",
"$\\begin{aligned}& B &=&\\ \\begin{bmatrix}b_{1,2}\\\\b_{1,3}\\\\\\vdots \\\\\\ \\ b_{n,n-2}\\ \\ \\\\\\ \\ b_{n,n-1}\\ \\ \\end{bmatrix}\\subset \\big \\lbrace -1,\\ 0,\\ 1\\big \\rbrace ^{N\\times n},\\\\& y^{\\prime } &=&\\ \\ \\big \\lbrace \\ y_{1,2},\\ y_{2,1},\\ \\dots ,\\ y_{n-1,n},\\ y_{n,n-1}\\ \\big \\rbrace ^\\top \\\\& &=&\\ \\ \\big \\lbrace \\ 1,\\ 1,\\ \\dots ,\\ 1,\\ 1\\ \\big \\rbrace ^\\top .\\end{aligned}$ Besides injecting the toxic data, the attacker also needs to disguise himself/herself.",
"It means that the adversary needs to coordinate a poisoned $w=\\lbrace w_{ij}\\rbrace $ associated with $w^{\\prime }_0$ .",
"Intuitively, the adversary could not obtain $w$ through the drastic changes, neither on each $w_{ij}$ nor $\\sum _{(i,j)} w_{ij}$ .",
"Such limitations lead to the following constraints for the adversary's action.",
"First, the total difference between $w^{\\prime }_0$ and $w$ would be smaller than $b$ , namely, $\\Vert \\ w\\ -\\ w^{\\prime }_0\\ \\Vert _1\\ \\le \\ b,\\ \\ b\\ \\in \\ \\mathbb {N}_+.$ Here the positive integer $b$ bounds the total number of malicious samples thereby limiting the capabilities of the attacker.",
"Furthermore, the adversary could not alter the number of votes on each pairwise comparison $c\\in \\mathcal {D}$ obviously.",
"This constraint on the adversary leads to the following condition: $\\Vert \\ w\\ -\\ w^{\\prime }_0\\ \\Vert _{\\infty }\\le \\ l,\\ l\\in \\mathbb {N}_+,\\ l\\le \\textbf {\\textit {min}}\\lbrace \\textbf {\\textit {max}}(w^{\\prime }_0),\\ b\\rbrace .$ The positive integer $l$ leads the conservative perturbations on the observed samples.",
"To summarize, the adversary‘s action set $\\Omega _1$ is $\\begin{aligned}\\Omega _1\\ \\ =\\ \\ \\left\\lbrace \\ \\ w\\ \\ \\left|\\ \\begin{matrix}w\\ \\in \\ \\mathbb {N}^{N},\\ \\ l,\\ b\\ \\in \\ \\mathbb {N},\\\\\\ \\Vert \\ w\\ -\\ w^{\\prime }_0\\ \\Vert _1\\ \\le \\ b,\\\\\\ \\Vert \\ w\\ -\\ w^{\\prime }_0\\ \\Vert _{\\infty }\\le \\ l,\\\\\\ \\ l\\ \\le \\ \\textbf {\\textit {min}}\\lbrace \\textbf {\\textit {max}}(w^{\\prime }_0),\\ b\\rbrace \\ \\ \\end{matrix}\\right.\\right\\rbrace .\\end{aligned}$ Furthermore, the attacker must pay for his/her malicious behaviors.",
"Let $s:\\mathbb {N}^N\\times \\mathbb {N}^N\\rightarrow \\mathbb {R}$ is a “cost” function measured the overhead of the perturbation as changing $w_0$ into $w$ .",
"The attacker hopes that the toxic weight $w$ will represent the lowest cost option.",
"Let $\\Omega _2$ be the budget set of the adversary $\\Omega _2\\ \\ =\\ \\ \\left\\lbrace w\\ \\Big |\\ w\\in \\underset{}{\\textbf {\\textit {arg min}}}\\ \\ s(w,\\ w^{\\prime }_0)\\right\\rbrace $ Finally, the action set is $\\Omega _0 = \\Omega _1\\cap \\Omega _1$ which figures out the capability of the adversary.",
"Poisoning Attack Strategies.",
"Here we specify the different poisoning strategies for the two attack scenarios.",
"Dynamic attack strategy.",
"Consider the goal and knowledge of attacker, we formulate the interaction between ranker and the adversary with full knowledge as a dynamic game.",
"In this game, information is assumed to be complete (i.e., the players’ payoff functions, as well as the constraint set $\\Omega _0$ and the flexible set of ranking parameter $\\Theta $ , are common knowledge) and perfect (i.e., the attacker knows the ranker's decision).",
"Having received the ranker's decision $\\hat{\\theta }$ , the attacker chooses a feasible decision $w\\in \\Omega _0$ that maximizes the ranker's loss function to increase the risk of the ranker's estimation based on $\\lbrace w,\\ B,\\ y^{\\prime }\\rbrace $ .",
"Such a dynamic game can be formulated into the following bi-level optimization problem: $& &\\underset{w\\ \\in \\ \\Omega _0}{\\textbf {\\textit {max}}}&\\ \\ \\ell (w;\\ \\hat{\\theta },\\ B,\\ y^{\\prime }), \\\\& &\\textbf {\\textit {subject to}}&\\ \\ \\hat{\\theta }\\in \\underset{\\theta \\ \\in \\ \\Theta }{\\textbf {\\textit {arg min}}}\\ \\ \\ell (\\theta ;\\ w^{\\prime }_0,\\ B,\\ y^{\\prime }).\\ $ The upper level optimization (REF ) amounts to selecting the toxic data $w$ to maximize the loss function of the ranker, while the lower level optimization () corresponds to calculate the ranking parameter $\\hat{\\theta }$ with original data $\\lbrace w_0,\\ B,\\ y^{\\prime }\\rbrace $ .",
"Once the adversary generates $w$ , he/she will deliver the toxic data to the ranker.",
"Then the poisoned parameter $\\bar{\\theta }$ will be obtained by $\\bar{\\theta }=\\underset{\\theta \\ \\in \\ \\Theta }{\\textbf {\\textit {arg min}}}\\ \\ \\ell (\\theta ;\\ w,\\ B,\\ y^{\\prime }).$ Static attack strategy.",
"This strategy is represented such a type of adversary whose ability is to inflict the highest possible risk of the ranker when no information about his/her interests is available.",
"It means that the two players make decisions simultaneously, and the attacker does not knows the ranker’s decision.",
"Such a static game can be formulated into the following min-max optimization problem: $\\begin{aligned}\\underset{\\theta \\ \\in \\ \\Theta }{\\textbf {\\textit {min}}}&\\ \\underset{w\\ \\in \\ \\Omega _0}{\\textbf {\\textit {max}}}\\ \\ell (\\theta ,\\ w;\\ B,\\ y^{\\prime }).\\end{aligned}$ The poisoned parameter $\\bar{\\theta }$ will be solved by (REF ).",
"However, solving the dynamic and static attack strategies from (REF ) and (REF ) are challenging.",
"On one hand, the bi-level optimization (REF ) and the min-max problem (REF )are both mixed-integer programming problem as the variable $w$ is restricted to be positive integers.",
"On the other hand, the feasible set $\\Omega _0$ corresponds to a non-linear constraint as it requires to find the perturbation in the neighborhood of $w^{\\prime }_0$ with the lowest cost.",
"It is well-known that linear integer programmings are NP-complete problems [36].",
"Such a non-linear constraint makes these problems even more complex.",
"Obviously, adopting the heuristic methods to solve the optimal attack strategies (REF ) and (REF ) is sub-optimal.",
"In this part, we will develop the other model based on ideas from distributionally robust optimization [54], [12], [22] that provides the tractable convex formulations for solving the optimal strategies in the dynamic and static scenarios."
],
[
"Distributional Perspective and Robust Game",
"In the above formulations (REF ) and (REF ), the attacker modifies the number of votes on each pairwise comparison with constraints $\\Omega _0$ .",
"This formulation leads to the mixed-integer programming problem.",
"Here we introduce a distributional perspective to establish the tractable optimization problem.",
"Generally speaking, the attacker and the ranker both access the original data $\\mathcal {D}$ to play the dynamic or static game.",
"The non-toxic pairwise comparisons $\\mathcal {D}=\\lbrace w^{\\prime }_0,\\ B,\\ y^{\\prime }\\rbrace $ are actually drawn from an empirical distribution $\\mathbb {P}_N$ $\\mathbb {P}_N\\ = \\ \\frac{1}{N}\\ \\underset{c\\ \\in \\ \\mathcal {C}}{\\sum }\\ \\delta ({w^0_{ij}}^{\\prime },\\ b_{i,j},\\ {y_{ij}}^{\\prime }),\\ \\ c=(i,\\ j),$ where $\\delta ({w^0_{ij}}^{\\prime },\\ b_{i,j},\\ {y_{ij}}^{\\prime })$ is the Dirac probability measure on $({w^0_{ij}}^{\\prime },\\ b_{i,j},\\ {y_{ij}}^{\\prime })$ .",
"With $B$ and $y^{\\prime }$ as (REF ), the marginal distribution of $w^{\\prime }_0$ plays a vital role in the sequel.",
"With some abuse of symbol, we treat the marginal distribution of $w_0$ as the distribution of the original data and $\\mathbb {P}_N=\\frac{1}{N}\\ \\underset{c\\ \\in \\ \\mathcal {C}}{\\sum }\\ \\delta ({w^0_{ij}}^{\\prime }),\\ \\ c=(i,\\ j).$ The attacker chooses a perturbation function $\\psi :\\mathbb {N}^{N}\\rightarrow \\mathbb {N}^{N}$ that changes the weight $w^{\\prime }_0$ to $w\\in \\Omega _0$ .",
"Such a perturbation $\\psi $ induces a transition from the empirical distribution $\\mathbb {P}_N$ to a poisoned distribution $\\mathbb {Q}$ .",
"If the attacker selects $\\mathbb {Q}$ in a sufficiently small neighborhood of $\\mathbb {P}_N$ , namely, the “distance” between the poisoned distribution $\\mathbb {Q}$ and the empirical distribution $P$ would be sufficiently small, the attacker could obtain a “local” solution and $\\mathbb {Q}$ is a “good” approximation of $\\mathbb {P}_N$ in the sense of such a “distance”.",
"Therefore, the poisoned sample $w$ would satisfy the constraints (REF ) and (REF ).",
"Here we directly work with the empirical distribution $\\mathbb {P}_N$ (or other nominal distribution) and consider $\\mathbb {Q}$ is close to the nominal distribution in terms of a certain statistical distance.",
"There exists some popular choices of the statistical distance, such as $\\phi $ -divergences [9], [32], [8], [75], [53], [54], [18], Prokhorov metric [21], Wasserstein distances [77], [11], [23], [51], [40] and maximum mean discrepancy [66].",
"For dynamic attack strategy (REF ), we adopt the $\\phi $ -divergence[41] as the discrepancy measure between the empirical distribution $\\mathbb {P}_n$ and the toxic distribution $\\mathbb {Q}$ .",
"Definition 1 ($\\phi $ -divergence and $\\chi ^2$ -divergence) Let $\\phi : \\mathbb {R}_+\\rightarrow \\mathbb {R}$ be a convex function with $\\phi (1) = 0$ .",
"Then the $\\phi $ -divergence between distributions $\\mathbb {Q}$ and $\\mathbb {P}$ defined on a measurable space $\\mathcal {X}$ is $\\begin{aligned}d_{\\phi }(\\mathbb {Q}\\ ||\\ \\mathbb {P})=\\int \\phi \\left(\\frac{d\\mathbb {Q}}{d\\mathbb {P}}\\right)\\mathrm {d}\\mathbb {P}=\\int _{\\mathcal {X}}\\phi \\left(\\frac{q(x)}{p(x)}\\right)p(x)\\mathrm {d}\\mu (x),\\end{aligned}$ where $\\mu $ is a $\\sigma $ -finite measure on $\\mathcal {X}$ satisfying $\\mathbb {Q}, \\mathbb {P}$ are absolutely continuous with respect to $\\mu $ , and $q=\\frac{\\mathrm {d}\\mathbb {Q}}{\\mathrm {d}\\mu }$ , $p=\\frac{\\mathrm {d}\\mathbb {P}}{\\mathrm {d}\\mu }$ are the Radon–Nikodym derivative with respect to $\\mu $ .",
"If $\\phi $ is adopted as $\\phi (t) = \\frac{1}{2}(t-1)^2$ , it is known as the $\\chi ^2$ -divergence.",
"Suppose that $\\mathfrak {X}(\\mathbb {P}_N)$ is a set of probability distributions from the empirical distribution with $\\chi ^2$ -divergence.",
"This $\\chi ^2$ ball with radius $\\alpha $ is given by $\\mathfrak {X}^{\\alpha }(\\mathbb {P}_N) = \\left\\lbrace \\ \\mathbb {Q}\\in \\mathcal {P}(\\Omega _1)\\ \\Big \\vert \\ d_{\\chi ^2}(\\mathbb {Q}\\ ||\\ \\mathbb {P}_N)\\le \\alpha \\ \\right\\rbrace ,$ where $\\mathcal {P}(\\Omega _1)$ denotes the set of all Borel probability measures on $\\Omega _1$ .",
"With carefully chosen $\\alpha $ , the adversary chooses $w$ from the toxic distribution $\\mathbb {Q}\\in \\mathfrak {X}^{\\alpha }(\\mathbb {P}_N)$ .",
"$w$ could satisfy the neighborhood constraints as (REF ) and (REF ).",
"Replacing the minimal `cost' constraint (REF ) by the neighborhood constraint defined with the $\\chi ^2$ ball, we formulate the following bi-level optimization to obtain the dynamic attack strategy $\\begin{aligned}& &\\underset{\\mathbb {Q}\\ \\in \\ \\mathfrak {X}^{\\alpha }(\\mathbb {P}_N)}{\\textbf {\\textit {max}}}&\\ \\ \\mathbb {E}_{w\\sim \\mathbb {Q}}\\big [\\ell (w;\\ \\hat{\\theta })\\big ],\\\\& &\\textbf {\\textit {subject to}}&\\ \\ \\hat{\\theta }=\\underset{\\theta \\ \\in \\ \\Theta }{\\textbf {\\textit {arg min}}}\\ \\ \\ell (\\theta ;\\ w^{\\prime }_0).\\end{aligned}$ The $\\chi ^2$ -divergence and the “local” neighborhood constraint $\\mathbb {Q}\\in \\mathfrak {X}^{\\alpha }(\\mathbb {P}_N)$ will help us to develop a tractable algorithm for the dynamic attack strategy.",
"Different with the dynamic attack strategy, the ranking parameter $\\hat{\\theta }$ would be unknown for the adversary in the static attack strategy.",
"The $\\chi ^2$ divergence will not help to simplify the min-max problem (REF ).",
"To sum up, we adopt the $p$ -Wasserstein distance [22] as the discrepancy measure between the empirical distribution $\\mathbb {P}_n$ and the toxic distribution $\\mathbb {Q}$ for the static attack strategy.",
"The $p$ -Wasserstein distance will help us to reformulate the min-max problem (REF ) into a single regularized problem.",
"Definition 2 ($p$ -Wasserstein distance) Let $p\\in [1,\\infty ]$ .",
"The $p$ -Wasserstein distance between distributions $\\mathbb {P},\\ \\mathbb {Q}\\in \\mathcal {P}(\\Omega )$ is defined as $1\\le p< \\infty $ $& \\mathcal {W}_p\\ (\\mathbb {P},\\ \\mathbb {Q})\\ \\ \\ \\ \\ =\\\\& \\left(\\underset{\\gamma \\in \\Gamma (\\mathbb {P},\\ \\mathbb {Q})}{\\textbf {\\textit {min}}}\\left\\lbrace {\\int }_{\\Omega \\times \\Omega }\\Big [d\\Big (w,\\ w^{\\prime }\\Big )\\Big ]^p\\ \\gamma \\Big (\\mathrm {d}w,\\ \\mathrm {d}w^{\\prime }\\Big )\\right\\rbrace \\right)^{\\frac{1}{p}}\\nonumber $ $p=\\infty $ $\\mathcal {W}_p\\ (\\mathbb {P},\\ \\mathbb {Q})\\ \\ \\ =\\underset{\\gamma \\in \\Gamma (\\mathbb {P},\\ \\mathbb {Q})}{\\textbf {\\textit {inf}}}\\ \\underset{\\vphantom{\\gamma \\in \\Gamma (\\mathbb {P},\\ \\mathbb {Q})}\\Omega \\times \\Omega }{\\gamma \\textnormal {-}\\textbf {\\textit {ess sup}}}\\ d\\Big (w,\\ w^{\\prime }\\Big )$ where $\\Gamma (\\mathbb {P},\\ \\mathbb {Q})$ denotes the set of all Borel probability distributions on $\\Omega \\times \\Omega $ with marginal distributions $\\mathbb {P}$ and $\\mathbb {Q}$ , $d:\\Omega \\times \\Omega \\rightarrow \\mathbb {R}_+$ is a nonnegative function, and $\\gamma \\textnormal {-}\\textbf {\\textit {ess sup}}$ expresses the essential supremum of $d(\\cdot ,\\ \\cdot )$ with respect to the measure $\\gamma $ .",
"The Wasserstein distance (REF ) and (REF ) arise in the problem of optimal transport [52], [72]: for any coupling $\\gamma \\in \\Gamma (\\mathbb {P},\\ \\mathbb {Q})$ , the conditional distribution $\\gamma _{w\\vert w^{\\prime }}$ can be viewed as a randomized overhead for ‘transporting’ a unit quantity of some material from a random location $w\\sim \\mathbb {P}$ to another location $w^{\\prime }\\sim \\mathbb {Q}$ .",
"If the cost of transportation from $w\\in \\Omega $ to $w^{\\prime }\\in \\Omega $ is given by $[d(w,w^{\\prime })]^p$ , $\\mathcal {W}_p\\ (\\mathbb {P},\\ \\mathbb {Q})$ will be the minimum expected transport cost [60].",
"Suppose that $\\mathfrak {W}_p(\\mathbb {P}_N)$ is a set of probability distributions constructed from the empirical distribution $\\mathbb {P}_N$ with $p$ -Wasserstein distance.",
"This Wasserstein ball of radius $\\alpha $ is given by $\\mathfrak {W}_p^{\\alpha }(\\mathbb {P}_N) = \\left\\lbrace \\ \\mathbb {Q}\\in \\mathcal {P}(\\Omega _1)\\ \\Big \\vert \\ \\mathcal {W}_p\\ (\\mathbb {P}_N,\\ \\mathbb {Q})\\le \\alpha \\ \\right\\rbrace .$ With local uncertainty set $\\mathfrak {W}^{\\alpha }_p(\\mathbb {P}_N)$ , the min-max optimization (REF ) could be expressed as the following distributionally robust optimization (DRO) problem: $\\begin{aligned}\\underset{\\theta \\ \\in \\ \\Theta }{\\vphantom{\\mathbb {Q}^{\\top }}\\textbf {\\textit {min}}}&\\ \\underset{\\mathbb {Q}\\ \\in \\ \\mathfrak {W}^{\\alpha }_p(\\mathbb {P}_N)}{\\textbf {\\textit {sup}}}\\ \\ \\mathbb {E}_{w\\sim \\mathbb {Q}}\\Big [\\ell (\\theta ,\\ w)\\Big ],\\end{aligned}$ where the supremum operation w.r.t.",
"$\\mathbb {Q}$ means that all players' optimal decision is based on the worst expected value of $\\ell $ from the set of distributions $\\mathfrak {W}^{\\alpha }_p(\\mathbb {P}_N)$ .",
"Here we replace the minimal `cost' constraint in (REF ) by the neighborhood constraint on the worst-case expectation.",
"With the local constraint $\\mathbb {Q}\\in \\mathfrak {W}^{\\alpha }_p(\\mathbb {P}_N)$ , the Wasserstein distance between the empirical distribution $\\mathbb {P}_N$ and the perturbed distribution $\\mathbb {Q}$ must be smaller than a given budget $\\alpha $ as $\\mathcal {W}_p\\ (\\mathbb {P},\\ \\mathbb {Q})\\le \\alpha $ .",
"It means that the attacker has a budget $\\alpha $ to implement his/her perturbation on the original data for ranking aggregation.",
"The robust game formulation (REF ) would relax the coarse-grid constraint as (REF ), and the analysis in the sequel reveals the central role played by this relaxation.",
"Actually, the bi-level problem (REF ) and the DRO problem (REF ) relate to a general robust game [1], [44], [45] between the attacker and the ranker as $\\underset{x_r\\ \\in \\ \\mathcal {X}_r}{\\textbf {\\textit {min}}}\\ \\underset{\\mathbb {Q}\\ \\in \\ \\mathfrak {U}}{\\textbf {\\textit {sup}}}\\ \\ \\mathbb {E}_{\\xi \\sim \\mathbb {Q}}\\Big [f_r(x_r,\\ x_{-r},\\ \\xi )\\Big ],\\ \\ r = 1,\\ 2\\ $ where $r$ indicates the role of the agent in the robust game, $x_r$ is the decision variable of the special player $r$ , and $x_{-r}$ denotes the decision variables of its rivals, and $\\mathcal {X}_r$ is the action set of player $r$ .",
"The random variable $\\xi $ illustrates the uncertainty or inaccuracy of distributional information to the players, and $\\mathfrak {U}$ is the uncertainty set of distribution of random variable $\\xi $ for all players (i.e., $\\mathfrak {X}^{\\alpha }(\\mathbb {P}_N)$ and $\\mathfrak {W}_p^{\\alpha }(\\mathbb {P}_N)$ ).",
"The pay-off function $f_r$ could be different for each player and the corresponding game is a non-zero sum game.",
"Comparing the general case (REF ) with (REF ) and (REF ), all players in (REF ) focus on the same pay-off function as $f_1 = f_2 = \\ell $ .",
"Moreover, the decision variable of the ranker $\\theta $ equals to $x_1$ .",
"The random variable $\\xi $ represents the distribution of pairwise comparison as $w$ .",
"So the decision variable of the attacker $x_2$ will be the constant (its role has been replaced by $\\xi $ ).",
"The robust game problem is first proposed by Bertsimas and Aghassi in [1].",
"It expands the boundaries of research of the classical Nash game [73], [55], [56] and the Bayesian game [26], [27], [28].",
"Different form the Nash and the Bayesian game[1], the only common knowledge of all participants in robust game is that all players being aware about an uncertainty set like $\\mathfrak {X}^{\\alpha }(\\mathbb {P}_N)$ and $\\mathfrak {W}_p^{\\alpha }(\\mathbb {P}_N)$ .",
"All possible parameters of payoff function are related to this set.",
"Here we investigate the existence of the equilibrium for distributionally robust Nash equilibrium of the proposed model (REF ).",
"First, we give the definition of the distributionally robust Nash equilibrium.",
"Definition 3 A pair of different players' action $\\lbrace x^*_1,\\ x^*_2\\rbrace $ is called a distributionally robust Nash equilibrium (DRNE) of (REF ) if they satisfy the following $x^*_r\\ \\in \\ \\underset{x_r\\ \\in \\ \\mathcal {X}_r}{\\vphantom{\\mathbb {Q}\\in \\mathcal {Q}^\\top _r}\\textbf {\\textit {arg min}}}\\ \\underset{\\mathbb {Q}\\ \\in \\ \\mathfrak {U}}{\\textbf {\\textit {sup}}}\\ \\ \\mathbb {E}_{\\xi \\sim \\mathbb {Q}}\\Big [f_r(x_r,\\ x_{-r},\\ \\xi )\\Big ],\\ r=1,2.$ Next, we can prove the existence of DRNE for the general robust game (REF ).",
"theoremequilibrium Let the pay-off function $f_r,\\ r=1,2$ be the weighted sum-of-squared loss $\\ell $ (REF ) in (REF ).",
"If the uncertainty set is $\\mathfrak {X}^{\\alpha }(\\mathbb {P}_N)$ or $\\mathfrak {W}_p^{\\alpha }(\\mathbb {P}_N)$ , the general robust game (REF ) has a DRNE.",
"To prove this existence result, we reformulate the problem (REF ) into a single optimization problem and show that the single problem has an optimal solution.",
"The detailed proof is provided in the Appendix ."
],
[
"Optimization",
"In this part we show our algorithms for computing the adversarial strategies.",
"Suppose the total number of pairwise comparison without perturbation is $M^0$ , and the frequencies of each type of the observed comparisons are $p = \\frac{1}{M^0}\\cdot w^{\\prime }_0,\\ \\ M^0 = \\sum _{(i,j)}\\ {w^0_{ij}}^{\\prime }.$ Let the maximum toxic dosage be $\\kappa $ .",
"It suggests that the number of toxic pairwise comparisons $M$ satisfies $M\\ =\\ \\underset{(i,j)}{\\sum }\\ w_{ij}\\ \\le \\ (1+\\kappa )\\cdot M^0.$ We replace the toxic weight $w$ with its frequency $q=\\lbrace q_{ij}\\rbrace \\in \\mathbb {R}^N_+$ when analyzing the equilibrium, studying the statistical nature of the worst-case estimator and solving the corresponding optimization problem.",
"We relax the integer programming problem into a general optimization by such a variable substitution.",
"Thus, the pay-off function (REF ) turns to be $\\ell (\\theta ,\\ q)=\\frac{1}{2N}\\ \\underset{(i,j)}{\\sum }\\ q_{ij}\\big (y_{ij}-\\theta _i+\\theta _j\\big )^2,$ and we still adopt $\\mathbb {P}_N$ and $\\mathbb {Q}$ as the distribution of the empirical data and the toxic data.",
"Furthermore, we can implement the integer attack with the optimal $q$ and $M$ .",
"Now we come to solve the bi-level optimization (REF ) and the distributionally robust optimization problem (REF ) with the variable substitution: $\\begin{aligned}& &\\underset{\\mathbb {Q}\\ \\in \\ \\mathfrak {X}^{\\alpha }(\\mathbb {P}_N)}{\\textbf {\\textit {max}}}&\\ \\ \\mathbb {E}_{q\\sim \\mathbb {Q}}\\big [\\ell (q;\\ \\hat{\\theta })\\big ],\\\\& &\\textbf {\\textit {subject to}}&\\ \\ \\hat{\\theta }=\\underset{\\theta \\ \\in \\ \\Theta }{\\textbf {\\textit {arg min}}}\\ \\ \\ell (\\theta ;\\ w^{\\prime }_0),\\end{aligned}$ and $\\underset{\\theta \\ \\in \\ \\Theta }{\\vphantom{\\mathbb {Q}\\ \\in \\ \\mathfrak {W}^{\\alpha }_p\\ (\\mathbb {P}_N)}\\textbf {\\textit {min}}}&\\ \\underset{\\mathbb {Q}\\ \\in \\ \\mathfrak {W}^{\\alpha }_p\\ (\\mathbb {P}_N)}{\\textbf {\\textit {sup}}}\\ \\ \\mathbb {E}_{q\\sim \\mathbb {Q}}\\ \\Big [\\ \\ell \\big (\\theta ;\\ q\\big )\\ \\Big ]$ For the dynamic attack strategy (REF ), a similar formulation has been studied for archiving a better variance-bias trade-off in maximum likelihood estimation[54].",
"Based on the $\\chi ^2$ -divergence, the bi-level problem (REF ) turns to be a convex problem.",
"We provide a detailed process of solving (REF ) in the supplementary material.",
"The distributionally robust optimization formulation (REF ) involves optimizing over the uncertainty set $\\mathfrak {W}^{\\alpha }_2(\\mathbb {P}_N)$ , which contains countless probability measures.",
"However, recent strong duality results of distributionally robust optimization involving Wasserstein uncertainty set [23] and [12]) ensure that the inner supremum in (REF ) admits an equivalent reformulation which would be a tractable, univariate optimization problem.",
"In the adversarial scenario of pairwise ranking, we have the following result.",
"The DRO problem (REF ) could be reformulated as a regularized regression problem.",
"theoremmainresult Let $\\mathcal {Z}=\\big \\lbrace p,\\ B,\\ y^{\\prime }\\big \\rbrace $ be the observed data set, where $B$ and $y^{\\prime }$ are defined as (REF ), $p$ is the frequency of each type of pairwise comparison as (REF ).",
"Consider the loss function of $z$ , and the distance function between $z_{c}$ , $z^{\\prime }_{c}$ are based on the $\\ell _2$ -norm.",
"In other words, we take $\\ell (\\theta ,\\ z)$ as (REF ) and $\\begin{aligned}& d(z_{c},\\ z^{\\prime }_{c})&=&\\ \\ \\big \\Vert \\big (p_{ij},\\ b_{i,j},\\ y^{\\prime }_{ij}\\big )-\\big (q_{ij},\\ b_{i,j},\\ y^{\\prime }_{ij}\\big )\\big \\Vert _2\\\\& &=&\\ \\ \\ \\big |\\ p_{ij}-q_{ij}\\ \\big |.\\end{aligned}$ Then, the DRO problem (REF ) has an equivalent form: $\\begin{aligned}& & & \\ \\ \\underset{\\theta \\ \\in \\ \\Theta }{\\vphantom{\\mathbb {Q}\\ \\in \\ \\mathfrak {W}^{\\alpha }_p\\ (\\mathbb {P}_N)}\\textbf {\\textit {min}}}\\ \\ \\underset{\\mathbb {Q}\\ \\in \\ \\mathfrak {W}^{\\alpha }_2(\\mathbb {P}_N)}{\\textbf {\\textit {sup}}}\\ \\ \\mathbb {E}_{q\\sim \\mathbb {Q}}\\Big [\\ \\ell \\big (\\theta ;\\ q\\big )\\ \\Big ]\\\\& &=&\\ \\ \\underset{\\theta \\ \\in \\ \\Theta }{\\vphantom{\\mathbb {Q}\\ \\in \\ \\mathfrak {W}^{\\alpha }_p(\\mathbb {P}_N)}\\textbf {\\textit {min}}}\\ \\ \\mathcal {L}(\\theta )\\ +\\ \\mathcal {R}(\\theta ),\\end{aligned}$ where $\\mathcal {L}(\\theta ) = \\frac{1}{2N}\\underset{(i,\\ j)}{\\sum }\\ p_{ij}(y^{\\prime }_{ij}-\\theta ^\\top b_{i,j})^2,$ and $\\mathcal {R}(\\theta ) = \\sqrt{\\frac{\\alpha }{4N}\\underset{(i,\\ j)}{\\sum }(y^{\\prime }_{ij}-\\theta ^\\top b_{i,j})^2}.$ We provide a detailed proof in the Appendix .",
"The example of linear regression with Wasserstein distance based uncertainty sets has been considered in [11].",
"The representation for regularized linear regression in Theorem REF can be seen as an extension of [11].",
"We adopt the weighted sum-of-squared loss and the “regularization” (REF ) here is not the $\\ell _2$ -norm of $\\theta $ .",
"(REF ) can be treated as a “regularization” which is the square root of the residual between $y^{\\prime }_{ij}$ and its estimation.",
"It represents a `worst' case in pairwise ranking: all possible comparisons appear and they have the same number of votes.",
"In this case, the pairwise ranking algorithm could not generate a reasonable ranking result.",
"The uncertainty budget $\\alpha $ play the role as the regularization parameter.",
"As $\\alpha $ increase, the ranking scores $\\theta $ obtained by (REF ) would come closer to the solution of (REF ).",
"The validity of the analysis above will be illustrated in the empirical studies.",
"With Theorem REF , we will have the following corollary which gives a tractable method to obtain the worst-case distribution $q^*_{\\alpha }$ .",
"If we have the worst-case solution, we can solve the corresponding dual variable from the optimal vale of the original DRO problem.",
"Corollary 1 () For $\\lambda \\ge 0$ and the weighted least square loss (REF ), we define $\\psi :\\mathbb {R}^N\\rightarrow \\mathbb {R}$ $\\begin{aligned}& & &\\ \\ \\ \\ \\psi _{\\lambda ,\\ \\ell }(p;\\ \\theta )\\\\& & =&\\ \\ \\underset{z^{\\prime }\\in \\mathbb {R}^{n+2}}{\\textbf {\\textit {sup}}}\\ \\frac{1}{N}\\underset{(i,j)}{\\sum }\\Big \\lbrace \\ell (\\theta ;\\ q_{ij})-\\lambda (p_{ij}-q_{ij})^2\\Big \\rbrace \\end{aligned}$ where $\\ell (\\theta ;\\ q_{ij}) = \\frac{q_{ij}}{2}\\big (y^{\\prime }_{ij}-\\theta _i+\\theta _j\\big )^2.$ Let $I_{\\textit {primal}}=\\underset{\\mathbb {Q}\\ \\in \\ \\mathfrak {W}^{\\alpha }_2(\\mathbb {P}_N)}{\\textbf {\\textit {sup}}}\\ \\mathbb {E}_{z^{\\prime }\\sim \\mathbb {Q}}\\ \\Big [\\ell \\big (\\theta ;\\ z^{\\prime }\\big )\\Big ],$ we have $I_{\\textit {primal}} = \\underset{\\lambda \\ge 0}{\\textbf {\\textit {inf}}}\\ \\Bigg \\lbrace \\ \\lambda \\alpha +\\mathbb {E}_{z^{\\prime }\\sim \\mathbb {Q}}\\Big [\\psi _{\\lambda ,\\ \\ell }(p;\\ \\theta )\\Big ]\\Bigg \\rbrace .$ Moreover, let $\\theta ^*_{\\alpha }$ be the optimal solution of the right hand side of (REF ) and the dual variable of $\\theta ^*_{\\alpha }$ is $\\lambda ^*_{\\alpha }$ will be a solution of (REF ): $\\lambda ^*_{\\alpha } = \\sqrt{\\frac{1}{16N\\alpha }\\cdot \\underset{(i,j)}{\\sum }\\big (y^{\\prime }_{ij}-(\\theta ^*_{\\alpha })^\\top b_{i,j}\\big )^2}.$ The optimal static attack strategy $q^*_{\\alpha }$ is a solution of (REF ) corresponding to $\\theta ^*_{\\alpha }$ and $\\lambda ^*_{\\alpha }$ : $q^*_{\\alpha } = \\underset{q\\ \\in \\ \\mathbb {R}^N_+}{\\textbf {\\textit {arg max}}}\\ \\underset{(i,j)}{\\sum }\\Bigg \\lbrace q_{ij}\\Big (y^{\\prime }_{ij}-(\\theta ^*_{\\alpha })^\\top a_{c}\\Big )^2-\\lambda ^*_{\\alpha }|p_{ij}-q_{ij}|^2\\Bigg \\rbrace .$ [h!]",
"InputInput OutputOutput Initialize the frequency of weights $p$ by (REF ) Obtain the worst-case ranking scoring $\\theta ^*_{\\alpha }$ under the uncertainty budget $\\alpha $ $\\scriptstyle \\theta ^*_{\\alpha }\\ \\in \\ \\underset{\\theta \\in \\Theta }{\\vphantom{\\mathbb {Q}\\in \\mathfrak {W}^{\\alpha }_p(\\mathbb {P}_N)}\\textbf {\\textit {arg min}}}\\ \\left(\\sqrt{\\frac{\\alpha }{4N}\\underset{(i,j)}{\\sum }(y^{\\prime }_{ij}-\\theta ^\\top b_{i,j})^2}+\\frac{1}{2N}\\underset{(i,j)}{\\sum }p_{ij}(y^{\\prime }_{ij}-\\theta ^\\top b_{i,j})^2\\right).$ Calculate the optimal dual variable through (REF ) $\\scriptstyle \\lambda ^*_{\\alpha }\\ =\\ \\sqrt{\\frac{1}{16N\\alpha }\\underset{(i,j)}{\\sum }\\big (y^{\\prime }_{ij}-(\\theta ^*_{\\alpha })^\\top b_{i,j}\\big )^2}.$ Obtain the toxic distribution $q^*_\\alpha $ corresponding to $\\theta ^*_{\\alpha }$ and $\\lambda ^*_{\\alpha }$ $\\scriptstyle q^*_\\alpha \\ \\in \\ \\underset{q\\in \\mathbb {R}^N_+}{\\textbf {\\textit {arg max}}}\\ \\underset{(i,j)}{\\sum }\\ \\Big \\lbrace q_{ij}(y^{\\prime }_{ij}-(\\theta ^*_{\\alpha })^\\top b_{i,j})^2-\\lambda ^*_{\\alpha }|p_{ij}-q_{ij}|^2\\Big \\rbrace .$ Assign the toxic weights with $q^*_\\alpha $ $\\scriptstyle w_\\alpha \\ =\\ M_0(1+\\kappa )\\cdot q^*_\\alpha .$ Round $w_\\alpha $ to obtain the $w^*_\\alpha $ as integer vector $\\scriptstyle w^*_\\alpha \\ =\\ \\textbf {\\textit {rounding}}(w_\\alpha ).$ the poisoned data $\\lbrace w^*_{\\alpha },\\ B,\\ y^{\\prime }\\rbrace $ .",
"Static Poisoning Attack on Pairwise Ranking.",
"Finally, we describe the whole optimization of the static poisoning attack on pairwise ranking with Algorithm REF.",
"First, the adversary changes the original weight $w^{\\prime }_0$ into the frequency $p$ as the initialization (line 1).",
"By Theorem REF, the attacker could obtain the worst-case estimation $\\theta ^*_{\\alpha }$ through (REF ) (line 2).",
"But the attacker cannot adopt $\\theta ^*_{\\alpha }$ as the attack operation.",
"Here we solve the dual variable $\\lambda ^*_{\\alpha }$ (line 4) to find the toxic distribution.",
"Then the toxic distribution $q^*_{\\alpha }$ with uncertainty budget $\\alpha $ is obtained by Corollary REF (line 4).",
"With some rounding operation (line 5 & 6), the adversary prepares the poisoned data $\\lbrace w^*_{\\alpha },\\ B,\\ y^{\\prime }\\rbrace $ .",
"Then the poisoned data is provided to the ranker who solved the ranking parameter by (REF ).",
"Then the whole poisoning process will be completed."
],
[
"Theoretical Analysis",
"In this section, we come back to (REF ) and give a couple of inequalities relating the local worst-case (or local minimax) risks and the usual statistical risks of the pairwise ranking under adversarial conditions.",
"In the traditional paradigm of statistical learning[71], we have a class of probability measures $\\mathcal {P}$ on a measurable instance space $\\mathcal {Z}$ and a class $\\mathcal {F}$ of measurable functions $\\ell :\\mathcal {Z}\\rightarrow \\mathbb {R}_+$ .",
"Each $\\ell \\in \\mathcal {F}$ quantifies the loss of a certain decision rule or a hypothesis.",
"With a slight abuse of terminology, we will refer to $\\mathcal {F}$ as the hypothesis space.",
"The (expected) risk of a hypothesis $\\ell $ on instances generated according to $\\mathbb {P}\\in \\mathcal {P}(\\mathcal {Z})$ is given by $R_{\\mathbb {P}}(\\ell ) := \\mathbb {E}_{z\\sim \\mathbb {P}}\\big [\\ell (z)\\big ] = {\\int }_{\\mathcal {Z}}\\ \\ell (z)\\ \\mathbb {P}(\\mathrm {d}z).$ Given an $N$ -tuple $\\lbrace z_1,\\dots ,z_N\\rbrace $ of i.i.d.",
"training examples drawn from an unknown distribution $\\mathbb {P}\\in \\mathcal {P}$ , the objective is to find a hypothesis $f\\in \\mathcal {F}$ whose risk $R(\\mathbb {P},\\ \\ell )$ is close to the minimum risk $R^*_{\\mathbb {P}}(\\mathcal {F}):=\\underset{\\ell \\ \\in \\ \\mathcal {F}}{\\textbf {\\textit {inf}}}\\ R_{\\mathbb {P}}(\\ell )$ with high probability.",
"Under some suitable regularity assumptions, this objective can be accomplished via Empirical Risk Minimization (ERM): $R_{\\mathbb {P}_N}(\\ell ) := \\frac{1}{N}\\ \\sum ^N_{i=1}\\ \\ell (z_c)$ and the minimum empirical risk is $R^*_{\\mathbb {P}_N}(\\mathcal {F}):=\\underset{\\ell \\ \\in \\ \\mathcal {F}}{\\textbf {\\textit {min}}}\\ R_{\\mathbb {P}_N}(\\ell ),$ where $\\mathbb {P}_N=\\frac{1}{N}\\sum ^N_{c=1}\\delta _{z_c}$ is the empirical distribution of the training examples.",
"Meanwhile, the minimax risk[40] can be defined as $\\hat{R}_{\\mathbb {P}_N}(\\mathcal {F}):=\\ \\underset{\\ell \\ \\in \\ \\mathcal {F}\\vphantom{\\mathbb {Q}\\ \\in \\ \\mathfrak {W}(\\mathbb {P}_N)}}{\\textbf {\\textit {min}}\\vphantom{\\textbf {\\textit {sup}}}}\\ \\ \\underset{\\mathbb {Q}\\ \\in \\ \\mathfrak {W}(\\mathbb {P}_N)}{\\textbf {\\textit {sup}}}\\ R_{\\mathbb {Q}}(\\ell )$ We assume that the instance space $\\mathcal {Z}$ is a Polish space (i.e., a complete separable metric space) with metric $d_{\\mathcal {Z}}$ .",
"We denote by $\\mathcal {P}(\\mathcal {Z})$ the space of all Borel probability measures on $\\mathcal {Z}$ , and by $\\mathcal {P}_m(\\mathcal {Z})$ with $m\\ge 1$ the space of all $\\mathbb {P}\\in \\mathcal {P}(\\mathcal {Z})$ with finite $m\\text{-\\textit {th}}$ moments.",
"The metric structure of $\\mathcal {Z}$ can be used to define a family of metrics on the spaces $\\mathcal {P}_m(\\mathcal {Z})$ .",
"We then define the local worst-case risk of $\\ell $ at $\\mathcal {P}$ , $R_{\\mathbb {P},\\alpha ,p}(\\ell ) := \\underset{\\mathbb {Q}\\ \\in \\ \\mathfrak {W}^{\\alpha }_p(\\mathbb {P}_N)}{\\textbf {\\textit {sup}}}\\ R_{\\mathbb {Q}}(\\ell )$ and the local minimax risk of $\\mathcal {P}$ , $R^*_{\\mathbb {P},\\alpha ,p}(\\mathcal {F}) := \\underset{\\ell \\ \\in \\ \\mathcal {F}}{\\textbf {\\textit {inf}}}\\ R_{\\mathbb {P},\\alpha ,p}(\\ell ).$ Next, we analyze the performance of the local minimax ERM procedure of the pairwise ranking, namely, $\\hat{\\ell }\\in \\underset{\\ell \\ \\in \\ \\mathcal {F}}{\\textbf {\\textit {arg min}}}\\ R_{\\mathbb {P}_N,\\alpha ,2}(\\ell ).$ theoremworstprob Consider the setting of pairwise ranking problem with the sum-of-squared loss, for any $t>0$ , it holds $\\Pr \\Big (\\ \\exists \\ \\ell \\in \\mathcal {F}: R_{\\mathbb {P}, \\alpha , 2}(\\ell )>\\varsigma _1\\ \\Big )\\ \\le \\ e^{-2t^2}$ and $\\ \\ \\Pr \\Big (\\exists \\ \\ell \\in \\mathcal {F}: R_{\\mathbb {P}_N, \\alpha , 2}(\\ell )>\\varsigma _2\\Big )\\ \\le \\ 2e^{-2t^2}$ where $\\begin{aligned}\\varsigma _1=\\underset{\\lambda \\ge 0}{\\textbf {\\textit {min}}}\\ \\ \\Bigg \\lbrace \\ \\lambda \\alpha ^2\\ +\\ \\mathbb {E}_{z\\sim \\mathbb {Q}}\\ \\big [\\ \\psi _{\\lambda ,\\ell }\\ (z)\\ \\big ]\\Bigg \\rbrace +\\frac{24\\mathcal {J}(\\mathcal {F})+t}{\\sqrt{N\\ }}.\\end{aligned}$ and $\\begin{aligned}& \\varsigma _2=\\underset{\\lambda \\ge 0}{\\textbf {\\textit {min}}}\\ \\ \\Bigg \\lbrace \\ (\\lambda +1)\\alpha ^2\\ +\\ \\mathbb {E}_{z\\sim \\mathbb {Q}}\\ \\big [\\ \\psi _{\\lambda ,\\ell }\\ (z)\\ \\big ]\\\\& {\\color {white}\\underset{\\lambda \\ge 0}{\\textbf {\\textit {min}}}\\ \\ \\Bigg \\lbrace }\\ \\ \\ \\ \\ \\ +\\frac{\\sqrt{\\textbf {\\textit {log}}(\\lambda +1)}}{\\sqrt{N\\ }}\\ \\Bigg \\rbrace +\\frac{24\\mathcal {J}(\\mathcal {F})+t}{\\sqrt{N\\ }},\\end{aligned}$ where $\\mathcal {J}(\\mathcal {F})$ is the Dudley’s entropy integral [20], which is served as the complexity measure of the hypothesis class $\\mathcal {F}$ .",
"Theorem REF is a type of data-dependent generalization bounds which is proposed for margin cost function class [38], [40].",
"By the strong duality results, we can establish this result from the dual representation of the Wasserstein DRO problem.",
"The detailed proof is provided in the Appendix .",
"Here we note that the hypothesis selected by the minimax ERM procedure (REF ) are uniform smoothness with respect to the underlying metric $d_{\\mathcal {Z}}(\\cdot ,\\cdot )$ .",
"Further, we have the following result.",
"Proofs are relegated to the Appendices .",
"theoremexcessrisk Consider the setting of pairwise ranking problem with the sum-of-squared loss, the following holds with probability as least $1-\\eta $ $\\begin{aligned}& & &\\ \\ \\ R_{\\mathbb {P},\\alpha ,2}(\\hat{\\ell })-R^*_{\\mathbb {P},\\alpha ,2}(\\mathcal {F})\\\\& &\\le &\\ \\ \\frac{48\\mathcal {J}(\\mathcal {F})}{\\sqrt{N}}+\\frac{48L\\big [\\text{diam}(\\mathcal {Z})\\big ]^2}{\\alpha \\sqrt{N}} + 3\\sqrt{\\frac{\\textbf {\\textit {log}}(\\frac{2}{\\eta })}{2N}},\\end{aligned}$ where $\\text{diam}(\\mathcal {Z})$ is the diameter of $\\mathcal {Z})$ $\\text{diam}(\\mathcal {Z}) = \\underset{z,z^{\\prime }\\in \\mathcal {Z}}{\\textbf {\\textit {sup}}} d_{\\mathcal {Z}}(z,\\ z^{\\prime }).$ In this section, four examples are exhibited with both simulated and real-world data to illustrate the validity of the proposed poisoning attack on pairwise ranking.",
"The first example is with simulated data while the latter three exploit real-world datasets involved crowdsourcing, election and recommendation."
],
[
"Simulated Study",
"Settings.",
"We first validate our poisoning attack framework on simulated data.",
"We create a random total ordering on set $V$ with $n$ candidates as the ground-truth ranking and generate the comparison matrix $B$ and the labels $y^{\\prime }$ as (REF ).",
"Next, we generate the ground-truth weight of each comparisons $w_0$ .",
"Notice that the original data $\\lbrace w_0,\\ B,\\ y^{\\prime }\\rbrace $ consists of some noisy comparisons.",
"In the simulation study, we can specify the percentage of noisy comparisons, denoted as $\\varrho $ .",
"We validate the proposed attack framework when $n$ , $w_0$ and $\\varrho $ vary.",
"Moreover, the maximum toxic dosage $\\kappa $ and the uncertainty budget $\\alpha $ are the hyper-parameters of the Algorithm REF .",
"Since the annotations of pairwise data are usually collected via crowdsourcing platforms where the attacker could produce hundreds of zombie accounts easily to inject the poisoned pairwise comparisons, we also vary $\\kappa $ and $\\alpha $ in our experiments.",
"At last, there exists a rounding operator in the Algorithm REF and we explore the results of different rounding functions, e.g.",
"ceiling, floor, and the nearest integer of each element in $w_{\\alpha }$ .",
"Competitors.",
"To the best of our knowledge, the proposed method is the first poisoning attack on pairwise ranking.",
"To see whether our proposed method could provide efficient perturbation data for misleading the pairwise ranking algorithm, we implement the random perturbation attack (referred to as `Random') and the Stackelberg or dynamic game attack (referred to as `Dynamic') as the competitors.",
"[leftmargin=*] The random perturbation attack modifies $w_0$ as $w_{\\text{random}}$ to manipulate the ranking result.",
"The random perturbation attack generates $w_{\\text{random}}$ and obeys the constraints (REF ) and (REF ) to hide his/her behaviors.",
"We vary $b$ and $l$ to explore the ability of random attack.",
"The random perturbation data is noted as $\\mathcal {Z}_{\\text{random}} = \\lbrace A,\\ y,\\ w_{\\text{random}}\\rbrace $ .",
"We assume this attacker is also lack of prior knowledge on the true ranking.",
"So the random perturbation attack also adopts the fixed label set $y$ .",
"The Stackelberg (dynamic) game attack comes from (REF ).",
"To execute this type of poisoning attack, the adversary would have the full knowledge of original training data $w_0$ and the corresponding relative ranking score $\\theta _{\\text{original}}$ .",
"With these advantages, the adversary can adjust his/her strategies to provide the optimal malicious action with the bi-level optimization like (REF ).",
"Without a doubt, the adversary endues with the privilege by such a hierarchical relation.",
"For the fair competition, we only perform one round of the leader-follower game as the other competitors.",
"Notice that this kind to attack is also proposed by this paper.",
"Due to the length limitation, we provide the details of this attack in the supplementary materials.",
"It is worth noting that the poisoning attack with dynamic game is not a practical attack method.",
"(REF ) is a bi-level optimization and the maximization process needs the solution of the minimization problem.",
"In other words, the attacker must obtain the relative ranking score $\\hat{\\theta }$ estimated from the original training data without perturbation.",
"This operation is much harder than injecting some modified training samples into the victim's training set.",
"Only the so-called “white-box” setting would satisfy its necessary requirements.",
"As the `Dynamic' method needs more exorbitant conditions, the `Dynamic' method only reflects the vulnerability of ranking aggregation algorithms but can not show the superiority of the `Static' method.",
"Evaluation Metrics.",
"We adopt the following measures for evaluating the ranking results aggregated by the different sets of pairwise comparisons.",
"[leftmargin=*] Kendall $\\tau $ Distance (Kendall-$\\tau $ ).",
"The Kendall rank correlation coefficient evaluates the degree of similarity between two sets of ranks given the same objects.",
"This coefficient depends upon the number of inversions of pairs of objects which would be needed to transform one rank order into the other.",
"Let $V=[n]$ be a set of $n$ candidates and $\\pi _1,\\ \\pi _2$ are two total orders or permutations on $V$ , the Kendall $\\tau $ distance is defined to be $d_K(\\pi _1,\\ \\pi _2) = \\frac{2}{n(n-1)}\\cdot \\vartheta ,$ where $\\vartheta = \\sum _{i=1}^{n-1}\\sum _{j=i+1}^n\\vartheta (\\pi _1(i),\\ \\pi _1(j),\\ \\pi _2(i),\\ \\pi _2(j))$ is the number of different pairs between these two ordered sets $\\pi _1, \\pi _2$ as $\\begin{aligned}& & &\\ \\ \\vartheta (\\pi _1(i),\\ \\pi _1(j),\\ \\pi _2(i),\\ \\pi _2(j))\\\\& &=&\\left\\lbrace \\begin{matrix}\\ \\ 1, & \\text{if\\ } (\\pi _1(i)-\\pi _1(j))(\\pi _2(i)-\\pi _2(j))>0,\\\\-1, & \\text{if\\ } (\\pi _1(i)-\\pi _1(j))(\\pi _2(i)-\\pi _2(j))<0,\\\\\\ \\ 0, & \\text{otherwise, }\\end{matrix}\\right.\\end{aligned}$ and $\\pi _1(i)$ represents the ranking score of the $i^{th}$ object in ranking list $\\pi _1$ .",
"Kendall $\\tau $ distance counts the number of pairwise mismatches between two rank orders.",
"Then this metric considers all candidates of $V$ .",
"However, Kendall-$\\tau $ ignores the importance of the top objects in a ranking list.",
"Reciprocal Rank (R-Rank).",
"The reciprocal rank is a statistic measure for evaluating any process that produces an order list of possible responses to a sample of queries, ordered by the probability of correctness or the ranking scores.",
"The reciprocal rank of a rank order is the multiplicative inverse of the rank of the first correct object: $RR = \\frac{1}{\\text{rank}_i},$ where ${\\text{rank}_i}$ refers to the rank position of the first candidates of the ground-truth ranking in the other list.",
"Precision at $K$ (P$@K$ ).",
"Precision at $K$ is the proportion of the top-$K$ objects in the other rank order that are consistent with the true ranking.",
"In this case, the precision and recall will be the same.",
"So we do not report the recall and F score for our poisoning attack method.",
"Average Precision at $K$ (AP$@K$ ).",
"Average precision at K is a weighted average of the precision.",
"If the top objects in the new ranking list are consistent with the true ranking, they will contribute more than the tail objects in this metric.",
"Normalized Discounted Cumulative Gain at K (NDCG$@K$ ).",
"Using a graded relevance scale of objects in ranking result, discounted cumulative gain (DCG) measures the usefulness, or gain, of the objects based on its position in the order list when recovering to the true ranking.",
"The gain is accumulated from the top to the bottom, with the gain of each result discounted at lower ranks.",
"Compared to DCG, NDCG will be normalized by the ideal DCG.",
"Table: Comparative results of different attack methods on simulated data.Figure: The amount of changed pairwise comparisons by the poisoning attack with static game.",
"The x-axis is the index of pairwise comparisons and the y-axis is the amount of change.",
"Note that the ranges of y-axis in each sub-figure are different.Figure: The number of correct pairwise comparisons and comparisons which conflict with the ground-truth ranking in the poisoned training set by `Static' method.Comparative Results.",
"We display the comparative results of different attack methods in Table REF .",
"There the number of candidates ranges from 10 to 100 ( $n=10, 20, 50, 100$ ).",
"The percentage of noisy comparisons is $\\varrho =0$ in the four cases.",
"We let the maximum toxic dosage to be 0 as $\\kappa =0$ to verify the effectiveness of the worst-case distribution in the Wasserstein ball with uncertainty budget $\\alpha $ .",
"We show the attack effect of `Static' and `Dynamic' methods with different budgets.",
"The performance of `Random' are affected by two parameters: the percentage of the new comparisons injected into the original training set, and the percentage of the existed comparisons deleted from the original training set.",
"Here we set these two parameters be $s_1 = s_2 = 0.05$ .",
"We obtain the following observations from Table REF .",
"The `Static' method can decrease the Kendall-$\\tau $ when the uncertainty budget $\\alpha $ increases.",
"Looking back on the Algorithm REF, the uncertainty budget $\\alpha $ is the weight of the second term in (REF ) and the two parts of (REF ) have the same monotonic respect to $\\theta $ .",
"With the increasing of $\\alpha $ , the impact of the second term (REF ) to the solution (REF ) becomes gradually.",
"The solution of (REF ) means that the algorithm will adopt all possible pairwise comparisons with same number of voting to aggregate the final ordered list.",
"There is no doubt that this case would be far away from the ground-truth ranking.",
"If $\\alpha $ approaches $\\infty $ , we would obtain this confusing solution.",
"This explains the behaviors of the `Static' methods when the Kendall-$\\tau $ is larger than 0.",
"In Figure REF , we see that the `Static' method does two things to perturb the training set: adding pairwise comparisons which conflict with the ground-truth ranking and removing the pairwise comparisons which is consistent with the ground-truth ranking.",
"The total amount of change enlarge when the uncertainty budget $\\alpha $ increase.",
"If the Kendall-$\\tau $ is smaller than 0, it means that the poisoned training dataset would support an opposite ranking list.",
"In Figure REF , each group corresponds to a poisoned data set by `Static' method with a certain uncertainty budget.",
"When the Kendall-$\\tau $ is smaller than 0 ($\\alpha \\ge 10^{-3}$ ), we observe that the number of comparisons which conflict with the ground-truth ranking is larger that the number of comparisons which is consistent with the ground-truth ranking.",
"Such training data could generate an arbitrarily ordered list.",
"If it happens, the Kendall-$\\tau $ could not monotonically decrease when we increase the uncertainty budget continuously.",
"Moreover, the uncertainty budget $\\alpha $ plays a totally different role in the `Dynamic' method.",
"The existing work [54], [18] reveal that such kind of min-max problem is a new type of regularization.",
"This regularization also carries out the `bias-variance' trade-off like the classical approaches like Tikhonov regularization.",
"In this case, the uncertainty budget $\\alpha $ can be explained as a regularization coefficient.",
"The Kendall-$\\tau $ of `Dynamic' method presents a `U'-type curve in our experiments.",
"Visualization.",
"We visualize the ranking list in Figure REF .",
"The visualization shows the same phenomenons as the numeric results in Table REF .",
"As the target ranking aggregation algorithm does not emphasize the top-K results and the adversary has no prior knowledge of the ranking results, the untrustworthy results of `Static' method only depend on the original data and the uncertainty budget.",
"So the proposed method is the non-target attack for pairwise ranking algorithm.",
"Manipulating the ranking list with specific goals, a.k.a the target attack, is the future work.",
"Figure: The ranking generated from the original data (Original), random attack data (Random), static poisoning attack data (Static) and dynamic poisoning attack data (Dynamic).Figure: The ranking generated from the original data (Original), random perturbation data (Random), poisoned data (Static and Dynamic) on Human Age dataset.",
"When the Kendall-τ\\tau is smaller than 0 (α≥10 -4 \\alpha \\ge 10^{-4}), we observe that the aggregated results would put the younger people at the top of the lists.",
"Moreover, the same phenomenons in the simulation are still observed.",
"The training data with more than 50%50\\% outliers could generate an arbitrarily ordered list.",
"If it happens, the Kendall-τ\\tau could not monotonically decrease when we increase the uncertainty budget continuously for the static attack strategies."
],
[
"Human Age",
"Description.",
"30 images from human age dataset FGNET are annotated by a group of volunteer users on ChinaCrowds platform.",
"The ground-truth age ranking is known to us.",
"The annotator is presented with two images and given a binary choice of which one is older.",
"Totally, we obtain $8,017$ pairwise comparisons from 94 annotators.",
"Comparative Results.",
"Notice that the real-world data has a high percentage of outliers (about $20\\%$ comparisons conflict with the correct age ranking).",
"We observe similar phenomenons as the simulation experiments.",
"When the uncertainty budget increase, the `Static' method would inject more comparisons which conflict with the true age ranking and delete the original comparisons which indicate the true ordered list.",
"Once the `wrong' samples overwhelm the `correct' samples, the ranking aggregation algorithm would like to generate a reversed list.",
"As there are only the `wrong' samples in the toxic training set by `Static' method, the final result could be arbitrary.",
"Table: Comparative results of different attack methods on human age data."
],
[
"Dublin Election",
"Description.",
"The Dublin election data sethttp://www.preflib.org/data/election/irish/ contains a complete record of votes for elections held in county Meath, Dublin, Ireland on 2002.",
"This set contains $64,081$ votes over 14 candidates.",
"These votes could be a complete or partial list over the candidate set.",
"The ground-truth ranking of 14 candidates are based on their obtained first preference voteshttps://electionsireland.org/result.cfm?election=2002cons=178sort=first.",
"The five candidates who receive the most first preference votes will be the winner of the election.",
"We are interested in the top-5 performance of the pairwise rank aggregation method.",
"Then these votes are converted into the pairwise comparisons.",
"The total number of the comparisons is $652,817$ .",
"Comparative Results.",
"In this experiment, we evaluate the ability of poisoning attack in election.",
"The election result is not obtained by pairwise ranking aggregation.",
"However, the ordered list aggregated from induced comparisons still shows positive correlation with the actual election result.",
"Different from the manipulation or strategic voting setting in election, the adversary could control the whole votes but with some constraints.",
"As a consequence, the poisoning attack could break the barrier of computational complexity [74], [69].",
"The proposed method focuses on the `non-target' attack on pairwise ranking aggregation.",
"The `Static' method could perturb the ranking list generated by the original algorithm with a sufficient uncertainty budget.",
"But the adversary is not able to manipulate the order with her/his preference as she/he can not decide the winner of election.",
"We call the problem as the `target' attack, where the adversary manipulates the order with her/his preference.",
"Our future work will study the `target' poisoning attack on pairwise ranking.",
"Moreover, the `Dynamic' method does not completely destroy the election result.",
"It indicates that the inaccurate supervision would mislead the adversary and the corresponding Nash equilibrium could show partiality for the ranking aggregation algorithm.",
"Table: Comparative results of different attack methods on Dublin election data.Table: Comparative results of different attack methods on Sushi election data."
],
[
"Sushi Preference",
"Description.",
"This dataset contains the results of a series of surveys which involves 5000 individuals for their preferences about various kinds of sushi.",
"The original survey provides 10 complete strict rank orders of 10 different kinds of sushi as 1) ebi (shrimp), 2) anago (sea eel), 3) maguro (tuna), 4) ika (squid), 5) uni (sea urchin), 6) sake (salmon roe), 7) tamago (egg), 8) toro (fatty tuna), 9) tekka-maki (tuna roll), and 10) kappa-maki (cucumber roll).",
"The complete strict rank orders are converted into the pairwise graph by [50].",
"We adopt the whole $221,670$ comparisons and the Hodgerank[34] method to aggregate a ranking list as the ground-truth.",
"Then 20 percent of pairwise comparisons are chosen to consist of the observation set.",
"The different attack approaches can manipulate the subset of data and induce the pairwise ranking algorithm to generate a different order list.",
"Comparative Results.",
"This experiment is a classic setting in recommendation and computational advertisement.",
"With the selected subset, the ranking aggregation method can produce a same ranking list as adopting with the whole preference data.",
"The random attack would not change this list in this experiment.",
"In addition, the `Dynamic' method is trapped with the inaccurate supervision and only shows a moderate destructive effect.",
"The `Static' method could generate a promise perturbation to mislead the ranking aggregation method as the Kendall-$\\tau $ would be $-1$ ."
],
[
"Computational Complexity Analysis",
"The computational complexity of the dynamic strategy depends on the number of turns of (REF ).",
"Given $n$ candidates, the complexity of the ranker is $\\mathcal {O}(n^6)$ for solving a least square problem and the complexity of the adversary is $\\mathcal {O}(n^2\\log (n^2)+\\log \\frac{1}{\\epsilon }\\cdot \\log (n^2))$ where $\\epsilon $ is the solution accuracy, $n^2\\log (n^2)$ is for sorting and the last part corresponds to the projection onto the $\\ell _2$ ball.",
"The computational complexity of the static strategy depends on the subroutines of Line 2 and Line 4 in Algorithm 1.",
"We solve the subroutine of Line 2 by gradient descent and evaluating the gradient needs $\\mathcal {O}(n^4)$ each time.",
"The complexity of Line 4 is $\\mathcal {O}(n^3+n^2\\log (n^2)+n^2)$ where $n^3$ is for the closed form, $n^2\\log (n^2)$ is for the sorting and $n^2$ for the projection onto the simplex.",
"We also display the computational complexity comparisons on the synthetic and the real-world datasets in Table REF and REF .",
"The results are mean of 100 trials with different pairwise comparisons or initialization.",
"All computation is done using MATLAB$^$ R2016b, on a Laptop PC with MacOS$^$ Big Sur, with 3.1GHz Intel$^{}$ Core i7 CPU, and 16GB 2133MHz DDR3 memory.",
"Table: Computational complexity (ms) comparisons on the synthetic dataset.",
"The results are the mean of 100 trials with different pairwise comparisons.Table: Computational complexity (ms) comparisons on the real-world datasets.",
"The results are the mean of 100 trials with different initialization.We initiate the first study of data poisoning attacks in the context of pairwise ranking.",
"We formulate the attack problem as a robust game between two players, the ranker and the adversary.",
"The attacker’s strategies are modeled as the distributionally robust optimization problems and some theoretical results are established, including the existence of distributionally robust Nash equilibrium and the generalization bounds.",
"Our empirical studies show that our attack strategies significantly break the performance of pairwise ranking in the sense that the correlation between the true ranking list and the aggregated result with toxic data can be decreased dramatically.",
"There are many avenues for further investigation – such as, providing the finite-sample and asymptotic results characterizing the theoretical performance of the estimator with adversarial learning, extending our attacks to more pairwise ranking algorithms such as spectral ranking, and trying to attack the ranking algorithms with defense paradigm.",
"We believe that a very interesting open question is to expand our understanding to better understand the role and capabilities of adversaries in pairwise ranking."
],
[
"Proof of Theorem ",
"Property 1 Let the pay-off function $f_r,\\ r=1,2$ be the weighted sum-of-squared loss $\\ell $ (REF ) in (REF ).",
"If the uncertainty set is $\\mathfrak {X}^{\\alpha }(\\mathbb {P}_N)$ or $\\mathfrak {W}_p^{\\alpha }(\\mathbb {P}_N)$ , we have $f_r$ is a continuous function, and for any fixed $\\lbrace x_{-r},\\ \\xi \\rbrace $ , $f_r(x_r,\\ x_{-r},\\ \\xi )$ is convex over $\\mathcal {X}_r$ .",
"$\\mathcal {X} = \\mathcal {X}_1\\times \\mathcal {X}_2$ is a compact set.",
"$\\mathbb {E}_{\\xi \\sim \\mathbb {Q}}[f_r(x_r,\\ x_{-r},\\ \\xi )]$ is finite-valued, $\\forall \\ x\\in \\mathcal {X}$ , $\\mathbb {Q}\\in \\mathfrak {U}$ .",
"$\\mathfrak {U}$ is a weakly compact set.",
"Proposition 1 Let $x=\\lbrace x_1,\\ x_2\\rbrace $ , $v=\\lbrace v_1,\\ v_2\\rbrace $ , we define $\\phi :\\mathcal {X}\\times \\mathcal {X}\\rightarrow \\mathbb {R}_+$ as $\\phi (v,\\ x)\\ =\\underset{\\mathbb {Q}\\ \\in \\ \\mathfrak {U}}{\\textbf {\\textit {sup}}}\\ \\mathbb {E}_{\\xi \\sim \\mathbb {Q}}\\ \\Big [\\ f_1(v_1,\\ x_{2},\\ \\xi )\\ \\Big ]+\\underset{\\mathbb {Q}\\ \\in \\ \\mathfrak {U}}{\\textbf {\\textit {sup}}}\\ \\mathbb {E}_{\\xi \\sim \\mathbb {Q}}\\ \\Big [\\ f_2(x_1,\\ v_{2},\\ \\xi )\\ \\Big ]\\nonumber $ With Property REF , $ x^*=\\lbrace x^*_1,\\ x^*_2\\rbrace $ is a distributional robust Nash equilibrium of (REF ) if and only if $\\lbrace x^*_1,\\ x^*_2\\rbrace \\ \\in \\ \\underset{v\\ \\in \\ \\mathcal {X}}{\\textbf {\\textit {arg\\ min}}}\\ \\phi (v,\\ x^*).$ The reformulation $\\phi $ is well known for deterministic Nash equilibrium, see for example [62].",
"The “if” part follows from the fact that if $\\lbrace x^*_1,\\ x^*_2\\rbrace $ is not an equilibrium of (REF ), there exists some $\\bar{x}_r$ , $r=1, 2$ , such that $\\underset{\\mathbb {Q}\\ \\in \\ \\mathfrak {U}}{\\textbf {\\textit {sup}}}\\ \\mathbb {E}_{\\xi \\sim \\mathbb {Q}}\\Big [\\ f_r(\\bar{x}_r,\\ x^*_{-r},\\ \\xi )\\ \\Big ]<\\underset{\\mathbb {Q}\\ \\in \\ \\mathfrak {U}}{\\textbf {\\textit {sup}}}\\ \\mathbb {E}_{\\xi \\sim \\mathbb {Q}}\\Big [\\ f_r(x^*_r,\\ x^*_{-r},\\ \\xi )\\ \\Big ]\\nonumber $ Let $\\bar{x}=\\lbrace \\bar{x}_r,\\ x^*_{-r}\\rbrace $ , we have $\\phi (\\bar{x},\\ x^*)<\\phi ( x^*,\\ x^*)$ .",
"This is a contradiction.",
"The “only if” part is obvious as $\\underset{\\mathbb {Q}\\ \\in \\ \\mathfrak {U}}{\\textbf {\\textit {sup}}}\\ \\mathbb {E}_{\\xi \\sim \\mathbb {Q}}\\Big [\\ f_r(x_r,\\ x^*_{-r},\\ \\xi )\\ \\Big ]>\\underset{\\mathbb {Q}\\ \\in \\ \\mathfrak {U}}{\\textbf {\\textit {sup}}}\\ \\mathbb {E}_{\\xi \\sim \\mathbb {Q}}\\Big [\\ f_r(x^*_r,\\ x^*_{-r},\\ \\xi )\\ \\Big ]\\nonumber $ Summing up each $r$ on both sides, the inequality shows that $\\lbrace x^*_1,\\ x^*_2\\rbrace $ is a global minimizer.",
"Based on the Proposition REF , we have the following existence result for distributional robust Nash equilibrium of REF .",
"* Based on the Proposition REF , each $\\mathbb {E}_{\\xi \\sim \\mathbb {Q}}[f_r(x_r,\\ x_{-r},\\ \\xi )]$ is continuous and convex for any $\\mathbb {Q}\\in \\mathfrak {U}$ .",
"The supremum preserves the convexity of $f_r$ and, under weakly compactness of $\\mathfrak {U}$ , the continuity of $f_r$ will hold.",
"Therefore $\\phi (v, x)$ is continuous and convex w.r.t.",
"$v$ on $\\mathcal {X}$ for any fixed $x\\in \\mathcal {X}$ .",
"The existence of an optimal solution to $\\underset{v\\ \\in \\ \\mathcal {X}}{\\textbf {\\textit {min}}}\\ \\phi (v,\\ x)$ follows from compactness of $\\mathcal {X}$ under the third condition in Assumption REF .",
"To complete the proof, we are left to show the existence of $x^*\\in \\mathcal {X}$ such that $x^*\\ \\in \\ \\underset{v\\ \\in \\ \\mathcal {X}}{\\textbf {\\textit {arg\\ min}}}\\ \\phi (v,\\ x^*).$ Let $\\Phi (x)$ be the set of optimal solutions to $\\underset{}{\\textbf {\\textit {min}}}\\ \\phi (v,\\ x)$ for each fixed $x\\in \\mathcal {X}$ .",
"Then $\\Phi (x)\\subset \\mathcal {X}$ holds.",
"By the convexity of $\\phi $ , $\\Phi (x)$ is a convex set.",
"Obviously, $\\Phi (x)$ is closed, namely, there exists a sequence $\\lbrace x_k\\rbrace $ with $\\textbf {\\textit {lim}}_{k\\rightarrow \\infty }x_k=\\bar{x}$ and $v_k\\in \\Phi (x_k)$ , if $\\textbf {\\textit {lim}}_{k\\rightarrow \\infty }v_k=\\bar{v}$ , we have $\\bar{v}\\in \\Phi (\\bar{x})$ .",
"Further, following by Theorem 4.2.1 in [4], $\\Phi $ is upper semi-continuous on $\\mathcal {X}$ .",
"By Kakutani’s fixed point theorem [35], , there exists $x^*\\in \\mathcal {X}$ such that $x^*\\in \\Phi (x^*)$ ."
],
[
"Proof of Theorem ",
"The following proposition shows the strong duality result for Wasserstein DRO [12], which ensures that the inner supremum in (REF ) admits a reformulation which is a simple, univariate optimization problem.",
"Note that there exists the other strong duality result of Wasserstein DRO [23].",
"Proposition 2 Let $d:\\mathbb {R}^{n+2}\\times \\mathbb {R}^{n+2}\\rightarrow [0,\\infty ]$ be a lower semi-continuous cost function satisfying $d(z,\\ z^{\\prime })=0$ whenever $z = z^{\\prime },\\ z = (p,\\ b,\\ y),\\ z^{\\prime } = (p^{\\prime },\\ b^{\\prime },\\ y^{\\prime })$ .",
"For $\\lambda \\ge 0$ and loss function $\\ell $ (REF ) that is upper semi-continuous in $(p,\\ b,\\ y)$ for each $\\theta $ , define $\\psi _{\\lambda ,\\ell }(z;\\theta ) :=\\underset{z^{\\prime }\\in \\mathbb {R}^{n+2}}{\\textbf {\\textit {sup}}}\\ \\ \\underset{(i,j)}{\\sum }\\ \\Bigg \\lbrace \\ \\ell (\\theta ;\\ z^{\\prime })-\\lambda d(z,\\ z^{\\prime })\\ \\Bigg \\rbrace .$ Then $\\underset{\\mathbb {Q}\\ \\in \\ \\mathfrak {W}^{\\alpha }_p(\\mathbb {P}_N)}{\\textbf {\\textit {sup}}}\\ \\ \\mathbb {E}_{z^{\\prime }\\sim \\mathbb {Q}}\\Big [\\ \\ell \\big (\\theta ,\\ z^{\\prime }\\big )\\ \\Big ] = \\underset{\\lambda \\ge 0}{\\textbf {\\textit {min}}}\\left\\lbrace \\ \\lambda \\alpha +\\frac{1}{N}\\sum _{z}\\psi _{\\lambda ,\\ell }(z;\\theta )\\ \\right\\rbrace $ * Let $\\Delta _{ij} = q_{ij}-p_{ij}$ .",
"The $\\psi _{\\lambda ,\\ell }$ function (REF ) has a new formulation as $\\begin{aligned}& & &\\ \\ \\ \\ \\psi _{\\lambda ,\\ell }(\\theta ,\\ p)\\\\& &=&\\ \\ \\underset{q\\ \\in \\ \\mathbb {R}^N_+}{\\textbf {\\textit {sup}}}\\ \\frac{1}{N}\\underset{c\\ \\in \\ \\mathcal {C}}{\\sum }\\ \\Big \\lbrace \\ell (\\theta ,\\ q_{ij})-\\lambda \\big [d(p_{ij},\\ q_{ij})\\big ]^2\\Big \\rbrace \\\\& &=&\\ \\ \\underset{q\\ \\in \\ \\mathbb {R}^N_+}{\\textbf {\\textit {sup}}}\\ \\frac{1}{N}\\underset{c\\ \\in \\ \\mathcal {C}}{\\sum }\\left\\lbrace \\frac{q_{ij}}{2}\\cdot \\left[(-\\theta ^\\top ,\\ 1)\\begin{pmatrix}a_{c}\\\\y_{ij}\\end{pmatrix}\\right]^2-\\lambda \\big |p_{ij}-q_{ij}\\big |^2\\right\\rbrace \\\\[1.5mm]& &=& \\ \\ \\ \\frac{1}{N}\\underset{c\\ \\in \\ \\mathcal {C}}{\\sum }\\ \\underset{\\Delta _{ij}\\ \\in \\ \\mathbb {R}}{\\textbf {\\textit {sup}}}\\Big (\\Delta _{ij} b_{ij}-\\lambda \\Delta _{ij}^2 + p_{ij}b_{ij}\\Big ),\\end{aligned}$ where $b_{ij} = (y_{ij}-\\theta ^\\top a_{c})^2/2$ , and the third equality holds due to $\\psi _{\\lambda , \\ell }$ is a decomposable function.",
"Expanding (REF ), we can simplify $\\psi _{\\lambda ,\\ell }$ as below: $\\begin{aligned}& & & \\ \\ \\ \\ \\psi _{\\lambda ,\\ell }(\\theta ,\\ p)\\\\[1.5mm]& &=&\\ \\ \\ \\frac{1}{N}\\langle p,\\ b\\rangle + \\frac{1}{N}\\underset{c\\ \\in \\ \\mathcal {C}}{\\sum }\\ \\underset{\\Delta _{ij}\\ \\in \\ \\mathbb {R}}{\\textbf {\\textit {sup}}}(\\Delta _{ij} b_{ij}-\\lambda \\Delta _{ij}^2)\\\\[1mm]& &=&\\ \\ \\left\\lbrace \\begin{array}{cc}\\langle p,\\ b\\rangle /N+\\Vert b\\Vert ^2_2/(4\\lambda N),\\ &\\ \\text{if }\\ \\lambda >0,\\\\[2mm]\\infty ,\\ &\\ \\text{if }\\ \\lambda =0.\\end{array}\\right.\\end{aligned}$ Next, we investigate the duality of (REF ) with Proposition REF .",
"As $\\psi _{\\lambda ,\\ell }(\\theta ,\\ z) = \\infty $ when $\\lambda =0$ , the dual formulation of the supremum in (REF ) would be $\\begin{aligned}& & &\\ \\ \\underset{\\mathbb {Q}\\ \\in \\ \\mathfrak {W}^{\\alpha }_p(\\mathbb {P}_N)}{\\textbf {\\textit {sup}}}\\ \\ \\mathbb {E}_{z^{\\prime }\\sim \\mathbb {Q}}\\Big [\\ \\ell \\big (\\theta ,\\ z^{\\prime }\\big )\\ \\Big ]\\\\& & &=\\ \\ \\underset{\\lambda \\ge 0}{\\textbf {\\textit {min}}}\\ \\ \\Bigg \\lbrace \\lambda \\alpha +\\psi _{\\lambda ,\\ell }(\\theta ,\\ p)\\Bigg \\rbrace \\\\& & &=\\ \\ \\underset{\\lambda >0}{\\textbf {\\textit {min}}}\\ \\left\\lbrace \\lambda \\alpha +\\frac{1}{N}\\langle p,\\ b\\rangle +\\frac{1}{4\\lambda N}\\Vert b\\Vert ^2_2\\right\\rbrace .\\end{aligned}$ By the definition of $b$ , we know that $\\ell (\\theta ,\\ p) = \\frac{1}{N}\\ \\big \\langle p,\\ b\\big \\rangle $ Moreover, notice that the right hand side of (REF ) is a convex function which approaches infinity when $\\lambda \\rightarrow \\infty $ , the global optimal of it can be obtained uniquely via the first order optimality condition as $\\frac{\\partial }{\\partial \\lambda } \\left\\lbrace \\ \\lambda \\alpha +\\frac{1}{N}\\langle p,\\ b\\rangle +\\frac{1}{4\\lambda N}\\Vert b\\Vert ^2_2\\ \\right\\rbrace = 0,$ and the optimal dual variable is $\\lambda ^*_{\\alpha } = \\frac{\\Vert b\\Vert _2}{2\\sqrt{\\alpha N}}.$ Substituting $\\lambda ^*_{\\alpha }$ and $b$ into (REF ), we have $\\begin{aligned}& & &\\ \\ \\underset{\\mathbb {Q}\\ \\in \\ \\mathfrak {W}^{\\alpha }_p(\\mathbb {P}_N)}{\\textbf {\\textit {sup}}}\\ \\ \\mathbb {E}_{z^{\\prime }\\sim \\mathbb {Q}}\\Big [\\ \\ell \\big (\\theta ,\\ z^{\\prime }\\big )\\ \\Big ]\\\\& &=&\\ \\ \\ \\sqrt{\\frac{\\alpha }{N}}\\cdot \\Vert b\\Vert _2+\\frac{1}{N}\\cdot \\langle p,\\ b\\rangle \\\\& &=&\\ \\ \\sqrt{\\frac{\\alpha }{4N}\\underset{c\\ \\in \\ \\mathcal {C}}{\\sum }(y_{ij}-\\theta ^\\top a_{c})^2}+\\frac{1}{2N}\\underset{c\\ \\in \\ \\mathcal {C}}{\\sum }p_{ij}(y_{ij}-\\theta ^\\top a_{c})^2.\\end{aligned}$"
],
[
"Some Propositions for Generalization Analysis.",
"propositionLipschitz Suppose that $\\ell $ is $L$ -Lipschitz function, i.e., $|\\ell (z)-\\ell (z^{\\prime })|\\le L\\cdot d_{\\mathcal {Z}}(z,z^{\\prime })$ for all $z,z^{\\prime }\\in \\mathcal {Z}$ .",
"Then, for any $\\mathbb {Q}\\in \\mathfrak {W}^{\\alpha }_p(\\mathbb {P}_N)$ , $R_\\mathbb {Q}(\\ell ) \\le R_{\\mathbb {P},\\alpha ,p}(\\ell ) \\le R_\\mathbb {Q}(\\ell ) + 2L\\alpha .$ For $p=1$ , the result follows immediately from the Kantorovich dual representation of $\\mathcal {W}_1(\\cdot ,\\ \\cdot )$ [72]: $\\mathcal {W}_1(\\mathbb {P},\\ \\mathbb {Q}) = \\textbf {\\textit {sup}}\\ \\Bigg \\lbrace \\ \\bigg |\\ \\mathbb {E}_{z\\sim \\mathbb {P}}\\big [h(z)\\big ]\\ -\\ \\mathbb {E}_{z\\sim \\mathbb {Q}}\\big [h(z)\\big ]\\ \\bigg |\\ \\underset{z,z^{\\prime }\\in \\mathcal {Z},z\\ne z^{\\prime }}{\\textbf {\\textit {sup}}}\\frac{\\big |\\ h(z)\\ -\\ h(z^{\\prime })\\ \\big |}{d_{\\mathcal {Z}}(z,z^{\\prime })}\\le 1\\Bigg \\rbrace $ with the triangle inequality: $\\mathcal {W}_1(\\mathbb {P},\\ \\mathbb {Q})\\le 2\\alpha ,\\ \\ \\ \\ \\forall \\ \\ \\mathbb {P},\\ \\mathbb {Q}\\ \\in \\ \\mathfrak {W}^{\\alpha }_1(\\mathbb {P}_{N}).$ For $p>1$ , the result follows from the fact that $\\mathcal {W}_1(\\mathbb {P},\\ \\mathbb {Q})\\ \\le \\ \\mathcal {W}_p(\\mathbb {P},\\ \\mathbb {Q}),\\ \\ \\ \\ \\forall \\ \\ \\mathbb {P},\\ \\mathbb {Q}\\ \\in \\ \\mathcal {P}_p(\\mathcal {Z}).$ Next we consider the case when the function $\\ell $ is smooth but not Lipschitz-continuous.",
"Since we are working with general metric spaces that may lack an obvious differentiable structure, we need to first introduce some concepts from metric geometry [2].",
"Definition 4 (Geodesic Space) A metric space $(\\mathcal {Z}, d_\\mathcal {Z})$ is a geodesic space if for every pair of points $z,\\ z^{\\prime }\\in \\mathcal {Z}$ there exists a constant-speed geodesic path $\\varrho :\\big [0,1\\big ]\\rightarrow \\mathcal {Z}$ , such that $\\varrho (0) = z$ , $\\varrho (1) = z^{\\prime }$ , and for all $0\\le s\\le t\\le 1$ $d_{\\mathcal {Z}}\\Big [\\varrho (s),\\ \\varrho (t)\\Big ]=(t-s)\\cdot d_{\\mathcal {Z}}\\Big [\\varrho (0),\\ \\varrho (1)\\Big ].$ Definition 5 (Geodesic convexity) A functional $\\ell :\\mathcal {Z}\\rightarrow \\mathbb {R}$ is geodesically convex if for any pair of points $z,\\ z^{\\prime }\\in \\mathcal {Z}$ there is a constant-speed geodesic $\\varrho $ , so that $\\begin{aligned}& \\ell \\Big (\\varrho (t)\\Big )&\\le &\\ \\ \\ (1-t)\\cdot \\ell \\Big (\\varrho (0)\\Big )\\ +\\ t\\cdot \\ell \\Big (\\varrho (1)\\Big )\\\\& &=&\\ \\ \\ (1-t)\\cdot \\ell (z)\\ +\\ t\\cdot \\ell (z^{\\prime }).\\end{aligned}$ Definition 6 (Upper Gradient) Suppose that $\\ell :\\mathcal {Z}\\rightarrow \\mathbb {R}$ is a Borel function.",
"The upper gradient of $\\ell $ is a functional $G_{\\ell }:\\mathcal {Z}\\rightarrow \\mathbb {R}_+$ satisfies that: for any pair of points $z,\\ z^{\\prime }\\in \\mathcal {Z}$ , there exist a constant-speed geodesic path $\\varrho $ : $|\\ell (z^{\\prime })-\\ell (z)|\\le \\int ^1_0G_{\\ell }(\\varrho (t))\\mathrm {d}t\\cdot d_{\\mathcal {Z}}(z,\\ z^{\\prime }).$ propositionuppergrad Suppose that $\\ell $ has a geodesically convex upper gradient $G_{\\ell }$ , we have $R_\\mathbb {Q}(\\ell ) \\le R_{\\mathbb {P},\\alpha ,p}(\\ell ) \\le R_\\mathbb {Q}(\\ell ) + 2\\alpha \\mu ,$ where $\\mu = \\underset{\\mathbb {Q}\\ \\in \\ \\mathfrak {W}^{\\alpha }_p(\\mathbb {P})}{\\textbf {\\textit {sup}}}\\Bigg (\\mathbb {E}_{z\\sim \\mathbb {Q}}\\Big [\\big |G_{\\ell }(z)\\big |^q\\Big ]\\Bigg )^{\\frac{1}{q}}$ and $1/p+1/q=1$ .",
"With fixed $\\mathbb {Q},\\ \\mathbb {Q}^{\\prime }\\in \\mathfrak {W}^{\\alpha }_p(\\mathbb {P})$ and let $\\gamma \\in \\Gamma (\\mathcal {Z}\\times \\mathcal {Z})$ achieve the infimum in (REF ) and (REF ) for $\\mathcal {W}_p(\\mathbb {Q},\\ \\mathbb {Q}^{\\prime })$ .",
"Then for any $(z,\\ z^{\\prime })\\sim \\gamma $ , we have $\\begin{aligned}& \\ell (z^{\\prime })-\\ell (z)&\\le &\\ \\ \\int ^1_0G_{\\ell }(\\varrho (t))\\mathrm {d}t\\cdot d_{\\mathcal {Z}}(z,\\ z^{\\prime })\\\\& &\\le &\\ \\ \\frac{1}{2}\\Big (G_{\\ell }(z)+G_{\\ell }(z^{\\prime })\\Big )\\cdot d_{\\mathcal {Z}}(z,\\ z^{\\prime }),\\end{aligned}$ where the first inequality is from the definition of the upper gradient (REF ) and the second one is by the assumed geodesic convexity of $G_{\\ell }$ .",
"Taking expectations of both sides with respect to $\\gamma $ and using Hölder inequality, we obtain $\\begin{aligned}& R_\\mathbb {Q}(\\ell )-R_{\\mathbb {Q}^{\\prime }}(\\ell )&\\le &\\ \\ \\ \\frac{1}{2}\\ \\Bigg (\\mathbb {E}_{(z,z^{\\prime })\\sim \\gamma }\\Big [\\big |G_{\\ell }(z)+G_{\\ell }(z^{\\prime })\\big |^q\\Big ]\\Bigg )^{\\frac{1}{q}}\\Bigg (\\mathbb {E}_{(z,z^{\\prime })\\sim \\gamma }\\big [d_{\\mathcal {Z}}(z,\\ z^{\\prime })\\big ]^p\\Bigg )^\\frac{1}{p}\\\\& &=&\\ \\ \\ \\frac{1}{2}\\ \\Bigg (\\mathbb {E}_{(z,z^{\\prime })\\sim \\gamma }\\Big [\\big |G_{\\ell }(z)+G_{\\ell }(z^{\\prime })\\big |^q\\Big ]\\Bigg )^{\\frac{1}{q}}\\cdot \\mathcal {W}_p(\\mathbb {Q},\\ \\mathbb {Q}^{\\prime }),\\nonumber \\end{aligned}$ where we adopt the $p$ -Wasserstein optimality of $\\gamma $ for $\\mathbb {Q}$ and $\\mathbb {Q}^{\\prime }$ .",
"By the triangle inequality, and since $z\\sim \\mathbb {Q}$ and $z^{\\prime }\\sim \\mathbb {Q}$ , $\\begin{aligned}& \\Bigg (\\mathbb {E}_{(z,z^{\\prime })\\sim \\gamma }\\Big [\\big |G_{\\ell }(z)+G_{\\ell }(z^{\\prime })\\big |^q\\Big ]\\Bigg )^{\\frac{1}{q}}&\\le &\\ \\ \\Bigg (\\mathbb {E}_{z\\sim \\mathbb {Q}}\\Big [\\big |G_{\\ell }(z)\\big |^q\\Big ]\\Bigg )^{\\frac{1}{q}}+\\ \\ \\Bigg (\\mathbb {E}_{z^{\\prime }\\sim \\mathbb {Q}^{\\prime }}\\Big [\\big |G_{\\ell }(z^{\\prime })\\big |^q\\Big ]\\Bigg )^{\\frac{1}{q}}\\\\& &\\le &\\ \\ 2\\underset{\\mathbb {Q}\\ \\in \\ \\mathfrak {W}^{\\alpha }_p(\\mathbb {P})}{\\textbf {\\textit {sup}}}\\ \\Bigg (\\mathbb {E}_{z\\sim \\mathbb {Q}}\\Big [\\big |G_{\\ell }(z)\\big |^q\\Big ]\\Bigg )^{\\frac{1}{q}}.\\end{aligned}$ Interchanging the roles of $\\mathbb {Q}$ and $\\mathbb {Q}^{\\prime }$ and proceeding with the same argument, we obtain the following estimation $\\begin{aligned}\\underset{\\mathbb {Q},\\ \\mathbb {Q}^{\\prime }\\ \\in \\ \\mathfrak {W}^{\\alpha }_p(\\mathbb {P})}{\\textbf {\\textit {sup}}}\\ \\Big |\\ R_\\mathbb {Q}(\\ell )\\ -\\ R_{\\mathbb {Q}^{\\prime }}(\\ell )\\ \\Big |\\ \\ \\le \\ \\ 2\\alpha \\underset{\\mathbb {Q}\\ \\in \\ \\mathfrak {W}^{\\alpha }_p(\\mathbb {P})}{\\textbf {\\textit {sup}}}\\ \\Bigg (\\mathbb {E}_{z\\sim \\mathbb {Q}}\\Big [\\big |G_{\\ell }(z)\\big |^q\\Big ]\\Bigg )^{\\frac{1}{q}}.\\end{aligned}$ Then $\\begin{aligned}& & &\\ \\ R_\\mathbb {Q}(\\ell )\\\\& &\\le &\\ \\ R_{\\mathbb {P},\\alpha ,p}(\\ell )\\\\& &=&\\ \\ 2\\alpha \\underset{\\mathbb {Q}^{\\prime }\\ \\in \\ \\mathfrak {W}^{\\alpha }_p(\\mathbb {P})}{\\textbf {\\textit {sup}}}\\ \\Big [R_{\\mathbb {Q}^{\\prime },\\alpha ,p}(\\ell )-R_\\mathbb {Q}(\\ell )+R_\\mathbb {Q}(\\ell )\\Big ]\\\\& &\\le &\\ \\ R_\\mathbb {Q}(\\ell ) + 2\\alpha \\underset{\\mathbb {Q}\\ \\in \\ \\mathfrak {W}^{\\alpha }_p(\\mathbb {P})}{\\textbf {\\textit {sup}}}\\ \\Bigg (\\mathbb {E}_{z\\sim \\mathbb {Q}}\\Big [\\big |G_{\\ell }(z)\\big |^q\\Big ]\\Bigg )^{\\frac{1}{q}}\\end{aligned}$ propositionregression Consider the setting of pairwise ranking problem with the sum-of-squared loss: let $\\mathcal {A}$ be a convex subset of $\\mathbb {R}^n$ , $\\mathcal {Y}=[-1,\\ 1]$ , and equip $\\mathcal {Z} = \\mathcal {A}\\times \\mathcal {Y}$ with the Euclidean metric $d_{\\mathcal {Z}}(z,z^{\\prime }) = \\sqrt{\\Vert a-a^{\\prime }\\Vert ^2_2+|y-y^{\\prime }|^2},\\ z=(a,\\ y).$ It means that we do not aggregate the pairwise comparisons into the same type and the weight.",
"Then, it holds that $R_{\\mathbb {Q}}(\\ell ) \\le R_{\\mathbb {P},\\alpha ,2}(\\ell ) \\le R_{\\mathbb {Q}}(\\ell ) + 4\\alpha (1+C)\\tau ,$ where $\\tau = \\left(1+L\\underset{\\mathbb {Q}\\ \\in \\ \\mathfrak {W}^{\\alpha }_2(\\mathbb {P}_N)}{\\textbf {\\textit {sup}}}\\mathbb {E}_{\\mathbb {Q}}\\Vert A\\Vert _2\\right),\\ \\ z=(a,\\ y)\\sim \\mathbb {Q},$ and $A=[a^\\top _1,\\dots ,a^\\top _N]$ .",
"As $\\mathcal {Z}\\subseteq \\mathbb {R}^{n+1}$ , $\\mathcal {Z}$ is a geodesic space as $\\gamma (t) = (1-t)\\cdot z+t\\cdot z^{\\prime },\\ \\ \\ \\forall \\ z,\\ z^{\\prime }\\in \\mathcal {Z}$ is the unique constant-speed geodesic path.",
"Moreover, the geodesically convex upper gradient of $\\ell $ is $\\begin{aligned}G_{\\ell }(z)=G_{\\ell }(a,\\ y)=2(B+C)(1+L\\Vert \\nabla h(a)\\Vert _2),\\ \\ \\forall \\ z\\in \\mathcal {Z}.\\end{aligned}$ where $\\ell (z)=\\ell (a,\\ y)=(y-h(a))^2$ .",
"In such a flat Euclidean setting, geodesic convexity coincides with the usual definition of convexity, and the map $z\\rightarrow G_{\\ell }(z)$ is convex evidently: for all pair $z,\\ z^{\\prime }\\in \\mathcal {Z}$ $G_{\\ell }((1-t)\\cdot z+t\\cdot z^{\\prime })\\le (1-t)\\cdot G_{\\ell }(z) + t\\cdot G_{\\ell }(z^{\\prime }).$ With the mean-value theorem $\\begin{aligned}& & &\\ \\ \\ \\ \\ell (z)-\\ell (z^{\\prime })\\\\& &\\le &\\ \\ \\int ^1_0\\big \\langle z-z^{\\prime },\\ \\nabla \\ell \\big ((1-t)\\cdot z+t\\cdot z^{\\prime }\\big )\\big \\rangle \\mathrm {d}t\\\\& &\\le &\\ \\ \\int ^1_0\\big \\Vert \\nabla \\ell \\big ((1-t)\\cdot z+t\\cdot z^{\\prime }\\big )\\big \\Vert _2 \\mathrm {d}t\\cdot \\Vert z-z^{\\prime }\\Vert _2\\\\& &=&\\ \\ \\int ^1_0\\big \\Vert \\nabla \\ell \\big ((1-t)\\cdot z+t\\cdot z^{\\prime }\\big )\\big \\Vert _2 \\mathrm {d}t\\cdot d_{\\mathcal {Z}}(z,\\ z^{\\prime })\\end{aligned}$ and a simple calculation $\\begin{aligned}\\Vert \\nabla \\ell (z)\\Vert ^2_2=4\\ell (z)(1+\\Vert \\nabla h(a)\\Vert ^2_2)\\le 4(B+C)^2(1+L^2\\Vert a\\Vert ^2_2),\\end{aligned}$ we have $\\Vert \\nabla \\ell (z)\\Vert _2\\le G_{\\ell }(z)$ for any $z\\in \\mathcal {Z}$ .",
"Thus, by Proposition , we have $\\begin{aligned}& R_\\mathbb {Q}(\\ell )&\\le &\\ \\ \\ R_{\\mathbb {P},\\alpha ,2}(\\ell )\\\\& &\\le &\\ \\ \\ R_\\mathbb {Q}(\\ell ) + 2\\alpha \\underset{\\mathbb {Q}\\ \\in \\ \\mathfrak {W}^{\\alpha }_2(\\mathbb {P}_N)}{\\textbf {\\textit {sup}}}\\ \\Bigg (\\mathbb {E}_{z\\sim \\mathbb {Q}}\\Big [\\big |G_{\\ell }(z)\\big |^2\\Big ]\\Bigg )^{\\frac{1}{2}}\\\\& &=&\\ \\ \\ R_\\mathbb {Q}(\\ell ) + 4\\alpha (B+C)\\Bigg (1+L\\underset{\\mathbb {Q}\\ \\in \\ \\mathfrak {W}^{\\alpha }_2(\\mathbb {P}_N)}{\\textbf {\\textit {sup}}}\\mathbb {E}_{z\\sim \\mathbb {Q}}\\Vert A\\Vert _2\\Bigg ).\\end{aligned}$"
],
[
"Proof of Theorem ",
"Assumption 1 $d:\\mathcal {Z}\\times \\mathcal {Z}\\rightarrow \\mathbb {R}_+$ in (REF ) and (REF ) is a nonnegative lower semi-continuous function satisfying $d(w, w^{\\prime })=0$ if and only if $w=w^{\\prime }$ .",
"Assumption 2 The loss function $\\ell \\in \\mathcal {F}\\subseteq L^1(\\mathrm {d}\\mathbb {Q})$ are upper semi-continuous, where $L^1(\\mathrm {d}\\mathbb {Q})$ denote the collection of Borel measurable functions $\\ell : \\mathcal {Z}\\rightarrow \\mathbb {R}$ such that $\\int |\\ell |\\ \\mathrm {d}\\mathbb {Q} < \\infty ,\\ \\forall \\ \\mathbb {Q}\\in \\mathcal {P}(\\mathcal {Z}).$ Assumption 3 The instance space $\\mathcal {Z}$ is bounded, namely, $\\text{diam}(\\mathcal {Z}) = \\underset{z,z^{\\prime }\\in \\mathcal {Z}}{\\textbf {\\textit {sup}}} d_{\\mathcal {Z}}(z,\\ z^{\\prime }) < \\infty .$ Assumption 4 $\\ell \\in \\mathcal {F}$ is uniformly bounded as $0\\le \\ell (z)\\le B < \\infty ,\\ \\ \\forall \\ \\ell \\in \\mathcal {F},\\ \\text{ and }\\ z\\in \\mathcal {Z}.$ Definition 7 Let $(\\mathcal {Z}, d_{\\mathcal {Z}})$ be a metric space.",
"For a function $\\ell :\\mathcal {Z}\\rightarrow \\mathbb {R}$ and a point $s\\in \\mathbb {R}$ , the upper contour set defined by $s$ is $\\ell ^{-1}\\big ([s,\\ \\infty )\\big )=\\big \\lbrace z\\in \\mathcal {Z}:\\ \\ell (z)\\ge s\\big \\rbrace ,$ and the corresponding lower contour set is $\\ell ^{-1}\\big ((\\infty ,\\ s]\\big )=\\big \\lbrace z\\in \\mathcal {Z}:\\ \\ell (z)\\le s\\big \\rbrace .$ We call a function $\\ell :\\mathcal {Z}\\rightarrow \\mathbb {R}$ is upper semi-continuous if and only if for any $s\\in \\mathbb {R}$ , $\\ell ^{-1}\\big ((\\infty ,\\ s]\\big )$ is an open set.",
"We adopt the Dudley’s entropy integral [20] as the complexity measure of the hypothesis class $\\mathcal {F}$ , $\\mathcal {J}(\\mathcal {F})=\\int _0^{\\infty }\\sqrt{\\textbf {\\textit {log}}\\ \\mathfrak {N}(\\mathcal {F},\\ \\Vert \\cdot \\Vert _{\\infty },\\ \\upsilon )}\\ \\mathrm {d}\\upsilon ,$ where $\\mathfrak {N}(\\mathcal {F},\\ \\Vert \\cdot \\Vert _{\\infty },\\ \\upsilon )$ is $\\upsilon $ -covering number of $\\mathcal {F}$ with respect to the uniform metric $\\Vert \\cdot \\Vert _{\\infty }$ , defined as the size of the smallest $\\upsilon $ -cover of $\\mathcal {F}$ $\\begin{aligned}& & &\\ \\ \\mathfrak {N}(\\mathcal {F},\\ \\Vert \\cdot \\Vert _{\\infty },\\ \\upsilon ) \\\\& &=&\\ \\ \\underset{m\\in \\mathbb {N}}{\\textbf {\\textit {min}}}\\ \\left\\lbrace \\exists \\ \\lbrace \\ell _1,\\ \\dots ,\\ \\ell _m\\rbrace \\subseteq \\mathcal {F}\\subseteq \\bigcup _{k=1}^m\\mathcal {B}^{\\Vert \\cdot \\Vert _\\infty }_\\upsilon (\\ell _k)\\right\\rbrace \\end{aligned}$ and $\\bigcup _{k=1}^m\\mathcal {B}^{\\Vert \\cdot \\Vert _\\infty }_\\upsilon (\\cdot )$ is a $\\upsilon $ -cover of $\\mathcal {F}$ with respect to $\\Vert \\cdot \\Vert _{\\infty }$ $\\Vert \\ \\ell -{\\ell }^{\\prime }\\ \\Vert _{\\infty } = \\underset{z\\ \\in \\ \\mathcal {Z}}{\\textbf {\\textit {sup}}}\\ |\\ \\ell (z)\\ -\\ {\\ell }^{\\prime }(z)\\ |.$ * This proof is a specialization of data-dependent generalization bounds for margin cost function class [38].",
"From the definition of the local minimax risk (REF ) and its duality form, $\\begin{aligned}& R_{\\mathbb {P},\\alpha ,p}(\\ell )&=&\\ \\ \\ \\underset{\\lambda >0}{\\textbf {\\textit {min}}}\\ \\Big \\lbrace \\ \\lambda \\alpha ^p\\ +\\ \\mathbb {E}_{z\\sim \\mathbb {P}}\\big [\\psi _{\\lambda ,\\ell }(z)\\big ]\\ \\Big \\rbrace \\\\& &\\le &\\ \\ \\ \\underset{\\lambda >0}{\\textbf {\\textit {min}}}\\ \\Big \\lbrace \\ \\lambda \\alpha ^p\\ +\\ \\mathbb {E}_{z\\sim \\mathbb {P}}\\big [\\psi _{\\lambda ,\\ell }(z)\\big ]\\ +\\ V_{\\lambda }\\Big \\rbrace \\end{aligned}$ where $V_{\\lambda }\\ \\ =\\ \\ \\underset{\\ell \\ \\in \\ \\mathcal {F}}{\\textbf {\\textit {sup}}}\\ \\Big \\lbrace \\ \\mathbb {E}_{z\\sim \\mathbb {P}}\\big [\\psi _{\\lambda ,\\ell }(z)\\big ]-\\mathbb {E}_{z\\sim \\mathbb {P}_N}\\big [\\psi _{\\lambda ,\\ell }(z)\\big ]\\ \\Big \\rbrace $ is a data-dependent random variable for any $\\lambda \\ge 0$ .",
"As $\\mathcal {F}$ and $\\mathbb {P}$ satisfy the Assumption REF and REF , we have $0\\le \\psi _{\\lambda ,\\ell }(z)\\le B,\\ \\forall \\ z\\in \\mathcal {Z}.$ Furthermore, known from McDiarmid’s inequality that, for any fixed $\\lambda \\ge 0$ $\\Pr \\Big (V_{\\lambda }\\ge \\mathbb {E}V_{\\lambda } + \\frac{Bt}{\\sqrt{N\\ }\\ \\ }\\Big )\\ \\le \\ 2e^{-2t^2}.$ Using a standard symmetrization argument, we have $\\mathbb {E}V_{\\lambda }\\le 2\\cdot \\mathbb {E}\\left[\\underset{\\ell \\ \\in \\ \\mathcal {F}}{\\textbf {\\textit {sup}}}\\ \\frac{1}{N}\\sum ^{N}_{i=1}\\ \\epsilon _i\\psi _{\\lambda ,\\ell }(z_i)\\right]$ where $\\epsilon _1,\\dots ,\\epsilon _N$ are i.i.d.",
"Rademacher random variables independent of $z_1,\\dots ,z_N$ .",
"To bound (REF ), we define the $\\mathcal {F}$ -indexed process $\\beta _\\mathcal {F}=\\lbrace \\beta _{\\ell }\\rbrace _{\\ell \\in \\mathcal {F}}$ as $\\beta _{\\ell } = \\frac{1}{N}\\sum ^{N}_{i=1}\\ \\epsilon _i\\psi _{\\lambda ,\\ell }(z_i).$ This is a zero-mean and sub-Gaussian process [70] with respect to the metric $\\Vert \\cdot \\Vert _{\\infty }$ as $\\begin{aligned}& & &\\ \\ \\mathbb {E}\\ \\Big [\\textbf {\\textit {exp}}\\big (t(\\beta _{\\ell }-\\beta _{{\\ell }^{\\prime }})\\big )\\Big ]\\\\& &=&\\ \\ \\mathbb {E}\\left[\\textbf {\\textit {exp}}\\left(\\frac{t}{\\sqrt{N}\\ \\ }\\sum ^{N}_{i=1}\\ \\epsilon _i\\big (\\psi _{\\lambda ,\\ell }(z_i)-\\psi _{\\lambda ,{\\ell }^{\\prime }}(z_i)\\big )\\right)\\right]\\\\& &=&\\ \\ \\Bigg \\lbrace \\mathbb {E}\\Bigg [\\textbf {\\textit {exp}}\\Bigg (\\frac{t}{\\sqrt{N\\ }\\ \\ }\\epsilon _i\\ \\underset{z^{\\prime }}{\\textbf {\\textit {sup}}}\\ \\underset{z^{\\prime \\prime }}{\\textbf {\\textit {inf}}}\\ \\Big \\lbrace \\ell (z^{\\prime })-\\lambda [d_{\\mathcal {Z}}(z_1,z^{\\prime })]^p-{\\ell }^{\\prime }(z^{\\prime \\prime })+\\lambda [d_{\\mathcal {Z}}(z_1,z^{\\prime \\prime })]^p\\Big \\rbrace \\Bigg )\\Bigg ]\\Bigg \\rbrace ^N\\\\& &\\le &\\ \\ \\Bigg \\lbrace \\mathbb {E}\\Bigg [\\textbf {\\textit {exp}}\\left(\\frac{t}{\\sqrt{N\\ }\\ \\ }\\epsilon _i\\ \\underset{z^{\\prime }}{\\textbf {\\textit {sup}}}\\ \\Big \\lbrace \\ell (z^{\\prime })-{\\ell }^{\\prime }(z^{\\prime })\\Big \\rbrace \\right)\\Bigg ]\\Bigg \\rbrace ^N\\\\& &\\le &\\ \\ \\ \\textbf {\\textit {exp}}\\left(\\frac{t^2\\Vert \\ell -{\\ell }^{\\prime }\\Vert ^2_{\\infty }}{2}\\right),\\end{aligned}$ where the second equation comes from the independence of $\\lbrace z_i\\rbrace _{i\\in [N]}$ and the definition of $\\psi _{\\lambda ,\\ell }(\\cdot )$ .",
"The last inequality follows the Hoeffding’s lemma [30].",
"With the $\\mathcal {F}$ -indexed process $\\beta _\\mathcal {F}$ and invoking Dudley’s entropy integral (REF ) [20] for the right-hand side of (REF ), we obtain $\\mathbb {E}V_{\\lambda }\\le 2\\cdot \\mathbb {E}\\left[\\underset{\\ell \\ \\in \\ \\mathcal {F}}{\\textbf {\\textit {sup}}}\\ \\beta _{\\ell }\\right]\\le \\frac{24}{\\sqrt{N\\ }\\ \\ }\\ \\mathcal {J}(\\mathcal {F}),\\ \\ \\ \\forall \\ \\lambda \\ge 0$ and $\\Pr \\Big (V_{\\lambda }\\ge \\frac{24\\mathcal {J}(\\mathcal {F})+Bt}{\\sqrt{N\\ }}\\Big )\\ \\le \\ 2e^{-2t^2}.$ In addition, the first part of the claims holds with ant fixed $\\lambda \\ge 0$ : $\\Pr \\left(\\ \\exists \\ \\ell \\in \\mathcal {F}: R_{\\mathbb {P}, \\alpha , p}(\\ell )>\\varsigma _1\\ \\right)\\ \\le \\ e^{-2t^2},\\ \\ \\forall \\ t>0,$ where $\\varsigma _1 = \\underset{\\lambda \\ge 0}{\\textbf {\\textit {min}}}\\ \\ \\Bigg \\lbrace \\ \\lambda \\alpha ^p\\ +\\ \\mathbb {E}_{z\\sim \\mathbb {Q}}\\ \\big [\\ \\psi _{\\lambda ,\\ell }\\ (z)\\ \\big ]\\Bigg \\rbrace +\\frac{24\\mathcal {J}(\\mathcal {F})+Bt}{\\sqrt{N\\ }}.$ For the second part, we start with two sequences: $\\lbrace \\lambda _k\\rbrace $ and $\\lbrace t_k\\rbrace $ $\\lambda _k = k,\\ \\ t_k = t+\\sqrt{\\textbf {\\textit {log}}(k)},\\ k=1,2,\\dots $ and (REF ) also holds as $\\begin{aligned}& & &\\ \\ \\Pr \\left(\\ \\exists \\ \\ell \\in \\mathcal {F}: R_{\\mathbb {P}, \\alpha , p}(\\ell )>\\underset{k}{\\textbf {\\textit {min}}}\\ \\ \\Bigg \\lbrace \\ \\lambda _k\\alpha ^p\\ +\\ \\mathbb {E}_{z\\sim \\mathbb {Q}}\\ \\big [\\ \\psi _{\\lambda _k,\\ell }\\ (z)\\ \\big ]\\Bigg \\rbrace +\\frac{24\\mathcal {J}(\\mathcal {F})+Bt_k}{\\sqrt{N\\ }}\\ \\right)\\\\& &\\le &\\ \\ \\sum _{k}\\ e^{-2t_k^2}\\\\& &\\le &\\ \\ \\sum _{k}\\ e^{-2\\textbf {\\textit {log}}(k)}\\ \\cdot \\ e^{-2t^2}\\\\& &\\le &\\ \\ 2e^{-2t^2}.\\end{aligned}$ Moreover, $\\begin{aligned}& & &\\ \\ \\underset{k}{\\textbf {\\textit {min}}}\\ \\ \\Bigg \\lbrace \\ \\lambda _k\\alpha ^p\\ +\\ \\mathbb {E}_{z\\sim \\mathbb {Q}}\\ \\big [\\ \\psi _{\\lambda _k,\\ell }\\ (z)\\ \\big ]\\Bigg \\rbrace +\\frac{24\\mathcal {J}(\\mathcal {F})+Bt_k}{\\sqrt{N\\ }}\\\\& &=&\\ \\ \\underset{k}{\\textbf {\\textit {min}}}\\ \\ \\Bigg \\lbrace \\ \\ k\\alpha ^p\\ \\ +\\ \\mathbb {E}_{z\\sim \\mathbb {Q}}\\ \\big [\\ \\psi _{\\lambda _k,\\ell }\\ (z)\\ \\big ]\\Bigg \\rbrace +\\frac{24\\mathcal {J}(\\mathcal {F})+Bt}{\\sqrt{N\\ }} + \\frac{B\\sqrt{\\textbf {\\textit {log}}(k)}}{\\sqrt{N\\ }}\\\\& &\\le &\\ \\ \\underset{\\lambda \\ge 0}{\\textbf {\\textit {min}}}\\ \\ \\Bigg \\lbrace \\ \\ (\\lambda +1)\\alpha ^p\\ \\ +\\ \\mathbb {E}_{z\\sim \\mathbb {Q}}\\ \\big [\\ \\psi _{\\lambda ,\\ell }\\ (z)\\ \\big ]\\Bigg \\rbrace +\\frac{24\\mathcal {J}(\\mathcal {F})+Bt}{\\sqrt{N\\ }} + \\frac{B\\sqrt{\\textbf {\\textit {log}}(\\lambda +1)}}{\\sqrt{N\\ }}\\\\\\end{aligned}$ where the last inequity holds since, for any $\\lambda \\ge 0$ , there exists $k\\in \\mathbb {N}_+$ such that $\\lambda \\le k\\le \\lambda +1$ , and $\\psi _{\\lambda _1,\\ell }\\le \\psi _{\\lambda _2,\\ell }$ holds whenever $\\lambda _1\\ge \\lambda _2$ as (REF ).",
"Notice that $R_{\\mathbb {P}_N, \\alpha ,p}(\\ell )\\le \\underset{\\lambda >0}{\\textbf {\\textit {min}}}\\ \\Big \\lbrace \\ \\lambda \\alpha ^p\\ +\\ \\mathbb {E}_{z\\sim \\mathbb {P}_N}\\big [\\psi _{\\lambda ,\\ell }(z)\\big ]\\ +\\ W_{\\lambda }\\Big \\rbrace ,$ where $W_{\\lambda }\\ \\ =\\ \\ \\underset{\\ell \\in \\mathcal {F}}{\\textbf {\\textit {sup}}}\\ \\Big \\lbrace \\ \\mathbb {E}_{z\\sim \\mathbb {P}_N}\\big [\\psi _{\\lambda ,\\ell }(z)\\big ]\\ -\\ \\mathbb {E}_{z\\sim \\mathbb {P}}\\big [\\psi _{\\lambda ,\\ell }(z)\\big ]\\ \\Big \\rbrace .$ Following the similar analysis of $R_{\\mathbb {P},\\alpha ,p}(\\ell )$ , the second part of the claims holds $\\Pr \\left(\\ \\exists \\ \\ell \\in \\mathcal {F}: R_{\\mathbb {P}_N, \\alpha , p}(\\ell )>\\varsigma _2\\ \\right)\\ \\le \\ 2e^{-2t^2},\\ \\forall \\ t>0$ where $\\begin{aligned}\\varsigma _2=\\underset{\\lambda \\ge 0}{\\textbf {\\textit {min}}}\\ \\ \\Bigg \\lbrace \\ (\\lambda +1)\\alpha ^p\\ +\\ \\mathbb {E}_{z\\sim \\mathbb {Q}}\\ \\big [\\ \\psi _{\\lambda ,\\ell }\\ (z)\\ \\big ]+\\frac{B\\sqrt{\\textbf {\\textit {log}}(\\lambda +1)}}{\\sqrt{N\\ }}\\ \\Bigg \\rbrace +\\frac{24\\mathcal {J}(\\mathcal {F})+Bt}{\\sqrt{N\\ }}.\\end{aligned}$"
],
[
"Proof of Theorem ",
"The common choice of the smoothness assumption is Lipschitz smoothness.",
"Next, we explore the behavior of the dual variable $\\lambda $ in (REF ) when the (REF ) archives the minimal.",
"The following lemma enables the control of its upper bound.",
"Assumption 5 The functions in $\\mathcal {F}$ are $L$ -Lipschitz, if they satisfy $\\underset{z,z^{\\prime }\\in \\mathcal {Z},z\\ne z^{\\prime }}{\\textbf {\\textit {sup}}}\\frac{\\ \\ell (z^{\\prime })-\\ell (z)\\ }{d_{\\mathcal {Z}}(z^{\\prime },\\ z)}\\le L,\\ \\forall \\ \\ell \\in \\mathcal {F}.$ lemmalambdacontrol Suppose that $\\mathbb {Q}\\in \\mathfrak {W}_p^{\\alpha }(\\mathbb {P}_n)\\subset \\mathcal {P}_{m}(\\mathcal {Z})$ and $\\tilde{\\ell }$ is the optimal solution of local worst-case risk with distribution $\\mathbb {Q}$ $\\tilde{\\ell }\\in \\underset{\\ell \\ \\in \\ \\mathcal {F}}{\\textbf {\\textit {arg min}}}\\ R_{\\mathbb {Q}, \\alpha , p}\\ (\\ell ),$ $\\tilde{\\lambda }$ is the infimum-archiving dual variable corresponding to $\\tilde{f}$ $\\tilde{\\lambda }\\in \\underset{\\lambda \\ge 0}{\\textbf {\\textit {min}}}\\ \\bigg \\lbrace \\ \\lambda \\alpha ^p\\ +\\ \\mathbb {E}_{z\\sim \\mathbb {Q}}\\ \\Big [\\psi _{\\lambda ,\\ \\tilde{\\ell }}(z)\\Big ]\\ \\bigg \\rbrace .$ Then under Assumption REF -REF , $\\tilde{\\lambda }$ satisfies $\\tilde{\\lambda }\\ \\le \\ L\\alpha ^{-(p-1)}.$ With the fixed $\\mathbb {Q}$ and the estimator $\\tilde{f}$ , we have $\\begin{aligned}\\tilde{\\lambda }\\alpha ^p\\le \\tilde{\\lambda }\\alpha ^p+\\mathbb {E}_{z\\sim \\mathbb {Q}}\\left[\\underset{z^{\\prime }\\in \\mathcal {Z}}{\\textbf {\\textit {sup}}}\\bigg \\lbrace \\tilde{f}(z^{\\prime })-\\tilde{f}(z)-\\tilde{\\lambda }\\Big [d_{\\mathcal {Z}}(z,z^{\\prime })\\Big ]^p\\bigg \\rbrace \\right]\\\\\\end{aligned}$ and the equality holds with $z^{\\prime }=z$ .",
"Due to the optimality of $\\tilde{\\lambda }$ and the dual formulation of local worst-case risk (REF ), (REF ) can be further bounder as $\\begin{aligned}&\\tilde{\\lambda }\\alpha ^p&\\le &\\ \\ \\lambda \\alpha ^p+\\mathbb {E}_{z\\sim \\mathbb {Q}}\\left[\\underset{z^{\\prime }\\in \\mathcal {Z}}{\\textbf {\\textit {sup}}}\\bigg \\lbrace \\tilde{f}(z^{\\prime })-\\tilde{f}(z)-\\lambda \\Big [d_{\\mathcal {Z}}(z,z^{\\prime })\\Big ]^p\\bigg \\rbrace \\right]\\\\& &\\le &\\ \\ \\lambda \\alpha ^p+\\mathbb {E}_{z\\sim \\mathbb {Q}}\\left[\\underset{z^{\\prime }\\in \\mathcal {Z}}{\\textbf {\\textit {sup}}}\\bigg \\lbrace L\\cdot d_{\\mathcal {Z}}(z,z^{\\prime })-\\lambda \\Big [d_{\\mathcal {Z}}(z,z^{\\prime })\\Big ]^p\\bigg \\rbrace \\right]\\\\& &\\le &\\ \\ \\lambda \\alpha ^p + \\underset{v\\ge 0}{\\textbf {\\textit {sup}}}\\ \\big \\lbrace Lv-\\lambda v^p\\big \\rbrace ,\\end{aligned}$ where the second line comes from the Lipschitz smoothness of $\\ell \\in \\mathcal {F}$ and the third line holds by substituting $v=d_{\\mathcal {Z}}(z,z^{\\prime })$ .",
"When $p=1$ , the result can be obtained by taking $\\lambda =L$ $\\tilde{\\lambda }\\alpha \\le L\\alpha + \\underset{z^{\\prime }\\in \\mathcal {Z}}{\\textbf {\\textit {sup}}}\\big \\lbrace Lv-Lv\\big \\rbrace =L\\alpha .$ If $p > 1$ , we can take the $v^*=\\left(\\frac{L}{\\lambda p}\\right)^{\\frac{1}{p-1}}$ which satisfies first-order optimal condition for $\\underset{v\\ge 0}{\\textbf {\\textit {sup}}}\\ \\big \\lbrace Lv-\\lambda v^p\\big \\rbrace $ and $\\tilde{\\lambda }\\alpha ^p\\le \\lambda \\alpha ^p + (p-1)L^{\\frac{p}{p-1}}p^{\\frac{p}{1-p}}\\lambda ^{\\frac{1}{1-p}}.$ Treating $\\lambda $ as a variable and minimizing the right-hand side of (REF ) by choosing $\\lambda = \\frac{L}{p\\alpha ^{p-1}}$ , the claim holds.",
"* Suppose that $\\ell ^*\\in \\mathcal {F}$ can archive the local minimax risk $R^*_{\\mathbb {P},\\alpha ,p}(\\mathcal {F})$ , we decompose the excess risk $\\begin{aligned}& R_{\\mathbb {P},\\alpha ,p}(\\hat{\\ell })-R^*_{\\mathbb {P},\\alpha ,p}(\\mathcal {F})&=&\\ \\ R_{\\mathbb {P},\\alpha ,p}(\\hat{\\ell })-R^*_{\\mathbb {P},\\alpha ,p}(\\ell ^*)\\\\& &\\le &\\ \\ R_{\\mathbb {P},\\alpha ,p}(\\hat{\\ell })-R_{\\mathbb {P}_N,\\alpha ,p}(\\hat{\\ell })+R^*_{\\mathbb {P}_N,\\alpha ,p}(\\ell ^*)-R^*_{\\mathbb {P},\\alpha ,p}(\\ell ^*),\\\\\\end{aligned}$ where the last equality stands by the optimality of $\\hat{\\ell }$ .",
"Next, we introduce $\\hat{\\lambda }$ and $\\lambda ^*$ as the corresponding dual variables of $\\hat{\\ell }$ and $\\ell ^*$ as $\\hat{\\ell }\\ \\in \\ \\ \\underset{\\lambda \\ge 0}{\\textbf {\\textit {min}}}\\ \\bigg \\lbrace \\ \\lambda \\alpha ^p\\ +\\ \\mathbb {E}_{z\\sim \\mathbb {P}_N}\\ \\Big [\\psi _{\\lambda ,\\ \\hat{\\ell }}(z)\\Big ]\\ \\bigg \\rbrace ,$ and $\\ell ^*\\ \\in \\ \\underset{\\lambda \\ge 0}{\\textbf {\\textit {min}}}\\ \\bigg \\lbrace \\ \\lambda \\alpha ^p\\ +\\ \\mathbb {E}_{z\\sim \\mathbb {P}}\\ \\Big [\\psi _{\\lambda ,\\ \\ell ^*}(z)\\Big ]\\ \\bigg \\rbrace .$ By the first part of Theorem REF , the right-hand side of (REF ) can be further bounded by $\\begin{aligned}& R_{\\mathbb {P},\\alpha ,p}(\\hat{\\ell })-R_{\\mathbb {P}_N,\\alpha ,p}(\\hat{\\ell })&=&\\ \\ \\underset{\\lambda \\ge 0}{\\textbf {\\textit {min}}}\\left\\lbrace \\ \\lambda \\alpha ^p\\ +\\ \\int _{\\mathcal {Z}}\\psi _{\\lambda ,\\hat{\\ell }}(z)\\mathbb {P}(\\mathrm {d}z)\\ \\right\\rbrace -\\left(\\ \\hat{\\lambda }\\alpha ^p\\ +\\ \\int _{\\mathcal {Z}}\\psi _{\\hat{\\lambda },\\hat{\\ell }}(z)\\mathbb {P}_N(\\mathrm {d}z)\\ \\right)\\\\& &\\le &\\ \\ \\int _{\\mathcal {Z}}\\psi _{\\hat{\\lambda },\\hat{\\ell }}(z)(\\mathbb {P}-\\mathbb {P}_N)(\\mathrm {d}z),\\end{aligned}$ and $\\begin{aligned}R^*_{\\mathbb {P}_N,\\alpha ,p}(\\ell ^*)-R^*_{\\mathbb {P},\\alpha ,p}(\\ell ^*)\\ \\le \\ \\int _{\\mathcal {Z}}\\psi _{{\\lambda ^*},{\\ell ^*}}(z)(\\mathbb {P}_N-\\mathbb {P})(\\mathrm {d}z).\\end{aligned}$ By Lemma , we know $\\hat{\\lambda }\\in \\Lambda :=\\big [\\ 0,\\ L\\alpha ^{-(p-1)}\\ \\big ]$ and define the function class $\\Psi = \\Big \\lbrace \\psi _{\\lambda ,\\ \\ell }\\ \\big \\vert \\ \\lambda \\in \\Lambda ,\\ \\ell \\in \\mathcal {F}\\Big \\rbrace ,$ (REF ) can be written as $\\begin{aligned}R_{\\mathbb {P},\\alpha ,p}(\\hat{\\ell })\\ -\\ R_{\\mathbb {P}_N,\\alpha ,p}(\\hat{\\ell })\\ \\ \\le \\ \\ \\underset{\\psi \\in \\Psi }{\\textbf {\\textit {sup}}}\\ \\left\\lbrace \\ \\int _{\\mathcal {Z}}\\ \\psi \\ \\mathrm {d}(\\ \\mathbb {P}-\\mathbb {P}_N\\ )\\ \\right\\rbrace .\\end{aligned}$ By Assumption REF , REF and the definition of $\\psi _{\\lambda ,\\ \\ell }$ as (REF ), we know that every $\\psi \\in \\Psi $ is bounded and take value in $[0,\\ B]$ .",
"Employing symmetrization, we have $R_{\\mathbb {P},\\alpha ,p}(\\hat{\\ell })-R_{\\mathbb {P}_N,\\alpha ,p}(\\hat{\\ell }) \\le 2\\mathfrak {R}_N(\\Psi ) + B\\sqrt{\\frac{\\textbf {\\textit {log}}(\\frac{2}{\\eta })}{N}}$ with probability at least $1-\\frac{\\eta }{2}$ , where $\\mathfrak {R}_N(\\Psi ) = \\mathbb {E}\\left[\\ \\underset{\\psi \\in \\Psi }{\\textbf {\\textit {sup}}}\\ \\frac{1}{N}\\sum ^{N}_{i=1}\\epsilon _i\\psi (z)\\ \\right]$ is the expected Rademacher average of $\\Psi $ , with i.i.d Rademacher random variables $\\lbrace \\epsilon _i\\rbrace $ which are independent of $\\lbrace z_i\\rbrace $ , $i\\in [N]$ .",
"Moreover, from Hoeffding’s inequality, it follows that $R^*_{\\mathbb {P}_N,\\alpha ,p}(\\ell ^*)-R^*_{\\mathbb {P},\\alpha ,p}(\\ell ^*)\\le B\\sqrt{\\frac{\\textbf {\\textit {log}}(\\frac{2}{\\eta })}{2N}}$ with probability at least $1-\\frac{\\eta }{2}$ .",
"Combining (REF ) and (REF ), and apply the Lemma from Appendix, we obtain the whole theorem."
],
[
"The Stackelberg Game Attack on Pairwise Ranking.",
"We study the poisoning attack on pairwise ranking, which injects the malicious pairwise comparisons into the training set of the ranking algorithm.",
"Meanwhile, the robust ranking algorithm could prune the outlier when leaning a consensus ranking with the noise observation.",
"Such an adversarial interaction between two opponents can is naturally a game.",
"One player will control the ranking algorithm, and the other player will manipulate the distribution of input data, especially the pairwise comparisons.",
"The optimal action for each player generally depends on both players’ strategies.",
"We adopt positive integers to index alternatives and users.",
"Henceforth, $V$ always is the set $\\lbrace 1, \\dots , n\\rbrace $ and denotes a set of alternatives to be ranked.",
"In our approach to attack pairwise ranking, we represent these candidates as vertices of a graph.",
"$U$ = $\\lbrace 1, \\dots , m\\rbrace $ denotes a set of voters or users.",
"For $i,\\ j\\in V$ , we write the pairwise comparison $i\\succ j$ or $(i,\\ j)$ to mean that alternative $i$ is preferred over alternative $j$ .",
"If we hope to emphasize the preference judgment of a particular user $u$ , we will write $i\\succ ^{u} j$ or $(u,\\ i,\\ j)$ .",
"For each user $u\\in U$ , the pairwise ranking matrix of user $u$ is a skew-symmetric matrix $Y^u=\\lbrace y^u_{ij}\\rbrace \\in \\mathbb {R}^{n\\times n},\\ i,\\ j\\in V,\\ u\\in U,$ i.e.",
"for any ordered pair $(i,\\ j)\\in V\\times V$ , we have $y^u_{ij} = -y^u_{ji}.$ Informally, $y^u_{ij}$ measures the “degree of preference” of the $i^{\\text{th}}$ alternative over the $j^{\\text{th}}$ alternative held by the $u^{\\text{th}}$ voter.",
"Here we focus on the “binary” case of $Y^u\\in \\lbrace -1,\\ 1\\rbrace ^{n\\times n}$ .",
"Here $y^u_{ij} = 1$ means there exist a particular preference judgment $(u,\\ i,\\ j)$ made by user $u$ .",
"Define the weight function $w:U\\times V\\times V\\rightarrow [0,\\ \\infty )$ as the indicator function $w^u_{ij}=w(u,i,j)=\\left\\lbrace \\begin{matrix}1, & \\text{if}\\ y^u_{ij} = 1,\\\\0, & \\text{otherwise.",
"}\\end{matrix}\\right.$ With the weight function $w$ , we can aggregate all users' pairwise comparison matrices $\\lbrace Y^u\\rbrace ,\\ u\\in U$ into a single comparison matrix $Y=\\left\\lbrace y_{ij}\\right\\rbrace $ with weights matrix $W^0=\\lbrace w^0_{ij}\\rbrace $ , where $y_{ij} = 1,\\ \\forall \\ (i,\\ j)\\in \\binom{V}{2},$ $\\binom{V}{2}$ is the set of all ordered pairs of elements of $V$ , and $w^0_{ij} = \\underset{u\\in U}{\\sum }w^u_{ij},\\ \\forall \\ (i,\\ j)\\in \\binom{V}{2}.$ A graph structure arises naturally from ranking data as follows.",
"Let $G = (V, E)$ be a directed graph whose vertex set is $V$ , the set of candidates to be ranked.",
"The edge set is $E := \\left\\lbrace e=(i,\\ j)\\ \\Big \\vert \\ (i,\\ j)\\in \\binom{V}{2}\\right\\rbrace .$ We call such $G$ a pairwise comparison graph.",
"One can further associate weights on the edges as (REF ).",
"Different from the general pairwise ranking setting, we do not prune the edges whose weights equal to 0.",
"As a consequence, the pairwise comparison graph $G$ is a complete graph.",
"The cardinality of the edge set is $|E|:= N = n(n-1).$ The comparison between $i$ and $j$ will be labeled by different annotators and their answers to the same question could be inconsistent, i.e., $y^{u_1}_{ij} = y^{u_2}_{ji} = 1,\\ u_1,\\ u_2\\in U.$ To obtain the true direction between vertex $i$ and $j$ , we define an estimator $\\hat{y}_{ij}$ of noise label ${y}_{ij}$ on edge $e=(i,\\ j)$ , $\\hat{y}_{e} = \\langle z_{e}, \\theta \\rangle + \\gamma _{e} + \\varepsilon _{e},\\ \\forall \\ e\\in E,$ where $Z=\\lbrace z_{e}\\rbrace \\in \\lbrace -1, 0, 1\\rbrace ^{N\\times n}$ , $e\\in E$ is the incident matrix of $G$ , $\\theta \\in \\mathbb {R}^n$ is some true scaling scores on $V$ , $\\varepsilon _{e}\\sim \\mathcal {N}(0,\\sigma ^2)$ is the Gaussian noise with zero mean and variance $\\sigma $ , and the outlier indicator variable $\\gamma _{e}\\in \\mathbb {R}$ is assumed to have a higher magnitude than $\\sigma $ .",
"Here the outliers are the aggregated edges whose directions conflict with the true ranking.",
"In order to estimate the $N + n$ unknown parameters ($N$ for $\\gamma $ and $n$ for $\\theta $ ), we aim to minimize the discrepancy between the annotation $y$ and the prediction $Z\\theta + \\gamma $ , as well as holding the outlier indicator $\\gamma $ sparse.",
"It gives us the following optimization problem: $\\begin{aligned}& &\\underset{\\theta ,\\ \\gamma }{\\textbf {\\textit {minimize}}}&\\ \\ \\ell _{w_0}(\\theta ,\\gamma ) +\\lambda \\cdot \\mathcal {R}_{w_0}(\\gamma ),\\end{aligned}$ where $\\begin{aligned}& \\ell _{w_0}(\\theta ,\\gamma ) &=& \\ \\ \\frac{1}{2}\\ \\Vert y - Z\\theta - \\gamma \\Vert ^2_{2,w_0}\\\\& &=&\\ \\ \\frac{1}{2}\\underset{e\\in E}{\\sum }w^0_{ij}(y_{ij}-\\gamma _{ij}-\\theta _i+\\theta _j)^2,\\end{aligned}$ $y = \\text{ver}(Y),\\ w^0 = \\text{ver}(W^0)$ is the vector form of $Y$ and $W^0$ , and the weighted regularization term $\\mathcal {R}_{w_0}$ is $\\mathcal {R}_{w_0}(\\gamma )=\\Vert \\gamma \\Vert _{1,w_0}=\\underset{e\\in E}{\\sum }w^0_{ij}|\\gamma _e|.$ In this situation, the weight $w^0$ and the label $y$ would be treated as the input data of the ranking problem (REF ).",
"Moreover, we introduce the variable $\\beta = (\\theta ,\\ \\gamma )^\\top $ to define the action space of the ranking algorithm.",
"We rewrite (REF ) as $\\begin{aligned}\\underset{\\beta \\in \\mathcal {B}_{\\lambda }}{\\textbf {\\textit {minimize}}}\\ \\ \\ell _{w_0}(\\beta ),\\end{aligned}$ where $\\ell _{w_0}(\\beta ) = \\frac{1}{2}\\left\\Vert y-\\begin{bmatrix}Z & \\\\& 1\\end{bmatrix}\\binom{\\theta }{\\gamma }\\right\\Vert ^2_{2,w_0}$ and $\\mathcal {B}_{\\lambda } = \\left\\lbrace \\beta \\ \\left|\\ \\left\\langle \\left(0,\\ w_0\\right),\\ \\binom{\\theta }{\\gamma }\\right\\rangle \\right.\\le \\varepsilon (\\lambda )\\right\\rbrace $ is the feasible set of (REF ) and the ranker's action space.",
"We model poisoning attack as a game between two players, the ranker and an attacker, where the latter wants to mislead its opponent into picking parameters to generate a difference order against the true ranking.",
"To disguise himself, the adversary needs to coordinate a poisoned $w$ associate with $y$ .",
"Intuitively, the adversary could not obtain $w$ through drastic changes, neither on each $w_{ij}$ nor $\\sum w_{ij}$ .",
"Such limitations lead to the following constraints for adversary's action.",
"First, the total difference between $w_0$ and $w$ would be smaller than $b$ , namely, $\\Vert w-w_0\\Vert _1\\le b,\\ \\ b\\in \\mathbb {Z}_+.$ Furthermore, the adversary could not alter the number of votes on any pairwise comparison $e\\in E$ obviously, $\\Vert w-w_0\\Vert _{\\infty }\\le l, \\ \\ l\\in \\mathbb {Z}_+,\\ \\ l\\le \\textbf {\\textit {min}}\\lbrace \\textbf {\\textit {max}}(w_0), b\\rbrace ,$ and the adversary‘s action space $\\mathcal {W}_{w_0}$ is $\\begin{aligned}\\mathcal {W}_{w_0}=\\ \\ \\left\\lbrace w\\ \\left|\\ \\begin{matrix}w\\in \\mathbb {Z}^{N}_+,\\ l,\\ b\\in \\mathbb {Z}_+,\\\\\\ \\Vert w-w_0\\Vert _1\\ \\le b,\\\\\\ \\Vert w-w_0\\Vert _{\\infty }\\le l,\\\\l\\le \\textbf {\\textit {min}}\\lbrace \\textbf {\\textit {max}}(w_0), b\\rbrace \\ \\end{matrix}\\right.\\right\\rbrace .\\end{aligned}$ The robust ranking algorithm (REF ) observes the poisoned training set sampling from $G$ , prunes the outlier and learns a ranking from the remaining data simultaneously.",
"Against the robust ranking algorithm that employs the defense described above, we can formulate the attacker’s goal as the following bi-level optimization problem: $& &\\underset{w}{\\textbf {\\textit {maximize}}}&\\ \\ \\ell _{w}(\\hat{\\theta },\\hat{\\gamma }) +\\lambda \\cdot \\mathcal {R}_{w}(\\hat{\\gamma }), \\\\& &\\textbf {\\textit {subject to}}&\\ \\ \\hat{\\theta },\\ \\hat{\\gamma }\\in \\underset{\\theta ,\\gamma }{\\textbf {\\textit {arg min}}}\\ \\ \\ell _{w}(\\theta ,\\ \\gamma ) +\\lambda \\cdot \\mathcal {R}_{w}(\\gamma ),\\\\& & &\\ \\ w\\in \\mathcal {W}_{w_0}.$ In (REF ), the $\\ell _1$ and $\\ell _{\\infty }$ distance constraints on $w$ correspond to the attacker only being able to find the perturbation in the neighborhood of $w^0$ .",
"The lower problem in (REF ) corresponds to the robust pairwise ranking algorithm.",
"With input data $\\lbrace w, y\\rbrace $ , the ranker obtain the relative score $\\theta $ by minimizing the discrepancy between the annotation $y$ and the prediction $Z\\theta + \\gamma $ while keeping $\\gamma $ to be sparse.",
"Unfortunately, the bilevel nature of (REF ) [5], [6]—maximizing the outer loss involves an inner minimization to find the parameters $\\theta , \\gamma $ —makes it difficult to solve, even less the discrete property of (REF ) and (REF ).",
"Next, we will discuss the poisoning attack on pairwise ranking in a different way.",
"Generally, we can look at the poison attack (REF ) from a distributional perspective.",
"The attacker and the ranker both access the weighted comparison graph $G$ to play a game as (REF ).",
"Actually, the non-toxic training data $\\lbrace w_0,\\ y\\rbrace $ are drawn according to a probability distribution $P$ $p(w_0,\\ y)=\\underset{e\\in E}{\\sum }\\ p(w^0_{ij},\\ y_{ij}).$ The attacker chooses a perturbation function $\\psi :\\mathbb {Z}^{N}_+\\rightarrow \\mathbb {Z}^{N}_+$ that change the weight $w_0$ to $w$ .",
"The attacker constructs the perturbation $\\psi $ with the limitation as (REF ).",
"Such a perturbation $\\psi $ induces a transition from empirical distribution $P$ to a poisoned distribution $Q$ $q(w, y) = q(\\psi (w_0), y)=\\underset{e\\in E}{\\sum }q(w_{ij}, y_{ij}).$ The attacker can only alter $b$ pairwise comparisons at most, increase or decrease the number of vote on any comparison less than $l$ , and formulate the poisoned training set $\\lbrace w,\\ y\\rbrace $ .",
"If the attacker selects $Q$ in a small enough neighborhood of $P$ , namely, the “distance” between the poisoned distribution $Q$ and the empirical distribution $P$ would be small, the attacker could obtain a good approximation of $P$ in the sense of such a “distance” and the poisoned sample $\\lbrace w,\\ y\\rbrace $ would be satisfied the constraints (REF ) and (REF ).",
"Let $\\phi : \\mathbb {R}_+\\rightarrow \\mathbb {R}$ be a convex function with $\\phi (1) = 0$ .",
"Then the $\\phi $ -divergence between distributions $Q$ and $P$ defined on a space $\\mathcal {X}$ is $\\begin{aligned}& d_{\\phi }(Q||P)&=&\\ \\ \\ \\int \\phi \\left(\\frac{dQ}{dP}\\right)dP\\\\& &=&\\ \\ \\int _{\\mathcal {X}}\\phi \\left(\\frac{q(x)}{p(x)}\\right)p(x)d\\mu (x),\\end{aligned}$ where $\\mu $ is a $\\sigma $ -finite measure with $Q, P \\ll \\mu $ , and $q=\\frac{dQ}{d\\mu }$ , $p=\\frac{dP}{d\\mu }$ .",
"Given $\\phi $ and sample $w_0$ , we reformulate the adversary's action space, the local neighborhood of the empirical distribution $P$ with radius $\\rho $ as $\\mathcal {Q}_{P}=\\lbrace \\text{distribution } Q \\text{ satisfies }d_{\\phi }(Q||P)\\le \\rho \\rbrace ,$ where $P$ is the empirical distribution of the pairwise comparisons, and $Q$ is the toxic distribution for poisoning attack.",
"Throughout this paper, we adopt $\\phi (t) = \\frac{1}{2}(t-1)^2,$ which gives the $\\chi ^2$ -divergence [68], [54], [18].",
"It means that $Q$ consists of discrete distributions supported on the observation $\\lbrace (w^0,\\ y)\\rbrace $ .",
"With opportunely chosen $\\rho $ , the adversary could obtain $w$ which satisfies the neighborhood constraints as (REF ) and (REF ).",
"The possible actions of two players $a = [\\beta _{\\lambda },\\ w]$ constitute the joint action space $\\mathcal {A}=\\mathcal {B}_{\\lambda }\\times \\mathcal {Q}_{P}$ which is assumed to be nonempty, compact, and convex.",
"Action spaces $\\mathcal {A}$ are parameters of the game (REF ).",
"Then the bi-level integer programming (REF ) can be written as a min-max optimization problem: $\\underset{q\\in \\mathcal {Q}_{P}}{\\sup }\\ \\underset{\\beta \\in \\mathcal {B}_{\\lambda }}{\\vphantom{Q\\in \\mathcal {Q}_{P}}\\inf }\\ \\mathbb {E}_{Q}[\\ell (\\beta ,\\ q(w,\\ y))]=\\underset{d_{\\phi }(Q||P)\\le \\rho }{\\sup }\\ \\ \\underset{\\Vert q(w)\\circ \\beta \\Vert _1\\le \\varepsilon (\\lambda )\\vphantom{Q\\in \\mathcal {Q}_{P}}}{\\inf }\\ \\ \\mathbb {E}_{Q}[\\ell (\\beta ,\\ q(w,\\ y))]\\nonumber $ where $q(w)\\circ \\beta =\\begin{bmatrix}0 & \\\\& q(W)\\end{bmatrix}\\begin{bmatrix}\\theta \\\\\\gamma \\end{bmatrix},$ $W = \\textbf {\\textit {diag}}(w)$ is a diagonal matrix.",
"Due to the $\\ell _1$ norm is decomposable, we can define a new set of loss function $f_{ij}:\\mathcal {B}_{\\lambda }\\times \\mathbb {Z}_+\\times \\lbrace -1,\\ 1\\rbrace ^{N}\\rightarrow \\mathbb {R}_+,\\ \\ \\forall \\ e\\in E$ $& & &\\ \\ f_{ij}(\\beta ,\\ q(w,\\ y))\\\\& &=&\\ \\ q(w_{ij})\\cdot \\frac{1}{2}(y_{ij}-\\gamma _{ij}-\\theta _i+\\theta _j)^2+\\lambda \\cdot q(w_{ij})|\\gamma _{ij}|\\nonumber \\\\& &=&\\ \\ q(w_{ij})\\cdot \\left[\\frac{1}{2}(y_{ij}-\\gamma _{ij}-\\theta _i+\\theta _j)^2+\\lambda \\cdot |\\gamma _{ij}|\\right]\\nonumber $ and the finite sum of $\\lbrace f_{ij}\\rbrace $ $f(\\beta ,\\ q(w,\\ y)) = \\underset{e\\in E}{\\sum }\\ f_{ij}(\\beta ,\\ q(w,\\ y))$ With fixed $\\lambda $ and some special form of $q$ , (REF ) could be a convex problem.",
"We swap the order of minimization and maximization in the min-max optimization problem (REF ) as $\\underset{\\beta \\in \\mathcal {B}_{\\lambda }}{\\vphantom{Q\\in \\mathcal {Q}_{P}}\\inf }\\ \\ \\underset{q\\in \\mathcal {Q}_{P}}{\\sup }\\ \\ \\ \\ \\mathbb {E}_{Q}[f(\\beta ,\\ q(w,\\ y))]=\\underset{\\beta \\in \\mathcal {B}_{\\lambda }}{\\vphantom{Q}\\inf }\\ \\ \\ \\underset{q}{\\sup }\\ \\left\\lbrace \\mathbb {E}_{Q}[f(\\beta ,\\ q(w,\\ y))],\\ \\textbf {\\textit {s.t.",
"}}\\ d_{\\phi }(Q||P)\\le \\rho \\vphantom{\\frac{1}{2}}\\right\\rbrace .$ In fact, the minimization of (REF ) and (REF ) correspond to the residual method and the Tikhonov regularization with discrepancy principle of the LASSO.",
"Indeed, it can be shown that the constrained minimization problem is equivalent to Tikhonov regularization, when the regularization parameter $\\lambda $ is chosen according to Morozov’s discrepancy principle [25].",
"Note that the objective function of (REF ) is a strictly convex function with respect to its arguments, then by [7], at least one Nash equilibrium exists.",
"From a game-theoretic viewpoint, (REF ) can be seen as a zero-sum game between two agents: the agent ranker (the infimum) seeks to incur the least possible loss, while the agent adversary (the supremum) seeks to obtain the worst possible objective function value – both given by $f(\\beta _{\\lambda }, w, y)$ .",
"For the supremum part of (REF ), the integer characteristic of $w$ obstructs obtaining a probability density function $q$ of the toxic distribution $Q$ .",
"Thanks to distributionally robust optimization, we reformulate the supremum part of (REF ) as a quadratically constrained linear maximization problem.",
"This tractable formulation can be solved by the probability simplex projection method.",
"Suppose the total number of pairwise comparison without toxic is $M^0 = \\underset{e\\in E}{\\sum }\\ w^0_{ij},$ and the frequencies of each comparison are $p = \\frac{1}{M^0}\\ w^0.$ Let the maximum toxic dosage be $\\kappa $ .",
"It suggests that the number of toxic pairwise comparisons $M$ satisfies $M = \\underset{e\\in E}{\\sum }\\ w_{ij} \\le (1+\\kappa )\\cdot M^0,$ Furthermore, we replace the toxic weight $w$ with its frequency $q = \\frac{w}{M}$ .",
"We relax the integer programming problem into a general optimization by such a variable substitution.",
"We note $z_{e} = \\frac{1}{2}(y_{ij}-\\gamma ^{\\lambda }_{ij}-\\theta ^{\\lambda }_i+\\theta ^{\\lambda }_j)^2+\\lambda \\cdot |\\gamma ^{\\lambda }_{ij}|,\\ e\\in E$ and $z=[z_1,\\dots ,z_N]\\in \\mathbb {R}^N_+$ .",
"The objective function with fixed $\\beta _\\lambda $ , maximizing the expectation $\\mathbb {E}_{Q}[f(\\beta _{\\lambda },\\ q(w,\\ y))]$ equals to compute the worst-case linear combination of $\\lbrace z_{ij}\\rbrace ,\\ e\\in E$ as $\\underset{q}{\\textbf {\\textit {maximize}}}\\ \\ \\left\\langle q,\\ z\\right\\rangle ,\\ \\textbf {s.t.",
"}\\ d_{\\phi }(Q||P)\\le \\rho .$ As $Q$ is a distribution, it requires that the combination coefficients $q$ should satisfy $\\sum _{e\\in E} q_e = 1\\ \\ \\text{or}\\ \\ \\langle 1,q\\rangle =1.$ It means that the distribution of $q$ is a probability simplex.",
"Furthermore, as $P$ and $Q$ are the discrete distributions and we choose $\\phi (t) = \\frac{1}{2}(t-1)^2$ in (REF ), the neighborhood constraint $d_{\\phi }(Q||P)\\le \\rho $ can be transformed as $\\frac{1}{2}\\Vert q-p\\Vert ^2_2\\le \\rho \\Vert p\\Vert ^2_2.$ Now we obtain the following quadratically constrained linear maximization problem which could be used to compute the supremum problem in (REF ): $\\begin{aligned}\\underset{q}{\\textbf {\\textit {maximize}}}&\\ \\ \\left\\langle q,z\\right\\rangle \\ \\ \\textbf {\\textit {s.t.",
"}}\\ \\ q\\in \\mathcal {Q}_{p}\\end{aligned}$ where $\\mathcal {Q}_{p}=\\left\\lbrace q\\ \\left|\\ \\frac{1}{2}\\Vert q-p\\Vert ^2_2\\le \\rho \\Vert p\\Vert ^2_2,\\ \\left\\langle 1,\\ q\\right\\rangle =1\\right.\\right\\rbrace .$ We reformulate the concave optimization (REF ) as a minimization problem for simplicity: $\\begin{aligned}\\underset{q}{\\textbf {\\textit {minimize}}}&\\ \\ \\left\\langle q,z\\right\\rangle \\ \\ \\textbf {\\textit {s.t.",
"}}\\ \\ q\\in \\mathcal {Q}_{p},\\end{aligned}$ and take a partial dual problem of this minimization, then maximize this dual problem to find the optimal $q$ .",
"First, we introduce the dual variable $\\mu \\ge 0$ for the quadratical constraint (REF ).",
"Notice that the strong duality exists for (REF ) because the Slater condition is satisfied by $q = p\\ \\ \\text{ and }\\ \\ 1^\\top p=1.$ Performing the standard min-max swap [13], it yields the following problem $\\underset{\\mu \\ge 0}{\\textbf {\\textit {minimize}}}\\ g(\\mu ) = \\\\\\underset{q}{\\inf }\\left\\lbrace \\frac{\\mu }{2}\\Vert q-p\\Vert ^2_2-\\mu \\rho \\Vert p\\Vert ^2_2+q^\\top z\\ |\\ q\\in \\mathbb {R}^N_+, q^\\top 1=1\\right\\rbrace \\nonumber .$ [tbh!]",
"InputInput OutputOutput Initialize the frequency of $w_0$ , $p$ by (REF ), Obtain the ranking parameters on the original data: $\\beta _{\\lambda }\\leftarrow \\textbf {\\textit {HodgeRank}}(w_0,y, \\tau ),$ Calculate the objective function value $z$ by (REF ): $w$ not converged Update the frequency: $q = \\textbf {\\textit {WorstCase}}(z, p, \\rho , \\varepsilon )$ , Assign the weight with $q$ , ${w}^{\\prime } = \\left[(1+\\kappa )\\cdot M^0\\right]q,$ Round ${w}^{\\prime }$ to obtain the $w$ as integer vector $w=\\textbf {\\textit {rounding}}({w}^{\\prime }),$ Update the ranking parameters: $\\beta _{\\lambda }\\leftarrow \\textbf {\\textit {HodgeRank}}(w,y, \\tau ),$ Update the objective function value: $z$ via (REF ), the poisoned data $\\lbrace w,y\\rbrace $ , the ranking parameters $\\beta _{\\lambda }=\\lbrace \\theta _{\\lambda }, \\gamma _{\\lambda }\\rbrace $ .",
"Poisoning Attack on Pairwise Ranking Given a collection of concave functions $\\lbrace g_q\\rbrace _{q\\in \\mathcal {Q}_{p}}$ , if it attains $\\inf g\\ =\\ \\underset{q\\in \\mathcal {Q}_{p}}{\\inf }\\ \\ g_q$ at some $q_0\\in \\mathcal {Q}_{p}$ , we know that $\\nabla g_{q_0}$ is the super-gradient of $g$ [29].",
"Suppose $q(\\mu )$ is the unique minimizer of the right hand side of (REF ), the dual function $g$ will be $g(\\mu ) = \\frac{\\mu }{2}\\Vert q(\\mu )-p\\Vert ^2_2-\\mu \\rho \\Vert p\\Vert ^2_2+q(\\mu )^\\top z$ and the derivative with respect to $\\mu $ (keeping $q(\\mu )$ fixed) is $g^{\\prime }(\\mu ) = \\frac{1}{2}\\Vert q(\\mu )-p\\Vert ^2_2-\\rho \\Vert p\\Vert ^2_2.$ As the constraints $q\\ge 0$ and $q^\\top 1=1$ require $q$ is on the probability simplex, we adopt the Euclidean projection of a vector to the probability simplex [19].",
"Such a projection provides an efficient solver of the infimum (REF ).",
"With no loss of generality, we assume that $z$ is an increasing sequence and the mean of $z$ is zero, $z_1\\le z_2\\le \\dots \\le z_N,\\ \\ \\ \\langle z, 1\\rangle = 0.$ Then we use $a,\\sigma \\in \\mathbb {R}^N_+$ , the cumulative summation of $z$ and $z^2$ as $a_i = \\underset{j\\le i}{\\sum }z_i,\\ \\ \\ \\sigma _i = \\underset{j\\le i}{\\sum }z_i^2,\\ \\ \\ i\\in [N].$ The infimum in (REF ) is equivalent to projecting the vector $v(\\mu )\\in \\mathbb {R}^N$ onto the probability simplex, $v_i = p_i - \\frac{1}{\\mu }z_i,\\ \\ \\ i\\in [N]$ According to [19], $q(\\mu )$ has the form as $q_i(\\mu )=(v_i - \\eta )_+$ for some $\\eta \\in \\mathbb {R}$ , where $\\eta $ is selected such that $\\sum q_i(\\mu ) = 1$ .",
"Finding such a value $\\eta $ is equivalent to finding the unique index $i$ such that $\\sum ^i_{j=1}(v_j-v_i)<1\\ \\ \\text{and}\\ \\ \\sum ^{i+1}_{j=1}(v_j-v_{i+1})>1.$ If no such index exists, we set $i=n$ as the sum $\\sum ^i_{j=1}(v_j-v_i)$ is increasing in $i$ and $v_1-v_1=0$ .",
"Given the index $i$ , $\\eta = p_i-\\frac{1}{i}-\\frac{1}{i\\mu }\\sum ^{i}_{j=1}z_i=p_i-\\frac{1}{i}-\\frac{1}{i\\mu }a_i$ satisfies $\\sum (v_i-\\eta )_+ = 1$ and $v_j-\\eta \\ge 0$ for any $j\\le i$ while $v_j-\\eta \\le 0$ for $j>i$ .",
"Meanwhile, the derivative $\\frac{\\partial }{\\partial \\mu }g(\\mu )$ (where $q(\\mu )$ is fixed) has a explicit form $\\begin{aligned}& g^{\\prime }(\\mu ) &=&\\ \\ \\frac{\\partial }{\\partial \\mu }\\left\\lbrace \\frac{\\mu }{2}\\Vert q(\\mu )-p\\Vert ^2-\\mu \\rho \\Vert p\\Vert ^2_2+q^\\top (\\mu )z\\right\\rbrace \\\\& &=&\\ \\ \\frac{1}{2}\\Vert q(\\mu )-p\\Vert ^2-\\rho \\Vert p\\Vert ^2\\\\& &=&\\ \\ \\frac{1}{2}\\sum ^{i}_{j=1}(v_j-\\eta -p_j)^2+\\frac{1}{2}\\sum ^{N}_{j=i+1}p^2_j-\\rho \\Vert p\\Vert ^2\\\\& &=&\\ \\ \\frac{1}{2}\\sum ^{i}_{j=1}\\left(\\frac{z_j}{\\mu }+\\eta \\right)^2+\\frac{1}{2}\\sum ^{N}_{j=i+1}p^2_j-\\rho \\Vert p\\Vert ^2\\\\& &=&\\ \\ \\frac{\\sigma _i}{2\\mu ^2}+\\frac{\\eta ^2i}{2}+\\frac{a_i\\eta }{\\mu }+\\sum _{j=i+1}^{N}p^2_j-\\rho \\Vert p\\Vert ^2\\end{aligned}$ The derivative $g^{\\prime }(\\mu )$ only needs $\\mathcal {O}(1)$ when $a$ and $\\sigma $ are known.",
"Binary search can calculate the optimal index $i$ and $q$ efficiently, which requires $\\mathcal {O}(\\log \\frac{1}{\\varepsilon }\\log N)$ to find $\\mu $ with accuracy $\\varepsilon $ .",
"We can get $\\eta $ through (REF ) if (REF ) are satisfied.",
"The solution $q(\\mu )$ is $q_i = \\left(p_i - \\frac{z_i}{\\mu }-\\eta \\right)_+, i\\in [N].$ Specifically, the computational complexity to obtain the sorted vector $z$ is ${\\cal O}(N\\log N)$ , and that of the estimate of the frequency $q$ is $\\mathcal {O}(\\log \\frac{1}{\\varepsilon }\\log N)$ .",
"The overall time computational complexity is $\\mathcal {O}(N\\log N + \\log \\frac{1}{\\varepsilon }\\log N)$ .",
"At last, we describe the whole optimization of the poison attack on pairwise ranking with Algorithm REF.",
"We summarize the complete optimization procedure of the supremum in (REF ) as Algorithm REF and Algorithm 3.",
"For the infimum part of (REF ), the agent HodgeRank finds $\\beta _{\\lambda }$ that minimizes the regularized loss on $\\lbrace w,\\ y\\rbrace $ where the hyper-parameter $\\lambda $ controls the regularization strength.",
"We include the solving process of HodgeRank as Algorithm REF for completeness.",
"[h!]",
"InputInput OutputOutput Make $z$ have the zero mean: $z \\leftarrow z - \\bar{z}$ , and sort $z$ .",
"Initialize $\\mu _{\\min } = 0$ , $a_i = \\sum _{j\\le i}z_{j}$ , and $\\sigma _i = \\sum _{j\\le i}z^2_{j}$ for all $i\\in [N]$ , $\\mu _{\\max } = \\mu _{\\infty } = \\max \\left\\lbrace \\Vert z\\Vert _{\\infty },\\ \\sqrt{\\frac{1}{\\rho \\Vert p\\Vert ^2_2}}\\Vert z\\Vert _2\\right\\rbrace $ $|\\mu _{\\max }-\\mu _{\\min }|>\\varepsilon \\mu _{\\infty }$ Set $\\mu = \\frac{1}{2}(\\mu _{\\min }+\\mu _{\\max })$ , and $(\\eta ,\\ i) = \\textbf {\\textit {FindShift}}(z,\\ p,\\ a,\\ \\mu ),$ Obtain the partial derivative ${g}^{\\prime }(\\mu )$ by (REF ) ${g}^{\\prime }(\\mu )>0$ $\\mu _{\\min } \\leftarrow \\mu $ $\\mu _{\\max } \\leftarrow \\mu $ Set $\\mu = \\frac{1}{2}(\\mu _{\\min }+\\mu _{\\max })$ , and $(\\eta ,\\ i) = \\textbf {\\textit {FindShift}}(z,\\ p,\\ a,\\ \\mu ),$ $q$ by (REF ).",
"$\\textbf {\\textit {WorstCase}}(z,\\ p,\\ \\rho ,\\ \\varepsilon )$ [bht!]",
"$\\textbf {\\textit {FindShift}}(z, p, a, \\mu )$ InputInput OutputOutput Initialize $i_{\\text{low}} = 1$ and $i_{\\text{high}} = N$ , $p_N - \\frac{1}{\\mu }z_N\\ge 0$ $\\eta = 0$ , $i=N$ , Break.",
"$i_{\\text{low}}\\ne i_{\\text{high}}$ $i = \\frac{1}{2}(i_{\\text{low}}+i_{\\text{high}})$ , $a_{\\text{left}} = \\sum ^i_{j=1}(v_j-v_i) = \\frac{1}{\\mu }(iz_i-a_i)$ , $a_{\\text{right}} = \\sum ^{i+1}_{j=1}(v_j-v_{i+1}) = \\frac{1}{\\mu }[(i+1)z_{i+1}-a_{i+1}]$ , $a_{\\text{right}} \\ge 1\\wedge a_{\\text{left}} < 1$ $\\eta = p_i-\\frac{1}{i}-\\frac{1}{i\\mu }a_i$ , Break.",
"$a_{\\text{left}}\\ge 1$ $i_{\\text{high}} = i-1$ $i_{\\text{low}} = i+1$ $i=i_{\\text{low}}$ , $\\eta = p_i-\\frac{1}{i}-\\frac{1}{i\\mu }a_i$ .",
"[h!]",
"InputInput OutputOutput Calculate the relative ranking score $\\hat{\\theta }$ $\\hat{\\theta } = (X^{\\top }X+\\delta I)^{-1}X^{\\top }\\sqrt{W}y$ where $X = \\sqrt{W}A$ , $\\sqrt{W}= \\textbf {\\textit {diag}}(\\sqrt{w})$ .",
"the corresponding ranking parameter $\\hat{\\theta }$ .",
"$\\textbf {\\textit {HodgeRank}}(w, A, y)$"
]
] | 2107.01854 |
[
[
"OPA: Object Placement Assessment Dataset"
],
[
"Abstract Image composition aims to generate realistic composite image by inserting an object from one image into another background image, where the placement (e.g., location, size, occlusion) of inserted object may be unreasonable, which would significantly degrade the quality of the composite image.",
"Although some works attempted to learn object placement to create realistic composite images, they did not focus on assessing the plausibility of object placement.",
"In this paper, we focus on object placement assessment task, which verifies whether a composite image is plausible in terms of the object placement.",
"To accomplish this task, we construct the first Object Placement Assessment (OPA) dataset consisting of composite images and their rationality labels.",
"We also propose a simple yet effective baseline for this task.",
"Dataset is available at https://github.com/bcmi/Object-Placement-Assessment-Dataset-OPA."
],
[
"Introduction",
"As a common image editing operation, image composition aims to generate a realistic-looking image by pasting the foreground object of one image on another image.",
"The composites can result in fantastic images that previously only exist in the imagination of artists.",
"However, it is challenging to insert a foreground object into a background image that satisfies the following requirements: 1) the foreground object has compatible color and illumination with the background image; 2) the inserted object may have an impact on the background image, like the reflection and shadow; 3) the foreground object should be placed at a reasonable location on the background considering location, size, occlusion, semantics, and etc.",
"To satisfy the above requirements, image harmonization [18], [2], shadow generation [10], [14], and object placement [16], [12] have been proposed to improve the quality of composite images from the above aspects, respectively.",
"In this paper, we focus on the third issue, object placement, aiming to paste foreground object on the background with suitable location, size, occlusion, etc.",
"As shown in Figure REF , the cases of unreasonable object placement [1] are including but not limited to: 1) the foreground object is too large or too small; 2) the foreground object does not have supporting force (e.g., hanging in the air); 3) the foreground object appears in a semantically unreasonable place (e.g., boat on the land); 4) unreasonable occlusion; 5) inconsistent perspectives between foreground and background.",
"The above unreasonable cases would significantly degrade the reality of composite images.",
"Considering a wide range of foreground objects and complicated scenarios, object placement is still a challenging task.",
"Figure: Some negative samples in our OPA dataset and the inserted foreground objects are marked with red outlines.",
"From left to right: (a) objects with inappropriate size; (b) objects hanging in the air; (c) objects appearing in the semantically unreasonable place; (d) unreasonable occlusion; (e) inconsistent perspectives.Some previous works attempted to learn reasonable object placement to generate realistic composite images.",
"One group of methods [6], [15], [19], [5] relied on explicit rules to find a reasonable location for the foreground object.",
"For example, the new background of inserted foreground should be close to its original background [5] or the foreground should be placed on a flat plane [6].",
"However, these explicit rules are only applicable to limited scenarios.",
"The other group of methods trained network to automatically learn the reasonable object placement, which can be further divided into supervised and unsupervised methods.",
"Supervised methods [16], [4], [21], [20], [11] leveraged the size/location of foreground object in the original image as ground-truth.",
"They predicted the bounding box or transformation of the foreground object based on the foreground and background features [16], [20].",
"Unsupervised methods like [17] did not use ground-truth size/location.",
"They learned reasonable transformation of foreground object, by pushing the generated composite images close to real images.",
"All the above works focus on generating reasonable composite images instead of object placement assessment.",
"In other words, they cannot automatically assess the rationality of a composite image in terms of object placement.",
"To evaluate the quality of generated composite images, the above works on learning object placement usually adopt the following three approaches.",
"1) [16] scored the correlation between the distributions of predicted boxes and ground-truth boxes.",
"[20] calculated the Frechet Inception Distance (FID) [9] between composite and real images to measure the placement plausibility.",
"However, they cannot evaluate each individual composite image.",
"2) [17], [5] utilized the improvement of downstream tasks (e.g., object detection) to evaluate the quality of composite images, where the training sets of the downstream tasks are augmented with generated composite images.",
"However, the evaluation cost is quite huge and the improvement in downstream tasks may not reliably reflect the quality of composite images, because [7] revealed that randomly generated composite images could also boost the performance of downstream tasks.",
"3) Another common evaluation strategy is user study, where people are asked to score the rationality of placement [11], [16].",
"User study complies with human perception and each composite image can be evaluated individually.",
"However, due to the subjectivity of user study, the gauge in different papers may be dramatically different.",
"There is no unified benchmark dataset and the results in different papers cannot be directly compared.",
"In summary, as far as we are concerned, no previous works focus on object placement assessment and no suitable dataset is available for this task.",
"In this work, we focus on the task of object placement assessment, that is, automatically assessing the rationality of a composite image in terms of object placement.",
"We build an Object Placement Assessment (OPA) dataset for this task, based on COCO [13] dataset.",
"First, we select unoccluded objects from multiple categories as our candidate foreground objects.",
"Then, we design a strategy to select compatible background images for each foreground object.",
"The foreground objects are pasted on their compatible background images with random sizes and locations to form composite images, which are sent to human annotators for rationality labeling.",
"Each image is labeled by four human annotators, where only the images with consistent labels are preserved in the dataset to ensure the annotation quality.",
"Finally, we split the collected dataset into training set and test set, in which the background images and foreground objects have no overlap between training set and test set.",
"More details about constructing the dataset will be elaborated in Section .",
"With the constructed dataset, we regard the object placement assessment task as a binary classification problem and any typical classification network can be applied to this task.",
"With the functionality of object placement assessment, our model can help obtain realistic composite images.",
"Particularly, given automatically (e.g., [17], [20]) or manually (e.g., by users) created composite images, we can apply object placement assessment model to select the composite images with high rationality scores."
],
[
"Dataset Construction",
"In this section, we describe the construction process of our Object Placement Asssessment (OPA) dataset, in which we first generate composite images and then ask human annotators to label these composite images w.r.t.",
"the rationality of object placement."
],
[
"Composite Image Generation",
"We select suitable foreground objects and background images from Microsoft COCO dataset [13], which are used to generate composite images.",
"Foreground object selection: There are 80 object categories in COCO [13] with annotated instance segmentation masks.",
"We only keep unoccluded foreground objects, because it is difficult to find reasonable placement for occluded objects.",
"We delete some categories according to the following rules: 1) the categories which usually appear at very specific locations, such as transportation-related categories (e.g., traffic light, stop sign) and human-centric categories (e.g., tie, snowboard); 2) the categories of large objects appearing in crowded space, such as large furniture (e.g., refrigerator, bed); 3) the categories with too few remaining objects after removing occluded and tiny foreground objects (e.g., toaster, hair drier); 4) the categories which are hard to verify the rationality, such as the flying object (e.g., kite, frisbee).",
"In summary, the above categories are either hard to find reasonable placement or hard to verify the rationality of object placement.",
"After filtering, 47 categories remain and the complete list is: airplane, apple, banana, bear, bench, bicycle, bird, boat, book, bottle, bowl, broccoli, bus, cake, car, cat, cellphone, chair, cow, cup, dog, donut, elephant, fire hydrant, fork, giraffe, horse, keyboard, knife, laptop, motorcycle, mouse, orange, person, pizza, potted plant, remote, sandwich, scissors, sheep, spoon, suitcase, toothbrush, truck, vase, wineglass, zebra.",
"With the annotated instance segmentation masks from COCO [13] dataset, we select 100 unoccluded foreground objects for each category.",
"Background image selection: For each foreground category, there should be a set of compatible background images.",
"For example, airplanes do not appear indoors and forks usually appear on the table.",
"In this work, we eliminate the burden of selecting compatible background images for object placement assessment task.",
"We fine-tune PlaceCNN [22] pretrained on places365 [22] to select a set of compatible background images for each category.",
"Specifically, for each category, we take the images containing the objects of this category as positive samples, and randomly sample an equal number of other images as negative samples.",
"Then, we fine-tune PlaceCNN [22] based on positive and negative samples to learn a binary classifier.",
"For each category, we apply the trained binary classifier to retrieve top 100 images which do not contain the objects of this category as a set of compatible background images.",
"Composite image generation: We generate a composite image by pasting one foreground object on another background image.",
"To avoid too much repetition, we limit the size and location of the foreground object according to some prior knowledge.",
"For each foreground category, we first calculate a reasonable range of its size ratio, which is defined as the ratio of foreground object size over its corresponding image size.",
"Given a foreground object and a compatible background image, we randomly sample 5 size ratios and 9 locations, leading to 45 composite images.",
"For size ratio, we divide the range of size ratio of foreground category into five bins based on 20%, 40%, 60%, 80% quantiles, and randomly sample one size ratio from each bin.",
"For location, we evenly divide the background image into 9 partitions and randomly sample one location from each partition.",
"We resize the foreground object according to certain size ratio and place it at certain location, producing a composite image.",
"Besides, we remove the composite images with incomplete foreground objects, e.g., half of the foreground object is out of the scope of the background image.",
"Figure: Some positive and negative samples in our OPA dataset and the inserted foreground objects are marked with red outlines.",
"Top row: positive samples; Bottom rows: negative samples, including objects with inappropriate size (e.g., f, g, h), without supporting force (e.g., i, j, k), appearing in the semantically unreasonable place (e.g., l, m, n), with unreasonable occlusion (e.g., o, p, q), and with inconsistent perspectives (e.g., r, s, t).Figure: The number of images per foreground category in our OPA dataset."
],
[
"Composite Image Labelling",
"Since the rationality of object placement is constrained by many complicated factors (e.g., location, size, occlusion, semantics), the number of negative images is significantly larger than the positive samples among the randomly generated composite images.",
"To achieve relatively balanced positive-negative ratio and save the human labor, we first fine-tune a ResNet-50 [8] classifier pretrained on ImageNet [3] to remove the obviously unreasonable composite images.",
"During fine-tuning, the real images are regarded as positive samples.",
"We additionally generate composite images via random copy-and-paste as negative samples, which have no overlap with the composite images in Section REF .",
"Although the generated composite images contain both positive samples and negative samples, negative samples are dominant and thus the learned binary classifier is useful.",
"To indicate the foreground object, we also feed foreground mask into ResNet-50 [8] classifier.",
"We apply the fine-tuned classifier to the composite images in Section REF and select the top 235,000 composite images with the highest scores for further labeling.",
"The selected composite images are supposed to have relatively higher ratio of positive samples.",
"To acquire the binary rationality label (1 for reasonable object placement and 0 for unreasonable object placement), we ask four human annotators to label the rationality for each composite image.",
"We purely focus on the object placement issues and ignore the other issues (e.g., inconsistent illumination between foreground and background, unnatural boundary between foreground and background).",
"Due to the subjectivity of this annotation task, we make detailed annotation guidelines (e.g., the reasonable range of sizes for each foreground category) and train human annotators for two weeks to make the annotations consistent across different annotators.",
"The detailed annotation guidelines are as follows, All foreground objects are considered as real objects instead of models or toys.",
"The foreground object placement conforms to the basic laws of physics.",
"Except for the flying objects (e.g., airplane), all the other objects should have reasonable supporting force.",
"The foreground object should appear in a semantically reasonable place.",
"We also make some specific rules for the ambiguous cases.",
"For example, for the container categories (e.g., bowl, bottle), we stipulate that they cannot be surrounded by fried dish.",
"If there is occlusion between the foreground object and background object, the rationality of occlusion should be considered.",
"The size of the foreground object should be judged based on its location and relative distance to other background objects.",
"We provide a reasonable range of size for each category and the estimated size of the foreground should be within the range of its category.",
"For animal categories (e.g., dog, sheep), we treat the sizes of animals of all ages (from baby animal to adult animal) as reasonable sizes.",
"The perspective of foreground object should look reasonable.",
"The inharmonious illumination and color, and unreasonable reflection and shadow are out of the scope of consideration.",
"Although some of the above rules may be arguable, which depends on the definition of rationality, our focus is making the annotation criterion as explicit as possible and the annotations across different images as consistent as possible, so that the constructed dataset is qualified for scientific study.",
"Besides, similar categories are labeled by the same group of human annotators to further mitigate the inconsistency.",
"Finally, we only keep the images for which four human annotators reach the agreement.",
"From the remaining images, we construct training set with 62,074 images and test set with 11,396 images, whose foreground objects and background images have no overlap.",
"We impose this constraint to better evaluate the generalization ability of different methods, because the foreground object and background image are generally out of the scope of training set in real-world applications."
],
[
"Dataset Statistics",
"After composite image generation and composite image labelling, there are 24,917 positive samples and 48,554 negative samples in our OPA dataset.",
"Our OPA dataset has 4,137 unrepeated foreground objects and 1,389 unrepeated background images.",
"We show some example positive and negative images in our dataset examples in Figure REF .",
"We also present the number of images (positive and negative) per foreground category in Figure REF .",
"We divide our OPA dataset into 62,074 training images and 11,396 test images, in which the foregrounds/backgrounds in training set and test set have no overlap.",
"The training (resp., test) set contains 21,351 (resp., 3,566) positive samples and 40,724 (resp., 7,830) negative samples.",
"Besides, the training (resp., test) set contains 2,701 (resp., 1,436) unrepeated foreground objects and 1,236 (resp., 153) unrepeated background images."
],
[
"Conclusion",
"In this work, we focus on the object placement assessment task, which verifies the rationality of object placement in a composite image.",
"To support this task, we have contributed an Object Placement Assessment (OPA) dataset.",
"This dataset will facilitate the research in automatic object placement, which can automatically forecast the diverse and plausible placement of foreground object on the background image."
]
] | 2107.01889 |
[
[
"Sets of Marginals and Pearson-Correlation-based CHSH Inequalities for a\n Two-Qubit System"
],
[
"Abstract Quantum mass functions (QMFs), which are tightly related to decoherence functionals, were introduced by Loeliger and Vontobel [IEEE Trans.",
"Inf.",
"Theory, 2017, 2020] as a generalization of probability mass functions toward modeling quantum information processing setups in terms of factor graphs.",
"Simple quantum mass functions (SQMFs) are a special class of QMFs that do not explicitly model classical random variables.",
"Nevertheless, classical random variables appear implicitly in an SQMF if some marginals of the SQMF satisfy some conditions; variables of the SQMF corresponding to these \"emerging\" random variables are called classicable variables.",
"Of particular interest are jointly classicable variables.",
"In this paper we initiate the characterization of the set of marginals given by the collection of jointly classicable variables of a graphical model and compare them with other concepts associated with graphical models like the sets of realizable marginals and the local marginal polytope.",
"In order to further characterize this set of marginals given by the collection of jointly classicable variables, we generalize the CHSH inequality based on the Pearson correlation coefficients, and thereby prove a conjecture proposed by Pozsgay et al.",
"A crucial feature of this inequality is its nonlinearity, which poses difficulties in the proof."
],
[
"Introduction",
"Graphical models like factor graphs [1], [2], [3] have been used to represent various statistical models.",
"In the following, we will call a factor graph consisting only of non-negative real-valued local functions a standard factor graph (S-FG).",
"S-FGs have many applications, in particular in communications and coding theory (see, e.g., [4], [5]) and statistical mechanics (see, e.g., [6]).",
"In these applications, factor graphs frequently represent the factorization of the joint probability mass functions (PMFs) of all the relevant random variables.",
"Quantities of interest can then be obtained by exactly or approximately computing marginals of this joint PMF and suitably processing these marginals.",
"Factor graphs have also been used to represent quantum-mechanical probabilities [7], [8].",
"In contrast to S-FGs, these factor graphs consist of complex-valued local functions satisfying some constraints.",
"In the following, we will call such factor graphs quantum-probability factor graphs (Q-FGs).",
"A Q-FG is typically used to represent the factorization of the joint quantum mass function (QMF) as introduced in [7].",
"In this paper, we first discuss similarities and differences between PMFs and QMFs.",
"Some of the features of QMFs will then motivate the study that is carried out in the rest of this paper."
],
[
"PMFs vs. QMFs",
"In this section, we highlight some similarities and crucial differences between PMFs and QMFs.",
"First, we consider a classical setup.",
"In particular, we assume that we are interested in a graphical model that represents the joint PMF $P_{Y_1, \\ldots , Y_n}(y_1, \\ldots , y_n)$ , where $Y_1, \\ldots , Y_n$ are some random variables of interest taking value in some alphabets $ \\mathcal {Y} _1, \\ldots , \\mathcal {Y} _n$ .For simplicity, in the following all alphabets will be finite.",
"(In a typical application, we might have observed $Y_1 = y_1, \\ldots , Y_{n-1} = y_{n-1}$ and would like to estimate $Y_n$ based on these observations.)",
"In most applications, the PMF $P_{Y_1, \\ldots , Y_n}(y_1, \\ldots , y_n)$ does not have a “nice” factorization in terms of simple factors.",
"However, frequently, with the introduction of suitable auxiliary variables $x_1, \\ldots , x_m$ taking values in some alphabets $ \\mathcal {X} _1, \\ldots , \\mathcal {X} _m$ , respectively, there is a function $p( \\mathbf {x} , \\mathbf {y} )$ , where $ \\mathbf {x} := (x_1, \\ldots , x_m)$ and $ \\mathbf {y} := (y_1, \\ldots , y_n)$ , such that $p( \\mathbf {x} , \\mathbf {y} )&\\in \\mathbb {R}_{\\ge 0} \\quad \\text{(for all $ \\mathbf {x} , \\mathbf {y} $)} \\ , \\\\\\quad \\sum _{ \\mathbf {x} , \\mathbf {y} }p( \\mathbf {x} , \\mathbf {y} )&= 1 \\ , \\\\\\sum _{ \\mathbf {x} }p( \\mathbf {x} , \\mathbf {y} )&= P_{ \\mathbf {Y} }( \\mathbf {y} ) \\quad \\text{(for all $ \\mathbf {y} $)} \\ ,$ and such that $p( \\mathbf {x} , \\mathbf {y} )$ has a “nice” factorization.",
"(For example, in a hidden Markov model, the joint PMF of the observations does not have a “nice” factorization, but the joint PMF of the hidden state process and the observations has a “nice” factorization.)",
"Note that the function $p( \\mathbf {x} , \\mathbf {y} )$ can, thanks to its properties, be considered as a joint PMF of some random variables $X_1, \\ldots , X_m, Y_1, \\ldots , Y_n$ .",
"Second, we consider a quantum-mechanical setup.",
"We assume, again, that we are interested in a graphical model representing the joint PMF $P_{Y_1, \\ldots , Y_n}(y_1, \\ldots , y_n)$ , where $Y_1, \\ldots , Y_n$ are some random variables of interest taking values in some alphabets $ \\mathcal {Y} _1, \\ldots , \\mathcal {Y} _n$ .",
"Such random variables can, for example, represent the measurements obtained when running some quantum-mechanical experiment, and we might be interested in estimating $Y_n$ based on the observations $Y_1 = y_1, \\ldots , Y_{n-1} = y_{n-1}$ .",
"As in the classical case, the PMF $P_{Y_1, \\ldots , Y_n}(y_1, \\ldots , y_n)$ usually does not have a “nice” factorization in terms of simple factors.",
"Moreover, standard physical modeling of quantum-mechanical systems shows that introducing a function $p( \\mathbf {x} , \\mathbf {y} )$ as defined above does usually not help toward obtaining a function with a “nice” factorization.",
"However, in many quantum-mechanical setups of interest, with the introduction of suitable auxiliary variables $x_1, \\ldots , x_m, x^{\\prime }_1, \\ldots , x^{\\prime }_m$ taking values in some alphabets $ \\mathcal {X} _1, \\ldots , \\mathcal {X} _m$ , $ \\mathcal {X} ^{\\prime }_1, \\ldots , \\mathcal {X} ^{\\prime }_m$ (with $ \\mathcal {X} ^{\\prime }_i = \\mathcal {X} _i$ , $i \\in \\lbrace 1, \\ldots , m \\rbrace $ ), there is a function $q( \\mathbf {x} , \\mathbf {x} ^{\\prime }, \\mathbf {y} )$ , called quantum mass function (QMF) [7], such that $q( \\mathbf {x} , \\mathbf {x} ^{\\prime }, \\mathbf {y} )&\\in \\mathbb {C}\\quad \\text{(for all $ \\mathbf {x} , \\mathbf {x} ^{\\prime }, \\mathbf {y} $)} \\ , \\\\\\quad \\sum _{ \\mathbf {x} , \\mathbf {x} ^{\\prime }, \\mathbf {y} }q( \\mathbf {x} , \\mathbf {x} ^{\\prime }, \\mathbf {y} )&= 1 \\ , \\\\q( \\mathbf {x} , \\mathbf {x} ^{\\prime }, \\mathbf {y} )&\\ \\text{is a PSD kernel in $( \\mathbf {x} , \\mathbf {x} ^{\\prime })$ for every $ \\mathbf {y} $} \\ , \\\\\\sum _{ \\mathbf {x} , \\mathbf {x} ^{\\prime }}q( \\mathbf {x} , \\mathbf {x} ^{\\prime }, \\mathbf {y} )&= P_{ \\mathbf {Y} }( \\mathbf {y} ) \\quad \\text{(for all $ \\mathbf {y} $)} \\ ,$ and such that $q( \\mathbf {x} , \\mathbf {x} ^{\\prime }, \\mathbf {y} )$ has a “nice” factorization.",
"The major difference between $p( \\mathbf {x} , \\mathbf {y} )$ and $q( \\mathbf {x} , \\mathbf {x} ^{\\prime }, \\mathbf {y} )$ is the fact that the former takes value in $\\mathbb {R}_{\\ge 0}$ , whereas the latter takes value in $\\mathbb {C}$ .",
"In particular, $\\sum _{ \\mathbf {y} } q( \\mathbf {x} , \\mathbf {x} ^{\\prime }, \\mathbf {y} )$ is in general not a PMF over $( \\mathbf {x} , \\mathbf {x} ^{\\prime })$ , thereby showing that $ \\mathbf {x} , \\mathbf {x} ^{\\prime }$ cannot be considered as random variables.",
"(See [7] for more details.)",
"In [8], the authors discussed an approach to QMFs where $ \\mathbf {y} $ does not appear explicitly anymore, but “emerges” from a QMF.",
"More precisely, they first introduced a simple quantum mass function (SQMF) $q( \\mathbf {x} , \\mathbf {x} ^{\\prime })$ that satisfies $q( \\mathbf {x} , \\mathbf {x} ^{\\prime })&\\in \\mathbb {C}_{\\ge 0} \\quad \\text{(for all $ \\mathbf {x} , \\mathbf {x} ^{\\prime }$)} \\ , \\\\\\quad \\sum _{ \\mathbf {x} , \\mathbf {x} ^{\\prime }}q( \\mathbf {x} , \\mathbf {x} ^{\\prime })&= 1 \\ , \\\\q( \\mathbf {x} , \\mathbf {x} ^{\\prime })&\\ \\text{is a PSD kernel in $( \\mathbf {x} , \\mathbf {x} ^{\\prime })$} \\ .$ Afterwards, they defined “classicable” variables.",
"Definition 1 Let $\\mathcal {I}$ be a subset of $\\lbrace 1, \\ldots , m \\rbrace $ and let $\\mathcal {I}^{\\mathrm {c}} := \\lbrace 1, \\ldots , m \\rbrace \\setminus \\mathcal {I}$ be its complement.",
"The variables $ \\mathbf {x} _{\\mathcal {I}}$ are called jointly classicable if the function $q( \\mathbf {x}_{\\mathcal {I}} , \\mathbf {x}_{\\mathcal {I}} ^{\\prime })&:= \\sum _{ \\mathbf {x}_{\\mathcal {I}^{\\mathrm {c}}} , \\mathbf {x}_{\\mathcal {I}^{\\mathrm {c}}} ^{\\prime }}q( \\mathbf {x} , \\mathbf {x} ^{\\prime })$ is zero for all $( \\mathbf {x}_{\\mathcal {I}} , \\mathbf {x}_{\\mathcal {I}} ^{\\prime })$ satisfying $ \\mathbf {x}_{\\mathcal {I}} \\ne \\mathbf {x}_{\\mathcal {I}} ^{\\prime }$ .It would be more precise to call this function $q_{\\mathcal {I}}$ .",
"However, for conciseness, we drop the index $\\mathcal {I}$ as it can be inferred from the arguments.",
"Note that if $ \\mathbf {x}_{\\mathcal {I}} $ are jointly classicable, then one can define the function $p( \\mathbf {x}_{\\mathcal {I}} ) := q( \\mathbf {x}_{\\mathcal {I}} , \\mathbf {x}_{\\mathcal {I}} )$ , for which it is straightforward, thanks to the properties of SQMFs, to show that it is a PMF.",
"It is in this sense that random variables $y_1, \\ldots , y_n$ that were omitted when going from QMFs to SQMFs can “emerge” again.Note that there is a strong connection of SQMFs to the so-called decoherence functional [9], [10], and via this also to the consistent-histories approach to quantum mechanics [11].",
"However, the starting point of our investigations is quite different.",
"Definition 2 Let $\\mathcal {K}$ be a collection of subsets $\\mathcal {I}$ of $\\lbrace 1, \\ldots , m \\rbrace $ such that $ \\mathbf {x}_{\\mathcal {I}} $ is classicable.",
"Example 3 Consider the Q-FG $ \\mathsf {N}_{4} $ in Fig.",
"REF , whose global function is an SQMF.",
"In that Q-FG, the matrix $\\rho $ represents a PSD matrix and the matrices $U_1$ , $U_2$ are unitary matrices.",
"One can show that for all choices of $\\rho $ , $U_1$ , and $U_2$ , the collection $\\mathcal {K}$ can be chosen to contain the sets $\\lbrace 1, 2 \\rbrace $ , $\\lbrace 1, 4 \\rbrace $ , $\\lbrace 2, 3 \\rbrace $ , and $\\lbrace 3, 4 \\rbrace $ .",
"Interestingly enough, the collection of functions $\\bigl \\lbrace p( \\mathbf {x}_{\\mathcal {I}} ) \\bigr \\rbrace _{\\mathcal {I} \\in \\mathcal {K}}$ is usually such that there is no PMF $p( \\mathbf {x} )$ such that for every $\\mathcal {I} \\in \\mathcal {K}$ , the function $p( \\mathbf {x}_{\\mathcal {I}} )$ can be obtained as a marginal of $p( \\mathbf {x} )$ .A similar observation is at the origin of the so-called “single-framework” rule in the consistent-histories approach to quantum mechanics.",
"In general, we can only guarantee that for two sets $\\mathcal {I}_1, \\ \\mathcal {I}_2 \\in \\mathcal {K}$ the following consistency constraint holds: $\\sum _{ \\mathbf {x} _{\\mathcal {I}_1 \\setminus \\mathcal {I}_2}} p( \\mathbf {x} _{\\mathcal {I}_1})&= \\sum _{ \\mathbf {x} _{\\mathcal {I}_2 \\setminus \\mathcal {I}_1}} p( \\mathbf {x} _{\\mathcal {I}_2})\\quad \\text{(for all $ \\mathbf {x} _{\\mathcal {I}_1 \\cap \\mathcal {I}_2}$)} \\ .$ Let us comment on these special properties of $\\bigl \\lbrace p( \\mathbf {x}_{\\mathcal {I}} ) \\bigr \\rbrace _{\\mathcal {I} \\in \\mathcal {K}}$ : It turns out that these special properties of $\\bigl \\lbrace p( \\mathbf {x}_{\\mathcal {I}} ) \\bigr \\rbrace _{\\mathcal {I} \\in \\mathcal {K}}$ are at the heart of quantum mechanical phenomena like Hardy's paradox [12] and the Frauchiger–Renner paradox [13].For a discussion of the latter in terms of SQMFs, see [8].",
"In fact, the Q-FG $ \\mathsf {N}_{4} $ in Fig.",
"REF can be used to analyze Hardy's paradox.",
"On the side, note that $ \\mathsf {N}_{4} $ also captures the essence of Bell's game [14].",
"Interestingly, these special properties of $\\bigl \\lbrace p( \\mathbf {x}_{\\mathcal {I}} ) \\bigr \\rbrace _{\\mathcal {I} \\in \\mathcal {K}}$ are very similar to the properties of the beliefs in the local marginal polytope of an S-FG (see, e.g., [15]).Local marginal polytopes are of relevance, for example, when characterizing locally operating message-passing iterative algorithms like the sum-product algorithm [16], [17].",
"The above observations motivate the systematic study of the collection $\\bigl \\lbrace p( \\mathbf {x}_{\\mathcal {I}} ) \\bigr \\rbrace _{\\mathcal {I} \\in \\mathcal {K}}$ for a given SQMF.",
"Indeed, one key contribution of this paper is to study this collection for the Q-FG $ \\mathsf {N}_{4} $ in Fig.",
"REF and compare this collection with other objects that can be associated with this Q-FG."
],
[
"Contributions",
"To better understand classicable variables' marginals, we define the set $ \\mathcal {M} ( \\mathsf {N}_{4} ) $ , which is the set of the marginals created by the classicable variables in the two-qubit system $ \\mathsf {N}_{4} $ , as shown in Fig.",
"REF .",
"One of our paper's main topics is to fully characterize $ \\mathcal {M} ( \\mathsf {N}_{4} ) $ .",
"For comparison, we introduce $ \\mathcal {LM}(\\mathcal {K}) $ (the local marginal polytope of the S-FG $ \\mathsf {N}_{1} $ in Fig.",
"REF ), $ \\mathcal {M} ( \\mathsf {N}_{1} ) $ (the set of realizable marginals of $ \\mathsf {N}_{1} $ ), $ \\mathcal {M} ( \\mathsf {N}_{2} ) $ (the set of realizable marginals of the Markov chain $ \\mathsf {N}_{2} $ in Fig.",
"REF ), and $ \\mathcal {M} ( \\mathsf {N}_{3} ) $ (the set of realizable marginals of $ \\mathsf {N}_{3} $ in Fig.",
"REF ).",
"We have the following results.",
"We prove the Venn diagram in Fig.",
"REF by showing that each part in the diagram is non-empty.",
"We can see that $ \\mathcal {M} ( \\mathsf {N}_{3} ) $ and $ \\mathcal {M} ( \\mathsf {N}_{4} ) $ are strict subsets of $ \\mathcal {LM}(\\mathcal {K}) $ ; both $ \\mathcal {M} ( \\mathsf {N}_{1} ) $ and $ \\mathcal {M} ( \\mathsf {N}_{2} ) $ have marginals that are not in $ \\mathcal {M} ( \\mathsf {N}_{4} ) $ ; the set $ \\mathcal {M} ( \\mathsf {N}_{4} ) $ consists of marginals that are not compatible with any joint PMF.",
"We generalize the Clauser-Horne-Shimony-Holt (CHSH) inequality [18] for Pearson correlation coefficients (PCCs), which resolves a conjecture proposed in [19].",
"Because PCCs are non-linear functions with respect to marginals, the inequality has a non-trivial proof.",
"We suspect that the proof approach is applicable for proving other non-linear Bell inequalities.",
"A violation of this inequality indicates that the associated marginals are not in $ \\mathcal {M} ( \\mathsf {N}_{3} ) $ .",
"We illustrate Hardy's paradox, Bell's game, and the maximum quantum violation of the PCC-based CHSH inequality by the classicable variables in $ \\mathsf {N}_{4} $ in Fig.",
"REF .",
"Besides these specific results, our paper is, more generally, about leveraging tools from factor graphs to understand certain quantities of interest in quantum information processing.",
"In particular, given that factor graphs have been proven very useful in classical information processing, but can also be used for doing quantum information processing, they allow one to understand and appreciate the similarities and the differences between classical and quantum information processing.",
"The rest of this paper is structured as follows.",
"Section reviews some basics of S-FGs.",
"In particular, Section REF proves the PCC-based CHSH inequality, and Section REF discusses the Markov chain in Fig.",
"REF .",
"Section introduces $ \\mathsf {N}_{4} $ , proves the Venn diagram in Fig.",
"REF and illustrates the maximum quantum violation of the PCC-based CHSH inequality.",
"Many details are left out due to space constraints; a more detailed discussion is given in [20]."
],
[
"Basic Notations and Definitions",
"The sets $\\mathbb {R}$ , $\\mathbb {R}_{\\ge 0}$ , $\\mathbb {R}_{>0}$ , and $\\mathbb {C}$ denote the field of real numbers, the set of nonnegative real numbers, the set of positive real numbers, and the field of complex numbers, respectively.",
"An overline denotes complex conjugation.",
"For any statement $ S $ , by the Iverson's convention, the function $ [S] $ is defined to be $ [ S ] := 1 $ if $ S $ is true and $ [ S ] := 0 $ otherwise."
],
[
"Standard Normal Factor Graphs (S-NFGs) ",
"In this section, we review some basic concepts and properties of an S-NFG.",
"The word “normal” refers to the fact that variables are arguments of only one or two local functions.",
"We use an example to introduce the fundamental concepts of an S-NFG first.",
"Example 4 [1], [3] Consider the multivariate function $& g_{ \\mathsf {N}_{1} }(x_{1},\\ldots ,x_{4}) \\\\&\\quad : = f_{1,2}(x_{1}, x_{2})\\cdot f_{1,4}(x_{1}, x_{4}) \\cdot f_{3,2}(x_{3}, x_{2})\\cdot f_{3,4}(x_{3}, x_{4}),$ where $g_{ \\mathsf {N}_{1} }$ , the so-called global function, is defined to be the product of the so-called local functions $f_{1,2}$ , $f_{1,4}$ , $f_{3,2}$ and $f_{3,4}$ .",
"We can visualize the factorization of $g$ with the help of the S-FG $ \\mathsf {N}_{1} $ in Fig.",
"REF .",
"Note that the S-FG $ \\mathsf {N}_{1} $ consists of four function nodes $f_{1,2}, \\ldots , f_{3,4}$ and four (full) edges with associated variables $x_{1}, \\ldots , x_{4}$ .",
"For an S-NFG, a half edge is an edge incident on one function node only and a full edge is an edge incident on two function nodes.",
"Definition 5 The S-NFG $\\mathsf {N}( \\mathcal {F} ( \\mathsf {N} ), \\mathcal {E} ( \\mathsf {N} ), \\mathcal {X} ( \\mathsf {N} ))$ consists of: The graph $( \\mathcal {F} ( \\mathsf {N} ), \\mathcal {E} ( \\mathsf {N} ) )$ with vertex set $ \\mathcal {F} ( \\mathsf {N} )$ and edge set $ \\mathcal {E} ( \\mathsf {N} )$ , where $ \\mathcal {E} ( \\mathsf {N} )$ consists of all full edges and half edges in $\\mathsf {N}$ .",
"With some slight abuse of notation, an $f \\in \\mathcal {F} ( \\mathsf {N} )$ will denote a function node and the corresponding local function.",
"The alphabet $ \\mathcal {X} ( \\mathsf {N} ) :=\\prod _{e\\in \\mathcal {E} ( \\mathsf {N} )} \\mathcal {X} _{e}$ , where $ \\mathcal {X} _{e}$ is the alphabet associated with the edge $e\\in \\mathcal {E} ( \\mathsf {N} )$ .",
"Definition 6 Given $ \\mathsf {N}( \\mathcal {F} ( \\mathsf {N} ), \\mathcal {E} ( \\mathsf {N} ), \\mathcal {X} ( \\mathsf {N} )) $ , we make the following definitions: For every function node $f\\in \\mathcal {F} ( \\mathsf {N} )$ , the set $\\partial f$ is the set of edges incident on $f$ .",
"An assignment $ \\mathbf {x} :=(x_{e})_{e\\in \\mathcal {E} ( \\mathsf {N} )}\\in \\mathcal {X} ( \\mathsf {N} )$ is called a configuration of the S-NFG.",
"The local function $f$ associated with function node $f\\in \\mathcal {F} ( \\mathsf {N} )$ denotes an arbitrary mapping $ f: \\prod _{e \\in \\partial f} \\mathcal {X} _{e} \\rightarrow \\mathbb {R}_{\\ge 0}.",
"$ The global function is $ g_{ \\mathsf {N} }( \\mathbf {x} ) :=\\prod _{ f\\in \\mathcal {F} (\\mathsf {N}) }f( \\mathbf {x}_{\\partial f} ).",
"$ The partition function is $ Z( \\mathsf {N} ) := \\sum _{ \\mathbf {x} } g_{ \\mathsf {N} }( \\mathbf {x} ), $ where $ \\sum _{ \\mathbf {x} } $ denotes $ \\sum _{ \\mathbf {x} \\in \\mathcal {X} ( \\mathsf {N} ) } $ .",
"The PMF induced on $ \\mathsf {N} $ is $ p_{ \\mathsf {N} } ( \\mathbf {x} ):= g_{ \\mathsf {N} }( \\mathbf {x} )/ Z( \\mathsf {N} ) .",
"$ Let $\\mathcal {I}$ be a subset of $ \\mathcal {E} ( \\mathsf {N} )$ and let $\\mathcal {I}^{\\mathrm {c}} := \\mathcal {E} ( \\mathsf {N} ) \\setminus \\mathcal {I}$ be its complement.",
"The marginal $ p_{\\mathsf {N},\\mathcal {I}}( \\mathbf {x} _{\\mathcal {I}}) $ is defined to be $ p_{\\mathsf {N},\\mathcal {I}}( \\mathbf {x} _{\\mathcal {I}}):= \\sum _{ \\mathbf {x}_{\\mathcal {I}^{\\mathrm {c}}} } p_{ \\mathsf {N} } ( \\mathbf {x} )$ .",
"Definition 7 Considering $ \\mathsf {N} \\in \\lbrace \\mathsf {N}_{1} , \\mathsf {N}_{2} , \\mathsf {N}_{3} \\rbrace $ , we make the following definitions: The alphabet $ \\mathcal {X} _{e} $ is $ \\mathcal {X} _{e} := \\lbrace 0, 1\\rbrace $ for all $ e \\in \\mathcal {E} ( \\mathsf {N} ) $ .",
"The set $ \\mathcal {K} $ is $ \\mathcal {K} := \\lbrace \\lbrace 1,2\\rbrace , \\lbrace 1,4\\rbrace , \\lbrace 2,3 \\rbrace , \\lbrace 3,4 \\rbrace \\rbrace $ .",
"For $ \\lbrace i, j \\rbrace \\in \\mathcal {K} $ , the marginal $ p_{\\mathsf {N}, i,j } $ is defined to be a $ | \\mathcal {X} _{e} | $ -by-$ | \\mathcal {X} _{e} | $ matrix with the entry $ p_{\\mathsf {N}, i,j } ( x_{i}, x_{j} ) $ and the marginal $ p_{\\mathsf {N}, i } $ is defined to be a $ | \\mathcal {X} _{e} | $ -by-$ | \\mathcal {X} _{e} | $ diagonal matrix with $ p_{\\mathsf {N}, i } ( x_{i} ) $ being the $ x_{i} $ -th diagonal term.",
"The collection of matrices $ \\mathbf {\\beta } $ is defined to be $ \\mathbf {\\beta } := \\bigl ( ( \\mathbf {\\beta } _{i,j} )_{\\lbrace i, j \\rbrace \\in \\mathcal {K}}, ( \\mathbf {\\beta } _{i} )_{i\\in \\mathcal {E}( \\mathsf {N}_{1} ) } \\bigr ) $ .",
"In particular, the matrix $ \\mathbf {\\beta } _{i,j} $ is defined to be a $ | \\mathcal {X} _{e} | $ -by-$ | \\mathcal {X} _{e} | $ matrix with entry $ \\beta _{i,j}( x_{i}, x_{j} ) \\in \\mathbb {R}_{\\ge 0} $ and the matrix $ \\mathbf {\\beta } _{i} $ is defined to be a $ | \\mathcal {X} _{e} | $ -by-$ | \\mathcal {X} _{e} | $ diagonal matrix with $ \\beta _{i}( x_{i} ) \\in \\mathbb {R}_{\\ge 0} $ being the $ x_{i} $ -th diagonal term.",
"The set of realizable marginals of $ \\mathsf {N} $ is defined to be $\\!\\!\\!\\!\\!\\!\\!\\!",
"\\mathcal {M} (\\mathsf {N}) :=\\left\\lbrace \\mathbf {\\beta } \\left|\\begin{array}{l}\\text{there exists an $ \\mathcal {F} (\\mathsf {N}) $such that}\\\\ \\mathbf {\\beta } _{i,j} = p_{\\mathsf {N}, i,j } , \\ \\mathbf {\\beta } _{i} = p_{\\mathsf {N}, i } , \\ \\lbrace i,j \\rbrace \\in \\mathcal {K}\\end{array}\\right.\\right\\rbrace .$ The set $ \\mathcal {LM}(\\mathcal {K}) $ is defined to be $\\!\\!\\!\\!\\!",
"\\mathcal {LM}(\\mathcal {K}) :=\\left\\lbrace \\mathbf {\\beta } \\left|\\begin{array}{l}0 \\le \\beta _{i,j}( x_{i}, x_{j} ) \\le 1,\\ \\forall x_{i}, x_{j}, i,j \\\\\\sum _{ x_{j} }\\beta _{i,j}( x_{i}, x_{j} ) = \\beta _{i}( x_{i} ),\\ \\forall x_{i}, i\\\\\\sum _{ x_{i} }\\beta _{i,j}( x_{i}, x_{j} ) = \\beta _{j}( x_{j} ),\\ \\forall x_{j}, j\\\\\\sum _{ x_{i} } \\beta _{i}( x_{i} ) = 1,\\ \\forall i\\end{array}\\right.\\right\\rbrace .$ The set $ \\mathcal {LM}(\\mathcal {K}) $ is essentially the local marginal polytope of the S-NFG $ \\mathsf {N}_{1} $ in Fig.",
"REF .",
"The definition of the local marginal polytope for an S-NFG is given in [15].",
"For each $ \\mathbf {\\beta } \\in \\mathcal {LM}(\\mathcal {K}) $ and $ \\lbrace i,j \\rbrace \\in \\mathcal {K} $ , each marginal $ \\mathbf {\\beta } _{i,j} $ can be used to represent the PMF for two random variables $ Y_{1}, Y_{2} \\in \\mathcal {X} _{e} $ by setting the probability $ \\mathrm {Pr} ( Y_{1} = x_{i}, Y_{2} = x_{j} )= \\beta _{i,j}(x_{i}, x_{j}) $ , $ x_{i}, x_{j} \\in \\mathcal {X} _{e} $ .",
"The functions $ \\mathrm {Cov} ( Y_{1}, Y_{2} ) $ , $ \\mathrm {Var} ( Y_{1} ) $ , $ \\mathrm {Var} ( Y_{2} ) $ are defined to be the covariance of $ Y_{1} $ and $ Y_{2} $ , and the variances of $ Y_{1} $ and $ Y_{2} $ , respectively.",
"When $ \\mathrm {Var} ( Y_{1} ), \\mathrm {Var} ( Y_{2} ) > 0$ , the PCC of $ Y_{1} $ and $ Y_{2} $ is defined to be $ \\mathrm {Corr} ( \\mathbf {\\beta } _{i,j} ) := \\mathrm {Cov} ( Y_{1}, Y_{2} )/ \\sqrt{ \\mathrm {Var} ( Y_{1} ) \\cdot \\mathrm {Var} ( Y_{2} ) }.",
"$ When there is no ambiguity, we use short-hands $ ( \\cdot )_{i,j} $ , $ ( \\cdot )_{i} $ , $ \\sum _{x_{i}} $ , and $\\lbrace \\cdot \\rbrace _{x_{i}} $ for $ ( \\cdot )_{\\lbrace i,j \\rbrace \\in \\mathcal {K} } $ , $ ( \\cdot )_{i\\in \\mathcal {E}( \\mathsf {N}_{1} ) } $ , $ \\sum _{x_{i}\\in \\mathcal {X} _{e} } $ , and $\\lbrace \\cdot \\rbrace _{x_{i}\\in \\mathcal {X} _{e} } $ , respectively.",
"Because $ \\mathcal {LM}(\\mathcal {K}) $ is a convex set by definition, Carathéodory's theorem [21] states that each element in $ \\mathcal {LM}(\\mathcal {K}) $ can be written as a convex combination of the vertices in $ \\mathcal {LM}(\\mathcal {K}) $ .",
"The full list of the vertices in $ \\mathcal {LM}(\\mathcal {K}) $ is given in [20].",
"Proposition 8 For $ \\lbrace i,j \\rbrace \\in \\mathcal {K} $ and $ 0 < \\beta _{i}(0), \\beta _{j}(0) < 1 $ , the PCC $ \\mathrm {Corr} ( \\mathbf {\\beta } _{i,j} ) $ satisfies $ \\mathrm {Corr} ( \\mathbf {\\beta } _{i,j} ) &= \\frac{ \\det ( \\mathbf {\\beta } _{i,j} )}{ \\sqrt{ \\det ( \\mathbf {\\beta } _{i} ) \\cdot \\det ( \\mathbf {\\beta } _{j} )} }.$ The requirement $ 0 < \\beta _{i}(0), \\beta _{j}(0) < 1 $ ensures that $ \\det ( \\mathbf {\\beta } _{i} ), \\det ( \\mathbf {\\beta } _{j} ) > 0 $ , and thus $ \\mathrm {Corr} ( \\mathbf {\\beta } _{i,j} ) $ is well-defined.",
"See the proof of [20].",
"Definition 9 Suppose that $ \\mathbf {\\beta } \\in \\mathcal {LM}(\\mathcal {K}) $ and $ 0 < \\beta _{i}(0) < 1,\\ i \\in \\mathcal {E}( \\mathsf {N}_{1} ) $ , we define $& \\mathrm {CorrCHSH}( \\mathbf {\\beta } ) := \\nonumber \\\\&\\quad \\ \\mathrm {Corr} ( \\mathbf {\\beta } _{1,2} ) + \\mathrm {Corr} ( \\mathbf {\\beta } _{1,4} )+ \\mathrm {Corr} ( \\mathbf {\\beta } _{3,2} )- \\mathrm {Corr} ( \\mathbf {\\beta } _{3,4} ).$"
],
[
"Properties for $ \\mathsf {N}_{3} $",
"In this subsection, we prove inequalities with respect to $ \\mathrm {CorrCHSH}( \\mathbf {\\beta } ) $ for $ \\mathbf {\\beta } \\in \\mathcal {M} ( \\mathsf {N}_{3} ) $ .",
"These inequalities genuinely are (nonlinear) Bell inequalities [22] in the usual sense.",
"By definition, it holds that $ \\mathcal {M} ( \\mathsf {N}_{1} ) \\subseteq \\mathcal {M} ( \\mathsf {N}_{3} ), \\qquad \\mathcal {M} ( \\mathsf {N}_{2} ) \\subseteq \\mathcal {M} ( \\mathsf {N}_{3} ),$ so any inequality that holds for all $ \\mathbf {\\beta } \\in \\mathcal {M} ( \\mathsf {N}_{3} ) $ also holds for all $ \\mathbf {\\beta } \\in \\mathcal {M} ( \\mathsf {N}_{1} ) \\cup \\mathcal {M} ( \\mathsf {N}_{2} ) $ .",
"Theorem 10 For any $ \\mathbf {\\beta } \\in \\mathcal {M} ( \\mathsf {N}_{3} ) $ such that $ 0 < \\beta _{i}(0) < 1 $ for all $ i \\in \\mathcal {E}( \\mathsf {N}_{1} ) $ , we have $| \\mathrm {CorrCHSH}( \\mathbf {\\beta } ) |&< 2\\sqrt{2}.$ We prove it by contradiction.",
"On the one hand, the set $ \\mathcal {M} ( \\mathsf {N}_{3} ) $ consists of marginals for binary random variables only.",
"On the other hand, to have $ \\mathrm {CorrCHSH}( \\mathbf {\\beta } ) = 2 \\sqrt{2} $ for some $ \\mathbf {\\beta } \\in \\mathcal {M} ( \\mathsf {N}_{3} ) $ , the PMF realizing $ \\mathbf {\\beta } $ needs to be the joint PMF for random variables with alphabet size greater than two.",
"For details, see the proof in [20].",
"The main idea in the proof of Theorem REF can be used to verify whether a proposed bound for a function with binary random variables is achievable.",
"It is different from the idea in the proof of the upcoming Theorem REF .",
"Theorem 11 For any $ \\mathbf {\\beta } \\in \\mathcal {M} ( \\mathsf {N}_{3} ) $ such that $ 0 < \\beta _{i}(0) < 1 $ for all $ i \\in \\mathcal {E}( \\mathsf {N}_{1} ) $ , we have $ | \\mathrm {CorrCHSH}( \\mathbf {\\beta } ) | \\le 5/2.",
"$ We give a proof sketch here.",
"For details, see the proof in [20].",
"Consider a subset of $ \\mathcal {LM}(\\mathcal {K}) $ such that in this subset, $ 0 < \\beta _{i}(0) < 1 $ for $ i \\in \\mathcal {E}( \\mathsf {N}_{1} ) $ , and the elements in $ \\mathbf {\\beta } $ satisfy the original linear CHSH inequality.",
"Denote this set as $ \\mathcal {LM}_{\\mathrm {CHSH}}(\\mathcal {K}) $ .",
"We have $ \\mathcal {M} ( \\mathsf {N}_{3} ) \\subsetneq \\mathcal {LM}_{\\mathrm {CHSH}}(\\mathcal {K}) $ .",
"Find a $ \\mathbf {\\beta } ^{*} \\in \\mathcal {M} ( \\mathsf {N}_{3} )$ such that $ \\mathrm {CorrCHSH}( \\mathbf {\\beta } ^{*}) = 5/2.",
"$ We formulate an optimization problem where $ \\mathrm {CorrCHSH}( \\mathbf {\\beta } ) $ is maximized over $ \\mathbf {\\beta } \\in \\mathcal {LM}_{\\mathrm {CHSH}}(\\mathcal {K}) $ such that $ \\mathbf {\\beta } $ has a similar structure as $ \\mathbf {\\beta } ^{*} $ , e.g., having the same number of zero entries in $ ( \\mathbf {\\beta } _{i,j})_{i,j} $ .",
"Note that this optimization problem has linear constraints only, which helps determine the optimal solution.",
"We prove $ \\mathrm {CorrCHSH}( \\mathbf {\\beta } ) \\le 5/2 $ in this case.",
"We generalize the proof for all $ \\mathbf {\\beta } \\in \\mathcal {LM}_{\\mathrm {CHSH}}(\\mathcal {K}) $ .",
"The proof of $ \\mathrm {CorrCHSH}( \\mathbf {\\beta } ) \\ge -5/2 $ is similar.",
"Theorem REF proves the conjecture stated in [19].",
"The key idea of the proof is that we consider $ \\mathcal {LM}_{\\mathrm {CHSH}}(\\mathcal {K}) $ instead of $ \\mathcal {M} ( \\mathsf {N}_{3} ) $ .",
"Suppose that we want to prove $ \\mathrm {CorrCHSH}( \\mathbf {\\beta } ) \\le 5/2 $ for $ \\mathbf {\\beta } \\in \\mathcal {M} ( \\mathsf {N}_{3} ) $ directly.",
"Because $ \\mathcal {M} ( \\mathsf {N}_{3} ) $ is a convex set, for any $ \\mathbf {\\beta } \\in \\mathcal {M} ( \\mathsf {N}_{3} ) $ , the marginal $ \\mathbf {\\beta } _{i,j} $ can be written as a convex combination of some joint PMF for $ X_{1},\\ldots ,X_{4} $ , i.e., $ \\lbrace p_{ \\mathsf {N}_{3} }( \\mathbf {x} ) \\rbrace _{ \\mathbf {x} } $ , which makes the expression of $ \\mathrm {CorrCHSH}( \\mathbf {\\beta } ) $ non-trivial.",
"By considering a superset of $ \\mathcal {M} ( \\mathsf {N}_{3} ) $ , i.e., $ \\mathcal {LM}_{\\mathrm {CHSH}}(\\mathcal {K}) $ , we can simplify $ \\mathrm {CorrCHSH}( \\mathbf {\\beta } ) $ .",
"We suspect that this idea can be generalized in the proof of other non-linear Bell inequalities."
],
[
"Markov Chain in Fig. ",
"In this subsection, we consider the Markov chain $ \\mathsf {N}_{2} $ in Fig.",
"REF .",
"Theorem 12 For the Markov chain $ \\mathsf {N}_{2} $ in Fig.",
"REF , we have $ \\mathrm {Corr} ( \\mathbf {\\beta } _{3,4} ) = \\mathrm {Corr} ( \\mathbf {\\beta } _{3,2} )\\cdot \\mathrm {Corr} ( \\mathbf {\\beta } _{1,2} )\\cdot \\mathrm {Corr} ( \\mathbf {\\beta } _{1,4} ).$ See [23].",
"Corollary 13 For the Markov chain $ \\mathsf {N}_{2} $ in Fig.",
"REF , it holds that $ | \\mathrm {Corr} ( \\mathbf {\\beta } _{3,4} ) | \\le | \\mathrm {Corr} ( \\mathbf {\\beta } _{1,2} ) | \\le 1.",
"$ It can be proven using Theorem REF and $ | \\mathrm {Corr} ( \\mathbf {\\beta } _{i,j} )| \\le 1 $ for $ \\lbrace i,j \\rbrace \\in \\mathcal {K} $ .",
"We prove another variation of the PCC-based CHSH inequality for $ \\mathsf {N}_{2} $ .",
"Corollary 14 For the Markov chain $ \\mathsf {N}_{2} $ in Fig.",
"REF , we have $\\bigl | & \\mathrm {Corr} ( \\mathbf {\\beta } _{1,2} ) + \\mathrm {Corr} ( \\mathbf {\\beta } _{2,4} )+ \\mathrm {Corr} ( \\mathbf {\\beta } _{1,3} ) - \\mathrm {Corr} ( \\mathbf {\\beta } _{3,4} ) \\bigr |\\le 2.$ See the proof of [20]."
],
[
"Quantum-Probability Normal Factor Graphs (Q-NFGs)",
"This section considers a quantum system represented by the Q-NFG $ \\mathsf {N}_{4} $ in Fig.",
"REF .",
"Such Q-NFGs have been discussed thoroughly in [7], [8].",
"Note that in Fig.",
"REF and Fig.",
"REF , the row index of a matrix is marked by a black dot.",
"The details of $ \\mathsf {N}_{4} $ are shown in [20].",
"Proposition 15 For any $ \\lbrace i, j \\rbrace \\in \\mathcal {K} $ , the variables $ \\tilde{x} _{i} $ and $ \\tilde{x} _{j} $ are jointly classicable, which implies that the marginals $ q_{i,j}( \\tilde{x} _{i}, \\tilde{x} _{j} ) $ and $ q_{i}( \\tilde{x} _{i} ) $ are non-negative real numbers for any $ \\tilde{x} _{i}, \\tilde{x} _{j} \\in \\mathcal {X} _{e} ^{2} $ .",
"It can be proven directly by Definition REF .",
"Then we define the set of realizable marginals of $ \\mathsf {N}_{4} $ based on the jointly classicable variables $ \\tilde{x} _{i} $ and $ \\tilde{x} _{j} $ for all $ \\lbrace i, j \\rbrace \\in \\mathcal {K} $ .",
"Definition 16 With $ \\tilde{0} := (0,0) $ , $ \\tilde{1} := (1,1) $ , and $ \\lbrace i, j \\rbrace \\in \\mathcal {K} $ , the matrices $ \\mathbf {q} _{i,j} $ and $ \\mathbf {q} _{i} $ induced by $ q_{ \\mathsf {N}_{4} } $ are defined to be $ \\mathbf {q} _{i,j} &:=\\begin{pmatrix}q_{i,j}( \\tilde{0} , \\tilde{0} )& q_{i,j}( \\tilde{0} , \\tilde{1} ) \\\\q_{i,j}( \\tilde{1} , \\tilde{0} )& q_{i,j}( \\tilde{1} , \\tilde{1} )\\end{pmatrix}, \\ \\mathbf {q} _{i} :=\\begin{pmatrix}q_{i}( \\tilde{0} ) & 0 \\\\0 & q_{i}( \\tilde{1} )\\end{pmatrix}.\\nonumber $ The set of realizable marginals of $ \\mathsf {N}_{4} $ is defined to be the set $ \\mathcal {M} ( \\mathsf {N}_{4} ) :=\\left\\lbrace \\mathbf {\\beta } \\left| \\mathbf {\\beta } _{i,j} = \\mathbf {q} _{i,j}, \\ \\mathbf {\\beta } _{i} = \\mathbf {q} _{i},\\ \\lbrace i, j \\rbrace \\in \\mathcal {K}\\right.",
"\\right\\rbrace $ , which is not the set of quantum correlations in the usual Bell nonlocality sense.",
"Proposition 17 For any $ \\mathbf {\\beta } \\in \\mathcal {M} ( \\mathsf {N}_{4} ) $ , there exist matrices $ \\rho $ , $ U_{1} $ , and $ U_{2} $ such that $\\!\\!\\!\\beta _{i,j}(x_{i}, x_{j}) &=( ( A_{i,x_{i}} \\otimes B_{j,x_{j}} )\\cdot \\rho \\cdot ( A_{i,x_{i}} \\otimes B_{j,x_{j}} )^{ \\mathsf {H} } ),$ for all $ x_{i}, x_{j} \\in \\mathcal {X} _{e} $ and $\\lbrace i,j \\rbrace \\in \\mathcal {K}$ , where $&A_{i,x_{i}} := E_{x_{i}} \\cdot U_{1}^{[i = 3]},\\ B_{j,x_{j}} := E_{x_{j}} \\cdot U_{2}^{[j = 4]}, \\\\&E_{x_{i}}( y_{i}, y_{i}^{\\prime } ) :=[ y_{i} = x_{i}] \\cdot [y_{i} = y_{i}^{\\prime } ], \\ x_{i}, y_{i}, y_{i}^{\\prime } \\in \\mathcal {X} _{e} .$ Note that the set $ \\lbrace E_{x_{i}} \\rbrace _{x_{i}} $ denotes the measurement of a single qubit in the computational basis.",
"Then we have $\\sum _{x_{i} \\in \\mathcal {X} _{e} } A_{i,x_{i}}^{ \\mathsf {H} }\\cdot A_{i,x_{i}} &=\\sum _{x_{j} \\in \\mathcal {X} _{e} } B_{j,x_{j}}^{ \\mathsf {H} }\\cdot B_{j,x_{j}} = I, \\quad \\lbrace i, j \\rbrace \\in \\mathcal {K},$ which shows that both $ \\lbrace A_{i,x_{i}} \\rbrace _{x_{i}} $ and $ \\lbrace B_{j,x_{j}} \\rbrace _{x_{j}} $ are sets of measurement matrices with binary outcomes $ x_{i} $ and $ x_{j} $ , respectively.",
"It can be proven directly.",
"After closing the dashed box in Fig.",
"REF , i.e., summing over the variables inside the box, we obtain (REF ).",
"Proposition 18 There exists a $ \\mathbf {\\beta } \\in \\mathcal {M} ( \\mathsf {N}_{4} ) $ such that $ | \\mathrm {Corr} ( \\mathbf {\\beta } _{3,4} )| > | \\mathrm {Corr} ( \\mathbf {\\beta } _{1,2} )| $ .",
"See [20].",
"Compared with Corollary REF , Proposition REF implies that $ \\mathcal {M} ( \\mathsf {N}_{4} ) $ provides extra $ \\mathbf {\\beta } $ that is not in $ \\mathcal {M} ( \\mathsf {N}_{2} ) $ .",
"Theorem 19 The Venn diagram in Fig.",
"REF holds.",
"See the proof of [20].",
"We make some remarks on the Venn diagram in Fig.",
"REF : On the one hand, the set $ \\mathcal {M} ( \\mathsf {N}_{4} ) $ provides extra marginals that are not in $ \\mathcal {M} ( \\mathsf {N}_{3} ) $ .",
"For example, by introducing entanglement in the quantum system, one can obtain a set of incompatible marginals (see, e.g., [20]).",
"On the other hand, the sets $ \\mathcal {M} ( \\mathsf {N}_{1} ) $ , $ \\mathcal {M} ( \\mathsf {N}_{2} ) $ , and $ \\mathcal {M} ( \\mathsf {N}_{3} ) $ also consist of marginals that are not in $ \\mathcal {M} ( \\mathsf {N}_{4} ) $ .",
"Proposition 20 For $ \\mathbf {\\beta } \\in \\mathcal {M} ( \\mathsf {N}_{4} ) $ satisfying $ 0 < \\beta _{i}(0) < 1 $ for all $ i \\in \\mathcal {E}( \\mathsf {N}_{1} ) $ , we have $ | \\mathrm {CorrCHSH}( \\mathbf {\\beta } ) | \\le 2\\sqrt{2} $ .",
"See [19].",
"Proposition 21 Hardy's paradox [12] and Bell's game [14] can be illustrated via the classicable variables induced by the SQMF $ q_{ \\mathsf {N}_{4} }( \\tilde{\\mathbf {x}} ) $ .",
"In Bell's game, we have $ \\mathrm {CorrCHSH}( \\mathbf {\\beta } ) = 2\\sqrt{2} $ , which also realizes the maximum quantum violation of $ \\mathrm {CorrCHSH}( \\mathbf {\\beta } ) $ as proven in Proposition REF .",
"See [20]."
]
] | 2107.01816 |
[
[
"Gauge Theory and the Analytic Form of the Geometric Langlands Program"
],
[
"Abstract We present a gauge-theoretic interpretation of the \"analytic\" version of the geometric Langlands program, in which Hitchin Hamiltonians and Hecke operators are viewed as concrete operators acting on a Hilbert space of quantum states.",
"The gauge theory ingredients required to understand this construction -- such as electric-magnetic duality between Wilson and 't Hooft line operators in four-dimensional gauge theory -- are the same ones that enter in understanding via gauge theory the more familiar formulation of geometric Langlands, but now these ingredients are organized and applied in a novel fashion."
],
[
"Introduction",
"Geometric Langlands duality as originally formulated by Beilinson and Drinfeld [1] is a relationship between categories associated to moduli spaces of fields on a Riemann surface $C$ .",
"Many ingredients that enter in formulating and analyzing this duality are familiar in quantum field theory.",
"In particular, two-dimensional conformal field theory plays a prominent role, as reviewed in [2].",
"The relation of geometric Langlands duality to quantum field theory can be understood more fully by formulating the subject in terms of a twisted version of ${\\mathcal {N}}=4$ super Yang-Mills theory [3].",
"From that point of view, geometric Langlands duality is deduced from electric-magnetic duality between the ${\\mathcal {N}}=4$ theory for a compact gauge group $G$ and the same theory based on the Langlands or GNO dual group $G^\\vee $ (the complexifications of these groups will be called $G_{\\mathbb {C}}$ and $G^\\vee _{\\mathbb {C}}$ ).",
"Twisting of ${\\mathcal {N}}=4$ super Yang-Mills theory produces a four-dimensional topological field theory, which naturally [5], [6], [7] assigns a number (the partition function) to a four-manifold, a vector space (the space of physical states) to a three-manifold, and a category of branes or boundary conditions to a two-manifold.This idealized description does not take into account the fact that the complex Lie groups $G_{\\mathbb {C}}$ and $G^\\vee _{\\mathbb {C}}$ and the associated moduli spaces are not compact.",
"Because of this noncompactness, one likely gets only a partial topological field theory.",
"Branes and spaces of physical states can be defined, but it is not clear that the integrals that formally would define partition functions associated to four-manifolds can really be defined satisfactorily.",
"(A somewhat similar situation arises in Donaldson theory of four-manifolds, though the details are quite different: one does not get a complete topological field theory, as the partition function cannot be suitably defined for all four-manifolds.)",
"The noncompactness also means that to define branes and spaces of physical states, one has to specify the allowed asymptotic behavior of a brane or a wavefunction.",
"Each (reasonable) choice leads to a different version of the duality.",
"Possibilities include the Betti and de Rham versions [4].",
"The usual geometric Langlands duality is a duality between the categories associated to two-manifolds.",
"Recently an analytic version of geometric Langlands duality has been discovered by Etingof, Frenkel, and Kazhdan [8], [9], [10], stimulated in part by a number of mathematical [11] and physical [12] developments.",
"Rather than categories and functors acting on categories, one considers a Hilbert space of quantum states and self-adjoint operators such as quantum Hitchin Hamiltonians acting on this Hilbert space.",
"Very roughly, the usual formulation of geometric Langlands duality involves deformation quantization of the algebra of holomorphic functions on the moduli space ${\\mathcal {M}}_H(G,C)$ of $G$ Higgs bundles on a Riemann surface $C$ , while the analytic version of the theory involves ordinary quantization of the same moduli space, viewed now as a real symplectic manifold.",
"As usual, quantization means that a suitable class of smooth functions – in general neither holomorphic nor antiholomorphic – become operators on a Hilbert space.",
"See also [14], [19], [15], [13], [16], [18], [20], [17] for prior work on the gauge theory interpretation of the spectrum of quantum Hitchin Hamiltonians.",
"The goal of the present article is to place the analytic version of geometric Langlands duality in the gauge theory framework.",
"The first step is simply to understand what one should do in that framework to study the quantization of ${\\mathcal {M}}_H$ viewed as a real symplectic manifold, as opposed to its deformation quantization when viewed as a complex symplectic manifold.",
"The basic idea here is that the problem of quantization of a real symplectic manifold $M$ is part of the $A$ -model of a suitable complexification $Y$ of $M$ (if such a $Y$ exists) [21].",
"We have explored this construction in more detail elsewhere as background to the present article [22].",
"For the application to geometric Langlands, we want to quantize the Higgs bundle moduli space ${\\mathcal {M}}_H$ viewed as a real symplectic manifold with one of its real symplectic structures.",
"A suitable complexification of ${\\mathcal {M}}_H$ is simply the product of two copies of ${\\mathcal {M}}_H$ with opposite complex structures.",
"With this as the starting point for understanding the quantization of ${\\mathcal {M}}_H$ , we will show that the analytic version of geometric Langlands can be understood by assembling in a novel fashion the same gauge theory ingredients that have been used previously for understanding the more traditional version of geometric Langlands.",
"The organization of this article is as follows.",
"In Section , we explain the basic setup for formulating the analytic version of geometric Langlands duality in terms of ${\\mathcal {N}}=4$ super Yang-Mills theory.",
"In Section , we explain the predictions of electric-magnetic duality for the joint eigenvalues of the Hitchin Hamiltonians, and in Section , we analyze Hecke or 't Hooft operators and the dual Wilson operators.",
"In Section , we show that the joint eigenfunctions of the Hitchin Hamiltonians satisfy a quantum-deformed WKB condition.",
"In Section , we discuss the quantization of real forms of the Higgs bundle moduli space.",
"Starting in Section , we use chiral algebras that arise at junctions between supersymmetric boundary conditions to explore the analytic version of geometric Langlands duality.",
"In Section , we study Hecke operators from the point of view of these chiral algebras.",
"In Section , we explore in detail an example with remarkable properties.",
"Some further issues are treated in appendices.",
"In Appendix , we explain that electric-magnetic duality together with positivity of the Hilbert space inner product in quantization of ${\\mathcal {M}}_H$ imply that the intersections of the varieties $L_{\\mathrm {op}}$ and $L_{\\overline{\\mathrm {op}}}$ that parametrize holomorphic and antiholomorphic opers must be isolated and transverse (as conjectured in [8] and proved in some cases).",
"We further point out that electric-magnetic duality implies a natural normalization for the joint eigenfunctions of the Hitchin Hamiltonians and argue that the Hilbert space norm of a normalized wavefunction is given by, roughly speaking, the torsion of the associated oper bundle.",
"In Appendix , we explore some examples of differential equations satisfied by line operators.",
"In Appendix , we construct a local model to analyze the singular behavior of the eigenfunctions of the Hitchin Hamiltonians along the divisor of not very stable bundles.",
"Throughout this article, $C$ is a Riemann surface of genus $g>1$ .",
"All considerations can be naturally extended to the case of bundles with parabolic structure, but for simplicity we will omit this generalization (which has been developed in [8], [9], [10]).",
"With sufficiently many parabolic points, there is a quite similar theory for genus $g=0,1$ .",
"The cases $g=0,1$ without parabolic structure (or with too few parabolic points for $g=0$ ) require a different treatment because any low energy description requires gauge fields, not just $\\sigma $ -model fields."
],
[
"Basic Setup",
"This section is devoted to an explanation of the basic framework in which we will study the analytic version of geometric Langlands, and a review of part of the background."
],
[
"Quantization Via Branes",
"Here we briefly summarize some things that have been described more fully elsewhere [21], [22].",
"Let $Y$ be a complex symplectic manifold, with complex structure $I$ and holomorphic symplectic form $\\Omega $ .",
"We view $Y$ as a real symplectic manifold with the real symplectic form $\\omega _Y={\\mathrm {Im}}\\,\\Omega $ .",
"We assume that $Y$ is such that a quantum $\\sigma $ -model with target $Y$ exists (as an ultraviolet-complete quantum field theory), and we consider the $A$ -model obtained by twisting this $\\sigma $ -model in a standard way.",
"It is not known in general for what class of $Y$ 's the $\\sigma $ -model does exist, but a sufficient condition is believed to be that the complex symplectic structure of $Y$ can be extended to a complete hyper-Kahler structure.",
"The example important for the present article is the case that $Y$ is the Higgs bundle moduli space ${\\mathcal {M}}_H$ , which indeed admits a complete hyper-Kahler metric [23].",
"In general, the $A$ -model of a symplectic manifold $Y$ , in addition to the usual Lagrangian branes whose support is middle-dimensional in $Y$ , can have coisotropic branes, supported on a coisotropic submanifold of $Y$ that is above the middle dimension [24].",
"The simplest case and the only case that we will need in the present article is a rank 1 coisotropic $A$ -brane whose support is all of $Y$ .",
"Let ${\\sf B}$ be the $B$ -field of the $\\sigma $ -model, and consider a brane with support $Y$ whose Chan-Paton or ${\\mathrm {CP}}$ bundle is a line bundle ${\\mathcal {L}}\\rightarrow Y$ , with a unitary connection ${\\sf A}$ of curvature ${\\sf F}=\\mathrm {d}{\\sf A}$ .",
"The Kapustin-Orlov condition for this data to define an $A$ -brane is that ${\\mathcal {I}}=\\omega ^{-1}({\\sf F}+{\\sf B})$ should be an integrable complex structure on $Y$ .",
"Two solutions of this condition immediately present themselves.",
"One choice is ${\\sf F}+{\\sf B}={\\mathrm {Re}}\\,\\Omega $ , ${\\mathcal {I}}=I$ .",
"The brane constructed this way is what we will call the canonical coisotropic $A$ -brane, ${\\mathcal {B}}_{\\mathrm {cc}}$ .",
"A second choice is ${\\sf F}+{\\sf B}=-{\\mathrm {Re}}\\,\\Omega $ , ${\\mathcal {I}}=-I$ .",
"This leads to what we will call the conjugate canonical coisotropic $A$ -brane, ${\\overline{\\mathcal {B}}}_{\\mathrm {cc}}$ .",
"To treat the two cases symmetrically, in the present article it is convenient to takeThis choice was made in [21].",
"However, when only one of ${\\mathcal {B}}_{\\mathrm {cc}}$ , ${\\overline{\\mathcal {B}}}_{\\mathrm {cc}}$ is relevant, it can be simpler to take ${\\sf F}=0$ , ${\\sf B}={\\mathrm {Re}}\\,\\Omega $ .",
"${\\sf B}=0$ , ${\\sf F}=\\pm {\\mathrm {Re}}\\,\\Omega $ .",
"This choice is only possible if there exists a complex line bundle ${\\mathcal {L}}\\rightarrow Y$ with curvature ${\\mathrm {Re}}\\,\\Omega $ .",
"In our application to the Higgs bundle moduli space, ${\\mathrm {Re}}\\,\\Omega $ is cohomologically trivial, so ${\\mathcal {L}}$ exists and can be assumed to be topologically trivial.",
"Now consider the algebra ${\\mathcal {A}}={\\mathrm {Hom}}({\\mathcal {B}}_{\\mathrm {cc}},{\\mathcal {B}}_{\\mathrm {cc}})$ (which in physical terms is the space of $({\\mathcal {B}}_{\\mathrm {cc}},{\\mathcal {B}}_{\\mathrm {cc}})$ open strings, with an associative multiplication that comes by joining of open strings).",
"Suppose that $\\Omega =\\Omega _0/\\hbar $ , where we keep $\\Omega _0$ fixed and vary $\\hbar $ .",
"For $\\hbar \\rightarrow 0$ , ${\\mathcal {A}}$ reduces to the commutative algebra ${\\mathcal {A}}_0$ of holomorphic functions on $Y$ in complex structure $I$ .",
"In order $\\hbar $ , the multiplication law in ${\\mathcal {A}}$ differs from the commutative multiplication law in ${\\mathcal {A}}_0$ by the Poisson bracket $\\lbrace f,g\\rbrace =(\\Omega ^{-1})^{ij}\\partial _i f\\partial _j g$ .",
"So ${\\mathcal {A}}$ can be viewed as a deformation quantization of ${\\mathcal {A}}_0$ .",
"$\\overline{\\mathcal {A}}={\\mathrm {Hom}}({\\overline{\\mathcal {B}}}_{\\mathrm {cc}},{\\overline{\\mathcal {B}}}_{\\mathrm {cc}})$ is related in the same way to the commutative algebra $\\overline{\\mathcal {A}}_0$ of holomorphic functions on $Y$ in complex structure $-I$ , or equivalently antiholomorphic functions in complex structure $I$ , and can be viewed as a deformation quantization of $\\overline{\\mathcal {A}}_0$ .",
"Generically, deformation quantization is a formal procedure that has to be defined over a ring of formal power series in $\\hbar $ .",
"When a quantum $\\sigma $ -model of $Y$ exists, it is expected that $\\hbar $ can be set to a complex value (such as 1), rather than being treated as a formal power series variable.",
"In the present example, one can give a more direct explanation of this.",
"The Higgs bundle moduli space ${\\mathcal {M}}_H(G,C)$ has a ${\\mathbb {C}}^*$ symmetry, rescaling the Higgs field by $\\varphi \\rightarrow \\lambda \\varphi $ , $\\lambda \\in {\\mathbb {C}}^*$ .",
"This operation rescales $\\Omega $ in the same way.",
"Of course, this is possible only because $\\Omega $ is cohomologically trivial.",
"The ring ${\\mathcal {A}}_0$ is generated by functions that scale with a definite degree (namely the Hitchin Hamiltonians) and the scaling symmetry implies that all the relations in the deformed ring ${\\mathcal {A}}$ are polynomials in the deformation parameter $\\hbar $ .",
"Hence it makes sense to set $\\hbar =1$ .",
"Similar remarks will apply when we consider quantization rather than deformation quantization, given that the important branes considered are ${\\mathbb {C}}^*$ -invariant: the scaling symmetry will imply that certain semiclassical formulas are actually exact.",
"Deformation quantization is particularly interesting if $Y$ is an affine variety, with lots of holomorphic functions.",
"The example of $Y={\\mathcal {M}}_H$ studied in the present article is far from being an affine variety.",
"The ring ${\\mathcal {A}}_0$ in this case is simply the ring of functions on the base of the Hitchin fibration; in other words, the global holomorphic functions on ${\\mathcal {M}}_H$ are simply the functions of Hitchin's Poisson-commuting Hamiltonians.",
"Hitchin's Poisson-commuting Hamiltonians can be quantized to commuting differential operators, acting on sections of the line bundle $K^{1/2}$ , where $K$ is the canonical bundle of ${\\mathcal {M}}_H$ .",
"This was shown by Hitchin [25] for $G={\\mathrm {SU}}(2)$ and by Beilinson and Drinfeld [1] in general.",
"In particular, the ring ${\\mathcal {A}}$ is commutative.",
"The fact that the Poisson-commuting classical Hamiltonians can be quantized to commuting differential operators can also be seen in the gauge theory language, as we will discuss in Section REF .",
"In the present article, we are really interested in quantization rather than deformation quantization.",
"How to modify the story just described to encompass quantization has been explained in [21], [22].",
"Suppose that $M$ is a real symplectic manifold, with symplectic form $\\omega _M$ , that we wish to quantize.",
"If $M$ has a complexification $Y$ that obeys certain conditions, then quantization of $M$ is part of the $A$ -model of $Y$ .",
"$Y$ should be a complex symplectic manifold with a holomorphic symplectic form $\\Omega $ whose restriction to $M$ is $\\omega _M$ .",
"Moreover, $Y$ should have an antiholomorphic involutionAn involution is an automorphism whose square is 1.",
"$\\tau $ with $M$ as a component of its fixed point set.",
"These conditions imply that $\\tau ^*\\Omega =\\overline{\\Omega }$ .",
"Finally, the quantum $\\sigma $ -model of $Y$ must exist.",
"Under these conditions, a choice of a prequantum line bundle ${{{\\mathfrak {L}}}}\\rightarrow M$ , in the sense of geometric quantization,A prequantum line bundle over a symplectic manifold $M$ with symplectic form $\\omega $ is a complex line bundle ${{\\mathfrak {L}}}\\rightarrow M$ with a unitary connection of curvature $\\omega $ .",
"determines an $A$ -brane ${\\mathcal {B}}$ with support $M$ .",
"(The details of this depend on the choice that was made in satisfying the condition ${\\sf F}+{\\sf B}={\\mathrm {Re}}\\,\\Omega $ to define the brane ${\\mathcal {B}}_{\\mathrm {cc}}$ .",
"We will be specific in Section REF .)",
"The $A$ -model answer for the Hilbert space obtained by quantizing $M$ with symplectic structure $\\omega _M$ and prequantum line bundle ${{\\mathfrak {L}}}$ is ${\\mathcal {H}}={\\mathrm {Hom}}({\\mathcal {B}},{\\mathcal {B}}_{\\mathrm {cc}})$ .",
"The definition of the hermitian inner product on ${\\mathcal {H}}$ is described in Section REF .",
"${\\mathcal {H}}$ is always a module for ${\\mathcal {A}}={\\mathrm {Hom}}({\\mathcal {B}}_{\\mathrm {cc}},{\\mathcal {B}}_{\\mathrm {cc}})$ .",
"This can be described by saying that those functions on $M$ that can be analytically continued to holomorphic functions on $Y$ are quantized to give operators on ${\\mathcal {H}}$ .",
"In addition, under mild conditions, a correspondenceA correspondence between $M$ and itself is simply a Lagrangian submanifold of $M_1\\times M_2$ , where $M_1$ and $M_2$ are two copies of $M$ , with respective symplectic structures $\\omega _M$ and $-\\omega _M$ .",
"A holomorphic correspondence between $Y$ and itself is defined similarly.",
"between $M$ and itself that can be analytically continued to a holomorphic correspondence between $Y$ and itself can be quantized in a natural way to give an operator on ${\\mathcal {H}}$ .",
"In the problem that we will be studying with $M={\\mathcal {M}}_H$ , the ring ${\\mathcal {A}}$ will be relatively small, consisting of polynomials in the holomorphic and antiholomorphic Hitchin Hamiltonians.",
"By way of compensation, there is an ample supply of correspondences – the Hecke correspondences – that will provide additional operators on ${\\mathcal {H}}$ .",
"Quantizing these correspondences is simple because of the ${\\mathbb {C}}^*$ scaling symmetry that was invoked in the discussion of deformation quantization.",
"We have described this in the context of two-dimensional $\\sigma $ -models, as is appropriate for quantization of a fairly general real symplectic manifold $M$ .",
"However, in the particular case that $M$ is the Higgs bundle moduli space ${\\mathcal {M}}_H(G,C)$ for gauge group $G$ on a Riemann surface $C$ , a four-dimensional picture is available and gives much more complete understanding.",
"For this, the starting point is ${\\mathcal {N}}=4$ super Yang-Mills theory, in four dimensions, with gauge group $G$ .",
"One restricts to four-manifolds of the form $\\Sigma \\times C$ , where $\\Sigma $ is an arbitrary two-manifold but $C$ is kept fixed.",
"At low energies, the four-dimensional gauge theory reduces for many purposes, assuming that $G$ has trivial center, to a supersymmetric $\\sigma $ -model on $\\Sigma $ with target ${\\mathcal {M}}_H(G,C)$ [26], [27].",
"A certain twisting of the ${\\mathcal {N}}=4$ theory in four dimensions produces a topological field theoryThis is really a partial topological field theory, as remarked in footnote REF .",
"that, when we specialize to four-manifolds of the form $\\Sigma \\times C$ , reduces to an $A$ -model on $\\Sigma $ with target ${\\mathcal {M}}_H(G,C)$ .",
"The brane ${\\mathcal {B}}_{\\mathrm {cc}}$ of the $\\sigma $ -model originates in the four-dimensional gauge theory as a boundary condition that is a simple deformation of Neumann boundary conditions for the gauge field, extended to the whole supermultiplet in a half-BPS fashion; see Section 12.4 of [3].",
"The four-dimensional lift of the ${\\mathcal {B}}_{\\mathrm {cc}}$ boundary condition has an interesting and important feature.",
"Though it is a brane in an $A$ -model that has full four-dimensional topological symmetry, the definition of the brane ${\\mathcal {B}}_{\\mathrm {cc}}$ depends on a choice of complex structure of $C$ .",
"The deformed Neumann boundary condition is a holomorphic-topological local boundary condition for the four-dimensional topological field theory.",
"Concretely, the boundary condition breaks some of the bulk supersymmetries of the physical theory.",
"As a consequence, some translation generators which are $Q$ -exact in the bulk cease to be $Q$ -exact in the presence of the boundary and the boundary condition is not topological in the four-dimensional sense.",
"Another explanation of the dependence of ${\\mathcal {B}}_{\\mathrm {cc}}$ on a choice of complex structure is simply that ${\\mathcal {B}}_{\\mathrm {cc}}$ is defined with ${\\sf B}=\\omega _J$ , whose definition depends on the complex structure of $C$ .",
"See the beginning of Section REF for a statement of which structures on ${\\mathcal {M}}_H$ do or do not depend on a choice of complex structure.",
"Boundary local operators supported at a point $p\\in C$ can depend holomorphically on $p$ , even though they depend topologically on the one remaining boundary direction.",
"This will be important whenever we discuss the four-dimensional lift of our constructions.",
"Although the $A$ -model has full four-dimensional topological invariance, in the presence of the brane ${\\mathcal {B}}_{\\mathrm {cc}}$ , only two-dimensional topological invariance is available.",
"For example, in proving the commutativity of the Hitchin Hamiltonians and the Hecke operators, we will use only two-dimensional topological invariance.",
"If $G$ has a nontrivial center ${\\mathcal {Z}}(G)$ , then the assertion about a reduction to a $\\sigma $ -model must be slightly modified.",
"A more precise statement is that upon compactification on $C$ , the four-dimensional gauge theory reduces, for many purposes, to the product of a $\\sigma $ -model with target ${\\mathcal {M}}_H(G,C)$ and a gauge theory with the finite gauge group ${\\mathcal {Z}}(G)$ (acting trivially on ${\\mathcal {M}}_H(G,C)$ ).",
"This finite gauge group will play no role until we want to compute the eigenvalues of Hitchin Hamiltonians and Hecke operators, so in much of the following it will not be mentioned.",
"A purely two-dimensional description via a $\\sigma $ -model with target ${\\mathcal {M}}_H(G,C)$ (possibly extended by the finite gauge group) is useful for many purposes.",
"But information is lost in the reduction to two dimensions, and the four-dimensional picture is needed for a complete account of the duality.To be more precise, a complete formulation of the duality is possible in four dimensions.",
"A fuller explanation of the duality comes from a certain starting point in six dimensions [28].",
"In standard mathematical treatments, an analogous statement is that geometric Langlands duality must be formulated in terms of the “stack” of $G_{\\mathbb {C}}$ -bundles or of $G_{\\mathbb {C}}^\\vee $ local systems, rather than in terms of a finite-dimensional moduli space.",
"There is a simple relation between the two statements.",
"It was shown by Atiyah and Bott [29] that the space of all $G$ -valued connections on a smooth $G$ -bundle $E\\rightarrow C$ , with the action of the group of complexified gauge transformations, provides a model of the stack of holomorphic $G_{\\mathbb {C}}$ -bundles.",
"That is because the $(0,1)$ part of any connection gives $E$ (or more precisely its complexification) a complex structure, making it a holomorphic $G_{\\mathbb {C}}$ bundle over $E$ .",
"A gauge field on a $G$ -bundle over $ \\Sigma \\times C$ determines, in particular, a family, parametrized by $\\Sigma $ , of gauge fields on $C$ .",
"So any such connection determines a map from $\\Sigma $ to the stack of $G_{\\mathbb {C}}$ bundles over $C$ .",
"Thus ${\\mathcal {N}}=4$ super Yang-Mills theory on $\\Sigma \\times C$ can be understood as a supersymmetric $\\sigma $ -model on $\\Sigma $ with the target being the stack of $G_{\\mathbb {C}}$ bundles over $C$ .",
"(In this formulation, the theory is a two-dimensional supersymmetric gauge theory on $\\Sigma $ coupled to matter fields.",
"The matter fields are gauge fields on $C$ and their superpartners, and the gauge group is the group $\\widehat{G}$ of maps of $C$ to the finite-dimensional group $G$ .",
"A gauge transformation in the two-dimensional theory is a map from $\\Sigma $ to $\\widehat{G}$ ; in the four-dimensional description, this is interpreted as a map of the four-manifold $\\Sigma \\times C$ to $G$ .",
"To keep these statements simple, we have assumed that all bundles are trivialized.)",
"The mathematical statement that the correct formulation involves stacks corresponds to the quantum field theory statement that the correct formulation is in four dimensions."
],
[
"Quantizing A Complex Manifold",
"We view ${\\mathcal {M}}_H(G,C)$ as a complex manifold in the complex structure, called $I$ by Hitchin [23], in which it parametrizes Higgs bundles over $C$ .",
"$I$ is one of a triple of complex structures $I,J,K$ that, along with the corresponding Kahler forms $\\omega _I,\\omega _J,\\omega _K$ , make a hyper-Kahler structure on $C$ .",
"In complex structure $J$ , ${\\mathcal {M}}_H(G,C)$ parametrizes flat bundles over $C$ with structure group $G_{\\mathbb {C}}$ .",
"Complex structure $J$ and the corresponding holomorphic symplectic form $\\Omega _J=\\omega _K+{\\mathrm {i}}\\omega _I$ are topological in the sense that they depend on $C$ only as an oriented two-manifold, while the other structures $I$ , $K$ , and $\\omega _J$ depend on a choice of complex structure on $C$ .",
"The complex symplectic form of ${\\mathcal {M}}_H(G,C)$ , in complex structure $I$ , is $\\Omega _I=\\omega _J+{\\mathrm {i}}\\omega _K$ .",
"We want to view ${\\mathcal {M}}_H$ as a real symplectic manifold with the real symplectic structure $\\omega =\\omega _J={\\mathrm {Re}}\\,\\Omega _I$ , and quantize it.",
"As we have explained in Section REF , the first step in quantizing any real symplectic manifold $M$ via branes is to pick a suitable complexification of it.",
"Here we are in a special situation: $M$ is actually a complex symplectic manifold $Y$ , with complex structure $I$ and complex symplectic form $\\Omega $ , and we want to quantize $Y$ with the real symplectic structure $\\omega _Y={\\mathrm {Re}}\\,\\Omega $ .",
"In such a case, there is a standard way to proceed, described in Section 5 of [22].",
"We set $\\widehat{Y}=Y_1\\times Y_2$ , where $Y_1$ and $Y_2$ are two copies of $Y$ , with opposite complex structures $I$ and $-I$ .",
"The complex structure of $\\widehat{Y}$ is thus a direct sum ${\\mathcal {I}}=I\\oplus (-I)$ .",
"The involution $\\tau :\\widehat{Y}\\rightarrow \\widehat{Y}$ that exchanges the two factors is antiholomorphic, and its fixed point set is a copy of $Y$ , embedded as the diagonal in $\\widehat{Y}=Y_1\\times Y_2$ .",
"We endow $\\widehat{Y}$ with the complex symplectic form $\\widehat{\\Omega }=\\frac{1}{2}\\Omega \\boxplus \\frac{1}{2}\\overline{\\Omega }$ ; in other words, the symplectic form of $\\widehat{Y}=Y_1\\times Y_2$ is $\\frac{1}{2}\\Omega $ on the first factor and $\\frac{1}{2}\\overline{\\Omega }$ on the second factor.",
"This definition ensures that the restriction of $\\widehat{\\Omega }$ to the diagonal is ${\\mathrm {Re}}\\,\\Omega $ .",
"Suppose that the complex symplectic structure of $Y$ can be extended to a complete hyper-Kahler metric.",
"Then the complex symplectic structure of $\\widehat{Y}$ can likewise be extended to a complete hyper-Kahler metric, namely a product hyper-Kahler metric on $\\widehat{Y}=Y_1\\times Y_2$ .",
"So all conditions are satisfied, and the $A$ -model of $\\widehat{Y}$ with real symplectic form ${\\mathrm {Im}}\\,\\widehat{\\Omega }=\\frac{1}{2}{\\mathrm {Im}}\\,\\Omega \\boxplus (-\\frac{1}{2}{\\mathrm {Im}}\\,\\Omega )$ is suitable for quantizing $Y$ with the symplectic structure $\\omega _Y={\\mathrm {Re}}\\,\\Omega $ .",
"To define coisotropic branes, we have to satisfy the Kapustin-Orlov condition that $\\omega ^{-1} ({\\sf F}+{\\sf B})$ should be an integrable complex structure.",
"In doing so, we will take ${\\sf B}=0$ , as this will make it possible to treat the two factors of $\\widehat{Y}$ symmetrically.",
"Since the complex structures of $Y_1$ and $Y_2$ are respectively $I$ and $-I$ and the $A$ -model symplectic structures are respectively $\\frac{1}{2}{\\mathrm {Im}}\\,\\Omega $ and $-\\frac{1}{2}{\\mathrm {Im}}\\,\\Omega $ , we can define canonical coisotropic branes ${\\mathcal {B}}_{{\\mathrm {cc}},1}$ and ${\\mathcal {B}}_{{\\mathrm {cc}},2}$ on $Y_1$ and $Y_2$ by taking in each case a ${\\mathrm {CP}}$ bundle ${\\mathcal {L}}$ with curvature ${\\sf F}=\\frac{1}{2}\\omega _J$ .",
"(In our application to the Higgs bundle moduli space, ${\\mathrm {Re}}\\,\\Omega $ is exact and $b_1(Y)=0$ , so a topologically trivial line bundle over $Y$ with curvature $\\frac{1}{2}\\omega _J$ exists and is unique up to isomorphism.)",
"One then defines on $\\widehat{Y}$ the product brane $\\widehat{{\\mathcal {B}}}_{\\mathrm {cc}}=\\widehat{{\\mathcal {B}}}_{{\\mathrm {cc}},1}\\times \\widehat{{\\mathcal {B}}}_{{\\mathrm {cc}},2}$ , with ${\\mathrm {CP}}$ bundle $\\widehat{{\\mathcal {L}}}={\\mathcal {L}}\\boxtimes {\\mathcal {L}}$ .",
"We also define a Lagrangian brane ${\\mathcal {B}}$ supported on $Y$ with trivial ${\\mathrm {CP}}$ bundle.",
"Following the general logic, the Hilbert space for quantization of $Y$ with the real symplectic structure ${\\mathrm {Re}}\\,\\Omega $ is ${\\mathcal {H}}={\\mathrm {Hom}}({\\mathcal {B}},\\widehat{{\\mathcal {B}}}_{\\mathrm {cc}})$ .",
"The prequantum line bundle in this situation is ${{\\mathfrak {L}}}=\\widehat{{\\mathcal {L}}}|_Y\\cong {\\mathcal {L}}^2$ .",
"Since ${\\mathcal {L}}$ has curvature $\\frac{1}{2}{\\mathrm {Re}}\\,\\Omega $ , ${\\mathcal {L}}^2$ has curvature ${\\mathrm {Re}}\\,\\Omega $ , so it is an appropriate prequantum line bundle for quantization of $Y$ with symplectic structure ${\\mathrm {Re}}\\,\\Omega $ .",
"Figure: (a) In the folded construction, we have two copies of a σ\\sigma -model on a strip.",
"The two copies are decoupled except on theright boundary, where they are glued together by a brane ℬ{\\mathcal {B}} that is supported on the diagonal in Y ^=Y 1 ×Y 2 \\widehat{Y}=Y_1\\times Y_2.",
"ℬ{\\mathcal {B}} has trivial CP {\\mathrm {CP}}bundle so this gluing of the two copies is its only effect.",
"(b) After unfolding,we have a single copy of the σ\\sigma -model on a strip of twice the width.",
"No trace of ℬ{\\mathcal {B}} remains.However, this definition has a useful variant.",
"To compute ${\\mathrm {Hom}}({\\mathcal {B}},\\widehat{{\\mathcal {B}}}_{\\mathrm {cc}})$ , we study the $\\sigma $ model with target $\\widehat{Y}$ on a strip $\\Sigma $ , with boundary conditions set by $\\widehat{{\\mathcal {B}}}_{\\mathrm {cc}}$ of the left boundary of the strip and by ${\\mathcal {B}}$ on the right boundary, as in fig.",
"REF (a).",
"The two factors of $\\widehat{Y}=Y_1\\times Y_2$ are decoupled in the bulk of the $\\sigma $ -model, since the metric on $\\widehat{Y}$ is a product; they are also decoupled on the left boundary, since the brane $\\widehat{{\\mathcal {B}}}_{\\mathrm {cc}}={\\mathcal {B}}_{{\\mathrm {cc}},1}\\times {\\mathcal {B}}_{{\\mathrm {cc}},2}$ is likewise a product.",
"So away from the right boundary of the strip, we can think of $\\Sigma $ as having two sheets, one of which is mapped to $Y_1$ and one to $Y_2$ , as in the figure.",
"The two sheets are coupled only on the right boundary, where, as ${\\mathcal {B}}$ is supported on the diagonal in $Y_1\\times Y_2$ , the two sheets are “glued together” and map to the same point in $Y$ .",
"This suggests that we should “unfold” the picture (fig.",
"REF (b)).",
"After this unfolding, we simply have a single sheet of twice the width that is mapped to a single copy of $Y$ .",
"Unfolding reverses the orientation of one of the two sheets of the folded picture, and this orientation reversal changes the sign of the $A$ -model symplectic form.",
"In the folded picture, the $A$ -model symplectic form was $\\frac{1}{2}\\omega _K$ on $Y_1$ and $-\\frac{1}{2}\\omega _K$ on $Y_2$ ; hence after unfolding, the $A$ -model symplectic form is $\\frac{1}{2}\\omega _K$ everywhere.",
"In other words, the unfolded picture involves the ordinary $A$ -model of a single copy of $Y$ with symplectic form $\\frac{1}{2}\\omega _K$ .",
"Before unfolding, the branes ${\\mathcal {B}}_{{\\mathrm {cc}},1}$ and ${\\mathcal {B}}_{{\\mathrm {cc}},2}$ both have ${\\mathrm {CP}}$ bundles with curvature $\\frac{1}{2} \\omega _J$ .",
"Reversing the orientation of $\\Sigma $ replaces the ${\\mathrm {CP}}$ bundle of a brane with its dual (or its inverse, in the rank 1 case), and so reverses the sign of the ${\\mathrm {CP}}$ curvature.",
"Hence in the unfolded picture, the boundaries are labeled by branes ${\\mathcal {B}}_{\\mathrm {cc}}$ and ${\\overline{\\mathcal {B}}}_{\\mathrm {cc}}$ whose respective ${\\mathrm {CP}}$ bundles are lines bundles ${\\mathcal {L}}$ and ${\\mathcal {L}}^{-1}$ with curvatures $\\frac{1}{2}\\omega _J$ and $-\\frac{1}{2}\\omega _J$ .",
"These are the conjugate canonical coisotropic branes that were introduced in Section REF .",
"The prequantum line bundle is still ${{\\mathfrak {L}}}={\\mathcal {L}}^2$ .",
"${\\mathcal {A}}={\\mathrm {Hom}}({\\mathcal {B}}_{\\mathrm {cc}},{\\mathcal {B}}_{\\mathrm {cc}})$ is a deformation quantization of the commutative algebra ${\\mathcal {A}}_0$ of holomorphic functions on $Y$ , and $\\overline{{\\mathcal {A}}}={\\mathrm {Hom}}({\\overline{\\mathcal {B}}}_{\\mathrm {cc}},{\\overline{\\mathcal {B}}}_{\\mathrm {cc}})$ is similarly a deformation quantization of the algebra $\\overline{\\mathcal {A}}_0$ of antiholomorphic functions on $Y$ .",
"What in the folded picture was ${\\mathcal {H}}={\\mathrm {Hom}}({\\mathcal {B}},\\widehat{{\\mathcal {B}}}_{\\mathrm {cc}})$ becomes in the unfolded picture ${\\mathcal {H}}={\\mathrm {Hom}}(\\overline{{\\mathcal {B}}}_{\\mathrm {cc}},{\\mathcal {B}}_{\\mathrm {cc}})$ .",
"With either description of ${\\mathcal {H}}$ , we need to define a hermitian product on ${\\mathcal {H}}$ .",
"For definiteness we use the folded language.For more detail on the following, see Section 2.7 of [22].",
"Topological field theory would give us in general a nondegenerate bilinear (not hermitian) pairing $(~,~)$ between ${\\mathcal {H}}={\\mathrm {Hom}}({\\mathcal {B}},\\widehat{{\\mathcal {B}}}_{\\mathrm {cc}})$ and its dual space ${\\mathcal {H}}^{\\prime }={\\mathrm {Hom}}(\\widehat{{\\mathcal {B}}}_{\\mathrm {cc}},{\\mathcal {B}})$ .",
"To get a hermitian pairing on ${\\mathcal {H}}$ , we need to compose this bilinear pairing with an antilinear map from ${\\mathcal {H}}$ to ${\\mathcal {H}}^{\\prime }$ .",
"Such a map in the underlying physical theory is provided by the $\\sf {CPT}$ symmetry $\\Theta $ .",
"But $\\Theta $ is not an $A$ -model symmetry; it maps the $A$ -model to a conjugate $A$ -model with the opposite sign of the symplectic form.",
"The involution $\\tau $ that exchanges the two factors of $\\widehat{Y}$ also exchanges the $A$ -model with its conjugate, since it is antisymplectic (it reverses the sign of the $A$ -model symplectic form), so $\\Theta _\\tau =\\Theta \\tau $ is an antilinear symmetry of the $A$ -model.",
"Finally, because the branes ${\\mathcal {B}}$ and ${\\mathcal {B}}_{\\mathrm {cc}}$ are $\\Theta _\\tau $ -invariant, $\\Theta _\\tau $ maps ${\\mathcal {H}}$ to ${\\mathcal {H}}^{\\prime }$ and we can define a nondegenerate hermitian pairing on ${\\mathcal {H}}$ by $\\langle \\psi ,\\psi ^{\\prime }\\rangle = (\\Theta _\\tau \\psi ,\\psi ^{\\prime }).$ For general $\\Theta _\\tau $ -invariant branes, such a pairing is not positive-definite.",
"For the specific case of quantizing a cotangent bundle, which is our main example on the $A$ -model side, one expects positivity.",
"The $B$ -model analog of this construction uses an antiholomorphic (not antisymplectic) involution $\\tau $ .",
"Positivity of the pairing in this case is subtle and is discussed in Appendix .",
"In the folded picture, $\\tau $ and therefore also $\\Theta _\\tau $ exchanges the two factors of $\\widehat{Y}=Y_1\\times Y_2$ ; in the unfolded picture, they exchange the two ends of the strip.",
"Exchanging the two ends of the strip reverses the orientation of the strip and therefore would change the sign of ${\\sf B}$ .",
"Hence in a description with ${\\sf B}\\ne 0$ , the definition of the inner product is less natural (one would need to accompany $\\Theta _\\tau $ with a $B$ -field gauge transformation).",
"That is why we took ${\\sf B}=0$ in solving the Kapustin-Orlov conditions for rank 1 coisotropic branes."
],
[
"Quantizing The Higgs Bundle Moduli Space",
"For our application to the case that $Y$ is the Higgs bundle moduli space ${\\mathcal {M}}_H(G,C)$ for some gauge group $G$ and Riemann surface $C$ , we really want to study the four-dimensional version of this construction.",
"This means that we study the ${\\mathcal {N}}=4$ super Yang-Mills theory, with gauge group $G$ , on $\\Sigma \\times C$ , where $\\Sigma $ is the strip of fig REF (b).",
"The boundary conditions on the left and right of the strip are set by the gauge theory versions of ${\\mathcal {B}}_{\\mathrm {cc}}$ and ${\\overline{\\mathcal {B}}}_{\\mathrm {cc}}$ .",
"A detailed description of ${\\mathcal {B}}_{\\mathrm {cc}}$ in four-dimensional gauge theory language was given in Section 12.4 of [3].",
"We can describe a Higgs bundle on $C$ by a pair $(A,\\phi )$ , where $A$ is gauge field, that is, a connection on a $G$ -bundle $E\\rightarrow G$ , and $\\phi $ is a 1-form valued in the adjoint bundle ${\\mathrm {ad}}(E)$ .",
"In this description, ${\\overline{\\mathcal {B}}}_{\\mathrm {cc}}$ is obtained from ${\\mathcal {B}}_{\\mathrm {cc}}$ by $(A,\\phi )\\rightarrow (A,-\\phi )$ (suitably extended to the rest of the four-dimensional supermultiplet).",
"This is a familiar involution of the Higgs bundle moduli space that acts holomorphically in complex structure $I$ and antiholomorphically in complex structures $J$ and $K$ .",
"Although we have used machinery of gauge theory and branes to construct a Hilbert space ${\\mathcal {H}}$ associated to quantization of ${\\mathcal {M}}_H(G,C)$ , the actual output of this construction is completely unsurprising.",
"A dense open set in ${\\mathcal {M}}_H(G,C)$ is a cotangent bundle $T^*{\\mathcal {M}}(G,C)$ , where ${\\mathcal {M}}$ is the moduli space of semistable holomorphic $G$ -bundles over $C$ .",
"Geometric quantization – or simply elementary quantum mechanics – suggests that the Hilbert space that we should associate to quantization of $T^*{\\mathcal {M}}(G,C)$ should be the space of ${\\mathrm {L}}^2$ half-densities on ${\\mathcal {M}}(G,C)$ .",
"The reason to speak of ${\\mathrm {L}}^2$ half-densities rather than ${\\mathrm {L}}^2$ functions is that on a general smooth manifold $N$ without some choice of a measure,The space ${\\mathcal {M}}(G,C)$ actually does have a natural measure, namely the one associated to its real symplectic structure when viewed as a moduli space of flat bundles over $C$ with compact structure group $G$ .",
"This is also the measure induced by its embedding in the hyper-Kahler manifold ${\\mathcal {M}}_H(G,C)$ .",
"However, this measure is not part of the $A$ -model and does not naturally appear in $A$ -model constructions such as the definition of the Hilbert space ${\\mathcal {H}}$ .",
"there is no way to integrate a function so there is no natural Hilbert space of ${\\mathrm {L}}^2$ functions.",
"A density on a manifold $N$ is a section of a trivial real line bundle ${\\sf K}$ and can be written locally in any coordinate system as $|\\mathrm {d}x^1\\mathrm {d}x^2\\cdots \\mathrm {d}x^w| \\,f(x^1,x^2,\\cdots , x^w)$ , where $f(x^1,\\cdots , x^w)$ is a function and $|\\mathrm {d}x^1\\mathrm {d}x^2\\cdots \\mathrm {d}x^w|$ is a measure, not a differential form.",
"${\\sf K}$ has a square root ${\\sf K}^{1/2}$ , also trivial, whose sections are locally described in a given coordinate system by functions $g(x^1,x^2,\\cdots , x^w) $ that transform under a change of coordinates in such a way that $|\\mathrm {d}x^1\\mathrm {d}x^2\\cdots \\mathrm {d}x^w| \\,g(x^1,x^2,\\cdots , x^w)^2$ is invariant.",
"It is convenient to formally write $ h=|\\mathrm {d}x^1\\mathrm {d}x^2\\cdots \\mathrm {d}x^w|^{1/2} g(x^1,x^2,\\cdots , x^w)$ for a section of ${\\sf K}^{1/2}$ .",
"Complex-valued half-densities, which are described by the same formula where locally $g$ is a complex-valued function, form a Hilbert space in an obvious way: $ ||h||^2=\\int |\\mathrm {d}x^1\\mathrm {d}x^2\\cdots \\mathrm {d}x^w|\\, |g(x^1,x^2\\cdots x^w)|^2.$ Now, motivated by the application to the complex manifold ${\\mathcal {M}}(G,C)$ , let us describe the bundle of densities or half-densities on a complex manifold $N$ .",
"If $N$ has complex dimension $n=w/2$ and local holomorphic coordinates $z^1,z^2,\\cdots , z^n$ , then $|\\mathrm {d}z^1 \\mathrm {d}z^2 \\cdots \\mathrm {d}z^n \\mathrm {d}\\overline{z}^1\\mathrm {d}\\overline{z}^2\\cdots \\mathrm {d}\\overline{z}^n|$ is a density on $N$ , in other words a section of ${\\sf K}\\rightarrow N$ .",
"On the other hand, $\\mathrm {d}z^1 \\mathrm {d}z^2 \\cdots \\mathrm {d}z^n $ is a section of the holomorphic canonical bundle $K\\rightarrow N$ , and $ \\mathrm {d}\\overline{z}^1\\mathrm {d}\\overline{z}^2\\cdots \\mathrm {d}\\overline{z}^n$ is a section of the complex conjugate line bundle $\\overline{K}\\rightarrow N$ (which can also be viewed as the canonical line bundle of $N$ if $N$ is endowed with the opposite complex structure).",
"So ${\\sf K}$ can be identified with $K\\otimes \\overline{K}$ ; more precisely $K\\otimes \\overline{K}\\cong {\\sf K}\\otimes _{\\mathbb {R}}{\\mathbb {C}}$ , that is, $K\\otimes \\overline{K}$ is the complexification of ${\\sf K}$ , the bundle of complex-valued densities.",
"Similarly $K$ always has a square root at least locally, and for any choice of local square root of $K$ , we have ${\\sf K}^{1/2}\\cong K^{1/2}\\otimes \\overline{K}{}^{1/2}$ ; more precisely $K^{1/2}\\otimes \\overline{K}^{1/2}\\cong {\\sf K}\\otimes _{\\mathbb {R}}{\\mathbb {C}}$ , that is, $K^{1/2}\\otimes \\overline{K}^{1/2}$ is the bundle of complex-valued half-densities.",
"As long as $\\overline{K}{}^{1/2}$ is the complex conjugate of $K^{1/2}$ , this relation holds for any choice of $K^{1/2}$ .",
"In Section 3 of [22], criteria were described under which brane quantization of $M=T^*N$ , with its standard symplectic structure as a cotangent bundle, leads to a Hilbert space of ${\\mathrm {L}}^2$ half-densities on $N$ .",
"Beyond requiring that $M=T^*N$ has a complexification $Y$ that is suitable for brane quantization, the necessary condition is that $Y$ should be the cotangent bundle of a complexification $W$ of $N$ (and $Y$ should have the natural complex symplectic structure of a cotangent bundle).",
"This condition is automatically satisfied if $M$ and $N$ are already complex manifolds and $Y$ is defined as the product of two copies of $M$ with opposite complex structures.",
"The Higgs bundle moduli space ${\\mathcal {M}}_H(G,C)$ contains $T^*{\\mathcal {M}}(G,C)$ as a dense open set, but is not actually isomorphic to $T^*{\\mathcal {M}}(G,C)$ .",
"One would not expect a measure zero discrepancy to be important in the definition of a Hilbert space of ${\\mathrm {L}}^2$ wavefunctions.",
"The construction in [22] maps the Hilbert space ${\\mathcal {H}}$ obtained in quantizing ${\\mathcal {M}}_H(G,C)$ to a space of half-densities on ${\\mathcal {M}}(G,C)$ without requiring that $T^*{\\mathcal {M}}(G,C)$ is literally all of ${\\mathcal {M}}_H(G,C)$ .",
"The Hilbert space ${\\mathcal {H}}$ of half-densities on ${\\mathcal {M}}(G,C)$ was already introduced in [8] without any reference to branes and $\\sigma $ -models or gauge theories.",
"The interpretation via branes makes it possible to apply electric-magnetic duality and other methods of gauge theory.",
"We will see an example next in discussing the Hitchin Hamiltonians."
],
[
"Hitchin Hamiltonians",
"As we have seen, in brane quantization of a complex manifold such as $Y={\\mathcal {M}}_H(G,C)$ , the Hilbert space has an unfolded description as ${\\mathcal {H}}={\\mathrm {Hom}}(\\overline{{\\mathcal {B}}}_{\\mathrm {cc}},{\\mathcal {B}}_{\\mathrm {cc}})$ .",
"${\\mathcal {H}}$ admits a left action ofIn the folded picture, we have instead a left action of both algebras ${\\mathrm {Hom}}({\\mathcal {B}}_{{\\mathrm {cc}},1},{\\mathcal {B}}_{{\\mathrm {cc}},1})$ and ${\\mathrm {Hom}}({\\mathcal {B}}_{{\\mathrm {cc}},2},{\\mathcal {B}}_{{\\mathrm {cc}},2})$ .",
"A left action of an algebra is the same as a right action of the opposite algebra.",
"(The notion of the opposite algebra is explained in Appendix C of [22].)",
"Unfolding reverses the orientation of one sheet in fig.",
"REF and hence replaces one of the two algebras with its opposite.",
"Of course, which algebra acts on the left and which on the right depends on some choices.",
"None of this will be important in the present article as ${\\mathcal {A}}$ and $\\overline{{\\mathcal {A}}}$ will be commutative and hence isomorphic to their opposites.",
"${\\mathcal {A}}={\\mathrm {Hom}}({\\mathcal {B}}_{\\mathrm {cc}},{\\mathcal {B}}_{\\mathrm {cc}})$ and a right action of $\\overline{{\\mathcal {A}}}={\\mathrm {Hom}}(\\overline{{\\mathcal {B}}}_{\\mathrm {cc}},\\overline{{\\mathcal {B}}}_{\\mathrm {cc}})$ .",
"${\\mathcal {A}}$ and $\\overline{{\\mathcal {A}}}$ are quantum deformations of the commutative rings ${\\mathcal {A}}_0$ and $\\overline{{\\mathcal {A}}}_0$ of holomorphic functions on $Y$ .",
"For a general $Y$ , these deformations can be noncommutative, but in the particular case of ${\\mathcal {M}}_H(G,C)$ , it turns out that ${\\mathcal {A}}$ and $\\overline{{\\mathcal {A}}}$ are commutative and hence there is no distinction between a left action and a right action.",
"Concretely, the ring ${\\mathcal {A}}_0$ of holomorphic functions on ${\\mathcal {M}}_H(G,C)$ in complex structure $I$ is simply the ring of functions on the base of the Hitchin fibration [23], [30].",
"Consider a solution $(A,\\phi )$ of Hitchin's equations, where $A$ is a connection on a $G$ -bundle $E\\rightarrow C$ and $\\phi $ is a 1-form valued in ${\\mathrm {ad}}(E)$ .",
"Let $\\varphi $ be the holomorphic Higgs field, that is, the $(1,0)$ part of $\\phi $ .",
"Hitchin's equations give $\\overline{\\partial }_A\\varphi =0$ .",
"So if ${\\mathcal {P}}$ be an invariant polynomial on the Lie algebra ${{\\mathfrak {g}}}_{\\mathbb {C}}$ of $G_{\\mathbb {C}}$ , homogeneous of some degree $s$ , then ${\\mathcal {P}}(\\varphi )$ is a holomorphic section of $K_C^s$ (with $K_C$ the canonical bundle of $C$ ; we will also set $T_C=K_C^{-1}$ ).",
"Given any $(0,1)$ -form $\\alpha $ on $C$ with values in $T_C^{s-1}$ , we can define $ H_{{\\mathcal {P}},\\alpha }=\\int _C \\alpha \\,{\\mathcal {P}}(\\varphi ).",
"$ This is a holomorphic function on ${\\mathcal {M}}_H$ and depends only on the cohomology class of $\\alpha $ in $H^1(C,T_C^{s-1})$ .",
"The dimension of $H^1(C,T_C^{s-1})$ is $(s+1)(g-1)$ , and this is the number of linearly independent functions $H_{{\\mathcal {P}},\\alpha }$ for a given ${\\mathcal {P}}$ .",
"For $G={\\mathrm {SU}}(2)$ , the ring ${\\mathcal {A}}_0$ of holomorphic functions on ${\\mathcal {M}}_H(G,C)$ is generated by the $H_{{\\mathcal {P}},\\alpha }$ where ${\\mathcal {P}}(\\varphi )={\\rm Tr}\\,\\varphi ^2$ .",
"More generally, a simple Lie group $G$ of rank $r$ has $r$ independent Casimir operators, corresponding to $r$ homogeneous polynomials ${\\mathcal {P}}_j$ , $j=1,\\cdots ,r$ of various degrees, and ${\\mathcal {A}}_0$ is generated by the $H_{{\\mathcal {P}}_j,\\alpha _j}$ .",
"For example, if $G={\\mathrm {SU}}(N)$ , we can take the generating polynomials to be ${\\rm Tr}\\,\\varphi ^{k}$ , $k=2,3,\\cdots ,N$ .",
"The functions $H_{{\\mathcal {P}}_j,\\alpha _j}$ are Poisson-commuting, since the holomorphic symplectic structure of ${\\mathcal {M}}_H(G,C)$ in complex structure $I$ is such that $\\varphi _z$ and $A_{\\overline{z}}$ are conjugate variables, and in particular any functions constructed from $\\varphi $ only (and not $A$ ) are Poisson-commuting.",
"These Poisson-commuting functions are the Hamiltonians of Hitchin's classical integrable system.",
"The quantum deformation from ${\\mathcal {A}}_0$ to ${\\mathcal {A}}$ is unobstructed in the sense that every element of ${\\mathcal {A}}_0$ can be quantum-deformed to an element of ${\\mathcal {A}}$ .",
"This statement means, concretely, that if ${\\mathcal {P}}$ is an invariant polynomial on ${\\mathfrak {g}}$ homogeneous of some degree $s$ , and $H_{{\\mathcal {P}},\\alpha }$ is a corresponding Hitchin Hamiltonian, then there is a differential operator $D_{{\\mathcal {P}},\\alpha }$ , acting on sections of $K^{1/2}\\rightarrow {\\mathcal {M}}(G,C)$ , whose leading symbol is equal to $H_{{\\mathcal {P}},\\alpha }$ .",
"The passage from $H_{{\\mathcal {P}},\\alpha }$ to $D_{{\\mathcal {P}},\\alpha }$ is not entirely canonical, since specifying the desired leading symbol of $D_{{\\mathcal {P}},\\alpha }$ leaves one free to add to $D_{{\\mathcal {P}},\\alpha }$ a globally-defined holomorphic differential operator of degree less than $s$ .",
"For $G_{\\mathbb {C}}={\\mathrm {SL}}(2,{\\mathbb {C}})$ , one can take ${\\mathcal {P}}$ to be of degree 2, and then the only globally-defined holomorphic differential operators of lower degree are the operators of degree 0 – the complex constants.",
"For groups of higher rank, in general there are more possibilities.",
"Mathematically, the fact that the deformation is unobstructed follows from the fact that, for any simple $G$ , $H^1({\\mathcal {M}}_H,{\\mathcal {O}})=0$ , by virtue of which there is no potential obstruction in the deformation.",
"A gauge theory version of this argument was given in Section 12.4 of [3].",
"From the point of view of the $\\sigma $ -model, or the underlying gauge theory, the deformation from $H_{{\\mathcal {P}},\\alpha }$ to $D_{{\\mathcal {P}},\\alpha }$ arises from an expansion in powers of $\\hbar $ .",
"This expansion terminates after finitely many steps, since we define ${\\mathcal {A}}_0$ to consist of functions whose restriction to a fiber of the cotangent bundle is a polynomial.",
"A specific definition of the $\\sigma $ -model or the gauge theory gives a specific recipe for passing from $H_{{\\mathcal {P}},\\alpha }$ to $D_{{\\mathcal {P}},\\alpha }$ , but this depends on data (such as a Riemannian metric on $C$ , not just a complex structure) that is not part of the $A$ -model.",
"Figure: (a) As a general statement in two-dimensional topological field theory, 𝒜= Hom (ℬ cc ,ℬ cc ){\\mathcal {A}}={\\mathrm {Hom}}({\\mathcal {B}}_{\\mathrm {cc}},{\\mathcal {B}}_{\\mathrm {cc}}) commutes with 𝒜 ¯= Hom (ℬ ¯ cc ,ℬ ¯ cc )\\overline{{\\mathcal {A}}}={\\mathrm {Hom}}(\\overline{{\\mathcal {B}}}_{\\mathrm {cc}},\\overline{{\\mathcal {B}}}_{\\mathrm {cc}})in acting on ℋ= Hom (ℬ ¯ cc ,ℬ cc ){\\mathcal {H}}={\\mathrm {Hom}}(\\overline{{\\mathcal {B}}}_{\\mathrm {cc}},{\\mathcal {B}}_{\\mathrm {cc}}), because elements a∈𝒜a\\in {\\mathcal {A}} and a ¯∈𝒜 ¯\\overline{a}\\in \\overline{{\\mathcal {A}}} are inserted on opposite boundaries.",
"Diffeomorphisminariance does not allow any natural notion of which is inserted “first.” (b) In general, in two-dimensional topological field theory, 𝒜{\\mathcal {A}} (and similarly𝒜 ¯\\overline{{\\mathcal {A}}}) can be noncommutative, because elements a,a ' ∈𝒜a,a^{\\prime }\\in {\\mathcal {A}} are inserted on the same boundary with a well-defined order, relativeto the boundary orientation.",
"However, in the present context there are two additional dimensions, comprising the Riemann surface CC, not drawnin the two-dimensional picture.",
"One can assume that aa and a ' a^{\\prime } have disjoint support in CC.",
"Hence they can be moved up and down pasteach other without singularity and must commute.Next we would like to explain why ${\\mathcal {A}}$ is commutative, like ${\\mathcal {A}}_0$ .",
"This was originally proved for ${\\mathrm {SU}}(2)$ by Hitchin [25] and for general simple $G$ by Beilinson and Drinfeld [1].",
"We will give a four-dimensional explanation similar to that in [3].",
"Of course the same considerations apply to $\\overline{{\\mathcal {A}}}$ .",
"As a warmup, we first explain why ${\\mathcal {A}}$ commutes with $\\overline{{\\mathcal {A}}}$ .",
"This is clear from the fact (fig.",
"REF (a)) that an element $a\\in {\\mathcal {A}}$ is inserted on the left boundary of the strip, while an element $\\overline{a}\\in \\overline{{\\mathcal {A}}}$ is inserted on the right boundary.",
"In two-dimensional topological field theory, we are free to slide these insertions up and down along the boundary independently.",
"There is no meaningful relative time-ordering between the two boundary insertions and they must commute.",
"By contrast, consider the insertion of a pair of elements $a,a^{\\prime }\\in {\\mathcal {A}}$ (fig REF (b)).",
"Here, as a general statement in two-dimensional topological field theory, there is a meaningful time-ordering between $a$ and $a^{\\prime }$ .",
"If we try to slide one of them past the other in time, there may be a discontinuity when they cross, and therefore in general we may have $aa^{\\prime }\\ne a^{\\prime }a$ .",
"However, in the present problem, we are really not in two dimensions but in four dimensions; there are two extra dimensions, making up the Riemann surface $C$ , that are not shown in the figure.",
"The definition of $H_{{\\mathcal {P}},\\alpha }$ in eqn.",
"(REF ) depended only on the cohomology class of $\\alpha $ in $H^1(C,T_C^{s-1})$ .",
"We can choose a representative with support in an arbitrarily selected small open ball in $C$ .",
"Therefore, when we consider a pair of elements $a,a^{\\prime }\\in {\\mathcal {A}}$ , we can assume that they are represented by operators that have disjoint support in $C$ .",
"Hence we can slide the two operators up and down past each other in the two-dimensional picture of fig.",
"REF (b) without any singularity.",
"Accordingly, they commute.",
"We can elaborate slightly on the four-dimensional origin of the quantum Hitchin hamiltonians.",
"The integrands ${\\mathcal {P}}(\\varphi )$ in the classical Hitchin Hamiltonians depend holomorphically on $C$ .",
"As we review in Section , the quantum Hitchin Hamiltonians $D_{{\\mathcal {P}},\\alpha }$ can also be written as $ D_{{\\mathcal {P}},\\alpha }=\\int _C \\alpha \\,{\\cal D}_{{\\mathcal {P}}} $ in terms of certain differential operators ${\\cal D}_{{\\mathcal {P}}}$ which act on the bundle locally at a point $p\\in C$ and depend holomorphically on $p$ .",
"We identify ${\\cal D}_{{\\mathcal {P}}}(p)$ as the action of a four-dimensional boundary local operator $O_{{\\mathcal {P}}}(p)$ .",
"The holomorphic-topological nature of the boundary condition insures that such boundary local operators commute in the topological direction and have non-singular operator product expansion (OPE) with each other.",
"As was explained in Section REF , the quantum Hilbert space ${\\mathcal {H}}={\\mathrm {Hom}}(\\overline{{\\mathcal {B}}}_{\\mathrm {cc}},{\\mathcal {B}}_{\\mathrm {cc}})$ is the space of ${\\mathrm {L}}^2$ half-densities on ${\\mathcal {M}}(G,C)$ , or equivalently the space of ${\\mathrm {L}}^2$ sections of $K^{1/2}\\otimes \\overline{K}{}^{1/2}$ .",
"We recall that this happens because brane quantization of ${\\mathcal {M}}_H(G,C)$ is equivalent to quantizing it as a cotangent bundle $T^*{\\mathcal {M}}(G,C)$ .",
"In quantizing a cotangent bundle $T^*W$ , a function whose restriction to a fiber of the cotangent bundle is a polynomial of degree $s$ becomes a differential operator of degree $s$ acting on half-densities on $W$ .",
"If, as in the case of interest here, $W$ is a complex manifold, then more specifically holomorphic functions on $T^*W$ become holomorphic differential operators on $W$ .",
"From a holomorphic point of view, one usually says that holomorphic functions on $T^*W$ (with polynomial dependence on the fibers) become holomorphic differential operators acting on sections of $K^{1/2}$ .",
"(The role of $K^{1/2}$ is explained from a $\\sigma $ -model point of view in [22], Section 3.2 and Appendix C.) In the case of the Hitchin Hamiltonians, the fact that they can be quantum deformed to differential operators acting on sections of $K^{1/2}$ , and not on sections of any other holomorphic line bundle, is part of the standard story [25], [1].",
"However, the antiholomorphic line bundle $\\overline{K}{}^{1/2}$ is invisible to a holomorphic differential operator, since its transition functions are antiholomorphic and commute with holomorphic differential operators.",
"So the holomorphic differential operators that are obtained by deformation quantization of holomorphic functions on $T^*W$ can naturally act on $K^{1/2}\\otimes \\overline{K}{}^{1/2}$ , or equivalently on the bundle ${\\sf K}^{1/2}$ of half-densities.",
"Similarly, under deformation quantization, antiholomorphic functions on $T^*W$ become antiholomorphic differential operators that can act on ${\\sf K}^{1/2}$ .",
"So ${\\mathcal {A}}$ and $\\overline{\\mathcal {A}}$ become, respectively, algebras of holomorphic and antiholomorphic differential operators acting on half-densities on ${\\mathcal {M}}(G,C)$ .",
"From this point of view, the statement that ${\\mathcal {A}}$ and $\\overline{\\mathcal {A}}$ commute just reflects the fact that holomorphic differential operators commute with antiholomorphic ones."
],
[
"The Duals Of The Coisotropic Branes",
"In order to be able to apply duality to this problem, we need one more ingredient.",
"We need to understand the duals of the $A$ -branes ${\\mathcal {B}}_{\\mathrm {cc}}$ and ${\\overline{\\mathcal {B}}}_{\\mathrm {cc}}$ in the $B$ -model of ${\\mathcal {M}}_H(G^\\vee ,C)$ .",
"To be specific, here we mean the $B$ -model in the complex structure that is called $J$ in [23], in which ${\\mathcal {M}}_H(G^\\vee ,C)$ parametrizes flat bundles over $C$ with structure group $G^\\vee _{\\mathbb {C}}$ .",
"We denote the connection on the flat bundle as ${\\mathcal {A}}=A+{\\mathrm {i}}\\phi $ , where $(A,\\phi )$ are the unitary connection and Higgs field that appear in Hitchin's equations.",
"A general $B$ -brane is a coherent sheaf, or a complex of coherent sheaves, on ${\\mathcal {M}}_H(G^\\vee ,C)$ .",
"However, the $A$ -branes ${\\mathcal {B}}_{\\mathrm {cc}}$ and ${\\overline{\\mathcal {B}}}_{\\mathrm {cc}}$ have additional properties that imply that the dual $B$ -branes must be rather special.",
"To explain this, we recall that the Higgs bundle moduli spaces are hyper-Kahler manifolds, with complex structures $I,J,K$ that obey the usual quaternion relations, and a triple of corresponding Kahler forms $\\omega _I,\\omega _J,\\omega _K$ and complex symplectic forms $\\Omega _I=\\omega _J+{\\mathrm {i}}\\omega _K$ , etc.",
"Geometric Langlands duality in general maps the $A$ -model of ${\\mathcal {M}}_H(G,C)$ with symplectic structure $\\omega _K$ to the $B$ -model of ${\\mathcal {M}}_H(G^\\vee ,C)$ in complex structure $J$ .",
"When we speak of the $A$ -model or the $B$ -model without further detail, these are the models we mean.",
"A generic brane in either of these models is merely an $A$ -brane or $B$ -brane of the appropriate type.",
"However, many of the branes that are most important in geometric Langlands have additional properties.",
"For example, a brane supported on a point in ${\\mathcal {M}}_H(G^\\vee ,C)$ is a brane of type $(B,B,B)$ , that is, it is a $B$ -brane in each of complex structures $I,J$ , and $K$ (or any linear combination).",
"The dual of a brane of type $(B,B,B)$ is a brane of type $(B,A,A)$ ; in the case of the Higgs bundle moduli space, the dual of a rank 1 brane supported at a point is a brane supported on a fiber of the Hitchin fibration, with a rank 1 flat ${\\mathrm {CP}}$ bundle.",
"These branes are the Hecke eigensheaves which are central objects of study in the geometric Langlands program; they will be discussed in Section .",
"In the case at hand, ${\\mathcal {B}}_{\\mathrm {cc}}$ and ${\\overline{\\mathcal {B}}}_{\\mathrm {cc}}$ are branes of type $(A,B,A)$ ; they are $A$ -branes of types $I$ and $K$ , by virtue of the Kapustin-Orlov conditions for coisotropic branes, and they are $B$ -branes of type $J$ , because the curvature $\\pm \\frac{1}{2}\\omega _J$ of their ${\\mathrm {CP}}$ bundles is of type $(1,1)$ in complex structure $J$ , so that those bundles are holomorphic in complex structure $J$ .",
"In general, the dual of a brane of type $(A,B,A)$ is a brane of type $(A,B,A)$ , so the duals of ${\\mathcal {B}}_{\\mathrm {cc}}$ and ${\\overline{\\mathcal {B}}}_{\\mathrm {cc}}$ will be branes of that type.",
"The simplest kind of brane of type $(A,B,A)$ is given by the structure sheaf of a complex Lagrangian submanifold in complex structure $J$ .",
"In more physical language, these are branes supported on a complex Lagrangian submanifold with a trivial ${\\mathrm {CP}}$ bundle.",
"And indeed, the duals of ${\\mathcal {B}}_{\\mathrm {cc}}$ and ${\\overline{\\mathcal {B}}}_{\\mathrm {cc}}$ are of this type.",
"These duals were first identified (in a different formulation) by Beilinson and Drinfeld [1], with the help of conformal field theory at critical level $k=-h^\\vee $ .",
"A gauge-theory explanation involves duality between the D3-NS5 and D3-D5 systems of string theory [31].",
"The complex Lagrangian submanifold supporting the dual of ${\\mathcal {B}}_{\\mathrm {cc}}$ parametrizes flat $G^\\vee _C$ bundles which satisfy a “holomorphic oper” condition.",
"We will denote it as $L_{\\mathrm {op}}$ .",
"Similarly, the dual of ${\\overline{\\mathcal {B}}}_{\\mathrm {cc}}$ is supported on a complex Lagrangian submanifold $L_{\\overline{\\mathrm {op}}}$ that parametrizes flat $G^\\vee _C$ bundles which satisfy an “antiholomorphic oper” condition.",
"The holomorphic oper condition can be stated rather economically as a global constraint on the holomorphic type of the bundle, i.e.",
"on the $(0,1)$ part of ${\\mathcal {A}}$ , as we will do here, or in a more local way, as we will do in Section REF .",
"Both formulations are standard mathematically.",
"In a four-dimensional gauge theory, $S$ -duality maps a deformed Neumann boundary condition to a deformed “Nahm pole” boundary condition, which imposes directly the local constraints [31].",
"One general way to define a complex Lagrangian submanifold of ${\\mathcal {M}}_H(G^\\vee ,C)$ is to consider all flat $G^\\vee _C$ bundles $E^\\vee \\rightarrow C$ with some fixed holomorphic type.",
"Specifying the holomorphic type of a bundle is equivalent to specifying the $(0,1)$ part of ${\\mathcal {A}}$ .",
"Here and in several analogous cases considered momentarily, we place no constraint on the $(1,0)$ part of the connection except that the full connection should be flat.",
"While preserving the flatness of the connection on $E^\\vee $ , we are free to add to the $(1,0)$ part of the connection an arbitrary $\\overline{\\partial }_{\\mathcal {A}}$ -closed form representing an element of $H^0(C,K_C\\otimes {\\mathrm {ad}}(E^\\vee ))$ .",
"The dimension of this space is half the dimension of ${\\mathcal {M}}_H(G^\\vee ,C)$ , so flat $G^\\vee _{\\mathbb {C}}$ bundles of a specified holomorphic type are a middle-dimensional submanifold $L$ of ${\\mathcal {M}}_H(G^\\vee ,C)$ .",
"$L$ is a complex Lagrangian submanifold, because the holomorphic symplectic structure $\\Omega _J$ of ${\\mathcal {M}}_H(G^\\vee ,C)$ in complex structure $J$ is a pairing between the $(1,0)$ and $(0,1)$ parts of ${\\mathcal {A}}$ and vanishes if the $(0,1)$ part is specified.",
"Once one picks a base point in $L$ , $L$ is isomorphic to the vector space $H^0(C,K_C\\otimes {\\mathrm {ad}}(E^\\vee ))$ .",
"We can define a second family of complex Lagrangian submanifolds, in the same complex structure on ${\\mathcal {M}}_H(G^\\vee ,C)$ , by specifying the antiholomorphic structure of a flat $G^\\vee _{\\mathbb {C}}$ bundle.",
"This amounts to specifying the $(1,0)$ part of ${\\mathcal {A}}$ , and letting the $(0,1)$ part vary.",
"It leads to a complex Lagrangian submanifold for the same reasons as before.",
"It may come as a slight surprise that fixing either the $(1,0)$ or the $(0,1)$ part of ${\\mathcal {A}}$ is a holomorphic condition in complex structure $J$ .",
"Indeed, complex structure $J$ on the Higgs bundle moduli space is not sensitive to the complex structure of the two-manifold $C$ , and treats the $(1,0)$ and $(0,1)$ parts of ${\\mathcal {A}}$ in a completely symmetric way.",
"The submanifolds $L_{\\mathrm {op}}$ and $L_{\\overline{\\mathrm {op}}}$ can be defined by specifying a particular choice of the holomorphic or antiholomorphic structure of a flat $E^\\vee $ bundle.",
"First we explain the definition for the case $G_{\\mathbb {C}}={\\mathrm {SL}}(2,{\\mathbb {C}})$ .",
"For this group, an oper is a flat bundle $E^\\vee \\rightarrow C$ that, as a holomorphic bundle, is a nontrivial extension of $K_C^{-1/2}$ by $K_C^{1/2}$ : $0\\rightarrow K_C^{1/2}\\rightarrow E^\\vee \\rightarrow K_C^{-1/2}\\rightarrow 0.",
"$ There is a unique such nontrivial extension, up to isomorphism.",
"The family of flat bundles of this holomorphic type is therefore a complex Lagrangian submanifold that we will call $L_{\\mathrm {op}}$ .",
"In making this definition, we have made a choice of $K_C^{1/2}$ , or equivalently a choice of spin structure on $C$ .",
"Indeed, for ${\\mathrm {SL}}(2,{\\mathbb {C}})$ , the definition of an oper depends on such a choice of spin structure (though we do not indicate this in the notation for $L_{\\mathrm {op}}$ ).",
"We return to this point in Section REF .",
"It is possible to give a simple description of $L_{\\mathrm {op}}$ once one picks a base point, that is, a particular ${\\mathrm {SL}}(2,{\\mathbb {C}})$ bundle $E^\\vee $ of oper type with flat connection ${\\mathcal {A}}_0$ .",
"In deforming $E^\\vee $ as an oper, we may as well keep the $(0,1)$ part of ${\\mathcal {A}}_0$ fixed, since we are required to keep it fixed up to a complex gauge transformation.",
"But we can modify the $(1,0)$ part of ${\\mathcal {A}}_0$ .",
"To do this while preserving the flatness of ${\\mathcal {A}}_0$ , we add to ${\\mathcal {A}}_0$ a $\\overline{\\partial }_{{\\mathcal {A}}_0}$ -closed $(1,0)$ -form, that is, an element of $H^0(C,K_C\\otimes {\\mathrm {ad}}(E^\\vee ))$ .",
"Using the exact sequence (REF ), one can show that $H^0(C,K_C\\otimes {\\mathrm {ad}}(E^\\vee ))$ is isomorphic to the space of quadratic differentials on $C$ .",
"This is the base of the Hitchin fibration for ${\\mathrm {SL}}(2,{\\mathbb {C}})$ , so $L_{\\mathrm {op}}$ is isomorphic to the base of the Hitchin fibration.",
"This isomorphism is not entirely canonical as it depends on the choice of a base point in $L_{\\mathrm {op}}$ .",
"A similar reasoning applies for other groups.",
"Similarly, an antiholomorphic oper for $G_{\\mathbb {C}}={\\mathrm {SL}}(2,{\\mathbb {C}})$ , or anti-oper for short, is a flat bundle $E^\\vee $ that, antiholomorphically, is a nontrivial extension of $\\overline{K}_C^{-1/2}$ by $\\overline{K}_C^{1/2}$ : $ 0\\rightarrow \\overline{K}_C^{1/2}\\rightarrow E^\\vee \\rightarrow \\overline{K}_C^{-1/2}\\rightarrow 0.",
"$ The family of such flat bundles is another complex Lagrangian submanifold, which we will call $L_{\\overline{\\mathrm {op}}}$ .",
"It is noncanonically isomorphic to the base of the Hitchin fibration, with the opposite complex structure.",
"In general, if $G_{\\mathbb {C}}^\\vee $ is a simple complex Lie group, there is a notion of a “principal embedding” of Lie algebras ${{\\mathfrak {su}}}(2)\\rightarrow {{\\mathfrak {g}}^\\vee }$ .",
"For example, if $G^\\vee ={\\mathrm {SL}}(n,{\\mathbb {C}})$ , the principal embedding is such that the $n$ -dimensional representation of ${\\mathfrak {g}}^\\vee $ transforms irreducibly under ${\\mathfrak {su}}(2)$ ; the corresponding principal subgroup is a copy of ${\\mathrm {SL}}(2,{\\mathbb {C}})$ or ${\\mathrm {SO}}(3,{\\mathbb {C}})$ depending on whether $n$ is even or odd.",
"For brevity we will sometimes ignore this subtlety and refer simply to a principal ${\\mathrm {SL}}(2,{\\mathbb {C}})$ subgroup, though the global form of the group is sometimes ${\\mathrm {SO}}(3,{\\mathbb {C}})$ .",
"A $G_{\\mathbb {C}}^\\vee $ oper is a flat $G_{\\mathbb {C}}^\\vee $ bundle $E_{\\mathbb {C}}^\\vee $ that, as a holomorphic bundle, is equivalent to a principal embedding of an ${\\mathrm {SL}}(2,{\\mathbb {C}})$ oper bundle, that is, a principal embedding of a rank two bundle that is a nontrivial extension of the form in eqn.",
"(REF ).",
"For $G^\\vee _{\\mathbb {C}}={\\mathrm {SL}}(n,{\\mathbb {C}})$ , this means that an oper bundle is, holomorphically, the $(n-1)^{th}$ symmetric tensor power of such a nontrivial extension, and therefore has a subbundle isomorphic to $K_C^{(n-1)/2}$ : $0\\rightarrow K_C^{(n-1)/2}\\rightarrow E^\\vee \\rightarrow \\cdots $ (and a filtration by powers of $K_C$ ).",
"Again, the $(1,0)$ part of the connection on $E_{\\mathbb {C}}^\\vee $ is not restricted except by requiring the full connection to be flat.",
"Similarly an antiholomorphic $G_{\\mathbb {C}}^\\vee $ oper is a flat $G_{\\mathbb {C}}^\\vee $ bundle that, as an antiholomorphic bundle, is equivalent to a principal embedding of an antiholomorphic ${\\mathrm {SL}}(2,{\\mathbb {C}})$ oper bundle.",
"At the $\\sigma $ -model level, the duals of ${\\mathcal {B}}_{\\mathrm {cc}}$ and ${\\overline{\\mathcal {B}}}_{\\mathrm {cc}}$ are the structure sheaves of $L_{\\mathrm {op}}$ and $L_{\\overline{\\mathrm {op}}}$ ; that is, they are rank 1 branes ${\\mathcal {B}}_{\\mathrm {op}}$ and ${\\mathcal {B}}_{\\overline{\\mathrm {op}}}$ supported on $L_{\\mathrm {op}}$ and $L_{\\overline{\\mathrm {op}}}$ with trivial ${\\mathrm {CP}}$ bundles."
],
[
"The Local Constraints",
"To describe a more local characterization of an oper, we consider first the case $G^\\vee _{\\mathbb {C}}={\\mathrm {SL}}(2,{\\mathbb {C}})$ .",
"The extension structure of $E^\\vee $ implies the existence of a global holomorphic section $s$ of $E^\\vee \\otimes K_C^{-1/2}$ .",
"Denote as $D$ the $(1,0)$ part of the connection.",
"The ${\\mathrm {SL}}(2,{\\mathbb {C}})$ -invariant combination $s \\wedge D s$ is a global holomorphic function on $C$ .",
"This function must be non-zero: if it vanished, we could write $D s= a s$ and $a$ would define a holomorphic flat connection on $K_C^{-1/2}$ , which does not exist (for $C$ of genus greater than 1 or in lower genus in the presence of parabolic structure).",
"Without loss of generality, we can normalize $s$ so that $s \\wedge D s=1$ .",
"This fixes $s$ up to sign.",
"The local version of the holomorphic oper condition for $G^\\vee _{\\mathbb {C}}={\\mathrm {SL}}(2,{\\mathbb {C}})$ is precisely the condition that $E^\\vee \\otimes K_C^{-1/2}$ admits a globally defined holomorphic section such that $s \\wedge D s=1$ .",
"Taking a derivative, we have $s \\wedge D^2 s =0$ and thus $s$ satisfies a second order differential equation $D^2 s + t s=0$ for some “classical stress tensor” $t$ on $C$ .",
"Under a change of local coordinate on $C$ , $t$ transforms as a stress tensor, not as a quadratic differential.",
"Not coincidentally, eqn.",
"(REF ) can be viewed as a classical limit of the Belavin-Polyakov-Zamolodchikov (BPZ) differential equations for the correlator of a degenerate field in two-dimensional conformal field theory.",
"The classical stress tensor can be used to define a set of generators of the algebra of holomorphic functions on the oper manifold, consisting of functions of the form $ f_{t,\\alpha }=\\int _C \\alpha \\,t ,$ with $\\alpha $ being a $(0,1)$ -form with values in $T_C$ .",
"The case of $G^\\vee _{\\mathbb {C}}={\\mathrm {SL}}(n,{\\mathbb {C}})$ can be analyzed similarly.",
"In this case, the oper structure of $E^\\vee $ (eqn.",
"(REF )) implies the existence of a global holomorphic section $s$ of $E^\\vee \\otimes K_C^{(1-n)/2}$ .",
"Then $s \\wedge D s \\wedge \\cdots D^{n-1} s$ is a global holomorphic function on $C$ which cannot vanish, for a similar reason to what we explained for $n=2$ .",
"We can normalize $s$ so that $s \\wedge D s \\wedge \\cdots D^{n-1} s=1$ ; this uniquely fixes $s$ , up to the possibility of multiplying by an $n^{th}$ root of 1.",
"Since $0=D(s \\wedge D s \\wedge \\cdots D^{n-1} s)=s\\wedge Ds \\wedge \\cdots \\wedge D^{n-2}s\\wedge D^ns$ , we learn that $s$ satisfies a degree $n$ differential equation $D^n s + t_2 D^{n-2} s+ \\cdots + t_n s =0.$ We can define a set of generators of the algebra of holomorphic functions on the oper manifold, consisting of functions of the form $ f_{t_k,\\alpha }=\\int _C \\alpha \\,t_k $ with $\\alpha $ being a $(0,1)$ -form with values in $T_C^{k-1}$ , $k=2,\\cdots ,n$ .",
"For general $G^\\vee _{\\mathbb {C}}$ , there is no distinguished representation as convenient as the $n$ -dimensional representation of ${\\mathrm {SL}}(n,{\\mathbb {C}})$ .",
"However, given an oper bundle $E^\\vee $ , we can consider associated bundles $E^\\vee _R$ in any irreducible representation $R$ of $G^\\vee _{\\mathbb {C}}$ .",
"By the definition of an oper, the structure group of $E^\\vee _R$ as a holomorphic bundle reduces to a rank 1 subgroup $H_{\\mathbb {C}}\\subset G^\\vee _{\\mathbb {C}}$ ; this subgroup is a copy of either ${\\mathrm {SL}}(2,{\\mathbb {C}})$ or ${\\mathrm {SO}}(3,{\\mathbb {C}})$ , depending on $G^\\vee _{\\mathbb {C}}$ and $R$ .",
"Let $R_n$ be the $n$ -dimensional irreducible representation of $H_{\\mathbb {C}}$ ($n$ is any positive integer or any odd positive integer for ${\\mathrm {SL}}(2,{\\mathbb {C}})$ or ${\\mathrm {SO}}(3,{\\mathbb {C}})$ , respectively).",
"As a representation of $H_{\\mathbb {C}}$ , we have $R\\cong \\oplus _{n=0}^\\infty Q_n\\otimes R_n$ , where $Q_n$ are some vector spaces, almost all of which vanish.",
"Actually, if $N$ is the largest integer for which $Q_n$ is nonzero, then $Q_N$ is 1-dimensional and we can replace $Q_N\\otimes R_N$ with $R_N$ .",
"So $R\\cong R_N\\oplus _{n=0}^{N-1}Q_n\\otimes R_n$ .",
"In this decomposition, a highest weight vector of $R_N$ with respect to a Borel subgroup $B_H$ of $H$ is a highest weight vector of $G^\\vee $ with respect to the Borel subgroup $B_{G^\\vee }$ of $G^\\vee $ that contains $B_H$ .",
"The associated bundle $E^\\vee _R$ has a similar decomposition as holomorphic bundle $ E^\\vee _{R}= E^\\vee _{R,N}\\oplus \\left(\\oplus _{n=1}^{N-1} Q_n\\otimes E^\\vee _{R_n}\\right), $ where $E^\\vee _{R_n}$ is the holomorphic bundle associated to an $H_{\\mathbb {C}}$ oper in the $n$ -dimensional representation.",
"For each $n$ we get from eqn.",
"(REF ) a canonical image $s_{R,n}$ of the vector space $Q_n$ into the space of global holomorphic sections of $E^\\vee _{R} \\otimes K_C^{(1-n)/2}$ , or equivalently a holomorphic map $ s_{R,n}: Q_n \\otimes K_C^{(n-1)/2} \\rightarrow E^\\vee _{R}.",
"$ Of particular importance here is the “highest weight” object $s_{R,N}$ , which we will just denote as $s_R$ : $s_R:K_C^{(N-1)/2}\\rightarrow E^\\vee _R.",
"$ Here $N$ is defined by saying that a highest weight vector of the representation $R$ (for some Borel subgroup $B$ ) is in an $N$ -dimensional representation of a principal ${\\mathrm {SL}}(2,{\\mathbb {C}})$ subgroup (which has a Borel subgroup contained in $B$ ).",
"For $G^\\vee ={\\mathrm {SL}}(n,{\\mathbb {C}})$ and $R$ the $n$ -dimensional representation or its dual, $N=n$ .",
"A partial analogue to the $s \\wedge D s=1$ condition is the condition that the collection $D^m s_{R,n}$ for $m< n\\le N$ should span $E^\\vee _{R}$ at each point of $C$ .",
"The derivatives $D^n s_{R,n}$ can then be expanded out in terms of the $D^m s_{R,n}$ with $m<n$ , giving rise to an intricate collection of differential equations.",
"(See Appendix for some examples.)",
"The coefficients of the differential equations are holomorphic functions on the oper manifold and can be expressed as polynomials in derivatives of a collection of observables $T_{\\cal P}$ which have the same labels as the integrands for the quantum Hitchin Hamiltonians for $G^\\vee _{\\mathbb {C}}$ .",
"Tensor products of the form $D^m s_{R,n} \\otimes D^{m^{\\prime }} s_{R^{\\prime },n^{\\prime }}$ can also be expanded in the basis of $D^{m^{\\prime \\prime }} s_{R^{\\prime \\prime },n^{\\prime \\prime }}$ for all $R^{\\prime \\prime }$ that appear in the decomposition of the tensor product $R \\otimes R^{\\prime }$ .",
"Coefficients in this expansion are also holomorphic functions on the oper manifold and can be expressed as polynomials in derivatives of a collection of observables $T_{\\cal P}$ .",
"The collection of observables $T_{\\cal P}$ is sometimes called the classical $W$ -algebra for $G^\\vee _{\\mathbb {C}}$ .",
"In that language, the differential equations satisfied by the $s_{R,n}$ are a classical analogue of the BPZ equations, and the tensor product expansion is analogous to the operator product expansion (OPE) of degenerate fields.",
"In four-dimensional gauge theory, the deformed Neumann boundary condition is $S$ -dual to the deformed Nahm pole boundary condition, which is also holomorphic-topological.",
"This boundary condition involves a certain prescribed singularity for the gauge theory fields at the boundary.",
"Effectively, the singular boundary conditions of the physical theory impose an oper boundary condition in the topologically twisted theory.",
"The gauge-invariant information contained in the subleading behaviour of the fields is captured by boundary local operators which match the $T_{\\cal P}$ observables and are $S$ -dual to the corresponding local operators at the deformed Neumann boundary condition which are employed in the definition of the quantum Hitchin Hamiltonians.",
"The $s_{R,n}$ also appear naturally in gauge theory, as we will illustrate in Section REF .",
"Finally, in order to gain further intuition on the various relations satisfied by the $s_{R,n}$ , we note that the oper manifold has a simpler “classical” cousin given by Hitchin's section of the Hitchin fibration.",
"For ${\\mathrm {SL}}(2,{\\mathbb {C}})$ , the Hitchin section parametrizes Higgs bundles $(E,\\varphi )$ such that $E$ is a direct sum $K_C^{1/2} \\oplus K_C^{-1/2}$ .",
"In other words, the Hitchin section is what we get if we work in complex structure $I$ (rather than $J$ ) and ask for the extension in eqn.",
"(REF ) to be split (as opposed to a non-split extension, leading to an oper bundle).",
"For any $G^\\vee _{\\mathbb {C}}$ , the Hitchin section parametrizes pairs $(E,\\varphi )$ such that $E$ is induced by the principal embedding of $K_C^{1/2} \\oplus K_C^{-1/2}$ .",
"For the Hitchin section, one can deduce local conditions analogous to what we have explained for opers, but using the Higgs field $\\varphi $ instead of the holomorphic derivative $D$ .",
"For example, the ${\\mathrm {SL}}(2,{\\mathbb {C}})$ Hitchin section of Higgs bundle moduli space is characterized locally by the existence of a holomorphic section $s$ of $E$ which satisfies $s \\wedge \\varphi s=1$ along with $\\varphi ^2 s = \\frac{1}{2} {\\rm Tr}\\, \\varphi ^2 s$ (the latter equation holds simply because $\\varphi ^2=\\frac{1}{2} {\\rm Tr}\\, \\varphi ^2$ for ${\\mathrm {SL}}(2,{\\mathbb {C}})$ ).",
"Comparing to the oper case, $D$ is replaced by $\\varphi $ and the stress tensor $t$ is replaced by the quadratic differential $\\frac{1}{2}{\\rm Tr}\\,\\varphi ^2$ .",
"In general, the associated bundle $E^\\vee _R$ in a representation $R$ of $G^\\vee $ will have sections $s_{R,n}$ , $n\\le N$ , such that $\\varphi ^m s_{R,n}$ for $m<n$ span $E^\\vee _{R}$ .",
"Hence $\\varphi ^n s_{R,n}$ can be expanded in terms of $\\varphi ^m s_{R,n}$ for $m<n$ ; likewise for two representations $R$ , $R^{\\prime }$ , $\\varphi ^m s_{R,n} \\otimes \\varphi ^{m^{\\prime }} s_{R^{\\prime },n^{\\prime }}$ can be expanded out in terms of $\\varphi ^{m^{\\prime \\prime }} s_{R^{\\prime \\prime },n^{\\prime \\prime }}$ for all $R^{\\prime \\prime }$ in $R \\otimes R^{\\prime }$ , with coefficients built from the gauge-invariant polynomials ${\\mathcal {P}}(\\varphi )$ .",
"The oper relations are a deformation of these.",
"With an extension of this analysis, one can recover the assertion of section REF that $L_{\\mathrm {op}}$ is noncanonically isomorphic to the base of the Hitchin fibration."
],
[
"Some Topological Subtleties",
"The definition of a $G_{\\mathbb {C}}$ oper depends on the choice of a spin structure on $C$ if the image of the principal embedding is an ${\\mathrm {SL}}(2,{\\mathbb {C}})$ subgroup of $G_{\\mathbb {C}}^\\vee $ , but not if it is an ${\\mathrm {SO}}(3,{\\mathbb {C}})$ subgroup.",
"(For example, for $G_{\\mathbb {C}}^\\vee ={\\mathrm {SL}}(n,{\\mathbb {C}})$ , the notion of an oper depends on a choice of spin structure precisely if $n$ is even.)",
"This dependence on spin structure for some groups is in tension with the claim that ${\\mathcal {B}}_{\\mathrm {op}}$ is the dual of ${\\mathcal {B}}_{\\mathrm {cc}}$ , since the definition of ${\\mathcal {B}}_{\\mathrm {cc}}$ did not seem to depend on a choice of spin structure.",
"The resolution of this point was essentially described in Section 8 of [32].",
"We will explain the details for groups of rank 1.",
"In the context of the twisted version of ${\\mathcal {N}}=4$ super Yang-Mills theory that is relevant to geometric Langlands, the electric-magnetic dual of ${\\mathrm {SO}}(3)$ gauge theory is not standard ${\\mathrm {SU}}(2)$ gauge theory, but what is sometimes called ${\\mathrm {Spin}}\\cdot {\\mathrm {SU}}(2)$ gauge theory.",
"For any $d>0$ , the group ${\\mathrm {Spin}}(d)\\cdot {\\mathrm {SU}}(2)$ is a double cover of ${\\mathrm {SO}}(d)\\times {\\mathrm {SO}}(3)$ that restricts to a nontrivial double cover of either factor.",
"In particular, ${\\mathrm {Spin}}(d)\\cdot {\\mathrm {SU}}(2)$ has projections to ${\\mathrm {SO}}(d)$ and to ${\\mathrm {SO}}(3)$ : $ \\begin{matrix}& & {\\mathrm {Spin}}(d)\\cdot {\\mathrm {SU}}(2) && \\cr &\\swarrow &&\\searrow &\\cr {\\mathrm {SO}}(d) &&&& {\\mathrm {SO}}(3).",
"\\end{matrix}$ By a ${\\mathrm {Spin}}(d)\\cdot {\\mathrm {SU}}(2)$ structure on a $d$ -manifold $M$ , we mean a principal bundle over $ M$ with that structure group such that the projection to the first factor gives the frame bundle of $M$ (the principal bundle associated to the tangent bundle $TM$ of $M$ ).",
"Likewise a ${\\mathrm {Spin}}(d)\\cdot {\\mathrm {SU}}(2)$ connection on a Riemannian manifold $M$ is a connection with that structure group that when restricted to the first factor is the Levi-Civita connection of the tangent bundle of $M$ .",
"Assuming that $M$ is spin, a down-to-earth description of a ${\\mathrm {Spin}}(d)\\cdot {\\mathrm {SU}}(2)$ structure on $M$ is as follows: once a spin structure is picked on $M$ , a ${\\mathrm {Spin}}(d)\\cdot {\\mathrm {SU}}(2)$ bundle is equivalent to an ${\\mathrm {SU}}(2)$ bundle $E^\\vee \\rightarrow M$ ; if the spin structure of $M$ is twisted by a line bundle $\\ell $ such that $\\ell ^2$ is trivial, then $E^\\vee $ is replaced by $E^\\vee \\otimes \\ell $ .",
"With this characterization, it is evident that although the variety $L_{\\mathrm {op}}$ of opers is not canonically defined in ${\\mathrm {SU}}(2)$ gauge theory, it is canonically defined in ${\\mathrm {Spin}}(4)\\cdot {\\mathrm {SU}}(2)$ gauge theory.",
"Indeed, bearing in mind that $\\ell \\cong \\ell ^{-1}$ , if $E^\\vee $ appears in the exact sequence defining an oper with some choice of $K^{1/2}$ , then $E^\\vee \\otimes \\ell $ appears in a similar exact sequence with $K^{1/2}$ replaced by $K^{1/2}\\otimes \\ell $ .",
"What is the dual of ${\\mathrm {SU}}(2)$ gauge theory, as opposed to ${\\mathrm {Spin}}(4)\\cdot {\\mathrm {SU}}(2)$ gauge theory?",
"The answer [32] is that the dual is ${\\mathrm {SO}}(3)$ gauge theory, but with an extra factor $\\Delta =(-1)^{\\int _M w_2(M) w_2(E)}$ included in the definition of the path integral.",
"Here $w_2(M)$ and $w_2(E)$ are respectively the second Stieffel-Whitney classes of $TM$ and of an ${\\mathrm {SO}}(3)$ bundle $E\\rightarrow M$ .",
"If the $B$ -model description is by ${\\mathrm {SU}}(2)$ gauge theory, and therefore the definition of $L_{\\mathrm {op}}$ requires a spin structure on $C$ , then the $A$ -model description is by ${\\mathrm {SO}}(3)$ gauge theory with the additional factor $\\Delta $ , and this must ensure that the definition of ${\\mathcal {B}}_{\\mathrm {cc}}$ similarly requires a choice of spin structure on $C$ .",
"That happens as follows.",
"If $M=\\Sigma \\times C$ where $\\Sigma $ and $C$ are oriented two-manifolds without boundary, then $w_2(M)=0$ and $\\Delta =1$ , so the factor of $\\Delta $ in the path integral has no consequence.",
"But suppose $M$ has a boundary $\\partial M$ with ${\\mathcal {B}}_{\\mathrm {cc}}$ boundary conditions.",
"To define the topological invariant $\\int _M w_2(M) w_2(E)$ , one needs a trivialization of the class $w_2(M) w_2(E)$ along $\\partial M$ .",
"${\\mathcal {B}}_{\\mathrm {cc}}$ boundary conditions are a version of free boundary conditions for the gauge field, so with ${\\mathcal {B}}_{\\mathrm {cc}}$ boundary conditions, there is no restriction on $w_2(E)$ along $\\partial M$ .",
"But we can trivialize $w_2(M)w_2(E)$ along $\\partial M$ by trivializing $w_2(M)$ , that is, by picking a spin structure along $\\partial M$ .",
"In our application, $\\partial M$ is the product of a Riemann surface $C$ with a contractible one-manifold (the boundary of the strip) and what is needed is a spin structure on $C$ .",
"Thus the $A$ -model dual of ${\\mathrm {SU}}(2)$ gauge theory is an ${\\mathrm {SO}}(3)$ gauge theory in which, despite appearances, the definition of ${\\mathcal {B}}_{\\mathrm {cc}}$ (or similarly $\\overline{{\\mathcal {B}}}_{\\mathrm {cc}}$ ) requires a choice of spin structure on $C$ .",
"These remarks have analogs for all groups such that the definition of an oper requires a choice of spin structure.",
"They do not have analogs in the usual physics of ${\\mathcal {N}}=4$ super Yang-Mills theory because they only come into play after topological twisting.",
"Untwisted ${\\mathcal {N}}=4$ super Yang-Mills theory has fermion fields whose definition requires a spin structure on $M$ .",
"When a spin structure is present, the difference between ${\\mathrm {Spin}}(4)\\cdot {\\mathrm {SU}}(2)$ gauge theory and ${\\mathrm {SU}}(2)$ gauge theory disappears.",
"Likewise, the choice of a spin structure trivializes $\\Delta $ ."
],
[
"Topological Aspects of the Oper Boundary Condition In Gauge Theory",
"When the center ${\\mathcal {Z}}(G^\\vee )$ of $G^\\vee $ is nontrivial, the description of the dual of ${\\mathcal {B}}_{\\mathrm {cc}}$ as the brane ${\\mathcal {B}}_{\\mathrm {op}}$ (and the analogous statement for $\\overline{{\\mathcal {B}}}_{\\mathrm {cc}}$ ) needs a slight refinement.",
"${\\mathcal {B}}_{\\mathrm {op}}$ is the dual of ${\\mathcal {B}}_{\\mathrm {cc}}$ in the $\\sigma $ -model of ${\\mathcal {M}}_H(G^\\vee ,C)$ , but we should recall that the low energy description also contains a ${\\mathcal {Z}}(G^\\vee )$ gauge field.",
"Along a boundary labeled by ${\\mathcal {B}}_{\\mathrm {op}}$ , the ${\\mathcal {Z}}(G^\\vee )$ gauge field is trivialized.",
"As explained momentarily, this condition ensures that when quantized on $I\\times C$ , where $I$ is an interval with ${\\mathcal {B}}_{\\mathrm {op}}$ and ${\\mathcal {B}}_{\\overline{\\mathrm {op}}}$ boundary conditions, the theory supports a discrete electric charge along $I$ .",
"Following the logic of Section 7 of [3], this condition is dual to the fact that on the $A$ -model side, a $G$ -bundle over $C$ has a characteristic class $\\zeta \\in H^2(C,\\pi _1(G))$ and is classified by $\\int _C\\zeta $ (for $G={\\mathrm {SO}}(3)$ , $\\zeta $ is the second Stieffel-Whitney class $w_2(E)$ ).",
"The Nahm pole boundary condition, which is a gauge theory version of ${\\mathcal {B}}_{\\mathrm {op}}$ [31], reduces the gauge group along the boundary from $G^\\vee $ to its center ${\\mathcal {Z}}(G^\\vee )$ .",
"The trivialization of the center along the boundary is an additional condition.",
"For $G^\\vee ={\\mathrm {SL}}(n,{\\mathbb {C}})$ , the oper condition, or the Nahm pole boundary condition, ensures the existence of the object $s$ that was introduced in Section REF , and was normalized there, up to an $n^{th}$ root of 1, by the condition condition $s\\wedge Ds \\wedge \\cdots \\wedge D^{n-1}s=1$ .",
"An $n^{th}$ root of 1 is an element of ${\\mathcal {Z}}(G^\\vee )$ , so a convenient way to express the fact that the ${\\mathcal {Z}}(G^\\vee )$ gauge invariance is trivialized along the boundary is to say that the boundary is equipped with a particular choice of normalized section $s$ .",
"When we quantize the theory on a strip ${\\mathbb {R}}\\times I$ (times the Riemann surface $C$ ), with oper and anti-oper boundary conditions, we have such trivializations $s_\\ell $ and $s_r$ at the left and right boundaries of the strip.",
"We are free to make a global gauge transformation by an element $b$ of ${\\mathcal {Z}}(G^\\vee )$ .",
"This acts on the pair of trivializations by $(s_\\ell ,s_r)\\rightarrow (b s_\\ell , b s_r)$ , so pairs differing in that way should be considered equivalent.",
"However, ${\\mathcal {Z}}(G^\\vee )$ acts on the equivalence classes of pairs $(s_\\ell ,s_r)$ by $(s_\\ell ,s_r)\\rightarrow (b s_\\ell ,s_r)$ , $b\\in {\\mathcal {Z}}(G^\\vee )$ , and this leads to an action of ${\\mathcal {Z}}(G^\\vee )$ on the physical Hilbert space.",
"This action of ${\\mathcal {Z}}(G^\\vee )$ on ${\\mathcal {H}}$ in the $B$ -model description is dual to the fact that, on the $A$ -model side, ${\\mathcal {H}}$ is graded by $\\int _C \\zeta $ .",
"For any $G^\\vee $ , the trivialization of the ${\\mathcal {Z}}(G^\\vee )$ gauge field on the boundary can be expressed in terms of the objects $s_{R,n}$ that were introduced in Section REF , but this is less simple than for ${\\mathrm {SL}}(n,{\\mathbb {C}})$ ."
],
[
"The Case That The Center Is Trivial",
"Now we can start to deduce interesting consequences of electric-magnetic duality.",
"Once one identifies the $B$ -model dual of ${\\mathcal {B}}_{\\mathrm {cc}}$ as ${\\mathcal {B}}_{\\mathrm {op}}$ , as we have done in Section REF , one immediately has a dual description of ${\\mathcal {A}}={\\mathrm {Hom}}({\\mathcal {B}}_{\\mathrm {cc}},{\\mathcal {B}}_{\\mathrm {cc}})$ : it is ${\\mathrm {Hom}}({\\mathcal {B}}_{\\mathrm {op}},{\\mathcal {B}}_{\\mathrm {op}})$ .",
"Since ${\\mathcal {B}}_{\\mathrm {op}}$ is a rank 1 Lagrangian brane supported on $L_{\\mathrm {op}}$ , ${\\mathrm {Hom}}({\\mathcal {B}}_{\\mathrm {op}},{\\mathcal {B}}_{\\mathrm {op}})$ is just the sum of the $\\overline{\\partial }_{\\mathcal {A}}$ cohomology groups $H^i(L_{\\mathrm {op}}, {\\mathcal {O}})$ .",
"In Section REF , we learned that $L_{\\mathrm {op}}$ is noncanonically isomorphic to a vector space.",
"Hence the cohomology $H^i(L_{\\mathrm {op}},{\\mathcal {O}})$ vanishes for $i>0$ , and ${\\mathrm {Hom}}(L_{\\mathrm {op}},L_{\\mathrm {op}})$ is simply the (undeformed!)",
"commutative algebra of holomorphic functions on $L_{\\mathrm {op}}$ .",
"Thus duality with the $B$ -model gives another explanation that ${\\mathcal {A}}$ must be commutative.",
"Moreover it shows that the “spectrum” of the algebra ${\\mathcal {A}}$ , in the abstract sense of the space of its 1-dimensional complex representations, is the “variety”For our purposes, “variety” is just a synonym for “complex manifold.” $L_{\\mathrm {op}}$ that parametrizes opers, as originally shown in [1].",
"In the language of previous sections, this is a canonical identification between differential operators $D_{\\mathcal {P}}$ and functions $T_{\\mathcal {P}}$ .",
"Precisely the same argument shows that $\\overline{{\\mathcal {A}}}={\\mathrm {Hom}}({\\mathcal {B}}_{\\mathrm {op}},{\\mathcal {B}}_{\\mathrm {op}})$ is the algebra of holomorphic functions on the variety $L_{\\overline{\\mathrm {op}}}$ of antiholomorphic opers.",
"The variety of opers is noncanonically isomorphic to the base of the Hitchin fibration, as explained in Section REF .",
"So the fact that ${\\mathcal {A}}$ is the algebra of holomorphic functions on $L_{\\mathrm {op}}$ is a sort of quantum deformation of the fact that ${\\mathcal {A}}_0$ is the algebra of holomorphic functions on the base of the Hitchin fibration.",
"A similar statement holds for $\\overline{{\\mathcal {A}}}$ , of course.",
"However, we want to understand the spectrum of ${\\mathcal {A}}\\times \\overline{\\mathcal {A}}$ not in the abstract sense already indicated but as concrete operators on ${\\mathcal {H}}={\\mathrm {Hom}}({\\overline{\\mathcal {B}}}_{\\mathrm {cc}},{\\mathcal {B}}_{\\mathrm {cc}})$ .",
"The dual theory gives a dual description by ${\\mathcal {H}}={\\mathrm {Hom}}({\\mathcal {B}}_{\\overline{\\mathrm {op}}},{\\mathcal {B}}_{\\mathrm {op}})$ .",
"If the center of $G^\\vee $ is trivial, this can be analyzed just in a $\\sigma $ -model (rather than a $\\sigma $ -model with ${\\mathcal {Z}}(G^\\vee )$ gauge fields).",
"Let us consider this case first.",
"Matters are simple because the branes involved are rank 1 Lagrangian branes, supported on the complex Lagrangian manifolds $L_{\\mathrm {op}}$ and $L_{\\overline{\\mathrm {op}}}$ .",
"In analyzing the problem, we will assume that $L_{\\mathrm {op}}$ and $L_{\\overline{\\mathrm {op}}}$ have only transverse intersections at isolated points.",
"This is known to be true for ${\\mathrm {SL}}(2,{\\mathbb {C}})$ and in general is one of the conjectures of Etingof, Frenkel, and Kazhdan [8], [9].",
"For the intersections to be isolated and transverse is actually a prediction of the duality; it is needed in order for the hermitian form on ${\\mathcal {H}}$ to be positive-definite, as expected from the $A$ -model construction in which ${\\mathcal {H}}$ is a Hilbert space of ${\\mathrm {L}}^2$ half-densities.",
"Unfortunately, to explain this requires a fairly detailed discussion of $B$ -model quantum mechanics, which has been relegated to Appendix .",
"(In this appendix, we learn that there is actually a further, unproved necessary condition for positivity.)",
"Let $\\Upsilon =L_{\\mathrm {op}}\\cap L_{\\overline{\\mathrm {op}}}$ .",
"Assuming that the intersection points are isolated and transverse, ${\\mathcal {H}}={\\mathrm {Hom}}({\\mathcal {B}}_{\\overline{\\mathrm {op}}},{\\mathcal {B}}_{\\mathrm {op}})$ simply has a basis with one basis vector $\\psi _u$ for every $u\\in \\Upsilon $ .",
"That is a general statement about intersections of Lagrangian branes in the $B$ -model.",
"Concretely, since $L_{\\mathrm {op}}$ is the subvariety of ${\\mathcal {M}}_H(G,C)$ that parametrizes flat bundles that are holomorphic opers, and $L_{\\overline{\\mathrm {op}}}$ is the subvariety that parametrizes flat bundles that are antiholomorphic opers, it follows that an intersection point represents a flat $G_{\\mathbb {C}}^\\vee $ bundle that is an oper both holomorphically and antiholomorphically.",
"We recall that the definition of a hermitian form on ${\\mathrm {Hom}}({\\mathcal {B}}_{\\overline{\\mathrm {op}}},{\\mathcal {B}}_{\\mathrm {op}})$ makes use of an antiholomorphic involution $\\tau $ that acts by $(A,\\phi )\\rightarrow (A,-\\phi )$ .",
"Hence $\\tau $ transforms a complex flat connection ${\\mathcal {A}}=A+{\\mathrm {i}}\\phi $ to the complex conjugate flat connection $\\overline{{\\mathcal {A}}}=A-{\\mathrm {i}}\\phi $ .",
"Recall that $A$ is a gauge field in a theory in which the gauge group is the compact form $G^\\vee $ .",
"Mathematically, the involution of $G^\\vee _{\\mathbb {C}}$ that leaves fixed $G^\\vee $ is called the Chevalley involution, so $\\tau $ acts on ${\\mathcal {A}}$ via the Chevalley involution (up to an inner automorphism, which here means a $G^\\vee $ -valued gauge transformation).",
"In the folded construction of the state space ${\\mathcal {H}}$ , $\\tau $ acts antiholomorphically on $\\widehat{Y}=Y_1\\times Y_2$ by exchanging the two factors.",
"That means that in the unfolded construction, $\\tau $ exchanges the two ends of the strip of fig.",
"REF (b).",
"It is not difficult to see explicitly why this happens.",
"If ${\\mathcal {A}}$ is a complex flat connection that is a holomorphic oper, then $\\overline{{\\mathcal {A}}}$ is a complex flat connection that is an antiholomorphic oper, and similarly, if ${\\mathcal {A}}$ is an antiholomorphic oper, then $\\overline{{\\mathcal {A}}}$ is a holomorphic one.",
"Thus $\\tau $ exchanges $L_{\\mathrm {op}}$ with $L_{\\overline{\\mathrm {op}}}$ and likewise exchangesThe statement that $\\tau $ exchanges ${\\mathcal {B}}_{\\mathrm {op}}$ with ${\\mathcal {B}}_{\\overline{\\mathrm {op}}}$ holds in the underlying physical $\\sigma $ -model.",
"Since $\\tau $ acts antiholomorphically on ${\\mathcal {M}}_H(G^\\vee ,C)$ in the relevant complex structure, it exchanges the $B$ -model with a conjugate $B$ -model and is not a $B$ -model symmetry (${\\mathcal {B}}_{\\mathrm {op}}$ and ${\\mathcal {B}}_{\\overline{\\mathrm {op}}}$ are valid branes in both the $B$ -model and its conjugate).",
"The $B$ -model symmetry that exchanges ${\\mathcal {B}}_{\\mathrm {op}}$ with ${\\mathcal {B}}_{\\overline{\\mathrm {op}}}$ and is used in defining the hermitian structure (eqn.",
"(REF )) is $\\Theta _\\tau =\\Theta \\tau $ , where $\\Theta =\\sf {CPT}$ .",
"${\\mathcal {B}}_{\\mathrm {op}}$ with ${\\mathcal {B}}_{\\overline{\\mathrm {op}}}$ .",
"Suppose that a point $u\\in L_{\\mathrm {op}}\\cap L_{\\overline{\\mathrm {op}}}$ corresponds to a complex flat bundle $E^\\vee _{\\mathbb {C}}$ that is an oper both holomorphically and antiholomorphically.",
"Then its complex conjugate $\\overline{E}^\\vee _{\\mathbb {C}}$ is also an oper both holomorphically and antiholomorphically.",
"If $\\overline{E}^\\vee _{\\mathbb {C}}$ is not gauge-equivalent to $E^\\vee _{\\mathbb {C}}$ as a flat bundle, then $\\overline{E}^\\vee _{\\mathbb {C}}$ corresponds to a point $\\overline{u}\\in L_{\\mathrm {op}}\\cap L_{\\overline{\\mathrm {op}}}$ that is distinct from $u$ .",
"If so, $u$ and $\\overline{u}$ will correspond to distinct basis vectors $\\psi _u$ and $\\psi _{\\overline{u}}$ of ${\\mathcal {H}}$ , and moreover these will be exchanged by $\\Theta _\\tau $ .",
"The natural $B$ -model pairing is diagonal in the basis of intersection points: the basis vectors can be normalized so that for $u,u^{\\prime }\\in \\Upsilon $ , $(\\psi _u,\\psi _{u^{\\prime }})=\\delta _{uu^{\\prime }}$ .",
"Therefore, if $\\Theta _\\tau $ exchanges two distinct basis vectors $\\psi _u$ and $\\psi _{\\overline{u}}$ , then $\\psi _u$ and $\\psi _{\\overline{u}}$ are both null vectors for the hermitian inner product that was defined in eqn.",
"(REF ).",
"The duality predicts that this hermitian inner product should be positive-definite, since on the $A$ -model side, ${\\mathcal {H}}$ is obtained by quantizing a cotangent bundle and is a Hilbert space of half-densities.",
"So we expect that a flat bundle that is an oper both holomorphically and antiholomorphically is actually real.",
"This was conjectured in [8], [9] and was proved by an explicit (but surprisingly non-trivial) computation for $G^\\vee ={\\mathrm {U}}(1)$ ; the result is also known for $G^\\vee ={\\mathrm {SU}}(2)$ [33], [34].",
"Finally, we can use the duality to predict the spectrum of the holomorphic and antiholomorphic Hitchin Hamiltonians as operators on ${\\mathcal {H}}$ .",
"Let $H_{{\\mathcal {P}},\\alpha }$ be a quantized Hitchin Hamiltonian, that is, an element of ${\\mathcal {A}}={\\mathrm {Hom}}({\\mathcal {B}}_{\\mathrm {cc}},{\\mathcal {B}}_{\\mathrm {cc}})$ .",
"The duality identifies ${\\mathcal {A}}$ with ${\\mathrm {Hom}}({\\mathcal {B}}_{\\mathrm {op}},{\\mathcal {B}}_{\\mathrm {op}})$ and therefore identifies $H_{{\\mathcal {P}},\\alpha }$ with a holomorphic function $f_{{\\mathcal {P}},\\alpha }$ on $L_{\\mathrm {op}}$ .",
"Acting on a basis vector $\\psi _u$ that corresponds to a point $u\\in L_{\\mathrm {op}}\\cap L_{\\overline{\\mathrm {op}}}$ , $H_{{\\mathcal {P}},\\alpha }$ simply acts by multiplication by the corresponding value $f_{{\\mathcal {P}},\\alpha }(u)$ .",
"Similarly, if $H_{\\overline{{\\mathcal {P}}},\\overline{\\alpha }}\\in \\overline{{\\mathcal {A}}}={\\mathrm {Hom}}(\\overline{{\\mathcal {B}}}_{\\mathrm {cc}},\\overline{{\\mathcal {B}}}_{\\mathrm {cc}})$ is an antiholomorphic quantized Hitchin Hamiltonian, then it corresponds under the duality to a holomorphic function $f_{\\overline{{\\mathcal {P}}},\\overline{\\alpha }}$ on $L_{\\overline{\\mathrm {op}}}$ , and it acts on $\\psi _u$ as multiplication by $f_{\\overline{{\\mathcal {P}}},\\overline{\\alpha }}(u)$ .",
"This completes the description of the eigenvalues of the quantized Hitchin Hamiltonians."
],
[
"Including the Center",
"It is not difficult to modify this description to take into account the center of $G_{\\mathbb {C}}^\\vee $ .",
"Consider as usual the $B$ -model on $M=\\Sigma \\times C$ .",
"It localizes on flat bundles over $\\Sigma \\times C$ .",
"In our application, $\\Sigma ={\\mathbb {R}}\\times I$ is contractible, so a flat bundle on $M$ is the pullback of a flat bundle on $C$ .",
"In the case of oper and anti-oper boundary conditions at the two ends of the strip, the flat bundle on $C$ is an oper both holomorphically and antiholomorphically; thus it is a real oper.",
"An oper bundle, real or not, is irreducible and its automorphism group consists only of the center ${\\mathcal {Z}}(G^\\vee )$ of the gauge group.",
"However, as explained in Section REF , the boundary conditions also give trivializations $s_\\ell $ and $s_r$ of the ${\\mathcal {Z}}(G^\\vee )$ gauge symmetry on the two boundaries, modulo gauge transformations that act by $(s_\\ell , s_r)\\rightarrow (b s_\\ell ,b s_r)$ , $b\\in {\\mathcal {Z}}(G^\\vee )$ .",
"For a given real oper corresponding to a point $u\\in \\Upsilon $ , let ${\\mathcal {T}}_u$ be the set of pairs $s_\\ell ,s_r$ modulo the action of ${\\mathcal {Z}}(G^\\vee )$ .",
"The $B$ -model localizes on the isolated set of points $u,\\varepsilon $ with $u\\in \\Upsilon $ , $\\varepsilon \\in {\\mathcal {T}}_u$ .",
"So the Hilbert space ${\\mathcal {H}}$ in the general case with a nontrivial center has a basis $\\psi _{u,\\varepsilon }$ for such $u,\\varepsilon $ .",
"One can think of $\\varepsilon \\in {\\mathcal {T}}_u$ as a sort of global holonomy between the left and right boundaries of the strip.",
"This refinement involving the torsor ${\\mathcal {T}}_u$ is not important in the dual description of the algebras ${\\mathcal {A}}$ and $\\overline{\\mathcal {A}}$ via holomorphic functions on $L_{\\mathrm {op}}$ or $L_{\\overline{\\mathrm {op}}}$ , since each algebra acts on only one side of the strip.",
"It is similarly not important in the determination of the eigenvalues of the Hitchin Hamiltonians, which only depends on the interpretation of ${\\mathcal {A}}$ and $\\overline{\\mathcal {A}}$ in terms of functions on $L_{\\mathrm {op}}$ and $L_{\\overline{\\mathrm {op}}}$ , and is not sensitive to global holonomy across the strip.",
"It does affect the multiplicity of the eigenvalues, since eigenvectors $\\psi _{u,\\varepsilon }$ with the same $u$ and different $\\varepsilon $ have the same eigenvalues of the Hitchin Hamiltonians.",
"And it will be relevant in describing the eigenvalues of the 't Hooft or Hecke operators, to which we turn next."
],
[
"Line Operators",
"In the usual formulation of geometric Langlands [1], the main objects of study include the Hecke functors acting on the category of $A$ -branes and the “eigenbranes” of these Hecke functors.",
"In the gauge theory picture [3], the Hecke functors are interpreted in terms of 't Hooft line operators.",
"Electric-magnetic duality maps 't Hooft line operators to Wilson line operators, leading to some of the usual statements about geometric Langlands duality.",
"In general, in two-dimensional topological field theory, line operators give functors acting on the category of boundary conditions because a line operator $T$ that runs parallel to a boundary labeled by a brane ${\\mathcal {B}}$ can be moved to the boundary, making a composite boundary condition $T{\\mathcal {B}}$ (fig.",
"REF (a)).",
"Here we assume that the two-manifold and the line operator (or more precisely the one-manifold on which it is supported) are oriented and that the orientation of the line operator agrees with the orientation of the boundary on which it acts.",
"The same figure also makes clear the notion of the adjoint of a line operator.",
"The adjoint $T^{\\prime }$ of a line operator is the same line operator with opposite orientation.",
"In fig.",
"REF (a), we could move the line operator $T$ to the right of the figure.",
"As its orientation is opposite to the orientation of the right boundary, this gives an action of the dual line operator $T^{\\prime }$ on the brane ${\\mathcal {B}}^{\\prime }$ that defines the boundary condition on the right boundary.",
"So we get ${\\mathrm {Hom}}({\\mathcal {B}}^{\\prime },T{\\mathcal {B}})={\\mathrm {Hom}}(T^{\\prime }{\\mathcal {B}}^{\\prime },{\\mathcal {B}})$ for any ${\\mathcal {B}},{\\mathcal {B}}^{\\prime }$ .",
"(Some line operators have the property that $T$ is isomorphic to $T^{\\prime }$ ; their support can be an unoriented 1-manifold.)",
"These statements hold in any two-dimensional topological field theory.",
"Our actual application involves a four-dimensional theory with two additional dimensions that comprise a Riemann surface $C$ .",
"Although it is possible to consider an 't Hooft operator (or a dual Wilson operator) whose support is an arbitrary curve $\\gamma $ in the four-manifold $\\Sigma \\times C$ , we will only consider the special case that $\\gamma =\\ell \\times p$ , where $p$ is a point in $C$ and $\\ell $ is a curve in $\\Sigma $ .",
"So our line operators will be defined in part by the choice of $p$ .",
"In addition, in the application to geometric Langlands, an 't Hooft operator is labeled by a finite-dimensional irreducible representation $R$ of $G^\\vee $ (or equivalently of $G^\\vee _{\\mathbb {C}}$ ).",
"When we want to indicate this data, we denote the 't Hooft operator as $T_{R,p}$ .",
"Similarly the dual Wilson operator depending on the representation $R$ and the point $p$ will be denoted as $W_{R,p}$ .",
"Figure: (a) A line operator TT parallel to the left boundary of the strip, and oriented compatibly.",
"Moving TT to the left, it mapsthe boundary condition labeled ℬ{\\mathcal {B}} to a composite boundary condition TℬT{\\mathcal {B}}.",
"This is a line operator viewed as a functor on the categoryof branes or boundary conditions, as in the usual formulation of geometric Langlands.",
"(b) In the analytic approach to geometricLanglands, the same line operator TT, running horizontally across the strip, and with some additional data at the endpoints, becomesan operator acting on physical states.",
"(c) and (d) The purpose of these drawings is to elucidate the additional datathat is needed at the left and right endpoints in (b).",
"At the left endpoint, we have an element α∈ Hom (ℬ,Tℬ)\\alpha \\in {\\mathrm {Hom}}({\\mathcal {B}},T{\\mathcal {B}}), and at the right endpoint,an element β∈ Hom (Tℬ ' ,ℬ ' )\\beta \\in {\\mathrm {Hom}}(T{\\mathcal {B}}^{\\prime },{\\mathcal {B}}^{\\prime }).Figure: This picture illustrates an algebraic manipulation described in the text.",
"Reading the drawing on the right from bottom to top,one first encounters the operations sketched in figs.",
"(c,d), followed by the fusion of the product TT ' TT^{\\prime } of line operators to the “identity,” that is,to a trivial line operator.In the analytic approach to geometric Langlands [8], [9], Hecke operators becomes ordinary operators acting on a Hilbert space of quantum states, rather than more abstract functors acting on a category.",
"Not surprisingly, the gauge theory interpretation of Hecke operators in this sense is based on the same 't Hooft line operators as before, used somewhat differently.",
"In fig.",
"REF (b), we consider the same line operator as before, but now running from left to right of the figure.",
"Some additional data must be supplied at the left and right endpoints where the line operator terminates on a boundary of the strip.",
"Let us assume for the moment that this has been done.",
"Then the line operator becomes an ordinary operator acting on quantum states.",
"Reading the figure from bottom to top, an element of ${\\mathrm {Hom}}({\\mathcal {B}}^{\\prime },{\\mathcal {B}})$ enters at the bottom and after the action of the line operator, a possibly different element of ${\\mathrm {Hom}}({\\mathcal {B}}^{\\prime },{\\mathcal {B}})$ emerges at the top.",
"(If we read the figure from top to bottom, we see the transpose operator acting on the dual vector space ${\\mathrm {Hom}}({\\mathcal {B}},{\\mathcal {B}}^{\\prime })$ .)",
"It is because line operators that are supported on a one-manifold in space at a fixed time can act in this way as ordinary quantum operators that they are traditionallyIn traditional applications in particle physics, there are no boundaries and the support of the line operator is taken to be a closed loop.",
"The operator is then often called a loop operator.",
"called line “operators.” The purpose of figs.",
"REF (c,d) is to explain what is happening where the line operator of fig.",
"REF (b) ends on the left or right boundary.",
"In fig.",
"REF (c), we see that the left endpoint of the line operator corresponds to an element $\\alpha \\in {\\mathrm {Hom}}({\\mathcal {B}},T{\\mathcal {B}})$ , and in fig.",
"REF (d), we see that the right endpoint corresponds to an element $\\beta \\in {\\mathrm {Hom}}(T{\\mathcal {B}}^{\\prime },{\\mathcal {B}}^{\\prime })$ .",
"Algebraically, the operator $\\widehat{T}:{\\mathrm {Hom}}({\\mathcal {B}}^{\\prime },{\\mathcal {B}})\\rightarrow {\\mathrm {Hom}}({\\mathcal {B}}^{\\prime },{\\mathcal {B}})$ associated to a line operator $T$ with the additional data $\\alpha ,\\beta $ can be described as follows.",
"For $\\psi \\in {\\mathrm {Hom}}({\\mathcal {B}}^{\\prime },{\\mathcal {B}})$ , we have $\\alpha \\circ \\psi \\circ \\beta \\in {\\mathrm {Hom}}(T{\\mathcal {B}}^{\\prime },T{\\mathcal {B}})={\\mathrm {Hom}}({\\mathcal {B}}^{\\prime },T^{\\prime }T{\\mathcal {B}})$ .",
"Then using the fact that line operators form an algebra and that the trivial line operator appears in the product $T^{\\prime }T$ , we get a map $w:{\\mathrm {Hom}}({\\mathcal {B}}^{\\prime },T^{\\prime }T{\\mathcal {B}})\\rightarrow {\\mathrm {Hom}}({\\mathcal {B}}^{\\prime } ,{\\mathcal {B}})$ .",
"Finally $\\widehat{T}(\\psi )=w\\circ \\alpha \\circ \\psi \\circ \\beta $ .",
"This sequence of algebraic manipulations corresponds to the picture of fig.",
"REF .",
"We will sometimes write $\\widehat{T}_{\\alpha ,\\beta }$ or $\\widehat{T}_{R,p,\\alpha ,\\beta }$ for the operator on ${\\mathcal {H}}$ that is constructed from a line operator $T$ or $T_{R,p}$ with endpoint data $\\alpha ,\\beta $ .",
"Figure: The argument showing that quantized Hitchin Hamiltonians commute can be adapted to show that line operators (viewed as actual operatorson a space of quantum states) commute with the quantized Hitchin Hamiltonians and with each other.",
"In each case, as sketched in (a) and (b) respectively,the key point is that because of the existence of additional dimensions, the two operators can slide up and down past each other without singularity.Figure: To compute the eigenvalues of the 't Hooft/Hecke operators, one considers dual Wilson operators that describe parallel transport froma×pa\\times p to b×pb\\times p, where pp is a point in CC and a,ba,b are points on the left and right boundaries of Σ\\Sigma .We will be particularly interested in elements $(\\alpha ,\\beta )$ which originate from local endpoints of a four-dimensional line defect onto four-dimensional boundary conditions which lift ${\\mathcal {B}}$ and ${\\mathcal {B}}^{\\prime }$ .",
"As remarked in Section REF , the four-dimensional lifts of the boundary conditions we are considering are not topological.",
"Instead, they are respectively holomorphic-topological and antiholomorphic-topological.",
"As a consequence, the local endpoint lifting $\\alpha $ will depend holomorphically on $p$ while the local endpoint lifting $\\beta $ will depend antiholomorphically on $p$ .",
"Notice that the actual path of the line defect in four-dimensions is immaterial, as long as it is topologically equivalent to a straight path.",
"Only the positions of the endpoints in $C$ matters.",
"One of the main properties of 't Hooft or Hecke operators, when regarded as in [8], [9] as operators on quantum states, is that they commute with each other and with the quantized Hitchin Hamiltonians.",
"This follows from the same reasoning that we used to show that the Hitchin Hamiltonians commute with each other.",
"An 't Hooft operator $T_{R,p}$ commutes with a Hitchin Hamiltonian $H_{{\\mathcal {P}},\\alpha }$ because one can assume that the support of $\\alpha $ is disjoint from the point $p$ , so that one can slide $T_{R,p}$ and $H_{{\\mathcal {P}},\\alpha }$ up and down past each other (fig.",
"REF (a)) without singularity.",
"Likewise, for distinct points $p,p^{\\prime }\\in C$ , 't Hooft operators $T_{R,p}$ and $T_{R^{\\prime },p^{\\prime }}$ commute (fig.",
"REF (b)).",
"Taking the limit $p^{\\prime }\\rightarrow p$ , it follows that $T_{R,p}$ and $T_{R^{\\prime },p}$ commute as well, even for $R\\ne R^{\\prime }$ ."
],
[
"Wilson Operators And Their Eigenvalues",
"Since the 't Hooft operators commute with the Hitchin Hamiltonians, they can be diagonalized in the same basis, namely the basis of states $\\psi _{u,\\varepsilon },$ $u\\in \\Upsilon ,$ $\\varepsilon \\in {\\mathcal {T}}_u$ , where $\\Upsilon =L_{\\mathrm {op}}\\cap L_{\\overline{\\mathrm {op}}}$ (Section REF ).",
"In fact, we can use electric-magnetic duality to determine the eigenvalues of the 't Hooft operators.",
"An 't Hooft operator $T_{R,p}$ is dual to a Wilson operator $W_{R,p}$ , labeled by the same representation $R$ of $G^\\vee _{\\mathbb {C}}$ and supported at the same point $p\\in C$ .",
"While the 't Hooft operator is a “disorder” operator, whose microscopic definition involves a certain sort of singularity, Wilson operators are defined classically in terms of holonomy, as follows.",
"In the relevant gauge theory on a four-manifold $M$ (for our purposes, $M=\\Sigma \\times C$ ), one has a $G^\\vee _{\\mathbb {C}}$ bundle $E^\\vee _{\\mathbb {C}}\\rightarrow M$ , with connection ${\\mathcal {A}}=A+{\\mathrm {i}}\\phi $ , to which we can associate a vector bundle $E^\\vee _R=E^\\vee _{\\mathbb {C}}\\times _{G^\\vee _{\\mathbb {C}}} R$ .",
"We denote the induced connection on this bundle simply as ${\\mathcal {A}}$ .",
"The Wilson operator is constructed from the holonomy of the connection ${\\mathcal {A}}$ on $E^\\vee _R$ , integrated in general along some oriented path $\\gamma \\subset M$ .",
"If $\\gamma $ is a closed loop, we take the trace of the holonomy around $\\gamma $ , and this gives a version of the Wilson operator that is important in many physical applications.",
"However, to compute the eigenvalues of an operator defined by an 't Hooft line operator that stretches across the strip (fig.",
"REF (b)), we need to consider a dual Wilson operator that similarly stretches across the strip (fig.",
"REF ).",
"This is a Wilson operator supported, not on a closed loop, but on a path $\\gamma \\subset \\Sigma \\times p$ from $a\\times p$ on the left boundary of the strip to $b\\times p$ on the right boundary.",
"In this case, the holonomy is best understood as a linear transformation from the fiber $E^\\vee _R$ at $a\\times p$ to the fiber of this bundle at $b\\times b$ .",
"Thus with an obvious notation for these fibers, $W_{R,p}$ is a linear transformation $ W_{R,p}:E^\\vee _{R,a\\times p}\\rightarrow E^\\vee _{R, b\\times p}.$ In order to treat the left and right edges of the strip more symmetrically, it is convenient to introduce the representation $R^{\\prime }$ dual to $R$ and view $W_{R,p}$ as a linear function on a representation.",
"Then we have $ W_{R,p}\\in {\\mathrm {Hom}}(E^\\vee _{R,a\\times p}\\otimes E^\\vee _{R^{\\prime },b\\times p},{\\mathbb {C}}).",
"$ So far, we have a linear function on a vector space, rather than a complex-valued function of connections, which could be quantized to get a quantum operator.",
"To get a complex-valued function of connections, we need to supply vectors $v\\in E^\\vee _{R,a\\times p}$ , $w\\in E^\\vee _{R^{\\prime },b\\times p}$ .",
"Then $W_{R,p}(v\\otimes w)$ is a complex valued function that can be quantized to get an operator.",
"A natural construction of suitable vectors was described in Section REF .",
"In eqn.",
"(REF ), we described, for a holomorphic oper with associated bundle $E^\\vee _R$ , a “highest weight section” $s_R:K_C^{(N-1)/2}\\rightarrow E^\\vee _R$ .",
"Thus, if we are given a vector $v\\in K_{C,p}^{(N-1)/2}$ , then we can define $s_R(v)\\in E^\\vee _{R,p}$ .",
"Similarly, if $E^\\vee _{R^{\\prime }}$ is an antiholomorphic oper, we have $\\overline{s}_{R^{\\prime }}:\\overline{K}_C^{(N-1)/2}\\rightarrow E^\\vee _{R^{\\prime }}$ , and hence, for $w\\in K_{C,p}^{(N-1)/2}$ , we have $\\overline{s}_{R^{\\prime }}(w)\\in E^\\vee _{R^{\\prime },p}$ .",
"Note that a dual pair of representations $R,R^{\\prime }$ have the same value of $N$ .",
"In the case of a bundle $E^\\vee _R\\rightarrow \\Sigma \\times C$ that is a holomorphic oper on the left boundary and an antiholomorphic oper on the right boundary, we can apply the holomorphic version of this construction on the left boundary and the antiholomorphic version on the right boundary, to get $W_{R,p,v\\otimes w}=W_{R,p}(s_{R}(v)\\otimes \\overline{s}_{R^{\\prime }}(w)).",
"$ This finally is a complex-valued function of connections that can be quantized to get a Wilson operator on physical states.",
"We will call this operator $W_{R,p,v\\otimes w}$ .",
"In the notation, we make use of the fact that the right hand side of eqn.",
"(REF ) depends on $v$ and $w$ only in the combination $v\\otimes w$ .",
"Hopefully it will cause no serious confusion that we use the same notation for a classical holonomy (or its matrix element) and the corresponding quantum operator.",
"Because of the way the $B$ -model localizes on flat connections, it is trivial to diagonalize this operator.",
"The flat connections that satisfy the boundary conditions, with trivializations of the center on the boundary and modulo gauge transformations, are in one-to-one correspondence with the usual basis of states $\\psi _{u,\\varepsilon }$ , $u\\in \\Upsilon ,$ $\\varepsilon \\in {\\mathcal {T}}_u$ that diagonalize the Hitchin Hamiltonians.",
"The Wilson operators are diagonal in this basis.",
"The eigenvalue of the quantum operator $ W_{R,p,v\\otimes w}$ on a given basis vector $\\psi _{u,\\varepsilon }$ is just the value of the corresponding classical function on the classical solution corresponding to $\\psi _{u,\\varepsilon }$ .",
"That value is simply the natural dual pairing $(s_R(v), \\overline{s}_{R^{\\prime }}(w))$ of the vectors $s_{R}(v)$ and $\\overline{s}_{R^{\\prime }}(w)$ in the dual vector spaces $E^\\vee _{R,p}$ and $E^\\vee _{R^{\\prime },p}$ .",
"This is true because a flat connection on $\\Sigma \\times C$ is actually a pullback from $C$ .",
"The center ${\\mathcal {Z}}(G^\\vee )$ acts on the physical Hilbert space ${\\mathcal {H}}$ by $(s_\\ell , s_r)\\rightarrow (b s_\\ell , s_r)$ (Section REF ).",
"This transforms the eigenvalue of $\\widehat{W}_{R,p}$ by a root of unity, which is simply the value of the central element $b$ in the representation $R$ .",
"For example, if $G^\\vee ={\\mathrm {SL}}(n,{\\mathbb {C}})$ and $R$ is the $n$ -dimensional representation, eigenvectors of $\\widehat{W}_{R,p}$ come in $n$ -plets of the form $\\lambda \\exp (2\\pi {\\mathrm {i}}k/n)$ , $\\lambda \\in {\\mathbb {C}}$ , $k=0,1,\\cdots , n-1$ .",
"This is dual to the fact that on the $A$ -model side, the Higgs bundle moduli space has components labeled by a characteristic class $\\zeta \\in H^2(C,\\pi _1(G))$ .",
"For $G^\\vee ={\\mathrm {SL}}(n,{\\mathbb {C}})$ , there are $n$ components, which are cyclically permuted by the 't Hooft operator $\\widehat{T}_{R,p}$ dual to $\\widehat{W}_{R,p}$ , leading to the same structure of the spectrum.",
"In general, the action of ${\\mathcal {Z}}(G^\\vee )$ on $R$ mirrors the way $\\widehat{T}_{R,p}$ permutes the components of the moduli space.",
"The holomorphic and antiholomorphic corner data needed to define the operators $W_{R,p}$ has consisted precisely of the vectors $v\\in K_C^{(N-1)/2}$ , $w\\in \\overline{K}_C^{(N-1)/2}$ .",
"For the $n$ -dimensional representation of ${\\mathrm {SL}}(n,{\\mathbb {C}})$ , we have $N=n$ .",
"We expect the same data to be needed to define holomorphic and antiholomorphic corner data for the dual 't Hooft operators.",
"These are arguably the simplest Wilson operators and we will call them principal Wilson operators.",
"However, if the representation $R$ is reducible when restricted to a principal ${\\mathfrak {su}}(2)$ subalgebra of ${\\mathfrak {g}}^\\vee $ , then it is possible to use the more general objects $s_{R,n}:Q_n\\otimes K_C^{(n-1)/2}\\rightarrow E^\\vee _R$ (eqn.",
"(REF )).",
"So picking $v\\in Q_n \\otimes K_C^{(n-1)/2}$ , $w\\in \\overline{Q}_{n^{\\prime }}\\otimes K_C^{(n^{\\prime }-1)/2}$ (where $n$ and $n^{\\prime }$ can be chosen independently), we can define $W_{R,p,n,n^{\\prime },v\\otimes w}=W_{R,p}(s_{R,n}(v)\\otimes \\overline{s}_{R^{\\prime },n^{\\prime }}(w))$ , which can again be interpreted as a quantum mechanical operator.",
"Even more generally, we can consider holomorphic and antiholomorphic derivatives with respect to $p$ of $W_{R,p,n,n^{\\prime },v\\otimes w} $ .",
"It is enough to consider $n-1$ holomorphic derivatives and $n^{\\prime }-1$ antiholomorphic ones; this suffices to define a complete set of Wilson operators, since it amounts to applying the linear form $W_{R,p}:E^\\vee _{R,a\\times p}\\otimes E^\\vee _{R^{\\prime },b\\times p}\\rightarrow {\\mathbb {C}}$ (eqn.",
"(REF )) to a set of vectors that according to the analysis in Section REF form a basis of the finite-dimensional vector space on which $W_{R,p}$ is acting.",
"What happens if we continue to differentiate?",
"With $n$ holomorphic derivatives or $n^{\\prime }$ antiholomorphic ones, we will run into differential equations satisfied by the Wilson operators $W_{R,p,n,n^{\\prime },v\\otimes w} $ .",
"The holomorphic and antiholomorphic sections $ s_{R,r}$ and $\\overline{s}_{R^{\\prime },n^{\\prime }}$ that were used to define these Wilson operators obey certain holomorphic and antiholomorphic differential equations (Section REF and Appendix ) as a function of $p$ , and the corresponding Wilson operators obey the same equations.",
"Having defined operators that act on the physical Hilbert space ${\\mathcal {H}}$ , it is natural to ask what algebra they obey.",
"In bulk, the product of line operators mimics the tensor product of representations of $G^\\vee $ .",
"Thus, if $R_i\\otimes R_j\\cong \\oplus _k N^k_{ij}R_k$ , with vector spaces $N^k_{ij}$ , then the corresponding decomposition of parallel Wilson operators is $W_{R_i} W_{R_j}=\\oplus _k N^k_{ij} W_k$ .",
"This is the appropriate relation for Wilson operators understood as functors acting on boundary conditions, as illustrated in fig.",
"REF (a).",
"From this algebra, the structure of the nonabelian group $G^\\vee $ can be reconstructed, in principle.",
"However, for Wilson operators as operators on quantum states, as we are discussing here, the picture is different.",
"Matters are simplest if we multiply two principal Wilson operators, associated to highest weight vectors in the corresponding representations.",
"Since the tensor product of highest weight vectors in two representations $R_1$ and $R_2$ is a highest weight vector in the tensor product $R_1\\otimes R_2$ , the product of two principal Wilson line operators is another principal Wilson line operator, for a representation $R_3$ whose highest weight is the sum of the highest weights of $R_1$ and $R_2$ .",
"We will see the structure of the nonabelian group if we multiply the more general Wilson operators $W_{R,p,n,n^{\\prime },v\\otimes w}$ and their derivatives.",
"A dual 't Hooft operator defined using the $S$ -dual data will, of course, have the same eigenvalues as the Wilson operators.",
"The main challenge is to identify precisely the appropriate $A$ -model endpoints.",
"We turn to that problem in Section REF .",
"We conclude this discussion of Wilson operators with the following remark.",
"The most illuminating realization of the oper boundary condition in four-dimensional gauge theory involves the Nahm pole [31].",
"Compared to a more direct approach that was assumed earlier [3] (in which the boundary condition is defined by just specifying the $(0,1)$ or $(1,0)$ part of ${\\mathcal {A}}$ along the boundary), the Nahm pole description of the oper boundary condition has two advantages: it leads directly to the local constraints discussed in Section REF ; and it also leads to a simple explanation of the duality with the $A$ -model description via ${\\mathcal {B}}_{\\mathrm {cc}}$ .",
"If one uses the Nahm pole description of the oper boundary condition, then the complex connection ${\\mathcal {A}}$ is singular along the boundary, and some renormalization is involved in defining the classical holonomy $W_R$ across the strip and the corresponding quantum operator.",
"The renormalization amounts to a complex gauge transformation that removes the Nahm pole singularity."
],
[
"'t Hooft Operators and Hecke Modifications",
"An 't Hooft operator $T_{R,p}$ (fig.",
"REF (a)) will produce a jump in the fields $(A,\\phi )$ and in the associated Higgs bundle $(E,\\varphi )$ , in the sense that the Higgs bundle $(E,\\varphi )$ just below the 't Hooft operator is generically not isomorphic to the Higgs bundle $(E^{\\prime },\\varphi ^{\\prime })$ just above it.",
"They differ by what is known as a Hecke modification.",
"The type of Hecke modification is determined by the magnetic singularity of the 't Hooft operator, which is classified by a choice of irreducible representation $R$ of the dual group $G^\\vee $ .",
"Hecke modifications of bundles and Higgs bundles were described for physicists in [3] and in more detail in Section 4 of [35].",
"Here we will give a very brief synopsis.",
"A Hecke modification of a holomorphic $G_{\\mathbb {C}}$ bundle $E$ at a point $p$ is a new bundle $E^{\\prime }$ that is presented with an isomorphism to $E$ away from $p$ , but such that this isomorphism does not extend over $p$ .",
"A section $w$ of $E^{\\prime }$ is a section of $E$ that is allowed to have poles of a specified type at $p$ , or that is constrained so that some components have zeroes of a specified type, or both.",
"For example, if $E$ is a rank 2 holomorphic vector bundle, trivialized near $p$ so that a section of $E$ is a pair of holomorphic functions $\\begin{pmatrix}f\\cr g\\end{pmatrix}$ , then an example of a Hecke modification of $E$ at $p$ is a new bundle $E^{\\prime }$ whose sections are a pair $\\begin{pmatrix}f\\cr g\\end{pmatrix}$ , where $f$ is holomorphic at $p$ but $g$ is allowed to have a simple pole at $p$ .",
"This example can be slightly generalized to give a family of Hecke modifications of $E$ at $p$ that are parametrized by ${\\mathbb {CP}}^1$ .",
"We simply pick a pair of complex numbers $u,v$ , not both zero, representing a point in ${\\mathbb {CP}}^1$ , and allow a section of $E^{\\prime }$ to have a polar part proportional to this pair: $ w= \\frac{1}{z}\\begin{pmatrix}u\\cr v\\end{pmatrix}+{\\mathrm {regular}}, $ where $z$ is a local parameter at $p$ .",
"The relation between $E$ and $E^{\\prime }$ is reciprocal: instead of saying that we obtain $E^{\\prime }$ from $E$ by allowing a pole of a certain type, we could say that we obtain $E$ from $E^{\\prime }$ by requiring a certain type of zero.",
"If an 't Hooft operator $T_{R,p}$ can map $E$ to $E^{\\prime }$ , then the dual 't Hooft operator (which is associated to the dual representation) can map $E^{\\prime }$ to $E$ .",
"A Hecke modification of a Higgs bundle $(E,\\varphi )$ actually does “nothing” to $\\varphi $ .",
"This means the following.",
"A Hecke modification of $(E,\\varphi )$ is just a Hecke modification $E^{\\prime }$ of $E$ such that $\\varphi :E\\rightarrow E\\otimes K_C$ is holomorphic as a map $E^{\\prime }\\rightarrow E^{\\prime }\\otimes K_C$ .",
"In the example of the rank 2 bundle, the Higgs field is locally $\\varphi =\\varphi _z\\,\\mathrm {d}z$ , where $\\varphi _z$ is a $2\\times 2$ matrix of holomorphic functions.",
"For $\\varphi $ to be holomorphic as a map $E^{\\prime }\\rightarrow E^{\\prime }\\otimes K_C$ , the necessary and sufficient condition is that, if $w$ is a section of $E^{\\prime }$ as characterized in eqn.",
"(REF ), then the polar part of $\\varphi w$ should be a multiple of $\\begin{pmatrix}u\\cr v\\end{pmatrix}$ ; in other words, $\\begin{pmatrix}u\\cr v\\end{pmatrix}$ must be an eigenvector of the matrix $\\varphi _z(p)$ .",
"Generically, this condition is not satisfied and hence most Hecke modifications of $E$ are not valid as Hecke modifications of $(E,\\varphi )$ .",
"If $\\varphi _z(p)$ is not nilpotent, it has two distinct eigenvectors and the Higgs bundle $(E,\\varphi )$ has two possible Hecke modifications of this type at the point $p$ ; if $\\varphi _z(p)$ is nilpotent but not zero, it has only one eigenvector and there is just one possible Hecke modification of this type; only if $\\varphi _z(p)=0$ does $(E,\\varphi )$ has the same ${\\mathbb {CP}}^1$ family of possible Hecke modifications of this type at $p$ that $E$ would have by itself.",
"Hecke modifications of the type just described can be viewed in two different ways.",
"They are dual to the two-dimensional representation of $G^\\vee ={\\mathrm {U}}(2)$ .",
"This group is self-dual, so the discussion is applicable in gauge theory of $G={\\mathrm {U}}(2)$ .",
"Alternatively, the same Hecke modification is dual to the two-dimensional representation of $G^\\vee ={\\mathrm {SU}}(2)$ .",
"In this case, the dual group is $G={\\mathrm {SO}}(3)$ .",
"In this application, since the rank two bundles $E$ and $E^{\\prime }$ do not have ${\\mathrm {SO}}(3)$ structure group, the preceding discussion should be restated in terms of the corresponding adjoint bundles ${\\mathrm {ad}}(E)$ and ${\\mathrm {ad}}(E^{\\prime })$ In this example, let $\\delta $ be the eigenvalue of $\\varphi (p)$ acting on $\\begin{pmatrix}u\\cr v\\end{pmatrix}$ : $ \\varphi (p) \\begin{pmatrix}u\\cr v\\end{pmatrix}=\\delta \\begin{pmatrix}u\\cr v\\end{pmatrix}.$ As we vary the choice of $E$ and $E^{\\prime }$ , $\\delta $ varies holomorphically.",
"It defines a holomorphic function on the space of Hecke modifications of this type.",
"Since the eigenvalues of $\\varphi (p)$ are $\\pm \\delta $ , we have $ \\delta ^2=-\\det \\varphi (p),$ where $\\det \\varphi (p)$ is a linear combination of the Hitchin Hamiltonians.",
"The sign of $\\delta $ distinguishes the two choices of Hecke modifications compatible with a given Higgs field."
],
[
"Hecke Correspondences",
"The 't Hooft operator $T_{R,p}$ can be viewed as an interface between the $A$ -model of ${\\mathcal {M}}_H(G,C)$ and itself.",
"This is a tautology; in fig.",
"REF (a), we see the $A$ -model of ${\\mathcal {M}}_H(G,C)$ above and below $T_{R,p}$ , so we can view $T_{R,p}$ as an interface between two copies of this $A$ -model.",
"In fact, this interface is of type $(B,A,A)$ , because of the supersymmetric properties of $T_{R,p}$ .",
"It is convenient to use a folding trick similar to the one of fig.",
"REF .",
"Instead of associating to $T_{R,p}$ an interface in the $A$ -model of ${\\mathcal {M}}_H(G,C)$ , we can associate to it a brane or boundary condition ${\\mathcal {B}}_{R,p}$ of type $(B,A,A)$ in the $A$ -model of a product ${\\mathcal {M}}_H(G,C)\\times {\\mathcal {M}}_H(G,C)$ (fig.",
"REF (b)).",
"Here the symplectic structure of ${\\mathcal {M}}_H(G,C)\\times {\\mathcal {M}}_H(G,C)$ is $\\omega _K\\boxplus (-\\omega _K)$ , with a minus sign in one factor because folding reverses the sign of the symplectic structure.",
"Let us consider explicitly what ${\\mathcal {B}}_{R,p}$ will look like for the basic example, described in Section REF , that $R$ is the two-dimensional representation of $G^\\vee ={\\mathrm {SU}}(2)$ .",
"What is its complex dimension?",
"If $C$ has genus $g$ , then the choice of a $G_{\\mathbb {C}}$ bundle $E$ depends on $3g-3$ complex parameters.",
"Choosing $E^{\\prime }$ 's that can be made from $E$ by action of $T_{R,p}$ adds one more complex parameter.",
"But we have to constrain the Higgs field $\\varphi $ so that $\\begin{pmatrix}u\\cr v\\end{pmatrix}$ is an eigenvector of $\\varphi (p)$ .",
"So the choice of $\\varphi $ involves $3g-4$ parameters, not $3g-3$ .",
"The upshot is that the support $Z_{R,p}$ of ${\\mathcal {B}}_{R,p}$ has dimension $6g-6$ , and thus $Z_{R,p}$ is middle-dimensional in ${\\mathcal {M}}_H(G,C)\\times {\\mathcal {M}}_H(G,C)$ .",
"Because $Z_{R,p}$ is middle-dimensional and is the support of a brane of type $(B,A,A)$ , it must be a complex Lagrangian submanifold.",
"The brane ${\\mathcal {B}}_{R,p}$ has rank 1 because the Hecke transformation by which $T_{R,p}$ produces $E^{\\prime }$ from $E$ is generically unique, if it exists.",
"In short, ${\\mathcal {B}}_{R,p}$ is a rank 1 Lagrangian brane of type $(B,A,A)$ .",
"We can be more specific, because ${\\mathcal {B}}_{R,p}$ is manifestly invariant under scaling of $\\varphi $ , which corresponds to the ${\\mathbb {C}}^*$ symmetry of ${\\mathcal {M}}_H(G,C)\\times {\\mathcal {M}}_H(G,C)$ .",
"We can give a simple description in the same sense in which ${\\mathcal {M}}_H(G,C)$ can be approximated by its dense open set $T^*{\\mathcal {M}}(G,C)$ .",
"(As in the general discussion of quantization, we expect that this approximation is sufficient in an ${\\mathrm {L}}^2$ theory.)",
"The intersection of $Z_{R,p}$ with ${\\mathcal {M}}(G,C)\\times {\\mathcal {M}}(G,C)\\subset T^*{\\mathcal {M}}(G,C)\\times T^*{\\mathcal {M}}(G,C)$ is the variety $X_{R,p}$ that parametrizes pairs $E,E^{\\prime }$ such that $E^{\\prime }$ can be reached from $E$ by a Hecke transformation of type $R$ at the point $p$ .",
"${\\mathbb {C}}^*$ invariance of $Z_{R,p} $ means that it can identified as the conormal bundleIf a submanifold $U\\subset M$ is defined locally by vanishing of some coordinates $q_1,\\cdots , q_r$ , then its conormal bundle in $T^*M$ is defined by setting to zero those coordinates and the momenta that Poisson-commute with them.",
"of $X_{R,p}$ .",
"$X_{R,p}$ is called the Hecke correspondence in this situation, and $Z_{R,p}$ is the Hecke correspondence for Higgs bundles.",
"The particular example of an 't Hooft operator dual to the two-dimensional representation of $G^\\vee ={\\mathrm {SU}}(2)$ is relatively simple because the space of possible Hecke modifications of a given bundle $E$ at a given point $p$ is 1-dimensional.",
"In general, the space of Hecke modifications of $E$ that can be made at $p$ by $T_{R,p}$ has a dimension that depends on $R$ and becomes arbitrarily large if $R$ is a representation of $G^\\vee $ of large highest weight.",
"When the dimension is sufficiently large, every $E^{\\prime }\\in {\\mathcal {M}}(G,C)$ can be made from $E$ by $T_{R,p}$ and the ways to do so form a complex manifold $\\Phi _{E^{\\prime },E,R,p}$ of positive dimension.",
"In such a situation, $Z_{R,p}$ is rather complicated.",
"It has a component on which the Higgs field vanishes (if $E^{\\prime }$ is produced from $E$ by a generic Hecke modification of very high weight, then no nonzero $\\varphi :E\\rightarrow E\\otimes K$ is holomorphic as a map $E^{\\prime }\\rightarrow E^{\\prime }\\otimes K$ ), and other components with nonzero Higgs field (it is possible to pick a Hecke modification of $E$ of arbitrarily high weight such that $\\varphi :E^{\\prime }\\rightarrow E^{\\prime }\\otimes K$ is holomorphic).",
"The ${\\mathrm {CP}}$ bundle of ${\\mathcal {B}}_{R,p}$ restricted to the various components is not just of rank 1.",
"For example, restricted to the component on which the Higgs field vanishes, this ${\\mathrm {CP}}$ bundle is formally (that is, modulo a proper treatment of singularities), the cohomology of $\\Phi _{E^{\\prime },E,R,p}$ ."
],
[
"'t Hooft Line Operators as Operators on Quantum States",
"Under fairly general conditions, if a symplectic manifold $M$ can be quantized by branes to get a Hilbert space ${\\mathcal {H}}$ , an $A$ -brane ${\\mathcal {B}}$ in $M\\times M$ (with additional data at the “corners,” as discussed presently) can be interpreted as a quantum operator on ${\\mathcal {H}}$ .",
"This is discussed in general in Section 4 of [22].",
"In general it is difficult to get an explicit description of such an operator.",
"But here we are in a special situation with a simple answer.",
"That is because $M={\\mathcal {M}}_H(G,C)$ is effectively a cotangent bundle $T^*{\\mathcal {M}}(G,C)$ , and the brane of interest is supported on the conormal bundle of a subvariety $X_{R,p}\\subset {\\mathcal {M}}(G,C) \\times {\\mathcal {M}}(G,C)$ .",
"A fairly general operator ${\\mathcal {O}}$ acting on the quantization of $T^*{\\mathcal {M}}(G,C)$ can be represented by an integral kernel $F(x,y)$ which is a half-density on ${\\mathcal {M}}(G,C)\\times {\\mathcal {M}}(G,C)$ , The action of ${\\mathcal {O}}$ on a state $\\Psi $ is $ {\\mathcal {O}}\\Psi (x)=\\int _{{\\mathcal {M}}(G,C)}\\mathrm {d}y \\, F(x,y) \\Psi (y).",
"$ What sort of integral kernel should we expect for the quantum operator $\\widehat{T}_{R,p}$ associated to the 't Hooft operator $T_{R,p}$ ?",
"The points $y$ and $x$ in eqn.",
"(REF ) correspond in fig.",
"REF (a) to the fields $(A,\\phi )$ just below and just above the line operator $T_{R,p}$ .",
"So they correspond to bundles $E,$ $E^{\\prime }$ , such that $E^{\\prime }$ can be reached from $E$ by a Hecke modification of type $R$ at $p$ .",
"Hence in a classical limit, $F(x,y)$ is a distribution supported on $X_{R,p}$ , which parametrizes such pairs $(E,E^{\\prime })$ .",
"The simplest case, which we will consider first, is that $F(x,y)$ is a delta function in the directions normal to $X_{r,p}$ .",
"More generally, in its dependence on the normal directions, $F(x,y)$ can be proportional to arbitrary derivatives of a delta function in the normal variables.",
"In general, one would expect quantum corrections to the claim that $F(x,y)$ is supported on the underlying classical correspondence $X_{R,p}$ .",
"However, in the present situation, there are no corrections, because the symplectic manifold that is being quantized is a cotangent bundle $T^*{\\mathcal {M}}(G,C)$ , and the correspondence $Z_{R,p}$ is a conormal bundle in $T^*{\\mathcal {M}}(G,C)\\times T^*{\\mathcal {M}}(G,C)$ .",
"The scale invariance of the cotangent bundle and the conormal bundle imply that the kernel $F(x,y)$ cannot depend on $\\hbar $ and can be evaluated in a semiclassical limit.",
"We should add a note on why it is valid here to argue based on scaling symmetry.",
"Brane quantization is based, as always, on studying $T^*{\\mathcal {M}}(G,C)$ in the context of a suitable complexification.",
"Similarly to study $Z_{R,p}$ in brane quantization involves complexifying it in a complexification of $T^*{\\mathcal {M}}(G,C)\\times T^*{\\mathcal {M}}(G,C)$ .",
"A scaling argument in brane quantization really involves scaling of the complexifications.",
"Such an argument is valid in the present setting because the structure of $T^*{\\mathcal {M}}(G,C)$ as a cotangent bundle and of $Z_{R,p}$ as a conormal bundle do extend to holomorphic stuctures of the same type for their complexifications.",
"We should also clarify what we mean by “semiclassical limit.” A one-loop correction is built into the assertion that a wavefunction is a half-density rather than a function and that $F(x,y)$ is correspondingly a half-density on ${\\mathcal {M}}(G,C) \\times {\\mathcal {M}}(G,C)$ .",
"The assertion that $F(x,y)$ can be computed semiclassically means that there is no quantum correction beyond this fact.",
"According to Etingof, Frenkel, and Kazhdan [9], the Hecke operator dual to the two-dimensional representation of ${\\mathrm {SU}}(2)$ is defined by an integral kernel that can be factored as the product of holomorphic and antiholomorphic factors.",
"Such a holomorphic factorization is expected in the $A$ -model.",
"A holomorphic factor $f$ will come from the left endpoint of $T_{R,p}$ in fig.",
"REF (a), or equivalently the lower left corner in fig.",
"REF (b), and an antiholomorphic factor $\\widetilde{f}$ will come from the right endpoint or the lower right corner.",
"We view the 't Hooft operator as a rank 1 brane of type $(B,A,A)$ in ${\\mathcal {M}}_H(G,C)\\times {\\mathcal {M}}_H(G,C)$ , with trivial ${\\mathrm {CP}}$ bundle.",
"In general, the space of corners between a brane of this type, supported on a Lagrangian submanifold $L$ , and the canonical coisotropic $A$ -brane is $H^0(L,K_L^{1/2})$ .",
"The 't Hooft operator corresponds to a brane whose support is the Hecke correspondence $Z_{R,p}$ .",
"So in this case, a holomorphic corner is a holomorphic section $f\\in H^0(Z_{R,p},K^{1/2}_{Z_{R,p}})$ .",
"Similarly, an antiholomorphic corner is an element $\\widetilde{f}\\in H^0(\\overline{Z}_{R,p},\\overline{K}^{1/2}_{\\overline{Z}_{R,p}})$ , where $\\overline{Z}_{R,p}$ is $Z_{R,p}$ with opposite complex structure.",
"The product $\\mu = f \\widetilde{f}$ of $f$ and $\\widetilde{f}$ will be a half-density on $Z_{R,p}$ .",
"We will show that this data is precisely what is needed to define a distributional kernel $F(x,y)$ .",
"If $f$ and $\\widetilde{f}$ are pull-backs from $X_{R,p}$ , we will get a delta function kernel.",
"If they have a polynomial dependence on the fiber of $Z_{R,p} \\rightarrow X_{R,p}$ , we will get a linear combination of derivatives of a delta function.",
"In order to describe a delta function distribution, we pick some local coordinates.",
"We parametrize the input to the Hecke transformation by coordinates $\\vec{x}=x_1,\\cdots , x_{3g-3}$ on ${\\mathcal {M}}(G,C)$ .",
"We will write $|\\mathrm {d}\\vec{x}|$ for the half-density $(\\mathrm {d}x_1 \\cdots \\mathrm {d}x_{3g-3} \\mathrm {d}\\overline{x}_1\\cdots \\mathrm {d}\\overline{x}_{3g-3})^{1/2}$ , and similarly for other variables introduced momentarily.",
"For a given $\\vec{x}$ , the output of the Hecke transformation ranges over a copy of ${\\mathbb {CP}}^1$ that we will call ${\\mathbb {CP}}^1_x$ ; we parametrize it by a complex variable $z$ .",
"${\\mathbb {CP}}^1_x $ is of complex codimension $3g-4$ in ${\\mathcal {M}}(G,C)$ .",
"${\\mathbb {CP}}^1_x$ can be defined locally by a condition $\\vec{n}=0$ , where $\\vec{n}=(n_1,\\cdots , n_{3g-4})$ are local holomorphic coordinates on the normal bundle $N$ to ${\\mathbb {CP}}^1_x$ in ${\\mathcal {M}}(G,C)$ .",
"We can write the kernel as $F(x,y)= b(z,\\overline{z},x, \\overline{x})|\\mathrm {d}\\vec{x} ||\\mathrm {d}z \\mathrm {d}\\vec{n} |\\delta (\\vec{n},\\vec{\\overline{n}})$ where $\\mathrm {d}\\vec{n}=\\mathrm {d}n_1 \\mathrm {d}n_2\\cdots \\mathrm {d}n_{g-4}$ and the delta function is defined by $\\int \\mathrm {d}\\vec{n} \\mathrm {d}\\vec{ \\overline{n}}\\delta (\\vec{n},\\vec{\\overline{n}})=1$ .",
"In view of that last relation, the delta function transforms under a change of coordinates on the normal bundle as $(\\mathrm {d}\\vec{n} \\mathrm {d}\\vec{\\overline{n}})^{-1}$ , which means that the possible kernels are in one-to-one correspondence with objects $ \\mu = b(z,\\overline{z},x, \\overline{x})|\\mathrm {d}\\vec{x}| |\\mathrm {d}z(\\mathrm {d}\\vec{n})^{-1}|.",
"$ The Hecke correspondence $Z_{R,p}$ for Higgs bundles is parametrized locally by the coordinates $\\vec{x}$ and $z$ , introduced above, which parametrize the Hecke correspondence $X_{R,p}$ for bundles, and additional variables $\\vec{m}$ that parametrize the choice of Higgs field.",
"Since $Z_{R,p}$ is the conormal bundle of $X_{R,p}$ , the variables $\\vec{m}$ are dual to the normal bundle coordinates $\\vec{n}$ that appear in eqn.",
"(REF ).",
"Therefore, we can replace $(\\mathrm {d}\\vec{n})^{-1}$ with $\\mathrm {d}\\vec{m}$ , and the possible delta function kernels are in one-to-one correspondence with half-densities $ \\mu = b(z,\\overline{z},x, \\overline{x})|\\mathrm {d}\\vec{x} \\mathrm {d}z\\mathrm {d}\\vec{m}|^2.",
"$ on $Z_{R,p}$ such that $b$ is independent of $\\vec{m}$ .",
"This discussion is immediately generalized to linear combinations of normal derivatives of a delta function.",
"A kernel that involves normal derivatives of a delta function $F(x,y)= b(x, \\overline{x},z,\\overline{z}, \\partial _{\\vec{n}}, \\partial _{\\vec{\\overline{n}}}) |\\mathrm {d}\\vec{x} | |\\mathrm {d}z \\mathrm {d}\\vec{n} |\\delta (\\vec{n},\\vec{\\overline{n}})$ corresponds to a half-density $ \\mu = b(x, \\overline{x}, z,\\overline{z},\\vec{m}, \\vec{\\overline{m}})|\\mathrm {d}\\vec{x} \\mathrm {d}z\\mathrm {d}\\vec{m}|.",
"$ on $Z_{R,p}$ such that $b$ depends polynomially on $\\vec{m}$ .",
"A holomorphically factorized kernel will take the form $\\mu = f \\widetilde{f}$ where $f$ and $\\widetilde{f}$ are respectively holomorphic and antiholomorphic.",
"We can now see what kind of holomorphic object $f$ must be.",
"$f$ must be a half-density on the Hecke correspondence $Z_{R,p}$ in the holomorphic sense: $f=v(\\vec{x}, z,\\vec{m}) (\\mathrm {d}\\vec{x}\\mathrm {d}z\\mathrm {d}\\vec{m})^{1/2}$ .",
"In more standard language, $f$ must be an element of $H^0(Z_{R,p}, K^{1/2}_{Z_{R,p}})$ .",
"As explained earlier, this is the expected form of the answer in the $A$ -model for a left endpoint of $T_{R,p}$ .",
"Similarly, $\\widetilde{f}$ is an antiholomorphic section of the anticanonical bundle of $Z_{R,p}$ , again in accord with the $A$ -model expectation.",
"If and only if $Z_{R,p}$ is a Calabi-Yau manifold, there is a particular holomorphic section $\\lambda _0$ of $K_{Z_{R,p}}^{1/2}$ that is everywhere nonzero.",
"If such a $\\lambda _0$ exists, then the data that defines any other holomorphic corner is $f= g \\lambda _0$ , where $g$ is a holomorphic function on $Z_{R,p}$ .",
"In particular, such a function $g$ (with only polynomial growth) is a polynomial in the Hitchin Hamiltonians $H_{{\\mathcal {P}},\\alpha }$ and the holomorphic function $\\delta $ that was defined in eqn.",
"(REF ), up to the relation $\\delta ^2 = -\\det \\varphi (p)$ , which expresses $\\delta ^2$ as a linear combination of the $H_{{\\mathcal {P}},\\alpha }$ .",
"A nonconstant polynomial $g(H_{{\\mathcal {P}},\\alpha },\\delta )$ has nontrivial zeroes, so $g\\lambda _0$ is everywhere nonzero only if $g$ is a constant, showing that $\\lambda _0$ is unique, up to a constant multiple, if it exists.More generally, if $Z$ is a complex manifold with $b_1(Z)=0$ , then an everywhere nonzero holomorphic function $g$ on $Z$ , with no exponential growth, is constant.",
"Indeed, since $b_1(Z)=0$ , the closed 1-form $\\mathrm {d}g/g$ is exact, $\\mathrm {d}g/g=\\mathrm {d}w$ for some $w$ , so $g=C e^w$ (with a nonzero constant $C$ ) and $g$ has exponential growth unless it is constant.",
"The Hecke correspondence for $G_{\\mathbb {C}}$ -bundles satisfies $b_1(X_{R,p})=0$ .",
"The Hecke correspondence for $G_{\\mathbb {C}}$ Higgs bundles is the conormal bundle of $X_{R,p}$ and hence $b_1(Z_{R,p})=0$ .",
"Corners of the form $g \\lambda _0$ , where $g$ is a polynomial in the Hitchin Hamiltonians and $\\delta $ , precisely match the corners for Wilson lines, built from $s$ , $Ds$ and polynomials in the observables dual to the Hitchin Hamiltonians.",
"Such a $\\lambda _0$ does indeed exist, by virtue of a result of Beilinson and Drinfeld [1] that was important in the work of Etingof, Frenkel, and Kazhdan [9].",
"The properties of $\\lambda _0$ mirror those of the simplest Wilson line corner $s$ of Section REF .",
"In particular, according to the result of Beilinson and Drinfeld, $\\lambda _0$ , like $s$ , varies with $p$ as a section of $K_p^{-1/2}$ .",
"Of course, the existence of such a mirror of $s$ is expected from electric-magnetic duality.",
"A slightly different formulation was useful in [9].",
"To explain this, let us go back to eqn.",
"(REF ), from which we see that if $\\mu $ can be holomorphically factorized, then the holomorphic factor is a holomorphic form $k = w(\\vec{x},z) (\\mathrm {d}\\vec{x} \\mathrm {d}z)^{1/2} (\\mathrm {d}\\vec{n})^{-1/2}$ .",
"We can replace $\\mathrm {d}z^{1/2} (\\mathrm {d}\\vec{n})^{-1/2}$ with $\\mathrm {d}z (\\mathrm {d}\\vec{y})^{-1/2}$ where $\\vec{y}=(z,\\vec{n})$ parametrizes the output of the Hecke transformation.",
"So in other words a holomorphically factorized kernel will come from a holomorphic objectSince $\\vec{y}$ is determined by $\\vec{x}$ and $z$ , and reciprocally $\\vec{x}$ is determined by $\\vec{y}$ and $z$ , we could equally well write $w(\\vec{y},z)$ instead of $w(\\vec{x},z)$ .",
"$ k=w(\\vec{x},z) (\\mathrm {d}\\vec{x})^{1/2} (\\mathrm {d}\\vec{y})^{-1/2}\\mathrm {d}z.",
"$ If we multiply $k$ by its complex conjugate, we get $|k|^2= |w(\\vec{x},z)|^2 |\\mathrm {d}\\vec{x}| |\\mathrm {d}z\\mathrm {d}\\overline{z}| |\\mathrm {d}\\vec{y}|^{-1}.$ This leads directly to the definition used in [9].",
"The quantity $|k|^2$ can be regarded as a map from half-densities in $\\vec{y}$ to half-densities in $\\vec{x}$ , valued in differential forms $|\\mathrm {d}z \\mathrm {d}\\overline{z}|$ that can be integrated over ${\\mathbb {CP}}^1_x$ .",
"That integral gives the Hecke operator at the point $p\\in C$ : $ H_p=\\int _{{\\mathbb {CP}}^1_x} |k|^2.",
"$ In Section , we will interpret some of these statements via two-dimensional chiral algebras.",
"In particular, the considerations about adjoint-valued chiral fermions in Section REF are a physicist's interpretation of the original analysis of Beilinson and Drinfeld.",
"The chiral algebra approach is local on $C$ and and can potentially be extended to situations where the space of Hecke modifications relating two given bundles has positive dimension."
],
[
"The Affine Grassmannian",
"So far, we have considered the simplest examples of Hecke modifications.",
"But to develop the theory further, one wants a more systematic approach.",
"As motivation, we consider first the case of a holomorphic vector bundle of rank $n$ .",
"Let $z$ be a local holomorphic parameter that vanishes at a point $p\\in C$ .",
"Let $U$ be a small neighborhood of $p$ .",
"$C$ has an open cover with two open sets, namely $U$ and $C^{\\prime }=C\\backslash p$ ($C$ with $p$ removed).",
"Pick a trivialization of $E$ in a small neighborhood $U$ of the point $p$ and restrict $E$ to $C^{\\prime }$ .",
"Let $E_0\\rightarrow U$ be a trivial rank $n$ vector bundle.",
"We have an open cover of $C$ by open sets $C^{\\prime }, U$ with vector bundles $E\\rightarrow C^{\\prime }$ and $E_0\\rightarrow U$ .",
"So we can define a new vector bundle $E^{\\prime }\\rightarrow C$ by gluing together $E$ and $E_0$ over $U^{\\prime }=C^{\\prime }\\cap U$ via a gauge transformation.",
"For example, we can use the diagonal gauge transformation from $E_0$ to $E$ $ g(z) =\\begin{pmatrix} z^{d_1} &&&\\cr & z^{d_2} &&\\cr &&\\ddots & \\cr &&& z^{d_n}\\end{pmatrix},$ with integers $d_1,\\cdots , d_n$ .",
"For $G={\\mathrm {U}}(n)$ , with suitable choices of the $d_i$ , this gives an example of a Hecke modification of a $G_{\\mathbb {C}}$ -bundle dual to an arbitrary irreducible finite-dimensional representation of $G={\\mathrm {U}}(n)$ .",
"For $G={\\mathrm {SU}}(n)$ , one modifies this by requiring $\\sum _i d_i=0$ so that $g(z)$ is valued in ${\\mathrm {SL}}(n,{\\mathbb {C}})$ , and for $G={\\mathrm {PSU}}(n)$ , one considers the $d_i$ to be valued in $\\frac{1}{n}{\\mathbb {Z}}$ , with $d_i-d_j\\in {\\mathbb {Z}}$ (this is equivalent to saying that if $g(z)$ is written in a representation of ${\\mathrm {PSU}}(n)$ , then only integer powers of $z$ appear).",
"A constant shift of all $d_i$ by $d_i\\rightarrow d_i+c$ , where $c\\in {\\mathbb {Z}}$ (or $c\\in \\frac{1}{n}{\\mathbb {Z}}$ for $G={\\mathrm {PSU}}(n)$ ) does not affect the space of Hecke modifications, since one can compensate for it by $E^{\\prime }\\rightarrow E^{\\prime }\\otimes {\\mathcal {O}}(p)^c$ .",
"For any $G$ , one can make a similar construction replacing diagonal matrices whose entries are powers of $z$ with a homomorphism $g(z)= z^{\\sf m}:{\\mathbb {C}}^*\\rightarrow T_{\\mathbb {C}}$ where ${\\mathbb {C}}^*$ is the punctured $z$ -plane, $T_{\\mathbb {C}}$ is a complex maximal torus of $G_{\\mathbb {C}}$ , and ${\\sf m}$ , which generalizes the $d$ -plet of integers $(d_1,\\cdots , d_n)$ in the previous paragraph, is an integral weight of the dual group $G$ , corresponding physically to the magnetic charge of an 't Hooft operator.",
"What we have described so far is a standard example of a Hecke transformation dual to an arbitrary finite-dimensional representation $R$ of $G$ ; $R$ is encoded in the integers $d_i$ or the choice of ${\\sf m}$ .",
"This construction depended on the initial choice of a trivialization of $E$ over $U$ .",
"By varying the choice of trivialization, one can obtain a whole space of Hecke modifications of the same type.",
"Once we pick a reference trivialization, any other trivialization of $E$ over the set $U$ would be obtained from the reference one by applying some gauge transformation $g^{\\prime }(z)$ that is holomorphic in $U$ .",
"The standard Hecke modification associated to this alternative trivialization of $E$ is described in the reference trivialization by the modified singular gauge transformation $g^{\\prime }(z) z^{\\sf m}$ .",
"Two singular gauge transformations lead to the same $E^{\\prime }$ if they can be related by composition from the right with a gauge transformation $g^{\\prime \\prime }(z)$ defined on $U$ .",
"Two trivializations of $E$ will thus give the same standard Hecke modification if the corresponding gauge transformations $g_1^{\\prime }(z)$ and $g_2^{\\prime }(z)$ satisfy $g_1^{\\prime }(z)z^{\\sf m}= g_2^{\\prime }(z)z^{\\sf m}g^{\\prime \\prime }(z)$ for some $g^{\\prime \\prime }(z)$ .",
"These relations can be formalized with the help of the affine Grassmannian ${\\mathrm {Gr}}_{G_{\\mathbb {C}}}$ .",
"It is customary to denote as $G_{\\mathbb {C}}[{\\cal K}]$ the (infinite-dimensional) space of $G_{\\mathbb {C}}$ -valued gauge transformations defined on $U^{\\prime }$ and as $G_{\\mathbb {C}}[{\\cal O}]\\subset G_{\\mathbb {C}}[{\\cal K}]$ the subspace of such gauge transformations which extend holomorphically to $U$ .",
"The affine Grassmannian ${\\mathrm {Gr}}_{G_{\\mathbb {C}}} \\equiv G_{\\mathbb {C}}[{\\cal K}]/G_{\\mathbb {C}}[{\\cal O}]$ parameterizes equivalent singular gauge transformations.",
"We define ${\\mathrm {Gr}}^{\\sf m}_{G_{\\mathbb {C}}}$ to be the orbit in ${\\mathrm {Gr}}_{G_{\\mathbb {C}}}$ of the standard Hecke modification by the gluing function $z^{\\sf m}$ : ${\\mathrm {Gr}}^{\\sf m}_{G_{\\mathbb {C}}} \\equiv \\left[G_{\\mathbb {C}}[{\\cal O}] z^{\\sf m}\\right].$ Every point in ${\\mathrm {Gr}}^{\\sf m}_{G_{\\mathbb {C}}}$ is on such an orbit, for some ${\\sf m}$ .",
"For generic choices of $G$ and ${\\sf m}$ , one runs into a phenomenon of “monopole bubbling” in which a downward jump can occur in the magnetic charge of an 't Hooft or Hecke operator (this was introduced in [36], [37]; see Section 10.2 of [3] for a short introduction).",
"Essentially, the orbit ${\\mathrm {Gr}}^{\\sf m}_{G_{\\mathbb {C}}}$ is not closed in the affine Grassmannian, and the (possibly singular) closure of the orbit includes other orbits ${\\mathrm {Gr}}^{{\\sf m}^{\\prime }}_{G_{\\mathbb {C}}}$ with smaller charge.",
"This can considerably complicate the analysis.",
"For $G={\\mathrm {U}}(n)$ or a related group ${\\mathrm {SU}}(n)$ or $\\mathrm {PSU}(n)$ , the condition to avoid monopole bubbling is that $|d_i-d_j|\\le 1$ for all $i,j$ .",
"Up to a constant shift of all the $d_i$ (which does not affect the space of Hecke modifications, as explained in the discussion of eqn.",
"(REF )), to avoid monopole bubbling we can assume that $k$ of the $d_i$ are $-1$ and the others 0.",
"Explicitly, this corresponds to a Hecke modification of the following sort.",
"For $E\\rightarrow C$ a rank $n$ holomorphic vector bundle and $p\\in C$ , one chooses a $k$ -dimensional subspace $V\\subset E_p$ and defines a new vector bundle $E^{\\prime }\\rightarrow C$ whose sections are sections of $E$ that are allowed to have a simple pole at $p$ with residue in $V$ .",
"The space of Hecke modifications of this type is parametrized by the choice of $V$ , that is, by the Grassmannian of $k$ -dimensional subspaces of $E_p$ .",
"Such Hecke modifications are dual to the $k^{th}$ antisymmetric power of the fundamental representation of $G={\\mathrm {U}}(n)$ or ${\\mathrm {SU}}(n)$ .",
"In this situation, $\\varphi $ will act as a $k \\times k$ matrix on the polar part of $w$ .",
"The characteristic polynomial of the $k \\times k$ restriction of $\\varphi $ takes the general form $u(x) = x^k + \\delta _1 x^{k-1} + \\cdots \\delta _k$ , for some holomorphic functions $\\delta _1,\\cdots ,\\delta _k$ .",
"These functions satisfy polynomial relationships with the Hitchin Hamiltonians ${\\mathcal {H}}_{{\\mathcal {P}},\\alpha }(\\varphi )$ which encode the constraint that $u(x)$ divides the characteristic polynomial of $\\varphi $ ."
],
[
"Quantum States And The Hitchin Fibration",
"A slightly different way to think about an eigenfunction of the Wilson operators and the Hitchin Hamiltonians is as follows.",
"Let $x$ be a point in ${\\mathcal {M}}_H(G^\\vee ,C)$ corresponding to a flat $G^\\vee _{\\mathbb {C}}$ bundle $E^\\vee _x\\rightarrow C$ , and let ${\\mathcal {B}}_x$ be a $B$ -brane supported at $x$ , with a rank 1 (and inevitably trivial) ${\\mathrm {CP}}$ bundle.",
"Then the part of ${\\mathrm {Hom}}({\\mathcal {B}}_x,{\\mathcal {B}}_{\\mathrm {op}})$ of degree zeroThe $B$ -model has a conserved fermion number symmetry, with the differential $Q$ having fermion number or degree 1.",
"When $x\\in L_{\\mathrm {op}}$ , ${\\mathrm {Hom}}({\\mathcal {B}}_x,{\\mathcal {B}}_{\\mathrm {op}})$ is also nonzero in positive degrees, but the positive degree states do not contribute in this discussion because ${\\mathrm {Hom}}({\\mathcal {B}}_{\\overline{\\mathrm {op}}},{\\mathcal {B}}_{\\mathrm {op}})$ is entirely in degree 0.",
"A similar remark applies later when we discuss the $A$ -model.",
"is a copy of ${\\mathbb {C}}$ if $x\\in L_{\\mathrm {op}}$ , that is if $E^\\vee _x$ is a holomorphic oper.",
"Otherwise ${\\mathrm {Hom}}({\\mathcal {B}}_x,{\\mathcal {B}}_{\\mathrm {op}})=0$ .",
"Similarly, the degree 0 part of ${\\mathrm {Hom}}({\\mathcal {B}}_{\\overline{\\mathrm {op}}},{\\mathcal {B}}_x)$ is ${\\mathbb {C}}$ if $x\\in L_{\\overline{\\mathrm {op}}}$ , that is if $E^\\vee _x$ is an antiholomorphic oper, and otherwise zero.",
"If and only if $E^\\vee $ is both an oper and an anti-oper, we can pick nonzero elements $\\alpha \\in {\\mathrm {Hom}}({\\mathcal {B}}_x,{\\mathcal {B}}_{\\mathrm {op}})$ , $\\beta \\in {\\mathrm {Hom}}({\\mathcal {B}}_{\\overline{\\mathrm {op}}},{\\mathcal {B}}_x)$ , and then define the element $\\alpha \\circ \\beta \\in {\\mathcal {H}}={\\mathrm {Hom}}({\\mathcal {B}}_{\\overline{\\mathrm {op}}},{\\mathcal {B}}_{\\mathrm {op}})$ .",
"A picture representing this situation is fig.",
"REF (a).",
"The brane ${\\mathcal {B}}_x$ is used to provide a boundary condition at the bottom of the strip; $\\alpha $ and $\\beta $ provide the “corner data” needed to define boundary conditions at the corners of the picture.",
"The path integral with the “initial conditions” set by ${\\mathcal {B}}_x,\\alpha ,\\beta $ defines a physical state of the system.",
"This state is the desired eigenstate of the Hitchin Hamiltonians and the Wilson operators.",
"Via electric-magnetic duality, we can get a dual picture.",
"After compactification on an oriented two-manifold $C$ , electric-magnetic duality of ${\\mathcal {N}}=4$ super Yang-Mills theory reduces at low energies to a mirror symmetry of the Higgs bundle moduli space [26], [27].",
"As explained in [38], this is a rare instance in which the SYZ interpretation [39] of mirror symmetry as $T$ -duality on the fibers of a family of Lagrangian tori can be made very explicit.",
"The Hitchin fibration is the map that takes a Higgs bundle $(E,\\varphi )$ to the characteristic poiynomial of $\\varphi $ .",
"The fibers of the map are abelian varieties that are complex Lagrangian submanifolds in complex structure $I$ .",
"This in particular means that they are Lagrangian submanifolds from the point of view of the real symplectic structure $\\omega _K={\\mathrm {Im}}\\,\\Omega _I$ of the Higgs bundle moduli space.",
"Hence, in the $A$ -model with symplectic structure $\\omega _K$ , the Hitchin fibration can be viewed as an SYZ fibration by Lagrangian submanifolds that generically are tori.",
"$T$ -duality on the fibers of this fibration maps the $A$ -model of symplectic structure $\\omega _K$ to the $B$ -model of complex structure $J$ .",
"The relation of electric-magnetic duality to this instance of mirror symmetry was an important input in [3].",
"In particular, mirror symmetry in this situation maps a rank 1 brane ${\\mathcal {B}}_x$ supported at a point $x$ to a brane ${\\mathcal {B}}_F$ supported on a fiber $F$ of the Hitchin fibration, with a ${\\mathrm {CP}}$ bundle that is a flat line bundleIn general, the ${\\mathrm {CP}}$ bundle of a rank 1 brane is more canonically a ${\\mathrm {Spin}}_c$ structure rather than a line bundle.",
"In the present context, as $F$ is an abelian variety and so has a canonical spin structure, the distinction is not important.",
"${\\mathcal {S}}\\rightarrow F$ .",
"The duals of $\\alpha \\in {\\mathrm {Hom}}({\\mathcal {B}}_x,{\\mathcal {B}}_{\\mathrm {op}})$ and $\\beta \\in {\\mathrm {Hom}}({\\mathcal {B}}_{\\overline{\\mathrm {op}}},{\\mathcal {B}}_x)$ are elements $\\alpha ^{\\prime }\\in {\\mathrm {Hom}}({\\mathcal {B}}_F,{\\mathcal {B}}_{\\mathrm {cc}})$ and $\\beta ^{\\prime }\\in {\\mathrm {Hom}}(\\overline{{\\mathcal {B}}}_{\\mathrm {cc}},{\\mathcal {B}}_F)$ .",
"The element $\\alpha ^{\\prime }\\circ \\beta ^{\\prime }\\in {\\mathcal {H}}$ , which corresponds to the picture of fig.",
"REF (b), represents an element of ${\\mathcal {H}}$ in the magnetic description, in which the Hitchin Hamiltonians are differential operators acting on half-densities on ${\\mathcal {M}}(G,C)$ and Wilson operators are replaced by 't Hooft operators.",
"The branes ${\\mathcal {B}}_x$ and ${\\mathcal {B}}_F$ both have compact support, consisting of either a point $x$ or a fiber $F$ of the Hitchin fibration.",
"Compact support makes it manifest that the states created by these branes (plus corner data) are normalizable.",
"More than that, compact support means that these states have well-defined pairings with states that are constructed similarly using an arbitrary brane ${\\mathcal {B}}^{\\prime }$ , possibly with noncompact support, again with suitable corner data.",
"In other words, the pairing constructed from the rectangle of fig.",
"REF (c) is always well-defined, regardless of the brane at the top of the rectangle, as long as the brane at the bottom of the rectangle has compact support.",
"This means roughly that an arbitrary brane, with a choice of corner data, defines a distributional state, not necessarily a normalizable vector in the Hilbert space ${\\mathcal {H}}$ , while a brane of compact support defines a vector that can be paired with any distribution.",
"The Hilbert space is contained in the space of distributional states, and contains a subspace, roughly analogous to a Schwartz space, spanned by states associated to branes of compact support.",
"The eigenfunctions of the Hitchin Hamiltonians lie in this subspace.",
"The $B$ -model gives a clear answer to the question of which points $x\\in {\\mathcal {M}}_H(G^\\vee ,C)$ are associated to physical states: these are points in $\\Upsilon =L_{\\mathrm {op}}\\cap L_{\\overline{\\mathrm {op}}}$ .",
"It is more difficult to extract directly from the $A$ -model a prediction for which pairs $F,{\\mathcal {S}}$ are similarly associated to physical states.",
"We will only be able to get a sort of semiclassical answer, which we expect to be valid asymptotically, in a sense that will be explained.",
"We recall that the ${\\mathrm {CP}}$ bundles of the branes ${\\mathcal {B}}_{\\mathrm {cc}}$ and $\\overline{{\\mathcal {B}}}_{\\mathrm {cc}}$ are dual line bundles ${\\mathcal {L}}$ and ${\\mathcal {L}}^{-1}$ , and that the prequantum line bundle of ${\\mathcal {M}}_H(G,C)$ , in the sense of geometric quantization, is ${{\\mathfrak {L}}}={\\mathcal {L}}^2$ .",
"We want the condition on $F$ and ${\\mathcal {S}}\\rightarrow F$ such that corners $\\alpha ^{\\prime }\\in {\\mathrm {Hom}}({\\mathcal {B}}_F,{\\mathcal {B}}_{\\mathrm {cc}})$ and $\\beta ^{\\prime }\\in {\\mathrm {Hom}}(\\overline{{\\mathcal {B}}}_{\\mathrm {cc}},{\\mathcal {B}}_F)$ exist.",
"We will describe the condition for $\\alpha ^{\\prime }$ and $\\beta ^{\\prime }$ to exist to lowest order in $\\sigma $ -model perturbation theory, and we will also explain to what extent higher order corrections can or cannot change the picture, in the regime where they are small.",
"So let us first explain the regime in which $\\sigma $ -model perturbation theory is valid.",
"This is the case that the Higgs field $\\varphi $ is parametrically large and far away from the discriminant locus (on which the spectral curve becomes singular).",
"Concretely if $(E,\\varphi )$ is any Higgs bundle with a smooth spectral curve, and we rescale $\\varphi $ by a large factor $ \\varphi \\rightarrow t\\varphi ,~~|t|\\gg 1,$ with any fixed value of $\\mathrm {Arg}\\,t$ , then $\\sigma $ -model perturbation theory becomes valid.",
"For $t\\rightarrow \\infty $ , the Higgs bundle moduli space has a concrete “semi-flat” description [40], leading to semiclassical results and asymptotic expansions as $t\\rightarrow \\infty $ for a variety of questions, including what we will consider here.",
"We will call the region $t\\rightarrow \\infty $ the WKB limit, for reasons that will emerge.",
"The brane ${\\mathcal {B}}_F$ is of type $(B,A,A)$ .",
"For such a brane, with support $F$ and ${\\mathrm {CP}}$ bundle ${\\mathcal {S}}$ , the leading $\\sigma $ -model approximation to ${\\mathrm {Hom}}({\\mathcal {B}}_F,{\\mathcal {B}}_{\\mathrm {cc}})$ , where ${\\mathcal {B}}_{\\mathrm {cc}}$ has ${\\mathrm {CP}}$ bundle ${\\mathcal {L}}$ , is the $\\overline{\\partial }$ cohomology of $F$ with values inAs remarked in footnote REF , ${\\mathcal {S}}$ and likewise ${\\mathcal {S}}^{-1}$ is canonically a ${\\mathrm {Spin}}_c$ structure on $F$ , not a line bundle.",
"Likewise, $K_F^{1/2}$ is not canonically defined as a line bundle (and for general may not exist at all as a line bundle) but on any complex manifold, $K_F^{1/2}$ is canonically defined as a ${\\mathrm {Spin}}_c$ structure.",
"Hence the product ${\\mathcal {S}}^{-1}\\otimes K_F^{1/2}$ exists canonically as an ordinary line bundle and the answer stated in the text makes sense.",
"In our application, $K_F$ is trivial, and we can likewise take $K_F^{1/2}$ to be trivial and define ${\\mathcal {S}}$ as a line bundle.",
"${\\mathcal {L}}|_F\\otimes {\\mathcal {S}}^{-1}\\otimes K_F^{1/2}$ , where $K_F^{1/2}$ is a square root of the canonical bundle of $F$ (see Appendix B of [22]).",
"This comes about because the leading $\\sigma $ -model approximation to the $A$ -model differential $Q$ is $Q\\sim \\overline{\\partial }+a , $ where $a$ is a $(0,1)$ -form on $F$ that defines the complex structure of the line bundle ${\\mathcal {L}}\\otimes {\\mathcal {S}}^{-1}\\otimes K_F^{1/2}$ .",
"The line bundle ${\\mathcal {L}}$ is flat when restricted to $F$ , because $F$ is a complex Lagrangian submanifold.",
"The line bundle ${\\mathcal {S}}$ is flat, because it is the ${\\mathrm {CP}}$ bundle of a Lagrangian brane.",
"And as $F$ is a complex torus, we can take $K_F^{1/2}$ to be trivial and omit this factor.",
"The $\\overline{\\partial }$ cohomology of a complex torus with values in a flat line bundle ${\\mathcal {L}}|_F\\otimes {\\mathcal {S}}^{-1}$ vanishes unless this line bundle is trivial.",
"So in the leading $\\sigma $ -model approximation, ${\\mathrm {Hom}}({\\mathcal {B}}_F,{\\mathcal {B}}_{\\mathrm {cc}})$ vanishes unless we pick ${\\mathcal {S}}\\cong {\\mathcal {L}}|_F$ .",
"For ${\\mathcal {S}}={\\mathcal {L}}|_F$ ,we can pick a nonzero $\\alpha ^{\\prime } \\in H^0(F,{\\mathcal {L}}|_F\\otimes {\\mathcal {S}}^{-1})\\cong {\\mathbb {C}}$ .",
"Similarly, as the ${\\mathrm {CP}}$ bundle of ${\\mathcal {B}}_{\\overline{\\mathrm {op}}}$ is ${\\mathcal {L}}^{-1}$ , the leading $\\sigma $ -model approximation to ${\\mathrm {Hom}}({\\mathcal {B}}_{\\overline{\\mathrm {op}}},{\\mathcal {B}}_F)$ is the $\\overline{\\partial }$ cohomology of $F$ with values in ${\\mathcal {S}}\\otimes {\\mathcal {L}}|_F\\otimes K_F^{1/2}$ , or, taking $K_F^{1/2}$ to be trivial, just ${\\mathcal {S}}\\otimes {\\mathcal {L}}|_F$ .",
"This cohomology vanishes if ${\\mathcal {S}}\\otimes {\\mathcal {L}}|_F$ is nontrivial; if it is trivial, which is so precisely if ${\\mathcal {S}}={\\mathcal {L}}^{-1}|_F$ , we can choose a nonzero $\\beta ^{\\prime }\\in H^0(F, {\\mathcal {S}}\\otimes {\\mathcal {L}}|_F)\\cong {\\mathbb {C}}$ .",
"In short, we can use the brane ${\\mathcal {B}}_F$ with suitable corner data to define a state in ${\\mathcal {H}}$ if and only if we can choose ${\\mathcal {S}}$ to be isomorphic to both ${\\mathcal {L}}|_F$ and ${\\mathcal {L}}^{-1}|_F$ .",
"In other words, the condition is that ${\\mathcal {L}}^2|_F$ must be trivial.",
"As the prequantum line bundle over ${\\mathcal {M}}_H(G,C)$ is ${{\\mathfrak {L}}}={\\mathcal {L}}^2$ , the condition is that ${{\\mathfrak {L}}}$ must be trivial when restricted to $F$ .",
"This is actually the WKB condition of elementary quantum mechanics, which also is part of the theory of geometric quantization.",
"To put the condition in a more familiar form, recall that the symplectic form $\\omega _K$ of ${\\mathcal {M}}_H(G,C)$ is cohomologically trivial, so it can be written as $\\omega _K=\\mathrm {d}\\lambda _K$ for a 1-form $\\lambda _K$ .",
"The prequantum line bundle ${{\\mathfrak {L}}}$ is supposed to be a unitary line bundle with a connection of curvature $\\omega _K$ .",
"We can take ${{\\mathfrak {L}}}$ to be a trivial line bundle with the connection $D=\\mathrm {d}+{\\mathrm {i}}\\lambda _K$ .",
"${{\\mathfrak {L}}}$ is flat when restricted to $F$ because $F$ is a Lagrangian submanifold.",
"The condition that ${{\\mathfrak {L}}}$ is trivial is that its global holonomies vanish.",
"In other words, the condition is that if $\\gamma \\subset F$ is a 1-cycle, then the holonomy of ${{\\mathfrak {L}}}$ around $\\gamma $ must vanish: $\\exp ({\\mathrm {i}}\\oint _\\gamma \\lambda _K)=1$ or in other words $\\oint _\\gamma \\lambda _K\\in 2\\pi {\\mathbb {Z}}$ .",
"To put this condition in a perhaps more familiar form, we can approximate ${\\mathcal {M}}_H(G,C)$ as a cotangent bundle $T^*{\\mathcal {M}}(G,C)$ and choose $\\lambda _K=\\sum _i p_i\\mathrm {d}q^i$ , where $q^i$ are coordinates on the base of the cotangent bundle and $p_i$ are fiber coordinates.",
"Then the condition is that $\\sum _i\\oint _\\gamma p_i \\mathrm {d}q^i\\in 2\\pi {\\mathbb {Z}},$ which may be recognizable as the WKB condition for associating a quantum state to the Lagrangian submanifold $F$ .",
"We have reached this conclusion to lowest order in $\\sigma $ -model (or gauge theory) perturbation theory, and we do not claim that the result is exact.",
"However, it is possible to argue that, at least sufficiently near the WKB limit, there is a quantum-corrected WKB condition that leads to qualitatively similar results.",
"Consider correcting the computation of ${\\mathrm {Hom}}(\\overline{{\\mathcal {B}}}_{\\mathrm {cc}},{\\mathcal {B}}_F)$ or ${\\mathrm {Hom}}({\\mathcal {B}}_F,{\\mathcal {B}}_{\\mathrm {cc}})$ in perturbation theory in inverse powers of the parameter $t$ that was introduced in eqn.",
"(REF ).",
"This has the effect of shifting the $(0,1)$ -form $a$ by a $(0,1)$ -form $c_1/t+c_2/t^2+\\cdots $ , leading to $Q=\\overline{\\partial }+a +\\frac{c_1}{t}+\\frac{c_2}{t^2}+\\cdots .$ Whatever the $c_k$ are, we can compensate for them by shifting $a$ .",
"An arbitrary shift in $a$ can be interpreted as the sum of a $\\overline{\\partial }$ -exact term, which does not affect the cohomology of $Q$ , plus a term that can be interpreted as resulting from a shift in the line bundle ${\\mathcal {S}}$ .",
"Thus, instead of needing ${\\mathcal {L}}|_F\\otimes {\\mathcal {S}}^{-1}$ and ${\\mathcal {L}}|_F\\otimes {\\mathcal {S}}$ to be trivial in order to associate a quantum state with $F$ , we need ${\\mathcal {S}}$ to satisfy conditions that are asymptotically close to these.",
"Correspondingly, the classical WKB condition for ${{\\mathfrak {L}}}|_F$ to be trivial is modified, at least for sufficiently large $t$ , to a quantum WKB condition that determines which fibers of the Hitchin fibration are associated to quantum states."
],
[
"WKB Condition and Special Geometry",
"In order to better understand the quantization condition, it is useful to recall some facts about the special geometry which governs the structure of ${\\mathcal {M}}_H(G,C)$ , as well as complex integrable systems that arise in Seiberg-Witten theory [41], [42], [43], [44], [45].",
"The basic ingredients of the geometry are The base ${{\\mathcal {B}}}$ of the Hitchin fibration, of complex dimension $r$ .",
"We denote a point in the base as $u$ and the discriminant locus as ${\\cal D}$ .",
"A local system $\\Gamma $ of lattices of rank $2r$ defined over ${\\cal B}\\backslash {\\cal D}$ ($\\cal B$ with $\\cal D$ removed), equipped with a symplectic form $\\langle \\cdot ,\\cdot \\rangle $ .In a true Seiberg-Witten geometry the symplectic form is integer-valued and $\\Gamma $ is self-dual.",
"The Seiberg-Witten geometry is self-mirror.",
"Hitchin systems are related to Seiberg-Witten geometries by discrete orbifold operations which relax these conditions.",
"We will denote a charge (an element of $\\Gamma $ ) as $\\gamma $ .",
"A collection of central charges $Z: \\Gamma \\rightarrow {\\mathbb {C}}$ which vary holomorphically on ${\\cal B}\\backslash {\\cal D}$ .",
"We will denote the central charge evaluated on a charge $\\gamma $ as $Z_\\gamma $ .",
"The $Z_\\gamma $ are also identified with periods of the canonical 1-form on the spectral curve of the Hitchin system.",
"Real angular coordinates $\\theta : \\Gamma \\rightarrow S^1$ on the fibers of the complex integrable system.",
"We will denote the coordinates evaluated on a charge $\\gamma $ as $\\theta _\\gamma $ .",
"They are dual to the $Z_\\gamma $ under the complex Poisson bracket $\\lbrace Z_\\gamma , \\theta _{\\gamma ^{\\prime }} \\rbrace = \\langle \\gamma , \\gamma ^{\\prime } \\rangle $ The complex symplectic form in complex structure $I$ is defined with the help of the inverse pairing: $\\Omega = \\langle \\mathrm {d}Z, \\mathrm {d}\\theta \\rangle $ Correspondingly, we have a 1-form $\\lambda = \\langle Z, \\mathrm {d}\\theta \\rangle \\equiv \\lambda _J + {\\mathrm {i}}\\lambda _K$ satisfying $\\mathrm {d}\\lambda =\\Omega $ , with periods $2 \\pi Z_\\gamma $ .",
"We will use this special coordinate system for ${\\mathcal {M}}_H(G^\\vee ,C)$ .",
"The parameters $u$ specify a fiber $F$ of the Hitchin fibration, which is an abelian variety, and the angles $\\theta _\\gamma $ parameterize the choice of a flat line bundle ${\\mathcal {S}}\\rightarrow F$ .",
"The WKB conditions for the existence of corners are thus that $\\theta _\\gamma = \\pm \\mathrm {Im}\\,Z_\\gamma $ and the quantization condition becomes $\\mathrm {Im}\\, Z_\\gamma \\in \\pi \\mathbb {Z}$ .",
"The ${\\mathcal {B}}_F$ branes as $A$ -branes are supposed to depend on the data $(u, \\theta )$ holomorphically in complex structure $J$ , as the $A_K$ twist on ${\\mathcal {M}}_H(G,C)$ is mirror to the $B_J$ twist on ${\\mathcal {M}}_H(G^\\vee ,C)$ .",
"Writing functions of $(u, \\theta )$ which are holomorphic in complex structure $J$ is essentially as challenging as computing the hyper-Kähler metric on the moduli space.",
"In the WKB region, though, the functions $ X_\\gamma = \\exp \\left(\\mathrm {Re}\\,Z_\\gamma + i \\theta _\\gamma \\right)$ are an excellent “semiflat” approximation to $J$ -holomorphic functions.",
"The Cauchy-Riemann equations fail by corrections suppressed exponentially in the WKB region [45].One can define locally some corrected $X_\\gamma $ which are truly $J$ -holomorphic, but non-trivial “wall crossing” coordinate transformations are required in different patches [40].",
"This subtlety will not be important here.",
"These statements have a transparent $A$ -model interpretation.",
"The semiclassical $A$ -brane moduli combine the ${\\mathrm {CP}}$ data with the deformation data associated to the same 1-forms on the brane support to give the $({\\mathbb {C}}^*)^{2r}$ coordinates $X_\\gamma $ .",
"The only corrections are non-perturbative and due to disk instantons ending on a cycle $\\gamma $ .",
"These only exist at codimension one loci where $Z_\\gamma $ is real and lead to the wall-crossing transformations.",
"The complex symplectic form in complex structure $J$ is approximately $\\Omega _J = \\langle \\mathrm {d}\\log X, \\mathrm {d}\\log X \\rangle $ Stated in this language, the WKB condition for the existence of a corner becomes $ X_\\gamma = e^{Z_\\gamma (u)}$ This can be interpreted as the parametric definition of an $r$ -dimensional complex Lagrangian submanifold, as expected.",
"Surprisingly, the parameters coincide with the coordinates $u$ on ${\\cal B}\\backslash {\\cal D}$ and thus we get an approximate holomorphic identification between ${\\cal B}\\backslash {\\cal D}$ and the space of branes equipped with a corner.",
"This identification is only valid in a semiclassical approximation.",
"To make such an expansion, as in Section REF , we replace $u$ with $tu$ , and take $t$ to be large.",
"The corner condition receives perturbative corrections, which will take a systematic form $X_\\gamma = e^{t Z_\\gamma (\\widetilde{u}) + t^{-1} c_{1,\\gamma }(\\widetilde{u}) + \\cdots }$ We introduced parameters $\\widetilde{u}$ that cannot be taken to coincide with the coordinates $u$ that are holomorphic in complex structure $I$ .",
"The relation between the two follows from the comparison of (REF ) and (REF ): $\\mathrm {Re}\\,Z_\\gamma (u) = \\mathrm {Re}\\,Z_\\gamma (\\widetilde{u}) + t^{-2} \\mathrm {Re}\\,c_{1,\\gamma }(\\widetilde{u}) + \\cdots $ The perturbatively-corrected quantization condition to have both types of corners will become $\\mathrm {Im}\\,Z_\\gamma (\\widetilde{u}) + t^{-2} \\mathrm {Im}\\,c_{1,\\gamma }(\\widetilde{u}) + \\cdots \\in \\pi \\mathbb {Z}.$"
],
[
"Match with real WKB opers",
"There is another natural occurrence of the periods $Z_\\gamma $ of the canonical 1-form on the spectral curve.",
"If we attempt a WKB analysis on $C$ of the oper differential equation, the monodromy data of the oper flat connection will be computed at the leading order in terms of the exponentiated periods $e^{Z_\\gamma (\\widetilde{u}) }$ [46], [47], [48], [49], [50].",
"Here we employed a non-canonical identification between $L_{{\\mathrm {op}}}$ and ${{\\mathcal {B}}}$ .",
"For example, for $G^\\vee = {\\mathrm {SU}}(2)$ we would write the oper differential operator as a reference operator deformed by a large quadratic differential $U(z;\\widetilde{u})$ $\\partial ^2_z - t_2(z) \\equiv \\partial ^2 - t^2 U(z;\\widetilde{u})- t_2^{0}(z)$ and the leading WKB approximation would involve the periods $Z_\\gamma (\\widetilde{u})$ of the WKB 1-form $\\sqrt{U(z;\\widetilde{u})} \\mathrm {d}z$ .",
"Here $t_2(z)$ is the classical stress tensor, $t_2^0(z)$ a reference choice of classical stress tensor and $t$ is the scaling parameter.",
"The WKB calculation is really a combination of a topological and an analytic problem.",
"The topological problem involves a careful Stokes analysis of the asymptotic behaviour of the flat sections of the connection.",
"The analytic problem involves the computation of the Voros symbols, which are periods of the all-orders WKB 1-form.",
"Remarkably, the Voros symbols precisely compute the corrected $X_\\gamma $ coordinates of the oper in the space of flat $G^\\vee $ connections [50]: $X_\\gamma = e^{t Z_\\gamma (\\widetilde{u}) + t^{-1} c_{1,\\gamma }(\\widetilde{u}) + \\cdots }$ The WKB analysis of the oper differential equation thus gives directly the $B$ -model analogue of the WKB corner condition in the mirror $A$ model."
],
[
"The Setup",
"So far we have studied the quantization of ${\\mathcal {M}}_H(G,C)$ as a real symplectic manifold.",
"An alternative is to view ${\\mathcal {M}}_H(G,C) $ as a complex symplectic manifold with complex structure $I$ and holomorphic symplectic structure $\\Omega _I=\\omega _J+{\\mathrm {i}}\\omega _K$ , and quantize a real symplectic submanifold ${\\mathcal {M}}_H^{\\mathbb {R}}\\subset {\\mathcal {M}}_H$ .",
"For this, as explained in Section REF , we look for an antiholomorphic involution $\\tau $ of ${\\mathcal {M}}_H(G,C)$ that satisfies $\\tau ^*\\Omega _I=\\overline{\\Omega }_I$ .",
"In our application, ${\\mathcal {M}}_H^{\\mathbb {R}}$ will be a cotangent bundle, leading to a simple description of the Hilbert space and of the action of the Hecke operators, and a real integrable system, equipped with a real version of the Hitchin fibration.",
"The classification of the possible antiholomorphic involutions is the same whether one considers holomorphic $G_{\\mathbb {C}}$ bundles as in [51], [52] or Higgs bundles, as in [53], [54], [55].",
"Those references provide much more detail than we will explain here.",
"We also note that the three-manifold $U_\\tau =(\\Sigma \\times \\widehat{I})/\\lbrace 1,\\tau \\rbrace $ that we will use in studying the duality was introduced in Section 11 of [53].",
"That paper also contains a duality proposal based on the structure of ${\\mathcal {M}}_H^{\\mathbb {R}}(G,C)$ as a real integrable system.",
"A suitable involution of ${\\mathcal {M}}_H$ can be constructed starting with an antiholomorphic involution $\\tau $ of the Riemann surface $C$ , which exists for suitable choices of $C$ .",
"An antiholomorphic map reverses the orientation of $C$ ; conversely, if $\\tau $ is an orientation-reversing smooth involution of an oriented two-manifold $C$ , then one can pick a complex structure on $C$ such that $\\tau $ acts antiholomorphically.Pick any Riemannian metric $\\mathrm {g}$ on $C$ .",
"Then $\\mathrm {g}^{\\prime }=({\\mathrm {g}}+\\tau ^*({\\mathrm {g}}))/2$ is a $\\tau $ -invariant metric, and $\\tau $ acts antiholomorphically in the complex structure determined by $\\mathrm {g}^{\\prime }$ .",
"Topologically, the possible choices of $\\tau $ can be classified as follows.",
"If $C$ has genus $g$ , then its Euler characteristic is $2-2g$ and the quotient $C^{\\prime }=C/\\lbrace 1,\\tau \\rbrace $ will have Euler characteristic $1-g$ .",
"$C^{\\prime }$ can be any possibly unorientable two-manifold, possibly with boundary, of Euler characteristic $1-g$ .",
"The boundary of $C^{\\prime }$ comes from the fixed points of $\\tau :C\\rightarrow C$ .",
"These fixed points make up a certain number of circles; any integer number of circles from 0 to $g+1$ is possible.",
"For example, if $g=0$ , $C^{\\prime }$ can be a disc or $\\mathbb {RP}^2$ ; if $g=1$ , $C^{\\prime }$ can be a cylinder, a Mobius strip, or a Klein bottle.",
"Having fixed $\\tau $ , we then choose the topological type of a lift of $\\tau $ to act on a smooth $G$ -bundle $E\\rightarrow C$ .",
"We choose the lift to preserve $\\tau ^2=1$ .",
"If $s$ is a fixed point of $\\tau :C\\rightarrow C$ , then $\\tau $ acts on the fiber $E_s$ of $E$ over $s$ by an automorphism $x_s$ of $G$ .",
"Here $x_s$ can be an inner automorphism, that is, conjugation by an element $g_s$ of $G$ , but more generally, if $G$ has non-trivial outer automorphisms, $x_s$ may be an outer automorphism.",
"The conjugacy class of $x_s$ is constant on each fixed circle $S$ , and we denote it as $x_S$ .",
"The condition $\\tau ^2=1$ places a condition on $x_S^2$ .",
"Since a Higgs bundle is defined by fields $(A,\\phi )$ that are adjoint-valued, purely to define an involution of the Higgs bundle moduli space ${\\mathcal {M}}_H(G,C)$ , we require only that $x_S^2=1$ in the adjoint form of $G$ ; as an element of $G$ , $x_S^2$ might be a central element not equal to 1.",
"The precise condition that should be imposed on $x_S^2$ depends on topological subtleties that were reviewed in Section REF ; we will return to this point in Section REF .",
"If $\\tau $ acts freely and $G$ is not simply-connected, other issues come into play in lifting $\\tau $ to act on $E$ .",
"For example, for $G={\\mathrm {SO}}(3)$ , a lift of $\\tau $ to act on $E$ only exists if $\\int _C w_2(E)=0$ .",
"In that case, there are two topologically inequivalent lifts, parametrized by $\\int _{C^{\\prime }}w_2(E^{\\prime })$ , where $E^{\\prime }\\rightarrow C^{\\prime }$ is the quotient of $E\\rightarrow C$ by $\\lbrace 1,\\tau \\rbrace $ .",
"A Riemann surface $C$ endowed with antiholomorphic involution $\\tau $ can be viewed as an algebraic curve defined over ${\\mathbb {R}}$ .",
"The boundary points of $C^{\\prime }$ , if any, correspond to the real points of $C$ over ${\\mathbb {R}}$ .",
"Once it has been lifted to act on the smooth bundle $E$ , $\\tau $ also acts on ${\\mathcal {M}}(G,C)$ , the moduli space of flat connections on $E$ , and on the corresponding Higgs bundle moduli space ${\\mathcal {M}}_H(G,C)$ .",
"Whether or not there are fixed points in the action of $\\tau $ on $C$ , there always are fixed points in the action of $\\tau $ onIf $A$ is any connection on $E$ , then $\\frac{1}{2}(A+\\tau ^*(A))$ is a $\\tau $ -invariant connection.",
"Generically, the $(0,1)$ part of this connection defines a stable $G_{\\mathbb {C}}$ -bundle, which corresponds to a $\\tau $ -invariant point in ${\\mathcal {M}}(G,C)$ .",
"The same connection with zero Higgs field defines a $\\tau $ -invariant point in ${\\mathcal {M}}_H(G,C)$ .",
"${\\mathcal {M}}(G,C)$ and on ${\\mathcal {M}}_H(G,C)$ .",
"Each component of the fixed point set of $\\tau $ on ${\\mathcal {M}}(G,C)$ or ${\\mathcal {M}}_H(G,C)$ is middle-dimensional.",
"This is a general property of antiholomorphic involutions of complex manifolds.",
"We will write ${\\mathcal {M}}^{\\mathbb {R}}(G,C)$ or ${\\mathcal {M}}_H^{\\mathbb {R}}(G,C)$ for a component of the fixed point set of $\\tau $ acting on ${\\mathcal {M}}(G,C)$ or ${\\mathcal {M}}_H(G,C)$ , respectively.",
"The components in general are classified by additional data that we have not introduced so far.",
"In particular, viewing ${\\mathcal {M}}(G,C)$ or ${\\mathcal {M}}_H(G,C)$ as moduli spaces of flat $G$ -valued or $G_{\\mathbb {C}}$ -valued connections, the monodromy $h_S$ of a flat bundle around $S$ will be invariant under $x_S$ , so it will lie in the $x_S$ -invariant subgroup $H_S$ or $H_{S,{\\mathbb {C}}}$ of $G$ or $G_{\\mathbb {C}}$ .",
"$H_S$ and $H_{S,{\\mathbb {C}}}$ are connected if $G$ is simply-connected, but in general not otherwise (for example, if $G={\\mathrm {SO}}(3)$ and $g_S=\\mathrm {diag}(1,-1,-1)$ , then $H_S$ contains the component of the identity and another component that contains the element $\\mathrm {diag}(-1,-1,1)$ ).",
"So in general, to specify a component of the fixed point sets requires an additional choice for each $S$ .",
"Associated to each fixed circle $S$ is a real form $G_{{\\mathbb {R}},S}$ of the complex Lie group $G_{\\mathbb {C}}$ .",
"Writing $g\\rightarrow \\overline{g}$ for the antiholomorphic involution of $G_{\\mathbb {C}}$ that leaves fixed the compact form $G$ , $G_{{\\mathbb {R}},S}$ is defined by the condition $ g=x_S(\\overline{g}).",
"$ In case $x_S$ is conjugation by $g_S\\in G$ , the condition becomes $ g=g_S \\overline{g} g_S^{-1}.$ ${\\mathcal {M}}_H^{\\mathbb {R}}(G,C)$ , with symplectic structure $\\omega _J$ , is the real symplectic manifold that we want to quantize.",
"The definition of ${\\mathcal {M}}_H^{\\mathbb {R}}(G,C)$ has depended on various choices which we are not indicating in the notation, but regardless of those choices, ${\\mathcal {M}}_H^{\\mathbb {R}}(G,C)$ has ${\\mathcal {M}}_H(G,C)$ as a complexification and this complexification has the appropriate properties for brane quantization of ${\\mathcal {M}}_H^{\\mathbb {R}}(G,C)$ .",
"${\\mathcal {M}}_H^{\\mathbb {R}}(G,C)$ has a real polarization that is defined as follows: a leaf of the polarization consists of $\\tau $ -invariant Higgs pairs $(A,\\phi )$ with fixed $A$ , but varying $\\phi $ .",
"The same condition, without the requirement of $\\tau $ -invariance, defines a holomorphic polarization of the complexification ${\\mathcal {M}}_H(G,C)$ of ${\\mathcal {M}}^{\\mathbb {R}}_H(G,C)$ .",
"Hence brane quantization of ${\\mathcal {M}}^{\\mathbb {R}}(G,C)$ is equivalent to its quantization using this real polarization.",
"Concretely, in this real polarization, ${\\mathcal {M}}_H^{\\mathbb {R}}(G,C)$ can be approximated by the cotangent bundle $T^*{\\mathcal {M}}^{\\mathbb {R}}(G,C)$ for the same reason that such a statement holds for the full Higgs bundle moduli space: if $(A,\\phi )$ is a $\\tau $ -invariant Higgs bundle representing a point in ${\\mathcal {M}}_H^{\\mathbb {R}}(G,C)$ , then generically $\\phi $ represents a cotangent vector to ${\\mathcal {M}}^{\\mathbb {R}}(G,C)$ at the point in ${\\mathcal {M}}^{\\mathbb {R}}(G,C)$ corresponding to $A$ .",
"Therefore, the Hilbert space ${\\mathcal {H}}_\\tau $ that arises in quantization of ${\\mathcal {M}}^{\\mathbb {R}}_H(G,C)$ is the space of ${\\mathrm {L}}^2$ half-densities on ${\\mathcal {M}}^{\\mathbb {R}}(G,C)$ , the usual answer for geometric quantization of a cotangent bundle.",
"We can also restrict the Hitchin fibration to the $\\tau $ -invariant locus.",
"$\\tau $ acts on the base ${{\\mathcal {B}}}$ of the Hitchin fibration with a fixed point set ${{\\mathcal {B}}}^{\\mathbb {R}}$ .",
"Functions on ${{\\mathcal {B}}}^{\\mathbb {R}}$ are Poisson-commuting (with respect to the real symplectic structure $\\omega _J$ of ${\\mathcal {M}}_H^{\\mathbb {R}}(G,C)$ ).",
"So ${\\mathcal {M}}_H^{\\mathbb {R}}(G,C)$ is a real integrable system.",
"The generic fiber of the map ${\\mathcal {M}}_H^{\\mathbb {R}}(G,C)\\rightarrow {{\\mathcal {B}}}^{\\mathbb {R}}$ is a real torus.",
"Baraglia and Schaposnik [53] proposed to define a mirror symmetry between ${\\mathcal {M}}_H^{\\mathbb {R}}(G,C)$ and a similar moduli space with $G$ replaced by $G^\\vee $ by $T$ -duality on the fibers of the real integrable system.",
"In general, one would expect such a definition to give reliable results at least asymptotically, far from the discriminant locus of the real integrable system.",
"At least when $\\tau $ acts freely, one can be more precise, as we discuss next."
],
[
"Four-Dimensional Picture And Duality",
"To learn something interesting about this construction, one wants to apply duality, and for this, as usual, a four-dimensional picture is helpful.",
"For a four-dimensional picture, we start with the usual four-manifold $M=\\Sigma \\times C$ , where $\\Sigma ={\\mathbb {R}}\\times I$ is a strip.",
"Here ${\\mathbb {R}}$ labels the “time” and $I$ is an interval, say the interval $0\\le w\\le 1$ .",
"We then modify $M$ by imposing an equivalence relation on the right boundary at $w=1$ : we declare that, for $p\\in C$ , points $(t,1,p)$ and $(t,1,\\tau (p))$ are equivalent.",
"(No equivalence is imposed except at $w=1$ .)",
"Imposing this equivalence relation amounts to requiring that the data at $w=1$ should be $\\tau $ -invariant (up to a gauge transformation).",
"So this is a way to implement in four dimensions what in the two-dimensional picture of Section REF was the brane ${\\mathcal {B}}_{\\mathbb {R}}$ at the right boundary.",
"We write $M_\\tau $ for the quotient of $M$ by this equivalence relation.",
"It is convenient to factor out the time and define $M_\\tau ={\\mathbb {R}}\\times U_\\tau $ .",
"There is an alternative construction of $U_\\tau $ as a quotient by the ${\\mathbb {Z}}_2$ group generated by $\\tau $ .",
"For this, we start with a doubled interval $\\widehat{I}: 0\\le w\\le 2$ .",
"Then we divide $\\widehat{U}= \\widehat{I}\\times C$ by the symmetry that acts by $(w,p)\\rightarrow (2-w,\\tau (p))$ .",
"A fundamental domain is the region $0\\le w\\le 1$ , so the quotient is the same as before.",
"Simplest is the case that $\\tau $ acts freely on $C$ .",
"$U_\\tau $ is then actually an orientable manifold, whose boundary is a single copy of $C$ , at $w=0$ .",
"$U_\\tau $ is the total space of a real line bundle over $C^{\\prime }=C/\\lbrace 1,\\tau \\rbrace $ .",
"$C^{\\prime }$ itself and the real line bundle are both unorientable, but the total space $U_\\tau $ is orientable.",
"The four-dimensional picture associated to quantization of ${\\mathcal {M}}_H^{\\mathbb {R}}(G,C)$ is just the $A$ -model on $M_\\tau ={\\mathbb {R}}\\times U_\\tau $ , with ${\\mathcal {B}}_{\\mathrm {cc}}$ boundary conditions at $w=0$ .",
"There is no need for an explicit boundary condition at $w=1$ , as there is no boundary there.",
"What in the two-dimensional description was a Lagrangian brane ${\\mathcal {B}}_{\\mathbb {R}}$ supported on ${\\mathcal {M}}_H^{\\mathbb {R}}(G,C)$ has been absorbed into the geometry of $M_\\tau $ .",
"Therefore, without further ado we can describe a dual description.",
"The dual is just the $B$ -model with gauge group $G^\\vee $ on the same four-manifold $M_\\tau $ , but now with ${\\mathcal {B}}_{\\mathrm {op}}$ boundary conditions at $w=0$ .",
"In a two-dimensional language, the brane ${\\mathcal {B}}^\\vee _{\\mathbb {R}}$ dual to ${\\mathcal {B}}_{\\mathbb {R}}$ is supported on ${\\mathcal {M}}_H^{\\mathbb {R}}(G^\\vee ,C)$ .",
"In particular, it is supported on the same locus ${{\\mathcal {B}}}_{\\mathbb {R}}$ on the base of the Hitchin fibration.",
"This is in accord with the duality proposal of Baraglia and Schaposnik [53], which appears to be valid at least when $\\tau $ acts freely on $C$ .",
"As usual, it is relatively easy to describe the physical states of the $B$ -model in quantization on $U_\\tau $ and the eigenvalues of the Hitchin Hamiltonians.",
"The first step is localization of the $B$ -model on complex-valued flat connections, which here means $G^\\vee _{\\mathbb {C}}$ -valued flat connections on $U_\\tau $ that satisfy the oper boundary condition.",
"To put it differently, the localization is on flat $G_{\\mathbb {C}}$ bundles $E^\\vee \\rightarrow U_\\tau $ whose restriction to $C=\\partial U_\\tau $ is a holomorphic oper.",
"Let $\\Upsilon _\\tau $ be the set of isomorphism classes of such bundles.",
"For positivity of the Hilbert space inner product, we expect $\\Upsilon _\\tau $ to be a discrete set of nondegenerate points (see Appendix ), similarly to the analogous set $\\Upsilon =L_{\\mathrm {op}}\\cap L_{\\overline{\\mathrm {op}}}$ encountered in the quantization of the full Higgs bundle moduli space.",
"Assuming this, the Hilbert space ${\\mathcal {H}}_\\tau $ has a basis $\\psi _u$ labeled by $u\\in \\Upsilon _\\tau $ .",
"The $\\psi _u$ are eigenfunctions of the Hitchin Hamiltonians.",
"The eigenvalue of a Hitchin Hamiltonian $H_{{\\mathcal {P}},\\alpha }$ on $\\psi _u$ is the value of the corresponding function $f_{{\\mathcal {P}},\\alpha }:L_{\\mathrm {op}}\\rightarrow {\\mathbb {C}}$ at the point in $L_{\\mathrm {op}}$ that corresponds to the boundary values of the flat bundle associated to $u$ .",
"If an oper bundle over $C=\\partial U_\\tau $ extends over $U_\\tau $ as a flat bundle, then in particular this implies that the antihomomorphic involution $\\tau :C\\rightarrow C$ lifts to an action on $E^\\vee $ and therefore that $E^\\vee $ is an antiholomorphic oper, as well as a holomorphic one.",
"So the set $\\Upsilon _\\tau $ has a natural map to $\\Upsilon =L_{\\mathrm {op}}\\cap L_{\\overline{\\mathrm {op}}}$ .",
"However, in general the map from $\\Upsilon _\\tau $ to $\\Upsilon $ is not an embedding, since a $\\tau $ -invariant flat bundle on the boundary of $U_\\tau $ that is a holomorphic oper may have more than one extension as a flat bundle over the interior of $U_\\tau $ .",
"To analyze this situation, note that flat $G^\\vee _{\\mathbb {C}}$ bundles over $U_\\tau $ correspond to homomorphisms $\\varrho :\\pi _1(U_\\tau )\\rightarrow G^\\vee _{\\mathbb {C}}$ , up to conjugation.",
"As $U_\\tau $ is contractible to $C^{\\prime }=C/\\lbrace 1,\\tau \\rbrace $ , we can equally well consider $\\varrho :\\pi _1(C^{\\prime })\\rightarrow G^\\vee _{\\mathbb {C}}$ .",
"The group $\\pi _1(C^{\\prime })$ has an index 2 subgroup $\\pi _{1,+}(C^{\\prime })$ consisting of orientation-preserving loops in $C^{\\prime }$ .",
"Such loops can be deformed to the boundary of $U_\\tau $ , so once a flat $G^\\vee _{\\mathbb {C}}$ bundle is given on the boundary of $U_\\tau $ , the restriction of the corresponding homomorphism $\\varrho $ to $\\pi _{1,+}(C^{\\prime })$ is uniquely determined.",
"However, there is some freedom in the extension of $\\varrho $ to the rest of $\\pi _1(C^{\\prime })$ .",
"$\\pi _1(C^{\\prime })$ is generated by the index 2 subgroup $\\pi _{1,+}(C^{\\prime })$ together with any orientation-reversing element $\\sigma \\in \\pi _1(C^{\\prime })$ .",
"To complete the description of $\\varrho $ , we need to specify $\\varrho (\\sigma )$ .",
"$\\varrho $ is supposed to be a homomorphism, so we require $\\varrho (\\sigma )^2=\\varrho (\\sigma ^2)$ , where, since $\\sigma ^2\\in \\pi _{1,+}(C^{\\prime })$ , $\\varrho (\\sigma ^2)$ is determined by the boundary data.",
"If $\\varrho (\\sigma ^2)$ is a regular element of $G^\\vee _{\\mathbb {C}}$ , then there are only finitely many choices for $\\varrho (\\sigma )$ , but $\\varrho (\\sigma )$ is not uniquely determined.",
"In particular, we are always free to transform $\\varrho (\\sigma )\\rightarrow \\varepsilon \\varrho (\\sigma )$ , where $\\varepsilon $ is an element of order 2 of the center of $G^\\vee $ .",
"Thus the subgroup ${\\mathcal {Z}}_2(G^\\vee )$ of the center of $G^\\vee $ consisting of elements of order 2 acts freely on $\\Upsilon _\\tau $ .",
"(For simple $G^\\vee $ , ${\\mathcal {Z}}_2(G^\\vee )$ is 1, ${\\mathbb {Z}}_2$ , or ${\\mathbb {Z}}_2\\times {\\mathbb {Z}}_2$ , depending on $G^\\vee $ .)",
"The $A$ -model dual of the action of ${\\mathcal {Z}}_2(G^\\vee )$ is the following.",
"Topologically, for simple $G$ , a $G$ -bundle on the three-manifold $U_\\tau $ is classified by $H^2(U_\\tau ,\\pi _1(G))$ .",
"In the present instance, $U_\\tau $ is contractible to $C^{\\prime }=C/\\tau $ so $H^2(U_\\tau ,\\pi _1(G))=H^2(C^{\\prime },\\pi _1(G))$ .",
"As $C^{\\prime }$ is unorientable, one has $H^2(C^{\\prime },\\pi _1(G))=\\pi _{1;2}(G)$ , where $\\pi _{1;2}(G)$ is the subgroup of $\\pi _1(G)$ consisting of elements of order 2.",
"Thus on the $A$ -model side, there is a grading of the Hilbert space by $\\pi _{1;2}(G)$ .",
"As $\\pi _{1;2}(G)={\\mathcal {Z}}_2(G^\\vee )$ , this matches the ${\\mathcal {Z}}_2(G^\\vee )$ action in the $B$ -model.",
"Finally let us discuss the natural line operators in this problem.",
"As in Section , we could in principle consider Wilson and 't Hooft line operators supported on an arbitrary 1-manifold $\\gamma \\subset M_\\tau $ .",
"But a natural special case is the following.",
"Pick a point $p\\in C$ .",
"In $\\widehat{I}\\times C $ , there is a natural path $\\widehat{\\gamma }_p$ between the points $0\\times p$ and $2\\times p$ on the left and right boundaries: we simply set $\\widehat{\\gamma }_p = \\widehat{I}\\times p$ .",
"Upon dividing by the group generated by $\\tau $ , $\\widehat{\\gamma }_p$ descends to a path $\\gamma _p\\subset U_\\tau $ , between the boundary points $0\\times p$ and $0\\times \\tau (p)$ .",
"In a purely two-dimensional description, one would have needed endpoint or corner data associated to the real brane ${{\\mathcal {B}}}_{\\mathbb {R}}$ , but in the four-dimensional description, this is not needed as $\\gamma $ has no endpoint at $w=1$ .",
"As in Section REF , we can choose an arbitrary representation $R$ of $G^\\vee $ and consider the holonomy $W_{R,\\gamma }$ of the bundle $E^\\vee _R=E^\\vee \\times _{G^\\vee _{\\mathbb {C}}}R$ on the curve $\\gamma _p$ .",
"However, to turn this holonomy into a quantum operator $ W_{R,p}$ , we need, as before, to supply endpoint data.",
"For example, if we choose canonical endpoints $s_{R,n}$ at $0\\times p$ and $s_{\\overline{R}, m}$ at $0\\times \\tau (p)$ , the Wilson operator will take the form of the inner product between $s_{R,n}$ transported to $\\tau (p)$ and $s_{\\overline{R}, m}$ .",
"As in Section REF , the Wilson operators $W_{R,p}$ constructed this way are diagonal on the basis of states $\\psi _u$ , with eigenvalues given by evaluating the inner product on the concrete canonical sections in the flat bundle corresponding to $u$ .",
"The duality predicts that a dual 't Hooft operator associated to the same representation $R$ , supported on the same curve $\\gamma _p$ , and with $S$ -dual endpoints, has the same eigenvalues.",
"The dual 't Hooft operator is associated to a real Hecke correspondence $Z^{\\mathbb {R}}_{R,p}$ of ${\\mathcal {M}}_H^{\\mathbb {R}}(G,C)$ with itself; $Z^{\\mathbb {R}}_{R,p}$ is just the $\\tau $ -invariant locus of the ordinary Hecke correspondence $Z_{R,p;\\overline{R},\\tau (p)}$ of ${\\mathcal {M}}_H(G,C)$ with itself, for a $\\tau $ -conjugate pair of points labeled by the conjugate (or dual) pair of representations $R,\\overline{R}$ .",
"As in Section REF , because of the scaling symmetry of the cotangent bundle, the quantum operator associated to the Hecke correspondence can be defined by a semiclassical formula.",
"Beyond the ingredients that were used in Section REF , one needs one further fact.",
"Suppose that $X$ is a complex manifold with canonical bundle $K_X$ ; let $\\tau :X\\rightarrow X$ be an antiholomorphic involution with fixed point set $X^{\\mathbb {R}}$ , and assume thast $X^{\\mathbb {R}}$ is orientable.",
"Then a holomorphic section of $K_X^{1/2}\\rightarrow X$ restricts on $X^{\\mathbb {R}}$ to a complex-valued half-density.",
"Hence a holomorphic endpoint or corner that one would use (together with an antiholomorphic one) in defining an 't Hooft operator in the quantization of ${\\mathcal {M}}_H$ restricts on the real locus to a half-density that defines an 't Hooft operator in the quantization of ${\\mathcal {M}}_H^{\\mathbb {R}}$ .",
"In our application, $Z^{\\mathbb {R}}_{R,p}$ is orientable because $Z_{R,p;\\overline{R},\\tau (p)}$ is a Calabi-Yau manifold, as discussed in Section REF ; the Calabi-Yau form of a Calabi-Yau manifold $X$ that has a real structure can be chosen to be real and restricts on the real locus to a top-degree differential form that defines an orientation of $X^{\\mathbb {R}}$ ."
],
[
"The Case That $\\tau $ Does Not Act Freely",
"Now we consider the case that $\\tau $ does not act freely on $C$ .",
"Suppose that the action of $\\tau $ leaves fixed a circle $S\\subset C$ .",
"Then $U_\\tau =\\widehat{U}/{\\mathbb {Z}}_2$ contains $S$ as a locus of ${\\mathbb {Z}}_2$ orbifold fixed points.",
"The local behavior near $S$ looks like $ S\\times {\\mathbb {R}}^2/\\lbrace 1,\\tau \\rbrace , $ where $\\tau $ acts on ${\\mathbb {R}}^2$ as a $\\pi $ rotation.",
"In the four-manifold $M_\\tau ={\\mathbb {R}}\\times U_\\tau $ , the fixed point set is ${\\mathbb {R}}\\times S$ .",
"We recall that in general the $\\tau $ action on $S$ is accompanied by an action of an automorphism $x_S$ that satisfies $x_S^2=1$ at least in the adjoint form of $G$ .",
"In $G$ gauge theory, one might expect to require $x_S^2=1$ in $G$ , but one has to take into account the topological subtleties that were reviewed in Section REF .",
"To illustrate the issues, we consider the case that $G$ has rank 1 and thus is ${\\mathrm {SU}}(2)$ or ${\\mathrm {SO}}(3)$ .",
"In this case, $G$ has no outer automorphisms and $x_S$ is conjugation by an element $g_S$ of $G$ .",
"As discussed in Section REF , there are two versions of ${\\mathrm {SU}}(2)$ gauge theory.",
"In ordinary ${\\mathrm {SU}}(2)$ gauge theory, the most obvious condition is to require $g_S^2=1$ acting on an ${\\mathrm {SU}}(2)$ bundle $E\\rightarrow M$ .",
"Then the only options are $g_S=1$ and $g_S=-1$ .",
"On the other hand, in ${\\mathrm {Spin}}\\cdot {\\mathrm {SU}}(2)$ gauge theory, the most obvious condition is to ask for $g_S^2=1$ acting on ${\\mathcal {S}}\\otimes E$ , where ${\\mathcal {S}}$ is a $\\tau $ -invariant spin bundle on $M$ (defined at least locally near the fixed point set) and $E$ is an ${\\mathrm {SU}}(2)$ bundle (defined wherever ${\\mathcal {S}}$ is).",
"Since $\\tau $ acts as a $\\pi $ rotation of the normal plane at the fixed point set, $\\tau ^2$ is a $2\\pi $ rotation and acts as $-1$ on ${\\mathcal {S}}$ .",
"This would suggest that we require $\\tau ^2=-1$ on $E$ , leading to $g_S=\\mathrm {diag}({\\mathrm {i}},-{\\mathrm {i}})$ , up to conjugation.A role for this conjugacy class was suggested by D. Baraglia.",
"However, we believe that it may also be possible to reverse these choices.",
"For example, in ${\\mathrm {SU}}(2)$ , before trying to divide by $\\tau $ , we could assume that there is a monodromy defect with monodromy $-1$ along the fixed point locus of $\\tau $ .",
"Then in taking the quotient we would want $g_S^2=-1$ .",
"Similarly, including such a monodromy defect before taking the quotient would motivate $g_S^2=+1$ for ${\\mathrm {Spin}}\\cdot {\\mathrm {SU}}(2)$ .",
"Similarly, there are two versions of ${\\mathrm {SO}}(3)$ gauge theory, with or without a factor $\\Delta =(-1)^{\\int _M w_2(M) w_2(E)}$ in the integrand of the path integral.",
"With or without this factor, we want $g_S^2=1$ , which gives two possibilities, namely $g_S=1$ and $g_S=\\mathrm {diag}(-1,-1,1)$ .",
"In the presence of a codimension 2 singularity with $g_S=\\mathrm {diag}(-1,-1,1)$ , $\\Delta $ is not well-defined topologically, so it appears that in this case we want $g_S=1$ .",
"Without the factor $\\Delta $ , both possibilities for $g_S$ are viable.",
"For ${\\mathrm {SO}}(3)$ , as the center is trivial, we do not have the option of including a defect with central monodromy, but we can include the dual of this, which is a defect that senses the topology of the gauge bundle $E$ restricted to the fixed point set.",
"For ${\\mathrm {SO}}(3)$ , this defect is a factor in the path integral of $(-1)^{\\int _W w_2(E)}$ , where $W$ is the fixed point set.",
"In general, it is a subtle question to find the dual of a singularity of this nature.",
"Part of the reason for the subtlety is that one cannot assume that the dual of an ${\\mathbb {R}}^2/{\\mathbb {Z}}_2$ orbifold singularity, defined by a condition of $\\tau $ -invariance, is another ${\\mathbb {R}}^2/{\\mathbb {Z}}_2$ orbifold singularity, defined by a dual condition of $\\tau $ -invariance.",
"In general, one only knows that the dual of an ${\\mathbb {R}}^2/{\\mathbb {Z}}_2$ orbifold singularity is a codimension 2 defect that preserves the same supersymmetry as the ${\\mathbb {R}}^2/{\\mathbb {Z}}_2$ orbifold singularity.",
"In a somewhat similar problem of rigid surface defects, it proved difficult to get a general understanding of the action of duality [56].",
"It is tempting to claim a simple answer if $g_S$ is central for all $S$ , on the following grounds.",
"Suppose that $g_S=1$ .",
"Then the singularity is only in the geometry, not the gauge field.",
"The ${\\mathbb {R}}^2/{\\mathbb {Z}}_2$ orbifold singularity is not a singularity at all topologically, as ${\\mathbb {R}}^2/{\\mathbb {Z}}_2$ is equivalent topologically to ${\\mathbb {R}}^2$ .",
"So if $g_S=1$ for all fixed circles, $U_\\tau $ is actually a manifold topologically, and one can “round off” the orbifold singularities to give it a smooth geometry.",
"Let us call the rounded version $\\widetilde{U}_\\tau $ .",
"If it is correct in the $A$ -model to replace the orbifold with the smooth manifold $\\widetilde{U}_\\tau $ , then the dual is the $B$ -model on the same manifold.",
"This reasoning has a potential analog for the more general case that $g_S$ is central but not equal to the identity.",
"We can still round off the defect to get the smooth manifold $\\widetilde{U}_\\tau $ , but now $\\widetilde{U}_\\tau $ contains a defect with central monodromy, supported on the orbifold locus $W$ .",
"As remarked earlier, the dual of this defect is a defect that senses the topology of the bundle, for example a defect defined by a factor $(-1)^{\\int _Ww_2(E)}$ in the case of gauge group ${\\mathrm {SO}}(3)$ .",
"The following is a strategy, in principle, to analyze the general case.",
"Surround the fixed point locus by a two-torus.",
"Then the orbifold defect defines a boundary condition for the 2d theory which arises from $T^2$ compactification of four-dimensional super Yang-Mills theory.",
"This is not quite a $\\sigma $ -model, because of the large unbroken gauge symmetry.",
"If we can identify the mirror of the boundary condition, it will provide boundary conditions for the $G^\\vee _{\\mathbb {C}}$ flat connection restricted to the two-torus on the dual side.",
"This would be sufficient to characterize how the oper flat connection at the boundary of $U_\\tau $ can extend to the interior and thus determine the spectrum on the $B$ -model side of the duality."
],
[
"Real Hecke Operators",
"The most important novelty of the case that $\\tau $ does not act freely on $C$ may be the existence of line operators supported on a real point in $C$ , as opposed to the line operators considered in Section REF that are supported on a $\\tau $ -conjugate pair of points.",
"First we describe a gauge theory picture for an 't Hooft operator supported at a real point $p\\in C$ .",
"We work on the four-manifold $M={\\mathbb {R}}\\times \\widehat{I}\\times C$ , with ${\\mathbb {R}}$ parametrized by $t$ , and $\\widehat{I}$ by $w$ .",
"A local picture suffices, so we take $C$ to be simply the complex $z$ -plane ${\\mathbb {C}}$ , and we consider the involution $\\tau $ that acts by $(t,w,z)\\rightarrow (-t,w,\\overline{z})$ .",
"Acting just on ${\\mathbb {C}}$ , $\\tau $ has the fixed line $S$ defined by ${\\mathrm {Im}\\, z}=0$ ; we acccompany $\\tau $ with an automorphism $x_S$ satisfying $x_S^2=1$ (or a slightly more general condition discussed in Section REF ).",
"An 't Hooft operator supported at $z=t=0$ is described by a Dirac monopole solution of the $G$ gauge theory.",
"The solution has a structure group that reduces to a maximal torus $T\\subset G$ and can be characterized by its curvature: $ F=\\frac{{\\sf m}}{2}\\star _3 \\mathrm {d}\\frac{1}{(t^2+|z|^2)^{1/2}}, $ where $\\star _3 $ is the Hodge star for the metric $\\mathrm {d}t^2+|\\mathrm {d}z|^2$ on ${\\mathbb {R}}\\times {\\mathbb {C}}$ , and ${\\sf m}$ is a constant element of the Lie algebra $\\mathfrak {t}$ of $T$ .",
"For a connection with this curvature to exist, ${\\sf m}$ must be an integral coweight, dual to a representation $R$ of $G^\\vee $ (it coincides with the object that was called ${\\sf m}$ in Section REF ).",
"Now we ask whether this solution is invariant under $\\tau $ , accompanied by the automorphism $x_S$ .",
"Since $\\star _3$ is odd under $\\tau $ , a necessary and sufficient condition is that ${\\sf m}$ should be odd under $x_S$ : $ x_S( {\\sf m})=-{\\sf m}.",
"$ When and only when it is possible to choose ${\\sf m}$ in its conjugacy class so that it is odd under $x_S$ , the solution constructed this way is $\\tau $ -invariant and descends to a solution on $M_\\tau =M/\\lbrace 1,\\tau \\rbrace $ .",
"It describes a real 't Hooft operator, supported on a real point in $C$ and associated to the representation $R$ of $G^\\vee $ .",
"Using this model solution, we can define a space of “real” Hecke modifications which can be implemented by a real 't Hooft operator.",
"There are real versions $G_{{\\mathbb {R}},S}({\\cal K})$ and $G_{{\\mathbb {R}},S}({\\cal O})$ of $G_{{\\mathbb {C}}}({\\cal K})$ and $G_{{\\mathbb {C}}}({\\cal O})$ which consist of gauge transformations which lie in $G_{{\\mathbb {R}},S}$ along $S$ .",
"We can thus define a real version ${\\mathrm {Gr}}_{G_{{\\mathbb {R}},S}} = G_{{\\mathbb {R}},S}({\\cal K})/G_{{\\mathbb {R}},S}({\\cal O})$ of the affine Grassmannian and the orbits $\\left[G_{{\\mathbb {R}},S}({\\cal O}) z^{\\sf m}\\right]$ of real Hecke modifications of type ${\\sf m}$ .",
"A knowledge of which real 't Hooft operators are possible for a given $x_S$ puts a very strong constraint on the dual of a ${\\mathbb {Z}}_2$ orbifold singularity with a given $x_S$ .",
"Geometrically, a real 't Hooft operator stretches from the boundary of $U_\\tau $ to a fixed point $p$ in the interior.",
"A full analysis of real 't Hooft operators, which we will not attempt here, would include a discussion of the possible endpoints of the 't Hooft operator on a fixed point and a derivation of the corresponding integral operators.",
"On the $S$ -dual side, we will have a Wilson operator stretched from the boundary to $p$ .",
"A specific endpoint of the Wilson line will give a vector $v_{\\overline{R},m}$ in the space of flat sections of the gauge bundle in a neighborhood of $p$ .",
"We cannot characterize this vector more precisely without knowing the $S$ -dual of the orbifold singularity; in particular, the flat bundle in the $B$ -model may not extend over the fixed point set.",
"The Wilson operator expectation value will take the form of a pairing $(v_{\\overline{R},m}, s_{R,n})$ , selecting a specific solution of the oper differential equations.",
"More generally, one can consider an arbitrary oriented three-manifold $U$ with boundary $C$ , and study the dual $A$ - and $B$ -models on ${\\mathbb {R}}\\times U$ .",
"Modulo technical difficulties (the moduli space of flat bundles on $U$ may be very singular), one can hope to define a space ${\\mathcal {H}}_U$ of physical states, with an action of the Hitchin Hamiltonians on the $A$ -model side and a prediction for their eigenvalues in terms of classical data on the $B$ -model side.",
"One can also study line operators, though in general there will be no close analogs of the ones that we have considered in this article.",
"The state space ${\\mathcal {H}}_U$ will have a hermitian inner product, but it is not clear that this inner product will be positive-definite in general, since it is no longer obtained by quantizing a cotangent bundle."
],
[
"Four-Dimensional Avatars of BAA Boundary Conditions and Corners",
"Four-dimensional super Yang-Mills theory admits many half-BPS boundary conditions [57] which are topological in the four-dimensional $A$ -twist and descend to BAA boundary conditions upon twisted compactification on $C$ [3], [58], [59].",
"Such a boundary condition, along with its “corners” with ${\\mathcal {B}}_{\\mathrm {cc}}$ and $\\overline{{\\mathcal {B}}}_{\\mathrm {cc}}$ , can be used to define a quantum state in ${\\mathcal {H}}={\\mathrm {Hom}}(\\overline{{\\mathcal {B}}}_{\\mathrm {cc}},{\\mathcal {B}}_{\\mathrm {cc}})$ .",
"We have already made use of this construction; see fig.",
"REF of Section .",
"Apart from studying additional examples, what we will add in the present section is the use of two-dimensional chiral algebras to study the corners and the associated quantum states.",
"In four dimensions, a junction or corner between two boundary conditions occurs on a two-manifold, which for our purposes is a copy of the Riemann surface $C$ .",
"In the case of a brane of type BAA that can be engineered in four-dimensional gauge theory, its corners with ${\\mathcal {B}}_{\\mathrm {cc}}$ , if they can likewise be engineered in four-dimensional gauge theory, are frequently holomorphic-topological and support holomorphic chiral algebras.",
"Chiral algebras that arise this way were studied in [58], [32].",
"Corners with $\\overline{{\\mathcal {B}}}_{\\mathrm {cc}}$ , if they can be engineered in four dimensions, likewise typically support antiholomorphic chiral algebras.",
"This is the situation that we will study in the present section."
],
[
"The Analytic Continuation Perspective",
"We begin by recalling a construction that simplifies the analysis of the relevant junctions.",
"The brane ${\\mathcal {B}}_{\\mathrm {cc}}$ can be derived from a deformed Neumann boundary condition in four dimensions.Ordinary Neumann boundary conditions for a gauge field assert that $n^i F_{ij}=0$ , where $F$ is the Yang-Mills curvature and $n$ is the normal vector to the boundary.",
"Deformed Neumann boundary conditions express $n^i F_{ij}$ in terms of the boundary values of some other fields.",
"Note that a different, undeformed, Neumann boundary condition, with a different extension to the rest of the supermultiplet, will enter the story in Section REF .",
"We will call this the deformed Neumann or ${\\mathcal {B}}_{\\mathrm {cc}}$ boundary condition.",
"The path integral of the $A$ -twisted 4d gauge theory in the presence of such a deformed Neumann boundary can be interpreted as a slightly exotic path integral for a three-dimensional theory defined on the boundary [61], [60].",
"The action of this three-dimensional theory is a holomorphic function of complex variables, and the path integral is taken on a middle-dimensional integration cycle in the space of fields.",
"The integration cycle is defined by $A$ -model localization in four dimensions, but the details of this are not important for our purposes.",
"We will only discuss properties that do not depend on the choice of integration cycle.",
"We will call this type of path integral loosely a contour path integral.",
"The relevant three-dimensional auxiliary theory is most familiar not in the case of conventional geometric Langlands but for what is known mathematically as “quantum” geometric Langlands.",
"This corresponds in gauge theory to working at a generic value of the canonical parameter $\\Psi $ that was introduced in [3].",
"At generic $\\Psi $ , the boundary theory associated to suitably deformed Neumann boundary conditions is a Chern-Simons theory with a complex connection ${\\mathcal {A}}$ , with curvature ${\\mathcal {F}}=\\mathrm {d}{\\mathcal {A}}+{\\mathcal {A}}\\wedge {\\mathcal {A}}$ , and action $ I_{\\mathrm {CS}}=\\frac{\\Psi }{4\\pi }\\int _N \\left( {\\rm Tr}\\,{\\mathcal {A}}\\wedge \\mathrm {d}{\\mathcal {A}}+\\frac{2}{3}{\\mathcal {A}}^3\\right).",
"$ Here $N$ is the boundary or a portion of the boundary of a four-manifold $M$ .",
"The boundary condition that leads to the theory $I_{\\mathrm {CS}}$ on $N$ is “topological” in the sense that the only structure of $N$ that is required to define it is an orientation.",
"If $N=\\partial M$ , then the $A$ -model path integral on $M$ with boundary condition that leads to the boundary coupling $I_{\\mathrm {CS}}$ is a Chern-Simons path integral on $N$ with a non-standard integration cycle (which depends on $M$ ).",
"More generally, $N$ itself may have a boundary; along $\\partial N$ , we consider a junction or corner between the deformed Neumann boundary condition that leads to $I_{\\mathrm {CS}}$ and some other boundary condition.",
"In this case, as in conventional Chern-Simons theory on a three-manifold with boundary, for suitable choices of the second boundary condition, a current algebra or Kac-Moody symmetry will appear along $\\partial N$ .",
"The level of these currents is $\\Psi -h$ , where $h$ is the dual Coxeter number of $G$ .",
"The contribution $\\Psi $ to the level can be computed classically from the failure of $I_{\\mathrm {CS}}$ to be gauge-invariant on a manifold with boundary, and the $-h$ is a 1-loop correction, which will be described in Section REF .",
"For the present article, we are interested in “ordinary” geometric Langlands at $\\Psi =0$ .",
"We will not get anything sensible if we simply set $\\Psi =0$ in the action (REF ), since a contour path integral with zero action will not make sense, and in fact there is no way to take the limit $\\Psi \\rightarrow 0$ while preserving topological invariance along $N$ .",
"That is one way to understand the fact that the ${\\mathcal {B}}_{\\mathrm {cc}}$ boundary condition that has been important in the present article is holomorphic-topological rather than topological.",
"To take the limit $\\Psi \\rightarrow 0$ to get a holomorphic-topological boundary condition, we can do the following.",
"Let $C$ be a complex Riemann surface with local complex coordinate $z$ , and assume that $N=S\\times C$ , where $S$ is a 1-manifold parametrized by $t$ .",
"Then take the limit $\\Psi \\rightarrow 0$ keeping fixed $\\varphi =\\frac{\\Psi }{4\\pi } {\\mathcal {A}}_z\\mathrm {d}z$ .",
"The Chern-Simons action goes over to $I_{\\varphi {\\mathcal {F}}}=\\int _N{\\rm Tr}\\, \\varphi _z {\\mathcal {F}}_{t \\overline{z}} \\mathrm {d}t \\mathrm {d}^2z.$ The degeneration of the Chern-Simons action (REF ) to the action of eqn.",
"(REF ) is somewhat analogous to the degeneration from a complex flat connection to a Higgs bundle as the complex structure of the Higgs bundle moduli space ${\\mathcal {M}}_H(G,C)$ is varied.",
"The action $I_{\\varphi {\\mathcal {F}}} $ describes what can be interpreted as a topological gauged quantum mechanics on the cotangent bundle to the space of $(0,1)$ connections on $C$ .",
"The Higgs field $\\varphi $ is the momentum conjugate to ${\\mathcal {A}}_{\\overline{z}}$ .",
"The analogous statement in two-dimensional terms is that the $A$ -model of ${\\mathcal {M}}_H(G,C)$ , with a ${\\mathcal {B}}_{\\mathrm {cc}}$ boundary, is related to an analytically continued quantum mechanics on ${\\mathcal {M}}_H(G,C)$ [61].",
"This formulation makes it obvious that the deformed Neumann boundary condition is not topological in the $A$ -twist.",
"Instead, it depends on a choice of complex structure on $C$ and it admits local operators which vary holomorphically along $C$ and topologically along $S$ : it is a holomorphic-topological boundary condition.",
"The equations of motion derived from $I_{\\varphi {\\mathcal {F}}}$ imply that $\\varphi $ is holomorphic in $z$ and independent of $t$ .",
"Away from the boundary of $N$ , the gauge-invariant local operators on $N$ are the gauge-invariant polynomials ${\\cal P}[\\varphi ](z)$ of $\\varphi $ , which descend to the Hitchin Hamiltonians in the 2d $A$ -model.",
"(At generic $\\Psi $ , there are no gauge-invariant local operators on a boundary characterized by the Chern-Simons action $I_{\\mathrm {CS}}$ .)",
"What happens if $N$ has a boundary?",
"A junction between the deformed Neumann boundary condition that leads to the $\\varphi {\\mathcal {F}}$ theory and a topological 4d boundary condition can in many cases be described by a boundary condition in the $\\varphi {\\mathcal {F}}$ theory, encoding both the topological boundary condition and the choice of junction.",
"In many important examples, the topological boundary condition is a half-BPS boundary condition of type BAA.",
"With suitable choices, as we discuss further in Section REF , $\\varphi $ can behave along $\\partial N$ as a holomorphic current generating a Kac-Moody symmetry.",
"But now the level of the Kac-Moody symmetry comes entirely from the 1-loop correction and is $-h$ .",
"In general, the boundary conditions which appear in the $\\varphi {\\mathcal {F}}$ auxiliary gauge theory are holomorphic as well and may support holomorphic local operators.",
"This reflects the same property of the corresponding junctions between the deformed Neuman boundary and the half-BPS boundary: they are holomorphic in the $A$ -twisted theory.",
"The appearance of holomorphic junctions in the GL-twisted theory at general $\\Psi $ and the relation to the Chern-Simons level were analyzed in [62]."
],
[
"The Role of Chiral Algebras",
"A holomorphic junction between a holomorphic-topological boundary condition and a topological one may support local operators which depend holomorphically on their position on $C$ .",
"Essentially by definition, these operators define a chiral algebra.",
"Although the chiral algebra depends on the choice both of the topological boundary condition and of the junction, we will suppress that dependence for notational convenience and denote the chiral algebra simply as ${\\cal V}$ .",
"The chiral algebra is akin to the chiral algebras of holomorphic local operators which can be found in a 2d CFT, or at the boundary of a 3d topological field theory (TFT) such as Chern-Simons theory.",
"There are some differences [63] due to the fact that the $\\varphi {\\mathcal {F}}$ theory is holomorphic-topological.",
"A single junction in the $A$ -twisted theory can be used to build a variety of different corners in the 2d $A$ -model, depending on the choice of local operators $O_i(z_i)$ placed at points $z_i$ in $C$ .",
"These corners are not all independent: they depend on the $z_i$ holomorphically, with singularities as $z_i \\rightarrow z_j$ controlled by the OPE of the chiral algebra.",
"The OPE or the associated Ward identities imply recursion relations between different corners.A solution of the Ward identities for a chiral algebra is usually called a conformal block.",
"In a physical 2d CFT, conformal blocks can usually be obtained by some sewing procedure on the Riemann surface.",
"This may not be possible for a general chiral algebra such as ${\\cal V}$ , but the notion of conformal blocks is still available and can be used to characterize the space of $A$ -model corners which can be produced from a given junction.",
"Recall the gauge-invariant local operators ${\\cal P}[\\varphi ](z)$ on the deformed Neumann boundary condition which give rise to the Hitchin Hamiltonians.",
"These operators can be brought to the junction along the topological direction $t$ along the boundary.",
"The resulting boundary-to-junction OPE is not singular and produces a collection of operators $S_{\\cal P}(z)$ in ${\\cal V}$ .",
"These operators are central, i.e.",
"they have non-singular OPE with the other operators in ${\\cal V}$ , for the same reason that the ${\\cal P}[\\varphi ](z)$ have non-singular OPE with each other: they can be freely displaced along the $t$ direction.",
"That property has an important corollary: $A$ -model corners labelled by a collection of operators $S_{\\cal P}(z) O_1(z_1) \\cdots O_n(z_n)$ satisfy the same Ward identities as a function of the $z_i$ as corners labelled only by $O_1(z_1) \\cdots O_n(z_n)$ .",
"This insures that the the action of ${\\cal P}[\\varphi ](z)$ on this space of $A$ -model corners is well-defined.",
"Figure: An 't Hooft operator TT, of type BAA, acting on the state created by a brane ℬ{\\mathcal {B}} of type BAA, together with suitable chiral corners.We can fuse TT with ℬ{\\mathcal {B}} tomake a new brane TℬT{\\mathcal {B}}, again of type BAA.",
"This involves the same action of line operators on branes that we started with in fig.",
"(a).",
"In additionwe have to consider the composition of the chiral corners at the left and right endpoints or boundaries of TT and ℬ{\\mathcal {B}}.We can also discuss the chiral algebra ingredients which occur in a four-dimensional description of the action of 't Hooft operators on the states created by branes with chiral algebra corners.",
"In fig.",
"REF , we sketch an 't Hooft operator $T$ , of type BAA, acting on the state created by a brane ${\\mathcal {B}}$ of the same type, with suitable chiral and antichiral corners.",
"The action of $T$ on the state can be described as the fusion of the interface represented by $T$ with the boundary ${\\mathcal {B}}$ , accompanied by a composition of the corresponding corners both with ${\\mathcal {B}}_{\\mathrm {cc}}$ and with $\\overline{{\\mathcal {B}}}_{\\mathrm {cc}}$ .",
"In four dimensions, in contrast to the two-dimensional picture of fig.",
"REF , there is an obvious difference between the 't Hooft line defect and the half-BPS boundary: the former is supported at a point $p\\in C$ while the latter wraps the whole $C$ .",
"So the composition $T{\\mathcal {B}}$ coincides with ${\\mathcal {B}}$ away from $p$ ; it can be described as the brane ${\\mathcal {B}}$ enriched with a line defect.",
"In order to describe the action of 't Hooft operators we should thus first generalize the discussion in this section to allow for boundary line defects ending on the junction at some point $p\\in C$ .",
"As usual, this setup will depend holomorphically on $p$ .",
"It depends only on $p$ because in the A-twisted theory, the boundary line defect itself is the image of a topological 't Hooft line defect and is thus also topological.",
"The presence of a boundary line defect $\\ell $ and its endpoint does not affect the choice of chiral algebra operators available elsewhere on $C$ , nor the Ward identities they satisfy away from $p$ .",
"It affects, though, their behavior near $p$ .",
"For each $\\ell $ , there is a vector space of possible endpoints $O_{\\ell ,i}(p)$ and they form a module ${\\cal V}_\\ell $ for ${\\cal V}$ : the module structure encodes the OPE between chiral algebra operators and endpoints and controls the Ward identities satisfied at $p$ by the ${\\cal V}_\\ell $ insertions.",
"As we bring an 't Hooft operator with endpoint $\\alpha _{R,n}(p)$ to the half-BPS boundary, we will produce a boundary defect $\\ell $ as well as a specific endpoint $S_{R,n}(p) \\in {\\cal V}_\\ell $ at the junction.",
"We can predict two general properties of such endpoints: they will have non-singular OPE with the chiral algebra ${\\cal V}$ , just like the $S_{\\cal P}(z)$ associated to Hitchin Hamiltonians, and they will satisfy the same differential equations in $p$ as $\\alpha _{R,n}(p)$ do, with coefficients controlled by the $S_{\\cal P}(z)$ .",
"These properties can be derived immediately by separating the 't Hooft line from the boundary along the topological direction and transforming $S_{R,n}(p)$ back to $\\alpha _{R,n}(p)$ .",
"Again, the centrality property guarantees that $A$ -model corners labelled by a collection of operators $S_{R,n}(p) O_1(z_1) \\cdots O_n(z_n)$ satisfy the same Ward identities as a function of the $z_i$ as corners labelled only by $O_1(z_1) \\cdots O_n(z_n)$ .",
"This insures that the the action of the 't Hooft operator on the state created by the brane is well-defined.",
"We conclude this general discussion with some comments on the description of the 't Hooft lines in the auxiliary $\\varphi F$ theory.",
"The 't Hooft lines perpendicular to $N$ can be interpreted as monopole local operators in the $\\varphi F$ theory.",
"Operators such as ${\\cal P}[\\varphi ](z)$ appear as local operators both in the $\\varphi F$ theory and in the four-dimensional theory because they are polynomial of the elementary fields and their definition does not depend on a choice of contour for the path integral.",
"The definition of disorder operators such as monopole operators, instead, requires one to modify the space of field configurations allowed in the path integral and thus affects the possible choices of integration contours.",
"This modification is encoded in the presence of an actual line defect in the four-dimensional theory, ending on $p$ in $N$ .",
"These considerations also apply to boundary disorder operators at $\\partial N$ , which will appear as $O_{\\ell ,i}(p)$ endpoints of a boundary line defect in the four-dimensional theory.",
"It would be interesting to analyze directly the space of monopole operators available in the $\\varphi F$ theory, as well as their images at the boundary.",
"Some of the tools were developed in [59] and applied to Chern-Simons theory there.",
"They involve cohomology calculations on the affine Grassmanian which are likely to give a local version of the analysis in Section REF .",
"We leave this exercise to an enthusiastic reader."
],
[
"From Corners to States",
"Now consider a strip with ${\\mathcal {B}}_{{\\mathrm {cc}}}$ and $\\overline{{\\mathcal {B}}}_{{\\mathrm {cc}}}$ at the two ends, and with boundary conditions set at the bottom of the strip by some other brane ${\\mathcal {B}}$ (fig.",
"REF ).",
"At the “corners,” the construction that we have just described produces chiral algebras ${\\mathcal {V}}$ and $\\overline{{\\mathcal {V}}}$ .",
"With suitable operator insertions at the corners, the path integral on the strip produces a (possibly distributional) state $\\chi = \\left|O_1(z_1) \\cdots O_n(z_n) \\overline{O}_{n+1}(\\overline{z}_{n+1}) \\cdots \\overline{O}_{n+\\overline{n}}(\\overline{z}_{n+ \\overline{n}})\\right\\rangle $ in the usual Hilbert space ${\\mathcal {H}}={\\mathrm {Hom}}(\\overline{{\\mathcal {B}}}_{\\mathrm {cc}},{\\mathcal {B}}_{\\mathrm {cc}})$ .",
"An important special case is that the chiral algebras ${\\mathcal {V}}$ and $\\overline{{\\mathcal {V}}}$ at the two corners are complex conjugate, though we are not restricted to this case.",
"Operators sitting on different corners cannot have OPE singularities, so the $\\overline{O}_j(\\overline{z}_j)$ operators at the $\\overline{{\\mathcal {B}}}_{{\\mathrm {cc}}}$ corner do not affect the Ward identities for the $O_i(z_i)$ and vice versa.",
"We can also consider the pairing of $\\chi $ with a test state $\\Psi \\in {\\cal H}$ .",
"The resulting inner product may in general be ill-defined, but it is well-defined if $\\Psi $ is sufficiently nice, for example if $\\Psi $ is created in a similar way using a brane of compact support at the top of the strip (fig.",
"REF (c)).",
"If the inner product is well-defined for all choices of the operator insertions, it gives a collection of correlation functions $\\langle \\Psi | O_1(z_1) \\cdots O_n(z_n) \\overline{O}_{n+1}(\\overline{z}_{n+1}) \\cdots \\overline{O}_{n+\\overline{n}}(\\overline{z}_{n+ \\overline{n}}) \\rangle $ on $C$ which satisfy the Ward identities for ${\\cal V}$ and $\\overline{\\cal V}$ .",
"Half-BPS boundary conditions decorated by boundary line defects will produce a larger collection of states $\\left|\\prod _i O_i(z_i) \\prod _j \\overline{O}_{j}(\\overline{z}_{j}) \\prod _k O_{\\ell _k,k}(p_k) \\overline{O}_{\\ell _k,k}(\\overline{p}_k) \\right\\rangle ,$ with corner data including endpoints for the boundary line defects.",
"In the remainder of this section we will make these structures explicit for some basic examples of half-BPS boundary conditions."
],
[
"Dirichlet Boundary Conditions",
"Four-dimensional half-BPS Dirichlet boundary conditions fix the gauge connection $A$ to vanish at the boundary, or more generally to take some specified value at the boundary, extended to other fields in a supersymmetric fashion.",
"In particular, three of the six scalar fields satisfy Dirichlet boundary conditions, and other three, which include the Higgs field $\\phi $ of the 2d $\\sigma $ -model, satisfy Neumann boundary conditions.",
"From the perspective of the auxiliary $\\varphi {\\mathcal {F}}$ theory on a portion $N$ of $\\partial M$ , a junction between the half-BPS Dirichlet boundary condition and the deformed Neumann boundary is represented by a Dirichlet boundary condition on the field ${\\mathcal {A}}_{\\overline{z}}$ along $\\partial N$ .",
"Gauge transformations are restricted to be trivial at the boundary; otherwise it would not be possible to specify the value of ${\\mathcal {A}}_{\\overline{z}}$ along the boundary.",
"We do not impose any boundary condition on $\\varphi =\\varphi _z\\mathrm {d}z$ , the conjugate of ${\\mathcal {A}}_{\\overline{z}}$ .",
"Since gauge transformations are constrained to be trivial along $\\partial N$ , local operators on $\\partial N$ , that is, on the junction, are not required to be gauge-invariant.",
"Instead, there is a $G$ global symmetryA global symmetry transformation by an element $g\\in G$ is a gauge transformation $g(x):N\\rightarrow G$ whose restriction to $\\partial N$ is constant, $g(x)|_{\\partial N}=g$ .",
"This preserves the condition ${\\mathcal {A}}_{\\overline{z}}|_{\\partial N}=0$ , so it is a symmetry of the theory defined with that boundary condition.",
"How the constant $g$ is extended over the interior of $N$ as a gauge transformation does not matter.",
"Any two choices differ by a gauge transformation that is trivial on the boundary and acts trivially on physical observables.",
"acting on operators on $\\partial N$ ; the boundary value $a$ of ${\\mathcal {A}}$ can be interpreted as a background connection for that global $G$ symmetry.",
"In the 4d setup, the $G$ symmetry acts on the whole half-BPS Dirichlet boundary condition, but in the $A$ -twist there are no local operators it can act on at interior points of $N$ .",
"The global symmetry acts on junction operators only.",
"The scalar field $\\varphi $ is a valid local operator at the junction, identified with the same operator in the auxiliary $\\varphi {\\mathcal {F}}$ theory.",
"In the $A$ -twisted theory, the operator $\\varphi _z$ is actually the conserved current associated to the boundary $G$ symmetry.",
"One way to derive this result is as follows.",
"If we vary the action $I_{\\varphi {\\mathcal {F}}}=\\int _N {\\rm Tr}\\,\\varphi _z F_{t\\overline{z}}\\mathrm {d}t \\mathrm {d}^2z$ with “free” boundary conditions, we find that the variation of $I_{\\varphi {\\mathcal {F}}}$ contains a boundary term $\\int _{\\partial N} {\\rm Tr}\\,\\varphi _z\\delta A_{\\overline{z}}$ , and therefore the Euler-Lagrange equations include a boundary condition $\\varphi _z|_{\\partial N}=0$ , with no restriction on $A_{\\overline{z}}$ .",
"If instead we want the boundary condition to be $A_{\\overline{z}}=a_{\\overline{z}}$ (where $a$ is some specified connection on $C$ ), we can add a boundary term to the action, so that the full action becomes $ I^{\\prime }_{\\varphi {\\mathcal {F}}}=I_{\\varphi {\\mathcal {F}}}-\\int _{\\partial N}{\\rm Tr}\\,\\varphi _z(A_{\\overline{z}}-a_{\\overline{z}}).$ Then the Euler-Lagrange equations give a boundary condition $(A_{\\overline{z}}-a_{\\overline{z}})|_{\\partial N}=0$ , with no constraint on $\\varphi $ .",
"But now the current is $J_z=\\partial I^{\\prime }_{\\varphi {\\mathcal {F}}}/\\partial a_{\\overline{z}}=\\varphi _z$ , as claimed.",
"As explained in Section REF and in [62], in a generalization of this problem with generic $\\Psi $ , the fact that $J_z$ is a Kac-Moody current can be seen classically, though a 1-loop computation similar to what we are about to describe is needed to show that the level is $\\Psi -h$ rather than $\\Psi $ .",
"At $\\Psi =0$ , the Kac-Moody level comes entirely from a 1-loop calculation.",
"The necessary computation can be done very easily with the help of a simple shortcut, which has an analogue in gauge theory in any dimension.",
"We place the $\\varphi {\\mathcal {F}}$ theory on a slab $I\\times C$ , where $I$ is an interval $[0,L]$ with the same Dirichlet boundary conditions at each end.",
"Anomalies can always be computed from the low energy limit of a theory, so in this case we can drop the modes that are nonconstant along $I$ and reduce to a purely two-dimensional theory.",
"This is equivalent to taking a naive $L\\rightarrow 0$ limit.",
"The resulting 2d theory has action $\\int {\\rm Tr}\\, \\varphi _z D_{\\overline{z}} {\\mathcal {A}}_t$ and is a bosonic $\\beta \\gamma $ system with fields $\\varphi _z$ and ${\\mathcal {A}}_t$ (which now depend only on $z$ and $\\overline{z}$ , not on $t$ ) of spins 1 and 0, valued in the adjoint representation.",
"A similar fermionic chiral $bc$ system with adjoint-valued fields has an anomaly coefficient $2h$ , so the bosonic $\\beta \\gamma $ system has anomaly $-2h$ .",
"In gauge theory on the slab $I\\times C$ , the anomaly is localized on the boundaries of the slab since the bulk theory is not anomalous.",
"By symmetry, half the anomaly comes from one end of the slab and half from the other end, so the anomaly coefficient at either end is $-h$ .",
"This value of the level has very special properties, as we will review momentarily.",
"The value $-h$ of the Kac-Moody level is often called the critical level and denoted as $\\kappa _c$ .",
"We will now determine the image in ${\\cal V}$ of the bulk local operators ${\\cal P}[\\varphi ](z)$ of the $\\varphi {\\mathcal {F}}$ gauge theory.",
"Naively, the image would just be ${\\cal P}[J](z)$ .",
"This expression, though, is ill-defined because of the OPE singularity of $J_z$ with itself.",
"We can regularize this expression by point splitting, carefully subtracting singular terms.",
"This is actually a familiar exercise in 2d CFT.",
"The simplest example is the image of the quadratic Hamiltonian ${\\rm Tr}\\, \\varphi _z^2$ .",
"The regularized version of the operator is the Sugawara operator $S_2(z)$ .",
"Recall that for general level $\\kappa $ , the Sugawara operator is proportional to the stress tensor: $T(z) = \\frac{1}{\\kappa + h} S_2(z)$ .",
"In particular, the singular part of the OPE of $S_2(z)$ $S_2(z) J(w) \\sim (\\kappa + h)\\frac{J(w)}{(z-w)^2} + (\\kappa + h) \\frac{D_w J(w)}{z-w} + \\cdots $ is proportional to the critically-shifted level $\\kappa +h$ .",
"When $\\kappa =\\kappa _c=-h$ , $S_2(z)$ has non-singular OPE with the Kac-Moody currents, i.e.",
"it is central.",
"This is precisely the property we expect for the image of ${\\rm Tr}\\varphi _z^2$ in ${\\cal V}$ and fully characterizes it among operators with the same scaling dimension.",
"So $S_2(z)$ corresponds to the quadratic Hitchin Hamiltonians.",
"The regularization of higher Hamiltonians takes more work, but we can invoke a general theorem [1]: the center of the critical Kac-Moody algebra for $G$ is generated by a collection of central elements $S_{{\\mathcal {P}}}(z)$ which regularize ${\\cal P}[J](z)$ .",
"As an algebra, the center of the critical Kac-Moody algebra is isomorphic to the space of holomorphic functions on the oper manifold $L_{\\mathrm {op}}$ for the Langlands dual group $G^\\vee $ .",
"We conclude that $S_{{\\mathcal {P}}}(z)$ is the image at the junction of the 3d operators ${\\cal P}[\\varphi ](z)$ , which are also identified via $S$ -duality with the generators of the algebra of holomorphic functions on $L_{\\mathrm {op}}$ .",
"The presence of a global $G$ symmetry at a Dirichlet boundary adds an extra ingredient to the construction of holomorphic-topological BAA branes.",
"As already remarked, we can generalize Dirichlet boundary conditions by setting the boundary value of the connection to any fixed connection $a$ along $C$ , rather than setting it to zero.",
"We thus produce a whole family of BAA branes $\\mathrm {Dir}(a)$ parameterized by the choice of background $G$ connection $a$ .",
"In the $A$ -twisted theory, the system only depends on $a_{\\overline{z}}$ at the ${\\mathcal {B}}_{\\mathrm {cc}}$ corner and on $a_z$ at the $\\overline{{\\mathcal {B}}}_{\\mathrm {cc}}$ corner.",
"The dependence is encoded respectively in the holomorphic currents $J_z$ at the ${\\mathcal {B}}_{\\mathrm {cc}}$ corner and anti-holomorphic currents $\\overline{J}_{\\overline{z}}$ at the $\\overline{{\\mathcal {B}}}_{\\mathrm {cc}}$ corner.",
"In the absence of local operator insertions at the junctions, Dirichlet boundary conditions thus define a family of (distributional) states $\\Delta (a) \\in {\\mathcal {H}}$ .",
"The insertion of Kac-Moody currents $J_z(z_i)$ or $\\overline{J}_{\\overline{z}}(\\overline{z}_j)$ at the two corners gives functional derivatives $\\prod _i \\frac{\\delta }{\\delta a_{\\overline{z}}}(z_i) \\prod _j \\frac{\\delta }{\\delta a_{ z}}(\\overline{z}_j) \\Delta (a)$ of the state with respect to the background connection.",
"The Kac-Moody nature of the holomorphic and anti-holomorphic connections has an important consequence: the two currents are separately conserved.",
"Recall the anomalous conservation laws: $D_{\\overline{z}} J_z &= - \\frac{\\kappa _c}{2 \\pi } f_{z \\overline{z}} \\cr D_z J_{\\overline{z}} &= \\frac{\\kappa _c}{2 \\pi } f_{z \\overline{z}}$ where $f$ is the curvature of $a$ .",
"These imply that the state $\\Delta (a)$ transforms covariantly but anomalously under infinitesimal complexified gauge transformations of $a$ .",
"To first order in $\\lambda $ , $\\Delta (a_z + D_z \\overline{\\lambda }, a_{\\overline{z}} + D_{\\overline{z}} \\lambda )=\\left[1+\\frac{\\kappa _c}{2 \\pi } \\int _C {\\rm Tr}\\left(\\overline{\\lambda }- \\lambda \\right)f_{z \\overline{z}} \\right] \\Delta (a_z,a_{\\overline{z}})$ A nice enough abstract state $\\Psi \\in {\\mathcal {H}}$ paired with Dirichlet boundary conditions at $t=0$ gives a functional $\\Psi (a)$ that transforms similarly under infinitesimal complexified gauge transformations of $a$ : $\\Psi (a_z + D_z \\overline{\\lambda }, a_{\\overline{z}} + D_{\\overline{z}} \\lambda )=\\left[1+\\frac{\\kappa _c}{2 \\pi } \\int _C {\\rm Tr}\\left(\\overline{\\lambda }- \\lambda \\right)f_{z \\overline{z}} \\right] \\Psi (a)$ Note that $\\Psi $ is invariant under real gauge transformations, which correspond to the special case $\\lambda =\\overline{\\lambda }$ .",
"Functional derivatives of $\\Psi (a)$ with respect to $a$ give correlation functions of critical Kac-Moody currents coupled to $a$ ."
],
[
"Connections vs. Bundles",
"Let $a$ be a connection on a Riemann surface $C$ with structure group the compact gauge group $G$ .",
"Consider a function $\\Psi (a)$ which is invariant not just under $G$ -valued gauge transformations, but under $G_{\\mathbb {C}}$ -valued gauge transformations, acting at the infinitesimal level by $\\delta a_{\\overline{z}}=-D_{\\overline{z}}\\lambda , ~~\\delta a_z=-D_z\\overline{\\lambda }.",
"$ Such a function determines a function on ${\\mathcal {M}}(G,C)$ , because ${\\mathcal {M}}(G,C)$ can be viewed as the quotient of the space of all $G$ -valued connections by the group of complex gauge transformations.This description is slightly imprecise as one needs to take account of considerations of stability to realize ${\\mathcal {M}}(G,C)$ as a quotient.",
"But actually, there is no difficulty: as long as the function $\\Psi (A)$ is continuous as well as invariant under $G_{\\mathbb {C}}$ -valued gauge transformations, it does descend to a function on ${\\mathcal {M}}(G,C)$ .",
"To get from the space of all connections to ${\\mathcal {M}}(G,C)$ , one throws away connections that define unstable holomorphic bundles, imposes an equivalence relation on the semistable ones, and then takes the quotient.",
"A function $\\Psi (a)$ that is continuous and $G_{\\mathbb {C}}$ -invariant is always invariant under the equivalence relation.",
"In any event, these considerations are unimportant for an ${\\mathrm {L}}^2$ theory.",
"Somewhat similar remarks apply in the next paragraph.",
"Conversely, given a function $f$ on ${\\mathcal {M}}(G,C)$ , to define a function $\\Psi (a)$ on the space of connections, we simply declare $\\Psi (a)$ , for a given $a$ , to equal $f$ at the point in ${\\mathcal {M}}(G,C)$ that is associated to the holomorphic bundle $E\\rightarrow C$ that is determined by the $(0,1)$ part of $a$ .",
"The function $\\Psi (a)$ defined this way is automatically invariant under $G_{\\mathbb {C}}$ -valued gauge transformations.",
"This correspondence between functions on ${\\mathcal {M}}(G,C)$ and $G_{\\mathbb {C}}$ -invariant functions of connections can be extended to functions $\\Psi (a)$ that are not $G_{\\mathbb {C}}$ -invariant, but rather transform covariantly under $G_{\\mathbb {C}}$ -valued gauge transformations, with an anomaly.",
"These functionals can be identified with sections of some line bundle over ${\\mathcal {M}}(G,C)$ .",
"For us, the most important case is a function $\\Psi (a)$ that transforms with holomorphic and antiholomorphic anomaly coefficients $\\kappa _c=-h$ , as in eqn.",
"(REF ).",
"In this case, the line bundle is actually the bundle of half-densities on ${\\mathcal {M}}(G,C)$ , as we will show momentarily.",
"More generally, if $\\kappa _c$ is replaced by some other level $\\kappa $ (the same both holomorphically and antiholomorphically), $\\Psi (a)$ would represent a section of $|K_{{\\mathcal {M}}(G,C)}|^{\\frac{\\kappa }{\\kappa _c}}$ .",
"There are many ways to demonstrate that a function that transforms with holomorphic and antiholomorphic anomaly coefficient $\\kappa _c$ correponds to a half-density on ${\\mathcal {M}}(G,C)$ (as shown originally by Beilinson and Drinfeld [1]).",
"We will proceed by showing that a function that transforms with the anomaly coefficient $2\\kappa _c$ is a density on ${\\mathcal {M}}(G,C)$ .",
"A density on ${\\mathcal {M}}(G,C)$ is something that can be integrated over ${\\mathcal {M}}(G,C)$ in a natural way, without using any structure of ${\\mathcal {M}}(G,C)$ beyond the fact that it is the quotient of the space of connections by the group of $G_{\\mathbb {C}}$ -valued gauge transformations.",
"To decide what kind of object $\\Psi (a)$ can be integrated over ${\\mathcal {M}}(G,C)$ , we will use a construction which is somewhat analogous to the definition of the bosonic string path integral.",
"Consider a two-dimensional $G$ gauge theory with connection $a$ , coupled to some matter system with holomorphic and anti-holomorphic Kac-Moody symmetry at levels $(\\kappa , \\kappa )$ .",
"As the path integral of this theory is formally invariant under complexified gauge transformations, we may hope to gauge-fix the path integral to an integral over ${\\mathcal {M}}(G,C)$ .",
"In order to do so, we need a family of gauge-fixing conditions.",
"We simply pick a representative 2d connection $a[m, \\overline{m}] = (a_z[\\overline{m}], a_{\\overline{z}}[m])$ for every point $m$ in ${\\mathcal {M}}(G,C)$ and gauge-fix $a = a[m, \\overline{m}]$ .",
"The aim is to reduce the integral over $a$ to an integral over $m,\\overline{m}$ .",
"We introduce Faddeev-Popov ghosts for this gauge-fixing in the customary manner.",
"For this, we introduce adjoint-valued ghosts $c$ , $\\overline{c}$ associated to complexified gauge parameters $\\lambda $ and $\\overline{\\lambda }$ , and an adjoint-valued 1-form $(b_z, \\overline{b}_{\\overline{z}})$ associated to the gauge-fixing condition.",
"The ghost action is $\\int \\left( {\\rm Tr}\\,b_z D_{\\overline{z}} c + {\\rm Tr}\\,b_{\\overline{z}} D_z c\\right)\\mathrm {d}^2z.$ This ghost system has holomorphic and anti-holomorphic Kac-Moody symmetries, with levels $(2h,2h)=(-2\\kappa _c, -2\\kappa _c)$ .",
"The BRST current is known to be nilpotent if and only if the total anomaly of matter plus ghosts vanishes, that is, if and only if $\\kappa - 2 \\kappa _c=0$ .",
"The integrand over ${\\mathcal {M}}(G,C)$ is prepared with the help of the $b$ zero modes.",
"If we denote the matter partition function as $\\Psi (a)$ , the gauge-fixed path integral becomes $\\int _{{\\mathcal {M}}(G,C)} \\Psi \\left(a[m,\\overline{m}]\\right) \\left\\langle \\prod _i \\left[\\int _C b_z \\frac{\\partial a_{\\overline{z}}[m]}{\\partial m_i} \\mathrm {d}m_i \\right] \\left[\\int _C b_{\\overline{z}} \\frac{\\partial a_{z}[\\overline{m}]}{\\partial \\overline{m}_i} \\mathrm {d}\\overline{m}_i \\right] \\right\\rangle $ Here we have simply imitated the usual definition of the path integral of the bosonic string coupled to a conformal field theory of holomorphic and antiholomorphic central charge $c=26$ .",
"We thus learn that gauge-covariant functionals with level $2 \\kappa _c$ correspond to densities on ${\\mathcal {M}}(G,C)$ .",
"Gauge-covariant functionals with level $\\kappa _c$ , as in (REF ), thus correspond to half-densities on ${\\mathcal {M}}(G,C)$ .",
"We can write down explicitly the Hilbert space inner product in this presentation: $\\left( \\Psi ^{\\prime }, \\Psi \\right) \\equiv \\int _{{\\mathcal {M}}(G,C)} \\overline{\\Psi }^{\\prime }\\left(a[m,\\overline{m}]\\right) \\Psi \\left(a[m,\\overline{m}]\\right) \\left\\langle \\prod _i \\left[\\int _C b_z \\frac{\\partial a_{\\overline{z}}[m]}{\\partial m_i} \\mathrm {d}m_i \\right] \\left[\\int _C b_{\\overline{z}} \\frac{\\partial a_{z}[\\overline{m}]}{\\partial \\overline{m}_i} \\mathrm {d}\\overline{m}_i \\right] \\right\\rangle $ We now have two ways to associate a functional $\\Psi (a)$ to a (nice enough) state $\\Psi \\in {\\mathcal {H}}$ : we identify an abstract state with a half-density on ${\\mathcal {M}}(G,C)$ and promote it to a functional of connections, or we contract $\\Psi $ with the states $\\Delta (a)$ produced by the shifted Dirichlet boundary condition $\\mathrm {Dir}(a)$ .",
"To show that these two procedures are equivalent, we can reason as follows.",
"Classically, the BAA brane $\\mathrm {Dir}(a)$ is supported on the fiber of $T^*{\\mathcal {M}}(G,C)$ at the bundle $E$ defined by $a_{\\overline{z}}$ .",
"This is the simplest type of conormal Lagrangian submanifold and the corresponding state $\\Delta (a)$ has delta function support at $E$ .",
"So the pairing $ (\\Delta (a), \\Psi )$ just evaluates the functional corresponding to $\\Psi $ at the connection $a$ .",
"As a final exercise, we can return to the definition of the quantum Hitchin Hamiltonians.",
"We would like to derive a formula expressing the functional $[D_{\\mathcal {P}}(z) \\circ \\Psi ](a)$ resulting from the action of a Hamiltonian on some $\\Psi $ in terms of the functional $\\Psi (a)$ associated to $\\Psi $ .",
"We consider a strip with ${\\mathcal {B}}_{\\mathrm {cc}}$ boundary conditions on the left boundary and some initial condition at the bottom of strip that, together with data at the corners, defines the state $\\Psi $ .",
"(This setup was sketched in fig.",
"REF (b), where the brane at the bottom of the strip is called ${\\mathcal {B}}_F$ .)",
"By definition, the functional $[D_{\\mathcal {P}}(z) \\circ \\Psi ](a)$ is computed by inserting ${\\mathcal {P}}[\\varphi _z]$ along the left boundary and moving it to the lower left corner of the strip.",
"When we do this, ${\\mathcal {P}}[\\varphi _z]$ is converted to the central element $S_{{\\mathcal {P}}}(z)$ of the chiral algebra, which is a regularized polynomial in $J_z$ and its derivatives.",
"In turn, the $J_z$ insertions can be traded for functional derivatives with respect to $a_{\\overline{z}}$ .",
"As a result, $[D_{\\mathcal {P}}(z) \\circ \\Psi ](a)$ is expressed as a certain differential operator $D^{(a)}_{\\mathcal {P}}(z)$ acting on $\\Psi (a)$ .",
"The operator $D^{(a)}_{\\mathcal {P}}(z)$ is a regularization of ${\\mathcal {P}}\\left[\\frac{\\delta }{\\delta a_{\\overline{z}}} \\right]$ .",
"It maps gauge-covariant functionals to gauge-covariant functionals precisely because $S_{{\\mathcal {P}}}(z)$ is central: the $S_{{\\mathcal {P}}}(z)$ insertion does not modify the Ward identities of the currents and thus the differential operator ${\\cal D}^{(a)}_{{\\mathcal {P}}}(z)$ commutes with the gauge-covariance constraints (REF ).",
"This is actually how the quantum Hitchin Hamiltonians are defined mathematically [1]: they encode the effect of an $S_{{\\mathcal {P}}}(z)$ insertion in a conformal block for the critical Kac-Moody algebra.",
"A similar presentation of Hecke operators requires a discussion of boundary 't Hooft operators at Dirichlet boundary conditions and their endpoints at the junction.",
"In the auxiliary 3d perspective, the boundary 't Hooft operators map to boundary monopole operators.",
"The classical moduli space of such disorder operators was discussed in a similar setting in [63]: it coincides with the affine Grassmannian ${\\mathrm {Gr}}_{G_{\\mathbb {C}}}$ .",
"In the presence of the disorder operator, the gauge bundle at some small distance from the boundary is a specific Hecke modification of whatever fixed bundle is determined by the boundary value of the connection.",
"Correspondingly, in order for $\\varphi _z$ to be non-singular at some distance from the boundary it must have some prescribed poles and zeroes at the boundary.",
"This bare boundary monopole configuration can be dressed by local functionals of $\\varphi _z$ .",
"In Section we will discuss the “spectral flow operators” $\\Sigma _g$ in the chiral algebra, which are labelled by a point in ${\\mathrm {Gr}}_{G_{\\mathbb {C}}}$ and enforce an appropriate version of the constraint on $J_z$ .",
"The spectral flow operators and their Kac-Moody descendants can play the role of endpoints of boundary 't Hooft operators.",
"We will observe the existence of certain (continuous) linear combinations of spectral flow operators which are central and can thus play the role of the images $S_{R,n}(z)$ of endpoints of bulk 't Hooft operators.",
"This will allow us to formulate Hecke operators in a 2d chiral algebra language."
],
[
"Nahm Pole Boundary Conditions",
"Half-BPS Dirichlet boundary conditions can be generalized to a larger collection of Nahm pole boundary conditions labelled by an embedding $\\rho :\\mathfrak {su}(2)\\rightarrow {\\mathfrak {g}}$ .",
"These boundary conditions allow for a choice of background connection $a_\\rho $ whose structure group commutes with $\\rho $ .",
"Following the analogy with the $\\Psi \\ne 0$ results in [62], we will tentatively identify the chiral algebra associated to these boundary conditions with the critical level limit of the ${\\cal W}^G_{\\rho ;\\kappa }$ chiral algebras, which are in turn defined as the Drinfeld-Sokolov reduction associated to $\\rho $ of a $G$ Kac-Moody algebra at level $\\kappa $ .",
"As a basic check of this proposal, we observe that ${\\cal W}_{\\rho ;\\kappa _c}$ has the same large center as critical Kac-Moody, generated by appropriate $S_{\\cal P}(z)$ .",
"A particularly interesting case is the Nahm pole associated to a regular embedding $\\rho $ .",
"The corresponding chiral algebra is the classical limit of a $W$ -algebra and is completely central.",
"It is generated by the $S_{\\cal P}(z)$ : all local operators on the junction are specializations of local operators on the deformed Neumann boundary.",
"The regular Nahm pole is associated to a BAA brane ${\\mathcal {B}}_N$ supported on the Hitchin section of the Hitchin fibration.",
"This section is a complex Lagrangian submanifold of ${\\mathcal {M}}_H$ , of type BAA, but it lies completely outside $T^*{\\mathcal {M}}(G,C)$ .",
"If an eigenstate $\\Psi $ of the Hitchin Hamiltonians is viewed purely as a square-integrable half-density on ${\\mathcal {M}}(G,C)$ , then it would appear not to make any sense to compute the inner product of $\\Psi $ with a state created by ${\\mathcal {B}}_N$ (with appropriate corners), as the space of square-integrable half-densities comes by quantization of $T^*{\\mathcal {M}}(G,C)$ , which is completely disjoint from the support of ${\\mathcal {B}}_N$ .",
"However, as discussed in Section , $\\Psi $ is actually associated to a brane in ${\\mathcal {M}}_H$ of compact support, and therefore should have a well-defined pairing with the state created by any brane.",
"In fact, in the dual $B$ -model, the computation is straightforward.",
"We return to this point at the end of Appendix .",
"The regular Nahm pole boundary supports boundary 't Hooft lines which were studied in [60].",
"They are in natural correspondence with bulk 't Hooft lines, and they are indeed the image of bulk 't Hooft lines brought to the boundary.",
"Nahm pole boundary conditions decorated by boundary 't Hooft operators thus give rise to BAA branes supported on the Hecke modification of the Hitchin section at a collection of points.",
"If the number of points is large enough, these BAA branes are nice submanifolds in $T^*{\\mathcal {M}}(G,C)$ .",
"The associated states should play a role in the separation of variables analysis of [12]."
],
[
"Enriched Neumann Boundary Conditions",
"A basic BAA boundary condition in the 2d $\\sigma $ -model of ${\\mathcal {M}}_H(G,C)$ is the Lagrangian boundary condition associated to the Lagrangian submanifold ${\\mathcal {M}}(G,C)\\subset {\\mathcal {M}}_H(G,C)$ .",
"In 4d terms, this comes from a half-BPS boundary condition of type BAA in in which the gauge field $A$ satisfies Neumann boundary conditions and the Higgs field $\\phi $ satisfies Dirichlet boundary conditions.The brane ${\\mathcal {B}}_{\\mathrm {cc}}$ comes instead from a deformation of Neumann boundary conditions for $A$ , in a sense described in footnote REF , extended to the rest of the supermultiplet in a different fashion and preserving a different symmetry (ABA rather than BAA).",
"We will refer to this boundary condition as BAA Neumann.",
"In terms of the $\\varphi {\\mathcal {F}}$ theory, this is simply the boundary condition defined by $\\varphi |_{\\partial N}=0$ , with no constraint on $A_{\\overline{z}}|_{\\partial N}$ .",
"We showed in Section REF that this is the boundary condition one gets from the Euler-Lagrange equations of the action $I_{\\varphi {\\mathcal {F}}}$ , with “free” variations of all fields.",
"With this boundary condition, no restriction is placed on a gauge transformation on the boundary.",
"That is consistent, because the boundary condition $\\varphi |_{\\partial N}=0$ is gauge-invariant.",
"However, there is a gauge anomaly on a boundary of $N$ that has this boundary condition.",
"The anomaly coefficient is $+h$ .",
"An easy way to see this is to consider the $\\varphi {\\mathcal {F}}$ theory on a slab $N=I\\times C$ , with the ${\\mathcal {A}}_{\\overline{z}}=0$ boundary condition at the left end of the slab and the $\\varphi _z=0$ boundary condition at the right end.",
"These boundary conditions are invariant under constant gauge transformations by an element $g\\in G$ .",
"At the left end of the slab, a constant gauge transformation is interpreted as a global symmetry.",
"This gives an action of $G$ as a group of global symmetries of the theory on the slab.",
"The theory with $\\varphi _z=0$ at one end of the slab and ${\\mathcal {A}}_{\\overline{z}}=0$ at the other end is completely trivial: up to a gauge transformation, the only classical solution is $\\varphi _z={\\mathcal {A}}_{\\overline{z}}=0$ everywhere, and there are no low energy excitations.",
"So the $G$ action is anomaly free.",
"As it acts by a constant gauge transformation, its anomaly coefficient is the sum of the anomaly coefficient of the global symmetry at the left end of the slab and of the gauge symmetry at the right end.",
"We learned in Section REF that the global symmetry has an anomaly coefficient $-h$ at the left end of the slab.",
"So the gauge symmetry must have an anomaly coefficient $+h$ at the right end.",
"So in short, the $\\varphi _z=0$ boundary condition in the $\\varphi {\\mathcal {F}}$ theory has anomaly $+h$ .",
"A possible cure for the anomaly is to add extra degrees of freedom at the junction with anomaly $-h$ , the critical level.",
"Unitary degrees of freedom at the junction will not help, as they have a positive anomaly coefficient.",
"Instead, we can do the following.",
"BAA Neumann boundary conditions can be enriched, preserving the supersymmetry of type BAA, by adding to the boundary 3d matter degrees of freedom that make a 3d superconformal quantum field theory (SQFT) with ${\\mathcal {N}}=4$ supersymmetry.",
"We will call the resulting boundary condition an enriched Neumann boundary condition (of type BAA, if it is necessary to specify this).",
"The interesting case is that the SQFT has $G$ symmetry and is coupled to the gauge field $A$ of the bulk ${\\cal N}=4$ theory; this is possible, because with Neumann boundary conditions, $A$ is unconstrained on the boundary.",
"The $A$ -twist of the bulk 4d theory induces an $A$ -twist of the boundary 3d SQFT.",
"The twisted boundary theory can contribute a negative amount to the anomaly at the junction.",
"For our application, we want holomorphic boundary conditions for the 3d SQFT that support a $G$ Kac-Moody algebra at critical level $\\kappa _c=-h$ .",
"For this purpose, we can employ one of the holomorphic boundary conditions defined in [64].",
"The effect of “enrichment” is that the boundary condition for the $\\varphi {\\mathcal {F}}$ theory ending on an enriched Neumann boundary is no longer $\\varphi _z|_{\\partial N}=0$ .",
"Rather, $\\varphi _z|_{\\partial N}$ equals the critical Kac-Moody currents of the boundary chiral algebra of the SQFT.",
"In particular, the images $S_{{\\mathcal {P}}}(z)$ of ${\\mathcal {P}}[\\varphi _z]$ are identified with the central elements built from the critical Kac-Moody currents for the matter.",
"The simplest example is the case that the SQFT is a theory of free 3d hypermultiplets transforming in a symplectic representation $R$ of $G$ .",
"Let $Z$ be the bosonic field in the hypermultiplets.",
"Twisting turns the components of $Z$ into spinors, still valued in the representation $R$ .",
"As analyzed in [58], with the appropriate sort of boundary condition, the twisted hypermultiplet path integral on a three-manifold with boundary is a 2d contour path integral, with a holomorphic action, of the general sort described in Section REF .",
"In this case, the holomorphic action is $S[Z,{\\mathcal {A}}] = \\int _C \\langle Z, \\overline{\\partial }_{\\mathcal {A}}Z \\rangle .$ where $\\langle \\cdot ,\\cdot \\rangle $ denotes the symplectic pairing on the representation $R$ , and we have included a coupling to the complex gauge field ${\\mathcal {A}}$ .",
"Fields $Z$ with such an action are sometimes called symplectic bosons and do have a negative Kac-Moody level; see Section REF for more about them.",
"The simplest possibility is to select an $R$ for which the level is precisely$-h$ .",
"It is also possible to select an $R$ for which the level is more negative and make up the difference with some extra 2d chiral fermions in a real representation $R_f$ of $G$ placed at the junction.",
"We will discuss the simplest possibility here and briefly comment on the general case at the end.",
"Now consider a junction between the deformed Neumann boundary condition that supports the $\\varphi {\\mathcal {F}}$ theory and the BAA Neumann boundary condition enriched by hypermultiplets.",
"The appropriate holomorphic action is the sum of $I_{\\varphi {\\mathcal {F}}}$ and $S[Z,{\\mathcal {A}}]$ : $\\widehat{I}=\\int _N {\\rm Tr}\\,\\varphi {\\mathcal {F}}+\\int _{C=\\partial N} \\langle Z, \\overline{\\partial }_{\\mathcal {A}}Z \\rangle .$ The Euler-Lagrange equation for ${\\mathcal {A}}_{\\overline{z}}$ gives a boundary condition $\\varphi _z|_{\\partial N}=\\mu (Z), $ where $\\mu (Z)$ is the holomorphic moment map for the action of $G_{\\mathbb {C}}$ on the representation $R$ .",
"Here the components of $\\mu (Z)$ become (after quantization) the Kac-Moody currents of the matter system, so this formula illustrates the statement that after enrichment, the appropriate boundary condition sets $\\varphi $ equal to the Kac-Moody currents.",
"We also have the classical equations of motion $0=\\overline{\\partial }_{\\mathcal {A}}\\varphi =\\overline{\\partial }_{\\mathcal {A}}Z.$ Triples $({\\mathcal {A}},\\varphi ,Z)$ satisfying these conditions along with eqn.",
"(REF ) describe a brane over ${\\mathcal {M}}_H(G,C)$ of type BAA.",
"The simplest case is that for given ${\\mathcal {A}},\\varphi $ , there is at most one $Z$ satisfying the conditions.",
"If so, the pairs $({\\mathcal {A}},\\varphi )$ for which such a $Z$ does exist furnish a complex Lagrangian submanifold of ${\\mathcal {M}}_H(G,C)$ , in complex structure $I$ , corresponding to a brane of type BAA.",
"These Lagrangian submanifolds are of conormal type, since if a suitable $Z$ exists for one Higgs pair $({\\mathcal {A}},\\varphi )$ , then a suitable $Z$ likewise exists after any rescaling of $\\varphi $ .",
"The natural quantization of these BAA branes is a path integral over $Z$ [59]: $\\Psi (a) = \\int DZ D\\overline{Z} \\,e^{\\int _C \\left[\\langle Z, \\overline{\\partial }_a Z \\rangle - \\langle \\overline{Z}, \\partial _a \\overline{Z} \\rangle \\right]}$ possibly modified by the insertion of a non-trivial corner in the form of a collection of $Z$ and $\\overline{Z}$ insertions in the path integral [64].",
"From the point of view of the present paper, the meaning of this formula is as follows.",
"We place the enriched Neumann brane at the bottom of a strip, playing the role of the brane denoted as ${\\mathcal {B}}_x$ in fig.",
"REF (b).",
"Assuming no operator insertions are made at the bottom corners of the strip, the state in ${\\mathcal {H}}={\\mathrm {Hom}}(\\overline{{\\mathcal {B}}}_{\\mathrm {cc}},{\\mathcal {B}}_{\\mathrm {cc}})$ defined by this picture is $\\Psi (a)$ .",
"The statement makes sense, because the $Z$ and $\\overline{Z}$ fields support current algebras at critical level $\\kappa _c=-h$ , so that the path integral of these fields does indeed define a half-density on ${\\mathcal {M}}(G,C)$ .",
"The chiral algebra at the junction in this construction consists of the subalgebra of gauge-invariant operators within the boundary chiral algebra of the 3d matter theory, i.e.",
"it consists of operators built from the $Z$ 's and their derivatives which have trivial OPE with the critical Kac-Moody currents.",
"One can modify the construction just described by including chiral and antichiral operators at the bottom corners of the strip; to describe the resulting state, one just includes the corresponding factors in eqn.",
"(REF ).",
"We will discuss this construction further in Section .",
"Figure: (a) A rectangle with an enriched Neumann brane ℬ{\\mathcal {B}} at the bottom and a generalized Dirichlet brane ℬ ' {\\mathcal {B}}^{\\prime } at the top.",
"(b) In topological fieldtheory, the “height” and “width” of the rectangle are arbitrary.",
"In a limit in which the height is small, we reduce to a purely three-dimensional computation ona product I×CI\\times C. As always, CC is not drawn.There is an alternative way to understand (REF ) directly in 4d.",
"The alternative perspective can be applied as well to a more general situation where the corresponding BAA brane has a non-trivial ${\\mathrm {CP}}$ bundle or where extra chiral fermions are added at the junctions.",
"In order to read off $\\Psi (a)$ , we can contract the state $\\Psi $ created by the enriched Neumann boundary with the state $\\Delta (a)$ created by a Dirichlet boundary condition $\\mathrm {Dir}(a)$ .",
"The inner product between these two states is represented by the path integral on the rectangle of fig.",
"REF (a) with $\\mathrm {Dir}(a)$ boundary conditions at the top and enriched Neumann at the bottom.",
"In two-dimensional topological field theory, the “height” and “width” of the rectangle are arbitrary.",
"Take the limit that the height is much less than the width (fig.",
"REF (b)).",
"In this limit, the path integral reduces to a path integral in a 3d theory on $I\\times C$ .",
"The 3d theory is produced by compactification from four to three dimensions on an interval with Dirichlet boundary conditions at one end and enriched Neumann boundary conditions at the other end.",
"This compactification gives a simple answer, because the 4d fields are all frozen at one boundary or the other: one just gets back the same 3d theory which was employed to construct the enriched Neumann boundary conditions.",
"In our example, this is the theory of the same free hypermultiplets that we started with, with the global $G$ symmetry now identified with the $G$ symmetry that acts at the Dirichlet boundary.",
"The answer of (REF ) is just the partition function of the 3d theory on $I\\times C $ , with the boundary conditions which give rise to the symplectic bosons or their complex conjugates.",
"We can thus apply (REF ) to a situation where the corresponding BAA brane is complicated, bypassing the 2d derivation.",
"Any (anti)chiral fermions added at the junctions would just contribute their partition function, i.e.",
"$\\Psi (a) = \\int DZ D\\overline{Z} D\\psi D\\overline{\\psi }\\,e^{\\int _C \\left[\\langle Z, \\overline{\\partial }_a Z \\rangle - \\langle \\overline{Z}, \\partial _a \\overline{Z} \\rangle \\right]+\\left[( \\psi , \\overline{\\partial }_a \\psi ) - (\\overline{\\psi }, \\partial _a \\overline{\\psi })\\right]}$ As long as the combined level of the symplectic bosons and fermions is $-h$ , this represents a half-density on ${\\mathcal {M}}(G,C)$ .",
"In this section we described states associated to elementary boundary conditions.",
"The construction can be easily generalized to describe operators associated to analogous elementary interfaces.",
"The composition of elementary interfaces can produce a vast collection of BAA boundary conditions and interfaces, which are associated to the composition of the corresponding operators.",
"This would allow, among other things, the calculation of $C \\times [0,1]$ partition functions for A-twisted 3d ${\\cal N}=4$ gauge theories with chiral and antichiral boundary conditions at the two ends of the segment.",
"We leave a detailed analysis of this problem, as well as the B-model analogue, to future work."
],
[
"Preliminaries",
"The quantization of BAA branes associated to enriched Neumann boundary conditions has given us examples (REF ) of wavefunctions which are defined as partition functions of 2d CFTs with chiral and antichiral critical Kac-Moody symmetry.",
"In this section we describe how to compute the action of quantum Hitchin Hamiltonians and Hecke operators on such partition functions, directly in a 2d CFT language.",
"At the same time, we will gain a better appreciation of the mathematical results we invoked in Section to define the Hecke operators.",
"We have already discussed briefly the 2d CFT interpretation of the quantum Hitchin Hamiltonians.",
"The critical Kac-Moody chiral algebra has a large center, generated by certain local operators $S_{\\cal P}(z)$ which have non-singular OPE with the currents.",
"The transformation of a correlation function under complexified gauge transformations is described by the Ward identities for the currents.",
"The statement that $S_{\\cal P}(z)$ has non-singular OPE with the currents means that a correlation function with insertions of such operators only $\\langle S_{{\\cal P}_1}(z_1) \\cdots S_{{\\cal P}_n}(z_n) \\rangle _a$ satisfies the same transformation properties (REF ) as a partition function.",
"It thus also defines a half-density on ${\\mathcal {M}}(C,G)$ .",
"Furthermore, the $S_{\\cal P}(z)$ are assembled from Kac-Moody currents, which can be traded for functional derivatives with respect to the connection.",
"We can thus expand recursively $\\langle S_{{\\cal P}_1}(z_1) \\cdots S_{{\\cal P}_n}(z_n) \\rangle _a = {\\cal D}^{(a)}_{{\\mathcal {P}}_1}(z_1 )\\langle S_{{\\cal P}_2}(z_2) \\cdots S_{{\\cal P}_n}(z_n) \\rangle _a$ and the final answer will be independent of the order of the operators to which we apply the recursion.",
"The differential operators ${\\cal D}^{(a)}_{{\\mathcal {P}}_1}(z_1 )$ thus commute.",
"Although here we referred to correlation functions of some 2d CFT, this is unnecessary: given a half-density on ${\\mathcal {M}}(C,G)$ represented by a gauge-covariant functional $\\Psi (a)$ on the space of connections, the functional derivatives with respect to $a$ behave just as Kac-Moody currents.",
"The differential operators ${\\cal D}^{(a)}_{{\\mathcal {P}}}(z)$ represent in a gauge-covariant manner the action of the quantum Hitchin Hamiltonians ${\\cal D}_{{\\mathcal {P}}}(z)$ on the half-density $\\Psi $ .",
"When doing calculations in a neighborhood $U$ of a point $p$ in $C$ , it is usually helpful to choose a representative connection for the bundle which vanishes on $U$ .",
"This is always possible because ${\\mathcal {A}}_{\\overline{z}}$ can be set to zero locally by a complex-valued gauge transformation.",
"That amounts to trivializing the bundle over $U$ , as we did in discussing general Hecke transformations in Section REF .",
"Then the Kac-Moody currents are meromorphic on $U$ and satisfy the Kac-Moody OPE in a standard form $J^a(z) J^b(w) \\sim \\frac{\\kappa _c \\delta ^{ab}}{(z-w)^2} + \\frac{f^{ab}_d J^d(w)}{z-w}.$ Recall the definition of the Fourier modes of the Kac-Moody algebra $J^a_n \\equiv \\oint _{|z|=\\epsilon } \\frac{\\mathrm {d}z}{2 \\pi {\\mathrm {i}}} z^n J^a(z)$ The insertion of such a Fourier mode represents an infinitesimal deformation of $a_{\\overline{z}}$ supported on the loop $|z|=\\epsilon $ , or a deformation of the bundle which modifies the gluing of a bundle over $U$ to a bundle over the rest of the surface by an infinitesimal gauge transformation in $U^{\\prime }=U\\backslash p$ .",
"In the absence of other operator insertions in the disk $|z|<\\epsilon $ (or in the presence of central operator insertions) correlation functions with insertions of the non-negative Fourier modes vanish.",
"The corresponding infinitesimal gauge transformations can be extended to $U$ and do not change the bundle.",
"They represent changes in the original trivialization over $U$ .",
"In the presence of a generic operator insertion in the disk, the non-negative modes act non-trivially: the insertion of a general local operator requires some choice of trivialization of the bundle and the result depends on the choice.",
"The negative Fourier modes can act non-trivially even in the absence of other operator insertions and represent infinitesimal gauge transformations which can change the bundle.",
"Repeated action of the negative modes builds the image at $z=0$ of the vacuum module for the Kac-Moody algebra.",
"The operator $S_{\\cal P}(0)$ and other central elements in the chiral algebra correspond by the operator-state correspondence to the vectors in the vacuum module that are annihilated by all the non-negative Fourier modes of the currents.",
"For example, the Sugawara vector is $|S_2\\rangle \\equiv {\\rm Tr}\\,J_{-1} J_{-1} |0 \\rangle ,$ with similar formulas for other central elements."
],
[
"Hecke Operators as Central Vertex Operators",
"The Hecke integral operators can also be analyzed with 2d chiral algebra technology.",
"We would like to lift the Hecke operators to operators acting on gauge-covariant functionals and give them a 2d chiral algebra interpretation in terms of the insertion of local operators which have trivial OPE with the Kac-Moody currents.",
"Such a formulation immediately guarantees that the Hecke operators commute with the quantum Hitchin Hamiltonians and with other Hecke operators.",
"We can follow verbatim the definition of Hecke modifications from Section REF .",
"First, we trivialize the bundle $E$ on a small neighborhood $U$ of a point $p$ .",
"We can then think of $E\\rightarrow C$ as built by gluing a trivial bundle over $U$ to the bundle $E$ over $C\\backslash p$ with a trivial gluing map.",
"Then we produce a new bundle $E^{\\prime }$ by modifying the gluing map to $z^{\\sf m}$ (where ${\\sf m}$ is an integral weight of the dual group and $z$ is a local parameter at $p$ ).",
"The bundles $E$ and $E^{\\prime }$ can be described by the same connection away from $U$ .",
"The connection $a$ which describes $E$ vanishes on $U$ , while the connection $a^{\\prime }$ which describes $E^{\\prime }$ coincides with $a$ outside of $U$ and can be taken in $U$ to be some specific reference connection supported on an annulus in $U^{\\prime }$ , and proportional to ${\\sf m}$ .",
"Take the functional $\\Psi $ which represents the input wavefunction, and evaluate it on $a^{\\prime }$ .",
"This gives a new functional $\\Psi _{{\\sf m}}(a)$ .",
"Crucially, $\\Psi _{\\sf m}(a)$ is not covariant under complexified gauge transformations: the new bundle $E^{\\prime }$ depends on the original choice of trivialization of $E$ .",
"Formally, $\\Psi _{\\sf m}(a)$ and its functional derivatives can be interpreted as correlation functions of Kac-Moody currents in the presence of a “spectral flow operator” $\\Sigma _{{\\sf m}}(0)$ .",
"The term “spectral flow” refers to a certain automorphism of the Kac-Moody algebra: $J^\\alpha _n &\\rightarrow J^{\\alpha }_{n+({\\sf m}, \\alpha )} \\cr J^h_n &\\rightarrow J^h_n - {\\sf m}\\kappa \\delta _{n,0}$ where $J^\\alpha $ is the current associated to a root $\\alpha $ and $J^h$ are the Cartan currents.",
"This is precisely the effect of a $z^{\\sf m}$ gauge transformation on the Fourier modes of the currents.",
"By definition, a spectral flow module is the image of the vacuum module under the spectral flow.",
"In particular, the image of the vacuum vector under spectral flow is annihilated by $J^{\\alpha }_{n+({\\sf m}, \\alpha )}$ with non-negative $n$ and is an eigenvector of $J^h_0$ with a nontrivial eigenvalue.",
"Correspondingly, a spectral flow operator $\\Sigma _{{\\sf m}}(0)$ is a local operator such that the OPE with the Kac-Moody currents become non-singular after a $z^{\\sf m}$ gauge transformation.",
"The $J^{\\alpha }(z)$ will have a pole/zero of order $({\\sf m}, \\alpha )$ at $z=0$ and $J^h(z)$ will have a simple pole of residueBosonization offers a convenient way to describe $\\Sigma _{{\\sf m}}$ .",
"Schematically, if the Cartan currents are bosonized as $J^h = \\partial \\phi ^h$ and the remaining currents as vertex operators $J^\\alpha = e^{\\frac{\\alpha }{\\kappa } \\cdot \\varphi }$ , then the spectral flow operator can be represented by a vertex operator $\\Sigma _{{\\sf m}} = e^{{\\sf m}\\cdot \\varphi }$ as well.",
"This representation can be useful for some calculations, but behaves poorly under general $G_{\\mathbb {C}}$ gauge transformations.",
"${\\sf m}\\kappa \\Sigma _{{\\sf m}}(0)$ .",
"We stress again that the functional $\\Psi _{{\\sf m}}(a)$ does not represent a half-density on ${\\mathcal {M}}(G,C)$ , as it depends on the choice of trivialization of $E$ .",
"The properties of the spectral flow operator $\\Sigma _{{\\sf m}}(0)$ characterize the precise failure of the gauge-covariance constraints (REF ).",
"Our objective is to build from $\\Sigma _{{\\sf m}}(0)$ some local operator insertion which is central and can thus represent the action of a Hecke operator on $\\Psi (a)$ .",
"Before continuing with the general discussion, we present the reference example of $G={\\mathrm {SO}}(3)$ and minimal charge.",
"The basic spectral flow automorphism is $J^\\pm _n \\rightarrow J^{\\pm }_{n \\pm 1} \\qquad \\qquad J^0_n \\rightarrow J^0_n - \\delta _{n,0}$ The spectral flow module is built from a vector $|1\\rangle $ which satisfies $J^{\\pm }_{n \\pm 1} |1\\rangle &=0 \\qquad \\qquad n\\ge 0 \\cr J^0_n |1\\rangle &=0 \\qquad \\qquad n> 0 \\cr J^0_0|1\\rangle &= |1\\rangle .$ For this case of the basic “charge 1” spectral flow operator, we will write $\\Sigma _1(0)$ for $\\Sigma _{\\sf m}(0)$ .",
"Following our discussion of the affine Grassmannian in Section REF , we can replace the gluing map $z^{\\sf m}$ by another gluing map $g$ in the same orbit ${\\mathrm {Gr}}^{\\sf m}$ .",
"The same construction with $z^{\\sf m}$ replaced by $g$ produces a functional $\\Psi _{g}(a)$ .",
"The functional derivatives of $\\Psi _{g}(a)$ can be interpreted as correlation functions in the presence of a modified spectral flow operator $\\Sigma _{g}(0)$ .",
"As a change of trivialization is implemented by the non-negative modes of the currents, we can express the action of these modes on $\\Sigma _{g}(0)$ as certain differential operators along ${\\mathrm {Gr}}^{\\sf m}$ .",
"We should stress that the definition of $\\Psi _{g}(a)$ really requires a choice of reference connection supported within $U^{\\prime }=U\\backslash p$ which realizes the gluing map $g$ .",
"Different connections describing the same $g$ are related by complex gauge transformations and thus may lead to a different normalization for $\\Psi _{g}(a)$ and $\\Sigma _{g}(0)$ .",
"As a result, $\\Psi _{g}(a)$ and $\\Sigma _{g}(0)$ are actually sections of a certain line bundle on ${\\mathrm {Gr}}^{\\sf m}$ .",
"We will indentify this line bundle in Section REF .",
"The non-negative modes of the currents will act as vector fields on sections of this line bundle.",
"This is the chiral algebra manifestation of the mismatch between the bundles of half-densities before and after the Hecke modification.",
"The line bundle on ${\\mathrm {Gr}}^{\\sf m}$ is controlled by the level of the Kac-Moody algebra.",
"In the next section, we will show that at critical level, this line bundle coincides with the bundle of densities on ${\\mathrm {Gr}}^{\\sf m}$ .",
"This means that in a theory that has holomorphic and antiholomorphic Kac-Moody levels that are both critical, the spectral flow operator is a density on ${\\mathrm {Gr}}^{\\sf m}$ and can be naturally integrated: $\\widehat{\\Sigma }_{\\sf m}(0) \\equiv \\int _{{\\mathrm {Gr}}^{\\sf m}}\\Sigma _{g}(0) |\\mathrm {d}g|^2.$ $\\widehat{\\Sigma }_{\\sf m}(0)$ has the appropriate properties for the Hecke operator of charge ${\\sf m}$ dual to a Wilson operator with minimal corners $s_{R}$ in the language of Section REF (that is, a Wilson operator defined using wavefunctions built from highest weight vectors).",
"The integral over ${\\mathrm {Gr}}^{\\sf m}$ generalizes the integral over ${\\mathbb {CP}}^1_x$ in eqn.",
"(REF ).",
"The action of a non-negative mode of the currents on $\\widehat{\\Sigma }_{\\sf m}(0)$ can be traded for a Lie derivative of $\\Sigma _{g}(0)$ along the corresponding vector field on ${\\mathrm {Gr}}^{\\sf m}$ .",
"As long as no boundary terms appear upon integration by parts (this may require a technical analysis when monopole bubbling is possible), $\\widehat{\\Sigma }_{\\sf m}(0)$ will be annihilated by the non-negative modes of the Kac-Moody algebra and is thus central.",
"Correspondingly, the averaged functional $ \\int _{{\\mathrm {Gr}}^{\\sf m}}\\Psi _{g}(a)|\\mathrm {d}g|^2$ obtained by acting with $\\widehat{\\Sigma }_{\\sf m}(0)$ on $\\Psi (a)$ is gauge-covariant and can represent the action of the principal Hecke operator of charge ${\\sf m}$ .",
"We can readily apply this construction to our illustrative example of $G={\\mathrm {SO}}(3)$ and minimal charge.",
"We can define a ${\\mathbb {CP}}^1$ family of spectral flow operators $\\Sigma _{1;\\mu }(0)$ as a global ${\\mathrm {SO}}(3,{\\mathbb {C}})$ rotation of $\\Sigma _1(0)$ .",
"Formally, we can write the corresponding states as $|1;\\mu \\rangle = e^{\\mu J^+_0 }|1\\rangle $ It is straightforward to express the action of the non-negative Fourier modes on $|1;\\mu \\rangle $ as differential operators in $\\mu $ and verify that they are total derivatives.",
"It is clear that $J^+_0 |1;\\mu \\rangle = \\partial _\\mu e^{\\mu J^+_0 }|1\\rangle = \\partial _\\mu |1;\\mu \\rangle $ The action of $J^0_0$ is also straightforward $J^0_0 |1;\\mu \\rangle = e^{\\mu J^+_0 }(J^0_0 + \\mu J^+_0)|1\\rangle = \\partial _\\mu \\left(\\mu |1;\\mu \\rangle \\right)$ Computing the action of $J^-_0$ requires only a bit more work: $J^-_0 |1;\\mu \\rangle = e^{\\mu J^+_0 }(J^-_0 + 2 \\mu J^0_0 + \\mu ^2 J^+_0)|1\\rangle = \\partial _\\mu \\left(\\mu ^2 |1;\\mu \\rangle \\right)$ This makes the insertion of $\\int \\Sigma _{1;\\mu , \\overline{\\mu }}(0)|\\mathrm {d}\\mu |^2$ central, as long as boundary terms for the integration by parts vanish.",
"The natural way to show that boundary terms vanish is to show that (REF ) is really the integral of a density on ${\\mathbb {CP}}^1$ .",
"In order to do so, we need to cover ${\\mathbb {CP}}^1$ with a second patch, starting from the opposite spectral flow operator $\\Sigma _{-1}(0)$ and deforming it to $\\Sigma _{-1;\\mu }(0)$ as $|-1;\\mu \\rangle = e^{-\\mu ^{-1} J^-_0 }|-1\\rangle $ The action of the non-negative Fourier modes on this family involves the same differential operators in $\\mu $ as for $\\mu ^2 |1;\\mu \\rangle $ .",
"Including the antichiral modes we find that we can consistently identify $\\Sigma _{-1;\\mu }(0) = |\\mu |^4 \\Sigma _{1;\\mu }(0)$ and combine them into a density $\\Sigma _{1;\\mu }(0)$ defined on the whole ${\\mathbb {CP}}^1$ .",
"We explain a different and more general approach to this result in Section REF .",
"The $\\Sigma _{1;\\mu }(0)$ insertion, by construction, corresponds to a very specific modification $a \\rightarrow a^{\\prime }[a;\\mu ]$ of the background connection.",
"Recall that we work in a gauge where $a$ vanishes inside the open patch $U$ and $a^{\\prime }[a;0]$ differs from $a$ by some reference connection supported on an annular region in $U^{\\prime }$ .",
"The insertion of the exponentiated Fourier mode $e^{\\mu J^+_0}$ adds a further specific modification to the connection on a wider annular region, producing $a^{\\prime }[a;\\mu ]$ .",
"The integral operator (REF ) corresponding to (REF ) is thus $\\int _{{\\mathbb {CP}}^1}\\Psi (a^{\\prime }[a;\\mu ])|\\mathrm {d}\\mu |^2$ Compare this with (REF ).",
"We should write $\\Psi (y)$ there as $\\Psi (a(y))$ here, with $a(x)$ denoting our gauge-fixing choice of a representative connection for every bundle $x$ .",
"There is no reason for $a^{\\prime }[a(x);\\mu ]$ to be already in a gauge-fixed form.",
"A complexified gauge transformation will be needed to bring it to the gauge-fixed form $a(x_\\mu )$ for the modified bundle $x_\\mu $ .",
"The anomaly will give some rescaling factor which we can write as the absolute value of an holomorphic quantity $\\omega $ : $ \\Psi (a^{\\prime }[a(x);\\mu ]) = |\\omega (x;\\mu )|^2 \\Psi (a(x_\\mu ))$ The integral operator becomes $\\int _{{\\mathbb {CP}}^1}|\\omega (x;\\mu )|^2 \\Psi (a(x_\\mu ))|\\mathrm {d}\\mu |^2$ We obtain: $F(x,y) = \\int _{{\\mathbb {CP}}^1} |\\omega (x;\\mu )|\\delta (y;x_\\mu )|\\mathrm {d}\\mu |^2$ where $\\delta (y;x)$ is a delta function supported on the diagonal in ${\\mathcal {M}}\\times {\\mathcal {M}}$ .",
"The left hand side of (REF ) is a half-density in $x$ and a density in $\\mu $ .",
"The right hand side involves a half-density in $y=x_\\mu $ .",
"We can thus identify $\\omega (x;\\mu )$ with $w(\\vec{x};\\mu )$ in $(\\ref {conf})$ .",
"The factor $\\omega (x;\\mu )$ encodes the anomalous rescaling of $\\Psi $ under a complexified gauge transformation and is thus non-vanishing.",
"This allows us to identify it with the holomorphic factor $k$ introduced by [1].We will see momentarily that $\\omega (x;\\mu )$ could be computed in the theory of adjoint free fermions.",
"We identify (REF ) with the Hecke operator $H_{p=0}$ associated to the two-dimensional representation of $G^\\vee $ with a minimal choice of corners corresponding to $|k|^2$ , as in eqn.",
"(REF )."
],
[
"Free Fermion Trick",
"In the last section, we observed that in a CFT with Kac-Moody symmetry, the operator $\\Sigma _g(0)$ , where $g$ is a gauge transformation associated to a Hecke transformation of weight ${\\sf m}$ , is a section of a line bundle over ${\\mathrm {Gr}}^{\\sf m}$ .",
"This line bundle, since it is determined by the anomaly, depends only on the central charge $\\kappa $ of the CFT.",
"We would like to compute this line bundle for a CFT of critical level $\\kappa _c=-h$ , but it turns out that it is particularly simple to compute it for a CFT whose level is $-\\kappa _c=+h$ .",
"This will give us the inverse of the line bundle over ${\\mathrm {Gr}}^{\\sf m}$ that we actually want.",
"After picking a spin structure on $C$ , or equivalently a choice of $K_C^{1/2}$ , we consider a system of chiral (Majorana-Weyl) fermions $\\psi ^a$ of spin 1/2 valued in the adjoint representation of the gauge group.",
"The Kac-Moody currents are constructed as normal ordered fermion bilinears and have anomalous gauge transformation due to the normal ordering.",
"The central charge is exactly $h=-\\kappa _c$ .",
"We claim that the spectral flow operators in the theory of adjoint free fermions are sections of $K^{-1}_{{\\mathrm {Gr}}^{\\sf m}}$ .",
"This means that the spectral flow operators in a CFT at the critical level $\\kappa _c$ are sections of the inverse of this or $K_{{\\mathrm {Gr}}^{\\sf m}}$ .",
"We will illustrate the case of $G={\\mathrm {SO}}(3)$ and minimal ${\\sf m}$ , and briefly indicate the generalization to other $G$ and ${\\sf m}$ .",
"After picking a Cartan subalgebra of ${\\mathrm {SO}}(3)$ , we have chiral fermions $\\psi ^\\pm $ and $\\psi ^0$ .",
"The basic Hecke modification at $z=0$ results in $\\psi ^+$ having a pole at $z=0$ and $\\psi ^-$ having a zero.",
"This is implemented simply by a $\\psi ^-$ insertion at $z=0$ .",
"A Hecke modification associated to a point $(u,v)\\in {\\mathbb {CP}}^1$ is implemented by an ${\\mathrm {SO}}(3,{\\mathbb {C}})$ rotation of $\\psi ^-$ , i.e.",
"by $\\psi ^-_{(u,v)} \\equiv u^2 \\psi ^- + 2 u v \\psi ^0 + v^2 \\psi ^+$ .",
"An insertion of $\\psi ^-_{(y,v)}(0)$ imposes the vanishing of $\\psi ^-_{(u,v)}$ at $z=0$ , while giving a pole to other linear combinations of the components of $\\psi $ .",
"In the theory of adjoint fermions, we thus have $\\Sigma _{1;(u,v)}(0) = \\psi ^-_{(u,v)}$ .",
"This is quadratic in homogeneous coordinates of ${\\mathbb {CP}}^1={\\mathrm {Gr}}^{\\sf m}$ , so it is a global section of ${\\mathcal {O}}(2) = K_{{\\mathrm {Gr}}^{\\sf m}}^{-1}$ , as claimed.",
"Hence at critical level, the spectral flow operator in this example is a section of $K_{{\\mathrm {Gr}}^{\\sf m}}$ , a fact that was exploited in Section REF .",
"For a general gauge group and charge, the reference Hecke modification results in the fermions labelled by a root $\\alpha $ having extra poles or zeroes of order $(\\lambda , \\alpha )$ at $z=0$ .",
"This is implemented by a very simple vertex operator: $\\prod _{\\alpha | (\\lambda , \\alpha )<0} \\prod _{n_\\alpha =0}^{-(\\lambda , \\alpha )-1}\\partial _z^{n_{\\alpha }}\\psi ^{\\alpha }$ It is straightforward to see that this product transforms as a section of $K_{{\\mathrm {Gr}}^{\\sf m}}^{-1}$ : each fermion derivative in the product matches one of the non-negative Fourier modes $J_{n_\\alpha }^\\alpha $ which act non-trivially on $\\Sigma _{\\sf m}$ ; these modes provide a basis of the tangent bundle to ${\\mathrm {Gr}}^{\\sf m}$ .",
"This computation could be expressed as a comparison of the Pfaffian of the Dirac operator acting on $\\psi $ , before and after the Hecke modification.",
"This Pfaffian is analyzed in detail in [1].",
"The operator $\\Sigma _{g}(z,\\overline{z})$ fails to be a true function of $z$ because of the gauge anomaly.",
"Indeed, even a rescaling of the local coordinate $z \\rightarrow \\lambda z$ changes the singular gauge transformation from $z^{\\sf m}$ to $(\\lambda z)^{\\sf m}$ and thus results in the action of the Cartan zero modes ${\\sf m}\\cdot J_0$ on the spectral flow operator, resulting in a non-trivial scaling dimension proportional to $({\\sf m},{\\sf m})$ .",
"We can study this anomalous dependence on $z$ with the help of the free fermion trick.",
"For example, for ${\\mathrm {SO}}(3)$ and minimal ${\\sf m}$ we have an insertion $\\psi ^-$ which behaves as a section of $K_C^{1/2}$ .",
"Correspondingly, for critical level the spectral flow operator is a section of $K_C^{-1/2}\\otimes \\overline{K}_C^{-1/2}=|K_C|^{-1}$ .",
"This remains true for the averaged Hecke operator $\\widehat{\\Sigma }_{\\sf m}(0)$ because in this example, the coordinates on ${\\mathrm {Gr}}^{\\sf m}={\\mathbb {CP}}^1$ have scaling dimension 0 and thus the measure $\\mathrm {d}\\mu $ does not contribute to the scaling dimension.",
"The fact that $\\widehat{\\Sigma }_{\\sf m}(0)$ is a section of $|K_C|^{-1}$ is expected from $S$ -duality.",
"It matches the fact that the holomorphic and antiholomorphic sections $s$ and $\\overline{s}$ used to define the dual Wilson operator (Section REF ) are sections of $K_C^{-1/2}$ and $\\overline{K}_C^{-1/2}$ , respectively.",
"For general groups and representations, matching the scaling dimension of $\\widehat{\\Sigma }_{\\sf m}(0)$ with the behavior of the corresponding Wilson operator is more subtle.",
"The scaling dimension of the fermionic insertion grows quadratically in the charge, but so does the negative scaling dimension of the measure $\\mathrm {d}\\mu $ on ${\\mathrm {Gr}}^{\\sf m}$ .",
"There is a nice cancellation between the derivatives on the fermions and the scaling dimension of the measure, so that the scaling dimension of $\\widehat{\\Sigma }_{\\sf m}$ is linear in ${\\sf m}$ .",
"The scaling dimensions of dual Wilson operators were described in Section REF ."
],
[
"Integral-differential Hecke Operators and the Oper Differential Equation",
"In Section REF , as well as integral Hecke operators, whose kernel has delta function support on the Hecke correspondence, we considered integral-differential Hecke operators, whose kernel is a derivative of a delta function.",
"The natural way to build such more general Hecke operators is to consider the insertion of Kac-Moody descendants of $\\Sigma _{g}$ , which represent functional derivatives $\\frac{\\delta }{\\delta a}$ taken in a neighbourhood of the location of the Hecke modification.",
"We can restrict ourselves to descendants by the negative modes of the Kac-Moody currents, as the non-negative modes can be traded for $g$ derivatives which would be integrated by parts.",
"The action of non-negative modes on a descendant of $\\Sigma _{g}$ will produce some linear combination of $g$ derivatives of other descendants.",
"We need some $g$ -dependent combination of descendants which transform as a density on $\\mathrm {Gr}^{\\sf m}$ and such that the action of non-negative Kac-Moody modes will produce total $g$ derivatives.",
"We can produce a simple example of that: $\\partial _z \\widehat{\\Sigma }_{\\sf m}(z, \\overline{z})$ .",
"Indeed, the $z$ derivative of a basic spectral flow operator $\\Sigma _{\\sf m}(z)$ coincides with the Cartan Kac-Moody descendant : $ \\partial _z \\Sigma _{\\sf m}(z)= \\frac{1}{2 \\pi {\\mathrm {i}}} \\oint \\frac{\\mathrm {d}w}{w-z} {\\sf m}\\cdot J (w) \\Sigma _{\\sf m}(z, \\overline{z})\\equiv {\\sf m}\\cdot J_{-1} \\circ \\Sigma _{\\sf m}(z, \\overline{z}).",
"$ To demonstrate this relation, recall that the insertion of $\\Sigma _{\\sf m}(z)$ in a correlation function represents a specific modification of the background connection in a neighbourhood $U$ of $z$ .",
"As we vary $z$ , the modified connection changes.",
"The change is supported in the annular region $U^{\\prime }$ and can be described by the insertion of a current integrated against the variation of the background connection.",
"The entire comparison occurs within the ${\\mathrm {U}}(1)$ subgroup of the gauge group determined by ${\\sf m}$ .",
"As a small shortcut, we can compare the effect of the $z$ derivative and of the integrated current insertion at the level of the bundle modifications they implement.",
"If $\\Sigma _{\\sf m}(z)$ implements the gauge transformation $g(w) = (w-z)^{\\sf m}$ on $U^{\\prime }$ , the $z$ derivative $\\partial _z \\Sigma _{\\sf m}(z)$ implements $\\partial _z (w-z)^{\\sf m}= \\frac{{\\sf m}}{z-w} (w-z)^{\\sf m}$ .",
"The $\\frac{{\\sf m}}{z-w}$ part is identified with the gauge transformation produced by the ${\\sf m}\\cdot J_{-1}$ Fourier mode and $(w-z)^{\\sf m}$ represents $\\Sigma _{\\sf m}(z)$ again.",
"As we are working at the level of the bundle modification instead of the connection, we could be missing effects due to the anomaly.",
"A simple check in the free fermion theory can exclude that.The bosonized description of $\\Sigma _{\\sf m}(z)$ is also an effective way to verify the computation.",
"Inserting this relation into the definition of $\\widehat{\\Sigma }_{\\sf m}(z, \\overline{z})$ , we find that $\\partial _z \\widehat{\\Sigma }_{\\sf m}(z, \\overline{z})$ can be written as an integral over ${\\mathrm {Gr}}^{\\sf m}$ of a specific Kac-Moody descendant of $\\Sigma _{g}(z, \\overline{z})$ .",
"Another natural way to produce well-defined integral-differential operators of this type is to consider descendants of $\\widehat{\\Sigma }_{\\sf m}$ by modes of the Sugawara vector or other central elements.",
"The resulting local operators are clearly gauge-invariant.",
"We expect that the classification of $g$ -dependent combinations of Kac-Moody descendants of $\\Sigma _{g}(z, \\overline{z})$ which are a total derivative on $\\mathrm {Gr}^{\\sf m}$ will match the corresponding classification of 't Hooft line endpoints $\\alpha _{R_{\\sf m},n}$ ."
],
[
"Wakimoto Realization",
"We will give here an alternative derivation of the properties of $\\Sigma _1$ with the help of the Wakimoto construction at critical level.",
"As a bonus, we will recover in a different way the oper differential equation.",
"The critical-level Wakimoto construction presents the Kac-Moody currents as the symmetry currents for a twisted $\\beta \\gamma $ system for ${\\mathbb {CP}}^1$ : $J^+ &= \\beta \\cr J^0 &= - \\beta \\gamma + \\partial \\alpha \\cr J^- &= -\\beta \\gamma ^2 - 2 \\partial \\gamma + \\partial \\alpha \\gamma \\cr $ where $\\alpha $ is a locally-defined holomorphic function.",
"The Sugawara vector simplifies to a Miura form $S_2 = (\\partial \\alpha )^2 + \\partial ^2 \\alpha $ and is thus manifestly a multiple of the identity operator, with trivial OPE with the currents.",
"The spectral flow automorphism extends naturally to the $\\beta \\gamma $ system, so that the spectral flow operator gives a zero to $\\gamma $ and a pole to $\\beta $ .",
"An operator that does this is usually indicated as $\\delta (\\gamma )$ .",
"Comparison with the expected form of $\\partial \\Sigma _1$ from eqn.",
"(REF ) gives $\\Sigma _1 = e^{\\alpha + \\overline{\\alpha }} \\delta (\\gamma )\\delta (\\overline{\\gamma }).$ The exponential prefactor provides the $\\partial \\alpha $ part of $J^0_{-1} \\circ \\Sigma _1$ .",
"An ${\\mathrm {SO}}(3,{\\mathbb {C}})$ rotation of this expression gives $\\Sigma _1(z;\\mu ) = e^{\\alpha + \\overline{\\alpha }} \\delta (\\gamma -\\mu )\\delta (\\overline{\\gamma }- \\overline{\\mu }).$ where $\\mu $ is an inhomogeneous coordinate on ${\\mathbb {CP}}^1$ .",
"The integral over $\\mu $ is easily done, resulting in $\\widehat{\\Sigma }_1(z) = e^{\\alpha + \\overline{\\alpha }} .$ This is a multiple of the identity and thus annihilated by all non-negative modes of the currents.",
"Furthermore, the oper differential equation manifestly holds: $\\partial _z^2 \\widehat{\\Sigma }_1(z) = S_2(z)\\widehat{\\Sigma }_1(z)$ .",
"We expect this pattern to persist for all $G$ and ${\\sf m}$ .",
"The critical Wakimoto realization gives central elements which take the form of a Miura oper built from the Cartan-valued $\\alpha $ .",
"The spectral flow operators will take the form of spectral flow operators for the $\\beta \\gamma $ system combined with some function of $\\alpha $ .",
"The averaged spectral flow operators will give multiples of the identity for the $\\beta \\gamma $ system, multiplied by certain functions of $\\alpha $ which give the Miura expression for solutions of the oper differential equation."
],
[
"Basics of Symplectic Bosons",
"As we discussed in the Section REF , Neumann boundary conditions enriched by 3d hypermultiplets create states described by a path integral $\\Psi [a] = \\int DZ D\\overline{Z}\\, e^{\\int _C \\left[\\langle Z, \\overline{\\partial }_a Z \\rangle - \\langle \\overline{Z}, \\partial _a \\overline{Z} \\rangle \\right]}$ This is a non-chiral version of the path integral for symplectic bosons.",
"Chiral symplectic bosons are the Grassmann-even analogue of chiral fermions.",
"They are a special case of $\\beta \\gamma $ systems where the conformal dimension of both $\\beta $ and $\\gamma $ is set to $1/2$ .",
"Concretely, chiral symplectic bosons are a collection of $2 n$ two-dimensional spin $1/2$ chiral bosonic fields $Z^a$ with action $\\int _{C}\\left( \\omega _{ab} Z^a \\overline{\\partial }Z^b + {\\cal A}_{ab} Z^a Z^b\\right)$ where $\\omega _{ab}$ is a constant symplectic form and we included a coupling to a background connection ${\\cal A}_{ab}$ of type $(0,1)$ defining an ${\\mathrm {Sp}}(2n)$ bundle on the Riemann surface $C$ .As the symplectic bosons are spinors, we do not strictly need to separately define a spin structure and an ${\\mathrm {Sp}}(2n)$ bundle.",
"Instead, we can specify a ${\\mathrm {Spin}}\\cdot {\\mathrm {Sp}}(2n)$ bundle, a notion that is precisely analogous to the ${\\mathrm {Spin}}\\cdot {\\mathrm {SU}}(2)$ bundles of Section REF .",
"We will employ Einstein summation convention in this section unless otherwise noted.",
"The analogy to chiral fermions is somewhat imperfect.",
"Chiral fermions are a well-defined two-dimensional (spin)CFT.",
"Chiral symplectic bosons are mildly anomalous.",
"The anomaly manifests itself as a sign ambiguity of the chiral partition function $\\int DZ \\,e^{\\int _{C}\\left( \\omega _{ab} Z^a \\overline{\\partial }Z^b + {\\cal A}_{ab} Z^a Z^b\\right)} = \\frac{1}{\\sqrt{\\det \\overline{\\partial }_{\\cal A} }}$ where we denote as $\\overline{\\partial }_{\\cal A}$ the $\\overline{\\partial }$ operator acting on sections of $K_C^{1/2} \\otimes E$ and $E$ is the rank $2n$ bundle associated to the ${\\mathrm {Sp}}(2n)$ bundle.Notice that generically $\\overline{\\partial }_{\\cal A}$ has no zero modes and the functional determinant is well-defined.",
"The partition function diverges for special choices of ${\\mathrm {Sp}}(2n)$ bundle where zero modes appear.",
"The non-chiral partition function, though, $\\int DZ D\\overline{Z}\\, e^{\\int _C \\left[\\langle Z, \\overline{\\partial }_A Z \\rangle - \\langle \\overline{Z}, \\partial _A \\overline{Z} \\rangle \\right]} = \\frac{1}{|\\det \\overline{\\partial }_{\\cal A} |}$ is unambiguous: it can be defined by an actual the path integral along the cycle $\\overline{Z}{}^a = (Z^a)^*$ (that is, the integration cycle is defined by saying $\\overline{Z}{}^a$ is the complex conjugate of $Z^a$ ).",
"We record here the OPE $Z^a(z) Z^b(w) \\sim \\frac{\\omega ^{ab}}{z-w},$ where $\\omega ^{ab}$ is the inverse symplectic form.",
"The corresponding algebra of modes is $[Z^a_m, Z^b_n] = \\omega ^{ab} \\delta _{n+m,0}.$ Because of the half-integral spin, the mode indices $n$ , $m$ are half-integral in the Neveu-Schwarz sector of the chiral algebra and in particular in the vacuum module.",
"The vacuum satisfies $Z^a_n |0\\rangle =0 \\qquad \\qquad n>0.$ The mode indices are integral in Ramond sector modules, which we will discuss in Section REF .",
"The variation of the action with respect to the ${\\mathrm {Sp}}(2n)$ connection ${\\cal A}$ gives Kac-Moody currents $J^{ab} = \\frac{1}{2} :Z^a Z^b:$ of level $- \\frac{1}{2}$ .",
"The fractional level is another manifestation of the global anomaly of the chiral theory.The $J^{ab}$ currents actually generate a quotient of the $\\widehat{\\mathfrak {sp}}(2n)_{-\\frac{1}{2}}$ chiral algebra: some linear combinations of current bilinears and derivatives of the currents vanish.",
"The number of level 2 descendants in the symplectic boson vacuum module is smaller than the number of level 2 descendants in the Kac-Moody vacuum module.",
"The partition function of the non-chiral theory defines a well-defined section of a bundle $|{\\cal L}|$ on the space of ${\\mathrm {Sp}}(2n,{\\mathbb {C}})$ bundles, where ${\\cal L}$ is the line bundle corresponding to Kac-Moody level $-1$ .",
"Once we specialize to the gauge group $G \\subset {\\mathrm {Sp}}(2n)$ of the 4d theory, we will obtain $G$ currents of a level which may not be critical.",
"This signals a gauge anomaly obstructing the existence of a 2d junction between the deformed Neumann boundary condition and the enriched Neumann boundary condition.",
"If the level is more negative than the critical level, we may attempt to cancel the anomaly by some auxiliary 2d system, such as a collection of free fermions.",
"In the example we discuss momentarily, the anomaly will be absent from the outset."
],
[
"The Trifundamental Example",
"We now specialize to $n=4$ and focus on a $G\\equiv {\\mathrm {SL}}(2) \\times {\\mathrm {SL}}(2)\\times {\\mathrm {SL}}(2)$ subgroup of ${\\mathrm {Sp}}(8)$ .",
"In other words, we identify $\\mathbb {C}^8$ with $\\mathbb {C}^2 \\otimes \\mathbb {C}^2 \\otimes \\mathbb {C}^2$ and we couple the theory to a connection $a$ for ${\\mathrm {SL}}(2) \\times {\\mathrm {SL}}(2)\\times {\\mathrm {SL}}(2)$ .",
"This gives enriched Neumann boundary conditions which are conjecturally $S$ -dual to a tri-diagonal interface, a BBB brane supported on the diagonal of ${\\mathcal {M}}_H \\times {\\mathcal {M}}_H \\times {\\mathcal {M}}_H$ with trivial ${\\mathrm {CP}}$ bundle [58], [65].The $S$ -duality statement can be generalized to other $G$ or diagonal interfaces between more than three copies.",
"The corresponding enriched Neumann boundary conditions employ the theories defined in [65].",
"The boundary chiral algebras for these theories are known from work by Arakawa [66] and have $G$ currents of critical level.",
"The corresponding states would be the partition function of a non-chiral 2d CFT built from Arakawa's chiral algebras, which is currently unknown.",
"The basic consequence of this $S$ -duality identification is that the state $\\Psi [a]$ produced by the partition function should intertwine the action of the three copies of ${\\mathrm {SL}}(2)$ Hitchin Hamiltonians and quantum Hecke operators.",
"Our goal in the rest of this section is to confirm this.",
"We denote the symplectic boson fields as $Z^{\\alpha \\beta \\gamma }(z)$ and the symplectic form as $\\epsilon _{\\alpha \\alpha ^{\\prime }}\\epsilon _{\\beta \\beta ^{\\prime }}\\epsilon _{\\gamma \\gamma ^{\\prime }}$ .",
"We get three copies of $\\widehat{\\mathfrak {sl}}(2)_{-2}$ Kac-Moody currents such as $J^{\\alpha \\alpha ^{\\prime }} =\\frac{1}{2} \\epsilon _{\\beta \\beta ^{\\prime }}\\epsilon _{\\gamma \\gamma ^{\\prime }} :Z^{\\alpha \\beta \\gamma }Z^{\\alpha ^{\\prime } \\beta ^{\\prime } \\gamma ^{\\prime }}:$ We will denote the three sets of currents simply as $J$ , $J^{\\prime }$ , $J^{\\prime \\prime }$ , avoiding indices when possible.",
"Crucially, these currents have critical level.",
"Accordingly, the Sugawara vectors are central.",
"A remarkable observation is that the three Sugawara vectors actually coincide here: $ :JJ: = :J^{\\prime }J^{\\prime }: = :J^{\\prime \\prime } J^{\\prime \\prime }:$ As the three Sugawara vectors coincide, the correlation functions of Sugawara vectors also coincide.",
"These are obtained from the action of the $\\mathfrak {sl}(2)$ quantum Hitchin Hamiltonians on the partition function $\\Psi (a)$ , seen as a half-density on ${\\mathcal {M}}\\times {\\mathcal {M}}\\times {\\mathcal {M}}$ .",
"Concretely, the the kinetic operator of the symplectic bosons $Z$ is the $\\overline{\\partial }_{a}$ operator acting on the bundle $E \\otimes E^{\\prime } \\otimes E^{\\prime \\prime } \\otimes K_C^{\\frac{1}{2}}$ .",
"The partition function is: $\\Psi (a) = \\frac{1}{|\\det \\overline{\\partial }_a|}.$ We have thus given a chiral algebra derivation of an intertwining property $\\boxed{H_i \\Psi = H^{\\prime }_i \\Psi = H^{\\prime \\prime }_i \\Psi }$ where $H_i$ run over the $\\mathfrak {sl}(2)$ quantum Hitchin Hamiltonians acting on the three spaces of ${\\mathrm {SL}}(2)$ bundles.",
"Similarly $\\boxed{\\overline{H}_i \\Psi = \\overline{H}^{\\prime }_i \\Psi = \\overline{H}^{\\prime \\prime }_i \\Psi }$ for the conjugate quantum Hitchin Hamiltonians, acting as antiholomorphic differential operators on the space of ${\\mathrm {SL}}(2)$ bundles.",
"These relations match the relations expected on the B-model side for the tri-diagonal interface.",
"Our next objective is to demonstrate the analogous intertwining relations for Hecke operators.",
"For simplicity, we will work with Hecke operators of minimal charge for ${\\mathrm {SO}}(3)$ , even though the symplectic bosons are coupled to ${\\mathrm {SL}}(2)$ bundles rather than ${\\mathrm {PSL}}(2)$ .",
"Minimal Hecke modifications map ${\\mathrm {SL}}(2)$ bundles to ${\\mathrm {SL}}(2)$ bundles twisted by a gerbe and viceversa, so we can consistently describe the action of pairs of Hecke operators on half-densities on ${\\mathcal {M}}(C,{\\mathrm {SU}}(2))$ .",
"A minimal Hecke operator will create the endpoint of a $Z \\rightarrow -Z$ cut for the symplectic bosons.",
"This leads us to consider Ramond vertex operators."
],
[
"The Ramond Sector",
"The symplectic boson chiral algebra admits Ramond modules which are associated to a circle with non-bounding spin structure.",
"In such a module, the mode expansion of the $Z^a$ fields involves modes $Z^a_n$ with integral $n$ .",
"The corresponding vertex operators introduce a cut across which the $Z^a$ flip sign.",
"The zero modes $Z^a_0$ form a Weyl algebra.",
"There is a rich collection of highest weight Ramond modules for the chiral symplectic bosons which is induced from a module for the zero mode Weyl algebra.",
"Every element of the Weyl module is promoted to a highest weight vector/vertex operator which is annihilated by the positive Fourier modes $Z^a_n$ .",
"The negative Fourier modes act freely and the zero modes act as in the Weyl module.",
"The Kac-Moody current zero modes $J_0^{ab}$ act on a highest weight vector as ${\\mathrm {Sp}}(2n)$ generators $Z^{(a}_0 Z^{b)}_0$ .",
"All Weyl modules break to some degree the ${\\mathrm {Sp}}(2n)$ symmetry of the VOA.",
"In other words, there is no Weyl module equipped with an ${\\mathrm {Sp}}(2n)$ -invariant vector.",
"Thus the insertion of any such vertex operator into a correlation function will always reduce ${\\mathrm {Sp}}(2n)$ gauge invariance at that point.",
"The simplest way to produce a Ramond vertex operator is to consider a spectral flow operator of minimal charge in ${\\mathrm {PSp}}(2n)={\\mathrm {Sp}}(2n)/{\\mathbb {Z}}_2$ .",
"Select a Lagrangian splitting $\\mathbb {C}^{2n} = V \\oplus V^\\vee $ .",
"Pick a singular gauge transformation which acts as $z^{\\frac{1}{2}}$ on $V$ and $z^{-\\frac{1}{2}}$ on $V^\\vee $ .",
"The resulting spectral flow operator $S_V(0)$ is a Ramond module.",
"Linear combinations of $Z^a$ in $V$ vanish as $z^{\\frac{1}{2}}$ as they approach $S_V(0)$ , while linear combinations in $V^\\vee $ diverge as $z^{-\\frac{1}{2}}$ .",
"This means that $S_V(0)$ is a highest-weight Ramond module annihilated by linear combinations of $Z_0^a$ in $V$ .",
"In particular, it only depends on $V$ .",
"$S_V(0)$ , for any $V$ , can be obtained by an ${\\mathrm {Sp}}(2n)$ rotation from some particular $S_{V_0}(0)$ , which we choose as a reference.",
"Without loss of generality, pick a basis where $\\omega ^{ab} = \\delta ^{a-b-n}- \\delta ^{b-a-n},~~a,b=1,\\cdots , 2n,$ and choose the reference vertex operator $S_{V_0}(z)$ to be annihilated by $Z^{n+1}_0, \\cdots , Z^{2n}_0$ .",
"Denote the remaining zero modes, which act as creation operators, as $u^a=Z_0^a$ , $a\\le n$ and denote the corresponding descendants of $S_{V_0}(z)$ as $S_{V_0}[u^a](z)$ , $S_{V_0}[u^a u^b](z)$ , etcetera.",
"Annihilation zero modes act as $Z_0^{a+n}=\\partial _{u^a}$ .",
"Consider a coherent state in the Weyl module: $S_{V_0}[e^{\\frac{1}{2} B_{ab} u^a u^b}](z)$ This is annihilated by linear combinations $Z^{a+n}_0 - B_{ac} Z^c_0$ .",
"That condition defines a rotated Lagrangian subspace $V = B \\circ V_0$ .",
"We thus identify $S_{B \\circ V_0}(z) = S_{V_0}[e^{\\frac{1}{2} B_{ab} u^a u^b}](z)$ Indeed, we have $\\partial _{B_{ab}}S_{B \\circ V_0}(z) = J_0^{ab}S_{B \\circ V_0}(z)$ ."
],
[
"Non-chiral Ramond Modules",
"Next, we can consider the combined theory of chiral and antichiral symplectic bosons.",
"In the Ramond sector, we now have an action of the chiral zero modes $Z_0^a$ and the antichiral zero modes $\\overline{Z}_0^a$ .",
"As the path integration contour relates $\\overline{Z}$ to the conjugate of $Z$ , it is natural to consider a space of Ramond states such that the zero modes are adjoint to each other.",
"Furthermore, the 2d theory in the zero momentum sector is essentially a quantum mechanics with target $\\mathbb {C}^{2n}$ .",
"It is thus natural to pick a polarization and set the zero mode Hilbert space to be ${\\mathrm {L}}^2(\\mathbb {C}^{n})$ .",
"This answer is actually independent of the choice of polarization, as we can use generalized Fourier transform operations to relate different polarizations.The metaplectic anomaly cancels out because we act simultaneously on holomorphic and anti-holomorphic variables.",
"The full Ramond Hilbert space ${\\mathcal {R}}$ is induced from ${\\mathrm {L}}^2(\\mathbb {C}^{n})$ in the usual way, by having $Z^a_k$ and $\\overline{Z}^a_k$ annihilate vectors in ${\\mathrm {L}}^2(\\mathbb {C}^{n})$ when $k>0$ and act freely with $k<0$ .",
"A variety of different highest weight Ramond vertex operators are realized as distributions on $\\mathbb {C}^{n}$ .",
"Pick the same polarization as in the definition of $S_{V_0}(0)$ .",
"Then the reference spectral flow operator $S_{V_0}(0)$ is represented by the distribution “1”.",
"The rotated $S_{B \\circ V_0}(z)$ is represented by the Gaussian $e^{\\frac{1}{2} B_{ab} u^a u^b-\\frac{1}{2} \\overline{B}_{ab} \\overline{u}^a \\overline{u}^b }$ The distribution $\\delta ^{(2n)}(u)$ , instead, represents a Ramond vertex operator annihilated by $Z^{1}_0, \\cdots Z^{n}_0$ , which is a spectral flow operator with charge opposite to $S_{V_0}(0)$ .",
"We can now specialize to $n=4$ and to the spectral flow operators associated to the ${\\mathrm {SL}}(2) \\times {\\mathrm {SL}}(2) \\times {\\mathrm {SL}}(2)$ subgroup of ${\\mathrm {Sp}}(8)$ .",
"We pick our polarization of $\\mathbb {C}^{8}$ to be invariant under the second and third ${\\mathrm {SL}}(2)$ groups in the product.",
"Concretely, we identify $Z_0^{2 \\beta \\gamma } = u^{\\beta \\gamma }$ and $Z_0^{1 \\beta \\gamma }=\\epsilon ^{\\beta \\beta ^{\\prime }}\\epsilon ^{\\gamma \\gamma ^{\\prime }}\\frac{\\partial }{\\partial u^{\\beta ^{\\prime }\\gamma ^{\\prime }}} .$ In this framework, the spectral flow operators $\\Sigma _{1;\\mu , \\overline{\\mu }}$ , $\\Sigma ^{\\prime }_{1;\\mu ^{\\prime }, \\overline{\\mu }}$ and $\\Sigma ^{\\prime \\prime }_{1;\\mu ^{\\prime \\prime }, \\overline{\\mu }}$ for the three ${\\mathrm {SL}}(2)$ 's are all special cases of ${\\mathrm {Sp}}(8)$ spectral flow operators of minimal charge, i.e.",
"of highest weight Ramond vertex operators.",
"They are represented by distributions, which we can average over $\\mu $ to obtain representations of $\\widehat{\\Sigma }_{1}$ , $\\widehat{\\Sigma }^{\\prime }_{1}$ , $\\widehat{\\Sigma }^{\\prime \\prime }_{1}$ .",
"The calculations are straightforward: $\\Sigma _1$ is represented by the distribution “1”.",
"$\\Sigma _{1;\\mu , \\overline{\\mu }}$ is represented by the Gaussian $e^{\\mu \\epsilon _{\\beta \\beta ^{\\prime }}\\epsilon _{\\gamma \\gamma ^{\\prime }} u^{2\\beta \\gamma } u^{2 \\beta ^{\\prime } \\gamma ^{\\prime }}- \\mathrm {c.c.",
"}}.$ Averaging over $\\mu $ , $\\widehat{\\Sigma }_{1}$ is represented by the distribution $\\delta ^{(2)}(\\epsilon _{\\beta \\beta ^{\\prime }}\\epsilon _{\\gamma \\gamma ^{\\prime }} u^{2\\beta \\gamma } u^{2 \\beta ^{\\prime } \\gamma ^{\\prime }}).$ $\\Sigma ^{\\prime }_1$ is represented by the distribution $\\delta ^{(2)}(u^{211})\\delta ^{(2)}(u^{212}).$ $\\Sigma ^{\\prime }_{1;\\mu ^{\\prime }, \\overline{\\mu }^{\\prime }}$ is represented by the distribution $\\delta ^{(2)}(u^{211}- \\mu ^{\\prime } u^{221})\\delta ^{(2)}(u^{212}- \\mu ^{\\prime } u^{222}).$ Averaging over $\\mu ^{\\prime }$ , we find that $\\widehat{\\Sigma }^{\\prime }_{1}$ is represented by the same distribution as $\\widehat{\\Sigma }_{1}$ .",
"$\\Sigma ^{\\prime \\prime }_1$ is represented by the distribution $\\delta ^{(2)}(u^{211})\\delta ^{(2)}(u^{221}).$ Similarly, $\\Sigma ^{\\prime \\prime }_{1;\\mu ^{\\prime \\prime }, \\overline{\\mu }^{\\prime \\prime }}$ is represented by the distribution $\\delta ^{(2)}(u^{211}- \\mu ^{\\prime \\prime } u^{212})\\delta ^{(2)}(u^{221}- \\mu ^{\\prime \\prime } u^{222}).$ Averaging over $\\mu ^{\\prime \\prime }$ , we find that $\\widehat{\\Sigma }^{\\prime \\prime }_{1}$ is represented by the same distribution as $\\widehat{\\Sigma }_{1}$ We conclude that $\\widehat{\\Sigma }_{1} = \\widehat{\\Sigma }^{\\prime }_{1} =\\widehat{\\Sigma }^{\\prime \\prime }_{1}$ in the theory of trifundamental symplectic bosons.",
"These relations play an analogous role to (REF ): inserted in correlation functions they prove that the partition function $\\Psi (a)$ intertwines the action of minimal Hecke operators for the three ${\\mathrm {SL}}(2)$ groups: $\\boxed{H_z \\Psi = H^{\\prime }_z \\Psi = H^{\\prime \\prime }_z \\Psi }$"
],
[
"A Marvelous Module",
"It is worth elaborating on the properties of the Weyl module generated by the distribution $M=\\delta ^{(2)}(\\epsilon _{\\beta \\beta ^{\\prime }}\\epsilon _{\\gamma \\gamma ^{\\prime }} u^{2\\beta \\gamma } u^{2 \\beta ^{\\prime } \\gamma ^{\\prime }}).$ We learned some surprising properties which follow from $M$ representing averaged spectral flow operators: $M$ is invariant under ${\\mathrm {SL}}(2) \\times {\\mathrm {SL}}(2) \\times {\\mathrm {SL}}(2)$ and the module treats the three ${\\mathrm {SL}}(2)$ groups in a completely symmetric manner.",
"The latter property is somewhat hidden in the analysis, so we can spell it out in detail here: we can change polarization by a Fourier transform and go to representations of the Weyl module in terms of functions of $u^{\\alpha 2 \\gamma }$ or $u^{\\alpha \\beta 2}$ .",
"The Fourier transform of $M$ produces distributions $\\delta ^{(2)}(\\epsilon _{\\beta \\beta ^{\\prime }}\\epsilon _{\\gamma \\gamma ^{\\prime }} u^{\\beta 2 \\gamma } u^{\\beta ^{\\prime } 2\\gamma ^{\\prime }})$ and $\\delta ^{(2)}(\\epsilon _{\\beta \\beta ^{\\prime }}\\epsilon _{\\gamma \\gamma ^{\\prime }} u^{\\beta \\gamma 2} u^{\\beta ^{\\prime } \\gamma ^{\\prime } 2})$ respectively.",
"Acting with the Weyl algebra on $M$ we generate a remarkable module with an explicit action of ${\\mathrm {SL}}(2) \\times {\\mathrm {SL}}(2) \\times {\\mathrm {SL}}(2)$ .",
"As a vector space, the module decomposes as $\\oplus _{d=1}^\\infty V_d\\otimes V_d \\otimes V_d$ where $V_d$ is the $d$ -dimensional irreducible representation of ${\\mathrm {SL}}(2)$ .",
"The Weyl generators act as a sum of two terms: one term raises $d$ by 1 and the other lowers it by 1."
],
[
"Some Generalizations",
"The ${\\mathrm {SU}}(2) \\times {\\mathrm {SU}}(2) \\times {\\mathrm {SU}}(2)$ gauge group can be seen as a special case of ${\\mathrm {Sp}}(2n) \\times \\mathrm {Spin}(2n+2)$ .",
"Trifundamental hypermultiplets are a special case of bifundamental hypermultiplets for ${\\mathrm {Sp}}(2n) \\times \\mathrm {Spin}(2n+2)$ .",
"Bifundamental hypermultiplets engineer an “NS5” interface between ${\\mathrm {Sp}}(2n)$ and $\\mathrm {Spin}(2n+2)$ 4d gauge theories.",
"The levels of the corresponding Kac-Moody currents in the theory of bifundamental symplectic bosons are critical, so the NS5 interface has non-anomalous corners with deformed Neumann boundaries and our construction applies.",
"The NS5 interface is $S$ -dual to a “D5” interface between $\\mathrm {{\\mathrm {Spin}}(2n+1)}$ and $\\mathrm {Spin}(2n+2)$ 4d gauge theories, at which the $\\mathrm {Spin}(2n+2)$ gauge group is reduced to $\\mathrm {{\\mathrm {Spin}}(2n+1)}$ .",
"The D5 interface will descend in the B-model to a BBB interface supported on the space of $\\mathrm {Spin}(2n+1)$ flat connections embedded in the space of $\\mathrm {Spin}(2n+2)$ flat connections.",
"This gives simple predictions for the action of quantum Hitchin Hamiltonians and 't Hooft operators on the NS5 interface.",
"Our calculations in this section should be generalized to verify these predictions.",
"Bifundamental hypermultiplets can also be used to engineer an “NS5” interface between ${\\mathrm {U}}(n)$ 4d gauge theories which has a relatively simple dual and should be amenable to a 2d chiral algebra analysis.",
"The Kac-Moody currents for the ${\\mathrm {SU}}(n)$ subgroups are critical, and the ${\\mathrm {U}}(1)$ anomalies can be cancelled by a single complex chiral fermion of charge $(1,-1)$ under the diagonal ${\\mathrm {U}}(1)$ gauge symmetries.",
"Again, our calculations in this section should be generalized to verify these predictions.",
"For example, the minimal spectral flow operators are represented by the distribution $\\delta ^{(2)}(\\det u)$ .",
"Acknowledgment Research at Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Research $\\&$ Innovation.",
"Research of EW supported in part by NSF Grant PHY-1911298.",
"We thank D. Baraglia, P. Etingof, E. Frenkel, D. Kazhdan, and L. Schaposnik for discussions.",
"We also thank P. Etingof for a careful reading of the manuscript and suggesting a number of clarifications."
],
[
"The $B$ -Model Hilbert Space",
"Here we will describe some properties of the physical state space of the $B$ -model.",
"The goal is to describe the conditions needed for the hermitian metric on this space to be positive.",
"First we describe what we will call $B$ -model quantum mechanics.",
"This is, most simply, the quantum mechanics that arises in compactification of the $B$ -model on a circle, though our application involves a different occurrence of the same model.",
"The input is a Calabi-Yau manifold $X$ of complex dimension $n$ with a holomorphic function $W$ (the superpotential) and a holomorphic volume form $$ .",
"We generally assume that $W$ has only isolated, though possibly degenerate, critical points.",
"The full $B$ -model quantum mechanics is defined with a Kahler metric $g$ on $X$ , but the $B$ -model (which describes only a topological sector of the theory) does not depend on $g$ .",
"Actually, because of $B$ -model localization, one does not lose much by specializing to $X={\\mathbb {C}}^n$ , but there is no need to do so.",
"We locally parametrize $X$ by holomorphic coordinates $x^i$ , $i=1,\\cdots , n$ .",
"The model also has two sets of canonically conjugate fermion variablesTo compare to more complete descriptions of this model, note that in [67], for example, our $\\chi ^{\\overline{i}}$ is denoted $\\psi ^{\\overline{i}}+\\overline{\\psi }^{\\overline{i}}$ , and our $\\theta _i$ is $-{\\mathfrak {g}}_{i\\overline{i}}\\overline{\\psi }^{\\overline{i}}$ .",
"A more complete description can also be found in [68].",
"$ \\lbrace \\chi ^{\\overline{i}},\\widetilde{\\chi }_{\\overline{j}}\\rbrace =\\delta ^i_j,~~~~ \\lbrace \\theta _i,\\widetilde{\\theta }^j\\rbrace =\\delta ^j_i.",
"$ $\\chi $ and $\\widetilde{\\theta }$ have fermion number 1 and the conjugate variables $\\widetilde{\\chi },\\,\\theta $ have fermion number $-1$ .",
"We represent the anticommutators by $\\widetilde{\\chi }_{\\overline{j}}=\\partial /\\partial \\chi ^{\\overline{j}}$ and $\\widetilde{\\theta }_i=\\partial /\\partial \\theta ^i$ .",
"Then a quantum wavefunction is a function $\\Psi (x,\\overline{x}, \\chi , \\theta )$ .",
"A wavefunction that is homogeneous in $\\chi ,\\theta $ of degree $(p,q)$ takes the form $\\chi ^{\\overline{i}_1}\\chi ^{\\overline{i}_2}\\cdots \\chi ^{\\overline{i}_p}\\theta _{i_i}\\theta _{i_2}\\cdots \\theta _{i_q} \\Phi _{\\overline{i}_1\\overline{i}_2\\cdots \\overline{i}_p} ^{i_1 i_2 \\cdots i_q}(x,\\overline{x})$ , and can be interpreted geometrically as a $(0,p)$ -form valued in $\\wedge ^q TX$ , $TX$ being the holomorphic tangent bundle of $X$ .",
"Acting on such a wavefunction, the differential of the $B$ -model is the operator $ Q=\\sum _j \\chi ^{\\overline{j}}\\frac{\\partial }{\\partial \\overline{x}^{ \\overline{j}}} +\\sum _i \\frac{\\partial W}{\\partial x^i}\\frac{\\partial }{\\partial \\theta _i} $ of fermion number 1.",
"Evidently $Q^2=0$ .",
"The space of physical states of the $B$ -model is defined as the cohomology of $Q$ .",
"This cohomology is typically easy to describe explicitly.",
"In the basic case of an affine variety, the cohomology can be represented by holomorphic functions of the $x^i$ modulo the ideal generated by the partial derivatives of $W$ .",
"If $X={\\mathbb {C}}^n$ and $W$ is a polynomial, it suffices to consider polynomial functions: $ {\\mathcal {H}}={\\mathbb {C}}[x_1,\\cdots ,x_n]/(\\partial W/\\partial x^1,\\cdots ,\\partial W/\\partial x^n).",
"$ The description of the state space in terms of holomorphic functions modulo an ideal makes it clear that the state space is a ring ${\\mathcal {R}}$ , which in the application to the closed string sector of a Landau-Ginzburg model is called the chiral ring of the model.",
"Assuming that the critical points of $W$ are isolated, ${\\mathcal {R}}$ has a decomposition as a direct sum of subrings associated to critical points $p_s$ : $ {\\mathcal {R}}=\\oplus {\\mathcal {R}}_s.$ In the context of the closed string $B$ -model, ${\\mathcal {R}}_s$ is the chiral ring of the theory in the vacuum associated to the critical point $p_s$ .",
"As a first step, we want to define the natural $B$ -model pairing on this space of states.",
"First, we need a way to integrate a wavefunction.",
"This will depend on the Calabi-Yau volume form $$ .",
"The antiholomorphic variables $\\overline{x}^{\\overline{i}}, \\chi ^{\\overline{i}}$ have opposite statistics and transform similarly in a change of coordinates, so they possess a natural integration measure ${\\mathrm {D}}(\\overline{x},\\chi )=\\mathrm {d}\\overline{x}^{\\overline{1}}\\cdots \\ \\overline{x}^{\\overline{n}}\\mathrm {d}\\chi ^{\\overline{1}}\\cdots \\mathrm {d}\\chi ^{\\overline{n}}$ .",
"A choice of $$ gives an integration measure ${\\mathrm {D}}x=\\frac{1}{n!}",
"_{i_1\\cdots i_n}\\mathrm {d}x^{i_1}\\cdots \\mathrm {d}x^{i_n}$ for $x$ .",
"Because the $\\theta _i$ have opposite statistics to $x^i$ but also transform under coordinate changes in a dual fashion, a natural measure ${\\mathrm {D}}\\theta $ is likewise proportional to $$ .",
"The combined system therefore has a measure ${\\mathrm {D}}(x,\\overline{x}, \\chi ,\\theta )$ .",
"This measure is proportional to $^2$ , a standard fact about the $B$ -model.",
"We use this measure to define integration.",
"If $\\alpha $ is any wavefunction that vanishes sufficiently rapidly at infinity, one can integrate by parts, in the sense that $ \\int {\\mathrm {D}}(x,\\overline{x},\\chi ,\\theta )\\, Q\\alpha =0.$ One cannot immediately apply this definition to physical states characterized as in eqn.",
"(REF ) as holomorphic functions of the $x^i$ , because such functions do not vanish sufficiently rapidly at infinity.",
"However, consider the $Q$ -exact quantity $ V=\\left\\lbrace Q, \\sum _i \\theta _i g^{i\\overline{j}}\\frac{\\partial \\overline{W}}{\\partial \\overline{x}^{\\overline{j}}}\\right\\rbrace =\\left|\\mathrm {d}W\\right|^2 +\\sum _{i,\\overline{j},\\overline{k}} \\chi ^{\\overline{k}}\\theta _i g^{i\\overline{j}}\\frac{\\partial ^2\\overline{W}}{\\partial \\overline{x}^{\\overline{j}}\\partial \\overline{x}^{\\overline{k}} } .",
"$ The quantity $\\exp \\left(-\\frac{1}{2\\epsilon }V\\right)$ , for any positive $\\epsilon $ , is of the form $1+\\lbrace Q,\\cdots \\rbrace $ , and vanishes rapidly at infinity assuming that $|\\mathrm {d}W|^2$ grows at infinity.",
"So instead of representing a physical state as in eqn.",
"(REF ) by a holomorphic function $f_0(x)$ (usually taken to be a polynomial) we can use the representative $f =f_0(x)\\cdot \\exp \\left(-\\frac{1}{2\\epsilon }V\\right)$ of the same physical state.",
"Assuming that $|\\mathrm {d}W|^2$ grows at infinity, this means that we can assume that physical states are represented by wavefunctions that vanish rapidly at infinity.",
"Hence one can define the natural $B$ -model pairing on ${\\mathcal {H}}$ , $ (f,g)=\\int {\\mathrm {D}}(x,\\overline{x},\\chi ,\\theta )\\, f g .",
"$ This is the usual nondegenerate, bilinear pairing of the $B$ -model, compactified on a circle and reduced to quantum mechanics.",
"Because $f =f_0\\cdot \\exp \\left(-\\frac{1}{2\\epsilon }V\\right)$ , for small $\\epsilon $ , is strongly localized near critical points of $W$ , that is points with $\\mathrm {d}W=0$ , the evaluation of $(f,g)$ can be expressed as a sum over contributions of critical points.",
"With respect to the decomposition of eqn.",
"(REF ), $(~,~)$ is block diagonal: it restricts to a nondegenerate pairing $(~,~):{\\mathcal {R}}_p\\otimes {\\mathcal {R}}_p\\rightarrow {\\mathbb {C}}$ for each $p$ , and annihilates ${\\mathcal {R}}_{p^{\\prime }}\\otimes {\\mathcal {R}}_p$ for $p^{\\prime }\\ne p$ .",
"However, we want a hermitian inner product in the $B$ -model, not a bilinear one.",
"For this, we assume that $X$ has an antiholomorphic involution $\\tau $ under which $W$ is real in the sense that $\\tau ^*(W)=\\overline{W}$ .",
"An equivalent statement is that $W$ is real when restricted to the fixed points of $\\tau $ , or that for $p\\in W$ , $\\overline{W}(\\overline{p})=W(p)$ , where $\\overline{p}=\\tau (p)$ .",
"We assume that $$ is likewise real, meaning that $\\tau ^*=\\overline{}$ , and we pick the local holomorphic coordinates $x^i$ to similarly satisfy $\\tau ^*x =\\overline{x}$ .",
"Let $\\Theta $ be complex conjugation (which descends from the $\\sf {CPT}$ symmetry of an underlying two-dimensional $\\sigma $ -model) and set $\\Theta _\\tau =\\Theta \\tau $ .",
"Then $\\Theta _\\tau $ commutes with $Q$ , so we can define a natural hermitian pairing on the physical state space of the $B$ -model by $\\langle f,g\\rangle =(\\Theta _\\tau f, g).",
"$ This is a hermitian pairing because it is linear in the second variable and antilinear in the first.",
"In this generality, the inner product $\\langle ~,~\\rangle $ is nondegenerate, but it is not positive-definite.",
"First of all, the inner product is never positive-definite if $W$ has critical points that are not real.",
"Indeed, since the original bilinear pairing annihilates ${\\mathcal {R}}_{p^{\\prime }}\\otimes {\\mathcal {R}}_p$ for $p^{\\prime }\\ne p$ , the hermitian pairing $\\langle ~,~\\rangle $ annihilates ${\\mathcal {R}}_{p^{\\prime }}\\otimes {\\mathcal {R}}_p$ for $p^{\\prime }\\ne \\overline{p}$ .",
"In particular, if $\\overline{p}\\ne p$ , then the pairing $\\langle ~,~\\rangle $ annihilates ${\\mathcal {R}}_p \\otimes {\\mathcal {R}}_p$ , so elements of ${\\mathcal {R}}_p$ are null vectors and the pairing is not positive-definite.",
"Even if all critical points are real, the inner product is never positive-definite if there are degenerate critical points.",
"To understand why, consider a simple example with $X={\\mathbb {C}}$ , $=\\mathrm {d}x$ , and $W(x)=\\lambda x^N/N$ , for some integer $N\\ge 2$ and real $\\lambda $ .",
"The critical point at $x=0$ is degenerate for $N>2$ and nondegenerate for $N=2$ .",
"For representatives of the space of physical states, we can take $f_k=x^k \\exp (-V/2\\epsilon )$ , $k=0,\\cdots , N-2$ .",
"One then has $\\langle f_r,f_s\\rangle =&\\frac{\\lambda (N-1)}{\\epsilon }\\int _{\\mathbb {C}}\\frac{|\\mathrm {d}x\\mathrm {d}\\overline{x}|\\mathrm {d}\\chi \\mathrm {d}\\theta }{2\\pi }\\, \\,x^{r+s}\\overline{x}^{N-2}\\theta \\chi \\, \\exp (-\\lambda ^2|x|^{2N-2}/\\epsilon ) \\cr =&\\frac{\\lambda (N-1)}{\\epsilon }\\int _{\\mathbb {C}}\\frac{|\\mathrm {d}x\\mathrm {d}\\overline{x}|}{2\\pi } \\,x^{r+s}\\overline{x}^{N-2}\\exp (-\\lambda ^2|x|^{2N-2}/\\epsilon ),$ where we included a convenient normalization factor $1/2\\pi $ .",
"This expression is independent of $\\epsilon $ , as expected because the cohomology class of $f_k$ does not depend on $\\epsilon $ .",
"The integral vanishes unless $r+s=N-2$ .",
"Therefore, $f_k$ is a null vector for this inner product unless $2k=N-2$ .",
"In particular, the inner product $\\langle ~,~\\rangle $ is not positive-definite if $N>2$ , that is, if $W$ has a degenerate critical point.",
"The general case of a degenerate critical point is similar.",
"It is a standard result in the $B$ -model that, if $p$ is a degenerate critical point, the elementThe element $1_p$ can be represented by any polynomial that is 1 to sufficiently high order near $p$ and vanishes to sufficiently high order near other critical points.",
"$1_p\\in {\\mathcal {R}}$ which is the identity in ${\\mathcal {R}}_p$ and zero in the other subrings is a null vector for $(~,~)$ .",
"If $p$ is real, then $1_p$ is $\\Theta _\\tau $ -invariant and so is also a null vector for $\\langle ~,~\\rangle $ ; $1_p$ is also a null vector if $p$ is not real, since then ${\\mathcal {R}}_p$ consists entirely of null vectors.",
"Even if all critical points of $W$ are real and nondegenerate, we do not get positivity for free.",
"Indeed, for $N=2$ , eq.",
"(REF ) gives $\\langle f,f\\rangle =1/\\lambda $ , so it is only positive for one sign of $\\lambda $ .",
"As long as there is only one critical point, the sign of $\\lambda $ is not really important, because there was an arbitrary overall sign in the definition of the inner product.",
"However, if there are multiple nondegenerate critical points $p_1,p_2,\\cdots , p_s$ , then for $r=1,\\cdots , s$ , we can pick a function $f_{r}$ that vanishes at all critical points except $p_r$ and is nonvanishing at $p_r$ .",
"Still with $X={\\mathbb {C}}$ and assuming the $p_r$ are real, the evaluation of $\\langle f_r,f_r\\rangle $ will reduce to the $N=2$ version of the above computation, with $\\lambda $ replaced by $W^{\\prime \\prime }(p_r)$ .",
"Thus to achieve positivity, we would want $W^{\\prime \\prime }(p_r)$ to have the same sign for all $r$ .",
"For real critical points in one dimension, it is impossible to satisfy this condition, since inevitably the sign of $W^{\\prime \\prime }$ alternates from one critical point of $W$ to the next.",
"However, with more variables, positivity is possible for any number of critical points.",
"Suppose that $W(x_1,x_2,\\cdots , x_n)$ is a holomorphic function on ${\\mathbb {C}}^n$ that is real when restricted to ${\\mathbb {R}}^n$ , and has real, nondegenerate critical points.",
"At any given critical point $p$ , we can pick real local coordinates $x_{i,p}$ such that $ W=\\frac{1}{2}\\sum _{i=1}^n\\lambda _{i,p} x_{i,p}^2 $ with real, nonzero $\\lambda _{i,p}$ .",
"We restrict the function $W$ (or equivalently ${\\mathrm {Re}}\\,W$ ) to ${\\mathbb {R}}^n$ and view it as a Morse function on ${\\mathbb {R}}^n$ .",
"The number of $\\lambda _{i,p}$ that are negative is defined to be the Morse index $d_p$ of $W$ at $p$ .",
"The generalization of the above one-variable computation shows that, for a state $f_p$ localized near $p$ , and normalized so that $f_p(p)=1$ , $ \\langle f_p,f_p\\rangle = \\frac{1}{\\lambda _{1,p}\\lambda _{2,p}\\cdots \\lambda _{n,p}}=\\frac{1}{\\det H_p}, $ where $H_p$ is the Hessian (or matrix of second derivatives) of the function $W$ at its critical point $p$ .",
"Thus, $\\langle f_p,f_p\\rangle $ is positive or negative depending on whether $d_p$ is even or odd.",
"If there are multiple critical points, then $\\langle ~,~\\rangle $ can be defined to be positive if and only if the Morse indices at critical points are all even or all odd.",
"For a simple example in two dimensions showing that such behavior is possible with any number of critical points, consider the case that $ W(x,y)=y U(x).",
"$ Assuming that there are no solutions of $U=\\mathrm {d}U=0$ , the critical points are at $y=U(x)=0$ .",
"They are nondegenerate if and only if $U(x)$ has only simple zeroes, and if so all critical points have Morse index 1.",
"The $B$ -model quantum mechanics that we have described has four supercharges, though we have emphasized just one that plays the role of a differential in a topological version of the model.",
"This quantum mechanics is the low energy limit of the closed string sector $B$ -model, with target $X$ and superpotential $W$ .",
"In the present paper, we are interested in open strings, not closed strings; that is, we are interested in the $B$ -model formulated on an interval $I$ , with boundary conditions set by branes.",
"This breaks at least half of the supersymmetry, and accordingly $B$ -model open strings, with a generic Calabi-Yau target, are not governed by the sort of quantum mechanics that we have described.",
"However, if $X$ is hyper-Kahler, this doubles the amount of supersymmetry.",
"For hyper-Kahler $X$ , open strings that end on complex Lagrangian branes are governed by the same $B$ -model quantum mechanics that we have described.",
"We consider the $B$ -model of $X$ in one of its complex structures.",
"In our application, $X$ is the Higgs bundle moduli space ${\\mathcal {M}}_H(G,C)$ , and we choose the complex structure $J$ , with holomorphic symplectic form $\\Omega _J=\\omega _K+{\\mathrm {i}}\\omega _I$ .",
"(The holomorphic volume form of the above discussion is then $=\\frac{1}{d!",
"}\\Omega _J^d$ , with $d=n/2$ .)",
"Locally, we can pick coordinates on $X$ with $\\Omega =\\sum _i \\mathrm {d}p_i \\mathrm {d}q^i$ .",
"Because of $B$ -model localization, this local description is enough for many purposes; we make a more global statement presently.",
"We start by considering closed strings in a way that uses the complex symplectic structure of $X$ .",
"Let ${\\mathcal {L}}X$ be the free loop space of $X$ , parametrizing maps $\\Phi :S^1\\rightarrow X$ .",
"We can describe $\\Phi $ by functions $p_i(\\sigma )$ , $q^j(\\sigma )$ , and we define a holomorphic function on ${\\mathcal {L}}X$ by $ W(\\Phi )=\\oint _{S^1} p_i \\mathrm {d}q^i.",
"$ It turns out that the supersymmetric $\\sigma $ -model with target $X$ can be understood as $B$ -model quantum mechanics on ${\\mathcal {L}}X$ with this superpotential.",
"For example, with this choice of $W$ , the usual potential energy $|\\mathrm {d}W|^2$ of $B$ -model quantum mechanics becomes $\\oint \\mathrm {d}\\sigma \\sum _i\\left(\\left|\\frac{\\mathrm {d}p_i}{\\mathrm {d}\\sigma }\\right|^2+\\left|\\frac{\\mathrm {d}q^i}{\\mathrm {d}\\sigma }\\right|^2\\right)$ , which is the usual kinetic energy of a $\\sigma $ -model with target space coordinates $p_i$ , $ q^j$ .",
"The condition for a critical point of $W(\\Phi )$ gives $\\frac{\\mathrm {d}p_i}{\\mathrm {d}\\sigma }=\\frac{\\mathrm {d}q^i}{\\mathrm {d}\\sigma }=0.$ Since $B$ -model quantum mechanics localizes on critical points, as described above, this means that the $B$ -model localizes on constant maps to $X$ .",
"That is a standard result.",
"Now let us consider open strings.",
"We consider an open-string model in which the superpotential is the same integral as in eqn.",
"(REF ) plus some boundary terms.",
"To see what boundary terms should be considered, first consider without boundary terms the functional $W_0(\\Phi )=\\int _I p_i \\mathrm {d}q^i =\\int _0^1 p_i \\frac{\\mathrm {d}q^i}{\\mathrm {d}t} \\mathrm {d}t$ on the interval $I:0\\le t\\le 1$ .",
"Looking for a critical point, we have the same condition $\\mathrm {d}p_i/\\mathrm {d}t= \\mathrm {d}q^i/\\mathrm {d}t =0$ as before, but now there are also endpoint conditions $p_i(0)=p_i(1)=0$ .",
"Since $p$ vanishes at the endpoints in the unperturbed theory, in adding boundary terms to $W_0(\\Phi )$ , we can assume that these boundary terms are functions of $q$ only.",
"Thus we choose holomorphic functions $P_\\ell (q^i)$ , $P_r(q^i)$ , and define the superpotential $W(\\Phi )=\\int _0^1 p_i \\frac{\\mathrm {d}q^i}{\\mathrm {d}t}\\mathrm {d}t - P_r(q^i(1)) +P_\\ell (q^i(0)).",
"$ The endpoint conditions for a critical point of $W$ are now $ p_i(1) &=\\frac{\\partial P_r(q(1))}{\\mathrm {d}q^i}\\cr p_i(0) &=\\frac{\\partial P_\\ell (q(0))}{\\mathrm {d}q^i}.$ In general, if $P(q^i)$ is a holomorphic function, the condition $p_i=\\partial P/\\partial q^i$ defines a complex Lagrangian submanifold of $X$ .",
"Conversely a generic complex Lagrangian submanifold can be described locally in this form; $P(q)$ is called the generating function of the canonical transformation that maps the complex Lagrangian submanifold $p_i=0$ to the complex Lagrangian submanifold $p_i=\\partial _i P$ .",
"So eqn.",
"(REF ) describes two complex Lagrangian submanifolds $L_\\ell $ and $L_r$ that are defining the boundary condition at $t=0$ and $t=1$ .",
"What we have found then is a version of $B$ -model quantum mechanics for open strings propapagating on $X$ with left and right boundary conditions set by complex Lagrangian submanifolds $L_\\ell $ and $L_r$ , respectively.",
"The construction can also be slightly modified to incorporate nongeneric complex Lagrangians, such as the one defined by $q^i=0$ , which cannot be put in the form $p_i=\\partial _i P$ for some $P$ .",
"In this construction, critical points of $W$ correspond to constant maps from $I$ to $X$ that map $I$ to a point in $L_\\ell $ (because of the boundary condition at $t=0$ ) and in $L_r$ (because of the boundary condition at $t=1$ ).",
"So in short, the localization is on constant maps of $I$ to $L_\\ell \\cap L_r$ .",
"Now we make a few remarks of a more global nature.",
"In general, the superpotential $W$ of eqn.",
"(REF ) is well-defined if $\\Omega $ is cohomologically trivial, meaning that $\\Omega =\\mathrm {d}\\lambda $ for a holomorphic 1-form $\\lambda $ , and the Lagrangians $L_\\ell $ and $L_r$ are exact, meaning that $\\lambda $ can be chosen to be exact (and not just closed) when restricted to $L_\\ell $ or to $L_r$ .",
"When these conditions are not satisfied, $W$ is in general multi-valued – it is only well-defined modulo periods of $\\Omega $ in $X$ and of $\\lambda $ in $L_\\ell $ and $L_r$ .",
"That actually does not prevent the $B$ -model from being well-defined, as the Lagrangian of the model only involves derivatives of $W$ and is not sensitive to a constant shift in $W$ .",
"In our application, we take $X$ to be the Higgs bundle moduli space, and $L_\\ell $ , $L_r$ to be the varieties $L_{\\mathrm {op}}$ and $L_{\\overline{\\mathrm {op}}}$ of holomorphic and antiholomorphic opers.",
"Based on our finite-dimensional analysis, we can immediately state necessary conditions for the $B$ -model hermitian form on ${\\mathcal {H}}={\\mathrm {Hom}}({\\mathcal {B}}_{\\overline{\\mathrm {op}}},{\\mathcal {B}}_{\\mathrm {op}})$ to be positive-definite.",
"The intersection $\\Upsilon =L_{\\mathrm {op}}\\cap L_{\\overline{\\mathrm {op}}}$ must consist only of real points.",
"And those intersections must be transverse, since that is the condition that makes the critical point of $W$ corresponding to an intersection nondegenerate.",
"This accounts for the claim in section REF that for positivity of the hermitian inner product of the physical state space of the $B$ -model, the intersections must be transverse.",
"These are necessary conditions, but the finite-dimensional discussion showed that in general they are not sufficient.",
"In the finite-dimensional case, the remaining necessary condition is that, if ${\\mathrm {Re}}\\,W$ is viewed as a Morse function on the real locus, then the differences in Morse indices of critical points should be even.",
"In our application, ${\\mathrm {Re}}\\,W$ (but not ${\\mathrm {Im}}\\,W$ ) is actually single-valued.",
"That is because, in the case of the Higgs bundle moduli space, ${\\mathrm {Re}}\\,\\Omega _J=\\omega _K$ is exact; moreover, as $L_{\\mathrm {op}}$ and $L_{\\overline{\\mathrm {op}}}$ (being topologically equivalent to ${\\mathbb {C}}^{n/2}$ ) are contractible, they are automatically exact.",
"${\\mathrm {Re}}\\,W$ is a Morse function on an infinite-dimensional space, but nevertheless it is possible to make sense of the Morse index of a critical point, in a regularized sense.",
"In the context of ${\\mathcal {N}}=4$ super Yang-Mills theory compactified on a Riemann surface $C$ , ${\\mathrm {Re}}\\,W$ is naturally written as a Chern-Simons integral on the three-manifold $I\\times C$ : $ {\\mathrm {Re}}\\,W=\\int _{I\\times C}{\\rm Tr}\\,\\left(\\phi \\wedge F-\\frac{1}{3}\\phi \\wedge \\phi \\wedge \\phi \\right).$ This is the imaginary part of the Chern-Simons function $\\frac{1}{2}\\int {\\rm Tr}\\,\\left({\\mathcal {A}}\\mathrm {d}{\\mathcal {A}}+\\frac{2}{3}{\\mathcal {A}}^3\\right)$ of the complex connection ${\\mathcal {A}}=A+{\\mathrm {i}}\\phi $ ; $F=\\mathrm {d}A+A\\wedge A$ is the Yang-Mills curvature.",
"At a critical point of ${\\mathrm {Re}}\\,W$ , its Hessian is an elliptic differential operator modulo the gauge group, and the Atiyah-Patodi-Singer (APS) $\\eta $ -invariant of this operator is a regularized version of the Morse index.",
"It is possible to use index theory to show that the differences between the values of $\\eta $ at critical points are integers.",
"For this, one uses the fact that the gradient flow equation for the function ${\\mathrm {Re}}\\,W$ is actually, and rather exceptionally, an elliptic differential equation (the KW equation [3]) in four dimensions [69].",
"The APS index theorem implies that the difference between the values of $\\eta $ at critical points is the index of the linearized KW equation for a four-dimensional field (not necessarily a solution of the KW equation) that interpolates between the two critical points.",
"The index is an integer, so the differences between $\\eta $ -invariants are integers.",
"However, for positivity of the hermitian form on ${\\mathcal {H}}$ , we need the differences to be even integers.",
"A proof of this is not immediately apparent.",
"If it is true that the differences of $\\eta $ -invariants are even integers, then ${\\mathrm {Re}}\\,W$ can be viewed, in the sense of [29], as an equivariantly perfect Morse function on the real locus, which here consists of complex gauge fields on $I\\times C$ that satisfy ${\\mathcal {A}}(t,p)=\\overline{{\\mathcal {A}}}(1-t,p)$ , up to a gauge transformation, for $p\\in C$ , along with the oper boundary condition at $t=0$ .",
"Alternatively, letting $I^{\\prime }$ be the smaller interval $0\\le t\\le 1/2$ , the real locus consists of complex gauge fields on $I^{\\prime }\\times C$ that satisfy the oper boundary condition at $t=0$ and which are real at $t=1/2$ , in the sense that the structure group of the complex connection ${\\mathcal {A}}=A+{\\mathrm {i}}\\phi $ reduces at $t=1/2$ to a real form of $G_{\\mathbb {C}}$ .",
"The claim is that for gauge fields on $I^{\\prime }\\times C$ with complex gauge group $G_{\\mathbb {C}}$ satisfying the indicated boundary conditions at the two ends, the function ${\\mathrm {Re}}\\,W$ is an equivariantly perfect Morse function.",
"Apart from the question of whether the inner product is positive, the formula (REF ) for the normalization of the state $f_p$ is noteworthy.",
"The formula says that if $f_p$ is a $B$ -model state localized at $p$ and normalized by $f_p(p)=1$ , then its norm is $\\langle f_p,f_p\\rangle =1/\\det \\,H_p$ , where $H_p$ is the Hessian of the Morse function at $p$ .",
"For the case of gauge theory on $I^{\\prime }\\times C$ with the boundary conditions that we have imposed, the critical points are flat $G_{\\mathbb {C}}$ bundles over $C$ that are real opers.",
"Consider a real oper that corresponds to a transverse intersection point $p\\in L_{\\mathrm {op}}\\cap L_{\\overline{\\mathrm {op}}}$ .",
"Associated to such a point is a 1-dimensional space of states in the $B$ -model; this is simply the space of complex-valued functions at the point $p$ .",
"There is a distinguished vector $\\Psi _p$ in this space, corresponding to the function 1.",
"The point $p$ determines a real oper bundle $E\\rightarrow I^{\\prime }\\times C$ ; let $E_{\\mathrm {ad}}\\rightarrow I^{\\prime }\\times C$ be the flat bundle associated to $E$ in the adjoint representation.",
"The analog of $1/\\det H_p$ in this situation is the Ray-Singer analytic torsionAnalytic torsion was originally defined for flat bundles with compact structure group but the definition can be generalized to flat bundles with noncompact structure group, for example structure group $G_{\\mathbb {C}}$ .",
"of the flat bundle $E_{\\mathrm {ad}}\\rightarrow I^{\\prime }\\times C$ , with the relevant boundary conditions at the ends.",
"Thus this is a conjectural formula for the Hilbert space norm of the state $\\Psi _p$ , with its natural $B$ -model normalization.",
"It would be desirable to find a combinatorial analog of the analytic torsion in this situation, but we do not immediately know how to do this with oper boundary conditions.",
"Figure: A triangle with the bottom corner labeled by Ψ p ∈ Hom (ℬ op ¯ ,ℬ op )\\Psi _p\\in {\\mathrm {Hom}}({\\mathcal {B}}_{\\overline{\\mathrm {op}}},{\\mathcal {B}}_{\\mathrm {op}}) and the other corners labeled by the canonicalcorners between the branes ℬ op {\\mathcal {B}}_{\\mathrm {op}} and ℬ op ¯ {\\mathcal {B}}_{\\overline{\\mathrm {op}}} and the dual ℬ{\\mathcal {B}} of the regular Nahm pole.",
"The path integral on this trianglecomputes (χ,Ψ p )(\\chi ,\\Psi _p).The normalization condition that we have imposed on $\\Psi _p$ has another interpretation.",
"We explained in Section REF that it makes sense to define the inner product of $\\Psi _p$ with a state created in the $A$ -model by the brane ${\\mathcal {B}}_N$ associated to the regular Nahm pole (with suitable corners), even though the support of this brane is entirely outside $T^*{\\mathcal {M}}(G,C)$ .",
"This computation is actually quite straightforward in the $B$ -model.",
"The $B$ -model dual of ${\\mathcal {B}}_N$ is simply the structure sheaf of ${\\mathcal {M}}_H$ ; that is, it is a rank 1 brane ${\\mathcal {B}}$ whose support is all of ${\\mathcal {M}}_H$ , with trivial ${\\mathrm {CP}}$ bundle.",
"${\\mathcal {B}}$ has a canonical corner with ${\\mathcal {B}}_{\\mathrm {cc}}$ , associated to the constant function 1 on the intersection of the support of the two branes, namely on ${\\mathcal {M}}_H\\cap L_{\\mathrm {op}}=L_{\\mathrm {op}}$ .",
"Likewise, ${\\mathcal {B}}$ has a canonical corner with $\\overline{{\\mathcal {B}}}_{\\mathrm {cc}}$ , represented by the function 1 on ${\\mathcal {M}}_H\\cap L_{{\\overline{\\mathrm {op}}}}=L_{{\\overline{\\mathrm {op}}}}$ .",
"Let $\\chi \\in {\\mathrm {Hom}}(\\overline{{\\mathcal {B}}}_{\\mathrm {cc}},{\\mathcal {B}}_{\\mathrm {cc}})$ be the state created by the brane ${\\mathcal {B}}$ with these canonical corners.",
"The $B$ -model pairing $(\\chi ,\\Psi _p)$ is computed by a path integral on a triangle (fig.",
"REF ) whose sides are labeled by the branes ${\\mathcal {B}}$ , ${\\mathcal {B}}_{\\mathrm {cc}}$ , and $\\overline{{\\mathcal {B}}}_{{\\mathrm {cc}}}$ and whose corners are labeled by the canonical corners of ${\\mathcal {B}}$ and the state $\\Psi _p$ .",
"The $B$ -model localizes on constant maps, and in this case the only constant map that satisfies the necessary conditions is a constant map of the triangle to the point $p$ .",
"We have chosen all the wavefunctions to equal 1.",
"So $(\\chi ,\\Psi _p)=1$ .",
"In other words, the normalization condition on the eigenstate $\\Psi _p$ of the Hitchin Hamiltonians that leads to the prediction stated previously for its Hilbert space norm is $(\\chi ,\\Psi _p)=1$ .",
"If it is possible to compute $(\\chi ,\\Psi _p)$ on the $A$ -model side, this would give a prediction for the Hilbert space norm of $\\Psi _p$ purely in $A$ -model terms.",
"The pairing $(\\chi ,\\Psi _p)$ is somewhat analogous to the Whittaker coefficient in the categorical approach to geometric Langlands."
],
[
"The Case of ${\\mathrm {SL}}(2,{\\mathbb {C}})$",
"The fundamental representation has a basis of canonical sections $s_2 \\equiv s$ , $Ds$ , with $D^2 s = t s$ .",
"Irreducible representations of ${\\mathrm {SL}}(2,{\\mathbb {C}})$ can be presented as symmetric products of the fundamental representation.",
"The symmetric powers $s^{\\otimes n}$ give canonical sections $s_{n+1}$ in the $(n+1)$ dimensional representation.",
"What differential equation is satisfied by $s_{n+1}$ ?",
"After replacing every $D^2 s$ with $t s$ , the object $D^r s_{n+1}$ , for $r\\le n$ , can be expressed as a linear combination of symmetrized forms of $s^{\\otimes (n-k)} Ds ^{\\otimes k}$ , for $k\\le r$ .",
"In order $n+1$ , no new object appears and one gets a differential equation.",
"For example, $D^3 s_3= D^2(s \\otimes Ds + Ds \\otimes s) =2 D(Ds \\otimes Ds + 2 t s \\otimes s) = 6 t Ds_3 + 4 Dt s_3.$ As we multiply representations, antisymmetrizations can be simplified by $s \\wedge Ds=1$ .",
"For example, we have $s_2 \\otimes Ds_2 = \\frac{1}{2} D s_3 + \\frac{1}{2} s_1$ where $s_1=1$ and we left implicit the embedding maps of $2 \\otimes 2 = 3\\oplus 1$ .",
"We could even write an OPE: $s_2(z) \\otimes s_2(w) = s_3(w) + \\frac{1}{2}(z-w)Ds_3(w) + \\cdots - \\frac{1}{2} (z-w) s_1(w)+ \\cdots $"
],
[
"The Case of ${\\mathrm {SL}}(3,{\\mathbb {C}})$",
"The fundamental representation has a basis of sections $s_3 \\equiv s$ , $Ds$ , $D^2s$ , with $D^3 s = t_2 Ds +t_3 s$ .",
"The exterior product $s_{\\overline{3}} = s_3 \\wedge Ds_3$ is a canonical section in the anti-fundamental representation.",
"We indeed have $D^3s_{\\overline{3}} = D^2(s_3 \\wedge D^2s_3) = D(D s_3 \\wedge D^2s_3+t_2 s_3 \\wedge D s_3) = t_2 Ds_{\\overline{3}} + (D t_2-t_3) s_{\\overline{3}}$ The square $s_{6,5} = s_3 \\otimes s_3$ is a canonical section of the symmetric square representation.",
"The first five derivatives $D^n s_{6,5}$ are independent.",
"The sixth derivative can be expressed in terms of the lower derivatives, but with coefficients which are rational functions in $t_2$ , $t_3$ and their derivatives.",
"It is better to define a second basis element $s_{6,1} \\equiv s_3 \\otimes D^2 s_3 + D^2 s_3 \\otimes s_3 - D s_3 \\otimes Ds_3- t_2 s_3 \\otimes s_3$ We then get relations of the form $&D^5 s_{6,5}- 5 t_2 D^3 s_{6,5}+ (7 t_3 + 4 D t_2)D^2 s_{6,5} + \\cdots = (4 t_3 - 2 D t_2) s_{6,1} \\cr &D s_{6,1} = (2 t_3- D t_2) s_{6,5}$ We can capture these relations and more in an OPE $s_3(z) \\otimes s_3(w) \\sim s_{6,5} + \\cdots + \\frac{1}{2} (z-w) s_{\\overline{3}} +\\cdots + \\frac{1}{4} (z-w)^2 s_{6,1} + \\cdots $ Notice that $s_3$ and $s_{\\overline{3}}$ are orthogonal to each other, so $s_{8,5} = s_3 \\otimes s_{\\overline{3}}$ is a canonical adjoint section and $s_{8,3} = s_3 \\otimes D s_{\\overline{3}}-D s_3 \\otimes s_{\\overline{3}}$ is a second one.",
"These satisfy the expected differential equations of the form $D^5 s_{8,5} = \\cdots $ and $D^3 s_{8,3}= \\cdots $ , with derivatives of both sections on both right hand sides."
],
[
"The Wobbly Divisor",
"An eigenfunction $\\Psi $ of the Hecke operators or the Hitchin Hamiltonians is naturally represented, as discussed in Section , by a rank 1 Lagrangian brane ${\\mathcal {B}}_F$ supported on a fiber $F$ of the Hitchin fibration.",
"$F$ must satisfy a quantum-corrected WKB condition.",
"To represent $\\Psi $ as an explicit half-density on ${\\mathcal {M}}(G,C)$ , one can take its inner product with a delta function state $\\delta (x,x_0)$ to get $\\Psi (x_0)=\\langle \\delta (x,x_0),\\Psi \\rangle $ .",
"The delta function state $\\delta (x,x_0)$ is represented by a Lagrangian brane ${\\mathcal {B}}_{x_0}$ supported on the fiber $L_{x_0}$ of the cotangent bundle $T^*{\\mathcal {M}}(G,C)\\rightarrow {\\mathcal {M}}(G,C)$ over the point $x_0\\in {\\mathcal {M}}(G,C)$ .",
"Though the evaluation of the inner product $\\langle \\delta (x,x_0),\\Psi \\rangle $ is a difficult problem that is not likely to admit any fairly explicit answer, it is well-defined and will produce a result that varies with $x_0$ in a real-analytic fashion as long as the brane ${\\mathcal {B}}_{x_0}$ in the $A$ -model of ${\\mathcal {M}}_H(G,C)$ is well-defined and varies with $x_0$ in a real-analytic fashion.",
"This in turn is true as long as $L_{x_0}$ is closed in ${\\mathcal {M}}_H(G,C)$ .",
"It is known, however, that this fails along a divisor ${\\mathcal {D}}\\subset {\\mathcal {M}}(G,C)$ that is sometimes called the wobbly divisor (see for example [70], [71], [72], [73]).",
"A stable bundle $E\\rightarrow C$ is said to be very stable if any Higgs field $\\varphi :E\\rightarrow E\\otimes K$ that is nilpotent is actually 0.",
"Conversely, if there is some nilpotent and nonzero $\\varphi :E\\rightarrow E\\otimes K$ , then $E$ is said to be wobbly.",
"The divisor ${\\mathcal {D}}$ parametrizes wobbly bundles.",
"The support of a brane must be closed, so when $L_{x_0}$ fails to be closed, it is not the support of a brane.",
"Hence the wavefunction $\\Psi (x_0)$ may become singular along ${\\mathcal {D}}$ .",
"In this appendix, we will analyze the behavior along ${\\mathcal {D}}$ for the case that $G={\\mathrm {SU}}(2)$ ; with minor changes, the discussion also applies for $G={\\mathrm {SO}}(3)$ .",
"Explicit computations for the behavior of eigenfunctions of the Hecke operators and the Hitchin Hamiltonians are available in genus 0 for bundles with parabolic structure [10].",
"Though we have not explicitly incorporated parabolic structure in the present article, and will not do so in the present appendix, all of our considerations extend naturally for bundles with parabolic structure.",
"(In gauge theory, parabolic structure is described by incorporating a certain class of surface operators, associated to codimension 2 singularities [74].)",
"Therefore, we will compare our results to what has been found in [10].",
"In what follows, $C$ is a Riemann surface of genus $g$ with canonical bundle $K$ and tangent bundle $T$ , and $E\\rightarrow C$ is a holomorphic vector bundle of rank 2 with $\\det \\,E$ trivial.",
"If $E$ is not very stable, then there is a nonzero Higgs field $\\varphi :E\\rightarrow E\\otimes K$ with $\\varphi ^2=0$ .",
"Then $E$ has a line subbundle ${\\mathcal {L}}=\\mathrm {ker}\\,\\varphi $ and so $E$ is an extension $ 0\\rightarrow {\\mathcal {L}}\\rightarrow E\\rightarrow {\\mathcal {L}}^{-1}\\rightarrow 0.$ We are interested in a nontrivial extension, but we will want to compare to the direct sum $E_0={\\mathcal {L}}\\oplus {\\mathcal {L}}^{-1}$ .",
"The main case that we are interested in will be that $E_0$ is unstable but $E$ is stable.",
"This is true generically ifA rank 2 bundle $E\\rightarrow C$ of degree 0 is stable if and only if any rank 1 subbundle ${\\mathcal {M}}\\subset E$ has negative degree.",
"If $E$ has a line subbundle of degree zero but none of positive degree, it is said to be semistable.",
"By definition, $E$ in the exact sequence (REF ) has the subbundle ${\\mathcal {L}}$ ; if ${\\mathcal {L}}$ has negative degree, this does not destabilize $E$ (and an $E$ appearing in such an exact sequence is generically stable).",
"However, for ${\\mathcal {L}}$ of negative degree, ${\\mathcal {L}}^{-1}$ has positive degree and the direct sum $E_0={\\mathcal {L}}\\oplus {\\mathcal {L}}^{-1}$ is unstable.",
"$c_1({\\mathcal {L}})<0.",
"$ For generic ${\\mathcal {L}}$ , the automorphism group of $E_0$ , as an ${\\mathrm {SL}}(2,{\\mathbb {C}})$ bundle, is the group of diagonal matrices, with we will call $P$ .",
"The nilpotent Higgs field in this situation is $\\varphi _0\\in H^0(C,{\\mathrm {Hom}}({\\mathcal {L}}^{-1},{\\mathcal {L}}\\otimes K))=H^0(C, {\\mathcal {L}}^2\\otimes K)$ .",
"We may expect that at a generic point of the wobbly divisor ${\\mathcal {D}}$ the space of nilpotent Higgs fields is 1-dimensional, so we anticipate that we will wantThe dimension of $H^q(C,{\\mathcal {M}})$ is denoted as $h^q(C,{\\mathcal {M}})$ or just $h^q({\\mathcal {M}})$ .",
"$h^0({\\mathcal {L}}^2\\otimes K)= 1$ .",
"For simplicity, we will study a component of ${\\mathcal {D}}$ with the property that ${\\mathcal {L}}$ can be generically deformed (keeping fixed only $c_1({\\mathcal {L}})$ ) within ${\\mathcal {D}}$ .",
"(There are components of ${\\mathcal {D}}$ such that this is not possible; they can be studied similarly, with similar results.)",
"Under this assumption, asking for 1 to be the generic value of $h^0({\\mathcal {L}}^2\\otimes K)$ , the generic value of $h^1({\\mathcal {L}}^2\\otimes K)$ must be 0 and the degree of ${\\mathcal {L}}$ is determined by the Riemann-Roch formula: $ 1=h^0({\\mathcal {L}}^2\\otimes K)-h^1({\\mathcal {L}}^2\\otimes K) = 1-g+ c_1({\\mathcal {L}}^2\\otimes K)=g-1 +2c_1({\\mathcal {L}})$ so $c_1({\\mathcal {L}})=1-\\frac{g}{2},$ Therefore, this construction as stated is only possible for even $g$ .",
"However, for odd $g$ , we can do something very similar: the half-integral value of $c_1({\\mathcal {L}})$ means that we should take the gauge group to be ${\\mathrm {SO}}(3)$ instead of ${\\mathrm {SU}}(2)$ and replace $E$ with ${\\mathrm {ad}}(E)$ everywhere.",
"As it is notationally simpler, we will continue with the language of ${\\mathrm {SU}}(2)$ gauge theory, but the following remarks also apply to the other case.",
"For $g>2$ , $c_1({\\mathcal {L}})<0$ , so an extension $E$ as in eqn.",
"(REF ) can be stable but the direct sum $E_0$ is unstable.",
"For $g=2$ , $E$ can be stable while $E_0$ is semistable.",
"The condition $h^0({\\mathcal {L}}^2\\otimes K)=1$ is Serre dual to $h^1({\\mathcal {L}}^{-2})=1$ .",
"Since $H^1(C,{\\mathcal {L}}^{-2})$ parametrizes deformations of $E_0$ that do not preserve the triangular structure (REF ) (and the counting of deformations of $E$ that do not preserve this structure is the same), the condition $h^1({\\mathcal {L}}^{-2})=1$ is indeed what we want so that the structure we are discussing occurs in codimension 1 in ${\\mathcal {M}}(G,C)$ , and thus can represent the generic behavior along ${\\mathcal {D}}$ .",
"We can also now count the deformations of $E_0$ that do have the triangular structure of eqn.",
"(REF ).",
"The dimension of this space of deformations is $h^1({\\mathcal {L}}^2)=g-1-2c_1({\\mathcal {L}}) =2g-3$ .",
"As a check, the dimension of the family of stable bundles that we make by deforming $E_0$ is $h^1({\\mathrm {ad}}(E_0)) - h^0({\\mathrm {ad}}(E_0))-\\mathrm {dim}\\,P = g + (2g-3)+1 -1 =3g-3$ , the expected value.",
"(We included $g$ diagonal deformations of $E_0$ .",
"In counting the parameter space of deformations of $E_0$ , we subtract ${\\mathrm {dim}}\\,P=1$ to account for the fact that generic deformations break the $P$ -invariance of $E_0$ .)",
"For what follows, define $ N = h^1({\\mathcal {L}}^2)= 2g-3,$ the number of upper triangular deformations of the unstable bundle $E_0$ along the divisor ${\\mathcal {D}}$ .",
"Thus the construction as stated gives only positive odd values of $N$ , though positive even values can be obtained by a similar construction with parabolic structure or by dropping the assumption that ${\\mathcal {L}}$ can be deformed arbitrarily while remaining within ${\\mathcal {D}}$ .",
"Now we are going to look at the local behavior of the Higgs bundle moduli space in this situation.",
"In the basis with $E_0=\\begin{pmatrix}{\\mathcal {L}}\\cr {\\mathcal {L}}^{-1}\\end{pmatrix}, $ the unstable bundle $E_0$ can be described explicitly by a $\\overline{\\partial }$ operator $\\overline{D}=\\overline{\\partial }+\\begin{pmatrix} a_0 & 0 \\cr 0 & -a_0\\end{pmatrix}.",
"$ In what follows, the “diagonal” deformations that change ${\\mathcal {L}}$ are not interesting because we have assumed that any generic ${\\mathcal {L}}$ (of the appropriate degree) can lead to the situation we are considering.",
"In other words, our assumptions are such that deformations of ${\\mathcal {L}}$ just move us along ${\\mathcal {D}}$ .",
"For the same reason, we will not be interested in diagonal Higgs fields.",
"By constrast, we have to look at the “above diagonal” and “below diagonal” deformations, which move us away from ${\\mathcal {D}}$ .",
"By our assumptions, the space of above diagonal bundle deformations has dimension $N$ , and the space of below diagonal bundle deformations has dimension 1.",
"We pick a basis $b_1,\\cdots , b_n$ of $H^1(C,{\\mathcal {L}}^2)$ and write a generic element of this group as $\\sum t_i b_i$ with complex parameters $t_i$ .",
"Similarly, we pick a nonzero $c\\in H^1(C,{\\mathcal {L}}^{-2})$ and write a generic element as $uc$ with a complex parameter $u$ .",
"So a generic off-diagonal perturbation of $\\overline{D}$ will give us $\\overline{D}^{\\prime }=\\overline{D} +\\begin{pmatrix} 0 & \\sum _{i=1}^n t_i b_i \\cr uc & 0 \\end{pmatrix}.",
"$ Similarly, for the Higgs field, we introduce a nonzero element $e$ of $H^0(C, K\\otimes {\\mathcal {L}}^2)$ and a basis $f_1,f_2,\\cdots ,f_N$ of $H^0(C,K\\otimes {\\mathcal {L}}^{-2})$ , and then, with additional complex parameters $s$ and $r_1,r_2,\\cdots , r_N$ , the off-diagonal part of the Higgs field is $\\varphi _\\perp =\\begin{pmatrix} 0 & se \\cr \\sum _{i=1}^N r_i f_i&0 \\end{pmatrix}.",
"$ On this data, we need to impose the Higgs bundle equation $[\\overline{D}^{\\prime },\\varphi _\\perp ]=0$ .",
"Since we already have $[\\overline{D},\\varphi _\\perp ]=0$ by virtue of the definition of $\\varphi _\\perp $ , the condition reduces to $\\mu =0$ with $ \\mu =\\sum _{i=1}^N r_i t_i -su.",
"$ (Some constants were set to 1 here by suitably normalizing the wavefunctions.)",
"To get a local description of the Higgs bundle moduli space, or more exactly a slice of it transverse to the uninteresting diagonal deformations, we also have to divide by $P\\cong {\\mathbb {C}}^*$ , the automorphism group of the unstable bundle $E_0$ .",
"$P$ acts on these variables with weights 1 for above diagonal parameters $t_i$ and $s$ , and $-1$ for below diagonal parameters $u$ and $r_i$ .",
"The combined operation of dividing by $P$ and imposing the condition (REF ) has a simple interpretation.",
"Introducing the symplectic form $\\Omega = \\sum _{i=1}^N \\mathrm {d}r_i\\, \\mathrm {d}t_i +\\mathrm {d}s \\,\\mathrm {d}u, $ we see that $\\mu $ is the moment map for the action of $P$ on these parameters.",
"The combined operation of setting $\\mu =0$ and dividing by $P$ is a complex symplectic quotient and the result will be a complex symplectic manifold, as expected for the Higgs bundle moduli space.",
"However, we have to be careful about what we mean by the quotient.",
"A Higgs bundle in which the weight 1 parameters $t_i$ and $s$ all vanish is unstable.Let $E$ be a rank 2 bundle with trivial determinant.",
"A Higgs bundle $(E,\\varphi )$ is unstable if $E$ has a $\\varphi $ -invariant subbundle of positive degree.",
"If $E$ has a $\\varphi $ -invariant subbundle of zero degree but none of positive degree, then $(E,\\varphi )$ is said to be semistable.",
"If $E$ has a subbundle of positive degree, but any $\\varphi $ -invariant subbundle of $E$ has negative degree, then the bundle $E$ is unstable but the Higgs bundle $(E,\\varphi )$ is stable.",
"In the present example, if $t_i=s=0$ , then $E$ has the $\\varphi $ -invariant subbundle ${\\mathcal {L}}^{-1}$ of positive degree.",
"So we want to take the quotient with a Mumford stability condition such that $t_i$ and $s$ are not allowed to all vanish.",
"Modulo the action of $P$ , the $N+1$ variables $t_i$ and $s$ therefore define a point in ${\\mathbb {CP}}^N$ and the local model for the Higgs bundle moduli space that we get from this construction is therefore $ M_H= T^*{\\mathbb {CP}}^N.",
"$ To be more exact, this is a transverse slice that captures the relevant aspects of the Higgs bundle moduli space near ${\\mathcal {D}}$ .",
"What about a corresponding local model for the moduli space $M$ of semistable bundles?",
"For this we forget the Higgs parameters $s$ and $r_i$ and just remember the bundle parameters $t_i$ and $u$ .",
"If we set all $t_i$ to vanish, we get an unstable bundle.",
"So the $t_i$ are taken to not all vanish and, modulo the action of $P$ , they define an element of ${\\mathbb {CP}}^{N-1}$ .",
"Taking the weight $-1$ variable $u$ into account, we see that the local model of $M$ is the total space of the line bundle ${\\mathcal {O}}(-1)\\rightarrow {\\mathbb {CP}}^{N-1}$ .",
"This is the same as the blowup of ${\\mathbb {C}}^N$ at a point.",
"The divisor ${\\mathcal {D}}$ of not very stable bundles is defined in this model by the equation $u=0$ ; in other words, it is the exceptional divisor ${\\mathbb {CP}}^{N-1}\\subset M$ .",
"It is convenient to define $x_i= u t_i$ .",
"So $M$ is parametrized by the $x_i$ with the point $\\vec{x}=0$ blown up.",
"For variables canonically conjugate to the $x_i$ , we can take $p_i=r_i/u$ , since $\\Omega =\\sum _{i=1}^N \\mathrm {d}p_i \\mathrm {d}x_i.",
"$ Of course, these coordinates are not good near the exceptional divisor.",
"The case $N=1$ is exceptional in many ways.",
"For $N=1$ , the bundle $E_0$ is semistable rather than unstable, since ${\\mathcal {L}}$ has degree 0.",
"Moreover, for $N=1$ , there is actually no blowup in the construction just described.",
"Only for $N>1$ is the divsor ${\\mathcal {D}}\\cong {\\mathbb {CP}}^{N-1}\\subset M$ wobbly.",
"For $N=1$ , it is semistable; that is, it parametrizes a family of semistable bundles.",
"We can now explain the assertion made at the beginning of this appendix that along a wobbly divisor, the fiber of the cotangent bundle to the moduli space of bundles fails to be closed in the Higgs bundle moduli space.",
"In this analysis, we replace the moduli space of bundles ${\\mathcal {M}}(G,C)$ and ${\\mathcal {M}}_H(G,C)$ by their local models $M$ and $M_H$ .",
"First we consider a point in $M$ that is not in ${\\mathcal {D}}$ .",
"The complement to ${\\mathcal {D}}$ in $M$ consists of nonzero $N$ -plets $\\vec{x}=(x_1,x_2,\\cdots , x_N)$ .",
"For example, take the point $p$ defined by $x_1\\ne 0$ , $x_2=\\cdots = x_N=0$ .",
"Since $x_1=ut_1$ , $x_1\\ne 0$ implies that $u$ and $t_1$ are both nonzero.",
"We can fix the action of $P$ to set $t_1=1$ .",
"The Higgs field is constrained by $ r_1-su=0, $ with no constraint on $r_2,\\cdots , r_N$ .",
"After using eqn.",
"(REF ) to solve for $r_1$ in terms of $s$ , we see that the space of Higgs fields at $p$ is parametrized by $s,r_2, \\cdots , r_N$ .",
"This is the expected copy of ${\\mathbb {C}}^N$ ; it is the fiber at $p$ of the cotangent bundle of $M$ , and it is closed in ${\\mathcal {M}}_H$ .",
"We will denote it as $L_p$ .",
"Now instead let us look at the fiber of the cotangent bundle at a point $p^{\\prime }\\in {\\mathcal {D}}$ .",
"For this, we can simply take $u=0$ , still with $t_1=1$ and $t_i=0,$ $i>1$ .",
"Eqn.",
"(REF ) still holds and tells us to set $r_1=0$ , with no constraint on $u$ .",
"Thus the cotangent fiber $L_{p^{\\prime }}$ is in this case still parametrized by $s$ and $r_2,\\cdots , r_N$ .",
"But $L_{p^{\\prime }}$ is not closed in $M_H$ .",
"To see this, consider the copy of ${\\mathbb {C}}\\subset L_{p^{\\prime }} $ defined by $(t_1,s)=(1,\\lambda ),\\,\\lambda \\in {\\mathbb {C}}$ and $r_2=\\cdots = r_N=0$ .",
"By the action of $P$ , the condition $(t_1,s)=(1,\\lambda )$ is equivalent to $(t_1,s)=(1/\\lambda ,1)$ .",
"Evidently, the limit $\\lambda \\rightarrow \\infty $ exists.",
"Therefore this copy of ${\\mathbb {C}}$ is not closed in $M_H$ ; its closure contains a point with $(t_1,s)=(0,1)$ , compactifying ${\\mathbb {C}}$ to ${\\mathbb {CP}}^1$ .",
"More generally, the closure of $L_{p^{\\prime }}$ in $M_H$ is parametrized by $(t_1,s,r_2,\\cdots , r_N)$ modulo the action of $P$ , with the constraint that $t_1$ and $s$ are not allowed to be both zero.",
"It is a ${\\mathbb {C}}^{n-1}$ bundle over ${\\mathbb {CP}}^1$ .",
"The points with $t_1=0$ , $s\\ne 0$ correspond to Higgs bundles $(E_0,\\varphi )$ such that the underlying bundle $E_0$ is unstable but the pair $(E_0,\\varphi )$ is stable.",
"These points are contained in $M_H$ but not in $T^*M$ .",
"If we set $t_1=s=0$ , we get an unstable Higgs bundle.",
"Now let us return to a generic point $p\\in M$ , not contained in ${\\mathcal {D}}$ , and ask what happens to $L_p$ as $p$ approaches ${\\mathcal {D}}$ .",
"We consider the same point $p$ as before, but since we now know that it is going to be important to allow the possibility $t_1=0$ , we write the constraint (REF ) without setting $t_1=1$ : $ t_1 r_1-su=0.",
"$ We also have the definition of the point $p$ : $ ut_1= x_1.$ $L_p $ is parametrized by $t_1,u,s$ and $r_1,r_2\\cdots , r_N$ , satisfying these equations, and modulo the action of $P$ .",
"To take the limit of $L_p$ as $p\\rightarrow p^{\\prime }$ , we just set $x_1=0$ in eqn.",
"(REF ).",
"The resulting variety has two components.",
"On one component, $u=r_1=0$ and $t_1,s$ are generically nonzero.",
"This component, which we will call $L_{p^{\\prime },1}$ , is the closure of $L_{p^{\\prime }}$ in $M_H$ , as described earlier; it is a ${\\mathbb {C}}^{N-1}$ bundle over ${\\mathbb {CP}}^1$ .",
"On the second component, $t_1=0$ , so that $s$ must be nonzero (for stability of $(E,\\varphi )$ ) and hence eqn.",
"(REF ) implies that $u=0$ .",
"The second component is a copy of ${\\mathbb {C}}^N$ , parametrized by $r_1,\\cdots , r_N$ .",
"Since $t_1=u=0$ , this second component $L_{p^{\\prime },2}$ is entirely contained in the complement of $T^*M$ in $M_H$ .",
"For generic $p$ , there is a rank 1 $A$ -brane ${\\mathcal {B}}_p$ supported on $L_p$ ; it is unique, as $L_p \\cong {\\mathbb {C}}^N$ is simply-connected.",
"For $p\\rightarrow p^{\\prime }$ , $L_p$ splits up into two components, either of which can suppport an $A$ -brane.",
"Since ${\\mathcal {B}}_p$ only varies smoothly away from the wobbly divisor, an eigenfunction of the Hitchin Hamiltonians can potentially become singular along this divisor.",
"We investigate this question in Section REF .",
"For $N=1$ , there is no wobbly divisor and no blowup in this construction.",
"(The genus 2 moduli space does have a wobbly divisor, but not with the properties that we have assumed.)",
"The $N$ -plet $(t_1,\\cdots , t_N)$ collapses to a single variable $t$ , and the moduli space $M$ , in this local model, is a copy of ${\\mathbb {C}}$ parametrized by $x=ut$ .",
"The behavior at $x=0$ is exceptional, but for a different reason.",
"At $x=0$ , the bundle $E$ reduces to the direct sum $E_0={\\mathcal {L}}\\oplus {\\mathcal {L}}^{-1}$ , which for $N=1$ is semistable.",
"There is an unbroken ${\\mathrm {U}}(1)$ gauge symmetry along the divisor ${\\mathcal {D}}$ , which must be taken into account in understanding the behavior along this divisor."
],
[
"Singular Behavior of Wavefunctions",
"A generic Hitchin Hamiltonian is $\\int _C \\alpha {\\rm Tr}\\varphi ^2$ , for $\\alpha \\in H^1(C,T)$ .",
"Because of the form of the Higgs field $\\varphi $ in eqn.",
"(REF ), any such Hamiltonian is homogeneous in $s$ of degree 1 and likewise homogeneous in the $r_i$ of degree 1.",
"Since the number of linearly independent Hitchin Hamiltonians is the same as the number of Higgs bundle parameters, we can generically pick a set of Hitchin Hamiltonians such that $ H_i = s r_i,~~~i=1,\\cdots ,N.$ (Because there are also diagonal perturbations that we are ignoring, this is not the full set of Hitchin Hamiltonians.",
"It is the set of Hitchin Hamiltonians that depend on the off-diagonal perturbations in this approximation.)",
"In terms of the canonical variables $p_i, x_i$ that are good away from the origin, this is $H_i=p_i\\sum _{k=1}^N x_k p_k$ .",
"Upon quantization, this becomes $ H_i=\\frac{1}{2}\\sum _{k=1}^n \\left(\\frac{\\partial }{\\partial x_i} x_k \\frac{\\partial }{\\partial x_k}+\\frac{\\partial }{\\partial x_k} x_k\\frac{\\partial }{\\partial x_i}\\right).",
"$ Operator ordering was chosen to make these operators formally symmetric (the Hitchin Hamiltonians are known to have this property).",
"The $H_i$ act on half-densities $ F(x_1,x_2,\\cdots , x_N)(\\mathrm {d}x_1\\mathrm {d}x_2\\cdots \\mathrm {d}x_N)^{1/2}.",
"$ Of course, what we have in eqn.",
"(REF ) is not an exact formula for the Hitchin Hamiltonians.",
"It is only an asymptotic formula valid for small $\\vec{x}$ .",
"Consider a scaling in which $x, p$ have respectively weights $1,-1$ .",
"The operators $H_i$ in eqn.",
"(REF ) scale with weight $-1$ .",
"The claim is that the Hitchin Hamiltonians are given by these expressions modulo terms of nonnegative weight.",
"To use this model to predict the behavior of eigenfunctions of the Hitchin Hamiltonians near the exceptional divisor ${\\mathcal {D}}$ , we need to use coordinates that are good near ${\\mathcal {D}}$ .",
"A convenient choice is to use $x_1$ along with $y_k=x_k/x_1$ for $k=2,3,\\cdots ,N$ .",
"These are good coordinates near a large open set in ${\\mathcal {D}}$ .",
"As well as changing variables from $x_1,x_2,x_3,\\cdots ,x_N$ to $x_1,y_2,y_3,\\cdots , y_N$ , we also want to express a half-density in a way that is natural in these coordinates.",
"An appropriate form is $ G(x_1,y_2,\\cdots , y_N) (\\mathrm {d}x_1 \\mathrm {d}y_2 \\mathrm {d}y_3\\cdots \\mathrm {d}y_N)^{1/2}.",
"$ Thus the relation between $F$ and $G$ is to be $ F(x_1,x_2,\\cdots ,x_N)=x_1^{-(N-1)/2} G(x_1,y_2,\\cdots , y_N), $ so that $F(x_1,\\cdots , x_N)(\\mathrm {d}x_1\\cdots \\mathrm {d}x_N)^{1/2}=G(x_1,y_2,\\cdots , y_N) (\\mathrm {d}x_1 \\mathrm {d}y_2\\cdots \\mathrm {d}y_N)^{1/2}$ .",
"The Hitchin Hamiltonians acting on $G$ turn out to be in the new coordinates $ H_1& = \\frac{\\partial }{\\partial x_1}x_1\\frac{\\partial }{\\partial x_1} -\\frac{1}{2}\\sum _{k=2}^N\\left( y_k \\frac{\\partial }{\\partial y_k}+\\frac{\\partial }{\\partial y_k} y_k\\right)\\frac{\\partial }{\\partial x_1} \\cr H_k& =\\frac{\\partial ^2}{\\partial x_1 \\partial y_k},~~~~~~~~ k>1.",
"$ Looking for a joint eigenfunction that behaves as $x_1^\\alpha $ for $x_1\\rightarrow 0$ , one sees that for even $N$ , the possibilities are $\\alpha =0$ and $\\alpha =(N-1)/2$ , in accord with computations [10] for parabolic bundles in genus 0.",
"In the case of a singular solution $x_1^{(N-1)/2}g(x_1,\\vec{y})$ with $g(x_1,\\vec{y})$ holomorphic at $x_1=0$ , $g(x_1,\\vec{y})$ is actually independent of $\\vec{y}$ at $x_1=0$ .",
"For odd $N$ , the corresponding statement is that, in addition to holomorphic solutions, there are solutions of the form $f(x_1,\\vec{y})+g(x_1,\\vec{y})x_1^{(N-1)/2}\\log x_1$ , where $f(x_1,\\vec{y})$ and $g(x_1,\\vec{y})$ are holomorphic at $x_1=0$ and again $g(x_1,\\vec{y})$ is independent of $\\vec{y}$ at $x_1=0$ .",
"These formulas only capture the most singular terms near the divisor $D$ and have a rather nongeneric behavior.",
"In particular, the eigenvalue equations $H_s\\Psi =\\lambda _s\\Psi $ , $s=1,\\cdots , N$ , do not have the expected $2^N$ linearly independent joint solutions.",
"The expected behavior is restored if one adds generic higher order terms to the Hitchin Hamiltonians, preserving their commutativity.",
"For example (abbreviating $\\partial /\\partial x_1$ as $\\partial _1$ and $\\partial /\\partial y_k$ as $\\partial ^{\\prime }_k$ ), a simple perturbation of the Hitchin Hamiltonians that preserves their commutativity to first order in $\\alpha $ is to replace the above formulas by $ \\widetilde{H}_1& = \\partial _1 x_1\\partial _1 -\\frac{1}{2}\\sum _{k=2}^N(y_k \\partial ^{\\prime }_k+\\partial ^{\\prime }_k y_k)\\partial _1+2 \\sum _{l,m=2}^N \\alpha _{klm}\\partial ^{\\prime }_l y^k \\partial ^{\\prime }_m \\cr \\widetilde{H}_k&= \\partial _1\\partial ^{\\prime }_k +\\sum _{l,m=2}^N \\alpha _{klm}\\partial ^{\\prime }_l\\partial ^{\\prime }_m,~~~~~k>1, $ with generic complex coefficients $\\alpha _{klm}=\\alpha _{kml}$ .",
"The singular behavior is similar to before."
]
] | 2107.01732 |
[
[
"Propagation-invariant vortex Airy beam whose singular point follows its\n main lobe"
],
[
"Abstract We propose and demonstrate a novel vortex Airy beam which is a superposition of an Airy beam and its laterally sheared beam with a $\\pi/2$ phase shift.",
"This new-type of vortex Airy beam exhibits stable propagation dynamics, wherein its singular point closely follows its main lobe, unlike conventional vortex Airy beams.",
"Notably, the orbital angular mode purity of this new vortex Airy beam is up to 10% better than that of a conventional vortex Airy beam.",
"We anticipate that this new type of vortex Airy beam, which combines the characteristics of an optical vortex and a diffraction-free Airy beam, will facilitate new directions in applications such as microscopy, material processing and nonlinear optics."
],
[
"Introduction",
"An Airy beam is a class of diffraction-free beams which include Bessel and Mathieu beams [1].",
"The first theoretical investigation of such beams was reported in 1979.",
"Berry and Balazs showed that a 1D-Airy wave packet is a solution to the potential-free Schrödinger equation [2], following this, Besieris et. al.",
"also suggested a 2D-Airy wave packet as a solution in 1994 [3].",
"After many years, in 2007, the first finite-energy Airy beam was experimentally demonstrated by Siviloglou et. al.",
"[4].",
"Having unique features like propagation-invariance and a self-accelerating parabolic trajectory [1], [5], the Airy beam has been utilized in applications including selective plane illumination microscopy (SPIM) [6], [7], [8], [9], rapid three-dimensional volumetric imaging [10], optical coherence tomography [11], material processing [12], [13], and optical micromanipulation [14].",
"Soon after the experimental demonstration of an Airy beam, a vortex Airy beam was investigated [15].",
"Conventional vortex Airy beams have a characteristic wherein the optical vortex imposed on the main lobe of the beam easily deforms spatially as the Airy beam propagates away from its focal point.",
"This is because the singular point of the main vortex lobe (the main singular point) deviates from the parabolic trajectory of the vortex Airy beam [16], [17].",
"It is difficult to make the singular point follow the parabolic course of the Airy beam and as such, this has been seen as a barrier to their use in practical applications.",
"One such promising application is stimulated emission depletion selective plane illumination microscopy (STED-SPIM) [18], [19], [20], which uses the combination of an Airy beam for the excitation beam and a vortex Airy beam for the STED beam.",
"This next-generation of STED microscopy would yield fast and high-resolution imaging with an unparalleled field of view.",
"In this manuscript, we propose a new vortex Airy beam, whose singular point follows its main lobe.",
"This new vortex Airy beam is composed of two conventional Airy beams which are the laterally- and phase-shifted with respect to one another.",
"Herein, we refer to this as the new-type vortex Airy beam.",
"We present our research as follows; first, we introduce the basic concept of the new-type vortex Airy beam, following which we theoretically examine the propagation dynamics and the orbital angular momentum (OAM) distribution of the beam.",
"We then detail our experimental generation of the new-type vortex Airy beam, and investigate its propagation dynamics.",
"This is followed by discussion and conclusions of our work.",
"In order to introduce a new-type vortex Airy beam, we start by examining the paraxial equation of diffraction, $\\left(2\\mathrm {i}\\partial _{\\tilde{z}} + \\partial ^2_{\\tilde{x}} + \\partial ^2_{\\tilde{y}} \\right) \\varphi = 0,$ where $\\tilde{s}\\!=\\!s/s_0\\,(s\\!=\\!x,y,z)$ is a normalized axis, $x_0(=\\!y_0)$ is a scaling factor of the transverse plane, $z_0(=\\!kx_0^2)$ is a scaling factor of the propagation axis, $k=2\\pi n/\\lambda $ is the wavenumber with the wavelength $\\lambda $ and the refractive index $n$ .",
"The electric field envelope of a 2D-Airy beam $\\varphi _\\mathrm {Airy}$ is a solution to Eq.",
"(REF ) [4], [21]: $\\varphi _\\mathrm {Airy}(\\tilde{x}, \\tilde{y},\\tilde{z}; \\tilde{x}_\\mathrm {d}, \\tilde{y}_\\mathrm {d}) = &\\prod _{\\tilde{s}=\\tilde{x}+\\tilde{x}_\\mathrm {d},\\tilde{y} + \\tilde{y}_\\mathrm {d}} \\mathrm {Ai} [\\tilde{s}-\\tilde{z}^2/4 +\\mathrm {i}a_0 \\tilde{z}]\\nonumber \\\\&\\quad \\times \\exp \\left[ a_0 \\left( \\tilde{s}-\\tilde{z}^2/2 \\right) -\\mathrm {i}\\frac{\\tilde{z}}{2}\\left( \\frac{\\tilde{z}^2}{6}-a_0^2 - \\tilde{s}\\right) \\right],$ where ($\\tilde{x}_\\mathrm {d}$ ,$\\tilde{y}_\\mathrm {d}$ ) gives the lateral constant shift of the Airy beam, $\\mathrm {Ai}(\\cdot )$ represents the Airy function, and $a_0$ is an exponential truncation factor.",
"The proposed Airy vortex beam is composed of two conventional Airy beams which are superimposed with one another.",
"As shown in Fig.",
"REF (a), the Laguerre–Gaussian mode with the radial index $p\\!=\\!0$ and the azimuthal index $\\ell \\!=\\!1$ (LG$_{01}$ mode) can be expressed as the superposition of Hermite–Gaussian modes of order $(i,j)\\!=\\!",
"(1,0)$ and $(0,1)$ (HG$_{10}$ and HG$_{01}$ ) with the phase shift of $\\pi /2$ [22].",
"Now, we can regard parts of an Airy beam as a higher order Hermite–Gaussian mode.",
"For example, the part of an Airy beam which comprises the main lobe and its left neighbor lobe can be approximated as the Hermite–Gaussian mode of order $(i,j)\\!=\\!",
"(1,0)$ .",
"Thus, the superposition of two Airy beams which have a relative phase of $\\pi /2$ and a lateral shift, is expected to be a vortex Airy beam carrying $\\ell \\!=\\!+1$ OAM.",
"In fact, there will be an azimuthal phase shift of $2\\pi $ around the main singular point (the singular point of the main vortex lobe).",
"In this manuscript, we call this beam a new-type vortex Airy beam with $\\ell \\!=\\!+1$ OAM (Fig.",
"REF (b)).",
"Similarly, the new-type vortex Airy beam with $\\ell \\!=\\!-1$ OAM can be generated via the superposition of two Airy beams with both a lateral shift and a $-\\pi /2$ -phase shift.",
"Figure: (a) The superposition of HG 10 _{10} and HG 01 _{01} modes to form an LG 01 _{01} mode and the superposition of two Airy beams into a new-type vortex Airy beam with a vortex-containing lobe.",
"(b) Plots of the phase distributions of the new-type fields.",
"The white dot indicates the location of the main singular point.We here express a new-type vortex Airy beam with $\\ell \\!=\\!\\pm 1$ OAM as $\\varphi ^\\pm _\\mathrm {new-type}(\\tilde{x}, \\tilde{y},\\tilde{z}) = \\varphi _\\mathrm {Airy}(\\tilde{x}, \\tilde{y},\\tilde{z}; b_1, b_1^{\\prime }) \\pm \\mathrm {i}\\varphi _\\mathrm {Airy}(\\tilde{x}, \\tilde{y},\\tilde{z}; b_1^{\\prime }, b_1) ,$ where $b_k$ and $b^{\\prime }_k$ , respectively, represent the $k$ th real zeros of $\\mathrm {Ai}(\\cdot )$ and $\\mathrm {Ai}^{\\prime }(\\cdot )$ ($b_1\\simeq -2.34$ and $b^{\\prime }_1\\simeq -1.02$ ) [23].",
"The total power of the new-type vortex Airy beam in the beam cross section is derived from Parseval's theorem as $\\int _{-\\infty }^{\\infty } \\int _{-\\infty }^{\\infty } |\\varphi ^\\pm _\\mathrm {new-type}(\\tilde{x}, \\tilde{y},\\tilde{z})|^2 \\mathrm {d}\\tilde{x} \\mathrm {d}\\tilde{y} = \\frac{1}{4\\pi a_0}\\exp \\left( \\frac{4}{3}a_0^3 \\right).$ An $a_0\\!=\\!0$ new-type vortex Airy beam, while being a non-real solution to the paraxial equation owing to its infinite power, is truly propagation-invariant.",
"We consider this beam to be a perfect vortex Airy beam.",
"As expected, when $a_0\\!>\\!0$ , the new-type vortex Airy beam has a finite energy distribution.",
"New-type vortex Airy beams with higher-order OAM can be obtained through the superposition of more than two Airy beams.",
"For example, a new-type vortex Airy beam with $\\ell \\!=\\!\\pm 2$ OAM is expressed as follows: $\\varphi ^{\\pm 2}_\\mathrm {new-type}(\\tilde{x}, \\tilde{y},\\tilde{z}) = &\\varphi _\\mathrm {Airy}(\\tilde{x}, \\tilde{y},\\tilde{z}; b_2^{\\prime }, b_1^{\\prime }) \\pm \\mathrm {i}\\varphi _\\mathrm {Airy}(\\tilde{x}, \\tilde{y},\\tilde{z}; b_1, b_1)\\nonumber \\\\ &- \\varphi _\\mathrm {Airy}(\\tilde{x}, \\tilde{y},\\tilde{z}; b_1^{\\prime }, b_2^{\\prime }).$"
],
[
"Propagation dynamics of the main singular point",
"We show that the new-type vortex Airy beam has a vortex lobe which remains stationary (i.e.",
"the position of the singular point does not change) as the distance from the focus ($\\tilde{z}\\!=\\!0$ ) changes.",
"This is in contrast to a conventional vortex Airy beam; the characteristic of which is shown in Fig.",
"REF (a).",
"The deformation of the intensity distribution of the conventional vortex Airy beam is attributed to the main singular point leaving the parabolic trajectory $(\\tilde{x},\\tilde{y}) = (\\tilde{z}^2/4,\\tilde{z}^2/4)$ with respect to propagation distance.",
"In the case of the perfect new-type vortex Airy beam (with $a_0\\!=\\!0$ ) the main singular point does follow a parabolic trajectory; this can be seen through solution of Eq.",
"(REF ) as follows: $\\varphi ^\\pm _\\mathrm {new-type}(\\tilde{x}, \\tilde{y},\\tilde{z};a_0=0) \\propto \\varphi ^\\pm _\\mathrm {new-type}\\left(\\tilde{x}-\\frac{\\tilde{z}^2}{4}, \\tilde{y}-\\frac{\\tilde{z}^2}{4},\\tilde{z}=0;a_0=0 \\right).",
"$ Moreover, the perfect new-type vortex Airy beam maintains the same intensity distribution for any propagation distance; this is shown in Fig.",
"REF (b).",
"Figure: Propagation dynamics of new-type vortex Airy beams with ℓ=+1\\ell \\!=\\!+1 OAM and conventional vortex Airy beams with ℓ=+1\\ell \\!=\\!+1 OAM.",
"White dots represent the positions of the ℓ=+1\\ell \\!=\\!",
"+1 main singular points.",
"(a) Intensity distribution of the perfect conventional vortex Airy beam.",
"(b) Intensity distribution of the perfect new-type vortex Airy beam.",
"(c) Intensity distribution of the a 0 =0.04a_0\\!=\\!0.04 conventional vortex Airy beam.",
"(d) Intensity distribution of the a 0 =0.04a_0\\!=\\!0.04 new-type vortex Airy beam.While the intensity distribution of the perfect new-type vortex Airy beam can be theoretically modeled, it is a non-real solution to the paraxial equation.",
"Real solutions with $a_0\\!>\\!0$ have also been investigated in this work.",
"In such cases, we observe that there is some deviation of the main singular point from the parabolic trajectory, as the beam propagates away from focus.",
"The amount of deviation $\\tilde{d}$ (as described in Appendix B) is however smaller than that observed in a conventional vortex Airy beam in cases where $a_0\\!<\\!0.27$ .",
"Plots of deviation as a function of axial position ($\\tilde{z}$ ) for different values of $a_0$ are shown in Fig.",
"REF .",
"When $a_0\\!=\\!0.27$ , the amplitude envelope of the new-type vortex Airy beam decays to $1/e$ times in the main vortex lobe since $b_1 > -a_0^{-1} > b_2(\\!\\simeq -4.09)$ .",
"Usually, the exponential truncation factor is experimentally made small ($a_0 \\ll 1$ ).",
"The smaller $a_0$ is, the smaller the amount of deviation $\\tilde{d}$ is at the same propagation distance for the new-type vortex Airy beams, in comparison with conventional vortex Airy beams.",
"If we consider the case where $a_0\\!=\\!0.04$ , the $1/e$ decay of the new-type vortex Airy beam is in its 26th side lobe since $-a_0^{-1}\\simeq b_{27}(\\simeq -25.1)$ .",
"Here, the new-type vortex Airy beam preserves the ring shape of the main vortex lobe (as shown in Fig.",
"REF (d)) as it propagates, whereas the main vortex lobe of the conventional vortex Airy beam separates as it propagates (as shown in Fig.",
"REF (c)).",
"We find that for $a_0 \\tilde{z}$ values $\\lesssim 0.1$ , the ring shape of the main vortex lobe is well-preserved upon propagation and the singular point follows the parabolic trajectory.",
"This is detailed in Appendix B.",
"Figure: The amount of deviation d ˜\\tilde{d} of the main singular point from the parabolic trajectory."
],
[
"Orbital angular momentum spectrum",
"As vortex Airy beams are not symmetric around the main singular point, they carry not only $\\ell \\!=\\!+1$ OAM, but also the other OAM across their profile.",
"Here we examine the OAM spectrum of these vortex Airy beams.",
"The OAM spectrum of the electric field envelope $\\varphi $ at $\\tilde{z}\\!=\\!0$ is defined by $D_m(\\tilde{r}) = \\frac{1}{2\\pi } \\int _0^{2\\pi } \\varphi \\left(\\tilde{x},\\tilde{y}, \\tilde{z} = 0\\right) e^{-\\mathrm {i}m\\phi } \\mathrm {d}\\phi ,$ where $m$ represents the topological charge (or OAM in a reduced Planck constant $\\hbar $ ), $\\tilde{r} \\!=\\!",
"\\sqrt{\\tilde{x}^2+ \\tilde{y}^2}$ and $\\phi \\!=\\!\\arctan (\\tilde{y}/\\tilde{x})$ are the normalized radius and the azimuthal angle of the polar coordinates in the transverse plane respectively [24].",
"Figure REF shows the absolute amplitude distributions of of the OAM spectra of the perfect conventional vortex Airy beam with $\\ell \\!=\\!+1$ OAM and the perfect new-type Airy vortex beam with $\\ell \\!=\\!+1$ OAM.",
"In the inner part of the main vortex lobe ($\\tilde{r} \\!\\le \\!",
"b_1-b_2 \\!\\simeq \\!",
"1.75$ ), both of the beams mainly contain $m\\!=\\!1$ optical vortex modes.",
"Figure REF shows a plot of the OAM spectrum of the inner part of the main vortex lobe of both conventional and new-type vortex Airy beams, where the Intensity ($I_m$ ) is derived as $I_m\\!=\\!",
"\\int _0^{b_1-b_2}|D_m|^2\\tilde{r}\\mathrm {d}\\tilde{r} / \\sum _n \\int _0^{b_1-b_2}|D_n|^2\\tilde{r}\\mathrm {d}\\tilde{r}$ .",
"The mode purity of the perfect new-type vortex Airy beam is $91\\,\\%$ , while that of the perfect conventional Airy vortex beam is $81\\,\\%$ .",
"Thus, the new-type vortex Airy beam is superior in terms of OAM mode purity as well as beam propagation characteristics.",
"Figure: The absolute amplitude distributions of OAM spectra of (a) the perfect conventional Airy beam with ℓ=+1\\ell \\!=\\!+1 OAM and (b) the perfect new-type vortex Airy vortex beam with ℓ=+1\\ell \\!=\\!+1 OAM at z ˜=0\\tilde{z}\\!=\\!0.",
"To clearly resolve the OAM modes, including unwanted modes, the spectral distributions of their absolute amplitude are plotted.",
"The intensity distributions in the real space are also displayed next to the OAM spectra.",
"The white dotted lines indicate r ˜=b 1 -b 2 \\tilde{r} \\!=\\!",
"b_1-b_2.Figure: OAM spectra of the inner part (0≤r ˜≤b 1 -b 2 0 \\!\\le \\!",
"\\tilde{r} \\!\\le \\!",
"b_1-b_2) of the main vortex lobe of the perfect new-type vortex Airy beam with ℓ=+1\\ell \\!=\\!+1 OAM and the perfect conventional Airy vortex beam with ℓ=+1\\ell \\!=\\!+1 OAM at z ˜=0\\tilde{z}\\!=\\!0."
],
[
"Experimental setup",
"We experimentally investigated the propagation dynamics of new-type vortex Airy beams, generated from an in-house-built Ti:sapphire regenerative amplifier pulsed laser system.",
"The output from this laser was horizontally polarized and had a Gaussian spatial profile with a beam radius of 3 mm.",
"Spectral purity was maintained by passing the beam through a bandpass filter (central wavelength, 800 nm; bandwidth, 5 nm).",
"The laser beam was then incident on a spatial light modulator (SLM) which acted as a phase mask.",
"The details of the phase mask displayed on the SLM can be found in Appendix C. The spatially phase-modulated laser beam was then transformed into a new-type vortex Airy beam by a converging lens ($f\\!=\\!300$ mm).",
"Using a CMOS imaging sensor in conjunction with a mechanical stage, we recorded the profiles of the generated vortex Airy beams at different propagation distances, in the vicinity of the focus.",
"For comparison purposes, we generated a conventional vortex Airy beam by implementing a phase mask with a sum of a cubic and a spiral phase distribution on the SLM [25]."
],
[
"Results and Discussion",
"Figure REF shows the experimental propagation dynamics of a finite-energy new-type vortex Airy beam with $\\ell \\!=\\!+1$ OAM and a finite-energy conventional vortex Airy beam with $\\ell \\!=\\!+1$ OAM.",
"The factors of the transverse plane, the propagation axis and the exponential truncation were evaluated to be $x_0\\!=\\!y_0\\!=\\!0.06\\,$ mm, $z_0\\!=\\!30\\,$ mm and $a_0\\!=\\!0.04$ , respectively.",
"The propagation dynamics of the conventional vortex Airy beam (Fig.",
"REF (a)) and the new-type vortex Airy beam (Fig.",
"REF (b)) clearly agree well with the numerical simulations shown in Figs.",
"REF (c) and REF (d), respectively.",
"The main vortex lobe of the finite-energy conventional vortex Airy beam deformed at $\\tilde{z}=1$ and finally divided into two spots at $\\tilde{z}=2$ , which as mentioned is attributed to the main singular point deviating from the parabolic trajectory.",
"In contrast, the main vortex lobe of the new-type vortex Airy beam maintained its ring shape even at $\\tilde{z}=2$ .",
"This is consistent the main singular point of this new-type vortex Airy beam following the parabolic trajectory (as expected, given $a_0\\tilde{z}$ was small).",
"Figure: Experimentally acquired intensity distributions of (a) the a 0 =0.04a_0\\!=\\!0.04 conventional vortex Airy beam with ℓ=+1\\ell \\!=\\!+1 OAM and (b) a 0 =0.04a_0\\!=\\!0.04 new-type vortex Airy beam with ℓ=+1\\ell \\!=\\!+1 OAM at z ˜=0,1,2\\tilde{z}\\!=\\!0,1,2.In order to show that a new-type vortex Airy beam has an $\\ell \\!=\\!1$ singular point in the main vortex lobe, we implemented an interference measurement at $\\tilde{z}=0$ with a reference beam (Fig.",
"REF ).",
"This was done using the random mask encoding method [15] wherein both object and reference beams were simultaneously generated from the same phase mask.",
"The interference image had a two-pronged fork pattern in the main vortex lobe (Fig.",
"REF (a)) and this was consistent with that predicted via numerical simulation (Fig.",
"REF (b)).",
"Thus, the dominant topological charge of the main vortex lobe was +1 [26], [27], [28], which was consistent with the OAM spectrum shown in Fig.",
"REF .",
"These results indicate that these new-type vortex Airy beams comprise a new family of vortex Airy beams.",
"Figure: (a) Observed image and (b) numerical calculation image of interference pattern at z ˜=0\\tilde{z}\\!=\\!0.",
"The white dot in (b) represents the main singular point.",
"(c) Observed image without a reference beam at z ˜=0\\tilde{z}\\!=\\!0.We comment on future applications of the new-type vortex Airy beams.",
"The dark spot of the singular point is well-preserved in the region of $a_0 |\\tilde{z}| \\lesssim 0.1$ , although the shape of the main vortex lobe is not perfectly symmetric especially on propagation.",
"The new-type vortex Airy beams are expected to possess the self-healing properties since they are indeed composed of Airy beams, while it is needed to examine of their self-healing properties in future work.",
"Thus, they can be useful for the STED beam in STED-SPIM.",
"Light-sheet imaging [6], [7], [8], [9], material processing [29], [30], [31] and nonlinear optics [32] may be another fruitful direction."
],
[
"Conclusion",
"In conclusion, we have presented theoretical and experimental investigations into the generation of a new-type vortex Airy beam.",
"These beams exhibit very stable propagation dynamics compared to conventional vortex Airy beams, wherein the position of the singular point within the beam intensity profile does not vary significantly on propagation from focus.",
"This is in contrast to conventional vortex Airy beams which exhibit significant movement of the singular point with beam propagation.",
"We anticipate that the propagation-insensitivity of the singular point in these new-type vortex Airy beams may herald new innovations in applications such as STED microscopy, light-sheet imaging, material processing and nonlinear optics.",
"This work was partially supported by Core Research for Evolutional Science and Technology program (No.",
"JPMJCR1903) of the Japan Science and Technology Agency (JST) and Kakenhi Grants-in-Aid (Nos.",
"JP16H06506, JP17K05069, JP20H02645) from the Japan Society for the Promotion of Science (JSPS)."
],
[
"Conventional vortex Airy beam",
"In this section, we derive a formula which describes a conventional vortex Airy beam that introduced in Ref. [15].",
"The electric field envelope of the Airy beam $\\varphi _\\mathrm {Airy}(\\tilde{x}, \\tilde{y},\\tilde{z};\\tilde{x}_\\mathrm {d},\\tilde{y}_\\mathrm {d}) = &\\prod _{\\tilde{s}=\\tilde{x}+\\tilde{x}_\\mathrm {d},\\tilde{y}+\\tilde{y}_\\mathrm {d}} \\mathrm {Ai} [\\tilde{s}-\\tilde{s}_m(z) +\\mathrm {i}a \\tilde{z}]\\nonumber \\\\&\\quad \\times \\exp \\left[ a \\left( \\tilde{s}-2\\tilde{s}_m(z) \\right) -\\mathrm {i}\\frac{\\tilde{z}}{2}\\left( \\frac{\\tilde{z}^2}{6}-a^2 - \\tilde{s}\\right) \\right]$ satisfies the normalized paraxial equation of monochromatic electromagnetic waves with wavenumber $k$ $\\left(2\\mathrm {i}\\partial _{\\tilde{z}} + \\partial ^2_{\\tilde{x}} + \\partial ^2_{\\tilde{y}} \\right) \\varphi = 0,$ where $\\tilde{s}\\!=\\!s/s_0 (s\\!=\\!x,y)$ is a normalized axis in the beam cross section, $x_0(\\!=\\!y_0)$ is a scale factor of the transverse plane, $\\tilde{z} = z/(kx_0^2)$ is a normalized propagation axis, $\\tilde{s}_m(z) = \\tilde{z}^2/4$ defines the lateral shift of the Airy beam, and $a$ is a truncation factor.",
"When an orbital angular momentum operator $\\hat{L}^\\pm \\equiv \\partial _{\\tilde{x}} \\pm i\\partial _{\\tilde{y}}$ [33] commutes with the operators $\\partial _{\\tilde{z}}$ , $\\partial ^2_{\\tilde{x}}$ , and $\\partial ^2_{\\tilde{y}}$ on both sides of Eq.",
"(REF ), we get $\\left(2\\mathrm {i}\\partial _{\\tilde{z}} + \\partial ^2_{\\tilde{x}} + \\partial ^2_{\\tilde{y}} \\right) \\hat{L}^\\pm \\varphi = 0,$ thus $\\hat{L}^\\pm \\varphi $ is also a solution of Eq.",
"(REF ).",
"The explicit form of $\\hat{L}^\\pm \\varphi _\\mathrm {Airy}$ is given by $\\hat{L}^\\pm \\varphi _\\mathrm {Airy} &= (1\\pm \\mathrm {i})\\left( a+\\mathrm {i}\\frac{\\tilde{z}}{2} \\right)\\varphi _\\mathrm {Airy} + \\varphi _\\mathrm {array}^\\pm , \\\\\\varphi _\\mathrm {array}^\\pm &= \\left[ \\left(\\partial _{\\tilde{x}}\\mathrm {Ai} [\\tilde{x} +\\tilde{x}_\\mathrm {d}-\\tilde{x}_m(z) +\\mathrm {i}a \\tilde{z}] \\right)\\mathrm {Ai} [\\tilde{y} + \\tilde{y}_\\mathrm {d}-\\tilde{y}_m(z) +\\mathrm {i}a \\tilde{z}] \\right.",
"\\nonumber \\\\&\\quad \\pm \\left.",
"i\\mathrm {Ai} [\\tilde{x}+\\tilde{x}_\\mathrm {d}-\\tilde{x}_m(z) +\\mathrm {i}a \\tilde{z}]\\left( \\partial _{\\tilde{y}}\\mathrm {Ai} [\\tilde{y} + \\tilde{y}_\\mathrm {d}-\\tilde{y}_m(z) +\\mathrm {i}a \\tilde{z}] \\right)\\right] \\nonumber \\\\&\\quad \\times \\prod _{\\tilde{s}=\\tilde{x}+\\tilde{x}_\\mathrm {d},\\tilde{y}+\\tilde{y}_\\mathrm {d}}\\exp \\left[ a \\left( \\tilde{s}-2\\tilde{s}_m(z) \\right) -\\mathrm {i}\\frac{\\tilde{z}}{2}\\left( \\frac{\\tilde{z}^2}{6}-a^2 - \\tilde{s}\\right) \\right],$ where $\\varphi _\\mathrm {array}^\\pm $ is a vortex array imposed on an Airy beam (Fig.",
"REF ).",
"The emergence of a vortex array was reported in the first study of vortex Airy beams [15].",
"In general, the orbital angular momentum operator $L^\\pm $ adds an orbital angular momentum $\\ell = \\pm 1$ .",
"For example, when the orbital angular momentum operator acts on a Laguerre–Gaussian mode with the radial index $p$ and azimuthal index $\\ell $ (LG$_{p\\ell }$ mode), the mode will be converted into the LG$_{p(\\ell \\pm 1)}$ mode [33].",
"$\\hat{L}^\\pm \\varphi _\\mathrm {Airy}$ (Eq.",
"(REF )), however, has the terms of an Airy beam as well as that of a vortex array Airy beam, the characteristic of which causes a degradation of the beam profile with distance from its focus ($\\tilde{z}\\!=\\!0$ ).",
"Figure: (a) Intensity and (b) phase profiles of a vortex array Airy beam ϕ array + (z ˜=0;a 0 =0,x ˜ d =0,y ˜ d =0)\\varphi _\\mathrm {array}^+ (\\tilde{z}\\!=\\!0; a_0\\!=\\!0,\\tilde{x}_\\mathrm {d}\\!=\\!0,\\tilde{y}_\\mathrm {d}\\!=\\!0).Here we show a beam described by Eq.",
"(REF ) which is generated by the spatial Fourier transformation of an LG$_{01}$ mode modulated by a cubic phase.",
"The inverse spatial Fourier transform of the Airy beam at $\\tilde{z}\\!=\\!0$ is defined as $\\mathcal {F}^{-1}[\\varphi _\\mathrm {Airy}(\\tilde{x},\\tilde{y},\\tilde{z}=0;\\tilde{x}_\\mathrm {d},\\tilde{y}_\\mathrm {d})] = \\prod _{s=x,y} \\exp (-a\\tilde{k}_s^2)\\exp \\left[ \\frac{\\mathrm {i}}{3}\\lbrace \\tilde{k}_s^3-3(a^2 + \\tilde{s}_\\mathrm {d})\\tilde{k}_s-\\mathrm {i}a^3\\rbrace \\right], \\nonumber \\\\$ where $\\tilde{k}_{s}\\!=\\!",
"s_0k_{s}$ ($s\\!=\\!x,y$ ) is a normalized wavenumber in the transverse plane [15].",
"When $a \\ll 1$ , we get $\\mathcal {F}^{-1}[\\varphi _\\mathrm {Airy}(\\tilde{x},\\tilde{y},\\tilde{z}=0;\\tilde{x}_\\mathrm {d},\\tilde{y}_\\mathrm {d})] = \\prod _{s=x,y} \\exp (-a\\tilde{k}_s^2)\\exp \\left[ \\frac{\\mathrm {i}\\tilde{k}_s^3}{3}-\\mathrm {i}\\tilde{s}_\\mathrm {d}\\tilde{k}_s \\right].",
"$ The spatial Fourier transform of Eq.",
"(REF ) is $\\varphi _\\mathrm {Airy}(\\tilde{x},\\tilde{y},\\tilde{z}=0;\\tilde{x}_\\mathrm {d},\\tilde{y}_\\mathrm {d}) \\nonumber \\\\\\qquad = \\frac{1}{2\\pi }\\int \\!\\int \\prod _{s=x,y} \\exp [-a \\tilde{k}_s^2]\\exp \\left[ \\frac{\\mathrm {i}\\tilde{k}_s^3}{3} \\right] \\exp [-\\mathrm {i}\\tilde{k}_s(\\tilde{s}+\\tilde{s}_\\mathrm {d})]\\mathrm {d}\\tilde{k}_x\\mathrm {d}\\tilde{k}_y, $ which physically means that a Gaussian beam whose beam radius is $a^{-1/2}$ with a cubic phase $(\\tilde{k}_x^3+\\tilde{k}_y^3)/3$ , propagating through a Fourier lens, generates a finite energy Airy beam at its focus.",
"From Eq.",
"(REF ), a beam described by $\\hat{L}^\\pm \\varphi _\\mathrm {Airy}$ is obtained by Fourier lens transformation of LG$_{01}$ mode beam with a cubic phase, $\\hat{L}^\\pm \\varphi _\\mathrm {Airy}(\\tilde{x},\\tilde{y},\\tilde{z}=0;\\tilde{x}_\\mathrm {d},\\tilde{y}_\\mathrm {d}) \\nonumber \\\\\\quad = \\frac{1}{2\\pi }\\int \\!\\int \\prod _{s=x,y} \\exp [-a \\tilde{k}_s^2]\\exp \\left[ \\frac{\\mathrm {i}\\tilde{k}_s^3}{3} \\right] \\hat{L}^\\pm \\exp [-\\mathrm {i}\\tilde{k}_s(\\tilde{s}+\\tilde{s}_\\mathrm {d})]\\mathrm {d}\\tilde{k}_x\\mathrm {d}\\tilde{k}_y\\nonumber \\\\\\quad =\\frac{-\\mathrm {i}}{2\\pi }\\int \\!\\int (\\tilde{k}_x\\pm \\mathrm {i}\\tilde{k}_y)\\prod _{s=x,y} \\exp [-a \\tilde{k}_s^2]\\exp \\left[ \\frac{\\mathrm {i}\\tilde{k}_s^3}{3} \\right] \\exp [-\\mathrm {i}\\tilde{k}_s(\\tilde{x}+\\tilde{x}_\\mathrm {d})]\\mathrm {d}\\tilde{k}_x\\mathrm {d}\\tilde{k}_y.\\nonumber \\\\$ A conventional vortex Airy beam is usually generated by applying the sum of a cubic phase and a spiral phase to a Gaussian beam [15], [16], [25].",
"We assume that the radius of the Gaussian beam is $\\tilde{w}_0$ .",
"When $\\tilde{w}_0 = (2/a)^{1/2}$ , $\\sim 96$ % of the phase modulated Gaussian beam is the phase modulated LG$_{01}$ mode beam in terms of energy ratio.",
"Most parts of the conventional vortex Airy beam is described by Eq.",
"(REF ).",
"Thereby, obtaining the simple expression of conventional vortex Airy beam, we regard $\\varphi _\\mathrm {conv.",
"}^\\pm (\\tilde{x},\\tilde{y},\\tilde{z})$ as $\\hat{L}^\\pm \\varphi _\\mathrm {Airy}(\\tilde{x},\\tilde{y},\\tilde{z}; \\tilde{x}_\\mathrm {d}\\!=\\!b_1^{\\prime },\\tilde{y}_\\mathrm {d}\\!=\\!b_1^{\\prime }, a\\!=\\!2a_0)$ ."
],
[
"Propagation dynamics of the main vortex lobe of the new-type vortex Airy beam",
"The position of the singular point $(\\tilde{x}_\\mathrm {sp},\\tilde{y}_\\mathrm {sp})$ of the main vortex lobe of a finite energy new-type beam shifts from the parabolic trajectory $(\\tilde{x},\\tilde{y})\\!=\\!",
"(\\tilde{z}^2/4,\\tilde{z}^2/4)$ .",
"The shift $(\\tilde{x}_\\mathrm {shift},\\tilde{y}_\\mathrm {shift})$ can be numerically fitted by hyperbolic tangent functions.",
"$\\left( \\begin{array}{c} \\tilde{x}_\\mathrm {shift}(z) \\\\ \\tilde{y}_\\mathrm {shift}(z) \\end{array} \\right) &=\\left( \\begin{array}{c} \\tilde{x}_\\mathrm {sp}(z) - \\tilde{x}_\\mathrm {sp}(0) \\\\ \\tilde{y}_\\mathrm {sp}(z) - \\tilde{y}_\\mathrm {sp}(0) \\end{array} \\right) -\\left( \\begin{array}{c} \\tilde{z}^2/4 \\\\ \\tilde{z}^2/4 \\end{array} \\right) \\nonumber \\\\&=\\left( \\begin{array}{c} 0.86\\cdot \\mathrm {arctanh} (1.1\\cdot a_0\\tilde{z} )\\\\ 0.69\\cdot \\mathrm {arctanh} (-1.4\\cdot a_0\\tilde{z} ) \\end{array} \\right)\\quad (0 \\le a_0\\tilde{z} \\lesssim 0.61),\\nonumber \\\\\\left( \\begin{array}{c} \\tilde{x}_\\mathrm {shift}(z) \\\\ \\tilde{y}_\\mathrm {shift}(z) \\end{array} \\right) &=\\left( \\begin{array}{c} \\tilde{y}_\\mathrm {shift}(-z) \\\\ \\tilde{x}_\\mathrm {shift}(-z) \\end{array} \\right)\\quad (-0.61 \\lesssim a_0\\tilde{z} <0).$ $\\ell \\!= \\!\\pm 1$ and $\\ell \\!= \\!\\mp 1$ singular points appear at $\\tilde{z} \\!\\sim \\!",
"-0.61/a_0$ .",
"The former one is the singular point of the main vortex lobe.",
"This singular point collides with another $\\ell \\!= \\!\\mp 1$ singular point, following which these points vanish at $\\tilde{z} \\!\\sim \\!",
"0.61/a_0$ (Fig.",
"REF ).",
"Figure: Trajectories of singular points of the new-type vortex Airy beam with ℓ=1\\ell \\!=\\!1 OAM.The main vortex lobe of a finite energy new-type vortex Airy beam deforms with propagation distance.",
"Figure.",
"REF depicts propagation dynamics of various new-type vortex Airy beams at $a_0\\tilde{z} = 0.1, 0.3$ and $0.5$ .",
"The shape of the vortex lobe resembles a closed ring at $a_0\\tilde{z}=0.1$ , and it becomes progressively more open as the value of $a_0\\tilde{z}$ increases, as seen for $a_0\\tilde{z}=0.3$ and $a_0\\tilde{z}=0.5$ .",
"This characteristics is due to the deviation of the main singular point from the parabolic trajectory ($\\tilde{d} = \\sqrt{\\tilde{x}_\\mathrm {shift}^2+\\tilde{y}_\\mathrm {shift}^2}=0.8$ ), which is significant with respect to the size of the vortex main lobe.",
"Figure: Propagation dynamics of ℓ=1\\ell \\!=\\!1 a 0 =0.01,0.04,0.16a_0\\!=\\!0.01, 0.04, 0.16 and 0.270.27 new-type vortex Airy beams at a 0 z ˜=0.1,0.3a_0\\tilde{z} = 0.1, 0.3 and 0.50.5.",
"White dots represent the position of the main singular point."
],
[
"Phase mask for new-type vortex Airy beam",
"Here we detail the characteristics of the phase mask used to tailor a new-type vortex Airy beam.",
"The Fourier transform of a new-type vortex Airy beam is described by the following expression $\\mathcal {F}^{-1}[\\varphi _\\mathrm {new-type}^\\pm (\\tilde{x},\\tilde{y},\\tilde{z}=0)] \\nonumber \\\\\\quad = \\mathcal {F}^{-1}[\\varphi _\\mathrm {Airy}(\\tilde{x}, \\tilde{y},\\tilde{z}=0; b_1, b_1^{\\prime })] \\pm \\mathrm {i}\\mathcal {F}^{-1}[\\varphi _\\mathrm {Airy}(\\tilde{x}, \\tilde{y},\\tilde{z}=0; b_1^{\\prime }, b_1)]\\nonumber \\\\\\quad =( \\exp [-\\mathrm {i}(b_1\\tilde{k}_x+b_1^{\\prime }\\tilde{k}_y)]\\pm \\mathrm {i}\\exp [-\\mathrm {i}(b_1^{\\prime }\\tilde{k}_x+b_1\\tilde{k}_y)])\\prod _{s=x,y} \\exp (-a_0\\tilde{k}_s^2)\\exp \\left[ \\frac{\\mathrm {i}\\tilde{k}_s^3}{3}\\right]\\nonumber \\\\\\quad = \\sqrt{2}(1\\pm \\mathrm {i}) \\sin \\left[ \\frac{\\pi }{4} \\mp \\frac{(b_1-b_1^{\\prime })(\\tilde{k}_x-\\tilde{k}_y)}{2} \\right] \\nonumber \\\\\\quad \\qquad \\qquad \\qquad \\times \\prod _{s=x,y} \\exp \\left[ -a_0\\tilde{k}_s^2+ \\mathrm {i}\\left( \\frac{\\tilde{k}_s^3}{3} - \\frac{(b_1+b_1^{\\prime })\\tilde{k}_s}{2} \\right) \\right],$ where $b_k$ and $b^{\\prime }_k$ represent the $k$ th real zeros of $\\mathrm {Ai}(\\cdot )$ and $\\mathrm {Ai}^{\\prime }(\\cdot )$ ($b_1\\simeq -2.34$ and $b^{\\prime }_1\\simeq -1.02$ ) respectively [23].",
"We assume the input beam is a Gaussian beam with beam radius $\\tilde{w}_0\\!=\\!a_0^{-1/2}$ in the $(\\tilde{k}_x,\\tilde{k}_y)$ plane.",
"Since we need both phase modulation and amplitude modulation through a phase mask, we calculated the phase mask pattern by using the Davis's method [34], [35], [36].",
"Figure REF shows the phase distribution that we displayed on the SLM.",
"When $a_0\\!=\\!0.04$ , the radius of the Gaussian beam is $\\tilde{w}_0\\!=\\!a_0^{-1/2}\\!=\\!5$ in the $(\\tilde{k}_x,\\tilde{k}_y)$ plane or $w_0\\!=\\!5/x_0$ in the $(k_x, k_y)\\!=\\!",
"(\\tilde{k}_x/x_0,\\tilde{k}_y/y_0)$ plane.",
"In experiments, the beam radius $w_0$ in the $(k_x, k_y)$ plane is usually a constant value, so $a_0(\\!=\\!w_0^{-2}x_0^{-2})$ is determined by the scale factor $x_0$ .",
"Figure: Phase distribution of the phase mask without the carrier phase modulation for a new-type vortex Airy beam with ℓ=1\\ell \\!=\\!1 OAM."
]
] | 2107.01812 |
[
[
"Three-Dimensional Stationary Spherically Symmetric Stellar Dynamic\n Models DEpending on Local Energy"
],
[
"Abstract Three-Dimensional Stationary Spherically Symmetric Stellar Dynamic Models Depending on the Local Energy.",
"Juergen Batt, Enno Joern, Alexander L. Skubachevskii The stellar dynamic models considered here are triples (f,rho,U) of three functions: the distribution function f=f(r,u), the local density rho=rho(r) and the Newtonian potential U=U(r), where r:=|x|, u:=|v| ((x,v) in R^3xR^3 are the space-velocity coordinates), and f is a function q of the local energy E=U(r)+u^2/2.",
"Our first result is an answer to the following question: Given a (positive) function p=p(r) on a bounded interval [0,R], how can one recognize p as the local density of a stellar dynamic model of the given type (\"inverse problem\")?",
"If this is the case, we say that p is \"extendable\" (to a complete stellar dynamic model).",
"Assuming that p is strictly decreasing, we reveal the connection between p and F, which appears in the nonlinear integral equation p=FU[p] and the solvability of Eddington's equation between F and q.",
"Second, we investigate the following question (\"direct problem\"): Which q induce distribution functions f of the form f=q(-E(r,u)-E0) of a stellar dynamic model?",
"This leads to the investigation of the nonlinear equation p=FU[p] in an approximate and constructive way by mainly numerical methods.",
"The paper extends preceding work on flat galaxies to the three-dimensional case.",
"In particular, the present answer to the extendability problem is completely different as in [1].",
"The present paper also opens the way to further explicit solutions of the Vlasov-Poisson system beyond the classical known examples which are for instant given in [4].",
"Keywords: Vlasov-Poisson system, stationary solutions, numerical approximation, mathematical physic, galaxy astrophysics."
],
[
"Introduction",
"The Vlasov–Poisson System (VPS) in 3 dimensions (stellar dynamic version) has the following form: $\\dfrac{\\partial {f}}{\\partial {t}}+v\\cdot \\dfrac{\\partial {f}}{\\partial {x}}-\\dfrac{\\partial }{\\partial {x}}\\, U(t,x)\\cdot \\frac{\\partial {f}}{\\partial {v}}&=0,\\\\\\Delta U(t,x)&=4\\pi \\rho (t,x) \\\\\\text{or} \\quad U(t,x)&=-\\int \\dfrac{\\rho (t,y)}{|x-y|}\\,dy, \\\\\\rho (t,x)&=\\int f(t,x,v)\\,dv.$ Here $f=f(t,x,v)\\ge 0$ is the distribution function of the gravitating matter, $U=U(t,x)\\le 0$ the Newtonian potential and $\\rho (t,x)\\ge 0$ the local density.",
"The system has been intensively investigated in many directions.",
"For the case of time-dependent functions (initial value problem), [9] gives a survey until 2007.",
"The stationary spherically symmetric functions are characterized by the property $f(x,v)=f(A_1x,A_2v)$ for all $A_1,A_2\\in S0(3)$ ; for a short account of this class, relevant for our work, see [1], also for references.",
"The aim of the present paper is twofold.",
"Our first problem is known as the “inverse problem”: to identify those functions $p$ , defined on a bounded interval $[0,R]$ , as the local density of a stationary spherically symmetric stellar dynamic model, in which $f$ depends on the local energy: $f(r,u)=q(-E-E_0), \\qquad \\text{where} \\quad E_0>0 \\quad \\text{is a constant}.$ This question occurs if one wants to determine the three quantities $f$ , $\\rho $ , $U$ from observation.",
"The result of an observation generally is a brightness profile, which, by certain strategies, can be turned into a mass profile.",
"The question which follows is the determination of the potential $U$ and the distribution $f$ (Sections –).",
"Our second problem is called the direct problem.",
"It is known that the distribution $f$ of a stationary spherically symmetric stellar dynamic model is a function of the local energy $E$ and the angular momentum $F:=x^2v^2-(xv)^2$ (this fact is called Jeans' theorem) [2].",
"The direct problem partially poses the opposite question, namely: which functions $q$ admit finding functions $\\rho (r)$ and $U(r)$ together with a constant $E_0>0$ such that $f(r,u)=q(-E-E_0)$ , $\\rho $ and $U$ form a triple of a stationary spherically symmetric stellar dynamic model (Sections –).",
"We give a short overview over the different sections.",
"Section : Introduction of the potential operator $U=Lp$ on its domain $\\mathcal {D}(L)$ (Definition ) with its elementary properties (Lemma ).",
"Each strictly decreasing function $p\\in \\mathcal {D}(L)$ satisfies a nonlinear integral equation $p=FLp$ with an appropriate $F=F[p]$ (Lemma ).",
"Section : Definition of the stationary spherically symmetric solutions depending on the local energy and proofs of their properties, Equivalence Lemma and Eddington's equation (Lemma ).",
"Section : The inverse problem: its formulation and its solution (Theorem ).",
"Section : Presentation of examples, which illustrate Theorem , and the concept of extendability.",
"Section : Formulation of the direct problem and its conversion into the equivalent problem of solving a nonlinear integral equation of the form $Lp-E_0=G_0(p).$ Section : Construction of an approximating nonlinear system (ANS) of the form $\\sum _{k=0}^{n-1} A_{ik}x_k:=\\sum _{k=0}^{n-1} B_{ik}x_{k}-C_kx_k=G_0(x_i)$ and calculation of the matrix ($A_{ik}$ ).",
"Section : Numerical analysis of the (ANS), description of the approximation and convergence, examples.",
"Section (Appendix): Contains Tonelli's work on Abel's and Eddington's equations with full proofs.",
"Section : Contains suggestions for further research."
],
[
"The potential operator in spherical symmetry",
"We define the potential operator $Lp(x):=\\int _{\\mathbb {R}^3}\\frac{p(y)}{|x-y|}\\,dy, \\qquad x\\in \\mathbb {R}^3,$ for certain functions $p$ on $\\mathbb {R}^3$ , which are spherically symmetric.",
"This means (by abuse of notation) that $p(x)=p(r)$ , $r:=|x|$ .",
"We first conclude that $Lp$ is also spherically symmetric .",
"In fact, if $A\\in SO(3)$ , then, assuming that $y=Az$ , we have $Lp(Ax)&=\\int _{\\mathbb {R}^3}\\frac{p(y)}{|Ax-y|}\\,dy=\\int _{\\mathbb {R}^3}\\frac{p(y)}{\\big |A(x-A^{-1}y)\\big |}\\,dy= \\int _{\\mathbb {R}^3}\\frac{p(Az)}{|x-z|}\\,dz\\\\&=\\int _{\\mathbb {R}^3}\\frac{p(z)}{|x-z|}\\,dz =Lp(x).$ We define $\\mathbb {R}_{0+}:=\\big \\lbrace r\\in \\mathbb {R}\\colon $ $r\\ge 0\\big \\rbrace $ and $\\mathbb {R}_{+}:=\\big \\lbrace r\\in \\mathbb {R}\\colon r>0\\big \\rbrace $ .",
"Let $\\mathcal {D}(L)$ be the set of functions $p\\colon \\mathbb {R}_{0+}\\rightarrow \\mathbb {R}_{0+}\\cup \\lbrace \\infty \\rbrace $ with the following properties: (a) $p\\in C(\\mathbb {R}_+)$ , (b) for all $r>0$ we have $\\displaystyle \\int _0^r p(s)s^2\\,ds<\\infty $ , $\\displaystyle \\int _r^\\infty p(s)s\\,ds<\\infty $ , (c) there exists a $\\delta >0$ such that for all $r\\in (0,\\delta )$ , we have $p(r)>0$ .",
"For $p\\in \\mathcal {D}(L)$ , we have: $& 1) \\ Lp(r)=4\\pi \\left[\\frac{1}{r}\\int _0^rp(s)s^2\\,ds+\\int _r^\\infty p(s)s\\,ds\\right], \\qquad r>0, \\\\& 2) \\ Lp\\in C^2(\\mathbb {R}_+), \\quad \\text{and} \\\\&\\phantom{2)}\\ (Lp)^{\\prime }(r)=-\\frac{4\\pi }{r^2}\\!\\int _0^r p(s)s^2\\,ds, \\quad r>0,\\\\&\\phantom{2)}\\ (Lp)^{\\prime \\prime }(r)=-\\frac{2}{r}(Lp)^{\\prime }(r)-4\\pi p(r), \\qquad r>0.",
"\\\\& 3) \\ Lp>0 \\ \\text{and} \\ (Lp)^{\\prime }<0, \\ \\text{that is}, Lp \\ \\text{is strictly decreasingon} \\ \\mathbb {R}_+.",
"\\ \\text{Because }\\\\&\\phantom{2)}\\ \\text{the limits}$ $Lp(0):=\\lim _{r\\rightarrow 0} Lp(r), \\qquad Lp\\,(\\infty )=\\lim _{r\\rightarrow \\infty } Lp(r)$ exist, the function $Lp$ has a strictly decreasing inverse $(Lp)^{-1}\\colon \\big (Lp(\\infty ), Lp(0)\\big )\\rightarrow (0,\\infty ).$ Proof.",
"1) Using spherical coordinates $x=(r\\sin \\psi \\cos \\varphi , r\\sin \\psi \\sin \\varphi , r\\cos \\psi ),$ we have $Lp(x)=\\int _0^\\pi \\!\\int _0^{2\\pi }\\!\\int _0^\\infty \\frac{p(s)\\sin \\psi }{\\sqrt{r^2+s^2-2rs\\cdot \\cos \\psi }}\\,s^2\\,ds\\,d\\varphi \\,d\\psi \\\\=2\\pi \\int _0^\\infty \\,\\int _0^\\pi \\frac{\\sin \\psi }{\\sqrt{r^2+s^2-2rs\\cdot \\cos \\psi }}\\, d\\psi p(s)s^2\\,ds.$ In the inner integral we substitute $u:=\\sqrt{r^2+s^2-2rs\\cos \\psi }$ and get $\\int _0^\\pi \\frac{\\sin \\psi }{\\sqrt{r^2+s^2-2rs\\cdot \\cos \\psi }}\\,d\\psi &=\\!\\!\\!\\int _{\\sqrt{r^2+s^2-2rs}}^{\\sqrt{r^2+s^2+2rs}}\\!\\!\\!1\\cdot du\\cdot \\frac{1}{rs}\\\\&=\\frac{(r+s)-|r-s|}{rs}={\\left\\lbrace \\begin{array}{ll}\\dfrac{2}{r}\\quad &\\text{for}\\quad s\\le r,\\\\[0,7em]\\dfrac{2}{s} \\quad &\\text{for}\\quad s\\ge r,\\end{array}\\right.",
"}$ and (REF ) follows.",
"2) () results from differentiating (REF ), and () follows from differentiating ().",
"3) Inequality $Lp>0$ is a consequence of Definition c), and $(Lp)^{\\prime }<0$ results from ().",
"The existence of the limits and of the inverse operator are direct consequences of these facts.$\\square $ Most of our functions $p\\in \\mathcal {D}(L)$ will have compact support.",
"We define $\\mathcal {D}_R(L):= \\lbrace p\\in \\mathcal {D}(L)\\colon p>0$ on $[0,R)$ , $p=0$ on $[R, \\infty )$ }.",
"$\\mathcal {D}_R^-(L):=\\lbrace p\\in \\mathcal {D}_R(L)\\colon p$ strictly decreasing on$[0,R)\\rbrace $ .",
"The functions $p\\in \\mathcal {D}_R^-(L)$ are solutions of a nonlinear integral equation, as the following lemma shows.",
"Figure: for F(h)=p∘(Lp) -1 (h)F(h)=p\\circ (Lp)^{-1}(h), (E 0 :=Lp(R)E_0:=Lp(R)).$~$ Let $p\\in \\mathcal {D}_R^-(L)$ .",
"Then there exists a unique strictly increasing function $F:=F[p]\\colon \\big [Lp(R), Lp(0)\\big )\\rightarrow p\\big ((0,R]\\big )$ such that $p(r)=F\\circ Lp(r), \\qquad r\\in (0,R].$ Proof.",
"Lemma 3) says that $Lp\\colon (0,R]\\rightarrow \\big [Lp(R), Lp(0)\\big )$ is strictly decreasing and has a strictly decreasing inverse $(Lp)^{-1}\\colon \\big [Lp(R), Lp(0)\\big )\\rightarrow (0,R].$ Because $p$ is strictly decreasing on $(0,R]$ , the composition $F:=p\\circ (Lp)^{-1}\\colon \\big [Lp(R), Lp(0)\\big ) \\rightarrow p((0,R])$ exists and is strictly increasing (see Diagram 2.1) Then $F(h)=p\\circ (Lp)^{-1}(h),\\qquad h\\in [Lp(R),Lp(0))$ implies $F\\circ Lp(r)=p(r), \\qquad r\\in (0,R].$ The uniqueness of $F$ is immediate: If $G$ satisfies $p=G\\circ Lp$ , then $G=p\\circ (Lp)^{-1}=F$ .",
"$\\square $ Let $F(h):=0$ for $h\\in (0,Lp(R))$ .",
"Then under the conditions of Lemma we have $p(r)=F\\circ Lp(r) \\ \\text{on} \\ (0,R]\\longleftrightarrow p(r)=F\\circ Lp(r) \\ \\text{on} \\ \\mathbb {R}_{+}.$ Proof.",
"Let $p(r)=F\\circ Lp(r)$ on $(0,R]$ .",
"In this case, if $R<r$ , then $p(r)=0$ , and $Lp(R)>Lp(r)$ implies that $F\\circ Lp(r)=0$ , i. e. $p(r)=F\\circ Lp(r)$ on $\\mathbb {R}_+$ .",
"The inverse statement is trivial.$\\square $ Often we use the abbreviation $P:=Lp$ ."
],
[
"Stationary spherically symmetric solutions depending\non the local energy and their properties",
"A triple $(f,\\rho ,U)$ of functions $f=f(r,u)$ , $\\rho =\\rho (r)$ , $U=U(r)$ is called a stationary spherically symmetric $E$ -dependent solution of the (VPS) if $\\rho \\in \\mathcal {D}_R^-(L)$ and there exists a function $q=q(s)$ with the following property: $&(Q)\\quad q\\in L_{\\rm loc}^1(\\mathbb {R}), \\;\\; q(s)=0 \\;\\; \\text{for} \\ s\\in (-\\infty ,0], \\quad q(s)>0 \\;\\; \\text{for} \\; s\\in (0,P(0)\\!-\\!E_0)\\\\& \\text{such that}\\\\&(V)\\quad f(r,u)=q\\big (-E(r,u)-E_0\\big ),\\\\&\\qquad \\;\\; E(r,u):=U(r)+\\frac{u^2}{2}, \\ u:=|v|,\\\\&\\qquad \\;\\; (\\text{we write} \\ f=f_q),\\\\&(P)\\quad U(r)=-L\\rho (r), \\ L\\rho (R)=:E_0,\\\\&(D)\\quad \\rho (r)=\\int _{\\mathbb {R}^3}f(r,|v|)\\, dv, \\qquad r\\in \\mathbb {R}_{+}.$ We note that $(Q)$ states the properties of $q$ , $(V)$ refers to $f\\,^{\\prime }s$ being an integral of Vlasov's equation (i.e., being constant along the characteristics), $(P)$ is the integrated form $(P_2$ ) of Poisson's equation, $(D)$ is the definition of the local density.",
"As a preparation for the following important lemma we prove a crucial identity.",
"Let $p\\in \\mathcal {D}(L)$ , $E_0>0$ and $q$ satisfy $(Q)$ .",
"Then for $E_0\\le h<Lp(0)$ the following equality holds: $\\int _{\\mathbb {R}^3}\\!q\\left(h-E_0-\\frac{v^2}{2}\\right)dv= 4\\pi \\sqrt{2}\\int _0^{h-E_0}\\!\\!\\!q(s)\\sqrt{h-E_0-s}\\,ds.$ Proof.",
"Since $q(s)=0$ for $s\\in (-\\infty ,0]$ , we have $\\int _{\\mathbb {R}^3}\\!\\!q\\left(h-E_0-\\frac{v^2}{2}\\right)dv&= \\int _{|v|<\\sqrt{2(h-E_0)}}\\;q\\left(h-E_0-\\frac{v^2}{2}\\right)dv\\\\&=\\int _0^\\pi \\!\\!\\int _0^{2\\pi }\\!\\!\\int _0^{\\sqrt{2(h-E_0)}}\\!\\!\\!q\\left(h-E_0\\!-\\!\\frac{u^2}{2}\\right) u^2\\,du\\, d\\varphi \\sin \\psi \\,d\\psi \\\\&=4\\pi \\int _0^{\\sqrt{2(h-E_0)}}\\!\\!q\\left(h-E_0-\\frac{u^2}{2}\\right)u^2\\,du.$ Passing to the new variable $s:=h-E_0-\\dfrac{u^2}{2}$ , we have $u=\\sqrt{2(h-E_0-s)}$ .",
"Hence we get (REF ).$\\square $ (Equivalence Lemma) (a) Let $(f_q,\\rho , U)$ be a stationary spherically symmetric $E$ -depending solution of the (VPS).",
"Let $F:=F[p\\,]$ (Lemma ), where $p=\\rho $ .",
"Then $F(h)=4\\pi \\sqrt{2}\\int _0^{h-E_0}\\!q(s)\\sqrt{h-E_0-s}\\,ds \\quad \\text{for} \\quad h\\in \\big [E_0, P(0)\\big ).$ (b) Let $q$ satisfy $(Q)$ , and let $F(h):=4\\pi \\sqrt{2}\\int _0^{h-E_0}\\!\\!q(s)\\sqrt{h-E_0-s}\\,ds \\quad \\text{for} \\quad h\\in \\big [E_0, P(0)\\big ).$ Assume the integral equation $p(r)=F\\circ Lp(r)$ has a solution $p\\in \\mathcal {D}_R^-(L)$ on $\\mathbb {R}_{+}$ .",
"We define $\\rho :=p$ , $U(r):=-L\\rho (r)$ , $E_0:=Lp(R)$ , and $f_q(r,u):=q(-E(r,u)-E_0)$ .",
"Then $(f_q,\\rho ,u)$ is a stationary spherically symmetric $E$ -depending solution of the (VPS).",
"Proof.",
"$~$ a) Let $(f_q,\\rho , U)$ be a stationary spherically symmetric $E$ -depending solution.",
"Then, by virtue of Lemma , $(D)$ , $(V)$ , and (REF ), we have $F\\circ L\\rho (r)=\\rho (r)&=\\int _{\\mathbb {R}^3}\\!f_q\\big (r, |v|\\big )\\,dv=\\int _{\\mathbb {R}^3}\\!q\\left(L\\rho (r)-E_0-\\frac{v^2}{2}\\right)dv\\\\&= 4\\pi \\sqrt{2}\\int _0^{L\\rho (r)-E_0}\\!\\!q(s)\\sqrt{L\\rho (r)-E_0-s}\\,ds \\quad \\text{for}\\quad r\\in (0,R],$ and (REF ) follows.",
"b) Our assumptions imply that $(Q)$ , $(P)$ , $(V)$ are satisfied.",
"Furthermore, by virtue of (REF ), (REF ), (REF ), and $(V)$ , we have $\\rho (r)=p(r)=F\\circ Lp(r)&=4\\pi \\sqrt{2}\\int _0^{Lp(r)-E_0}\\!\\!q(s) \\sqrt{Lp(r)-E_0-s}\\, ds\\\\&= \\int _{\\mathbb {R}^3} q\\left(Lp(r)-E_0-\\frac{v^2}{2}\\right)dv\\\\&=\\int _{\\mathbb {R}^3} \\!f(r,u)\\, dv, \\qquad r\\in (0,R].$ Hence $(D)$ is also satisfied.$\\square $ In the sequel, we will use the definition $F_0(h):=F(h+E_0) \\quad \\text{on} \\quad [\\,0,P(0)-E_0).$ Then (REF ) has the form $F_0(h)=4\\pi \\sqrt{2}\\int _0^h q(s)\\sqrt{h-s} \\, ds, \\quad h\\in [0, P(0)-E_0).$ This is an equation of the form $g(x)=\\int _0^x\\!\\!f(s)\\sqrt{x-s} \\, ds,$ which is called Eddington's equation.",
"The results on its solvability are based on the theory for an equation of the form $g(x)=\\int _0^x\\!\\!\\frac{f(s)}{\\sqrt{x-s}} \\, ds,$ which is called Abel's equation.",
"It was Tonelli [12], who has given existence proofs for these equations (a review of his work is given in [6]).",
"For the sake of the completeness of the present work, the main results and their proofs are given in the Appendix."
],
[
"The inverse problem",
"In this section we consider and solve the following question: Given a function $p\\in \\mathcal {D}_R^-(L)$ , under which conditions is $p$ the local density of a stationary spherically symmetric $E$ -dependent solution?",
"In this case we say “$p$ is extendable” (by $f$ and $U$ to a stationary spherically symmetric $E$ -dependent solution).",
"The following proposition gives a first necessary and sufficient criterion that a given $p\\in \\mathcal {D}_R^-(L)$ is extendable.",
"Let $p\\in \\mathcal {D}_R^-(L)$ .",
"Then $p$ is extendable if and only if Eddington's equation (REF ) has a solution $q$ with $(Q)$ for $F:=F[p]$ from Lemma and $F_0(h):=F(h+E_0)$ .",
"Proof.",
"Necessity: If $p$ is extendable, then there exists $q$ with $(Q)$ such that $f(r,u)&=q\\left(-U(r)-E_0+\\frac{u^2}{2}\\right), &(V)&\\qquad \\text{with}\\\\U(r)&=-Lp(r),\\quad E_0=Lp(R), &(P)& ,\\\\p(r)&=\\int _{\\mathbb {R}^3} f(r,u)\\,dv.",
"&(D)&$ Lemma part (a) shows then that Eddington's equation (REF ) has the solution $q$ with $(Q)$ .",
"Sufficiency: If Eddington's equation (REF ) has a solution $q$ with $(Q)$ for $F:=F[p]$ , then $f:=f_q$ satisfies $(V)$ with $U(r):=-Lp(r)$ $(P)$ and $E_0=Lp(R)$ .",
"Therefore, by virtue of $(V)$ , (REF ), and (REF ) we have $\\int _{\\mathbb {R}^3}\\!\\!f_q\\big (r,|v|\\big )\\, dv=\\int _{\\mathbb {R}^3}\\!\\!q\\left(Lp(r)-E_0-\\frac{v^2}{2}\\right)dv\\\\=4\\pi \\sqrt{2}\\int _0^{Lp(r)-E_0}\\!\\!\\!q(s) \\sqrt{Lp(r)-E_0-s}\\, ds=F\\circ Lp(r)=p(r),$ that is, $(D)$ is also valid, and $p$ is extendable.$\\square $ In the next theorem, we investigate the solvability of Eddington's equation in the form $F_0(h)=4\\pi \\sqrt{2}\\int _0^h q(s)\\sqrt{h-s}\\, ds$ for given $F_0(h):=F(h+E_0)$ , $F:=F[p]$ , in more detail.",
"The following theorem gives different conditions of extendability for a function $p\\in \\mathcal {D}_R^-(L)$ in explicit form.",
"The spaces $L_{\\rm loc}^1[0,T]$ and $AC[0,T)$ are defined in the Appendix.",
"Let $p\\in \\mathcal {D}_R^-(L)$ , $p\\,\\big |_{(0,R]}\\in C^2(0,R]$ , $F:=F[p]$ , $E_0:=P(R)$ and $F_0(\\,\\cdot \\,)=F(\\,\\cdot \\,+E_0)$ .",
"Then the following statements hold.",
"1) Eddington's equation $F_0(h)=4\\pi \\sqrt{2}\\int _0^h\\!q(s)\\sqrt{h-s}\\,ds, \\qquad 0\\le h<P(0)-E_0$ has a unique real-valued solution $q\\in L_{\\rm loc}^1[0,P(0)-E_0)$ , which is given by $q(h):=\\frac{1}{4\\pi \\sqrt{2}}\\,\\frac{2}{\\pi }\\,\\frac{d}{dh}\\, H_{F^{\\prime }_0}(h), \\qquad 0\\le h<P(0)-E_0,$ where $H_{F_0^{\\prime }}(h):=\\int _0^h\\frac{F_0^{\\prime }(s)}{\\sqrt{h-s}}\\,ds \\;\\;\\;\\text{lies in}\\;\\; AC\\big [0, P(0)-E_0\\big ), \\quad F_0\\in C^2\\big [0, P(0)-E_0\\big ).$ 2) $p$ is extendable if and only if $q>0$ on $(0, P(0)-E_0)$ , that is ${\\begin{array}{c}\\frac{d}{dh}\\,H_{F_0^{\\prime }}(h)=\\frac{1}{\\sqrt{h}}\\,F_0^{\\prime }(0)+\\int _0^h\\frac{F_0^{\\prime \\prime }(s)}{\\sqrt{h-s}}\\,ds>0 \\quad \\text{on} \\quad \\big (0,P(0)-E_0\\big )\\\\[0ex]\\end{array}}$ $\\left(F_0^{\\prime }(0)=\\frac{p^{\\prime }(R)}{P^{\\prime }(R)}\\ge 0\\right)$ .",
"3) Sufficient conditions for the extendability of $p$ are: (a) $F_0^{\\prime \\prime }(s)>0$ on $\\big (0, P(0)-E_0\\big )$ , (b) $X(r):=p^{\\prime }(r)\\cdot P^{\\prime \\prime }(r)-p^{\\prime \\prime }(r)P^{\\prime }(r)>0$ on $(0,R)$ , (c) $\\dfrac{2}{r}\\,p^{\\prime }(r)+p^{\\prime \\prime }(r)>0$ on $(0,R)$ , where (a) and (b) are equivalent and (c) implies (a) and (b).",
"Proof.",
"The assumptions on $p$ and Lemma imply that $P\\colon (0,R]\\rightarrow [E_0,P(0))$ is a strictly decreasing bijection in $C^2(0,R]$ with strictly decreasing inverse $P^{-1}\\colon [E_0, P(0))\\rightarrow (0,R]$ in $C^2[E_0,P(0))$ .",
"The composition with $p\\in C^2(0,R]$ : $F:=p\\circ P^{-1}\\colon [E_0,P(0))\\rightarrow [0,p(0))$ is strictly increasing and $F\\in C^2[E_0,P(0))$ , $F_0\\in C^2[0, P(0)-E_0)$ , $F_0^{\\prime }(\\,\\cdot \\,)=F^{\\prime }(\\,\\cdot \\,+E_0)$ , and $F_0^{\\prime \\prime }(\\,\\cdot \\,)=F^{\\prime \\prime }(\\,\\cdot \\,+E_0)$ .",
"1) To show that (REF ) has a unique real-valued solution $q\\in L_{\\rm loc}^1[0,P(0)-E_0)$ , we need to verify that for $g:=\\dfrac{F_0}{4\\pi \\sqrt{2}}$ the assumptions (a), (i) and (ii) of Lemma (in the Appendix) are satisfied.",
"It is sufficient to do this for $g:=F_0$ .",
"Obviously, $F_0\\in C^2[0, P(0)-E_0)\\subset AC[0,P(0)-E_0)$ and $F_0(0)=F(E_0)=pP^{-1}(E_0)=p(R)=0.$ For $H_{F_0^{\\prime }}(h):=\\int _0^h\\frac{F_0^{\\prime }(s)}{\\sqrt{h-s}}\\, ds$ (a) (i) means that we have to show $H_{F_0^{\\prime }}\\in AC[0, P(0)-E_0)$ .",
"We observe that we have $F_0^{\\prime }\\in C^1[0, P(0)-E_0)$ and that $F_0^{\\prime }(0)=F^{\\prime }(E_0)=p^{\\prime }\\big (P^{-1}(E_0)\\big )\\cdot (P^{-1})^{\\prime }(E_0)= \\frac{p^{\\prime }(R)}{P^{\\prime }(R)}\\ge 0.$ Integrating by parts, we get $H_{F_0^{\\prime }}(h)&=-2\\sqrt{h-s}\\, F_0^{\\prime }(s)\\big |_0^h+2\\int _0^h\\!\\!F_0{^{\\prime \\prime }}(s)\\sqrt{h-s}\\, ds\\\\&= 2\\sqrt{h}\\, F_0^{\\prime }(0)+2\\int _0^h\\!\\!F_0^{\\prime \\prime }(s)\\sqrt{h-s}\\,ds,$ and $H_{F_0^{\\prime }}\\in AC[0,P(0)-E_0)$ follows, i.e.",
"(i) is satisfied.",
"Also we get $H_{F_0^{\\prime }}(0)=0$ , which is (a) (ii).",
"It follows from Lemma A.5 (a) that (REF ) has a unique realvalued solution $q$ which is given by $q(h):=\\frac{1}{4\\pi \\sqrt{2}}\\,\\frac{2}{\\pi }\\,\\frac{d}{dh}\\, H_{F_0^{\\prime }}(h)= \\frac{1}{4\\pi \\sqrt{2}}\\,\\frac{2}{\\pi }\\left[\\frac{1}{\\sqrt{h}}\\, F_0^{\\prime }(0)+\\int _0^h\\!\\!\\frac{F_0^{\\prime \\prime }(s)}{\\sqrt{h-s}}\\,ds\\right].$ 2) Since $H_{F_0^{\\prime }}\\in AC\\big [0, P(0)-E_0\\big )$ , we have $q\\in L_{\\rm loc}^1[0, P(0)-E_0)$ .",
"By Theorem REF , it satisfies $(Q)$ and $p$ is extendable if and only if $q>0$ on $(0, P(0)-E_0)$ .",
"3) The proof of 3) is based on a change of variables in the integral $\\int _0^h\\!\\!\\frac{F_0^{\\prime \\prime }(s)}{\\sqrt{h-s}}\\, ds.$ We define a $C^2$ -diffeomorphism $\\Phi $ $\\Phi \\colon \\big [0, P(0)-E_0\\big )\\rightarrow (0,R]$ as the composition of the shift $T\\colon \\big [0, P(0)-E_0\\big )\\rightarrow \\big [E_0, P(0)\\big ), \\qquad s\\mapsto s+E_0,$ with $P^{-1}\\colon \\big [E_0, P(0)\\big )\\rightarrow (0,R], \\qquad s\\mapsto P^{-1}(s),$ that is, $\\Phi :=P^{-1}\\circ T$ , $s\\mapsto r=P^{-1}(s+E_0)$ (see Diagram REF ).",
"Figure: Action of Φ\\Phi and Ψ\\Psi The inverse of $\\Phi $ is $\\Psi :=T^{-1}\\circ P$ : $\\Psi \\colon (0,R]\\rightarrow \\big [0, P(0)-E_0\\big ), \\qquad r\\mapsto s=P(r)-E_0.$ We now represent $F_0^{\\prime }(s)$ , $F_0^{\\prime \\prime }(s)$ as functions of $p^{\\prime }(r)$ , $p^{\\prime \\prime }(r)$ , $P^{\\prime }(r)$ , $P^{\\prime \\prime }(r)$ as follows: for $s=\\Psi (r)$ , we have $&F_0(s)=F_0\\big (\\Psi (r)\\big )=F\\big (P(r)-E_0+E_0\\big )=F\\big (P(r)\\big )=pP^{-1}\\big (P(r)\\big )=p(r),\\\\&\\frac{d}{dr}\\, F_0\\big (\\Psi (r)\\big )=F^{\\prime }\\big (P(r)\\big )\\cdot P^{\\prime }(r)=p^{\\prime }(r).$ Since $P^{\\prime }(r)<0$ , we obtain for $s=\\Psi (r)$ $F_0^{\\prime }(s)&=F_0^{\\prime }\\big (\\Psi (r)\\big )=F^{\\prime }\\big (P(r)\\big )=\\frac{p\\,^{\\prime }(r)}{P^{\\prime }(r)},\\\\\\frac{d}{dr}\\,F_0^{\\prime }\\big (\\Psi (r)\\big )&=\\frac{d}{dr}\\,F^{\\prime }\\big (P(r)-E_0+E_0\\big )=\\frac{d}{dr}F^{\\prime }\\big (P(r)\\big )\\\\&= F^{\\prime \\prime }\\big (P(r)\\big )\\cdot P^{\\prime }(r)=\\frac{d}{dr}\\,\\frac{p\\,^{\\prime }(r)}{P^{\\prime }(r)}\\\\&=\\frac{P^{\\prime }(r)\\,p\\,^{\\prime \\prime }(r)-p\\,^{\\prime }(r)\\cdot P^{\\prime \\prime }(r)}{P^{\\prime }(r)^2}.$ Hence, for $s=\\Psi (r)$ , we have $F_0^{\\prime \\prime }(s)=F_0^{\\prime \\prime }\\big (\\Psi (r)\\big )=F^{\\prime \\prime }\\big (P(r)\\big )=\\frac{p\\,^{\\prime }(r)P^{\\prime \\prime }(r)-p\\,^{\\prime \\prime }(r)P^{\\prime }(r)}{\\big |P^{\\prime }(r)\\big |^3}\\,.$ Changing coordinates $s\\rightarrow r$ in the integral of (REF ) by $\\Phi (s):=P^{-1}\\circ T(s)=P^{-1}(s+E_0)=r$ , $\\Phi (0)=R$ , $\\Phi (P(0)-E_0)=0$ , $\\Phi (h)=P^{-1}\\circ T(h)=P^{-1}(h+E_0)$ , $ds=P^{\\prime }(r)\\,dr$ , and observing the negative sign of $P^{\\prime }(r)$ , we get $\\int _0^h\\!\\!\\frac{F_0^{\\prime \\prime }(s)}{\\sqrt{h-s}}\\, ds=\\int _{P^{-1}(h+E_0)}^R\\!\\frac{p\\,^{\\prime }(r)P^{\\prime \\prime }(r)-p\\,^{\\prime \\prime }(r)P^{\\prime }(r)}{\\big |P^{\\prime }(r)\\big |^3\\sqrt{h-\\big (P(r)-E_0\\big )}}\\,\\big |P^{\\prime }(r)\\big |\\,dr,$ and 3) (a) or 3) (b) imply $q>0$ in view of (REF ), (REF ).",
"Let 3) (c) hold.",
"Then, by virtue of Lemma , we have $p\\,^{\\prime }P^{\\prime \\prime }-p\\,^{\\prime \\prime }P^{\\prime }&=p\\,^{\\prime }\\left(-\\frac{2}{r}\\,P^{\\prime }-4\\pi p\\right)-p\\,^{\\prime \\prime }P^{\\prime }\\\\&=4\\pi p(-p\\,^{\\prime })+(-P^{\\prime }) \\left(\\frac{2}{r} \\,p\\,^{\\prime }+p\\,^{\\prime \\prime }\\right)>0,$ and 3) (b) is fulfilled.$\\square $ For later numerical calculations it is useful to write (REF ) in another form.",
"Because $\\Phi (h)=P^{-1}(h+E_0)$ , we have $P\\big (\\Phi (h)\\big )=h+E_0$ and therefore $h=P\\big (\\Phi (h)\\big )-P(R),$ and with (REF ) and (REF ) we get for (REF ) $\\frac{d}{dh}\\,H_{F_0^{\\prime }}(h) =\\frac{1}{\\sqrt{P\\big (\\Phi (h)\\big )-P(R)}}\\\\\\times \\left(\\frac{p\\,^{\\prime }(R)}{P^{\\prime }(R)}+\\int _{\\Phi (h)}^R\\!\\!\\frac{p\\,^{\\prime }(r)P^{\\prime \\prime }(r)-p\\,^{\\prime \\prime }(r)P^{\\prime }(r)}{\\big |P^{\\prime }(r)\\big |^2}\\sqrt{\\frac{P\\big (\\Phi (h)\\big )\\!-\\!P(R)}{P\\big (\\Phi (h)\\big )\\!-\\!P(r)}}\\, dr\\right)$ The integrand has a singularity at $r=\\Phi (h)$ .",
"Examples in the following section will show that the conditions 3) (a), (b), and (c) are not necessary for the extendability.",
"$~$"
],
[
" Examples",
"This example allows to compute explicitly the other functions involved in the theory: $P$ , $P^{-1}$ , $F$ , $q$ , $G_0=:F_0^{-1}$ (which is the right hand side of the approximating nonlinear system (REF ) occurring in Section ).",
"For $0\\le r\\le R$ , we have $p\\,^{\\prime }(r)=-\\frac{2}{R^2}\\,r, \\qquad p\\,^{\\prime \\prime }(r)=-\\frac{2}{R^2},$ hence $\\dfrac{2}{r}\\, p\\,^{\\prime }(r)+p\\,^{\\prime \\prime }=-\\dfrac{6}{R^2}<0$ .",
"Therefore the sufficient condition 3) (c) in Theorem does not hold.",
"Substituting $p$ into (REF ), we obtain $P(r)&=4\\pi \\left[\\frac{1}{r}\\int _0^r\\left(1-\\frac{s^2}{R^2}\\right)s^2\\,ds+\\int _r^R\\!\\left(1-\\frac{s^2}{R^2}\\right)s\\,ds\\right]\\\\&=4\\pi \\left[\\frac{r^2}{3}-\\frac{1}{5}\\,\\frac{r^4}{R^2}+\\frac{1}{2}\\big (R^2-r^2\\big ) -\\frac{1}{4}\\,\\frac{1}{R^2}\\big (R^4-r^4\\big )\\right]\\\\&=4\\pi R^2\\left[\\frac{1}{20}\\left(\\frac{r}{R}\\right)^4\\!-\\!\\frac{1}{6}\\left(\\frac{r}{R}\\right)^2\\!+\\!\\frac{1}{4}\\right] =\\pi R^2\\left[\\frac{1}{5}\\left(\\frac{r}{R}\\right)^4-\\frac{2}{3}\\left(\\frac{r}{R}\\right)^2\\!+\\!1\\right]\\!:=\\!",
"h,\\\\P^{\\prime }(r)&=4\\pi R^2\\left[\\frac{1}{5}\\,\\frac{r^3}{R^4}-\\frac{1}{3}\\,\\frac{r}{R^2}\\right],\\\\P^{\\prime \\prime }(r)&=4\\pi R^2\\left[\\frac{3}{5}\\,\\frac{r^2}{R^4}-\\frac{1}{3}\\,\\frac{1}{R^2}\\right],\\\\P(R)&=E_0=\\frac{8}{15}\\,\\pi R^2$ and it follows $X(r)&=p\\,^{\\prime }P^{\\prime \\prime }-p\\,^{\\prime \\prime }P^{\\prime }=-\\frac{2}{R^2}\\,r\\cdot 4\\pi R^2\\left[\\frac{3}{5}\\, \\frac{r^2}{R^4}-\\frac{1}{3}\\, \\frac{1}{R^2}\\right]\\\\&\\qquad \\qquad \\qquad \\quad \\;\\;+ \\frac{2}{R^2}\\cdot 4\\pi R^2\\left[\\frac{1}{5}\\, \\frac{r^3}{R^4}-\\frac{1}{3}\\,\\frac{r}{R^2}\\right]=-\\frac{16\\pi }{5}\\,\\frac{r^3}{R^4}<0,$ and sufficient condition 3) (b) in Theorem is not fulfilled.",
"Since $P(r)=h$ and $P\\colon [0,R]\\rightarrow \\big [E_0, P(0)\\big ]=\\Big [\\dfrac{8}{15}\\,\\pi R^2, \\pi R^2\\Big ]$ is a biquadratic form in $\\dfrac{r}{R}$ , we can calculate $P^{-1}$ : $\\left(\\frac{r}{R}\\right)^2&=\\frac{5}{3}-\\sqrt{\\frac{25}{9}-5+\\frac{5}{\\pi R^2}\\,h}\\,,\\\\r&=P^{-1}(h)=R\\sqrt{\\frac{5}{3}-\\sqrt{\\frac{5}{\\pi R^2}\\,h-\\frac{20}{9}}}\\,.$ Hence we obtain $F(h)&=p( P^{-1}(h))=1-\\left(\\frac{P^{-1}(h)}{R}\\right)^2=1-\\left(\\frac{5}{3}-\\sqrt{\\frac{5}{\\pi R^2}\\,h-\\frac{20}{9}}\\,\\right)\\\\&=\\sqrt{\\frac{5}{\\pi R^2}\\,h-\\frac{20}{3}}-\\frac{2}{3}\\,, \\qquad h\\in \\big [E_0, P(0)\\big ],\\\\F_0(h)&=F(h+E_0)=\\sqrt{\\frac{5}{\\pi R^2}\\left(h+\\frac{8}{15}\\,\\pi R^2\\right)-\\frac{20}{9}}-\\frac{2}{3}\\\\&=\\sqrt{ah+\\frac{4}{9}}-\\frac{2}{3}\\,, \\quad \\text{with} \\quad a:=\\frac{5}{\\pi R^2}, \\ h\\in \\big [0,P(0)-E_0\\big ].$ We have $&F_0(0)=0,\\qquad F_0(P(0)-E_0)=F_0\\Big (\\dfrac{7}{15}\\,\\pi R^2\\Big )= \\sqrt{\\dfrac{7}{3}+\\dfrac{4}{9}}-\\dfrac{2}{3}\\,=1,\\\\&F_0^{\\prime }(h)=\\dfrac{a}{2}\\dfrac{1}{\\sqrt{ah+\\dfrac{4}{9}}},\\\\&F_0^{\\prime }(0)=\\dfrac{3}{4}\\,a,\\qquad F_0^{\\prime }(P(0)-E_0)=\\dfrac{3a}{10},\\\\&F_0^{\\prime \\prime }(h)=-\\dfrac{a^2}{4}\\dfrac{1}{\\Big (\\sqrt{ah+\\dfrac{4}{9}}\\Big )^3}.$ From [5] it follows that $H_{F_0^{\\prime }}(h)&=\\int _0^h\\!\\!\\frac{F_0^{\\prime }(s)}{\\sqrt{h-s}}\\,ds=\\frac{a}{2}\\int _0^h\\!\\!\\frac{ds}{\\sqrt{as+\\dfrac{4}{9}}\\,\\sqrt{h-s}}\\\\&= \\frac{a}{2}\\,\\frac{2}{\\sqrt{a}}\\,\\mathop {\\rm arctg}\\sqrt{\\dfrac{(as+4/9)}{a(h-s)}}\\,\\Biggl |_{s=0}^{s=h}=\\sqrt{a}\\left(\\frac{\\pi }{2}-\\mathop {\\rm arctg}\\left(\\frac{2}{3}\\,\\frac{1}{\\sqrt{a}}\\dfrac{1}{\\sqrt{h}}\\right)\\right).$ Differentiating the last expression, we obtain $\\frac{d}{dh}\\,H_{F_0^{\\prime }}(h)&=-\\frac{\\sqrt{a}}{1+\\dfrac{4}{9}\\, \\dfrac{1}{a}\\,\\dfrac{1}{h}}\\,\\frac{2}{3}\\cdot \\frac{1}{\\sqrt{a}}\\,\\Big (-\\frac{1}{2}\\Big )\\,\\frac{1}{h^{3/2}}\\\\&=\\frac{1}{3}\\,\\dfrac{1}{h+\\dfrac{4}{9}\\,\\dfrac{1}{a}}\\cdot \\frac{1}{\\sqrt{h}}>0, \\qquad h\\in (0,P(0)-E_0).$ We conclude that $H_{F_0^{\\prime }}$ is in $AC\\big [0,P(0)-E_0\\big )$ , $H_{F_0^{\\prime }}(0)=0$ and $\\dfrac{d}{dh} H_{F_0^{\\prime }}(h)>0$ on $(0, P(0)-E_0)$ .",
"Hence, by virtue of Theorem 1) and 2), $p$ is extendable and $q(h):=\\frac{1}{4\\pi \\sqrt{2}}\\,\\frac{2}{\\pi }\\, \\frac{d}{dh}\\, H_{F_0^{\\prime }}(h)=\\frac{\\sqrt{2}}{4\\pi ^2}\\, \\frac{1}{3}\\,\\frac{1}{h+\\dfrac{4}{9}\\,\\dfrac{1}{a}}\\cdot \\frac{1}{\\sqrt{h}}>0$ is the unique solution of Eddington's equation (REF ).",
"This example illustrates as well the inverse problem (with known $p$ ) as the direct problem (with known $q$ — in Section ).",
"It will be expanded further in Section .",
"$p(r):={\\left\\lbrace \\begin{array}{ll}\\Big (1-\\dfrac{r}{R}\\Big )^2, \\ &0\\le r\\le R,\\\\0, \\ & R< r, \\; R>0.\\end{array}\\right.",
"}$ This is an example, which allows to decide about its extendability easily by means of Theorem 3) (b).",
"For $0\\le r\\le R$ , we have $p\\,^{\\prime }(r)=-\\frac{2}{R}+2\\,\\frac{r}{R^2}, \\qquad p\\,^{\\prime \\prime }(r)=\\frac{2}{R^2}\\,.$ The function $\\dfrac{2}{r}\\,p\\,^{\\prime }(r)+p\\,^{\\prime \\prime }(r)=\\dfrac{6}{R^2}-\\dfrac{4}{rR}$ changes its sign at the point $\\dfrac{2R}{3}\\in (0,R)$ .",
"Thus Theorem 3) (c) is not applicable.",
"From (REF ) we obtain $P(r)&=4\\pi \\left[\\frac{1}{r}\\int _0^r\\!\\!\\left(s^2-\\frac{2s^3}{R}+\\frac{s^4}{R^2}\\right) ds+\\int _r^R\\!\\!\\left(s-\\frac{2s^2}{R}+\\frac{s^3}{R^2}\\right) ds\\right]\\\\&=4\\pi \\left[\\frac{1}{3}\\,r^2\\!-\\!\\frac{1}{2}\\,\\frac{r^3}{R}\\!+\\!\\frac{1}{5}\\,\\frac{r^4}{R^2}\\!+\\!\\frac{1}{2}\\big (R^2\\!-\\!r^2\\big )\\!-\\!\\frac{2}{3}\\,\\frac{1}{R}\\big (R^3\\!-\\!r^3\\big )+\\frac{1}{4}\\,\\frac{1}{R^2}\\big (R^4\\!-\\!r^4\\big )\\right]\\\\&=4\\pi \\left[-\\frac{1}{6}\\,r^2+\\frac{1}{6}\\,\\frac{r^3}{R}-\\frac{1}{20}\\,\\frac{r^4}{R^2}+\\frac{3}{4}\\,R^2\\right],\\\\[0.3ex]P^{\\prime }(r)&=4\\pi \\left[-\\frac{1}{3}\\,r+\\frac{1}{2}\\,\\frac{r^2}{R}-\\frac{1}{5}\\,\\frac{r^3}{R^2}\\right],\\\\[0.3ex]P^{\\prime \\prime }(r)&=4\\pi \\left[-\\frac{1}{3}+\\frac{r}{R}-\\frac{3}{5}\\,\\frac{r^2}{R^2}\\right],\\\\[2ex]X(r)&=p\\,^{\\prime }(r)P^{\\prime \\prime }(r)-p\\,^{\\prime \\prime }(r)P^{\\prime }(r)\\\\&=4\\pi \\left[\\left(-\\frac{2}{R}\\!+\\!2\\frac{r}{R^2}\\right)\\left(-\\frac{1}{3}\\!+\\!\\frac{r}{R}-\\frac{3}{5}\\,\\frac{r^2}{R^2}\\right)\\!-\\!\\frac{2}{R^2}\\left(-\\frac{1}{3}\\,r+\\frac{1}{2}\\,\\frac{r^2}{R}-\\frac{1}{5}\\,\\frac{r^3}{R^2}\\right)\\right]\\\\&=\\frac{4\\pi }{R}\\left(\\frac{2}{3}-2\\alpha +\\frac{11}{5}\\,\\alpha ^2-\\frac{4}{5}\\,\\alpha ^3\\right)=:\\frac{4\\pi }{R}f(\\alpha ) \\ \\text{with} \\ \\alpha :=\\frac{r}{R}\\le 1.$ On $[0,1]$ we have $f^{\\prime }(\\alpha )=-2+\\dfrac{22}{5}\\,\\alpha -\\dfrac{12}{5}\\,\\alpha ^2$ .",
"It is easy to see that $f$ is decreasing on $\\Big [0,\\dfrac{5}{6}\\Big ]$ , increasing on $\\Big [\\dfrac{5}{6},1\\Big ]$ , and $f(5/6)=\\dfrac{7}{108}$ .",
"Hence $f(\\alpha )>0$ on $[0,1]$ and $X(r)>0$ on $(0,R)$ .",
"Hence, by virtue of Theorem 3) (b) $p$ is extendable.",
"In this case it is not possible to calculate $P^{-1}$ explicitly (as in Example ), because $P(r)$ is a monotone non special polynomial of degree 4. .",
"$~$ $p(r):={\\left\\lbrace \\begin{array}{ll}e^{-r}-e^{-R}, \\ &0\\le r\\le R\\\\0, \\ & R< r, \\; R>0.\\end{array}\\right.",
"}$ We have for $0<r\\le R$ $p\\,^{\\prime }(r)&{\\phantom{:}}=-e^{-r}, \\qquad p\\,^{\\prime \\prime }(r)=e^{-r},\\\\P(r)& :=4\\pi \\left[\\frac{1}{r}\\int _0^r\\!\\!\\big (e^{-s}-e^{-R}\\big )s^2\\,ds+\\int _r^R\\!\\!\\big (e^{-s}-e^{-R}\\big )s\\,ds \\right]\\\\&{\\phantom{:}}=4\\pi \\left[\\frac{1}{r}\\left(e^{-s}\\big (-s^2-2s-2\\big )-e^{-R}\\frac{s^3}{3}\\right)\\bigg |_0^r+\\left(e^{-s}(-s-1)-e^{-R}\\frac{s^2}{2}\\right)\\bigg |_r^R\\,\\right]\\\\&{\\phantom{:}}=4\\pi \\left[-e^{-r}\\left(r+2+\\frac{2}{r}\\right)+\\frac{2}{r}-\\frac{1}{3}\\,r^2e^{-R}-e^{-R}\\left(R+1+\\frac{R^2}{2}\\right) \\right.\\\\&\\left.\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\; +e^{-r}(r+1)+e^{-R}\\,\\frac{r^2}{2}\\right]\\\\&{\\phantom{:}}=4\\pi \\left[-e^{-r}\\left(1+\\frac{2}{r}\\right)+\\frac{2}{r}-e^{-R}\\left(1+R+\\frac{R^2}{2}-\\frac{r^2}{6}\\right)\\right],\\\\P^{\\prime }(r)&{\\phantom{:}}=4\\pi \\left[e^{-r}\\left(1+\\frac{2}{r}+\\frac{2}{r^2}\\right)-\\frac{2}{r^2}+\\frac{1}{3}\\,e^{-R}r\\right],\\\\P^{\\prime \\prime }(r)&{\\phantom{:}}=4\\pi \\left[-e^{-r}\\left(1+\\frac{2}{r}+\\frac{4}{r^2}+\\frac{4}{r^3}\\right)+\\frac{4}{r^3}+\\frac{1}{3}\\,e^{-R}\\right].$ We get on the triangle $\\lbrace (r,R); \\; 0 < R,\\; r \\in (0,R]\\rbrace $ $X(r,R)&=p\\,^{\\prime }(r)P^{\\prime \\prime }(r)-p\\,^{\\prime \\prime }(r)P^{\\prime }(r)=-e^{-r}\\Big (P^{\\prime }(r)+P^{\\prime \\prime }(r)\\big )\\\\&=-4\\pi e^{-r}\\left[e^{-r}\\left(1+\\frac{2}{r}+\\frac{2}{r^2}\\right)-\\frac{2}{r^2}+\\frac{1}{3}\\,e^{-R}r-e^{-r}\\left(1+\\frac{2}{r}+\\frac{4}{r^2}+\\frac{4}{r^3}\\right)\\right.",
"\\\\& \\left.",
"\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;+\\frac{4}{r^3}+\\frac{1}{3}\\,e^{-R}\\right]\\\\&=4\\pi e^{-r}\\left[ \\Big \\lbrace e^{-r}\\left(\\frac{2}{r^2}+\\frac{4}{r^3}\\right)+\\frac{2}{r^2} \\Big \\rbrace -\\Big \\lbrace \\frac{4}{r^3}+\\frac{e^{-R}}{3}\\,(r+1) \\Big \\rbrace \\right]\\\\&=\\;\\;\\;4\\pi \\,\\frac{e^{-r}}{r^3}\\left[ \\lbrace e^{-r}(2r+4)+2r\\rbrace - \\lbrace 4+\\frac{1}{3}\\,e^{-R}(r^4+r^3)\\rbrace \\right].$ Because $\\frac{\\partial X}{\\partial R} (r,R) =\\frac{4\\pi }{3}\\; \\frac{r+1}{e^{r+R}} > 0 \\;\\;\\;\\; \\text{for} \\;\\; 0<R, \\;\\; r\\in (0,R]$ X is strictly increasing in direction of growing values of $R > 0$ and each $r$ fixed in $\\big [0 , R\\big ]$ .",
"It seems that $X, P, P^{\\prime }, P^{\\prime \\prime }$ have singularities at $r=0$ .",
"But they can be repaired, as shown for $X$ in (REF ) and (REF ).",
"For $ P, P^{\\prime }, P^{\\prime \\prime }$ it can be done in the same way.",
"The last formula of $X(r,R)$ is not useful to calculate values of $X(r,R)$ with $r$ near zero, because X is the difference of two large, positive terms, who have nearly equal values.",
"Therefore we make the following transformation: If $\\;\\;\\;\\tilde{X}(r,R):= e^{2r} r^3 (4\\pi )^{-1}\\cdot X \\left(r,R\\right),\\;\\;$ then we get $\\tilde{X} \\left(r,R\\right) & =\\left[ (2r+4) -\\frac{e^{r-R}}{3} r^3 (1+r) + (2r-4) e^r \\right]\\\\& =\\left[ -\\frac{e^{r-R}}{3} r^3 (1+r) + ( 2r+4) + (2r-4) \\left(1+r+ \\frac{r^2}{2!",
"}+ \\frac{r^3}{3!",
"}+ \\frac{r^4}{4!",
"}+ \\sum _{k=5}^{\\infty } \\frac{r^k}{k!}",
"\\right)\\right]\\\\&=\\left[ -\\frac{e^{r-R}}{3} r^3 (1+r)+ (2r+4)+( 2r+2r^2+ \\frac{2r^3}{2!",
"}+ \\frac{2r^4}{3!",
"}+ \\frac{2r^5}{4!}",
") \\right.",
"\\\\&\\left.\\;\\;\\;\\;\\;\\;\\; + (-4-4r- \\frac{4r^2}{2!",
"}- \\frac{4r^3}{3!",
"}- \\frac{4r^4}{4!})",
"+ (2r-4) \\sum _{k=5}^{\\infty } \\frac{r^k}{k!}",
"\\right]\\\\&=\\left[ -\\frac{e^{r-R}}{3} r^3 (1+r) +r^3 ( 1-\\frac{4}{6}) + r^4 ( \\frac{2}{6}- \\frac{1}{6})+ r^5\\frac{2}{4!",
"}+ (2r-4) \\sum _{k=5}^{\\infty } \\frac{r^k}{k!}",
"\\right]\\\\& = r^3 \\left[ -\\frac{e^{r-R}}{3} (1+r) + \\frac{1}{3} + \\frac{r}{6}+ \\frac{r^2}{12}+ (2r-4) r^2 \\sum _{k=5}^{\\infty } \\frac{r^{k-5}}{k!}",
"\\right].$ Dividing $\\tilde{X}$ by $e^{2r} r^3 (4\\pi )^{-1}$ we have $X(r,R)=\\frac{4\\pi }{e^{2r}} \\left[ -\\frac{e^{r-R}}{3} (1+r) + \\frac{1}{3} + \\frac{r}{6}+ \\frac{r^2}{12}+ (2r-4) r^2 \\sum _{k=5}^{\\infty } \\frac{r^{k-5}}{k!}",
"\\right].$ It follows $\\lim \\limits _{r\\rightarrow 0} X(r,R)=4\\pi \\left[ -\\frac{1}{3 e^R} + \\frac{1}{3} \\right] \\in \\left(0,\\frac{4\\pi }{3}\\right] \\;\\;\\; \\text{for} \\; R>0.$ First we calculate $\\; X(r,r)$ $ X(r,r)&=\\frac{4\\pi }{e^{2r}} \\left[ -\\frac{e^{r-r}}{3} (1+r) + \\frac{1}{3} + \\frac{r}{6}+ \\frac{r^2}{12}+ (2r-4) r^2 \\sum _{k=5}^{\\infty } \\frac{r^{k-5}}{k!}",
"\\right]\\nonumber \\\\&=\\frac{4\\pi }{e^{2r}} r \\left[ \\frac{r}{12} - \\frac{1}{6} + (2r-4) \\sum _{k=5}^{\\infty } \\frac{r^{k-4}}{k!}",
"\\right] \\nonumber \\\\& {\\left\\lbrace \\begin{array}{ll}=0 & \\text{\\quad for }r=2, \\\\>0 & \\text{\\quad for }2<r,\\\\<0 & \\text{\\quad for }r<2.\\end{array}\\right.",
"}$ A possible application of Theorem 3) (b) requires the determination of the sign of $X(r,R)$ for $0<r<R$ .",
"We distiguish between two cases $(1)\\; R\\ge 2 \\text{\\quad and \\quad } (2)\\; R < 2$ In case (1) and for $0<r<2$ we know from (REF ) and Chart (REF ) (Diagram REF ), that $ X(r,R)>X(r,2) > 0,$ for $2 \\le r < R$ we have with (REF ) and (REF ) that $ X(r,R) > X(r,r) \\ge 0.",
"$ It follows that for case (1) $R\\ge 2$ we have $X(r,R)>0$ for $0<r<R$ and p is extendable by Theorem 3) (b).",
"In the case (2) with $0< R < 2$ , we know from (REF ) that $X(r,r) < 0$ , on the other hand $X(r,2)>0$ for $r \\in [0,2)$ .",
"By (REF ) there exist a unique zero of $X(r,R)$ between $R$ and 2 for each r fixed in $(0,2)$ .",
"The zeros ly on a monotone curve produced by $X(r,R)=0$ in the neigborhood above the diagonal (see Diagram REF ) : $R(r)=-ln \\left( 6 \\frac{( r+2) e^{- r} + r- 2}{(r+ 1) r^3} \\right) \\;\\; \\text{for}\\;\\; 0<r\\le R<2.$ Therefore the extendability of $p$ can be only decided with Theorem 2) or (REF ) of Remark .",
"The integral is not calculable explicitly and the integrand is singular at the left end of the integration interval.",
"But the integral can be approximated by a Newton-Cotes formula avoiding border points (for instance the Midpoint Rule).",
"The error occurring by using approximation formulas has to be carefully estimated against the preceding term $p\\,^{\\prime }(R)/P^{\\prime }(r)$ to claim the extendability of $p$ by Theorem 2).",
"Such a case will be treated in Example and we leave the details here to the reader.",
"Table: X(r,2))X(r,2)) for 0≤r≤20\\le r\\le 2$~$ Figure: Ranges of positive and negative values of X(r,R)X(r,R)$~$ $p(r):={\\left\\lbrace \\begin{array}{ll}r^{-b}-R^{-b}, \\ &0< r\\le R\\\\0, \\ & R < r, \\; R>0, \\ 0<b<3.\\end{array}\\right.",
"}$ First, we note that, if $b<3$ , then $p(r)\\in \\mathcal {D}_R^-(L)$ .",
"For $0<r\\le R$ ,we have $p\\,^{\\prime }(r)=-br^{-b-1}, \\qquad p\\,^{\\prime \\prime }(r)=b(b+1)r^{-b-2}.$ For $1<b<3$ we get $\\frac{2}{r}\\,p\\,^{\\prime }(r)+p\\,^{\\prime \\prime }(r)=-2br^{-b-2}+b(b+1)r^{-b-2}=b(b-1)r^{-b-2}>0,$ and it follows from Theorem 3) (c) that $p$ is extendable.",
"For $0<b<3$ , $b\\ne 2$ and $0<r<R$ , we have $P(r)&=4\\pi \\left[\\frac{1}{r}\\int _0^r\\!\\!\\big (s^{-b+2}-R^{-b}s^2\\big )\\,ds+ \\int _r^R\\!\\!\\big (s^{-b+1}-R^{-b}s\\big )\\,ds\\right]\\\\&=4\\pi \\left[\\frac{1}{-b+3}\\,r^{-b+2}-\\frac{1}{3}\\,R^{-b}r^2+\\left(\\frac{1}{-b+2}\\,s^{-b+2}-R^{-b}\\frac{s^2}{2}\\right)\\bigg |_r^R\\right]\\\\&=4\\pi \\left[\\frac{r^{-b+2}}{-b+3}-R^{-b}\\frac{r^2}{3}+\\frac{R^{-b+2}}{-b+2}-\\frac{r^{-b+2}}{-b+2}-\\frac{R^{-b+2}}{2}+\\frac{R^{-b}r^2}{2}\\right]\\\\&=4\\pi \\left[\\frac{-r^{-b+2}}{(-b+3)(-b+2)}+\\frac{1}{6}\\,R^{-b}r^2+ \\frac{b}{2(-b+2)}\\,R^{-b+2}\\right].$ For $0<r<R$ , we have $\\frac{X(r)}{4\\pi }&=(-b)r^{-b\\!-\\!1}\\left(\\frac{b-1}{-b+3}\\,r^{-b}\\!+\\!\\frac{1}{3}\\,R^{-b}\\right)\\!-\\!b(b\\!+\\!1)r^{-b-2}\\left(\\frac{-r^{-b+1}}{-b+3}\\!+\\!\\frac{1}{3}\\,R^{-b}r\\right)\\\\&=\\frac{2b}{-b+3}\\,r^{-2b-1}-\\frac{1}{3}(b^2+2b)R^{-b}r^{-b-1}\\\\&=\\frac{R^{-b}r^{-b-1}}{-b+3}\\left(2b\\left(\\frac{R}{r}\\right)^b-\\frac{b^2}{3}(3-b)-\\frac{2b}{3}\\,(3-b)\\right)\\\\&=\\frac{R^{-b}r^{-b-1}}{-b+3}\\cdot b\\cdot \\left(2\\left(\\bigg (\\frac{R}{r}\\bigg )^b-1\\right)+\\frac{1}{3}(b^2-b)\\right).$ For $1\\le b<3, b\\ne 2$ , we get $X(r)>0$ and we conclude from Theorem 3) (b) that $p$ is extendable.",
"For $0<b<1$ , $X$ has a zero at $r_0$ on $(0,R)$ , which is a solution of the equation $\\left(\\frac{R}{r}\\right)^b-1-\\frac{1}{6}\\,(b-b^2)=0.$ It follows $\\frac{r_0}{R}=\\left(\\frac{6}{6+(b-b^2)}\\right)^{\\tfrac{1}{b}}.$ We have $\\dfrac{r_0}{R}=0,9216$ for $b=\\dfrac{1}{2}$ , $\\dfrac{r_0}{R}=1$ for $b=1$ , and $\\lim \\limits _{b\\rightarrow 0}\\dfrac{r_0}{R}\\,(b)=e^{-1/6}\\approx 0,8465$ .",
"The existence of a zero of $X$ requires the application of Theorem 2).",
"We omit the details.",
"$~$ This example is intended to illustrate the direct problem with given $q$ and $p$ calculated approximately in Section .",
"Let $q(s):=c\\sqrt{s}$ , where $c>0$ .",
"Then, changing the variable $s=h\\sin ^2 t$ , we have $F_0(h):=4\\pi \\sqrt{2}\\cdot c\\!\\int _0^h\\!\\!\\!\\sqrt{s}\\, \\sqrt{h-s}\\,ds= \\frac{\\pi ^2c}{\\sqrt{2}}\\, h^2.$ see [5].",
"$~$ $~$ An unextendable $p$ .",
"The aim of this example is to show numerically that not all $p$ , which satisfy the assumptions of Theorem , are automatically extendable.",
"That is, we construct a $p$ that will differ in important details from the extendable examples so far given: whereas all our examples $p$ where either convex or concave on $[0,R]$ , this example will be concave in in the subinterval $[0,w]$ and convex in $[w,R]$ for some $0<w<R$ ($p^{\\prime \\prime }(w)=0$ ).",
"Inequality (REF ) and formula (REF ) will play an important role.",
"We shall construct a function $p(r)=\\left\\lbrace \\begin{array}{lc}a_0+a_1r+a_2r^2+a_3r^3+a_4r^4, & r\\in [0,R],\\\\0, & r>R,\\end{array}\\right.$ such that $p\\in {\\cal D}_R^-(L)$ and $\\dfrac{d}{dh}H_{F^{\\prime }_0}(h)<0$ for some $h\\in (0,P(0)-E_0)$ .",
"Then, by virtue of Theorem 2), $p$ will be not extendable.",
"The condition $p\\in {\\cal D}_R^-(L)$ requires $& p(0)>0,\\\\& p(R)=0,\\\\\\hspace{-79.66771pt} \\text{We add the properties} \\nonumber \\\\& p^{\\prime }(0)=0,\\\\& p^{\\prime }(R)=0,\\\\& p^{\\prime \\prime }(w)=0\\quad \\text{for some}\\, w\\in (0,R).$ The equalities () and (REF ) imply that $F^{\\prime }_0(0)=0$ and this simplifies (REF ).",
"A moments' reflexion shows then that (5.5), (5.6), (5.7), (5.8), (5.9) hold if and only if $a_0>0, \\;\\; a_1=0$ and $&a_2R^2+a_3R^3+a_4R^4=-a_0,\\\\&2a_2R+3a_3R^2+4a_4R^3=0,\\\\&2a_2+6a_3w+12a_4w^2=0.$ $~$ Further we assume that $R=2,\\qquad w=\\dfrac{13}{10},\\qquad a_0=2.$ This choise of $R, w, a_0$ cannot be made arbitrarily.",
"Tests of different values of R show that for $w\\le R/4$ or $3R/4\\le w$ $p$ will never be strictly monotone decreasing whatever $a_0>0$ may be.",
"For $R/4 < w \\le R/2$ p is strictly decreasing, but X is nowhere zero in $(0,R)$ .",
"For $R/2 < w < 3R/4$ $p$ is strictly decreasing and X has a zero in $(0,R)$ .",
"The determinant of linear algebraic system (REF )–() relatively to $a_2, a_3$ , and $a_4$ is not zero.",
"Therefore there exists exactly one solution of system (REF )–() $a_2=-\\dfrac{39}{146},\\qquad a_3=-\\dfrac{107}{146},\\qquad a_4=\\dfrac{45}{146}.$ We now have $& p(r)=2+\\dfrac{1}{146}\\left(-39r^2-107r^3+45r^4\\right),\\\\& p^{\\prime }(r)=\\dfrac{1}{146}\\left(-78r-321r^2+180r^3\\right),\\\\& p^{\\prime \\prime }(r)=\\dfrac{1}{146}\\left(-78-642r+540r^2\\right),$ (see Diagram REF ).",
"Figure: for p,p ' p, p^{\\prime } and p '' p^{\\prime \\prime }$~$ From () it follows that $p^{\\prime }(r)<0$ for $r\\in (0,2)$ .",
"This implies that $p\\in {\\cal D}_R^-(L)$ .",
"In order to calculate $P(r)$ , we remark that it is easy to see that $L(r^l)=4\\pi \\left(-\\dfrac{r^{l+2}}{(l+2)(l+3)}+\\dfrac{R^{l+2}}{l+2}\\right),\\qquad l=0,1,2,\\dots .,$ wich directly implies for $l=0,2,3,4$ the bracket terms in the following formular $\\hspace{-8.5pt}P(r)=Lp(r)\\\\=4\\pi \\biggl [a_0\\left(-\\dfrac{r^2}{6}+\\dfrac{R^2}{2}\\right)+a_2\\left(-\\dfrac{r^4}{20}+\\dfrac{R^4}{4}\\right)+a_3\\left(-\\dfrac{r^5}{30}+\\dfrac{R^5}{5}\\right)+a_4\\left(-\\dfrac{r^6}{42}+\\dfrac{R^6}{6}\\right)\\biggr ]\\\\=4\\pi \\left[-\\dfrac{1}{3}r^2+\\dfrac{39}{2920}r^4+\\dfrac{107}{4380}r^5-\\dfrac{15}{2044}r^6+\\dfrac{558}{365} \\right].$ Calculating $P^{\\prime }(r)$ and $P^{\\prime \\prime }(r)$ and substituting these derivatives into the equality $X(r)=p^{\\prime }(r)P^{\\prime \\prime }(r)-p^{\\prime \\prime }(r)P^{\\prime }(r)$ , we have $X(r)=\\dfrac{4\\pi }{746060}\\bigl [-1093540r^2+1183812r^3-233688r^4\\\\-330515r^5+329025r^6-81000r^7\\bigr ].$ It is easy to see that the function $X=X(r)$ has the zero at $r_0=0$ and $X(1)<0$ , $X(2)>0$ .",
"Therefore there is a zero of $X(r)$ at $r_1$ , $1<r_1<2$ with the approximate value of $r_1\\approx w+0.0575585$ (see Diagram REF ).",
"(Occasionally, we work with equivalent fractions of entire numbers to avoid rounding errors of decimal representations.)",
"Figure: for X,P,P ' X, P, P^{\\prime } and P '' P^{\\prime \\prime }Our aim is to show that there exists a point $h\\in (0, P(0)-E_0)$ such that $\\dfrac{d}{dh}H_{F^{\\prime }_0}(h)<0$ .",
"By virtue of Theorem 2), this means that $p$ is not extendable.",
"Because $p^{\\prime }(r)=0$ , (REF ) shows that the sign of $\\dfrac{d}{dh}H_{F^{\\prime }_0}(h)$ is equal to the sign of the integral on the right side of (REF ).",
"Therefore we have to investigate its integrand $ J(r,h):=\\dfrac{X(r)}{\\left|P^{\\prime }(r)\\right|^2}\\sqrt{\\dfrac{P(\\Phi (h))-P(R)}{P(\\Phi (h))-P(r)} }.$ As it was shown in the proof of Theorem , $ \\Phi :[0,P(0)-E_0)\\rightarrow (0,R], h\\mapsto \\Phi (h)=P^{-1}(h+E_0) $ is a $C^2$ –diffeomorphysm.",
"If $r\\in (\\Phi (h),R)$ , then $P(\\Phi (h))>P(r)>P(R)$ .",
"Hence $P(\\Phi (h))-P(r)<P(\\Phi (h))-P(R)$ , i.e.",
"$\\dfrac{P(\\Phi (h))-P(R)}{P(\\Phi (h))-P(r)} >1, \\quad \\text{and} \\quad \\lim \\limits _{r\\rightarrow \\Phi (h) }\\dfrac{P(\\Phi (h))-P(R)}{P(\\Phi (h))-P(r)}= \\infty ,$ and because $J(R)$ and $X(r)$ always have the same sign, that is, $J(r)<0$ for $\\Phi (h)<r<r_1$ and $J(r)>0$ for $r_1<r<R$ .",
"Since $\\Phi $ is a bijection from $[0,P(0)-E_0)$ onto $(0,R]$ , for a given $\\tilde{r}:=0.01\\in (0,R)$ , there exists a unique $\\tilde{h}\\in [0,P(0)-E_0)$ such that $\\Phi (\\tilde{h})=\\tilde{r}=0.01$ .",
"By virtue of Lemma , $P(r)\\in C^2({\\mathbb {R}}_+)$ .",
"Therefore, using Taylor's formula, we have $P(\\Phi (\\tilde{h}))-P(r)=P^{\\prime }(r+\\theta (\\Phi (\\tilde{h})-r))(\\Phi (\\tilde{h})-r),$ where $0<\\theta <1$ .",
"We denote $c_0:=\\min \\limits _{r\\in [\\tilde{r},R]} |P^{\\prime }(r)|,\\quad c_1:=\\left|P(\\Phi (\\tilde{h}))-P(R)\\right|,\\quad c_2:=\\max \\limits _{r\\in [\\tilde{r},R]} |X(r)|.$ Clearly, $0<c_i<\\infty \\, (i=0,1,2)$ .",
"Then we have $J(r,\\tilde{h})\\le \\dfrac{\\sqrt{c_1}c_2}{c_0^2}\\cdot \\dfrac{1}{\\sqrt{\\Phi (\\tilde{h})-r}}\\,.$ Therefore the function $J(r,\\tilde{h})$ is integrable with respect to $r$ over the interval $(\\Phi (\\tilde{h}),R)$ .",
"Now we are going to show that $\\dfrac{d}{dh}H_{F^{\\prime }_0}(h)|_{h=\\tilde{h}}<0$ by proving the following inequality: $\\int \\limits _{\\Phi (\\tilde{h})}^R \\, J(r,\\tilde{h})\\, dr\\, <0.$ To this end, we introduce two rectangular triangles $T_1:=\\lbrace (0.01;0),(0.01;-4),(0.31;0)\\rbrace \\quad \\text{and}\\quad T_2:=\\lbrace (1.3;0),(2;1.21),(2;0)\\rbrace $ with their hypotenuses $g_1(r)&=\\dfrac{4}{0.3}(r-0.31)\\;\\text{for}\\; r\\in [0.01,0.31]\\;\\;\\text{and}\\;\\;g_2(r)=\\dfrac{1.21}{0.7}(r-1.3)\\;\\text{for}\\; r\\in [1.3, 2]$ (see Diagram REF ).",
"Figure: for integrand JJ, hypotenuses g 1 ,g 2 g_1, g_2 and triangles T 1 ,T 2 T_1, T_2.",
"Table: for g 1 (r)-J(r,h ˜)g_1(r)-J(r,\\tilde{h})Table: for g 2 (r)-J(r,h ˜)g_2(r)-J(r,\\tilde{h})$~$ Numerical calculations show (see Charts REF and REF ) that $ 0>g_1(r)>J(r,\\tilde{h})\\; \\text{for}\\; r\\in [0.01,0.31]\\quad \\text{and}\\quad 0< J(r,\\tilde{h})<g_2(r) \\;\\text{for}\\; r\\in [1.3,2].$ The area of $T_1$ is $0.60$ and that of $T_2$ is $0.4235$ .",
"Therefore, substituting the values $\\Phi (\\tilde{h})=0.01$ and $R=2$ , we obtain $\\int \\limits _{\\Phi (\\tilde{h})}^R \\, J(r,\\tilde{h})\\, dr\\, =\\int \\limits _{0.01}^{0.31} \\, J(r,\\tilde{h})\\, dr\\, +\\,\\int \\limits _{0.31}^w \\, J(r,\\tilde{h})\\, dr\\, +\\,\\int \\limits _{w}^2 \\, J(r,\\tilde{h})\\, dr\\, \\\\\\le \\int \\limits _{0.01}^{0.31} \\, g_1(r)\\, dr\\, +\\, 0\\, +\\,\\int \\limits _{w}^2 \\, g_2(r)\\, dr\\,=-0.60+0.4235=-0.1765<0.$ This finishes the proof of the unextendability of $p$ .",
"Figure: for F 0 ,F 0 ' ,F 0 '' F_0, F_0^{\\prime }, F_0^{\\prime \\prime }It is interesting to compare for $R=2$ $F_0$ , $F_0^{\\prime }$ , $F_0^{\\prime \\prime }$ of Example 5.1 (p extendable) on $h\\in [0,\\;28/15\\pi ]$ and 5.6 (p unextendable) on $h\\in [0,\\; \\approx 10]$ .",
"Since in Example 5.1 $F_0^{\\prime } > 0$ and $F_0\" < 0$ , then $F_0 > 0$ is strictly increasing and $F_0^{\\prime }$ is strictly decreasing.",
"$F_0^{\\prime \\prime }$ is strictly decreasing, as shows its formula.",
"In Example 5.6 it is not possible to calculate $P^{-1}$ explicitly (as in Example), because $P(r)$ is a monotone, nonelemantary polynomial of degree 4.",
"In such a case there still exists the following possibility: The set $\\lbrace ( P(r_k), r_{k}) : 0\\le r_k\\le R; \\, k=1,\\dots ,l ;\\, l>0\\rbrace $ is contained in the graph of $P^{-1}$ , therefore one can approximate $P^{-1}$ by a polynomial of degree $m \\; (l>2m+1)$ with the method of least squares.",
"With this approximation, we can calculate $ F, F_0, F_0^{\\prime }, F_0^{\\prime \\prime }$ (here we omit the calculations).",
"Diagram REF shows $F_0$ and $F_0^{\\prime }, F_0\"$ enlarged with the factor 5, because the value sets of $F_0$ and $F_0^{\\prime }, F_0\"$ are very different.",
"We see $F_0\\ge 0$ and strictly monotone, $F_0^{\\prime }\\ge 0$ but not monotone, and $F_0^{\\prime \\prime } $ is not monotone and changes its sign.",
"$~$"
],
[
"The direct problem and its equivalent formulation",
"We have already mentioned in the introduction in connection with Jeans' theorem, what is called the direct problem: Given a function $q\\in L_{\\rm loc}^1[0,T)$ , $q>0$ on $[0,T)$ for a sufficiently large interval $[0,T)$ — under which conditions do there exist functions $(f_q,\\rho ,U)$ , which form a stationary spherically symmetric $E$ -dependent solution of the (VPS)?",
"A first answer is given by Lemma b): These functions exist if for the function $F(h)&:=4\\pi \\sqrt{2}\\int _0^{h-E_0}\\!\\!\\!q(s)\\sqrt{h-E_0-s}\\,ds, \\qquad h\\in [E_0,P(0)),\\\\F&\\phantom{:}=0 \\quad \\text{on} \\ (-\\infty ,E_0),$ the integral equation $p(r)=F\\circ Lp(r) \\quad \\text{on} \\ \\mathbb {R}_{+}$ has a solution $p\\in \\mathcal {D}_R^-(L)$ .",
"The following Lemma exhibits further properties of the functions involved.",
"We may assume $T>P(0)-E_0$ and define $F_0(h):=F(h+E_0)=4\\pi \\sqrt{2}\\int _0^h\\!\\!q(s)\\sqrt{h-s}\\,ds, \\qquad 0\\le h<P(0)-E_0.$ Then $F_0$ is strictly increasing on $[0, P(0)-E_0),$ and so is $F$ on $[E_0, P(0))$ .",
"The solution $p\\in \\mathcal {D}_R$ of (REF ) such that $Lp(R)=E_0$ is in Proof.",
"It follows from Lemma (b), that $F_0\\in AC[0,P(0)-E_0)$ and $F_0$ is strictly increasing because $F_0^{\\prime }(h)=\\frac{4\\pi }{\\sqrt{2}}\\int _0^h\\!\\!\\frac{q(s)}{\\sqrt{h-s}}\\, ds>0 \\quad \\text{a.e.on} \\ [0, P(0)-E_0).$ We show that $p\\in \\mathcal {D}_R^-(L)$ .",
"If $0<r_1<r_2<R$ , then $Lp(r_1)>Lp(r_2)$ (Lemma 3)) and $p(r_1)=F\\circ Lp(r_1)>F\\circ Lp(r_2)=p(r_2)$ , so that $p$ is strictly decreasing on $(0,R)$ .",
"Using the relations $F(h)=0$ for $h\\in (-\\infty , E_0)$ , $Lp(R)=E_0$ , and $Lp(r)<Lp(R)$ for $r>R$ , we have $p(r)=F\\circ Lp(r)=0$ $(r>R)$ .$\\square $ We see that the solution of the direct problem is closely connected with solving the nonlinear integral equation $p(r)=F\\circ Lp(r) \\ \\text{on} \\ \\mathbb {R}_{+},$ or — what is the same after Corollary — on $(0,R]$ .",
"To make this equation accessible for numerical investigation, we are going to transform it into a slightly different form.",
"Because $F_0\\colon [0, P(0)-E_0)\\rightarrow [0, p(0))$ is a strictly increasing bijection (Lemma ), it has a strictly increasing inverse $G_0\\colon [0,p(0))\\rightarrow [0,P(0)-E_0)$ and $F\\colon [E_0,P(0))\\rightarrow [0,p(0))$ has an inverse $G\\colon [0,p(0))\\rightarrow [E_0,P(0)),$ which has the form $G(h)=G_0(h)+E_0$ .",
"Then we get two equivalent statements in the following Lemma.",
"Let $p\\in \\mathcal {D}_R^-(L)$ as in (REF ), and $Lp(R)=E_0$ .",
"Then $p=F\\circ Lp \\Longleftrightarrow Lp-E_0=G_0p \\ \\text{on} \\ [0,R].$ Proof.",
"$\\Longrightarrow :$ Because $Lp(R)=E_0$ , $r\\le R$ implies $Lp(r)\\ge E_0$ .",
"We apply $G$ to both parts of the equation $p=F\\circ Lp$ .",
"Then we get $Gp=Lp$ or $Lp=G_0p+E_0$ .",
"$\\Longleftarrow :$ If $Lp-E_0=G_0p$ , then $Lp=G_0p+E_0=G(p)$ , and application of $F$ yields $F\\circ Lp=F\\circ Gp=p$ .$\\square $ Now let the functions $q$ and $F$ be as in the beginning of this section.",
"Our aim is to solve the integral equation (REF ), or, what is equivalent according to Lemma : $Lp(r)-Lp(R)=G_0p(r) \\ \\text{on} \\ [0,R].$ Our fist aim is to consider (REF ) at the points $r=R_k$ of the equidistant partition $0=R_0<R_1<R_2<\\dots <\\dots <R_n:=R, \\qquad R_k:=k\\,\\frac{R}{n}\\,, \\quad k=0,\\dots ,n, \\ \\text{of} \\ R.$ To this end, we introduce the space of piecewise linear functions $p_{R_0R_1\\dots R_{n-1}R_n}^{x_0x_1\\dots x_{n-1}x_n}(r):=-\\frac{r-R_k}{R_k-R_{k-1}}\\,x_{k-1}+\\frac{r-R_{k-1}}{R_k-R_{k-1}}\\cdot x_k \\ \\text{on}\\ [R_{k-1}, R_k],$ $k=1,\\dots n$ , with the property $p_{R_0R_1\\dots R_{n-1}R_n}^{x_0x_1\\dots x_{n-1}x_n} (R_k)=x_k, \\qquad k=0,\\dots ,n.$ A basis for this space is the set $\\left\\lbrace p_{R_{k-1}R_kR_{k+1}}^{010}, \\qquad k=0,1,\\dots ,n-1\\right\\rbrace ,$ where for the sake of unified notation, we have defined $p_{R_{k-1}R_kR_{k+1}}^{010}\\big |_{k=0}:=p_{R_0R_1}^{10}$ ($R_{-1}$ is not defined).",
"Any piecewise linear $p$ with $0=p(R)=p(R_n)=x_n$ can be represented in the form $p(r)=\\sum _{k=0}^{n-1}x_k p_{R_{k-1}R_kR_{k+1}}^{010}$ and its image under $L$ is $Lp(r)=\\sum _{k=0}^{n-1}x_kLp_{R_{k-1}R_kR_{k+1}}^{010}(r).$ If $(x_0,x_1,\\dots ,x_{n-1})$ is a solution of the system $ \\sum _{k=0}^{n-1}\\left(Lp_{R_{k-1}R_kR_{k+1}}^{010}(R_i)-Lp_{R_{k-1}R_kR_{k+1}}^{010}(R)\\right)x_k=G_0(x_i),\\qquad \\mathrm {(ANS)}$ $i=0,1,\\dots ,n-1$ , and if we define $p$ by (REF ) and $Lp(R)=:E_0$ , then $Lp(r)-E_0=G_0\\big (p(r)\\big ) \\;\\; \\text{for} \\; r=R_0,R_1,\\dots ,R_{n-1}.$ The system is also satisfied for $r=R$ because $Lp(R)=E_0$ and $G_0\\big (p(R)\\big )=0$ .",
"We call (REF ) the “approximating nonlinear system”.",
"It will be the subject of the next section."
],
[
"The approximating nonlinear system (ANS)",
"The approximating nonlinear system (REF ) has the form $Ax=G_0(x),$ where for a vector $x\\in \\mathbb {R}^n$ , we define $G_0(x):=\\big (G_0(x_i)\\big )_{i=0}^{n-1}$ and $A$ is the $n\\times {n}$ matrix with coefficients $A_{ik}$ , $A_{ik}:=Lp_{R_{k-1}R_kR_{k+1}}^{010}(R_i)-Lp_{R_{k-1}R_kR_{k+1}}^{010}(R)=:B_{ik}-C_k,\\\\i,k=0,1,\\dots ,n-1.$ The expressions $Lp_{R_{k-1}R_kR_{k+1}}^{010}(R_i)$ are composed of terms of the form (i) $L_{\\rm I}p_{RS}^{01}(r):=Lp_{RS}^{01}(r), \\qquad 0\\le r\\le R<S$ , (ii) $L_{\\rm I}p_{ST}^{10}(r):=Lp_{ST}^{10}(r), \\qquad 0\\le r\\le S<T$ , (iii) $L_{\\rm II}p_{RS}^{01}(r):=Lp_{RS}^{01}(r), \\hspace*{18.49428pt} 0\\le R<S\\le r$ , (iv) $L_{\\rm II}p_{ST}^{10}(r):=Lp_{ST}^{10}(r), \\hspace*{18.49428pt} 0\\le S< T\\le r$ , where $p_{RS}^{01}(s)=\\dfrac{s-R}{S-R} \\; \\chi _{[R,S]}(s)$ , $p_{ST}^{10}(s)=\\dfrac{s-T}{S-T}\\,\\chi _{[S,T]}(s)$ .",
"If $p\\in L^\\infty (R_+)$ and $sp\\in L^1(R_+)$ , we have $Lp(0):=\\lim _{r\\downarrow 0}Lp(r)=4\\pi \\cdot \\lim _{r\\downarrow 0}\\left[\\frac{1}{r}\\int _0^r\\!\\!p(s)s^2\\,ds+\\int _r^\\infty \\!\\!p(s)s\\,ds\\right]=4\\pi \\int _0^\\infty \\!\\!p(s)s\\,ds.$ For $a,b\\in \\mathbb {R}$ and $n\\ge 2$ , we shall apply the formula $a^n-b^n=(a-b)\\big (a^{n-1}+a^{n-2}b+\\dots ab^{n-2}+b^{n-1}\\big ).$ The calculation of the expressions (i)—(iv) requires the following lemma.",
"$~$ (i) Let $0\\leqq r\\leqq R<S$ .",
"Then we have $L_{\\rm I}p_{RS}^{01}(r)&=\\frac{4\\pi }{S-R}\\int _R^S\\!\\!",
"(s-R)s\\,ds=\\frac{4\\pi }{S-R}\\left(\\frac{s^3}{3}-R\\,\\frac{s^2}{2}\\,\\Big |_R^S\\right)\\\\&=\\frac{4\\pi }{S-R}\\left[\\frac{1}{3}\\big (S^3-R^3\\big )-\\frac{R}{2}\\big (S^2-R^2\\big )\\right]\\\\&=4\\pi \\left[\\frac{1}{3}\\big (S^2+SR+R^2\\big )-\\frac{R}{2}(S+R)\\right]\\text{\\quad with (\\ref {e7.2})}\\\\&=4\\pi \\left[\\frac{1}{3}\\,S^2-\\frac{1}{6}\\,SR-\\frac{1}{6}\\,R^2\\right]=\\frac{4\\pi }{3}\\left[S^2-\\frac{1}{2}\\,R(S+R)\\right].$ (ii) Let $0\\le r\\le S<T$ .",
"Then we have $L_{\\rm I}p_{ST}^{10}(r)&=4\\pi \\cdot \\frac{(-1)}{T-S}\\int _S^T\\!\\!",
"(s-T)s\\,ds=-\\frac{4\\pi }{T-S}\\left(\\frac{s^3}{3}-T\\,\\frac{s^2}{2}\\right)\\bigg |_S^T\\\\&=-\\frac{4\\pi }{T-S}\\left[\\frac{1}{3}\\,\\big (T^3-S^3\\big )-\\frac{1}{2}\\,T\\big (T^2-S^2\\big )\\right]\\\\&=-4\\pi \\left[\\frac{1}{3}\\,\\big (T^2+TS+S^2\\big )-\\frac{1}{2}\\,T(T+S)\\right]=\\frac{4\\pi }{3}\\left[\\frac{1}{2}\\,T(T+S)-S^2\\right].$ (iii) Let $0\\le R<S\\le r$ .",
"Then we have $L_{\\rm II}p_{RS}^{01}(r)&=\\frac{4\\pi }{r}\\,\\frac{1}{S-R} \\int _R^{S}\\!\\!",
"(s-R)s^2\\,ds=\\frac{4\\pi }{r}\\, \\frac{1}{S-R}\\left(\\frac{s^4}{4}-R\\,\\frac{s^3}{3}\\right)\\bigg |_R^S\\\\&=\\frac{4\\pi }{r}\\,\\frac{1}{S-R}\\left[\\frac{1}{4}\\,\\big (S^4-R^4\\big )-\\frac{R}{3}\\,\\big (S^3-R^3\\big )\\right]\\\\&=\\frac{4\\pi }{r}\\left[\\frac{1}{4}\\,\\big (S^3+S^2R+SR^2+R^3\\big )-\\frac{R}{3}\\,\\big (S^2+RS+R^2\\big )\\right]\\\\&=\\frac{\\pi }{r}\\left[S^3-\\frac{1}{3}\\,\\big (S^2R+SR^2+R^3\\big )\\right].$ (iv) Let $0\\le S<T\\le r$ .",
"Then we have $L_{\\rm II}p_{ST}^{10}(r)&=\\frac{-4\\pi }{r}\\,\\frac{1}{T-S}\\int _S^T\\!\\!",
"(s-T)s^2\\,ds=-\\frac{4\\pi }{r}\\, \\frac{1}{T-S}\\left(\\frac{s^4}{4}-\\frac{Ts^3}{3}\\bigg |_S^T\\right)\\\\&=-\\frac{4\\pi }{r}\\,\\frac{1}{T-S}\\left[\\frac{1}{4}\\,\\big (T^4-S^4\\big )-\\frac{T}{3}\\,\\big (T^3-S^3\\big )\\right]\\\\&=-\\frac{4\\pi }{r}\\left[\\frac{1}{4}\\,\\big (T^3+ST^2+S^2T+S^3\\big )-\\frac{T}{3}\\,\\big (T^2+ST+S^2\\big )\\right]\\\\&=\\frac{\\pi }{r}\\left[\\frac{1}{3}\\,\\big (T^3+T^2S+TS^2\\big )-S^3\\right].$ $~$ Figure: NO_CAPTION$~$ In order to calculate the $B_{ik}$ and $C_k$ we divide the square $\\lbrace (i,k):i, k=0,1,2,\\dots ,n-1\\rbrace $ into subsections $J_0,\\dots ,J_5$ : $J_0:=\\lbrace (0,0)\\rbrace $ , $J_1:=\\lbrace (i,0)\\colon i=1,\\dots ,n-1\\rbrace $ , $J_2:=\\lbrace (0,k)\\colon k=1,\\dots ,n-1\\rbrace $ , $J_3:=\\lbrace (i,k)\\colon 2\\le i\\le n-1$ , $1\\le k\\le i-1\\rbrace $ , $J_4:=\\lbrace (i,k)\\colon i=k=1,\\dots ,n-1\\rbrace $ , $J_5:=\\lbrace (i,k)\\colon 1\\le i\\le k-1, \\ 2\\le k\\le n-1\\rbrace $ .",
"(Calculation of the $B_{ik}=Lp_{R_{k-1}R_kR_{k+1}}^{010}(R_i)$ ).",
"$J_0=\\lbrace 0,0\\rbrace $ .",
"By Lemma ,(ii), we get $B_{00}=L_{\\rm I}p_{\\,0R_1}^{10}(0)=\\frac{4\\pi }{3}\\,\\frac{1}{2}\\,R_1^2=\\frac{2\\pi }{3}\\left(\\frac{R}{n}\\right)^2.$ $J_1=\\lbrace (i,0): i=1,\\dots ,n-1\\rbrace $ .",
"By Lemma ,(iv), we obtain $B_{i0}=L_{\\rm II}p_{\\,0R_1}^{10}(R_i)=\\frac{\\pi }{i\\dfrac{R}{n}}\\,\\frac{1}{3}\\left(\\frac{R}{n}\\right)^3=\\frac{\\pi }{3}\\,\\frac{1}{i}\\left(\\frac{R}{n}\\right)^2.$ $J_2=\\lbrace (0,k): k=1,\\dots ,n-1\\rbrace $ .",
"By Lemma ,(i),(ii), we get $B_{0k}&=Lp_{R_{k-1}R_kR_{k+1}}^{010}(0)=L_{\\rm I}p_{R_{k-1}R_k}^{01}(0)+L_{\\rm I}p_{R_kR_{k+1}}^{10}(0)\\\\&=\\frac{4\\pi }{3}\\left[R_k^2-\\frac{1}{2}\\,R_{k-1}\\big (R_k+R_{k-1}\\big )\\right]+\\frac{4\\pi }{3}\\left[\\frac{1}{2}\\,R_{k+1}\\big (R_{k+1}+R_k\\big )-R_k^2\\right]\\\\&=\\frac{4\\pi }{3}\\left(\\frac{R}{n}\\right)^2\\left[k^2-\\frac{1}{2}\\,(k-1)(2k-1)+\\frac{1}{2}\\,(k+1)(2k+1)-k^2\\right]\\\\&=\\frac{2\\pi }{3}\\left(\\frac{R}{n}\\right)^2\\left[-\\big (2k^2-k-2k+1\\big )+\\big (2k^2+k+2k+1\\big )\\right]=4\\pi k\\left(\\frac{R}{n}\\right)^2.$ $J_3=\\lbrace (i,k): 2\\le i\\le n-1,$ $1\\le k\\le i-1\\rbrace $ .",
"By virtue of Lemma ,(iii),(iv), we obtain $B_{ik}&=Lp_{R_{k-1}R_kR_{k+1}}^{010}(R_i)=L_{\\rm II}p_{R_{k-1}R_k}^{01}(R_i)+ L_{\\rm II}p_{R_kR_{k+1}}^{10}(R_i)\\\\&=\\frac{\\pi }{R_i}\\Bigg [R_k^3-\\frac{1}{3}\\left(R_k^2R_{k-1}+R_kR_{k-1}^2+ R_{k-1}^3\\right)\\\\&\\qquad \\qquad +\\frac{1}{3}\\left(R_{k+1}^3+R_{k+1}^2R_k+R_{k+1}R_k^2\\right)-R_k^3\\Bigg ]\\\\&=\\frac{\\pi }{i}\\,\\frac{1}{3}\\left(\\frac{R}{n}\\right)^2\\big [-\\left(k^2(k-1)+k(k-1)^2+(k-1)^3\\right)+(k+1)^3\\\\&\\qquad \\qquad +(k+1)^2k+(k+1)k^2\\big ]\\\\&=\\frac{\\pi }{i}\\,\\frac{1}{3}\\left(\\frac{R}{n}\\right)^2\\left[12k^2+2\\right]=4\\pi \\left(\\frac{R}{n}\\right)^2\\frac{1}{i}\\,\\left(k^2+\\frac{1}{6}\\right).$ $J_4=\\lbrace (i,k): i=k=1,\\dots ,n-1\\rbrace $ .",
"By virtue of Lemma ,(iii),(ii), we have $B_{kk}&=Lp_{R_{k-1}R_kR_{k+1}}^{010}(R_k) =L_{\\rm II}p_{R_{k-1}R_k}^{01} (R_k)+L_{\\rm I}p_{R_kR_{k+1}}^{10}(R_k)\\\\&=\\frac{\\pi }{R_k}\\left[R_k^3-\\frac{1}{3}\\left(R_k^2R_{k-1}+R_kR_{k-1}^2+R_{k-1}^3\\right)\\right]\\\\&\\qquad \\qquad + \\frac{4\\pi }{3}\\left[\\frac{1}{2}\\,R_{k+1}\\big (R_{k+1}+R_k\\big )-R_k^2\\right]\\\\&=\\frac{\\pi }{k}\\left(\\frac{R}{n}\\right)^2\\left[k^3-\\frac{1}{3}\\left(k^2(k-1)+k(k-1)^2+(k-1)^3\\right)\\right]\\\\&\\qquad \\qquad +\\frac{4\\pi }{3}\\left(\\frac{R}{n}\\right)^2\\left[\\frac{1}{2}(k+1)\\left((k+1)+k\\right)-k^2\\right]\\\\&=\\frac{\\pi }{k}\\left(\\frac{R}{n}\\right)^2\\left[2k^2-\\frac{4}{3}\\,k+\\frac{1}{3}\\right]+4\\pi \\left(\\frac{R}{n}\\right)^2\\frac{1}{3}\\left[\\frac{3}{2}\\,k+\\frac{1}{2}\\right]\\\\&=4\\pi \\left(\\frac{R}{n}\\right)^2\\left[k-\\frac{1}{6}+\\frac{1}{12k}\\right].$ $J_5=\\lbrace (i,k)\\colon 1\\le i\\le k-1$ , $2\\le k\\le n-1\\rbrace $ .",
"By virtue of Lemma ,(i),(ii), we obtain $B_{ik}&=Lp_{R_{k-1}R_kR_{k+1}}^{010}(R_i)=L_{\\rm I}p_{R_{k-1}R_k}^{01}(R_i) +L_{\\rm I}p_{R_kR_{k+1}}^{10}(R_i)\\\\&=\\frac{4\\pi }{3}\\left[R_k^2-\\frac{1}{2}\\,R_{k-1}\\big (R_k+R_{k-1}\\big )\\right]+\\frac{4\\pi }{3}\\left[\\frac{1}{2}\\big (R_{k+1}\\big (R_{k+1}+R_k\\big )-R_k^2\\right]\\\\&=\\frac{4\\pi }{3}\\left(\\frac{R}{n}\\right)^2\\left[k^2-\\frac{1}{2}(k-1)(2k-1)+\\frac{1}{2}\\big (k^2+2k+1+k^2+k\\big )-k^2\\right]\\\\&=\\frac{4\\pi }{3}\\left(\\frac{R}{n}\\right)^2\\left[k^2-\\frac{1}{2}\\,\\big (2k^2-k-2k+1\\big )+\\frac{1}{2}\\,\\big (2k^2+3k+1\\big )-k^2\\right]=4\\pi \\left(\\frac{R}{n}\\right)^2k.$ (Calculation of the $C_k:=Lp_{R_{k-1}R_kR_{k+1}}^{010}(R)$ ).",
"From Lemma , (iv), it follows that $C_0=L_{\\rm II}p_{R_0R_1}^{10}(R)=\\frac{\\pi }{3R}\\cdot \\left(\\frac{R}{n}\\right)^3.\\qquad \\qquad \\qquad \\qquad \\quad $ By virtue of Lemma , (iii),(iv), we obtain $C_k&=Lp_{R_{k-1}R_kR_{k+1}}^{010}(R)=L_{\\rm II}p_{R_{k-1}R_k}^{01}(R) + L_{\\rm II}p_{R_kR_{k+1}}^{10}(R)\\\\&=\\frac{\\pi }{R}\\bigg [R_k^3-\\frac{1}{3}\\left(R_k^2R_{k-1}+R_{k-1}^2R_k+R_{k-1}^3\\right)\\\\&\\qquad \\qquad +\\frac{1}{3}\\left(R_{k+1}^3+R_{k+1}^2R_k+R_{k+1}R_k^2\\right)-R_k^3\\bigg ]\\\\&=\\frac{\\pi }{R}\\left(\\frac{R}{n}\\right)^3\\!\\bigg [-\\frac{1}{3}\\left(k^2(k-1)+k(k-1)^2\\!+\\!",
"(k-1)^3\\right)\\\\&\\qquad \\qquad \\quad \\;\\;\\;+\\frac{1}{3}\\left((k+1)^3+(k+1)^2k+(k+1)k^2\\right)\\bigg ]\\\\&=\\frac{4\\pi }{n}\\left(\\frac{R}{n}\\right)^2\\left[k^2+\\frac{1}{6}\\right].$ With $A_{ik}=B_{ik}-C_k$ we have $A_{ik}\\big ([0,R]\\big )=R^2A_{ik}\\big ([0,1]\\big ).$"
],
[
"The numerical analysis of the (ANS)",
"The aim of this section is to indicate some of the numerical procedures for solving the system $ Ax=G_0(x)\\qquad \\mathrm {(ANS)}$ with the matrix $A=(A_{ik})_{i,k=0}^{n-1}$ , $A_{ik}:=B_{ik}-C_k$ , $i,k=0,\\dots ,n-1$ , where $G_0(x):=\\big (G_0(x_i)\\big )_{i=0}^{n-1}$ , $x\\in (\\mathbb {R}_{+})^n$ for the scalar function $G_0\\colon [0,p(0))\\rightarrow [0,P(0)-E_0)$ .",
"The matrix $A$ depends on $n$ $(A=A(n))$ and can be calculated with the formulas developed in Section .",
"The solution $x=(x_0,x_1,\\dots ,x_{n-1})^T$ represents the values $x_0,x_1,\\dots ,x_{n-1}$ of the approximation polygon $p_{R_0R_1\\dots R_{n-1}}^{x_0x_1\\dots x_{n-1}}$ at $R_0<R_1<\\dots <R_{n-1}$ with $p_{R_0R_1\\dots R_{n-1}}^{x_0x_1\\dots x_{n-1}}(R)=0$ and $x_0, x_1, \\dots , x_{n-1}, x_n=0$ , $R_k:=\\dfrac{k}{n}\\,R$ , $k=0,1,\\dots ,n-1$ .",
"To determine the solution $x$ of the (REF ) we use Newton's method for the equation $Ax-G_0(x)=0$ .",
"For the convergence of Newton's method a suitable choice of the starting point $x^{(0)}$ is crucial.",
"In the sequence of partitions $\\pi _{2^i}:=\\left\\lbrace \\frac{k}{2^i}\\,R, \\ k=0,\\dots ,2^i\\right\\rbrace , \\qquad i=0,1,2,\\dots ,$ we use the solution $p_{\\pi _{2^i}}$ as starting point for the next partition $\\pi _{2^{i+1}}$ for $n=2, 4, 8, 16$ , $32, 64, 128$ , and thus we are able to show that Kantorovich's criterion [11] (which much depends of $x^{(0)}$ and $\\Vert x^{(0)}-x^{(1)}\\Vert $ ) yields convergence of the sequences in the following two examples.",
"For $n=2^0=1$ , $Ax-G_0(x)=0$ is a one-dimensional nonlinear equation.",
"To find a solution of this equation, the method of nested intervals by bisection can be used.",
"Example of Section .",
"$~$ In this example we can calculate the operator $G_0$ explicitly.",
"In fact, in Example it was shown that $&F\\colon [E_0,P(0)]=\\left[\\frac{8}{15}\\,\\pi R^2, \\pi R^2\\right]\\rightarrow [0,1], \\ \\text{where}\\\\&F(h)=\\sqrt{ah-\\frac{20}{9}}-\\frac{2}{3}\\,,\\qquad a:=\\frac{5}{\\pi R^2}\\,.$ It was also proved that $&F_0\\colon \\left[0, P(0)-E_0\\right]=\\left[0,\\frac{7}{15}\\pi R^2\\right]\\rightarrow [0,1], \\ \\text{where}\\\\&F_0(h)=\\sqrt{ah+\\frac{4}{9}}-\\frac{2}{3}\\,.$ Therefore we have $&G_0:=F_0^{-1}\\colon [0,1]\\rightarrow \\left[0,\\frac{7}{15}\\,\\pi R^2\\right], \\ \\text{where}\\\\&G_0(t)=\\frac{\\pi R^2}{5}\\left(t^2+\\frac{4}{3}\\,t\\right).$ With the described choice of starting points $x^{(0)}$ Newton's method converges for $n=2,4$ ; for $n=8,16,32,64,128$ Kantorovich's criterion guarantees the convergence.",
"For each $n$ , we have carried out the iterations until the norm of Newton's improvement shows a relative error with respect to the last iteration of magnitude $10^{-9}$ — this had been achieved by less than 5 iterations.",
"Chart REF illustrates the convergence of the polygons $p_n$ as the solutions of the (REF ) on the interval $[0,R]$ for $R=8$ towards the strict solution $p$ of the equation $p(r)=FLp(r)$ or $Lp-E_0=G_0(p)$ , which in this case is known as $p(r)=1-\\left(\\dfrac{r}{R}\\right)^2$ .",
"In the center of the chart there are listed the values $p_n(r_k)$ $(r_k=0,0.5,1,\\dots ,7.5,8)$ of the approximation polygons for $n=2,4,\\dots ,128$ .",
"The last column contains the differences $p(r_k)-p_{128}(r_k)$ , which are smaller than $5\\cdot 10^{-5}$ , and show the pointwise convergence $p_n(r_k)\\rightarrow p(r_k)$ .",
"The third line from last gives the $L_2$ -norms of the differences $p-p_n$ , showing their convergence to 0 and in particular revealing the fact that doubling the number of supporting points results in dividing the $L_2$ -norm by 4.",
"The last two lines list the values of the $E_{0n}$ and the relative error $|E_0-E_{0n}|\\big /|E_0|$ in %.",
"A doubling of supporting point results in a convergence factor of $1/4$ also here.",
"Table: Example of Section :~ Approximations p n (r)p_n(r) of p(r)=1-r R 2 p(r)=1-\\left(\\dfrac{r}{R}\\right)^2for R=8R=8 and n=2n=2, 4, 8, 16.",
"32, 64, 128$~$ Figure: to Chart 8.1: Polygon p 128 p_{128}$~$ Example of Section .",
"$~$ First we have to compute $G_0(t)=F_0^{-1}(h)$ .",
"The equality $F_0(h)=\\dfrac{\\pi ^2c}{\\sqrt{2}}\\,h^2$ implies that $G_0(t)=\\dfrac{\\@root 4 \\of {2}}{\\pi \\sqrt{c}}\\,\\sqrt{t}$ .",
"To obtain comparable and clearly representable values $x$ of the solutions in Chart REF we choose $c=\\sqrt{2}/(4^2 \\pi ^4 1000)$ .",
"In this example we calculate the solutions $p_n$ as in the first example with Newton's method.",
"Again Kantorovich's criterion guarantees its convergence for $n=4,8,\\dots ,128$ .",
"The results are shown in Chart REF , which differ from Chart REF in the following: 1.",
"The last column shows the relative error (in %) of $|p_{128}(r_k)-p_{64}(r_k)|\\big / p_{128}(r_k)\\cdot 100$ .",
"2.",
"The third line from last reads Norm-Error% $=100\\cdot \\big |\\Vert p_n\\Vert _2-\\Vert p_{2n}\\Vert _2\\big |\\big /\\Vert p_{2n}\\Vert _2$ .",
"3.",
"The last line reads $E_0-\\text{Error \\%}=100\\cdot |E_{0,n}-E_{0,2n}|\\big /E_{0,2n}$ .",
"$~$ $~$ Table: Example of Section :~ Approximations of p(r)p(r) forG 0 (t)=4π1000tG_0(t)=4 \\pi \\sqrt{1000} \\sqrt{t} for R=8R=8 andn=2 n=2, 4, 8, 16, 32, 64, 128Figure: to Chart 8.2: Polygon p 128 p_{128}"
],
[
"Appendix: Abel's and Eddington's equations",
"The goal of this section is the existence proof for Eddington's equation.",
"It is based on the existence proof for Abel's equation, which we treat first.",
"We shall use the following notation.",
"For an interval $[0,T)$ , we let $L_{\\rm loc}^1[0,T)&:=\\left\\lbrace p\\colon [0,T)\\rightarrow \\mathbb {R}; \\quad p|_{[0,a]}\\in L^1[0,a] \\ \\text{for all} \\ a\\in [0,T)\\right\\rbrace ,\\\\AC[0,T)&:=\\big \\lbrace p\\colon [0,T)\\rightarrow \\mathbb {R}; \\quad p|_{[0,a]} \\ \\text{is absolutelycontinuous on} \\ [0,a]\\\\&\\hspace*{116.6563pt}\\text{ for all} \\ a\\in [0,T) \\big \\rbrace \\; \\text{ \\cite [ p. 106]{8}}.$ We begin with two lemmas and then turn to Abel's equation.",
"Let $f\\in L_{\\rm loc}^1[0,T)$ .",
"Then the function $g(x):=\\int _0^x\\frac{f(s)\\,ds}{\\sqrt{x-s}}\\,ds, \\qquad x\\in [0,T)$ belongs to $L_{\\rm loc}^1[0,T)$ .",
"Proof.",
"We may assume that $f\\ge 0$ .",
"Using the equality $\\int _a^b\\!\\!\\frac{d\\sigma }{\\sqrt{b-\\sigma }\\,\\sqrt{\\sigma -a}}=\\pi , \\qquad 0\\le a<b,$ and Fubini's Theorem, we obtain $\\infty >\\int _0^x\\!\\!f(s)\\,ds&=\\frac{1}{\\pi }\\int _0^x\\!\\!f(s)\\int _s^x\\!\\!\\frac{1}{\\sqrt{x-\\sigma }\\,\\sqrt{\\sigma -s}}\\,d\\sigma \\,ds\\\\&= \\frac{1}{\\pi }\\int _0^x \\left(\\int _0^\\sigma \\frac{f(s)}{\\sqrt{\\sigma -s}}\\,ds\\right)\\frac{1}{\\sqrt{x-\\sigma }}\\,d\\sigma \\\\[1mm]&=\\frac{1}{\\pi }\\int _0^x\\!\\!g(\\sigma )\\frac{1}{\\sqrt{x-\\sigma }}\\,d\\sigma .$ It follows $\\left(\\sigma \\rightarrow g(\\sigma )\\dfrac{1}{\\sqrt{x-\\sigma }}\\right)\\in L_{\\rm loc}^1[0,T)$ and $g\\in L_{\\rm loc}^1[0,T)$ .$\\square $ Let $f,g\\in L_{\\rm loc}^1[0,T)$ .",
"Then we have $\\int _0^x\\!\\!\\frac{f(s)}{\\sqrt{x-s}}\\,ds=g(x) \\quad \\text{a.e.",
"on} \\ (0,T)$ if and only if $\\int _0^x\\!\\!f(s)\\,ds=\\frac{1}{\\pi }\\int _0^x\\!\\!\\frac{g(s)}{\\sqrt{x-s}}\\,ds \\quad \\text{on} \\ (0,T).$ Proof.",
"Let (REF ) hold.",
"Then $\\int _0^x\\!\\!\\frac{g(s)}{\\sqrt{x-s}}\\,ds&=\\int _0^x\\left(\\int _0^s\\!\\!\\frac{f(\\sigma )}{\\sqrt{s-\\sigma }}\\,d\\sigma \\right)\\frac{1}{\\sqrt{x-s}}\\,ds\\\\[1mm]&=\\int _0^x\\!\\!f(\\sigma )\\left(\\int _\\sigma ^x\\!\\!\\frac{1}{\\sqrt{x-s}\\,\\sqrt{s-\\sigma }}\\,ds\\right)\\,d\\sigma =\\pi \\int _0^x\\!\\!f(\\sigma )d\\sigma ,$ i.e., (REF ) is fulfilled.",
"Now assume (REF ).",
"Let $h(x):=\\displaystyle \\int _0^x\\dfrac{f(s)}{\\sqrt{x-s}}\\,ds-g(x)$ .",
"We need to show $h=0$ .",
"Lemma gives $h\\in L_{\\rm loc}^1[0,T)$ .",
"As in the first part of the proof, using (REF ), we have $\\int _0^x\\!\\!\\frac{h(s)}{\\sqrt{x-s}}\\,ds&=\\int _0^x\\!\\!\\frac{1}{\\sqrt{x-s}}\\int _0^s\\!\\!\\frac{f(\\sigma )}{\\sqrt{s-\\sigma }}\\,d\\sigma \\,ds-\\int _0^x\\!\\!\\frac{g(s)}{\\sqrt{x-s}}\\,ds\\\\[1mm]&=\\int _0^x\\left(\\int _\\sigma ^x\\frac{ds}{\\sqrt{x-\\sigma }\\,\\sqrt{\\sigma -s}}\\right)f(\\sigma )\\,d\\sigma -\\int _0^x\\!\\!\\frac{g(s)}{\\sqrt{x-s}}\\,ds=\\\\[1mm]&=\\pi \\int _0^x\\!\\!f(\\sigma )\\,d\\sigma -\\int _0^x\\!\\!\\frac{g(s)}{\\sqrt{x-s}}\\,ds=0.$ Hence $\\pi \\int _0^x\\!\\!",
"h(s)\\,ds&=\\int _0^x\\!\\!",
"h(s)\\int _s^x\\!\\!\\frac{1}{\\sqrt{x-\\sigma }\\,\\sqrt{\\sigma -s}}\\,d\\sigma \\,ds=\\\\[1mm]&=\\int _0^x\\!\\!\\frac{1}{\\sqrt{x-\\sigma }}\\left(\\int _0^\\sigma \\!\\!\\frac{h(s)}{\\sqrt{\\sigma -s}}\\, ds\\right)d\\sigma =0, \\qquad x\\in [0,T),$ and $h=0$ follows.$\\square $ We now consider the solvability of Abel's equation.",
"(Existence and uniqueness for Abel's equation).",
"(a) Let $g\\in L_{\\rm loc}^1[0,T)$ and assume (i) $G\\in AC[0,T)$ , where $\\displaystyle G(x):=\\int _0^x\\!\\!\\dfrac{g(s)}{\\sqrt{x-s}}\\,ds$ , $~$ (ii) $G(0)=0$ .",
"Then $f$ defined by $f(x):=\\frac{1}{\\pi }\\,G^{\\prime }(x)$ is the unique solution of Abel's equation $g(x)=\\int _0^x\\!\\!\\frac{f(s)}{\\sqrt{x-s}}\\,ds, \\qquad x\\in [0,T).$ (b) Conversely, if $f\\in L_{\\rm loc}^1[0,T)$ and $g$ satisfies (REF ), then $g\\in AC[0,T)$ , (i), (ii) hold, and $f(x)=\\frac{1}{\\pi }\\,G^{\\prime }(x), \\qquad x\\in [0,T).$ Proof.",
"(a) By assumption, $f\\in L_{\\rm loc}^1[0,T)$ , and we have $\\int _0^x\\!\\!f(s)\\,ds=\\frac{1}{\\pi }\\int _0^x\\!\\!G^{\\prime }(s)\\,ds=\\frac{1}{\\pi }\\big (G(x)-0\\big )=\\frac{1}{\\pi }\\int _0^x\\frac{g(s)}{\\sqrt{x-s}}\\, ds, \\qquad x\\in (0,T).$ Lemma then shows that $f$ solves (REF ).",
"The uniqueness follows from (b).",
"(b) It follows from Lemma that $g\\in L_{\\rm loc}^1[0,T)$ .",
"We define $G(x):=\\int _0^x\\!\\!\\frac{g(s)}{\\sqrt{x-s}}\\,ds.$ Since $f$ satisfies (REF ), Lemma gives $\\int _0^x\\!\\!f(s)\\,ds=\\frac{1}{\\pi }\\, G(x), \\qquad x\\in [0,T).$ Hence $G\\in AC[0,T)$ , $G(0)=0$ , and $f(x)=\\dfrac{1}{\\pi }\\,G^{\\prime }(x)$ , $x\\in (0,T)$ , which proves the uniqueness assertion in (a).$\\square $ The example $g(s)=\\dfrac{1}{\\sqrt{s}}$ , $\\displaystyle G(x)=\\int _0^x\\!\\!\\dfrac{ds}{\\sqrt{s}\\,\\sqrt{x-s}}=\\pi $ shows that (ii) does not necessarily follow from (i).",
"Now we treat the solvability of Eddington's equation.",
"(Existence and uniqueness for Eddington's equation).",
"(a) Let $g\\in AC[0,T)$ , $g(0)=0$ , and assume that (i) $H_{g^{\\prime }}\\in AC[0,T)$ , where $\\displaystyle H_{g^{\\prime }}(x):=\\int _0^x\\!\\!\\dfrac{g^{\\prime }(s)}{\\sqrt{x-s}}\\,ds$ , (ii) $H_{g^{\\prime }}(0)=0$ .",
"Then $f$ defined by $f(x):=\\frac{2}{\\pi }\\,H_{g^{\\prime }}^{\\prime }(x)$ is the unique solution of Eddington's equation $g(x)=\\int _0^x\\!\\!f(s)\\sqrt{x-s}\\, ds.$ (b) Conversely, if $f\\in L_{\\rm loc}^1[0,T)$ , and $g$ satisfies (REF ), then $g\\in AC[0,T)$ , $g(0)=0$ , (i), (ii) hold, and $f$ has the representation (REF ).",
"In addition, $\\displaystyle g^{\\prime }(x)=\\frac{1}{2}\\int _0^x\\frac{f(s)}{\\sqrt{x-s}}\\, ds$ .",
"Proof.",
"(a) Equation (REF ), partial integration, (i) (ii), and Fubini's theorem yield $\\int _0^x\\!\\!f(s)\\sqrt{x-s}\\, ds&=\\frac{2}{\\pi }\\int _0^x\\!\\!",
"H_{g^{\\prime }}^{\\prime }(s)\\sqrt{x-s}\\,ds=\\frac{1}{\\pi }\\int _0^x\\!\\!\\frac{H_{g^{\\prime }}(s)}{\\sqrt{x-s}}\\,ds\\\\[1mm]&=\\frac{1}{\\pi }\\int _0^x\\!\\!\\frac{1}{\\sqrt{x-s}}\\left(\\int _0^s\\frac{g^{\\prime }(\\sigma )}{\\sqrt{s-\\sigma }}\\, d\\sigma \\right)ds\\\\[1mm]&=\\frac{1}{\\pi }\\int _0^x\\left(\\int _\\sigma ^x\\!\\!\\frac{ds}{\\sqrt{x-s}\\,\\sqrt{s-\\sigma }}\\right)g^{\\prime }(\\sigma )d\\sigma =g(x),$ that is, (REF ).",
"Uniqueness follows from (b).",
"(b) Let $f\\in L_{\\rm loc}^1[0,T)$ , and $g$ satisfy (REF ).",
"We show that $g\\in AC[0,T)$ and $g(0)=0$ .",
"We formally define $h(x):=\\int _0^x\\!\\!\\frac{f(s)}{\\sqrt{x-s}}\\,ds.$ By Lemma , $h$ exists a.e.",
"on $[0,T)$ and $h\\in L_{\\rm loc}^1[0,T)$ .",
"In fact, equality (REF ) and Fubini's theorem imply $2g(x)&=2\\int _0^x\\!\\!f(\\sigma )\\sqrt{x-\\sigma }\\,d\\sigma =\\int _0^x\\!\\!f(\\sigma )\\left(\\int _\\sigma ^x\\!\\!\\frac{ds}{\\sqrt{s-\\sigma }}\\right)d\\sigma \\\\&=\\int _0^x\\left(\\int _0^s\\!\\!\\frac{f(\\sigma )}{\\sqrt{s-\\sigma }}\\, d\\sigma \\right)ds=\\int _0^x\\!\\!",
"h(s)\\,ds.$ Hence $g\\in AC[0,T)$ , and $g(0)=0$ .",
"Furthermore, $2g^{\\prime }(x)=h(x)=\\int _0^x\\!\\!\\frac{f(s)}{\\sqrt{x-s}}\\,ds,$ and from Lemma it follows that $\\int _0^x\\!\\!f(s)\\,ds=\\frac{2}{\\pi }\\int _0^x\\!\\!\\frac{g^{\\prime }(s)}{\\sqrt{x-s}}\\,ds=\\frac{2}{\\pi }\\,H_{g^{\\prime }}(x).$ Hence $H_{g^{\\prime }}\\in AC[0,T)$ , and $H_{g^{\\prime }}(0)=0$ .",
"Differentiating the last equality, we get (REF ), which also shows the uniqueness in (a).",
"$\\square $"
],
[
"Suggestions for further work",
"In Example it was constructed a function $p$ satisfying the assumptions of Theorem such that according to numerical calculations the function $q$ can have negative values, i.e.",
"$p$ is nonextendable.",
"It would be interesting to give a rigorous proof of this result.",
"A further question is the extension of the present work to the case of cylindrical symmetry."
],
[
"Acknowledgements",
"The work of the third autor was supported by the Ministry of Science and Education of Russian Federation, project number FSSF–2020–0018.",
"The autors are grateful to Academician of RAS Prof. V.V.",
"Kozlow for recommending the publication of a preview of the present work in Doklady Mathematics (see [3]) .",
"$~$"
]
] | 2107.01898 |
[
[
"NOTE: Solution for KDD-CUP 2021 WikiKG90M-LSC"
],
[
"Abstract WikiKG90M in KDD Cup 2021 is a large encyclopedic knowledge graph, which could benefit various downstream applications such as question answering and recommender systems.",
"Participants are invited to complete the knowledge graph by predicting missing triplets.",
"Recent representation learning methods have achieved great success on standard datasets like FB15k-237.",
"Thus, we train the advanced algorithms in different domains to learn the triplets, including OTE, QuatE, RotatE and TransE.",
"Significantly, we modified OTE into NOTE (short for Norm-OTE) for better performance.",
"Besides, we use both the DeepWalk and the post-smoothing technique to capture the graph structure for supplementation.",
"In addition to the representations, we also use various statistical probabilities among the head entities, the relations and the tail entities for the final prediction.",
"Experimental results show that the ensemble of state-of-the-art representation learning methods could draw on each others strengths.",
"And we develop feature engineering from validation candidates for further improvements.",
"Please note that we apply the same strategy on the test set for final inference.",
"And these features may not be practical in the real world when considering ranking against all the entities."
],
[
"Introduction",
"Knowledge graphs are directed multi-relational graphs about facts, usually expressed in the form of $(h, r, t)$ triplets, where $h$ and $t$ represent head entity and tail entity respectively, and $r$ is the relation between head entity and tail entity, e.g.",
"(Geoffrey Hinton, citizen of, Canada).",
"Large encyclopedic knowledge graphs, like Wikidata [1] and Freebase [2], can provide rich structured information about entities and benefit a wide range of applications, such as recommender systems, question answering and information retrieval.",
"However, large knowledge graphs usually face the challenge of incompleteness.",
"For example, 71% of people in Freebase have no birth place and 75% have no nationality [3].",
"Therefore, predicting these missing facts in a knowledge graph is a crucial task, also named as knowledge graph completion task.",
"We can see Figure REF for a clear understanding.",
"In order to address the issue of knowledge graph completion on large knowledge graphs, the 2021 KDD cup releases the WikiKG90M-LSC Task, which focuses on imputing missing facts in a knowledge graph extracted from the entire Wikidata knowledge base.",
"Our method for this task consists of two stages: On the first stage, we propose an ensemble method for different knowledge embedding methods to build a strong model, in which, knowledge embedding methods, like TransE [4] and RotatE [5], focus on embedding entities and relations into vectors and then we can use these embeddings to predict missing relations.",
"On the second stage, we adopt several statistical features of the WikiKG90M dataset to help improve the final ensemble model performance.",
"We conducted several experiments on WikiKG90M dataset to demonstrate the superiority of our method, and our team is currently among the awardees of the WikiKG90M-LSC track of the OGB-LSC, where we achieve 0.9727 MRR result in the final test set.",
"Some features from test candidates are used in final inference, which are not practical when considering ranking against all the entities.",
"And its limitation will be discussed in Section REF .",
"Figure: Impute missing triplets in knowledge graphs.",
"[1]"
],
[
"Methodology",
"[1]https://ogb.stanford.edu/assets/img/ogblsctaskoverview.png"
],
[
"Triplet Embedding",
"The majority of knowledge graph representation algorithms relies on the triplets and they remain good interpretability in graph reasoning.",
"In this competition, we adopt advance algorithms in different domains to encode the entities and relations, including NOTE, QuatE, RotatE and TransE.",
"Considering the properties of each model, we ensemble their score results for the final prediction."
],
[
"NOTE",
"[6] proposes OTE to model the symmetry/antisymmetry, inversion, and composition patterns.",
"It takes the relations as an orthogonal transform in a high dimensional space.",
"Relation matrix is orthogonalized so its inverse matrix can be obtained by simple transposing.",
"The full model can be seen as an ensemble of $K$ OTE models.",
"The score functions are defined as Eq.",
"REF and REF .",
"$d((h,r),t)=\\sum _{i=1}^K(\\Vert s_r^h(i)\\phi (M_r(i))e_h(i)-e_t(i)\\Vert ),$ $d(h,(r,t))=\\sum _{i=1}^K(\\Vert s_r^t(i)\\phi (M_r(i))^Te_t(i)-e_h(i)\\Vert ),$ where $s_r^h(i)=diag({exp}(s_r(i)))$ and $s_r^t(i)=diag({exp}(-s_r(i)))$ are the weights of relation matrix, $\\phi $ is the Gram-Schmidt process.",
"However, though [6] has scaled the $L_2$ norm of relation embeddings through scalar tensors $s_r(i) \\in \\mathcal {R}_{d_s}$ , the convergence is still unstable in our experiments.",
"The $exp$ operation in $s_r^h(i)$ and $s_r^t(i)$ enlarges values and could lead to this issue.",
"Therefore, we further use $L_2$ norm to regularize the scalar tensors.",
"In this case, the weights of relation matrix are modified into Eq.",
"REF and REF .",
"We denote such modified version as NOTE (short for Norm-OTE).",
"$s_r^h(i)=\\frac{diag({exp}(s_r(i)))}{\\Vert diag({exp}(s_r(i)))\\Vert },$ $s_r^t(i)=\\frac{diag({exp}(-s_r(i)))}{\\Vert diag({exp}(-s_r(i)))\\Vert }.$"
],
[
"QuatE",
"[7] extend ComplEx [8] into quaternion space for better geometrical interpretations and more latent inter-dependencies.",
"It proves that the Hamilton product in quaternion space can model both symmetry/antisymmetry and inversion patterns except the composition pattern."
],
[
"RotatE",
"RotatE can be view as an orthogonal transform in complex domain.",
"Each relation is taken as a rotation from the head entity to the tail entity.",
"[5] has proved that the Hadamard product is able to model various relation patterns.",
"In theory, it has equivalent model capability as OTE, so we also take it as a basic method."
],
[
"TransE",
"[4] interprets relationships as translations operating on entity embeddings in a real field.",
"Although such assumption is not able to model complex relationships such as 1-N, N-1 and N-N, experiments on different datasets have proved its robustness and it can model composition pattern.",
"Thus, we use results of this model to ensemble in our experiments."
],
[
"Graph Context",
"Besides the specific triplets, the structure of sub-graphs comprised of triplets also reflect some semantic information.",
"For example, Geoffrey Hinton in Figure REF is connected with King's College Cambridge through relation Graduated from.",
"We can infer that Geoffrey Hinton should be a person and is probable to be born in UK, as King's College Cambridge is located in UK.",
"Entities and relations in this sub-graph influence each other.",
"To supplement information of the graph structure, we use representative techniques such as Post-Smoothing [9] and DeepWalk[10]."
],
[
"Relation-based Post Smoothing",
"We propose a two-stage method to capture the relationships among the entities in sub-graphs.",
"Recent end-to-end models always use graph neural networks as encoder and triplet-based methods as decoder.",
"But in our implementation, we take the opposite approach.",
"In the first stage, we train TransE to encode the entities and relations according to the triplet context.",
"In the second stage, we take the learned representations as entity embeddings.",
"Then we propagate them through the entity adjacent matrix.",
"A hyperparameter $\\alpha $ decides the weight of the entity itself while $1-\\alpha $ denote weights of its neighborhood.",
"The final updated embeddings are used for prediction.",
"For example, for a given entity u, the final embedding $\\mathbf {u}^{\\prime }$ is represented in Eq.",
"REF .",
"$\\mathbf {u}^{\\prime } = \\alpha \\mathbf {u} + (1-\\alpha ) \\sum _{v \\in \\mathcal {N}(u)}{f(\\mathbf {v}, \\mathbf {r})},$ where $\\mathbf {u}$ and $\\mathbf {v}$ are the embedding learned in the first stage.",
"$f$ is depending on the knowledge embedding algorithm."
],
[
"DeepWalk",
"[10] propose DeepWalk to learn latent representations of nodes in homogeneous networks.",
"In our solution, the relations in knowledge graph are ignored.",
"We focus on the entity structure and use skip-gram technique to learn the semantic and structural correlations between entities along generated paths."
],
[
"Manual Feature Engineering",
"In addition to embedding models, manual feature engineering is also a key part of our work.",
"Since our goal is to predict tail through head and relation, our manual features include two parts, head to tail feature and relation to tail feature.",
"Feature selection needs to be performed after obtaining the features."
],
[
"Head to Tail Features",
"Head to tail feature is to predict the probability of tail by the current head.",
"We start to walk from the head, and calculate the probability of walking through different paths to reach tail.",
"For a head, relation and tail triple, it has 6 different walk directions $direct$ , including head to tail(HT), head to relation(HR), relation to head(RH), relation to tail(RT), tail to head(TH), tail to relation(TR).",
"The probability of $e_1$ to $e_2$ in direction $direct$ is as fallow.",
"$P_{direct}(e_1, e_2)=\\frac{S_{direct}(e_1, e_2)}{\\sum _{e \\sim N_{direct}(e_1)}S_{direct}(e_1, e)}$ $S_{direct}(e_1, e_2)$ is the frequency of $e_1$ to $e_2$ in the direction of $direct$ .",
"When developing our model, we calculated all features from the training triplets and validation candidates.",
"It is worth nothing that we only calculate the $F_{HT}$ and $F_{RT}$ from test data at the final inference time, without modifying the weight of our developed models.",
"And we apply the same rule strategies developed from validation for final test prediction.",
"That is to say, test data is only touched at the final inference time.",
"We define 7 manual head to tail feature as follow.",
"Figure: The walk paths of head to tail features.",
"The yellow arrow is a direction of head entity to tail entity.",
"HT is walk in the direction of head to tail, while TH is in direction of tail to head.$F_{HT}(h, t)= P_{HT}(h, t)$ $F_{TH}(h, t)= P_{TH}(h, t)$ $F_{TH-HT}(h, t)=\\sum _{e \\sim N_{TH}(h) \\cap N_{TH}(t)}P_{TH}(h, e) * P_{HT}(e, t)$ $F_{HT-HT}(h, t)=\\sum _{e \\sim N_{HT}(h) \\cap N_{TH}(t)}P_{HT}(h, e) * P_{HT}(e, t)$ $F_{HT-TH}(h, t)=\\sum _{e \\sim N_{HT}(h) \\cap N_{HT}(t)}P_{HT}(h, e) * P_{TH}(e, t)$ $F_{TH-TH}(h, t)=\\sum _{e \\sim N_{TH}(h) \\cap N_{HT}(t)}P_{TH}(h, e) * P_{TH}(e, t)$ $F_{HT-HT-TH}(h, t)=\\sum _{e_1 \\sim N_{HT}(h)} \\sum _{e_2 \\sim N_{HT}(t)}P_{HT}(h, e_1) *P_{HT}(e_1, e_2) * P_{TH}(e_2, t)$"
],
[
"Relation to Tail Features",
"We also define 5 manual relation to tail feature as fallow.",
"$F_{RT}(r, t)=P_{RT}(r, t)$ $F_{RH}(r, t)=P_{RH}(r, t)$ $F_{RT-TR-RT}(r, t)=\\sum _{e_1 \\sim N_{RT}(r)} \\sum _{e_2 \\sim N_{TR}(t)}P_{RT}(r, e_1) *P_{TR}(e_1, e_2) * P_{RT}(e_2, t)$ $F_{RH-HR-RT}(r, t)=\\sum _{e_1 \\sim N_{RH}(r)} \\sum _{e_2 \\sim N_{HR}(t)}P_{RH}(r, e_1) *P_{HR}(e_1, e_2) * P_{RT}(e_2, t)$ $F_{RT-HR-RT}(r, t)=\\sum _{e_1 \\sim N_{RT}(r)} \\sum _{e_2 \\sim N_{HR}(t)}P_{RT}(r, e_1) *P_{HR}(e_1, e_2) * P_{RT}(e_2, t)$"
],
[
"Feature Selection",
"In order to combine the above-mentioned features and models, we use grid search for feature selection.",
"The grid search will output the weights of the embedding models and manual features."
],
[
"Limitation Discussion.",
"In practical scene, the tail entities follow the long tail distribution.",
"Finding candidates from all entities involves a very complicated process with strategies like rules and approximate nearest neighbor searching.",
"However, different from our actual application, in this competition, candidates are provided from uniform distribution together with long tail ground truth according to the task description paper from [11].",
"Therefore, it comes a simple strategy to narrow the candidate choices by dropping tail entities with low frequencies counting from candidates.",
"But the long tail relations still depend on the Knowledge Embedding strategies, which can be found in our experimental results.",
"To narrow the gap between competition and practical application in the real scene, we suggest that the candidates should not be provided."
],
[
"Experimental Details",
"Original WikiKG90M dataset contains three time-stamps: September, October, and November of 2020, for training, validation, and testing, respectively, and only entities and relation types that appear in the earliest September knowledge graph are retained.",
"The default parameter settings all models are batch_size=1000, learning rate of the mlp (mlp_lr)=2e-5, learning rate decay step (lrd_step)=1e-5, learning rate of the embedding (lr)=0.1, gamma=12 and hidden_size=200.",
"Specifically, ote_size=20 in NOTE model.",
"For our final submission, we mix the training data and validation data."
],
[
"Experimental Results",
"Table REF shows the specific structure of all the models we use for ensemble, and report the final validation and test results of our ensemble method.",
"Table: Experimental Results: \"-\" means default parameter setting."
],
[
"Conclusion",
"In this paper, we present our solution for the final test.",
"First, we ensemble different models for representation learning.",
"Specifically, we propose the NOTE model to make the training process steady.",
"And the DeepWalk and the post-smoothing technique are used to capture the graph structure information among learned embeddings.",
"We also use recent advance models including QuatE, RotatE, TransE.",
"Second, we detail the manual feature engineering.",
"The selected features are used to adjust the predictions.",
"The experimental results show that our solution achieves excellent performance on the WikiKG90M dataset.",
"For the final submission, we apply the same strategy on the test candidates in final inference, and these features are not practical when considering ranking against all the entities."
]
] | 2107.01892 |
[
[
"Combining Orthology and Xenology Data in a Common Phylogenetic Tree"
],
[
"Abstract A rooted tree $T$ with vertex labels $t(v)$ and set-valued edge labels $\\lambda(e)$ defines maps $\\delta$ and $\\varepsilon$ on the pairs of leaves of $T$ by setting $\\delta(x,y)=q$ if the last common ancestor $\\text{lca}(x,y)$ of $x$ and $y$ is labeled $q$, and $m\\in \\varepsilon(x,y)$ if $m\\in\\lambda(e)$ for at least one edge $e$ along the path from $\\text{lca}(x,y)$ to $y$.",
"We show that a pair of maps $(\\delta,\\varepsilon)$ derives from a tree $(T,t,\\lambda)$ if and only if there exists a common refinement of the (unique) least-resolved vertex labeled tree $(T_{\\delta},t_{\\delta})$ that explains $\\delta$ and the (unique) least resolved edge labeled tree $(T_{\\varepsilon},\\lambda_{\\varepsilon})$ that explains $\\varepsilon$ (provided both trees exist).",
"This result remains true if certain combinations of labels at incident vertices and edges are forbidden."
],
[
"Introduction",
"An important task in evolutionary biology and genome research is to disentangle the mutual relationships of related genes.",
"The evolution of a gene family can be understood as a tree $T$ whose leaves are genes and whose inner vertices correspond to evolutionary events, in particular speciations (where genomes are propagated into different lineages that henceforth evolve independently), duplications (of genes within the same genome) and horizontal gene transfer (where copies of an individual genes are transferred into an unrelated species) [4].",
"Mathematically, these concepts are described in terms of rooted trees $T$ with vertex labels $t$ representing event types and edge labels $\\lambda $ distinguishing vertical and horizontal inheritance.",
"On the other hand, orthology (descent from a speciation) or xenology (if the common history involves horizontal transfer events) can be regarded as binary relation on the set $L$ of genes.",
"Given the orthology or xenology relationships, one then asks whether there exists a vertex or edge labeled tree $T$ with leaf set $L$ that “explains” the relations [9], [5].",
"Here, we ask when such relational orthology and xenology data are consistent.",
"A conceptually similar question is addressed in a very different formal setting in [15].",
"Instead of considering a single binary orthology or xenology relation, we consider here multiple relations of each type.",
"This is more conveniently formalized in terms of maps that assign finite sets of labels.",
"Two types of maps are of interest: Symbolic ultrametrics, i.e., symmetric maps determined by a label at the last common ancestor of two genes [2] generalize orthology.",
"Fitch maps, i.e., non-symmetric maps determined by the union of labels along the path connecting two genes [12], form a generalization of xenology.",
"For both types of maps unique least-resolved trees (minimal under edge-contraction) exist and can be constructed by polynomial time algorithms [2], [12].",
"Here we consider the problem of finding trees that are simultaneously edge- and vertex-labeled and simultaneously explain both types of maps.",
"We derive a simple condition for the existence of explaining trees and show that there is again a unique least-resolved tree among them.",
"We then consider a restricted version of problem motivated by concepts of observability introduced in [17]."
],
[
"Trees and Hierarchies",
"Let $T$ be a rooted tree with vertex set $V(T)$ , leaf set $L(T)\\subseteq V(T)$ , set of inner vertices $V^0(T)V(T)\\setminus L(T)$ , root $\\rho \\in V^0(T)$ , and edge set $E(T)$ .",
"An edge $e=\\lbrace u,v\\rbrace \\in E(T)$ is an inner edge if $u,v\\in V^0(T)$ .",
"The ancestor partial order on $V(T)$ is defined by $x\\preceq _T y$ whenever $y$ lies along the unique path connecting $x$ and the root.",
"We write $x \\prec _T y$ if $x\\preceq _T y$ and $x\\ne y$ .",
"For $v\\in V(T)$ , we set $\\operatorname{child}(v)\\lbrace u\\mid \\lbrace v,u\\rbrace \\in E(T),\\, u\\prec _T v\\rbrace $ .",
"All trees $T$ considered here are phylogenetic, i.e., they satisfy $|\\operatorname{child}(v)|\\ge 2$ for all $v\\in V^0(T)$ .",
"The last common ancestor of a vertex set $W\\subseteq V(T)$ is the unique $\\preceq _T$ -minimal vertex $\\operatorname{lca}_T(W)\\in V(T)$ satisfying $w\\preceq _T\\operatorname{lca}_T(W)$ for all $w\\in W$ .",
"For brevity, we write $\\operatorname{lca}_T(x,y)\\operatorname{lca}_T(\\lbrace x,y\\rbrace )$ .",
"Furthermore, we will sometimes write $vu\\in E(T)$ as a shorthand for “$\\lbrace u,v\\rbrace \\in E(T)$ with $u\\prec _T v$ .” We denote by $T(u)$ the subtree of $T$ rooted in $u$ and write $L(T(u))$ for its leaf set.",
"Furthermore, $L^T_v\\lbrace (x,y)\\mid x,y\\in L(T), \\operatorname{lca}_T(x,y)=v\\rbrace $ denotes the set of pairs of leaves that have $v$ as their last common ancestor.",
"By construction, $L^T_v\\cap L^T_{v^{\\prime }}=\\emptyset $ if $v\\ne v^{\\prime }$ .",
"Since $T$ is phylogenetic, we have $L^T_v\\ne \\emptyset $ for all $v\\in V^0(T)$ , i.e., $\\mathcal {L}(T)\\lbrace L^T_v\\mid v\\in V^0(T)\\rbrace $ is a partition of the set of distinct pairs of vertices.",
"A hierarchy on $L$ is set system $\\mathcal {H}\\subseteq 2^L$ such that (i) $L\\in \\mathcal {H}$ , (ii) $A\\cap B\\in \\lbrace A,B,\\emptyset \\rbrace $ for all $A,B\\in \\mathcal {H}$ , and (iii) $\\lbrace x\\rbrace \\in \\mathcal {H}$ for all $x\\in L$ .",
"There is a well-known bijection between rooted phylogenetic trees $T$ with leaf set $L$ and hierarchies on $L$ , see e.g.",
"[19].",
"It is given by $\\mathcal {H}(T) \\lbrace L(T(u)) \\mid u\\in V(T) \\rbrace $ ; conversely, the tree $T_{\\mathcal {H}}$ corresponding to a hierarchy $\\mathcal {H}$ is the Hasse diagram w.r.t.",
"set inclusion.",
"Thus, if $v=\\operatorname{lca}_T(A)$ for some $A\\subseteq L(T)$ , then $L(T(v))$ is the inclusion-minimal cluster in $\\mathcal {H}(T)$ that contains $A$ [11].",
"Let $T$ and $T^*$ be phylogenetic trees with $L(T)=L(T^*)$ .",
"We say that $T^*$ is a refinement of $T$ if $T$ can be obtained from $T^*$ by contracting a subset of inner edges or equivalently if and only if $\\mathcal {H}(T)\\subseteq \\mathcal {H}(T^*)$ .",
"Lemma 1 Let $T^*$ be a refinement of $T$ and $u^*v^*\\in E(T^*)$ .",
"Then there is a unique vertex $w\\in V(T)$ such that $L(T(w))\\in \\mathcal {H}(T)$ is inclusion-minimal in $\\mathcal {H}(T)$ with the property that $L(T^*(v^*))\\subsetneq L(T(w))$ .",
"In particular, if $\\operatorname{lca}_{T^*}(x,y)=u^*$ , then $\\operatorname{lca}_T(x,y)=w$ .",
"Let $u^*v^*\\in E(T^*)$ .",
"Since $\\mathcal {H}(T)\\subseteq \\mathcal {H}(T^*)$ , $L(T)=L(T^*)\\in \\mathcal {H}(T)$ and $v^*$ is not the root of $T^*$ , there is a unique inclusion-minimal $A\\in \\mathcal {H}(T)$ with $L(T^*(v^*))\\subsetneq A$ , which corresponds to a unique vertex $w\\in V(T)$ that satisfies $L(T(w))=A$ .",
"In the following, we denote with $w^*\\in V(T^*)$ the unique vertex that satisfies $A=L(T^*(w^*))$ , which exists since $A\\in \\mathcal {H}(T)\\subseteq \\mathcal {H}(T^*)$ .",
"Now let $x,y\\in L(T)$ be two leaves with $\\operatorname{lca}_{T^*}(x,y)=u^*$ .",
"From $v^*\\prec _{T^*} u^*$ , we obtain $L(T^*(v^*))\\subsetneq L(T^*(u^*))$ and $L(T^*(u^*))\\subseteq L(T^*(w^*))=L(T(w))$ .",
"Hence, we have $L(T^*(u^*))\\subseteq L(T(w))$ , which implies $x,y\\in L(T(w))$ and thus also $z\\operatorname{lca}_T(x,y)\\preceq _T w$ .",
"Denote by $z^*\\in V(T^*)$ the unique vertex in $T^*$ with $L(T^*(z^*))=L(T(z))$ .",
"Since $z\\preceq _T w$ , it satisfies $L(T^*(z^*))\\subseteq L(T^*(w^*))$ .",
"Since $x,y\\in L(T^*(z^*))\\cap L(T^*(u^*))\\ne \\emptyset $ , we either have $L(T^*(u^*))\\subseteq L(T^*(z^*))$ or $L(T^*(z^*))\\subsetneq L(T^*(u^*))$ .",
"In the second case, we obtain $\\operatorname{lca}_{T^*}(x,y)\\preceq _{T^*} z^*\\prec _{T^*} u^*$ , a contradiction to $\\operatorname{lca}_{T^*}(x,y)=u^*$ .",
"In the first case, we have $L(T^*(v^*))\\subsetneq L(T^*(u^*))\\subseteq L(T(z))\\subseteq L(T(w))$ .",
"Due to inclusion minimality of $L(T(w))$ we have $L(T(z)) = L(T(w))$ .",
"Thus $\\operatorname{lca}_{T}(x,y) = z = w$ .",
"Lemma REF ensures that, for every $u^*\\in V^0(T^*)$ , there is a unique $w\\in V(T)$ such that $\\operatorname{lca}_T(x,y)=w$ for all $(x,y)\\in L_{u^*}^{T^*}$ , and thus $L_{u^*}^{T^*}\\subseteq L_w^T$ .",
"Thus we have Corollary 1 If $T^*$ is a refinement of $T$ , then the partition $\\mathcal {L}(T^*)$ is a refinement of $\\mathcal {L}(T)$ ."
],
[
"Symbolic Ultrametrics",
"We write $L^{(2)}\\lbrace (x,y)\\mid x,y\\in L,\\, x\\ne y\\rbrace $ for the “off-diagonal” pairs of leafs and let $M$ be a finite set.",
"Definition 1 A tree $T$ with leaf set $L$ and labeling $t:V^0(T)\\rightarrow M$ of its inner vertices explains a map $\\delta :L^{(2)}\\rightarrow M$ if $t(\\operatorname{lca}(x,y)) = \\delta (x,y)$ for all distinct $x,y\\in L$ .",
"Such a map must be symmetric since $\\operatorname{lca}_T(x,y)=\\operatorname{lca}_T(y,x)$ for all $x,y\\in L$ .",
"A shown in [2], a map $\\delta :L^{(2)}\\rightarrow M$ can be explained by a labeled tree $(T,t)$ if and only if $\\delta $ is a symbolic ultrametric, i.e., iff, for all pairwise distinct $u,v,x,y\\in L$ holds (i) $\\delta (x,y)=\\delta (y,x)$ (symmetry), (ii) $\\delta (x,y)=\\delta (y,u)=\\delta (u,v)\\ne \\delta (y,v)=\\delta (x,v)=\\delta (x,u)$ is never satisfied (co-graph property), and (iii) $|\\lbrace \\delta (u,v),\\delta (u,x),\\delta (v,x)\\rbrace |\\le 2$ (exclusion of rainbow triangles).",
"In this case, there exists a unique least-resolved tree $(T_{\\delta },t_{\\delta })$ (that explains $\\delta $ ) with a discriminating vertex labeling $t_{\\delta }$ , i.e., $t_{\\delta }(x)\\ne t_{\\delta }(y)$ for all $ xy\\in E(T_{\\delta })$ [2], [9].",
"This tree $(T_{\\delta },t_{\\delta })$ is also called a discriminating representation of $\\delta $ [2].",
"The construction of symbolic ultrametrics could also be extended to maps $\\tilde{\\delta }:L^{(2)}\\rightarrow 2^M$ , i.e, to allow multiple labels at each vertex.",
"However, this does not introduce anything new.",
"To see this, we note that the sets of vertex pairs $L^T_v$ that share the same last common ancestor are pairwise disjoint.",
"In particular, $\\tilde{\\delta }$ thus must be a fixed element in $2^M$ on each $L^T_v$ , $v\\in V^0$ , and thus we think of the images $\\tilde{\\delta }(x,y)$ simply as single labels “associated to” elements in $2^M$ rather than sets of labels.",
"Lemma 2 Let $\\delta :L^{(2)}\\rightarrow M$ be a symbolic ultrametric with least-resolved tree $(T_{\\delta },t_{\\delta })$ .",
"Then there is a map $t:V(T)\\rightarrow M$ such that $(T,t)$ explains $\\delta $ if and only if $T$ is a refinement of $T_{\\delta }$ .",
"In this case, the map $t$ is uniquely determined by $T$ and $\\delta $ .",
"Suppose $(T,t)$ explains $\\delta $ and let $e=vu\\in E(T)$ be an edge with $\\delta (u)=\\delta (v)$ and $u\\prec v$ .",
"Note that both $u$ and $v$ must be inner vertices.",
"Let $T/e$ denote the tree obtained from $T$ by contracting the edge $e$ , i.e., removing $e$ from $T$ and identifying $u$ and $v$ .",
"We will keep the vertex $v$ in $T/e$ as placeholder for the identified vertices $u$ and $v$ .",
"By construction, $T/e$ has the clusters $\\mathcal {H}(T/e)=\\mathcal {H}(T)\\setminus \\lbrace L(T(u))\\rbrace $ .",
"Set $t_{T/e}(x)=t(x)$ for all $x\\in V^0(T)\\setminus \\lbrace u\\rbrace $ .",
"Clearly, $v$ is the unique vertex in $T/e$ such that $L((T/e)(v))$ is inclusion-minimal with property $L(T(u^{\\prime }))\\subsetneq L((T/e)(v))$ for any $u^{\\prime }\\operatorname{child}_{T}(u)$ .",
"Therefore, by Lemma REF , $\\operatorname{lca}_{T}(x,y)=u$ implies $\\operatorname{lca}_{T/e}(x,y)=v$ , and thus, we have $t(\\operatorname{lca}_T(x,y))=t_{T/e}(\\operatorname{lca}_{T/e}(x,y))$ for all $(x,y)\\in L^{(2)}$ , and thus $(T/e,t_{T/e})$ explains $\\delta $ .",
"Stepwise contraction of all edges whose endpoints have the same label eventually results in a tree $T^{\\prime }$ and a map $t^{\\prime }$ such that $t^{\\prime }(x)\\ne t^{\\prime }(y)$ for all edges of $T^{\\prime }$ .",
"Thus $(T^{\\prime },t^{\\prime })$ coincides with the unique discriminating representation of $\\delta $ , i.e., $(T^{\\prime },t^{\\prime })=(T_{\\delta },t_{\\delta })$ .",
"By construction, $T$ is a refinement of $T_{\\delta }$ .",
"Conversely, let $\\delta $ be a symbolic ultrametric with (unique) discriminating representation $(T_{\\delta },t_{\\delta })$ and let $T$ be a refinement of $T_{\\delta }$ .",
"By Cor.",
"REF , $\\mathcal {L}(T)$ is a refinement $\\mathcal {L}(T_{\\delta })$ .",
"Hence, the map $t:V^0(T)\\rightarrow M$ specified by $t(\\operatorname{lca}_T(x,y))t_{\\delta }(\\operatorname{lca}_{T_{\\delta }}(x,y))$ for all $(x,y)\\in L^{(2)}$ is well-defined.",
"By construction, therefore, $(T,t)$ explains $\\delta $ .",
"In particular, therefore, every refinement $T$ of $T_{\\delta }$ admits a vertex labeling $t$ such that $(T,t)$ explains $\\delta $ .",
"The choice of $t$ is unique since every inner vertex of a phylogenetic tree is the last common ancestor of at least one pair of vertices, and thus no relabeling of an inner vertex preserves the property that the resulting tree explains $\\delta $ ."
],
[
"Fitch Maps",
"Definition 2 A tree $T$ with edge labeling $\\lambda :E(T)\\rightarrow 2^N$ , with finite $N$ , explains a map $\\varepsilon :L^{(2)}\\rightarrow 2^N$ if for all $k\\in N$ holds: $k\\in \\varepsilon (x,y)$ iff $k\\in \\lambda (e)$ for some edge along the unique path in $T$ that connects $\\operatorname{lca}_T(x,y)$ and $y$ .",
"A map $\\varepsilon :L^{(2)}\\rightarrow 2^N$ that is explained by a tree $(T,\\lambda )$ in this manner is a Fitch map [12].",
"A Fitch map is called monochromatic if $|N|=1$ .",
"Like symbolic ultrametrics, Fitch maps are explained by unique least resolved trees.",
"The key construction is provided by the sets $U_{\\lnot m}[y]\\lbrace x\\in L\\setminus \\lbrace y\\rbrace \\mid m\\notin \\varepsilon (x,y)\\rbrace \\cup \\lbrace y\\rbrace $ for $y\\in L$ and $m\\in N$ .",
"Let us write $\\mathcal {N}_{\\varepsilon }\\lbrace U_{\\lnot m}[y] \\mid y\\in L,\\, m\\in N\\rbrace $ .",
"Then $\\varepsilon $ is a Fitch map if and only if (i) $\\mathcal {N}_{\\varepsilon }$ is hierarchy-like, i.e., $A\\cap B\\in \\lbrace A,B,\\emptyset \\rbrace $ for all $A,B\\in \\mathcal {N}_{\\varepsilon }$ and (ii) $|U_{\\lnot m}[y^{\\prime }]|\\le |U_{\\lnot m}[y]|$ for all $y\\in L$ , $m\\in N$ , and $y^{\\prime }\\in U_{\\lnot m}[y]$ [12].",
"Fitch maps allow some freedom in distributing labels on the edge set.",
"The precise notion of “least-resolved” thus refers to the fact that it is neither possible to contract edges nor to remove subsets of labels from an edge.",
"The unique least-resolved tree for a Fitch map $\\varepsilon $ , called the $\\varepsilon $ -tree $(T_{\\varepsilon },\\lambda _{\\varepsilon })$ , is determined by the hierarchy $\\mathcal {H}(T_{\\varepsilon })=\\mathcal {N}_{\\varepsilon }\\cup \\lbrace L\\rbrace \\cup \\big \\lbrace \\lbrace x\\rbrace \\mid x\\in L\\big \\rbrace $ and the labeling $\\lambda _{\\varepsilon }(\\operatorname{parent}(v),v)\\lbrace m\\in N \\mid \\exists y\\in L\\text{ s.t.\\ } L(T_{\\varepsilon }(v))=U_{\\lnot m}[y]\\rbrace $ for all $e=\\lbrace \\operatorname{parent}(v),v\\rbrace \\in E(T_{\\varepsilon })$ [12].",
"Let $(T,\\lambda )$ and $(T^{\\prime },\\lambda ^{\\prime })$ be two edge-labeled trees on the same leaf set and with $\\lambda :E(T)\\rightarrow 2^N$ and $\\lambda ^{\\prime }:E(T^{\\prime })\\rightarrow 2^N$ .",
"Then $(T,\\lambda )$ is a refinement of $(T^{\\prime },\\lambda ^{\\prime })$ , in symbols $(T^{\\prime },\\lambda ^{\\prime })\\le (T, \\lambda )$ if (i) $\\mathcal {H}(T^{\\prime })\\subseteq \\mathcal {H}(T)$ and (ii) if $L(T(v))=L(T^{\\prime }(v^{\\prime }))$ , then $\\lambda ^{\\prime }(\\operatorname{parent}_{T^{\\prime }}(v^{\\prime }),v^{\\prime })\\subseteq \\lambda (\\operatorname{parent}_{T}(v),v)$ .",
"Proposition 1 [12] If $(T,\\lambda )$ explains $\\varepsilon $ , then $(T_{\\varepsilon },\\lambda _{\\varepsilon })\\le (T,\\lambda )$ .",
"Furthermore, $(T_{\\varepsilon },\\lambda _{\\varepsilon })$ is the unique least-resolved tree that explains $\\varepsilon $ .",
"In particular, $(T_{\\varepsilon },\\lambda _{\\varepsilon })$ minimizes $\\ell _{\\min }\\sum _{e\\in E(T_{\\varepsilon })}|\\lambda _{\\varepsilon }(e)|$ .",
"Lemma 3 Let $\\varepsilon :L^{(2)}\\rightarrow 2^N$ be a Fitch map with least-resolved tree $(T_{\\varepsilon },\\lambda _{\\varepsilon })$ .",
"Then there exists an edge labeling $\\lambda :E(T)\\rightarrow 2^N$ such that $(T,\\lambda )$ explains $\\varepsilon $ if and only if $T$ is a refinement of $T_{\\varepsilon }$ .",
"Suppose $(T,\\lambda )$ explains $\\varepsilon $ .",
"By Prop.",
"REF , this implies $(T_{\\varepsilon },\\lambda _{\\varepsilon })\\le (T,\\lambda )$ , i.e., $T$ is a refinement of $T_{\\varepsilon }$ .",
"Conversely, let $\\varepsilon $ be a Fitch map with least-resolved tree $(T_{\\varepsilon },\\lambda _{\\varepsilon })$ and let $T$ be a refinement of $T_{\\varepsilon }$ .",
"Define, for all edges $\\lbrace \\operatorname{parent}_{T}(v),v\\rbrace \\in E(T)$ , the edge labeling $\\lambda (\\lbrace \\operatorname{parent}_{T}(v),v\\rbrace ) {\\left\\lbrace \\begin{array}{ll}\\lambda _{\\varepsilon }(\\operatorname{parent}_{T_{\\varepsilon }}(v^{\\prime }),v^{\\prime })& \\text{ if } L(T(v))=L(T_{\\varepsilon }(v^{\\prime })), \\\\\\emptyset & \\text{ otherwise.}\\end{array}\\right.",
"}$ The map $\\lambda $ is well-defined, since there is at most one $v^{\\prime }\\in V(T_{\\varepsilon })$ with $L(T(v))=L(T_{\\varepsilon }(v^{\\prime }))$ .",
"Claim.",
"$(T,\\lambda )$ and $(T_{\\varepsilon },\\lambda _{\\varepsilon })$ explain the same Fitch map $\\varepsilon $ .",
"By assumption, $(T_{\\varepsilon },\\lambda _{\\varepsilon })$ explains $\\varepsilon $ .",
"Let $(a,b)\\in L^{(2)}$ , $k\\in N$ , and let $\\varepsilon ^{\\prime }$ be the Fitch map explained by $(T,\\lambda )$ .",
"First, suppose $k\\in \\varepsilon (a,b)$ , i.e., there is an edge $e^{\\prime }=\\lbrace \\operatorname{parent}_{T_{\\varepsilon }}(w^{\\prime }),w^{\\prime }\\rbrace $ with $k\\in \\lambda _{\\varepsilon }(e^{\\prime })$ such that $w^{\\prime }\\prec _{T_{\\varepsilon }}\\operatorname{lca}_{T_{\\varepsilon }}(a,b)$ by the definition of Fitch maps.",
"We have $a\\notin L(T_{\\varepsilon }(w^{\\prime }))$ .",
"Since $T$ is a refinement of $T_{\\varepsilon }$ , there is a vertex $w\\in V(T)$ with $L(T(w))=L(T_{\\varepsilon }(w^{\\prime }))$ .",
"In particular, therefore, $\\lambda (\\lbrace \\operatorname{parent}_{T}(w),w\\rbrace )=\\lambda _{\\varepsilon }(e^{\\prime })$ .",
"This together with the fact that $a\\notin L(T_{\\varepsilon }(w^{\\prime }))=L(T(w))$ immediately implies $k\\in \\varepsilon ^{\\prime }(a,b)$ .",
"Now suppose $k\\in \\varepsilon ^{\\prime }(a,b)$ .",
"Hence, there is an edge $e=\\lbrace \\operatorname{parent}_{T}(v),v\\rbrace $ with $v\\prec _{T}\\operatorname{lca}_{T}(a,b)$ and $k\\in \\lambda (e)$ .",
"By construction of $\\lambda $ , the latter implies that there is a vertex $v^{\\prime }\\in V(T_{\\varepsilon })$ with $L(T(v))=L(T_{\\varepsilon }(v^{\\prime }))$ and, in particular, $k\\in \\lambda _{\\varepsilon }(\\operatorname{parent}_{T_{\\varepsilon }}(v^{\\prime }),v^{\\prime })$ .",
"The latter together with $a\\notin L(T(v))=L(T_{\\varepsilon }(v^{\\prime }))$ implies that $k\\in \\varepsilon (a,b)$ .",
"Since $(a,b)\\in L^{(2)}$ and $k\\in N$ were chosen arbitrarily, we conclude that $\\varepsilon =\\varepsilon ^{\\prime }$ , and thus, $(T,\\lambda )$ also explains $\\varepsilon $ .",
"The labeling $\\lambda $ defined in Eq.",
"(REF ) satisfies $\\ell _{\\min } = \\sum _{e\\in T(e)} |\\lambda (e)|$ by construction and Prop.",
"REF .",
"Furthermore, we observe that $(T^*,\\lambda ^*)$ is obtained from $(T,\\lambda )$ by contracting only edges with $\\lambda (e)=\\emptyset $ .",
"More precisely, $e$ is contracted if and only if $e$ is an inner edge with $\\lambda (e)=\\emptyset $ .",
"This implies Corollary 2 Suppose $(T,\\lambda ^{\\prime })$ explains the Fitch map $\\varepsilon $ .",
"Then $\\lambda : E(T)\\rightarrow 2^N$ given by Eq.",
"(REF ) is the unique labeling such that $(T,\\lambda )$ explains $\\varepsilon $ and $\\sum _{e\\in E(T)}|\\lambda (e)|=\\ell _{\\min }$ .",
"Suppose $(T,\\lambda ^{\\prime \\prime })$ explains $\\varepsilon $ and $\\sum _{e\\in E(T)}|\\lambda ^{\\prime \\prime }(e)|=\\ell _{\\min }$ .",
"By Prop.",
"REF , we have $(T_{\\varepsilon },\\lambda _{\\varepsilon })\\le (T,\\lambda ^{\\prime \\prime })$ and thus $\\lambda _{\\varepsilon }(\\operatorname{parent}_{T_{\\varepsilon }}(v^{\\prime }),v^{\\prime })\\subseteq \\lambda ^{\\prime \\prime }(\\operatorname{parent}_{T}(v),v)$ if $L(T_{\\varepsilon }(v^{\\prime }))=L(T(v))$ .",
"Since, moreover, $\\lambda _{\\varepsilon }(\\operatorname{parent}_{T_{\\varepsilon }}(v^{\\prime }),v^{\\prime })=\\lambda (\\operatorname{parent}_{T}(v),v)$ if $L(T_{\\varepsilon }(v^{\\prime }))=L(T(v))$ by Eq.",
"(REF ), minimality of $\\lambda ^{\\prime \\prime }$ implies $\\lambda ^{\\prime \\prime }=\\lambda $ ."
],
[
"Tree-like Pairs of Maps",
"Symbolic ultrametrics and Fitch maps on $L^{(2)}$ derive from trees in very different ways by implicitly leveraging information about inner vertices and edges of the a priori unknown tree.",
"It is of interest, therefore, to know when they are consistent in the sense that they can be simultaneously explained by a tree.",
"Definition 3 An ordered pair $(\\delta ,\\varepsilon )$ of maps $\\delta :L^{(2)}\\rightarrow M$ and $\\varepsilon :L^{(2)}\\rightarrow 2^N$ is tree-like if there is a tree $T$ endowed with a vertex labeling $t:V^0(T)\\rightarrow M$ and edge labeling $\\lambda :L^{(2)}\\rightarrow 2^N$ such that $(T,t)$ explains $\\delta $ and $(T,\\lambda )$ explains $\\varepsilon $ .",
"Naturally, we ask when $(\\delta ,\\varepsilon )$ is explained by a vertex and edge labeled tree $(T,t,\\lambda )$ , i.e., when $(\\delta ,\\varepsilon )$ is a tree-like pair of maps on $L^{(2)}$ .",
"Furthermore, we ask whether a tree-like pair of maps is again explained by a unique least-resolved tree $(T^*,t^*,\\lambda ^*)$ .",
"Theorem 1 Let $\\delta :L^{(2)}\\rightarrow M$ and $\\varepsilon :L^{(2)}\\rightarrow 2^N$ .",
"Then $(\\delta ,\\varepsilon )$ is tree-like if and only if $\\delta $ is a symbolic ultrametric.",
"$\\varepsilon $ is a Fitch map.",
"$\\mathcal {H}^*\\mathcal {H}(T_{\\delta })\\cup \\mathcal {H}(T_{\\varepsilon })$ is a hierarchy.",
"In this case, there is a unique least-resolved vertex and edge labeled tree $(T^*,t^*,\\lambda ^*)$ explaining $(\\delta ,\\varepsilon )$ .",
"The tree $T^*$ is determined by $\\mathcal {H}(T^*)=\\mathcal {H}^*$ , the vertex labeling $t^*$ is uniquely determined by $t_{\\delta }$ and the edge labeling $\\lambda ^*$ with minimum value of $\\sum _{e\\in E(T^*)} |\\lambda ^*(e)|$ is uniquely determined by $\\lambda _{\\varepsilon }$ .",
"Suppose $(\\delta ,\\varepsilon )$ is tree-like, i.e., there is a tree $(T,t,\\lambda )$ such that $(T,t)$ explains $\\delta $ and $(T,\\lambda )$ explains $\\varepsilon $ .",
"Thus $\\delta $ is a symbolic ultrametric and $\\varepsilon $ is a Fitch map.",
"Furthermore, $T$ is a refinement of least-resolved trees $T_{\\delta }$ and $T_{\\varepsilon }$ because of the uniqueness of these least-resolved trees, and we have $\\mathcal {H}(T_{\\delta })\\subseteq \\mathcal {H}(T)$ and $\\mathcal {H}(T_{\\varepsilon })\\subseteq \\mathcal {H}(T)$ and thus $\\mathcal {H}^*\\subseteq \\mathcal {H}(T)$ .",
"Since $\\mathcal {H}(T)$ is a hierarchy and the subset $\\mathcal {H}^*$ contains both $L$ and all singletons $\\lbrace x\\rbrace $ with $x\\in L$ , $\\mathcal {H}^*$ is a hierarchy.",
"Conversely, suppose conditions (1), (2), and (3) are satisfied.",
"The first two conditions guarantee the existence of the least-resolved tree $(T_{\\delta },t_{\\delta })$ and $(T_{\\varepsilon },\\lambda _{\\varepsilon })$ explaining $\\delta $ and $\\varepsilon $ , respectively.",
"Thus $\\mathcal {H}^*=\\mathcal {H}(T_{\\delta })\\cup \\mathcal {H}(T_{\\varepsilon })$ is well-defined.",
"Condition (3) stipulates that $\\mathcal {H}^*$ is a hierarchy and thus there is a unique tree $T^*$ such that $\\mathcal {H}(T^*)=H^*$ , which by construction is a refinement of both $T_{\\delta }$ and $T_{\\varepsilon }$ .",
"By Lemmas REF and REF , $T^*$ can be equipped with a vertex-labeling $t^*$ and an edge-labeling $\\lambda ^*$ such that $(T^*,t^*)$ explains $\\delta $ and $(T^*,\\lambda ^*)$ explains $\\varepsilon $ , respectively.",
"Thus $(\\delta ,\\varepsilon )$ is tree-like.",
"We now show that $(T^*,t^*,\\lambda ^*)$ is least-resolved w.r.t.",
"$(\\delta ,\\varepsilon )$ and thus that for every $e\\in E(T^*)$ , the tree $T^{\\prime }T^*/e$ does not admit a vertex labeling $t^{\\prime }:V^0(T^{\\prime })\\rightarrow M$ and an edge-labeling $\\lambda ^{\\prime }:E(T^{\\prime })\\rightarrow 2^N$ such that $(T^{\\prime },t^{\\prime },\\lambda ^{\\prime })$ explains $(\\delta ,\\varepsilon )$ .",
"Let $e= \\lbrace \\operatorname{parent}(v),v\\rbrace \\in E(T^*)$ .",
"Hence, $L(T^*(v)) \\in \\mathcal {H}(T^*)$ .",
"If $v\\in L(T^*)$ , then we have $L(T^*)\\ne L(T^{\\prime })$ and the claim trivially holds.",
"Thus suppose that $v\\in V^0(T)$ in the following.",
"Since the edge $e$ is contracted in $T^{\\prime }$ , we have $\\mathcal {H}(T^{\\prime }) = \\mathcal {H}(T^*)\\setminus \\lbrace L(T(v))\\rbrace $ and thus, $\\mathcal {H}(T_\\delta )\\lnot \\subseteq \\mathcal {H}(T^{\\prime })$ or $\\mathcal {H}(T_{\\varepsilon })\\lnot \\subseteq \\mathcal {H}(T^{\\prime })$ .",
"Thus $T^{\\prime }$ is not a refinement of $T_{\\delta }$ or $T_{\\varepsilon }$ .",
"By Lemma REF and REF , respectively, this implies that there is no $t^{\\prime }$ such that $(T^{\\prime },t^{\\prime })$ explains $\\delta $ or no $\\lambda ^{\\prime }$ such that $(T^{\\prime },\\lambda ^{\\prime })$ explains $\\varepsilon $ , respectively.",
"Thus $(T^*,t^*,\\lambda ^*)$ is least-resolved w.r.t.",
"$(\\delta ,\\varepsilon )$ .",
"It remains to show that $(T^*,t^*,\\lambda ^*)$ is unique.",
"Since $T^*$ is uniquely determined by $\\mathcal {H}^*$ , it suffices to show that the labeling of $T^*$ is unique.",
"This, however, follows immediately from Lemma REF and Cor.",
"REF , respectively.",
"We note that every refinement $T$ of the least-resolved tree $(T^*,t^*,\\lambda ^*)$ admits a vertex labeling $t:V^0(T)\\rightarrow M$ and an edge labeling $\\lambda : E(T)\\rightarrow 2^N$ such that $(T,t,\\lambda )$ explains $(\\delta ,\\varepsilon )$ .",
"Theorem 2 Given two maps $\\delta :L^{(2)}\\rightarrow M$ and $\\varepsilon :L^{(2)}\\rightarrow 2^N$ it can be decided in $O(|L|^2 |N|)$ whether $(\\delta ,\\varepsilon )$ is tree-like.",
"In the positive case, the unique least-resolved tree $(T^*,t^*,\\lambda ^*)$ can be obtained with the same effort.",
"Based on Theorem REF , a possible algorithm consists of three steps: (i) check whether $\\delta $ is a symbolic ultrametric, (ii) check whether $\\varepsilon $ is a Fitch map and, if both statements are true, (iii) compute $\\mathcal {H}^*\\mathcal {H}(T_{\\delta })\\cup \\mathcal {H}(T_{\\varepsilon })$ and use this information to compute the unique least-resolved vertex and edge labeled tree $(T^*,t^*,\\lambda ^*)$ .",
"By [12], the decision whether $\\varepsilon $ is a Fitch map and, in the positive case, the construction of the least-resolved tree $(T_{\\varepsilon },\\lambda _{\\varepsilon })$ can be achieved in $O(|L|^2 |N|)$ time.",
"Moreover, it can be verified in $O(|L|^2)$ whether or not a given map $\\delta $ is a symbolic ultrametric, and, in the positive case, the discriminating tree $(T_{\\delta },t_{\\delta })$ can be computed within the same time complexity (cf.",
"[14]).",
"The common refinement $T$ with $\\mathcal {H}(T)=\\mathcal {H}(T_{\\delta })\\cup \\mathcal {H}(T_{\\varepsilon })$ can be computed in $O(|L|)$ time using LinCR [18].",
"The edge labels $\\lambda ^*$ are then carried over from $(T_{\\varepsilon },\\lambda _{\\varepsilon })$ using the correspondence between $u^*v^*\\in E(T^*)$ and $uv\\in E(T_{\\varepsilon })$ iff $L(T^*(v^*))=L(T_\\varepsilon (v))$ , otherwise $\\lambda ^*(\\lbrace u^*,v^*\\rbrace )=\\emptyset $ .",
"This requires $O(|L|\\cdot |N|)$ operations.",
"The vertex labels can then be assigned by computing, for all $(x,y)\\in L^{(2)}$ , the vertex $v=\\operatorname{lca}_{T^*}(x,y)$ and assigning $t^*(v)=\\delta (x,y)$ in quadratic time using a fast last common ancestor algorithm [7].",
"Thus we arrive at a total performance bounds of $O(|L|^2|N|))$ ."
],
[
"Tree-like Pairs of Maps with Constraints",
"One interpretation of tree-like pairs of maps $(\\delta ,\\varepsilon )$ is to consider $\\delta $ as the orthology relation and $\\varepsilon $ as the xenology relation.",
"In such a setting, certain vertex labels $t(v)$ preclude some edge labels $\\lambda (\\lbrace v,u\\rbrace )$ with $u\\prec v$ .",
"For example, a speciation vertex cannot be the source of a horizontal transfer edge.",
"We use the conventional notations $t(u)=$ and $t(v)=\\square $ for speciation and duplication vertices [6], respectively, set $t(u)=\\triangle $ for a third vertex type, and consider the monochromatic Fitch map $\\varepsilon \\colon L^{(2)}\\rightarrow \\lbrace \\emptyset ,\\mathbb {I}\\rbrace $ .",
"Thus, we require that $\\lambda (\\lbrace v,u\\rbrace )=\\mathbb {I}$ and $u\\prec _T v$ implies $t(v)=\\triangle $ [17], [20], [1].",
"This condition simply states that neither a speciation nor a gene duplication is the source of a horizontal transfer.",
"In [17], we considered evolutionary scenarios that satisfy another rather stringent observability condition: (C) For every $v\\in V^0(T)$ , there is a child $u\\in \\operatorname{child}(v)$ such that $\\lambda (\\lbrace v,u\\rbrace )=\\emptyset $ .",
"We call a Fitch map $\\lambda $ that satisfies (C) a type-C Fitch map.",
"In this case, for every $v\\in V^0(T)$ , there is a leaf $x\\in L(T(v))$ such that $\\lambda (e)=\\emptyset $ for all edges along the path from $v$ to $x$ .",
"As an immediate consequence of (C), we observe that, given $|L|\\ge 2$ , for every $x\\in L$ there is a $y\\ne x$ such that $\\varepsilon (x,y)=\\emptyset $ .",
"This condition is not sufficient, however, as the following example shows.",
"Consider the tree $((x,y),(x^{\\prime },y^{\\prime }))$ in Newick notation, with the edges in the two cherries $(x,y)$ and $(x^{\\prime },y^{\\prime })$ being labeled with $\\emptyset $ , and two $\\mathbb {I}$ -labeled edges incident to the root.",
"Then, for every $z\\in L$ , we have $\\varepsilon (z,z^{\\prime })=\\emptyset $ , where $z^{\\prime }$ is the sibling of $z$ , but condition (C) is not satisfied.",
"In a somewhat more general setting, we formalize these two types of labeling constraints as follows: Definition 4 Let $\\delta :L^{(2)}\\rightarrow M$ and $\\varepsilon :L^{(2)}\\rightarrow 2^N$ be two maps and $M_{\\emptyset }\\subseteq M$ .",
"Then, $(\\delta ,\\varepsilon )$ is $M_{\\emptyset }$ -tree-like if there is a tree $(T,t,\\lambda )$ that explains $(\\delta ,\\varepsilon )$ and the labeling maps $t:V^0(T)\\rightarrow M$ and $\\lambda :E(T)\\rightarrow 2^N$ satisfy (C) and (C1) If $t(v)\\in M_{\\emptyset }$ , then $\\lambda (\\lbrace v,u\\rbrace )=\\emptyset $ for all $u\\in \\operatorname{child}(v)$ .",
"Hence, $M_{\\emptyset }$ puts extra constraints to the vertex and edge labels on trees that satisfy (C) and explain $(\\delta ,\\varepsilon )$ .",
"Note, an $\\emptyset $ -tree-like ($M_{\\emptyset } = \\emptyset $ ) must only satisfy (C) and (C1) can be omitted.",
"Theorem 3 Let $\\delta :L^{(2)}\\rightarrow M$ and $\\varepsilon :L^{(2)}\\rightarrow 2^N$ be two maps and $M_{\\emptyset }\\subseteq M$ .",
"Then, $(\\delta ,\\varepsilon )$ is $M_{\\emptyset }$ -tree-like if and only if $(\\delta ,\\varepsilon )$ is tree-like and its least-resolved tree $(T^*,t^*,\\lambda ^*)$ satisfies (C) and (C1).",
"If $(\\delta ,\\varepsilon )$ is tree-like and its least-resolved tree $(T^*,t^*,\\lambda ^*)$ satisfies (C) and (C1), then $(\\delta ,\\varepsilon )$ is $M_{\\emptyset }$ -tree-like by definition.",
"For the converse, suppose $(\\delta ,\\varepsilon )$ is $M_{\\emptyset }$ -tree-like and let $(T,t,\\lambda )$ be a vertex and edge labeled tree that explains $(\\delta ,\\varepsilon )$ and satisfies (C) and (C1).",
"Let $\\lambda ^{\\prime }$ be the edge labeling for $T$ as specified in Eq.",
"(REF ) where $(T_{\\varepsilon },\\lambda _{\\varepsilon })$ is replaced by $(T^*,\\lambda ^*)$ .",
"By the arguments in the proof of Lemma REF , $(T,\\lambda ^{\\prime })$ still explains $\\varepsilon $ and hence, $(T,t,\\lambda ^{\\prime })$ explains $(\\delta ,\\varepsilon )$ .",
"Moreover, since $\\ell _{\\min }\\sum _{e\\in E(T^*)} |\\lambda ^*(e)|$ and by construction of $\\lambda ^{\\prime }$ , we have $\\ell _{\\min } = \\sum _{e\\in E(T)} |\\lambda ^{\\prime }(e)|$ .",
"Since $(T^*,\\lambda ^*)\\le (T,\\lambda ^{\\prime })$ , it must hold $\\lambda _{\\varepsilon }(e^{\\prime })\\subseteq \\lambda ^{\\prime }(e)$ for all $e^{\\prime }=\\operatorname{parent}(v^{\\prime })v^{\\prime }\\in E(T_{\\varepsilon })$ and $e=\\operatorname{parent}(v)v\\in E(T)$ with $L(T(v)) = L(T_{\\varepsilon }(v^{\\prime }))$ .",
"Since $\\lambda ^{\\prime }$ is minimal by construction, we have $\\lambda _{\\varepsilon }(e^{\\prime }) = \\lambda ^{\\prime }(e)$ for all corresponding edges $e$ and $e^{\\prime }$ .",
"In particular, it must hold that $\\lambda (e)=\\emptyset $ implies $\\lambda ^{\\prime }(e) = \\emptyset $ for all $e\\in E(T)$ .",
"To see this, assume for contradiction there is some edge $e=uv\\in E(T)$ with $\\lambda (e)=\\emptyset $ but $\\lambda ^{\\prime }(e) \\ne \\emptyset $ .",
"Since $(T,\\lambda )$ satisfies (C), there is a path from $u$ to some leaf $y\\in L(T)$ that consists of edges $f$ with label $\\lambda (f) = \\emptyset $ only and that contains the edge $e$ .",
"Hence, for $x\\in L(T(u))\\setminus L(T(v))$ , we have $\\operatorname{lca}_T(x,y) = u $ and thus, $\\varepsilon (x,y)=\\emptyset $ .",
"However, since we assume that $\\lambda ^{\\prime }(e)=N^{\\prime }\\ne \\emptyset $ , we obtain $N^{\\prime }\\subseteq \\varepsilon (x,y)\\ne \\emptyset $ ; a contradiction.",
"Now it is easy to verify that $(T,t,\\lambda ^{\\prime })$ still satisfies (C) and (C1).",
"Figure: Effect of an edge contraction on paths in TT.",
"All pathstraversing the contracted edge e=uve=uv in TT correspond to paths inT/eT/e in which ee is contracted.",
"All other path remainunchanged.",
"Furthermore w e =u ' w_e=u^{\\prime }, i.e., the edge contractioncorresponds to the deletion of L(T(v))L(T(v)) from ℋ(T)\\mathcal {H}(T).Now consider edge contractions, Fig.",
"REF .",
"To obtain $T^*$ we are only allowed to contract edges $e = uv\\in E(T)$ that satisfy $t(u)=t(v)$ and $\\lambda ^{\\prime }(e)=\\emptyset $ .",
"The latter follows from the fact that edges $uv$ with $t(u)\\ne t(v)$ cannot be contracted without losing the information of at least one of the labels $t(u)$ or $t(v)$ and minimality of $\\lambda ^{\\prime }$ , since otherwise the labels $\\lambda ^{\\prime }(e)$ do not contribute to the explanation of the Fitch map and thus would have been removed in the construction of $\\lambda ^{\\prime }$ .",
"For such an edge $e$ , the tree $(T/e,t_{T/e},\\lambda ^{\\prime }_{T/e})$ is obtained by contracting the edge $e=uv$ to a new vertex $w_e$ and assigning $t_{T/e}(w_e)=t(v)=t(u)$ and keeping the edge labels of all remaining edges.",
"The tree $(T/e,t_{T/e},\\lambda ^{\\prime }_{T/e})$ then explains $(\\delta ,\\varepsilon )$ .",
"To see this, we write $y\\operatorname{lca}_T(a,b)$ and $y^{\\prime }\\operatorname{lca}_{T/e}(a,b)$ for distinct $a,b\\in L$ and compare for $c\\in \\lbrace a,b\\rbrace $ the path $P_{yc}$ in $T$ and $P^{\\prime }_{y^{\\prime }c}$ in $T/e$ .",
"If $y=u$ or $y=v$ then $y^{\\prime }=w_e$ .",
"The paths therefore either consist only of corresponding edges, in which case the edge labels are the same, or they differ exactly by the contraction of $e$ .",
"The latter does not affect the explanation of $\\varepsilon (a,b)$ because $\\lambda ^{\\prime }(e)=\\emptyset $ .",
"Since $t(u)=t(v)$ , contraction of $uv$ also does not affect $\\delta $ .",
"In particular, therefore, neither $u$ nor $v$ is a leaf, i.e., $e$ is an inner edge.",
"Condition (C) is trivially preserved under contraction of inner edges.",
"Suppose $t(v)=t(u)\\in M_{\\emptyset }$ and thus $t_{T/e}(w_e)\\in M_{\\emptyset }$ .",
"Since $(T,t,\\lambda ^{\\prime })$ satisfies (C1) we have $\\lambda ^{\\prime }(\\lbrace v,u^{\\prime }\\rbrace )=\\lambda ^{\\prime }(\\lbrace u,u^{\\prime \\prime }\\rbrace ) = \\emptyset $ for all $u^{\\prime }\\in \\operatorname{child}(v)$ and all $u^{\\prime \\prime }\\in \\operatorname{child}(u)$ and thus after contracting $e$ it holds that $\\lambda ^{\\prime }_{T/e}(w_e,w^{\\prime })=\\emptyset $ for all $w^{\\prime }\\in \\operatorname{child}_{T/e}(w_e)=\\operatorname{child}_{T}(v)\\mathbin {\\mathchoice{\\leavevmode \\vtop {{\\hfil \\m@th \\displaystyle #\\hfil \\cr \\cup \\cr \\cdot \\crcr }}}{\\leavevmode \\vtop {{\\hfil \\m@th \\textstyle #\\hfil \\cr \\cup \\cr \\cdot \\crcr }}}{\\leavevmode \\vtop {{\\hfil \\m@th \\scriptstyle #\\hfil \\cr \\cup \\cr \\cdot \\crcr }}}{\\leavevmode \\vtop {{\\hfil \\m@th \\scriptscriptstyle #\\hfil \\cr \\cup \\cr \\cdot \\crcr }}}}\\operatorname{child}_{T}(u)$ .",
"Otherwise, $t(u)=t(v)\\notin M_{\\emptyset }$ and thus by construction $t_{T/e}(w_e)\\notin M_{\\emptyset }$ .",
"In summary, $(T/e,t^{\\prime },\\lambda )$ satisfies (C) and (C1).",
"Repeating this coarse graining until no further contractible inner edges are available results in the unique least-resolved tree $(T^*,t^*,\\lambda ^*)$ .",
"Since the unique least-resolved tree $(T^*,t^*,\\lambda ^*)$ can be computed in quadratic time by Thm.",
"REF , and it suffices by Thm.",
"REF to check (C) and (C1) for $(T^*,t^*,\\lambda ^*)$ , the same performance bound applies to the recognition of constrained tree-like pairs of maps.",
"We note that an analogous result holds if only (C) or only (C1) is required for $(T,t,\\lambda )$ .",
"Furthermore, one can extend (C1) in such a way that for a set $\\mathcal {Q}$ of pairs $(q,m)$ with $q\\in M$ and $m\\in N$ of labels that are incompatible at a vertex $v$ and an edge $vv^{\\prime }$ with $v^{\\prime }\\in \\operatorname{child}(v)$ .",
"The proof of Thm.",
"REF still remains valid since also in this case no forbidden combinations of vertex an edge colors can arise from contracting an edge $e=uv$ with $t(u)=t(v)$ .",
"In the special case $\\delta (x,y)=1\\notin M_{\\emptyset }$ for all $(x,y)\\in L^{(2)}$ , one obtains $t^*(u)=1$ for all $u\\in V(T^*)$ and thus $(T^*,\\lambda ^*)=(T_{\\varepsilon },\\lambda _{\\varepsilon })$ and (C1) imposes no constraint.",
"Hence, Thm.",
"REF specializes to Corollary 3 A Fitch map $\\varepsilon $ is type-C if and only if its least-resolved tree $(T_{\\varepsilon },\\lambda _{\\varepsilon })$ satisfies (C).",
"In [17] a stronger version of condition (C) has been considered: (C2) If $\\lambda (\\lbrace v,u\\rbrace )\\ne \\emptyset $ for some $u\\in \\operatorname{child}(v)$ , then $\\lambda (\\lbrace v,u^{\\prime }\\rbrace )=\\emptyset $ for all $u^{\\prime }\\in \\operatorname{child}(v)\\setminus \\lbrace u\\rbrace $ .",
"This variant imposes an additional condition on the edges $e=uv$ that can be contracted.",
"More precisely, an inner edge of $(T,t,\\lambda )$ can be contracted without losing the explanation of $(\\delta ,\\varepsilon )$ and properties (C1) and (C2) if and only if (i) $t(u)=t(v)$ , (ii) $\\lambda (e)=\\emptyset $ and (iii) at most one of the the edges $uu^{\\prime }$ , $u^{\\prime }\\in \\operatorname{child}(u)$ and $vv^{\\prime }$ , $v^{\\prime }\\in \\operatorname{child}(v)$ has a non-empty label.",
"Now consider two consecutive edges $uv$ and $vw$ with $t(u)=t(v)=t(w)$ , $\\lambda (\\lbrace u,v\\rbrace )=\\lambda (\\lbrace v,w\\rbrace )=\\emptyset $ and suppose there is $u^{\\prime }\\operatorname{child}(u)$ with $\\lambda (\\lbrace u,u^{\\prime }\\rbrace )\\ne \\emptyset $ , $w^{\\prime }\\in \\operatorname{child}(w)$ with $\\lambda (\\lbrace w,w^{\\prime }\\rbrace )\\ne \\emptyset $ , and $\\lambda (\\lbrace v,v^{\\prime }\\rbrace )=\\emptyset $ for all $v^{\\prime }\\in \\operatorname{child}(v)$ .",
"Then one can contract either $uv$ or $vw$ but not both edges.",
"Thus least-resolved trees explaining $(\\delta ,\\varepsilon )$ and satisfying (C1) and (C2) are no longer unique."
],
[
"Concluding Remark",
"Here we have shown that symbolic ultrametrics and Fitch maps can be combined by the simple and easily verified condition that $\\mathcal {H}(T_{\\delta })\\cup \\mathcal {H}(T_{\\varepsilon })$ is again a hierarchy (Thm.",
"REF ), i.e., that the two least-resolved trees have a common refinement.",
"The least-resolved tree $(T^*,t^*,\\lambda ^*)$ that simultaneously explains both $\\delta $ and $\\varepsilon $ is unique in this case and can be computed in quadratic time if the label set $N$ is bounded and $O(|L|^2 |N|)$ time in general.",
"The closely related problem of combining a hierarchy and symmetrized Fitch maps, defined by $m\\in \\varepsilon (x,y)$ iff there is an edge $e$ with $m\\in \\lambda (e)$ along the path from $x$ to $y$ [10], is NP-complete [13].",
"It appears that the main difference is the fact that symmetrized Fitch maps do not have a unique least-resolved tree as explanation.",
"The distinction between much simpler problems in the directed setting and hard problems in the undirected case is also reminiscent of the reconciliation problem for trees, which are easy for rooted trees and hard for unrooted trees, see e.g.",
"[3].",
"We have also seen that certain restrictions on the Fitch maps that are related to the “observability” of horizontal transfer do not alter the complexity of the problem.",
"These observability conditions are defined in terms of properties of the explaining trees, raising the question whether these constraints also have a natural characterization as properties of the Fitch maps.",
"On a more general level, both symbolic ultrametrics and Fitch maps arise from evolutionary scenarios comprising an embedding of the gene tree $T$ into a species tree, with labeling functions $t$ and $\\lambda $ on $T$ encoding event-types and distinctions in the evolutionary fate of offsprings, respectively.",
"Here we have focused entirely on gene trees with given labels.",
"The embeddings into species trees are known to impose additional constraints [8], [16].",
"Acknowledgments.",
"This work was supported in part by the Deutsche Forschungsgemeinschaft."
]
] | 2107.01893 |
[
[
"Robust Online Convex Optimization in the Presence of Outliers"
],
[
"Abstract We consider online convex optimization when a number k of data points are outliers that may be corrupted.",
"We model this by introducing the notion of robust regret, which measures the regret only on rounds that are not outliers.",
"The aim for the learner is to achieve small robust regret, without knowing where the outliers are.",
"If the outliers are chosen adversarially, we show that a simple filtering strategy on extreme gradients incurs O(k) additive overhead compared to the usual regret bounds, and that this is unimprovable, which means that k needs to be sublinear in the number of rounds.",
"We further ask which additional assumptions would allow for a linear number of outliers.",
"It turns out that the usual benign cases of independently, identically distributed (i.i.d.)",
"observations or strongly convex losses are not sufficient.",
"However, combining i.i.d.",
"observations with the assumption that outliers are those observations that are in an extreme quantile of the distribution, does lead to sublinear robust regret, even though the expected number of outliers is linear."
],
[
"Introduction",
"Methods for online convex optimization (OCO) are designed to work even in the presence of adversarially generated data [22], [43], [7], but this is only possible because strong boundedness assumptions are imposed on the losses that limit the influence of individual data points.",
"On the other hand, the most practically successful methods do not enforce an a priori specified bound on the losses, but instead adapt to the norms of the observed gradients or to the observed loss range.",
"For example, the regret bound for AdaGrad adapts to the ranges of the gradient components per dimension [19], the regret bound for online ridge regression scales with the largest observed loss [45], the regret bound for AdaHedge in the prediction with experts setting scales with the observed loss range of the experts [14], the regret bound for online gradient descent on strongly convex losses scales with the maximum gradient norm squared [24], etc.",
"In all such cases a small number of outliers with large gradients among an otherwise benign dataset can significantly worsen performance.",
"This is also clear directly from the algorithms themselves, where we see that large gradients have the effect of significantly decreasing the effective step size for all subsequent data points, leading to slower learning.",
"Extreme outliers may occur naturally, for instance because of heavy-tailed distributions or sensor glitches, but if each loss is based on the input of a user, then we may also be concerned that a small number of adversarial users may try to poison the data stream [30].",
"We formally capture the robustness of OCO methods by modifying the standard setting to measure performance only on the rounds that are not outliers.",
"The goal of the learner is to perform as well as if the outliers were not present, up to some overhead that is incurred for filtering out the outliers.",
"As in standard OCO, learning proceeds in $T$ rounds, and at the start of each round $t$ the learner needs to issue a prediction $_t$ from a bounded convex domain.",
"The environment then reveals a convex loss function $f_t$ with (sub)gradient $_t := \\nabla f_t(_t)$ at $_t$ , and performance is measured by the cumulative difference between the learner's losses and the losses of the best fixed parameters $.",
"Unlike in thestandard OCO setting, however, we only sum up losses over the subsetof inlier rounds $ S{1,2,...,T}$ that are notoutliers, leading to the following notion of \\emph {robust regret}:\\begin{equation}R_T(\\mathcal {S}) := \\sum _{t \\in \\mathcal {S}} \\big (f_t(_t) - f_t(\\big ).\\end{equation}The challenge for the learner is to guarantee small robust regretwithout knowing $ S$.",
"Importantly, we aim for robust regret boundsthat scale with the loss range or gradient norms of the roundsin~$ S$, but not with the size of the outliers, so even extremeoutliers should not be able to confuse the learner.$ In Section , we first consider the fully adversarial case where the only thing the learner knows is that there are at most $k$ outliers, so $T - |\\mathcal {S}| \\le k$ , and both the inliers and the outliers are generated adversarially, without any bound on the range of the outliers, and with the range of the inliers also unknown a priori.",
"We introduce a simple filtering approach that filters out some of the largest gradients, and passes on the remaining rounds to a standard online learning algorithm ALG.",
"When the losses are linear, this approach is able to guarantee that $R_T(\\mathcal {S}) = R_T^\\text{ALG}( + O\\big (G(\\mathcal {S}) k\\big )\\qquad \\text{for all $\\mathcal {S}$ such that $T - |\\mathcal {S}| \\le k$ simultaneously,}$ where $G(\\mathcal {S})$ is the norm of the largest gradient among the rounds in $\\mathcal {S}$ and $R_T^\\text{ALG}($ is the regret of ALG on a subset of rounds under the guarantee that their gradient norms are at most $2G(\\mathcal {S})$ .",
"The extension to general convex losses then follows from a standard reduction to the linear loss case.",
"We follow up by showing that (REF ) is unimprovable, not just for adversarial losses, but even if the losses are independent and identically distributed (i.i.d.)",
"according to a fixed probability distribution or if the losses are strongly convex.",
"This fixes the dependence on the number of outliers $k$ to be linear in $k$ in quite some generality.",
"Nevertheless, in Section we identify sufficient conditions to get around the linear dependence: if the gradients are i.i.d., and we take $\\mathcal {S}= \\mathcal {S}_p$ to be the rounds in which $\\Vert _t\\Vert _*$ is at most the $p$ -quantile $G_p$ of the common distribution of their norms, then there exists a method based on approximating $G_p$ by its empirical counterpart on the available data that guarantees that the expected robust regret is at most $\\operatornamewithlimits{\\mathbb {E}}*{R_T(\\mathcal {S}_p)} = O*{G_p *{\\sqrt{p T} + \\sqrt{p(1-p)T \\ln T} +\\ln ^2 T}}.$ Since $O*{G_p \\sqrt{p T}}$ would be expected if $\\mathcal {S}_p$ were known in advance, we see that the overhead grows sublinearly in $T$ and is even asymptotically negligible for outlier proportion $1-p = o(1/\\ln (T))$ .",
"More generally, we extend this result such that the gradients do not need to be i.i.d.",
"themselves, but it is sufficient if there exist i.i.d.",
"random variables $_t$ and a constant $L$ such that $\\Vert _t\\Vert _* \\le L\\Vert _t\\Vert _*$ .",
"We then define the quantile with respect to the distribution of the $_t$ .",
"This covers nonlinear losses of the form $f_t() = h_t(^\\intercal _t)$ for convex functions $h_t$ that are $L$ -Lipschitz, like the logistic loss $f_t() = \\ln (1+\\exp (-Y_t ^\\intercal _t))$ and the hinge loss $f_t() =\\max \\lbrace 1-Y_t^\\intercal _t\\rbrace $ for $Y_t \\in \\lbrace -1,+1\\rbrace $ , both with $L=1$ ."
],
[
"Related Work",
"The definition of robust regret may remind the reader of the adaptive regret [23], [13], which measures regret on a contiguous interval of rounds $\\mathcal {I}$ that is unknown to the learner.",
"Since adaptive regret can be controlled by casting it into the framework of specialist (sleeping) experts [20], [12], it is natural to ask whether the same is possible for the robust regret.",
"To apply the specialist experts framework, we would assign a separate learner (specialist) to each possible subset of rounds $\\mathcal {S}$ that would then be active only on $\\mathcal {S}$ , and such a pool of $m$ learners would be aggregated using a meta-algorithm.",
"Computational issues aside, this approach runs into two problems: the first is that all existing meta-algorithms assume the losses to be bounded within a known range, and therefore cannot be applied since we do not assume that even the range of the inliers is known.",
"Second, even if the range issue could be resolved, the specialist regret would incur a $\\Omega (\\sqrt{T \\log m}) =\\Omega (\\sqrt{k T \\log (T/k)}$ overhead, already if we only consider all $m = \\binom{T}{T-k} \\ge (T/k)^k$ possible subsets with exactly $k$ outliers.",
"We see that $k$ now multiplies $T$ , which is much worse than the optimal additive dependence on $k$ in (REF ).",
"Reducing the dependence on the largest gradient norm has previously been considered in the context of adaptive online and stochastic convex optimization [19], [47].",
"However, these methods still depend on the average of all (squared) gradient norms, and therefore require these norms to be finite.",
"In contrast to these adaptive methods, our method can handle a small number of adversarial samples, with large or even infinite norm, while our robust regret analysis still guarantees a sub-linear bound.",
"In the context of stochastic optimization, [39] propose a robust version of mirror descent based on truncating the gradients returned by a stochastic oracle.",
"Their main goal is to establish a sub-Gaussian confidence bound on the optimization error under weak assumptions about the tails of the noise distribution.",
"Contrary to our setup, they control the smoothness of the objective and the variance of the noise, so that already a vanilla (non-robust) version of SGD would achieve a vanishing optimization error in expectation (but not with a sub-Gaussian confidence).",
"[17] propose a robust meta-algorithm for stochastic optimization that repeatedly trains a standard algorithm as a base learner and filters out the outliers.",
"This approach is conceptually similar to our filtering method, but it is designed to work in a batch setting, with the data (sample functions) given in advance.",
"[41] provide a robust batch algorithm for stochastic optimization by applying the ideas from robust mean estimation to robustify stochastic gradient estimates in a (batch) gradient descent algorithm.",
"In the online learning and bandit literature, interesting results were obtained for dealing with adversarial corruptions of data that are otherwise generated i.i.d., to still benefit from the stochastic setting [37], [21], [1].",
"[46] and [48] further consider data poisoning attacks on an online learner, but the focus is on the optimization of the adversary, while the learner remains fixed.",
"In all these works, contrary to ours, the corrupted data is still assumed to lie in the same range as the non-corrupted data.",
"A notable exception is the very recent work of [10], which proposes online algorithms for contextual bandits and linear regression in a framework in which the linear model is realizable (well-specified) up to small noise, and a fixed, randomly selected, fraction of examples is arbitrarily corrupted (as in the Huber $\\epsilon $ -contamination model [26]), but still remains bounded.",
"In contrast, we avoid strong distributional assumptions such as model realizability, and do not make any probabilistic assumptions about the corruption mechanism or impose any constraints on the magnitude of the outliers.",
"Starting with pioneering works of Tukey and Huber [44], [26] there has been a tremendous amount of past work in the area of robust statistics, which concerns the basic tasks of classical statistics in the presence of outliers and heavy-tailed distributions [27].",
"A more recent line of research building on the work of [6], [38], [34], [36] concerns estimation with sub-Gaussian-style confidence for heavy-tailed distributions.",
"Finally, our setup is different from, but conceptually related to, a line of research on machine learning and statistical problems in the presence of adversarial data corruptions [9].",
"This has been studied, for instance, in the context of parameter estimation [31], [11], [16], [35], [42], robust PCA [5], regression [28], [18], [32], classification [29], [33], [2] and many other cases.",
"See the in-depth survey by [15] for an overview of recent advances in this direction."
],
[
"Outline",
"We start by summarizing our setting and notation in the next section.",
"Then, in Section , we prove the upper bound (REF ) for adversarial losses, and show matching lower bounds both for i.i.d.",
"losses and for strongly convex losses.",
"As a further example, we show how robust regret can be used to bound the excess risk in the Huber $\\epsilon $ -contaminated setting via online-to-batch-conversion.",
"In Section we turn to the quantile case and establish (REF ).",
"Finally, Section concludes with a discussion of possible directions for future work.",
"Some proofs are deferred to the appendix."
],
[
"Setting and Notation",
"Formally, we consider the following online learning protocol.",
"In each round $t = 1,2,\\ldots $ the learner first predicts $_t \\in \\mathcal {W}$ , where the domain $\\mathcal {W}$ is a non-empty, compact and convex subset of $\\mathbb {R}^d$ .",
"The adversary then reveals a convex loss function $f_t: \\mathcal {W}\\rightarrow \\mathbb {R}$ , and the learner suffers loss $f_t(_t)$ .",
"We assume throughout that there always exists a gradient or, more generally, a subgradient $_t := \\nabla f_t(_t)$ at the learner's prediction, which is implied by convexity of $f_t$ whenever $_t$ lies in the interior of $\\mathcal {W}$ and also on the boundary if there exists a finite convex extension of all $f_t$ to a larger domain that contains $\\mathcal {W}$ in its interior.",
"The performance of the learner with respect to any fixed parameters $\\mathcal {W}$ is measured by the robust regret $R_T(\\mathcal {S})$ over the rounds $\\mathcal {S}\\subseteq [T] :=\\lbrace 1,2,\\ldots ,T\\rbrace $ that are not outliers, as defined in ().",
"The definition of subgradients implies that $f_t(_t) - f_t( \\le (_t - ^\\intercal _t$ , which implies that $R_T(\\mathcal {S})$ is bounded from above by the linearized robust regret $\\widetilde{R}_T(\\mathcal {S}) := \\sum _{t \\in \\mathcal {S}} (_t - ^\\intercal _t.$ We will state our main results for an arbitrary norm $\\left\\Vert \\cdot \\right\\Vert $ on $\\mathcal {W}$ and measure gradient lengths in terms of the dual norm $\\left\\Vert _t \\right\\Vert _* = \\sup _{\\in \\mathbb {R}^d : \\left\\Vert \\right\\Vert \\le 1} ^\\intercal _t$ .",
"Let $D = \\max _{\\in \\mathcal {W}} \\left\\Vert - \\Vert \\right.$ denote the diameter of the domain.",
"For the analysis of the robust regret, we need a Lipschitz bound for the gradients that are in the set $\\mathcal {S}$ , which we denote by $G(\\mathcal {S}) := \\max _{t \\in \\mathcal {S}} \\left\\Vert _t \\right\\Vert _*.$"
],
[
"Robustness to Adversarial Outliers",
"In this section we derive matching upper and lower bounds of the form in (REF )."
],
[
"Upper Bounds",
"Let ALG be any Lipschitz-adaptive algorithm, which we will use as our base online learning algorithm.",
"Our general approach is to add a filtering meta-algorithm FILTER that examines (the norm of) incoming gradients and decides whether to filter them or pass them on to ALG for learning.",
"If $\\mathcal {S}$ were known in advance, then FILTER could filter out all outliers and pass on only the rounds in $\\mathcal {S}$ , but since $\\mathcal {S}$ is not known, FILTER needs to learn which rounds to pass on.",
"Although most online learning algorithms base their updates only on gradients, we note that we do allow ALG to use the full loss function $f_t$ to update its state when FILTER passes on round $t$ to ALG.",
"When a round $t$ is filtered, we assume that ALG behaves as if that round had not happened, so we will have $_{t+1} = _t$ .",
"Our FILTER for this section is displayed in Algorithm REF .",
"[htb] Top-$k$ Filter: Filtering for Adversarial Setting Maximum number of outliers $k$ Initialize: Let $\\mathcal {L}_0 = \\lbrace 0,0,\\ldots ,0\\rbrace $ be an ordered list of length $k+1$ .",
"$t = 1,2,\\ldots $ Maintain invariant that $\\mathcal {L}_t$ contains $k+1$ largest gradients $\\Vert _t\\Vert _* > \\min \\mathcal {L}_{t-1}$ Obtain $\\mathcal {L}_t$ from $\\mathcal {L}_{t-1}$ by removing the smallest item in $\\mathcal {L}_{t-1}$ and inserting $\\Vert _t\\Vert _*$ Set $\\mathcal {L}_t$ equal to $\\mathcal {L}_{t-1}$ Filter with factor 2 slack $\\Vert _t\\Vert _* > 2 \\min \\mathcal {L}_t$ Filter round $t$ Pass round $t$ on to ALG Theorem 1 Suppose ALG is any Lipschitz-adaptive algorithm that guarantees linearized regret bounded by $B_T(G)$ on the rounds that it is passed by FILTER, if the gradients in those rounds have length at most $G$ , and let FILTER be Algorithm REF with parameter $k$ .",
"Then the linearized robust regret of ALG+FILTER is bounded by $\\widetilde{R}_T(\\mathcal {S})\\le B_T\\big (2 G(\\mathcal {S})\\big ) + 4D(\\mathcal {S}) G(\\mathcal {S}) (k+1)\\qquad \\text{for any $\\mathcal {S}: T - |\\mathcal {S}| \\le k$ and $\\mathcal {W}$,}$ where $D(\\mathcal {S}) = \\max _{t : \\Vert _t\\Vert _* \\le 2 G(\\mathcal {S})} \\Vert _t - $ .",
"There are two main ideas to the proof.",
"First, since the list $\\mathcal {L}_t$ in Algorithm REF contains $k+1$ elements and there are at most $k$ outliers, at least one of the elements of $\\mathcal {L}_t$ must be one of the inliers from $\\mathcal {S}$ .",
"It follows that the smallest element of $\\mathcal {L}_t$ is a lower bound on $G(\\mathcal {S})$ .",
"The second idea is that, instead of filtering on this lower bound directly, we filter with factor 2 slack.",
"Since every filtered gradient is also added to $\\mathcal {L}_t$ , this factor 2 ensures that the minimum of $\\mathcal {L}_t$ must at least double for every $k+1$ rounds that are filtered.",
"The resulting exponential growth of the filtered rounds means that the contribution to the robust regret of all filtered rounds is dominated by the last $k+1$ rounds, and therefore does not grow with $T$ .",
"Let $\\mathcal {F}\\subset [T]$ denote the rounds filtered out by Algorithm REF , and let $\\mathcal {P}= [T] \\setminus \\mathcal {F}$ denote the rounds that are passed on to ALG.",
"Then the linearized robust regret splits as follows: $\\widetilde{R}_T(\\mathcal {S})= \\sum _{t \\in \\mathcal {S}\\cap \\mathcal {P}} (_t - ^\\intercal _t+ \\sum _{t \\in \\mathcal {S}\\cap \\mathcal {F}} (_t - ^\\intercal _t.$ We will show that Algorithm REF guarantees that the gradients on the passed rounds are bounded as follows: $\\Vert _t\\Vert _* \\le 2 G(\\mathcal {S})\\qquad \\text{for all $t \\in \\mathcal {P}$,}$ which implies that $\\sum _{t \\in \\mathcal {S}\\cap \\mathcal {P}} (_t - ^\\intercal _t&= \\sum _{t \\in \\mathcal {P}} (_t - ^\\intercal _t- \\sum _{t \\in \\mathcal {P}\\setminus \\mathcal {S}} (_t - ^\\intercal _t\\\\&\\le B_T(2 G(\\mathcal {S})) + 2 D(\\mathcal {S}) G(\\mathcal {S}) |\\mathcal {P}\\setminus \\mathcal {S}|\\le B_T(2 G(\\mathcal {S})) + 2 D(\\mathcal {S}) G(\\mathcal {S}) k,$ where the first inequality uses the assumption on ALG and Hölder's inequality, and the second inequality uses that $|\\mathcal {P}\\setminus \\mathcal {S}| \\le |[T] \\setminus \\mathcal {S}| \\le k$ .",
"We proceed to prove (REF ).",
"During the first $k$ rounds, $\\min \\mathcal {L}_t = 0$ , so (REF ) is trivially satisfied.",
"In all later rounds, $\\mathcal {L}_t \\subseteq \\lbrace \\Vert _s\\Vert _* : s \\le t\\rbrace \\subseteq \\lbrace \\Vert _s\\Vert _* : s \\le T\\rbrace $ .",
"Consequently, $\\mathcal {L}_t$ must contain at least one element $\\Vert _t\\Vert _*$ with $t \\in \\mathcal {S}$ , because $T - |\\mathcal {S}| \\le k$ and $|\\mathcal {L}_t| = k+1 > k$ .",
"It follows that $\\min \\mathcal {L}_t \\le G(\\mathcal {S})$ , so all passed gradients satisfy (REF ).",
"Let $G_\\text{min} = \\min \\lbrace \\Vert _t\\Vert _* \\mid t \\in \\mathcal {F}\\rbrace > 0$ be the length of the shortest filtered gradient.",
"To complete the proof, we will show that $\\sum _{t \\in \\mathcal {S}\\cap \\mathcal {F}} (_t - ^\\intercal _t\\le D(\\mathcal {S}) \\!\\sum _{t \\in \\mathcal {S}\\cap \\mathcal {F}} \\!\\!\\Vert _t\\Vert _*\\le D(\\mathcal {S}) \\hspace{-20.0pt}\\sum _{\\begin{array}{c}t \\in \\mathcal {F}\\\\ G_\\text{min} \\le \\Vert _t\\Vert _* \\le G(\\mathcal {S})\\end{array}} \\hspace{-20.0pt} \\Vert _t\\Vert _*\\le 2D(\\mathcal {S}) G(\\mathcal {S}) (k+1).$ The first of these inequalities follows from Hölder's inequality, and the second from the definition of $G(\\mathcal {S})$ .",
"To establish the last inequality, we proceed by induction: since $\\mathcal {L}_t$ contains the $k+1$ largest observed gradient norms, we observe that there can be at most $k+1$ filtered rounds in which $G(\\mathcal {S})/2^{i+1} < \\Vert _t\\Vert _* \\le G(\\mathcal {S})/2^i$ , because after $k+1$ such rounds we will have $\\min \\mathcal {L}_t > G(\\mathcal {S})/2^{i+1}$ forever.",
"It follows that we have the following induction step: $\\hspace{-20.0pt}\\sum _{\\begin{array}{c}t \\in \\mathcal {F}\\\\ G_\\text{min} \\le \\Vert _t\\Vert _* \\le G(\\mathcal {S})/2^i\\end{array}} \\hspace{-20.0pt} \\Vert _t\\Vert _*\\le (k+1) G(\\mathcal {S})/2^i \\quad +\\hspace{-20.0pt}\\sum _{\\begin{array}{c}t \\in \\mathcal {F}\\\\ G_\\text{min} \\le \\Vert _t\\Vert _* \\le G(\\mathcal {S})/2^{i+1}\\end{array}} \\hspace{-20.0pt} \\Vert _t\\Vert _*\\qquad \\text{for $i=0,1,2,\\ldots $}$ Unrolling the induction, we therefore obtain $\\hspace{-20.0pt}\\sum _{\\begin{array}{c}t \\in \\mathcal {F}\\\\ G_\\text{min} \\le \\Vert _t\\Vert _* \\le G(\\mathcal {S})\\end{array}} \\hspace{-20.0pt} \\Vert _t\\Vert _*\\le (k+1) G(\\mathcal {S}) \\!\\!\\sum _{i=0}^{{\\log _2\\frac{G(\\mathcal {S})}{G_{\\text{min}}}}} \\!\\!2^{-i}\\le (k+1) G(\\mathcal {S}) \\sum _{i=0}^\\infty 2^{-i}= (k+1) G(\\mathcal {S}) 2,$ which is what remained to be shown.",
"As for the run-time, one may maintain the $k$ largest gradient norms encountered in a priority queue.",
"The time used by Algorithm REF on top of ALG is $O(\\ln k) \\le O(\\ln T)$ per round.",
"This may be pessimistic in practise, as FILTER only performs work if the current gradient is among the $k+1$ largest seen so far."
],
[
"Examples",
"To make the result from Theorem REF more concrete, let us instantiate ALG as online gradient descent (OGD), which starts from any $_1 \\in \\mathcal {W}$ and updates according to $_{t+1} = \\Pi _\\mathcal {W}(_t - \\eta _t _t),$ where $\\Pi _\\mathcal {W}()$ denotes Euclidean projection of $$ onto $\\mathcal {W}$ , and $\\eta _t > 0$ is a hyperparameter called the step size.",
"Tuning the step size for general convex losses, we find that we can tolerate at most $k = O(\\sqrt{T})$ outliers without suffering in the rate: Corollary 2 (General Convex Losses) Let $\\left\\Vert \\cdot \\right\\Vert $ be the $\\ell _2$ -norm, let ALG be OGD with step size $\\eta _t = D/\\sqrt{2 \\sum _{s=1}^t \\Vert _s\\Vert _2^2}$ and let FILTER be Algorithm REF with parameter $k$ .",
"Then the robust regret is bounded by $\\begin{split}R_T(\\mathcal {S})&\\le 2 D \\sqrt{\\sum _{t \\in \\mathcal {S}} \\Vert _t\\Vert _2^2}+ 2D G(\\mathcal {S}) \\Big (2k + \\sqrt{k} + 2\\Big )\\\\&\\le 2 D G(\\mathcal {S}) \\Big (\\sqrt{T} + 2k + \\sqrt{k} + 2\\Big )\\qquad \\text{for any $\\mathcal {S}: T - |\\mathcal {S}| \\le k$ and $\\mathcal {W}$.",
"}\\end{split}$ The proof of the corollary is in Appendix .",
"The step size of OGD may also be tuned for $\\sigma $ -strongly convex losses, which are guaranteed to be curved in all directions, and satisfy the requirement that $f_t( \\ge f_t() + ( )^\\intercal \\nabla f_t() +\\frac{\\sigma }{2} \\Vert \\Vert _2^2\\qquad \\text{for all $\\in \\mathcal {W}$.",
"}$ In this case, we obtain the following guarantee on the robust regret, which is proved in Appendix : Corollary 3 (Strongly Convex Losses) Suppose the loss functions $f_t$ are $\\sigma $ -strongly convex.",
"Let $\\left\\Vert \\cdot \\right\\Vert $ be the $\\ell _2$ -norm, let ALG be OGD with step size $\\eta _t =\\frac{1}{\\sigma t}$ and let FILTER be Algorithm REF with parameter $k$ .",
"Then the robust regret is bounded by $R_T(\\mathcal {S})\\le \\frac{2 G(\\mathcal {S})^2}{\\sigma } \\big (\\ln T + 1\\big )+ \\frac{5\\widetilde{G}(\\mathcal {S})^2}{2\\sigma }(k+1)\\qquad \\text{for any $\\mathcal {S}: T - |\\mathcal {S}| \\le k$ and $\\mathcal {W}$,}$ where $\\widetilde{G}(\\mathcal {S}) = 2 G(\\mathcal {S}) + \\max _{t : \\Vert _t\\Vert _2 \\le 2G(\\mathcal {S})} \\Vert \\nabla f_t(\\Vert _2$ .",
"The standard regret bound of OGD for strongly convex losses is of order $\\frac{G^2}{\\sigma } \\log T$ [24], so in this case we can tolerate $k = O(\\log T)$ outliers without suffering in the rate, under the additional assumption that $\\widetilde{G}(\\mathcal {S}) =O(G(\\mathcal {S}))$ .",
"This seems like a reasonable assumption if we think of the condition $\\Vert _t\\Vert _2 \\le 2 G(\\mathcal {S})$ as expressing that round $t$ is not too extreme."
],
[
"Huber $\\epsilon $ -Contamination",
"As a final example, we consider the Huber $\\epsilon $ -contamination setting [26].",
"In this case losses are of the form $f_t() =f(,\\xi _t)$ , where the random variables $\\xi _t$ are sampled i.i.d.",
"from a mixture distribution $P_\\epsilon $ defined by $\\xi \\sim {\\left\\lbrace \\begin{array}{ll}P & \\text{if $M = 0$}\\\\Q & \\text{if $M = 1$}\\end{array}\\right.",
"}\\qquad \\text{where} \\qquad M \\sim \\operatorname{Bernoulli}(\\epsilon )$ for some $\\epsilon \\in [0,1)$ .",
"The interpretation is that $P$ is the actual distribution of interest, which is contaminated by outliers drawn from $Q$ .",
"The hidden variable $M$ is not observed by the learner, so it is not known which observations are outliers.",
"Let $\\mathcal {S}^* \\subseteq [T]$ denote the set of inlier rounds in which $M_t = 0$ .",
"Then the robust regret $R_T(\\mathcal {S}^*)$ may be viewed as the ordinary regret on a modified loss function $\\tilde{f}(,M,\\xi )$ that is equal to $f(,\\xi )$ on samples from $P$ but zero on samples from $Q$ , i.e.",
"$\\tilde{f}(,M,\\xi )= \\mathbf {1}\\lbrace M = 0\\rbrace f(,\\xi )$ and $R_T(\\mathcal {S}^*) = \\sum _{t=1}^T *{\\tilde{f}(_t,M_t,\\xi _t) - \\tilde{f}(M_t,\\xi _t)}.$ Let the risk with respect to the inlier distribution $P$ be defined as $\\operatorname{Risk}_P() = \\operatornamewithlimits{\\mathbb {E}}_{\\xi \\sim P} *{f(,\\xi )}.$ Then, applying online-to-batch conversion [8] to the modified loss $\\tilde{f}$ , we obtain the following result, which bounds the excess risk under $P$ by the robust regret when the observations are drawn from the contaminated mixture $P_\\epsilon $ , without requiring any assumptions about the outliers coming from $Q$ : Lemma 4 (Huber $\\epsilon $ -Contamination) Suppose the losses $f_t$ are i.i.d.",
"according to the mixture distribution $P_\\epsilon $ , and let $P \\in \\operatorname{arg\\,min}_{\\in \\mathcal {W}}\\operatorname{Risk}_P()$ be the optimal parameters for the distribution of the inliers.",
"Let $\\bar{}_T = \\frac{1}{T} \\sum _{t=1}^T _t$ , where $_1,\\ldots ,_T$ are the predictions of the learner.",
"Then $\\operatornamewithlimits{\\mathbb {E}}_{P_\\epsilon } *{\\operatorname{Risk}_P(\\bar{}_T) - \\operatorname{Risk}_P(P)}\\le \\frac{\\operatornamewithlimits{\\mathbb {E}}_{P_\\epsilon } *{R_T(P,\\mathcal {S}^*)}}{(1-\\epsilon )T}.$ Moreover, if $|f(,\\xi ) - f(P,\\xi )| \\le B$ almost surely when $\\xi \\sim P$ is an inlier, then for any $0 < \\delta \\le 1$ $\\operatorname{Risk}_P(\\bar{}_T) - \\operatorname{Risk}_P(P)\\le \\frac{R_T(P,\\mathcal {S}^*)}{(1-\\epsilon )T}+ \\frac{2B}{1-\\epsilon } \\sqrt{\\frac{2}{T} \\ln \\frac{1}{\\delta }}$ with $P_\\epsilon $ -probability at least $1-\\delta $ .",
"(Details of the proof are given in Appendix .)",
"We see that, if we can control the robust regret with respect to the unknown set $\\mathcal {S}^*$ of inlier rounds, then we can also control the excess risk with respect to the inlier distribution $P$ .",
"For example, instantiating the learner as in Corollary REF leads to the following specialization of Lemma REF .",
"Corollary 5 In the setting of Lemma REF , suppose that $\\Vert \\nabla f(,\\xi )\\Vert \\le G$ for all $\\in \\mathcal {W}$ almost surely when $\\xi \\sim P$ is an inlier, and that $\\epsilon \\le 1/2$ .",
"Let the learner be instantiated as in Corollary REF with $k = {\\epsilon T + \\sqrt{2 T \\epsilon (1-\\epsilon ) \\ln (2/\\delta )}+ \\tfrac{1}{3}(1-\\epsilon ) \\ln (2/\\delta )}$ for any $0 <\\delta \\le 1$ .",
"Then $\\operatorname{Risk}_P(\\bar{}_T) - \\operatorname{Risk}_P(P)\\le 12 D G \\epsilon + \\frac{2DG \\big (5 \\sqrt{2\\ln (2/\\delta )}+2\\big )}{\\sqrt{T}}+ \\frac{2 D G \\big (\\ln (2/\\delta ) + 10\\big )}{T}$ with $P_\\epsilon $ -probability at least $1-\\delta $ .",
"Here $\\nabla f(,\\xi )$ should be read as the gradient of $f(,\\xi )$ with respect to $$ .",
"The constant dependence on $DG\\epsilon $ , which does not go to zero with increasing $T$ , is unavoidable because $P$ is non-identifiable based on samples from $P_\\epsilon $ .",
"For instance, consider the linear loss $f(w,\\xi ) = \\xi w$ with $\\mathcal {W}= [-D/2,+D/2]$ and $P_\\epsilon $ such that $\\xi = -G$ and $\\xi = +G$ both with probability $\\epsilon $ , and $\\xi = 0$ with probability $1-2\\epsilon $ .",
"Then we cannot distinguish the case that $P = P_\\epsilon (\\cdot \\mid \\xi \\le 0)$ and $Q$ is a point-mass on $+G$ from the case that $P=P_\\epsilon (\\cdot \\mid \\xi \\ge 0)$ with $Q$ a point-mass on $-G$ .",
"No matter what the output of the learner is, its excess risk under $P$ will always be at least $DG\\epsilon $ in one of these two cases.",
"The proof of Corollary REF is postponed to Appendix .",
"It is a straightforward combination of Lemma REF and Corollary REF , with the only point of attention being the tuning of the number of outliers $k$ .",
"In expectation, the number of outliers is $\\epsilon T$ , but we choose $k$ slightly larger so that the probability that the number of outliers exceeds $k$ is negligible."
],
[
"Lower Bounds",
"We now show that the bounds obtained in the previous part of this section are non-improvable in general.",
"First note that one can always choose $\\mathcal {S}=[T]$ (no outliers) and apply a standard lower bound for online learning algorithms which guarantees expected regret $\\Omega (\\sqrt{T})$ for general losses and $\\Omega (\\ln T)$ for strongly-convex losses.",
"This matches the first term in the bound of Theorem REF .",
"Therefore, we will only show a bound $\\Omega (k)$ , which, combined with the standard one, leads to a $\\Omega (\\max \\lbrace \\sqrt{T},k\\rbrace ) = \\Omega (\\sqrt{T} + k)$ lower bound on the regret for general convex losses and $\\Omega (\\ln T + k)$ for strongly convex losses.",
"Consider a learning task over domain $\\mathcal {W}= [-W,W]$ for some $W > 0$ .",
"To prove a lower bound for general convex losses, we choose the loss sequence to be $f_t(w) = G\\xi _t w$ , where $\\xi _t \\in \\lbrace -1,+1\\rbrace $ are i.i.d.",
"Rademacher random variables with $\\Pr (\\xi _t = -1) = \\Pr (\\xi _t = +1) = \\frac{1}{2}$ , while $G > 0$ controls the size of the gradients/losses.",
"Theorem 6 (Lower Bound with I.I.D.",
"Losses) For any $k$ and any online learning algorithm run on the sequence defined above, there exist adversarial choices of $\\mathcal {S}$ with $T-|\\mathcal {S}| \\le k$ and $u \\in \\mathcal {W}$ such that $\\operatornamewithlimits{\\mathbb {E}}_{f_1,\\ldots ,f_T} *{ R_T(u, \\mathcal {S}) } \\ge \\frac{D G(\\mathcal {S}) k}{4},$ where $f_1,\\ldots ,f_T$ are i.i.d.",
"as described above.",
"Let $S_1 = \\lbrace t \\in [k] \\colon \\xi _t = 1\\rbrace $ and $S_{-1} = \\lbrace t \\in [k] \\colon \\xi _t = -1\\rbrace $ .",
"The adversary will choose $u = -W \\zeta $ and $\\mathcal {S}= S_{\\zeta } \\cup \\lbrace k+1,\\ldots ,T\\rbrace $ , where $\\zeta \\in \\lbrace -1,1\\rbrace $ is a Rademacher random variable independent of $\\xi _1,\\ldots ,\\xi _T$ .",
"The expected regret jointly over $\\xi _1,\\ldots ,\\xi _T,\\zeta $ is then given by $\\operatornamewithlimits{\\mathbb {E}}*{ R_T(u, \\mathcal {S}) } &=\\operatornamewithlimits{\\mathbb {E}}*{ G \\sum _{t=1}^T \\mathbf {1}_{t \\in \\mathcal {S}} w_t \\xi _t - G \\sum _{t=1}^T\\mathbf {1}_{t \\in \\mathcal {S}} u \\xi _t} \\\\&=G \\sum _{t=1}^k \\underbrace{\\operatornamewithlimits{\\mathbb {E}}*{\\mathbf {1}_{\\zeta = \\xi _t} w_t \\xi _t }}_{=0}+ G \\sum _{t=k+1}^T \\underbrace{\\operatornamewithlimits{\\mathbb {E}}*{w_t \\xi _t}}_{=0}+ GW \\sum _{t=1}^k \\underbrace{\\operatornamewithlimits{\\mathbb {E}}*{\\mathbf {1}_{\\zeta = \\xi _t} \\zeta \\xi _t }}_{=1/2}+ GW \\sum _{t=k+1}^T \\underbrace{\\operatornamewithlimits{\\mathbb {E}}*{\\zeta \\xi _t}}_{=0} \\\\&= GW \\frac{k}{2},$ where we used the independence of $\\xi _t$ and $\\zeta $ in the second and the fourth sum, while $\\operatornamewithlimits{\\mathbb {E}}*{\\mathbf {1}_{\\zeta = \\xi _t} w_t \\xi _t }= \\operatornamewithlimits{\\mathbb {E}}*{\\; \\operatornamewithlimits{\\mathbb {E}}*{\\mathbf {1}_{\\zeta = \\xi _t} w_t \\xi _t \\, |\\, \\xi _t}\\;}= \\operatornamewithlimits{\\mathbb {E}}*{w_t \\xi _t/ 2} = 0, \\quad \\text{and} \\;\\operatornamewithlimits{\\mathbb {E}}*{\\mathbf {1}_{\\zeta = \\xi _t} \\zeta \\xi _t } = \\operatornamewithlimits{\\mathbb {E}}*{\\mathbf {1}_{\\zeta = \\xi _t}} = \\frac{1}{2}.$ As the bound holds for the random choice of $\\zeta $ it also holds for the worst-case choice of $\\zeta $ .",
"The theorem now follows from $D = \\max _{u,w \\in \\mathcal {W}} |w - u| = 2 W$ and $G(\\mathcal {S}) = \\max _{t \\in \\mathcal {S}} |g_t| = \\max _{t \\in \\mathcal {S}} G |\\xi _t| = G$ .",
"A similar bounding technique leads to a lower bound for $\\sigma $ -strongly convex losses, except that the distribution of the losses differs between the first $k$ rounds and the later rounds.",
"This still implies a lower bound for adversarially generated data, but not for i.i.d.",
"losses.",
"In this case, we will choose the domain $\\mathcal {W}= [-W,W]$ , the loss sequence based on the $\\sigma $ -strongly convex squared loss, $f_t(w) = \\frac{\\sigma }{2} (w - W \\xi _t)^2$ , for $t \\le k$ , and $f_t(w) = \\frac{\\sigma }{2} (w-W\\zeta )^2$ for $t \\ge k$ , where $\\xi _1,\\ldots \\xi _k$ and $\\zeta $ are again i.i.d.",
"Rademacher variables.",
"Theorem 7 (Lower Bound for Strongly Convex Losses) For any $k$ and any online learning algorithm, there exist adversarial choices of $\\mathcal {S}$ with $T-|\\mathcal {S}| \\le k$ and $u \\in \\mathcal {W}$ such that $\\operatornamewithlimits{\\mathbb {E}}_{f_1,\\ldots ,f_T} *{ R_T(u, \\mathcal {S}) } \\ge \\frac{G^2(\\mathcal {S}) k}{16 \\sigma },$ where $f_1,\\ldots ,f_T$ are the $\\sigma $ -strongly convex losses described above.",
"Using the same notation as in the proof of Theorem REF , the adversary will choose $u = W \\zeta $ and $\\mathcal {S}= S_{\\zeta } \\cup \\lbrace t+1,\\ldots ,T\\rbrace $ .",
"The expected regret jointly over $\\xi _1,\\ldots ,\\xi _k,\\zeta $ is given by $\\operatornamewithlimits{\\mathbb {E}}*{ R_T(u, \\mathcal {S}) } &=\\frac{\\sigma }{2} \\sum _{t=1}^k \\underbrace{\\operatornamewithlimits{\\mathbb {E}}*{\\mathbf {1}_{\\zeta = \\xi _t} (w_t - W\\xi _t)^2}}_{\\ge W^2/2}+ \\frac{\\sigma }{2} \\sum _{t=k+1}^T \\underbrace{\\operatornamewithlimits{\\mathbb {E}}*{(w_t - W\\zeta )^2}}_{\\ge 0} \\\\&\\quad - \\frac{\\sigma }{2} \\sum _{t=1}^k \\underbrace{\\operatornamewithlimits{\\mathbb {E}}*{\\mathbf {1}_{\\zeta = \\xi _t} (\\zeta - W\\xi _t)^2}}_{=0}\\ge \\frac{\\sigma W^2 k}{4},$ where to bound the first sum we used $\\operatornamewithlimits{\\mathbb {E}}*{\\mathbf {1}_{\\zeta = \\xi _t} (w_t - W\\xi _t)^2 }&= \\operatornamewithlimits{\\mathbb {E}}*{\\; \\operatornamewithlimits{\\mathbb {E}}*{\\mathbf {1}_{\\zeta = \\xi _t} (w_t - W\\xi _t)^2 \\, |\\, \\xi _t}\\;}= \\operatornamewithlimits{\\mathbb {E}}*{(w_t - W\\xi _t)^2 / 2} \\\\&= \\operatornamewithlimits{\\mathbb {E}}*{w_t^2/2 - W\\xi _t w_t + W^2/2} = w_t^2/2 + W^2/2 \\ge W^2/2.$ To finish the proof note that $|\\nabla f_t(w_t)| = \\sigma |w_t - W\\xi _i| \\le 2 \\sigma W$ so that $G(\\mathcal {S}) \\le 2 \\sigma W$ ."
],
[
"Robustness for Quantiles",
"In this section we consider robust online linear optimization in the stochastic i.i.d.",
"setting.",
"That is, we consider i.i.d.",
"gradients $_t \\sim \\operatornamewithlimits{\\mathbb {P}}$ that are in particular independent of the learner's prediction $_t$ .",
"Let $G_p q_p({}_*)$ be the $p$ -quantile of the gradient in dual norm ${\\cdot }_*$ .",
"To keep things simple, we will assume that $\\operatornamewithlimits{\\mathbb {P}}$ does not have an atom at $G_p$ , so that $\\operatornamewithlimits{\\mathbb {P}}{{}_* \\le G_p} = p$ exactly.",
"We call a gradient $_t$ an outlier if ${_t}_* > G_p$ .",
"Fix a domain $\\mathcal {W}$ of diameter $D$ in the norm ${\\cdot }$ .",
"We are interested in algorithms that know $\\mathcal {W}$ and $p$ but not $G_p$ , play $_t \\in \\mathcal {W}$ , and we aim to bound their expected robust regret on the (random!)",
"set of inliers $\\mathcal {S}= {t \\in [T] : {_t}_* \\le G_p}$ .",
"That is, we aim to control $\\bar{R}_T~~\\operatornamewithlimits{\\mathbb {E}}*{\\max _{\\mathcal {W}}R_T( \\mathcal {S})}~=~\\operatornamewithlimits{\\mathbb {E}}*{\\max _{\\mathcal {W}}\\sum _{t \\in [T] : {_t}_* \\le G_p} {_t - _t}}.$ Note that a bound on the expected robust regret implies a robust pseudo-regret bound, where the data-dependent maximum is replaced by the fixed minimiser of the expected loss on inliers, i.e.",
"$* \\in \\arg \\min _{\\mathcal {W}} \\intercal \\operatornamewithlimits{\\mathbb {E}}{_t\\,}{\\,{_t}_* \\le G_p}$ .",
"Our FILTER algorithm for the stochastic setting is shown as Algorithm .",
"The main idea is that it only passes rounds to the base ALG for which it is virtually certain that they are inliers.",
"To this end our FILTER computes a lower confidence bound $\\operatorname{LCB}_t$ on the quantile $G_p$ .",
"Smaller gradients are included, while larger ones are discarded.",
"The crux of the robust regret bound proof is then dealing with the inlier gradients that end up being dropped.",
"We will find it instructive to state our algorithms and confidence bounds with a free confidence parameter $\\delta $ .",
"Tuning our approach will then lead us to set $\\delta =T^{-2}$ .",
"[htb] Quantile level $p \\in (0,1)$ , confidence $\\delta $ , online learner ALG $t = 1,2,\\ldots $ Have ALG produce $_t$ .",
"Receive gradient $_t$ Let $\\hat{q}_{t-1}$ be the empirical quantile function of past gradients $_1, \\ldots , _{t-1}$ .",
"Compute $\\operatorname{LCB}_{t-1} = \\hat{q}_{t-1}(p - u_{t-1})$ at threshold $u_{t-1} = \\sqrt{t^{-1} 2 p(1-p) \\ln \\frac{1}{\\delta }}+ \\frac{1}{3} t^{-1} \\ln \\frac{1}{\\delta }$ ${_t}_* \\le \\operatorname{LCB}_{t-1}$ Pass round $t$ on to ALG Ignore round $t$ Filtering meta algorithm for Robust Quantile Regret We now show that the expected robust regret is small.",
"Theorem 8 Let ALG have individual sequence regret bound $B_T(G)$ for $T$ rounds with gradients of dual norm at most $G$ , and which is concave in $T$ .",
"Let $D$ be the diameter of the domain.",
"Then the FILTER Meta-Algorithm with $\\delta =T^{-2}$ has expected robust regret bounded by $\\bar{R}_T~\\le ~B_{p T}(G_p)+D G_p *{4 \\sqrt{2 p (1-p) T \\ln T}+ \\frac{13}{3} (\\ln T)^2+ 3}.$ If ALG does its job, the first term is the minimax optimal regret for when the outlier rounds were known.",
"The other terms quantify the cost of being robust.",
"When $p$ is not extreme, this cost is of order $G_p D \\sqrt{T \\ln T}$ , rendering it the dominant term overall (escalating the minimax regret by a mild log factor).",
"When $p$ tends to 1 or 0, the robustness overhead gracefully reduces to the $(\\ln T)^2$ regime.",
"The proof can be found in Appendix .",
"The main ideas are as follows.",
"As we have no control over outlier gradients (they may be astronomical), we must assume that ALG gets confused without recourse if FILTER ever passes it any outlier.",
"Note that FILTER is not evaluated on outlier rounds, so it does not suffer from this gradient's magnitude.",
"But its effect is that, for all we know, ALG is rendered forever useless, upon which FILTER may incur the maximum possible regret of $G_p D T$ .",
"Our approach will be to choose our threshold for inclusion conservatively, and to apply concentration in all rounds simultaneously, to ensure this bad event is rare (this is the source of the $\\ln T$ factor).",
"A second concentration allows us to deal with the discarded inliers.",
"We conclude the section with a selection of remarks.",
"Examples The examples of Section REF also apply here.",
"Depending on the setting, and hence the appropriate base algorithm ALG, the dominant regret term can be either the $D G_p \\sqrt{p (1-p) T \\ln T}$ term, or the $B_{p T}(G_p)$ term.",
"The former case applies for OGD, while the latter case happens in the $K$ -experts setting with many experts and few rounds, i.e.",
"$K \\gg T$ .",
"There adding robustness comes essentially for free.",
"Anytime Robust Regret As stated, the algorithm needs to know the horizon $T$ up front to set the confidence parameter $\\delta $ in the deviation width $u_t$ .",
"We can use a standard doubling trick on $T$ to get an anytime algorithm.",
"Anytime concentration One may wonder how much the analysis can be improved by replacing our union bound over time steps with a time-uniform Bernstein concentration inequality, as e.g.",
"developed by [25].",
"Sadly, the best we can hope for is to be able to use $\\delta = \\frac{1}{T}$ , which would lead to a constant factor $\\sqrt{2}$ improvement on the dominant term.",
"We cannot tolerate a higher overall failure probability, for we have to pacify the regret upon failure, which may be of order $T$ .",
"High Probability Version Going into the proof, we see that a high probability robust regret bound is also possible.",
"We would need to change the analysis of $P^{(2)}$ , as we currently analyse it in expectation.",
"Observing that it is a sum of $T$ conditionally independent increments, we may use martingale concentration to find that, with probability at least $1-T^{-1}$ , this sum is at most its mean (which features in the expected regret bound) plus a deviation of order $\\sqrt{T \\ln T}$ .",
"We obtain a high-probability analogue of Theorem REF with slightly inflated constant.",
"Large-Feature-Vectors-as-Outliers We may also deal with non-i.i.d.",
"gradients using exactly the same techniques developed above, as follows.",
"We assume that $f_t() = h_t(^\\intercal _t)$ , where $_t \\in \\mathbb {R}^d$ is a feature vector available at the beginning of round $t$ , and $h_t$ is a scalar Lipschitz convex loss function, revealed at the end of round $t$ .",
"This setting includes e.g.",
"linear classification with hinge or logistic loss.",
"Upon assuming that feature vectors $_1, _2, \\ldots $ are drawn i.i.d.",
"from $\\operatornamewithlimits{\\mathbb {P}}$ (while the $h_t$ are arbitrary, possibly adversarially chosen), we can take the $p$ -quantile $X_p q_p({}_*)$ of the dual norm of the feature vectors.",
"We may then measure the robust expected regret (REF ) on the inlier rounds $\\mathcal {S}= *{t \\in [T] : {_t}_* \\le X_p}$ , and obtain the analogue of Theorem REF , where the only subtlety is using the gradient bound on $h_t$ to transfer from inlier $_t$ to small loss.",
"Proposition 9 Consider a joint distribution on sequences of feature vectors and scalar Lipschitz convex functions $(_1, h_1), (_2, h_2), \\ldots $ such that the feature vectors $_1, _2, \\ldots $ are i.i.d.",
"with distribution $\\operatornamewithlimits{\\mathbb {P}}$ on $\\mathbb {R}^d$ .We do not constrain the distribution of $h_1,h_2,\\ldots $ , so we can model adversarial loss functions that are correlated with the feature vectors.",
"Let $X_p = q_p({}_*)$ be the $p$ -quantile of the feature dual norm.",
"Let ALG be an algorithm for online-convex optimisation over a domain of diameter $D$ and loss functions $f_t() = h_t(^\\intercal _t)$ that guarantees individual-sequence regret bounded by $B_T(X)$ in any $T$ -round interaction with ${_t}_* \\le X$ , without having to know $X$ up front.",
"Consider FILTER Meta-Algorithm with $_t$ replaced by $_t$ .",
"Then the expected robust regret on inlier rounds $\\mathcal {S}= {t \\in [T] : {_t}_* \\le X_p}$ is bounded by $\\bar{R}_T=\\operatornamewithlimits{\\mathbb {E}}*{\\max _{\\mathcal {W}}\\sum _{t \\in \\mathcal {S}} *{f_t(_t) - f_t(}}\\le B_{p T}(X_p)+D X_p *{4 \\sqrt{2 p (1-p) T \\ln T}+ \\frac{13}{3} (\\ln T)^2+ 3}.$ The proof follows that of Theorem REF , with one extra (standard) step.",
"Namely, to bound the loss on inlier rounds (for the dropped rounds term $P^{(2)}$ in the proof, and the concentration failure term $P^{(3)}$ in the proof), we use convexity, Hölder and bounded derivative to obtain $f_t(_t) - f_t(*)~\\le ~h_t(_t^\\intercal _t) - h_t({*}^\\intercal _t )~\\le ~h_t^{\\prime }(_t^\\intercal _t) (_t - *)^\\intercal _t~\\le ~D X_p.$"
],
[
"Online-to-Batch Example",
"We now discuss an example where the standard theory for stochastic gradient descent does not apply, but the iterate average of online gradient descent with quantile-based filtering still gives risk convergence guarantees.",
"To keep things simple, we work in the one-dimensional setting with $\\mathcal {W}=[-1,+1]$ .",
"To stay within the assumptions of Proposition REF , we take $f_t$ to be the logistic loss $f_t(w) = h_t(w _t)$ with $h_t(z) = \\ln (1+e^{- y_t z})$ for $y_t \\in {-1,+1}$ .",
"To make things interesting, we take $_t \\in \\mathbb {R}$ to have a distribution with heavy tails, with $\\operatornamewithlimits{\\mathbb {P}}({_t} > x)$ of order $x^{-(1+\\gamma )}$ for large enough $x$ , for some $\\gamma \\in (0,1)$ .",
"Taking $\\gamma > 0$ ensures that the expected loss $\\operatornamewithlimits{\\mathbb {E}}[f_t(w)]$ is finite (as $f_t(w) \\approx (-_t y_t w)_+$ for large $_t$ ), and hence has a bonafide minimiser (which can be in the interior or on the boundary, depending on the details of the distribution).",
"Taking $\\gamma < 1$ ensures that the tails are so heavy that $\\operatornamewithlimits{\\mathbb {E}}[f_t^{\\prime }(w)^2] = \\infty $ (as $f_t^{\\prime }(w) \\approx (-y_t _t)_+$ for large $_t$ ), and hence standard theory for SGD does not apply.",
"Instead we will use Lemma REF and Proposition REF to argue that the filtered iterate average $\\bar{w}_T$ approximates the minimiser of the risk $u^*$ in the sense that $\\operatornamewithlimits{\\mathbb {E}}\\nolimits _{\\operatornamewithlimits{\\mathbb {P}}} *{\\operatorname{Risk}_{\\operatornamewithlimits{\\mathbb {P}}}(\\bar{w}_T) - \\operatorname{Risk}_{\\operatornamewithlimits{\\mathbb {P}}}(u^*)}~\\rightarrow ~ 0\\quad \\text{as}\\quad T \\rightarrow \\infty .$ To bound the risks above, we will decompose $\\operatornamewithlimits{\\mathbb {P}}= p P + (1-p) Q$ where $p$ is a quantile level chosen below, $P = \\operatornamewithlimits{\\mathbb {P}}[\\big ]{\\cdot }{{_t} \\le X_p}$ and $Q = \\operatornamewithlimits{\\mathbb {P}}[\\big ]{\\cdot }{{_t} > X_p}$ .",
"We will bound the $\\operatornamewithlimits{\\mathbb {P}}$ -risks in terms of $P$ -risks, then we will use Lemma REF to bound the $P$ -risk difference in terms of the robust regret, and we will use Proposition REF to bound that regret.",
"We will settle on picking $p = 1-\\frac{1}{\\sqrt{T}}$ .",
"This has the effect that the $p$ -quantile is $X_p \\propto T^{\\frac{1}{2(1+\\gamma )}}$ (by inverting the tail probability).",
"On the one hand, for any $w \\in \\mathcal {W}$ , the bias, i.e.",
"the difference in risk on $P$ (inliers only) and on $\\operatornamewithlimits{\\mathbb {P}}$ (full distribution), is at most of order ${\\operatorname{Risk}_{\\operatornamewithlimits{\\mathbb {P}}}(w)-p \\operatorname{Risk}_{P}(w)}&~\\le ~\\operatornamewithlimits{\\mathbb {E}}\\nolimits _{\\operatornamewithlimits{\\mathbb {P}}} *{ f_t(w) \\mathbf {1}_{{_t} > X_p}}~\\approx ~\\operatornamewithlimits{\\mathbb {E}}\\nolimits _{\\operatornamewithlimits{\\mathbb {P}}} *{ (- _t y_t w) \\mathbf {1}_{{_t} > X_p}}\\\\&~\\le ~\\operatornamewithlimits{\\mathbb {E}}\\nolimits _{\\operatornamewithlimits{\\mathbb {P}}} *{ {_t} \\mathbf {1}_{{_t} > X_p}}~=~\\int _{X_p}^\\infty \\operatornamewithlimits{\\mathbb {P}}({_t} > x) x~\\propto ~X_p^{-\\gamma }\\propto T^{-\\frac{\\gamma }{2(1+\\gamma )}}.$ On the other hand, the regret bound for $T$ -round online gradient descent with gradient norms bounded by $X$ is $B_T(X) = O(D X \\sqrt{T})$ .",
"Hence for our choice of $p$ , the first term in the bound from Proposition REF is dominant and of order $X_p \\sqrt{T}$ .",
"Dividing by $T$ to plug in to Lemma REF results in $\\frac{X_p\\sqrt{T}}{T} \\propto T^{-\\frac{\\gamma }{2(1+\\gamma )}}$ .",
"Both contributions (bias and regret) are of the same order and converge to zero, indicating that quantile-filtered online gradient descent achieves (REF )."
],
[
"Conclusion and Future Work",
"We have shown that the robust regret can be controlled for adversarial data when there are at most $k$ outliers.",
"A general question that we leave open is whether it is possible to get a bound for adversarial losses that does not depend on the number of outliers $k$ , but on some other natural property of the losses.",
"For instance, we may try to incorporate prior knowledge about the size of the gradients by specifying a prior $\\pi $ on gradient norms and bounding the robust regret in terms of the prior probability $\\pi (G(\\mathcal {S}))$ of the size of the inlier gradients.",
"A possible way to approach this might be to introduce specialist experts for different thresholds $G$ and then aggregate these.",
"This runs into severe difficulties, however, because we only find out whether a specialist should be active or not in round $t$ after making our prediction $_t$ and observing $_t$ .",
"Moreover, specialists would have different loss ranges and the robust regret can only depend on the loss range $G(\\mathcal {S})$ of the correct specialist.",
"We also provided a sublinear bound on the robust regret for i.i.d.",
"gradients when the outliers are defined as rounds in which the gradients exceed their $p$ -quantile, or when they can be bounded in terms of an i.i.d.",
"variable $_t$ .",
"Alternatively, outliers might be defined as gradients with norms exceeding their empirical $p$ -quantile at the end of $T$ rounds.",
"For i.i.d.",
"gradients, the empirical $p$ -quantile after $T$ rounds is close to the actual $p$ -quantile with high probability, so this case can be handled by running the method from Section for a slightly inflated $p$ .",
"However, the empirical quantile formulation continues to make sense even when gradients are not i.i.d., so it would be interesting to know whether a linear number of outliers can be tolerated in any such non-i.i.d.",
"cases."
],
[
"Acknowledgments",
"Van Erven and Sachs were supported by the Netherlands Organization for Scientific Research (NWO) under grant number VI.Vidi.192.095.",
"Kotłowski was supported by the Polish National Science Centre under grant No.",
"2016/22/E/ST6/00299."
],
[
"Proofs for Examples from Section ",
"[Proof of Corollary REF ] Let $\\mathcal {P}\\subset [T]$ denote the rounds that are passed on to OGD.",
"Then the linearized regret of OGD on the rounds in $\\mathcal {P}$ is bounded by $B_T(2 G(\\mathcal {S}))\\le 2 D \\sqrt{\\sum _{t \\in \\mathcal {P}} \\Vert _t\\Vert _2^2},$ as follows from arguments similar to those by [19] (see e.g.",
"Corollary 2 by [40]).",
"Bounding further, we obtain $B_T(2 G(\\mathcal {S}))&\\le 2 D \\sqrt{\\sum _{t \\in \\mathcal {S}}\\Vert _t\\Vert _2^2 + \\sum _{t \\in \\mathcal {P}\\setminus \\mathcal {S}}\\Vert _t\\Vert _2^2}\\le 2 D \\sqrt{\\sum _{t \\in \\mathcal {S}}\\Vert _t\\Vert _2^2} + 2 D \\sqrt{\\sum _{t \\in \\mathcal {P}\\setminus \\mathcal {S}}\\Vert _t\\Vert _2^2}\\\\&\\le 2 D \\sqrt{\\sum _{t \\in \\mathcal {S}} \\Vert _t\\Vert _2^2}+ 2 D G(\\mathcal {S}) \\sqrt{k},$ where the last step uses that $|\\mathcal {P}\\setminus \\mathcal {S}| \\le |[T]\\setminus \\mathcal {S}| \\le k$ by assumption on $\\mathcal {S}$ .",
"Plugging this into (REF ) and bounding $R_T(\\mathcal {S}) \\le \\widetilde{R}_T(\\mathcal {S})$ and $D(\\mathcal {S}) \\le D$ , the first inequality in (REF ) follows.",
"Finally, using that $\\Vert _t\\Vert _2 \\le G(\\mathcal {S})$ for all $t \\in \\mathcal {S}$ , we see that the second inequality holds as well.",
"[Proof of Corollary REF ] Let $\\mathcal {P}\\subset [T]$ denote the rounds that are passed on to ALG.",
"By the proof of Theorems 2.1 and 4.1 of [3], the linearized regret of ALG on the rounds in $\\mathcal {P}$ is bounded by $\\widetilde{R}_T(\\mathcal {P})\\le \\frac{1}{2} \\sum _{t \\in \\mathcal {P}} \\frac{\\Vert _t\\Vert _2^2}{\\sigma t}+ \\frac{\\sigma }{2} \\sum _{t \\in \\mathcal {P}} \\Vert _t - _2^2\\le \\frac{2 G(\\mathcal {S})^2}{\\sigma } \\big (\\ln T + 1\\big )+ \\frac{\\sigma }{2} \\sum _{t \\in \\mathcal {P}} \\Vert _t - _2^2.$ Plugging this into Theorem REF and applying the definition of strong convexity, we get that the robust regret is bounded by $R_T(\\mathcal {S})&\\le \\widetilde{R}_T(\\mathcal {S})- \\frac{\\sigma }{2} \\sum _{t \\in \\mathcal {S}} \\Vert _t - _2^2\\\\&\\le \\frac{2 G(\\mathcal {S})^2}{\\sigma } \\big (\\ln T + 1\\big )+ 4D(\\mathcal {S}) G(\\mathcal {S}) (k+1)+ \\frac{\\sigma }{2} \\sum _{t \\in \\mathcal {P}} \\Vert _t - _2^2- \\frac{\\sigma }{2} \\sum _{t \\in \\mathcal {S}} \\Vert _t - _2^2\\\\&\\le \\frac{2 G(\\mathcal {S})^2}{\\sigma } \\big (\\ln T + 1\\big )+ 4D(\\mathcal {S}) G(\\mathcal {S}) (k+1)+ \\frac{\\sigma }{2} \\sum _{t \\in \\mathcal {P}\\setminus \\mathcal {S}} \\Vert _t -_2^2\\\\&\\le \\frac{2 G(\\mathcal {S})^2}{\\sigma } \\big (\\ln T + 1\\big )+ 4D(\\mathcal {S}) G(\\mathcal {S})(k+1)+ \\frac{\\sigma D(\\mathcal {S})^2}{2} k.$ From this the desired result follows because $G(\\mathcal {S}) \\le \\widetilde{G}(\\mathcal {S})/2$ and $D(\\mathcal {S}) \\le \\max _{t : \\Vert _t\\Vert _2 \\le 2 G(\\mathcal {S})} \\frac{\\Vert _t\\Vert _2 +\\Vert \\nabla f_t(\\Vert _2}{\\sigma } \\le \\frac{\\widetilde{G}(\\mathcal {S})}{\\sigma }$ by Lemma REF below.",
"Lemma 10 Suppose $f_t$ is $\\sigma $ -strongly convex.",
"Then $\\Vert - _2 \\le \\frac{\\Vert \\nabla f_t()\\Vert _2 + \\Vert \\nabla f_t(\\Vert _2}{\\sigma }$ for all $,\\mathcal {W}$ .",
"Applying the definition of $\\sigma $ -strong convexity twice, we have $(- ^\\intercal \\nabla f_t( + \\frac{\\sigma }{2} \\Vert - _2^2\\le f_t() - f_t( \\le (- ^\\intercal \\nabla f_t() -\\frac{\\sigma }{2} \\Vert \\Vert _2^2,$ which leads to $\\sigma \\Vert - _2^2&\\le (- ^\\intercal \\big (\\nabla f_t() - \\nabla f_t(\\big )\\\\&\\le \\Vert - _2 \\Vert \\nabla f_t() - \\nabla f_t(\\Vert _2\\le \\Vert - _2 \\big (\\Vert \\nabla f_t()\\Vert _2 + \\Vert \\nabla f_t(\\Vert _2\\big ),$ from which the result follows.",
"[Proof of Lemma REF ] Let $\\widetilde{\\operatorname{Risk}}_{P_\\epsilon }() = \\operatornamewithlimits{\\mathbb {E}}_{(M,\\xi ) \\sim P_\\epsilon } *{\\tilde{f}(,M,\\xi )}$ denote the risk for the modified loss under the mixture distribution $P_\\epsilon $ .",
"Then the key to both results is to observe that $\\operatorname{Risk}_P(\\bar{}_T) - \\operatorname{Risk}_P(P)= \\frac{\\widetilde{\\operatorname{Risk}}_{P_\\epsilon }(\\bar{}_T) - \\widetilde{\\operatorname{Risk}}_{P_\\epsilon }(P)}{1-\\epsilon }.$ This allows us to apply standard results for online-to-batch conversion to the modified losses $\\tilde{f}$ under distribution $P_\\epsilon $ : the first inequality follows by combining $\\operatornamewithlimits{\\mathbb {E}}_{P_\\epsilon } *{\\widetilde{\\operatorname{Risk}}_{P_\\epsilon }(\\bar{}_T) - \\widetilde{\\operatorname{Risk}}_{P_\\epsilon }(P)}\\le \\frac{\\operatornamewithlimits{\\mathbb {E}}_{P_\\epsilon } *{\\sum _{t=1}^T *{\\tilde{f}(_t,M_t,\\xi _t) - f(P,M_t,\\xi _t)}}}{T},$ with (REF ), and the second result follows by applying Corollary 2 of [8] to the modified excess losses $\\tilde{f}(,M,\\xi ) - \\tilde{f}(P,M,\\xi )$ .",
"By the boundedness assumption on the original excess loss a.s. under $P$ , these are bounded in $[-B,B]$ a.s. under $P_\\epsilon $ , and we obtain $\\widetilde{\\operatorname{Risk}}_{P_\\epsilon }(\\bar{}_T) - \\widetilde{\\operatorname{Risk}}_{P_\\epsilon }(P)\\le \\frac{\\sum _{t=1}^T *{\\tilde{f}(_t,M_t,\\xi _t) - \\tilde{f}(P,M_t,\\xi _t)}}{T} + 2B \\sqrt{\\frac{2}{T} \\ln \\frac{1}{\\delta }}$ with $P_\\epsilon $ -probability at least $1-\\delta $ .",
"The second result then follows by plugging in (REF ) again.",
"[Proof of Corollary REF ] Let $\\mathcal {A}$ be the event that (REF ) holds with $B=DG$ and $\\delta $ replaced by $\\delta /2$ , and let $\\mathcal {B}$ be the event that the number of outliers $\\sum _{t=1}^T M_t$ as at most $k$ .",
"Then Corollary REF implies that (REF ) holds on the intersection of $\\mathcal {A}$ and $\\mathcal {B}$ , because $&\\operatorname{Risk}_P(\\bar{}_T) - \\operatorname{Risk}_P(P)\\le \\frac{2 D G \\Big (\\sqrt{T} + 2k + \\sqrt{k} + 2\\Big )}{(1-\\epsilon )T}+ \\frac{2DG}{1-\\epsilon } \\sqrt{\\frac{2}{T} \\ln \\frac{2}{\\delta }}\\\\&\\le \\frac{2 D G \\Big (\\sqrt{T} + 3k + 2\\Big )}{(1-\\epsilon )T}+ \\frac{2DG}{1-\\epsilon } \\sqrt{\\frac{2}{T} \\ln \\frac{2}{\\delta }}\\\\&\\le \\frac{6 D G \\epsilon }{1-\\epsilon }+\\frac{2 D G\\left(1 + 3 \\sqrt{2 \\epsilon (1-\\epsilon )\\ln (2/\\delta )}\\right)}{(1-\\epsilon ) \\sqrt{T}}+ \\frac{2 D G \\big ((1-\\epsilon )\\ln (2/\\delta )+ 5\\big )}{(1-\\epsilon )T}+ \\frac{2DG}{1-\\epsilon } \\sqrt{\\frac{2}{T} \\ln \\frac{2}{\\delta }}\\\\&\\le 12 D G \\epsilon +\\frac{2DG \\big (2 + 5\\sqrt{2\\ln (2/\\delta )}\\big )}{\\sqrt{T}}+ \\frac{2 D G \\big (\\ln (2/\\delta )+ 10\\big )}{T},$ where the last inequality uses the assumption that $\\epsilon \\le 1/2$ to obtain a simpler expression.",
"Now the probability of $\\mathcal {A}$ is at least $1-\\delta /2$ by the second result of Lemma REF , which applies because $-DG \\le -(-P)^\\intercal \\nabla f(P,\\xi ) \\le f(,\\xi ) - f(P,\\xi ) \\le (-P)^\\intercal \\nabla f(,\\xi ) \\le DG$ $P$ -almost surely.",
"And the probability of $\\mathcal {B}$ is at least $1-\\delta /2$ by Bernstein's inequality [4].",
"Hence, by the union bound, it follows that both $\\mathcal {A}$ and $\\mathcal {B}$ hold simultaneously with probability at least $1-\\delta $ , as required."
],
[
"Proof of Theorem ",
"The point of departure is that, by definition, $\\mathbf {1}*{{_t}_* \\le G_p}$ is i.i.d.",
"Bernoulli-$p$ .",
"By Bernstein's concentration inequality (for fixed time $t$ ), we have that w.p.",
"$\\ge 1-\\delta $ $\\frac{1}{t} \\sum _{s=1}^t \\mathbf {1}*{{_s}_* \\le G_p}~\\ge ~p- \\sqrt{\\frac{2 p(1-p) \\ln \\frac{1}{\\delta }}{t}}- \\frac{\\ln \\frac{1}{\\delta }}{3 t}$ Rephrasing this event with the empirical quantile function $\\hat{q}_t$ , we see that w.p.",
"$\\ge 1-\\delta $ , $G_p~\\ge ~\\operatorname{LCB}_t~~\\hat{q}_t*{ p- u_t}\\qquad \\text{where}\\qquad u_t\\sqrt{\\frac{2 p(1-p) \\ln \\frac{1}{\\delta }}{t}}+ \\frac{\\ln \\frac{1}{\\delta }}{3 t}.$ We apply the analogous concentration in the other direction to find that with probability at least $1-\\delta $ , $G_{p - 2 u_t - 3/(2 t) \\ln \\frac{1}{\\delta }} \\le \\operatorname{LCB}_t$ where the extra margin $\\frac{3 \\ln \\frac{1}{\\delta }}{2 t}$ is necessary to correct for the fact that $p - 2 u_t$ may be closer to $1/2$ than $p$ , and hence require a slightly enlarged confidence width.",
"In the remainder of the proof, we fix $\\delta = T^{-2}$ , to ensure that the probability of failure of either event (REF ) or (REF ) over the course of $T$ rounds is at most $\\frac{2}{T}$ .",
"Let $\\mathcal {E}_t$ denote the event that (REF ) and (REF ) hold at round $t$ , and let $\\mathcal {E} = \\bigcap _{t=1}^T \\mathcal {E}_t$ .",
"We split the expected robust regret in three parts $\\bar{R}_T \\le P^{(1)} + P^{(2)} + P^{(3)}$ spelled out below, depending on whether the desired concentration event $\\mathcal {E}$ holds or not, and within $\\mathcal {E}$ we split the rounds in those where FILTER passes the gradient on to ALG and those were it was ignored: $P^{(1)}&~~\\operatornamewithlimits{\\mathbb {E}}*{\\mathbf {1}_{\\mathcal {E}}\\max _{\\mathcal {W}} \\sum _{t \\in [T] : {_t}_* \\le \\operatorname{LCB}_{t-1}} {_t - _t}}\\\\P^{(2)}&~~\\operatornamewithlimits{\\mathbb {E}}*{\\mathbf {1}_{\\mathcal {E}}\\max _{\\mathcal {W}} \\sum _{t \\in [T] : \\operatorname{LCB}_{t-1} < {_t}_* \\le G_p} {_t - _t}}\\\\P^{(3)}&~~\\operatornamewithlimits{\\mathbb {E}}*{\\mathbf {1}_{\\mathcal {E}^c}\\max _{\\mathcal {W}} \\sum _{t \\in [T] : {_t}_* \\le G_p} {_t - _t}}$ For part $P^{(1)}$ , we apply the individual-sequence regret bound $B_{\\hat{T}}(G_p)$ of ALG, where $\\hat{T}$ is the random number of rounds, for which we have $\\operatornamewithlimits{\\mathbb {E}}{\\hat{T}} \\le p T$ .",
"We then drop the indicator and Jensen the expectation inside to get $P^{(1)} \\le B_{p T}(G_p)$ .",
"For part $P^{(2)}$ , we use that $\\mathcal {E}$ and $\\operatorname{LCB}_{t-1} < {_t}_*$ imply that $G_{p - 2 u_t - 3/(2 t) \\ln \\frac{1}{\\delta }} < {_t}_* < G_p$ to find $\\mathbf {1}_{\\mathcal {E}}\\max _{\\mathcal {W}}\\sum _{t \\in [T] : \\operatorname{LCB}_{t-1} < {_t}_* \\le G_p} {_t - _t}~\\le ~\\mathbf {1}_{\\mathcal {E}}\\sum _{t \\in [T] : G_{p - 2 u_t - 3/(2 t) \\ln \\frac{1}{\\delta }} < {_t}_* \\le G_p} D G_p$ Dropping the indicator, taking expectation and using the definition of quantile yields $P^{(2)}&~\\le ~D G_p\\sum _{t=1}^T\\operatornamewithlimits{\\mathbb {P}}*{G_{p - 2 u_t - 3/(2 t) \\ln \\frac{1}{\\delta }} < {_t}_* < G_p}\\\\&~\\le ~D G_p \\sum _{t=1}^T (2 u_t + 3/(2 t) \\ln \\frac{1}{\\delta })\\\\&~\\le ~D G_p *{4 \\sqrt{2 p (1-p) T \\ln T}+ \\frac{13}{3} (\\ln T)^2 + 1}$ Finally, for $P^{(3)}$ we use that the integrand is bounded by $T D G_p$ , and the error probability by $\\operatornamewithlimits{\\mathbb {P}}(\\mathcal {E}^c) \\le \\frac{2}{T}$ to find $P^{(3)} \\le 2 D G_p$ ."
]
] | 2107.01881 |
[
[
"Towards Scheduling Federated Deep Learning using Meta-Gradients for\n Inter-Hospital Learning"
],
[
"Abstract Given the abundance and ease of access of personal data today, individual privacy has become of paramount importance, particularly in the healthcare domain.",
"In this work, we aim to utilise patient data extracted from multiple hospital data centres to train a machine learning model without sacrificing patient privacy.",
"We develop a scheduling algorithm in conjunction with a student-teacher algorithm that is deployed in a federated manner.",
"This allows a central model to learn from batches of data at each federal node.",
"The teacher acts between data centres to update the main task (student) algorithm using the data that is stored in the various data centres.",
"We show that the scheduler, trained using meta-gradients, can effectively organise training and as a result train a machine learning model on a diverse dataset without needing explicit access to the patient data.",
"We achieve state-of-the-art performance and show how our method overcomes some of the problems faced in the federated learning such as node poisoning.",
"We further show how the scheduler can be used as a mechanism for transfer learning, allowing different teachers to work together in training a student for state-of-the-art performance."
],
[
"Introduction",
"Federated learning is a field that has emerged recently due to the abundance of data available today and the risks that this poses to individuals.",
"Privacy (particularly of personal information) is of great importance and should be protected by researchers working in machine learning.",
"In parallel, the emergence of electronic health records (EHRs) has allowed the digitisation of much personal information pertaining to the health conditions of individuals.",
"EHRs are often used in many machine learning research projects [23].",
"This often involves the transfer and storage of very sensitive information which increases the risk of data leakage.",
"As a result, federated learning can minimise this risk by utilising the data at it's source rather than transferring it to the researchers servers to be processed.",
"Machine learning researchers working with EHR data will be all too familiar with the difficulty of gaining access to this information in the first place.",
"It can be a very lengthy and exhausting process (for good reason) to gain access and utilise the EHRs of one healthcare institution let alone accessing the datasets of many.",
"Federated learning offers an alternative in that it allows the data of the patients to be utilised while reducing the risk of their privacy being compromised.",
"Federated learning is not without its own limitations however.",
"The different datasets that are stored in the different nodes may have different underlying distributions due to their data collection processes which can make machine learning across multiple domains difficult.",
"There is also the possibility of data at each node being corrupted either maliciously or accidentally, leading to data that is undesirable to use for training.",
"These issues can all lead to difficulties in training and convergence of the overall model being trained.",
"To overcome these limitations, in this work we propose the following setup.",
"Firstly, We use federated learning to i) protect the privacy of patients by minimising the movement of their data and ii) improve the quality of our machine learning model by utilising a diverse dataset sourced from different hospitals.",
"As we are `blind' to the data at the nodes, we propose the use of a student-teacher network setup.",
"The teacher (a reinforcement learning agent) will have access to the local servers and be able to select the appropriate data for the `student' (our model) to be trained on at that time.",
"The `scheduler' will be responsible for directing the teacher to a given data centre to select data for training on.",
"To summarise: (The Student:) - the machine learning model we are training.",
"(The Teacher:) - a reinforcement learning agent that selects data from a data centre based on the state of the student.",
"This essentially defines a curriculum at each training step for the student.",
"(The Scheduler:) - directs the teacher to the appropriate data centre for training.",
"This is also based on the state of the student at each training iteration.",
"Section discusses the related work that has been carried out and Section provides a detailed description of how our algorithm works.",
"Section details the datasets we benchmark our method against.",
"We then present the results in Section and discuss their significance and interesting behaviours of our model in Section ."
],
[
"Related Work",
"Federated learning has been used by researchers to exploit larger pools of data for training [4], preserve data privacy [31] and distributing computational resource requirements [32].",
"Federated learning has also been used for healthcare applications to simultaneously utilise multiple datasets to train a model on patient data.",
"In this work we create a model that learns in a federal fashion through the interaction of a scheduler that is trained using meta-gradients and a student-teacher algorithm that is trained using reinforcement learning.",
"Meta-learning has been used effectively in [25] where the loss of a student model on a validation set was used as a signal to update the weights of a generative model.",
"This work demonstrated the rapid and effective training of methods that exploit meta-gradients.",
"Meta-learning was also used in [33], where the meta-gradients are used to tune the parameters of an actor-critic algorithm.",
"As a result of the efficacy of this method in these domains we choose to use meta-gradients in order to schedule which data centre the gradients to update are student model will come from.",
"The meta-learned scheduler chooses a node representing a data centre where a student-teacher algorithm is used to sample data.",
"Student-teacher algorithms have been used in multiple works, with the general premise that one algorithm (teacher) is trained to train another (student) [6], [18].",
"These methods have also been used with a curriculum [2], where the curriculum is either pre-defined and exploited by the teacher [5] or implicitly learned by the teacher during training [9].",
"Federated learning is a method of training a model (in our case a deep neural network) by using data from multiple centres, without having central access to each of them [20].",
"Local models at each of the data centres are iteratively updated and aggregated to form a global model.",
"At each round of iteration, a central coordinator samples a subset, $m$ , of local models, $S_m$ , and sends them the current global model $G^t$ .",
"Each member of $S_m$ then updates this global model using their local data to create an updated model $L^{t+1}$ .",
"These models are then aggregated and are sent back to update the global model as: $G^{t+1} = G^t + \\frac{\\eta }{n}\\sum ^{m}_{i=1}\\left(L^{t+1}_{i} - G^t\\right)$ where $n$ is the number of nodes (i.e., data centres) and $\\eta $ acts as a learning rate for replacing the global model with the aggregate of the local models.",
"While this has been shown to work in many cases, [28] make the argument that there is inherent difficulty in updating neural networks in this manner.",
"They argue that the permutation invariance of the summation operand renders averaging in the parameter space a naive approach.",
"For meaningful averaging to be done, the permutation must first be undone."
],
[
"Compromising Federated Learning",
"One of the vulnerabilities of federated learning is that nodes being compromised can significantly affect the training of the global model [3].",
"Attacks of these sort can either `poison' the data found at one of the nodes (known as an adversarial attack) or bias the model that is trained at one of the nodes significantly, leading it to highly skew the aggregation step [26].",
"There is also the possibility of the attack being a single-shot attack or a repeated attack [7].",
"In the single shot case, only one of the nodes is compromised whereas in the repeated case, multiple nodes can be compromised at any given time.",
"Many works have been produced in discussing how federated learning can be compromised by introducing a backdoor into the training process [10], [1].",
"A backdoor is an attack that causes a classifier to produce unexpected behaviour if a specific trigger is added to the input space.",
"An example is a sticker being added to an image and associating this with the incorrect label [10].",
"Defences against these attacks have been developed with some authors using pruning of redundant neurons for the core classification task [17], using outlier detection to detect potential triggers [27], and re-training and preprocessing inputs [19].",
"In this work we aim to overcome these limitations and build defence into the training procedure through the use of a student-teacher network that actively selects which data to train on."
],
[
"Methodology",
"Our method is comprised of three agents in the training setup, the student, the teacher and the scheduler."
],
[
"The Overall Setup",
"The overall setup of our federated learning training routine is as follows.",
"We have a scheduler that controls which node we will be learning from (this can be one-hot or we can select multiple nodes).",
"The teacher at the node can then select a batch of data according to the state of the student.",
"The student at the node is a copy of the global student.",
"We use the student to forward pass the batch of data selected by the teacher and return the loss.",
"In the one-hot scheduler scenario, we send back the loss to the global student model to update the weights via backpropagation.",
"In the multi-node learning scenario, we aggregate the losses from all nodes selected and feed these back to the global model for updating."
],
[
"Data Preprocessing",
"The first step we must take in order to exploit our teacher setup is to rank our data according to some metric.",
"Using [30] as a guide, we choose to use the Mahalanobis distance expressed as: $d\\left({\\mathbf {x}_n}\\right) = \\left(\\left(\\mathbf {x_n} - \\mu \\right)^T \\mathbf {S^{-1 }}\\left({\\mathbf {x_n} - \\mu }\\right)\\right)^\\frac{1}{2}$ for medical datasets, and the cosine similarity as our similarity metric for image datasets.",
"As the tabular data found in electronic health record systems consist of multiple data types, we encode these using a denoising autoencoder.",
"This trained encoder is distributed to all the nodes so that the data in each node is processed in the same way for consistency."
],
[
"The Teacher",
"For the student-teacher interaction we follow the setup in [5].",
"The task of the teacher is to select a batch of data from the curriculum by selecting the index along the curriculum and the `width' around that index to include in the selection.",
"The following sequential steps are implemented: The data at each node is organised into $N$ curriculum batches according to some metric $H$ .",
"The teacher selects one or more batches for feeding into the student.",
"A pre-trained autoencoder is used to create a latent representation of the batch.",
"The student is trained on this batch and it's performance on a separate validation set is recorded.",
"In this work we use the Mahalanobis distance for $H$ for medical data, and cosine similarity for image data as summarised in [30].",
"The teacher is a reinforcement learning agent and therefore is tasked with minimising the Bellman loss function given by: $\\mathcal {L}(\\theta _i) = \\left(r + \\gamma \\max _{a^{\\prime }}Q(s^{\\prime },a^{\\prime };\\theta ^{-}_i) - Q(s,a;\\theta _i) \\right)^2$ where $r$ is the reward of a state-action $(s,a)$ tuple, $\\gamma $ the discount factor, $Q$ is the q-value defining the value of taking an action given a state and $\\theta $ and $\\theta ^{-}$ are the parameters of the prediction and target (the version of the teacher that is held constant for $K$ steps to stabilise training as described in [21]) networks respectively.",
"As we also choose to use an actor-critic setup for the teacher, the action space and Q-function are separately parameterised.",
"This allows a continuous action space and the actor that selects actions is updated using the following loss: $\\nabla \\theta ^{\\mu } J \\approx \\frac{1}{N} \\sum _i \\nabla _a Q\\left(s,a \\:\\vert \\:\\theta ^Q\\right)\\:\\vert \\:_{s=s_i, a=\\mu \\left(s_i\\right)} \\nabla _{\\theta ^{\\mu }} \\mu \\left(s\\:\\vert \\:\\theta ^{\\mu }\\right)$ where $Q$ is the Q-function and $\\mu $ is the policy.",
"The teacher can either be pre-trained, or jointly trained with the scheduler.",
"The teacher can also either be trained on the dataset of one node and distributed to the rest or independently trained at each node.",
"The latter is preferable due to the ability of the teacher to adapt to the dataset at hand.",
"However, the former is useful when not all nodes in the federated system have access to computational power.",
"The intuition is that the curriculum strategy learned by the teacher should be general for the task at hand and thereby provide strong performance."
],
[
"The Student",
"The input to the teacher is the current state of the student.",
"The student in this work is a feedforward neural network that is tasked with classification.",
"The state of the student is defined as a representation of the weights of the student.",
"Given a matrix of weights, $W^{ij}$ , between layers $i$ and $j$ of the network, for each row, $W^{ij}_{n:}$ , we take the inner product of the row with a fixed reference vector $a$ .",
"From this inner product we extract $\\vert \\langle W_{n:}^{ij}, a \\rangle \\vert $ and $\\angle \\left(W_{n:}^{ij}, a\\right)$ for $n = 1,2, \\dots , M_i$ where $M_i$ is the number of hidden nodes in layer $i$ .",
"These values are concatenated to represent the row and this process is repeated for all rows to build the vector.",
"For more hidden layers the process is repeated until we have one vector representing the network.",
"This provides us with a representative vector, $\\mathbf {v} \\in \\mathbb {R}^{2\\left(\\sum _{l}^{h} M_l\\right)}$ , where $h$ is the number of hidden layers.",
"This vector is what is fed to the teacher to understand the state of the student."
],
[
"The Scheduler",
"The scheduler is the last of our agents in the training setup.",
"This is also a neural network that takes the student state as input and selects which of the nodes the training data should come from at the current iteration of training.",
"This agent is trained using the meta-gradients generated from validation losses similarly to how they are employed in [25].",
"We have an `inner' loop of training whereby the student-teacher interaction takes place.",
"In the `outer' loop, we aggregate the losses on validation sets at each node and use this aggregated loss as the signal to update the weights of our scheduler.",
"Figure: A diagram displaying what happens in the training routine every iteration.",
"The black arrows indicate selection, the blue indicate the extraction of state, the dashed indicate the transfer of a model and the orange indicate the movement of losses.Figure REF shows diagrammatically the training procedure at every iteration of training.",
"In the outer loop the scheduler selects the node(s) to use for data selection.",
"The global student model is sent to the node and the teacher at the node then selects the data in the inner loop and trains the student network on this.",
"The student is then sent back to the central node and distributed to all nodes.",
"This student is tested on separate validation sets at each node and their losses are aggregated by summing them.",
"They are then used as the loss to update the scheduler.",
"The inner loop loss function is dependent upon the target task with crossentropy used for classification and mean squared error used for regression.",
"The scheduler is updated as: $\\theta _{sc}^{t+1} = \\theta _{sc}^t + \\omega \\sum _{0 \\le t^{\\prime } \\le t} \\alpha ^{t-t^{\\prime }} \\nabla \\left(\\mathcal {L}_{inner} \\right)$ where $\\mathcal {L}_{inner}$ is given by: $\\ell _{inner}\\left(T\\left(\\mathbf {s}_{t}; \\theta _{te}\\right),\\mathbf {y}^*_{t}; \\theta _{st}\\right) \\\\+ \\sum _{n=1}^{N-1} \\ell _{inner}\\left(\\mathbf {x}^n_{v}, \\mathbf {y}^n_{v}; \\theta _{st}\\right)$ where $\\theta _{sc}$ are scheduler weights, $\\omega $ is a learning rate for stochastic gradient descent, $\\alpha $ is a momentum hyperparameter, $\\ell _{inner}$ is our local task loss function, $T$ is the teacher network taking as input the student state at iteration $t$ , $\\mathbf {s}_t$ , $\\mathbf {y^*_t}$ is the ground truth associated with the teacher selection and $\\theta _{te}$ and $\\theta _{st}$ the teacher and student parameterisations respectively.",
"$\\mathbf {x}^n_v$ and $\\mathbf {y}^n_v$ are the features and labels of the validation set of node $n$ respectively."
],
[
"Entropy Loss",
"As the scheduler is trained using meta-gradients, the model may converge after a set number of iterations.",
"While we would like convergence of the student (i.e., the task solving model), we do not necessarily need the scheduler to converge.",
"In fact using the loss on the validation set as the signal to update the scheduler automatically prevents the scheduler from converging for very long.",
"This is because, should the scheduler converge on selecting a particular data centre, the validation scores on the other data centres will deteriorate thereby increasing the aggregated loss on the validations and providing an update signal to the scheduler.",
"This however was found in practice to require many iterations of training and so in order to encourage the exploration further we add an entropic loss term to the scheduler.",
"This takes the form: $\\ell _{ent} = \\frac{1}{H\\left[S\\left(s_t;\\theta ^t_{sc}\\right)\\right] + \\epsilon }$ where $H$ is entropy, $S\\left(s_t;\\theta ^t_{sc}\\right)$ is the softmax output of the scheduler and $\\epsilon $ is a small positive value (we use $10^{-5}$ ) to prevent potential division by zero.",
"$\\ell _{ent}$ is then added on to the end of the expression shown in Equation REF to discourage fast convergence of the scheduler.",
"For tabular classifications, we define the student to be a feedforward neural network consisting of 2 hidden layers and 50 nodes in each layer.",
"These are all activated with ReLU activations.",
"For the image recognition tasks we initialise a student that has 4 convolutional layers with 32 filters of size 3x3 in the first two layers and 64 of these in the second two.",
"These are all activated by ReLU and maxpooled and are followed by 3 feedforward layers of size 50 nodes each.",
"For the results reported in Section , we use a pre-trained teacher (i.e., the teacher has been trained using reinforcement learning on different students for the same classification problem).",
"The teacher has 2 hidden layers and 150 nodes each activated by ReLU apart from the final layer (of size 2) which is activated by a tanh function.",
"For the scheduler we use a feedforward neural network with 2 hidden layers and 100 nodes in each layer activated by ReLU.",
"The output is activated by a softmax of size the number of nodes in the federated system."
],
[
"Datasets",
"In this study we considered the patient data collected in the electronic health records (EHR) of the Other Unknown Hospital (OUH), which the authors are associated with.",
"The data is split randomly in $N$ datasets so that each can act as a separate machine in a federated system.",
"The features include demographic, physiological and medical information (such as age, heart rate upon entry and any medical tests requested by clinical staff who greet the patient).",
"We aim to predict which department in the hospital the patient will consume resource from (i.e., which department in the hospital will ultimately be responsible for treating the patient) rendering this a seven-class classification.",
"In carrying out this classification, this allows hospitals to predict their resource requirements ahead of time and update scheduling and planning accordingly.",
"Only patients who were admitted in an emergency were considered providing a dataset of 14,324 patients.",
"A training set of 60% of the dataset was used and was balanced, leaving 8,589 patients for training on.",
"The validation set was 20% of the dataset and testing was also 20%.",
"These sets are then evenly divided according to the number of nodes in the federated system.",
"The full feature set is included in the supplementary material."
],
[
"eICU",
"In order to validate our results on real-world data collected from different hospitals, we introduce the eICU dataset [22] also hosted on Physionet [8].",
"The task here is mortality prediction (binary classification) based on features extracted from admission to the ICU as is done in [24].",
"As this dataset contains identifiers for individual hospitals, we are able to create nodes corresponding to each hospital.",
"The features selected are as outlined in the appendix.",
"We choose to learn from the eight hospitals with the largest populations in the dataset leaving us with 8,594 instances.",
"We sample 60% from each node to keep as the training set, and keep 20% for the validation set and the final 20% as the test set.",
"As per usual, the validation set is kept on the local node for performance aggregation during the scheduler training.",
"To test our methodology on the image space we also report results on the CIFAR-10 image recognition dataset [15].",
"We use 40,000 training examples for the training set and 10,000 each for validation and test set examples.",
"We once again randomly divide the dataset into $N$ datasets to mimic a federated learning system.",
"It should be noted that in Table REF the CIFAR-10 dataset has been split into three samples of size 15000, 15000 and 10000 due to the possibility of the teacher selecting a full training set batchsize and memory constraints.",
"As before, we utilise the MNIST dataset [16] as another publicly available dataset to assess our results against.",
"We use 30,000 examples for training, 10,000 for validation and 10,000 for the test set.",
"Once again the dataset is partitioned into $N$ datasets to emulate the federated learning approach."
],
[
"Results",
"As our work lies in the intersection of two research areas within the field of machine learning (namely federated learning and student-teacher learning), we choose to use baselines from both of these fields as comparators.",
"From the student-teacher learning side, we will assess how our method compares in terms of final model performance only.",
"For the federated learning comparison we will compare not only the final model performance but also the robustness of the method to attack."
],
[
"Final Model Performance",
"Table REF shows how the performance of our federated learning method (FLST) compares to other state-of-the-art classification methods.",
"The baselines we use are the reinforcement learning trained student-teacher setup without scheduling [5] (RLST), two state-of-the-art methods used for classifying tabular data (DeepFM [11] and Deep+CrossNet [29]) and two state-of-the-art classifiers for image recognition (GPipe [13] and DenseNet [12]).",
"As baselines we train a standard feedforward neural network (a convolutional neural network for the image datasets) using stochastic mini-batch training (SMBT) and a curriculum (CURRIC) for comparison.",
"We see that our federated system is capable of producing a performance that is competing with state-of-the-art models that are trained in a centralised manner.",
"Figure REF shows the scheduler outputs during training for the hospital admission problem and Figure REF shows the performance of the student and the actions of the teacher during training.",
"When training on the eICU dataset, we utilised data from individual hospitals as the separate nodes in the federated system.",
"A natural question that arises is how the increasing the number of nodes in the system affects the final performance of the student.",
"Figure: The number of nodes in the federated system (i.e., number of hospitals) versus the student performance at test time.",
"Note that in this figure the seed of the scheduler is held constant and the error bars correspond to the scores generated from using different hospitals for training.",
"So for N=k nodes, the error bars correspond to training only using data from ten different combinations of k hospitals.In Figure REF we carry out an ablation on the number of nodes in the system and how it affects performance.",
"In this plot, the scheduler is held at constant seed and the error bars are generated due to selecting different hospitals to train on.",
"There are a total of 208 hospitals in the eICU dataset, but we only select from the top 20 in terms of volume of data recorded.",
"As a result, Figure REF shows error bars for the difference in performance when ten different combinations of hospital data are used.",
"We can see that with more data being used in the system, the final performance generally increases.",
"We also see that the variance in the performance starts to decrease with the increase in the number of nodes.",
"Given the increase in the volume of data being trained with as nodes are added, this aligns with expectations.",
"The performance seems to plateau indicating that adding more nodes to the system may not necessarily be beneficial for performance.",
"This could be useful for real-world application as it will allow practitioners to prioritise data centres with the highest quality data for their federated systems.",
"Table REF shows how our method performs when compared to other federated learning algorithms.",
"For our baselines we use FedAvg [20] where the local models at each node are aggregated before being averaged, as well as FedMA [28], which constructs a shared global model in a layer-wise manner by matching and averaging hidden elements (such as neurons and hidden states).",
"We investigate how performance deteriorates when exposed to different backdoor attacks.",
"We see that our model performs equivalently to state-of-the art federated training setups in terms of test-time performance but outperforms these models when exposed to attack.",
"Through the use of the scheduler, our approach provides an added layer of redundancy in the system thereby allowing attacks to be avoided after their implicit detection through degraded performance on the validation sets stored at all nodes.",
"Figure: A four-node federated system being scheduled for training a student on the hospital admission location prediction problem.",
"The different colours represent the different nodes.Figure: The performance of the student and actions taken by the teachers (scheduled according to Figure ) at each node to train the student.",
"The orange `x' and blue bar indicate the first and second outputs of the teacher respectively (index of data along the curriculum that is selected and how much data around this to include in that batch).",
"The red line shows the performance of the student on the held-out test set on the hospital admission problem.In our approach we choose to exploit local models to select data and therefore extract gradients that we use to update a global model.",
"The advantage of this approach is that it provides flexibility for a poisoned node within the federated system to be discounted or unused.",
"This can also be done on the fly without the need to inspect the local models after each training run or the data stored at each node.",
"In the following section we look at the ways that a node can be compromised and see how our setup may be able to avoid the global model being poisoned by these scenarios."
],
[
"Implicit Defensive Setup",
"There are various ways in which a federated system can be attacked.",
"In this section, we first show that the scheduler chooses appropriate nodes for training.",
"We then aim to show how using a teaching setup, we can avoid some of the issues that could be faced by a federated learning system under attack."
],
[
"Selecting the Right Teacher",
"In our first set of experiments we aimed to see if the scheduler would be able to select the appropriate teacher for the learning task at hand.",
"Our scheduler's task is to select a teacher from three different nodes to train the student.",
"We pre-trained three teachers for separate tasks (hospital admission, CIFAR-10 and MIMIC-III mortality prediction [14]) and allowed our scheduler to choose from these in order to source the data for training.",
"Figure REF shows how with training, the scheduler learns to assign the teaching job to the node that contains the teacher trained to teach CIFAR-10 learning.",
"As training progresses, the response from the scheduler becomes entirely dominated by the node in the federated system that corresponds to the appropriate teacher for the task.",
"From this we see that our approach allows for robust teacher selection (further examples of this are included in the supplementary material).",
"Figure: Scheduler selection as the student is trained.",
"The student is being trained on CIFAR-10 and the scheduler learns to use the CIFAR-10 teacher to teach the student."
],
[
"Compromised Data at Nodes",
"Our next set of experiments investigated the robustness of our method to attack through poisoning of data at local nodes in the federated system.",
"By scheduling training through the use of meta-gradients, we hypothesised that there would be an extra layer of redundancy which would prevent immediate poisoning of the model.",
"In this experimental setup we split our dataset randomly (the size of the splits is also random).",
"We only keep one node clean, with the rest of the datasets on the other nodes being replaced with random values.",
"Figure REF shows learned scheduling for a student being trained on the MNIST dataset.",
"We see how the scheduler begins with selecting a corrupted dataset before quickly transitioning to selecting another corrupted dataset.",
"The scheduler then selects a dataset with clean data and this selection dominates for the rest of training.",
"In Figure REF we see that due to the initial training on corrupted data, the test-set performance degrades.",
"However, as soon as the scheduler learns to use the clean data, the performance improves rapidly.",
"It can also be noticed that the performance achieved is below state-of-the-art.",
"This is likely due to there being a much smaller diversity in trainin data due to node corruption.",
"Figure: The scheduler selection for the different nodes in the federated system.",
"We see that the scheduler learns to select the only clean dataset for training.",
"The magnitudes of the different colours indicate the softmax output of the scheduler.Figure: The teacher data selection as well as the performance of the student on the held-out test set for the MNIST digit recognition problem.",
"The orange `x' and blue bars represent the first and second outputs of the teacher respectively.",
"The dashed line shows the maximum accuracy achieved.",
"The red line is the performance of the student on the held-out test set.The next experiment we investigate is how our system trains when there are compromised local models (whether it is the student or the teacher).",
"We do this by replacing one of the weights of the teacher on one of the local nodes with random values.",
"Due to the setup, the effect of having a compromised student or teacher is the same: a high loss which encourages the scheduler to change its selection.",
"Figure REF shows how the scheduler selects nodes to train from.",
"We see initially nodes 1 and 0 are used to train before the scheduler attempts to use the poisoned teacher.",
"After repeated reductions in the federated validation sets, the scheduler rapidly changes its selection favouring nodes 4 and increasingly 3.",
"With further training, we see this rapid removal of training using node 1 continue whenever it is encountered.",
"Figure: Scheduler selection for a four-node federated system.",
"The hosp1 node (orange) corresponds to the poisoned teacher (randomised weights).",
"We see that after brief selection, the scheduler reduces the contribution of this node.",
"The student is being trained on the MNIST task.Due to the use of a scheduler, there is potential for exploiting transfer learning in order to train students on tasks that there are no trained teachers for.",
"We hypothesise that through exploiting the various teacher's skills in succession, we can provide the gradients that the student needs in order to master the unseen task.",
"Experiments to test this hypothesis are included in the supplementary material."
],
[
"Discussion and Conclusion",
"In this work we have shown that using a federated system, with teachers at the nodes that select the data to provide gradient updates to a central model, can achieve state-of-the-art performance as well as protect the privacy of patients.",
"We have further shown that the setup provides some protection against attacks on data stored at each node or the local models being used at each node.",
"However, there remain some challenges associated with this approach that need to be addressed in order for it to become practicable.",
"The first is that the training of the centralised model (student) is inherently unstable due to the continual training that this setup expects.",
"It is expected that the student performance will converge with training, however should data poisoning occur, the scheduler will need to continue a few iterations of training in order to recover a well-performing model.",
"However, posing the problem in this way also provides flexibility for growing datasets at each node.",
"All that would need to happen would be the re-sorting of the curriculum at each node and the scheduler and teachers could be used as before.",
"Another limitation is the need for centralised control.",
"It is important in this setup that all the hospitals communicate their responses to the central node for actions to be taken by the scheduler.",
"In the case of large institutions such as hospitals, this may be acceptable, but is unlikely to be for faster-paced learning environments such as learning from mobile phones, where interruptions to communication can be frequent.",
"However, upon re-connection to the federated system, any reductions in performance to the whole system will be used as signals to improve the scheduler selection and with training the performance should recover.",
"Furthermore, in order to ensure diversity in selections by the scheduler we introduced the entropy loss that discouraged convergence on one selection.",
"This may be a naive way of encouraging diversity in selection and we believe that there may be better additive losses and regularisation terms that can be used to design a loss function that will serve the purposes of the scheduler better.",
"For further protection against attack, sentry agents (much like the teachers) could also be trained to detect any anomalies or designed attacks within the batches selected by the teachers before the losses are passed onto the central node.",
"This would reduce the burden of scanning the entire dataset at the node before training.",
"To conclude, we believe we have presented a promising direction for federated learning between large institutions such as hospitals.",
"With further work, we believe that we can develop this into a robust system that can continually learn from growing datasets while maintaining a state-of-the-art performance for the task at hand."
]
] | 2107.01707 |
[
[
"Minimal norm Hankel operators"
],
[
"Abstract Let $\\varphi$ be a function in the Hardy space $H^2(\\mathbb{T}^d)$.",
"The associated (small) Hankel operator $\\mathbf{H}_\\varphi$ is said to have minimal norm if the general lower norm bound $\\|\\mathbf{H}_\\varphi\\| \\geq \\|\\varphi\\|_{H^2(\\mathbb{T}^d)}$ is attained.",
"Minimal norm Hankel operators are natural extremal candidates for the Nehari problem.",
"If $d=1$, then $\\mathbf{H}_\\varphi$ has minimal norm if and only if $\\varphi$ is a constant multiple of an inner function.",
"Constant multiples of inner functions generate minimal norm Hankel operators also when $d\\geq2$, but in this case there are other possibilities as well.",
"We investigate two different classes of symbols generating minimal norm Hankel operators and obtain two different refinements of a counter-example due to Ortega-Cerd\\`{a} and Seip."
],
[
"Introduction",
"Let $\\mathbb {T}^d$ denote the $d$ -dimensional torus and equip $\\mathbb {T}^d$ with its Haar measure.",
"The Hardy space $H^p(\\mathbb {T}^d)$ is the subspace of $L^p(\\mathbb {T}^d)$ comprised of functions whose Fourier coefficients are supported on $\\mathbb {N}_0^d$ , where $\\mathbb {N}_0=\\lbrace 0,1,2,\\ldots \\rbrace $ .",
"Let $\\overline{H^2}(\\mathbb {T}^d)$ be the subspace of $L^2(\\mathbb {T}^d)$ comprised of the complex conjugates of functions in $H^2(\\mathbb {T}^d)$ .",
"The orthogonal projections from $L^2(\\mathbb {T}^d)$ to $H^2(\\mathbb {T}^d)$ and from $L^2(\\mathbb {T}^d)$ to $\\overline{H^2}(\\mathbb {T}^d)$ will be denoted $P$ and $\\overline{P}$ , respectively.",
"For a symbol $\\varphi $ in $H^2(\\mathbb {T}^d$ ), we consider the associated (small) Hankel operator $\\mathbf {H}_\\varphi f = \\overline{P}(\\overline{\\varphi } f)$ which maps $H^2(\\mathbb {T}^d)$ to $\\overline{H^2}(\\mathbb {T}^d)$ .",
"The lower and upper norm estimates $\\Vert \\varphi \\Vert _{H^2(\\mathbb {T}^d)} \\le \\Vert \\mathbf {H}_\\varphi \\Vert \\le \\Vert \\varphi \\Vert _{H^\\infty (\\mathbb {T}^d)}$ are both well-known and trivial.",
"We say that the Hankel operator $\\mathbf {H}_\\varphi $ has minimal norm if it attains the lower bound in (REF ).",
"Recall that a function $I$ in $H^2(\\mathbb {T}^d)$ is called inner whenever $|I(z)|=1$ for almost every $z$ in $\\mathbb {T}^d$ .",
"If $\\varphi = C I$ for a constant $C$ and an inner function $I$ , then clearly $\\Vert \\varphi \\Vert _{H^2(\\mathbb {T}^d)} = \\Vert \\varphi \\Vert _{H^\\infty (\\mathbb {T}^d)} = |C|$ and consequently $\\mathbf {H}_\\varphi $ has minimal norm by (REF ).",
"It turns out that there are no other minimal norm Hankel operators on the one-dimensional torus.",
"Theorem 1 Suppose that $\\varphi $ is in $H^2(\\mathbb {T})$ .",
"Then $\\mathbf {H}_\\varphi $ has minimal norm if and only if $\\varphi $ is a constant multiple of an inner function.",
"By orthogonality, the upper bound in (REF ) can be improved to $\\Vert \\mathbf {H}_\\varphi \\Vert \\le \\inf \\left\\lbrace \\Vert \\psi \\Vert _{L^\\infty (\\mathbb {T}^d)}\\,:\\, P\\psi = \\varphi \\right\\rbrace .$ If the Hankel operator $\\mathbf {H}_\\varphi $ is bounded, then the Nehari problem is to find a function $\\psi $ attaining the infimum on the right hand side of (REF ).",
"By (REF ) and (REF ), it is clear that if $\\varphi $ is a constant multiple of an inner function, then a solution to the Nehari problem is trivially $\\psi =\\varphi $ .",
"Nehari [10] established that on the one-dimensional torus, the problem always has a solution $\\psi $ which satisfies $\\Vert \\mathbf {H}_\\varphi \\Vert =\\Vert \\psi \\Vert _{L^\\infty (\\mathbb {T})}$ .",
"In general, let $C_d\\ge 1$ denote the smallest real number such that $\\inf \\left\\lbrace \\Vert \\psi \\Vert _{L^\\infty (\\mathbb {T}^d)}\\,:\\, P\\psi = \\varphi \\right\\rbrace \\le C_d \\Vert \\mathbf {H}_\\varphi \\Vert $ for every $\\varphi $ in $H^2(\\mathbb {T}^d)$ .",
"The non-trivial part of Nehari's theorem is that $C_1=1$ .",
"Ortega-Cerdà and Seip [11] found a sequence of polynomials which demonstrates that if $d$ is even, then $C_d \\ge \\left(\\frac{\\pi ^2}{8}\\right)^\\frac{d}{4}.$ The arguments in [11] also imply that every polynomial in the sequence generates a minimal norm Hankel operator.",
"In hindsight, this is perhaps not very surprising.",
"If $\\mathbf {H}_\\varphi $ has minimal norm, then we in a sense minimize the right hand side of (REF ).",
"The present paper grew out of a desire to put the polynomials from [11] in context.",
"Another source of motivation is the fact that characterizations of inner functions in dimension one, which in our case is provided by Theorem REF , often lead to a rich theory in higher dimensions.",
"A similar phenomenon can be encountered in the recent paper [3].",
"We will study two different classes of symbols generating minimal norm Hankel operators, both inspired by $\\varphi (z)=z_1+z_2$ which is the basic case in the construction of [11].",
"In the first class, we think of $\\varphi $ as a sum of two inner functions in separate variables.",
"In the second class, we consider $\\varphi $ as a 1-homogeneous polynomial.",
"Our first main result provides sufficient conditions on when the product or sum of symbols generating minimal Hankel norm operators again will generate minimal norm Hankel operators.",
"Theorem 2 Suppose that $\\varphi _1$ and $\\varphi _2$ in $H^2(\\mathbb {T}^d)$ depend on separate variables and that both $\\mathbf {H}_{\\varphi _1}$ and $\\mathbf {H}_{\\varphi _2}$ have minimal norm.",
"$\\mathbf {H}_{\\varphi _1 \\varphi _2}$ has minimal norm.",
"If additionally $\\varphi _1(0)=\\varphi _2(0)=0$ , then $\\mathbf {H}_{\\varphi _1+\\varphi _2}$ has minimal norm.",
"Theorem REF suggests the following recipe for constructing symbols generating minimal norm Hankel operators.",
"Choose any number of (not necessarily distinct) inner functions vanishing at the origin.",
"If necessary, rename the variables to ensure that the inner functions depend on mutually separate variables.",
"Combine these functions using linear combinations and multiplications, but make sure to use each function only once.",
"The polynomials used by Ortega-Cerdà and Seip fit into this framework as follows.",
"Choose $2d$ copies of the inner function $I(z)=z$ in $H^2(\\mathbb {T})$ and rename the variables $z_1,z_2,\\ldots ,z_{2d}$ .",
"Take the pairwise sum of these functions, obtaining $z_1+z_2$ , $z_3+z_4$ , all the way up to $z_{2d-1}+z_{2d}$ .",
"Finally, multiply together these $d$ functions to obtain $\\varphi _d(z) = \\prod _{j=1}^{d} (z_{2j-1}+z_{2j}).$ In view of Theorem REF , we know that the resulting Hankel operator $\\mathbf {H}_{\\varphi _d}$ has minimal norm.",
"Consequently, $\\Vert \\mathbf {H}_{\\varphi _d}\\Vert =\\Vert \\varphi _d\\Vert _{H^2(\\mathbb {T}^{2d})}=2^{d/2}$ .",
"In [11], this fact has to be established using the Schur test.",
"Since $\\pi \\ge 3$ , we see from (REF ) that $C_d \\rightarrow \\infty $ as $d\\rightarrow \\infty $ .",
"By a contradiction to the Closed Graph Theorem this demonstrates that there are $\\varphi $ in $H^2(\\mathbb {T}^\\infty )$ such that $\\mathbf {H}_\\varphi $ is bounded, but for which the corresponding Nehari problem has no solution $\\psi $ in $L^\\infty (\\mathbb {T}^\\infty )$ .",
"This allowed the authors of [11] to complete a research program initiated by Helson [7], [8], [9].",
"Using the recipe outlined above, we can revisit the counter-example from [11] and exhibit an explicit symbol $\\varphi $ in $H^2(\\mathbb {T}^\\infty )$ for which the Nehari problem has no solution in the following strong sense.",
"Theorem 3 Consider $\\varphi (z) = \\frac{\\sqrt{6}}{\\pi }\\sum _{k=1}^\\infty \\frac{1}{k} \\prod _{j=(k-1)k/2+1}^{k(k+1)/2} \\frac{z_{2j-1}+z_{2j}}{\\sqrt{2}}.$ It holds that $\\Vert \\mathbf {H}_\\varphi \\Vert =\\Vert \\varphi \\Vert _{H^2(\\mathbb {T}^\\infty )}=1$ , but for no $2<p\\le \\infty $ is there an element $\\psi $ in $L^p(\\mathbb {T}^\\infty )$ such that $P\\psi = \\varphi $ .",
"The first statement of Theorem REF is a direct consequence of Theorem REF .",
"For the second statement, we argue by duality and borrow a simple estimate from [1].",
"The fact that there are $\\varphi $ in $H^2(\\mathbb {T}^\\infty )$ such that $\\mathbf {H}_\\varphi $ is bounded, but such that there is no $\\psi $ in $L^p(\\mathbb {T}^\\infty )$ with $P\\psi = \\varphi $ when $2<p\\le \\infty $ can also be deduced from the method in [11] and said estimate (Lemma REF below).",
"The main novelty of Theorem REF is therefore that we provide an explicit example.",
"Let us now turn to our second class of symbols.",
"Recall that a function $f$ in $L^2(\\mathbb {T}^d)$ is called $m$ -homogeneous if its Fourier coefficients are supported on the frequencies $\\alpha $ in $\\mathbb {Z}^d$ which satisfy the equation $\\alpha _1+\\alpha _2+\\cdots +\\alpha _d=m$ .",
"Let $H^2_m(\\mathbb {T}^d)$ be the subspace of $H^2(\\mathbb {T}^d)$ comprised of $m$ -homogeneous functions.",
"The search for $m$ -homogeneous symbols generating minimal norm Hankel operators is facilitated by our second main result.",
"The proof is rather easy, but we believe that the result may be of some independent interest in due to the prominence played by $m$ -homogeneous expansions in function theory on polydiscs (see [5], [12]).",
"Theorem 4 Suppose that $\\varphi $ is in $H^2_m(\\mathbb {T}^d)$ .",
"Let $\\mathbf {H}_{\\varphi ,k}$ denote the restriction of the Hankel operator $\\mathbf {H}_\\varphi $ to $H^2_k(\\mathbb {T}^d)$ and let $\\mathbf {0}$ denote the zero operator.",
"If $0 \\le k \\le m$ , then $\\mathbf {H}_{\\varphi ,k}$ maps $H^2_k(\\mathbb {T}^d)$ to $\\overline{H^2_{m-k}}(\\mathbb {T}^d)$ .",
"Moreover, $\\mathbf {H}_\\varphi $ enjoys the orthogonal decomposition $\\mathbf {H}_\\varphi = \\left(\\bigoplus _{k=0}^m \\mathbf {H}_{\\varphi ,k}\\right)\\oplus \\mathbf {0}.$ If $0\\le k \\le m$ , then $\\mathbf {H}_{\\varphi ,k}^\\ast $ is unitarily equivalent to $\\mathbf {H}_{\\widetilde{\\varphi },m-k}$ for $\\widetilde{\\varphi }(z)=\\overline{\\varphi (\\overline{z})}$ .",
"Using Theorem REF will find polynomial symbols generating minimal norm Hankel operators, but which cannot be obtained by the recipe discussed above.",
"As a byproduct we also obtain the following improvement on the lower bound (REF ).",
"Theorem 5 Let $C_d$ denote the optimal constant in (REF ).",
"If $d$ is even, then $C_d \\ge \\left(\\frac{5\\pi }{\\pi +6\\sqrt{3}}\\right)^\\frac{d}{2}.$ The lower bound in Theorem REF can improved slightly by testing against a better function in the proof below.",
"Conversely, the lower bound in (REF ) is the best possible which can be obtained from the symbol $\\varphi (z)=z_1+z_2$ .",
"As explained in [2], the optimal solution to the Nehari problem is in this case $\\psi (z) = \\sum _{k\\in \\mathbb {Z}} \\frac{(-1)^k}{1-2k} z_1^{1-k} z_2^{k}$ for $z$ on $\\mathbb {T}^2$ .",
"It also follows from the arguments in [2] that $\\Vert \\psi \\Vert _{L^\\infty (\\mathbb {T}^2)}=\\pi /2$ .",
"The present paper is comprised of two additional sections.",
"In Section we establish Theorem REF , Theorem REF and Theorem REF .",
"Section is devoted to the study of $m$ -homogeneous symbols of Hankel operators and contains the proof of Theorem REF and Theorem REF ."
],
[
"Symbols generated by inner functions",
"In the proof of Theorem REF we will use the inner-outer factorization of functions in $H^p(\\mathbb {T})$ , for which our standard reference is Duren's monograph [6].",
"Every non-trivial function $f$ in $H^p(\\mathbb {T})$ can be written as $f = I F$ , where $I$ is inner and $F$ is outer.",
"The factorization is unique up to a unimodular constant.",
"In particular, it holds that $\\Vert f\\Vert _{H^p(\\mathbb {T})}=\\Vert F\\Vert _{H^p(\\mathbb {T})}$ and $F$ may be represented as $F(z) = \\exp \\left(\\int _0^{2\\pi } \\frac{e^{i\\theta }+z}{e^{i\\theta }-z}\\log |f(e^{i\\theta })|\\,\\frac{d\\theta }{2\\pi } \\right).$ We stress that $F$ does not vanish in the unit disc $\\mathbb {D}$ , so $\\sqrt{F}$ will be analytic in $\\mathbb {D}$ .",
"We explained in the introduction that if $\\varphi = C I$ for a constant $C$ and an inner function $I$ , then $\\mathbf {H}_\\varphi $ is easily seen to have minimal norm by (REF ).",
"Our job is therefore to establish the converse statement.",
"Suppose therefore that $\\varphi $ is a non-trivial element in $H^2(\\mathbb {T})$ and that $\\mathbf {H}_\\varphi $ has minimal norm.",
"Factor $\\varphi = I \\Phi ,$ where $I$ is inner and $\\Phi $ is outer, so that $\\Vert \\Phi \\Vert _{H^2(\\mathbb {T})}=\\Vert \\varphi \\Vert _{H^2(\\mathbb {T})}.$ Then $f = I \\sqrt{\\Phi }$ and $g = \\sqrt{\\Phi }$ satisfy $\\Vert f\\Vert _{H^2(\\mathbb {T})}=\\Vert g\\Vert _{H^2(\\mathbb {T})}=\\Vert \\Phi \\Vert _{H^1(\\mathbb {T})}^{1/2}.$ By our assumption that $\\mathbf {H}_\\varphi $ has minimal norm, we find that $\\Vert \\Phi \\Vert _{H^2(\\mathbb {T})}=\\Vert \\varphi \\Vert _{H^2(\\mathbb {T})} = \\Vert \\mathbf {H}_\\varphi \\Vert \\ge \\frac{\\left|\\left\\langle \\overline{P}(\\overline{\\varphi } f), \\overline{g}\\right\\rangle \\right|}{\\Vert f\\Vert _{H^2(\\mathbb {T})}\\Vert g\\Vert _{H^2(\\mathbb {T})}} = \\frac{\\Vert \\Phi \\Vert _{H^2(\\mathbb {T})}^2}{\\Vert \\Phi \\Vert _{H^1(\\mathbb {T})}},$ where we used that $\\overline{P}$ is self-adjoint and $\\overline{P}(\\overline{g})=\\overline{g}$ in the final equality.",
"This shows that $\\Vert \\Phi \\Vert _{H^2(\\mathbb {T})} \\le \\Vert \\Phi \\Vert _{H^1(\\mathbb {T})}$ , which by the Cauchy–Schwarz inequality implies that there is some constant $C>0$ such that $|\\Phi (e^{i\\theta })|=C$ for almost every $e^{i\\theta } \\in \\mathbb {T}$ .",
"By the representation (REF ) we conclude that $\\varphi = C I$ (up to a unimodular constant).",
"Remark The proof of Theorem REF presented above is inspired by the modern proof of Nehari's theorem attributed to Helson (see [13]).",
"This argument exploits the inner-outer factorization to demonstrate that if $\\mathbf {H}_\\varphi $ is bounded, then $\\varphi $ defines a bounded linear functional on $H^1(\\mathbb {T})$ with $\\Vert \\varphi \\Vert _{(H^1(\\mathbb {T}))^\\ast }=\\Vert \\mathbf {H}_\\varphi \\Vert $ .",
"The Hahn–Banach Theorem and the Riesz Representation Theorem can now be combined to show that there is some $\\psi $ in $L^\\infty (\\mathbb {T})$ with $P\\psi = \\varphi $ and $\\Vert \\psi \\Vert _{L^\\infty (\\mathbb {T})}=\\Vert \\varphi \\Vert _{(H^1(\\mathbb {T}))^\\ast }$ .",
"We require two preliminary results for the proof of Theorem REF .",
"The first is a special case of [4], which contains the corresponding result for all Schatten norms.",
"A simpler proof of the present special case can be found in [14].",
"Lemma 6 Suppose that $\\varphi _1$ and $\\varphi _2$ in $H^2(\\mathbb {T}^d)$ depend on separate variables.",
"If both $\\mathbf {H}_{\\varphi _1}$ and $\\mathbf {H}_{\\varphi _2}$ are bounded, then $\\Vert \\mathbf {H}_{\\varphi _1\\varphi _2}\\Vert = \\Vert \\mathbf {H}_{\\varphi _1}\\Vert \\Vert \\mathbf {H}_{\\varphi _2}\\Vert $ .",
"Lemma 7 Suppose that $\\varphi _1$ and $\\varphi _2$ in $H^2(\\mathbb {T}^d)$ depend on separate variables and that $\\varphi _1(0)=\\varphi _2(0)=0$ .",
"If both $\\mathbf {H}_{\\varphi _1}$ and $\\mathbf {H}_{\\varphi _2}$ are bounded, then $\\Vert \\mathbf {H}_{\\varphi _1+\\varphi _2}\\Vert ^2 \\le \\Vert \\mathbf {H}_{\\varphi _1}\\Vert ^2 + \\Vert \\mathbf {H}_{\\varphi _2}\\Vert ^2.$ To avoid trivialities, we assume that $\\Vert \\varphi _j\\Vert _{H^2(\\mathbb {T}^d)}\\ne 0$ for $j=1,2$ .",
"Every $f$ in $H^2(\\mathbb {T}^d)$ with $\\Vert f\\Vert _{H^2(\\mathbb {T}^d)}=1$ can be orthogonally decomposed as $f = \\frac{t_1}{\\Vert \\varphi _1\\Vert _{H^2(\\mathbb {T}^d)}} \\varphi _1 + \\frac{t_2}{\\Vert \\varphi _2\\Vert _{H^2(\\mathbb {T}^d)}} \\varphi _2 + t_3 g.$ Here $t_1,t_2,t_3$ are nonnegative real numbers satisfying $t_1^2+t_2^2+t_3^2=1$ and $g$ is a function which is orthogonal to both $\\varphi _1$ and $\\varphi _2$ and which satisfies $\\Vert g\\Vert _{H^2(\\mathbb {T}^d)}=1$ .",
"A direct computation based on (REF ) shows that $\\mathbf {H}_{\\varphi _j} f = t_j \\Vert \\varphi _j\\Vert _{H^2(\\mathbb {T}^d)} + t_3 \\mathbf {H}_{\\varphi _j} g,$ for $j=1,2$ , since $\\varphi _1$ and $\\varphi _2$ depend on separate variables and $\\varphi _1(0)=\\varphi _2(0)=0$ .",
"We now have the orthogonal decomposition $\\mathbf {H}_{\\varphi _1+\\varphi _2} f = \\big (t_1 \\Vert \\varphi _1\\Vert _{H^2(\\mathbb {T}^d)}+t_2\\Vert \\varphi _2\\Vert _{H^2(\\mathbb {T}^d)}\\big )+t_3\\mathbf {H}_{\\varphi _1}g + t_3\\mathbf {H}_{\\varphi _2} g.$ By orthogonality and the fact that $\\Vert g\\Vert _{H^2(\\mathbb {T}^d)}=1$ , we get $\\Vert \\mathbf {H}_{\\varphi _1+\\varphi _2} f\\Vert _{H^2(\\mathbb {T}^d)}^2 \\le \\big (t_1 \\Vert \\varphi _1\\Vert _{H^2(\\mathbb {T}^d)}+t_2\\Vert \\varphi _2\\Vert _{H^2(\\mathbb {T}^d)}\\big )^2 + t_3^2 \\big (\\Vert \\mathbf {H}_{\\varphi _1}\\Vert ^2 + \\Vert \\mathbf {H}_{\\varphi _1}\\Vert ^2\\big ).$ Using the Cauchy–Schwarz inequality on the first term and exploiting the general lower bound $\\Vert \\varphi _j\\Vert _{H^2(\\mathbb {T}^d)}\\le \\Vert \\mathbf {H}_{\\varphi _j}\\Vert $ from (REF ) for $j=1,2$ , we get $\\Vert \\mathbf {H}_{\\varphi _1+\\varphi _2} f\\Vert _{H^2(\\mathbb {T}^d)}^2 \\le (t_1^2+t_2^2+t_3^2)\\big (\\Vert \\mathbf {H}_{\\varphi _1}\\Vert ^2 + \\Vert \\mathbf {H}_{\\varphi _2}\\Vert ^2\\big ) = \\Vert \\mathbf {H}_{\\varphi _1}\\Vert ^2 + \\Vert \\mathbf {H}_{\\varphi _2}\\Vert ^2.$ This completes the proof since $f$ is an arbitrary norm 1 element in $H^2(\\mathbb {T}^d)$ .",
"We begin with (a), where Lemma REF and the assumption that $\\mathbf {H}_{\\varphi _1}$ and $\\mathbf {H}_{\\varphi _2}$ have minimal norm imply that $\\Vert \\mathbf {H}_{\\varphi _1\\varphi _2}\\Vert = \\Vert \\mathbf {H}_{\\varphi _1}\\Vert \\Vert \\mathbf {H}_{\\varphi _2}\\Vert = \\Vert \\varphi _1\\Vert _{H^2(\\mathbb {T}^d)} \\Vert \\varphi _2\\Vert _{H^2(\\mathbb {T}^d)} = \\Vert \\varphi _1 \\varphi _2\\Vert _{H^2(\\mathbb {T}^d)}.$ The final equality is a trivial consequence of the fact that $\\varphi _1$ and $\\varphi _2$ depend on separate variables.",
"Hence $\\mathbf {H}_{\\varphi _1\\varphi _2}$ is has minimal norm.",
"In the case (b), we similarly get from Lemma REF and the assumption that $\\mathbf {H}_{\\varphi _1}$ and $\\mathbf {H}_{\\varphi _2}$ have minimal norm that $\\Vert \\mathbf {H}_{\\varphi _1+\\varphi _2}\\Vert ^2 \\le \\Vert \\mathbf {H}_{\\varphi _1}\\Vert ^2 + \\Vert \\mathbf {H}_{\\varphi _2}\\Vert ^2 = \\Vert \\varphi _1\\Vert _{H^2(\\mathbb {T}^d)}^2 +\\Vert \\varphi _2\\Vert _{H^2(\\mathbb {T}^d)}^2 = \\Vert \\varphi _1+\\varphi _2\\Vert _{H^2(\\mathbb {T}^d)}^2.$ The final equality holds because $\\varphi _1 \\perp \\varphi _2$ .",
"Hence $\\mathbf {H}_{\\varphi _1+\\varphi _2}$ has minimal norm.",
"We require following estimate in the proof of the second part of Theorem REF .",
"Lemma 8 Suppose that $1 \\le q \\le 2$ .",
"Then $\\left\\Vert \\frac{z_1+z_2}{\\sqrt{2}}\\right\\Vert _{H^q(\\mathbb {T}^2)}^{-1} \\ge 1 + \\frac{2\\log {2}-1}{8}(2-q).$ We first extract from the proof of [1] the estimate $\\Vert \\varphi \\Vert _{H^p(\\mathbb {T}^2)}^{-1} \\ge \\frac{1}{\\sqrt{2}} \\left(1+\\frac{q}{2}\\right)^{\\frac{1}{q}}.$ The proof is completed by using Taylor's theorem at $q=2$ .",
"For every positive integer $k$ , let $\\varphi _k(z) = \\prod _{j=(k-1)k/2+1}^{k(k+1)/2} \\frac{z_{2j-1}+z_{2j}}{\\sqrt{2}}$ and note that $\\Vert \\varphi _k\\Vert _{H^2(\\mathbb {T}^\\infty )}=1$ .",
"By the recipe outlined after Theorem REF , it is clear that $\\Vert \\mathbf {H}_\\varphi \\Vert =\\Vert \\varphi \\Vert _{H^2(\\mathbb {T}^\\infty )}=1$ if $\\varphi (z) = \\frac{\\sqrt{6}}{\\pi } \\sum _{k=1}^\\infty \\frac{\\varphi _k(z)}{k}.$ It remains to establish the second claim, where we shall argue by contradiction.",
"Fix $2<p\\le \\infty $ and assume that there is some $\\psi $ in $L^p(\\mathbb {T}^\\infty )$ such that $P\\psi = \\varphi $ .",
"Since $P$ is self-adjoint, we get from Hölder's inequality that $\\frac{|\\langle f,\\varphi \\rangle |}{\\Vert f\\Vert _{H^q(\\mathbb {T}^\\infty )}} \\le \\Vert \\psi \\Vert _{L^p(\\mathbb {T}^\\infty )}<\\infty $ for every non-trivial function $f$ in $H^q(\\mathbb {T}^\\infty )$ , where $q = p/(p-1)$ .",
"The fact that $2<p\\le \\infty $ means that $1 \\le q <2$ .",
"Choosing the $f = \\varphi _k$ , we see that $\\lim _{k \\rightarrow \\infty } \\frac{\\langle \\varphi _k, \\varphi \\rangle }{\\Vert \\varphi _k\\Vert _{H^q(\\mathbb {T}^\\infty )}} = \\frac{\\sqrt{6}}{\\pi }\\lim _{k \\rightarrow \\infty } \\frac{1}{k} \\frac{\\Vert \\varphi _k\\Vert _{H^2(\\mathbb {T}^\\infty )}^2}{\\Vert \\varphi _k\\Vert _{H^q(\\mathbb {T}^\\infty )}} = \\frac{\\sqrt{6}}{\\pi } \\lim _{k \\rightarrow \\infty } \\frac{1}{k}\\frac{1}{\\Vert \\varphi _1\\Vert _{H^q(\\mathbb {T}^2)}^k} = \\infty ,$ by Lemma REF .",
"This contradicts (REF ) and hence our assumption that there is some $\\psi $ in $L^q(\\mathbb {T}^\\infty )$ with $P\\psi = \\varphi $ must be wrong."
],
[
"$m$ -homogeneous symbols",
"To prepare for the proof of Theorem REF , we first orthogonally decompose $H^2(\\mathbb {T}^d)$ and $\\overline{H^2}(\\mathbb {T}^d)$ using $m$ -homogeneous functions.",
"It is clear that $H^2(\\mathbb {T}^d) = \\bigoplus _{m=0}^\\infty H^2_m(\\mathbb {T}^d) \\qquad \\text{and} \\qquad \\overline{H^2}(\\mathbb {T}^d) = \\bigoplus _{m=0}^\\infty \\overline{H^2_m}(\\mathbb {T}^d).$ Note that the functions in $\\overline{H^2_m}(\\mathbb {T}^d)$ are $-m$ -homogeneous, since they are precisely the complex conjugates of functions from $H^2_m(\\mathbb {T}^d)$ .",
"To establish (a), decompose a function $f$ in $H^2(\\mathbb {T}^d)$ as $f = \\sum _{k=0}^\\infty f_k$ in view of (REF ).",
"By assumption, our symbol $\\varphi $ is $m$ -homogeneous.",
"Hence we have $\\overline{\\varphi } f = \\sum _{k=0}^\\infty \\overline{\\varphi } f_k,$ and $\\overline{\\varphi } f_k$ is $(-m+k)$ -homogeneous.",
"Since homogenity is preserved under $\\overline{P}$ , this shows that $\\mathbf {H}_\\varphi $ maps $H_k^2(\\mathbb {T})$ to $\\overline{H^2_{m-k}}(\\mathbb {T}^d)$ when $0 \\le k \\le m$ .",
"This completes the proof of the first claim.",
"If $k \\ge m+1$ , then $\\overline{\\varphi }f_k$ has positive homogeneity and hence $\\overline{P}(\\overline{\\varphi } f_k)=0$ .",
"Combining what we have done with (REF ) shows that $\\mathbf {H}_\\varphi $ enjoys the stated orthogonal decomposition $\\mathbf {H}_\\varphi = \\left(\\bigoplus _{k=0}^m \\mathbf {H}_{\\varphi ,k}\\right)\\oplus \\mathbf {0}.$ For the proof of (b), we first check that if $f$ is in $H^2(\\mathbb {T}^d)$ and is $g$ in $\\overline{H^2}(\\mathbb {T}^d)$ , then $\\langle \\mathbf {H}_\\varphi f, g \\rangle = \\langle \\overline{P}(\\overline{\\varphi }f), g \\rangle = \\langle f, \\varphi g \\rangle = \\langle f, P(\\varphi g) \\rangle ,$ which shows that $\\mathbf {H}_\\varphi ^\\ast g = P(\\varphi g)$ .",
"This also shows that $\\mathbf {H}_\\varphi ^\\ast $ is unitarily equivalent to the Hankel operator $\\mathbf {H}_{\\widetilde{\\varphi }}$ where $\\widetilde{\\varphi }(z)=\\overline{\\varphi (\\overline{z})}$ .",
"The decomposition (REF ) therefore applies to $\\mathbf {H}_{\\widetilde{\\varphi }}$ , which means that $\\mathbf {H}_{\\varphi ,k}^\\ast $ must be unitarily equivalent to $\\mathbf {H}_{\\widetilde{\\varphi },k-m}$ .",
"The following result illustrates how Theorem REF pertains to minimal norm Hankel operators.",
"Part (a) allows us to focus on the restricted Hankel operators and part (b) reduces the number of restricted Hankel operators we need to consider.",
"Corollary 9 Let $\\varphi $ be in $H^2_m(\\mathbb {T}^d)$ .",
"Then $\\mathbf {H}_\\varphi $ has minimal norm if and only if $\\max _{0<k\\le \\lfloor m/2 \\rfloor } \\Vert \\mathbf {H}_{\\varphi ,k} \\Vert \\le \\Vert \\varphi \\Vert _{H^2(\\mathbb {T}^d)}$ where $\\mathbf {H}_{\\varphi ,k}$ denotes the restriction of $\\mathbf {H}_\\varphi $ to $H^2_k(\\mathbb {T}^d)$ .",
"It is clear from Theorem REF (a) that $\\mathbf {H}_\\varphi $ has minimal norm if and only if $\\max _{0 \\le k \\le m} \\Vert \\mathbf {H}_{\\varphi ,k} \\Vert \\le \\Vert \\varphi \\Vert _{H^2(\\mathbb {T}^d)},$ so our goal is to demonstrate that the set we take the maxima over may be decreased to obtain (REF ).",
"From Theorem REF (b) we find that $\\Vert \\mathbf {H}_{\\varphi ,k}\\Vert = \\Vert \\mathbf {H}_{\\varphi ,k}^\\ast \\Vert =\\Vert \\mathbf {H}_{\\widetilde{\\varphi },m-k}\\Vert =\\Vert \\mathbf {H}_{\\varphi ,m-k}\\Vert $ for $0 \\le k \\le m$ , which allows us to decrease the set in (REF ) to $0 \\le k \\le \\lfloor m/2 \\rfloor $ .",
"It remains to exclude the case $k=0$ .",
"Since $H^2_0(\\mathbb {T}^d)$ is comprised of constant functions, it follows at once from the definition of $\\mathbf {H}_\\varphi $ that $\\Vert \\mathbf {H}_{\\varphi ,0}\\Vert =\\Vert \\varphi \\Vert _{H^2(\\mathbb {T}^d)}$ .",
"Hence the desired inequality is automatically satisfied for $k=0$ .",
"Remark Since the maximum in (REF ) is taken over an empty set of integers if $m=1$ , Corollary REF ensures that $\\varphi _1(z) = \\sum _{j=1}^d c_j z_j$ generates a minimal norm Hankel operator for any choice of coefficients.",
"This can also be seen from the recipe inspired by Theorem REF .",
"The 1-homogeneous symbols (REF ) are used in [4] to extend the result of [11] to certain Schatten classes.",
"We can put Corollary REF to use and easily obtain the following concrete examples.",
"It is clear that if $a$ and $b$ are positive, then we cannot construct the polynomials $\\varphi _2$ and $\\varphi _3$ using the recipe inspired by Theorem REF .",
"Theorem 10 Consider the polynomials $\\varphi _2(z) = z_1^2+ a z_1 z_2 + z_2^2 \\qquad \\text{and}\\qquad \\varphi _3(z) = z_1^3 + b z_1^2 z_2 + b z_1 z_2^2 + z_2^3$ where $a$ and $b$ are nonnegative real numbers.",
"The Hankel operator $\\mathbf {H}_{\\varphi _2}$ has minimal norm if and only if $a\\le 1/2$ , $\\mathbf {H}_{\\varphi _3}$ has minimal norm if and only if $b \\le \\sqrt{2}-1$ .",
"The function $\\varphi _2$ is 2-homogeneous.",
"By Corollary REF , it is sufficient to check which coefficients $a$ ensure that the inequality $\\Vert \\mathbf {H}_{\\varphi _2,1}\\Vert \\le \\sqrt{1+a^2}$ is satisfied.",
"The matrix representation of the operator $\\mathbf {H}_{\\varphi _2,1}\\colon H^2_1(\\mathbb {T}^2) \\rightarrow \\overline{H^2_1}(\\mathbb {T}^2)$ with respect to the standard basis is $M_{\\varphi _2,1} =\\begin{pmatrix}1 & a \\\\a & 1\\end{pmatrix}.$ The norm of this matrix is seen to be $1+a$ , since $a\\ge 0$ by assumption.",
"The requirement (REF ) becomes $1+a \\le \\sqrt{2+a^2}$ , which simplifies to $a\\le 1/2$ .",
"The function $\\varphi _3$ is 3-homogeneous.",
"By Corollary REF , it is sufficient to check which nonnegative coefficients $b$ ensure that $\\Vert \\mathbf {H}_{\\varphi _3,1}\\Vert \\le \\sqrt{2+2b^2}.$ The matrix representation of the operator $\\mathbf {H}_{\\varphi _3,1}\\colon H^2_1(\\mathbb {T}^2) \\rightarrow \\overline{H^2_2}(\\mathbb {T}^2)$ with respect to the standard basis is $M_{\\varphi _3,1} =\\begin{pmatrix}1 & b \\\\b & b \\\\b & 1\\end{pmatrix}\\qquad \\text{and hence} \\qquad M_{\\varphi _3,1}^\\ast M_{\\varphi _3,1} =\\begin{pmatrix}1+2b^2 & 2b+b^2 \\\\2b+b^2 & 1+2b^2\\end{pmatrix}.$ Since $b\\ge 0$ it is easy to see that the norm of the latter matrix is $1+2b+3b^2$ .",
"The requirement (REF ) becomes $1+2b+3b^2 \\le 2+2b^2$ , which simplifies to $b \\le \\sqrt{2}-1$ .",
"A simple argument based on Lemma REF shows that if $d$ is an even integer, then $C_d \\ge C_2^{d/2}$ .",
"It is therefore sufficient to establish that $C_2 \\ge \\frac{5\\pi }{\\pi +6\\sqrt{3}}.$ Starting from the definition of $C_2$ from (REF ) and arguing as in the proof of the second part of Theorem REF , we get that $C_2 \\ge \\frac{|\\langle f, \\varphi \\rangle |}{\\Vert \\mathbf {H}_\\varphi \\Vert \\, \\Vert f\\Vert _{H^1(\\mathbb {T}^2)}}$ for any pair of non-trivial functions $f$ in $H^1(\\mathbb {T}^2)$ and $\\varphi $ in $H^2(\\mathbb {T}^d)$ .",
"We will choose $f(z)= z_1^2 + z_1z_2+z_2^2 \\qquad \\text{and} \\qquad \\varphi (z) = z_1^2+\\frac{z_1 z_2}{2}+z_2^2.$ Clearly $\\langle f, \\varphi \\rangle = 5/2$ and by Theorem REF (a) we know that $\\Vert \\mathbf {H}_\\varphi \\Vert =\\Vert \\varphi \\Vert _{H^2(\\mathbb {T}^2)}=3/2$ .",
"Since the coefficients of $f$ are real and since $f$ is 2-homogeneous, we can simplify $\\Vert f\\Vert _{H^1(\\mathbb {T}^2)} = \\int _0^{2\\pi }\\int _0^{2\\pi } \\big |f(e^{i\\theta _1},e^{i\\theta _2})\\big |\\,\\frac{d\\theta _1}{2\\pi }\\frac{d\\theta _2}{2\\pi } = \\int _0^\\pi \\big |e^{i\\theta }+1+e^{-i\\theta }\\big |\\,\\frac{d\\theta }{\\pi }.$ Using that $e^{i\\theta }+e^{-i\\theta }=2\\cos {\\theta }$ and that the solution to the equation $2\\cos {\\theta }+1=0$ on the interval $0\\le \\theta \\le \\pi $ is $\\theta = 2\\pi /3$ , we find that $\\Vert f\\Vert _{H^1(\\mathbb {T}^2)} = \\int _0^{2\\pi /3} (2\\cos {\\theta }+1)\\,\\frac{d\\theta }{\\pi }-\\int _{2\\pi /3}^\\pi (2\\cos {\\theta }+1)\\,\\frac{d\\theta }{\\pi } = \\frac{2}{3}+\\frac{\\sqrt{3}}{\\pi }-\\left(\\frac{1}{3}-\\frac{\\sqrt{3}}{\\pi }\\right).$ Inserting everything into (REF ) and tidying up yields the stated lower bound (REF ).",
"Remark Some cursory numerical experiments indicate that it might be optimal to choose $a=1/2$ in Theorem REF (a).",
"For this choice of symbol $\\varphi $ , it is optimal to choose $f(z)=z_1^2+c z_1z_2 + z_2^2$ for some $0.8<c<0.9$ ."
]
] | 2107.01680 |
[
[
"Advanced turning maneuver of a multi-legged robot using pitchfork\n bifurcation"
],
[
"Abstract Legged robots have excellent terrestrial mobility for traversing diverse environments and thus have the potential to be deployed in a wide variety of scenarios.",
"However, they are susceptible to falling and leg malfunction during locomotion.",
"Although the use of a large number of legs can overcome these problems, it makes the body long and leads to many legs being constrained to contact with the ground to support the long body, which impedes maneuverability.",
"To improve the locomotion maneuverability of such robots, the present study focuses on dynamic instability, which induces rapid and large movement changes, and uses a 12-legged robot with a flexible body axis.",
"Our previous work found that the straight walk of the robot becomes unstable through Hopf bifurcation when the body axis flexibility is changed, which induces body undulations.",
"Furthermore, we developed a simple controller based on the Hopf bifurcation and showed that the instability facilitates the turning of the robot.",
"In this study, we newly found that the straight walk becomes unstable through pitchfork bifurcation when the body-axis flexibility is changed in a way different from that in our previous work.",
"In addition, the pitchfork bifurcation induces a transition into a curved walk, whose curvature can be controlled by the body-axis flexibility.",
"We developed a simple controller based on the pitchfork-bifurcation characteristics and demonstrated that the robot can perform a turning maneuver superior to that with the previous controller.",
"This study provides a novel design principle for maneuverable locomotion of many-legged robots using intrinsic dynamic properties."
],
[
"Introduction",
"Legged locomotion, such as that of animals, allows excellent terrestrial mobility for traversing diverse environments.",
"Legged robots thus have potential to be deployed in a wide variety of scenarios, such as search and rescue [18], [29], hazardous environment operation and exploration [8], [42], and planetary exploration [5], [46].",
"Various legged robots with agile animal-like locomotion have recently been developed [1], [4], [19], [21], [20], [25], [27], [30], [31], [34], [38].",
"However, most of these robots have four legs and falling, which may result in the breakdown of mechanical and electrical components and from which it is difficult to recover, is inevitable during locomotion.",
"Furthermore, damage to even one leg greatly degrades their locomotive performance [11].",
"The use of a large number of legs prevents falling and allows a certain level of leg malfunction to be tolerated [22], [28].",
"Although the use of a large number legs has advantages for legged robots, it makes the body long and motion planning and control difficult due to the many intrinsic degrees of freedom and complex interaction with the environment.",
"In particular, many contact legs are physically constrained on the ground to support the long body, which can impede maneuverability.",
"The underlying mechanism of agile locomotion using a large number of legs remains unclear from biological and engineering viewpoints [16].",
"Maneuverable locomotion for robots with a large number of legs remains challenging.",
"Conventional controllers precisely plan the motion of all degrees of freedom of the robot (e.g., how the long body is bent, where each foot touches the ground, and in what order the legs move) and control the robot to stabilize the desired motion.",
"However, this approach has huge computational and energy costs, making it inefficient.",
"To design a simple and efficient controller with high locomotor performance, the fundamental dynamic principles embedded in the robot dynamics including the interaction with the environment should be fully utilized [1], [9], [24], [25].",
"For maneuverable locomotion of many-legged robots that overcomes the above difficulties and the limitations of conventional approaches, the present study focuses on dynamic instability, which induces rapid and large movement changes, and uses a 12-legged robot whose body axis is flexible.",
"Our previous work [2] showed that although many contact legs can impede maneuverability, they induce straight walk instability and body undulations through Hopf bifurcation when the body-axis flexibility is changed.",
"Stability refers to the capability to resist and recover from disturbances; straight walk instability is thus expected to allow the robot to easily change walking direction.",
"Therefore, we developed a simple controller based on the straight walk instability induced by the Hopf bifurcation to change walking direction without precise motion planning and control, and demonstrated that the straight walk instability facilitates the turning of the robot [3].",
"In the present study, we show that the pitchfork bifurcation of the straight walk is caused by changes in the body-axis flexibility in a different way from that in previous work [2], [3].",
"The pitchfork bifurcation not only destabilizes straight walking, but also causes curved walking, where the flexible body axis forms a curved shape.",
"Furthermore, we found that the curvature of curved walking can be controlled by the body-axis flexibility.",
"We developed a simple control strategy based on the pitchfork-bifurcation characteristics, which improved the turning maneuver compared to that achieved using Hopf bifurcation.",
"This study provides a design principle for a simple and efficient control scheme to create maneuverable locomotion for many-legged robots using intrinsic dynamic properties."
],
[
"Robot",
"We used the many-legged robot developed in [2] and improved in [3].",
"The total length and mass are 135 cm and 8.5 kg, respectively.",
"The robot consists of 6 body segment modules (modules 1–6), as shown in Fig.",
"REF .",
"Each module is composed of a single body and one pair of legs and has the same length.",
"The body segments are passively connected by yaw joints (yaw joints 1–5) onto which torsional springs and potentiometers are installed.",
"The yaw joint angles are zero when the body segments were aligned.",
"Each leg has two links connected by pitch joints.",
"The legs in the first module (module 1) have an additional link connected by a yaw joint to supplement the control of the walking direction during turning tasks.",
"Each leg joint is manipulated by an encoder-equipped motor.",
"The first module has a laser range scanner (Hokuyo, URG-04LX) to find the relative position of a target for turning.",
"The robot was controlled by an external host computer (Intel Pentium 4 2.8 GHz, RT-Linux) with 2-ms intervals, and walked on a wooden flat floor with a vinyl floor mat to suppress slipping.",
"The computer control signals and electric power were provided via external cables, which were kept slack and suspended to avoid influencing the locomotor behavior of the robot.",
"Figure: (A) Photograph and (B) schematic model of many-legged robot.",
"The robot consists of 6 modules, each of which has one body segment and one pair of legs.",
"The legs are controlled by two pitch joints so that the leg tips follow a periodic trajectory, including the anterior extreme position (AEP) and the posterior extreme position (PEP).",
"Body segments are passively connected by yaw joints with installed torsional springs.",
"The legs in the first module have additional yaw joints to change walking direction.",
"The laser range scanner is installed on the first module to find a position relative to a target.To make the robots walk in a straight line, we controlled the legs using the two pitch joints of each leg to follow the desired movement, which consists of two parts, namely half of an elliptical curve that starts from the posterior extreme position (PEP) and ends at the anterior extreme position (AEP), and a straight line from the AEP to the PEP (Fig.",
"REFB).",
"In the straight line, the leg tips moved from the AEP to the PEP at a constant speed parallel to the body.",
"We set the duration of the half elliptical curve to $0.29$ s, that of the straight line to $0.31$ s, and the distance between the AEP and the PEP in each leg to 3 cm.",
"The left and right legs in each module moved in antiphase, and the relative phase between the ipsilateral legs on adjacent modules was set to $2\\pi /3$ rad.",
"When the leg yaw joint angles of the first module were fixed so that the leg tip trajectories were parallel to the body, the robot was expected to walk in a straight line while keeping the body segments parallel to each other because torsional springs were installed on the body-segment yaw joints and all support-leg tips moved parallel to the body segments at an identical speed."
],
[
"Experimental results",
"Our previous work [2], [3] revealed that when we used torsional springs with the same spring constant for all body-segment yaw joints (yaw joints 1–5) and changed the spring constant uniformly among the joints, the straight walk became unstable through Hopf bifurcation, which induced body undulations.",
"This bifurcation was verified by a Floquet analysis with a simple physical model.",
"In this study, we performed robot experiments of walking in a straight line, where we changed the body axis flexibility in a way that was different from that in our previous work.",
"Specifically, we used the same spring constant for yaw joints 2–5 ($k_i=41$ Nmm/deg, $i=2,\\dots ,5$ ) and used various spring constants for yaw joint 1 ($k_1=15$ , 17, 21, 28, 41, and 75 Nmm/deg).",
"We set all the body segments parallel to each other as the initial conditions.",
"The leg yaw joints in the first module were fixed during the experiments.",
"Figure: Characteristics of curved walk for k 1 k_1 values below threshold value.",
"Yaw joint angles for (A) straight walk with k 1 =41k_1=41 Nmm/deg and (B) curved walk with k 1 =15k_1=15 Nmm/deg (see Movies 1–3).",
"(C) Average angle of yaw joint 1 during curved walk for 1/k 1 1/k_1 that indicates pitchfork bifurcation.",
"The data points and error bars correspond to the means and standard errors, respectively, of the results of five experiments.",
"(D) Radius of curvature of body axis for 1/k 1 1/k_1.",
"The data points and error bars correspond to the means and standard errors, respectively, of the results of ten experiments.",
"Photographs for (E) straight walk with k 1 >k ^ 1 k_1>\\hat{k}_1, (F) curved walk with small curvature with k 1 ∼k ^ 1 k_1\\sim \\hat{k}_1, and (G) curved walk with large curvature with k 1 <k ^ 1 k_1<\\hat{k}_1.When we used large spring constants for $k_1$ , the robot kept walking in a straight line as expected, and the body segments were aligned, with all body-segment yaw joint angles being almost zero (Figs.",
"REFA and E, see Movie 1).",
"However, when $k_1$ was set to below a threshold value, the body-segment yaw joints showed non-zero angles with the same sign; that is, the body axis was curved and the robot walked in a curved line (Figs.",
"REFB, F, and G, see Movies 2 and 3).",
"Specifically, the robot walked in a curved line for $k_1=15$ , 17, 21, and 28 Nmm/deg, but not for $k_1=41$ and 75 Nmm/deg.",
"Furthermore, both left- and right-curved walking could be achieved depending on the initial robot conditions.",
"Figure REFC shows the angles of yaw joint 1 for $1/k_1$ averaged over 5 s during a curved walk (the angles for the other body-segment yaw joints are shown in Fig.",
"REF ).",
"The magnitude of these angles increases with $1/k_1$ .",
"These results suggest that the presence of pitchfork bifurcation depends on $k_1$ .",
"These angle data were fitted by the square root of $1/k_1$ [39].",
"The bifurcation point was estimated to be $k_1=34$ Nmm/deg ($1/k_1=0.030$ deg/Nmm).",
"Figure: Average absolute angles of body-segment yaw joints during curved walk for 1/k 1 1/k_1.",
"The data points and error bars correspond to the means and standard errors, respectively, of the results of ten experiments.The dependence of the body-segment yaw joint angles on $1/k_1$ (Fig.",
"REFC and Fig.",
"REF ) indicates the change of the curved shape of the body axis for the curved walk.",
"Figure REFD shows the radius of curvature $r$ of the body axis for $1/k_1$ calculated as $r=5L/\\sum _{i=1}^{5}|\\theta _i|$ , where $\\theta _i$ is the angle of yaw joint $i$ ($i=1,\\dots ,5$ ) and $L$ is the length of the body segments.",
"This figure shows that we can control the curvature of the body axis to perform a curved walk by adjusting $k_1$ through pitchfork bifurcation."
],
[
"Verification by Floquet analysis with simple physical model",
"The robot experiments suggested that the presence of pitchfork bifurcation in the straight walk depends on the spring constant $k_1$ (Figs.",
"REFC and REF ).",
"We verified this bifurcation from a theoretical viewpoint using a Floquet analysis with a simple physical model, as done in our previous work [2].",
"The model was simplified from the original high-dimensional mechanical model to extract the fundamentals of locomotion dynamics (Fig.",
"REFA).",
"In particular, the model was two-dimensional because the movements were designed to make the robots walk without up-and-down, roll, or pitch motions of the body segments.",
"Furthermore, because an important role of legs in locomotion is to receive reaction forces from the floor, we neglected the inertial force of the legs and used the geometric constraint forces of the leg tips.",
"The equations of motion were linearized around the state of a straight walk, and the Floquet exponents were investigated on the complex plane.",
"In previous work, when all the spring constants for the body-segment yaw joints were decreased uniformly, one pair of exponents crossed the imaginary axis from the left-half plane and entered the right-half plane, which implied Hopf bifurcation.",
"Figure REFB shows all 16 Floquet exponents when $k_1$ was varied, with the other spring constants $k_i$ ($i=2,\\dots ,5$ ) fixed, as done in the robot experiments.",
"Except for the zero exponents, all exponents lie in the left-half plane for large $k_1$ .",
"However, with decreasing $k_1$ , one exponent moves along the real axis and enters the right-half plane.",
"This indicates that the straight walk becomes unstable and pitchfork bifurcation occurs above a critical value of $k_1$ .",
"Furthermore, the components of the destabilizing eigenvector at the bifurcation point were $0.31$ , $0.16$ , $0.21$ , $0.21$ , and $0.11$ for yaw joints 1–5, respectively.",
"These components had the same sign, and yaw joints 1 and 5 had large and small components, respectively, which is consistent with the robot experiments (Fig.",
"REF ).",
"These results verify the destabilization of the straight walk and the emergence of a curved walk through pitchfork bifurcation observed in the robot experiments.",
"Figure: Floquet analysis using simple two-dimensional model.",
"(A) simple model.",
"(B) Floquet exponents when k 1 k_1 was varied."
],
[
"Turning strategy based on pitchfork bifurcation",
"To investigate the maneuverability of the robots achieved with the aid of pitchfork bifurcation, we focused on a turning task in which the robot approached a target located on the floor in a direction different from that where the robot was oriented, as performed in our previous work [3].",
"For a target at any location (relative angle $\\psi $ and distance $R$ ), there exists a unique radius of curvature $\\hat{r}$ of the curved walk with which the robot will approach the target (Fig.",
"REFA).",
"Because the radius of curvature $r$ of the body axis induced by the pitchfork bifurcation monotonically decreases with $1/k_1$ (Fig.",
"REFD), $k_1=\\hat{k}_1$ is uniquely determined so that $r=\\hat{r}$ .",
"This means that when we use $k_1=\\hat{k}_1$ , the robot spontaneously approaches the target due to the pitchfork bifurcation characteristics, which is an optimal strategy for turning.",
"However, this strategy is feedforward, depending only on the initial relative position between the robot and the target, and the direction in which the robot turns (left or right) depends on the initial robot conditions, and thus does not guarantee the success of the turning tasks.",
"Therefore, we also used a supplementary turning controller to approach the target by means of the laser range scanner and leg yaw joints of the first module developed in our previous work [3] (see Appendix ).",
"This supplementary controller allowed the robots to approach the targets even when $k_1\\ne \\hat{k}_1$ .",
"Figure: Turning task.",
"(A) Radius of curvature r ^\\hat{r} of curved walk with which the robot approaches a target (relative angle ψ\\psi and distance RR).",
"(B) Trajectory of the first module on the floor, (C) target distance, and (D) relative target angle for five experiments for three spring constants with ψ=45 ∘ \\psi =45^\\circ , R=1.3R=1.3 m, r ^=0.88\\hat{r}=0.88 m, and 1/k ^ 1 =0.0481/\\hat{k}_1=0.048 deg/Nmm (see Movies 4–6).",
"(E) Evaluation criteria ε 1 \\varepsilon _1 and ε 2 \\varepsilon _2 for 1/k 1 1/k_1.",
"The data points and error bars correspond to the means and standard errors, respectively, of the results of five experiments.",
"Photographs for (F) initial conditions, (G) unsuccessful approach with k 1 >k ^ 1 k_1>\\hat{k}_1, (H) successful approach with k 1 ∼k ^ 1 k_1\\sim \\hat{k}_1, and (I) unsuccessful approach with k 1 <k ^ 1 k_1<\\hat{k}_1."
],
[
"Experimental results",
"For the initial conditions, we used $\\psi =45^\\circ $ and $R=1.3$ m for the relative angle and distance between the first module and the target, respectively, which yielded $\\hat{r}=0.88$ m and $\\hat{k}_1=21$ Nmm/deg ($1/\\hat{k}_1=0.048$ deg/Nmm), and set all body-segment yaw joint angles to zero (Fig.",
"REFF).",
"Figure REFB shows the trajectory of the first module on the floor during the turning task for three torsional spring constants, namely $k_1=15$ ($<\\hat{k}_1$ ), 21 ($\\sim \\hat{k}_1$ ), and 41 Nmm/deg ($>\\hat{k}_1$ ).",
"Figures REFC and D show the time profiles of the target distance and relative target angle with respect to the walking direction, respectively, for these three spring constants.",
"When the distance was less than 0.15 m, we assumed that the robot reached the target and this task was successfully completed.",
"For $k_1=41$ Nmm/deg ($>\\hat{k}_1$ ), the robot hardly changed walking direction and the first module trajectory bulged outward.",
"As a result, the robot could not reach the target (Fig.",
"REFG, see Movie 4).",
"For $k_1=15$ Nmm/deg ($<\\hat{k}_1$ ), although the robot could quickly change walking direction, it moved in directions away from the target due to the small radius of curvature created by pitchfork bifurcation and could not reach the target (Fig.",
"REFI, see Movie 5).",
"In contrast, for $k_1=21$ Nmm/deg ($\\sim \\hat{k}_1$ ), the robot reached the target through the optimal curved walk generated by pitchfork bifurcation (Fig.",
"REFH, see Movie 6).",
"To quantitatively clarify the turning performance dependence on $k_1$ , we employed two evaluation criteria, namely $\\varepsilon _1$ and $\\varepsilon _2$ .",
"For the criterion $\\varepsilon _1$ , we used the distance of the target at 23 s to evaluate how quickly and successfully the robot approached the target.",
"For the criterion $\\varepsilon _2$ , we used the absolute value of the relative target angle with respect to the walking direction to evaluate how quickly and successfully the robot was oriented to the target.",
"Figure REFE shows the results for $1/k_1$ .",
"Both criteria showed minimum values around $k_1=\\hat{k}_1$ , which means that the turning strategy using pitchfork bifurcation achieved the best performance and that the robot made the best use of the curved walk induced by pitchfork bifurcation to complete the turning task.",
"To verify the performance of the proposed controller using pitchfork bifurcation, we additionally performed the same experiment as that in Fig.",
"REFB but using different initial conditions of the target, namely $\\psi =40^\\circ $ and $R=1.5$ m, which yielded $\\hat{r}=1.2$ m and $\\hat{k}_1=26$ Nmm/deg ($1/\\hat{k}_1=0.039$ deg/Nmm).",
"Figures REFA and B show the evaluation criteria $\\varepsilon _1$ and $\\varepsilon _2$ , respectively, for $1/k_1$ .",
"Both criteria show minimum values around $k_1=\\hat{k}_1$ , which means that the turning strategy using pitchfork bifurcation achieved the best performance, in the same way as shown in Fig.",
"REFE.",
"The results show similar trends, which verifies the performance of the proposed controller.",
"Figure: Evaluation criteria (A) ε 1 \\varepsilon _1 and (B) ε 2 \\varepsilon _2 for 1/k 1 1/k_1 for different condition (1/k ^ 1 =0.0391/\\hat{k}_1=0.039 deg/Nmm).",
"The data points and error bars correspond to the means and standard errors, respectively, of the results of five experiments."
],
[
"Comparison with previous strategy",
"To examine how the turning performance was improved by pitchfork bifurcation, we also performed experiments using the turning strategy based on Hopf bifurcation used in our previous work [3] and compared the performance.",
"For Hopf bifurcation, we used the same spring constant among the body-segment yaw joints and employed five spring constants ($k_i=8.7$ , 11, 15, 21, and 41 Nmm/deg, $i=1,\\dots ,5$ ) to evaluate the turning performance for $k_i$ , where the Hopf bifurcation point is about $k_i=18$ Nmm/deg ($1/\\hat{k}_i=0.057$ deg/Nmm), as obtained in our previous work [3].",
"The experimental conditions were identical to those in Fig.",
"REFE except for the spring constants of the body-segment yaw joints.",
"Figures REFA and B compare the turning performance in terms of the criteria $\\varepsilon _1$ and $\\varepsilon _2$ , respectively, between the strategies based on pitchfork and Hopf bifurcations.",
"Both criteria for Hopf bifurcation showed minimum values in the unstable region, as observed in our previous work [3].",
"The minimum values of pitchfork bifurcation are lower than those of Hopf bifurcation for both criteria.",
"This means that the turning strategy based on pitchfork bifurcation created by tuning the body-axis flexibility is superior to that based on Hopf bifurcation.",
"Figure: Comparison of evaluation criteria (A) ε 1 \\varepsilon _1 and (B) ε 2 \\varepsilon _2 between the turning strategies based on pitchfork bifurcation and Hopf bifurcation.",
"The data points and error bars correspond to the means and standard errors, respectively, of the results of five experiments."
],
[
"Conclusion and discussion",
"In this study, we found that the straight walk of a many-legged robot with flexible body axis becomes unstable through pitchfork bifurcation when the body-axis flexibility is changed.",
"The straight walk transitioned into curved walk, whose curvature depended on the body-axis flexibility.",
"We developed a simple controller based on the pitchfork-bifurcation characteristics and demonstrated that the robot achieves high turning maneuver superior to the previous controller based on the Hopf bifurcation.",
"Maneuverability is related to the ability to change movement direction.",
"When the movement direction is destabilized during locomotion, the instability provides driving forces to rapidly change the movement direction and thus enhances maneuverability.",
"Some fighter aircrafts, such as the F-16, are designed to be aerodynamically unstable to enhance maneuverability [7], [23].",
"The use of dynamic instability is thus useful from an engineering viewpoint.",
"The strategy of using movement direction instability to enhance maneuverability is also used by animals.",
"Because the instability is determined by the body dynamics during interaction with the environment, it is prominent in locomotion generated through aerodynamics and hydrodynamics, such as the locomotion of flying insects [13], [32], [41] and sea animals [14], [15], [44].",
"It also appears in legged locomotion.",
"When the center of mass is high, as in mammals, whose legs are under the body, leaning the body to the left or right induces instability and helps turning [10], [35].",
"However, when the center of mass is low, as in reptiles and arthropods, whose legs are out to the side of the body, locomotor behavior is almost two-dimensional because the center of mass moves in a horizontal plane.",
"Therefore, the effect of body leaning is small and thus such a turning strategy cannot be used, which implies that the stability of the walking direction in the horizontal plane becomes more crucial.",
"It has been suggested that cockroaches manipulate the position of the reaction forces from the floor entering the body to control the stability of a straight walk in a horizontal plane and that the straight walk instability helps their turning [33], [35].",
"Various bio-inspired robots that use their body axis for propulsion, such as snake robots [6], [37] and fish robots [12], [26], [36], have high maneuverability.",
"However, legged robots still have difficulty in achieving highly maneuverable locomotion.",
"This is partly because their interaction with the environment (i.e., foot contact with the ground) is intermittent due to the repetition of foot-contact and foot-off phases in leg movement.",
"Although this intermittency allows the traversal of diverse environments, it can make the robot lose balance.",
"Therefore, the control design of legged robots has focused on the avoidance of balance loss using dynamic criteria, such as a supporting polygon [17] and a zero-moment point [43], and maneuverability has not been well investigated.",
"Although increasing the number of legs prevents balance loss, it also increases the number of contact legs, which impedes maneuverability.",
"Moreover, the number of degrees of freedom to be controlled increases, making both motion planning and control difficult.",
"In addition, general many-legged robots use actuators for controlling not only the leg joints but also body-segment joints [40], [45], which has huge computational and energy costs.",
"In contrast, our robot has passive body-segment joints and turning controller is simple, which does not directly control the movement of the body axis and instead determines the body-axis flexibility based on the pitchfork bifurcation characteristics inherent in the robot dynamics.",
"The generation of robot movements not by actuators but by dynamics is crucial for efficient locomotion [9], and our strategy greatly reduces the computational and energy costs.",
"This study provides a design principle for a simple and efficient control scheme to create maneuverable locomotion for many-legged robots using intrinsic dynamic properties.",
"The proposed turning strategy can be further improved.",
"For example, variable stiffness can be used for the body-segment yaw joints to change the stability characteristics depending on the task and conditions."
],
[
"Supplementary movies",
"We recorded 6 supplementary movies to show the pitchfork bifurcation of a straight walk and turning performance in the robot experiments: Straight walk using a large spring constant for the torsional spring in yaw joint 1.",
"Curved walk with a small curvature using a small spring constant for the torsional spring in yaw joint 1.",
"Curved walk with a large curvature using very small spring constant for the torsional spring in yaw joint 1.",
"Unsuccessful approach using a spring constant larger than the optimal value for yaw joint 1.",
"Unsuccessful approach using a spring constant smaller than the optimal value for yaw joint 1.",
"Successful approach using a spring constant close to the optimal value."
],
[
"Supplementary turning control by leg joints",
"The optimal turning strategy is feedforward, depending only on the initial relative position between the robot and target.",
"In addition, the direction in which the robot turns (left or right) depends on the initial robot conditions because of the property of pitchfork bifurcation.",
"To guarantee a successful approach to the target, we used a supplementary feedback-based turning controller, which was developed in our previous work [3].",
"Specifically, we used the relative target angle $\\psi $ of the first module measured by the laser range scanner and the leg yaw joints $\\theta _1$ and $\\theta _2$ of the first module.",
"We determined the desired angles $\\hat{\\theta }_1$ and $\\hat{\\theta }_2$ of $\\theta _1$ and $\\theta _2$ for each gait cycle ($t_i^n\\le t<t_i^n+T$ , $t=t_i^n$ is the time when the desired leg tip is at the PEP for the $n$ th gait cycle, and $T$ is the gait cycle duration ($=0.6$ s)) using $\\hat{\\theta }_i(t)&\\hspace{-8.53581pt}=&\\hspace{-8.53581pt}\\left\\lbrace \\hspace{-2.84526pt}\\begin{array}{ll}\\hat{\\theta }_i(t_i^n) & \\hspace{-2.84526pt}t_i^n\\le t<t_i^n+t_{\\scriptsize {\\mbox{start}}} \\\\\\hat{\\theta }_i(t_i^n)+\\Delta _i\\displaystyle \\frac{t-t_i^n-t_{\\scriptsize {\\mbox{start}}}}{t_{\\scriptsize {\\mbox{end}}}-t_{\\scriptsize {\\mbox{start}}}} & \\hspace{-2.84526pt}t_i^n+t_{\\scriptsize {\\mbox{start}}}\\le t\\le t_i^n+t_{\\scriptsize {\\mbox{end}}} \\\\\\hat{\\theta }_i(t_i^n+t_{\\scriptsize {\\mbox{end}}}) & \\hspace{-2.84526pt}t_i^n+t_{\\scriptsize {\\mbox{end}}}<t<t_i^n+T\\end{array}\\right.",
"\\nonumber \\\\\\Delta _i&\\hspace{-8.53581pt}=&\\hspace{-8.53581pt}\\left\\lbrace \\hspace{-2.84526pt}\\begin{array}{l}\\psi (t_i^n+t_{\\scriptsize {\\mbox{start}}})-\\hat{\\theta }_i(t_i^n+t_{\\scriptsize {\\mbox{start}}}) \\\\\\hspace{56.9055pt} |\\psi (t_i^n+t_{\\scriptsize {\\mbox{start}}})-\\hat{\\theta }_i(t_i^n+t_{\\scriptsize {\\mbox{start}}})|<5^\\circ \\\\5^\\circ \\\\\\hspace{56.9055pt} \\mbox{otherwise}\\end{array}\\right.",
"\\nonumber $ where $t_{\\scriptsize {\\mbox{start}}}$ and $t_{\\scriptsize {\\mbox{end}}}$ were set to 40% and 80%, respectively, of the duration of the half elliptical curve of the leg tip trajectory ($=0.12$ and $0.23$ s).",
"This means that each leg changed its yaw direction toward the target only during the swing phase with $5^\\circ $ of the maximum turning angle for one gait cycle.",
"We also limited the maximum angle of the leg yaw joint to $5^\\circ $ during the turning task.",
"This supplementary control did not aim to make the robots follow the optimal curved path generated by the turning strategy using pitchfork bifurcation.",
"Instead, it was designed so that the first module modulated the walking direction based on the target direction, which solves the problems related to the feedforward property of the optimal turning strategies and the turning direction due to initial robot conditions, and furthermore allows the robots to approach the target even when $k_1\\ne \\hat{k}_1$ ."
],
[
"Acknowledgment",
"This study was supported in part by JSPS KAKENHI Grant Numbers JP17H04914 and JP19KK0377 and the Inamori Foundation."
]
] | 2107.01837 |
[
[
"Extensions of ADMM for Separable Convex Optimization Problems with\n Linear Equality or Inequality Constraints"
],
[
"Abstract The alternating direction method of multipliers (ADMM) proposed by Glowinski and Marrocco is a benchmark algorithm for two-block separable convex optimization problems with linear equality constraints.",
"It has been modified, specified, and generalized from various perspectives to tackle more concrete or complicated application problems.",
"Despite its versatility and phenomenal popularity, it remains unknown whether or not the ADMM can be extended to separable convex optimization problems with linear inequality constraints.",
"In this paper, we lay down the foundation of how to extend the ADMM to two-block and multiple-block (more than two blocks) separable convex optimization problems with linear inequality constraints.",
"From a high-level and methodological perspective, we propose a unified framework of algorithmic design and a roadmap for convergence analysis in the context of variational inequalities, based on which it is possible to design a series of concrete ADMM-based algorithms with provable convergence in the prediction-correction structure.",
"The proposed algorithmic framework and roadmap for convergence analysis are eligible to various convex optimization problems with different degrees of separability, in which both linear equality and linear inequality constraints can be included.",
"The analysis is comprehensive yet can be presented by elementary mathematics, and hence generically understandable."
],
[
"Introduction",
"The alternating direction method of multipliers (ADMM) was proposed originally in [13] by Glowinski and Marrocco for solving nonlinear elliptic problems, and it has become a benchmark algorithm for solving various convex optimization problems with linear equality constraints and separable objective functions without coupled variables.",
"Methodologically, it can be regarded as a splitting version of the classic augmented Lagrangian method (ALM) proposed in [24], [27].",
"It has found applications in an extremely broad range of areas, particularly in fields related to data science such as machine learning, computer vision, and distributed/centralized optimization.",
"When a concrete application is considered, the original ADMM may need to be modified or specified appropriately from various perspectives to better capture the underlying structures and properties of the specific model.",
"Some such examples include its linearized/proximal versions as proposed in [17], [18].",
"It has also inspired many more generalized versions for solving more complicated problems, among which are a series of ADMM variants for solving multiple-block separable convex optimization problems whose objective functions consist of more than two blocks of components without coupled variables.",
"We refer to, e.g.",
"[3], [12], [8], [16], for some survey papers about the ADMM, among a large volume of literatures.",
"Despite the versatility and phenomenal popularity of ADMM, it remains unknown whether or not it can be extended to separable convex optimization problems with linear inequality constraints, even for two-block separable convex optimization problems.",
"Let us start with the canonical two-block separable convex optimization problem with linear equality constraints $ \\min \\big \\lbrace \\theta _1(x) + \\theta _2(y) \\;|\\; Ax+By=b, x\\in {\\cal X}, y\\in {\\cal Y} \\big \\rbrace , $ where $\\theta _i: {\\Re }^{n_i}\\rightarrow {\\Re } \\;(i=1,2)$ are closed proper convex functions and they are not necessarily smooth; ${\\cal X}\\subseteq \\Re ^{n_1}$ and $ {\\cal Y}\\subseteq \\Re ^{n_2}$ are closed convex sets; $A\\in \\Re ^{m\\times n_1}$ and $B\\in \\Re ^{m\\times n_2}$ are given matrices; $b\\in \\Re ^m$ is a given vector.",
"Let $\\lambda \\in \\Re ^m$ be the Lagrange multiplier and consider the Lagrangian function of the problem (REF ) $ L_E(x,y,\\lambda )= \\theta _1(x) + \\theta _2(y) - \\lambda ^T (Ax + By -b ), \\quad (x,y,\\lambda )\\in {\\cal X}\\times {\\cal Y}\\times \\Re ^m.$ Then, the iterative scheme of the ADMM for solving (REF ) reads as $ \\hbox{(ADMM)}\\quad \\left\\lbrace \\begin{array}{l}x^{k+1} \\in \\arg \\min \\bigl \\lbrace L_E(x, y^k,\\lambda ^k) + \\frac{\\beta }{2} \\Vert Ax+By^k-b\\Vert ^2 \\;|\\; x\\in {\\cal X} \\bigr \\rbrace ,\\\\[0.15cm]y^{k+1} \\in \\arg \\min \\bigl \\lbrace L_E(x^{k+1}, y,\\lambda ^k) + \\frac{\\beta }{2} \\Vert Ax^{k+1}+By-b\\Vert ^2 \\;|\\; y\\in {\\cal Y}\\bigr \\rbrace ,\\\\[0.15cm]{\\lambda }^{k+1} = \\arg \\max \\bigl \\lbrace L_E(x^{k+1}, y^{k+1},\\lambda ) - \\frac{1}{2\\beta } \\Vert \\lambda -\\lambda ^k\\Vert ^2 \\;|\\; \\lambda \\in {\\Re ^m} \\bigr \\rbrace ,\\end{array} \\right.$ where $\\beta >0$ is the penalty parameter.",
"That is, the ADMM (REF ) generates the new output $(x^{k+1},y^{k+1},\\lambda ^{k+1})$ with the input $(y^k, \\lambda ^k)$ .",
"Note that the update of $\\lambda ^{k+1}$ in (REF ) can be explicitly expressed as ${\\lambda }^{k+1} = \\lambda ^k - \\beta (A x^{k+1} + By^{k+1} -b).$ We refer to, e.g., [10], [11], [14], [21], [22], [30], for some convergence study of the ADMM (REF ).",
"The ADMM (REF ) updates the variables $x$ and $y$ by treating the functions $\\theta _1$ and $\\theta _2$ separately in its iterations, and the subproblems in (REF ) are usually much easier than the original problem (REF ).",
"For many application problems, the subproblems in (REF ) could be easy enough to have closed-form solutions or be solved up to high precisions.",
"This feature mainly accounts for the versatility and efficiency of the ADMM (REF ) in various areas.",
"Certainly, how difficult the resulting subproblems in (REF ) are still depends on the corresponding functions, coefficient matrices, and constraint sets; and many variants of the ADMM (REF ) have been proposed in the literatures for more meticulous studies.",
"But we only concentrate on the foundational case (REF ), and for succinctness, we do not further elaborate on more detailed cases such as how to solve the resulting subproblems.",
"If the linear equality constraints in (REF ) are changed to linear inequality constraints while all the other settings are remained, we obtain the following model: $ \\min \\big \\lbrace \\theta _1(x) + \\theta _2(y) \\;|\\; A x + By \\ge b , x\\in {\\cal X}, y\\in {\\cal Y}\\big \\rbrace .$ The two-block separable convex optimization model (REF ) with linear inequality constraints captures particular applications such as the support vector machine with a linear kernel in [6], [29] and its variants in [25], [26].",
"To solve (REF ), it is easy to see that it can be reformulated as the following three-block separable model with linear equality constraints: $ \\min \\big \\lbrace \\theta _1(x) + \\theta _2(y) \\;|\\; A x + By - z= b, x\\in {\\cal X}, y\\in {\\cal Y}, z\\in \\Re ^m_+\\big \\rbrace ,$ where $z\\in \\Re ^m_+$ is an auxiliary variable.",
"Then, a direct extension of the ADMM (REF ) can be applied to the reformulated model (REF ).",
"More specifically, let $\\lambda \\in \\Re ^m$ be the Lagrange multiplier and the Lagrangian function of (REF ) be $ L_{E}(x,y,z,\\lambda )= \\theta _1(x) + \\theta _2(y) - \\lambda ^T (Ax + By -z-b ), \\quad (x,y,z,\\lambda )\\in {\\cal X}\\times {\\cal Y}\\times \\Re ^m_+\\times \\Re ^m.$ Directly extending the ADMM (REF ) to (REF ) results in the scheme $ \\hbox{(EADMM)}\\quad \\left\\lbrace \\begin{array}{l}x^{k+1} \\in \\arg \\min \\bigl \\lbrace L_{E}(x, y^k,z^k,\\lambda ^k) + \\frac{\\beta }{2} \\Vert Ax+By^k-z^k-b\\Vert ^2 \\;|\\; x\\in {\\cal X} \\bigr \\rbrace ,\\\\[0.1cm]y^{k+1} \\in \\arg \\min \\bigl \\lbrace L_{E}(x^{k+1}, y,z^k,\\lambda ^k) + \\frac{\\beta }{2} \\Vert Ax^{k+1}+By-z^k-b\\Vert ^2 \\;|\\; y\\in {\\cal Y} \\bigr \\rbrace ,\\\\[0.1cm]z^{k+1} \\in \\arg \\min \\bigl \\lbrace L_{E}(x^{k+1}, y^{k+1},z,\\lambda ^k) + \\frac{\\beta }{2} \\Vert Ax^{k+1}+By^{k+1}-z-b\\Vert ^2 \\;|\\; z\\in {\\Re ^m_+}\\bigr \\rbrace ,\\\\[0.1cm]{\\lambda }^{k+1} = \\arg \\max \\bigl \\lbrace L_{E}(x^{k+1}, y^{k+1},z^{k+1},\\lambda ) - \\frac{1}{2\\beta } \\Vert \\lambda -\\lambda ^k\\Vert ^2 \\;|\\; \\lambda \\in {\\Re ^m} \\bigr \\rbrace ,\\end{array} \\right.$ where $\\beta >0$ is also the penalty parameter.",
"According to [5], however, convergence of the direct extension of ADMM (REF ) is not guaranteed unless more restrictive conditions on the objective functions, coefficient matrices, as well as the penalty parameter, are additionally posed.",
"Alternatively, the scheme (REF ) should be revised appropriately to render the convergence.",
"For example, the output of (REF ) should be further corrected by those correction steps studied in [19], [20], [23].",
"The number of blocks really matters for extensions of the ADMM (REF ), from both theoretical and numerical perspectives.",
"In the literatures, there are numerous numerical studies showing that, when the ADMM (REF ) and its variants are applied, it is generally not preferred to artificially create more blocks of variables/functions by introducing auxiliary variables.",
"One reason is the mentioned lack of theoretical guarantee of convergence as rigorously analyzed in [5].",
"Another more subtle reason is that if the underlying augmented Lagrangian function is decomposed by more than twice (which is usually for the sake of obtaining subproblems as easy as those in (REF )), then the approximation to the underlying augmented Lagrangian function might be too inaccurate and accordingly the resulting scheme may become numerically slower or even divergent.",
"One more consequence is that tuning the penalty parameter $\\beta $ usually becomes more challenging when the underlying augmented Lagrangian function is decomposed into too many blocks.",
"This consequence is certainly based on experience and empirical study, instead of rigorous theory.",
"Hence, for the generic two-block convex optimization model with linear inequality constraints (REF ), it is interesting to discuss whether or not we can extend the original ADMM (REF ) in some senses that all the major features and structures of the original ADMM (REF ) can be kept.",
"That is, the underlying augmented Lagrangian function should be decomposed only twice at each iteration, the resulting subproblems should be as easy as those in (REF ), and the convergence can be rigorously guaranteed without extra conditions.",
"To the best of our knowledge, this question remains unknown and our main purpose is to answer this question.",
"Let us combine both (REF ) and (REF ) in our discussion, and consider the general two-block separable convex optimization model with linear equality or inequality constraints $ \\min \\big \\lbrace \\theta _1(x) + \\theta _2(y) \\;|\\; A x + By=b\\ (\\hbox{or} \\ge b) , x\\in {\\cal X}, y\\in {\\cal Y} \\big \\rbrace ,$ in which the settings are the same as those in (REF ) and (REF ).",
"The solution set of the model (REF ) is assumed to be nonempty.",
"Our main interest is certainly the case of (REF ) with linear inequality constraints, i.e., (REF ).",
"The reason for considering (REF ) is that the algorithmic framework and the roadmap for convergence analysis to be presented are eligible to both (REF ) and (REF ).",
"Another reason is that, as mentioned, we prefer to keep the features and structures of the original ADMM (REF ) when the linear inequality constraints are considered in (REF ) because of both theoretical and numerical purposes.",
"Hence, treating (REF ) and (REF ) uniformly can help us discern the difference of the to-be-proposed algorithms from the original ADMM (REF ) more clearly.",
"From a high-level and methodological perspective, we will propose a unified framework of algorithmic design and convergence analysis for the model (REF ), with which a series of specific algorithms can be easily designed and their convergence can be proved uniformly by following a common roadmap without any additional conditions on the functions, coefficient matrices, or the penalty parameter.",
"We aim at laying down the foundation of algorithmic design and convergence analysis for extensions of the ADMM (REF ) from the canonical two-block model (REF ) to the more general one (REF ), as well as to the even more complicated multiple-block one (REF ).",
"The rest of this paper is organized as follows.",
"In Section , we review the variational inequality characterization of the model (REF ).",
"Our analysis will be mainly conducted in the variational inequality context.",
"Then, we extend the ADMM (REF ) and propose an prototypical algorithmic framework in Section ; its convergence is also proved in this section.",
"In Sections and , we specify the prototypical algorithmic framework as two concrete algorithms for the model (REF ).",
"In Sections -, we consider a multiple-block generalized model of (REF ), i.e., (REF ), and parallelize the analysis in Sections -, respectively.",
"In Section , we give an overview of how the proposed algorithmic frameworks can be unified for convex optimization problems with different degrees of separability.",
"Finally, some conclusions are drawn in Section ."
],
[
"Variational inequality characterization",
"In this section, we summarize some preliminaries for further analysis.",
"In particular, the variational inequality (VI) characterizations of the optimization problems appearing in our discussion are crucial.",
"As analyzed in our previous works such as [21], [23], the VI approach appears to be a convenient and powerful tool for us to look into the structure of the problem under discussion, as well as to conduct convergence analysis.",
"Our analysis in this paper will also be conducted in the context of variational inequalities.",
"We start from the VI representation of the optimality condition of a convex optimization problem.",
"The following lemma will be frequently used in our following analysis.",
"Its proof is elementary and it can be found in, e.g., [2].",
"Lemma 2.1 Consider the optimization problem $\\min \\big \\lbrace \\theta (z) + f(z) \\;|\\; z\\in {\\cal Z}\\big \\rbrace ,$ where ${\\cal Z}\\subset \\Re ^n$ is a closed convex set, $\\theta (z)$ and $f(z)$ are convex functions.",
"If $f$ is differentiable on an open set which contains ${\\cal Z}$ , and the solution set of this problem is nonempty, then we have that $ z^* \\in \\arg \\min \\big \\lbrace \\theta (z) + f(z) \\;|\\; z\\in {\\cal Z}\\big \\rbrace $ if and only if $z^*\\in {\\cal Z}, \\quad \\theta (z) - \\theta (z^*) + (z-z^*)^T\\nabla f(z^*) \\ge 0, \\quad \\forall \\, z\\in {\\cal Z}.$ Now, let us focus on the model (REF ) and derive its optimality condition in terms of the VI formulation.",
"Without ambiguity of notation, let us reuse $\\lambda $ for the Lagrange multiplier and consider the Lagrangian function of the problem (REF ) $ L(x,y,\\lambda )= \\theta _1(x) + \\theta _2(y) - \\lambda ^T (Ax + By -b ), \\quad (x,y,\\lambda )\\in {\\cal X}\\times {\\cal Y}\\times \\Lambda ,$ where $ \\Lambda =\\left\\lbrace \\begin{array}{ll}\\Re ^m, & \\hbox{if $Ax+ By = b$} , \\\\[0.1cm]\\Re ^m_+, & \\hbox{if $Ax + By \\ge b$}.\\end{array} \\right.$ Furthermore, let $\\Omega :={\\cal X}\\times {\\cal Y}\\times \\Lambda $ .",
"We call a point $(x^*,y^*,\\lambda ^*)$ defined on $\\Omega $ a saddle point of the Lagrangian function (REF ) if it satisfies the inequalities $ L_{\\lambda \\in \\Lambda }(x^*,y^*,\\lambda ) \\le L(x^*,y^*,\\lambda ^*) \\le L_{x\\in {\\cal X}, y\\in {\\cal Y}}(x, y,\\lambda ^*).",
"$ Obviously, a saddle point can be characterized by $(x^*, y^*, \\lambda ^*)\\in \\Omega , \\quad \\left\\lbrace \\begin{array}{rl}L(x,y^*,\\lambda ^*) -L(x^*,y^*,\\lambda ^*) \\ge 0, & \\forall \\, x\\in {\\cal X}, \\\\[0.1cm]L(x^*,y,\\lambda ^*) -L(x^*,y^*,\\lambda ^*) \\ge 0, & \\forall \\, y\\in {\\cal Y}, \\\\[0.1cm]L(x^*,y^*,\\lambda ^*) - L(x^*,y^*,\\lambda ) \\ge 0, & \\forall \\,\\lambda \\in \\Lambda .\\end{array} \\right.$ Alternatively, according to Lemma REF , the inequalities above can be written as the following VIs: $ (x^*, y^*, \\lambda ^*)\\in \\Omega , \\quad \\left\\lbrace \\begin{array}{rl}\\theta _1(x) - \\theta _1(x^*) + (x-x^*)^T(- {A}^T\\lambda ^*) \\ge 0, & \\forall \\, x\\in {\\cal X}, \\\\[0.1cm]\\theta _2(y) - \\theta _2(y^*) + (y-y^*)^T(- B^T\\lambda ^*) \\ge 0, & \\forall \\, y\\in {\\cal Y}, \\\\[0.1cm](\\lambda -\\lambda ^*)^T( Ax^* + By^* -b)\\ge 0, & \\forall \\,\\lambda \\in \\Lambda .\\end{array} \\right.$ More compactly, (REF ) can be rewritten as $ w^*\\in \\Omega , \\quad \\theta (u) - \\theta (u^*) + (w-w^*)^TF(w^*) \\ge 0, \\quad \\forall \\, w\\in \\Omega ,$ where $ w = \\left(\\begin{array}{c}x\\\\y\\\\[0.1cm]\\lambda \\end{array} \\right),\\;\\; u = \\left(\\begin{array}{c}x\\\\y\\end{array} \\right), \\;\\; \\theta (u)=\\theta _1(x) +\\theta _2(y), \\;\\;F(w) =\\left(\\begin{array}{c}- {A}^T\\lambda \\\\- {B}^T\\lambda \\\\{A}x + By -b \\end{array} \\right).",
"$ It is clear that the function $\\theta (u)$ is convex and the operator $F$ in (REF ) is affine with a skew-symmetric matrix.",
"Thus, we have $ (w-\\tilde{w})^T(F(w)-F(\\tilde{w}))\\equiv 0 , \\;\\; \\forall \\, w, \\tilde{w}.",
"$ The solution set of the VI () is denoted by $\\Omega ^*$ , which is also the set of the saddle points of the Lagrangian function (REF ) defined on $\\Omega $ ."
],
[
"Prototypical algorithmic framework",
"In this section, we focus on the VI reformulation (), and propose an algorithmic framework conceptually in the context of variational inequalities.",
"This algorithmic framework will be the prototype for various specific algorithms and we will show how to specify the prototypical algorithmic framework as concrete algorithms for the model (REF ).",
"We shall also prove the convergence of the algorithmic framework, and establish a roadmap for the convergence proof.",
"With this roadmap, proving the convergence for different algorithms specified from the prototypical algorithmic framework can simply be reduced to identifying two matrices and then verifying the positive definiteness of another matrix.",
"This prototypical algorithm framework and its roadmap for convergence analysis can enable us to treat a series of different algorithms uniformly from a high-level perspective, and help us discern their respective difference from the original ADMM (REF ) clearly."
],
[
"Algorithmic framework",
"First of all, for any $w=(x, y, \\lambda )\\in \\Re ^{n_1}\\times \\Re ^{n_2}\\times \\Re ^m$ , we define $\\xi \\in \\Re ^{3m\\times 3m}$ by $ \\xi := Pw, \\qquad \\hbox{where} \\qquad P = \\left(\\begin{array}{ccc}\\sqrt{\\beta } A & 0 & 0 \\\\0 & \\sqrt{\\beta } B & 0 \\\\0 & 0 & \\frac{1}{\\sqrt{\\beta }} I_m\\end{array} \\right) .",
"$ Accordingly, we define ${\\Xi }= \\big \\lbrace \\xi \\;|\\; \\xi = Pw,\\, w\\in \\Omega \\big \\rbrace \\qquad \\hbox{and} \\qquad {\\Xi ^*}= \\big \\lbrace \\xi ^* \\;|\\; \\xi ^*= Pw^*,\\, w^*\\in \\Omega ^*\\big \\rbrace .",
"$ Then, we propose a prototypical algorithmic framework for the VI reformulation (), which is in a prediction-correction structure.",
"A Prototypical Algorithmic Framework for VI $(\\ref {VI-ID})$.",
"(Prediction Step) With $\\xi ^k \\in \\Xi $ , find $\\tilde{w}^k \\in \\Omega $ such that $ \\tilde{w}^k \\in \\Omega , \\;\\;\\theta (u) - \\theta (\\tilde{u}^k) + (w - \\tilde{w}^k)^T F(\\tilde{w}^k) \\ge (\\xi -\\tilde{\\xi }^k)^T{{\\mbox{${\\cal Q}$}}}(\\xi ^k-\\tilde{\\xi }^k), \\;\\; \\forall \\, w \\in {\\Omega },$ with ${{\\mbox{${\\cal Q}$}}} \\in \\Re ^{3m\\times 3m}$ , and the matrix ${{\\mbox{${\\cal Q}$}}}^T+{{\\mbox{${\\cal Q}$}}}$ is positive definite.",
"(Correction Step) With $\\tilde{w}^k$ solved by (REF ) and thus ${\\tilde{\\xi }}^k:= P{\\tilde{w}}^k$ , generate $\\xi ^{k+1}$ by $ {\\xi }^{k+1} = {\\xi }^k - {{\\mbox{${\\cal M}$}}}(\\xi ^k - \\tilde{\\xi }^k), $ where ${{\\mbox{${\\cal M}$}}} \\in \\Re ^{3m\\times 3m}$ is non-singular."
],
[
"Roadmap for convergence analysis",
"Now, we prove the convergence of the prototypical algorithmic framework (REF ).",
"This is the unified analysis of convergence for various algorithms that can be specified from the prototypical algorithmic framework (REF ).",
"The roadmap for convergence analysis will become clear based on the following analysis.",
"Theorem 3.1 For the matrices ${{\\mbox{${\\cal Q}$}}}$ in (REF ) and ${\\mbox{${\\cal M}$}}$ in (REF ), if there is a positive definite matrix ${\\mbox{${\\cal H}$}}\\in \\Re ^{3m\\times 3m}$ such that $ {\\mbox{${\\cal H}$}}{\\mbox{${\\cal M}$}}={{\\mbox{${\\cal Q}$}}} $ and $ {\\mbox{${\\cal G}$}}:= {{\\mbox{${\\cal Q}$}}}^T + {{\\mbox{${\\cal Q}$}}} - {\\mbox{${\\cal M}$}}^T{\\mbox{${\\cal H}$}}{\\mbox{${\\cal M}$}}\\succ 0, $ then we have $ \\Vert {\\xi }^{k+1} -{\\xi }^*\\Vert _{{\\mbox{${\\cal H}$}}}^2\\le \\Vert {\\xi }^k -{\\xi }^*\\Vert _{{\\mbox{${\\cal H}$}}}^2 - \\Vert {\\xi }^k - \\tilde{\\xi }^k \\Vert _{{\\mbox{${\\cal G}$}}}^2,\\quad \\forall \\, \\xi ^*\\in {\\Xi ^*}.",
"$ Proof.",
"Setting $w$ in (REF ) as any fixed $w^*\\in \\Omega ^*$ , and using (REF ) $(\\tilde{w}^k-w^*)^T F(\\tilde{w}^k)\\equiv (\\tilde{w}^k-w^*)^T F(w^*),$ we get $ (\\tilde{\\xi }^k-\\xi ^*)^T{{\\mbox{${\\cal Q}$}}}(\\xi ^k-\\tilde{\\xi }^k)\\ge \\theta (\\tilde{u}^k)-\\theta (u^*) + (\\tilde{w}^k-w^*)^T F(w^*), \\quad \\forall \\,w^*\\in \\Omega ^*.",
"$ The right-hand side of the last inequality is non-negative.",
"Thus, we have $ ({\\xi }^k-\\xi ^*)^T{{\\mbox{${\\cal Q}$}}}(\\xi ^k-\\tilde{\\xi }^k) \\ge (\\xi ^k-\\tilde{\\xi }^k)^T {{\\mbox{${\\cal Q}$}}}(\\xi ^k-\\tilde{\\xi }^k), \\quad \\forall \\, \\xi ^*\\in \\Xi ^*.",
"$ Then, by simple manipulations, we obtain ${ \\Vert {\\xi }^k -{\\xi }^*\\Vert _{{\\mbox{${\\cal H}$}}}^2 - \\Vert {\\xi }^{k+1} -{\\xi }^*\\Vert _{{\\mbox{${\\cal H}$}}}^2 } \\nonumber \\\\&\\stackrel{(\\ref {M-COR})}{=} & \\Vert {\\xi }^k -{\\xi }^*\\Vert _{{\\mbox{${\\cal H}$}}}^2 - \\Vert ({\\xi }^{k} -{\\xi }^*) -{\\mbox{${\\cal M}$}}(\\xi ^k - \\tilde{\\xi }^k) \\Vert _{{\\mbox{${\\cal H}$}}}^2 \\nonumber \\\\& \\stackrel{(\\ref {M-HMQ})}{=}& 2 ({\\xi }^{k} -{\\xi }^*) ^T{{\\mbox{${\\cal Q}$}}}(\\xi ^k - \\tilde{\\xi }^k)-\\Vert {\\mbox{${\\cal M}$}}(\\xi ^k - \\tilde{\\xi }^k) \\Vert _{{\\mbox{${\\cal H}$}}}^2 \\nonumber \\\\&\\stackrel{(\\ref {M-XiQ})}{\\ge } & 2 ({\\xi }^{k} -\\tilde{\\xi }^k)^T {{\\mbox{${\\cal Q}$}}}(\\xi ^k - \\tilde{\\xi }^k)-\\Vert {\\mbox{${\\cal M}$}}(\\xi ^k - \\tilde{\\xi }^k) \\Vert _{{\\mbox{${\\cal H}$}}}^2 \\nonumber \\\\& = & (\\xi ^k - \\tilde{\\xi }^k)^T[( {{\\mbox{${\\cal Q}$}}}^T + {{\\mbox{${\\cal Q}$}}}) -{\\mbox{${\\cal M}$}}^T{\\mbox{${\\cal H}$}}{\\mbox{${\\cal M}$}}] (\\xi ^k - \\tilde{\\xi }^k) \\nonumber \\\\& \\stackrel{(\\ref {M-HMG})}{=} & \\Vert {\\xi }^k - \\tilde{\\xi }^k \\Vert _{{\\mbox{${\\cal G}$}}}^2.$ The assertion of this theorem is proved.",
"$\\Box $ Theorem 3.2 Let $\\lbrace \\xi ^k\\rbrace $ and $\\lbrace \\tilde{w}^k\\rbrace $ be the sequences generated by the algorithmic framework (REF ).",
"If the conditions (REF ) and (REF ) are satisfied, then the sequence $\\lbrace \\xi ^k\\rbrace $ converges to some $\\xi ^{\\infty } \\in \\Xi ^*$ .",
"Proof.",
"First of all, it follows from (REF ) that the sequence $\\lbrace \\xi ^k\\rbrace $ is bounded and $ \\lim _{k\\rightarrow \\infty } \\Vert {\\xi }^k - \\tilde{\\xi }^k \\Vert _{{\\mbox{${\\cal G}$}}}^2 =0.$ Thus, the sequence $\\lbrace \\tilde{\\xi }^k\\rbrace $ is also bounded.",
"Let $\\xi ^{\\infty }$ be a cluster point of $\\lbrace \\tilde{\\xi }^k\\rbrace $ and $\\lbrace \\tilde{\\xi }^{k_j}\\rbrace $ be a subsequence converging to $\\xi ^{\\infty }$ .",
"Let $\\lbrace \\tilde{\\xi }^k\\rbrace $ and $\\lbrace \\tilde{\\xi }^{k_j}\\rbrace $ be the induced sequences by $\\lbrace \\tilde{w}^k\\rbrace $ and $\\lbrace \\tilde{w}^{k_j}\\rbrace $ , respectively.",
"It follows from (REF ) that $ \\tilde{w}^{k_j}\\in \\Omega , \\;\\; \\theta (u) - \\theta (\\tilde{u}^{k_j}) + (w - \\tilde{w}^{k_j})^T F(\\tilde{w}^{k_j}) \\ge (\\xi -\\tilde{\\xi }^{k_j})^T{{\\mbox{${\\cal Q}$}}}(\\xi ^{k_j}-\\tilde{\\xi }^{k_j}), \\quad \\forall \\, w \\in {\\Omega }.",
"$ Since the matrix ${{\\mbox{${\\cal Q}$}}}$ is non-singular, it follows from (REF ), the continuity of $\\theta (u)$ , and $F(w)$ that $ w^{\\infty }\\in \\Omega , \\;\\; \\theta (u) - \\theta (u^{\\infty }) + (w - w^{\\infty })^T F(w^{\\infty }) \\ge 0, \\quad \\forall \\, w \\in {\\Omega }.",
"$ This VI indicates that $w^{\\infty }$ is a solution point of (), and thus $\\xi ^\\infty =Pw^\\infty \\in \\Xi ^\\ast $ .",
"Moreover, it follows from (REF ) and $\\lim _{j\\rightarrow \\infty } \\tilde{\\xi }^{k_j} =\\xi ^{\\infty }$ that the subsequence $\\lbrace \\xi ^{k_j}\\rbrace $ also converges to $\\xi ^{\\infty }$ .",
"Finally, because of (REF ), we have $ \\Vert \\xi ^{k+1} - \\xi ^{\\infty } \\Vert _{{\\mbox{${\\cal H}$}}}^2 \\le \\Vert \\xi ^k - \\xi ^{\\infty }\\Vert _{{\\mbox{${\\cal H}$}}}^2, $ and thus $\\lbrace \\xi ^k\\rbrace $ converges to $\\xi ^{\\infty }$ .",
"The proof is complete.",
"$\\Box $ The convergence of the prototypical algorithmic framework (REF ) is thus proved in Theorem REF .",
"As just shown, the proof essentially requires to verify the conditions (REF ) and (REF )."
],
[
"Remarks",
"Based on the analysis above, concrete algorithms for the model (REF ) can be constructed by choosing specific matrices ${\\mbox{${\\cal Q}$}}$ in (REF ) and ${\\mbox{${\\cal M}$}}$ in (REF ), and then their convergence can be proved by simply verifying the conditions (REF ) and (REF ).",
"The proposed prototypical algorithmic framework (REF ) opens a door to designing various specific algorithms that are tailored for specific applications of the model (REF ), and the proposed roadmap for convergence analysis provides a unified and simplified way to prove the convergence of various algorithms.",
"Here, we give the prototypical algorithmic framework (REF ) and the roadmap for convergence analysis from a methodological perspective, rather than discussing how to choose the matrices ${\\mbox{${\\cal Q}$}}$ and ${\\mbox{${\\cal M}$}}$ optimally, which should vary from case to case when a specific application is under consideration.",
"In Sections and , we will present some such algorithms and illustrate how to follow the proposed roadmap for convergence analysis.",
"For each algorithm, we will still use the same letters to denote these matrices but with some subscripts."
],
[
"Primal-dual extension of the ADMM (",
"First of all, let us revisit the original ADMM (REF ) for the model (REF ), and introduce an auxiliary notation $ {\\lambda }^{k+\\frac{1}{2}} := \\lambda ^k - \\beta (Ax^k+By^k-b).",
"$ Then, ignoring some constant terms in the objective functions of the corresponding subproblems, we can rewrite the ADMM (REF ) as $ \\left\\lbrace \\begin{array}{l}x^{k+1} \\in \\arg \\!\\min \\bigl \\lbrace \\theta _1(x) - x^TA^T\\lambda ^{k+\\frac{1}{2}} +{\\textstyle {\\frac{\\beta }{2}}}\\Vert A(x-x^k)\\Vert ^2 \\;|\\; {x\\in {\\cal X}} \\bigr \\rbrace , \\\\[0.15cm]y^{k+1} \\in \\arg \\!\\min \\bigl \\lbrace \\theta _2(y) - y^TB^T\\lambda ^{k+\\frac{1}{2}}+{\\textstyle {\\frac{\\beta }{2}}} \\Vert B(y-y^k)+A(x^{k+1}-x^k)\\Vert ^2 \\;|\\; y\\in {\\cal Y} \\bigr \\rbrace , \\\\[0.15cm]{\\lambda }^{k+1} = \\arg \\!\\max \\bigl \\lbrace - \\lambda ^T\\bigl (Ax^{k+1} +B{y}^{k+1}-b\\bigr ) - {\\textstyle {\\frac{1}{2\\beta }}}\\Vert \\lambda -\\lambda ^k\\Vert ^2 \\;|\\; \\lambda \\in \\Re ^m\\bigr \\rbrace .\\end{array} \\right.$ This is a reformulation of the ADMM (REF ) with some terms that are meticulously regrouped.",
"It will be the reference for us to discern the difference of the new algorithms from the original ADMM (REF ) more conveniently."
],
[
"Algorithm",
"The first algorithm for (REF ) is presented below.",
"Since the primal variables $x$ and $y$ are updated first before the dual variable $\\lambda $ , it is called a primal-dual extension of the ADMM (REF ) for (REF ).",
"A Primal-Dual Extension of the ADMM (REF ) for (REF ).",
"(Prediction Step) With given $(Ax^k, By^k, \\lambda ^k)$ , find $\\tilde{w}^k=(\\tilde{x}^k, \\tilde{y}^k,\\tilde{\\lambda }^k)$ via $ \\left\\lbrace \\begin{array}{l}\\tilde{x}^k \\in \\hbox{argmin}\\bigl \\lbrace \\theta _1(x) - x^TA^T{\\lambda }^k + \\frac{1}{2}\\beta \\Vert A(x-x^k)\\Vert ^2 \\;|\\; x\\in {\\cal X}\\bigr \\rbrace , \\\\[0.2cm]\\tilde{y}^k \\in \\hbox{argmin}\\bigl \\lbrace \\theta _2(y) - y^TB^T{\\lambda }^k + \\frac{1}{2}\\beta \\Vert B(y-y^k) + A(\\tilde{x}^k-x^k)\\Vert ^2 \\;|\\; y\\in {\\cal Y}\\bigr \\rbrace , \\\\[0.2cm]\\tilde{\\lambda }^k= \\arg \\!\\max \\bigl \\lbrace -\\lambda ^T\\bigl (A\\tilde{x}^k +B\\tilde{y}^k-b\\bigr ) - \\frac{1}{2\\beta }\\Vert \\lambda -\\lambda ^k\\Vert ^2 \\;|\\; \\lambda \\in {\\Lambda }\\bigr \\rbrace .\\end{array} \\right.$ (Correction Step) Correct the predictor $\\tilde{w}^k$ solved by (REF ), and generate the new iterate $(Ax^{k+1}, By^{k+1}, \\lambda ^{k+1})$ with $\\nu \\in (0,1)$ by $ \\left(\\begin{array}{c}Ax^{k+1} \\\\[0.1cm]By^{k+1} \\\\[0.1cm]\\lambda ^{k+1}\\end{array}\\right) = \\left(\\begin{array}{c}Ax^{k} \\\\[0.1cm]By^{k} \\\\[0.1cm]\\lambda ^{k}\\end{array}\\right) -\\left(\\begin{array}{ccc}{\\nu }I_m & -{\\nu } I_m & 0 \\\\[0.1cm]0 & {\\nu } I_m & 0 \\\\[0.1cm]- \\nu \\beta I_m & 0 & I_m\\end{array}\\right) \\left(\\begin{array}{c}Ax^{k} -A\\tilde{x}^{k} \\\\[0.1cm]By^{k} - B \\tilde{y}^{k}\\\\[0.1cm]\\lambda ^{k} -\\tilde{\\lambda }^k\\end{array}\\right).$ Remark 4.1 Comparing with the reformulated iterative scheme of the ADMM (REF ), we see that the only difference in the prediction step (REF ) is the constant vector $\\lambda ^k$ in the crossing terms (equivalently, constant vectors in the corresponding quadratic terms), while all major features and structures of the ADMM (REF ) are remained in (REF ).",
"This very minor difference does not essentially change the difficulty of the resulting $x$ - and $y$ -subproblems.",
"That is, the $x$ - and $y$ -subproblems in (REF ) are of the same difficulty as those in the original ADMM (REF ) (i.e., (REF )).",
"For the $\\lambda $ -subproblem in (REF ), it can be specified respectively as $ \\tilde{\\lambda }^k = \\lambda ^k -\\beta (A\\tilde{x}^k + B\\tilde{y}^k -b) \\qquad \\hbox{or}\\qquad \\tilde{\\lambda }^k = [\\lambda ^k -\\beta (A\\tilde{x}^k + B\\tilde{y}^k -b)]_+,$ when the model (REF ) or (REF ) is considered, either of which is easy to compute.",
"Remark 4.2 The correction step (REF ) requires extremely simple computation.",
"Looking into the implementation of the ADMM (REF ), we know that it is the sequence $\\lbrace Ax^k, By^k,\\lambda ^k\\rbrace $ , instead of $\\lbrace x^k, y^k,\\lambda ^k\\rbrace $ , that are essentially required for executing the iterations.",
"Hence, when the ADMM (REF ) and its variants are implemented, one advantage is that $Ax^k$ (rather than $x^k$ ) and $By^k$ (rather than $y^k$ ) can be treated together for recursions.",
"The correction step (REF ) exactly has this advantage too, and it treats $Ax^k$ , $A{\\tilde{x}}^k$ , $Bx^k$ and $B{\\tilde{y}}^k$ aggregately with very cheap computation for updating them.",
"Indeed, only few floating-point additions are required.",
"Overall, the algorithm (REF ) maintains all major structures and features of the original ADMM (REF ); the resulting subproblems are of the same difficulty; and the additional computation required by the correction step (REF ) is ignorable."
],
[
"Specification of the prototype algorithmic framework (",
"Now, we show that the algorithm (REF ) can be obtained by specifying the prototype algorithmic framework (REF ).",
"That is, we identify the specific matrices ${\\mbox{${\\cal Q}$}}$ in (REF ) and ${\\mbox{${\\cal M}$}}$ in (REF ) such that (REF ) and (REF ) can be reduced to the prediction step (REF ) and the correction step (REF ), respectively.",
"The specified matrices corresponding to the algorithm (REF ) are denoted by ${\\mbox{${\\cal Q}$}}_{{\\mbox{\\footnotesize $_{\\!",
"{P\\!D}}$}}}$ and ${{\\mbox{${\\cal M}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{P\\!D}}$}}}$ , respectively.",
"Accordingly, we divide the discussion into two subsections."
],
[
"Analysis for the prediction step (",
"According to Lemma REF , the predictor generated by (REF ) satisfies $\\tilde{w}^k\\in \\Omega $ and $ \\left\\lbrace \\begin{array}{l} \\theta _1(x) - \\theta _1(\\tilde{x}^k) + (x - \\tilde{x}^k)^T\\lbrace -A^T\\lambda ^k +\\beta A^T\\!A(\\tilde{x}^k-x^k)\\rbrace \\ge 0, \\;\\; \\forall \\, x\\in {\\cal X},\\\\[0.1cm]\\theta _2(y) - \\theta _2(\\tilde{y}^k) + (y - \\tilde{y}^k)^T\\lbrace -B^T\\lambda ^k +\\beta B^T\\!A(\\tilde{x}^k-x^k)+ \\beta B^T\\!B(\\tilde{y}^k-y^k) \\rbrace \\ge 0, \\;\\; \\forall \\, y\\in {\\cal Y}, \\\\[0.1cm]\\hspace{79.6678pt}(\\lambda - \\tilde{\\lambda }^k)^T\\lbrace \\frac{1}{\\beta } (\\tilde{\\lambda }^k-\\lambda ^k) + (A\\tilde{x}^k + B\\tilde{y}^k-b) \\rbrace \\ge 0, \\;\\; \\forall \\, \\lambda \\in {\\Lambda }.\\end{array}\\right.$ Using the VI form (), we can rewrite it as $ \\left\\lbrace \\begin{array}{l}\\theta _1(x) - \\theta _1(\\tilde{x}^k) + (x - \\tilde{x}^k)^T\\lbrace \\underline{-A^T\\tilde{\\lambda }^k} \\\\ \\hspace{133.72786pt}+ \\beta A^T\\!A(\\tilde{x}^k-x^k) + A^T\\!",
"(\\tilde{\\lambda }^k-\\lambda ^k)\\rbrace \\ge 0, \\;\\; \\forall \\, x\\in {\\cal X}, \\\\[0.1cm]\\theta _2(y) - \\theta _2(\\tilde{y}^k) + (y - \\tilde{y}^k)^T\\lbrace \\underline{-B^T\\tilde{\\lambda }^k}+ \\beta B^TA(\\tilde{x}^k-x^k) \\\\\\hspace{133.72786pt}+\\beta B^T\\!B(\\tilde{y}^k-y^k) + B^T\\!",
"(\\tilde{\\lambda }^k-\\lambda ^k)\\rbrace \\ge 0, \\;\\; \\forall \\, y\\in {\\cal Y}, \\\\[0.1cm]\\hspace{71.13188pt}(\\lambda - \\tilde{\\lambda }^k)^T\\lbrace (\\underline{A\\tilde{x}^k + B\\tilde{y}^k-b}) +(1/\\beta ) \\; (\\tilde{\\lambda }^k-\\lambda ^k)\\rbrace \\ge 0, \\;\\; \\forall \\, \\lambda \\in {\\Lambda }.\\end{array} \\right.$ It is easy to verify that the sum of the three underlining terms in (REF ) is precisely $F(\\tilde{w}^k)$ , where $F(\\cdot )$ is defined in (REF ).",
"Hence, we have the following lemma.",
"Lemma 4.1 With the given $(Ax^k, By^k, \\lambda ^k)$ , the predictor $\\tilde{w}^k\\in \\Omega $ generated by (REF ) satisfies $ \\tilde{w}^k\\in \\Omega , \\;\\;\\; \\theta (u) - \\theta (\\tilde{u}^k) + (w- \\tilde{w}^k )^T \\lbrace F(\\tilde{w}^k)+ Q_{{\\mbox{\\footnotesize $_{\\!",
"{P\\!D}}$}}}(\\tilde{w}^k-w^k) \\rbrace \\ge 0, \\quad \\forall \\, w\\in \\Omega ,$ where $ Q_{{\\mbox{\\footnotesize $_{\\!",
"{P\\!D}}$}}}= \\left(\\begin{array}{ccc}\\beta A^TA & 0 & A^T \\\\\\beta B^TA & \\beta B^TB & B^T \\\\0 & 0 & \\frac{1}{\\beta } I_m\\end{array} \\right).$ Proof.",
"The assertion directly follows from (REF ).",
"$\\Box $ Using the notation in (REF ), we can rewrite the matrix $Q_{{\\mbox{\\footnotesize $_{\\!",
"{P\\!D}}$}}}$ in (REF ) as $ Q_{{\\mbox{\\footnotesize $_{\\!",
"{P\\!D}}$}}}= P^T {{\\mbox{${\\cal Q}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{P\\!D}}$}}} P\\qquad \\hbox{where} \\qquad {{\\mbox{${\\cal Q}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{P\\!D}}$}}} = \\left(\\begin{array}{ccc}I_m & 0 & I_m \\\\I_m & I_m & I_m \\\\0 & 0 & I_m\\end{array} \\right).$ Note that the matrices $Q_{{\\mbox{\\footnotesize $_{\\!",
"{P\\!D}}$}}}\\in \\Re ^{(n_1+n_2+m)\\times (n_1+n_2+m)}$ and ${{\\mbox{${\\cal Q}$}}_{{\\mbox{\\footnotesize $_{\\!",
"{P\\!D}}$}}}}\\in \\Re ^{3m\\times 3m}$ are different.",
"Recall the notation in (REF ).",
"It follows from (REF ) that $ \\tilde{w}^k\\in \\Omega , \\;\\;\\; \\theta (u) - \\theta (\\tilde{u}^k) + (w- \\tilde{w}^k )^T F(\\tilde{w}^k)\\ge (\\xi -\\tilde{\\xi }^k)^T{{\\mbox{${\\cal Q}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{P\\!D}}$}}}(\\xi ^k - \\tilde{\\xi }^k), \\quad \\forall \\, w\\in \\Omega .$ Thus, the prediction step (REF ) can be specified by the prototypical prediction step (REF ) with ${\\mbox{${\\cal Q}$}}:= {{\\mbox{${\\cal Q}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{P\\!D}}$}}}$ as defined in (REF )."
],
[
"Analysis for the correction step (",
"Left-multiplying the matrix $\\hbox{diag}(\\sqrt{\\beta }I_m,\\sqrt{\\beta }I_m, \\frac{1}{\\sqrt{\\beta }}I_m)$ to both sides of the correction step (REF ), we get $ \\left(\\begin{array}{c}\\sqrt{\\beta } Ax^{k+1} \\\\[0.1cm]\\sqrt{\\beta } By^{k+1} \\\\[0.1cm]\\frac{1}{\\sqrt{\\beta }}\\lambda ^{k+1}\\end{array}\\right) = \\left(\\begin{array}{c}\\sqrt{\\beta }Ax^{k} \\\\[0.1cm]\\sqrt{\\beta }By^{k} \\\\[0.1cm]\\frac{1}{\\sqrt{\\beta }} \\lambda ^{k}\\end{array}\\right) -\\left(\\begin{array}{ccc}\\sqrt{\\beta } {\\nu }I_m & - \\sqrt{\\beta }{\\nu } I_m & 0 \\\\[0.1cm]0 & \\sqrt{\\beta } {\\nu } I_m & 0 \\\\[0.1cm]- \\nu \\sqrt{\\beta } I_m & 0 & \\frac{1}{\\sqrt{\\beta }} I_m\\end{array}\\right) \\left(\\begin{array}{c}Ax^{k} -A\\tilde{x}^{k} \\\\[0.1cm]By^{k} - B \\tilde{y}^{k}\\\\[0.1cm]\\lambda ^{k} -\\tilde{\\lambda }^k\\end{array}\\right).$ Then, we have $ \\left(\\begin{array}{c}\\sqrt{\\beta } Ax^{k+1} \\\\[0.1cm]\\sqrt{\\beta } By^{k+1} \\\\[0.1cm]\\frac{1}{\\sqrt{\\beta }}\\lambda ^{k+1}\\end{array}\\right) = \\left(\\begin{array}{c}\\sqrt{\\beta }Ax^{k} \\\\[0.1cm]\\sqrt{\\beta }By^{k} \\\\[0.1cm]\\frac{1}{\\sqrt{\\beta }} \\lambda ^{k}\\end{array}\\right) -\\left(\\begin{array}{ccc}{\\nu }I_m & - {\\nu } I_m & 0 \\\\[0.1cm]0 & {\\nu } I_m & 0 \\\\[0.1cm]- \\nu I_m & 0 & I_m\\end{array}\\right) \\left(\\begin{array}{c}\\sqrt{\\beta }(Ax^{k} -A\\tilde{x}^{k}) \\\\[0.1cm]\\sqrt{\\beta } (By^{k} - B \\tilde{y}^{k})\\\\[0.1cm]\\frac{1}{\\sqrt{\\beta }} (\\lambda ^{k} -\\tilde{\\lambda }^k)\\end{array}\\right).$ It follows from (REF ) that (REF ) can be written as $ {\\xi }^{k+1} = {\\xi }^k - {{\\mbox{${\\cal M}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{P\\!D}}$}}}(\\xi ^k - \\tilde{\\xi }^k), $ where $ {{\\mbox{${\\cal M}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{P\\!D}}$}}} =\\left(\\begin{array}{ccc}{\\nu }I_m & -{\\nu } I_m & 0 \\\\[0.2cm]0 & {\\nu } I_m & 0 \\\\[0.2cm]- \\nu I_m & 0 & I_m\\end{array}\\right) \\quad \\hbox{with} \\quad \\nu \\in (0,1).",
"$ Thus, the correction step (REF ) can be specified by the prototypical correction step (REF ) with ${\\mbox{${\\cal M}$}}:= {{\\mbox{${\\cal M}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{P\\!D}}$}}}$ as defined in (REF )."
],
[
"Convergence",
"Then, according to the roadmap presented in Section REF , proving the convergence of the algorithm (REF ) can be reduced to verifying the conditions (REF ) and (REF ) with the specified matrices ${\\mbox{${\\cal Q}$}}_{{\\mbox{\\footnotesize $_{\\!",
"{P\\!D}}$}}}$ and ${{\\mbox{${\\cal M}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{P\\!D}}$}}}$ .",
"Let us first identify the corresponding matrix ${{\\mbox{${\\cal H}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{P\\!D}}$}}}$ with the specified matrices ${\\mbox{${\\cal Q}$}}_{{\\mbox{\\footnotesize $_{\\!",
"{P\\!D}}$}}}$ and ${{\\mbox{${\\cal M}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{P\\!D}}$}}}$ .",
"Lemma 4.2 For any $\\nu \\in (0,1)$ , the matrix $ {{\\mbox{${\\cal H}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{P\\!D}}$}}} = \\left(\\begin{array}{ccc}(1+\\frac{1}{\\nu }) I_m & (1+\\frac{1}{\\nu }) I_m & I_m \\\\[0.2cm](1+\\frac{1}{\\nu }) I_m & (1+ \\frac{2}{\\nu }) I_m & I_m \\\\[0.2cm]I_m & I_m & I_m\\end{array}\\right) $ is positive definite, and it holds that $ {{\\mbox{${\\cal H}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{P\\!D}}$}}} {{\\mbox{${\\cal M}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{P\\!D}}$}}}= {{\\mbox{${\\cal Q}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{P\\!D}}$}}} ,$ where ${{\\mbox{${\\cal M}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{P\\!D}}$}}}$ and $ {{\\mbox{${\\cal Q}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{P\\!D}}$}}}$ are defined in (REF ) and (REF ), respectively.",
"Proof.",
"It is easy to see that ${{\\mbox{${\\cal H}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{P\\!D}}$}}}$ is positive definite.",
"In addition, we have ${{\\mbox{${\\cal H}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{P\\!D}}$}}} {{\\mbox{${\\cal M}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{P\\!D}}$}}}&=&\\left(\\begin{array}{ccc}(1+\\frac{1}{\\nu }) I_m & (1+\\frac{1}{\\nu }) I_m & I_m \\\\[0.2cm](1+\\frac{1}{\\nu }) I_m & (1+ \\frac{2}{\\nu }) I_m & I_m \\\\[0.2cm]I_m & I_m & I_m\\end{array}\\right)\\left(\\begin{array}{ccc}{\\nu }I_m & -{\\nu } I_m & 0 \\\\[0.2cm]0 & {\\nu } I_m & 0 \\\\[0.2cm]- \\nu I_m & 0 & I_m\\end{array}\\right) \\nonumber \\\\& =& \\left(\\begin{array}{ccc}I_m & 0 & I_m \\\\I_m & I_m & I_m \\\\0 & 0 & I_m\\end{array} \\right) \\;=\\; {{\\mbox{${\\cal Q}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{P\\!D}}$}}}.$ The assertion (REF ) is proved.",
"$\\Box $ Lemma 4.3 For the matrices ${{\\mbox{${\\cal Q}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{P\\!D}}$}}}$ , $ {{\\mbox{${\\cal M}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{P\\!D}}$}}}$ and ${{\\mbox{${\\cal H}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{P\\!D}}$}}}$ defined in (REF ), (REF ) and (REF ), respectively, the matrix $ {{\\mbox{${\\cal G}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{P\\!D}}$}}}:= ({{\\mbox{${\\cal Q}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{P\\!D}}$}}}^T + {{\\mbox{${\\cal Q}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{P\\!D}}$}}}) -{{\\mbox{${\\cal M}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{P\\!D}}$}}}^T{{\\mbox{${\\cal H}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{P\\!D}}$}}}{{\\mbox{${\\cal M}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{P\\!D}}$}}} $ is positive definite.",
"Proof.",
"First, by elementary matrix multiplications, we have ${{\\mbox{${\\cal M}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{P\\!D}}$}}}^T{{\\mbox{${\\cal H}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{P\\!D}}$}}}{{\\mbox{${\\cal M}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{P\\!D}}$}}} & = & {{\\mbox{${\\cal M}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{P\\!D}}$}}}^T{{\\mbox{${\\cal Q}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{P\\!D}}$}}} ={{\\mbox{${\\cal Q}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{P\\!D}}$}}}^T {{\\mbox{${\\cal M}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{P\\!D}}$}}} \\nonumber \\\\& =& \\left(\\begin{array}{ccc}I_m & I_m & 0 \\\\0 & I_m & 0 \\\\I_m & I_m & I_m\\end{array} \\right) \\left(\\begin{array}{ccc}{\\nu }I_m & -{\\nu } I_m & 0 \\\\[0.1cm]0 & {\\nu } I_m & 0 \\\\[0.1cm]- \\nu I_m & 0 & I_m\\end{array}\\right) = \\left(\\begin{array}{ccc}{\\nu }I_m & 0 & 0 \\\\[0.1cm]0 & {\\nu } I_m & 0 \\\\[0.1cm]0 & 0 & I_m\\end{array}\\right).$ Then, it follows that ${\\cal G}_{{\\mbox{\\footnotesize $_{\\!",
"{P\\!D}}$}}} &= & ( {{\\mbox{${\\cal Q}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{P\\!D}}$}}}^T + {{\\mbox{${\\cal Q}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{P\\!D}}$}}}) -{{\\mbox{${\\cal M}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{P\\!D}}$}}}^T{{\\mbox{${\\cal H}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{P\\!D}}$}}}{{\\mbox{${\\cal M}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{P\\!D}}$}}} \\nonumber \\\\& = & \\left(\\begin{array}{ccc}2 I_m & I_m & I_m \\\\I_m & 2 I_m & I_m \\\\I_m & I_m & 2I_m\\end{array} \\right) - \\left(\\begin{array}{ccc}{\\nu }I_m & 0 & 0 \\\\[0.1cm]0 & {\\nu } I_m & 0 \\\\[0.1cm]0 & 0 & I_m\\end{array}\\right)= \\left(\\begin{array}{ccc}(2-{\\nu })I_m & I_m & I_m \\\\[0.1cm]I_m & (2-{\\nu }) I_m & I_m \\\\[0.1cm]I_m & I_m & I_m\\end{array}\\right).$ Thus, the matrix $ {{\\mbox{${\\cal G}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{P\\!D}}$}}}$ is positive definite for any $\\nu \\in (0,1)$ .",
"$\\Box $ For the matrices ${{\\mbox{${\\cal Q}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{P\\!D}}$}}}$ in (REF ) and ${{\\mbox{${\\cal M}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{P\\!D}}$}}}$ in (REF ), the assertions in Lemmas REF and REF hold.",
"Consequently, we have the following theorem which essentially implies the convergence of the algorithm (REF ).",
"The proof of this theorem follows directly from Theorems REF and REF .",
"Theorem 4.1 Let $\\lbrace \\xi ^k\\rbrace $ be the sequence generated by the algorithm (REF ).",
"Then, we have $ \\Vert {\\xi }^{k+1} -{\\xi }^*\\Vert _{ {{\\mbox{${\\cal H}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{P\\!D}}$}}}}^2\\le \\Vert {\\xi }^k -{\\xi }^*\\Vert _{ {{\\mbox{${\\cal H}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{P\\!D}}$}}}}^2 - \\Vert {\\xi }^k - \\tilde{\\xi }^k \\Vert _{{{\\mbox{${\\cal G}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{P\\!D}}$}}}}^2,\\quad \\forall \\, \\xi ^*\\in {\\Xi ^*},$ and thus the sequence $\\lbrace \\xi ^k\\rbrace $ converges to some $\\xi ^* \\in \\Xi ^*$ ."
],
[
"Dual-primal extension of ADMM (",
"In this section, we extend the ADMM (REF ) in the way that the dual variable $\\lambda $ is updated first and then the primal variables $x$ and $y$ are updated.",
"We show that this new algorithm can also be obtained by specifying the prototypical algorithmic framework (REF ).",
"Hence, we also follow the roadmap in Section REF to prove its convergence."
],
[
"Algorithm",
"The second algorithm for (REF ) is presented below; it is called a dual-primal extension of the ADMM (REF ).",
"A Dual-Primal Extension of the ADMM (REF ) for (REF ).",
"(Prediction Step) With given $(Ax^k, By^k, \\lambda ^k)$ , find $\\tilde{w}^k=(\\tilde{x}^k, \\tilde{y}^k,\\tilde{\\lambda }^k)$ via $ \\left\\lbrace \\begin{array}{l}\\tilde{\\lambda }^k= \\arg \\!\\max \\bigl \\lbrace -\\lambda ^T \\bigl (Ax^k +By^k-b\\bigr ) - \\frac{1}{2\\beta }\\Vert \\lambda -\\lambda ^k\\Vert ^2 \\;|\\; \\lambda \\in {\\Lambda }\\bigr \\rbrace , \\\\[0.2cm]\\tilde{x}^k \\in \\hbox{argmin}\\bigl \\lbrace \\theta _1(x) - x^TA^T\\tilde{\\lambda }^k+ \\frac{1}{2}\\beta \\Vert A(x-x^k)\\Vert ^2\\;|\\; x\\in {\\cal X}\\bigr \\rbrace , \\\\[0.2cm]\\tilde{y}^k \\in \\hbox{argmin}\\bigl \\lbrace \\theta _2(y) - y^TB^T\\tilde{\\lambda }^k+ \\frac{1}{2}\\beta \\Vert B(y-y^k) + A(\\tilde{x}^k-x^k)\\Vert ^2\\;|\\; y\\in {\\cal Y}\\bigr \\rbrace .\\end{array} \\right.$ (Correction Step) Correct the predictor $\\tilde{w}^k$ generated by (REF ), and generate the new iterate $(Ax^{k+1}, By^{k+1}, \\lambda ^{k+1})$ with $\\nu \\in (0,1)$ by $ \\left(\\begin{array}{c}Ax^{k+1} \\\\[0.1cm]By^{k+1} \\\\[0.1cm]\\lambda ^{k+1}\\end{array}\\right) = \\left(\\begin{array}{c}Ax^{k} \\\\[0.1cm]By^{k} \\\\[0.1cm]\\lambda ^{k}\\end{array}\\right) -\\left(\\begin{array}{ccc}{\\nu }I_m & -{\\nu } I_m & 0 \\\\[0.1cm]0 & {\\nu } I_m & 0 \\\\[0.1cm]- \\beta I_m & - \\beta I_m & I_m\\end{array}\\right) \\left(\\begin{array}{c}Ax^{k} -A\\tilde{x}^{k} \\\\[0.1cm]By^{k} - B \\tilde{y}^{k}\\\\[0.1cm]\\lambda ^{k} -\\tilde{\\lambda }^k\\end{array}\\right).$ Remark 5.1 The algorithm (REF ) essentially shares the same features as the algorithm (REF ), despite their only difference in the order of updating the primal and dual variables, as well as their very slight difference of the matrices in their respective correction steps."
],
[
"Specification of the prototypical algorithmic framework (",
"Similarly, we analyze the prediction step (REF ) and the correction step (REF ), and show that they can be obtained by specifying the prototypical prediction and correction steps (REF ) and (REF ), respectively.",
"Hence, the algorithm (REF ) can also be specified by the prototypical algorithmic framework (REF ).",
"The specified matrices are denoted by ${{\\mbox{${\\cal Q}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{D\\!P}}$}}}$ and $ {{\\mbox{${\\cal M}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{D\\!P}}$}}}$ , respectively."
],
[
"Analysis for the prediction (",
"According to Lemma REF , the predictor generated by (REF ) satisfies $\\tilde{w}^k\\in \\Omega $ and $\\left\\lbrace \\begin{array}{l} \\theta _1(x) - \\theta _1(\\tilde{x}^k) + (x - \\tilde{x}^k)^T\\lbrace -A^T\\tilde{\\lambda }^k +\\beta A^T\\!A(\\tilde{x}^k-x^k)\\rbrace \\ge 0, \\;\\; \\forall \\, x\\in {\\cal X},\\\\[0.1cm]\\theta _2(y) - \\theta _2(\\tilde{y}^k) + (y - \\tilde{y}^k)^T\\lbrace -B^T\\tilde{\\lambda }^k+\\beta B^T\\!A(\\tilde{x}^k-x^k)+ \\beta B^T\\!B(\\tilde{y}^k-y^k) \\rbrace \\ge 0, \\;\\; \\forall \\, y\\in {\\cal Y}, \\\\[0.1cm]\\hspace{45.52458pt}(\\lambda - \\tilde{\\lambda }^k)^T\\lbrace \\frac{1}{\\beta } (\\tilde{\\lambda }^k-\\lambda ^k) + (A{x}^k + B{y}^k-b)\\rbrace \\ge 0, \\;\\; \\forall \\, \\lambda \\in {\\Lambda }.\\end{array}\\right.$ Using the VI form (), we have that $ \\left\\lbrace \\begin{array}{l}\\theta _1(x) - \\theta _1(\\tilde{x}^k) + (x - \\tilde{x}^k)^T\\lbrace \\underline{-A^T\\tilde{\\lambda }^k} + \\beta A^T\\!A(\\tilde{x}^k-x^k) \\rbrace \\ge 0, \\quad \\forall \\, x\\in {\\cal X},\\\\[0.1cm]\\theta _2(y) - \\theta _2(\\tilde{y}^k) + (y - \\tilde{y}^k)^T\\lbrace \\underline{-B^T\\tilde{\\lambda }^k}+ \\beta B^T\\!A(\\tilde{x}^k-x^k) \\\\\\hspace{173.56198pt}+\\beta B^T\\!B(\\tilde{y}^k-y^k) \\rbrace \\ge 0, \\quad \\forall \\, y\\in {\\cal Y}, \\\\[0.1cm]\\hspace{28.45274pt}(\\lambda - \\tilde{\\lambda }^k)^T\\lbrace (\\underline{A\\tilde{x}^k + B\\tilde{y}^k-b}) -A(\\tilde{x}^k - x^k)-B(\\tilde{y}^k - y^k) \\\\\\hspace{176.407pt}+ (1/\\beta ) (\\tilde{\\lambda }^k-\\lambda ^k)\\rbrace \\ge 0, \\quad \\forall \\, \\lambda \\in {\\Lambda }.\\end{array} \\right.$ The sum of the underling parts of (REF ) is exactly $F(\\tilde{w}^k)$ , where $F(\\cdot )$ is defined in (REF ).",
"Thus, we have the following lemma.",
"Lemma 5.1 With the given $(Ax^k, By^k, \\lambda ^k)$ , the predictor $\\tilde{w}^k\\in \\Omega $ produced by (REF ) satisfies $ \\tilde{w}^k\\in \\Omega , \\;\\;\\; \\theta (u) - \\theta (\\tilde{u}^k) + (w- \\tilde{w}^k )^T\\lbrace F(\\tilde{w}^k) + Q_{{\\mbox{\\footnotesize $_{\\!",
"{D\\!P}}$}}}(\\tilde{w}^k-w^k)\\rbrace \\ge 0, \\quad \\forall \\, w \\in \\Omega ,$ where $ Q_{{\\mbox{\\footnotesize $_{\\!",
"{D\\!P}}$}}}= \\left(\\begin{array}{ccc}\\beta A^TA & 0 & 0 \\\\\\beta B^TA & \\beta B^TB & 0 \\\\-A & -B & \\frac{1}{\\beta } I_m\\end{array} \\right).$ Proof.",
"The assertion directly follows from (REF ).",
"$\\Box $ Using the notation in (REF ), we can rewrite the matrix $Q_{{\\mbox{\\footnotesize $_{\\!",
"{D\\!P}}$}}}$ in (REF ) as $ Q_{{\\mbox{\\footnotesize $_{\\!",
"{D\\!P}}$}}}= P^T {{\\mbox{${\\cal Q}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{D\\!P}}$}}} P\\qquad \\hbox{where} \\qquad {{\\mbox{${\\cal Q}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{D\\!P}}$}}} = \\left(\\begin{array}{ccc}I_m & 0 & 0 \\\\I_m & I_m & 0 \\\\-I_m & -I_m & I_m\\end{array} \\right).$ Also, note that the matrices $Q_{{\\mbox{\\footnotesize $_{\\!",
"{D\\!P}}$}}}$ and ${{\\mbox{${\\cal Q}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{D\\!P}}$}}}$ are different.",
"It follows from (REF ) that $ \\tilde{w}^k\\in \\Omega , \\;\\;\\; \\theta (u) - \\theta (\\tilde{u}^k) + (w- \\tilde{w}^k )^T F(\\tilde{w}^k)\\ge (\\xi -\\tilde{\\xi }^k)^T{{\\mbox{${\\cal Q}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{D\\!P}}$}}}(\\xi ^k - \\tilde{\\xi }^k), \\quad \\forall \\, w\\in \\Omega .$ Thus, the prediction step (REF ) can be specified by the prototypical prediction step (REF ) with ${\\mbox{${\\cal Q}$}}:= {{\\mbox{${\\cal Q}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{D\\!P}}$}}}$ as defined in (REF )."
],
[
"Analysis for the correction procedure (",
"Left-multiplying the matrix $\\hbox{diag}(\\sqrt{\\beta }I_m,\\sqrt{\\beta }I_m, \\frac{1}{\\sqrt{\\beta }}I_m)$ to both sides of the correction step (REF ), we get $ \\left(\\begin{array}{c}\\sqrt{\\beta } Ax^{k+1} \\\\[0.1cm]\\sqrt{\\beta } By^{k+1} \\\\[0.1cm]\\frac{1}{\\sqrt{\\beta }}\\lambda ^{k+1}\\end{array}\\right) = \\left(\\begin{array}{c}\\sqrt{\\beta }Ax^{k} \\\\[0.1cm]\\sqrt{\\beta }By^{k} \\\\[0.1cm]\\frac{1}{\\sqrt{\\beta }} \\lambda ^{k}\\end{array}\\right) -\\left(\\begin{array}{ccc}\\sqrt{\\beta } {\\nu }I_m & - \\sqrt{\\beta }{\\nu } I_m & 0 \\\\[0.1cm]0 & \\sqrt{\\beta } {\\nu } I_m & 0 \\\\[0.1cm]- \\nu \\sqrt{\\beta } I_m & - \\nu \\sqrt{\\beta } I_m & \\frac{1}{\\sqrt{\\beta }} I_m\\end{array}\\right) \\left(\\begin{array}{c}Ax^{k} -A\\tilde{x}^{k} \\\\[0.1cm]By^{k} - B \\tilde{y}^{k}\\\\[0.1cm]\\lambda ^{k} -\\tilde{\\lambda }^k\\end{array}\\right).$ Then, we have $ \\left(\\begin{array}{c}\\sqrt{\\beta } Ax^{k+1} \\\\[0.1cm]\\sqrt{\\beta } By^{k+1} \\\\[0.1cm]\\frac{1}{\\sqrt{\\beta }}\\lambda ^{k+1}\\end{array}\\right) = \\left(\\begin{array}{c}\\sqrt{\\beta }Ax^{k} \\\\[0.1cm]\\sqrt{\\beta }By^{k} \\\\[0.1cm]\\frac{1}{\\sqrt{\\beta }} \\lambda ^{k}\\end{array}\\right) -\\left(\\begin{array}{ccc}{\\nu }I_m & - {\\nu } I_m & 0 \\\\[0.1cm]0 & {\\nu } I_m & 0 \\\\[0.1cm]- \\nu I_m & - \\nu I_m & I_m\\end{array}\\right) \\left(\\begin{array}{c}\\sqrt{\\beta }(Ax^{k} -A\\tilde{x}^{k}) \\\\[0.1cm]\\sqrt{\\beta } (By^{k} - B \\tilde{y}^{k})\\\\[0.1cm]\\frac{1}{\\sqrt{\\beta }} (\\lambda ^{k} -\\tilde{\\lambda }^k)\\end{array}\\right).$ It follows from (REF ) that (REF ) can be written as $ {\\xi }^{k+1} = {\\xi }^k - {{\\mbox{${\\cal M}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{D\\!P}}$}}}(\\xi ^k - \\tilde{\\xi }^k), $ where $ {{\\mbox{${\\cal M}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{D\\!P}}$}}} =\\left(\\begin{array}{ccc}{\\nu }I_m & -{\\nu } I_m & 0 \\\\[0.2cm]0 & {\\nu } I_m & 0 \\\\[0.2cm]-I_m & -I_m & I_m\\end{array}\\right) \\quad \\hbox{with} \\quad \\nu \\in (0,1).$ Thus, the correction step (REF ) can be specified by the prototypical correction step (REF ) with ${{\\mbox{${\\cal M}$}}}:= {{\\mbox{${\\cal M}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{D\\!P}}$}}}$ as defined in (REF )."
],
[
"Convergence",
"Since it is shown that the algorithm (REF ) can be specified by the prototypical algorithmic framework (REF ), its convergence can be guaranteed if the conditions (REF ) and (REF ) are satisfied by the just specified matrices ${{\\mbox{${\\cal Q}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{D\\!P}}$}}}$ and ${{\\mbox{${\\cal M}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{D\\!P}}$}}}$ .",
"Let us identify the corresponding matrix ${{\\mbox{${\\cal H}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{D\\!P}}$}}}$ with the specified matrices ${{\\mbox{${\\cal Q}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{D\\!P}}$}}}$ and ${{\\mbox{${\\cal M}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{D\\!P}}$}}}$ .",
"Lemma 5.2 For any $\\nu \\in (0,1)$ , the matrix $ {{\\mbox{${\\cal H}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{D\\!P}}$}}} = \\left(\\begin{array}{ccc}\\frac{1}{\\nu } I_m & \\frac{1}{\\nu } I_m & 0 \\\\[0.2cm]\\frac{1}{\\nu } I_m & \\frac{2}{\\nu } I_m & 0 \\\\[0.2cm]0 & 0 & I_m\\end{array}\\right) $ is positive definite, and it holds that $ {{\\mbox{${\\cal H}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{D\\!P}}$}}} {{\\mbox{${\\cal M}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{D\\!P}}$}}}= {{\\mbox{${\\cal Q}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{D\\!P}}$}}} , $ where ${{\\mbox{${\\cal M}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{D\\!P}}$}}}$ and $ {{\\mbox{${\\cal Q}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{D\\!P}}$}}}$ are defined in (REF ) and (REF ), respectively.",
"Proof.",
"It is easy to see that ${{\\mbox{${\\cal H}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{D\\!P}}$}}}$ is positive definite.",
"In addition, we have ${{\\mbox{${\\cal H}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{D\\!P}}$}}} {{\\mbox{${\\cal M}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{D\\!P}}$}}}&=& \\left(\\begin{array}{ccc}\\frac{1}{\\nu } I_m & \\frac{1}{\\nu } I_m & 0 \\\\[0.2cm]\\frac{1}{\\nu } I_m & \\frac{2}{\\nu } I_m & 0 \\\\[0.2cm]0 & 0 & I_m\\end{array}\\right) \\left(\\begin{array}{ccc}{\\nu }I_m & -{\\nu } I_m & 0 \\\\[0.2cm]0 & {\\nu } I_m & 0 \\\\[0.2cm]-I_m & -I_m & I_m\\end{array}\\right) \\nonumber \\\\& =& \\left(\\begin{array}{ccc}I_m & 0 & 0 \\\\I_m & I_m & 0 \\\\-I_m & -I_m & I_m\\end{array} \\right) \\;=\\; {{\\mbox{${\\cal Q}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{D\\!P}}$}}}.$ The assertion (REF ) is proved.",
"$\\Box $ Lemma 5.3 Let $ {{\\mbox{${\\cal Q}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{D\\!P}}$}}}$ , ${{\\mbox{${\\cal M}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{D\\!P}}$}}}$ and ${{\\mbox{${\\cal H}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{D\\!P}}$}}} $ be the matrices defined in (REF ), (REF ) and (REF ), respectively.",
"Then the matrix $ {{\\mbox{${\\cal G}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{D\\!P}}$}}}:=( {{\\mbox{${\\cal Q}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{D\\!P}}$}}}^T + {{\\mbox{${\\cal Q}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{D\\!P}}$}}}) -{{\\mbox{${\\cal M}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{D\\!P}}$}}}^T{{\\mbox{${\\cal H}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{D\\!P}}$}}}{{\\mbox{${\\cal M}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{D\\!P}}$}}}$ is positive definite.",
"Proof.",
"First, by elementary matrix multiplications, we know that ${{\\mbox{${\\cal M}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{D\\!P}}$}}}^T{{\\mbox{${\\cal H}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{D\\!P}}$}}}{{\\mbox{${\\cal M}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{D\\!P}}$}}} & = & {{\\mbox{${\\cal M}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{D\\!P}}$}}}^T {{\\mbox{${\\cal Q}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{D\\!P}}$}}} ={{\\mbox{${\\cal Q}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{D\\!P}}$}}}^T {{\\mbox{${\\cal M}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{D\\!P}}$}}} \\nonumber \\\\& =& \\left(\\begin{array}{ccc}I_m & I_m & -I_m \\\\0 & I_m & -I_m \\\\0 & 0 & I_m\\end{array} \\right) \\left(\\begin{array}{ccc}{\\nu }I_m & -{\\nu } I_m & 0 \\\\[0.1cm]0 & {\\nu } I_m & 0 \\\\[0.1cm]-I_m & -I_m & I_m\\end{array}\\right) \\nonumber \\\\& = & \\left(\\begin{array}{ccc}(1+{\\nu })I_m & I_m & -I_m \\\\[0.1cm]I_m &(1 + {\\nu }) I_m & -I_m \\\\[0.1cm]-I_m & -I_m & I_m\\end{array}\\right).$ Then, we have ${{\\mbox{${\\cal G}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{D\\!P}}$}}} &= & ( {{\\mbox{${\\cal Q}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{D\\!P}}$}}}^T + {{\\mbox{${\\cal Q}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{D\\!P}}$}}}) -{{\\mbox{${\\cal M}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{D\\!P}}$}}}^T{{\\mbox{${\\cal H}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{D\\!P}}$}}}{{\\mbox{${\\cal M}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{D\\!P}}$}}} \\nonumber \\\\& = & \\left(\\begin{array}{ccc}2 I_m & I_m & - I_m \\\\I_m & 2 I_m & - I_m \\\\- I_m & -I_m & 2I_m\\end{array} \\right) - \\left(\\begin{array}{ccc}(1+{\\nu })I_m & I_m & -I_m \\\\[0.1cm]I_m &(1 + {\\nu }) I_m & -I_m \\\\[0.1cm]-I_m & -I_m & I_m\\end{array}\\right) \\nonumber \\\\& =& \\left(\\begin{array}{ccc}(1-{\\nu })I_m & 0 & 0 \\\\[0.2cm]0 & (1-{\\nu }) I_m & 0 \\\\[0.2cm]0 & 0 & I_m\\end{array}\\right).$ Thus, the matrix $ {{\\mbox{${\\cal G}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{D\\!P}}$}}}$ is positive definite for any $\\nu \\in (0,1)$ .",
"$\\Box $ For the matrices ${{\\mbox{${\\cal Q}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{D\\!P}}$}}}$ in (REF ) and ${{\\mbox{${\\cal M}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{D\\!P}}$}}}$ in (REF ), the assertions of Lemmas REF and REF hold.",
"Consequently, we have the following theorem which essentially implies the convergence of the algorithm (REF ).",
"Its proof follows directly from Theorems REF and REF .",
"Theorem 5.1 Let $\\lbrace \\xi ^k\\rbrace $ be the sequence generated by the algorithm (REF ).",
"Then, we have $ \\Vert {\\xi }^{k+1} -{\\xi }^*\\Vert _{ {{\\mbox{${\\cal H}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{D\\!P}}$}}}}^2\\le \\Vert {\\xi }^k -{\\xi }^*\\Vert _{ {{\\mbox{${\\cal H}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{D\\!P}}$}}}}^2 - \\Vert {\\xi }^k - \\tilde{\\xi }^k \\Vert _{{{\\mbox{${\\cal G}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{D\\!P}}$}}}}^2,\\quad \\forall \\, \\xi ^*\\in {\\Xi ^*},$ and thus the sequence $\\lbrace \\xi ^k\\rbrace $ converges to some $\\xi ^* \\in \\Xi ^*$ ."
],
[
"Extensions to multi-block separable convex optimization problems with linear equality or inequality constraints",
"Recently, there are many intensive discussions on how to extend the ADMM (REF ) from the two-block separable convex optimization model (REF ) to its generalized multiple-block models, from both theoretical and algorithmic perspectives.",
"We refer to, e.g., [19], [20], [23], for a few works.",
"Because of the work [5], it is known that the convergence is not guaranteed if the ADMM (REF ) is directly extended.",
"This means we should be cautious in both algorithmic design and convergence analysis when considering extensions of the ADMM (REF ) to multiple-block separable convex optimization problems.",
"As mentioned, this fact also discourages us to reformulate a separable model with linear inequality constraints as another separable model with only linear equality constraints but with more blocks of auxiliary variables, and then consider applying some existing ADMM type algorithms that are eligible to models with linear equality constraints.",
"In the following three sections, we consider natural extensions from the model (REF ) to its multiple-block generalized one: $ \\min \\Bigl \\lbrace \\sum _{i=1}^{p} \\theta _i(x_i) \\;\\big |\\; \\sum _{i=1}^{p} A_ix_i=b\\ (\\hbox{or} \\ge b) , \\;\\; x_i\\in {\\cal X}_i \\Bigr \\rbrace ,$ where $\\theta _i: {\\Re }^{n_i}\\rightarrow {\\Re }, \\, i=1,\\ldots , p$ , are closed proper convex functions and they are not necessarily smooth; ${\\cal X}_i\\subseteq \\Re ^{n_i},\\, i=1,\\ldots , p$ , are closed convex sets; $A_i\\in \\Re ^{m\\times n_i},\\, i=1,\\ldots , p$ , are given matrices; $b\\in \\Re ^m$ is a given vector.",
"The model (REF ) can be regarded as a special case of (REF ) with $p=2$ .",
"Let us focus on the multiple-block case of (REF ) with $p\\ge 3$ , and parallelize the discussions in Sections - for this multiple-block case.",
"Unlike the failure of the direct extension of the ADMM (REF ), we will show that the proposed algorithms (REF ) and (REF ) can both be directly extended from the two-block model (REF ) to the multiple-block generalized model (REF ).",
"We first summarize some notations and results similar as those in Sections - for the multiple-block model (REF ).",
"Without ambiguity, some notations are denoted by the same letters as previous sections."
],
[
"VI characterization",
"Let $\\lambda \\in \\Re ^m$ be the Lagrange multiplier of (REF ) and the Lagrangian function of the problem (REF ) be $ L(x_1,\\ldots ,x_p,\\lambda ) = \\sum _{i=1}^{p}\\theta _i(x_i) -\\lambda ^T\\Bigl (\\sum _{i=1}^{p} A_ix_i-b\\Bigr ).$ The optimality condition of (REF ) can be written as the following VI: $ w^*\\in \\Omega , \\quad \\theta (x) - \\theta (x^*) + (w-w^*)^T F(w^*) \\ge 0, \\quad \\forall \\, w\\in \\Omega ,$ where $ w=\\left(\\!\\!\\begin{array}{c}x_1\\\\\\vdots \\\\x_{p} \\\\[0.1cm]\\lambda \\end{array}\\!\\!",
"\\right), \\quad x=\\left(\\!\\!\\begin{array}{c}x_1\\\\\\vdots \\\\x_{p} \\\\\\end{array}\\!\\!",
"\\right),\\quad \\theta (x) = \\sum _{i=1}^{p} \\theta _i(x_i), \\quad F(w) = \\left(\\!\\!\\begin{array}{c}- A_1^T\\lambda \\\\\\vdots \\\\-A_{p}^T\\lambda \\\\[0.1cm]\\sum _{i=1}^{p} A_ix_i-b\\end{array}\\!\\!",
"\\right),$ and $ \\Omega = \\prod _{i=1}^p {\\cal X}_i \\times \\Lambda \\quad \\hbox{with}\\quad \\Lambda =\\left\\lbrace \\begin{array}{ll}\\Re ^m, & \\hbox{if $\\sum _{i=1}^{p} A_ix_i = b$} , \\\\[0.2cm]\\Re ^m_+, & \\hbox{if $\\sum _{i=1}^{p} A_ix_i\\ge b$}.\\end{array} \\right.",
"$ Again, we denote by $\\Omega ^*$ the solution set of the VI (REF )."
],
[
"Prototypical algorithm framework for VI (",
"Similar as the VI $(\\ref {VI-ID})$ , we also present a prototypical algorithmic framework for the VI (REF ), from which concrete algorithms for the model (REF ) can be specified.",
"Let us further denote the following notations: $ P =\\left(\\begin{array}{ccccc}\\sqrt{\\beta } A_1 & \\qquad 0 \\qquad & \\qquad \\cdots \\qquad & \\qquad \\cdots \\qquad & 0 \\\\[0.1cm]0 & \\sqrt{\\beta } A_2 & \\ddots & & \\vdots \\\\[0.1cm]\\vdots & \\ddots & \\ddots & \\ddots & \\vdots \\\\[0.1cm]\\vdots & & \\ddots &\\sqrt{\\beta } A_{p} & 0\\\\[0.1cm]0 & \\cdots & \\cdots & 0 & \\frac{1}{\\sqrt{\\beta }}I_{m}\\end{array}\\!\\!\\right), \\qquad \\xi = Pw= \\left(\\begin{array}{c}\\sqrt{\\beta } A_1x_1 \\\\[0.1cm]\\sqrt{\\beta } A_2x_2 \\\\[0.1cm]\\vdots \\\\[0.1cm]\\sqrt{\\beta } A_px_p \\\\[0.1cm]\\frac{1}{\\sqrt{\\beta }} \\lambda \\end{array} \\right).$ Accordingly, we define ${\\Xi }= \\big \\lbrace \\xi \\;|\\; \\xi = Pw, \\; w\\in \\Omega \\big \\rbrace \\quad \\hbox{and} \\quad {\\Xi ^*}= \\big \\lbrace \\xi ^* \\;|\\; \\xi ^*= Pw^*, \\; w^*\\in \\Omega ^*\\big \\rbrace .",
"$ Then, the prototypical algorithm framework for the VI (REF ) is presented as follows.",
"A Prototypical Algorithmic Framework for VI (REF ).",
"(Prediction Step) With given $w^k$ and thus $\\xi ^k= Pw^k$ , find $\\tilde{w}^k \\in \\Omega $ such that $ \\tilde{w}^k \\in \\Omega , \\;\\;\\theta (x) - \\theta (\\tilde{x}^k) + (w - \\tilde{w}^k)^T F(\\tilde{w}^k) \\ge (\\xi -\\tilde{\\xi }^k)^T{{\\mbox{${\\cal Q}$}}}(\\xi ^k-\\tilde{\\xi }^k), \\;\\; \\forall \\, w \\in {\\Omega },$ with ${\\mbox{${\\cal Q}$}}\\in \\Re ^{(p+1)m\\times (p+1)m}$ , and the matrix ${{\\mbox{${\\cal Q}$}}}^T+{{\\mbox{${\\cal Q}$}}}$ is positive definite.",
"(Correction Step) With $\\tilde{w}^k$ solved by (REF ) and thus $\\tilde{\\xi }^k=P\\tilde{w}^k$ , generate $\\xi ^{k+1}$ by $ {\\xi }^{k+1} = {\\xi }^k - {\\mbox{${\\cal M}$}}(\\xi ^k - \\tilde{\\xi }^k), $ where ${\\mbox{${\\cal M}$}}\\in \\Re ^{(p+1)m\\times (p+1)m}$ is a non-singular matrix."
],
[
"Roadmap for convergence analysis",
"Similar as Section REF , we prove the convergence of prototype algorithmic framework (REF ) for the VI (REF ) and set up a roadmap for the convergence analysis.",
"Theorem 6.1 For the matrices ${{\\mbox{${\\cal Q}$}}}$ in (REF ) and ${\\mbox{${\\cal M}$}}$ in (REF ), if there is a positive definite matrix ${\\mbox{${\\cal H}$}}\\in \\Re ^{(p+1)m\\times (p+1)m}$ such that $ {\\mbox{${\\cal H}$}}{\\mbox{${\\cal M}$}}={{\\mbox{${\\cal Q}$}}} $ and $ {\\mbox{${\\cal G}$}}: = {{\\mbox{${\\cal Q}$}}}^T + {{\\mbox{${\\cal Q}$}}} - {\\mbox{${\\cal M}$}}^T{\\mbox{${\\cal H}$}}{\\mbox{${\\cal M}$}}\\succ 0, $ then we have $ \\Vert {\\xi }^{k+1} -{\\xi }^*\\Vert _{{\\mbox{${\\cal H}$}}}^2\\le \\Vert {\\xi }^k -{\\xi }^*\\Vert _{{\\mbox{${\\cal H}$}}}^2 - \\Vert {\\xi }^k - \\tilde{\\xi }^k \\Vert _{{\\mbox{${\\cal G}$}}}^2,\\quad \\forall \\, \\xi ^*\\in {\\Xi ^*}.",
"$ Then, analogous to the analysis in Section REF , convergence of the prototype algorithmic framework (REF ) can be established easily.",
"We summarize the convergence result in the following theorem, and skip the proof.",
"Theorem 6.2 Let $\\lbrace \\xi ^k\\rbrace $ be the sequence generated by the prototype algorithmic framework (REF ).",
"If the conditions (REF ) and (REF ) are satisfied, then the sequence $\\lbrace \\xi ^k\\rbrace $ converges to some $\\xi ^{\\infty } \\in \\Xi ^*$ ."
],
[
"Some useful matrices",
"In order to simplify the notations to be used, we define the following $p\\times p$ block matrices: $ {\\cal L} = \\left(\\begin{array}{cccc}I_m & 0 & \\cdots & 0 \\\\[0.1cm]I_m & I_m & \\ddots & \\vdots \\\\[0.1cm]\\vdots & & \\ddots & 0 \\\\[0.1cm]I_m & I_m & \\cdots & I_m\\end{array}\\!\\!\\right)\\qquad \\;\\hbox{and}\\; \\qquad {\\cal I} =\\left(\\begin{array}{cccc}I_m & 0 & \\cdots & 0 \\\\[0.1cm]0 & I_m & \\ddots & \\vdots \\\\[0.1cm]\\vdots & \\ddots & \\ddots & 0 \\\\[0.1cm]0 & \\cdots & 0 & I_m\\end{array}\\!\\!\\right).$ We also define the $1\\times p$ block matrix $ {\\cal E} = \\left(\\!\\begin{array}{cccc}I_m & I_m & \\cdots & I_m\\end{array}\\!\\!\\right).",
"$ Then, it is easy to see the following properties: $ {\\cal L}^{-1} = \\left(\\begin{array}{cccc}I_m & 0 & \\cdots & 0 \\\\[0.1cm]-I_m & I_m & \\ddots & \\vdots \\\\[0.1cm]0 & \\ddots & \\ddots & 0 \\\\[0.1cm]0 & 0 & -I_m & I_m\\end{array}\\!\\!\\right)\\quad \\hbox{and} \\quad {\\cal L}^T + {\\cal L} = {\\cal I} + {\\cal E}^T {\\cal E}.",
"$ These matrices will be used in our analysis."
],
[
"Primal-dual extension of the ADMM (",
"This section is parallel to Section .",
"We consider a concrete algorithm for the multiple-block model (REF ), in which the primal variables $x_i$ ($i=1,\\ldots ,p$ ) are updated first before the dual variable $\\lambda $ .",
"We show that it can be specified from the prototype algorithmic framework (REF ).",
"The penalty parameter is still denoted by $\\beta >0$ ."
],
[
"Algorithm",
" A Primal-Dual Extension of the ADMM (REF ) for (REF ).",
"(Prediction Step) With given $(A_1x_1^{k},A_2x_2^{k},\\cdots , A_px_p^{k}, \\lambda ^{k})$ , find $\\tilde{w}^k\\in \\Omega $ via $\\left\\lbrace \\begin{array}{l}\\tilde{x}_1^k \\in \\arg \\min \\bigl \\lbrace \\theta _1(x_1) -x_1^TA_1^T\\lambda ^k +\\frac{\\beta }{2} \\Vert A_1(x_1-x_1^k)\\Vert ^2 \\;|\\; x_1\\in {\\cal X}_1 \\bigr \\rbrace ;\\\\[0.2cm]\\tilde{x}_2^k \\in \\arg \\min \\bigl \\lbrace \\theta _2(x_2) -x_2^TA_2^T\\lambda ^k +\\frac{\\beta }{2} \\Vert A_1(\\tilde{x}_1^k-x_1^k)+ A_2(x_2-x_2^k)\\Vert ^2 \\;|\\; x_2\\in {\\cal X}_2 \\bigr \\rbrace ;\\\\[0.1cm]\\qquad \\qquad \\vdots \\\\\\tilde{x}_i^k\\in \\arg \\min \\bigl \\lbrace \\theta _i(x_i) -x_i^TA_i^T\\lambda ^k +\\frac{\\beta }{2} \\Vert \\sum _{j=1}^{i-1}A_j(\\tilde{x}_j^k-x_j^k)+ A_i(x_i-x_i^k)\\Vert ^2 \\;|\\; x_i\\in {\\cal X}_i \\bigr \\rbrace ;\\\\[0.1cm]\\qquad \\qquad \\vdots \\\\\\tilde{x}_{p}^k \\in \\arg \\min \\bigl \\lbrace \\theta _{p}(x_{p}) -x_{p}^TA_{p}^T\\lambda ^k +\\frac{\\beta }{2} \\Vert \\sum _{j=1}^{p-1}A_j(\\tilde{x}_j^k-x_j^k)+ A_{p}(x_{p}-x_{p}^k)\\Vert ^2\\;|\\; x_p\\in {\\cal X}_{p} \\bigr \\rbrace ; \\\\[0.3cm]\\tilde{\\lambda }^k =\\arg \\max \\bigl \\lbrace - \\lambda ^T\\bigl (\\sum _{j=1}^{p} A_j\\tilde{x}_j^k -b\\bigr ) -\\frac{1}{2\\beta }\\Vert \\lambda -\\lambda ^k\\Vert ^2 \\;|\\; \\lambda \\in \\Lambda \\bigr \\rbrace .\\end{array}\\right.$ (Correction Step) Correct the predictor $\\tilde{w}^k$ solved by (REF ), and generate the new iterate $(A_1x_1^{k+1}, A_2x_2^{k+1}, \\cdots , A_px_p^{k+1}, \\lambda ^{k+1})$ with $\\nu \\in (0,1)$ by $ \\left(\\!\\!\\begin{array}{c}A_1x_1^{k+1} \\\\[0.1cm]A_2x_2^{k+1} \\\\[0.1cm]\\vdots \\\\[0.1cm]A_px_p^{k+1} \\\\[0.1cm]\\lambda ^{k+1}\\end{array}\\!\\!\\right) =\\left(\\!\\!\\begin{array}{c}A_1x_1^{k} \\\\[0.1cm]A_2x_2^{k} \\\\[0.1cm]\\vdots \\\\[0.1cm]A_px_p^{k} \\\\[0.1cm]\\lambda ^{k}\\end{array}\\!\\!\\right)-\\left(\\!\\!\\begin{array}{ccccc}\\nu I_m & -\\nu I_m & 0 & \\cdots & 0 \\\\[0.1cm]0 & \\nu I_m & \\ddots & \\ddots & \\vdots \\\\[0.1cm]\\vdots & \\ddots & \\ddots & -\\nu I_m & 0\\\\[0.1cm]0 & \\cdots & 0 &\\nu I_m & 0\\\\[0.1cm]- \\nu \\beta I_m & 0& \\cdots & 0 & I_m\\end{array}\\!\\!\\right)\\left(\\!\\!\\begin{array}{c}A_1x_1^{k} -A_1\\tilde{x}_1^{k} \\\\[0.1cm]A_2x_2^{k} - A_2\\tilde{x}_2^{k}\\\\[0.1cm]\\vdots \\\\[0.1cm]A_px_p^{k} - A_p\\tilde{x}_p^{k}\\\\[0.1cm]\\lambda ^{k} -\\tilde{\\lambda }^k\\end{array}\\!\\!\\right).$ Remark 7.1 The algorithm (REF ) keeps the main features and structures of various ADMM's extensions in the literature for multiple-block separable convex optimization problems with linear equality constraints, see, e.g., [19], [20], [23].",
"The subproblems in the prediction step (REF ) treat each $\\theta _i$ individually; they are of the same form as those in (REF ) or (REF ).",
"The correction step (REF ) also treats $A_ix_i$ and $A_i{\\tilde{x}}_i$ , $i=1,\\cdots ,p$ , aggregately.",
"Hence, it also requires ignorable computation with only few floating-point additions."
],
[
"Specification of the prototypical algorithmic framework (",
"Now, we show that the algorithm (REF ) can be obtained by specifying the prototype algorithmic framework (REF ).",
"That is, we identify the specific matrices ${\\mbox{${\\cal Q}$}}$ and ${\\mbox{${\\cal M}$}}$ in (REF ) and (REF ) such that (REF ) and (REF ) can be reduced to the prediction step (REF ) and the correction step (REF ), respectively.",
"The specified matrices corresponding to the algorithm (REF ) are denoted by ${\\mbox{${\\cal Q}$}}_{{\\mbox{\\footnotesize $_{\\!",
"{P\\!D}}$}}}$ and ${{\\mbox{${\\cal M}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{P\\!D}}$}}}$ , respectively.",
"We also divide the discussion into two subsections."
],
[
"Analysis for the prediction step (",
"Similar as Section REF , for the predictor $\\tilde{w}^k$ generated by (REF ), we have $\\theta (x) - \\theta (\\tilde{x}^k) +(w- \\tilde{w}^k)^T F(\\tilde{w}^k) \\ge (w- \\tilde{w}^k)^TQ_{{\\mbox{\\footnotesize $_{\\!",
"{P\\!D}}$}}}(w^k - \\tilde{w}^k),\\quad \\forall \\, w\\in {\\Omega },$ where $ Q_{{\\mbox{\\footnotesize $_{\\!",
"{P\\!D}}$}}} = \\left(\\begin{array}{ccccc}\\beta A_1^TA_1 & 0 & \\cdots & 0 & A_1^T \\\\[0.3cm]\\beta A_2^TA_1 & \\beta A_2^TA_2 & \\ddots & \\vdots & A_2^T \\\\[0.3cm]\\vdots & & \\ddots & 0 & \\vdots \\\\[0.3cm]\\beta A_{p}^TA_1 & \\beta A_{p}^TA_2 & \\cdots &\\beta A_{p}^TA_{p} & A_{p}^T\\\\[0.3cm]0 & 0& \\cdots & 0 & \\frac{1}{\\beta }I_{m}\\end{array}\\!\\!\\right).",
"$ Using the notations $P$ and $\\xi $ in (REF ), and the notations ${\\cal L}$ and ${\\cal E}$ in (REF ) and (REF ), we can rewrite the matrix $Q_{{\\mbox{\\footnotesize $_{\\!",
"{P\\!D}}$}}}$ in (REF ) as $ Q_{{\\mbox{\\footnotesize $_{\\!",
"{P\\!D}}$}}}= P^T {{\\mbox{${\\cal Q}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{P\\!D}}$}}} P,\\quad \\hbox{where} \\quad {{\\mbox{${\\cal Q}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{P\\!D}}$}}} = \\left(\\begin{array}{ccccc}I_m & 0 & \\cdots & 0 & I_m \\\\[0.1cm]I_m & I_m & \\ddots & \\vdots & I_m \\\\[0.1cm]\\vdots & & \\ddots & 0 & \\vdots \\\\[0.1cm]I_m & I_m & \\cdots & I_m & I_m\\\\[0.1cm]0 & 0& \\cdots & 0 & I_m\\end{array}\\!\\!\\right) =\\left(\\begin{array}{cc}{\\cal L} & {\\cal E}^T\\\\[0.1cm]0 & I_m\\end{array}\\!\\!\\right).$ Then, it follows from (REF ) that we have the following inequality similar as (REF ): $ \\theta (x) - \\theta (\\tilde{x}^k) + (w - \\tilde{w}^k)^T F(\\tilde{w}^k) \\ge (\\xi -\\tilde{\\xi }^k)^T{{\\mbox{${\\cal Q}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{P\\!D}}$}}}(\\xi ^k-\\tilde{\\xi }^k), \\quad \\forall \\, w \\in {\\Omega }.$ Thus, the prediction step (REF ) can be specified by the prototypical prediction step (REF ) with ${\\mbox{${\\cal Q}$}}:= {{\\mbox{${\\cal Q}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{P\\!D}}$}}}$ as defined in (REF )."
],
[
"Analysis for the correction step (",
"Left-multiplying the matrix $\\hbox{diag}(\\sqrt{\\beta }I_m,\\ldots ,\\sqrt{\\beta }I_m, \\frac{1}{\\sqrt{\\beta }}I_m)$ to both sides of the correction step (REF ), we get $\\left(\\!\\!\\!\\begin{array}{c}\\sqrt{\\beta }A_1x_1^{k+1} \\\\[0.1cm]\\sqrt{\\beta }A_2x_2^{k+1} \\\\[0.1cm]\\vdots \\\\[0.1cm]\\sqrt{\\beta }A_px_p^{k+1} \\\\[0.1cm]\\frac{1}{\\sqrt{\\beta }}\\lambda ^{k+1}\\end{array}\\!\\!\\!\\right) =\\left(\\!\\!\\!\\begin{array}{c}\\sqrt{\\beta }A_1x_1^{k} \\\\[0.1cm]\\sqrt{\\beta }A_2x_2^{k} \\\\[0.1cm]\\vdots \\\\[0.1cm]\\sqrt{\\beta }A_px_p^{k} \\\\[0.1cm]\\frac{1}{\\sqrt{\\beta }}\\lambda ^{k}\\end{array}\\!\\!\\!\\right)-\\left(\\!\\!\\!\\begin{array}{ccccc}\\nu \\sqrt{\\beta } I_m & -\\nu \\sqrt{\\beta } I_m & 0 & \\cdots & 0 \\\\[0.1cm]0 & \\nu \\sqrt{\\beta } I_m & \\ddots & \\ddots & \\vdots \\\\[0.1cm]\\vdots & \\ddots & \\ddots & -\\nu \\sqrt{\\beta } I_m & 0\\\\[0.1cm]0 & \\cdots & 0 &\\nu \\sqrt{\\beta } I_m & 0\\\\[0.1cm]- \\nu \\sqrt{\\beta } I_m & 0& \\cdots & 0 & \\frac{1}{\\sqrt{\\beta }} I_m\\end{array}\\!\\!\\!\\right)\\left(\\!\\!\\!\\begin{array}{c}A_1x_1^{k} -A_1\\tilde{x}_1^{k} \\\\[0.2cm]A_2x_2^{k} - A_2\\tilde{x}_2^{k}\\\\[0.2cm]\\vdots \\\\[0.2cm]A_px_p^{k} - A_p\\tilde{x}_p^{k}\\\\[0.2cm]\\lambda ^{k} -\\tilde{\\lambda }^k\\end{array}\\!\\!\\!\\right).$ It can be written as $ \\left(\\!\\!\\begin{array}{c}\\sqrt{\\beta }A_1x_1^{k+1} \\\\[0.1cm]\\sqrt{\\beta }A_2x_2^{k+1} \\\\[0.1cm]\\vdots \\\\[0.1cm]\\sqrt{\\beta }A_px_p^{k+1} \\\\[0.1cm]\\frac{1}{\\sqrt{\\beta }}\\lambda ^{k+1}\\end{array}\\!\\!\\right) =\\left(\\!\\!\\begin{array}{c}\\sqrt{\\beta }A_1x_1^{k} \\\\[0.1cm]\\sqrt{\\beta }A_2x_2^{k} \\\\[0.1cm]\\vdots \\\\[0.1cm]\\sqrt{\\beta }A_px_p^{k} \\\\[0.1cm]\\frac{1}{\\sqrt{\\beta }}\\lambda ^{k}\\end{array}\\!\\!\\right)-\\left(\\begin{array}{ccccc}\\nu I_m & -\\nu I_m & 0 & \\cdots & 0 \\\\[0.1cm]0 & \\nu I_m & \\ddots & \\ddots & \\vdots \\\\[0.1cm]\\vdots & \\ddots & \\ddots & -\\nu I_m & 0\\\\[0.1cm]0 & \\cdots & 0 &\\nu I_m & 0\\\\[0.1cm]- \\nu I_m & 0& \\cdots & 0 & I_m\\end{array}\\!\\!\\right)\\left(\\!\\!\\begin{array}{c}\\sqrt{\\beta }(A_1x_1^{k} -A_1\\tilde{x}_1^{k}) \\\\[0.1cm]\\sqrt{\\beta }(A_2x_2^{k} - A_2\\tilde{x}_2^{k})\\\\[0.1cm]\\vdots \\\\[0.1cm]\\sqrt{\\beta }(A_px_p^{k} - A_p\\tilde{x}_p^{k})\\\\[0.1cm]\\frac{1}{\\sqrt{\\beta }} (\\lambda ^{k} -\\tilde{\\lambda }^k)\\end{array}\\!\\!\\right).$ Recall the respective definitions ${\\cal L}$ and ${\\cal E}$ in (REF ) and (REF ).",
"We have ${\\cal L}^{-T} = \\left(\\begin{array}{cccc}I_m & - I_m & 0 & 0 \\\\[0.1cm]0 & I_m & \\ddots & 0 \\\\[0.1cm]\\vdots & \\ddots & \\ddots & - I_m \\\\[0.1cm]0 & \\cdots & 0 & I_m \\\\[0.1cm]\\end{array}\\!\\!\\right) \\qquad \\hbox{and} \\qquad {\\cal E} {\\cal L}^{-T}= \\left(\\!\\begin{array}{cccc}I_m & 0 & \\cdots & 0\\end{array}\\!\\!\\right).",
"$ Thus, using the notations in (REF ), we can rewrite the correction step (REF ) as $ {\\xi }^{k+1} = {\\xi }^k - {{\\mbox{${\\cal M}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{P\\!D}}$}}}(\\xi ^k - \\tilde{\\xi }^k), $ where $ {{\\mbox{${\\cal M}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{P\\!D}}$}}} = \\left(\\begin{array}{cc}{\\nu }{\\cal L}^{-T} & 0\\\\[0.2cm]-\\nu {\\cal E} {\\cal L}^{-T} & I_m\\end{array}\\right).$ Thus, the correction step (REF ) can be specified by the prototypical correction step (REF ) with ${\\mbox{${\\cal M}$}}:= {{\\mbox{${\\cal M}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{P\\!D}}$}}}$ as defined in (REF )."
],
[
"Convergence",
"Then, according to the roadmap presented in Section REF , proving the convergence of the algorithm (REF ) can be reduced to verifying the conditions (REF ) and (REF ) with the specified matrices ${\\mbox{${\\cal Q}$}}_{{\\mbox{\\footnotesize $_{\\!",
"{P\\!D}}$}}}$ and ${{\\mbox{${\\cal M}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{P\\!D}}$}}}$ .",
"That is, the remaining task is to find a positive definite matrix ${{\\mbox{${\\cal H}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{P\\!D}}$}}}$ such that $ {{\\mbox{${\\cal H}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{P\\!D}}$}}}{{\\mbox{${\\cal M}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{P\\!D}}$}}}={{\\mbox{${\\cal Q}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{P\\!D}}$}}} \\qquad \\hbox{and}\\qquad {{\\mbox{${\\cal Q}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{P\\!D}}$}}}^T + {{\\mbox{${\\cal Q}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{P\\!D}}$}}} - {{\\mbox{${\\cal M}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{P\\!D}}$}}}^T{{\\mbox{${\\cal H}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{P\\!D}}$}}}{{\\mbox{${\\cal M}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{P\\!D}}$}}} \\succ 0, $ where ${{\\mbox{${\\cal Q}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{P\\!D}}$}}}$ and ${{\\mbox{${\\cal M}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{P\\!D}}$}}}$ are given by (REF ) and (REF ), respectively.",
"Lemma 7.1 For the matrices ${{\\mbox{${\\cal Q}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{P\\!D}}$}}}$ and ${{\\mbox{${\\cal M}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{P\\!D}}$}}}$ given by (REF ) and (REF ), respectively, the matrix $ {{\\mbox{${\\cal H}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{P\\!D}}$}}} = \\left(\\begin{array}{cc}\\frac{1}{\\nu }{\\cal L}{\\cal L}^T + {\\cal E}^T{\\cal E} & {\\cal E}^T \\\\[0.2cm]{\\cal E} & I_m\\end{array}\\right) \\quad \\hbox{with} \\quad \\nu \\in (0,1) $ is positive definite, and it satisfies ${{\\mbox{${\\cal H}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{P\\!D}}$}}}{{\\mbox{${\\cal M}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{P\\!D}}$}}} = {{\\mbox{${\\cal Q}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{P\\!D}}$}}} $ .",
"Proof.",
"It is easy to check the positive definiteness of $ {{\\mbox{${\\cal H}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{P\\!D}}$}}}$ .",
"In addition, for the block matrix ${{\\mbox{${\\cal Q}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{P\\!D}}$}}}$ in (REF ), we have ${{\\mbox{${\\cal H}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{P\\!D}}$}}} {{\\mbox{${\\cal M}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{P\\!D}}$}}} &= & \\left(\\begin{array}{cc}\\frac{1}{\\nu }{\\cal L}{\\cal L}^T + {\\cal E}^T{\\cal E} & {\\cal E}^T \\\\[0.2cm]{\\cal E} & I_m\\end{array}\\right) \\left(\\begin{array}{cc}{\\nu }{\\cal L}^{-T} & 0\\\\[0.2cm]-\\nu {\\cal E} {\\cal L}^{-T} & I_m\\end{array}\\right) \\nonumber \\\\&= &\\left(\\begin{array}{cc}{\\cal L} & {\\cal E}^T\\\\[0.1cm]0 & I_m\\end{array}\\!\\!\\right)={{\\mbox{${\\cal Q}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{P\\!D}}$}}}.$ The assertions of this lemma are proved.",
"$\\Box $ Lemma 7.2 Let $ {{\\mbox{${\\cal Q}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{P\\!D}}$}}}$ , ${{\\mbox{${\\cal M}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{P\\!D}}$}}}$ and ${{\\mbox{${\\cal H}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{P\\!D}}$}}}$ be defined in (REF ), (REF ) and (REF ), respectively.",
"Then the matrix $ {{\\mbox{${\\cal G}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{P\\!D}}$}}} := ({{\\mbox{${\\cal Q}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{P\\!D}}$}}}^T + {{\\mbox{${\\cal Q}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{P\\!D}}$}}}) -{{\\mbox{${\\cal M}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{P\\!D}}$}}}^T{{\\mbox{${\\cal H}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{P\\!D}}$}}}{{\\mbox{${\\cal M}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{P\\!D}}$}}} $ is positive definite.",
"Proof.",
"By elementary matrix multiplications, we know that ${{\\mbox{${\\cal M}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{P\\!D}}$}}}^T{{\\mbox{${\\cal H}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{P\\!D}}$}}}{{\\mbox{${\\cal M}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{P\\!D}}$}}} = {{\\mbox{${\\cal Q}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{P\\!D}}$}}}^T {{\\mbox{${\\cal M}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{P\\!D}}$}}} =\\left(\\begin{array}{cc}{\\cal L}^T & 0\\\\[0.1cm]{\\cal E} & I_m\\end{array}\\!\\!\\right) \\left(\\begin{array}{cc}{\\nu }{\\cal L}^{-T} & 0\\\\[0.2cm]-\\nu {\\cal E} {\\cal L}^{-T} & I_m\\end{array}\\right) = \\left(\\begin{array}{cc}{\\nu }{\\cal I} & 0 \\\\[0.1cm]0 & I_{m}\\end{array}\\right).$ Then, it follows from $ {\\cal L}^T +{\\cal L} = {\\cal I} + {\\cal E}^T{\\cal E}$ (see (REF )-(REF )) that ${{\\mbox{${\\cal G}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{P\\!D}}$}}} &= & ( {{\\mbox{${\\cal Q}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{P\\!D}}$}}}^T + {{\\mbox{${\\cal Q}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{P\\!D}}$}}}) -{{\\mbox{${\\cal M}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{P\\!D}}$}}}^T{{\\mbox{${\\cal H}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{P\\!D}}$}}}{{\\mbox{${\\cal M}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{P\\!D}}$}}} \\nonumber \\\\[0.1cm]& = & \\left(\\begin{array}{cc}{\\cal L}^T +{\\cal L} & {\\cal E}^T\\\\[0.1cm]{\\cal E} & 2 I_m\\end{array}\\!\\!\\right)-\\left(\\begin{array}{cc}{\\nu }{\\cal I} & 0 \\\\[0.1cm]0 & I_{m}\\end{array}\\right) = \\left(\\begin{array}{cc}(1-\\nu ){\\cal I} + {\\cal E}^T {\\cal E} & {\\cal E}^T\\\\[0.1cm]{\\cal E} & I_m\\end{array}\\!\\!\\right).$ Thus, the matrix $ {{\\mbox{${\\cal G}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{P\\!D}}$}}}$ is positive definite for any $\\nu \\in (0,1)$ .",
"$\\Box $ Then, according to Theorems REF and REF , the convergence of the algorithm (REF ) can be obtained.",
"We skip the proof for succinctness."
],
[
"Dual-primal extension of the ADMM (",
"Similar as Section , we can also consider an extension of the ADMM (REF ) which updates the dual variables $\\lambda $ first and then updates the primal variables $x_i$ , $i=1,\\ldots ,p$ .",
"The resulting algorithm is called a dual-primal extension of the ADMM (REF ) for (REF ).",
"We show that it can also be obtained by specifying the prototype algorithmic framework (REF )."
],
[
"Algorithm",
"A dual-primal extension of the ADMM (REF ) for (REF ) is presented as follows.",
"A Dual-Primal Extension of the ADMM (REF ) for (REF ).",
"(Prediction Step) With given $(A_1x_1^{k},A_2x_2^{k},\\cdots , A_px_p^{k}, \\lambda ^{k})$ , generate $\\tilde{w}^k\\in \\Omega $ via $\\left\\lbrace \\begin{array}{l}\\tilde{\\lambda }^k =\\arg \\max \\bigl \\lbrace - \\lambda ^T\\bigl (\\sum _{j=1}^{p} A_j{x}_j^k -b\\bigr ) - \\frac{1}{2\\beta }\\Vert \\lambda -\\lambda ^k\\Vert ^2 \\;|\\; \\lambda \\in \\Lambda \\bigr \\rbrace ; \\\\[0.2cm]\\tilde{x}_1^k \\in \\arg \\min \\bigl \\lbrace \\theta _1(x_1) -x_1^TA_1^T\\tilde{\\lambda }^k +\\frac{\\beta }{2} \\Vert A_1(x_1-x_1^k)\\Vert ^2 \\;|\\; x_1\\in {\\cal X}_1 \\bigr \\rbrace ; \\\\[0.2cm]\\tilde{x}_2^k \\in \\arg \\min \\bigl \\lbrace \\theta _2(x_2) -x_2^TA_2^T\\tilde{\\lambda }^k +\\frac{\\beta }{2} \\Vert A_1(\\tilde{x}_1^k-x_1^k) + A_2(x_2-x_2^k)\\Vert ^2 \\;|\\; x_2\\in {\\cal X}_2 \\bigr \\rbrace ; \\\\[0.1cm]\\qquad \\qquad \\vdots \\\\\\tilde{x}_i^k\\in \\arg \\min \\bigl \\lbrace \\theta _i(x_i) -x_i^TA_i^T\\tilde{\\lambda }^k +\\frac{\\beta }{2} \\Vert \\sum _{j=1}^{i-1}A_j(\\tilde{x}_j^k-x_j^k)+ A_i(x_i-x_i^k)\\Vert ^2 \\;|\\; x_i\\in {\\cal X}_i \\bigr \\rbrace ;\\\\[0.1cm]\\qquad \\qquad \\vdots \\\\\\tilde{x}_{p}^k \\in \\arg \\min \\bigl \\lbrace \\theta _{p}(x_{p}) -x_{p}^TA_{p}^T\\tilde{\\lambda }^k +\\frac{\\beta }{2} \\Vert \\sum _{j=1}^{p-1}A_j(\\tilde{x}_j^k-x_j^k)+ A_{p}(x_{p}-x_{p}^k)\\Vert ^2 \\;|\\; x_p\\in {\\cal X}_{p} \\bigr \\rbrace .\\end{array}\\right.$ (Correction Step) Correct the predictor $\\tilde{w}^k$ solved by (REF ), and generate the new iterate $(A_1x_1^{k+1}, A_2x_2^{k+1}, \\cdots , A_px_p^{k+1}, \\lambda ^{k+1})$ with $\\nu \\in (0,1)$ by $ \\left(\\!\\begin{array}{c}A_1x_1^{k+1} \\\\[0.1cm]A_2x_2^{k+1} \\\\[0.1cm]\\vdots \\\\[0.1cm]A_px_p^{k+1} \\\\[0.1cm]\\lambda ^{k+1}\\end{array}\\!\\right) =\\left(\\!\\begin{array}{c}A_1x_1^{k} \\\\[0.1cm]A_2x_2^{k} \\\\[0.1cm]\\vdots \\\\[0.1cm]A_px_p^{k} \\\\[0.1cm]\\lambda ^{k}\\end{array}\\!\\right)-\\left(\\!\\!\\begin{array}{ccccc}\\nu I_m & -\\nu I_m & 0 & \\cdots & 0 \\\\[0.1cm]0 & \\nu I_m & \\ddots & \\ddots & \\vdots \\\\[0.1cm]\\vdots & \\ddots & \\ddots & -\\nu I_m & 0\\\\[0.1cm]0 & \\cdots & 0 &\\nu I_m & 0\\\\[0.1cm]- \\beta I_m & -\\beta I_m & \\cdots & -\\beta I_m & I_m\\end{array}\\!\\!\\right)\\left(\\!\\begin{array}{c}A_1x_1^{k} -A_1\\tilde{x}_1^{k} \\\\[0.1cm]A_2x_2^{k} - A_2\\tilde{x}_2^{k}\\\\[0.1cm]\\vdots \\\\[0.1cm]A_px_p^{k} - A_p\\tilde{x}_p^{k}\\\\[0.1cm]\\lambda ^{k} -\\tilde{\\lambda }^k\\end{array}\\!\\right).$ Remark 8.1 The algorithm (REF ) differs from the algorithm (REF ) slightly in the order of the update of variables.",
"All subproblems in the prediction step (REF ) are of the same difficulty as those in (REF ).",
"The correction step (REF ) also differs from (REF ) slightly in some entries of their corresponding matrices, and it also only requires ignorable computation."
],
[
"Specification of the prototypical algorithmic framework (",
"Now, we show that the algorithm (REF ) can also be obtained by specifying the prototype algorithmic framework (REF ).",
"The specified matrices ${\\mbox{${\\cal Q}$}}$ and ${\\mbox{${\\cal M}$}}$ in (REF ) and (REF ) are denoted by ${\\mbox{${\\cal Q}$}}_{{\\mbox{\\footnotesize $_{\\!",
"{D\\!P}}$}}}$ and ${{\\mbox{${\\cal M}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{D\\!P}}$}}}$ , respectively.",
"Again, we divide the discussion into two subsections."
],
[
"Analysis for the prediction step (",
"Similar as the analysis in Section REF , for the predictor $\\tilde{w}^k$ generated by (REF ), we have $\\theta (x) - \\theta (\\tilde{x}^k) +(w- \\tilde{w}^k)^T F(\\tilde{w}^k) \\ge (w- \\tilde{w}^k)^TQ_{{\\mbox{\\footnotesize $_{\\!",
"{D\\!P}}$}}}(w^k - \\tilde{w}^k),\\; \\;\\forall \\; w\\in {\\Omega },$ where $ Q_{{\\mbox{\\footnotesize $_{\\!",
"{D\\!P}}$}}} = \\left(\\begin{array}{ccccc}\\beta A_1^TA_1 & 0 & \\cdots & 0 & 0 \\\\[0.3cm]\\beta A_2^TA_1 & \\beta A_2^TA_2 & \\ddots & \\vdots & 0 \\\\[0.3cm]\\vdots & & \\ddots & 0 & \\vdots \\\\[0.3cm]\\beta A_{p}^TA_1 & \\beta A_{p}^TA_2 & \\cdots &\\beta A_{p}^TA_{p} & 0\\\\[0.3cm]- A_1 & - A_2& \\cdots & - A_p & \\frac{1}{\\beta }I_{m}\\end{array}\\!\\!\\right).",
"$ Using the notations $P$ and $\\xi $ in (REF ), and the notations ${\\cal L}$ and ${\\cal E}$ in (REF ) and (REF ), we can rewrite the matrix $Q_{{\\mbox{\\footnotesize $_{\\!",
"{D\\!P}}$}}}$ in (REF ) as $ Q_{{\\mbox{\\footnotesize $_{\\!",
"{D\\!P}}$}}}= P^T {{\\mbox{${\\cal Q}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{D\\!P}}$}}} P,\\quad \\hbox{where} \\quad {{\\mbox{${\\cal Q}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{D\\!P}}$}}} = \\left(\\begin{array}{ccccc}I_m & 0 & \\cdots & 0 & 0 \\\\[0.1cm]I_m & I_m & \\ddots & \\vdots & 0 \\\\[0.1cm]\\vdots & & \\ddots & 0 & \\vdots \\\\[0.1cm]I_m & I_m & \\cdots & I_m & 0\\\\[0.1cm]- I_m & - I_m & \\cdots & - I_m & I_m\\end{array}\\!\\!\\right) =\\left(\\begin{array}{cc}{\\cal L} & 0\\\\[0.1cm]-{\\cal E} & I_m\\end{array}\\!\\!\\right).$ Then, it follows from (REF ) that we have the following inequality similar as (REF ): $ \\theta (x) - \\theta (\\tilde{x}^k) + (w - \\tilde{w}^k)^T F(\\tilde{w}^k) \\ge (\\xi -\\tilde{\\xi }^k)^T{{\\mbox{${\\cal Q}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{D\\!P}}$}}}(\\xi ^k-\\tilde{\\xi }^k), \\quad \\forall \\, w \\in {\\Omega }.$ Thus, the prediction step (REF ) can be specified by the prototypical prediction step (REF ) with ${\\mbox{${\\cal Q}$}}:= {{\\mbox{${\\cal Q}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{D\\!P}}$}}}$ as defined in (REF )."
],
[
"Analysis for the correction step (",
"Left-multiplying the matrix $\\hbox{diag}(\\sqrt{\\beta }I_m,\\ldots ,\\sqrt{\\beta }I_m, \\frac{1}{\\sqrt{\\beta }}I_m)$ to both sides of the correction step (REF ), we get $\\left(\\!\\!\\!\\begin{array}{c}\\sqrt{\\beta }A_1x_1^{k+1} \\\\[0.1cm]\\sqrt{\\beta }A_2x_2^{k+1} \\\\[0.1cm]\\vdots \\\\[0.1cm]\\sqrt{\\beta }A_px_p^{k+1} \\\\[0.1cm]\\frac{1}{\\sqrt{\\beta }}\\lambda ^{k+1}\\end{array}\\!\\!\\!\\right) =\\left(\\!\\!\\!\\begin{array}{c}\\sqrt{\\beta }A_1x_1^{k} \\\\[0.1cm]\\sqrt{\\beta }A_2x_2^{k} \\\\[0.1cm]\\vdots \\\\[0.1cm]\\sqrt{\\beta }A_px_p^{k} \\\\[0.1cm]\\frac{1}{\\sqrt{\\beta }}\\lambda ^{k}\\end{array}\\!\\!\\!\\right)-\\left(\\!\\!\\!\\begin{array}{ccccc}\\nu \\sqrt{\\beta } I_m & -\\nu \\sqrt{\\beta } I_m & 0 & \\cdots & 0 \\\\[0.1cm]0 & \\nu \\sqrt{\\beta } I_m & \\ddots & \\ddots & \\vdots \\\\[0.1cm]\\vdots & \\ddots & \\ddots & -\\nu \\sqrt{\\beta } I_m & 0\\\\[0.1cm]0 & \\cdots & 0 &\\nu \\sqrt{\\beta } I_m & 0\\\\[0.1cm]- \\sqrt{\\beta } I_m & 0& \\cdots & - \\sqrt{\\beta } I_m & \\frac{1}{\\sqrt{\\beta }} I_m\\end{array}\\!\\!\\!\\right)\\left(\\!\\!\\!\\begin{array}{c}A_1x_1^{k} -A_1\\tilde{x}_1^{k} \\\\[0.2cm]A_2x_2^{k} - A_2\\tilde{x}_2^{k}\\\\[0.2cm]\\vdots \\\\[0.2cm]A_px_p^{k} - A_p\\tilde{x}_p^{k}\\\\[0.2cm]\\lambda ^{k} -\\tilde{\\lambda }^k\\end{array}\\!\\!\\!\\right).$ It can be written as $ \\left(\\!\\!\\begin{array}{c}\\sqrt{\\beta }A_1x_1^{k+1} \\\\[0.1cm]\\sqrt{\\beta }A_2x_2^{k+1} \\\\[0.1cm]\\vdots \\\\[0.1cm]\\sqrt{\\beta }A_px_p^{k+1} \\\\[0.1cm]\\frac{1}{\\sqrt{\\beta }}\\lambda ^{k+1}\\end{array}\\!\\!\\right) =\\left(\\!\\!\\begin{array}{c}\\sqrt{\\beta }A_1x_1^{k} \\\\[0.1cm]\\sqrt{\\beta }A_2x_2^{k} \\\\[0.1cm]\\vdots \\\\[0.1cm]\\sqrt{\\beta }A_px_p^{k} \\\\[0.1cm]\\frac{1}{\\sqrt{\\beta }}\\lambda ^{k}\\end{array}\\!\\!\\right)-\\left(\\!\\!\\begin{array}{ccccc}\\nu I_m & -\\nu I_m & 0 & \\cdots & 0 \\\\[0.1cm]0 & \\nu I_m & \\ddots & \\ddots & \\vdots \\\\[0.1cm]\\vdots & \\ddots & \\ddots & -\\nu I_m & 0\\\\[0.1cm]0 & \\cdots & 0 &\\nu I_m & 0\\\\[0.1cm]- I_m & 0& \\cdots & -I_m & I_m\\end{array}\\!\\!\\right)\\left(\\!\\!\\begin{array}{c}\\sqrt{\\beta }(A_1x_1^{k} -A_1\\tilde{x}_1^{k}) \\\\[0.1cm]\\sqrt{\\beta }(A_2x_2^{k} - A_2\\tilde{x}_2^{k})\\\\[0.1cm]\\vdots \\\\[0.1cm]\\sqrt{\\beta }(A_px_p^{k} - A_p\\tilde{x}_p^{k})\\\\[0.1cm]\\frac{1}{\\sqrt{\\beta }} (\\lambda ^{k} -\\tilde{\\lambda }^k)\\end{array}\\!\\!\\right).$ It follows from (REF ) and (REF ) that $ \\left(\\begin{array}{ccccc}\\nu I_m & -\\nu I_m & 0 & \\cdots & 0 \\\\[0.1cm]0 & \\nu I_m & \\ddots & \\ddots & \\vdots \\\\[0.1cm]\\vdots & \\ddots & \\ddots & -\\nu I_m & 0\\\\[0.1cm]0 & \\cdots & 0 &\\nu I_m & 0\\\\[0.1cm]- I_m & \\cdots & -I_m & -I_m & I_m\\end{array}\\!\\!\\right) = \\left(\\begin{array}{cc}{\\nu }{\\cal L}^{-T} & 0\\\\[0.2cm]- {\\cal E} & I_m\\end{array}\\right).$ Using the notations in (REF ), we can rewrite the correction (REF ) as $ {\\xi }^{k+1} = {\\xi }^k - {{\\mbox{${\\cal M}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{D\\!P}}$}}}(\\xi ^k - \\tilde{\\xi }^k), $ where $ {{\\mbox{${\\cal M}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{D\\!P}}$}}} = \\left(\\begin{array}{cc}{\\nu }{\\cal L}^{-T} & 0\\\\[0.2cm]- {\\cal E} & I_m\\end{array}\\right).$ Thus, the correction step (REF ) can be specified by the prototypical correction step (REF ) with ${\\mbox{${\\cal M}$}}:= {{\\mbox{${\\cal M}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{D\\!P}}$}}}$ as defined in (REF )."
],
[
"Convergence",
"Also, according to the roadmap presented in Section REF , proving the convergence of the algorithm (REF ) can be reduced to verifying the conditions (REF ) and (REF ) with the specified matrices ${{\\mbox{${\\cal Q}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{D\\!P}}$}}}$ and ${{\\mbox{${\\cal M}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{D\\!P}}$}}}$ .",
"That is, the remaining task is to find a positive definite matrix ${{\\mbox{${\\cal H}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{D\\!P}}$}}}$ such that $ {{\\mbox{${\\cal H}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{D\\!P}}$}}}{{\\mbox{${\\cal M}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{D\\!P}}$}}}={{\\mbox{${\\cal Q}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{D\\!P}}$}}} \\qquad \\hbox{and}\\qquad {{\\mbox{${\\cal Q}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{D\\!P}}$}}}^T + {{\\mbox{${\\cal Q}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{D\\!P}}$}}} - {{\\mbox{${\\cal M}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{D\\!P}}$}}}^T{{\\mbox{${\\cal H}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{D\\!P}}$}}}{{\\mbox{${\\cal M}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{D\\!P}}$}}} \\succ 0, $ where the matrices $ {{\\mbox{${\\cal Q}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{D\\!P}}$}}}$ and ${{\\mbox{${\\cal M}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{D\\!P}}$}}}$ are given by (REF ) and (REF ), respectively.",
"Lemma 8.1 For the matrices $ {{\\mbox{${\\cal Q}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{D\\!P}}$}}}$ and ${{\\mbox{${\\cal M}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{D\\!P}}$}}}$ given by (REF ) and (REF ), respectively, the matrix $ {{\\mbox{${\\cal H}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{D\\!P}}$}}} = \\left(\\begin{array}{cc}\\frac{1}{\\nu }{\\cal L}{\\cal L}^T & 0 \\\\[0.2cm]0 & I_m\\end{array}\\right) \\quad \\hbox{with} \\quad \\nu \\in (0,1) $ is positive definite, and it satisfies ${{\\mbox{${\\cal H}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{D\\!P}}$}}} {{\\mbox{${\\cal M}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{D\\!P}}$}}} = {{\\mbox{${\\cal Q}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{D\\!P}}$}}} $ .",
"Proof.",
"It is clear that ${{\\mbox{${\\cal H}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{D\\!P}}$}}}$ is positive definite.",
"In addition, for the block matrix ${{\\mbox{${\\cal Q}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{D\\!P}}$}}}$ in (REF ), we have ${{\\mbox{${\\cal H}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{D\\!P}}$}}} {{\\mbox{${\\cal M}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{D\\!P}}$}}} &= & \\left(\\begin{array}{cc}\\frac{1}{\\nu }{\\cal L}{\\cal L}^T & 0 \\\\[0.2cm]0 & I_m\\end{array}\\right) \\left(\\begin{array}{cc}{\\nu }{\\cal L}^{-T} & 0\\\\[0.2cm]- {\\cal E} & I_m\\end{array}\\right) \\;=\\; \\left(\\begin{array}{cc}{\\cal L} & 0 \\\\[0.1cm]-{\\cal E} & I_m\\end{array}\\!\\!\\right)={{\\mbox{${\\cal Q}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{D\\!P}}$}}}.$ The assertions of this lemma are proved.",
"$\\Box $ Lemma 8.2 Let $ {{\\mbox{${\\cal Q}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{D\\!P}}$}}}$ , ${{\\mbox{${\\cal M}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{D\\!P}}$}}}$ and ${{\\mbox{${\\cal H}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{D\\!P}}$}}}$ be defined in (REF ), (REF ) and (REF ), respectively.",
"Then the matrix $ {{\\mbox{${\\cal G}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{D\\!P}}$}}} := ({{\\mbox{${\\cal Q}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{D\\!P}}$}}}^T + {{\\mbox{${\\cal Q}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{D\\!P}}$}}}) -{{\\mbox{${\\cal M}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{D\\!P}}$}}}^T{{\\mbox{${\\cal H}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{D\\!P}}$}}}{{\\mbox{${\\cal M}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{D\\!P}}$}}} $ is positive definite.",
"Proof.",
"First, by elementary matrix multiplications, we get ${{\\mbox{${\\cal M}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{D\\!P}}$}}}^T{{\\mbox{${\\cal H}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{D\\!P}}$}}}{{\\mbox{${\\cal M}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{D\\!P}}$}}} = {{\\mbox{${\\cal Q}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{D\\!P}}$}}}^T {{\\mbox{${\\cal M}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{D\\!P}}$}}} \\nonumber =\\left(\\begin{array}{cc}{\\cal L}^T & - {\\cal E}^T\\\\[0.1cm]0 & I_m\\end{array}\\!\\!\\right) \\left(\\begin{array}{cc}{\\nu }{\\cal L}^{-T} & 0\\\\[0.2cm]-{\\cal E} & I_m\\end{array}\\right) = \\left(\\begin{array}{cc}{\\nu }{\\cal I} + {\\cal E}^T{\\cal E} & -{\\cal E}^T \\\\[0.1cm]- {\\cal E} & I_{m}\\end{array}\\right).$ Then, using $ {\\cal L}^T +{\\cal L} = {\\cal I} + {\\cal E}^T{\\cal E}$ (see (REF )-(REF )), we have ${{\\mbox{${\\cal G}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{D\\!P}}$}}} &= & ( {{\\mbox{${\\cal Q}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{D\\!P}}$}}}^T + {{\\mbox{${\\cal Q}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{D\\!P}}$}}}) -{{\\mbox{${\\cal M}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{D\\!P}}$}}}^T{{\\mbox{${\\cal H}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{D\\!P}}$}}}{{\\mbox{${\\cal M}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{D\\!P}}$}}} \\nonumber \\\\[0.1cm]& = & \\left(\\begin{array}{cc}{\\cal L}^T +{\\cal L} & - {\\cal E}^T\\\\[0.1cm]- {\\cal E} & 2 I_m\\end{array}\\!\\!\\right)- \\left(\\begin{array}{cc}{\\nu }{\\cal I} + {\\cal E}^T{\\cal E} & -{\\cal E}^T \\\\[0.1cm]- {\\cal E} & I_{m}\\end{array}\\right)= \\left(\\begin{array}{cc}(1-\\nu ){\\cal I} & 0\\\\[0.1cm]0 & I_m\\end{array}\\!\\!\\right).$ Thus, the matrix $ {{\\mbox{${\\cal G}$}}}_{{\\mbox{\\footnotesize $_{\\!",
"{D\\!P}}$}}}$ is positive definite for any $\\nu \\in (0,1)$ .",
"Then, according to Theorems REF and REF , the convergence of the algorithm (REF ) can be obtained.",
"We skip the proof for succinctness."
],
[
"Panorama",
"The ALM in [24], [27] was proposed for the nonseparable generic convex optimization problem with linear equality constraints which can be regarded as a one-block model, and as mentioned, the original ADMM (REF ) is an extension of the ALM by splitting the underlying augmented Lagrangian function twice when the two-block separable model (REF ) is considered.",
"However, as proved in [5], the same extension may fail in guaranteeing the convergence if a multiple-block generalized model of (REF ) is considered (recall (REF )).",
"Hence, it seems impossible to unify the algorithmic design and convergence analysis for the original ALM in [24], [27], the original ADMM (REF ) and its direct extensions, when the number of separable blocks of a convex optimization model with linear equality constraints increases from $p=1$ , $p=2$ , to $p\\ge 3$ .",
"On the other hand, it is easy to see that the algorithms (REF ) and (REF ) are direct extensions of the algorithms (REF ) and (REF ), respectively, when the model under discussion is changed from the two-block model (REF ) to its multiple-block generalized model (REF ).",
"Alternatively, the algorithms (REF ) and (REF ) are just special cases of the algorithms (REF ) and (REF ) with $p=2$ , respectively.",
"Hence, the algorithms (REF ) and (REF ) are eligible to the model (REF ) with different cases of $p\\ge 2$ .",
"Indeed, we can show that the algorithms (REF ) and (REF ) can also be applied to the following nonseparable generic convex optimization problem with linear equality or inequality constraints: $ \\min \\big \\lbrace \\theta (x) \\;|\\; A x =b\\ (\\hbox{or} \\ge b) , x\\in {\\cal X} \\big \\rbrace ,$ which can alternatively be regarded as the special case of (REF ) with $p=1$ .",
"For the model (REF ), let us define $ \\Lambda =\\left\\lbrace \\begin{array}{ll}\\Re ^m, & \\hbox{if $Ax = b$} , \\\\[0.1cm]\\Re ^m_+, & \\hbox{if $Ax \\ge b$}.\\end{array} \\right.$ Then, we can propose an algorithmic framework similar as (REF ) and (REF ) for the generic model (REF ), as well as a roadmap for convergence analysis similar as those in Sections REF and REF .",
"Specific algorithms can also be obtained similarly as what we have done in Sections , , and .",
"For succinctness, we only present some specific algorithms and skip other details.",
"A Primal-Dual Variant of the ALM for (REF ).",
"(Prediction Step) With given $(Ax^k, \\lambda ^k)$ , find $\\tilde{w}^k=(\\tilde{x}^k,\\tilde{\\lambda }^k)$ via $ \\left\\lbrace \\begin{array}{l}\\tilde{x}^k \\in \\hbox{argmin}\\bigl \\lbrace \\theta (x) - x^TA^T{\\lambda }^k + \\frac{1}{2}\\beta \\Vert A(x-x^k)\\Vert ^2 \\;|\\; x\\in {\\cal X}\\bigr \\rbrace , \\\\[0.2cm]\\tilde{\\lambda }^k= \\arg \\!\\max \\bigl \\lbrace -\\lambda ^T\\bigl (A\\tilde{x}^k-b\\bigr ) - \\frac{1}{2\\beta }\\Vert \\lambda -\\lambda ^k\\Vert ^2 \\;|\\; \\lambda \\in {\\Lambda }\\bigr \\rbrace .\\end{array} \\right.$ (Correction Step) Correct the predictor $\\tilde{w}^k$ solved by (REF ), and generate the new iterate $(Ax^{k+1}, \\lambda ^{k+1})$ with $\\nu \\in (0,1)$ by $ \\left(\\begin{array}{c}Ax^{k+1} \\\\[0.1cm]\\lambda ^{k+1}\\end{array}\\right) = \\left(\\begin{array}{c}Ax^{k} \\\\[0.1cm]\\lambda ^{k}\\end{array}\\right) -\\left(\\begin{array}{cc}{\\nu } I_m & 0 \\\\[0.1cm]- \\nu \\beta I_m & I_m\\end{array}\\right) \\left(\\begin{array}{c}Ax^{k} -A\\tilde{x}^{k} \\\\[0.1cm]\\lambda ^{k} -\\tilde{\\lambda }^k\\end{array}\\right).$ A Dual-Primal Variant of the ALM for (REF ).",
"(Prediction Step) With given $(Ax^k, \\lambda ^k)$ , find $\\tilde{w}^k=(\\tilde{x}^k, \\tilde{\\lambda }^k)$ via $ \\left\\lbrace \\begin{array}{l}\\tilde{\\lambda }^k= \\arg \\!\\max \\bigl \\lbrace -\\lambda ^T \\bigl (Ax^k +By^k-b\\bigr ) - \\frac{1}{2\\beta }\\Vert \\lambda -\\lambda ^k\\Vert ^2 \\;|\\; \\lambda \\in {\\Lambda }\\bigr \\rbrace , \\\\[0.2cm]\\tilde{x}^k \\in \\hbox{argmin}\\bigl \\lbrace \\theta (x) - x^TA^T\\tilde{\\lambda }^k+ \\frac{1}{2}\\beta \\Vert A(x-x^k)\\Vert ^2\\;|\\; x\\in {\\cal X}\\bigr \\rbrace .\\end{array} \\right.$ (Correction Step) Correct the predictor $\\tilde{w}^k$ generated by (REF ), and generate the new iterate $(Ax^{k+1}, \\lambda ^{k+1})$ with $\\nu \\in (0,1)$ by $ \\left(\\begin{array}{c}Ax^{k+1} \\\\[0.1cm]\\lambda ^{k+1}\\end{array}\\right) = \\left(\\begin{array}{c}Ax^{k} \\\\[0.1cm]\\lambda ^{k}\\end{array}\\right) -\\left(\\begin{array}{ccc}{\\nu } I_m & 0 \\\\[0.1cm]- \\beta I_m & I_m\\end{array}\\right) \\left(\\begin{array}{c}Ax^{k} -A\\tilde{x}^{k} \\\\[0.1cm]\\lambda ^{k} -\\tilde{\\lambda }^k\\end{array}\\right).$ It is easy to see that, when the special case of (REF ) with linear equality constraints is considered, the algorithms () and () differ from the classic ALM in [24], [27] only slightly in their prediction steps with some constant vectors and in their correction steps with ignorable computation.",
"They maintain all major features and structures of the ALM, but they can be used for the cases of (REF ) with both linear equality and inequality constraints.",
"In addition, the algorithms () and () can be rendered from the algorithms (REF ) and (REF ), respectively, by removing the $x_2$ -subproblems in (REF ) and (REF ) as well as the second rows and columns of the matrices in (REF ) and (REF ) correspondingly.",
"Thus, they are also included as special cases by the algorithms (REF ) and (REF ) with $p=1$ .",
"In a nutshell, the proposed algorithmic framework and roadmap for convergence analysis are uniformly eligible to the nonseparable generic convex optimization model (REF ), the two-block separable model (REF ), and its multiple-block generalized model (REF ) with an arbitrary $p$ .",
"The resulting algorithms maintain the same features and structures from stem to stern for various convex optimization models with different degrees of separability, in which both linear equality and inequality constraints can be included; and the convergence analysis can be unified by a common roadmap.",
"In this sense, our philosophy of algorithmic design and the roadmap for convergence analysis are panoramic and consistent."
],
[
"Conclusions",
"The classic alternating direction method of multipliers (ADMM) has been widely used for various convex optimization problems with linear equality constraints and two-block separable objective functions without coupled variables.",
"It is known that the ADMM cannot be directly extended to multiple-block (more than two blocks) separable convex optimization problems with linear equality constraints, while it is unknown whether or not it can be extended to two-block or multiple-block separable convex optimization problems with linear inequality constraints.",
"In this paper, we focus on extensions of the ADMM to both two-block and multiple-block separable convex optimization problems with either linear inequality or linear equality constraints, and propose prototypical algorithmic frameworks which can be specified as concrete algorithms for the targeted models.",
"The specified algorithms keep the major structures and features of the original ADMM, and only require very simple additional steps to guarantee the convergence.",
"We also establish standard roadmaps to prove the convergence of the proposed prototypical algorithmic frameworks without any extra conditions.",
"We show that, if we follow the roadmaps to derive the convergence of any algorithm specified from the proposed prototypical algorithmic frameworks, then essentially it only requires to specify two matrices and then to check the positive definiteness of another matrix.",
"Our analysis is comprehensive enough to uniformly cover the nonseparable generic model as well as the two-block and multiple-block separable convex optimization models, in which both the linear equality and linear inequality constraints can be included.",
"Our analysis only uses very elementary mathematics and hence it is understandable for laymen.",
"Our aim is to study possible extensions of the original ADMM from a high-level and methodological perspective; thus we do not present any experiment results.",
"As mentioned, the proposed prototypical algorithmic frameworks basically maintain all the major structures and features of the original ADMM (REF ) which account for its versatility and efficiency, while the additional correction steps are extremely simple in computation.",
"It is easy to empirically verify the efficiency of the algorithms specified from the proposed algorithmic frameworks.",
"For instance, we have tested more than ten benchmark application problems in various fields, including the least absolute shrinkage and selection operator [28], the $L_1$ regularized logistic regression problem [15], some basic total-variation-based image reconstruction problems in [4], the support vector machine in [9], the sparse inverse covariance selection model in [1], as well as a number of basic optimization models in [3] (including linear and quadratic programming problems, and the least absolute deviations problem).",
"These application problems can all be modelled as concrete applications of the model (REF ) and they have been well solved by the original ADMM (REF ) in the literatures.",
"For comparison purpose, we implemented the original codes provided by the respective authors and kept their respective well-tuned settings, including the values of the penalty parameter $\\beta $ , for implementing the prediction steps (REF ) and (REF ), and then simply set $\\nu =0.99$ for the correction steps (REF ) and (REF ).",
"It has been affirmatively verified by our experiments that the proposed algorithms (REF ) and (REF ) perform nearly the same as the original ADMM (REF ).",
"That is, the versatility and efficiency of the original ADMM (REF ) are completely maintained by the specified algorithms (REF ) and (REF ) if the special model (REF ) is considered.",
"Here, we opt to skip the tedious descriptions of various numerical results for succinctness.",
"The conclusion is that algorithms specified from the proposed prototypical algorithms frameworks are eligible to the more general models (REF ) and (REF ), while they can work as well as the original ADMM (REF ) if the special case (REF ) is considered.",
"We would like to emphasize that we mainly initiate the foundation of algorithmic design and convergence analysis on the ground of the original ADMM, and our target models are the most generic and abstract separable convex optimization models with linear equality or inequality constraints.",
"We do not further discuss how to modify, specify, or generalize an algorithm that can be specified from the proposed prototypical algorithmic frameworks for the sake of better taking advantage of the structures and properties of a specific application.",
"Hence, we do not discuss how to solve the resulting subproblems more efficiently or how to find better step sizes; nor do we investigate sharper convergence results such as worst-case convergence rates in terms of iteration complexity, various asymptotical convergence rates under different conditions, or other more challenging issues under additional assumptions on the objective functions, coefficient matrices, and/or others.",
"When a specific application problem is considered, it is possible to specify the proposed prototypical algorithmic frameworks as more application-tailored algorithms.",
"It is also possible to discuss how to combine other techniques with the prototypical algorithmic frameworks to obtain more attractive numerical schemes; such examples include acceleration schemes, inertial schemes, neural networks, stochastic/randomized techniques, and so on.",
"All these more detailed discussions are excluded in our discussion for succinctness.",
"Our focus is exclusively the discussion of extensions of the most fundamental ADMM (REF ) from the canonical two-block model (REF ) to its generalized two-block model (REF ) and multiple-block model (REF ), which can include both linear equality and inequality constraints."
]
] | 2107.01897 |
[
[
"GraspME -- Grasp Manifold Estimator"
],
[
"Abstract In this paper, we introduce a Grasp Manifold Estimator (GraspME) to detect grasp affordances for objects directly in 2D camera images.",
"To perform manipulation tasks autonomously it is crucial for robots to have such graspability models of the surrounding objects.",
"Grasp manifolds have the advantage of providing continuously infinitely many grasps, which is not the case when using other grasp representations such as predefined grasp points.",
"For instance, this property can be leveraged in motion optimization to define goal sets as implicit surface constraints in the robot configuration space.",
"In this work, we restrict ourselves to the case of estimating possible end-effector positions directly from 2D camera images.",
"To this extend, we define grasp manifolds via a set of key points and locate them in images using a Mask R-CNN backbone.",
"Using learned features allows generalizing to different view angles, with potentially noisy images, and objects that were not part of the training set.",
"We rely on simulation data only and perform experiments on simple and complex objects, including unseen ones.",
"Our framework achieves an inference speed of 11.5 fps on a GPU, an average precision for keypoint estimation of 94.5% and a mean pixel distance of only 1.29.",
"This shows that we can estimate the objects very well via bounding boxes and segmentation masks as well as approximate the correct grasp manifold's keypoint coordinates."
],
[
"Introduction",
"As humans share tasks with robots that are increasingly more autonomous, it will become essential to provide user interfaces or robot behaviors that allow to flexibly define the task objectives.",
"Hence in a human-robot collaborative manipulation task, knowledge of the entire object's grasp manifold (i.e.",
"suitable grasp candidates), provides a step in this direction (e.g.",
"shared autonomy).",
"Note that the rapid detection of grasp manifolds in image space can have other applications ranging from robot motion planning to character animation in video games.",
"In this paper, we a present a grasp manifold estimator GraspME, based on the Detectron2 framework [2].",
"Our model estimates the grasp manifolds, classifies the objects and computes their bounding boxes and segmentation masks all at the same time from a 2D image.",
"A outcomes of such a grasp manifold estimation is depicted in Fig.",
"REF .",
"We train our model by supervised learning on simulation data from an environment we develeopped in PyBullet [3].",
"We devised two sets of objects: the first with simple geometry and the second with more complex geometry from the 50 category subset of 3DNet [4].",
"Our simulation environment generates RGB, depth and segmentation images together with bounding boxes and grasp manifold keypoints for each object in a scene.",
"Figure: Predicted grasp manifoldsfor complex objects using our approach, depicted as black lines.Object detection and semantic segmentation as well as grasp point localization have been improved steadily by the vision community which has led to a large list of baselines, e.g.",
"Mask R-CNN [1] and several models based on its framework.",
"Solutions based on grasp point detection often either rely on predefined grasp points or trial and error learning, which often proves to be rather unstable on unseen objects.",
"An important aspect is often neglected, namely that for most objects, infinitely many grasp points exist instead of just a few predefined.",
"This amount of grasp points can typically be defined by a manifold on a given object, mostly depending on the object geometry.",
"Thus, our contribution consists of the introduction of the new problem setting of grasp manifold estimation on objects, GraspME, a framework for object detection and grasp manifold keypoints estimation from 2D images and a simulation environment to generate suitable scenes and data for this task.",
"Object's grasp manifold provide more knowledge about the scene than simple grasp points.",
"In human-robot collaboration this means providing more solutions for a handover.",
"In space sharing scenario this grasp manifold may lead to more reactive behavior fallingback to different grasp solutions if the human moves and thus minimally disrupting the human.",
"Another area that could benefit from grasp manifolds is Task and Motion Planning (TAMP).",
"For example, it could be used in Logic Geometric Programming [5], where optimization over continuously many grasp locations in a manifold - in contrast to fixing a specific grasp - could lead to a better trajectory with respect to the used control cost or even lead to more feasible and stable solutions of the problem.",
"This paper is structured as follows: In Section , we present relevant related work.",
"We then formulate the problem of grasp manifold estimation from images and introduce notation in Section .",
"Before introducing our GraspME framework and implementation in Section , we present the dataset we work with to train our estimator in Section .",
"Finally, Sections and present our experiments, results and conclusions."
],
[
"Related Work",
"Our work merges the two research fields of object detection and grasp point detection.",
"The first part is done by following Mask R-CNN [1] while we combine it with a keypoint detection approach for the second.",
"Mask R-CNN: Mask R-CNN is a successor of Faster R-CNN [6] and Fast R-CNN [7] and addresses the problem of object detection.",
"It consists of two parts, a backbone to generate region proposals and a head to solve the actual task.",
"To detect objects, Mask R-CNN's head is composed of two computation branches of which one is responsible for classifying the object and generating an axis aligned bounding box while the second branch estimates the object's segmentation mask.",
"Mask R-CNN has also been used before to detect unseen objects in simulation in [8].",
"Regarding keypoint estimation, Mask R-CNN suggested the extension of their framework with a third branch to estimate human poses using keypoints.",
"Our own framework extends this approach by applying it on a new problem of predicting grasp manifolds and their corresponding keypoints.",
"Grasp Point Detection: A common approach to detect grasp points is to predefine fixed points on objects and localize them to plan a grasp trajectory, e.g.",
"[9].",
"These methods often lack in generalizability, as they cannot work properly on unseen objects without grasp point labels.",
"Another possibility is to let the model learn to differentiate between good and bad grasp points and extrapolate the knowledge to unseen objects.",
"To be able to do this distinction, the model can either rely on predefined grasp points or more common by executing random grasps on the objects and learn by trial-and-error [10], [11], [12], [13].",
"A benchmark for object affordances has been recently introduced [14] to evaluate point cloud deep learning networks.",
"Our framework complements the aforementioned approaches by introducing manifolds that contain possible grasp points from which an algorithm can sample and execute a grasp.",
"Keypoint Detection: The topic of keypoint detection is often associated to Human Pose Estimation.",
"There are also several datasets for this problem, e.g.",
"MPII [15] or COCO [16].",
"Several approaches address Human Pose Estimation via keypoint detection and incorporating domain knowledge, e.g.",
"the COCO 2016 keypoint detection winner [17].",
"However, keypoints detection has lately also being used to tackle other problems such as improving the quality of image generation [18] or object detection by estimating the object's center as keypoint [19].",
"We use keypoints to describe the grasp manifolds and estimate them during inference.",
"Figure: The three object models in blue and transparent with the corresponding grasp manifold in red: cuboid, cylinder, capsule (left to right)."
],
[
"Definition",
"We formalize the concept of an object's grasp manifold as a region $GM$ with a continuous closed border.",
"Any point $p \\in GM$ defines a potential grasp on the object, such that the closing point of the gripper is the same as this point $p$ .",
"To simplify the problem, we approximate such a grasp manifold by a set of keypoints, to which we will refer to as grasp manifold keypoints $kp_i$ .",
"Typically, they are chosen as corner points that span the corresponding manifold, thus approximating it as close as possible."
],
[
"Object and Gripper Geometry",
"Depending on the object's geometry, the corresponding grasp manifold can be defined as a line or a whole surface (see next subsections).",
"We assume the usage of a parallel two finger gripper, i.e.",
"to perform a grasp using an object's grasp manifold, the gripper should be aligned parallel to the manifold such that the closing point of the gripper would be on the chosen grasp point on the manifold.",
"We expect that this can easily be extended to other types of gripper."
],
[
"Simple Objects",
"We assume that the simple objects have a lengthy shape, i.e.",
"one of the sides along the axes is longer than the other, i.e.",
"the object's main axis.",
"In most cases, the main axis defines already the grasp manifold, e.g.",
"for cylinders and capsules.",
"If we neglect any other situational circumstances, it is possible to perform a grasp on a cylinder or a capsule if the gripper is aligned parallel to the main axis of the object.",
"Therefore, the grasp manifold for capsules and cylinders is defined by the starting and the ending keypoint of the object's main axis going through the object's center, resulting in a line.",
"An example of the grasp manifold for these two object types can be seen in Fig.",
"REF as a red line.",
"For the cuboids, the manifold containing possible grasp points is much larger.",
"The main axis of the cuboid is still the key to define it.",
"However, it can expand in the upper and lower direction as well, creating a plane parallel to the cuboid's faces, as can be seen on the left in Fig.",
"REF in red.",
"For simplicity, we reduced the grasp manifold for cuboids to a line (the main axis), such that only two keypoints define the grasp manifold for simple objects."
],
[
"Complex Objects",
"For objects with less trivial geometries we defined the grasp manifold manually by specifying the corresponding keypoints.",
"To simplify the problem, we restrict ourselves to a maximum of $K$ keypoints per object which approximates the real grasp manifold.",
"The approximated grasp manifold is defined by connecting the sequence of keypoints $\\lbrace kp_i\\rbrace _{i=1}^{k}$ with $k \\in [2, K]$ , conditioned by the number of keypoints used to approximate the grasp manifold for the corresponding object.",
"The grasp manifold of objects like bananas or bottles is a line, while it is a surface for objects like cameras and guitars.",
"Figure: The complex object models taken from 3DNet .Our proposed model estimates the grasp manifold from 2D images via keypoints localization instead of directly computing the object's pose, as mentioned in .",
"Since this procedure is analog to human pose estimation, we use the method of Mask R-CNN [1] and the corresponding Detectron2 framework [2] as basis for our model."
],
[
"Architecture",
"Similar to Mask R-CNN, our framework also consists of two main parts.",
"First, a backbone model, the Region Proposal Network (RPN) from Faster R-CNN [6], is used on the whole image to generate region proposals.",
"These are fed into the network's region of interest (ROI) head for the actual task: the classification, the bounding box detection, the mask prediction and the keypoints estimation.",
"The overall framework architecture can be seen in Fig.",
"REF .",
"For the backbone, we use the ResNet-FPN variant from Mask R-CNN [1].",
"It consists of a ResNet [20] of depth 50, denoted as ResNet-50, or of depth 101, denoted as ResNet-101, with a Feature Pyramid Network (FPN) [21] on top.",
"The rest of the architecture follows the suggestions from Mask R-CNN and the Detectron2 framework [2] for the Human Pose Estimation.",
"For our experiments, we use only the 2D RGB images as input.",
"Depending on the object types, we set the number of keypoints to detect to $K = 2$ for the simple objects and to $K = 10$ for the complex objects.",
"We chose this number because we found that we do not need more than 10 to approximate manifolds for our objects.",
"Extending to more keypoints would be possible.",
"Since not every complex object type needs 10 keypoints to define the corresponding grasp manifold, we add extra keypoints at the object's origin with a visibility flag equal to 0 such that each object type has a set of 10 keypoints and these additional keypoints will be neglected during the training.",
"Since our approach should be able to detect unknown objects, all object types belong to the same category “object” which leads to a class-agnostic object detection task.",
"However, we additionally train a network with different object classes for comparison.",
"Figure: GraspME framework for grasp manifold estimation."
],
[
"Training Data Generation",
"Our simulation environment is based on PyBullet [3].",
"The generated data should be diverse enough such that the trained model generalizes to unseen poses and objects, and to real data.",
"We consider the scenario where a robot grasps objects from a flat table surface while observing the scene from above.",
"The camera's position is sampled from a hemisphere around the tabletop's center.",
"The camera records RGB and depth data together with corresponding segmentation images of the observed scenes.",
"The segmentation images are automatically generated by the PyBullet simulation.",
"Additionally, we store the axis aligned bounding boxes per object and the keypoints that define the grasp manifold for the corresponding object."
],
[
"Bounding Boxes",
"The bounding boxes are computed by using the minimal and maximal x- and y-coordinates of the object's segmentation mask.",
"This way, we define the bounding box with the lower point $(x_{min}, y_{min})$ , its width $w = x_{max} - x_{min}$ and its height $h = y_{max} - y_{min}$ .",
"Due to occlusions, the segmentation mask and the bounding box are computed only for the object's visible part."
],
[
"Keypoints",
"We define the keypoints for each object type beforehand, and project them during the simulation on the image plane using the full projection matrix of the camera, giving us the absolute position of a grasp manifold keypoint $kp_i = (x_i, y_i)$ .",
"Since a keypoint could be invisible due to being occluded or outside of the camera's view, we set a visibility flag $v_i$ for them using the COCO format [16] for keypoint detection, i.e.",
"$v_i = 0$ if the keypoint does not exist on the object, $v_i = 1$ if the keypoint exists but is not visible and $v_i = 2$ if the keypoint is visible.",
"Furthermore, for the simple objects only, if $kp_1$ is not visible but $kp_2$ is, we swap $kp_1$ and $kp_2$ and their corresponding visibility flags such that $kp_1$ should always be visible.",
"This is possible due to the symmetric properties of those objects."
],
[
"Object Shapes and Randomization",
"For cuboids, cylinders and capsules, the sizes are randomized for each object instance before rendering them, i.e.",
"the cuboid's length, width and height are chosen such that the length has the largest value with a small probability of generating cubes.",
"The same holds for the capsule's and the cylinder's length and radius.",
"Typical samples during the simulation can be seen in Fig.",
"REF .",
"For complex objects, we use a small subset of 11 objects from the 50 category subset of 3DNet [4]: apple, banana, bottle, camera, can, grenade, guitar, gun, maglite, mug and pliers.",
"These synthetic 3D mesh models are rescaled and transformed from the original versions to fit our simulation.",
"The object models are depicted in Fig.",
"REF while some samples from the simulation with complex objects can be seen in Fig.",
"REF and REF .",
"To increase the diversity of the simulated dataset, we incorporate domain randomization techniques which have been proposed by Tobin et al.",
"[22].",
"Hence, we randomize for each scene separately, the camera's view point, the lighting conditions with one light source, the table's color, the total amount of objects and the objects' colors, positions and orientations before dropping them from above the tabletop using simulated physics.",
"Figure: Results of the predictions (bottom) in comparison to the ground-truth (top) for simple (left) and complex objects (right).",
"The corresponding grasp manifolds are depicted as black lines.Table: An overview over the training parameters chosen for the different architectures and experiments.",
"The model name is a concatenation of abbreviations of the training parameters in the following order: backbone (R50/R101), training data (S/C/P), class agnostic (merged classes = M) or classification (C), iterations (40/80)"
],
[
"Training",
"The training procedure for the RPN, the classification, the bounding box detection, the mask prediction and the keypoint estimation are adopted from the Mask R-CNN which is why we refer readers for further details to [1] and its predecessors Fast R-CNN [7] and Faster R-CNN [6].",
"During training, we apply some randomly chosen online augmentations to improve the generalizability of our model.",
"We use the implemented augmentations from the Detectron2 framework [2] like flipping and changes of lighting, saturation, brightness and contrast, from which 2 augmentations are chosen at random for each batch."
],
[
"Dataset",
"We generated 40,000 synthetic scenes per object set.",
"The model is then trained on 80% of the data, i.e.",
"on 32,000 data points, while the remaining data is withheld for validation and testing, each containing 4,000 data points.",
"The images are of size $512 \\times 512$ pixels which we keep fixed as input for our models.",
"We collect these amounts of data for three datasets: the first one contains only simple objects (called “Simple”), the second one contains only complex objects (called “Complex”) and the third contains 8 out of 11 complex objects (called “Part”).",
"A fourth dataset including 4,000 data points each for validation and for testing contains the remaining three objects that did not appear in any scene in “Part” (called “Unseen”)."
],
[
"Hyper parameters",
"We chose an image batch size of 10 while the batch size per image is 64 for the RPN and 128 for the ROI head due to memory restrictions.",
"As we cannot compare our method to any baseline due to lacking one, we use different training parameters as follows to get a better overview over the effects of these parameters.",
"The models are trained for 40k and 80k iterations with a base learning rate of 0.001.",
"We decrease the learning rate by a factor of 10 after 30k iterations if trained for 40k iterations and additionally after 60k if trained for 80k iterations.",
"Due to the similar approach, we use the two pretrained models from Mask R-CNN for the Human Pose Estimation experiments as initialization and finetune them on our datasets and problem setting.",
"The training of the models is performed on two GeForce GTX 1080 Ti GPU.",
"An overview over all model configurations can be found in Table REF .",
"Table: The results on the simple and complex test data w.r.t.",
"AP in percent (%), IoU of the grasp manifold in percent (%) and mean pixel distance between keypoints on the three main tasks of bounding box, mask and keypoint estimation."
],
[
"Experiments",
"The experiments are conducted on the two scenarios with simple objects and with complex objects.",
"Since there does not exist any baseline yet that estimates whole grasp manifolds from 2D images, we can only compare different architectures and training setups of our models.",
"An overview of the performed experiments for all of the model's outputs can be seen in Table REF .",
"Overall, we achieve an average speed of 11.5 frames per second on one of the before mentioned GPUs which makes our framework suitable for real time applications."
],
[
"Metrics",
"We evaluate our models using the standard COCO metrics [16] regarding average precision (AP) for the three outputs bounding box (bb), segmentation mask (segm) and keypoints (kp).",
"As the metric for the evaluation of the latter output is optimized for human pose estimation, we additionally compute the mean Intersection over Union (IoU) of the ground-truth grasp manifold with the predicted one by using the same number of keypoints as the ground-truth (clip) or the full set of predicted keypoints (full) as well as its standard deviation.",
"We also measure the mean pixel distance (mDist) of the predicted keypoints to the ground-truth ones by matching them first with the closest ground-truth keypoint set per object.",
"Afterwards, we conduct some ablation studies on the different model architectures and training setups of our models.",
"Due to a lacking baseline, we compare our model to the Random baseline, i.e.",
"we randomly sample keypoints from the predicted bounding boxes from the models R50-S-M-40 for the “Simple\" test dataset and R50-C-M-40 for the “Complex”, “Part” and “Unseen” test datasets.",
"Table: The results on the complex test data including unseen objects w.r.t.",
"average precision (AP) in percent (%) on the three main tasks of bounding box, mask and keypoint estimation."
],
[
"Simple Objects",
"As can be seen from the upper part of Table REF , we achieve a very high accuracy with all our models for the bounding box detection and the segmentation mask estimation while the AP for keypoints is rather low.",
"The reason for this is the objects' symmetry, which makes it harder to estimate the correct keypoints without any additional information.",
"Flipping the keypoint labels during training could solve this problem, by specializing on certain image regions, i.e.",
"$kp_1$ could always be rather on the left of the object while $kp_2$ could always be on the right.",
"As can be seen in Fig.",
"REF , if we detect an object, we indeed predict the keypoints and the corresponding grasp manifolds very well in comparison with its ground-truth in Fig.",
"REF .",
"The models with the ResNet-101-FPN backbone achieve the highest accuracies as well as the best values for our own metrics.",
"We get values around 40% for the grasp manifold IoUs, which proves that our assumptions about the low AP$^{kp}$ values is correct.",
"This might also be the reason for the rather high mean pixel distances between the ground-truth and the predicted keypoints.",
"As all of the objects had two ground-truth keypoints which were fully used, the values for IoU$_{clip}$ and IoU$_{full}$ are the same.",
"Our model outperforms the Random baseline by far in all of the keypoint related metrics."
],
[
"Complex Objects",
"The results of the bounding box detection and the segmentation mask estimation, reported in the lower part of Table REF , are off similar quality as for simple objects.",
"However, due to the unique shape of the objects, the AP$^{kp}$ values are much higher as the keypoints are much easier to identify on the objects.",
"This is also reflected in the low mean pixel distance between ground-truth and prediction of around 1.6 pixels which also leads to a grasp manifold IoU of 65%.",
"Though using the full set of available keypoints does not seem to be helpful as the IoU is much smaller.",
"Objects with less keypoints do not benefit from the additional keypoints as their grasp manifold is approximated already well enough.",
"Due to the larger number of keypoints, the Random baseline achieves higher scores with our metrics but is still a lot worse than our framework.",
"Overall, the best results are again achieved by the models with the ResNet-101-FPN backbone.",
"These accurate results can also be seen in Fig.",
"REF , where we predict all of the keypoints nearly perfect in comparison with its ground-truth in Fig.",
"REF .",
"Figure: Results of the predictions (bottom) in comparison to the ground-truth (top) for unseen objects on R50-P-M-40.",
"The corresponding grasp manifolds are depicted as black lines."
],
[
"Unseen Objects",
"For the third experiment, we want to evaluate the generalizability of our models by using the dataset “Part” to train on a subset of the objects and test on the dataset “Unseen” containing some withheld objects.",
"Additionally, we provide results on the full dataset “Complex” with all objects.",
"We report the corresponding overview in Table REF .",
"As can be seen, the models trained and evaluated on the “Part” dataset achieve similar performance to the models trained on the “Complex” dataset with respect to the COCO metrics while getting even better scores with our own metrics in terms of mean IoU with nearly 70% and mean pixel distance to the ground-truth keypoints of around 1.3 pixels.",
"This could be due to the withheld objects that might belong to the objects that are more difficult.",
"Even though the model has never encountered objects from the “Unseen” dataset, it could still segment most of them from the images and estimate corresponding bounding boxes.",
"However, computing the expected keypoints from the ground-truth was not possible, following the COCO metrics and the high pixel distance.",
"There is some intersection of the grasp manifolds though, as can be seen from the IoU values, from which we can assume that a grasp manifold has still been found by the models.",
"Using the full amount of keypoints decreases the quality of the results.",
"We assume that these low values overall come from the very different object shapes in comparison to the known object's shapes and that it was difficult for the model to approximate the expected grasp manifold and the corresponding keypoints.",
"However, the model might have predicted another unintentional grasp manifold that is still a valid grasp manifold.",
"By extending our approach to find several grasp manifolds or using more than one ground-truth grasp manifold per object, we might be able to achieve better results.",
"To emphasize this hypothesis, we compare some of the results on the unseen objects in Fig.",
"REF .",
"As can be seen, the result's quality depends on the object.",
"The grasp manifolds for the maglites are quite accurate while the grasp manifolds for the apples also seem to be rather close to the ground-truth.",
"As the guitar is the most difficult of these objects regarding shape, the results are not so good in comparison with the ground-truth.",
"However, the model still often predicts a grasp manifold by focusing on the guitar's corpus which seems to be valid, even if it is different from the expected one.",
"We conclude that our framework is able to estimate grasp manifolds also on unseen objects.",
"As the “Complex” dataset contains both seen and unseen objects, the COCO scores are obviously lower as for the models trained on all objects but we still achieve very good results in all three categories and also estimating the grasp manifolds well enough as can be seen from the IoU.",
"Therefore, having some unseen objects mixed with known objects does not decrease the results too much, except for the mean pixel distance.",
"The model might even benefit from having these mixed scenes and hence, achieve better results.",
"Even for the “Unseen” dataset as well as for the “Part” and the “Complex” datasets, our framework outperforms the Random baseline.",
"Figure: Results of the predictions for unseen real data on R101-C-M-40.",
"The corresponding grasp manifolds are depicted as black lines."
],
[
"Ablation Studies",
"To further evaluate our framework, we compare different model architectures and training setups as ablation studies.",
"The first part is about the model's architecture in terms of the backbone.",
"Regarding the results of our experiments, we can see that models with the ResNet-101-FPN backbone achieved better results than with the ResNet-50-FPN backbone.",
"Training the model by additional 40k iterations does not seem to improve the results as much as expected regarding the COCO metrics.",
"For the ResNet-101-FPN backbone, the model trained for only 40k instead of 80k iterations achieves even better results.",
"The higher number of training iterations is only noticeable with respect to the IoU and the mDist.",
"The difference between using separate classes for each object type or simply having one for all, i.e.",
"having the class-agnostic case, is rather unsignificant.",
"For the complex objects, the class-agnostic model achieves better results by nearly 2 points in all three categories of the COCO metrics and slightly better values for the IoU and the mDist metrics.",
"Therefore, we recommend the class-agnostic model as it can also be used for unseen objects without additional training."
],
[
"Real Data",
"We present initial results on real camera images.",
"However, as ground truth labels were not available, we could only evaluate the resulting images.",
"Some of the better results are depicted in Fig.",
"REF , which seems promising regarding the usage in real applications, as in most cases the object's main axis is predicted."
],
[
"Conclusion",
"Overall, we showed that our models achieve good results in terms of object detection for both simple and complex objects.",
"Even though the values for evaluating the models on unseen objects are low, we can see that our framework could still partially generalize to these shapes and predict a grasp manifold.",
"Thus, our model can support other methods for finding suitable grasp points on objects by spanning a whole manifold of possibilities.",
"By having a frame rate of 11.5 fps, we expect to use our approach for real time applications.",
"We plan to provide a proof of concept by integrating it into a trajectory optimization framework and demonstrate our model's usage to perform human-robot-collaboration tasks, e.g.",
"via shared autonomy and autonomous environment interactions in real scenarios.",
"The sampled grasps in these scenarios will also be compared to those provided by other algorithms mentioned in Section .",
"Furthermore, we want to extend our framework in taking advantage of additional depth data to gain valuable information about the scene.",
"As this problem setting is unknown yet, we hope to draw interest to this scenario and encourage other researchers to approach it and use our framework as baseline."
],
[
"Acknowledgment",
"We want to thank the authors of [1] and [2] for making the code of their framework publicly available.",
"This work was conducted while Ruben Bauer was performing his Masters dissertation in the Machine Learning and Robotics Lab, University of Stuttgart, Germany.",
"This work is partially funded by the research alliance “System Mensch”."
]
] | 2107.01836 |
[
[
"The Ultraviolet Deep Imaging Survey of Galaxies in the Bootes Void I:\n catalog, color-magnitude relations and star formation"
],
[
"Abstract We present a deep far and near-ultraviolet (FUV and NUV) wide-field imaging survey of galaxies in the Bootes Void using Ultra-Violet Imaging Telescope onboard {\\em AstroSat}.",
"Our data reach $5\\sigma$ limiting magnitudes for point sources at 23.0 and 24.0 AB mag in FUV and NUV respectively.",
"We report a total of six star-forming galaxies residing in the Bootes Void alongside the full catalog, and of these, three are newly detected in our FUV observation.",
"Our void galaxy sample spans a range of UV colors $(-0.35\\, \\leq$ FUV$-$NUV $\\leq\\, 0.68)$ and absolute magnitudes $(-14.16\\, \\leq\\, \\mathrm{M_{NUV}}\\, \\leq\\, -18.65)$.",
"In addition, {\\em Sloan Digital Sky Survey} and {\\em Two-micron All Sky Survey} archival data are being used to study UV, optical, and infrared color-magnitude relations for our galaxies in the void.",
"We investigate the nature of bi-modal color distribution, morphologies, and star formation of the void galaxies.",
"Most of the galaxies in our sample are fainter and less massive than L$^{\\ast}$ galaxies, with M$_\\mathrm{r} > -20$.",
"Our analysis reveals a dominant fraction of bluer galaxies over the red ones in the void region probed.",
"The internal and Galactic extinction corrected FUV star formation rates (SFRs) in our void galaxy catalog varies in a large range of $0.05$ to $51.01$ M$_{\\odot} yr^{-1}$, with a median $3.96$ M$_{\\odot} yr^{-1}$.",
"We find a weak effect of the environment on the SFRs of galaxies.",
"Implications of our findings are discussed."
],
[
"INTRODUCTION",
"Cosmic web, largely composed of voids, filaments and wall-like structures, is observed to be inhomogeneous at mega-parsec scale.",
"The large-scale structures that we see in our present day universe are a manifestation of the primordial gravitational density fluctuations [76].",
"The cosmic voids occupy $\\sim $ 77% of the cosmic volume and they represent $\\sim $ 15% of the total halo mass content implying that the average density of void is around 20% of the average cosmic density [19].",
"The formation and evolution of the voids depend on two processes, i$.$ e$.$ , small voids merge to shape into a larger under density and due to collapse of over-densities around a region in space [79], [89].",
"They usually tend to exist within the cosmic web [49] in a spherical foam-like structure.",
"Such voids can be populated by substructures such as mini-sheets and filaments that run through the voids.",
"As these voids grow older they become progressively empty and possess less substructures within them [71].",
"Typical size of a large scale void ranges from 20h$^{-1}$ to 50h$^{-1}$ Mpc, but its depth remains unclear with an under-density of ${\\delta }_\\mathrm {{v}} = \\frac{{\\rho }_v}{<{\\rho }>}-1 \\approx -0.8$ .",
"The voids were first discovered in observation by [28], [37].",
"Later [43], [44] discovered the Bootes Void, one of the largest void present in the northern hemisphere.",
"Much recently, Sloan Digital Sky Survey (SDSS) provided a detailed structure of cosmic voids using large-scale structure galaxy catalog from Baryon Oscillation Spectroscopic Survey (BOSS) [50].",
"Voids are thought to provide a pristine environment for understanding the secular evolution and dynamics of galaxies [90] as they are devoid of phenomena typically active in a denser medium like groups and clusters, e$.$ g$.$ , ram pressure stripping [32], gas strangulation [60] (galaxy nurture).",
"As a result, the evolution of void galaxies is thought to be slower than those in denser medium leading to an abundance of young galaxies at primitive stages of evolution, thus, studying void galaxies may unearth key features of the early stages of galaxy formation scenario.",
"In fact, a number of objective prism surveys [55], imaging and spectroscopic observations [22], [84], [92] of the Bootes Void show remarkable similarity in the overall void galaxy properties.",
"The galaxies present in voids tend to be bluer than wall and field galaxies with large specific star formation rates (sSFRs) [56].",
"Voids are mainly populated with late-type galaxies although the presence of active galactic nuclei and early-type galaxies have also been reported recently [8].",
"Not only that, void galaxies have also shown evidences of recent merger interaction [46], [30].",
"Some of these galaxies show unusual morphological features such as knots, asymmetries, apparent one-spiral arm, and offset nucleus [23].",
"Based on these recent reports, one would expect to find a complete spectrum of galaxy morphology at different evolutionary phases in a void environment.",
"It has been shown that the global properties such as morphology, color, and star formation rates (SFRs) of galaxies depend predominantly on their local environment and internal driven mechanism rather than their global environment [84], [86], [61].",
"Based on a comparative analysis of the emission line galaxies (ELGs) situated in sparse and dense environments, it has been suggested that the local environment density around a galaxy may have no effect on its chemical evolution [91].",
"On the other hand, a void environment has been shown to affect the size and stellar masses of the galaxies inside them [8].",
"The evolutionary history of galaxies is shown to depend on whether a galaxy resides deep inside the void or on the periphery as well as on the size of the host void [65], [66].",
"It remains unclear at what scale and which galaxy properties are affected by the environment which would require multi-wavelength deep imaging and spectroscopic surveys of the void region.",
"SDSS has already done a great job in this aspect.",
"Deep imaging observation of the void in the far and near ultraviolet (FUV and NUV) bands with SDSS like spatial resolution is, however, missing since Galaxy Evolution Explorer (GALEX) [51] did shallow surveys ($\\sim 205$ sec) of the void region (e$.$ g$.$ , of the Bootes Void in the northern hemisphere).",
"In this work, we present a deep imaging survey of about 615 sq.",
"arcmin region of the Bootes Void in FUV and NUV filters of the Ultra-Violet Imaging Telescope (UVIT) on-board AstroSat Satellite [85].",
"Since the galaxies in voids show bluer color and are star-forming, the stellar population would contain a significant fraction of young stars (O, B-type) of intermediate masses (2-5 M$_{\\odot }$ which actively emit in FUV and NUV [29], [68], [67].",
"The FUV emission in a galaxy arises from the photosphere of massive O- and B-type stars and therefore, it traces star formation going on in a galaxy over a timescale of ${\\sim }$ 10$^{8}$ yr [15], [48].",
"Using our UV deep imaging survey, we produce a catalog of void galaxies with fluxes from FUV to near-infrared (NIR) and study their morphology, UV-optical and NIR color-magnitude relations [95], [27], [4], their star formation properties and compare them with non-void galaxies.",
"The work is organized as follows: in section , we describe our data used for analysis, and in section , we explain the procedure adopted for data reduction and analysis.",
"We briefly discuss our methodology for catalog preparation and photometric redshift calculation in section whereas in section , we deduce the reliability of our UVIT detection followed by section , and , where we discuss about the internal dust obscuration, stellar masses and quantify UV emission of galaxies.",
"In section , we discuss properties of void galaxies detected in the analysis on the basis of various color-magnitude diagrams (CMDs).",
"Finally, in section , we conclude our findings and examine the future prospects of our research.",
"A standard $\\Lambda $ CDM cosmology with ${\\Omega }_M$ = 0.3, ${\\Omega }_{\\Lambda }$ = 0.7, and H$_{0}$ = 70 km s$^{-1}$ Mpc$^{-1}$ is assumed in this work.",
"AB magnitude system [59] has been followed throughout the work.",
"We have converted 2MASS Vega magnitudes to AB using conversions given in [11] to maintain uniformity of the magnitude scale.",
"Figure: Sky map comprises of a section in the Bootes Void.",
"The UVIT observation covers the area marked by the blue circle.",
"The red open rectangle denotes the area observed by the KISS survey .",
"The rectangular area extends to RA range 178.5 ∘ ^{\\circ } – 244.4 ∘ ^{\\circ }, DEC range 42.55 ∘ ^{\\circ } – 44.35 ∘ ^{\\circ }.",
"Each black dot represents a galaxy from the SDSS archival catalog.Figure: Top panel: Bootes Void observed in the FUV and NUV filters of AstroSat/UVIT centered at (α,δ)=(14 h 08 m 27.8 s ,+48 d 55 m 56.6 s )(\\alpha , \\delta )\\, =\\, (14^h\\ 08^m\\ 27.8^s,\\ +48^d\\ 55^m\\ 56.6^s) with a diameter of 28 ' ^{\\prime } each.",
"Bottom panel: Color composite images of some peculiar galaxies in the Bootes Void region marked with alphabets in the FoV; (red: SDSS r-filter, green: UVIT NUV filter, blue: UVIT FUV filter)."
],
[
"The AstroSat is India's first dedicated multi-wavelength space satellite launched by Indian Space Research Organization (ISRO) in September, 2015.",
"UVIT on-board AstroSat satellite observes primarily in three channels: FUV (${\\lambda }=$ 1300 - 1800 ${\\mbox{\\normalfont Å}}$ ), NUV (${\\lambda =}$ 2000 - 3000 ${\\mbox{\\normalfont Å}}$ ) and visible (${\\lambda }=$ 3200 - 5500 ${\\mbox{\\normalfont Å}}$ ) wavelength bands.",
"The field of view (FoV) of each channel is about 28$^\\prime $ in diameter with pixel size of $\\approx 0^{\\prime \\prime }.417$ and spatial resolution of $< 1^{\\prime \\prime }.8$ in FUV and NUV channels [85].",
"The angular resolution of UVIT is 3-4 times higher as compared to previously launched UV space telescope GALEX.",
"We proposed to explore an area of 615 sq$.$ arcminutes in the Bootes Void to observe with UVIT.",
"The aforementioned area is centered at $\\alpha $ = 212.115$^{\\circ }$ / 14$^h$ 08$^m$ 27.8$^s$ and $\\delta $ = 48.932$^{\\circ }$ / 48$^d$ 55$^m$ 56.6$^s$ .",
"Observations were taken in BaF2 ($\\lambda _\\mathrm {eff} = 1541\\, {\\mbox{\\normalfont Å}}$ ) and Silica-1 ($\\lambda _\\mathrm {eff} = 2418\\, {\\mbox{\\normalfont Å}}$ ) filters of UVIT.",
"The total on-source exposure time assigned to BaF2 (F154W) and Silica-1 (N242W) filters $\\approx $ 10000 sec each (PI: Kanak Saha).",
"Figure REF shows the field of observation of a recent KPNO International Spectroscopic Survey (KISS) of ELGs in direction of Bootes Void [91] and our UVIT FoV.",
"The top panel of Figure REF shows FUV/NUV FoV observed by UVIT; in the bottom panel, we have shown the color composite images of five peculiar galaxies detected in our FoV.",
"The images are color coded as follows- red: SDSS r filter, green : UVIT NUV , and blue: UVIT FUV.",
"Two of the five galaxies (third and fourth image) in the figure are void members while the others lie outside the void.",
"To extend this survey further into the optical and infrared parts of the electromagnetic spectrum, we have included SDSS and 2MASS observations.",
"SDSS has five filters u, g, r, i, z having mean wavelengths 3560 ${\\mbox{\\normalfont Å}}$ , 4680 ${\\mbox{\\normalfont Å}}$ , 6180 ${\\mbox{\\normalfont Å}}$ , 7500 ${\\mbox{\\normalfont Å}}$ , 8870 ${\\mbox{\\normalfont Å}}$ , respectively, spanning over the optical and infrared bands of the electromagnetic spectrum [98].",
"We have used well calibrated archival imaging data from the SDSS Data Release 12 (DR12) [1] for the same patch of sky in the Bootes Void as observed by UVIT.",
"Similarly, imaging archival data from 2MASS have also been taken up which observes in NIR wavelength filters J ,H, Ks with mean wavelengths 1.24 ${\\mu }$ m, 1.66 ${\\mu }$ m, 2.16 ${\\mu }$ m, respectively [81].",
"Figure: Top panels: Surface brightness distribution of the PSF is fitted with a circular Moffat function (see Eq. )",
"in FUV and NUV filters.",
"FWHM refers to the full width at half maxima for the Moffat function.",
"Inset images show the stacked PSF.",
"Bottom panels: Encircled energy curve for FUV (left) and NUV (right).",
"Vertical dashed lines in either case denote the radii containing 80% encircled energy.",
"one pixel=0 '' ^{\\prime \\prime }.417."
],
[
"DATA REDUCTION AND ANALYSIS",
"The FUV and NUV observations were carried out by AstroSat/UVIT in the photon counting mode with a frame rate of $\\sim 34 $ frames per second.",
"This would accumulate about $\\sim 45000 - 50000$ frames in a typical good dump-orbit.",
"The orbit-wise dataset was processed using the official L2 pipeline in which we removed frames that are affected by the cosmic-ray shower and these were not included in the final science-ready images and the subsequent calculation of the photometry.",
"This results in an average loss of $\\sim 20\\%$ data to science-ready images.",
"The final science-ready image had a total exposure time of $t_{exp}=8600$ sec in FUV and $t_{exp}=7513$ sec NUV bands.",
"Astrometric correction was performed using the GALEX FUV/NUV tiles and SDSS r-band image as references.",
"We have used an IDL program that takes an input set of matched xpixel/ ypixel (from UVIT images) and RA/ DEC (from GALEX FUV/NUV and SDSS r band image) and perform a TANGENT-Plane astrometric plate solution similar to ccmap task of IRAF [87].",
"The astrometric accuracy in NUV was found to be $\\sim 0^{\\prime \\prime }.2$ while for FUV, the RMS was found to be $\\sim 0^{\\prime \\prime }.24$ , approximately half a pixel size.",
"The photometric calibration was performed with a white dwarf star Hz4; the photometric zero-point for F154W band is 17.78 mag and 19.81 mag for N242W [85].",
"Once photometric calibration and astrometric correction are successfully applied, we run Source Extractor (SExtractor) software [6] on the science-ready images to extract sources and estimate the background.",
"We use following extraction parameters to detect sources in FUV and NUV images: DETECT_THRESH = 3- and 5$\\sigma $ and DETECT_MINAREA = 16/ 9 pixels for FUV/ NUV filters depending on their angular resolution (See, section REF )."
],
[
"Point Spread Function and Background estimation",
"We perform a robust calculation of full width half maxima (FWHM) measurements of the point spread function (hereafter, PSF) on the UVIT NUV and FUV science-ready images.",
"We start with stacking a few unsaturated, isolated point sources of varying magnitudes to get an unbiased profile of a point source with a high signal-to-noise ratio (SNR) (See, inset image Figure REF ).",
"The isolated sources were selected from our UVIT FUV/NUV 3$\\sigma $ catalog with an additional criteria of CLASS_STAR (discussed in section ) $\\ge $ $0.9$ .",
"Also, we visually examine the sources to check their symmetry around its centroid.",
"In this work, we have adopted two methods: firstly, isophotal ellipse fitting is performed on the stacked images using IRAF STSDAS packageSTSDAS is product of the Space Telescope Science Institute, which is operated by AURA for NASA.. We fit the one-dimensional circular Moffat function over the surface brightness distribution $I(r)$ as a function of radius to obtain the parameters required for calculating FWHM [54].",
"The Moffat function used to model the PSF is given by $I(r) = I_0 \\left[1 + \\bigg (\\frac{ r }{ \\alpha }\\bigg )^2 \\right]^{-\\beta }$ with FWHM = $ 2\\alpha \\sqrt{2^\\frac{1}{\\beta }-1}$ .",
"Here, $I_{0}$ is the central surface brightness and $\\beta $ and $\\alpha $ are the free parameters.",
"${\\beta }$ is the seeing parameter that determines the spread of Moffat function.",
"Here, we use Moffat function for simplicity as Gaussian function alone doesn't accurately fit the wings of a stellar profile [88] but see [69] for the wing modelling in F154W band.",
"The Gaussian function is a limiting case of Moffat function (${\\beta } \\rightarrow {\\infty }$ ).",
"The second method for determining PSF involves creating encircled energy (EE) curve as a function of radius for the same stacked source.",
"As per our calculation, the radius corresponding to a circular area enclosing 80% of the total normalised energy of the stacked stellar profile came out to be close to PSF FWHM obtained using the first method.",
"The results from both methods are in a good agreement as shown in Figure REF .",
"The resulting values of the PSF FWHM for F154W and N242W filters from the fitting are 3.83 pixels (1$^{\\prime \\prime }$ .59) and 2.58 pixels (1$^{\\prime \\prime }$ .08).",
"We perform the background subtraction over the science-ready images in both filters for accurate flux estimation from the sources.",
"For this, we run SExtractor [6] with a detection threshold = 2$\\sigma $ on both images.",
"Subsequently, we mask all sources at and above 2${\\sigma }$ from the entire FUV and NUV images.",
"Thereafter, we measure integrated flux due to background by randomly placing multiple square boxes ($\\approx $ 1000) over various parts of the masked image avoiding the source locations.",
"The size of boxes (5${\\times }$ 5 pixels for NUV/ 7${\\times }$ 7 pixels for FUV) were chosen such that the area enclosed within the boxes were close to the area bounded by a PSF-size point source.",
"Figure REF shows the background flux histograms for the FUV and NUV images.",
"These histograms are fitted with a Gaussian function with a mean ($\\mu $ ) and standard deviation ($\\sigma $ ).",
"From the fitting, we obtain a sky surface brightness of 27.99 mag arcsec$^{-2}$ and 27.55 mag arcsec$^{-2}$ in the FUV and NUV observations, respectively.",
"Measured mean background flux per pixel was subtracted prior to photometry.",
"We follow Kron photometric technique [47] in this work.",
"The Kron apertures are elliptical or circular depending on the intensity distribution of the source, and the apertures capture 80-90% of the total flux radiated by a source.",
"We calculate the point source detection limit using the estimated background noise ($\\sigma _\\mathrm {sky}$ ), number of pixels in a circular aperture (r), and desired detection threshold.",
"The detection limits are calculated at 3${\\sigma }$ and 5${\\sigma }$ threshold considering aperture radius r = 4 pixels ($\\sim $ 1$^{\\prime \\prime }$ .6) and 3 pixels ($\\sim $ 1$^{\\prime \\prime }$ .2) for FUV and NUV images, respectively.",
"We use $\\sigma _{sky}$ corresponding to our FUV/NUV sky histograms (Figure REF ) $\\approx $ $8.4$ $\\times $ $10^{-6}$ /$3.23$ $\\times $ $10^{-5}$ count s$^{-1}$ (cps).",
"Note that these values are about a factor of $\\approx $ 3 times lowered than their counterparts obtained using the SExtractor (FUV/NUV $\\approx $ $2.2$ $\\times $ $10^{-5}$ cps/$9.5$ $\\times $ $10^{-5}$ cps).",
"Nevertheless, the mean background obtained from both the methods are in good agreement.",
"Using our $\\sigma _{sky}$ from the histograms, the 3${\\sigma }$ point source detection limit for FUV/NUV observations are found to be 25.02/26.22 mag.",
"Similarly, our FUV/NUV survey reaches a detection limit of 24.46/25.66 mag at 5$\\sigma $ .",
"Although we realize that the actual 3- and 5$\\sigma $ detection limits depend on aperture size or the number of pixels within; shape of the object (point-like or extended), we consider this to be a good approximation of the actual UVIT detection limits.",
"Figure REF shows magnitude histogram for all the extracted sources from FUV/NUV observation.",
"From the magnitude histogram, the limiting magnitude for FUV/NUV observations at and above 3${\\sigma }$ $\\approx $ 24.0/25.0 mag.",
"Whereas for 5${\\sigma }$ threshold, the limiting magnitude for FUV/NUV $\\approx $ 23.0/24.0 mag.",
"Typical uncertainty in FUV/NUV magnitudes within the limit magnitudes is 0.3/0.2 mag.",
"Figure: The figure comprises of background flux histograms for UVIT FUV/NUV filters.",
"Background flux per pixel is calculated by fitting a single peak Gaussian function over the integrated flux distribution of 1000 boxes spread across the entire image.Figure: The magnitude histograms for 3σ\\sigma and 5σ\\sigma FUV/NUV sources extracted by SExtractor and are not extinction corrected.",
"Typical foreground dust extinction values for this FoV are A F 154W=0.24A_\\mathrm {F154W}=0.24 mag and A 242 W=0.22A_\\mathrm {242W}=0.22 mag.UV fluxes are highly susceptible to both internal and Galactic dust attenuation.",
"[78] full-sky 100 $\\mu $ m map gives us Galactic dust reddening $E(B - V)$ along a given line of sight.",
"The values of extinction parameter $A_{F154W}/E(B-V) = 8.104$ , $A_{N242W}/E(B-V) = 7.746$ were calculated using [18] extinction law.",
"We correct SDSS u, g, r, i, z and 2MASS J, H, and Ks-band fluxes for Galactic extinction in a similar manner.",
"We use the method given by [20] for redshift K-correction (K$_\\mathrm {z}$ ) of the catalogue fluxes to $z = 0$ .",
"Consequently, we calculate the absolute magnitude (M$_\\mathrm {\\lambda }$ ) of a galaxy at luminosity distance D$_\\mathrm {L}$ in a given passband (denoted by its wavelength $\\lambda $ ) using the following equation( REF ).",
"$M_\\mathrm {\\lambda } = m_\\mathrm {\\lambda } - 5(\\log D_\\mathrm {L} - 1.0) - A_\\mathrm {\\lambda } - K_\\mathrm {z}$ The cross-matching of sources over several observational surveys is a non-trivial task as morphologies and resolution may change over various wavelengths and filters used.",
"Particularly, in our case, PSF FWHM for UVIT, SDSS and 2MASS survey are $\\approx $ 1$^{\\prime \\prime }$ .5, 1$^{\\prime \\prime }$ .3 and 2$^{\\prime \\prime }$ .8, respectively.",
"The minimum cross-matching radius that we use for matching UVIT to SDSS catalogs $\\sim $ 1$^{\\prime \\prime }$ .5 whereas to 2MASS catalog $\\sim $ 2$^{\\prime \\prime }$ .8.",
"Such values for cross-matching radius were selected as our field is a void field and to a large extent non-crowded.",
"We find no multiple matches for any source present in our UVIT 3$\\sigma $ catalog on cross matching with other catalogs from different surveys.",
"We employ topcathttp://www.starlink.ac.uk/topcat/ software for the purpose.",
"While we did the cross-matching, we also visually examined the sources in the UVIT FoV.",
"Figure: NUV--g vs.. g--i and FUV--NUV vs.. g--r color-color diagrams used for STAR/GALAXYSTAR/GALAXY classification of 159 sources with z phot z_\\mathrm {{phot}} and CLASS_STAR ≤\\le 0.1 from our UVIT 3σ\\sigma catalog are shown in left and right panels of the figure, respectively.",
"The black markers represent galaxies from our UVIT 3σ\\sigma catalog with spectroscopic observation.",
"The background point sources are taken from the overlap of archival GALEX AIS catalog and SDSS DR12 spectrocopic catalog whereas extended sources are from GALEX-SDSS-WISE Legacy catalog ."
],
[
"Catalog construction and Source classification",
"The total number of sources extracted at 3$\\sigma $ depth from FUV/NUV images are 709/4,931 and at 5$\\sigma $ depth, 146/2,050 UVIT 3$\\sigma $ and 5$\\sigma $ source catalog could be made available pertaining to an online request.",
"These catalogs are going to be published online soon.. As we are probing void galaxies in our FoV, we segregate the extended objects from the rest of the sources.",
"At a rudimentary level, we use CLASS_STAR (hereafter, CS) parameter given by SExtractor.",
"The parameter gives a probabilistic value between zero (= $GALAXY$ ) to one (= $STAR$ ).",
"In the current work, CS corresponding to SDSS r-band image is considered for the analysis where an object with CS $ \\le $ 0.1 is regarded as an extended source.",
"As a result, we identify 184 and 74 extended sources with 3$\\sigma $ and 5$\\sigma $ detection, respectively.",
"For all these extended sources, we have FUV and NUV observations.",
"We are exclusively interested in sources with measured redshifts (spectroscopic/photometric).",
"Therefore, on matching the extended sources (CS $\\le $ 0.1) present in UVIT FUV 3$\\sigma $ catalog with SDSS spectrocopic/photometric catalog taken from SDSS DR12, we split the 184 extended sources in the following categories: Objects with photo-$z$ only = 159 Objects with spec-$z$ = 8 Objects absent in either of the catalogs (spectrocopic/photometric) = 17 Visual examination of the sources which were not a part of any of the two catalogs reveals that some of them are bright and saturated stars, part of an extended source, or faint sources ($u \\sim $ 23 mag).",
"Hence, we reject the objects in this category for any further analysis.",
"All 8 extended sources with spec-$z$ were already classified as $GALAXY$ by $SDSS$ .",
"In addition, we find a pea shaped/compact object having spec-$z$ in our FoV with CS $>$ 0.1, classified as $GALAXY$ by $SDSS$ .",
"The 159 extended sources present in UVIT 3$\\sigma $ catalog with photo-$z$ are subjected to rigorous classification methodology.",
"For the identification of 159 objects as galaxies, we use color-color diagrams described in [10] that classifies photometrically selected sources into various categories of astrophysical origin.",
"This method involves broad band photometric data combining seven passbands from GALEX (FUV/NUV) to SDSS (u, g, r, i, z).",
"We use NUV$-$ g vs$.$ g$-$ i and FUV$-$ NUV vs$.$ g$-$ r color-color diagrams as extended sources (mostly galaxies) are well separated from point-like (Star/QSO) sources on these color-color planes (Figure REF ).",
"For this purpose, we curate a catalog of point sources combining archival GALEX all-sky imaging survey (AIS) observations with archival SDSS DR12 spectroscopic catalog.",
"The point sources present in the catalog are spectroscopically classified as $STAR$ by SDSS.",
"On the other hand, the extended source catalog comprising of galaxies are taken from GALEX-SDSS-WISE Legacy catalog [73] (Hereafter, $Salim\\text{ }Catalog$ ).",
"We plot our 159 objects along with the sources present in the point source and extended catalogs on the color-color diagrams to observe that our UVIT sources mostly overlap the region corresponding to extended sources in both color-color diagrams and therefore, we consider these objects as extended sources(or galaxies) for further analysis in the work.",
"On matching SDSS spectroscopic catalog with our UVIT $3\\sigma $ catalog, we confirm two galaxy members of the Bootes Void.",
"The photometric redshifts given by SDSS ($z^\\prime $ ) have comparable (of the same order) errors ($z_\\mathrm {{err}}^\\prime $ ) associated with them.",
"Hitherto, $z^{\\prime }$ for none of the galaxies fall within the redshift of the Bootes Void i$.$ e$.$ , 0.04 $\\le z \\le $ 0.06 [44], [43].",
"However, nine galaxies are identified out of 159 photometrically selected extended sources such that $z{^\\prime \\pm z{_\\mathrm {{err}}^\\prime }}$ falls within the void's redshift range.",
"In the following section, we determine our own set of photometric redshifts for galaxies with $z^\\prime $ by modelling their broadband spectral energy distribution to select the void galaxy candidates with increased certainty."
],
[
"Determination of Photometric Redshifts and assigning candidature to the Bootes Void",
"ccccccccccc Void candidature determination based on cumulative distribution function for EAZY redshifts G No.RADECFUV$_\\mathrm {GALEX}$ NUV$_\\mathrm {GALEX}$ FUV$_\\mathrm {UVIT}$ NUV$_\\mathrm {UVIT}$$z^\\prime $$z$ nP($z$ ) G114:07:25.6348:50:43.420.32$\\pm $ 0.1419.94$\\pm $ 0.0920.29$\\pm $ 0.05 19.80$\\pm $ 0.010.029$\\pm $ 0.230.055$\\pm $ 0.01210.54 G214:07:39.5548:57:33.1--23.31$\\pm $ 0.2223.07$\\pm $ 0.070.449$\\pm $ 0.1540.057$\\pm $ 0.06420.12 G314:08:43.4448:54:10.8-22.55$\\pm $ 0.4122.27$\\pm $ 0.1422.47$\\pm $ 0.050.102$\\pm $ 0.0470.043$\\pm $ 0.01110.56 G414:07:53.3349:01:13.121.70$\\pm $ 0.3221.74$\\pm $ 0.2521.52$\\pm $ 0.0921.91$\\pm $ 0.030.177$\\pm $ 0.0940.057$\\pm $ 0.00310.77 G514:09:09.3449:08:49.2-21.97$\\pm $ 0.2822.20$\\pm $ 0.1322.05$\\pm $ 0.040.253$\\pm $ 0.1820.053$\\pm $ 0.05520.14 G614:07:40.0849:05:10.521.10$\\pm $ 0.1920.73$\\pm $ 0.1420.87$\\pm $ 0.0720.58$\\pm $ 0.020.078$\\pm $ 0.0230.055$\\pm $ 0.01410.50 Col ID: (1) Galaxy No., (2) RA J2000, (3) DEC J2000, (4) GALEX FUV mag, (5) GALEX NUV mag, (6) UVIT FUV mag, (7) UVIT NUV mag, (8) photometric redshift given by SDSS, (9) photometric redshift calculated using EAZY, (10) n = $\\frac{| z-{\\mu _z}|}{\\sigma _z}$ , (11) cumulative probability of the galaxy at redshift $z$ to be located in the redshift range of the void P($0.04 \\le z \\le 0.06$ ).",
"Magnitudes in the table are not corrected for Galactic extinction.",
"We determine the photometric redshifts of all 159 photometrically selected galaxies using the photo-$z$ code called EAZY [13].",
"In the process, the photometric fluxes of seven broadband filters were utilized namely, UVIT (FUV, NUV) and SDSS (u, g, r, i, z).",
"EAZY provides us ${\\chi }^2$ minimized redshift for the best fit linear combination of all galaxy templates.",
"We use six standard galaxy templates in our calculations and derive photometric redshifts ($z_\\mathrm {phot}$ ) for all galaxies.",
"The procedure is inclusive of the wavelength dependent template error function.",
"Figure: Left Panel: The probability density function P(z) as a function of redshift color-coded for four void galaxies with z phot z_{\\rm phot} based on our EAZY redshift estimation with P(z) ≥\\ge 0.5.",
"Redshift extent of the Bootes Void marked with black dotted lines.",
"The galaxies are numbered according to Table .",
"Right Panel: EAZY best fit spectral energy distributions with observed fluxes for the four void galaxies.",
"The galaxies are color-coded and numbered same as the Left panel.In order to find the quality of our fit, we deduce photometric redshifts for the two void galaxies with $z_\\mathrm {spec}$ in a similar manner with an equal number of broadband filter magnitudes and calculate $\\Delta z = |z_\\mathrm {phot}-z_\\mathrm {spec}|/(1+z_\\mathrm {spec})$ .",
"The quantity ${\\Delta z}$ for the two void galaxies averages to $\\approx $ 0.01 wherein conventionally one discards $z_\\mathrm {phot}$ with $\\Delta z$ $>$ 0.1 [80].",
"The result indicates fair agreement between the EAZY photo-z and SDSS spectroscopic redshifts.",
"EAZY also provides us with 1-, 2- and 3$\\sigma $ confidence intervals computed from posterior probability distribution for each galaxy.",
"The mean redshift ($\\mu _z$ ) and sigma ($\\sigma _z$ ) for the probability distributions were calculated post assuming these distributions as a single peak Gaussian function.",
"Nearly, all $z_\\mathrm {phot}$ fall within 1$\\sigma $ confidence limits of the posterior probability distribution (see $n$ in Table REF ) and the uncertainty in $z_\\mathrm {phot}$ were given by $|z_\\mathrm {phot}-{\\mu _z}|$ .",
"EAZY photometric redshift of six galaxies lie in the redshift range of the Bootes Void.",
"The cumulative probabilities for all six galaxies to exist inside the void corresponding to their EAZY redshift (P($0.04 \\le z \\le 0.06$ )) and the output redshifts are given in Table REF .",
"Evidently, P($z$ ) gives a clear depiction of the void candidature among the six galaxies.",
"With a cut of P($z$ ) $\\gtrsim $ 0.5, we further narrow down our potential void candidates sample to four galaxies (G1, G3, G4, and G6) with $z_\\mathrm {phot}$ which we intend to study further in the work.",
"Furthermore, the probability distribution function P($z$ ) as a function of $z$ , and the model spectral energy distributions and observed fluxes for the four galaxies are shown in the left and right panel of Figure REF , respectively.",
"With this work, we report four newly detected void galaxies with $z_\\mathrm {phot}$ (G1, G3, G4 and G6) along with two previously identified void galaxies with $z_\\mathrm {spec}$ .",
"Figure REF shows the redshift distribution of galaxies in the direction of the Bootes Void.",
"The void is roughly considered as spherical in shape.",
"The red circle marks a radius of $\\sim $ 46 Mpc from the center and roughly denotes the boundary of the void.",
"As described in [42], we assume the center of the void at $\\alpha $ = 14$^h$ 48$^m$ , $\\delta $ = +47$^d$ and mean redshift of $\\sim $ 0.05.",
"Most of our UVIT detected void galaxies lie close to the boundary of the void.",
"A follow up spectroscopic survey would be required to confirm void membership of the four galaxies with z$_\\mathrm {phot}$ .",
"Figure: The cone diagram shows the redshift distribution of galaxies in the direction of Bootes Void.",
"The black dots in the wedge diagram are sources taken from SDSS DR12 spectroscopic catalog and the pair of black dashed lines encloses the solid angle subtended by our UVIT pointing.",
"The red open circle marks the boundary of the Bootes Void; green open circles are the confirmed void galaxy members (with z spec z_\\mathrm {{spec}}); blue open circles represents void galaxies with z phot z_\\mathrm {{phot}}.",
"The likelihood for each galaxy with z phot z_\\mathrm {{phot}} to be located in the void is higher than 50% (see Table )."
],
[
"UVIT Detections",
"Prior to UVIT, GALEX has observed our FoV in both FUV and NUV passbands as part of its all-sky imaging survey (AIS).",
"We compare the SNRs of the void galaxies with $z_\\mathrm {phot}$ in GALEX and UVIT FUV observations to showcase the enhanced sensitivity of the UVIT deep imaging survey.",
"We use the following equation for our calculation of SNRs [70].",
"$SNR = \\frac{F_\\mathrm {g} t \\epsilon }{\\sqrt{F_\\mathrm {g} t \\epsilon + f_\\mathrm {b} n_\\mathrm {pix} t \\epsilon } } ,$ where $f_\\mathrm {b}$ denotes the background noise per pixel (estimated from the final science-ready images) and $t$ is exposure time for UVIT FUV observation as mentioned in section .",
"$F_\\mathrm {g}$ denotes the total number of detected photons from the source alone within a given aperture containing $n_\\mathrm {pix}$ pixels measured from the science-ready images.",
"In other words, $F_\\mathrm {g}$ denotes the total number of detected photons minus the number of background photons from the same aperture.",
"For comparison with GALEX observation, we consider a circular aperture of $r$ $\\approx $ 2$^{\\prime \\prime }$ .5 at a fixed position corresponding to the RA/DEC of sources present in the SDSS catalog placed on GALEX and UVIT FUV images to estimate the total detected photons from the galaxies ($F_\\mathrm {g}$ ).",
"We have visually checked that there are no other sources within this circular aperture.",
"The values of $f_\\mathrm {b}$ and $t$ for the particular GALEX FUV tile are $3 {\\times }$ 10$^{-4}$ cps and 205 sec, respectively.",
"Often, there is a small to moderate variation in the exposure time across the FoV due to the fact that edges receive less exposure than the center of the FoV.",
"Therefore, we correct our SNRs for this effect by introducing a factor $\\epsilon $ in Equation REF .",
"We determine $\\epsilon $ by taking the ratio of mean effective exposure across the image to the maximum exposure received at the center of the image.",
"In the case of GALEX, we use a high resolution relative response map described in [57] corresponding to our FoV and evaluate $\\epsilon $ $=$ 0.81.",
"When we closely examine the UVIT exposure map generated from the L2 pipeline, we do not find such a gradual decrease in the effective exposure time as we move outwards at least upto 13$^{\\prime }$ from the center.",
"However, beyond 13$^{\\prime }$ radius, the exposure time falls off sharply.",
"On considering the entire FoV, we deduce $\\epsilon $ $=$ 0.98 for UVIT.",
"Interestingly, within 13$^{\\prime }$ from the centre of the FoV, we note that there can be a variation of $\\simeq 2$ % in the exposure time over 8 pixels (exposure map appears to have a Moire pattern).",
"This variation induces $\\simeq 1$ % uncertainty in the reported SNRs, see Figure REF .",
"Our estimates of the SNRs are labeled in Figure REF with the cutout images of the four void galaxies from SDSS r, UVIT FUV and GALEX FUV images.",
"We observe that the UVIT SNRs are much higher than the GALEX SNRs.",
"Only one galaxy (G1 in Figure REF ) cross the limiting SNR = 3 in GALEX.",
"Based on our SNR calculation and photo-$z$ estimation, we find three new void galaxies in our FUV observation i$.$ e$.$ , G3, G4 and G6.",
"In addition, we refer to GALEX merged catalog for procuring FUV/NUV magnitudes with errors and for the size of Kron apertures used for photometry for all six void galaxy candidates [9].",
"Table REF lists GALEX and UVIT FUV/NUV magnitudes for all void galaxy candidates.",
"Of these, three galaxies fainter than 22 mag in UVIT FUV observation are not detected in GALEX catalog (G2, G3 and G5).",
"It is worth mentioning here that GALEX AIS reaches a typical 5$\\sigma $ depth of $\\sim $ 20 AB mag in FUV [9].",
"Moreover, on overlaying GALEX FUV Kron apertures on SDSS cutouts, we find a few nearby sources within the apertures in case of G1 and G4 making them unfit in the subsequent analysis.",
"Figure: The figure shows the cutout images of four void galaxies with z phot z_\\mathrm {phot} in three filters (top row:SDSS r, middle row: UVIT FUV and bottom row: GALEX FUV).",
"The blue colored numbers in the top row are assigned to the galaxies according to Table .",
"The center of circles inscribing the sources is as per the RA/DEC given in SDSS catalog.",
"The radius of each circle shown in the figure is 2 '' ^{\\prime \\prime }.5.",
"The signal-to-noise ratios (SNRs) for all sources estimated from UVIT and GALEX FUV observations are written beneath the middle and bottom row, respectively.Figure REF shows a color composite image of a portion of our FoV using SDSS i, g and UVIT NUV filters.",
"We highlight all four void galaxies discussed earlier in this section.",
"The result underlines the importance of using the UVIT deep observations over GALEX AIS data for this analysis.",
"Figure: The color composite image (Red: SDSS i-filter, Green: SDSS g-filter, Blue: UVIT NUV filter) highlights the four void galaxies out of which three (G3, G4, and G6 in Figure ) are undetected in GALEX FUV observation.",
"The serial numbers given to the galaxies are in reference to Table ."
],
[
"EXTINCTION IN THE UV CONTINUUM",
"Internal dust present within a galaxy scatters and/or absorbs UV photons which makes it strenuous to estimate the absolute UV flux emitted from a galaxy.",
"Several factors such as the geometry of a galaxy, amount of dust and its components affects the intensity of UV flux attenuation in a galaxy.",
"There are various dust attenuation laws available for local and high redshift star-forming galaxies [17], [16], [63].",
"Different methods could be used to solve for dust obscuration of UV photons which are based on two principles: Using slope (${\\beta }$ ) of a power-law function ($f_{\\lambda } = {\\lambda }^{\\beta }$ ) followed by UV continuum emission of galaxies over the wavelength range 1300 - 2600 Å [17].",
"The other method is based on total energy budget of a galaxy and it represents a combination of FUV and IR luminosities [33].",
"UV ${\\beta }$ slope efficiently works as a diagnostic for internal dust attenuation [93], [53] and far-IR luminosities are unavailable for our entire sample of void galaxies.",
"Therefore, we use a method based on UV spectral slope ${\\beta }$ .",
"The values of $\\beta $ are calculated using the following relation [58]: ${\\beta } = -\\frac{m({\\lambda }_{1})-m({\\lambda }_{2})}{2.5\\log (\\frac{\\lambda _{1}}{\\lambda _{2}})}-2$ Here, ${\\lambda }_{1}$ and ${\\lambda }_{2}$ are the effective wavelengths corresponding to UVIT FUV and UVIT NUV filters.",
"The slope, thus, calculated can be used to find color excess $E(B-V)$ using the following relation REF [64]: ${\\beta } = - 2.616 + 4.684 E ( B - V )$ The relation is derived using Calzetti $+$ 00 dust curve [16] on BPASS galaxy model [64].",
"The dust attenuation law established by [16] is used to find the value of k(${\\lambda }$ ) for F154W filter of UVIT.",
"The extinction relation, A$_\\mathrm {\\lambda } = k({\\lambda })E(B-V)$ (where $k(1541\\, \\mbox{\\normalfont Å}) = 10.18$ ) gives us total extinction in the FUV filter.",
"The resultant A$_\\mathrm {FUV}$ vs$.$ ${\\beta }$ curve obtained by the above discussed method is less steeper than [53] curve.",
"The slope $\\beta $ provides us rough estimates of the ongoing star formation and internal dust obscuration of a galaxy [64].",
"Lesser negative values of $\\beta $ symbolises either the abundance of old stellar type or high internal dust concentration within a galaxy.",
"${\\beta }$ for our sample ranges from $-$ 2.72 to $-$ 0.60 with median $\\approx $ $-$ 1.35 indicating active ongoing star formation with low to moderate internal dust obscuration [96].",
"Henceforth, the intrinsic FUV luminosities of galaxies are used to calculate the FUV SFRs as described in the next section .",
"In the following part of the work, unless mentioned otherwise, all colors and absolute magnitudes are corrected for Galactic extinction only, while the SFRs reported are corrected also for internal extinction."
],
[
"Stellar mass estimation",
"Stellar masses (M$_{*}$ ) of galaxies are widely considered as one of the fundamental parameters that drive galaxy evolution over cosmic time.",
"Not only galaxy evolution, unbiased, robust estimate of stellar masses can play a crucial role to constrain models of galaxy formation as well [74].",
"We estimate stellar masses of the void galaxies and the remaining non-void galaxies upto z $\\le $ 0.1 present in the FoV using two methods.",
"In our first method, we perform broad-band (from AstroSat/UVIT far-UV to SDSS z-band) SED modelling using Code Investigating GALaxy Emission (CIGALE) [12]; similar to previous section REF .",
"Our SED modelling proceeds with standard assumptions for the star formation histories (SFH), initial mass function (IMF), dust attenuation, etc$.$ .",
"We adopt a double exponential function for the SFH with $SFR(t)\\text{ }{\\propto }\\text{ }exp(-t/{\\tau })$ forming bulk of the stellar mass and another exponential function to accommodate the recent burst of star formation.",
"In the previous expression, $t$ is the time since onset of star formation and ${\\tau }$ is e-folding timescale.",
"The young and old stellar populations are separated by 10 Myr.",
"The intrinsic stellar population in the galaxy is modelled with a [14] stellar population library.",
"We choose [75] IMF with a range of masses varying from 0.1 - 100 M$_{\\odot }$ for determining the intrinsic population.",
"The metallicity for each galaxy was given as a free parameter (to chose from an array of values [0.0004, 0.004,0.008,0.02]) in the fitting.",
"For the dust attenuation, we adopt the module, dustatt_modified_starburst based on [16] starburst attenuation curve.",
"The input parameters for color excess or reddening of stellar continuum and nebular lines are provided in accordance with our dust attenuation calculation in previous section .",
"We fix the power law slope ($\\delta $ ) of the dust attenuation curve to $-0.5$ which is steeper than the [16] curve ($\\delta $ = 0) and the UV bump amplitude to $1.0$ , respectively.",
"In addition, we use [24] module to model polycyclic aromatic hydrocarbons emission.",
"Under this module, we consider no AGN contribution and IR power law slope is set to $2.0$ .",
"The above mentioned modules and input parameters remain unchanged throughout the process.",
"In the second method, color-based stellar masses (M$_{*color}$ ) for individual galaxies are obtained from the relation between $g-r$ color and stellar mass-to-light ratio corresponding to optical luminosity (L$_\\mathrm {r}$ ) using [5] which is based on `diet' Salpeter IMF.",
"Later we multiply M$_{*color}$ by a factor of $0.7$ to scale it to normal Salpeter IMF for appropriate comparison with SED-based stellar masses (M$_{*SED}$ ).",
"Figure REF shows one to one relation between M$_{*SED}$ and M$_{*color}$ .",
"Both set of stellar masses are in close agreement with each other.",
"Henceforth, we use M$_{*SED}$ throughout the work for analysis.",
"Void galaxies with z$_{phot}$ reported in the work are low-mass systems (M$_*$ $\\lesssim $ 10$^{9}$ M$_{\\odot }$ ) and evidently, M$_*$ for most the void galaxies lies below the stellar mass of a L$_{*}$ galaxy, i$.$ e$.$ , $3 \\times 10^{10}$ M$_{\\odot }$ .",
"Figure: Comparison between SED-derived and color-derived stellar masses for void and non-void galaxies (zz ≤\\le 0.1) in our FoV.",
"Dashed line represents one to one relation while dotted lines represents 1σ\\sigma scatter."
],
[
"FUV STAR FORMATION RATE",
"The star formation rate provides key insight into the assembly history of a galaxy's stellar mass.",
"Far-UV fluxes emitted by young, massive stars (typically O, B type) amounts to the instantaneous star formation in a galaxy.",
"In other words, far-UV fluxes (if internal extinction corrected) can provide one of the best estimates of the recent star formation (over $\\sim 100$ Myr) in a galaxy.",
"The FUV emission and the associated SFR has been estimated in galaxies with different Hubble types ranging from late-types to early-types [15], [97].",
"We have calculated the FUV star formation rate (in units of M$_{\\odot }$ yr$^{-1}$ ) using the following relation given by [41].",
"$SFR_\\mathrm {FUV}=1.4 \\times 10^{-28} L_\\mathrm {FUV} (ergs\\ s^{-1}\\ Hz^{-1})$ Where L$_{\\mathrm {FUV}}$ is the intrinsic FUV luminosity of a galaxy.",
"The FUV SFRs are calibrated assuming that the star formation history of a galaxy is constant for the last $\\sim 100$ Myr.",
"In Table , we show the SFR along with UV magnitudes (FUV/NUV), stellar masses, absolute magnitudes (M$_\\mathrm {r}$ ), optical color g$-$ r, and UV$-$ optical color NUV$-$ r for our sample of void galaxies.",
"The FUV SFRs for the void galaxies detected in the FoV spans a wide range from 0.05 M$\\odot $yr$^{-1}$ to 51.01 M$\\odot $yr$^{-1}$ with median SFR$_\\mathrm {FUV}$ $\\sim $ 3.96 M$\\odot $yr$^{-1}$ .",
"The FUV SFRs for most of our sample galaxies are comparable to that of a normal spiral galaxy within the local volume and are higher than that of a low-mass, star-forming dwarf galaxy.",
"In Figure REF , we show the distribution of void and non-void galaxies on the FUV sSFR-M$_*$ plane.",
"The background galaxies comprise of $Salim\\text{ }Catalog$ with $z$ $\\le $ 0.1.",
"We compute the internal dust corrected FUV SFRs and color-based stellar masses for the background sample using the same recipe as described in the preceding sections.",
"The background galaxies from $Salim\\text{ }Catalog$ are well distributed over the sSFR$-$ M$_{*}$ plane.",
"However, as Figure REF shows, most of the void galaxies with photo-z lie on the low-mass end of the distribution and they are basically vigorously star-forming galaxies with $\\log (\\mathrm {sSFR})$ ranging from $-$ 9.5 yr$^{-1}$ to $-$ 7.7 yr$^{-1}$ with a median $\\approx $ $-$ 9.09 yr$^{-1}$ .",
"These values signify that all the void galaxies detected in our work are star-forming in nature.",
"Even the most massive galaxy in our sample belongs to the star-forming cloud.",
"Interestingly, the sSFR for the non-void galaxies detected in our FoV are comparable to those of void galaxies.",
"ccccccccccc Photometric details, SFR$_\\mathrm {FUV}$ and stellar masses of six void galaxies reported in the work.",
"S No.",
"RA DEC FUV$_\\mathrm {AB}$ NUV$_\\mathrm {AB}$ $z$ SFR$_\\mathrm {FUV}$ M$_{*}$ M$_\\mathrm {r}$ g$-$ r NUV$-$ r J2000 J2000 mag mag M$\\odot $ yr$^{-1}$ 10$^{10}$ M$\\odot $ mag mag mag G1 14:07:25.63 48:50:43.4 20.29$\\pm $ 0.05 19.80$\\pm $ 0.01 0.055$\\pm $ 0.012 8.874 0.044 $-$ 18.47 0.13 1.13 G3 14:08:43.44 48:54:10.8 22.27$\\pm $ 0.14 22.47$\\pm $ 0.05 0.043$\\pm $ 0.011 0.053 0.011 $-$ 16.00 0.35 1.84 G4 14:07:53.33 49:01:13.1 21.52$\\pm $ 0.10 21.91$\\pm $ 0.04 0.057$\\pm $ 0.004 0.088 0.029 $-$ 16.74 0.26 1.44 G6 14:07:40.08 49:05:10.5 20.88$\\pm $ 0.07 20.58$\\pm $ 0.02 0.055$\\pm $ 0.014 2.253 0.093 $-$ 18.44 0.34 1.91 S1 14:08:13.59 48:51:44.7 19.06$\\pm $ 0.03 18.40$\\pm $ 0.01 0.0518$\\pm $ 0.0001 51.010 7.009 $-$ 22.08 0.60 3.35 S2 14:08:11.40 48:53:44.4 19.36$\\pm $ 0.04 19.15$\\pm $ 0.01 0.0511$\\pm $ 0.0002 5.668 0.631 $-$ 20.16 0.39 2.30 Colors and absolute magnitudes are K-corrected and extinction corrected.",
"SFR$_{FUV}$ are also corrected for internal extinction.",
"G1, G3, G4 and G6 - void galaxies with $z_\\mathrm {phot}$ ; S1 and S2 - void galaxies with $z_\\mathrm {spec}$ .",
"Figure: FUV sSFR vs.. Stellar mass for void and non-void galaxies (zz ≤\\le 0.1).",
"The trend of local galaxies shown using SalimCatalogSalim\\text{ }Catalog in the background.",
"Dotted line represents galaxies with SFR = 1 M ⊙ _{\\odot }yr -1 ^{-1}."
],
[
"COLOR-MAGNITUDE DIAGRAMS",
"In this section, we summarize the results from the UV/optical/NIR color-magnitude diagrams (CMDs) to study the properties of our sample void galaxies.",
"Our void galaxies are divided in two categories, i$.$ e$.$ with $z_\\mathrm {spec}$ and z$_\\mathrm {photo}$ , based on the means of their redshift determination.",
"The galaxies detected outside the Bootes Void having either redshifts (photometric/ spectroscopic) are termed as non-void galaxies with $z$ $\\le $ 0.1 in the subsequent CMDs."
],
[
"UV color$-$ magnitude diagram",
"The FUV$-$ NUV color for a large sample galaxies varies across $\\sim $ 2 mag (see Figure REF ).",
"In general, star-forming galaxies are found to have an average FUV$-$ NUV color $\\approx $ 0.4 mag and the color peaks at 0.9 mag where the transition from late (young) to early (old) type galaxies takes place [27].",
"In Figure REF , we have shown UV CMD distribution for all galaxies detected in our FoV upto $z$ ${\\le }$ 0.1 wherein the side color bar represents their internal dust corrected FUV SFRs.",
"The FUV$-$ NUV color of the void galaxies (with z$_\\mathrm {spec}$ or z$_\\mathrm {photo}$ ) are inclined towards the bluer end of the color scale with an average value of $\\approx $ 0.2 mag indicating recent star formation in these systems along with late type or irregular morphological features.",
"Based on the FUV$-$ NUV colors and FUV SFRs, it is apparent that void galaxies comprise of a significant amount of young stellar population.",
"However, we do not observe any strong correlation between FUV SFRs and FUV$-$ NUV color for our entire sample of galaxies which is in agreement with [36].",
"[94] derived UV luminosity function for local galaxies ($z$ $\\le $ 0.1) for which the characteristic NUV magnitude (M$^{*}_{\\mathrm {NUV}}$ ) came out to be $-$ 18.23 mag whereas M$_{\\mathrm {NUV}}\\text{ }\\epsilon $ [$-$ 14.16, $-$ 18.65] mag for our sample of void galaxies implying that the distribution of our void sample traverses both the galaxy population type.",
"Figure REF shows no major difference in FUV SFRs of the void and non-void galaxies.",
"Previously reported work such as [91], [7], [21] deduce similar results where impact of the environment on the SFRs of galaxies were found to be insignificant.",
"However, the total fraction of blue/red galaxies is strongly dependent on the environment at a given stellar mass range [25], [3].",
"Figure: FUV--NUV vs.. M NUV _{\\mathrm {NUV}} CMD for all galaxies (z ≤\\le 0.1) detected in the UVIT FoV.",
"Each symbol represents a galaxy colour coded with FUV SFR.",
"Filled diamonds are the void galaxies with z phot _\\mathrm {phot} detected in our FoV whereas filled stars denotes void galaxies with z spec z_{\\mathrm {spec}}.",
"The open circles represent non-void galaxies detected in our survey."
],
[
"UV$-$ NIR color-magnitude diagram",
"In Figure REF , the background galaxies are from $Salim\\text{ }Catalog$ ($z$ $\\le $ 0.1).",
"We refer to 2MASS all-sky Extended Source Catalog (XSC) [38] for procuring K-band magnitudes for all galaxies present in $Salim\\text{ }Catalog$ .",
"The 2MASS XSC magnitudes are converted to AB magnitude system using the relation given in [11].",
"On NUV$-$ K vs$.$ M$_\\mathrm {K}$ color$-$ magnitude plane, the distribution of galaxies is bivariate as can be seen in Figure REF .",
"The NUV$-$ K color provides a range of ${\\approx }$ 8 mag which can be used efficiently to distinguish between galaxies based on their morphologies and stellar population type (early/late).",
"Also, K$-$ band luminosity is a tracer for total stellar mass of a galaxy [5].",
"As most of our photometrically verified void galaxies are absent in the NIR observations, therefore, we only study properties of void galaxies with spectroscopic observations using this CMD (see Figure REF ).",
"We scale (NUV$-$ K)$_\\mathrm {AB-Vega}$ color from [27] to (NUV$-$ K)$_\\mathrm {AB-AB}$ magnitude system following prescriptions given by [11] and find that the blue sequence comprising of spirals and irregular galaxies peak at 3.55 mag.",
"The two void galaxies belong to the blue sequence as seen in Figure REF .",
"Here, the absolute magnitudes, M$_\\mathrm {K}$ of these galaxies show a striking difference of two magnitude implying a significant variation in their total stellar masses.",
"Figure: NUV--K vs.. M K _\\mathrm {K} CMD of galaxies detected in UVIT.",
"The background galaxies (grey circles) are from SalimCatalogSalim\\text{ }Catalog upto (zz ≤\\le 0.1) .",
"Black open circles represent galaxies residing outside the Bootes Void in our FoV (zz ≤\\le 0.1).",
"Void galaxies with z spec z_{\\mathrm {spec}} are shown by green star-shaped symbols"
],
[
"Galaxy Bimodality using optical colors",
"Optical colors have been quite successful in classifying galaxies in the local Universe [82].",
"Galaxies present in local Universe can be broadly classified into two categories, i$.$ e$.$ , star formation quenched galaxies which are dominated with elliptical and S0s, likely to be found in denser environments and actively star-forming galaxies with spiral, disc-like and irregular morphologies mostly residing in the sparse environment [39].",
"These galaxies tend to separate themselves into two groups based on UV$-$ optical, optical$-$ optical, UV$-$ NIR colors up to $z$ $\\sim $ 1 [4], [97], [95].",
"In Figure REF , we show g$-$ r vs$.$ M$_\\mathrm {r}$ color-magnitude distribution that is circumcentered around two modes: Blue Cloud peaking at g$-$ r = 0.5 mag and Red Sequence peaks at g$-$ r = 0.9 mag.",
"Galaxies which fall in between the two groups are said to be Green Valley galaxies [72].",
"Figure REF show optical CMD of UVIT identified void and non-void galaxies present in our FoV.",
"In the background, we use a magnitude limited sample of 1,16,010 galaxies brighter than r $<$ 17.77 mag from SDSS upto $z$ ${\\lesssim }$ 0.1 to construct the color magnitude contours.",
"Nearly all our UVIT detected void galaxies belong to the Blue Cloud population, which fits the conventional understanding of galaxy formation and evolution.",
"Thereby, the red counterpart of the bimodal distribution is unseen in our void sample.",
"The two spectroscopically verified void galaxies belong to two different population type, i$.$ e$.$ , the Blue Cloud (image labelled as d in Figure REF ) and the Green Valley (image labelled as c in Figure REF ).",
"The remaining void galaxies with z$_{\\rm phot}$ are blue in color with late type morphologies.",
"In totality, our sample follows a similar trend on the given optical CMD as shown by the galaxies present in Void Galaxy Survey (VGS) [45] (see Figure REF ).",
"The absolute magnitudes, M$_\\mathrm {r}$ , for most of our sample and the VGS is fainter than ${\\approx }$ $-$ 20 mag.",
"A few of the galaxies from VGS are the members of the Red Sequence as seen in Figure REF .",
"However, we find none such galaxies for our sample.",
"We observe that the non-void galaxies detected in our FoV belong to both the population type; spanning a wide range of optical color and luminosity while void galaxies majorly confine to the bluer and fainter end of the optical CMD."
],
[
"UV $-$ Optical color-magnitude diagram",
"We have shown NUV$-$ r vs$.$ M$_\\mathrm {r}$ CMD for galaxies observed in our FoV in the left panel of Figure REF .",
"Background sample comprises galaxies in $Salim\\text{ }Catalog$ ($z$ $\\le $ 0.1).",
"The bivariate distribution of galaxies as a function of NUV$-$ optical color and optical absolute magnitude is clearly visible.",
"We fit the following relations to the peak color as a function of the absolute magnitude in the red sequence REF and blue sequence REF , respectively [95]: $(NUV - r) = 1.897- 0.175M_{r}$ $(NUV - r) = 2.39 + 0.075(M_{r} + 20) - 0.808\\tanh {\\frac{M_r + 20.32}{1.81}}.$ The UV$-$ optical CMD have been extensively used in literature to follow the evolution of galaxies from the blue sequence to the red sequence, to study the evolution of early-type galaxies, and to deduce the mechanism responsible for star formation quenching [95], [52], [40].",
"The NUV$-$ r color is a tracer of minimal amounts ($\\sim $ 1% mass fraction) of recent star formation ($\\le $ 1Gyr) (RSF) [77].",
"[40] suggest that galaxies with NUV$-$ r $<$ 5.5 mag are likely to have undergone RSF confirming episodes of RSF for our void galaxies.",
"The non-void galaxies in our FoV are distributed among both the population type, but we do not observe such a bimodality within the UVIT identified void galaxies.",
"The NUV$-$ r color histogram on the right panel of Figure REF shows the color distribution for our sample and for galaxies present in $Salim\\text{ }Catalog$ .",
"The galaxies from $Salim\\text{ }Catalog$ show a clear bivariate distribution which fits well with a double peaked Gaussian function.",
"The mean NUV$-$ r colors for the blue and red sequences are $\\mu _{Salim}^{blue} = 3.02$ mag and $\\mu _{Salim}^{red} = 5.36$ mag, respectively whereas mean $\\mu _{UVIT}^{VG}$ for our sample calculated by fitting a single component Gaussian profile equals $1.99$ mag.",
"The spread in the NUV$-$ r color for our UVIT detected void galaxies is unimodal, and centered below $\\mu _{Salim}^{blue}$ .",
"Moreover, we perform Kolmogorov-Smirnov (KS) and Anderson-Darling (AD) tests on the NUV$-$ r color distribution of our void galaxies and the blue sequence of $Salim\\text{ }Catalog$ (NUV$-$ r $\\le $ 4) to find whether the distribution of NUV$-$ r color for the void galaxies are a subset of a larger sample of local galaxies.",
"With p-value = $0.007$ , high KS statistic (= $0.64$ ) and AD statistic (= $6.40$ ), we reject null hypothesis at a significance level = $0.05$ and infer that both sets of color belong to different parent populations.",
"We acknowledge that our sample size for void galaxies is not significant enough for a strong statistical inference.",
"The blue-ward shift in the NUV$-$ r color of our void galaxies could be seen as a consequence of their low-density environment.",
"Intriguingly, we detect a few older (red) galaxies (NUV$-$ r $>$ 4) outside the Bootes Void using UVIT observation, however, none of the void galaxies is seen to be passive, red and dead.",
"Based on various CMDs studied in this work, we show that the star-forming void galaxies in our sample are fainter than their counterparts present in the field/ dense environment.",
"Our sample of void galaxies lacks faint early-type galaxies such as dwarf ellipticals.",
"Perhaps, one needs to have a dedicated, high-sensitivity infrared survey of galaxies in these sparse environments."
],
[
"Discussion and conclusions",
"The work primarily focuses on the photometric properties of the void galaxies detected in the Bootes Void, for which we have an ongoing survey covering a larger fraction of void using UVIT/AstroSat.",
"The science-ready images are created first by processing the Level 1 data provided by ISRO using the official L2 pipeline.",
"The end-product of this pipeline is an L2 image which is further corrected for astrometry.",
"We use the appropriate GALEX-tiles and SDSS r-band images to correct for the astrometry in the L2 image (both in FUV and NUV).",
"The difference in morphological features of a galaxy in various wavebands may have induced a slight offset ($\\sim 0^{\\prime \\prime }.2 - 0^{\\prime \\prime }.3$ ) in the centroid (RA/ DEC) of sources in the final UVIT images (but see the color composite in Figure REF ).",
"Most of the void galaxies reported by us lack spectroscopic observations.",
"SDSS spectroscopic target selection criteria depend on the r-band apparent magnitude, and mean surface brightness [83] along with several other parameters.",
"Our analysis and previous reports on void or isolated galaxies suggest that these systems have low optical luminosities and surface brightness [45], [34], [26].",
"Therefore, one must reset the desired observational limits while surveying a void field.",
"We encounter a few false detections and discrepancies in $STAR/ GALAXY$ classification in the archival SDSS photometric catalog.",
"Hence, we perform $STAR/ GALAXY$ classification of our detected sources using UV-optical color-color diagrams.",
"We work with SDSS photometric redshifts due to the absence of spectroscopic observation for all objects detected in our FoV.",
"The error associated with SDSS photometric redshifts were significant enough to be included in our analysis.",
"Thereafter, we use EAZY for determining photo-$z$ with better precision to assign void membership to the galaxies.",
"Most of the galaxies with $z_\\mathrm {phot}$ were either absent in 2MASS images or detected with poor SNR ($\\lesssim $ 3).",
"In the process, we only use photometric fluxes of seven wavebands.",
"Hence, the lack of IR fluxes may induce slight inaccuracy in our photo-$z$ calculations.",
"Spectroscopic observations of the final sample of four void galaxies with z$_\\mathrm {phot}$ would confirm their candidature in the Bootes Void.",
"In a similar manner, we exclude IR fluxes in the SED fitting process for determining M$_\\mathrm {*SED}$ that may incur certain discrepancies in our calculations, although, we verify our results with M$_\\mathrm {*color}$ and find good correspondence between the two stellar masses in most of the cases.",
"UV emission from galaxies are subjected to extensive internal dust extinction.",
"We calculate $A_\\mathrm {FUV}$ with the help of two extreme UV broadband fluxes.",
"This method tends to be erroneous as the UV continuum may get altered by some spectral features, and by the presence of old population [62].",
"Other techniques to calculate $A_\\mathrm {FUV}$ require Balmer series line ratios - classic Balmer decrement method [31], or total IR imaging observations [33] which are not available for our entire sample.",
"The work discusses about the effect of the global environment on the FUV SFRs and sSFRs of galaxies.",
"The local affects such as galaxy interactions are not taken into account in our analysis.",
"We argue that the global environment weakly impacts the ongoing star formation in galaxies which is supported by similar studies done previously.",
"We stress on the fact that our sample size is small to provide a conclusive evidence to our findings.",
"Quantities such as, SFRs and dispersion in sSFRs distribution depend on the stellar mass range of the galaxies taken under consideration [35], [45].",
"We further plan to investigate the problem with a large and diverse sample in terms of stellar mass for a concrete understanding of the environmental effects.",
"Following are the primary scientific outcomes from our multi-wavelength analysis of star-forming galaxies present in Bootes Void: 1.",
"We present a total of six void galaxies having FUV observation based on the deep UV imaging survey carried out by AstroSat/UVIT.",
"Of these, three are new detections within the UVIT FoV.",
"2.",
"Our sample spans quite a range of stellar masses, even though, it is predominated by low-mass systems as most of them have stellar masses below L$_*$ galaxies.",
"3.",
"The SFRs are corrected for the internal dust extinction using UV spectral slope $\\beta $ .",
"The resultant values of $\\beta $ suggest low to moderate dust obscuration in the void galaxies.",
"4.",
"The median SFR$_\\mathrm {FUV}$ for the reported void galaxy sample $3.96$ M$\\odot $yr$^{-1}$ .",
"The FUV SFRs of void galaxies are comparable to non-void low-mass, star-forming galaxies present in our sample.",
"The ongoing moderate to high SFRs indicate the abundance of young massive O, B-type stars.",
"Void galaxies show high values of sSFRs with median log(sSFR) $\\approx $ $-9.09$ yr$^{-1}$ signifying on-going star formation at rapid timescales.",
"5.",
"The UV, optical and NIR color-magnitude diagrams show that our void galaxies are bluer in color and possess disc-like, irregular morphologies, in some cases with spiral features.",
"The most of our void galaxies have optical and UV luminosities less than L$^{\\ast }$ galaxies.",
"6.",
"The color distribution of our void sample is confined to the blue sequence as seen in all the CMDs.",
"In particular, we found a distinct shift in the NUV$-$ r color distribution (Figure REF ) in our sample when compared to the blue sequence of a larger sample of local galaxies.",
"This implies that galaxies present in voids are bluer than their counterpart present in the field or denser environment.",
"7.",
"Galaxies belonging to the red sequence are missing from our sample.",
"Perhaps, a deeper infrared observation of the void region is in need to reach a firm conclusion.",
"It could also be possible that a handful of galaxies in the low density environment are recently formed and are not matured yet.",
"This remains to be investigated.",
"Acknowledgements: We thank the referee for providing constructive suggestions/comments.",
"The authors, DP and ACP thank Inter University center for Astronomy and Astrophysics (IUCAA), Pune, India for providing facilities to carry out this work.",
"The UVIT project is a collaboration between IIA, IUCAA, TIFR, ISRO from Indian side and CSA from Canadian side.",
"This publication uses UVIT data from the AstroSat mission of the Indian Space Research Organisation (ISRO), archived at the Indian Space Science Data Center (ISSDC).",
"Astropy [2], IRAF [87], SExtractor [6], EAZY [13], CIGALE [12]"
]
] | 2107.01774 |
[
[
"Remark on using quantum states prepared by the adiabatic quantum\n computation"
],
[
"Abstract We indicate that there are points to keep in mind in utilizing quantum states prepared by the adiabatic quantum computation.",
"Even if an instantaneous expectation value of a physical quantity for the adiabatically prepared quantum state is close to an expectation value for the true vacuum, this does not assure us that the prepared vacuum is close to the true vacuum.",
"In general time average of the expectation value tend to systematically differ from the true value.",
"Using a simple model we discuss how to diminish this systematic difference."
],
[
"We indicate that there are points to keep in mind in utilizing quantum states prepared by the adiabatic quantum computation.",
"Even if an instantaneous expectation value of a physical quantity for the adiabatically prepared quantum state is close to an expectation value for the true vacuum, this does not assure us that the prepared vacuum is close to the true vacuum.",
"In general time average of the expectation value tend to systematically differ from the true value.",
"Using a simple model we discuss how to diminish this systematic difference.",
"PACS numbers:03.65.-w, 03.67.-a Recently, several quantum systems are analyzed by quantum computers or quantum simulators with not so many qubits[1].",
"The quantum annealing[2] gives a fundamental principle of the D-wave.",
"The adiabatic quantum computation, advocated by Farhi et.al.",
"[3] more than two decades ago, can also be carried out for quantum systems with not so many number of qubits.",
"For some quantum field theories the adiabatic quantum computation have been used for preparing ground states[4], [5].",
"After preparing ground states, it has been observed that an expectation value of certain physical quantity varies significantly under a constant Hamiltonian[4].",
"This oscillation originates from the deviation of the prepared vacuum from the true vacuum.",
"It also has been observed that a time average of the expectation value of the physical quantity systematically differs from the exact value computed by another method.",
"The purpose of this paper is to indicate that, even if an instantaneous expectation value of a physical quantity for the approximate vacuum prepared by the adiabatic method is close to the expectation value for the true vacuum by chance, it is inevitable in general that an expectation value of a physical quantity for the approximate vacuum significantly oscillates in time around a point that slightly differs from an expectation value for the true vacuum.",
"We also discuss how to diminish this systematic difference.",
"According to the adiabatic quantum computation[3], we start from a simple Hamiltonian $H_{0}$ that has a non-generate trivial vacuum.",
"We gradually change the Hamiltonian in time to a target Hamiltonian $H_{T}$ that we should analyze.",
"The quantum adiabatic theorem[6], [7] assures us that the trivial vacuum of the initial Hamiltonian $H_{0}$ approaches the vacuum of the target Hamiltonian $H_{T}$ if the change of the Hamiltonian is very moderate and there is a sufficient energy gap between the vacuum and excited states of the time varying Hamiltonian.",
"We simulate quantum adiabatic computation using the quantum simulator by IBM for the simplest one-qubit case.",
"We examine the quantum state prepared by the quantum adiabatic computation.",
"First, we choose the initial Hamiltonian ${\\hat{H}}_{0}=-JZ, J>0$ , and the target Hamiltonian ${\\hat{H}}_{T}=-JX$ .",
"The initial ground state is $|0\\rangle $ and the desired final state is $|+\\rangle $ .",
"The adiabatic Hamiltonian ${\\hat{H}}_{A}(s)$ that connects ${\\hat{H}}_{0}$ and ${\\hat{H}}_{T}$ is given by ${\\hat{H}}_{A}(s)=(1-s){\\hat{H}}_{0}+s{\\hat{H}}_{T}, \\qquad 0 \\le s \\le 1,$ where for example $s={t \\over T}, 0 \\le t \\le T$ for an adequate time period $T$ .",
"The quantum adiabatic computation starts at the time $t=0$ and finishes at the time $t=T$ .",
"For the ideal case, the ground state of the target Hamiltonian ${\\hat{H}}_{T}$ has been prepared at the time $t=T$ .",
"After the time $t=T$ , we observe a physical quantity under the constant Hamiltonian ${\\hat{H}}_{T}$ .",
"If the true ground state $|+\\rangle $ has been prepared, an expectation value of the physical quantity is constant.",
"In the actual adiabatic quantum computation we have a quantum state that is slightly different from the true vacuum $|+\\rangle $ .",
"Let us represent the state we have at the time $t=T$ as $|\\psi (t=T)\\rangle =\\alpha |+\\rangle +\\beta |-\\rangle , \\quad |\\alpha |^{2}+|\\beta |^{2}=1,$ where $|\\beta |^{2}$ is supposed to be small.",
"Under the total Hamiltonian ${\\hat{H}}_{T}=-JX$ , this quantum state time develops as $|\\psi (t)\\rangle ={\\alpha }e^{iJ(t-T)}|+\\rangle +{\\beta }e^{-iJ(t-T)}|-\\rangle ,$ where we have set the Plank constant as $\\hbar =1$ for simplicity.",
"For this state we measure the observable $Z$ .",
"The expectation value of $Z$ time develops as $\\langle \\psi (t)|Z|\\psi (t)\\rangle =2|\\alpha \\beta |\\cos (2J(t-T)+\\theta ),$ where the angle $\\theta $ is defined by $\\alpha \\beta ^{*}=|\\alpha \\beta ^{*}|e^{i\\theta }$ .",
"Thus at the time $t=T$ we get the expectation value $2|\\alpha \\beta |\\cos {\\theta }$ , which may be a good approximation of the desired value $\\langle +|Z|+\\rangle =0$ by chance.",
"The deviation, however, reaches up to $2|\\alpha \\beta |$ in the time development.",
"We can obtain the precise value by time averaging $\\langle \\psi (t)|Z|\\psi (t)\\rangle $ over a period.",
"Fig.1(a) shows one of our quantum simulation results.",
"From the peak to peak value $4|\\alpha \\beta |$ we compute the varance $2|\\alpha \\beta |^{2}$ as $0.000730$ , and we get $2|\\beta |^{2}=0.000730$ .",
"By another simulation shots we observe $-X$ that commutes with the Hamiltonian.",
"After the time $t=T$ the expectation value of $-X$ is almost constant(Fig.1(b)).",
"Its average is $-0.999320$ and its variance is $1.35 \\times 10^{-9}$ for one of our simulation result with $10^{6}$ shots.",
"The value $-0.999320$ almost agrees with the previous value $-1+2|\\beta |^{2}=-0.999270$ .",
"Second, we examine another simple one-qubit model.",
"We take the initial Hamiltonian as ${\\hat{H}}_{0}=-JZ, J>0$ , and we take the target Hamiltonian as ${\\hat{H}}_{T}=-JH$ , where $H$ is the Hadamard gate.",
"We again observe the physical quantity $Z$ .",
"We represent the eigenstates of the target Hamiltonian ${\\hat{H}}_{T}=-JH$ as $|h\\pm \\rangle $ , where they satisfy $H|h\\pm \\rangle =\\pm |h\\pm \\rangle $ .",
"The explicit expressions of $|h\\pm \\rangle $ are $|h+\\rangle ={1 \\over \\sqrt{4-2\\sqrt{2}}}(|0\\rangle +(\\sqrt{2}-1)|1\\rangle ),$ $|h-\\rangle ={1 \\over \\sqrt{4+2\\sqrt{2}}}(|0\\rangle -(\\sqrt{2}+1)|1\\rangle ),$ After the adiabatic state preparation process, the observable $Z$ time develops as $e^{i{\\hat{H}}_{T}t}Ze^{-i{\\hat{H}}_{T}t}=e^{-iJ{H}t}Ze^{iJ{H}t}={1 \\over \\sqrt{2}}H-{1 \\over \\sqrt{2}}Y\\sin {2Jt}+{1 \\over 2}(Z-X)\\cos {2Jt}.$ At the time $t=T$ if we have a state $|\\psi (t=T)\\rangle =\\alpha |h+\\rangle +\\beta |h-\\rangle , |\\alpha |^{2}+|\\beta |^{2}=1$ , instead of the desired state $|h+\\rangle $ , we have at a time $t(\\ge T)$ $\\langle \\psi (t)|Z|\\psi (t)\\rangle ={1 \\over \\sqrt{2}}(1-2|\\beta |^{2})+\\sqrt{2}|\\alpha \\beta |\\cos (2J(t-T)+\\theta ),$ where we have again set $\\alpha \\beta ^{*}= |\\alpha \\beta ^{*}|e^{i\\theta }$ .",
"Since the physical quantity $Z$ does not anti-commute with the target Hamiltonian ${\\hat{H}}_{T}=-JH$ , the expectation value $\\langle \\psi (t)|Z|\\psi (t)\\rangle $ oscillate in time around the value ${1 \\over \\sqrt{2}}(1-2|\\beta |^{2})$ that slightly less than the desired value ${1 \\over \\sqrt{2}}=0.707107$ .",
"Fig.2 shows one of our simulation results.",
"We have used the second order Suzuki-Trotter formula[8], [9].",
"In the result, a time average of $\\langle \\psi (t)|Z|\\psi (t)\\rangle $ is 0.706690, which we have computed from the average of the maximum value and the minimum value.",
"We can find the value $|\\beta |^{2}$ from the variance of the values of $\\langle \\psi (t)|Z|\\psi (t)\\rangle $ in the range $t \\ge T$ .",
"In the result, the variance $|\\alpha |^{2}|\\beta |^{2}$ is $0.0003222$ and we find $|\\beta |^{2}=0.0003223$ .",
"Thus the expectation value of $Z$ is slightly improved to $0.707145$ .",
"Thus the systematic error for the expectation value of $Z$ that is obtained from the time average has been diminished.",
"We have studied the quantum state preparation by the adiabatic quantum computation.",
"We have examined two simple 1-qubit models.",
"For the first case, the prepared quantum state is supposed to be a superposition of the true vacuum and the excited state.",
"The expectation value oscillates in time around the expectation value of the true vacuum.",
"This is rather special case that the physical quantity anti-commute with the target Hamiltonian.",
"The second model will represent rather general case.",
"We observe the physical quantity that does not anti-commute with the target Hamiltonian.",
"In this case the time average of the expectation value differs from the expectation value for the true vacuum.",
"We can diminish this difference from the time behavior of the expectation value.",
"Although our models may be simple, our analysis would be useful to grasp properties of adiabatically prepared quantum states for more complicated systems, such as $(1+1)-$ dimensional Schwinger model[4].",
"Figure Captions Fig.1(a) Simulation result of the adiabatic state preparation for the Hamiltonian ${\\hat{H}}_{T}=-JX$ by IBM qasm-simulator.",
"We have started from the ground state of ${\\hat{H}}_{0}=-JZ$ and we have observed $Z$ .",
"We have set $J=1$ , the adiabatic time period $T=36$ , and one time-step width $\\delta {t}={1 \\over 8}$ .",
"The number of shots is $10^{6}$ .",
"The orange line represent the theoretical value.",
"After the time $T$ , a time average over a period precisly leads to 0.",
"Fig.1(b) An expectation value of $-X$ .",
"We have used another $10^{6}$ shots.",
"Fig.2 Simulation result of the adiabatic state preparation for the Hamiltonian ${\\hat{H}}_{T}=-JH$ by IBM qasm-simulator.",
"We have started from the ground state of ${\\hat{H}}_{0}=-JZ$ and we have observed $Z$ .",
"We have set $J={\\pi \\over 4}$ , the adiabatic time period $T=36$ , and one time-step width $\\delta {t}={1 \\over 24}$ .",
"The number of shots is $10^{6}$ .",
"The orange line represent the theoretical value.",
"After the time $T$ , a time average over a period slightly less than the theoretical value ${1 \\over \\sqrt{2}}$ .",
"Figure: NO_CAPTION Figure: NO_CAPTION Figure: NO_CAPTION"
]
] | 2107.01743 |
[
[
"Plasmonic resonances of slender nanometallic rings"
],
[
"Abstract We develop an approximate quasi-static theory describing the low-frequency plasmonic resonances of slender nanometallic rings and configurations thereof.",
"First, we use asymptotic arguments to reduce the plasmonic eigenvalue problem governing the geometric (material- and frequency-independent) modes of a given ring structure to a 1D-periodic integro-differential problem in which the eigenfunctions are represented by azimuthal voltage and polarization-charge profiles associated with each ring.",
"Second, we obtain closed-form solutions to the reduced eigenvalue problem for azimuthally invariant rings (including torus-shaped rings but also allowing for non-circular cross-sectional shapes), as well as coaxial dimers and chains of such rings.",
"For more general geometries, involving azimuthally non-uniform rings and non-coaxial structures, we solve the reduced eigenvalue problem using a semi-analytical scheme based on Fourier expansions of the reduced eigenfunctions.",
"Third, we used the asymptotically approximated modes, in conjunction with the quasi-static spectral theory of plasmonic resonance, to study and interpret the frequency response of a wide range of nanometallic slender-ring structures under plane-wave illumination."
],
[
"Introduction",
"Localized-surface-plasmon resonance, namely the excitation of collective electric-field and electron-charge-density oscillations in metallic nanoparticles and nanostructures, has been instrumental over the past several decades in enabling new techniques for manipulating visible and near-infrared electromagnetic waves on nanometric scales, with applications including sensing, targeted heating and metamaterials [1], [2].",
"An important motif in fundamental research as well as applications has been the use of nearly singular, i.e., multiple-scale, geometries to achieve tunable and high-Q resonances, field enhancement in nanoscale `hotspots' and for probing light-matter interactions and non-classical phenomena [3], [4], [5], [6], [7].",
"The class of multiple-scale nanometallic structures most extensively studied in nanoplasmonics consists of closely spaced particles and similarly particles very near to a metallic substrate [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22].",
"Such structures exhibit a rich spectrum comprised of several distinct families of modes characterized by their symmetries, as well as behavior in the limit where the aspect ratio $\\kappa $ , say the ratio of particle radius to gap width, is large.",
"Particular emphasis has been given to bonding-gap modes, which are characterized by localization of the electric field to the gap, with the field directed across the gap.",
"For large $\\kappa $ , the characteristic angular frequencies $\\omega $ associated with these modes behave roughly like $\\omega /\\omega _p\\simeq 1/\\kappa ^{1/4}$ , wherein $\\omega _p$ denotes the plasma angular frequency [17].",
"For large $\\kappa $ , the first few bonding-gap modes strongly couple with incident radiation, giving rise to low-frequency resonances associated with a giant field enhancement in the gap.",
"Another class of multiple-scale nanometallic structures that is commonly employed consists of slender particles such as high-aspect-ratio nanorods and spheroids [23], [24], [25], [26], [27], [28], [29], [30].",
"Parallels can be drawn between the bonding-gap modes of closely spaced particles and the longitudinal modes of slender particles, which are characterized by polarization-charge distributions that for sufficient slenderness vary mainly along the particle axis.",
"The latter longitudinal modes redshift as the particle becomes more and more slender, analogously to the redshift of the bonding-gap modes described above.",
"In the slender-particle case, however, the plasmon-frequency redshift is far more singular: $\\omega /\\omega _p\\simeq 1/\\kappa $ , where now $\\kappa $ represents the ratio of length to thickness [25].",
"Accordingly, the first few longitudinal modes of a slender particle typically give rise to remarkably low-frequency plasmonic resonances (down to the near-infrared regime), which are typically associated with high Q factors and distinctive directional characteristics.",
"Slender particles with a curved centerline, such as slender rings and helixes, have the advantage of being more spatially compact relative to the free-space wavelengths corresponding to their longitudinal resonances; the latter resonances are therefore less affected by radiation damping.",
"Indeed, let the length and thickness of the particle be $2l$ and $2b$ , respectively; the scaling $\\omega /\\omega _p\\simeq 1/\\kappa $ implies longitudinal resonances at wavelengths $\\lambda $ roughly proportional to $(l/b) \\lambda _p$ , where $\\lambda _p=2\\pi c/\\omega _p$ and $c$ is the speed of light in vacuum.",
"Hence, if the characteristic linear dimension of the particle is $2a$ , we have that $a/\\lambda \\simeq (a/l)(b/\\lambda _p)$ .",
"One avenue to reducing the latter size factor and thence radiation damping is to reduce $b$ , though this is ultimately limited by manufacturing capabilities as well as non-classical effects at subnanometric scales.",
"An alternative avenue is to coil the centerline so as to decrease $a/l$ .",
"In particular, $a/l$ is unity for a straight particle and $1/\\pi $ for a circular ring.",
"The above characteristics of high-aspect-ratio nanometallic particles can be indirectly inferred from classical analytical solutions for cylinder and sphere dimers, ellipsoids and tori, which are exact in the quasi-static approximation [31], [32], [2].",
"While these solutions allow for arbitrary aspect ratios, they are usually cumbersome, such that often it is not straightforward to extract the singular behavior of these solutions in the high-aspect-ratio limit.",
"For more general geometries, analytical solutions are generally unavailable, while brute-force finite-element simulations become expensive at high aspect ratios.",
"An alternative, more insightful and versatile, theoretical approach is to consider the high-aspect-ratio limit — in which plasmonic resonance is expected to be most pronounce — from the outset, using tools of asymptotic analysis and singular-perturbation theory [33].",
"The latter tools are classical, though their application to plasmonics is relatively new and so far the focus has been on near-touching geometries [17], [20].",
"Recently, we took a first step in applying an asymptotic approach to the analysis of the plasmonic properties of slender nanometallic structures [34].",
"In particular, we developed an approximate theory of the longitudinal plasmonic resonances of slender bodies of revolution, with the thickness profile along the symmetry axis being essentially arbitrary.",
"Besides key scalings, new closed-form approximations for special geometries and physical insight into the role of slenderness and shape, this work furnished a versatile semi-analytical scheme which allows one to rapidly calculate the plasmonic response of a certain family of complex 3D geometries.",
"The framework in [34] is underpinned by a spectral theory which is exact in the quasi-static approximation; it is based on the so-called plasmonic eigenvalue problem [35], [36], [2], [37], which defines a set of material- and frequency-independent modes of a nanometallic structure based on its geometry, with the relative permittivity of the structure playing the eigenvalue role.",
"Given this set of modes, calculating the optical response of the structure for arbitrarily prescribed radiation sources and physical values of the particle permittivity amounts to the evaluation of normalization and overlap integrals involving the modes and the incident radiation field.",
"At near-resonance frequencies, often a single mode dominates the spectral expansion; this modal approach is accordingly both efficient and insightful in that it provides a linkage between physical resonances and mathematical eigenfunctions.",
"In [34], we used asymptotic tools to calculate modes as well as overlap integrals.",
"In particular, to analyze the plasmonic eigenvalue problem, we relied mainly on the method of matched asymptotic expansions, where the physical domain is decomposed into distinguished regions which are separately analyzed and then matched together; the use of such methodology in the context of slender particles is especially popular in fluid dynamics, where it is known as `slender-body theory' [38], [39], [40], [41], [42], [43].",
"This allowed us to reduce the plasmonic eigenvalue problem for the body of revolution (restricted to the longitudinal — in this case axisymmetric — modes) to an asymptotically equivalent 1D problem whose domain is a finite line segment corresponding to the body's centerline.",
"In this reduced problem, the eigenvalue is the original permittivity eigenvalue scaled by $\\kappa ^2$ .",
"With that rescaling, the reduced problem involves $\\kappa $ only weakly, namely through its logarithm.",
"The eigenfunctions in the reduced problem represent effective 1D voltage and polarization-charge profiles.",
"From these profiles, the corresponding mode distributions can be evaluated in 3D, everywhere in the interior and exterior of the particle; they can also be used to easily evaluate the normalization and overlap integrals appearing in the spectral formulation.",
"The reduced eigenvalue problem consists of a differential equation which is coupled to an integral equation.",
"The differential equation represents an effective Gauss law for an infinitesimal longitudinal segment of the body.",
"(The boundary conditions are dependent on the local geometry of the tips.)",
"The integral equation represents a spatially nonlocal capacity relation accounting for electrostatic interactions between different longitudinal segments of the body.",
"The purpose of the present work is to extend the plasmonic slender-body in [34] to slender nanometallic rings and configurations thereof.",
"By a slender ring we mean a body whose thickness about a circular curve is small relative to the radius of that circle; we will allow azimuthal variations in thickness and non-circular cross-sectional shapes, thus going considerably beyond the idealized case of a slender torus.",
"Furthermore, we will also consider ring dimers and chains, the separation between the rings being comparable to their radii.",
"An obvious source of motivation for focusing on ring structures is that they are commonly employed in nanoplasmonics [44], [45], [46], [47], [48], [49], [50], [51], [52]; beyond their moderate compactness, as discussed above, nanometallic rings are relatively easy to fabricate in a tunable fashion and their hollow center is advantageous for some sensing applications [53].",
"There are also more technical reasons to focus on ring geometries.",
"Thus, the absence of tips eliminates the need to derive boundary conditions to close the reduced eigenvalue problem, which is generally quite a subtle aspect of the theory.",
"Furthermore, the fact that rings degenerate to circular curves in the high-aspect-ratio limit will be seen to enable convenient diagonalizations of the integral operators appearing in the reduced problem.",
"The rest of the paper is structured as follows.",
"In §, we formulate the general problem of scattering of a plane wave from a nanometallic structure in the quasi-static approximation and then review the plasmonic eigenvalue problem and its use for solving that scattering problem.",
"In §, we employ asymptotic arguments in the high-aspect-ratio limit to derive a reduced eigenvalue problem in the case of a single ring; at this stage, we assume circular cross sections, although the thickness may vary in the azimuthal direction.",
"To emphasize the physics, the asymptotic arguments are described intuitively and in dimensional notation; readers requiring a more formal justification of the approximations can refer to the derivation of the reduced problem in [34] and textbooks describing the method of matched asymptotic expansions and slender-body theory [33].",
"In the remainder of §, we discuss the singular scaling of the permittivity eigenvalues, the accuracy of the approximation scheme and develop closed-form solutions for torus-shaped rings and a semi-analytical scheme for azimuthally non-uniform rings.",
"In §, we generalize the reduced eigenvalue problem to the case of ring dimers, develop closed-form solutions for coaxial dimers formed of torus-shaped rings and a semi-analytical scheme for more general dimer geometries.",
"In § we briefly discuss further geometrical extensions to non-circular cross sections and chains of rings; the latter extension is illustrated by considering the consequences of a defect in a coaxial chain of ring dimers.",
"In § we demonstrate the application of the asymptotic modes to calculate absorption cross sections for ring configurations illuminated by a plane-wave.",
"We give concluding remarks in §.",
"Consider a homogeneous nanometallic structure in vacuum.",
"We assume that the characteristic linear dimension of the structure is small relative to the free-space wavelength such that the quasi-static approximation is applicable.",
"In what follows, we formulate a scattering problem governing the electric near-field in the vicinity of the structure in the scenario where the structure is illuminated by a plane wave; for simplicity, we do not include in the formulation the possibility of near-field external sources.",
"In the quasi-static approximation, the electric field is irrotational in the vicinity of the structure.",
"Defining this near-field to be the real part of $\\mathbf {E}e^{-i\\omega t}$ , wherein $\\omega $ is angular frequency and $t$ time, we may accordingly introduce an electric potential $\\varphi $ such that $\\mathbf {E}=-\\nabla \\varphi $ .",
"Given the absence of near-field external sources, the potential $\\varphi $ satisfies $\\nabla \\cdot \\left(\\epsilon \\nabla \\varphi \\right)=0,$ where $\\epsilon $ denotes the permittivity relative to vacuum.",
"In the exterior of the structure, $\\epsilon =1$ .",
"In the interior of the structure, $\\epsilon $ is given by the complex-valued and frequency dependent relative permittivity of the metal, say $\\epsilon _r(\\omega )$ , for which empirical data is available (e.g., [54]).",
"For the sake of discussion, it is useful to refer to the Drude model [1] $\\epsilon _r(\\omega ) = 1- \\frac{\\omega _p^2}{\\omega ^2+i\\gamma \\omega },$ wherein $\\omega _p$ is the plasma frequency and $\\gamma $ is a parameter representing ohmic losses.",
"The Drude model is mostly appropriate for frequencies considerably lower than $\\omega _p$ , which as we shall see is the case for the longitudinal resonances of slender-ring structures.",
"This model will be used for the sake of illustration in §.",
"The quasi-static formulation is closed by the far-field condition $\\mathbf {E}\\rightarrow \\mathbf {E}_{\\infty } \\quad \\text{as} \\quad |\\mathbf {x}|\\rightarrow \\infty ,$ where the constant vector $\\mathbf {E}_{\\infty }$ represents the incident plane-wave.",
"Specifically, keeping in mind that the near-field domain is small relative to the free-space wavelength, $\\mathbf {E}_{\\infty }$ is the electric-field phasor of the incoming wave evaluated at the location of the structure."
],
[
"Plasmonic eigenvalue problem",
"Upon setting $\\mathbf {E}_{\\infty }$ to zero, (REF ) and (REF ) give the so-called `plasmonic eigenvalue problem' governing the localized-surface-plasmon resonances of the nanometallic structure.",
"Since frequency enters this problem solely via the metal permittivity $\\epsilon _r$ , it is natural to take $\\epsilon _r$ as the eigenvalue, henceforth denoted $\\mathcal {E}$ , rather than the frequency.",
"The permittivity eigenvalues are scale invariant and independent of material and frequency; they are determined solely by the geometry of the structure.",
"For a smooth geometry, there are infinitely many negative-real permittivity eigenvalues which accumulate at $-1$ [35], [2], [36], [55].",
"The associated field distributions, namely eigenfunctions, are also scale-invariant up to a linear stretching and can always be chosen real-valued.",
"Physically, the eigenfunctions can be interpreted as perpetual localized-surface-plasmon oscillations in the absence of material losses and external forcing.",
"(Radiation losses are effectively neglected in the quasi-static approximation.)",
"Mathematically, the set of eigenfunctions possesses completeness and orthogonality properties that can be used to explicitly solve arbitrary scattering problems.",
"In particular, the solution to the scattering problem formulated above can be written as the spectral expansion (see [56], [2], [36], [57], [37]) $\\varphi (\\mathbf {x}) = -\\mathbf {E}_{\\infty }\\cdot \\mathbf {x} + \\sum _{I \\in \\mathcal {I}}\\frac{\\epsilon _r(\\omega )-1}{\\epsilon _r(\\omega )-\\mathcal {E}^{(I)}}\\frac{\\int dV\\,\\mathbf {E}_{\\infty }\\cdot \\nabla \\varphi ^{({I})}}{\\int dV\\, \\nabla \\varphi ^{({I})} \\cdot \\nabla \\varphi ^{({I})}}\\,\\varphi ^{({I})}(\\mathbf {x}),$ where $\\mathcal {I}$ is an index set for the eigenvalues and eigenfunctions, $dV$ is a volume element and the integrals are over the interior of the structure.",
"The integrals in the numerator and denominator are called overlap and normalization integrals, respectively.",
"We note that scattering problems involving near-field sources can be treated similarly, with $\\mathbf {E}_{\\infty }$ replaced by the total field in the absence of the structure.",
"From (REF ), quasi-static approximations for the optical cross sections can be readily calculated, as we shall see in §.",
"For frequencies such that the complex-valued permittivity $\\epsilon _r(\\omega )$ is close to a permittivity eigenvalue, the corresponding term in the expansion (REF ) — or terms, in the case of perfect or near-perfect degeneracy — may become dominant.",
"In that case, we say that the associated eigenfunctions (or eigenfunctions) are resonantly excited by the incident radiation.",
"Thus, for such near-resonance frequencies, the spectral solution can be asymptotically simplified, revealing a connection between the physical and mathematical perspectives described above.",
"The existence and characteristics of such resonances in practice, however, depend on several additional factors including the relative smallness of the imaginary component of $\\epsilon _r(\\omega )$ , the overlap between the incident field and the eigenfunctions participating in the resonance, as well as interference effects.",
"All these dependencies can be studied using (REF ) once the eigenvalues and eigenfunctions of the geometry have been calculated."
],
[
"Geometry",
"In this section we consider the plasmonic eigenvalue problem (cf.",
"§REF ) in the case of a single nano-sized ring of radius $a$ and characteristic thickness $b$ .",
"For now, we assume that the ring is formed of circular cross sections of radius $bf(\\phi )$ , which are centered about a circular centerline of radius $a$ ; the thickness profile $f(\\phi )$ is a dimensionless function of the azimuthal angle $\\phi $ .",
"(Non-circular cross-sections are treated in §REF .)",
"In Fig.",
"REF , we define the Cartesian $(x,y,z)$ , cylindrical $(\\rho ,z,\\phi )$ and `cross-sectional' $(r,\\theta ,\\phi )$ coordinate systems.",
"Accordingly, the position measured from the center of the ring can be written $\\mathbf {x}=\\mathbf {y}+r\\mathbf {\\hat{e}}_r$ , where $\\mathbf {\\hat{e}}_r=\\mathbf {\\hat{e}}_\\rho \\cos \\theta +\\mathbf {\\hat{e}}_z\\sin \\theta $ and $\\mathbf {y}(\\phi )=a\\mathbf {\\hat{e}}_{\\rho }$ is the circular centerline, with $\\mathbf {\\hat{e}}_{\\rho }=\\mathbf {\\hat{e}}_x\\cos \\phi +\\mathbf {\\hat{e}}_y\\sin \\phi $ .",
"Furthermore, the ring boundary can be written $r=bf(\\phi )$ .",
"Figure: Schematic of a slender ring with circular cross sections as considered in §."
],
[
"Reduced eigenvalue problem",
"Henceforth, we consider slender rings for which the aspect ratio $\\kappa =a/b$ is large and $f(\\phi )$ is of order unity.",
"In particular, our interest is in modes, i.e., solutions of the plasmonic eigenvalue problem, for which $\\mathcal {E}\\rightarrow -\\infty $ as $\\kappa \\rightarrow \\infty $ .",
"(According to the Drude model (REF ), large-negative permittivity eigenvalues imply low frequencies $\\omega \\ll \\omega _p$ .)",
"For any mode with this asymptotic property, scaling arguments suggest that, as $\\kappa \\rightarrow \\infty $ , cross-sectional variations of the interior potential become negligible relative to azimuthal variations.",
"We accordingly label such modes `longitudinal'.",
"We note that in the case of a body of revolution [34], those modes can be identified based on symmetry as they simply correspond to the axisymmetric modes.",
"In light of the above, we approximate the interior potential as $\\varphi (\\mathbf {x})= v(\\phi ),$ where the azimuthal function $v(\\phi )$ represents the voltage relative to infinity.",
"(Without loss of generality, we take the potential to vanish at infinity.)",
"Another key azimuthal function is the polarization-charge line density $q(\\phi )=\\oint \\limits _{r=bf(\\phi )} dl\\,\\sigma ,$ where $\\sigma =\\epsilon _0[\\mathbf {E}\\cdot \\mathbf {\\hat{n}}]_i^e$ is the polarization-charge surface density, $dl$ is a differential length element and the integral is over the circular cross-sectional interface $r=bf(\\phi )$ at constant $\\phi $ ; in the expression for $\\sigma $ , $\\epsilon _0$ is the permittivity of vacuum, $\\mathbf {\\hat{n}}$ an outward normal unit vector and the square brackets stand for the jump across the interface in that direction, the subscript $i$ and superscript $e$ indicating the interior and exterior sides, respectively.",
"For $|\\mathcal {E}|\\gg 1$ , continuity of electrical displacement implies that the interior normal field is negligible compared to the exterior normal field.",
"We therefore make the approximation $\\sigma = \\epsilon _0 (\\mathbf {E}\\cdot \\mathbf {\\hat{n}})_e,$ the $e$ subscript indicating evaluation on the exterior side of the interface.",
"Consider now the potential distribution in the exterior of the ring, specifically at distances from the centerline comparable with the characteristic thickness $b$ .",
"In that neighborhood of the ring boundary, the exterior potential can be approximated as $\\varphi (\\mathbf {x})=-\\frac{q(\\phi )}{2\\pi \\epsilon _0}\\ln \\frac{r}{bf(\\phi )}+v(\\phi ).$ Namely, in any given cross-sectional plane, the exterior potential is locally provided by the potential distribution around an infinite, perfectly conducting, cylinder, which coincides with the ring in that plane and is held at a potential $v(\\phi )$ .",
"In light of (REF ), the electrostatic surface-charge density at the cylinder boundary is nothing but the polarization surface-charge density $\\sigma $ , thence the cylinder's net apparent surface charge, per unit length, is $q(\\phi )$ .",
"Lastly, the absence of $\\theta $ -dependent solutions in (REF ) can be justified based on the circular shape of the boundary, the uniform potential prescribed on that boundary, as well as the anticipation that the potential gradient is negligible in the azimuthal direction and decays away from the ring's centerline.",
"We shall now derive two relations between $v(\\phi )$ and $q(\\phi )$ .",
"The first is based on Gauss law, which in the absence of free charge reduces to (cf.",
"(REF )) $\\oint dA\\,\\epsilon _0\\epsilon \\mathbf {E}\\,\\cdot \\mathbf {\\hat{n}}=0,$ where $dA$ is a differential area element, $\\mathbf {\\hat{n}}$ is an outward normal unit vector and the integral is over an arbitrary closed surface.",
"Consider that closed surface to be the boundary of a curved tube, which closely encloses a segment of the ring between the cross-sectional planes $\\phi =\\phi ^{\\prime }$ and $\\phi =\\phi ^{\\prime }+\\Delta \\phi $ .",
"From (REF ) and (REF ), the contribution to the integral from the part of the tube boundary that lies outside the ring is $aq(\\phi ^{\\prime }) \\Delta \\phi $ .",
"Since the interior azimuthal field is approximately uniform over the cross section, we find that the contribution from the part of the tube boundary that lies inside the ring, namely the end faces, is $\\epsilon _0\\mathcal {E}\\left\\lbrace (A\\mathbf {E}\\cdot \\mathbf {\\hat{e}}_{\\phi })_{\\phi ^{\\prime }+\\Delta \\phi }-(A\\mathbf {E}\\cdot \\mathbf {\\hat{e}}_{\\phi })_{\\phi ^{\\prime }}\\right\\rbrace ,$ where $A$ denotes the cross-sectional area.",
"By taking the limit $\\Delta \\phi \\rightarrow 0$ , using (REF ), we find the effective Gauss law $\\frac{q}{\\epsilon _0}=\\frac{\\mathcal {E}}{\\kappa ^2}\\frac{d}{d\\phi }\\left(\\bar{A}\\frac{dv}{d\\phi }\\right),$ where we define the scaled cross-sectional area $\\bar{A}=A/b^2$ .",
"In the present case, where the cross-sectional shape is circular, $\\bar{A}=\\pi f^2$ .",
"The second relation arises from the need to resolve the logarithmic growth of the exterior potential away from the centerline of the ring.",
"This is done by matching (REF ) with an approximation for the exterior potential that holds at distances from the centerline that are comparable to $a$ .",
"Thus, on the latter scale, the finite thickness of the ring is indiscernible and the potential is approximated as that of a circular wire of charge line density $q(\\phi )$ , viz., $\\varphi (\\mathbf {x}) = \\frac{a}{4\\pi \\epsilon _0}\\int _0^{2\\pi } d\\phi ^{\\prime }\\,\\frac{q(\\phi ^{\\prime })}{|\\mathbf {x}-\\mathbf {y}(\\phi ^{\\prime })|}.$ We then compare (REF ) with the behavior of (REF ) close to the centerline, which is derived in Appendix , to derive the matching condition $v(\\phi )=\\frac{q(\\phi )}{2\\pi \\epsilon _0}\\ln \\frac{8\\kappa }{f(\\phi )}+\\frac{1}{4\\pi \\epsilon _0}\\int _0^{2\\pi }d\\phi ^{\\prime }\\,\\frac{q(\\phi ^{\\prime })-q(\\phi )}{2\\sin \\frac{|\\phi -\\phi ^{\\prime }|}{2}},$ which is an integral capacitance relation between the voltage at a given azimuthal angle $\\phi $ and the azimuthal distribution of polarization charge over the entire ring.",
"The differential Gauss law (REF ) and the integral capacitance relation (REF ) together constitute a `reduced eigenvalue problem' for the `reduced eigenvalue' $\\mathcal {E}/\\kappa ^2$ and `reduced eigenfunctions' $v(\\phi )$ and $q(\\phi )/\\epsilon _0$ , which are $2\\pi $ -periodic functions of $\\phi $ .",
"Like the exact plasmonic eigenvalue problem, this reduced problem is purely geometric; in particular, the geometry enters through the thickness profile $f(\\phi )$ and the logarithm of the aspect ratio $\\kappa $ ."
],
[
"`Logarithmically approximated' solutions to the reduced eigenvalue problem",
"Before attempting to solve the reduced eigenvalue problem in its present form, it is insightful to discuss a coarse `logarithmic' approximation of the reduced problem which is based on the formal largeness of $\\ln \\kappa $ as $\\kappa \\rightarrow \\infty $ .",
"In this approximation, the integral capacitance relation (REF ) reduces to the algebraic relation $q/\\epsilon _0\\approx \\frac{2\\pi }{\\ln \\kappa }v.$ Substituting (REF ) into the Gauss law (REF ) then gives $-\\frac{\\mathcal {E}}{\\kappa ^2}\\frac{d}{d\\phi }\\left(\\bar{A}\\frac{dv}{d\\phi }\\right)+\\frac{2\\pi }{\\ln \\kappa }v\\approx 0,$ to be solved together with the periodic boundary conditions on $v$ .",
"The simplified model (REF ) clearly implies the asymptotic scaling $\\mathcal {E}\\simeq \\frac{\\kappa ^2}{\\ln \\kappa }\\quad \\text{as} \\quad \\kappa \\rightarrow \\infty ,$ which slightly differs from the usually quoted scaling for slender geometries mentioned in the introduction by the logarithmic factor; we note that the same scaling was found in [34] for the longitudinal modes of slender bodies of revolution.",
"In particular, consider the case of a torus-shaped ring, with $b$ identified as the cross-sectional radius.",
"Then $\\bar{A}(\\phi )\\equiv \\pi $ and solving (REF ) yields the approximate eigenvalues $\\mathcal {E}^{({m})}\\approx -\\frac{2\\kappa ^2}{m^2\\ln \\kappa }, \\quad m=1,2,\\ldots ,$ with corresponding pairs of independent eigenfunctions $v^{({m,c})} \\approx \\cos m\\phi , \\quad v^{({m,s})} \\approx \\sin m\\phi ,\\qquad \\mathrm {{(18{\\mathrm {a},\\!\\mathrm {b}})}}$ the degeneracy being induced by symmetry.",
"The corresponding polarization-charge eigenfunctions $q^{({m,c})}$ and $q^{({m,s})}$ can readily be deduced from (REF ).",
"The above scheme corresponds to a leading-order approximation in a perturbative expansion in inverse powers of $\\ln \\kappa $ .",
"In particular, the relative error in (REF ) is on the order of $1/\\ln \\kappa $ .",
"In slender-body theory, such logarithmic approximations are often referred to as `local slender-body theory'.",
"This should be contrasted with the non-approximated reduced eigenvalue problem, which constitutes a `nonlocal slender-body theory,' the term `nonlocal' here referring to the integral capacitance relation (REF ).",
"It can be shown that the reduced eigenvalue problem effectively sums the logarithmic expansion mentioned above, resulting in solutions which are `algebraically' accurate, i.e., involving relative errors which are asymptotically smaller than some negative power of $\\kappa $ .",
"The distinction between local and nonlocal slender-body theory should not be confused with the notion of spatial nonlocality of the metal's dielectric function, which we do not address in this paper."
],
[
"Diagonalization of the self-interaction integral operator",
"We now proceed to derive `exact' solutions to the reduced eigenvalue problem (cf.",
"(REF ) and (REF )), i.e., without exploiting the formal largeness of $\\ln \\kappa $ .",
"An essential step is to note that the integral operator appearing in the capacitance relation (REF ) is diagonalized by the set of Fourier eigenfunctions.",
"Specifically, for any integer $m$ , $\\int _0^{2\\pi }d\\phi ^{\\prime }\\,\\frac{e^{im\\phi ^{\\prime }}-e^{im\\phi }}{2\\sin \\frac{|\\phi ^{\\prime }-\\phi |}{2}}=\\lambda _m e^{im\\phi },$ with $\\lambda _0=0$ and $\\lambda _m= -4\\sum _{k=1}^{|m|}\\frac{1}{2k-1} \\quad \\text{for} \\quad m=\\pm 1,\\pm 2,\\ldots $ A proof of this result is given in Appendix ."
],
[
"Torus-shaped rings",
"We first consider the case of torus-shaped rings, for which $f$ and $\\bar{A}$ are independent of $\\phi $ .",
"Identifying the cross-sectional radius as $b$ , we have $f(\\phi )\\equiv 1$ and $\\bar{A}(\\phi )\\equiv \\pi $ .",
"(Henceforth, we will always adopt this convention when referring to torus-shaped rings.)",
"For constant $\\bar{A}$ , the same set of Fourier eigenfunctions that diagonalizes the integral operator in the capacitance relation (REF ) as in (REF ) also diagonalizes the differential operator in the Gauss law (REF ), since $d^2e^{im \\phi }/d\\phi ^2 = -m^2e^{im \\phi }$ .",
"Furthermore, for constant $f$ , the logarithmic factor multiplying $q(\\phi )$ in (REF ) reduces to a constant.",
"Accordingly, it is immediate to deduce the eigenvalues $\\mathcal {E}^{({m})} = -\\frac{2\\kappa ^2}{m^2}\\left(\\ln 8 \\kappa - 2\\sum _{k=1}^m\\frac{1}{2k-1}\\right)^{-1}, \\quad \\text{for} \\quad m=1,2,\\ldots ,$ with associated voltage eigenfunctions $v^{({m,c})} =\\cos m\\phi , \\quad v^{({m,\\,s})} =\\sin m\\phi .\\qquad \\mathrm {{(22{\\mathrm {a},\\!\\mathrm {b}})}}$ The closed-form expression (REF ) constitutes an algebraically accurate slender-body approximation for the reduced eigenvalues of a torus-shaped ring.",
"It is asymptotically consistent with, and improves on, the corresponding logarithmically accurate approximation (REF ).",
"Surprisingly, the algebraically accurate voltage eigenfunctions (REF ) are of the same form as the logarithmically accurate eigenfunctions (REF ).",
"The algebraically accurate relation between the voltage and polarization-charge eigenfunctions, however, reads $q^{({m,c})}/\\epsilon _0=2\\pi \\left(\\ln 8 \\kappa - 2\\sum _{k=1}^m\\frac{1}{2k-1}\\right)^{-1}v^{({m,c})},$ which should be compared with the corresponding logarithmically accurate relation (REF ).",
"In the present case of a torus-shaped ring, exact solutions of the plasmonic eigenvalue problem can be obtained by separation of variables in toroidal coordinates [31].",
"In Fig.",
"REF , we compare the first three eigenvalues, not counting multiplicities, computed in this manner with the corresponding values predicted by the slender-body approximation (REF ).",
"As expected, the slender-body approximations approach the computed eigenvalues as $\\kappa \\rightarrow \\infty $ .",
"At the same time, the comparison in Fig.",
"REF demonstrates the fact that, for fixed $\\kappa $ , the slender-body approximation deteriorates in accuracy with increasing mode number; conversely, as the mode number increases, accuracy can be retained only by increasing $\\kappa $ .",
"This is not surprising given that our slender-body approximation corresponds to a limit process where $\\kappa \\rightarrow \\infty $ for a fixed mode; there is therefore no reason to expect the approximation to hold in the double limit where both $\\kappa $ and $m$ are large [34].",
"In fact, the scale on which the voltage and charge eigenfunctions vary along the ring — assumed large compared to the ring thickness in the slender-body approximation — becomes comparable to the ring thickness for $m\\simeq \\kappa $ ; moreover, for such large $m$ , (REF ) gives $|\\mathcal {E}^{({m})}|\\simeq 1$ , whereas the theory relies on the permittivity contrast being large.",
"We note that this limitation of the theory does not interfere with our goal of describing the low-order (low-frequency) longitudinal resonances of slender nanometallic rings.",
"Figure: The three lowest permittivity eigenvalues, not counting multiplicities, of a torus-shaped ring as a function of the reciprocal of its aspect ratio κ\\kappa .",
"The solid black lines correspond to exact solutions of the plasmonic eigenvalue problem obtained by separation of variables in toroidal coordinates .",
"The dot-dashed blue lines depict the slender-body approximation ()."
],
[
"Rings of non-uniform thickness",
"Consider now rings for which the thickness profile $f$ depends on the azimuthal angle $\\phi $ .",
"In this more general case, we solve the reduced eigenvalue problem using a semi-analytical method, which is based on approximating the reduced eigenfunctions $v(\\phi )$ and $q(\\phi )$ by the truncated Fourier series $v = \\sum _{k = 0}^{K}\\left\\lbrace \\alpha _k \\cos k\\phi + \\beta _{k} \\sin k\\phi \\right\\rbrace ,\\quad \\frac{q}{\\epsilon _0} = \\sum _{k = 1}^{K}\\left\\lbrace \\tilde{\\alpha }_k \\cos k\\phi + \\tilde{\\beta }_k \\sin k\\phi \\right\\rbrace ,\\qquad \\mathrm {{(24{\\mathrm {a},\\!\\mathrm {b}})}}$ where $K\\in \\mathbb {N}$ is a truncation parameter discussed below.",
"Note that the zeroth harmonic is omitted from the charge representation (REF b) in accordance with the zero-net-charge constraint $\\int _{0}^{2\\pi }d\\phi \\,q = 0$ , which readily follows from integration of Gauss law (REF ).",
"With (REF ), Fourier projection of the capacitance relation (REF ) and Gauss law (REF ) yields a $2K\\times 2K$ generalized eigenvalue problem of the form $\\mathsf {q} =-\\bar{\\mathcal {E}}\\,\\mathsf {M}\\cdot \\mathsf {v},\\quad \\mathsf {v} = \\mathsf {U}\\cdot \\mathsf {q},\\qquad \\mathrm {{(25{\\mathrm {a},\\!\\mathrm {b}})}}$ where $\\mathsf {v} = (\\alpha _1,\\ldots ,\\alpha _K,\\beta _1,\\ldots ,\\beta _K)^T$ and $\\mathsf {q} = (\\tilde{\\alpha }_1,\\ldots ,\\tilde{\\alpha }_K,\\tilde{\\beta }_1,\\ldots ,\\tilde{\\beta }_K)^T$ , with $T$ denoting the vector transpose; we have introduced the notation $\\bar{\\mathcal {E}}=\\mathcal {E}/\\kappa ^2$ for the reduced eigenvalue; and the matrices $\\mathsf {M}$ and $\\mathsf {U}$ are provided in Appendix , along with an expression for the coefficient $\\alpha _0$ , which is computed a posteriori.",
"As already mentioned, the slender-body approximation cannot be expected to correctly capture thickness-scale variations in the longitudinal direction.",
"In fact, the reduced formulation can be shown to be ill posed for $K\\simeq \\kappa $ [58].",
"Accordingly, the truncation parameter $K$ should be chosen much smaller than $\\kappa $ .",
"Of course, $K$ should also be chosen larger than the number of modes to be resolved.",
"In contrast to the case of a torus-shaped ring, it is clear from the above scheme that for a non-uniform ring each mode generally contains multiple Fourier harmonics.",
"In particular, whereas for a torus-shaped ring only modes $(1,c)$ and $(1,s)$ , which share the degenerate eigenvalue $\\mathcal {E}^{({1})}$ , are `dipolar', i.e., include a first Fourier harmonic, for non-uniform rings generally all modes include a dipolar component.",
"As we shall see in §, this observation is important for interpreting the differences between the resonant response of uniform and non-uniform rings under plane-wave illumination.",
"As an example, we consider an azimuthally non-uniform ring whose thickness profile is $f=1+0.5\\cos \\phi $ .",
"The voltage eigenfunctions can be classified as being either even or odd with respect to the symmetry plane of the geometry which is normal to the plane of the ring; referring to (REF a), these modes are respectively comprised of either only cosine or only sine Fourier components.",
"Consistently with our notation for the longitudinal modes of a torus-shaped ring (cf.",
"(REF )), we denote the $m$ th even mode by the superscript $(m,c)$ ; similarly, the $m$ th odd mode by the superscript $(m,s)$ .",
"In Fig.",
"REF , we show for $\\kappa =10$ the `dipolar' $(1,c)$ and `quadrupolar' $(2,c)$ modes of a torus-shaped ring, along with the corresponding modes of the azimuthally non-uniform ring.",
"In the latter case, the modes are computed by solving (REF ) with $K = 6$ .",
"Figure: Slender-body approximations for the `dipolar' (1,c)(1,c) and `quadrupolar' (2,c)(2,c) voltage eigenfunctions and associated eigenvalues of a torus-shaped ring and an azimuthally non-uniform ring with thickness profile f(φ)=1+0.5cosφf(\\phi )=1+0.5\\cos \\phi ; for both geometries κ=10\\kappa =10."
],
[
"Longitudinal models of ring dimers",
"In this section we consider the longitudinal modes of ring dimers.",
"In §REF , we generalize the reduced eigenvalue problem derived in §REF for a single ring to the dimer case.",
"In §REF , we obtain exact closed-form solutions to the generalized reduced problem in the case of a dimer formed of coaxial, generally dissimilar, torus-shaped rings.",
"In §REF , we consider more general ring dimers using a semi-analytical scheme (we also present an ad hoc approximation which is in some cases suitable).",
"As we will demonstrate in §, configurations of more than two rings can be handled similarly."
],
[
"Reduced eigenvalue problem for ring dimers",
"We adopt the same notation as in §, only with subscripts added to indicate to which ring a given quantity is associated with.",
"By revisiting the derivation in §, we find that in the thickness-scale vicinity of the rings we have the local approximation $\\varphi (\\mathbf {x})=-\\frac{q_n(\\phi )}{2\\pi \\epsilon _0}\\ln \\frac{r}{b_nf_n(\\phi )}+v_n(\\phi ), \\quad n=1,2,$ for the exterior potential, which generalizes (REF ).",
"Similarly, the effective Gauss law (REF ) now applies to each ring separately: $\\frac{q_n}{\\epsilon _0} = \\frac{\\mathcal {E}}{\\kappa _n^2}\\frac{d}{d\\phi }\\left(\\bar{A}_n\\frac{d v_n}{d\\phi }\\right), \\quad n=1,2.$ It remains to generalize the integral capacitance relation (REF ).",
"To this end, consider the ring-scale exterior potential (REF ), which generalizes as $\\varphi (\\mathbf {x}) = \\sum _{n=1,2}\\frac{a_n}{4\\pi \\epsilon _0}\\int _0^{2\\pi } d\\phi ^{\\prime }\\,\\frac{q_n(\\phi ^{\\prime })}{|\\mathbf {x}-\\mathbf {y}_n(\\phi ^{\\prime })|}.$ Using the results of Appendix to match (REF ) and (REF ), we find the coupled pair of integral capacitance relations $v_n(\\phi ) = \\frac{q_n(\\phi )}{2\\pi \\epsilon _0}\\ln {\\frac{8\\kappa _n}{f_n(\\phi )}} + \\frac{1}{4\\pi \\epsilon _0}\\int _{0}^{2\\pi }d\\phi ^{\\prime }\\, \\frac{q_n(\\phi ^{\\prime })-q_n(\\phi )}{2\\sin \\frac{\\left|\\phi ^{\\prime }-\\phi \\right|}{2}} \\\\ +\\frac{a_k}{4\\pi \\epsilon _0}\\int _{0}^{2 \\pi }d\\phi ^{\\prime }\\,\\frac{q_k(\\phi ^{\\prime })}{|\\mathbf {y}_n(\\phi )-\\mathbf {y}_k(\\phi ^{\\prime })|}, $ for $(n,k)=(1,2)$ and $(2,1)$ .",
"The differential equations (REF ) and integral equations (REF ) together constitute a generalized reduced eigienvalue problem for ring dimers.",
"We see that the rings interact solely through the last `coupling' integral on the right-hand side of (REF ), whereby the polarization-charge distribution of one ring induces a voltage disturbance in the other ring, and vice versa.",
"For coaxial rings, the coupling-integral operator in (REF ) is diagonalized by the same Fourier basis that diagonalizes the self-interaction integral operator as in (REF ).",
"Indeed, in this case, the distance between the point on the centerline of ring 1 at azimuthal angle $\\phi _1$ and the point on the centerline of ring 2 at azimuthal angle $\\phi _2$ can be written $|\\mathbf {y}_1(\\phi _1)-\\mathbf {y}_2(\\phi _2)|=D(\\phi _1-\\phi _2)$ , where $D(u) = \\sqrt{h^2 + a_1^2 + a_2^2 - 2a_1a_2\\cos u}$ , $h$ being the vertical distance between the rings.",
"Since $D(u)$ is even, periodic and positive, its reciprocal can be expanded as a cosine Fourier series.",
"It then readily follows that $\\int _{0}^{2 \\pi }d\\phi ^{\\prime }\\,\\frac{e^{i m\\phi ^{\\prime }}}{|\\mathbf {y}_1(\\phi )-\\mathbf {y}_2(\\phi ^{\\prime })|} = \\tau _me^{i m\\phi }, \\quad \\text{for} \\quad m=0,\\pm 1,\\pm 2,\\ldots $ where $\\tau _m = \\int _{0}^{2 \\pi }du\\,\\frac{\\cos mu}{D(u)}.$ Figure: Schematic of a bilayer dimer of torus-shaped rings.",
"The rings have centerline radii a 1 a_1 and a 2 a_2, and aspect ratios κ 1 =a 1 /b 1 \\kappa _1 = a_1/b_1 and κ 2 =a 2 /b 2 \\kappa _2 = a_2/b_2.",
"The horizontal and vertical separation between the rings is dd and hh, respectively.",
"The coaxial case considered in § corresponds to the case d=0d=0.",
"The non-coaxial case d≠0d\\ne 0 is considered in §.With (REF ), it is straightforward to generalize the semi-analytical scheme of §REF to the case of arbitrary coaxial ring dimers.",
"Instead, we shall focus in the present subsection on the case of coaxial dimers formed of torus-shaped, not necessarily identical, rings (see Fig.",
"REF for $d=0$ ), where (REF ) actually facilitates the derivation of closed-form solutions.",
"Later, in §REF , we will present a more general semi-analytical scheme, not relying on (REF ), that applies to arbitrary ring dimers, including non-coaxial dimers formed of non-uniform rings."
],
[
"Coaxial homodimers",
"Consider first the case of a coaxial homodimer, namely a coaxial pair of identical torus-shaped rings ($a_1 = a_2 = a$ , $\\kappa _1 = \\kappa _2 =\\kappa $ ).",
"Given the symmetries of the geometry, we anticipate eigenvalues $\\mathcal {E}^{({m,\\pm })}$ , for $m=1,2,\\ldots $ , with corresponding eigenfunctions $\\begin{pmatrix}v_{1}^{({m,c,\\pm })}\\\\v_{2}^{({m,c,\\pm })}\\end{pmatrix} = \\begin{pmatrix}1\\\\\\pm 1\\end{pmatrix}\\cos m\\phi ,\\quad \\begin{pmatrix}v_{1}^{({m,s,\\pm })}\\\\v_{2}^{({m,s,\\pm })}\\end{pmatrix} = \\begin{pmatrix}1\\\\\\pm 1\\end{pmatrix}\\sin m\\phi .\\qquad \\mathrm {{(32{\\mathrm {a},\\!\\mathrm {b}})}}$ The $\\pm $ modes are, respectively, even and and odd about the plane equidistance between the rings; we shall also refer to these as in- and out-of-phase modes, respectively, as this will allow a generalized interpretation in the heterodimers case considered next.",
"Writing the eigenfunctions as in (REF ), use of the diagonalization identities (REF ) and (REF ) readily yields $\\mathcal {E}^{({m,\\pm })} = -\\frac{2 \\kappa ^2}{m^2}\\left(\\ln 8 \\kappa - 2\\sum _{k=1}^{m}\\frac{1}{2k-1} \\pm \\frac{\\Delta _m}{2} \\right)^{-1},$ where $\\Delta _m=a\\tau _m$ is a dimensionless function of the ratio $h^{\\prime }=h/a$ defined by the quadrature (cf.",
"(REF )) $\\Delta _m(h^{\\prime })=\\int _0^{2\\pi }\\,du\\frac{\\cos mu}{\\sqrt{h^{\\prime 2}+2-2\\cos u}},$ which determines the eigenvalue splitting induced by the interaction between the rings (cf.",
"(REF )).",
"The functions $\\Delta _m(h^{\\prime })$ are positive.",
"Hence, for any $m$ , the in-phase modes are higher energy (less negative permittivity), as one would expect.",
"Furthermore, the functions $\\Delta _m(h^{\\prime })$ are monotonically decreasing, asymptotically like $\\Delta _m=O(1/h^{2m+1})$ as $h^{\\prime }\\rightarrow \\infty $ ; this represents the approach to the eigenvalues (REF ) of a single torus-shaped ring.",
"We also note that $\\Delta _m(h^{\\prime })$ is logarithmically singular as $h^{\\prime }\\rightarrow 0$ , which is acceptable given that the theory only holds for $h\\gg b$ , i.e., $h^{\\prime }\\gg 1/\\kappa $ .",
"In particular, for $m=1$ we find from (REF ) the behaviours $\\Delta _1\\sim -2\\ln h^{\\prime }+6\\ln 2-4 \\quad \\text{as} \\quad h^{\\prime }\\rightarrow 0, \\quad \\Delta _1\\sim \\frac{\\pi }{h^{\\prime 3}} \\quad \\text{as} \\quad h^{\\prime }\\rightarrow \\infty ,\\qquad \\mathrm {{(35{\\mathrm {a},\\!\\mathrm {b}})}}$ In Fig.",
"REF , the eigenvalues $\\mathcal {E}^{({1,\\pm })}$ of the in- and out-of-phase dipolar modes, calculated using (REF ), are depicted by the solid curves as a function of $h/a$ , for $\\kappa =10$ .",
"Also shown are the asymptotic behaviors of the eigenvalues for small and large $h/a$ , which follow from the behaviors (REF )."
],
[
"Coaxial heterodimers",
"The above results are easily generalized to allow for dissimilar torus-shaped rings.",
"Since the geometry remains azimuthally invariant, we look for voltage eigenfunctions of the form $\\lbrace v_1,v_2\\rbrace = \\lbrace c_1,c_2\\rbrace \\times \\cos m\\phi $ , in which $c_1$ and $c_2$ are constant prefactors and $m=1,2,\\ldots $ There are also $\\pi /2$ -rotations of these modes having the same eigenvalues.",
"Using the diagonalization identities (REF ) and (REF ), we find that, for $m=1,2,\\ldots $ , the reduced eigenvalue problem of §REF is transformed into the $2\\times 2$ matrix eigenvalue problem $\\begin{pmatrix}c_{1}\\\\c_{2}\\end{pmatrix} =-\\mathcal {E}\\begin{pmatrix}\\gamma _{1,1} & \\gamma _{1,2} \\\\\\gamma _{2,1} & \\gamma _{2,2}\\end{pmatrix}\\begin{pmatrix}c_{1}\\\\c_{2}\\end{pmatrix},$ in which the matrix on the right-hand side has the diagonal and non-diagonal elements $\\gamma _{n,n} = \\frac{m^2}{2\\kappa _n^2}\\left(\\ln 8\\kappa _n - 2\\sum _{k=1}^{m}\\frac{1}{2k-1}\\right), \\quad \\gamma _{n,k} = \\frac{m^2}{4\\kappa _k^2}(a_k\\tau _m),\\qquad \\mathrm {{(37{\\mathrm {a},\\!\\mathrm {b}})}}$ respectively, where the products $a_k\\tau _m$ are dimensionless geometric factors which determine the coupling between the rings (cf.",
"(REF )).",
"Solving the above system gives the eigenvalues $\\mathcal {E}^{({m,\\pm })} = -\\frac{2}{\\gamma _{1,1}+\\gamma _{2,2} \\pm \\sqrt{(\\gamma _{1,1}-\\gamma _{2,2})^2 + 4\\gamma _{1,2}\\gamma _{2,1}}}$ and corresponding eigenvectors $\\begin{pmatrix}c_{1}^{({m,\\pm })}\\\\c_{2}^{({m,\\pm })}\\end{pmatrix} = \\begin{pmatrix}\\gamma _{1,2}\\\\\\frac{\\gamma _{2,2}-\\gamma _{1,1} \\pm \\sqrt{(\\gamma _{1,1}-\\gamma _{2,2})^2 + 4\\gamma _{1,2}\\gamma _{2,1}}}{2}\\end{pmatrix}.$ In summary, for $m=1,2,\\ldots $ , there are two degenerate plasmonic eigenvalues $\\mathcal {E}^{({m,\\pm })}$ , given by (REF ), with corresponding eigenfunctions $\\begin{pmatrix}v_{1}^{({m,c,\\pm })}\\\\v_{2}^{({m,c,\\pm })}\\end{pmatrix} = \\begin{pmatrix}c_{1}^{({m,\\pm })}\\\\c_{2}^{({m,\\pm })}\\end{pmatrix}\\cos m\\phi ,\\quad \\begin{pmatrix}v_{1}^{({m,s,\\pm })}\\\\v_{2}^{({m,s,\\pm })}\\end{pmatrix} = \\begin{pmatrix}c_{1}^{({m,\\pm })}\\\\c_{2}^{({m,\\pm })}\\end{pmatrix}\\sin m \\phi , \\qquad \\mathrm {{(40{\\mathrm {a},\\!\\mathrm {b}})}}$ where $c_1^{({m,\\pm })}$ and $c_2^{({m,\\pm })}$ are given by (REF ).",
"It can be verified from the above results that $\\mathcal {E}^{({m,+})}>\\mathcal {E}^{({m,-})}$ , namely that for any given $m$ the $+$ mode is higher energy than the $-$ mode.",
"Furthermore, $c_1^{({m,\\pm })}$ is positive, whereas $c_2^{({m, +})}$ and $c_2^{({m, -})}$ are positive and negative, respectively.",
"The latter observation suggests referring to the $\\pm $ modes as in- and out-of-phase modes, respectively, consistently with the coaxial homodimer case.",
"In contrast to the latter case, where the in- and out-of-phase labeling is associated with a mirror symmetry, here the voltage profile in one ring is not simply the same as or negative of that in the other.",
"We shall see in § that this distinction is important for interpreting the resonant response of coaxial dimers.",
"As an example, consider a coaxial heterodimer with $a_1/a_2=2$ and $b_1=b_2$ (i.e., $\\kappa _1=10$ and $\\kappa _2=5$ ).",
"In Fig.",
"REF a, the dashed curves depict the eigenvalues $\\mathcal {E}^{({1,\\pm })}$ , calculated using (REF ), of the in- and out-of-phase dipolar modes of this coaxial heterodimer as a function of $h/a_1$ .",
"The voltage profiles for the corresponding cosine-dipolar modes, for $h/a_1=0.3$ , are shown in Fig.",
"REF b.",
"Unlike in the coaxial-homodimer case also presented in that figure, the rings of the heterodimer do not become arbitrarily close as $h\\rightarrow 0$ , hence the eigenvalues of the heterodimer approach finite values in that limit.",
"Of course, we could have also considered a coaxial heterodimer formed of rings of similar radius yet different thickness, in which case the eigenvalues would be singular as $h\\rightarrow 0$ , as in the coaxial-homodimer case.",
"Figure: (a) Eigenvalues of in- and out-of-phase dipolar modes of a a coaxial homodimer (κ=10\\kappa =10) and heterodimer (κ 1 =10\\kappa _1=10, κ 2 =5\\kappa _2=5, a 1 /a 2 =2a_1/a_2=2) as a function of the scaled vertical separation h/a 1 h/a_1.",
"The eigenvalues are calculated based on formulas () and () for the homodimer and heterodimer, respectively.",
"The dash-dotted lines depict the asymptotic behaviors implied by ().",
"(b) Corresponding cosine-dipolar voltage eigenfunctions in the heterodimer case, for h/a 1 =0.3h/a_1=0.3."
],
[
"Semi-analytical scheme",
"We now present a semi-analytical scheme to solve the reduced eigenvalue problem of §REF in the case of an arbitrary ring dimer, meaning that the rings need not be coaxial, identical nor azimuthally uniform.",
"Following the derivation of the single-ring semi-analytical scheme in §REF , we represent the voltage and polarization-charge distributions in each ring by truncated Fourier series as in (REF ), with the Fourier coefficients now denoted $\\alpha _{n,k},\\beta _{n,k}$ , etc., with the first and second subscripts corresponding to the ring number and Fourier harmonic, respectively.",
"With that representation, projection of the reduced eigenvalue problem on the truncated Fourier basis yields the $4K\\times 4K$ generalized matrix-eigenvalue problem $\\begin{pmatrix}\\mathsf {q}_1\\\\\\mathsf {q}_2\\end{pmatrix} =-{\\bar{\\mathcal {E}}}\\begin{pmatrix}\\dfrac{\\kappa _2}{\\kappa _1}\\mathsf {M}_{1} & 0\\\\0 & \\dfrac{\\kappa _1}{\\kappa _2}\\mathsf {M}_{2}\\end{pmatrix}\\cdot \\begin{pmatrix}\\mathsf {v}_1\\\\\\mathsf {v}_2\\end{pmatrix},\\quad \\begin{pmatrix}\\mathsf {v}_1\\\\\\mathsf {v}_2\\end{pmatrix} = \\begin{pmatrix}\\mathsf {U}_{1} & \\sqrt{\\frac{a_2}{a_1}}\\mathsf {V}\\\\\\sqrt{\\frac{a_1}{a_2}}\\mathsf {V}^T & \\mathsf {U}_{2}\\end{pmatrix}\\cdot \\begin{pmatrix}\\mathsf {q}_1\\\\\\mathsf {q}_2\\end{pmatrix},\\qquad \\mathrm {{(41{\\mathrm {a},\\!\\mathrm {b}})}}$ where $\\mathsf {v}_n = (\\alpha _{n,1},\\ldots ,\\alpha _{n,K},\\beta _{n,1},\\ldots ,\\beta _{n,K})^T$ and $\\mathsf {q}_n = (\\tilde{\\alpha }_{n,1},\\ldots ,\\tilde{\\alpha }_{n,K},\\tilde{\\beta }_{n,1},\\ldots ,\\tilde{\\beta }_{n,K})^T$ ; $\\bar{\\mathcal {E}}=\\mathcal {E}/(\\kappa _1\\kappa _2)$ is a rescaled eigenvalue; $\\mathsf {M}_n$ and $\\mathsf {U}_n$ are identical to the matrices $\\mathsf {M}$ and $\\mathsf {U}$ , respectively, used in the single-ring scheme (REF ), with the geometric parameters being those of ring $n$ ; and $\\mathsf {V}$ is a coupling matrix whose form is provided in Appendix §REF ."
],
[
"Non-coaxial homodimers",
"As an example of a non-coaxial dimer configuration, we consider the bilayer configuration shown in Fig.",
"REF .",
"It consists of a pair of torus-shaped rings whose centerlines define parallel planes separated by the vertical distance $h$ , and whose symmetry axes are separated by the horizontal distance $d$ .",
"The case where $d$ vanishes corresponds to the coaxial configuration considered in §REF , while the limit $h^2+d^2\\rightarrow \\infty $ corresponds to that of non-interacting rings.",
"We have seen that both of these extreme cases are analytically solvable, with each longitudinal mode involving only a single Fourier harmonic.",
"In contrast, in the general case each mode is expected to consist of a combination of Fourier harmonics.",
"Symmetry still allows, however, the modes of the bilayer geometry to be classified based on whether the voltage eigenfunctions are even or odd about the $x$ –$z$ mirror plane.",
"With the azimuthal angle $\\phi $ measured from the $x$ direction, these even and odd modes involve only cosine or sine Fourier harmonics, respectively.",
"Thus, the semi-analytical scheme (REF ) is reduced in this case to two uncoupled $2K\\times 2K$ matrix problems.",
"We first consider the homodimer case where the rings are identical ($a_1=a_2=a$ , $\\kappa _1=\\kappa _2=\\kappa $ ).",
"In particular, we focus attention on the two modes continuated, as $d$ is increased from zero, from the degenerate in-phase dipolar modes $(1,c,+)$ and $(1,s,+)$ found in the coaxial case (§REF ).",
"Fig.",
"REF depicts the variation with $d/a$ of the respective eigenvalues, say $\\mathcal {E}^{({1,c,+})}$ and $\\mathcal {E}^{({1,s,+})}$ , calculated using the semi-analytical scheme (REF ) for two values of $h/a$ .",
"Note that the `$c$ ' and `$s$ ' modes are even and odd about the $x$ –$z$ plane, respectively, as in the coaxial case, but are no longer degenerate rotations of each other.",
"The insets show the voltage eigenfunctions for the indicated values of $d/a$ and $h/a$ .",
"A numerical study based on the semi-analytical scheme (REF ) suggests that the modes of this bilayer homodimer configuration are, by visual inspection, dominated by a single Fourier harmonic.",
"This harmonic corresponds to that of the mode of the corresponding coaxial (or isolated-ring) configuration from which the bilayer mode is continuated from.",
"This observation suggests an intuitive and ad hoc approximation in which the voltage profiles in both rings are constructed from just the apparently dominant harmonic.",
"With that assumption, we obtain $\\mathcal {E}^{({m,c,\\pm })} \\approx -\\frac{2 \\kappa ^2}{m^2 }\\left(\\ln 8 \\kappa - \\sum _{k=1}^{m}\\frac{2}{2k-1} \\pm \\frac{a}{2\\pi }\\int _{0}^{2 \\pi }\\int _{0}^{2 \\pi }d\\phi _1d\\phi _2\\,\\frac{\\cos m\\phi _1\\cos m\\phi _2}{|\\mathbf {y}_1(\\phi _1)-\\mathbf {y}_2(\\phi _2)|} \\right)^{-1}$ along with a similar expression for $\\mathcal {E}^{({m,s,\\pm })}$ where $\\cos m\\phi _1\\cos m\\phi _2$ is replaced by $\\sin m\\phi _1\\sin m\\phi _2$ .",
"In Fig.",
"REF , the dash-dotted curves depict approximation (REF ) for $\\mathcal {E}^{({1,c,+})}$ and $\\mathcal {E}^{({1,s,+})}$ .",
"Despite the approximation being heuristic, the agreement with the semi-analytical slender-body scheme (REF ) is reasonably good for all $d/a$ and especially at moderately large $d/a$ .",
"It is clear that this approximation is exact for $d=0$ and asymptotically correct as $d/a\\rightarrow \\infty $ ."
],
[
"Non-coaxial heterodimers",
"Consider now a bilayer heterodimer where the torus-shaped rings are not identical.",
"Unlike in the bilayer homodimer scenario considered above, now the spectrum of Fourier harmonics comprising each mode undergoes significant evolution as the horizontal displacement $d$ is increased from the coaxial case $d=0$ , or, alternatively, decreased from the isolated-ring case $d=\\infty $ .",
"Moreover, we find that this evolution can continuously link two different Fourier harmonics in the latter limiting cases, in which each mode is composed of a single Fourier harmonic.",
"We demonstrate this in Fig.",
"REF by considering the evolution of the two modes continuated from the coaxial in-phase dipolar mode $({1,c,+})$ and out-of-phase quadrupolar mode $({2,c,-})$ , respectively, with the geometric parameters of the two rings chosen such that the eigenvalues are close in the coaxial configuration.",
"In the coaxial configuration, the mode $(2,c,-)$ is dominated by a quadrupolar distribution in the larger ring, whereas the mode $(1,c,+)$ is dominated by a dipolar distribution in the smaller ring.",
"As $d$ is continuously increased to $\\infty $ , the mode $(2,c,-)$ is ultimately dominated by the dipolar mode $(1,c)$ of the smaller ring; similarly, the mode $(1,c,+)$ is ultimately dominated by the quadrupolar mode $(2,c)$ of the larger ring.",
"This evolution is seen to involve multiple stages during which the Fourier harmonics of each mode undergo mixing and the eigenvalues split apart and then re-approach each other twice.",
"We will see in § some of the implications of this evolution in the context of the scattering problem.",
"Figure: Evolution as a function of d/a 1 d/a_1 of two eigenvalues of the bilayer heterodimer configuration schematically shown in Fig.",
", in the case κ 1 =10\\kappa _1 = 10, κ 2 =5\\kappa _2=5, a 1 /a 2 =2a_1/a_2 = 2 and h/a 1 =0.3h/a_1=0.3.",
"The modes are those continuated from the in-phase dipolar mode (1,c,+)(1,c,+) and out-of-phase quadrupolar mode (2,c,-)(2,c,-), respectively, of the coaxial configuration obtained for d=0d=0.",
"Also shown are the voltage eigenfunctions at the indicated values of d/a 1 d/a_1."
],
[
"Arbitrary cross-sectional shapes",
"It is straightforward to extend our theoretical framework to allow for non-circular cross-sectional shapes.",
"In particular, let us briefly revisit the derivation in §REF of the single-ring reduced eigenvalue problem.",
"Given our focus on longitudinal modes of slender rings, we still expect that the interior potential is approximately uniform over the ring's cross section and varies mainly in the azimuthal direction.",
"Thus, the representation (REF ) of the interior potential by an azimuthal voltage profile still holds.",
"The exterior potential in the vicinity of the ring, however, can no longer be approximated as in (REF ), since that radially symmetric distribution assumes that the cross sections are circular.",
"Nonetheless, it can be shown that (REF ) still holds at intermediate radial distances from the centerline, i.e., $b\\ll r\\ll a$ , if only the cross-sectional radius $bf(\\phi )$ is replaced by the so-called `conformal radius', say $bf^*(\\phi )$ , of the cross-sectional geometry at the azimuthal angle $\\phi $ .",
"Working in the corresponding cross-sectional plane, the conformal radius $bf^*(\\phi )$ can be extracted from a conformal mapping from the exterior of a circle of that radius to the domain exterior to the true cross section (see, e.g., [33] and [59]).",
"In particular, for elliptical cross sections with semi-diameters $b\\sigma _1(\\phi )$ and $b\\sigma _2(\\phi )$ , one finds $bf^*(\\phi )=(b\\sigma _1(\\phi )+b\\sigma _2(\\phi ))/2$ .",
"Expressions for several other geometries can be found in [60].",
"As a consequence, $f(\\phi )$ should be replaced by $f^*(\\phi )$ in the capacitance relation (REF ), while (REF ) remains unchanged, with $\\bar{A}(\\phi )$ still denoting the scaled cross-sectional area $A(\\phi )/b^2$ .",
"The significance of the present extension is that, in contrast to the case of circular cross sections, $\\bar{A}(\\phi )$ can now be tuned independently from ${f^*}(\\phi )$ .",
"The generalization of this extension to the case of ring dimers is evident.",
"Consider for example an azimuthally uniform ring whose cross-sectional shape is arbitrary.",
"The eigenvalues follow from the result (REF ) for a torus-shaped ring as $\\mathcal {E}^{({m})} = -\\frac{2\\pi \\kappa ^2}{m^2\\bar{A}}\\left(\\ln \\frac{8 \\kappa }{f^*} - 2\\sum _{k=1}^m\\frac{1}{2k-1}\\right)^{-1}, \\quad \\text{for} \\quad m=1,2,\\ldots $ Note the logarithmically weak influence of the ring's cross-sectional shape, which we emphasize is a specific feature of the longitudinal modes considered herein.",
"This prediction is consistent with the experimental and numerical results in [44], which show only slight differences between the plasmon-resonance frequencies of rings of different cross-sectional shape yet similar cross-sectional area."
],
[
"Chain of rings",
"For clarity of exposition, we have so far considered either single rings or ring dimers.",
"It is straightforward, however, to extend our approximation scheme to a system consisting of an arbitrary number of interacting rings.",
"In particular, here we consider a coaxial chain of $N$ not necessarily identical azimuthally invariant rings.",
"(In light of the preceding generalization, azimuthally invariant does not necessarily imply torus-shaped.)",
"In this case, the appropriate reduced eigenvalue problem can be treated analytically.",
"The analysis closely follows the derivation in §§REF for a coaxial dimer of torus-shaped rings.",
"Similar to the latter case, symmetry implies collective modes, with azimuthal number $m=1,2,\\ldots $ , in which the voltage eigenfunction in the $n$ th ring is $v_n = c_n\\cos m\\phi $ .",
"(There are also $\\pi /2$ rotations of these collective modes having the same eigenvalues.)",
"Following the steps in §§REF , for given $m$ we find the $N\\times N$ generalized matrix-eigenvalue problem (cf.",
"(REF )) $\\mathsf {c} =-\\mathcal {E}\\,\\mathsf {G}\\cdot \\mathsf {c}$ where $\\mathsf {c} = (c_1,c_2,\\ldots ,c_N)^T$ and $\\mathsf {G}$ is a $m$ -dependent $N\\times N$ matrix whose components are defined analogously to (REF ).",
"Figure: (a) Eigenvalues of the `dipolar' (m=1m=1) modes of a coaxial chain of N=43N=43 identical torus-shaped rings (κ=10\\kappa =10).",
"The chain consists of two identical sub-chains, each formed of 10 dimers (spacing between rings and dimers being 0.30.3 and 0.70.7 times the ring radius, respectively), which are linked by a trimer (spacings similar to dimer sub-chains).",
"Each eigenvalue is doubly degenerate, corresponding to a π/2\\pi /2 rotation of the eigenfunctions about the symmetry axis.",
"The spectrum is calculated in the slender-body approximation by solving ().",
"It consists of two nearly continuous bands that cross the eigenvalues ℰ (1,±) \\mathcal {E}^{({1,\\pm })} of the dimers in isolation.",
"Additionally, there are two isolated eigenvalues which are associated with localized `defect modes'.",
"The defect mode marked in (a) is visualized in terms of the dipole intensities c n c_n (cf.",
"()) in (b) and voltage profiles in (c).",
"The horizontal lines in (a) mark the eigenvalues ℰ (1,-) \\mathcal {E}^{({1,-})} and ℰ (1,+) \\mathcal {E}^{({1,+})} of the dimers in isolation.As an example, we employ (REF ) to calculate the eigenfunctions of a finite coaxial chain of torus-shaped rings.",
"Specifically, we consider a chain consisting of a series of identical, equally spaced, homodimers, where the center dimer is replaced by a homotrimer.",
"In Fig.",
"REF , we focus our attention on the dipolar modes of the chain ($m=1)$ , for which the permittivity-eigenvalue spectrum is seen to consist of two disjoint, nearly continuous, bands formed of a large number of densely distributed eigenvalues; the lower- and higher-energy bands respectively cross the eigenvalues $\\mathcal {E}^{({1,-})}$ and $\\mathcal {E}^{({1,+})}$ of the dimers in isolation (§REF ).",
"Additionally, there are two isolated eigenvalues, one in the spectral gap between the two bands and one at an energy lower than the lower-energy band.",
"As demonstrated in the figure, the isolated eigenvalues correspond to modes that are localized around the defect.",
"We remark that (REF ) predicts localized modes for all $m$ , not only for the dipolar modes.",
"In retrospect, the existence of localized modes rationalizes the application of our quasi-static theory to the case of an extended chain of rings, whose total length in any realistic scenario would be at least comparable to the free-space wavelength.",
"Accordingly, we expect non-localized modes of the chain to be significantly affected by retardation [61]."
],
[
"Plane-wave illumination",
"Armed with our slender-body approximations for the longitudinal modes of slender-ring structures, we now return to the scattering problem formulated in §REF .",
"In that problem, the near-field potential $\\varphi (\\mathbf {x})$ generically possesses the asymptotic far-field behavior [62] $\\varphi (\\mathbf {x}) \\sim -\\mathbf {E}_{\\infty }\\cdot \\mathbf {x} + \\mathbf {E}_{\\infty }\\cdot \\alpha \\cdot \\frac{\\mathbf {x}}{4\\pi |\\mathbf {x}|^3} \\quad \\text{as}\\quad |\\mathbf {x}|\\rightarrow \\infty ,$ in which $\\alpha $ is the polarizability tensor of the structure.",
"In terms of that tensor, the absorption cross section in the direction of the applied field, say $\\hat{\\textbf {\\textit {\\i }}}$ , is given by [32] $C_{\\text{abs}} = \\frac{2\\pi }{\\lambda }\\,\\hat{\\textbf {\\textit {\\i }}}\\hat{\\textbf {\\textit {\\i }}}:\\text{Im}\\,\\alpha ,$ where $\\lambda $ denotes the wavelength of the incident plane-wave.",
"We shall use the approximation scheme developed in this paper to calculate $\\alpha $ and hence $C_{\\text{abs}}$ for a range of slender-ring geometries.",
"A slender-body approximation for $\\alpha $ can be extracted by considering the large $|\\mathbf {x}|$ expansion of the spectral solution (REF ) and substituting the slender-body approximations for the voltage eigenfunctions and eigenvalues.",
"For a structure formed of $N$ arbitrarily shaped rings, we find $\\alpha = \\frac{1}{\\epsilon _0}\\sum _{I \\in \\mathcal {I}}\\frac{\\epsilon _r(\\omega )-1}{\\epsilon _r(\\omega )-\\mathcal {E}^{(I)}}\\frac{\\left(\\sum _{n=1}^N a_n \\int _0^{2\\pi }d\\phi \\,\\mathbf {y}_nq_n^{({I})}\\right)\\left(\\sum _{n=1}^N a_n \\int _0^{2\\pi }d\\phi \\,\\mathbf {y}_nq_n^{({I})}\\right)}{\\sum _{n=1}^{N}a_n\\int _0^{2\\pi }d\\phi \\,q_n^{({I})}v_n^{({I})}}.$ A derivation of this result is given in Appendix .",
"We note that it only includes the low-frequency longitudinal modes studied in this paper, which are expected to dominate the plasmonic response as further discussed in [34].",
"To evaluate (REF ) for general ring geometries, we employ the semi-analytical scheme (REF ), whose generalization from the case of two to $N$ rings is straightforward; furthermore, non-circular cross sections can be included in that scheme as discussed in §REF .",
"In the special cases of single azimuthally invariant rings, as well as azimuthally invariant coaxial dimers, it is possible to evaluate (REF ) in closed form.",
"Indeed, in those cases symmetry dictates that the polarizability tensor possesses the form $\\alpha = (\\mathbf {\\hat{e}}_x\\mathbf {\\hat{e}}_x+\\mathbf {\\hat{e}}_y\\mathbf {\\hat{e}}_y)\\alpha $ , where $\\mathbf {\\hat{e}}_x$ and $\\mathbf {\\hat{e}}_y$ are orthogonal unit vectors parallel to the plane defined by the rings and $\\alpha $ is a scalar polarizability.",
"Furthermore, since for these geometries each mode involves a single Fourier harmonic, and since the centerlines $\\mathbf {y}_n$ are circular curves, the overlap integrals in the numerator of (REF ) vanish for all except the dipolar modes.",
"Assuming for simplicity that the azimuthally invariant rings are torus-shaped, we use the results of § to find $\\alpha = \\frac{\\epsilon _r(\\omega )-1}{\\epsilon _r(\\omega )-\\mathcal {E}^{(1)}} \\frac{2\\pi ^2a^3}{\\ln 8\\kappa -2},$ with $\\mathcal {E}^{({1})}$ provided by (REF ).",
"Similarly, using the results of §REF , we find for a coaxial dimer of torus-shaped rings ${\\alpha } = \\sum _{\\pm }\\frac{1-\\epsilon _r(\\omega )}{\\epsilon _r(\\omega )-\\mathcal {E}^{({1,\\pm })}}\\frac{\\pi \\mathcal {E}^{({1,\\pm })}\\left(A_1c_1^{({1,\\pm })}+A_2c_2^{({1,\\pm })}\\right)^2}{\\frac{A_1}{a_1}\\left(c_1^{({1,\\pm })}\\right)^2 + \\frac{A_2}{a_2}\\left(c_2^{({1,\\pm })}\\right)^2},$ with $\\mathcal {E}^{({1,\\pm })}$ , $c_1^{({1,\\pm })}$ and $c_1^{({2,\\pm })}$ provided by (REF ) and (REF ).",
"In the more specific coaxial homodimer case, the numerator in (REF ) vanishes identically for the out-of-phase dipolar mode, namely because the induced dipole excited in the two rings cancel.",
"Thus, (REF ) degenerates to $\\alpha = \\frac{\\epsilon _r(\\omega )-1}{\\epsilon _r(\\omega )-\\mathcal {E}^{(1,+)}} \\frac{4\\pi ^2a^3}{\\ln 8\\kappa -2+\\frac{1}{2}\\Delta _1},$ where $\\mathcal {E}^{({1,+})}$ now possesses the simplified form (REF ) and $\\Delta _1$ is given by (REF ).",
"We now apply the above results for the scattering problem to several ring structures whose plasmonic eigenvalues and eigenfunctions we have analyzed earlier in the paper.",
"In particular, we consider single torus-shaped and azimuthally non-uniform rings, as well as coaxial and non-coaxial dimers of torus-shaped rings.",
"Figure: Absorption cross section (), normalized by the ring volume VV, for the torus-shaped and azimuthally non-uniform rings defined in the caption of Fig. .",
"The incident-field direction ı ^\\hat{\\imath } is in the plane of the ring, pointing in the maximum-thickness direction in the case of the azimuthally non-uniform ring.",
"Solid blue line: exact solution for the torus-shaped ring .",
"Dash-dotted blue line: slender-body approximation () for the torus-shaped ring.",
"Dashed red line: numerically evaluated slender-body approximation () for the azimuthally nonuniform ring.",
"We assume the Drude model () with ω p =1.96×10 16 rad/s\\omega _p= 1.96 \\times 10^{16} \\,\\text{rad}/\\text{s} and γ=9.05×10 13 rad/s\\gamma = 9.05\\times 10^{13}\\,\\text{rad}/\\text{s}.Figure: Absorption cross section (), normalized by the dimer volume VV, for the coaxial homodimer of Fig.",
"; the coaxial heterodimer of Fig.",
"and Fig.",
"for d=0d=0; and the bilayer heterodimer of Fig.",
"for d/a 1 =0.5d/a_1=0.5.",
"The incident-field direction ı ^\\hat{\\imath } is in the plane of the rings, pointing in the direction of the horizontal displacement of the smaller ring in the non-coaxial case.",
"Solid black line: slender-body approximation () for the coaxial homodimer.",
"Dashed blue line: slender-body approximation () for the coaxial heterodimer.",
"Dash-dotted red line: numerically evaluated slender-body approximation () for the non-coaxial heterodimer.",
"Vertical lines: frequencies corresponding to the eigenvalues ℰ (1) \\mathcal {E}^{({1})} of the larger and smaller torus-shaped rings in isolation.We assume the Drude model () with ω p =1.96×10 16 rad/s\\omega _p = 1.96\\times 10^{16}\\,\\text{rad}/\\text{s} and γ=9.05×10 13 rad/s\\gamma = 9.05\\times 10^{13}\\,\\text{rad}/\\text{s}.In Fig.",
"REF , we show $C_{\\text{abs}}$ divided by volume for a torus-shaped ring alongside that for the azimuthally non-uniform ring from Fig.",
"REF .",
"The incident-field direction $\\hat{\\imath }$ is parallel to the plane of the ring; in the case of the azimuthally non-uniform ring, $\\hat{\\imath }$ is further specified as pointing towards the direction of maximum thickness of the ring.",
"For the torus-shaped ring, the slender-body approximation (REF ) is compared with the exact computations given in [63] and good agreement is found despite the aspect ratio $\\kappa =10$ being only moderately large.",
"The torus-shaped ring exhibits a single resonance peak, which is associated with the excitation of the cosine-dipolar mode $(1,c)$ shown at the top-right corner of Fig.",
"REF .",
"(With $\\phi $ measured from $\\hat{\\imath }$ , the corresponding sine mode is not excited.)",
"In contrast, for the azimuthally non-uniform ring, we observe multiple resonance peaks.",
"As explained in §REF , in that case all modes generally include a dipolar component (first Fourier harmonic).",
"Accordingly, the overlap integrals in (REF ) do not vanish identically as they do in the torus case for $m>1$ .",
"In particular, the lowest- and second-lowest-frequency peaks seen in Fig.",
"REF are due to excitation of the `dipolar' mode $(1,c)$ and `quadrupolar' mode $(2,c)$ shown on the left of Fig.",
"REF .",
"In Fig.",
"REF , we show $C_{\\text{abs}}$ divided by volume for several bilayer dimer configurations formed of torus-shaped rings: a coaxial homodimer (same as in Fig.",
"REF ), a coaxial heterodimer (same as in Fig.",
"REF and Fig.",
"REF for $d=0$ ) and a bilayer non-coaxial heterodimer (same as in Fig.",
"REF for $d/a_1=0.5$ ).",
"The incident-field direction $\\hat{\\imath }$ is taken parallel to the planes of the rings; in the non-coaxial-heterodimer case, $\\hat{\\imath }$ is further specified as pointing along the direction of the horizontal displacement of the smaller ring.",
"For the coaxial homodimer, we observe a single resonant peak, a prediction which is clearly inferred from the closed-form slender-body approximation (REF ).",
"This is because only the in-phase cosine-dipolar mode $(1,c,+)$ is excited by the plane wave.",
"(With $\\phi $ measured from $\\hat{\\imath }$ , the corresponding sine mode is not excited.)",
"While this is not demonstrated here, we know from the analytical expression (REF ) for $\\mathcal {E}^{({1,+})}$ , or from Fig.",
"REF a, to expect this peak to blueshift with decreasing distance between the rings.",
"This is opposite to the trend observed for the analogous low-frequency bonding-gap modes of sphere dimers [11], [15], [17], [20].",
"For the coaxial heterodimer, we observe an additional resonance peak at higher frequency.",
"This is because the net dipole induced in the rings for the out-of-phase dipolar modes does not cancel out between the two rings as it does in the coaxial-homodimer case.",
"This feature is easily inferred from the slender-theory approximation (REF ), which explicitly shows two resonances associated with the in-phase and out-of-phase dipolar modes.",
"(With $\\phi $ measured from $\\hat{\\imath }$ , it is again only the `cosine' dipolar modes that are excited.)",
"As seen in Fig.",
"REF b, for this configuration the out-of-phase dipolar mode is dominated by a dipolar distribution in the larger ring, whereas the in-phase dipolar mode is dominated by a dipolar distribution in the smaller ring.",
"From Fig.",
"REF a, we expect the in-phase and out-of-phase resonance to respectively blueshift and redshift as the vertical distance between the rings is reduced.",
"Consider next the non-coaxial-heterodimer configuration.",
"Similarly to the azimuthally non-uniform ring, we observe multiple resonance peaks owing to the absence of axial symmetry.",
"In particular, we note that the higher-frequency resonance in the coaxial-heterodimer case, which is associated with the in-phase-dipolar mode $(1,c,+)$ of that configuration, is replaced by two distinct peaks.",
"This can be understood from the study carried out in Fig.",
"REF .",
"Thus, this pair of resonances are associated with the excitation of the two modes continuated from the in-phase-dipolar $(1,c,+)$ and out-of-phase-quadrupolar $(2,c,-)$ modes of the corresponding coaxial-heterodimer configuration.",
"While, owing to symmetry, in the coaxial case only dipolar modes are excited, in the non-coaxial case both modes include dipolar components and can therefore be excited.",
"Specifically, the non-coaxial configuration is as shown in insets (b) and (g) of Fig.",
"REF , in which case the eigenvalues are farthest apart as a function of the horizontal displacement, and both modes have a visible dipolar component in the smaller ring."
],
[
"Concluding remarks",
"We have developed an approximate theory to describe the longitudinal localized-surface-plasmon resonances of slender metallic nanorings of virtually arbitrary shape, as well as more involved nanometallic structures formed of two or more such rings, the separation between the rings being comparable with their centerline radii.",
"At the heart of the theory is an asymptotic reduction of the 3D plasmonic eigenvalue problem governing the material- and frequency-independent longitudinal modes of the structure to a reduced 1D formulation, in which the plasmonic eigenmodes are represented by azimuthal voltage and polarization-charge profiles attached to each ring.",
"Once this reduced eigenvalue problem has been solved for a given geometry, approximations for the near-field eigenfunctions in 3D can be extracted from their associated 1D eigenfunctions.",
"When joined with the standard spectral theory of plasmonic resonance, this provides the means to obtain any quantity of interest in a physical scattering problem (that involves the same geometry), such as scattering cross sections.",
"Our approximation approach heavily draws on a transfer of knowledge from the field of fluid dynamics, where approximations of the sort derived here are known as slender-body theory.",
"We stress that the slender-body theory developed herein is of the more accurate `nonlocal' type, which in the present context means that we account for electrostatic interactions between any two azimuthal segments of any of the rings forming the structure.",
"The inclusion of these interactions can be shown to ensure an `algebraic' rather than `logarithmic' asymptotic accuracy of the scheme in the large-aspect-ratio limit pertinent to slender rings.",
"This claimed accuracy is demonstrated by the good agreement in Figs.",
"REF and REF with known solutions for single torus-shaped rings.",
"To demonstrate the versatility of the theory, we have applied it to the calculation of the longitudinal modes and absorption cross sections of a number of configurations.",
"Still, it is clear that we only considered a small sample of the wide range of geometries that can be studied.",
"For several families of geometries, namely azimuthally invariant rings (not necessarily torus-shaped) and coaxial dimers and chains thereof, the reduced formulation was solved in closed form, thus generating several apparently new analytical approximations.",
"We also considered a range of other geometries, including azimuthally nonuniform rings and non-coaxial multiple-ring configurations.",
"In the latter cases, we solved the reduced formulation using straightforward semi-analytical schemes, in which the 1D eigenfunctions are represented by their Fourier coefficients.",
"Overall, the present theoretical framework enables one to rapidly gauge the plasmonic properties of unprecedentedly complex 3D structures, which hopefully will foster the study of new plasmonic structures and phenomena.",
"With that in mind, several generalizations to our theory are desirable.",
"These include the asymptotic description of non-longitudinal modes (see discussion in [34]); adapting the theory to Maxwell’s equations so as to treat metallic structures comparable in size to the free-space wavelength; and accounting for the spatial nonlocality of the metal's dielectric response, which should be important for rings of subnanometric thickness.",
"Acknowledgements.",
"This work was supported by the Engineering and Physical Sciences Research Council through the New Investigator Award grant EP/R041458/1."
],
[
"Matching the ring and cross-sectional scales",
"In this appendix, we shall derive the small $r$ behavior of the ring-scale potential (REF ).",
"A preliminary step is to write that solution in the form $\\varphi = \\frac{q(\\phi )}{4\\pi \\epsilon _0}\\int _{0}^{2\\pi }\\,a d\\phi ^{\\prime }\\, \\frac{1}{|\\mathbf {x}(r,\\theta ,\\phi )-\\mathbf {y}(\\phi ^{\\prime })|} +\\frac{1}{4\\pi \\epsilon _0} \\int _{0}^{2\\pi }\\,ad\\phi ^{\\prime }\\,\\frac{q(\\phi ^{\\prime })-q(\\phi )}{|\\mathbf {x}(r,\\theta ,\\phi )-\\mathbf {y}(\\phi ^{\\prime })|},$ where the geometry gives $|\\mathbf {x}(r,\\theta ,\\phi )-\\mathbf {y}(\\phi ^{\\prime })| = \\sqrt{r^2 + 4(a^2+ar\\cos \\theta )\\sin ^2\\frac{\\phi -\\phi ^{\\prime }}{2}}.$ Using (REF ), it is readily seen that the second integral in (REF ) is regular as $r\\rightarrow 0$ : $\\int _{0}^{2\\pi }\\,ad\\phi ^{\\prime }\\,\\frac{q(\\phi ^{\\prime })-q(\\phi )}{|\\mathbf {x}(r,\\theta ,\\phi )-\\mathbf {y}(\\phi ^{\\prime })|}= \\int _{0}^{2\\pi }\\,d\\phi ^{\\prime }\\, \\frac{q(\\phi ^{\\prime })-q(\\phi )}{2\\sin \\frac{\\left|\\phi -\\phi ^{\\prime }\\right|}{2}} + o(1) \\quad \\text{as}\\quad r\\rightarrow 0.$ To treat the first integral in (REF ), which is singular as $r\\rightarrow 0$ , we use the method of splitting the range of integration [33].",
"Thus, we write $\\int _{0}^{2\\pi }\\,a d\\phi ^{\\prime }\\, \\frac{1}{|\\mathbf {x}(r,\\theta ,\\phi )-\\mathbf {y}(\\phi ^{\\prime })|} =\\left\\lbrace \\int _0^\\delta \\,dt+\\int _{\\delta }^{\\pi }\\,dt\\right\\rbrace \\frac{2}{\\sqrt{\\bar{r}^2+4(1+\\bar{r}\\cos \\theta )\\sin ^2\\frac{t}{2}}},$ where we introduce the normalized radial coordinate $\\bar{r}=r/a$ and an auxiliary parameter in the range $\\bar{r}\\ll \\delta \\ll 1$ .",
"Denote the first integral on the right-hand side of (REF ) by $I_1$ .",
"Making the change of variables $\\tau =t/\\bar{r}$ , we have $I_1=2\\int _0^{\\delta /\\bar{r}} \\frac{d\\tau }{\\sqrt{1+4(1+\\bar{r}\\cos \\theta )\\bar{r}^{-2}\\sin ^2(\\bar{r}\\tau /2)}}.$ Using $\\bar{r}\\tau <\\delta \\ll 1$ , the integrand can be expanded to show that $I_1 = 2\\int _0^{\\delta /\\bar{r}}\\frac{d\\tau }{(1+\\tau ^2)^{1/2}}+o(1) = 2\\ln \\frac{2\\delta }{\\bar{r}} +o(1)\\quad \\text{as} \\quad \\bar{r}\\rightarrow 0,$ where for the last step we used $\\delta /\\bar{r}\\gg 1$ .",
"Similarly, denote the second integral on the right-hand side of (REF ) by $I_2$ .",
"Using $t>\\delta \\gg \\bar{r}$ , the integrand of that integral can be expanded to show that $I_2 = \\int _\\delta ^\\pi \\frac{dt}{\\sin \\frac{t}{2}}+o(1) = 2\\ln \\frac{4}{\\delta }+o(1) \\quad \\text{as} \\quad \\bar{r}\\rightarrow 0.$ Combining (REF ), (REF ) and (REF ), we find the requisite behavior $\\varphi (\\mathbf {x}) = \\frac{q(\\phi )}{2\\pi \\epsilon _0}\\ln \\frac{8a}{r}+\\frac{1}{4\\pi \\epsilon _0}\\int _0^{2\\pi }d\\phi ^{\\prime }\\,\\frac{q(\\phi ^{\\prime })-q(\\phi )}{2\\sin \\frac{|\\phi -\\phi ^{\\prime }|}{2}} + o(1)\\quad \\text{as}\\quad r\\rightarrow 0.$"
],
[
"Proof of identity (",
"In this appendix we prove identity (REF ), which constitutes a diagonalization of the integral operator appearing in the capacitance relation (REF ).",
"To this end, it is convenient to denote that integral operator as $\\mathcal {N}[q(\\phi )] = \\int _{0}^{2\\pi }d\\phi ^{\\prime }\\, \\frac{q(\\phi ^{\\prime })-q(\\phi )}{2\\sin \\frac{\\left|\\phi -\\phi ^{\\prime }\\right|}{2}}.$ We shall show that, for any non-negative integer $n$ , there exists a constant $\\lambda _n$ such that $\\mathcal {N}[\\cos n\\phi ] = \\lambda _n\\cos n\\phi , \\quad \\mathcal {N}[\\sin n\\phi ] = \\lambda _n\\sin n\\phi .\\qquad \\mathrm {{(\\mathrm {60}{\\mathrm {a},\\!\\mathrm {b}})}}$ Our method of proof will also furnish the constants, which constitute the eigenvalues of the operator $\\mathcal {N}$ .",
"In what follows, we consider only the cosine eigenfunctions, the corresponding result for the sine eigenfunctions readily following from a straightforward change of variables.",
"Direct calculation shows that $\\mathcal {N}[1] = 0$ and $\\mathcal {N}[\\cos \\phi ] = -4\\cos \\phi $ , thence that (REF a) holds for $n=0$ and $n=1$ with $\\lambda _0=0$ and $\\lambda _1=-4$ .",
"Now, let $m$ be any integer $>1$ and assume that (REF a) also holds for $n=1,\\ldots ,m-1$ , with $\\lambda _n=-\\frac{4}{2n-1}+\\lambda _{n-1}.$ This assumed difference relation is consistent with the result for $n=1$ and implies that $\\lambda _n= -4\\sum _{k=1}^n\\frac{1}{2k-1} \\quad \\text{for} \\quad n=1,\\ldots ,m-1.$ Next, we use the definition (REF ) and (REF a) for $n=m-2$ to write $\\mathcal {N}[\\cos m\\phi ] &=& 2\\mathcal {N}[\\cos \\left((m-1)\\phi \\right)\\cos \\phi ] - \\lambda _{m-2}\\cos \\left((m-2)\\phi \\right).$ Using (REF ) (REF a) for $n=m-1$ , we have $\\mathcal {N}[\\cos \\left((m-1)\\phi \\right)\\cos \\phi ] =\\int _{0}^{2\\pi }d\\phi ^{\\prime }\\,\\frac{\\cos \\left((m-1)\\phi ^{\\prime }\\right)\\cos \\phi ^{\\prime } - \\cos \\left((m-1)\\phi \\right)\\cos \\phi }{2\\sin \\frac{|\\phi ^{\\prime } - \\phi |}{2}}\\\\= \\int _{0}^{2\\pi }d\\phi ^{\\prime }\\,\\frac{\\cos \\left((m-1)\\phi ^{\\prime }\\right)\\left(\\cos \\phi ^{\\prime } -\\cos \\phi \\right)}{2\\sin \\frac{|\\phi ^{\\prime } - \\phi |}{2}} + \\cos \\phi \\,\\mathcal {N}[\\cos \\left((m-1)\\phi \\right)]\\\\= -\\int _{0}^{2\\pi }d\\phi ^{\\prime }\\,\\frac{\\cos \\left((m-1)\\phi ^{\\prime }\\right)\\sin {\\frac{\\phi ^{\\prime }-\\phi }{2}}\\sin {\\frac{\\phi ^{\\prime }+\\phi }{2}}}{\\sin \\frac{|\\phi ^{\\prime } - \\phi |}{2}} + \\lambda _{m-1}\\cos \\phi \\cos \\left((m-1)\\phi \\right)\\\\= 2\\frac{\\cos ((m-2)\\phi )}{2m-3} - 2\\frac{\\cos m\\phi }{2m-1} + \\frac{1}{2}\\lambda _{m-1}\\left(\\cos m\\phi + \\cos ((m-2)\\phi )\\right).$ So that combining (REF ) and (REF ) gives $\\mathcal {N}[\\cos m\\phi ] = \\left(\\lambda _{m-1}-\\frac{4}{2m-1} \\right)\\cos m\\phi + \\left( \\lambda _{m-1} - \\lambda _{m-2}+\\frac{4}{2m-3}\\right)\\cos ((m-2)\\phi ),$ where the second term on the right-hand side vanishes owing to (REF ), while the first term on the right-hand side shows that (REF ), and thence (REF ), holds also for $n=m$ .",
"It follows by induction that (REF ) holds for any non-negative integer $n$ , with $\\lambda _0=0$ and $\\lambda _n$ given by (REF ) for $n>0$ ."
],
[
"Single ring",
"For a single ring of non-uniform thickness, the reduced eigenvalue problem of §REF is approximated by the generalized matrix-eigenvalue problem (REF ).",
"In that problem, $\\mathsf {M} = \\lbrace M_{n,k}\\rbrace $ and $\\mathsf {U} = \\lbrace U_{n,k}\\rbrace $ are $2K\\times 2K$ matrices with components $M_{n,k} = \\frac{nk}{\\pi } \\times \\begin{dcases}\\int _0^{2\\pi }d\\phi \\,\\bar{A}(\\phi )\\sin n\\phi \\sin k\\phi , \\quad n \\le K,\\, k\\le K\\\\ -\\int _0^{2\\pi }d\\phi \\,\\bar{A}(\\phi )\\sin n\\phi \\cos k\\phi , \\quad n \\le K,\\, k> K\\\\ -\\int _0^{2\\pi }d\\phi \\,\\bar{A}(\\phi )\\cos n\\phi \\sin k\\phi , \\quad n > K,\\, k\\le K\\\\ \\int _0^{2\\pi }d\\phi \\,\\bar{A}(\\phi )\\cos n\\phi \\cos k\\phi , \\quad n > K,\\, k > K\\end{dcases}$ and $U_{n,k} = -\\frac{1}{\\pi }\\delta _{n,k}\\sum _{k=1}^n\\frac{1}{2k-1} +\\frac{1}{2\\pi ^2}\\times \\begin{dcases}\\int _0^{2\\pi }d\\phi \\,\\cos n\\phi \\cos k\\phi \\ln {\\frac{8\\kappa }{f(\\phi )}}, \\quad n \\le K,\\, k\\le K\\\\ \\int _0^{2\\pi }d\\phi \\,\\cos n\\phi \\sin k\\phi \\ln {\\frac{8\\kappa }{f(\\phi )}}, \\quad n \\le K,\\, k> K\\\\ \\int _0^{2\\pi }d\\phi \\,\\sin n\\phi \\cos k\\phi \\ln {\\frac{8\\kappa }{f(\\phi )}}, \\quad n > K,\\, k\\le K\\\\ \\int _0^{2\\pi }d\\phi \\,\\sin n\\phi \\sin k\\phi \\ln {\\frac{8\\kappa }{f(\\phi )}}, \\quad n > K,\\, k > K.\\end{dcases}$ Furthermore, the coefficient $\\alpha _0$ can be calculated a posteriori as $\\alpha _0 = \\frac{1}{4\\pi ^2}\\sum _{k=1}^{K}\\int _0^{2\\pi }d\\phi \\,\\left(\\tilde{\\alpha }_k\\cos k\\phi + \\tilde{\\beta }_k\\sin k\\phi \\right)\\ln {\\frac{8\\kappa }{f(\\phi )}}.$"
],
[
"Ring dimer",
"For an arbitrary ring dimer, the generalized eigenvalue problem of §REF is approximated by the generalized matrix-eigenvalue problem (REF ).",
"In that problem, the $2K\\times 2K$ matrices $\\mathsf {M}_n$ and $\\mathsf {U}_n$ are like above with $n=1,2$ indicated the ring number, whereas $\\mathsf {V}=\\lbrace V_{n,k}\\rbrace $ is a $2K\\times 2K$ coupling matrix with components $V_{n,k} = \\frac{\\sqrt{a_1a_2}}{4\\pi ^2}\\begin{dcases}\\int _{0}^{2 \\pi }\\int _{0}^{2 \\pi }d\\phi _1 d\\phi _2\\,\\frac{\\cos n\\phi _1 \\cos k\\phi _2}{|\\mathbf {y}_1(\\phi _1)-\\mathbf {y}_2(\\phi _2)|}, \\quad n \\le K,\\, k\\le K\\\\ \\int _{0}^{2 \\pi }\\int _{0}^{2 \\pi }d\\phi _1 d\\phi _2\\,\\frac{\\cos n\\phi _1 \\sin k\\phi _2}{|\\mathbf {y}_1(\\phi _1)-\\mathbf {y}_2(\\phi _2)|}, \\quad n \\le K,\\, k> K\\\\ \\int _{0}^{2 \\pi }\\int _{0}^{2 \\pi }d\\phi _1 d\\phi _2\\,\\frac{\\sin n\\phi _1 \\cos k\\phi _2}{|\\mathbf {y}_1(\\phi _1)-\\mathbf {y}_2(\\phi _2)|}, \\quad n > K,\\, k\\le K\\\\ \\int _{0}^{2 \\pi }\\int _{0}^{2 \\pi }d\\phi _1 d\\phi _2\\,\\frac{\\sin n\\phi _1 \\sin k\\phi _2}{|\\mathbf {y}_1(\\phi _1)-\\mathbf {y}_2(\\phi _2)|}, \\quad n > K,\\, k > K.\\\\\\end{dcases}$"
],
[
"Far-field expansion",
"In this appendix we derive the far field behavior of the scattered field starting from the spectral solution (REF ), with the eigenvalues and eigenmodes approximated by their slender-body approximations.",
"As in §, we only include longitudinal modes in this calculation.",
"For generality, we consider a system of $N$ rings.",
"Consider first the far-field behavior of the eigenpotentials $\\varphi ^{({I})}(\\mathbf {x})$ , whose slender-body approximations are (cf.",
"(REF )) $\\varphi ^{({I})}(\\mathbf {x}) = \\sum _{n=1}^N\\frac{a_n}{4\\pi \\epsilon _0}\\int _0^{2\\pi } d\\phi ^{\\prime }\\,\\frac{q_n^{({I})}(\\phi ^{\\prime })}{|\\mathbf {x}-\\mathbf {y}_n(\\phi ^{\\prime })|}.$ Using the asymptotic behavior $\\frac{1}{|\\mathbf {x} - \\mathbf {y}_n|} \\sim \\frac{1}{|\\mathbf {x}|} + \\frac{\\mathbf {x}\\cdot \\mathbf {y}_n}{|\\mathbf {x}|^3}\\quad \\text{as}\\quad |\\mathbf {x}|\\rightarrow \\infty ,$ for $n=1,2,\\ldots ,N$ , and the zero-charge constraints $\\int _0^{2\\pi }d\\phi \\,q_n^{({\\mathcal {I}})} = 0$ , (REF ) gives $\\varphi ^{({\\mathcal {I}})}(\\mathbf {x}) \\sim \\frac{\\mathbf {x}}{4\\pi \\epsilon _0|\\mathbf {x}|^3}\\cdot \\sum _{n=1}^Na_n\\int _0^{2\\pi }d\\phi \\,\\mathbf {y}_n\\,q_n^{({\\mathcal {I}})} \\quad \\text{as}\\quad |\\mathbf {x}|\\rightarrow \\infty .$ We next note that, for the sake of evaluating the overlap and normalization integrals in (REF ), we may approximate the eigenfield inside the $n$ th ring as $\\nabla \\varphi _n^{({\\mathcal {I}})}= \\frac{1}{a_n^2}\\frac{dv_n^{({I})}}{d\\phi }\\frac{\\partial \\mathbf {y}_n}{\\partial \\phi }.$ Thus, using integration by parts together with the effective Gauss law (REF ), we find $\\frac{\\int dV\\, \\mathbf {E}_{\\infty }\\cdot \\nabla \\varphi ^{({I})}}{\\int dV\\, \\nabla \\varphi ^{({I})} \\cdot \\nabla \\varphi ^{({I})}} = \\frac{\\sum _{n=1}^N a_n\\int _0^{2\\pi }d\\phi \\,\\mathbf {E}_{\\infty }\\cdot \\mathbf {y}_n\\, q_n^{({I})}}{\\sum _{n=1}^N a_n\\int _0^{2\\pi }d\\phi \\,v_n^{({I})} q_n^{({I})}}.$ We emphasize that this result is valid even for azimuthaly non-uniform rings and arbitrary cross-sectional shapes.",
"With the above slender-body approximations, the spectral solution (REF ) gives the far-field behavior (REF ), with the polarization tensor $\\alpha $ given by (REF )."
]
] | 2107.01716 |
[
[
"Multi-View Correlation Distillation for Incremental Object Detection"
],
[
"Abstract In real applications, new object classes often emerge after the detection model has been trained on a prepared dataset with fixed classes.",
"Due to the storage burden and the privacy of old data, sometimes it is impractical to train the model from scratch with both old and new data.",
"Fine-tuning the old model with only new data will lead to a well-known phenomenon of catastrophic forgetting, which severely degrades the performance of modern object detectors.",
"In this paper, we propose a novel \\textbf{M}ulti-\\textbf{V}iew \\textbf{C}orrelation \\textbf{D}istillation (MVCD) based incremental object detection method, which explores the correlations in the feature space of the two-stage object detector (Faster R-CNN).",
"To better transfer the knowledge learned from the old classes and maintain the ability to learn new classes, we design correlation distillation losses from channel-wise, point-wise and instance-wise views to regularize the learning of the incremental model.",
"A new metric named Stability-Plasticity-mAP is proposed to better evaluate both the stability for old classes and the plasticity for new classes in incremental object detection.",
"The extensive experiments conducted on VOC2007 and COCO demonstrate that MVCD can effectively learn to detect objects of new classes and mitigate the problem of catastrophic forgetting."
],
[
"Introduction",
"Object detection is a basic computer vision task in many multimedia applications, such as autonomous driving, object tracking and so on.",
"Modern object detection methods based on Convolution Neural Networks (CNNs) have achieved state-of-the-art results, which are usually trained on predefined datasets with a fixed number of classes.",
"In many practical applications, new object classes often emerge after the detectors have been trained.",
"Due to the privacy of data and limited storage of the devices, sometimes the old data can not be available for training the detectors from scratch.",
"Even if the old data are available, this procedure will take a long training time.",
"Fine-tuning is a commonly used method to transfer the pretrained model on new data.",
"However, directly fine-tuning on new classes will severely decrease the performance on old classes [17], which is known as catastrophic forgetting [7] [9] [26].",
"Therefore, improving the ability of object detectors to learn new object classes continuously is necessary.",
"Recently, incremental learning has been paid more attention to classification, which aims to continuously learn to address new tasks from new data while preserving the learned knowledge from the old data.",
"Based on the regularization methods to overcome catastrophic forgetting, the incremental learning methods can be divided into two categories [14]: parameter-based [1] [17] [36] and distillation-based [2] [16] [21] [29] [30].",
"Due to the difficulty of designing a reasonable metric to evaluate the importance of all parameters, we follow the distillation-based regularization methods to preserve the learned knowledge from the old classes when adapting the old model on the data of new object classes.",
"Figure: Illustration of three views for correlation distillation.Different from image classification, object detection involves distinguishing foreground from complex background and the precise localization of objects, which is more challenging for incremental learning.",
"Existing incremental object detection methods [5] [10] [11] [19] [32] [37] mainly adopt knowledge distillation to regularize the behavior of the incremental model to be similar to the old model for preserving the learned knowledge.",
"The typical way is to minimize the distance between features or output logits of the old and incremental models.",
"However, due to the inherent difference in the categories to be detected, we should preserve the stability to detect old classes and the plasticity to learn new classes.",
"Directly enforcing the incremental model to imitate all activations in the feature maps of the old model (denoted as first-order distillation) makes the incremental model confusing about which knowledge is important and should be transferred.",
"The sufficiently learned valuable knowledge may not be well preserved, instead some unimportant and misleading knowledge may be preserved.",
"As shown in Figure REF (a), the relative relations between the important activations are broken in the distillation procedure.",
"In the perspective of the linguistic structuralism [25], the meaning of a sign depends on its relations with other signs within the system [27].",
"Analogously, the meaning of an activation value depends on its relation with other activation values within the feature map.",
"An activation value will be meaningless without regard to its context activation values.",
"Compared with the first-order distillation, the correlation distillation (second-order distillation) transfers a similarity correlation matrix as shown in Figure REF (b), which explores intra-feature structural relations rather than individual activations and transfers a high-level representation of the activations in the feature maps.",
"As studied in the representational similarity analysis in neuroscience [18], the transformed feature contains more information than the original feature, which is the high-level abstractions of the activation behavior in the features of neural network [4].",
"For object detection, the relations within the activations in feature maps contain more information, such as image-level relations, foreground-background relations and intra-instance relations.",
"Exploring and transferring the relative similarity of the discriminative patterns in the feature space can preserve the stability and plasticity for incremental object detection.",
"In this paper, we propose a novel multi-view correlation distillation based incremental object detection method (MVCD), which mainly focuses on the design of distillation losses in the feature space of the two-stage object detector Faster R-CNN [31].",
"It is a dual network including the old model and the incremental model, which cooperate for transferring the old model trained on old classes to incrementally detect new classes.",
"To trade off between the stability to preserve the learned knowledge from the old data and the plasticity to learn new knowledge from new classes, we design the correlation distillation losses from three views in the feature space of the object detector, which consists of the channel-wise, point-wise and instance-wise views as shown in Figure REF .",
"Here, the three views for the feature maps can be seen as three abstractions of activation behaviors obtained from the stimuli from the different parts of the input.",
"The channel-wise view explores the correlation among the feature map channels in the image-level feature.",
"The point-wise view explores the knowledge among the discriminative regions corresponding to the foreground and the background.",
"The instance-wise view explores the correlation among the intra-instance patches to preserve the discriminability of features for detecting the old classes.",
"The contributions of our work are as follows: We propose a novel incremental object detection method, which is the first attempt to explore the multi-view correlations (second-order distillation) in the feature space of the two-stage object detector.",
"To get a good trade-off between the stability and the plasticity of the incremental model, we design correlation distillation losses from three views for regularizing the learning in feature space, which transfers the learned channel-wise, point-wise and instance-wise correlations to the incremental model.",
"A new metric called Stability-Plasticity-mAP (SPmAP) is proposed to quantize Stability and Plasticity, which is integrated with the original mAP to measure the performance of incremental object detector comprehensively.",
"Extensive experiments are conducted on VOC2007 [6] and COCO [23].",
"The results demonstrate the effectiveness of the proposed method to learn to detect new classes continuously, and it also achieves promising results compared with previous methods."
],
[
"Related Work",
"Incremental learning aims to develop machine learning systems to continuously deal with streams of new data while preserving the learned knowledge from the old data.",
"The main challenge is to mitigate catastrophic forgetting and find a good trade-off between the stability and the plasticity of the incremental model.",
"According to the optimization directions to preserve the learned knowledge, existing works can be divided into two categories [14]: parameter-based and distillation-based.",
"The parameter-based methods aim to preserve important parameters and penalize the changes of these parameters, such as EWC [17] and MAS [1].",
"However, designing a metric to evaluate the importance of all parameters is also a tough task.",
"Therefore, we mainly focus on distillation-based methods in our work.",
"Distillation-based Incremental Learning: Knowledge distillation is a commonly used technique to transfer knowledge from one network to another network.",
"Hinton et al.",
"[13] propose to transfer the knowledge from a large network to a small network using distillation by encouraging the responses of these two networks to be similar.",
"For incremental learning, distillation-based methods utilize the learned knowledge from the old model to guide the learning of the new model by minimizing the distillation losses.",
"LwF [21] utilizes a modified cross-entropy loss to preserve original knowledge with only examples from the new task.",
"iCaRL [30] combines representation learning and knowledge distillation for jointly learning feature representation and classifiers, and a small set of exemplars is selected to perform nearest-mean-of-exemplars classification.",
"Rannen et al.",
"[29] propose an auto-encoder based method to retain the knowledge from old tasks, which prevents the reconstructions of the features from changing and leaves space for the features to adjust.",
"Sun et al.",
"[33] [34] propose to maintain a lifelong dictionary, which is used to transfer knowledge to learn each new metric learning task.",
"Recently, several novel knowledge distillation methods have explored the relationships between samples or instances to transfer the knowledge from teacher model to student model [20] [24] [27].",
"Liu et al.",
"[24] construct the instance relationship matrix.",
"Park et al.",
"[27] propose the distance-wise and angle-wise distillation losses to minimize the difference in relations.",
"Li et al.",
"[20] propose to explore the local correlations to transfer the knowledge.",
"Inspired by these methods, we believe that transferring the correlations in feature space for incremental learning may not only preserve the learned knowledge of the old model but also maintain the scalability to learn new knowledge, which can get a balance between stability and plasticity of the incremental model.",
"Incremental Object Detection: The first incremental object detector [32] is based on Fast R-CNN [8].",
"It uses EdgeBoxes [38] and MCG [3] to precompute proposals, and knowledge distillation is used to regularize the outputs of the final classification and regression layers in order to preserve the performance on old classes.",
"Recently, several end-to-end incremental object detection methods [5] [10] [11] [19] are proposed.",
"Chen et al.",
"[5] propose to use L2 loss to minimize the difference between the feature maps of the old and the incremental models, which is referred to hint loss.",
"Hao et al.",
"[10] introduce a hierarchical large-scale retail object detection dataset called TGFS and presents a class-incremental object detector that utilizes an exemplar set with a fixed size of old data for training.",
"Hao et al.",
"[11] use a frozen duplication of RPN to preserve the knowledge gained from the old classes, and a feature-changing loss (L2 Loss) is proposed to reduce the difference of the feature maps between the old and new classes.",
"Li et al.",
"[19] extract three types of knowledge from the original model, which is based on RetinaNet [22], and it uses smooth L1 loss to minimize the feature difference.",
"A dual distillation training function is proposed in [37] that pre-trains a separate model only for the new classes, such that a student model can learn from two teacher models simultaneously.",
"In addition, several novel works on incremental few-shot object detection are proposed [28] [35].",
"However, the few-shot setting is more challenging than the many-shot setting in incremental object detection, and the problem of incremental object detection on the many-shot setting has not been well resolved.",
"In our work, we mainly focus on general incremental learning for object detection.",
"The typical way of the above-mentioned incremental object detection methods to preserve the learned knowledge is to imitate the important activations of the original model by minimizing the first-order distillation losses.",
"However, it is hard for the incremental model to fully understand the transferred knowledge due to the inherent difference in the categories to be detected.",
"Different from these methods, we explore the important correlations in the feature space of the object detector and only transfer the correlations instead of the values in the feature maps, which can preserve the relative relations within the important learned knowledge and maintain the capability to learn to detect new classes."
],
[
"Overview",
"The proposed multi-view correlation distillation mechanism for incremental object detection is shown in Figure REF , which is a dual network.",
"A frozen copy of the old model trained on the old data provides the learned knowledge of the old classes, such as the activations on feature maps, detection results and the logits from the output layers.",
"The incremental model is adapted to detect both old and new classes on new data with the annotations of new classes as well as the learned knowledge from the old model.",
"Samples of the new data are input into both the old model and the incremental model, then the detection results of the old model are integrated with the ground-truth of new classes to guide the learning of the incremental model.",
"We also use the commonly used distillation loss (L1 loss) on the final output layers (classification layer and regression layer) to penalize the difference between the logits from the old model and the incremental model.",
"In addition to these techniques, in our work, we mainly focus on the distillation in the feature space of the object detector to better preserve the learned knowledge.",
"Different from image classification, the feature space in object detectors can be divided into image-level features and instance-level features, so we elaborately design the distillation losses for both of them to maintain the important knowledge."
],
[
"Multi-view Correlation Distillation",
"The typical way to preserve the learned knowledge from the old model in feature space is to minimize the distance such as L1 loss between the activations of the old model and the incremental model.",
"However, it is difficult for the incremental model to fully understand the transferred knowledge in feature space, which may result in the preservation of unimportant information instead of the useful knowledge for minimizing the overall loss.",
"Meanwhile, this constraint may also restrict the plasticity of the incremental model for learning new classes.",
"Incremental object detection aims to not only preserve the learned knowledge but also maintain the scalability for learning new classes.",
"Therefore, we design a novel correlation distillation mechanism, which explores and transfers the important correlations from channel-wise, point-wise and instance-wise views in the feature space of the old object detector.",
"The channel-wise view explores the correlation between the important feature maps in the image-level feature.",
"The point-wise view explores the correlation between the discriminative background and foreground regions.",
"The instance-wise view explores the correlation between the intra-instance patches, which aims to preserve the discriminability of the features for precisely detecting the old classes.",
"The total loss function is defined as: $\\begin{aligned}\\mathcal {L}=\\mathcal {L}_{frcnn}+\\mathcal {D}_{out}+\\lambda (\\mathcal {D}_{cc}+\\mathcal {D}_{pc}+\\mathcal {D}_{ic})\\end{aligned}$ where $\\mathcal {L}_{frcnn}$ is the standard loss function in Faster R-CNN, and $\\mathcal {D}_{out}$ is the commonly used distillation loss on the final classification and regression layers, and here we use L1 loss.",
"$\\mathcal {D}_{cc}$ , $\\mathcal {D}_{pc}$ and $\\mathcal {D}_{ic}$ are the proposed channel-wise, point-wise and instance-wise correlation distillation losses.",
"We set $\\lambda =1$ in our experiments."
],
[
"Channel-wise Correlation Distillation",
"The convolution kernels are responsible for extracting different patterns, so the channel-wise importances are different for each sample, which is the indispensable knowledge to transfer.",
"However, the correlations between the feature distribution along channels are seldom considered in previous first-order-distillation-based methods.",
"Intuitively, to preserve the plasticity of the incremental object detector, only the important channel-wise activations learned on the old data should be transferred to the incremental model and the rest unimportant channels can be left for learning new classes.",
"Due to the disadvantage of first-order distillation, we propose a channel-wise correlation distillation loss.",
"It constrains the specific inter-channel relations for different samples, and the consistent correlations of feature distribution along channels between the old and incremental model are preserved.",
"It is achieved by distilling the correlations within the important channels of each image rather than restricting the overall activation values to be similar.",
"Figure: SE block in RPN.The squeeze-and-excitation module (SE module) [15], as a widely used channel attention module, can generate channel-wise weights for each image.",
"SE module consists of squeeze and excitation operations.",
"The original feature maps are aggregated across spatial dimensions, and a channel descriptor is obtained through a squeeze operation.",
"Then, the sample-specific activations are learned for each channel (channel-wise attention vectors $\\textbf {v}$ ) through an excitation operation.",
"The original feature maps are then reweighted to generate the output of the SE block.",
"To measure the importance of channels in image-level features, an SE module is added after the convolution layer in RPN which has higher-level features for discriminating foreground and background.",
"The structure is shown in Figure REF .",
"After the channel-wise attention vector $\\textbf {v} = \\left\\lbrace v_1,v_2,...,v_C\\right\\rbrace $ from the old model is obtained for each image, we normalize the vector to $[0,1]$ .",
"The important channels $F^{cc}\\in \\mathbb {R}^{N^{cc}\\times W \\times H}$ are classified by a threshold (0.5), where $N^{cc}$ is the number of important channels.",
"The loss can be written as: $\\begin{aligned}S^{cc}(i,j)=\\psi (F^{cc}(i),F^{cc}(j))&,\\quad S^{cc^{\\prime }}(i,j)=\\psi (F^{cc^{\\prime }}(i),F^{cc^{\\prime }}(j))\\\\\\mathcal {D}_{cc}=\\frac{1}{N^{cc}\\times N^{cc}}&\\sum _{i=1}^{N^{cc}}\\sum _{j=1}^{N^{cc}}|S^{cc}(i,j)-S^{cc^{\\prime }}(i,j)|\\end{aligned}$ where $S^{cc}$ is the channel-wise correlation matrix and $F$ and $F^{^{\\prime }}$ represent the features of the incremental model and the old model respectively.",
"$i$ and $j$ are the indexes of the channels.",
"$\\psi (\\cdot ,\\cdot )$ is cosine similarity between two vectors.",
"The channel-wise correlation matrices of the old model and the incremental model both use the indexes of important channels obtained from the old model.",
"The channel-wise correlations represent the relative distribution of specific patterns learned in different channels, which is the abstractions of channel-level activation behaviors.",
"The channel-wise correlation distillation only transfers correlations between the important channels of old classes and removes redundant channel-wise information, which leaves room for learning new classes."
],
[
"Point-wise Correlation Distillation",
"RPN is used to discriminate the object-like region proposals and background region proposals.",
"The discriminative points in the activation-based spatial attention map of the feature in RPN correspond to the obvious regions of the background or foreground.",
"Extracting the feature vectors of these points and only transferring the correlation between these point-wise feature vectors can preserve the knowledge of obvious foreground and background learned on the old data.",
"The rest points are left to be optimized on new data.",
"The obvious regions are the points with high or low responses on the activation-based spatial attention map, which can be obtained by thresholding the attention map.",
"We use $F_{att}=\\sum _{i=1}^{C}{|F_i|}$ to get the attention map, where $C$ is the number of channels.",
"Then, the activation-based spatial attention map is further normalized for selecting the point-wise feature vectors with high or low responses.",
"Here, we use two thresholds $\\theta _{high}$ and $\\theta _{low}$ to select these point-wise feature vectors of discriminative regions.",
"The correlation matrix is constructed between them, which also uses the cosine similarity to describe the correlation.",
"In the dual network, the activation-based spatial attention map is obtained from the image-level feature of the old model ($F^{^{\\prime }}$ ), then the attention map is used to select points with high and low responses.",
"After the indexes of these points are obtained $P^{high}=\\left\\lbrace (x_1,y_1),(x_2,y_2)...,(x_{N^h},y_{N^h})\\right\\rbrace $ and $P^{low}=\\left\\lbrace (x_1,y_1),(x_2,y_2)...,(x_{N^l},y_{N^l})\\right\\rbrace $ , where $N^h$ and $N^l$ are the numbers of the points with high and low responses respectively, we extract the point-wise feature vectors from the features of the old model and the incremental model ($F^{^{\\prime }}$ and $F$ ) respectively.",
"The point-wise correlation matrices are also calculated for these two models respectively, and the distance between these two matrices is minimized.",
"The loss is written as Equation REF .",
"$\\begin{aligned}S^{pc}(i,j)\\!=\\!\\psi (F^{high}(i),F^{low}(j)),&\\quad S^{pc^{\\prime }}(i,j)\\!=\\!\\psi (F^{high^{\\prime }}(i),F^{low^{\\prime }}(j))\\\\\\mathcal {D}_{pc}=&||S^{pc}-S^{pc^{\\prime }}||_F^2\\end{aligned}$ where $F^{high}\\in \\mathbb {R}^{N^h \\times C}$ and $F^{low}\\in \\mathbb {R}^{N^l \\times C}$ are the extracted point-wise feature vectors corresponding to the points with high and low responses from the incremental model respectively.",
"Similarly, $F^{high^{\\prime }}\\in \\mathbb {R}^{N^h \\times C}$ and $F^{low^{\\prime }}\\in \\mathbb {R}^{N^l \\times C}$ represent the corresponding feature vectors from the old model.",
"The point-wise correlations represent the abstractions of the activation behaviors about the relative responses between foreground and background.",
"The point-wise correlation distillation can preserve the consistent correlations between the feature distributions of obvious foreground and background, and the indistinct regions can be left to learn new classes."
],
[
"Instance-wise Correlation Distillation",
"Inspired from [20], the local features and their correlation in each instance also contain many details and discriminative patterns.",
"The old model can generate more discriminative local features with sufficient old data, while the incremental model is hard to achieve that due to the lack of old data.",
"The old model trained on old classes can make right predictions for different categories of objects with similar appearances based on the discriminative local regions of each instance.",
"Therefore, learning the local knowledge of each instance from the old model is an important way to maintain the stability to detect the old classes.",
"The intra-instance correlation is not considered when simply imitating the global activations of instance-level features, which will degrade the discriminability for detecting old classes due to the loss of distinctive local patterns.",
"To transfer the correlation of the local regions for each instance from the old model to the incremental model, we compute the correlation matrix of the local regions for each instance-level feature, which is the pooled feature after the RoI-Pooling and convolution layers in the detection head.",
"The pooled feature of each instance $F^l \\in \\mathbb {R}^{pc \\times ph \\times pw}$ ($pc=2048$ , $ph=4$ and $pw=4$ ) are divided into $k \\times k$ ($k=2$ ) patches, and each patch has a shape of $pc \\times \\frac{ph}{k} \\times \\frac{pw}{k}$ ($2048\\times 2\\times 2$ ).",
"The instance-wise correlation distillation loss is defined as: $\\begin{aligned}S^{ic}(i,j)=\\psi (F^l(i),F^l(j))&,\\quad S^{ic^{\\prime }}(i,j)=\\psi (F^{l^{\\prime }}(i),F^{l^{\\prime }}(j))\\\\\\mathcal {D}_{ic}=\\frac{1}{k\\times k}\\sum _{i=1}^{k}&\\sum _{j=1}^{k}|S^{ic}(i,j)-S^{ic^{\\prime }}(i,j)|\\end{aligned}$ where $F^l(\\cdot )$ and $F^{l^{\\prime }}(\\cdot )$ are the vectorized features of the local patches from the incremental model and the old model respectively.",
"The instance-wise correlations represent the abstractions of activation behaviors within an instance, which explores the correlations between the local parts of an instance.",
"The instance-wise correlation distillation can transfer the relative relationship between the response values of class-specific local features learned from the old model.",
"For incremental object detection, due to the different numbers of old and new classes and the different difficulties of learning each class, mAP is not very suitable for measuring the performance of the incremental model on handling the stability-plasticity dilemma.",
"Therefore, to quantize Stability and Plasticity, we propose a new metric called Stability-Plasticity-mAP (SPmAP), as written in Equation REF .",
"Because incremental object detection aims to reach the performance of the model trained on all data with only the data of new classes, we use the model train on all classes as the up-bound model to measure stability and plasticity.",
"Stability is the average difference of precisions on old classes, and Plasticity is the average difference of precisions on new classes.",
"We also integrate the overall mAP difference ($mAP_{dif}$ ) representing the overall performance of all classes into the metric to measure the performance comprehensively.",
"$\\begin{aligned}SPmAP=((Stability&+Plasticity)/2+mAP_{dif})/2 \\\\Stability=\\frac{1}{N^{o}}&\\sum \\nolimits _{i=1}^{N^{o}}(UP(i)-INC(i)) \\\\Plasticity=\\frac{1}{N^{n}}&\\sum \\nolimits _{i=N^{o}+1}^{N}(UP(i)-INC(i)) \\\\mAP_{dif}=\\frac{1}{N}&\\sum \\nolimits _{i=1}^{N}\\!",
"(UP(i)\\!-\\!INC(i)) \\end{aligned}$ where $N$ is the number of all classes.",
"$N^{o}$ and $N^{n}$ represent the numbers of old and new classes respectively.",
"$UP$ and $INC$ are the average precisions of the up-bound model and the incremental model."
],
[
"Experiment Setup",
"Datasets.",
"The proposed method is evaluated on two benchmark datasets Pascal VOC 2007 and Microsoft COCO.",
"VOC2007 has 20 object classes, and we use the trainval subset for training and the test subset for evaluation.",
"COCO has 80K images in the training set and 40K images in the validation set for 80 object classes, and the minival (the first 5K images from the validation set) split is used for evaluation.",
"There are two schemes to add new classes for evaluation: addition at once and sequential addition.",
"Evaluation Metrics.",
"The compared methods are fine-tuning and some recent related works [32] [5] [10] [11] [19].",
"We reproduce the distillation methods and evaluate their performance under the same settings as our proposed method.",
"We also design a baseline (Plain L1) that directly minimizes the L1 loss between the activations in the features of the old model and the incremental model.",
"The basic object detector is Faster R-CNN for all methods.",
"We use both mean average precision (mAP) at 0.5 IoU threshold and the proposed “SPmAP” to measure the performance.",
"Implementation Details.",
"The old model is trained for 20 epochs, and the initial learning rate is set to 0.001 ($lr=0.001$ ), and decays every 5 epochs with $gamma=0.1$ .",
"The momentum is set to 0.9.",
"The incremental model is trained for 10 epochs with $lr=0.0001$ and decays to 0.00001 after 5 epochs.",
"The confidence and IoU threshold for NMS are set to 0.5 and 0.3 respectively.",
"The thresholds in Section REF are set to $\\theta _{high}=0.8$ and $\\theta _{low}=0.1$ , and $k$ in Section REF is set to 2.",
"ResNet-50 [12] is used as the backbone.",
"We conduct all experiments on a single NVIDIA GeForce RTX 2080 Ti."
],
[
"Addition of Classes at Once",
"In the first experiment, we evaluate the performance on adding new classes at once.",
"We take 19, 15 and 10 classes from VOC2007 sorted in alphabetical order as the old classes, and the remaining 1, 5, 10 classes are the corresponding new classes as described in [32].",
"For COCO, we take the first 40 classes as the old classes and the remaining 40 classes as the new classes.",
"In these settings, if the image contains the categories to be detected, it will be selected for training or testing, so there is an overlap between the old data and the new data.",
"However, the annotations of old classes in the new data are not available.",
"Table REF lists the per-category average precision on VOC2007 test subset.",
"Old($\\cdot $ ) represents the model trained on the old data, and Up-bound($\\cdot $ ) represents the model trained on all data of both old and new classes.",
"On the first setting, the mAP of fine-tuning gets only 26.2%, which has caused severely catastrophic forgetting.",
"Different from the original fine-tuning, which randomly initializes the classification layer for a new task, in order to preserve the learned knowledge, we initialize the parameters in the classification layer and the regression layer of the incremental model with those of the old model learned from the old classes.",
"However, the performance of fine-tuning still degrades a lot.",
"As can be seen, when only add one new class (“tv monitor\"), the mAP and SPmAP of MVCD can reach 69.7% and 3.4% respectively, outperforming other L1/L2-distillation-based incremental object detection methods by a large margin.",
"The mAP of MVCD exceeds the suboptimal Plain L1 about 0.8%.",
"It represents our method can better balance stability and plasticity.",
"For the second setting, we take 15 classes as the old classes, and the remaining 5 classes are added at once.",
"MVCD also performs well compared with other methods.",
"The mAP increases by about 2.0% compared with Plain L1.",
"With the increasing number of new classes, the mAP of fine-tuning is improved, however, the phenomenon of catastrophic forgetting is not mitigated.",
"On the third setting, when 10 classes are added at once, the mAP of MVCD gets 66.1%, outperforming the suboptimal Plain L1 about 0.5%.",
"Similarly, MVCD achieves the best SPmAP compared with other methods.",
"We also evaluate the performance on adding more classes as shown in Table REF .",
"We take 40 classes from COCO training dataset as the old classes and the remaining 40 classes as the new classes.",
"Both mAP and SPmAP of MVCD outperform Plain L1 and exceeds fine-tuning by a large margin.",
"The above results demonstrate that MVCD can effectively mitigate catastrophic forgetting on the setting of addition at once.",
"The comparisons with other methods with the metric “SPmAP” also verify the superiority of MVCD on maintaining the stability and plasticity of the incremental model.",
"As can be seen, the designed “Plain L1” achieves comparable performance with other first-order-distillation-based methods.",
"Therefore, in the following experiments, we use “Plain L1” as the baseline for comparison."
],
[
"Sequential Addition of Multiple Classes",
"In this experiment, we evaluate the performance of our method by adding classes sequentially for incremental learning.",
"For the first setting, we also take 15 and 10 classes from VOC2007 sorted in alphabetical order as old classes, and the remaining 5 and 10 classes are as new classes.",
"Table REF lists the mAP(%) when adding 5 and 10 classes sequentially.",
"As can be seen, MVCD in the setting of sequential addition of 5 classes outperforms Plain L1 in all incremental learning steps, and it can reach 51.89% after the 5th incremental learning step.",
"The average improvements over all steps is 1.6%, and the max difference can reach 3.25% in the 2th step.",
"We also evaluate the performance on adding 10 new classes sequentially with ten-step and five-step incremental learning respectively.",
"In the ten-step learning, we add one new class at a time step, and in the five-step learning, we add two new classes at a time step.",
"As shown in the ten-step setting result, the proposed MVCD has consistent improvements in all learning steps.",
"After the 6th learning step, MVCD still exceeds Plain L1 6.78% (40.05% vs. 33.27%).",
"Due to the small number of samples in some categories, the performance is decreased slightly.",
"However, MVCD is still better than Plain L1, which demonstrates the effectiveness of multi-view correlation distillation.",
"In the five-step setting, the mAP of MVCD can still reach 48.23% after the 5th incremental learning step, and it outperforms Plain L1 by a large margin in all learning steps.",
"These experiments demonstrate that the proposed MVCD can mitigate catastrophic forgetting better than the first-order distillation even after many incremental learning steps.",
"We also split the training set of VOC2007 and COCO into four groups: A, B, C and D as described in [11].",
"For fair comparisons, we also use ResNet-50 [12] in this experiment.",
"For each group, images that only contain the objects of classes in this group are selected, which means that there are no overlaps in these four groups.",
"The results are shown in Table REF , the performance of MVCD is better than Plain L1 in all incremental learning steps.",
"On VOC2007, MVCD improves about 6.01% compared with Plain L1 after the last learning step.",
"On COCO, MVCD is consistently better than Plain L1 in all learning steps."
],
[
"Ablation Study",
"As listed in Table REF , the proposed multi-view correlation distillation losses $\\mathcal {D}_{cc}$ , $\\mathcal {D}_{pc}$ and $\\mathcal {D}_{ic}$ on three settings are evaluated separately.",
"The baseline only uses the distillation on the final classification and regression layers as shown in the first row in Table REF .",
"“$+$ \" represents the increased mAP(%) compared with the baseline.",
"Firstly, these three distillation losses are evaluated individually.",
"As can be seen, the accuracy increases by about 0.85% on average by using $\\mathcal {D}_{cc}$ .",
"The average increments of 1.85% and 1.99% are obtained when $\\mathcal {D}_{pc}$ and $\\mathcal {D}_{ic}$ are individually utilized.",
"It verifies that these three correlation distillation losses are all useful for incremental object detection.",
"The performance on SPmAP also show the effectiveness of these losses on preserving stability and plasticity.",
"Then, we test different combinations of arbitrarily two losses, and the results on mAP show that the performances of these combinations are a little decreased compared with the combination of three correlation distillation losses.",
"The alternative choices of hyper-parameters $\\theta $ , $k$ and distillation ways on channel-wise, point-wise and instance-wise features are also tested as shown in Table REF .",
"For MVCD in the last row, we use the settings as described in implementation details.",
"For point-wise correlation distillation, we replace the high and low thresholds with a single threshold $\\theta =0.5$ to divide the point-wise feature vectors into the vectors with high responses and low responses.",
"Compared with our final setting $\\theta _{high}=0.8$ and $\\theta _{low}=0.1$ , the performance decreases a lot, which verifies that only preserving the correlation between the most discriminative point-wise features can maintain the stability and plasticity of the incremental model better.",
"For the instance-wise correlation distillation, the instance-level feature is divided into $4\\times 4$ ($k=4$ ) patches, and the result shows $k=2$ is better than $k=4$ .",
"We also replace the channel-wise, point-wise and instance-wise correlation distillation losses with L1 loss to minimize the distance between the selected features.",
"The performance on SPmAP is worse than preserving the correlations, which demonstrates correlation distillation is more appropriate to get a tradeoff between stability and plasticity for incremental object detection.",
"We also compare the training time of the proposed incremental learning method with training the detector from scratch using the similar GPU memory as listed in Table REF .",
"When adding a few new classes, the proposed incremental object detection method has absolute superiority in training time with just a minor accuracy loss."
],
[
"Conclusion",
"In this paper, we propose a novel multi-view correlation distillation based incremental object detection method, which transfers the correlations from the channel-wise, point-wise and instance-wise views in the feature space of the two-stage object detector.",
"The channel-wise and point-wise correlations are designed for image-level features, and the instance-wise correlation is designed for instance-level features, which can get a good trade-off between the stability and the plasticity of the incremental model.",
"Experimental results on VOC2007 and COCO with the new metric “SPmAP” demonstrate the effectiveness of the proposed method on incrementally learning to detect objects of new classes without severely forgetting originally learned knowledge."
]
] | 2107.01787 |
[
[
"Phase-field study of surface diffusion enhanced break-ups of nanowire\n junctions"
],
[
"Abstract Using a phase-field model which incorporates enhanced diffusion at the nanowire surfaces, we study the effect of different parameters on the stability of intersecting nanowires.",
"Our study shows that at the intersection of nanowires, sintering (curvature driven material flow) leads to the formation of junctions.",
"These junctions act the initiators of nanowire break-up.",
"The subsequent break-ups take place due to Rayleigh instability at the arms away from these junctions.",
"Finally, at long time scales, the fragments coarsen due to the differences in sizes.",
"The radii of the nanowires that form the junction, the difference in size of the intersecting nanowires and the angle of intersection play a dominant role in determining the kinetics of break-up while the density of intersections has little or no effect on the kinetics.",
"We rationalise our results using maps of (i) mean curvatures (and, hence, chemical potentials), and, (ii) Interfacial Shape Distributions (ISDs) (which are based on probability densities associated with different combinations of the two principal curvatures).",
"Finally, we use the moment of inertia tensor to characterise the (non-spherical) shapes and morphologies of (central) nanowire fragments at the junctions."
],
[
"Introduction",
"Metallic nanowires are used in a wide range of applications: for example, their ductility and bendability allows for their use in solar cells, flexible and transparent electronic devices, light emitting diodes, and so on – see, for example, , , , .",
"Regular ordered array of metallic nanodots have been used in biosensing applications .",
"In particular, network of Ag nanowires are considered as the next generation of transparent conducting electrodes due to their enhanced electrical and optical properties , .",
"Given such widespread use, their stability at elevated temperatures under standard operating conditions is an important area of study.",
"Thus, an understanding of the driving forces for morphological changes and control of the parameters which affect those driving forces are of significant academic and technological importance.",
"The effect of nanowire size, network density and temperature on the stability of metallic nanowires have been investigated experimentally , , .",
"The structural stability of the metallic nanowires at the nanoscale is governed by numerous factors — the most dominant factors being Rayleigh instability , , and anisotropy in interfacial energy , , , , .",
"Thermally accelerated surface diffusion plays a central role in the resulting change in morphology of the metallic nanowires due to Rayleigh instability, and their subsequent break-up into nanodots , , , , , , .",
"Rayleigh instability mediated nanowire fragmentation has also been observed in non-metallic nanowires , .",
"Quantum effects at the nanoscale also play a role in morphological evolution in conjunction with Rayleigh instability and have been investigated both experimentally and numerically , .",
"It has been observed experimentally that nanowire junctions have a drastic effect on the break-up kinetics of Au and Ag nanowires when annealed at elevated temperatures much below their melting point .",
"Presence of such junctions between the nanowires lead to initial break-up preferentially at the junction, followed by break-up at remainder of the nanowires.",
"Such fragmentation behaviour of nanowires can be exploited to produce ordered array of nanodots for different technological applications , .",
"In the past, various modelling approaches (mostly atomistic) have been adopted to simulate the nanoparticle morphologies and various effects at the nanoscale — a review of such approaches can be found in and references therein.",
"Linear stability analysis of Rayleigh instability is also well understood; for example, a linear stability analysis of the effect of general surface energy anisotropy on Rayleigh instability has been carried out in (extending the work of Cahn ).",
"Phase-field modelling is an efficient numerical technique for simulating microstructural evolution at the mesoscopic length scales and diffusive time scales , .",
"There exist several phase-field models to study variable mobility; for example, variable mobilities have been introduced in the past to study phenomena such as domain growth in binary mixtures , late-stage coarsening in phase separating systems , , and for determining the coarsening kinetics of bulk-diffusion-controlled and interface-diffusion-controlled growth in systems with interconnected phases .",
"The use of variable mobility phase-field models has also been made for effectively incorporating the effect surface diffusion, assuming isotropic interfacial energy , , .",
"Such variable mobility phase-field models have been used in simulating the Rayleigh instability in the solid-state , destabilisation of nanoporus membranes by grain boundary grooving , thermal stability of nanoporus aggregates , and, instability in multi-layer nanocrystalline thin-films due to Rayleigh instability driven by grain boundary .",
"Phase-field models have been developed to incorporate anisotropy in interfacial energy using either trigonometric functions for the interfacial energy coefficient , , or using higher order tensor terms which has benefits over the former approach .",
"Phase-field model for stability of nanowire fragmentation with regularized trigonometric function of interfacial energy along with finite amplitude axisymmetric perturbations has been developed previously .",
"Higher order tensor terms in the free energy functional of phase-field models have been used to study the faceting of precipitates due to interfacial energy anisotropy .",
"Our aim, in this paper, is to implement a continuum model (based on the Cahn-Hilliard equation ) for long time evolution of the morphologies of the nanowires with enhanced surface diffusion.",
"Further, we want to study the systems in which the surface energy is anisotropic.",
"Specifically, we use the extended Cahn-Hilliard equation which consists of a fourth order tensor term in order to incorporate cubic surface energy anisotropy, and, following the approach adopted in some of the previous phase-field studies mentioned above, we define an order-parameter dependent mobility function.",
"This helps us incorporate enhanced surface diffusion in our model.",
"Using this model, we carry out a systematic study of various factors such as wire diameters, the angles of intersection, density of intersections in the simulation cell and interfacial energy anisotropy on fragmentation of intersecting nanowires.",
"At this point, we want to note that there exist phase-field studies pertaining to the phenomena of solid-state dewetting in nanowires , , .",
"Different phenomena like the formation of nanoparticles through solid-state dewetting of a thin-film on a substrate , stress effects on solid-state dewetting of thin-films , solid-state dewetting of Au aggregates on titanium oxide nanorods , and effect of surface energy anisotropy on Rayleigh-like solid-state dewetting have been studied in the past.",
"However, unlike these models which include the substrate on which dewetting takes place, in our model, the wires are free-standing.",
"The rest of this paper is organized as follows: we describe (albeit briefly) the formulation and numerical implementation of the phase-field model in section , which also contains the simulation details.",
"Results and discussion follow in section where we discuss the effect of various parameters on the kinetics of junction break-up systematically, followed by a summary of our salient conclusions in section ."
],
[
"Phase-field model",
"As indicated in the introduction, our phase-field model is a combination of extended Cahn-Hilliard model for cubic anisotropy in interfacial energy coupled with enhanced surface diffusion implemented using a variable mobility .",
"Since these models are well known in the literature, we briefly describe the models and other details in this section, for the sake of completion."
],
[
"Formulation",
"We consider a conserved, non-dimensionalised order parameter, denoted by $c(x,t)$ to describe our system.",
"This order parameter takes a value of zero in vacuum and unity in the (nanowire) material albeit across a flat interface; in the circular cross-section nanowire-vacuum geometry, the order parameter value shifts from unity and zero to account for Gibbs-Thomson effect.",
"The free energy functional of the system is given by the expression: $F = N_{V} \\int _{\\Omega } \\left\\lbrace f_0(c) + \\frac{\\kappa _{c}}{2} (|\\nabla c|)^2 + \\frac{\\gamma _{\\langle hkl \\rangle }}{2} (\\nabla ^2 c)^2 \\right\\rbrace d\\Omega ,$ where, $N_V$ denotes the number of atoms per unit volume (assumed to be constant), $f_0(c)$ is the bulk free energy density per atom, $\\kappa _{c}$ is the gradient energy coefficient (assumed to be a scalar and hence gives rise to an isotropic interfacial free energy), and, the third term accounts for cubic interfacial free energy anisotropy as explained below.",
"The cubic anisotropy in the interfacial energy of the nanowires is incorporated into the model using the extended Cahn-Hilliard formulation , which is the third term in the equation.",
"In Eq.",
"REF , $\\gamma _{\\langle hkl \\rangle }$ is the coefficient of a fourth order term which is defined for a particular crystallographic orientation as: $\\gamma _{\\langle hkl \\rangle } = \\gamma _{I} + \\gamma _{A}\\left(h^{4} + k^{4} + l^{4}\\right),$ where, $\\gamma _{A}$ and $\\gamma _I$ are the anisotropic and isotropic contributions from the fourth rank term, and $h,\\;k,\\;l$ denote the Miller Indices of the normal to an interface.",
"The values of $\\gamma _{A}$ and $\\gamma _{I}$ can be calculated from the scaling curve in , if the ratio of interfacial energies along two directions is known.",
"In Eq.",
"REF , the function $f_0(c)$ represents the bulk free energy density per atom, which is given by the polynomial, $f_0(c)=A_{c}\\left[c^2(1-c)^2\\right].$ This polynomial produces a double well potential with the energy minima at c=0 (vacuum) and c=1 (nanowire material).",
"$A_c$ is a coefficient that determines the height of the potential energy barrier.",
"Together with the gradient energy coefficient $\\kappa _c$ , $A_c$ determines the interfacial energy and width in the system (in the absence of the $\\gamma _{<hkl>}$ term; when present, $\\gamma _{<hkl>}$ along with $A_c$ and $\\kappa _c$ determine the interfacial properties, namely, energy and width).",
"The chemical potential is derived from the variational derivative of the free energy functional Eq.REF : $\\mu = \\frac{1}{N_V}\\frac{\\delta F}{\\delta c} = \\left[ \\frac{\\partial f_0(c)}{\\partial c} - \\kappa _{c} \\nabla ^2 c + \\gamma _{<hkl>}\\nabla ^4 c \\right].$ The evolution of the system is described by the modified Cahn-Hilliard equation, which is given by the expression: $\\displaystyle {\\frac{\\partial c}{\\partial t} = \\nabla \\cdot \\left[ M(c) \\cdot \\nabla \\left(\\frac{df}{dc} - \\kappa _{c}\\nabla ^2c + \\gamma _{<hkl>}\\nabla ^4 c \\right)\\right]},$ where, $M(c)$ denotes the mobility as a function of the order parameter, and is used to capture the effect of surface diffusion.",
"Various polynomials have been suggested in the literature for use as variable mobility functions in the Cahn-Hilliard equation.",
"Most commonly used mobility functions in previous studies have been quadratic and quartic function , , , , of order parameter, to incorporate the dominating role of surface diffusion in the models.",
"We have used a mobility function in our model which serves the same purpose, but with an added advantage of reducing the stiffness of the equations in conjunction with the semi-implicit Fourier spectral numerical method.",
"The function is defined as: $M(c) = \\left[c(1-c)\\right]^\\frac{1}{2}.$ The above function gives rise to the highest mobility of atoms along the interface ($c=0.5$ ) and defines zero mobility of atoms in the bulk ($c=1$ ) and the mobility is restricted to the solid (that is, mobility in the vacuum $c=0 $ is also zero).",
"Thus, in these simulations, surface diffusion plays a dominant role in material transport.",
"Substituting (REF ) into (REF ), the evolution equation can be written as: $\\displaystyle {\\frac{\\partial c}{\\partial t} = \\nabla \\cdot \\left[ \\left\\lbrace c(1-c)\\right\\rbrace ^\\frac{1}{2} \\cdot \\nabla \\left(\\frac{df}{dc} - \\kappa _{c}\\nabla ^2c + \\gamma _{<hkl>}\\nabla ^4 c \\right)\\right]}.$"
],
[
"Numerical implementation",
"The evolution equation (Eq.",
"REF ) is solved numerically in order to track the morphological evolution of both finite and infinite cylindrical nanowires (in both 2-D and 3-D).",
"We use the semi-implicit Fourier spectral technique for solving the evolution equation.",
"This method is known to be efficient for solving non-linear partial differential equations and also eliminates the severe time-step constraint , .",
"We follow the same method of discretisation used in .",
"We transform equation (REF ) to Fourier space and perform first order forward finite-difference discretisation in time.",
"$\\frac{{\\tilde{c}({\\bf k })}^{t+\\Delta t} - {\\tilde{c}({\\bf k })}^t}{\\Delta t} = i {\\bf k } \\lbrace \\lbrace c(1-c)\\rbrace ^\\frac{1}{2} \\cdot [i {\\bf k }^{\\prime } \\left(\\tilde{g}(c^t)+ \\tilde{c}({\\bf k^{\\prime }})^t\\chi \\right) ]_{r}\\rbrace _{k},$ where, $i$ is the pure imaginary number, $\\chi = \\left(\\kappa _{c} k^{\\prime 2} + \\gamma _{I}k^{\\prime 4} + \\gamma _{A}\\left(k_{x}^{\\prime 4} + k_{y}^{\\prime 4} + k_{z}^{\\prime 4}\\right)\\right),$ $\\tilde{c}({\\bf k }, t)$ is the non-dimensionalised order parameter field in the Fourier space and, ${\\bf k}$ and ${\\bf k^{\\prime }}$ are the Fourier space vectors.",
"${\\bf |k|} = k = \\sqrt{k_{x}^{2} + k_{y}^{2} + k_{z}^{2}}$ , where $k_{x}$ , $k_{y}$ and $k_{z}$ are the three components of the Fourier space vector.",
"The operation $[\\cdot ]_{r}$ represents the inverse spatial Fourier transform of the quantity in square brackets to the real space, while $\\lbrace \\cdot \\rbrace _{k}$ represents the forward spatial Fourier transform of the quantity in curly brackets.",
"The function $g(c)$ represents the derivative of the bulk free energy density function with respect to $c$ and is defined as: $g(c) = \\frac{\\partial f_0(c)}{\\partial c} = 2A_{c}\\left\\lbrace c(1-c)(1-2c)\\right\\rbrace .$ The severe time step constraint associated with the explicit solution of the equation is circumvented by introducing a suitable stabilizing constant, which separates the mobility function into two parts: $\\xi $ and $\\lbrace c(1-c)\\rbrace ^\\frac{1}{2} - \\xi $ , after Zhu et al.",
".",
"Therefore, the modified evolution equation becomes: $\\frac{{\\tilde{c}({\\bf k })}^{t+\\Delta t} - {\\tilde{c}({\\bf k })}^t}{\\Delta t} = i {\\bf k } \\left\\lbrace \\left[\\xi + \\left\\lbrace \\lbrace c(1-c)\\rbrace ^\\frac{1}{2} - \\xi \\right\\rbrace \\right] \\cdot [i {\\bf k }^{\\prime }(\\tilde{g}(c^t) + \\tilde{c}({\\bf k^{\\prime } })^t \\chi ) ]_r\\right\\rbrace _{k}.$ After some algebraic manipulation, the evolution equation can be written as: $\\beta \\tilde{c}({\\bf k})^{t+\\Delta t} = \\beta \\tilde{c}({\\bf k })^{t} + i {\\bf k } \\Delta t\\lbrace {c(1-c)}^\\frac{1}{2}\\cdot [i {\\bf k }^{\\prime } \\left(\\tilde{g}(c^t) + \\tilde{c}({\\bf k^{\\prime }}\\right)^t \\chi ]_{r}\\rbrace _{k}$ where, $\\beta = \\left(1+\\xi \\Delta t\\chi \\right)$ .",
"Thus, knowing the $c$ at a given time $t$ , the $c$ at time $t+\\Delta t$ can be obtained in the Fourier space using Eq.",
"REF ."
],
[
"Simulation details",
"We have carried out simulations of finite and infinite nanowires in both 2- and 3-D.",
"In this paper, we present 3-D results for wires with isotropic interfacial energy and 2-D results for wires with (cubic) anisotropic interfacial energy.",
"Note that the employment of Fourier spectral technique for the numerical solution implies imposition of periodic boundary conditions.",
"We introduce a small noise in the scaled order parameter, which adequately simulates the thermal noise in the system.",
"We used the software package Fastest Fourier Transform in the West (FFTW3) for computing the discrete Fourier transforms.",
"We use the same non-dimensionalisation as described in Abinandanan and Haider .",
"The non-dimensional simulation parameters used in our simulations are tabulated in Table1.",
"Table: Non-dimensional simulation parametersThe angle of intersection of the nanowires, the radii of the nanowires and the size difference in the radii between the nanowires that form the intersections, the density of intersections and the anisotropy in the surface energy of the nanowires are the parameters that are of interest to us.",
"In this section, we present our results for all these scenarios."
],
[
"Effect of relative orientation",
"We consider a system with isotropic interfacial energy (in 3D) with a single intersection of the nanowires (of circular cross-section) in the simulation cell.",
"We assume periodic boundary conditions.",
"As seen from the schematic top view of the relative orientations in Fig1, in the case of an intersection angle of $90^{\\circ }$ , both the wires are infinite.",
"However, for all the other three angles, the nanowire parallel to the $x$ -axis of the simulation cell is infinite while the other one is finite.",
"Having said that, we have chosen big enough system sizes so that the break-up at the edges of the finite nanowire does not affect the break-up at the intersection.",
"In Fig2, we show the time evolution of the nanowires of equal initial radii ($\\mathrm {R_1} = \\mathrm {R_2} = 12$ (non-dimensional) length units) with an angle of $90^{\\circ }$ at the intersection to begin with.",
"Note that in 3D, one nanowire (in this case, the one along the x-axis) is at the bottom and the other on top (in this case, the one along the y-axis); in this and the subsequent cases, $\\mathrm {R_1}$ represents the radius of the nanowire at the bottom and $\\mathrm {R_2}$ represents the radius of the nanowire on top.",
"Figure: Schematic (top view) of the relative orientation between the wires used in the current studyFigure: t = 2900As can be seen from the figure, due to the high curvatures at the point of contact at the intersection, initially, a junction forms at the intersection (fig2a, corresponding to a (non-dimensional) time of 1000 units).",
"The material accumulation at the junction leads to the break-up of the nanowire (fig2b, corresponding to a (non-dimensional) time of 1750 units).",
"This, in turn, leads to subsequent break-up of the nanowire as seen in Fig2(c)-(e), corresponding to a (non-dimensional) times of 2500, 2750 and 2800 units.",
"As can be seen from fig2f, corresponding to a (non-dimensional) time of 2900 units, the broken pieces of the nanowire coarsen due, primarily to, the difference in sizes between the central particle and the others along the wire length.",
"In order to better understand the effect of orientation, in Fig3, we show the morphological evolution in the case of an angle of intersection of 45$^{\\circ }$ between the two nanowires (of the same radii, namely, 12 units).",
"Figure: t = 2900In this case also, there is junction formation at the intersection; however, unlike the previous case where the contact between the nanowires is at a point, in this case, the contact between the two wires is along a line.",
"Hence, the differences in curvatures at different points at the intersection lead to more material filling in at the sites which make smaller angle with the wire along the $x$ -axis.",
"Thus, when the central break-up takes place, the morphology of the central particle is not spherical; it is elongated and is aligned closer to (the infinite wire along) the $x$ -axis.",
"The subsequent break-ups and the coarsening are similar to the earlier case.",
"In Fig4 and Fig5, we show the morphological evolution for the cases of nanowires (of radii 12 units) which make angles of intersection of $30^{\\circ }$ and $60^{\\circ }$ , respectively.",
"These morphologies are qualitatively similar to that in Fig3 in terms of the morphology of the central particle.",
"Figure: t = 2950Figure: t = 2800These microstructural features are in good agreement with experimental observations.",
"For example, all the features noticed in the simulations above, namely, the formation of junction, the first break-up at the junction, and elongated central particle when the angle of intersection of wires is not 90$^{\\circ }$ are seen in experiments – specifically, see the Figures 3 and 4 of .",
"Interestingly, our simulation results also resemble nano-welds generated experimentally using different techniques like furnace annealing , and laser nano-welding of long Ag nanowires .",
"However, it is not clear to us if these welding are a result of local melting at the junction.",
"In our simulations, however, there is no local melting and all the morphological changes are through mass transport by enhanced surface diffusion.",
"Phase-field modelling has been used previously to simulate the sintering of Ag nanoparticles .",
"The effect of local sintering at the junction of nanowires have also been studied using atomistic simulation methods, where it was observed that the nanowires undergo self-limiting rotation during neck growth, which is a result of complex interaction of surface diffusion and dislocation growth .",
"We also observe the nanowire junction break-up in all cases described above as a result of local sintering at the junctions of nanowires.",
"In isolated nanowires, the primary cause of fragmentation is the Rayleigh instability.",
"But, the presence of junctions lead to modification of curvature and chemical potential in the intersection region, leading to junction formation followed by preferential break-up at the junctions.",
"Our simulation results are also in good agreement not only with the experimental results but also with the kinetic Monte Carlo simulation studies of Vigonski et al .",
"They reported a similar observation of preferential fragmentation of nanowires initially at the junction.",
"Vigonski et al, proposed that intersecting surfaces of nanowires act as sites of defects promoting atomic diffusion and attributed break-up at the junction primarily to the mechanism of diffusion of surface atoms.",
"In this study, we show that the geometric factors in terms of the curvature leads to sintering at the junctions, of course, assisted by faster surface diffusion.",
"Further, unlike the results from the Monte Carlo simulations, the phase-field model allow us to explore long time dynamics and morphological evolution such as the subsequent nanodot formation and the coarsening of the dots.",
"In order to better understand the junction formation and the subsequent break-up at the junctions, we have mapped the Gaussian curvature and chemical potential in these systems.",
"As an example, we show the chemical potential maps at two different time steps — one at the onset of junction formation and the other after break-up — and the corresponding Gaussian curvature maps in Fig6 for the first case, namely, when the wires are orthogonal to each other.",
"In this plot, the Gaussian curvature is visualized as a colour map (at the order parameter isosurface of $c=0.5$ ) overlaid on the nanowire assembly.",
"The chemical potential is visualized as a colour map in a plane which cuts through the cross section of nanowires (the nanowires are superimposed for visual clarity).",
"The colour maps for the chemical potential correspond to excess chemical potential above its mean value.",
"Not surprisingly, the chemical potential maps have one-to-one correspondence with the Gaussian curvature maps indicating that the driving forces for atomic diffusion at different regions of the nanowire assembly are due the the surface energy.",
"From the Gaussian curvature maps it can be seen that the two principal curvatures assume values of opposite sign giving rise to a saddle shaped geometry when the nanowires sinter at the junction.",
"Initially, due to the high curvature at the constrictions formed near the junction (fig6a), chemical potential is substantially high in this region as can be seen in fig6c, leading to higher atomic transport.",
"Hence, atoms diffuse from these narrow regions towards the central nanoparticle and arms of the nanowires.",
"Therefore, these constrictions get narrower finally causing detachment of nanowires from the junction.",
"Thereafter, the free ends of the nanowires retract due to high curvature at the tip of the broken junction.",
"As these nanowire free ends retract, more matter accumulates and these tips get blunted.",
"Subsequently these tips acquire a spherical shape and get detached; this mechanism is the same as observed in Rayleigh instability driven break-up.",
"This process continues leading to formation of further constrictions along the arms of the nanowires, which are regions of higher chemical potential as can be seen in fig6d.",
"This leads to subsequent formation of nanodots.",
"These nanodots have a positive Gaussian curvature as can be seen in fig6b, which occurs due to the spheriodisation of the nanoparticles in order to reduce the (isotropic) interfacial energy.",
"Figure: t = 2800The angle between the nanowires does not affect the sequence of the nanowire break-up; the first fragmentation always occurs at the junction; but, it does affect the break-up kinetics.",
"The time to first break-up (at the junction) is calculated for different orientations, and plotted in Fig7.",
"Here, we have assumed that there is a break-up at the junction if two or more of the arms at a junction (consisting of four arms) detach.",
"The data points are obtained from a set of three different simulations for each configuration (that is, by using three different seed values of pseudo-random number generator).",
"It is observed that the system is relatively more stable as the acute angle between the nanowires decrease.",
"The time to first junction break-up is highest for the $30^{\\circ }$ configuration system, and the least for $90^{\\circ }$ configuration system, when only the effect of relative wire orientation is considered in isolation.",
"This observation can be correlated with the average distance between nearest nanodots from the central nanoparticle along the two nanowires, formed subsequently after the junction break-up (values given in Table2).",
"From the values presented in Table2, it can be inferred that the average separation between the nanodots from the central nanoparticle along the two wires increases as the angle between the nanowires decrease.",
"These values were calculated by averaging the results from three different set of simulations.",
"The correlation between time to first junction break-up and average separation between the nanodots from the junction center can be explained with the Gaussian curvature maps; as the angle between the nanowire decrease, the curvature increases at the junction which leads to enhanced atomic transport in between the two wires.",
"Therefore, more material accumulates at the central agglomerate before constriction occurs at the junction, leading to nanowire junction break-up.",
"For this reason, the central nanoparticle acquires an oblate spheroidal shape as the the angle between nanowire is reduced.",
"The end result is the junction break-up substantially delayed along with the particles being placed afar from the center of the junction.",
"Figure: Plot showing the dependence of time to first break-up at the junction on therelative wire orientation.",
"The line connecting the data points is drawn only as a guide to the eye.Table: Distance of first nanodots from the central nanoparticle (wire 1 corresponds to the infinite length nanowire at the bottom, and wire 2 corresponds to the finite length inclined nanowire on top of the infinite length nanowire)."
],
[
"Effect of relative wire diameter",
"It has been observed experimentally that there is on average about 25% variation in the diameter of the fabricated metallic nanowires .",
"Therefore, in order to study the effect of relative variation in diameters of the nanowires, we use three different assemblies of nanowires in $90^{\\circ }$ configuration, with initial radii of $\\mathrm {R_1},\\mathrm {R_2} = 12,14,16$ .",
"We study the three different combinations of radii —the first combination with $\\mathrm {R_1}=14, \\mathrm {R_2}=16$ , the second combination with $\\mathrm {R_1}=12, \\mathrm {R_2}=14$ , and the final combination with $\\mathrm {R_1}=12, \\mathrm {R_2}=16$ ; with the relative radius variations of 14%, 17%, and 33%, respectively.",
"We have carried out a set of three different simulations for each case, and the average from these simulations are presented here.",
"It is observed that for all three combinations, the nanowires with smaller radius breaks up at the junction and the central agglomerate becomes part of the larger radius nanowires.",
"The time to first junction break-up (values given in Table3) within a given margin of error is dependent primarily on the radius of the smaller diameter nanowire.",
"Having said that, the difference in the radii of the nanowires has a small but definite effect on kinetics.",
"The kinetics when the radii are different is slightly faster than when the radii are the same; for example, for the cases of $R_1=12$ ,$R_2=14$ and $R_1=12$ ,$R_2=16$ are slightly faster than $R_1=R_2=12$ case.",
"Table: References"
]
] | 2107.01801 |
[
[
"Field analogue of the Ruijsenaars-Schneider model"
],
[
"Abstract We suggest a field extension of the classical elliptic Ruijsenaars-Schneider model.",
"The model is defined in two different ways which lead to the same result.",
"The first one is via the trace of a chain product of $L$-matrices which allows one to introduce the Hamiltonian of the model and to show that the model is gauge equivalent to a classical elliptic spin chain.",
"In this way, one obtains a lattice field analogue of the Ruijsenaars-Schneider model with continuous time.",
"The second method is based on investigation of general elliptic families of solutions to the 2D Toda equation.",
"We derive equations of motion for their poles, which turn out to be difference equations in space with a lattice spacing $\\eta$, together with a zero curvature representation for them.",
"We also show that the equations of motion are Hamiltonian.",
"The obtained system of equations can be naturally regarded as a field generalization of the Ruijsenaars-Schneider system.",
"Its lattice version coincides with the model introduced via the first method.",
"The limit $\\eta \\to 0$ is shown to give the field extension of the Calogero-Moser model known in the literature.",
"The fully discrete version of this construction is also discussed."
],
[
"Introduction",
"Our main purpose in this paper is to introduce (1+1)-dimensional field theory generalization of the elliptic $N$ -body Ruijsenaars-Schneider model [1], [2] which is usually regarded as a relativistic extension of the Calogero-Moser system.",
"This is done in two different ways, so the paper consists of two main parts.",
"In the first part (sections 2–4) we define a discrete space classical Ruijsenaars-Schneider chain starting from the classical homogeneous elliptic ${\\rm GL}_N$ spin chain on $n$ -sites.",
"Assuming periodic boundary conditions in (discrete) space direction, it is a finite-dimensional integrable system of classical mechanics.",
"By construction, it is gauge equivalent to the elliptic spin chain (or lattice version of the generalized Landau-Lifshitz model) with some special choice of level of the Casimir functions at each site.",
"In the second part (sections 5 and 6) we introduce a (1+1)-dimensional field analogue of the Ruijsenaars-Schneider model with continuous space variable whose natural finite-dimensional reduction turns out to be equivalent to the Ruijsenaars-Schneider chain introduced in the first part.",
"Our method is based on investigation of general elliptic solutions (called elliptic families) to the difference version of the 2D Toda equation.",
"We derive equations of motion for their poles together with a zero-curvature representation for them and show that they are Hamiltonian.",
"The continuum limit in space direction is shown to give the field extension of the Calogero-Moser model introduced in [3] via analyzing elliptic families of solutions to the Kadomtsev-Petviashvili equation.",
"Below we describe the contents of the both parts of the paper in more details.",
"The classical homogeneous elliptic ${\\rm GL}_N$ spin chain on $n$ -site is a widely known integrable system.",
"It is an integrable ${\\rm GL}_N$ -generalization of the lattice Landau-Lifshitz equation [4]–[8].",
"It is defined via the (classical) monodromy matrix depending on a spectral parameter $z$ as a product of the Lax matrices at each site: $\\begin{array}{c}\\displaystyle {T(z)={{\\mathcal {L}}}^{1}(z){\\mathcal {L}}^{2}(z)\\ldots {\\mathcal {L}}^{n}(z)\\,, \\quad {\\mathcal {L}}^i(z)\\in {\\rm Mat}(N,\\mbox{C}) .",
"}\\end{array}$ Each Lax matrix depends on a set of dynamical variables (coordinates in the phase space), which are combined into a matrix $S^i\\in {\\rm Mat}(N,\\mbox{C}) $ , so that ${\\mathcal {L}}^i(z)={\\mathcal {L}}^i(z,S^i)$ .",
"The trace of the monodromy matrix $t(z)={\\rm tr}\\, T(z)$ is a generating function of Hamiltonians.",
"They are in involution with respect to the classical quadratic $r$ -matrix structure (with the Belavin-Drinfeld elliptic classical $r$ -matrix [9] and $c$ be an arbitrary constant) $\\begin{array}{c}\\displaystyle {\\lbrace {\\mathcal {L}}_1^{i}(z),{\\mathcal {L}}_2^{j}(w)\\rbrace =\\frac{1}{c}\\,\\delta ^{ij}[{\\mathcal {L}}_1^{i}(z){\\mathcal {L}}_2^{i}(w),r_{12}(z-w)]\\,,}\\end{array}$ which is equivalent to $n$ copies of the classical generalized Sklyanin algebras at each site.",
"In this paper we use a modified description of the classical Sklyanin's elliptic Lax matrix.",
"Namely, following [10] we define ${\\mathcal {L}}^i(z,S^i)$ as $\\begin{array}{c}\\displaystyle {{\\mathcal {L}}^i(z,S^i)={\\rm tr}_2(R_{12}^\\eta (z)S_2^i)\\,, \\quad S_2^i=1_{N}\\otimes S^i\\in {\\rm Mat}(N,\\mbox{C})^{\\otimes 2}\\,,}\\end{array}$ where $R_{12}^\\eta (z)\\in {\\rm Mat}(N,\\mbox{C})^{\\otimes 2}$ is the quantum Baxter-Belavin elliptic $R$ -matrix [11], [12] and we use the standard convention on numbering the spaces where the matrices act.",
"Let us stress that although $R_{12}^\\eta (z)$ is quantum, the Lax matrix (REF ) is classical.",
"The parameter $\\eta $ usually plays the role of the Planck constant since in the classical limit $\\eta \\rightarrow 0$ we have $R_{12}^\\eta (z)=\\eta ^{-1}1_{N}\\otimes 1_N+r_{12}(z)+\\ldots $ , where $r_{12}(z)$ is the classical $r$ -matrix entering (REF ).",
"At the same time in (REF ) $\\eta $ is regarded as the relativistic deformation parameter, similarly to what happens in the Ruijsenaars-Schneider modelAn explanation of the presence of quantum $R$ -matrix in a classical model comes from associative Yang-Baxter equation which is fulfilled by the quantum Baxter-Belavin elliptic $R$ -matrix.",
"This equation unifies classical and quantum integrable structures.",
"See [13], [14], [15] and references therein..",
"In fact, the explicit dependence on $\\eta $ can be removed by some simple re-definitions.",
"However, we keep it since the form (REF ) has the following important property [16] (see also [17]–[22],[13]).",
"In the case when $S^i$ is a rank 1 matrix ($S^i=\\xi ^i\\otimes \\psi ^i$ , $\\xi ^i,\\psi ^i\\in \\mbox{C}^N$ ), the Lax matrix can be represented in the factorized form $\\begin{array}{c}\\displaystyle {{\\mathcal {L}}^i(z,S^i)=g(z+N\\eta ,q^i)e^{P^i/c}g^{-1}(z,q^i)\\in {\\rm Mat}(N,\\mbox{C}) \\,,\\quad P^i={\\rm diag}(p_1^i,\\ldots ,p_N^i)}\\end{array}$ (up to a scalar factor), where $q^i$ denotes a set of $N$ coordinate variables $q^i=\\lbrace q_1^i,\\ldots ,q_N^i\\rbrace $ and the explicit form of the matrix $g(z, q^i)$ is given below in the main text.",
"It is known as the intertwining matrix entering the IRF-Vertex correspondence [23]–[26].",
"The factorization (REF ) provides an explicit parametrization of the matrix $S^i$ through the canonical variables $p^i_k$ , $q^i_k$ , $i=1,\\ldots ,n$ , $k=1,\\ldots ,N$ , thus providing the classical analogue for representation of the generalized Sklyanin algebra by difference operators.",
"Moreover, the gauge transformed Lax matrix $\\begin{array}{c}\\displaystyle {g^{-1}(z,q^i){\\mathcal {L}}^i(z,S^i)g(z,q^i)=g^{-1}(z,q^i)g(z+N\\eta ,q^i)e^{P^i/c}:=L^{RS}(z,p^i,q^i)}\\end{array}$ is equal to the Lax matrix of the classical $N$ -body elliptic Ruijsenaars-Schneider model with momenta $p^i$ , coordinates of particles $q^i$ and the relativistic deformation parameter $\\eta $ .",
"The “velocity of light” $c$ enters (REF ) as a normalization factor behind momenta.",
"On the spin chain side it is the constant in the r.h.s.",
"of (REF ).",
"We restrict ourselves to the case when all matrices of dynamical variables $S^i$ are of rank one.",
"Then, taking into account (REF )–(REF ), we represent the transfer matrix $t(z)$ in the form $\\begin{array}{l}\\displaystyle {t(z)={\\rm tr}\\Big ( {\\tilde{L}}^1(z) {\\tilde{L}}^2(z)\\ldots {\\tilde{L}}^n(z) \\Big )\\,,}\\end{array}$ where $\\begin{array}{l}\\displaystyle {{\\tilde{L}}^i(z)=g^{-1}(z,q^{i-1})g(z+N\\eta ,q^i)\\,e^{P^i/c}\\,,\\quad i=1,\\ldots ,n\\ \\ \\hbox{and}\\ \\ q^0=q^n\\,.",
"}\\end{array}$ The Lax matrices ${\\tilde{L}}^i(z)$ can be found explicitly.",
"Then we derive equations of motion generated by a special Hamiltonian flow.",
"It is the one which has continuous limit to the (1+1)-dimensional field theory (the generalized Landau-Lifshitz equation) for the elliptic spin chain.",
"More precisely, we prove that the transfer-matrix (REF ) provides the Hamiltonian $\\begin{array}{c}\\displaystyle {H=c\\sum \\limits _{k=1}^n \\log h_{k-1,k}\\,,\\qquad h_{k-1,k}=\\sum \\limits _{j=1}^N b^k_j\\,,\\qquad b_j^k=\\frac{\\prod \\limits _{l=1}^N\\vartheta ({\\bar{q}}^k_j-{\\bar{q}}^{k-1}_l-\\eta ) }{\\vartheta (-\\eta )\\prod \\limits _{l: l\\ne j}\\vartheta ({ q}^{k}_j-{ q}^{k}_l) }\\,e^{p^k_j/c}\\,,}\\end{array}$ where $\\vartheta (z)$ is the odd Jacobi theta-function (REF ), and ${\\bar{q}}^k_j=q^k_j-\\sum _{i=1}^N q^k_i/N$ are coordinates “in the center of masses frame” at each site.",
"This Hamiltonian generates equations of motion $\\begin{array}{c}\\displaystyle {\\frac{{\\ddot{q}}^k_i }{ {\\dot{q}}^k_i }=-\\sum \\limits _{l=1}^N {\\dot{ q}}_l^{k+1}E_1({\\bar{q}}_i^k-{\\bar{q}}_l^{k+1}+\\eta )-\\sum \\limits _{l=1}^N {\\dot{ q}}_l^{k-1}E_1({\\bar{q}}_i^k-{\\bar{q}}_l^{k-1}-\\eta )+2\\sum \\limits _{l\\ne i}^N {\\dot{q}}_l^{k}E_1({q}_i^k-{q}_l^{k})+}\\\\\\displaystyle {+\\sum \\limits _{m,l=1}^N {\\dot{ q}}_m^{k}{\\dot{ q}}_l^{k+1}E_1({\\bar{q}}_m^k-{\\bar{q}}_l^{k+1}+\\eta )-\\sum \\limits _{m,l=1}^N {\\dot{ q}}_l^{k}{\\dot{ q}}_m^{k-1}E_1({\\bar{q}}_m^{k-1}-{\\bar{q}}_l^k+\\eta )\\,.",
"}\\end{array}$ With some simple normalization factor the Lax matrix (REF ) turns into $\\displaystyle {{L^{\\prime }}^k_{ij}(z)=\\phi (z,{\\bar{q}}^{k-1}_i-{\\bar{q}}^{k}_j+\\eta ){\\dot{q}}_j^k\\,,}$ where $\\phi (z, q)$ is the Kronecker function (REF ).",
"Equations of motion (REF ) are equivalently written in the form of the semi-discrete zero curvature (Zakharov-Shabat) equation $\\begin{array}{c}\\displaystyle {\\frac{d}{dt}\\,{{ L^{\\prime }}^k}(z)={L^{\\prime }}^k(z){M^{\\prime }}^k(z)-{ M^{\\prime }}^{k-1}(z){ L^{\\prime }}^k(z)\\,.",
"}\\end{array}$ with $M$ -matrices $\\begin{array}{c}\\displaystyle {{M^{\\prime }}^k_{ij}(z)=-(1-\\delta _{ij})\\phi (z,q_i^k-q_j^k)\\,{\\dot{q}}_j^k+\\delta _{ij}\\sum \\limits _{m,l=1}^N {\\dot{ q}}_l^{k+1}{\\dot{ q}}_m^{k}E_1({\\bar{q}}_m^{k}-{\\bar{q}}_l^{k+1}+\\eta )+}\\\\\\displaystyle {+\\delta _{ij}\\Big (-E_1(z){\\dot{q}}^k_i\\sum \\limits _{m\\ne i}^N{\\dot{q}}_m^kE_1(q_i^k-q_m^k)-\\sum \\limits _{m=1}^N{\\dot{q}}_m^{k+1}E_1({\\bar{q}}^k_i-{\\bar{q}}^{k+1}_m+\\eta )\\Big )\\,.",
"}\\end{array}$ Here $E_1(z)$ is the logarithmic derivative of the odd Jacobi theta-function and $\\phi (z, q)$ is the Kronecker function given by (REF ).",
"At this stage the model is discrete and finite-dimensional.",
"To proceed to field generalization we use another approach.",
"We will see that the 1+1 version corresponds to straightforward field extension of the described Ruijsenaars-Schneider chain.",
"The idea of another approach is to exploit the close connection between elliptic solutions to nonlinear integrable equations and many-body systems.",
"The investigation of dynamics of poles of singular solutions to nonlinear integrable equations was initiated in the seminal paper [27], where elliptic and rational solutions to the Korteweg-de Vries and Boussinesq equations were studied.",
"As it was proved later in [28], [29], poles of rational solutions to the Kadomtsev-Petviashvili (KP) equation as functions of the second hierarchical time $t_2$ move as particles of the integrable Calogero-Moser system [30], [31], [32], [33].",
"The method suggested by Krichever [34] for elliptic solutions of the KP equation consists in substituting the solution not in the KP equation itself but in the auxiliary linear problem for it (this implies a suitable pole ansatz for the wave function).",
"This method allows one to obtain the equations of motion together with their Lax representation.",
"Dynamics of poles of elliptic solutions to the 2D Toda lattice and modified KP (mKP) equations was studied in [35], see also [36].",
"It was proved that the poles move as particles of the integrable Ruijsenaars-Schneider many-body system which is a relativistic generalization of the Calogero-Moser system.",
"In the paper [3] elliptic families of solutions to the KP equation were studied.",
"In this more general case the solution is assumed to be an elliptic function not of $x=t_1$ , as it was assumed before, but of a general linear combination of higher times of the KP hierarchy.",
"It was shown that poles of such solutions as functions of $x=t_1$ and $t_2$ move according to equations of motion of the field generalization of the Calogero-Moser system.",
"In this paper we extend this result to elliptic families of solutions to the 2D Toda hierarchy.",
"We derive equations of motion for such solutions.",
"These equations of motion can be naturally thought of as a field generalization of the Ruijsenaars-Schneider system.",
"In the limit when the parameter $\\eta $ having the meaning of the inverse velocity of light tends to 0, the obtained equations of motion become those dealt with in the paper [3].",
"We also consider elliptic families of solutions to the fully difference integrable version of the 2D Toda lattice equation and derive equations of motion for the poles.",
"Let us say a few words about the nature of the elliptic families.",
"These solutions belong to a particular class of algebraic-geometrical solutions associated with an algebraic curve $\\Gamma $ of genus $g$ with some additional data.",
"An algebraic-geometrical solution is elliptic with respect to some variable $\\lambda $ if there exists a $g$ -dimensional vector ${\\bf W}$ such that it spans an elliptic curve ${\\cal E}$ embedded in the Jacobian of the curve $\\Gamma $ .",
"The tau-function of such solution is $\\tau (x, {\\bf t}, \\lambda )=e^{Q(x, {\\bf t})}\\Theta \\Bigl ({\\bf V}_0 x/\\eta +\\sum _{k\\ge 1}{\\bf V}_k t_k +{\\bf W}\\lambda +{\\bf Z}\\Bigr ),$ where $\\Theta $ is the Riemann theta-function and $Q$ is a quadratic form in the hierarchical times ${\\bf t}=\\lbrace t_1, t_2, t_3, \\ldots \\rbrace $ of the 2D Toda hierarchy (we put all “negative” times equal to zero for simplicity).",
"The vectors ${\\bf V}_k$ are $b$ -periods of certain normalized meromorphic differentials on $\\Gamma $ .",
"The existence of a $g$ -dimensional vector ${\\bf W}$ such that it spans an elliptic curve ${\\cal E}$ embedded in the Jacobian is a nontrivial transcendental constraint.",
"If such a vector ${\\bf W}$ exists, then the theta-divisor intersects the shifted elliptic curve $\\displaystyle {{\\cal E}+{\\bf V}_0 x/\\eta +\\sum _k {\\bf V}_k t_k}$ at a finite number of points $\\lambda _i =\\lambda _i(x,{\\bf t})$ .",
"Therefore, for elliptic families we have: $\\Theta \\Bigl ({\\bf V}_0 x/\\eta +\\sum _{k\\ge 1}{\\bf V}_k t_k +{\\bf W}\\lambda +{\\bf Z}\\Bigr )=f(x, {\\bf t})e^{\\gamma _1\\lambda +\\gamma _2\\lambda ^2}\\prod _{i=1}^N\\sigma (\\lambda -\\lambda _i(x, {\\bf t})).$ with a function $f(x, {\\bf t})$ and some constants $\\gamma _1, \\gamma _2$ .",
"Here $\\sigma (\\lambda )$ is the Weierstrass $\\sigma $ -function defined in the Appendix A.",
"The zeros $\\lambda _i$ of the tau-function are poles of the elliptic solutions.",
"We show that the equations of motion of the poles $\\lambda _i =\\lambda _i(x,t)$ , where $t=t_1$ , are given by $\\begin{array}{c}\\displaystyle {\\ddot{\\lambda }_i(x)+\\sum _k \\Bigl (\\dot{\\lambda }_i(x)\\dot{\\lambda }_k(x-\\eta )\\zeta (\\lambda _i(x)-\\lambda _k(x-\\eta ))+\\dot{\\lambda }_i(x)\\dot{\\lambda }_k(x+\\eta )\\zeta (\\lambda _i(x)-\\lambda _k(x+\\eta ))\\Bigr )}\\\\ \\\\\\displaystyle {-\\, 2 \\sum _{k\\ne i}\\dot{\\lambda }_i(x)\\dot{\\lambda }_k(x )\\zeta (\\lambda _i(x)-\\lambda _k(x))+(c(x-\\eta , t)-c(x, t))\\dot{\\lambda }_i(x)=0.",
"}\\end{array}$ Here $c(x, t)=\\frac{1}{\\beta }\\sum _{i,k}\\dot{\\lambda }_i(x)\\dot{\\lambda }_k(x+\\eta )\\zeta (\\lambda _i(x)-\\lambda _k(x+\\eta )), \\qquad \\beta =\\sum _i \\dot{\\lambda }_i(x),$ and $\\zeta (\\lambda )$ is the Weierstrass $\\zeta $ -function (a close relative of the function $E_1$ in (REF )).",
"Equations (REF ) are represented in the zero-curvature form $\\dot{L}(x)+L(x)M(x)-M(x+\\eta )L(x)=0\\,,$ and the Lax pair is obtained explicitly in section 5.3.",
"Then, the equations (REF ) can be naturally restricted to the lattice by setting $\\lambda _i^k=\\lambda _i(x_0+k\\eta )$ and rewritten as $\\begin{array}{c}\\displaystyle {\\ddot{\\lambda }_i^k+\\sum _j \\Bigl (\\dot{\\lambda }_i^k\\dot{\\lambda }_j^{k-1}\\zeta (\\lambda _i^k-\\lambda _j^{k-1})+\\dot{\\lambda }_i^k\\dot{\\lambda }_j^{k+1}\\zeta (\\lambda _i^k-\\lambda _j^{k+1})\\Bigr )}\\\\ \\\\\\displaystyle {-\\, 2 \\sum _{j\\ne i}\\dot{\\lambda }_i^k\\dot{\\lambda }_j^k\\zeta (\\lambda _i^k-\\lambda _j^k)+(c^{k-1}(t)-c^k(t))\\dot{\\lambda }_i^k=0}\\end{array}$ with $c^k(t)=\\frac{1}{\\beta }\\sum _{i,j}\\dot{\\lambda }_i^k\\dot{\\lambda }_j^{k+1}\\zeta (\\lambda _i^k-\\lambda _j^{k+1}),$ in which form they can be shown to be equivalent to equations (REF ).",
"Details of the equivalence between (REF )-(REF ) and (REF )-(REF ) are given in section 5.4.",
"In section 5.6 we also describe the continuum limit to the (1+1)-dimensional Calogero-Moser field theory discussed in [3].",
"The fully discrete version of the equations (REF ) is obtained in section 6.",
"In Appendix A the necessary definitions and properties of elliptic functions are given.",
"In Appendix B we describe properties of the elliptic $R$ -matrix which are used in the derivation of the Ruijsenaars-Schneider spin chain.",
"In Appendix C, using the factorization formulae for the Lax matrix, we obtain the explicit change of variables between the Ruijsenaars-Schneider model and the relativistic top."
],
[
"Ruijsenaars-Schneider model in the form of relativistic top",
"In this section we recall the necessary preliminaries related to the Ruijsenaars-Schneider model and relativistic top.",
"From the point of view of the next sections this case corresponds to the “Ruijsenaars-Schneider chain” on one site."
],
[
"The standard Hamiltonian and equations of motion.",
"The elliptic Ruijsenaars-Schneider model is defined by the Lax matrix [2] $\\begin{array}{l}\\displaystyle {L^{\\rm RS}_{ij}(z)=\\phi (z,q_{ij}+\\eta )\\,b_j\\,,\\ i,j=1,\\ldots ,N\\,,}\\end{array}$ where $\\phi (z, q)$ is the Kronecker function defined in (REF ) and $\\begin{array}{l}\\displaystyle {b_j=\\prod _{k:k\\ne j}^N\\frac{\\vartheta (q_{j}-q_k-\\eta )}{\\vartheta (q_{j}-q_k)}\\,e^{p_j/c}\\,,\\quad c={\\rm const}\\in \\mbox{C}\\,.",
"}\\end{array}$ Here $\\vartheta (z)$ is the odd Jacobi theta-function (REF ).",
"Note that original definition of $b_j$ in [2] is different from (REF ).",
"This is due to a freedom in the definition of (REF )–(REF ) coming from the canonical map $\\begin{array}{c}\\displaystyle {p_j\\ \\rightarrow \\ p_j+c_1\\log \\prod \\limits _{k\\ne j}^N\\frac{\\vartheta (q_{j}-q_k+\\eta )}{\\vartheta (q_{j}-q_k-\\eta )}}\\end{array}$ with arbitrary constant $c_{1}$ .",
"The Hamiltonian $\\begin{array}{l}\\displaystyle {H^{\\rm RS}=c\\frac{{\\rm tr}L^{\\rm RS}(z)}{\\phi (z,\\eta )}=c\\sum \\limits _{j=1}^N b_j(p,q)}\\end{array}$ with the canonical Poisson brackets $\\begin{array}{c}\\displaystyle {\\lbrace p_i,q_j\\rbrace =\\delta _{ij}\\,,\\quad \\lbrace p_i,p_j\\rbrace =\\lbrace q_i,q_j\\rbrace =0}\\end{array}$ provides the following equations of motion: $\\begin{array}{l}\\displaystyle {{\\dot{q}}_j=\\lbrace H^{\\rm RS},q_j\\rbrace =\\partial _{p_j}H^{\\rm RS}=b_j=\\prod _{k\\ne j}\\frac{\\vartheta (q_{j}-q_k-\\eta )}{\\vartheta (q_{j}-q_k)}\\,e^{p_j/c}\\,.",
"}\\end{array}$ We see that the Hamiltonian is proportional to the sum of velocities: $\\begin{array}{l}\\displaystyle {\\frac{1}{c}\\,H^{\\rm RS} =\\sum \\limits _{j=1}^N{\\dot{q}}_j}\\end{array}$ and the Lax matrix (REF ) takes the form $\\begin{array}{l}\\displaystyle {L^{\\rm RS}_{ij}(z)=\\phi (z,q_{ij}+\\eta )\\,{\\dot{q}}_j\\,.",
"}\\end{array}$ The Hamiltonian equations for momenta are as follows: $\\begin{array}{c}\\displaystyle {\\frac{1}{c}\\,{\\dot{p}}_i=\\frac{1}{c}\\,\\lbrace H^{\\rm RS},p_i\\rbrace =-\\frac{1}{c}\\,\\partial _{q_i}H^{\\rm RS}}\\\\ \\ \\\\\\displaystyle {=\\sum \\limits _{l\\ne i} ({\\dot{q}}_i+{\\dot{q}}_l)E_1(q_{il})-{\\dot{q}}_i E_1(q_{il}-\\eta )-{\\dot{q}}_l E_1(q_{il}+\\eta )\\,,}\\end{array}$ where $q_{ij}=q_i-q_j$ , and $E_1(w)=\\vartheta ^{\\prime }(w)/\\vartheta (w)$ (REF ).",
"By differentiating both parts of (REF ) with respect to time we get $\\begin{array}{c}\\displaystyle {\\frac{{\\ddot{q}}_i}{{\\dot{q}}_i}=\\frac{1}{c}\\,{\\dot{p}}_i+\\sum \\limits _{l\\ne i}({\\dot{q}}_i-{\\dot{q}}_l)(E_1(q_{il}-\\eta )-E_1(q_{il}))\\,.",
"}\\end{array}$ Plugging (REF ) into (REF ) we get the well known equations of motion of the elliptic Ruijsenaars-Schneider model in the Newtonian form: $\\begin{array}{c}\\displaystyle {{\\ddot{q}}_i=\\sum \\limits _{k\\ne i}{\\dot{q}}_i{\\dot{q}}_k(2E_1(q_{ik})-E_1(q_{ik}+\\eta )-E_1(q_{ik}-\\eta ))\\,,\\quad i=1, \\ldots ,N\\,.",
"}\\end{array}$ Equations of motion (REF ) are equivalent to the Lax equation $\\begin{array}{c}\\displaystyle {{\\dot{L}}^{\\rm RS}(z)\\equiv \\lbrace H^{\\rm RS},L^{\\rm RS}(z)\\rbrace =[L^{\\rm RS}(z),M^{\\rm RS}(z)]}\\end{array}$ with the $M$ -matrix $\\begin{array}{c}\\displaystyle {M^{\\rm RS}_{ij}(z)=-(1-\\delta _{ij})\\phi (z,q_i-q_j)\\,{\\dot{q}}_j}\\\\ \\ \\\\\\displaystyle {-\\delta _{ij}\\Big ({\\dot{q}}_i\\,(E_1(z)+E_1(\\eta )) +\\sum \\limits _{k\\ne i} {\\dot{q}}_k\\,(E_1(q_{ik}+\\eta )-E_1(q_{ik}))\\Big ).",
"}\\end{array}$ This follows from a direct calculation with the help of (REF ) and (REF )."
],
[
"Logarithm of the Hamiltonian.",
"Alternatively, one can use the following Hamiltonian: $\\begin{array}{c}\\displaystyle {H^{\\prime }=c\\log H^{\\rm RS}=c\\log \\sum \\limits _{j=1}^N b_j\\,.",
"}\\end{array}$ Then $\\begin{array}{c}\\displaystyle {{\\dot{q}}_j=\\frac{\\partial H^{\\prime }}{\\partial p_j}=\\frac{b_j}{H^{\\rm RS}}\\,,\\qquad \\frac{1}{c}\\,{\\dot{p}}_i=-\\frac{1}{c}\\frac{\\partial H^{\\prime }}{\\partial q_i}=-\\frac{1}{H^{\\rm RS}}\\frac{\\partial H^{\\rm RS}}{\\partial q_i}\\,,}\\end{array}$ so that (cf.",
"(REF )) $\\begin{array}{c}\\displaystyle {\\sum \\limits _{j=1}^N{\\dot{q}}_j=1\\,.",
"}\\end{array}$ The Lax matrix (REF ) becomes now $\\begin{array}{c}L^{\\rm RS}_{ij}(z)=\\phi (z,q_{ij}+\\eta )\\,{\\dot{q}}_jH^{\\rm RS}\\end{array}$ instead of (REF ) but this makes no difference since $H^{\\rm RS}$ is a conserved quantity.",
"It is easy to see that the equations of motion in the Newtonian form (REF ) remain the same with the Hamiltonian (REF )."
],
[
"Classical relativistic top",
"Let us consider the elliptic ${\\rm GL}_N$ spin chain on a single site.",
"It is an integrable system called relativistic top [10].",
"The Lax matrix is as follows: $\\begin{array}{c}\\displaystyle {{\\mathcal {L}}^\\eta (z)=\\sum \\limits _{a\\in \\,\\mbox{Z}}_{ N}\\times \\mbox{Z}}_{ N}\\end{array} T_a { S}_a\\varphi _a(z,\\omega _a+\\eta )\\,,$ where $S_a$ are dynamical variables (classical spins) numbered by the index $a=(a_1,a_2)\\in \\mbox{Z}_N\\times \\mbox{Z}_N$ in the special matrix basis $T_a\\in {\\rm Mat}(N,\\mbox{C}) $ (REF ), which is often used for elliptic quantum $R$ -matrices.",
"The set of functions $\\varphi _a(z,\\omega _a+\\eta )$ and the quantities $\\omega _a$ are given in (REF ).",
"The dynamical variables are combined into a matrix $S\\in {\\rm Mat}(N,\\mbox{C}) $ : $\\begin{array}{c}\\displaystyle {S=\\sum \\limits _{i,j=1}^N S_{ij}E_{ij}=\\sum \\limits _{a_1,a_2=0}^{N-1} S_{a}T_a\\,,}\\end{array}$ where $E_{ij}$ are matrix units.",
"Using the property (REF ) let us rewrite the Lax matrix (REF ) in terms of the Baxter-Belavin elliptic $R$ -matrix $R_{12}^\\eta (z)$ [11], [37] in the form (REF ): $\\begin{array}{c}\\displaystyle {R_{12}^\\eta (z)=\\frac{1}{N}\\sum \\limits _{a\\in \\,\\mbox{Z}}_{ N}\\times \\mbox{Z}}_{ N}\\end{array}T_a\\otimes T_{-a} \\varphi _a(z,\\omega _a+\\eta )\\in {\\rm Mat}(N,\\mbox{C})^{\\otimes 2}\\,.$ Alternative equivalent forms are given in the Appendix B.",
"In terms of the $R$ -matrix, the Lax matrix (REF ) acquires the following compact form: $\\begin{array}{c}\\displaystyle {{\\mathcal {L}}^\\eta (z)={\\rm tr}_2(R_{12}^\\eta (z)S_2)\\,,\\quad S_2=1_N\\otimes S,}\\end{array}$ where the trace is over the second tensor component.",
"We emphasis that the $R$ -matrix is quantum, while the Lax matrix is classical.",
"The parameter $\\eta $ plays the role of the Planck constant in the $R$ -matrix since the classical $r$ -matrix comes from (REF ) in the classical limit $\\begin{array}{c}\\displaystyle {R_{12}^\\eta (z)=\\frac{1}{N\\eta }+r_{12}(z)+O(\\eta )\\,,}\\end{array}$ $\\begin{array}{c}\\displaystyle {r_{12}(z)=\\frac{1}{N}\\,1_N\\otimes 1_N\\, E_1(z)+\\frac{1}{N}\\sum \\limits _{a\\ne 0} T_a\\otimes T_{-a} \\varphi _a(z,\\omega _a)\\in {\\rm Mat}(N,\\mbox{C})^{\\otimes 2}\\,.",
"}\\end{array}$ At the same time $\\eta $ plays the role of the relativistic deformation parameter in the relativistic top model likewise it appears in the Ruijsenaars-Schneider model.",
"In the standard approach [4], [6] the parameter $\\eta $ is absent in the Lax matrix and so it does not enter the classical (Poisson) Sklyanin algebra.",
"Below we explain how this parameter can be eliminated by some re-definitions.",
"However, it is important for us to keep it in the Lax matrix because in this form the latter has a nice property of factorization."
],
[
"Classical Sklyanin algebra.",
"The classical quadratic $r$ -matrix Poisson structure isThe coefficient $1/c$ in (REF ) is introduced here in order to match the relation with the Ruijsenaars-Schneider model.",
": $\\begin{array}{c}\\displaystyle {\\lbrace {\\mathcal {L}}_1^{\\eta }(z),{\\mathcal {L}}_2^{\\eta }(w)\\rbrace =\\frac{1}{c}\\,[{\\mathcal {L}}_1^{\\eta }(z){\\mathcal {L}}_2^{\\eta }(w),r_{12}(z-w)]\\,,}\\end{array}$ where the classical $r$ -matrix is given by (REF ).",
"Plugging the Lax matrix (REF ) into (REF ) and using identity (REF ) one gets $\\begin{array}{c}\\displaystyle {\\lbrace S_\\alpha ,S_\\beta \\rbrace =\\!\\frac{1}{c}\\sum \\limits _{\\xi \\ne 0} \\kappa _{\\alpha -\\beta ,\\xi }S_{\\alpha -\\xi }S_{\\beta +\\xi }\\Big ( E_1(\\omega _\\xi )\\!-\\!E_1(\\omega _{\\alpha -\\beta -\\xi })\\!+\\!E_1(\\omega _{\\alpha -\\xi }+\\eta )\\!-\\!E_1(\\omega _{\\beta +\\xi }+\\eta ) \\Big )\\,,}\\end{array}$ which is the classical Sklyanin algebra.",
"The constants $\\kappa _{\\alpha ,\\beta }$ are defined in (REF )."
],
[
"Eliminating the parameter $\\eta $ .",
"Let us comment on the form of the classical Lax matrix ${\\mathcal {L}}^{\\eta }(z)$ (REF ), (REF ).",
"Usually [4], [6] the classical Lax matrix of the top is written as $\\begin{array}{c}\\displaystyle {{\\mathcal {L}}(z,{\\tilde{S}})=1_N {\\tilde{S}}_0+\\sum \\limits _{a\\ne 0} T_a {\\tilde{S}}_a\\varphi _a(z,\\omega _a)\\,.",
"}\\end{array}$ It is known to satisfy (REF ), which provides the classical Sklyanin Poisson algebra for $N^2$ generators ${\\tilde{S}}_a$ .",
"Writing (REF ), we assume that it is also fulfilled for ${\\mathcal {L}}^{\\eta }(z)$ .",
"It happens for the following reason [38], [10].",
"First of all, this can be verified by a direct calculation, so that the classical Sklyanin algebra for $S_a$ contains additional parameter $\\eta $ .",
"However, this dependence is artificial.",
"Using the relation $\\begin{array}{c}\\displaystyle {\\frac{\\varphi _a(z-\\eta ,\\omega _a+\\eta )}{\\phi (z-\\eta ,\\eta )}=\\frac{\\varphi _a(z,\\omega _a)}{\\varphi _a(\\eta ,\\omega _a)},}\\end{array}$ one easily obtains $\\begin{array}{c}\\displaystyle {\\frac{1}{\\phi (z-\\eta ,\\eta )}\\,{\\mathcal {L}}^\\eta (z-\\eta ,S)={\\mathcal {L}}(z,{\\tilde{S}})}\\end{array}$ if $\\begin{array}{c}\\displaystyle {S={\\mathcal {L}}(\\eta ,{\\tilde{S}})\\,.",
"}\\end{array}$ Using (REF ) in the basis $T_\\alpha $ (REF ) we may write (REF ) explicitly: $\\begin{array}{c}\\displaystyle {S_0={\\tilde{S}}_0\\,,\\qquad S_\\alpha ={\\tilde{S}}_\\alpha \\varphi _\\alpha (\\eta ,\\omega _\\alpha )\\ {\\rm for}\\ \\alpha \\ne 0\\,.",
"}\\end{array}$ Let us remark that a similar phenomenon with the same change of variables take place in quantum Sklyanin algebra generated by exchange relation $R_{12}^\\hbar (z-w){\\hat{{\\mathcal {L}}}}^\\eta _1(z){\\hat{{\\mathcal {L}}}}^\\eta _2(w)={\\hat{{\\mathcal {L}}}}^\\eta _2(w){\\hat{{\\mathcal {L}}}}^\\eta _1(z)R_{12}^\\hbar (z-w)$ .",
"Then it contains two parameters $\\hbar $ and $\\eta $ , but the latter can be removed by (REF ) or fixed somehow.",
"For example, in the case $\\hbar =\\eta $ the Sklyanin algebra has representation ${\\hat{S}}_a=T_{-a}$ since the exchange relation turns into the Yang-Baxter equation in this case.",
"So that the second parameter is artificial.",
"We see that the two Lax matrices ${\\mathcal {L}}^\\eta (z)$ and (REF ) are related by the explicit change of variables (REF ) or (REF ), and the shift of the spectral parameter $z\\rightarrow z-\\eta $ does not effect (REF ) because $r_{12}(z-w)$ depends on the difference of spectral parameters.",
"In what follows we need the explicit dependence on $\\eta $ in ${\\mathcal {L}}^\\eta (z)$ for establishing its relation to the Ruijsenaars-Schneider model.",
"For this purpose we will consider $S$ to be a rank one matrix (this is not true for $\\tilde{S}$ )."
],
[
"Lax pair.",
"The Lax equation follows from (REF ) in the following way.",
"Since $S=\\mathop {\\hbox{Res}}\\limits \\limits _{w=0}{\\mathcal {L}}^\\eta (w)$ , then the residue at $w=0$ of both parts of (REF ) yields $\\begin{array}{c}\\displaystyle {\\lbrace {\\mathcal {L}}_1^{\\eta }(z),S_2\\rbrace =[{\\mathcal {L}}_1^{\\eta }(z)S_2,\\frac{1}{c}\\,r_{12}(z)]\\,.",
"}\\end{array}$ Taking trace of both parts of (REF ) in the second tensor component, we get the Lax equation $\\begin{array}{c}\\displaystyle {{\\dot{{\\mathcal {L}}}}^{\\eta }(z)=\\lbrace H^{top},{\\mathcal {L}}_1^{\\eta }(z)\\rbrace =[{\\mathcal {L}}_1^{\\eta }(z),M(z)]\\,,\\qquad M(z)=-{\\rm tr}_2\\Big (r_{12}(z)S_2\\Big )\\,,}\\end{array}$ where the Hamiltonian is $\\begin{array}{c}\\displaystyle {H^{top}=c\\,{\\rm tr}\\, S=c\\,\\frac{{\\rm tr}{\\mathcal {L}}^\\eta (z)}{\\phi (z,\\eta )}\\,.",
"}\\end{array}$ More precisely, $\\begin{array}{c}\\displaystyle {M(z)=-S_0 1_N E_1(z)-\\sum \\limits _{\\alpha \\ne 0}T_\\alpha S_\\alpha \\varphi _\\alpha (z,\\omega _\\alpha )\\,.",
"}\\end{array}$ Equations of motion take the form: $\\begin{array}{c}\\displaystyle {\\dot{S}=[S,J^\\eta (S)]\\,,}\\end{array}$ $\\begin{array}{c}\\displaystyle {J^\\eta (S)=1_N S_0 E_1(\\eta )+\\sum \\limits _{\\alpha \\ne 0}T_\\alpha S_\\alpha J_\\alpha ^\\eta \\,,\\quad J_\\alpha ^\\eta =E_1(\\eta +\\omega _\\alpha )-E_1(\\omega _\\alpha )\\,.",
"}\\end{array}$ They follow from the Lax equation (REF ) under the substitution (REF ), (REF ) and usage of (REF ) and the identity (REF )."
],
[
"Factorization of Lax matrices and relation between the models",
"Following [23], [24], [26], we introduce the intertwining matrix $\\begin{array}{l}\\displaystyle {g(z,q)=\\Xi (z,q)\\left(d^{0}\\right)^{-1}}\\end{array}$ with $\\begin{array}{c}\\displaystyle {\\Xi _{ij}(z,q)=\\vartheta \\left[ \\begin{array}{c}\\frac{1}{2}-\\frac{i}{N} \\\\ \\frac{N}{2}\\end{array} \\right] \\left(z-Nq_j+\\sum \\limits _{m=1}^Nq_m\\left.\\right|N\\tau \\right)\\,,}\\end{array}$ and the diagonal matrix $\\begin{array}{c}\\displaystyle {d^0_{ij}(z,q)=\\delta _{ij}d^0_{j}=\\delta _{ij}{\\prod \\limits _{k\\ne j}\\vartheta (q_j-q_k)}\\,,}\\end{array}$ where the theta function with characteristics is defined in (REF ).",
"The matrix (REF ) is the intertwining matrix entering the relations of the IRF-Vertex correspondence.",
"Its properties are described in the Appendix B."
],
[
"Factorization formula.",
"It was observed in [16] (at quantum level) that the Lax matrix (REF )–(REF ) can be represented in the factorized form $\\begin{array}{l}\\displaystyle {L^{\\rm RS}_{ij}(z)=\\frac{\\vartheta ^{\\prime }(0)}{\\vartheta (\\eta )}\\sum \\limits _{k=1}^Ng^{-1}_{ik}(z,q)g_{kj}(z+N\\eta ,q)\\,e^{p_j/c}\\,,}\\end{array}$ or $\\begin{array}{l}\\displaystyle {L^{\\rm RS}(z)=\\frac{\\vartheta ^{\\prime }(0)}{\\vartheta (\\eta )}\\,g^{-1}(z,q)g(z+N\\eta ,q)\\,e^{P/c}\\,,\\quad P={\\rm diag}(p_1,\\ldots ,p_N)\\,.",
"}\\end{array}$ Moreover, the gauge transformed Lax matrix $\\begin{array}{l}\\displaystyle {{\\mathcal {L}}^\\eta (z)=g(z,q)L^{\\rm RS}(z)g^{-1}(z,q)=\\frac{\\vartheta ^{\\prime }(0)}{\\vartheta (\\eta )}\\,g(z+N\\eta ,q)\\,e^{P/c}g^{-1}(z,q)}\\end{array}$ is the Lax matrix of type (REF ) since it has the same quasi-periodic properties (see (REF ) below) and a simple pole at $z=0$ .",
"In contrast to (REF ) a special choice of the residue $S$ is assumed in (REF ).",
"It is of rank one, i.e.",
"it corresponds to some special choice of values of the Casimir functions (in the classical Sklyanin algebra), likewise the spinless Calogero-Moser model is related to the coadgoint orbit of minimal dimension.",
"Relation (REF ) can be viewed as the classical version of the IRF-Vertex relation (REF ).",
"It provides the change $S=S(p,q,\\eta ,c)$ from canonical variables to spin variables, which will be discussed in detail in the next subsection.",
"Let us compute the residue of both parts of (REF ) at $z=0$ .",
"For this purpose we need the properties of the matrix $g(z,q)$ (REF )–(REF ).",
"In particular, it is degenerated at $z=0$ , and the residue $\\breve{g}(0,q)=\\mathop {\\hbox{Res}}\\limits \\limits _{z=0}g^{-1}(z,q)$ is a rank one matrix.",
"In this way we get parametrizations of $S$ matrix in the form $S=\\xi \\otimes \\psi $ : $\\begin{array}{l}\\displaystyle {S=\\frac{\\vartheta ^{\\prime }(0)}{\\vartheta (\\eta )}\\,g(N\\eta ,q)\\,e^{P/c} \\breve{g}(0,q)}\\end{array}$ or $\\begin{array}{c}\\displaystyle {S=\\xi \\otimes \\psi \\,,\\qquad \\xi =\\frac{\\vartheta ^{\\prime }(0)}{\\vartheta (\\eta )}\\,g(N\\eta )\\,e^{P/c}\\,\\rho \\,,\\qquad \\psi =\\frac{1}{N}\\,\\rho ^T{\\breve{g}}(0)\\,.",
"}\\end{array}$ with $\\rho $ from (REF ) and $\\breve{g}(0,q)$ from (REF )."
],
[
"Factorization from IRF-Vertex relations.",
"Notice that on one hand we deal with the Lax matrix ${\\mathcal {L}}^\\eta (z)$ in the form (REF ), and on the other hand we use its factorized form (REF ) for a special choice of $S$ .",
"A connection between these two representations come from the IRF-Vertex relation, which includes both $R$ -matrix and the matrix $g(z,q)$ .",
"We review it in Appendix B.",
"The easiest way is to use the identity (REF ) in $ {\\rm Mat}(N,\\mbox{C}) ^{\\otimes 2}$ , which includes a special matrix $\\mathcal {O}_{12}\\in {\\rm Mat}(N,\\mbox{C}) ^{\\otimes 2}$ (REF ) with the property (REF ).",
"Following [22], multiply both parts of (REF ) by $S_2=1_N\\otimes S$ with $S$ -matrix presented in the form (REF ) $\\begin{array}{c}\\displaystyle {\\frac{\\vartheta ^{\\prime }(0)}{\\vartheta (\\eta )}\\,g_2(N\\eta )e^{P_2/c}{\\breve{g}}_2(0,q)\\,R^\\eta _{12}(z)=}\\\\\\displaystyle {=\\frac{\\vartheta ^{\\prime }(0)}{\\vartheta (\\eta )}\\,g_2(N\\eta )e^{P_2/c}g_1(z+N\\eta ,q)\\,\\mathcal {O}_{12}\\, g_2^{-1}(N\\eta ,q)\\,g_1^{-1}(z,q)\\,.",
"}\\end{array}$ Next, compute the trace over the second tensor component of both parts (REF ).",
"The property (REF ) simplifies the r.h.s.",
"of (REF ) since ${\\rm tr}_2(\\mathcal {O}_{12}e^{P_2/c})=e^{P/c}$ , and therefore $\\begin{array}{c}\\displaystyle {{\\mathcal {L}}^\\eta (z,S)={\\rm tr}_2(R^\\eta _{12}(z)S_2)=\\frac{\\vartheta ^{\\prime }(0)}{\\vartheta (\\eta )}\\,g(z+N\\eta ,q)\\,e^{P/c}g^{-1}(z,q)\\,.",
"}\\end{array}$ In what follows we also need degeneration of the factorized form.",
"By comparing (REF ) and (REF ) in the $\\eta \\rightarrow 0$ limit we get $\\begin{array}{l}\\displaystyle {\\left(g^{-1}(z)g^{\\prime }(z)\\right)_{ij}=\\frac{1}{N}\\,\\delta _{ij}\\left(E_1(z)-\\sum \\limits _{k\\ne i}E_1(q_{ik})\\right)+\\frac{1}{N}\\,(1-\\delta _{ij})\\phi (z,q_{ij})\\,.",
"}\\end{array}$ In Section 4 we also use degeneration of (REF ) coming from (REF ) to derive the accompany $M$ -matrix."
],
[
"Explicit change of variables.",
"The explicit change of variables $S_a=S_a(p,q,\\eta ,c)$ can be found in [16] (see also [20], [21]) in the elliptic case.",
"(The trigonometric and rational cases were addressed in [10].)",
"In Appendix C we derive this formula in the elliptic case.",
"In our notation it takes the form $\\begin{array}{c}\\displaystyle {S_a=\\frac{(-1)^{a_1+a_2}}{N}\\,e^{\\pi \\imath a_2\\omega _a}\\sum \\limits _{m=1}^N e^{p_m/c} e^{2\\pi \\imath a_2(\\eta -{\\bar{q}}_m)}\\frac{\\vartheta (\\eta +\\omega _\\alpha )}{\\vartheta (\\eta )}\\prod \\limits _{l:\\,l\\ne m}^N\\frac{\\vartheta (q_m-q_l-\\eta -\\omega _a)}{\\vartheta (q_m-q_l)}\\,,}\\end{array}$ where $\\bar{q}_m$ is the coordinate in the center of masses frame.",
"The classical Sklyanin generators $S_a$ are dynamical variables in the relativistic top model described above, and (REF ) provides its relation to the Ruijsenaars-Schneider model in the special case ${\\rm rk}(S)=1$ generated by the gauge equivalence (REF ).",
"Put it differently, (REF ) is a classical analogue of the representation of the generalized Sklyanin algebra by difference operators (in the classical limit the shift operators are substituted by exponents of momenta).",
"Therefore, we have the following statement.",
"The set of functions $S_a=S_a(p,q,\\eta ,c)$ satisfy the classical Sklyanin algebra Poisson brackets (REF ) computed by means of the canonical Poisson brackets (REF ).",
"The Lax matrix (REF ) satisfy the classical exchange relations (REF ).",
"The proof of a similar statement was proposed in [21] by a direct gauge transformation relating the $r$ -matrix structure (REF ) with the dynamical one known for the Ruijsenaars-Schneider model [19].",
"We also claim that the matrix of dynamical variables $S$ with components (REF ) is represented in the form (REF ).",
"The proof and explicit expressions for $\\xi $ and $\\psi $ are given in Appendix C."
],
[
"Classical ${\\rm GL}_N$ elliptic spin chain",
"Here we review properties of the Lax matrices for elliptic classical spin chains and describe the Hamiltonian flow which is then used for constructing the Ruijsenaars-Schneider chain in the next section.",
"We deal with the classical version [4], [6] of the generalized elliptic (anisotropic) homogeneous spin chain on $n$ sites associated with ${\\rm GL}_N$ .",
"It is described by the elliptic Baxter-Belavin $R$ -matrix [11], [37].",
"The generating function of the Hamiltonians is given by trace $t(z)$ of the monodromy matrix $T(z)$ : $\\begin{array}{c}\\displaystyle {t(z)={\\rm tr}T(z)\\,,\\qquad T(z)={\\mathcal {L}}^{1}(z){\\mathcal {L}}^{2}(z)...{\\mathcal {L}}^{n}(z)\\,,}\\end{array}$ where ${\\mathcal {L}}^i(z)$ is the classical Sklyanin's Lax matrix on $i$ th site of the chain.",
"It is fixed by the quasi-periodic properties on the lattice $\\mbox{Z}\\oplus \\tau \\mbox{Z}$ in the complex plane (defining the elliptic curve $\\Sigma _\\tau ={\\mbox{C}}/(\\mbox{Z}\\oplus \\tau \\mbox{Z})$ ) and the residue at a simple pole $z=0$ : $\\begin{array}{c}\\displaystyle {\\mathop {\\hbox{Res}}\\limits \\limits _{z=0}{\\mathcal {L}}^i(z)=S^i=\\sum \\limits _{k,j=1}^N E_{kj}S_{kj}^i\\in {\\rm Mat}(N,\\mbox{C})}\\end{array}$ (it is the only pole in the fundamental domain).",
"Here $E_{kj}$ is the standard matrix basis in ${\\rm Mat}(N,\\mbox{C})$ (matrix units) and $S^i_{kj}$ are the classical Sklyanin's generators at $i$ th site.",
"The monodromy properties are as follows: $\\begin{array}{c}\\displaystyle {{\\mathcal {L}}^i(z+1)= Q_1^{-1}{\\mathcal {L}}^{i}(z)Q_1\\,,\\quad \\quad {\\mathcal {L}}^{i}(z+\\tau )=\\exp (-2\\pi \\imath \\eta )Q_2^{-1} {\\mathcal {L}}^{i}(z)Q_2\\,,}\\end{array}$ where $Q_{1,2}\\in {\\rm Mat}(N,\\mbox{C})$ are finite-dimensional representations for generators of the Heisenberg group given by (REF ).",
"More explicitly, $\\begin{array}{c}\\displaystyle {{\\mathcal {L}}^{i}(z)=\\sum \\limits _{a\\in \\,\\mbox{Z}}_{ N}\\times \\mbox{Z}}_{ N}\\end{array} S^i_a T_a \\varphi _a(z,\\omega _a+\\eta )\\,,$ where $T_a$ is the special basis (REF ) in ${\\rm Mat}(N,\\mbox{C})$ constructed by means of the matrices $Q_1$ , $Q_2$ .",
"Similarly to (REF ), we can write the Lax matrices (REF ) in the compact form: $\\begin{array}{c}\\displaystyle {{\\mathcal {L}}^i(z)={\\rm tr}_2(R_{12}^\\eta (z)S_2^i)\\,,\\qquad S_2^i=1_{N\\times N}\\otimes S^i\\in {\\rm Mat}(N,\\mbox{C})^{\\otimes 2}\\,.",
"}\\end{array}$ Consider the lattice version of the generalized Landau-Lifshitz model, i.e.",
"the classical elliptic spin chain [4].",
"It is defined by the monodromy matrix $T(z)$ (REF ) with the Lax matrices ${\\mathcal {L}}^i(z)$ (REF ) or (REF ).",
"The Poisson structure is given by $n$ copies of (REF ): $\\begin{array}{c}\\displaystyle {\\lbrace {\\mathcal {L}}_1^{i}(z),{\\mathcal {L}}_2^{j}(w)\\rbrace =\\frac{1}{c}\\,\\delta ^{ij}[{\\mathcal {L}}_1^{i}(z){\\mathcal {L}}_2^{i}(w),r_{12}(z-w)]\\,.",
"}\\end{array}$ In order to have a local Hamiltonian (when only neighbouring sites interact), the residues $S^i$ $\\begin{array}{c}\\displaystyle {S^i=\\mathop {\\hbox{Res}}\\limits \\limits _{z=0}{\\mathcal {L}}^i(z)\\,,\\quad i=1\\,,\\ldots \\,,n}\\end{array}$ should be rank one matrices: $\\begin{array}{c}\\displaystyle {S^i=\\xi ^i\\otimes \\psi ^i\\,,}\\end{array}$ where $\\xi ^i\\in \\mbox{C}^N$ are column-vectors and $\\psi ^i\\in \\mbox{C}^N$ are row-vectors.",
"Then the local Hamiltonian is defined as follows.",
"Let us compute the coefficient of $t(z)$ (REF ) in front of $1/z^n$ .",
"It equals $\\begin{array}{c}\\displaystyle {\\exp (H/c)=\\mathop {\\hbox{Res}}\\limits \\limits _{z=0}z^{n-1}t(z)={\\rm tr}(S^1S^2...S^n).",
"}\\end{array}$ Plugging (REF ) into (REF ) and taking its logarithm, we get $\\begin{array}{c}\\displaystyle {H=c\\log {\\rm tr}(S^1S^2...S^n)=c\\sum \\limits _{k=1}^n \\log h_{k,k+1}\\,,}\\end{array}$ $\\begin{array}{c}\\displaystyle {h_{k,k+1}=(\\psi ^k,\\xi ^{k+1})=\\sum \\limits _{l=1}^N \\psi ^k_l\\xi ^{k+1}_l\\,,}\\end{array}$ where $\\xi ^{n+1}=\\xi ^{1}$ and the notation $(\\psi ^k,\\xi ^{k+1})$ means the standard scalar product.",
"To get equations of motion, consider $\\begin{array}{c}\\displaystyle {{\\rm tr}_2\\lbrace {\\mathcal {L}}_1^k(z),T_2(w)\\rbrace \\stackrel{(\\ref {b22})}{=}-{\\mathcal {L}}^k(z) M^k(z,w)+M^{k-1}(z,w){\\mathcal {L}}^k(z)\\,,}\\end{array}$ where $\\begin{array}{c}\\displaystyle {M^k(z,w)=-\\frac{1}{c}\\,{\\rm tr}_2\\Big ( {\\mathcal {L}}^1_2(w)\\,...\\,{\\mathcal {L}}_2^k(w)r_{12}(z-w){\\mathcal {L}}_2^{k+1}(w)\\,...\\,{\\mathcal {L}}_2^n(w) \\Big )}\\end{array}$ By taking the coefficient of the $n$ th order pole at $w=0$ in (REF )–(REF ) and dividing both parts of (REF ) by $\\exp (H/c)$ (REF ), we see that the Lax matrices ${\\mathcal {L}}^k(z)$ satisfy a set of the semi-discrete Zakharov-Shabat equationsThe Lax equation holds for the monodromy matrix $T(z)$ .",
"From (REF ) it follows that ${\\dot{T}}(z)=[T(z),M_n(z)]$ .",
"$\\begin{array}{c}\\displaystyle {{\\dot{{\\mathcal {L}}}}^k(z)=\\lbrace H,{\\mathcal {L}}^k(z)\\rbrace ={\\mathcal {L}}^k(z)M^k(z)-M^{k-1}(z){\\mathcal {L}}^k(z)\\,,}\\end{array}$ where $\\begin{array}{c}\\displaystyle {M^k(z)=-{\\rm tr}_2\\Big ( r_{12}(z) {\\mathcal {S}}^{k+1,k}_2\\Big )\\,,\\quad \\mathop {\\hbox{Res}}\\limits \\limits _{z=0}M^k(z)=-{\\mathcal {S}}^{k+1,k}\\,,}\\end{array}$ and $\\begin{array}{c}\\displaystyle {\\quad {\\mathcal {S}}^{k+1,k}=\\frac{\\xi ^{k+1}\\otimes \\psi ^{k}}{h_{k,k+1}}\\,.",
"}\\end{array}$ The second order pole at $z=0$ in the r.h.s.",
"of (REF ) is cancelled out since $S^k{\\mathcal {S}}^{k+1,k}=S^k$ , i.e.",
"$\\begin{array}{c}\\displaystyle {\\frac{1}{h_{k,k+1}} (\\xi ^k\\otimes \\psi ^{k})(\\xi ^{k+1}\\otimes \\psi ^{k})-\\frac{1}{h_{k-1,k}} (\\xi ^{k}\\otimes \\psi ^{k-1}) (\\xi ^k\\otimes \\psi ^{k})=0\\,.",
"}\\end{array}$ Similarly to (REF ) we have the following explicit expression for $M^k(z)$ : $\\begin{array}{c}\\displaystyle {M^k(z)=-{\\mathcal {S}}^{k+1,k}_0 1_N E_1(z)-\\sum \\limits _{\\alpha \\ne 0}T_\\alpha {\\mathcal {S}}^{k+1,k}_\\alpha \\varphi _\\alpha (z,\\omega _\\alpha )\\,.",
"}\\end{array}$ The equations of motion are of the form $\\begin{array}{c}\\displaystyle {\\dot{S}^k=S^kJ^\\eta ({\\mathcal {S}}^{k+1,k})-{\\mathcal {S}}^{k+1,k}J^\\eta (S^k)\\,,}\\end{array}$ where $J^\\eta $ is the linear map (REF )."
],
[
"The Ruijsenaars-Schneider chain",
"This section is organized as follows.",
"First, we define the lattice field analogue of the Ruijsenaars-Schneider model (the Ruijsenaars spin chain) and find its Lax matrix.",
"In subsection 4.2 the Hamiltonian and equations of motion are derived similarly to those for the elliptic spin chain described in the previous section.",
"In subsection 4.3, using a set of IRF-Vertex type relations, we compute the $M$ -matrices entering the semi-discrete zero curvature (Zakharov-Shabat) equations.",
"Finally, we explain how the obtained Lax pair can be modified in order to have a form similar to the ordinary Ruijsenaars-Schneider model."
],
[
"Classical $L$ -matrix",
"Let us parameterize all the $L$ -matrices of the elliptic spin chain in (REF ) by $n$ sets of canonical variables $p^{k}_i,q^{k}_j$ , $i,j=1,\\ldots ,N$ , $k=1,\\ldots ,n$ $\\begin{array}{l}\\displaystyle {\\lbrace p^k_i,q_j^l\\rbrace =\\delta ^{kl}\\delta _{ij}}\\end{array}$ as in (REF ), so that $\\begin{array}{l}\\displaystyle {{\\mathcal {L}}^k(z)=\\frac{\\vartheta ^{\\prime }(0)}{\\vartheta (\\eta )}\\,g(z+N\\eta ,q^k)\\,e^{P^k/c}g^{-1}(z,q^k)\\,,\\quad P^k={\\rm diag}(p^k_1,\\ldots ,p^k_N)}\\end{array}$ and $\\begin{array}{l}\\displaystyle {S^k=S^k(p^k,q^k)=\\xi ^k\\otimes \\psi ^k\\,,\\ \\ k=1,\\ldots ,n}\\end{array}$ with $\\begin{array}{l}\\displaystyle {\\xi ^k=\\xi ^k(p^k,q^k)=\\frac{\\vartheta ^{\\prime }(0)}{\\vartheta (\\eta )}\\,g(N\\eta ,q^k)\\,e^{P^k/c}\\,\\rho \\,,\\qquad \\psi ^k=\\psi ^k(q^k)=\\frac{1}{N}\\,\\rho ^T{\\breve{g}}(0,q^k)\\,.",
"}\\end{array}$ Plugging (REF ) into (REF ), we get $\\begin{array}{l}\\displaystyle {T(z)=\\Big (\\frac{\\vartheta ^{\\prime }(0)}{\\vartheta (\\eta )}\\Big )^ng(z+N\\eta ,q^1)\\,e^{P^1/c}g^{-1}(z,q^1)g(z+N\\eta ,q^2)\\,e^{P^2/c}g^{-1}(z,q^2)\\ldots \\,,}\\end{array}$ and, therefore, $t(z)$ (REF ) can be equivalently rewritten in the form $\\begin{array}{l}\\displaystyle {t(z)={\\rm tr}\\Big ( {\\tilde{L}}^1(z) {\\tilde{L}}^2(z)\\ldots {\\tilde{L}}^n(z) \\Big )}\\end{array}$ (by identifying $q^0=q^n$ ), where $\\begin{array}{l}\\displaystyle {{\\tilde{L}}^k(z)=g^{-1}(z,q^{k-1}){L}^k(z)g(z,q^{k})}\\end{array}$ or (from (REF )) $\\begin{array}{l}\\displaystyle {{\\tilde{L}}^k(z)=\\frac{\\vartheta ^{\\prime }(0)}{\\vartheta (\\eta )}\\,g^{-1}(z,q^{k-1})g(z+N\\eta ,q^k)\\,e^{P^k/c}\\,.",
"}\\end{array}$ To obtain explicit an expression for ${\\tilde{L}}^k(z)$ , we need to compute the matrix $g^{-1}(z,q^{k-1})g(z+N\\eta ,q^k)$ : $\\begin{array}{l}\\displaystyle {g^{-1}(z,q^{k-1})g(z+N\\eta ,q^k)\\stackrel{(\\ref {a21})}{=}d^{0}(q^{k-1})\\Xi ^{-1}(z,{\\bar{q}}^{k-1})\\Xi (z+N\\eta ,{\\bar{q}}^k)\\Big (d^0(q^k)\\Big )^{-1}\\,,}\\end{array}$ where we have introduced the notation $\\begin{array}{l}\\displaystyle {{\\bar{q}}^k_i=q^k_i-\\frac{1}{N}\\sum \\limits _{j=1}^N q^k_j\\,,}\\end{array}$ i.e.",
"each $g$ -matrix depends on the coordinates in the center of masses frame.",
"It is necessary for the following reason.",
"The Lax matrix should have a pole at some fixed point ($z=0$ ), and the latter comes from (REF ).",
"Coming back to the calculation (REF ), we use the following formula proved in [16]: $\\begin{array}{c}\\displaystyle {\\Big ( -\\vartheta ^{\\prime }(0)\\, \\Xi ^{-1}(z,{\\bar{q}}^{k-1})\\Xi (z+N\\eta ,{\\bar{q}}^k)\\Big )_{ij}=\\phi (z,{\\bar{q}}^{k-1}_i-{\\bar{q}}^{k}_j+\\eta )\\frac{\\prod \\limits _{l=1}^N\\vartheta ({\\bar{q}}^k_j-{\\bar{q}}^{k-1}_l-\\eta ) }{\\prod \\limits _{l: l\\ne i}\\vartheta ({\\bar{q}}^{k-1}_i-{\\bar{q}}^{k-1}_l) }\\,.",
"}\\end{array}$ Plugging also the matrices $d^0$ (REF ) into (REF ), we get $\\begin{array}{c}\\displaystyle {\\Big ( -\\vartheta ^{\\prime }(0)\\, g^{-1}(z,{\\bar{q}}^{k-1})g(z+N\\eta ,{\\bar{q}}^k)\\Big )_{ij}=\\phi (z,{\\bar{q}}^{k-1}_i-{\\bar{q}}^{k}_j+\\eta )\\frac{\\prod \\limits _{l=1}^N\\vartheta ({\\bar{q}}^k_j-{\\bar{q}}^{k-1}_l-\\eta ) }{\\prod \\limits _{l: l\\ne j}\\vartheta ({\\bar{q}}^{k}_j-{\\bar{q}}^{k}_l) }\\,.",
"}\\end{array}$ Note that under the identification ${\\bar{q}}^{k-1}:={\\bar{q}}^{k}$ the upper product in the r.h.s.",
"acquires the factor $\\vartheta (-\\eta )$ .",
"Dividing by it the both sides, we reproduce the Lax matrix of the Ruijsenaars-Schneider model (REF )–(REF ) or (REF ).",
"Finally, for the $L$ -matrices (REF ) entering the transfer matrix (REF ) we have: $\\begin{array}{c}\\displaystyle {{\\tilde{L}}^k_{ij}(z)=\\phi (z,{\\bar{q}}^{k-1}_i-{\\bar{q}}^{k}_j+\\eta )\\frac{\\prod \\limits _{l=1}^N\\vartheta ({\\bar{q}}^k_j-{\\bar{q}}^{k-1}_l-\\eta ) }{\\vartheta (-\\eta )\\prod \\limits _{l: l\\ne j}\\vartheta ({\\bar{q}}^{k}_j-{\\bar{q}}^{k}_l) }\\,e^{p^k_j/c}\\,.",
"}\\end{array}$"
],
[
"The Hamiltonian.",
"The Hamiltonian can be obtained from $t(z)$ (REF ) in the same way as in the spin chain case (see (REF )–(REF )).",
"For this purpose compute the residue of ${\\tilde{L}}^k_{ij}(z)$ : $\\begin{array}{c}\\displaystyle {\\mathop {\\hbox{Res}}\\limits \\limits _{z=0}{\\tilde{L}}^k(z)=\\rho ^T\\otimes b^k\\,,}\\end{array}$ where $\\rho $ is taken from (REF ) and $b^k$ is a row-vector, so that $\\begin{array}{c}\\displaystyle {\\mathop {\\hbox{Res}}\\limits \\limits _{z=0}{\\tilde{L}}^k_{ij}(z)=b_j^k\\,,\\quad b_j^k=\\frac{\\prod \\limits _{l=1}^N\\vartheta ({\\bar{q}}^k_j-{\\bar{q}}^{k-1}_l-\\eta ) }{\\vartheta (-\\eta )\\prod \\limits _{l: l\\ne j}\\vartheta ({\\bar{q}}^{k}_j-{\\bar{q}}^{k}_l) }\\,e^{p^k_j/c}\\,.",
"}\\end{array}$ Then $\\begin{array}{c}\\displaystyle {\\exp (H/c)=\\mathop {\\hbox{Res}}\\limits \\limits _{z=0}z^{n-1}t(z)={\\rm tr}\\Big ( (\\rho ^T\\otimes b^1)(\\rho ^T\\otimes b^2)\\ldots (\\rho ^T\\otimes b^n) \\Big )\\,.",
"}\\end{array}$ Finally, the Hamiltonian is of the form $\\begin{array}{c}\\displaystyle {H=c\\sum \\limits _{k=1}^n \\log h_{k,k+1}\\,,\\quad h_{k,k+1}=(\\rho ^T,b^{k+1})}\\end{array}$ and $\\begin{array}{c}\\displaystyle {h_{k-1,k}=(\\rho ^T,b^{k})=\\sum \\limits _{j=1}^N b^k_j=\\sum \\limits _{j=1}^N\\frac{\\prod \\limits _{l=1}^N\\vartheta ({\\bar{q}}^k_j-{\\bar{q}}^{k-1}_l-\\eta ) }{\\vartheta (-\\eta )\\prod \\limits _{l: l\\ne j}\\vartheta ({\\bar{q}}^{k}_j-{\\bar{q}}^{k}_l) }\\,e^{p^k_j/c}\\,.",
"}\\end{array}$ By construction, the trace $t(z)$ (REF ) coincides with the one for the elliptic spin chain (REF ) under the substitution (REF )–(REF ).",
"To see this, we mention that the terms $h_{k,k+1}$ entering (REF )–(REF ) and those from (REF )–(REF ) are equal to each other: $\\begin{array}{c}\\displaystyle {h_{k-1,k}=(\\rho ^T,b^{k})=(\\psi ^{k-1},\\xi ^k)}\\end{array}$ for $\\xi ^k$ and $\\psi ^k$ defined in (REF ).",
"In order to verify (REF ), one should compare the trace of the residue of ${\\tilde{L}}^k(z)$ computed from (REF ) and (REF )."
],
[
"The\nHamiltonian equations of motion.",
"Let us proceed to the equations of motion.",
"From (REF )–(REF ) we have $\\begin{array}{c}\\displaystyle {{\\dot{q}}_i^k=\\frac{\\partial H}{\\partial p_i^k}=\\frac{b_i^k}{h_{k-1,k}}.",
"}\\end{array}$ The latter yields $\\begin{array}{c}\\displaystyle {\\sum \\limits _{i=1}^N{\\dot{q}}_i^k=1\\quad \\hbox{for}\\ \\hbox{all}\\ k\\,.",
"}\\end{array}$ With (REF ) the Lax matrix (REF ) takes the form: $\\begin{array}{c}\\displaystyle {{\\tilde{L}}^k_{ij}(z)=\\phi (z,{\\bar{q}}^{k-1}_i-{\\bar{q}}^{k}_j+\\eta )b_j^k\\,,\\quad b_j^k=h_{k-1,k}\\,{\\dot{q}}_j^k\\,.",
"}\\end{array}$ Next, $\\begin{array}{c}\\displaystyle {\\frac{1}{c}\\, {\\dot{p}}_i^k=-\\frac{1}{c}\\, \\frac{\\partial H}{\\partial q_i^k}=-\\frac{1}{h_{k-1,k}}\\,\\frac{\\partial }{\\partial q_i^k }\\,h_{k-1,k}-\\frac{1}{h_{k,k+1}}\\,\\frac{\\partial }{\\partial q_i^k }\\,h_{k,k+1}.",
"}\\end{array}$ Its r.h.s is evaluated from the explicit expression (REF ): $\\begin{array}{c}\\displaystyle {\\frac{1}{h_{k-1,k}}\\,\\frac{\\partial }{\\partial q_i^k }\\,h_{k-1,k}={\\dot{q}}_i^k\\sum \\limits _{l=1}^N E_1({\\bar{q}}^{k}_i-{\\bar{q}}^{k-1}_l-\\eta )-{\\dot{q}}_i^k\\sum \\limits _{l\\ne i}E_1(q_i^k-q_l^k)+\\sum \\limits _{l\\ne i} {\\dot{q}}_l^kE_1(q_l^k-q_i^k)}\\\\ \\ \\\\\\displaystyle {-\\frac{1}{N}\\sum \\limits _{l=1}^N {\\dot{q}}_l^k \\sum \\limits _{m=1}^NE_1({\\bar{q}}^{k}_l-{\\bar{q}}^{k-1}_m-\\eta )\\,,}\\end{array}$ $\\begin{array}{c}\\displaystyle {\\frac{1}{h_{k,k+1}}\\,\\frac{\\partial }{\\partial q_i^k }\\,h_{k,k+1}=-\\sum \\limits _{l=1}^N {\\dot{q}}_l^{k+1}E_1({\\bar{q}}_l^{k+1}-{\\bar{q}}_i^k-\\eta )+\\frac{1}{N}\\sum \\limits _{l=1}^N {\\dot{q}}_l^{k+1}\\sum \\limits _{m=1}^N E_1({\\bar{q}}_l^{k+1}-{\\bar{q}}_m^k-\\eta )\\,,}\\end{array}$ where the last terms (double sums) come from dependence on the center of masses coordinates (REF ).",
"Summing up (REF ) and (REF ), we get the following equation for momenta (REF ): $\\begin{array}{c}\\displaystyle {\\frac{1}{c}\\, {\\dot{p}}_i^k=-{\\dot{q}}_i^k\\sum \\limits _{l=1}^N E_1({\\bar{q}}^{k}_i-{\\bar{q}}^{k-1}_l-\\eta )-\\sum \\limits _{l=1}^N {\\dot{q}}_l^{k+1}E_1({\\bar{q}}_i^k-{\\bar{q}}_l^{k+1}+\\eta )+\\sum \\limits _{l\\ne i}({\\dot{q}}_i^k+{\\dot{q}}_l^k)E_1(q_i^k-q_l^k)}\\\\\\displaystyle {+\\frac{1}{N}\\sum \\limits _{l=1}^N {\\dot{q}}_l^k \\sum \\limits _{m=1}^NE_1({\\bar{q}}^{k}_l-{\\bar{q}}^{k-1}_m-\\eta )-\\frac{1}{N}\\sum \\limits _{l=1}^N {\\dot{q}}_l^{k+1}\\sum \\limits _{m=1}^NE_1({\\bar{q}}_l^{k+1}-{\\bar{q}}_m^k-\\eta )\\,.",
"}\\end{array}$ The second line of this equation is independent of the index $i$ .",
"It has appeared from the dependence of ${\\bar{q}}_l^k$ on the center of masses coordinates $\\sum _l q_l^k$ at each ($k$ th) site."
],
[
"The Newtonian form.",
"Let us represent the Hamiltonian equations of motion in the Newtonian form.",
"By differentiating both parts of (REF ) with respect to the time $t$ , we get $\\begin{array}{c}\\displaystyle {{\\ddot{q}}^k_i=\\frac{ {\\dot{b}}^k_i }{h_{k-1,k}}-\\frac{ {\\dot{h}}_{k-1,k} }{h_{k-1,k}}\\,{\\dot{q}}^k_i={\\dot{q}}^k_i\\Big ( \\frac{ {\\dot{b}}^k_i }{b^k_i}-\\frac{ {\\dot{h}}_{k-1,k} }{h_{k-1,k}} \\Big )\\,,}\\end{array}$ where $\\begin{array}{c}\\displaystyle {\\partial _t\\log h_{k-1,k}=\\frac{ {\\dot{h}}_{k-1,k} }{h_{k-1,k}}=\\frac{ 1 }{h_{k-1,k}}\\sum \\limits _{l=1}^N {\\dot{b}}^k_l=\\sum \\limits _{l=1}^N {\\dot{q}}^k_l\\,\\frac{{\\dot{b}}^k_l }{b^k_l}\\,.",
"}\\end{array}$ We see that we need to compute ${{\\dot{b}}^k_i }/{b^k_i}$ .",
"From the definition of $b^k_i$ (REF ) we have $\\begin{array}{c}\\displaystyle {\\frac{{\\dot{b}}^k_i }{b^k_i}=\\sum \\limits _{l=1}^N ({\\dot{\\bar{q}}}_i^{k}-{\\dot{\\bar{q}}}_l^{k-1})E_1({\\bar{q}}_i^k-{\\bar{q}}_l^{k-1}-\\eta )-\\sum \\limits _{l\\ne i}^N ({\\dot{ q}}_i^{k}-{\\dot{q}}_l^{k})E_1({q}_i^k-{q}_l^{k})+\\frac{1}{c}\\, {\\dot{p}}_i^k\\,.",
"}\\end{array}$ Note that we can remove “bar” from velocities ${\\dot{\\bar{q}}}$ in the first sum since $\\begin{array}{c}\\displaystyle {{\\dot{\\bar{q}}}^k_i-{\\dot{\\bar{q}}}^m_j={\\dot{ q}}^k_i-{\\dot{ q}}^m_j}\\end{array}$ for any values of indices due to (REF ).",
"Using (REF ) and plugging (REF ) into (REF ), we get $\\begin{array}{c}\\displaystyle {\\frac{{\\dot{b}}^k_i }{b^k_i}=-\\sum \\limits _{l=1}^N {\\dot{ q}}_l^{k+1}E_1({\\bar{q}}_i^k-{\\bar{q}}_l^{k+1}+\\eta )-\\sum \\limits _{l=1}^N {\\dot{ q}}_l^{k-1}E_1({\\bar{q}}_i^k-{\\bar{q}}_l^{k-1}-\\eta )+2\\sum \\limits _{l\\ne i}^N {\\dot{q}}_l^{k}E_1({q}_i^k-{q}_l^{k})+}\\\\\\displaystyle {+\\frac{1}{N}\\sum \\limits _{l=1}^N {\\dot{q}}_l^k \\sum \\limits _{m=1}^NE_1({\\bar{q}}^{k}_l-{\\bar{q}}^{k-1}_m-\\eta )-\\frac{1}{N}\\sum \\limits _{l=1}^N {\\dot{q}}_l^{k+1}\\sum \\limits _{m=1}^NE_1({\\bar{q}}_l^{k+1}-{\\bar{q}}_m^k-\\eta )\\,.",
"}\\end{array}$ Therefore, from (REF ) we obtain the following result: $\\begin{array}{c}\\displaystyle {\\frac{{\\ddot{q}}^k_i }{ {\\dot{q}}^k_i }=-\\sum \\limits _{l=1}^N {\\dot{ q}}_l^{k+1}E_1({\\bar{q}}_i^k-{\\bar{q}}_l^{k+1}+\\eta )-\\sum \\limits _{l=1}^N {\\dot{ q}}_l^{k-1}E_1({\\bar{q}}_i^k-{\\bar{q}}_l^{k-1}-\\eta )+2\\sum \\limits _{l\\ne i}^N {\\dot{q}}_l^{k}E_1({q}_i^k-{q}_l^{k})+}\\\\\\displaystyle {+\\frac{1}{N}\\sum \\limits _{l=1}^N {\\dot{q}}_l^k \\sum \\limits _{m=1}^NE_1({\\bar{q}}^{k}_l-{\\bar{q}}^{k-1}_m-\\eta )-\\frac{1}{N}\\sum \\limits _{l=1}^N {\\dot{q}}_l^{k+1}\\sum \\limits _{m=1}^NE_1({\\bar{q}}_l^{k+1}-{\\bar{q}}_m^k-\\eta )-\\partial _t\\log h_{k-1,k}\\,.",
"}\\end{array}$ The last term $\\partial _t\\log h_{k-1,k}$ can be found using (REF ) and (REF ).",
"We have: $\\begin{array}{c}\\displaystyle {\\partial _t\\log h_{k-1,k}=-\\sum \\limits _{m,l=1}^N {\\dot{ q}}_m^{k}{\\dot{ q}}_l^{k+1}E_1({\\bar{q}}_m^k-{\\bar{q}}_l^{k+1}+\\eta )+\\sum \\limits _{m,l=1}^N {\\dot{ q}}_l^{k}{\\dot{ q}}_m^{k-1}E_1({\\bar{q}}_m^{k-1}-{\\bar{q}}_l^k+\\eta )}\\\\\\displaystyle {+\\frac{1}{N}\\sum \\limits _{l=1}^N {\\dot{q}}_l^k \\sum \\limits _{m=1}^NE_1({\\bar{q}}^{k}_l-{\\bar{q}}^{k-1}_m-\\eta )-\\frac{1}{N}\\sum \\limits _{l=1}^N {\\dot{q}}_l^{k+1}\\sum \\limits _{m=1}^NE_1({\\bar{q}}_l^{k+1}-{\\bar{q}}_m^k-\\eta )\\,,}\\end{array}$ where for the last line we also used (REF ).",
"Note also that the latter expression can be represented in the form $\\begin{array}{c}\\displaystyle {\\partial _t\\log h_{k-1,k}={\\tilde{c}}^{k-1}-{\\tilde{c}}^{k}\\,,}\\end{array}$ where $\\begin{array}{c}\\displaystyle {{\\tilde{c}}^{k-1}=\\sum \\limits _{m,l=1}^N {\\dot{ q}}_l^{k}{\\dot{ q}}_m^{k-1}E_1({\\bar{q}}_m^{k-1}-{\\bar{q}}_l^k+\\eta )+\\frac{1}{N}\\sum \\limits _{l=1}^N {\\dot{q}}_l^k \\sum \\limits _{m=1}^NE_1({\\bar{q}}^{k}_l-{\\bar{q}}^{k-1}_m-\\eta )\\,.",
"}\\end{array}$ Finally, we obtain the equations of motion by plugging (REF ) into (REF ): $\\begin{array}{c}\\displaystyle {\\frac{{\\ddot{q}}^k_i }{ {\\dot{q}}^k_i }=-\\sum \\limits _{l=1}^N {\\dot{ q}}_l^{k+1}E_1({\\bar{q}}_i^k-{\\bar{q}}_l^{k+1}+\\eta )-\\sum \\limits _{l=1}^N {\\dot{ q}}_l^{k-1}E_1({\\bar{q}}_i^k-{\\bar{q}}_l^{k-1}-\\eta )+2\\sum \\limits _{l\\ne i}^N {\\dot{q}}_l^{k}E_1({q}_i^k-{q}_l^{k})+}\\\\\\displaystyle {+\\sum \\limits _{m,l=1}^N {\\dot{ q}}_m^{k}{\\dot{ q}}_l^{k+1}E_1({\\bar{q}}_m^k-{\\bar{q}}_l^{k+1}+\\eta )-\\sum \\limits _{m,l=1}^N {\\dot{ q}}_l^{k}{\\dot{ q}}_m^{k-1}E_1({\\bar{q}}_m^{k-1}-{\\bar{q}}_l^k+\\eta )\\,.",
"}\\end{array}$"
],
[
"Semi-discrete Zakharov-Shabat equation",
"As is seen above, the Ruijsenaars-Schneider chain is the gauge transformed elliptic spin chain together with the change of variables (REF )–(REF ).",
"With this identification, the Hamiltonians (REF ) and (REF ) coincide.",
"From to the relation (REF ) between the Lax matrices and the semi-discrete Zakharov-Shabat equation (REF ) we conclude that we also have the semi-discrete Zakharov-Shabat representation for the Ruijsenaars-Schneider chain $\\begin{array}{c}\\displaystyle {\\frac{d}{dt}\\,{{\\tilde{L}}^k}(z)=\\lbrace H,{\\tilde{L}}^k(z)\\rbrace ={\\tilde{L}}^k(z){\\tilde{M}}^k(z)-{\\tilde{M}}^{k-1}(z){\\tilde{L}}^k(z)\\,,}\\end{array}$ with the Lax matrices (REF ) and the $M$ -matrices ${\\tilde{M}}^k(z)$ : $\\begin{array}{c}\\displaystyle {{\\tilde{M}}^k(z)=g^{-1}(z,q^k) M^k(z) g(z,q^k) +g^{-1}(z,q^k) {\\dot{g}}(z,q^k)\\,.",
"}\\end{array}$ Here $M^k(z)$ is given by (REF ), and the variables $\\xi ^k,\\psi ^k$ in (REF ) are taken from (REF ).",
"The aim of this subsection is to obtain explicit expression for ${\\tilde{M}}^k(z)$ using (REF ).",
"We follow the strategy used in [22] to reproduce the Ruijsenaars-Schneider $M$ -matrix (REF ) from the IRF-Vertex relations.",
"First, let us express $M^k(z)$ (REF )–(REF ) through the canonical variables $\\begin{array}{c}\\displaystyle {M^k(z)=-\\frac{1}{h_{k,k+1}}{\\rm tr}_2\\Big ( \\Big ( \\xi ^{k+1}(p^{k+1},q^{k+1})\\otimes \\psi ^{k}(q^k) \\Big )_2 r_{12}(z)\\Big )\\,.",
"}\\end{array}$ Plugging $\\xi ^{k+1}$ and $\\psi ^{k}$ from (REF ), we have $\\begin{array}{c}\\displaystyle {M^k(z)=-\\frac{1}{h_{k,k+1}}\\frac{\\vartheta ^{\\prime }(0)}{\\vartheta (\\eta )}\\,\\frac{1}{N}\\,{\\rm tr}_2\\Big ( (\\rho \\otimes \\rho ^T)_2\\, {\\breve{g}}_2(0,q^k)\\,r_{12}(z)\\,g_2(N\\eta ,q^{k+1})\\,e^{P_2^{k+1}/c} \\Big )\\,.",
"}\\end{array}$ Next, we substitute ${\\breve{g}}(0,q^k)\\,r_{12}(z)$ from (REF ), where all matrices in the r.h.s.",
"depend on $q^k$ .",
"Using $(\\rho \\otimes \\rho ^T){\\mathcal {O}}_{12}=N{\\mathcal {O}}_{12}$ , we obtain: $\\begin{array}{c}\\displaystyle {M^k(z)=-\\frac{1}{h_{k,k+1}}\\frac{\\vartheta ^{\\prime }(0)}{\\vartheta (\\eta )}\\,{\\rm tr}_2\\Big ( \\Big (g_1^{\\prime }(z,q^k)\\, \\mathcal {O}_{12}\\, {\\breve{g}}_2(0,q^k)\\, g_1^{-1}(z,q^k)}\\\\ \\ \\\\\\displaystyle {+g_1(z,q^k)\\, \\mathcal {O}_{12}\\,A_2(q^k)\\, g_1^{-1}(z,q^k)\\Big )g_2(N\\eta ,q^{k+1})\\,e^{P_2^{k+1}/c} \\Big )\\,.",
"}\\end{array}$ Then the gauged transformed $M$ -matrix (REF ) takes the form $\\begin{array}{c}\\displaystyle {{\\tilde{M}}^k(z)=-g^{-1}(z)g^{\\prime }(z)G-F+g^{-1}(z,q^k) {\\dot{g}}(z,q^k)\\,,}\\end{array}$ where $\\begin{array}{c}\\displaystyle {G=\\frac{1}{h_{k,k+1}}\\, {\\rm tr}_2\\left({\\mathcal {O}}_{12}\\frac{\\vartheta ^{\\prime }(0)}{\\vartheta (\\eta )}\\,{\\breve{g}}_2(0,q^k)g_2(N\\eta ,q^{k+1})\\,e^{P^{k+1}_2/c}\\right)}\\end{array}$ and $\\begin{array}{c}\\displaystyle {F=\\frac{1}{h_{k,k+1}}\\, {\\rm tr}_2\\left({\\mathcal {O}}_{12}\\frac{\\vartheta ^{\\prime }(0)}{\\vartheta (\\eta )}\\,A_2(q^k)\\,g_2(N\\eta ,q^{k+1})\\,e^{P_2^{k+1}/c}\\right).",
"}\\end{array}$ Let us compute the matrices $F$ and $G$ .",
"Consider the residue of ${\\tilde{L}}^{k+1}(z)$ at $z=0$ .",
"On the one hand it comes from (REF ), and on the other hand it can be found from (REF ): $\\begin{array}{c}\\displaystyle {{\\tilde{L}}^{k+1}[-1]=\\mathop {\\hbox{Res}}\\limits \\limits _{z=0}{\\tilde{L}}^{k+1}(z)=\\frac{\\vartheta ^{\\prime }(0)}{\\vartheta (\\eta )}\\,{\\breve{g}}(0,q^{k})g(N\\eta ,q^{k+1})\\,e^{P^{k+1}/c}=\\rho \\otimes b^{k+1}\\,.",
"}\\end{array}$ Plugging it into (REF ) and taking also into account (REF ), we see that $G$ is the identity matrix: $\\begin{array}{c}\\displaystyle {G=1_N\\,.",
"}\\end{array}$ In order to compute the matrix $F$ , consider the $z^0$ -term in the expansion of ${\\tilde{L}}^{k+1}(z)$ near $z=0$ .",
"Using the factorized form (REF ) and the expansion (REF ), we obtain: $\\begin{array}{c}\\displaystyle {{\\tilde{L}}^{k+1}[0]=\\frac{\\vartheta ^{\\prime }(0)}{\\vartheta (\\eta )}\\,{\\breve{g}}(0,q^{k})g^{\\prime }(N\\eta ,q^{k+1})\\,e^{P^{k+1}/c}+\\frac{\\vartheta ^{\\prime }(0)}{\\vartheta (\\eta )}\\,A(q^k)\\,g(N\\eta ,q^{k+1})\\,e^{P^{k+1}/c}.",
"}\\end{array}$ The first term in the r.h.s.",
"of (REF ) is obtained by differentiating both sides of (REF ) with respect to $\\eta $ : $\\begin{array}{c}\\displaystyle {\\partial _\\eta {\\tilde{L}}^{k+1}[-1]=-E_1(\\eta ){\\tilde{L}}^{k+1}[-1]+N\\frac{\\vartheta ^{\\prime }(0)}{\\vartheta (\\eta )}\\,{\\breve{g}}(0,q^{k})g^{\\prime }(N\\eta ,q^{k+1})\\,e^{P^{k+1}/c}\\,,}\\end{array}$ so that $\\begin{array}{c}\\displaystyle {\\frac{\\vartheta ^{\\prime }(0)}{\\vartheta (\\eta )}\\,A(q^k)\\,g(N\\eta ,q^{k+1})\\,e^{P^{k+1}/c}={\\tilde{L}}^{k+1}[0]-\\frac{1}{N}\\Big (\\partial _\\eta {\\tilde{L}}^{k+1}[-1]+E_1(\\eta ){\\tilde{L}}^{k+1}[-1]\\Big )\\,.",
"}\\end{array}$ Expressions ${\\tilde{L}}^{k+1}[-1]$ and ${\\tilde{L}}^{k+1}[0]$ are known explicitly from (REF ) and (REF ): $\\begin{array}{c}\\displaystyle {{\\tilde{L}}^{k+1}_{ij}[-1]=b_j^{k+1}\\,,\\quad {\\tilde{L}}^{k+1}_{ij}[0]=b_j^{k+1}E_1({\\bar{q}}^k_i-{\\bar{q}}^{k+1}_j+\\eta )\\,,}\\\\ \\ \\\\\\displaystyle {\\Big (\\partial _\\eta {\\tilde{L}}^{k+1}[-1]+E_1(\\eta ){\\tilde{L}}^{k+1}[-1]\\Big )_{ij}=-b_j^{k+1}\\sum \\limits _{l=1}^NE_1({\\bar{q}}^{k+1}_j-{\\bar{q}}^{k}_l-\\eta )\\,.",
"}\\end{array}$ Plugging all this into (REF ) and dividing both parts by $h_{k,k+1}$ , we obtain (using also the definition (REF )): $\\begin{array}{c}\\displaystyle {\\frac{1}{h_{k,k+1}}\\frac{\\vartheta ^{\\prime }(0)}{\\vartheta (\\eta )}\\Big (A(q^k)\\,g(N\\eta ,q^{k+1})\\,e^{P^{k+1}/c}\\Big )_{ij}}\\\\ \\ \\\\\\displaystyle {={\\dot{q}}_j^{k+1}E_1({\\bar{q}}^k_i-{\\bar{q}}^{k+1}_j+\\eta )+\\frac{1}{N}\\,{\\dot{q}}_j^{k+1}\\sum \\limits _{l=1}^NE_1({\\bar{q}}^{k+1}_j-{\\bar{q}}^{k}_l-\\eta )\\,.",
"}\\end{array}$ Using the property (REF ), we find the (diagonal) matrix $F$ (REF ): $\\begin{array}{c}\\displaystyle {F_{ij}=\\delta _{ij}\\sum \\limits _{m=1}^N{\\dot{q}}_m^{k+1}E_1({\\bar{q}}^k_i-{\\bar{q}}^{k+1}_m+\\eta )+\\delta _{ij}\\frac{1}{N}\\sum \\limits _{l,m=1}^N{\\dot{q}}_m^{k+1}E_1({\\bar{q}}^{k+1}_m-{\\bar{q}}^{k}_l-\\eta )\\,.",
"}\\end{array}$ To get the final answer for ${\\tilde{M}}^k(z)$ (REF ), let us simplify its last term $g^{-1}(z,q^k) {\\dot{g}}(z,q^k)$ .",
"Using its definition (REF )–(REF ), we have: $\\begin{array}{c}\\displaystyle {g^{-1}(z){\\dot{g}}(z)=g^{-1}(z)g^{\\prime }(z)\\left(-N\\,{\\rm diag}(\\dot{q})+1_{N}\\sum \\limits _k{\\dot{q}}_k\\right)-{\\dot{d}}^0(d^0)^{-1}\\,.",
"}\\end{array}$ Substitute (REF ) and (REF ) into (REF ).",
"Due to (REF ) the term $-g^{-1}g^{\\prime }(z)G$ is canceled with the one proportional to $\\sum \\limits _k{\\dot{q}}_k$ in (REF ): $\\begin{array}{c}\\displaystyle {{\\tilde{M}}^k(z)=-F-Ng^{-1}(z)g^{\\prime }(z){\\rm diag}(\\dot{q})-{\\dot{d}}^0(d^0)^{-1}\\,.",
"}\\end{array}$ The quantity $g^{-1}(z)g^{\\prime }(z)$ is known from (REF ) and $\\begin{array}{c}\\displaystyle {\\Big ({\\dot{d}}^0(d^0)^{-1}\\Big )_{ii}=\\sum \\limits _{m\\ne i}({\\dot{q}}_i^k-{\\dot{q}}_m^k)E_1(q_i^k-q_m^k)\\,.",
"}\\end{array}$ Therefore, $\\begin{array}{c}\\displaystyle {{\\tilde{M}}_{ij}^k(z)=-(1-\\delta _{ij})\\phi (z,q_i^k-q_j^k)\\,{\\dot{q}}_j^k-\\delta _{ij}E_1(z){\\dot{q}}^k_i+}\\end{array}$ $\\displaystyle {+\\delta _{ij}\\Big (\\sum \\limits _{m\\ne i}^N{\\dot{q}}_m^kE_1(q_i^k-q_m^k)-\\sum \\limits _{m=1}^N{\\dot{q}}_m^{k+1}E_1({\\bar{q}}^k_i-{\\bar{q}}^{k+1}_m+\\eta )-\\frac{1}{N}\\sum \\limits _{l,m=1}^N{\\dot{q}}_m^{k+1}E_1({\\bar{q}}^{k+1}_m-{\\bar{q}}^{k}_l-\\eta )\\Big )\\,.",
"}$ To summarize, we have proved that the semi-discrete Zakharov-Shabat equation (REF ) holds for the matrices (REF ) and (REF ) on the equations of motion of the Ruijsenaars-Schneider chain (REF ), (REF ) or (REF )–(REF ).",
"This can be also verified by direct substitution using identities (REF ) and (REF ) similarly to verification of the Lax pair for the Ruijsenaars-Schneider model (REF ), (REF )."
],
[
"Modified Lax pair.",
"All Lax matrices (REF ) can be simultaneously divided by $h_{k-1,k}$ .",
"Then the resultant Lax matrix depend on the velocities (REF ).",
"From the point of view of the ordinary Ruijsenaars-Schneider model it is similar to transition to the logarithm of Hamiltonian (REF ).",
"Consider $\\begin{array}{c}\\displaystyle {{L^{\\prime }}^k_{ij}(z)={\\tilde{L}}^k_{ij}(z)\\frac{1}{h_{k-1,k}}=\\phi (z,{\\bar{q}}^{k-1}_i-{\\bar{q}}^{k}_j+\\eta ){\\dot{q}}_j^k\\,.",
"}\\end{array}$ This can be done since the transfer matrix is divided by conserved quantity (REF ): $\\begin{array}{l}\\displaystyle {t^{\\prime }(z)={\\rm tr}\\Big ( { L^{\\prime }}^1(z) { L^{\\prime }}^2(z)...{ L^{\\prime }}^n(z) \\Big )=\\frac{t(z)}{h_{1,2}h_{2,3}...h_{n,1}}=t(z)\\exp (-H/c)\\,,}\\end{array}$ The Lax equation for ${ L^{\\prime }}^k(z)$ is of the form: $\\begin{array}{c}\\displaystyle {\\frac{d}{dt}\\,{{ L^{\\prime }}^k}(z)={L^{\\prime }}^k(z){\\tilde{M}}^k(z)-{\\tilde{M}}^{k-1}(z){ L^{\\prime }}^k(z)-{ L^{\\prime }}^k(z)\\partial _t\\log h_{k-1,k}\\,,}\\end{array}$ Using (REF ) the last term in (REF ) can be removed by redefining ${\\tilde{M}}^k(z)$ : $\\begin{array}{c}\\displaystyle {{M^{\\prime }}^k(z)={\\tilde{M}}^k(z)+{\\tilde{c}}^k 1_N\\,.",
"}\\end{array}$ Then $\\begin{array}{c}\\displaystyle {\\frac{d}{dt}\\,{{ L^{\\prime }}^k}(z)={L^{\\prime }}^k(z){M^{\\prime }}^k(z)-{ M^{\\prime }}^{k-1}(z){ L^{\\prime }}^k(z)\\,.",
"}\\end{array}$ Explicit form of the $M$ -matrix (REF ) is as follows: $\\begin{array}{c}\\displaystyle {{M^{\\prime }}^k_{ij}(z)=-(1-\\delta _{ij})\\phi (z,q_i^k-q_j^k)\\,{\\dot{q}}_j^k-\\delta _{ij}E_1(z){\\dot{q}}^k_i+}\\end{array}$ $\\displaystyle {+\\delta _{ij}\\Big (\\sum \\limits _{m\\ne i}^N{\\dot{q}}_m^kE_1(q_i^k-q_m^k)-\\sum \\limits _{m=1}^N{\\dot{q}}_m^{k+1}E_1({\\bar{q}}^k_i-{\\bar{q}}^{k+1}_m+\\eta )+\\sum \\limits _{m,l=1}^N {\\dot{ q}}_l^{k+1}{\\dot{ q}}_m^{k}E_1({\\bar{q}}_m^{k}-{\\bar{q}}_l^{k+1}+\\eta )\\Big )\\,.",
"}$ To summarize, the Lax pair (REF ) and (REF ) satisfy the semi-discrete zero curvature equation (REF ) and provide equations of motion of the Ruijsenaars-Schneider chain (REF )."
],
[
"Field analogue of the elliptic Ruijsenaars-Schneider system from elliptic families\nof solutions to the 2D Toda lattice",
"In this section we derive equations of motion for poles of general elliptic solutions (which we call elliptic families) to the 2D Toda lattice hierarchy and show that they are Hamiltonian and equivalent to (REF ) under some simple substitutions and re-definitions."
],
[
"The 2D Toda lattice hierarchy",
"Following [39], we briefly review the 2D Toda lattice hierarchy.",
"The sets of independent variables are two infinite sets of continuous time variables ${\\bf t}=\\lbrace t_1, t_2, t_3, \\ldots \\rbrace $ , $\\bar{\\bf t}=\\lbrace \\bar{t}_1, \\bar{t}_2, \\bar{t}_3, \\ldots \\rbrace $ and a discrete integer-valued variable $n$ which is sometimes denoted as $t_0$ .",
"The main objects are two pseudo-difference Lax operators ${\\bf L}=e^{\\partial _n}+\\sum _{k\\ge 0}U_{k,n} e^{-k\\partial _n}, \\quad \\bar{\\bf L}=a_n e^{-\\partial _n}+\\sum _{k\\ge 0}\\bar{U}_{k,n} e^{k \\partial _n},$ where $e^{\\partial _n}$ is the shift operator acting as $e^{\\pm \\partial _n}f(n)=f(n\\pm 1 )$ and the coefficient functions are functions of ${\\bf t}$ , $\\bar{\\bf t}$ .",
"The equations of the hierarchy are differential equations for the functions $a_n$ , $U_{k,n}$ , $\\bar{U}_{k,n}$ .",
"They are encoded in the Lax equations $\\partial _{t_m}{\\bf L}=[{\\cal B}_m, {\\bf L}], \\quad \\partial _{t_m}\\bar{\\bf L}=[{\\cal B}_m, \\bar{\\bf L}]\\qquad {\\cal B}_m=({\\bf L}^m)_{\\ge 0},$ $\\partial _{\\bar{t}_m}{\\bf L}=[\\bar{\\cal B}_m, {\\bf L}], \\quad \\partial _{\\bar{t}_m}\\bar{\\bf L}=[\\bar{\\cal B}_m, \\bar{\\bf L}]\\qquad \\bar{\\cal B}_m=(\\bar{\\bf L}^m)_{< 0},$ where $\\displaystyle {\\Bigl (\\sum _{k\\in \\mbox{Z}}} U_{k,n} e^{k \\partial _n}\\Bigr )_{\\ge 0}=\\sum _{k\\ge 0} U_{k,n} e^{k\\partial _n}$$,$ (k$\\mbox{Z}$ Uk,n ekn)< 0= k<0 Uk,n ekn$For example, $ B1=en+bn$, $ B1=ane-n$,where we have denoted $ U0,n=bn$.",
"It can be shown that the zerocurvature (Zakharov-Shabat) equations\\begin{equation}\\partial _{t_n}{\\cal B}_m -\\partial _{t_m}{\\cal B}_n +[{\\cal B}_m, {\\cal B}_n]=0,\\end{equation}\\begin{equation}\\partial _{\\bar{t}_n}{\\cal B}_m -\\partial _{t_m}\\bar{\\cal B}_n +[{\\cal B}_m, \\bar{\\cal B}_n]=0,\\end{equation}\\begin{equation}\\partial _{\\bar{t}_n}\\bar{\\cal B}_m -\\partial _{\\bar{t}_m}\\bar{\\cal B}_n +[\\bar{\\cal B}_m,\\bar{\\cal B}_n]=0\\end{equation}provide an equivalent formulation of the hierarchy.$ The 2D Toda equation is the first member of the hierarchy.",
"It is obtained from () at $m=n=1$ which is equivalent to the system of equations $\\left\\lbrace \\begin{array}{l}\\partial _{t_1}\\log a_n=b_n-b_{n-1}\\\\ \\\\\\partial _{\\bar{t}_1} b_n =a_n -a_{n+1}.\\end{array}\\right.$ Excluding $b_n$ from this system, we get the differential equation for $a_n$ : $\\partial _{t_1}\\partial _{\\bar{t}_1}\\log a_n=2a_n-a_{n+1}-a_{n-1}.$ It is one of the forms of the 2D Toda equation.",
"In terms of the function $\\varphi _n$ introduced through the relation $a_n=e^{\\varphi _n-\\varphi _{n-1}}$ it acquires the familiar form $\\partial _{t_1}\\partial _{\\bar{t}_1}\\varphi _n=e^{\\varphi _n-\\varphi _{n-1}}-e^{\\varphi _{n+1}-\\varphi _n}.$ The universal dependent variable of the hierarchy is the tau-function $\\tau _n =\\tau _n ({\\bf t}, \\bar{\\bf t})$ .",
"The change of the dependent variables from $a_n, b_n$ to the tau-function, $a_n=\\frac{\\tau _{n+1}\\tau _{n-1}}{\\tau ^2_n}, \\qquad b_n=\\partial _{t_1}\\log \\frac{\\tau _{n+1}}{\\tau _n},$ brings the 2D Toda equation to the form [40] $\\partial _{t_1}\\partial _{\\bar{t}_1}\\log \\tau _n=-\\frac{\\tau _{n+1}\\tau _{n-1}}{\\tau ^2_n}.$ At fixed $n$ and $\\bar{\\bf t}$ the 2D Toda lattice hierarchy is reduced to the Kadomtsev-Petviashvili (KP) hierarchy with the independent variables ${\\bf t}=\\lbrace t_1, t_2, t_3,\\ldots \\rbrace $ , with the KP equation (the first member of the hierarchy) being satisfied by $u_n =\\partial ^2_{t_1}\\log \\tau _n.$ An important class of solutions to the 2D Toda lattice hierarchy is the algebraic-geometrical solutions constructed from a smooth algebraic curve $\\Gamma $ of genus $g$ with some extra data.",
"The tau-function for such solutions is given by $\\tau _n ({\\bf t}, \\bar{\\bf t})=e^{Q(n, {\\bf t}, \\bar{\\bf t})}\\Theta \\Bigl ({\\bf V}_0 n +\\sum _{k\\ge 1}{\\bf V}_k t_k +\\sum _{k\\ge 1}\\bar{\\bf V}_k \\bar{t}_k+{\\bf Z}\\Bigr ),$ where $Q$ is a quadratic form of its variables and $\\Theta $ is the Riemann theta-function with the Riemann's matrix being the period matrix of holomorphic differentials on the curve $\\Gamma $ .",
"Components of the $g$ -dimensional vectors ${\\bf V}_k$ , $\\bar{\\bf V}_k$ are $b$ -periods of certain normalized meromorphic differentials on $\\Gamma $ .",
"When one considers algebraic-geometrical solutions, it is natural to treat $t_0=n$ as a continuous rather than discrete variable.",
"This is also helpful for passing to the continuum limit.",
"Namely, let us introduce the continuous variable $x=x_0+\\eta n$ , where $\\eta $ is a constant (a lattice spacing), then the Toda equation becomes a difference equation in $x$ : $\\partial _{t_1}\\partial _{\\bar{t}_1}\\log a(x)=2a(x)-a(x+\\eta )-a(x-\\eta ).$ It is equivalent to the zero curvature equation $\\partial _{\\bar{t}_1}{\\cal B}_1 -\\partial _{t_1}\\bar{\\cal B}_1 +[{\\cal B}_1, \\bar{\\cal B}_1]=0$ for the difference operators ${\\cal B}_1=e^{\\eta \\partial _x}+b(x), \\qquad \\bar{\\cal B}_1 =a(x)e^{-\\eta \\partial _x},$ which is the compatibility condition of the linear problems $\\begin{array}{l}\\partial _{t_1}\\psi (x)=\\psi (x+\\eta )+b(x)\\psi (x),\\\\ \\\\\\partial _{\\bar{t}_1}\\psi (x)=a(x)\\psi (x-\\eta )\\end{array}$ for a wave function $\\psi $ ."
],
[
"Elliptic families",
"Let us fix the variables $\\bar{\\bf t}$ and consider the dependent variables as functions of $x, {\\bf t}$ .",
"A general solution to the 2D Toda and KP equations is known to be of the form $\\begin{array}{l}\\displaystyle {a(x, {\\bf t})=\\frac{\\tau (x+\\eta , {\\bf t})\\tau (x-\\eta , {\\bf t})}{\\tau ^2 (x, {\\bf t})},}\\\\ \\\\\\displaystyle {b(x, {\\bf t})=\\partial _{t_1}\\log \\frac{\\tau (x+\\eta , {\\bf t})}{\\tau (x, {\\bf t})},}\\\\ \\\\\\displaystyle {u(x, {\\bf t})=\\partial ^2_{t_1}\\log \\tau (x, {\\bf t}).",
"}\\end{array}$ We are going to consider solutions that are elliptic functions with respect to some variable $t_k$ or a linear combination $\\displaystyle {\\lambda =\\beta _0x+\\sum _k \\beta _k t_k}$ .",
"We call them elliptic families.",
"The elliptic families form a subclass of algebraic-geometrical solutions.",
"As it was already mentioned in Introduction, an algebraic-geometrical solution is elliptic with respect to some direction if there exists a $g$ -dimensional vector ${\\bf W}$ such that it spans an elliptic curve ${\\cal E}$ embedded in the Jacobian of the curve $\\Gamma $ : $\\tau (x, {\\bf t}, \\lambda )=e^{Q(x, {\\bf t})}\\Theta \\Bigl ({\\bf V}_0 x/\\eta +\\sum _{k\\ge 1}{\\bf V}_k t_k +{\\bf W}\\lambda +{\\bf Z}\\Bigr ),$ This is a nontrivial transcendental constraint.",
"The space of corresponding algebraic curves has codimension $g-1$ in the moduli space of all the curves.",
"If such a vector ${\\bf W}$ exists, then the theta-divisor intersects the shifted elliptic curve $\\displaystyle {{\\cal E}+{\\bf V}_0 x/\\eta +\\sum _k {\\bf V}_k t_k}$ at a finite number of points $\\lambda _i =\\lambda _i(x,{\\bf t})$ .",
"Therefore, for elliptic families we have: $\\Theta \\Bigl ({\\bf V}_0 x/\\eta +\\sum _{k\\ge 1}{\\bf V}_k t_k +{\\bf W}\\lambda +{\\bf Z}\\Bigr )=f(x, {\\bf t})e^{\\gamma _1\\lambda +\\gamma _2\\lambda ^2}\\prod _{i=1}^N\\sigma (\\lambda -\\lambda _i(x, {\\bf t})).$ Here $\\gamma _1, \\gamma _2$ are constants and $\\sigma (\\lambda )$ is the Weierstrass $\\sigma $ -function defined in the Appendix.",
"The form of the exponential factor in the right hand side of (REF ) follows from monodromy properties of the theta-function.",
"Having in mind the discrete version, we will also denote $\\lambda _i^k=\\lambda (x)$ for $x=x_0+k\\eta $ .",
"In what follows we denote $t_1=t$ .",
"From (REF ) we conclude that if $b(x, t, \\lambda )$ is an elliptic family of solutions to the 2D Toda equation, then it has the form $b(x, t, \\lambda )=\\sum _{i=1}^N \\Bigl (\\dot{\\lambda }_i (x)\\zeta (\\lambda -\\lambda _i(x))-\\dot{\\lambda }_i (x+\\eta )\\zeta (\\lambda -\\lambda _i(x+\\eta ))\\Bigr ) +c(x,t),$ where dot means the $t$ -derivative and $c(x,t)$ is some function.",
"Since $a(x,t, \\lambda )$ , $b(x, t, \\lambda )$ and $u(x,t,\\lambda )$ given by (REF ) are elliptic functions of $\\lambda $ , one should have $\\sum _{i}\\Bigl (\\lambda _i(x+\\eta )+\\lambda _i(x-\\eta )-2\\lambda _i(x)\\Bigr )=0,$ $\\sum _i \\dot{\\lambda }_i(x+\\eta )=\\sum _i \\dot{\\lambda }_i(x),$ $\\sum _i \\ddot{\\lambda }_i(x)=0.$ From (REF ), (REF ), (REF ) it follows that $\\sum _i \\lambda _i(x)= \\alpha x +\\beta t+\\alpha _0, \\qquad \\dot{\\alpha }=\\dot{\\beta }=0.$ Here $\\alpha _0, \\alpha , \\beta $ are $\\eta $ -periodic functions of $x$ .",
"We can say that the requirement of ellipticity implies that the “center of masses” of the points $\\lambda _i$ moves linearly in time.",
"A meromorphic function $f(\\lambda )$ is called a double-Bloch function if it satisfies the following monodromy properties: $f(\\lambda +2\\omega _{\\alpha })=B_{\\alpha }f(\\lambda ), \\quad \\alpha =1,2.$ The complex constants $B_{\\alpha }$ are called Bloch multipliers.",
"Our goal is to find $b(x, t, \\lambda )$ such that the equation $\\partial _{t}\\psi (x)-\\psi (x+\\eta )-b(x)\\psi (x)=0$ has sufficiently many double-Bloch solutions.",
"The existence of such solutions turn out to be a very restrictive condition.",
"The double-Bloch functions with simple poles $\\lambda _i$ can be represented in the form $\\psi (x)=\\sum _i c_i(x)\\Phi (\\lambda -\\lambda _i(x), z ),$ where $c_i$ are residues at the poles $\\lambda _i$ and the function $\\Phi (\\lambda , z)$ is defined in (REF ).",
"The variable $z$ has the meaning of the spectral parameter.",
"In what follows we often suppress the second argument of $\\Phi $ writing simply $\\Phi (\\lambda , z):=\\Phi (\\lambda )$ ."
],
[
"Equations of motion of the field analogue of the elliptic\nRuijsenaars-Schneider system",
"Our strategy is similar to that of the work [3].",
"We are going to substitute (REF ), (REF ) into (REF ) and cancel all the poles which are at $\\lambda =\\lambda _i(x)$ and $\\lambda =\\lambda _i(x+\\eta )$ .",
"The substitution gives: $\\sum _i \\dot{c}_i(x)\\Phi (\\lambda -\\lambda _i(x))-\\sum _i c_i(x)\\dot{\\lambda }_i(x)\\Phi ^{\\prime } (\\lambda -\\lambda _i(x))-\\sum _i \\dot{c}_i(x+\\eta )\\Phi (\\lambda -\\lambda _i(x+\\eta ))$ $-\\sum _i \\Bigl ((\\dot{\\lambda }_i (x)\\zeta (\\lambda -\\lambda _i(x))-\\dot{\\lambda }_i (x+\\eta )\\zeta (\\lambda -\\lambda _i(x+\\eta ))\\Bigr ) \\sum _jc_j(x)\\Phi (\\lambda -\\lambda _j(x))$ $-c(x,t)\\sum _i c_i(x)\\Phi (\\lambda -\\lambda _i(x))=0.$ The cancellation of poles yields the following system of equations: $c_i(x+\\eta )=\\dot{\\lambda }_i(x+\\eta )\\sum _j c_j(x)\\Phi (\\lambda _i(x+\\eta )-\\lambda _j(x)):=\\sum _j L_{ij}(x)c_j(x),$ $\\begin{array}{c}\\displaystyle { \\dot{c}_i(x)=\\dot{\\lambda }_i(x)\\sum _{j\\ne i}c_j(x)\\Phi (\\lambda _i(x)-\\lambda _j(x)) +c_i(x)\\sum _{j\\ne i}\\dot{\\lambda }_j(x)\\zeta (\\lambda _i(x)-\\lambda _j(x))}\\\\ \\\\\\displaystyle {-c_i(x) \\sum _j \\dot{\\lambda }_j(x+\\eta )\\zeta (\\lambda _i(x)-\\lambda _j (x+\\eta ))+c_i(x)c (x,t):=\\sum _j M_{ij}(x)c_j(x)}.\\end{array}$ Here the matrices $L$ , $M$ are $L_{ij}(x, z)=\\dot{\\lambda }_i(x+\\eta )\\Phi (\\lambda _i (x+\\eta )-\\lambda _j(x), z),$ $\\begin{array}{c}M_{ij}(x, z)=(1-\\delta _{ij})\\dot{\\lambda }_i(x)\\Phi (\\lambda _i (x)-\\lambda _j(x), z)\\\\ \\\\\\displaystyle {+\\, \\delta _{ij}\\Bigl (\\sum _{k\\ne i}\\dot{\\lambda }_k (x)\\zeta (\\lambda _i(x)-\\lambda _k(x))-\\sum _{k}\\dot{\\lambda }_k (x+\\eta )\\zeta (\\lambda _i(x)-\\lambda _k(x+\\eta ))+c(x,t)\\Bigr )}.\\end{array}$ Let us introduce the matrices $\\begin{array}{l}A_{ij}^+(x)=\\Phi (\\lambda _i(x+\\eta )-\\lambda _j(x)),\\\\ \\\\A_{ij}^0(x)=(1-\\delta _{ij})\\Phi (\\lambda _i(x)-\\lambda _j(x))\\end{array}$ and diagonal matrices $\\begin{array}{l}\\Lambda _{ij}(x)=\\delta _{ij}\\lambda _i(x),\\\\ \\\\\\displaystyle {D_{ij}^0(x)=\\delta _{ij}\\sum _{k\\ne i}\\dot{\\lambda }_k(x)\\zeta (\\lambda _i(x)-\\lambda _k(x))},\\\\ \\\\\\displaystyle {D_{ij}^\\pm (x)=\\delta _{ij}\\sum _{k\\ne i}\\dot{\\lambda }_k(x\\pm \\eta )\\zeta (\\lambda _i(x)-\\lambda _k(x\\pm \\eta ))}.\\end{array}$ In terms of these matrices, the matrices $L$ and $M$ read: $L(x)=\\dot{\\Lambda }(x+\\eta )A^+(x), \\quad M(x)=\\dot{\\Lambda }(x)A^0(x)+D^0(x)-D^+(x)+c(x, t)I,$ where $I$ is the unity matrix.",
"The compatibility condition of the overdetermined system (REF ), (REF ) is the semi-discrete zero curvature (Zakharov-Shabat) equation $R(x):=\\dot{L}(x)+L(x)M(x)-M(x+\\eta )L(x)=0.$ The matrices $L$ , $M$ here depend on the spectral parameter $z$ .",
"We have: $R(x)=\\ddot{\\Lambda }(x+\\eta )A^+(x)+\\dot{\\Lambda }(x+\\eta )\\Bigl (S(x)+A^+(x)(D^0(x)-D^+(x))$ $-(D^0(x+\\eta )-D^+(x+\\eta ))A^+(x)+(c(x,t)-c(x+\\eta , t))A^+(x)\\Bigr ),$ where $S(x)=\\dot{A}^+(x)+A^+(x)\\dot{\\Lambda }(x)A^0(x)-A^0(x+\\eta )\\dot{\\Lambda }(x+\\eta )A^+(x).$ Using (REF ), (REF ), we calculate: $\\dot{A}^+_{ij}(x)=(\\dot{\\lambda }_i(x+\\eta )-\\dot{\\lambda }_j(x))\\Phi (\\lambda _i(x+\\eta )-\\lambda _j(x))$ $\\times \\Bigl (\\zeta (\\lambda _i(x+\\eta )-\\lambda _j(x)+\\mu )-\\zeta (\\lambda _i(x+\\eta )-\\lambda _j(x))-\\zeta (\\mu )\\Bigr ) ,$ and $\\Bigl (A^+(x)\\dot{\\Lambda }(x)A^0(x)-A^0(x+\\eta )\\dot{\\Lambda }(x+\\eta )A^+(x)\\Bigr )_{ij}$ $=\\sum _{k\\ne j}\\Phi (\\lambda _i(x+\\eta )-\\lambda _k(x))\\Phi (\\lambda _k(x)-\\lambda _j(x))\\dot{\\lambda }_k(x)$ $-\\sum _{k\\ne i}\\Phi (\\lambda _i(x+\\eta )-\\lambda _k(x+\\eta ))\\Phi (\\lambda _k(x+\\eta )-\\lambda _j(x))\\dot{\\lambda }_k(x+\\eta )$ $=\\Phi (\\lambda _i(x+\\eta )-\\lambda _j(x))\\left(\\sum _{k\\ne j}\\dot{\\lambda }_k(x)\\zeta (\\lambda _i(x+\\eta )-\\lambda _k(x))-\\sum _{k\\ne j}\\dot{\\lambda }_k(x)\\zeta (\\lambda _j(x)-\\lambda _k(x))\\right)$ $-\\Phi (\\lambda _i(x+\\eta )-\\lambda _j(x))\\left(\\sum _{k\\ne i}\\dot{\\lambda }_k(x+\\eta )\\zeta (\\lambda _i(x+\\eta )-\\lambda _k(x+\\eta ))\\right.$ $-\\left.\\sum _{k\\ne i}\\dot{\\lambda }_k(x+\\eta )\\zeta (\\lambda _j(x)-\\lambda _k(x+\\eta ))\\right)$ $+(\\dot{\\lambda }_i(x+\\eta )-\\dot{\\lambda }_j(x))\\Phi (\\lambda _i(x+\\eta )-\\lambda _j(x))\\Bigl (\\zeta (\\mu )-\\zeta (\\lambda _i(x+\\eta )-\\lambda _j(x)+\\mu )\\Bigr ).$ In the calculation, the condition (REF ) was taken into account.",
"Therefore, we have: $S_{ij}(x)=\\Phi (\\lambda _i(x+\\eta )-\\lambda _j(x))\\Bigl (D_{ii}^-(x+\\eta )+D_{jj}^+(x)-D_{jj}^0(x)-D_{ii}^0(x+\\eta )\\Bigr )$ and, combining everything together, we obtain the matrix identity $R(x)=\\Bigl (\\ddot{\\Lambda }(x+\\eta )\\dot{\\Lambda }^{-1}(x+\\eta )+D^-(x+\\eta )+D^+(x+\\eta )-2D^0(x+\\eta )+(c(x,t)-c(x+\\eta , t))I\\Bigr ) L(x),$ from which we see that the compatibility condition is equivalent to vanishing of the diagonal matrix in front of $L(x)$ : $\\ddot{\\Lambda }(x+\\eta )\\dot{\\Lambda }^{-1}(x+\\eta )+D^-(x+\\eta )+D^+(x+\\eta )-2D^0(x+\\eta )+(c(x,t)-c(x+\\eta , t))I=0.$ This results in the equations of motion $\\begin{array}{c}\\displaystyle {\\ddot{\\lambda }_i(x)+\\sum _k \\Bigl (\\dot{\\lambda }_i(x)\\dot{\\lambda }_k(x-\\eta )\\zeta (\\lambda _i(x)-\\lambda _k(x-\\eta ))+\\dot{\\lambda }_i(x)\\dot{\\lambda }_k(x+\\eta )\\zeta (\\lambda _i(x)-\\lambda _k(x+\\eta ))\\Bigr )}\\\\ \\\\\\displaystyle {-\\, 2 \\sum _{k\\ne i}\\dot{\\lambda }_i(x)\\dot{\\lambda }_k(x )\\zeta (\\lambda _i(x)-\\lambda _k(x))+(c(x-\\eta , t)-c(x, t))\\dot{\\lambda }_i(x)=0.",
"}\\end{array}$ These equations resemble the Ruijsenaars-Schneider equations of motion and provide their field generalization.",
"If $\\lambda _i(x)=x_i(t)+x$ , then equations (REF ) become the equations of motion for the elliptic Ruijsenaars-Schneider system with coordinates of particles $x_i$ (with $c(x,t)=c(x+\\eta , t)$ ).",
"The condition (REF ) allows us to find the explicit form of the function $c(x,t)$ .",
"Summing equations (REF ) over $i=1, \\ldots , N$ and using (REF ), we get $c(x, t)=\\frac{1}{\\beta }\\sum _{i,k}\\dot{\\lambda }_i(x)\\dot{\\lambda }_k(x+\\eta )\\zeta (\\lambda _i(x)-\\lambda _k(x+\\eta ))$ (up to an arbitrary function of $t$ and an $\\eta $ -periodic function of $x$ which do not affect the equations of motion).",
"Let us also present the lattice version of equations (REF ) which are obtained from them after the substitution $\\lambda _i^k=\\lambda _i (k\\eta +x_0)$ : $\\begin{array}{c}\\displaystyle {\\ddot{\\lambda }_i^k+\\sum _j \\Bigl (\\dot{\\lambda }_i^k\\dot{\\lambda }_j^{k-1}\\zeta (\\lambda _i^k-\\lambda _j^{k-1})+\\dot{\\lambda }_i^k\\dot{\\lambda }_j^{k+1}\\zeta (\\lambda _i^k-\\lambda _j^{k+1})\\Bigr )}\\\\ \\\\\\displaystyle {-\\, 2 \\sum _{j\\ne i}\\dot{\\lambda }_i^k\\dot{\\lambda }_j^k\\zeta (\\lambda _i^k-\\lambda _j^k)+(c^{k-1}(t)-c^k(t))\\dot{\\lambda }_i^k=0}\\end{array}$ with $c^k(t)=\\frac{1}{\\beta }\\sum _{i,j}\\dot{\\lambda }_i^k\\dot{\\lambda }_j^{k+1}\\zeta (\\lambda _i^k-\\lambda _j^{k+1}).$"
],
[
"Equivalence to the equations of sections ",
"It is not difficult to see that with the conditions (REF )–(REF ) these equations (with $\\beta =1$ ) are equivalent to (REF ).",
"Indeed, identifying $\\lambda _i^k=q_i^k$ and passing to the center of masses frame, we see that $\\begin{array}{l}\\lambda _i^k-\\lambda _j^{k-1}=\\bar{q}_i^k -\\bar{q}_j^{k-1}+\\alpha \\eta /N,\\\\ \\\\\\lambda _i^k-\\lambda _j^{k+1}=\\bar{q}_i^k -\\bar{q}_j^{k+1}-\\alpha \\eta /N,\\end{array}$ so that the arguments of the $\\zeta $ - and $E_1$ -functions in (REF ) and (REF ) coincide if $\\alpha =-N$ .",
"Next, if one chooses the periods to be $2\\omega _1=1$ , $2\\omega _2=\\tau $ , the $\\zeta (z)$ - and $E_1(z)$ -functions differ by a term linear in $z$ (see (REF )).",
"The corresponding contribution to $c^k(t)$ (REF ) is $\\frac{1}{\\beta }\\sum _i \\dot{\\lambda }_i^k \\lambda _i^k \\sum _j \\dot{\\lambda }_j^{k+1}-\\frac{1}{\\beta }\\sum _i \\dot{\\lambda }_i^{k+1}\\lambda _i^{k+1}\\sum _j \\dot{\\lambda }_j^{k}=\\sum _i \\dot{\\lambda }_i^k \\lambda _i^k -\\sum _i \\dot{\\lambda }_i^{k+1}\\lambda _i^{k+1}$ since $\\displaystyle {\\sum _j \\dot{\\lambda }_j^k =\\beta }$ .",
"Writing equations (REF ) as $\\begin{array}{c}\\displaystyle {\\frac{\\ddot{\\lambda }_i^k}{\\dot{\\lambda }_i^k}+\\sum _j \\Bigl (\\dot{\\lambda }_j^{k-1}\\zeta (\\lambda _i^k-\\lambda _j^{k-1})+\\dot{\\lambda }_j^{k+1}\\zeta (\\lambda _i^k-\\lambda _j^{k+1})\\Bigr )- 2 \\sum _{j\\ne i}\\dot{\\lambda }_j^k\\zeta (\\lambda _i^k-\\lambda _j^k)+c^{k-1}(t)-c^k(t)=0},\\end{array}$ we find the corresponding contribution from the sums over $j$ to be $\\sum _j \\dot{\\lambda }_j^{k-1}(\\lambda _i^k -\\lambda _j^{k-1})+\\sum _j \\dot{\\lambda }_j^{k+1}(\\lambda _i^k -\\lambda _j^{k+1})-2\\sum _j \\dot{\\lambda }_j^{k}(\\lambda _i^k -\\lambda _j^{k})$ $=\\lambda _i^k\\underbrace{\\sum _j (\\dot{\\lambda }_j^{k-1}+\\dot{\\lambda }_j^{k+1}-2\\dot{\\lambda }_j^k)}_{\\mbox{$=0$ due to (\\ref {el2a})}}-\\sum _j (\\dot{\\lambda }_j^{k-1}\\lambda _j^{k-1}+\\dot{\\lambda }_j^{k+1}\\lambda _j^{k+1}-2\\dot{\\lambda }_j^{k}\\lambda _j^{k}),$ so this contribution cancels with the one coming from $c^k(t)$ (REF ).",
"The $L-M$ pair (REF ), (REF ) discussed in sections REF , REF is also equivalent to the $L-M$ pair (REF ), (REF ).",
"Indeed, one can straightforwardly check that with the identification of the $L$ - and $M$ -matrices $\\tilde{L}_{ij}^k(z)=-h_{k-1, k}e^{E_1(z)(\\lambda _i^{k-1}-\\lambda _j^k)}L_{ji}^{k-1}(-z),$ $\\tilde{M}_{ij}^k(z)=e^{E_1(z)(\\lambda _i^{k}-\\lambda _j^k)}M_{ji}^{k}(-z)-\\delta _{ij}(E_1(z)\\dot{\\lambda }_i^k +\\tilde{c}^k)$ the semi-discrete Zakharov-Shabat equations (REF ) and (REF ) become equivalent.",
"In the right hand sides of (REF ), (REF ) the matrices $L^k$ , $M^k$ are given by (REF ), (REF ) under the identification $x=k\\eta +x_0$ : $L^k =L(k\\eta +x_0)$ , $M^k =M(k\\eta +x_0)$ .",
"For the modified Lax pair (REF ), (REF ) the relations (REF ), (REF ) slightly simplify: ${L^{\\prime }}_{ij}^k(z)=-e^{E_1(z)(\\lambda _i^{k-1}-\\lambda _j^k)}L_{ji}^{k-1}(-z),$ ${M^{\\prime }}_{ij}^k(z)=e^{E_1(z)(\\lambda _i^{k}-\\lambda _j^k)}M_{ji}^{k}(-z)-\\delta _{ij}E_1(z)\\dot{\\lambda }_i^k\\,.$"
],
[
"Hamiltonian structure",
"Let us show that the equations (REF ) with $c^k(t)$ given by (REF ) are Hamiltonian.",
"We fix the canonical Poisson brackets $\\lbrace p_i^k, p_j^l\\rbrace =\\lbrace \\lambda _i^k, \\lambda _j^l\\rbrace =0, \\quad \\lbrace \\lambda _i^k, p_j^l\\rbrace =\\delta _{ij}\\delta _{kl}.$ The Hamiltonian is ${\\cal H}=\\frac{\\beta }{\\eta }\\sum _k \\log H_k$ with $H_k=\\sum _i e^{\\eta p_i^k}\\frac{\\prod \\limits _{j}\\sigma (\\lambda _i^k-\\lambda _j^{k-1})}{\\prod \\limits _{j\\ne i}\\sigma (\\lambda _i^k-\\lambda _j^k)}.$ The first set of Hamiltonian equations is $\\dot{\\lambda }_i^k=\\frac{\\partial {\\cal H}}{\\partial p_i^k}=\\frac{\\beta }{H_k}\\,\\eta e^{\\eta p_i^k}\\frac{\\prod \\limits _{j}\\sigma (\\lambda _i^k-\\lambda _j^{k-1})}{\\prod \\limits _{j\\ne i}\\sigma (\\lambda _i^k-\\lambda _j^k)}.$ Taking the time derivative of (logarithm of) this equation, we get $\\eta \\dot{p}_i^k=\\frac{\\ddot{\\lambda }_i^k}{\\dot{\\lambda }_i^k}-\\sum _j(\\dot{\\lambda }_i^k-\\dot{\\lambda }_j^{k-1})\\zeta (\\lambda _i^k-\\lambda _j^{k-1})+\\sum _{j\\ne i}(\\dot{\\lambda }_i^k-\\dot{\\lambda }_j^k)\\zeta (\\lambda _i^k-\\lambda _j^k)+\\partial _t \\log H_k.$ The second set of Hamiltonian equations is $\\dot{p}_i^k(x)=-\\frac{\\partial {\\cal H}}{\\partial \\lambda _i^k}.$ The variation of the Hamiltonian is $\\eta \\delta {\\cal H}=\\beta \\sum _k \\frac{\\delta H_k}{H_k}=\\sum _k \\sum _{i,l} \\dot{\\lambda }_i^k\\zeta (\\lambda _i^k-\\lambda _l^{k-1})(\\delta \\lambda _i^k-\\delta \\lambda _l^{k-1})-\\sum _k \\sum _{i\\ne l} \\dot{\\lambda }_i^k\\zeta (\\lambda _i^k-\\lambda _l^{k})(\\delta \\lambda _i^k-\\delta \\lambda _l^{k}).$ Changing the summation indices $k\\rightarrow k+1$ and $i\\leftrightarrow l$ when necessary, we have: $\\eta \\delta {\\cal H}=\\sum _k \\sum _{i,l} \\dot{\\lambda }_i^k\\zeta (\\lambda _i^k-\\lambda _l^{k-1})\\delta \\lambda _i^k +\\sum _k \\sum _{i,l} \\dot{\\lambda }_l^{k+1}\\zeta (\\lambda _i^k-\\lambda _l^{k+1})\\delta \\lambda _i^k$ $-\\sum _k \\sum _{i\\ne l} (\\dot{\\lambda }_i^k+\\dot{\\lambda }_l^k)\\zeta (\\lambda _i^k-\\lambda _l^{k})\\delta \\lambda _i^k.$ From here we see that $\\begin{array}{c}\\displaystyle {\\eta \\dot{p}_i^k=-\\sum _{l} \\dot{\\lambda }_i^k\\zeta (\\lambda _i^k-\\lambda _l^{k-1}) -\\sum _{l} \\dot{\\lambda }_l^{k+1}\\zeta (\\lambda _i^k-\\lambda _l^{k+1})+\\sum _{l\\ne i} (\\dot{\\lambda }_i^k+\\dot{\\lambda }_l^k)\\zeta (\\lambda _i^k-\\lambda _l^{k}).",
"}\\end{array}$ Comparing with (REF ), we obtain: $\\frac{\\ddot{\\lambda }_i^{k}}{\\dot{\\lambda }_i^k}+\\sum _{l} \\dot{\\lambda }_l^{k+1}\\zeta (\\lambda _i^k-\\lambda _l^{k+1})+\\sum _{l} \\dot{\\lambda }_l^{k-1}\\zeta (\\lambda _i^k-\\lambda _l^{k-1})-2\\sum _{l\\ne i} \\dot{\\lambda }_l^k\\zeta (\\lambda _i^k-\\lambda _l^{k})+\\partial _t \\log H_k=0.$ The calculation of $\\partial _t \\log H_k =\\dot{H}_k/H_k$ is straightforward using (REF ) and (REF ).",
"The result is $\\partial _t \\log H_k = c^{k-1}(t)-c^k(t),$ where $c_k(t)$ is given by (REF ).",
"Therefore, the equations of motion (REF ) are reproduced."
],
[
"The continuum limit to 1+1 Calogero-Moser filed theory",
"In this section we show that the continuum ($\\eta \\rightarrow 0$ ) limit of the field Ruijsenaars-Schneider model yields the field Calogero-Moser model as it appears in [3].",
"Instead of the lattice version (REF ) it is convenient to work with the equivalent $x$ -dependent Hamiltonian density ${\\cal H}(x)=\\frac{\\beta }{\\eta }\\, \\log \\left(\\sum _i e^{\\eta p_i(x)}\\sigma (\\lambda _i(x)-\\lambda _i(x-\\eta ))\\prod _{l\\ne i}\\frac{\\sigma (\\lambda _i(x)-\\lambda _l(x-\\eta ))}{\\sigma (\\lambda _i(x)-\\lambda _l(x))}\\right)$ and the canonical Poisson brackets $\\lbrace p_i(x), p_j(y)\\rbrace =\\lbrace \\lambda _i(x), \\lambda _j(y)\\rbrace =0, \\quad \\lbrace \\lambda _i(x), p_j(y)\\rbrace =\\delta _{ij}\\delta (x-y).$ We are interested in the $\\eta $ -expansion of (REF ) as $\\eta \\rightarrow 0$ .",
"We have: ${\\cal H}(x)=\\frac{\\beta }{\\eta }\\, \\log \\left[\\eta \\sum _i \\Bigl (1+\\eta p_i+\\frac{1}{2}\\, \\eta ^2 p_i^2+O(\\eta ^3)\\Bigr )(\\lambda _i^{\\prime }-\\frac{1}{2}\\, \\eta \\lambda _i^{\\prime \\prime } +\\frac{1}{6}\\, \\eta ^2 \\lambda _i^{\\prime \\prime \\prime }+O(\\eta ^3)\\Bigr )\\right.$ $\\left.\\times \\exp \\Bigl (\\sum _{j\\ne i}\\Bigl (\\eta \\lambda _j^{\\prime } \\zeta (\\lambda _i-\\lambda _j)-\\frac{1}{2}\\, \\eta ^2 \\lambda _j^{\\prime \\prime }\\zeta (\\lambda _i-\\lambda _j)-\\frac{1}{2}\\, \\eta ^2 \\lambda _j^{\\prime }{}^{2}\\wp (\\lambda _i-\\lambda _j)+O(\\eta ^3)\\Bigr )\\Bigr )\\right],$ where prime denotes the $x$ -derivative.",
"Equation (REF ) implies that $\\sum _i \\lambda _i^{\\prime }(x)=\\alpha $ is a constant.",
"In the continuum limit the $x$ -flow tends to the $t_1$ -flow, and so the limit of $\\alpha $ as $\\eta \\rightarrow 0$ is equal to $\\beta $ .",
"Therefore, the first few terms of the $\\eta $ -expansion of ${\\cal H}(x)$ are ${\\cal H}(x)=\\mbox{const} \\, +(1+O(\\eta ))H_1^{\\rm CM}(x)-\\frac{\\eta }{2}\\,H_2^{\\rm CM}(x) +O(\\eta ^2),$ where $H_1^{\\rm CM}(x)=\\sum _i \\tilde{p}_i\\lambda _i^{\\prime }$ is the first Hamiltonian density of the field Calogero-Moser model (a field analogue of the total momentum) and $\\begin{array}{c}\\displaystyle {H_2^{\\rm CM}(x)=-\\sum _i \\tilde{p}_i^2 \\lambda _i^{\\prime }-\\frac{1}{4}\\sum _i \\frac{\\lambda _i^{\\prime \\prime }{}^2}{\\lambda _i^{\\prime }}-\\frac{1}{3}\\sum _i \\lambda _i^{\\prime \\prime \\prime }+\\frac{1}{\\beta }\\Bigl (\\sum _i \\tilde{p}_i \\lambda _i^{\\prime }\\Bigr )^2}\\\\ \\\\\\displaystyle {-\\frac{1}{2}\\sum _{i\\ne j}(\\lambda _i^{\\prime \\prime }\\lambda _j^{\\prime }-\\lambda _j^{\\prime \\prime }\\lambda _i^{\\prime })\\zeta (\\lambda _i-\\lambda _j)+\\frac{1}{2}\\sum _{i\\ne j}(\\lambda _i^{\\prime }\\lambda _j^{\\prime }{}^2 +\\lambda _j^{\\prime }\\lambda _i^{\\prime }{}^2)\\wp (\\lambda _i-\\lambda _j)}\\end{array}$ is the second (standard) Hamiltonian density.",
"The Hamiltonian of the model is ${\\cal H}_2^{\\rm CM}=\\int H_2^{\\rm CM}(x)dx.$ Up to a full derivative and the canonical transformation $p_i\\longrightarrow \\tilde{p}_i=p_i-\\frac{\\lambda _i^{\\prime \\prime }}{2\\lambda _i^{\\prime }}+\\sum _{j\\ne i}\\lambda _j^{\\prime }\\zeta (\\lambda _i-\\lambda _j)$ the Hamiltonian density (REF ) coincides with the Hamiltonian density for the field Calogero-Moser model presented in [3].",
"The fact that the transformation (REF ) is canonical can be verified straightforwardly.",
"The only nontrivial calculation is required to show that $\\lbrace \\tilde{p}_i(x), \\tilde{p}_j(y)\\rbrace =0$ .",
"This can be done using the identities $f(x)\\delta ^{\\prime \\prime }(x-y)-f(y)\\delta ^{\\prime \\prime }(y-x)=-\\Bigl (f^{\\prime }(x)\\delta ^{\\prime }(x-y)-f^{\\prime }(y)\\delta ^{\\prime } (y-x)\\Bigr ),$ $f(x)\\delta ^{\\prime }(x-y)+f(y)\\delta ^{\\prime }(y-x)=-f^{\\prime }(x)\\delta (x-y)$ for the delta-function and its derivatives."
],
[
"Fully discrete version",
"The fully discrete (or difference) version of the above construction can be obtained by considering elliptic families of solutions to the Hirota bilinear difference equation [41] for the tau-function $\\tau ^{l,m}(x)$ , where $l,m$ are discrete times: $\\tau ^{l,m}(x+\\eta )\\tau ^{l+1,m+1}(x)-\\kappa \\tau ^{l,m+1}(x+\\eta )\\tau ^{l+1,m}(x)+(\\kappa -1)\\tau ^{l+1,m}(x+\\eta )\\tau ^{l,m+1}(x)=0.$ Here $\\kappa $ is a parameter.",
"This equation is known to provide an integrable time discretization of the 2D Toda equation.",
"One of the auxiliary linear problems for the equation (REF ) is [42] $\\psi ^{m+1}(x)=\\psi ^m(x+\\eta )-\\kappa \\frac{\\tau ^m(x)\\tau ^{m+1}(x+\\eta )}{\\tau ^{m+1}(x)\\tau ^{m}(x+\\eta )}\\, \\psi ^m(x),$ where the index $l$ is supposed to be fixed and the same for all entries.",
"The elliptic families of solutions with elliptic parameter $\\lambda $ are given by $\\tau ^{l,m}(x)=\\rho ^{l,m}(x)e^{c_1\\lambda +c_2\\lambda ^2}\\prod _j \\sigma (\\lambda -\\lambda _j^{l,m}(x)),$ where $\\rho ^{l,m}(x)$ is some function which does not depend on $\\lambda $ and $c_1, c_2$ are constants.",
"If the constraint $\\sum _j \\Bigl (\\lambda ^{m+1}_j(x+\\eta )-\\lambda ^{m+1}_j(x)\\Bigr )=\\sum _j \\Bigl (\\lambda ^{m}_j(x+\\eta )-\\lambda ^{m}_j(x)\\Bigr )$ is satisfied, then the coefficient in front of the second term in the right hand side of (REF ) is an elliptic function of $\\lambda $ and we can find double-Bloch solutions of the form $\\psi ^m (x)=\\sum _i c_i^m(x)\\Phi (\\lambda -\\lambda _i^m(x), z ).$ The substitution of (REF ) into (REF ) yields: $\\sum _ic_i^{m+1}(x)\\Phi (\\lambda -\\lambda _i^{m+1}(x))-\\sum _ic_i^{m}(x+\\eta )\\Phi (\\lambda -\\lambda _i^{m}(x+\\eta ))$ $+\\, \\kappa _m(x)\\frac{\\prod \\limits _{j}\\sigma (\\lambda -\\lambda _j^{m}(x))\\sigma (\\lambda -\\lambda _j^{m+1}(x+\\eta ))}{\\prod \\limits _{j}\\sigma (\\lambda -\\lambda _j^{m+1}(x))\\sigma (\\lambda -\\lambda _j^{m}(x+\\eta ))}\\sum _i c_i^{m}(x)\\Phi (\\lambda -\\lambda _i^{m}(x))=0,$ where $\\kappa _m(x)=\\kappa \\frac{\\rho ^m(x)\\rho ^{m+1}(x+\\eta )}{\\rho ^{m+1}(x)\\rho ^{m}(x+\\eta )}.$ It is enough to cancel poles in the left hand side at $\\lambda = \\lambda _i^{m+1}(x)$ and $\\lambda = \\lambda _i^{m}(x+\\eta )$ .",
"A direct calculation shows that the conditions of cancellation of the poles read $c_i^{m}(x+\\eta )=f_i^m(x)\\sum _j c_j^m(x)\\Phi (\\lambda _i^{m}(x+\\eta )-\\lambda _j^m(x)):=\\sum _j L_{ij}^m(x)c_j^m(x),$ $c_i^{m+1}(x)=g_i^m(x)\\sum _j c_j^m(x)\\Phi (\\lambda _i^{m+1}(x)-\\lambda _j^m(x)):=\\sum _j M_{ij}^m(x)c_j^m(x),$ where $f_i^m(x)=\\kappa _m(x)\\frac{\\prod \\limits _{j}\\sigma (\\lambda _i^{m}(x+\\eta )-\\lambda _j^m(x))\\sigma (\\lambda _i^{m}(x+\\eta )-\\lambda _j^{m+1}(x+\\eta ))}{\\prod \\limits _{j}\\sigma (\\lambda _i^{m}(x+\\eta )-\\lambda _j^{m+1}(x))\\prod \\limits _{j\\ne i}\\sigma (\\lambda _i^{m}(x+\\eta )-\\lambda _j^{m}(x+\\eta ))},$ $g_i^m(x)=-\\kappa _m(x)\\frac{\\prod \\limits _{j}\\sigma (\\lambda _i^{m+1}(x)-\\lambda _j^m(x))\\sigma (\\lambda _i^{m+1}(x)-\\lambda _j^{m+1}(x+\\eta ))}{\\prod \\limits _{j}\\sigma (\\lambda _i^{m+1}(x)-\\lambda _j^{m}(x+\\eta ))\\prod \\limits _{j\\ne i}\\sigma (\\lambda _i^{m+1}(x)-\\lambda _j^{m+1}(x ))}.$ Note that $\\sum _i (f_i^m(x)-g_i^m(x))=0$ as the sum of residues of the elliptic function $\\varphi (\\lambda )=\\frac{\\prod \\limits _{j}\\sigma (\\lambda -\\lambda _j^{m}(x))\\sigma (\\lambda -\\lambda _j^{m+1}(x+\\eta ))}{\\prod \\limits _{j}\\sigma (\\lambda -\\lambda _j^{m+1}(x))\\sigma (\\lambda -\\lambda _j^{m}(x+\\eta ))}.$ The matrices $L^m(x)$ , $M^m(x)$ are $L^m_{ij}(x, z)=f_i^m(x)\\Phi \\Bigl (\\lambda _i^{m}(x+\\eta )-\\lambda _j^m(x), z\\Bigr ),$ $M^m_{ij}(x, z)=g_i^m(x)\\Phi \\Bigl (\\lambda _i^{m+1}(x)-\\lambda _j^m(x), z\\Bigr ).$ They depend on the spectral parameter $z$ .",
"The compatibility condition of the linear problems (REF ), (REF ) has the form of the fully discrete zero curvature equation $R^m(x):=L^{m+1}(x)M^m(x)-M^m (x+\\eta )L^m(x)=0.$ We have: $\\begin{array}{c}\\displaystyle {R^m_{ij}(x)=f_i^{m+1}(x)\\sum _k g_k^m(x)\\Phi (\\lambda _i^{m+1}(x+\\eta )-\\lambda _k^{m+1}(x))\\Phi (\\lambda _k^{m+1}(x)-\\lambda _j^{m}(x))}\\\\ \\\\\\displaystyle {-\\, g_i^{m}(x+\\eta )\\sum _k f_k^m(x) \\Phi (\\lambda _i^{m+1}(x+\\eta )-\\lambda _k^{m}(x+\\eta ))\\Phi (\\lambda _k^{m}(x+\\eta )-\\lambda _j^{m}(x))}.\\end{array}$ Cancellation of the leading singularity at $\\mu =0$ leads to the condition $f_i^{m+1}(x)\\sum _k g_k^m(x)-g_i^{m}(x+\\eta )\\sum _k f_k^m(x)=0$ Taking into account (REF ) we see that it is equivalent to $f_i^{m+1}(x)=g_i^{m}(x+\\eta ).$ Now we are going to prove that if (REF ) is satisfied, then $R_{ij}^m(x)=0$ , so the zero curvature equation is fulfilled.",
"The proof is along the lines of ref.",
"[42].",
"Using the identity (REF ), we rewrite (REF ) as $\\begin{array}{c}\\displaystyle {R_{ij}^m(x)=\\Phi (\\lambda _i^{m+1}(x+\\eta )-\\lambda _j^m(x))f_i^{m+1}(x)\\sum _k g_k^m(x)}\\\\ \\\\\\displaystyle {\\times \\Bigl (\\zeta (\\lambda _i^{m+1}(x+\\eta )-\\lambda _k^{m+1}(x))+\\zeta (\\lambda _k^{m+1}(x)-\\lambda _j^m(x))+ \\zeta (\\mu )-\\zeta (\\lambda _i^{m+1}(x+\\eta )-\\lambda _j^m(x)\\Bigr )}\\\\ \\\\\\displaystyle {-\\, \\Phi (\\lambda _i^{m+1}(x+\\eta )-\\lambda _j^m(x))g_i^{m}(x+\\eta )\\sum _k f_k^m(x)}\\\\ \\\\\\displaystyle {\\times \\Bigl (\\zeta (\\lambda _i^{m+1}(x+\\eta )\\!",
"-\\!",
"\\lambda _k^{m}(x+\\eta ))\\!",
"+\\!",
"\\zeta (\\lambda _k^{ m}(x+\\eta )\\!",
"-\\!",
"\\lambda _j^m(x))\\!",
"+ \\!",
"\\zeta (\\mu )\\!-\\!\\zeta (\\lambda _i^{m+1}(x+\\eta )\\!",
"-\\!",
"\\lambda _j^m(x)\\Bigr )}\\end{array}$ Using (REF ) and (REF ), we can represent it in the form $R_{ij}^m(x)=g_i^m(x+\\eta )\\Phi (\\lambda _i^{m+1}(x+\\eta )-\\lambda _j^m(x))G_{ij}^m(x),$ where $G_{ij}^m(x)=\\sum _k \\Bigl (g_k^m(x)\\zeta (\\lambda _i^{m+1}(x+\\eta )-\\lambda _k^{m+1}(x))+g_k^m(x)\\zeta (\\lambda _k^{m+1}(x)-\\lambda _j^{m}(x))$ $\\phantom{aaaaaaaaa}-f_k^m(x)\\zeta (\\lambda _i^{m+1}(x+\\eta )-\\lambda _k^{m}(x+\\eta ))-f_k^m(x)\\zeta (\\lambda _k^{m}(x+\\eta )-\\lambda _j^{m}(x))\\Bigr ).$ But $G_{ij}^m(x)$ is the sum of residues of the elliptic function $F(\\lambda )=\\Bigl (\\zeta (\\lambda _i^{m+1}(x+\\eta )-\\lambda )+\\zeta (\\lambda -\\lambda _j^m(x))\\Bigr )\\prod _j\\frac{\\sigma (\\lambda -\\lambda _j^{m}(x))\\sigma (\\lambda -\\lambda _j^{m+1}(x+\\eta ))}{\\sigma (\\lambda -\\lambda _j^{m+1}(x))\\sigma (\\lambda -\\lambda _j^{m}(x+\\eta ))}$ and, therefore, $G_{ij}^m(x)=0$ .",
"Finally, let us write down equations (REF ) explicitly: $\\prod _j \\frac{\\sigma (\\lambda _i^m(x)-\\lambda _j^{m}(x-\\eta ))\\sigma (\\lambda _i^m(x)-\\lambda _j^{m+1}(x))\\sigma (\\lambda _i^m(x)-\\lambda _j^{m-1}(x+\\eta ))}{\\sigma (\\lambda _i^m(x)-\\lambda _j^{m}(x+\\eta ))\\sigma (\\lambda _i^m(x)-\\lambda _j^{m-1}(x))\\sigma (\\lambda _i^m(x)-\\lambda _j^{m+1}(x-\\eta ))}=-\\frac{\\kappa _{m}(x\\!",
"-\\!",
"\\eta )}{\\kappa _{m-1}(x)}.$ Note that $\\frac{\\kappa _{m}(x\\!",
"-\\!",
"\\eta )}{\\kappa _{m-1}(x)}=\\frac{\\rho ^{m-1}(x)\\rho ^m(x+\\eta )\\rho ^{m+1}(x-\\eta )}{\\rho ^{m-1}(x+\\eta )\\rho ^m(x-\\eta )\\rho ^{m+1}(x)}.$ This is the field analog of the doubly discrete Ruijsenaars-Schneider system.",
"Similar equations were obtained in [43]."
],
[
"Conclusion",
"In this paper we have introduced two integrable models one of which is a natural lattice version of the other.",
"The first one is a finite-dimensional system which we call the Ruijsenaars-Schneider chain.",
"We show that it is gauge equivalent to a special case of the homogenous classical elliptic (XYZ) spin chain when residues of all Lax matrices in the chain are of rank one.",
"The second one is the field analogue of the Ruijsenaars-Schneider model with continuous space and time variables.",
"The definition of this model is the main result of the paper.",
"This is a (1+1)-dimensional model which admits a semi-discrete zero curvature (Zakharov-Shabat) representation for elliptic Lax pair with spectral parameter.",
"This model is obtained through a multi-pole ansatz for general elliptic solutions (elliptic families) of the 2D Toda lattice hierarchy.",
"Then we show that a natural space discretization of this model coincides with the Ruijsenaars-Schneider chain.",
"The fully discrete version of the model (i.e.",
"discrete in both space and time) is also introduced.",
"It is based on studying elliptic families of solutions to the Hirota bilinear difference equation [41] which is known to provide the integrable discretization of the 2D Toda equation.",
"The corresponding equations of motion for poles of the elliptic solutions are very similar to those obtained in [43] from a general ansatz for elliptic $L$ -$M$ pair.",
"We also discuss the continuum limit of the model, which coincides with the field analogue of the Calogero-Moser system introduced in [44] and reproduced in [3] as a dynamical system for poles of elliptic families of solutions to the Kadomtsev-Petviashvili equation.",
"Note that by construction the Ruijsenaars-Schneider chain is gauge equivalent to the elliptic spin chain.",
"A similar gauge equivalence exists between the (1+1)-dimensional Calogero-Moser field theory and the continuous Landau-Lifshitz equation [18], [45].",
"The exact relation between the obtained (1+1)-dimensional field analogue of the Ruijsenaars-Schneider model and the (semi-discrete) equations of the Landau-Lifshitz type will be discussed elsewhere."
],
[
"Weierstrass $\\sigma $ -, {{formula:178f3c7c-a57b-4d88-956c-603cfcd29f96}} - and {{formula:41e24c79-d960-488c-8b00-52ebde21f8c5}} -functions.",
"The $\\sigma $ -function with quasi-periods $2\\omega _1$ , $2\\omega _2$ such that ${\\rm Im} (\\omega _2/ \\omega _1 )>0$ is defined by the infinite product $\\sigma (x)=\\sigma (x |\\, \\omega _1 , \\omega _2)=x\\prod _{s\\ne 0}\\Bigl (1-\\frac{x}{s}\\Bigr )\\, e^{\\frac{x}{s}+\\frac{x^2}{2s^2}},\\quad s=2\\omega _1 m_1+2\\omega _2 m_2 \\quad \\mbox{with integer $m_1, m_2$}.$ It is connected with the Weierstrass $\\zeta $ - and $\\wp $ -functions by the formulas $\\zeta (x)=\\sigma ^{\\prime }(x)/\\sigma (x)$ , $\\wp (x)=-\\zeta ^{\\prime }(x)=-\\partial _x^2\\log \\sigma (x)$ .",
"We also need the function $\\Phi =\\Phi (x, z )$ defined as $\\Phi (x, z )=\\frac{\\sigma (x+z )}{\\sigma (z )\\sigma (x)}\\,e^{-\\zeta (z )x}.$ It has a simple pole at $x=0$ with residue 1 and the expansion $\\Phi (x, z )=\\frac{1}{x}-\\frac{1}{2}\\, \\wp (z ) x +\\ldots , \\qquad x\\rightarrow 0.$ The quasiperiodicity properties of the function $\\Phi $ are $\\Phi (x+2\\omega _{\\alpha } , z )=e^{2(\\zeta (\\omega _{\\alpha } )z -\\zeta (z )\\omega _{\\alpha } )}\\Phi (x, z ).$ In the main text we often suppress the second argument of $\\Phi $ writing simply $\\Phi (x, z )=\\Phi (x)$ .",
"We will also need the $x$ -derivative $\\Phi ^{\\prime }(x, z )=\\partial _x \\Phi (x, z )$ .",
"The function $\\Phi $ satisfies the following identities: $\\Phi ^{\\prime }(x)=\\Phi (x)\\Bigl (\\zeta (x+z )-\\zeta (x)-\\zeta (z )\\Bigr ),$ $\\Phi (x) \\Phi (y)=\\Phi (x+y)\\Bigl (\\zeta (x)+\\zeta (y)+\\zeta (z )-\\zeta (x+y+z )\\Bigr )$ which are used in the main text."
],
[
"Theta-functions.",
"The theta-functions with characteristics $a,b$ are defined as follows: $\\begin{array}{c}\\displaystyle {\\theta {\\left[\\begin{array}{c}a\\\\b\\end{array}\\right]}(z|\\, \\tau ) =\\sum _{j\\in \\mbox{Z}}}\\exp \\left(2\\pi \\imath (j+a)^2\\frac{\\tau }{2}+2\\pi \\imath (j+a)(z+b)\\right)\\,.\\end{array}$ In our paper, we consider the case of rational characteristics $a\\,,b\\in \\frac{1}{N}\\,\\mbox{Z}$ .",
"In particular, the odd theta function used in the paper ($\\theta _1(z)$ in the Jacobi notation) is $\\begin{array}{c}\\displaystyle {\\vartheta (z)=\\vartheta (z,\\tau )\\equiv -\\theta {\\left[\\begin{array}{c}1/2\\\\1/2\\end{array}\\right]}(z|\\, \\tau )\\,.",
"}\\end{array}$ The following quasi-periodicity properties hold.",
"For $a,b,a^{\\prime }\\in (1/N)\\mbox{Z}$ $\\begin{array}{c}\\displaystyle {\\theta {\\left[\\begin{array}{c}a\\\\b\\end{array}\\right]}(z+1|\\, \\tau )={\\bf {e}}(a)\\,\\theta {\\left[\\begin{array}{c}a\\\\b\\end{array}\\right]}(z|\\, \\tau )\\,,}\\end{array}$ $\\begin{array}{c}\\displaystyle {\\theta {\\left[\\begin{array}{c}a+1\\\\b\\end{array}\\right]}(z|\\, \\tau )=\\theta {\\left[\\begin{array}{c}a\\\\b\\end{array}\\right]}(z|\\, \\tau )\\,,}\\end{array}$ $\\begin{array}{c}\\displaystyle {\\theta {\\left[\\begin{array}{c}a\\\\b\\end{array}\\right]}(z+a^{\\prime }\\tau |\\, \\tau )={\\bf {e}}\\Big ( -{a^{\\prime }}^2\\frac{\\tau }{2}-a^{\\prime }(z+b) \\Big )\\,\\theta {\\left[\\begin{array}{c}a+a^{\\prime }\\\\b\\end{array}\\right]}(z|\\, \\tau )\\,,}\\end{array}$ where we denote $\\begin{array}{c}\\displaystyle {{\\bf {e}}(x):=\\exp (2\\pi \\imath x)}\\end{array}$ for brevity."
],
[
"The Kronecker function and the function $E_1$ .",
"We also use the following set of $N^2$ functions: $\\begin{array}{c}\\displaystyle {\\varphi _a(z,\\omega _a+\\eta )={\\bf {e}}(a_2z/N)\\,\\phi (z,\\omega _a+\\eta )\\,,\\quad \\omega _a=\\frac{a_1+a_2\\tau }{N}\\,,}\\end{array}$ where $a=(a_1, a_2)\\in \\mbox{Z}_N\\times \\mbox{Z}_N$ and $\\begin{array}{l}\\displaystyle {\\phi (z,u)=\\frac{\\vartheta ^{\\prime }(0)\\vartheta (z+u)}{\\vartheta (z)\\vartheta (u)}}\\end{array}$ is the Kronecker function.",
"It has a simple pole at $z=0$ with residue 1: $\\begin{array}{l}\\displaystyle {\\mathop {\\hbox{Res}}\\limits \\limits _{z=0}\\phi (z,u)=1\\,.",
"}\\end{array}$ The quasi-periodicity properties are as follows: $\\begin{array}{l}\\displaystyle {\\phi (z+1,u)= \\phi (z,u)\\,,\\qquad \\phi (z+\\tau ,u)= {\\bf {e}}(-u)\\phi (z,u)\\,.",
"}\\end{array}$ The expansion of $\\phi (z,u)$ near $z=0$ has the form $\\begin{array}{l}\\displaystyle {\\phi (z,u)=\\frac{1}{z}+E_1(u)+\\frac{E_1^2(u)-\\wp (u)}{2}+O(z^2),}\\end{array}$ where $\\begin{array}{l}\\displaystyle {E_1(u)=\\frac{\\vartheta ^{\\prime }(u)}{\\vartheta (u)}.",
"}\\end{array}$ It follows from the definition (REF ) that $\\begin{array}{l}\\displaystyle {\\partial _z\\phi (z,u)=(E_1(z+u)-E_1(z))\\phi (z,u)\\,,}\\\\ \\ \\\\\\displaystyle {\\partial _u\\phi (z,u)=(E_1(z+u)-E_1(u))\\phi (z,u)\\,.",
"}\\end{array}$ We also use a set of widely known addition formulae: $\\begin{array}{c}\\displaystyle {\\phi (z,u_1)\\phi (z,u_2)=\\phi (z,u_1+u_2)\\Big (E_1(z)+E_1(u_1)+E_1(u_2)-E_1(z+u_1+u_2)\\Big )\\,,}\\end{array}$ $\\begin{array}{c}\\displaystyle {\\phi (z_1, u_1) \\phi (z_2, u_2) = \\phi (z_1, u_1 + u_2) \\phi (z_2 - z_1, u_2) + \\phi (z_2, u_1 + u_2) \\phi (z_1 - z_2, u_1)}\\end{array}$ and $\\begin{array}{c}\\displaystyle {\\phi (z,u_1-v)\\phi (w,u_2+v)\\phi (z-w,v)-\\phi (z,u_2+v)\\phi (w,u_1-v)\\phi (z-w,u_1-u_2-v)=}\\\\ \\ \\\\\\displaystyle {=\\phi (z,u_1)\\phi (w,u_2)\\Big ( E_1(v)-E_1(u_1-u_2-v)+E_1(u_1-v)-E_1(u_2+v) \\Big )\\,.",
"}\\end{array}$"
],
[
"Relation to the Weierstrass functions.",
"The above definitions of the Weierstrass functions (REF )–(REF ) are easily related to those given in terms of theta-functions (REF )–(REF ) if we choose the periods to be $2\\omega _1=1$ , $2\\omega _2=\\tau $ : $\\displaystyle {\\zeta (z)=E_1(z)+2\\eta _0 z\\,,\\quad \\eta _0=-\\frac{1}{6}\\frac{\\vartheta ^{\\prime \\prime \\prime }(0)}{\\vartheta ^{\\prime }(0)}\\,,}$ $\\displaystyle {\\sigma (z)=\\frac{\\vartheta (z)}{\\vartheta ^{\\prime }(0)}\\,e^{\\eta _0 z^2}\\,,}$ $\\displaystyle {\\Phi (z,u)=\\phi (z,u)\\,e^{-z E_1(u)}\\,,}$ Under the substitution (REF ) the identities (REF )–(REF ) are transformed into (REF ) and (REF ) respectively.",
"The Weierstrass $\\wp $ -function appearing in (REF ) is $\\begin{array}{l}\\displaystyle {\\wp (u)=-\\partial ^2_u\\log \\vartheta (u)+\\frac{1}{3}\\frac{\\vartheta ^{\\prime \\prime \\prime }(0)}{\\vartheta ^{\\prime }(0)}\\,.",
"}\\end{array}$"
],
[
"Some relations for\ntheta-functions with rational characteristics.",
"Using definition (REF ), one cane rewrite the set of functions (REF ) in a slightly different form.",
"Set $\\begin{array}{c}\\displaystyle {\\theta _{\\alpha }(z,\\tau )=\\theta {\\left[\\begin{array}{c}\\frac{\\alpha _2}{N}+\\frac{1}{2}\\\\\\frac{\\alpha _1}{N}+\\frac{1}{2}\\end{array}\\right]}(z,\\tau )\\,,\\qquad \\alpha \\in \\mbox{Z}_N\\times \\mbox{Z}_N}\\end{array}$ Then for any $\\alpha $ we have $\\begin{array}{c}\\displaystyle {\\frac{\\theta _{\\alpha }(z+\\eta ,\\tau )}{\\theta _{\\alpha }(\\eta ,\\tau )}={\\bf {e}}(\\alpha _2z/N)\\frac{\\vartheta (z+\\eta +\\omega _\\alpha )}{\\vartheta (\\eta +\\omega _\\alpha )}}\\end{array}$ and, therefore, $\\begin{array}{c}\\displaystyle {\\varphi _\\alpha (z,\\eta +\\omega _\\alpha )=\\frac{\\vartheta ^{\\prime }(0)\\theta _{\\alpha }(z+\\eta ,\\tau )}{\\vartheta (z)\\theta _{\\alpha }(\\eta ,\\tau )}\\,.",
"}\\end{array}$ Introduce also $\\begin{array}{c}\\displaystyle {\\theta ^{(j)}(z)=\\theta {\\left[\\begin{array}{c}\\frac{1}{2}-\\frac{j}{N}\\\\\\frac{1}{2}\\end{array}\\right]}(z,N\\tau )\\,,\\qquad j\\in \\mbox{Z}_N\\,.",
"}\\end{array}$ Then for $-\\vartheta (z)$ we have $\\begin{array}{c}\\displaystyle {\\theta {\\left[\\begin{array}{c}\\frac{1}{2}\\\\\\frac{1}{2}\\end{array}\\right]}(z,\\tau )=C(\\tau )\\prod \\limits _{j=0}^{N-1}\\theta ^{(j)}(z)\\,,\\qquad \\displaystyle {C(\\tau )=\\frac{\\vartheta ^{\\prime }(0,\\tau )}{\\vartheta ^{\\prime }(0,N\\tau )}\\,\\frac{1}{\\prod \\limits _{j=1}^{N-1}\\theta ^{(j)}(0)}\\,,}}\\end{array}$ so that the following relation holds: $\\begin{array}{c}\\displaystyle {\\frac{\\vartheta ^{\\prime }(0,\\tau )}{\\vartheta (z,\\tau )}\\,\\frac{\\prod \\limits _{j=0}^{N-1}\\theta ^{(j)}(z)}{\\prod \\limits _{j=1}^{N-1}\\theta ^{(j)}(0)}=-\\vartheta ^{\\prime }(0,N\\tau )\\,.",
"}\\end{array}$ Consider the matrix $\\begin{array}{c}\\displaystyle {X_{ij}(x_j)=\\vartheta \\left[ \\begin{array}{c}\\frac{1}{2}-\\frac{i}{N} \\\\ \\frac{N}{2}\\end{array} \\right] \\left(Nx_j\\left.\\right|N\\tau \\right)\\,.",
"}\\end{array}$ Then the following determinant of the Vandermonde type formula holds [16]: $\\begin{array}{c}\\displaystyle {\\det X=C_N(\\tau )\\,\\vartheta (\\sum \\limits _{k=1}^Nx_k)\\prod \\limits _{i<j}\\vartheta (x_j-x_i)\\,,\\quad \\quad C_N(\\tau )=\\frac{(-1)^{N}}{(\\imath \\eta (\\tau ))^{\\frac{(N-1)(N-2)}{2}}}\\,,}\\end{array}$ where $\\eta (\\tau )$ is the Dedekind eta-function: $\\begin{array}{c}\\displaystyle {\\eta (\\tau )=e^{\\frac{\\pi \\imath \\tau }{12}}\\prod \\limits _{k=1}^\\infty (1-e^{2\\pi \\imath \\tau k})=\\Big (\\frac{\\vartheta ^{\\prime }(0)}{2\\pi }\\Big )^{1/3}\\,.",
"}\\end{array}$"
],
[
"Finite Fourier transformation on $\\mbox{Z}_N$ .",
"For any $m\\in \\mbox{Z}$ and $N\\in \\mbox{Z}_+$ $\\begin{array}{c}\\displaystyle {e^{2\\pi \\imath m\\eta }\\phi (N\\eta ,z+m\\tau |N\\tau )=\\frac{1}{N}\\sum \\limits _{k=0}^{N-1}e^{-2\\pi \\imath m\\frac{k}{N}}\\phi (z,\\eta +\\frac{k}{N}|\\tau )\\,,}\\end{array}$ $\\begin{array}{c}\\displaystyle {\\phi (z,\\eta |\\tau )=\\sum \\limits _{k=0}^{N-1}e^{2\\pi \\imath zk}\\phi (Nz,\\eta +k\\tau |N\\tau )\\,.",
"}\\end{array}$"
],
[
"Matrix basis.",
"Consider the pair of $N\\times N$ matrices $\\begin{array}{c}\\displaystyle {(Q_1)_{kl}=\\delta _{kl}\\exp (\\frac{2\\pi \\imath }{{ N}}k)\\,,\\ \\ \\ (Q_2)_{kl}=\\delta _{k-l+1=0\\,{\\hbox{\\tiny {mod}}}\\,{ N}}\\,.",
"}\\end{array}$ They satisfy the properties $\\begin{array}{c}\\displaystyle {Q_2^{a_2} Q_1^{a_1}=\\exp \\left(\\frac{2\\pi \\imath }{{ N}}\\,a_1a_2\\right)Q_1^{a_1} Q_2^{a_2}\\,,\\ a_{1,2}\\in \\mbox{Z};\\qquad Q_1^{ N}=Q_2^{ N}=1_{{ N}\\times { N}}\\,,}\\end{array}$ so that these matrices represent the generators of the Heisenberg group.",
"Let us construct a special basis in ${\\rm Mat}(N,\\mbox{C})$ in terms of (REF ) in the following way: $\\begin{array}{c}\\displaystyle {T_a=T_{a_1 a_2}=\\exp \\left(\\frac{\\pi \\imath }{{ N}}\\,a_1a_2\\right)Q_1^{a_1}Q_2^{a_2}\\,,\\quad a=(a_1,a_2)\\in \\mbox{Z}_{ N}\\times \\mbox{Z}_{ N}\\,.",
"}\\end{array}$ In particular, $T_0=T_{(0,0)}=1_N$ .",
"For the product we have $\\begin{array}{c}\\displaystyle {T_\\alpha T_\\beta =\\kappa _{\\alpha ,\\beta } T_{\\alpha +\\beta }\\,,\\quad \\kappa _{\\alpha ,\\beta }=\\exp \\left(\\frac{\\pi \\imath }{N}(\\beta _1\\alpha _2-\\beta _2\\alpha _1)\\right)\\,,\\quad \\alpha +\\beta =(\\alpha _1+\\beta _1,\\alpha _2+\\beta _2)}\\end{array}$ Also $\\begin{array}{c}\\displaystyle {{\\rm tr}(T_\\alpha T_\\beta )=N\\delta _{\\alpha +\\beta ,(0,0)}\\,.",
"}\\end{array}$ Let us perform the transformation relating the standard matrix basis $E_{ij}$ in $ {\\rm Mat}(N,\\mbox{C}) $ , given by $(E_{ij})_{kl}=\\delta _{ik}\\delta _{jl}$ , with (REF ).",
"For the pair of matrices $Q_{1,2}$ (REF ) and integer numbers $a_1,a_2$ we have $\\begin{array}{c}\\displaystyle {Q_1^{a_1}=\\sum \\limits _{k=1}^N E_{kk}\\,{\\bf {e}}(\\frac{k a_1}{N})\\,,\\qquad Q_2^{a_2}=\\sum \\limits _{k=1}^N E_{k-a_2,k}\\,,}\\end{array}$ where in the last sum we assume the value of index $k-a_2$ modulo $N$ .",
"Then for the basis matrix $T_a$ (REF ) one gets $\\begin{array}{c}\\displaystyle {T_a={\\bf {e}}(-\\frac{a_1 a_2}{2N})\\sum \\limits _{k=1}^N E_{k-a_2,k}\\,{\\bf {e}}(\\frac{k a_1}{N})\\,.",
"}\\end{array}$ For an arbitrary matrix $B=\\sum _{i,j=1}^N E_{ij}B_{ij}\\in {\\rm Mat}(N,\\mbox{C}) $ its components $B_a=B_{(a_1,a_2)}$ in the basis $T_a$ can be found using (REF ) and (REF ): $\\begin{array}{c}\\displaystyle {B_a=\\frac{1}{N}\\,{\\rm tr}(B T_{-a})=\\frac{1}{N}\\,{\\bf {e}}(-\\frac{a_1a_2}{2N})\\sum \\limits _{k=1}^N B_{k,k+a_2}\\,{\\bf {e}}(-\\frac{a_1 k}{N})\\,.",
"}\\end{array}$ Similarly, given a set of components $B_{(\\alpha _1,\\alpha _2)}$ , $\\alpha _{1,2}\\in \\mbox{Z}_N$ for a matrix $B\\in {\\rm Mat}(N,\\mbox{C}) $ in the basis $\\lbrace T_\\alpha \\rbrace $ we have the following expression for its components $B_{ij}$ in the standard basis: $\\begin{array}{c}B_{ij}=\\left\\lbrace \\begin{array}{l}\\displaystyle {\\sum \\limits _{\\alpha _1=0}^{N-1}B_{(\\alpha _1,j-i)}{\\bf {e}}\\Big (\\frac{\\alpha _1(i+j)}{2N}\\Big )\\,,\\quad j\\ge i\\,,}\\\\\\displaystyle {\\sum \\limits _{\\alpha _1=0}^{N-1}B_{(\\alpha _1,j-i+N)}{\\bf {e}}\\Big (\\frac{\\alpha _1(i+j-N)}{2N}\\Big )\\,,\\quad j<i\\,.",
"}\\end{array}\\right.\\end{array}$"
],
[
"The Baxter-Belavin $R$ -matrix.",
"We use the Baxter-Belavin elliptic quantum $R$ -matrix in the following form: $\\begin{array}{c}\\displaystyle {R_{12}^\\hbar (z)=\\frac{1}{N}\\sum \\limits _{a\\in \\,\\mbox{Z}_{ N}\\times \\mbox{Z}_{ N}} T_a\\otimes T_{-a} \\varphi _a(z,\\omega _a+\\hbar )\\in {\\rm Mat}(N,\\mbox{C})^{\\otimes 2}\\,.",
"}\\end{array}$ It is equivalently written in the standard basis as follows [37]: $\\begin{array}{c}\\displaystyle {R_{12}^\\hbar (z)=\\sum \\limits _{i,j,k,l=1}^N R_{ij,kl}\\, E_{ij}\\otimes E_{kl}\\,,}\\end{array}$ $\\begin{array}{c}\\displaystyle {R_{ij,kl}=-\\vartheta ^{\\prime }(0,N\\tau )\\frac{ \\theta ^{(i-k)}(z+N\\hbar ) }{ \\theta ^{(j-k)}(z)\\theta ^{(i-j)}(N\\hbar ) }\\,\\delta _{i+k=j+l\\ {\\rm mod}\\ N}\\,.",
"}\\end{array}$ It is also convenient to write it in terms of the Kronecker function (REF ).",
"For this purpose we need the identity $\\begin{array}{c}\\displaystyle {-\\vartheta ^{\\prime }(0,N\\tau )\\frac{ \\theta ^{(a+b)}(z+u) }{ \\theta ^{(a)}(z)\\theta ^{(b)}(u) }={\\bf {e}}\\Big ( \\frac{ab\\tau -au-bz}{N} \\Big )\\phi (z-a\\tau ,u-b\\tau |N\\tau )}\\\\ \\ \\\\\\displaystyle {={\\bf {e}}(-u\\frac{a}{N})\\,\\varphi _{(0,-\\frac{b}{N})}(z-a\\tau ,u-b\\tau |N\\tau )\\,.",
"}\\end{array}$ Then $\\begin{array}{c}\\displaystyle {R_{ij,kl}=\\delta _{i+k=j+l\\ {\\rm mod}\\ N}{\\bf {e}}\\Big ( \\frac{(k-j)(j-i)\\tau +(k-j)N\\hbar +(j-i)z}{N} \\Big )}\\\\ \\ \\\\\\displaystyle {\\times \\phi (z+(k-j)\\tau ,N\\hbar +(j-i)\\tau |N\\tau )\\,.",
"}\\end{array}$ Equivalence between different representations can be shown by using the relation between the bases $E_{ij}$ , $T_a$ and the Fourier formulae (REF )–(REF )."
],
[
"IRF-Vertex correspondence.",
"The matrix (REF ) participates in the IRF-Vertex relation $\\begin{array}{c}\\displaystyle {g_2(z_2,q)\\,g_1(z_1,q+N\\hbar ^{(2)})\\,R^{\\hbox{\\tiny {F}}}_{12}(\\hbar ,z_1-z_2|\\,q)=R^\\hbar _{12}(\\hbar ,z_1-z_2)\\,g_1(z_1,q)\\, g_2(z_2,q+N\\hbar ^{(1)})}\\end{array}$ between the (vertex type) Baxter-Belavin $R$ -matrix (REF ) and the (IRF-type) Felder's dynamical $R$ -matrix [46]: $\\begin{array}{c}\\displaystyle {R^{\\hbox{\\tiny {F}}}_{12}(\\hbar ,z_1-z_2|\\,q)}\\\\ \\ \\\\\\displaystyle {=\\sum \\limits _{i\\ne j}E_{ii}\\otimes E_{jj}\\, \\phi (N\\hbar ,-q_{ij})+\\sum \\limits _{i\\ne j}E_{ij}\\otimes E_{ji}\\, \\phi (z_1-z_2,q_{ij})+\\phi (N\\hbar ,z_1-z_2)\\sum \\limits _{i}E_{ii}\\otimes E_{ii}\\,.",
"}\\end{array}$ The shift of argument $g_1(z_1,q+N\\hbar ^{(2)})$ in (REF ) is understood as $\\begin{array}{c}\\displaystyle {g_1(z_1,q+N\\hbar ^{(2)})=P_2^{N\\hbar }\\,g_1(z_1,q) P_2^{-N\\hbar } \\,,\\quad P_2^\\hbar =\\sum \\limits _{k=1}^N 1_{N\\times N}\\otimes E_{kk}\\exp (\\hbar \\frac{\\partial }{\\partial q_k})\\,.",
"}\\end{array}$"
],
[
"Properties of the intertwining matrix.",
"The matrix $g(z,q)$ is degenerated at $z=0$ due to (REF ): $\\begin{array}{c}\\displaystyle {\\det \\Xi (z,q)=C_N(\\tau )\\,\\vartheta (z)\\prod \\limits _{i<j}\\vartheta (q_i-q_j)\\,,}\\end{array}$ and the factor $\\vartheta (z)$ comes from the fact that the sum of coordinates (in the center of masses frame) equals zero.",
"The matrix $g(z)$ (REF ) satisfies the following properties (see [22] for a review): 1.",
"The matrix $g(z,q)$ is degenerated at $z=0$ (REF ).",
"2.",
"The matrix $g(0,q)$ has one-dimensional kernel generated by the vector-column $\\rho $ : $\\begin{array}{c}\\displaystyle {g(0,q)\\rho =0\\,,\\quad \\rho =(1,1,...,1)^T\\in \\mbox{C}^N\\,.",
"}\\end{array}$ Properties of this type were described in [37].",
"Their proof can be also found in [18].",
"Let us consider $g^{-1}(z,q)$ near $z=0$ : $\\begin{array}{c}\\displaystyle {g^{-1}(z,q)=\\frac{1}{z}\\,{\\breve{g}}(0,q)+A(q)+O(z)\\,,\\quad {\\breve{g}}(0,q)=\\mathop {\\hbox{Res}}\\limits \\limits _{z=0}\\,g^{-1}(z,q)\\,.",
"}\\end{array}$ Then the matrix ${\\breve{g}}(0)$ is of rank oneLocally, in some basis $g(z,q)$ is represented in the form ${\\rm diag}(z,1,...,1)$ .",
"Therefore, ${\\breve{g}}(0,q)$ has $N-1$ zero eigenvalues.",
"$\\begin{array}{c}\\displaystyle {{\\breve{g}}(0)=\\rho \\otimes \\upsilon \\,,\\quad \\upsilon =\\frac{1}{N}\\,\\rho ^T{\\breve{g}}(0,q)\\in \\mbox{C}^N\\,.",
"}\\end{array}$ Below we derive an explicit expression for the inverse of the matrix $g(z,q)$ ."
],
[
"IRF-Vertex correspondence for semidynamical $R$ -matrix.",
"In [47] the following (semidynamical) $R$ -matrix was used for quantization of the Ruijsenaars-Schneider model: $\\begin{array}{c}\\displaystyle {R^{\\hbox{\\tiny {ACF}}}_{12}(\\hbar ,z_1,z_2|\\,q)=\\sum \\limits _{i\\ne j}E_{ii}\\otimes E_{jj}\\, \\phi (N\\hbar ,-q_{ij})+\\sum \\limits _{i\\ne j}E_{ij}\\otimes E_{ji}\\, \\phi (z_1-z_2,-q_{ij})-}\\\\ \\ \\\\\\displaystyle {-\\sum \\limits _{i\\ne j}E_{ij}\\otimes E_{jj}\\, \\phi (z_1+N\\hbar ,-q_{ij})+\\sum \\limits _{i\\ne j}E_{jj}\\otimes E_{ij}\\, \\phi (z_2,-q_{ij})+}\\\\ \\ \\\\\\displaystyle {+\\Big (E_1(N\\hbar )+E_1(z_1-z_2)+E_1(z_2)-E_1(z_1+N\\hbar )\\Big )\\sum \\limits _{i}E_{ii}\\otimes E_{ii}\\,,}\\end{array}$ where $E_1$ is defined in (REF ).",
"This $R$ -matrix satisfies the quantum Yang-Baxter equation with shifted spectral parameters.",
"Following [48] let us write down its relation to the Baxter-Belavin $R$ -matrix (REF ) in the form of type (REF ): $\\begin{array}{c}\\displaystyle {R^\\hbar _{12}(z_1-z_2)=g_1(z_1+N\\hbar ,q)\\,g_2(z_2,q)\\,R^{\\hbox{\\tiny {ACF}}}_{12}(\\hbar ,z_1,z_2|\\,q)\\, g_2^{-1}(z_2+N\\hbar ,q)\\,g_1^{-1}(z_1,q)\\,.",
"}\\end{array}$ Multiplying both sides by $g_2^{-1}(z_2,q)$ and evaluating residue at $z_2=0$ , we get the following useful formula Note that in the $N=1$ case (REF ) boils down to the definition (REF ) of the Kronecker function.",
"A similarity of the quantum $R$ -matrix (REF ) with the Kronecker function underlies the so-called associative Yang-Baxter equation.",
"See [48] and references therein.",
": $\\begin{array}{c}\\displaystyle {{\\breve{g}}_2(0,q)\\,R^\\hbar _{12}(z)=g_1(z+N\\hbar ,q)\\,\\mathcal {O}_{12}\\, g_2^{-1}(N\\hbar ,q)\\,g_1^{-1}(z,q)\\,,}\\end{array}$ where ${\\breve{g}}(0)$ is given by (REF ) and $\\begin{array}{c}\\displaystyle {\\mathcal {O}_{12}=\\sum \\limits _{i,j=1}^N E_{ii}\\otimes E_{ji}\\,.",
"}\\end{array}$ For an arbitrary matrix $T=\\sum _{i,j}E_{ij}T_{ij}\\in {\\rm Mat}(N,\\mbox{C}) $ we have $\\begin{array}{c}\\displaystyle {{\\rm tr}_2 \\left(\\mathcal {O}_{12}T_2\\right)=\\sum \\limits _{i=1}^NE_{ii}\\sum \\limits _{j=1}^N T_{ij}\\,.",
"}\\end{array}$ Besides (REF ), we use its degeneration (see the classical limit (REF ) below) $\\hbar \\rightarrow 0$Relation (REF ) appears in the $\\hbar ^0$ order, while in $\\hbar ^{-1}$ order one has ${\\breve{g}}_2(0)=g_1(z){\\mathcal {O}}_{12}\\,g_1^{-1}(z)\\,{\\breve{g}}_2(0)$ , which is true due to the property (REF ).",
": $\\begin{array}{c}\\displaystyle {{\\breve{g}}_2(0,q)\\,r_{12}(z)=g_1^{\\prime }(z)\\, \\mathcal {O}_{12}\\, {\\breve{g}}_2(0)\\, g_1^{-1}(z)+g_1(z)\\, \\mathcal {O}_{12}\\,A_2\\, g_1^{-1}(z)\\,,}\\end{array}$ where $A$ comes from the expansion (REF ) and $g^{\\prime }(z)$ is the derivative of $g(z)$ with respect to $z$ ."
],
[
"Change of variables.",
"Here we show how to obtain (REF ) using the factorization formula (REF ) for the Ruijsenaars-Schneider Lax matrix (REF ).",
"Let us compute the $a=(a_1, a_2)$ -component of the Lax matrix (REF ) $\\begin{array}{l}\\displaystyle {{\\mathcal {L}}^\\eta _{ij}(z)=\\frac{\\vartheta ^{\\prime }(0)}{\\vartheta (\\eta )}\\sum \\limits _{m=1}^N\\Xi _{im}(z+N\\eta ,q)e^{p_m/c}\\,\\Xi _{mj}^{-1}(z,q)\\,.",
"}\\end{array}$ Plugging it into (REF ), we get $\\begin{array}{l}\\displaystyle {{\\mathcal {L}}^\\eta _{a}(z)=\\frac{1}{N}\\,{\\bf {e}}(-\\frac{a_1a_2}{2N})\\frac{\\vartheta ^{\\prime }(0)}{\\vartheta (\\eta )}\\sum \\limits _{k,m=1}^N\\Xi _{km}(z+N\\eta ,q)e^{p_m/c}\\,\\Xi _{m,k+a_2}^{-1}(z,q)\\,{\\bf {e}}(-\\frac{a_1 k}{N})\\,.",
"}\\end{array}$ From (REF ) we know that ${\\mathcal {L}}^\\eta _{a}(z)=S_a\\varphi _a(z,\\omega _a+\\eta )$ .",
"Therefore, we could find $S_a$ from ${\\mathcal {L}}^\\eta _{a}(z)$ , which we are going to compute.",
"Let us represent (REF ) in the form $\\begin{array}{l}\\displaystyle {{\\mathcal {L}}^\\eta _{a}(z)=\\sum \\limits _{m=1}^N e^{p_m/c}\\,{\\mathcal {L}}^\\eta _{a;m}(z)\\,,}\\end{array}$ $\\begin{array}{l}\\displaystyle {{\\mathcal {L}}^\\eta _{a;m}(z)=\\frac{1}{N}\\,{\\bf {e}}(-\\frac{a_1a_2}{2N})\\frac{\\vartheta ^{\\prime }(0)}{\\vartheta (\\eta )}\\sum \\limits _{k=1}^N\\Xi _{km}(z+N\\eta ,q)\\Xi _{m,k+a_2}^{-1}(z,q)\\,{\\bf {e}}(-\\frac{a_1 k}{N})\\,.",
"}\\end{array}$ Our aim now is to evaluate the latter expression.",
"For this purpose we need the properties (REF )–(REF ).",
"Using explicit form of the matrix $\\Xi $ (REF ), it easy to see from (REF ) that $\\begin{array}{l}\\displaystyle {\\Xi _{km}(z+N\\eta +a_1,q)=(-1)^{a_1}{\\bf {e}}(-\\frac{a_1 k}{N})\\,\\Xi _{km}(z+N\\eta ,q)\\,.",
"}\\end{array}$ Therefore, $\\begin{array}{l}\\displaystyle {{\\mathcal {L}}^\\eta _{a;m}(z)=\\frac{1}{N}\\,{\\bf {e}}(-\\frac{a_1a_2}{2N})\\,(-1)^{a_1}\\frac{\\vartheta ^{\\prime }(0)}{\\vartheta (\\eta )}\\sum \\limits _{k=1}^N\\Xi _{km}(z+N\\eta +a_1,q)\\Xi _{m,k+a_2}^{-1}(z,q)\\,.",
"}\\end{array}$ Next, add and subtract $a_2\\tau $ to the argument of $\\Xi _{km}(z+N\\eta +a_1,q)$ .",
"Then using (REF ) with $a^{\\prime }=-a_2/N$ we obtain $\\begin{array}{c}\\displaystyle {\\Xi _{km}(z+N\\eta +a_1,q)=\\vartheta \\left[ \\begin{array}{c}\\frac{1}{2}-\\frac{k}{N} \\\\ \\frac{N}{2}\\end{array} \\right] \\left(z-N{\\bar{q}}_m+N\\eta +a_1+a_2\\tau -\\frac{a_2}{N}\\,N\\tau \\left.\\right|N\\tau \\right)}\\\\ \\ \\\\\\displaystyle {={\\bf {e}}\\Big ( -\\frac{a_2^2}{2N}\\tau +\\frac{a_2}{N}(z-N{\\bar{q}}_m+N\\eta +a_1+a_2\\tau +\\frac{N}{2}) \\Big )\\,\\Xi _{k+a_2,m}(z+N(\\eta +\\omega _a),q)\\,,}\\end{array}$ where the notation $\\omega _a$ (REF ) is used.",
"Plugging it into (REF ), we arrive at $\\begin{array}{c}\\displaystyle {{\\mathcal {L}}^\\eta _{a;m}(z)=\\frac{1}{N}\\,{\\bf {e}}(\\frac{a_1a_2}{2N}+\\frac{a_2^2}{2N}\\tau )\\,(-1)^{a_1+a_2}{\\bf {e}}(a_2(\\eta -{\\bar{q}}_m)){\\bf {e}}(z\\frac{a_2}{N})}\\\\ \\ \\\\\\displaystyle {\\times \\frac{\\vartheta ^{\\prime }(0)}{\\vartheta (\\eta )}\\sum \\limits _{k=1}^N\\Xi _{m,k+a_2}^{-1}(z,q)\\Xi _{k+a_2,m}(z+N(\\eta +\\omega _a),q)\\,.",
"}\\end{array}$ Finally, we use (REF ) for the Ruijsenaars-Schneider Lax matrix (REF )–(REF ).",
"Namely, we need the $i=j=m$ diagonal element of (REF ) with $\\eta $ being replaced by $\\eta +\\omega _a$ (except for the common factor $\\vartheta ^{\\prime }(0)/\\vartheta (\\eta )$ ).",
"This yields $\\begin{array}{c}\\displaystyle {{\\mathcal {L}}^\\eta _{a;m}(z)=\\frac{1}{N}\\,{\\bf {e}}(\\frac{a_2}{2}\\,\\omega _a)\\,(-1)^{a_1+a_2}}\\\\ \\ \\\\\\displaystyle {\\times {\\bf {e}}(a_2(\\eta -{\\bar{q}}_m))\\varphi _a(z,\\omega _a+\\eta )\\frac{\\vartheta (\\eta +\\omega _\\alpha )}{\\vartheta (\\eta )}\\prod \\limits _{l:\\,l\\ne m}^N\\frac{\\vartheta (q_m-q_l-\\eta -\\omega _a)}{\\vartheta (q_m-q_l)}\\,.",
"}\\end{array}$ Returning back to (REF ) and canceling $\\varphi _a(z,\\omega _a+\\eta )$ , we find the final answer $\\begin{array}{c}\\displaystyle {S_a=\\frac{(-1)^{a_1+a_2}}{N}\\,{\\bf {e}}(\\frac{a_2}{2}\\,\\omega _a)\\,\\sum \\limits _{m=1}^N e^{p_m/c}{\\bf {e}}(a_2(\\eta -{\\bar{q}}_m))\\frac{\\vartheta (\\eta +\\omega _\\alpha )}{\\vartheta (\\eta )}\\prod \\limits _{l:\\,l\\ne m}^N\\frac{\\vartheta (q_m-q_l-\\eta -\\omega _a)}{\\vartheta (q_m-q_l)}\\,.",
"}\\end{array}$"
],
[
"Inverse of the\nmatrix $\\Xi (z,q)$ .",
"Consider the set of matrices with components (REF ) in the basis $T_a$ : $\\begin{array}{c}\\displaystyle {{\\mathcal {L}}_{;m}^\\eta (z)=\\sum \\limits _{a}{\\mathcal {L}}_{a;m}^\\eta (z)T_a\\in {\\rm Mat}(N,\\mbox{C}) \\,,\\quad m=1,...,N\\,.",
"}\\end{array}$ It follows from its initial definition (REF ), (REF ) that the matrix elements in the standard basis are of the form: $\\begin{array}{l}\\displaystyle {{\\mathcal {L}}^\\eta _{ij;m}(z)=\\frac{\\vartheta ^{\\prime }(0)}{\\vartheta (\\eta )}\\Xi _{im}(z+N\\eta ,q)\\Xi _{mj}^{-1}(z,q)\\,.",
"}\\end{array}$ Therefore, $\\begin{array}{c}\\displaystyle {\\Xi _{mj}^{-1}(z,q)=\\frac{\\vartheta (\\eta )}{\\vartheta ^{\\prime }(0)}\\frac{{\\mathcal {L}}^\\eta _{ij;m}(z)}{\\Xi _{im}(z+N\\eta ,q)}\\,.",
"}\\end{array}$ To get an explicit expression, we need to compute the matrices ${\\mathcal {L}}_{;m}^\\eta (z)$ , $m=1,...,N$ in the standard basis.",
"For this purpose substitute (REF ) into (REF ) with $B_a={\\mathcal {L}}^\\eta _{a;m}(z)$ .",
"Both cases in the r.h.s.",
"of (REF ) provide the same answer (the latter is verified directly using the transformation properties (REF )–(REF ) for the theta-function (REF )): $\\begin{array}{c}\\displaystyle {\\frac{\\vartheta (\\eta )}{\\vartheta ^{\\prime }(0)}{\\mathcal {L}}^\\eta _{ij;m}(z)=\\frac{1}{N}\\sum \\limits _{a_1=0}^{N-1}{\\bf {e}}\\Big (\\frac{a_1}{2N}(i+j)\\Big ){\\bf {e}}\\Big (\\frac{j-i}{2}\\,\\omega _{(a_1,j-i)}\\Big )\\,(-1)^{a_1+j-i}}\\\\ \\ \\\\\\displaystyle {\\times {\\bf {e}}\\Big ((j-i)(\\eta -{\\bar{q}}_m)\\Big ){\\bf {e}}\\Big (z\\frac{j-i}{N}\\Big )\\frac{\\vartheta (z+\\eta +\\omega _{(a_1,j-i)})}{\\vartheta (z)}\\prod \\limits _{l:\\,l\\ne m}^N\\frac{\\vartheta (q_m-q_l-\\eta -\\omega _{(a_1,j-i)})}{\\vartheta (q_m-q_l)}\\,,}\\end{array}$ where $\\omega _{(a_1,j-i)}=\\frac{a_1+(j-i)\\tau }{N}$ .",
"Dividing this expression by $\\Xi _{im}(z+N\\eta ,q)$ we obtain $\\Xi _{mj}^{-1}(z,q)$ (REF ).",
"Notice that by construction the r.h.s.",
"of (REF ) is independent of $\\eta $ , so we put $\\eta =0$ in the final answer since all entering functions are regular in $\\eta $ .",
"Also, the r.h.s.",
"of (REF ) is independent of index $i$ .",
"We fix it as $i=N$ .",
"Finally, using $\\Xi _{Nm}(z+N\\eta ,q)=-\\vartheta (z+\\frac{N-1}{2}+N\\eta -N{\\bar{q}}_m|N\\tau )$ we obtain $\\begin{array}{c}\\displaystyle {\\Xi _{ij}^{-1}(z,q)=\\frac{(-1)^{j+1}}{N\\vartheta (z+\\frac{N-1}{2}-N{\\bar{q}}_i|N\\tau )}}\\\\ \\ \\\\\\displaystyle {\\times \\sum \\limits _{a_1=0}^{N-1}{\\bf {e}}\\Big (\\frac{a_1j}{2N}+j\\,\\frac{a_1+j\\tau }{2N}\\Big ){\\bf {e}}(-j{\\bar{q}}_i){\\bf {e}}\\Big (z\\frac{j}{N}\\Big )\\frac{\\vartheta (z+\\frac{a_1+j\\tau }{N})}{\\vartheta (z)}\\prod \\limits _{l:\\,l\\ne i}^N\\frac{\\vartheta (q_i-q_l-\\frac{a_1+j\\tau }{N})}{\\vartheta (q_i-q_l)}\\,.",
"}\\end{array}$ Equivalently, for the matrix $g(z,q)$ (REF ) we have $\\begin{array}{c}\\displaystyle {g_{ij}^{-1}(z,q)=\\frac{(-1)^{j+1}}{N\\vartheta (z+\\frac{N-1}{2}-N{\\bar{q}}_i|N\\tau )}}\\\\ \\ \\\\\\displaystyle {\\times \\sum \\limits _{a_1=0}^{N-1}{\\bf {e}}\\Big (\\frac{a_1j}{2N}+j\\,\\frac{a_1+j\\tau }{2N}\\Big ){\\bf {e}}(-j{\\bar{q}}_i){\\bf {e}}\\Big (z\\frac{j}{N}\\Big )\\frac{\\vartheta (z+\\frac{a_1+j\\tau }{N})}{\\vartheta (z)}\\prod \\limits _{l:\\,l\\ne i}^N\\vartheta \\Big (q_i-q_l-\\frac{a_1+j\\tau }{N}\\Big )\\,.",
"}\\end{array}$"
],
[
"Derivation of $S=\\xi \\otimes \\psi $ .",
"The matrix ${\\breve{g}}(0,q)$ (REF ) is easily calculated from (REF ): $\\begin{array}{c}\\displaystyle {{\\breve{g}}_{ij}(0,q)=\\frac{(-1)^{j}}{N\\vartheta (\\frac{N-1}{2}-N{\\bar{q}}_i|N\\tau )}}\\\\ \\ \\\\\\displaystyle {\\times {\\bf {e}}\\Big (\\frac{j^2\\tau }{2N}\\Big ){\\bf {e}}(-j{\\bar{q}}_i)\\frac{1}{\\vartheta ^{\\prime }(0)}\\sum \\limits _{a_1=0}^{N-1}{\\bf {e}}\\Big (\\frac{a_1j}{N}\\Big )\\prod \\limits _{l=1}^N\\vartheta \\Big (q_i-q_l-\\frac{a_1+j\\tau }{N}\\Big )\\,.",
"}\\end{array}$ Note that due to (REF ) the r.h.s.",
"of (REF ) is independent of the index $i$ , so that the functions $\\begin{array}{c}\\displaystyle {f_j(x,q)=\\frac{{\\bf {e}}(-jx)}{\\vartheta (\\frac{N-1}{2}-Nx|N\\tau )}\\sum \\limits _{a_1=0}^{N-1}{\\bf {e}}\\Big (\\frac{a_1j}{N}\\Big )\\prod \\limits _{l=1}^N\\vartheta \\Big (x-{\\bar{q}}_l-\\frac{a_1+j\\tau }{N}\\Big )}\\end{array}$ obey the property $\\begin{array}{c}\\displaystyle {f_j(q)=f_j(q_i,q)=f_j(q_k,q)\\quad \\mbox{for all $i,j,k$}\\,.",
"}\\end{array}$ In this notation $\\begin{array}{c}\\displaystyle {{\\breve{g}}_{ij}(0,q)=\\frac{(-1)^{j}}{N\\vartheta ^{\\prime }(0)}\\,{\\bf {e}}\\Big (\\frac{j^2\\tau }{2N}\\Big )f_j(q)\\,.",
"}\\end{array}$ Finally, using (REF )-(REF ), we find the change of variables (REF ) in the form $S=\\xi \\otimes \\psi $ with $\\begin{array}{c}\\displaystyle {\\xi _i=\\frac{\\vartheta ^{\\prime }(0)}{\\vartheta (\\eta )}\\sum \\limits _{k=1}^Ng_{ik}(N\\eta )\\,e^{p_k/c}\\,,\\qquad \\psi _j=\\frac{(-1)^{j}}{N\\vartheta ^{\\prime }(0)}\\,{\\bf {e}}\\Big (\\frac{j^2\\tau }{2N}\\Big )f_j(q)\\,.",
"}\\end{array}$ The normalization can be chosen in a different way since (REF ) is defined up to $\\xi _i\\rightarrow \\lambda \\xi _i$ , $\\psi _j\\rightarrow \\psi _j/\\lambda $ .",
"Let us also mention that in the rational case $\\psi _j$ are elementary symmetric functions of coordinates (see [10]), so that (REF ) provides its elliptic analogue."
],
[
"Acknowledgments",
"tocsection Acknowledgments The work of A. Zabrodin has been funded within the framework of the HSE University Basic Research Program.",
"The work of A. Zotov was performed at the Steklov International Mathematical Center and supported by the Ministry of Science and Higher Education of the Russian Federation (agreement no.",
"075-15-2019-1614)."
]
] | 2107.01697 |
[
[
"Nonequilibrium thermodynamic process with hysteresis and metastable\n states -- A contact Hamiltonian with unstable and stable segments of a\n Legendre submanifold"
],
[
"Abstract In this paper, a dynamical process in a statistical thermodynamic system of spins exhibiting a phase transition is described on a contact manifold, where such a dynamical process is a process that a metastable equilibrium state evolves into the most stable symmetry broken equilibrium state.",
"Metastable and the most stable equilibrium states in the symmetry broken phase or ordered phase are assumed to be described as pruned projections of Legendre submanifolds of contact manifolds, where these pruned projections of the submanifolds express hysteresis and pseudo-free energy curves.",
"Singularities associated with phase transitions are naturally arose in this framework as has been suggested by Legendre singularity theory.",
"Then a particular contact Hamiltonian vector field is proposed so that a pruned segment of the projected Legendre submanifold is a stable fixed point set in a region of a contact manifold, and that another pruned segment is a unstable fixed point set.",
"This contact Hamiltonian vector field is identified with a dynamical process departing from a metastable equilibrium state to the most stable equilibrium one.",
"To show the statements above explicitly an Ising type spin model with long-range interactions, called the Husimi-Temperley model, is focused, where this model exhibits a phase transition."
],
[
"Introduction",
"Contact geometry is known as an odd-dimensional analogue of symplectic geometry [1], [2], [3], and has been studied from viewpoints of pure and applied mathematics [4].",
"From the pure mathematics side, contact topology[5], contact Riemannian geometry [6], and so on [7] are studied.",
"From the applied mathematics side, geometrization of thermodynamics [8], [9], statistical mechanics [10], [11], applications to information geometry [12], [13], [14], and so on [15], [16], [17], [18], [19] are studied.",
"In particular contact geometric approaches to thermodynamics [20], [21] and dissipative mechanics [22], [23], [24] are intensively studied.",
"Although the both sides are close in some sense [25], we feel that some gap between them exists, and that profound theorems found in pure mathematics should be applied to such application areas.",
"By filling such a gap, undiscovered notions and facts are expected to be found as in other previous contacts between physics and geometry [26], [27] Nonequilibrium statistical mechanics and thermodynamics are developing branches of physics, interesting in their own right [28], [29], and their development should prove useful in various research areas, since these branches are closely related to nanotechnology [30], mathematical engineering including Markov chain Monte Carlo methods[31], and so on [32], [33].",
"Nonequilibrium phenomena are time-dependent thermodynamic phenomena, and some simple cases have successfully been addressed [34], [35].",
"Although intricate systems are never fully appreciated, some progress in understanding such has been made in proper frameworks.",
"An example is on a dynamical process from a metastable state into a most stable equilibrium state [36].",
"Another example is to deal with hysteresis curves in magnetic systems [37].",
"Beyond these, one should expect further progress.",
"As stated above, although considerable activity is being devoted to formulate nonequilibrium statistical thermodynamics, there is little consensus in the literature on its foundation.",
"To establish a foundation of nonequilibrium theory, one might focus on a canonical example as a first step.",
"This is because the Ising model, a canonical model, played a central role in developing equilibrium theory [38].",
"Choosing a simple toy model appropriately many quantities are evaluated analytically, and insights can be gained from its simplicity.",
"One of such a good model is the Husimi-Temperley model.",
"This is based on the Ising model, by modifying the interaction range from the nearest neighbor one to the global or mean field type.",
"This model, the Husimi-Temperley model, exhibits a phase transition, and several quantities can be calculated analytically [39], [40], [41].",
"On the one hand, one might be concerned that such long-range interactions are physically irrelevant.",
"On the other hand, one might think that systems with long-range interactions are ubiquitous in the world.",
"Examples of such systems include self-gravitating particles, two-dimensional fluid, and so on [36].",
"These examples together with the assumption that the existence of a universality class of statistical systems, the investigation of statistical systems of spins with long-range interactions is physically relevant.",
"In this paper this later perspective is adapted.",
"To formulate nonequilibrium statistical mechanics one might take the attitude that a contact geometric approach is employed [42], [12], [43].",
"One of the questions in constructing such a formulation is how to deal with phase transitions [44], and another one is how to introduce dynamics describing nonequilibrium phenomena.",
"In this paper both of these questions are addressed for the case of the Husimi-Temperley model, that is an Ising type model.",
"This model enables many quantities to be expressed analytically, and because of its simplicity physical insights can be gained.",
"One advantage of the use of contact geometry is that Legendre singularity theory [45] is expected to provide a sophisticated tool set to elucidate mechanisms of phase transitions [46], [47]."
],
[
"Outline of this contribution",
"In this paper analysis of the Husimi-Temperley model is summarized to make this paper self-contained, and then its geometric description is proposed.",
"In this proposal stable and unstable segments of the hysteresis and pseudo-free energy curves are considered.",
"Each curve is identified with a union of pruned segments of a projected Legendre submanifold, where this Legendre submanifold is a 1-dimensional submanifold of a 3-dimensional contact manifold.",
"In this framework a thermodynamic phase space is identified with a contact manifold where the contact form restricts vector fields so that the first law of thermodynamics holds, and the time-development of contact Hamiltonian systems is identified with the time-development of thermodynamic processes in thermodynamic phase spaces.",
"The main theorem in this paper and its physical interpretation are informally stated as follows.",
"Claim 1.1 (Informal version of Theorem REF ).",
"The integral curves of a contact Hamiltonian vector field on a 3-dimensional contact manifold connect a unstable segment of a Legendre submanifold and a stable one.",
"Physically this contact Hamiltonian vector field expresses the dynamical process departing from a unstable branch of the hysteresis curve to a stable one.",
"Simultaneously this vector field expresses the dynamical process departing from a unstable branch of the free-energy curve to a stable one.",
"To explain how to arrive at this claim, a procedure together with calculations of this paper is summarized below.",
"Since this summary is an abstraction of the calculations for the specific model, this summary is seen as a generalization from the specific procedure.",
"Introduce a statistical model (of spins) with a parameter $J_{\\,0}\\in \\mathbb {R}$ , let $x\\,\\in \\mathbb {R}$ and $y\\,\\in \\mathbb {R}$ denote a dimensionless applied external field and the negative of magnetization, respectively.",
"Then $x$ and $y$ form a pair of thermodynamic conjugate variables.",
"In general, the main task for elucidating thermodynamic properties of a microscopic model is to calculate the corresponding partition function by integrating all the degrees of freedom with some measure.",
"This measure is often chosen as the canonical measure, and the partition function yields the free-energy.",
"Consider the case that a dimensionless free-energy per degree of freedom $\\psi _{\\,J_0}$ is obtained with the so-called saddle point method: $\\psi _{\\,J_0}(x,y^{\\,*})\\simeq \\min _{y}\\ \\psi _{\\,J_0}(x,y)=\\min _{\\mu }\\ \\psi _{\\,J_0}(x,y_{\\,\\mu }^{\\,*}),$ where $\\mu \\,\\in \\mathbb {N}$ is a label for discriminating various local minima of $\\psi _{\\,J_0}$ with respect to $y$ , and $y_{\\,\\mu }^{\\,*}$ denotes a local minimum point.",
"In this paper the following form of $\\psi _{\\,J_0}$ is focused: $\\psi _{\\,J_0}(x,y)=\\frac{J_{\\,0}}{2}y^{\\,2}-\\int \\check{s}(\\Delta )\\,\\mathrm {d}\\,\\Delta ,\\qquad \\Delta :=2J_{\\,0}y+x,$ where $\\check{s}$ is a function, and $\\check{s}(\\Delta )=\\tanh (\\Delta )$ for the Husimi-Temperley model.",
"Introduce the 3-dimensional contact manifold $(T^{\\,*}\\mathbb {R}\\times \\mathbb {R},\\lambda )$ whose coordinates are $(x,y,z)$ so that $\\lambda =\\mathrm {d}z+y\\,\\mathrm {d}x$ is a contact 1-form.",
"Then a union of (metastable) equilibrium states are identified with a Legendre submanifold.",
"The coordinate expression of a (metastable) equilibrium state is $(x,y_{\\mu }^{\\,*}(x),z(x,y_{\\,\\mu }^{\\,*}(x)))$ labeled by $\\mu $ , where $\\frac{\\partial \\psi _{\\,J_0}}{\\partial y_{\\,\\mu }^{\\,*}}=\\left(y_{\\,\\mu }^{\\,*}-\\check{s}(\\Delta _{\\,\\mu }^{\\,*})\\,\\right) J_{\\,0}=0,\\quad \\frac{\\partial \\psi _{\\,J_0}}{\\partial x}=-y_{\\,\\mu }^{\\,*},\\quad z=\\psi _{\\,J_0}(x,y_{\\,\\mu }^{\\,*}),\\qquad \\Delta _{\\,\\mu }^{\\,*}:=2J_{\\,0}y_{\\,\\mu }^{\\,*}+x,$ with $\\check{s}$ constituting a self-consistent equation $y_{\\,\\mu }^{\\,*}=\\check{s}(\\Delta _{\\,\\mu }^{\\,*})$ .",
"Self-consistent equations often appear in the study of systems with phase transitions, where a phase transition is equivalent to a bifurcation of the solution of the self-consistent equation.",
"From this construction, the hysteresis curve is nothing but the projection of the Legendre submanifold onto the $(x,y)$ -plane up to sign convention.",
"In addition, the pseudo-free energy curve is the projection of the Legendre submanifold onto the $(x,z)$ -plane.",
"The set of multiple branches of the hysteresis curve is recognized as a multi-valued function of $x$ , $x\\mapsto y_{\\,\\mu }^{\\,*}=y_{\\,\\mu }^{\\,*}(x)$ .",
"If this multi-valued function is invertible, then $x=x(y_{\\,\\mu }^{\\,*})$ exists, where the function $y_{\\,\\mu }^{\\,*}\\mapsto x(y_{\\,\\mu }^{\\,*})$ may be a single-valued function.",
"In this model $x$ can explicitly be written in terms of $y_{\\mu }^{\\,*}$ on the Legendre submanifold, and the multiple branches of the hysteresis curve can be drawn by varying the value of $y_{\\,\\mu }^{\\,*}$ continuously in the recognition that the graph $(x,y_{\\,\\mu }^{\\,*}(x))$ is depicted by $(x(y_{\\,\\mu }^{\\,*}),y_{\\,\\mu }^{\\,*})$ , where $x:y_{\\,\\mu }^{\\,*}\\mapsto x(y_{\\,\\mu }^{\\,*})$ is a single-valued function.",
"Thus the Legendre submanifold whose projections are labeled by $\\mu $ can be treated as a one submanifold, rather than multiple submanifolds.",
"The projection of the Legendre submanifold onto the $(x,z)$ -plane expresses a pseudo-free energy as a multi-valued function.",
"This multi-valued function expresses a set of metastable, unstable, and most stable equilibrium states.",
"These projections in the symmetry broken phase are depicted in Fig.",
"REF .",
"Note that $\\psi _{\\,J_0}$ is not convex with respect to $x$ (see Remark REF and Ref.",
"[48]), whereas $\\psi _{\\,J_{0}}$ is convex with respect to $y$ in the high temperature phase (see Remark REF together with Fig.",
"REF ).",
"Figure: Unpruned projections of the Legendre submanifoldin the low temperature phase (symmetry broken phase).",
"(Left) The (x,z)(x,z)-plane.",
"(Right) The (x,y)(x,y)-plane.",
"Prune the top branch on the $(x,z)$ -plane of the projection (see Fig.",
"REF ).",
"This pruning procedure is equivalent to prune the middle segment passing through $(0,0)$ on the $(x,y)$ -plane.",
"Then the resultant disconnected segments of the projection of the Legendre submanifold yield disconnected hysteresis and pseudo-free energy curves that are expected to be observed in experiments.",
"Figure: Pruned segments of the projections of the Legendre submanifoldin the low temperature phase (symmetry broken phase).The regions ℐ ∓ {\\cal I}^{\\,\\mp } are such thatℐ - :={x∈ℝ <0 |ψ 1 (x)<ψ 2 (x)}⊂ℝ{\\cal I}^{\\,-}:=\\lbrace \\, x\\in \\mathbb {R}_{\\,<0}\\,|\\,\\psi _{\\,1}(x)<\\psi _{\\,2}(x)\\,\\rbrace \\subset \\mathbb {R} andℐ + :={x∈ℝ >0 |ψ 1 (x)<ψ 2 (x)}⊂ℝ{\\cal I}^{\\,+}:=\\lbrace \\, x\\in \\mathbb {R}_{\\,>0}\\,|\\,\\psi _{\\,1}(x)<\\psi _{\\,2}(x)\\,\\rbrace \\subset \\mathbb {R},where ψ μ (x)=ψ J 0 (x,y μ * (x))\\psi _{\\,\\mu }(x)=\\psi _{\\,J_0}(x,y_{\\,\\mu }^{\\,*}(x))is an abbreviation.",
"(Left) The (x,z)(x,z)-plane.",
"(Right) The (x,y)(x,y)-plane.",
"Choose a contact Hamiltonian $h(x,y,z)=\\psi _{\\,0}(x)(z-\\psi _{\\,1}(x))(z-\\psi _{\\,2}(x)),$ where $\\psi _{\\,0}$ is a positive function of $x$ .",
"Its corresponding contact Hamiltonian vector field expresses the dynamical process on the $(x,y)$ - and $(x,z)$ -planes.",
"The pruned projections of the Legendre submanifold are shown to be stable and unstable (see Fig.",
"REF ).",
"This gives Claim REF .",
"Figure: Contact Hamiltonian vector field that expresses the dynamical processin the low temperature phase (symmetry broken phase).The fixed point setsare pruned segments of the projections of theLegendre submanifold.",
"(Left) The (x,z)(x,z)-plane.",
"(Right) The (x,y)(x,y)-plane.",
"In the case where there is only one single-valued function of $x$ one can find a contact Hamiltonian such that the corresponding segment of the projection of the Legendre submanifold is attractor as has been argued in Refs.",
"[12], [25].",
"As a corollary of Claim REF , the following is obtained.",
"Claim 1.2 (Informal version of Corollary REF ).",
"When pruning the unstable segments of the projected Legendre submanifold, as the set of attractors of the contact Hamiltonian systems, the cusp of the shape $\\wedge $ appears on the $(x,z)$ -plane, and the kink of the shape $\\hspace*{-1.5649pt}$ appears on the $(x,y)$ -plane (see Fig.",
"REF ).",
"Figure: Union of the stable pruned projections of the Legendre submanifoldin the low temperature phase (symmetry broken phase).",
"(Left) The (x,z)(x,z)-plane.",
"The cusp of the shape ∧\\wedge appears, andthe existence of this cusp corresponds to the existence of the1st-order phase transition.",
"(Right) The (x,y)(x,y)-plane.",
"The kink of the shape \\hspace*{-1.5649pt} appears, andthis expresses the experimental observationwhere the hysteresis curve is ruined by perturbation.From Claim REF , the long-time evolution of the proposing contact Hamiltonian system plays a similar role of the Maxwell construction discussed in thermodynamics and the role of the convexification by the Legendre transforms.",
"The rest of this paper is organized as follows.",
"In Section , some preliminaries are provided in order to keep this paper self-contained.",
"They are basics of contact geometry including projections of Legendre submanifolds, contact Hamiltonian systems, and so on.",
"In addition the so-called Husimi-Temperley model and its thermodynamics are briefly summarized.",
"In Section , after arguing that metastable and unstable equilibrium states are described as a Legendre submanifold in a contact manifold, pruned segments of the projected Legendre submanifold are introduced.",
"Then, a contact Hamiltonian vector field is introduced where this vector field expresses the dynamical process under the case that the unstable and stable segments of the hysteresis curve exist in the symmetry broken phase.",
"Section summarizes this paper and discusses some future works.",
"In Section of Appendix, the case of unpruned projection of the Legendre submanifold is briefly studied."
],
[
"Preliminaries",
"This section is intended to provide a brief summary of the necessary background, and consists of 2 parts.",
"They are about contact geometry, and about thermodynamic properties of the Husimi-Temperley model."
],
[
"Contact and symplectic geometries",
"To argue contact geometry of nonequilibrium thermodynamics, some known facts on contact and symplectic geometries are summarized and notations are fixed here [1], [2].",
"Various formulas and tools developed in differential geometry are known [26], [27].",
"For example, the Lie derivative of a $k$ -form $\\alpha $ along a vector field $X$ can be written as ${\\cal L}_{X}\\alpha =\\mathrm {d}\\imath _{\\,X}\\alpha +\\imath _{\\,X}\\mathrm {d}\\alpha $ , where $\\mathrm {d}$ is the exterior derivative and $\\imath _{\\,X}$ the interior product with $X$ .",
"This is known as the Cartan formula.",
"Let ${\\cal C}$ be a $(2n+1)$ -dimensional manifold ($n=1,2,\\ldots $ ), and $\\lambda $ a 1-form on ${\\cal C}$ such that $\\lambda \\wedge \\underbrace{\\mathrm {d}\\lambda \\wedge \\cdots \\wedge \\mathrm {d}\\lambda }_{n}\\ne 0,\\quad \\mbox{at any point on ${\\cal C}$}.$ Then this $\\lambda $ is referred to as a contact form.",
"Notice that another 1-form $f\\,\\lambda $ with $f$ being a non-vanishing function is also a contact form if $\\lambda $ is contact.",
"If a $2n$ -dimensional vector space $E\\subset T_{\\,p}{\\cal C}$ is written by $E=\\ker \\lambda $ with $\\ker \\lambda =\\lbrace \\ X\\in T_{p}{\\cal C}\\ |\\ \\lambda (X)=0\\ \\rbrace $ around $p\\in {\\cal C}$ , then the pair $({\\cal C},\\ker \\lambda )$ is referred to as a $(2n+1)$ -dimensional contact manifold (in the wider sense), where $\\lambda (X)$ denotes the pairing between $\\lambda $ and $X$ .",
"According to the Darboux theorem, there exist coordinates $(x,y,z)$ such that $\\lambda =\\mathrm {d}z-\\sum _{a=1}^{n}y_{\\,a}\\,\\mathrm {d}x^{\\,a},\\quad \\mbox{or}\\quad \\lambda =\\mathrm {d}z+\\sum _{a=1}^{n}y_{\\,a}\\,\\mathrm {d}x^{\\,a},$ where $x=(x^{\\,1},\\ldots ,x^{\\,n})$ and $y=(y_{\\,1},\\ldots ,y_{\\,n})$ .",
"These coordinates are referred to as canonical or Darboux coordinates.",
"If there exists such a contact form globally over ${\\cal C}$ , then the pair $({\\cal C},\\lambda )$ is referred to as a contact manifold (in the narrower sense).",
"One typical contact manifold is given as $T^{\\,*}\\mathbb {R}^{n}\\times \\mathbb {R}$ with some 1-form.",
"On a contact manifold $({\\cal C},\\lambda )$ , there exists a vector field $R$ that satisfies $\\imath _{\\,R}\\,\\mathrm {d}\\lambda =0,\\qquad \\mbox{and}\\qquad \\imath _{\\,R}\\lambda =1.$ This $R$ is referred to as the Reeb vector field, and is uniquely determined for a fixed $\\lambda $ .",
"This is written as $R=\\partial /\\partial z$ in the canonical coordinates such that $\\lambda $ is written as (REF ).",
"A contact vector field $X$ is a vector field on a contact manifold $({\\cal C},\\lambda )$ that preserves the contact structure $\\ker \\lambda $ , ${\\cal L}_{\\,X}\\lambda =f\\lambda $ with $f$ being a non-vanishing function.",
"There is a way to specify a contact vector field with a function described below.",
"The contact Hamiltonian vector field $X_{\\,h}$ associated with a function $h$ is the uniquely determined vector field such that $\\imath _{X_{\\,h}}\\lambda =h,\\qquad \\lambda \\wedge {\\cal L}_{X_{\\,h}}\\lambda =0,$ where the second equation reduces to $\\imath _{\\,X_{h}}\\mathrm {d}\\lambda =-\\,(\\mathrm {d}h-(Rh)\\lambda ),$ which is shown by applying $\\imath _{\\,R}$ and the Cartan formula.",
"The function $h$ is referred to as a contact Hamiltonian.",
"Note that there are some sign conventions on defining contact Hamiltonian vector fields.",
"The coordinate expression of $X_{\\,h}$ is obtained as follows.",
"Let $(x,y,z)$ be canonical coordinates such that $\\lambda =\\mathrm {d}z-\\sum _{a=1}^ny_{\\,a}\\mathrm {d}x^{\\,a}$ in (REF ).",
"Then from (REF ), one has $X_{\\,h}=\\sum _{a=1}^{n}\\left(\\dot{x}^{\\,a}\\frac{\\partial }{\\partial x^{\\,a}}+\\dot{y}_{\\,a}\\frac{\\partial }{\\partial y_{\\,a}}\\right)+\\dot{z}\\frac{\\partial }{\\partial z},$ where $\\dot{x}$ , $\\dot{y}$ , and $\\dot{z}$ are the functions $\\dot{x}^{\\,a}=-\\,\\frac{\\partial h}{\\partial y_{\\,a}},\\quad \\dot{y}_{\\,a}=\\frac{\\partial h}{\\partial x^{\\,a}}+y_{\\,a}\\frac{\\partial h}{\\partial z},\\quad \\dot{z}=h-\\sum _{b=1}^{n}y_{\\,b}\\frac{\\partial h}{\\partial y_{\\,b}},\\qquad a=1,\\ldots ,n.$ Identifying $\\dot{\\ }=\\mathrm {d}/\\mathrm {d}t$ and $t\\in {\\cal I}$ , $({\\cal I}\\subseteq \\mathbb {R})$ , one has that $X_{\\,h}$ expresses a dynamical system.",
"The $t$ will be identified with time in Section .",
"This dynamical system is referred to as a contact Hamiltonian system associated with $h$ .",
"For 3-dimensional contact manifolds, one can drop the subscripts and superscripts as $\\dot{x}=-\\,\\frac{\\partial h}{\\partial y},\\quad \\dot{y}=\\frac{\\partial h}{\\partial x}+y\\,\\frac{\\partial h}{\\partial z},\\quad \\dot{z}=h-y\\,\\frac{\\partial h}{\\partial y}.$ In the case of $\\lambda =\\mathrm {d}z+y\\,\\mathrm {d}x$ for 3-dimensional manifolds, one derives $\\dot{x}=\\frac{\\partial h}{\\partial y},\\quad \\dot{y}=-\\,\\frac{\\partial h}{\\partial x}+y\\,\\frac{\\partial h}{\\partial z},\\quad \\dot{z}=h-y\\,\\frac{\\partial h}{\\partial y}.$ On contact manifolds, some special submanifolds play important roles.",
"Given a $(2n+1)$ -dimensional contact manifold $({\\cal C},\\lambda )$ , an $n$ -dimensional submanifold such that $\\phi ^{\\,*}\\lambda =0$ is referred to as a Legendre submanifold ( Legendrian submanifold ), where $\\phi ^{\\,*}$ is the pullback of an embedding $\\phi $ .",
"Such a submanifold is generated by a function, and an example is shown in Example REF .",
"If the dimension of a Legendre submanifold is unity, then this submanifold is referred to as a Legendre curve.",
"Example 2.1 Let $(\\mathbb {R}^{\\,3},\\lambda )$ be a 3-dimensional contact manifold, $(x,y,z)$ its coordinates, and $\\lambda =\\mathrm {d}z\\mp y\\,\\mathrm {d}x$ .",
"In addition, let $\\psi _{\\,\\mathrm {R}}$ be a function of $x$ .",
"Then $\\phi \\,{\\cal A}_{\\,\\psi _{\\mathrm {R}}}=\\left\\lbrace \\ (x,y,z)\\in \\mathbb {R}^{\\,3}\\ \\bigg |\\ z=\\psi _{\\,\\mathrm {R}}(x),\\quad \\mbox{and}\\quad y=\\pm \\frac{\\mathrm {d}\\psi _{\\,\\mathrm {R}}}{\\mathrm {d}x}\\ \\right\\rbrace $ is a Legendre submanifold generated by $\\psi _{\\,\\mathrm {R}}$ due to $\\phi ^{\\,*}\\lambda =0$ and $\\dim (\\phi \\,{\\cal A}_{\\psi _{\\mathrm {R}}})=1$ .",
"Another example being relevant to this paper is as follows.",
"Example 2.2 Let $(\\mathbb {R}^{\\,3},\\lambda )$ be a 3-dimensional contact manifold, $(x,y,z)$ its coordinates, and $\\lambda =\\mathrm {d}z- y\\,\\mathrm {d}x$ .",
"In addition, let $\\psi _{\\,I}$ , $f_{\\,I}$ , and $\\Delta $ be the functions $\\psi _{\\,I}(x,y)=y^{\\,2}-f_{\\,I}(\\Delta ),\\quad \\Delta (x,y)=2 y- x.$ Then the embedded manifold $\\phi \\,{\\cal A}_{\\,\\psi _{\\,I}}=\\left\\lbrace \\ (x,y,z)\\in \\mathbb {R}^{\\,3}\\ \\bigg |\\ z=\\psi _{\\,I}(x,y),\\quad \\mbox{and}\\quad y=\\frac{\\mathrm {d}f_{\\,I}}{\\mathrm {d}\\Delta }(\\Delta (x,y)),\\quad \\mbox{where}\\ x\\in {\\cal I}\\ \\right\\rbrace ,$ is a Legendre submanifold, where $y=\\mathrm {d}f_{\\,I}/\\mathrm {d}\\Delta $ can be treated as an algebraic equation for $y$ with $x$ being a continuous parameter, and ${\\cal I}\\subset \\mathbb {R}$ is a region in which the real solution for $y$ exists.",
"The submanifold (REF ) is verified to be Legendrian as $\\dim (\\phi \\,{\\cal A}_{\\,\\psi _{\\,I}})=1$ and $\\phi ^{\\,*}\\lambda &=&\\mathrm {d}\\, (\\phi ^{\\,*}z) - (\\phi ^{\\,*}y)\\,\\mathrm {d}(\\phi ^{\\,*}x)\\nonumber \\\\&=&\\frac{\\partial \\psi _{\\,I}}{\\partial x}\\mathrm {d}x+\\frac{\\partial \\psi _{\\,I}}{\\partial y}\\mathrm {d}y-\\frac{\\mathrm {d}f_{\\,I}}{\\mathrm {d}\\Delta }\\mathrm {d}x\\nonumber \\\\&=&-\\frac{\\mathrm {d}f_{\\,I}}{\\mathrm {d}\\Delta }\\frac{\\partial \\Delta }{\\partial x}\\mathrm {d}x+\\left(2y-\\frac{\\mathrm {d}f_{\\,I}}{\\mathrm {d}\\Delta }\\frac{\\partial \\Delta }{\\partial y}\\right)\\mathrm {d}y- \\frac{\\mathrm {d}f_{\\,I}}{\\mathrm {d}\\Delta }\\mathrm {d}x\\nonumber \\\\&=&0.\\nonumber $ Similarly, for the case that $\\lambda =\\mathrm {d}z+y\\,\\mathrm {d}x,\\quad \\psi _{\\,I}=y^{\\,2}-f_{\\,I}(\\Delta ),\\quad \\Delta =-2\\,y-x,$ the embedded submanifold $\\phi \\,{\\cal A}_{\\,\\psi _{\\,I}}=\\left\\lbrace \\ (x,y,z)\\in \\mathbb {R}^{\\,3}\\ \\bigg |\\ z=\\psi _{\\,I}(x,y),\\quad \\mbox{and}\\quad y=- \\frac{\\mathrm {d}f_{\\,I}}{\\mathrm {d}\\Delta }(\\Delta (x,y)),\\quad \\mbox{where}\\ x\\in {\\cal I}\\ \\right\\rbrace ,$ is Legendrian.",
"In some physical context, the algebraic equation above appears as a self-consistent equation for determining the value of an order parameter in statistical mechanics (see around (REF )).",
"In Example REF with $\\lambda =\\mathrm {d}z-y\\,\\mathrm {d}x$ , it follows that $\\frac{\\partial \\psi _{\\,I}}{\\partial y}=2y-2\\frac{\\mathrm {d}f_{\\,I}}{\\mathrm {d}\\Delta }=0\\qquad \\mbox{on $\\phi {\\cal A}_{\\,\\psi _{I}}$ },$ implying that $y$ on the Legendre submanifold is written by the solution to $\\partial \\psi _{\\,I}/\\partial y=0$ , and that the solution is written in terms of the derivative $y= \\mathrm {d}f_{\\,I}/\\mathrm {d}\\Delta $ .",
"This structure motivates the following generalization from Example REF .",
"Example 2.3 Let $(T^{\\,*}\\mathbb {R}\\times \\mathbb {R},\\lambda )$ be a 3-dimensional contact manifold with $\\lambda =\\mathrm {d}z+y\\,\\mathrm {d}x$ where $(x,y,z)$ its coordinates.",
"In addition, let $\\psi _{\\,\\mathrm {II}}$ be a function of $(x,y)$ .",
"Then the embedded manifold $\\phi \\,{\\cal A}_{\\,\\psi _{\\,\\mathrm {II}}}=\\left\\lbrace \\ (x,y,z)\\in T^{\\,*}\\mathbb {R}\\times \\mathbb {R}\\ \\bigg |\\ z=\\psi _{\\,\\mathrm {II}}(x,y),\\ y=-\\frac{\\partial \\psi _{\\,\\mathrm {II}}}{\\partial x},\\ \\frac{\\partial \\psi _{\\,\\mathrm {II}}}{\\partial y}=0.\\quad \\mbox{where}\\ x\\in {\\cal I}\\ \\right\\rbrace ,$ is a Legendre submanifold, where $\\partial \\psi _{\\,\\mathrm {II}}/\\partial y=0$ can be treated as an algebraic equation for $y$ with $x$ being a parameter, and ${\\cal I}\\subset \\mathbb {R}$ is a region in which the real solution for $y$ exists.",
"This submanifold is Legendrian due to $\\dim (\\phi \\,{\\cal A}_{\\,\\psi _{\\,\\mathrm {II}}})=1$ and $\\phi ^{\\,*}\\lambda =0$ .",
"In what follows some projections of Legendre submanifolds are defined.",
"Consider first the cotangent bundle $T^{\\,*}\\mathbb {R}$ .",
"Let $x$ be a coordinate of $\\mathbb {R}$ , and $y$ a coordinate of $T_{\\,x}^{\\,*}\\mathbb {R}$ .",
"The so-called Liouville 1-form is expressed as $\\alpha =y\\,\\mathrm {d}x$ , inducing a symplectic form expressed as $\\omega =\\mathrm {d}\\alpha =\\mathrm {d}y\\wedge \\mathrm {d}x$ .",
"Second, let $z$ be a coordinate of another $\\mathbb {R}$ .",
"Then take the 3-dimensional contact manifold, $(T^{\\,*}\\mathbb {R}\\times \\mathbb {R},\\lambda )$ where $\\lambda =\\mathrm {d}z\\mp \\alpha $ .",
"On this contact manifold a Legendre submanifold, or Legendre curve, is identified with an embedded 1-dimensional curve in the 3-dimensional manifold, and its projection onto a plane could yield some singularities.",
"The projection of the Legendre curve onto the $(x,y)$ -plane is referred to as a Lagrange map, and that onto the $(x,z)$ -plane as a Legendre map.",
"The image of a Legendre map is referred to as the wave front of the Legendre curve.",
"Example 2.4 For the case of $\\lambda =\\mathrm {d}z\\mp y\\,\\mathrm {d}x$ , the wave front of the Legendre curve generated by $\\psi _{\\,\\mathrm {R}}$ being a (single-valued) function depending on $x$ $\\phi {\\cal A}_{\\,\\psi _{\\mathrm {R}}}=\\left\\lbrace \\ (x,y,z)\\in T^{\\,*}\\mathbb {R}\\times \\mathbb {R}\\ \\bigg |\\ y=\\pm \\frac{\\mathrm {d}\\psi _{\\,\\mathrm {R}}}{\\mathrm {d}x}(x),\\quad \\mbox{and}\\quad z=\\psi _{\\,\\mathrm {R}}(x)\\ \\right\\rbrace $ is the graph $(x,\\psi _{\\,\\mathrm {R}}(x))$ on the $(x,z)$ -plane.",
"Here there is no singularity associated with this projection provided that $\\psi _{\\,\\mathrm {R}}$ is smooth.",
"Example 2.5 For the case of $\\lambda =\\mathrm {d}z-y\\,\\mathrm {d}x$ , the wave front of the Legendre curve $\\phi {\\cal A}_{\\,\\psi _{\\,I}^{\\,\\prime }}=\\left\\lbrace \\ (x,y,z)\\in T^{\\,*}\\mathbb {R}\\times \\mathbb {R}\\ \\bigg |\\ z=y^2-\\frac{\\Delta ^{\\,3}}{3},\\ y=\\Delta ^{\\,2},\\quad \\mbox{where}\\ \\Delta (x,y)=2y-x,\\ \\mbox{and}\\ \\frac{-1}{8}\\le x\\ \\right\\rbrace $ is obtained as follows (see Example REF , and take $f(\\Delta )=\\Delta ^{\\,3}/3$ with $\\lambda =\\mathrm {d}z-y\\,\\mathrm {d}x$ ).",
"First the conditions $y=\\Delta ^{\\,2}=(2y-x)^{\\,2}$ , and $-1/8\\le x$ yield on the $(x,y)$ -plane as the image of the Lagrange curve $y_{\\,\\pm }(x)=\\frac{4x+1\\pm \\sqrt{1+8x}}{8},\\quad \\mbox{for}\\quad -\\frac{1}{8} \\le x,$ which can be seen as a multi-valued function of $x$ .",
"Each branch of this multi-valued function is jointed at $x=-1/8$ , as can be verified from $y_{\\,+}(-1/8)=y_{\\,-}(-1/8)=1/16$ .",
"Second, $z=y^{\\,2}-\\Delta ^{\\,3}/3$ is drawn on the $(x,z)$ -plane as $z(x,y_{\\,\\pm }(x))=y_{\\,\\pm }(x)^{\\,2}-\\frac{y_{\\,\\pm }(x)^{\\,3/2}}{3}.$ Hence, the wave front is $(x,z_{\\,+}(x))\\cup (x,z_{\\,-}(x))$ , where $z_{\\,\\pm }(x)=z(x,y_{\\,\\pm }(x))$ .",
"In Fig.REF , the projections of the Legendre curve onto various planes are shown.",
"A singular point appears at $x=-1/8$ on the wave front.",
"The projection onto the $(x,z)$ -plane can be seen as a double-valued function.",
"This view will be used in analyzing a dynamical process in Section .",
"Figure: Projection of (),where ++ and -- denote the lines obtained by y + (x)y_{\\,+}(x)and y - (x)y_{\\,-}(x), respectively.",
"(Left) Projection onto the (x,z)(x,z)-plane, the wave front.",
"(Middle) Projection onto the (x,y)(x,y)-plane, the image of the Lagrange map.",
"(Right) Projection onto the (y,z)(y,z)-plane."
],
[
"Thermodynamics of the Husimi-Temperley model",
"In this subsection an Ising type spin system is introduced, and then its thermodynamic properties derived with canonical statistical mechanics are summarized.",
"The aim of this subsection is to introduce a toy model, where that model should be appropriate in the sense that most of quantities are analytically obtained and introduced quantities are physically interpretable.",
"In Section , geometric analysis of this model will be shown.",
"Consider a lattice whose total number of lattice points is $N$ .",
"At the lattice point specified by $i\\in \\lbrace 1,\\ldots ,N\\rbrace \\subset \\mathbb {N}$ put a spin variable $\\sigma _{\\,i}=\\pm 1$ , and then $\\sigma :=(\\sigma _{\\,1},\\ldots ,\\sigma _{\\,N})$ .",
"The space of the spin variables is denoted ${\\cal S}=\\lbrace \\pm 1\\rbrace ^{\\,N}$ , so that $\\sigma \\in {\\cal S}$ .",
"The total energy defined for this system is introduced as ${\\cal H}(\\sigma )=-\\frac{J_{\\,0}}{N}\\sum _{i=1}^{N}\\sum _{j=1}^{N}\\sigma _{\\,i}\\sigma _{\\,j}-H\\sum _{i=1}^{N}\\sigma _{\\,i},$ where $J_{\\,0}\\in \\mathbb {R}$ is constant expressing the strength of spin interactions, and $H\\in \\mathbb {R}$ constant expressing an externally applied magnetic field.",
"Equation (REF ) is seen as a function, ${\\cal H}:{\\cal S}\\rightarrow \\mathbb {R}$ .",
"This model is referred to as the Husimi-Temperley model.",
"To elucidate thermodynamic properties of the model, introduce $m:{\\cal S}\\rightarrow \\mathbb {R}$ such that $m(\\sigma )=\\frac{1}{N}\\sum _{i=1}^N\\sigma _{\\,i},$ which is an order parameter.",
"The variables $m$ and $H$ form a thermodynamic conjugate pair.",
"The canonical statistical mechanics is then applied to the Husimi-Temperley model so that thermodynamic properties of this model are elucidated, where the heat bath temperature is denoted by $T>0$ .",
"The main task is to calculate the partition function $Z=\\sum _{\\sigma _{1}=\\pm 1}\\cdots \\sum _{\\sigma _{N}=\\pm 1}\\mathrm {e}^{\\ -\\,\\beta {\\cal H}(\\sigma )}=\\sum _{\\sigma _{1}=\\pm 1}\\cdots \\sum _{\\sigma _{N}=\\pm 1}\\ \\mathrm {e}^{\\ \\beta J_{\\,0}\\left(\\sum _{l=1}^{N}\\sigma _{l}\\right)^{\\,2}/\\,N}\\,\\mathrm {e}^{\\,\\beta H\\sum _{l=1}^{N}\\sigma _{\\,l}},$ where $\\beta $ has been defined by $\\beta =1/(k_{\\mathrm {B}}T)$ with $k_{\\,\\mathrm {B}}$ being the Boltzmann constant.",
"In the following the so-called saddle point method for Gaussian integral is applied so that an approximate expression of $Z$ is obtained for $N\\gg 1$ .",
"Under this approximation the free energy obtained from that $Z$ yields various thermodynamic quantities by differentiation.",
"The expression of $Z$ reduces as follows.",
"The identity $\\mathrm {e}^{\\ bs^{\\,2} }=\\int _{\\mathbb {R}}\\frac{\\mathrm {d}\\varpi }{\\sqrt{2\\pi }}\\mathrm {e}^{\\ -\\varpi ^{\\ 2}+2\\varpi \\sqrt{b}\\,s},\\quad s\\in \\mathbb {R}$ with substitutions $b=\\beta J_{\\,0}/N$ and $s=\\sum _{a=1}^{N}\\sigma _{a}$ yields $Z=\\int _{\\mathbb {R}}\\frac{\\mathrm {d}\\varpi }{\\sqrt{2\\pi }}\\mathrm {e}^{\\ -\\varpi ^{\\ 2}}\\left[2\\cosh \\left(2\\varpi \\sqrt{\\frac{\\beta J_{\\,0}}{N}}+\\beta H\\right)\\right]^{\\,N}.$ Furthermore, the change of variables from $\\varpi $ to $\\xi $ so that $\\varpi =\\sqrt{N}\\xi $ , one has $Z=\\sqrt{\\frac{N}{2\\pi }}\\int _{\\mathbb {R}}\\,\\mathrm {d}\\xi \\exp \\left[\\,-\\,N\\left[\\,\\xi ^{\\,2}-\\ln \\left(2\\,\\cosh \\left(2\\xi \\sqrt{\\beta J_{\\,0}}+\\beta H\\,\\right)\\,\\right)\\,\\right]\\right].$ In the limit $N\\gg 1$ , the expression of $Z$ above reduces to $Z\\simeq \\exp \\left[\\,-\\,N\\min _{\\xi \\in \\mathbb {R}}\\left[\\,\\xi ^{\\,2}-\\ln \\left(2\\,\\cosh \\left(2\\xi \\sqrt{\\beta J_{\\,0}}+\\beta H\\,\\right)\\,\\right)\\,\\right]\\right],$ where some irrelevant constants have been omitted.",
"This approximation is known as the saddle point method.",
"Under this approximation the free energy ${\\cal F}=-k_{\\,\\mathrm {B}}T\\ln Z$ is ${\\cal F}\\simeq {\\cal F}^{\\,\\mathrm {saddle}},\\quad {\\cal F}^{\\,\\mathrm {saddle}}(T,H;\\beta ,J_{\\,0})=k_{\\,\\mathrm {B}}T N\\min _{\\xi \\in \\mathbb {R}}\\left[\\,\\xi ^{\\,2}-\\ln \\left(2\\,\\cosh \\left(2\\xi \\sqrt{\\beta J_{\\,0}}+\\beta H\\,\\right)\\,\\right)\\,\\right].$ This expression is rewritten by introducing the variable $y$ satisfying $\\xi =y\\sqrt{\\beta J_{\\,0}}$ as ${\\cal F}^{\\,\\mathrm {saddle}}(T,H;\\beta ,J_{\\,0})&=&k_{\\,\\mathrm {B}}T N\\min _{y\\in \\mathbb {R}}\\left[\\,\\beta J_{\\,0}y^{\\,2}-\\ln \\left(2\\,\\cosh \\left(2 \\beta J_{\\,0}y+\\beta H\\,\\right)\\,\\right)\\,\\right]\\\\&=&Nf_{\\,\\beta ,J_{\\,0}}(x,y^{\\,*}),$ where $x&=&H,\\\\y^{\\,*}&=&\\arg \\min _{y\\in \\mathbb {R}} f_{\\,\\beta ,J_{\\,0}}(x,y),\\\\f_{\\,\\beta ,J_{\\,0}}(x,y)&=&J_{\\,0}y^{\\,2}-\\frac{1}{\\beta }\\ln \\left(2\\,\\cosh \\left(2\\beta J_{\\,0}\\,y+\\beta x\\,\\right)\\,\\right),$ From (), the physical meaning of $f_{\\,\\beta ,J_{\\,0}}(x,y^{\\,*})$ is the value of the free energy per degree of freedom.",
"From () and (), $f_{\\,\\beta ,J_{\\,0}}(x,y)$ can be interpreted as a relaxation or extension of $f_{\\,\\beta ,J_{\\,0}}(x,y^{\\,*})$ , where $y^{\\,*}$ is relaxed to $y\\in \\mathbb {R}$ .",
"This function $f_{\\,\\beta ,J_{\\,0}}:\\mathbb {R}^{\\,2}\\rightarrow \\mathbb {R}$ is referred to as a pseudo-free energy (per degree of freedom) in this paper.",
"The dissimilarity between pseudo-free energy and free energy is that the convexity of pseudo-free energy is not guaranteed.",
"The $y^{\\, *}$ obtained as () is chosen from $\\lbrace y_{\\,\\mu }^{\\,*}\\,|\\,\\mu =1,2,3\\rbrace $ , where $y_{\\,\\mu }^{\\,*}$ satisfies $\\left.\\frac{\\partial f_{\\,\\beta ,J_{\\,0}}}{\\partial y}\\right|_{y_{\\,\\mu }^{\\,*}}=0,$ with $\\mu \\in \\mathbb {N}$ denoting a label for a solution to (REF ).",
"The reason for introducing labels $\\mu $ is to take into account the possibility that the algebraic equation (REF ) has at most countably many solutions.",
"This equation, (REF ), is explicitly expressed as $y_{\\,\\mu }^{\\,*}-\\tanh (2\\beta J_{\\,0}y_{\\,\\mu }^{\\,*}+\\beta x\\,)=0,\\qquad \\mu =1,2,\\ldots .$ Notice that $y^{\\,*}$ satisfies (REF ), since $y^{\\,*}$ is chosen from possible $y_{\\,\\mu }^{\\,*}$ , $(\\mu =1,2,\\ldots )$ .",
"One way to solve (REF ) is to find intersection points of the curve and the line on the $(y,s)$ -plane, $s_{\\,x;\\beta ,J_{\\,0}}(y)=\\tanh (2\\beta J_{\\,0}y+\\beta x\\,),\\qquad \\mbox{and}\\qquad s(y)=y,$ In Fig.",
"REF , the intersection points discussed above are shown.",
"Figure: Intersection points of ()are the solutions to().",
"(Left) There are 3 intersection points in the low temperature phase(β=1.0\\beta =1.0, J 0 =1.0J_{\\,0}=1.0, x=0.1x=0.1), whichare denoted by y 1 * ,y 2 * ,y 3 * y_{\\,1}^{\\,*}, y_{\\,2}^{\\,*}, y_{\\,3}^{\\,*}.",
"(Right) There is 1 intersection point in the high temperature phase(β=0.4\\beta =0.4, J 0 =1.0J_{\\,0}=1.0, x=0.1x=0.1), which is denoted by y 1 * y_{\\,1}^{\\,*}.Although an explicit expression for $y_{\\,\\mu }^{\\,*}$ as a function of $x,\\beta ,J_{\\,0}$ is not obtained, a condition when the number of solutions changes is found, that is obtained from $\\left.\\frac{\\mathrm {d}s_{\\,x;\\beta ,J_{\\,0}}}{\\mathrm {d}y}\\right|_{\\,y=\\widetilde{y}^{\\,*}}\\quad \\left\\lbrace \\begin{array}{cl}>1&\\mbox{the number of solutions is $3$,}\\\\=1&\\mbox{the critical point,}\\\\<1&\\mbox{the number of solutions is $1$ }\\end{array}\\right.,$ where $\\widetilde{y}^{\\,*}$ is the solution of (REF ) near $y=0$ .",
"Hence the critical point is determined by the tangency condition $\\mathrm {d}s_{\\,x;\\beta ,J_{\\,0}}/\\mathrm {d}y=1$ at $y=\\widetilde{y}^{\\,*}$ , and is expressed as $\\left.\\frac{\\mathrm {d}s_{\\,x;\\beta ,J_{\\,0}}}{\\mathrm {d}y}\\right|_{\\,y=\\widetilde{y}^{\\,*}}=\\left.\\frac{2\\beta J_{\\,0}}{\\cosh ^{\\,2}(2\\beta J_{\\,0}y+\\beta x)}\\right|_{\\,y=\\widetilde{y}^{\\,*}}=1.$ For example, consider the systems without external magnetic field, $x=0$ , from which $\\widetilde{y}^{\\,*}=0$ .",
"This and (REF ) yield that $\\beta =1/(2J_{\\,0})$ is the critical point, at which the number of the solutions changes.",
"This critical point associated with a phase transition with respect to temperature divides 1-dimensional region $\\mathbb {R}_{>0}$ into two, where this domain is the totality of $\\beta $ .",
"One is the low temperature phase, and the other one the high temperature phase.",
"The low temperature phase is also referred to as the ordered phase and as symmetry broken phase.",
"For each phase, $f_{\\,\\beta ,J_{\\,0}}(x,y)$ as a function of $y$ is shown in Fig.REF .",
"Figure: Graph of f β,J 0 (x,y)f_{\\,\\beta ,J_{\\,0}}(x,y) as a function of yy withxx kept fixed.",
"The point y * (x)y^{\\,*}(x) as a particular point of the yy-axisassigns the equilibriumthermodynamic state for each xx, wherey * (x)=argmin μ f β,J 0 (x,y μ * )y^{\\,*}(x)=\\arg \\min _{\\mu }f_{\\,\\beta ,J_{\\,0}}(x,y_{\\,\\mu }^{\\,*}) withy μ * y_{\\,\\mu }^{\\,*} being solutions to ∂f β,J 0 /∂y=0\\partial f_{\\,\\beta ,J_{\\,0}}/\\partial y=0,(see ()).",
"(Left)There are 3 solutions y μ * y_{\\,\\mu }^{\\,*} in the low temperature phase(β=1.0\\beta =1.0, J 0 =1.0J_{\\,0}=1.0, x=0.1x=0.1), whichare denoted by y 1 * ,y 2 * ,y 3 * y_{\\,1}^{\\,*}, y_{\\,2}^{\\,*}, y_{\\,3}^{\\,*}.",
"(Right) There is 1 solution y μ * y_{\\,\\mu }^{\\,*} in the high temperature phase(β=0.4\\beta =0.4, J 0 =1.0J_{\\,0}=1.0, x=0.1x=0.1), which is denoted by y 1 * y_{\\,1}^{\\,*}.The solution found around $y=0$ of $s_{\\,x;\\beta ,J_{\\,0}}(y)=s(y)$ in (REF ) is approximately expressed as $y\\sim \\frac{\\beta x}{1-2\\beta J_{\\,0}},\\quad \\mbox{around $y=0$}$ with the Taylor expansion of $\\tanh (\\cdot )$ around the origin.",
"The variable $y^{\\,*}$ is interpreted as the negative of the canonical average of $m$ , denoted $\\left\\langle \\,{m}\\, \\right\\rangle _{\\mathrm {can}}$ , $\\left\\langle \\,{m}\\, \\right\\rangle _{\\mathrm {can}}=\\sum _{\\sigma _{\\,1}=\\pm 1}\\cdots \\sum _{\\sigma _{\\,N}=\\pm 1}m(\\sigma )\\,\\mathrm {e}^{\\,-\\,\\beta {\\cal H}(\\sigma )}.$ To verify this, one first expresses $\\left\\langle \\,{m}\\, \\right\\rangle _{\\mathrm {can}}$ in terms of a derivative of ${\\cal F}$ .",
"Then, the resultant expression is shown to be written in terms of $y^{\\,*}$ .",
"First, it follows that $\\left\\langle \\,{m}\\, \\right\\rangle _{\\mathrm {can}}=\\frac{1}{N}\\left\\langle \\,{\\sum _{i=1}^{N}\\sigma _{\\,i}}\\, \\right\\rangle _{\\mathrm {can}}=\\frac{1}{N}\\frac{\\partial \\ln Z}{\\partial (\\beta H)}=\\frac{-1}{N}\\frac{\\partial (\\beta {\\cal F})}{\\partial (\\beta H)}.$ Under the approximation $\\beta {\\cal F}\\simeq \\beta {\\cal F}^{\\,\\mathrm {saddle}}=\\beta N f_{\\,\\beta ,J_{\\,0}}(x,y^{\\,*}),$ together with () and (REF ), one has the desired relation, $\\left\\langle \\,{m}\\, \\right\\rangle _{\\mathrm {can}}\\simeq -\\,\\frac{\\partial (\\beta f_{\\,\\beta ,J_{\\,0}}(x,y^{\\,*}))}{\\partial (\\beta x)}=-\\,\\tanh (2\\beta J_{\\,0}y^{\\,*}+\\beta x)=-\\,y^{\\,*}.$ Since $\\left\\langle \\,{m}\\, \\right\\rangle _{\\mathrm {can}}\\simeq - y^{\\,*}$ and $-1\\le \\left\\langle \\,{m}\\, \\right\\rangle _{\\mathrm {can}}\\le 1,$ the $y^{\\,*}$ is physically interpretable if $y^{\\,*}\\in \\overline{\\Upsilon },\\quad \\Upsilon =(-1,1),$ where $\\overline{\\Upsilon }=[-1,1]$ .",
"To discuss properties of the pseudo-free energy, restrict ourselves to the case that $\\lbrace \\, y_{\\,\\mu }^{\\,*}\\,|\\,\\mu =1,2,3\\, \\rbrace \\subset \\Upsilon $ and that $y_{\\,\\mu }^{\\,*}\\in \\Upsilon $ depends on $x$ smoothly for each $\\mu $ .",
"In this case, from (REF ) and the relation $\\tanh ^{\\,-1}(\\varsigma )=\\frac{1}{2}\\ln \\frac{1+\\varsigma }{1-\\varsigma }, \\qquad -1<\\varsigma <1,$ that this $x$ can be written as a function of $y_{\\,\\mu }^{\\,*}$ , $x:\\Upsilon \\rightarrow \\mathbb {R}$ such that $x(y_{\\,\\mu }^{\\,*})=-2J_{\\,0} y_{\\,\\mu }^{\\,*} + \\frac{1}{2\\beta }\\ln \\left(\\frac{1+y_{\\,\\mu }^{\\,*}}{1-y_{\\,\\mu }^{\\,*}}\\right),\\qquad y_{\\,\\mu }^{\\,*}\\in \\Upsilon .$ The curve $y_{\\,\\mu }^{\\,*} \\mapsto x(y_{\\,\\mu }^{\\,*})$ is a single-valued function of $y_{\\,\\mu }^{\\,*}$ , and has the property that $x(-y_{\\,\\mu }^{\\,*})=-x(y_{\\,\\mu }^{\\,*})$ .",
"Around $y_{\\,\\mu }^{\\,*}=0$ , this curve is approximately expressed as the line $x(y_{\\,\\mu }^{\\,*})\\sim \\frac{1-2 \\beta J_{\\,0}}{\\beta } y_{\\,\\mu }^{\\,*},\\qquad \\quad \\mbox{around $y_{\\,\\mu }^{\\,*}=0$}$ with the Taylor expansion of $\\ln (\\cdot )$ .",
"To discuss various quantities without any physical dimension, one introduces $\\psi _{\\,\\overline{J_{0}}}(\\overline{x},\\overline{y}):=\\overline{J_{\\,0}} \\overline{y}^{\\,2} -\\ln (2\\cosh (2\\overline{J_{\\,0}}\\overline{y}+\\overline{x})),\\qquad \\overline{J_{\\,0}}:=\\beta J_{\\,0},\\quad \\overline{x}:=\\beta x,\\quad \\overline{y}(\\overline{x}):= y(\\beta x),$ from (REF ), (), and ().",
"Note that $\\psi _{\\,\\overline{J_{0}}}(\\overline{x},\\overline{y})$ can be written as $\\beta f_{\\,\\overline{J_{0}}}(\\overline{x},\\overline{y})$ , and that $\\overline{y}=y$ due to the property that $y$ is dimensionless.",
"Similarly, $\\overline{y_{\\,\\mu }^{\\,*}}(\\overline{x}):= y_{\\,\\mu }^{\\,*}(\\beta x)$ , and from the definition of $\\overline{y_{\\,\\mu }^{\\,*}}$ and (REF ) it follows that $\\overline{y_{\\,\\mu }^{\\,*}}-\\tanh (2\\overline{J_{\\,0}}\\overline{y_{\\,\\mu }^{\\,*}} +\\overline{x})=0,$ which can be written as $\\overline{x}(\\overline{y_{\\,\\mu }^{\\,*}})=-\\, 2\\overline{J_{\\,0}} \\overline{y_{\\,\\mu }^{\\,*}} + \\frac{1}{2}\\ln \\left(\\frac{1+\\overline{y_{\\,\\mu }^{\\,*}}}{1-\\overline{y_{\\,\\mu }^{\\,*}}}\\right),\\qquad \\overline{y_{\\,\\mu }^{\\,*}}\\in \\Upsilon .$ The graph $(\\overline{x},\\overline{y_{\\,\\mu }^{\\,*}}(\\overline{x}))$ can be depicted with (REF ), and this curve passes from $(-\\infty ,-1)$ , via $(0,0)$ , to $(+\\infty ,+1)$ on the $(\\overline{x},\\overline{y})$ -plane.",
"It is convenient to introduce the function $s_{\\,\\overline{x};\\overline{J_{\\,0}}}(\\overline{y})=\\tanh (2\\overline{J_{\\,0}}\\overline{y}+\\overline{x}),$ that corresponds to $s_{\\,x;\\beta ,J_{\\,0}}(y)$ in (REF ).",
"Differentiation of the above equations yields the following: $-\\frac{\\partial \\psi _{\\,\\overline{J_{0}}}(\\overline{x},\\overline{y_{\\,\\mu }^{\\,*}})}{\\partial \\overline{x}}&=&\\tanh (2\\overline{J_{\\,0}}\\overline{y_{\\,\\mu }^{\\,*}}+\\overline{x})=\\overline{y_{\\,\\mu }^{\\,*}},\\\\\\frac{\\partial \\psi _{\\,\\overline{J_{0}}}(\\overline{x},\\overline{y})}{\\partial \\overline{y}}&=&2\\overline{J_{\\,0}}\\left(\\,\\overline{y}-\\tanh (2\\overline{J_{\\,0}}\\overline{y}+\\overline{x})\\,\\right),\\\\\\frac{\\mathrm {d}\\overline{x}}{\\mathrm {d}\\overline{y_{\\,\\mu }^{\\,*}}}&=&-\\left(2\\overline{J_{\\,0}} - \\frac{1}{1-\\overline{y_{\\,\\mu }^{\\,*}}^{\\,2}}\\right).$ Remark 2.1 Observe from (REF )–(REF ) and (REF )–() the following.",
"For each $\\overline{y_{\\,\\mu }^{\\,*}}$ , the function $\\psi _{\\,\\overline{J_{0}}}(\\overline{x},\\overline{y_{\\,\\mu }^{\\,*}})$ is not convex with respect to $\\overline{x}$ due to the derivative of (REF ), $\\frac{\\partial ^2 \\psi _{\\,\\overline{J_{0}}}(\\overline{x},\\overline{y_{\\,\\mu }^{\\,*}})}{\\partial \\overline{x}^2}=\\frac{-1}{\\cosh ^{2}(2\\overline{J_{\\,0}}\\overline{y_{\\,\\mu }^{\\,*}}+\\overline{x})}<0.$ In the low temperature phase, the function $\\psi _{\\,\\overline{J_{0}}}(\\overline{x},\\overline{y})$ is not convex with respect to $\\overline{y}$ , due to the derivative of (), $\\frac{\\partial ^2 \\psi _{\\,\\overline{J_{0}}}(\\overline{x},\\overline{y})}{\\partial \\overline{y}^2}=2\\overline{J_{\\,0}}\\left(\\,1-\\frac{2\\overline{J_{\\,0}}}{\\cosh ^2(2\\overline{J_{\\,0}}\\overline{y}+\\overline{x})}\\,\\right).$ When $2\\overline{J_{\\,0}}\\ge 1$ , it follows from $1/(1-\\overline{y_{\\,\\mu }^{\\,*}}^{\\,2})\\ge 1$ in () that $\\mathrm {d}\\overline{x}/\\mathrm {d}\\overline{y_{\\,\\mu }^{\\,*}}$ can vanish, $\\frac{\\mathrm {d}\\overline{x}}{\\mathrm {d}\\overline{y_{\\,\\mu }^{\\,*}}}=0\\quad \\mbox{at}\\quad \\overline{y_{\\,\\mu }^{\\,*}}_{\\,\\pm },\\quad \\overline{y_{\\,\\mu }^{\\,*}}_{\\,\\pm }:=\\pm \\sqrt{1-\\frac{1}{2\\overline{J_{\\,0}}}}.$ Hence in the region $2\\overline{J_{\\,0}}\\ge 1$ , the quantity $\\mathrm {d}\\overline{y_{\\,\\mu }^{\\,*}}/\\mathrm {d}\\overline{x}$ can diverge, where the $\\mathrm {d}\\overline{y_{\\,\\mu }^{\\,*}}/\\mathrm {d}\\overline{x}$ is the negative of the normalized magnetic susceptibility.",
"To avoid cumbersome notation we drop the bar, $\\overline{\\cdots }$ , in the following.",
"Remark 2.2 Observe that the $y_{\\,\\mu }^{\\,*}$ is a solution to the algebraic equation $\\partial \\psi _{\\,J_{0}}/\\partial y=0$ , due to ().",
"This solution $y_{\\,\\mu }^{\\,*}$ is written as the negative of the derivative $-\\partial \\psi _{\\,J_{0}}/\\partial x$ , due to (REF ).",
"This structure for $x$ and $y$ has also appeared in Example REF .",
"Physical interpretations of states specified with $y^{\\,*}$ and $y_{\\,\\mu }^{\\,*}$ are assumed.",
"Postulate 2.1 (Metastable and unstable equilibrium states).",
"Fix $J_{\\,0}(\\ne 0)$ and $x$ .",
"When $y=y_{\\,\\mu }^{\\,*}(x)$ and $z=\\psi _{\\,J_{\\,0}}(x,y_{\\,\\mu }^{\\,*}(x))$ with $y_{\\,\\mu }^{\\,*}\\ne y^{\\,*}=\\arg \\min _{\\mu ^{\\prime }}\\psi _{\\,J_{\\,0}}(x,y_{\\,\\mu ^{\\prime }}^{\\,*})$ , the state specified by $(x,y_{\\,\\mu }^{\\,*}(x),\\psi _{\\,J_{\\,0}}(x,y_{\\,\\mu }^{\\,*}(x)))$ is assumed to express a metastable or unstable equilibrium state labeled with $\\mu $ .",
"In addition, when $y=y^{\\,*}$ and $z=\\psi _{\\,J_{\\,0}}(x,y^{\\,*}(x))$ , the state $(x,y^{\\,*}(x),\\psi _{\\,J_{\\,0}}(x,y^{\\,*}(x)))$ is assumed to express the (most-stable) equilibrium state.",
"In Postulate REF , the terms “metastable equilibrium state” and “unstable equilibrium state” have been written, and they are briefly explained here.",
"In this paper, equilibrium states are special states where pairs of thermodynamic variables can be described as the derivatives of a (multi-valued) potential.",
"By definition, there is a potential function defined at equilibrium states.",
"Equilibrium states are then classified with these potential functions as follows.",
"If the potential function is a single-valued function and convex, then this function expresses the most stable equilibrium state.",
"If it is not the case, then such an equilibrium state is either a metastable equilibrium state or a unstable equilibrium state.",
"There is little consensus in the literature on how to define or to distinguish between metastable and unstable equilibrium states.",
"In this paper, the dissimilarity of the metastable and unstable equilibrium states is that the unstable equilibrium states are not observed in experiments.",
"In the conventional thermodynamics, the most stable equilibrium state is constructed by the convexification with the Legendre transform.",
"Definitions of metastable, unstable, and the most stable equilibrium states for the Husimi-Temperley model will be given in the language of contact geometry in the following section (see Definition REF ).",
"From Postulate REF , the discussions in this subsection have been about unstable and metastable equilibrium states and the equilibrium state.",
"So far no dynamical property of the system has been discussed.",
"In the next section, dynamical equations will be proposed by giving a contact Hamiltonian."
],
[
"Geometry of dynamical process in symmetry broken phase",
"In this section, a contact geometric description of the thermodynamic variables derived from the Husimi-Temperley model is given, and a physically appropriate dynamical system is proposed.",
"To this end, physically allowed process are discussed in terms of contact geometry first.",
"Consider a possible thermodynamic state specified by $(2n+1)$ variables, where the even number $2n$ is due to the pair of thermodynamic conjugate variables, and the 1 due to the free energy value.",
"During a change of thermodynamic states, the first law of thermodynamics should hold.",
"To discuss a smooth change of a state in time in terms of differential geometry, one introduces a $(2n+1)$ -dimensional manifold, a 1-form, and a vector field on the manifold.",
"This 1-form is used for restricting vector fields so that the first law of thermodynamics holds.",
"From this discussion, a contact manifold $({\\cal C},\\lambda )$ , or $({\\cal C},\\ker \\lambda )$ in a wider sense, and a class of vector fields are introduced for describing thermodynamics.",
"In this context, ${\\cal C}=T^{\\,*}Q\\times \\mathbb {R}$ is a natural manifold with $Q$ being a manifold.",
"On this setting an infinitesimal contact transform, that is a contact vector field, gives physically allowed processes as curves by integrating the vector field $X\\in \\ker \\lambda $ .",
"Thus, in this paper a thermodynamic phase space is identified with a contact manifold, a thermodynamic state is identified with a point of the manifold, a dynamical thermodynamic process in a certain class of nonequilibrium processes is identified with an integral curve of a contact vector field.",
"Beyond this formal procedure, for describing a particular thermodynamic process or phenomenon, one specifies an appropriate contact vector field on the contact manifold.",
"Choosing such an appropriate vector field from various allowed contact vector fields is not straightforward in general.",
"Instead, rather than a vector field, one can alternatively choose a function, since there is a correspondence between a function and a contact vector field, and such a function is a contact Hamiltonian.",
"For the Husimi-Temperley model, the thermodynamic phase space is specified as follows in this paper.",
"Definition 3.1 (Thermodynamic phase space and contact manifold).",
"Let $x$ be a coordinate for $\\mathbb {R}$ , $y$ that for $T_{\\,x}^{\\,*}\\mathbb {R}$ , and $z$ that for another $\\mathbb {R}$ .",
"Take the 3-dimensional manifold ${\\cal C}=T^{\\,*}\\mathbb {R}\\times \\mathbb {R}$ , and $\\lambda $ the 1-form $\\lambda =\\mathrm {d}z+y\\,\\mathrm {d}x$ .",
"This ${\\cal C}$ is referred to as the thermodynamic phase space for the Husimi-Temperley model.",
"In addition the pair $({\\cal C},\\lambda )$ is referred to as the contact manifold for the Husimi-Temperley model.",
"The coordinates in Definition REF at the most stable equilibrium are interpreted as $x=\\beta H$ , $y\\simeq -\\,\\left\\langle \\,{m}\\, \\right\\rangle _{\\mathrm {can}}$ , and $z$ is the lowest value of the dimensionless free-energy, $\\beta f_{\\,\\beta ,J_{\\,0}}$ , where $\\simeq $ is due to the saddle point approximation.",
"Notice that entropy and temperature are not included in Definition REF , and the magnetization and the externally applied magnetic field are focused in this paper so that the dimension of the manifold is 3, which renders various discussions on geometric properties simple.",
"Temperature is then treated as a parameter, and thus all the curves in the thermodynamic phase space express isothermal processes in this paper."
],
[
"Equilibrium",
"Equilibrium states are the most fundamental states in thermodynamic systems since they form the backbone of various thermodynamic states.",
"At equilibrium a thermodynamic quantity as a function can be obtained by differentiating a potential with respect to the corresponding thermodynamic conjugate variable.",
"In case of a gas system with constant temperature and volume environment, this potential is the Helmholtz free energy.",
"In case of systems of spins on lattices, an appropriate potential is $\\psi =\\beta {\\cal F}$ with ${\\cal F}$ being the Gibbs free energy.",
"From some arguments in thermodynamics, there is a set of correspondences between a fluid system contained in a box and a spin system.",
"A magnetization and an applied external magnetic field in the spin system correspond to a volume and the negative of pressure in the fluid system.",
"In the framework of contact geometric thermodynamics, an equilibrium state is described as a Legendre submanifold generated by a function, where such a function is identified with a thermodynamic potential.",
"For the Husimi-Temperley model, the metastable, unstable, and the most stable equilibrium states are defined as in a special case of Example REF .",
"In the high temperature phase, the projection of the Legendre submanifold onto the $(x,z)$ -plane can be expressed as a (single-valued) function, where the number of the labels is unity and thus the label can be omitted.",
"Meanwhile in the symmetry broken phase, the projection of the Legendre submanifold onto the $(x,z)$ -plane can be expressed as a 3-valued function.",
"To discriminate these branches of this 3-valued function, introduce the single-valued functions $\\psi _{\\,3}$ , $\\psi _{\\,1}$ , and $\\psi _{\\,2}$ such that $\\psi _{\\,J_{\\,0}}(x,y)=\\left\\lbrace \\begin{array}{ll}\\psi _{\\,3}(x,y)&\\mbox{the top branch on the $(x,z)$-plane}\\\\\\psi _{\\,1}(x,y)&\\mbox{the bottom branch on the $(x,z)$-plane}\\\\\\psi _{\\,2}(x,y)&\\mbox{the other branch on the $(x,z)$-plane}\\end{array}\\right.\\qquad \\mbox{in the symmetry broken phase,}$ where the suffix $J_{\\,0}$ for $\\psi _{\\,1}$ , $\\psi _{\\,2}$ , and $\\psi _{\\,3}$ has been omitted.",
"Definition 3.2 (Metastable, unstable, and most stable equilibrium states).",
"On the thermodynamic phase space ${\\cal C}$ of $({\\cal C},\\lambda )$ in Definition REF , let $\\psi _{\\,J_{\\,0}}$ be the function of $(x,y)$ as in (REF ).",
"Then in the high temperature phase, the submanifold specified by $z=\\psi _{\\,J_{\\,0}}$ , $y=y^{\\,*}=-\\,\\partial \\psi _{\\,J_{\\,0}}/\\partial x$ , and $\\partial \\psi _{\\,J_{\\,0}}/\\partial y|_{\\,y=y^{\\,*}}=0$ is the Legendre submanifold generated by $\\psi _{\\,J_{\\,0}}$ , where $y^{\\,*}$ is the unique solution that is written as the derivative of $\\psi _{\\,J_{\\,0}}$ , (see (REF )).",
"This Legendre submanifold is referred to as the equilibrium state.",
"In the symmetry broken phase, the Legendre submanifold with $z=\\psi _{\\,1}$ , $y=y_{\\,1}^{\\,*}=-\\,\\partial \\psi _{\\,1}/\\partial x$ , and $\\partial \\psi _{\\,1}/\\partial y|_{\\,y=y_{\\,1}^{\\,*}}=0$ is referred to as the most stable equilibrium state.",
"The the Legendre submanifold with $z=\\psi _{\\,3}$ , $y=y_{\\,3}^{\\,*}=-\\,\\partial \\psi _{\\,3}/\\partial x$ , and $\\partial \\psi _{\\,3}/\\partial y|_{\\,y=y_{\\,3}^{\\,*}}=0$ is referred to as the unstable equilibrium state.",
"The Legendre submanifold with $z=\\psi _{\\,2}$ , $y=y_{\\,2}^{\\,*}=-\\,\\partial \\psi _{\\,2}/\\partial x$ , and $\\partial \\psi _{\\,2}/\\partial y|_{\\,y=y_{\\,2}^{\\,*}}=0$ is referred to as the metastable equilibrium state.",
"Notice in Definition REF that, although the number of the Legendre submanifold is 1, there are 2 non-most-stable equilibrium states and 1 the most stable equilibrium state for the Husimi-Temperley model.",
"The metastable, unstable, and most stable equilibrium states are originated from the Legendre submanifold, and are yielded by a classification and partition of the submanifold.",
"Several projections of Legendre submanifolds are defined in contact geometry as have briefly been summarized in Section REF .",
"In some cases singular points are described in a lower dimensional space and some multi-valued functions can be described.",
"To detect phase transition and to characterize transitions in terms of contact geometry, such projections are applied to the equilibrium states of the Husimi-Temperley model.",
"In the physics literature it is common to draw graphs on the $(y,z)$ -plane.",
"These graphs on the $(y,z)$ -plane correspond to the graphs in Fig.",
"REF with some scaling factor.",
"In the following other projections are focused.",
"In Fig.",
"REF , the 2 cases of the wave front are depicted.",
"From this set of the cases, as known in the literature, the one in the lower temperature phase and the one in high temperature phase are distinguished.",
"Such a difference is due to a phase transition.",
"In the framework of standard thermodynamics [49], the branch having the lowest value of the free energy is observed, and the ones having higher values are not observed.",
"Hence the cusp of the wedge shape $\\wedge $ , obtained by pruning the branches forming $\\triangledown $ in the left and middle panels, should appear in perturbed or noisy systems in experiments.",
"A physical interpretation of the branches having non-lowest values varies, and ours is that those branches represent metastable and unstable equilibrium states (see Fig.2 of Ref.[46]).",
"In the case where the shape $\\wedge $ appears, the phase transition with respect to the externally applied field $H$ is classified as the 1st-order phase transition, since the free energy $z$ as a function is not differentiable at this cusp point.",
"Figure: Projections of the Legendre submanifold generated byψ J 0 (x,y)\\psi _{\\,J_{\\,0}}(x,y) onto the (x,z)(x,z)-plane (wave front).These were drawn with the use of()and ()by varying the value of yy in (-1,1)(-1,1).",
"(Left) Far from the critical point in the low temperature phase(J 0 =1.0J_{\\,0}=1.0 in the dimensionless variable, obtained fromβ=1.0\\beta =1.0 and J 0 =1.0J_{\\,0}=1.0 as the dimensional variables),(Middle) Near the critical point in the low temperature phase(J 0 =0.6J_{\\,0}=0.6 in the dimensionless variable, obtained fromβ=0.6\\beta =0.6 and J 0 =1.0J_{\\,0}=1.0 as the dimensional variables),(Right) the high temperature phase(J 0 =0.4J_{\\,0}=0.4 in the dimensionless variable, obtained fromβ=0.4\\beta =0.4 and J 0 =1.0J_{\\,0}=1.0 as the dimensional variables).In Fig.",
"REF , the images of the Lagrange map are drawn.",
"As in the case of Fig.",
"REF , the lower temperature phase differs from the higher temperature one, and forms a multi-valued function with the shape of $\\mathcal {S}$ .",
"In perturbed or noisy systems, such a multi-valued function does not appear.",
"One of observed structures in such experiments is a kink structure of the shape $\\hspace*{-1.5649pt}$ .",
"Another one, which we focus on first, is a pair of the disconnected curves that are obtained by pruning the middle segment passing through the origin $(0,0)$ , since such middle segment is physically unstable.",
"In this case, a hysteresis phenomenon takes place.",
"As will be discussed, from Corollary REF , the kink $\\hspace*{-1.5649pt}$ will be obtained as a stable fixed point set in the contact manifold.",
"Figure: Projections of the Legendre submanifold generated byψ J 0 (x,y)\\psi _{\\,J_{\\,0}}(x,y) onto the (x,y)(x,y)-plane (Images of the Lagrange map).These were drawn with the use of()by varying the value of yy in (-1,1)(-1,1).",
"(Left) Far from the critical point in the low temperature phase(J 0 =1.0J_{\\,0}=1.0 in the dimensionless variable constructed fromβ=1.0\\beta =1.0 and J 0 =1.0J_{\\,0}=1.0 in the dimensional variables),(Middle) Near the critical point in the low temperature phase(J 0 =0.6J_{\\,0}=0.6 in the dimensionless variable, obtained fromβ=0.6\\beta =0.6 and J 0 =1.0J_{\\,0}=1.0 as the dimensional variables),(Right) The high temperature phase(J 0 =0.4J_{\\,0}=0.4 in the dimensionless variable, obtained fromβ=0.4\\beta =0.4 and J 0 =1.0J_{\\,0}=1.0 as the dimensional variables).In Fig.",
"REF , points of the projections shown in Figs.",
"REF and REF are plotted for the low temperature phase.",
"A spinodal point is the point where $\\mathrm {d}x/\\mathrm {d}y=0$ in general, and in this case they are $\\mathrm {ii}$ and $\\mathrm {iv}$ on the $(x,y)$ -plane.",
"In Ref.",
"[49] the Van der Waals gas system is considered, and the corresponding branch is identified with being “unphysical”.",
"Then, in the Husimi–Temperley model, the segment in between the spinodal points is unphysical.",
"In this paper unphysical states are assumed to be invisible or ruined.",
"In this sense the present projection of the curve does not reflect correct thermodynamics.",
"To render this segment non-existent, introduce a pruned projection of the Legendre curve by removing such an invisible segment.",
"The resultant pruned projection of the Legendre curve onto the $(x,y)$ -plane consists of disconnected curves.",
"Similarly the resultant projection onto the $(x,z)$ -plane consists of disconnected curves.",
"Figure: Points of the projections shown inFigs.",
"and.The points 0,..., vi 0,\\ldots ,\\mathrm {vi} in the left figure correspond tothe points 0,..., vi 0,\\ldots ,\\mathrm {vi} in the right one.",
"(Left) The curve was drawn with the use of()and ()by varying the value of yy in (-1,1)(-1,1).The undirected curvesv- vi ¯\\overline{\\mathrm {v-vi}}, i- ii ¯\\overline{\\mathrm {i-ii}}, ii - iii ¯\\overline{\\mathrm {ii-iii}} in theleft figure arethe images of the functions ψ 1 \\psi _{\\,1}, ψ 2 \\psi _{\\,2}, and ψ 3 \\psi _{\\,3}, argued inSection .",
"(Right) The line was drawn with the use of()by varying the value of yy in (-1,1)(-1,1).The points ii \\mathrm {ii} and iv \\mathrm {iv} are spinodal points, and they are expressed as(x(y μ- * ),y μ- * )(x(y_{\\,\\mu \\,-}^{\\,*}),y_{\\,\\mu \\,-}^{\\,*})and (x(y μ+ * ),y μ+ * )(x(y_{\\,\\mu \\,+}^{\\,*}),y_{\\,\\mu \\,+}^{\\,*}).",
"These spinodal pointsy μ± * y_{\\,\\mu \\,\\pm }^{\\,*} and xx as a function of yy have been defined in()and (),respectively.In Fig.",
"REF the pair of the disconnected curves is shown on the $(x,y)$ -plane.",
"This pair of the curves is obtained by pruning the middle segment passing through the origin $(x,y)=(0,0)$ .",
"Edges of the pruned segments are located at the spinodal points $(x(y_{\\,\\mu \\,-}^{\\,*}),y_{\\,\\mu \\,-}^{\\,*})$ and $(x(y_{\\,\\mu \\,+}^{\\,*}),y_{\\,\\mu \\,+}^{\\,*})$ , where $y_{\\,\\mu \\,\\pm }^{\\,*}$ have been defined in (REF ) and $x$ as a function of $y_{\\,\\mu }^{\\,*}$ has been defined in (REF ).",
"The corresponding pruned projections onto the $(x,z)$ -planes form double-valued functions.",
"Figure: Pruned projections of the Legendre submanifold generated byψ J 0 (x,y)\\psi _{\\,J_{\\,0}}(x,y) onto the (x,y)(x,y)- and (x,z)(x,z)-planes.These were depicted with the use of()by varying the value of yy in (-1,1)(-1,1), and pruning the middle segments.Such middle parts on the (x,y)(x,y)-planeare from (x(y μ- * ),y μ- * )(x(y_{\\,\\mu \\,-}^{\\,*}),y_{\\,\\mu \\,-}^{\\,*})to (x(y μ+ * ),y μ+ * )(x(y_{\\,\\mu \\,+}^{\\,*}),y_{\\,\\mu \\,+}^{\\,*}), wherey μ± * y_{\\,\\mu \\,\\pm }^{\\,*} and xx as a function of yy have been defined in()and (),respectively.The curves expressing lower values of yy in the lower panels of the leftand right figures correspond to the curves of the shape of //in the upper panels.",
"(Left) Far from the critical point in the low temperature phase(J 0 =1.0J_{\\,0}=1.0in the dimensionless variable constructed fromβ=1.0\\beta =1.0 and J 0 =1.0J_{\\,0}=1.0 in the dimensional variables).",
"(Right) Near the critical point in the low temperature phase(J 0 =0.6J_{\\,0}=0.6 in the dimensionless variable, obtained fromβ=0.6\\beta =0.6 and J 0 =1.0J_{\\,0}=1.0 as the dimensional variables).Singularities associated with the phase transition with respect to the external field do not appear in the 1-dimensional Legendre submanifold $\\phi {\\cal A}$ embedded in the 3-dimensional contact manifold.",
"To verify this, first recall that in general a singular point of a curve is a point where its tangent vector vanishes.",
"Second, focus on the curve on the Legendre submanifold $\\gamma _{\\,xyz}:\\Upsilon \\rightarrow \\phi {\\cal A}$ , ($y\\mapsto (x(y),y,z(y))$ ), where $y$ is the abbreviation of $y_{\\,\\mu }^{\\,*}$ .",
"Then calculate the tangent vector along $\\gamma _{\\,xyz}$ , from which one has $\\gamma _{\\,xyz\\,*}\\left(\\frac{\\mathrm {d}}{\\mathrm {d}y}\\right)&=&\\frac{\\mathrm {d}x}{\\mathrm {d}y}\\frac{\\partial }{\\partial x}+\\frac{\\partial }{\\partial y}+\\frac{\\mathrm {d}z}{\\mathrm {d}y}\\frac{\\partial }{\\partial z}\\nonumber \\\\&=&\\left(-2J_{\\,0}+\\frac{1}{1-y^{\\,2}}\\right)\\frac{\\partial }{\\partial x}+\\frac{\\partial }{\\partial y}\\ne 0,\\qquad \\mbox{at any point of $y\\in \\Upsilon $,}\\nonumber $ where (REF ), (), () have been used, and $\\gamma _{\\,xyz\\,*}$ is the push-forward of $\\gamma _{\\,xyz}$ .",
"Thus, there is no singular point on the Legendre curve.",
"Meanwhile such singularities appear on the $(x,z)$ -plane as the result of the projection.",
"To show this, one calculates the tangent vector of the curve $\\gamma _{\\,xz}:\\Upsilon \\rightarrow \\mathbb {R}\\times \\mathbb {R}$ , ($y\\mapsto (x(y),z(y))$ ), $\\gamma _{\\,xz\\,*}\\left(\\frac{\\mathrm {d}}{\\mathrm {d}y}\\right)&=&\\frac{\\mathrm {d}x}{\\mathrm {d}y}\\frac{\\partial }{\\partial x}+\\frac{\\mathrm {d}z}{\\mathrm {d}y}\\frac{\\partial }{\\partial z}\\nonumber \\\\&=&\\left(-2J_{\\,0}+\\frac{1}{1-y^{\\,2}}\\right)\\frac{\\partial }{\\partial x}.\\nonumber $ From this calculation and (REF ), one verifies that there are singular points at the spinodal points, $y_{\\,\\mu \\,\\pm }^{\\,*}$ ."
],
[
"Nonequilibrium",
"Nonequilibrium processes are time-dependent thermodynamic processes, and their geometric descriptions have been proposed in the literature.",
"There are a variety of classes of nonequilibrium states, and our nonequilibrium thermodynamic states are such that thermodynamic variables can uniquely specify thermodynamic states where such variables are initially defined at the equilibrium state.",
"In the contact geometric framework, such a description of a thermodynamic process is to choose a suitable contact Hamiltonian system.",
"Among various nonequilibrium thermodynamic processes, relaxation processes have mainly been investigated, where such a process describes a time-development of a state towards the equilibrium state.",
"In this section, such an appropriate contact Hamiltonian is introduced for describing the dynamical process from metastable equilibrium states to the most stable equilibrium one.",
"We focus on the low temperature phase (or symmetry broken phase), since the system in the high temperature phase is equivalent to systems with no-phase transitions and has been addressed [12].",
"To discuss system in Fig.",
"REF , label branches of the 2-valued function of $x$ as in Fig.",
"REF (left, low temperature phase), where the function $\\psi _{\\,J_{0}}$ does not depend on $y$ on the Legendre submanifold due to ().",
"Then, on the Legendre submanifold generated by $\\psi _{\\,J_{0}}$ , the abbreviation $\\psi _{\\,\\mu }(x)=\\psi _{\\,J_{0}}(x,y_{\\,\\mu }^{\\,*})$ is introduced for each $\\mu $ .",
"The region ${\\cal I}^{\\,+}\\subset \\mathbb {R}_{\\,>0}$ is defined such that there are two (single-valued) functions, in particular $\\psi _{\\,1}$ and $\\psi _{\\,2}$ are labeled such that $\\psi _{\\,1}(x)<\\psi _{\\,2}(x)$ , $(x\\in {\\cal I}^{\\,+})$ .",
"That is, ${\\cal I}^{\\,+}:=\\lbrace \\, x\\in \\mathbb {R}_{\\,>0}\\,|\\,\\psi _{\\,1}(x)<\\psi _{\\,2}(x)\\,\\rbrace $ .",
"In the high temperature phase, the (single-valued) function appears.",
"Then, decompose the subset of the Legendre submanifold $\\phi {\\cal A}_{\\,\\psi _{\\,J_{0}}}$ into the ones in ${\\cal I}^{\\,+}$ $\\phi {\\cal A}_{\\,\\mu }^{\\,{\\cal I}^{\\,+}}=\\left\\lbrace \\ (x,y,z)\\in {\\cal C}\\ \\bigg |\\ y=-\\,\\frac{\\mathrm {d}\\psi _{\\,\\mu }}{\\mathrm {d}x} ,\\ z=\\psi _{\\,\\mu }(x),\\quad x\\in {\\cal I}^{\\,+}\\ \\right\\rbrace ,\\quad \\mu =1,2.$ Figure: Wave front.",
"(Left) Low temperature phase.The label μ\\mu for ψ μ \\psi _{\\,\\mu } is chosen so thatψ 1 (x)<ψ 2 (x)\\psi _{\\,1}(x)<\\psi _{\\,2}(x).",
"(Right) High temperature phase, the (single-valued) function appears.One then can show the main theorem of this paper as below.",
"Notice that no explicit expression of $\\psi _{\\,\\mu }$ defined on ${\\cal I}^{\\,+}$ is needed for each $\\mu $ .",
"Theorem 3.1 (Attractor as a segment of the hysteresis curve in the symmetry broken phase).",
"On the thermodynamic phase space for the Husimi-Temperley model, choose a contact Hamiltonian $h$ as $h(x,z)= \\psi _{\\,0}(x)(z-\\psi _{\\,1}(x))(z-\\psi _{\\,2}(x)).$ where $\\psi _{\\,0}$ is an arbitrary function of $x$ such that $\\psi _{\\,0}(x)>0$ .",
"Then it follows that The space $\\phi {\\cal A}_{\\,1}^{\\,{\\cal I}^{\\,+}}$ is asymptotically stable in ${\\cal D}_{\\,1}^{\\,+}$ where ${\\cal D}_{\\,1}^{\\,+}= \\lbrace \\ (x,y,z)\\in {\\cal C}\\ | \\ x\\in {\\cal I}^{\\,+},\\ z<\\psi _{\\,2}(x)\\ \\rbrace .$ Proof Our strategy for proving this is to show the existence of a Lyapunov function [50] for the dynamical system obtained from substituting the contact Hamiltonian (REF ) into (REF ).",
"The details are as follows.",
"First, a point of departure for this proof is to express the explicit form of the dynamical system written in terms of the coordinates $(x,y,z)$ .",
"From (REF ) and (REF ), the dynamical system is explicitly written as $\\dot{x}&=&0,\\\\\\dot{y}&=&-\\,\\frac{\\mathrm {d}\\psi _{\\,0}}{\\mathrm {d}x}(z-\\psi _{\\,1})(z-\\psi _{\\,2})+\\psi _{\\,0}\\left(y+\\frac{\\mathrm {d}\\psi _{\\,1}}{\\mathrm {d}x}\\right)(z-\\psi _{\\,2})+\\psi _{\\,0}\\left(y+\\frac{\\mathrm {d}\\psi _{\\,2}}{\\mathrm {d}x}\\right)(z-\\psi _{\\,1})\\\\\\dot{z}&=&h= \\psi _{\\,0}(x)(z-\\psi _{\\,1}(x))(z-\\psi _{\\,2}(x)).$ The next step is to find fixed point sets.",
"From $\\dot{x}|_{\\,\\phi {\\cal A}_{\\mu }^{{\\cal I}^{+}}}=0,\\quad \\dot{y}|_{\\,\\phi {\\cal A}_{\\mu }^{{\\cal I}^{+}}}=0,\\quad \\dot{z}|_{\\,\\phi {\\cal A}_{\\mu }^{{\\cal I}^{+}}}=0,\\qquad \\mu =1,2$ one has that $\\phi {\\cal A}_{\\,\\mu }^{{\\cal I}^{+}}\\subset {\\cal C}$ , $(\\mu =1,2)$ forms a fixed point set for each $\\mu $ .",
"Here a phase portrait of the dynamical system is roughly discussed.",
"It follows from (REF ) that $x$ is constant in time, and thus $\\psi _{\\,\\mu }(x)$ does not depend on time.",
"Third, to prove the theorem, a Lyapunov function is constructed [50].",
"Define the function $V_{\\,1}$ in ${\\cal D}_{\\,1}^{\\,+}$ such that $V_{\\,1}(x,z)=\\frac{1}{2}(z-\\psi _{\\,1}(x))^{\\,2},\\qquad (x,y,z)\\in {\\cal D}_{\\,1}^{\\,+}$ Then, it follows that $V_{\\,1}(x,z)\\ge 0,\\quad \\frac{\\mathrm {d}V_{\\,1}}{\\mathrm {d}t}=(z-\\psi _{\\,1})h(x,z)=\\psi _{\\,0}(z-\\psi _{\\,1})^{\\,2}(z-\\psi _{\\,2})\\le 0,\\qquad (x,y,z)\\in {\\cal D}_{\\,1}^{\\,+},$ where the equality holds on the fixed point set $\\phi {\\cal A}_{\\,1}^{\\,{\\cal I}^{\\,+}}$ .",
"Hence $V_{\\,1}$ is a Lyapunov function in ${\\cal D}_{\\,1}^{\\,+}$ .",
"According to the theorem of Lyapunov, one completes the proof.",
"$\\Box $ Theorem REF shows that the proposed contact Hamiltonian vector field is such that the pruned segment of the projected Legendre submanifold is an attractor in a region of a contact manifold.",
"The global behavior for $z$ is understood from Fig.",
"REF .",
"One then deduces from Fig.",
"REF that, given $x$ , $\\lim _{t\\rightarrow \\infty }z(t)=\\psi _{\\,1}(x)$ in ${\\cal D}_{\\,1}^{\\,+}$ .",
"In the case where there is only one function $\\psi $ of $x$ defined on a region in $\\mathbb {R}$ one can find a contact Hamiltonian such that the corresponding segment of the projection of the Legendre submanifold is an attractor as has been argued in Refs.",
"[12], [25].",
"Figure: Phase space ofthe dynamical system consisting of z ˙=h(x,z)\\dot{z}=h(x,z) and x ˙=0\\dot{x}=0(() and(), respectively).From z ˙=h\\dot{z}=h, it follows that the zeros ofhh are the fixed points.From hh in ()its zeros are the set z=ψ 1 (x)z=\\psi _{\\,1}(x) and the set z=ψ 2 (x)z=\\psi _{\\,2}(x).In addition, from the sign of hh, the set z=ψ 1 (x)z=\\psi _{\\,1}(x)is stable in 𝒟 1 + {\\cal D}_{\\,1}^{\\,+}.This contact Hamiltonian vector field expresses the dynamical process departing from metastable equilibrium states to the most stable equilibrium one.",
"To grasp local flow around the fixed point sets, the integral curves of the linearized equations are shown below.",
"For the point $(y_{\\mu },z_{\\mu })=(- \\psi _{\\,\\mu }^{\\,\\prime },\\psi _{\\,\\mu })$ , introduce $Y_{\\,\\mu }$ and $Z_{\\,\\mu }$ such that $y_{\\,\\mu }(t)=- \\psi _{\\,\\mu }^{\\,\\prime }(x)+Y_{\\,\\mu }(t),\\quad z_{\\,\\mu }(t)=\\psi _{\\,\\mu }(x)+Z_{\\,\\mu }(t),\\quad \\mbox{where}\\quad \\psi _{\\,\\mu }^{\\,\\prime }(x):=\\frac{\\mathrm {d}\\psi _{\\,\\mu }}{\\mathrm {d}x}(x),\\quad \\mu =1,2$ which yield linearized equations.",
"To avoid cumbersome notations, introduce $\\psi _{\\,21}(x)=\\psi _{\\,2}(x)-\\psi _{\\,1}(x)\\ >0, \\quad $ for each point $x$ .",
"Then the linearized equations are obtained as $\\dot{Z}_{\\,1}&=&-\\,\\underbrace{\\psi _{\\,0}\\,\\psi _{\\,21}}_{>\\ 0}\\, Z_{\\,1},\\qquad \\dot{Z}_{\\,2}=\\ \\ \\underbrace{\\psi _{\\,0}\\,\\psi _{\\,21}}_{>\\ 0}\\, Z_{\\,2},\\nonumber \\\\\\dot{Y}_{\\,1}&=&-\\psi _{\\,0}\\,\\psi _{\\,21}\\, Y_{\\,1}+(\\,\\psi _{\\,0}^{\\,\\prime }\\,\\psi _{\\,21}+\\psi _{\\,0}\\,\\psi _{\\,21}^{\\,\\prime }\\,)\\,Z_{\\,1},\\qquad \\dot{Y}_{\\,2}=\\ \\psi _{\\,0}\\,\\psi _{\\,21}\\, Y_{\\,2}+(\\psi _{\\,0}^{\\,\\prime }\\,\\psi _{\\,21}+\\psi _{\\,0}\\,\\psi _{\\,21}^{\\,\\prime }\\,)\\,Z_{\\,2},\\nonumber $ where $\\psi _{\\,21}^{\\,\\prime }(x)=\\mathrm {d}\\psi _{\\,21}/\\mathrm {d}x$ and $\\psi _{\\,0}^{\\,\\prime }=\\mathrm {d}\\psi _{\\,0}/\\mathrm {d}x$ that are constants in time.",
"To solve this linear system of equations, letting $c_{\\,\\mu }$ and $d_{\\,\\mu }$ be the constants such that $c_{\\,1}=\\psi _{\\,0}\\,\\psi _{\\,21},\\quad c_{\\,2}=-\\,c_{\\,1},\\qquad d_{\\,1}=\\psi _{\\,0}^{\\,\\prime }\\,\\psi _{\\,21}+\\psi _{\\,0}\\,\\psi _{\\,21}^{\\,\\prime },\\quad d_{\\,2}=-d_{\\,1},$ one can write $\\dot{Z}_{\\,\\mu }=-\\,c_{\\,\\mu } Z_{\\,\\mu },\\quad \\dot{Y}_{\\,\\mu }=-\\,c_{\\,\\mu } Y_{\\,\\mu }+d_{\\,\\mu }Z_{\\,\\mu },\\quad \\mu =1,2.$ The solution of this system is $Z_{\\,\\mu }(t)=Z_{\\,\\mu }(0)\\,\\mathrm {e}^{\\,-c_{\\mu }\\,t},\\quad Y_{\\,\\mu }(t)=(\\,Y_{\\,\\mu }(0)+d_{\\,\\mu }Z_{\\,\\mu }(0)\\, t\\,)\\,\\mathrm {e}^{\\,- c_{\\mu }\\,t}.$ From this, the inequalities $c_{\\,1}>0$ and $c_{\\,2}<0$ , one has that the fixed point set $\\phi {\\cal A}_{\\,1}^{{\\cal I}^{\\,+}}$ is linearly stable, and the $\\phi {\\cal A}_{\\,2}^{{\\cal I}^{\\,+}}$ linearly unstable.",
"Observe from (REF ) that the strength of instability is large when the value of $\\psi _{\\,21}$ is large.",
"The condition when $\\psi _{\\,21}(x)$ is large can be read off from Fig.",
"REF .",
"The strength of such an instability is small near the critical point, and it is large far from the critical point.",
"So far the phase space ${\\cal D}_{\\,1}^{\\,+}$ of the dynamical system is focused, and a similar claim can be stated for the region ${\\cal D}_{\\,1}^{\\,-}$ , where ${\\cal D}_{\\,1}^{\\,-}$ is defined with some ${\\cal I}^{\\,-}\\subset \\mathbb {R}_{<0}$ .",
"By combining these and refining it, one has the following.",
"Corollary 3.1 (Reconstruction of the stability of the hysteresis and pseudo-free energy curves as the Legendre submanifold).",
"Similar to the set ${\\cal D}_{\\,1}^{\\,+}=\\lbrace (x,y,z)\\in {\\cal C}\\,|\\,x\\in {\\cal I}^{\\,+},z<\\psi _{\\,2}(x)\\rbrace $ , where ${\\cal I}^{\\,+}$ has been defined as ${\\cal I}^{\\,+}=\\lbrace \\, x\\in \\mathbb {R}_{\\,>0}\\,|\\,\\psi _{\\,1}(x)<\\psi _{\\,2}(x)\\,\\rbrace \\subset \\mathbb {R}$ in the caption to Fig.",
"REF , introduce ${\\cal D}_{\\,1}^{\\,-}=\\lbrace (x,y,z)\\in {\\cal C}\\,|\\,x\\in {\\cal I}^{\\,-},z<\\psi _{\\,2}(x)\\rbrace $ , where ${\\cal I}^{\\,-}$ has been defined as ${\\cal I}^{\\,-}=\\lbrace \\, x\\in \\mathbb {R}_{\\,<0}\\,|\\,\\psi _{\\,1}(x)<\\psi _{\\,2}(x)\\,\\rbrace \\subset \\mathbb {R}$ in the caption to Fig.",
"REF .",
"In the joined region ${\\cal D}_{\\,1}^{\\,+}\\cup {\\cal D}_{\\,1}^{\\,-}$ in the contact manifold, one has the contact Hamiltonian vector fields where the undirected curves $\\overline{\\mathrm {0-i}}$ and $\\overline{\\mathrm {v-vi}}$ are stable fixed point sets, and $\\overline{\\mathrm {i-ii}}$ and $\\overline{\\mathrm {iv-v}}$ are unstable fixed point sets of the curve without the curve $\\overline{\\mathrm {ii-iv}}$ in Fig.",
"REF .",
"Notice that the $(y,z)$ -plane at $x=0$ has been removed from ${\\cal C}$ in Corollary REF .",
"On this removed plane, the double-valued function becomes a single valued function, and thus the present contact Hamiltonian is not relevant.",
"In Fig.",
"REF , the projected contact Hamiltonian vector fields on the 2 regions of ${\\cal C}$ in Corollary REF are simultaneously shown.",
"From this corollary, one has the following.",
"Remark 3.1 The cusp of the shape $\\wedge $ on the $(x,z)$ -plane is obtained in the long-time limit of the time-development of the contact Hamiltonian system, where such a shape is expected to be observed in experiments under perturbation.",
"The kink structure of the shape $\\hspace*{-1.5649pt}$ on the $(x,y)$ -plane is obtained in the long-time limit of the time-development of the contact Hamiltonian system, where such a kink shape is expected to be observed in experiments under perturbation.",
"Figure: Projected contact Hamiltonian vector fields on the 2 regions of 𝒞{\\cal C}stated inCorollary ,where the values of β\\beta and J 0 J_{\\,0} were chosen to expressa thermodynamic phase space beingfar from the critical point in the low temperature phase(J 0 =1.0J_{\\,0}=1.0in the dimensionless variable constructed fromβ=1.0\\beta =1.0 and J 0 =1.0J_{\\,0}=1.0 in the dimensional variables).The pruned segments of the projections of the Legendre submanifold arestable and unstable fixed point sets.",
"(Left) The (x,z)(x,z)-plane.",
"(Right) The (x,y)(x,y)-plane."
],
[
"Discussions and conclusions",
"This paper offers a contact geometric approach to thermodynamic systems that exhibit a phase transition.",
"One key in this paper has been that the set of metastable, unstable, and the most stable equilibrium states is identified with a Legendre submanifold whose projections form multi-valued functions.",
"As the main theorem of this paper unstable and stable segments of a hysteresis curve have been described as unstable and stable fixed point sets for a contact Hamiltonian vector field.",
"Simultaneously the pseudo-free energy curve has also been described similarly, where this simultaneity is ascribed to the different projections of the unique Legendre submanifold.",
"On this 1-dimensional Legendre submanifold there is no singularity even in the symmetry broken phase.",
"Meanwhile there are singularities on the 2-dimensional plane as the result of the projection.",
"Although these calculations have been for the so-called Husimi-Temperley model, calculations for other models and those for the present model are expected to be similar.",
"The series of calculations has been summarized as a procedure, and then this procedure has been summarized in Introduction of this paper.",
"A significance of this study is to provide a unified geometric manner in which a contact Hamiltonian and a single Legendre submanifold together with an associated pruning process lead to various notions and tools in thermodynamics.",
"Such notions and tools are non-most-stable and most stable equilibrium states, hysteresis and pseudo-free energy curves, and rules similar to the Maxwell construction and the convexification.",
"There remain unsolved problems that have not been addressed in this paper.",
"They include derive the contact Hamiltonian vector field from a dynamical system that describes microscopic spins [40], apply the present approach to various statistical mechanical models, thermodynamic systems, and electric circuits [17], extend the present or similar analysis to high-dimensional systems, rather than the present 3-dimensional contact manifold, apply Legendre singularity and cobordism theories intensively to thermodynamic systems [46], [51], clarify the relation between a contact version of the (graph) selector [52], [53] and the pruning introduced in this paper.",
"and so on.",
"By addressing these, it is expected that a relevant and sophisticated geometric methodology will be established for dealing with various intricate systems and critical phenomena."
],
[
"Acknowledgment",
"The author was partially supported by JSPS (KAKENHI) grant number JP19K03635, and thanks Lenonid Polterovich for discussions on applications of contact geometry to nonequilibrium statistical mechanics.",
"In addition the author thanks Minoru Koga for giving various suggestions and fruitful discussions on this study."
],
[
"Appendix",
"For the sake of completeness, a system with the unpruned projection of the Legendre submanifold is argued in this section."
],
[
"Three branch system",
"To discuss system in Fig.",
"REF , label branches of the 3-valued function of $x$ as in Fig.",
"REF (left, low temperature phase), where the function $\\psi _{\\,J_{0}}$ does not depend on $y$ on the Legendre submanifold due to ().",
"Then, as in the case of Section REF , on the Legendre submanifold generated by $\\psi _{\\,J_{0}}$ , the abbreviation $\\psi _{\\,\\mu }(x)=\\psi _{\\,J_{0}}(x,y_{\\,\\mu }^{\\,*})$ is introduced for each $\\mu $ .",
"In the region ${\\cal I}\\subset \\mathbb {R}$ , there are three (single-valued) functions $\\psi _{\\,1},\\psi _{\\,2},\\psi _{\\,3}$ labeled such that $\\psi _{\\,1}(x)<\\psi _{\\,2}(x)<\\psi _{\\,3}(x)$ , $(x\\in {\\cal I})$ .",
"In the high temperature phase, the (single-valued) function appears.",
"Figure: Wave front.",
"(Left) Low temperature phase.The label μ\\mu for ψ μ \\psi _{\\,\\mu } is chosen so thatψ 1 (x)<ψ 2 (x)<ψ 3 (x)\\psi _{\\,1}(x)<\\psi _{\\,2}(x)<\\psi _{\\,3}(x).",
"(Right) High temperature phase, the (single-valued) function appears.We focus on the low temperature phase again as in the case of the 2-valued function.",
"To discuss the low temperature phase, decompose the subset of the Legendre submanifold $\\phi {\\cal A}_{\\,\\psi _{\\,J_{0}}}$ into the ones in ${\\cal I}$ $\\phi {\\cal A}_{\\,\\mu }^{\\,{\\cal I}}=\\left\\lbrace \\ (x,y,z)\\in {\\cal C}\\ \\bigg |\\ y=-\\,\\frac{\\mathrm {d}\\psi _{\\,\\mu }}{\\mathrm {d}x} ,\\ z=\\psi _{\\,\\mu }(x),\\quad x\\in {\\cal I}\\ \\right\\rbrace ,\\quad \\mu =1,2,3.$ One can show the following theorem.",
"Notice, as in the case of 2-valued function, that no explicit expression of $\\psi _{\\,\\mu }$ defined on ${\\cal I}$ is needed for each $\\mu $ .",
"Theorem A.1 On the thermodynamic phase space for the Husimi-Temperley model, choose a contact Hamiltonian $h$ as $h(x,z)=- \\psi _{\\,0}(x)(z-\\psi _{\\,1}(x))(z-\\psi _{\\,2}(x))(z-\\psi _{\\,3}(x)).$ where $\\psi _{\\,0}$ is an arbitrary function of $x$ such that $\\psi _{\\,0}(x)>0$ .",
"Then it follows that The space $\\phi {\\cal A}_{\\,1}^{\\,{\\cal I}}$ is asymptotically stable in ${\\cal D}_{\\,1}$ where ${\\cal D}_{\\,1}= \\lbrace \\ (x,y,z)\\in {\\cal C}\\ | \\ x\\in {\\cal I},\\ z<\\psi _{\\,2}(x)\\ \\rbrace .$ The space $\\phi {\\cal A}_{\\,3}^{\\,{\\cal I}}$ is asymptotically stable in ${\\cal D}_{\\,3}$ where ${\\cal D}_{\\,3}= \\lbrace \\ (x,y,z)\\in {\\cal C}\\ | \\ x\\in {\\cal I},\\ z>\\psi _{\\,2}(x)\\ \\rbrace .$ Proof Our strategy for proving this is to show the existence of a Lyapunov function [50] for the dynamical system obtained from substituting the contact Hamiltonian (REF ) into (REF ).",
"The details are as follows.",
"First, a point of departure for this proof is to express the explicit form of the dynamical system written in terms of the coordinates $(x,y,z)$ .",
"From (REF ) and (REF ), the dynamical system is explicitly written as $\\dot{x}&=&0,\\\\\\dot{y}&=&\\frac{\\mathrm {d}\\psi _{\\,0}}{\\mathrm {d}x}(z-\\psi _{\\,1})(z-\\psi _{\\,2})(z-\\psi _{\\,3})-\\psi _{\\,0}\\left(y+\\frac{\\mathrm {d}\\psi _{\\,1}}{\\mathrm {d}x}\\right)(z-\\psi _{\\,2})(z-\\psi _{\\,3})\\nonumber \\\\&&-\\psi _{\\,0}\\left(y+\\frac{\\mathrm {d}\\psi _{\\,2}}{\\mathrm {d}x}\\right)(z-\\psi _{\\,1})(z-\\psi _{\\,3})-\\psi _{\\,0}\\left(y+\\frac{\\mathrm {d}\\psi _{\\,3}}{\\mathrm {d}x}\\right)(z-\\psi _{\\,1})(z-\\psi _{\\,2})\\\\\\dot{z}&=&h=-\\, \\psi _{\\,0}(x)(z-\\psi _{\\,1}(x))(z-\\psi _{\\,2}(x))(z-\\psi _{\\,3}(x)).$ The next step is to find fixed point sets.",
"From $\\dot{x}|_{\\,\\phi {\\cal A}_{\\mu }^{{\\cal I}}}=0,\\quad \\dot{y}|_{\\,\\phi {\\cal A}_{\\mu }^{{\\cal I}}}=0,\\quad \\dot{z}|_{\\,\\phi {\\cal A}_{\\mu }^{{\\cal I}}}=0.\\qquad \\mu =1,2,3$ one has that $\\phi {\\cal A}_{\\,\\mu }^{{\\cal I}}\\subset {\\cal C}$ , $(\\mu =1,2,3)$ forms a fixed point set for each $\\mu $ .",
"Here a phase portrait of the dynamical system is roughly discussed.",
"It follows from (REF ) that $x$ is constant in time, and thus $\\psi _{\\,\\mu }(x)$ does not depend on time.",
"Third, to prove the theorem, Lyapunov functions are constructed [50].",
"Define the functions $V_{\\,1}$ in ${\\cal D}_{\\,1}$ and $V_{\\,3}$ in ${\\cal D}_{\\,3}$ such that $V_{\\,1}(x,z)&=&\\frac{1}{2}(z-\\psi _{\\,1}(x))^{\\,2},\\qquad (x,y,z)\\in {\\cal D}_{\\,1}\\nonumber \\\\V_{\\,3}(x,z)&=&\\frac{1}{2}(z-\\psi _{\\,3}(x))^{\\,2},\\qquad (x,y,z)\\in {\\cal D}_{\\,3}.\\nonumber $ Then, one has In ${\\cal D}_{\\,1}$ , it follows that $V_{\\,1}(x,z)\\ge 0,\\quad \\frac{\\mathrm {d}V_{\\,1}}{\\mathrm {d}t}=(z-\\psi _{\\,1})h(x,z)=-\\psi _{\\,0}(z-\\psi _{\\,1})^{\\,2}(z-\\psi _{\\,2})(z-\\psi _{\\,3})\\le 0,\\qquad (x,y,z)\\in {\\cal D}_{\\,1},$ where the equality holds on the fixed point set $\\phi {\\cal A}_{\\,1}^{\\,{\\cal I}}$ .",
"Hence $V_{\\,1}$ is a Lyapunov function in ${\\cal D}_{\\,1}$ .",
"In ${\\cal D}_{\\,3}$ , it follows that $V_{\\,3}(x,z)\\ge 0,\\quad \\frac{\\mathrm {d}V_{\\,3}}{\\mathrm {d}t}=(z-\\psi _{\\,3})h(x,z)=-\\psi _{\\,0}(z-\\psi _{\\,1})(z-\\psi _{\\,2})(z-\\psi _{\\,3})^{\\,2}\\le 0,\\qquad (x,y,z)\\in {\\cal D}_{\\,3},$ where equality holds on the fixed point set $\\phi {\\cal A}_{\\,3}^{\\,{\\cal I}}$ .",
"Hence $V_{\\,3}$ is a Lyapunov function in ${\\cal D}_{\\,3}$ .",
"According to the theorem of Lyapunov, one completes the proof.",
"$\\Box $ The global behavior for $z$ is understood from Fig.",
"REF .",
"One then deduces from Fig.",
"REF that, given $x$ , $\\lim _{t\\rightarrow \\infty }z(t)=\\psi _{\\,1}(x)$ in ${\\cal D}_{\\,1}$ , and $\\lim _{t\\rightarrow \\infty }z(t)=\\psi _{\\,3}(x)$ in ${\\cal D}_{\\,3}$ .",
"Figure: Phase space ofthe dynamical system consisting of z ˙=h(x,z)\\dot{z}=h(x,z) and x ˙=0\\dot{x}=0(() and()).From z ˙=h\\dot{z}=h, it follows that the zeros ofhh are the fixed points.From hh in ()its zeros are the set z=ψ 1 (x)z=\\psi _{\\,1}(x), z=ψ 2 (x)z=\\psi _{\\,2}(x) andthe set z=ψ 3 (x)z=\\psi _{\\,3}(x).In addition, from the sign of hh, the set z=ψ 1 (x)z=\\psi _{\\,1}(x)and the set z=ψ 3 (x)z=\\psi _{\\,3}(x) are stable in some domains.To elucidate the behavior of the contact Hamiltonian vector field in Theorem REF on the lower dimensional spaces that have been used for the projections, see Fig.",
"REF .",
"This Theorem states that $\\phi {\\cal A}_{\\,1}^{\\,{\\cal I}}$ and $\\phi {\\cal A}_{\\,3}^{\\,{\\cal I}}$ are stable.",
"This is equivalent to say that the part of Legendre curves $\\overline{\\mathrm {v-vi}}$ and $\\overline{\\mathrm {ii-iii}}$ are stable in some domains.",
"Although it is not immediately clear how the stability of the curve $\\overline{\\mathrm {ii-iii}}$ plays a role in physical context, the role of stability of the curve $\\overline{\\mathrm {v-vi}}$ is clear.",
"That stability for $\\overline{\\mathrm {v-vi}}$ is consistent with the thermodynamic stability.",
"To grasp local flow around the fixed point sets, integral curves of the linearized equations are shown below.",
"For the point $(y_{\\mu },z_{\\mu })=(- \\psi _{\\,\\mu }^{\\,\\prime },\\psi _{\\,\\mu })$ , introduce $Y_{\\,\\mu }$ and $Z_{\\,\\mu }$ such that $y_{\\,\\mu }(t)=- \\psi _{\\,\\mu }^{\\,\\prime }(x)+Y_{\\,\\mu }(t),\\quad z_{\\,\\mu }(t)=\\psi _{\\,\\mu }(x)+Z_{\\,\\mu }(t),\\quad \\mbox{where}\\quad \\psi _{\\,\\mu }^{\\,\\prime }(x):=\\frac{\\mathrm {d}\\psi _{\\,\\mu }}{\\mathrm {d}x}(x),\\quad \\mu =1,2,3$ which yield linearized equations.",
"To avoid cumbersome notations, introduce $\\psi _{\\,\\mu \\mu ^{\\prime }}=\\psi _{\\,\\mu }(x)-\\psi _{\\,\\mu ^{\\prime }}(x),\\quad \\mu ,\\mu ^{\\,\\prime }=1,2,3,\\qquad \\psi _{\\,\\mu \\mu ^{\\prime }}>0,\\quad \\qquad (\\mu >\\mu ^{\\,\\prime }).$ Then the linearized equations are obtained as $\\dot{Z}_{\\,1}=-\\,\\underbrace{\\psi _{\\,0}\\,\\psi _{\\,21}\\psi _{\\,31}}_{>\\ 0}\\, Z_{\\,1},\\qquad \\dot{Z}_{\\,2}=\\ \\ \\underbrace{\\psi _{\\,0}\\,\\psi _{\\,21}\\psi _{\\,32}}_{>\\ 0}\\, Z_{\\,2},\\qquad \\dot{Z}_{\\,3}=-\\,\\underbrace{\\psi _{\\,0}\\psi _{\\,31}\\psi _{\\,32}}_{>\\ 0}\\, Z_{\\,3},$ and $\\dot{Y}_{\\,1}&=&-\\psi _{\\,0}\\,\\psi _{\\,21}\\psi _{\\,31}\\, Y_{\\,1}+(\\,\\psi _{\\,0}^{\\,\\prime }\\,\\psi _{\\,21}\\,\\psi _{\\,31}+\\psi _{\\,0}\\,\\psi _{\\,21}^{\\,\\prime }\\,\\psi _{\\,31}+\\psi _{\\,0}\\,\\psi _{\\,21}\\,\\psi _{\\,31}^{\\,\\prime })\\,Z_{\\,1},\\nonumber \\\\\\dot{Y}_{\\,2}&=&\\ \\psi _{\\,0}\\,\\psi _{\\,21}\\psi _{\\,32}\\, Y_{\\,2}-(\\,\\psi _{\\,0}^{\\,\\prime }\\,\\psi _{\\,21}\\,\\psi _{\\,32}+\\psi _{\\,0}\\,\\psi _{\\,21}^{\\,\\prime }\\,\\psi _{\\,32}+\\psi _{\\,0}\\,\\psi _{\\,21}\\,\\psi _{\\,32}^{\\,\\prime })\\,Z_{\\,2},\\nonumber \\\\\\dot{Y}_{\\,3}&=&-\\psi _{\\,0}\\,\\psi _{\\,31}\\psi _{\\,32}\\, Y_{\\,3}+(\\,\\psi _{\\,0}^{\\,\\prime }\\,\\psi _{\\,31}\\,\\psi _{\\,32}+\\psi _{\\,0}\\,\\psi _{\\,31}^{\\,\\prime }\\,\\psi _{\\,32}+\\psi _{\\,0}\\,\\psi _{\\,31}\\,\\psi _{\\,32}^{\\,\\prime })\\,Z_{\\,3}.\\nonumber $ To solve this linear system of equations, letting $c_{\\,\\mu }$ and $d_{\\,\\mu }$ be some constants, one can write the system as $\\dot{Z}_{\\,\\mu }=-\\,c_{\\,\\mu } Z_{\\,\\mu },\\quad \\dot{Y}_{\\,\\mu }=-\\,c_{\\,\\mu } Y_{\\,\\mu }+d_{\\,\\mu }Z_{\\,\\mu }.$ The solution of this system is $Z_{\\,\\mu }(t)=Z_{\\,\\mu }(0)\\,\\mathrm {e}^{\\,-c_{\\mu }\\,t},\\quad Y_{\\,\\mu }(t)=(\\,Y_{\\,\\mu }(0)+d_{\\,\\mu }Z_{\\,\\mu }(0)\\, t\\,)\\,\\mathrm {e}^{\\,-c_{\\mu }\\,t}.$ From this, the inequalities $c_{\\,1}>0$ , $c_{\\,3}>0$ , $c_{\\,2}<0$ , one has that the fixed point sets $\\phi {\\cal A}_{\\,1}^{{\\cal I}}$ and $\\phi {\\cal A}_{\\,3}^{{\\cal I}}$ are linearly stable, and the $\\phi {\\cal A}_{\\,2}^{{\\cal I}}$ is linearly unstable.",
"Observe from (REF ) and (REF ) that the strength of instability is large when the value of $\\psi _{\\,\\mu \\mu ^{\\,\\prime }}$ is large.",
"The condition when $\\psi _{\\,\\mu \\mu ^{\\,\\prime }}(x)$ is large can be read off from Fig.",
"REF .",
"The strength of such an instability is small near the critical point, and it is large far from the critical point."
]
] | 2107.01758 |
[
[
"Effective continuum models for the buckling of non-periodic architected\n sheets that display quasi-mechanism behaviors"
],
[
"Abstract In this work, we construct an effective continuum model for architected sheets that are composed of bulky tiles connected by slender elastic joints.",
"Due to their mesostructure, these sheets feature quasi-mechanisms -- low-energy local kinematic modes that are strongly favored over other deformations.",
"In sheets with non-uniform mesostructure, kinematic incompatibilities arise between neighboring regions, causing out-of-plane buckling.",
"The effective continuum model is based on a geometric analysis of the sheets' unit cells and their energetically favorable modes of deformation.",
"Its major feature is the construction of a strain energy that penalizes deviations from these preferred modes of deformation.",
"The effect of non-periodicity is entirely described through the use of spatially varying geometric parameters in the model.",
"Our simulations capture the out-of-plane buckling that occurs in non-periodic specimens and show good agreement with experiments.",
"While we only consider one class of quasi-mechanisms, our modeling approach could be applied to a diverse set of shape-morphing systems that are of interest to the mechanics community."
],
[
"Introduction",
"Advanced manufacturing and synthesis technologies have given engineers the ability to design media with complex micro- and mesostructures that strongly influence bulk constitutive properties [1], [2], [3].",
"For example, the micro/mesoscale geometry can be designed to attain extreme or unconventional global mechanical behaviors such as high stiffness-to-weight ratios [1] and bistable auxeticity [4].",
"These fabrication processes have considerably expanded the design space for shape-shifting media [5], [6], [7] and deployable structures [8], [9].",
"In this context, mesoscale design has been used to create compliant features that replace conventional hinges, extensional elements and flexures [10], [11], [12], or to create structures whose mechanical behaviors can be tailored by adjusting the geometry of a pattern [13], [14], [15], [16], [17], [18], [19], [20], [21].",
"In structured media, the mesoscale geometry can be designed to energetically favor desired local modes of deformation [22], [23].",
"We refer to these behaviors as “quasi-mechanisms” when they accompany a non-negligible change in the system's energetic state.",
"This distinguishes quasi-mechanisms from pure mechanisms, which are zero-energy kinematic modes.",
"We emphasize that quasi-mechanisms are local behaviors: these energetic preferences can be spatially modulated by designing non-uniform mesostructures.",
"Within this context, origami [10], [14], [24], [25], kirigami [26], [15], [27], [28] and auxetic motifs [29], [30], [31], [4], [32], [16] are the most popular classes of mesostructures that lead to quasi-mechanisms.",
"However, demonstrations of shape-shifting materials have also been achieved using thermally responsive bilayer lattices [20] and in 3D structures such as snapology origami [33].",
"Quasi-mechanisms can be used to attain non-homogeneous strain field objectives (even under uniform loading conditions) by relying on non-uniform internal structures that spatially modulate local effective material properties.",
"Morphing from a planar state to a doubly curved 3D geometry is an example of where this non-uniformity is important: Gauss' Theorema Egregium tells us that changing a surface's Gaussian curvature requires a non-isometric mapping [34], which in turn requires mesostructural non-uniformity if the actuation is driven by a spatially uniform stimulus [9], [20].",
"However, optimally designing non-uniform micro/mesostructures that lead to desired global behaviors can be challenging.",
"The presence of geometric features at disparate length scales means that conventional finite element approaches become computationally expensive due to the need for meshes that resolve the finest features and yet span the entire structure.",
"Homogenization theory provides a way to determine effective properties of periodic structures [35], but in practice it is often only viable in the limited context of linear elasticity, as the presence of non-linearity and instabilities significantly complicates the methods [36].",
"In light of this, engineers have used a variety of reduced order modeling techniques to investigate forward elastic equilibrium and stability problems, as well as to inversely design non-uniform mesostructures at a lesser computational expense.",
"These techniques range from bar-and-hinge [37], [38], [39] and structural frame [40] models that capture the mechanics of folded sheets, to representations of structural element networks that are based on effective springs [41], equivalent lattices [42], Chebyshev nets [43], discrete elastic rods [43], [44] and Kirchhoff rods [45].",
"Despite the above-mentioned advancements in modeling using networks of reduced order elements, there are limitations to the existing approaches.",
"They can be computationally expensive in cases where the structure is much larger than the mesoscale unit cell size and a reduced order element (such as a discrete elastic rod) is needed for every constituent of the physical network (e.g., in hierarchical systems).",
"Additionally, some of these models lack the generality needed to make themselves useful to the study of other systems.",
"For example, bar-and-hinge origami models would not be suitable for extensional spring networks.",
"It can also be challenging to calibrate constants such that accurate results are achieved using these models.",
"For these reasons, the mechanics community has pursued the development of effective continuum models.",
"These models are powerful approaches to capturing the behavior of structures with internal geometric patterns in instances where there is a sufficient separation of length scales between the local geometric parameters and the global behaviors [46].",
"When this separation of scales exists, an energy density function can be constructed to capture the mechanical behaviors of the structure as if it were a bulk material, thus removing the need to resolve the geometric features at the smaller length scales with a fine mesh.",
"This coarse meshing allows for significantly faster finite element simulations of complex physical behaviors.",
"To this end, effective continuum models have been used to understand the behavior of periodic structured media that display quasi-mechanism behaviors [47] and can capture their responses to non-uniform loading conditions [48], [49].",
"However, these effective continuum modeling frameworks have not been applied to modeling the quasi-mechanism behaviors of graded media.",
"Figure: (a-b) A sheet with a periodic cut pattern that displays a quasi-mechanism mode of deformation: rotation of tiles about slender elastic joints.",
"As the tiles rotate, the unit cell dimensions change from L α i L_\\alpha ^i to L α f L_\\alpha ^f.",
"Although tile rotations are low-energy kinematic modes compared to other deformations, the energetic cost associated with the deformation of the joints is not negligible.",
"(c) Introducing a gradient in the cut pattern modulates the quasi-mechanism kinematics over the sheet.",
"The scale bar represents 3 cm 3~\\mathrm {cm}.",
"(d) The mesostructural non-uniformity shown in (c) affects the extent to which tiles can rotate in different regions of the sheet, creating kinematic incompatibilities between the quasi-mechanism behaviors of different regions.",
"Here, λ x \\lambda _x is the maximum stretch a unit cell can attain in the direction of loading through quasi-mechanism behaviors.",
"(e) These in-plane kinematic incompatibilities lead to out-of-plane buckling.",
"The design of the buckling sheets shown in (c-e) was first discussed in our prior work .This article demonstrates how geometric analyses of unit cells can be used to construct effective continuum models for architected sheets with graded mesostructures.",
"We illustrate this approach by studying generalizations of the auxetic sheets introduced by Grima et al.",
"[29] to spatially varying distributions of diamond-shaped cuts [16], [50], [51].",
"The tessellated unit cells consist of bulky tiles connected by slender joints, and display two elastic regimes: a soft regime that occurs when the tiles rotate about the joints (as shown in Fig.",
"REF a-b), and a stiff regime when the joints are subjected to tension.",
"We design heterogeneous cut patterns to provoke in-plane kinematic incompatibilities under simple point-loading scenarios, which leads to out-of-plane buckling in a region of the structure [16] (shown in Fig.",
"REF e).",
"This article is organized as follows.",
"In Section , we discuss our effective continuum model for non-periodically patterned sheets that display quasi-mechanism behaviors.",
"Our modeling approach entails first performing a geometric analysis of unit cells to derive their energetically favorable kinematic modes.",
"Specifically, we derive the effect of geometric parameters on the rotational behavior of the tiles about the joints.",
"Next, we begin constructing our strain energy density function by attributing an energy penalty to deviations from the above-mentioned kinematic modes, which may occur due to kinematic incompatibilities between neighboring regions of the sheets.",
"Since the joints are not ideal pins, the rotation of tiles is an elastic process, albeit softer than deviations from this preferred local behavior.",
"We use a common constitutive model for elastic materials to approximate the elastic energy associated with the tile rotations.",
"We extract the value of a few non-geometric constants from tensile experiments on periodically patterned structures and these parameters are then used to simulate the non-periodic structure.",
"This type of effective material modeling enables us to use a coarse mesh to solve for pre-buckled equilibrium, the onset of instabilities, and post-buckled equilibrium.",
"The numerical approach is discussed in Section , and we compare these numerical results to a new set of experiments in Section , highlighting the good agreement between coarse mesh finite element simulations and experiments.",
"Our concluding remarks and perspective for future work are presented in Section .",
"While our modeling method is demonstrated for the class of quasi-mechanisms discussed above, we believe it would be straightforward to apply it to many other quasi-mechanisms that are of interest to the mechanics community, such as origami tessellations [25] and shape-shifting bilayer lattices [20]."
],
[
"Modeling approach",
"In this section, we discuss how a strain energy density function can be extracted by modeling the effect that mesoscale geometric features have on a structure's energetically favorable local modes of deformation.",
"Our approach is presented for modeling effective continua within the context of initially flat sheets with diamond-shaped cut patterns, although it could be generalized to other types of 2D or 3D architected media."
],
[
"Quasi-mechanism kinematics",
"Our aim is to create an effective continuum model that captures the quasi-mechanism kinematics of sheets with diamond-shaped cut patterns (Fig.",
"REF a).",
"These sheets are tessellations of unit cells that are composed of four bulky tiles connected by slender elastic joints (Fig.",
"REF b).",
"The structures may be either periodic or non-periodic tessellations of unit cells (as in Fig.",
"REF a or Fig.",
"REF c, respectively).",
"In either case, the quasi-mechanism local modes of deformation can be derived from a simple geometric analysis relating unit cell geometry to the rigid body rotations of the bulky tiles about the joints (Fig.",
"REF b-c).",
"Figure: (a) An example of a sheet with a uniform pattern of diamond-shaped cuts.",
"(b) A unit cell (shaded) consists of four tiles (boxed).",
"(c) The quasi-mechanism kinematics consist of tile rotations about the slender elastic joints.",
"This deformation mode can be entirely described by the projection of the tile diagonals onto the fixed orthogonal coordinate frame.",
"This rotational mode has a non-negligible energetic cost, but one that is still much lesser than deformations where the joints are under tension or shear.",
"(d) The reference configuration of the boxed tile shown in (b).",
"Five parameters define the geometry of a unit cell: l 1 l_1 and l 2 l_2 are the reference configuration lengths of the unit cell grid spacing in the 𝐞 1 \\mathbf {e}_1 and 𝐞 2 \\mathbf {e}_2 directions, δ\\delta is the width of the slender joints, and w 1 w_1 and w 2 w_2 are the half-widths of the two diamond-shaped cuts that define the tiles' inclinations.",
"The diagonals d v d_v and d h d_h and the angle γ\\gamma between these two can be computed from those parameters.",
"Finally, θ\\theta is the angle between the red diagonal, d h d_h, and the 𝐞 1 \\mathbf {e}_1 direction.",
"As the tile rotates from one configuration to another, this angle varies (as shown in b-c).",
"The projected lengths of the tile's deformed configuration in the 𝐞 1 \\mathbf {e}_1 and 𝐞 2 \\mathbf {e}_2 directions are d h cos(θ)d_h\\cos (\\theta ) and d v sin(γ+θ)d_v\\sin (\\gamma +\\theta ), respectively.",
"This allows us to compute the unit cell stretches: only the rotation of one tile about a joint needs to be analyzed to determine the quasi-mechanism kinematics of the unit cell.",
"(a-d) Adapted from by permission of The Royal Society of Chemistry.Five spatially varying geometric parameters constitute a geometry vector field $\\phi (x_\\alpha )$ and define the quasi-mechanism kinematics of our sheets.",
"Namely $\\phi =\\lbrace l_1,l_2,\\delta ,w_1,w_2\\rbrace $ , where $l_1(x_\\alpha )$ and $l_2(x_\\alpha )$ are the lengths of the unit cell grid spacing in the $\\mathbf {e}_1$ and $\\mathbf {e}_2$ directions, $\\delta (x_\\alpha )$ is the width of the slender joints, and $w_1(x_\\alpha )$ and $w_2(x_\\alpha )$ are the half-widths of the two diamond-shaped cuts that define the tiles' inclinations.",
"These parameters are illustrated in Fig.",
"REF d. A few geometric parameters that are functions of the five mentioned above are also shown in Fig.",
"REF d and will be discussed below.",
"We seek to identify a function $g(\\mathbf {C}, \\phi )$ such that the local quasi-mechanisms are described by the implicit relation $g(\\mathbf {C}, \\phi )=0$ .",
"Here, $\\mathbf {C}$ is the right Cauchy-Green strain tensor.",
"To do so, we first define a unit cell as a $2 \\times 2$ arrangement of quadrilateral tiles.",
"Due to the symmetry of the unit cell, we can fully describe its quasi-mechanism kinematics by analyzing the geometry and rotation of a single tile.",
"We use the bottom left tile in the unit cell, such as the one boxed in Fig.",
"REF b-c. For a unit cell located at $x_\\alpha $ with geometry defined by $\\phi (x_\\alpha )=\\lbrace l_1(x_\\alpha ),l_2(x_\\alpha ),\\delta (x_\\alpha ),w_1(x_\\alpha ),w_2(x_\\alpha )\\rbrace $ , the respective lengths $d_h$ and $d_v$ of the diagonals illustrated in Fig.",
"REF d in red and blue are $ d_h(\\phi )=\\sqrt{l_1^2+(l_2-2w_2-\\delta )^2}\\,\\,\\,\\,\\,\\,\\mathrm {and}\\,\\,\\,\\,\\,\\,d_v(\\phi )=\\sqrt{l_2^2+(l_1-2w_1-\\delta )^2}\\,.$ The angle $\\gamma $ between these two diagonals is given in terms of the geometric parameters $\\phi $ as $\\gamma (\\phi ) = \\frac{\\pi }{2}-\\arctan \\left(\\frac{l_2-2w_2-\\delta }{l_1}\\right)-\\arctan \\left(\\frac{2w_1+\\delta -l_1}{l_2}\\right)\\,.$ As the tile rotates about the joint, the angle $\\theta $ between the diagonal $d_h$ and the $\\mathbf {e}_1$ direction varies, as shown in Fig.",
"REF b-c. During this tile rotation, the projected lengths of the tile diagonals on the fixed orthogonal frame $\\mathbf {e}_i$ change, and the unit cell will have effective stretches $\\lambda _1$ and $\\lambda _2$ of $\\lambda _1(\\theta )=\\frac{d_h\\cos \\theta }{l_1}\\,\\,\\,\\,\\,\\,\\mathrm {and}\\,\\,\\,\\,\\,\\,\\lambda _2(\\theta )=\\frac{d_v\\sin (\\gamma +\\theta )}{l_2}\\,.$ We can invert the function for $\\lambda _1(\\theta )$ to obtain $\\theta (\\lambda _1)$ as $\\theta (\\lambda _1)=\\arccos \\left( \\frac{\\lambda _1 l_1}{d_h} \\right)\\,.$ Substituting (REF ) into the expression for $\\lambda _2(\\theta )$ in (REF ) leads to the following explicit formula for $\\lambda _2(\\lambda _1)$ : $\\lambda _2(\\lambda _1)=\\frac{d_v}{l_2}\\sin \\left[\\gamma +\\arccos \\left( \\frac{\\lambda _1 l_1}{d_h} \\right)\\right]\\,.$ We first derived this explicit function for the quasi-mechanism kinematics in our prior work [16].",
"Through trigonometric identities and algebraic manipulation, this can be written in implicit form: $\\Bigg (\\frac{l_1\\lambda _1}{d_h}\\Bigg )^2+\\Bigg (\\frac{l_2\\lambda _2}{d_v}\\Bigg )^2-2\\sin (\\gamma )\\frac{l_1 \\lambda _1}{d_h}\\frac{l_2 \\lambda _2}{d_v}-\\cos ^2(\\gamma )=0$ In our reference frame, the implicit function (REF ) can be rewritten using the components of $\\mathbf {C}$ , since $\\mathbf {C_{11}}=\\lambda _1^2 \\mathbf {e}_1\\otimes \\mathbf {e}_1$ and $\\mathbf {C_{22}}=\\lambda _2^2\\mathbf {e}_2\\otimes \\mathbf {e}_2$ : $g(\\mathbf {C}, \\phi )=\\frac{l_1^2\\mathbf {C_{11}}}{d_h^2(\\phi )}+\\frac{l_2^2\\mathbf {C_{22}}}{d_v^2(\\phi )}-2\\sin \\Big (\\gamma (\\phi )\\Big )\\frac{l_1 l_2}{d_h(\\phi ) d_v(\\phi )}\\sqrt{\\mathbf {\\det C}}-\\cos ^2\\Big (\\gamma (\\phi )\\Big )=0$ The quasi-mechanism kinematics expressed in (REF ) describe the unit cells' preferred modes of local deformation as a function of geometric parameters.",
"We emphasize that a unit cell may not deform according to this function.",
"For example, this may occur if neighboring unit cells have a different geometry and cause kinematic incompatibility or if global loading conditions make these modes of deformation energetically unfavorable.",
"In these cases, $g(\\mathbf {C},\\phi )\\ne 0$ .",
"In Section REF , we will model the stiffening that occurs when (REF ) cannot be satisfied by embedding this kinematic description as a penalty term in our strain energy function."
],
[
"Kinematics of a thin elastic plate",
"Our aim is to embed the quasi-mechanism behavior described by (REF ) into an effective continuum model.",
"We consider a thin elastic plate whose material particle positions of the mid-plane in an initially flat reference configuration are $\\mathbf {X}=x_\\alpha \\mathbf {e}_\\alpha $ .",
"The indices $\\alpha $ and $\\beta $ in this subsection relate to the mid-plane of the plate (we use the Einstein summation convention for repeated indices), and the index `3' corresponds to the direction normal to the reference surface.",
"The coordinate frame $\\lbrace \\mathbf {e}_i\\rbrace $ is fixed and orthonormal.",
"The domains for the material coordinates $x_\\alpha $ are $x_1\\in [0,a]$ and $x_2\\in [0,b],$ where $a$ and $b$ are constants.",
"The thickness $t$ is much smaller than the other material domain dimensions, and we seek the mid-surface mapping $\\mathbf {\\chi }(x_\\alpha )$ : $\\chi (x_\\alpha )=\\Big (x_\\alpha +u_\\alpha (x_\\beta )\\Big )\\mathbf {e}_\\alpha +w(x_\\beta )\\mathbf {e}_3,$ where $u_\\alpha $ and $w$ are the in-plane and out-of-plane components of the mid-plane displacement vector, respectively.",
"The deformation gradient tensor $\\mathbf {\\tilde{F}}=\\nabla \\chi $ can be expressed in terms of the gradients of $u_\\alpha $ and $w$ .",
"We label $\\mathbf {F}$ as the in-plane component of the deformation gradient tensor ($\\mathbf {F}\\equiv \\mathbf {I}+\\nabla u_\\alpha )$ .",
"Since we have two material coordinates embedded in three spatial dimensions, the deformation gradient assumes the following form: $\\mathbf {\\tilde{F}}=\\begin{bmatrix}1+u_{1,1} & u_{1,2} \\\\u_{2,1} & 1+u_{2,2} \\\\w_{,1} & w_{,2} \\\\\\end{bmatrix} =\\begin{bmatrix}\\mathbf {F}\\\\\\nabla w\\\\\\end{bmatrix}$ We use the right Cauchy-Green deformation tensor, $\\mathbf {C}$ , as our measure for in-plane strain, and the Laplacian of the out-of-plane deflections, $\\Delta w$ , as our bending strain measure: $\\mathbf {C}=\\mathbf {\\tilde{F}^T \\tilde{F}}=\\mathbf {F^T F} + \\nabla w \\otimes \\nabla w,\\ \\ \\ \\ \\ \\ \\Delta w = \\frac{\\partial ^2 w}{\\partial x_1^2}+\\frac{\\partial ^2 w}{\\partial x_2^2}$"
],
[
"Strain energy",
"Now that we have an implicit function (REF ) describing the quasi-mechanism behavior and a formulation of thin plate kinematics, we can construct a strain energy density function for our sheets.",
"The first step is to attribute an energy penalty $\\Psi _p$ for deviations from the quasi-mechanism behavior.",
"As discussed in Section REF , $g(\\mathbf {C},\\phi )=0$ when local deformations correspond to quasi-mechanism behaviors, and $g(\\mathbf {C},\\phi )\\ne 0$ when there is a deviation from these energetic preferences.",
"Therefore we can write our energy penalty $\\Psi _p$ as $\\Psi _p=\\frac{1}{2\\eta }g^2(\\mathbf {C}, \\phi )\\ ,$ where $\\eta $ is a small parameter.",
"For our perforated sheets, $g(\\mathbf {C},\\phi )$ is given in (REF ).",
"Therefore, $\\Psi _p=\\frac{1}{2\\eta }\\Bigg ( \\frac{l_1^2\\mathbf {C_{11}}}{d_h^2}+\\frac{l_2^2\\mathbf {C_{22}}}{d_v^2}-2\\sin (\\gamma )\\frac{l_1 l_2}{d_h d_v}\\sqrt{\\mathbf {\\det C}}-\\cos ^2(\\gamma ) \\Bigg )^2\\ .$ For elastic bodies, deforming according to these preferential modes will still entail non-zero energy.",
"Thus, we must also assign a soft elastic energy density $\\Psi _s$ to this scenario (this softness is relative to the energy expense of deviating from quasi-mechanism behaviors).",
"A compressible Neo-Hookean model provides the flexibility to approximate our experimental data from tensile tests well while using only two material parameters.",
"Therefore, the total membrane strain energy density function $\\Psi _m(\\mathbf {C}, \\phi ) = \\Psi _p(\\mathbf {C}, \\phi ) + \\Psi _s(\\mathbf {C})$ is $\\Psi _m(\\mathbf {C}, \\phi ) = \\Psi _p(\\mathbf {C}, \\phi )+\\frac{\\mu }{2}(\\bar{I}_1-2) + \\frac{\\lambda }{2}(J-1)^2\\ ,$ where $J=\\sqrt{\\det (\\mathbf {C})}$ , $\\bar{I}_1=\\mathrm {tr} (\\mathbf {C}) J^{-1}$ , $\\mu $ and $\\lambda $ are the Lamé parameters and $\\Psi _p$ is given in (REF ).",
"Our bending energy density functions is $\\Psi _b= \\frac{B (\\Delta w)^2}{2}\\ ,$ where $B$ is a bending stiffness constant.",
"Our strain energy per unit thickness is the sum of $\\Psi _m$ and $\\Psi _b$ , integrated over the 2D domain spanned by the mid-plane of the sheet, $\\Omega $ : $ \\mathcal {E}(\\mathbf {u}, w) =\\int _{\\Omega } \\bigg (\\Psi _m(\\mathbf {C}, \\phi ) + \\Psi _b(\\Delta w)\\bigg ) \\ dA$ All of the parameters in the energy function are either geometric or can be extracted from three simple tensile experiments: one on a dogbone specimen of the bulk rubber with no cut patterns, and two (conducted in orthogonal directions) on a sheet with periodic but anisotropic cuts."
],
[
"Contact model",
"In the case where the sheet lies on a rigid surface, we wish to enforce the contact condition $w \\ge 0$ .",
"While techniques such as the active set method directly impose this constraint, we opt to relax this condition and instead use a rather simple penalty-based contact model.",
"Thus, for problems where the sheet is lying on a flat surface, we consider a contact penalty energy for negative out-of-plane deflections: $\\Psi _c(w) = \\frac{P}{2}(dw^-)^2, \\qquad dw^- = \\min (0, w+\\varepsilon ),$ where $P$ is the penalty stiffness and $\\varepsilon > 0$ is a small tolerance length.",
"Notice that the contact energy is nonzero only when $w < -\\varepsilon $ .",
"This ensures that the contact condition does not interfere with the stability of the initially flat, unbuckled plate, and only becomes active post-bifurcation.",
"We add this contact energy onto (REF ) to give the total energy functional $\\mathcal {E}(\\mathbf {u}, w) = \\int _\\Omega \\Psi _m(\\mathbf {C},\\phi ) + \\Psi _b(\\Delta w) + \\Psi _c(w) \\ dA.$ We will discuss the variations of this energy to compute equilibrium and stability in Section ."
],
[
"Finite element implementation",
"In this section, we present the equilibrium conditions for the system.",
"Using a mixed formulation, we compute the solution using standard first order Lagrange polynomial finite elements.",
"More details for our solution procedure and stability analysis are provided in the appendices.",
"We implement this formulation in the deal.II open source finite element library [52].",
"Figure: An example of a domain and of a set of boundary conditions used in our simulations.",
"In-plane displacements are prescribed on a portion of the boundary and in-plane traction-free edges are observed on the remainder.",
"Additionally, we constrain out-of-plane displacements and have no applied moments on the entire boundary.",
"This drawing displays the boundary conditions used to model the sheet with non-uniform cut patterns shown in Fig c-e.We consider a rectangular domain in a displacement-controlled setting.",
"The in-plane displacements $\\mathbf {u}$ are prescribed on $\\partial _u \\Omega \\subset \\partial \\Omega $ and we have in-plane traction free edges on the remainder, $\\partial _f \\Omega = \\partial \\Omega \\backslash \\partial _u \\Omega $ .",
"Additionally, we constrain out-of-plane displacements $w$ and have moment-free edges on the entire boundary.",
"Fig REF shows an example of a domain and of a set of boundary conditions used in some of our simulations.",
"While the boundary conditions may be altered for a more general case, the mixed formulation discussed in Subsection REF may not be appropriate for situations such as clamped boundaries."
],
[
"Equilibrium and mixed formulation",
"The equilibrium condition is the stationarity of our energy functional from (REF ) in both $\\mathbf {u}$ and $w$ , $\\frac{\\mathrm {d}}{\\mathrm {d} \\kappa } \\Big [ \\mathcal {E}(\\mathbf {u} + \\kappa \\delta \\mathbf {u}, w + \\kappa \\delta w) \\Big ]_{\\kappa = 0} = 0 \\qquad \\text{for all} \\quad \\delta \\mathbf {u} \\in \\mathcal {U}_0,\\quad \\delta w \\in H^2_0(\\Omega ),$ where $\\mathcal {U}_0$ is the set of kinematically admissible in-plane displacement variations $\\mathcal {U}_0 = \\left\\lbrace \\mathbf {u} \\in \\left( H^1(\\Omega ) \\right)^2, \\ \\mathbf {u} = 0 \\ \\text{on} \\ \\partial _u \\Omega \\right\\rbrace ,$ and we search for solutions $\\mathbf {u} \\in \\mathcal {U}$ and $w \\in \\mathcal {W}$ where $\\mathcal {U} = \\left\\lbrace \\mathbf {u} \\in \\left( H^1(\\Omega ) \\right)^2, \\ \\mathbf {u} = \\mathbf {u}_0 \\ \\text{on} \\ \\partial _u \\Omega \\right\\rbrace , \\quad \\mathcal {W} = \\left\\lbrace w \\in H^2(\\Omega ), \\ w = w_0 \\ \\text{on} \\ \\partial \\Omega \\right\\rbrace .$ A common issue for plate problems is the bi-harmonic operator on $w$ that arises from the Gateaux derivative of the bending energy.",
"In this case, the weak form contains a product of the second derivative of $w$ and its variation, so that the usual Galerkin finite element method with even quadratic Lagrange polynomial shape functions is not appropriate.Standard Lagrange polynomial shape functions have discontinuous first-derivatives at the boundaries of elements.",
"This would result in integrating the product of two Dirac delta functions, which is undefined.",
"Therefore, we turn to a mixed formulation that is widely used for linear biharmonic problems [53].",
"We introduce a scalar function $v \\in H^1_0(\\Omega )$ and set it equal to $\\Delta w$ by considering an augmented energy $\\widehat{\\mathcal {E}}(\\mathbf {u}, w) = \\sup _{v \\in H^1_0(\\Omega )} \\ \\int _{\\Omega } \\Psi _m(\\mathbf {C}) + \\Psi _c(w) - B \\left( \\nabla w \\cdot \\nabla v - \\frac{1}{2} {v}^2 \\right) \\ dA.$ Stationarity of $\\widehat{\\mathcal {E}}$ in both $\\mathbf {u}$ and $w$ , along with the suprema condition on $v$ , gives the weak form of equilibrium $ \\begin{aligned}0 &= \\int _{\\Omega } \\left( 2 \\mathbf {F} {\\Psi _m}{\\mathbf {C}} \\right) : \\nabla \\delta \\mathbf {u} \\ dA \\qquad && \\forall \\delta \\mathbf {u} \\in \\mathcal {U}_0, \\\\0 &= \\int _{\\Omega } \\left( 2 {\\Psi _m}{\\mathbf {C}}\\nabla w \\right) \\cdot \\nabla \\delta w + {\\Psi _c}{w} - B \\nabla v \\cdot \\nabla \\delta w \\ dA \\qquad && \\forall \\delta w \\in H^1_0 (\\Omega ), \\\\0 &= \\int _{\\Omega } - B \\nabla w \\cdot \\nabla \\delta v - B v \\delta v \\ dA \\qquad && \\forall \\delta v \\in H^1_0(\\Omega ).",
"\\\\\\end{aligned}$ The first two lines in (REF ) are the equilibrium relations for in-plane and out-of-plane displacements, respectively.",
"The final line is the constraint that $v = \\Delta w$ weakly.",
"The strong form of these relations can be found in .",
"Notice that (REF ) only contains first derivatives of the displacements and their variations.",
"It is shown in [53] that we may now consider $w \\in H^1(\\Omega )$ .",
"Therefore, we use a Galerkin finite element formulation with p = 1 shape functions for the fields $\\mathbf {u}$ , $w$ and $v$ .",
"We solve the nonlinear system with typical Newton-Raphson iterations.",
"Details on the finite element formulation and solution procedure can be found in ."
],
[
"Stability analysis",
"To probe the stability of an equilibrium configuration, it is common practice to calculate the eigenvalues of the tangent stiffness matrix.",
"A negative eigenvalue implies an instability, and the equilibrium solution can then be perturbed in the direction of the corresponding eigenvector to explore the buckled solution.",
"However, the mixed formulation complicates this procedure.",
"To assess stability, we must restrict the eigenvectors to the subspace upon which the constraint $v = \\Delta w$ is satisfied.",
"To this end, we consider an effective stiffness matrix on this subspace.",
"By solving the linear constraint explicitly, we can condense $v$ out of the system matrix.",
"Then, we calculate eigenvalues of this reduced stiffness matrix to asses stability.",
"We use the linear constraint to map the corresponding eigenvector back to the full variable set and perturb the system.",
"The magnitude of the perturbation is chosen to be on the same order as the displacement increment.",
"The direction of the perturbation is decided such that the $w$ component at the middle of the sheet is positive.",
"The full details of the stability analysis can be found in ."
],
[
"Results",
"In this section, we discuss the extraction of effective material model constants from experiments on sheets with uniform cut patterns and we compare experimental and numerical results on the post-buckling behavior of sheets with non-periodic mesostructure."
],
[
"Extracting model constants from experiments on sheets with uniform cut patterns",
"As discussed in Section REF , our energy given in (REF ) requires the extraction of four parameters from experiments: the Lamé moduli ($\\lambda $ and $\\mu $ ), the energy penalty parameter ($\\eta $ ), and the bending stiffness ($B$ ).",
"We obtained $\\lambda $ , $\\mu $ and $\\eta $ from tensile tests on the specimen with uniform cut patterns shown in Fig.",
"REF a, where $l_1=l_2=6~\\mathrm {mm}$ , $\\delta =l_1/8$ , $w_1=(l_1-\\delta )/2$ , and $w_2=0~\\mathrm {mm}$ .",
"The sheets have a thickness of $t=1.55~\\mathrm {mm}$ , width dimensions of $108~\\mathrm {mm}$ in each direction and are made of natural rubber gum.",
"The diamond-shape cuts were made using a laser cutter.",
"The specimen was placed on a custom apparatus that grips the edges with roller pins, thus allowing free sliding in the direction perpendicular to the tension.",
"To obtain $\\lambda $ and $\\mu $ , the sheet was loaded in the direction that induces quasi-mechanism behavior (rotation of the tiles about the elastic joints).",
"Since the sheet's cut pattern is uniform, no kinematic incompatibilities arise and only the soft elastic mode is present.",
"The values of $\\mu =17~\\mathrm {kPa}$ and $\\lambda = 0.1~\\mathrm {kPa}$ provided a good fit to our data, as shown in Fig.",
"REF .",
"To attain $\\eta $ , the sheet was loaded in the perpendicular direction, where tiles do not rotate because their diagonals are aligned in the direction of loading and the elastic joints are in tension.",
"We attain a good fit of our data by setting $\\eta =0.002~\\mathrm {kPa}^{-1}$ .",
"Fig.",
"REF shows a comparison of effective continuum simulations of the in-plane elastic behaviors with experiments and Abaqus/Standard simulations from prior work [16], where the mesh fully resolves the fine features of the specimen geometry.",
"Figure: Effective stress vs. stretch for a sheet with a periodic cut pattern.",
"The insets show four unit cells of this structure, see Fig.",
"a for an image of the entire sheet.",
"We compare our effective continuum model (solid red and blue lines) represented by () to experiments (solid black lines) and fine-grain finite element simulations (gray dashes) that fully resolve the small geometric features in our sheets.",
"These experiments and the fine-grain simulations (using Abaqus/Standard) were conducted in our prior work ).",
"The experimental curve for the soft loading direction does not start at λ=1\\lambda =1 due to the effect of gravity in a vertically loaded tensile testing machine.",
"The inset on the bottom left of the figure shows a small region of the mesh used in the Abaqus simulations to capture the geometry of the elastic joints.",
"The large number of elements needed for these fine grain simulations motivates the usage of effective continuum models.",
"The insets in this image were adapted from by permission of The Royal Society of Chemistry.We adjust the classic bending stiffness for a Kirchhoff-Love plate [54] by including a scaling factor $\\alpha (f)$ that accounts for the reduced bending stiffness of a sheet with porosity $f$ .",
"Therefore, the bending stiffness of the patterned sheet can be written in the following form: $B = \\frac{\\alpha (f) E t^2}{12(1-\\nu ^2)}\\ .$ Here, $E=2~\\mathrm {MPa}$ is Young's modulus (obtained from linear regime tensile tests on a $55~\\mathrm {mm}\\times 9.2~\\mathrm {mm}\\times 1.5~\\mathrm {mm}$ dogbone sample of natural rubber), $t$ is the sheet thickness, and $\\nu =0.5$ is Poisson's ratio.",
"A recent paper by Shrimali, et al.",
"[55] showed that the effective bending stiffness of thin perforated plates is much more dependent on the plate's porosity $f$ than on the shape or size of the perforations.",
"This holds both for plates where the sheet thickness is much smaller than the unit cell dimension, and vice versa.",
"Given the porosity of our sheets ($f\\approx 0.5$ ), we adopt a scaling value of $\\alpha (f)=0.25$ , as suggested by the results in [55].",
"Their results also justify our use of a uniform bending stiffness.",
"Again, (REF ) is the strain energy per unit thickness, hence the scaling of $B$ with $t^2$ .",
"Based on these considerations, no additional experiment is required to obtain the bending stiffness."
],
[
"Out-of-plane buckling of sheets with graded mesostructure",
"We now consider a more interesting pattern of cuts that is non-periodic, and where spatial variations in the local quasi-mechanism behavior lead to kinematic incompatibilities.",
"To model the behavior of these sheets, we update the geometry vector $\\phi (x_\\alpha )=\\lbrace l_1(x_\\alpha ),l_2(x_\\alpha ),\\delta (x_\\alpha ),w_1(x_\\alpha ),w_2(x_\\alpha )\\rbrace $ .",
"We have three specimens of equal thickness $t=1.55$ mm, but varying aspect ratios.",
"Now, $l_1=\\lbrace 4.5~\\mathrm {mm},\\ 6~\\mathrm {mm},\\ 7.5~\\mathrm {mm}\\rbrace $ for the three sheets (the overall width dimensions of the square sheets scale linearly with $l_1$ to $162~\\mathrm {mm}$ , $216~\\mathrm {mm}$ , and $270~\\mathrm {mm}$ , respectively).",
"The other parameters are $l_2=2l_1$ , $\\delta = l_1/8$ , $w_1=(l_1-\\delta )/2$ , and $w_2(x_\\alpha )=\\frac{l_1-\\delta }{2}\\Big (1-\\sin \\frac{\\pi x_2}{18l_2}\\Big )$ .",
"The non-uniform geometry is accounted for by considering spatially varying $w_2(x_\\alpha )$ in the finite element formulation.",
"We note that although the geometric parameter $w_2(x_\\alpha )$ is non-uniform, we still use a uniform soft elastic energy density, $\\Psi _s$ , because it represents the energetic cost of the non-ideal mechanism and the joint density is still uniform.",
"The geometric gradation of the mesostructure leads to variations in the local quasi-mechanism behavior over the extent of the sheet.",
"This causes in-plane kinematic incompatibilities, which lead to out-of plane buckling after each sheet's critical stretch is reached, as shown in Fig.",
"REF a-b.",
"We show the buckled mode nucleation and the evolution of the post-buckled height of the central point in the sheets as a function of boundary point displacement in Fig.",
"REF c. We compare simulations of our effective continuum model (computed using the deal.II finite element library [52] on a $36 \\times 36$ uniform quadrilateral mesh) to measurements of the physical samples (using a level-calibrated mounted caliper) and see excellent agreement between the two, especially at larger boundary displacements.",
"As expected, the stretch at which buckling occurs is delayed by increasing the thickness-to-width ratio.",
"The difference between the computational predictions and experimental measurements of buckling nucleation and height at lower stretch values can be partially attributed to the fact that our simulations do not account for friction with the table or gravity.",
"These two physical processes are important since the material is soft and bending is a low-energy deformation for shells with small gaussian curvature.",
"As the dome height increases, the structure becomes less susceptible to the effect of gravity.",
"Finally, to better visualize how the post-buckling behavior evolves and is affected by the aspect ratio of the sheet, we show laser scans of the physical specimens and deformed simulation meshes at three different boundary point displacements in Fig.",
"REF .",
"Accurate quantitative comparisons are challenging due to the manual stitching process that follows the acquisition of laser scan data patches, which introduces slight distortions and puts certain regions of the scanned sheet at an inclined plane relative to the rest of the structure.",
"As expected, the post-buckled domes are wider (relative to the overall width of the sheets) for specimens that have larger thickness-to-width ratios, showing good qualitative agreement between experiments and simulations.",
"Furthermore, the onset of buckling occurs at greater stretches as $t/l_1$ increases.",
"Figure: Buckling behavior of sheets with non-uniform cut patterns.",
"(a) Up to a certain stretch λ\\lambda , point displacements lead to in-plane deformations.",
"(b) Following a critical value of λ\\lambda , the in-plane kinematic incompatibilities will lead to out-of-plane buckling.",
"The scale bar represents 3 cm.",
"(c) Comparison of dome height between effective continuum simulations (solid lines) and experiments (dots) for sheets of three aspect ratios.",
"Here, h mid h_{mid} is the height of a sheet's center point, λ\\lambda is the stretch of the sheet's center line in the 𝐞 1 \\mathbf {e}_1 direction, tt is the sheet thickness, and l 1 l_1 is the length of the unit cell grid spacing in the 𝐞 1 \\mathbf {e}_1 direction.Figure: Post-buckling behavior of sheets with three thickness-to-width ratios.",
"These are the same three sheets represented in Fig. c.",
"Here, tt is the sheet thickness, l 1 l_1 is the length of the unit cell grid spacing in the 𝐞 1 \\mathbf {e}_1 direction, and λ\\lambda is the applied stretch at the midpoint of the sheet edge.",
"In each entry of the stretch vs. aspect ratio grid, the laser scans are plotted directly above the simulated deformed meshes.",
"As expected, we see that sheets with higher thickness-to-width ratios will nucleate at larger stretches and will buckle into wider domes relative to the overall sheet width.These results show that this effective continuum modeling framework is a powerful tool for understanding the physics of quasi-mechanisms in non-periodic media.",
"In our previous work [16], we only captured in-plane deformation mappings using standard, fine-grained finite element procedures since the large number of elements needed to resolve the small mesostructural features (in the range between $10^5$ and $10^6$ elements depending on the structure being simulated) caused the calculation of out-of-plane buckling modes to have an inviable computational cost.",
"Using the effective continuum approach we can get accurate results merely using a $36\\times 36$ uniform quadrilateral mesh, a reduction of two to three orders of magnitude in the number of elements used.",
"Each of the bifurcation curves in Fig REF took roughly 5 minutes to compute running on a single core of a Intel® Xeon® 5218 processor.",
"Meanwhile, we could not make simulations for the post-buckling behavior of our sheets converge in a reasonable amount of time using a standard fine-grained FEM approach."
],
[
"Conclusions",
"We present an effective continuum modeling framework for architected media that display quasi-mechanism behaviors and demonstrate its validity on sheets that are patterned with diamond-shaped cuts.",
"The model incorporates a penalty for deviations from quasi-mechanism behaviors and relies on material model parameters extracted directly from experiments.",
"We show that the approach correctly predicts the mechanical behavior of non-periodic media, even when the model's parameters are derived from experiments on periodic specimens.",
"Our approach permits accurate and efficient simulations of mechanical behaviors that would otherwise be impractical to model using fine-grained simulations that fully resolve the material's small geometric features.",
"We note that the implicit relation (REF ) does not define the function $g(\\mathbf {C},\\phi )$ uniquely, implying that other choices of the functions $\\Psi _p$ from (REF ) and $\\Psi _m$ from (REF ) are possible.",
"A good agreement with experiments is still attained, suggesting that the buckling behavior of the sheet is robust with respect to the choice of the function $g$ .",
"There are a few limitations to this approach.",
"First, it requires a sufficient separation of length scales between the global deformation mode dimensions and the unit cell size.",
"Therefore, it would not be able to capture the local buckling modes observed in some kirigami sheets [56] or handle the dome kinking that occurs in our systems if they are fabricated from extremely thin sheets [16].",
"Furthermore, although we believe that this modeling approach could be applied to a broad range of architected media that display quasi-mechansims, extracting the material model constants from experiments may be more challenging in other systems in comparison to the perforated sheets we have discussed.",
"Finding a suitable soft elastic energy density $\\Psi _s$ that is appropriate for the quasi-mechanism regime also requires the modeler to have an intuition for which constitutive models can be appropriately tailored to fit experimental data attained from experiments on their system.",
"In the future, this modeling framework could be adapted to 3D media and materials with temporally varying mechanical properties, provided that they also display quasi-mechanisms."
],
[
"Acknowledgements",
"C.M.",
"and C.D.",
"were supported by the US Army Research Office Grant W911NF-17-1-0147.",
"This work was also supported by a NASA Space Technology Research Fellowship to C.M.",
"We thank Andrei Constantinescu and Kaushik Bhattacharya for helpful discussions, and Paul Stovall for assistance with fabrication."
],
[
"Strong form of equilibrium",
"The strong form of the equilibrium relations under the mixed formulation are $ \\begin{aligned}-\\nabla \\cdot \\left( 2 \\mathbf {F} {\\Psi _m}{\\mathbf {C}} \\right) &= 0 \\qquad && \\text{in } \\Omega , \\\\-\\nabla \\cdot \\left( 2 {\\Psi _m}{\\mathbf {C}} \\nabla w \\right) + {\\Psi _c}{w} + B \\Delta v &= 0 && \\text{in } \\Omega , \\\\B(\\Delta w - v) &= 0 \\qquad && \\text{in } \\Omega , \\\\\\end{aligned}$ with boundary conditions $\\begin{aligned}\\left( 2 \\mathbf {F} {\\Psi _m}{\\mathbf {C}} \\right) \\cdot n &= 0 \\qquad && \\text{on } \\partial _f \\Omega , \\\\u &= u_0 && \\text{on } \\partial _u \\Omega , \\\\w = w_0, \\ v &= 0 && \\text{on } \\partial \\Omega .\\end{aligned}$ The first two equations in (REF ) are the in-plane and out-of-plane momentum balance equations, respectively.",
"The last equation is the constraint that $v = \\Delta w$ ."
],
[
"Finite element formulation and Solution Procedure",
"The fields $\\mathbf {u}$ , $w$ , and $v$ are $H^1(\\Omega )$ , so we may consider a Galerkin finite element formulation with p = 1 shape functions for them.",
"Therefore, $\\mathbf {u} = \\sum _{i = 0}^{n_u} u_i \\mathbf {\\Phi }^u_i, \\qquad w = \\sum _{i = 0}^{n_w} w_i \\Phi ^w_i, \\qquad v = \\sum _{i = 0}^{n_v} v_i \\Phi ^v_i,$ where $\\lbrace \\mathbf {\\Phi }^u_i\\rbrace $ is the set of vector-valued shape functions for the in-plane displacements.",
"$\\lbrace \\Phi ^w_i\\rbrace $ and $\\lbrace \\Phi ^v_i\\rbrace $ are the scalar-valued sets of shape functions for $w$ and $v$ , respectively.",
"Because we assume homogeneous boundary conditions for both of these fields, we can then consider $\\lbrace \\Phi ^w_i\\rbrace = \\lbrace \\Phi ^v_i\\rbrace $ .",
"Then, using these shape functions for the variations in (REF ), the discrete equilibrium equations can be written as $\\begin{bmatrix}\\mathbf {R}^u \\\\\\mathbf {R}^w \\\\\\mathbf {R}^v \\\\\\end{bmatrix} = \\mathbf {R} = \\mathbf {0}\\ ,$ where $\\begin{aligned}R^u_i &= \\int _\\Omega \\left( 2 \\mathbf {F} {\\Psi _m}{\\mathbf {C}} \\right) : \\nabla \\mathbf {\\Phi }^u_i \\ dA, \\\\R^w_i &= \\int _{\\Omega } \\left( 2 {\\Psi _m}{\\mathbf {C}} \\nabla w - B \\nabla v \\right) \\cdot \\nabla \\Phi ^w_i + {\\Psi _c}{w} \\Phi ^w_i \\ dA, \\\\R^v_i &= \\int _{\\Omega } - B \\, v \\, \\Phi ^v_i - B \\nabla w \\cdot \\nabla \\Phi ^v_i \\ dA.",
"\\\\\\end{aligned}$ To solve for this equilibrium, we use Newton-Raphson updates of the form $\\mathbf {K}(\\mathbf {x}) \\Delta \\mathbf {x} = - \\mathbf {R}(\\mathbf {x}),$ where $\\mathbf {x} = [ u_0, \\ldots , u_{n_u}, w_0, \\ldots , w_{n_w}, v_0, \\ldots , v_{n_v} ]$ is the vector of degrees of freedom, $\\Delta \\mathbf {x}$ are their updates, and $\\mathbf {K}$ is the tangent stiffness matrix $\\mathbf {K} = \\begin{bmatrix}\\mathbf {K}^{uu} & \\mathbf {K}^{uw} & \\mathbf {0} \\\\\\mathbf {K}^{wu} & \\mathbf {K}^{ww} & \\mathbf {K}^{wv} \\\\\\mathbf {0} & \\mathbf {K}^{vw} & \\mathbf {K}^{vv}\\end{bmatrix},$ where $\\begin{aligned}K^{uu}_{ij} &= \\int _\\Omega \\nabla \\mathbf {\\Phi }^u_i : {^2 \\Psi _m}{\\mathbf {F} \\partial \\mathbf {F}} : \\nabla \\mathbf {\\Phi }^u_j \\ dA, \\\\K^{ww}_{ij} &= \\int _\\Omega \\nabla \\Phi ^w_i \\cdot {^2 \\Psi _m}{\\nabla w \\partial \\nabla w } \\cdot \\nabla \\Phi ^w_j \\ dA, \\\\K^{vv}_{ij} &= \\int _{\\Omega } -B\\, \\Phi ^v_i \\Phi ^v_j \\ dA, \\\\K^{uw}_{ij} &= K^{wu}_{ji} = \\int _\\Omega \\nabla \\mathbf {\\Phi }^u_i : {^2 \\Psi _m}{\\mathbf {F} \\partial \\nabla w } \\cdot \\nabla \\Phi ^w_j \\ dA, \\\\K^{w v}_{ij} &= K^{v w}_{ji} = \\int _\\Omega -B \\nabla \\Phi ^w_i \\cdot \\nabla \\Phi ^{v}_j \\ dA.",
"\\\\\\end{aligned}$ The displacements $\\mathbf {u}_0$ on the boundary are incremented, and Newton-Raphson is used to reach an equilibrium configuration.",
"The previous equilibrium configuration is used as an initial guess for the subsequent iterations."
],
[
"Stability analysis with mixed method constraint",
"To probe the stability of an equilibrium configuration, it is common practice to calculate the eigenvalues of the tangent stiffness matrix.",
"A negative eigenvalue implies an instability, and the equilibrium solution can be perturbed in the direction of the corresponding eigenvector to explore the buckled solution.",
"In our case, we must restrict ourselves to eigenvectors in the subspace where the constraint $v = \\Delta w$ is satisfied.",
"To this end, we consider an effective stiffness matrix from the quadratic form, upon which the constraint is satisfied.",
"Consider the discrete constraint equation: $\\mathbf {R}^{v} = \\mathbf {K}^{v w} \\mathbf {w} + \\mathbf {K}^{v v} \\mathbf {v} = \\mathbf {0}.$ This can also be written in the following form: $\\mathbf {v} = -\\left( \\mathbf {K}^{v v} \\right)^{-1} \\mathbf {K}^{v w} \\mathbf {w}.$ We can then use a reduced variable set $\\mathbf {x}_r$ under which the constraint is satisfied, as $\\mathbf {x} = \\begin{bmatrix}\\mathbf {u} \\\\\\mathbf {w} \\\\\\mathbf {v}\\end{bmatrix} = \\begin{bmatrix}\\mathbf {I}_{n_u \\times n_u} & \\mathbf {0} \\\\\\mathbf {0} & \\mathbf {I}_{n_w \\times n_w}\\\\\\mathbf {0} & -\\left( \\mathbf {K}^{v v} \\right)^{-1} \\mathbf {K}^{v w}\\end{bmatrix} \\begin{bmatrix}\\mathbf {u} \\\\\\mathbf {w}\\end{bmatrix} = \\mathbf {P} \\mathbf {x}_r.$ Then, the quadratic form gives $\\mathbf {x}^T \\mathbf {K} \\mathbf {x} = \\mathbf {x}_r^T \\, \\widetilde{\\mathbf {K}} \\, \\mathbf {x}_r,$ where $\\widetilde{\\mathbf {K}} = \\mathbf {P}^T \\mathbf {K} \\mathbf {P} =\\begin{bmatrix}\\mathbf {K}^{uu} & \\mathbf {K}^{uw} \\\\\\mathbf {K}^{wu} & \\left( \\mathbf {K}^{ww} - \\mathbf {K}^{w v} \\left( \\mathbf {K}^{vv} \\right)^{-1} \\mathbf {K}^{v w} \\right)\\end{bmatrix}.$ Then to assess stability, we probe the eigenvalues of this effective stiffness matrix $\\widetilde{\\mathbf {K}}$ .",
"An eigenvalue passing through zero along the principle deformation path implies an instability.",
"The corresponding eigenvector can then be used to produce a perturbation, using $\\mathbf {P}$ to map back to the full variable set.",
"The magnitude of the perturbation is chosen to be on the same order as the displacement increment.",
"The direction of the perturbation is decided such that the $w$ component at the middle of the sheet is positive."
]
] | 2107.01704 |
[
[
"Cusp in the Symmetry Energy, Speed of Sound in Neutron Stars and\n Emergent Pseudo-Conformal Symmetry"
],
[
"Abstract We review how the \"cusp\" predicted in the nuclear symmetry energy generated by a topology change at density $n_{1/2}\\gsim 2 n_0$ can have a surprising consequence, so far unrecognized in nuclear physics and astrophysics communities, on the structure of dense compact-star matter.",
"The topology change, when translated into nuclear EFT with \"effective\" QCD degrees of freedom in terms of hidden local and scale symmetries duly taken into account, predicts an EoS that is soft below and stiff above $n\\gsim n_{1/2}$, involving no low-order phase transitions, and yields the macrophysical properties of neutron stars consistent -- so far with no tension -- with the astrophysical observations, including the maximum mass $ 2.0\\lsim M/ M_\\odot\\lsim 2.2$ as well as the GW data.",
"Furthermore it describes the interior core of the massive stars populated by baryon-charge-fractionalized quasi-fermions that are neither baryonic nor quarkonic.",
"It is argued that the cusp \"buried\" in the symmetry energy resulting from strong correlations with hidden heavy degrees of freedom leads, at $n\\gsim n_{1/2}$, to what we dubbed \"pseudo-conformal\" sound speed, $v^2_{pcs}/c^2\\approx 1/3$, precociously converged from below at $n_{1/2}$.",
"It is not strictly conformal since the trace of energy-momentum tensor is not zero even in the chiral limit.",
"This observation with the topology change identified with the putative hadron-quark continuity, taking place at at density $\\gsim 2 n_0$, implies that the quantities accurately measured at $\\sim n_0$ cannot give a stringent constraint for what takes place at the core density of compact stars $\\sim (3-7) n_0$.",
"This is because the change of degrees of freedom in effective field theory is involved.",
"We discuss the implication of this on the recent PREX-II \"dilemma\" in the measured skin thickness of $^{208}$Pb."
],
[
"introduction",
"In accessing dense neutron-star matter in terms of a topology change for the putative hadron-quark continuity, it was discovered in 2011 [1] that a cusp is present in the nuclear symmetry energy $E_{\\rm sym}$ at a density $\\sim (2-3)$ times the normal nuclear matter density $n=n_0\\simeq 0.16$ fm$^{-3}$ .",
"This cusp structure has been found to play the most important role in the approach to the EoS of dense compact-star matter developed entirely independently of other on-going approaches in nuclear astrophysics.",
"When the idea of [1] was first submitted to Phys.",
"Rev.",
"Lett., it was summarily rejected by the DA Editor of PRL at the time on the ground that “the skyrmion approach to nuclear physics was never considered as a viable alternative to more traditional ab initio calculations ... and is now more or less abandoned ...\" The DAE's decision followed two concurrent events that took place then.",
"First, all the mathematically astute – mostly particle – theorists who contributed to resuscitating, in the light of QCD, the Skyrme model which had been left totally ignored for two decades in nuclear physics community, quickly moved over to string theory which was then beginning to undergo its first “string revolution for theory of everything.\"",
"In the absence of those string theorists, making progress with highly intricate mathematics involved in skyrmion physics as is now being abundantly recognized in other areas of physics such as condensed matter, string theory etc.",
"became much too difficult for many nuclear theorists – except for a few diehard enthusiasts.",
"The second event was that at the time the paper [1] was written, chiral EFT was becoming very popular in the nuclear theory community.",
"It was also a lot easier to access numerically for quick results for publication.",
"What transpired in this paper and is continuing to transpire in current developments, as briefly mentioned in Conclusion regarding the BPS skyrmion and holographic dual approaches to nuclear physics, aptly illustrates an unfortunately prevalent short-sightedness on truly novel and unconventional ideas in the field too often dismissed as “conjectural.\"",
"Formulated with the minimum number of degrees of freedom available it has the power to go beyond the standard chiral effective field theory (S$\\chi $ EFT), currently heralded as as a possible “first-principles approach\" to nuclear theory at low energy and densitiy, and gives extremely simple predictions that have the merit to be unambiguously confronted by experiments in the density regime inaccessible by S$\\chi $ EFT.",
"It has thus far accounted with no unsurmountable tension for all macro-physical observables available in both terrestrial and astrophysical laboratories.",
"See for the current status, e.g., [2].",
"In this paper, we show that this cusp structure zeroes in on the recent issue raised by the PREX-II measurement of the neutron skin thickness of $^{208}$ Pb [3] and the impact on the equation of state (EoS) of massive compact stars.",
"An analysis using the new $R_{\\rm skin}^{208}$ and certain correlations with the symmetry energy $J$ and its slope $L$ (to be defined) at $n=n_0$ led to the 1 $\\sigma $ intervals [4] $J=(38.1\\pm 4.7)\\ {\\rm MeV}, \\ L=(106\\pm 37)\\ {\\rm MeV}.$ These values seemingly overshoot greatly the currently “accepted\" values [5] We will elaborate on these “accepted\" values below, $J=(31.7\\pm 1.1)\\ {\\rm MeV}, \\ L=(59.8\\pm 4.1)\\ {\\rm MeV}.$ This result means the EoS must be a lot stiffer at normal nuclear matter density than what has been considered up to date.",
"A similar observation termed as a “dilemma\" is arrived at by Piekarewicz from the electric dipole polarizability of neutron-rich nuclei [6].",
"Naively extrapolated to the massive compact-star density, the $R_{\\rm skin}^{208}$ data could rule out most of, or at least put in serious tension, the EoS' currently available in the literature for compact-star physics.",
"The stiff EoS implied by the dilemma turns out, as we will discuss later, to have a dramatic effect on the properties of massive stars such as the composition of the star core and sound speed.",
"We will show that the cusp structure discovered in [1] gives rise to a totally different picture.",
"This has a close connection to the lore popularly accepted in certain nuclear astrophysics circles that the EoS determined accurately at low density, say, as $\\sim n_0$ , should make an indispensable constraint to the EoS at higher densities.",
"Put differently, the lore that we shall refer to as nuclear-astrophysics lore (nLORE for short) states that what happens in the core of compact stars must be constrained by what happens in nuclear matter.",
"This of course must be true in a uniguely given theory, namely, QCD.",
"However given that QCD can directly access neither nuclear matter nor compact-star matter, what's available is effective field theory (EFT) in the sense defined by Weinberg's Folk Theorem.",
"In EFT, this nLORE cannot be valid if there are phase changes or crossovers.",
"In fact, we will argue the presence of the cusp in our approach debunks the nLORE on constraints on EoS.",
"What turns out to importantly figure in our argument is the existence of that cusp at a density $\\mathrel {\\unknown.",
"{\\hspace{1.0pt}\\sim }}$ >$$ (2-3)n0$ in the symmetry energy induced by a (what we consider to be robust) topology change in dense matter that effectively encodes the putative hadron-quark continuity expected in QCD.",
"This point is signaled also in a different context in \\cite {fraction}.",
"It aptly reconciles a soft EoS at $ n $\\sim $ $<$ n1/2$ to a hard EoS at $ n > n1/2$ accounting notably, among others, for massive $$\\sim $ $>$ 2 M$ compact-stars and other macroscopic star properties including the recent gravity wave data.\\section {The cusp in \\mathbf {E_{\\rm sym}}}The quantity that plays the most important role in the EoS for compact-star matter~\\cite {Steiner:2004fi} is the symmetry energy $ Esym$ in the energy per nucleon given by\\begin{eqnarray}E(n, \\alpha ) & = & E(n, \\alpha = 0) + E_{\\rm sym}(n)\\alpha ^2 + O(\\alpha ^4) + \\cdots ,\\end{eqnarray}where $ =(N-P)/(N+P)$ is the neutron-proton asymmetry with $ P$ ($ N$) standing for the number of protons (neutrons) in $ A=N+P$ nucleon system.",
"The $ J$ and $ L$ concerned are\\begin{eqnarray}J=E_{\\rm sym}(n_0), \\ L=3 \\frac{\\partial E_{\\rm sym}(n)}{\\partial n}|_{n=n_0}.\\end{eqnarray}The issue associated with the Pb skin thickness puzzle involves this symmetry energy on which our discussion will be focused.",
"In standard nuclear physics approaches (SNPA) anchored on effective density functionals such as the Skyrme potential, relativistic mean field (RMF) and varieties thereof as well as S$$PT up to manageable chiral order, equipped with a certain number of parameters fit to available empirical data, the $ E(n,)$ can be more or less reliably determined in the vicinity of the nuclear matter equilibrium density $ n0$.",
"It has also been extended, with albeit significant uncertainty, up to slightly above $ n0$ from heavy-ion collision experiments.",
"Thus one can say that the nuclear symmetry energy $ Esym$ is fairly well determined up to $ n0$ in SNPAs.",
"It should, however, be stressed that its slope in density, namely, $ L$ and higher derivatives remain uncertain, say in S$$PT, unless chiral-order terms up to N$ m$LO for $ m$\\sim $ $>$ 4$ are fully included.",
"This is closely tied to the fact that the chiral power expansion (say, in S$$EFT) is bound to break down for $ kF$ for $$\\sim $ $>$ 5$ ((e.g., \\cite {holt-rho-weise}) as the hadron-quark crossover density is approached.",
"So the problem is how $ Esym$ and its derivatives behave beyond the equilibrium density $ n0$.",
"This is where heavy degrees of freedom (HDsF) need to enter.\\subsection {Cusp in Skyrmion Crystal at O(1/N_c)}We address this problem exploiting a topological structure of dense baryonic matter.",
"This is because in the large $ Nc$ limit in QCD, the only known non-perturbative tool available in strong interaction physics applicable to baryonic matter at large density -- in the absence of lattice QCD -- is putting skyrmions (or instantons in holographic QCD) on crystal lattice~\\cite {crystal}.",
"Application of the crystal skyrmion lattice method to nuclear matter and dense matter has been around for some time (see for an early review \\cite {PV-lattice}) but only recently is the power of skyrmion approach beginning to be recognized in nuclear physics, contrary to condensed matter as well as string theory where the skyrmion structure in various spatial dimensions has been having remarkable impacts~\\cite {multifacet}.",
"This is because of the extreme mathematical subtlety involved in the skyrmion physics.",
"Furthermore, the condition for the validity of lattice skyrmions in particular, i.e.",
"large $ Nc$ and large density, is not met at the density where there is a wealth of experimental data, namely finite nuclei\\footnote {There is a striking recent development that we will refer to below as the potential power of the skyrmion structure in finite nuclei relative to the nLORE.",
"It will exhibit the indispensable role of HDsF.}.",
"However the cusp structure in question that takes place at relatively high density -- relative to normal nuclear matter -- seems to meet the two conditions as indicated by the quasi-scale invariance seen in the crystal simulation in the half-skyrmion phase~\\cite {PKLMR}.$ To illustrate the basic idea, we first take the Skyrme model [14] stabilized by the (Skyrme) quartic term for skyrmions supplemented by a scalar dilaton as first shown in [1].",
"What is crucial is that the Skyrme model encodes the necessary topological structure.",
"But by itself, it misses certain nontrivial crucial dynamical characteristics encoded in QCD.",
"We will implement these missing ingredients with hidden local symmetry (HLS for short) supplemented with hidden scale symmetry (HSS) and incorporate them for quantitative discussions in the generalized chiral effective field theory (EFT) approach that is dubbed $Gn$ EFT [2].",
"We will present the argument that the HLS and HSS (combined, referred to as sHLS), the degrees of freedom associated with which are identified with the HDsF involved at high density, are “dual\" to QCD (gluons and quarks)This notion of hadron-quark duality will be specified below.",
"in the density regime relevant to compact stars.",
"It will be argued that the density involved is located far below asymptotic density at which hardon-quark continuity presumably does break down (to be specified below).",
"Following [1], we calculate the symmetry energy by quantizing the crystal as a whole object through a collective rotation in iso-space with the rotation angle $C(t)$ acting on the relevant chiral fields $U=\\xi ^2$ (in unitary gauge) as $\\xi _c(\\mathbf {x}) & \\rightarrow & \\xi (\\mathbf {x}, x) = C(t) \\xi _c(\\mathbf {x}) C^\\dagger (t),\\nonumber \\\\V_{\\mu ,c}(\\mathbf {x}) & \\rightarrow & V_{\\mu }(\\mathbf {x},t) = C(t) V_{\\mu ,c}(\\mathbf {x}) C^\\dagger (t) ,$ where the subindex “$c$ \" means the static configuration with the lowest energy for a given crystal size $L$ and $C(t)$ is a time-dependent unitary $SU(2)$ matrix in isospace.",
"We define the angular velocity through ${\\mathbf {\\Omega }}$ $\\frac{i}{2}\\mathbf {\\tau }\\cdot \\mathbf {\\Omega } & = & C^\\dagger (t)\\partial _0 C(t) .$ The energy of the $n$ -nucleon system can be written as $M_{\\rm tot} & = & M_{\\rm static} + \\frac{1}{2}\\lambda _{I}^{\\rm tot} \\mathbf {\\Omega }^2.$ By regarding the angular momentum in isospace, $\\mathbf {J} = \\delta M_{\\rm tot}/\\delta \\mathbf {\\Omega }$ , as the isospin operator, one can write the total energy of the system as $M_{\\rm tot} & = & n M_{\\rm sol} + \\frac{1}{2n \\lambda _{I}}I^{\\rm tot}(I^{\\rm tot} + 1),$ where $M_{\\rm sol}$ , $\\lambda _{I}$ and $I^{\\rm tot}$ are, respectively, the mass and moment of inertia of the single skyrmion in the system, and the total isospin of the $n$ -nucleon.",
"Given that the $n$ -nucleon system is taken a nearly pure neutron system, $I^{\\rm tot} \\le n/2$ , to the leading order of $n$ for $n \\rightarrow \\infty $ , the energy per baryon takes the form $E & = & M_{\\rm sol} + \\frac{1}{8 \\lambda _{I}}\\alpha ^2.$ Thus the symmetry energy is $E_{\\rm sym} & = & \\frac{1}{8 \\lambda _{I}} +O(1/N_c^2).$ The moment of inertia $\\lambda _I\\sim O(N_c)$ can be computed in the leading $N_c$ order as the integral over the single cell and takes the form $\\lambda _{I} & = & \\frac{f_\\pi ^2}{6}\\left\\langle \\left(4 - 2 \\phi _0^2\\right)\\right\\rangle + \\delta \\lambda _{I} +\\cdots ,$ where the first term comes from the quadratic current algebra term and the second stands for the contribution from the Skyrme quartic term which consists of four terms involving $\\phi _0$ and space derivatives of the chiral field $\\xi $ .",
"Here $\\phi _0$ , proportional to the quark condensate $\\langle \\bar{q}q\\rangle $ , plays a crucial role in the whole development in [2].",
"Figure: Panorama of the symmetry energy E sym E_{\\rm sym}.",
"Left panel (copied from ): Wilderness in both various nuclear models and Sχ\\chi EFTs and bounds given by neutron star (up-to-date) observations (solid blue lines).",
"Right panel: Schematic form of the cusp (dotted line) in skyrmion crystal .",
"The solid line caricatures the effect of smearing by heavy degrees of freedom (HDsF).",
"The interval between n ' ∼n^\\prime \\mathrel {\\unknown.",
"{\\hspace{1.0pt}\\sim }}>n0and and n1/2isthedensityregimethatisarguablythemostdifficulttoaccessbystandard is the density regime that is arguably the most difficult to access by standard EFTfrombelowandbyQCDproperfromaboveasdiscussedinthetextinconnectionwiththesoundspeedandthetidaldeformation.EFT from below and by QCD proper from above as discussed in the text in connection with the sound speed and the tidal deformation.In the skyrmion crystal formalism, the topology change is associated with the behavior of the quark condensate at a density labeled $n_{1/2}$ which should, and generically does, lie above $n_0$ .",
"The quark condensate $\\Sigma \\equiv \\langle \\bar{q}q\\rangle $ , nonzero both globally and locally for $n <n_{1/2}$ , goes to zero at $n_{1/2}$ when space averaged, $\\phi _0\\equiv \\bar{\\Sigma }\\rightarrow 0$ .",
"But it is locally non-zero, thus generating chiral density wave and giving rise to a non-vanishing pion decay constant, $f_\\pi \\ne 0$ .",
"This transition triggers a skyrmion in the matter to fractionalize into 2 half-skyrmions.",
"Since the order parameter, here the pion decay constant, is non-zero in the changeover, there is no low-order phase transition.",
"This half-skyrmion “phase\"Lacking a better terminology, we will continue to (mis)use this term.",
"resembles what is referred to as “pseudo-gap phase\" in condensed matter physics, e.g., in high-T superconductivity.",
"An important – and most crucial – property of the symmetry energy in this formulation, $\\propto 1/\\lambda _{I}$ , is that it develops a cusp at $O(1/N_c)$ at the density $n_{1/2}$ where $\\phi _0\\rightarrow 0$ .",
"The cusp structure seen in the skyrmion lattice simulation [1] is schematically depicted in Fig.",
"REF (right panel).",
"The exact location of the cusp depends on the parameters of the Lagrangian which are à priori unknown, so it is arbitrary.",
"It will be determined later from neutron-star observations to lie within the range $2 \\mathrel {\\unknown.",
"{\\hspace{1.0pt}\\sim }}$ <$$ n1/2/n0 < 4$.",
"This cusp form comes from an interplay involving the behavior of $ 0$ between the quadratic derivative current algebra term and the countering contribution from the Skyrme quartic derivative term.",
"Roughly what happens is that the increase of $$ from the quadratic term as $ 0$ goes to zero is stopped by the quartic term at $ n1/2$ and starts dropping, causing the cusp in $ 1/$.",
"It will be shown that this picture will be modified in nature by, among others, two observations.",
"First, the skyrmion crystal simulation cannot be trusted at low density below $ n/12$, and next, the Skyrme quartic term can be taken as what results from integrating out HDsF from the skyrmion Lagrangian.",
"The large $ Nc$ consideration gives a remarkably simple $ Esym$.$ To illustrate what is captured in this cusp, we quote in Fig.",
"REF (left panel) the recent illuminating summary by B.A.",
"Li et al.",
"[15] of the up-to-date experimental and theoretical status of the symmetry energy.",
"It presents a giant wilderness.",
"All the theoretical models available up to date, e.g., various energy density functionals, $\\chi $ EFTs etc,, fit $\\$E_{\\rm sym}$ by fiat to what's given in nature at $\\sim n_0$ .",
"There are ample parameters available to allow it.",
"The swamp sets in beyond $n_0$ .",
"Given the absence of trustful models – not to mention theories, there is no guidance how the $E_{\\rm sym}$ should move at higher densities.",
"There is nothing to prevent it from going up or down, even plunging below zero.",
"The current experimental observations such as neutron stars (and heavy-ion data limited to only a few times $n_0$ ) do not fare any better as indicated by the solid (blue) lines in the left panel of Fig.",
"REF .",
"What is certain is that he cusp is buried in this jungle.",
"It may not be absurd to think that the cusp structure could be just an artifact of the lattice simulation.",
"But it turns out, we will see, that it is not.",
"When the jungle is cleared up by the symmetries assumed to be involved, the cusp yields an extremely simple and portent mechanism needed for the EoS for massive stars.",
"In particular, we will argue, the cusp represents the hadron-quark “duality\" expressed in topology change [2].",
"It will lead to what will be termed “pseudo-conformal sound speed\" and baryon-charge fractionalized “confined\" fermions in the core of neutron stars.",
"We should stress that what's involved in our approach is “hadron-quark duality,\" not just hadron-quark continuity that captures crossover from hadronic degrees of freedom to quark/gluon degrees of freedom.",
"In fact the notion of hadron-quark duality is a lot more general in the sense elaborated recently in the Cheshire-Cat Principle [16].",
"It represents the necessity at densities exceeding $n_0$ of the “heavy degrees of freedom (HDsF).\"",
"Those HDsF are to encode the quarks/gluons degrees of QCD at some high density without explicit presence of quark/gluons.",
"How to do this precisely is presently unknown in the density regimes relevant to compact stars.",
"This is because the densities involved are too far from the asymptotic regime where perturbative QCD is applicable and the only nonperturbative tool known, lattice QCD, is famously inaccessible at high density.",
"So the question is: How does one proceed?",
"In [1], the cusp was reproduced by the role played by the pions and the vector mesons in the nuclear tensor force in standard nuclear structure physics.",
"There the vector mesons were identified as hidden local fields and the scalar $\\sigma $ as a dilaton.",
"The key idea there was to exploit the vacuum-change-induced density dependence in the hHLS Lagrangian in the presence of baryonic matter [21].",
"Here we repeat essentially the same arguments to bring out certain characteristics of sHLS hidden in the discussions, namely, the “duality\" assumed to hold à la Seiberg between hidden local gauge fields and QCD gluons and a hadrons-quarks/gluons duality.",
"The objective is to link it to what S$\\chi $ EFT does at low energy (and density) and to extend it to higher densities where S$\\chi $ EFT is to break down.",
"This would make our approach to compact-star matter in line with the spirit of the Folk Theorem on EFT.",
"At present, the duality assumed is only a conjecture, but there are several compelling indications that such duality does most likely hold at high density (and perhaps also at high temperature) [17], [18], [19], [20].",
"In the absence of a rigorous proof, we take this as our working assumption.",
"Figure: Tensor force vs. density n=(1-3)n 0 n=(1-3) n_0: Without topology change (left panel) and with topology change at n 1/2 ≈2n 0 n_{1/2}\\approx 2n_0 (right panel)Our reasoning relies on two well-known (established) facts in nuclear physics in the presence of the HDsF.",
"The first is that the symmetry energy is predominantly controlled by the nuclear tensor force, and the second is that the nuclear tensor force gets principal contributions from the exchange of the pseudo-Nambu-Goldstone pion $\\pi $ and the $\\rho $ meson and coming with an opposite sign, they tend to destructively interfere.",
"It has also been established, within the framework of $Gn$ EFT with density-scaling hadron masses [21], that the net tensor force is to decrease with increasing density in the effective range of force in nuclear medium with short-range correlations suitably taken into account.",
"What figures here are the “vector manifestation\" of the $\\rho $ meson at high density, the dilaton condensate controlling the hadron masses in dense medium and the interplay of the $\\omega $ -nucleon coupling with the nucleon mass [2].",
"The resulting tensor force is depicted in Fig.",
"REF .",
"The left panels shows the decreasing tensor force at increasing density in the absence of topology changeThis dropping of the tensor force at increasing density is manifested in various nuclear phenomena, the validity of which has been amply supported.",
"A most spectacular case is the simple and elegant explanation of the long lifetime of C-14 beta-decay [22].",
"There are some ab initio calculations using three-body forces that seem to explain more or less equally well, but this should not be considered belittling the beauty of the simple tensor-force mechanism.",
"Correctly done, both are equally correct in physics.. (We note for later discussion that the net tensor force would vanish in the relevant range at $n\\sim 3n_0$ ).",
"However if there intervenes the topology change at $n_{1/2}$ , the tensor force undergoes a dramatic change as seen Fig.",
"REF (right panel).",
"For $n\\mathrel {\\unknown.",
"{\\hspace{1.0pt}\\sim }}$ >$$ n1/2$, the $$ tensor gets suppressed more or less completely so that the net tensor gets abruptly recovered to that of the pionic strength.$ How this changeover comes about is quite involved requiring a series of arguments [2], but it is not hard to understand what'a at work in the mechanism with two assumptions.",
"The assumptions are that (A) the vector mesons introduced as HDsF are hidden local symmetric subject to “composite gauge symmetry\" [23], [24] and (B) the scalar that provides an attractive nuclear force is the dilaton $\\sigma $ of the “genuine dilaton\" structure [25].",
"Now the assumption (A) asserts that at some high density, the vector mesons become massless, in particular with the gauge coupling $g_\\rho $ going to zero [23] associated with the vector manifestation fixed point mentioned above, and the assumption (B) admits a (precocious) emergence of spontaneously broken scale symmetry with $f_\\sigma \\approx f_\\pi \\ne 0$ , accommodating massive matter fields, in particular, light-quark baryons, à la genuine dilaton scenario with the dilaton condensate dictating how hadron masses scale in density [21].",
"The two effects entail the abrupt changeover at $n_{1/2}$ in the tensor force in Fig.",
"REF from the left panel to the right panel.",
"To see how this changeover produces the cusp in $E_{\\rm sym}$ , one recalls that the symmetry energy is predominantly controlled by the tensor force.",
"A quick and simple way to estimate the dominant tensor-force contribution to $E_{\\rm sym}$ is to do the closure-sum approximation of the iterated tensor force terms [26].",
"This exploits that the ground state is strongly coupled by the tensor force (subject, however, to the decreasing strength with density described above) to the states of excitation energies $\\sim 200$ MeV, so $E_{\\rm sym}\\approx C\\frac{\\langle V_T^2\\rangle }{200\\ {\\rm MeV}}$ with $C> 0$ a known constant.",
"With the NN interactions duly screened by short-range correlations (for which the $\\omega $ meson enters), it can be seen that $\\langle V_T^2\\rangle $ decreases as density goes toward $n_{1/2}$ and then increases afterwards in the precise way as in the skyrmion lattice simulation, thus reproducing the cusp Fig.",
"REF at $n_{1/2}$ .",
"While this argument holds more reliably on the right side of the cusp, namely in the halfs-skrymion phase, it is not expected to to hold well in the skyrmion phase away from the cusp.",
"This is because there the effects well described by S$\\chi $ EFT that include complicated configurations at high chiral orders involving other components of the force than the tensor-force are missing in this treatment.",
"This will become visible in the $Gn$ EFT result shown below."
],
[
"Smoothed Cusp",
"This calculation for the cusp (with the nuclear tensor force affected by the topology change) smoothed by the HDsF corresponds to the large $N_c$ and quasi-classical approximation in standard nuclear physics calculations.",
"In the formulation of $Gn$ EFT, this is equivalent to the mean-field approximation with the sHLS Lagrangian [2] which corresponds to the Landau Fermi-liquid fixed point approximation in the large $N_c$ and large $\\bar{N}\\equiv k_F/(\\Lambda -k_F)$ limit [27].",
"As shown in Fig.",
"REF (right panel) this cusp is made to smoothly cross over in the “$V_{lowk}$ renormalized group (RG) approach\" going beyond the Fermi-liquid fixed point approximation in $Gn$ EFT employed in [13], [2].",
"It takes into account $1/\\bar{N}$ corrections in the “ring-diagram\" approximation.",
"It corresponds to a generalized Fermi-liquid theory applicable to the relevant range of densities from $n_0$ to the compact-star matter density $\\sim (5-7)n_0$ with the topology change incorporated at $n_{1/2}$ .",
"It is strictly valid in the large $N_c$ limit but has been verified to work well for nuclear matter, arguably as well as the S$\\chi $ EFT to N$^3$ LO.",
"The power of this approach is that while the S$\\chi $ EFT most likely breaks down at $n_{1/2}$ , it becomes more reliable at higher densities as the Fermi-liquid fixed point is approached, that is as $1/\\bar{N}\\rightarrow 0$ .",
"The role sHLS plays here in smoothing the cusp is analogous to eliminating the cusp singularity in the $\\eta ^\\prime $ potential term for the $\\eta ^\\prime $ EFT with the HLS fields becoming topological Chern-Simons fields, giving rise to the fractional quantum Hall droplet baryon [18], [20].",
"Although the cusp is smoothed, it makes the symmetry energy that is soft below $n_{1/2}$ to stiffen above $n_{1/2}$ .",
"This not only accounts for the observed massive neutron stars but as we will show, will render moot the nLORE, hence the possible PREX-II dilemma.",
"What's even more striking is that the cusp impacts via the $E_{\\rm sym}$ so constructed the sound speed $v_{pcs}$ as $E_{\\rm sym}(n)\\rightarrow v^2_{pcs} (n)/c^2\\rightarrow 1/3\\ {\\rm for }\\ n\\mathrel {\\unknown.",
"{\\hspace{1.0pt}\\sim }}>$ n1/2.",
"This is because $E_{sym} (n)=E(n,\\alpha =1)-E(n,\\alpha =0)$ from Eq.",
"() and the trace of the energy-momentum tensor (TEMT) given by $E(n,\\alpha )$ is density-independent for $\\alpha =0$ and $\\alpha =1$ for $n\\mathrel {\\unknown.",
"{\\hspace{1.0pt}\\sim }}$ >$$ n1/2$~\\cite {MR-review}.",
"Hence the crossover in the latter directly impacts the bump in $ vpcs$.",
"These matters will be taken up in Sect.~\\ref {bump}.$ In order to give credence to the $E_{\\rm sym}$ obtained in $Gn$ EFT that we will rely on, we summarize what the $Gn$ EFT treated in $V_{lowk}$ RG predicts for the EoS for nuclear matter and how it fares in nature.",
"As stressed in [2], the possible topology change density is constrained to the range $2\\mathrel {\\unknown.",
"{\\hspace{1.0pt}\\sim }}$ <$$ n1/2/n0 < 4$.",
"For simplicity we take $ n1/2$\\sim $ $>$ 2 n0$ as representing our prediction within a small range of uncertainty.$ In what follows, the strangeness flavor degrees of freedom, hyperons as well as kaons, do not enter in the density regime involved, say, $n\\sim (5-6) n_0$ .",
"The reason for this is explained in Conclusion Section.",
"We divide the density regime into two: (A) $n\\mathrel {\\unknown.",
"{\\hspace{1.0pt}\\sim }}$ >$$ n0$ and (B) $ n$\\sim $ $>$ n1/2=2n0$.\\begin{itemize}\\item (A) Up to the topology change density n_{1/2}\\mathrel {\\unknown.",
"{\\hspace{1.0pt}\\sim }}>\\end{itemize}$ 2 n0$, there is only one parameter that is completely determined by how the pion decay constant $ f$ scales with density; it is known up to $ nn0$.",
"Within a bit of fine-tuning on this scale parameter, all the EoS properties come out fully consistent with the accepted values : They are $ n0=0.16$ fm$ -3$, BE =16.7 MeV, $ K0=250 (24020)$ MeV, $ J=Esym(n0)=30.2 (31.73.2)$ MeV, $ L=67.8 (58.728.1)$ MeV.Given in the parenthesis are quoted -- for the illustrative purpose -- from the recent compilation by Zhang and Li~\\cite {BAL}.",
"The same analysis gives the comparison at $ n=2n0$: $ Esym(2n0)= 56.4 (50.555.99)$ MeV.",
"This will be relevant for our argument given below.\\item (B) For $ n > n1/22 n0$, there are effectively two additional scaling parameters, one for the coupling constant $ gA$ and the other for the $$-meson gauge coupling which differs from the $$ gauge coupling that flows to the vector manifestation fixed point $ g=0$.",
"Both are intricately correlated with the emergent scale symmetry~\\cite {MR-review}.",
"This does not affect what follows below, so we won^{\\prime }t go into details here.$ The predicted star properties areThere is a possible caveat in what is quoted as our predictions for the relation between the radii $R$ and masses $M$ , particularly for GW data.",
"It is argued [29] that to make a quantitatively reliable analysis on $R$ vs. $M$ , the EoSs of the core and crust should be treated thermodynamically consistently.",
"This consistency has not been imposed in [13] from which we are quoting the predicted values where the crust-core transition was taken at $n_{\\rm core-crust}\\approx 0.5 n_0$ .",
"This caveat might be relevant to the quantities $\\Lambda _{1.4}$ and $R_{1.4}$ but most likely less for other macrophysical quantities of massive stars.",
": Maximum star mass $2.05 \\mathrel {\\unknown.",
"{\\hspace{1.0pt}\\sim }}$ >$$ Mmax/M$\\sim $ $>$ 2.23$ for $ 2.0$\\sim $ $<$ n1/2/n0 < 4.0$, radius $ R2.05M12.0$ km, $ 1.4$\\sim $ $<$ 650$, $ R1.4 12.8 $ km.$ Figure: E sym E_{\\rm sym} (in MeV) (left panel) and v pcs 2 /c 2 v_{pcs}^2/c^2 (right panel) for neutron matter (α=1\\alpha =1) calculated in GnGnEFT for n 1/2 ≈2.5n 0 n_{1/2}\\approx 2.5 n_0.",
"The bump samples the density range between n ' n^\\prime and n 1/2 n_{1/2} in Fig.",
"(right panel).",
"Note that v pcs v_{pcs} follows directly and entirely from E sym E_{\\rm sym} for n∼n\\mathrel {\\unknown.",
"{\\hspace{1.0pt}\\sim }}>n1/2asexplainedinthetext.",
"as explained in the text."
],
[
"Heavy degrees of freedom as dual to gluons and quarks: Hadron-quark continuity",
"The issue of possible resolution to the PREX-II dilemma in our approach which is closely linked to also other issues currently in discussion in the literature is encapsulated in $E_{\\rm sym}$ in Fig.",
"REF (left panel).",
"It is given by the generalized $Gn$ EFT that involves only one Lagrangian with the HDsF suitably incorporated together with the topology change.",
"It contains no phase transitions in the sense of Ginzburg-Landau-Wilsonian paradigm, but it is taken to simulate hadron-quark/gluon continuity.",
"Here we are quoting the result obtained for the crossover density $n_{1/2} \\approx 2.5 n_0$ .",
"For the semi-quantitative aspect we are addressing here, the conclusion we arrive at is essentially the same for the range $2\\mathrel {\\unknown.",
"{\\hspace{1.0pt}\\sim }}$ <$$ n1/2/n0 <4$.\\subsection {Heavy Degrees of Freedom and Correlated Fermions}\\begin{figure*}\\centering \\includegraphics [width=0.4]{2.5-solar.pdf}\\includegraphics [width=0.52]{strong-bump.pdf}\\caption {Plethora of bumps, spikes, skates and what not in the sound speed c_s\\equiv v_s: (left panel) Wilderness for massive star with M > 2.5 M_\\odot with and without phase transitions assuming it is a stable neutron star instead of a black hole~\\cite {bump-orgy}; (right panel) strongly interacting baryonic matter in the core~\\cite {bump-impact}.",
"}\\end{figure*}In addressing the issues involved, there are two important points to note:$ First, the heavy-degrees of freedom smoothen (or do away with) the cusp “singularity\" with the vector mesons playing the (dual) role of the gluons and induce the crossover from soft-to-hard in the EoS at the transition region.",
"As mentioned, the maximum that can be reached in $Gn$ EFT is $M_{\\rm max}\\approx 2.23 M_\\odot $ .",
"At this crossover density, however, the maximum of the bump/spike in the sound speed exceeds the causality bound at $n\\ge 4n_0$ , so may not be physically acceptable although no other star properties seem to go haywire.",
"This implies that our approach will get into tension with $\\mathrel {\\unknown.",
"{\\hspace{1.0pt}\\sim }}$ >$$ 2.5 M$ stars should they be confirmed to be stable neutron stars.$ Second, totally distinctive from all currently available ones in the literature, the present EoS unambiguously predicts [2] what we call “pseudo-conformal sound speed\" $v_{pcs}^2/c^2\\approx 1/3$ for density $n\\mathrel {\\unknown.",
"{\\hspace{1.0pt}\\sim }}$ >$$ n1/2$ depicted in Fig.~\\ref {EsymGnEFT} (right lane).",
"It is the solid line in Fig.~\\ref {EsymGnEFT} (left lane) that connects the numerically obtained $ Vlowk$RG ``data\" that precisely gives the sound speed $ vpcs2/c2$ that converges to 1/3 at $ n3n0$\\sim $ $>$ n1/2$ and stays at 1/3 beyond the central density $ 5n0$ of the star.$ It should be noted that $v_{pcs}^2/c^2=1/3$ here does not represent the conformal sound speed associated with the vanishing trace of the energy-momentum tensor (TEMT).",
"It is not the “conformal sound-speed bound\" that is referred to in the literature in addressing the role of “deconfined quarks\" in the core of dense neutron stars.",
"In the system we are dealing with here, the TEMT cannot go to zero in the density regime involved since it is still far from the (putative) IR fixed point [25].",
"We underline here that $v_{pcs}^2/c^2=1/3$ reflects pseudo-conformality emergent from strong correlations involving the degrees of freedom including the HDsF that give rise to nearly non-interacting quasi-fermions [2], [2].",
"It embodies hidden scale symmetry.",
"It is far from the state of nearly free “deconfined\" quarks discussed in the literature where first-order phase transitions are invoked.",
"It depicts a strongly correlated matter in a way resembling what takes place in certain condensed matter physics."
],
[
"Bumps of Sound Speed",
"There are a great deal of discussions currently in the literature on the impact on the EoS of dense baryonic matter in terms of the structure of the sound speed in the crossover region in density $\\mathrel {\\unknown.",
"{\\hspace{1.0pt}\\sim }}$ >$$ 2 n0$.",
"From the point of view of nuclear physics, the problem here as mentioned above is that from the crossover region indicated between $ n$ and $$\\sim $ $>$ n1/2$ in Fig.~\\ref {Esym} for hadrons-to-quarks/gluons to the core density of massive stars is the density regime which is the hardest to access theoretically, both bottom-up and top-down in density.",
"Chiral effective field theory S$$EFT works reliably for nuclear matter properties, but it is very likely to break down at this crossover region.",
"Top-down, the perturbative QCD must also break down at the star-matter density and will certainly be inapplicable at the crossover region.",
"Thus it is not absurd to come up with various wild effects taking place in the region involved.$ In fact, much discussed are the plethora of bumps, spikes, kinks etc.",
"of various sizes ranging from the crossover to the core density region of star with or without phase transitions.",
"Two illustrative cases are given in Fig. .",
"The left panel shows the possibilities of massive stars $M > 2.5 M_\\odot $ with bumps of the sound speed all the way from zero to violating the causality bound, typically involving phase changes [30].",
"Our approach, as mentioned above, cannot access this mass star within the framework we are working with.",
"It will require a major revamping to accommodate such massive stars.",
"On the contrary, the right panel illustrates the case without phase changes (or with smooth crossover) that displays the sound speed largely violating the conformal bound $v_s^2/c^2=1/3$ starting from the crossover region [31].",
"Since the issue of the PREX-II is related to what's treated in [31], this case is highly relevant.",
"The analysis of [31] relies on what is called “non-parametric model based on Gaussian processes, un-tied to specific nuclear models, not subject to systematic errors and possesses wide-range intra-density correlations and targets wide-range of densities.\"",
"While it is not clear to us what this model means with respect to our approach, there is a striking difference between the two.",
"It is in the structure of the constituents in the core of massive stars.",
"For comparison with our prediction, we make a list of some of the results reached by the analysis [31] on the most massive star known so far, i.e., J0740+6620 (NICER+XMM-Newton): $M_{max}=2.24^{+0.34}_{-1.06} M_\\odot $ , $R_{1.4} =12.54^{+1.01}_{-1.06}$ km, $ \\Delta R=R_{2.0}-R_{1.4}=-0.04^{+0.81}_{-0.83}$ km and $n_{\\rm cent}=3.0^{+1.6}_{-1.6} n_0$ .",
"Based on these and other considerations, the authors of [31] arrive at the conclusion that the conformal sound speed bound is strongly violated as depicted in Fig.",
"(right panel) reaching the maximum $v_s^2/c^2=0.79 ^{+0.21}_{-0.20}.$ This strong deviation from the conformal sound speed is attributed by the authors to “strongly interacting hadronic degrees of freedom\" that the authors interpret as “disfavoring\" the appearance of “explicit\" QCD degrees of freedom in the core of stars.",
"This property is consistent with the low central density $\\sim 3 n_0$ found in the analysis.",
"The PREX-II dilemma (REF ) would belong to this class of scenario.",
"It should be admitted that given the total paucity of theoretical tools applicable in that density regime, perhaps one cannot rule out this possibility.",
"However what $Gn$ EFT has predicted is strongly at odds with the conclusion of [31].",
"As summarized in [2], our pseudo-conformal structure yields the following: $M_{max}\\approx 2.24 M_\\odot $ , $R_{1.4} \\approx 12.8$ km, $\\Delta R=R_{2.0}-R_{1.4}\\approx - 0.08$ km and $n_{\\rm cent}\\approx 5.1 n_0$ .",
"Thus apart from the central density which signals the structure of the constituents of the core, the overall macrophysical properties predicted are essentially the same as those of [31].",
"But the sound speed of the star (see Fig.",
"REF (right panel) and Fig.",
"(right panel)) is drastically different.",
"The closeness, but not equal, of the pseudo-conformal speed to the conformal sound speed bound is consistent with the higher central density predicted, signaling fractionalized quasi-fermions different from baryonic matter."
],
[
"Topology Encodes Microscopic Dynamics",
"The principal advantage of our approach on the contraryWe believe that what we are discussing here would be little, if any, affected by the caveat associated with the crust.",
"is that it relies on (potentially robust) topological structure which provides a coarse-grained macroscopic description of what is presumably taking place in the density regime more or less uncontrolled by theoryThe crucial role of topology played here has an analogy in condensed matter physics.",
"For instance in the fractional quantum Hall effects Chern-Simons topological field theory captures the microscopic structure of, say, Kohn-Sham density functional theory as pointed out (with condensed-matter references) in [32]..",
"In our approach the sound speed does produce a simple bump structure of Fig.",
"REF (right lane) caused by the intervention of the HDsF dual to QCD degrees of freedom [2] with its characteristics capturing the crossover density $n_{1/2}$ .",
"For the case of $n_{1/2}\\simeq 2.0 (3.0) n_0$ , it is a bump reaching $v_s^2/c^2\\sim 0.7 (0.8)$ .",
"But for $n_{1/2}\\sim 4n_0$ , as mentioned, it is a spike with the maximum of which going out of the causality bound.",
"Yet despite the different bump heights in the range $2\\mathrel {\\unknown.",
"{\\hspace{1.0pt}\\sim }}$ <$$ n1/2/n0 $\\sim $ $<$ 4$ (even violating causality bound for the case $ n1/2=4 n0$), the sound speed $ vpcs2/c2$ converges in all cases to 1/3 slightly above $ n1/2$ with the global star properties not noticeably different between them.",
"The positive aspect of our prediction~\\cite {MR-review} is that it is an extremely economical description -- coarse-graining the microscopic details of what^{\\prime }s found in \\cite {bump-orgy,bump-impact} -- that could be capturing the underlying physics.",
"We are arguing that it is precisely the {\\it correlated strong interactions} leading to the Landau Fermi-liquid quasiparticle structure which becomes more valid with increasing density after the topology change as $ N=kF/(F-kF)$~\\cite {shankar}, {\\it manifesting pseudo-conformal symmetry} in the sound speed.$ The question that can be raised is how can the physics of the complexity in the sound speed favored by [31] be reproduced by the extremely simple structure driven by the emerging hidden scale symmetry that leads to Fig.",
"REF (right lane)?",
"The possible answer to this could be that the macroscopic properties of massive neutron stars are in some way insensitive to the microscopic details of the bump structure of the sound speed with the emergent symmetries manifesting in the sound speed related to what's operative in the “quenching of $g_A$ \" in baryonic matter mentioned below.",
"If so, the question is: Are there any physically meaningful observable probes for them?",
"To the best we are aware, there seems to be none at present.",
"Needless to say, as coarse-grained, there can be fluctuations on top of 1/3 coming from corrections to the underlying scale symmetry.",
"That the sound speed converges precociously to $v_{pcs}^2/c^2\\simeq 1/3$ could be an oversimplification.",
"First of all the density involved $< 10 n_0$ is far from the density at which the vector manifestation limit and/or the dilaton limit fixed point is approached [2], so the scale symmetry should be broken (as indicated by the effective dilaton mass which must be substantial counterbalancing the $\\omega $ mass).",
"However there is an indication that scale symmetry can be “emergent,\" even if not intrinsic, in certain highly correlated nuclear dynamics.",
"One prominent evidence for it was seen in the so-called “quenched $g_A$ \" phenomenon in nuclear beta decay [33].",
"The effective $g_A$ in nuclear Gamow-Teller transition in light nuclei $g_A^\\ast \\approx 1$ arises due to strong nuclear correlations influenced by hidden scale symmetry reflected in low-energy theorems.",
"The approach $g_V\\rightarrow g_A=1$ at high density $\\mathrel {\\unknown.",
"{\\hspace{1.0pt}\\sim }}$ >$$ 25n0$ as the dilaton limit fixed point is approached is closely correlated to how the quenched $ gA$ results in finite nuclei~\\cite {multifarious}.",
"Furthermore that the simple sound-speed structure directly governed by the $ Esym(n)$ setting in slightly above the crossover density with none of the compact-star properties significantly disagreeing with observations is another indication that the hidden scale symmetry is manifested in the density regime of compact stars.\\subsection {The PREX-II ``Dilemma\" and Hadron-Quark Duality}We are now equipped with what^{\\prime }s needed to address the PREX-II dilemma and the issue of whether in EFT the EoS at low density near $ n0$ must necessarily constrain what happens at higher densities relevant to massive stars.$ One can read off from the HDsF-driven $E_{sym}$ ( Fig.",
"REF ) that $J\\approx 30.2$ MeV and $L\\approx 67.8$ MeV.",
"$J$ is “soft\" consistent with (REF ) but the slope $L$ comes higher than the central value of (REF ) by $\\sim 10$ MeV.",
"Reliable S$\\chi $ EFT calculations to N$^3$ LO converge to the central value of $\\sim (52-56)$ MeV [35] which is consistent with (REF ).",
"What does this difference of $\\sim 10$ MeV mean?",
"This can be understood as that the “soft\" EoS at $n\\mathrel {\\unknown.",
"{\\hspace{1.0pt}\\sim }}$ <$$ n0$ starts to stiffen as the density approaches $ n1/2$\\sim $ $>$ 2n0$ reflecting the cusp smoothed by the HDsF.",
"This leads to $ Esym(2n0)56.4$ MeV consistent with what is indicated in nature~\\cite {BAL}.",
"This reflects that the S$$EFT defined with the cutoff $$\\sim $ $<$ mV$ starts breaking down at $ 2n0$ precisely due to the {\\it necessity} of the HDsF signaling the emergence -- vial Seiberg-type duality -- of QCD degrees of freedom.",
"This crossover not only accounts for the massive star masses but also provides a simple mechanism to bring -- with additional help with the crust consistently treated thermodynamically -- $ 1.4650$ (which is still consistent with the presently accepted with the upper bound) to a lower value in the vicinity of $ 400$ which may be favored should the tidal deformability bound be further tightened in the future measurements.",
"Again there is a logically simple reason for this.",
"The heavy-meson-induced smoothing tends to locate the central density of the $ 1.4 M$ star for $ 1.4$ in the density regime $ < n1/2$, i.e., the lower side of the cusp -- which is soft -- that could be in principle accurately calculated by S$$EFTs (and $ Gn$EFT) by fine-tuning the crossover density $ n1/2$ within the range involved.$ We are then led to suggest that the “strong $R_{\\rm skin}$ -$L$ in the PREX-II measurement could not constrain the EoS of the core of compact stars.",
"There is, in fact, nothing special about arriving at this sort of conclusion in effective field theories for QCD, the presumed fundamental theory of strong interaction physics.",
"An apt example, perhaps not widely recognized in nuclear physics community, is the applicability of the skyrmion approach – as an EFT – to nuclear physics.",
"Given that the skyrmion theory should be a good low-energy effective field theory of QCD at large $N_c$ limit, it should in principle describe various different low-energy properties of nuclear physics valid at large $N_c$ .",
"Indeed in some cases, it works extremely well.",
"For instance, the BPS skyrmion is seen to give an excellent description of nuclear binding energies [36] and radii [37] for a wide range of nuclei from light to heavy nuclei $A > 200$ .",
"But the same BPS Lagrangian by itself does not satisfy the soft-pion theorems, the hall-mark of current algebra and chiral symmetry.",
"This seems at odds with the general belief that nuclear phenomena are governed by chiral symmetry which in the skyrmion theory is encoded in the current algebra term in the Lagrangian.",
"But it does not necessarily mean that soft theorems naively interpreted must give the constraint in the domain where the BPS structure is more appropriate.",
"It is now understood that the infinite tower of vector mesons, say, HDsF generalized from what we have been discussing, subsumed in the BPS Lagrangian is at work for the particular nuclear properties concerned [38].",
"Furthermore one can write down [39] a skyrmion model as a sum of BPS submodels, each of which has its own characteristics applicable to different regions of scales and dynamics.",
"How to go from one to others is of course an open issue that is to be worked out.",
"It is clear however that it is not necessarily constrained by nLORE.",
"The recent discovery of Seiberg-type dualities for sHLS [17], [18], [19], [20] indicates that there may intervene more than the skyrmion-half-skyrmion topology change we have been exploiting in the phase structure of dense hadronic matter in going to high densities in the core of massive stars, a notable current example being the phase where the $\\eta ^\\prime $ ring singularity is exposed, say, in the vicinity of the putative IR fixed point density [40].",
"Such a multiple phase structure involving “hadrons\" in place of quarks and gluons could persist all the way to the density where the hadron-quark continuity breaks down [41]."
],
[
"Strangeness Plays No Role",
"In what's treated in this paper and elsewhere, the strangeness degrees of freedom played no role.",
"This seems at odds with the genuine dilaton scheme [25] where kaons figure on the same footing as the dilaton.",
"This point is discussed in [2].",
"It turns out however to be feasible to argue that the so-called “hyperon problem\" is absent in the density regime relevant to the core of massive neutron stars.",
"The argument was based on the RG approach to interacting protons and neutrons coupled to the HDsF and the kaons on the Fermi surface [42].",
"Invoking the same large $N_c\\rightarrow \\infty $ and large $\\bar{N}=\\frac{k_F}{\\Lambda - k_F}\\rightarrow \\infty $ limit that underlies the results obtained in this paper (and more generally [2]), it was shown [43] that (1) kaons condense and hyperons appear at the same density and (2) the kaon condensation threshold density $n_K$ satisfies the bound $n_K > \\bar{N} n_0.$ This implies that $n_K$ should be considerably higher than the core density of the stars $\\sim (5-6)n_0$ .",
"We note that the bound (REF ) with $n_K >7 n_0$ was arrived at in a different but related consideration – short-range correlations – a long time ago [44].",
"A rigorous justification for this could be given in the $V_{lowk}$ RG approach to $Gn$ EFT $\\in SU(3)_f$ , which unfortunately is not feasible at the moment."
],
[
"Concluding remarks",
"Starting with the cusp structure found in a skyrmion-crystal simulation, with the incorporation of heavy degrees of freedom considered to be dual to the gluons/quarks in the EoS for dense matter, we have arrived at the symmetry energy $E_{\\rm sym}(n)$ that contraries the nLORE (standard nuclear astrophysics lore), hence the PREX-II “dilemma,\" and gives rise to the pseudo-conformal sound speed for $n > n_{1/2}$ .",
"This result if confirmed could bring about a potential paradigm change in nuclear physics.",
"There are several remarks to make to support this proposal.",
"The first is that the topological cusp structure renders moot the necessity to constrain the high-density EoS for massive compact stars by that fixed at normal nuclear matter density.",
"This of course does not mean that a potentially unified theory with no patching cannot be connected from $n$ below to above $n_{1/2}$ .",
"In fact in our approach with a single Lagrangian they are connected via topological change in a single Lagrangian.",
"The corollary to the first remark, perhaps equally unorthodox, is that the dense star core populated by the fractionalized fermionic constituents [7] as precisely predicted by $E_{\\rm sym}$ in the present formulation [2] renders the pseudo-conformal sound speed $v_{pcs}^2/c^2\\approx 1/3$ consistent with the picture of “deconfined\" quarks with the polytropic index $\\gamma $ approaching 1 [45].",
"Note however our system is not at the IR fixed point, hence the constituents of the core are not deconfined.",
"This revamps the common notion of “deconfined quarks\" as the only genuine signal for QCD degrees of freedom.",
"The structure of dense matter that emerges from our work is that what results from highly nonperturbative correlation of QCD degrees of freedom, quarks and gluons, at the crossover density regime has a dual description via topology change in terms of hidden local gauge fields and hidden genuine dilaton scalar field in chiral symmetric environment.",
"We suggest that such a dual description stays viable and applicable up to the density at which the notion of hadron-quark continuity [41] as well as generalized Fermi-liquid structure of weakly interacting fractionalized quasi-fermions, neither baryonic nor quarkonic, break down, possibly taking place only at asymptotic density way outside of the range of most massive compact stars stable against gravitational collapse."
],
[
"Acknowledgments",
"The work of YLM was supported in part by the National Science Foundation of China (NSFC) under Grant No.",
"11875147 and 11475071."
]
] | 2107.01879 |
[
[
"Two tractable models of non-stationary light scattering by subwavelength\n particles and their application to Fano resonances"
],
[
"Abstract We introduce two tractable analytical models to describe dynamic effects at resonant light scattering by subwavelength particles.",
"One of them is based on generalization of the temporal coupled-mode theory, and the other employs the normal mode approach.",
"We show that sharp variations in the envelope of the incident pulse may initiate unusual, counterintuitive dynamics of the scattering associated with interference of modes with fast and slow relaxation.",
"To exhibit the power of the models, we apply them to explain the dynamic light scattering of a square-envelope pulse by an infinite circular cylinder made of $GaP$, when the pulse carrier frequency lies in the vicinity of the destructive interference at the Fano resonances.",
"We observe and explain intensive sharp spikes in scattering cross section just behind the leading and trailing edges of the incident pulse.",
"The latter occurs when the incident pulse is over and is explained by the electromagnetic energy released in the particle at the previous scattering stages.",
"The accuracy of the models is checked against their comparison with results of the direct numerical integration of the complete set of Maxwell's equations and occurs very high.",
"The models' advantages and disadvantages are revealed, and the ways to apply them to other types of dynamic resonant scattering are discussed."
],
[
"Introduction",
"High-$Q$ resonances are of utmost importance in a wide diversity of problems.",
"It is explained by the fact that the corresponding resonance should have a high $Q$ -factor to obtain strong resonant effects.",
"However, the characteristic relaxation time for a resonance is inversely proportional to its $Q$ -factor.",
"Thus, the price one must pay for making use of high-$Q$ resonances is long-lasting transient effects.",
"On the other hand, the frontier of modern photonics moves toward short and ultrashort pulses.",
"Nowadays, these two factors together make typical the situation, when the duration of a laser pulse becomes comparable or even shorter than the relaxation times of the resonant effects initiated by this pulse.",
"It means that the theoretical description of these resonant phenomena based on steady-state approximations becomes erroneous and should be revised thoroughly.",
"It is well-known that non-steady resonant scattering may qualitatively differ from its steady-state realizations and exhibits new effects, which do not exist in the steady-state case, see, e.g., Ref.",
"[1].",
"However, the corresponding study in subwavelength optics has begun only recently [2-4].",
"This paper presents a general approach to the theoretical description of non-steady resonant light scattering by subwavelength particles based on the extension of the Mie solution to this case.",
"We apply it to inspect the light scattering by a cylinder.",
"We show that non-steady resonant scattering of light may exhibit counterintuitive new effects, and the scattering field patterns, in this case, may have very little in common with the one for the steady-state scattering.",
"Besides the purely academic interest, the results obtained may be employed to design a new generation of fast, multifunctional nanodevices.",
"The ab initio description of the non-steady resonant light scattering is possible only by numerical integration of the complete set of Maxwell's equations.",
"Nowadays, due to the existence of plenty of pieces of software, both free and commercial, created to perform this integration, the integration has become a more or less routine procedure.",
"However, to find in the range of the problem parameters a windows, where a desired effect is the most pronounced, the dependence of the scattering on these parameters in a wide domain of their variations is required.",
"To this end, numerical methods are not appropriate, and analytically tractable models are required.",
"To the best of our knowledge, for the time being, none of them exists.",
"Here, we present two such models, apply them to describe non-steady Fano resonances, and compare the results with direct numerical integration of the complete set of Maxwell's equations discussed in Ref. [1].",
"The comparison indicates the high accuracy of both models and reveals their mutual advantages and disadvantages.",
"The models unveil the physical grounds for the counterintuitive spikes in the scattered radiation observed behind the leading and trailing edges of the incident pulse.",
"The first model is based on the temporal coupled-mode theory (TCMT) [2] generalized to applications to essentially non-steady scattering.",
"The second model mimics non-steady resonant vibrations by a superposition of dynamics of driven harmonic oscillators (HO).",
"The latter approach looks similar to the harmonic inversion, see, e.g., Ref.",
"[3], [4], [5].",
"However, in our case, the mode selection for the approximation and, most importantly, the choice of the values of the model parameters are based on the system in question's physical properties.",
"Thus, it reveals the role of different excitations in the system dynamic and sheds light on the physical nature of the system as a whole.",
"Besides, this makes it possible to reduce the number of modes to be studied just to a few with non-trivial dynamics.",
"The paper has the following structure: In Sec.",
"2 the problem formulation is presented.",
"Sec.",
"3 is devoted to the problem analysis and discussion of the obtained results.",
"In Sec.",
"4 we formulate conclusions.",
"Cumbersome details and expressions as well as several plots illustrating the developed approach are moved to Appendix.",
"The simplest exactly solvable light scattering problems correspond to the steady-state scattering of a linearly polarized plane wave by a homogeneous sphere (the Mie solution) or infinite circular cylinder.",
"In these solutions, the scattered field is presented as an infinite series of partial waves (dipole, quadrupole, etc.",
"), also called multipoles [6].",
"Here, we generalize this approach to dynamic light scattering.",
"It makes it possible to study quite intricate transient features.",
"One of the most typical resonant responses in light scattering by finite obstacles is the Fano resonance.",
"A characterizing it asymmetric lineshape is explained by either constructive (the maximal scattering) or destructive (the minimal scattering) interference occurring close to each other in the frequency domain.",
"The interfering parties are the so-called resonant (narrow line) and background (broad line) partitions of the same multipole.",
"The narrow-line and broad-line partitions correspond to excitations with slow and fast relaxation time in the time-domain, respectively.",
"Naturally, the procedure of splitting a single partial wave into the two partitions is not unique.",
"Accordingly, there are two main equivalent approaches to it.",
"In the first approach, a partial wave is presented as a sum of the radiation of the conventional electric and toroidal multipoles [7].",
"However, in what follows, it will be more convenient for us to employ another approach.",
"The resonant partition is associated with the corresponding electromagnetic mode excited in the bulk of the particle (volume polariton).",
"In contrast, the background partition corresponds to the radiation of the surface current induced by the same incident wave scattered by the particle with the same geometrical shape but made of a hypothetical material called the perfect electric conductor (PEC) [8].",
"At steady-state scattering, at a point of the destructive interference, the two partitions cancel each other.",
"As a result, the contribution of the corresponding multipole is suppressed.",
"For a subwavelength particle, when just a few first multipoles produce the overwhelming contribution to the overall scattering, the suppression even of one of them may reduce the scattering cross section dramatically [8].",
"However, since, as has been mentioned above, the resonant and background partitions are characterized by substantially different relaxation times, it is evident that during transient regimes, the mutual cancelation does not occur.",
"Therefore, the violation of the destructive interference conditions must give rise to a considerable increase in the scattering intensity during the transient and other unusual effects.",
"Some of them have already been discussed in our previous publications [9], [10].",
"We stress that though this increase in the scattering intensity looks similar to the well-known overshot effect when a driven high-$Q$ oscillator exhibits oscillatory relaxation to the steady-state, the physical grounds for the former is entirely different: If the overshot is related to vibrations of a single oscillator, the discussed effect is explained by a superposition of two different oscillations.",
"This distinction in the physical nature of the two cases gives rise to the corresponding difference in their manifestations.",
"In particular, the amplitude of the scattering spike relative to that of the steady-state may be in orders of magnitude larger than that exhibited by the overshot.",
"Thus, the dynamic resonant light scattering is a new and, practically, untouched subfield.",
"Plenty of exciting effects hidden there are still undiscovered.",
"In the present paper, we continue to explore this appealing topic.",
"To understand typical main features of the phenomenon, we consider the simplest problem formulation, namely the scattering of a rectangular incident pulse with duration $\\tau $ , carrier frequency $\\omega $ and temporal dependence of the fields $\\sim \\exp (-i\\omega t)$ by an infinite circular cylinder with the base radius $R$ and complex refractive index $m=n+i\\kappa $ .",
"The amplitudes of the electromagnetic fields $A(t)=A_0= const \\ne 0$ inside the pulse and zero outside it.",
"The refractive index of the surrounding cylinder medium equals unity.",
"The cylinder is nonmagnetic, so its permeability $\\mu = 1$ .",
"For further simplification, just the TE polarization and normal incidence are considered, see Fig.",
"REF (a)."
],
[
"Instantaneous scattering cross section",
"In the conventional steady-state case, the scattering is quantitatively described by the cross section $C_{\\rm sca}$ calculated per unit of length of the cylinder.",
"$C_{\\rm sca}$ is defined as the ratio of the integral power flux through a closed remote surface surrounding the scatterer to the intensity of the incident light.",
"At non-steady scattering, both these quantities are time-dependent, and their ratio is not a constant anymore.",
"Moreover, the ratio depends on the shape and position of the surface used to calculate the flux since the speed of light is finite.",
"To describe non-steady scattering, we introduce the instantaneous scattering cross section $C_{\\rm sca}(t)$ as the ratio of the instantaneous value of the power flux through a cylindrical surface, coaxial with the scattering cylinder and lying in the far wave zone, calculated per a unit of length of the cylinder, to the constant intensity of the rectangular incident pulse $I_0$ [1].",
"In a more general case, when the pulse envelope has a time-dependent shape $I(t)$ , the scattered flux may be normalized over, e.g., the maximal value of $I(t)$ , i.e., $I_0 = \\mathop {\\rm Max}\\limits _t\\lbrace I(t)\\rbrace $ .",
"The corresponding dimensionless scattering efficiency, $Q_{\\rm sca}(t)$ is connected with $C_{\\rm sca}(t)$ by the usual relation $Q_{\\rm sca}(t) = C_{\\rm sca}(t)/(2R)$ , where $R$ is the radius of the base of the cylinder.",
"We also do not perform the time averaging of the Poynting vector over the period of oscillations.",
"The exact solution describing the steady-state scattering is built as an infinite series of partial waves (multipoles).",
"For the problem in question, the complex amplitudes of the multipoles (scattering coefficients) associated with the outgoing partial waves $a_\\ell $ and the field within the cylinder $d_\\ell $ are given by the well-known expressions presented in Appendix Eq.",
"(), see also, e.g., [6].",
"Here $\\ell $ designates the multipole order (dipole, quadrupole, etc.).",
"For the given problem, the scattering coefficients satisfy the identity [11]: $a_\\ell \\equiv a_\\ell ^{(\\rm PEC)} - \\frac{J^\\prime _\\ell (mx)}{H^{(1)\\prime }_\\ell (x)}d_\\ell ;\\; a_\\ell ^{(\\rm PEC)} \\equiv \\frac{J^\\prime _\\ell (x)}{H^{(1)\\prime }_\\ell (x)},$ where $x = kR$ stands for the size parameter; $k=\\omega /c$ ; $c$ is the speed of light in a vacuum; $J_\\ell (z)$ and $H^{(1)}_\\ell (z)$ designate the Bessel and Hankel functions, respectively; prime denotes the derivative over the entire argument of a function; and $a_\\ell ^{(\\rm PEC)}$ is the scattering coefficient of the same cylinder made of the perfect electric conductor.",
"Routine calculations result in the following expressions [1]: $\\!\\!\\!\\!\\!\\!\\!\\!& & Q_{\\rm sca}=Q_{\\rm sca}^{(0)} + Q_{\\rm sca}^{(\\rm osc)} = \\sum _{\\ell =-\\infty }^{\\infty }\\left\\lbrace Q^{(0)}_{{\\rm sca}\\,(\\ell )}+ Q_{{\\rm sca}\\,(\\ell )}^{(\\rm osc)}\\right\\rbrace , \\\\\\!\\!\\!\\!\\!\\!\\!\\!& & Q_{{\\rm sca}\\,(\\ell )}^{(0)}\\!",
"=\\!",
"\\frac{2}{x}|a_\\ell |^2;\\; Q_{{\\rm sca}\\,(\\ell )}^{(\\rm osc)}\\!",
"= \\!-\\frac{i}{x}\\!\\left[a_\\ell ^2 e^{2i(kr-\\omega t)}\\!-\\!c.c.\\right]\\!\\!, $ Here $Q_{\\rm sca}^{(0)}$ is the conventional scattering efficiency, while $Q_{\\rm sca}^{(\\rm osc)}$ is an additional rapidly oscillating in time and space term with zero average."
],
[
"Fano resonances",
"The Fano resonances [12], [13], [14] are a good example demonstrating unusual, counterintuitive effects in transient processes of resonant light scattering.",
"For the steady-state scattering, their detailed discussion is presented, e.g., in Ref. [11].",
"Though in that paper, a spherical particle is considered, generalization to the case of a cylinder is a straightforward matter.",
"As it has been mentioned above, a key point of the Fano resonances is a presentation of the scattered wave as a sum of two partitions: resonant and background.",
"In the proximity of the resonance the amplitude and phase of the former have a sharp $\\omega $ -dependence, while for the latter its $\\omega $ -dependence is weak.",
"An important conclusion following from the results of Ref.",
"[11] is that the splitting of $a_\\ell $ into the two terms, given by Eq.",
"(REF ), actually, is the singling out the background ($a_\\ell ^{(\\rm PEC)}$ ) and resonant ($- \\frac{J^\\prime _\\ell (mx)}{H^{(1)\\prime }_\\ell (x)}d_\\ell $ ) partitions.",
"Our goal is to recover the full time-dependence $Q_{\\rm sca}(t)$ .",
"For high-$Q$ resonances, which we are interested in, the characteristic time scales of the transients should be large relative to the period of the field oscillations $2\\pi /\\omega $ .",
"Then, a quasi-steady approximation may be employed.",
"It implies the same structure of the solution as that for steady-state scattering.",
"However, now the scattering coefficients are regarded as slowly-varying functions of time.",
"Let us apply this assumption to the TCMT and HO."
],
[
"TCMT",
"In the specified case, the TCMT equations read as follows, see, e.g., Ref.",
"[15]: $& & \\frac{\\mathrm {d} p(t)}{\\mathrm {d} t} =-\\left({i} \\omega _{0}-\\gamma \\right) p(t)+\\kappa s^{+}(t) \\\\& & s^{-}(t) =B s^{+}(t)+\\zeta p(t) $ Here $s^+(t)$ and $s^-(t)$ are the amplitudes of the incoming (converging) and outgoing (diverging) cylindrical waves, respectively; $B$ is the background reflection coefficient; $\\kappa $ and $\\zeta $ are coupling constants; $p(t)$ describes the internal resonant mode excitation; and $\\hat{\\omega }_{0}\\equiv \\omega _{0}+i\\gamma $ is the nearest pole of the scattering coefficients in the plane of complex $\\hat{\\omega }$ .",
"It is important to stress that for the selected temporal dependence $\\sim \\exp (-i\\omega t)$ decaying modes must have $\\gamma <0$ .",
"Then, the corresponding poles are situated in the lower semiplane.",
"The analysis of Eqs.",
"(REF ), () performed in Ref.",
"[15] for the steady-state scattering indicates that $\\kappa =\\zeta =\\sqrt{2|\\gamma |}\\exp (i\\theta )$; $B=\\exp (i\\phi )$ and $\\theta =(\\phi + \\pi )/2 + n\\pi $ , where $n$ is an arbitrary integer.",
"In this case the steady-state scattering coefficient $a_\\ell ^{\\rm (TCMT)}$ is defined as $[s^{-}(t)-s^{+}(t)]/({2s^{+}(t)})$ .",
"Eventually, it gives rise to a certain expression for $a_{\\ell }^{\\rm (TCMT)}$ , where phase $\\phi $ remains undefined yet.",
"The authors of Ref.",
"[15] fix it by fitting the profile $|a_{\\ell }^{\\rm (TCMT)}(\\omega )|^2$ to $|a_{\\ell }(\\omega )|^2$ obtained from the steady-state exact solution.",
"However, any fitting procedure is ambiguous since its results depend on the fitting window's size.",
"Meanwhile, there are other ways to fix $\\phi $ , free from this disadvantage.",
"In this paper we fix $\\phi $ from the condition $|a_{\\ell }^{\\rm (TCMT)}(\\omega )|=|a_{\\ell }(\\omega )|$ at the carrier frequency of the incident pulse.",
"This gives rise to a quadratic equation, whose solution results in two values of $\\phi $ in the non-trivial domain $-\\pi \\le \\phi \\le \\pi $.",
"The final choice is made based on the better overall coincidence of the two profiles.",
"Such a choice is a straightforward matter, see Appendix.",
"The next difficulty is that the employed expression for $a_\\ell ^{\\rm (TCMT)}$ is valid for the steady-state scattering solely.",
"The latter is evident if we consider, e.g., the case when the incident pulse is already over, i.e., $s^+(t)=0$ .",
"At the same time, the particle still radiates the accumulated electromagnetic energy, so that $s^-(t)\\ne 0$ .",
"In this case, the discussed expression diverges.",
"To avoid this difficulty, we have to redefine $a_\\ell ^{\\rm (TCMT)}$ .",
"For the considered rectangular pulse, it may be done as follows: $a_\\ell ^{\\rm (TCMT)}(t)=\\frac{1}{2A_{0(\\ell )}\\exp (-i\\omega t)}[s^{-}(t)-s^{+}(t)]$ , where $A_{0(\\ell )}$ is a constant amplitude of a converging incident cylindrical wave, corresponding to a given multipolarity.",
"This definition coincides with the above one for the steady-state scattering but remains finite at $s^+(t)=0$ .",
"In an arbitrary pulse shape, the role of $A_{0(\\ell )}$ may play the corresponding maximal value.",
"Regarding $Q_{\\rm sca}(t)$ , since in the discussed quasi-steady approximation the solution retains the same structure as that for the steady-sate; i.e., Eqs.",
"(REF ), () still remain valid but the replacement $a_\\ell \\rightarrow a_\\ell ^{\\rm (TCMT)}(t)$ is required.",
"As for the dependence $a_\\ell ^{\\rm (TCMT)}(t)$ , it is readily obtained by integration of Eq.",
"(REF ) with the initial condition $p(0)=0$ : $& & a_\\ell ^{\\rm (TCMT)}(t) = \\frac{i \\gamma \\left[1-e^{i \\phi } \\left(2 e^{it (\\omega -\\omega _0)+\\gamma t}-1\\right)\\right]+\\left(e^{i \\phi }-1\\right) (\\omega -\\omega _0)}{2 (\\omega -\\omega _0 - i \\gamma )}$ at $0\\le t\\le \\tau $ and $a_\\ell ^{\\rm (TCMT)}(t)=\\frac{i \\gamma \\left(e^{-\\gamma \\tau - i(\\omega -\\omega _0)\\tau }-1\\right) e^{ i[(\\omega - \\omega _0)t + \\phi ]+\\gamma t}}{\\omega -\\omega _0 -i\\gamma },$ at $t>\\tau $ (remember that $\\gamma <0$ ).",
"Note, that $a_\\ell ^{\\rm (TCMT)}(t)$ given by Eqs.",
"(REF ), (REF ) are indeed slowly-varying relative to $\\exp (-i\\omega t)$ since in the vicinity of a high-$Q$ resonance $|\\omega -\\omega _0| \\ll \\omega $ and $|\\gamma | \\ll \\omega $.",
"Another point to be stressed is that $a_\\ell ^{\\rm (TCMT)}(t)$ given by Eqs.",
"(REF ), (REF ) is not continuous at $t=0$ and $t=\\tau $ : At $t=0$ it has a jump from zero at $t=-0$ to $(\\exp [i\\phi ]-1)/2$ at $t=+0$ .",
"At $t=\\tau $ the jump has the same value but the opposite sign.",
"These discontinuities are a direct consequence of Eq.",
"(), which implies that the background scattering follows the variations of $s^+(t)$ instantaneously, without any delay."
],
[
"Harmonic oscillators",
"Another model is based on the well-known fact that any linear oscillatory dynamic may be approximated by that of a system of driven coupled harmonic oscillators.",
"We just have to apply it to the problem in question.",
"The general solution of the equations for driven coupled HO has the form (see, e.g.",
"[16]) $z_{k}(t) = z_{ks}(t)+\\sum _{n}{C_n \\Delta _{nk} e^{-i\\hat{\\omega }_n t}} ,$ where $z_{k}(t)$ is a complex coordinate of the $k$ -th oscillator; $\\Delta _{nk} e^{-i\\hat{\\omega }_n t}$ and $\\hat{\\omega }_n$ stand for the corresponding eigenvector and complex eigenfrequency, respectively; $z_{ks}(t)$ is a particular solution, for a given drive, and $C_n$ are the constants of integration defined by the initial conditions.",
"An important point is that if the eigenvectors are selected as new variables (normal modes), the corresponding system of equations is diagonalized, i.e., each term in the sum in Eq.",
"(REF ) evolves independently of the others and its dynamic is described by that of a single oscillator [16].",
"The main idea of the adaptation of Eq.",
"(REF ) for a drive with a carrier frequency $\\omega $ is that only the dynamics of the resonant eigenmodes with the frequency mismatch $|\\omega - {\\rm Re}\\,\\hat{\\omega }_n|$ of the order of $|{\\rm Im}\\,\\hat{\\omega }_n|$ or smaller than that are modeled.",
"All other off-resonant eigenmodes are supposed to follow the drive adiabatically, obeying the quasi-steady approximation.",
"This approach is reasonable, provided the contribution of the resonant modes to the overall dynamic is overwhelming.",
"Fortunately, this is the case in most resonant phenomena in subwavelength optics and related problems.",
"Next, according to what just has been said, for the problem in question the eigenmode dynamic is described by the equation: $\\ddot{f} - 2\\gamma \\dot{f} +\\omega _{0}^2 f = A_0[\\theta (t)-\\theta (t-\\tau )]\\exp [-i\\omega t];\\;\\; (\\gamma <0),$ supplemented with the initial conditions $f(0)=\\dot{f}(0)=0$ , where dot stands for d/d$t$ and $\\theta (z)$ is the Heaviside step function.",
"Eq.",
"(REF ) has the following exact solution: $f(t) &=& A_0 e^{-i\\omega t} \\frac{e^{(i\\omega +\\gamma )t}\\left[\\omega _{0\\gamma }\\cos (\\omega _{0\\gamma }t)-\\left(\\gamma +i\\omega \\right)\\sin (\\omega _{0\\gamma }t)+\\right]-\\omega _{0\\gamma }}{\\left(\\omega ^2-\\omega _0^2-2i\\omega \\gamma \\right)\\omega _{0\\gamma }},$ at $0\\le t \\le \\tau $ and $f(t) &=& \\frac{e^{\\gamma (t-\\tau )}}{\\omega _{0\\gamma }}\\Bigg \\lbrace \\left[\\dot{f}(\\tau )-\\gamma f(\\tau )\\right]\\sin \\omega _{0\\gamma }(t-\\tau )+ \\omega _{0\\gamma }f(\\tau )\\cos \\omega _{0\\gamma }(t-\\tau )\\Bigg \\rbrace , $ at $t>\\tau $ .",
"Here $\\omega _{0\\gamma } \\equiv \\sqrt{\\omega _0^2-\\gamma ^2}$ .",
"In what follows, we employ Eqs.",
"(REF ), (REF ) to model the dynamics of the scattering coefficients.",
"Naturally, for different coefficients the values of $A_0,\\;\\omega _0$ and $\\gamma $ are also different and will be defined below.",
"The key equation now is Eq.",
"(REF ), where the steady-state $a_\\ell ^{\\rm (PEC)}$ and $d_\\ell $ are replaced by $a_\\ell ^{\\rm (PEC)}(t)$ and $d_\\ell (t)$ and regarded as independent eigenmodes.",
"Let us stress the dramatic difference in the steady-state profiles $|d_\\ell (x)|$ and $|a_\\ell ^{\\rm (PEC)}(x)|$ .",
"If for the former the characteristic scale is of the order of $1/(mx) \\ll 1$ , for the latter it is just $1/x = O(1)$ , see Eq.",
"(REF ) and Ref. [11].",
"As a result, though in the vicinity of the maxima, the profiles $|d_\\ell (x)|$ may be well-approximated by Lorentzians, this is not the case for $|a_\\ell ^{\\rm (PEC)}(x)|$ .",
"Therefore, the actual dynamic of each PEC-mode may be quite far from that of a harmonic oscillator.",
"Fortunately, since the characteristic scale in the time-domain is inverse of that in the $\\omega $ -domain, the specified hierarchy of the scales means that the transients of the PEC-modes to the quasi-steady-state scattering is fast.",
"In contrast, the ones for the resonant partitions are relatively slow.",
"Accordingly, the approximation of the latter requires maximal accuracy.",
"Regarding the possible errors in the approximation of the dynamics of the PEC-modes, they are not crucial to the approach since due to the fastness of the PEC-modes, they affect just the very initial stage of the transient (a few periods $2\\pi /\\omega $ , see below).",
"To employ Eq.",
"(REF ) for modeling the dynamics of the eigenmodes, we have to fix the values of the following four parameters: Re$[A_0]$ , Im$[A_0],\\;\\omega _0$ and $\\gamma $ .",
"The approximation details are as follows: For the resonant d-modes the values of $\\omega _0$ and $\\gamma $ are given by the corresponding poles of $d_\\ell (\\omega )$ in the same manner as that in TCMT.",
"Then, we require that at the drive frequency $\\omega $ , the complex amplitude of the oscillator coincides with that for $d_\\ell (\\omega )$ of the exact solution.",
"The profile of a PEC-mode is far from a narrow resonant line.",
"Therefore, the poles in the complex plane might have nothing to do with the dynamic of the mode.",
"Then, $\\omega _0$ and $\\gamma $ become free parameters, and we need two more conditions to complete the problem.",
"For them, we select (i) the equality of the frequencies maximizing the profile $|a^{\\rm (PEC)}(\\omega )|$ and the corresponding profile of the oscillator and (ii) the equality of the maxima themselves.",
"This procedure fixes all four parameters of the HO-model unambiguously.",
"For more details see Appendix."
],
[
"Results and Discussions",
"To illustrate the accuracy of the two models, their mutual advantages and disadvantages, we perform a comparison of their applications to the problem numerically studied in Ref. [1].",
"It is convenient to transfer to the dimensionless time: $t_{\\rm new}=t_{\\rm old}c/R$ .",
"Then, $\\omega _{\\rm new}=\\omega _{\\rm old}R/c \\equiv x$.",
"Since below only the dimensionless quantities are in use the subscript \"new\" will be dropped.",
"In Ref.",
"[1] the following values of the parameters are employed: $\\tau =191.1$, $m=m_{\\rm sim}=3.125$ and $x=x_{\\rm sim}=\\omega _{\\rm sim}=1.702$ .",
"The choice is done since this pair of $m$ and $x$ corresponds to a local minimum of $Q_{\\rm sca}^{(0)}$ associated with the destructive Fano resonances at $\\ell =0,\\;\\pm 2$ , both of which are situated at the close vicinity of $x=x_{\\rm sim}$ ($x \\approx 1.695$ for $\\ell = 0$ and $x \\approx 1.759$ for $\\ell = \\pm 2$ ).",
"For all other multipoles, $x_{\\rm sim}$ corresponds to the off-resonant regions.",
"Thus, only the dynamics of the modes with $\\ell =0,\\;\\pm 2$ should be approximated.",
"Moreover, since the scattering coefficients differing only by the sign of $\\ell $ are identical, the modes with $\\ell =\\pm 2$ may be regarded as a single one.",
"Figure: (a) The mutual orientation of the cylinder, coordinate frame, and vectors 𝐤\\mathbf {k}, 𝐄\\mathbf {E} and 𝐇\\mathbf {H} of the incident linearly polarized plane wave.",
"Spikes in scattering behind the leading (b) and trailing (c) edges of the rectangular incident pulse.",
"Note the large amplitudeof the spikes relative to that of the steady-state scattering.",
"The latter is shown by the part of the plots (c) at t<τt<\\tau .",
"A certainscattering at t<0t<0 exhibiting by the numerics in (a) is related to a smoothing of the rectangular envelope required for the codestabilization.The poles of the scattering coefficients adjacent to $\\omega _{\\rm sim}$ are $\\hat{\\omega }_0^{\\ell =0}\\approx 1.741-0.097i;\\;\\;\\hat{\\omega }_0^{\\ell =2}\\approx 1.535-0.0614i;$ For TCMT $\\phi ^{\\ell =0}(\\omega _{\\rm sim})\\approx -2.263,\\;\\; \\phi ^{\\ell =2}(\\omega _{\\rm sim})\\approx 0.471,$ For HO $d$ -modes: $\\hat{\\omega }_\\ell $ are given by Eq.",
"(REF ).",
"Regarding $A_0$ , $A^{\\ell =0}_{0 (d)}\\!",
"\\approx \\!",
"-0.263 \\!+\\!0.554i;\\;\\; A^{\\ell =2}_{0 (d)}\\!",
"\\approx \\!",
"0.555 \\!+\\!",
"0.142i.$ PEC-modes: $& &\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!",
"\\hat{\\omega }_{0{\\rm (PEC)}}^{\\ell =0}\\!",
"\\approx \\!",
"2.495\\!",
"-\\!",
"0.837i;\\;\\;\\hat{\\omega }_{0{\\rm (PEC)}}^{\\ell =2}\\!\\approx \\!",
"2.137\\!",
"-\\!",
"0.532i; \\\\& &\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!",
"A^{\\ell =0}_{0{\\rm (PEC)}}\\!",
"\\approx \\!",
"3.811\\!",
"- \\!0.979i;\\;\\; A^{\\ell =2}_{0{\\rm (PEC)}} \\!",
"\\approx \\!",
"-0.392\\!",
"-\\!0.793i.$ Note that, despite the error in the approximation of $a^{\\rm (PEC)}$ , the approximation of $a_\\ell (\\omega )$ in the vicinity of the Fano resonances is quite accurate, see Appendix.",
"The quantitative comparison of the two models with the numerics of Ref.",
"[1] is shown in Fig.",
"REF (b,c).",
"The origin of the $t$ -axis is shifted so that $t=0$ corresponds to the moment when the scattered radiation for the first time is detected by the measuring monitors situated in the far wave zone.",
"The comparison exhibits the high accuracy of both models.",
"To get a quantitative criterion of the accuracy of the models in a given window $t_1 \\le t \\le t_2$ we do the following: The time in the window is sampled with a step $\\delta t$ .",
"Then, the root mean square error (RMSE) for the difference $Q_{\\rm sac}^{\\rm num}(t)-Q_{\\rm sac}^{\\rm mod}(t)$ is calculated for the given sampling.",
"Here $Q_{\\rm sac}^{\\rm mod}(t)$ designates the corresponding dependences for each model.",
"At $\\delta t \\rightarrow 0$ the RMSE converges to a certain constant $\\delta Q_{\\rm sac}(t_1,t_2)$ , regarded as a measure of the model accuracy in the given window.",
"For the two windows with essentially non-steady dynamics adjacent to the leading and trailing edges of the incident pulse ($t_1=0,\\;t_2=37.5$ and $t_1=\\tau ,\\;t_2=\\tau +37.5$ , see Fig.",
"REF (b,c)) $\\delta Q^{\\rm (TCMT)}_{\\rm sac}=0.196$ , $\\delta Q^{\\rm (HO)}_{\\rm sac}=0.169$ , and $\\delta Q^{\\rm (TCMT)}_{\\rm sac}=0.157$ , $\\delta Q^{\\rm (HO)}_{\\rm sac}=0.129$ , respectively The selected width of the window $\\Delta t = 37.5$ satisfies the condition $\\exp (-|\\gamma | \\Delta t) \\approx 0.1$ , where $|\\gamma |$ is the smallest decrement corresponding to $\\ell =2$ , see Eq.",
"(REF )..",
"Here superscripts HO and TCMT designate the corresponding models."
],
[
"Conclusions",
"Thus, though both models demonstrate high accuracy, the one of the HO-model is a little better than that of TCMT.",
"In addition, while the TCMT-model just describes the complicated dynamics of the resonant light scattering, the HO-model also elucidates the physical grounds for the sharp intensive spikes in the scattering observed in the numerics behind the leading and trailing edges of the incident pulse: The spikes during the transient are explained by the violation of the balance between the resonant and background excitations.",
"The violation, in turn, is coursed by the difference in the corresponding relaxation times.",
"Then, the uncompensated part of the interfering modes is exhibited as a spike.",
"Note that if the \"basement\" of the scattered pulse is cut off at the level above the intensity of the steady-state scattering, the duration of the remaining parts of the spikes occurs very small.",
"This procedure may be used as a new method of (ultra)short pulse generation at nanoscale.",
"Regarding possible extensions of the models, while the governing equations of TCMT are simpler than those of HO, the accuracy of the HO-model is higher relative to the one of TCMT.",
"Besides, the implementation of the HO model is more straightforward and does not require a sophisticated procedure to connect the model's parameters with those of the initial underlying problem.",
"If just a single resonant excitation is a concern, the HO-model may be used based even on fitting an experimentally obtained spectrum of the modules of the output signal.",
"On the other hand, when the interference of several excitations is essential, its application implies knowledge of the phases, obtained either from an analytical solution or measured experimentally.",
"While the latter is challenging in the optical range, it is a routine procedure at radio frequencies; see, e.g., [17], [18] and references therein.",
"Thus, both models complement each other and may be very useful in descriptions of a wide variety of resonant phenomena.",
"The authors are very grateful to Boris Y. Rubinstein for his valuable help in symbolic computer calculations.",
"M.I.T.",
"acknowledges the financial support of the Russian Foundation for Basic Research (Projects No.",
"20-02-00086) for the analytical study, the Moscow Engineering Physics Institute Academic Excellence Project (agreement with the Ministry of Education and Science of the Russian Federation of 27 August 2013, Project No.",
"02.a03.21.0005) for the modeling of the resonant light scattering, as well as the contribution of the Russian Science Foundation for the computer simulation (Project No.",
"21-12-00151) and the provision of user facilities (Project No.",
"19-72-30012)."
],
[
"Scattering coefficients",
"The scattering coefficient for the problem under consideration read as follows: $a_\\ell &=& \\frac{mJ_\\ell (mx)J^{\\prime }_\\ell (x) - J_\\ell (x)J_\\ell ^{\\prime }(mx)}{mJ_\\ell (mx)H^{(1)\\prime }_\\ell (x) - H^{(1)}_\\ell (x)J_\\ell ^{\\prime }(mx)}, \\\\d_\\ell &=& \\frac{{2i}/{(\\pi x)}}{mJ_\\ell (mx)H^{(1)\\prime }_\\ell (x) - H^{(1)}_\\ell (x)J_\\ell ^{\\prime }(mx)}\\;,$ Note that $a_\\ell = a_{-\\ell }$ , $d_\\ell = d_{-\\ell }$, see Eqs.",
"(REF )–()."
],
[
"phase in the TCMT-model",
"To obtain the value of $\\phi $ , note that according to Ref.",
"[15] $a_{\\ell }^{\\rm (TCMT)}$ may be presented as $\\begin{aligned} a_{\\ell }^{\\rm (TCMT)} =\\frac{1}{2} \\frac{{i}\\left(\\omega _{0}-\\omega \\right) ({e}^{{i} \\phi }-1 )+\\gamma \\left(1+{e}^{{i} \\phi }\\right)}{{i}\\left(\\omega _{0}-\\omega \\right)-\\gamma }.",
"\\end{aligned}$ (the same expression may be obtained from Eq.",
"(REF ), if we formally consider its limit at $t \\rightarrow \\infty $ ).",
"Then, after some algebra Eq.",
"(REF ) gives rise to the following expression for $|a_{\\ell }^{\\rm (TCMT)}|^2$ : $|a_{\\ell }^{\\rm (TCMT)}|^2 =\\sin ^2\\frac{\\phi }{2}\\left| \\frac{\\cot \\frac{\\phi }{2}+\\frac{\\omega -\\omega _0}{\\gamma }}{1+i\\frac{\\omega -\\omega _0}{\\gamma }}\\right|^2 =\\frac{1}{1+q^2}\\frac{(q+\\epsilon )^2}{1+\\epsilon ^2},$ where $\\epsilon =-(\\omega -\\omega _0)/\\gamma $ and $ q=-\\cot (\\phi /{2})$ .",
"We remind that $\\gamma <0$ .",
"The last formula in Eq.",
"(REF ) is the conventional Fano profile with the asymmetry parameter $q$ [12], [13], [14].",
"Following the procedure described in the main text, we have to equalize $|a_{\\ell }^{\\rm (TCMT)}(\\omega _{\\rm sim})|^2$ to $|a_{\\ell }(\\omega _{\\rm sim})|^2$ , where $a_{\\ell }(\\omega _{\\rm sim})$ is given by Eq.",
"(REF ) at $x=\\omega _{\\rm sim}$ (remember that in the selected dimensionless variables $x$ and $\\omega $ numerically are equal to each other).",
"This brings about a quadratic equation for $q$ , whose solutions are $q_{_{(\\pm )}} = \\frac{\\epsilon _{\\rm sim}\\pm (1+\\epsilon _{\\rm sim}^2)|a_{\\ell }(\\omega _{\\rm sim})|\\sqrt{1-|a_{\\ell }(\\omega _{\\rm sim})|^2}}{(1+\\epsilon _{\\rm sim}^2)|a_{\\ell }(\\omega _{\\rm sim})|^2-1},$ where $\\epsilon _{\\rm sim}=-(\\omega _{\\rm sim}-\\omega _0)/\\gamma $ .",
"Note that for the problem in question $|a_\\ell |$ always is smaller than unity, see, e.g., [19].",
"Therefore, the roots in Eq.",
"(REF ) are always real.",
"Plots of $|a_{\\ell }(\\omega )|$ as well as their approximations at $q=q_{_{(+)}},\\;q=q_{_{(-)}}$ and $\\ell =0,\\;2$ at the values of $\\omega _0,\\;\\gamma $ given by Eq.",
"((REF )) are shown in Fig.",
"REF .",
"Thus, for $\\ell =0$ the better overall approximation is at $q=q_{_{(-)}}$ .",
"For the case in question its numerical value is 0.4700, which corresponds to $\\phi \\approx -2.263$ .",
"In contrast, for $\\ell =2$ the better overall approximation is at $q=q_{_{(+)}} \\approx -4.172$ , corresponding to $\\phi \\approx 0.471$ .",
"Figure: The approximation of the profile |a 0 (ω)||a_0(\\omega )| shown as a blue line (unmarked) by Eq.",
"() at q=q (-) q=q_{_{(-)}} and q=q (+) q=q_{_{(+)}}.",
"The general view (a) and the vicinity of ω=ω sim \\omega =\\omega _{\\rm sim} (b).",
"Though all plots have the same value at ω=ω sim \\omega =\\omega _{\\rm sim}, the overall approximation is better at q=q (-) q=q_{_{(-)}}.Figure: The same as that in Fig.",
"(a) for ℓ=2\\ell =2.",
"The better overall approximation is for q=q (+) q=q_{_{(+)}}Figure: dd-modes.",
"Exact steady-state scattering coefficients (solid lines) and their approximations in the HO-model (dashed lines).",
"Abs, Re and Im designate the modula, real and imaginary parts of the coefficients, respectively.",
"The larger the difference between ω sim \\omega _{\\rm sim} and the resonant frequency maximizing |d ℓ ||d_\\ell |, the lager the approximation error, cf.",
"panels (a) and (b).Figure: The same as that in Fig.",
"for PEC-modes.Figure: The same as that in Fig.",
"for aa-modes."
],
[
"The approximation procedure for the HO-model",
"A steady-state solution of Eq.",
"(REF ) with the r.h.s equals $A_0\\exp [-i\\omega t]$ is $f_s(t) = F_s(\\omega )e^{-i\\omega t};\\;\\; F_s(\\omega )\\equiv -\\frac{|A_0|e^{i\\varphi }}{\\omega ^2-\\omega _0^2-2i\\gamma \\omega }.$ Thus, to apply the HO-model to the discussed light scattering problem, the following model parameters should be fixed: the complex drive amplitude $A_0\\equiv |A_0|e^{i\\varphi }$ , the frequency $\\omega _0$ and the damping factor $\\gamma $ .",
"In the case of the $d$ -modes the values of $\\omega _0$ and $\\gamma $ are given by the position of the nearest to $\\omega _{\\rm sim}$ complex pole of the steady-state coefficient $d_\\ell (\\omega )$ .",
"The remaining undefined parameters are $A_0$ is readily obtained from the equality $F_s(\\omega _{\\rm sim}) = d_\\ell (\\omega _{\\rm sim})$ .",
"The case of the PEC-modes is more tricky since neither $\\omega _0$ , nor $\\gamma $ can be obtained in the same easy manner as that for the $d$ -modes.",
"Once again, it is convenient to present the complex amplitude $A_0$ in the form $A_0 = |A_0|e^{i\\varphi }$ .",
"Then, the maximum of $|F_s(\\omega )|^2$ equals $\\mathop {\\rm Max}\\limits _\\omega \\lbrace |F_s(\\omega )|^2\\rbrace \\equiv |F_s|^2_{\\rm max}= \\frac{|A_0|^2}{4\\gamma ^2\\omega _{0\\gamma }^2},$ and is achieved at $\\omega =\\omega _{\\rm max}\\equiv \\sqrt{\\omega ^2_0-2\\gamma ^2}.$ We remind that $\\omega _{0\\gamma } = \\sqrt{\\omega _0^2-\\gamma ^2}=\\sqrt{\\omega _{\\rm max}^2+\\gamma ^2}$ , see Eqs.",
"(REF )–(REF ), (REF ).",
"Let us consider $\\omega _{\\rm max}$ as a new independent parameter instead of $\\omega _0$ .",
"Next, according to the procedure described in the main text, we require that $\\omega _{\\rm max} = \\omega ^{\\rm (PEC)}_{{\\rm max}(\\ell )}$ , where $|a^{\\rm (PEC)}_\\ell (\\omega ^{\\rm (PEC)}_{{\\rm max}(\\ell )})|^2_{\\rm max} \\equiv |a^{\\rm (PEC)}_\\ell |^2_{\\rm max}$ is the maximal value of the corresponding quantity.",
"This fixes the value of $\\omega _{\\rm max}$ by the shape of the profile $|a^{\\rm (PEC)}_\\ell (\\omega )|^2$ .",
"The condition $|F_s|^2_{\\rm max} = |a^{\\rm (PEC)}_\\ell |^2_{\\rm max}$ expresses $|A_0|^2$ in terms of $\\gamma $ .",
"To fix $\\gamma $ we require that $|F_s(\\omega _{\\rm sim})|^2 = |a^{\\rm (PEC)}_\\ell (\\omega _{\\rm sim})|^2$ , where $\\omega _{\\rm sim}$ is the carrier frequency of the incident pulse.",
"It gives rise to a biquadratic equation for $\\gamma $ .",
"After some algebra, its only solution satisfying the condition Re$\\,\\gamma <0$ , Im$\\,\\gamma =0$ may be presented in the following form: $\\gamma \\!",
"=-\\frac{\\omega _{\\rm max}}{\\sqrt{2}} \\left\\lbrace -1 +\\!\\sqrt{1+\\frac{(\\omega _{\\rm max}^2-\\omega _{\\rm sim}^2)^2|a^{\\rm (PEC)}_\\ell |^2_{\\rm sim}}{\\omega _{\\rm max}^4(|a^{\\rm (PEC)}_\\ell |^2_{\\rm max}\\!-\\!|a^{\\rm (PEC)}_\\ell |^2_{\\rm sim})}} \\right\\rbrace ^{1/2}$ where $|a^{\\rm (PEC)}_\\ell |^2_{\\rm sim}\\equiv |a^{\\rm (PEC)}_\\ell (\\omega _{\\rm sim})|^2$ .",
"Nonnegativity of the expressions under the signs of radicals is seen straightforwardly.",
"Now, the last unfixed parameter $\\varphi $ is readily obtained from the condition that the phase of $a^{\\rm (PEC)}_\\ell (\\omega _{\\rm sim})$ equals the one of $F_s(\\omega _{\\rm sim})$ .",
"The values of $|a^{\\rm (PEC)}_\\ell |^2_{\\rm max,\\,sim}$ and $\\omega =\\omega ^{\\rm (PEC)}_{{\\rm max}\\,(\\ell )}$ are obtained numerically according to the definition of $a^{\\rm (PEC)}_\\ell $ , see Eq.",
"(REF ).",
"The application of this procedure results in the values of the parameters presented in the main text, see Eqs.",
"(REF ) and ().",
"Note that while the approximation errors for the PEC modes is large, the final approximations for $a_\\ell $ in the proximity of $\\omega _{\\rm sim}$ are quite accurate, cf.",
"Figs.",
"REF and REF .",
"Next, there are some points, where $|a^{\\rm (PEC)}_\\ell (\\omega )|$ vanishes, see, e.g., Fig REF (a).",
"If $\\omega _{\\rm sim}$ coincides with such a point, i.e., $|a^{\\rm (PEC)}_\\ell (\\omega )|=0$ , the phase of $a^{\\rm (PEC)}_\\ell (\\omega )$ is indeterminate, and our method to fix the complete set of the HO-model parameters seemingly fails.",
"However, in this case, $A_0$ in the r.h.s.",
"of Eq.",
"(REF ) should be set to zero, the PEC-mode does not contribute to the model dynamic, and the corresponding parameters merely are not required.",
"Another point of a special interest is the one, when $\\omega _{\\rm sim}=\\omega _{\\rm max}$ .",
"In this case the straightforward application of Eq.",
"(REF ) gives rise to an indeterminate form $0/0$ .",
"As usual, it means that we have to consider the limit $\\omega _{\\rm sim}\\rightarrow \\omega _{\\rm max}$ .",
"Expanding in Eq.",
"(REF ) $|a^{\\rm (PEC)}_\\ell (\\omega )|^2$ about the point $\\omega _{\\rm sim}=\\omega _{\\rm max}$ in powers of $\\delta \\omega =\\omega _{\\rm sim}-\\omega _{\\rm max}$ we readily obtain that $\\lim _{\\omega _{\\rm sim}\\rightarrow \\omega _{\\rm max}}\\!\\!\\!\\!",
"\\!\\!\\!\\!\\gamma =-\\frac{\\omega _{\\rm max}}{\\sqrt{2}} \\left\\lbrace -1+\\sqrt{\\frac{4}{\\alpha ^2 \\omega _{\\rm max}^2}+1}\\right\\rbrace ^{1/2},$ where $\\alpha ^2\\equiv -\\frac{1}{2|a^{\\rm (PEC)}_\\ell |^2_{\\rm max}}\\left(\\frac{\\partial ^2 |a^{\\rm (PEC)}_\\ell |^2_{\\rm max}}{\\partial \\omega ^2}\\right)_{\\!\\!\\omega _{\\rm max}}\\!\\!\\!\\!\\!\\!>0.$"
],
[
"Overall dynamic",
"The comparison of the overall dynamic of the scattering obtained by the numerics [1] with that described by the models is presented in Figs.",
"REF and REF .",
"Figure: Q sac (t)Q_{\\rm sac}(t) obtained by the direct numeric integration of the complete set of Maxwell's equations and that described by the TCMT-model; t=0t=0 corresponds to the moment, when the scattered radiation for the first time is detected by the measuring monitors.",
"The moment t=τt=\\tau is clearly seen by the abrupt change of the dynamic from almost sinusoidal to essentially non-sinusoidal.",
"Note the overshot of the first oscillation behind both edges of the incident pulse (i.e., behind the points t=0t=0 and t=τt=\\tau ) for the model relative to that for the numerics, explained by the instantaneous excitation of the background partition for the TCMT-model, see the discussion after Eq.",
"() in the main text.Figure: The same as that in Fig.",
"for the HO-model.",
"The overshot does not occur."
]
] | 2107.01822 |
[
[
"Tuning and Amplifying the Interactions in Superconducting Quantum\n Circuits with Subradiant Qubits"
],
[
"Abstract We propose a tunable coupler consisting of N fixed-frequency qubits, which can tune and even amplify the effective interaction between two superconducting quantum circuits.",
"The tuning range of the interaction is proportional to N, with a minimum value of zero and a maximum that can exceed the physical coupling rates between the coupler and the circuits.",
"The effective coupling rate is determined by the collective magnetic quantum number of the qubit ensemble, which takes only discrete values and is free from collective decay and decoherence.",
"Using single-photon pi-pulses, the coupling rate can be switched between arbitrary choices of the initial and final values within the dynamic range in a single step without going through intermediate values.",
"A cascade of the couplers for amplifying small interactions or weak signals is also discussed.",
"These results should not only stimulate interest in exploring the collective effects in quantum information processing, but also enable development of applications in tuning and amplifying the interactions in a general cavity-QED system."
],
[
"Effective Hamiltonian with one layer",
"We consider a system, $H=H_0+V$ , where two general circuit components are coupled via a qubit ensemble H0 = m=12m xmxm + n=1Nn2nz + {n, n'} gn,n' (n+n'- + n-n'+), V = m=12n=1Ngm,n( xmn- + xmn+ )V1 + gm,n(xmn++xmn-)V2.",
"Here, $\\omega _{m}$ , $x_m$ , and $x_m^{\\dagger }$ are the resonant frequency, raising and lowering operators of the $m$ th system component $\\rm X_m$ .",
"Furthermore, $\\omega _{n}$ and $\\sigma _{n}^{\\alpha }$ with $\\alpha =x,y,z$ are the characteristic frequency and standard Pauli operators of the $n$ th qubit in the ensemble with a total qubit number of $N$ , $g_{\\alpha ,\\beta }$ is the coupling rate between the two components $\\alpha $ and $\\beta $ , and $\\lbrace n, n^{\\prime }\\rbrace $ takes all possible pairs in the ensemble.",
"To transform the component-coupler interaction into an effective component-component interaction to the second order of $g_{\\alpha ,\\beta }$ , we apply the following unitary transformation to the original Hamiltonian [1] $U = \\exp \\Bigg [\\overbrace{\\sum _{m=1}^{2}\\sum _{n=1}^{N}-\\frac{g_{m,n}}{\\Delta _{m,n}}\\left( x_m^{\\dagger }\\sigma _{n}^{-} - x_m \\sigma _{n}^{+} \\right)}^{X_1}+ \\overbrace{\\sum _{m=1}^{2}\\sum _{n=1}^{N}-\\frac{g_{m,n}}{\\Sigma _{m,n}}\\left( x_m^{\\dagger }\\sigma _{n}^{-} - x_m \\sigma _{n}^{+} \\right)}^{X_2}\\Bigg ].$ On the one hand, we have $\\llbracket H_0,X_1+X_2\\rrbracket =-V$ .",
"The transformation can be simplified as $U^{\\dagger }HU=H_0 + (1/2)\\llbracket V, X_1+X_2 \\rrbracket $ to the second-order accuracy of $g_{m,n}/\\Delta _{m,n}$ and $g_{m,n}/\\Sigma _{m,n}$ .",
"On the other hand, we have V1, X1= -gm,nm,n{ gm',nnz( x1x2 + x1x2 ) + gm,n(nz{xm,xm } + [xm,xm] ) + gm,n'[xm,xm]( n+n'- + n-n'+ )}, V1, X2= gm,nm,n{gm',nnz( x1x2 + x1x2 ) + gm,nnz (xm2 + xm2 ) - gm,n'[xm,xm]( n+n'+ + n-n'- )}, V2, X1= -gm,nm,n{gm',nnz( x1x2 + x1x2 ) + gm,nnz (xm2 + xm2 ) + gm,n'[xm,xm]( n+n'+ + n-n'- )}, V2, X2= gm,nm,n{gm',nnz( x1x2 + x1x2 ) + gm,n(nz{xm,xm } - [xm,xm] ) - gm,n'[xm,xm]( n+n'- + n-n'+ )}.",
"Here, we have omitted the summation symbols over $m$ and $n$ for the simplicity of notation, $m^{\\prime }$ and $n^{\\prime }$ indicate all the different numbers from $m$ and $n$ .",
"In total, we obtain the effective Hamiltonian H = m xmxm + n2nz + gn,n'2 nxn'x + gm,ngm',n2(1m,n- 1m,n) nz( r1 + r1 )( r2 + r2 ) + gm,n22(1m,n-1m,n)nz(x1 + x1)2 - gm,n22(1m,n+1m,n)xm, xm(n- + n+)2 - gm,ngm,n'2(1m,n+1m,n)xm, xmnxn'x.",
"For the cases with two resonators, one resonator and one qubit, and two qubits, respectively, we have the following effective Hamiltonians HR-R = m=12n=1N [m + gm,n2( 1m,n-1m,n)nz ]rmrm + g1,ng2,n2(1m,n- 1m,n)nz( r1 + r1 )( r2 + r2 ) + 12[n + gm,n2( 1m,n-1m,n) (rmrm+rm rm + 1)] nz + n=1Nn'n12[gn,n'-gm,ngm,n'(1m,n+1m,n)]nxn'x, HR-Q = m=12n=1N [1 + g1,n2( 11,n-11,n)nz ]r1r1 + [2+g2,n2(12,n+12,n)]2z2 + g1,ng2,n2(1m,n- 1m,n)nz( r1 + r1) 2x + 12[ n + g1,n2( 11,n-11,n) (r1r1+r1 r1 + 1) + g2,n2(12,n - 12,n) ]nz + n=1Nn'n12[ gn,n' - g1,ng1,n'(11,n+ 11,n) + g2,ng2,n'(12,n+ 12,n)2z ]nxn'x, HQ-Q = m=12n=1N [m + gm,n2(1m,n + 1m,n)]mz2 + g1,ng2,n2(1m,n- 1m,n)nz1x2x + 12[n+gm,n2(1m,n-1m,n)]nz + n=1Nn'n12[gn,n' + gm,ngm,n'(1m,n+1m,n)mz]nxn'x.",
"If we further assume the homogeneity among the coupler qubits, we use the collective angular momentum operators to describe the whole qubit ensemble and arrive at a more compact form of the effective Hamiltonian.",
"They are HR-R = m=12[m + gm2( 1m-1m)Jz ]rmrm + g1g22(1m- 1m)Jz( r1 + r1 )( r2 + r2 ) + 12[q + gm2( 1m-1m) (rmrm+rm rm + 1)]Jz +12[gq-gm2(1m+1m)](Jx)2, HR-Q = m=12[1 + g12( 11-11)Jz ]r1r1 + [2+g22(12+12)(Jx)2]2z2 +g1g22(1m- 1m)Jz( r1 + r1) 2x + 12[q + g12( 11-11) (r1r1+r1 r1 + 1) + g22(12 - 12) ]Jz +12[ gq - g12(11+ 11) ](Jx)2, HQ-Q = m=12[m + gm2(1m + 1m)(Jx)2]mz2 +g1g22(1m- 1m) Jz1x2x +12[q+gm2(1m-1m)]Jz +12gq (Jx)2."
],
[
"Effective Hamiltonian with multiple layers",
"We consider a system with $D$ layers of $N$ homogeneous qubits, where any two adjacent layers are coupled by an XY-type interaction H = m=12 m xmxm + d=1Dq2Jdz + d=1D-1gq (Jd+Jd+1-+Jd-Jd+1+) + g1( x1+x1 )J1x + g2(x2+x2)Jdx.",
"By assuming that $\\langle J_{d}^{z} \\rangle \\approx -N/2$ , we introduce the following replacement of variables for large $N$ [2], [3], [4], [5], [6] Jd+ Nad, Jd- Nad, Jdz = 2adad-N.",
"This gives H = m=12m xmxm + Ng1( x1 + x1 )( b1 + b1 ) + Ng2( x2+ x2 )( ad + ad ) + d=1D q adad + d=1D-1 Ngq (adad+1+adad+1).",
"We note that the last term should be written as $\\sum _{d=1}^{D-1} Ng_{\\rm q} \\left(a_{d}^{\\dagger }+a_{d}\\right)\\left(a_{d+1}^{\\dagger }+ a_{d+1}\\right)$ for an Ising-type interaction, described by $g_{q}J_{d}^{x}J_{d+1}^{x}$ , between two adjacent layers.",
"Similar to the definition of magnons in a XY-type spin chain [7], [8], [9], [10], we define ak = 2D+1d=1D(skD+1)ad.",
"The Hamiltonian can be written as H0 = m=12m xmxm + k=1D(q+2gk)akak, V = m=12k=1N gm,k( xmak + xm ak )V1 + gm,k( xmak + xm ak )V2, where $g_{k}=Ng_{\\rm q}\\cos \\left[k\\pi /(D+1)\\right]$ , $g_{m,k}=\\sqrt{N}g_{m}\\sin \\left[mk\\pi /(D+1)\\right]\\sqrt{2/(D+1)}$ .",
"For an Ising-type interaction, one may add $\\sum _{k=1}^{D} g_{k} \\left(a_{k}^{\\dagger 2} + a_{k}^{2}\\right)$ in $H_0$ .",
"To derive the effective Hamiltonian, we apply the following unitary transformation U = [m=12k=1Dm,k-( xmak - xm ak )X1 +m=12k=1Dm,k+( xmak - xm ak )X2].",
"We obtain H0, X1+X2 = m,k-(m,n - 2gk) ( xm+ak + xm-ak ) + m,k+(m,n + 2 gk) ( xm+ak + xm-ak ), H0, X1+X2 = [m,k-m,n + 2(m,k+-m,k-) gk ] ( xm+ak + xm-ak ) + [m,k+m,n + 2(m,k+-m,k-) gk] ( xm+ak + xm-ak ), for the XY- and Ising-type interactions, respectively.",
"The rest of the commutators are V1, X1 = -m,k-{gm',k ( x1x2 + x1x2 ) + gm,k {xm,xm} - (gm,k{ak,ak } + gm,k'{ akak' + akak' }) xm,xm} V1, X2 = m,k+{gm',k ( x1x2 + x1x2 ) + gm,k {xm2 + xm2} + (gm,k{ak+ 2+ak- 2 } + gm,k'{ akak' + akak' }) xm,xm}, V2, X1 = -m,k- {gm',k ( x1x2 + x1x2 ) + gm,k {xm2 + xm2} - (gm,k{ak+ 2+ak- 2 } + gm,k'{ akak' + akak' }) xm,xm}, V2, X2 = m,k+{ gm',k ( x1x2 + x1x2 ) + gm,k {xm,xm} + (gm,k{ak,ak } + gm,k'{ akak' + akak' }) xm,xm}.",
"As before, we have omitted the summation symbol in the above formulae for the simplicity of notation.",
"The component-coupler interaction can be eliminated to the second-order accuracy if m,k- = -gm,n(m,n - 2gk ), m,k+ = -gm,n(m,n + 2gk ), or m,k-m,n + 2(m,k+-m,k-) gk = -gm,n, m,k+m,n + 2(m,k+-m,k-) gk = -gm,n, which gives the following effective Hamiltonian H = m xmxm + (q+2gk) ak ak + gk (ak2 + ak2) + 12(m,k+-m,k-)gm',k (x1+x1)(x2+x2) + 12(m,k+-m,k-)gm,k(xm + xm)2 + 12(m,k++m,k-)gm,k(ak+ak)2 xm,xm+ 12(m,k++m,k-)gm,k'(ak + ak )(ak' + ak' )xm,xm.",
"Here, the underlined term exists only for an Ising-type interaction."
]
] | 2107.01842 |
[
[
"A System for Traded Control Teleoperation of Manipulation Tasks using\n Intent Prediction from Hand Gestures"
],
[
"Abstract This paper presents a teleoperation system that includes robot perception and intent prediction from hand gestures.",
"The perception module identifies the objects present in the robot workspace and the intent prediction module which object the user likely wants to grasp.",
"This architecture allows the approach to rely on traded control instead of direct control: we use hand gestures to specify the goal objects for a sequential manipulation task, the robot then autonomously generates a grasping or a retrieving motion using trajectory optimization.",
"The perception module relies on the model-based tracker to precisely track the 6D pose of the objects and makes use of a state of the art learning-based object detection and segmentation method, to initialize the tracker by automatically detecting objects in the scene.",
"Goal objects are identified from user hand gestures using a trained a multi-layer perceptron classifier.",
"After presenting all the components of the system and their empirical evaluation, we present experimental results comparing our pipeline to a direct traded control approach (i.e., one that does not use prediction) which shows that using intent prediction allows to bring down the overall task execution time."
],
[
"Introduction",
"Intelligent robots can substitute or assist humans to accomplish complicated and laborious tasks.",
"They are becoming present in our lives from production lines to hospitals and our homes.",
"However, many applications remain challenging for robots to function in full autonomy.",
"Teleoperation is an intermediate solution for controlling robots in scenarios where the task objectives have to be decided in real-time, such as disaster relief [1], autonomous driving [2], or assistive devices [3], [4].",
"Shared control has been investigated to effectively blend user and autonomous control during teleoperation.",
"The linear blending paradigm introduced by Dragan et.",
"al [5] is still widely applied in many shared control frameworks [4], [6], [7].",
"In the approach, the amount of arbitration is dependent on the confidence of user prediction.",
"However, the user loses control authority when the robot predicts the user's intent with high confidence.",
"Some works allocate maximum control authority to the user by providing minimal assistance only when it is necessary.",
"Broad et.",
"al.",
"[8] introduced minimum intervention shared control that computes whether the control signal leads to an unsafe state and replaces the user control if so.",
"Our recent work [9] formulates shared control as an optimization problem, which can conveniently balance control authority and optimality when a complete robot policy is available.",
"While these works are relevant to the teleoperation of simple manipulation task where direct control is not optimal, they are generally limited to controlling the end-effector of the robot by blending between direct and autonomous control.",
"They rely on a semantic mapping of the workspace but they do not let the autonomy take complete advantage of these models, so as to maximize control authority.",
"Additionally when using interfaces such as hand gestures controllers, direct teleoperation is often nearly impossible as the mismatch between the kinematics of the robot and hand gestures is too large to produce fluid movements.",
"Figure: Overview of the teleoperation systemHence in this systems paper, we demonstrate a complete traded control teleoperation implementation, where the user specifies the task objectives and executes the motion autonomously.",
"Contrarily to the aforementioned approaches, it does not blend between direct and autonomous control.",
"Our system makes use of available models, in terms of object poses and shapes to plan robot motion trajectories.",
"We present and evaluate all components needed for such a system, which can be decomposed into three parts: 1) a perception pipeline capable of identifying and tracking objects, 2) an intent estimation system that can identify which objects to grab and how, 3) a motion planning system that can produce accurate manipulation motion in accordance to the human intent.",
"$\\endcsname $Video available at https://sites.google.com/view/ohyn-teleoperation-pipeline/home After presenting the individual components, we assess the accuracy of the the different modules on dedicated tasks.",
"We evaluate the object localization and tracking module on several objects in simulation, test the accuracy of our grasp intent prediction module using a dataset of trajectories.",
"Finally, we present results using our grasp intent inference module, where various users are simulated using degraded user trajectories collected using the hand gesture controller.",
"We summarize our main contributions as the following: A teleoperation system capable of traded control using hand gestures Simulated user experiment assessing the capacity of our grasp intent prediction module to perform teleoperation of pick and place motions A solution for automatic initialization of an existing object tracking module using Mask R-CNN [10] This paper is structured as follows: we present related work in Section .",
"Section presents our user interface.",
"Section presents our object tracking pipeline, including the combination of Mask R-CNN and a model-based object tracker.",
"Section presents the assessment of the modules in our pipeline.",
"Conclusions are drawn in Section .",
"Traded control is a discrete switching mechanism between high-level robot autonomy and low-level control depending on predefined circumstances.",
"It is also referred to as control switching, as the system allocates all-or-none assistance rather than a blended spectrum between user and robot controls.",
"The operator initiates a sub-task or behavior for the robot and the robot performs the sub-task autonomously while the operator monitors the robot [11], [1].",
"[12] showed that intent-based traded control can improve teleoperation performance and alleviate difficulties in high-latency teleoperation scenarios."
],
[
"Hand Gesture Recognition for Robot Control",
"The Leap Motion controller (Ultra Leap, https://www.ultraleap.com/) is a consumer-grade, marker-less motion capture sensor that tracks hand gestures and finger movements up to 200 Hz.",
"[13] showed that its accuracy is below 2.5mm, however, the controller shows inconsistent performance due to its limited sensory range [14].",
"Nevertheless, its simplicity and its capability to track the hand in 6-Dof are the reasons for its application.",
"Prior works used deep learning to improve the accuracy of the gesture recognition, such as SVMs and random forests [15], or neural networks using radial basis functions (RBF) [16].",
"Similar to [17], we propose to train a gesture classifier (i.e., which object is intended) for hand motion recognition rather than mapping hand features directly to robot configurations.",
"Achieving higher accuracy is easier on classification than regression (i.e., predicting accurate positions) tasks, which is one of the justifications for our traded control approach."
],
[
"Depth Based Object Tracking (DBOT)",
"We utilize the implementation of depth-based object tracking methods described in [18] (“particle tracker”) and [19] (“Gaussian tracker”) to acquire the 6D pose of objects during teleoperation.",
"Compared to recent learning-based methods such as PoseCNN [20] and DenseFusion [21], the methods take a model-based approach.",
"The particle tracker in DBOT tracks objects by computing a posterior distribution over the object using a dynamic Bayesian network for inference [18], while the Gaussian tracker improves the performance of a Gaussian filter using a robustification method as well as reducing the filter's computational complexity [19].",
"This approach has the advantage of being robust without requiring any extra tuning or pre-training."
],
[
"Hand Gesture Based Robot Control",
"The user provides grasping intentions and commands by performing reach-and-grasp motions with the right hand as if the user naturally reaches and grasps an object while looking at the environment from the robot's perspective.",
"The hand motion is captured using a Leap Motion controller.",
"Features are captured and published via ROS topics at a frame rate of 180Hz.",
"Since the user is actually reaching towards an invisible object, the grabbing positions vary significantly as shown in REF .",
"We resolve this issue with a traded control paradigm and learning a classifier to distinguish how the user is intending to grab the object.",
"Figure: (a) Grab positions from users tracked with the Leap Motion controller, top grab (red)/right grab (blue).",
"(b), (c) User interface for reach and grab motion in a setting with three objects."
],
[
"Traded Control",
"To alleviate the inconsistent hand tracking performance, we adopt a traded control method rather than a continuous shared control paradigm.",
"This also relieves the problems that arise from the physical difference between the human arm and the robot arm.",
"Once objects are identified as described in Section , we predict the user's intent of the target object and in which direction the user is intending to grasp.",
"As soon as the intention is identified, the robot controller executes the object reach and grab motion.",
"The user still maintains control authority by having the ability to decide in which order to grab the set of objects, that is, we rely on the human user for high-level decision making and the robot takes care of the low-level control and motion planning."
],
[
"Grasp Intention Prediction",
"We train a multi-layer perceptron using supervised learning to classify the goal object and the grasp direction.",
"We assume a fixed set of objects ($m$ =3) along with their positions and two possible grab directions (top/right, $n$ =2), as shown in Figures REF and REF .",
"The input includes eight features: distances from the hand objects, x-component of the hand position, x-component of the hand direction, x,y-components of the palm normal vector, and y-rotation of the hand are selected through experience.",
"The model consists of three dense layers of 64 hidden units that are connected to two separate layers of two units and outputs the class labels.",
"Figure: Overview of the object tracking pipeline and an image of the simulated robot environment."
],
[
"Object Tracking Pipeline",
"The object tracking pipeline is automatically initialized and tracks the 6D pose of rigid objects.",
"We make use of the existing object tracking library [18], [19] and provide a solution to alleviate the burden of manual initialization.",
"The pipeline consists of two modules: object tracker initializer (DBOT initializer) and the object tracker (DBOT) as shown in Figure REF .",
"The DBOT initializer receives camera images of the environment (see Figure REF ) and predicts the initial pose as well as the semantic label of observed objects."
],
[
"Automated Object Tracker Initialization",
"DBOT assumes that the initial pose of the object is given.",
"In practice, this initialization is done by the user by manually positioning a marker over the object's depth image in a 3D visualization of the depth image.",
"We use Mask R-CNN [10], a state of the art instance segmentation method, to automate the initialization process by identifying the masks and the labels using transfer learning.",
"Once the depth pixels are segmented from the depth image using the object mask, we use point cloud registration to compute the object's 6D pose.",
"Utilizing the labels from the Mask R-CNN, the corresponding mesh model of the object is loaded and a set of points are sampled from the mesh as a reference point cloud for registration.",
"Rigid registration is performed using the coherent-point-drift (CPD) algorithm [22], and we use the average position of the masked point cloud as a rough initialization during the registration.",
"The output of the CPD is a 4$\\times $ 4 homogeneous transformation matrix.",
"We refer to this estimated pose as mesh_pose.",
"The process takes approximately 1-16 seconds, depending on the number of tracked objects and the number of iterations during point cloud registration.",
"The step is performed once to initialize the DBOT tracker.",
"Figure: Snapshot of our object tracking with real images"
],
[
"DBOT Tracker",
"Once the DBOT trackers are successfully initialized, the simulation or a real robot system can subscribe to the refined poses from DBOT and use the information to perform precise object manipulation.",
"The DBOT tracker runs with 10Hz for up to 7 objects on CPU.",
"The performance can be further improved by utilizing GPU.",
"Figure: bowl"
],
[
"Simulated Robot Environment",
"The environment consists of the Baxter robot with graspable objects on a table as shown in Figure REF right.",
"A virtual depth camera is added on Baxter's head display to generate first-person view images (RGB-D).",
"The simulated environment is created using the RAI$\\endcsname $https://github.com/MarcToussaint/rai interface.",
"RAI includes a physics-simulated environment as well as a robot motion optimization solver for k-Order Motion Optimization (KOMO) problem[23].",
"The interface provides simple functionality to define motion optimization problems, by specifying the list of optimization objectives that represent cost terms or in-/equality constraints."
],
[
"Mask R-CNN Transfer Learning on a Synthetic Dataset",
"We use transfer learning to tune the Mask R-CNN to ensure the detection of custom objects.",
"We collect images of the simulated robot environment including the following objects: a cube, a sphere, a toy, a teapot, a cup, a jug, and a bowl, as shown in Figure REF .",
"The simulated images are not like real images as they do not include noise, shadows, irregular lighting conditions, or texture.",
"The images are augmented during training to ensure the network generalizes towards real or non-perfect images.",
"Augmentations include flipping the image, affine transformation, light contrast, blur/sharpen, and color modifications.",
"The dataset is also collected with arbitrary robot arm joint positions included in the image so that Mask R-CNN learns to neglect robot arms during detection.",
"We obtained detection rates from the Mask R-CNN ranging from 92% to 95% with an error rate between 0.16% to 0.74% based on different training settings such as the number of epochs (10 to 80 epochs) and the dataset size (107 to 50K samples).",
"The detection rate is defined as the number of classified objects with respect to the number of ground truth objects.",
"The error rate is defined as the number of wrong class predictions with respect to the number of classified objects.",
"The rest consists of undetected objects, presumably simply not detected, out of sight, or just partially visible.",
"We do not report detailed results for each setting, as its effect turned out to be marginal when comparing the results of the whole initialization pipeline.",
"As long as the detected label is correct, a decent mask is adequate for estimating the initial pose described in Section REF .",
"The model used during the evaluation of the pose initialization shown in Table REF was trained with unmodified images with 40 epochs.",
"The detection rate of the Mask R-CNN was 92.73% with an error rate of 0.16% out of a set of 4,798 random samples.",
"The images of the samples were not included during the Mask R-CNN training and did not contain the arms of the Baxter robot."
],
[
"Accuracy of the Initial Object Pose Estimation",
"We evaluate the accuracy of the pose estimation using the metric proposed in [20].",
"The average distance is computed using the closest point distance of the pairwise distances between two 3D models with ground truth transformation (translation $t$ , rotation $R$ ) and estimated transformation (translation $\\hat{t}$ , rotation $\\hat{R}$ ): $\\text{ADD-S} &= \\frac{1}{m} \\sum _{x_1 \\in M} \\min _{x_2 \\in M} || (R x_1 + t) - (\\hat{R} x_2 + \\hat{t}) ||$ in a set $\\mathcal {M}$ with $m$ number of points, for both symmetric and asymmetric objects.",
"Following prior works [20], [21], we report the area under the ADD-S curve (AUC) with a threshold up to 0.1m by computing the pose accuracy while increasing the threshold.",
"Similarly, we also measure the percentage of ADD-S below a threshold of 2cm, which is the minimum tolerance for robot grasping manipulation.",
"Figure REF and Table REF show the accuracy of different object tracking methods including the method described in Section REF .",
"The mask_pose(or mask in Table REF ) indicates the center position of the masked point cloud with an offset (2cm in the z-axis of the camera coordinates) added to compensate the bias in the point cloud.",
"The mask_pose does not include rotation which explains the low AUC, but is a good baseline when comparing translation.",
"The mesh_pose shows better precision in translation compared to the mask_pose, nonetheless, its main role is to provide an initial rotation estimation to initialize DBOT."
],
[
"Accuracy of the DBOT Tracker",
"We report the result of the estimated pose once the DBOT tracker is tracking the object after receiving the initial mesh_pose.",
"The object pose from the DBOT is captured after 1 second and 3 seconds after initialization while the object is kept static.",
"Table REF shows that both trackers outperform the initial mesh_pose.",
"This indicates that the tracker was able to refine the pose towards the correct object pose after receiving the estimated pose.",
"The particle tracker refined the pose faster and is more accurate than the Gaussian tracker.",
"An assumption is that the Gaussian tracker is less robust to inaccurate initialization.",
"The authors of DBOT mentioned in their paper [19] that the particle tracker is slightly more robust, but the Gaussian tracker is more precise.",
"The Gaussian tracker can tolerate distortions in the input point cloud as well as occluded settings where the particle tracker is not able to track.",
"We compare our objects to similar objects in the YCB dataset in terms of size and form, as shown in Table REF .",
"A direct comparison of the results to prior work in pose detection [21], [20] is not completely fair, due to the different datasets used for evaluation and that we utilized simulated images.",
"However, it justifies the feasibility of our approach and its applicability in robot manipulation.",
"Table: Evaluation of 6D pose (ADD-S) on YCB-Video dataset.The mean AUC of DenseFusion for the four objects is 93.03%.",
"The mean AUC of the mesh_pose is 87.93% and for the particle tracker after 3s is 92.6%.",
"As already mentioned, the direct comparison is not totally fair, but as an interpretation that the object tracking pipeline is robust enough to be applicable in a robot manipulation setting."
],
[
"Accuracy of the Grasp Intent Prediction",
"We collected reach-and-grab motion trajectories from two users (1 male, 1 female).",
"Each trajectory consisted of around 2$\\mathtt {\\sim }$ 5 seconds and we collected a total of 350 trajectories for training in four different environment settings.",
"The users started with their right hand above the Leap Motion controller and reached forward to grab a target object in a specific direction (right or top) while looking at the display similar to Figures REF and REF .",
"The start and termination of the trajectories were defined by a key press.",
"Figure REF shows the average prediction accuracy over 18 grasp episodes of one environment using trajectories excluded during training.",
"The overall prediction accuracies are 79.4% and 77.4% for target object prediction and grab direction prediction.",
"The goal object prediction accuracy reached 100% before reaching 70% of the episode duration, and the average accuracy for predicting the direction reached up to 89% at termination.",
"The low accuracy during the first 20% of the episode resulted from the time gap between the start of recording and the start of the movement.",
"The reader may note that the prediction results are not optimal and optimization of the hyperparameters can be carried out for better results.",
"Utilization of recurrent neural networks may also help with improving the early prediction accuracy.",
"Figure: Prediction accuracy of the grasp intention prediction averaged over the duration of the episode"
],
[
"Teloperation Task Experiment Setup",
"We designed a teleoperation task for manipulation to test the efficacy of the system.",
"We hypothesize that the earlier the goal object is identified, the earlier the robot can start planning the motion, which would lead to faster task execution in a traded control setting.",
"We carried out a simulated user study by simulating different behaviors of users.",
"We collected trajectories from three different types of virtual users Normal user consists of trajectories collected from human users Noisy user by injecting a Gaussian noise to the Normal user at each time step Biased random offset over the Normal user trajectory to simulate imperfect perception during teleoperation, e.g., recognizing the object as closer than it actually is.",
"The difference between a Noisy user and a Biased user is that the Biased user has the same random noise over the trajectory whereas the noise in the Noisy user changes every time step.",
"The task is to perform a sequence of picking motions, to grab three objects from a table.",
"The user decides which object to grab and demonstrates the picking motion.",
"We assume that the poses of all objects are known by fusing the framework presented in Section , but the robot must infer in which order the objects are grabbed.",
"A goal object is identified when the robot predicts the same target for $t$ consecutive time steps ($t$ =80).",
"The prediction of the first $k$ time steps are neglected to reduce the prediction error in the beginning of the episode ($k$ =300)."
],
[
"Evaluation of Control Modes",
"We denote Early mode as the control mode in which the robot starts to plan its grasping trajectory towards the predicted object during user demonstration.",
"We compare this mode with Late mode, where the robot does not start motion planning until the user finishes the demonstration.",
"The system was evaluated according to the following criteria: time taken to predict goal object (time until execution), episode duration, prediction accuracy (goal prediction, direction prediction) when the robot identified the goal.",
"Table REF shows the results averaged over 12 episodes.",
"The time until execution is summed up over three object grasps and the episode duration indicates the total time for picking three objects including robot motion planning time.",
"Although it shows a compromise in the prediction accuracy, early motion planning and execution based on goal prediction resulted in shorter episode duration, as hypothesized.",
"It is shown that it was approx.",
"5 seconds faster than when the robot started motion planning once the user finished the trajectory.",
"The Noisy user and the Biased error took longer before the robot confidently identified the goal.",
"However, there was no penalty in the prediction accuracy except in the direction prediction for the Biased user in Early mode.",
"The prediction model was robust enough to tolerate the noisy settings.",
"Overall, the results show that the proposed traded control system can improve teleoperation performance while using noisy hand gestures to control the robot.",
"Table: Teleoperation results for different simulated users and control modes"
],
[
"Conclusions",
"We presented a teleoperation system that utilizes intuitive human grabbing hand gestures to perform sequential manipulation tasks.",
"To mitigate the issues that arise when using hand gestures, the robot autonomously generates a grasping or retrieving motion using trajectory optimization as soon as the robot identifies the user's intention.",
"For the object tracking pipeline, we proposed the combination of Mask R-CNN [10] and the model-based object tracker DBOT [19] for automatic initialization and object localization.",
"In addition, we trained a prediction model to identify the user intent from grabbing hand gestures during traded control so that the robot can start planning its trajectory in advance.",
"The simulated user study indicated that using intent prediction brought down the overall task execution time.",
"As the majority of our work is done in a simulated environment, limitations may arise during the application of the system in a real robot setting.",
"We will focus on the application of the system in a real robot setting for future work."
],
[
"Acknowledgment",
"This work is partially funded by the research alliance “System Mensch”.",
"The authors thank the International Max Planck Research School for Intelligent Systems (IMPRS-IS) for supporting Yoojin Oh."
]
] | 2107.01829 |
[
[
"Tits groups of Iwahori-Weyl groups and presentations of Hecke algebras"
],
[
"Abstract Let $G$ be a connected reductive group over a non-archimedean local field $F$ and $I$ be an Iwahori subgroup of $G(F)$.",
"Let $I_n$ is the $n$-th Moy-Prasad filtration subgroup of $I$.",
"The purpose of this paper is two-fold: to give some nice presentations of the Hecke algebra of connected, reductive groups with $I_n$-level structure; and to introduce the Tits group of the Iwahori-Weyl group of groups $G$ that split over an unramified extension of $F$.",
"The first main result of this paper is a presentation of the Hecke algebra $\\mathcal H(G(F),I_n)$, generalizing the previous work of Iwahori-Matsumoto on the affine Hecke algebras.",
"For split $GL_n$, Howe gave a refined presentation of the Hecke algebra $\\mathcal H(G(F),I_n)$.",
"To generalize such a refined presentation to other groups requires the existence of some nice lifting of the Iwahori-Weyl group $W$ to $G(F)$.",
"The study of a certain nice lifting of $W$ is the second main motivation of this paper, which we discuss below.",
"In 1966, Tits introduced a certain subgroup of $G(\\mathbf k)$, which is an extension of $W$ by an elementary abelian $2$-group.",
"This group is called the Tits group and provides a nice lifting of the elements in the finite Weyl group.",
"The \"Tits group\" $\\mathcal T$ for the Iwahori-Weyl group $W$ is a certain subgroup of $G(F)$, which is an extension of the Iwahori-Weyl group $W$ by an elementary abelian $2$-group.",
"The second main result of this paper is a construction of Tits group $\\mathcal T$ for $W$ when $G$ splits over an unramified extension of $F$.",
"As a consequence, we generalize Howe's presentation to such groups.",
"We also show that when $G$ is ramified over $F$, such a group $\\mathcal T$ of $W$ may not exist."
],
[
"Presentations of Hecke algebras",
"Let $G$ be a connected reductive group over a non-archimedean local field $F$ .",
"Let $I$ be an Iwahori subgroup of $G(F)$ and $W$ be the Iwahori-Weyl group of $G(F)$ .",
"Then $G(F)=\\sqcup _{w \\in W} I w I$ .",
"The group $W$ is a quasi-Coxeter group, namely, it is a semidirect product of an affine Weyl group $W_{\\text{af}}$ with a group $\\Omega $ of length-zero elements.",
"The Iwahori-Hecke algebra $\\mathcal {H} _0=\\mathcal {H} (G(F), I)$ is the $\\mathbb {Z} $ -algebra of the compactly supported, $I$ -biinvariant functions on $G(F)$ .",
"The Iwahori-Matsumoto presentation of $\\mathcal {H} _0$ reflects the quasi-Coxeter group structure of $W$ : the generators of $\\mathcal {H} _0$ are the characteristic functions $\\mathbb {1}_{I w I}$ , where $w$ runs over elements in $W$ , and the relations are given by multiplications of the characteristic functions via the condition on the length functions of $W$ .",
"See Theorem REF for the precise statement.",
"The representations of $G(F)$ which are generated by the Iwahori-fixed vectors gives to the representations of the Iwahori-Hecke algebra $\\mathcal {H} _0$ .",
"Let $n \\in \\mathbb {N} $ and $I_n$ be the $n$ -th congruence subgroup of $I$ .",
"Let $\\mathcal {H} _n=\\mathcal {H} (G(F), I_n)$ be the $\\mathbb {Z} $ -algebra of the compactly supported, $I_n$ -biinvariant functions on $G(F)$ .",
"It plays a role in the study of representations of $G(F)$ with deeper level structure.",
"One main purpose of this paper is to establish some nice presentations of $\\mathcal {H} _n$ .",
"The first main result is the generalization of the Iwahori-Matsumoto presentation to $\\mathcal {H} _n$ : the generators are the characteristic functions on the $I_n$ -double cosets on $G(F)$ and the multiplications of the characteristic functions are given via the conditions on the length function of $W$ .",
"We refer to Theorem REF for the precise statement.",
"As a consequence, we show that the algebra $\\mathcal {H} _n$ is finitely generated.",
"In [13], Howe discovered a nice presentation of $\\mathcal {H} _n$ when $G=GL_n(F)$ .",
"Here the generators are the characteristic functions $\\mathbb {1}_{g I_n}$ for $g \\in I/I_n$ and $\\mathbb {1}_{I_n m(w) I_n}$ , where $w$ runs over elements of $W$ of length 0 and 1, and $m(w)$ is a nice representative of $w$ in $G(F)$ .",
"This presentation is a refinement of the Iwahori-Matsumoto presentation and has some nice applications to the representation theory of $p$ -adic groups.",
"Howe's presentation was later generalized by the first-named author to split groups.",
"We observed that such refined presentation requires the existence of the nice lifting of the Iwahori-Weyl group $W$ to $G(F)$ .",
"Such a lifting, which we introduce in §, is motivated by Tits work on finite Weyl groups.",
"We call such a lifting the Tits group of the Iwahori-Weyl group $W$ and call the refined presentation of $\\mathcal {H} _n$ the Howe-Tits presentation.",
"In Theorem REF , we show that if the Tits group for $W$ exists, then the algebra $\\mathcal {H} _n$ admits the Howe-Tits presentation."
],
[
"Tits groups of the finite Weyl groups and Iwahori-Weyl groups",
"Now we come to the second main purpose of this paper: the study of the Tits groups.",
"We first make a short digression and discuss Tits groups of finite Weyl groups.",
"Let $G$ be a connected reductive group split over a field $\\mathfrak {F}$ and $W_0$ be its absolute Weyl group.",
"Tits in [23] introduced the Tits group $\\mathcal {T} $ of $W_0$ .",
"It is a subgroup of $G(\\mathfrak {F})$ , which is an extension of $W_0$ by $T_2$ , where $T_2$ is the elementary abelian subgroup generated by $\\alpha ^\\vee (-1)$ , where $\\alpha $ runs over all the roots in $G$ .",
"Moreover, for any $w \\in W_0$ , there exists a nice lifting $n_w \\in \\mathcal {T} $ .",
"These liftings have nice properties: $n_{s_\\alpha }^2=\\alpha ^\\vee (-1)$ for any simple root $\\alpha $ .",
"The set $\\lbrace n_s\\rbrace $ for simple reflections $s$ satisfies the Coxeter relations, i.e., for any simple reflections $s$ and $s^{\\prime }$ , we have $n_{s} n_{s^{\\prime }} \\cdots =n_{s^{\\prime }} n_s \\cdots ,$ where each side of the expression above has $k(s, s^{\\prime })$ factors.",
"Here $k(s, s^{\\prime })$ is the order of $s s^{\\prime }$ .",
"We refer to the recent work of Reeder, Levy, Yu and Gross [17], Adams and the second-named author [3] and Rostami [19] for some further study of the elements $n_w$ and its applications to supercuspidal representations of $p$ -adic groups.",
"Now let us come back to the group $G(F)$ .",
"Our second main result of this paper is the construction of a Tits group $\\mathcal {T} $ of the Iwahori-Weyl group of a connected, reductive group $G$ that is ${\\breve{F}}$ -split.",
"We establish in Theorem REF that Theorem 1.1 We have the short exact sequence $1 {@display}{\\longrightarrow }{\\rightarrow }S_2 {@display}{\\longrightarrow }{\\rightarrow }\\mathcal {T} {@display}{\\longrightarrow }{\\rightarrow }W {@display}{\\longrightarrow }{\\rightarrow }1.$ Moreover, for any $w \\in W$ , there exists a lifting $n_w \\in \\mathcal {T} $ such that For any affine simple reflection $s_a$ , $n_{s_a}^2=b^\\vee (-1)$ where $b$ is the gradient of $a$ .",
"We have $n_w=n_{s_{i_1}} n_{s_{i_2}} \\cdots n_{s_{i_n}} n_\\tau $ for any reduced expression $w=s_{i_1} s_{i_2} \\cdots s_{i_n} n_\\tau $ , where $\\tau \\in \\Omega $ and $s_{i_1}, \\ldots , s_{i_n}$ are simple reflections.",
"We refer to §REF for the definition of the elementary abelian 2-group $S_2$ .",
"As a consequence, we have the Howe-Tits presentation of $\\mathcal {H} _n$ for groups that are ${\\breve{F}}$ -split.",
"It is also worth pointing out that for ramified groups, such a $\\mathcal {T} $ may not exist.",
"We give an example in §REF ."
],
[
"The difficulty and strategy",
"In this subsection, we describe the strategy that goes into the construction of the Tits group of the Iwahori-Weyl group $W$ of $G$ over $F$ .",
"The Tits group of the finite absolute Weyl group is constructed via a “pinning” of $G(\\mathfrak {F})$ .",
"Roughly speaking, a pinning gives a collection of isomorphisms from additive group $_a$ to the simple root subgroups of $G$ .",
"Given a pinning, one may define the lifting of simple reflections $n_s$ and check that the conditions (1) & (2) in §REF are satisfied.",
"The Tits group of the finite Weyl group is generated by the $n_s$ where $s$ varies over the finite simple reflections.",
"When $G_F$ is not quasi-split, the group need not admit a “pinning\" analogous to the one discussed above, and hence there is no natural choice of representatives for the elements of the relative or affine Weyl group over $F$ .",
"We construct the Tits group of the Iwahori-Weyl group of $G$ over $F$ in two steps.",
"We first construct the Tits group of Iwahori-Weyl group over ${\\breve{F}}$ , where ${\\breve{F}}$ is the completion of the maximal unramified extension of $F$ contained in a fixed separable closure of $F$ .",
"Next, we “descend\" this construction down to $F$ .",
"The advantage of this approach is that the group $G_{\\breve{F}}$ is always quasi-split and admits a nice system of pinnings analogous to the one discussed in the preceding paragraph.",
"We now explain these two steps in more detail.",
"Let $G$ be a connected, reductive group over $F$ such that $G_{\\breve{F}}$ is ${\\breve{F}}$ -split and let $T$ be a maximal $F$ -torus in $G$ that is ${\\breve{F}}$ -split.",
"Let ${\\breve{{a}}}$ be a $\\sigma $ -stable alcove in the apartment $\\mathcal {A} (T, {\\breve{F}})$ and let $\\breve{\\mathbb {S}}$ be the set of affine simple reflections through the walls of ${\\breve{{a}}}$ .",
"To choose representatives of the elements of $\\breve{\\mathbb {S}}$ , we introduce an affine pinning; for each affine simple root ${\\breve{a}}$ with gradient ${\\breve{b}}$ , this is a homomorphism $x_{\\breve{a}}: _a \\rightarrow U_{\\breve{b}}$ such that the image of $m(s_{{\\breve{a}}}) = x_{\\breve{a}}(1) x_{-{\\breve{a}}}(1) x_{\\breve{a}}(1)$ in the affine Weyl group is $s_{\\breve{a}}$ .",
"We then show that this set of representatives satisfy Coxeter relations and furthermore, $m(s_{{\\breve{a}}})^2 = {\\breve{b}}^\\vee (-1)$ for each affine simple reflection ${\\breve{a}}$ .",
"We show that the group generated by $\\lbrace \\breve{\\lambda }(\\varpi _F)\\;|\\; \\breve{\\lambda }\\in X_*(T)\\rbrace $ , the $\\lbrace m({\\breve{s}})\\;|\\; {\\breve{s}}\\in \\breve{\\mathbb {S}}\\rbrace $ , and the group $\\breve{S}_2 = \\langle {\\breve{b}}^\\vee (-1)\\;|\\; {\\breve{b}}\\in \\Phi (G,T)\\rangle $ yields a Tits group of the Iwahori-Weyl group over ${\\breve{F}}$ .",
"We also include an example here of a wildly ramified unitary group over ${\\breve{F}}$ for which the Tits group of the Iwahori-Weyl group over ${\\breve{F}}$ does not exist.",
"This is done in §.",
"We now explain the descent step.",
"Let $\\sigma $ denote the Frobenius morphism on $G_{\\breve{F}}$ such that the $F$ -structure it yields is $G$ .",
"Let ${{{a}}}= {\\breve{{a}}}^\\sigma $ and let $\\mathbb {S} $ be the set of reflections through the walls of ${{{a}}}$ .",
"Then $\\mathbb {S} $ generates the Coxeter group $W_{\\text{af}}$ and $W = W_\\text{af}\\rtimes \\Omega _{{{{a}}}}$ where $\\Omega _{{{a}}}$ is the stabilizer of the alcove ${{{a}}}$ .",
"By the work of Lusztig [16] it is known that the elements of $\\mathbb {S} $ correspond to certain “nice\" $\\sigma $ -orbits in $\\breve{\\mathbb {S}}$ .",
"We construct an affine pinning over ${\\breve{F}}$ such that the set of representatives $\\lbrace m({\\breve{s}})\\;|\\; {\\breve{s}}\\in \\mathcal {X} \\rbrace $ obtained using this pinning is $\\sigma $ -stable for each of these nice $\\sigma $ -orbits $\\mathcal {X} $ .",
"This descent argument yields a set of representatives in $G(F)$ for the elements of $\\mathbb {S} $ that satisfy Coxeter relations.",
"This is done in §REF .",
"Let $\\breve{\\mathcal {T}}$ be as in (1), but with the representatives $\\lbrace m({\\breve{s}})\\;|\\; {\\breve{s}}\\in \\breve{\\mathbb {S}}\\rbrace $ as in the preceding paragraph.",
"Then $\\breve{\\mathcal {T}}$ is $\\sigma $ -stable.",
"We need to show that $\\breve{\\mathcal {T}}^{\\sigma } \\subset G(F)$ is a Tits group of $W$ over $F$ .",
"The most difficult part of the argument is to carry out the descent step for the elements of $\\Omega _{{\\breve{{a}}}}$ ; since $\\Omega _{{{a}}}= \\Omega _{{\\breve{{a}}}}^\\sigma $ , we need to show that for $\\breve{\\tau }\\in \\Omega _{\\breve{{a}}}^\\sigma $ , there is a representative $m({\\breve{\\tau }})$ of ${\\breve{\\tau }}$ in $\\breve{\\mathcal {T}}$ with $\\sigma (m({\\breve{\\tau }})) = m({\\breve{\\tau }})$ .",
"The results from our previous construction allow us to choose a representative $m({\\breve{\\tau }})$ of ${\\breve{\\tau }}$ in $\\breve{\\mathcal {T}}$ with the property that $\\sigma (m({\\breve{\\tau }})) = cm({\\breve{\\tau }})$ for a suitable $c \\in \\breve{S}_2$ , and a priori, we do not have any control over this element $c \\in \\breve{S}_2$ .",
"We carry out this step by constructing a Frobenius morphism $\\sigma $ associated to the $F$ -isomorphism class of $G$ , and then construct the Tits group $\\breve{\\mathcal {T}}\\subset G({\\breve{F}})$ of ${\\breve{W}}$ over ${\\breve{F}}$ so that $\\breve{\\mathcal {T}}^\\sigma \\subset G({\\breve{F}})^\\sigma $ is a Tits group of $W$ over $F$ .",
"This is done in §REF - §REF and some parts of the argument are based on a case-by-case analysis."
],
[
"Acknowledgments:",
"The authors would like to thank T. Haines, G. Lusztig and M.-F. Vigneras for useful discussions.",
"R.G.",
"would like to thank the Infosys foundation for their support through the Young Investigator award.",
"X.H.",
"is partially supported by a start-up grant and by funds connected with Choh-Ming Chair at CUHK, and by Hong Kong RGC grant 14300220."
],
[
"Notation",
"Let $F$ be a non-archimedean local field with $\\mathfrak {O} _F$ its ring of integers, $\\mathfrak {p} _F$ its maximal ideal, $\\varpi _F$ a uniformizer, and $\\mathbf {k}=\\mathbb {F} _q$ its residue field.",
"Let $p$ be the characteristic of $\\mathbf {k}$ .",
"Let $\\bar{F}$ be the completion of a separable closure of $F$ .",
"Let ${\\breve{F}}$ be the completion of the maximal unramified subextension with valuation ring $\\mathfrak {O} _{\\breve{F}}$ and residue field $\\bar{\\mathbf {k}}$ .",
"Note that $\\varpi _F$ is also a uniformizer of ${\\breve{F}}$ .",
"Let $\\Gamma = \\operatorname{Gal}(\\bar{F}/F)$ and $\\Gamma _0=\\operatorname{Gal}(\\bar{F}/{\\breve{F}})$ .",
"Let $G$ be a connected, reductive group over $F$ .",
"By Steinberg's Theorem (see [22]), $G_{\\breve{F}}$ is quasi-split.",
"Let $\\sigma $ denote the Frobenius action on $G({\\breve{F}})$ such that $G(F) = G({\\breve{F}})^\\sigma $ .",
"Let $A$ be a maximal $F$ -split torus of $G$ and $S$ be a maximal ${\\breve{F}}$ -split $F$ -torus of $G$ containing $A$ .",
"Let $T = Z_G(S)$ .",
"Then $T$ is defined over $F$ and is a maximal $F$ -torus of $G$ containing $S$ .",
"Let ${\\widetilde{F}}$ be the field of invariants of the kernel of the representation of $\\Gamma _0$ on $X^*(T)$ .",
"This extension is Galois over ${\\breve{F}}$ .",
"Hence $T$ and $G$ are split over ${\\widetilde{F}}$ .",
"By [21], there exists a uniformizer $\\varpi _{\\widetilde{F}}$ of ${\\widetilde{F}}$ with $\\operatorname{Nm}_{{\\widetilde{F}}/{\\breve{F}}}(\\varpi _{\\widetilde{F}}) = \\varpi _F$ , where $\\operatorname{Nm}_{{\\widetilde{F}}/{\\breve{F}}}$ is the norm map.",
"Fix one such.",
"Let $\\tilde{\\Phi }(G,T)$ be the set of roots of $T_{\\widetilde{F}}$ in $G_{\\widetilde{F}}$ .",
"Then the set of relative roots of $S$ in $G_{\\breve{F}}$ , denoted by $\\breve{\\Phi }(G,S)$ , is the set of the restrictions of the elements in $\\tilde{\\Phi }(G,T)$ to $S$ .",
"Let ${\\breve{W}}_0$ denote the relative Weyl group of $G$ with respect to $S$ and let $W(G,T)$ denote the absolute Weyl group of $G$ .",
"Let $\\mathcal {B} (G, {\\breve{F}})$ (resp.",
"$\\mathcal {B} (G,F)$ ) denote the enlarged Bruhat-Tits building of $G({\\breve{F}})$ (resp.",
"$G(F)$ ).",
"Then $\\mathcal {B} (G, {\\breve{F}})$ carries an action of $\\sigma $ and $\\mathcal {B} (G,F)= \\mathcal {B} (G, {\\breve{F}})^\\sigma $ .",
"Let $\\mathcal {A} (S, {\\breve{F}})$ be the apartment in $\\mathcal {B} (G, {\\breve{F}})$ corresponding to $S$ .",
"Let ${\\breve{{a}}}$ be a $\\sigma $ -stable alcove in $\\mathcal {A} (S,{\\breve{F}})$ .",
"Let ${\\breve{v}}_0$ be a special vertex contained in the closure of ${\\breve{{a}}}$ .",
"Set ${{{a}}}= {\\breve{{a}}}^\\sigma $ ; this is an alcove in the apartment $\\mathcal {A} (A, F)$ (see [8]).",
"Let $\\breve{\\Phi }_\\text{af}(G,S)$ denote the set of affine roots of $G({\\breve{F}})$ relative to $S$ .",
"Let $V = X_*(S) \\otimes _\\mathbb {Z} \\mathbb {R} $ .",
"The choice of ${\\breve{v}}_0$ also allows us to identify $\\mathcal {A} (S, {\\breve{F}})$ with $V$ via ${\\breve{v}}_0 {@display}{\\longmapsto }{\\mapsto }0 \\in V$ , which we now do.",
"We then view ${\\breve{{a}}}\\subset V$ .",
"Let $\\breve{\\Delta }\\subset \\breve{\\Phi }_\\text{af}(G,S)$ be the set of affine roots such that the corresponding vanishing hyperplanes form the walls of ${\\breve{{a}}}$ .",
"The Weyl chamber in $V$ that contains ${\\breve{{a}}}$ then yields a set of simple roots for $\\breve{\\Phi }(G,S)$ which we denote as $\\breve{\\Delta }_0$ .",
"Clearly $\\breve{\\Delta }_0 \\subset \\breve{\\Delta }$ ."
],
[
"Iwahori-Weyl group over $\\breve{F}$",
"Let ${\\breve{I}}$ be the Iwahori subgroup associated to ${\\breve{{a}}}$ .",
"Let $\\kappa _{T, {\\breve{F}}}: T({\\breve{F}}) \\rightarrow X_*(T)_{\\Gamma _0}$ denote the Kottwitz homomorphism.",
"The map $\\kappa _{T, {\\breve{F}}}$ is surjective and its kernel $T({\\breve{F}})_1$ is the unique parahoric subgroup of $T({\\breve{F}})$ .",
"By [14], we have the following commutative diagram $\\begin{tikzcd}T({\\widetilde{F}}) {r}{\\kappa _{T,{\\widetilde{F}}}} {d}{\\operatorname{Nm}_{{\\widetilde{F}}/{\\breve{F}}}}&X_*(T) {d}{pr}\\\\T({\\breve{F}}){r}{\\kappa _{T, {\\breve{F}}}} &X_*(T)_{\\Gamma _0}.\\end{tikzcd}$ Let ${\\breve{W}}= N_G(S)({\\breve{F}})/T({\\breve{F}})_1$ be the Iwahori-Weyl group of $G({\\breve{F}})$ with length function ${\\breve{l}}$ .",
"This group fits into an exact sequence $1 \\rightarrow X_*(T)_{\\Gamma _0} \\rightarrow \\tilde{W}\\rightarrow {\\breve{W}}_0 \\rightarrow 1.$ Recall that we have chosen a special vertex ${\\breve{v}}_0$ .",
"With this, we have a semi-direct product decomposition ${\\breve{W}}\\cong X_*(T)_{\\Gamma _0} \\rtimes {\\breve{W}}_0.$ Let $\\breve{\\mathbb {S}}= \\lbrace s_{\\breve{a}}\\;|\\; {\\breve{a}}\\in \\breve{\\Delta }\\rbrace $ be the set of simple reflections with respect to the walls of ${\\breve{{a}}}$ .",
"Let $\\breve{\\mathbb {S}}_0 =\\lbrace s_{\\breve{a}}\\;|\\; {\\breve{a}}\\in \\breve{\\Delta }_0\\rbrace $ .",
"Let ${\\breve{W}}_{\\text{af}} \\subset {\\breve{W}}$ be the Coxeter group generated by $\\breve{\\mathbb {S}}$ .",
"Let $T_{{\\mathrm {sc}}}, N_{{\\mathrm {sc}}}$ denote the inverse images of $T\\cap G_\\text{der}$ , resp.",
"$N_G(S)\\cap G_\\text{der}$ in $G_{\\mathrm {sc}}$ .",
"Let $S_{{\\mathrm {sc}}}$ denote the split component of $T_{{\\mathrm {sc}}}$ .",
"Then ${\\breve{W}}_{\\text{af}}$ may be identified with the Iwahori-Weyl group of $G_{{\\mathrm {sc}}}$ .",
"It fits into the exact sequence $1 \\rightarrow {\\breve{W}}_{\\text{af}} \\rightarrow {\\breve{W}}\\rightarrow X^*(Z(\\hat{G})^{\\Gamma _0}) \\rightarrow 1.$ Let $\\Omega _{{\\breve{{a}}}}$ be the stabilizer of ${\\breve{{a}}}$ in ${\\breve{W}}$ .",
"Then $\\Omega _{{\\breve{{a}}}}$ maps isomorphically to $X^*(Z(\\hat{G})^{\\Gamma _0})$ and we have a $\\sigma $ -equivariant semi-direct product decomposition ${\\breve{W}}\\cong {\\breve{W}}_{\\text{af}} \\rtimes \\Omega _{\\breve{{a}}}.$ Let ${\\breve{l}}$ be the length function on ${\\breve{W}}$ .",
"Then ${\\breve{l}}(s) = 1$ for all $s \\in \\breve{\\mathbb {S}}$ and $\\Omega _{\\breve{{a}}}$ is the set of elements of length 0 in ${\\breve{W}}$ ."
],
[
"Iwahori-Weyl group over $F$",
"Let $I$ be the Iwahori subgroup of $G(F)$ associated to ${{{a}}}$ .",
"Then $I = {\\breve{I}}^\\sigma $ .",
"Let $M = Z_G(A)$ and $M(F)_1$ be the unique parahoric subgroup of $M(F)$ .",
"We may identify $M(F)_1$ with the kernel of the Kottwitz homomorphism $M(F) \\rightarrow X^*(Z(\\hat{M})^{\\Gamma _0})^\\sigma $ .",
"Let $W= N_G(A)(F)/M(F)_1$ denote the Iwahori-Weyl group of $G(F)$ with length function $l$ .",
"By [18], we have a natural isomorphism $W \\cong \\breve{W} \\,^{\\sigma }$ .",
"It is proved in [18] that (a) for $w, w^{\\prime } \\in W$ , $\\breve{\\ell }(w w^{\\prime })=\\breve{\\ell }(w)+\\breve{\\ell }(w^{\\prime })$ if and only if $\\ell (w w^{\\prime })=\\ell (w)+\\ell (w^{\\prime })$ .",
"The semi-direct product decomposition of ${\\breve{W}}$ in (REF ) is $\\sigma $ -equivariant and yields a decomposition $W \\cong {\\breve{W}}_{\\text{af}}^\\sigma \\rtimes \\Omega _{{\\breve{{a}}}}^\\sigma .$ Let $W_{\\text{af}} = W_{\\text{af}}^\\sigma $ and let $\\mathbb {S} $ be the set of reflections through the walls of ${{{a}}}$ .",
"Then $(W_\\text{af}, \\mathbb {S})$ is a Coxeter system.",
"The group $\\Omega _{{{a}}}$ , which is the stabilizer of the alcove ${{{a}}}$ , is isomorphic to $\\Omega _{{\\breve{{a}}}}^\\sigma $ and is the set of length 0 elements is $W$ .",
"The simple reflections $\\mathbb {S} $ of $W_{\\text{af}}$ are certain elements in ${\\breve{W}}_{\\text{af}}$ .",
"The explicit description is as follows.",
"For any $\\sigma $ -orbit $\\mathcal {X} $ of $\\breve{\\mathbb {S}}$ , we denote by ${\\breve{W}}_{\\mathcal {X}}$ the parabolic subgroup of ${\\breve{W}}_{\\text{af}}$ generated by the simple reflections in $\\mathcal {X} $ .",
"If moreover, ${\\breve{W}}_{\\mathcal {X}}$ is finite, we denote by ${\\breve{w}}_{\\mathcal {X}}$ the longest element in ${\\breve{W}}_{\\mathcal {X}}$ .",
"It is proved by Lusztig [16] that there exists a natural bijection $s {@display}{\\longmapsto }{\\mapsto }\\mathcal {X} $ from $\\mathbb {S} $ to the set of $\\sigma $ -orbits of $\\breve{\\mathbb {S}}$ with ${\\breve{W}}_{\\mathcal {X}}$ finite such that the element $s \\in W_{\\text{af}} \\subset {\\breve{W}}_{\\text{af}}$ equals to ${\\breve{w}}_{\\mathcal {X}}$ ."
],
[
"Moy-Prasad filtration subgroups",
"Let ${\\breve{I}}$ be the Iwahori subgroup of $G({\\breve{F}})$ associated to the alcove ${\\breve{{a}}}$ .",
"Recall that we have chosen a special point ${\\breve{v}}_0$ in $\\mathcal {A} (S, {\\breve{F}})$ , using which we have identified $\\mathcal {A} (S, {\\breve{F}})$ with $V$ .",
"Let $(\\phi _{\\breve{a}})_{{\\breve{a}}\\in \\breve{\\Phi }(G,S)}$ be the corresponding valuation of root datum of $(T, (U_{\\breve{a}})_{{\\breve{a}}\\in \\breve{\\Phi }(G,S)})$ (see [7]).",
"For ${\\breve{v}}\\in {\\breve{{a}}}$ , ${\\breve{a}}\\in \\breve{\\Phi }(G,S)$ and $r \\in \\mathbb {R} $ , let $U_{{\\breve{a}}}({\\breve{F}})_{{\\breve{v}}, r}$ denote the filtration of the root subgroup $U_a$ (see [8]).",
"More precisely, $U_{{\\breve{a}}}({\\breve{F}})_{{\\breve{v}}, r} = \\lbrace u \\in U_{\\breve{a}}({\\breve{F}})\\;|\\; \\langle {\\breve{a}}, {\\breve{v}}\\rangle +\\phi _{\\breve{a}}(u) \\ge r\\rbrace .$ The subgroup $U_{{\\breve{a}}}({\\breve{F}})_{ {\\breve{v}}, 0}$ does not depend on the choice of ${\\breve{v}}\\in {\\breve{{a}}}$ and we may denote it as $ U_{{\\breve{a}}}({\\breve{F}})_{ {\\breve{{a}}}, 0}$ .",
"Note that ${\\breve{I}}$ is generated by $T({\\breve{F}})_1$ and $U_{{\\breve{a}}}({\\breve{F}})_{ {\\breve{{a}}}, 0}, {\\breve{a}}\\in \\breve{\\Phi }(G,S)$ .",
"Let ${\\breve{I}}_n$ be the $n$ -th Moy-Prasad filtration subgroup of ${\\breve{I}}$ .",
"In particular, for $n \\geqslant 1$ , $ {\\breve{I}}_n$ is a normal subgroup of ${\\breve{I}}$ .",
"Let $\\mathcal {T} ^{NR}$ denote the Neron-Raynaud model of $T$ , a group scheme of finite type over $\\mathfrak {O} _F$ with connected geometric fibers such that $\\mathcal {T} ^{NR}(\\mathfrak {O} _{\\breve{F}}) = T({\\breve{F}})_1$ .",
"Let $\\breve{T}_n = \\operatorname{Ker}(\\mathcal {T} ^{NR}(\\mathfrak {O} _{\\breve{F}}) \\rightarrow \\mathcal {T} ^{NR}(\\mathfrak {O} _{\\breve{F}}/\\mathfrak {p} _{\\breve{F}}^n))$ .",
"Then ${\\breve{I}}_n$ is generated by $\\breve{T}_n$ and $U_{{\\breve{a}}}({\\breve{F}})_{{\\breve{v}}_{\\breve{{a}}}, n}, {\\breve{a}}\\in \\breve{\\Phi }(G,S)$ , where ${\\breve{v}}_{\\breve{{a}}}$ is the barycenter of ${\\breve{{a}}}$ .",
"Let $I_n=\\breve{I}_n ^\\sigma $ .",
"Then for $n \\geqslant 1$ , $I_n$ is a normal subgroup of $I$ ."
],
[
"The subgroup ${\\breve{P}}_s$",
"Let $s \\in \\mathbb {S} $ .",
"Let $\\mathcal {X} $ be the $\\sigma $ -stable orbit in $\\breve{\\mathbb {S}}$ corresponding to $s$ (see §REF ).",
"Let $\\breve{P}_s=\\sqcup _{{\\breve{w}}\\in {\\breve{W}}_\\mathcal {X}} \\breve{I} \\dot{{\\breve{w}}}\\breve{I} \\supset \\breve{I}$ be the parahoric subgroup of $G({\\breve{F}})$ associated to $\\mathcal {X} $ .",
"This is the parahoric subgroup attached to ${\\breve{{a}}}_s = \\overline{{\\breve{{a}}}}^{\\breve{W}_\\mathcal {X}}$ , where $\\overline{{\\breve{{a}}}}$ is the closure of the alcove ${\\breve{{a}}}$ .",
"Then ${\\breve{P}}_s$ is generated by $T({\\breve{F}})_1$ and $U_{{\\breve{a}}}({\\breve{F}})_{{\\breve{{a}}}_s, 0},\\; {\\breve{a}}\\in \\breve{\\Phi }(G,S)$ .",
"Let ${\\breve{P}}_{s,n}$ be the $n$ -th Moy-Prasad filtration subgroup of ${\\breve{P}}_s$ .",
"It is generated by ${\\breve{T}}_n$ and $U_{{\\breve{a}}}({\\breve{F}})_{{\\breve{v}}_s, n}, {\\breve{a}}\\in \\breve{\\Phi }(G,S)$ where ${\\breve{v}}_{s}$ is the barycenter of ${\\breve{{a}}}_s$ ."
],
[
"The Hecke algebra $\\mathcal {H} (G(F), I_n)$",
"Let $\\mathcal {H} _n = \\mathcal {H} (G(F), I_n)$ be the Hecke algebra of compactly supported, $I_n$ -biinvariant $\\mathbb {Z} $ -valued functions on $G(F)$ .",
"Note that $I_0=I$ .",
"The algebra $\\mathcal {H} _0$ is the Iwahori-Hecke algebra."
],
[
"Tits group associated to an absolute Weyl groups",
"In this subsection, we assume that $\\mathfrak {F}$ is any field and $G$ is a reductive group split over $\\mathfrak {F}$ .",
"Let $T$ denote a maximal $F$ -split torus in $G$ .",
"We follow [23].",
"For any root $a$ , we denote by $a^\\vee $ the corresponding coroot.",
"Let $S_2$ be the elementary abelian two-group generated by $\\lbrace a^\\vee (-1)\\rbrace $ for all roots $a$ .",
"Associated to any pinning of $G$ , we have the Tits group $\\mathcal {T} _{\\text{fin}}$ .",
"This is a subgroup of $N_G(T)$ , generated by $\\lbrace n_s\\rbrace $ , where $s$ runs over the simple reflections in the absolute Weyl group $W(G, T)$ and $n_s$ is a certain lift of $s$ to $N_G(T)$ .",
"Below are some properties on the Tits group $\\mathcal {T} _{\\text{fin}}$ : $n_{s_a}^2=a^\\vee (-1)$ for any simple root $a$ .",
"The set $\\lbrace n_s\\rbrace $ for simple reflections $s$ satisfies the Coxeter relations, i.e., for any simple reflections $s$ and $s^{\\prime }$ , we have $n_{s} n_{s^{\\prime }} \\cdots =n_{s^{\\prime }} n_s \\cdots ,$ where each side of the expression above has $k(s, s^{\\prime })$ factors.",
"Here $k(s, s^{\\prime })$ is the order of $s s^{\\prime }$ in $W(G, T)$ .",
"The map $n_s {@display}{\\longmapsto }{\\mapsto }s$ induces a short exact sequence $1 {@display}{\\longrightarrow }{\\rightarrow }T_2 {@display}{\\longrightarrow }{\\rightarrow }\\mathcal {T} _{\\text{fin}} {@display}{\\longrightarrow }{\\rightarrow }W(G, T) {@display}{\\longrightarrow }{\\rightarrow }1.$ For any $w \\in W(G, T)$ , we may define $n_w=n_{s_1} \\cdots n_{s_k} \\in \\mathcal {T} _{\\text{fin}}$ , where $s_1 \\cdots s_k$ is a reduced expression of $w$ .",
"As a consequence of (2), the definition of $n_w$ is independent of the choice of the reduced expression of $w$ .",
"We call the liftings $\\lbrace n_w\\rbrace _{w \\in W(G, T)}$ a Tits cross-section of $W(G, T)$ in $\\mathcal {T} _{\\text{fin}}$ ."
],
[
"A Tits group of Iwahori-Weyl group over $\\breve{F}$",
"Motivated by the construction of the Tits group of the absolute Weyl group, we introduce the Tits groups of Iwahori-Weyl groups.",
"For each ${\\breve{b}}$ in the relative root system $\\breve{\\Phi }(G,S)$ , we set ${\\breve{b}}_* ={\\left\\lbrace \\begin{array}{ll} {\\breve{b}}, & \\text{ if } {\\breve{b}}\\text{ is reduced}; \\\\ {\\breve{b}}/2, & \\text{ otherwise.}",
"\\end{array}\\right.",
"}$ Note that any element ${\\breve{w}}\\in {\\breve{W}}$ can be written as ${\\breve{w}}={\\breve{s}}_{i_1} \\cdots {\\breve{s}}_{i_n} \\breve{\\tau }$ , where ${\\breve{s}}_{i_1}, \\cdots , {\\breve{s}}_{i_n} \\in \\breve{\\mathbb {S}}$ and $\\breve{\\tau }\\in \\Omega _{{\\breve{{a}}}}$ .",
"If $n=\\breve{\\ell }({\\breve{w}})$ , then we say that ${\\breve{w}}={\\breve{s}}_{i_1} \\cdots {\\breve{s}}_{i_n} \\breve{\\tau }$ is a reduced expression of ${\\breve{w}}$ in ${\\breve{W}}$ .",
"Definition 3.1 Let $\\breve{S}_2$ be the elementary abelian two-group generated by $\\breve{b}^\\vee (-1)$ for $\\breve{b} \\in \\breve{\\Phi }(G, S)$ .",
"A Tits group of ${\\breve{W}}$ is a subgroup $\\breve{\\mathcal {T}}$ of $N_G(S)(\\breve{F})$ such that The natural projection $\\breve{\\phi }: N_G(S)(\\breve{F}) {@display}{\\longrightarrow }{\\rightarrow }{\\breve{W}}$ induces a short exact sequence $1 {@display}{\\longrightarrow }{\\rightarrow }\\breve{S}_2 {@display}{\\longrightarrow }{\\rightarrow }\\breve{\\mathcal {T}}\\xrightarrow{} {\\breve{W}}{@display}{\\longrightarrow }{\\rightarrow }1.$ There exists a Tits cross-section $\\lbrace m({\\breve{w}})\\rbrace _{{\\breve{w}}\\in {\\breve{W}}}$ of ${\\breve{W}}$ in $\\breve{\\mathcal {T}}$ such that for ${\\breve{a}}\\in \\breve{\\Delta }$ , $m({\\breve{s}}_{{\\breve{a}}})^2 = {\\breve{b}}_*^\\vee (-1)$ , where ${\\breve{b}}$ is the gradient of ${\\breve{a}}$ .",
"for any reduced expression ${\\breve{w}}={\\breve{s}}_{i_1} \\cdots {\\breve{s}}_{i_n} \\breve{\\tau }$ in ${\\breve{W}}$ , we have $m({\\breve{w}})=m({\\breve{s}}_{i_1}) \\cdots m({\\breve{s}}_{i_n}) m(\\breve{\\tau })$ .",
"It is easy to see that the condition (2) (b) in Definition REF is equivalent to Condition (2)(b)$^{\\dagger }$ : $m({\\breve{w}}{\\breve{w}}^{\\prime })=m({\\breve{w}}) m({\\breve{w}}^{\\prime })$ for any ${\\breve{w}}\\in {\\breve{W}}_{\\text{af}}$ and ${\\breve{w}}^{\\prime } \\in {\\breve{W}}$ with $\\breve{\\ell }({\\breve{w}}{\\breve{w}}^{\\prime })=\\breve{\\ell }({\\breve{w}})+\\breve{\\ell }({\\breve{w}}^{\\prime })$ .",
"Suppose that a Tits group $\\breve{\\mathcal {T}}$ of ${\\breve{W}}$ exists and $\\breve{\\phi }: \\breve{\\mathcal {T}}{@display}{\\longrightarrow }{\\rightarrow }{\\breve{W}}$ is the projection map.",
"Let $\\breve{\\mathcal {T}}_{\\text{af}}=\\breve{\\phi }^{-1}(\\breve{W}_{\\text{af}})$ .",
"This is the subgroup of $\\breve{\\mathcal {T}}$ generated by $\\breve{S}_2$ and $m({\\breve{w}})$ for $w \\in {\\breve{W}}_{\\text{af}}$ .",
"We have the following commutative diagram $\\begin{tikzcd}1 [r] & \\breve{S}_2 [d, equal] [r] & \\breve{\\mathcal {T}}_{\\text{af}} [d, hook] [r,\"\\breve{\\phi }\"] & {\\breve{W}}_{\\text{af}} [d, hook] [r] & 1 \\\\1 [r] & \\breve{S}_2 [r] & \\breve{\\mathcal {T}}[r,\"\\breve{\\phi }\"] & {\\breve{W}}[r] & 1.\\end{tikzcd}$ For ${\\breve{\\tau }}\\in \\Omega _{\\breve{{a}}}$ , any lifting $m^{\\prime }({\\breve{\\tau }})$ of ${\\breve{\\tau }}$ in $G({\\breve{F}})$ lies in the normalizer of ${\\breve{I}}$ , where ${\\breve{I}}$ is the Iwahori subgroup attached to the alcove ${\\breve{{a}}}$ .",
"This a special case of the fact that for $g \\in G({\\breve{F}})$ and $x \\in \\mathcal {A} (S, {\\breve{F}})$ , $g {\\breve{P}}_x g^{-1}= {\\breve{P}}_{g \\cdot x}$ , where ${\\breve{P}}_x$ is the parahoric subgroup attached to $x$ .",
"Let us add some comments on $\\breve{b}_*$ .",
"In this paper, we will construct the Tits group for connected reductive groups split over $\\breve{F}$ .",
"For these groups, there is no difference between $\\breve{b}_*$ and $\\breve{b}$ .",
"We expect that Tits groups (in the Definition REF ) exist for tamely ramified groups.",
"Then one needs to use $\\breve{b}_*$ instead of $\\breve{b}$ in condition (2) (a), e.g., for the tamely ramified unitary groups."
],
[
"Tits groups over $F$",
"The Tits group of $W$ is defined as follows.",
"Note that any element $w \\in W$ can be written as $w=s_{i_1} \\cdots s_{i_n} \\tau $ , where $s_{i_1}, \\cdots , s_{i_n} \\in \\mathbb {S} $ and $\\tau \\in \\Omega _{{{{a}}}}$ .",
"If $n=\\ell (w)$ , then we say that $w=s_{i_1} \\cdots s_{i_n} \\tau $ is a reduced expression of $w$ in $W$ .",
"Definition 3.2 Let $S_2=\\breve{S}_2^\\sigma $ .",
"A Tits group of $W$ is a subgroup $\\mathcal {T} $ of $N_G(A)(F)$ such that The natural projection $\\phi : N_G(A)(F) {@display}{\\longrightarrow }{\\rightarrow }W$ induces a short exact sequence $1 {@display}{\\longrightarrow }{\\rightarrow }S_2 {@display}{\\longrightarrow }{\\rightarrow }\\mathcal {T} \\xrightarrow{} W {@display}{\\longrightarrow }{\\rightarrow }1.$ There exists a Tits cross-section $\\lbrace m(w)\\rbrace _{w \\in W}$ of $W$ in $\\mathcal {T} $ such that for $a \\in \\Delta $ , $m(s_{a})^2 = b^\\vee (-1)$ , where $b$ is the gradient of $a$ .",
"for any reduced expression $w=s_{i_1} \\cdots s_{i_n} \\tau $ in $W$ , we have $m(w)=m(s_{i_1}) \\cdots m(s_{i_n}) m(\\tau )$ .",
"Note that in general, $S_2$ is larger than than the subgroup generated by $b^\\vee (-1)$ for $b \\in \\Phi (G, A)$ .",
"Suppose that a Tits group $\\mathcal {T} $ of $W$ exists and $\\phi : \\mathcal {T} {@display}{\\longrightarrow }{\\rightarrow }W$ is the projection map.",
"Let $\\mathcal {T} _{\\text{af}}=\\phi ^{-1}(W_{\\text{af}})$ .",
"This is the subgroup of $\\mathcal {T} $ generated by $S_2$ and $m(w)$ for $w \\in W$ .",
"We have the following commutative diagram $\\begin{tikzcd}1 [r] & \\breve{S}_2^\\sigma [d, equal] [r] & \\mathcal {T} _{\\text{af}} [d, hook] [r,\"\\phi \"] & W_{\\text{af}} [d, hook] [r] & 1 \\\\1 [r] & \\breve{S}_2^\\sigma [r] & \\mathcal {T} [r,\"\\phi \"] & W [r] & 1.\\end{tikzcd}$ We would like to point out that unlike Tits groups of absolute Weyl groups for split reductive groups, the Tits groups of ${\\breve{W}}$ and $W$ may not exist in general.",
"See §REF ."
],
[
"Two presentations of the Hecke algebra $\\mathcal {H} (G(F), I_n)$",
"We first recall the Iwahori-Matsumoto presentation of the Iwahori-Hecke algebras: Theorem 4.1 For $w \\in W$ , let $\\dot{w}$ be any representative of $w$ in $G({\\breve{F}})$ .",
"The Hecke algebra $\\mathcal {H} _0$ is a free module with basis $\\lbrace \\mathbb {1}_{I \\dot{w} I}\\rbrace _{w \\in W}$ and the multiplication is given by the following formulas: (1) $\\mathbb {1}_{I \\dot{w} I} \\mathbb {1}_{I \\dot{w}^{\\prime } I}=\\mathbb {1}_{I \\dot{w} \\dot{w}^{\\prime } I}$ if $\\ell (w w^{\\prime })=\\ell (w)+\\ell (w^{\\prime })$ .",
"(2) $\\mathbb {1}_{I \\dot{s} I} \\mathbb {1}_{I \\dot{w} I}=(q^{\\breve{\\ell }(s)}-1) \\mathbb {1}_{I \\dot{w} I}+q^{\\breve{\\ell }(s)} \\mathbb {1}_{I \\dot{s} \\dot{w} I}$ for $s \\in \\mathbb {S} $ and $w \\in W$ with $s w<w$ .",
"The first main result of this section is the following similar presentation for $\\mathcal {H} _n$ for $n \\geqslant 1$ .",
"We call it the Iwahori-Matsumoto presentation for $\\mathcal {H} _n$ .",
"Theorem 4.2 Let $n \\geqslant 1$ .",
"The algebra $\\mathcal {H} _n$ is generated by $\\mathbb {1}_{I_n g I_n}$ for $g \\in G(F)$ subject to the following relations: (0) If $g$ and $g^{\\prime }$ are in the same $I_n \\times I_n$ -coset of $G(F)$ , then $\\mathbb {1}_{I_n g I_n}=\\mathbb {1}_{I_n g^{\\prime } I_n}$ .",
"(1) If $\\ell (\\pi (g g^{\\prime }))=\\ell (\\pi (g))+\\ell (\\pi (g^{\\prime }))$ , then $\\mathbb {1}_{I_n g I_n} *\\mathbb {1}_{I_n g^{\\prime } I_n}=\\mathbb {1}_{I_n g g^{\\prime } I_n}.$ (2) If $\\pi (g)=s \\in \\mathbb {S} $ , then $\\mathbb {1}_{I_n g I_n} *\\mathbb {1}_{I_n g^{\\prime } I_n}={\\left\\lbrace \\begin{array}{ll} q^{\\breve{\\ell }(s)} \\mathbb {1}_{I_n g g^{\\prime } I_n}, & \\text{ if } \\pi (g g^{\\prime })=\\pi (g^{\\prime }), \\\\ q^{\\breve{\\ell }(s)} \\sum _{u \\in P_{s, n}/I_n} \\mathbb {1}_{I_n u g g^{\\prime } I_n}, & \\text{ if } \\pi (g g^{\\prime })<\\pi (g^{\\prime }).\\end{array}\\right.",
"}$ In the above, $P_{s,n} = {\\breve{P}}_{s, n}^\\sigma $ with ${\\breve{P}}_{s,n}$ as in §REF .",
"Note that if $s w<w$ , then $(I \\dot{s} I) (I \\dot{w} I)=I \\dot{w} I \\sqcup I \\dot{s} \\dot{w} I$ .",
"For any $g \\in I \\dot{s} I$ and $g^{\\prime } \\in I \\dot{w} I$ , there are two possibilities: either $(I_n g I_n) (I_n g^{\\prime } I_n) \\subset I \\dot{w} I$ or $(I_n g I_n) (I_n g^{\\prime } I_n) \\subset I \\dot{s} \\dot{w} I$ .",
"Thus there are two cases for the multiplication $\\mathbb {1}_{I_n g I_n} * \\mathbb {1}_{I_n g^{\\prime } I_n}$ ."
],
[
"Collection of some results from {{cite:fea5c6a5bf7222fd3e121cccace2c92350e77394}}",
"We define the map $\\pi : G(F) {@display}{\\longrightarrow }{\\rightarrow }W, \\quad g {@display}{\\longmapsto }{\\mapsto }w \\text{ for } g \\in I \\dot{w} I.$ It is proved in [12] that for any $g \\in G(F)$ , $(\\breve{I}_n g \\breve{I}_n/\\breve{I}_n)^\\sigma =I_n g I_n/I_n$ and $\\sharp I_n g I_n/I_n=q^{\\breve{\\ell }(\\pi (g))}$ .",
"Moreover, it is proved in [12] that for $g, g^{\\prime } \\in G(F)$ with $\\ell (\\pi (g g^{\\prime }))=\\ell (\\pi (g))+\\ell (\\pi (g^{\\prime }))$ , the multiplication map in $G(F)$ induces a bijection $I_n g I_n \\times _{I_n} I_n g^{\\prime } I_n \\cong I_n g g^{\\prime } I_n.$ Here $I_n g I_n \\times _{I_n} I_n g^{\\prime } I_n$ be the quotient of $I_n g I_n \\times I_n g^{\\prime } I_n$ by the action of $I_n$ defined by $a \\cdot (z, z^{\\prime })=(z a ^{-1}, a z^{\\prime })$ .",
"In this case, $\\mathbb {1}_{I_n g I_n} * \\mathbb {1}_{I_n g^{\\prime } I_n}=\\mathbb {1}_{I_n g g^{\\prime } I_n}.$ Corollary 4.3 Let $n \\geqslant 1$ .",
"The algebra $\\mathcal {H} _n$ is generated by $\\mathbb {1}_{I_n g I_n}$ for $g \\in I$ and $\\mathbb {1}_{I_n \\dot{w} I_n}$ for $w \\in W$ with $\\ell (w) \\leqslant 1$ .",
"In particular, $\\mathcal {H} _n$ is finitely generated.",
"Note that $\\mathcal {H} _n$ is spanned by $\\mathbb {1}_{I_n g I_n}$ for $g \\in G(F)$ .",
"Suppose that $g \\in I \\dot{w} I$ and $w=s_{i_1} \\cdots s_{i_n} \\tau $ for $s_{i_1}, \\ldots , s_{i_n} \\in \\mathbb {S} $ and $\\tau \\in W$ with $\\ell (\\tau )=0$ .",
"Then $g=g_1 \\cdots g_n g^{\\prime }$ , with $g_j \\in I \\dot{s}_{i_j} I$ for $1 \\leqslant j \\leqslant n$ and $g^{\\prime } \\in I \\dot{\\tau }I$ .",
"Then $\\mathbb {1}_{I_n g I_n}=\\mathbb {1}_{I_n g_1 I_n} *\\cdots *\\mathbb {1}_{I_n g_n I_n}* \\mathbb {1}_{I_n g^{\\prime } I_n}$ .",
"For any $s \\in \\mathbb {S} $ and $g \\in I \\dot{s} I$ , we have $g=i_1 \\dot{s} i_2$ for some $i_1, i_2 \\in I$ .",
"Then $\\mathbb {1}_{I_n g I_n}=\\mathbb {1}_{I_n i_1 I_n} *\\mathbb {1}_{I_n \\dot{s} I_n} *\\mathbb {1}_{I_n i_2 I_n}$ .",
"For any $\\tau \\in W$ with $\\ell (\\tau )=0$ and $g \\in I \\dot{\\tau }I$ , we have $g=i \\tau $ for some $i \\in I$ .",
"Then $\\mathbb {1}_{I_n g_1 I_n}=\\mathbb {1}_{I_n i I_n} *\\mathbb {1}_{I_n \\dot{\\tau }I_n}$ .",
"Note that $\\Omega _{{{a}}}$ is finitely generated.",
"Let $\\lbrace \\tau _1, \\cdots , \\tau _l\\rbrace $ be a generating set of $\\Omega _{{{a}}}$ .",
"Then $\\mathcal {H} _n$ is generated by $\\mathbb {1}_{I_n g I_n}$ for $g \\in I$ , $\\mathbb {1}_{I_n \\dot{s} I_n}$ for $s \\in \\mathbb {S} $ and $\\mathbb {1}_{I_n \\dot{\\tau }_i I_n}$ for $1 \\leqslant i \\leqslant l$ .",
"Thus $\\mathcal {H} _n$ is finitely generated."
],
[
"The subgroup ${\\breve{P}}_{s,n}$",
"Let ${\\breve{P}}_{s,n}$ be as in §REF .",
"Lemma 4.4 For $n \\geqslant 1$ , $\\breve{P}_{s, n}$ is a normal subgroup of $\\breve{I}$ .",
"Note that $\\breve{P}_{s, n}$ is a normal subgroup of the parahoric subgroup $\\breve{P}_{s}$ .",
"Since $\\breve{I} \\subset \\breve{P}_{s}$ , we have that $\\breve{P}_{s, n}$ is stable under the conjugation action of $\\breve{I}$ .",
"It remains to show that $\\breve{P}_{s, n} \\subset \\breve{I}$ .",
"Recall that ${\\breve{v}}_s$ is the barycenter of the facet ${\\breve{{a}}}_s$ in the closure of the base alcove ${\\breve{{a}}}$ .",
"Let $\\breve{\\Phi }^+(G,S)$ be the set of positive roots in $\\breve{\\Phi }(G,S)$ .",
"Then, using [8], it follows that $0 \\leqslant \\langle {\\breve{a}}, {\\breve{v}}_s\\rangle \\leqslant 1$ for any $a \\in \\breve{\\Phi }^+(G,S)$ .",
"By definition, $\\breve{P}_{s, n}$ is generated by $T_n$ and $U_{\\breve{a}}({\\breve{F}})_{{\\breve{v}}_s, n}$ .",
"Let $u \\in U_{\\breve{a}}({\\breve{F}})_{{\\breve{v}}_s, n}$ .",
"If ${\\breve{a}}\\in \\Phi ^+(G,S)$ , then the condition $\\langle {\\breve{a}}, {\\breve{v}}_s\\rangle + \\phi _{\\breve{a}}(u) \\ge n$ implies that $\\phi _{\\breve{a}}(u) \\geqslant n-1 \\geqslant 0$ .",
"If $-a \\in \\Phi ^+(G,S)$ , then the condition $\\langle {\\breve{a}}, {\\breve{v}}_s\\rangle +\\phi _{\\breve{a}}(u) \\geqslant n$ implies that $ \\phi _{\\breve{a}}(u) \\geqslant n \\geqslant 1$ .",
"In both cases, $ U_{\\breve{a}}({\\breve{F}})_{{\\breve{v}}_s, n}\\subset {\\breve{I}}$ .",
"Therefore $\\breve{P}_{s, n} \\subset {\\breve{I}}$ .",
"Lemma 4.5 Let $g \\in G$ with $\\pi (g)=s \\in \\mathbb {S} $ .",
"Set $P_{s, n}=\\breve{P}_{s, n}^\\sigma $ .",
"Then for $n \\geqslant 1$ , $g I_n g ^{-1}I_n=I_n g I_n g ^{-1}=P_{s, n}$ and it is a normal subgroup of $I$ .",
"For $n \\geqslant 1$ , $\\breve{I}_n$ is stable under the conjugation action of $\\dot{s} \\breve{I}_n \\dot{s} ^{-1}$ since $\\dot{s} \\breve{I}_n \\dot{s} ^{-1}\\subset {\\breve{I}}$ .",
"Therefore $\\breve{I}_n (\\dot{s} \\breve{I}_n \\dot{s} ^{-1})=(\\dot{s} \\breve{I}_n \\dot{s} ^{-1}) \\breve{I}_n$ .",
"On the other hand, $T_n \\subset \\breve{I}_n$ .",
"Let ${\\breve{a}}\\in \\breve{\\Phi }(G,S)$ and let $u \\in U_{\\breve{a}}({\\breve{F}})_{{\\breve{v}}_s,n}$ .",
"So $\\langle {\\breve{a}}, {\\breve{v}}_s\\rangle + \\phi _{\\breve{a}}(u) \\ge n$ .",
"We claim that $ u \\in {\\breve{I}}_n \\cup \\dot{{\\breve{s}}}{\\breve{I}}_n \\dot{{\\breve{s}}}^{-1}$ .",
"Suppose $ u \\notin {\\breve{I}}_n$ .",
"Then ${\\breve{a}}+ \\phi _{\\breve{a}}(u)-n$ is a negative affine root, so $\\langle {\\breve{a}}, {\\breve{v}}_s\\rangle + \\phi _{\\breve{a}}(u) -n \\le 0$ .",
"Hence $\\langle {\\breve{a}}, {\\breve{v}}_s\\rangle + \\phi _{\\breve{a}}(u)- n=0$ .",
"Then ${\\breve{a}}+\\phi _{\\breve{a}}(u) - n$ belongs to $\\breve{\\Phi }_{{\\breve{v}}_s}$ , the set of affine roots that vanish at ${\\breve{v}}_s$ .",
"Write $\\breve{\\Phi }_{{\\breve{v}}_s} = (\\Phi _{{\\breve{v}}_s} \\cap \\Phi ^+_{\\text{af}}(G,S)) \\sqcup (\\Phi _{{\\breve{v}}_s} \\cap \\Phi _{\\text{af}}^-(G,S))$ .",
"Then, since ${\\breve{W}}_\\mathcal {X} $ is the Weyl group of $\\Phi _{{\\breve{v}}_s}$ and $s$ is the longest element of ${\\breve{W}}_{\\mathcal {X}}$ , it follows that $s({\\breve{a}}+\\phi _{\\breve{a}}(u)-n)=s({\\breve{a}})+\\phi _{\\breve{a}}(u)-n$ is a positive affine root.",
"Then $u \\in \\dot{s} \\breve{I}_n \\dot{s} ^{-1}$ .",
"Therefore $\\breve{P}_{s, n}=\\breve{I}_n (\\dot{s} \\breve{I}_n \\dot{s} ^{-1})=(\\dot{s} \\breve{I}_n \\dot{s} ^{-1}) \\breve{I}_n.$ We have $g \\breve{I}_n g ^{-1}=(i \\dot{s}) \\breve{I}_n (i s) ^{-1}=i (\\dot{s} \\breve{I}_n \\dot{s} ^{-1}) i ^{-1}$ for some $i \\in \\breve{\\mathcal {I}}$ .",
"Thus $(g \\breve{I}_n g ^{-1}) \\breve{I}_n =i (\\dot{s} \\breve{I}_n \\dot{s} ^{-1}) i ^{-1}\\breve{I}_n =i (\\dot{s} \\breve{I}_n \\dot{s} ^{-1})\\breve{I}_n i ^{-1}=i \\breve{P}_{s, n} i ^{-1}=\\breve{P}_{s, n}.$ Thus $(\\breve{P}_{s, n}/\\breve{I}_n)^\\sigma =(g \\breve{I}_n g ^{-1}\\breve{I}_n/\\breve{I}_n)^\\sigma =g (\\breve{I}_n g ^{-1}\\breve{I}_n/\\breve{I}_n)^\\sigma =g (I_n g ^{-1}I_n/I_n).$ Here the last equality follows from §REF .",
"Thus $g I_n g ^{-1}I_n=\\breve{P}_{s, n}^\\sigma =P_{s, n}$ .",
"This is a normal subgroup of $I$ since $\\breve{P}_{s, n}$ is a normal subgroup of $\\breve{I}$ .",
"As $P_{s, n} \\subset I$ and $I_n$ is a normal subgroup of $I$ , we also have $(g I_n g ^{-1}) I_n=I_n (g I_n g ^{-1})$ .",
"Proposition 4.6 Let $n \\geqslant 1$ .",
"Let $g, g^{\\prime } \\in G$ with $\\pi (g)=\\pi (g^{\\prime })=s \\in \\mathbb {S} $ .",
"The multiplication map on $G$ induces a surjective map $I_n g I_n \\times _{I_n} I_n g^{\\prime } I_n {@display}{\\longrightarrow }{\\rightarrow }P_{s, n} g g^{\\prime } I_n.$ Moreover, each fiber contains exactly $q^{\\breve{\\ell }(s)}$ elements.",
"By Lemma REF , $I_n g I_n g^{\\prime } I_n=(I_n g I_n g ^{-1}) g g^{\\prime } I_n=P_{s, n} g g^{\\prime } I_n$ .",
"If $\\pi (g g^{\\prime })=s$ , then $P_{s, n}=I_n (g g^{\\prime }) I_n (g g^{\\prime }) ^{-1}$ and $P_{s, n} g g^{\\prime } I_n=I_n (g g^{\\prime }) I_n (g g^{\\prime }) ^{-1}(g g^{\\prime }) I_n=I_n g g^{\\prime } I_n.$ By §REF , $\\sharp I_n g g^{\\prime } I_n/I_n=q^{\\breve{\\ell }(s)}$ .",
"Since the map $I_n g I_n \\times _{I_n} I_n g^{\\prime } I_n/I_n {@display}{\\longrightarrow }{\\rightarrow }P_{s, n} g g^{\\prime } I_n/I_n$ is equivariant under the left action of $I_n$ and $I_n$ acts transitively on $P_{s, n} g g^{\\prime } I_n/I_n$ , all the fibers have the same cardinality and the cardinality equals to $\\sharp (I_n g I_n \\times _{I_n} I_n g^{\\prime } I_n/I_n)/\\sharp (P_{s, n} g g^{\\prime } I_n/I_n)=q^{2\\breve{\\ell }(s)}/q^{\\breve{\\ell }(s)}=q^{\\breve{\\ell }(s)}.$ If $\\pi (g g^{\\prime })=1$ , then $g^{\\prime }=g ^{-1}i$ for some $i \\in I$ and we have the following commutative diagram ${I_n g I_n \\times _{I_n} I_n g^{\\prime } I_n [r] [d] & P_{s, n} g g^{\\prime } I_n [d] \\\\I_n g I_n \\times _{I_n} I_n g ^{-1}I_n i [r] & P_{s, n} i.",
"}$ Thus it suffices to consider the case where $g^{\\prime }=g ^{-1}$ .",
"Since $n \\geqslant 1$ , by Lemma REF and Lemma REF , $g I_n g ^{-1}\\subset P_{s, n} \\subset I$ .",
"Since $I_n$ is a normal subgroup of $I$ , $I_n$ is stable under the conjugation action of $g I_n g ^{-1}$ .",
"Thus $g ^{-1}I_n g$ is stable under the conjugation action of $I_n$ .",
"Conjugation by $g ^{-1}$ , we have that $g ^{-1}I_n g \\cap I_n$ is a normal subgroup of $I_n$ .",
"Thus for any $p \\in P_{s, n}$ , the inverse image of $I_n p I_n$ in $I_n g I_n \\times _{I_n} I_n g^{\\prime } I_n$ equals to $\\lbrace (I_n g a, b g ^{-1}I_n); a, b \\in I_n/(g ^{-1}I_n g \\cap I_n), g a b g ^{-1}\\in I_n p I_n\\rbrace /I_n.$ Let $p^{\\prime } \\in P_{s, n}$ .",
"Then since $P_{s, n}=(g I_n g ^{-1}) I_n$ , we have $p^{\\prime } I_n=(g i g ^{-1}) p I_n$ for some $i \\in I_n$ .",
"Note that $I_n$ is stable under the conjugation action of $g i g ^{-1}\\in P_{s, n} \\subset I$ .",
"Hence $(g i g ^{-1}) I_n p I_n=I_n (g i g ^{-1}) p I_n=I_n p^{\\prime } I_n$ and the inverse image of $I_n p^{\\prime } I_n$ in $I_n g I_n \\times _{I_n} I_n g^{\\prime } I_n$ equals to $\\lbrace (I_n g i a, b g ^{-1}I_n); a, b \\in I_n/(g ^{-1}I_n g \\cap I_n), g a b g ^{-1}\\in I_n p I_n\\rbrace /I_n.$ In particular all the fibers have the same cardinality.",
"So the cardinality of each fiber equals to $\\sharp (I_n g I_n \\times _{I_n} I_n g^{\\prime } I_n/I_n)/\\sharp (P_{s, n}/I_n)=q^{2\\breve{\\ell }(s)}/q^{\\breve{\\ell }(s)}=q^{\\breve{\\ell }(s)}.$ The statement is proved."
],
[
"Proof of Theorem ",
"We choose a representative $g$ for each $I_n \\times I_n$ -orbit on $G(F)$ .",
"We denote the set of representatives by $\\mathcal {Y} $ .",
"Then the set $\\lbrace \\mathbb {1}_{I_n g I_n}; g \\in \\mathcal {Y} \\rbrace $ is a basis of $\\mathcal {H} _n$ as a free $\\mathbb {Z} $ -module.",
"In particular, the set $\\lbrace \\mathbb {1}_{I_n g I_n}; g \\in \\mathcal {Y} \\rbrace $ generates $\\mathcal {H} _n$ as an algebra.",
"For any $g_1, g_2 \\in \\mathcal {Y} $ , we have $\\mathbb {1}_{I_n g_1 I_n} * \\mathbb {1}_{I_n g_1 I_n}=\\sum _{g_3 \\in \\mathcal {Y}} c_{g_1, g_2, g_3} \\mathbb {1}_{I_n g_3 I_n} \\in \\mathcal {H} _n$ for some $c_{g_1, g_2, g_3} \\in \\mathbb {Z} $ .",
"We denote this relation by $(*_{g_1, g_2})$ .",
"It is tautological that the equalities $(*_{g_1, g_2})$ for $g_1, g_2 \\in \\mathcal {Y} $ form a set of relations for the algebra $\\mathcal {H} _n$ .",
"By Corollary REF , the algebra $\\mathcal {H} _n$ is generated by $\\mathbb {1}_{I_n g I_n}$ for $g \\in \\mathcal {I} $ with $\\ell (\\pi (g)) \\leqslant 1$ .",
"Thus the equalities $(*_{g_1, g_2})$ for $g_1, g_2 \\in \\mathcal {Y} $ with $\\ell (\\pi (g_1)) \\leqslant 1$ form a set of relations for the algebra $\\mathcal {H} _n$ .",
"If $\\ell (\\pi (g_1))=0$ , then for any $g_2 \\in \\mathcal {Y} $ , $\\ell (\\pi (g_1 g_2))=\\ell (\\pi (g_2))$ .",
"Thus the equality $(*_{g_1, g_2})$ is obtained from the relations $(0)$ and $(1)$ in Theorem REF .",
"If $\\ell (\\pi (g_1))=1$ , then $\\pi (g_1)=s$ for some $s \\in \\mathbb {S} $ .",
"Let $w=\\pi (g_2)$ .",
"If $s w>w$ , then the equality $(*_{g_1, g_2})$ is obtained from the relations $(0)$ and $(1)$ in Theorem REF .",
"If $s w<w$ , then $\\pi (g_1 g_2)=\\pi (g_2)$ or $\\pi (g_1 g_2)<\\pi (g_2)$ .",
"In either case, the equality $(*_{g_1, g_2})$ is obtained from the relations $(0)$ and $(2)$ in Theorem REF .",
"Theorem REF is proved."
],
[
"The Howe-Tits presentation of $\\mathcal {H} _n$",
"In [13], Howe discovered a nice presentation for the Hecke algebra $\\mathcal {H} _n$ .",
"This presentation was later generalized to split groups by the first-named author in [10].",
"This nice presentation of $\\mathcal {H} _n$ has found applications in the representation theory of $p$ -adic groups.",
"For instance, this presentation was used to establish a variant of a Hecke algebra isomorphism of Kazhdan for sufficiently close local fields, which in turn was used to study the local Langlands correspondence for connected reductive groups in characteristic $p$ with an understanding of the local Langlands correspondence of such groups in characteristic 0 (see [5], [15], [10], [2], [11]).",
"Before stating the theorem, we first introduce some structure constants.",
"For $\\tau , \\tau ^{\\prime } \\in \\Omega _{{{a}}}$ and $s \\in \\mathbb {S} $ , let $c_{\\tau , \\tau ^{\\prime }} &= m(\\tau )m(\\tau ^{\\prime }) m(\\tau \\tau ^{\\prime })^{-1}\\\\\\nonumber c_{\\tau , s} &= m(\\tau )m(s)m(\\tau )^{-1}m(\\tau s\\tau ^{-1})^{-1}.\\nonumber $ Recall that the Tits' axiom (T3) (see [7]) says that $m(s)Im(s)^{-1} \\subset I \\cup I m(s) I$ .",
"In particular, if $g \\in I$ but $g \\notin I\\cap m(s)Im(s)^{-1}$ , this axiom implies that $m(s)gm(s)^{-1}\\in Im(s)I$ .",
"Hence there exist $g_1, g_2$ in $I$ such that $m(s)g m(s)^{-1} =g_1 m(s) g_2$ .",
"We have the following theorem.",
"Theorem 4.7 Let $\\mathcal {T} $ be a Tits group of $W$ and $\\lbrace m(w)\\rbrace _{w \\in W}$ is a Tits cross-section of $W$ in $\\mathcal {T} $ .",
"The Hecke algebra $\\mathcal {H} _n$ has generators $\\mathbb {1}_{I_n m(s) I_n}, s \\in \\mathbb {S} $ , $\\mathbb {1}_{I_n m(\\tau ) I_n},\\; \\tau \\in \\Omega _{{{a}}}$ , $\\mathbb {1}_{I_n g I_n}, g \\in I$ , subject to the following relations: For $s, s^{\\prime }$ distinct elements of $\\mathbb {S} $ with $s \\cdot s^{\\prime }$ of order $k(s,s^{\\prime })$ , $ \\underbrace{\\mathbb {1}_{I_n m(s) I_n}*\\mathbb {1}_{I_n m(s^{\\prime }) I_n} *\\cdots }_{k(s, s^{\\prime }) \\text{ factors}} = \\underbrace{\\mathbb {1}_{I_n m(s^{\\prime }) I_n}*\\mathbb {1}_{I_n m(s) I_n} *\\cdots .",
"}_{k(s, s^{\\prime }) \\text{factors}} $ For $s \\in \\mathbb {S} $ , $\\mathbb {1}_{I_n m(s) I_n} * \\mathbb {1}_{I_n m(s) I_n} * \\mathbb {1}_{I_n m(s)^2 I_n} = q^{\\breve{l}(s)} \\displaystyle {\\sum _{x \\in P_{s,n}/I_n} \\mathbb {1}_{I_n x I_n}}.$ For $\\tau , \\tau ^{\\prime } \\in \\Omega _{{{a}}}$ , $\\mathbb {1}_{I_n m(\\tau ) I_n} * \\mathbb {1}_{I_n m(\\tau ^{\\prime }) I_n} = \\mathbb {1}_{I_n c_{\\tau , \\tau ^{\\prime }} I_n}* \\mathbb {1}_{I_n m(\\tau \\tau ^{\\prime }) I_n}$ .",
"For $\\tau \\in \\Omega _{{{a}}}$ and $s \\in \\mathbb {S} $ , $\\mathbb {1}_{I_n m(\\tau ) I_n} * \\mathbb {1}_{I_n m(s) I_n} *\\mathbb {1}_{I_n m(\\tau ^{-1}) I_n} * \\mathbb {1}_{I_n c_{\\tau , \\tau ^{-1}} I_n} = \\mathbb {1}_{I_n c_{\\tau ,s }I_n}* \\mathbb {1}_{I_n m(\\tau s \\tau ^{-1}) I_n}.$ For $\\tau \\in \\Omega _{{{a}}}$ and $g \\in I$ , $\\mathbb {1}_{I_n m(\\tau ) I_n} * \\mathbb {1}_{I_n g I_n} * \\mathbb {1}_{I_n m(\\tau ^{-1}) I_n} * \\mathbb {1}_{I_n (c_{\\tau , \\tau ^{-1}})^{-1}I_n} = \\mathbb {1}_{I_n m(\\tau )g m(\\tau )^{-1}I_n}.$ $\\mathbb {1}_{I_n}$ is the identity element of $\\mathcal {H} (G(F), I_n)$ .",
"For $g,g^{\\prime } \\in I$ , $\\mathbb {1}_{I_n g I_n}* \\mathbb {1}_{I_n g^{\\prime } I_n} = \\mathbb {1}_{I_n gg^{\\prime } I_n}.$ For $s \\in \\mathbb {S} $ and for $g \\in I \\cap m(s)Im(s)^{-1}$ , $ \\mathbb {1}_{I_n m(s) I_n}* \\mathbb {1}_{I_n g I_n} = \\mathbb {1}_{I_n m(s)gm(s)^{-1} I_n} * \\mathbb {1}_{I_n m(s) I_n}$ For $s \\in \\mathbb {S} $ and for $g \\in I \\backslash (I\\cap m(s)Im(s)^{-1})$ , let $g_1, g_2$ are elements of $I$ such that $m(s)g m(s)^{-1} =g_1 m(s) g_2$ .",
"Then $ \\mathbb {1}_{I_n m(s) I_n}* \\mathbb {1}_{I_n g I_n} * \\mathbb {1}_{I_n m(s) I_n} * \\mathbb {1}_{I_n m(s)^2 I_n}= q^{\\breve{l}(s)} (\\mathbb {1}_{I_n g_1 I_n} * \\mathbb {1}_{I_n m(s) I_n}* \\mathbb {1}_{I_n g_2 I_n}).$ Let $\\hat{\\mathcal {H}}_n$ be the quotient of the free $\\mathbb {Z} $ -algebra generated by the elements (1) - (3), by the subalgebra generated by relations (A) - (C) stated in the theorem.",
"For clarity, to distinguish the elements of $\\hat{\\mathcal {H}}_n$ from the elements of $\\mathcal {H} _n$ , we denote the generators of $\\hat{\\mathcal {H}}_n$ as $\\mathbb {\\hat{1}}_{I_n m(s) I_n}, s\\in \\mathbb {S} $ , $\\mathbb {\\hat{1}}_{I_n m(\\tau ) I_n}, \\tau \\in \\Omega _{{{a}}}$ , $\\mathbb {\\hat{1}}_{I_n g I_n}, g \\in I$ .",
"By Theorem REF , the relations (A)-(C) in Theorem REF are satisfied for $\\mathcal {H} _n$ .",
"Thus we have an algebra homomorphism $\\hat{\\mathcal {H}}_n \\rightarrow \\mathcal {H} _n$ .",
"This map is surjective by Corollary REF .",
"For $w \\in W$ , write $w = w_1 \\tau $ for $w_1 \\in W_{\\text{af}}$ and $\\tau \\in \\Omega _{{{a}}}$ .",
"Let $m(w_1) = m(s_{i_1}) \\cdots m(s_{i_c})$ for a chosen reduced expression of $w_1$ and let $m(w) = m(w_1)m(\\tau ) \\in \\mathcal {T} $ .",
"Define $\\mathbb {\\hat{1}}_{I_nm(w)I_n}= \\mathbb {\\hat{1}}_{I_n m(s_{i_1})I_n}* \\cdots \\mathbb {\\hat{1}}_{I_n m(s_{i_c})I_n} * \\mathbb {\\hat{1}}_{I_n m(\\tau )I_n}$ .",
"Then relation (A)(i) implies that this expression is independent of the choice of reduced expression for $w_1$ .",
"For $g \\in G(F)$ , write $g = x m(w) y$ for $w \\in W$ and $x, y \\in I$ .",
"Define $\\mathbb {\\hat{1}}_{I_n g I_n}= \\mathbb {\\hat{1}}_{I_n x I_n} *\\mathbb {\\hat{1}}_{I_nm(w)I_n} *\\mathbb {\\hat{1}}_{I_n y I_n}$ .",
"We show that (a) Let $w \\in W$ and $x, y, x_1, y_1 \\in I$ such that $x m(w) y=x_1 m(w) y_1$ .",
"Then $\\mathbb {\\hat{1}}_{I_nxI_n} *\\mathbb {\\hat{1}}_{I_nm(w)I_n} *\\mathbb {\\hat{1}}_{I_n yI_n} = \\mathbb {\\hat{1}}_{I_nx_1I_n} *\\mathbb {\\hat{1}}_{I_nm(w)I_n}*\\mathbb {\\hat{1}}_{I_ny_1I_n}.$ Since $yy_1^{-1} = m(w)^{-1}(x^{-1}x_1)m(w)$ , $yy_1^{-1} \\in \\displaystyle {I \\cap m(w)^{-1}I m(w)}$ .",
"Therefore, $\\mathbb {\\hat{1}}_{I_n m(w)I_n} * \\mathbb {\\hat{1}}_{I_nyy_1^{-1}I_n} = \\mathbb {\\hat{1}}_{I_n m(s_{i_1})I_n}* \\ldots \\mathbb {\\hat{1}}_{I_n m(s_{i_c})I_n} * \\mathbb {\\hat{1}}_{I_n m(\\tau ) yy_1^{-1}m(\\tau )^{-1}I_n} * \\mathbb {\\hat{1}}_{I_n m(\\tau )I_n}.$ using relations B(iii) and C(i).",
"It is easy to check that $a \\in {\\mathrm {Ad}}(m(s_{i_1})\\ldots m(s_{i_c}))^{-1} (I) \\cap I$ implies that $a \\in {\\mathrm {Ad}}(m(s_{i_c}))( I) \\cap I$ and ${\\mathrm {Ad}}(m(s_{i_c}))(a) \\in I \\cap {\\mathrm {Ad}}(m(s_{i_1})\\ldots m(s_{i_{c-1}}))^{-1} (I).$ Using the above and relation C(iii) repeatedly, we have $\\mathbb {\\hat{1}}_{I_nm(w)I_n} * \\mathbb {\\hat{1}}_{I_nyy_1^{-1}I_n} & =\\mathbb {\\hat{1}}_{I_n m(s_{i_1})I_n}* \\mathbb {\\hat{1}}_{I_n m(s_{i_2})I_n}*\\ldots .",
"\\mathbb {\\hat{1}}_{I_n m(s_{i_c})m(\\tau ) yy_1^{-1}m(\\tau )^{-1}m(s_{i_c})^{-1}I_n}*\\\\& \\quad * \\mathbb {\\hat{1}}_{I_n m(s_{i_c})I_n}* \\mathbb {\\hat{1}}_{I_n m(\\tau )I_n}\\\\& = \\mathbb {\\hat{1}}_{I_n m(w) yy_1^{-1}m(w)^{-1}I_n}* \\mathbb {\\hat{1}}_{I_nm(w)I_n}\\\\& = \\mathbb {\\hat{1}}_{I_nx^{-1}x_1I_n} * \\mathbb {\\hat{1}}_{I_nm(w)I_n}.$ Now (a) follows from C(i) and C(ii).",
"In particular, the element $\\mathbb {\\hat{1}}_{I_n g I_n}$ is well-defined.",
"We prove that Relation (0) of Theorem REF holds for $\\hat{\\mathcal {H}}_n$ .",
"Let $g=x m(w) y$ with $x, y \\in I$ and $g^{\\prime } \\in I_n g I_n$ .",
"Then $g^{\\prime }=x^{\\prime } m(w) y^{\\prime }$ for some $x^{\\prime }, y^{\\prime } \\in I$ with $x^{\\prime } \\in I_n x$ and $y^{\\prime } \\in y I_n$ .",
"By (a) and relation (C)(i), (ii), we have $\\mathbb {\\hat{1}}_{I_n g I_n}= \\mathbb {\\hat{1}}_{I_n x I_n} *\\mathbb {\\hat{1}}_{I_nm(w)I_n} *\\mathbb {\\hat{1}}_{I_n y I_n}=\\mathbb {\\hat{1}}_{I_n x^{\\prime } I_n} *\\mathbb {\\hat{1}}_{I_nm(w)I_n} *\\mathbb {\\hat{1}}_{I_n y^{\\prime } I_n}=\\mathbb {\\hat{1}}_{I_n g^{\\prime } I_n}.$ We prove that Relation (1) of Theorem REF holds for $\\hat{\\mathcal {H}}_n$ .",
"We need to show that if $l(\\pi (gg^{\\prime })) = l(\\pi (g)) + l(\\pi (g^{\\prime }))$ , then $\\mathbb {\\hat{1}}_{I_n g I_n} * \\mathbb {\\hat{1}}_{I_ng^{\\prime }I_n} = \\mathbb {\\hat{1}}_{I_ngg^{\\prime }I_n}.$ To prove this claim, we may easily reduce ourselves to the case when $l(\\pi (g)) \\le 1$ .",
"If $l(\\pi (g))=0$ , then (REF ) follows from relations B(i), B(ii), B(iii), C(i) and C(ii).",
"If $l(\\pi (g) = 1$ , we may assume $g = m(s)$ for a suitable $s \\in \\mathbb {S} $ .",
"Let $\\pi (g^{\\prime }) = w$ .",
"Write $g^{\\prime } = x^{\\prime } m(w) y^{\\prime }$ for $x^{\\prime }, y^{\\prime } \\in I$ .",
"Then $gg^{\\prime } = m(s)x^{\\prime }m(w)y^{\\prime }$ .",
"Since $l(sw) = l(s)+1$ , we see that $m(s) I m(w) \\subset I m(s)m(w) I$ .",
"In particular, $I \\subset m(s)^{-1}I m(s) m(w) I m(w)^{-1}$ .",
"Write $x^{\\prime }= x_1^{\\prime }x_2^{\\prime }$ for $x_1^{\\prime } \\in m(s)^{-1}I m(s)$ and $x_2^{\\prime } \\in m(w) I m(w)^{-1}$ .",
"Then $gg^{\\prime } = m(s) x_1^{\\prime } m(s)^{-1}m(s) m(w) m(w)^{-1}x_2^{\\prime } m(w) y^{\\prime }$ and $\\mathbb {\\hat{1}}_{I_n m(s) g^{\\prime } I_n} &= \\mathbb {\\hat{1}}_{I_n m(s) x_1 m(s)^{-1}I_n} *\\mathbb {\\hat{1}}_{I_n m(s) m(w) I_n} * \\mathbb {\\hat{1}}_{I_n m(w)^{-1}x_2 m(w) y^{\\prime } I_n}.$ In this case, (REF ) holds using relations C(i), C(ii) and C(iii).",
"We prove that relation (2) of Theorem REF holds for $\\hat{\\mathcal {H}}_n$ .",
"We may assume $g = m(s)$ .",
"Let $\\pi (g^{\\prime }) = w$ .",
"In this case, $l(sw)<l(w)$ .",
"Write $g^{\\prime }= x^{\\prime }m(w)y^{\\prime }$ .",
"Then $m(s)x^{\\prime }m(w) \\in I m(w) I \\sqcup I m(s) m(w) I$ .",
"Now $m(s)x^{\\prime }m(w) \\in I m(s) m(w) I$ if and only if $\\pi (gg^{\\prime }) <\\pi (g^{\\prime })$ .",
"Further, writing $x^{\\prime }= x_1^{\\prime }x_2^{\\prime }$ for $x_1^{\\prime } \\in m(s)^{-1}I m(s)$ and $x_2^{\\prime } \\in m(w) I m(w)^{-1}$ , using C(iii) and C(ii), we have $\\mathbb {\\hat{1}}_{I_n m(s) I_n} * \\mathbb {\\hat{1}}_{I_n x^{\\prime } I_n} * \\mathbb {\\hat{1}}_{I_n m(w) I_n } * \\mathbb {\\hat{1}}_{I_n y^{\\prime } I_n }&= \\mathbb {\\hat{1}}_{I_n m(s) x_1 m(s)^{-1}I_n} *\\mathbb {\\hat{1}}_{I_n m(s) I_n}\\\\& * \\mathbb {\\hat{1}}_{I_n m(w) I_n} * \\mathbb {\\hat{1}}_{I_n m(w)^{-1}x_2 m(w) y^{\\prime } I_n}.",
"\\nonumber $ Since $l(sw)<l(w)$ , we have $w = sw^{\\prime }$ for a suitable $w^{\\prime } \\in W$ .",
"Then $\\mathbb {\\hat{1}}_{I_n m(w) I_n } = \\mathbb {\\hat{1}}_{I_n m(s) I_n }* \\mathbb {\\hat{1}}_{I_n m(w_1) I_n }$ .",
"Using this in (REF ) and using Relation A(ii) finishes the proof of Relation (2) when $\\pi (gg^{\\prime }) <\\pi (g^{\\prime })$ .",
"Next, $m(s)x^{\\prime }m(w) \\in I m(w) I$ if and only if $\\pi (gg^{\\prime }) = \\pi (g^{\\prime })$ .",
"In this case, we may write $x^{\\prime } = x_1^{\\prime } m(s)^{-1}x_2^{\\prime }$ where $x_1^{\\prime } \\in m(s)^{-1}I m(s)$ and $x_2^{\\prime } \\in m(w) I m(w)^{-1}$ .",
"Then $m(s)g^{\\prime } = (m(s) x_1^{\\prime }m(s)^{-1}) m(w) (m(w)^{-1}x_2^{\\prime } m(w) y^{\\prime })$ Now $\\mathbb {\\hat{1}}_{I_n m(s) I_n} * \\mathbb {\\hat{1}}_{I_n g^{\\prime } I_n } &= q^{l(s)} (\\mathbb {\\hat{1}}_{I_n m(s) x_1^{\\prime } m(s)^{-1}I_n} *\\mathbb {\\hat{1}}_{I_n x_2^{\\prime } I_n} * \\mathbb {\\hat{1}}_{I_n m(w) I_n } * \\mathbb {\\hat{1}}_{I_n y^{\\prime } I_n })\\\\&=q^{l(s)}( \\mathbb {\\hat{1}}_{I_n m(s) x_1 m(s)^{-1}I_n} * \\mathbb {\\hat{1}}_{I_n m(w) I_n } * \\mathbb {\\hat{1}}_{I_n m(w)^{-1}x_2^{\\prime } m(w) I_n} *\\mathbb {\\hat{1}}_{I_n y^{\\prime } I_n})\\\\& = q^{l(s)} (\\mathbb {\\hat{1}}_{I_n m(s) g^{\\prime } I_n}).$ In the above, the first equality uses C(iv) and C(i), the second one uses C(iii) and C(i).",
"We have verified that relations (0) - (2) of Theorem REF hold for $\\hat{\\mathcal {H}}_n$ .",
"This concludes the proof of the theorem."
],
[
"The Tits group of the relative Weyl group over ${\\breve{F}}$",
"We begin with a discussion on the Tits group of the relative Weyl group of $G$ over ${\\breve{F}}$ , which is probably well-known, but a reference discussing this does not seem to be available in literature.",
"Let $G$ be a connected, reductive group over $F$ .",
"In this section, we prove the existence of the Tits group of a finite relative Weyl group of $G_{\\breve{F}}$ .",
"We consider a Steinberg pinning $(x_{\\widetilde{a}})_{{\\widetilde{a}}\\in \\tilde{\\Delta }_0}$ of $G_{\\widetilde{F}}$ relative to $S$ (see [8]).",
"It has the following properties: $x_{\\widetilde{a}}: _a \\rightarrow U_{\\widetilde{a}}$ is a ${\\widetilde{F}}$ -isomorphism.",
"For each ${\\widetilde{a}}\\in \\tilde{\\Delta }_0$ and $\\gamma \\in \\operatorname{Gal}({\\widetilde{F}}/{\\breve{F}})$ , $x_{\\gamma ({\\widetilde{a}})} = \\gamma \\circ x_{\\widetilde{a}}\\circ \\gamma ^{-1}$ .",
"This extends to a Chevalley-Steinberg system of pinnings $x_{\\widetilde{a}}: _a \\xrightarrow{} U_{\\widetilde{a}}$ for all ${\\widetilde{a}}\\in \\tilde{\\Phi }(G,T)$ , which is compatible with the action of $\\operatorname{Gal}({\\widetilde{F}}/{\\breve{F}})$ .",
"From this, we get a set of pinnings for ${\\breve{a}}\\in \\breve{\\Phi }(G,S)$ , which we briefly recall.",
"Let $U_{\\breve{a}}$ be the root subgroup of the root $a$ .",
"When $2a$ is not a root, we have $x_{\\breve{a}}: \\operatorname{Res}_{{\\breve{F}}_{{\\breve{a}}}/{\\breve{F}}}_a \\xrightarrow{} U_{\\breve{a}}$ .",
"If $2{\\breve{a}}$ is a root, let $H_0({\\breve{F}}_{\\breve{a}}, {\\breve{F}}_{2{\\breve{a}}}) = \\lbrace (u,v) \\in {\\breve{F}}_{\\breve{a}}\\times {\\breve{F}}_{\\breve{a}}\\;|\\; u \\cdot \\gamma (u) = v+\\gamma _0(v)\\rbrace $ , where $\\gamma _0$ is the non-trivial ${\\breve{F}}_{2{\\breve{a}}}$ -automorphism of ${\\breve{F}}_{\\breve{a}}$ .",
"We have $x_{\\breve{a}}: Res_{{\\breve{F}}_{2{\\breve{a}}}/{\\breve{F}}} H_0({\\breve{F}}_{\\breve{a}}, {\\breve{F}}_{2{\\breve{a}}})\\xrightarrow{}U_{\\breve{a}}.$ For any ${\\widetilde{a}}\\in \\tilde{\\Delta }_0$ , let $n_{s_{\\widetilde{a}}}:=x_{\\widetilde{a}}(1) x_{-{\\widetilde{a}}}(1) x_{{\\widetilde{a}}}(1).$ We note here that we have used the convention of [8], and $n_{s_{\\widetilde{a}}} \\in N_G(S)({\\breve{F}})$ (This is different from the convention used in [20] where $n_{s_{\\widetilde{a}}}:=x_{\\widetilde{a}}(1) x_{-{\\widetilde{a}}}(-1) x_{{\\widetilde{a}}}(1)$ ).",
"Now, let ${\\breve{a}}\\in \\breve{\\Delta }_0$ .",
"If $2{\\breve{a}}$ is not a root, we set $n_{s_{\\breve{a}}} := x_{\\breve{a}}(1)x_{-{\\breve{a}}}(1)x_{\\breve{a}}(1).$ Next, suppose $2{\\breve{a}}$ is a root.",
"By [21], there exists $c \\in {\\breve{F}}_{\\breve{a}}$ such that $c \\gamma _0(c)=2$ .",
"Set $n_{s_{\\breve{a}}} = x_{\\breve{a}}(c,1) x_{-{\\breve{a}}}(c,1)x_{\\breve{a}}(c,1).$ Note that if the residue characteristic of $F$ is not 2, such a $c$ in fact lies in $\\mathfrak {O} _{\\breve{F}}^\\times $ , and when the characteristic of $F$ is 2, $c =0$ .",
"By [8], we have $n_{s_{\\breve{a}}}= \\prod n_{s_{\\widetilde{a}}}^{-1} n_{s_{{\\widetilde{a}}^{\\prime }}}n_{s_{{\\widetilde{a}}}}^{-1}.$ where the product is indexed by the family of sets $\\lbrace {\\widetilde{a}}, {\\widetilde{a}}^{\\prime }\\rbrace $ with ${\\widetilde{a}}, {\\widetilde{a}}^{\\prime } \\in \\tilde{\\Phi }(G,T)$ such that ${\\widetilde{a}}+{\\widetilde{a}}^{\\prime }$ is a root and ${\\widetilde{a}}|_S ={\\widetilde{a}}^{\\prime }|_S = {\\breve{a}}$ .",
"Let $\\breve{\\mathcal {T}}_{\\text{fin}}$ be the group generated by the elements $\\lbrace n_{s_{\\breve{a}}}\\;|\\; {\\breve{a}}\\in \\breve{\\Delta }_0\\rbrace $ .",
"Note that $S_2 = \\langle {\\breve{a}}^\\vee (-1)\\;|\\; {\\breve{a}}\\in \\breve{\\Delta }_0\\rangle $ is contained in $\\mathcal {T} _\\text{fin}$ .",
"Then the elements $\\lbrace n_{s_{{\\breve{a}}}}\\;|\\; {\\breve{a}}\\in \\breve{\\Delta }_0\\rbrace $ satisfy Coxeter relations is a consequence of [7] (see [4]).",
"Furthermore, we have that $n_{s_{\\breve{a}}}^2 ={\\left\\lbrace \\begin{array}{ll} {\\breve{a}}^\\vee (-1), & \\text{ if $2{\\breve{a}}$ is not a root}; \\\\ 1, & \\text{ if $2{\\breve{a}}$ is a root}.\\end{array}\\right.",
"}$ Let $\\breve{S}_2 = \\langle {\\breve{a}}^\\vee (-1)\\; |\\; {\\breve{a}}\\in \\breve{\\Phi }(G,S) \\rangle $ .",
"Then $\\mathcal {T} _{\\text{fin}}$ fits into a short exact sequence $1 \\rightarrow \\breve{S}_2 \\rightarrow \\breve{\\mathcal {T}}_{\\text{fin}} \\rightarrow W(G,S) \\rightarrow 1.",
"$"
],
[
"An example of $G_{\\breve{F}}$ for which {{formula:9b96a13a-c7dd-492a-aa37-bb908cefdb18}} does not exist",
"In this subsection, we will give an example of a wildly ramified unitary group over ${\\breve{F}}$ for which the Tits group $\\breve{\\mathcal {T}}$ does not exist.",
"Let $F = \\mathbb {Q} _2, {\\breve{F}}= \\mathbb {Q} _2^{\\text{un}}, {\\widetilde{F}}= {\\breve{F}}(\\sqrt{-}1)$ .",
"Then ${\\widetilde{F}}$ is a wildly ramified quadratic extension of ${\\breve{F}}$ .",
"Let $G$ be a connected reductive group over ${\\breve{F}}$ with $G_{\\breve{F}}= \\mathrm {U}_6\\subset \\operatorname{Res}_{{\\widetilde{F}}/{\\breve{F}}}\\mathrm {GL}_6$ .",
"Let $\\gamma $ denote the generator of $\\operatorname{Gal}({\\widetilde{F}}/{\\breve{F}})$ .",
"Then $ \\breve{\\Phi }(G,S) = \\lbrace \\pm e_i \\pm e_j\\;|\\; 1\\le i<j\\le 3\\rbrace \\cup \\lbrace \\pm 2 e_i\\;|\\; 1 \\le i \\le 3\\rbrace , \\\\\\breve{\\Phi }_{\\text{af}}(G,S) = \\lbrace \\pm e_i \\pm e_j +\\frac{1}{2}\\mathbb {Z} \\;|\\;1 \\le i<j \\le 3\\rbrace \\cup \\lbrace \\pm 2e_i+\\mathbb {Z} \\;|\\; 1 \\le i\\le 3\\rbrace .$ The hyperplanes with respect to the roots ${\\breve{a}}_1:= e_1-e_2, {\\breve{a}}_2:=e_2-e_3, {\\breve{a}}_3:= 2e_3, {\\breve{a}}_0:=-e_1 - e_2 +\\frac{1}{2}$ form a alcove in $\\mathcal {A} (S,{\\breve{F}})$ which we denote as ${\\breve{{a}}}$ .",
"Let $\\breve{\\Delta }_0 =\\lbrace {\\breve{a}}_i\\;|\\; 1\\le i \\le 3\\rbrace $ and $\\breve{\\Delta }= \\breve{\\Delta }_0 \\cup \\lbrace {\\breve{a}}_0\\rbrace $ .",
"Let ${\\breve{s}}_i = s_{{\\breve{a}}_i}, 0 \\le i \\le 3$ .",
"Suppose $\\breve{\\mathcal {T}}$ can be defined.",
"Then $\\breve{\\mathcal {T}}$ contains representatives $m({\\breve{s}}_i), 0 \\le i\\le 3$ that satisfy Coxeter relations and such that $m({\\breve{s}}_i)^2 = ({\\breve{b}}_i)_*^\\vee (-1) \\in \\breve{S}_2$ .",
"The Coxeter relations involving the reflection ${\\breve{s}}_0$ are ${\\breve{s}}_0{\\breve{s}}_1= {\\breve{s}}_1{\\breve{s}}_0$ , ${\\breve{s}}_0{\\breve{s}}_2{\\breve{s}}_0= {\\breve{s}}_2{\\breve{s}}_0{\\breve{s}}_2$ and ${\\breve{s}}_0{\\breve{s}}_3={\\breve{s}}_3{\\breve{s}}_0$ .",
"Additionally, we have ${\\breve{s}}_0^2=1$ .",
"Let $t \\in T({\\breve{F}})$ .",
"We may write $t = \\operatorname{diag}(d_1, d_2, d_3, \\gamma (d_3)^{-1}, \\gamma (d_2)^{-1}, \\gamma (d_1)^{-1})$ .",
"Then ${\\breve{s}}_1(t) &= \\operatorname{diag}(d_2, d_1, d_3, \\gamma (d_3)^{-1}, \\gamma (d_1)^{-1}, \\gamma (d_2)^{-1}),\\\\{\\breve{s}}_2(t) &= \\operatorname{diag}(d_1, d_3, d_2, \\gamma (d_2)^{-1}, \\gamma (d_3)^{-1}, \\gamma (d_1)^{-1}),\\\\{\\breve{s}}_3(t) &= \\operatorname{diag}(d_1, d_2, \\gamma (d_3)^{-1}, d_3, \\gamma (d_2)^{-1}, \\gamma (d_1)^{-1}),\\\\{\\breve{s}}_0(t) &= \\operatorname{diag}(\\gamma (d_2)^{-1}, \\gamma (d_1)^{-1}, d_3, \\gamma (d_3)^{-1}, d_1, d_2).$ To see the last equality, we note that a reduced expression for the image of ${\\breve{s}}_0$ in $W(G,S)$ is ${\\breve{s}}_2{\\breve{s}}_3{\\breve{s}}_2{\\breve{s}}_1{\\breve{s}}_2{\\breve{s}}_3{\\breve{s}}_2$ .",
"This equality can also be seen by noting that the image of ${\\breve{s}}_0$ in $W(G,S) \\subset W(G,T)$ represents the permutation $(1,5)(2,6)$ in the symmetric group $S_6$ .",
"For $\\sigma $ a permutation in the symmetric group $S_6$ , let $g_\\sigma $ denote the corresponding permutation matrix in $\\mathrm {GL}_6({\\widetilde{F}})$ whose entries are all 0 or 1.",
"The element $m({\\breve{s}}_0) \\in \\mathrm {U}_6({\\breve{F}}) \\subset \\mathrm {GL}_6({\\widetilde{F}})$ can be written as a product $t_0 \\cdot g_{(1,5)(2,6)}$ where $t_0 = \\operatorname{diag}(d_1, d_2, d_3, \\gamma (d_3)^{-1}, \\gamma (d_2)^{-1}, \\gamma (d_1)^{-1}).$ Now $m({\\breve{s}}_0)^2 \\in \\breve{S}_2$ implies that $d_2 = \\pm \\gamma (d_1)$ .",
"Next, the relation $m({\\breve{s}}_0)m({\\breve{s}}_1) = m({\\breve{s}}_1)m({\\breve{s}}_0)$ implies that $t_0 g_{(1,5)(2,6)} = {\\breve{s}}_1(t_0) m({\\breve{s}}_1) g_{(1,5)(2,6)} m({\\breve{s}}_1)^{-1}$ .",
"So ${\\breve{s}}_1(t_0) t_0^{-1}= g_{(1,5)(2,6)} m({\\breve{s}}_1) g_{(1,5)(2,6)}m({\\breve{s}}_1)^{-1}.$ Since the right side of the expression above maps to the trivial element of ${\\breve{W}}$ , it is an element of $\\breve{S}_2$ ; in particular, it is a diagonal matrix with entries $\\pm 1$ .",
"Since ${\\breve{s}}_1(t_0)t_0^{-1} = \\operatorname{diag}(\\pm \\gamma (d_1) d_1^{-1}, \\pm d_1 \\gamma (d_1)^{-1}, 1,1,\\pm \\gamma (d_1) d_1^{-1},\\pm d_1 \\gamma (d_1)^{-1})$ we have that $\\gamma (d_1) = \\pm d_1$ .",
"Similarly, $m({\\breve{s}}_0) m({\\breve{s}}_3) = m({\\breve{s}}_3) m({\\breve{s}}_0)$ implies that $d_3 =\\pm 1$ .",
"The action of $m({\\breve{s}}_0)$ on $V$ is given by $s_{e_1+e_2}(v) - val(d_1)(e_1+e_2)^\\vee (v)$ , where $val$ is normalized so that $val({\\breve{F}})=\\mathbb {Z} $ .",
"Hence $val(d_1) = \\frac{1}{2}$ (since ${\\breve{s}}_0$ is the reflection with respect to the vanishing hyperplane of the affine root $-e_1-e_2 + \\frac{1}{2}$ ).",
"Hence $\\gamma (d_1) \\ne d_1$ .",
"So $\\gamma (d_1) = -d_1$ .",
"In conclusion, the assumption that $\\breve{\\mathcal {T}}$ can be defined implies that there exists $d_1 \\in {\\widetilde{F}}$ with $val(d_1)=\\frac{1}{2}$ and $Tr_{{\\widetilde{F}}/{\\breve{F}}}(d_1) = 0$ .",
"However, there does not exist an element of ${\\widetilde{F}}$ with valuation $\\frac{1}{2}$ and with trace 0.",
"To see this, note that any element of ${\\widetilde{F}}$ is of the form $a+b \\sqrt{-}1$ for $a, b \\in {\\breve{F}}$ .",
"Then $\\gamma (a+b\\sqrt{-}1) = a - b \\sqrt{-}1$ .",
"If $\\gamma (a+b\\sqrt{-}1) = -a-b\\sqrt{-}1$ then $a=0$ .",
"Further $\\sqrt{-}1$ is a unit in ${\\widetilde{F}}$ , so $val( b\\sqrt{-}1) = val(b) \\in \\mathbb {Z} $ .",
"This gives a contradiction and proves that $\\breve{\\mathcal {T}}$ cannot be defined.",
"In the rest of this section, we will show that the Tits group of ${\\breve{W}}$ exists if $G$ splits over $\\breve{F}$ ."
],
[
"Affine pinning",
"Recall that we have chosen a special vertex ${\\breve{v}}_0$ and we view ${\\breve{{a}}}\\subset V$ .",
"For each reflection ${\\breve{s}}$ in $\\breve{\\mathbb {S}}$ there is a unique affine root such that the reflection in the hyperplane with respect to this affine root is ${\\breve{s}}$ .",
"Let $\\breve{\\Delta }$ denote this collection of affine roots.",
"The Weyl chamber in $V$ that contains ${\\breve{{a}}}$ determines a set of simple roots of $\\breve{\\Phi }(G,T)$ which we denote as $\\breve{\\Delta }_0$ .",
"Note that $\\breve{\\Delta }_0 \\subset \\breve{\\Delta }$ .",
"We consider a pinning $\\lbrace x_{\\breve{a}}\\rbrace _{{\\breve{a}}\\in \\tilde{\\Delta }_0}$ of $G_{\\breve{F}}$ relative to $T$ (see [8]), that is, for each ${\\breve{a}}\\in \\breve{\\Delta }_0$ , we fix a ${\\breve{F}}$ -isomorphism $x_{\\breve{a}}: _a \\rightarrow U_{\\breve{a}}$ , where $U_{\\breve{a}}$ is the root subgroup of the root ${\\breve{a}}$ .",
"This extends to a Chevalley system of pinnings $x_{\\breve{a}}: _a \\xrightarrow{} U_{\\breve{a}}$ for all ${\\breve{a}}\\in \\breve{\\Phi }(G,T)$ .",
"Let ${\\breve{a}}\\in \\breve{\\Delta }\\backslash \\breve{\\Delta }_0$ and let ${\\breve{b}}\\in \\breve{\\Phi }(G,T)$ be the gradient of ${\\breve{a}}$ .",
"Let $m_{\\varpi _F}: _a \\rightarrow _a$ denote ${\\breve{F}}$ -morphism given by multiplication by $\\varpi _{F}$ , where $\\varpi _F$ is a uniformizer of $F$ .",
"Define $x_{{\\breve{a}}}:= x_{{\\breve{b}}} \\circ m_{\\varpi _{F}}$ .",
"Note that $x_{{\\breve{a}}}$ is a ${\\breve{F}}$ -isomorphism from $_a$ to $U_{{\\breve{b}}}$ .",
"The set $\\lbrace x_{\\breve{a}}\\;|\\; {\\breve{a}}\\in \\breve{\\Delta }\\rbrace $ is called an affine pinning of $G_{{\\breve{F}}}$ .",
"For ${\\breve{a}}\\in \\breve{\\Delta }$ define $n_{s_{\\breve{a}}} := x_{\\breve{a}}(1)x_{-{\\breve{a}}}(1)x_{\\breve{a}}(1).$ We note here again that we use the convention of [8] and $n_{s_{\\breve{a}}} \\in N_G(T)({\\breve{F}})$ .",
"Lemma 5.1 The set $\\lbrace n_{\\breve{s}}\\;|\\; {\\breve{s}}\\in \\breve{\\mathbb {S}}\\rbrace $ satisfies the Coxeter relations.",
"Remark 5.2 For a different proof of the Coxeter relations for the affine Weyl group of a split reductive group, see [10].",
"Let ${\\breve{s}}= s_{\\breve{a}}$ and ${\\breve{s}}^{\\prime } = s_{{\\breve{a}}^{\\prime }}$ for ${\\breve{a}}, {\\breve{a}}^{\\prime } \\in \\breve{\\Delta }$ with gradients ${\\breve{b}},{\\breve{b}}^{\\prime }$ respectively.",
"Let $\\breve{\\Phi }_{{\\breve{b}},{\\breve{b}}^{\\prime }} \\subset \\breve{\\Phi }(G,T)$ denote the rank 2 root system spanned by ${\\breve{b}},{\\breve{b}}^{\\prime }$ .",
"If $\\breve{\\Phi }_{{\\breve{b}},{\\breve{b}}^{\\prime }}$ is a product of rank 1 root systems, then the Coxeter relation is obvious, so we may and do assume that $\\breve{\\Phi }_{{\\breve{b}},{\\breve{b}}^{\\prime }}$ is irreducible.",
"Set $k = k({\\breve{s}},{\\breve{s}}^{\\prime })$ .",
"We put the elements of $\\breve{\\Phi }_{{\\breve{b}},{\\breve{b}}^{\\prime }}$ in “circular order\" as required in [7], that is, we can enumerate the elements of $\\breve{\\Phi }_{{\\breve{b}},{\\breve{b}}^{\\prime }}$ as ${\\breve{b}}_1, {\\breve{b}}_2, \\cdots , {\\breve{b}}_{2k}$ so that ${\\breve{b}}_1={\\breve{b}}, {\\breve{b}}_k={\\breve{b}}^{\\prime }$ , and for $1<i<2k$ , $\\breve{\\Phi }_{{\\breve{b}},{\\breve{b}}^{\\prime }} \\cap (\\mathbb {Q} _+{\\breve{b}}_{i-1} + \\mathbb {Q} _+ {\\breve{b}}_{i+1}) = \\lbrace {\\breve{b}}_{i-1}, {\\breve{b}}_i, {\\breve{b}}_{i+1}\\rbrace .$ By [7], for any $u \\in U_{-{\\breve{b}}}({\\breve{F}})\\backslash \\lbrace 1\\rbrace $ and $u^{\\prime } \\in U_{-{\\breve{b}}^{\\prime }}({\\breve{F}})\\backslash \\lbrace 1\\rbrace $ , there exists unique triples $(u_1, m(u), u_2) \\in U_{{\\breve{b}}}({\\breve{F}}) \\times N_G(S)({\\breve{F}}) \\times U_{{\\breve{b}}}({\\breve{F}})$ and $(u^{\\prime }_1, m(u^{\\prime }), u^{\\prime }_2) \\in U_{{\\breve{b}}^{\\prime }}({\\breve{F}}) \\times N_G(S)({\\breve{F}}) \\times U_{{\\breve{b}}^{\\prime }}({\\breve{F}})$ such that $u = u_1m(u)u_2$ and $u^{\\prime }=u^{\\prime }_1 m(u^{\\prime }) u^{\\prime }_2$ .",
"By [7], $m(u) \\cdot m(u^{\\prime }) ^{-1}\\cdots = m(u^{\\prime }) ^{-1}\\cdot m(u) \\cdots ,$ where each side has $k$ factors.",
"It is clear from equation (REF ) that there exist $u \\in U_{-{\\breve{b}}_*}$ and $u^{\\prime } \\in U_{-{\\breve{b}}^{\\prime }_*}$ such that $n_{s_{\\breve{a}}} = m(u)$ and $n_{s_{{\\breve{a}}^{\\prime }}} = m(u^{\\prime })^{-1}$ .",
"Now the statement follows from (REF ).",
"Now we prove the following existence result of Tits group over $\\breve{F}$ .",
"Proposition 5.3 Suppose that $G$ is split over $\\breve{F}$ .",
"Let $\\lbrace x_{{\\breve{a}}}\\;|\\; {\\breve{a}}\\in \\breve{\\Delta }\\rbrace $ be an affine pinning of $G_{\\breve{F}}$ and $\\lbrace n_{s_{\\breve{a}}}\\;|\\; {\\breve{a}}\\in \\breve{\\Delta }\\rbrace $ be as in (REF ).",
"Let $\\breve{\\mathcal {T}}$ be the group generated by $\\breve{S}_2$ , $\\lbrace n_{s_{\\breve{a}}}\\;|\\; {\\breve{a}}\\in \\breve{\\Delta }\\rbrace $ and ${\\breve{\\lambda }}(\\varpi _F)$ for ${\\breve{\\lambda }}\\in X_*(T)$ .",
"Then $\\breve{\\mathcal {T}}$ is a Tits group of ${\\breve{W}}$ .",
"By direct calculation, $n_{{\\breve{s}}_{\\breve{a}}}^2={\\breve{b}}^\\vee (-1)$ , where ${\\breve{b}}$ is the gradient of ${\\breve{a}}$ .",
"Now we define the lifting $n_{{\\breve{w}}}$ .",
"If ${\\breve{w}}\\in {\\breve{W}}_{\\text{af}}$ , then we set $n_{{\\breve{w}}}=n_{{\\breve{s}}_{i_1}} \\cdots n_{{\\breve{s}}_{i_k}}$ , where ${\\breve{s}}_{i_1} \\cdots {\\breve{s}}_{i_k}$ is a reduced expression of ${\\breve{w}}$ in ${\\breve{W}}$ .",
"By Lemma REF , the element $n_{{\\breve{w}}}$ is independent of the choice of reduced expression.",
"If $\\breve{\\tau }\\in \\Omega _{{\\breve{{a}}}}$ , we may write $\\breve{\\tau }$ as $t_{\\breve{\\lambda }} {\\breve{y}}$ with $\\breve{\\lambda }\\in X_*(T)$ and ${\\breve{y}}\\in {\\breve{W}}_0$ .",
"We then set $n_{\\breve{\\tau }}=\\breve{\\lambda }(\\varpi _F) n_{{\\breve{y}}}$ .",
"Note that any element ${\\breve{w}}\\in {\\breve{W}}$ is of the form ${\\breve{w}}={\\breve{w}}_1 \\breve{\\tau }$ for some ${\\breve{w}}_1 \\in {\\breve{W}}_{\\text{af}}$ and $\\breve{\\tau }\\in \\Omega _{{\\breve{{a}}}}$ .",
"We set $n_{{\\breve{w}}}=n_{{\\breve{w}}_1} n_{\\breve{\\tau }}$ .",
"The collection $\\lbrace n_{{\\breve{w}}}\\;|\\; {\\breve{w}}\\in {\\breve{W}}\\rbrace $ satisfies condition (2) in Definition REF .",
"Now we check condition (1).",
"Note that $\\breve{S}_2$ is a normal subgroup of $\\breve{\\mathcal {T}}$ .",
"For any ${\\breve{w}}\\in {\\breve{W}}$ and $\\breve{\\lambda }\\in X_*(T)$ , we have $n_{{\\breve{w}}} \\breve{\\lambda }(\\varpi _F) n_{{\\breve{w}}} ^{-1}=\\breve{\\lambda }^{\\prime }(\\varpi _F)$ , where $\\breve{\\lambda }^{\\prime }={\\breve{w}}(\\breve{\\lambda }) \\in X_*(T)$ .",
"Let $\\breve{\\mathcal {T}}^{\\prime }$ be the subgroup of $\\breve{\\mathcal {T}}$ generated by $n_{s_{{\\breve{a}}}}$ for ${\\breve{a}}\\in \\breve{\\Delta }$ .",
"Then any element in $\\breve{\\mathcal {T}}$ is of the form $t_1 n \\breve{\\lambda }(\\varpi _F)$ for some $t_1 \\in \\breve{S}_2$ , $n \\in \\breve{\\mathcal {T}}^{\\prime }$ and $\\breve{\\lambda }\\in X_*(T)$ .",
"If $\\breve{\\phi }(t_1 n \\breve{\\lambda }(\\varpi _F))=1$ , then $\\breve{\\lambda }\\in \\mathbb {Z} \\breve{\\Phi }^\\vee (G, T)$ .",
"Note that $\\mathbb {Z} \\breve{\\Phi }^\\vee (G, T)$ equals to the lattice spanned by ${\\breve{w}}({\\breve{b}})$ , where ${\\breve{w}}\\in {\\breve{W}}_0$ and ${\\breve{b}}$ is the gradient of some ${\\breve{a}}\\in \\breve{\\Delta }\\backslash \\breve{\\Delta }_0$ .",
"By direct calculation, $n_{s_{\\breve{a}}} = {\\breve{b}}^\\vee (\\varpi _F) n_{s_{\\breve{b}}}$ .",
"In particular, ${\\breve{b}}^\\vee (\\varpi _F) \\in \\breve{\\mathcal {T}}^{\\prime }$ and hence $\\breve{\\lambda }(\\varpi _F) \\in \\breve{\\mathcal {T}}^{\\prime }$ for all $\\breve{\\lambda }\\in \\mathbb {Z} \\breve{\\Phi }^\\vee (G, T)$ .",
"Therefore $\\ker \\breve{\\phi }$ is contained in the subgroup generated by $\\breve{S}_2$ and $\\breve{\\mathcal {T}}^{\\prime }$ .",
"Any element of $\\breve{\\mathcal {T}}^{\\prime }$ is of the form $n_{{\\breve{s}}_{i_1}}^{\\pm 1} \\cdots n_{{\\breve{s}}_{i_k}}^{\\pm 1}$ .",
"Since $n_{{\\breve{s}}}^2 \\in \\breve{S}_2$ , we have $n_{{\\breve{s}}_{i_1}}^{\\pm 1} \\cdots n_{{\\breve{s}}_{i_k}}^{\\pm 1} \\in n_{{\\breve{s}}_{i_1}} \\cdots n_{{\\breve{s}}_{i_k}} \\breve{S}_2$ .",
"It remains to show that (a) If ${\\breve{s}}_{i_1} \\cdots {\\breve{s}}_{i_k}=1$ , then $n_{{\\breve{s}}_{i_1}} \\cdots n_{{\\breve{s}}_{i_k}} \\in \\breve{S}_2$ .",
"We argue by induction on $k$ .",
"By the deletion condition of Coxeter groups (see [6]), there exists ${\\breve{s}}_{i^{\\prime }_1}, \\ldots , {\\breve{s}}_{i^{\\prime }_k}$ such that $n_{{\\breve{s}}_{i_1}} \\cdots n_{{\\breve{s}}_{i_k}}=n_{{\\breve{s}}_{i^{\\prime }_1}} \\cdots n_{{\\breve{s}}_{i^{\\prime }_k}}$ and ${\\breve{s}}_{i^{\\prime }_l}={\\breve{s}}_{i^{\\prime }_{l+1}}$ for some $l$ .",
"We have $n_{{\\breve{s}}_{i^{\\prime }_1}} \\cdots n_{{\\breve{s}}_{i^{\\prime }_k}}\\in n_{{\\breve{s}}_{i^{\\prime }_1}} \\cdots n_{{\\breve{s}}_{i^{\\prime }_{l-1}}} \\breve{S}_2 n_{{\\breve{s}}_{i^{\\prime }_{l+2}}} \\cdots n_{{\\breve{s}}_{i^{\\prime }_k}} =n_{{\\breve{s}}_{i^{\\prime }_1}} \\cdots \\hat{n}_{{\\breve{s}}_{i^{\\prime }_l}} \\hat{n}_{{\\breve{s}}_{i^{\\prime }_{l+1}}} \\cdots n_{{\\breve{s}}_{i^{\\prime }_k}} \\breve{S}_2$ .",
"Note that ${\\breve{s}}_{i^{\\prime }_1} \\cdots \\hat{{\\breve{s}}}_{i^{\\prime }_l} \\hat{{\\breve{s}}}_{i^{\\prime }_{l+1}} \\cdots {\\breve{s}}_{i^{\\prime }_k}=1$ .",
"Since there are only $k-2$ simple reflections involved, by inductive hypothesis, $n_{{\\breve{s}}_{i^{\\prime }_1}} \\cdots \\hat{n}_{{\\breve{s}}_{i^{\\prime }_l}} \\hat{n}_{{\\breve{s}}_{i^{\\prime }_{l+1}}} \\cdots n_{{\\breve{s}}_{i^{\\prime }_k}} \\in \\breve{S}_2$ .",
"Hence $n_{{\\breve{s}}_{i_1}} \\cdots n_{{\\breve{s}}_{i_k}} \\in \\breve{S}_2$ .",
"Condition (1) of Definition REF is verified."
],
[
"The strategy",
"The main result of this section is the existence of Tits groups over $F$ for any connected reductive group defined over $F$ and splits over $\\breve{F}$ .",
"The strategy is as follows.",
"Let $G$ connected, reductive group over $F$ and let $\\sigma $ be the Frobenius morphism on $G$ with $G({\\breve{F}})^\\sigma = G(F)$ .",
"We first construct an affine pinning such that the set $\\lbrace n_{s_{\\breve{a}}}\\;|\\; {\\breve{a}}\\in \\mathcal {X} \\rbrace $ is $\\sigma $ -stable for any $\\sigma $ -orbit $\\mathcal {X} $ of $\\breve{\\Delta }$ with ${\\breve{W}}_{\\mathcal {X}}$ finite.",
"This result has two consequences.",
"First, when $G$ is semisimple and simply connected, it yields a definition of the Tits group of its Iwahori-Weyl group, which is the affine Weyl group, over $F$ .",
"Second, we may construct a Tits group $\\breve{\\mathcal {T}}$ over $\\breve{F}$ that is stable under the action of a given quasi-split Frobenius morphism $\\sigma $ ; We then choose a suitable Frobenius morphism $\\sigma ^*$ for each inner form and show that there exists a Tits cross-section in $\\breve{\\mathcal {T}}$ that is “compatible” with the Frobenius morphism $\\sigma ^*$ ; Finally, we use the descent argument to show that $\\breve{\\mathcal {T}}^{\\sigma ^*} \\subset G({\\breve{F}})^{\\sigma ^*}$ is a Tits group of the Iwahori-Weyl group over $F$ of the group $G({\\breve{F}})^{\\sigma ^*}$ ."
],
[
"Affine pinnings and Frobenius morphisms",
"We have proved in Lemma REF that given any affine pinning $\\lbrace x_{\\breve{a}}\\;|\\; {\\breve{a}}\\in \\breve{\\Delta }\\rbrace $ , the set $\\lbrace n_{\\breve{s}}\\;|\\; {\\breve{s}}\\in \\breve{\\mathbb {S}}\\rbrace $ satisfies the Coxeter relations, where $n_{s_{\\breve{a}}}= x_{\\breve{a}}(1)x_{-{\\breve{a}}}(1)x_{\\breve{a}}(1)$ .",
"By §REF , for any $s \\in \\mathbb {S} $ , there exists a $\\sigma $ -orbit $\\mathcal {X} $ of $\\breve{\\Delta }$ with ${\\breve{W}}_{\\mathcal {X}}$ finite such that $s={\\breve{w}}_{\\mathcal {X}} \\in {\\breve{W}}$ .",
"In this section, we show the following.",
"Proposition 6.1 Let $G$ be a connected reductive group defined over $F$ that splits over ${\\breve{F}}$ .",
"Let $\\sigma $ be a Frobenius morphism on $G_{\\breve{F}}$ .",
"There exists an affine pinning $\\lbrace x_{\\breve{a}}\\;|\\; {\\breve{a}}\\in \\breve{\\Delta }\\rbrace $ such that the set $\\lbrace n_{s_{\\breve{a}}}\\;|\\; {\\breve{a}}\\in \\mathcal {X} \\rbrace $ is $\\sigma $ -stable for any $\\sigma $ -orbit $\\mathcal {X} $ of $\\breve{\\Delta }$ with ${\\breve{W}}_{\\mathcal {X}}$ finite.",
"Remark 6.2 (1) If $\\sigma $ is a quasi-split Frobenius, that is, if $G_{\\breve{F}}^\\sigma $ is quasi-split, then ${\\breve{W}}_{\\mathcal {X}}$ is finite for any $\\sigma $ -orbit $\\mathcal {X} $ in $\\breve{\\Delta }$ .",
"(2) Assuming that $G$ is absolutely simple, the finiteness assumption on ${\\breve{W}}_\\mathcal {X} $ fails only for inner forms of type $A$ , that is, if $G_{\\breve{F}}$ is split of type $A$ , and if $\\sigma $ is such that $G_{\\breve{F}}^\\sigma $ is a group over $F$ whose adjoint group is ${\\mathrm {PGL}}_1(D)$ for a suitable division algebra $D$ .",
"Such a group is anisotropic over $F$ with trivial affine Weyl group and its Iwahori-Weyl group has only length zero elements.",
"(3) Recall from §REF that the elements of $\\mathbb {S} $ are in bijection with $\\sigma $ -orbits $\\mathcal {X} $ such that ${\\breve{W}}_{\\mathcal {X}}$ is finite.",
"So, it suffices to consider such orbits to construct the Tits group over $F$ .",
"However, while the proof below uses the assumption that the $\\sigma $ -orbit $\\mathcal {X} $ is such that ${\\breve{W}}_{\\mathcal {X}}$ is finite, we will show in Proposition REF through a different argument that the finiteness assumption on ${\\breve{W}}_{\\mathcal {X}}$ can be dropped.",
"Let $\\lbrace x_{\\breve{a}}\\;|\\; {\\breve{a}}\\in \\breve{\\Delta }\\rbrace $ be an affine pinning and $\\mathcal {X} $ be a $\\sigma $ -orbit in $\\breve{\\Delta }$ such that ${\\breve{W}}_\\mathcal {X} $ is finite.",
"Let $k = \\# \\mathcal {X} $ .",
"Fix ${\\breve{a}}\\in \\mathcal {X} $ and let ${\\breve{b}}$ be the gradient of ${\\breve{a}}$ .",
"Since $W_{\\mathcal {X}}$ is finite, we have ${\\breve{w}}_{\\mathcal {X}} \\in {\\breve{W}}_{\\mathcal {X}}^\\sigma $ .",
"In particular, $W_\\mathcal {X} ^\\sigma \\ne 1$ .",
"Thus ${\\breve{b}}|_A \\ne 0$ and hence is a root $b$ in $\\Phi (G,A)$ .",
"We show that (a) There exists $u \\in \\mathfrak {O} _{{\\breve{F}}}^\\times $ such that $\\sigma ^k(x_{\\breve{a}}(u)) = x_{\\breve{a}}(u)$ .",
"Let $v \\in \\mathcal {A} (A,F) \\subset \\mathcal {A} (T, {\\breve{F}})$ and $r \\in \\mathbb {R} $ .",
"For ${\\breve{b}}\\in \\breve{\\Phi }(G,T)$ , let $U_{\\breve{b}}({\\breve{F}})_{v,r} \\subset U_{\\breve{b}}({\\breve{F}})$ denote the filtration of root subgroup $U_{{\\breve{b}}}({\\breve{F}})$ as in [8].",
"We recall the definition of the filtration of the root subgroup $U_b(F)$ (cf.",
"[8]).",
"Let $\\breve{\\Phi }^b: = \\lbrace {\\breve{c}}\\in \\breve{\\Phi }(G,T)\\;|\\; {\\breve{c}}|_A = b \\text{ or } 2b\\rbrace $ .",
"This is a $\\sigma $ -stable positively closed subset of $\\breve{\\Phi }(G,T)$ ; that is if ${\\breve{c}}_1.",
"{\\breve{c}}_2 \\in \\breve{\\Phi }^b$ such that ${\\breve{c}}+{\\breve{c}}^{\\prime }$ is a root, then ${\\breve{c}}+{\\breve{c}}^{\\prime } \\in \\breve{\\Phi }^b$ .",
"For any fixed ordering, the subset $U_b({\\breve{F}})_{v,r} := \\prod _{{\\breve{c}}\\in \\breve{\\Phi }^b, {\\breve{c}}|_A = b} U_{\\breve{c}}({\\breve{F}})_{v,r} \\prod _{{\\breve{c}}\\in \\breve{\\Phi }^b_{nd}, {\\breve{c}}|_A = 2b} U_{\\breve{c}}({\\breve{F}})_{v,2r}$ is a subgroup of $U_b({\\breve{F}})$ .",
"Let $U_b(F)_{v,r} := U_b({\\breve{F}})_{v,r} \\cap U_b(F)$ .",
"We have $U_{\\breve{b}}({\\breve{F}})_{v,r} = U_{{\\breve{b}}}({\\breve{F}})_{{\\breve{v}}_0, r+ {\\breve{b}}(v - {\\breve{v}}_0)},$ where ${\\breve{v}}_0$ is the special vertex in $\\mathcal {A} (S,{\\breve{F}})$ fixed in §REF .",
"The pinning $x_{\\breve{a}}: _a \\rightarrow U_{{\\breve{b}}}({\\breve{F}})$ satisfies $x_{{\\breve{a}}}(\\mathfrak {O} _{{\\breve{F}}}) = U_{\\breve{b}}({\\breve{F}})_{{\\breve{v}}_0,r_0}$ and $x_{{\\breve{a}}}(\\mathfrak {p} _{{\\breve{F}}}) = U_{{\\breve{b}}}({\\breve{F}})_{{\\breve{v}}_0,s_0}$ for a suitable $r_0<s_0$ .",
"Using (REF ) and then adjusting $r$ if necessary, we ensure $x_{{\\breve{a}}}(\\mathfrak {O} _{{\\breve{F}}}) = U_{{\\breve{b}}}({\\breve{F}})_{v,r},\\;\\; x_{{\\breve{a}}}(\\mathfrak {p} _{{\\breve{F}}}) = U_{{\\breve{b}}}({\\breve{F}})_{ v, r+}$ for a suitable $r \\in \\mathbb {R} $ .",
"In particular, $U_{{\\breve{b}}}({\\breve{F}})_{v,r}\\ne U_{{\\breve{b}}}({\\breve{F}})_{v,r+}$ .",
"Now [8] implies that $U_b(F)_{v,r} \\ne U_b(F)_{v,r+}$ .",
"In other words, there exists $u^{\\prime } \\in U_b(F)_{v,r}$ such that $ u^{\\prime } \\notin \\prod _{{\\breve{c}}\\in \\breve{\\Phi }^b, {\\breve{c}}|_A = b} U_{\\breve{c}}({\\breve{F}})_{v,r+} \\prod _{{\\breve{c}}\\in \\breve{\\Phi }^b_{nd}, {\\breve{c}}|_A = 2b} U_{\\breve{c}}({\\breve{F}})_{v,2r}.$ Note that $\\sigma ^k$ fixes every element of $\\mathcal {X} $ (and hence also every element of $\\Phi ^b$ ) and $\\sigma ^k(u^{\\prime }) = u^{\\prime }$ .",
"By (REF ), we have $u^{\\prime } \\in \\prod _{{\\breve{c}}\\in \\breve{\\Phi }^b, {\\breve{c}}|_A = b} U_{\\breve{c}}({\\breve{F}})_{v,r}^{\\sigma ^k} \\prod _{{\\breve{c}}\\in \\breve{\\Phi }^b_{nd}, {\\breve{c}}|_A = 2b} U_{\\breve{c}}({\\breve{F}})_{v,2r}^{\\sigma ^k},$ and $u^{\\prime } \\notin \\prod _{{\\breve{c}}\\in \\breve{\\Phi }^b, {\\breve{c}}|_A = b} U_{\\breve{c}}({\\breve{F}})_{v,r+}^{\\sigma ^k} \\prod _{{\\breve{c}}\\in \\breve{\\Phi }^b_{nd}, {\\breve{c}}|_A = 2b} U_{\\breve{c}}({\\breve{F}})_{v,2r}^{\\sigma ^k}.$ Thus there exists ${\\breve{c}}\\in \\Phi ^b, {\\breve{c}}|_A = b$ such that $ U_{\\breve{c}}({\\breve{F}})_{v,r+}^{\\sigma ^k} \\subsetneq U_{\\breve{c}}({\\breve{F}})_{v,r}^{\\sigma ^k}$ .",
"Since ${\\breve{c}}= \\sigma ^i({\\breve{b}})$ for a suitable $i<k$ , we also have $U_{\\breve{b}}({\\breve{F}})_{v,r+}^{\\sigma ^k} \\subsetneq U_{\\breve{b}}({\\breve{F}})_{v,r}^{\\sigma ^k}.$ Let $u^{\\prime \\prime } \\in U_{\\breve{b}}({\\breve{F}})_{v,r}^{\\sigma ^k}\\backslash U_{\\breve{b}}({\\breve{F}})_{v,r+}^{\\sigma ^k}$ and $u= x_{\\breve{a}}^{-1}(u^{\\prime \\prime })$ .",
"Then $\\sigma ^k(x_{\\breve{a}}(u)) = x_{\\breve{a}}(u)$ .",
"By (REF ), $u\\in \\mathfrak {O} _{{\\breve{F}}}^\\times $ .",
"(a) is proved.",
"Since $x_{\\breve{a}}(u) x_{-{\\breve{a}}}(u^{-1}) x_{\\breve{a}}(u) \\in N_G(T)({\\breve{F}})$ , we have $x_{\\breve{a}}(u) \\sigma ^k(x_{-{\\breve{a}}}(u^{-1}))x_{\\breve{a}}(u)=\\sigma ^k(x_{\\breve{a}}(u)) \\sigma ^k(x_{-{\\breve{a}}}(u^{-1})) \\sigma ^k(x_{\\breve{a}}(u))\\in N_G(S)({\\breve{F}}).$ The uniqueness assertion in [8] implies that $\\sigma ^k(x_{-{\\breve{a}}}(u^{-1})) = x_{-{\\breve{a}}}(u^{-1})$ .",
"Let $x_{\\breve{a}}^{\\prime } = x_{\\breve{a}}\\circ m_u$ , where $m_u$ is the multiplication by $u$ .",
"We consider the pinning $\\lbrace x_{\\breve{a}}^{\\prime }, \\sigma \\circ x_{\\breve{a}}^{\\prime }, \\cdots \\sigma ^{k-1} \\circ x_{{\\breve{a}}}^{\\prime }\\rbrace $ .",
"Then $x_{{\\breve{a}}}^{\\prime }(1) =x_{{\\breve{a}}}(u)$ and $x_{-{\\breve{a}}}^{\\prime }(1) = x_{-{\\breve{a}}}(u^{-1})$ .",
"For ${\\breve{c}}\\in \\mathcal {X} $ , let $n_{s_{\\breve{c}}}^{\\prime } = x_{{\\breve{c}}}^{\\prime }(1)x_{-{\\breve{c}}}^{\\prime }(1)x_{{\\breve{c}}}^{\\prime }(1)$ be the representative in $N_G(S)({\\breve{F}})$ of $s_{\\breve{c}}$ obtained using this pinning.",
"Then $\\sigma ^{k}(n_{s_{\\breve{c}}}^{\\prime } )= n_{s_{\\breve{c}}}^{\\prime }$ and the set $\\lbrace n_{s_{\\breve{c}}}^{\\prime }\\;|\\; {\\breve{c}}\\in \\mathcal {X} \\rbrace = \\lbrace n_{s_{\\breve{a}}}^{\\prime }, \\sigma (n_{s_{\\breve{a}}}^{\\prime }), \\cdots , \\sigma ^{k-1}(n_{s_{\\breve{a}}}^{\\prime })\\rbrace $ is $\\sigma $ -stable."
],
[
"The Frobenius morphism for each inner form",
"In this rest of this section, let $G$ denote a connected, reductive group over $F$ that is quasi-split over $F$ and split over ${\\breve{F}}$ .",
"Let $\\sigma $ denote the Frobenius morphism on $G_{\\breve{F}}$ so that the $F$ -structure it yields is $G$ .",
"We will later construct for each $F$ -isomorphism class of inner twists of $G$ a suitable Frobenius morphism $\\sigma ^*$ and let $G^* = G_{\\breve{F}}^{\\sigma ^*}$ be the $F$ -group in the given isomorphism class of inner twists.",
"By Proposition REF , there exists an affine pinning $\\lbrace x_{\\breve{a}}\\;|\\; {\\breve{a}}\\in \\breve{\\Delta }\\rbrace $ such that the set $\\lbrace n_{{\\breve{s}}}\\;|\\; {\\breve{s}}\\in \\breve{\\mathbb {S}}\\rbrace $ is $\\sigma $ -stable.",
"For ${\\breve{\\lambda }}\\in X_*(T)$ , let $n_{\\breve{\\lambda }}= {\\breve{\\lambda }}(\\varpi _F)$ .",
"Then $\\sigma (n_{{\\breve{\\lambda }}})=n_{\\sigma ({\\breve{\\lambda }})}$ .",
"Let $\\breve{\\mathcal {T}}$ be the Tits group of ${\\breve{W}}$ generated by $\\breve{S}_2$ , $\\lbrace n_{{\\breve{s}}}\\;|\\; {\\breve{s}}\\in \\breve{\\mathbb {S}}\\rbrace $ and $\\lbrace n_{{\\breve{\\lambda }}}\\;|\\;{\\breve{\\lambda }}\\in X_*(T)\\rbrace $ .",
"Then $\\breve{\\mathcal {T}}$ is stable under the action of $\\sigma $ .",
"We will choose a suitable Frobenius morphism $\\sigma ^*$ for each $F$ -isomorphism class of inner twists of $G$ such that $\\breve{\\mathcal {T}}$ is stable under the action of $\\sigma ^*$ .",
"Finally, we will show that $\\breve{\\mathcal {T}}^{\\sigma ^*}$ is a Tits group over $F$ for $G^*$ ."
],
[
"The group $(\\Omega _{{\\breve{{a}}},{\\mathrm {ad}}})_{\\sigma }$",
"The $F$ -isomorphism classes of inner twists of $G$ is parametrized by $H^1(\\langle \\sigma \\rangle , G_{{\\mathrm {ad}}}({\\breve{F}}))$ .",
"By [9], we have $H^1(\\langle \\sigma \\rangle , G_{{\\mathrm {ad}}}({\\breve{F}})) \\cong H^1(\\langle \\sigma \\rangle , \\Omega _{{\\breve{{a}}}, {\\mathrm {ad}}})=(\\Omega _{{\\breve{{a}}},{\\mathrm {ad}}})_{\\sigma }.$ Now, we describe the group $\\Omega _{{\\breve{{a}}},{\\mathrm {ad}}}$ in more detail.",
"We may assume that $G_{{\\breve{F}}, {\\mathrm {ad}}}$ is ${\\breve{F}}$ -simple.",
"We will use the same labeling of the roots in $\\Phi (G,T)$ as in [6] and we denote the indexing set of simple reflections by $I$ .",
"With this, the set $\\lbrace s_{\\breve{a}}\\;|\\; {\\breve{a}}\\in \\breve{\\Delta }_0\\rbrace $ is identified with $\\lbrace {\\breve{s}}_i\\;|\\; i \\in I\\rbrace $ .",
"Note that we have used the letter $I$ for the indexing set for the simple reflections; this should not cause any confusion, since the Iwahori subgroup will not be mentioned in the rest of this paper.",
"For $J \\subset I$ , let ${\\breve{y}}_J$ be the maximal element in the subgroup generated by ${\\breve{s}}_i, i \\in J$ .",
"Let $\\breve{\\rho }^\\vee $ be the half-sum of the positive coroots in any positive system.",
"Let $i \\in I$ .",
"If $\\omega _i$ is minuscule, we denote by ${\\breve{\\nu }}_{{\\mathrm {ad}}, (i)}=t_{\\omega ^\\vee _i} {\\breve{y}}_{(i)} \\in \\Omega _{{\\breve{{a}}}, {\\mathrm {ad}}}$ the corresponding element.",
"Here ${\\breve{y}}_{(i)}={\\breve{y}}_{I \\backslash \\lbrace i\\rbrace } {\\breve{y}}_I$ .",
"Note that if ${\\breve{\\nu }}_{{\\mathrm {ad}}, (i)}={\\breve{\\nu }}_{{\\mathrm {ad}}, (j)}^k$ for some $k \\in \\mathbb {N} $ , then we also have that ${\\breve{y}}_{(i)}={\\breve{y}}_{(j)}^k$ .",
"The description of $\\Omega _{{\\breve{{a}}},ad}$ is given in the following table.",
"We list according to the type of the local Dynkin diagram of $G_{{\\breve{F}},{\\mathrm {ad}}}$ .",
"We only list the types for which $\\Omega _{{\\breve{{a}}},{\\mathrm {ad}}}$ is non-trivial.",
"In the last column, we make a choice of generator ${{\\breve{\\nu }}_{{\\mathrm {ad}},0}}$ in the case where $\\Omega _{{\\breve{{a}}}, {\\mathrm {ad}}}$ is cyclic.",
"Such element ${{\\breve{\\nu }}_{{\\mathrm {ad}},0}}$ will be used later.",
"Table: The group Ω a ˘, ad \\Omega _{{\\breve{{a}}},{\\mathrm {ad}}}If $\\sigma $ acts trivially on $\\Omega _{{\\breve{{a}}},{\\mathrm {ad}}}$ , then $(\\Omega _{{\\breve{{a}}},{\\mathrm {ad}}})_{\\sigma } \\cong \\Omega _{{\\breve{{a}}},{\\mathrm {ad}}}$ .",
"If the action of $\\sigma $ on $\\Omega _{{\\breve{{a}}},{\\mathrm {ad}}}$ is nontrivial, then $(\\Omega _{{\\breve{{a}}},{\\mathrm {ad}}})_{\\sigma }={\\left\\lbrace \\begin{array}{ll} \\mathbb {Z}/2 \\mathbb {Z}, & \\text{ if } G_{{\\breve{F}},{\\mathrm {ad}}} \\text{ is of type } A_{2n+1} \\text{ or } D_n; \\\\ 1, & \\text{ otherwise}.",
"\\end{array}\\right.",
"}$"
],
[
"The construction of suitable Frobenius morphism $\\sigma ^*$",
"Let $j: G_{\\breve{F}}\\rightarrow G_{{\\breve{F}}, {\\mathrm {ad}}}$ denote the adjoint quotient.",
"This induces maps $T \\rightarrow T_{{\\mathrm {ad}}}$ , ${\\breve{W}}\\rightarrow {\\breve{W}}_{{\\mathrm {ad}}}$ and $\\Omega _{{\\breve{{a}}}} \\rightarrow \\Omega _{{\\breve{{a}}},{\\mathrm {ad}}}$ , and we will denote all these maps by $j$ as well.",
"The exact sequence $1 \\rightarrow Z \\rightarrow T \\rightarrow T_{{\\mathrm {ad}}} \\rightarrow 1$ induces exact sequences $1 \\rightarrow X_*(Z^0) \\rightarrow X_*(T) \\xrightarrow{} X_*(T_{{\\mathrm {ad}}})$ and $1 \\rightarrow X_*(Z^0) \\rightarrow \\Omega _{\\breve{{a}}}\\xrightarrow{} \\Omega _{{\\breve{{a}}},{\\mathrm {ad}}},$ where $Z$ is the center of $G$ and $Z^0$ is the maximal torus in the center of $G$ .",
"We will construct a suitable Frobenius morphism $\\sigma ^*$ associated to each $F$ -isomorphism class of inner twists of $G$ .",
"It suffices to consider the case where $G_{F, {\\mathrm {ad}}}$ is $F$ -simple.",
"We first discuss the case where $G_{{\\breve{F}}, {\\mathrm {ad}}}$ is ${\\breve{F}}$ -simple.",
"We choose as follows the element ${{\\breve{\\nu }}_{{\\mathrm {ad}}}}$ in $\\Omega _{{\\breve{{a}}}, {\\mathrm {ad}}}$ whose image in $(\\Omega _{{\\breve{{a}}},{\\mathrm {ad}}})_\\sigma $ the parametrizes the inner twist $G^*$ of $G$ .",
"If $G_{{\\breve{F}}, {\\mathrm {ad}}}$ is of type $A_{2n+1}$ or $D_{2n+1}$ for some $n \\in \\mathbb {N} $ and the $\\sigma $ -action on $\\Omega _{{\\breve{{a}}}, {\\mathrm {ad}}}$ is nontrivial, then $\\Omega _{{\\breve{{a}}},{\\mathrm {ad}}}$ is nontrivial, then $(\\Omega _{{\\breve{{a}}},{\\mathrm {ad}}})_{\\sigma }=\\mathbb {Z}/2 \\mathbb {Z} $ .",
"In this case, we take ${{\\breve{\\nu }}_{{\\mathrm {ad}}}}=1$ if $G^*$ is quasi-split over $F$ and ${{\\breve{\\nu }}_{{\\mathrm {ad}}}}={{\\breve{\\nu }}_{{\\mathrm {ad}},0}}$ if $G^*$ is not quasi-split over $F$ .",
"Here ${{\\breve{\\nu }}_{{\\mathrm {ad}},0}}$ is the generator of $\\Omega _{{\\breve{{a}}}, {\\mathrm {ad}}}$ listed in Table REF .",
"In other cases, we may take ${{\\breve{\\nu }}_{{\\mathrm {ad}}}}$ to be any element in $\\Omega _{{\\breve{{a}}}, {\\mathrm {ad}}}$ that corresponds to the $F$ -isomorphism class of $G^*$ .",
"The choice of ${{\\breve{\\nu }}_{{\\mathrm {ad}}}}$ is not essential, but will simplify some calculations in the rest of this section.",
"Let ${{\\breve{\\nu }}_{{\\mathrm {ad}}}}=t_{{\\breve{\\eta }}_{{\\mathrm {ad}}}}{\\breve{z}}$ .",
"We construct suitable liftings of $t_{{{\\breve{\\eta }}_{{\\mathrm {ad}}}}}$ and ${\\breve{z}}$ .",
"The lifting of $t_{{{\\breve{\\eta }}_{{\\mathrm {ad}}}}}$ is constructed as follows.",
"Note that the quotient $X_*(T_{\\mathrm {ad}})/j(X_*(T))$ is finite.",
"Consider the element $t_{{{\\breve{\\eta }}_{{\\mathrm {ad}}}}} \\in X_*(T_{{\\mathrm {ad}}})$ .",
"Let $k\\ge 1$ be the smallest integer such that $k{{\\breve{\\eta }}_{{\\mathrm {ad}}}}\\in j(X_*(T))$ .",
"Write $k{{\\breve{\\eta }}_{{\\mathrm {ad}}}}= j({\\breve{\\eta }}) \\text{ for some } {\\breve{\\eta }}\\in X_*(T).$ Let $ {\\breve{\\nu }}= t_{\\breve{\\eta }}{\\breve{z}}.$ Note that ${\\breve{\\nu }}\\in {\\breve{W}}$ , but need not lie in $\\Omega _{{\\breve{{a}}}}$ .",
"We know that $n_{{\\breve{\\eta }}_{{\\mathrm {ad}}}}={{\\breve{\\eta }}_{{\\mathrm {ad}}}}(\\varpi _F)$ .",
"Set $g_{\\breve{\\eta }}= {\\breve{\\eta }}(\\varpi _F^{1/k}).$ Note that $g_{\\breve{\\eta }}\\in T(\\bar{F}) \\subset G(\\bar{F})$ and $j(g_{\\breve{\\eta }}) = n_{{\\breve{\\eta }}_{{\\mathrm {ad}}}}.$ Note that for each root ${\\breve{a}}\\in \\breve{\\Phi }(G,T)$ , we have $\\frac{\\langle {\\breve{a}}, {\\breve{\\eta }}\\rangle }{k} \\in \\mathbb {Z} $ because $j({\\breve{a}}) = {\\breve{a}}$ and $\\langle {\\breve{a}}, {\\breve{\\eta }}\\rangle = \\langle j({\\breve{a}}), j({\\breve{\\eta }}) \\rangle = k \\langle {\\breve{a}}, {{\\breve{\\eta }}_{{\\mathrm {ad}}}}\\rangle \\in k\\mathbb {Z} $ .",
"In particular, conjugation by $g_{\\breve{\\eta }}$ preserves $G(\\breve{F})$ .",
"Now we construct a lifting $g_{{\\breve{z}}}$ of ${\\breve{z}}$ in $\\breve{\\mathcal {T}}$ .",
"If $G_{{\\breve{F}},{\\mathrm {ad}}}$ is of type $D_n$ with $n$ even, then we set $g_{{\\breve{z}}}=n_{{\\breve{z}}}$ .",
"Otherwise, $\\Omega _{{\\breve{{a}}}, {\\mathrm {ad}}}$ is a cyclic group.",
"From our construction, ${{\\breve{\\nu }}_{{\\mathrm {ad}}}}={{\\breve{\\nu }}_{{\\mathrm {ad}},0}}^i$ for $0 \\leqslant i<|(\\Omega _{{\\breve{{a}}}, {\\mathrm {ad}}})_\\sigma |$ .",
"We then have ${\\breve{z}}={\\breve{z}_0}^i$ .",
"Set $g_{{\\breve{z}}}=n_{{\\breve{z}_0}}^i$ .",
"Let $g_{\\breve{\\nu }}= g_{\\breve{\\eta }}g_{\\breve{z}}\\in G(\\bar{F})$ .",
"We set $\\sigma ^*={\\mathrm {Ad}}(g_{{\\breve{\\nu }}}) \\circ \\sigma .$ Next, suppose $G_{{\\breve{F}},{\\mathrm {ad}}}$ is not simple.",
"By our assumption $G_{F, {\\mathrm {ad}}}$ is simple.",
"We may write $G_{{\\mathrm {ad}}} =\\operatorname{Res}_{L_k/F} G_{{\\mathrm {ad}}}^{\\prime }$ , where $L_k$ is a finite unramified extension of $F$ of degree $k$ contained in ${\\breve{F}}$ and $G^{\\prime }_{{\\breve{F}}, {\\mathrm {ad}}}$ is ${\\breve{F}}$ -simple.",
"Then $G_{{\\breve{F}}, {\\mathrm {ad}}} =G_{{\\breve{F}}, {\\mathrm {ad}}}^{(1)} \\times \\cdots \\times G_{{\\breve{F}}, {\\mathrm {ad}}}^{(k)},$ where $G_{{\\breve{F}}, {\\mathrm {ad}}}^{(1)} \\cong \\cdots \\cong G_{{\\breve{F}}, {\\mathrm {ad}}}^{(k)}\\cong G^{\\prime }_{{\\breve{F}}, {\\mathrm {ad}}}$ and the action of $\\sigma $ permutes transitively the simple factors $G_{{\\breve{F}}, {\\mathrm {ad}}}^{(1)}, \\ldots , G_{{\\breve{F}}, {\\mathrm {ad}}}^{(k)}$ .",
"We may also write $\\Omega _{{\\breve{{a}}}, {\\mathrm {ad}}} = \\Omega _{{\\breve{{a}}}, {\\mathrm {ad}}}^{(1)} \\times \\cdots \\times \\Omega _{{\\breve{{a}}}, {\\mathrm {ad}}}^{(k)}$ , where $\\Omega _{{\\breve{{a}}}, {\\mathrm {ad}}}^{(1)} \\cong \\cdots \\cong \\Omega _{{\\breve{{a}}},{\\mathrm {ad}}}^{(k)}$ are as in Table REF .",
"Then the projection map $\\Omega _{{\\breve{{a}}}, {\\mathrm {ad}}}^{(1)} {@display}{\\longrightarrow }{\\rightarrow }(\\Omega _{{\\breve{{a}}}, {\\mathrm {ad}}})_{\\sigma }$ is surjective.",
"In fact, $(\\Omega _{{\\breve{{a}}}, {\\mathrm {ad}}}^{(1)})_{\\sigma ^k} \\xrightarrow{} (\\Omega _{{\\breve{{a}}}, {\\mathrm {ad}}})_{\\sigma }$ .",
"Let ${{\\breve{\\nu }}_{{\\mathrm {ad}}}}= t_{{\\breve{\\eta }}_{{\\mathrm {ad}}}}{\\breve{z}}\\in \\Omega _{{\\breve{{a}}}, {\\mathrm {ad}}}^{(1)}$ such that its image in $(\\Omega _{{\\breve{{a}}}, {\\mathrm {ad}}})_{\\sigma }$ parametrizes the isomorphism class of $G^*$ .",
"We construct $g_{{\\breve{\\nu }}} \\in G(\\bar{F})$ as above.",
"More precisely, write ${{\\breve{\\nu }}_{{\\mathrm {ad}}}}= t_{{\\breve{\\eta }}_{{\\mathrm {ad}}}}{\\breve{z}}$ .",
"Then $t_{\\breve{\\eta }}$ and $g_{\\breve{\\eta }}$ have been constructed in (REF ).",
"If $G_{{\\breve{F}},{\\mathrm {ad}}}^{(1)}$ is of type $D_n$ with $n$ even, then we set $g_{{\\breve{z}}}=n_{{\\breve{z}}}$ .",
"Otherwise we have ${{\\breve{\\nu }}_{{\\mathrm {ad}}}}={{\\breve{\\nu }}_{{\\mathrm {ad}},0}}^i$ for $0 \\leqslant i<|\\left(\\Omega _{{\\breve{{a}}}, {\\mathrm {ad}}}^{(1)}\\right)_{\\sigma ^k}|$ .",
"We also have ${\\breve{z}}={\\breve{z}_0}^i$ .",
"Set $g_{{\\breve{z}}}=n_{{\\breve{z}_0}}^i$ .",
"Let $g_{\\breve{\\nu }}= g_{\\breve{\\eta }}g_{\\breve{z}}$ and let $\\sigma ^* = Ad(g_{\\breve{\\nu }}) \\circ \\sigma $ ."
],
[
"The action of $\\sigma ^*$ on {{formula:9d5963b0-7be0-4fed-bceb-1c3c38a12a07}}",
"It is easy to see that $\\breve{S}_2$ is stable under the action of $\\sigma ^*$ .",
"For each ${\\breve{\\lambda }}\\in X_*(T)$ , we have $\\sigma ^*(n_{\\breve{\\lambda }}) = {\\mathrm {Ad}}(g_{\\breve{z}})(n_{\\sigma (\\lambda )}) = n_{{\\breve{z}}(\\sigma (\\lambda ))} = n_{\\sigma ^*({\\breve{\\lambda }})}.$ Note that $\\sigma ^*$ acts as ${\\mathrm {Ad}}({{\\breve{\\eta }}_{{\\mathrm {ad}}}}) \\circ \\sigma $ on ${\\breve{W}}_{{\\mathrm {ad}}}$ .",
"So for any ${\\breve{y}}\\in {\\breve{W}}_0$ , $\\sigma ^*({\\breve{y}}) = t_{{{\\breve{\\eta }}_{{\\mathrm {ad}}}}- {\\breve{y}}^{\\prime }({{\\breve{\\eta }}_{{\\mathrm {ad}}}})} {\\breve{y}}^{\\prime }$ , where ${\\breve{y}}^{\\prime } = {\\mathrm {Ad}}({\\breve{z}})(\\sigma ({\\breve{y}}))$ .",
"Note that ${{\\breve{\\eta }}_{{\\mathrm {ad}}}}- {\\breve{y}}^{\\prime }({{\\breve{\\eta }}_{{\\mathrm {ad}}}})\\in \\mathbb {Z} \\Phi ^\\vee (G,T)$ .",
"Since $\\breve{\\mathcal {T}}$ is generated by $\\breve{S}_2$ , $m({\\breve{y}})$ for ${\\breve{y}}\\in {\\breve{W}}_0$ and $n_{{\\breve{\\lambda }}}$ for ${\\breve{\\lambda }}\\in X_*(T)$ , by the following lemma, we have $\\sigma ^*(\\breve{\\mathcal {T}})=\\breve{\\mathcal {T}}.$ Lemma 6.3 Let ${\\breve{y}}\\in {\\breve{W}}_0$ .",
"Then for any lifting $m({\\breve{y}}) \\in \\breve{\\mathcal {T}}$ , we have $\\sigma ^*(m({\\breve{y}})) = n_{{{\\breve{\\eta }}_{{\\mathrm {ad}}}}- {\\breve{y}}^{\\prime }({{\\breve{\\eta }}_{{\\mathrm {ad}}}})} g_{\\breve{z}}\\sigma (m({\\breve{y}}))g_{\\breve{z}}^{-1}.$ We have ${\\mathrm {Ad}}({\\breve{\\nu }})(\\sigma ({\\breve{y}})) = t_{{\\breve{\\eta }}- {\\breve{y}}^{\\prime }({\\breve{\\eta }})} {\\breve{z}}\\sigma ({\\breve{y}}) {\\breve{z}}^{-1}$ .",
"Now $ j({\\breve{\\eta }}-{\\breve{y}}^{\\prime }({\\breve{\\eta }})) &=j({\\breve{\\eta }})-j({\\breve{y}}^{\\prime }({\\breve{\\eta }}))=k({{\\breve{\\eta }}_{{\\mathrm {ad}}}})-{\\breve{y}}^{\\prime }(j({\\breve{\\eta }}))=k({{\\breve{\\eta }}_{{\\mathrm {ad}}}})-k {\\breve{y}}^{\\prime }({{\\breve{\\eta }}_{{\\mathrm {ad}}}}) \\\\&=k({{\\breve{\\eta }}_{{\\mathrm {ad}}}}- {\\breve{y}}^{\\prime }({{\\breve{\\eta }}_{{\\mathrm {ad}}}})) = k(j({{\\breve{\\eta }}_{{\\mathrm {ad}}}}- {\\breve{y}}^{\\prime }({{\\breve{\\eta }}_{{\\mathrm {ad}}}}))).$ Here the last equality follows from the fact that ${{\\breve{\\eta }}_{{\\mathrm {ad}}}}- {\\breve{y}}^{\\prime }({{\\breve{\\eta }}_{{\\mathrm {ad}}}}) \\in \\mathbb {Z} \\breve{\\Phi }^\\vee (G,T)$ and the restriction of the map $j$ to $\\mathbb {Z} \\breve{\\Phi }^\\vee (G,T)$ (which is just $X_*(T_{sc})$ ) is the identity map.",
"Hence there exists ${\\breve{\\mu }}\\in X_*(Z^0)$ such that ${\\breve{\\eta }}- {\\breve{y}}^{\\prime }({\\breve{\\eta }}) = k({{\\breve{\\eta }}_{{\\mathrm {ad}}}}- {\\breve{y}}^{\\prime }({{\\breve{\\eta }}_{{\\mathrm {ad}}}})) +{\\breve{\\mu }}.$ Since ${\\breve{\\mu }}\\in X_*(Z^0)$ , ${\\breve{y}}^{\\prime }({\\breve{\\mu }})={\\breve{\\mu }}$ and thus $({\\breve{y}}^{\\prime })^{i}({\\breve{\\eta }}) - ({\\breve{y}}^{\\prime })^{i+1}({\\breve{\\eta }}) = k(({\\breve{y}}^{\\prime })^{i}({{\\breve{\\eta }}_{{\\mathrm {ad}}}}) - ({\\breve{y}}^{\\prime })^{i+1}({{\\breve{\\eta }}_{{\\mathrm {ad}}}}))+{\\breve{\\mu }}$ for any $i$ .",
"Let $l$ be the order of ${\\breve{y}}^{\\prime }$ .",
"Then $l {\\breve{\\mu }}=\\sum _{i=0}^{l-1} \\bigl (k(({\\breve{y}}^{\\prime })^{i}({{\\breve{\\eta }}_{{\\mathrm {ad}}}}) - ({\\breve{y}}^{\\prime })^{i+1}({{\\breve{\\eta }}_{{\\mathrm {ad}}}}))+{\\breve{\\mu }}\\bigr )=\\sum _{i=0}^{l-1} \\bigl (({\\breve{y}}^{\\prime })^{i}({\\breve{\\eta }}) - ({\\breve{y}}^{\\prime })^{i+1}({\\breve{\\eta }})\\bigr )=0.$ So ${\\breve{\\mu }}=0$ and ${\\breve{\\eta }}- {\\breve{y}}^{\\prime }({\\breve{\\eta }}) = k({{\\breve{\\eta }}_{{\\mathrm {ad}}}}- {\\breve{y}}^{\\prime }({{\\breve{\\eta }}_{{\\mathrm {ad}}}}))$ .",
"Then $\\sigma ^*(m({\\breve{y}})) = g_{\\breve{\\eta }}{\\mathrm {Ad}}({\\breve{y}}^{\\prime })(g_{{\\breve{\\eta }}}^{-1}) g_{\\breve{z}}\\sigma (m({\\breve{y}}))g_{{\\breve{z}}}^{-1}.$ It remains to show that $g_{\\breve{\\eta }}{\\mathrm {Ad}}({\\breve{y}}^{\\prime })(g_{{\\breve{\\eta }}}^{-1}) =n_{{{\\breve{\\eta }}_{{\\mathrm {ad}}}}- {\\breve{y}}^{\\prime }({{\\breve{\\eta }}_{{\\mathrm {ad}}}})}$ .",
"With the definition of $g_{\\breve{\\eta }}$ in (REF ), we have $g_{\\breve{\\eta }}{\\mathrm {Ad}}({\\breve{y}}^{\\prime })(g_{{\\breve{\\eta }}}^{-1}) &= ({\\breve{\\eta }}- {\\breve{y}}^{\\prime }({\\breve{\\eta }}))(\\varpi _F^{1/k})= (\\tilde{\\eta }_{{\\mathrm {ad}}} - {\\breve{y}}^{\\prime }(\\tilde{\\eta }_{\\mathrm {ad}}))(\\varpi _F) = n_{{{\\breve{\\eta }}_{{\\mathrm {ad}}}}- {\\breve{y}}^{\\prime }({{\\breve{\\eta }}_{{\\mathrm {ad}}}})}.$ This finishes the proof of the lemma.",
"Now we state the main result of this section.",
"Theorem 6.4 Let $G$ be a connected reductive group, quasi-split over $F$ and split over $\\breve{F}$ .",
"Let $\\sigma ^*$ be the Frobenius morphism associated to a given $F$ -isomorphism class of inner twists of $G$ .",
"Then $\\breve{\\mathcal {T}}^{\\sigma ^*}$ is a Tits group of the Iwahori-Weyl group of $G(\\breve{F})^{\\sigma ^*}$ .",
"In the rest of this section, we will prove this theorem.",
"The proof involves, among other things, some identities on the finite Tits groups, which we now summarize."
],
[
"Some identities in finite Tits groups",
"In this subsection, let $\\mathfrak {F}$ be any field and let $G$ be a split, connected, reductive group over $\\mathfrak {F}$ .",
"Let $\\mathcal {T} _{\\text{fin}}$ be the Tits group of the absolute Weyl group $W_0$ of $G(\\mathfrak {F})$ and $\\lbrace n_w\\rbrace _{w \\in W_0}$ is a Tits cross-section of $W$ on $\\mathcal {T} _{\\text{fin}}$ .",
"In the application to the proof of Theorem REF , $\\mathfrak {F}$ is the field $\\breve{F}$ and $\\mathcal {T} _{\\text{fin}}$ is the subgroup of $\\breve{\\mathcal {T}}$ generated by $n_{{\\breve{s}}}$ for ${\\breve{s}}\\in \\breve{W}_0$ .",
"However, the identities on the finite Tits group hold in the general setting.",
"Let $\\lbrace s_i\\;|\\; i \\in I\\rbrace $ be the set of simple reflections of the absolute Weyl group and $\\lbrace a^\\vee _i\\;|\\;i \\in I\\rbrace $ be the set of simple coroots.",
"Then $n_{s_i}^2=a^\\vee _i(-1)$ .",
"For any subset $J \\subset I$ , let $\\rho ^\\vee _J$ be the half sum of positive coroots spanned by $\\lbrace a_j^\\vee \\;|\\; j \\in J\\rbrace $ and $y_J$ be the maximal element in the subgroup generated by $s_i$ for $i \\in J$ .",
"We will simply write $\\rho ^\\vee $ for $\\rho ^\\vee _{I}$ .",
"For any $i \\in I$ , we set $y_{(i)}=y_{I\\backslash \\lbrace i\\rbrace } y_{I}.$ We are interested in the power of $n_{y_{(i)}}$ when $\\omega ^\\vee _i$ is a minuscule coweight.",
"This is calculated using the following result.",
"Proposition 6.5 (Proposition 3.2.1 of [19]) Let $u, v \\in W_0$ .",
"Then $n_u n_v=n_{u v} \\; \\Pi _{a>0, v(a)<0, u v(a)>0} a^\\vee (-1).$ The following corollary is an easy consequence of the proposition above and some results in [1].",
"Corollary 6.6 Suppose that $G_{\\mathfrak {F}, {\\mathrm {ad}}}$ is $\\mathfrak {F}$ -simple.",
"Let $\\omega ^\\vee _i$ be a minuscule coweight and $k$ be the order of $y_{(i)}$ in $W_0$ .",
"Then $n_{y_{(i)}}^k$ is the center of $G(\\mathfrak {F})$ .",
"Suppose that $G$ is of type $A_n$ .",
"For $0 \\leqslant i \\leqslant n$ , $n_{y_{(1)}}^{i+1} = {\\left\\lbrace \\begin{array}{ll} n_{y_{(1)}^{i+1}}, & \\text{ if } i \\text{ is even},\\\\(a_1 ^\\vee + a_3^{\\vee } + \\cdots + a_{i}^\\vee )(-1) n_{y_{(1)}^{i+1}}, & \\text{ if } i \\text{ is odd}.\\end{array}\\right.}",
"$ Suppose that $G$ is of type $D_n$ with $n$ odd.",
"Then (a) $n_{y_{(n)}}^2 = {\\left\\lbrace \\begin{array}{ll}(a_2^\\vee +a_4^\\vee + \\cdots +a_{n-1}^\\vee )(-1) n_{y_{(1)}}, & \\text{ if } n \\equiv 1 \\mod {4}, \\\\(a_2^\\vee +a_4^\\vee + \\cdots +a_{n-3}^\\vee +a_n^\\vee )(-1) n_{y_{(1)}}, & \\text{ if } n \\equiv 3 \\mod {4}.\\end{array}\\right.}",
"$ (b) $n_{y_{(n)}}^3 = {\\left\\lbrace \\begin{array}{ll}(a_{n-1}^\\vee +a_n^\\vee )(-1) n_{y_{(n-1)}}, & \\text{ if } n \\equiv 1 \\mod {4}, \\\\n_{y_{(n-1)}}, & \\text{ if } n \\equiv 3 \\mod {4}.\\end{array}\\right.}",
"$ (c) $n_{y_{(n)}}^4=(a^\\vee _{n-1}+a^\\vee _n)(-1)$ .",
"Suppose that $G_{\\mathfrak {F}, {\\mathrm {ad}}}$ is of type $D_{2n}$ and $\\lbrace i, j, k\\rbrace =\\lbrace 1, 2n-1, 2n\\rbrace $ .",
"Then there exists a central element $z$ of $G(\\mathfrak {F})$ such that $n_{y_{(i)}}n_{y_{(j)}} = z\\cdot n_{y_{(k)}}= n_{y_{(j)}}n_{y_{(i)}}.$ All the parts of the corollary are consequences of Proposition REF and some explicit calculations.",
"In the case where $G$ is almost simple over $\\mathfrak {F}$ , Adrian showed in [1] that $n_{y_{(i)}}^k=1$ , where $k$ is the order of $y_{(i)}$ in $W_0$ .",
"This implies (1).",
"Parts (2), (3) are direct consequences of Proposition REF .",
"Part (4) is deduced from [1]."
],
[
"The $\\sigma ^*$ -stable liftings of {{formula:f6426599-8036-473c-8401-a63bd0e64408}}",
"In this subsection, we will prove the following result.",
"Proposition 6.7 Let ${\\breve{s}}\\in \\breve{\\mathbb {S}}_0$ and $\\mathcal {X} $ be the $\\sigma ^*$ -orbit of ${\\breve{s}}$ .",
"Then $(\\sigma ^*)^{|\\mathcal {X} |}(n_{\\breve{s}})=n_{\\breve{s}}.$ As a consequence, we obtain the following stronger version of Proposition REF .",
"Corollary 6.8 There exists a set of representatives $\\lbrace m({\\breve{s}})\\;|\\; {{\\breve{s}}\\in \\breve{\\mathbb {S}}}\\rbrace $ in $\\breve{\\mathcal {T}}$ that is $\\sigma ^*$ -stable.",
"It suffices to consider the case where $G_{F, {\\mathrm {ad}}}$ is $F$ -simple.",
"In this case, $\\sigma ^*$ acts transitively on the set of connected components of the affine Dynkin diagram of $G_F$ .",
"The case $\\sigma ^*=\\sigma $ is already proved.",
"Now we assume that $\\sigma ^* \\ne \\sigma $ .",
"Then each $\\sigma ^*$ -orbit on $\\breve{\\mathbb {S}}$ contains a simple reflection in $\\breve{\\mathbb {S}}_0$ .",
"For each $\\sigma ^*$ -orbit $\\mathcal {X} $ , we fix a representative ${\\breve{s}}_\\mathcal {X} $ such that ${\\breve{s}}_\\mathcal {X} \\in \\mathcal {X} \\cap \\breve{\\mathbb {S}}_0$ .",
"Then any element ${\\breve{s}}\\in \\breve{\\mathbb {S}}$ is of the form ${\\breve{s}}=(\\sigma ^*)^l({\\breve{s}}_\\mathcal {X})$ for a unique $\\sigma ^*$ -orbit $\\mathcal {X} $ and a unique $l$ with $0 \\leqslant l<|\\mathcal {X} |$ .",
"We then set $m({\\breve{s}})=(\\sigma ^*)^l n_{{\\breve{s}}_{\\mathcal {X}}} \\in \\breve{\\mathcal {T}}$ .",
"By Proposition REF , $\\lbrace m({\\breve{s}})\\;|\\; {{\\breve{s}}\\in \\breve{\\mathbb {S}}}\\rbrace $ is $\\sigma ^*$ -stable."
],
[
"Reduction step",
"We first explain how to reduce ourselves to the case when $G_{{\\breve{F}}, {\\mathrm {ad}}}$ is ${\\breve{F}}$ -simple.",
"We keep notations as in §REF .",
"Note that ${\\breve{W}}_{\\text{af}} ={\\breve{W}}_{\\text{af}}^{(1)} \\times {\\breve{W}}_{\\text{af}}^{(2)} \\cdots \\times {\\breve{W}}_{\\text{af}}^{(k)},$ with ${\\breve{W}}_{\\text{af}}^{(1)} \\cong {\\breve{W}}_{\\text{af}}^{(2)} \\cong \\cdots \\cong {\\breve{W}}_{\\text{af}}^{(k)} $ and $\\sigma $ permutes these factors transitively.",
"Write $\\breve{\\mathbb {S}}= \\breve{\\mathbb {S}}^{(1)} \\times \\cdots \\times \\breve{\\mathbb {S}}^{(k)}$ .",
"The element ${\\breve{s}}= ({\\breve{s}}^{(1)}, {\\breve{s}}^{(2)}\\cdots , {\\breve{s}}^{(k)})$ and $n_{{\\breve{s}}} = n_{{\\breve{s}}^{(1)}}\\cdot n_{{\\breve{s}}^{(2)}} \\cdots \\cdot n_{{\\breve{s}}^{(k)}}$ .",
"Note that $\\sigma ^k$ stabilizes the set $\\breve{\\mathbb {S}}^{(1)}$ and that $(\\sigma ^*)^k$ acts as ${\\mathrm {Ad}}({{\\breve{\\nu }}_{{\\mathrm {ad}}}}) \\circ \\sigma ^k$ on $\\breve{\\mathbb {S}}^{(1)}$ .",
"Let $\\mathcal {X} ^{(1)}$ be the $({\\mathrm {Ad}}({{\\breve{\\nu }}_{{\\mathrm {ad}}}}) \\circ \\sigma ^k)$ -orbit of ${\\breve{s}}^{(1)}$ in $\\breve{\\mathbb {S}}^{(1)}$ .",
"Note that $|\\mathcal {X} | = k|\\mathcal {X} ^{(1)}|$ .",
"Thus $(\\sigma ^*)^{|\\mathcal {X} |}(n_{{\\breve{s}}}) = n_{{\\breve{s}}}$ if and only if $({\\mathrm {Ad}}(g_{\\breve{\\nu }}) \\circ \\sigma ^k)^{|\\mathcal {X} ^{(1)}|}(n_{{\\breve{s}}^{(1)}}) = n_{{\\breve{s}}^{(1)}}$ .",
"In particular, we may reduce ourselves to the case when $k=1$ , i.e., the case when $G_{{\\breve{F}}, {\\mathrm {ad}}}$ is ${\\breve{F}}$ -simple.",
"In this case, $\\Omega _{{\\breve{{a}}}, {\\mathrm {ad}}}$ is as in Table REF and we drop all the superscripts in the rest of the argument.",
"Next we show that it suffices to prove the equality (REF ) below.",
"This equality only involves the elements from the finite Tits group.",
"If ${{\\breve{\\nu }}_{{\\mathrm {ad}}}}=1$ , then $\\sigma ^*=\\sigma $ and (a) follows from the fact that the set $\\lbrace n_{{\\breve{s}}}\\;|\\; {\\breve{s}}\\in \\breve{\\mathbb {S}}\\rbrace $ is $\\sigma $ -stable.",
"Now we assume that ${{\\breve{\\nu }}_{{\\mathrm {ad}}}}\\ne 1$ .",
"Recall that ${{\\breve{\\nu }}_{{\\mathrm {ad}}}}=t_{{\\breve{\\eta }}_{{\\mathrm {ad}}}}{\\breve{z}}\\in \\Omega _{{\\breve{{a}}},{\\mathrm {ad}}}$ .",
"We have $({{\\breve{\\nu }}_{{\\mathrm {ad}}}}\\circ \\sigma )^{|\\mathcal {X} |}=t_{\\breve{\\xi }_{{\\mathrm {ad}}}} ({\\breve{z}}\\circ \\sigma )^{|\\mathcal {X} |} \\in {\\breve{W}}_{{\\mathrm {ad}}} \\rtimes \\langle \\sigma \\rangle $ for some $\\breve{\\xi }_{{\\mathrm {ad}}}\\in X_*(T_{{\\mathrm {ad}}})$ .",
"Since $(\\sigma ^*)^{|\\mathcal {X} |}({\\breve{s}})={\\breve{s}}$ , we have $({\\mathrm {Ad}}({\\breve{z}}) \\circ \\sigma )^{|\\mathcal {X} |}({\\breve{s}})={\\breve{s}}$ and ${\\breve{s}}(\\breve{\\xi }_{{\\mathrm {ad}}})=\\breve{\\xi }_{{\\mathrm {ad}}}$ .",
"Recall in §REF , we have $k \\breve{\\eta }_{{\\mathrm {ad}}}=j(\\breve{\\eta })$ for some $\\breve{\\eta }\\in X_*(T)$ .",
"Since $\\breve{\\xi }_{{\\mathrm {ad}}}$ is an integral linear combination of the ${\\breve{W}}_0$ -orbit of $\\breve{\\eta }_{{\\mathrm {ad}}}$ , we have $k \\breve{\\xi }_{{\\mathrm {ad}}}=j(\\breve{\\xi })$ for some $\\breve{\\xi }\\in X_*(T)$ .",
"By (REF ) and Lemma REF , we have $(\\sigma ^*)^{|\\mathcal {X} |}(n_{\\breve{s}})={\\mathrm {Ad}}(\\breve{\\xi }(\\varpi _F^{1/k})) ({\\mathrm {Ad}}(g_{{\\breve{z}}}) \\circ \\sigma )^{|\\mathcal {X} |}(n_{\\breve{s}}).$ By the proof of Lemma REF , we have ${\\mathrm {Ad}}(\\breve{\\xi }(\\varpi _F^{1/k})(n_{\\breve{s}})=n_{\\breve{s}}$ .",
"Thus it remains to prove that $({\\mathrm {Ad}}(g_{{\\breve{z}}}) \\circ \\sigma )^{|\\mathcal {X} |}(n_{\\breve{s}})=n_{\\breve{s}}.$"
],
[
"The case where $\\sigma ^*({\\breve{s}})={\\breve{s}}$",
"If $G_{{\\breve{F}}, {\\mathrm {ad}}}$ is of type $D_{2n}$ , $g_{\\breve{z}}= n_{\\breve{z}}$ .",
"Otherwise $g_{\\breve{z}}= n_{{\\breve{z}}_0}^i$ for a suitable $0 \\leqslant i <|(\\Omega _{{\\breve{{a}}}, {\\mathrm {ad}}})_{\\sigma }|$ .",
"For type $D_{2n}$ and for the other types with $i=1$ , the statement follows from Coxeter relations.",
"In more detail, since $\\sigma ^*({\\breve{s}}) = {\\breve{s}}$ we know that $\\sigma ^*({\\breve{b}}) = \\pm {\\breve{b}}$ .",
"Since $\\sigma ^*$ preserves $\\breve{\\Delta }$ , we see that $\\sigma ^*({\\breve{b}}) = {\\breve{b}}$ .",
"But $\\sigma ^*({\\breve{b}}) = {\\breve{z}}(\\sigma ({\\breve{b}}))$ .",
"Now, since $\\sigma ({\\breve{b}}) \\in \\breve{\\Delta }_0$ and ${\\breve{z}}(\\sigma ({\\breve{b}})) \\in \\breve{\\Delta }_0$ , we see that $l({\\breve{z}}s_{\\sigma ({\\breve{b}})}) = l({\\breve{z}})+1 = l(s_{\\breve{b}}{\\breve{z}})$ .",
"By Condition (2)(b)$^\\dagger $ of §REF , $g_{{\\breve{z}}}n_{s_{\\sigma ({\\breve{b}})}} = n_{{\\breve{z}}s_{\\sigma ({\\breve{b}})}} =n_{s_{\\breve{b}}{\\breve{z}}}= n_{s_{\\breve{b}}}g_{{\\breve{z}}}$ .",
"If $G_{{\\breve{F}}, {\\mathrm {ad}}}$ is not of type $D_{2n}$ and $i>1$ , then the action of $\\sigma $ on $\\Omega _{{\\breve{{a}}}, {\\mathrm {ad}}}$ is trivial and $G_{{\\breve{F}}, {\\mathrm {ad}}}$ is of type $A_n$ , $D_{2n+1}$ or $E_6$ .",
"We have ${\\mathrm {Ad}}({\\breve{z}}_0^i)({\\breve{s}}) = {\\breve{s}}$ for $1<i <|\\Omega _{{\\breve{{a}}}, {\\mathrm {ad}}}|$ and we need to prove that $n_{{\\breve{z}}_0}^i n_{\\breve{s}}n_{{\\breve{z}}_0}^{-i} = n_{\\breve{s}}$ .",
"If $G_{{\\breve{F}}, {\\mathrm {ad}}}$ is of type $A_n$ , then since ${\\breve{z}}_0^i$ does not have any fixed points on $\\breve{\\mathbb {S}}$ , there is no such ${\\breve{s}}$ and the statement is trivial.",
"If $G_{{\\breve{F}}, {\\mathrm {ad}}}$ is of type $E_6$ , then $|\\Omega _{{\\breve{{a}}}, {\\mathrm {ad}}}|=3$ .",
"If $i>1$ , then $i=2$ and ${\\breve{z}}_0=({\\breve{z}}_0^2)^2$ .",
"If ${\\breve{z}}_0^2$ fixes ${\\breve{s}}$ , then necessarily ${\\breve{z}}_0$ fixes ${\\breve{s}}$ .",
"By Condition (2)(b)$^\\dagger $ of §REF , $n_{{\\breve{z}}_0} n_{{\\breve{s}}} = n_{{\\breve{s}}} n_{{\\breve{z}}_0}$ .",
"Hence $g_{\\breve{z}}n_{\\breve{s}}g_{{\\breve{z}}}^{-1}= n_{{\\breve{z}}_0}^2 n_{{\\breve{s}}} n_{{\\breve{z}}_0}^{-2} = n_{{\\breve{s}}}$ .",
"If $G_{{\\breve{F}}, {\\mathrm {ad}}}$ is of type $D_{2n+1}$ , then $|\\Omega _{{\\breve{{a}}}, {\\mathrm {ad}}}|=4$ .",
"The element ${\\breve{z}}_0^3$ has no fixed elements in $\\breve{\\mathbb {S}}$ .",
"For $i=2$ , since ${\\breve{z}}_0^2$ fixes ${\\breve{s}}$ , we have ${\\breve{s}}={\\breve{s}}_k$ with $1<k<2n$ .",
"In this case, ${\\breve{z}}_0({\\breve{s}})={\\breve{s}}_{2n+1-k}$ .",
"By Condition (2)(b)$^\\dagger $ of §REF , we have $n_{{\\breve{z}}_0} n_{{\\breve{s}}_k} n_{{\\breve{z}}_0}^{-1} = n_{{\\breve{s}}_{2n+1-k}}$ and $n_{{\\breve{z}}_0} n_{{\\breve{s}}_{2n+1-k}} n_{{\\breve{z}}_0}^{-1} = n_{{\\breve{s}}_k}$ .",
"So $n_{{\\breve{z}}_0}^2 n_{{\\breve{s}}} n_{{\\breve{z}}_0}^{-2} = n_{{\\breve{s}}}$ ."
],
[
"The remaining cases",
"In this subsection, we assume that $\\sigma ^* \\ne \\sigma $ .",
"So in particular, we have $(\\Omega _{{\\breve{{a}}}, {\\mathrm {ad}}})_\\sigma \\ne 1$ .",
"We first discuss the case where the $\\sigma $ -action on $\\Omega _{{\\breve{{a}}}, {\\mathrm {ad}}}$ is trivial.",
"If $G_{{\\breve{F}}, {\\mathrm {ad}}}$ is of type $A_n$ , then $g_{\\breve{z}}= n_{{\\breve{z}}_0}^i$ for some $i$ with $1 \\leqslant i<|(\\Omega _{{\\breve{{a}}}, {\\mathrm {ad}}})|$ .",
"We have $({\\mathrm {Ad}}({{\\breve{z}}_0}^{i|\\mathcal {X} |})({\\breve{s}})={\\breve{s}}$ .",
"Since ${\\breve{z}}_0$ acts transitively on the gradients of elements of $\\breve{\\Delta }$ , we see that $|(\\Omega _{{\\breve{{a}}}, {\\mathrm {ad}}})|$ divides $i |\\mathcal {X} |$ .",
"By Corollary REF (2), we have $({\\mathrm {Ad}}(g_{{\\breve{z}}}) \\circ \\sigma )^{|\\mathcal {X} |}(n_{\\breve{s}})=({\\mathrm {Ad}}(g_{{\\breve{z}}}))^{|\\mathcal {X} |}(n_{\\breve{s}})=({\\mathrm {Ad}}(n_{{\\breve{z}}_0}^{i|\\mathcal {X} |})(n_{\\breve{s}})=n_{\\breve{s}}$ .",
"If $G_{{\\breve{F}}, {\\mathrm {ad}}}$ is not of type $A$ and the $\\sigma $ -action on $\\Omega _{{\\breve{{a}}}, {\\mathrm {ad}}}$ is trivial, then each $\\sigma ^*$ -orbit on $\\breve{\\mathbb {S}}$ is of size 1 or the order $l$ of ${\\breve{z}}$ in ${\\breve{W}}_0$ .",
"The case where $\\sigma ^*({\\breve{s}})={\\breve{s}}$ is handled in §REF .",
"If $\\sigma ^*({\\breve{s}})\\ne {\\breve{s}}$ , then by Corollary REF (1), $(\\sigma ^*)^l(n_{\\breve{s}})={\\mathrm {Ad}}(g_{{\\breve{z}}})^l \\sigma ^l(n_{\\breve{s}})={\\mathrm {Ad}}(g_{{\\breve{z}}})^l(n_{\\breve{s}})=n_{\\breve{s}}.$ Next we discuss the case where the $\\sigma $ -action on $\\Omega _{{\\breve{{a}}}, {\\mathrm {ad}}}$ is non-trivial.",
"Since $(\\Omega _{{\\breve{{a}}}, {\\mathrm {ad}}})_\\sigma \\ne 1$ , $G_{{\\breve{F}}, {\\mathrm {ad}}}$ is of type $A_{2n+1}$ or type $D_n$ .",
"If $G_{{\\breve{F}}, {\\mathrm {ad}}}$ is of type $A_{2n+1}$ , then $g_{{\\breve{z}}}=n_{{\\breve{y}}_{(1)}}$ and $\\sigma (g_{{\\breve{z}}})=n_{{\\breve{y}}_{(2n+1)}}$ .",
"We have $g_{{\\breve{z}}} \\sigma (g_{{\\breve{z}}})=({\\breve{a}}^\\vee _1+{\\breve{a}}^\\vee _3+\\cdots +{\\breve{a}}^\\vee _{2n+1})(-1) \\in Z(\\breve{F})$ .",
"Note that any $\\sigma ^*$ -orbit on $\\breve{\\mathbb {S}}$ is of size 1 or 2.",
"The case where $\\sigma ^*({\\breve{s}})={\\breve{s}}$ is handled in §REF .",
"If $\\sigma ^*({\\breve{s}})\\ne {\\breve{s}}$ , then $(\\sigma ^*)^2(n_{\\breve{s}})={\\mathrm {Ad}}(g_{{\\breve{z}}} \\sigma (g_{{\\breve{z}}})) \\sigma ^2(n_{\\breve{s}})={\\mathrm {Ad}}(({\\breve{a}}^\\vee _1+{\\breve{a}}^\\vee _3+\\cdots +{\\breve{a}}^\\vee _{2n+1})(-1)) n_{\\breve{s}}=n_{\\breve{s}}.$ If $G_{{\\breve{F}}, {\\mathrm {ad}}}$ is of type $D_n$ with $n$ odd, then $g_{{\\breve{z}}}=n_{{\\breve{y}}_{(n)}}$ and $\\sigma (g_{{\\breve{z}}})=n_{{\\breve{y}}_{(n-1)}}$ .",
"By Corollary REF (3), $g_{{\\breve{z}}} \\sigma (g_{{\\breve{z}}})=1$ or $({\\breve{a}}^\\vee _{n-1}+{\\breve{a}}^\\vee _n)(-1)$ .",
"In either case, $g_{{\\breve{z}}} \\sigma (g_{{\\breve{z}}}) \\in Z(\\breve{F})$ .",
"We then follow the same argument as the type $A_{2n+1}$ case above.",
"If $G_{{\\breve{F}}, {\\mathrm {ad}}}$ is of type $D_n$ with $n$ even and $g_{{\\breve{z}}}=n_{{\\breve{y}}_{(1)}}$ , then $\\sigma (g_{{\\breve{z}}})=n_{{\\breve{y}}_{(1)}}$ .",
"By Corollary REF (4), $g_{{\\breve{z}}} \\sigma (g_{{\\breve{z}}})=({\\breve{a}}^\\vee _{n-1}+{\\breve{a}}^\\vee _n)(-1) \\in Z(\\breve{F})$ .",
"We then follow the same argument as the type $A_{2n+1}$ case above.",
"If $G_{{\\breve{F}}, {\\mathrm {ad}}}$ is of type $D_n$ with $n$ even and $g_{{\\breve{z}}}=n_{{\\breve{y}}_{(n-1)}}$ or $n_{{\\breve{y}}_{(n)}}$ , then by Corollary REF (4), $g_{{\\breve{z}}} \\sigma (g_{{\\breve{z}}})=z n_{{\\breve{y}}_{(1)}}$ for some $z \\in Z(\\breve{F})$ .",
"Note that each $\\sigma ^*$ -orbit on $\\breve{\\mathbb {S}}$ is of size 1 or 4.",
"We have $({\\mathrm {Ad}}(g_{\\breve{z}}) \\circ \\sigma )^4 &={\\mathrm {Ad}}(g_{{\\breve{z}}} \\sigma (g_{{\\breve{z}}})) {\\mathrm {Ad}}(\\sigma ^2(g_{{\\breve{z}}} \\sigma (g_{{\\breve{z}}})) \\circ \\sigma ^4 \\\\ &={\\mathrm {Ad}}(z n_{{\\breve{y}}_{(1)}} \\sigma ^2(z n_{{\\breve{y}}_{(1)}})) \\circ \\sigma ^4 \\\\ &={\\mathrm {Ad}}(z \\sigma ^2(z) n_{{\\breve{y}}_{(1)}}^2) \\circ \\sigma ^4=\\sigma ^4.$ Thus $(\\sigma ^*)^4(n_{\\breve{s}})=({\\mathrm {Ad}}(g_{\\breve{z}}) \\circ \\sigma )^4(n_{\\breve{s}})=\\sigma ^4(n_{\\breve{s}})=n_{\\breve{s}}.$ This finishes the verification of (REF ) in all the remaining cases and thus finishes the proof of Proposition REF ."
],
[
"The $\\sigma ^*$ -fixed liftings of {{formula:f770f899-b4be-477f-8a69-4d1421663083}}",
"Let $\\breve{\\tau }=t_{\\breve{\\lambda }}{\\breve{y}}\\in \\Omega _{{\\breve{{a}}}}^{\\sigma ^*}$ .",
"We will set $m({\\breve{\\tau }}) = n_{\\breve{\\lambda }}m({\\breve{y}})$ for a suitable $m({\\breve{y}}) \\in \\breve{\\mathcal {T}}$ .",
"Let us first choose $m({\\breve{y}})$ when $G_{{\\breve{F}},{\\mathrm {ad}}}$ is ${\\breve{F}}$ -simple.",
"If $G_{{\\breve{F}}, {\\mathrm {ad}}}$ is of type $D_n$ with $n$ even, then we set $m({\\breve{y}})=n_{{\\breve{y}}}$ .",
"Otherwise, $\\Omega _{{\\breve{{a}}}, {\\mathrm {ad}}}$ is a cyclic group and ${\\breve{y}}={\\breve{z}_0}^j$ for $0 \\leqslant j<|\\Omega _{{\\breve{{a}}}, {\\mathrm {ad}}}|$ .",
"Set $m({\\breve{y}})=n_{{\\breve{z}_0}}^j$ and $m(\\breve{\\tau })=n_{\\breve{\\lambda }}m({\\breve{y}})$ .",
"If $G_{{\\breve{F}}, {\\mathrm {ad}}}$ is not ${\\breve{F}}$ -simple, then $W(G, T)=W(G, T)^{(1)} \\times \\cdots \\times W(G, T)^{(k)},$ where $W(G, T)^{(1)} \\cong \\cdots \\cong W(G, T)^{(k)}$ are irreducible finite Weyl groups and the action of $\\sigma $ permutes transitively the irreducible factors $W(G, T)^{(1)}, \\ldots , W(G, T)^{(k)}$ .",
"There exist ${\\breve{y}}^{(1)} \\in W(G, T)^{(1)}$ such that ${\\breve{y}}={\\breve{y}}^{(1)} \\sigma ({\\breve{y}}^{(1)}) \\cdots \\sigma ^{k-1}({\\breve{y}}^{(1)})$ .",
"Define $m({\\breve{y}}^{(1)})$ as in the preceding paragraph.",
"More precisely, if $G_{{\\breve{F}}, {\\mathrm {ad}}}^{(1)}$ is of type $D_n$ with $n$ even, then we set $m({\\breve{y}}^{(1)})=n_{{\\breve{y}}^{(1)}}$ .",
"Otherwise, $\\Omega _{{\\breve{{a}}}, {\\mathrm {ad}}}^{(1)}$ is a cyclic group and ${\\breve{y}}^{(1)}={\\breve{z}_0}^j$ for $0 \\leqslant j<|\\Omega _{{\\breve{{a}}}, {\\mathrm {ad}}}^{(1)}|$ .",
"Set $m({\\breve{y}}^{(1)})=n_{{\\breve{z}_0}}^j$ .",
"Let $m({\\breve{y}}) = m({\\breve{y}}^{(1)}) \\sigma (m({\\breve{y}}^{(1)})) \\cdots \\sigma ^{k-1}(m({\\breve{y}}^{(1)})).$ The main result of this subsection is the following.",
"Proposition 6.9 Let $\\breve{\\tau }\\in \\Omega _{{\\breve{{a}}}}^{\\sigma ^*}$ .",
"Then $\\sigma ^*(m(\\breve{\\tau }))=m(\\breve{\\tau })$ ."
],
[
"Reduction step",
"We begin with a simple lemma.",
"Lemma 6.10 For each ${\\breve{\\tau }}\\in \\Omega _{{\\breve{{a}}}}$ , we have $\\sigma ^*({\\breve{\\tau }}) = \\sigma ({\\breve{\\tau }})$ .",
"Let ${\\breve{\\tau }}=t_{{\\breve{\\lambda }}} {\\breve{y}}$ .",
"Then $\\sigma ^*({\\breve{\\tau }}) = t_{\\sigma ^*({\\breve{\\lambda }})}t_{{{\\breve{\\eta }}_{{\\mathrm {ad}}}}- y^{\\prime }({{\\breve{\\eta }}_{{\\mathrm {ad}}}})}{\\breve{y}}^{\\prime }$ , where ${\\breve{y}}^{\\prime }=Ad({\\breve{z}})(\\sigma ({\\breve{y}}))$ .",
"Since $\\Omega _{{\\breve{{a}}}, {\\mathrm {ad}}}$ is abelian, ${\\breve{y}}^{\\prime } = \\sigma ({\\breve{y}})$ and $\\sigma ^*({\\breve{\\lambda }}) -\\sigma ({\\breve{\\lambda }}) = {\\breve{z}}(\\sigma ({\\breve{\\lambda }}))-\\sigma ({\\breve{\\lambda }})={{\\breve{\\eta }}_{{\\mathrm {ad}}}}- y^{\\prime }({{\\breve{\\eta }}_{{\\mathrm {ad}}}})+{\\breve{\\mu }}$ for a suitable ${\\breve{\\mu }}\\in X_*(Z^0)$ .",
"In particular, we have ${\\breve{z}}$ commutes with ${\\breve{y}}^{\\prime }=\\sigma ({\\breve{y}})$ .",
"Let $l$ be the order of ${\\breve{z}}$ .",
"Then ${\\breve{z}}^{i+1}(\\sigma ({\\breve{\\lambda }})) -{\\breve{z}}^{i}(\\sigma ({\\breve{\\lambda }})) = {\\breve{z}}^{i}({{\\breve{\\eta }}_{{\\mathrm {ad}}}}) - y^{\\prime } {\\breve{z}}^{i}({{\\breve{\\eta }}_{{\\mathrm {ad}}}})+ {\\breve{\\mu }}$ for any $i$ .",
"Since ${{\\breve{\\nu }}_{{\\mathrm {ad}}}}^l=1$ , we have ${{\\breve{\\eta }}_{{\\mathrm {ad}}}}+ {\\breve{z}}({{\\breve{\\eta }}_{{\\mathrm {ad}}}}) + \\cdots +{\\breve{z}}^{l-1}({{\\breve{\\eta }}_{{\\mathrm {ad}}}}) =0$ .",
"Thus $l {\\breve{\\mu }}= \\sum _{i=0}^{l-1} \\bigl ({\\breve{z}}^{i}({{\\breve{\\eta }}_{{\\mathrm {ad}}}}) - y^{\\prime } {\\breve{z}}^{i}({{\\breve{\\eta }}_{{\\mathrm {ad}}}})+ {\\breve{\\mu }}\\bigr )=\\sum _{i=0}^{l-1} \\bigl ({\\breve{z}}^{i+1}(\\sigma ({\\breve{\\lambda }})) -{\\breve{z}}^{i}(\\sigma ({\\breve{\\lambda }}))\\bigr )=0$ and hence ${\\breve{\\mu }}=0$ .",
"So $\\sigma ^*({\\breve{\\tau }}) = t_{\\sigma ({\\breve{\\lambda }})} \\sigma ({\\breve{y}}) = \\sigma ({\\breve{\\tau }})$ .",
"Let $\\breve{\\tau }=t_{\\breve{\\lambda }}{\\breve{y}}\\in \\Omega _{{\\breve{{a}}}}^{\\sigma ^*}=\\Omega _{{\\breve{{a}}}}^{\\sigma }$ .",
"Then we have $\\sigma ({\\breve{y}})={\\breve{y}}$ and ${\\breve{\\lambda }}=\\sigma ^*({\\breve{\\lambda }})+{{\\breve{\\eta }}_{{\\mathrm {ad}}}}-{\\mathrm {Ad}}({\\breve{z}})(\\sigma ({\\breve{y}}))({{\\breve{\\eta }}_{{\\mathrm {ad}}}})$ .",
"By (REF ) and Lemma REF , $ \\sigma ^*(m(\\breve{\\tau })) &=\\sigma ^*(n_{\\breve{\\lambda }}m({\\breve{y}}))=\\sigma ^*(n_{{\\breve{\\lambda }}}) \\sigma ^*(m({\\breve{y}})) \\\\ &=n_{\\sigma ^*({\\breve{\\lambda }})+{{\\breve{\\eta }}_{{\\mathrm {ad}}}}-{\\mathrm {Ad}}({\\breve{z}})(\\sigma ({\\breve{y}})){{\\breve{\\eta }}_{{\\mathrm {ad}}}}} {\\mathrm {Ad}}(g_{{\\breve{z}}}) \\sigma (m({\\breve{y}})) \\\\ &=n_{{\\breve{\\lambda }}} {\\mathrm {Ad}}(g_{{\\breve{z}}}) \\sigma (m({\\breve{y}})).",
"$ To verify $\\sigma ^*(m(\\breve{\\tau }))=m(\\breve{\\tau })$ , it remains to show $\\sigma (m({\\breve{y}}))=m({\\breve{y}})$ ; ${\\mathrm {Ad}}(g_{{\\breve{z}}})(m({\\breve{y}})) = m({\\breve{y}})$ .",
"Now we show that it suffices to check the case where $G_{{\\breve{F}}, {\\mathrm {ad}}}$ is ${\\breve{F}}$ -simple.",
"Lemma 6.11 We have $\\sigma (m({\\breve{y}}))=m({\\breve{y}})$ if and only if $\\sigma ^k(m({\\breve{y}}^{(1)})) = m({\\breve{y}}^{(1)})$ .",
"${\\mathrm {Ad}}(g_{{\\breve{z}}})(m({\\breve{y}})) = m({\\breve{y}})$ if and only if ${\\mathrm {Ad}}(g_{{\\breve{z}}})(m({\\breve{y}}^{(1)})) = m({\\breve{y}}^{(1)})$ .",
"Note that $\\sigma ({\\breve{y}}) ={\\breve{y}}$ if and only if $\\sigma ^k({\\breve{y}}^{(1)}) = {\\breve{y}}^{(1)}$ .",
"By the definition of $m({\\breve{y}})$ , we have $\\sigma (m({\\breve{y}})) = \\sigma (m({\\breve{y}}^{(1)})) \\sigma ^2(m({\\breve{y}}^{(1)})) \\cdots \\sigma ^{k}(m({\\breve{y}}^{(1)}))$ .",
"Further, (REF ) implies that $\\breve{\\mathcal {T}}_{\\text{fin}} \\cong \\breve{\\mathcal {T}}_{\\text{fin}}^{(1)} \\times \\breve{\\mathcal {T}}_{\\text{fin}}^{(2)} \\times \\cdots \\times \\breve{\\mathcal {T}}_{\\text{fin}}^{(k)},$ where $ \\breve{\\mathcal {T}}_{\\text{fin}}^{(i)}$ is the finite Tits group attached to $W(G,T)^{(i)}$ .",
"Hence $\\sigma (m({\\breve{y}})) = \\sigma ^{k}(m({\\breve{y}}^{(1)})) \\sigma (m({\\breve{y}}^{(1)})) \\sigma ^2(m({\\breve{y}}^{(1)})) \\cdots \\sigma ^{k-1}(m({\\breve{y}}^{(1)})).$ Now it follows that $\\sigma (m({\\breve{y}}) = m({\\breve{y}})$ if and only if $ \\sigma ^{k}(m({\\breve{y}}^{(1)}))= m({\\breve{y}}^{(1)})$ .",
"For (2), since $g_{\\breve{z}}\\in \\breve{\\mathcal {T}}_{\\text{fin}}^{(1)}$ , and $\\sigma ^i(m({\\breve{y}}^{(1)}) \\in \\breve{\\mathcal {T}}_{\\text{fin}}^{(i)}$ for each $0 \\leqslant i \\leqslant k-1$ , we see using (REF ) that $g_{\\breve{z}}$ commutes with $\\sigma ^i(m({\\breve{y}}^{(i)})$ for all $i \\ge 1$ .",
"Hence ${\\mathrm {Ad}}(g_{{\\breve{z}}})(m({\\breve{y}})) = m({\\breve{y}})$ if and only if ${\\mathrm {Ad}}(g_{{\\breve{z}}})(m({\\breve{y}}^{(1)})) = m({\\breve{y}}^{(1)})$ ."
],
[
"Proof of Proposition ",
"We assume that $G_{{\\breve{F}}, {\\mathrm {ad}}}$ is ${\\breve{F}}$ -simple and we drop the subscripts in the discussion below.",
"In particular, we may assume $\\Omega _{{\\breve{{a}}}, {\\mathrm {ad}}}$ , ${{\\breve{\\nu }}_{{\\mathrm {ad}},0}}$ are as in Table REF .",
"So $g_{\\breve{z}}= n_{\\breve{z}}$ if $\\Omega _{{\\breve{{a}}}, {\\mathrm {ad}}}$ is of type $D_n$ with $n$ even.",
"Otherwise, $g_{\\breve{z}}= n_{{\\breve{z}}_0}^i$ for a suitable $0 \\leqslant i <|(\\Omega _{{\\breve{{a}}}, {\\mathrm {ad}}})_{\\sigma }|$ .",
"Also $m({\\breve{y}}) = n_{\\breve{y}}$ if $\\Omega _{{\\breve{{a}}}, ad}$ is of type $D_n$ with $n$ even.",
"Otherwise, $m({\\breve{y}}) = n_{{\\breve{z}}_0}^j$ for a suitable $0 \\leqslant j <|(\\Omega _{{\\breve{{a}}}, {\\mathrm {ad}}}|$ .",
"We show that ${\\mathrm {Ad}}(g_{\\breve{z}})(m({\\breve{y}})) = m({\\breve{y}})$ .",
"When $G_{{\\breve{F}}, {\\mathrm {ad}}}$ is of type $D_n, n$ even, this is a consequence of Corollary REF (4).",
"Otherwise, $g_{{\\breve{z}}}$ and $m({\\breve{y}})$ are both powers of $n_{{\\breve{z}}_0}$ and the claim is obvious.",
"We show that $\\sigma (m({\\breve{y}})) = m({\\breve{y}})$ .",
"The proof involves a detailed case-by-case analysis.",
"Recall that the representatives $\\lbrace n_{s_{\\breve{a}}}\\;|\\; {\\breve{a}}\\in \\breve{\\Delta }\\rbrace $ satisfies $H(\\breve{\\Delta }, \\sigma )$ .",
"When $G_{{\\breve{F}}, {\\mathrm {ad}}}$ is of type $D_n, n$ even, we have $\\sigma (m({\\breve{y}}))=\\sigma (n_{\\breve{y}})=n_{\\breve{y}}=m({\\breve{y}})$ .",
"Next we consider the case where $\\Omega _{{\\breve{{a}}}, {\\mathrm {ad}}}$ is cyclic.",
"If the action of $\\sigma $ on $\\Omega _{{\\breve{{a}}},ad}$ is trivial, then $\\sigma ({\\breve{z}}_0) = {\\breve{z}}_0$ and $\\sigma (n_{{\\breve{z}}_0}) = n_{{\\breve{z}}_0}$ .",
"In this case, $\\sigma (m({\\breve{y}})) = \\sigma (n_{{\\breve{z}}_0}^j) = \\sigma (n_{{\\breve{z}}_0})^j = n_{{\\breve{z}}_0}^j = m({\\breve{y}})$ .",
"It remains to prove the claim when $\\Omega _{{\\breve{{a}}}, {\\mathrm {ad}}}$ is cyclic and the action of $\\sigma $ on $\\Omega _{{\\breve{{a}}}, {\\mathrm {ad}}}$ is non-trivial.",
"Recall that $j: G_{\\breve{F}}\\rightarrow G_{{\\breve{F}}, {\\mathrm {ad}}}$ is the adjoint quotient.",
"Let ${\\breve{\\tau }}\\in \\Omega _{{\\breve{{a}}}}^{\\sigma ^*}$ .",
"If $j({\\breve{\\tau }})=1$ , then ${\\breve{y}}=1$ and $m({\\breve{y}})=1$ .",
"In this case, $\\sigma (m({\\breve{y}}))=1=m({\\breve{y}})$ .",
"Now we assume that $j({\\breve{\\tau }}) \\ne 1$ .",
"This happens when $G_{{\\breve{F}}, {\\mathrm {ad}}}$ is of type $A_{2n+1}$ and $j=n+1$ or $G_{{\\breve{F}}, {\\mathrm {ad}}}$ is of type $D_{2n+1}$ and $j=2$ .",
"In both these cases $\\sigma ({{\\breve{\\nu }}_{{\\mathrm {ad}},0}}) = {{\\breve{\\nu }}_{{\\mathrm {ad}},0}}^{-1}$ and $\\sigma ({\\breve{z}}_0) = {\\breve{z}}_0^{-1}$ .",
"If $G_{{\\breve{F}}, {\\mathrm {ad}}}$ is of type $A_{2n+1}$ , then $m({\\breve{y}})=n_{{\\breve{z}}_0}^{n+1}$ and ${\\breve{z}}_0^{n+1}={\\breve{y}}_{(n+1)}$ .",
"By Corollary REF (2), we have $n_{{\\breve{z}}_0}^{n+1} = {\\left\\lbrace \\begin{array}{ll} n_{{\\breve{z}}_0^{n+1}}, & \\text{ if } n \\text{ is even},\\\\(a_1 ^\\vee + a_3^{\\vee } + \\cdots + a_{n}^\\vee )(-1) n_{{\\breve{z}}_0^{n+1}}, & \\text{ if } n \\text{ is odd}.\\end{array}\\right.",
"}$ We have $\\sigma ({\\breve{y}}_{(n+1)})={\\breve{y}}_{(n+1)}$ .",
"Thus $\\sigma (n_{{\\breve{z}}_0}^{n+1}) = {\\left\\lbrace \\begin{array}{ll} n_{{\\breve{z}}_0}^{n+1}, & \\text{ if } n \\text{ is even},\\\\(a_1 ^\\vee + a_3^{\\vee } + \\cdots + a_{2n+1}^\\vee )(-1) n_{{\\breve{z}}_0}^{n+1}, & \\text{ if } n \\text{ is odd}.\\end{array}\\right.",
"}$ We identify $\\Omega _{{\\breve{{a}}}, {\\mathrm {ad}}}$ with $X_*(T_{{\\mathrm {ad}}})/X_*(T_{sc})$ .",
"Under this identification, $j({\\breve{\\tau }}) &= (n+1){{\\breve{\\nu }}_{{\\mathrm {ad}},0}}=(n+1) \\frac{{\\breve{a}}_1^\\vee +2{\\breve{a}}_2^\\vee +3 {\\breve{a}}_3^\\vee +\\cdots +(2n+1){\\breve{a}}_{2n+1}^\\vee }{2n+2} \\\\ &=\\frac{{\\breve{a}}_1^\\vee +3 {\\breve{a}}_3^\\vee + 5{\\breve{a}}_5^\\vee +\\cdots + (2n+1){\\breve{a}}_{2n+1}^\\vee }{2} + {\\breve{a}}_2^\\vee +2 {\\breve{a}}_4^\\vee +\\cdots +n {\\breve{a}}_{2n}^\\vee \\\\ & \\equiv \\frac{{\\breve{a}}_1^\\vee +{\\breve{a}}_3^\\vee + {\\breve{a}}_5^\\vee +\\cdots +{\\breve{a}}_{2n+1}^\\vee }{2}\\mod {X}_*(T_{sc}).$ Since ${\\breve{\\tau }}\\in X_*(T)/X_*(T_{sc})$ and $j$ acts as identity on $X_*(T_{sc})$ , $\\frac{{\\breve{a}}_1^\\vee +{\\breve{a}}_3^\\vee + {\\breve{a}}_5^\\vee +\\cdots +{\\breve{a}}_{2n+1}^\\vee }{2} \\in X_*(T)$ .",
"Hence $({\\breve{a}}_1^\\vee +{\\breve{a}}_3^\\vee +\\cdots +{\\breve{a}}_{2n+1}^\\vee )(-1)= \\left(\\left(\\frac{{\\breve{a}}_1^\\vee +{\\breve{a}}_3^\\vee + {\\breve{a}}_5^\\vee +\\cdots +{\\breve{a}}_{2n+1}^\\vee }{2}\\right)(-1)\\right)^2=1 \\in G(\\breve{F}).$ Therefore we have $\\sigma (m({\\breve{y}}))=\\sigma (n_{{\\breve{z}}_0}^{n+1})=n_{{\\breve{z}}_0}^{n+1}=m({\\breve{y}})$ .",
"If $G_{{\\breve{F}}, {\\mathrm {ad}}}$ is of type $D_{2n+1}$ , we have $m({\\breve{y}}) = n_{{\\breve{z}}_0}^2$ , and by Corollary REF (3), $\\sigma (m({\\breve{y}})) = tm({\\breve{y}})$ where $t = ({\\breve{a}}_{2n}^\\vee + {\\breve{a}}_{2n+1}^\\vee )(-1)$ .",
"We claim that $t=1$ in $G({\\breve{F}})$ .",
"The argument is similar to type $A_{2n+1}$ .",
"Consider the element $j({\\breve{\\tau }})\\in \\Omega _{{\\breve{{a}}}, {\\mathrm {ad}}} =X_*(T_{{\\mathrm {ad}}})/X_*(T_{sc})$ .",
"Then ${{\\breve{\\nu }}_{{\\mathrm {ad}},0}}= \\frac{{\\breve{a}}_1^\\vee +2{\\breve{a}}_2^\\vee +\\cdots + (2n-1) {\\breve{a}}_{2n-1}^\\vee + \\frac{1}{2} (2n-1) {\\breve{a}}_{2n}^\\vee +\\frac{1}{2}(2n+1){\\breve{a}}_{2n+1}^\\vee }{2} \\mod {X}_*(T_{sc}).$ Then $j({\\breve{\\tau }}) \\equiv 2{{\\breve{\\nu }}_{{\\mathrm {ad}},0}}\\equiv \\frac{1}{2} {\\breve{a}}_{2n}^\\vee +\\frac{1}{2}{\\breve{a}}_{2n+1}^\\vee \\mod {X}_*(T_{sc})$ .",
"Since $2{\\breve{\\tau }}\\in X_*(T_{sc})$ and $j$ acts as identity on $X_*(T_{sc})$ , we see that ${\\breve{\\tau }}&\\equiv \\frac{1}{2} {\\breve{a}}_{2n}^\\vee +\\frac{1}{2}{\\breve{a}}_{2n+1}^\\vee \\mod {X}_*(T_{sc}).$ Since ${\\breve{\\tau }}\\in X_*(T)/X_*(T_{sc})$ , we see that $ \\frac{1}{2} {\\breve{a}}_{2n}^\\vee +\\frac{1}{2}{\\breve{a}}_{2n+1}^\\vee \\in X_*(T)$ .",
"Hence $t &= ({\\breve{a}}_{2n}^\\vee + {\\breve{a}}_{2n+1}^\\vee )(-1)= \\left(\\left(\\frac{1}{2} {\\breve{a}}_{2n}^\\vee +\\frac{1}{2}{\\breve{a}}_{2n+1}^\\vee \\right)(-1)\\right)^2=1.$ Hence $\\sigma (m({\\breve{y}}))=m({\\breve{y}})$ .",
"This finishes the proof of Proposition REF ."
],
[
"Proof of Theorem ",
"For ${\\breve{s}}\\in \\breve{\\mathbb {S}}$ , let $m({\\breve{s}}) \\in \\breve{\\mathcal {T}}$ be the lifting of ${\\breve{s}}$ in Corollary REF .",
"For any $\\breve{\\tau }\\in \\Omega _{{\\breve{{a}}}}^{\\sigma ^*}$ , let $m(\\breve{\\tau }) \\in \\breve{\\mathcal {T}}$ be the lifting of $\\breve{\\tau }$ constructed in §REF .",
"Then $\\sigma ^*(m(\\breve{\\tau }))=m(\\breve{\\tau })$ for all $\\breve{\\tau }\\in \\Omega _{{\\breve{{a}}}}^{\\sigma ^*}$ and $\\sigma ^*(m({\\breve{s}}))=m(\\sigma ^*({\\breve{s}}))$ for all ${\\breve{s}}\\in \\breve{\\mathbb {S}}$ .",
"We set $\\mathcal {T} =\\breve{\\mathcal {T}}^{\\sigma ^*}$ .",
"For any $s \\in \\mathbb {S} $ , we have $s={\\breve{w}}_{\\mathcal {X}}$ for some $\\sigma $ -orbit $\\mathcal {X} $ in $\\breve{\\Delta }$ with ${\\breve{W}}_{\\mathcal {X}}$ finite (see §REF ).",
"Let ${\\breve{w}}_{\\mathcal {X}}={\\breve{s}}_{i_1} \\cdots {\\breve{s}}_{i_n}$ be a reduced expression of ${\\breve{w}}_{\\mathcal {X}}$ in ${\\breve{W}}_{\\text{af}}$ .",
"Then ${\\breve{w}}_{\\mathcal {X}}=\\sigma ({\\breve{s}}_{i_1}) \\cdots \\sigma ({\\breve{s}}_{i_n})$ is again a reduced expression of ${\\breve{w}}_{\\mathcal {X}}$ in ${\\breve{W}}_{\\text{af}}$ .",
"We have $ m({\\breve{w}}_{\\mathcal {X}}) &=m({\\breve{s}}_{i_1}) \\cdots m({\\breve{s}}_{i_n})=m(\\sigma ({\\breve{s}}_{i_1})) \\cdots m(\\sigma ({\\breve{s}}_{i_n}))=\\sigma (m({\\breve{s}}_{i_1})) \\cdots \\sigma (m({\\breve{s}}_{i_n})) \\\\ &=\\sigma (m({\\breve{w}}_{\\mathcal {X}})).$ In particular, $m(s)=m({\\breve{w}}_{\\mathcal {X}}) \\in \\mathcal {T} =\\breve{\\mathcal {T}}^{\\sigma ^*}$ .",
"Let $w \\in W_{\\text{af}}$ and $s_{i_1} \\cdots s_{i_n}$ be a reduced expression of $w$ in $W$ .",
"We set $m(w)=m(s_{i_1}) \\cdots m(s_{i_n})$ .",
"Then $m(w) \\in \\mathcal {T} $ .",
"Suppose that $s^{\\prime }_{i_1} \\cdots s^{\\prime }_{i_n}$ is another reduced expression of $w$ in $W$ .",
"By §REF (a), $\\breve{\\ell }(w)=\\breve{\\ell }(s_{i_1})+\\cdots +\\breve{\\ell }(s_{i_n})=\\breve{\\ell }(s^{\\prime }_{i_1})+\\cdots +\\breve{\\ell }(s^{\\prime }_{i_n})$ .",
"Since $\\lbrace m({\\breve{s}})\\;|\\; {\\breve{s}}\\in \\breve{\\mathbb {S}}\\rbrace $ satisfies the Coxeter relations, by condition (2)(b) $^\\dagger $ in §REF , $m(s_{i_1}) \\cdots m(s_{i_n})=m(s^{\\prime }_{i_1}) \\cdots m(s^{\\prime }_{i_n})$ .",
"In other words, $m(w)$ is independent of the choice of reduced expression in $W$ .",
"Finally for $w \\in W$ , we have $w=w_1 \\tau $ for a unique $w_1 \\in W_{\\text{af}}$ and $\\tau \\in \\Omega _{{{{a}}}}=\\Omega _{{\\breve{{a}}}}^{\\sigma ^*}$ .",
"We set $m(w)=m(w_1) m(\\tau )$ .",
"Then $m(w) \\in \\mathcal {T} $ .",
"In other words, the map $\\phi : \\mathcal {T} {@display}{\\longrightarrow }{\\rightarrow }W$ is surjective.",
"We have $\\ker (\\phi )=\\ker (\\breve{\\phi }) \\cap \\mathcal {T} _{\\text{af}}=\\breve{S}_2 \\cap \\mathcal {T} _{\\text{af}}=S_2.$ It remains to show that for each $a \\in \\Delta $ , we have $m(s_a)^2 = b^\\vee (-1) $ where $b$ is the gradient of $a$ .",
"By [8], we know that the elements of $\\Delta $ and in bijection with $\\sigma ^*$ -orbits $\\mathcal {X} $ of $\\breve{\\Delta }$ with ${\\breve{a}}|_{\\mathcal {A} (A,F)}$ non-constant for ${\\breve{a}}\\in \\mathcal {X} $ .",
"Let $\\mathcal {F} $ denote the $\\mathbb {R} $ -vector space of affine linear functions on $\\mathcal {A} (A,F)$ .",
"Then $\\mathcal {F} $ may be identified with the $\\sigma ^*$ -invariants of $\\breve{\\mathcal {F}}$ where $\\breve{\\mathcal {F}}$ is the $\\mathbb {R} $ -vector space of affine functions on $\\mathcal {A} (T,{\\breve{F}})$ (which we have identified with $V = X_*(T) \\otimes \\mathbb {R} $ after choosing a special point).",
"Under this identification, we have for $a \\in \\Delta $ , $ a {@display}{\\longmapsto }{\\mapsto }\\frac{1}{|\\mathcal {X} |}\\sum _{{\\breve{a}}\\in \\mathcal {X}} {\\breve{a}},$ where $\\mathcal {X} $ is the $\\sigma ^*$ -orbit on $\\breve{\\Delta }$ that corresponds to $a$ .",
"Fix a $\\sigma ^*$ -invariant scalar product $\\langle \\cdot , \\cdot \\rangle $ on $V = X_*(T) \\otimes \\mathbb {R} $ and we identify $V$ with $V^*$ via this inner product.",
"We may extend this to a scalar product on $\\breve{\\mathcal {F}}$ by setting $\\langle \\breve{f}, \\breve{g} \\rangle = \\langle D\\breve{f}, D\\breve{g} \\rangle $ , where $D\\breve{f} \\in V$ is the gradient of $\\breve{f}$ .",
"For any $\\breve{f}\\in \\breve{\\mathcal {F}}$ with $D\\breve{f} \\ne 0$ , let $\\breve{f}^\\vee = \\frac{2\\breve{f}}{\\langle \\breve{f}, \\breve{f} \\rangle }$ .",
"Then for $a \\in \\Delta $ , we have $a^\\vee = \\frac{2a}{\\langle a, a\\rangle }$ .",
"Fix ${\\breve{a}}\\in \\breve{\\Delta }$ whose restriction to $\\mathcal {A} (A, F)$ is the affine root $a$ .",
"Then $\\langle a, a \\rangle = \\frac{1}{|\\mathcal {X} |} \\sum _{{\\breve{a}}^{\\prime } \\in \\mathcal {X}} \\langle {\\breve{a}}, {\\breve{a}}^{\\prime }\\rangle .$ This implies that $a^\\vee = c_a \\sum _{{\\breve{a}}^{\\prime } \\in \\mathcal {X}} {\\breve{a}}^{\\prime \\vee }, \\;\\; b^\\vee = c_a \\sum _{{\\breve{a}}^{\\prime } \\in \\mathcal {X}} {\\breve{b}}^{\\prime \\vee }$ where $c_a = \\frac{\\langle {\\breve{a}}, {\\breve{a}}\\rangle }{\\sum _{{\\breve{a}}^{\\prime } \\in \\mathcal {X}} \\langle {\\breve{a}}, {\\breve{a}}^{\\prime }\\rangle } = \\frac{\\langle {\\breve{b}}, {\\breve{b}}\\rangle }{\\sum _{{\\breve{a}}^{\\prime } \\in \\mathcal {X}} \\langle {\\breve{b}}, {\\breve{b}}^{\\prime }\\rangle }.$ Now, let us prove that $m(s_a)^2 = b^\\vee (-1)$ .",
"We may easily reduce ourselves to the case where $G_{\\breve{F}}$ is ${\\breve{F}}$ -simple and simply connected.",
"Via a simple case-by-case analysis, each $\\sigma ^*$ -orbit $\\mathcal {X} $ consists of simple roots in $\\breve{\\Delta }$ whose corresponding Dynkin diagram is either a product of $A_1$ 's or is a single copy of $A_2$ .",
"In the former case, $c_a=1$ and by (REF ), $ m(s_a)^2 = m(s_{{\\breve{a}}_1})^2 m(s_{{\\breve{a}}_2}^2) \\cdots m(s_{{\\breve{a}}_k})^2 = {\\breve{b}}_1^\\vee (-1){\\breve{b}}_2^\\vee (-1)\\cdots {\\breve{b}}_k^\\vee (-1) = b^\\vee (-1),$ where $\\mathcal {X} = \\lbrace {\\breve{a}}_1, {\\breve{a}}_2, \\cdots , {\\breve{a}}_k\\rbrace $ .",
"In the latter case, $\\mathcal {X} = \\lbrace {\\breve{a}}_1, {\\breve{a}}_2\\rbrace $ and ${\\breve{a}}_1+{\\breve{a}}_2$ is an affine root.",
"In this case, $c_a=2$ .",
"By (REF ), $m(s_a)^2 = (m(s_{{\\breve{a}}_1})m(s_{{\\breve{a}}_2})m(s_{\\breve{a}}))^2 = 1 = b^\\vee (-1).$ Thus $\\mathcal {T} $ is a Tits group of $W$ and $\\lbrace m(w)\\;|\\; w \\in W\\rbrace $ is a Tits cross-section of $W$ in $\\mathcal {T} $ ."
]
] | 2107.01768 |
[
[
"Randomized Dimensionality Reduction for Facility Location and\n Single-Linkage Clustering"
],
[
"Abstract Random dimensionality reduction is a versatile tool for speeding up algorithms for high-dimensional problems.",
"We study its application to two clustering problems: the facility location problem, and the single-linkage hierarchical clustering problem, which is equivalent to computing the minimum spanning tree.",
"We show that if we project the input pointset $X$ onto a random $d = O(d_X)$-dimensional subspace (where $d_X$ is the doubling dimension of $X$), then the optimum facility location cost in the projected space approximates the original cost up to a constant factor.",
"We show an analogous statement for minimum spanning tree, but with the dimension $d$ having an extra $\\log \\log n$ term and the approximation factor being arbitrarily close to $1$.",
"Furthermore, we extend these results to approximating solutions instead of just their costs.",
"Lastly, we provide experimental results to validate the quality of solutions and the speedup due to the dimensionality reduction.",
"Unlike several previous papers studying this approach in the context of $k$-means and $k$-medians, our dimension bound does not depend on the number of clusters but only on the intrinsic dimensionality of $X$."
],
[
"Introduction",
"Clustering is a fundamental problem with many applications in machine learning, statistics, and data analysis.",
"Although many formulations of clustering are NP-hard in the worst case, many heuristics and approximation algorithms exist and are widely deployed in practice.",
"Unfortunately, many of those algorithms suffer from large running times, especially if the input data sets are high-dimensional.",
"In order to improve the performance of clustering algorithms in high-dimensional spaces, a popular approach is to project the input point set into a lower-dimensional space and perform the clustering in the projected space.",
"Reducing the dimension (say, from $m$ to $d \\ll m$ ) has multiple practical and theoretical advantages, including (i) lower storage space, which is linear in $d$ as opposed to $m$ ; (ii) lower running time of the clustering procedure - the running times are often dominated by distance computations, which take time linear in the dimension; and (iii) versatility: one can use any algorithm or its implementation to cluster the data in the reduced dimension.",
"Because of its numerous benefits, dimensionality reduction as a tool for improving algorithm performance has been studied extensively, leading to many theoretical tradeoffs between the projected dimension and the solution quality.",
"A classic result in this area is the Johnson-Lindenstrauss (JL) lemma originalJL which (roughly) states that a random projection of a dataset $X \\subseteq \\mathbb {R}^m$ of size $n$ onto a dimension of size $O(\\log n)$ approximately preserves all pairwise distances.",
"This tool has been subsequently applied to many clustering and other problems (see [24] and references therein).",
"Although the JL lemma is known to be tight [18] in general, better tradeoffs are possible for specific clustering problems.",
"Over the last few years, several works [4], [7], [3], [19] have shown that combining random dimensionality reduction with $k$ -means leads to better guarantees than implied by the JL lemma.",
"In particular, a recent paper by Makarychev, Makarychev, and Razenshteyn ilyapaper shows that to preserve the $k$ -means cost up to an arbitrary accuracy, it suffices to project the input set $X$ onto a dimension of size $O(\\log k)$ , as opposed to $O(\\log n)$ guaranteed by the JL lemma.",
"Since $k$ can be much smaller than $n$ , the improvement to the dimension bound can be substantial.",
"However, when $k$ is comparable to $n$ , the improvement is limited.",
"This issue is particularly salient for clustering problems with a variable number of clusters, where no a priori bound on the number of clusters exists.",
"In this paper we study randomized dimensionality reduction over Euclidean space $\\mathbb {R}^m$ in the context of two fundamental clustering problems with a variable number of clusters.",
"In particular: Facility location (FL): given a set of points $X \\subset \\mathbb {R}^m$ and a facility opening cost, the goal is to open a subset $\\mathcal {F} \\subseteq X$ of facilities in order to minimize the total cost of opening the facilities plus the sum of distances from points in $X$ to their nearest facilities (see Section for a formal definition).",
"Such cost functions are often used when the “true” number of clusters $k$ is not known, see e.g., [20], section 16.4.1.",
"Single-linkage clustering, or (equivalently) Minimum Spanning Tree (MST): given a set of points $X \\subset \\mathbb {R}^m$ , the goal is to connect them into a tree in order to minimize the total cost of the tree edges.",
"This is a popular variant of Hierarchical Agglomerative Clustering (HAC) that creates a hierarchy of clusters, see e.g., [20], section 17.2.",
"We remark that some papers, e.g., [1] define approximate HAC operationally, by postulating that each step of the clustering algorithm must be approximately correct.",
"However, there are other theoretical formulations of approximate HAC as well, e.g., [9], [23].",
"Since single-linkage clustering has a natural objective function induced by MST, defining approximate single-linkage clustering as approximate MST is a natural, even if not unique, choice."
],
[
"Our Results",
"Our main results show that, for both FL and MST, it is possible to project input point sets into low (sometimes even constant) dimension while provably preserving the quality of the solutions.",
"Specifically, our theorems incorporate the doubling dimension $d_X$ of the input datasets $X$ .",
"This parameterWe formally define it in Section .",
"measures the “intrinsic dimensionality” of $X$ and can be much lower than its ambient dimension $m$ .",
"If $X$ has size $n$ , the doubling dimension $d_X$ is always at most $\\log n$ , and is often much smaller.",
"We show that random projections into dimension roughly proportional to $d_X$ suffice in order to approximately preserve the solution quality.",
"The specific bounds are listed in Table REF .",
"Figure: Number of dimensions dd required for a random projection to provide a good clustering approximationWe distinguish between two types of guarantees.",
"The first type states that the minimum cost of FL or MST is preserved by a random projection (with high probability) up to the specified factor.",
"This guarantee is useful if the goal is to quickly estimate the optimal value.",
"The second type states that a solution computed in the projected space induces a solution in the original space which approximates the best solution (in the original space) up to the specified approximation factor.",
"This guarantee implies that one can find an approximately optimal clustering by mapping the data into low dimensions and clustering the projected data.",
"To obtain the second guarantee, we need to assume that the solution in the projected space is either globally optimal (for MST) or locally optimalInformally, a solution is locally optimal if opening any new facility does not decrease its cost.",
"The formal definition is slightly more general, and is given in Section .",
"Note that any solution found by local search algorithms such as that in [22] satisfies this condition.",
"(for FL).",
"We note that these two types of guarantees are incomparable.",
"In fact, for FL, our proofs of the cost and of the solution guarantees are substantially different.",
"We also prove analogous theorems for the “squared” version of FL, where the distance between points is defined as the square of the Euclidean distance between them.",
"We complement the above results by showing that the conditions and assumptions in our theorem cannot be substantially reduced or eliminated.",
"Specifically, for both FL and MST, we show that: The bounds on the projected dimension $d$ in the theorems specified in the table must be at least $\\Omega (d_X)$ , as otherwise the approximation factors for both the cost and the solution become super-constant (Theorems REF , REF , REF ) The assumptions that the solution in the projected space is (locally) optimal cannot be relaxed to “approximately optimal” (Lemmas REF , REF ).",
"Also, we show that, in contrast to facility location and MST, one must project to $\\Omega (\\log k)$ dimensions for preserving both the cost and solution for $k$ -means and $k$ -medians clustering, even if the doubling dimension $d_X$ is $O(1)$ .",
"Finally, we present an experimental evaluation of the algorithms suggested by our results.",
"Specifically, we show that both FL and MST, solving these problems in reduced dimension can reduce the running time by 1-2 orders of magnitude while increasing the solution cost only slightly.",
"We also give empirical evidence that the doubling dimension of the input point set affects the quality of the approximate solutions.",
"Specifically, we study two simple point sets of size $n$ that have similar structure but very different doubling dimension values ($O(1)$ and $O(\\log n)$ , respectively).",
"We empirically show that a good approximation of the MST can be found for the former point set by projecting it into much fewer dimensions than the latter point set."
],
[
"Related Work",
"There is a long line of existing work on approximating the solution of various clustering problems in metric spaces with small doubling dimensions (see [10], [11], [5], [26]).",
"The state of the art result is given in [25] where a near linear $(1+\\epsilon )$ -approximation algorithm is given for a variety of clustering problems.",
"However, these runtimes have a doubly-exponential dependence on $d$ which is proven to be unavoidable unless P = NP [25].",
"For MST in spaces of doubling dimension $d_X$ , it is known that an $(1+\\epsilon )$ -approximate solution can be computed in time $2^{O(d_X)} n \\log n + \\epsilon ^{-O(d_X)} n$ [12].",
"To the best of our knowledge, none of these algorithms have been implemented.",
"In addition, the notion of doubling dimension has also been previously used to study algorithms for high dimensional problems such as the nearest neighbor search, see e.g., [14], [6], [13].",
"The paper [14] is closest in spirit to our work, as it shows that, for a fixed point $q$ and a data set $X$ , a random projection into $O(d_X)$ dimensions approximately preserves the distance from $q$ to its nearest neighbor in $X$ with a “good” probability.",
"If the probability of success was of the form $1-1/{2n}$ , we could apply this statement to all (up to $n$ ) facilities in the solution simultaneously, which would prove our results.",
"Unfortunately, the probability of failure is much higher than $1/n$ , and therefore this approach fails.",
"Nevertheless, our proofs use some of the lemmas developed in that work, as discussed in Section ."
],
[
"Problem Definitions",
"The Euclidean Facility Location problem is defined as follows: We are given a dataset $X \\subset \\mathbb {R}^m$ of $n$ points and a nonnegative function $c: X \\rightarrow \\mathbb {R}$ that represents the cost of opening a facility at a particular point.",
"The goal is to find a subset $\\mathcal {F} \\subseteq X$ that minimizes the objective $\\operatorname{cost}(\\mathcal {F}) = \\sum _{f \\in \\mathcal {F}} c(f) + \\sum _{x \\in X} D(x, \\mathcal {F}),$ where $D(x, \\mathcal {F}) = \\min _{f \\in \\mathcal {F}} \\Vert x-f\\Vert $ .",
"In this work we restrict our attention to the case that $\\Vert \\cdot \\Vert $ is the Euclidean $(\\ell _2)$ metric.",
"The first term $\\sum _{f \\in \\mathcal {F}} c(f)$ is referred to as the opening costs and the second term $ \\sum _{x \\in X} D(x, \\mathcal {F})$ is referred to as the connection costs.",
"In this work, we also focus on the uniform version of facility location where all opening costs are the same.",
"By re-scaling the points, we can further assume that $f(x) = 1$ for all $x \\in X$ .",
"Therefore, throughout the paper, we focus on minimizing the following objective function: $\\operatorname{cost}(\\mathcal {F}) =| \\mathcal {F}| + \\sum _{x \\in X} \\, \\min _{f \\in \\mathcal {F}}\\Vert x-f\\Vert .$ A set $\\mathcal {F}$ of facilities is also referred to as a solution to the facility location problem.",
"The Euclidean Minimum Spanning Tree problem is defined as follows.",
"Given a dataset $X \\subset \\mathbb {R}^m$ of $n$ points, we wish to find a set $\\mathcal {M}$ of edges $(x, y)$ that forms a spanning tree of $X$ and minimizes the following objective function: $\\operatorname{cost}(\\mathcal {M}) = \\sum _{(x, y) \\in \\mathcal {M}}\\Vert x-y\\Vert .$"
],
[
"Properties of Doubling Dimension",
"We parameterize our dimensionality reduction using doubling dimension, a measure of the intrinsic dimensionality of the dataset.",
"The notion of doubling dimension holds for a general metric space $X$ and is defined as follows.",
"Let $B(x,r)$ denote the ball of radius $r$ centered at $x \\in X$ , intersected with the points in $X$ .",
"Then the doubling constant $\\lambda _X$ is the smallest constant $\\lambda $ such that for all $x \\in X$ and for all $r > 0$ , there exists $S\\subseteq X$ with $|S| \\le \\lambda $ such that $B(x,r) \\subseteq \\bigcup _{s \\in S} B(s, r/2)$ .",
"The doubling dimension of $X$ is is defined as $d_X := \\log \\lambda _X$ .",
"One can see that $\\lambda _X \\le |X|,$ so $d_X \\le \\log |X|$ .",
"In this paper, we focus on the case that $X$ is a subset of Euclidean space $\\mathbb {R}^m$ ."
],
[
"Dimension Reduction",
"In this paper we define a random projection as follows.",
"Definition 2.1 A random projection from $\\mathbb {R}^m$ to $\\mathbb {R}^d$ is a linear map $G$ with i.i.d.",
"entries drawn from $\\mathcal {N}(0, 1/d)$ .",
"The following dimensionality reduction result related to doubling dimension was proven in [14].",
"Informally, the lemma below states that a random projection of $X$ onto a dimension $O(d_X)$ subspace does not `expand' $X$ very much.",
"Lemma 2.2 (Lemma $4.2$ in [14]) Let $X \\subseteq B(0,1)$ be a subset of the $m$ -dimensional Euclidean unit ball, and let $G$ be a random projection from $\\mathbb {R}^m$ to $\\mathbb {R}^d$ .",
"Then there exist universal constants $c, C > 0$ such that for $d \\ge C \\cdot d_X + 1$ and $t > 2$ , $\\Pr (\\exists x \\in X, \\, \\Vert Gx\\Vert \\ge t) \\le \\exp (-cdt^2)$ .",
"For our proofs, we will need some additional preliminary results on random projections, which are deferred to Appendix ."
],
[
"Local Optimality for Facility Location",
"We now define the notion of a locally optimal solution for facility location.",
"As stated in the introduction, this notion plays a key role in our approximation guarantees.",
"Before we present our criterion for local optimality, we begin by discussing the Mettu Plaxton (MP) algorithm, an approximation algorithm for the facility location problem.",
"The MP approximation algorithm gives a useful geometric quantity to understand the facility location problem."
],
[
"Approximating the Cost of Facility Location",
"For each $p \\in X$ , we associate with it a radius $r_p > 0$ which satisfies the relation $\\sum _{q \\in B(p, r_p) \\cap X} (r_p - \\Vert p-q\\Vert ) = 1.$ It can be checked that a unique value $r_p$ satisfying $1/n \\le r_p \\le 1$ exists for every $p$ .",
"The geometric interpretation of $r_p$ is shown in Figure $\\ref {fig:circle_radii}$ .",
"This quantity was first defined by Mettu and Plaxton MPalg, who proved that a simple greedy procedure of iteratively selecting facilities that lie in balls of radii $2r_p$ gives a 3 factor approximation algorithm for the facility location problem.",
"For completeness, their algorithm is given in Appendix .",
"Figure: r p r_p is defined such that the dotted lines add to 1.One of the main insights from Mettu and Plaxton's algorithm is that the sum of the radii $r_p$ is a constant factor approximation to the cost of the optimal solution.",
"This insight was first stated in [2] where it was used to design a sublinear time algorithm to approximate the cost of the facility location problem.",
"In particular, we have the following result from [2] about the approximation property of the radii values.",
"Lemma 3.1 (Lemma 2 in [2]) Let $C_{OPT}$ denote the cost of the optimal facility location solution.",
"Then $\\frac{1}{4} \\cdot C_{OPT} \\le \\sum _{p \\in X} r_p \\le 6 \\cdot C_{OPT}$ .",
"For our purposes, we use the radii values to define a local optimality criterion for a solution to the facility location problem.",
"Our local optimality criterion states that each point $p$ must have a facility that is within distance $3r_p$ .",
"Definition 3.2 A solution $\\mathcal {F}$ to the facility location problem is locally optimal if for all $p \\in X$ , $B(p, 3r_p) \\cap \\mathcal {F} \\ne \\emptyset $ .",
"We show in Lemma REF that a solution that is not locally optimal can be improved, i.e.",
"the objective function given in Eq.",
"(REF ) can be improved, by adding $p$ to the set of facilities.",
"This implies that any global optimal solution must also be locally optimal, so requiring a solution of the facility location problem to be locally optimal is a less restrictive condition than requiring a solution to be globally optimal.",
"Lemma 3.3 Let $\\mathcal {F}$ be any collection of facilities.",
"If there exists a $p \\in X$ such that $B(p, 3r_p) \\cap \\mathcal {F} = \\emptyset $ then $\\textup {cost}(\\mathcal {F} \\cup \\lbrace p\\rbrace ) < \\textup {cost}(\\mathcal {F})$ , i.e., we can improve the solution.",
"The proof of Lemma REF is deferred to Appendix ."
],
[
"Approximating the Optimal Facility Location Cost",
"In this subsection we show that we can estimate the cost of the global optimal solution for a point set $X$ by computing the value of the radii after a random projection onto dimension $d = O(d_X)$ .",
"We do this by showing that for each $p$ , the value of $r_p$ can be approximated up to a constant multiplicative factor in $\\mathbb {R}^d$ , the lower dimension.",
"For each $p \\in X$ , let $r_p$ and $\\tilde{r}_p$ be the radius of $p$ and $Gp$ in $\\mathbb {R}^m$ and $\\mathbb {R}^d$ , respectively, computed according to Eq.",
"(REF ).",
"Then we prove that $\\mathbb {E}[\\tilde{r}_p] = \\Theta (r_p)$ , where the expectation is over the randomness of the projection $G$ .",
"This proof can be divided into showing $\\mathbb {E}[\\tilde{r}_p] = O(r_p)$ and $\\mathbb {E}[\\tilde{r}_p] = \\Omega (r_p).$ Our proof strategy for the former is to use the concentration inequality in Lemma REF to roughly say that points in $B(p, r_p) \\cap X$ cannot get `very far' away from $p$ after a random projection.",
"In particular, they must all still be at a distance $O(r_p)$ of $p$ after the random projection.",
"Then using the geometric definition of $r_p$ given in (REF ) and Figure REF , we can say that the corresponding radii of $Gp$ in $\\mathbb {R}^d$ denoted as $\\tilde{r}_p$ must then be upper bounded by $O(r_p)$ .",
"Our proof strategy for the latter is different in that our challenge is to show that points do not `collapse' closer together.",
"In more detail, for a fixed point $p$ , we need to show that after a dimension reduction, many new points do not come inside a ball of radius $O(r_p)$ around the point $Gp$ .",
"An application of Theorem REF in Appendix , due to Indyk and Naor indyknaor, deals with this event.",
"By adding these expectations over each point $p$ and applying Lemma REF , we can prove that the facility location cost is preserved under a random projection.",
"Formally, we obtain the following theorem: Theorem 4.1 Let $X \\subseteq \\mathbb {R}^m$ and let $G$ be a random projection from $\\mathbb {R}^m$ to $\\mathbb {R}^d$ for $d = O(d_X)$ .",
"Let $\\mathcal {F}_m$ be the optimal solution in $\\mathbb {R}^m$ and let $\\mathcal {F}_d$ be the optimal solution for the dataset $GX \\subseteq \\mathbb {R}^d$ .",
"Then there exist constants $c, C >0$ such that $ c \\cdot \\textup {cost}(\\mathcal {F}_m) \\le \\mathbb {E}[\\textup {cost}(\\mathcal {F}_d)] \\le C \\cdot \\textup {cost}(\\mathcal {F}_m)$ .",
"The full proof of Theorem REF and the lemmas bounding $\\mathbb {E}[\\tilde{r}_p]$ are deferred to Appendix ."
],
[
"Obtaining Facility Location Solution in Larger Dimension",
"As discussed in the introduction, for many applications, it is not enough to be able to approximate the cost of the optimal solution, but rather obtain a good solution.",
"In particular, we would like to perform dimensionality reduction on a dataset $X$ , use some algorithm to solve facility location, and then have the guarantee that the quality of the solution we found is a good indicator of the quality of the solution in the original dimension.",
"Furthermore, since optimally solving facility location in the smaller dimension might still be a challenging task, it is desirable to have a guarantee that a good solution (not necessarily the global optimum) will be a good solution in the larger dimension.",
"We show in this section that this is indeed the case for locally optimal solutions.",
"Specifically, we show that the cost of a locally optimal solution found in $\\mathbb {R}^d$ does not increase substantially when evaluated in the larger dimension.",
"More formally, we prove the following theorem: Theorem 4.2 Let $X \\subset \\mathbb {R}^m$ and $G$ be a random projection from $\\mathbb {R}^m$ to $\\mathbb {R}^d$ for $d = O(d_X \\cdot \\log (1/\\epsilon )/\\epsilon ^2)$ .",
"Let $\\mathcal {F}_d$ be a locally optimal solution for the dataset $GX$ .",
"Then, the cost of $\\mathcal {F}_d$ evaluated in $\\mathbb {R}^m$ , denoted as $\\textup {cost}_m(\\mathcal {F}_d)$ , satisfies $ \\mathbb {E}[\\textup {cost}_m(\\mathcal {F}_d)] \\le |\\mathcal {F}_d| + O\\bigg (\\sum _{p \\in X} r_p\\bigg ) \\le \\textup {cost}_d(\\mathcal {F}_d) + O(F),$ where $F$ is the optimal facility location cost of $X$ in $\\mathbb {R}^m$ .",
"To describe the proof intuition, first note that the cost function defined in Eq.",
"(REF ) has two components.",
"One is the number of facilities opened, and the other is the connection cost.",
"The first term is automatically preserved in the larger dimension since the number of facilities stays the same.",
"Therefore, the main technical challenge is to show that if a facility is within distance $O(\\tilde{r}_p)$ of a fixed point $p$ in $\\mathbb {R}^d$ (note that $\\tilde{r}_p$ is calculated according to Eq.",
"(REF ) in $\\mathbb {R}^d$ ), then the facility must be within distance $O(r_p)$ in $\\mathbb {R}^m$ , the larger dimension.",
"From here, one can use Lemma REF to bound $\\sum _{p \\in X} r_p$ by $O(F)$ , and the simple fact that $|\\mathcal {F}_d| \\le \\operatorname{cost}_d(\\mathcal {F}_d)$ .",
"The proof of our main technical challenge relies on the careful balancing of the following two events.",
"First, we control the value of the radius $\\tilde{r}_p$ and show that $\\tilde{r}_p \\approx r_p$ .",
"In particular, we show that the probability of $\\tilde{r}_p \\ge k r_p$ for any constant $k$ is exponentially decreasing in $k$ .",
"Next, we need to bound the probability that a `far' point comes `close' to $p$ after the dimensionality reduction.",
"While there exists a known result on this (e.g., Theorem REF in Appendix ), we need a novel, more detailed result to quantify how close far points can come after the dimension reduction.",
"To study this in a more refined manner, we bucket the points in $X \\setminus \\lbrace p\\rbrace $ according to their distance from $p$ , with the $i$ th level representing distance approximately $i$ from $p$ .",
"We show that points in $X\\setminus \\lbrace p\\rbrace $ that are in `level' $i$ do not shrink to a `level' smaller than $O(\\sqrt{i})$ .",
"Note that we need to control this even across all levels.",
"To do this requires a chaining type argument which crucially depends on the doubling dimension of $X$ .",
"Finally, a careful combination of probabilities gives us our result.",
"The proof of Theorem REF is deferred to Appendix .",
"Remark 4.3 Our proof of Theorem REF generalizes to the case of arbitrary opening costs $c_p$ by changing the definition of $r_p$ to be $ \\sum _{q \\in B(p, r_p)} (r_p - \\Vert x-q\\Vert ) = c_p$ ."
],
[
"Facility Location with Squared Costs",
"Facility location problem with squared costs is the following variant of facility location.",
"Given a dataset $X \\subset \\mathbb {R}^m$ , our goal is to find a subset $\\mathcal {F} \\subseteq X$ that minimizes the objective $\\operatorname{cost}(\\mathcal {F}) =| \\mathcal {F}| + \\sum _{x \\in X} \\, \\min _{f \\in \\mathcal {F}}\\Vert x-f\\Vert ^2.$ In contrast to (REF ), we are adding the squared distance from each point to its nearest facility in $\\mathcal {F}$ , rather than just the distance.",
"This is comparable to $k$ -means, whereas standard facility location is comparable to $k$ -medians.",
"For the facility location problem with squared costs, we are again able to show that a random projection of $X$ into $O(d_X)$ dimensions preserves the optimal cost up to an $O(1)$ factor, and that any locally optimal solution in the reduced dimension has its cost preserved in the original dimension.",
"The formal statements and proofs are very similar to those of the standard facility location problem, and are deferred to Appendix ."
],
[
"Dimension Reduction for MST",
"In this section we demonstrate the effectiveness of dimensionality reduction for the minimum spanning tree (MST) problem.",
"As in the case of facility location, we show that we can estimate the cost of the optimum MST solution by computing the MST in a lower dimension, and that the minimum spanning tree in the lower dimension is an approximate solution to the high-dimensional MST problem.",
"This time our approximations, both to the optimum cost and the optimum solution, can be $(1 + \\epsilon )$ -approximations for any $\\epsilon > 0$ , as opposed to the constant factor approximations that we could guarantee for facility location.",
"To formally state our theorem, for some spanning tree $T$ of $X$ , let $\\operatorname{cost}_X(T)$ be the sum of the lengths of the edges in $T$ .",
"Likewise, let $\\operatorname{cost}_{GX}(T)$ be the sum of the lengths of the edges in $T$ , where distances are measured in the projected tree $GX$ .",
"Our main result is the following theorem: Theorem 5.1 For some positive integers $m, d$ , let $X \\subset \\mathbb {R}^m$ be a point set of size $n$ and let $G: \\mathbb {R}^m \\rightarrow \\mathbb {R}^d$ be a random projection.",
"Let $\\mathcal {M}$ represent the minimum spanning tree of $X$ , with $M = \\operatorname{cost}_X(\\mathcal {M})$ and $\\widetilde{\\mathcal {M}}$ represent the minimum spanning tree of $GX$ , with $\\widetilde{M} = \\operatorname{cost}_{GX}(\\widetilde{\\mathcal {M}})$ .",
"Then, for some sufficiently large constant $C_6,$ if $d \\ge C_6 \\cdot \\epsilon ^{-2} \\cdot (\\log \\epsilon ^{-1} \\cdot d_X + \\log \\log n)$ , the following are true: The cost of the MST is preserved under projection with probability at least $\\frac{9}{10}.$ In other words, $\\widetilde{M} \\in [1-\\epsilon , 1+\\epsilon ] \\cdot M$ .",
"The optimal projected MST $\\widetilde{\\mathcal {M}}$ is still an approximate MST in the original dimension with probability at least $\\frac{9}{10}.$ In other words, $\\operatorname{cost}_{X}(\\widetilde{\\mathcal {M}}) \\in [1, 1+\\epsilon ] \\cdot M$ .",
"Hence, we obtain a significantly stronger theoretical guarantee for preserving the MST than $d = \\Theta (\\epsilon ^{-2} \\log n)$ , which is promised by the Johnson-Lindenstrauss Lemma, assuming that $d_X$ and $\\epsilon ^{-1}$ are constant or very small.",
"Our main technical result in establishing Theorem REF is the following crucial lemma, which will in fact allow us to prove both parts of the above theorem simultaneously.",
"Lemma 5.2 For all notation as in Theorem REF , $\\mathbb {E}[\\operatorname{cost}_{X}(\\widetilde{\\mathcal {M}}) - \\operatorname{cost}_{GX}(\\widetilde{\\mathcal {M}})] \\le O(\\epsilon ) \\cdot M.$ The proof strategy for Lemma REF involves first dividing the edges of $\\widetilde{\\mathcal {M}}$ into levels based on their lengths, and bounding the difference between edge lengths (pre- and post- projection) in each level separately.",
"To analyze a level consisting of the edges of length approximately $t$ , we first partition the point set $X$ (in the original dimension $\\mathbb {R}^m$ ) into balls $B_1, \\dots , B_r$ of radius $c \\cdot t$ for a small constant $c$ , and show via chaining-type arguments that not too many pairs of balls that were originally far apart come close together after the random projection.",
"Moreover, using Lemma REF , we show that almost all of the balls do not expand by much.",
"Therefore, there are not many bad pairs of balls $(B_i, B_j)$ , where $(B_i, B_j)$ is bad if there exists $p \\in B_i, q \\in B_j$ where $\\Vert p-q\\Vert $ is much bigger than $t$ but $\\Vert Gp-Gq\\Vert $ is approximately $t$ .",
"Now, assuming that none of the balls expand by much in the random projection, for any bad pair $(B_i, B_j)$ and edges $(p, q)$ and $(p^{\\prime }, q^{\\prime })$ with $p, p^{\\prime } \\in B_i$ and $q, q^{\\prime } \\in B_j,$ we cannot have both edges in the minimum spanning tree of $GX$ .",
"This is because $\\Vert Gp-Gq\\Vert , \\Vert Gp^{\\prime }-Gq^{\\prime }\\Vert \\approx t,$ but since $B_i$ and $B_j$ have radius $c \\cdot t$ and do not expand by much, we can improve the spanning tree by replacing $(Gp, Gq)$ with either $(Gp, Gp^{\\prime })$ or $(Gq, Gq^{\\prime })$ .",
"So, each bad pair can have at most 1 edge in $\\widetilde{\\mathcal {M}}$ , the MST of $GX$ .",
"Overall, in each level, not too many edges in $\\widetilde{\\mathcal {M}}$ can shrink by much after the projection.",
"The full proofs of Lemma REF and Theorem REF are given in Appendix ."
],
[
"Lower Bounds for Projection Dimension",
"In this section, we state various lower bounds for the projection dimension $d$ for both facility location clustering and minimum spanning tree.",
"We also show that, in contrast to facility location, low doubling dimension does not actually help with dimensionality reduction for $k$ -means or $k$ -medians clustering.",
"All proofs are deferred to Appendix .",
"In all results of this section, we think of $X$ as a point set of size $n$ in Euclidean space $\\mathbb {R}^m$ , and $G$ as a random projection sending $X$ to $GX \\subset \\mathbb {R}^d$ .",
"In this section, for FL, we always let $\\mathcal {F}$ be the optimal set of facilities in $X$ , with cost $F$ , and $\\widetilde{\\mathcal {F}}$ be the optimal set of facilities in $GX$ , with cost $\\widetilde{F}$ .",
"We define $\\mathcal {M}, M, \\widetilde{\\mathcal {M}}, \\widetilde{M}$ analogously for MST.",
"We use $o(1)$ to denote functions going to 0 as $n \\rightarrow \\infty ,$ and $\\omega (1)$ to denote functions going to $\\infty $ as $n \\rightarrow \\infty $ , where $n = |X|$ .",
"First, we show that the dependence of the projected dimension $d$ on the doubling dimension $d_X$ in Theorems REF , REF , and REF are all required to obtain constant factor approximations for either the cost or the pullback solution.",
"Namely, we show the following three theorems: Theorem 6.1 (FL) Let $d = o(\\log n)$ .",
"There exists $X$ with doubling dimension $\\Theta (\\log n)$ , such that with at least $2/3$ probability over $G: \\mathbb {R}^m \\rightarrow \\mathbb {R}^d$ , $\\widetilde{F} = o(1) \\cdot F$ .",
"Moreover, with probability at least $2/3,$ $\\widetilde{\\mathcal {F}}$ , when pulled back to $X$ , has cost $\\omega (1) \\cdot F$ in the original dimension.",
"Theorem 6.2 (MST) Let $d = o(\\log n).$ There exists $X$ with doubling dimension $\\Theta (\\log n)$ , such that with probability at least $2/3,$ $\\widetilde{M} = o(1) \\cdot M$ .",
"Theorem 6.3 (MST) Let $d = o(\\log n).$ There exists $X$ with doubling dimension $\\Theta (\\log n)$ , such that with probability at least $2/3,$ $\\widetilde{\\mathcal {M}},$ when pulled back to $X$ , will have cost $\\omega (1) \\cdot M$ .",
"Next, we show that (local) optimality is required for Theorems REF and REF , and cannot be replaced with approximate optimality.",
"In other words, random projections to $o(\\log n)$ dimensions do not necessarily preserve the set of approximate solutions for either facility location or MST, even for point sets of low doubling dimension.",
"Namely, we show the following two lemmas: Lemma 6.4 (FL) Let $d = o(\\log n)$ .",
"There exists $X$ with constant doubling dimension, such that with at least $2/3$ probability, there exists a $(1+O(m^{-1/2d})) = (1+o(1))$ -approximate solution $\\mathcal {F}^{\\prime }$ for $GX$ whose total cost when pulled back to $X$ is at least $\\Omega (m^{1/2d}) \\cdot F = \\omega (1) \\cdot F$ .",
"Lemma 6.5 (MST) Let $d = o(\\log n)$ but $d = \\omega (\\log \\log n)$ .",
"There exists $X$ with constant doubling dimension, such that with at least $2/3$ probability, there exists a $(1+o(1))$ -approximate MST $\\mathcal {M}^{\\prime }$ for $GX$ whose total cost whose total cost when pulled back to $X$ is at least $\\omega (1) \\cdot M$ .",
"Finally, we show that the guarantees of facility location are in fact not maintained for $k$ -means and $k$ -medians clustering.",
"In other words, the bound of $O(\\log k)$ by [19] is optimal even for sets of doubling dimension $O(1)$ .",
"Theorem 6.6 ($k$ -means/$k$ -medians) Let $k < n$ and $d = o(\\log k)$ .",
"Then, there exists $X$ with constant doubling dimension, such that with probability at least $2/3$ , the $k$ -means (resp., medians) cost of $GX$ is $o(1)$ times the $k$ -means (resp., medians) cost of $X$ .",
"Moreover, the optimal choice of $k$ centers in $GX$ , when pulled back to $X$ , will be an $\\omega (1)$ -approximate solution in the original dimension $\\mathbb {R}^m$ .",
"At a first glance, Theorem REF may appear to contradict our upper bounds for facility location.",
"However, in our counterexamples for $k$ -means and $k$ -medians, the cost (both initially and after projection) is substantially smaller than $k$ .",
"Facility location adds a cost of $k$ for the $k$ facilities that are created, and since these facilities now make up the bulk of the cost, the facility location cost is still approximately preserved under random projection."
],
[
"Experiments",
"We use the following datasets in our experiments for Subsections REF and REF .",
"Faces Dataset: This dataset is used in the influential ISOMAP paper and consists of 698 images of faces in dimension 4096 [27].",
"From [16], we can estimate that the doubling dimension of this dataset is a small constant.",
"MNIST `2' Dataset: 1000 randomly chosen images from the MNIST dataset (dimension 784) restricted to the digit 2.",
"We picked 2 since it is considered in the original ISOMAP paper [27].",
"All of our experimental results are averaged over 20 independent trials and the projection dimension $d$ ranges from 5 to 20 inclusive.All of our experiments were done on a CPU with i5 2.7 GHz dual core and 8 GB RAM.",
"Figure: Facility Location Experiments.",
"(a) Blue: Ratio of solution costs with/without dimensionality reduction, as a function of dd.",
"Orange: Running time (in secs) as a function of dd.",
"(b) Same plot as (a) but for MNIST `2' dataset.Figure: Minimum Spanning Tree Experiments.",
"(a) Blue: Ratio of solution costs with/without dimensionality reduction, as a function of dd.",
"Orange: Running time (in secs) as a function of dd.",
"(b) Same plots as (a) but for MNIST `2' dataset.",
"(c) Dataset 1 (low doubling dimension) can be projected into a much smaller dimension than Dataset 2 for MST computation."
],
[
"Facility Location: Cost versus Accuracy Analysis",
"In this section we compare the accuracy of the MP algorithm with/without dimensionality reduction for various number of centers opened."
],
[
"Experimental Setup ",
"We project our datasets and compute a facility location clustering with the opening costs scaled so that $n/2, n/5,$ and $n/10$ facilities are opened respectively.",
"We then take this solution and evaluate its cost in the original dimension.",
"We also perform a clustering in the original dimension with the same prescribed number of facilities opened and plot the ratio of the cost of the solution found in the lower dimension (but evaluated in the larger dimension) to the solution found in the larger dimension.",
"We also plot the time taken for the clustering algorithm in the projected dimension.",
"We use the MP algorithm to perform our clustering due to the intractability of finding the exact optimum and also because the MP algorithm is fast and quite practical to use."
],
[
"Results",
"Our results are plotted in Figures REF -REF .",
"Our experiments empirically demonstrate that the dimensionality reduction step does not significantly decrease the accuracy of the solution.",
"Furthermore, we get a substantial reduction in the runtime since the average runtime was at least 20 seconds for Faces and around $6.5$ seconds for MNIST `2' in the original dimension for all the values of $k$ tested, which is 1-2 orders of magnitude higher than the runtime when random projections are used.",
"Note that the runtime includes the time taken to perform the random projection.",
"Overall, our experiments demonstrate that the method of performing dimensionality reduction to perform facility location clustering is well-founded."
],
[
"MST: Cost versus Accuracy Analysis",
"We empirically show the benefits of using dimensionality reduction for minimum spanning tree computation."
],
[
"Experimental Setup",
"We project our datasets and compute a MST.",
"We then take the tree found in the lower dimension and compare its cost in the higher dimension against the actual MST.",
"Our MST algorithm is a variant of the Boruvka algorithm from [21] that is suitable for point sets in large dimensions and is implemented in the popular `mlpack' machine learning library [8]."
],
[
"Results",
"Our results are plotted in Figures REF -REF .",
"In the blue plots of these figures, the ratio of the cost of the tree found in the projected dimension, but evaluated in the original dimension, to the cost of the actual MST is shown.",
"We see that indeed as projection dimension increases, the ratio approaches 1.",
"However even for very low values of $d$ , such as 10, the tree found in the projected space serves as a good approximate for the actual MST.",
"Conversely, we see that as $d$ increases, the cost of computing the MST also increases as shown in the orange plots of the Figures REF and REF .",
"Note that the time taken to perform the projection is also included.",
"The time taken to compute the MST in the original dimension was approximately $3.2$ seconds for the Faces dataset and $7.1$ seconds for the MNIST `2' dataset.",
"Therefore, projection to dimension $d=20$ gives us approximately 80x improvement in speed for the Faces dataset and 30x improvement in speed for the MNIST `2' dataset while having a low cost distortion."
],
[
"Large versus Small Doubling Dimension ",
"In this section we present two datasets in $\\mathbb {R}^n$ where one dataset has doubling dimension $O(1)$ and the other has doubling dimension at least $\\Omega (\\log n)$ which is asymptotically the largest doubling dimension of any set of size $n$ .",
"We empirically show that the second dataset requires larger projection dimension than the first to guarantee that the MST found in the projected space induces a good solution in the original space.",
"Our two datasets are the following.",
"Let $e_i$ denote the standard basis vectors in $\\mathbb {R}^n$ .",
"We first draw $n$ standard Gaussians $g_1, \\cdots , g_n \\in \\mathbb {R}$ .",
"Our datasets are: Dataset 1: $\\lbrace g_1 \\cdot e_1, g_1 \\cdot e_1 + g_2 \\cdot e_2, \\ldots , g_1 \\cdot e_1 + \\cdots + g_n \\cdot e_n\\rbrace $ .",
"Dataset 2: $\\lbrace g_1 \\cdot e_1, g_1 \\cdot e_2, \\ldots , g_n \\cdot e_n \\rbrace $ .",
"Note that we use the same $g_i$ 's for both datasets.",
"The above datasets appear to be similar, but it can be shown that their respective doubling dimensions are $O(1)$ and $\\Omega (\\log n)$ ."
],
[
"Experimental Setup",
"We let $n = 1000$ and construct the two datasets.",
"We project our datasets and find the MST for each dataset in the projected space.",
"Then we evaluate the cost of this tree in the larger dimension and compare this cost to the cost of the actual MST for each dataset."
],
[
"Results",
"Figure REF demonstrates that we can find a high quality approximation of the MST by finding the MST in a much smaller dimension for Dataset 1 compared to Dataset 2.",
"For example, Dataset 1 required only $d = 10$ dimensions to approximate the true MST within $10\\%$ relative error while Dataset 2 needed $d = 38$ to get within $10\\%$ relative error of the true MST."
],
[
"Acknowledgments",
"This research was supported in part by the NSF TRIPODS program (awards CCF-1740751 and DMS-2022448); NSF award CCF-2006798; MIT-IBM Watson collaboration; Simons Investigator Award; and NSF Graduate Research Fellowship Program."
],
[
"Omitted Preliminaries",
"In this section, we state all of the preliminary results needed relating to random projections that were omitted in Section .",
"In all of the following results, we treat $G$ as a random projection from $\\mathbb {R}^m$ to $\\mathbb {R}^d$ .",
"First, if $x \\in S^{m-1}$ then the following statements hold about the distribution of $\\Vert Gx\\Vert $ [14]: $\\Pr ( | \\Vert Gx\\Vert -1 | \\ge t) &\\le \\exp (-dt^2/8), \\\\\\Pr ( \\Vert Gx\\Vert \\le 1/t) &\\le \\left( \\frac{3}{t} \\right)^d.",
"$ The following serves as a converse to Equation ().",
"Proposition A.1 If $x \\in S^{m-1}$ and $t \\ge 1$ then the following is true about the distribution of $\\Vert Gx\\Vert $ : $\\Pr ( \\Vert Gx\\Vert \\le 1/t) \\ge \\left( \\frac{1}{et} \\right)^d.$ Since $x \\in S^{m-1}$ , $d \\cdot \\Vert G x\\Vert ^2$ is a chi-squared random variable with $d$ degrees of freedom so it has density $\\frac{1}{2^{d/2} \\Gamma (d/2)} x^{d/2-1} \\exp (-x/2).$ Thus, for all $t \\le 1,$ the probability that $\\Vert Ga\\Vert ^2$ is less than $1/t^2$ is at least $&\\hspace{14.22636pt} \\frac{1}{2^{d/2} \\Gamma (d/2)} \\int _0^{d/t^2} x^{d/2-1} \\exp (-x/2) dx \\\\&\\ge \\frac{\\exp (-d/2t^2)}{2^{d/2} \\Gamma (d/2)} \\int _0^{d/t^2} x^{d/2-1} dx \\\\&\\ge \\frac{\\exp (-d)}{2^{d/2}\\Gamma (d/2) (d/2)} \\cdot \\left(\\frac{d}{t^2}\\right)^{d/2} \\\\&\\ge \\frac{\\exp (-d)}{2^{d/2} \\cdot (d/2)^{d/2}} \\cdot \\left(\\frac{d}{t^2}\\right)^{d/2} \\\\&= \\left(\\frac{1}{e \\cdot t}\\right)^d,$ where we used the well-known fact that $\\Gamma (x) \\cdot x \\le x^x$ for all $x \\ge 1$ .",
"We will need the following lemma to prove some of our lower bound results from Section .",
"Lemma A.2 Let $C \\ge 1$ and fix some point $v$ of norm at most $C$ in $\\mathbb {R}^d$ .",
"Then, if $x \\sim \\frac{1}{\\sqrt{d}} \\cdot \\mathcal {N}(0, I_d)$ is a $d$ -dimensional scaled multivariate Normal, then $\\Pr (\\Vert x-v\\Vert \\le \\frac{1}{C}) \\ge n^{-1/10},$ if $d \\le \\log n/(10 C^2)$ and $n$ is sufficiently large.",
"By the rotational symmetry of the multivariate normal, assume $v = (r, 0, \\dots , 0),$ where $0 \\le r \\le C.$ Then, if $x = (x_1, y)$ for $x_1 \\in \\mathbb {R}, y \\in \\mathbb {R}^{d-1}$ , then if $r-\\frac{1}{2C} \\le x_1 \\le r$ and $\\Vert y\\Vert \\le \\frac{1}{2C},$ then we indeed have $\\Vert x-v\\Vert \\le \\frac{1}{C}.$ Since $\\sqrt{d} x_1 \\sim \\mathcal {N}(0, 1)$ and $r \\le C$ , the probability that $r - \\frac{1}{2C} \\le x_1 \\le r$ equals the probability that $\\mathcal {N}(0, 1) \\in [(r-1/2C) \\sqrt{d}, r \\sqrt{d}],$ which is at least $\\frac{\\sqrt{d}}{2C} \\cdot \\frac{1}{\\sqrt{2 \\pi }} \\cdot e^{-C^2 d/2}.$ Moreover, the probability that $\\Vert y\\Vert \\le \\frac{1}{2C}$ is at least $\\left(\\frac{1}{2eC}\\right)^d$ by Proposition REF .",
"Therefore, $\\Pr \\left(\\Vert x-v\\Vert \\le \\frac{1}{C}\\right) &\\ge \\frac{\\sqrt{d}}{2C} \\cdot \\frac{1}{\\sqrt{2 \\pi }} \\cdot e^{-C^2 d/2} \\cdot \\left(\\frac{1}{2 e C}\\right)^d \\\\&\\ge n^{-1/10},$ where the last inequality is true because $d \\le \\log n/(10 C^2)$ and that $n$ is sufficiently large.",
"We recall Lemma REF , due to Indyk and Naor indyknaor.",
"Lemma A.3 (Lemma REF ) Let $X \\subseteq B(0,1)$ be a subset of the $m$ -dimensional Euclidean unit ball.",
"Then there exist universal constants $c, C > 0$ such that for $d \\ge C \\cdot d_X + 1$ and $t > 2$ , $\\Pr (\\exists x \\in X, \\, \\Vert Gx\\Vert \\ge t) \\le \\exp (-cdt^2)$ .",
"Indyk and Naor also prove the following result about the distance to the nearest neighbor after a random projection.",
"Theorem A.4 (Theorem $4.1$ in [14]) Let $G$ be a random projection from $\\mathbb {R}^m$ to $\\mathbb {R}^d$ for $d = O(d_X \\cdot \\log (1/\\epsilon )/\\epsilon ^2 \\log (1/\\delta ))$ .",
"Then for every $x \\in X$ , with probability at least $1-\\delta $ , the following statements hold: $D(Gx, G(X \\setminus \\lbrace x\\rbrace )) \\le (1+\\epsilon )D(x, X \\setminus \\lbrace x \\rbrace )$ Every $y \\in X$ with $\\Vert x-y\\Vert > (1+2\\epsilon )D(x, X\\setminus \\lbrace x\\rbrace )$ satisfies $\\Vert Gx-Gy\\Vert > (1+\\epsilon )D(x, X\\setminus \\lbrace x\\rbrace )$ where $D(x, X) = \\min _{y \\in X} \\Vert x-y\\Vert .$"
],
[
"The Mettu-Plaxton (MP) Algorithm and Local Optimality",
"First, we give the pseudocode for the Mettu-Plaxton (MP) algorithm for facility location, described in Section .",
"boxruled [H] InputInput OutputOutput Set $\\mathcal {F}$ of facilities $\\mathcal {F} \\leftarrow \\emptyset $ $i = 1$ to $n$ Compute $r_i$ satisfying: $ \\sum _{q \\in B(p_i, r_i)} (r_i - \\Vert p_i-q\\Vert ) = 1$ Sort such that $r_1 \\le \\ldots \\le r_n$ $i=1$ to $n$ $B(p_i, 2r_i) \\cap \\mathcal {F} = \\emptyset $ $\\mathcal {F} \\leftarrow \\mathcal {F} \\cup \\lbrace p_i\\rbrace $ Output $\\mathcal {F}$ $\\textsc {MP Algorithm}$ Next, we prove Lemma REF , which roughly stated that a globally optimal solution for facility location is always locally optimal.",
"[Proof of Lemma REF ] Consider an arbitrary point $p \\in X$ .",
"We first establish a lower bound on the number of points in $B(p, r_p) \\cap X$ .",
"Note that by definition of $r_p$ , we have $|B(p, r_p) \\cap X|r_p \\ge \\sum _{q \\in B(p,r_p)} (r_p-\\Vert p-q\\Vert ) = 1 $ so it follows that $|B(p, r_p) \\cap X| \\ge 1/r_p.$ Now suppose that $B(p, 3r_p) \\cap \\mathcal {F} = \\emptyset $ and let $m$ be the number of points in $|B(p, r_p) \\cap X|$ excluding $p$ .",
"The total connection cost of all these points to their nearest facility must be at least $2mr_p$ .",
"Accounting for point $p$ , the total connection costs of points in $B(p, r_p) \\cap X$ is at least $2mr_p + 3r_p$ .",
"Now if we open a new facility at point $p$ , then the connection costs of these points is at most $mr_p$ but we also incur an additional cost for opening a facility at $p$ .",
"Therefore, the total cost of the solution decreases by at least $(2mr_p + 3r_p) - (1 + mr_p) = (m+3)r_p -1.",
"$ Now from Eq.",
"(REF ), we have that $(m+3)r_p > 1$ , which means that the total cost decreases if we open a new facility at $p$ ."
],
[
"Approximating the Optimal Facility Location Cost",
"In this subsection, we prove Theorem REF .",
"As stated in Subsection REF , our proof involves computing an upper bound and a lower bound for $\\mathbb {E}[\\tilde{r}_p]$ .",
"We first proceed with an upper bound in Lemma REF .",
"Lemma C.1 Let $X \\subseteq \\mathbb {R}^m$ and let $p \\in X$ .",
"Let $G$ be a random projection from $\\mathbb {R}^m$ to $\\mathbb {R}^d$ for $d = O(\\log \\lambda _X \\cdot \\log (1/\\epsilon )/\\epsilon ^2)$ .",
"Let $r_p$ and $\\widetilde{r}_p$ be the radius of $p$ and $Gp$ in $\\mathbb {R}^m$ and $\\mathbb {R}^d$ respectively, computed according to Eq.",
"(REF ).",
"Then $\\mathbb {E}[\\tilde{r}_p] \\le (2+O(\\epsilon ))r_p.$ Let $\\delta > 0$ be fixed and let $\\mathcal {E}_k$ be the event that $\\max _{x \\in B(p, r_p) \\cap X} \\Vert G(x-p)\\Vert \\in [(k-1)(1+\\delta )r_p, k(1+\\delta )r_p)$ Note that $\\mathcal {E}_k$ implies that there exists an $x \\in B(p, r_p) \\cap X$ such that $\\Vert G(x-p)\\Vert \\ge (k-1)(1+\\delta )r_p$ , so by Lemma REF we have $\\Pr (\\mathcal {E}_k) &\\le \\Pr (\\exists x \\in B(p, r_p) \\cap X, \\, \\Vert G(x-p)\\Vert \\ge k(1+\\delta )r_p) \\nonumber \\\\&\\le \\exp (-c(k-1)^2(1+\\delta )^2d)$ for some constant $c$ .",
"We now show that conditioned on $\\mathcal {E}_k$ , we have $\\tilde{r}_p \\le (k+1)r_p(1+\\delta )$ .",
"This is because conditioning on $\\mathcal {E}_k$ gives us $\\sum _{Gq \\in B(Gp, (k+1)(1+\\delta )r_p)} ((k+1)r_p(1+\\delta )-\\Vert Gp-Gq\\Vert ) \\\\\\ge \\sum _{q \\in B(p, r_p)} (1+\\delta )r_p,$ where $Gq \\in B(Gp, (k+1)(1+\\delta )r_p)$ is interpreted as summing over the points in the set $GX \\cap B(Gp, (k+1)(1+\\delta )r_p).$ Furthermore, $\\sum _{q \\in B(p, r_p)} r_p(1+\\delta ) &\\ge \\sum _{q \\in B(p, r_p)} r_p \\\\&\\ge \\sum _{q \\in B(p, r_p)} (r_p - \\Vert p-q\\Vert ) = 1.$ Therefore, by the observation that the function $f(r) = \\sum _{q \\in B(p, r)} (r - \\Vert p-q\\Vert )$ is increasing in $r$ , it follows that $(k+1)(1+\\delta )r_p \\ge \\tilde{r}_p$ .",
"Therefore, we have $\\mathbb {E}[\\tilde{r}_p \\mid \\mathcal {E}_k] \\le (k+1)(1+\\delta )r_p.$ Now using Eq.",
"(REF ) $\\mathbb {E}[\\tilde{r}_p] &= \\sum _{k=1}^{\\infty } \\mathbb {E}[\\tilde{r}_p \\mid \\mathcal {E}_k]\\Pr (\\mathcal {E}_k) \\\\&\\le (2+\\delta )r_p \\\\&\\hspace{28.45274pt}+ r_p\\sum _{k=2}^{\\infty }(k+1) \\exp (-c(k-1)^2(1+\\delta )^2d) \\\\&\\le (2+\\delta )r_p \\\\&\\hspace{28.45274pt}+ (1+\\delta )r_p\\int _0^{\\infty }(x+2)\\exp (-C^{\\prime }x^2) \\ dx$ where $C^{\\prime } = c(1+\\delta )^2d$ .",
"We can explicitly evaluate that $ \\int _0^{\\infty } (x+2)\\exp (-C^{\\prime }x^2) \\ dx = \\sqrt{\\frac{\\pi }{C^{\\prime }}} + \\frac{1}{2C^{\\prime }}.$ Noting that $d = \\Omega (1/\\epsilon ^2)$ , we have that $ \\mathbb {E}[\\tilde{r}_p] \\le (2+O(\\epsilon ))r_p $ by picking $\\delta = O(\\epsilon )$ .",
"We now show the corresponding lower bound.",
"Lemma C.2 Let $X \\subseteq \\mathbb {R}^m$ and let $p \\in X$ .",
"Let $G$ be a random projection from $\\mathbb {R}^m$ to $\\mathbb {R}^d$ for $d = O(\\log \\lambda _X \\cdot \\log (1/\\epsilon )/\\epsilon ^2)$ .",
"Let $r_p$ and $\\widetilde{r}_p$ be the radius of $p$ and $Gp$ in $\\mathbb {R}^m$ and $\\mathbb {R}^d$ respectively, computed according to Eq.",
"(REF ).",
"Then $ \\mathbb {E}[\\tilde{r}_p] \\ge \\frac{(1-\\epsilon )r_p}{4}.$ Let $k$ be the size of the set $|B(p, r_p/2) \\cap X|$ .",
"By definition of $r_p$ , the following inequality holds: $1 &= \\sum _{q \\in B(p, r_p)}(r_p-\\Vert p-q\\Vert ) \\nonumber \\\\&\\ge \\sum _{q \\in B(p, r_p/2)}(r_p-\\Vert p-q\\Vert ) \\ge \\frac{k r_p}{2}.$ Now let $\\mathcal {E}$ be the event that the ball $B(Gp, (1-\\epsilon )r_p/2)$ contains at most $k$ points.",
"By invoking Theorem REF , we will show that $\\Pr (\\mathcal {E}) \\ge 1/2$ .",
"Consider the set $X$ without the $k-1$ points in $B(p,r_p/2) - \\lbrace p\\rbrace $ , and with an extra point $q$ at distance $(1-\\epsilon )r_p/2$ from $p$ .",
"The added point $q$ becomes a nearest neighbor of $p \\in X$ .",
"By Theorem REF part (2) applied to an appropriately chosen $\\epsilon ^{\\prime }= O(\\epsilon )$ , with probability at least $1/2$ , no point outside of $B(p,r_p/2)$ is mapped within $(1-\\epsilon )r_p/2$ of $p$ .",
"After removing $q$ , only the originally removed $k-1$ points (and $p$ ) can lie in $B(p,r_p(1-\\epsilon )/2)$ .",
"Conditioning on $\\mathcal {E}$ , it follows that $\\sum _{Gq \\in B(Gp, (1-\\epsilon )r_p/2)}\\left(\\frac{(1-\\epsilon ) r_p}{2}-\\Vert Gp-Gq\\Vert \\right) \\\\\\le \\frac{(1-\\epsilon )kr_p}{2} .$ Combining Eq.",
"(REF ) with (REF ), we have that $ \\sum _{Gq \\in B(Gp, (1-\\epsilon )r_p/2)}\\left(\\frac{(1-\\epsilon ) r_p}{2}-\\Vert p-q\\Vert \\right) \\le 1-\\epsilon < 1.",
"$ Therefore, conditional on $\\mathcal {E}$ , it follows that $\\widetilde{r}_p \\ge (1-\\epsilon )r_p/2$ .",
"Hence, $\\mathbb {E}[\\tilde{r}_p] \\ge \\frac{ \\mathbb {E}[\\tilde{r}_p \\mid \\mathcal {E}]}{2} \\ge \\frac{(1-\\epsilon )r_p}{4}.$ Combining Lemma REF and REF gives us the complete proof of Theorem REF .",
"[Proof of Theorem REF ] The theorem follows from combining the result given in Lemma REF , that the sum of the radii $r_p$ is a constant factor approximation to the global optimal solution, and Lemmas REF and REF , which state that $\\mathbb {E}[\\tilde{r}_p]$ is a constant factor approximation to $r_p$ ."
],
[
"Obtaining a Solution to Facility Location in Larger Dimension",
"Recall that the main technical challenge is to show that if a facility is within distance $O(\\tilde{r}_p)$ of a fixed point $p$ in $\\mathbb {R}^d$ (note that $\\tilde{r}_p$ is calculated according to Eq.",
"(REF ), in $\\mathbb {R}^d$ ), then the facility must also be within distance $O(r_p)$ in $\\mathbb {R}^m$ , the larger dimension.",
"We prove this claim formally in Theorem REF .",
"Before presenting Theorem REF , we need the following technical result later on for our probability calculations.",
"Lemma C.3 Denote $\\textup {erf}(x)$ to be the error function defined as $\\textup {erf}(x) = \\frac{2}{\\sqrt{\\pi }} \\int _0^x \\exp (-t^2) \\ dt.$ Then, $1-\\textup {erf}(x) \\le \\exp (-x^2) $ for all $x \\ge 1$ .",
"Note that $f(x) \\exp (-t^2)/\\sqrt{\\pi }$ is a valid probability density function over $\\mathbb {R}$ so that $ \\text{erf}(x) = 1- \\Pr (|Z| \\ge x)$ where $Z$ is distributed according to the density $f$ .",
"Now $\\Pr (Z \\ge x) &= \\frac{1}{\\sqrt{\\pi }} \\int _x^{\\infty } \\exp (-t^2) \\ dt\\\\&\\le \\frac{1}{\\sqrt{\\pi }} \\int _x^{\\infty } \\frac{t}{x} \\exp (-t^2) \\ dt \\\\&= \\frac{\\exp (-x^2)}{2x\\sqrt{\\pi }}$ where the inequality follows from the fact that $t \\ge x$ .",
"By symmetry, we have $ \\text{erf}(x) + \\exp (-x^2) -1 \\ge \\exp (-x^2)\\left(1 - \\frac{1}{x\\sqrt{\\pi }} \\right) \\ge 0 $ for $x \\ge 1$ .",
"The proof of Theorem REF relies on the careful balancing of the following two events.",
"First, we control the value of the radius $\\tilde{r}_p$ and show that $\\tilde{r}_p \\approx r_p$ .",
"In particular, we show that the probability of $\\tilde{r}_p \\ge k r_p$ for any constant $k$ is exponentially decreasing in $k$ .",
"The argument for this part follows similarly to the argument in Lemma REF .",
"Next, we need to bound the probability that a `far' point comes `close' to $p$ after the dimensionality reduction.",
"While Theorem REF roughly states that `far' points do not come too `close', we need a more detailed result to quantify how close far points can come after the dimension reduction.",
"To study this in a more refined manner, we bucket the points in $X \\setminus \\lbrace p\\rbrace $ according to their distance from $p$ .",
"The distance spacing between buckets will be a linear scale.",
"We show that points in $X\\setminus \\lbrace p\\rbrace $ that are in `level' $i$ do not shrink to a `level' smaller than $O(\\sqrt{i})$ .",
"Note that we need to control this even across all levels.",
"To do this requires a chaining type argument which crucially depends on the doubling dimension of $X$ .",
"Finally, a careful combination of probabilities gives us our result.",
"Theorem C.4 Let $X\\subseteq \\mathbb {R}^m$ and let $G$ be a random projection from $\\mathbb {R}^m$ to $\\mathbb {R}^d$ for $d = O(\\log \\lambda _X \\cdot \\log (1/\\epsilon )/\\epsilon ^2)$ .",
"Fix $p \\in X$ and let $x \\in X$ be the point that maximizes $\\Vert p-x\\Vert $ subject to the condition $Gx \\in B(Gp, C \\tilde{r}_p)$ where $C$ is a fixed constant.",
"Then $ \\mathbb {E}\\Vert p-x\\Vert \\le 2C(1+O(\\epsilon )) r_p.",
"$ For simplicity, let $r = r_p, \\tilde{r} = \\tilde{r}_p,$ and define $t_{-1} = 0, t_0 = 1,$ and $t_i = 1+2\\epsilon + \\frac{\\epsilon (i-1)}{4} $ for all $i \\ge 1$ .",
"Define $\\mathcal {E}_i$ to be the event that $2Crt_i \\le \\Vert p-x\\Vert \\le 2Crt_{i+1}$ (the range $2Crt_i$ to $2Crt_{i+1}$ are our `buckets' from the discussion preceding the proof).",
"Then $\\mathbb {E}\\Vert p-x\\Vert = \\sum _{i \\ge -1} \\mathbb {E}[\\Vert p-x\\Vert \\mid \\mathcal {E}_i] \\Pr (\\mathcal {E}_i).$ We first bound $\\Pr (\\mathcal {E}_i)$ in two different ways.",
"By conditioning on the value of $\\tilde{r}$ , we can write this probability as $\\Pr (\\mathcal {E}_i) &= \\sum _{j \\ge -1}\\Pr (\\mathcal {E}_i \\text{ and } 2rt_j \\le \\tilde{r} \\le 2rt_{j+1}) \\\\&= \\sum _{j \\ge -1}\\big [\\Pr (\\mathcal {E}_i \\mid 2rt_j \\le \\tilde{r} \\le 2rt_{j+1}) \\nonumber \\\\&\\hspace{56.9055pt}\\cdot \\Pr (2rt_j \\le \\tilde{r}\\le 2rt_{j+1})\\big ] .$ In the first of our two bounds for $\\Pr (\\mathcal {E}_i)$ , we proceed by bounding $\\Pr (2rt_j \\le \\tilde{r}\\le 2rt_{j+1})$ .",
"Heuristically, the event $2rt_j \\le \\tilde{r}\\le 2rt_{j+1}$ would mean that some point in $B(p, r)$ will be very far away from $p$ after the random projection and the probability of this event can be controlled very well.",
"More formally, we first claim that the event $2rt_j \\le \\tilde{r}\\le 2rt_{j+1}$ implies that there exists a point $z$ in $B(p, r)$ such that $\\Vert G(z-p)\\Vert \\ge rt_j$ .",
"This is because otherwise, we have $\\Vert Gp-Gq\\Vert < rt_j$ for all $q \\in B(p,r)$ .",
"This means that $\\sum _{q \\in B(Gp, 2rt_j)} (2rt_j - \\Vert Gp-Gq\\Vert ) &> \\sum _{q \\in B(p, rt_j)} (2rt_j -rt_j) \\\\&\\ge |B(p, r) \\cap X| \\cdot rt_j \\\\&\\ge |B(p, r) \\cap X| \\cdot r.$ We also know that $|B(p, r) \\cap X| \\cdot r \\ge 1$ from (REF ).",
"Altogether, we have that $\\sum _{q \\in B(Gp, 2rt_j)} (2rt_j - \\Vert Gp-Gq\\Vert ) > 1$ which cannot happen by definition of $\\tilde{r}$ and our assumption that $2rt_j \\le \\tilde{r}$ (see Figure REF ).",
"Therefore by Lemma REF , we have $\\Pr (2rt_j \\le \\tilde{r}\\le 2rt_{j+1}) \\le \\exp (-C_1dt_j^2)$ for some constant $C_1$ .",
"Summing over the variable $j$ in inequality (REF ) gives us a bound on $\\Pr (\\mathcal {E}_i)$ .",
"We will only end up using this bound for $j \\ge \\Omega (\\sqrt{i})$ , and will use the second bound for small $j$ .",
"We now give a second bound on $\\Pr (\\mathcal {E}_i)$ by controlling $\\Pr (\\mathcal {E}_i \\text{ and } 2rt_j \\le \\tilde{r} \\le 2rt_{j+1})$ .",
"Note that the event $\\mathcal {E}_i \\text{ and } 2rt_j \\le \\tilde{r} \\le 2rt_{j+1}$ together imply that there exists some $x$ that satisfies $2Crt_i \\le \\Vert p-x\\Vert \\le 2Crt_{i+1} \\ \\text{ and } \\Vert G(x-p)\\Vert \\le 2Crt_{j+1} $ due to the fact that $Gx$ is a point in $B(Gp, C \\tilde{r}_p)$ .",
"Therefore, $&\\Pr (\\mathcal {E}_i \\text{ and } 2rt_j \\le \\tilde{r} \\le 2rt_{j+1}) \\\\\\le &\\Pr (\\exists x, \\, 2Crt_i \\le \\Vert p-x\\Vert \\le 2Crt_{i+1} \\\\&\\hspace{56.9055pt} \\text{ and } \\Vert G(x-p)\\Vert \\le 2Crt_{j+1}).$ We bound the right hand side of the above probability for the range $j = O(\\sqrt{i})$ .",
"Let $X_i = \\lbrace x \\in X \\mid 2Crt_{i} \\le \\Vert x-p\\Vert < 2Crt_{i+1} \\rbrace .$ By the definition of doubling dimension, we can find a covering of $X_i$ with at most $\\lambda ^{O(\\log (t_i/\\epsilon ))}$ balls of radius $ 2Cr\\epsilon /4$ centered at points in some set $S \\subseteq X$ .",
"Then by Lemma REF , we have $\\hspace{-5.69046pt}\\Pr \\bigg (\\exists s \\in S \\ \\exists x \\in B(s, 2Cr\\epsilon /4) \\cap X_i, \\\\\\hspace{34.14322pt}\\Vert Gs-Gx\\Vert \\ge \\frac{2Cr\\epsilon \\sqrt{i}}{8}\\bigg ) \\le \\exp (-O(di))$ if $d \\ge \\Omega (\\log (\\lambda ) \\log (1/\\epsilon )/\\epsilon ^2)$ .",
"Now fix $s \\in S$ .",
"If $\\Vert G(s-p)\\Vert < 2Cr(1 + \\epsilon + \\epsilon \\sqrt{i}/4)$ then $\\frac{\\Vert G(s-p)\\Vert }{\\Vert s-p\\Vert } &\\le \\frac{1+\\epsilon + \\epsilon \\sqrt{i}/4}{1+2\\epsilon + \\epsilon i/4} \\\\&\\le {\\left\\lbrace \\begin{array}{ll}1-\\epsilon /4, &\\text{for} \\ 0 \\le i \\le 1/\\epsilon ^2 \\\\O(1)/\\sqrt{i}, &\\text{for} \\ i > 1/\\epsilon ^2\\end{array}\\right.",
"}.$ Hence by applying the two inequalities (REF ) and () to the unit vector $(s-p)/\\Vert s-p\\Vert $ , we have $&\\Pr \\left(\\exists s \\in S, \\Vert G(s-p)\\Vert \\le 2Cr\\left(1 + \\epsilon + \\frac{\\epsilon \\sqrt{i}}{4} \\right) \\right) \\\\&\\le {\\left\\lbrace \\begin{array}{ll}\\exp (-c^{^{\\prime \\prime }}d \\epsilon ^2), &\\text{for} \\ 0 \\le i \\le 1/\\epsilon ^2 \\\\i^{-c^{^{\\prime \\prime }}d}, &\\text{for} \\ i > 1/\\epsilon ^2\\end{array}\\right.",
"}.$ Note that we used the inequality (REF ) for the bound $i \\le 1/\\epsilon ^2$ and the inequality () for $i > 1/\\epsilon ^2$ .",
"Combining the above bound with the inequality in (REF ) gives us $&\\Pr \\left(\\exists x \\in X_i, \\Vert G(x-p)\\Vert \\le 2Cr\\left(1+ \\epsilon + \\frac{\\epsilon \\sqrt{i}}{8}\\right) \\right) \\\\&\\le {\\left\\lbrace \\begin{array}{ll}2 \\exp (-c^{^{\\prime \\prime }}d \\epsilon ^2), &\\text{for} \\ 0 \\le i \\le 1/\\epsilon ^2 \\\\2i^{-c^{^{\\prime \\prime }}d}, &\\text{for} \\ i > 1/\\epsilon ^2\\end{array}\\right.",
"}.$ Thus for $j \\le C_2\\sqrt{i}$ , we have $&\\Pr (\\exists x, 2Crt_{j+1} > \\Vert G(x-p)\\Vert \\\\&\\hspace{56.9055pt} \\text{ and } 2Crt_{i} \\le \\Vert x-p\\Vert < 2Crt_{i+1}) \\\\&\\le {\\left\\lbrace \\begin{array}{ll}2 \\exp (-C_3d \\epsilon ^2), &\\text{for} \\ 0 \\le i \\le 1/\\epsilon ^2 \\\\2i^{-C_3d}, &\\text{for} \\ i > 1/\\epsilon ^2\\end{array}\\right.",
"}$ where $C_2, C_3$ are fixed constants.",
"Using the representation given in () for $\\Pr (\\mathcal {E}_i)$ along with (REF ), we see that for $0 \\le i \\le 1/\\epsilon ^2$ , we can bound $ \\Pr (\\mathcal {E}_i) \\le 4\\exp (-C_2d \\epsilon ^2) + \\sum _{j \\ge 1} \\exp (-C_1dj^2\\epsilon ^2)$ while for $i > 1/\\epsilon ^2$ , we instead use the following stronger bound $ \\Pr (\\mathcal {E}_i) \\le 2C_2\\sqrt{i} \\cdot i^{-C_3d} + \\sum _{j \\ge C_2\\sqrt{i}} \\exp (-C_1dj^2\\epsilon ^2)$ which comes from using (REF ) for $j \\le C_2 \\sqrt{i}$ and () for larger $j$ .",
"Combing these bounds with (REF ), we have $\\mathbb {E}\\Vert p-x\\Vert \\le 2C(1+O(\\epsilon ))r + \\sum _{i \\ge 0} 2Crt_{i+1}\\Pr (\\mathcal {E}_i).$ Our task is to now bound the sum $\\sum _{i \\ge 0} t_{i+1}\\Pr (\\mathcal {E}_i)$ .",
"In the rest of the proof, we will show that this sum is $O(\\epsilon )$ .",
"We split the sum into two terms depending on if $i \\le 1/\\epsilon ^2$ or if $i > 1/\\epsilon ^2$ .",
"Using the bounds (REF ) and (REF ) gives us $&\\hspace{14.22636pt}\\sum _{i \\ge 0}t_{i+1}\\Pr (\\mathcal {E}_i) \\nonumber \\\\&\\le C_4\\sum _{i \\le 1/\\epsilon ^2} i \\bigg ( \\exp (-C_2 d \\epsilon ^2) + \\sum _{j \\ge 1} \\exp (-C_1dj^2\\epsilon ^2) \\bigg ) \\\\&+C_4 \\sum _{i > 1/\\epsilon ^2} \\epsilon i \\bigg ( i^{-C_2 d + 1/2} + \\sum _{j \\ge C_2\\sqrt{i}} \\exp (-C_1dj^2\\epsilon ^2) \\bigg ) $ for some constant $C_4$ .",
"In (REF ), we are using the fact that $t_{i+1} = O(i)$ and for (), we are instead using $t_{i+1} = O(\\epsilon i)$ .",
"We can bound (REF ) $\\frac{\\exp (-C_2 d \\epsilon ^2)}{\\epsilon ^4} + \\frac{1}{\\epsilon ^2} \\sum _{j \\ge 1} \\exp (-C_1 d j^2 \\epsilon ^2) \\le O(\\epsilon )$ by using the fact that $d = \\Omega (\\log (1/\\epsilon )/\\epsilon ^2)$ .",
"We now focus on bounding ().",
"As a first step, we have the estimate $ \\epsilon \\sum _{i > 1/\\epsilon ^2} i^{-C_2 d + 3/2} = O(\\epsilon ) $ which holds for large enough constant $d$ .",
"Finally, bounding the remaining sum of () by an integral gives us $&\\hspace{14.22636pt} \\epsilon \\sum _{i > 1/\\epsilon ^2} i\\sum _{j \\ge C_2 \\sqrt{i}}\\exp (-C_1dj^2\\epsilon ^2) \\\\&\\le \\epsilon \\int _{1}^{\\infty } x \\int _{\\sqrt{x}}^{\\infty } \\exp (-C_5d t^2 \\epsilon ^2) \\ dt \\ dx$ for some constant $C_5$ .",
"Now using the definition of the complementary error function, we can compute that $&\\hspace{14.22636pt} \\epsilon \\int _{1}^{\\infty } x \\int _{\\sqrt{x}}^{\\infty } \\exp (-C_5d t^2 \\epsilon ^2) \\ dt \\ dx \\\\&\\le O(d^{-1/2}) \\int _1^{\\infty } x \\cdot \\text{erfc}(\\epsilon \\sqrt{C_5d \\cdot x}) \\ dx \\\\&\\le O(\\epsilon ) \\int _1^{\\infty } x \\cdot \\text{erfc}(\\epsilon \\sqrt{C_5d \\cdot x}) \\ dx.$ From Lemma REF , we have $\\int _1^{\\infty } x \\cdot \\text{erfc}(\\epsilon \\sqrt{C_5d \\cdot x}) \\ dx \\le \\int _1^{\\infty }x \\cdot \\exp (-C_5^2\\epsilon ^2 d \\cdot x) \\ dx \\\\= O(\\epsilon )$ using the fact that $d = \\Omega (\\log (1/\\epsilon )/\\epsilon ^2)$ .",
"Altogether, the bounds (REF ) and (REF ) allow us to bound the right hand side of (REF ) and () and therefore, bound the sum $\\sum _{i \\ge 0} t_{i+1}\\Pr (\\mathcal {E}_i)$ as $O(\\epsilon )$ .",
"Finally, using (REF ), we end up with $\\mathbb {E}\\Vert p-x\\Vert \\le 2C(1+O(\\epsilon ))r. $ As a corollary, we can prove Theorem REF .",
"[Proof of Theorem REF ] Let $\\mathcal {F}_d$ be a locally optimal solution in $GX$ .",
"When we evaluate the cost of $\\mathcal {F}_d$ in the larger dimension $\\mathbb {R}^m$ , the number of facilities stays the same.",
"Now since $\\mathcal {F}_d$ is a locally optimal solution in $\\mathbb {R}^d$ , each point $p$ has a facility that is within distance $C\\tilde{r}_p$ in $\\mathbb {R}^d$ .",
"Then by Theorem REF , the connection cost of $p$ in the larger dimension is bounded by $C^{\\prime }r_p$ , for some constant $C^{\\prime }$ , in expected value.",
"Summing over all points $p \\in X$ gives us $\\mathbb {E}[\\operatorname{cost}_m(\\mathcal {F}_d)] \\le |\\mathcal {F}_d|+O\\bigg (\\sum _{p \\in X}r_p\\bigg ).$ Finally, since $|\\mathcal {F}_d| \\le \\operatorname{cost}_d(\\mathcal {F}_d)$ by definition, and since $\\sum _{p \\in X} r_p = O(F)$ by Lemma REF , we have that $|\\mathcal {F}_d|+O\\bigg (\\sum _{p \\in X}r_p\\bigg ) \\le \\operatorname{cost}_d(\\mathcal {F}_d)+O(F).$ Together, these prove the theorem."
],
[
"Dimension Reduction for MST: Omitted Proofs",
"In this section, we prove Lemma REF and Theorem REF ."
],
[
"Proof of Theorem ",
"In this subsection, we prove that Lemma REF implies Theorem REF .",
"To see why, first note that $\\operatorname{cost}_X(\\widetilde{\\mathcal {M}}) \\ge \\operatorname{cost}_X(\\mathcal {M})$ and $\\operatorname{cost}_{GX}(\\mathcal {M}) \\ge \\operatorname{cost}_{GX}(\\widetilde{\\mathcal {M}})$ , since $\\mathcal {M}$ is the minimum spanning tree on $X$ and $\\widetilde{\\mathcal {M}}$ is the minimum spanning tree on $GX$ .",
"Moreover, for each edge $e = (x, y) \\in \\mathcal {M},$ $\\Vert Gx-Gy\\Vert $ has distribution $\\chi _d/\\sqrt{d} \\cdot \\Vert x-y\\Vert ,$ where $\\chi _d$ is the square root of a chi-square with $d$ degrees of freedom.",
"This has mean $\\mu = \\Vert x-y\\Vert \\cdot \\frac{1}{\\sqrt{2d}} \\cdot \\frac{\\Gamma ((d+1)/2)}{\\Gamma (d/2)} = \\Vert x-y\\Vert \\cdot \\left(1 - O\\left(\\frac{1}{d}\\right)\\right)$ and variance $\\Vert x-y\\Vert ^2-\\mu ^2 = \\Vert x-y\\Vert ^2 \\cdot O(1/d)$ [28].",
"Therefore, the standard deviation of $\\Vert G(x-y)\\Vert $ is at most $\\epsilon \\cdot \\Vert x-y\\Vert $ since $d = \\Omega (\\epsilon ^{-2})$ .",
"Therefore, the expectation of $\\operatorname{cost}_{GX}(\\mathcal {M})$ is $\\sum _{e = (x, y) \\in \\mathcal {M}} \\Vert x-y\\Vert \\cdot (1-O(1/d)) = M \\cdot (1-O(1/d))$ .",
"Also, using the well known fact that for any (possibly correlated) random variables $X_1, \\dots , X_n,$ $\\sqrt{Var(X_1+\\dots +X_n)} \\le \\sum \\sqrt{Var(X_i)},$ we have that the standard deviation of $\\operatorname{cost}_{GX}(\\mathcal {M})$ is at most $\\sum _{e = (x, y) \\in \\mathcal {M}} \\epsilon \\cdot \\Vert x-y\\Vert = \\epsilon \\cdot M$ .",
"To finish, define random variables $Z_1 = \\operatorname{cost}_X(\\widetilde{\\mathcal {M}})-\\operatorname{cost}_X(\\mathcal {M}),$ $Z_2 = \\operatorname{cost}_X(\\mathcal {M}) - \\operatorname{cost}_{GX}(\\mathcal {M}),$ and $Z_3 = \\operatorname{cost}_{GX}(\\mathcal {M}) - \\operatorname{cost}_{GX}(\\widetilde{\\mathcal {M}})$ .",
"Our observations from the previous paragraph tell us that $Z_1$ and $Z_3$ are nonnegative, and $Z_2$ has nonnegative expectation and standard deviation bounded by $O(\\epsilon ) \\cdot M$ .",
"Finally, Lemma REF tells us that $\\mathbb {E}[Z_1+Z_2+Z_3] \\le O(\\epsilon ) \\cdot M$ .",
"However, this means that $\\mathbb {E}[Z_1] \\le O(\\epsilon ) \\cdot M,$ so $0 \\le Z_1 \\le O(\\epsilon ) \\cdot M$ with high probability by Markov's inequality.",
"Therefore, $\\operatorname{cost}_X(\\widetilde{\\mathcal {M}}) \\le (1+O(\\epsilon )) \\cdot M$ with high probability, so the pullback is a $1+O(\\epsilon )$ approximation with high probability.",
"Likewise, we also have that $0 \\le Z_3 = O(\\epsilon )$ with high probability, and since $\\mathbb {E}[Z_2], \\sqrt{Var(Z_2)} \\le O(\\epsilon ) \\cdot M$ , we also have that $|Z_2| = O(\\epsilon )$ with high probability.",
"Thus, $|Z_2+Z_3| = O(\\epsilon )$ with high probability, which means $\\operatorname{cost}_{GX}(\\widetilde{\\mathcal {M}}) \\in [1-O(\\epsilon ), 1+O(\\epsilon )] \\cdot M$ .",
"As a result, the MST cost is preserved under dimensionality reduction with high probability as well."
],
[
"Proof of Lemma ",
"In this subsection, we prove prove Lemma REF .",
"In fact, we show the following stronger statement.",
"$ \\hspace{-8.5359pt}\\mathbb {E}_G\\left[\\sum _{e = (x, y) \\in \\widetilde{\\mathcal {M}}} \\max (0, \\Vert x-y\\Vert -(1+5\\epsilon )\\Vert Gx-Gy\\Vert )\\right] \\\\\\le \\epsilon \\cdot M.$ To see why this implies Lemma REF , by removing the maximum with 0, Equation (REF ) implies that $\\mathbb {E}_G[\\operatorname{cost}_{X}(\\widetilde{\\mathcal {M}}) - (1+5 \\epsilon ) \\cdot \\operatorname{cost}_{GX}(\\widetilde{\\mathcal {M}})] \\le \\epsilon \\cdot M.$ But $\\mathbb {E}_G[\\operatorname{cost}_{GX}(\\widetilde{\\mathcal {M}})] \\le \\mathbb {E}_G[\\operatorname{cost}_{GX}(\\mathcal {M})] \\le (1+\\epsilon ) \\cdot M,$ which means that $\\mathbb {E}_G[\\operatorname{cost}_{X}(\\widetilde{\\mathcal {M}}) - \\operatorname{cost}_{GX}(\\widetilde{\\mathcal {M}})] \\le \\epsilon \\cdot M + 5 \\epsilon \\cdot (1+\\epsilon ) \\cdot M = O(\\epsilon ) \\cdot M.$ [Proof of Equation (REF )] Consider some range $A_i = \\left[(1+\\epsilon )^i, (1+\\epsilon )^{i+1}\\right).$ We will bound the expectation of $K_i := \\sum _{\\begin{array}{c}e = (x, y) \\in \\widetilde{\\mathcal {M}} \\\\ \\Vert Gx-Gy\\Vert \\in A_i\\end{array}} \\max \\big (0, \\Vert x-y\\Vert -(1+5\\epsilon )\\Vert Gx-Gy\\Vert \\big )$ and sum our upper bounds for $\\mathbb {E}_G[K_i]$ over a range of $i$ .",
"For $K_i$ to be nonzero, we need there to exist $(x, y)$ such that $\\Vert x-y\\Vert \\ge \\Vert Gx-Gy\\Vert $ and $\\Vert Gx-Gy\\Vert \\in A_i,$ so $\\Vert x-y\\Vert \\ge (1+\\epsilon )^i.$ Therefore, we only need to sum $\\mathbb {E}[K_i]$ over integers $i$ such that $(1+\\epsilon )^i \\le \\operatorname{diam}(X)$ .",
"To do this, first consider some fixed $i$ and some sufficiently large constant $C_1$ , and define $t := t_i := \\frac{\\epsilon }{C_1} \\cdot (1+\\epsilon )^i$ .",
"Consider the following greedy procedure of selecting a partition of $X$ .",
"First, pick some point $x_1$ arbitrarily, then pick some point $x_2$ of distance more than $t$ from $x_1$ (in the original space), then some point $x_3$ of distance more than $t$ from $x_1$ and $x_2$ , and so on until we have some $x_1, \\dots , x_r$ and can no longer pick any more points.",
"Finally, we partition $X$ into subsets $X_1, \\dots , X_r$ so that each $x \\in X$ is in $X_p$ if $x_p$ is the closest point to $x$ (breaking ties arbitrarily).",
"Note that the partitioning is deterministic (independent of $G$ ).",
"We show the following proposition: Proposition D.1 The MST cost $M$ of the dataset $X$ (in the original space $\\mathbb {R}^m$ ) is at least $\\frac{r \\cdot t}{2}.$ By a known result on Steiner Trees [17], $M$ is at least $\\frac{r}{2(r-1)}$ times the MST cost of the set $\\lbrace x_1, \\dots , x_r\\rbrace \\subset X$ , assuming $r \\ge 2$ .",
"As the distance between any $x_p, x_q$ is at least $r$ , the MST cost of $\\lbrace x_1, \\dots , x_r\\rbrace $ is at least $(r-1) \\cdot t,$ so $M \\ge \\frac{r}{2(r-1)} \\cdot (r-1) \\cdot t = \\frac{r \\cdot t}{2}.$ Finally, as $(1+\\epsilon )^i \\le \\operatorname{diam}(X)$ , we have $t \\le \\operatorname{diam}(X)/C_1$ , so if $C_1 > 2$ , then the greedy procedure of partitioning $X$ cannot end with just $x_1$ , so indeed $r \\ge 2$ .",
"Now, we consider partitioning each $X_p$ into subsets $X_{p, 1}, \\dots , X_{p, s}$ as follows.",
"Since the radius of $X_p$ is at most $t$ , by definition of the doubling dimension, for each $k \\ge 1$ we can split $X_p$ into at most $\\lambda _X^k$ balls of radius at most $t/2^k$ .",
"We choose the smallest integer $k$ so that all of these balls have diameter at most $2t$ when projected by $G$ , and let $s_p = s$ be the number of subsets $X_{p, q}$ formed for each $p$ .",
"(Note: this partitioning $X_{p, q}$ is now dependent on $G$ .)",
"We claim the following: Proposition D.2 For any fixed $p$ and all integers $k \\ge 1,$ $\\Pr (s_p > \\lambda _X^{k}) \\le \\exp \\left(-c d 2^{k}\\right)$ .",
"For any fixed $k$ , we split $X_p$ into at most $\\lambda _X^k$ balls of radius at most $t/2^k$ : this process is independent of $G$ .",
"Now, fix a small ball: when we apply the random projection $G$ , the probability that it has radius more than $t$ when projected is at most $\\exp \\left(-c d 2^{2k}\\right)$ , by Lemma REF .",
"But there are $\\lambda _X^k \\le \\exp \\left(c d k\\right)$ such balls if $d$ is at least $c^{-1} \\log \\lambda _X$ , so the probability that even one of $G X_{p, q}$ has radius more than $t$ is at most $\\exp \\left(-c d 2^{2k}\\right) \\cdot \\exp \\left(c d k \\right) = \\exp \\left(-c d (2^{2k}-k)\\right) \\le \\exp \\left(-c d 2^k\\right)$ .",
"We also make the following observations: If $x \\in X_p,$ then $\\Vert x-x_p\\Vert \\le t$ , so the diameter of each $X_p$ is at most $2t$ .",
"Likewise, the diameter of each $X_{p, q}$ is at most $2t$ in both the original space and the reduced space.",
"By properties of the doubling dimension, for any $x_p$ and all $k \\ge 1,$ there are at most $\\lambda _X^{C_2 \\cdot k}$ points $\\lbrace x_{p^{\\prime }}\\rbrace _{p^{\\prime }= 1}^{r}$ within $2^k \\cdot t$ of $x_p$ for some $C_2$ , since $x_1, \\dots , x_r$ are all at least $t$ apart.",
"Recall that $D(X_p, X_{p^{\\prime }})$ is the maximum distance between points in $X_p$ and $X_{p^{\\prime }}$ (in the original space), as opposed to $d(X_p, X_{p^{\\prime }})$ which is the minimum distance.",
"Now, for any fixed $i$ , we bound the expectation of $L_i := \\sum _{\\begin{array}{c}p, p^{\\prime } \\\\ d(GX_p, GX_{p^{\\prime }}) < (1+\\epsilon )^{i+1} \\\\ D(X_p, X_{p^{\\prime }}) \\ge (1+5 \\epsilon ) \\cdot (1+\\epsilon )^i\\end{array}} D(X_p, X_{p^{\\prime }}) \\cdot s_p s_{p^{\\prime }},$ where the sum is over all pairs $p, p^{\\prime } \\in [r]$ .",
"First, we make the following claim.",
"Lemma D.3 For all $i$ and any fixed $G$ , $L_i \\ge K_i$ .",
"For any edge $e \\in \\widetilde{\\mathcal {M}},$ if $e$ has length in range $A_i$ (in the projected space), then this length is greater than $2C_1 \\cdot t$ (assuming $\\epsilon < 1/2$ ).",
"Then, $e$ is some edge $(Gx, Gy)$ where $x \\in X_{p, q}, y \\in X_{p^{\\prime }, q^{\\prime }}$ , where $(p, q) \\ne (p^{\\prime }, q^{\\prime })$ by Observation REF .",
"So, if edge $e$ contributes toward the sum in $K_i,$ then $D(X_p, X_{p^{\\prime }}) \\ge \\Vert x-y\\Vert \\ge (1+5 \\epsilon ) \\cdot \\Vert Gx-Gy\\Vert \\ge (1+5 \\epsilon ) \\cdot (1+\\epsilon )^i.$ At the same time, $d(GX_p, GX_{p^{\\prime }}) \\le \\Vert Gx-Gy\\Vert < (1+\\epsilon )^{i+1}.$ Thus, this pair $(p, p^{\\prime })$ contributes toward the sum in $L_i$ .",
"Moreover, $D(X_p, X_{p^{\\prime }}) \\ge \\Vert x-y\\Vert \\ge \\max (0, \\Vert x-y\\Vert -(1+5 \\epsilon ) \\Vert Gx-Gy\\Vert ).$ This will be useful since $L_i$ is a sum over $D(X_p, X_{p^{\\prime }})$ (multiplied by $s_p s_{p^{\\prime }}$ ) and $K_i$ is a sum over $\\max (0, \\Vert x-y\\Vert -(1+5 \\epsilon ) \\Vert Gx-Gy\\Vert )$ .",
"Finally, it is impossible for two pairs $(Gx, Gy)$ and $(Gx^{\\prime }, Gy^{\\prime })$ to both be edges in $\\widetilde{\\mathcal {M}}$ that contribute to the sum $K_i$ , if $x, x^{\\prime } \\in X_{p, q}$ and $y, y^{\\prime } \\in X_{p^{\\prime }, q^{\\prime }}$ .",
"If there were such pairs $(Gx, Gy), (Gx^{\\prime }, Gy^{\\prime })$ , this means that the edges $(Gx, Gy)$ and $(Gx^{\\prime }, Gy^{\\prime })$ have length in $A_i$ , and therefore have length at least $2C_1 \\cdot t$ .",
"However, the diameters of $G X_{p, q}$ and $G X_{p^{\\prime }, q^{\\prime }}$ are at most $2 t$ , so it would be better to replace edge $(Gx^{\\prime }, Gy^{\\prime })$ with either edge $(Gx, Gx^{\\prime })$ or edge $(Gy, Gy^{\\prime })$ : exactly one of these replacements will preserve the spanning tree property, and either replacement reduces the total cost.",
"Thus, for each pair $(p, p^{\\prime })$ contributing to the sum in $L_i,$ at most $s_p \\cdot s_{p^{\\prime }}$ corresponding pairs $(x, y)$ can contribute to the sum in $K_i$ , and since $D(X_p, X_{p^{\\prime }}) \\ge \\max (0, \\Vert x-y\\Vert -(1+5 \\epsilon ) \\Vert Gx-Gy\\Vert )$ whenever $x \\in X_{p, q}, y \\in X_{p^{\\prime }, q^{\\prime }}$ , this finishes the proof.",
"We will now bound the expectation of $L_i$ .",
"Lemma D.4 For any fixed $i$ , $\\mathbb {E}[L_i] \\le \\frac{\\epsilon ^2}{10 \\log n} \\cdot M$ .",
"For each $j \\ge 1,$ define $B_{i, j}$ to be the interval $[(1+5 \\epsilon ) \\cdot (1+\\epsilon )^{i+j-1}, (1+5 \\epsilon ) \\cdot (1+\\epsilon )^{i+j})$ .",
"Fix some $p, p^{\\prime }$ such that $D(X_p, X_{p^{\\prime }}) \\in B_{i, j}$ (note: this is independent of $G$ ).",
"Since all points in $X_p$ are at most $t$ from $x_p$ (and similar for $X_{p^{\\prime }}$ ), we have that $\\Vert x_p-x_{p^{\\prime }}\\Vert \\ge (1+5 \\epsilon ) \\cdot (1+\\epsilon )^{i+j-1} - 2 t \\ge (1+3 \\epsilon ) \\cdot (1+\\epsilon )^{i+j}.$ Now, if $d(GX_p, GX_{p^{\\prime }}) < (1+\\epsilon )^{i+1},$ then one of the following three events must be true: $\\Vert x_p - x_{p^{\\prime }}\\Vert \\le (1+\\epsilon )^{i+(j/2)} \\cdot (1+3\\epsilon )$ $\\operatorname{diam}(GX_p) \\ge \\epsilon \\cdot (1+\\epsilon )^{i+(j/2)}$ $\\operatorname{diam}(GX_{p^{\\prime }}) \\ge \\epsilon \\cdot (1+\\epsilon )^{i+(j/2)}$ .",
"Indeed, if all three were false, then $d(GX_p, GX_{p^{\\prime }}) \\ge \\Vert x_p - x_{p^{\\prime }}\\Vert - \\operatorname{diam}(GX_p) - \\operatorname{diam}(GX_{p^{\\prime }}) \\ge (1+\\epsilon )^{i+(j/2)} \\cdot (1+\\epsilon ) \\ge (1+\\epsilon )^{i+1}$ .",
"Now, the probability of the first event (over the randomness of $G$ ) is at most the probability that a random projection shrinks $x_p-x_{p^{\\prime }}$ by a factor of at least $(1+\\epsilon )^{j/2}$ .",
"By Equation (REF ), if $j \\le \\epsilon ^{-1},$ then this happens with probability at most $\\exp \\left(-d(j \\epsilon )^2/100\\right)$ , and by Equation (), if $j > \\epsilon ^{-1},$ then this happens with probability at most $(1+\\epsilon )^{-(j/2) \\cdot d/20} \\le \\exp \\left(-d (j \\epsilon )/100\\right).$ The probability of each of the second and third events occurring, since $\\operatorname{diam}(X_p), \\operatorname{diam}(X_{p^{\\prime }}) \\le \\epsilon \\cdot (1+\\epsilon )^i/C_1,$ is at most $\\exp \\left(-c d \\cdot C_1^2 (1+\\epsilon )^{j}\\right) \\le \\exp \\left(-d (j \\epsilon )/100\\right)$ by Lemma REF .",
"Next, note that by Proposition REF , for some constant $C_3,$ $\\Pr (s_p \\ge \\lambda _X^{C_3 \\cdot k}) \\le \\exp (-d \\cdot 2^k/100)$ for all real $k \\ge 1$ , and the same is true for $s_{p^{\\prime }}$ .",
"Again consider some fixed $j$ and some $p, p^{\\prime }$ with $D(X_p, X_{p^{\\prime }}) \\in B_{i, j}.$ Define the random variable $Z_{p, p^{\\prime }} := s_p s_{p^{\\prime }} \\cdot \\mathbb {I}\\left(d(GX_p, GX_{p^{\\prime }}) < (1+\\epsilon )^{i+1}\\right),$ where $\\mathbb {I}$ represents an indicator random variable.",
"Then, if $j \\le \\epsilon ^{-1}$ , $d(GX_p, GX_{p^{\\prime }}) < (1+\\epsilon )^{i+1}$ occurs with probability at most $3 \\cdot \\exp \\left(-d(j \\epsilon )^2/100\\right) \\le 3 \\cdot \\exp \\left(-d \\cdot \\epsilon ^2/100\\right)$ , so $\\Pr (Z_{p, p^{\\prime }} > 0) \\le 3 \\cdot \\exp \\left(-d \\cdot \\epsilon ^2/100\\right)$ .",
"Next, for any $k \\ge 1$ , if $Z_{p, p^{\\prime }} \\ge \\lambda _X^{2k \\cdot C_3},$ then either $s_p$ or $s_{p^{\\prime }}$ is at least $\\lambda _X^{k \\cdot C_3}$ , which occurs with probability at most $2 \\exp \\left(-d \\cdot 2^k/100\\right)$ by Proposition REF .",
"Hence, $\\mathbb {E}[Z_{p, p^{\\prime }}] &\\le 3 \\cdot \\exp \\left(-\\frac{d \\epsilon ^2}{100}\\right) \\cdot \\lambda _X^{2 \\cdot C_3} \\\\&\\hspace{42.67912pt}+ \\sum _{k = 1}^{\\infty } 2 \\cdot \\exp \\left(-\\frac{d \\cdot 2^k}{100}\\right) \\cdot \\lambda _X^{2(k+1) \\cdot C_3} \\\\&\\le 10 \\cdot \\exp \\left(-\\frac{d \\epsilon ^2}{200}\\right)$ by our choice of the dimension $d$ .",
"However, if $j > \\epsilon ^{-1},$ then $d(GX_p, GX_{p^{\\prime }})$ occurs with probability at most $3 \\cdot \\exp \\left(-d(j \\epsilon )/100\\right)$ , so $\\Pr (Z_{p, p^{\\prime }} > 0) \\le 3 \\cdot \\exp \\left(-d \\cdot (j \\epsilon )/100\\right)$ .",
"But for any $k \\ge 1,$ if $Z_{p, p^{\\prime }} \\ge \\lambda _X^{2(k+\\log (j \\epsilon )) \\cdot C_3},$ then either $s_p$ or $s_{p^{\\prime }}$ is at least $\\lambda _X^{(k+\\log (j \\epsilon )) \\cdot C_3},$ which occurs with probability at most $2 \\exp \\left(-d \\cdot 2^k \\cdot (j \\epsilon )/100\\right).$ Hence, $\\mathbb {E}[Z_{p, p^{\\prime }}] &\\le 3 \\cdot \\exp \\left(-\\frac{d(j \\epsilon )}{100}\\right) \\cdot \\lambda _X^{2(1+\\log (j \\epsilon )) \\cdot C_3} \\\\&\\hspace{14.22636pt}+ \\sum _{k = 1}^{\\infty } 2 \\cdot \\exp \\left(-\\frac{d \\cdot (j \\epsilon ) \\cdot 2^k}{100}\\right) \\cdot \\lambda _X^{2(k+1+\\log (j \\epsilon )) \\cdot C_3} \\\\&\\le 10 \\cdot \\exp \\left(-\\frac{d(j \\epsilon )}{200}\\right)$ by our choice of the dimension $d$ .",
"Next, note that for each $p$ , the number of $p^{\\prime }$ with $D(X_p, X_{p^{\\prime }}) \\le (1+5 \\epsilon ) \\cdot (1+\\epsilon )^{i+j} \\le \\left[C_1 \\cdot \\epsilon ^{-1} \\cdot (1+\\epsilon )^{5+j}\\right] \\cdot t$ is at most $\\lambda _X^{C_2 \\cdot (\\log C_1 + \\log \\epsilon ^{-1} + \\epsilon \\cdot (5+j))}$ by Observation REF .",
"Hence, the total number of pairs $(p, p^{\\prime })$ with $D(X_p, X_{p^{\\prime }}) \\in B_{i, j}$ is at most $r \\cdot \\lambda _X^{C_2 \\cdot (\\log C_1 + \\log \\epsilon ^{-1} + \\epsilon \\cdot (5+j))} \\le r \\cdot \\lambda _X^{C_4 \\cdot (\\log \\epsilon ^{-1} + j \\epsilon )}$ for some constant $C_4.$ Combining everything together, we have that $\\mathbb {E}[L_i] &= \\sum _{j \\ge 1} \\sum _{p, p^{\\prime }: D(X_p, X_{p^{\\prime }}) \\in B_{i, j}} D(X_p, X_{p^{\\prime }}) \\cdot \\mathbb {E}[Z_{p, p^{\\prime }}] \\nonumber \\\\&\\le \\sum _{j = 1}^{\\epsilon ^{-1}} \\bigg (r \\cdot \\lambda _X^{C_4 \\cdot (\\log \\epsilon ^{-1} + j \\epsilon )} \\cdot (1+5 \\epsilon ) \\cdot (1+\\epsilon )^{i+j} \\nonumber \\\\&\\hspace{56.9055pt} \\cdot 10 \\cdot \\exp \\bigg (-\\frac{d \\epsilon ^2}{200}\\bigg )\\bigg ) \\nonumber \\\\&+ \\sum _{j > \\epsilon ^{-1}} \\bigg (r \\cdot \\lambda _X^{C_4 \\cdot (\\log \\epsilon ^{-1} + j \\epsilon )} \\cdot (1+5 \\epsilon ) \\cdot (1+\\epsilon )^{i+j} \\nonumber \\\\&\\hspace{56.9055pt} \\cdot 10 \\cdot \\exp \\bigg (-\\frac{d (j \\epsilon )}{200}\\bigg )\\bigg ) \\nonumber \\\\&\\le 20 C_1 r t \\cdot \\Biggr (\\sum _{j = 1}^{\\epsilon ^{-1}} \\lambda _X^{C_5 (\\log \\epsilon ^{-1})} \\exp \\left(-\\frac{d \\epsilon ^2}{200}\\right)\\\\&\\hspace{14.22636pt}+\\sum _{j > \\epsilon ^{-1}} \\lambda _X^{C_5 (\\log \\epsilon ^{-1} + j \\epsilon )} \\exp \\left(-\\frac{d (j \\epsilon )}{200}\\right)\\Biggr ) $ for some constant $C_5$ .",
"Above, the first equality follows by definition of $L_i$ .",
"The next inequality follows from our bound on $\\mathbb {E}[Z_{p, p^{\\prime }}]$ , our bound the number of $(p, p^{\\prime })$ with $D(X_p, X_{p^{\\prime }}) \\in B_{i, j},$ and since $D(X_p, X_{p^{\\prime }}) \\in B_{i, j}$ implies $D(X_p, X_{p^{\\prime }}) \\le (1+5 \\epsilon ) \\cdot (1+\\epsilon )^{i+j}.$ The final inequality follows from simple factorization and the facts that $C_1 t \\le (1+\\epsilon )^i$ and $1+5\\epsilon \\le 2$ .",
"Now, if we choose $d = C_6 \\cdot (\\log \\log n + \\log \\epsilon ^{-1} \\log \\lambda _X) \\cdot \\epsilon ^{-2}$ for some sufficiently large constant $C_6,$ we have that $\\lambda _X^{C_5 (\\log \\epsilon ^{-1})} \\cdot \\exp (-d \\epsilon ^2/200) \\le \\frac{\\epsilon ^3}{1000 C_1 \\log n}$ for all $j \\le \\epsilon ^{-1}$ , and $\\lambda _X^{C_5(\\log \\epsilon ^{-1}+j \\epsilon )} \\cdot \\exp \\left(-d(j \\epsilon )/200\\right) \\le \\exp \\left(j \\cdot \\epsilon ^{-1}\\right) \\cdot \\exp \\left(-d(j \\epsilon )/400\\right) \\le \\frac{\\exp (-j)}{1000 C_1 \\log n}$ for all $j > \\epsilon ^{-1}$ .",
"Hence, Equation (REF ) can be upper bounded by $20 C_1 \\cdot rt \\cdot \\left(\\epsilon ^{-1} \\cdot \\frac{\\epsilon ^3}{1000 C_1 \\log n} + \\frac{\\sum _{j \\ge \\epsilon ^{-1}}e^{-j}}{1000 C_1 \\log n}\\right) \\\\\\le \\frac{\\epsilon ^2}{20 \\log n} \\cdot rt.",
"$ Proposition REF tells us that $rt \\le 2 M$ .",
"Hence, Equation (REF ) is at most $\\frac{\\epsilon ^2}{10 \\log n} \\cdot M,$ as desired.",
"In sum, we have that $\\mathbb {E}[K_i] \\le \\frac{\\epsilon ^2}{10 \\log n} \\cdot M$ for all $i$ .",
"Moreover, for small $i$ , Equation (REF ) tells us that $\\mathbb {E}[K_i] \\le \\frac{\\epsilon ^2}{20 \\log n} \\cdot rt \\le \\frac{\\epsilon ^2}{20 \\log n} \\cdot n \\cdot (1+\\epsilon )^i,$ since $r \\le n$ and $t \\le (1+\\epsilon )^i$ .",
"Therefore, the LHS of Equation (REF ) is at most $\\sum _{i: (1+\\epsilon )^i \\le \\operatorname{diam}(X)} \\mathbb {E}[K_i] \\\\\\le \\frac{\\epsilon ^2}{20 \\log n} \\cdot \\sum _{i: (1+\\epsilon )^i \\le \\operatorname{diam}(X)} \\min \\left(2 M, n \\cdot (1+\\epsilon )^i\\right).$ Using the bound $\\frac{\\epsilon ^2}{10 \\log n} \\cdot M$ for $i$ with $\\frac{\\operatorname{diam}(X)}{n} < (1+\\epsilon )^i \\le \\operatorname{diam}(X)$ and the bound $\\frac{\\epsilon ^2}{20 \\log n} \\cdot n \\cdot (1+\\epsilon )^i$ for $i$ with $(1+\\epsilon )^i \\le \\frac{\\operatorname{diam}(X)}{n},$ we can bound this by $\\frac{\\epsilon ^2}{20 \\log n} \\cdot \\left(2 M \\cdot \\log _{1+\\epsilon } n + \\frac{\\operatorname{diam}(X)}{1 - \\frac{1}{1+\\epsilon }}\\right) \\\\\\le \\frac{\\epsilon \\cdot M}{5} + \\frac{\\operatorname{diam}(X) \\cdot \\epsilon }{10 \\log n} < \\epsilon \\cdot M,$ since $\\operatorname{diam}(X) \\le M.$ This concludes the proof."
],
[
"Dependence on the Doubling Dimension",
"In this subsection, we prove Theorems REF , REF , and REF .",
"We begin with Theorem REF .",
"To do so, we construct a set $X$ of $m$ points in $\\mathbb {R}^m$ such that if we randomly project $X$ to $o(\\log m)$ dimensions, then with high probability, the facility location cost is not preserved up to a constant factor.",
"Moreover, the optimal set of facility centers in the projected space, with high probability, is not a constant-factor approximation to facility location in the original space.",
"The point set $X$ we choose will just be a scaled set of identity vectors in $\\mathbb {R}^m$ : it is simple to see that this point set has $\\lambda _X = m$ .",
"These points have the convenient property that each point's projection is independent of each other.",
"[Proof of Theorem REF ] As mentioned previously, the points in $X$ will just be $R e_1, \\dots , Re_m$ , the $m$ identity unit vectors in $\\mathbb {R}^m$ scaled by a factor $R \\ge 1$ .",
"Since these points each have distance $R \\sqrt{2} \\ge \\sqrt{2}$ from each other, the optimum set of facilities is all of them, which has cost $m$ .",
"Now, consider a random projection $G$ down to $d = o(\\log m)$ dimensions, and define $C = \\sqrt{\\frac{\\log n}{10 d}}$ and $R = \\sqrt{C}$ .",
"Note that $R, C = \\omega (1)$ .",
"Our goal will be to show that with at least $\\frac{2}{3}$ probability, for all but $\\frac{3m}{C}$ points $p \\in GX$ , $\\tilde{r}_p \\le \\frac{2}{R}$ , where we recall that $\\tilde{r}_p$ is the positive real number such that $ \\sum _{q \\in B(p, \\tilde{r}_p) \\cap GX} (\\tilde{r}_p - \\Vert p-q\\Vert ) = 1.$ We trivially have the bound $\\tilde{r}_p \\le 1$ for all $p \\in GX,$ which means that if we show our goal, then $\\sum _{p \\in GX} \\tilde{r}_p \\le m \\cdot \\frac{2}{R} + \\frac{3m}{C} \\cdot 1 \\le \\frac{5m}{R} = o(m)$ .",
"However, $\\sum _{p \\in GX} \\tilde{r}_p$ is a constant-factor approximation to the optimum facility location cost by Lemma REF , which proves that the facility location cost of $GX$ is $o(m)$ .",
"Now, for each $e_i,$ by Equation (REF ), we have for any $C \\ge 6,$ $\\Pr (\\Vert Ge_i\\Vert \\le C) &\\ge 1 - \\exp \\left(-d(C-1)^2/8\\right) \\\\&\\ge 1 - \\exp \\left(-(C-1)^2/8\\right) \\\\&\\ge 1 - \\frac{1}{2C}.$ Moreover, conditioned on $\\Vert Ge_i\\Vert \\le C,$ by Lemma REF , we have that for each $j \\ne i,$ $\\Pr (\\Vert Ge_j-Ge_i\\Vert \\le \\frac{1}{C}) \\ge n^{-1/10}$ if $n$ is sufficiently large.",
"Therefore, if $Ge_i: \\Vert Ge_i\\Vert \\le C$ is fixed, since the $Ge_j$ 's are independent vectors, we can apply the Chernoff bound to say that with probability at least $1-n^{-10},$ at least $\\log n \\ge R$ values of $j \\ne i$ satisfy $\\Vert Ge_j-Ge_i\\Vert \\le \\frac{1}{C}$ , or equivalently, $\\Vert G(Re_j)-G(Re_i)\\Vert \\le \\frac{R}{C} = \\frac{1}{R}$ .",
"By removing our conditioning on $Ge_i$ , we have that with probability at least $1 - \\frac{1}{2C} - n^{-10} \\ge 1 - \\frac{1}{C},$ there are at least $R$ points in $GX$ that are within $\\frac{1}{R}$ of $G(Re_i),$ in which case we have that for $p = G(Re_i)$ , $\\tilde{r}_p \\le \\frac{2}{R}.$ Therefore, in expectation, at most $\\frac{m}{C}$ of the points in $GX$ have $\\tilde{r}_p > \\frac{2}{R}.$ Thus, by Markov's inequality, with probability at least $\\frac{2}{3},$ at most $\\frac{3m}{C}$ of the points in $GX$ have $\\tilde{r}_p > \\frac{2}{R}$ .",
"This proves the first part of the theorem.",
"To prove the second part of the theorem, note that the optimal facility location cost over $GX$ is $o(m)$ with probability at least $2/3$ , which implies that the number of open facilities in any optimal solution $\\mathcal {F}_d$ is $o(m)$ .",
"But then, each point in $X$ which is not an open facility center is at least $R \\sqrt{2}$ away from the nearest open facility center in the original space $X$ , so the facility location cost in $X$ is at least $(m-o(m)) \\cdot R \\sqrt{2} = \\omega (m)$ .",
"Next, we prove Theorems REF and REF .",
"These results prove that the dependence on doubling dimension $d_X$ is required in the projected dimension $d$ , both to approximate the cost of the minimum spanning tree and to produce a minimum spanning tree in the lower dimension that is still an approximate MST in the original dimension.",
"[Proof of Theorem REF ] Let $X = \\lbrace 0, e_1,\\dots ,e_m\\rbrace $ , where $m = n-1,$ 0 is the origin in $\\mathbb {R}^m$ and $e_i$ is the $i$ th identity vector for each $1 \\le i \\le m$ .",
"Clearly, the minimum spanning tree connects 0 to all of the $e_i$ 's and has cost $M = m$ .",
"Now, we show that for $C = \\sqrt{\\frac{\\log n}{10d}} = \\omega (1)$ , the MST cost of $GX$ is at most $\\frac{10 m}{C} = o(m)$ for sufficiently large $m$ with at least $2/3$ probability.",
"To do so, note that since $G$ 's entries are independent, $Ge_1, \\dots , Ge_m$ are all i.i.d.",
"$\\frac{1}{\\sqrt{d}} \\cdot \\mathcal {N}(0, I_d)$ .",
"Consider some $e_i, e_j$ and suppose that $\\Vert G e_i\\Vert , \\Vert G e_j\\Vert \\le C$ but $\\Vert G(e_i-e_j)\\Vert \\ge \\frac{4}{C}.$ Then, if we let $v = G(e_i+e_j)/2,$ for each $k \\ne i, j$ , $\\Pr (\\Vert G e_k - v\\Vert \\le \\frac{1}{C}) \\ge n^{-1/10}$ by Lemma REF .",
"By the independence of $Ge_1, \\dots , Ge_m$ , with probability at least $1-n^{-10},$ there is some $k \\ne i, j$ in $[m]$ such that $\\Vert G e_k - v\\Vert \\le \\frac{1}{C}$ .",
"In this case, the minimum spanning tree of $GX$ would not have the edge $(Ge_i, Ge_j),$ as this edge could be replaced by either the edge $(Ge_i, Ge_k)$ or $(Ge_k, Ge_j),$ both of which are shorter.",
"Thus, with probability at least $1-n^{-8},$ if we just connect the points $Ge_i$ over all $i$ with $\\Vert Ge_i\\Vert \\le C$ in an MST, every edge has length at most $\\frac{4}{C}.$ We can create a possibly suboptimal spanning tree by connecting all $Ge_i$ with norm at most $C$ in an MST, connecting one of these vertices arbitrarily to $0 = G \\cdot 0$ , and finally connecting $Ge_i$ to 0 for all $i$ with $\\Vert Ge_i\\Vert > C$ .",
"The first part has total cost at most $m \\cdot \\frac{4}{C}$ with probability at least $1-n^{-8}$ .",
"The second part has total cost at most $C$ with probability at least $1-n^{-8}$ (as long as some $\\Vert Ge_i\\Vert \\le C$ ).",
"Finally, the third part has total expected cost $m \\cdot \\mathbb {E}[\\Vert Ge_i\\Vert \\cdot \\mathbb {I}(\\Vert Ge_i\\Vert \\ge C)],$ since each edge $e_i$ contributes to the third part only if $\\Vert Ge_i\\Vert \\ge C$ , and there are $m$ potential vertices $Ge_1, \\dots , Ge_m.$ However, by the Cauchy-Schwarz inequality, we know that $\\mathbb {E}\\left[\\Vert Ge_i\\Vert \\cdot \\mathbb {I}(\\Vert Ge_i\\Vert \\ge C)\\right] &\\le \\sqrt{\\mathbb {E}\\left[\\Vert Ge_i\\Vert ^2\\right] \\cdot \\Pr (\\Vert Ge_i\\Vert \\ge C)} \\\\&\\le \\sqrt{1 \\cdot \\exp \\left(-d \\cdot (C-1)^2/8\\right)} \\\\&\\le \\exp \\left(-(C-1)^2/16\\right) \\le \\frac{1}{C},$ with the final inequality true if $C \\ge 7$ .",
"Therefore, with probability at least $\\frac{4}{5},$ the third part has cost at most $\\frac{5 m}{C}$ by Markov's inequality.",
"So, with probability at least $\\frac{4}{5} - 2 n^{-8} \\ge \\frac{2}{3},$ the total cost of this spanning tree in $GX$ (which may not even be minimal) is at most $\\frac{4}{C} \\cdot m + C + \\frac{5}{C} \\cdot m \\le \\frac{10 m}{C}$ assuming $m$ is sufficiently large.",
"[Proof of Theorem REF ] As in our proof of Theorem REF , let $C = \\sqrt{\\frac{\\log n}{10 d}} = \\omega (1)$ .",
"Consider $n = C \\cdot m+1$ and let $X = \\lbrace 0\\rbrace \\cup \\lbrace e_i \\cdot k/C\\rbrace $ for $1 \\le i \\le m, 1 \\le k \\le C$ .",
"The minimum spanning tree connects 0 to $e_i/C$ to $2e_i/C$ to so on, so each edge has length $1/C$ and the total MST cost is $M = m$ .",
"Now, by Equations (REF ) and (), for each $e_i$ , the probability that $\\Vert Ge_i\\Vert \\in [1/10, 100]$ is at least $1 - \\exp (-d/10) - (3/100)^d > 0.06$ for all $d \\ge 1.$ Thus, with exponential failure probability in $m$ , among $e_1, \\dots , e_{m/2}$ , at least $0.02 m$ of the $Ge_i$ 's have norm between $1/10$ and 100.",
"Now, for some $i \\le m/2$ with $1/10 \\le \\Vert Ge_i\\Vert \\le 100$ , since $d = o(\\log n)$ , by Lemma REF , the probability that $\\Vert Ge_j-Ge_i\\Vert \\le \\frac{1}{100 C}$ for any $j > m/2$ is at least $n^{-1/10}$ .",
"Hence, with exponential failure probability, for each $i$ with $1/10 \\le \\Vert Ge_i\\Vert \\le 100$ , there is some $j > m/2$ with $\\Vert Ge_j-Ge_i\\Vert \\le \\frac{1}{100 C}.$ Let $I$ be the set of $i$ such that $\\Vert Ge_i\\Vert \\ge \\frac{1}{10}$ and there is some $j$ with $\\Vert Ge_j-Ge_i\\Vert \\le \\frac{1}{100 C}.$ For each $i \\in I,$ the distance between $Ge_i \\cdot k/C$ and $Ge_i \\cdot \\ell /C$ for any $\\ell \\ne k$ is at least $\\frac{1}{10C}$ but the distance between $Ge_i \\cdot k/C$ and $Ge_j \\cdot k/C$ is at most $\\frac{1}{100C}.$ This means that the closest point to $Ge_i \\cdot k/C$ in $GX$ is of the form $Ge_j \\cdot k^{\\prime }/C$ for some $j \\ne i$ and $k^{\\prime }$ which may or may not equal $k$ .",
"However, for every $Gx \\in GX,$ the minimum spanning tree of $GX$ must contain the edge connecting $Gx$ to its closest neighbor, so for each $i \\in I$ and $1 \\le k \\le C$ , $\\widetilde{\\mathcal {M}}$ must connect $Ge_i \\cdot k/C$ to $Ge_j \\cdot k^{\\prime }/C$ , which has length at least $k/C$ in the original space $\\mathbb {R}^m$ .",
"Therefore, the pullback of the MST has length at least $\\sum _{i \\in I} \\sum _{k = 1}^{C} \\frac{k}{C} \\ge \\frac{C}{2} \\cdot |I|,$ which with exponential failure probability in $m$ is at least $\\frac{C}{100} \\cdot m = \\frac{C}{100} \\cdot M = \\omega (M)$ ."
],
[
"Approximate Solutions Cannot be Pulled Back",
"In this subsection, we prove Lemmas REF and REF .",
"In other words, we give a simple example showing that our definition of locally optimal (for FL) and that optimal (for MST) is necessary, if we want dependence on $d_X = \\log \\lambda _X$ as opposed to $\\log n$ .",
"In particular, our lemmas give examples showing that pulling back of any approximately optimal solution found in the projected space to the original space does not work.",
"[Proof of Lemma REF ] Consider the following set of points: $Y = \\lbrace b_1,b_2, \\ldots , b_m\\rbrace = \\lbrace e_1, e_1 + e_2, \\ldots , e_1 + e_2 + \\cdots + e_m \\rbrace $ where $e_i$ is the $i$ th standard basis vector.",
"We refer to this dataset as the `walk' dataset.",
"Using the definition of doubling dimension (see Section ), we can compute that the doubling dimension of $Y$ is some constant independent of $m$ .",
"Now construct the dataset $X$ by scaling all the points in $Y$ by the factor $m^{1+1/2d}$ .",
"This does not affect the doubling dimension.",
"Consider the projection of $X$ into $\\mathbb {R}^d$ where $d = O(1)$ .",
"Before projection, the optimum solution is to open all facilities, costing $m$ .",
"Now consider applying a random projection $G$ and note that the projection of the differences $G(b_i - b_{i+1})$ are independent.",
"Therefore, by Proposition REF , there is a pair of consecutive points $b_i, b_{i+1}$ such that $\\Vert G(b_i-b_{i+1})\\Vert $ shrinks by a factor of $C_1/m^{1/d}$ with probability at least $9/10$ .",
"Furthermore, by Equation (), we have that all the differences $\\Vert G(b_i-b_{i+1})\\Vert $ do not shrink by a factor worse than $C_2/m^{1/d}$ with probability at least $9/10$ .",
"Hence, with some constant probability, both the following events occur: There exists some $i^*$ such that $\\Vert G(b_{i^*}-b_{i^*+1})\\Vert = O(m^{1-1/2d})$ $\\Vert G(b_i-b_{i+1})\\Vert = \\Omega (m^{1-1/2d})$ for all $i$ .",
"In this case, the optimal solution in the projected space is to include all facilities, which has total cost $m$ .",
"However, a solution that is within a $1+O(m^{-1/2d})$ multiplicative factor of the optimal solution is to include all facilities except for $Gb_i^*$ .",
"However, evaluating this solution in the original dimension incurs a cost at least $\\Omega (m^{1+1/2d})$ , whereas the optimal cost is still $m$ .",
"Hence, the approach has approximation ratio of at least $m^{1/2d}$ , which is $\\omega (1)$ , i.e., superconstant unless $d=\\Omega (\\log m)$ .",
"[Proof of Lemma REF ] Assume WLOG that $n = 2k^2$ for some $k$ , that $X$ lies in $\\mathbb {R}^{m}$ for $m = k+1$ , and that $d = \\epsilon \\cdot \\log n$ for some $\\epsilon = o(1)$ .",
"Now, let $e_1, e_2, \\dots , e_k$ represent the identity vectors in $\\mathbb {R}^k$ .",
"Now, we will choose our $n$ points as follows.",
"First, we will choose the $k^2$ points $X^{\\prime } = \\lbrace (0, \\textbf {0}), (\\frac{1}{k}, \\textbf {0}), \\dots , (\\frac{k^2-1}{k}, \\textbf {0})\\rbrace ,$ where $\\textbf {0}$ represents the last $k$ coordinates all being 0.",
"For the remaining $k^2$ points, for each $0 \\le i \\le k-1$ we add the set $X_i = \\lbrace (i, e_i), (i + \\frac{1}{k}, e_i) \\dots , (i+\\frac{k-1}{k}, e_i)\\rbrace $ .",
"We let $X = X^{\\prime } \\cup X_0 \\cup \\dots \\cup X_{k-1}$ .",
"First, we show that the doubling dimension of $X$ , $\\lambda _X,$ is at most $O(1)$ .",
"First, note that $X^{\\prime }$ and each $X_i$ is trivially embeddable into one dimension, because the points in $X^{\\prime }$ and in each $X_i$ only vary on one coordinate, so each of these individually have doubling dimension $O(1)$ .",
"Therefore, for any ball $B = B(r, p)$ of radius $r \\le 10$ around some point $p$ , $B \\cap X$ is contained in some union of $O(1)$ of $X^{\\prime }, X_0, \\dots , X_{k-1}$ .",
"Consequently, the points in $B \\cap X$ can be decomposed into $O(1)$ balls of radius $r/2$ , since $B \\cap X^{\\prime }$ and $B \\cap X_i$ each have doubling dimension bounded by a constant.",
"Now, if we consider some ball $B = B(r, p)$ of radius $r > 10,$ suppose that $p = (a_0, a_1, \\dots , a_k) \\in \\mathbb {R}^{k+1}$ .",
"Now, consider the 5 points $\\lbrace (a_0+\\frac{j}{2} \\cdot r, \\textbf {0})\\rbrace _{j = -2}^{2},$ where the $\\textbf {0}$ represents the last $k$ coordinates all being 0.",
"For every point $x$ in $X \\cap B,$ $x$ 's first coordinate must be in the range $[a_0-r, a_0+r]$ and $x$ 's remaining coordinates have total magnitude at most 1.",
"With these two observations, it is immediate that every point in $X \\cap B$ is within $r/2$ of some point $\\lbrace (a_0 + \\frac{j}{2} \\cdot r, \\textbf {0})\\rbrace $ for some integer $-2 \\le j \\le 2$ .",
"Therefore, if $r > 10$ , $B \\cap X$ can be covered by 5 balls of radius $r/2.$ Thus, $\\lambda _X = O(1)$ , so $X$ has doubling dimension $\\log \\lambda _X = O(1)$ .",
"Now, a straightforward verification tells us that for any $i \\ne j,$ the points in $X_i$ and the points in $X_j$ are at least $\\sqrt{2}$ away from each other.",
"Moreover, each point $(i+\\frac{j}{k}, e_i)$ 's closest point in $X^{\\prime }$ is the corresponding point $(i+\\frac{j}{k}, \\textbf {0}),$ and this distance is 1.",
"Therefore, the minimum spanning trees of $X$ are as follows.",
"First, connect the points in $X^{\\prime }$ in a line and all of the points in each $X_i$ in a line.",
"Finally, for each $0 \\le i \\le k-1,$ choose some arbitrary $j$ and connect $(i+\\frac{j}{k}, e_i)$ and $(i+\\frac{j}{k}, \\textbf {0}).$ The total MST cost $M$ is $\\frac{k^2-1}{k} + k \\cdot \\frac{k-1}{k} + k \\cdot 1 = 3 k - 1 - \\frac{1}{k} = (3-o(1)) k$ .",
"Now, when the random projection $G: \\mathbb {R}^{k+1} \\rightarrow \\mathbb {R}^d$ is applied, we have that each vector $(0, e_i)$ is independently mapped to some vector $(a_{i1}, \\dots , a_{id})$ , where each $a_{ij}$ for $1 \\le i \\le k, 1 \\le j \\le d$ is an i.i.d.",
"$\\mathcal {N}(0, 1/d)$ .",
"So for any $\\epsilon = o(1)$ and $n$ sufficiently large, if we choose $\\delta = e^{-1/(100 \\epsilon )},$ we have that $\\Pr (|a_{i1}|, \\dots , |a_{id}| \\le \\delta /\\sqrt{d}) = \\Theta (\\delta )^{d} \\le e^{-\\log n/4} < 1/\\sqrt{2k},$ where we used the fact that $d = \\varepsilon \\log n$ .",
"Hence, a simple Chernoff bound tells us that with $1-o(1)$ probability, at least $\\sqrt{k}/2$ of the $(0, e_i)$ 's get mapped to some $(a_{i1}, \\dots , a_{id})$ with norm at most $\\delta $ .",
"Now, consider the following $\\omega (1)$ -approximate MST for $X$ .",
"Let $A = \\epsilon ^{-1}$ , and choose some set $I = \\lbrace i_1, \\dots , i_A\\rbrace $ .",
"Our “approximate” MST will be as follows.",
"For each $i \\in I,$ remove the $k-1$ edges connecting $X_i$ together, and for each $1 \\le j \\le k,$ connect $(i + \\frac{j}{k}, 0)$ with $(i + \\frac{j}{k}, e_i)$ .",
"Each time this is done, we remove $k-1$ edges of length $1/k$ and add $k-1$ edges of length 1 (recall that one of these edges of length 1 was already in the MST), so the MST cost increases by $\\epsilon ^{-1} ((k-1) 1 - (k-1)/k) = \\epsilon ^{-1} k \\cdot (1-o(1)).$ Hence, regardless of what set $A$ we chose, the approximate MST is a $\\omega (1)$ -approximation, as the true MST has cost $M = O(k)$ .",
"However, we claim that with high probability, we can choose $A$ so that this becomes a $(1+o(1))$ -approximation in the projected space.",
"Indeed, since $\\epsilon \\ge \\frac{1}{\\log n}$ , with $1-o(1)$ probability, at least $\\sqrt{k}/2 \\ge \\epsilon ^{-1}$ values $e_i$ get mapped to some point with norm at most $\\delta $ .",
"So, we choose $A$ to be of size $\\epsilon ^{-1}$ so that for all $i \\in A,$ $e_i$ gets mapped to a point with norm at most $\\delta $ .",
"Recall that $\\mathcal {M}$ denote the true MST for $X$ , and let $\\mathcal {M}^{\\prime }$ be this poor-approximation spanning tree.",
"Note that the only edges in $\\mathcal {M}^{\\prime } \\backslash \\mathcal {M}$ connect $(i + \\frac{j}{k}, 0)$ to $(i + \\frac{j}{k}, e_i)$ for $i \\in I, 0 \\le j \\le k-1$ .",
"Since there are $\\varepsilon ^{-1} \\cdot k$ such edges, and each edge has size at most $\\delta $ when projected, we have that $\\operatorname{cost}_{GX}(\\mathcal {M}^{\\prime }) &\\le \\operatorname{cost}_{GX}(\\mathcal {M}) + \\delta \\cdot \\varepsilon ^{-1} \\cdot k \\\\&\\le \\operatorname{cost}_{GX}(\\mathcal {M}) + \\varepsilon ^{-1} \\cdot e^{-\\varepsilon ^{-1}/100} \\cdot k \\\\&= \\operatorname{cost}_{GX}(\\mathcal {M}) + o(k).$ Now, let's suppose that $d \\ge \\omega (\\log \\log n)$ .",
"We saw in subsection REF that $\\operatorname{cost}_{GX}(\\mathcal {M})$ had expectation at most $M = \\operatorname{cost}_X(\\mathcal {M})$ and standard deviation $O(M/\\sqrt{\\log \\log n})$ , regardless of the dataset $X$ .",
"So, with $9/10$ probability, $\\operatorname{cost}_{GX}(\\mathcal {M}^{\\prime }) \\le \\operatorname{cost}_{GX}(\\mathcal {M}) + o(k) = (1+o(1)) M$ .",
"Moreover, by Theorem REF , with $9/10$ probability, $\\widetilde{M},$ the cost of the MST in the reduced space $GX$ , is within a $1 \\pm o(1)$ factor of $M$ .",
"Therefore, with at least $4/5-o(1)$ probability, $\\operatorname{cost}_{GX}(\\mathcal {M}^{\\prime }) \\le (1+o(1)) \\cdot \\widetilde{M},$ so $\\mathcal {M}^{\\prime }$ is an $\\omega (1)$ -approximate MST in $X$ but a $1+o(1)$ -approximate MST in $GX$ ."
],
[
"Lower Bounds for $k$ -means and {{formula:0b4e4b13-67c0-47c4-ab8e-5a5ee6b5c990}} -medians",
"In this subsection, we prove Theorem REF , which shows the tightness of the bounds of [19] for $k$ -means and $k$ -medians clustering even in the case of constant doubling dimension.",
"We remark that [19] showed tightness of their result if doubling dimension is ignored.",
"Namely, they showed the existence of such a point set $X$ that may have large doubling dimension.",
"Hence, our contribution is making such a set that also has doubling dimension $O(1)$ .",
"[Proof of Theorem REF ] We start with the case where $n = 2t$ and $k = 2t-1$ for some $t$ .",
"As in [19], we wish to consider $t$ pairs of points where each pair is of distance 1 from each other, but all other distances are larger.",
"Namely, we do the following.",
"First, define $D = t^{1/d}/10,$ and let $R = \\sqrt{D}$ .",
"We have that $D, R = \\omega (1)$ , since $d = o(\\log n) = o(\\log t)$ .",
"Now, for $1 \\le i \\le t,$ let $a_i = (2 \\cdot i, \\textbf {0}),$ meaning that $a_i$ 's first coordinate is $2 \\cdot i$ and the remaining $t = m-1$ coordinates are 0.",
"Next, for each $1 \\le i \\le t-1,$ define $b_i = a_i + e_{i+1}$ , i.e., $b_i$ has first coordinate $2 \\cdot i$ , $(i+1)$ th coordinate 1, and all remaining coordinates $0.$ However, define $b_t = a_t + \\frac{1}{R} \\cdot e_{i+1}$ .",
"Our set $X$ will be the union of the $a_i$ 's and $b_i$ 's.",
"Now, since $k = n-1,$ the $k$ -medians cost of $X$ is just the distance between the closest pair of points in $X$ , which is $\\frac{1}{R}.$ The $k$ -means cost of $X$ is just the squared distance between the closest pair of points in $X$ .",
"However, by Proposition REF , for each $i$ , $\\Pr \\left(\\Vert G b_i - G a_i\\Vert \\le \\frac{10}{t^{1/d}}\\right) &= \\Pr \\left(\\Vert G e_{i+1}\\Vert \\le \\frac{10}{t^{1/d}}\\right) \\\\&\\ge \\left(\\frac{10}{e \\cdot t^{1/d}}\\right)^d \\ge \\frac{3}{t}.$ Moreover, since $e_2, \\dots , e_{t}$ are all distinct unit vectors, the vectors $Ge_2, \\dots , Ge_t$ are independent, which means that with probability at least $1 - (1 - 3/t)^{t-1} \\ge 0.9$ (for $t$ sufficiently large), some $1 \\le i \\le t-1$ will have $\\Vert Gb_i-Ga_i\\Vert \\le 10/t^{1/d} = 1/D.$ Thus, some pair of points $(a_i, b_i)$ satisfy $\\Vert Ga_i-Gb_i\\Vert \\le 1/D$ , whereas the closest distance between two points in $X$ was only $1/R$ .",
"Therefore, with at least $9/10$ probability, the $k$ -medians cost has multiplied by a $R/D = o(1)$ factor after projection, and likewise, the $k$ -means cost has multiplied by a $R^2/D^2 = o(1)$ factor.",
"Now, let $p, q \\in X$ be the pair of points minimizing $\\Vert Gp-Gq\\Vert $ .",
"With probability at least $4/5$ , $\\Vert Ga_t-Gb_t\\Vert \\ge 1/(20 R) > 1/D$ , which means that either $p$ or $q$ is not in $\\lbrace a_t, b_t\\rbrace $ : assume WLOG that $p \\notin \\lbrace a_t, b_t\\rbrace $ .",
"Thus, an optimal choice of $k$ centers (for either $k$ -means or $k$ -medians) is choosing all points in $X$ , except $p$ .",
"But then, in the original space, these centers have $k$ -medians cost equal to the distance from $p$ to its closest point in $X$ , which is at least 1.",
"Likewise, the $k$ -means cost is also at least 1.",
"However, the optimal $k$ -medians and $k$ -means costs are $1/R$ and $1/R^2$ , respectively, so the optimal choice in $GX$ is an $R = \\omega (1)$ or $R^2 = \\omega (1)$ approximation for $k$ -medians and $k$ -means, respectively.",
"This finishes the proof in the case that $k = n-1$ .",
"For general values of $k < n,$ we can simply consider having $n^{\\prime } = k+1$ points in the configuration as above, but with exactly one of the points replicated $n-k$ times.",
"In this case, the cost of $k$ -medians clustering is still the distance of the closest pair of distinct points, and the cost of $k$ -medians clustering is still the square of the distance of the closest pair of distinct points.",
"So, the lower bound of $\\Omega (\\log k)$ still holds."
],
[
"Facility Location with Squared Costs",
"Recall that the facility location with squared costs problem is defined as follows.",
"Given a dataset $X \\subset \\mathbb {R}^m$ , our goal is to find a subset $\\mathcal {F} \\subseteq X$ that minimizes the objective $\\operatorname{cost}(\\mathcal {F}) =| \\mathcal {F}| + \\sum _{x \\in X} \\, \\min _{f \\in \\mathcal {F}}\\Vert x-f\\Vert ^2.$ Similar to Equation (REF ), we give a geometric expression that is a constant factor approximation to the cost of the objective presented in (REF ).",
"For each $p \\in X$ , associate it with a radius $r_p > 0$ that satisfies the relation $\\sum _{q \\in B(p,r)} ( r_p^2 - \\Vert p-q\\Vert ^2) = 1.$ We generalize the results in [22] and [2] to give an analogue of Lemma REF for the squared objective (REF ).",
"Lemma F.1 Let $C_{OPT}$ denote the cost of the optimal solution to the objective given in (REF ).",
"Then $\\frac{1}{8} \\cdot C_{OPT} \\le \\sum _{p \\in X} r_p^2 \\le 24 \\cdot C_{OPT}.",
"$ To prove Lemma REF , we first given an algorithm for (REF ) inspired by the MP algorithm.",
"Our algorithm, which we denote as the `Squared MP Algorithm,' is the following.",
"[H] InputInput OutputOutput Set $\\mathcal {F}$ of facilities $\\mathcal {F} \\leftarrow \\emptyset $ $i = 1$ to $n$ Compute $r_i$ satisfying: $ \\sum _{q \\in B(p_i, r_i)} (r_i^2 - \\Vert p_i-q\\Vert ^2) = 1$ Sort such that $r_1 \\le \\ldots \\le r_n$ $i=1$ to $n$ $B(p_i, 2r_i) \\cap \\mathcal {F} = \\emptyset $ $\\mathcal {F} \\leftarrow \\mathcal {F} \\cup \\lbrace p_i\\rbrace $ Output $\\mathcal {F}$ $\\textsc {Squared MP Algorithm}$ We first claim that the set of facilities returned by Algorithm is a constant factor approximation to the optimal set.",
"Theorem F.2 Let $C_{OPT}$ denote the cost of the optimal solution to the objective given in (REF ) and let $\\mathcal {F}$ denote the set of facilities returned by Algorithm .",
"Then $\\textup {cost}(\\mathcal {F}) \\le 6 \\cdot C_{OPT}$ .",
"The proof follows similarly to Theorem 1 in [22] with some adaptations.",
"Let $\\mathcal {F}^{\\prime }$ denote any set of facilities.",
"For any point $x \\in X$ , let $ \\text{charge}(x, \\mathcal {F}^{\\prime }) = d(x, \\mathcal {F}^{\\prime })^2 + \\sum _{p \\in \\mathcal {F}^{\\prime }} \\max (0, r_p^2 - \\Vert p-x\\Vert ^2)$ where $d(x, \\mathcal {F}^{\\prime })$ denotes the distance between $x$ and the closest point to $x$ in $\\mathcal {F}^{\\prime }$ and $r_p$ is defined as in (REF ).",
"We first show that $\\sum _{x \\in X} \\text{charge}(x, \\mathcal {F}^{\\prime }) = \\operatorname{cost}(F^{\\prime })$ .",
"Indeed, this follows from swapping the order of summation: $&\\hspace{14.22636pt} \\sum _{x \\in X} \\text{charge}(x, \\mathcal {F}^{\\prime }) \\\\&= \\sum _{x \\in X} \\sum _{p \\in \\mathcal {F}^{\\prime }} \\max (0, r_p^2 - \\Vert p-x\\Vert ^2) + \\sum _{x \\in X} d(x, \\mathcal {F}^{\\prime })^2 \\\\&= \\sum _{p \\in \\mathcal {F}^{\\prime }} \\sum _{x \\in X} \\max (0, r_p^2 - \\Vert p-x\\Vert ^2)+ \\sum _{x \\in X} d(x, \\mathcal {F}^{\\prime })^2 \\\\&= \\sum _{p \\in \\mathcal {F}^{\\prime }} 1 + \\sum _{x \\in X} d(x, \\mathcal {F}^{\\prime })^2 = \\operatorname{cost}(\\mathcal {F}^{\\prime }).$ Now denote $F^*$ as the set of facilities for the optimal solution.",
"We first study the individual term $\\text{charge}(x, \\mathcal {F}^*)$ .",
"We first give a lower bound for $\\text{charge}(x, \\mathcal {F}^*)$ .",
"Let $q^*$ be the closest point to $x \\in \\mathcal {F}^*$ .",
"If $x \\notin B(q^*, r_{q^*})$ then $\\text{charge}(x, \\mathcal {F}^*) \\ge \\Vert x-q^*\\Vert ^2 > r_{q^*}^2$ .",
"Otherwise, $\\text{charge}(x, \\mathcal {F}^*) &\\ge \\Vert x-q^*\\Vert ^2 + r_{q^*}^2 - \\Vert x-q^*\\Vert ^2 \\\\&= r_{q^*}^2 \\ge \\Vert x-q^*\\Vert ^2$ so altogether, $\\text{charge}(x, \\mathcal {F}^*) \\ge \\max (r_{q^*}^2, \\Vert x-q^*\\Vert ^2).$ Now let $\\mathcal {F}$ denote the set of solutions returned by Algorithm .",
"We now upper bound $\\text{charge}(x, \\mathcal {F})$ in terms of the quantities $r_{q^*}^2, \\Vert x-q^*\\Vert ^2$ .",
"Recall that $q^* \\in \\mathcal {F}^*$ is the closest point to $x$ in $\\mathcal {F}^*$ .",
"We note that there must be a point $q \\in \\mathcal {F}$ such that $r_q \\le r_{q^*}$ and $\\Vert q-q^*\\Vert \\le 2r_{q*}$ due to how Algorithm selects the set of facilities in step 6.",
"Now if $x \\in B(q, r_q)$ then $d(x, \\mathcal {F}) \\le \\Vert x-q\\Vert $ and thus $ \\text{charge}(x, \\mathcal {F}) \\le r_q^2$ since step 6 of Algorithm insures that $x \\notin B(q^{\\prime }, r_{q^{\\prime }})$ for any other $q^{\\prime } \\in \\mathcal {F}$ .",
"Otherwise, $x \\notin B(q, r_q)$ in which case we claim that $\\text{charge}(x, \\mathcal {F}) \\le \\Vert x-q\\Vert ^2$ .",
"This claim is immediate unless there exists some $q^{\\prime } \\in \\mathcal {F}$ such that $x \\in B(q^{\\prime }, r_{q^{\\prime }})$ .",
"However in this case, a similar reasoning as above means $\\text{charge}(x, \\mathcal {F}) \\le r_{q^{\\prime }}^2$ but $\\Vert x-q\\Vert \\ge \\Vert q-q^{\\prime }\\Vert -\\Vert x-q^{\\prime }\\Vert > 2r_{q^{\\prime }} - r_{q^{\\prime }} = r_{q^{\\prime }}$ where the second inequality again follows from step 6 of Algorithm .",
"Therefore, $\\text{charge}(x, \\mathcal {F}) \\le \\Vert x-q\\Vert ^2 &\\le (\\Vert x-q^*\\Vert + \\Vert q^*-q\\Vert )^2 \\nonumber \\\\&\\le 2\\Vert x-q^*\\Vert ^2 + 2\\Vert q^*-q\\Vert ^2 \\nonumber \\\\&\\le 2\\Vert x-q^*\\Vert ^2 + 4r_{q^*}^2.",
"$ Comparing (REF ) to (REF ), we can compute that the ratio of $2\\Vert x-q^*\\Vert ^2 + 4r_{q^*}^2$ to $\\max (r_{q^*}^2, \\Vert x-q^*\\Vert ^2)$ is at most 6 from which it follows that $\\text{charge}(x, \\mathcal {F}) \\le 6 \\cdot \\text{charge}(x, \\mathcal {F}^*).$ Summing over $x \\in X$ completes the proof.",
"Using Theorem REF , we are now in position to prove Lemma REF .",
"The proof of Lemma REF follows similarly to the proof of Lemma 2 in [2] with some modifications to suit our alternate objective function given in (REF ).",
"[Proof of Lemma REF ] We first prove the lower bound.",
"Note that for every $p_i \\in X$ , Algorithm will open a facility within distance at most $2r_p$ .",
"Hence, $4 \\sum _{p \\in X} r_p^2$ is an upper bound on the cost to connect the points to their nearest facility.",
"Now from similar reasoning as in the proof of Theorem REF , we note that each $p$ is in at most one ball $B(q, r_q)$ for some $q \\in \\mathcal {F}$ , where $\\mathcal {F}$ denotes the set of facilities returned by Algorithm .",
"Therefore, $\\sum _{p \\in X} r_p^2 \\ge \\sum _{q \\in \\mathcal {F}} \\sum _{p \\in B(q, r_q)} r_p^2.$ Now if $p \\in B(q, r_q)$ for some $q \\in \\mathcal {F}$ then we must have $r_q \\le 2r_p$ because otherwise, step 6 of Algorithm would not have chosen $q$ as a facility center.",
"Thus, $\\sum _{p \\in X} r_p^2 \\ge \\sum _{q \\in \\mathcal {F}} \\sum _{p \\in B(q, r_q)} r_p^2 \\ge \\frac{1}{4} \\sum _{q \\in \\mathcal {F}} r_q^2 \\cdot |B(q, r_q)|.$ Finally, we know that $ 1 = \\sum _{p \\in B(q, r_q)}( r_q^2 - \\Vert p-q\\Vert ^2) \\le r_q^2 \\cdot |B(q, r_q)| $ from which it follows that $4\\sum _{p \\in X} r_p^2 \\ge |\\mathcal {F}|$ .",
"Altogether, we see that $8 \\sum _{p \\in X} r_p^2$ is an upper bound to the cost of the solution returned by Algorithm so the lower bound follows.",
"For the upper bound, we will show that the sum of the radii squared is not too large compared to $\\operatorname{cost}(\\mathcal {F})$ where $\\mathcal {F}$ is the set of facilities returned by Algorithm .",
"Consider $p \\notin \\mathcal {F}$ and let $q$ be the closest facility to $p$ .",
"First, we must have $r_p^2 \\le 2(\\Vert p-q\\Vert ^2 + r_q^2)$ because otherwise, $r_p^2 > (\\Vert p-q\\Vert + r_q)^2$ which implies that $B(q, r_q) \\subseteq B(p, r_p)$ .",
"Furthermore, $&\\hspace{14.22636pt}\\sum _{p^{\\prime } \\in B(p, r_p)} (r_p^2 - \\Vert p-p^{\\prime }\\Vert ^2) \\\\&\\ge \\sum _{p^{\\prime } \\in B(q, r_q)} (r_p^2 - \\Vert p-p^{\\prime }\\Vert ^2) \\\\&> \\sum _{p^{\\prime } \\in B(q, r_q)}(2r_q^2 +2\\Vert p-q\\Vert ^2 - \\Vert p-p^{\\prime }\\Vert ^2) \\\\&\\ge \\sum _{p^{\\prime } \\in B(q, r_q)}(r_q^2 +2\\Vert p-q\\Vert ^2 + \\Vert p^{\\prime }-q\\Vert ^2- \\Vert p-p^{\\prime }\\Vert ^2) \\\\& \\ge \\sum _{p^{\\prime } \\in B(q, r_q)} (r_q^2 - \\Vert q-p^{\\prime }\\Vert ^2) = 1$ which contradicts (REF ).",
"To summarize, if $p \\notin \\mathcal {F}$ and $q$ is the closest facility in $\\mathcal {F}$ to $p$ , then $r_p^2 \\le 2(\\Vert p-q\\Vert ^2 + r_q^2).$ Going back to the upper bound, recall the definition of $\\text{charge}(p, \\mathcal {F})$ used in the proof of Theorem REF : $ \\text{charge}(p, \\mathcal {F}) = d(p, \\mathcal {F})^2 + \\sum _{q \\in \\mathcal {F}} \\max (0, r_q^2 - \\Vert q-p\\Vert ^2).$ We also showed there that $\\sum _{p \\in X} \\text{charge}(p, \\mathcal {F}) = \\operatorname{cost}(\\mathcal {F})$ .",
"Now $\\operatorname{cost}(\\mathcal {F}) &= \\sum _{p \\in X} \\text{charge}(p, \\mathcal {F}) \\\\&\\ge \\sum _{q \\in \\mathcal {F}} r_q^2 + \\sum _{p \\in X \\setminus \\mathcal {F}} \\max (r_{\\delta (p)}^2, \\Vert p-\\delta (p)\\Vert ^2)$ where $\\delta (p)$ denotes the closest element in $\\mathcal {F}$ to $p$ .",
"From (REF ), we know that $r_p^2 \\le 2(\\Vert p-q\\Vert ^2 + r_q^2)$ so $\\max (r_{\\delta (p)}^2, \\Vert p-\\delta (p)\\Vert ^2) \\ge r_p^2/4$ which gives us $ 6 \\cdot C_{OPT} \\ge \\operatorname{cost}(\\mathcal {F}) \\ge \\frac{1}{4} \\cdot \\sum _{p \\in X} r_p^2, $ as desired.",
"We can prove the following statements about the expected value of $r_p$ , defined as in (REF ), after a random projection to a suitable dimension depending on the doubling dimension of the set $X$ .",
"The following lemma is analogous to Lemmas REF and REF and omit its proof since the proof follows identically from the proofs in Lemmas REF and REF .",
"Lemma F.3 Let $X \\subseteq \\mathbb {R}^m$ and let $p \\in X$ .",
"Let $G$ be a random projection from $\\mathbb {R}^m$ to $\\mathbb {R}^d$ for $d = O(\\log \\lambda _X)$ .",
"Let $r_p$ and $\\tilde{r}_p$ be the radius of $p$ and $Gp$ in $\\mathbb {R}^m$ and $\\mathbb {R}^d$ respectively, computed according to Eq.",
"(REF ).",
"Then there exist constants $c, C > 0$ such that $ cr_p^2 \\le \\mathbb {E}[\\tilde{r}_p^2] \\le C r_p^2.$ Combining Lemma REF , which states that $\\sum _p r_p^2$ is a constant factor approximation to thelb optimal solution of the objective given in (REF ), with Lemma REF , we obtain the following theorem that is analogous to Theorem REF .",
"Theorem F.4 Let $X \\subseteq \\mathbb {R}^m$ and let $p \\in X$ .",
"Let $G$ be a random projection from $\\mathbb {R}^m$ to $\\mathbb {R}^d$ for $d = O(\\log \\lambda _X)$ .",
"Let $\\mathcal {F}_m$ be the optimal solution in $\\mathbb {R}^m$ and let $\\mathcal {F}_d$ be the optimal solution for the dataset $GX \\subseteq \\mathbb {R}^d$ .",
"Then there exists constants $c,C>0$ such that $ c \\cdot \\textup {cost}(\\mathcal {F}_m) \\le \\mathbb {E}[\\textup {cost}(\\mathcal {F}_d)] \\le C \\cdot \\textup {cost}(\\mathcal {F}_m).$ Note that the crucial ingredient in the proof of Theorem REF that allowed us to connect properties of the doubling dimension to facility location clustering was the relation given in Equation (REF ).",
"The analogous relation for our new objective function in (REF ) is given in (REF ) and one can easily check that the steps in the proof of Theorem REF transfer.",
"Therefore, we have the following theorem.",
"Theorem F.5 Let $X\\subseteq \\mathbb {R}^m$ and let $G$ be a random projection from $\\mathbb {R}^m$ to $\\mathbb {R}^d$ for $d = O(\\log \\lambda _X \\cdot \\log (1/\\epsilon )/\\epsilon ^2)$ .",
"Fix $p \\in X$ and let $Gx$ be any point in $B(Gp, C\\tilde{r}_p)$ in $\\mathbb {R}^d$ where $C$ is a fixed constant and $\\tilde{r}_p$ is computed according to Eq.",
"(REF ) in $\\mathbb {R}^d$ .",
"Then $ \\mathbb {E}\\Vert p-x\\Vert \\le 2C(1+O(\\epsilon )) r_p.",
"$ To derive a statement analogous to Theorem REF for our alternate objective function, we need a notion of a locally optimal solution.",
"This task also follows from using Section as a blue print.",
"In particular, we can define local optimality of a solution to (REF ) as follows.",
"Definition F.6 A solution $\\mathcal {F}$ to the objective given in (REF ) is locally optimal if for all $p \\in X$ , we have $B(p, 3r_p) \\cap \\mathcal {F}\\ne \\emptyset $ where $r_p$ is computed as in (REF ).",
"Then the following lemma follows similarly to Lemma REF .",
"Lemma F.7 Let $\\mathcal {F}$ be an any collection of facilities.",
"If there exists a $p \\in X$ such that $B(p, 3r_p) \\cap \\mathcal {F} = \\emptyset $ , then $ \\textup {cost}(\\mathcal {F} \\cup \\lbrace p\\rbrace ) < \\textup {cost}(\\mathcal {F})$ , i.e., we can improve the solution.",
"Finally, as a corollary to Lemma REF and Theorem REF , we have the following corollary.",
"Corollary F.8 Let $X \\subset \\mathbb {R}^m$ and let $G$ be a random projection from $\\mathbb {R}^m$ to $\\mathbb {R}^d$ for $d = O(\\log \\lambda _X \\cdot \\log (1/\\epsilon )/\\epsilon ^2)$ .",
"Let $\\mathcal {F}_d$ be a locally optimal solution for the dataset $GX$ for the objective function given in (REF ).",
"Then, the cost of $\\mathcal {F}_d$ evaluated in $\\mathbb {R}^m$ , denoted as $\\textup {cost}_m(\\mathcal {F}_d)$ , satisfies $ \\mathbb {E}[\\textup {cost}_m(\\mathcal {F}_d)] \\le |\\mathcal {F}_d| + C^{\\prime } \\cdot \\sum _{p \\in X} r_p$ for some constant $C^{\\prime } > 0$ .",
"Remark F.9 We can compute that a constant smaller than 3 works for Definition REF and consequently Lemma REF but this choice is inconsequential since we already incur a multiplicative constant factor in Theorem REF .",
"Finally, we argue that the lower bound of Theorem REF also carries over to our new objective function, meaning that the dimension we project to must depend on the doubling dimension.",
"We define the connection cost of the objective (REF ) as the second portion.",
"Theorem F.10 Let $d = o(\\log n)$ and let $G$ be be a random projection from $\\mathbb {R}^m$ to $\\mathbb {R}^d$ .",
"There exists $X \\subseteq \\mathbb {R}^m$ where $|X| = n$ such that with at least $2/3$ probability, the optimal cost multiplies by $o(1)$ when projected.",
"In addition, there exists an optimal solution $\\widetilde{\\mathcal {F}}$ in $\\mathbb {R}^d$ that is only an $\\omega (1)$ -approximate solution in the original space $\\mathbb {R}^m$ .",
"[Proof Sketch] The proof follows similarly as in the proof of Theorem REF .",
"We again define $X = \\lbrace Re_1, \\dots , Re_m\\rbrace $ , where $R = \\sqrt{C}$ and $C = \\sqrt{\\frac{\\log n}{10 d}}$ .",
"As in the proof of Theorem 6.1, we again have for any fixed $p = R e_i$ , with probability at least $1 - \\frac{1}{C},$ there are at least $R$ points in $GX$ within $\\frac{1}{R}$ distance of $Gp$ .",
"For any such point $p$ , letting $\\tilde{r}_p$ be the associated radius for $GX$ around $Gp$ as computed by Equation (REF ), we have that $\\tilde{r}_p \\le \\frac{2}{\\sqrt{R}}$ .",
"So, with at least $2/3$ probability, at most $\\frac{3m}{C}$ of the points have $\\tilde{r}_p > \\frac{2}{\\sqrt{R}} = o(1)$ .",
"As in the proof of Theorem 6.1, this shows that the optimal cost multiplies by a $o(1)$ factor, by using Lemma REF this time.",
"In the original space $X \\subset \\mathbb {R}^m$ , the optimal squared facility location cost is $m$ , which is achievable by setting every point in $X$ as a facility.",
"However, since the optimal facility cost in $GX$ is $o(m),$ the optimal solution $\\widetilde{\\mathcal {F}}$ in the reduced space $\\mathbb {R}^d$ assigns at most $o(m)$ points to be facilities.",
"Therefore, for the remaining $m-o(m)$ points, the connection cost in the original space is at least $(R \\sqrt{2})^2 \\ge R^2,$ so the cost of $\\widetilde{\\mathcal {F}}$ in the original space $X$ is at least $R^2 \\cdot (m-o(m)) = \\omega (1) \\cdot m$ .",
"Thus, any optimal solution $\\widetilde{\\mathcal {F}}$ is an $\\omega (1)$ -approximate solution in the original space $\\mathbb {R}^m$ ."
]
] | 2107.01804 |
[
[
"Calibrating generalized predictive distributions"
],
[
"Abstract In prediction problems, it is common to model the data-generating process and then use a model-based procedure, such as a Bayesian predictive distribution, to quantify uncertainty about the next observation.",
"However, if the posited model is misspecified, then its predictions may not be calibrated -- that is, the predictive distribution's quantiles may not be nominal frequentist prediction upper limits, even asymptotically.",
"Rather than abandoning the comfort of a model-based formulation for a more complicated non-model-based approach, here we propose a strategy in which the data itself helps determine if the assumed model-based solution should be adjusted to account for model misspecification.",
"This is achieved through a generalized Bayes formulation where a learning rate parameter is tuned, via the proposed generalized predictive calibration (GPrC) algorithm, to make the predictive distribution calibrated, even under model misspecification.",
"Extensive numerical experiments are presented, under a variety of settings, demonstrating the proposed GPrC algorithm's validity, efficiency, and robustness."
],
[
"Introduction",
"Prediction of future observations is a common goal in applications and is a fundamental problem in statistics and machine learning.",
"Motivated by the “all models are wrong” part of the famous quote attributed to George Box, many researchers advocate for model-free methods that can make point predictions in complex problems without the risk of model misspecification biases.",
"However, without specification of a statistical model, quantification of prediction uncertainty—via valid prediction intervals or, in our present case, valid predictive distributions—can be a challenge.",
"Therefore, motivated by the practical need for prediction uncertainty quantification [42], [54] and the “but some models are useful” part of the famous quote, many researchers advocate for the use of (carefully specified) statistical models and the associated model-based methods.",
"But despite the data analyst's best efforts to specify a sound model, misspecification biases are unavoidable.",
"Is it possible for a model-based method to have a built-in correction that will adjust, and in a data-driven way, the prediction intervals/predictive distribution to account for potential model misspecification biases?",
"If so, then this would provide a sort of “best of both worlds,” that is, data analysts can work in a familiar model-based framework that readily provides prediction uncertainty quantification with the added comfort that their predictive inference will, in a certain sense, remain valid even when the model is misspecified.",
"Development of this built-in correction of model-based methods—in particular, of Bayesian predictive distributions—to achieve the aforementioned “best of both worlds” is the goal of the present paper.",
"Our motivation for this work was thinking about insurance applications where the insurance company's goal is to predict the loss (total claim amount) in the next time unit based on observed losses in previous time units, often treated as independent and identically distributed (iid).",
"In these applications, it is common for the loss distribution to be heavy-tailed, and a host of different models, and the associated model-based (likelihood and Bayesian) methods have been developed specifically for this [3], [17], [30], [36].",
"Of course, even a carefully-specified heavy-tailed model could be misspecified, so some authors have proposed more flexible—and more complicated—nonparametric methods that let the data decide the distributional form [25], [26].",
"At the end of the day, however, practitioners want their models/methods to be exactly as complicated as necessary.",
"Could, for example, the tail of a simple, thin-tailed predictive distribution be fattened if necessary, in a data-driven way, to adjust for possibly heavy-tailed data?",
"Similar questions arise in other applications, such as spatial data analysis, especially when interest is in spatially and temporally dependent extremes; see Section REF for details and references.",
"Our starting point is a Bayesian formulation wherein, if the model is correctly specified and satisfies certain regularity conditions, then the posterior distribution will concentrate around the true parameter value asymptotically and, consequently, the corresponding Bayesian predictive distribution will merge with the true data-generating distribution.",
"In this case, we say that the Bayesian predictive distribution is valid or calibrated, at least asymptotically, in the sense that the upper-$\\alpha $ quantile of the predictive distribution is an approximate $100(1-\\alpha )$ % upper prediction limit.",
"When the Bayesian model is misspecified, however, the situation is not so straightforward [6], [28], [29], [21], [45].",
"To start, there is no “true parameter value” so the posterior distribution cannot possibly concentrate there.",
"Moreover, since the model is wrong, it is not possible for the Bayesian predictive distribution to merge with the true distribution, so the latter generally will not be calibrated in the sense defined above.",
"To help overcome this model misspecification bias, a number of authors have recently been using a so-called generalized posterior, which amounts to using a power likelihood in the Bayesian update [37], [1], [53].",
"This power, denoted by $\\eta $ , is called the learning rate and must be chosen using the data.",
"But note that $\\eta $ is not a model parameter with a true value that can be “learned” in any meaningful way, e.g., using a prior and Bayesian updates.",
"Instead, $\\eta $ is a tuning parameter and must be chosen to achieve some desirable operating characteristic.",
"A number of learning rate selection procedures have been developed in the literature recently, including [24], [33], [21], and [51]; see [59] for a comparison.",
"For sure, whatever operating characteristics are achieved for the generalized posterior through these choices of $\\eta $ do not automatically carry over to the corresponding predictive distribution, so it is necessary to reconsider things when prediction is the goal.",
"Here we make two contributions: first, we define a suitable $\\eta $ -generalized predictive distribution—see (REF )—for which calibration is possible; second, inspired by [51], we develop a learning rate selection procedure designed so that calibration in the above sense is achieved.",
"The remainder of this paper is organized as follows.",
"First, in Section , we provide some background on generalized posterior distributions and the recently developed methods for learning rate selection.",
"In Section , we define our generalized predictive distribution, introduce our predictive distribution calibration method, and formulate the corresponding generalized predictive calibration algorithm, or GPrC for short.",
"The performance of our GPrC method is demonstrated in a variety of examples with different kinds of model misspecification.",
"First, some relatively simple illustrations using independent and identically distributed (iid) models are presented in Section to highlight the GPrC algorithm's ability to adjust a relatively simple, thin-tailed model to accommodate heavy-tailed data.",
"Then, in Section , we extend our generalized predictive distribution and GPrC algorithm formulation to adjust for model misspecification in more complex, non-iid models used for analyzing time series and spatial data.",
"In particular, in Section REF , we consider prediction of the response evaluated at a new spatial location, where we posit a simple Gaussian process model and, using the GPrC algorithm, are able to achieve valid predictions, even in the extreme tails and under very non-Gaussian data-generating processes.",
"Finally, some concluding remarks are given in Section ."
],
[
"Generalized posterior distributions",
"For simplicity, suppose that we have iid data $Y_1,\\ldots ,Y_n$ , taking values in a space $\\mathbb {Y}$ , with common marginal distribution $P^\\star $ ; later we will consider more general cases with dependence and/or covariates.",
"The primary goal is to predict the next observation, $Y_{n+1}$ .",
"For the purpose of analyzing data and ultimately making predictions, it is common to introduce a statistical model, ${P}= \\lbrace P_\\theta : \\theta \\in \\Theta \\rbrace $ , a family of distributions on $\\mathbb {Y}$ , indexed by a parameter $\\theta $ in the parameter space $\\Theta $ ; let $p_\\theta $ denote the density of $P_\\theta $ with respect to some dominating $\\sigma $ -finite measure on $\\mathbb {Y}$ , such as Lebesgue or counting measure.",
"The prediction problem basically amounts to learning $P^\\star $ from data, but the introduction of a model shifts the focus to the “true value” $\\theta ^\\star $ of $\\theta $ .",
"Then the likelihood function $L_n$ —which, in this iid setting, is $L_n(\\theta ) = \\prod _{i=1}^n p_\\theta (Y_i)$ —plays an important role in any model-based approach.",
"One such approach is Bayesian, which proceeds by introducing a prior distribution $\\Pi $ on $\\Theta $ that quantifies the a priori uncertainty about the unknown value of $\\theta $ .",
"This prior is then combined with the likelihood, via Bayes's formula, to get a posterior distribution $ \\Pi _n(d\\theta ) \\propto L_n(\\theta ) \\, \\Pi (d\\theta ), \\quad \\theta \\in \\Theta .",
"$ If the model is sufficiently regular and correctly specified, i.e., if $P^\\star \\in {P}$ , then the Bayesian posterior distribution has nice asymptotic properties.",
"That is, the Bernstein–von Mises theorem [56] implies that $\\Pi _n$ is asymptotically normal with mean equal to $\\hat{\\theta }_n$ , the maximum likelihood estimator, and covariance matrix proportional to the inverse of the Fisher information matrix.",
"Since $\\hat{\\theta }_n$ is typically consistent, this implies that $\\Pi _n$ will be centered around $\\theta ^\\star $ and that the corresponding posterior credible sets are asymptotically correct confidence sets.",
"In practice, however, one can never be sure that the model is correctly specified, so it is practically relevant to consider what happens if the model is misspecified, i.e., if $P^\\star \\notin {P}$ .",
"An immediate consequence is that there is no “true” $\\theta ^\\star $ , so it is not clear what the Bayesian posterior distribution $\\Pi _n$ might be learning about.",
"However, under certain regularity conditions, a misspecified version of the aforementioned Bernstein–von Mises theorem holds [29], which states that $\\Pi _n$ is still asymptotically normal, centered at $\\hat{\\theta }_n$ , with covariance matrix $n^{-1} V$ ; the specific form of $V$ is known—it depends on the form of $P_\\theta $ —but the expression is not needed for what follows.",
"Moreover, $\\hat{\\theta }_n$ is a consistent estimator of the Kullback–Leibler minimizer $ \\theta ^\\dagger = \\arg \\min _{\\theta \\in \\Theta } \\int \\log (dP^\\star / dP_\\theta ) \\, dP^\\star , $ but its asymptotic covariance matrix is generally different from $n^{-1} V$ .",
"Therefore, while inference on $\\theta ^\\dagger $ would be meaningful—since $P_{\\theta ^\\dagger }$ is the “best approximation” of $P^\\star $ in ${P}$ —the covariance matrix mismatch implies that posterior credible sets derived from $\\Pi _n$ could have arbitrarily low coverage probability.",
"Is there anything that can be done to correct for the effects of model misspecification, short of starting over from scratch with a different ${P}$ ?",
"An idea that has gained some traction in the recent literature is the use of a so-called generalized posterior, which uses Bayes's formula but with a power likelihood: $\\Pi _n^{(\\eta )}(d\\theta ) \\propto L_n^\\eta (\\theta ) \\, \\Pi (d\\theta ), \\quad \\theta \\in \\Theta , \\quad \\eta > 0.$ The power $\\eta $ is commonly referred to as the learning rate.",
"Even in correctly specified models, there is an advantage to working with the generalized posterior, since, for any $\\eta < 1$ , the asymptotic concentration properties of $\\Pi _n^{(\\eta )}$ can be established without entropy conditions [57], [61], [22]; see [35] for a different use of power likelihood.",
"When the model is potentially misspecified, the likelihood function $L_n$ does not hold the same stature as it does in well-specified cases, so its role in the Bayesian update is less clear.",
"But if one interprets the Kullback–Leibler minimizer $\\theta ^\\dagger $ as the quantity of interest, the problem then can be viewed as one of “risk-minimization,” with $\\ell _\\theta (y) = -\\log p_\\theta (y)$ as the loss.",
"From this perspective, the generalized posterior (REF ) is also a so-called Gibbs posterior [27], [52], which has been shown to be the proper coherent updating of prior information under misspecification; see [58] and [2].",
"Therefore, the learning rate is a fundamental part of Bayesian inference in misspecified models.",
"Roughly speaking, $\\eta $ controls the spread of the generalized posterior.",
"That is, if the learning rate is too large, then the generalized posterior will be tightly concentrated around $\\hat{\\theta }_n$ ; conversely, if the learning rate is small, then the prior gets more weight and generally this will make the generalized posterior less concentrated around $\\hat{\\theta }_n$ .",
"So, clearly, the choice of $\\eta $ affects the practical, finite-sample properties of the generalized posterior.",
"This begs the question: how to choose $\\eta $ ?"
],
[
"Learning rate selection methods",
"A number of different data-driven methods for selecting the learning rate $\\eta $ have been proposed in the recent literature.",
"Here we give just a brief summary of these; a more thorough review can be found in [59].",
"Motivated by the developments in [2], [24] proposed to choose $\\eta $ by matching the prior-to-posterior expected information gain between ordinary Bayesian and generalized posterior.",
"By measuring this gain in terms of the Fisher divergence, they are able to solve this equation for $\\eta $ .",
"This expression involves certain unknowns, but they provide simple, Monte Carlo estimates for these unknowns, leading to a data-driven learning rate $\\hat{\\eta }$ .",
"Motivated by the prequential view of Bayesian model selection in [11], [21] proposed the SafeBayes algorithm that chooses $\\eta $ by minimizing a minor variation on the cumulative log-loss, i.e., $ \\hat{\\eta }= \\arg \\min _\\eta \\sum _{i=1}^n \\int \\lbrace -\\log p_\\theta (Y_i)\\rbrace \\, \\Pi _{i-1}^{(\\eta )}(d\\theta ).",
"$ The learning rate selection method is viewed as the covariance mismatch problem in [33].",
"Motivated by the asymptotic properties of weighted likelihood bootstrap method [41] under well-specified model, they developed a loss-likelihood bootstrap suitable for cases involving misspecified models.",
"By the asymptotic distribution properties of the loss-likelihood bootstrap samples, [33] proposed to match its covariance matrix with the asymptotic covariance matrix of the generalized Bayesian posterior.",
"[51] aimed specifically to choose the learning rate $\\eta $ to partially correct for the aforementioned covariance matrix mismatch, so that $\\eta $ -generalized posterior credible sets were calibrated in the sense that the frequentist coverage probability is approximately the nominal level.",
"They achieved this through a generalized posterior calibration algorithm, or GPC, which approximates the coverage probability function using bootstrap and then uses a version of stochastic approximation (see Section REF below) to match that coverage to the nominal level.",
"While both GPC and Lyddon et al.",
"use bootstrap, the methods are, in fact, quite different; in particular, GPC more directly targets calibration of posterior credible regions.",
"The overall conclusion drawn in [59] is that only the GPC algorithm provides satisfactory calibration of the generalized posterior credible sets in general.",
"This is not surprising, given that the other methods are designed to achieve other properties.",
"However, if the generalized posterior's purpose is to provide valid, data-driven uncertainty quantification about the unknowns, then this kind of calibration is essential.",
"Therefore, in what follows, we aim to develop a prediction-focused analogue of calibration and the GPC algorithm developed in [51]."
],
[
"Objective: calibration",
"Suppose we have data $Y^n = (Y_1,\\ldots ,Y_n)$ generated from a true distribution $P^\\star $ , and the primary goal is prediction of the next observation, $Y_{n+1}$ .",
"While prediction is the primary goal, rather than inference, common practice is to introduce a statistical, ${P}= \\lbrace P_\\theta : \\theta \\in \\Theta \\rbrace $ .",
"From here, as described in Section REF below, we will construct a model-based predictive distribution for $Y_{n+1}$ , depending of course on the observed data $Y^n$ .",
"Here we explain what operating characteristics we hope this predictive distribution to have, whether the model is correctly specified or not.",
"For $\\alpha \\in (0,1)$ , let $Q_\\alpha (y^n)$ denote the upper-$\\alpha $ quantile of this predictive distribution, which is a function of $y^n$ .",
"Then, following [11], [20], and [54], we say that the predictive distribution is calibrated (at level $\\alpha $ ) if $P^\\star \\lbrace Q_\\alpha (Y^n)\\ge Y_{n+1}\\rbrace \\ge 1-\\alpha ,$ that is, if the set to which the predictive distribution assigns probability $1-\\alpha $ has frequentist coverage probability at least $1-\\alpha $ .",
"Obviously, if $P^\\star $ were known, then we could take the upper-$\\alpha $ quantile, $Q_\\alpha ^\\star (y^n)$ , of the corresponding conditional distribution $P^\\star (\\cdot \\mid y^n)$ , and it would be calibrated in the sense of (REF ); in the iid case, this would simplify because $Q_\\alpha ^\\star $ would be a constant, independent of $y^n$ .",
"In practice, however, the true distribution $P^\\star $ is unknown, so calibration is non-trivial.",
"In what follows, we develop a framework in which a model-based predictive distribution can be tuned to accommodate potential model misspecification in such a way that calibration is achieved, at least approximately.",
"This will rely on ideas that parallel those described above for generalized posteriors and learning rate selection.",
"At least at a high level, our setup closely resembles that in [20] in the following sense: we are starting with a simple, pragmatic Bayesian model, which we readily acknowledge may not be correctly specified, and then develop a data-driven adjustment to our pragmatic model's predictive distribution so that it is reliable—or, in Grünwald's words, “safe”—at least in the sense that (REF ) is satisfied."
],
[
"Definition",
"There are a number of ways one might consider defining a generalized predictive distribution.",
"A relative general umbrella that these different ideas fall under is to define a predictive distribution indexed by a trio of positive scalars $(a,b,c)$ as follows: $ f_n^{(a,b,c)}(y) \\propto \\Bigl \\lbrace \\int p_\\theta ^a(y) \\, \\Pi _n^{(b)}(d\\theta ) \\Bigr \\rbrace ^{1/c}.",
"$ Here, $p_\\theta ^a$ is the model density to power $a > 0$ , $\\Pi _n^{(b)}$ is the generalized posterior in (REF ) with learning rate $b > 0$ , and the proportionality constant is determined by integrating the right-hand side with respect to $y$ .",
"Then different ideas for constructing a generalized predictive distribution correspond to different configurations of $(a,b,c)$ .",
"An ordinary Bayes predictive distribution corresponds to $a=b=c=1$ .",
"A natural generalization of the Bayesian predictive distribution is to take $a=c=1$ and let $b=\\eta $ be a free learning rate parameter to be chosen using, e.g., one of the procedures described in Section REF .",
"Given that the generalized posterior has the model density $p_\\theta $ to a power $\\eta $ , it also makes sense to consider the same power on the model density when forming the predictive distribution.",
"This corresponds to $c=1$ and $a=b=\\eta $ , with $\\eta $ a tuning parameter to be selected.",
"[8], [9] considered the case where $b=1$ and $a=c=\\frac{1}{2}(1-\\beta )$ , where $\\beta $ indexes the user's choice of divergence measure.",
"With this choice of $(a,b,c)$ , the corresponding $f_n^{(a,b,c)}$ is the Bayes estimator, i.e., posterior risk minimizer, under the so-called “$\\beta $ -divergence” loss, where $\\beta =-1$ corresponds to Kullback–Leibler divergence, $\\beta =0$ corresponds to Hellinger distance, etc.",
"The specific form of this divergence is not important for us here.",
"A further generalization was presented in [60], which basically takes $b=\\frac{1}{2}(1+\\beta )$ ; we say “basically” because they actually take the ordinary Bayes posterior density to that power $b$ , as opposed to using $b$ as the learning rate in a generalized posterior.",
"The key point is that none of these can reliably be tuned to achieve calibration.",
"First, the ordinary Bayes predictive in Item 1 has no free tuning parameters, so it will only be calibrated when the model is correctly specified.",
"Second, adjusting the divergence measure with respect to which the predictive density referred to in Item 4 above is the Bayes estimator will provide no calibration guarantees.",
"Third, the proposal in Item 2 has similar issues because the generalized posterior will, under certain conditions, concentrate around $\\theta ^\\dagger $ when $n$ is large, which implies that the $(a,b,c)=(1,\\eta ,1)$ predictive density would be roughly $p_{\\theta ^\\dagger }$ .",
"Since this is afflicted by model misspecification bias, i.e., $P_{\\theta ^\\dagger } \\ne P^\\star $ , and has no parameters left to be tuned, it cannot be successfully calibrated.",
"The proposal in Item 3 above would avoid the criticism that model misspecification bias remains when $n$ is large.",
"However, there is an even simpler proposal that can accomplish the same thing.",
"Indeed, consider $a=\\eta $ and $b=c=1$ in the above framework.",
"That is, define the $\\eta $ -generalized predictive distribution as $f_n^{(\\eta )}(y) \\propto \\int p_\\theta ^\\eta (y) \\, \\Pi _n(d\\theta ),$ where $\\Pi _n=\\Pi _n^{(1)}$ is the ordinary Bayes posterior.",
"In the case where $\\Pi _n$ concentrates around $\\theta ^\\dagger $ for large $n$ , it is clear from the expression in (REF ) that we end up with $f_n^{(\\eta )} \\propto p_{\\theta ^\\dagger }^\\eta $ , approximately.",
"Since the dependence on a tunable $\\eta $ remains, even with $n \\rightarrow \\infty $ , we still have the flexibility to calibrate the predictive distribution.",
"The key question is how can $\\eta $ be tuned in order to achieve at least approximate calibration, and we address this question in Section REF below.",
"To see that (REF ) corresponds to a well-defined density, note that typically the learning rate would be between 0 and 1.",
"By Jensen's inequality, if $\\eta < 1$ , then $ \\int p_\\theta ^\\eta (y) \\, \\Pi _n(d\\theta ) \\le \\Bigl \\lbrace \\int p_\\theta (y) \\, \\Pi _n(d\\theta ) \\Bigr \\rbrace ^\\eta < \\infty , \\quad \\text{for (almost) all $y$}, $ and the integral in the upper bound above is simply the ordinary Bayes predictive density, say, $p_n$ .",
"If $p_n$ 's tails are not too heavy, then $y \\mapsto p_n^\\eta (y)$ would be integrable, hence the right-hand side of (REF ) defines a proper predictive distribution.",
"Note that the “not too heavy” condition concerns only the model—not the true $P^\\star $ —so it is entirely within the data analyst's control and can be readily checked in specific examples.",
"Before we move on to the calibration algorithm, we should comment on the choice to work with the predictive distribution in (REF ) as opposed to the one mentioned in Item 3 above that works with the $\\eta $ -generalized posterior $\\Pi _n^{(\\eta )}$ .",
"Recall that the primary role played by $\\eta $ was to control the spread of the generalized posterior, with small $\\eta $ leading to wider spread.",
"This is what motivated [51] to tune $\\eta $ so that the nominal coverage could be achieved—the covariance mismatch, due to misspecification, could be (conservatively) overcome by stretching the posterior's contours sufficiently far.",
"In the prediction setting, however, we integrate over $\\theta $ with respect to $\\Pi _n$ , so the shape of its contours is less important.",
"That is, in prediction, the covariance mismatch has a $o(1)$ effect in the sense that $\\Pi _n$ concentrates at $\\theta ^\\dagger $ as $n \\rightarrow \\infty $ , regardless of any (reasonable) adjustments that may have been made to the posterior contours.",
"On the other hand, misspecification in the model density $p_{\\theta ^\\dagger }$ remains, even asymptotically, so its effect is $O(1)$ .",
"Given that the generalized predictive in (REF ), with the ordinary posterior that ignores the $o(1)$ effect of covariance mismatch, is much simpler (see Section REF ), we opt for this instead of the more complicated method described in Item 3 above."
],
[
"The GPrC algorithm",
"From the generalized predictive distribution $f_n^{(\\eta )}$ in (REF ), with a particular $\\eta $ value, we obtain an upper prediction limit as the solution $q=Q_\\alpha (\\eta ; Y^n)$ of the equation $ \\int _{-\\infty }^q f_n^{(\\eta )}(y) \\, dy = 1-\\alpha .",
"$ That is, $Q_\\alpha (\\eta ;Y^n)$ is the upper-$\\alpha $ quantile of the $\\eta $ -generalized predictive distribution.",
"The goal is to select a value of the learning rate $\\eta $ so that (REF ) holds for $Q_\\alpha (\\eta ; Y^n)$ .",
"Of course, all of what follows can be modified in the obvious way if a prediction lower limit or a prediction interval is desired instead of a prediction upper limit.",
"Towards this, define the coverage probability of the prediction upper limit: $ c_\\alpha (\\eta ) = P^\\star \\lbrace Q_\\alpha (\\eta ; Y^n) \\ge Y_{n+1}\\rbrace .",
"$ Note that this is a probability with respect to the joint distribution of $(Y^n,Y_{n+1})$ under $P^\\star $ .",
"The fact our notation $Q_\\alpha $ includes “$\\alpha $ ” does not guarantee that $Q_\\alpha (\\eta ; Y^n)$ is a valid prediction upper limit; calibration in the sense of (REF ) is what needs to be shown.",
"That is, we aim to find $\\eta $ to solve the equation $c_\\alpha (\\eta ) = 1-\\alpha .$ If $P^\\star $ were known, then the coverage probability function $\\eta \\mapsto c_\\alpha (\\eta )$ could at least be evaluated numerically, to any desired accuracy, using Monte Carlo, and then the equation (REF ) could be solved using stochastic approximation (see below).",
"In practice, however, $P^\\star $ is unknown so a different strategy is required.",
"Here we will make use of the bootstrap (in one form of another) to approximate the $P^\\star $ probabilities using the observed data.",
"To see more clearly where we are going, it may help to re-express the coverage probability using the familiar iterated expectation formula: $c_\\alpha (\\eta ) = E^\\star \\bigl [ P^\\star \\lbrace Y_{n+1} \\le Q_\\alpha (\\eta ; Y^n) \\mid Y^n\\rbrace \\bigr ],$ where $E^\\star $ is expectation with respect to $P^\\star $ .",
"This reveals that there are effectively two expectations that need to be approximated: one is over $Y_{n+1}$ with $Y^n$ fixed, and the other is over $Y^n$ .",
"Our approximations of these two expectations will be easiest to describe in the case of iid data; we will extend the idea to non-iid cases in Section below.",
"Let $\\lbrace Y_b^n\\rbrace _{b=1}^B$ be the $B$ bootstrap samples, each of size $n$ , generated from $Y^n$ .",
"That is, each $Y_b^n$ is a random sample of size $n$ , with replacement, from the observed data $Y^n$ .",
"Next, for each $b=1,\\ldots ,B$ , let $Q_\\alpha (\\eta ; Y_b^n)$ denote the quantile of the $\\eta $ -generalized predictive distribution in (REF ) based on data $Y_b^n$ ; more details about this predictive quantile computation below.",
"In the iid case, we have $ P^\\star (Y_{n+1} \\le x \\mid Y^n) = P^\\star (Y_{n+1} \\le x), $ and the right-hand side can be readily estimated using the empirical distribution function from $Y^n$ .",
"This immediately leads to an empirical version of the expression in (REF ), $ \\hat{c}_\\alpha (\\eta ) = \\frac{1}{B} \\sum _{b=1}^B \\Bigl [ \\frac{1}{n} \\sum _{i=1}^n 1\\lbrace Y_i \\le Q_\\alpha (\\eta ; Y_b^n)\\rbrace \\Bigr ], $ where $1\\lbrace A\\rbrace $ denotes the indicator function of the event $A$ .",
"For the iid case considered here, the inner average over $i=1,\\ldots ,n$ approximates the conditional probability in (REF ), given $Y^n$ , and the outer average over $b=1,\\ldots ,B$ approximates the outer expectation with respect to the distribution of $Y^n$ .",
"The idea behind the GPrC algorithm is to solve $\\hat{c}_\\alpha (\\eta ) = 1-\\alpha $ for $\\eta $ instead of (REF ), which leads to a data-driven choice, $\\hat{\\eta }$ , of the learning rate $\\eta $ .",
"To properly solve the equation $\\hat{c}_\\alpha (\\eta )=1-\\alpha $ , we need a root-finding procedure that accommodates the Monte Carlo variability in $\\hat{c}_\\alpha $ .",
"As in [51], we adopt the stochastic approximation method of [46]; see, also, [32].",
"In particular, for a vanishing, deterministic sequence $\\lbrace \\kappa _t: t \\ge 1\\rbrace $ , such that $ \\sum _{t=1}^\\infty \\kappa _t = \\infty \\quad \\text{and} \\quad \\sum _{t=1}^\\infty \\kappa _t^2 < \\infty , $ and a starting value $\\eta ^{(1)}> 0$ , define the sequence of candidate solutions $\\eta ^{(t+1)} = \\eta ^{(t)} + \\kappa _t \\lbrace \\hat{c}_\\alpha (\\eta ^{(t)}) - (1-\\alpha )\\rbrace , \\quad t \\ge 1.$ If, instead of bootstrap samples, we could approximate the coverage probability function $c_\\alpha $ using Monte Carlo samples from $P^\\star $ , then it could be checked using the standard convergence theory for stochastic approximation sequences (e.g., [47] that $\\eta ^{(t)}$ converges $P^\\star $ -almost surely to a solution of the equation in (REF ).",
"Given that bootstrap is a generally reliable computational tool for approximating sampling distributions, and that there is no reason to expect our present setup to be atypical (see Section below), we propose to update $\\eta ^{(t)}$ according to the rule (REF ) until it converges, and we denote the limit as $\\hat{\\eta }$ .",
"This makes up what we call the generalized predictive calibration, or GPrC, procedure; see Algorithm REF .",
"[t] InputInput OutputOutput A learning rate $\\eta $ that calibrates the generalized predictive distribution in terms of a desired coverage probability.",
"$|\\hat{c}_\\alpha (\\eta ^{(t)})-(1-\\alpha )|>\\epsilon $ $\\eta ^{(t+1)} \\leftarrow \\eta ^{(t)}+\\kappa _t\\lbrace \\hat{c}_\\alpha (\\eta ^{(t)})-(1-\\alpha )\\rbrace $ $b=1,\\dots ,B$ Estimate the upper-$\\alpha $ quantile of each bootstrap sample, $Q_\\alpha (\\eta ^{(t+1)}; Y^n_b)$ Evaluate $\\hat{c}_\\alpha (\\eta ^{(t+1)})$ using bootstrap samples $t \\leftarrow t + 1$ Generalized Predictive Calibration (GPrC) In our implementation of the GPrC algorithm, we recommend a starting value $\\eta ^{(0)}=0.5$ .",
"The idea behind this choice is that we are anticipating some degree of model misspecification, in which case calibration would require $\\eta < 1$ , so we want to take a “warm start” in order to accelerate convergence.",
"Other choices of starting values perform similarly, however.",
"The convergence tolerance $\\epsilon = 0.01 \\times \\alpha $ is intended to balance the quality of the coverage probability approximation versus the speed of convergence.",
"We suggest using a cutoff that is an increasing function of $\\alpha $ because calibrating at the extreme quantiles—small $\\alpha $ values—is more challenging, hence a smaller tolerance is recommended in order to encourage more iterations.",
"The convergence tolerance should also depend on the precision of the estimated coverage probabilities.",
"In the present case of iid data, $\\hat{c}_\\alpha (\\eta )$ is evaluated as an average of $nB$ indicators, so there is no point to make the tolerance $\\epsilon $ less than $(nB)^{-1}$ .",
"But for the dependent data cases discussed later, there is less information available in the data, so the number of indicators being averaged to evaluate $\\hat{c}_\\alpha (\\eta )$ is much smaller, hence less precision.",
"In the spatial case, for example, the precision is bounded by $B^{-1}$ , so we recommend a tolerance value of $\\epsilon = \\max \\lbrace 0.01 \\times \\alpha , B^{-1}\\rbrace $ .",
"It remains to say a few words about the computation of $Q_\\alpha (\\eta ; Y^n)$ for a given $\\alpha $ , $\\eta $ , and data set $Y^n$ .",
"The situations we have in mind (see the subsequent sections) are those where a simple model ${P}$ is used with the GPrC algorithm there to correct for any misspecification bias.",
"When the posited model is relatively simple, it may be possible to evaluate the posterior distribution and, hence, the ordinary Bayes predictive distribution in closed-form, e.g., if the prior is conjugate.",
"In that case, evaluating the quantile $Q_\\alpha (\\eta ; Y^n)$ can be solved using standard numerical methods.",
"For more complicated models, Monte Carlo-based methods may be needed to evaluate the quantiles.",
"Assuming one can obtain samples $\\lbrace \\theta ^{(m)}: m=1,\\ldots ,M\\rbrace $ from the posterior distribution $\\Pi _n$ , the $\\eta $ -generalized predictive density in (REF ) can be approximated as $ \\hat{f}_n^{(\\eta )}(y) \\propto \\frac{1}{M} \\sum _{m=1}^M p_{\\theta ^{(m)}}^\\eta (y), $ and the normalizing constant and quantile can be found via quadrature.",
"These Monte Carlo approximations would be required for each of the $B$ bootstrap samples, but not for each individual update of the learning rate $\\eta $ in the GPrC algorithm.",
"This is where we find a computational advantage compared to the algorithm in [51].",
"In the latter reference, they are concerned with the posterior distribution, so the aforementioned covariance mismatch is crucial and cannot be ignored.",
"As we explained above, the bias resulting covariance mismatch is a lower-order term in the prediction setting, and can be ignored.",
"By not requiring the posterior to change with $\\eta $ , we can use the same $B$ sets of posterior samples in each of the $\\eta $ updates in (REF ).",
"Therefore, updating $Q_\\alpha (\\cdot ; Y_b^n)$ for a given bootstrap sample $b$ to reflect a change in the learning rate along the GPrC sequence is relatively inexpensive, so the computational cost is of the order $B$ .",
"Compare this to the original GPC algorithm in [51], where an update of $\\eta $ would have computational cost of the order $B \\times M$ , where $M$ is the desired number of Monte Carlo samples.",
"Therefore, our particular choice of the generalized posterior density in (REF ) leads to a much faster and efficient algorithm than one which focuses on adjusting the learning rate in the posterior.",
"And in terms of computational time, in our examples here, at least for the iid setting in Section below, a run of GPrC takes only a matter of seconds to complete, even with $B=200$ or more.",
"The GPrC algorithm can be extended to independent but not iid data cases, and even certain dependent data cases, with or without covariates.",
"This only requires the use of suitable variations on the basic bootstrap approach described above designed to accommodate the assumed data structure.",
"We make this extension for time series and spatial data applications in Sections REF and REF , respectively."
],
[
"Further details about GPrC",
"Here we highlight a few important features of our proposed procedure, in particular, what the tuned $\\eta $ -generalized predictive does and what values of $\\eta $ lead to the key calibration property in (REF ).",
"Simple, numerical examples will be given to illustrate these points.",
"First, note that model misspecification bias cannot be corrected simply by adjusting a tuning parameter like $\\eta $ in (REF ), at least not in general.",
"So we have to be clear: we make no claims that there exists $\\eta $ such that the $\\eta $ -generalized predictive distribution $f_n^{(\\eta )}$ closely approximates $P^\\star $ or, more generally, $P^\\star (\\cdot \\mid Y^n)$ , in any global sense.",
"Since a complete correction of the model misspecification bias is generally out of reach, our goal instead is a more modest one —to ensure that the predictive distribution achieves the calibration property (REF ), at least at a particular level $\\alpha $ , even when the model is misspecified; cf. [20].",
"This amounts to adjusting $\\eta $ so that the tails of $f_n^{(\\eta )}$ match those of $P^\\star $ in a certain sense to be made clear below.",
"The best place to start is with a correctly specified model.",
"That is, let $P^\\star = P_{\\theta ^\\star }$ for some “true” parameter value $\\theta ^\\star \\in \\Theta $ , and let $p_{\\theta ^\\star }$ denote the corresponding density.",
"Using the asymptotic setting as a guide, recall that $f_n^{(\\eta )}(y) \\approx p_{\\theta ^\\star }^\\eta (y)$ when $n$ is large.",
"Since the true distribution is calibrated at every level $\\alpha $ , we should take $\\eta \\approx 1$ , which is what the GPrC algorithm does.",
"For illustration, let ${\\sf Gamma}(a,b)$ denote a gamma distribution with shape parameter $a > 0$ and rate parameter $b > 0$ ; the density function is $ y \\mapsto y^{a-1} e^{-b y}, \\quad y > 0.",
"$ Consider a gamma model, $P_\\theta = {\\sf Gamma}(3,\\theta )$ , where the shape parameter is fixed at 3 but the rate parameter $\\theta > 0$ is free to vary.",
"The true value is $\\theta ^\\star =2$ in our experiment.",
"With a conjugate gamma prior, $\\theta \\sim {\\sf Gamma}(a,b)$ , and iid data $Y^n=(Y_1,\\ldots ,Y_n)$ , the posterior distribution is $\\Pi _n = {\\sf Gamma}(a_n, b_n)$ , where $a_n = a + 3n$ and $b_n = b + \\sum _{i=1}^n Y_i$ .",
"Finally, the $\\eta $ -generalized predictive distribution is the so-called generalized beta prime distribution [38] with density $f_n^{(\\eta )}(y) \\propto \\frac{(y/d)^{cp-1}}{\\lbrace 1+(y/d)^c\\rbrace ^{p+q}},$ where $(c,d,p,q)=(1, b_n/\\eta , 2\\eta +1, a_n+\\eta -1)$ .",
"For this illustration, we simulated 1000 data sets, each of size $n=400$ , from the above gamma model, and ran the GPrC algorithm in each case to identify a learning rate $\\hat{\\eta }$ such that calibration at different $\\alpha $ levels is achieved, at least approximately.",
"Figure REF shows the the distribution of the $\\hat{\\eta }$ values over the 1000 replications, at the three levels $\\alpha \\in \\lbrace 0.01, 0.05, 0.10\\rbrace $ .",
"Notice that the $\\hat{\\eta }$ values tend to concentration around $\\eta =1$ , as expected.",
"Figure: Histogram of the η ^\\hat{\\eta } values selected by the GPrC algorithm in the correctly specified gamma example, based on 1000 replications, each with sample size n=400n=400.Above we mentioned that the model misspecification bias generally cannot be corrected, i.e., there is no learning rate $\\eta $ such that $f_n^{(\\eta )}$ accurately approximates $P^\\star $ in a global sense.",
"There are, however, certain cases where the model misspecification bias is sufficiently mild that it can be completely corrected by tuning $\\eta $ .",
"Suppose the true distribution is normal, $P^\\star = {\\sf N}(\\mu ^\\star , \\sigma ^{\\star 2})$ , and the model is $P_\\theta = {\\sf N}(\\theta , \\sigma ^2)$ , where $\\theta $ is unknown and to be inferred while $\\sigma $ is fixed and generally different from $\\sigma ^\\star $ .",
"This is relatively mild misspecification because the tails of the model basically match those of $P^\\star $ , so correcting for it may not be out of the question.",
"With iid data $Y^n=(Y_1,\\ldots ,Y_n)$ and a conjugate prior $\\theta \\sim {\\sf N}(m, v)$ , the posterior distribution is $\\Pi _n = {\\sf N}(m_n, v_n)$ , with $ m_n = \\frac{\\sigma ^2 m + v \\sum _{i=1}^n Y_i}{\\sigma ^2 + n v} \\quad \\text{and} \\quad v_n = \\frac{\\sigma ^2 v}{\\sigma ^2 + n v}, $ respectively.",
"Then the $\\eta $ -generalized predictive distribution is also normal, i.e., $f_n^{(\\eta )}$ is a ${\\sf N}(m_n, v_n + \\eta ^{-1} \\sigma ^2)$ density.",
"When $n$ is large, $m_n \\approx \\mu ^\\star $ and $v_n \\approx 0$ , so to achieve calibration, we would need $\\eta \\approx (\\sigma /\\sigma ^\\star )^2$ .",
"Table REF compares the $\\hat{\\eta }$ values selected by the GPrC algorithm with $\\eta ^\\dagger = (\\sigma /\\sigma ^\\star )^2$ , for different values of $n$ and $\\alpha $ .",
"Clearly, GPrC is tending to select $\\hat{\\eta }$ values near $\\eta ^\\dagger $ .",
"Moreover, the empirical coverage probabilities of the $\\hat{\\eta }$ -generalized predictive distribution quantiles shown in Table REF are all near the nominal level, suggesting that calibration in the sense of (REF ) is achieved, at least approximately, across all $n$ and $\\alpha $ .",
"Therefore, the GPrC algorithm successfully corrects for the model misspecification bias, which is relatively mild in this case.",
"Table: Average η ^\\hat{\\eta } values from the GPrC algorithm and (in parentheses) the empirical coverage probability of the corresponding η ^\\hat{\\eta }-generalized predictive distribution quantiles, at three different levels α∈{0.01,0.05,0.10}\\alpha \\in \\lbrace 0.01, 0.05, 0.10\\rbrace , in the mildly misspecified normal example, based on 1000 replications.Next, what do we expect the GPrC algorithm to do under more severe model misspecification?",
"For example, suppose that we use the same gamma model described above but it happens that the true distribution is log-normal, say, $Y^n$ consists of iid observations coming from ${\\sf logN}(\\mu ^\\star , \\sigma ^{\\star 2})$ , with $\\mu ^\\star = \\sigma ^\\star = 1$ .",
"Based on how the GPrC algorithm is defined, we would expect that it would choose $\\eta $ so that the upper-$\\alpha $ quantile of the $\\eta $ -generalized predictive distribution given in (REF ) agrees with the upper-$\\alpha $ quantile of the true distribution, ${\\sf logN}(\\mu ^\\star , \\sigma ^{\\star 2})$ .",
"That is, GPrC is aiming to solve the equation $Q_\\alpha (\\eta ; Y^n) = Q_\\alpha ^\\star ,$ where $Q_\\alpha ^\\star =\\exp \\lbrace \\mu ^\\star +\\sigma ^\\star \\, \\Phi (1-\\alpha )\\rbrace $ .",
"In practice, however, GPrC can only find an approximate solution, because an exact solution would require knowledge of $P^\\star $ or, in this case, $(\\mu ^\\star ,\\sigma ^\\star )$ , which is information the algorithm does not have.",
"But this is a simulation study, where $P^\\star $ is known, so it is possible to solve the equation (REF ) exactly.",
"We did precisely this and the results are summarized in Figure REF .",
"That is, for each of 1000 replications, with sample size $n=400$ , we evaluate both the $\\hat{\\eta }$ value from the GPrC algorithm and the solution of the equation (REF ), for $\\alpha \\in \\lbrace 0.01, 0.05, 0.10\\rbrace $ , and the plot shows histograms of the former compared to the average (red vertical line) of the latter.",
"The key observation is that the GPrC estimates are centered around the average of those solutions to the equation (REF ), which confirms our claim that GPrC aims to match the quantile of the $\\eta $ -generalized predictive distribution to that of the true distribution.",
"Figure: Histogram of the η ^\\hat{\\eta } values selected by the GPrC algorithm across 1000 replicates, each with n=400n=400, in the misspecified log-normal–gamma example.",
"The average learning rates that solve () across the replications are shown by red vertical lines at 0.670, 0.506, and 0.322 in the α=0.10\\alpha =0.10, α=0.05\\alpha =0.05, and α=0.01\\alpha =0.01 cases, respectively.For a closer look at the GPrC algorithm's performance, we consider comparisons with three other methods.",
"The first, denoted by gamma, is where we stick with the ordinary Bayes predictive distribution, with $\\eta \\equiv 1$ , under the misspecified model.",
"The second is the proposed method, denoted by gamma + GPrC, where the misspecified gamma model is assumed but the learning rate $\\eta $ chosen according to GPrC.",
"Third, is a (version of the) Bayesian nonparametric formulation based on a Dirichlet process mixture model, which we denote here by DP Mixture.",
"The variation being employed here is the fast, recursive approximation, originally motivated by [40] and [39]—see, also, [55] and [34]—and developed fully for the prediction setting in [23].",
"Finally, the last method, denoted by log-normal, is not really a method, it is the oracle that uses the true distribution for prediction.",
"We simulate data from the log-normal model and compare the coverage probability of the $100(1-\\alpha )$ % prediction upper limits or, equivalently, the upper-$\\alpha $ quantiles of the various predictive distributions.",
"Note that, despite the non-trivial model misspecification bias, the gamma + GPrC is able to calibrate its prediction limits, while the simple gamma model cannot.",
"In this case, the DP mixture method too is able to successfully calibrate its prediction limits; but see Section .",
"Table: Empirical coverage probabilities for the upper 100(1-α)100(1-\\alpha )% prediction limits, for four methods in the misspecified log-normal–gamma example, based on 1000 replications.We conclude this section with one last example that is simple enough to do the relevant calculations in closed-form.",
"Suppose that the model $P_\\theta $ is ${\\sf N}(\\mu ,\\sigma ^2)$ , where $\\theta =(\\mu ,\\sigma ^2)$ , but that the true distribution $P^\\star $ is a Laplace distribution, denoted by ${\\sf Lap}(\\mu ^\\star , \\lambda ^\\star )$ , where $\\mu ^\\star $ is the mean and $\\lambda ^\\star $ is the scale parameter; that is, $P^\\star $ has density $ p^\\star (y) = (2\\lambda ^{\\star })^{-1} e^{-|y-\\mu ^\\star |/\\lambda ^\\star }, \\quad y \\in \\mathbb {R}.",
"$ For a Bayesian analysis, we proceed by introducing a conjugate normal–inverse gamma prior for $(\\mu , \\sigma ^2)$ , where the conditional prior for $\\mu $ , given $\\sigma ^2$ , is ${\\sf N}(m, k\\sigma ^2)$ and the marginal prior for $(1/\\sigma ^2)$ is ${\\sf Gamma}(a,b)$ .",
"The prior hyperparameters, $(m,k,a,b)$ , are taken to be fixed constants.",
"Then it is not too difficult to show that the $\\eta $ -generalized predictive density (REF ) is a location-scale transformation of a Student-t density; that is, $ (Y_{n+1} \\mid Y^n,\\eta ) \\sim m_n + \\Bigl \\lbrace \\Bigl ( \\frac{1}{\\eta } + \\frac{k}{nk+1} \\Bigr ) \\frac{2b_n}{2a_n+\\eta -1} \\Bigr \\rbrace ^{1/2} \\, {\\sf t}_{2a_n+\\eta -1}, $ where $ m_n = \\tfrac{nk}{nk+1} \\, \\hat{\\mu }_n + \\tfrac{1}{nk + 1} \\, m, \\qquad a_n = a + \\tfrac{n}{2}, \\qquad b_n = b + \\tfrac{n}{2} \\hat{\\sigma }_n^2 + \\tfrac{n}{2(nk+1)} (\\hat{\\mu }_n - m)^2, $ with $\\hat{\\mu }_n = n^{-1} \\sum _{i=1}^n Y_i$ and $\\hat{\\sigma }_n^2 = n^{-1} \\sum _{i=1}^n (Y_i - \\hat{\\mu }_n)^2$ .",
"Since the mean and variance of ${\\sf Lap}(\\mu ^\\star , \\lambda ^\\star )$ are $\\mu ^\\star $ and $2\\lambda ^{\\star 2}$ , respectively, we find that $\\hat{\\mu }_n \\rightarrow \\mu ^\\star $ and $\\hat{\\sigma }_n^2 \\rightarrow 2\\lambda ^{\\star 2}$ in $P^\\star $ -probability.",
"Since $a_n \\rightarrow \\infty $ , it follows that, for large $n$ , the $\\eta $ -generalized predictive density can be approximated by $ f_n^{(\\eta )}(y) \\approx {\\sf N}(y \\mid \\mu ^\\star , 2\\eta ^{-1} \\lambda ^{\\star 2}).",
"$ Incidentally, the minimizer of the Kullback–Leibler divergence of $P_\\theta $ from $P^\\star $ is $\\theta ^\\dagger = (\\mu ^\\star , 2\\lambda ^{\\star 2})$ , so the right-hand side of the above approximation agrees with that mentioned above based on the posterior concentration properties of $\\Pi _n$ under model misspecficiation, i.e., $f_n^{(\\eta )}(y) \\propto p_{\\theta ^\\dagger }^\\eta (y)$ , approximately.",
"Since the upper-$\\alpha $ quantile of ${\\sf Lap}(\\mu ^\\star , \\lambda ^\\star )$ is $ Q_\\alpha ^\\star = \\mu ^\\star - \\lambda ^\\star \\log (2\\alpha ), $ using the above approximation, we find that $\\eta $ should be chosen such that $ \\mu ^\\star + (2\\eta ^{-1} \\lambda ^{\\star 2})^{1/2} \\, \\Phi ^{-1}(1-\\alpha ) = Q_\\alpha ^\\star , $ or, equivalently, we want to use $ \\eta _\\alpha = 2 \\lbrace \\Phi ^{-1}(1-\\alpha ) / \\log (2\\alpha )\\rbrace ^{2}.",
"$ Note, again, that this “ideal” choice of $\\eta $ depends on $\\alpha $ ; however, in this case, since both the model and the true distribution are location-scale families, $\\eta _\\alpha $ does not depend on features of $P^\\star $ , so this value is accessible in real applications.",
"To confirm that, indeed, the GPrC algorithm selects $\\eta $ close to the $\\eta _\\alpha $ described above, we do a brief simulation study.",
"The boxplots in Figure REF summarize the $\\hat{\\eta }$ values chosen by the GPrC algorithm, the red dots correspond to the $\\eta _\\alpha $ value defined above, and the blue dots correspond to the average $\\eta $ value obtained matching the actual, data-dependent $\\eta $ -generalized predictive distribution—not the asymptotic approximation—to the true quantile $Q_\\alpha ^\\star $ .",
"As expected, the red and blue dots are indistinguishable, and the GPrC algorithm's learning rate choices tightly concentrate around these “ideal” $\\eta $ values, confirming its effectiveness.",
"Interestingly, note that at least for moderate $\\alpha $ levels, the ideal $\\eta _\\alpha $ and those that tend to be chosen by GPrC are larger than 1.",
"The reason is that the Student-t predictive density—but not the normal limit—is wider than the true Laplace density, so GPrC becomes more aggressive, adaptively shrinking the prediction intervals while maintaining calibration.",
"Figure: Boxplot of the learning rate selected by the GPrC algorithm in the misspecified Laplace–normal example (μ ☆ =0,λ ☆ =1)(\\mu ^\\star =0,\\lambda ^\\star =1), based on 1000 replications, each with n=400n=400.",
"Red dots denote η α \\eta _\\alpha and blue dots denote the average η\\eta to match the predictive distribution quantiles to Q α ☆ Q_\\alpha ^\\star ."
],
[
"Setup and take-away messages",
"Our original motivation behind the GPrC algorithm was that in, e.g., actuarial science of finance applications, often the goal is to predict observations when the underlying distribution is skewed and heavy-tailed.",
"It can be difficult to specify good, finite-dimensional parametric models to handle such data; moreover, nonparametric methods are more complicated and the most commonly used versions—Dirichlet process mixtures of normal kernels with a thin-tailed base measure for the prior—are suited only for cases where the tails are not too heavy.",
"Therefore, it would be beneficial if one could take a simple, pragmatic model, which is not assumed to be correctly specified, and let the data determine what kind of adjustments (if any) are needed.",
"So, for our first set of illustrations, we consider heavy-tailed data, like what often manifests in financial applications, and use the GPrC algorithm to adjust so that the predictive distribution is calibrated.",
"In particular, we consider two different $P^\\star $ s, namely, the Pareto and generalized extreme value distributions.",
"Both are supported on $[0,\\infty )$ , hence are skewed, and have a shape parameter that controls the heaviness of the tail; the specific distributional forms are given below.",
"In both cases, we consider a simple log-normal model, $P_\\theta = {\\sf logN}(\\mu , \\sigma ^2)$ , with a conjugate prior for $\\theta =(\\mu , \\sigma ^2)$ .",
"Since log-normal has relatively thin tails, it can be severely misspecified depending on the Pareto or generalized extreme value distribution's shape parameter, so makes for a good test of the GPrC methodology.",
"We compare the GPrC results with the misspecified Bayesian predictive distributions and the aforementioned variation on the Bayesian nonparametric Dirichlet process mixture model.",
"The comparison will be based on two metrics, namely, empirical coverage probability of $100(1-\\alpha )$ % upper prediction limits and a one-sided version of the empirical interval score suggested by [18].",
"To make this precise, we need a bit more notation.",
"Let $R$ denote the number of replications, which we take to be $R=1000$ in our experiments.",
"For each $r=1,\\ldots ,R$ , we simulate data $(Y_r^n,Y_{r,n+1})$ from $P^\\star $ , and obtain the prediction upper limit $\\widehat{Q}_\\alpha (Y_r^n)$ based on each method.",
"Then the empirical coverage probability is defined as $ \\frac{1}{R} \\sum _{r=1}^R 1\\lbrace \\widehat{Q}_\\alpha (Y_r^n) \\ge Y_{r,n+1}\\rbrace .",
"$ Of course, the empirical coverage probability should be close to the nominal level $1-\\alpha $ .",
"Similarly, we consider a relative interval score $S/S^\\star $ , where $S$ is given by $ S = \\frac{1}{R} \\sum _{r=1}^R \\bigl [ \\widehat{Q}_\\alpha (Y_r^n) + \\alpha ^{-1} \\lbrace Y_{r,n+1} - \\widehat{Q}_\\alpha (Y_r^n)\\rbrace \\, 1\\lbrace Y_{r,n+1} > \\widehat{Q}_\\alpha (Y_r^n)\\rbrace \\bigr ], $ and $S^\\star $ is the same but with the true quantile $Q_\\alpha ^\\star $ of $P^\\star $ : $ S^\\star = \\frac{1}{R} \\sum _{r=1}^R \\bigl [ Q_\\alpha ^\\star + \\alpha ^{-1} ( Y_{r,n+1} - Q_\\alpha ^\\star ) \\, 1\\lbrace Y_{r,n+1} > Q_\\alpha ^\\star \\rbrace \\bigr ].",
"$ Small interval scores are better and, since the true quantile $Q_\\alpha ^\\star $ is “best” in the sense of having the smallest interval score, we expect $S/S^\\star \\ge 1$ and closer to 1 is better.",
"In our experiments we vary $n \\in \\lbrace 100, 200, 400\\rbrace $ and $\\alpha \\in \\lbrace 0.01, 0.05, 0.10\\rbrace $ .",
"The take-away messages here are two-fold.",
"First, as expected, the greater the disparity between the posited model and true distribution, and further out in the tails of the distribution one aims to predict, the performance of a naive method that does not adjust for possible model misspecification gets worse.",
"In particular, the coverage probability of an unadjusted model-based prediction upper bounds can be well below the nominal level.",
"Second, our DP mixture does well overall in terms of coverage, but tends to be less efficient in terms of interval score compared to the GPrC algorithm.",
"Moreover, the GPrC solution can be readily extended to cases beyond the simple iid prediction problems considered here in this section, while the DP mixture formulation is far less straightforward."
],
[
"Pareto data",
"As a first illustration involving skewed, heavy-tailed data, suppose the true distribution $P^\\star $ is Pareto, with distribution function given by $ P^\\star (Y \\le y) = 1 - (1 + y)^{-a}, \\quad y \\ge 0, $ where $a > 0$ is shape parameter that controls the heaviness of the tails.",
"In particular, smaller $a$ means the distribution function approaches 1 more slowly as $y \\rightarrow \\infty $ , hence a heavier tail.",
"So the “degree of misspecification”—of the Pareto with shape parameter $a$ compared to log-normal—is increasing as $a$ decreases.",
"In our experiments, we considered three such degrees of misspecification, namely, $a=4$ , $a=3$ , and $a=2$ .",
"In all cases, the standard Bayesian solution that ignores the potential model misspecification performs poorly, especially so in the case with the highest degree of misspecification.",
"For the two moderate cases, $a \\in \\lbrace 3,4\\rbrace $ , the log-normal + GPrC and the DP mixture methods perform similarly in terms of relative interval score and coverage probability, so we omit the detailed comparisons.",
"We focus here on the case $a=2$ with highest degree of misspecification, which is arguably the most interesting.",
"Figure REF shows the relative interval scores and empirical coverage probabilities of the various $100(1-\\alpha )$ % upper prediction limits, as functions of $n$ , for various $\\alpha $ levels.",
"The take-away message is that both log-normal + GPrC and DP mixture are able to achieve the nominal coverage probability across $n$ and $\\alpha $ , but that the former does so with a slightly better relative interval score, suggesting a benefit in terms of overall interval efficiency.",
"Clearly, the standard Bayes solution that ignores the misspecification is not competitive in this very heavy-tailed situation with a high degree of misspecification.",
"Figure: Summary of prediction limit performance in the Pareto–log-normal example, with a=2a=2.",
"Left column shows the relative interval score; right column shows the coverage probability.",
"Top to bottom, the rows correspond to α=0.10\\alpha =0.10, α=0.05\\alpha =0.05, and α=0.01\\alpha =0.01.",
"Dashed horizontal lines in the right column correspond to two Monte Carlo standard errors around the target coverage probability 1-α1-\\alpha ."
],
[
"Generalized extreme value data",
"For our second illustration involving skewed, heavy-tailed data, suppose $P^\\star $ is a generalized extreme value distribution, with distribution function $ P^\\star (Y \\le y) = \\exp \\bigl [ -\\lbrace 1 + \\xi (y - 2)\\rbrace ^{-1/\\xi } \\bigr ], \\quad y \\ge 2 - \\xi ^{-1}, $ where $\\xi > 0$ is the shape parameter.",
"Like in the Pareto example above, the shape parameter controls the heaviness of the generalized extreme value distribution's tails.",
"However, in this case, heaviness of the tails, or the “degree of misspecification,” is increasing in $\\xi $ .",
"For our experiments we consider $\\xi =0.2$ , $\\xi =0.5$ , and $\\xi =0.7$ .",
"For the moderate degrees of misspecification, namely, $\\xi =0.2$ and $\\xi =0.5$ , both log-normal + GPrC and DP mixture perform comparably, so we omit the detailed results and focus our attention on the most interesting case, $\\xi =0.7$ , corresponding to a very heavy-tailed $P^\\star $ , which makes the thin-tailed log-normal model significantly misspecified.",
"Like above, Figure REF shows the relative interval scores and empirical coverage probabilities of the various $100(1-\\alpha )$ % upper prediction limits, as functions of $n$ , for various $\\alpha $ levels.",
"In this case, at the less extreme $\\alpha $ levels, namely, $\\alpha =0.10$ and $\\alpha =0.05$ , both the Bayes solution that ignores misspecification and the log-normal + GPrC that adjusts for it perform well in terms of interval score and coverage.",
"The DP mixture method appears to be producing too wide of prediction intervals, as indicated by the large interval score.",
"For the extreme quantile, $\\alpha =0.01$ , the separation between the methods becomes more clear and the benefits of GPrC's adjustments to specifically achieve coverage emerge, as we see by its ability to cover within an acceptable range of the nominal 99% rate and have the smallest interval score.",
"Figure: Summary of prediction limit performance in the generalized extreme value–log-normal example, with ξ=0.7\\xi =0.7.",
"Left column shows the relative interval score; right column shows the coverage probability.",
"Top to bottom, the rows correspond to α=0.10\\alpha =0.10, α=0.05\\alpha =0.05, and α=0.01\\alpha =0.01.",
"Dashed horizontal lines in the right column correspond to two Monte Carlo standard errors around the target coverage probability 1-α1-\\alpha ."
],
[
"Calibration in regression problems",
"Here we extend the GPrC methodology to cases where the $Y_i$ 's are accompanied by predictor variables $X_i \\in \\mathbb {R}^d$ , for some $d \\ge 1$ .",
"Here we focus on the setting in which the pairs $D_i = (X_i, Y_i) \\in \\mathbb {R}^{d+1}$ are iid; the setting in which the predictors are non-random can be handled similarly, and is discussed briefly at the end of this section.",
"Let $P^\\star $ denote the true distribution of $D=(X,Y)$ , and suppose that $D_i = (X_i,Y_i)$ are iid copies of $D$ , for $i=1,\\ldots ,n$ .",
"The data analyst would typically opt to model only the conditional distribution of $Y$ , given $X=x$ , by a distribution $P_{\\theta ;x}$ , with a density $p_\\theta (\\cdot \\mid x)$ , which depends on a model parameter $\\theta \\in \\Theta $ .",
"The most common example of this, which we will adopt here, is the textbook linear regression model with $ P_{\\theta ; x} = {\\sf N}\\bigl ( g_\\beta (x), \\sigma ^2 \\bigr ), \\quad \\theta = (\\beta ,\\sigma ^2) \\in \\mathbb {R}^{q+1}, $ where $g_\\beta : \\mathbb {R}^d \\rightarrow \\mathbb {R}$ belongs to a given parametric class of functions indexed by $\\beta \\in \\mathbb {R}^q$ ; the common linear model corresponds to $g_\\beta (x) = x^\\top \\beta $ .",
"This choice to model only the conditional distribution is equivalent to assuming a joint model for $(X,Y)$ but assuming the marginal distribution for $X$ is known and does not depend on $\\theta $ .",
"In any case, a Bayesian approach can be carried out, which leads to a posterior distribution, $\\Pi _n$ , of $\\theta $ that depends on data $D^n$ .",
"The presence of the predictor variables makes the predictive distribution in this case is slightly different from before.",
"Indeed, the $\\eta $ -generalized predictive distribution here is $ f_n^{(\\eta )}(y \\mid x) \\propto \\int p_\\theta ^\\eta (y \\mid x) \\, \\Pi _n(d\\theta ).",
"$ That is, our prediction of $Y_{n+1}$ depends on a value of the associated $X_{n+1}$ , which is available to the data analyst at the time prediction is to be carried out.",
"From this $\\eta $ -generalized predictive distribution comes an upper-$\\alpha $ quantile, which we will denote by $Q_\\alpha (\\eta ; x, D^n)$ , which makes its dependence on the value $x$ of the predictor variable explicit.",
"Define the coverage probability as $ c_\\alpha (\\eta ) = P^\\star \\lbrace Q_\\alpha (\\eta ; X_{n+1}, D^n) \\ge Y_{n+1}\\rbrace , $ and note that this probability is taken with respect to the true joint distribution of $D^{n+1} = \\lbrace (X_i,Y_i): i=1,\\ldots ,n+1\\rbrace $ under $P^\\star $ .",
"The GPrC algorithm can be applied here in almost exactly the same way as in the basic iid setup in Section REF .",
"That is, take $B$ many bootstrap samples $D_b^n$ from $D^n$ , which amounts to $B$ distinct samples of pairs $(X_i,Y_i)$ with replacement from the original sample—this is the so-called paired bootstrap [12], [16], [15], [44].",
"Then approximate the above coverage probability by $ \\hat{c}_\\alpha (\\eta ) = \\frac{1}{B} \\sum _{b=1}^B \\Bigl [ \\frac{1}{n} \\sum _{i=1}^n 1\\lbrace Q_\\alpha (\\eta ; X_i, D_b^n) \\ge Y_i\\rbrace \\Bigr ], $ where each term in the inner sum is based on applying the $\\eta $ -generalized predictive distribution to predict $Y_i$ at the given $X_i$ .",
"Note that this formula does not actually use the value $X_{n+1}$ .",
"That value would, however, be used for forming the actual predictive distribution, $f_n^{(\\hat{\\eta })}(y \\mid X_{n+1})$ and associated quantile $Q_\\alpha (\\hat{\\eta }; X_{n+1}, D^n)$ that would be used to predict $Y_{n+1}$ in the real application.",
"We use $Q_\\alpha (\\hat{\\eta }; D^n)$ in our simulations below to check the coverage probability of the proposed GPrC-based method.",
"Here we consider linear regression model misspecification in terms of the distribution of the error terms.",
"In particular, we consider two skewed error distributions, namely ${\\sf ChiSq}(2)$ and generalized extreme value distribution with shape $\\xi =0.5$ , both centered to have mean zero.",
"Note that the latter is both skewed and heavy-tailed.",
"The rows of the $X$ matrix is sampled from a mean-zero multivariate normal distribution with a unit variance and first-order autoregressive structure, i.e., $\\mathsf {E}(x_{ij} x_{ik}) = \\rho ^{|j-k|}$ , with correlation $\\rho =0.5$ .",
"We also use $\\beta =(2,2,2,2,2)\\top $ to sample $Y_i$ , and compare the GPrC results with the plug-in predictive interval using maximum likelihood estimator with a standard normal error distribution.",
"Empirical coverage probabilities are presented in Tables REF –REF .",
"Both GPrC and plug-in methods perform well in terms of coverage at $\\alpha =0.10$ , so these details are not shown.",
"At the more extreme $\\alpha $ levels, especially $\\alpha =0.01$ , the plug-in method's predictive intervals are too narrow to cover within an acceptable range of the 99% target, even with a relatively large sample size.",
"Table: Empirical coverage probability of the corresponding η ^\\hat{\\eta }-generalized predictive distribution quantiles, at two confidence levels α∈{0.05,0.01}\\alpha \\in \\lbrace 0.05, 0.01\\rbrace , in the centered chi-square error distribution example, based on 1000 replications.Table: Empirical coverage probability of the corresponding η ^\\hat{\\eta }-generalized predictive distribution quantiles, at confidence levels α=0.01\\alpha =0.01, in the centered generalized extreme value error distribution example, based on 1000 replications.Regression problems involving fixed/non-random covariates require a slightly different formulation.",
"Suppose that $Y^n = (Y_1,\\ldots ,Y_n)$ consists of independent observations, where $Y_i$ has an $i$ -specific marginal distribution $P_{\\theta ; x_i}$ , depending on a fixed covariate $x_i$ .",
"The most common example of this is in a designed study where the $Y_i$ measurements are taken under pre-determined experimental settings.",
"Then the goal is to predict $Y_{n+1}$ under another pre-determined setting $x_{n+1}$ .",
"The same construction of an $\\eta $ -generalized predictive distribution described above can be applied here.",
"The only difference in the GPrC formulation comes in the bootstrap approximation of the coverage probability: in this case, the paired bootstrap is replaced by the residual boostrap [13]."
],
[
"Time series",
"Here we consider the problem of calibrating predictive distributions when the data are dependent.",
"We start here with the simplest case of time series—or temporally dependent—data; the next section considers spatially dependent data.",
"Suppose we have data $Y^n = (Y_1,\\ldots ,Y_n)$ and the goal is to predict $Y_{n+1}$ .",
"Naturally, if the data are believed to be temporally dependent, then that information could be used to improve prediction.",
"However, developing a sound model for dependent data can be a challenge and model-based inference/prediction could be severely biased when based on a misspecified model.",
"Ideally, the data analyst could work with a relatively simple model for temporally dependent data and, if necessary, the data would suggest when some adjustments might help to accommodate model misspecification.",
"This is what the GPrC algorithm aims to provide.",
"For the process $Y_1,Y_2,\\ldots $ , let $P^\\star $ denote the true distribution and $P_\\theta $ a posited model; note the slight abuse of notation letting, e.g., $P^\\star $ , denote here the full joint distribution whereas $P^\\star $ stood for the marginal distribution of an individual $Y_i$ in the previous sections.",
"For the model $P_\\theta $ , we will be considering a simple, Gaussian, first-order autoregressive process that posits $ (Y_{i+1} \\mid Y_i=y^{\\prime }) \\sim {\\sf N}(\\rho y^{\\prime }, \\sigma ^2), \\quad i=0,1,\\ldots , $ where $\\theta =(\\rho ,\\sigma ) \\in (-1,1) \\times (0,\\infty )$ .",
"Of course, other more sophisticated models are possible; we opt for a simple and concrete model here to showcase the GPrC algorithm's ability to overcome model misspecification biases.",
"As before, given the posited model and the observed data $Y^n=(Y_1,\\ldots ,Y_n)$ , one can carry out a Bayesian analysis that leads to a posterior distribution, $\\Pi _n$ , for $\\theta $ .",
"Given that the model assumes a Markov or one-step temporal dependence, the $\\eta $ -generalized predictive distribution in this case has a density of the form $ f_n^{(\\eta )}(y \\mid y^{\\prime }) \\propto \\int p_\\theta ^\\eta (y \\mid y^{\\prime }) \\, \\Pi _n(d\\theta ), $ where $p_\\theta (\\cdot \\mid y^{\\prime })$ is the ${\\sf N}(\\rho y^{\\prime }, \\sigma ^2)$ density.",
"Of course, this predictive density has quantiles, and we denote the upper-$\\alpha $ quantile by $ Q_\\alpha (\\eta ; Y_n, Y^n), \\quad \\alpha \\in (0,1).",
"$ Note here the dependence on $Y_n$ in two places: first, as a component in the data $Y^n$ and, second, in that the predictive density for $Y_{n+1}$ depends explicitly on $Y_n$ .",
"For the GPrC algorithm, all that is left to specify is the coverage probability and a bootstrap approximation.",
"Of course, the coverage probability function is $ c_\\alpha (\\eta ) = P^\\star \\lbrace Q_\\alpha (\\eta ; Y_n, Y^n) \\ge Y_{n+1}\\rbrace , $ where the probability is with respect to the joint distribution of $Y^{n+1}$ determined by $P^\\star $ .",
"For a bootstrap approximation, it is important that the particular choice of bootstrap respects the temporal dependence.",
"To achieve this, we apply the block bootstrap strategy proposed by [31]; see, also, [43] and [10].",
"The basic idea behind the block boostrap is, as the name suggests, to resample blocks of observations with the goal of retaining the temporal dependence structure.",
"The version we employ here in our illustration is as follows.",
"Select a fix block length parameter $\\ell < n$ such that the number of blocks of a time series, $k=n/\\ell $ , is an integer.",
"For each bootstrap sample, we construct the bootstrap sample $Y_b^n$ by concatenating the results of sampling $k$ overlapping blocks of subsequences $ \\lbrace (Y_{s_j+1},\\ldots , Y_{s_j+\\ell }): j=1,\\ldots ,k\\rbrace , $ where $\\lbrace s_1,\\dots ,s_k\\rbrace $ are the starting points of each block, and are generated from a discrete uniform distribution on $\\lbrace 1,2,\\dots ,n-l+1\\rbrace $ .",
"Here we set the block length $\\ell \\sim n^{1/3}$ according to the recommendations in [4]; see, also [19] and [5].",
"Given the block bootstrap samples $Y_b^n$ for $b=1,\\ldots ,B$ , the empirical coverage probability is approximated as $\\hat{c}_\\alpha (\\eta ) = \\frac{1}{B} \\sum _{b=1}^B \\Bigl [ \\frac{1}{n-1} \\sum _{i=1}^{n-1} 1\\lbrace Q_\\alpha (\\eta ; Y_i, Y_b^n) \\ge Y_{i+1}\\rbrace \\Bigr ].$ Note that the inner sum in (REF ) ranges over $i=1,\\ldots ,n-1$ , covering all the consecutive pairs $(Y_i,Y_{i+1})$ in the data, so $Y_n$ is never used as a direct argument in the $Q_\\alpha (\\eta ; \\cdot , Y_b^n)$ function.",
"However, after $\\hat{\\eta }$ is determined and it is time to predict $Y_{n+1}$ , we would take $Q_\\alpha (\\hat{\\eta }; Y_n, Y^n)$ as our $100(1-\\alpha )$ % upper limit, with $Y_n$ plugged in.",
"More generally, if the model posited lag-$m$ dependence, then the inner sum in (REF ) would range over $i=1,\\ldots ,n-m$ , covering all consecutive $m$ -tuples $(Y_i,\\ldots ,Y_{i+m-1})$ .",
"Next, we investigate the performance of the GPrC-modified predictive distribution in in three different simulation scenarios: First-order autoregressive with Laplace errors, i.e., $Y_{i+1} = 0.9Y_i + \\varepsilon _i$ , for $i=1,\\ldots ,n-1$ , where the $\\varepsilon _i$ 's are iid ${\\sf Lap}(0,1)$ ; Nonlinear time series with Laplace errors, i.e., $Y_{i+1} = \\sin (Y_i) + \\varepsilon _i$ , for $i=1,\\ldots ,n-1$ , where the $\\varepsilon _i$ 's are iid ${\\sf Lap}(0,1)$ ; Nonlinear time series with heteroscedastic Laplace errors, i.e., $Y_{i+1} = \\sin (Y_i) + (0.5+0.25Y_i^2)^{1/2}\\varepsilon _i$ , for $i=1,\\ldots ,n-1$ , where the $\\varepsilon _i$ 's are iid ${\\sf Lap}(0,1)$ ; The first scenario is one where the temporal dependence structure is correctly specified but the error distribution has heavier-than-normal tails.",
"The second is one where both the temporal dependence structure and the error distribution of the posited model are misspecified.",
"Finally, the third scenario is one where the temporal dependence structure of the posited model is misspecified, and the distribution of the errors is heteroskedastic with heavy-tailed distribution.",
"The choice of “0.90” in the first scenario ensures that there is relatively strong temporal dependence in the true data-generating process.",
"For comparison, we consider two standard methods.",
"The first method, which we call the plug-in method, is quite basic and is a natural choice when the auto-regressive model is assumed to be correct.",
"That is, the plug-in method produces maximum likelihood estimates, $\\hat{\\theta }= (\\hat{\\rho }, \\hat{\\sigma })$ , for the model parameter $\\theta =(\\rho ,\\sigma )$ and returns the $100(1-\\alpha )$ % prediction limit for $Y_{n+1}$ as $\\hat{\\rho }Y_n + \\Phi ^{-1}(1-\\alpha ) \\, \\hat{\\sigma }$ .",
"Of course, if the model is correctly specified, then this would be approximately calibrated.",
"The second method is the proposed GPrC algorithm, that starts with a simple Bayesian model and then uses the data to tune $\\eta $ as described above.",
"In particular, we take a conjugate normal–inverse gamma to get the Bayesian posterior.",
"Table REF summarizes the coverage probability for the two different methods, with $n \\in \\lbrace 100,200,400\\rbrace $ at level $\\alpha =0.01$ .",
"The plug-in method under covers at this extreme $\\alpha $ level, whereas the GPrC method is close to the target coverage even with a sample size as low as $n = 100$ .",
"Table: Empirical coverage probability for the two methods' predictive distribution quantiles, at level α=0.01\\alpha =0.01, for the three time series model scenarios described in the text, based on 1000 replications."
],
[
"Spatial data",
"Let $\\mathcal {S}\\subseteq \\mathbb {R}^2$ denote a spatial region on which a stochastic process $Y = \\lbrace Y(s): s \\in \\mathcal {S}\\rbrace $ is defined.",
"For example, $\\mathcal {S}$ may denote a set of geographical locations (expressed in some coordinate system) and $Y(s)$ denotes the temperature, precipitation level, etc.",
"at location $s \\in \\mathcal {S}$ .",
"Data consists of a finite collection $s^n = (s_1,\\ldots ,s_n)$ of locations at which observations are made, along with the vector of $Y$ -process measurements $ Y(s^n) = \\bigl ( Y(s_1), \\ldots , Y(s_n) \\bigr ).",
"$ The goal is to predict the value of $Y(s_{n+1})$ at a new spatial location $s_{n+1} \\in \\mathcal {S}$ .",
"Model-based predictions are common in spatial applications [50].",
"A model $P_\\theta $ would consist of certain assumptions about the distribution of the stochastic process $Y$ .",
"A common choice is to model $Y$ as a Gaussian process and the parameter $\\theta $ characterizes its mean and covariance functions.",
"To make this characterization relatively simple, it is tempting to make rather strong assumptions, e.g., stationarity and/or isotropy, in addition to Gaussianity.",
"Such assumptions can be hard to justify, so working with such a simple model opens the data analyst up to risk of model misspecification bias.",
"Therefore, it would be interesting to see if a GPrC adjustment on top of a simple model could calibrate predictions and alleviate some of the data analyst's risk.",
"In particular, we consider a simple Gaussian process model [14] for $Y$ , generically denoted by $P_\\theta $ , that posits a constant mean function $E\\lbrace Y(s)\\rbrace \\equiv \\mu $ an exponential covariance function $\\theta (s,t) = \\sigma ^2 \\bigl ( e^{-\\Vert s-t\\Vert /\\hat{\\phi }} + \\hat{\\tau }1\\lbrace s=t\\rbrace \\bigr ), \\quad s,t \\in \\mathcal {S},$ where $\\theta =(\\mu , \\sigma ^2)$ is treated as the unknown model parameter, while (variogram-based) plug-in estimates $\\hat{\\phi }$ and $\\hat{\\tau }$ of the range and scaled nugget parameters are treated as fixed and known.",
"(Some applications might have covariates that could be incorporated into the model's mean function, but we will not consider this here.)",
"Different from our previous sections with little or no dependence built into the model, here the model itself involves non-negligible dependence.",
"So the in-model conditional density function for $Y(s_{n+1})$ , given $s_{n+1}$ and data $D^n=\\lbrace s^n, Y(s^n)\\rbrace $ , is normal with $m_\\theta (s_{n+1}) & = \\mu + \\gamma _\\theta (s^{n+1})^\\top \\Gamma _\\theta ^{-1}(s^n) \\lbrace Y(s^n) - \\mu 1_n\\rbrace \\\\v_\\theta (s_{n+1}) & = C_\\theta (s_{n+1}, s_{n+1}) - \\gamma _\\theta (s^{n+1})^\\top \\Gamma _\\theta ^{-1}(s^n) \\gamma _\\theta (s^{n+1}),$ where $1_n$ is an $n$ -vector of unity, $\\gamma _\\theta (s^{n+1})$ is an $n$ -vector with $i^\\text{th}$ entry $C_\\theta (s_i, s_{n+1})$ , and $\\Gamma _\\theta (s^n)$ is an $n \\times n$ matrix with $(i,j)^\\text{th}$ entry $C_\\theta (s_i, s_j)$ .",
"Obviously, the density function of the aforementioned conditional distribution depends on the unknown $\\theta $ , so we denote this by $p_\\theta (\\cdot \\mid s_{n+1}, D^n)$ .",
"We complete the Bayesian model formulation by introducing a conjugate normal–inverse gamma prior for $\\theta $ , from which we immediately obtain a posterior distribution $\\Pi _n$ for $\\theta $ depending on data $D^n$ .",
"Then we define our $\\eta $ -generalized predictive density for $Y(s_{n+1})$ , given $s_{n+1}$ and data $D^n$ , as $ f_n^{(\\eta )}(y \\mid s_{n+1}) \\propto \\int p_\\theta ^\\eta (y \\mid s_{n+1}, D^n) \\, \\Pi _n(d\\theta ).",
"$ The GPrC method aims to use information in the available data to tune the learning rate $\\eta $ in order to calibrate this predictive distribution in the event that the simple conjugate Gaussian process model is misspecified.",
"Of course, the GPrC algorithm relies on the bootstrap.",
"Since these spatial data models are more complex than those in previous sections, naturally an appropriate bootstrap procedure will be similarly more complex.",
"The procedure we consider here is the so-called semi-parametric bootstrap procedure developed in [49].",
"This procedure is “semi-parametric” in the sense that it assumes the model's mean and covariance functions are correctly specified, but does not rely on any distributional assumptions, such as Gaussianity.",
"Start by obtaining suitable estimates $\\hat{\\theta }$ of the model parameter $\\theta $ ; the scale $\\sigma ^2$ is typically estimated through a sample variogram, while $\\mu $ can be estimated using a simple average of the observed $Y(s^n)$ 's.",
"Next, define the $n$ -vector of residuals $ \\varepsilon (s^n) = \\Gamma _{\\hat{\\theta }}^{-1/2}(s^n) \\lbrace Y(s^n) - \\hat{\\mu }1_n\\rbrace .",
"$ The idea is that the residuals are roughly free of any spatial dependence, i.e., that they are approximately iid.",
"Let $B$ be the desired number of bootstrap samples.",
"Now, for $b=1,\\ldots ,B$ , take a sample of size $n+1$, with replacement from the $n$ -vector of residuals and denote these as $ \\varepsilon _b(s^{n+1}) = \\bigl ( \\varepsilon _b(s_1), \\ldots , \\varepsilon _b(s_n), \\varepsilon _b(s_{n+1}) \\bigr )^\\top , \\quad b=1,\\ldots ,B.",
"$ Then we just add back the mean and spatial dependence in the natural way, $ Y_b(s^{n+1}) = \\hat{\\mu }1_n + \\Gamma _{\\hat{\\theta }}^{1/2}(s^{n+1}) \\, \\varepsilon _b(s^{n+1}), \\quad b=1,\\ldots ,B, $ where $ \\Gamma _\\theta (s^{n+1}) = \\begin{pmatrix} \\Gamma _\\theta (s^n) & \\gamma _\\theta (s^{n+1}) \\\\ \\gamma _\\theta (s^{n+1})^\\top & C_{\\hat{\\theta }}(s_{n+1}, s_{n+1})\\end{pmatrix} $ Finally, at a candidate $\\eta $ , and for each $b=1,\\ldots ,B$ , the GPrC will use the first $n$ components, $Y_b(s^n)$ , of $Y_b(s^{n+1})$ , to construct the generalized predictive distribution and extract the quantile $Q_\\alpha (\\eta ; s_{n+1}, Y_b(s^n))$ .",
"And then the coverage probability is approximated as $ \\hat{c}_\\alpha (\\eta ) = \\frac{1}{B} \\sum _{b=1}^B 1\\lbrace Q_\\alpha (\\eta ; s_{n+1}, Y_b(s^n)) \\ni Y_b(s_{n+1})\\rbrace , $ where, note, the predictive distribution quantile is tested against the corresponding last entry, $Y_b(s_{n+1})$ , of $Y_b(s^{n+1})$ .",
"This process is repeated on the sequence of $\\eta $ values as defined the GPrC algorithm until convergence to some $\\hat{\\eta }$ .",
"Other versions of the spatial bootstrap are possible.",
"For example, if the model's mean function were non-constant, say a parametric function of the spatial coordinates and/or covariates, then this could be readily accommodated by adjusting how the residuals above are calculated.",
"Different variations on the bootstrap procedure itself are also available [7] whose use in the GPrC algorithm will be explored in subsequent work.",
"To investigate the performance of our proposed GPrC for prediction in this spatial data context, we consider three different data-generating processes, as described below; the first two scenarios were investigated in [49], while the third is from [48].",
"For each scenario, the $n$ spatial locations $s^n$ at which the $Y$ process is measured are uniformly distributed in a disc of radius $r = 20$ around $(0, 0)$ , and the target location is set at $s_{n+1} = (0, 0)$ .",
"A (correctly specified) Gaussian process model as described above, with $\\mu ^\\star =0$ , $\\sigma ^\\star =1$ , $\\phi ^\\star =3$ , and $\\tau ^\\star =0$ .",
"Following [49], let $L_{n+1}$ denote a vector of iid ${\\sf logN}(0,1)$ random variables and define the response $Y$ as $ Y(s^{n+1}) = \\log \\Bigl [ \\Gamma _\\theta ^{1/2} \\bigl \\lbrace L_{n+1} / (e^2-e^1)^{1/2} \\bigr \\rbrace \\Bigr ], $ where $\\mu ^\\star =0$ , $\\sigma ^\\star =1$ , $\\phi ^\\star =3$ , and $\\tau ^\\star =0$ , with log taken component-wise.",
"Following [48], let the response $Y$ be defined through the following hierarchical generalized extreme value process.",
"Start with two independent Gaussian processes, say, $Z_1$ and $Z_2$ , with covariance functions having the same exponential form as in (REF ), with associated covariance parameters $(\\mu _1, \\sigma _1, \\phi _1, \\tau _1) = (10,1,4,0)$ and $(\\mu _2, \\sigma _2, \\phi _2, \\tau _2) = (0,1, 1.4, 0)$ .",
"Let $\\Phi $ and $G$ denote the distribution functions of the standard normal and standard Fréchet distributions, respectively, and then the define the process $W$ as $ W(s) = Z_1(s) + \\sigma \\xi ^{-1} \\bigl ( [G^{-1} \\circ \\Phi \\lbrace Z_2(s)\\rbrace ]^\\xi - 1 \\bigr ), $ where $(\\sigma , \\xi ) = (3, 0.5)$ .",
"Like the log-normal setup in Scenario 2 above, since the mean $\\mu _1$ of $Z_1$ is rather large, it is virtually impossible for $W(\\cdot )$ to be negative.",
"Therefore, we take the response process as $Y(s) = \\log W(s)$ .",
"We compare the performance of GPrC with that of two alternative approaches.",
"One is a simple plug-in approach that starts with variogram-based estimates $\\hat{\\theta }$ of the basic model parameters and constructs an upper prediction limit for the response at a new location $s_{n+1}$ as $ \\widehat{Q}_\\alpha (D^n) = m_{\\hat{\\theta }}(s_{n+1}) + \\Phi ^{-1}(1-\\alpha ) \\, \\lbrace v_{\\hat{\\theta }}(s_{n+1}) \\rbrace ^{1/2}.",
"$ Of course, if the process is Gaussian, then this would be an approximately valid $100(1-\\alpha )$ % prediction upper limit.",
"However, if the proposed Gaussian process model is misspecified, the plugin predictive interval can result in a narrower predictive interval.",
"Second, we consider a more robust semi-parametric bootstrap method based directly on the output from the bootstrap procedure in [49].",
"That is, we let $\\widehat{Q}_\\alpha (D^n)$ be the upper-$\\alpha $ quantile of the empirical distribution of $\\lbrace Y_b(s_{n+1}): b=1,\\ldots ,B\\rbrace $ .",
"The three prediction limits are compared based on coverage probability and relative interval scores as described in Section REF .",
"The results are presented in Table REF .",
"Table: Empirical coverage probabilities and relative interval scores based on 1000 data sets of size n=100n=100, for each of the three scenarios, in the spatial data example.In Scenario 1, the correctly specified Gaussian process, all three methods perform well in terms of coverage and interval score, as expected.",
"Perhaps GPrC has a slight advantage over bootstrap in terms of interval score at the extreme 99% quantile, but both are comparable to plug-in method which would be (at least close to) optimal in this correctly specified model setting.",
"In Scenarios 2 and 3, both where the Gaussian process model is misspecified, the plug-in method suffers in terms of under-coverage, as expected.",
"Surprisingly, both GPrC and the bootstrap are able to overcome the model misspecification and reach nearly the target coverage across the board, while maintaining some amount of efficiency as indicated by the interval scores.",
"In particular, although GPrC is making use of the semi-parametric bootstrap to approximate coverage probabilities, its predictions are still effectively model-based.",
"So the fact that calibration can be (approximately) achieved using a model-based Bayesian method in a challenging application under fairly severe model misspecification is remarkable."
],
[
"Conclusion",
"Following up on recent work that tunes the learning rate parameter in generalized posterior distributions, in this paper we developed a procedure to calibrate generalized predictive distributions.",
"The idea is the upper-$\\alpha $ quantiles of one's (subjective) predictive distribution ought to have an alternative (objective) interpretation as a valid, $100(1-\\alpha )$ % prediction upper limit.",
"Our proposal—the GPrC algorithm—is to construct a generalized predictive distribution that depends on a tuning parameter $\\eta $ , approximate the coverage probability of its $\\eta $ -dependent prediction upper limit, and then tune $\\eta $ in order to match the target coverage.",
"The key step is the coverage probability approximation, which we do via bootstrap, and we demonstrated numerically that calibration can be achieved via the proposed GPrC algorithm in a variety of settings using appropriate bootstrap procedures.",
"In Section we presented a combination of heuristic and numerical arguments to support the claim that GPrC does, indeed, achieve calibration, at least approximately.",
"The challenge to providing a rigorous mathematical proof of this conjecture is that the combination of at least two powerful computational tools—bootstrap, stochastic approximation, and sometimes Monte Carlo sampling—each theoretically sound on its own, adds a level of complexity that makes the GPrC algorithm's dynamics very difficult to analyze.",
"The same is true for the GPC algorithm of [51], so further theoretical understanding of how these methods work is an interesting open problem.",
"The focus of this paper was on developing a general method for calibrating Bayesian-like predictive distributions under model misspecification and demonstrating that this method can be used in a relatively wide range of applications.",
"It would be interesting to investigate a particular application, e.g., in the spatial domain, to tailor GPrC to that specific application, and push the limits of how complex the models can be while still achieving approximate calibration."
],
[
"Acknowledgments",
"The authors thank Nicholas Syring for helpful comments on an initial draft.",
"This work is partially supported by the U.S. National Science Foundation, grants DMS–1811802 and SES–2051225."
]
] | 2107.01688 |
[
[
"Influence of rotation on axisymmetric plasma equilibria: double-null DTT\n scenario"
],
[
"Abstract We study the dependence of some relevant tokamak equilibrium quantities on the toroidal plasma rotation.",
"The Grad-Shafranov equation generalised to the rotating case is analytically solved employing two different representations for the homogenous solution.",
"Using an expression in terms of polynomials, we describe the separatrix shape by a few geometrical parameters, reproducing different plasma scenarios such as double-null and inverse triangularity.",
"In this setting, the introduction of toroidal rotation corresponds to variations on relevant plasma quantities, most notably an enhancement of the poloidal beta.",
"Using a more general expression in terms of Bessel functions, we reconstruct the full plasma boundary of the double-null configuration proposed for the upcoming DTT experiment, demonstrating how said configuration is compatible with different values of the plasma velocity."
],
[
"introduction",
"The basic concept at the ground of any operational regime of a tokamak device [1] is the theoretical existence of an axisymmetric plasma equilibrium [2].",
"In a real machine, this equilibrium can exist for a time which is inherently limited by the duration of the discharge.",
"The duration is usually much longer than the characteristic timescale on which magnetohydrodynamic instabilities develop, leading to abrupt losses of confinement, and much shorter than dissipation timescales due to resistivity or other non ideal effects, leading to slow losses of confinement.",
"The description of a tokamak equilibrium is based on the balance of the ideal MHD forces, i.e., pressure gradients versus magnetic pressure and tension, resulting in the well-known Grad-Shafranov equation (GSE) [3], in which the presence of steady matter flux is neglected.",
"This assumption can be motivated by the conditions of operation of tokamak machines, which discharge is, in general, associated to a flux-free quasi-ideal plasma.",
"Nonetheless, the emergence of a spontaneous rotation in Tokamak devices has been observed since the early nineties [4], [5], both in the toroidal and poloidal directions.",
"Many proposals have been argued in order explain this phenomenon, which can be interpreted as a result of a self–organization of the plasma in the transition from turbulent to laminar flow.",
"Indeed, the transition between turbulent and laminar regimes is an interchange phenomenon, due to the unavoidable linear and nonlinear instability of the rotating plasma.",
"Another important operation condition of a tokamak leading to important rotation profiles is the heating of the plasma via hot neutral beam injection: the beam injected in the tangential direction, tranferring angular momentum into the plasma, can trigger rotation flows inside the configuration [6].",
"According to these considerations, the inclusion of rotation in the computation of a tokamak equilibrium is a relevant topic that may require increasing attention in the years to come.",
"The theory of rotating tokamak equilibria has been developed by many authors (e.g.",
"see [7], [8] and citing articles), with most studies mainly relying on the introduction of a Bernoulli–like function (as in traditional fluid dynamics) in order to generalize the GSE to a plasma with flow, while keeping its mathematical form mostly intact [9].",
"Here, we investigate the case of a tokamak equilibrium in the presence of a toroidal velocity field and we address its description through the introduction of a generalized pressure function, as in [7].",
"We first construct simple semi–analytical solutions of the obtained equation, generalizing the well-known Solov'ev scenario [10].",
"Then, we implement our model to study how the double–null configuration at $5\\,$ MA of the DTT Italian proposal [11] is modified by the presence of toroidal rotation.",
"We are able to characterize the dependence of some basic plasma quantities on the toroidal velocity, such as the poloidal beta $\\beta _\\text{pol}$ , the plasma current $I_\\text{p}$ , the profile of the safety factor $q$ , the position of the magnetic axis and the morphology of the separatrix with respect to the isobar surface at zero pressure, taken as the plasma boundary.",
"Clearly, the introduction of a toroidal rotational field in a Tokamak equilibrium can also be studied via a performing numerical code, see for instance [12].",
"However, our semi–analytical study, based on the construction of a generic solution for a linear GSE, is a powerful tool to establish precise relations among the model parameters.",
"The choice of a linear equilibrium allows to individualize the basic eigenfunctons of the configurational problem and it is justified by the expansion of the unkwnon functions of the magnetic flux function up to the lowest order of approximation.",
"In this respect, our correlation between the parameter governing the rotation intensity and the $\\beta $ value of the plasma must be regarded as a general feature of the considered family of plasma configurations.",
"The manuscript is structured as follows.",
"In section , we recall the fundamental equations from the known literature, we outline the basis for our study introducing the necessary assumptions, and we provide a convenient form for the particular solution of the equilibrium.",
"In section , we solve the homogeneous problem using a purely polynomial expansion of the magnetic flux function $\\psi $ .",
"We show how this simple solution is able to represent different plasma scenarios, characterized only by few constraints, and what is the impact of plasma rotation on the equilibrium properties.",
"In section , we provide a different, more general solution to the homogeneous problem, which allows for a more precise determination of the plasma separatrix while still mantaining a flexible fitting procedure.",
"We study the specific case of the DTT double-null scenario, illustrating the fitting procedure and the impact of rotation on some relevant equilibrium properties.",
"Concluding remarks follow."
],
[
"basic equations",
"The equilibrium of magnetically confined plasmas can be described by few basic equations: (v)v = -P+JB , B = 0 , 0J = B , which express the conservation of the momentum of a charged fluid with density $\\rho $ , velocity field $\\mathbf {v}$ and pressure $P$ in the presence of self-consistent current density $\\mathbf {J}$ and magnetic field $\\mathbf {B}$ .",
"Working in cylindrical coordinates $(R,\\phi ,Z)$ and assuming axisymmetry, i.e., $\\partial _\\phi f=0$ for any quantity $f$ , the magnetic field can be expressed as $\\mathbf {B}=_0I{\\phi }+{\\psi }\\times {\\phi }$ , in terms of the two scalar functions $\\psi $ and $I$ , which are related to the magnetic flux and toroidal magnetic field inside the plasma, respectively.",
"Furthermore in the static case, $\\mathbf {v}=0$ , the equilibrium problem reduces to the well-known Grad-Shafranov equation: $\\Delta ^*{\\psi } = -_0^2 II^{\\prime } -_0 P^{\\prime } R^2\\,,$ where $\\Delta ^*{}\\equiv \\partial _R^2-\\partial _R/R+\\partial _Z^2$ , and the prime denotes differentiation of the arbitrary functions $I$ and $P$ with respect to $\\psi $ , the fundamental degree of freedom of the system.",
"The solutions of this equation for a confined plasma correspond to nested tori of constant $\\psi $ , and have been extensively studied in the literature [refs].",
"Earliest analytical studies focus on the Solov'ev scenario, in which the right-hand side of the equation is made independent on $\\psi $ by the assumptions: 0 P' = S1 , 02II' = S2 , * = -S2 -S1 R2 , with $S_{1,2}=const$ .",
"Other choices can be made while still preserving the linearity of the equation, like quadratic source function scenario with $P^{\\prime },II^{\\prime }\\sim \\psi $ , or the dissimilar source function scenario, with $P^{\\prime } \\sim const.,\\,II^{\\prime }\\sim \\psi $ .",
"The analysis is more subtle in the case of a plasma configuration rotating in the toroidal direction with velocity $\\mathbf {v} = \\omega R^2 {\\phi }$ .",
"In this case Eq.",
"(REF ) can be generalised as $\\left(\\Delta ^*{\\psi }+_0^2 II^{\\prime }\\right){\\psi }=-_0 R^2\\left({P}+ \\rho R \\omega ^2 {R}\\right)\\,,$ where two difficulties arise: the plasma density enters the equilibrium balance, and the pressure is no longer a pure function of $\\psi $ .",
"However, it is clear from ideal Ohm's law, $\\mathbf {E}+\\mathbf {v}\\times \\mathbf {B}=0$ , combined with stationary Faraday's law, ${E}=0$ , that the rotation frequency $\\omega $ is a new surface function, a result also known as corotation theorem [13].",
"The set of equations must be closed introducing an equation of state for the fluid, with many possible choices [14]; here we consider the ideal gas law $P=\\rho k T$ , where $k$ is the Boltzmann constant divided by the ion mass and $T$ is the plasma temperature, which can be safely assumed to be a surface function in tokamak equilibrium configurations, due to the high parallel transport in these devices.",
"In view of these assumptions, Eq.",
"(REF ) can be rewritten introducing an auxiliary function $\\theta (\\psi )$ as: () kT0-2 R22 , * = -02 II' - 0 R2 [ ' + R2' +(1-0 ) kT' ] .",
"This expression is further simplified defining a generalized pressure $P_\\text{T}(\\psi )=\\rho _0kT\\exp (\\theta /kT)=P \\exp (-\\omega ^2R^2/2kT)$ which is a source function itself, and coincides with the thermodynamic pressure in the $\\omega \\rightarrow 0$ limit [7].",
"Finally we have: $\\Delta ^*{\\psi } = -_0^2 II^{\\prime } - _0 R^2\\left[ P_\\text{T}^{\\prime } + P_\\text{T} R^2 \\left( \\frac{\\omega ^2}{2kT} \\right)^{\\prime }\\right]e^{\\frac{\\omega ^2R^2}{2kT}} \\,,$ which gives the equilibrium of a rotating plasma once the arbitrary functions $I$ , $P_\\text{T}$ , $\\omega $ and $T$ are assigned.",
"Before continuing our analysis, let us introduce the following normalizations: defining the plasma major radius as $R_0$ and the toroidal magnetic field at the major radius as $B_0$ , we normalize length with $R_0$ , magnetic field with $B_0$ , magnetic flux with $B_0R_0^2$ , pressure with $B_0^2/2_0$ , $I$ with $B_0R_0/_0$ and current density with $B_0/_0R_0$ .",
"All quantities are to be meant adimensional from now on, unless stated otherwise."
],
[
"Solov'ev-like configuration",
"Similarly to the Solov'ev assumption in the static scenario, we can make the right-hand side of Eq.",
"(REF ) independent on $\\psi $ by setting $\\frac{P_\\text{T}^{\\prime }}{2} = P_1 \\,, \\quad II^{\\prime } = I_1 \\,, \\quad \\frac{\\omega ^2R_0^2}{2kT}=M^2 \\,,$ where $P_1,I_1,M$ are assumed as contants.",
"The latter expresses the ratio of plasma velocity to thermal velocity at the plasma major radius, and it serves as a parameter to introduce rotation in the equilibrium computation.",
"The resulting expressions for the equilibrium equation and the relevant quantities are: * = -R2 P1 eM2R2 - I1 , PT() = 2P1 , P(,R)=2P1eM2R2 , I() = 2I1+I0 , () = MR02kT() , where $I_0$ is an integration constant introduced to take into account the vacuum toroidal magnetic field.",
"As usual in the theory of linear differential equations, the full solution is given by the sum of a particular solution $\\psi _\\text{P}$ , plus the general homogeneous solution defined by $\\Delta ^*{\\psi _\\text{H}}=0$ .",
"It is easy to verify by substitution that the former can be written as: $\\psi _\\text{P} = \\frac{P_1}{4M^4}\\left[1+M^2R^2-e^{M^2R^2} \\right] -\\frac{I_1}{2}Z^2\\,.$ In this form, we naturally recover the static Solov'ev solution $-P_1R^4/8-I_1Z^2/2$ in the $M\\rightarrow 0$ limit."
],
[
"polynomial solution",
"Concerning the solution of $\\Delta ^*{\\psi _\\text{H}}=0$ , the usual strategy is to employ separation of variables and assume an expression like $\\psi _\\text{H}\\sim f(R)g(Z)$ .",
"The linearity of the equation allows to consider a sum of any number of such terms.",
"For intance, many authors consider polynomials in the $Z$ variable, and in the special case of up-down symmetry, corresponding to even power only, the following representation can be used [15]: $\\psi _\\text{H}=\\sum _{n=0,2,...}\\sum _{k=0}^{n/2} f_{n,k}(R) Z^{n-2k}\\,.$ It can be verified by substitution that the functions $f_{n,k}(R)$ are given recursively by the relations (R2-R/R)fn,0 = 0 , (R2-R/R)fn,k=-(n-2k+1)(n-2k+2)fn,k-1 .",
"This representation of the homogeneous solution in terms of the lowest even powers of $Z$ is suitable for describing up–down symmetric configurations in terms of the minimum number of parameters.",
"For our first analysis, we consider Eq.",
"(REF ) truncated at a maximum index $n=4$ , and further simplified setting some integration constants to 0 to get rid of terms $\\propto \\ln (R)$ , resulting in the following expression: H = C0+C2R2+C4(R4-4R2Z2) +C6(R6-12R4Z2+8R2Z4) +C8( R8-24R6Z2+48R4Z4-64R2Z6/5 ) , where the constants $C_i$ , $i=0,2,4,6,8$ , offer enough freedom to fix the plasma minor radius $a$ , the triangularity $\\delta $ , the elongation $\\kappa $ and the boundary curvature at the outermost point $c$ , via the conditions: (1-a,0) = 0 , (1+a,0) = 0 , (1-a,a) = 0 , R(1-a,a) = 0 , 2Z2/R(1+a,0)=c .",
"Concerning the constants $P_1,I_1,I_0$ contained in Eqs.",
"() and (REF ), we set their values according to the desired plasma poloidal beta $\\beta _\\text{p}$ , current $I_\\text{p}$ and toroidal magnetic field on axis, given by: pol = Pss = P1S R(R,Z) eM2R2B2s , Ip = Js = R P1 eM2R2 + I1R s , B(R0,0) = 2I1(R0,0)+I0R0 .",
"The integrals in the above equations are performed over the confined plasma region inside the magnetic separatrix, defined by $\\psi (R,Z)=0$ and corresponding to the boundary between closed and open magnetic lines.",
"We observe here that, even though in the presence of plasma motion the pressure is not constant on magnetic surfaces, like in the static case, according to Eq.",
"() we still have $P=0$ on the separatrix just defined.",
"Hence the magnetic and matter boundaries of the plasma coincide.",
"The same region can also be described by the points $(R,Z)\\in \\lbrace 1-a,1+a\\rbrace \\times \\lbrace -Z_\\text{m}(R),Z_\\text{m}(R)\\rbrace $ , with $\\psi (R,Z_\\text{m}(R))=0$ .",
"The function $Z_\\text{m}(R)$ can be calculated explicitly, describing the plasma upper boundary (or lower, with minus sign) in terms of $a$ , $\\delta $ , $\\kappa $ , $c$ , $P_1$ , $I_1$ and $M$ , however we omit its cumbersome expression for brevity.",
"Figure: Analytic plasma shape defined as the curve ψ(R,Z)=0\\psi (R,Z)=0, parametrized through the constants aa, δ\\delta , κ\\kappa and cc, for some arbitrary values of the physical constants P 1 P_1, I 1 I_1 and MM.In general, closed–form expressions for the integrals cannot be found, hence we resort to standard numerical recipes for their calculation.",
"In the practical implementation, we find that a simple guess and check strategy leads to satisfying results after a single iteration."
],
[
"Study of rotation influence",
"We note that the rotation velocity is treated as a free parameter so far, via the constant $M$ , with $M=0$ corresponding to the static plasma case.",
"Within this framework, we are able to evaluate its direct impact on the other equilibrium features.",
"In Fig.REF we show the capabilities of solution Eq.",
"() of reproducing plasma shapes with different values of the parameters, as in Table REF .",
"Table: Values of the minor radius aa, triangularity δ\\delta , elongation κ\\kappa and curvature cc for the plasma configurations of Fig..Figure: Contours of constant magnetic flux ψ\\psi (left) and pressure PP (right) in the (R,Z) plane, calculated for the M=0M=0 (solid) and M=0.3M=0.3 (dashed) cases and corresponding to the parameters reported in Table 1.With regard to the second row, the pointy shape of the profile at its top and bottom suggests the presence of x-points.",
"However, it must be noted that the solution used here has not enough free constants to impose the proper null condition on the magnetic field at a desired location; the x-points can only emerge at a certain location, fixed by parameters $\\delta $ and $\\kappa $ , for specific choices of $a$ and $c$ .",
"We will see in the next Section how to address this shortcoming.",
"Fig.REF shows the dependence on $M$ of $\\beta _\\text{pol}$ , $I_\\text{p}$ and $q_{95}$ , the safety factor at 95% plasma volume, normalized to their respective values in the static case.",
"The other parameters used for the fit correspond to the second row scenario of Table 1.",
"While the entity of the variations differ for other choices of the parameters, however, the general qualitative behaviour is consistently that of an enhancement of both $\\beta _\\text{pol}$ and $I_\\text{p}$ , while the safety factor is suppressed.",
"In all the considered cases this has never resulted in breaking the Kruskal-Shafranov stability condition $q>1$ over the whole plasma profile, even for unrealistically high rotation velocities.",
"These results suggest that, when modeling real plasma equilibria using a static analytical solution or numerical code, the errors commited can get increasingly large with plasma rotation.",
"The evaluation of $M$ can thus give quantitative insight on the necessity to employ an exact solution or equilibrium solver which take the plasma rotation into account.",
"Figure: Variation of poloidal beta, plasma current and safety factor at 95% plasma volume, with respect to the parameter MM, each normalized by their respective value in the static case.",
"The other parameters of the configuration are kept fixed and correspond to Table 1, second row."
],
[
"general solution",
"Suppose that the plasma boundary curve is known, either analytically or numerically, and one wants an accurate fit reproducing its shape.",
"The polynomial solution Eq.",
"() studied in the previous section is easy to implement, requiring only few contraints in order to obtain the equilibrium, but it evidently fails in such situations.",
"As discussed above, the position of x-points, i.e., boundary points with vanishing magnetic field gradient, can be fixed exactly only at the cost of neglecting other constraints.",
"This issue could be overcome by extending the expression in Eq.",
"() up to a suitable higher power of $Z$ , thus generating new free constants in the solution.",
"However, we show here that the general solution of the homogeneous equation $\\Delta ^*{\\psi _\\text{H}}=0$ can be written in a compact form, without any truncation, and that it can be used to solve this kind of problem.",
"To find the general solution, we express $\\psi _\\text{H}$ as a Fourier transform in the $Z$ variable: $\\psi _\\text{H}(R,Z)=\\int _{-\\infty }^{\\infty }\\chi (R,k)\\mathrm {e}^{ikZ}{k}.$ Its reality is ensured by the condition $\\chi (R,-k)=\\overline{\\chi (R,k)}$ .",
"Plugging Eq.",
"(REF ) into $\\Delta ^*{\\psi _\\text{H}}=0$ , we obtain an ordinary differential equation in the variable $R$ for each $k$ .",
"By making the change of variables $x_k=|k|R$ and $\\chi (R,k)=R\\epsilon (x_k,k)$ (for $k\\ne 0$ ), it is easy to verify that: $x_k^2\\epsilon (x_k,k)^{\\prime \\prime }+x_k\\epsilon (x_k,k)^{\\prime }-\\left(1+x_k^2\\right)\\epsilon (x_k,k)=0\\;,$ where the prime denotes differentiation with respect to $x_k$ .",
"This is known as Bessel's modified equation with index 1, and its solution is readily available in mathematical literature: $\\epsilon (x_k,k)=a_{k}I_1(x_k)+b_{k}K_1(x_k) \\,,$ where $a_{k}, b_{k}$ are functions of $k$ .",
"By substitution back into Eq.",
"(REF ), we obtain: $\\psi _\\text{H}(R,Z)=R\\int _{-\\infty }^{\\infty } \\left[ a_k I_1(|k|R) + b_k K_1(|k|R) \\right] \\mathrm {e}^{ikZ}{k}\\,,$ which is the general solution of the homogenous problem.",
"This expression can be adapted to a given scenario by imposing specific boundary conditions.",
"In this respect, for the sake of simplicity, we represent the functions $a_k,\\,b_k$ as a sum of sufficiently narrow gaussians (i.e., delta functions), centered around arbitrarily given wave vectors $k_i$ and weighted by amplitudes $\\bar{a}_i,\\,\\bar{b}_i$ , thus obtaining: $\\psi _\\text{H}(R,Z) = R \\sum _{i=1}^N \\left[\\bar{a}_i I_1(R|k_i|) + \\bar{b}_i K_i(R|k_i|) \\right] \\cos (k_i Z)\\,,$ where the term $\\cos (k_iZ)$ is the reduction of the complex exponential to the real, up-down symmetric case.",
"Then, a given set of points $\\lbrace r_l,z_l\\rbrace $ lying along the boundary curve of the addressed plasma configuration generates an associated set of algebraic equations of the form $\\psi (r_l,z_l)=0$ , which can be solved to determine the arbitrary constants."
],
[
"DTT double-null configuration",
"We illustrate this procedure in the practical case of the double-null plasma scenario predicted for the upcoming DTT experiment.",
"Its main parameters are reported in Table REF , and are available in Ref.",
"[11] along with the predicted separatrix shape.",
"Table: Main plasma parameters of the DTT double-null scenario, taken from .We proceed as follows: firstly, we model the desired separatrix as an analytic curve, using a piecewise rational expression (e.g.",
"quadratic).",
"Secondly, we extract a set of boundary points chosen at random but equally distributed around the plasma region.",
"Thirdly, the set of wavenumbers $k_i$ is chosen as an equally distributed grid of values close to the scale length of the configuration, estimated as $\\pi /(a\\kappa )$ .",
"The solution of the resulting set of algebraic equations gives the constants $\\bar{a}_i,\\,\\bar{b}_i$ as functions of $P_1$ , $I_1$ and $M$ .",
"The former two are still obtained according to Eqs.",
"(), () and (), and we can study the behaviour of the equilibrium for different values of $M$ .",
"Figure: Contours of constant ψ\\psi (left) and PP (right) in the (R,Z) plane for the DTT double-null plasma, in the static case (M=0M=0, solid blue) and rotating case (M=0.6M=0.6, dashed red).Fig.REF shows the fitted magnetic configuration and the curves of constant pressure, where we highlighted the correspondence between the set of boundary fitting points and the obtained separatrix.",
"In the presence of rotation, the qualitative behaviour of the plasma is still that of an outward shift of magnetic and pressure lines, while the separatrix is kept fixed by the imposed constraints and has no major modifications.",
"Figure: Safety factor profile of the DTT double-null scenario in the static case (M=0M=0, solid blue) and rotating case (M=0.6M=0.6, dashed red).Concerning the safety factor profile, plotted in Fig.REF , we predict a slight suppression in the core region, while closer to the plasma boundary $q$ actually increases, contrary to the general behaviour observed in the previous section using solution ().",
"Figure: Contours of constant toroidal velocity ωR\\omega R over the whole configuration (left), and contours of constant pressure in the vicinity of the x-point (right, M=0M=0 solid blue, M=0.6M=0.6 dashed red).Finally, we can plot the curves of constant toroidal speed $\\omega (\\psi ) R$ by assuming a simple form for the temperature, taken as $T(\\psi ) = T_\\text{edge} + \\frac{\\psi }{\\psi _\\text{axis}}(T_\\text{core}-T_\\text{edge})$ , with the temperature values according to [11].",
"The result is shown in Fig.REF , along with the morphology of pressure lines in the vicinity of the x-point.",
"In this formalism, we don't expect serious modifications to the shape of the plasma in this region, having imposed our constraints along the separatrix itself.",
"However, observing how the pressure (and its gradient) are suppressed in the presence of rotation might provide useful information when considering transport dynamics.",
"For example, it is common practice to feed equilibrium data obtained from a given solver into a separate code which simulates particle transport.",
"A scenario in which the parameter $M$ is measured to be consistently far from 0, giving rise to noticeable modifications of the equilibrium, would need to take plasma rotation into account.",
"Of course, the present analysis is aimed at providing a simple semi-analytical tool to gain insight in this direction, while accurate equilibrium solvers with plasma flow should be used for more elaborate analysis (e.g.",
"[12])."
],
[
"concluding remarks",
"In this work, we studied the equilibrium of an axisymmetric plasma in the presence of rotation along the toroidal direction.",
"After recalling the mathematical basic formalism, we adopted suitable assumptions on the arbitrary functions in order to obtain analytic plasma profiles, with enough freedom to represent a variety of plasma settings.",
"In particular, the polynomial expression of section requires to fix only few basic plasma parameters, such as the minor radius and the triangularity.",
"This simplicity allows to find analytical expressions for the plasma separatrix and to deal with a variety of scenarios, e.g.",
"double-null and negative triangularity.",
"Then, in section , we presented the general solution of the considered problem, and illustrated a suitable fitting procedure when dealing with a known plasma separatrix (either analytically or numerically).",
"As a practical implementation of this framework, we studied the double-null plasma scenario proposed for the upcoming Italian experiment DTT, estimating the impact of plasma rotation on the equilibrium properties and highlighting some points of interest such as the modification of the plasma pressure gradient morphology in the vicinity of the x-point, with possible effects on particle transport dynamics in that region.",
"Of course, the analysis performed here has the merit of simplicity due to its analytic nature, but needs to be confirmed by more detailed numerical studies when dealing with more realistic situations.",
"Moreover, many physical constraints here neglected (e.g., the specifics of the given tokamak magnetic coils, or its current drive mechanism) would need to be taken into account.",
"Nevertheless, the two approaches of section (reduced) and (general) agree on the qualitative behaviour of the plasma parameters as functions of the rotation velocity, hence they can both be used as quick investigative tools concerning the introduction of toroidal rotation in tokamak plasma equilibria.",
"In experimental situations, the parameter $M$ can be estimated from Eq.",
"(14) providing direct measurements of ion rotation speed and temperature, e.g.",
"through diagnostics like charge exchange recombination spectroscopy [16].",
"Depending on the value of $M$ , quantitative estimates on the relevance of plasma rotation can be argued by the methods outlined here."
]
] | 2107.01890 |
[
[
"Protected probabilistic classification"
],
[
"Abstract This paper proposes a way of protecting probabilistic prediction models against changes in the data distribution, concentrating on the case of classification and paying particular attention to binary classification.",
"This is important in applications of machine learning, where the quality of a trained prediction algorithm may drop significantly in the process of its exploitation.",
"Our techniques are based on recent work on conformal test martingales and older work on prediction with expert advice, namely tracking the best expert."
],
[
"Introduction",
"A common problem in applications of machine learning is that, soon after a predictor is trained, the distribution of the data may change, and the predictor may need to be retrained.",
"There are efficient ways of online detection of a change in distribution, such as using conformal test martingales [16], but there are inevitably awkward gaps between the change in distribution and its detection and between the detection of the change and the deployment of a retrained predictor.",
"This paper proposes a way of preventing a catastrophic drop in the quality of the trained predictor when the data distribution changes.",
"Given a base predictor, our procedure gives an enhanced predictor that is more robust to changes in the data distribution.",
"To use Anscombe's [1] insurance metaphor (repeatedly mentioned in [8]), our procedure provides an insurance policy (hopefully not too expensive) against such changes.",
"The case of regression was discussed in an earlier paper [15], and in this note we concentrate on the simpler case of binary classification.",
"We will assume that the label space is $\\lbrace 0,1\\rbrace $ (except for Section , in which we will use a dataset, Bank Marketing, with label space $\\lbrace 1,2\\rbrace $ ).",
"Suppose we are given a predictive system that maps past data and an object $x$ to a number $p\\in [0,1]$ , interpreted as the predicted probability that $x$ 's label is 1.",
"We will refer to it as our base predictive system.",
"We will be interested in two seemingly different questions about the base predictive system: Online testing Can we gamble successfully against the base predictive system?",
"We are interested in online testing [16], i.e., in constructing test martingales with respect to the base predictive system that take large values on the actual sequence of observations.",
"Online prediction Can we improve the base predictive system, modifying its predictions $p_n$ to better predictions $p^{\\prime }_n$ ?",
"If the quality of online prediction is measured using the log-loss function [6], the difference between the two questions almost disappears, as we will see in Sections and .",
"After discussing online testing in Section and online prediction in Section , we will give an example of a theoretical performance guarantee for our prediction procedure (a straightforward application of a known result) in Section .",
"In Section we report encouraging experimental results, and Section concludes."
],
[
"Testing predictions by betting",
"We consider a potentially sequence of actual observations $z_1,z_2,\\dots $ , each consisting of two components: $z_n=(x_n,y_n)$ , where $x_n\\in \\mathbf {X}$ is an object chosen from an object space $\\mathbf {X}$ , and $y_n\\in \\lbrace 0,1\\rbrace $ is a binary label.",
"A predictive system is a function that maps any object $x$ and any finite sequence of observations $z_1,\\dots ,z_i$ (intuitively, the past data) for any $i\\in \\lbrace 0,1,\\dots \\rbrace $ to a number $p\\in [0,1]$ (intuitively, the probability that the label of $x$ is 1).",
"Fix a base predictive system, and let $p_1,p_2,\\dots $ be its predictions for the actual observations: $p_n$ is the prediction output by the base predictive system on $x_n$ and $z_1,\\dots ,z_{n-1}$ ; it is interpreted as the predicted probability that $y_n=1$ .",
"(In this note we do not need any measurability assumptions; in particular, $\\mathbf {X}$ is not supposed to be a measurable space.)",
"In this paper we are mostly interested in the special case where the output $p$ of the base predictive system depends only on $x$ and not on $z_1,\\dots ,z_i$ .",
"In this case we will say that our predictive system is a prediction rule.",
"A typical way in which prediction rules appear in machine learning is as result of training a prediction algorithm.",
"In Section we will be only interested in a prediction rule, but for now we do not impose this restriction.",
"[bt] Jumper betting martingale ($(p_1,p_2,\\dots )\\mapsto (S_1,S_2,\\dots )$ ) [1] $C_{\\epsilon }:=1/\\left|\\mathbf {E}\\right|$ for all $\\epsilon \\in \\mathbf {E}$ $C:=1$ $n=1,2,\\dots $ : $\\epsilon \\in \\mathbf {E}$ : $C_{\\epsilon } := (1-J)C_{\\epsilon } + (J/\\left|\\mathbf {E}\\right|)C$ $\\epsilon \\in \\mathbf {E}$ : $C_{\\epsilon } := C_{\\epsilon } B_{f_{\\epsilon }(p_n)}(\\lbrace y_n\\rbrace ) / B_{p_n}(\\lbrace y_n\\rbrace )$ $S_n := C := \\sum _{\\epsilon \\in \\mathbf {E}} C_{\\epsilon }$ Our online testing procedure is given as Algorithm .",
"One of its two parameters is a finite family $f_{\\epsilon }:[0,1]\\rightarrow [0,1]$ , $\\epsilon \\in \\mathbf {E}$ , of calibrating functions.",
"The intuition behind $f_{\\epsilon }$ is that we are trying to improve the base predictions $p_n$ , or calibrate them; the idea is to use a new prediction $f_{\\epsilon }(p_n)$ instead of $p_n$ .",
"In the experimental Section we will use a subset of the family $f_{\\epsilon }(p) := p + \\epsilon p(1-p),$ where $\\epsilon \\in [-1,1]$ .",
"For $\\epsilon >0$ we are correcting for the forecasts $p$ being underestimates of the true probability of 1, while for $\\epsilon <0$ we are correcting for $p$ being overestimates.",
"Our family is required to be finite, and we choose $\\mathbf {E}:=\\lbrace -1,-0.5,0,0.5,1\\rbrace $ .",
"We do not know in advance which $f_{\\epsilon }$ will work best, and moreover, it seems plausible that suitable values of $\\epsilon $ will change over time.",
"Therefore, we use the idea of “tracking the best expert” [7].",
"Algorithm uses the notation $B_p$ , $p\\in \\lbrace 0,1\\rbrace $ , for the Bernoulli distribution on $\\lbrace 0,1\\rbrace $ with parameter $p$ : $B_p(\\lbrace 1\\rbrace )=p$ .",
"To each sequence $\\theta =(\\theta _1,\\theta _2,\\dots )$ of elements of $\\mathbf {E}$ corresponds the elementary test martingale $\\prod _{i=1}^n\\frac{B_{f_{\\theta _i}(p_i)}(\\lbrace y_i\\rbrace )}{B_{p_i}(\\lbrace y_i\\rbrace )},\\quad n=0,1,\\dots .$ The other parameter of the Jumper betting martingale of Algorithm is $J\\in (0,1]$ , the jumping rate.",
"This martingale is obtained by “derandomizing” (to use the terminology of [13]) the stochastic test martingale corresponding to the probability measure $\\mu $ on $[0,1]^{\\infty }$ defined as the probability distribution of the following Markov chain with state space $\\mathbf {E}$ .",
"The initial state $\\theta _1$ is chosen from the uniform probability measure on $\\mathbf {E}$ (line of Algorithm ), and the transition function prescribes maintaining the same state with probability $1-J$ and, with probability $J$ , choosing a new state from the uniform probability measure on $\\mathbf {E}$ (line ).",
"We derandomize the stochastic test martingale by averaging, $ S_n:=\\int \\prod _{i=1}^n\\frac{B_{f_{\\theta _i}(p_i)}(\\lbrace y_i\\rbrace )}{B_{p_i}(\\lbrace y_i\\rbrace )}\\mu (\\mathrm {d}\\theta ),$ which gives us a deterministic test martingale.",
"Figure: Examples of calibration functions.In Section we will see an example where already the simple choice (REF ) leads to very successful betting for a benchmark dataset.",
"However, there are numerous other natural calibration functions, some of which are shown in Figure REF .",
"The function in blue is in the quadratic family (REF ); these functions are fully above, fully below, or (for $\\epsilon =0$ ) situated on the bisector of the first quadrant (shown as the dotted line).",
"In many situations other calibration functions will be more suitable.",
"For example, it is well known that untrained humans tend to be overconfident [9].",
"An example of a calibration function correcting for overconfidence is the cubic function $f_{a,b}(p) := p + a p(p-b)(p-1),$ where $(a,b)\\in [0,1]^2$ .",
"An example of such a function is shown in Figure REF in orange.",
"The meaning of the parameters is that $b$ is the value of $p$ (such as $0.5$ ) that we believe does not need correction, and that $a$ indicates how aggressively we want to correct for overconfidence ($a<0$ meaning that in fact we are correcting for underconfidence).",
"If the predictor predicts a $p$ that is close to 0 or 1, we correct for his overconfidence (assuming $a>0$ ) by moving $p$ towards the neutral value $b$ .",
"Alternatively, we could use Cox's [2] calibration functions, one-parameter $f_{\\beta }(p):=\\frac{p^{\\beta }}{p^{\\beta }+(1-p)^{\\beta }},$ where $\\beta \\in \\mathbb {R}$ , or two-parameter $f_{\\alpha ,\\beta }(p):=\\frac{p^{\\beta }\\exp (\\alpha )}{p^{\\beta }\\exp (\\alpha )+(1-p)^{\\beta }},$ where $\\alpha ,\\beta \\in \\mathbb {R}$ .",
"An example of a function in the class (REF ) is shown in Figure REF in green."
],
[
"Prediction algorithms based on adaptive calibration",
"For any predictive system, we define its gale [12] as a function mapping any finite sequence of observations to the product $B_{p_1}(y_1)\\cdots B_{p_n}(y_n)$ , where $n$ is the number of observations, $y_1,\\dots ,y_n$ are their labels, and $p_1,\\dots ,p_n$ are the predictions for those observations.",
"We regard the gale as the capital process of a player playing an extremely challenging game: his capital cannot go up, and for it not to go down he has to predict with the probability measure concentrated on the true outcome.",
"The gale of the base predictive system will be referred to as the base gale.",
"Remark 1 The notion of a gale is very similar to Cox's [3] notion of partial likelihood, but we cannot say that a gale is partial in any sense (since it is not part of a fuller likelihood function: there is no probability measure on the objects [12]).",
"[bt] Jumper predictor ($(p_1,p_2,\\dots )\\mapsto (p^{\\prime }_1,p^{\\prime }_2,\\dots )$ ) [1] $C_{\\epsilon }:=1$ for all $\\epsilon \\in \\mathbf {E}$ $n=1,2,\\dots $ : $C := \\sum _{\\epsilon \\in \\mathbf {E}} C_{\\epsilon }$ $\\epsilon \\in \\mathbf {E}$ : $C_{\\epsilon } := C_{\\epsilon } / C$ $\\epsilon \\in \\mathbf {E}$ : $C_{\\epsilon } := (1-J)C_{\\epsilon } + J/\\left|\\mathbf {E}\\right|$ $p^{\\prime }_n := \\sum _{\\epsilon \\in \\mathbf {E}} f_{\\epsilon }(p_n) C_{\\epsilon }$ $\\epsilon \\in \\mathbf {E}$ : $C_{\\epsilon } := C_{\\epsilon } B_{f_{\\epsilon }(p_n)}(\\lbrace y_n\\rbrace )$ For simplicity, we will discuss only positive gales and martingales (i.e., those that do not take zero values).",
"This will be sufficient for the considerations of Section .",
"Each test martingale with respect to the base predictive system is the ratio of a gale to the base gale, and vice versa.",
"This establishes a bijection between test martingales and gales.",
"Algorithm is the predictive system whose gale corresponds to the test martingale of Algorithm .",
"Algorithm is a special case the Aggregating Algorithm (AA) [13] corresponding to the log-loss function $\\lambda (y,p):={\\left\\lbrace \\begin{array}{ll}-\\log p & \\text{if $y=1$}\\\\-\\log (1-p) & \\text{if $y=0$}\\end{array}\\right.",
"}$ (the logarithm is typically natural, but in Section we will consider decimal logarithms).",
"Analogously to (REF ), to each sequence $\\theta =(\\theta _1,\\theta _2,\\dots )$ corresponds the elementary predictor that outputs, at each step $n$ , $ p^{\\prime }_n:=f_{\\theta _n}(p_n),\\quad n=1,2,\\dots ,$ as its prediction.",
"The AA is described in [13], and in our case of the log-loss function the optimal in a natural sense exponential learning rate is $\\beta :=\\exp (-1)$ , and the AA coincides with the APA (“Aggregating Pseudo-Algorithm”).",
"The prior distribution $\\mu $ on the elementary predictors is as described in the previous section.",
"At the beginning of step $n$ , the prior weight of the elementary predictor $\\theta $ is multiplied by $\\prod _{i=1}^{n-1}B_{f_{\\theta _i}(p_i)}(\\lbrace y_i\\rbrace ),$ in the sense that the posterior distribution is $\\mu ^{\\prime }(\\mathrm {d}\\theta )=\\mu (\\mathrm {d}\\theta )\\prod _{i=1}^{n-1}B_{f_{\\theta _i}(p_i)}(\\lbrace y_i\\rbrace ).$ Let $D_n(\\epsilon )$ be the total posterior (unnormalized) weight of the elementary predictors $\\theta $ that are in state $\\epsilon $ at the beginning of step $n$ , i.e., $\\theta _n=\\epsilon $ .",
"We start from $D_1(\\epsilon )=1/\\left|\\mathbf {E}\\right|$ (for all $\\epsilon $ ), and the recursion is $D_n(\\epsilon ):=(1-J)D^{\\prime }_n(\\epsilon )+\\frac{J}{\\left|\\mathbf {E}\\right|}\\sum _{\\epsilon ^{\\prime }\\in \\mathbf {E}}D^{\\prime }_n(\\epsilon ^{\\prime }),$ where $D^{\\prime }_n(\\epsilon ^{\\prime }):=B_{f_{\\epsilon ^{\\prime }}(p_{n-1})}(\\lbrace y_{n-1}\\rbrace )D_{n-1}(\\epsilon ^{\\prime })$ for all $\\epsilon ^{\\prime }\\in \\mathbf {E}$ .",
"We can see that $D_n(\\epsilon )\\propto C_{\\epsilon }$ , where $C_{\\epsilon }$ is computed in line of Algorithm at iteration $n$ , and $\\propto $ means coincidence to within a positive factor independent of $\\epsilon $ .",
"The AA prediction can now be computed as $\\frac{\\sum _{\\epsilon }f_{\\epsilon }(p_n)D_n(\\epsilon )}{\\sum _{\\epsilon }D_n(\\epsilon )}.$ This again results in the prediction algorithm given as Algorithm .",
"The variables $C_{\\epsilon }$ in it are different from the $C_{\\epsilon }$ in Algorithm , but the difference is not essential (a positive factor independent of $\\epsilon $ ); the former are the normalized versions of the latter."
],
[
"An example of a theoretical guarantee",
"Theoretical performance guarantees for Algorithm and related procedures is potentially a big topic, but we will give only a very simple result, a special case of a known result for the AA.",
"In the context of the AA, a gale is $\\exp (-L)$ , where $L$ is a loss process.",
"The following lemma is the main property of the AA.",
"Lemma 1 The gale of the AA is the average of the elementary predictors' gales.",
"For a proof, see [14].",
"Let us use the notation $\\operatorname{Loss}(p_1,\\dots ,p_n\\mid y_1,\\dots ,y_n):=\\sum _{i=1}^n\\lambda (y_i,p_i)$ for the cumulative log-loss of predictions $p_i\\in [0,1]$ on labels $y_i\\in \\lbrace 0,1\\rbrace $ , where $\\lambda $ is defined by (REF ).",
"The simplest performance guarantee for Algorithm is $\\operatorname{Loss}(p^{\\prime }_1,\\dots ,p^{\\prime }_n\\mid y_1,\\dots ,y_n)\\le \\operatorname{Loss}(f_{\\theta _1}(p_1),\\dots ,f_{\\theta _n}(p_n)\\mid y_1,\\dots ,y_n)\\\\+\\log \\left|\\mathbf {E}\\right|+k \\log (\\left|\\mathbf {E}\\right|-1)+k \\log \\frac{1}{J}+(n-k-1) \\log \\frac{1}{1-J},$ for any $n$ and any sequence $f_{\\theta _1},\\dots ,f_{\\theta _n}$ of calibrating functions (from the family $(f_{\\epsilon })$ ), where $k=k(\\theta _1,\\dots ,\\theta _n)$ is the number of switches, $k:=\\left|\\left\\lbrace i\\in \\lbrace 1,\\dots ,n-1\\rbrace : \\theta _i \\ne \\theta _{i+1}\\right\\rbrace \\right|.$ If we further average Algorithm over the uniform probability measure over $\\epsilon \\in [0,1]$ , we obtain the guarantee $\\operatorname{Loss}(p^{\\prime }_1,\\dots ,p^{\\prime }_n\\mid y_1,\\dots ,y_n)\\le \\operatorname{Loss}(f_{\\theta _1}(p_1),\\dots ,f_{\\theta _n}(p_n)\\mid y_1,\\dots ,y_n)\\\\+\\log \\left|\\mathbf {E}\\right|+k \\log (\\left|\\mathbf {E}\\right|-1)+(k+1) \\log n,$ again for any $n$ and any sequence of calibrating functions, with $k$ defined in the same way.",
"This follows from the results of [13] (such as Theorem 2) and [7], and can be easily deduced from Lemma REF ."
],
[
"Experimental results",
"In this section we report results of our experiments with the Bank Marketing dataset (the only dataset in the top twelve most popular datasets at the UC Irvine Machine Learning Repository that fits the scenario of Section ; we will use, however, the full version of this dataset as given at the openml.org repository, since it is easy to do from scikit-learn).",
"The dataset consists of 45,211 observations representing telemarketing calls for selling long-term deposits offered by a Portuguese retail bank, with data collected from 2008 to 2013 [10].",
"The labels are 1 or 2, with the 2s (indicating a successful sale) comprising only 12% of all labels.",
"The observations are listed in chronological order.",
"We took the first 10,000 observations as the training set and trained a random forest with default parameters and random seed 2021 on it (the random forest method gives the best results for this dataset in our preliminary experiments; this element of data snooping appears harmless since we are interested in improving the base predictive system, and it is natural to expect similar or better results for less successful prediction algorithms).",
"The random forest often outputs probabilities of success that are equal to 0 or 1, and when such a prediction turns out to be wrong (which happens repeatedly), the log-loss is infinite.",
"It is natural to truncate a probability $p\\in [0,1]$ of 2 to the interval $[\\epsilon ,1-\\epsilon ]$ replacing $p$ by $p^*:={\\left\\lbrace \\begin{array}{ll}\\epsilon & \\text{if $p\\le \\epsilon $}\\\\p & \\text{if $p\\in (\\epsilon ,1-\\epsilon )$}\\\\1-\\epsilon & \\text{if $p\\ge 1-\\epsilon $},\\end{array}\\right.",
"}$ where we set $\\epsilon :=0.1$ (in scikit-learn, $\\epsilon =10^{-15}$ , but $\\epsilon :=0.1$ leads to significantly better results).",
"The resulting prediction rule is our base predictive system.",
"After we find it, we never use the training set again, and the numbering of observations starts from the first element of the test set (i.e., the dataset in the chronological order without the training set).",
"Figure: The Jumper test martingale.Figure REF shows the trajectory of $\\log _{10}S_n$ , $n=1,\\dots ,35211$ , where $S_n$ is the value of the Jumper test martingale over the test set with the jumping rate $J:=0.01$ and the family (REF ) with $\\epsilon \\in \\mathbf {E}:=\\lbrace -1,-0.5,0.0.5,1\\rbrace $ .",
"It is interesting that the steepest growth of the test martingale (on the log scale) starts towards the end of the dataset, long after the financial crisis of 2007–2008 ended.",
"The final value of the test martingale in Figure REF is approximately $10^{919.3}$ .",
"Figure: The ROC curve for the Jumper enhancement.Figure REF gives the ROC curve for the random forest and the random forest enhanced by Algorithm .",
"We can see that the improvement is substantial.",
"In terms of the log-loss function and decimal logarithms, the loss goes down from $5684.1$ to $4764.8$ (the difference between these two numbers being, predictably, the exponent $919.3$ in the final value of the test martingale in Figure REF ).",
"Figure: The analogues of Figure for J:=0.1J:=0.1 on the left and J:=0.001J:=0.001 on the right.The value $J=0.01$ is the one that has been used most commonly in the existing papers.",
"However, the dependence of our results on the values of parameters is weak: see, e.g., Figure REF , which shows results for jumping rates $0.1$ and $0.001$ .",
"However, using a specific value of $J$ may be risky in that the Jumper test martingale loses capital exponentially quickly if the base prediction algorithm is already ideal (which makes the insurance policy discussed in Section expensive).",
"A safer option is to use the Mean Jumper procedure averaging the Jumper test martingales over a small set of $J$ including $J=1$ [15]."
],
[
"Conclusion",
"The methods of adaptive calibration that we propose in this note need to be validated on other datasets and for other calibrating functions, such as (REF ), (REF ), and (REF ).",
"Notice that calibrating functions may depend not only on the current predicted probability $p$ but also on the current object $x$ .",
"(So that “calibration” may be understood in a very wide sense, as in [4], and include elements of “resolution” [5].)",
"A natural direction of further research is to extend our methods and results to multiclass classification.",
"Notice that Cox's calibrating functions (REF ) and (REF ) immediately extend to the multiclass case."
],
[
"Acknowledgments",
"We are grateful to Glenn Shafer for his advice.",
"This research has been partially supported by Stena Line.",
"In our computational experiments we used scikit-learn [11]."
]
] | 2107.01726 |
[
[
"Online and Offline Robot Programming via Augmented Reality Workspaces"
],
[
"Abstract Robot programming methods for industrial robots are time consuming and often require operators to have knowledge in robotics and programming.",
"To reduce costs associated with reprogramming, various interfaces using augmented reality have recently been proposed to provide users with more intuitive means of controlling robots in real-time and programming them without having to code.",
"However, most solutions require the operator to be close to the real robot's workspace which implies either removing it from the production line or shutting down the whole production line due to safety hazards.",
"We propose a novel augmented reality interface providing the users with the ability to model a virtual representation of a workspace which can be saved and reused to program new tasks or adapt old ones without having to be co-located with the real robot.",
"Similar to previous interfaces, the operators then have the ability to program robot tasks or control the robot in real-time by manipulating a virtual robot.",
"We evaluate the intuitiveness and usability of the proposed interface with a user study where 18 participants programmed a robot manipulator for a disassembly task."
],
[
"Introduction",
"Industrial robots play a vital role in manufacturing tasks such as welding and assembly due to their ability to perform these tasks with a high degree of precision and reliability.",
"Nevertheless, despite their importance in manufacturing, the programming of such robots remains a costly and time-consuming task often requiring operators to have a certain degree of knowledge of robotics and programming [1], [2], [3], [4], [5], [6].",
"The most popular methods for programming industrial robots are referred to as online and offline.",
"The online method requires the use of the real physical robot and consists of recording the path designed by an operator by controlling the joints of the robot via a teaching pendant with a joystick and/or keypad.",
"The offline method on the other hand does not require the actual robot and consists of constructing a virtual representation of the robot and its workspace to simulate tasks prior to applying it on the real robots [7], [8].",
"Various simulators and 3D engines have recently been extended to support robotics platforms, making the offline programming of robots possible.",
"These methods present however multiple disadvantages.",
"For online programming, the control of a robot via a teaching pendant often requires the robot in question to be removed from production during programming or, under certain circumstances, a shutdown of the production line [2].",
"In the case of offline programming, the reconstruction of the robot's workspace in a virtual environment can be time consuming.",
"Furthermore, the transition from the simulation to the execution on the real robot may suffer from inaccuracies, with additional costs required to overcome them [7].",
"Moreover, offline methods require a certain degree of familiarity with programming and simulators.",
"The significance of these disadvantages increases when robots need to be reprogrammed frequently.",
"This is especially true in High-Mix Low-Volume (HMLV) manufacturing, where there is a large variety of items to be produced in small quantities [7].",
"As such, research has been conducted to design intuitive and efficient programming interfaces for users with little to no experience in the fields of robotics or programming [1].",
"To suit these needs, Augmented Reality (AR) technology has been proposed as a solution due to its ability to superimpose information in various forms onto the real world.",
"Indeed, AR has been growing increasingly popular in recent years and has been used in various fields such as architecture, construction, education, manufacturing and engineering [9].",
"More specifically, in the field of robotics, AR interfaces provide a new medium for interaction with the robot and enable the exchange of information during tasks [10] in addition to providing users with the ability to preview a robot's intended actions.",
"As such, several augmented or mixed reality interfaces have been proposed to enable users not only to control a robot [6] but also to program it by defining trajectories and tasks via the manipulation of a virtual robot in the interface [5], [8], [11], [12], [13].",
"These interfaces provide the operators with an intuitive way of programming the robot without an extensive knowledge about programming and combines some of the advantages of current online and offline methods by enabling users to simulate a task directly in the robot's workspace, making the transition from simulation to real execution easier.",
"However, the proposed solutions require the users to share the same workspace as the robot such that the robot in question cannot be used for production during the programming process.",
"To further reduce the time during which the robots are unavailable for production, we propose an AR interface providing users with the ability to program the robot from a virtual representation of the workspace which can be built and saved by the user such that tasks can be programmed from different locations (Fig.",
"REF ).",
"Adjustments to the trajectory, if necessary, could then be made by displaying the designed task in the real robot's workspace resulting in less time during which the robot in question would be unavailable for production.",
"In Section , we first present a state of the art of robot control and programming via augmented reality.",
"This is followed by a description of our proposed AR interface and its components in Section .",
"In Section , we describe the experiments conducted to evaluate our interface.",
"In Section , we present and give a brief discussion on the results of our experiments.",
"Finally, we conclude our paper in Section with some potential future work descriptions."
],
[
"Related work",
"Augmented reality has been used in combination with robotics for various use cases allowing users to not only interact with or control the robot and visualise their intentions but also program the robot by defining target points in the actual workspace that the robot should reach.",
"This enables inexperienced users with little to no programming skills to design complex robot operations and ensure that the trajectory to be taken by the real robot is collision free [2].",
"Various AR interfaces displayed either via a Head Mounted Device (HMD), smartphone, tablet or screen have been proposed in recent years for industrial settings as well as for collaborative tasks where the user can perform the task alongside the operator.",
"For instance, in [8], [11], [12], [13], the authors present mixed reality interfaces with a Microsoft HoloLens to program tasks by defining waypoints directly in the robot's workspace and previewing the trajectory via a virtual robot before sending the commands to a real robot.",
"To place the waypoints and interact with the other features each respective interface offers, the interfaces employ features available on the HoloLens, namely gesture recognition as well as speech recognition and head pose in some cases [11].",
"The main drawbacks of such interfaces are that the selection and dragging features in the HoloLens may not always be reliable due to imperfect hand tracking [13] such that the trajectory may not be optimal and it may be difficult to perform tasks that demand a high degree of precision.",
"To overcome this issue, in [8], the authors present an interface providing users with the ability to scale the path for a more accurate planning during such tasks, as well as the selection of the method by which the trajectory between waypoints should be interpolated (line, arc, etc.).",
"However, despite allowing users to be hands-free, the HoloLens may be less intuitive than a 2D interface for novice users and HMDs are not necessarily available to most users.",
"Additionally, it has been reported that such devices may cause discomfort and sickness which may be affecting their industrial acceptance [14].",
"In recent years, various platforms and toolkits such as ARCore, ARToolKit and Mixed Reality Toolkit (MRTK) have been introduced allowing users to develop augmented or mixed reality applications for certain smartphones and tablets making the technology more accessible.",
"Such toolkits have been used to develop interfaces enabling users to perform pick-and-place tasks in collaboration with a robot [14], define waypoints on a 2D surface for a tool to pass through [5] as well as control the joints of the robot via the interface and visualise a trajectory demonstrated to the robot via kinesthetic teaching [4].",
"However, in the case of [14], [5], the interfaces do not provide a visualisation of the whole robot which may be important in some environments such that the operators can check for eventual collisions.",
"Whilst this is not the case with the interface presented in [4], the teaching mode presented in the interface involves demonstrating a task to the robot such that the robot is backdrivable through a gravity compensation controller, which is not available to most industrial robots.",
"In addition to some of the drawbacks mentioned above, most of the interfaces presented in this section require the operators to employ the interface whilst being next to the robot's workspace where the operators can interact directly with the physical objects present in the environment.",
"Due to safety issues, similar to teaching a robot a task via a pendant, this would require shutting down the robots in the production line.",
"In this paper, we propose an AR interface available on smartphone and tablet providing users with the ability to not only control and program a robot in its workspace but also model the workspace via virtual objects.",
"This provides the advantage of being able to use the interface to program robot tasks from different locations and adapt previously saved tasks to new situations, environments, tools, or robots."
],
[
"Methodology",
"The interface proposed in this work was developed with Google's ARCore software library on Android Studio such that it can currently be run on most AR compatible Android deviceshttps://developers.google.com/ar/devices.",
"The main functionality of the interface is that it provides non-expert users with an offline programming solution by means of a virtual robot in a simulation environment, as well as an online programming solution by controlling the real robot with the provided augmented reality tools.",
"The virtual robot can be placed at a desired location within the environment and can then be connected to the real one, if required, to visualise what has been programmed offline or to directly control the robot online.",
"We implemented several mechanisms to control the robot to achieve tasks in end-effector space.",
"Additionally, the interface provides various options for the user to add/modify objects and obstacles, which can be moved, rotated and scaled in the workspace of the robot.",
"The workspace created virtually can also be saved for later programming without requiring the real robot to be operated at the same time.",
"Videos of the AR interface and the experiments described in Section are available at: https://sites.google.com/view/idiap-ar-robot-interface/"
],
[
"Simulation of the robot via ARCore",
"We first describe the procedure to create a virtual robot in the desired workspace by explaining the connection to the real robot along with their calibration and communication.",
"Upon launch, ARCore detects surfaces and planes which allow the users to interact with the virtual objects placed in the environment.",
"The virtual robot is created by following the same approach as KDLhttps://www.orocos.org/kdl.html.",
"It uses the description of a robot in the form of Unified Robot Description Format (URDF) to transform it into visual kinematic chains representing the real robot.",
"The interface provides two methods with which the virtual robot can be placed in the workspace: the manual method and the marker calibration method.",
"The manual method lets the user select a location on a plane detected by ARCore upon which to place the virtual robot (Fig.",
"REF ).",
"The robot's orientation and position can then be adjusted with the constraint of moving on the plane upon which the robot is placed with the help of the blue and red translation axes and the green rotation ring as seen in Fig.",
"REF .",
"The marker calibration method exploits ARCore's Augmented Images APIshttps://developers.google.com/ar/develop/java/augmented-images to superimpose the virtual robot onto the real robot as illustrated in Fig.",
"REF .",
"It first detects the pose of a previously placed 2D image in the workspace with respect to the ARCore coordinate system.",
"Using the homogeneous transformation between the image and the base of robot coordinate systems, the pose of the robot base with respect to ARCore is determined and is used to place the virtual robot.",
"Figure: Fully virtual workspace with objects and obstacles (in red) and the various ways of controlling a robot: (a) control of the end-effector position, (b) control of the end-effector orientation, (c) control of the end-effector along a plane and (d) control of the elbow."
],
[
"Communication with the real robot",
"The virtual robot is connected to the real robot via Robot Operating Systemhttps://www.ros.org/ (ROS) which allows interactions between the real and virtual robots.",
"As ROS is already available for most of the available industrial and collaborative robots, it provides a generic interface that can work with any robot and that enables the use of multiple sources of sensory information together for the robot to work with.",
"Our interface thus allows the ARCore device to communicate with the server via ROS, as illustrated in Fig.",
"REF -left.",
"The robot is controlled via ROS through a server computer enabling its real-time communication with the robot motors using the robot's API provided by the manufacturer of the robot, as illustrated in Fig.",
"REF -right.",
"As the implementation of our interface is built on top of ROS and not on a specific robot's API, our proposed work here is robot agnostic.",
"We present here the methods available to control the robot's joints and end-effector as well as to plan trajectories.",
"As opposed to joystick or teaching pendant based methodologies of programming robots, we provide more diverse options to move the robot.",
"Users are able to reposition the robot's end-effector using the translation axes shown in Fig.",
"REF and orient it using the rotational rings, each describing the rotation around the translation axes, as shown in Fig.",
"REF .",
"These changes in the task space are applied to the joint space through a weighted inverse kinematics algorithm.",
"Internally, a joint impedance controller was used to reach new joint space targets.",
"The third option enables the user to move the end-effector constrained on a 2D plane defined by the device's orientation, as seen in Fig.",
"REF .",
"This option provides a more intuitive way to interact with the robot from the perspective of the user who may also be moving.",
"The final option is the direct control of the joints that are highlighted in the interface such as the elbow joint, as illustrated in Fig.",
"REF .",
"The application also provides the user with the tools to create, save, load and replay end-effector trajectories on the device.",
"To define a trajectory, the user can place keypoints in the workspace by controlling the robot via the techniques described above.",
"Fig.",
"REF shows four of these workspace keypoints, depicted by partially transparent robot grippers.",
"These locations are then sequentially connected by an iterative Linear Quadratic Regulator (iLQR) [15] used as a trajectory planner, which determines the corresponding joint trajectory from the task space locations.",
"From this trajectory, an equivalent task space trajectory is computed and depicted with a yellow curve as in Fig.",
"REF .",
"Additionally, the interface provides the ability to replace the keypoints with multivariate Gaussian distributions which are represented by ellipsoids in the scene (Fig.",
"REF ).",
"This enables the user to specify the precision and coordination required at each point by rotating the ellipsoid and modifying the scale (described in the following section) along each axis resulting in a smoother trajectory.",
"As all of these techniques can be performed solely on the virtual robot or by directly controlling the real robot, the proposed interface is compatible for online and offline programming.",
"Figure: Trajectory planning with (a) keypoints illustrated by transparent end-effectors and (b) Gaussians (green ellipsoids) where their scale represents the (co)variations allowed along each axis, which are then converted to full precision matrices within the iLQR optimal control technique employed for planning."
],
[
"Simulation of virtual objects and workspaces",
"In offline programming, the user employs a simulator to visualise a virtual robot and define its motions to achieve a task.",
"Industrial robotics tasks are often defined by the workspace where the robot is located and by the objects and tools that it can interact with.",
"To this end, we propose to simulate, via the AR interface, workspaces with all their objects/tools available, without requiring the control and/or presence of the real robot.",
"This provides operators with the ability to design trajectories in a virtual workspace without the need to occupy the real workspace (see Fig.",
"REF ).",
"During the testing phase, adjustments, if required, could then be performed directly in the real robot's workspace by using the marker calibration method to superimpose the virtual robot onto the real one and loading the saved trajectories.",
"The interface allows users to place virtual objects and obstacles (see Fig.",
"REF ) in the shape of rectangular prisms, spheres and cylinders of modifiable sizes.",
"The difference between the obstacles (in red in Fig.",
"REF ) and the objects (in black in Fig.",
"REF ) is that the former is mainly defined for the user to plan a collision-free trajectory of the robot.",
"This means that whereas the robot can apply actions on the objects using the physics engine of the interface, we preferred to exclude these interactions on the obstacles to facilitate the programming.",
"Note that here the collision-free trajectory is defined by the user interaction on the interface and the implementation of an obstacle avoidance planner is left as future work.",
"These objects and obstacles can be moved using the same methods, described in the previous section, to control the robot's end-effector and their scale along each axis can be modified using a similar representation to the one used to translate the end-effector.",
"The creation of such objects and obstacles helps the user to create a virtual workspace that matches at best to the real one (see Fig.",
"REF ) and use it as a template such as in Fig.",
"REF .",
"Trajectories can then also be planned in this virtual workspace (Fig.",
"REF ) using the functionalities described in the previous section and all the virtual components within the scene (i.e.",
"workspace, objects, obstacles and trajectories) can be saved in a file on the device so that they can be reused/adapted for future tasks.",
"Figure: (a) definition of an obstacle by scaling and positioning a red virtual box to encompass an object; (b) virtual representation of the workspace used for the offline condition."
],
[
"Experiments",
"To evaluate the intuitiveness of the proposed interface and the programming of a robot purely within a smartphone-based AR setup, we conducted a study where 18 participants programmed a peg disassembly task.",
"This study was approved by Idiap Research Institute’s Data and Research Ethics Committee."
],
[
"Experimental Setup",
"The participants programmed a 7-axis Franka Emika Panda robot manipulator, with a peg disassembly task of the National Institute of Standards and Technology (NIST) task board 1 [16], see Fig.",
"REF .",
"For the AR interface, we used an Android smartphone (6.39-inch touchscreen display with a resolution of $\\textit {1080} \\times \\textit {2340}$ pixels).",
"The participants of our study (age range 20–35) had varying knowledge of robotics systems but little to no experience of virtual, augmented and mixed reality."
],
[
"Conditions and Protocol",
"The participants were asked to perform the disassembly of a cylindrical peg from the NIST task board and place it into a disposal box, as shown in Fig.",
"REF .",
"Two conditions were tested in the study: the online and the offline programming of the disassembly task.",
"In the online condition, the participants directly controlled the real robot's motions from the AR interface and their inputs were immediately translated into robot motions.",
"Prior to the task, the virtual robot was aligned with the real one based on the marker calibration method described in Section and the interface was connected to the robot as illustrated in Fig.",
"REF .",
"Furthermore, the real and virtual robots were first set to a default joint configuration after which participants could then start the experiment by displacing the virtual end-effector via the options described in Section REF .",
"The task was considered successfully completed when the peg was correctly disposed inside of the box.",
"A failure was recorded when the peg was either incorrectly disposed, dropped or an action performed by the user resulted in the robot's safety stop.",
"In the offline condition, participants instead controlled, via the AR interface, a virtual robot acting on a pre-stored virtual copy of the workspace of the online condition.",
"In particular, the participants first placed the virtual workspace on a table different from the one of the real robot (see Fig.",
"REF ) and programmed the disassembly of the same cylindrical peg from the board and its placement into a yellow box.",
"The participants were given 5 minutes of time to define a robot trajectory by means of via-points, as shown in Fig.",
"REF .",
"The robot trajectory was then sent to the real robot and enacted on the real workspace.",
"If the peg was correctly disposed, the task was considered as successfully completed.",
"In case of failure, the participants were given one extra minute to perform adjustments to the programmed trajectory in the virtual workspace, according to the visual feedback they observed from the real robot.",
"This decision was made to provide a fair comparison with respect to the online condition where participants experienced real-time feedback while controlling the robot.",
"We adopted a within-subjects study design, with each participant operating the robot in both the online and the offline conditions.",
"The order of the conditions was counterbalanced.",
"Prior to the experiment, the participants were provided with instructions and were given 1-2 minute to familiarise with the AR interface and the disassembly task.",
"Figure: Experimental setup with Franka Emika Panda, NIST board, calibration marker and disposal box."
],
[
"Questionnaire and Logged Data",
"For each condition, we recorded whether the participants were successful in completing the task.",
"Furthermore, the completion time $\\mathbf {t_c}$ of the disassembly task was recorded (i.e., seconds between the beginning of the task and its success/failure).",
"At the end of the experiment, each participant filled a questionnaire with the following 6 Likert scale statements (1 – completely disagree, 5 – completely agree): The interface is easy to understand, I found the visualisation of the virtual robot useful, I found it easy to control the robot in real time, I found it easy to plan trajectories with the virtual robot, I found the AR interface useful to program the robot, I would use the AR interface for robot programming.",
"Each statement included an optional comment section where participants could provide feedback about their experience and provide suggestions on what could be improved.",
"Figure: Boxplots of the completion time 𝐭 𝐜 \\mathbf {t_c} acquired during the successful experiments for (a) entire population, (b) group A and (c) group B."
],
[
"Results",
"We hereafter present the analysis of the different metrics collected during the study.",
"When relevant, the data is presented and the analysis is performed by separating the participants into two groups, with the 9 participants who experienced the online condition first and then the offline condition assigned to group A.",
"The other 9 participants who experienced the conditions in the reverse order were assigned to group B.",
"Figure: Boxplots of the completion time 𝐭 𝐜 \\mathbf {t_c} for group A and B for the online condition.Table: Success rate on the disassembly task, reported separately by groups and conditions.We first computed the success rate of each group for each experiment as summarised in Table REF .",
"Our hypothesis entering the study was that the success rate in the offline condition for the participants in group A (that first operated the real-robot in the online condition) should be higher with respect to what observed for group B.",
"The hypothesis was motivated by the fact that participants from group A have a better understanding of the application and the setup when performing this experiment compared to those of group B.",
"Although we observed a 11% difference in success rate, no statistically significant difference was found (test on the Agresti-Coull interval, $ p > .05 $ ).",
"For the completion time $\\mathbf {t_c}$ , the descriptive statistics are presented in Fig.",
"REF as boxplots.",
"We tested the data for normality with the Shapiro-Wilk test, rejecting the null hypothesis ($ p < .05 $ ).",
"For each subset, we therefore ran a non-parametric Wilcoxon signed-rank test for differences between conditions.",
"Statistically significant differences were found, both for the overall population ($p < .01$ ), and for groups A and B ($p < .05$ ).",
"To test for potential effects of the ordering of conditions, we compared the completion times $\\mathbf {t_c}$ of groups A and B for each condition.",
"We therefore run a non-parametric Mann-Whitney U test and found the difference between the two groups for the online condition to be statistically significant ($p < .05$ ), as shown in Fig.",
"REF .",
"A possible explanation for this difference may be due to the discrepancy between the control of the virtual robot in comparison with the dynamics of the real robot.",
"As participants from group B had performed the offline condition first, we noticed that they took more time to adjust to the new dynamics than their counterparts.",
"Finally, no statistically significant difference between groups was observed for the offline condition ($p > .05$ )."
],
[
"User feedback",
" The scores of the Likert scale statements presented in Section REF are visualised in Fig.",
"REF for the whole study population.",
"As for the completion time $\\mathbf {t_c}$ , we looked for differences of questionnaire scores between the two groups with a Mann-Whitney U test, finding however no statistically significant differences ($p > .05$ ).",
"Figure: Average user ratings for the assertions provided in Section .Overall, the results indicate that the participants perceived the proposed AR interface as easy to understand and useful for robot programming.",
"However, out of the 18 participants, only 7 participants stated that they would use the interface for programming industrial robots.",
"While praising the easiness and quickness of use of the interface, the rest of the participants raised concerns about the interface's lack of accuracy, especially for manufacturing tasks such as insertion.",
"Most participants mentioned how it was difficult to perform precise robot motions by means of the dragging motion on the smartphone screen.",
"On the other hand, almost all participants stated that the virtual workspace provides a good representation of the real workspace and that the interface offers an intuitive way of planning trajectories as errors can easily be visualised and corrected."
],
[
"Discussion",
"The results presented in Section indicate a promising success rate for both online and offline programming with completion times $\\mathbf {t_c}$ ranging from 1 to 4 minutes.",
"Nevertheless, during the study we observed how certain aspects of the interface could have hindered the participants in accomplishing the task.",
"A general problem of AR interfaces is their weakness in the estimation and visualisation of depth [17], [18]; a limitation that requires the users to adopt coping strategies such as e.g., change their location to obtain different views of the scene and, consequently, a better perception of depth.",
"In our study, we indeed observed how the participants who adopted such strategies performed the disassembly tasks faster and more accurately than their counterparts.",
"This was especially true for the offline condition, where the issue of the AR interface with depth was particularly relevant.",
"Another issue indicated by the study's participants was in the perceived lack of accuracy during dragging motions across the screen.",
"This led to inaccurate or unexpected motions on the real robot, especially when the translation axis being manipulated was perpendicular to the surface of the device.",
"A potential solution to this issue would be to automatically disable the control of the robot along the axis being employed when the aforementioned condition is met.",
"While avoiding the problem of unexpected robot motions, this solution would also encourage the users to move around the workspace, in order to regain control of a certain axis, indirectly addressing the aforementioned issue of depth perception.",
"Finally, the user feedback also provided insights into various potential areas of research such as using the interface to control the real robot remotely, in a similar fashion to teleoperation.",
"Additionally, some participants stated that they would consider using the AR interface not only as a control tool but also as a visualisation/monitoring tool.",
"For example, in a learning from demonstration [1] scenario, the AR interface could be used to inspect the quality of the demonstrated trajectories or to visualise their variability with aptly placed Gaussians, in a way similar to [4]."
],
[
"Conclusion & Future work",
"In this paper, we presented an augmented reality interface for smartphones and/or tablets, enabling users to control a robot in real-time, to program it offline as well as to model a workspace by means of virtual objects.",
"The proposed interface aims to provide operators programming industrial robots with an alternative to common online programming methodologies, for which the presence of a real robot is not required and pre-stored workspaces and trajectories can be adapted to re-program a robot.",
"Future work will address the challenge of performing more accurate motions.",
"We will investigate the option of adapting the sensitivity of the dragging action on the interface, giving the required precision to the user when felt necessary.",
"Also, we will test the option of switching from the use of dragging motions to a button based system where users can affect the displacement along each axis individually.",
"This would enable users to perform rapid actions via the current dragging motions on the screen and then switch to the button based system when more accuracy is required.",
"Additionally, the ARCore software development kit currently lacks the ability to detect 3D objects.",
"Having such a feature would allow to automatically place virtual objects or obstacles, resulting in a faster creation of the virtual workspace.",
"We plan to integrate a 3D object detection pipeline in the interface to provide this capability.",
"Motivated by the promising results and the participants' feedback presented in Sections and , we also plan to investigate possible extensions of the approach to other robot applications beyond manufacturing."
]
] | 2107.01884 |
[
[
"Mirror Mirror on the Wall: Wireless Environment Reconfiguration Attacks\n Based on Fast Software-Controlled Surfaces"
],
[
"Abstract The intelligent reflecting surface (IRS) is a promising new paradigm in wireless communications for meeting the growing connectivity demands in next-generation mobile networks.",
"IRS, also known as software-controlled metasurfaces, consist of an array of adjustable radio wave reflectors, enabling smart radio environments, e.g., for enhancing the signal-to-noise ratio (SNR) and spatial diversity of wireless channels.",
"Research on IRS to date has been largely focused on constructive applications.",
"In this work, we demonstrate for the first time that the IRS provides a practical low-cost toolkit for attackers to easily perform complex signal manipulation attacks on the physical layer in real time.",
"We introduce the environment reconfiguration attack (ERA) as a novel class of jamming attacks in wireless radio networks.",
"Here, an adversary leverages the IRS to rapidly vary the electromagnetic propagation environment to disturb legitimate receivers.",
"The IRS gives the adversary a key advantage over traditional jamming: It no longer has to actively emit jamming signals, instead the IRS reflects existing legitimate signals.",
"In addition, the adversary doesn't need any knowledge about the legitimate channel.",
"We thoroughly investigate the ERA in wireless systems based on the widely employed orthogonal frequency division multiplexing (OFDM) modulation.",
"We present insights into the attack through analytical analysis, simulations, as well as experiments.",
"Our results show that the ERA allows to severely degrade the available data rates even with reasonably small IRS sizes.",
"Finally, we implement an attacker setup and demonstrate a practical ERA to slow down an entire Wi-Fi network."
],
[
"Introduction",
"Part of the ever-evolving digital landscape is growing demand for wireless connectivity at high data rates and low latency.",
"In addressing this need, increasingly sophisticated mobile communication networks are being deployed.",
"In particular, we are in the midst of the worldwide roll-out of 5G networks, which are the key-enablers for emerging applications such as, e. g., autonomous driving, smart cities, smart grids, and immersive entertainment [2], [19], [1].",
"Such applications will lead to an increased dependency on a wireless infrastructure with high availability and high attack resistance.",
"Specific to wireless networks is jamming of radio signals, which leads to denial of service and can pose a serious threat to, e. g., cellular networks such as 4G and 5G [26], [15], [3].",
"Figure: Illustration of the ERA setting where the attacker Eve uses an IRS to gain partial control over the wireless channel between legitimate parties Alice and Bob.",
"c i,k c_{i,k} and g i,k g_{i,k} are the channels to (and from) the IRS, d k d_k is the direct (non-IRS) channel, with the k th k^{th} OFDM subcarrier and i th i^{th} IRS element.Next-generation wireless networks make use of sophisticated communication technologies such as massive MIMO (massive multiple-input and multiple-output), which is now realized with 5G [6].",
"An even more recent example for a technological advance are intelligent reflecting surfaces (IRS) [43].",
"IRS consist of an array of electronically adjustable reflectors with respect to radio waves.",
"IRS enable smart radio environments [25], [36] to, e. g., enhance the wireless radio channel quality in terms of signal-to-noise ratio (SNR) [24] or spatial diversity [13].",
"However, the IRS is also a novel attacker tool for malicious purposes — an issue that has received only little attention as of yet.",
"In this work, we show that time-varying IRS allow to disrupt wireless communications by (smart) reflecting radio signals originating from the legitimate parties.",
"We introduce the environment reconfiguration attack (ERA), which can be viewed as a novel class of practical, low-cost, and low-complexity jamming attacks.",
"The essence of the ERA lies in high-speed IRS reconfigurations, which are digitally controlled by the attacker Eve.",
"In effect, the wireless propagation environment, i. e., the wireless channel, between the communication parties Alice and Bob (cf. Fig.",
"REF ) exhibits exceptionally fast and instantaneous changes that otherwise do not occur in nature.",
"In turn, severe variations are applied to signals coming from the legitimate transmitter which disturb the intended receiver.",
"A key difference to traditional jamming attacks is that the attacker does not actively emit a jamming signal but merely reflects signals generated by a victim party.",
"Accordingly, the ERA leads to correlated interference and dramatically simplifies the implementation of such attacks [27], as the attacker neither needs an RF transmitter nor a receiver.",
"Unlike previous work [29], the ERA does not require the attacker to have any channel knowledge and only rudimentary knowledge (such as the modulation scheme) about the communication system.",
"This crucial relaxation allows us to demonstrate the first real-world jamming attack based on IRS.",
"In this paper, we show that the IRS is a practical and low-cost attacker tool, enabling the ERA.",
"We investigate the attack using orthogonal frequency division multiplexing (OFDM) which is widely used in modern wireless networks, including 4G, 5G, and Wi-Fi.",
"We perform a thorough theoretical analysis to explain the fundamental attack mechanisms.",
"Furthermore, we show simulation results that allow us to characterize the attack requirements on signal power, distances and IRS dimensions.",
"Finally, we implement an attacker setup and demonstrate a practical ERA, slowing down an entire wireless network.",
"Our results show that the attack works with reasonably small IRS sizes, notably the used IRS has dimensions 40 $\\times $ 16.",
"Moreover, we provide a practical IRS optimization algorithm to enhance the attack performance.",
"In summery, building upon the advent of IRS, we introduce a new class of practical jamming attacks which are low-cost and can easily be deployed in many wireless scenarios.",
"The paper at hand contains the following key contributions: We propose the environment reconfiguration attack (ERA) as a novel class of jamming attacks, based on low-cost IRS.",
"We present a theoretical analysis explaining how the ERA affects OFDM communications.",
"We show comprehensive simulation results to determine the attacker requirements on signal power, distances and IRS dimensions.",
"We demonstrate a practical ERA on commodity Wi-Fi using a low-cost IRS prototype, allowing to substantially reduce the wireless throughput in the entire network.",
"We present an IRS optimization algorithm to further enhance the ERA jamming performance.",
"Background In this section, we provide technical background on the IRS, jamming attacks, and OFDM communications.",
"Intelligent Reflecting Surface An IRS is a synthetic planar structure with digitally reconfigurable reflection properties of electromagnetic (EM) waves.",
"In wireless communications, the IRS is a rather new concept that has evolved from physics research on metamaterials and metasurfaces [24] which are tailored to enable non-standard EM wave field manipulations.",
"More recently, the evolutionary step from the metasurface to the IRS has been made: Metasurface designs have been drastically simplified and became digitally controllable.",
"An IRS consists of many distributed identical unit cells, each of which reflects impinging EM waves.",
"Most importantly, the complex reflection coefficient of each element across the surface is individually programmable, allowing to influence the wireless channel of communication parties (see Fig.",
"REF ).",
"Practical IRS designs are often targeted to adjust only the signal phase with quantization as low as 1bit [48].",
"Thus, the IRS provides a simple digital interface towards the physical layer of wireless communications and enables what is coined smart radio environments [25] with novel applications such as, e. g., optimization of the signal-to-noise ratio (SNR) [5] or spatial diversity [13].",
"Since only ambient signals are reflected, the IRS is inherently energy efficient and does not require active RF chains.",
"Thus, IRS have low hardware complexity since manufacturing requires standard microstrip technology on low-cost printed circuit board (PCB) substrate.",
"Currently, the IRS is in discussion to complement future wireless infrastructure on a large scale in wireless networks beyond 5G [49].",
"Jamming Wireless communication relies on a broadcast medium that must be shared between many users.",
"In principle, each user is free to transmit at any time and thus, signals are by definition subject to interference.",
"Instead of just the desired signal, a receiver then additionally picks up an unwanted signal, disrupting the intended communication.",
"Despite regularly occurring interference from other user's communications, malicious parties can also launch jamming attacks.",
"Here, an attacker deliberately produces interference to disable the communication of targeted users.",
"Jamming attacks can be classified into a variety of different categories, including the type of interference and the strategy to trigger emission of the interfering signal [18].",
"A jammer may use noise signals, constant tones, or even valid waveforms.",
"Attackers can apply constant jamming or act reactively in order to disable only selected parts of the victim communication, such as physical control channels [15].",
"Orthogonal frequency division multiplexing (OFDM) Due to its unique properties, OFDM has become one of the most important and widely used modulation techniques in wireless networks [9], [16].",
"Most importantly, OFDM can cope with multipath signal propagation easily.",
"In order to push data rates, wide channel bandwidths need to be used.",
"However, when transmitting a wide-bandwidth signal over a wireless link, it will most likely experience some form of frequency selective attenuation due to fading from multipath signal propagation.",
"OFDM divides a wide bandwidth into numerous independent (say, orthogonal) narrowband channels, i. e., subcarriers, and can thus handle frequency selective channels at low computational complexity.",
"Taking the concept to the next level, OFDM based multiple access (OFDMA) schemes assign different subcarriers to different users.",
"Finally, the modulation and demodulation of OFDM are elegantly handled using an efficient (inverse) fast Fourier transform (FFT).",
"Today, OFDM has become the definitive transmission scheme for broadcasting, e. g., DAB and DVB, cellular systems, e. g., 4G and 5G, and personal networks, e. g., Wi-Fi.",
"Related Work In this section, we summarize the relevant literature on IRS and jamming attacks, and also describe how our work differs from previous proposals.",
"Intelligent reflecting surface.",
"The IRS has been widely recognized as a potential major innovation in wireless communications and has stimulated much research activity recently.",
"Hence, there is a manifold literature now.",
"Regarding key concepts and literature reviews, we refer to numerous overview works [5], [43], [44], [25].",
"To the best of our knowledge, previous works on IRS in a security context focus on theoretical aspects.",
"Most notably, Lyu et al.",
"[29] proposed the IRS for minimizing the signal power received by a victim party for jamming.",
"We further elaborate the similarities and differences to our work towards the end of this section.",
"Several works, e. g., [12] and [7], provide analytical and simulation results in the context of physical layer security assisted by an IRS.",
"Huang and Wang [21] discuss a pilot contamination attack using an IRS to increase signal leakage by reflecting pilot signals.",
"In [47], the authors pursue IRS to be used as a mitigation for active jamming attacks.",
"In the following we give examples for studies including practical IRS demonstrations with a focus on improving wireless communication.",
"An early work from 2014 is [24], where the authors demonstrate wave field shaping.",
"Work from 2019 [13] has shown that IRS are capable of enhancing spatial diversity.",
"Arun and Balakrishnan in 2020 [4] demonstrated a large prototype IRS with 3200 elements for passive beamforming applications.",
"In recent work of Pei et al.",
"[33], an IRS is used to achieve substantial channel improvements, enabling a long-range communication field trial over 500m.",
"Several works report practical IRS designs, e. g., [48], [22], [46].",
"Jamming attacks.",
"The literature widely recognizes jamming attacks as a risk to the reliability of wireless communications.",
"Several works have pointed out the threat of jamming against 4G [26], [15] and 5G [3] networks.",
"Grover et al.",
"[18] provide an overview on different jamming strategies, localization and detection techniques, and countermeasures.",
"However, the ERA does not fit any of the reported categories properly.",
"Poisel gives a highly comprehensive overview on all classes of jamming in his book [34].",
"Lichtman et al.",
"[27] provide a taxonomy for jamming attacks by defining four attacker capabilities time correlation, protocol awareness, ability to learn, and signal spoofing.",
"Following their categories, the ERA may be labeled as a partially time-correlated jammer.",
"However, unlike the author's category-based conjecture, the ERA is a low-complexity attack.",
"Hang et al.",
"[20] investigate repeater jamming against direct sequence spread spectrum (DSSS).",
"The ERA may indeed be seen as a special case of repeater jamming, as a reflection of the signal in fact is a time-varying copy of the legitimate signal.",
"Thus, the ERA is conceptually related.",
"In the ERA, however, the attacker eliminates RF receiver and transmitter chains and processing delays.",
"Pöpper et al.",
"[35] report a method to achieve jamming-resistant broadcast communications without shared keys.",
"The authors comment on the repeater jammer which could circumvent their security assumptions in some cases and also point to processing delays.",
"For our IRS-based approach, however, processing delays vanish.",
"Clancy [10] has pointed out that OFDM communications can be efficiently disrupted by jamming or nulling of pilot signals for channel estimation.",
"The ERA now provides a simple method to realize the manipulation of the OFDM equalizer.",
"Also, many works pursue detection of jamming, examples include [39], [8], [28].",
"A different body of work examines helpful aspects of jamming, e. g., to provide confidentiality [42].",
"However, Tippenhauer et al.",
"[40] have shown that jamming for confidentiality has fundamental security limitations.",
"Differentiation from previous work.",
"The general idea of maliciously using an IRS for jamming was first proposed by Lyu et al.",
"[29] in 2020, albeit in a very different manner that we believe results in a much lower practicality than the ERA.",
"The approach of [29] is based on an IRS to minimize the signal power received by a victim party – a method opposite to the classical IRS-based SNR improvement.",
"Here, the superposition of the direct signal and the malicious IRS signal shall result in destructive interference, i. e., the IRS signal is to be a phase-exact cancellation signal.",
"However, finding a specific IRS configuration to meet this goal is non-trivial.",
"Addressing this issue, the authors formulate an optimization scheme to obtain a corresponding IRS configuration from the channel states $c_{i,k}$ , $g_{i,k}$ , and $d_k$ , cf.",
"Fig.",
"REF .",
"Thus in this approach the attacker needs to have full knowledge of all involved channel states.",
"Unfortunately for an attacker, $d_k$ can only be found by the victim parties and obtaining $c_{i,k}$ and $g_{i,k}$ is infeasible (without a large number of additional RF receivers at the attacker's IRS), as recognized in the literature [5], [43], [44].",
"In contrast, the ERA approach presented in this paper works entirely different, thereby eliminating the unrealistic requirement of channel knowledge for the attacker.",
"Crucially, the attack leverages the IRS to rapidly toggle between (two) effective wireless channels.",
"In particular, we address OFDM receivers which get disturbed by the unnatural switching between channel states, e. g., partly due to adaptive behavior.",
"Our goal is not the minimization of the signal reception of one or both of the ERA channels.",
"Rather, the ERA exploits signal changes from the difference between the two ERA channels as a source of interference.",
"Thus, the attack neither requires synchronization or phase-exact knowledge of all channels, and thereby avoids a location-dependent attack performance (signal phase changes by movement), as our experimental results show.",
"In order to compare the two attack strategies, we would like to point out that a cancellation approach [29] is equivalent to reducing the SNR – an aspect that we readily cover in our simulations in Section REF , showing that the ERA can achieve substantially increased jamming performance.",
"Attack Overview Figure: Illustration of the ERA, indicating the legitimate communication and the adversarial IRS operation.",
"The attacker toggles the IRS configuration rapidly to disturb the legitimate receiver.Parties.",
"In this work, we consider a physical layer attacker Eve trying to disrupt the wireless radio communication of two legitimate parties Alice and Bob who deploy a conventional OFDM-based wireless communication system.",
"Thus, Alice and Bob may use Wi-Fi, 4G, or 5G and could represent a base-station and an end-user, respectively.",
"The attacker Eve has full control over an IRS which is part of the wireless propagation channel between Alice and Bob.",
"Eve is capable of applying custom configurations to the IRS at update rates comparably to the symbol rate used by Alice and Bob.",
"Apart from that, we grant the attacker basic wireless eavesdropping capabilities, i. e., the attacker possesses a wireless receiver and can receive and demodulate signals of Alice and Bob.",
"However, Eve does not have a wireless transmitter and thus cannot transmit any signals on itself.",
"Finally, our system and attacker model is illustrated in Fig.",
"REF .",
"Note that the attacker operates at the physical layer and therefore we do not need to take the cryptography applied at the upper layer of the user's communication into account.",
"Attack and overview of investigation.",
"In the ERA, the attacker Eve uses a software-controlled surface, i. e., an IRS, to rapidly vary the wireless radio channel between Alice and Bob.",
"This yields fast and instantaneous variations in the legitimate signals that normally would not occur in nature.",
"Disturbed by the anomalous signal behavior, the intended receiver fails to correctly demodulate the incoming signals, leading to a denial of service.",
"In this work, we design an ERA against OFDM communications by rapidly toggling between two distinct IRS configurations.",
"An illustration of the corresponding attacker action is shown in Fig.",
"REF .",
"Compared to classical jamming attacks, the ERA allows attackers to silently disable the wireless communications of victim parties, i. e., the attacker does not actively generate a jamming signal.",
"Instead, it manipulates signals transmitted by Alice and Bob during propagation.",
"We begin our investigations by examining the fundamental attack mechanisms in an analytical analysis (Section ).",
"Here, we lay the foundations of the attack and show that ERA-induced fast channel variations are harmful for wireless OFDM communication.",
"We then turn to a simulation model (Section ) of an end-to-end wireless OFDM link.",
"From the simulation, we deduce several key factors of the attack, such as, e. g., signal power and attacker distances.",
"For both theoretical analysis and simulations, we abstract the effect of the adversarial IRS as a time-varying signal component and omit the impact of specific IRS patterns.",
"Finally, we use a practical IRS implementation to design and evaluate real-world ERAs to demonstrate successful jamming attacks (Section ).",
"In the first and simplest variant, we rapidly toggle the IRS patterns by either setting all elements to '0' or '1'.",
"This attack is of remarkably low complexity and requires nothing more than a certain proximity between the attacker and a victim party.",
"The second attack variant is more advanced and includes an optional setup phase where the attacker optimizes the two IRS patterns to increase the jamming efficiency.",
"This procedure incorporates the channel state information (CSI) from Alice and Bob, as provided by CSI feedback signals in existing wireless standards.",
"Theoretical Analysis In this section, we present a theoretical analysis of the mechanisms underlying the ERA against OFDM communications.",
"We outline that the ERA affects channel equalization from outdated channel estimations and subcarrier orthogonality.",
"Modelling Preliminaries We begin our considerations by introducing the models for the legitimate OFDM communications and the IRS attacker.",
"OFDM We assume that Alice and Bob generate their RF transmit signals using a modulator fed by conventional complex-valued in-phase and quadrature (IQ) baseband signals [16].",
"The baseband signals for OFDM are generated by taking the inverse discrete Fourier transform of a block of $K$ complex modulated data symbols $X_k[n]$ for all $k= 0,\\ldots , K-1$ subcarriers, yielding the $n^{th}$ OFDM symbol.",
"For instance, the data symbols contained in $X_k[n]$ may be modulated using, e. g., binary phase shift keying (BPSK) or quadrature amplitude modulation (QAM) of arbitrary order.",
"Then, in the time domain, a cyclic prefix is prepended to each OFDM symbol.",
"At the receiver side (see Fig.",
"REF ), after time- and frequency synchronization, removal of the cyclic prefix, and discrete Fourier transform, the received baseband signal on the $k^{th}$ subcarrier of the $n^{th}$ OFDM symbol in the frequency domain is given by: $Y_k[n] = H_k[n]\\ X_k[n] + Z_k[n], $ where $H_k[n]$ is the complex channel gain of the link between Alice and Bob for the $k^{th}$ subcarrier, and $Z_k[n] \\sim \\mathcal {CN}(0,\\sigma ^2)$ is additive white Gaussian noise (AWGN).",
"Following the implementation of practical systems, we assume that (known) pilot symbols are transmitted with a preamble to allow channel estimation at the receiver side.",
"The pilot symbols are populated on each of the $K$ subcarriers of the $n^{th}$ OFDM symbol (i. e., block-type pilot arrangement [11]) and allow Alice and Bob to obtain CSI using, e. g., a standard Least-Squares (LS) channel estimator: $\\hat{H}_{k}[n] = \\frac{Y_k[n]}{X_k[n]} = H_k[n] + \\frac{Z_k[n]}{X_k[n]} = H_k[n] + \\tilde{Z}_k[n].",
"$ The channel estimate then is used to equalize the subsequently received OFDM symbols: $\\hat{X}_k[n] = \\frac{Y_k[n]}{\\hat{H}_{k}[n]} $ Intelligent Reflecting Surface We now establish the model for OFDM wireless communication in the presence of an IRS.",
"We assume an IRS consisting of $N$ identical sub-wavelength-sized elements, arranged in an array on a planar surface to reflect impinging waves with a programmable phase shift.",
"The generalized reflection coefficient for the $i^{th}$ IRS element can be expressed as: $r_i = \\alpha _i e^{j \\phi _i} \\qquad i = 1,...,N,$ where we assume $\\alpha _i = 1$ and $\\phi _i \\in [0, 2 \\pi )$ .",
"Note that the IRS used in the experiments in Section is a binary phase-tunable IRS, i. e., then $\\phi _i \\in \\lbrace 0, \\pi \\rbrace $ and $r_i \\in \\lbrace -1, 1\\rbrace $ which correspond to '0' and '1' states of the IRS control signal.",
"Next, following the illustration in Fig.",
"REF , we find an expression for the channel between Alice and Bob, taking the IRS contribution into account.",
"Here we assume that the non-IRS channel is static and therefore denote the IRS as only source of channel variation depending on $n$ .",
"The effective channel between Alice and Bob in (REF ) then is: $H_k[n] = H_k^{IRS}[n] + d_k = \\sum _{i=1}^N c_{i,k}\\, r_i[n]\\, g_{i,k} + d_k,$ where $c_{i,k}, g_{i,k}, d_k \\in \\mathbb {C}$ , respectively, are the complex channel gains of the link between Alice and the $i^{th}$ IRS element, Bob and the $i^{th}$ IRS element, the direct link between Alice and Bob for the $k^{th}$ subcarrier (cf.",
"Fig.",
"REF ).",
"Figure: Block-diagram of a typical OFDM receiver architecture.",
"Analytical Analysis We now proceed to show how the fast channel variations invoked by the ERA will impact OFDM wireless communication.",
"Channel Equalization A fundamental part of every OFDM receiver (cf.",
"Fig.",
"REF ) is the channel estimation that is mandatory to equalize the received data symbols [9].",
"As previously outlined, operating an IRS allows the attacker to alter the wireless channel between Alice and Bob which will thus likewise affect the channel equalization.",
"We assume the non-IRS channel $d_k$ is static and Eve switches between two IRS configurations $r^{(0)}_{i}$ and $r^{(1)}_{i}$ , corresponding to the channels $H^{(0)}_{k}$ and $H^{(1)}_{k}$ .",
"Now consider the pilot symbols for channel estimation have been transmitted with the malicious IRS configured as $r^{(0)}_{i}$ .",
"Using (REF ), the victim receiver obtains the following channel estimate: $\\hat{H}_k[n] = H^{(0)}_{k} + \\tilde{Z}_k[n].$ Now, Eve switches the IRS configuration to $r^{(1)}_{i}$ , changing the channel of the subsequent OFDM symbols to $H^{(1)}_{k}$ .",
"Thus, the victim receiver's equalizer, cf.",
"(REF ), will operate with an outdated channel estimation: $\\hat{X}_k[n] = \\frac{Y_k[n]}{\\hat{H}_k[n]} = \\frac{X_k[n]\\ H^{(1)}_{k} + Z_k[n]}{H^{(0)}_{k} + \\tilde{Z}_k[n]},$ leading to a symbol error of ek[n] = Xk[n] - Xk[n] = Xk[n] ( H(1)k - H(0)k - Zk[n] ) + Zk[n]H(0)k + Zk[n].",
"For high SNRs, which is a reasonable assumption when using LS channel estimation, the symbol error is approximated by $e_k[n] \\approx X_k[n] \\frac{ H^{(1)}_{k} - H^{(0)}_{k} }{H^{(0)}_{k}} = X_k[n] \\frac{ H^{IRS,(1)}_{k} - H^{IRS,(0)}_{k}}{ H^{IRS,(0)}_{k} + d_k }$ The resulting expression in (REF ) tells us that the IRS-induced symbol error is proportional to ($i$ ) the transmitted symbol, ($ii$ ) the difference between the two IRS channels, and ($iii$ ) is inversely proportional to the direct channel contribution.",
"Thus, the attacker can maximize its chance of causing a false symbol decision by producing a pair of IRS channels, e. g., ${H^{IRS,(1)}_{k} = -H^{IRS,(0)}_{k}}$ .",
"In particular, this can be achieved by inverting the sign of all IRS reflection coefficients $r_i$ .",
"Thus, we likewise adopt this approach in our simulations and experiments in Sections and .",
"Intercarrier Interference OFDM systems in general are susceptible inter-carrier interference (ICI) which is caused by a degradation of subcarrier orthogonality.",
"ICI usually results from imperfections such as Doppler shifts, frequency offsets, and channel variations during an OFDM symbol period [16], [9].",
"We emphasize that the time-varying IRS used in the ERA will deliberately introduce rapid and instantaneous channel variations at sub-symbol timing resulting in substantial ICI.",
"To model the ICI, (REF ) is modified to account for the interference $H_{k,k^{\\prime }}$ from other subcarriers $k^{\\prime } \\ne k$ to the received OFDM signal on the $k^{th}$ subcarrier [9]: $Y_{k}[n] = H_{k}[n] X_{k}[n] + \\underbrace{\\sum _{k^{\\prime } \\ne k}^{} H_{k,k^{\\prime }}[n] X_{k^{\\prime }}[n]}_{\\textrm {ICI}}\\ +\\ Z_{k}[n].$ In Appendix we show that if the ERA-induced fast channel variations are zero-mean over one OFDM symbol, the signal-to-interference ratio (SIR) on the $k^{th}$ subcarrier is given by $SIR_k = \\frac{S_k}{I_{IRS}}= \\frac{|d_{k}|^2}{P_{IRS}},$ which means that the IRS does not contribute to the direct signal power $S_k$ , but the total power received from the IRS, $P_{IRS}$ , completely translates into ICI, $I_{IRS}$ , only.",
"Most importantly, this result is valid even without any optimization of the IRS elements with respect to the channels of the legitimate parties.",
"Simulation Results After having analytically outlined the key mechanisms of the ERA affecting an OFDM system, we now strive to further explore the attack through simulations.",
"We give comprehensive results, identifying attack parameters, including signal power, attacker distance, and IRS dimensions.",
"Further, we show that the ERA leads to significant packet error rates (PER) and is way more efficient when compared with a classical jamming attack using noise signals.",
"As an example for general OFDM-based radio systems, we consider Wi-Fi here, since our experimental investigation following in Section also builds upon Wi-Fi devices.",
"As the underlying simulation environment, we choose the MATLAB WLAN toolbox due to the availability of end-to-end simulation capabilities for the entire IEEE 802.11n physical layer, including channel coding and standard-compliant channel models.",
"We summarize the essential simulation parameters in Table REF .",
"To mimic the adversarial IRS operation in the ERA, we add time-varying reflection, i. e., a complex square wave signal from the IRS, to one tap of the CIR.",
"Further, we randomize the time instant of the packet start with respect to the IRS modulation.",
"For fairness in comparing the error rates across different modulation and coding schemes (MCS), we adjust the packet payload sizes to always result in 16 entire OFDM data symbols, regardless of the MCS setting.",
"Wi-Fi uses an OFDM symbol duration of ${4}{}$ and thus, the data portion of transmitted packets has a duration of 64.",
"Like traditional jamming attacks, the ERA is subject to link budget constraints.",
"Thus, the attack efficiency depends on the signal power arriving at the receiver from the attacker.",
"Although in the ERA the attacker does not generate a jamming signal itself, we can still define a jamming-to-signal ratio (JSR) as the ratio of IRS signal to direct (non-IRS) signal powers $JSR = \\frac{P_{IRS}}{S}=\\frac{P_{IRS}}{\\sum _{k}S_k}.$ For our simulations below, we use the JSR to assess the attacker strength.",
"As an indication for the attacker's success, we leverage the PER.",
"Table: Summary of the simulation parametersAttacker Signal Power Figure: End-to-end PER simulation results for IEEE 802.11n Wi-Fi under an ERA with 30kHz over varying JSRs for various modulation and coding schemes.We investigate the victim PER performance as a function of the JSR for various MCS settings.",
"Therefore, we assume the attacker signal originating from the IRS to have constant power while periodically toggling the phase between 0 and $\\pi $ at a rate of 30kHz, as is the case when inverting the sign of all IRS reflection coefficients $r_i$ .",
"The legitimate receiver has a high SNR of 50dB.",
"We plot the PER results for MCS 0 - 7 (covering BPSK, QPSK, 16-QAM, and 64-QAM modulations on the subcarriers ) as a function of the JSR in Fig.",
"REF .",
"As expected, higher order modulations are more prone to interference from an ERA.",
"The results also highlight that the ERA indeed is capable of producing error rates which render reliable wireless communication impractical.",
"To relate the ERA performance to classical noise-based jamming or signal power reduction attacks [29], we compare the attack against an SNR reduction.",
"For the ERA, we now consider the legitimate receiver to have an otherwise noise-free channel.",
"For the SNR reduction, we consider the IRS to remain static while the attacker now deteriorates the SNR by adding noise with power equivalent to the IRS signal strength during the ERA.",
"We plot the PER simulation results in Fig.",
"REF , which indicates that the ERA achieves considerably better jamming performance when compared to a noise jammer at the same power.",
"Figure: End-to-end PER simulation results for IEEE 802.11n Wi-Fi to compare an ERA against SNR reduction, e. g., from noise jamming or signal power reduction.",
"For the ERA case, we assume a noise-free channel.",
"Channel Modulation Frequency To fully characterize the ERA, we vary the IRS modulation frequency.",
"We conduct the simulation for MCS indicies 0 - 7 at an SNR of 50dB for the channel between Alice and Bob and a JSR of -10dB.",
"We plot the PER simulation results in Fig.",
"REF against the IRS update frequency.",
"For the MCS indices 0 and 1, we observe particularly lower PERs due to the more robust modulation parameters.",
"Despite that, the PER clearly increases as a function of the modulation frequency for all MCS values.",
"The increasing PER at lower modulation frequencies can be explained by the increasing probability of an IRS reconfiguration taking place during packet transmission.",
"That is, the packet error rate resulting from an ERA with IRS pattern durations $T_{IRS}$ longer than the packet duration $T_p$ is upper bounded by $T_p/T_{IRS}$ .",
"As the PER for modulation frequencies above approximately 16kHz reaches a plateau, we conclude that at least one IRS reconfiguration during transmission of the data symbols suffices to achieve the maximum attack efficiency for a certain JSR.",
"Figure: End-to-end PER simulation results for IEEE 802.11n Wi-Fi for the ERA over channel modulation frequency for varying modulation and coding schemes at an SNR of 50dB with JSR of -10dB.",
"Surface Size Figure: Simulation of the minimum surface size requirement for to achieve a JSR of -10dB.",
"(a) Geometrical configuration used for the simulation, indicating the relative positions of Alice, Bob, and Eve's IRS.",
"(b) Minimum IRS size versus d AB d_{AB} for varying attacker distances d EA d_{EA}, assuming free-space path loss at 5.35GHz.We will now show that an ERA is feasible even for rather weak attacker configurations regarding the attacker distance and IRS dimensions.",
"Previously, we have determined the JSRs necessary for the attacker to degrade the PER of Alice and Bob (see Fig.",
"REF ).",
"Note that we define the JSR as the ratio of the signal power coming from the IRS and the direct (non-IRS) signal power.",
"Thus, the attacker generally seeks to pick up sufficient power from the legitimate users.",
"The attacker can either minimize the distance to one of the victim parties to minimize path loss or increase the IRS size.",
"Although both strategies are suitable, we assume the attacker must maintain a minimum distance and also cannot increase the IRS size arbitrarily without raising suspicion.",
"Hence, we derive a connection between JSR, attacker distance, and the surface size.",
"For the parties, we assume the geometrical configuration shown in Fig.",
"REF (a).",
"We start with the free-space path loss of the direct link between Alice and Bob [16], where the received power is proportional to $L_d = \\left( \\frac{\\lambda }{4 \\pi d_{AB}} \\right)^2,$ with the carrier frequency wavelength $\\lambda = c_0/f$ .",
"For an optimal surface configuration, the free-space path gain from Alice to Bob via the IRS is found by [32]: $L_{IRS} = \\left( \\frac{A_{IRS}}{4 \\pi d_{AE} d_{EB}} \\right)^2.$ Assuming Alice and Bob use omni-directional antennas, the JSR becomes $JSR = \\frac{L_{IRS}}{L_d} = \\left( \\frac{A_{IRS}\\ d_{AB}}{d_{AE} d_{EB} \\lambda } \\right)^2,$ which allows us to link the surface area $A_{IRS}$ to the JSR: $A_{IRS} = \\sqrt{JSR} \\frac{d_{AE} d_{EB} \\lambda }{d_{AB}}$ We use Equation (REF ) to plot the minimum IRS size required by an attacker to achieve a JSR of -10dB in Fig.",
"REF (b).",
"We show the result as a function of the distance between Alice and Bob and for distances 1m, 2m, 10m, and 20m of Eve to Alice.",
"Consider, for example, Alice and Bob are at a distance of 30 and Eve is at a distance of 10 to Alice.",
"Then, an IRS size of only 0.19 is sufficient to achieve a JSR of -10, which results in a severe PER degradation for Alice and Bob.",
"Experimental Evaluation After having approached the ERA through theoretical analysis and simulations in the previous sections, we now proceed with a practical evaluation of the ERA.",
"Therefore, we first describe our experimental setup comprising of a low-cost IRS prototype and commodity Wi-Fi devices.",
"Furthermore, we demonstrate that the ERA is capable of severe link quality degradation, leading to a significant reduction in the effective wireless data throughput.",
"Experimental Attack Setup In this section, we present our experimental attack setup consisting of a prototype IRS and two microcontrollers.",
"We estimate the cost of the setup to be around 10040 for microcontroller development boards, 30 for PCBs, 30 for surface-mount components.. IRS Prototype Figure: (a) Unit cell schematic and dimensions.",
"(b) Unit cell phase response over frequency.Figure: Intelligent reflecting surface prototype module.",
"(a) Front view with patch elements (20cm x 16cm).",
"(b) Back view with control lines, PIN diodes, and biasing circuitry.As the essential part of a first exploration of the ERA in practical experiments, we use two low-cost IRS prototype modules (see Fig.",
"REF (a)) with 128 binary-phase tunable unit-cell elements in total, arranged in a $16 \\times 8$ array on standard FR4 PCB substrate.",
"The elements are rectangular patch reflectors on top of a ground plane.",
"Attached to each element, there is a PIN diode which can switch a parasitic element to the reflector, allowing to shift its resonance frequency.",
"Thereby, the reflection coefficient of each element can be individually switched between two states, i. e., a '0' state and a '1' state, by turning the control voltage to the reflector element either on or off.",
"The unit cell circuitry and the reflector design are shown in Fig.",
"REF (a).",
"The IRS prototype used in our experiments is optimized to achieve a 180 phase difference in the reflected wave for the '0' and '1' states (see Fig.",
"REF (b)), i. e., $r_i \\in \\lbrace -1, 1\\rbrace $ in (REF ).",
"IRS Modulation As we strive for rather high IRS modulation frequencies, we drive the 128 IRS elements in parallel.",
"Therefore, we connect each of the 128 control lines to a GPIO pin of two STM32F407 microcontrollers, allowing us to achieve IRS modulation frequencies of up to 1.6.",
"The frequency and surface patterns used for the modulation are programmable from the host controller through an UART serial communication interface.",
"Like in the theoretical analysis and the simulations, cf.",
"Section , we apply a simple binary surface modulation.",
"That is, we periodically toggle between two IRS configurations and thereby maintain a low attack complexity.",
"For instance, we switch between all 128 IRS elements either set to the '0' or '1' state.",
"As discussed in Section , since $r_i \\in \\lbrace -1, 1\\rbrace $ , this leads to switching between two channels $H^{(0)}_{k}$ and $H^{(1)}_{k}$ , with ${H^{IRS,(1)}_{k} = -H^{IRS,(0)}_{k}}$ .",
"Wireless Throughput Measurement Figure: Floorplan of the office space used for throughput measurements, indicating the positions of the WLAN router (access point), the attacker setup, as well as each of the 37 throughput measurement positions.Figure: Throughput measurement results from testing download speeds at 37 positions in the office space with and without the ERA taking place.We now demonstrate that the ERA is capable of significant throughput reduction in entire wireless networks.",
"Therefore, we deploy a commercial off-the-shelf WLAN router to provide an IEEE 802.11ac network in an office space.",
"We position the attacker setup strategically at the router with distances of 1 and 2.",
"We detail and summarize the setup in Table REF .",
"For the experiment, we use a laptop connected to the Internet via the Wi-Fi network to measure the effective end-to-end speed of the connection .",
"We perform speed measurements without the ERA (the malicious IRS remains static) and with the ERA enabled (switching all IRS elements between '0' or '1' state).",
"We repeat this procedure for a total of 37 positions distributed throughout the office space, as indicated in Fig.",
"REF .",
"We show the results of the throughput measurements in Fig.",
"REF .",
"Here we can see that the ERA leads to an average throughput reduction of 78 % and 40 % for the attacker at 1 and 2 distance to the router, respectively.",
"Recall that the attacker does not actively emit any jamming signal to achieve this result.",
"Furthermore, the attacker does not perform any kind of synchronization to the legitimate signals or optimization of the IRS configurations.",
"Notably, the ERA also leads to substantial throughput reduction where the wireless channel between the client and the IRS is obstructed, i. e., in different rooms with walls in between.",
"Thus, we conclude that the ERA is a scalable attack, allowing the attacker to slow down the wireless network at many different places.",
"Table: Summary of the experimental setupFigure: Experimental ERA setup with WLAN router and attacker IRS.",
"Systematic Packet Error Rate Measurement We perform a second experiment to systematically assess the practical effectiveness of the ERA, aiming to obtain PER measurements similarly to our simulation result from Section REF .",
"Therefore, we deploy single-board computers equipped with ath9k-based network interface cards (NICs) [45] for IEEE 802.11n Wi-Fi at the legitimate parties Alice and Bob.",
"The NICs give us low level access to the Wi-Fi communication, i. e., we can transmit packets with defined length and MCS setting.",
"Here, we use a 2x2 MIMO configuration with off-the-shelf Wi-Fi antennas.",
"One of the parties provides a Wi-Fi network on channel 60 (at 5300), allocating 40 bandwidth.",
"We place the attacker setup attacker at distance 2 and 3 in line-of-sight to Alice and Bob, respectively.",
"The channel between Alice and Bob also has line-of-sight conditions.",
"For the whole duration of the experiment, the propagation environment remains static apart from the adversarial IRS operation.",
"In our setup, Alice transmits 20000 packets with randomized payload data to Bob.",
"For each transmission, we configure the payload size and the MCS setting.",
"Similarly to the simulation, we adjust the payload size to always result in 9 entire OFDM symbols (data symbol duration 3.6, packet duration 6.8).",
"On Bob's side, we count the number of successfully received packets to finally obtain the PER.",
"We plot the PER results as a function of the adversarial IRS modulation frequency in Fig.",
"REF (a).",
"Also, we indicate the previously discussed upper PER bound given by $T_p/T_{IRS}$ for $T_{IRS} > T_p$ .",
"Essentially, our measurement with standard Wi-Fi NICs confirms our previous simulation results, showing that higher-order modulations are more susceptible to the ERA.",
"However, instead of reaching a plateau, we observe a drop in the PER when increasing the IRS modulation frequency beyond 30kHz.",
"We believe that this effect is due to hardware imperfections on the IRS prototype which initially was not designed to operate at high modulation speeds.",
"As evident from the results, the upper PER bound based on the timing parameters holds.",
"However, despite the fixed packet time duration, it appears that our bound seems to be too optimistic for MCS values below 12.",
"We attribute this to reduced synchronization efforts, i. e., the receiver will barely be affected by an IRS change during the packet's preamble portion, reducing the effective ERA-sensitive packet length.",
"Figure: Measured PER over channel modulation frequency.",
"(a) Binary pattern modulation.",
"(b) Tailored pattern modulation.Surface Pattern Optimization Thus far, we have tested the simplest ERA strategy where the attacker switches all surface elements periodically between the '0' or '1' states.",
"However, this strategy can be further improved by matching the used IRS configurations to the wireless link under attack.",
"Thus, the attacker may prepend its jamming operation with a setup phase in order to optimize the IRS configurations used during the subsequent ERA.",
"The attacker therefore can incorporate eavesdropped CSI feedback of the victim parties to further enhance the attack efficiency.",
"For a first demonstration, we design and test an adaptive optimization algorithm to find IRS configurations well-suited for the ERA.",
"The intuition of the algorithm is to use the adversarial IRS for maximizing a dissimilarity measure between the pair of IRS-induced channel responses of the victim wireless link.",
"Following our analytical analysis in Section , we expect this to improve the attacker's success.",
"Algorithm REF outlines the procedure.",
"The result are two IRS configurations $r^{(0)}_{i}$ and $r^{(1)}_{i}$ .",
"Note that we here denote the binary surface control settings ('0' or '1') as a proxy for reflection coefficients.",
"[ht] Distinct IRS configurations $r^{(0)}_{i}$ , $r^{(1)}_{i}$ for ERA.",
"start with random $N$ -bit IRS configurations $r^{(0)}_{i}, r^{(1)}_{i}$ dissimilarity metric $d$ algorithm rounds $R=2$ $j = 0$ to $R$ configure IRS as $r^{(1)}_{i}$ $ref^{(1)} \\leftarrow H_k(r^{(1)}_{i})$ configure IRS as $r^{(0)}_{i}$ $i \\leftarrow 0$ to $N$ $ref^{(0)}_{i,0} \\leftarrow H_k(r^{(0)}_{i})$ $r^{(0)}_i \\leftarrow r^{(0)}_i \\oplus 1$ update IRS element $i$ $ref^{(0)}_{i,1} \\leftarrow H_k(r^{(0)}_{i})$ $d(\\textrm {ref}^{(1)}, ref^{(0)}_{i,0}) > d(\\textrm {ref}^{(1)}, ref^{(0)}_{i,1})$ $r^{(0)}_i \\leftarrow r^{(0)}_i \\oplus 1$ update IRS element $i$ swap($r^{(0)}_{i}$ , $r^{(1)}_{i}$ ) Adversarial binary surface optimization The randomly chosen initial IRS configurations in Algorithm REF are given below: $r^{(0)}_{i}$ = 0x5CC81D86E5DAB902B071665D1D7DC2F1 $r^{(1)}_{i}$ = 0xC859CCA60594481B193BF3D236E877AE The result of the algorithm are the updated IRS configurations: $r^{(0)}_{i}$ = 0xFFFF9F9F08089E08474721D92AC1B57A $r^{(1)}_{i}$ = 0x00006060E5D776A2F8B876020C034C05 Figure: Evolution of Euclidean distance between the channel responses during the iterative IRS optimization.Fig.",
"REF shows the evolution of the Euclidean distance between $|H_k(r^{(0)}_{i})|$ and $|H_k(r^{(1)}_{i})|$ over the iteration steps, clearly exhibiting the characteristic behaviour of our algorithm.",
"Finally, we also plot the pair of channel responses as observed by Alice and Bob before and after the optimization in Fig.",
"REF .",
"Here, we can see that our procedure indeed is highly effective in providing distinct channel responses designated to be used in the ERA.",
"Note that even though the reception for $|H_k(r^{(0)}_{i})|$ has improved after running the algorithm, the difference between the two channel states is maximized.",
"The result is a vivid example for the combination of inherent simplicity and possibilities of the IRS for previously infeasible attacks.",
"Figure: Effective normalized channel responses observed by Alice and Bob, before and after running the adversarial IRS optimization algorithm.Using the presented algorithm with the Euclidean distance as a metric and magnitude CSI information on the link between Alice and Bob, we obtain the adapted IRS configurations $r^{(0)}_{i}$ and $r^{(1)}_{i}$ , which we now use to conduct the ERA.",
"We repeat the PER measurement experiment from the previous section and plot the results in Fig.",
"REF (b).",
"Here it is evident that the optimization was able to improve the attacker efficiency.",
"Now, even the robust BPSK modulation for MCS 8 exhibits a significant PER induced by the ERA.",
"Further, the optimization has also led to substantially increased PERs for the remaining MCS values.",
"Discussion In this section, we discuss ($i$ ) the real-world applicability, ($ii$ ) the attacker capabilities, and ($iii$ ) reason about countermeasures and mitigation.",
"Also, we give directions for future work.",
"Real-world Applicability We assess the costs and complexity of an ERA to be low.",
"Our results show that a sub 100 attacker setup can have significant impact on the effective wireless throughput.",
"Once an attacker possesses a functional IRS, only basic microcontroller programming is required to rapidly vary a number of logic signals controlling the IRS.",
"Thus, the attack can be easily carried out by non-specialists.",
"While the commercial availability of IRS devices is currently still limited, several companies , are working on product-grade IRS implementations.",
"Besides that, many IRS designs are publicly available and can easily be reproduced by others using cheap PCB assemblies.",
"Instead of using an own IRS, an attacker could also hijack existing IRS infrastructure which may be deployed in future wireless networks [49], most likely already at strategically advantageous positions.",
"Attacker Capabilities To conduct an ERA, the attacker's IRS must be within the wireless propagation environment between the victim nodes.",
"As wireless communication is inherently supposed to bridge distances this will not be a hurdle for an attacker.",
"As discussed, the JSR is an important parameter bounding the attack performance.",
"In order to improve its JSR, the attacker can choose a favorable position or increase the IRS size.",
"Therefore, to compensate the small size of our IRS prototype, we have used rather short attacker distances in our experiments, which still represents a valid attacker model.",
"Our simulation results show that sufficient JSR values are, in principle, still possible for higher attacker distances and surface sizes.",
"However, this also reveals a limitation of ERA: the attacker is passive and cannot amplify the signals it reflects.",
"Hence, as it is generally the case for wireless communications (and jamming), the attack is limited by the available link budget.",
"Our simulation results show the underlying relationship between JSR and PER.",
"For this purpose, we have simplified the attacker's signal originating from the IRS to a time-varying signal component from alternating the sign of the IRS reflection coefficients.",
"Although finding a corresponding IRS configuration to meet a certain JSR is non-trivial, our practical tests tests show that even with a binary-phase tunable IRS and without optimized surface configurations, the ERA significantly disrupts the victim communication.",
"In Section REF , we have granted the attacker access to the CSI of Alice and Bob to demonstrate that an attacker can further optimize the IRS configurations used during the ERA.",
"In an actual attack, the attacker would rely on eavesdropping CSI feedback, e. g., from the user to the base station.",
"For instance, this is commonly used in IEEE 802.11 WLAN standards, 4G, and 5G to implement, e. g., transmit beamforming [15], [37], [23], [14].",
"Note that, in the standards mentioned, these signals are not encrypted.",
"Countermeasures The ERA is based on an IRS within the channel between Alice and Bob.",
"For the attack to work, a part of the transmitted signal must reach the receiver via the adversarial IRS.",
"Due to the broadcast nature of wireless signal propagation, it is likely that an ERA cannot generally be prevented.",
"The transmitter could use beamforming to diminish the attacker's success, trying to minimize the signal power reaching the IRS.",
"However, this requires a mechanism for attack detection and localization and an advanced attacker may even leverage beamforming to its favor by providing a preferred path via the IRS to the receiver.",
"Since the interference signal produced in the ERA is correlated to the useful signal, it may also be possible to find signal processing-based countermeasures at the receiver side.",
"However, we emphasize these considerations are speculative.",
"Countermeasures, if they exist, cannot be implemented immediately in end-user equipment because the very low-level signal processing of radio transceivers is usually implemented in hardware or is not updatable.",
"Finally, to mitigate the attack, wireless communication systems could apply encryption of physical layer control channels, i. e., to prevent the attacker to obtain CSI feedback.",
"However, this will not render the ERA infeasible, but would only impede an adversarial IRS optimization.",
"Moreover, this requires drastic changes to protocols and such measures can likely only be implemented within future standards.",
"Future work In this paper, we have presented a novel class of jamming attacks based on IRS-induced fast changes in the radio propagation environment of wireless communication parties.",
"Naturally, this work only represents a very first exploration of the ERA and, more broadly, the IRS as a toolkit for practical wireless physical layer attacks.",
"Therefore, our work may serve as a basis for future work studying, for example, the following aspects.",
"Improving the attack.",
"We have provided first insights into the optimization of the IRS configuration for an ERA, demonstrating the potential for increased attack efficiency.",
"The evaluation of improved optimization algorithms based on eavesdropping CSI feedback is left for future work.",
"Also, future work should investigate non-binary surface modulation signals where the attacker uses more than two IRS configurations.",
"Finally, there is room for hardware improvements to the attacker setup, perhaps through dedicated IRS designs for high modulation frequencies.",
"Attack detection and countermeasures.",
"More work is needed to examine whether existing jamming attack detection and mitigation strategies, e. g., [18], can be adapted to the ERA.",
"Also, we see a need to evaluate the possibility of signal processing based mitigation strategies that could be incorporated into future transmitter and receiver architectures.",
"Application to other modulations.",
"We have outlined the ERA against OFDM communications, as it is the preferred modulation scheme for modern wireless communication systems, including Wi-Fi, 4G, 5G.",
"Further studies should investigate the applicability of ERA to other modulation schemes.",
"Conclusion In this paper, we have first used the IRS as a cost-effective attacker tool to accomplish physical layer attacks in wireless radio networks.",
"Based on this observation, we introduce the Environment Reconfiguration Attack (ERA) as a novel wireless jamming attack primitive.",
"Without actively emitting a jamming signal, the ERA allows an attacker to significantly reduce or even disable the wireless communication capabilities of victim parties.",
"Our approach takes advantage of a time-varying IRS which we use to rapidly modulate the channel response of victim wireless communication parties.",
"Using the widespread OFDM modulation as an example, we have shown that exceptionally fast and instantaneous changes in the radio propagation environment disturb radio receivers substantially.",
"We have approached the ERA through analytical analysis, simulations, and experiments.",
"Our work breaks down the fundamental attack mechanisms and determines important attacker requirements before demonstrating multiple experimental attacks on actual wireless networks.",
"Our work highlights that the IRS must be considered as a powerful attacker tool for physical layer attacks against wireless communications.",
"The IRS is a striking example of how emerging technologies are causing attack taxonomies to shift as previously complex attacks become tractable.",
"Acknowledgements This work was supported in part by the German Federal Ministry of Education and Research (BMBF) within the project MetaSEC (Grant 16KIS1234K) and by the German Research Foundation (DFG) within the framework of the Excellence Strategy of the Federal Government and the States - EXC2092 CASA - 390781972.",
"Derivation of ICI Power We here derive the ICI arising from the ERA due to sub-symbol channel variations.",
"Fortunately, $H_{k,k^{\\prime }}[n]$ can be related to the complex time varying channel impulse response (CIR) $h_l[n, m]$ , at the $m^{th}$ sample of the $n^{th}$ OFDM-symbol for all $L$ , $l= 0,\\ldots , L-1$ , channel taps [9]: $H_{k,k^{\\prime }}[n] = \\frac{1}{K} \\sum _{l=0}^{L-1} \\underbrace{ \\sum _{m=0}^{K-1} h_l[n,m]\\ e^{-j 2 \\pi m (k- k^{\\prime })/K}}_{H_l[n,k- k^{\\prime }]} \\cdot e^{-j2\\pi l k^{\\prime } / K}$ where $H_l[n,k- k^{\\prime }]$ is the discrete Fourier transform (DFT) of the $l^{th}$ channel tap in time (sample) direction at the subcarrier offset $k - k^{\\prime }$ .",
"While static channels do not result in any ICI, the frequency contents of the fluctuating channel response during the OFDM symbol yield crosstalk from offset subcarriers $k^{\\prime }$ .",
"Note that for the desired signal, i. e., $k^{\\prime } = k$ , (REF ) yields the channel frequency response of the time-averaged CIR.",
"During the ERA, the attacker switches between IRS surface configurations.",
"Naturally, switching corresponds to abrupt changes within the channel response of Alice and Bob, and therefore we expect $H_l[n,k- k^{\\prime }]$ to contain significant high-frequency terms.",
"We now will continue showing that the ERA is capable of turning the complete signal power from the attacker to interference.",
"We account for the attacker's IRS by splitting the CIR into static direct (non-IRS) and IRS portions: $h_l[n,m] = h^{d}_{l} + h^{IRS}_{l}[n,m].$ Assuming that the attacker only affects a single channel tap $l = l_{IRS}$ , the IRS-induced ICI is thus found from (REF ), omitting the non-IRS taps: HIRSk,k'[n] = 1K HlIRS[n,k- k'] e-j2lIRS k' / K, with squared magnitude given by $\\left|H^{IRS}_{k,k^{\\prime }}[n] \\right|^2= \\frac{1}{K^2} \\left| H_{l_{IRS}}[n,k- k^{\\prime }] \\right|^2.$ For brevity and simplicity, we here consider the special case that the IRS is configured such that the sum of the IRS channel tap over one OFDM symbol is zero, namely $\\sum _{m=0}^{K-1} h_{l_{IRS}}[n,m] = H_{l_{IRS}}[n,0] =0 .$ Substituting this in () and setting $k’=k$ results in HIRSk[n] = HIRSk,k[n] = 1K HlIRS[n,0] e-j2lIRS k / K=0, which means that the IRS channel tap does not contribute to the useful signal but to the ICI only.",
"Using (REF ), the signal power of the useful signal $S_k$ is thus given by: $S_k = \\left|H_k[n]\\right|^2 = \\left|H_k^{IRS}[n] + d_k \\right|^2 = \\left|d_k\\right|^2.$ Assuming that all data symbols $X_{k}[n]$ on different subcarriers and OFDM symbols are independent and using (REF ) and (), the total ICI power due to the IRS is given by IIRS= k' k|HIRSk,k'[n] |2 = k'=0K-1 |HIRSk,k'[n] |2 = 1K2 k'=0K-1 | HlIRS[n,k'] |2 = 1Km=0K-1| hlIRS[n,m] |2, where we used Parseval's theorem for the DFT in the last step.",
"If the magnitude IRS channel tap is constant, i. e., the malicious IRS modulation results only in phase shifting, i. e., ${| h_{l_{IRS}}[n,m]| = |h_{l_{IRS}}|}$ , this can be simplified further to: $I_{IRS} = \\sum _{k^{\\prime } \\ne k}\\left|H^{IRS}_{k,k^{\\prime }}[n] \\right|^2 = |h_{l_{IRS}}|^2= P_{IRS}, $ which means that the total power received from the IRS, $P_{IRS}$ , completely translates into ICI, only.",
"Thus the signal-to-interference ratio (SIR) due to ICI on the $k^{th}$ subcarrier is given by $SIR_k = \\frac{S_k}{I_{IRS}}= \\frac{|d_{k}|^2}{|h_{l_{IRS}}|^2 } = \\frac{|d_{k}|^2}{P_{IRS}}.$"
],
[
"Background",
"In this section, we provide technical background on the IRS, jamming attacks, and OFDM communications."
],
[
"Intelligent Reflecting Surface",
"An IRS is a synthetic planar structure with digitally reconfigurable reflection properties of electromagnetic (EM) waves.",
"In wireless communications, the IRS is a rather new concept that has evolved from physics research on metamaterials and metasurfaces [24] which are tailored to enable non-standard EM wave field manipulations.",
"More recently, the evolutionary step from the metasurface to the IRS has been made: Metasurface designs have been drastically simplified and became digitally controllable.",
"An IRS consists of many distributed identical unit cells, each of which reflects impinging EM waves.",
"Most importantly, the complex reflection coefficient of each element across the surface is individually programmable, allowing to influence the wireless channel of communication parties (see Fig.",
"REF ).",
"Practical IRS designs are often targeted to adjust only the signal phase with quantization as low as 1bit [48].",
"Thus, the IRS provides a simple digital interface towards the physical layer of wireless communications and enables what is coined smart radio environments [25] with novel applications such as, e. g., optimization of the signal-to-noise ratio (SNR) [5] or spatial diversity [13].",
"Since only ambient signals are reflected, the IRS is inherently energy efficient and does not require active RF chains.",
"Thus, IRS have low hardware complexity since manufacturing requires standard microstrip technology on low-cost printed circuit board (PCB) substrate.",
"Currently, the IRS is in discussion to complement future wireless infrastructure on a large scale in wireless networks beyond 5G [49]."
],
[
"Jamming",
"Wireless communication relies on a broadcast medium that must be shared between many users.",
"In principle, each user is free to transmit at any time and thus, signals are by definition subject to interference.",
"Instead of just the desired signal, a receiver then additionally picks up an unwanted signal, disrupting the intended communication.",
"Despite regularly occurring interference from other user's communications, malicious parties can also launch jamming attacks.",
"Here, an attacker deliberately produces interference to disable the communication of targeted users.",
"Jamming attacks can be classified into a variety of different categories, including the type of interference and the strategy to trigger emission of the interfering signal [18].",
"A jammer may use noise signals, constant tones, or even valid waveforms.",
"Attackers can apply constant jamming or act reactively in order to disable only selected parts of the victim communication, such as physical control channels [15]."
],
[
"Orthogonal frequency division multiplexing (OFDM)",
"Due to its unique properties, OFDM has become one of the most important and widely used modulation techniques in wireless networks [9], [16].",
"Most importantly, OFDM can cope with multipath signal propagation easily.",
"In order to push data rates, wide channel bandwidths need to be used.",
"However, when transmitting a wide-bandwidth signal over a wireless link, it will most likely experience some form of frequency selective attenuation due to fading from multipath signal propagation.",
"OFDM divides a wide bandwidth into numerous independent (say, orthogonal) narrowband channels, i. e., subcarriers, and can thus handle frequency selective channels at low computational complexity.",
"Taking the concept to the next level, OFDM based multiple access (OFDMA) schemes assign different subcarriers to different users.",
"Finally, the modulation and demodulation of OFDM are elegantly handled using an efficient (inverse) fast Fourier transform (FFT).",
"Today, OFDM has become the definitive transmission scheme for broadcasting, e. g., DAB and DVB, cellular systems, e. g., 4G and 5G, and personal networks, e. g., Wi-Fi.",
"In this section, we summarize the relevant literature on IRS and jamming attacks, and also describe how our work differs from previous proposals.",
"Intelligent reflecting surface.",
"The IRS has been widely recognized as a potential major innovation in wireless communications and has stimulated much research activity recently.",
"Hence, there is a manifold literature now.",
"Regarding key concepts and literature reviews, we refer to numerous overview works [5], [43], [44], [25].",
"To the best of our knowledge, previous works on IRS in a security context focus on theoretical aspects.",
"Most notably, Lyu et al.",
"[29] proposed the IRS for minimizing the signal power received by a victim party for jamming.",
"We further elaborate the similarities and differences to our work towards the end of this section.",
"Several works, e. g., [12] and [7], provide analytical and simulation results in the context of physical layer security assisted by an IRS.",
"Huang and Wang [21] discuss a pilot contamination attack using an IRS to increase signal leakage by reflecting pilot signals.",
"In [47], the authors pursue IRS to be used as a mitigation for active jamming attacks.",
"In the following we give examples for studies including practical IRS demonstrations with a focus on improving wireless communication.",
"An early work from 2014 is [24], where the authors demonstrate wave field shaping.",
"Work from 2019 [13] has shown that IRS are capable of enhancing spatial diversity.",
"Arun and Balakrishnan in 2020 [4] demonstrated a large prototype IRS with 3200 elements for passive beamforming applications.",
"In recent work of Pei et al.",
"[33], an IRS is used to achieve substantial channel improvements, enabling a long-range communication field trial over 500m.",
"Several works report practical IRS designs, e. g., [48], [22], [46].",
"Jamming attacks.",
"The literature widely recognizes jamming attacks as a risk to the reliability of wireless communications.",
"Several works have pointed out the threat of jamming against 4G [26], [15] and 5G [3] networks.",
"Grover et al.",
"[18] provide an overview on different jamming strategies, localization and detection techniques, and countermeasures.",
"However, the ERA does not fit any of the reported categories properly.",
"Poisel gives a highly comprehensive overview on all classes of jamming in his book [34].",
"Lichtman et al.",
"[27] provide a taxonomy for jamming attacks by defining four attacker capabilities time correlation, protocol awareness, ability to learn, and signal spoofing.",
"Following their categories, the ERA may be labeled as a partially time-correlated jammer.",
"However, unlike the author's category-based conjecture, the ERA is a low-complexity attack.",
"Hang et al.",
"[20] investigate repeater jamming against direct sequence spread spectrum (DSSS).",
"The ERA may indeed be seen as a special case of repeater jamming, as a reflection of the signal in fact is a time-varying copy of the legitimate signal.",
"Thus, the ERA is conceptually related.",
"In the ERA, however, the attacker eliminates RF receiver and transmitter chains and processing delays.",
"Pöpper et al.",
"[35] report a method to achieve jamming-resistant broadcast communications without shared keys.",
"The authors comment on the repeater jammer which could circumvent their security assumptions in some cases and also point to processing delays.",
"For our IRS-based approach, however, processing delays vanish.",
"Clancy [10] has pointed out that OFDM communications can be efficiently disrupted by jamming or nulling of pilot signals for channel estimation.",
"The ERA now provides a simple method to realize the manipulation of the OFDM equalizer.",
"Also, many works pursue detection of jamming, examples include [39], [8], [28].",
"A different body of work examines helpful aspects of jamming, e. g., to provide confidentiality [42].",
"However, Tippenhauer et al.",
"[40] have shown that jamming for confidentiality has fundamental security limitations.",
"Differentiation from previous work.",
"The general idea of maliciously using an IRS for jamming was first proposed by Lyu et al.",
"[29] in 2020, albeit in a very different manner that we believe results in a much lower practicality than the ERA.",
"The approach of [29] is based on an IRS to minimize the signal power received by a victim party – a method opposite to the classical IRS-based SNR improvement.",
"Here, the superposition of the direct signal and the malicious IRS signal shall result in destructive interference, i. e., the IRS signal is to be a phase-exact cancellation signal.",
"However, finding a specific IRS configuration to meet this goal is non-trivial.",
"Addressing this issue, the authors formulate an optimization scheme to obtain a corresponding IRS configuration from the channel states $c_{i,k}$ , $g_{i,k}$ , and $d_k$ , cf.",
"Fig.",
"REF .",
"Thus in this approach the attacker needs to have full knowledge of all involved channel states.",
"Unfortunately for an attacker, $d_k$ can only be found by the victim parties and obtaining $c_{i,k}$ and $g_{i,k}$ is infeasible (without a large number of additional RF receivers at the attacker's IRS), as recognized in the literature [5], [43], [44].",
"In contrast, the ERA approach presented in this paper works entirely different, thereby eliminating the unrealistic requirement of channel knowledge for the attacker.",
"Crucially, the attack leverages the IRS to rapidly toggle between (two) effective wireless channels.",
"In particular, we address OFDM receivers which get disturbed by the unnatural switching between channel states, e. g., partly due to adaptive behavior.",
"Our goal is not the minimization of the signal reception of one or both of the ERA channels.",
"Rather, the ERA exploits signal changes from the difference between the two ERA channels as a source of interference.",
"Thus, the attack neither requires synchronization or phase-exact knowledge of all channels, and thereby avoids a location-dependent attack performance (signal phase changes by movement), as our experimental results show.",
"In order to compare the two attack strategies, we would like to point out that a cancellation approach [29] is equivalent to reducing the SNR – an aspect that we readily cover in our simulations in Section REF , showing that the ERA can achieve substantially increased jamming performance."
],
[
"Attack Overview",
"Parties.",
"In this work, we consider a physical layer attacker Eve trying to disrupt the wireless radio communication of two legitimate parties Alice and Bob who deploy a conventional OFDM-based wireless communication system.",
"Thus, Alice and Bob may use Wi-Fi, 4G, or 5G and could represent a base-station and an end-user, respectively.",
"The attacker Eve has full control over an IRS which is part of the wireless propagation channel between Alice and Bob.",
"Eve is capable of applying custom configurations to the IRS at update rates comparably to the symbol rate used by Alice and Bob.",
"Apart from that, we grant the attacker basic wireless eavesdropping capabilities, i. e., the attacker possesses a wireless receiver and can receive and demodulate signals of Alice and Bob.",
"However, Eve does not have a wireless transmitter and thus cannot transmit any signals on itself.",
"Finally, our system and attacker model is illustrated in Fig.",
"REF .",
"Note that the attacker operates at the physical layer and therefore we do not need to take the cryptography applied at the upper layer of the user's communication into account.",
"Attack and overview of investigation.",
"In the ERA, the attacker Eve uses a software-controlled surface, i. e., an IRS, to rapidly vary the wireless radio channel between Alice and Bob.",
"This yields fast and instantaneous variations in the legitimate signals that normally would not occur in nature.",
"Disturbed by the anomalous signal behavior, the intended receiver fails to correctly demodulate the incoming signals, leading to a denial of service.",
"In this work, we design an ERA against OFDM communications by rapidly toggling between two distinct IRS configurations.",
"An illustration of the corresponding attacker action is shown in Fig.",
"REF .",
"Compared to classical jamming attacks, the ERA allows attackers to silently disable the wireless communications of victim parties, i. e., the attacker does not actively generate a jamming signal.",
"Instead, it manipulates signals transmitted by Alice and Bob during propagation.",
"We begin our investigations by examining the fundamental attack mechanisms in an analytical analysis (Section ).",
"Here, we lay the foundations of the attack and show that ERA-induced fast channel variations are harmful for wireless OFDM communication.",
"We then turn to a simulation model (Section ) of an end-to-end wireless OFDM link.",
"From the simulation, we deduce several key factors of the attack, such as, e. g., signal power and attacker distances.",
"For both theoretical analysis and simulations, we abstract the effect of the adversarial IRS as a time-varying signal component and omit the impact of specific IRS patterns.",
"Finally, we use a practical IRS implementation to design and evaluate real-world ERAs to demonstrate successful jamming attacks (Section ).",
"In the first and simplest variant, we rapidly toggle the IRS patterns by either setting all elements to '0' or '1'.",
"This attack is of remarkably low complexity and requires nothing more than a certain proximity between the attacker and a victim party.",
"The second attack variant is more advanced and includes an optional setup phase where the attacker optimizes the two IRS patterns to increase the jamming efficiency.",
"This procedure incorporates the channel state information (CSI) from Alice and Bob, as provided by CSI feedback signals in existing wireless standards."
],
[
"Theoretical Analysis",
"In this section, we present a theoretical analysis of the mechanisms underlying the ERA against OFDM communications.",
"We outline that the ERA affects channel equalization from outdated channel estimations and subcarrier orthogonality."
],
[
"Modelling Preliminaries",
"We begin our considerations by introducing the models for the legitimate OFDM communications and the IRS attacker.",
"We assume that Alice and Bob generate their RF transmit signals using a modulator fed by conventional complex-valued in-phase and quadrature (IQ) baseband signals [16].",
"The baseband signals for OFDM are generated by taking the inverse discrete Fourier transform of a block of $K$ complex modulated data symbols $X_k[n]$ for all $k= 0,\\ldots , K-1$ subcarriers, yielding the $n^{th}$ OFDM symbol.",
"For instance, the data symbols contained in $X_k[n]$ may be modulated using, e. g., binary phase shift keying (BPSK) or quadrature amplitude modulation (QAM) of arbitrary order.",
"Then, in the time domain, a cyclic prefix is prepended to each OFDM symbol.",
"At the receiver side (see Fig.",
"REF ), after time- and frequency synchronization, removal of the cyclic prefix, and discrete Fourier transform, the received baseband signal on the $k^{th}$ subcarrier of the $n^{th}$ OFDM symbol in the frequency domain is given by: $Y_k[n] = H_k[n]\\ X_k[n] + Z_k[n], $ where $H_k[n]$ is the complex channel gain of the link between Alice and Bob for the $k^{th}$ subcarrier, and $Z_k[n] \\sim \\mathcal {CN}(0,\\sigma ^2)$ is additive white Gaussian noise (AWGN).",
"Following the implementation of practical systems, we assume that (known) pilot symbols are transmitted with a preamble to allow channel estimation at the receiver side.",
"The pilot symbols are populated on each of the $K$ subcarriers of the $n^{th}$ OFDM symbol (i. e., block-type pilot arrangement [11]) and allow Alice and Bob to obtain CSI using, e. g., a standard Least-Squares (LS) channel estimator: $\\hat{H}_{k}[n] = \\frac{Y_k[n]}{X_k[n]} = H_k[n] + \\frac{Z_k[n]}{X_k[n]} = H_k[n] + \\tilde{Z}_k[n].",
"$ The channel estimate then is used to equalize the subsequently received OFDM symbols: $\\hat{X}_k[n] = \\frac{Y_k[n]}{\\hat{H}_{k}[n]} $"
],
[
"Intelligent Reflecting Surface",
"We now establish the model for OFDM wireless communication in the presence of an IRS.",
"We assume an IRS consisting of $N$ identical sub-wavelength-sized elements, arranged in an array on a planar surface to reflect impinging waves with a programmable phase shift.",
"The generalized reflection coefficient for the $i^{th}$ IRS element can be expressed as: $r_i = \\alpha _i e^{j \\phi _i} \\qquad i = 1,...,N,$ where we assume $\\alpha _i = 1$ and $\\phi _i \\in [0, 2 \\pi )$ .",
"Note that the IRS used in the experiments in Section is a binary phase-tunable IRS, i. e., then $\\phi _i \\in \\lbrace 0, \\pi \\rbrace $ and $r_i \\in \\lbrace -1, 1\\rbrace $ which correspond to '0' and '1' states of the IRS control signal.",
"Next, following the illustration in Fig.",
"REF , we find an expression for the channel between Alice and Bob, taking the IRS contribution into account.",
"Here we assume that the non-IRS channel is static and therefore denote the IRS as only source of channel variation depending on $n$ .",
"The effective channel between Alice and Bob in (REF ) then is: $H_k[n] = H_k^{IRS}[n] + d_k = \\sum _{i=1}^N c_{i,k}\\, r_i[n]\\, g_{i,k} + d_k,$ where $c_{i,k}, g_{i,k}, d_k \\in \\mathbb {C}$ , respectively, are the complex channel gains of the link between Alice and the $i^{th}$ IRS element, Bob and the $i^{th}$ IRS element, the direct link between Alice and Bob for the $k^{th}$ subcarrier (cf.",
"Fig.",
"REF ).",
"Figure: Block-diagram of a typical OFDM receiver architecture.We now proceed to show how the fast channel variations invoked by the ERA will impact OFDM wireless communication."
],
[
"Channel Equalization",
"A fundamental part of every OFDM receiver (cf.",
"Fig.",
"REF ) is the channel estimation that is mandatory to equalize the received data symbols [9].",
"As previously outlined, operating an IRS allows the attacker to alter the wireless channel between Alice and Bob which will thus likewise affect the channel equalization.",
"We assume the non-IRS channel $d_k$ is static and Eve switches between two IRS configurations $r^{(0)}_{i}$ and $r^{(1)}_{i}$ , corresponding to the channels $H^{(0)}_{k}$ and $H^{(1)}_{k}$ .",
"Now consider the pilot symbols for channel estimation have been transmitted with the malicious IRS configured as $r^{(0)}_{i}$ .",
"Using (REF ), the victim receiver obtains the following channel estimate: $\\hat{H}_k[n] = H^{(0)}_{k} + \\tilde{Z}_k[n].$ Now, Eve switches the IRS configuration to $r^{(1)}_{i}$ , changing the channel of the subsequent OFDM symbols to $H^{(1)}_{k}$ .",
"Thus, the victim receiver's equalizer, cf.",
"(REF ), will operate with an outdated channel estimation: $\\hat{X}_k[n] = \\frac{Y_k[n]}{\\hat{H}_k[n]} = \\frac{X_k[n]\\ H^{(1)}_{k} + Z_k[n]}{H^{(0)}_{k} + \\tilde{Z}_k[n]},$ leading to a symbol error of ek[n] = Xk[n] - Xk[n] = Xk[n] ( H(1)k - H(0)k - Zk[n] ) + Zk[n]H(0)k + Zk[n].",
"For high SNRs, which is a reasonable assumption when using LS channel estimation, the symbol error is approximated by $e_k[n] \\approx X_k[n] \\frac{ H^{(1)}_{k} - H^{(0)}_{k} }{H^{(0)}_{k}} = X_k[n] \\frac{ H^{IRS,(1)}_{k} - H^{IRS,(0)}_{k}}{ H^{IRS,(0)}_{k} + d_k }$ The resulting expression in (REF ) tells us that the IRS-induced symbol error is proportional to ($i$ ) the transmitted symbol, ($ii$ ) the difference between the two IRS channels, and ($iii$ ) is inversely proportional to the direct channel contribution.",
"Thus, the attacker can maximize its chance of causing a false symbol decision by producing a pair of IRS channels, e. g., ${H^{IRS,(1)}_{k} = -H^{IRS,(0)}_{k}}$ .",
"In particular, this can be achieved by inverting the sign of all IRS reflection coefficients $r_i$ .",
"Thus, we likewise adopt this approach in our simulations and experiments in Sections and ."
],
[
"Intercarrier Interference",
"OFDM systems in general are susceptible inter-carrier interference (ICI) which is caused by a degradation of subcarrier orthogonality.",
"ICI usually results from imperfections such as Doppler shifts, frequency offsets, and channel variations during an OFDM symbol period [16], [9].",
"We emphasize that the time-varying IRS used in the ERA will deliberately introduce rapid and instantaneous channel variations at sub-symbol timing resulting in substantial ICI.",
"To model the ICI, (REF ) is modified to account for the interference $H_{k,k^{\\prime }}$ from other subcarriers $k^{\\prime } \\ne k$ to the received OFDM signal on the $k^{th}$ subcarrier [9]: $Y_{k}[n] = H_{k}[n] X_{k}[n] + \\underbrace{\\sum _{k^{\\prime } \\ne k}^{} H_{k,k^{\\prime }}[n] X_{k^{\\prime }}[n]}_{\\textrm {ICI}}\\ +\\ Z_{k}[n].$ In Appendix we show that if the ERA-induced fast channel variations are zero-mean over one OFDM symbol, the signal-to-interference ratio (SIR) on the $k^{th}$ subcarrier is given by $SIR_k = \\frac{S_k}{I_{IRS}}= \\frac{|d_{k}|^2}{P_{IRS}},$ which means that the IRS does not contribute to the direct signal power $S_k$ , but the total power received from the IRS, $P_{IRS}$ , completely translates into ICI, $I_{IRS}$ , only.",
"Most importantly, this result is valid even without any optimization of the IRS elements with respect to the channels of the legitimate parties."
],
[
"Simulation Results",
"After having analytically outlined the key mechanisms of the ERA affecting an OFDM system, we now strive to further explore the attack through simulations.",
"We give comprehensive results, identifying attack parameters, including signal power, attacker distance, and IRS dimensions.",
"Further, we show that the ERA leads to significant packet error rates (PER) and is way more efficient when compared with a classical jamming attack using noise signals.",
"As an example for general OFDM-based radio systems, we consider Wi-Fi here, since our experimental investigation following in Section also builds upon Wi-Fi devices.",
"As the underlying simulation environment, we choose the MATLAB WLAN toolbox due to the availability of end-to-end simulation capabilities for the entire IEEE 802.11n physical layer, including channel coding and standard-compliant channel models.",
"We summarize the essential simulation parameters in Table REF .",
"To mimic the adversarial IRS operation in the ERA, we add time-varying reflection, i. e., a complex square wave signal from the IRS, to one tap of the CIR.",
"Further, we randomize the time instant of the packet start with respect to the IRS modulation.",
"For fairness in comparing the error rates across different modulation and coding schemes (MCS), we adjust the packet payload sizes to always result in 16 entire OFDM data symbols, regardless of the MCS setting.",
"Wi-Fi uses an OFDM symbol duration of ${4}{}$ and thus, the data portion of transmitted packets has a duration of 64.",
"Like traditional jamming attacks, the ERA is subject to link budget constraints.",
"Thus, the attack efficiency depends on the signal power arriving at the receiver from the attacker.",
"Although in the ERA the attacker does not generate a jamming signal itself, we can still define a jamming-to-signal ratio (JSR) as the ratio of IRS signal to direct (non-IRS) signal powers $JSR = \\frac{P_{IRS}}{S}=\\frac{P_{IRS}}{\\sum _{k}S_k}.$ For our simulations below, we use the JSR to assess the attacker strength.",
"As an indication for the attacker's success, we leverage the PER.",
"Table: Summary of the simulation parameters"
],
[
"Attacker Signal Power",
"We investigate the victim PER performance as a function of the JSR for various MCS settings.",
"Therefore, we assume the attacker signal originating from the IRS to have constant power while periodically toggling the phase between 0 and $\\pi $ at a rate of 30kHz, as is the case when inverting the sign of all IRS reflection coefficients $r_i$ .",
"The legitimate receiver has a high SNR of 50dB.",
"We plot the PER results for MCS 0 - 7 (covering BPSK, QPSK, 16-QAM, and 64-QAM modulations on the subcarriers ) as a function of the JSR in Fig.",
"REF .",
"As expected, higher order modulations are more prone to interference from an ERA.",
"The results also highlight that the ERA indeed is capable of producing error rates which render reliable wireless communication impractical.",
"To relate the ERA performance to classical noise-based jamming or signal power reduction attacks [29], we compare the attack against an SNR reduction.",
"For the ERA, we now consider the legitimate receiver to have an otherwise noise-free channel.",
"For the SNR reduction, we consider the IRS to remain static while the attacker now deteriorates the SNR by adding noise with power equivalent to the IRS signal strength during the ERA.",
"We plot the PER simulation results in Fig.",
"REF , which indicates that the ERA achieves considerably better jamming performance when compared to a noise jammer at the same power.",
"Figure: End-to-end PER simulation results for IEEE 802.11n Wi-Fi to compare an ERA against SNR reduction, e. g., from noise jamming or signal power reduction.",
"For the ERA case, we assume a noise-free channel."
],
[
"Channel Modulation Frequency",
"To fully characterize the ERA, we vary the IRS modulation frequency.",
"We conduct the simulation for MCS indicies 0 - 7 at an SNR of 50dB for the channel between Alice and Bob and a JSR of -10dB.",
"We plot the PER simulation results in Fig.",
"REF against the IRS update frequency.",
"For the MCS indices 0 and 1, we observe particularly lower PERs due to the more robust modulation parameters.",
"Despite that, the PER clearly increases as a function of the modulation frequency for all MCS values.",
"The increasing PER at lower modulation frequencies can be explained by the increasing probability of an IRS reconfiguration taking place during packet transmission.",
"That is, the packet error rate resulting from an ERA with IRS pattern durations $T_{IRS}$ longer than the packet duration $T_p$ is upper bounded by $T_p/T_{IRS}$ .",
"As the PER for modulation frequencies above approximately 16kHz reaches a plateau, we conclude that at least one IRS reconfiguration during transmission of the data symbols suffices to achieve the maximum attack efficiency for a certain JSR.",
"Figure: End-to-end PER simulation results for IEEE 802.11n Wi-Fi for the ERA over channel modulation frequency for varying modulation and coding schemes at an SNR of 50dB with JSR of -10dB."
],
[
"Surface Size",
"We will now show that an ERA is feasible even for rather weak attacker configurations regarding the attacker distance and IRS dimensions.",
"Previously, we have determined the JSRs necessary for the attacker to degrade the PER of Alice and Bob (see Fig.",
"REF ).",
"Note that we define the JSR as the ratio of the signal power coming from the IRS and the direct (non-IRS) signal power.",
"Thus, the attacker generally seeks to pick up sufficient power from the legitimate users.",
"The attacker can either minimize the distance to one of the victim parties to minimize path loss or increase the IRS size.",
"Although both strategies are suitable, we assume the attacker must maintain a minimum distance and also cannot increase the IRS size arbitrarily without raising suspicion.",
"Hence, we derive a connection between JSR, attacker distance, and the surface size.",
"For the parties, we assume the geometrical configuration shown in Fig.",
"REF (a).",
"We start with the free-space path loss of the direct link between Alice and Bob [16], where the received power is proportional to $L_d = \\left( \\frac{\\lambda }{4 \\pi d_{AB}} \\right)^2,$ with the carrier frequency wavelength $\\lambda = c_0/f$ .",
"For an optimal surface configuration, the free-space path gain from Alice to Bob via the IRS is found by [32]: $L_{IRS} = \\left( \\frac{A_{IRS}}{4 \\pi d_{AE} d_{EB}} \\right)^2.$ Assuming Alice and Bob use omni-directional antennas, the JSR becomes $JSR = \\frac{L_{IRS}}{L_d} = \\left( \\frac{A_{IRS}\\ d_{AB}}{d_{AE} d_{EB} \\lambda } \\right)^2,$ which allows us to link the surface area $A_{IRS}$ to the JSR: $A_{IRS} = \\sqrt{JSR} \\frac{d_{AE} d_{EB} \\lambda }{d_{AB}}$ We use Equation (REF ) to plot the minimum IRS size required by an attacker to achieve a JSR of -10dB in Fig.",
"REF (b).",
"We show the result as a function of the distance between Alice and Bob and for distances 1m, 2m, 10m, and 20m of Eve to Alice.",
"Consider, for example, Alice and Bob are at a distance of 30 and Eve is at a distance of 10 to Alice.",
"Then, an IRS size of only 0.19 is sufficient to achieve a JSR of -10, which results in a severe PER degradation for Alice and Bob."
],
[
"Experimental Evaluation",
"After having approached the ERA through theoretical analysis and simulations in the previous sections, we now proceed with a practical evaluation of the ERA.",
"Therefore, we first describe our experimental setup comprising of a low-cost IRS prototype and commodity Wi-Fi devices.",
"Furthermore, we demonstrate that the ERA is capable of severe link quality degradation, leading to a significant reduction in the effective wireless data throughput."
],
[
"Experimental Attack Setup",
"In this section, we present our experimental attack setup consisting of a prototype IRS and two microcontrollers.",
"We estimate the cost of the setup to be around 10040 for microcontroller development boards, 30 for PCBs, 30 for surface-mount components.. As the essential part of a first exploration of the ERA in practical experiments, we use two low-cost IRS prototype modules (see Fig.",
"REF (a)) with 128 binary-phase tunable unit-cell elements in total, arranged in a $16 \\times 8$ array on standard FR4 PCB substrate.",
"The elements are rectangular patch reflectors on top of a ground plane.",
"Attached to each element, there is a PIN diode which can switch a parasitic element to the reflector, allowing to shift its resonance frequency.",
"Thereby, the reflection coefficient of each element can be individually switched between two states, i. e., a '0' state and a '1' state, by turning the control voltage to the reflector element either on or off.",
"The unit cell circuitry and the reflector design are shown in Fig.",
"REF (a).",
"The IRS prototype used in our experiments is optimized to achieve a 180 phase difference in the reflected wave for the '0' and '1' states (see Fig.",
"REF (b)), i. e., $r_i \\in \\lbrace -1, 1\\rbrace $ in (REF )."
],
[
"IRS Modulation",
"As we strive for rather high IRS modulation frequencies, we drive the 128 IRS elements in parallel.",
"Therefore, we connect each of the 128 control lines to a GPIO pin of two STM32F407 microcontrollers, allowing us to achieve IRS modulation frequencies of up to 1.6.",
"The frequency and surface patterns used for the modulation are programmable from the host controller through an UART serial communication interface.",
"Like in the theoretical analysis and the simulations, cf.",
"Section , we apply a simple binary surface modulation.",
"That is, we periodically toggle between two IRS configurations and thereby maintain a low attack complexity.",
"For instance, we switch between all 128 IRS elements either set to the '0' or '1' state.",
"As discussed in Section , since $r_i \\in \\lbrace -1, 1\\rbrace $ , this leads to switching between two channels $H^{(0)}_{k}$ and $H^{(1)}_{k}$ , with ${H^{IRS,(1)}_{k} = -H^{IRS,(0)}_{k}}$ .",
"We now demonstrate that the ERA is capable of significant throughput reduction in entire wireless networks.",
"Therefore, we deploy a commercial off-the-shelf WLAN router to provide an IEEE 802.11ac network in an office space.",
"We position the attacker setup strategically at the router with distances of 1 and 2.",
"We detail and summarize the setup in Table REF .",
"For the experiment, we use a laptop connected to the Internet via the Wi-Fi network to measure the effective end-to-end speed of the connection .",
"We perform speed measurements without the ERA (the malicious IRS remains static) and with the ERA enabled (switching all IRS elements between '0' or '1' state).",
"We repeat this procedure for a total of 37 positions distributed throughout the office space, as indicated in Fig.",
"REF .",
"We show the results of the throughput measurements in Fig.",
"REF .",
"Here we can see that the ERA leads to an average throughput reduction of 78 % and 40 % for the attacker at 1 and 2 distance to the router, respectively.",
"Recall that the attacker does not actively emit any jamming signal to achieve this result.",
"Furthermore, the attacker does not perform any kind of synchronization to the legitimate signals or optimization of the IRS configurations.",
"Notably, the ERA also leads to substantial throughput reduction where the wireless channel between the client and the IRS is obstructed, i. e., in different rooms with walls in between.",
"Thus, we conclude that the ERA is a scalable attack, allowing the attacker to slow down the wireless network at many different places.",
"Table: Summary of the experimental setupFigure: Experimental ERA setup with WLAN router and attacker IRS."
],
[
"Systematic Packet Error Rate Measurement",
"We perform a second experiment to systematically assess the practical effectiveness of the ERA, aiming to obtain PER measurements similarly to our simulation result from Section REF .",
"Therefore, we deploy single-board computers equipped with ath9k-based network interface cards (NICs) [45] for IEEE 802.11n Wi-Fi at the legitimate parties Alice and Bob.",
"The NICs give us low level access to the Wi-Fi communication, i. e., we can transmit packets with defined length and MCS setting.",
"Here, we use a 2x2 MIMO configuration with off-the-shelf Wi-Fi antennas.",
"One of the parties provides a Wi-Fi network on channel 60 (at 5300), allocating 40 bandwidth.",
"We place the attacker setup attacker at distance 2 and 3 in line-of-sight to Alice and Bob, respectively.",
"The channel between Alice and Bob also has line-of-sight conditions.",
"For the whole duration of the experiment, the propagation environment remains static apart from the adversarial IRS operation.",
"In our setup, Alice transmits 20000 packets with randomized payload data to Bob.",
"For each transmission, we configure the payload size and the MCS setting.",
"Similarly to the simulation, we adjust the payload size to always result in 9 entire OFDM symbols (data symbol duration 3.6, packet duration 6.8).",
"On Bob's side, we count the number of successfully received packets to finally obtain the PER.",
"We plot the PER results as a function of the adversarial IRS modulation frequency in Fig.",
"REF (a).",
"Also, we indicate the previously discussed upper PER bound given by $T_p/T_{IRS}$ for $T_{IRS} > T_p$ .",
"Essentially, our measurement with standard Wi-Fi NICs confirms our previous simulation results, showing that higher-order modulations are more susceptible to the ERA.",
"However, instead of reaching a plateau, we observe a drop in the PER when increasing the IRS modulation frequency beyond 30kHz.",
"We believe that this effect is due to hardware imperfections on the IRS prototype which initially was not designed to operate at high modulation speeds.",
"As evident from the results, the upper PER bound based on the timing parameters holds.",
"However, despite the fixed packet time duration, it appears that our bound seems to be too optimistic for MCS values below 12.",
"We attribute this to reduced synchronization efforts, i. e., the receiver will barely be affected by an IRS change during the packet's preamble portion, reducing the effective ERA-sensitive packet length.",
"Figure: Measured PER over channel modulation frequency.",
"(a) Binary pattern modulation.",
"(b) Tailored pattern modulation."
],
[
"Surface Pattern Optimization",
"Thus far, we have tested the simplest ERA strategy where the attacker switches all surface elements periodically between the '0' or '1' states.",
"However, this strategy can be further improved by matching the used IRS configurations to the wireless link under attack.",
"Thus, the attacker may prepend its jamming operation with a setup phase in order to optimize the IRS configurations used during the subsequent ERA.",
"The attacker therefore can incorporate eavesdropped CSI feedback of the victim parties to further enhance the attack efficiency.",
"For a first demonstration, we design and test an adaptive optimization algorithm to find IRS configurations well-suited for the ERA.",
"The intuition of the algorithm is to use the adversarial IRS for maximizing a dissimilarity measure between the pair of IRS-induced channel responses of the victim wireless link.",
"Following our analytical analysis in Section , we expect this to improve the attacker's success.",
"Algorithm REF outlines the procedure.",
"The result are two IRS configurations $r^{(0)}_{i}$ and $r^{(1)}_{i}$ .",
"Note that we here denote the binary surface control settings ('0' or '1') as a proxy for reflection coefficients.",
"[ht] Distinct IRS configurations $r^{(0)}_{i}$ , $r^{(1)}_{i}$ for ERA.",
"start with random $N$ -bit IRS configurations $r^{(0)}_{i}, r^{(1)}_{i}$ dissimilarity metric $d$ algorithm rounds $R=2$ $j = 0$ to $R$ configure IRS as $r^{(1)}_{i}$ $ref^{(1)} \\leftarrow H_k(r^{(1)}_{i})$ configure IRS as $r^{(0)}_{i}$ $i \\leftarrow 0$ to $N$ $ref^{(0)}_{i,0} \\leftarrow H_k(r^{(0)}_{i})$ $r^{(0)}_i \\leftarrow r^{(0)}_i \\oplus 1$ update IRS element $i$ $ref^{(0)}_{i,1} \\leftarrow H_k(r^{(0)}_{i})$ $d(\\textrm {ref}^{(1)}, ref^{(0)}_{i,0}) > d(\\textrm {ref}^{(1)}, ref^{(0)}_{i,1})$ $r^{(0)}_i \\leftarrow r^{(0)}_i \\oplus 1$ update IRS element $i$ swap($r^{(0)}_{i}$ , $r^{(1)}_{i}$ ) Adversarial binary surface optimization The randomly chosen initial IRS configurations in Algorithm REF are given below: $r^{(0)}_{i}$ = 0x5CC81D86E5DAB902B071665D1D7DC2F1 $r^{(1)}_{i}$ = 0xC859CCA60594481B193BF3D236E877AE The result of the algorithm are the updated IRS configurations: $r^{(0)}_{i}$ = 0xFFFF9F9F08089E08474721D92AC1B57A $r^{(1)}_{i}$ = 0x00006060E5D776A2F8B876020C034C05 Figure: Evolution of Euclidean distance between the channel responses during the iterative IRS optimization.Fig.",
"REF shows the evolution of the Euclidean distance between $|H_k(r^{(0)}_{i})|$ and $|H_k(r^{(1)}_{i})|$ over the iteration steps, clearly exhibiting the characteristic behaviour of our algorithm.",
"Finally, we also plot the pair of channel responses as observed by Alice and Bob before and after the optimization in Fig.",
"REF .",
"Here, we can see that our procedure indeed is highly effective in providing distinct channel responses designated to be used in the ERA.",
"Note that even though the reception for $|H_k(r^{(0)}_{i})|$ has improved after running the algorithm, the difference between the two channel states is maximized.",
"The result is a vivid example for the combination of inherent simplicity and possibilities of the IRS for previously infeasible attacks.",
"Figure: Effective normalized channel responses observed by Alice and Bob, before and after running the adversarial IRS optimization algorithm.Using the presented algorithm with the Euclidean distance as a metric and magnitude CSI information on the link between Alice and Bob, we obtain the adapted IRS configurations $r^{(0)}_{i}$ and $r^{(1)}_{i}$ , which we now use to conduct the ERA.",
"We repeat the PER measurement experiment from the previous section and plot the results in Fig.",
"REF (b).",
"Here it is evident that the optimization was able to improve the attacker efficiency.",
"Now, even the robust BPSK modulation for MCS 8 exhibits a significant PER induced by the ERA.",
"Further, the optimization has also led to substantially increased PERs for the remaining MCS values."
],
[
"Discussion",
"In this section, we discuss ($i$ ) the real-world applicability, ($ii$ ) the attacker capabilities, and ($iii$ ) reason about countermeasures and mitigation.",
"Also, we give directions for future work."
],
[
"Real-world Applicability",
"We assess the costs and complexity of an ERA to be low.",
"Our results show that a sub 100 attacker setup can have significant impact on the effective wireless throughput.",
"Once an attacker possesses a functional IRS, only basic microcontroller programming is required to rapidly vary a number of logic signals controlling the IRS.",
"Thus, the attack can be easily carried out by non-specialists.",
"While the commercial availability of IRS devices is currently still limited, several companies , are working on product-grade IRS implementations.",
"Besides that, many IRS designs are publicly available and can easily be reproduced by others using cheap PCB assemblies.",
"Instead of using an own IRS, an attacker could also hijack existing IRS infrastructure which may be deployed in future wireless networks [49], most likely already at strategically advantageous positions."
],
[
"Attacker Capabilities",
"To conduct an ERA, the attacker's IRS must be within the wireless propagation environment between the victim nodes.",
"As wireless communication is inherently supposed to bridge distances this will not be a hurdle for an attacker.",
"As discussed, the JSR is an important parameter bounding the attack performance.",
"In order to improve its JSR, the attacker can choose a favorable position or increase the IRS size.",
"Therefore, to compensate the small size of our IRS prototype, we have used rather short attacker distances in our experiments, which still represents a valid attacker model.",
"Our simulation results show that sufficient JSR values are, in principle, still possible for higher attacker distances and surface sizes.",
"However, this also reveals a limitation of ERA: the attacker is passive and cannot amplify the signals it reflects.",
"Hence, as it is generally the case for wireless communications (and jamming), the attack is limited by the available link budget.",
"Our simulation results show the underlying relationship between JSR and PER.",
"For this purpose, we have simplified the attacker's signal originating from the IRS to a time-varying signal component from alternating the sign of the IRS reflection coefficients.",
"Although finding a corresponding IRS configuration to meet a certain JSR is non-trivial, our practical tests tests show that even with a binary-phase tunable IRS and without optimized surface configurations, the ERA significantly disrupts the victim communication.",
"In Section REF , we have granted the attacker access to the CSI of Alice and Bob to demonstrate that an attacker can further optimize the IRS configurations used during the ERA.",
"In an actual attack, the attacker would rely on eavesdropping CSI feedback, e. g., from the user to the base station.",
"For instance, this is commonly used in IEEE 802.11 WLAN standards, 4G, and 5G to implement, e. g., transmit beamforming [15], [37], [23], [14].",
"Note that, in the standards mentioned, these signals are not encrypted."
],
[
"Countermeasures",
"The ERA is based on an IRS within the channel between Alice and Bob.",
"For the attack to work, a part of the transmitted signal must reach the receiver via the adversarial IRS.",
"Due to the broadcast nature of wireless signal propagation, it is likely that an ERA cannot generally be prevented.",
"The transmitter could use beamforming to diminish the attacker's success, trying to minimize the signal power reaching the IRS.",
"However, this requires a mechanism for attack detection and localization and an advanced attacker may even leverage beamforming to its favor by providing a preferred path via the IRS to the receiver.",
"Since the interference signal produced in the ERA is correlated to the useful signal, it may also be possible to find signal processing-based countermeasures at the receiver side.",
"However, we emphasize these considerations are speculative.",
"Countermeasures, if they exist, cannot be implemented immediately in end-user equipment because the very low-level signal processing of radio transceivers is usually implemented in hardware or is not updatable.",
"Finally, to mitigate the attack, wireless communication systems could apply encryption of physical layer control channels, i. e., to prevent the attacker to obtain CSI feedback.",
"However, this will not render the ERA infeasible, but would only impede an adversarial IRS optimization.",
"Moreover, this requires drastic changes to protocols and such measures can likely only be implemented within future standards."
],
[
"Future work",
"In this paper, we have presented a novel class of jamming attacks based on IRS-induced fast changes in the radio propagation environment of wireless communication parties.",
"Naturally, this work only represents a very first exploration of the ERA and, more broadly, the IRS as a toolkit for practical wireless physical layer attacks.",
"Therefore, our work may serve as a basis for future work studying, for example, the following aspects.",
"Improving the attack.",
"We have provided first insights into the optimization of the IRS configuration for an ERA, demonstrating the potential for increased attack efficiency.",
"The evaluation of improved optimization algorithms based on eavesdropping CSI feedback is left for future work.",
"Also, future work should investigate non-binary surface modulation signals where the attacker uses more than two IRS configurations.",
"Finally, there is room for hardware improvements to the attacker setup, perhaps through dedicated IRS designs for high modulation frequencies.",
"Attack detection and countermeasures.",
"More work is needed to examine whether existing jamming attack detection and mitigation strategies, e. g., [18], can be adapted to the ERA.",
"Also, we see a need to evaluate the possibility of signal processing based mitigation strategies that could be incorporated into future transmitter and receiver architectures.",
"Application to other modulations.",
"We have outlined the ERA against OFDM communications, as it is the preferred modulation scheme for modern wireless communication systems, including Wi-Fi, 4G, 5G.",
"Further studies should investigate the applicability of ERA to other modulation schemes."
],
[
"Conclusion",
"In this paper, we have first used the IRS as a cost-effective attacker tool to accomplish physical layer attacks in wireless radio networks.",
"Based on this observation, we introduce the Environment Reconfiguration Attack (ERA) as a novel wireless jamming attack primitive.",
"Without actively emitting a jamming signal, the ERA allows an attacker to significantly reduce or even disable the wireless communication capabilities of victim parties.",
"Our approach takes advantage of a time-varying IRS which we use to rapidly modulate the channel response of victim wireless communication parties.",
"Using the widespread OFDM modulation as an example, we have shown that exceptionally fast and instantaneous changes in the radio propagation environment disturb radio receivers substantially.",
"We have approached the ERA through analytical analysis, simulations, and experiments.",
"Our work breaks down the fundamental attack mechanisms and determines important attacker requirements before demonstrating multiple experimental attacks on actual wireless networks.",
"Our work highlights that the IRS must be considered as a powerful attacker tool for physical layer attacks against wireless communications.",
"The IRS is a striking example of how emerging technologies are causing attack taxonomies to shift as previously complex attacks become tractable."
],
[
"Acknowledgements",
"This work was supported in part by the German Federal Ministry of Education and Research (BMBF) within the project MetaSEC (Grant 16KIS1234K) and by the German Research Foundation (DFG) within the framework of the Excellence Strategy of the Federal Government and the States - EXC2092 CASA - 390781972."
],
[
"Derivation of ICI Power",
"We here derive the ICI arising from the ERA due to sub-symbol channel variations.",
"Fortunately, $H_{k,k^{\\prime }}[n]$ can be related to the complex time varying channel impulse response (CIR) $h_l[n, m]$ , at the $m^{th}$ sample of the $n^{th}$ OFDM-symbol for all $L$ , $l= 0,\\ldots , L-1$ , channel taps [9]: $H_{k,k^{\\prime }}[n] = \\frac{1}{K} \\sum _{l=0}^{L-1} \\underbrace{ \\sum _{m=0}^{K-1} h_l[n,m]\\ e^{-j 2 \\pi m (k- k^{\\prime })/K}}_{H_l[n,k- k^{\\prime }]} \\cdot e^{-j2\\pi l k^{\\prime } / K}$ where $H_l[n,k- k^{\\prime }]$ is the discrete Fourier transform (DFT) of the $l^{th}$ channel tap in time (sample) direction at the subcarrier offset $k - k^{\\prime }$ .",
"While static channels do not result in any ICI, the frequency contents of the fluctuating channel response during the OFDM symbol yield crosstalk from offset subcarriers $k^{\\prime }$ .",
"Note that for the desired signal, i. e., $k^{\\prime } = k$ , (REF ) yields the channel frequency response of the time-averaged CIR.",
"During the ERA, the attacker switches between IRS surface configurations.",
"Naturally, switching corresponds to abrupt changes within the channel response of Alice and Bob, and therefore we expect $H_l[n,k- k^{\\prime }]$ to contain significant high-frequency terms.",
"We now will continue showing that the ERA is capable of turning the complete signal power from the attacker to interference.",
"We account for the attacker's IRS by splitting the CIR into static direct (non-IRS) and IRS portions: $h_l[n,m] = h^{d}_{l} + h^{IRS}_{l}[n,m].$ Assuming that the attacker only affects a single channel tap $l = l_{IRS}$ , the IRS-induced ICI is thus found from (REF ), omitting the non-IRS taps: HIRSk,k'[n] = 1K HlIRS[n,k- k'] e-j2lIRS k' / K, with squared magnitude given by $\\left|H^{IRS}_{k,k^{\\prime }}[n] \\right|^2= \\frac{1}{K^2} \\left| H_{l_{IRS}}[n,k- k^{\\prime }] \\right|^2.$ For brevity and simplicity, we here consider the special case that the IRS is configured such that the sum of the IRS channel tap over one OFDM symbol is zero, namely $\\sum _{m=0}^{K-1} h_{l_{IRS}}[n,m] = H_{l_{IRS}}[n,0] =0 .$ Substituting this in () and setting $k’=k$ results in HIRSk[n] = HIRSk,k[n] = 1K HlIRS[n,0] e-j2lIRS k / K=0, which means that the IRS channel tap does not contribute to the useful signal but to the ICI only.",
"Using (REF ), the signal power of the useful signal $S_k$ is thus given by: $S_k = \\left|H_k[n]\\right|^2 = \\left|H_k^{IRS}[n] + d_k \\right|^2 = \\left|d_k\\right|^2.$ Assuming that all data symbols $X_{k}[n]$ on different subcarriers and OFDM symbols are independent and using (REF ) and (), the total ICI power due to the IRS is given by IIRS= k' k|HIRSk,k'[n] |2 = k'=0K-1 |HIRSk,k'[n] |2 = 1K2 k'=0K-1 | HlIRS[n,k'] |2 = 1Km=0K-1| hlIRS[n,m] |2, where we used Parseval's theorem for the DFT in the last step.",
"If the magnitude IRS channel tap is constant, i. e., the malicious IRS modulation results only in phase shifting, i. e., ${| h_{l_{IRS}}[n,m]| = |h_{l_{IRS}}|}$ , this can be simplified further to: $I_{IRS} = \\sum _{k^{\\prime } \\ne k}\\left|H^{IRS}_{k,k^{\\prime }}[n] \\right|^2 = |h_{l_{IRS}}|^2= P_{IRS}, $ which means that the total power received from the IRS, $P_{IRS}$ , completely translates into ICI, only.",
"Thus the signal-to-interference ratio (SIR) due to ICI on the $k^{th}$ subcarrier is given by $SIR_k = \\frac{S_k}{I_{IRS}}= \\frac{|d_{k}|^2}{|h_{l_{IRS}}|^2 } = \\frac{|d_{k}|^2}{P_{IRS}}.$"
]
] | 2107.01709 |
[
[
"Learning Delaunay Triangulation using Self-attention and Domain\n Knowledge"
],
[
"Abstract Delaunay triangulation is a well-known geometric combinatorial optimization problem with various applications.",
"Many algorithms can generate Delaunay triangulation given an input point set, but most are nontrivial algorithms requiring an understanding of geometry or the performance of additional geometric operations, such as the edge flip.",
"Deep learning has been used to solve various combinatorial optimization problems; however, generating Delaunay triangulation based on deep learning remains a difficult problem, and very few research has been conducted due to its complexity.",
"In this paper, we propose a novel deep-learning-based approach for learning Delaunay triangulation using a new attention mechanism based on self-attention and domain knowledge.",
"The proposed model is designed such that the model efficiently learns point-to-point relationships using self-attention in the encoder.",
"In the decoder, a new attention score function using domain knowledge is proposed to provide a high penalty when the geometric requirement is not satisfied.",
"The strength of the proposed attention score function lies in its ability to extend its application to solving other combinatorial optimization problems involving geometry.",
"When the proposed neural net model is well trained, it is simple and efficient because it automatically predicts the Delaunay triangulation for an input point set without requiring any additional geometric operations.",
"We conduct experiments to demonstrate the effectiveness of the proposed model and conclude that it exhibits better performance compared with other deep-learning-based approaches."
],
[
"Introduction",
"Delaunay triangulation (DT) finds a triangulation such that no point in $P$ is inside the circumcircle of any triangle when a point set $P$ is given [14].",
"Moreover, DT maximizes the minimum angle of the triangles and avoids skinny triangles with very large or very small angles.",
"In addition, DT has various applications in many different fields.",
"It is often used to generate triangular meshes when solving partial differential equations using the finite element method (FEM) or finite volume method (FVM) for a given geometric domain.",
"Further, DT generates triangles close to equilateral triangles for a given point set, which is preferred when using the FEM and FVM for solution accuracy and efficiency.",
"Additionally, DT is used for data clustering.",
"After connecting the data points using DT, data clustering is completed by removing the long edges of a certain length or longer and leaving only the remaining connected edges.",
"In addition, DT is also used when generating Voronoi diagrams because DT corresponds to the dual graph of the Voronoi diagram.",
"The circumcenter of the Delaunay triangle becomes the vertices of the Voronoi diagram, which has many applications in biology, ecology, computational fluid dynamics, and computational physics.",
"DT is an important combinatorial optimization problem involving geometry with various applications, but most of the existing algorithms are nontrivial algorithms, requiring the understanding of geometric requirements or performing geometric operations, such as the edge flip.",
"The combinatorial optimization problem is a significant problem in computer science.",
"The combinatorial optimization problem finds an optimal subset from a finite set of objects.",
"Several examples of combinatorial optimization problems include the convex hull, traveling salesman problem (TSP), knapsack, and vehicle routing problem [2], [21], [9].",
"Many combinatorial optimization problems are NP problems, and heuristic algorithms are often used to find nearly optimal solutions when solving these problems.",
"Recently, deep-learning-based techniques have been used to solve these complex combinatorial optimization problems.",
"Both supervised learning and reinforcement learning are used to solve these problems.",
"A representative supervised learning-based deep learning model for solving combinatorial optimization problems is the pointer network (Ptr-Net) [30].",
"The conventional sequence-to-sequence (seq2seq) model computes conditional probabilities from a fixed dictionary.",
"Therefore, it has limitations for solving combinatorial optimization problems in which the size of the output dictionary changes depending on the input size.",
"However, Ptr-Net solves this problem.",
"Vinyal et al.",
"proposed and applied Ptr-Net to find (approximate) solutions to the following combinatorial optimization problems with variable-sized inputs and output dictionaries: convex hull, DT, and TSP.",
"In addition, Ptr-Net demonstrates the possibility to solve these combinatorial optimization problems, but it has limitations for practical use, especially for solving DT and TSP.",
"Afterward, various studies on the TSP problem using supervised and reinforcement learning were conducted, but studies on the DT problem are scarce.",
"Recently, a transformer model has been proposed to solve the long-term dependency problem of the recurrent neural network (RNN) model [28].",
"The transformer model consists of self-attention, multi-head attention, and positional encoding and has the advantage that it does not involve recurrence.",
"The transformer model is widely used in natural language processing, voice recognition, and image recognition [32], [6], [7].",
"The concept of self-attention used in the transformer model indicates that attention is carried out toward itself and can be used in encoder and decoder models.",
"The transformer model learns by considering the relationships between all pairs in the input/output sequence.",
"In this paper, we propose a new neural network model to solve DT using self-attention and domain knowledge.",
"The proposed method is based on supervised learning.",
"It has the advantage of being simple and efficient because it automatically outputs DT for an input point set when the neural network model is sufficiently trained without requiring any additional geometric operations.",
"The sequence order of input and output has a significant effect on the performance of the seq2seq-based models.",
"However, the effect of applying input/output sequence ordering is not well studied.",
"We first propose a method of ordering input and output sequences that can significantly improve the model learning performance for the DT problem.",
"Second, we apply self-attention in the encoder to better learn the point-to-point relationship when generating DT.",
"Our model can effectively learn the topological relationship between distant points in the encoder by applying the self-attention mechanism.",
"Finally, we develop a new attention score function that can augment the existing attention mechanism using domain knowledge.",
"It is a newly developed penalty-based attention score function, making the conditional probability very small if the Delaunay condition is not satisfied.",
"Our main contributions are summarized as follows: We propose a novel neural network model that can learn DT based on self-attention and domain knowledge.",
"To the best of our knowledge, the proposed model exhibits the best performance among deep-learning-based techniques.",
"The proposed model is the first that applies a self-attention mechanism for learning DT in the encoder.",
"We also propose an efficient input/output sequence ordering method to learn the sequence more regularly and effectively.",
"We propose a new type of attention score function using domain knowledge, a flexible function that can be applied to other combinatorial optimization problems involving geometry.",
"We also develop several evaluation metrics that measure model performance.",
"Triangulation is a fundamental and essential problem in computer graphics and computational geometry.",
"Triangulation of a set of points $P$ is a partition of the convex hull into simplices, such that the union of all these simplices equals its convex hull and that any pair of them intersects in a common face.",
"In the plane, triangulations comprise triangles with their edges and vertices.",
"Among various triangulations, the triangulation of particular interest is DT.",
"Specifically, DT for a point set $P$ is a triangulation such that no point in $P$ is inside the circumcircle of any triangle [14].",
"Figure REF presents one example of DT with circumcircles.",
"Further, DT maximizes the minimum angle of all angles of the triangle in the triangulation and avoids sliver triangles [14].",
"Figure: Delaunay triangulation with circumcircles (dotted).Many algorithms have been proposed for generating DT with the algorithm complexity of $O\\left( n\\mathrm {log}(n) \\right)$ or $O\\left( n^2 \\right)$ [10], [18], [22].",
"One of the most popular DT algorithms is the flip algorithm, which constructs any triangulation from the points and then repeatedly flips the edges until every triangle meets the Delaunay condition.",
"When two triangles ABC and ABD share an edge AB, if the sum of the $\\alpha $ and $\\beta $ angles is less than or equal to $180^{\\circ }$ , the Delaunay condition is satisfied.",
"Figure REF displays two triangulations, where one triangulation (left) does not meet the Delaunay condition and the other triangulation (right) meets the Delaunay condition after performing an edge flip (swap).",
"There are incremental algorithms and divide conquer algorithms for generating DT [23], [3].",
"Recently, sweep-hull algorithm is also proposed [24].",
"However, most of the existing algorithms are non-trivial algorithms, which often require to perform repeated edge flips and often require to install extra software.",
"Figure: Triangulation that (a) does not meet the Delaunay condition and (b) meets the Delaunay condition after performing an edge flip."
],
[
"Delaunay Triangulation using Deep Learning",
"The DT problem belongs to combinatorial optimization problems involving geometry similar to the convex hull and TSP problems.",
"The problem finds subsets of triangles that satisfy the Delaunay condition in each point set.",
"Most of these combinatorial optimization problems are not trivial to solve and are often NP problems.",
"Research on generating DT using deep learning has been studied minimally for several reasons.",
"First, when the solution of DT is expressed as an input/output sequence, the input/output length varies depending on the problem size.",
"However, conventional seq2seq-based models are not effective for this variable-length dictionary.",
"Second, the mesh data structure is not a regular data structure, such as an image, but is an irregular data structure, making learning difficult for the model.",
"Applying unmodified deep learning methods, such as the convolutional neural network (CNN), is challenging [12].",
"Third, it is possible to predict the coordinates of a point when creating a mesh using a deep learning method, such as the RNN, but in this case, there is no limit to the feature space, which causes blurring [30].",
"Lastly, the elements constituting the triangular mesh consist of three points, which must be learned as a group, and it is difficult for the model to learn the relationships and dependencies between points in distant locations.",
"In particular, as the number of points forming a point set increases, the sequence length becomes longer, making it challenging to learn the dependencies between points in distant locations in the sequence.",
"To the best of our knowledge, two deep learning-based approaches can generate DT from a given input point set.",
"Both methods are based on supervised learning and use the encoder-decoder model.",
"These methods automatically predict DT for a given point set after the model sufficiently learns DT.",
"In [30], the authors used the Ptr-Net for generating DT.",
"It obtains 80.7% accuracy and 93.0% triangle coverage when the number of points ($m$ ) is 5.",
"However, when $m$ = 10, the accuracy dramatically decreases to 22.6%, and the triangle coverage decreases to 81.3%.",
"In addition, when $m$ = 50, it does not produce any precisely correct triangulation, and the triangle coverage is also significantly reduced to 52.8%.",
"In [11], the authors proposed the multi-Ptr-Net (M-Ptr-Net), a variant of the Ptr-Net, which uses multiple “pointers” to learn by combining the vertices that constitute a triangle into a group for the DT problem.",
"It uses multiple “pointers” using the multi-label classification idea [11].",
"It fits a loss function of the multi-label classification using the sigmoid function instead of the softmax function.",
"They reported that when $m$ = 5, the accuracy improves by 0.45% compared to the Ptr-Net.",
"However, two existing deep learning-based approaches for generating DT still have limitations for practical use due to the low accuracy and triangle coverage.",
"In addition, the model does not sufficiently learn point-to-point relationships forming triangles."
],
[
"Neural Combinatorial Optimization",
"In recent years, various efforts have been made to solve combinatorial optimization problems using state-of-the-art artificial intelligence techniques [19], [17], [15].",
"Both supervised learning and reinforcement learning are used to solve combinatorial optimization problems, but their effectiveness depends on the problem.",
"Supervised learning is effective when an optimal solution is available and sufficient data are given in the training data.",
"However, in many cases, it is not easy to obtain such a high-quality labeled dataset.",
"The supervised learning-based approach is complex, and scalability is poor when the problem is large or NP.",
"In such a large or NP problem, many reinforcement learning-based methods have been proposed.",
"Bello et al.",
"proposed a framework to solve combinatorial optimization problems using neural networks and reinforcement learning [2].",
"The method solves the TSP and demonstrates nearly optimal results on two-dimensional Euclidean graphs for up to 100 nodes.",
"Nazari et al.",
"presented a framework for solving the vehicle routing problem using reinforcement learning [21].",
"The framework provides a nearly optimal solution and outperforms the classical heuristics proposed for solving the vehicle routing problem.",
"Recently, several attempts have been made to solve the combinatorial optimization problem using a graph CNN [4]."
],
[
"Seq2seq Models",
"The RNN-based seq2seq models are efficient neural networks for the task of corresponding one sequence to another and are widely used in machine translation [25], [5].",
"The typical seq2seq models have two RNNs called the encoder and decoder.",
"The encoder reads the input sequence sequentially to create a context vector, and the decoder receives the context vector and sequentially outputs the output sequence.",
"For the input sequence $P$ and the corresponding output sequence $O^{P}=\\left( O_{1}, O_{2},...,O_{n} \\right)$ , the seq2seq model estimates the probability of sequence $O^{P}$ using the chain rule as follows: $p\\left( O^{P}|P;\\theta \\right)=\\prod _{i=1}^{n}p(O_i|O_1,...,O_{i-1},P;\\theta ),$ where $\\theta $ is a learnable parameter.",
"During the inference process, in decoder step $i$ , the model predicts the sequence by selecting $O_i$ , which maximizes $p(O_i|O_1,...,O_{i-1},P;\\theta )$ .",
"However, this seq2seq model contains all information in a context vector with a fixed size, resulting in the loss of information, and an attention mechanism is used to solve this problem [1].",
"The basic idea of the attention mechanism is that the decoder predicts the output and attention for the entire input sequence of the encoder once again at every time step.",
"However, even with the attention mechanism, there is still a problem that the output dictionary depends on the length of the input sequence [20]."
],
[
"Pointer Network (Ptr-Net)",
"In the existing seq2seq model, the model is difficult to use for problems where the size of the output dictionary changes according to the input size (e.g., combinatorial optimization problems) because the conditional probability, $p(O_i|O_1,...,O_{i-1},P;\\theta )$ is calculated and selected from a fixed dictionary.",
"Similar to the seq2seq model, the Ptr-Net consists of two RNNs: an encoder and a decoder.",
"For each time step, the encoder takes an element of the input sequence as input and outputs the embedding at that time.",
"The decoder outputs a pointer to the input data from the embedding received from the encoder through a modified attention mechanism.",
"The Ptr-Net calculates the probability as follows using the modified attention mechanism: $\\begin{aligned}u_j^i &=v^{T} \\mathrm {tanh} \\left(W_1e_j+W_2h_i \\right), j\\in \\left( 1,\\cdots ,m \\right) \\\\p\\left( O_{i} | O_{1},...,O_{i-1}, P \\right) &=\\mathrm {softmax} \\left(u^{i} \\right),\\end{aligned}$ where the hidden states of the encoder and decoder are $\\left( e_1,...,e_m \\right)$ and $\\left( h_1,...,h_n \\right)$ , respectively.",
"In addition, $u^i$ is a vector of length $m$ representing each score of the encoder step, and $W_1$ , $W_2$ , and $v$ are learnable parameters.",
"Unlike previous attention mechanisms that create a new context vector due to softmax $u^i$ , the Ptr-Net considers this to be a probability for each input element.",
"In a dictionary of input elements, $u^i$ is a vector of “pointers” to the input elements."
],
[
"Multi-Ptr-Net",
"For the DT problem, three points form a triangle when creating a triangulation.",
"However, the Ptr-Net has limitations in learning the topological relationship of the points, especially when the number of input points is large.",
"Moreover, the Ptr-Net has a limitation in learning the sequence as a pair; thus, multi-Ptr-Net (M-Ptr-Net) is proposed to improve these limitations [11].",
"The M-Ptr-Net is motivated by the multi-label classification method so that multiple embedding results in the encoder can be pointed at simultaneously in one step of the decoder.",
"At each time step of the decoder, it calculates the probability using the following equation: $p\\left( O_{i} | O_{1},...,O_{i-1}, P \\right) =\\mathrm {sigmoid} \\left(u^{i} \\right).$ The original Ptr-Net uses the softmax function for “pointing”, which is used for the multi-class classification problem.",
"In M-Ptr-Net, the sigmoid function used in the multi-label classification problem is employed instead of the softmax function.",
"Figure REF depicts an example of applying M-Ptr-Net to the DT problem.",
"The input includes five planar points, $P_1=\\left( x_1,y_1 \\right),...,P_5=\\left( x_5,y_5 \\right)$ with four elements where $\\left( x_i, y_i \\right)$ are the Cartesian coordinates of the points.",
"The output sequence represents the solution for the DT problem.",
"Unlike the Ptr-Net, the strength of the M-Ptr-Net is that it is not affected by the order of the points forming a triangle because it learns the three points forming a triangle as a group.",
"Figure: Example using M-Ptr-Net for the DT problem.",
"Input P=P 1 ,...,P 5 P=\\left\\lbrace P_1,..., P_5 \\right\\rbrace and the output=⇒,1,2,3,1,2,4,2,3,5,2,4,5,⇐\\left( \\Rightarrow , \\left\\lbrace 1, 2, 3 \\right\\rbrace ,\\left\\lbrace 1, 2, 4 \\right\\rbrace , \\left\\lbrace 2, 3, 5 \\right\\rbrace , \\left\\lbrace 2,4,5 \\right\\rbrace ,\\Leftarrow \\right).",
"The tokens ⇒\\Rightarrow and ⇐\\Leftarrow are beginning and end of sequence, respectively."
],
[
"Transformer and Self-attention",
"The conventional encoder-decoder models using the attention mechanism have problems in that the amount of learning increases rapidly as the number of steps increases.",
"Moreover, it is challenging to learn the dependence between words in distant locations.",
"To improve these problems, the transformer model is a neural network model that avoids recurrence and relies entirely on the attention mechanism to derive the global dependence of the input and output.",
"The transformer model also uses the encoder-decoder model and includes multi-head attention, self-attention, and positional encoding [27], [31].",
"Three types of attention exist in the original transformer model.",
"The first is the attention from the encoder to the encoder, the second is the attention from the decoder to the decoder, and finally, the third is the attention from the decoder to the encoder.",
"Among them, the first two inner attentions are called self-attention, and they occur inside the encoder or decoder.",
"Self-attention is a process of generating a new expression of a corresponding word in consideration of the relative positional relationship of words in a sentence and differs from the recurrence layer and convolutional layer.",
"First, the total computational complexity per layer and the minimum number of sequential operations are both small.",
"Second, it is possible to learn a wide range of dependencies in a short path.",
"Finally, each head presents a model that can be interpreted.",
"The transformer interprets the attention as receiving a query and a key-value pair and outputs an output vector that synthesizes the value vector corresponding to the query vector.",
"Single-head attention used in the transformer can be expressed by the following formula [27], [31]: $\\mathrm {Head=Attention}\\left( QW^{Q}, KW^{K}, VW^{V} \\right)=\\mathrm {softmax}\\left( \\frac{QW^{Q}\\left( \\left( KW^{K} \\right)^{T} \\right)}{\\sqrt{d_{k}}} \\right)VW^{V},$ where $Q$ is a $\\left( q, d_{\\mathrm {model}} \\right)$ matrix consisting of $q$ query vectors, $K$ is a $\\left( k, d_{\\mathrm {model}} \\right)$ .",
"Moreover, $W^Q$ and $W^K$ are learnable parameters with dimension $\\left( d_{\\mathrm {model}}, d_k \\right)$ .",
"$W^V$ is a learnable parameter with dimensions $\\left( d_{\\mathrm {model}}, d_v \\right)$ .",
"Self-attention is the case where $Q$ , $K$ , and $V$ are the same.",
"The result of reinterpreting each embedding is the output considering the relationship between the input embeddings.",
"The proposed neural network model consists of an encoder and decoder, and each consists of a long short-term memory (LSTM) cell [13].",
"The input sequence in the training data is the point set, $P=\\left(P_{1},...,P_{m} \\right)$ , where $P_i$ is the Cartesian coordinates of $m$ planar points.",
"The outputs, $O^{P}=\\left( O_1,...,O_n \\right)$ are sequences representing the solution to the corresponding DT problem.",
"Each $O_i$ refers to the index of $P$ having an integer value between 1 and $m$ , and three consecutive $O_i$ are combined to represent a triangle."
],
[
"Sequence Ordering",
"For the seq2seq-based neural network models, the data order has a significant influence on learning performance in the input and output sequences [30], [29].",
"Sequence ordering is simple and effective in that it only requires changing the sequence order but can significantly improve learning performance.",
"Vinayl et al.",
"found that restricting the equivalence class as much as possible for the output is always better [30].",
"The authors proposed a method to perform sequence ordering only for the output sequence.",
"In [29], the authors insisted that both the input and output sequence orders are important, but they do not suggest a specific sequence ordering method for the DT problem.",
"The proposed sequence ordering is motivated by the work in [30], [29].",
"We propose a new input/output sequence ordering method to improve model performance.",
"If the sequence is ordered properly and meaningful rules are set, the rules are created in the data, making the model easier to learn.",
"The proposed method consists of three steps.",
"First, we sorted the input sequence in lexicographic order when putting the input sequence into the encoder.",
"The lexicographic order in the input sequence refers to a method of sorting $P$ in the order of small $x$ -axis coordinates with respect to $P$ and sorting in the order of small $y$ -axis coordinates if the $x$ -axis coordinates are the same.",
"The two orders in the output sequence are the ordering between triangles and the ordering for the indices of the points that form each triangle.",
"Second, we sorted the index of the points forming a triangle by increasing the triangle representation.",
"For example, we used $\\left( 1, 2, 3 \\right)$ instead of $\\left( 2, 3, 1 \\right)$ to represent a triangle.",
"Finally, we sorted each triangle by its incenter coordinates.",
"We computed the incenter coordinates of each triangle and sort them in the order of the small $x$ -axis coordinates.",
"If the incenter coordinates of the $x$ -axis were the same, we sorted them in the order of the small $y$ -axis.",
"We also tested the other sequence ordering method for sorting input/output sequences, such as sorting the sequence by angle, but our preliminary results indicate that the proposed ordering method exhibits better performance than other sequence ordering methods.",
"Better input/output sequence ordering of the proposed model will be explored in future work.",
"Self-attention outputs a new embedding reinterpreted in consideration of the relationship between input embeddings as described earlier.",
"Figure REF presents the proposed neural network model based on self-attention.",
"In the proposed model, we apply self-attention to the embeddings for each point output of the encoder.",
"We expected that the model can learn about the relationship and dependence among points better than in the absence of self-attention.",
"We also expected that the model would output a well-analyzed embedding by considering the relationship between points with far distances in the step in the encoder.",
"Figure: Proposed neural net model.In the decoder, self-attention can be used like the transformer model, but we did not use the self-attention mechanism in the decoder for the following two reasons.",
"First, in the encoder, it is difficult to include the relationship between points with far distances.",
"However, in the decoder, the state of the surrounding step is considered more important than the step with a far distance.",
"Second, because the decoder has a step length of up to 5 to 6 times that of the encoder, the cost of self-attention with a complexity of $O(n^2)$ becomes higher.",
"Additional performance improvement can be achieved if the decoder also uses self-attention, but by using self-attention only in the encoder, the proposed model improves performance within a reasonable computational cost.",
"Similar to the transformer model [27], [31], the proposed model also uses the residual connection, but it uses a single head.",
"This is because even a single head creates enough complexity to improve performance greatly.",
"Figure 5 displays the model to which the proposed self-attention is applied for DT.",
"Both input/output sequences were sorted using the proposed sequence ordering method.",
"In this example, $P=\\left( P_1,...,P_5 \\right)$ and $O^{P}=\\left( \\Rightarrow , \\left( 1, 2, 3 \\right), \\left( 1, 2, 4 \\right), \\left( 2, 3, 5 \\right), \\left( 2, 4, 5 \\right), \\Leftarrow \\right)$ .",
"Figure: Example of input and output sequence representation for the DT using the proposed model.",
"Input P=P 1 ,...,P 5 P=\\left( P_1,...,P_5 \\right) and O P =⇒,1,2,3,1,2,4,2,3,5,2,4,5,⇐O^{P}=\\left( \\Rightarrow , \\left( 1, 2, 3 \\right), \\left( 1, 2, 4 \\right), \\left( 2, 3, 5 \\right), \\left( 2, 4, 5 \\right), \\Leftarrow \\right).",
"The tokens ⇒\\Rightarrow and ⇐\\Leftarrow .",
"The tokens ⇒\\Rightarrow and ⇐\\Leftarrow are beginning and end of sequence, respectively."
],
[
"Attention score function",
"The conventional attention mechanism does not consider domain knowledge and is thus difficult to apply to geometric combinatorial optimization problems, such as DT.",
"In this paper, we propose a novel attention score function that augments the existing attention mechanism.",
"This function is a new type of attention score function that forcibly sets the conditional probability to zero when the desired geometric requirements are not satisfied in each step in the decoder.",
"The proposed attention score function is described in Algorithm 1.",
"First it uses the definition of a triangle: each triangle consists of three points.",
"Therefore, it makes the model generate triangles by forcing the length of the output sequence to be a multiple of three.",
"In addition, at every third step of forming a triangle in the decoder, the function checks whether the Delaunay condition is satisfied for each (candidate) point of $P$ .",
"If any candidate point of $P$ violates the Delaunay condition, it adds $\\gamma $ to the attention score in Eq.",
"(REF ) such that the models cannot “point” to that candidate point.",
"For this case, the new attention score function is computed as: $\\bar{u_j}^i=u_{j}^{i}+\\gamma ,$ where $\\gamma $ (e.g., $-\\infty $ ) is a user-defined parameter.",
"Otherwise, $\\bar{u_j}^i=u_{j}^{i}$ .",
"The proposed attention score function is motivated by the definition of DT, and the geometric requirement is determined regarding whether it satisfies the Delaunay condition described in Section 2.",
"However, the proposed attention score function is a flexible method that can be extended to other geometric problems.",
"[t] Attention Score Function [1] Step $i$ , input sequence $P$ , predicted output sequence $O^{i}=\\left( O_1,...,O_{i-1} \\right)$ , attention score vector $u^i$ , parameter $\\gamma $ New attention score $\\bar{u_j}^i$ $i$ mod 3 $\\ne $ 1 add $\\gamma $ to a score of the end of sequence token $i$ mod 3 = 0 $j = 1,..., m$ triangle $T=\\left( P_{O_{i-2}}, P_{O_{i-1}}, P_{O_{j}} \\right)$ any point of $P$ is inside the circumcircle of $T$ $\\bar{u_j}^i=u_{j}^{i}+\\gamma $ $\\bar{u_j}^i=u_{j}^{i}$"
],
[
"Beam search decoding",
"Beam search is a method of improving efficiency by limiting the number of nodes to remember using the best-first search (BFS) method [1].",
"At every time step, the method keeps the most likely sequence elements corresponding to the beam size [26].",
"Finally, the decoder selects the sequence with the highest joint probability among the beam subsets.",
"This can be expressed as follows: $\\operatornamewithlimits{argmax}\\limits _{ O^{w}\\in O_{B}^{w}} \\prod _{i=1}^{w}P\\left(O_i | P, O_1,...,O_{i-1} \\right),$ where $O_{B}^{w}$ is a set of beam candidate sequences at time step $w$ ."
],
[
"Experiments",
"List of Experiments.",
"We conducted experiments to investigate the performance of our model.",
"We ran our tests by increasing the number of points ($m$ ) in the input point set as $m$ = 5, 10, 15, and 20.",
"All methods use the same hyper-parameters for all experiments.",
"The first experiment determines the effect of input/output sequence ordering on model performance.",
"For the proposed neural network model, random ordering and the proposed ordering method described in Section 4 are compared.",
"Second, we compared the proposed model with other existing deep-learning-based DT generation methods.",
"We compared our model to the Ptr-Net [30] and M-Ptr-Net [11] models.",
"The method for multi-label classification was not explicitly mentioned when using M-Ptr-Net.",
"For implementation, the cross-entropy loss was added after the sigmoid function, and then the top three were chosen from the output.",
"There was no mention of input/output sequence ordering for the M-Ptr-Net.",
"Therefore, the same output sequence ordering method mentioned in [30] was used for the M-Ptr-Net.",
"In the experiment, we used two different decoders: greedy and beam search (BS).",
"The greedy decoder selects the vertex with the highest probability at every decoding step [21].",
"The BS decoder selects a sequence with the highest joint probability among a subset of beams within a beam width size.",
"Our first two experiments employed greedy decoders, and a BS decoder was used for the third experiment.",
"When using BS, a beam width of four was used in all tasks.",
"The fourth experiment indicates the solution time (prediction time) of the proposed model as $m$ increases.",
"We investigated whether the proposed model can be practically used in terms of the solution time.",
"All models were implemented using the Tensorflow 2.0 library, and experiments were conducted using a single Intel Xeon Gold 6152 CPU 2.10 GHz and a single Nvidia Titan V100 PCIe 32 GB GPU.",
"The runtime can vary due to hardware and implementations.",
"Architecture and Hyper-parameters.",
"We used virtually the same architecture throughout all experiments and datasets.",
"Tanh was used as an activation function for the encoder and decoder of the models, and a single layer LSTM with 256 hidden units was used.",
"The decoder attention mechanism also has 256 hidden units.",
"The Adam optimizer was used [16] with a learning rate of 0.002, $\\beta _1$ of 0.9, and $\\beta _2$ of 0.999, and the Xavier method was employed for parameter initialization [8].",
"Training was performed until the training loss converges.",
"In Eq.",
"(REF ), the $\\gamma $ value was set to $-\\infty $ .",
"We did not tune all hyper-parameters to reach the best performance.",
"Datasets.",
"We generated 1M training example samples (point sets) of each task.",
"Of the total sample data, 90% of the data were used as the training dataset, and the rest were used for testing.",
"In all cases, we sampled from a uniform distribution in $\\left[ 0,1 \\right]\\times \\left[ 0,1 \\right]$ .",
"For a given point set, the ground truth data of DT were obtained using MATLAB.",
"We released all datasets at https://github.com/hunni10/DelaunayDataset.",
"Performance Measure.",
"We evaluated the performance of the proposed neural network model using four different metrics.",
"The first metric is the triangle coverage (TC), which is defined as the ratio of triangles that the model predicts correctly.",
"For two triangular elements consisting of three vertices, even if the permutation of the vertices is different, it is considered as the same triangle.",
"For example, $\\left( 1,2,3 \\right)$ and $\\left( 2,3,1 \\right)$ represent the same triangle.",
"Additionally, any permutation of the triangles in the output sequence represents the same triangulation.",
"We assumed that the total number of (testing) samples is $S$ .",
"We let $\\hat{O}_i$ and $O_i$ be the output sequence of the $i^{\\mathrm {th}}$ sample of the predictions and the ground truth, respectively.",
"The second metric is accuracy (ACC), which is defined as follows: $\\mathrm {ACC}=\\frac{\\sum _{i=1}^{S}S_{i}}{S},$ where $S_{i}=1$ if $\\hat{O}_i$ and $O_i$ represent the same triangulation.",
"Otherwise, $S_{i}=0$ .",
"For example, we assume two testing samples each consist of five triangles and that ground truth also has five triangles.",
"For one testing sample, if the model predicts the same triangulation as the ground truth, and for another testing sample, if only four out of five triangles are correctly predicted, then the TC is 0.9, and ACC is 0.5, respectively.",
"It is also essential to determine whether the number of triangles in the triangulation predicted by the model matches the actual number of triangles in the ground truth.",
"The third metric measures the similarity of the number of triangles predicted by the model to the actual number of triangles in the ground truth.",
"We let $\\hat{t}_i$ and $t_i$ be the number of triangles of the predictions and the ground truth of the $i^{\\mathrm {th}}$ sample, respectively.",
"The third metric, the triangle count accuracy (TCA), is defined as follows: $\\mathrm {TCA}=\\frac{\\sum _{i=1}^{S}T_{i}}{S},$ where $T_{i}=1$ if $t_i$ and $\\hat{t}_i$ .",
"Otherwise it is zero.",
"In practice, even the length of the output sequence predicted by the model may not be a multiple of three.",
"If the length of the output sequence $\\hat{O}_i$ is not a multiple of three, the output exceeding a multiple of three is excluded when calculating $\\hat{t}_i$ because it cannot form triangles during triangulation.",
"For example, if $\\hat{O}_i=\\left(O_1, O_2, O_3, O_4, O_5 \\right)$ , then $O_4$ and $O_5$ are excluded.",
"The final metric is the DT rate (DTR), which measures how well each triangle predicted by the model satisfies the Delaunay condition.",
"It is the ratio of triangles that satisfy the Delaunay condition among the triangles predicted by the model.",
"All four metrics have values between 0% and 100%, and the values closer to 100% indicate better results.",
"In terms of solution quality, both TC and ACC are the most critical metrics for evaluating the performance of the model and should be considered together.",
"This is because the accuracy metric is extremely strict (especially for large $m$ ) in that the accuracy is zero for one sample if even one triangle among the triangles predicted by the model is different from the ground truth."
],
[
"Experiment 1: Effect of Sequence Ordering",
"Figure REF illustrates the comparison results of the proposed input/output sequence ordering and random ordering.",
"Both methods use the same proposed model, and the only difference is whether the dataset is sorted or not.",
"The performance is much better in all metrics when the sequence was sorted using the proposed ordering method for all $m$ values.",
"This is consistent with the results of previous work in which the sequence order dramatically influences the performance of the Ptr-Net-based models [2, 13].",
"Sequence ordering must be performed because performance is greatly improved by simply changing the sequence order of the dataset without making any changes to the model.",
"As the $m$ value increases, the sorted and random ordering performance gap is much more significant in both TC and ACC metrics.",
"For $m$ = 5, the TC and ACC of the sorted ordering are 99.30% and 96.46%, respectively, whereas the TC and ACC of the random ordering are 97.10% and 87.79%.",
"When the sequences are not sorted in the case of $m$ = 20, TC is 76.61%, and ACC is 0.06%.",
"If the sequences are sorted, TC and ACC are 97.69% and 55.16%, respectively.",
"Interestingly, sorting the input/output sequence also has a significant effect on the TCA.",
"In sorting the input/output sequence for all $m$ values, the number of triangles predicted by the model is more consistent with the actual number of triangles of the ground truth.",
"No significant difference exists between the DTR ratio for all $m$ values, whether the sequences were sorted or not.",
"Figure: Performance of the proposed input/output sequence ordering compared to random ordering using four metrics: triangle coverage (TC), accuracy (ACC), triangle count accuracy (TCA), and Delaunay triangle rate (DTR).."
],
[
"Experiment 2: Comparison with Previous Work",
"Figure REF depicts the results of the proposed model compared with the existing deep learning-based DT generation methods according to $m$ values.",
"We observe that the performance of the proposed model outperforms other methods for all $m$ values in the TC, ACC, and DTR.",
"This performance is primarily for two reasons.",
"First, the model can learn the point-to-point relationship through self-attention in the encoder.",
"Second, the attention score function in the decoder provides a high penalty function to candidates who do not satisfy the Delaunay condition.",
"In particular, the performance gap between the proposed model and other methods increases as the value of $m$ increases.",
"Specifically, the proposed model achieves TC = 98.54% and ACC = 82.15% when $m$ is 10, whereas Ptr-Net achieves TC = 94.15% and ACC = 62.71%, and M-Ptr-Net achieves TC = 86.98% and ACC = 38.14%.",
"When the value of $m$ is 20, the proposed model achieves TC = 97.69% and ACC = 55.16%, whereas Ptr-Net achieves TC = 92.42% and ACC = 24.36%, and M-Ptr-Net achieves TC = 67.44% and ACC = 0.39%.",
"We observe that especially for large $m$ values, the length of the output sequence predicted by the Ptr-Net is slightly longer than that of the ground truth in many cases.",
"In these cases, the TCA may appear to be high because it excludes outputs exceeding a multiple of three from the output sequence.",
"Even if the predicted sequence length is not exactly correct, only the length of the valid sequence, which corresponds to the number of valid triangles, is only considered.",
"Among the compared models, the M-Ptr-Net has the worst performance for all metrics.",
"In particular, the ACC becomes zero when $m$ = 20.",
"Unlike the Ptr-Net, the M-Ptr-Net chooses three points simultaneously, so the model does not sufficiently learn the relationship among the three points to be selected.",
"Figure REF displays one example of the predicted triangulation of each model and the ground truth when $m$ = 15.",
"The blue elements represent elements in the ground truth, and red elements represent elements not in the ground truth.",
"In this example, only the proposed model exhibits the same triangulation as the ground truth, but other models fail to predict the ground truth.",
"We only demonstrate one example, but the proposed model performance is superior to other existing models in most testing samples.",
"Figure: Performance of the proposed model compared to the previous work using four metrics: triangle coverage (TC), accuracy (ACC), triangle count accuracy (TCA), and Delaunay triangle rate (DTR)..Figure: Predictions examples for each model with the greedy decoder and the ground truth when mm = 15.",
"Blue elements represent elements in the ground truth, and red elements represent elements not in the ground truth.",
"Only the proposed model predicts the ground truth.."
],
[
"Experiment 3: Effect of Beam Search Decoder",
"Figure REF shows the proposed model results when the BS decoder (BS = 4) is used instead of the greedy decoder.",
"In the previous two experiments, the greedy decoder was used.",
"We observe that the BS decoder significantly improves the performance compared to the greedy decoder in all four metrics.",
"In particular, the effect of using the BS decoder increases as the $m$ value increases.",
"For example, the performance gap between the greedy and BS decoder is only 0.03% for the TC metric when $m$ = 5, and it increases to 0.59% when $m$ = 20.",
"For the ACC metric, the performance gap is 0.12% when $m$ = 5, but it becomes 6.07% when $m$ = 20.",
"If the BS decoder is used, the performance can be improved by continuously tracking the most probable candidate points at the expense of prediction time.",
"If the BS decoder is used together with the proposed attention score function, the number of candidates of the first two steps (points) increases.",
"Therefore, using the proposed attention score function is more effective because it offers more chances to determine the correct triangulation in the decoder.",
"Figure REF presents one example of predictions of each model with the BS decoder and the ground truth when $m$ = 20.",
"Only the proposed model can predict the triangulation of the ground truth correctly.",
"Figure: Performance of the proposed model with greedy decoder and BS decoder using four metrics: Triangle coverage (TC), accuracy (ACC), triangle count accuracy (TCA), and Delaunay triangle rate (DTR).Figure: Example of predictions of each model using the greedy and beam search (BS) decoders when mm = 20.",
"Blue elements represent elements in the ground truth, and red elements represent elements not in the ground truth.",
"The dotted element is a replicated element.",
"Only the proposed model (BS) predicts the ground truth."
],
[
"Experiment 4: Solution Time",
"Figure 11 presents the average solution (prediction) time for the model to predict one testing sample.",
"Employing the BS decoder is a trade-off in terms of performance and computational time.",
"When the BS decoder is used, the performance improves compared to the greedy decoder, but the solution time increases.",
"Our results indicate that the performance improves significantly with only a slight increase in solution time.",
"The solution time for both the greedy and BS decoders increases almost linearly as $m$ increases.",
"When $m$ = 20, the solution time is 1 s for greedy and 2 s for BS decoders.",
"Among the compared methods, the M-Ptr-Net has the worst performance, but the solution time is the fastest because it uses multiple pointers.",
"The number of steps in the M-Ptr-Net decoder is fewer than other models because it predicts three points simultaneously in one step in the output sequence.",
"The proposed model with the BS decoder is the slowest, but the difference is not large.",
"Figure: Solution time (sec) with respect to the number of points (mm)."
],
[
"Discussion",
"The proposed model is based on supervised learning and therefore has the limitations of supervised learning.",
"It requires numerous high-quality labeled data for learning, and the performance of the model depends on such labeled data.",
"We plan to apply reinforcement learning as a future work for the DT generation problem.",
"The proposed model outperforms other existing deep-learning-based approaches, but it still fails in some cases especially for large $m$ values.",
"We present several cases of failure of the predictions of the proposed model.",
"First, the edge consisting of the first two predicted points is an edge that is not in the ground truth.",
"For this case, the proposed attention score function cannot exclude these points from the candidates because it evaluates the candidate points every three steps.",
"We observe that these failure cases often occur when the testing sample points have similar or identical values in the $x$ -axis.",
"For these cases, the model learning performance can deteriorate because the input sequence is lexicographically ordered.",
"The model also fails to predict the ground truth when it predicts the same triangles multiple times, especially for large $m$ values.",
"In addition, the number of triangles predicted by the model is sometimes more or less than the number of triangles in the ground truth.",
"Figure REF depicts two cases in which the model prediction fails.",
"In the first case, the first edge (red) predicted by the model is not in the ground truth, as described earlier.",
"It creates an initial triangle that is not in the ground truth.",
"In the second case, the model predicts the same triangles (hatched element) twice, but not the triangle in the ground truth (white element).",
"Future work will also study how to solve these cases in the decoder.",
"Figure: Two examples of when the model prediction fails.",
"(a) the first edge predicted by the model is not included in the ground truth.",
"(b) The duplicated triangle (dotted) is predicted by the model."
],
[
"Conclusions",
"We propose a novel approach for learning DT using new attention mechanisms based on self-attention and domain knowledge.",
"The experimental results reveal that the proposed model exhibits better performance in all metrics than the existing deep-learning- based DT generation methods and presents the possibility of using a deep-learning-based approach for DT generation.",
"Specifically, when $m$ = 10, the proposed model achieves 98.79% triangle coverage and 84.10% accuracy.",
"In terms of the solution time, the computational time used in practice is not long.",
"Our experimental results reveal that the performance improvement is significant when the input/output sequence is sorted using the proposed method.",
"We also observe that the model learns the topological relationship between points much better using self-attention in the encoder.",
"Significantly, the effect of using self-attention is more critical when the input sequence length becomes longer.",
"A significant performance improvement is obtained in the decoder using the proposed penalty-based attention score function, which excludes candidate points that do not satisfy the Delaunay condition.",
"The proposed penalty-based attention score function is robust in that it can be applied to other geometric-based combinatorial optimization problems.",
"As future work, we plan to apply the proposed attention score function to other geometry-based combinatorial optimization problems, such as convex hull and TSP problems.",
"We also plan to apply the multi-head attention technique used in the transformer model to the DT generation problem."
],
[
"CRediT authorship contribution statement",
"Jaeseung Lee: Software, Methodology, Investigation, Woojin Choi: Software, Investigation, Jibum Kim: Writing - Methodology, Supervison, Review & Editing The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.",
"This work was supported in part by the National Research Foundation of Korea (NRF) Grant funded by the Korea Government (MSIT) under Grant NRF-2020R1A2C1 007917, and in part by the Incheon National University Research Grant, in 2020."
]
] | 2107.01759 |
[
[
"A comparative study of eight human auditory models of monaural\n processing"
],
[
"Abstract A number of auditory models have been developed using diverging approaches, either physiological or perceptual, but they share comparable stages of signal processing, as they are inspired by the same constitutive parts of the auditory system.",
"We compare eight monaural models that are openly accessible in the Auditory Modelling Toolbox.",
"We discuss the considerations required to make the model outputs comparable to each other, as well as the results for the following model processing stages or their equivalents: Outer and middle ear, cochlear filter bank, inner hair cell, auditory nerve synapse, cochlear nucleus, and inferior colliculus.",
"The discussion includes a list of recommendations for future applications of auditory models."
],
[
"Introduction",
"Computational auditory models reflect our fundamental knowledge about hearing processes and have been accumulated during decades of research [1].",
"Models are used to derive conclusions, reproduce findings, and develop future applications.",
"Usually, models are build in stages that reflect basic parts of the auditory system, such as cochlear filtering, mechanoneural interface, and neural processing, by applying signal-processing methods such as bandpass filtering and envelope processing, among others [2].",
"Models of monaural processing often form a basis for binaural models [3] and more complex models of auditory-based multimodal cognition [4].",
"For this reason, combined with the increasing popularity of reproducible research [5], it is not surprising that there is an increasing number of auditory models available as software packages [6], [7], .",
"However, models must be used with caution because they approximate auditory processes and are designed and evaluated under a specific configuration for a specific set of input sounds.",
"While the evaluation conditions are selected to test the main properties of the simulated stages, models may provide different predictions when processing “unseen sounds.” Combined with the wide and low-threshold availability of model implementations, there is a chance of applying a model outside its specific signal or parameter range.",
"Thus, studies comparing models' properties and configurations are important to model users.",
"For example, Saremi et al.",
"[9] compared seven models of cochlear filtering with respect to various parameters describing the nonlinear filtering process of an active cochlea, and Lopez-Poveda [10] compared eight models of the auditory periphery based on the reproduction of auditory-nerve properties.",
"Most other related studies focus on a specific task [11], [12], [13] or an introduction to modelling frameworks [14], [15].",
"In the current study, we compare various monaural auditory models that approximate subcortical neural processing.",
"We provide insights into advantages and limitations, as well as into considerations that are relevant for binaural processing.",
"While our comparison provides guidelines for model selection, we also provide model configurations that reduce the heterogeneity across model outputs.",
"These configurations are evaluated using the same set of sound stimuli across models.",
"We analysed auditory models fulfilling two main criteria.",
"First, the selected models describe the auditory path beginning with the acoustic input up to subcortical neural stages, in the cochlear nucleus (brainstem) and the inferior colliculus (midbrain).",
"These neural processing stages are often used to simulate higher order processes such as speech encoding.",
"Consideration of these stages extends previous comparisons of auditory periphery models [10], [9].",
"Second, the model implementations are publicly available and validated to simulate psychoacoustic performance and/or physiological properties.",
"We use the implementations available in the Auditory Modelling Toolbox (AMT) [16], [8].",
"Based on our inclusion criteria, some models are excluded from the comparison, e.g., models that have only been evaluated at the level of cochlear filtering, such as models based on Hopf bifurcation [17] and the model of asymmetric resonators with fast-acting compression from [18].",
"Further, we did not include models focusing on specific psychoacoustic metrics [19], [20], despite the fact that such models are often based on comparable auditory stages as those described in this study.",
"Also, for the sake of simplicity, our analyses are focused on the comparison across models rather than on a comparison with experimental data.",
"Nevertheless, we provide experimental references to the simulations that are illustrated throughout this paper.",
"Additionally, to encourage reproducible research in auditory modelling, all our paper figures can be retrieved using AMT 1.0, including (raw) waveform representations of intermediate model outputs.",
"On the other hand, we provide insights relevant for the binaural system, including properties that can often be attributed to the effects of the monaural auditory processing, e.g., the temporal processing of interaural level differences [21].",
"To do so, the set of sounds and metrics used to evaluate the selected models were chosen to include fast and slow temporal aspects, such as temporal fine structure and temporal envelope, and to contain a wide range of presentation levels.",
"Table: List of selected models.",
"The model labels used in this study correspond with the model functions in AMT 1.0."
],
[
"Models",
"We define three model families, classified by their objectives [31], which translate into three different levels of detail in simulating the cochlear processing, as schematised in Fig.",
"REF .",
"The selected models are listed in Table REF and are labelled throughout this paper by the last name of the first author and the year of the corresponding publication.",
"This naming system directly reflects the corresponding model functions implemented in AMT 1.0 [8].",
"Figure: Block diagrams of the selected auditory models.",
"Vertical lines: Intermediate model outputs as the basis for the evaluation in the corresponding sections.",
"Blue: Type of hearing impairment that can be accounted for in the corresponding stage (see a brief overview in Sec.",
").We define the family of biophysical models (Fig.",
"REFa) that use a transmission line consisting of many resonant stages coupled by the cochlear fluid.",
"Biophysical models aim at exploring how the properties of the system emerge from biological-level mechanisms, needing a fine-grained description at this level.",
"The biophysical models are represented by verhulst2015 [25] and its extended version, verhulst2018 [26] (model version 1.2 [32], [33]).",
"We further define phenomenological models which are primarily concerned with predicting physiological properties of the system, using a more abstract level of detail than the biophysical models.",
"The phenomenological models considered here rely on dynamically adapted bandpass-filtering stages combined with nonlinear mappings (Fig.",
"REFb) and are represented by zilany2014 [23] and its extended version bruce2018 [27], both combined with the same-frequency inhibition-excitation (SFIE) stages for subcortical processing [34].",
"Further approximation is given by functional-effective models [35], which target the simulation of behavioural (perceptual) performance rather than the direct simulation of neural representations, and usually approximate the cochlear processing by using static bandpass filtering with an optional nonlinear mapping (Fig.",
"REFc).",
"The linear effective models are represented by dau1997 [22] and osses2021 [30] and the nonlinear effective models are represented by king2019 [28] and relanoiborra2019 [29].",
"Given that for each model a similar level of approximation has been generally used in the design of subsequent model stages, we use the described classification to reflect the nature of the entire model, from cochlear processing to the processing of higher stages.",
"The selected monaural models share common stages of signal processing, as indicated in the schematic diagrams of Fig.",
"REF .",
"Each model stage mimics, with greater or lesser detail, underlying hearing processes along the ascending auditory pathway.",
"The thick vertical lines in Fig.",
"REF indicate the intermediate model outputs which are the basis for our evaluation.",
"Note that these stages are, conceptually speaking, independent of each other, however because of nonlinear interactions between them, processing performed by these stages is not commutative, thus requires a step-by-step approach.",
"Next, we provide a brief description of each model stage."
],
[
"Outer ear",
"The listener's head, torso, and pinna filter incoming sounds.",
"The ear-canal resonance further emphasises frequencies around 3000 Hz [36].",
"Both effects can be accounted for by filtering the sound with a head-related transfer function [37] and then applying a headphone-to-tympanic-membrane transfer function, as used in relanoiborra2019 and osses2021.",
"The other six selected models do not include an outer-ear filter, implicitly assuming that either the outer ear does not introduce a significant effect in the subsequent sound processing chain, or that the sounds are presented near the tympanic membrane, as is the case for a sound presentation using in-ear earphones."
],
[
"Middle ear",
"Six of the eight evaluated models include a stage of middle-ear filtering.",
"The transfer functions of the middle-ear filters used in these models are shown in Fig.",
"REF .",
"The transfer functions have been designed to represent stapes velocity near the oval window of the cochlea.",
"The verhulst2015 and verhulst2018 models use an approximation of middle-ear forward pressure gain (“M1” in [38]).",
"The humanised zilany2014 and bruce2018 models use a linear middle-ear filter [39], [40] designed to match experimental data [41].",
"The relanoiborra2019 model uses the filter from [42], [43].",
"The osses2021 model also uses the filter from [42], [43] scaled to provide a 0-dB amplitude in the frequency range of the passband and a fixed group-delay compensation.",
"Middle-ear filtering not only introduces a bandpass characteristic to the incoming signal (Fig.",
"REF ), but also affects the operating range of cochlear compression in models relying on nonlinear cochlear processing, i.e., verhulst2015, verhulst2018, zilany2014, bruce2018, and relanoiborra2019.",
"The passband gains of the middle-ear filters are indicated in Table REF and range between $-66.9$ dB (relanoiborra2019) and $+24$ dB (verhulst2015).",
"In nonlinear models, lower and higher passband gains vary the actual input level to the filter bank, shifting the start of the compression to higher and lower knee points, respectively.",
"Figure: Amplitude spectra of the four middle-ear filters used in six of the evaluated models.",
"The lines were shifted vertically to display their individual maximum at 0 dB.",
"For relanoiborra2019 and osses2021, the grey dashed line shows the combined response of the outer- and middle-ear filters."
],
[
"Cochlear filtering",
"A cochlear filter bank performs a spectral analysis of incoming signals by simulating the mechanical oscillations of the basilar membrane (BM) in the cochlea.",
"Because of complex interactions between the BM, the cochlear fluid, and outer hair cells (OHCs), this analysis depends on the tonotopic position along the cochlea.",
"All approaches used to simulate cochlear filtering produce a set of $N$ temporal signals, for $N$ simulated characteristic frequencies (CFs).",
"Each cochlear section, having a CF expressed in Hz, is assumed to either have relatively sharp frequency tuning [44]: $\\mbox{Q\\textsubscript {ERB}} = 12.7 \\cdot \\left(\\mbox{CF}/1000 \\right)^{0.3}$ or broader tuning [45]: $\\mbox{Q\\textsubscript {ERB}}=\\mbox{CF}/\\left[ 24.7 \\cdot \\left(4.37\\cdot \\mbox{CF}/1000+1\\right)\\right]$ In verhulst2015 and verhulst2018, the cochlear filtering is simulated by a transmission-line model [46].",
"In zilany2014 and bruce2018, the filtering is based on a chirp filter bank [47] tuned to a human cochlea [39], [40], [48].",
"In these models, the cochlear filters are assumed to be tuned according to Eq.",
"REF .",
"In dau1997 and osses2021, the linear Gammatone filter bank from [49] is used.",
"King2019 uses the Gammatone filter bank from [49] followed by a compressive stage acting above a given knee point.",
"In relanoiborra2019, the cochlear processing is simulated by the dual-resonance nonlinear filter bank (DRNL) [42].",
"The cochlear filters of these models are assumed to be tuned according to Eq.",
"REF ."
],
[
"Inner hair cell",
"The inner hair cells (IHCs) transform the mechanical BM oscillations into receptor potentials, subsequently initiating neuronal discharges in the auditory nerve (AN) [50].",
"In the most simple approach, the IHC processing can be simulated as an envelope detector that removes phase information for high CFs, implemented as a half-wave rectification followed by a lowpass (LP) filter.",
"This approach is used in dau1997, king2019, relanoiborra2019, and osses2021, in which the LP filters have $-3$ -dB cut-off frequencies ($f$cut-off) between 1000 and 2000 Hz.",
"In zilany2014, bruce2018, and verhulst2015, a nonlinear transformation is applied to the output of the cochlear filter bank, followed by a cascade of LP filters with $f$cut-off of 3000 Hz (zilany2014 and bruce2018) and 1000 Hz (verhulst2015).",
"The resulting $f$cut-off of each model ranges between 642 Hz (verhulst2015) and 1000 Hz (dau1997, relanoiborra2019, king2019), as indicated in Table REF .",
"In verhulst2018, a more sophisticated IHC model is used [51], that is implemented as a three-channel Hodgkin-Huxley type model, with each of the channels representing mechanoelectrical and (fast and slow) potassium-gated processing [51], [26]."
],
[
"Auditory nerve",
"The transduction from receptor potentials into patterns of neural activity can be derived from the interaction between the IHC and AN.",
"Several AN synapse models have been inspired by the three-store diffusion model [50], assuming that the release of synaptic material is managed in three storage compartments.",
"For steady-sound inputs, this model predicts a rapid neural firing shortly after the sound onset with a decreasing rate towards a plateau discharge rate, a phenomenon called adaptation.",
"The AN synapse models in verhulst2015, verhulst2018, and zilany2014 are based on [50], but zilany2014 further incorporates a power-law adaptation following the diffusion model from [23].",
"The synapse model in bruce2018 uses a diffusion model based on [52] to: (1) have limited release sites, and (2) come after the power-law adaptation instead of before it [27].",
"The outputs of these models simulate the firing of neuron synapsesIn this study we adopted the spelling “neuron” instead of “neurone,” following the etymological argument from [105].",
"having a specific spontaneous rate of high-, medium-, and/or low-spontaneous rates.",
"The effective models, on the other hand, rely on a more coarse AN simulation, expressed in arbitrary units (a.u.).",
"In king2019, adaptation is simulated by applying a highpass filter with a cut-off frequency of 3 Hz [28].",
"In dau1997, relanoiborra2019, and osses2021, adaptation is simulated by so-called adaptation loops [35] that introduce a nearly logarithmic compression to stationary input signals and a linear transformation for fast signal fluctuations (Appendix B in [30]).",
"The arbitrary units of these transformed outputs are named model units (MUs)."
],
[
"Subcortical processing",
"AN firing patterns propagate to higher stages along the auditory pathway, first through the auditory brainstem, then towards more cortical regions [53].",
"On its way, AN spiking is mapped onto fluctuation patterns by neurons that are sensitive to the amplitude of low-frequency fluctuations [54].",
"This fluctuation sensitivity has been approximated using various approaches.",
"Our analyses focus on model approximations of the ventral cochlear nucleus (CN) and inferior colliculus (IC) [34], as well as on different modulation-filter-bank variants [22], [55].",
"As a result, we exclude the analysis of other subcortical structures such as those that play a particular role in the binaural interaction between ears (e.g., the dorsal cochlear nucleus and lateral superior olive) [56], [53].",
"The processing in the ventral CN and IC can be simulated using the same-frequency inhibition-excitation (SFIE) model, resulting in a widely tuned modulation filter (Q factor$\\approx $ 1) with a best-modulation frequency (BMF) depending on the parameters of the model [34], [24].",
"The SFIE model has already been used in combination with the biophysical and phenomenological models described here.",
"For example, zilany2014 has been combined with the SFIE model using between one and three modulation filters .",
"Or, verhulst2015 and verhulst2018 have been combined with the SFIE model tuned to one modulation filter centred at a BMF of 82.4 Hz (see Table REF ) [26], [32].",
"Further, bruce2018 can be combined with the SFIE model in the UR EAR 2020b toolbox [58].",
"Note that zilany2014, verhulst2015, and verhulst2018 have used the output of their mean firing rate generator –an output that can be conceptualised as peri-stimulus time histograms (PSTHs) [59]– as an input to the SFIE model.",
"In bruce2018, because of the stochastic processes in its spike generator, repeated processing of the same stimulus is recommended to obtain a faithful PSTH that can appropriately account for power-law adaptation properties (see Sec.",
"3 in [27]).",
"In the effective models, on the other hand, subcortical neural processing is further approximated based on the modulation-filter-bank concept [22], [55].",
"In dau1997, king2019, relanoiborra2019, and osses2021, linear modulation filter banks are used, covering a range of BMFs up to 1000 Hz.",
"In dau1997, twelve modulation filters with a Q-factor of 2 and overlapped at their $-3$ dB points are used.",
"The same modulation filters are used in relanoiborra2019 and osses2021, but an additional 150-Hz LP filter is applied [60] and the number of filters is limited so that the highest BMF is less than a quarter of the corresponding CF [61].",
"In king2019, the filter bank is used with a wider tuning (Q=1), using ten 50%-overlapped filters having a maximum BMF of 120 Hz [28]."
],
[
"Model configuration",
"We evaluated the intermediate model outputs that are indicated by thick vertical black lines in Fig.",
"REF .",
"The evaluation points are located after the cochlear filter bank (Stage 3), the IHC processing stage (Stage 4), the AN synapse stage or equivalent (Stage 5), and after the IC processing stage or equivalent (Stage 6).",
"Starting with the default parameters of each model, we introduced small adjustments to obtain the most comparable model outputs.",
"All the comparisons can be reproduced with the function exp_osses2022 from AMT 1.0 [8]."
],
[
"Level scaling",
"The same set of sound stimuli was used as input to all models.",
"The waveform amplitudes were assumed to represent sound pressure expressed in Pascals (Pa).",
"The models zilany2014, verhulst2015, verhulst2018, bruce2018, and relanoiborra2019 use this level convention and did not require further level scaling.",
"The models dau1997, king2019, and osses2021 interpret sound pressures between $-1$ and 1 Pa as amplitudes in the range $\\pm 0.5$ , thus a factor of 0.5 (attenuation by 6 dB) was applied to the generated stimuli to meet the level convention of these models.",
"For these latter models, which include mostly level-independent stages, such calibration is relevant because the adaptation loops (used in dau1997, osses2021, also extensible to relanoiborra2019) include level-dependent scaling (Eqs.",
"B1–B3 in [30]).",
"In king2019 a calibrated knee point (default of 30 dB) is used in its cochlear compression stage (Stage 3).",
"All signal levels are reported as root-mean-square (rms) values referenced to 20 $\\mu $ Pa, in dB sound pressure level (dB SPL)."
],
[
"Cochlear filtering",
"The phenomenological and effective models can be set to simulate any CF.",
"The biophysical models, however, because of the nature of the transmission-line structure, have a discrete tonotopy that translates into a discrete set of available CFs.",
"The models verhulst2015 and verhulst2018 were set to 401 cochlear sections spaced at $\\Delta x=$ 0.068 mm with tonotopic distances $x_n$ ranging between $x_1=3.74$ mm and $x_{401}=30.9$ mm, that are related to CFs between CF$_1=12010$ Hz and CF$_{401}=113$ Hz, according to the apex-to-base mapping of Eq.",
"REF [62], $\\mbox{CF}_n=A_0\\cdot \\left( 10^{-a \\cdot x_n/1000} \\right) - A \\cdot k$ where $x_n$ (in mm) can be obtained as $x_1+\\Delta x\\cdot \\left(n-1\\right)$ , and $A=165.4188$ Hz, $a=61.765$ 1/m, $k=0.85$ , and $A_0=20682$ Hz.",
"Note that when reporting results, we indicate the cochlear section number $n$ and its corresponding CF$_n$ .",
"The cochlear-filtering parameters of zilany2014 and bruce2018 were those adapted to a human cochlea [39], [40].",
"Moreover, in order to analyse separately the effects of cochlear filtering and IHC processing in zilany2014, the outputs from the chirp filters representing the static and OHC-controlled filters (C2 and C1 in [23]) were added together and analysed before the IHC nonlinear mapping was applied.",
"This analysis follows a similar rationale as analysing the main output of the DRNL filter bank in relanoiborra2019 (see Fig.",
"3a from [42]).",
"Finally, in king2019, we used a compression factor of 0.3 for all simulated CFs, which is different from the one-channel (on-CF) compression used in [28]."
],
[
"Inner hair cell and auditory nerve",
"Default parameters were used for the IHC and AN stages of the evaluated effective models.",
"However, the biophysical and phenomenological models require the choice of parameters to simulate a population of AN fibres.",
"For each CF we simulated 20 fibres, having either high- (HSR), medium- (MSR), or low-spontaneous rates (LSR), distributed in percentages of 60-20-20% [63], [64], resulting in a 12-4-4 configuration (HSR-MSR-LSR).",
"Note that for verhulst2015 and verhulst2018, this deviates from the standard 13-3-3 configuration [25], [26].",
"For verhulst2015 and verhulst2018, the spontaneous rates of each fibre type were 68.5, 10, and 1 spikes/s for HSR, MSR, and LSR, respectively, as used in human-tuned simulations [26].",
"For zilany2014, the spontaneous rates of each fibre type were 100, 4, and 0.1 spikes/s and for bruce2018 were 70, 4, and 0.1 spikes/s for HSR, MSR, and LSR, respectively.",
"We further disabled the random fractional noise generators in zilany2014 and bruce2018 [65], and the random spontaneous rates in bruce2018 (“std” from Tab.",
"I in [27] was set to zero).",
"With this configuration, the mean-rate synapse outputs of verhulst2015, verhulst2018, zilany2014, and bruce2018 are deterministic.",
"For this reason, to obtain population responses, we simulated the AN processing of each type of neuron only once and then weighted them by factors of 0.6, 0.2, and 0.2 for HSR, MSR, and LSR fibres, respectively.",
"In contrast, the PSTH outputs that are reported for zilany2014 and bruce2018 are not deterministic, requiring the simulation of each AN fibre for each CF.",
"Therefore, PSTH population responses were obtained by counting the average number of spikes in time windows of 0.5 ms across 100 repetitions of the corresponding stimuli."
],
[
"Subcortical processing",
"The default configuration of the model stages of subcortical processing (Stage 6, Fig.",
"REF ) differs in the number of modulation filters (from 1 to 12) and in their tuning across models.",
"In our study, we use only one modulation filter targeting a BMF of approximately 80 Hz (see “Theoretical BMF” in Table REF ) and a Q-factor of approximately 1 for zilany2014, verhulst2015, verhulst2018, bruce2018, and king2019, and a Q-factor of 2 for dau1997, relanoiborra2019, and osses2021.",
"For the biophysical and phenomenological models, we used the SFIE model [34], [24] using two different configurations.",
"The SFIE model [34] integrated in verhulst2015 and verhulst2018 has CN parameters with excitatory and inhibitory time constants of $\\tau \\textsubscript {exc}=0.5$ ms and $\\tau \\textsubscript {inh}=2$ ms, a delay $D=1$ ms, and a strength of inhibition of $S=0.6$ .",
"The IC stage uses $\\tau \\textsubscript {exc}=0.5$ ms, $\\tau \\textsubscript {inh}=2$ ms [26], $D=2$ ms, and $S=1.5$ [34], achieving a BMF of 82.4 Hz.",
"For zilany2014 and bruce2018, the SFIE model is a separate stage [24], implemented as carney2015 in AMT 1.0, where either the mean-rate (zilany2014) or the PSTH outputs (bruce2018) are used as inputs.",
"In our analysis, we only used the output of the band-enhanced IC cell, which corresponds to the SFIE model from [34].",
"The CN parameters were identical to those for the biophysical models.",
"The IC parameters were $\\tau \\textsubscript {exc}=1.11$ ms, $\\tau \\textsubscript {inh}=1.67$ ms, $D=1.1$ ms, and $S=0.9$ , achieving a BMF of 83.9 Hz [24].",
"Note the different inhibition strength $S$ between models.",
"In the biophysical models, the IC output is dominated by inhibitory responses ($S>1$ ) whereas in the phenomenological models the IC output is dominated by excitatory responses ($S<1$ ).",
"In this section we analyse the outputs of the eight selected auditory models in a number of test conditions, whose results are presented in Figs.",
"REF –REF .",
"We aimed at a comparison across models and thus, for the sake of clarity, we refrained from a direct comparison to ground-truth references from physiological data.",
"However, such a comparison is important and interesting.",
"For this reason, we provide references where similar experimental and/or simulation analyses have been presented.",
"These references are indicated as “Literature” in the caption of the corresponding figure.",
"Alternatively, the outputs of the biophysical and phenomenological models may be considered as referential because they have been primarily developed to reflect physiological responses to sounds.",
"Figure: Input-output (I/O) curves for pure tones.",
"Top (a–c): Stimulus frequency of 500 Hz.",
"Bottom (d–f): Stimulus frequency of 4000 Hz.",
"Left (a,d): On-frequency simulations, i.e., output of the cochlear filter with the CF tuned to that of the stimulus frequency.",
"Middle (b,e), right (c,f): Off-frequency simulations, one ERB below and above the on-frequency, respectively.",
"The exact simulated on- and off-frequency CFs are indicated in the title of each panel.",
"All I/O curves were shifted vertically by the reference gains given in Table (see the text for details).",
"Literature: Figs.",
"1–3 from and Fig.",
"3 from ."
],
[
"Cochlear filtering: Compressive growth",
"Sound processing in the cochlea depends not only on the frequency but also on the level of the input stimulus.",
"The amplitude of the vibration displacement increases for higher levels, following an amplitude growth that comprises linear and compressive regimes [66].",
"For this reason, we assessed the curve relating the input stimulus levels with levels at the output of the filter banks, known as input-output (I/O) curves for (1) the on-frequency CF tuned to the frequency of the input stimulus, and (2) the off-frequency responses of cochlear filters tuned to one equivalent rectangular bandwidth number (ERB$_N$ ) [45] below and above the stimulus frequency.",
"We report the I/O curves for pure tones of frequencies 500 and 4000 Hz, with a duration of 100 ms, presented at levels between 0 and 100 dB SPL (steps of 10 dB), gated on and off with a 10-ms raised-cosine ramp.",
"The obtained I/O curves are shown in Fig.",
"REF for on-frequency (left panels) and off-frequency simulations ($\\pm $ 1 ERB$_N$ , middle and right panels).",
"Note that the I/O curves were vertically shifted by the reference gains provided in Table REF which were derived for each model for a pure tone of frequency 1000 Hz and 100 dB SPL.",
"As expected for the level-independent Gammatone filters used in dau1997 and osses2021, the curves were linear in all panels of Fig.",
"REF .",
"For the remaining models, more compressive behaviour was observed for on-frequency curves (left panels) while more linear curves were obtained for off-frequency CFs (middle and right panels), except for relanoiborra2019 and king2019, that had on- and off-frequency compression.",
"For zilany2014/bruce2018, the I/O curves were fairly linear in response to 500-Hz tones (top panels) for both on- and off-frequency CFs.",
"For 4000-Hz tones, a prominent compressive behaviour was observed in the on-frequency curves (panel d) where, additionally, the curve for verhulst2018 turned from a compressive to a linear regime for signal levels above 80 dB.",
"The off-frequency I/O curves obtained for verhulst2018 were similar to those for verhulst2015 but had overall lower and higher amplitudes for the pure tones of 500 Hz (panels b–c) and 4000 Hz (panels e–f), respectively, as a consequence of the differences in their middle-ear filters (see Fig.",
"REF ).",
"The tendency to a more linear regime in off-frequency CFs has been shown previously [66].",
"This is in fact the basis for having compression only applied to the on-frequency channel in king2019 [28].",
"However, the default compression rate of 0.3 for the on-frequency channel with no compression for off-frequency channels leads to an unrealistic level balance between on- and off-frequency channels."
],
[
"Cochlear filtering: Frequency selectivity",
"The frequency selectivity of each filter bank was computed in response to Gaussian noises with a flat spectrum between 20 and 10000 Hz, presented at 40, 70, and 100 dB SPL.",
"Due to the stochasticity of these stimuli, we obtained model responses to a 3-s long noise that were subsequently analysed and averaged in the frequency domain using 500-ms sections.",
"The same 3-s noise was used for all test levels, after the corresponding level scaling."
],
[
"Filter tuning",
"The frequency response of thirty-two filters with CFs between 126 Hz ($n=396$ ) and 9587 Hz ($n=24$ , Eq.",
"REF ) at steps of $n=12$ bins was obtained.",
"For each filter response, a quality factor Q$-$ 3 dB=CF/BW was obtained, where BW is the bandwidth defined by the lower and upper 3-dB down points of each filter transfer function.",
"The frequency selectivity simulations for each of the filter banks are shown in Fig.",
"REF for the noises at 40 (panel a), 70 (panel b), and 100 dB SPL (panel c).",
"The analytical filter tuning curves given by Eqs.",
"REF and REF are indicated as light and dark grey traces in Fig.",
"REF .",
"Note that with this comparison we assume that the Q factors within one ERB are similar to Q$-3$ dB values.",
"The results for 40-dB noises show that the frequency selectivity follows either the analytical tuning of Eq.",
"REF (zilany2014, bruce2018, verhust2015, and verhulst2018) or the tuning of Eq.",
"REF (dau1997, relanoiborra2019, king2019, and osses2021).",
"When looking at the results for higher levels (Fig.",
"REFb–c), no change in tuning was observed for dau1997 and osses2021, as expected for linear models with no compression.",
"For the nonlinear models, the results in Fig.",
"REFb for 70-dB noises showed overall lower Q factors, but with only a small change for king2019 and relanoiborra2019.",
"The results for 100-dB noises in Fig.",
"REFc showed a further lowering of Q factors in the biophysical and phenomenological models, reaching values as low as Q $\\approx 2$ in verhulst2015, lower Q factors for frequencies up to about 4000 Hz in relanoiborra2019, and virtually unaffected Q factors in king2019.",
"A closer inspection to the outputs of king2019 revealed that there was a filter broadening as a consequence of its broken-stick nonlinearity stage, but this broadening predominantly affected the frequency responses outside the range defined by the 3-dB bandwidth used to derive the Q factors.",
"To illustrate the Q-factor transition when increasing the signal level in each model, the difference between Q factors obtained from 40- and 100-dB noises is shown in Fig.",
"REFd, where a decrease in Q factor with increasing signal level is represented by a positive Q difference.",
"Additionally, we observed that relanoiborra2019 and king2019 introduce a change in selectivity at overall higher levels compared to the biophysical and phenomenological models.",
"A closer look at this aspect revealed that this change occurs because relanoiborra2019 and king2019 only apply compression after the bandpass filtering and, therefore, lower level signals are used as input for their compression (broken-stick) module.",
"Figure: Filter tuning expressed as quality factors Q for noises of 40, 70, and 100 dB SPL (panels a-c), and Q-factor difference obtained from the results of 40- and 100-dB noises (panel d).",
"Literature: Fig.",
"4 from and Fig.",
"4B from .Figure: Simulated IHC responses to pure tones of different frequencies evaluated at the corresponding on-frequency bin.",
"The amplitudes were normalised with respect to their maximum value to allow a direct comparison across models.",
"Literature: Fig.",
"9 from and Fig.",
"7 from ."
],
[
"Number of filters",
"The number of filters in a filter bank is relevant for several model applications because too few filters can lead to a loss of signal information [69] and too many filters may unnecessarily increase the computational costs.",
"The number of filters is a free parameter in zilany2014/bruce2018, but is fixed for verhulst2015 and verhulst2018 to yield an accurate precision of the transmission-line solver [70].",
"The remaining models use by default one ERB-wide bands (dau1997, king2019, and osses2021), or have an overlap every 0.5 ERB (relanoiborra2019).",
"Here, we report the minimum number of filters that are required to obtain a filter bank with overlapping at $-3$ -dB points of the individual filter responses.",
"Using the empirical Q-factors of Fig.",
"REF , we assessed the number of filters that would be required to cover a frequency range between 126 Hz (bin $n=396$ , Eq.",
"REF ) and the first filter with its upper cut-off frequency equal or greater than 8000 Hz.",
"The number of filters derived from the 40-dB and 100-dB frequency tuning curves (Fig.",
"REFa,c) are shown in Table REF , including the average filter bandwidth in ERB for the corresponding model.",
"For the biophysical models, the filters were much wider at the higher level than for the other models, with average bandwidths being as wide as 3.05 ERB for verhulst2015 and 2.30 ERB for verhulst2018.",
"This contrasts with the 1.57 ERB for zilany2014 and bruce2018 and the 1.15 ERB or less for the remaining models.",
"These bandwidths are a consequence of the fast-acting (sample-by-sample) compression that is applied just before the transmission-line in the biophysical models and the slower-acting bandwidth control in zilany2014 (denoted as the “control path” in the chirp-filter bank).",
"While cochlear filters are generally wider at high sound levels [71], [72], the appropriate tuning must be evaluated depending on the species' characteristics, the tested CFs, and the type of evaluated excitation signals."
],
[
"IHC processing: Phase locking to temporal fine structure",
"To illustrate the loss in phase locking to temporal fine structure with increasing stimulus frequency, we simulated IHC responses to pure tones with frequencies between 150 Hz ($n=387$ ) and 4013 Hz ($n=112$ , Eq.",
"REF ) spaced at $n=25$ bins, resulting in twelve test frequencies.",
"The tones were generated at 80 dB SPL, with a duration of 100 ms, and were gated on and off with 5-ms raised-cosine ramps.",
"The simulated waveforms, that are assumed to approximate the IHC potential, are displayed and described in terms of AC (fast-varying) and DC (average bias) components, and the simulated resting potentials ($V\\textsubscript {rest}$ ).",
"The AC potential was assessed from the peak-to-peak amplitudes as $V\\textsubscript {AC}=V\\textsubscript {peak,max}-V\\textsubscript {peak,min}$ .",
"The DC potential was obtained as $V\\textsubscript {DC}=V\\textsubscript {AC}/2-V\\textsubscript {rest}$ [68], [67].",
"The obtained IHC waveforms are shown in Fig.",
"REF .",
"Within each panel, bottom to top waveforms represent on-frequency simulations for the test signals, from low to high frequency carriers, respectively.",
"For all model outputs, the four highest carriers (1870$\\le f_c\\le $ 4013 Hz) were amplified by factors between 1 and 3, as indicated in the figure insets.",
"The simulated voltages before the tone onset, i.e., the resting potential $V\\textsubscript {rest}$ , were equal to 0 for all models except for verhulst2018, where $V\\textsubscript {rest}$ was $-57.7$ mV (not schematised in Fig.",
"REF ).",
"It seems clear, however, that the decrease of peak-to-peak AC voltage available towards high frequencies –a measure of the residual amount temporal fine structure– is significantly different across models.",
"When increasing the CFs from 1099 to 4013 Hz, three models showed $V\\textsubscript {AC}$ reduced by less than 76.0% (king2019: 59.7%, decreased from $3.12\\cdot 10^{-3}$ to $1.26\\cdot 10^{-3}$ a.u.",
"; dau1997: 62.6%, decreased from 0.097 to 0.037 a.u.",
"; and verhulst2018: 76.0%, decreased from 39.2 to 9.4 mV), while the other five models showed $V\\textsubscript {AC}$ reductions of at least 92.5%.",
"From the low-frequency IHC waveforms (bottom-most waveforms in each panel), it can be seen that the simulated amplitudes of dau1997, king2019, relanoiborra2019, and osses2021 did not go below their $V\\textsubscript {rest}$ (horizontal grid lines in Fig.",
"REF ) as a result of the applied half-wave rectification process.",
"Furthermore, zilany2014/bruce2018 and verhulst2015 have $V\\textsubscript {peak,min}$ amplitudes of $-66$ mV and $-4.7$ mV, respectively.",
"Despite the different range in their minimum voltages, there is a strong qualitative resemblance between waveforms (green and red traces in the figure).",
"In fact both models use the same type of IHC nonlinearity (compare Eqs.",
"17–18 from [47] with Eqs.",
"4–5 from [25]).",
"In these two models also the same LP filter implementation was used, only differing by the filter order and cut-off frequencies (see Table REF ).",
"The obtained AC/DC ratios are shown in Fig.",
"REF , where a reduction in phase locking is related to a lower ratio.",
"For all models, the ratio decreased with increasing frequency.",
"All AC/DC curves, except that for verhulst2018, overlap well at low frequencies with ratios between 2.1 and 5.9 (below 1000 Hz), decreasing to ratios between 0.06 (osses2021) and 0.83 (dau1997) at 4013 Hz.",
"Although the AC/DC curve for verhulst2018 showed the highest values (ratios between 137.4 at 460 Hz down to 1.3 at 4013 Hz), we still observed the systematic decrease in ratio with increasing frequency.",
"If we further focus on the AC/DC curves in the frequency range between 600 and 1000 Hz, where the phase-locking is expected to start declining [67], all models showed monotonically decreasing curves starting from about 833 Hz (except for verhulst2018, that always showed a decreasing tendency).",
"The lowest ratios were observed for osses2021, followed by the similarly-steeped curve of zilany2014.",
"Finally, a similar AC/DC curve was obtained for relanoiborra2019 and verhulst2015.",
"Figure: Ratio between simulated AC and DC components (VAC/VDCV\\textsubscript {AC}/V\\textsubscript {DC}, see the text) in response to 80-dB pure tones.",
"Literature: Fig.",
"10 from and Fig.",
"8 from ."
],
[
"AN firing patterns",
"AN responses for a number of pure tones were simulated, including rate-level functions expressed as onset and steady-state responses, and the model responses to amplitude modulated (AM) tones.",
"With these benchmarks we attempt to characterise the model responses at the output of the AN synapse stage or their equivalent, with a particular interest on the phenomenon of adaptation [65], [73].",
"We comment on how adaptation is affected by the type of output of Stage 5, using either the approximations from the effective models, the average or instantaneous firing rate estimates of the phenomenological models (zilany2014, bruce2018), or the average rates of the biophysical models (verhulst2015, verhulst2018)."
],
[
"Adaptation",
" Figure: Simulated AN responses to a 4000-Hz pure tone of 70 dB SPL.",
"For ease of visualisation, the responses from osses2021, verhulst2015, and the PSTHs are horizontally shifted by 20 ms.",
"Literature: Fig.",
"1 from and Figs.",
"3 and 10 from .To illustrate the effect of auditory adaptation, we obtained AN model responses to a 4000-Hz pure tone of 70 dB SPL, duration of 300 ms, that was gated on and off with a cosine ramp of 2.5 ms.",
"The obtained AN responses are shown in Fig.",
"REF .",
"All responses had a prominent amplitude overshoot just after the tone onset which then decreased to a plateau (e.g., between 300 and 340 ms, grey dashed lines).",
"After the tone offset ($t=350$ ms), the AN responses showed an undershoot with decreased amplitudes that subsequently returned to their resting level.",
"This stereotypical behaviour is related to the AN adaptation process.",
"The waveforms from effective models using the adaptation loops (dau1997, relanoiborra2019, osses2021) are shown in Fig.",
"REFa, where their amplitudes expressed model units (MU) had values between $-230.5$ MU and 1440.2 MU (dau1997), with a strong onset overshoot and a resting position at 0 MU.",
"For king2019 (Fig.",
"REFb), a mild overshoot was observed, whose maximum amplitude (1.52$\\cdot 10^{-3}$ a.u.)",
"was higher in absolute value than that for the undershoot ($-1.08\\cdot 10^{-3}$ a.u.).",
"With an observed steady-state peak-to-peak amplitude of $0.87\\cdot 10^{-3}$ a.u.",
"king2019 is, at this stage, the model that preserves the most temporal fine structure.",
"For the phenomenological models (zilany2014 and bruce2018), the simulated waveforms using their two types of AN synapse outputs are shown in Fig.",
"REFc–d, based on a PSTH (dark green or brown curves) and mean-rate synapse output (light green or brown curves).",
"The obtained PSTH and mean rate responses in zilany2014 differ in their steady-state values (lower values for the PSTH estimate), while for bruce2018 the difference is in their onset responses, with almost no onset adaptation in the simulated mean-rate output.",
"For the biophysical models (Fig.",
"REFe), the AN synapse outputs represent mean firing rates where a stronger effect of adaptation was observed for verhulst2018 (sky blue), with a plateau after onset that was reached after about 150 ms (at $t\\approx 200$ ms) while for verhulst2015 (red) the plateau is reached shortly after the tone onset.",
"Figure: Simulated rate-level functions derived from the steady-state AN responses of 4000-Hz pure tones.",
"For all models, average responses are shown (coloured traces).",
"For the biophysical and phenomenological models, the responses for HSR, MSR, and LSR neurons are also shown (grey traces).",
"Literature: Fig.",
"7 from , Fig.",
"5A from , and Fig.",
"3 from .Figure: Simulated rate-level functions derived from the onset (maximum) AN responses of 4000-Hz pure tones.",
"The colour codes and legends are as in Fig. .",
"Literature: Fig.",
"3 from ."
],
[
"Rate-level functions",
"Rate-level functions were simulated for a 4000-Hz pure tone presented at levels between 0 and 100 dB SPL with a duration of 300 ms, gated on and off with 2.5-ms cosine ramps.",
"The obtained results are shown in Figs.",
"REF and REF for rate-level curves in the steady-state regime and for onset responses, respectively.",
"For all models, average rates are shown (coloured traces) while for the phenomenological and biophysical models (panels c–h), the simulated response for the three types of neurons (HSR, MSR, and LSR) are shown (grey traces).",
"For the phenomenological and biophysical models, the discharge curves in Fig.",
"REFc–h tend to saturate towards higher levels, which is in line with experimental evidence .",
"For the effective models (Fig.",
"REFa,b), with the exception of relanoiborra2019, the simulated rates did not show saturation as a function of level.",
"In relanoiborra2019, the simulated rates were between 70.2 and 83 MU for signal levels beyond 40 dB.",
"This saturation effect results from the combined action of the nonlinear cochlear filter (Stage 3) with the later expansion stage (Stage 5, Fig.",
"REF ) that precedes the adaptation loops.",
"Despite the overall lack of saturation in the evaluated effective models when looking at the steady-state outputs, a different situation is observed for the onset responses of Fig.",
"REF , where the responses of the models using adaptation loops had a prominent (onset) saturation (dau1997: 1443 MU for levels above 50 dB; relanoiborra2019: 1435 MU for levels above 30 dB; osses2021: 614 MU for levels above 50 dB).",
"Other interesting aspects to highlight are that: (1) almost no onset effect is observed in the mean-rate output of bruce2018; (2) king2019 does not account for any type of saturation as the signal level increases (Figs.",
"REFb, REFb).",
"It should be noted that although hard saturation (as in Fig REFa) has not been experimentally observed for onset AN responses, it is expected a drecrease in the growth of onset rate-curves with level [74], a condition that is not met in king2019 nor verhulst2015.",
"Figure: Simulated on-frequency AN responses to a 4000-Hz tone 100% amplitude-modulated at 100 Hz.",
"Left: Onset responses.",
"Right: Steady-state responses.",
"Literature: Fig.",
"12 from and Fig.",
"3C from ."
],
[
"AM model responses",
"Model responses were obtained for a 4000-Hz pure tone that was sinusoidally modulated in amplitude (modulation index of 100%) at a rate $f\\textsubscript {mod}=100$ Hz, presented at 60 dB SPL and a duration of 500 ms, including up/down ramps of 2.5 ms.",
"The initial (0-50 ms) and later (350-400 ms) portions of the simulated responses are shown in the left and right panels of Fig.",
"REF , respectively.",
"In all models the modulation rate of 100 Hz is visible as amplitude fluctuations with the corresponding periodicity of 10 ms.",
"In addition, adaptation was observed with stronger simulated responses immediately after the tone onset (left panels) than during the steady-state portion of the response (right panels).",
"For the effective models with adaptation loops (dau1997, relanoiborra2019, osses2021), the maximum amplitudes (Fig.",
"REFa, left) were much lower in osses2021 than for dau1997 and relanoiborra2019, due to the stronger overshoot limitation.",
"For these models it is also observed that their phases are not perfectly aligned due to the outer and middle ear filters that introduced a delay into relanoiborra2019 (black traces run “ahead” the blue traces of dau1997), while the group-delay compensation in osses2021 (Sec.",
"REF ) seemed to overcompensate the alignment of the simulated waveforms (purple traces run “behind” the blue traces).",
"In the right panel, the dynamic range of relanoiborra2019 (black traces) is lower than for osses2021 and dau1997, which have very similar steady-state amplitudes.",
"The reduced dynamic range in relanoiborra2019 is mainly due to the nonlinear cochlear compression of the filter bank that interacts further with the expansion stage.",
"In king2019 (Fig.",
"REFb), a small effect of adaptation was observed with a maximum onset response of 0.88$\\cdot 10^{-3}$ a.u.",
"(left panel) that decreases to a local maximum amplitude of 0.24$\\cdot 10^{-3}$ a.u.",
"during the steady-state response (right panel).",
"The AN responses produced by verhulst2015 and verhulst2018 (panel e) showed an overshoot reaching firing rates of 598.5 and 565.2 spikes/s, respectively.",
"After the onset, the overshoot effect quickly disappeared in verhulst2015, reaching a maximum local rate of 251 spikes/s during the second modulation cycle and 222 spikes/s between 370 and 400 ms.",
"In contrast, verhulst2018 adapted more slowly after the onset with a maximum rate of 319 spikes/s in response to the second modulation cycle, while the response continued adapting reaching a maximum rate of 176 spikes/s between times 370 and 400 ms. For zilany2014 (Fig.",
"REFc) and bruce2018 (Fig.",
"REFd), the mean-rate and PSTH outputs are shown as lighter and darker traces, respectively.",
"It can be observed that in zilany2014, the AM modulations showed a similar mean-rate and PSTH excursions of about 100 spikes/s (Fig.",
"REFb, right: mean rates between 194 and 295 spikes/s; PSTHs with rates between 94 and 196 spikes/s), but the PSTHs had overall lower rates.",
"In bruce2018, a greater AM fluctuation is observed for the PSTHs outputs (darker brown traces) with an excursion of 185 spikes/s (Fig.",
"REFd, right: rates between 56 and 241 spikes/s) compared with the 40 spikes/s (rates between 121 and 161 spikes/s) of its mean-rate output.",
"Additionally, bruce2018 showed a limited effect of adaptation in its mean-rate outputs, also showing a shallower AM response in comparison to the obtained PSTH.",
"We will not focus on the mean-rate output of this model, because (1) their authors validated the model primarily using PSTHs, recommending the use of that AN synapse output for further processing [27], (2) the model using PSTH outputs can be used as input for subcortical processing stages in the UR EAR toolbox [58], and (3) all the studies that we have so far identified using bruce2018 consistently used PSTHs outputs [75], [76].",
"It should be noted that the zilany2014 model, from the same model family, has been extensively validated using both mean-rate and PSTHs outputs.",
"In fact, for studies where psychoacoustic aspects have been investigated there is a tendency to use the mean-rate model outputs."
],
[
"Synchrony capture",
" Figure: Simulated AN responses to a complex tone with three frequency components at 414, 650, and 1000 Hz.",
"The model simulations were obtained at on- and off-frequency CFs spaced at 1 ERB.",
"Literature: Figs.",
"7 and 8 from and Fig.",
"1 from .Model responses were obtained for a complex tone of 50 dB SPL formed by three sinusoids of equal peak amplitude and frequencies of 414 Hz (9.6 ERB$_N$ ), 650 Hz (12.6 ERB$_N$ ), and 1000 Hz (15.6 ERB$_N$ ).",
"This type of complex tone with more carriers and greater range of frequencies are commonly used in studies of profile analysis and it is useful to explain an interesting AN property named “synchrony capture” [79], [54].",
"When synchrony capture occurs, the neural activity in on-frequency channels is driven primarily by one frequency component in the harmonic complex, such that there are minimal fluctuations due to the fundamental-frequency envelope, while off-frequency channels exhibit fluctuating AN patterns at the fundamental frequency.",
"To illustrate whether the evaluated models account for synchrony capture, the model outputs in response to the described complex tone were obtained for frequencies between 415 Hz ($n$ =320) and 1007 Hz ($n$ =245, Eq.",
"REF ) for CFs spaced at approximately 1 ERB ($\\Delta n=12$ or 13), resulting in three on-CF and four off-CF channels.",
"The obtained simulations are shown in Fig.",
"REF for a 30-ms window (between 220 and 250 ms).",
"For each waveform, a schematic metric of envelope fluctuation was obtained and shown as thick grey lines.",
"Those envelope fluctuations were constructed by connecting consecutive local maxima that had amplitudes above the mean responses (onset excluded) of each simulated channel.",
"Subsequently, the standard deviation of the obtained envelope estimate divided by the amplitude scales for each model is indicated in the insets of each panel (e.g., scale of 800 MU for dau1997, relanoiborra2019, and osses2021) were drawn as maroon circles that are connected with dashed lines along the right vertical axes in Fig.",
"REF , with higher values indicating greater envelope fluctuation variability.",
"The scaling used for this estimate allows for a direct comparison between models.",
"In Fig.",
"REF it can be observed that for all models, the on-frequency channels had nearly flat envelope fluctuations.",
"The variability estimate averaged across on-frequency bins ranged between 0.071 (king2019) and 1.14 (bruce2018).",
"The variability estimate across off-frequency bins ranged between 0.50 (king2019) and 2.72 (relanoiborra2019).",
"For all models the off-CF variability was greater than the on-CF variability, with king2019 being the least sensitive model to code envelope fluctuations."
],
[
"Subcortical processing",
"We show two sets of figures to schematise the subcortical processing of the evaluated models."
],
[
"Modulation transfer function",
" Figure: Modulation transfer functions (MTFs) of a modulation filter with a BMF≈80\\approx 80 Hz, assessed using 1000-Hz AM tones presented at 30 (panel a) or 70 dB SPL (panel b) that were sinusoidally modulated with ffmod frequencies between 10 and 130 Hz.",
"The MTFs are normalised to the maximum model response across the tested ffmod frequencies.",
"Literature: Figs.",
"4–6 from , Figs.",
"1 and 4 from .The first set of figures represents a modulation transfer function (MTF) in response to 100% AM tones modulated at $f$mod rates between 10 and 130 Hz (steps of 5 Hz).",
"The tones were centred at 1000 Hz, had a duration of 300 ms, included 5-ms up/down ramps, and were presented at 30 and 70 dB SPL.",
"For this processing, 100 ms in the last portion of the simulated responses were used (between times 190 and 290 ms).",
"The MTFs were assessed from the maximum of the simulated responses.",
"The responses were normalised to the corresponding maximum estimate over the set of tested $f\\textsubscript {mod}$ values, so that the MTF of each model had a maximum value of 1.",
"The resulting MTFs are shown in Fig.",
"REF .",
"The results in Fig.",
"REFa show that the models produce bandpass-shaped MTFs with estimated BMFs between 35 Hz (zilany2014) and 70 Hz (dau1997, relanoiborra2019, and osses2021) that are below the theoretical BMFs (see Table REF ).",
"It is interesting to observe that the sharpest MTFs were obtained not only for dau1997 and osses2021 (both designed with Q=2), but also for king2019 (which has a Q=1), while a wider tuning was observed for the remaining models, including relanoiborra2019 (which has a Q=2).",
"For the biophysical and phenomenological models, the MTFs obtained for the 70-dB AM tones (Fig.",
"REFb) were different than those obtained for 30 dB (Fig.",
"REFa).",
"For these models, the MTFs were no longer bell-shaped and seemed to act as lowpass filters, which is inline with physiological evidence indicating that regions of “enhancement” in MTFs of low level-signals can become regions of “suppression” for higher presentation levels (see, e.g., Fig.",
"4 from [80]).",
"The effective models were more insensitive to the change in presentation levels.",
"Perhaps the only exception to this is relanoiborra2019, where a narrower MTF was obtained in Fig.",
"REFb (compared with panel a).",
"The models dau1997, osses2021, and king2019 have MTFs that are qualitatively similar across presentation levels."
],
[
"Response to clicks of alternating polarity",
"The second set of figures focuses on simulating the response to a typical click train as used in the assessment of auditory brainstem responses (ABRs) [82].",
"We used a click train with a repetition rate of 10 Hz and a duration of 1 s (i.e., containing 10 clicks).",
"The clicks had an alternating polarity (amplitude $A$ or $-A$ ) and were presented at 70 dB peak-equivalent SPL (dB peSPL) [83], i.e., using $A=0.1789$ Pa. Each individual click had a duration of 100 $\\mu $ s. For this processing, the simulated outputs of model Stage 6 (see Fig.",
"REF ) were averaged across CFs obtaining a broadband representation, i.e., all simulated representations were added together and were then divided by the number of CFs [25], [26].",
"This type of output can be used to derive a peak-to-peak or peak-to-trough amplitude correlate of the wave-V ABR component [82].",
"For this processing, we used the default number of CFs for the biophysical and effective models, while for zilany2014 and bruce2018, 50 CFs were obtained between CF$_n=133.7$ Hz ($n=393$ , Eq.",
"REF ) and CF$_n=12010$ Hz ($n=1$ ), spaced at $n=8$ bins to roughly meet the number of filters from Table REF .",
"The obtained click responses are shown in Fig.",
"REF .",
"The biophysical models provided click responses that had positive and negative amplitudes (Fig.",
"REFe–f), which was not the case for the phenomenological models that also use the SFIE model.",
"This is because verhulst2015 and verhulst2018 assume that a population response can be obtained from the sum of single neuron activity (as, e.g., in [48]), with no half-wave rectification in the SFIE model (a non-explicit choice of the authors [25], [26]) that, after scaling [26], [32], results in a simplified neural representation that correlates with changes in electrical dipoles visible in scalp-recorded potentials [26].",
"The effective models, that use the modulation-filter-bank concept, showed only positive amplitudes for all filters with BMFs $\\ge 10$ Hz [22] due to their envelope extraction, a phase-insensitive (“venelope”) processing [84], [28].",
"For modulation frequencies below 10 Hz the perceptual models preserve the phase information, something that is not illustrated in Fig.",
"REF (nor in Fig.",
"REF ).",
"Finally, the simulated peak-to-peak amplitudes in response to the last positive and negative clicks of the pulse train (ninth and tenth click, shown in Fig.",
"REF ) are shown in the entries “Click $A$ ” and “Click $-A$ ” of Table REF .",
"From those amplitudes it can be observed that there are models that have higher peak-to-peak amplitudes in response to positive clicks (zilany2014, verhulst2015, bruce2018, relanoiborra2019) and others where higher amplitudes are observed in response to the clicks of negative polarity (dau1997, verhulst2018, king2019, osses2021.",
"Although we do not discuss the significance of this polarity sensitivity, this aspect has been a matter of discussion, in particular for electrical hearing, where it has been found that evoked potentials in response to positive and negative polarity clicks represent one of the differences between humans [85], [86] and other mammals [87], whose responses are more sensitive to stimulation with clicks of negative and positive polarities, respectively.",
"Figure: Simulated IC responses using one modulation filter (BMF≈\\approx 80 Hz) to a click train of alternating polarity with a total duration of 1 s, repetition rate of 10 Hz and click duration of 100 μ\\mu s. Literature: Fig.",
"1 from and Figs.",
"8 and 9 from ."
],
[
"Computational costs",
"The computational cost required to run each model was measured using the same click train as described in the previous section.",
"Therefore, it assesses the time required to process an input signal of 1-s duration between the model stages 1 to 6 (Fig.",
"REF ).",
"This metric aims at providing a relative notion of the processing times across models.",
"Note that some model implementations can use parallel processing, which was disabled in this evaluation.",
"The assessment was performed on a personal computer equipped with an Intel Core i5-10210UR, 1.6-GHz processor with 16 GB of RAM memory.",
"The results of the computational costs used by each model are given in the entry “Performance” of Table REF .",
"The time required by the models to process one frequency channel ranged between $\\sim $ 0.02 s (osses2021, dau1997) and 2.5 s (bruce2018).",
"For individual frequency channels, the biophysical models (verhulst2015 and verhulst2018) showed moderate calculation times between 0.3 and 0.8 s, however, these models always require (internally) the simulation of the whole discretised cochlea with 1000 cochlear sections, independent of the number of user-requested cochlear channels (default number of 401 for the Verhulst models).",
"This means for the current simulations, that the reported processing times of 122.9 and 319.5 s for verhulst2015 and verhulst2018, respectively, cannot be further reduced, even if the user requests the simulation of less CFs.",
"In contrast, in any model based on a parallel filter bank, including zilany2014 and bruce2018, each cochlear section is independent of each other, and a user-defined number of frequency channels can be simulated, which vastly reduces the computation time for different model configurations.",
"Due to the long processing time of the evaluated biophysical models, their implementations include an option of parallel processing (also available in the original implementation of bruce2018 [27]), where multiple input signals can be processed simultaneously.",
"The number of signals that can be processed in parallel will depend on the number of threads of the host computer.",
"As a further solution to the long processing time, the transmission-line, IHC, and AN modules of verhulst2018 have been approximated using a convolutional neural-network approach [88], [89]."
],
[
"Models in perspective",
"The stimuli and comparison measures used in our evaluation (Sec. )",
"were chosen to reflect relevant temporal and spectral properties of the models in a normal-hearing condition.",
"Our evaluation was meant to provide an objective view accompanied by a graphical representation of how the model responses could account for specific aspects of the hearing processing according to their model structure.",
"In the current section we provide a brief overview of the context in which each of the selected models has been used, including some general recommendations for further applications."
],
[
"Applicability of the models for monaural processing",
"Dau1997 is a monaural model that has been used to simulate a number of psychoacoustic tasks including tone-in-noise and AM detection experiments using a forced-choice paradigm , .",
"To enable the model for the comparison between two or more sounds, the output of Stage 6 (Fig.",
"REF ) is used as input to a decision back-end based on a signal-detection-theory (SDT) framework: the template-matching approach.",
"This framework, extended to adopt two templates, has been recently validated to account for the perceptual similarity between two sounds using osses2021 [30].",
"The models zilany2014 and bruce2018 can account for elevated hearing thresholds due to OHC (“Cochlear gain loss” in Stage 3) or IHC impairment (“IHC loss” in Stage 4) [47].",
"The AN stage (Stage 5) includes two types of outputs: an actual spike generator and an analytical mean-rate synapse output.",
"The spike generator has been primarily used to simulate physiological data, including the phenomenon of short- and long-term adaptation [65].",
"The mean-rate synapse output using zilany2014 has been used to simulate specific psychoacoustic tasks [90], [57].",
"The models verhulst2015 and verhulst2018 were initially designed to simulate otoacoustic emissions [46] and can account for elevated hearing thresholds due to OHC impairment (“Cochlear gain loss” in Stage 3).",
"Furthermore, they allow to study effects of the gradual disconnection of AN fibres, known as synaptopathy, on auditory brainstem responses [91], [26].",
"When coupled with a decision back-end, they can simulate psychoacoustic performance in simultaneous tone-in-noise and high-rate AM tasks ($f\\textsubscript {mod}\\sim $ 100–120 Hz) [92], [93].",
"The model relanoiborra2019 can predict speech intelligibility [29] when coupled with a decision back-end stage [29], [94].",
"Relying on the prediction power of earlier model implementations, relanoiborra2019 should be able to (1) account for elevated thresholds based on OHC and IHC impairment [95], and (2) to predict a number of psychoacoustic tasks including simultaneous and forward masking and amplitude modulation [96].",
"Our results showed that relanoiborra2019 accounts well for hearing properties such as nonlinearities in the cochlear processing and auditory adaptation, including a saturation behaviour similar to that of the AN physiological models.",
"The model king2019 was designed to simulate perceptual tasks of amplitude and frequency modulation detection, primarily at low modulation rates ($f\\textsubscript {mod}\\le 20$ Hz), including deterministic limitations (suboptimal template matching strategies) or stochastic limitations such as internal additive noise, multiplicative noise [97], and memory noise [98].",
"The model can be adapted to simulate hearing impairment by modifying its compression parameters (knee point and compression rate), and increasing the width of the underlying cochlear filters.",
"Despite the simplicity of this model –in fact one of its strengths– we have shown in this paper that the model can account for several of the comparison metrics, with the exception of the narrowing of cochlear filters at higher presentation levels (Fig.",
"REF ) and the adaptation saturation (Figs.",
"REF –REF )."
],
[
"Applicability of the models for binaural processing",
"Models of binaural processing rely on simulating the monaural processing up to the AN level [2].",
"For this reason it is not surprising that several of the tested models have been used as preprocessing stages for binaural back-ends.",
"The lowpass modulation filter (similar to dau1997) [35], [22] served as the basis for a model of binaural masking that uses a decision stage based on the equalisation-cancellation theory [99].",
"This model was later extended to predict perceptual attributes of room acoutics [100], [101].",
"The model zilany2014 has been used to predict the sensitivity to interaural time and level differences by estimating the disparity between left and right AN responses using a decision back-end based on shuffled cross-correlograms [102].",
"Finally, bruce2018 has been used to simulate the lateralisation of high-frequency stimuli in a coincidence-counting model [75]."
],
[
"Simplified auditory representations",
"In order to broaden our understanding of the auditory processes, computational models that are as complete as possible are required.",
"The more detailed knowledge comes at the price of more computationally-expensive implementations.",
"Thus, the development of models follows their objective: Biophysical and phenomenological models attempt to shed light on the mechanisms behind the auditory processing and for that purpose they need to be as complete as possible.",
"On the other hand, effective models have a more epistemic status providing an intelligible (simplified) representation of the process and can guide the design of new experiments or the development of listener-targeted products.",
"Such model simplification, however, potentially reduces the number of effects a model can account for, leading to an actual narrowing of its application field.",
"An example of a successful model simplification is presented in [55], where MTFs were simulated using only a stage of envelope extraction followed by a modulation filter bank omitting, thus, the stages of cochlear filtering and auditory adaptation.",
"This model, however, is not thought to predict the performance in listening conditions where the omitted model stages do play a role, as it is the case, in this example, for forward masking tasks.",
"Peripheral auditory models are often combined with a decision back-end module converting simulated responses into a behavioural response such as detectability or discriminability or to obtain perceptual metrics , .",
"For successful simulations, the decision stage should appropriately weight the information contained in the model representations.",
"An analysis of weighted time-frequency representations (time, audio frequency, and/or modulation frequency) can reveal what portions of the simulated responses are more relevant , [103].",
"It is important to note that the simplification of auditory models based on statistical methods or machine learning processes requires a careful interpretation.",
"While these approaches might be well suited to achieve goals as real-time processing in applications such as speech perception or in the prediction of evoked potentials [76], they limit the modular comprehension of each auditory stage, especially if multiple model stages are approximated [76], [104].",
"In a recent study [76], firing rates of cortical A1 neurons in ferrets were approximated using several time-frequency representations ranging from simple short-time Fourier transforms to more detailed models of AN synapses (including bruce2018) to which a linear-nonlinear (LNL) encoder was used.",
"Based on their separately-fitted encoders, the authors concluded that cortical processing in ferrets perform a “very simple signal transformation,” without even discussing how different the linear and nonlinear components in each of their encoders were.",
"For this reason, despite the success of the authors in approximating neural responses in ferrets, we believe that it is difficult to know whether the “simple transformation” is indeed related to the underlying system (the cortical processing in ferrets) and not related to the complexity of operations in the fitted encoders."
],
[
"Considerations for further modelling work",
"The following is a list of aspects that we recommend to keep in mind for further auditory modelling work, based on the general observations of this study: [itemsep=.0cm,leftmargin=*] We recommend to use an outer-ear module if the evaluated sounds are assumed to be reproduced via loudspeakers, or supra-aural or circumaural headphones.",
"The results in Figs.",
"REF and REF show nonlinear interactions between model stages as a function of level and for different types of signals.",
"This suggests that different sets of stimuli are required to characterise the behaviour of complex processes such as that of nonlinear filter banks.",
"In other words, models may not always act as a linear time invariant (LTI) system.",
"In Sec.",
"REF we suggested a minimum number of filters for each filter bank to roughly meet a $-3$ -dB filter crossing (Table REF , “40 dB: Number of bands”).",
"The actual number of bands that are needed may vary from application to application and depend on the type of sounds that are to be simulated.",
"This choice can be particularly critical in models where the number of bands are a free parameter (here zilany2014 and bruce2018).",
"For models that are used as front-ends to machine-learning applications, Lyon [18] suggested a “not-too-sparse set of channels” with about a 50% overlap between filters, i.e., twice the number of channels that we recommend in Table REF .",
"It is important to keep in mind, however, that our estimation was based on model responses to white noises, which are sustained signals in time and broadband in frequency.",
"At higher presentation levels, where nonlinear filter banks are compressive, similar estimations using sine tones (sustained narrowband signals, as in Fig.",
"REF ) or clicks (transient broadband sounds) may result in a different number of required bands.",
"Different simulation results can be expected when evaluating mean-rate and PSTH outputs of models including AN synapse stages as shown in Figs.",
"REF –REF .",
"The particular choice of the type of output depends on the target application of the model.",
"The spike generator is primarily used to simulate physiological data , , while the mean-rate synapse output is typically used to simulate specific psychoacoustic tasks , ."
],
[
"Conclusions",
"In this paper we compared eight models of auditory processing that simulate responses –with different levels of accuracy– up to the level of the inferior colliculus in the midbrain.",
"We described and quantified the similarities and differences among model implementations, and we derived a minimum number of filters required for those stages to ensure the preservation of auditory information based on our estimates of frequency selectivity.",
"We showed that despite the differences in model design that result in more physiologically- (biophysical and phenomenological models) or perceptually-plausible approximations (effective models), the models can still account for a considerable number of hearing properties.",
"Examples of these properties are the phase-locking reduction in inner-hair-cell processing and the phenomenon of auditory adaptation.",
"Still, an in-depth understanding of each of the model stages is required when selecting a model for an application and to compare the models to one another.",
"We encourage future users to be explicitly aware of the specific datasets of sounds and experimental paradigms upon which their models have been evaluated and the underlying model limitations.",
"The further use of back-end modules, e.g., based on machine learning, statistics, or signal detection theory, may help to better facilitate the information obtained from auditory models.",
"To this end, the comparison across model implementations provides a guideline for their selection and excellent way to challenge the capabilities of the different models.",
"We are grateful to several colleagues who participated in technical discussions during the writing process: Enrique Lopez-Poveda, Fotios Drakopoulos, Alessandro Altoè, Armin Kohlrausch, and Richard Lyon.",
"We are particularly grateful to Clara Hollomey, who provided immense support for the integration of all models into AMT 1.0.",
"The authors AOV and LV.",
"received support from ANR (project: 17-EURE-0017), SV received support from the ERC project RobSpear (Grant No.",
"678120), and PM received support from the H2020 project SONICOM (EC Grant No.",
"101017743).",
"Table: Model configurations and numeric results.",
"Middle ear: Details of frequency response of the middle-ear filters.",
"Cochlear filter bank: 40 dB and 100 dB refer to filter characteristics between 160 and 8000 Hz in response to white noises of 40 and 100 dB SPL, respectively.",
"IHC: Parameters and frequency response of the LP filter structures.",
"Subcortical processing: Theoretical and estimated BMF of the modulation filter with the closest BMF to 100 Hz and peak-to-peak amplitude of the simulated IC outputs in response to 70-dB-peSPL clicks of positive (AA) and negative (-A-A) (see Sec.",
"for more details).",
"Performance: Time required to process a 1-s long stimulus using each of the selected auditory models, between Stages 1 and 6 (Fig. ).",
"All fcut-offf\\textsubscript {cut-off} in this table were measured at the -3-3 dB points of the amplitude spectrum."
]
] | 2107.01753 |
[
[
"Adding Complex Fermions to the Grassmannian-like Coset Model"
],
[
"Abstract In the ${\\cal N}=2$ supersymmetric coset model, $\\frac{SU(N+M)_k \\times SO(2 N M)_1}{ SU(N)_{k+M} \\times U(1)_{ N M (N+M)(k+N+M)}}$, we construct the $SU(M)$ nonsinglet ${\\cal N}=2$ multiplet of spins $(1, \\frac{3}{2}, \\frac{3}{2}, 2)$ in terms of coset fields.",
"The next $SU(M)$ singlet and nonsinglet ${\\cal N}=2$ multiplets of spins $(2, \\frac{5}{2}, \\frac{5}{2}, 3)$ are determined by applying the ${\\cal N}=2$ supersymmetry currents of spin $\\frac{3}{2}$ to the bosonic singlet and nonsinglet currents of spin $3$ in the bosonic coset model.",
"We also obtain the operator product expansions(OPEs) between the currents of the ${\\cal N}=2$ superconformal algebra and above three kinds of ${\\cal N}=2$ multiplets.",
"These currents in two dimensions play the role of the asymptotic symmetry, as the generators of ${\\cal N}=2$ \"rectangular $W$-algebra\", of the $M \\times M$ matrix generalization of ${\\cal N}=2$ $AdS_3$ higher spin theory in the bulk.",
"The structure constants in the right hand sides of these OPEs are dependent on the three parameters $k, N$ and $M$ explicitly.",
"Moreover, the OPEs between $SU(M)$ nonsinglet ${\\cal N}=2$ multiplet of spins $(1, \\frac{3}{2}, \\frac{3}{2}, 2)$ and itself are analyzed in detail.",
"The complete OPE between the lowest component of the $SU(M)$ singlet ${\\cal N}=2$ multiplet of spins $(2, \\frac{5}{2}, \\frac{5}{2}, 3)$ and itself is described.",
"In particular, when $M=2$, it is known that the above ${\\cal N}=2$ supersymmetric coset model provides the realization of the extension of the large ${\\cal N}=4$ nonlinear superconformal algebra.",
"We determine the currents of the large ${\\cal N}=4$ nonlinear superconformal algebra and the higher spin-$\\frac{3}{2}, 2$ currents of the lowest ${\\cal N}=4$ multiplet for generic $k$ and $N$ in terms of the coset fields."
],
[
" Introduction",
"It is known that the most general coset in the bosonic theory can be described by [1] ${.",
"}1\\endcsname \\frac{SU(N+M)_k }{SU(N)_{k} \\times U(1)_{ N M (N+M)k}},$ where the three parameters, $k, M$ and $N$ are present The possibility of four parameters in the different coset model is studied in [2]..",
"According to the observation of [3], this coset model is dual to $M \\times M$ matrix generalization of $AdS_3$ Vasiliev higher spin theory [4], [5] by taking the appropriate limit on these parameters.",
"Note that for $M=1$ case, the Gaberdiel-Gopakumar conjecture [6] can be seen and see also [7], [8], [9] for the relevant works.",
"At the particular value of the level $k$ with generic $N$ and $M$ , the operator product expansion (OPE) between the charged spin-2 current and itself leads to the one of the “rectangular” $W$ -algebra [10] with $SU(M)$ symmetry which is the asymptotic symmetry of $AdS_3$ higher spin theory [3] and see also [11].",
"For generic $k, N$ and $M$ , this OPE is further studied in [12] and see also [13], [14] There is a previous work on the Grassmannian coset model in [15].. Why do we add the complex fermions into the above coset model (REF )?",
"This is one of the ways to make the bosonic theory to be the supersymmetric theory.",
"In other words, by using the fermionic operators of spin-$\\frac{1}{2}$ , we can construct the half integer currents including the supersymmetry generators of spin-$\\frac{3}{2}$ explicitly Without introducing the fermions, we can have the supersymmetric version of (REF ) by considering the special value of the level $k$ .",
"This is because we can realize the various spin-1 currents in terms of fermions.",
"For example, see also [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29].. We would like to discuss about the supersymmetric version of (REF ) and it is given by [1], [30] ${.",
"}2\\endcsname \\frac{SU(N+M)_k \\times SO(2 N M)_1}{SU(N)_{k+M} \\times U(1)_{ N M (N+M)(k+N+M)}}.$ Note that there exists an $ SO(2 N M)_1$ factor associated with complex fermions in the numerator and the levels in the denominator are changed appropriately.",
"For $M=1$ , by dividing (REF ) out the $SU(M=1)_{k+N}$ further, the ${\\cal N}=2$ $AdS_3$ higher spin gravity is related to the Kazama-Suzuki model [31], [32], according to [33].",
"See also the relevant works in [34], [35], [36], [37], [38], [39], [40], [41], [27], [42], [43], [44], [45].",
"For $M=2$ , the above coset arises in the context of the large ${\\cal N}=4$ holography [46].",
"See also previous works on this holography [47], [48], [49], [50], [51], [52], [53], [54], [55], [56], [57], [58], [59], [60], [61], [62], [63], [64], [65], [66], [67], [68], [69], [2], [70], [71], [72].",
"For generic $M > 2$ , in [30], the extension of ${\\cal N}=2$ superconformal algbra is described.",
"The generators of ${\\cal N}=2$ superconformal algebra are given by the spin-$1, \\frac{3}{2}, \\frac{3}{2}, 2$ currents and denoted by $K, G^+, G^-, T$ respectively The OPE between $G^+$ and $G^-$ does not produce the standard ${\\cal N}=2$ superconformal algebra for generic $M >1$ because the first order pole of this OPE has the additional terms.",
"However, we are using the same terminology “${\\cal N}=2$ superconformal algebra” in this paper..",
"The extra generators of $SU(M)$ nonsinglets are classified by the spin-1 currents, spin-$\\frac{3}{2}$ currents, and the spin-2 currents.",
"Their numbers are given by $(M^2-1)$ , $2(M^2-1)$ and $(M^2-1)$ respectively.",
"If we increase the spin by one, then there are the spin-2 currents, spin-$\\frac{5}{2}$ currents, and the spin-3 currents of $SU(M)$ singlets and nonsinglets.",
"Their numbers are given by $ M^2$ , $2M^2$ and $M^2$ respectively.",
"As we increase further, then the general features of $SU(M)$ singlets and nonsinglets go through: there are the $M^2$ spin-$s$ currents, the $2M^2$ spin-$(s+\\frac{1}{2})$ currents, and the $M^2$ spin-$(s+1)$ currents The ${\\cal N}=2$ “rectangular” $W$ -algebra appears as the asymptotic symmetry of the $M \\times M$ matrix generalization of ${\\cal N}=2$ $AdS_3$ higher spin theory [3].. We present them here, in the notation of ${\\cal N}=2$ multiplet Strictly speaking, when we are saying about “${\\cal N}=2$ multiplet” in this paper, the four component currents do not satisfy the standard ${\\cal N}=2$ primary conditions with the currents $(K,G^+,G^-,T)$ : there are some higher order poles appear and some structure constants appear differently., with the assignment of spin and the $SU(M)$ index $a=1,2, \\cdots , (M^2-1)$ as follows: ${.",
"}3\\endcsname && (W^{-(1), 0}, G^{+(\\frac{3}{2}), 0},G^{-(\\frac{3}{2}), 0}, W^{+(2), 0}) \\equiv (K, G^+, G^-, T),\\nonumber \\\\&& (W^{-(1), a}, G^{+(\\frac{3}{2}), a},G^{-(\\frac{3}{2}), a}, W^{+(2), a}),\\nonumber \\\\&& (W^{-(2), 0}, G^{+(\\frac{5}{2}), 0}, G^{-(\\frac{5}{2}), 0}, W^{+(3), 0}),\\nonumber \\\\&& (W^{-(2), a}, G^{+(\\frac{5}{2}), a}, G^{-(\\frac{5}{2}), a}, W^{+(3), a}),\\nonumber \\\\&& (W^{-(3), 0}, G^{+(\\frac{7}{2}), 0}, G^{-(\\frac{7}{2}), 0}, W^{+(4), 0}),\\nonumber \\\\&& (W^{-(3), a}, G^{+(\\frac{7}{2}), a}, G^{-(\\frac{7}{2}), a}, W^{+(4), a}),\\qquad \\cdots \\qquad ,\\nonumber \\\\&& (W^{-(s), 0}, G^{+(s+\\frac{1}{2}), 0}, G^{-(s+\\frac{1}{2}), 0}, W^{+(s+1), 0}),\\nonumber \\\\&& (W^{-(s), a}, G^{+(s+\\frac{1}{2}), a}, G^{-(s+\\frac{1}{2}), a}, W^{+(s+1), a}),\\qquad \\cdots \\qquad .$ In (REF ), the generators of ${\\cal N}=2$ superconformal algebra are denoted by $(K, G^+, G^-, T)$ .",
"Each current of half integer spins appears in the same ${\\cal N}=2$ multiplet while each current of integer spins appears in the different ${\\cal N}=2$ multiplets.",
"The coset realization on the above currents is done for the generators of $(K,G^+, G^-, T)$ and the next three kinds of $ (W^{-(1), a}, G^{+(\\frac{3}{2}), a},G^{-(\\frac{3}{2}), a})$ so far in [30] We are using the notation [30] of ${\\cal N}=2$ “rectangular” $W$ -algebra in the bulk for its dual CFT currents in (REF )..",
"In this paper, we would like to construct the coset realization for the following currents ${.",
"}4\\endcsname && (\\mbox{known}, \\mbox{known},\\mbox{known}, W^{+(2), a}),\\nonumber \\\\&& (W^{-(2), 0}, G^{+(\\frac{5}{2}), 0}, G^{-(\\frac{5}{2}), 0}, W^{+(3), 0}),\\nonumber \\\\&& (W^{-(2), a}, G^{+(\\frac{5}{2}), a}, G^{-(\\frac{5}{2}), a}, W^{+(3), a}).$ Here, the first three components of the first ${\\cal N}=2$ multiplet (REF ) are known in terms of coset fields.",
"Moreover, we describe some of the OPEs between the currents for low spins.",
"Then how we can construct the currents in (REF ) in terms of coset fields explicitly?",
"For the spin-2 current appearing in the first line of (REF ), we can use both the previous known currents which belong to the same ${\\cal N}=2$ multiplet and the generators of $(K,G^+,G^-,T)$ .",
"For the currents belonging to the second and third ${\\cal N}=2$ multiplets in (REF ), we can use the previous known currents of spin-$2,3$ currents in the bosonic coset model (REF ).",
"We figure out that the singlet spin-3 current $W^{(3)}$ will appear in the last component of the second ${\\cal N}=2$ multiplet while the nonsinglet spin-3 current $P^a$ will appear in the last component of the third ${\\cal N}=2$ multiplet and the nonsinglet spin-2 current $K^a$ will appear in the first component of the third ${\\cal N}=2$ multiplet, as the ${\\cal N}=2$ supersymmetric versions.",
"This is because the OPE between the spin-2 current $K^a$ and the spin-2 current $K^b$ leads to the spin-3 current $P^c$ in the bosonic coset model.",
"Furthermore, when we fix $M=1$ , the singlet spin-3 current arises in the last component of the corresponding ${\\cal N}=2$ multiplet (For example, [37], [39], [27]) and therefore it is natural to think about the above description for general $M$ case.",
"One way to determine the ${\\cal N}=2$ superpartners of the spin-$2,3$ currents found in the bosonic coset model is to consider, as a first step, the OPEs between the ${\\cal N}=2$ supersymmetry generators $G^{\\pm }$ and the spin-3 currents $W^{(3)}$ and $P^a$ of the bosonic theory.",
"Then we will obtain the intermediate spin-$\\frac{5}{2}$ currents in the specific poles which depend on the complex fermions as well as the bosonic currents.",
"Further computations for the OPEs between the above supersymmetry generators $G^{\\pm }$ and these intermediate spin-$\\frac{5}{2}$ currents obtained at the previous stage can be performed.",
"Then we will obtain the spin-2 currents which will contain both the complex fermions and the bosonic currents as before.",
"Then we have singlet spin-2 current $W^{-(2),0}$ and the nonsinglet spin-2 currents $W^{-(2),a}$ as in (REF ).",
"By calculating the OPEs between the spin-$\\frac{3}{2}$ currents $G^{\\pm }$ and these spin-2 currents obtained newly, we can determine the singlet spin-$\\frac{5}{2}$ currents $W^{\\pm (\\frac{5}{2}),0}$ and the nonsinglet spin-$\\frac{5}{2}$ currents $W^{\\pm (\\frac{5}{2}),a}$ .",
"In general, they are different from the above intermediate spin-$\\frac{5}{2}$ currents.",
"Finally, after further action of supersymmetry generators $G^{\\pm }$ on these spin-$\\frac{5}{2}$ currents, the singlet spin-3 current $W^{+(3),0}$ and the nonsinglet spin-3 currents $W^{+(3),a}$ can be determined completely.",
"So far, the OPEs we are considering are the ones between the currents of the ${\\cal N}=2$ superconformal algebra (the first line of (REF )) and the currents of the singlet and nonsinglet ${\\cal N}=2$ multiplets (REF ).",
"Therefore, the right hand sides of these OPEs look similar to the behavior of standard ${\\cal N}=2$ primary currents.",
"The difference is that we observe that there appear some other additional singular terms and other type of currents living in other ${\\cal N}=2$ multiplet.",
"The next things we should do, for the simplest cases, is to compute 1) the OPEs between the currents in the first ${\\cal N}=2$ multiplet in (REF ) and 2) the OPE between $W^{-(2),0}$ and itself.",
"In this case we do expect that the right hand sides of these OPEs will produce other ${\\cal N}=2$ multiplets nontrivially.",
"For the OPEs between the first nonsinglet ${\\cal N}=2$ multiplet in (REF ), there exist new primary spin-$\\frac{5}{2}$ currents having two free indices and primary spin-3 current also having two free indices.",
"For the OPE between the lowest component and itself of the second singlet ${\\cal N}=2$ multiplet in (REF ), there is no new primary current.",
"Although we consider the OPE between the singlet spin-2 current and itself, we do expect that the OPEs between the currents of the singlet ${\\cal N}=2$ multiplets $ (W^{-(s), 0}, G^{+(s+\\frac{1}{2}), 0},G^{-(s+\\frac{1}{2}), 0}, W^{+(s+1), 0})$ do not contain the currents from the nonsinglet ${\\cal N}=2$ multiplets, in the presence of the currents of the ${\\cal N}=2$ superconformal algebra with modified stress energy tensor.",
"Furthermore, we can study the extension of the large ${\\cal N}=4$ nonlinear superconformal algebra in the present context because we have some information on the coset fields via (REF ).",
"The question is how to reorganize the currents of ${\\cal N}=4$ multiplets in terms of the currents of ${\\cal N}=2$ multiplets and the coset fields.",
"In this case, the parameter $M$ is fixed by 2.",
"We can try to obtain the relevant currents of the above extension by recalling the defining OPEs in [73] between them starting from the low spins.",
"The nontrivial part is to obtain the currents of the the large ${\\cal N}=4$ nonlinear superconformal algebra because the stress energy tensor appears very nontrivially (For example, [74]) and we should fix the normalizations for the spin-1 currents and the spin-$\\frac{3}{2}$ currents correctly.",
"Once we identify the currents of the the large ${\\cal N}=4$ nonlinear superconformal algebra, then it is rather straightforward to determine the currents of the lowest ${\\cal N}=4$ multiplet by using the defining relations in the OPEs in [73].",
"In section 2, we review how we can add the complex fermions into the bosonic coset model and the coset realization for the ${\\cal N}=2$ superconformal algebra is reviewed.",
"We describe the known currents in (REF ).",
"In section 3, after finding the nonsinglet spin-2 currents $W^{+(2),a}$ , we compute the OPEs between the first ${\\cal N}=2$ multiplet in the second line of (REF ) and the ${\\cal N}=2$ stress energy tensor in the first line of (REF ).",
"In section 4, by considering the OPEs between the spin-$\\frac{3}{2}$ currents $G^{\\pm }$ and the spin-3 current $W^{(3)}$ and analyzing the OPEs between the spin-$\\frac{3}{2}$ currents $G^{\\pm }$ and the spin-$\\frac{5}{2}$ current found newly successively, the lowest current of the second ${\\cal N}=2$ multiplet in the third of (REF ) can be obtained.",
"Moreover, the remaining other three components of this ${\\cal N}=2$ multiplet can be determined.",
"As before, the OPEs between the second ${\\cal N}=2$ multiplet in the third line of (REF ) and the ${\\cal N}=2$ stress energy tensor in the first line of (REF ) are determined.",
"In section 5, we do this section by considering the third ${\\cal N}=2$ multiplet in the fourth line of (REF ) by following the procedure done in section 4.",
"In section 6, after analyzing the OPEs between the first ${\\cal N}=2$ multiplet, the arising of the new primary currents of spin-$\\frac{5}{2}, 3$ having the free indices $a b$ of $SU(M)$ is described.",
"In section 7, some of the OPEs between the second ${\\cal N}=2$ multiplet are analyzed.",
"In section 8, for the $M=2$ case, the coset realization gives us an extension of the large ${\\cal N}=4$ nonlinear superconformal algebra [75], [74], [76], [77] and some of the currents of the lowest ${\\cal N}=4$ multiplet are given explicitly.",
"In section 9, we conclude our work and the future directions are given.",
"The various Appendices are for the details in previous sections we describe."
],
[
" Review",
"Some known results are reviewed in this section."
],
[
"The role of complex fermions",
"In the coset ${.",
"}1\\endcsname \\frac{SU(N+M)_k \\times SO(2 N M)_1}{SU(N)_{k+M} \\times U(1)_{ N M (N+M)(k+N+M)}},$ the decomposition of $SU(N+M)$ into the $SU(N) \\times SU(M)$ in the numerator can be performed as in the bosonic case [3].",
"We use the generators $(t^{\\alpha }, t^a, t^{u(1)}, t^{(\\rho \\bar{i})},t^{(\\bar{\\sigma } j)})$ with the normalized metric.",
"The index $\\alpha $ runs over $\\alpha =1,2, \\cdots , (N^2-1)$ while the index $a$ runs over $a=1,2, \\cdots , (M^2-1)$ .",
"The fundamental indices $\\rho $ and $j$ run over $\\rho =1,2, \\cdots , N$ and $j=1,2,\\cdots , M$ while the antifundamental indices $\\bar{\\sigma }$ and $\\bar{i}$ run over $\\bar{\\sigma }=1,2, \\cdots , N$ and $\\bar{i}=1,2, \\cdots , M$ .",
"The $f$ and $d$ symbols can be obtained from the above generators.",
"The $SU(N+M)$ currents of spin-1 in the numerator satisfy the standard OPEs.",
"The $N M$ complex fermions of spin-$\\frac{1}{2}$ in the numerator satisfy the following OPE ${.",
"}2\\endcsname \\psi ^{(\\rho \\bar{i})}(z) \\, \\psi ^{(\\bar{\\sigma } j)}(w) = \\frac{1}{(z-w)} \\,\\delta ^{\\rho \\bar{\\sigma }} \\, \\delta ^{j \\bar{i}} + \\cdots .$ We introduce the $SU(N)$ currents, the $SU(M)$ currents and $U(1)$ current from these complex fermions living in the $SO(2 N M)_1$ factor as follows [30]: ${.",
"}3\\endcsname J^{\\alpha }_f \\equiv t^{\\alpha }_{\\rho \\bar{\\sigma }} \\, \\delta _{j \\bar{i}} \\,\\psi ^{(\\rho \\bar{i})} \\, \\psi ^{(\\bar{\\sigma } j)}, \\qquad J^{a}_f \\equiv -t^{a}_{j \\bar{i}} \\, \\delta _{\\rho \\bar{\\sigma }} \\,\\psi ^{(\\rho \\bar{i})} \\, \\psi ^{(\\bar{\\sigma } j)}, \\qquad J^{u(1)}_f \\equiv \\delta _{\\rho \\bar{\\sigma }} \\, \\delta _{j \\bar{i}} \\,\\psi ^{(\\rho \\bar{i})} \\, \\psi ^{(\\bar{\\sigma } j)}.$ The appropriate contractions between the indices are taken.",
"We can also choose the second currents with positive sign.",
"Then the currents in the denominator of the coset (REF ) are given by ${.",
"}4\\endcsname J^{\\alpha } + J^{\\alpha }_f, \\qquad \\sqrt{M N (M+N)} \\, J^{u(1)} + (M+N) \\, J_f^{u(1)},$ together with (REF ).",
"Note that the level for the $J^{\\alpha }_f$ is given by $M$ and the level for the $J_f^{u(1)}$ is given by $M N$ by using (REF ).",
"See also Appendix $A.2$ .",
"Compared with the bosonic coset (REF ), the levels in the denominator are increased by $M$ and $N M (N+M)^2$ respectively.",
"Note that the level for $J^{\\alpha }$ and $J^{u(1)}$ is given by $k$ .",
"The role of $J_f^a$ will appear in the next subsection.",
"Note that the OPEs between any two different currents in (REF ) are regular It is obvious that the OPEs between the bosonic currents $(J^{\\alpha }, J^a, J^{u(1)}, J^{(\\rho \\bar{i})},J^{(\\bar{\\sigma } j)})$ and the currents in (REF ) are regular.. See also Appendix $A.2$ .",
"With the help of complex fermions, we can construct the currents of half-integer spin which are necessary to obtain the supersymmetric theory."
],
[
"The ${\\cal N}=2$ superconformal algebra",
"The stress energy tensor by Sugawara construction is the difference between the stress energy tensor in the numerator and the one in the denominator and is given by ${.",
"}5\\endcsname T & = &\\frac{1}{2(k+M+N)} \\Bigg [ J^{\\alpha } J^{\\alpha } + J^a J^a + \\delta _{\\rho \\bar{\\sigma }} \\delta _{j \\bar{i}} \\,J^{(\\rho \\bar{i})} J^{(\\bar{\\sigma } j)} + \\delta _{\\rho \\bar{\\sigma }} \\delta _{j \\bar{i}}J^{(\\bar{\\sigma } j)} J^{(\\rho \\bar{i})} + J^{u(1)} J^{u(1)}\\Bigg ]\\nonumber \\\\&-& \\frac{1}{2} \\Bigg [\\delta _{\\rho \\bar{\\sigma }} \\, \\delta _{j \\bar{i}} \\,\\psi ^{(\\rho \\bar{i})} \\, \\partial \\, \\psi ^{(\\bar{\\sigma } j)} -\\delta _{\\rho \\bar{\\sigma }} \\, \\delta _{j \\bar{i}} \\,\\partial \\, \\psi ^{(\\rho \\bar{i})} \\, \\psi ^{(\\bar{\\sigma } j)}\\Bigg ]- \\frac{1}{2(k+M+N)} \\, \\Bigg [ (J^{\\alpha }+ J_f^{\\alpha }) \\,(J^{\\alpha } + J_f^{\\alpha }) \\nonumber \\\\& + &( J^{u(1)} + \\sqrt{\\frac{M+N}{M N}}\\, J^{u(1)}_f ) \\, (J^{u(1)}+\\sqrt{\\frac{M+N}{M N}} J^{u(1)}_f) \\Bigg ].$ Compared with the bosonic case [3] The bosonic stress energy tensor is given by $T_{boson} = \\frac{1}{2(k+N+M)} \\Bigg [ J^{\\alpha } J^{\\alpha } + J^a J^a +\\delta _{\\rho \\bar{\\sigma }} \\delta _{j \\bar{i}} \\,J^{(\\rho \\bar{i})} J^{(\\bar{\\sigma } j)} + \\delta _{\\rho \\bar{\\sigma }} \\delta _{j \\bar{i}}J^{(\\bar{\\sigma } j)} J^{(\\rho \\bar{i})} + J^{u(1)} J^{u(1)}\\Bigg ]- \\frac{1}{2(k+N)} \\, J^{\\alpha } \\, J^{\\alpha } - \\frac{1}{2k} \\,J^{u(1)} \\, J^{u(1)}$ ., due to the presence of the $SO(2 N M)_1$ factor in the numerator, its contribution appears in the second line of (REF ).",
"The quantity $(k+M+N)$ in the first line can be interpreted as the sum of the level $k$ and $(N+M)$ of $SU(N+M)$ .",
"There exists a common factor $\\frac{1}{2(k+M+N)}$ in the stress energy tensor of the denominator.",
"For the $SU(N)$ current in (REF ), the quantity $(k+M+N)$ can be regarded as the sum of the level $(k+M)$ and the $N$ of $SU(N)$ .",
"When we multiply $M N (M+N)$ in the third line of (REF ) and divide it, then the corresponding level provides the overall factor $\\frac{1}{2 N M (M+N)(k+M+N)}$ for the $U(1)$ current (REF ) in the denominator (REF ).",
"In other words, the level can be identified with $ N M (M+N)(k+M+N)$ .",
"Then it is straightforward to compute the central charge for the supersymmetric coset model ${.",
"}6\\endcsname c= \\frac{3 M N k}{(k+M+N)} + \\frac{(k+N)(M^2-1)}{(k+M+N)}=\\frac{(k M^2+3 k M N-k+M^2 N-N)}{(k+M+N)}.$ When we have further $SU(M)_{k+N}$ factor in the denominator of (REF ), then the central charge is given by the first contribution of the first relation in (REF ).",
"According to the observation of [31], this can be obtained from the level $k$ of $SU(N+M)$ , the dual Coxeter number $(N+M)$ of $SU(N+M)$ and the dimension $2 N M$ of $\\frac{SU(N+M)}{SU(N) \\times SU(M) \\times U(1)}$ .",
"See also [37], [39].",
"The corresponding current of $SU(M)_{k+N}$ factor is given by $(J^a + J^a_f)$ and the level of $J_f^a$ is $N$ .",
"Therefore, the total level is $(k+N)$ .",
"By adding $SU(M)_{k+N}$ factor in the previous coset, we obtain the above coset (REF ).",
"Then the central charge for the $SU(M)_{k+N}$ factor is given by the second contribution of the first relation in (REF ) We can compute the central charges for the group $G=SU(N+M)_k \\times SO(2NM)_1$ and for the group $H=SU(N)_{k+M} \\times U(1)_{NM(N+M)(k+M+N)}$ directly.",
"For the former, we have the contributions from each factor $c_G=\\frac{k ((M+N)^2-1)}{k +(M+N)}+\\frac{1}{1+(2NM-2)} \\frac{1}{2} 2 N M (2N M-1)$ .",
"For the latter $c_H=\\frac{(k+M)(N^2-1)}{(k+M)+N}+\\frac{NM(N+M)(k+M+N)}{NM(N+M)(k+M+N)+0}$ from each contribution.",
"Then we obtain that the central charge (REF ) is given by their difference as follows: $c=c_G-c_H$ .",
"By using the property of $T(z) \\, T_H(w) = 0 +\\cdots $ where $T=T_G-T_H$ , we confirm that the OPE $T(z) \\, T(w)$ can be reduced to the OPE of $T_G(z) \\, T_G(w)-T_H(z) \\, T_H(w)$ [79], [80].. By taking the product of the above complex fermions and the spin 1 currents transforming $({\\bf N},\\overline{\\bf {M}})$ or $(\\overline{\\bf {N}},{\\bf M})$ with the appropriate contractions between the indices, we obtain the spin-$\\frac{3}{2}$ currents [30] of ${\\cal N}=2$ superconformal algebra as follows: ${.",
"}7\\endcsname G^{+} \\equiv \\delta _{\\rho \\bar{\\sigma }} \\, \\delta _{j \\bar{i}} \\,J^{(\\rho \\bar{i})} \\, \\psi ^{(\\bar{\\sigma } j)}, \\qquad G^{-} \\equiv \\delta _{\\rho \\bar{\\sigma }} \\, \\delta _{j \\bar{i}} \\,\\psi ^{(\\rho \\bar{i})} \\, J^{(\\bar{\\sigma } j)}.$ It turns out that the remaining spin 1 current of ${\\cal N}=2$ superconformal algebra is described by ${.",
"}8\\endcsname K \\equiv \\frac{1}{(k+M+N)} \\, \\Bigg ( \\sqrt{ \\frac{M+N}{M N}}\\, M N \\, J^{u(1)}- k \\, J_f^{u(1)} \\Bigg ).$ Note that the currents (REF ), (REF ) and (REF ) of the ${\\cal N}=2$ superconformal algebra ${.",
"}9\\endcsname (K, G^{+}, G^{-}, T)$ are regular in the OPEs between these and the currents (REF ) of the denominator of the coset.",
"In Appendix $A$ , we present some OPEs between the spin $\\frac{1}{2}$ operators and the spin 1 operators, some OPEs between the spin 1 operators, the OPEs between the spin $\\frac{1}{2}, 1$ operators and the currents (REF ), and the complete OPEs of the ${\\cal N}=2$ superconformal algebra.",
"As noted by [30], there is a difference between the ${\\cal N}=2$ superconformal algebra we are using in this paper and the standard ${\\cal N}=2$ superconformal algebra for general $M (\\ne 1)$ : i) the normalizations in the spin-$1,\\frac{3}{2}$ currents are different and ii) in the OPE of $G^{+}(z) \\, G^{-}(w)$ , the stress energy tensor term has an additional term, $(J^a + J_f^a)^2$ .",
"For $M=1$ case, this additional term vanishes and we obtain the standard ${\\cal N}=2$ superconformal algebra."
],
[
"The so far known currents",
"In the bosonic coset model, the currents contain the $J^a$ of spin-1, the $K^a$ of spin-2, the $P^a$ of spin-3 and the $W^{(3)}$ of spin-3.",
"They are all primary under the corresponding stress energy tensor appearing in the footnote REF .",
"The question is whether they are primary under the above stress energy tensor (REF ) or not.",
"Recall that the bosonic stress energy tensor consists of the first line of (REF ) in addition to $J^{\\alpha }\\, J^{\\alpha }$ and $J^{u(1)} \\, J^{u(1)}$ terms.",
"By construction, the above primary currents do not have any singular terms in the OPEs between them and these quadratic terms.",
"Then we can regard the first line of (REF ) as the bosonic stress energy tensor.",
"Let us look at the second and third lines of (REF ).",
"They are either purely complex fermions dependent terms, $J^{\\alpha }$ dependent term, $J^{u(1)}$ dependent term, $J^{\\alpha }$ with complex fermions, or $J^{u(1)}$ with complex fermions.",
"It is clear to observe that the above primary currents consisting of purely bosonic operators do not produce any nontrivial singular terms when we calculate the OPEs between them and the second and third lines of (REF ).",
"This implies that the above primary currents with bosonic stress energy tensor are primary also under the above stress energy tensor (REF ).",
"Furthermore, because the above primary currents do not contain the complex fermions, it is obvious that the regular conditions for these currents with the operators (REF ) in the denominator hold.",
"Then all the (quasi)primary currents in the bosonic coset model can play the role of primary operators under the stress energy tensor (REF ).",
"However, we should check whether they transform under the spin-$\\frac{3}{2}$ currents in (REF ) and the spin-1 current (REF ) or not.",
"So far, there exist the following currents [3], [12], [30] ${.",
"}10\\endcsname && \\mbox{spin-1}: J^a, \\qquad \\mbox{spin-2}: K^a, \\qquad \\mbox{spin-3}: P^a, \\qquad \\cdots ,\\nonumber \\\\&& \\mbox{spin-1}: K, \\qquad \\mbox{spin-$\\frac{3}{2}$}: G^{+},G^{-}, \\qquad \\mbox{spin-2}: T, \\qquad \\mbox{spin-3}: W^{(3)}, \\qquad \\cdots .$ We would like to construct the additional higher spin currents of integer or half-integer spins in terms of the coset operators $(J^{\\alpha }, J^a, J^{u(1)}, J^{(\\rho \\bar{i})},J^{(\\bar{\\sigma } j)})$ and $(\\psi ^{(\\rho \\bar{i})}, \\psi ^{(\\bar{\\sigma } j)})$ .",
"Moreover, by using their OPEs, we determine the possible new higher spin currents.",
"We will obtain ${\\cal N}=2$ supersymmetric currents corresponding to the above $K^a, W^{(3)}$ and $P^a$ .",
"Although these currents do not belong to the extension of ${\\cal N}=2$ superconformal algebra we are describing in this paper, they can provide how we can construct their ${\\cal N}=2$ supersymmetric versions explicitly by using the currents of ${\\cal N}=2$ superconformal algebra."
],
[
"The nonsinglet\ncurrents of spins $(1, \\frac{3}{2}, \\frac{3}{2}, 2)$",
"In [30], the three currents of this multiplet were obtained.",
"We can consider the $SU(M)$ current in (REF ) as the lowest component of this multiplet.",
"This current has the adjoint index $a$ .",
"In order to have the spin-$\\frac{3}{2}$ current which has an index $a$ , we use the generator of $SU(M)$ by making the product of spin-1 current and spin-$\\frac{1}{2}$ operator with the appropriate contractions of the indices.",
"It turns out that in [30] there exist the nonsinglet spin-$\\frac{3}{2}$ currents ${.",
"}11\\endcsname G^{+,a} \\equiv -t^a_{j \\bar{i}} \\, \\delta _{\\rho \\bar{\\sigma }} \\,J^{(\\rho \\bar{i})} \\, \\psi ^{(\\bar{\\sigma } j)}, \\qquad G^{-,a} \\equiv -t^a_{j \\bar{i}}\\, \\delta _{\\rho \\bar{\\sigma }} \\,\\psi ^{(\\rho \\bar{i})} \\, J^{(\\bar{\\sigma } j)}.$ Then it is natural to describe the following${\\cal N}=2$ multiplet ${.",
"}12\\endcsname (J^a_f, G^{+\\, a}, G^{-\\, a}, ?",
").$ In next section, we will determine the last component of this multiplet."
],
[
"The regular condition",
"By construction, all the currents should satisfy the regular conditions in the OPEs between them and the currents in the denominator of the coset (REF ).",
"All the bosonic currents in the coset model of [3] satisfy automatically because the additional currents in (REF ) come from the complex fermions.",
"For example, the spin-1 current in (REF ) satisfies the trivial OPEs ${.",
"}13\\endcsname J^a(z) \\, ( J^{u(1)} + \\sqrt{\\frac{M+N}{M N}}\\, J^{u(1)}_f )(w) =0 + \\cdots ,\\nonumber \\\\J^a(z) \\, (J^{\\alpha } + J_f^{\\alpha })(w) =0 +\\cdots .$ If we determine the currents from the coset fields from the beginning by taking the multiple product between them and introducing the arbitrary coefficients to be fixed, then it is necessary to check the regular conditions like as (REF ) explicitly.",
"However, when we are calculating some OPEs between the known currents which satisfy the regular conditions already, we do not need to check them at each step because some composite operators appearing in the right hand sides of these OPEs satisfy these regular conditions automatically.",
"At the final stage, it is better to check these conditions for consistency checks.",
"Therefore, the lowest singlet and nonsinglet ${\\cal N}=2$ multiplets are classified by (REF ) and (REF ).",
"The former is the ${\\cal N}=2$ supersymmetric extension of the stress energy tensor of the bosonic coset model."
],
[
"The nonsinglet multiplet of spins\n$(1,\\frac{3}{2},\\frac{3}{2},2)$",
"After identifying the last component of the ${\\cal N}=2$ multiplet correctly, we present the OPEs between the generators of ${\\cal N}=2$ superconformal algebra and this ${\\cal N}=2$ multiplet.",
"Moreover, the OPEs between the supersymmetry generators and the spin-2 current $K^a$ obtained in the bosonic coset model are given."
],
[
"The spin-2 current",
"The simplest ${\\cal N}=2$ multiplet is described in (REF ).",
"In this section, we would like to obtain the last component of this multiplet in terms of coset fields explicitly.",
"Later we will also consider their OPEs and determine their algebraic structures in section 6.",
"How do we obtain the last component of the multiplet (REF )?",
"According to the primary condition of the ${\\cal N}=2$ multiplet under the multiplet of (REF ) in the ${\\cal N}=2$ superconformal algebra, we can use the OPEs between $G^{\\pm }(z)$ and $G^{\\mp ,a}(w)$ and look at the first order pole.",
"See Appendix $B$ .",
"Based on the explicit expressions of (REF ) and (REF ), we can compute the OPE between $G^{+}(z)$ and $G^{-,a}(w)$ explicitly.",
"After subtracting the descendant terms from the first order pole of the OPE $G^{+}(z)\\, G^{-,a}(w)$ , the primary spin-2 current is After extracting the bosonic spin-2 current from this spin-2 current, we obtain the following spin-2 current $\\hat{W}^{+(2),a} =k \\,t^a_{j \\bar{i}} \\, \\delta _{\\rho \\bar{\\sigma }} \\,\\psi ^{(\\rho \\bar{i})}\\,\\partial \\psi ^{(\\bar{\\sigma } j)}-\\sqrt{\\frac{M+N}{M N}} \\, J^{u(1)}\\, J_f^a+ t^{\\alpha }_{\\rho \\bar{\\sigma }} \\, t^a_{j \\bar{i}} \\, J^{\\alpha } \\,\\psi ^{(\\rho \\bar{i})}\\,\\psi ^{(\\bar{\\sigma } j)} -\\frac{1}{M} \\, J^a \\, J_f^{u(1)} +\\frac{1}{2} (i f +d)^{b a c} \\, J^b \\, J_f^c+ \\frac{k}{2}\\, \\partial \\, J_f^a + \\Big ( - \\frac{N}{2(2k+M)}\\,d^{a b c} \\, J^b \\, J^c + \\frac{N}{k}\\, \\sqrt{\\frac{M+N}{M N}}\\,J^a \\, J^{u(1)} \\Big )$ which is a primary.",
"In section 6, we will use the last relation of (REF ) in order to apply the results in [12].",
"${.",
"}1\\endcsname W^{+(2),a}& = & -t^a_{j \\bar{i}} \\, \\delta _{\\rho \\bar{\\sigma }} \\, J^{(\\rho \\bar{i})}\\,J^{(\\bar{\\sigma } j)} + k \\,t^a_{j \\bar{i}} \\, \\delta _{\\rho \\bar{\\sigma }} \\,\\psi ^{(\\rho \\bar{i})}\\,\\partial \\psi ^{(\\bar{\\sigma } j)}-\\sqrt{\\frac{M+N}{M N}} \\, J^{u(1)}\\, J_f^a\\nonumber \\\\&+ & t^{\\alpha }_{\\rho \\bar{\\sigma }} \\, t^a_{j \\bar{i}} \\, J^{\\alpha } \\,\\psi ^{(\\rho \\bar{i})}\\,\\psi ^{(\\bar{\\sigma } j)} -\\frac{1}{M} \\, J^a \\, J_f^{u(1)} +\\frac{1}{2} (i f +d)^{b a c} \\, J^b \\, J_f^c\\nonumber \\\\&- & \\frac{N}{2} \\, \\partial \\, J^a + \\frac{k}{2}\\, \\partial \\, J_f^a\\equiv -\\frac{1}{2} \\, K^a + \\hat{W}^{+(2),a},$ where the spin-2 current obtained in the bosonic coset model is given by [3] ${.",
"}2\\endcsname K^a & = & \\delta _{\\rho \\bar{\\sigma }} \\,t^a_{j\\bar{i}} \\, (J^{(\\rho \\bar{i})} \\, J^{(\\bar{\\sigma } j)} +J^{(\\bar{\\sigma } j)} \\, J^{(\\rho \\bar{i})})-\\frac{N}{(M+2k)} \\, d^{abc} \\, J^b\\, J^c\\nonumber \\\\& + & \\frac{2N}{k} \\sqrt{\\frac{M+N}{M N}} \\, J^a \\, J^{u(1)}.$ We realize that the field contents in (REF ) cannot be written in terms of the known currents in (REF ) and (REF ) due to the first five terms.",
"In particular, the fourth term of (REF ) looks like the first term of spin-3 current $P^a$ appearing in (REF ) in the sense that $J^{(\\rho \\bar{i})}\\,J^{(\\bar{\\sigma } j)}$ is replaced by $\\psi ^{(\\rho \\bar{i})}\\,\\psi ^{(\\bar{\\sigma } j)}$ and other factors remain unchanged.",
"The reason why we write down the above spin-2 current in terms of the known previous spin-2 current and other piece is that when we calculate some OPEs including this current, we can use the previous result found in [12].",
"For example, in the OPE between $W^{+(2),a}(z)$ and $W^{+(2),b}(w)$ , it is rather complicated to determine the OPE between the first term of (REF ) and itself.",
"Instead, we can change those first term by using (REF ) in terms of $J^a, J^{u(1)}$ and $K^a$ .",
"We will see the details in section 6."
],
[
" The OPEs with the currents of\n${\\cal N}=2$ superconformal algebra",
"Once we have determined the four currents in the multiplet, ${.",
"}3\\endcsname (J_f^a, G^{+,a}, G^{-,a}, W^{+(2),a}),$ we need to check whether they are really the components of ${\\cal N}=2$ multiplet or not.",
"It is straightforward to calculate the following OPEs with the help of (REF ), (REF ), (REF ) and (REF ) ${.",
"}4\\endcsname K(z) \\, J_f^a(w) & = & 0 +\\cdots ,\\nonumber \\\\K(z) \\, G^{\\pm ,a}(w) & = & \\pm \\frac{1}{(z-w)}\\, G^{\\pm ,a}(w) +\\cdots ,\\nonumber \\\\K(z) \\, W^{+(2),a}(w) & = & \\frac{1}{(z-w)^2}\\, \\Bigg [ N\\, J^a -k \\,J_f^a \\Bigg ](w) +\\cdots .$ When we compare the ones in Appendix $B$ with these OPEs (REF ), the last OPE contains the additional term $J^a$ in the second order pole of the right hand side.",
"There are two ways to fix this inconsistency by considering the above spin-2 current with some modifications in order to have $J^a_f(w)$ in the above second order pole or by taking the spin-1 current as $(N\\, J^a -k \\,J_f^a)$ rather than the previous $J_f^a$ .",
"Recall that the OPE between $K(z)$ and $J^a(w)$ is regular because it is obvious to see that the OPE between the spin-1 current $J^a$ and the spin-1 $J_f^{u(1)}$ having complex fermions does not have any singular terms and moreover it is known that $J^a(z) \\, J^{u(1)}(w) = 0 + \\cdots $ from the observation of [3].",
"This implies that the above linear combination of spin-1 current satisfies the corresponding OPE which is the first one in (REF ).",
"Then we will have the right property in Appendix $B$ .",
"However, we will take the lowest component of the multiplet as $J^a_f$ rather than $(N\\, J^a -k \\,J_f^a)$ as in (REF ).",
"One of the reasons why we do not consider this linear combination as the lowest component of the above multiplet is that it is better to treat $J^a$ and $J_f^a$ separately when we decide which composite operators are written in terms of the known currents or not.",
"Otherwise we should replace all the $J^a_f$ term by adding $J^a$ term in order to preserve the above linear combination in all the composite operators Note that all the ${\\cal N}=2$ multiplets in this paper do not satisfy the properties in Appendix $B$ .",
"In other words, some of the coefficients appearing in Appendix $B$ appear differently and the vanishing coefficients in the higher order poles in Appendix $B$ appear nontrivially.. We observe that the combination of $(N\\, J^a -k \\,J_f^a)$ will appear at many places of the OPEs we will calculate later.",
"Let us describe the OPEs between the spin-$\\frac{3}{2}$ current of ${\\cal N}=2$ superconformal algebra and the currents of the multiplet in (REF ) For convenience, we present the following OPEs $J^a(z) \\, G^{\\pm }(w)=\\pm \\, \\frac{1}{(z-w)}\\, G^{\\pm ,a}(w) +\\cdots $ , $J^{\\alpha }(z) \\, G^+(w)= \\frac{1}{(z-w)}\\, t^{\\alpha }_{\\rho \\bar{\\sigma }}\\, \\delta _{j \\bar{k}}\\,\\psi ^{(\\bar{\\sigma } j)}\\, J^{(\\rho \\bar{k})}(w) + \\cdots $ and $J^{\\alpha }(z) \\, G^-(w)= -\\frac{1}{(z-w)}\\, t^{\\alpha }_{\\rho \\bar{\\nu }}\\,\\delta _{m \\bar{m}} \\, \\psi ^{(\\rho \\bar{m})}\\,J^{( \\bar{\\nu } m)}(w) + \\cdots $ which will be used in the last OPE of (REF ).",
"${.",
"}5\\endcsname G^+(z) \\, J_f^a(w) & = & \\frac{1}{(z-w)}\\, G^{+,a}(w) + \\cdots ,\\nonumber \\\\G^{+}(z) \\, G^{+,a}(w) & = & 0 +\\cdots ,\\nonumber \\\\G^{+}(z) \\, G^{-,a}(w) & = & \\frac{1}{(z-w)^2} \\, \\Bigg [ N\\, J^a -k \\,J_f^a \\Bigg ](w) \\nonumber \\\\& + & \\frac{1}{(z-w)} \\Bigg [\\frac{1}{2}\\,\\Bigg ( N\\, \\partial \\, J^a -k \\,\\partial \\, J_f^a \\Bigg ) +W^{+(2),a} \\Bigg ](w) + \\cdots ,\\nonumber \\\\G^{+}(z) \\, W^{+(2),a}(w) & = & \\frac{1}{(z-w)^2}\\, \\frac{3}{2}\\, (k+N)\\,G^{+,a}(w)\\nonumber \\\\& + & \\frac{1}{(z-w)}\\, \\frac{1}{2}\\, (k+N)\\,\\partial \\, G^{+,a}(w) + \\cdots .$ By changing the sign of $J_f^a(w)$ (and the remaining currents are the same) we obtain the standard OPE of $G^{+}(z) \\, J_f^a(w)$ which leads to the minus sign of the right hand side of this OPE.",
"See also Appendix $B$ .",
"Note that there is a term of $J^a(w)$ in the second order pole of $G^{+}(z) \\, G^{-,a}(w)$ as before.",
"From this OPE, we have obtained the nonsinglet spin-2 current we mentioned before.",
"In the last OPE, the numerical value in the right hand side has an extra factor $(k+N)$ when we compare with the one in Appendix $B$ In the last OPE, we use the following identity $t^a_{j \\bar{i}} \\, \\delta _{k \\bar{l}}\\, \\delta _{\\rho \\bar{\\mu }}\\, \\delta _{\\tau \\bar{\\sigma }}\\,\\psi ^{(\\bar{\\sigma } j)}\\, \\psi ^{(\\rho \\bar{i})}\\, \\psi ^{(\\bar{\\mu } k)}\\, J^{(\\tau \\bar{l})}= \\frac{1}{2} (i \\, f + d)^{b a c}\\, G^{+,b} \\, J_f^c -\\frac{1}{M} \\, G^+ \\, J_f^a -\\frac{1}{M}\\, G^{+,a}\\, J_f^{u(1)}$ by using the rearrangement lemma in [78], [80].",
"Here the $J_f^{u(1)}$ term is canceled by other term in the first order pole of the last OPE.",
"Then we are left with the known currents finally..",
"Similarly, we obtain the following OPEs ${.",
"}6\\endcsname G^-(z) \\, J_f^a(w) & = & -\\frac{1}{(z-w)}\\, G^{-,a}(w) + \\cdots ,\\nonumber \\\\G^{-}(z) \\, G^{+,a}(w) & = & \\frac{1}{(z-w)^2} \\, \\Bigg [ -N\\, J^a +k \\,J_f^a \\Bigg ](w) \\nonumber \\\\& + & \\frac{1}{(z-w)} \\Bigg [\\frac{1}{2}\\,\\Bigg ( -N\\, \\partial \\, J^a + k \\,\\partial \\, J_f^a \\Bigg ) +W^{+(2),a} + i \\, f^{a b c}\\, J^b \\, J_f^c\\Bigg ](w) + \\cdots ,\\nonumber \\\\G^{-}(z) \\, G^{-,a}(w) & = & 0 +\\cdots ,\\nonumber \\\\G^{-}(z) \\, W^{+(2),a}(w) & = & \\frac{1}{(z-w)^2}\\,\\frac{1}{2}\\, (3k+2M+3N)\\, G^{-,a}(w) \\nonumber \\\\& + &\\frac{1}{(z-w)} \\, \\Bigg [ \\frac{1}{3} \\,\\frac{1}{2}\\, (3k+2M+3N)\\, \\partial \\, G^{-,a}+ i \\, f^{a b c}\\, J^b \\, G^{-,c}- i \\, f^{a b c}\\, G^{-,b}\\, J_f^c\\nonumber \\\\& - & \\frac{ M}{3} \\, \\partial \\, G^{-,a} \\Bigg ](w) + \\cdots .$ By changing the sign of $J_f^a(w)$ as before, we obtain the standard OPE of $G^{-}(z) \\, J_f^a(w)$ in Appendix $B$ .",
"There exists a term of $J^a(w)$ in the second order pole of $G^{-}(z) \\, G^{+,a}(w)$ .",
"In the last OPE of (REF ), we do not combine with the first and the fourth terms in the first order pole In all the OPEs in this paper, we intentionally put the exact coefficients coming from the descendant terms in the right hand sides of the OPEs without simplifying them..",
"The last three terms of the last OPE (which is primary under the stress energy tensor) are written in terms of the known currents in (REF ) In the last OPE, the following identity is used $t^a_{j \\bar{i}} \\, \\delta _{\\rho \\bar{\\mu }}\\, \\delta _{\\tau \\bar{\\sigma }}\\, \\delta _{k \\bar{l}}\\, \\psi ^{(\\rho \\bar{i})}\\, \\psi ^{(\\bar{\\sigma } j)}\\, \\psi ^{(\\tau \\bar{l})}\\,J^{(\\bar{\\mu } k)}= \\frac{1}{M}\\, J_f^a \\, G^- + \\frac{1}{M}\\, J_f^{u(1)}\\,G^{-,a}-\\frac{1}{2}(i \\, f+ d)^{a b c}\\, J_f^c \\, G^{-,b}$ with the help of [78], [80] again.",
"The $J_f^{u(1)}$ term we do not want to have can be canceled by other term appearing in the first order pole of the last OPE of (REF ).. We can check that each current in the multiplet we are considering in this subsection satisfies the primary condition under the stress energy tensor (REF ) as follows: ${.",
"}7\\endcsname T(z) \\, J_f^a(w) & = & \\frac{1}{(z-w)^2} \\, J_f^a(w) +\\frac{1}{(z-w)} \\, \\partial \\, J_f^a(w) + \\cdots ,\\nonumber \\\\T(z) \\, G^{\\pm ,a}(w) & = & \\frac{1}{(z-w)^2} \\, \\frac{3}{2} \\,G^{\\pm ,a}(w) +\\frac{1}{(z-w)} \\, \\partial \\, G^{\\pm ,a}(w) + \\cdots ,\\nonumber \\\\T(z) \\, W^{+(2),a}(w) & = & \\frac{1}{(z-w)^2} \\, 2\\, W^{+(2),a}(w) +\\frac{1}{(z-w)} \\, \\partial \\, W^{+(2),a}(w) + \\cdots .$ Of course, it is rather nontrivial to check these OPEs (REF ) for generic $k,N$ and $M$ by hands but we can check them for fixed $N$ and $M$ by using the Thielemans package [81].",
"When we notice the presence of higher order terms where the pole is greater than or equal to three, then we can try to do this checking by hands on the higher order terms.",
"Usually, it is not necessary to check the first and second order poles of the OPEs.",
"For the quasi primary operators, it is nontrivial to check the fourth order poles by hands.",
"As a consistency check, we can check that the currents satisfy the following regular conditions ${.",
"}8\\endcsname (J_f^a, G^{+,a}, G^{-,a}, W^{+(2),a})(z) \\,( J^{u(1)} + \\sqrt{\\frac{M+N}{M N}}\\, J^{u(1)}_f )(w) =0 + \\cdots ,\\nonumber \\\\(J_f^a, G^{+,a}, G^{-,a}, W^{+(2),a})(z) \\, (J^{\\alpha } + J_f^{\\alpha })(w) =0 +\\cdots .$ For example, the relative coefficients in (REF ) are fixed from these constraints and the primary condition under the stress energy tensor (REF ) (and its ${\\cal N}=2$ extension)."
],
[
" Some OPEs with the spin-2 current $K^a$",
"It is clear that the OPE between $K(z)$ and the spin-2 current $K^a(w)$ does not contain the singular terms because the OPE between the $J^{u(1)}(z)$ and $K^a(w)$ is regular and the $K^a(w)$ does not have the complex fermions.",
"Then the question is what are the OPEs between $G^{\\pm }(z)$ and $K^a(w)$ The OPE between $K(z)$ and $K^a(w)$ is regular and the OPE between $T(z)$ and $K^a(w)$ satisfies the standard OPE for a primary field mentioned before.. By using the explicit expressions of (REF ) and (REF ) we obtain ${.",
"}9\\endcsname G^{+}(z) \\, K^{a}(w) & = & -\\frac{1}{(z-w)^2} \\,\\frac{2(k^2-1)(2k+M+N)}{k(2k+M)} \\, G^{+,a}(w) \\nonumber \\\\& + &\\frac{1}{(z-w)} \\, \\Bigg [ -\\frac{1}{3}\\frac{2(k^2-1)(2k+M+N)}{k(2k+M)} \\, \\partial \\, G^{+,a}+ V^{+(\\frac{5}{2}),a} \\Bigg ](w) \\nonumber \\\\& + & \\cdots .$ Note that the first order pole provides the spin-$\\frac{5}{2}$ current which cannot be written in terms of the known currents (REF ) and (REF ).",
"The primary spin-$\\frac{5}{2}$ current under the stress energy tensor (REF ), after subtracting the descendant term in the first order pole is ${.",
"}10\\endcsname V^{+(\\frac{5}{2}),a} & = &-2 (k+N) \\, t^a_{j \\bar{i}} \\, \\delta _{\\rho \\bar{\\sigma }} \\, \\partial \\,J^{(\\rho \\bar{i})}\\,\\psi ^{(\\bar{\\sigma } j)}-\\frac{2(k+N)}{k} \\,\\sqrt{\\frac{M+N}{M N}} \\, J^{u(1)}\\, G^{+,a}\\nonumber \\\\&-& \\frac{2}{k M} \\, (k+M+N) \\, J^a \\, G^{+}-(i f - \\frac{2k+M+2N}{2k+M} \\, d)^{a b c} \\, J^b \\, G^{+,c}\\nonumber \\\\& + & 2 \\, t^{\\alpha }_{\\rho \\bar{\\sigma }} \\, t^a_{j \\bar{i}} \\, J^{\\alpha } \\,J^{(\\rho \\bar{i})}\\,\\psi ^{(\\bar{\\sigma } j)} -\\frac{2}{3}\\,\\frac{2(k^2-1)(2k+M+N)}{k(2k+M)} \\, \\partial \\, G^{+,a}.$ Note that the first, second and fifth terms are the reasons why we cannot write down this current in terms of the known currents.",
"Again, the field content of the fifth term looks like the fourth term of (REF ) in the sense that the spin-$\\frac{1}{2}$ operator is replaced by the spin-1 current.",
"Similarly, we can determine the following OPE ${.",
"}11\\endcsname G^{-}(z) \\, K^{a}(w) & = &-\\frac{1}{(z-w)^2} \\,\\frac{2(k^2-1)(2k+M+N)}{k(2k+M)} \\, G^{-,a}(w) \\nonumber \\\\& + &\\frac{1}{(z-w)} \\, \\Bigg [ -\\frac{1}{3}\\frac{2(k^2-1)(2k+M+N)}{k(2k+M)} \\, \\partial \\, G^{-,a}+ V^{-(\\frac{5}{2}),a} \\Bigg ](w) \\nonumber \\\\& + & \\cdots ,$ where the primary spin-$\\frac{5}{2}$ current is ${.",
"}12\\endcsname V^{-(\\frac{5}{2}),a} & = &-2 (k+N) \\, t^a_{j \\bar{i}} \\, \\delta _{\\rho \\bar{\\sigma }} \\, \\partial \\,J^{(\\bar{\\sigma } j)}\\,\\psi ^{(\\rho \\bar{i})}+\\frac{2(k+N)}{k} \\,\\sqrt{\\frac{M+N}{M N}} \\, J^{u(1)}\\, G^{-,a}\\nonumber \\\\&+& \\frac{2}{k M} \\, (k+M+N) \\, J^a \\, G^{-}-(i f + \\frac{2k+M+2N}{2k+M} \\, d)^{a b c} \\, J^b \\, G^{-,c}\\nonumber \\\\& - & 2 \\, t^{\\alpha }_{\\rho \\bar{\\sigma }} \\, t^a_{j \\bar{i}} \\, J^{\\alpha } \\,J^{(\\bar{\\sigma } j)}\\,\\psi ^{(\\rho \\bar{i})} -\\frac{2}{3}\\,\\frac{2(k^2-1)(2k+M+N)}{k(2k+M)} \\, \\partial \\, G^{-,a}.$ The field contents of (REF ) look similar to the one in (REF ).",
"They will appear at many places in the OPEs we will consider later.",
"Due to the presence of the new primary fields (REF ) and (REF ) for given OPEs, the bosonic spin-2 current $K^a$ by itself cannot play the role of the last component of the multiplet in (REF ).",
"In other words, the right spin-2 current in (REF ) contains both fermionic dependent terms and bosonic current dependent terms in addition to $K^a$ .",
"If we remember the OPEs in Appendix $B$ , the OPEs in (REF ) and (REF ) imply that the spin-2 current $K^a$ can be a candidate for the lowest component of other nonsinglet ${\\cal N}=2$ multiplet which will be explained later if we succeed to eliminate the second order poles of (REF ) and (REF ) Of course, if we add some terms to $K^a$ and can remove $V^{\\pm (\\frac{5}{2}), a}$ terms properly, then this modified spin-2 current can be the last component of the ${\\cal N}=2$ multiplet of this section.",
"See the last OPEs in (REF ) and (REF )..",
"In summary, the equations (REF ), (REF ), (REF ) and (REF ) imply that the right hand sides of these OPEs contain the currents (REF ), the nonsinglet spin-1 current $J^a$ and their composite operators.",
"As emphasized before, these OPEs do not satisfy Appendix $B$ ."
],
[
" The singlet\nmultiplet of spins $(2,\\frac{5}{2},\\frac{5}{2},3)$",
"We would like to construct this multiplet starting from the singlet spin-3 current found in the bosonic coset model [3]."
],
[
"Construction of lowest component",
"Once we know any component of this multiplet, then the other three components can be determined, in principle, by using the ${\\cal N}=2$ primary conditions described in Appendix $B$ .",
"Let us first consider how we obtain the lowest component of this ${\\cal N}=2$ multiplet Although we have realized that the spin-2 current $K^a$ will participate in the lowest component from the previous section, it is not clear how we can continue to calculate the additional terms by adding the possible composite operators of spin-2 to the $K^a$ with an appropriate contraction in the indices..",
"There exists a singlet spin-3 current $W^{(3)}$ in the bosonic coset model and its expression in terms of coset fields is described by [3] ${.",
"}1\\endcsname W^{(3)} & = &b_1 \\, d^{\\alpha \\beta \\gamma } \\, J^{\\alpha } J^{\\beta } J^{\\gamma }+ b_2 \\, d^{a b c} \\, J^a J^b J^c+b_3 \\, J^{u(1)} J^{u(1)} J^{u(1)} + b_4 \\, J^{\\alpha } J^{\\alpha } J^{u(1)}\\nonumber \\\\&+& b_5 \\, J^a \\, J^a \\, J^{u(1)} + b_6 \\,t^{\\alpha }_{\\rho \\bar{\\sigma }} \\, \\delta _{j\\bar{i}}\\,J^{\\alpha } \\, (J^{(\\rho \\bar{i})} \\, J^{(\\bar{\\sigma } j)} +J^{(\\bar{\\sigma } j)} \\, J^{(\\rho \\bar{i})})\\nonumber \\\\&+& b_7 \\, \\delta _{\\rho \\bar{\\sigma }} \\,t^a_{j\\bar{i}} \\, J^a (J^{(\\rho \\bar{i})} J^{(\\bar{\\sigma } j)} +J^{(\\bar{\\sigma } j)} J^{(\\rho \\bar{i})})+b_8 \\, \\delta _{\\rho \\bar{\\sigma }} \\,\\delta _{j\\bar{i}} \\, J^{u(1)} (J^{(\\rho \\bar{i})} J^{(\\bar{\\sigma } j)} +J^{(\\bar{\\sigma } j)} J^{(\\rho \\bar{i})})\\nonumber \\\\&+& b_{12} \\,\\delta _{\\rho \\bar{\\sigma }} \\,\\delta _{j\\bar{i}} \\, \\partial \\, J^{(\\rho \\bar{i})} \\,J^{(\\bar{\\sigma } j)}+ b_{13} \\,\\, \\delta _{\\rho \\bar{\\sigma }} \\,\\delta _{j\\bar{i}} \\, \\partial \\, J^{(\\bar{\\sigma } j)} \\, J^{(\\rho \\bar{i})}+ b_{14} \\, \\partial ^2 \\, J^{u(1)}.$ Here the relative coefficients are functions of $k, N$ and $M$ as follows: ${.",
"}2\\endcsname b_2 & = & -\\frac{ N (k+N) (k+2 N)}{M (k+M) (k+2 M)} \\, b_1,\\qquad b_3 = \\sqrt{\\frac{M+N}{M N}} \\,\\frac{ 2 (k+N) (k+2 N) (M+N)}{k^2 M} \\, b_1,\\nonumber \\\\b_4 & = & \\sqrt{\\frac{M+N}{M N}} \\, \\frac{ 6 (k+N)}{k} \\, b_1,\\qquad b_5 = \\sqrt{\\frac{ M+N}{M N}} \\,\\frac{ 6 N (k+N) (k+2 N)}{k M (k+2 M)}\\,b_1,\\nonumber \\\\b_6 & = & -\\frac{3 (k+N)}{M} \\, b_1,\\qquad b_7 = \\frac{3 (k+N) (k+2 N)}{M (k+2 M)} \\, b_1,\\nonumber \\\\b_8 & = & -\\sqrt{\\frac{M+N}{M N}}\\,\\frac{ 3 (k+N) (k+2 N)}{k M}\\, b_1,\\qquad b_{12} = \\frac{3 (k+N) (k+2 N)}{M}\\, b_1,\\nonumber \\\\b_{13} & = & -\\frac{3 (k+N) (k+2 N)}{M} \\, b_1,\\qquad b_{14} = - \\sqrt{\\frac{ M+N}{M N}}\\,N (k+N) (k+2 N) \\, b_1.$ Although we add the spin-1 currents coming from the complex fermions to the ones in the bosonic coset model studied in [3], the regular conditions with the coset fields in the coset (REF ) hold as follows: ${.",
"}3\\endcsname W^{(3)}(z) \\, ( J^{u(1)} + \\sqrt{\\frac{M+N}{M N}}\\, J^{u(1)}_f )(w) =0 + \\cdots ,\\nonumber \\\\W^{(3)}(z) \\, (J^{\\alpha } + J_f^{\\alpha })(w) =0 +\\cdots ,$ because the OPEs between the spin-3 current $W^{(3)}$ (consisting of purely bosonic fields) and these spin-1 currents (consisting of purely fermionic fields) do not have any singular terms as in (REF ).",
"We need to calculate the OPEs between the supersymmetry generators of the ${\\cal N}=2$ superconformal algebra and the above singlet spin-3 current in order to determine its superpartners.",
"Because the spin-$\\frac{3}{2}$ currents consist of the spin-$1, \\frac{1}{2}$ operators transforming as $({\\bf N}, \\overline{{\\bf M}})$ or $(\\overline{{\\bf N}},{\\bf M})$ , we should first calculate the OPEs between the spin-1 currents and the spin-3 current $W^{(3)}$ .",
"Note that it is obvious to see that the OPEs between the spin-$\\frac{1}{2}$ operator and $W^{(3)}$ are regular.",
"In Appendix $C$ , we present some relevant OPEs between the spin-1 current and the singlet spin-3 current.",
"It turns out that after extracting the OPE $W^{(3)}(z) \\, G^{\\pm }(w)$ and then changing the order of the two currents we have ${.",
"}4\\endcsname && G^{\\pm }(z) \\, W^{(3)}(w) =\\nonumber \\\\&&\\mp \\frac{1}{(z-w)^3} \\,\\frac{ (k^2-1) (k^2-4) (k+M+N) (2 k+M+N) (3 k+2 M+2 N)}{k^2 M (k+M) (k+2 M)} \\, b_1 \\, G^{\\pm }(w)\\nonumber \\\\&& \\pm \\frac{1}{(z-w)^2} \\,G^{\\pm (\\frac{5}{2}),0}(w) + {\\cal O} (\\frac{1}{(z-w)}).$ We do not specify the first order poles of (REF ) here.",
"They can be written explicitly from the results of Appendix $C$ .",
"Because there are no descendant terms associated with the spin-$\\frac{3}{2}$ currents at the second order pole, the primary singlet spin-$\\frac{5}{2}$ currents appear in the second order pole.",
"One of them is given by ${.",
"}5\\endcsname G^{+(\\frac{5}{2}),0} & = &\\frac{(k^2-4)(k+M+N)(3k+2M+2N)}{k M (k+2M)} \\, b_1 \\,\\Bigg [ - 3 \\, t^{\\alpha }_{\\sigma \\bar{\\sigma }} \\, \\delta _{j \\bar{i}} \\, J^{\\alpha }\\, \\psi ^{(\\bar{\\sigma } j)}\\, J^{(\\sigma \\bar{i})}\\nonumber \\\\&-& \\frac{3}{(k+M)} \\, (k+N) \\, J^a \\, G^{+,a}-\\frac{3}{k}\\, \\sqrt{\\frac{M+N}{M N}} \\, (k+N)J^{u(1)}\\, G^{+} \\nonumber \\\\&+& 3 \\, (k+N) \\, \\delta _{\\rho \\bar{\\sigma }}\\, \\delta _{j \\bar{i}} \\,\\psi ^{(\\bar{\\sigma } j)}\\, \\partial \\, J^{(\\rho \\bar{i})} -\\frac{(k^2-1)(2k+M+N)}{k(k+M)} \\, \\partial \\, G^{+} \\Bigg ].$ Due to the first, third and fourth terms, we cannot rewrite (REF ) in terms of previous known currents we have discussed so far The overall factor $b_1$ appears in this expression by using (REF ) for the $W^{(3)}$ ..",
"The field contents look similar to the one $V^{+(\\frac{5}{2}),a}$ (REF ).",
"There is no $f$ or $d$ symbols in (REF ) because this current should be a singlet field.",
"We can easily check that the first term comes from the sixth term of (REF ).",
"The other is given by ${.",
"}6\\endcsname G^{-(\\frac{5}{2}),0} & = &\\frac{(k^2-4)(k+M+N)(3k+2M+2N)}{k M (k+2M)} \\, b_1 \\,\\Bigg [ 3 \\, t^{\\alpha }_{\\sigma \\bar{\\sigma }} \\, \\delta _{j \\bar{i}} \\, J^{\\alpha }\\, \\psi ^{(\\sigma \\bar{i})}\\, J^{(\\bar{\\sigma } \\bar{j})}\\nonumber \\\\&+& \\frac{3}{(k+M)} \\, (k+N) \\, J^a \\, G^{-,a}+\\frac{3}{k}\\, \\sqrt{\\frac{M+N}{M N}} \\, (k+N)J^{u(1)}\\, G^{-} \\nonumber \\\\&+& 3 \\, (k+N) \\, \\delta _{\\rho \\bar{\\sigma }}\\, \\delta _{j \\bar{i}} \\,\\psi ^{(\\rho \\bar{i})}\\, \\partial \\, J^{(\\bar{\\sigma } j)} -\\frac{(k^2-1)(2k+M+N)}{k(k+M)} \\, \\partial \\, G^{-} \\Bigg ].$ The field contents of (REF ) look similar to the one $V^{-(\\frac{5}{2}),a}$ (REF ).",
"Again the first term comes from the sixth term of (REF ).",
"Of course, after we determine the lowest component of this multiplet later, we should check whether the above two primary spin-$\\frac{5}{2}$ currents are really the elements of this multiplet under the action of the generators of ${\\cal N}=2$ superconformal algebra based on Appendix $B$ .",
"In general, we expect that the spin-$\\frac{5}{2}$ currents obtained by $W^{(3)}$ from the bosonic coset model will be different from the ones determined by the corresponding currents in ${\\cal N}=2$ multiplet.",
"However, it turns out that they are equivalent to each other.",
"In next section, we will observe some examples where this is not the case.",
"From Appendix $B$ , the second order pole in the OPE of either $G^{-}(z) \\, G^{+(\\frac{5}{2}),0}(w)$ or $G^{+}(z) \\, G^{-(\\frac{5}{2}),0}(w)$ leads to the following primary singlet spin-2 current, which is the lowest component of ${\\cal N}=2$ multiplet in this section If we act $G^{\\pm }(z)$ on $T_{boson}(w)$ appearing in the footnote REF , then we obtain the spin-$\\frac{3}{2}$ current at the second order pole which is proportional to $G^{\\pm }(w)$ .",
"After acting on $G^{\\mp }(z)$ further, then we obtain the second order pole which is proportional to $K(w)$ : the lowest component of (REF ).",
"This is one way to observe that the multiplet of (REF ) is an ${\\cal N}=2$ extension of the stress energy tensor in the bosonic coset model.",
"${.",
"}7\\endcsname W^{-(2),0} & = &\\frac{(k^2-4) (k+M+N) (3 k+2 M+2 N)}{k M (k+2M)}\\, b_1 \\, \\Bigg [\\nonumber \\\\& - & \\frac{2(k^3+2 k^2 M+2 k^2 N+3 k M N+2 k+M+N)}{k (k+M)}\\,\\delta _{\\rho \\bar{\\sigma }} \\delta _{j \\bar{i}} \\,J^{(\\rho \\bar{i})} J^{(\\bar{\\sigma } j)}\\nonumber \\\\&+ &3 \\, M \\,J^{\\alpha }\\, J^{\\alpha }-\\frac{2 (k^2-1) (2 k+M+N) }{k (k+M) } \\ J^{\\alpha } \\, J_f^{\\alpha }\\nonumber \\\\&-&\\frac{2 (k^2-1) (2 k+M+N) }{k (k+M) } \\, \\sqrt{\\frac{M+N}{M N}}\\, J^{u(1)} J^{u(1)}_f\\nonumber \\\\& - & \\frac{2 (k^2-1) (2 k+M+N) }{k (k+M)}\\,J^a \\, J^a_f\\nonumber \\\\&+& \\,\\frac{ (k^2-1) (2 k+M+N) }{ (k+M) }\\,\\delta _{\\rho \\bar{\\sigma }} \\, \\delta _{j \\bar{i}} \\,\\partial \\, \\psi ^{(\\rho \\bar{i})} \\, \\psi ^{(\\bar{\\sigma } j)}+ \\frac{3 N (k+N) }{ (k+M) }\\,J^a \\, J^a\\nonumber \\\\&-&\\frac{ (k^2-1) (2 k+M+N) }{ (k+M) }\\, \\delta _{\\rho \\bar{\\sigma }} \\, \\delta _{j \\bar{i}} \\,\\psi ^{(\\rho \\bar{i})} \\, \\partial \\, \\psi ^{(\\bar{\\sigma } j)}\\nonumber \\\\&+&\\frac{3 (k+N) (M+N) }{k }\\, J^{u(1)} \\, J^{u(1)}\\nonumber \\\\&+&\\frac{ N M(k^3+2 k^2 M+2 k^2 N+3 k M N+2 k+M+N)}{k (k+M)}\\,\\sqrt{\\frac{M+N}{M N}}\\, \\partial \\, J^{u(1)} \\, \\Bigg ].$ Note that the second, fifth and ninth terms in (REF ) do not arise in the stress energy tensor $T$ (REF ) and we cannot express this in terms of the known currents obtained so far.",
"The fifth term is characteristic of this spin-2 current (or of modified stress energy tensor in Appendix $A$ ) in the sense that we do not see this term from the bosonic stress energy tensor also.",
"In doing this computation, we need to obtain the OPE between $G^{-,a}(z)$ and $G^{+,b}(w)$ when we consider the OPE between the $G^-(z)$ and the second term of (REF ) at the coordinate $w$ .",
"See also Appendix $D$ where the corresponding OPE is written in terms of the coset fields and section 6 where the right hand side of this OPE is expressed in terms of the known currents explicitly.",
"In section 7, we will calculate the OPE between this singlet spin-2 current and itself."
],
[
"Construction of second and third components",
"From the Thielemans package [81] with [82], we can check that the OPE between $G^+(z)$ and $W^{-(2),0}(w)$ does not contain the singular terms except the first order pole for fixed $N=5, M=4$ values.",
"It turns out that the first order pole of this OPE is proportional to $G^{+(\\frac{5}{2}),0}(w)$ .",
"Then we should determine the coefficient appearing in front of this spin-$\\frac{5}{2}$ current.",
"As described before, we can focus on the particular nonderivative term of $G^{+(\\frac{5}{2}),0}(w)$ .",
"That is, the first term of (REF ).",
"We can observe that the contribution from this term comes from the first three terms in $W^{-(2),0}(w)$ of (REF ) and by collecting all the contributions we can obtain the final coefficient appearing in the spin-$\\frac{5}{2}$ current $G^{+(\\frac{5}{2}),0}(w)$ in the above OPE.",
"This implies that the spin-$\\frac{5}{2}$ current $G^{+(\\frac{5}{2}),0}$ belongs to the second component of this ${\\cal N}=2$ multiplet.",
"See also Appendix $B$ .",
"Similarly, we should also check whether the spin-$\\frac{5}{2}$ current $G^{-(\\frac{5}{2}),0}$ belongs to the third component of this ${\\cal N}=2$ multiplet or not.",
"In this case, we also observe that the OPE between $G^-(z)$ and $W^{-(2),0}(w)$ does not contain the singular terms except the first order pole for fixed $N=5, M=4$ values.",
"By considering the first three terms in $W^{-(2),0}(w)$ of (REF ) and collecting all the contributions, the final coefficient appearing in the spin-$\\frac{5}{2}$ current $G^{-(\\frac{5}{2}),0}(w)$ in the above OPE can be fixed completely.",
"Therefore, the spin-$\\frac{5}{2}$ current $G^{-(\\frac{5}{2}),0}$ belongs to the third component of this ${\\cal N}=2$ multiplet.",
"Then we have the following currents obtained so far in this multiplet ${.",
"}8\\endcsname (W^{-(2),0}, G^{+(\\frac{5}{2}),0}, G^{-(\\frac{5}{2}),0}, ?",
"),$ with the coset field contents given in (REF ), (REF ) and (REF ).",
"The last component of (REF ) will be described in next subsection and we expect that the corresponding generalization of (REF ) will appear and further checks will be given later."
],
[
"Construction of last component",
"In this subsection, we would like to construct the unknown last component in (REF ).",
"First of all, in the previous description of (REF ), we have obtained the following result ${.",
"}9\\endcsname G^{-}(z) \\,G^{+(\\frac{5}{2}),0}(w)\\Bigg |_{\\frac{1}{(z-w)^2}} = W^{-(2),0}(w).$ According to the explanation of Appendix $B$ , the last component of the ${\\cal N}=2$ multiplet can be obtained from the next first order pole in the OPE between $G^{-}(z)$ and $G^{+(\\frac{5}{2}),0}(w)$ .",
"After subtracting the descendant term associated with (REF ) in the first order pole, we arrive at the following relation ${.",
"}10\\endcsname W^{+(3),0}(w) & \\equiv &G^{-}(z) \\, G^{+(\\frac{5}{2}),0}(w)\\Bigg |_{\\frac{1}{(z-w)}}-\\frac{1}{4}\\, \\partial \\, W^{-(2),0}(w).$ Then from the first order pole in the OPE between $G^{-}(z)$ and $G^{+(\\frac{5}{2}),0}(w)$ with derivative terms (REF ), we eventually obtain the following singlet spin-3 current as follows: ${.",
"}11\\endcsname W^{+(3),0}&=& \\frac{(k^2-4)(k+M+N)(3k+2M+2N)}{k M (k+2M)} \\, b_1 \\,\\Bigg \\lbrace - 3 \\Bigg [ t^{\\alpha }_{\\sigma \\bar{\\sigma }}\\,\\delta _{j \\bar{i}} \\,J^{(\\bar{\\sigma } j)} \\, J^{\\alpha } \\,J^{(\\sigma \\bar{i})}\\nonumber \\\\&-& t^{\\alpha }_{\\rho \\bar{\\nu }} \\,\\delta _{m \\bar{m}}\\, t^{\\alpha }_{\\sigma \\bar{\\sigma }}\\, \\delta _{j \\bar{i}}\\,\\psi ^{(\\bar{\\sigma } j)}\\, (( \\psi ^{(\\rho \\bar{m})}\\, J^{(\\bar{\\nu } m)})\\, J^{(\\sigma \\bar{i})})+ \\sqrt{\\frac{M+N}{M N}}\\,t^{\\alpha }_{\\sigma \\bar{\\sigma }}\\, \\delta _{j \\bar{i}}\\, \\psi ^{(\\bar{\\sigma } j)}\\,J^{\\alpha }\\, \\psi ^{(\\sigma \\bar{i})}\\, J^{u(1)}\\nonumber \\\\&-& (t^{\\beta } \\, t^{\\alpha })_{\\rho \\bar{\\sigma }}\\, \\delta _{j \\bar{i}}\\,J^{\\alpha }\\, J^{\\beta }\\, \\psi ^{(\\rho \\bar{i})}\\, \\psi ^{(\\bar{\\sigma } j)}+ t^a_{i \\bar{k}}\\, t^{\\alpha }_{\\sigma \\bar{\\sigma }} \\, J^{\\alpha }\\, J^a \\,\\psi ^{(\\sigma \\bar{k})}\\, \\psi ^{(\\bar{\\sigma } i)}\\nonumber \\\\&-& k \\, t^{\\alpha }_{\\sigma \\bar{\\sigma }} \\,\\delta _{j \\bar{i}}\\,\\psi ^{(\\bar{\\sigma } j)}\\, J^{\\alpha }\\, \\partial \\, \\psi ^{(\\sigma \\bar{i})}\\Bigg ] \\nonumber \\\\&-& \\frac{3}{(k+M)} \\, (k+N) \\, \\Bigg [\\frac{1}{2}\\, (-N \\, J^a \\, \\partial \\, J^a +k \\, J^a \\, \\partial \\, J_f^a)+ G^{-,a}\\, G^{+,a}+J^a \\, W^{+(2),a}\\nonumber \\\\& + & i \\, f^{a b c}\\, J^a \\, J^b J_f^c\\Bigg ]\\nonumber \\\\& - &\\frac{3}{k}\\, \\sqrt{\\frac{M+N}{M N}} \\, (k+N)\\Bigg [\\sqrt{\\frac{M+N}{M N}}\\, G^- \\, G^+ -\\frac{1}{2}\\,(k+M+N)\\, J^{u(1)}\\, \\partial \\, K \\nonumber \\\\& + & (k+M+N)\\, J^{u(1)}\\, T-\\frac{1}{2}\\, J^{u(1)}\\, (J^a+J_f^a)^2 \\Bigg ] \\nonumber \\\\&+& 3 \\, (k+N) \\, \\Bigg [\\delta _{\\rho \\bar{\\sigma }}\\, \\delta _{j \\bar{i}}\\, J^{(\\bar{\\sigma } j)}\\, \\partial \\,J^{(\\rho \\bar{i})} + \\sqrt{\\frac{M+N}{M N}}\\,\\delta _{\\rho \\bar{\\sigma }}\\, \\delta _{j \\bar{i}}\\,\\psi ^{(\\bar{\\sigma } j)}\\, \\partial \\,(\\psi ^{(\\rho \\bar{i})} \\, J^{u(1)})\\nonumber \\\\&+& t^{\\alpha }_{\\rho \\bar{\\sigma }}\\, \\delta _{j \\bar{i}}\\,\\psi ^{(\\bar{\\sigma } j)} \\,\\partial \\, (\\psi ^{(\\rho \\bar{i})} \\, J^{\\alpha })-t^a_{i \\bar{k}}\\, \\delta ^{i \\bar{i}}\\,\\delta _{\\rho \\bar{\\sigma }}\\, \\delta _{j \\bar{i}}\\,\\psi ^{(\\bar{\\sigma } j)} \\,\\partial \\, (\\psi ^{(\\rho \\bar{k})} \\, J^{a})\\nonumber \\\\&-& k \\,\\delta _{\\rho \\bar{\\sigma }}\\, \\delta _{j \\bar{i}}\\,\\psi ^{(\\bar{\\sigma } j)} \\, \\partial ^2 \\, \\psi ^{(\\rho \\bar{i})}\\Bigg ] \\nonumber \\\\& - &\\frac{(k^2-1)(2k+M+N)}{k(k+M)} \\, \\Bigg [-\\frac{1}{2}\\, (k+M+N)\\, \\partial ^2 \\, K +(k+M+N)\\,\\partial \\, T\\nonumber \\\\& - & \\frac{1}{2}\\, \\partial \\, (J^a +J_f^a)^2\\Bigg ] \\Bigg \\rbrace -\\frac{1}{4}\\, \\partial \\, W^{-(2),0}.$ We do not simplify this expression further because there are not too many common terms.",
"The second term can be further simplified by using the rearrangement lemma in [80].",
"Except the last term of (REF ), we can read off the five contributions denoted by each two brackets inside of curly bracket from (REF ).",
"Note that the first term of (REF ) originates from the $b_6$ term of (REF ) For convenience, we present, as in the footnote REF , some relevant OPEs $J^a(z) \\, G^{\\pm }(w) =\\pm \\, \\frac{1}{(z-w)}\\, G^{\\pm ,a}(w) +\\cdots $ , $J^{\\alpha }(z) G^+(w) =\\frac{1}{(z-w)}\\, t^{\\alpha }_{\\rho \\bar{\\sigma }}\\,\\delta _{j \\bar{k}}\\, \\psi ^{(\\bar{\\sigma } j)}\\, J^{(\\rho \\bar{k} )}(w) +\\cdots $ and $J^{\\alpha }(z) G^-(w) =-\\frac{1}{(z-w)}\\, t^{\\alpha }_{\\rho \\bar{\\sigma }}\\,\\delta _{j \\bar{k}}\\, \\psi ^{(\\rho \\bar{k})}\\, J^{(\\bar{\\sigma } j )}(w)+ \\cdots $ .",
"These OPEs will be used at various places of this paper.. We can understand this feature from the OPE between $G^{-}(z)$ and the first term of $G^{+(\\frac{5}{2}),0}(w)$ where the OPE between the former and the spin-$\\frac{1}{2}$ operator provides the spin-1 current transforming as $(\\overline{\\bf N},{\\bf M})$ .",
"In this sense, the singlet spin-3 current (REF ) is a generalization of the spin-3 current (REF ) in the ${\\cal N}=2$ supersymmetric coset model.",
"As mentioned before, some OPEs in Appendix $D$ (or in section 6) can be used.",
"Then we have the following currents of this multiplet we are considering in this section ${.",
"}12\\endcsname (W^{-(2),0}, G^{+(\\frac{5}{2}),0}, G^{-(\\frac{5}{2}),0}, W^{+(3),0}).$ They also satisfy the regular conditions as in (REF ).",
"It would be interesting to construct the lowest singlet spin-2 current from the beginning directly without using the information of the spin-3 current in the bosonic coset model and by collecting all the possible composite singlet spin-2 operators with arbitrary coefficients.",
"The nontrivial part in this direction is how to fix these coefficients which will depend on the three parameters $k, N$ and $M$ explicitly.",
"In next subsection, in order to check that this multiplet (REF ) is right ${\\cal N}=2$ multiplet, other OPEs with the generators of ${\\cal N}=2$ superconformal algebra are described Compared with the bosonic coset model description, the last component of (REF ) is an ${\\cal N}=2 $ supersymmetric version of previous spin-3 current (REF ) and the remaining three components are its superpartners.",
"In (REF ), the stress energy tensor (REF ) is the ${\\cal N}=2$ supersymmetric version of $T_{boson}$ of the footnote REF .",
"There are also three other components in (REF ).",
"It seems that there is no bosonic analog for the first ${\\cal N}=2$ multiplet of (REF ).."
],
[
"The OPEs with the currents of ${\\cal N}=2$ \nsuperconformal algebra",
"The OPEs between the spin-1 current of (REF ) and the currents of (REF ), by realizing the corresponding OPEs for fixed $N$ and $M$ , can be summarized by ${.",
"}13\\endcsname K(z) \\, W^{-(2),0}(w) &= &\\frac{1}{(z-w)^2}\\, \\Bigg [ \\frac{2 (k^2-1) (k^2-4)}{k^2 M (k+M) (k+2 M) } \\nonumber \\\\& \\times & (k+M+N)^2 (2 k+M+N) (3 k+2 M+2 N)\\,b_1 \\Bigg ] \\, K(w) + \\cdots ,\\nonumber \\\\K(z) \\, G^{\\pm (\\frac{5}{2}),0}(w) &= &\\mp \\frac{1}{(z-w)^2} \\,\\Bigg [ \\frac{ (k^2-1) (k^2-4) }{k^2 M (k+M) (k+2 M) } \\nonumber \\\\&\\times & (k+M+N) (2 k+M+N) (3 k+2 M+2 N) \\, b_1 \\Bigg ] \\,G^{\\pm }(w) \\nonumber \\\\& \\pm & \\frac{1}{(z-w)} \\,G^{\\pm (\\frac{5}{2}),0}(w) + \\cdots ,\\nonumber \\\\K(z) \\, W^{+(3),0}(w) &= & \\frac{1}{(z-w)^2}\\, \\Bigg [-\\, W^{-(2),0} \\nonumber \\\\& - & \\frac{ (k^2-1) (k^2-4) (k+M+N)^2 (2 k+M+N) (3 k+2 M+2 N)}{k^2 M (k+M) (k+2 M)} \\, b_1\\, \\nonumber \\\\& \\times & \\Big ( T-\\frac{1}{2(k+M+N)}\\, (J^a +J_f^a)^2 \\Big )\\Bigg ](w) + \\cdots .$ Note that in the last OPE of (REF ), the combination of stress energy tensor with $(J^a+J_f^a)^2$ term is exactly the same as the one in the OPE between the supersymmetry generators in Appendix $A$ .",
"Compared with the ones in Appendix $B$ , there are additional terms in the right hand sides of the OPEs.",
"Although we can check all the relevant terms in the right hand sides, the structure constants appearing in the composite operators of the right hand sides in (REF ) can be determined by focusing on the particular operators as before.",
"For example, in the last OPE of (REF ), we can focus on the $J^{\\alpha }\\, J^{\\alpha }$ term in the second order pole, which determines the coefficient of $W^{-(2),0}$ because we do not see this particular term in the remaining two terms.",
"After that by considering the $J^a \\, J_f^a$ term, which will appear in the first and the last terms of the second order pole, we can fix the coefficient of $(J^a+J_f^a)^2$ term because the coefficient of the first term is already known from the previous analysis.",
"Finally, from the $J^a \\, J^a$ term in the second order pole, which will appear in the second and the last terms, the coefficient appearing in the stress energy tensor (the second term) can be obtained because the coefficient of the last terms is known.",
"Next we summarize the following OPEs between the spin-$\\frac{3}{2}$ currents of the ${\\cal N}=2$ superconformal algebra and the currents of ${\\cal N}=2$ multiplet in this section ${.",
"}14\\endcsname && G^{\\pm }(z) \\, W^{-(2),0}(w) =\\frac{1}{(z-w)}\\, 2(k+M+N) \\, G^{\\pm (\\frac{5}{2}),0}(w) + \\cdots ,\\nonumber \\\\&& G^{\\pm }(z) \\, G^{\\pm (\\frac{5}{2}),0}(w) =0 + \\cdots ,\\nonumber \\\\&& G^{\\pm }(z) \\, G^{\\mp (\\frac{5}{2}),0}(w) =\\nonumber \\\\&& \\pm \\frac{1}{(z-w)^3} \\,\\frac{ (k^2-1) (k^2-4) (k+M+N)^2 (2 k+M+N)(3 k+2 M+2 N)}{k^2 M (k+M) (k+2 M)} \\, b_1\\,K(w)\\nonumber \\\\&& +\\frac{1}{(z-w)^2}\\, W^{-(2),0}(w)+\\frac{1}{(z-w)}\\, \\Bigg [ \\frac{1}{4}\\, \\partial \\, W^{-(2),0} \\mp W^{+(3),0}\\Bigg ](w) + \\cdots ,\\nonumber \\\\&& G^{\\pm }(z) \\, W^{+(3),0}(w) =\\nonumber \\\\&& \\mp \\frac{1}{(z-w)^3}\\frac{(k^2-1) (k^2-4) (k+M+N)^2 (2 k+M+N) (3 k+2 M+2 N)}{k^2 M (k+M) (k+2 M)} \\, b_1\\,G^{\\pm }(w)\\nonumber \\\\&& \\pm \\frac{1}{(z-w)^2} \\,\\frac{5}{2} (k+M+N) \\,\\, G^{\\pm (\\frac{5}{2}),0}(w)\\nonumber \\\\&& \\pm \\frac{1}{(z-w)} \\, \\frac{1}{5}\\,\\frac{5}{2} (k+M+N) \\, \\partial \\, G^{\\pm (\\frac{5}{2}),0}(w)+ \\cdots .$ As before, there are additional higher order terms in the right hand sides of these OPEs when we compare with the standard ${\\cal N}=2$ primary conditions described in Appendix $B$ .",
"In the third OPE of (REF ), we can also obtain the singlet spin-3 current by using the upper sign.",
"The lower sign was used in previous analysis.",
"In the last OPE of (REF ), we can fix the structure constant of the second order pole (where there are no descendant terms) by focusing on the first terms in (REF ) or (REF ).",
"In other words, in the computation of the left hand side of these OPEs, after we select the possible terms from $W^{+(3),0}$ (REF ) for fixed $(N,M)$ in the Thielemans package and add all the contributions from those terms for generic $(N,M)$ , we compute them manually and obtain $k,N$ and $M$ dependence explicitly.",
"It turns out that the first order pole is given by the descendant terms only.",
"Finally, we can compute the OPEs with stress energy tensor.",
"It turns out that ${.",
"}15\\endcsname && T(z) \\, W^{-(2),0}(w) =\\frac{1}{(z-w)^2}\\, 2\\, W^{-(2),0}(w) + \\frac{1}{(z-w)}\\, \\partial \\,W^{-(2),0}(w) + \\cdots ,\\nonumber \\\\&& T(z) \\, G^{\\pm (\\frac{5}{2}),0}(w) =\\frac{1}{(z-w)^2}\\, \\frac{5}{2} \\,G^{\\pm (\\frac{5}{2}),0}(w) + \\frac{1}{(z-w)}\\, \\partial \\,G^{\\pm (\\frac{5}{2}),0}(w) + \\cdots ,\\nonumber \\\\&& T(z) \\, W^{+(3),0}(w) = \\nonumber \\\\&& - \\frac{1}{(z-w)^4} \\,\\frac{3 (k^2-1) (k^2-4) (3 k+2 M+2 N) (k+M+N)^2 (2 k+M+N)}{2 k^2 M (k+M) (k+2 M)} \\, b_1 \\, K(w)\\nonumber \\\\&& +\\frac{1}{(z-w)^2}\\, 3 \\,W^{+(3),0}(w) + \\frac{1}{(z-w)}\\, \\partial \\,W^{+(3),0}(w)+\\cdots .$ The singlet spin-3 current $W^{+(3),0}$ in (REF ) is a quasi primary current under the stress energy tensor.",
"If we add the extra term to the $W^{+(3),0}$ , ${.",
"}16\\endcsname W^{+(3),0}+ \\frac{3}{4}\\, K\\, W^{-(2),0},$ then we can remove the fourth order pole in the last OPE of (REF ) with stress energy tensor because the OPE between $K(z)$ and $W^{-(2),0}(w)$ has only a second order pole from (REF ).",
"Then (REF ) becomes a primary field.",
"We can easily obtain the OPEs between the currents of ${\\cal N}=2$ superconformal algebra and the composite operator $ K\\, W^{-(2),0}$ and then we will obtain the corresponding OPEs containing (REF ) In the OPE between $K(z)$ and $ K\\, W^{-(2),0}(w)$ , the second order pole has $W^{-(2),0}$ and $K \\, K$ terms.",
"Similarly, the OPE between $G^{+}(z)$ and $ K\\, W^{-(2),0}(w)$ provides the second order pole with $G^{+(\\frac{5}{2}),0}$ and the first order pole together with $G^{+} \\, W^{-(2),0}$ and $K \\, G^{+(\\frac{5}{2}),0}$ .",
"Moreover, the OPE between $G^{-}(z)$ and $ K\\, W^{-(2),0}(w)$ leads to the second order pole with $G^{-(\\frac{5}{2}),0}$ and the first order pole together with $G^{-} \\, W^{-(2),0}$ and $K \\, G^{-(\\frac{5}{2}),0}$ .",
"We observe that there are several nonlinear terms..",
"Therefore, the singlet currents are given by (REF ), (REF ), (REF ) and (REF ) in the coset realization and they satisfy (REF ), (REF ), and (REF ) under the action of the generators of ${\\cal N}=2$ superconformal algebra.",
"The right hand sides of these OPEs contain the currents of ${\\cal N}=2$ multiplet, the composite spin-2 operator $(J^a+J_f^a)^2$ nonlinearly as well as the currents of ${\\cal N}=2$ superconformal algebra.",
"We will describe some features on the OPEs between this ${\\cal N}=2$ multiplet and itself in section 7."
],
[
" The nonsinglet\nmultiplet of spins $(2,\\frac{5}{2},\\frac{5}{2},3)$",
"We continue to construct the ${\\cal N}=2$ multiplet which has $SU(M)$ index $a$ .",
"From the experience of [12], we realize that the OPE between the spin-2 current $K^a(z)$ and itself $K^b(w)$ produces the spin-3 current $P^c(w)$ .",
"We can start with either this spin-2 current plus other terms as a candidate for the lowest component or this spin-3 current plus other terms as a candidate for the last component of the ${\\cal N}=2$ multiplet.",
"Let us take the latter."
],
[
"Construction of lowest component",
"Let us consider the spin-3 current found in bosonic coset model of [3] ${.",
"}1\\endcsname P^a & = & a_1 \\,t^{\\alpha }_{\\rho \\bar{\\sigma }} \\, t^a_{j\\bar{i}} \\, J^{\\alpha }J^{(\\rho \\bar{i})} \\, J^{(\\bar{\\sigma } j)}+ a_2 \\, J^{\\alpha } \\, J^{\\alpha } \\, J^a+a_3 \\, J^b \\, J^b \\, J^a+a_4 \\, J^a \\, J^{u(1)} \\, J^{u(1)}\\nonumber \\\\&+& a_5 \\, d^{abc} \\, \\delta _{\\rho \\bar{\\rho }} \\,t^b_{j\\bar{i}} \\, J^c (J^{(\\rho \\bar{i})} J^{(\\bar{\\rho } j)} +J^{(\\bar{\\rho } j)} J^{(\\rho \\bar{i})})+ a_7 \\, \\delta _{\\rho \\bar{\\sigma }} \\,t^a_{j\\bar{i}}\\, J^{u(1)} (J^{(\\rho \\bar{i})} J^{(\\bar{\\sigma } j)} +J^{(\\bar{\\sigma } j)} J^{(\\rho \\bar{i})})\\nonumber \\\\&+& a_8 \\, \\delta _{\\rho \\bar{\\sigma }} \\, \\delta _{j\\bar{i}}\\,J^a \\, (J^{(\\rho \\bar{i})} \\, J^{(\\bar{\\sigma } j)} +J^{(\\bar{\\sigma } j)} \\, J^{(\\rho \\bar{i})})+ a_9 \\, d^{abc} \\, J^b \\, J^c \\, J^{u(1)}+ a_{11} \\, i \\, f^{abc} \\, \\partial \\, J^b \\, J^c\\nonumber \\\\&+& a_{12} \\, \\delta _{\\rho \\bar{\\sigma }} \\,t^a_{j\\bar{i}} \\, \\partial \\, J^{(\\rho \\bar{i})} \\, J^{(\\bar{\\sigma } j)}+ a_{13} \\, \\delta _{\\rho \\bar{\\sigma }} \\,t^a_{j\\bar{i}} \\, \\partial \\, J^{(\\bar{\\sigma } j)} \\, J^{(\\rho \\bar{i})}(z)+a_{16} \\, \\partial ^2 \\, J^a\\nonumber \\\\&+& a_{17} \\, 6 \\, \\mbox{Tr} \\, (t^{a} \\,t^{\\left(b \\right.}",
"\\, t^c \\, t^{\\left.",
"d \\right)}) \\,J^b \\, J^c \\, J^d.$ Here the relative coefficients depend on $k, N$ and $M$ as follows: ${.",
"}2\\endcsname a_2 & = &\\frac{a_1}{k}, \\qquad a_3= \\frac{ N (k+2 N)}{k (k+M) (3 k+2 M)} \\, a_1,\\qquad a_4 = \\frac{ (k+2 N) (M+N)}{k^2 M} \\, a_1,\\nonumber \\\\a_5 & = & -\\frac{ (k+2 N)}{4 (k+M)} \\, a_1,\\qquad a_7 = \\frac{(k+2 N)}{2 k} \\, \\sqrt{\\frac{M+N}{M N}} \\, a_1,\\qquad a_8 = -\\frac{ (k+2 N)}{2 k M} \\, a_1,\\nonumber \\\\a_9 &=& - \\frac{(k+2 N)N}{2 k (k+M)} \\,\\sqrt{\\frac{ (M+N)}{M N}} \\, a_1,\\qquad a_{11} =\\frac{ (k^2-8) N (k+2 N)}{4 k (k+M) (3 k+2 M)} \\, a_1,\\nonumber \\\\a_{12} & = & -\\frac{1}{2} \\, (k+2 N) \\, a_1,\\qquad a_{13} =\\frac{1}{2} \\, (k+2 N) \\, a_1,\\\\a_{16} & = & -\\frac{ N (6 k^3+9 k^2 M+4 k M^2+12 M)(k+2 N)}{12 k (k+M) (3 k+2 M)} \\, a_1,\\qquad a_{17} = \\frac{ N (k+2 N)}{6 (k+M) (3 k+2 M)} a_1.\\nonumber $ First of all this nonsinglet spin-3 current satisfies ${.",
"}3\\endcsname P^{a}(z) \\, ( J^{u(1)} + \\sqrt{\\frac{M+N}{M N}}\\, J^{u(1)}_f )(w) =0 + \\cdots ,\\nonumber \\\\P^{a}(z) \\, (J^{\\alpha } + J_f^{\\alpha })(w) =0 +\\cdots .$ That is, the regular conditions in the bosonic coset model are valid in the ${\\cal N}=2$ supersymmetric coset model because the additional terms in (REF ) come from the complex fermions which commute with the spin-3 currents made by bosonic operators purely.",
"Let us describe the OPEs between the supersymmetry generators of the ${\\cal N}=2$ superconformal algebra and the nonsinglet spin-3 current of the bosonic coset model.",
"We use the OPEs between the spin-1 currents and the spin-3 current described in Appendix $C$ .",
"It turns out that with the help of (REF ) and (REF ) ${.",
"}4\\endcsname && G^{\\pm }(z) \\, P^a(w) = \\nonumber \\\\&&\\mp \\frac{1}{(z-w)^3} \\,\\frac{ (k^2-1) (k^2-4) (2 k+M+N) (3 k+2 M+2 N)}{2k^2 (k+M) (3k+2 M)} \\, b_1 \\, G^{\\pm ,a}(w)\\nonumber \\\\&& \\pm \\frac{1}{(z-w)^2} \\,\\hat{V}^{\\pm (\\frac{5}{2}),a}(w) + {\\cal O} (\\frac{1}{(z-w)}).$ The point here is that the second order pole of (REF ) possesses exactly the first, second and fifth terms of $V^{+(\\frac{5}{2}),a}$ (REF ) or $V^{-(\\frac{5}{2}),a}$ (REF ) with same relative coefficients.",
"Then this implies that we can express the above second order pole of (REF ) in terms of previous known operators.",
"We can absorb those three unwanted terms by using $V^{\\pm (\\frac{5}{2}),a}$ and recombine the remaining terms with the known operators appearing in the second order pole.",
"Then we obtain the following spin-$\\frac{5}{2}$ currents which are not new because they are previously known expressions ${.",
"}5\\endcsname \\hat{V}^{\\pm (\\frac{5}{2}),a} & = &\\frac{(k^2-4)(3k+2M+2N)}{2k(k+M)} \\, a_1 \\,\\Bigg [\\frac{1}{2}\\, V^{\\pm (\\frac{5}{2}),a}\\nonumber \\\\&\\pm & \\frac{(2k+M+N)}{k (3k+2M)} \\, J^a \\, G^{\\pm } +\\frac{(k^2-1) M (2k +M+N)}{3k(2k+M)(3k+2M)} \\, \\partial \\, G^{\\pm ,a}\\nonumber \\\\&+& \\frac{(2k+M+N)}{2(3k+2M)} \\, i \\, f^{a b c} \\, J^b\\, G^{\\pm ,c}\\mp \\frac{M(2k+M+N)}{2(2k+M)(3k+2M)} \\, d^{a b c} \\, J^b\\, G^{\\pm ,c}\\Bigg ],$ where $V^{\\pm (\\frac{5}{2}),a} $ are given by (REF ) and (REF ).",
"The last four terms in $\\hat{V}^{\\pm ,(\\frac{5}{2}),a}$ (REF ) are primary operators under the stress energy tensor $T$ (REF ) and they can be seen from (REF ) and (REF ).",
"As done in previous section, the second order pole in the OPEs of $G^{\\mp }(z) \\, \\hat{V}^{\\pm ,(\\frac{5}{2}),a}(w)$ leads to the following primary nonsinglet spin-2 current ${.",
"}6\\endcsname G^{\\mp }(z) \\, \\hat{V}^{\\pm ,(\\frac{5}{2}),a}(w)\\Bigg |_{\\frac{1}{(z-w)^2}}& = & \\pm W^{-(2),a }(w),$ where the lowest nonsinglet spin-2 current can be written as ${.",
"}7\\endcsname W^{-(2),a } & = & \\frac{(k^2-4)(3 k+2 M+2 N) }{ k (k+M) (3 k+2 M)}\\,a_1 \\,\\Bigg [\\frac{1}{4} \\, (2 k+M) (3 k+M+3 N)\\, K^a\\nonumber \\\\& + & \\frac{1}{k} \\, (k^2-1) (2 k+M+N)\\, \\Bigg ( W^{+(2),a} + \\frac{1}{2 } \\, i \\, f^{a b c} \\, J^b \\, J_f^c \\Bigg ) \\Bigg ].$ Here we have the relations $W^{+(2),a}$ (REF ) and $K^a$ (REF ) which are given by the coset fields.",
"Note that this is a new quantity which cannot be written in terms of the previous known currents obtained so far.",
"The spin-2 current $K^a$ obtained in the bosonic coset model is not an element of any ${\\cal N}=2$ multiplet.",
"When we compute the OPEs between the supersymmetry generators and the fourth or fifth terms of (REF ) in the computation of (REF ), we should use some property appearing in Appendix $D$ or section 6 together with the footnote REF .",
"Note that this (REF ) is a new current in the sense that although it contains both the known current $ W^{+(2),a}$ (REF ) and known operator $i \\, f^{a b c} \\, J^b \\, J_f^c$ which is primary, it also contains the spin-2 current $K^a$ (REF ) That is $K^a$ is one of the currents in the bosonic coset model but this is not an element of any ${\\cal N}=2$ multiplet in the ${\\cal N}=2$ supersymmetric coset model.. As soon as the spin-2 current $K^a$ appears in any OPEs we are considering, then we should replace it with other two currents $W^{\\pm (2),a}$ and other terms by using the relation (REF ).",
"Therefore, we have obtained the nonsinglet spin-2 current in (REF ) by acting the supersymmetry generators of the ${\\cal N}=2$ superconformal algebra on the spin-3 current successively.",
"In next subsections, we would like to construct its superpartners explicitly."
],
[
"Construction of second and third components",
"Because we have found the lowest component of the ${\\cal N}=2$ multiplet we consider in this section, it is straightforward to determine the other three components.",
"For given the lowest component in (REF ), we calculate the following OPEs ${.",
"}8\\endcsname G^{\\pm }(z) \\, W^{-(2),a }(w) & = &\\frac{1}{(z-w)}\\, G^{\\pm (\\frac{5}{2}),a}(w) +\\cdots .$ It is a good sign for the existence of the pole one term by realizing Appendix $B$ .",
"When we compute the above OPE for the first term of (REF ), we can use the previous relations in (REF ) and (REF ).",
"For the second term of (REF ), we can use the relations (REF ) and (REF ).",
"For the third term of (REF ), we obtain the composite operators.",
"After simplifying further, the first order pole provides the following primary nonsinglet spin-$\\frac{5}{2}$ currents ${.",
"}9\\endcsname G^{\\pm (\\frac{5}{2}),a} & = &\\frac{ (k^2-4)(3 k+2M+2N)}{ k (k+M) (3 k+2M)} \\, a_1 \\Bigg [\\frac{1}{4} \\, (2k+M)(3k+M+3N) \\, V^{\\pm (\\frac{5}{2}),a}\\nonumber \\\\&-&\\frac{1}{2k}\\, (k^2-1)(2k+M+N) \\, i\\, f^{a b c } \\, G^{\\pm ,b}\\, J_f^c+ \\frac{1}{2k}\\, (k^2-1)(2k+M+N) \\, i\\, f^{a b c } \\, J^{b}\\, G^{\\pm ,c}\\nonumber \\\\&-& \\frac{1}{6 k} \\, (k^2-1) \\, M \\, (2k+M+N) \\, \\partial G^{\\pm ,a} \\Bigg ].$ Due to the presence of the third term of (REF ), we have the second term of (REF ) which does not appear in (REF ).",
"We can check that the three terms of (REF ) except the first one are given by the known operators and are primary under the stress energy tensor.",
"Because the first term in (REF ) cannot be written in terms of the known currents, we have the new primary spin-$\\frac{5}{2}$ currents.",
"Note that in the OPE of (REF ), there are no higher order pole terms which can be checked for fixed $N$ and $M$ also."
],
[
"Construction of last component",
"As done in previous section, we construct the last component of this multiplet.",
"We can compute the following OPE ${.",
"}10\\endcsname G^{-}(z) \\,G^{+(\\frac{5}{2}),a}(w)\\Bigg |_{\\frac{1}{(z-w)^2}} = (2k+M+2N)\\,W^{-(2),a}(w).$ The structure constant of (REF ) can be determined by keeping track of the particular term in the nonsinglet spin-2 current $W^{-(2),a}(w)$ .",
"Let us introduce the final last component which is given by ${.",
"}11\\endcsname W^{+(3),a} & \\equiv &G^{-}(z) \\, G^{+(\\frac{5}{2}),a}(w)\\Bigg |_{\\frac{1}{(z-w)}}-\\frac{1}{4}\\, \\partial \\, \\Bigg (G^{-}(z) \\,G^{+(\\frac{5}{2}),a}(w)\\Bigg |_{\\frac{1}{(z-w)^2}}\\Bigg )\\nonumber \\\\&= &\\frac{ (k^2-4)(3 k+2M+2N)}{ k (k+M) (3 k+2M)} \\, a_1 \\Bigg \\lbrace \\frac{1}{4} \\, (2k+M)(3k+M+3N) \\, \\Bigg (\\nonumber \\\\& - &2(k+N) \\Bigg [ \\delta _{\\rho \\sigma } \\, t^a_{j \\bar{i}} \\, J^{(\\bar{\\sigma } j)}\\,\\partial \\, J^{(\\rho \\bar{i})}- k \\,\\delta _{\\rho \\bar{\\sigma }} \\, t^a_{j \\bar{i}} \\, \\psi ^{(\\bar{\\sigma } j)}\\, \\partial ^2 \\psi ^{(\\rho \\bar{i})} \\nonumber \\\\& + & \\sqrt{\\frac{M+N}{M N}}\\,\\delta _{\\rho \\sigma } \\, t^a_{j \\bar{i}} \\, \\psi ^{(\\bar{\\sigma } j)}\\,\\partial \\,(\\psi ^{(\\rho \\bar{i})}\\, J^{u(1)})+\\delta _{\\rho \\bar{\\sigma }} \\, t^a_{j \\bar{i}} \\,t^{\\alpha }_{\\sigma \\bar{\\rho }} \\,\\delta ^{\\rho \\bar{\\rho }}\\,\\psi ^{(\\bar{\\sigma } j)} \\, \\partial \\, (\\psi ^{(\\sigma \\bar{i})} \\, J^{\\alpha })\\nonumber \\\\&-& \\delta _{\\rho \\bar{\\sigma }} \\, (t^a \\, t^b)_{j \\bar{k}} \\, \\psi ^{(\\bar{\\sigma } j)}\\, \\partial \\, (\\psi ^{(\\rho \\bar{k})}\\, J^b)\\Bigg ]\\nonumber \\\\&-& \\frac{2}{k}\\, (k+N)\\, \\sqrt{\\frac{M+N}{M N}}\\,\\Bigg [ \\sqrt{\\frac{M+N}{M N}}\\, G^- \\, G^{+,a}-\\frac{N}{2}\\,J^{u(1)}\\, \\partial \\, J^a +\\frac{k}{2} \\, J^{u(1)}\\, \\partial \\, J_f^a\\nonumber \\\\&+& J^{u(1)}\\, W^{+(2),a} + i \\, f^{a b c}\\, J^{u(1)}\\, J^b \\, J_f^c\\Bigg ]\\nonumber \\\\&-& \\frac{2}{k M}\\, (k+M+N) \\, \\Bigg [ G^{-,a} \\, G^+ -\\frac{1}{2}\\, (k+M+N) \\, J^a \\, \\partial \\, K + (k+M+N)\\, J^a \\, T \\nonumber \\\\&-& \\frac{1}{2}\\, J^a \\, (J^b+J_f^b)^2 \\Bigg ]\\nonumber \\\\&-& \\Bigg [ i \\, f^{a b c} \\, G^{-,b}\\, G^{+,c} +\\frac{1}{2} (-N \\, J^b \\, \\partial \\, J^c + k \\, J^b \\, \\partial \\,J_f^c) \\, i\\, f^{a b c} + i \\, f^{a b c}\\, J^b \\, W^{+(2),c}\\nonumber \\\\&-& f^{a b c}\\, f^{c d e} \\, J^b \\, J^d \\,J^e_f\\Bigg ]\\nonumber \\\\&+& \\frac{(2k+M+2N)}{(2k+M)} \\Bigg [d^{a b c}\\, G^{-,b} \\, G^{+,c} -\\frac{N}{2} \\,d^{a b c} \\, J^b \\, \\partial \\, J^c + \\frac{k}{2}\\,d^{a b c} \\, J^b \\, \\partial \\, J_f^c + d^{a b c}\\, J^b\\,W^{+(2),c} \\nonumber \\\\&+& d^{a b c} \\, i\\, f^{c d e} \\, J^b \\, J^d \\, J^e_f\\Bigg ] \\nonumber \\\\&+& 2 \\Bigg [ t^{\\alpha }_{\\rho \\bar{\\sigma }} \\, t^a_{j \\bar{i}}\\,J^{(\\bar{\\sigma } j)} \\, J^{\\alpha }\\, J^{(\\rho \\bar{i})}- t^{\\alpha }_{\\rho \\bar{\\sigma }} \\, t^a_{j\\bar{i}}\\,t^{\\alpha }_{\\rho _1 \\bar{\\nu }} \\, \\delta _{m \\bar{m}} \\,\\psi ^{(\\bar{\\sigma } j)} ((\\psi ^{(\\rho _1 \\bar{m})} \\, J^{(\\bar{\\nu } m)})J^{(\\rho \\bar{i})})\\nonumber \\\\& - & k \\, t^{\\alpha }_{\\rho \\bar{\\sigma }} \\,t^a_{j \\bar{i}}\\,\\psi ^{(\\bar{\\sigma } j)} \\, J^{\\alpha }\\, \\partial \\, \\psi ^{(\\rho \\bar{i})}+ \\sqrt{\\frac{M+N}{M N}}\\, t^{\\alpha }_{\\rho \\bar{\\sigma }} \\,t^a_{j \\bar{i}}\\,\\psi ^{(\\bar{\\sigma } j)} \\, J^{\\alpha }\\,\\psi ^{(\\rho \\bar{i})}\\, J^{u(1)}\\ \\nonumber \\\\& + & (t^{\\beta } \\, t^{\\alpha })_{\\sigma \\bar{\\sigma }}\\, t^a_{j \\bar{i}}\\,\\psi ^{(\\bar{\\sigma } j)} \\, J^{\\alpha }\\,\\psi ^{(\\sigma \\bar{i})}\\, J^{\\beta }- t^{\\alpha }_{\\rho \\bar{\\sigma }}\\, (t^a \\, t^b)_{j \\bar{k}}\\,\\psi ^{(\\bar{\\sigma } j)} \\, J^{\\alpha }\\, \\psi ^{(\\rho \\bar{k})}\\, J^b\\Bigg ]\\nonumber \\\\&-& \\frac{2}{3}\\, \\frac{2(k^2-1)(2k+M+N)}{k(2k+M)} \\, \\Bigg [-\\frac{N}{2}\\, \\partial ^2 \\, J^a+\\frac{k}{2}\\, \\partial ^2 \\, J_f^a +\\partial \\, W^{+(2),a} \\nonumber \\\\& + & i \\, f^{a b c} \\, \\partial \\, (J^b \\, J_f^c)\\Bigg ]\\Bigg ) \\nonumber \\\\&-&\\frac{1}{2k}\\, (k^2-1)(2k+M+N) \\, i \\, f^{a b c}\\,\\Bigg ( -\\frac{N}{2} \\, \\partial \\, J^b \\, J_f^c +\\frac{k}{2}\\,\\partial \\, J^b_f \\, J_f^c + W^{+(2),b}\\, J_f^c \\nonumber \\\\&+& i \\, f^{b c d} (( J^c \\, J_f^d) \\, J_f^c)+G^{+,b} \\, G^{-,c}\\Bigg )\\nonumber \\\\& + & \\frac{1}{2k}\\, (k^2-1)(2k+M+N) \\, \\Bigg (i \\, f^{a b c} \\, G^{-,b}\\, G^{+,c} +\\frac{1}{2} (-N \\, J^b \\, \\partial \\, J^c + k \\, J^b \\, \\partial \\,J_f^c) \\, i\\, f^{a b c} \\nonumber \\\\& + & i \\, f^{a b c}\\, J^b \\, W^{+(2),c}- f^{a b c}\\, f^{c d e} \\, J^b \\, J^d \\,J^e_f\\Bigg )\\nonumber \\\\&-& \\frac{1}{6 k} \\, (k^2-1) \\, M \\, (2k+M+N) \\,\\Bigg (-\\frac{N}{2}\\, \\partial ^2 \\, J^a+\\frac{k}{2}\\, \\partial ^2 \\, J_f^a +\\partial \\, W^{+(2),a} \\nonumber \\\\& + & i \\, f^{a b c} \\, \\partial \\, (J^b \\, J_f^c)\\Bigg )\\Bigg \\rbrace \\nonumber \\\\& - & \\frac{1}{4}\\,(2k+M+2N)\\, \\partial \\,W^{-(2),a}.$ The first seventeen lines of (REF ) come from the spin-$\\frac{5}{2}$ current $V^{+(\\frac{5}{2}),a}$ consisting of seven independent terms in (REF ) and the remaining six lines come from the remaining three terms of (REF ).",
"Then we can read off the corresponding OPEs between the spin-$\\frac{3}{2}$ current and the spin-$\\frac{5}{2}$ current by looking at the seven pairs of brackets inside the curly bracket.",
"We observe that the $ t^{\\alpha }_{\\rho \\bar{\\sigma }} \\, t^a_{j \\bar{i}}\\,J^{(\\bar{\\sigma } j)} \\, J^{\\alpha }\\, J^{(\\rho \\bar{i})}$ term appearing inside of the sixth pair of bracket originates from the $a_1$ term of (REF ) and therefore the above nonsinglet spin-3 current is a generalization of (REF ) in the ${\\cal N}=2$ supersymmetric coset model we are describing in this paper.",
"As before, in the OPE between $G^{-}(z)$ and the sixth term of $V^{+(\\frac{5}{2}),a}(w)$ , the OPE between $G^-$ and the spin-$\\frac{1}{2}$ operator leads to the spin-1 current transforming as $(\\overline{\\bf N},{\\bf M})$ .",
"By combining this with the two other spin-2 currents, we obtain the first term inside the sixth pair of brackets.",
"Then we are left with the following currents ${.",
"}12\\endcsname (W^{-(2),a}, G^{+(\\frac{5}{2}),a}, G^{-(\\frac{5}{2}),a}, W^{+(3),a}).$ As described in the beginning of this section, it is an open problem to consider the nonsinglet spin-2 current $K^a$ first and obtain the nonsinglet spin-3 current after acting the spin-$\\frac{3}{2}$ currents of the ${\\cal N}=2$ superconformal algebra on $K^a$ successively.",
"Once we have obtained the nonsinglet spin-3 current, then it is straightforward to determine its superpartners step by step.",
"We can check whether this procedure reproduces the above ${\\cal N}=2$ multiplet in (REF ) or not.",
"In next subsections, further checks on the currents under the action of the currents of ${\\cal N}=2$ superconformal algebra are given."
],
[
"The OPEs with the currents of ${\\cal N}=2$ superconformal\nalgebra",
"Let us present the OPEs with the spin-1 current of the ${\\cal N}=2$ superconformal algebra ${.",
"}13\\endcsname && K(z) \\, W^{-(2),a}(w) =\\nonumber \\\\&& \\frac{1}{(z-w)^2}\\,\\frac{ (k^2-1) (k^2-4) (3 k+2 M+2 N) (2 k+M+N)}{k^2 (k+M) (3 k+2 M)}\\, a_1 \\, \\Bigg [N \\, J^a -k \\, J_f^a\\Bigg ](w) + \\cdots ,\\nonumber \\\\&& K(z) \\, G^{\\pm (\\frac{5}{2}),a}(w) = \\nonumber \\\\&& \\mp \\frac{1}{(z-w)^2}\\,\\frac{ (k^2-1) (k^2-4) (k+N) (2 k+M+N) (3 k+2 M+2 N)}{k^2 (k+M) (3 k+2 M)} \\, a_1 \\, G^{\\pm ,a}(w)\\nonumber \\\\&& \\pm \\frac{1}{(z-w)}\\,G^{\\pm (\\frac{5}{2}),a}(w) +\\cdots ,\\nonumber \\\\&& K(z) \\, W^{+(3),a}(w) = \\frac{1}{(z-w)^3} \\Bigg [\\nonumber \\\\&& \\frac{ (k^2-1) (k^2-4) M (2 k+M+N) (3 k+2 M+2 N)}{2 k^2 (k+M) (3 k+2 M)} \\, a_1\\, ( N \\, J^a - k \\, J_f^a )\\Bigg ](w)\\nonumber \\\\&& +\\frac{1}{(z-w)^2} \\Bigg [-\\frac{1}{2}\\,\\partial \\, (\\mbox{pole-3})-(2k+M+2N)\\, W^{-(2),a}\\nonumber \\\\&&-\\frac{ (k^2-1) (k^2-4) (k+N) (2 k+M+N) (3 k+2 M+2 N)}{k^2 (k+M) (3 k+2 M)} \\, a_1\\, i \\, f^{a b c } \\, J^b \\, J_f^c\\nonumber \\\\&&-\\frac{ (k^2-1) (k^2-4) (k+N) (2 k+M+N) (3 k+2 M+2 N)}{k^2 (k+M) (3 k+2 M)}\\, a_1\\,W^{+2(a)}\\Bigg ](w)+ \\cdots .$ As before, the easy way to determine the various structure constants appearing in the right hand sides of the above OPEs is to figure out how the various composite operators arise from which parts of each current.",
"For the last OPE of (REF ), it is rather nontrivial to keep track of the contributions from $ t^{\\alpha }_{\\rho \\bar{\\sigma }} \\, t^a_{j\\bar{i}}\\,t^{\\alpha }_{\\rho _1 \\bar{\\nu }} \\, \\delta _{m \\bar{m}} \\,\\psi ^{(\\bar{\\sigma } j)} ((\\psi ^{(\\rho _1 \\bar{m})} \\, J^{(\\bar{\\nu } m)})J^{(\\rho \\bar{i})})$ appearing in the sixth pair of brackets of the nonsinglet spin-3 current in (REF ) because of nontrivial normal ordering of the currents.",
"Moreover, the presence of $W^{+2(a)}$ (which belongs to other ${\\cal N}=2$ multiplet) in the second order pole can be understood from the fact that the above spin-3 current contains this current in various places of (REF ).",
"In doing this, we also need the OPEs between $J_f^a(z)$ and $G^{\\pm ,b}(w)$ which will be described in detail later.",
"Let us describe other OPEs as follows: ${.",
"}14\\endcsname && G^{+}(z) \\, W^{-(2),a}(w) = \\frac{1}{(z-w)}\\, G^{+(\\frac{5}{2}),a}(w)+ \\cdots ,\\nonumber \\\\&& G^{+}(z) \\, G^{+(\\frac{5}{2}),a}(w) = 0 + \\cdots ,\\nonumber \\\\&& G^{+}(z) \\, G^{-(\\frac{5}{2}),a}(w) =\\nonumber \\\\&&\\frac{1}{(z-w)^3} \\,\\frac{ (k^2-1) (k^2-4) (k+M+N) (2 k+M+N) (3 k+2 M+2 N)}{k^2 (k+M) (3 k+2 M)}\\,\\Bigg [ N \\, J^a - k \\, J_f^a \\Bigg ](w) \\nonumber \\\\&& +\\frac{1}{(z-w)^2} \\, (2k+M+2N)\\,W^{-(2),a}(w) +\\frac{1}{(z-w)} \\, \\Bigg [\\frac{1}{4} \\,\\partial \\, \\mbox{(pole-2)}\\nonumber \\\\&& + i\\, f^{a b c} \\, J^b \\, W^{-(2),c} + i \\,f^{a b c} \\, J_f^b \\, W^{-(2),c} +\\frac{1}{4} \\,(2 k+3 M+2 N)\\,\\partial \\, W^{-(2),a}- W^{+(3),a}\\Bigg ](w) \\nonumber \\\\&& + \\cdots ,\\nonumber \\\\&& G^{+}(z) \\, W^{+(3),a}(w) = \\nonumber \\\\&& -\\frac{1}{(z-w)^3}\\,\\frac{(k^2-1) (k^2-4) (k+N) (k+M+N) (2 k+M+N)(3 k+2 M+2 N)}{k^2 (k+M) (3 k+2 M)}\\, a_1\\,\\nonumber \\\\&& \\times \\, G^{+,a}(w) \\nonumber \\\\&& + \\frac{1}{(z-w)^2} \\,\\frac{1}{4} (10 k+7 M+10 N)\\, G^{+(\\frac{5}{2}),a}\\nonumber \\\\&& +\\frac{1}{(z-w)}\\, \\Bigg [\\frac{1}{5}\\, \\partial \\, \\mbox{(pole-2)}+i \\, f^{a b c}\\, J^b \\, G^{+(\\frac{5}{2}),c} +i \\, f^{a b c}\\, J_f^b \\, G^{+(\\frac{5}{2}),c}+ \\frac{2 M}{5}\\, \\partial \\,G^{+(\\frac{5}{2}),a}\\Bigg ](w) \\nonumber \\\\&& + \\cdots .$ In the third OPE of (REF ), it is rather nontrivial to observe the first order pole explicitly.",
"We should write down all the possible terms of spin-3 with free index $a$ .",
"It turns out that the free index appears in the $f$ symbols and the currents themselves.",
"This first order pole is another way to determine the nonsinglet spin-3 current.",
"For the last OPE of (REF ), we can fix the coefficient of the last term appearing in the first order pole by using the property of quasi primary condition.",
"That is, after calculating the third order pole in the OPEs between the stress energy tensor and the second and third terms of the first order pole and obtaining $-2M \\, G^{+(\\frac{5}{2}),a}(w)$ , we can fix the structure constant of $\\partial \\, G^{+(\\frac{5}{2}),a}(w)$ as $\\frac{2M}{5}$ because we can check that the third order pole in the OPE between the stress energy tensor and this derivative term leads to $5 \\, G^{+(\\frac{5}{2}),a}(w)$ .",
"Then the last three terms appearing in the first order pole of the last OPE satisfy the quasi primary condition as we expected In doing this, first we check the field contents for fixed $N$ and $M$ and then we need to figure out which parts of the currents of ${\\cal N}=2$ multiplet contribute to the right hand sides of the OPEs.",
"Next we calculate those contributions manually for generic $N$ and $M$ .. We continue to describe the next OPEs ${.",
"}15\\endcsname && G^{-}(z) \\, W^{-(2),a}(w) =\\frac{1}{(z-w)}\\, G^{-(\\frac{5}{2}),a}(w)+ \\cdots ,\\nonumber \\\\&& G^{-}(z) \\, G^{+(\\frac{5}{2}),a}(w) = \\nonumber \\\\&&-\\frac{1}{(z-w)^3} \\,\\frac{ (k^2-1) (k^2-4) (k+M+N) (2 k+M+N) (3 k+2 M+2 N)}{k^2 (k+M) (3 k+2 M)}\\,\\nonumber \\\\&& \\times \\, \\Bigg [ N \\, J^a - k \\, J_f^a \\Bigg ](w) \\nonumber \\\\&& +\\frac{1}{(z-w)^2} \\, (2k+M+2N)\\,W^{-(2),a}(w) +\\frac{1}{(z-w)} \\, \\Bigg [\\frac{1}{4} \\,\\partial \\, \\mbox{(pole-2)} + W^{+(3),a}\\Bigg ](w) \\nonumber \\\\&& + \\cdots ,\\nonumber \\\\&& G^{-}(z) \\, G^{-(\\frac{5}{2}),a}(w) = 0 + \\cdots ,\\nonumber \\\\&& G^{-}(z) \\, W^{+(3),a}(w) = \\nonumber \\\\&& \\frac{1}{(z-w)^3} \\,\\frac{(k^2-1) (k^2-4) (k+N) (k+M+N) (2 k+M+N)(3 k+2 M+2 N)}{k^2 (k+M) (3 k+2 M)}\\, a_1\\,\\nonumber \\\\&& \\times \\, G^{-,a}(w) \\nonumber \\\\&& - \\frac{1}{(z-w)^2} \\, \\frac{5}{4} (2k+M+2N) \\,G^{-(\\frac{5}{2}),a}(w)-\\frac{1}{(z-w)} \\,\\frac{1}{5} \\, \\partial \\, \\mbox{(pole-2)}+\\cdots .$ The second OPE of (REF ) provides the way we obtain the nonsinglet spin-3 current described before.",
"In the last OPE of (REF ) which will be rather complicated compared to other OPEs above, we should also calculate all the contributions from the nonsinglet spin-3 current.",
"Note that when we calculate the OPE between $G^-(z)$ and $i \\, f^{a b c}\\,G^{+,b} \\, G^{-,c}(w)$ which is one of the terms of $W^{+(3),a}(w)$ and appears in the sixth line from the below, we should obtain the first order pole of the OPE between $W^{+(2),b}(z)$ and $G^{-,c}(w)$ .",
"See also the second OPE of (REF ).",
"Then for example, the contribution of the fifth nonderivative term in $V^{-(\\frac{5}{2}),a}$ (REF ) with (REF ) arises in this particular pole.",
"The point here is, in general, the first order pole of the OPE between $W^{+(2),b}(z)$ and $G^{-,c}(w)$ produces a new quasi primary operator but with the contraction of indices $b$ and $c$ appearing in the $f$ symbol we can write down this first order pole in terms of the known currents.",
"Therefore, there will be no new (quasi)primary field in the second order pole of the last OPE in (REF ) This feature also appears in the previous last OPE of (REF ).. We will return to this issue around (REF ) in next section.",
"After collecting all the contributions on the fifth nontrivial term in $V^{-(\\frac{5}{2}),a}$ (REF ) in this calculation of the OPE between $G^-(z)$ and $W^{+(3),a}(w)$ , we are left with the structure constant in the second order pole as above.",
"Moreover, the first order pole has the descendant terms only which can be checked by fixed $N$ and $M$ .",
"Finally, the OPEs with the stress energy tensor are described by ${.",
"}16\\endcsname && T(z) \\, W^{-(2),a}(w) =\\frac{1}{(z-w)^2}\\, 2\\, W^{-(2),a}(w) + \\frac{1}{(z-w)}\\, \\partial \\,W^{-(2),a}(w) + \\cdots ,\\nonumber \\\\&& T(z) \\, G^{\\pm (\\frac{5}{2}),a}(w) =\\frac{1}{(z-w)^2}\\, \\frac{5}{2} \\,G^{\\pm (\\frac{5}{2}),a}(w) + \\frac{1}{(z-w)}\\, \\partial \\,G^{\\pm (\\frac{5}{2}),a}(w) + \\cdots ,\\nonumber \\\\&& T(z) \\, W^{+(3),a}(w) = \\nonumber \\\\&&\\frac{1}{(z-w)^4}\\, \\Bigg [\\nonumber \\\\&&-\\frac{3(k^2-1)(k^2-4)(k+M+N)(2k+M+N)(3k+2M+2N)}{2k^2(k+M)(3k+2M)}\\,a_1\\, ( N \\, J^a - k \\, J_f^a)\\Bigg ](w)\\nonumber \\\\&& +\\frac{1}{(z-w)^2}\\, 3 \\,W^{+(3),a}(w) + \\frac{1}{(z-w)}\\, \\partial \\,W^{+(3),a}(w)+ \\cdots .$ For the nonsinglet spin-3 current, there is a fourth order pole in (REF ).",
"This implies that the nonsinglet spin-3 current is quasi primary field.",
"By introducing the composite operator $ K \\, W^{-(2),a}$ , we can make the following nonsinglet spin-3 current ${.",
"}17\\endcsname W^{+(3),a} + \\frac{3}{2}\\, (k+M+N) \\, K \\, W^{-(2),a}$ be primary.",
"In doing this, the first OPE of (REF ) is used in (REF ) In the OPE between $K(z)$ and $ K\\, W^{-(2),a}(w)$ , the second order pole contains $W^{-(2),a}$ and $K \\, (N\\, J^a-k \\, J_f^a)$ terms.",
"The OPE between $G^{+}(z)$ and $ K\\, W^{-(2),a}(w)$ leads to the second order pole with $G^{+(\\frac{5}{2}),a}$ and the first order pole with $G^{+} \\, W^{-(2),a}$ and $K \\, G^{+(\\frac{5}{2}),a}$ .",
"Furthermore, the OPE between $G^{-}(z)$ and $ K\\, W^{-(2),a}(w)$ becomes the second order pole with $G^{-(\\frac{5}{2}),a}$ and the first order pole together with $G^{-} \\, W^{-(2),a}$ and $K \\, G^{-(\\frac{5}{2}),a}$ .",
"There exist several nonlinear terms..",
"Therefore, the nonsinglet currents are given by (REF ), (REF ) and (REF ) in the coset realization and they satisfy (REF ), (REF ), (REF ) and (REF ) under the action of the currents of ${\\cal N}=2$ superconformal algebra.",
"Also note that the right hand side of these OPEs contain the currents of ${\\cal N}=2$ multiplet in (REF ) as well as the currents of ${\\cal N}=2$ multiplet in (REF )."
],
[
"The OPEs between the nonsinglet\nmultiplet of spins $(1,\\frac{3}{2},\\frac{3}{2},2)$ \nand itself",
"The simplest nontrivial OPEs between the nonsinglet ${\\cal N}=2$ multiplets can be described in this section."
],
[
" The OPEs with lowest component ",
"From the explicit coset realizations in (REF ), (REF ) and (REF ), we determine the following OPEs (the OPEs between the lowest component and the four components in (REF )) ${.",
"}1\\endcsname J_f^a(z) \\, J_f^b(w) & = & \\frac{1}{(z-w)^2}\\, N \\, \\delta ^{a b} +\\frac{1}{(z-w)} \\, i \\, f^{a b c} \\, J_f^c(w) + \\cdots ,\\nonumber \\\\J_f^a(z) \\, G^{+,b}(w) & = &\\frac{1}{(z-w)} \\, \\Bigg [-\\frac{1}{M} \\, \\delta ^{a b} \\, G^{+}+ \\frac{1}{2} \\, ( i \\,f + d)^{a b c} \\, G^{+,c} \\Bigg ](w) + \\cdots ,\\nonumber \\\\J_f^a(z) \\, G^{-,b}(w) & = &\\frac{1}{(z-w)} \\, \\Bigg [\\frac{1}{M} \\, \\delta ^{a b} \\, G^{-}+ \\frac{1}{2} \\, ( i \\,f - d)^{a b c} \\, G^{-,c} \\Bigg ](w) + \\cdots ,\\nonumber \\\\J_f^a(z) \\, W^{+(2),b}(w) & = & \\frac{1}{(z-w)^2}\\,\\Bigg [ -\\frac{1}{M}\\,(k+M+N)\\,\\delta ^{a b}\\, K -\\frac{N}{2}\\, i\\, f^{a b c}\\, J^c \\nonumber \\\\& + & \\frac{1}{2} \\,d^{a b c} (N\\, J^c- k \\, J_f^c )\\Bigg ](w) \\nonumber \\\\& + &\\frac{1}{(z-w)}\\, \\Bigg [ \\frac{1}{2}\\, (i \\, f +d)^{c b d}\\, i \\, f^{a d e} \\,J^c \\, J_f^e \\nonumber \\\\& + & i \\, f^{a b c} \\Bigg (W^{+(2),c} -\\frac{1}{k M} \\, (k+M+N)\\, J^c K -\\frac{1}{2} \\, (i \\, f+ d)^{d c e} \\, J^d \\, J_f^e +\\frac{N}{2} \\, \\partial \\, J^c\\nonumber \\\\&+&\\frac{1}{2}\\,(K^c +\\frac{N}{(2k+M)} \\, d^{c d e} \\, J^d \\, J^e - N \\,\\partial \\, J^c)\\Bigg ) \\Bigg ](w) + \\cdots .$ In the last OPE of (REF ), we should replace the nonsinglet spin-2 current $K^c$ with the other spin-2 currents by using the nonsinglet spin-2 current (REF ) because the $K^c$ is not an element of ${\\cal N}=2$ multiplet we are considering in the ${\\cal N}=2$ supersymmetric coset model.",
"It is obvious to see $J_f^a(z) \\, K^{b}(w) = 0 +\\cdots $ from Appendix $A$ .",
"The presence of the spin-2 current $K^c$ can be understood from the fact that in doing this, the unwanted terms, consisting of the second, third, fourth, and sixth terms of $W^{+(2),c}$ (REF ) with $f$ symbol after simplifying, are replaced by the nonsinglet spin-2 current $W^{+(2),c}$ and other terms which contain the nonsinglet spin-2 current $K^c$ .",
"In the last line of (REF ), we replace the first term of $K^c$ (REF ) with other terms where $J^c \\, J^{u(1)}$ term is combined with $J^c \\, K$ term and to appear in the second line from the below.",
"From the last OPE of (REF ), we observe that the right hand side of the OPE contains also the lowest component of the ${\\cal N}=2$ multiplet in (REF )."
],
[
"The OPEs with the second component ",
"Now let us consider the following OPEs (the OPEs between the second component and the three components in (REF )) ${.",
"}2\\endcsname && G^{+,a}(z) \\, G^{+,b}(w) = 0 + \\cdots ,\\nonumber \\\\&& G^{+,a}(z) \\, G^{-,b}(w) = \\frac{1}{(z-w)^3} \\, k \\, N \\, \\delta ^{a b}\\nonumber \\\\&& +\\frac{1}{(z-w)^2} \\, \\Bigg [ \\frac{(k+M+N)}{M} \\,\\delta ^{a b} \\, K +\\frac{N}{2} \\, (i f - d)^{a b c} \\, J^c +\\frac{k}{2} \\,( i f + d)^{a b c} \\, J_f^c \\Bigg ](w) \\nonumber \\\\&& + \\frac{1}{(z-w)} \\, \\Bigg [ \\delta ^{a b} \\Bigg ( \\frac{1}{M} \\,(k+M+N) \\, T -\\frac{1}{2M} \\, J^a \\, J^a -\\frac{1}{2M}\\, J_f^a\\,J_f^a \\nonumber \\\\& & + \\frac{1}{2M} \\, (k+M+N) \\, \\partial \\, K \\Bigg )+ i \\, f^{a b c} \\Bigg ( -\\frac{1}{2} \\, W^{+(2),c} -\\frac{N}{4} \\, \\partial \\, J^c \\nonumber \\\\&& + \\frac{k}{4} \\, \\partial \\, J_f^c -\\frac{1}{2}\\,(K^c +\\frac{N}{(2k+M)} \\, d^{c d e} \\, J^d \\, J^e - N \\,\\partial \\, J^c)\\nonumber \\\\&& + \\frac{1}{4} \\, d^{c d e} \\, J^d \\, J_f^e -\\frac{1}{4} \\, i \\, f^{c d e} \\, J^d \\, J_f^e + \\frac{(k+M+N)}{k M}\\, J^c \\, K \\Bigg ) \\nonumber \\\\&& + d^{a b c} \\Bigg (-\\frac{1}{2} \\, W^{+(2),c} -\\frac{N}{4} \\, \\partial \\, J^c + \\frac{k}{4} \\, \\partial \\, J_f^c+ \\frac{1}{4} \\, d^{c d e} \\, J^d \\, J_f^e -\\frac{1}{4} \\, i \\, f^{c d e} \\, J^d \\, J_f^e \\Bigg )\\nonumber \\\\&& - \\frac{1}{M}\\, J^b \\, J_f^a -\\frac{1}{4} \\, (i f +d)^{e b c}\\, (i f +d)^{a c d} \\, J^e\\, J_f^d\\Bigg ](w) + \\cdots ,\\nonumber \\\\&& G^{+,a}(z) \\, W^{+(2),b}(w) = \\frac{1}{(z-w)^2}\\,\\Bigg [ \\frac{1}{2M}\\, (3k+2M+3N)\\, \\delta ^{a b}\\, G^+ +\\frac{1}{4} \\, (N-k)\\, i \\, f^{a b c}\\, G^{+,c}\\nonumber \\\\&& - \\frac{3}{4}\\, (k+N)\\, d^{a b c}\\, G^{+,c}\\Bigg ](w)+\\frac{1}{(z-w)}\\, \\Bigg [ \\frac{1}{3}\\, \\partial \\, \\mbox{(pole-2)} +R^{+(\\frac{5}{2}),a b} \\Bigg ](w) + \\cdots .$ In Appendix $D$ , we present the coset realization for the second OPE of (REF ).",
"The question is how we obtain the above result from Appendix $D$ ?",
"For the second order pole, we can check them without any difficulty.",
"For the first order pole, we can compute the $\\delta ^{a b}$ , $i \\, f^{a b c}$ and $d^{a b c}$ with the first order pole appearing in Appendix $D$ .",
"In other words, we have a free index $c$ for the last two cases while we do not have any free index for the first case.",
"From the expression of $\\delta ^{a b} \\, G^{+,a}(z) \\, G^{-,b}(w)\\Big |_{\\frac{1}{(z-w)}}$ , we can use the expression of stress energy tensor $T$ (REF ).",
"Then it turns out that we have four independent terms appearing in the first two lines of the first order pole having $\\delta ^{a b}$ in (REF ).",
"From the expression of $i \\, f^{a b c } \\, G^{+,a}(z) \\, G^{-,b}(w)\\Big |_{\\frac{1}{(z-w)}}$ , we can use the expressions $W^{+(2),c}$ (REF ) and $K^c$ (REF ) in order to remove the unwanted terms.",
"Then we have nine terms appearing in the second, third and fourth lines of the first order pole having $i \\, f^{a b c}$ as well as $J^c \\, J_f^{u(1)}$ term.",
"From the expression of $ d^{a b c } \\, G^{+,a}(z) \\, G^{-,b}(w)\\Big |_{\\frac{1}{(z-w)}}$ , the previous relation (REF ) can be used.",
"Then we have the five terms appearing in the fifth line of the first order pole having $ d^{a b c}$ as well as $J^c \\, J_f^{u(1)}$ term in (REF ).",
"Furthermore, the last term in Appendix $D$ having the factor $t^a \\, t^c \\, t^b$ can be simplified and will appear in the last line of the first order pole as well as $J^c \\, J_f^{u(1)}$ term.",
"However, this $J^c \\, J_f^{u(1)}$ -term can be cancelled by the above two contributions.",
"Therefore, we are left with the above final result where there is no $J^c \\, J_f^{u(1)}$ term For convenience, we present the OPEs $J^a(z) \\, G^{\\pm ,b}(w) =\\frac{1}{(z-w)}\\,\\Bigg [ \\pm \\frac{1}{M}\\, \\delta ^{a b}\\, G^{\\pm } \\mp \\frac{1}{2}( \\mp i \\, f +d)^{a b c}\\, G^{\\pm ,c}\\Bigg ](w) + \\cdots $ ..",
"In the last OPE of (REF ), there exist the following terms, after subtracting the descendant terms, which is a primary ${.",
"}3\\endcsname R^{+(\\frac{5}{2}),a b} & = &\\frac{(3 k+N)}{2 M}\\, \\delta ^{a b}\\, \\partial \\, G^++ \\frac{1}{4} (- k-N) \\, i \\, f^{a b c}\\, \\partial \\, G^{+,c} +\\frac{1}{4} (-3 k-N)\\, d^{a b c}\\, \\partial \\, G^{+,c}\\nonumber \\\\&- & \\sqrt{\\frac{M+N}{M N}}\\, i \\, f^{a b c}\\,J^{u(1)}\\, G^{+,c}-\\frac{(k M N+M+2 N)}{M N}\\, \\delta _{\\rho \\bar{\\sigma }}\\, \\delta ^{a b}\\, \\delta _{k \\bar{i}}\\, \\psi ^{(\\bar{\\sigma } k)}\\, \\partial \\,J^{(\\rho \\bar{i})}\\nonumber \\\\&- & \\frac{(k N+1)}{2 N}\\, i \\, f^{a b c}\\delta _{\\rho \\bar{\\sigma }}\\,\\, t^c_{k \\bar{i}}\\, \\psi ^{(\\bar{\\sigma } k)}\\, \\partial \\,J^{(\\rho \\bar{i})}-\\frac{(k M N+M+2 N)}{2 M N}\\,d^{a b c}\\delta _{\\rho \\bar{\\sigma }}\\,\\, t^c_{k \\bar{i}}\\, \\psi ^{(\\bar{\\sigma } k)}\\, \\partial \\,J^{(\\rho \\bar{i})}\\nonumber \\\\&+& i \\, f^{a b c}\\, t^c_{k \\bar{i}}\\,t^{\\alpha }_{\\rho \\bar{\\mu }}\\, J^{(\\rho \\bar{i})}\\,\\psi ^{(\\bar{\\mu } k)}\\, J^{\\alpha }+\\frac{1}{M}\\, J^a \\, G^{+,b}-\\frac{k}{M}\\,\\delta ^{a b}\\, \\delta _{\\tau \\bar{\\mu }} \\, \\delta _{j \\bar{k}}\\,J^{(\\tau \\bar{k})}\\, \\partial \\, \\psi ^{(\\bar{\\mu } j)}\\nonumber \\\\&+& \\frac{k}{2}\\,i \\, f^{a b c}\\,\\delta _{\\tau \\bar{\\mu }} \\, t^c_{j \\bar{k}}\\,J^{(\\tau \\bar{k})}\\, \\partial \\, \\psi ^{(\\bar{\\mu } j)}-\\frac{k}{2} \\, d^{a b c}\\,\\delta _{\\tau \\bar{\\mu }} \\, t^c_{j \\bar{k}}\\,J^{(\\tau \\bar{k})}\\, \\partial \\, \\psi ^{(\\bar{\\mu } j)}\\nonumber \\\\&- & \\frac{M+N}{M N}\\,t^a_{l \\bar{k}}\\, \\delta _{\\tau \\bar{\\mu }} ((\\psi ^{(\\bar{\\mu } l)}\\,J^{(\\tau \\bar{k})}) \\, J_f^b)-t^{\\alpha }_{\\rho \\bar{\\sigma }}\\, t^b_{j \\bar{i}}\\,t^a_{l \\bar{k}}\\, t^{\\alpha }_{\\mu \\bar{\\nu }}\\,\\psi ^{(\\bar{\\sigma } j)} \\, \\psi ^{(\\rho \\bar{i})}\\, \\psi ^{(\\bar{\\nu } l)} \\, J^{(\\mu \\bar{k})}\\nonumber \\\\&+& \\frac{1}{M^2}\\, \\delta ^{a b}\\, G^{+}\\, J_f^{u(1)}-\\frac{1}{2M}\\, i \\, f^{a b c}\\, G^{+,c}\\, J_f^{u(1)}-\\frac{1}{2M}\\, d^{a b c}\\, G^{+,c}\\, J_f^{u(1)}-\\frac{1}{M}\\, J^b \\, G^{+,a}\\nonumber \\\\&-& \\frac{1}{2M}\\, (i \\, f+ d)^{a b c}\\, G^{+}\\, J_f^c+ \\frac{1}{4}\\, (i \\, f+ d)^{a c e}(i \\, f +d)^{c b d}\\, G^{+,e}\\, J_f^d\\nonumber \\\\&+& \\frac{1}{2M} (i \\, f +d )^{c b a}\\, J^c \\, G^+-\\frac{1}{4}\\, (i \\, f+ d)^{d a e}(i \\, f +d)^{c b d}\\, J^c \\, G^{+,e}\\nonumber \\\\&-& \\frac{1}{2} \\, (i \\, f+ d)^{a c d} \\, (t^d \\, t^b)_{k \\bar{i}}\\,\\delta _{\\rho \\bar{\\sigma }}\\, J^{(\\rho \\bar{i})}\\, \\psi ^{(\\bar{\\sigma } k)} \\, J^{c}-\\frac{1}{3}\\, \\partial \\, \\Bigg (G^{+,a}(z) \\, W^{+(2),b}(w)\\Bigg |_{\\frac{1}{(z-w)^2}}\\Bigg ).$ The question is whether this can be written in terms of the known currents or not.",
"Let us look at $t^{\\alpha }_{\\rho \\bar{\\sigma }}\\, t^b_{j \\bar{i}}\\,t^a_{l \\bar{k}}\\, t^{\\alpha }_{\\mu \\bar{\\nu }}\\,\\psi ^{(\\bar{\\sigma } j)} \\, \\psi ^{(\\rho \\bar{i})}\\, \\psi ^{(\\bar{\\nu } l)} \\, J^{(\\mu \\bar{k})} $ term appearing in the sixth line of (REF ).",
"Although the index $\\alpha $ is summed but the product of the generators of $SU(M)$ has the free indices $a$ and $b$ with four different lower indices contracted with the coset fields.",
"As far as I know, there is no identity in the product of the two generators of $SU(M)$ with two different free indices There is a relation $t^c_{j \\bar{i}}\\, t^{d}_{l\\bar{m}}=\\delta ^{cd}\\, \\delta _{j \\bar{m}} \\, \\delta _{l \\bar{i}}+\\frac{M}{2}\\, (i \\, f+d)^{c d e}\\, t^e_{j \\bar{m}}\\, \\delta _{l \\bar{i}}-\\frac{1}{M} \\, \\delta ^{c d}\\, \\delta _{l \\bar{m}}\\, \\delta _{j \\bar{i}}-\\frac{1}{2} \\, (i \\, f+d)^{c d e}\\, t^e_{l \\bar{m}}\\, \\delta _{j\\bar{i}}-\\frac{1}{2} \\, (i \\, f+d)^{c d e}\\, t^e_{j \\bar{i}}\\, \\delta _{l \\bar{m}}-\\frac{M}{4} \\, (i \\, f +d)^{c a b} (i \\, f+d)^{a d e}\\,t^{b}_{j \\bar{i}}\\, t^e_{l \\bar{m}}$ .",
"From this we obtain $i \\, f^{a b c} \\, t^b_{j \\bar{i}}\\, t^a_{l \\bar{k}}=t^c_{j \\bar{k}}\\, \\delta _{l \\bar{i}}-t^c_{l \\bar{i}}\\, \\delta _{j \\bar{k}}$ and $d^{a b c}\\, t^b_{j \\bar{i}}\\, t^a_{l \\bar{k}} =t^c_{j \\bar{k}}\\, \\delta _{l \\bar{i}}-\\frac{2}{M}\\,t^c_{j \\bar{i}}\\, \\delta _{l \\bar{k}}-\\frac{2}{M}\\,t^c_{l \\bar{k}}\\, \\delta _{j \\bar{i}} + t^c_{l \\bar{i}}\\, \\delta _{j \\bar{k}}$ .",
"In other words, when we contract with $f$ or $d$ symbols, then the product of two generators leads to several single generators with appropriate indices.",
"We have seen these features in (REF ) and (REF ) also..",
"Therefore, we cannot express (REF ) in terms of the previous known currents obtained so far (although we have tried to rewrite it by using the various invariant tensors appearing in [12]).",
"According to (REF ), each ${\\cal N}=2$ current has a single $SU(M)$ index.",
"It would be interesting to study how we can obtain the superpartners of (REF ) explicitly if they exist.",
"It is natural to consider (REF ) as the second component of the ${\\cal N}=2$ multiplet and then the lowest, the third and last components are not known so far."
],
[
"The OPE with spin-2 current",
"Although the nonsinglet spin-2 current $K^b$ in the bosonic coset model does not belong to the component of the ${\\cal N}=2$ multiplet, it is very useful to calculate the OPE between the nonsinglet spin-$\\frac{3}{2}$ current $G^{+,a}$ of (REF ) and this spin-2 current.",
"It turns out, from Appendix $D$ where the coset field realizations are given, that we have ${.",
"}4\\endcsname && G^{+,a}(z) \\, K^{b}(w) = \\frac{1}{(z-w)^2} \\,\\frac{2(k^2-1)(2k+M+N)}{k(2k+M)} \\,\\Bigg [\\frac{1}{M} \\, \\delta ^{a b} \\, G^{+}- \\frac{1}{2} \\, ( i \\,f + d)^{a b c} \\, G^{+,c}\\Bigg ](w)\\nonumber \\\\&& +\\frac{1}{(z-w)}\\, \\Bigg [ -\\frac{1}{3}\\,\\frac{2(k^2-1)(2k+M+N)}{k(2k+M)} \\nonumber \\\\& & \\times \\Bigg (\\frac{1}{M} \\, \\delta ^{a b} \\, \\partial \\, G^{+}- \\frac{1}{2} \\, ( i \\,f + d)^{a b c} \\, \\partial \\, G^{+,c}\\Bigg )\\nonumber \\\\&& - \\frac{1}{2} \\, ( i f +d)^{a b c} \\,\\Bigg ( V^{+(\\frac{5}{2}),c}+\\frac{2}{k M} \\, (k+M+N) \\, J^c\\, G^+ \\nonumber \\\\& & +(i f -\\frac{(2k+M+2N)}{2k+M} \\, d)^{ c d e}\\,J^d \\, G^{+,e} + \\frac{2}{3}\\,\\frac{2(k^2-1)(2k+M+N)}{k(2k+M)} \\, \\partial \\, G^{+,c} \\Bigg )\\nonumber \\\\&& - \\frac{2}{k M}\\, (k+M+N)\\,J^b \\, G^{+,a} \\nonumber \\\\&& + \\frac{2k(k+2M)}{3(k^2-4)(k+M+N)(3k+2M+2N) b_1}\\,\\delta ^{a b} \\, G^{+(\\frac{5}{2}), 0} \\nonumber \\\\&& - (i f -\\frac{(2k+M+2N)}{2k+M} \\, d)^{b c d}\\,\\Bigg ( \\frac{1}{M} \\, \\delta ^{a d} \\, J^c\\, G^+ -\\frac{1}{2} \\, (i f +d)^{a d e} \\, J^c \\, G^{+,e} \\Bigg ) \\nonumber \\\\&& - \\frac{2}{3}\\,\\frac{2(k^2-1)(2k+M+N)}{k(2k+M)} \\,\\Bigg (\\frac{1}{M} \\, \\delta ^{a b} \\, \\partial \\, G^{+}- \\frac{1}{2} \\, ( i \\,f + d)^{a b c} \\, \\partial \\, G^{+,c}\\Bigg )\\Bigg ](w)+ \\cdots .$ The first two lines of the first order pole in (REF ) are the descendant terms associated with the currents in the second order pole.",
"Recall that the first, second and fifth terms in $V^{+(\\frac{5}{2}),c}$ (REF ) appear as unwanted terms in the sense that they cannot be written in terms of the known currents.",
"We can check that the second, fourth and last terms of the first order pole in the corresponding OPE in Appendix $D$ can be written in terms of $V^{+(\\frac{5}{2}),c}$ with $f$ and $d$ symbols plus other terms.",
"Moreover, the spin-$\\frac{5}{2}$ current $G^{+(\\frac{5}{2}),0}$ (REF ) contains similar first, third and fourth terms and they can be written in terms of $G^{+(\\frac{5}{2}),0}$ and other known operators.",
"Finally, we arrives at the above results of the first order pole in (REF ) by matching those unwanted terms with $V^{+(\\frac{5}{2}),c}$ , $G^{+(\\frac{5}{2}),0}$ and other known composite operators.",
"Of course, we can write down $V^{+(\\frac{5}{2}),c}$ in terms of $G^{+(\\frac{5}{2}),c}$ with other terms by using (REF ).",
"In summary, as before, from the second OPE in (REF ), we observe that the right hand side of the OPE contains the lowest component of the ${\\cal N}=2$ multiplet in (REF ) by noting the presence of the nonsinglet spin-2 current.",
"Moreover, from the last OPE, we observe that $i \\, f^{a b c}\\, t^c_{k \\bar{i}}\\,t^{\\alpha }_{\\rho \\bar{\\mu }}\\, J^{(\\rho \\bar{i})}\\,\\psi ^{(\\bar{\\mu } k)}\\, J^{\\alpha }$ term of (REF ) can be interpreted as $ i \\, f^{a b c}\\, V^{+(\\frac{5}{2}),c}$ (or $i \\, f^{a b c}\\, G^{+(\\frac{5}{2}),c}$ ) plus other terms.",
"This implies that the right hand side of the OPE contains the second component of the ${\\cal N}=2$ multiplet in (REF ) as we expect."
],
[
" The OPEs with the third component ",
"The remaining two OPEs are given by ${.",
"}5\\endcsname && G^{-,a}(z) \\, G^{-,b}(w) = 0 + \\cdots ,\\nonumber \\\\&& G^{-,a}(z) \\, W^{+(2),b}(w) = \\frac{1}{(z-w)^2}\\,\\Bigg [ \\frac{1}{2M}\\, (3k+3N)\\, \\delta ^{a b}\\, G^- -\\frac{1}{4} \\, (N-k)\\, i \\, f^{a b c}\\, G^{-,c}\\nonumber \\\\&& - \\frac{3}{4}\\, (k+N)\\, d^{a b c}\\, G^{-,c}\\Bigg ](w) +\\frac{1}{(z-w)}\\,\\Bigg [ \\frac{1}{3}\\, \\partial \\, \\mbox{(pole-2)}+R^{-(\\frac{5}{2}),a b}\\Bigg ](w) + \\cdots .$ The first order pole in the last OPE of (REF ), after subtracting the descendant terms, contains the following primary spin-$\\frac{5}{2}$ current with free indices $a$ and $b$ ${.",
"}6\\endcsname R^{-(\\frac{5}{2}),a b} &= &\\frac{(k+N)}{2 M}\\, \\delta ^{a b}\\, \\partial \\, G^{-}-\\sqrt{\\frac{M+N}{M N}}\\, i \\, f^{a b c}\\,J^{u(1)}\\, G^{-,c}\\nonumber \\\\&- & \\frac{(M+2 N)}{M^2 N}\\, \\delta ^{a b}\\, \\delta _{\\rho \\bar{\\sigma }}\\,\\delta _{i \\bar{k}}\\, \\psi ^{(\\sigma \\bar{k})}\\, \\partial \\, J^{(\\bar{\\rho } i)}\\nonumber \\\\&+& \\frac{(2 k N+1)}{2 N}\\, i \\,f^{a b c} \\, t^c_{k \\bar{i}}\\, \\delta _{\\rho \\bar{\\sigma }}\\,\\psi ^{(\\rho \\bar{i})}\\, \\partial \\, J^{(\\bar{\\sigma } k)}-\\frac{(M+2 N)}{2 M N}\\, d^{a b c} \\, t^c_{k \\bar{i}}\\, \\delta _{\\rho \\bar{\\sigma }}\\,\\psi ^{(\\rho \\bar{i})}\\, \\partial \\, J^{(\\bar{\\sigma } k)}\\nonumber \\\\&+& i \\, f^{a b c}\\, t^{c}_{i \\bar{k}}\\, t^{\\alpha }_{\\mu \\bar{\\rho }}\\,J^{(\\bar{\\rho } i)}\\, \\psi ^{(\\mu \\bar{k})}\\, J^{\\alpha }-\\frac{1}{M}\\, J^a \\, G^{-,b}+ \\frac{(M+N)}{M N}\\, t^a_{k \\bar{l}}\\, \\delta _{\\mu \\bar{\\tau }}\\, ((\\psi ^{(\\mu \\bar{l})} \\, J^{(\\bar{\\tau } k)})\\, J_f^b)\\nonumber \\\\&-& t^{\\alpha }_{\\sigma \\bar{\\rho }}\\, t^b_{i \\bar{j}}\\, t^a_{k \\bar{l}}\\, t^{\\alpha }_{\\nu \\bar{\\mu }} \\, \\psi ^{(\\sigma \\bar{j})}\\,\\psi ^{(\\bar{\\rho } i)}\\,\\psi ^{(\\nu \\bar{l})}\\, J^{(\\bar{\\mu } k)} -\\frac{1}{M^2}\\,\\delta ^{a b}\\, G^- \\, J_f^{u(1)}-\\frac{1}{2M}\\, i \\, f^{a b c}\\,G^{-,c}\\, J_f^{u(1)}\\nonumber \\\\& + &\\frac{1}{2M}\\,d^{a b c}\\,G^{-,c}\\, J_f^{u(1)}+\\frac{1}{M}\\, J^b \\, G^{-,a}+ \\frac{1}{2M} \\, (i \\, f +d)^{a b d}\\, G^- \\, J_f^d\\nonumber \\\\&-& \\frac{1}{4}\\, (i \\, f+d)^{c a e} (i\\, f +d)^{c b d}\\, G^{-,e}\\,J_f^d -\\frac{1}{2M}\\, (i \\, f+d)^{c b a} \\, J^c \\, G^-\\nonumber \\\\& - & \\frac{1}{4}\\, (i \\, f-d)^{d a e} (i\\, f +d)^{c b d}\\,J^c \\, G^{-,e}+\\frac{1}{4} (3 k+N)\\, i \\, f^{a b c}\\, \\partial \\, G^{-,c}\\nonumber \\\\&+&\\frac{1}{4} (-k-N)\\,d^{a b c}\\, \\partial \\, G^{-,c}+ \\frac{1}{2}\\, (i \\, f+ d)^{c a d}\\, t^b_{j\\bar{i}}\\,\\delta _{\\rho \\bar{\\sigma }}\\, ( t^b \\, t^d)_{j \\bar{l}}\\,J^{(\\bar{\\sigma } j)}\\, \\psi ^{(\\rho \\bar{l})}\\, J^c\\nonumber \\\\& - &\\frac{1}{3}\\, \\partial \\, \\Bigg (G^{-,a}(z) \\, W^{+(2),b}(w)\\Bigg |_{\\frac{1}{(z-w)^2}}\\Bigg ).$ Again, due to the existence of $t^{\\alpha }_{\\sigma \\bar{\\rho }}\\, t^b_{i \\bar{j}}\\, t^a_{k \\bar{l}}\\, t^{\\alpha }_{\\nu \\bar{\\mu }} \\, \\psi ^{(\\sigma \\bar{j})}\\,\\psi ^{(\\bar{\\rho } i)}\\,\\psi ^{(\\nu \\bar{l})}\\, J^{(\\bar{\\mu } k)}$ in (REF ), we cannot express this in terms of the known currents.",
"It is an open problem whether the above spin-$\\frac{5}{2}$ current (REF ) is a third component of any ${\\cal N}=2$ multiplet with free two indices.",
"It is natural to consider the OPE between the spin-$\\frac{3}{2}$ current $G^-$ of the ${\\cal N}=2$ superconformal algebra and the previous spin-$\\frac{5}{2}$ current with two indices $R^{+(\\frac{5}{2}), a b}$ (REF ) and obtain the possible lowest component of ${\\cal N}=2$ multiplet.",
"After that we need to check whether the OPE between the spin-$\\frac{3}{2}$ current $G^-$ and this lowest component (which will contain the quartic complex fermions with generators of $SU(N)$ and $SU(M)$ ) will give us the above spin-$\\frac{5}{2}$ current (REF ) plus other terms or not.",
"See also the footnote REF for the four product of complex fermions."
],
[
"The OPE with the spin-2 current",
"As before, the corresponding OPE can be described as ${.",
"}7\\endcsname && G^{-,a}(z) \\, K^{b}(w) =-\\frac{1}{(z-w)^2} \\,\\frac{2(k^2-1)(2k+M+N)}{k(2k+M)} \\,\\Bigg [\\frac{1}{M} \\, \\delta ^{ b a} \\, G^{-}- \\frac{1}{2} \\, ( i \\,f + d)^{b a c} \\, G^{-,c}\\Bigg ](w)\\nonumber \\\\&& +\\frac{1}{(z-w)}\\, \\Bigg [ -\\frac{1}{3}\\,\\frac{2(k^2-1)(2k+M+N)}{k(2k+M)} \\nonumber \\\\& & \\times \\Bigg (\\frac{1}{M} \\, \\delta ^{ b a} \\, \\partial \\, G^{-}- \\frac{1}{2} \\, ( i \\,f + d)^{ b a c} \\, \\partial \\, G^{-,c}\\Bigg )\\nonumber \\\\& & - \\frac{1}{2} \\, ( i f +d)^{ b a c} \\,\\Bigg ( V^{-(\\frac{5}{2}),c}-\\frac{2}{k M} \\, (k+M+N) \\, J^c\\, G^- \\nonumber \\\\& &+(i f +\\frac{(2k+M+2N)}{2k+M} \\, d)^{ c d e}\\,J^d \\, G^{-,e} + \\frac{2}{3}\\,\\frac{2(k^2-1)(2k+M+N)}{k(2k+M)} \\, \\partial \\, G^{-,c} \\Bigg )\\nonumber \\\\&& + \\frac{2}{k M}\\, (k+M+N)\\,J^b \\, G^{-,a} \\nonumber \\\\&& - \\frac{2k(k+2M)}{3(k^2-4)(k+M+N)(3k+2M+2N) b_1}\\,\\delta ^{a b} \\, G^{-(\\frac{5}{2}), 0} \\nonumber \\\\& & - (i f +\\frac{(2k+M+2N)}{2k+M} \\, d)^{b c d}\\,\\Bigg ( \\frac{1}{M} \\, \\delta ^{ d a } \\, J^c\\, G^- -\\frac{1}{2} \\, (i f +d)^{ d a e} \\, J^c \\, G^{-,e} \\Bigg ) \\nonumber \\\\&& - \\frac{2}{3}\\,\\frac{2(k^2-1)(2k+M+N)}{k(2k+M)} \\,\\Bigg (\\frac{1}{M} \\, \\delta ^{ b a } \\, \\partial \\, G^{-}- \\frac{1}{2} \\, ( i \\,f + d)^{ b a c} \\, \\partial \\, G^{-,c}\\Bigg )\\Bigg ](w)+ \\cdots .$ The second, fourth and last terms of the first order pole in the corresponding OPE in Appendix $D$ can be written in terms of $V^{-(\\frac{5}{2}),c}$ with $f$ and $d$ symbols plus other terms.",
"By using the first, second and fifth terms in $V^{-(\\frac{5}{2}),c}$ (REF ) appearing as unwanted terms, we can reexpress the corresponding terms in the first order pole in terms of $V^{-(\\frac{5}{2}),c}$ and other known operators.",
"From the first, third and fourth terms in the spin-$\\frac{5}{2}$ current $G^{-(\\frac{5}{2}),0}$ (REF ), the corresponding terms in the first order pole can be written in terms of $G^{-(\\frac{5}{2}),0}$ and other known operators by focusing on the $\\delta ^{a b}$ factor in the product of two $SU(M)$ generators with indices $a$ and $b$ .",
"Finally, we obtain the above results (REF ) where we can write down $V^{-(\\frac{5}{2}),c}$ in terms of $G^{-(\\frac{5}{2}),c}$ with other terms by using (REF )."
],
[
" The final OPE",
"Now we describe the final OPE between the spin-2 current and itself with Appendix $E$ ${.",
"}8\\endcsname && W^{+(2),a}(z) \\, W^{+(2),b}(w) = \\frac{1}{(z-w)^4}\\,\\frac{3}{2}\\, k \\, N \\, (k + N)\\, \\delta ^{a b} \\nonumber \\\\&& +\\frac{1}{(z-w)^3} \\, \\Bigg [\\frac{1}{2} N (2 k+N)\\, i \\, f^{a b c}\\, J^c +\\frac{1}{2} k (k+2 N)\\, i \\, f^{a b c}\\,J_f^c \\Bigg ](w) \\nonumber \\\\&& +\\frac{1}{(z-w)^2} \\, \\Bigg [ \\delta ^{a b} \\,2(k+N) \\Bigg ( \\frac{1}{M} \\, (k+M+N) \\, T -\\frac{1}{2M}\\,J^c \\, J^c-\\frac{1}{2M} \\, J_f^c \\, J_f^c \\Bigg )\\nonumber \\\\&& + d^{a b c}\\, \\Bigg ( -(k+N)\\, W^{+(2),c}+\\frac{1}{2}\\, (i \\, f +d)^{d c e}\\, J^d \\, J_f^e+\\frac{k}{2}\\, (1-k-N)\\, \\partial \\, J_f^c \\nonumber \\\\&& +\\frac{1}{2} N (k+N-1)\\, \\partial \\, J^c\\Bigg ) + i\\, f^{a b c}\\, \\Bigg ( \\frac{k}{4}\\, (k+2N)\\,\\partial \\, J_f^c \\nonumber \\\\&&+ \\frac{N(2 k^3 M+k^2 M^2+k^2 M N+2 k M+4 k N+M^2+M N)}{2 k M (2 k+M)}\\, \\partial \\, J^c\\Bigg )\\nonumber \\\\&& -\\frac{(k+2 M+N)}{M}\\,J^b \\, J^a_f-\\frac{(k+N)}{M}\\, J^a \\, J^b_f-\\frac{k}{4}\\,(i \\, f + d)^{d a e}\\, (i \\, f+d)^{c b d} \\, J^c \\,J_f^e\\nonumber \\\\&& -\\frac{k}{4}\\,(i \\, f + d)^{d b e}\\, (i \\, f+d)^{c a d} \\, J^c \\,J_f^e-\\frac{1}{4}\\, f^{d h f} \\, f^{e c g}\\,(i \\, f + d)^{h a c}\\, (i \\, f+d)^{d b e} \\, J^f \\,J_f^g\\nonumber \\\\&& -\\frac{k}{4}\\,i \\, f^{b c g}\\, (i \\, f+d)^{d a c} \\, J^d \\,J_f^g-\\frac{N}{4}\\,d^{e b c}\\, (i \\, f+d)^{e a d} \\, J^c \\,J_f^d\\nonumber \\\\&& -\\frac{k}{4}\\,i \\, f^{a c g}\\, (i \\, f+d)^{d b c} \\, J^d \\,J_f^g-\\frac{N}{4}\\,d^{e a c}\\, (i \\, f+d)^{e b d} \\, J^c \\,J_f^d\\Bigg ](w)\\nonumber \\\\&& +\\frac{1}{(z-w)} \\, \\Bigg [ \\frac{1}{2}\\, \\partial \\, \\mbox{(pole-2)}+ R^{+(3),a b}\\Bigg ](w) + \\cdots .$ In the second order pole of (REF ), we observe that there exists $d^{a b c}\\, W^{+(2),c}$ term.",
"Moreover, there are no quartic terms in the complex fermions by collecting all the contributions There is a relation $k \\, \\frac{M+N}{M N}\\, J_f^a \\, J_f^b + k \\, t^{\\alpha }_{\\rho \\bar{\\sigma }}\\, t^{\\alpha }_{\\mu \\bar{\\nu }}\\, t^a_{j \\bar{i}}\\, t^b_{k \\bar{l}}\\,(\\psi ^{(\\rho \\bar{i})}\\, \\psi ^{(\\bar{\\sigma } j)})(\\psi ^{(\\mu \\bar{l})}\\,\\psi ^{(\\bar{\\nu } k)})+\\frac{k}{M^2}\\,\\delta ^{a b}\\, J_f^{u(1)}\\, J_f^{u(1)}-\\frac{k}{M}\\, d^{a b c}\\, J_f^c \\,J_f^{u(1)}+ \\frac{k}{4}\\, (i \\, f+ d)^{d a c}\\,(i \\, f+ d)^{d b e}\\, J_f^c \\, J_f^e=0$ .",
"That is, the exact coefficients appearing in these five terms with the appropriate complex fermion terms lead to zero.",
"Note that the particular combination of this quartic fermions (with two indices $a$ and $b$ ) $ t^{\\alpha }_{\\rho \\bar{\\sigma }}\\, t^{\\alpha }_{\\mu \\bar{\\nu }}\\, t^a_{j \\bar{i}}\\, t^b_{k \\bar{l}}\\,\\psi ^{(\\rho \\bar{i})}\\, \\psi ^{(\\bar{\\sigma } j)}\\,\\psi ^{(\\mu \\bar{l})}\\,\\psi ^{(\\bar{\\nu } k)}$ is a candidate term of the lowest component of ${\\cal N}=2$ multiplet with two free indices..",
"In the first order pole of (REF ), after subtracting the descendant terms, we are left with $R^{+(3),a b}$ terms.",
"In particular, we observe that by checking the purely bosonic terms in $R^{+(3), a b}$ , there is a term ${.",
"}9\\endcsname \\frac{2k(k+M)(3k+2M)}{(k^2-4)(2k+M)(3k+2M+2N)(3k+M+3N)}\\,\\frac{1}{a_1}\\, i \\, f^{a b c}\\, W^{+(3),c}.$ This can be seen from the OPE between $K^a$ and $K^b$ and the $a_1$ term of $P^c$ appears in the inside of the sixth pair of bracket in (REF ).",
"The nonsinglet spin-3 current $ W^{+(3),c}$ in (REF ) is the last component of the ${\\cal N}=2$ multiplet in (REF ).",
"This is reasonable because the OPE between $K^a(z)$ and $K^b(w)$ leads to $i \\, f^{a b c}\\, P^c$ term at the first order pole in the bosonic coset model [12].",
"One of the reasons why we cannot write down this $R^{+(3), a b}$ in terms of the known currents is that there is a term from the OPE between the fourth term of $W^{+(2),a}$ (REF )and itself (which is presented in Appendix $E$ ) ${.",
"}10\\endcsname i \\, f^{\\alpha \\beta \\gamma } \\, t^{\\alpha }_{\\rho \\bar{\\sigma }}\\,t^{\\beta }_{\\sigma \\bar{\\nu }} \\, t^b_{j \\bar{i}}\\, t^a_{l \\bar{k}} \\,J^{\\gamma }\\, \\psi ^{(\\rho \\bar{i})} \\,\\psi ^{(\\bar{\\sigma } j)} \\,\\psi ^{(\\sigma \\bar{k})}\\,\\psi ^{(\\bar{\\nu } l)},$ which can be obtained by acting $G^-(z)$ on $t^{\\alpha }_{\\rho \\bar{\\sigma }}\\, t^b_{j \\bar{i}}\\,t^a_{l \\bar{k}}\\, t^{\\alpha }_{\\mu \\bar{\\nu }}\\,\\psi ^{(\\bar{\\sigma } j)} \\, \\psi ^{(\\rho \\bar{i})}\\, \\psi ^{(\\bar{\\nu } l)} \\, J^{(\\mu \\bar{k})}(w)$ .",
"This spin-$\\frac{5}{2}$ operator is the characteristic term for the $R^{+(\\frac{5}{2}), a b}$ in (REF ).",
"Then the OPE between $G^-$ and $R^{+(\\frac{5}{2}), a b}$ contains the above term (REF ).",
"Note from the footnote REF that the OPE between $G^{-}$ and $J^{\\gamma }$ leads to the composite field of spin-$\\frac{1}{2}$ and spin-1 operators transforming as $({\\bf N}, \\overline{\\bf M})$ and $(\\overline{\\bf N},{\\bf M})$ respectively.",
"Therefore we obtain (REF ) after contracting the indices properly.",
"In this way, there is a connection between $R^{+(\\frac{5}{2}), a b}$ and $R^{+(3), a b}$ via a supersymmetry generator.",
"It is an open problem to see whether there exists any ${\\cal N}=2$ multiplet of $(R^{-(2),a b}, R^{+(\\frac{5}{2}), a b},R^{-(\\frac{5}{2}),a b }, R^{+(3), a b})$ or not.",
"Probably, if we compute the OPEs between the currents of high spins, then the lowest current $R^{-(2), a b }$ can be seen Because we present all the OPEs between the nonsinglet spin-2 operators in Appendix $E$ , by reversing the orders of some OPEs we can write all the contributions on the first order poles in (REF ) and read off the nonsinglet spin-3 current $R^{+(3), a b}$ explicitly which covers a several pages..",
"In summary, we have the complete OPEs in (REF ), (REF ), (REF ) and (REF ).",
"The nontrivial part of these OPEs is that in the right hand side of these OPEs, the components of the third ${\\cal N}=2$ multiplet (REF ) as well as the components of the first ${\\cal N}=2$ multiplet (REF ) and the currents of ${\\cal N}=2$ superconformal algebra arise.",
"We have seen the new primary currents having two free indices of $SU(M)$ .",
"The various quasi primary operators in the first order pole of (REF ) will appear as in the bosonic case."
],
[
"Towards the OPEs between the singlet\nmultiplet of spins $(2,\\frac{5}{2},\\frac{5}{2},3)$ \nand itself",
"The simplest OPE between the singlet currents is given by the OPE between the singlet spin-2 current and itself.",
"We obtain the following OPE with the help of Appendix $F$ ${.",
"}1\\endcsname && W^{-(2),0}(z) \\, W^{-(2),0}(w) = \\nonumber \\\\&&\\frac{1}{(z-w)^4}\\,\\Bigg [ \\frac{ 6 (k^2-1) (k^2-4)^2 N (k+M+N)^3 (2 k+M+N) (3 k+2 M+2 N)^2}{k^3 M (k+M)^2 (k+2 M)^2}\\nonumber \\\\&& \\times (k^3+2 k^2 M+2 k^2 N+3 k M N+2 k+M+N) \\, b_1^2\\Bigg ] \\nonumber \\\\&& +\\frac{1}{(z-w)^2}\\, \\Bigg [\\frac{8 (k^2-1) (k^2-4)^2 (k+M+N)^4 (2 k+M+N)(3 k+2 M+2 N)^2}{k^4 M^2 (k+M)^2 (k+2 M)^2} \\nonumber \\\\&& \\times (k^3+2 k^2 M+2 k^2 N+3 k M N+2 k+M+N) \\, b_1^2 \\, \\Bigg (T-\\frac{1}{2(k+M+N)} \\, (J^a+J_f^a)^2\\Bigg )\\nonumber \\\\&&+\\frac{ 4 (k^2-4) (k+M+N)^2 (3 k+2 M+2 N)}{k^2 M (k+M) (k+2 M)}\\nonumber \\\\&& \\times (k^3-k^2 M-k^2 N-3 k M N-4 k-2 M-2 N) \\, b_1\\,W^{-(2),0} \\Bigg ](w)\\nonumber \\\\&& + \\frac{1}{(z-w)} \\, \\frac{1}{2} \\, \\partial \\, (\\mbox{pole-2})(w)+\\cdots .$ From the fourth order pole of (REF ), we can fix the normalization of the singlet spin-2 current by taking the unknown coefficient $b_1^2$ properly.",
"For example, one way to fix is such that the fourth order pole is given by $\\frac{c}{2}$ where $c$ is the central charge in (REF ).",
"The self coupling constant of the singlet spin-2 current which depends on the $k, N$ and $M$ explicitly appears in the second order pole of (REF ).",
"As we expected, the right sides of this OPE consist of 1) the central term, 2) stress energy tensor, 3) $(J^a+J_f^a)^2$ term, 4) the singlet spin-2 current and 5) their descendant terms It is useful to use the following relation $\\delta _{\\rho \\bar{\\sigma }}\\, \\delta _{j \\bar{i}}\\, J^{(\\rho \\bar{i})}\\,J^{(\\bar{\\sigma } j)} = (k+M+N)\\, T_{boson} + \\frac{M}{2(k+N)}\\, J^{\\alpha }\\,J^{\\alpha } +\\frac{(M+N)}{2k}\\, J^{u(1)} \\, J^{u(1)}-\\frac{1}{2}\\, J^a \\,J^a + \\frac{1}{2}\\, M N \\, \\sqrt{\\frac{M+N}{M N}}\\, \\partial \\,J^{u(1)}$ together with the footnote REF in order to calculate the OPE between the first term of $W^{-(2),0}$ and itself.",
"Then we know the OPE between $T_{boson}$ and itself and the four remaining composite spin-2 operators of above belong to $W^{-(2),0}$ and we can also compute the OPEs between them.. We can check that the singlet spin-$2, \\frac{5}{2}, 3$ currents, $W^{-(2),0}$ , $G^{\\pm (\\frac{5}{2}),0}$ and $W^{+(3),0}$ have the regular terms in the OPEs with $(J^a+J_f^a)$ .",
"Moreover, the OPE between the combination of $T-\\frac{1}{2(k+M+N)} \\, (J^a+J_f^a)^2$ and $(J^a+J_f^a)$ does not have any singular terms.",
"This implies that the singlet currents, ${.",
"}2\\endcsname T-\\frac{1}{2(k+M+N)} \\, (J^a+J_f^a)^2, \\qquad W^{-(2),0},\\qquad G^{\\pm (\\frac{5}{2}),0}, \\qquad W^{+(3),0},$ are decoupled from the spin-1 current $(J^a+J_f^a)$ .",
"It is easy to observe that the previous (quasi)primary conditions for these singlet spin-$2, \\frac{5}{2}, 3$ currents still hold under the modified stress energy tensor because the $(J^a+J_f^a)^2$ term does not produce any singular terms We can see that the OPEs between $(J^a+J_f^a)$ and $K, G^{\\pm }$ do not produce any singular terms.. After the OPEs between the currents of $(W^{-(s),0}, G^{+(s+\\frac{1}{2}),0},G^{-(s+\\frac{1}{2}),0},W^{+(s+1),0})$ in (REF ) where $s=2,3,4, \\cdots $ are obtained, we expect that the right hand sides of these OPEs will consist of the composite operators in terms of these singlet currents as well as the modified stress energy tensor (REF ) (and maybe $K$ and $G^{\\pm }$ also).",
"The new singlet (quasi)primary current will have the regular OPE with the spin-1 current $(J^a+J_f^a)$ .",
"Then the algebra from $(W^{-(s),0}, G^{+(s+\\frac{1}{2}),0},G^{-(s+\\frac{1}{2}),0},W^{+(s+1),0})$ will close by themselves up to the presence of the currents of ${\\cal N}=2$ superconformal algebra with modified stress energy tensor.",
"In other words, the right hand sides of these OPEs do not contain the $SU(M)$ nonsinglet fields appearing in (REF ).",
"On the other hands, the OPEs between the nonsinglet currents do contain the $SU(M)$ singlet fields."
],
[
" The\nextension of the large ${\\cal N}=4$ nonlinear superconformal algebra\nfor {{formula:91d36380-0602-4a04-96d1-ca80c557fa19}}",
"We describe in this section how we can realize the extension of the large ${\\cal N}=4$ nonlinear superconformal algebra for $M=2$ ."
],
[
"Four spin-$\\frac{3}{2}$ currents",
"The four supersymmetry generators of the large ${\\cal N}=4$ nonlinear superconformal algebra can be obtained by combining $G^{+,a}$ with $G^{-,a}$ and also combining $G^+$ and $G^-$ properly ${.",
"}1\\endcsname \\hat{G}_{11} & = & \\sqrt{\\frac{1}{k+N+2}} \\, \\Bigg [ G^{+,1}-i \\, G^{+,2}+ G^{-,1}-i \\, G^{-,2} \\Bigg ],\\nonumber \\\\\\hat{G}_{12} & = & -\\frac{1}{\\sqrt{2}} \\,\\sqrt{\\frac{1}{k+N+2}} \\, \\Bigg [ G^{+}+\\sqrt{2} \\, G^{+,3}- G^{-}+\\sqrt{2} \\, G^{-,3} \\Bigg ],\\nonumber \\\\\\hat{G}_{21} & = & \\frac{1}{\\sqrt{2}} \\,\\sqrt{\\frac{1}{k+N+2}} \\, \\Bigg [ G^{+}-\\sqrt{2} \\, G^{+,3}- G^{-}-\\sqrt{2} \\, G^{-,3} \\Bigg ],\\nonumber \\\\\\hat{G}_{22} & = & \\sqrt{\\frac{1}{k+N+2}} \\,\\Bigg [ G^{+,1}+i \\, G^{+,2}+ G^{-,1}+i \\, G^{-,2} \\Bigg ].$ These expressions (REF ) satisfy the fundamental relations Appendix $(D.1)$ of [73] or Appendix (REF ) in this paper We have the following six spin-1 currents $\\hat{A}_{\\pm }= \\frac{1}{\\sqrt{2}}\\, (-i\\, J^1 \\mp J^2)$ , $\\hat{A}_{3}= -\\frac{i}{\\sqrt{2}}\\, J^3$ and $\\hat{B}_{\\pm }= \\frac{1}{\\sqrt{2}}\\, (i\\, J_f^1 \\mp J_f^2)$ , $\\hat{B}_{3}= \\frac{i}{\\sqrt{2}}\\, J_f^3$ .",
"The defining relations for these $SU(2)$ currents are given in $(2.13)$ and $(2.16)$ of [73] where we can fix the normalizations of the spin-1 currents.",
"The defining OPEs between these spin-1 current and four spin-$\\frac{3}{2}$ currents are given in Appendix $C$ of [73].",
"Furthermore, the spin-$1, \\frac{3}{2}$ currents are primary under the stress energy tensor.",
"The central charge (REF ) gives us $c=\\frac{3(k+N+2 k N)}{(k+2+N)}$ for $M=2$ .. From the explicit expressions in $G^{\\pm }$ (REF ) and $G^{\\pm ,a}$ (REF ), the summation over the fundamental and antifundamental $SU(N)$ indices appears in the Kronecker delta with the coset fields.",
"On the other hands, the summation over the fundamental and antifundamental $SU(M=2)$ indices appears in the matrix elements of the $SU(M=2)$ generators with the coset fields.",
"For example, in the spin-$\\frac{3}{2}$ current $\\hat{G}_{11}$ , the index $j=1$ and the index $\\bar{i}=2$ survives in the first combination of $ (G^{+,1}-i \\, G^{+,2})$ and the index $j=2$ and the index $\\bar{i}=1$ survives in the second combination of $ (G^{-,1}-i \\, G^{-,2})$ .",
"This corresponds to the last $4N \\times 4N$ matrix in Appendix $(B.2)$ of [73] where the nonzero elements appear in the two $N \\times N $ identity matrices inside of this matrix.",
"We can analyze the other three currents similarly from $G^{\\pm }$ (REF ) and $G^{\\pm ,a}$ (REF )."
],
[
"Higher spin-$\\frac{3}{2}$ currents",
"Then the next spin-$\\frac{3}{2}$ currents can be determined by using the relations Appendix $(G.1)$ of [73] (or Appendix (REF ) in this paper) and it turns out that ${.",
"}2\\endcsname T_{+}^{(\\frac{3}{2})} & = &-\\frac{1}{\\sqrt{2}} \\,\\sqrt{\\frac{1}{k+N+2}} \\, \\Bigg [ G^{+}-\\sqrt{2} \\, G^{+,3} \\Bigg ],\\nonumber \\\\T_{-}^{(\\frac{3}{2})} & = &\\frac{1}{\\sqrt{2}} \\,\\sqrt{\\frac{1}{k+N+2}} \\, \\Bigg [ G^{-}-\\sqrt{2} \\, G^{-,3} \\Bigg ],\\nonumber \\\\U^{(\\frac{3}{2})} & = &-\\sqrt{\\frac{1}{k+N+2}} \\, \\Bigg [ G^{+,1}-i \\, G^{+,2} \\Bigg ],\\nonumber \\\\V^{(\\frac{3}{2})} & = &\\sqrt{\\frac{1}{k+N+2}} \\, \\Bigg [ G^{-,1}+ i \\, G^{-,2} \\Bigg ].$ Again, from (REF ) and (REF ), we observe that in the spin-$\\frac{3}{2}$ current $T_{+}^{(\\frac{3}{2})}$ , the index $j=1$ and the index $\\bar{i}=1$ survives and this corresponds to the nonzero elements in the $N \\times N $ identity matrix inside of the $4N \\times 4N$ matrix in the last expression of Appendix $B$ of [73].",
"Here, each of spin-$\\frac{3}{2}$ currents in (REF ) is reduced to further here.",
"Note that the eight rank two tensors which are $4N \\times 4N$ matrices and the nonzero elements appear in four $N \\times N$ identity matrices in [73].",
"In (REF ), we are left with these $4N \\times 4N$ matrices where only $N \\times N$ identity matrix arises.",
"Then the eight spin-$\\frac{3}{2}$ currents of the left hand sides in (REF ) and (REF ) for generic $k$ and $N$ can be realized by two $G^{\\pm }$ and six $G^{\\pm , a}$ (where $a=1,2,3$ ) in the right hand sides."
],
[
"Higher spin-2 currents",
"By using Appendix $(G.2)$ of [73] together with (REF ) and (REF ) (or Appendix (REF )), we can determine the following spin-2 currents for generic $k$ and $N$ as follows: ${.",
"}3\\endcsname U_{-}^{(2)} &=& -\\frac{1}{2(k+N+2)} \\Bigg [ -i \\, f^{3 1 c} \\, K^c + \\frac{(k+N+2)}{k}\\, i \\, f^{3 1 c} \\,J^c \\, K + 2 \\, J^1 \\, J_f^3 \\nonumber \\\\&- & f^{3 2 c} \\, K^c + \\frac{(k+N+2)}{k} \\, f^{3 2 c} \\,J^c \\, K - 2 i \\, J^2 \\, J_f^3 \\Bigg ],\\nonumber \\\\V_{+}^{(2)} &=& \\frac{1}{2(k+N+2)} \\Bigg [i \\, f^{3 1 c} \\, K^c - \\frac{(k+N+2)}{k}\\, i \\, f^{3 1 c} \\,J^c \\, K + 2 \\, J^1 \\, J_f^3 \\nonumber \\\\&- & f^{3 2 c} \\, K^c + \\frac{(k+N+2)}{k}\\, f^{3 2 c} \\,J^c \\, K + 2 i \\, J^2 \\, J_f^3 \\Bigg ],\\nonumber \\\\U_{+}^{(2)} &=& -\\frac{1}{2(k+N+2)} \\Bigg [2 \\, i\\, f^{1 3 c} \\, W^{+(2),c}+i \\, f^{1 3 c} \\, K^c - \\frac{1}{k}\\, i \\, f^{1 3 c} \\,J^c \\, K + 4 \\, J^3 \\, J_f^1\\nonumber \\\\& + &f^{a 3 c} \\, f^{1 c d} \\,J^a \\, J_f^d+2 \\, f^{2 3 c} \\, W^{+(2),c}\\nonumber \\\\&+ & f^{2 3 c} \\, K^c - \\frac{(k+N+2)}{k} \\, f^{2 3 c} \\,J^c \\, K - 4 i \\, J^3 \\, J_f^2- i\\, f^{a 3 c} \\, f^{2 c d} \\,J^a \\, J_f^d\\Bigg ],\\nonumber \\\\V_{-}^{(2)} &=& \\frac{1}{2(k+N+2)} \\Bigg [2\\, i\\, f^{3 1 c} \\, W^{+(2),c}+i \\, f^{3 1 c} \\, K^c - \\frac{(k+N+2)}{k}\\, i \\, f^{3 1 c} \\,J^c \\, K + 2 \\, J^1 \\, J_f^3 \\nonumber \\\\&- & 2 \\, f^{3 2 c} \\, W^{+(2),c}- f^{3 2 c} \\, K^c + \\frac{(k+N+2)}{k} \\, f^{3 2 c} \\,J^c \\, K + 2 i \\, J^2 \\, J_f^3\\Bigg ],\\nonumber \\\\T^{(2)} & = &-\\frac{1}{2(k+N+2)} \\Bigg [-\\frac{2 (k+N)(k+N+2)}{(k+N+2 k N)}\\, \\delta ^{3 3} \\, T+2 \\, J^3 \\, J_f^3+ \\sqrt{2} \\, i \\, f^{3 a b} \\, J^a \\, J_f^b\\nonumber \\\\& + &J^a \\, J^a + J_f^a \\, J_f^a+ 2 \\sqrt{2}\\, W^{+(2),3}\\Bigg ],\\nonumber \\\\W^{(2)} & = &\\frac{1}{2(k+N+2)} \\Bigg [-2 \\, J^1 \\, J_f^1- 2 \\, f^{2 1 c} \\, W^{+(2),c}- 2 \\, f^{2 1 c} \\, K^{c}\\nonumber \\\\& +& \\frac{2 (k+N+2)}{k} \\, f^{2 1 c} \\, J^c \\, K\\nonumber \\\\& + &2 i \\, J^1 \\, J_f^2 +i \\, f^{a 1 c} \\, f^{2 c d} \\,J^a \\, J_f^d+ 2 (k+N+2)\\, \\delta ^{2 2} \\, T- \\delta ^{2 2}\\, J^a \\, J^a\\nonumber \\\\& - &\\delta ^{2 2}\\,J_f^a \\, J_f^a+ f^{a 2 c} \\, f^{2 c d} \\,J^a \\, J_f^d\\Bigg ].$ The way we obtain (REF ) is that for fixed $(N,M)=(5,2)$ , we can determine the field contents explicitly.",
"After that, we read off the generic $(N,M)$ dependence manually.",
"If we compute the relevant OPEs manually from the beginning, then we will obtain different expressions.",
"However, we can check that eventually those become the above results (REF ) by using some identities in the structure constants.",
"In (REF ), although there exist the spin-2 current $K^a$ dependent terms, we can replace them by using (REF ) with $W^{\\pm (2),a}$ term and others.",
"Therefore, the six spin-2 currents for generic $k$ and $N$ can be realized by the six spin-2 currents $W^{\\pm (2),a}$ where $a=1,2,3$ and other composite operators By considering the second order poles of the last four OPEs in Appendix $(G.2)$ of [73], the spin-1 current of the lowest ${\\cal N}=4$ higher spin multiplet can be realized by $K$ in the ${\\cal N}=2$ superconformal algebra in (REF ).",
"Moreover, the stress energy tensor of the ${\\cal N}=4$ large nonlinear superconformal algebra is identified with the stress energy tensor in (REF ).",
"The stress energy tensor consists of purely bosonic part, purely fermionic part and boson fermionic part.",
"The purely fermionic part can be summarized by the first two terms in the second line of $T$ (REF ) and $J_f^a \\, J_f^a$ term after using the identity appearing in the last equation of Appendix $A$ .",
"The boson fermionic part is given by both $J^{\\alpha }\\, J^{\\alpha }_f$ term and $J^{u(1)}\\, J^{u(1)}_f$ term.",
"The purely bosonic part consists of the second, third and fourth terms of $T$ (REF ).",
"Then we can make the correspondences between the stress energy tensor in [73] and the one in (REF ) by focusing on these three parts.."
],
[
"Higher spin-$\\frac{5}{2},3$ currents",
"Now we can compute Appendix $(G.4)$ of [73] by using (REF ) and (REF ) and taking the first order poles (or Appendix (REF )), we will obtain the four spin-$\\frac{5}{2}$ currents.",
"Then we need to calculate the OPEs between $G^{\\pm ,a}(z)$ and $W^{\\pm (2),b}(w)$ as done in section 6.",
"In other words, we should compute the OPEs between the second ${\\cal N}=2$ multiplet and itself (and the OPEs between the first ${\\cal N}=2$ multiplet and the second ${\\cal N}=2$ multiplet).",
"For fixed $N$ with $M=2$ , we have checked that we can write down the spin-$\\frac{5}{2}$ currents explicitly.",
"See Appendix (REF ).",
"It turns out that they can be written in terms of $V^{\\pm (\\frac{5}{2}),a}$ (or $G^{\\pm (\\frac{5}{2}),a}$ ) plus other terms In the next lowest ${\\cal N}=4$ multiplet, there are also other four spin-$\\frac{5}{2}$ currents..",
"Eventually, the spin-3 current can be realized by the equation $(3.50)$ of [73].",
"That is, after determining the spin-$\\frac{5}{2}$ current $ W_{-}^{(\\frac{5}{2})}$ for generic $N$ and $M$ , we use ${.",
"}4\\endcsname W^{(3)} &=& \\hat{G}_{21}(z) \\, W_{-}^{(\\frac{5}{2})}(w) \\Bigg |_{\\frac{1}{(z-w)}}-\\Bigg [ \\frac{1}{4} \\, \\partial \\, \\Bigg (\\hat{G}_{21}(z) \\, W_{-}^{(\\frac{5}{2})}(w)\\Bigg |_{\\frac{1}{(z-w)^2}}\\Bigg )\\nonumber \\\\& + &\\frac{8 i N(3k+1)}{(N+k+2) ( 5N+4+6k N+5k )} \\,(-\\frac{i}{\\sqrt{2}})\\, \\Big ( T \\, J^3-\\frac{1}{2} \\partial ^2 \\, J^3 \\Big )\\nonumber \\\\& + &\\frac{8 i k (3N+1)}{(N+k+2) ( 5N+4+6 k N+5k )} \\,(\\frac{i}{\\sqrt{2}})\\, \\Big ( T \\, J_f^3-\\frac{1}{2} \\partial ^2 \\, J_f^3 \\Big )\\nonumber \\\\& + &\\frac{8(k-N)}{ (5N+4+6 k N+5k) }\\, \\Big ( T \\, K-\\frac{1}{2} \\partial ^2 K \\Big ) \\Bigg ].$ Of course, from the first line of (REF ), we should calculate the corresponding OPEs.",
"In the last three lines, the relations appearing in the footnotes REF and REF are used."
],
[
"For $M=3$",
"What happens for $M=3$ case?",
"In particular, how do the supersymmetry generators appear when we increase $M$ by 1?",
"The field contents for the spin-$\\frac{3}{2}$ currents $G^{\\pm , a}$ remain the same when we select the right choice for the index $a$ as $SU(2)$ .",
"The previous indices $1,2,3$ in (REF ) correspond to $1,4,7$ for $M=3$ case.",
"From the definition of $G^{\\pm ,a}$ (REF ), the summations over $j$ and $\\bar{i}$ contain the index 3.",
"However, any generators $t^a$ for indices $a=1,4,7$ do not have any rows and columns having an index 3.",
"Then we are left with the spin-$\\frac{3}{2}$ currents $G^{\\pm , a}$ with same field contents for $M=2$ case.",
"Now we look at the other spin-$\\frac{3}{2}$ currents in $G^{\\pm }$ (REF ).",
"In this case, there exist the summations over $j$ and $\\bar{i}$ having the index 3.",
"Therefore, as we increase the $M$ value, the field contents for these spin-$\\frac{3}{2}$ currents are increasing.",
"In other words, the field contents of spin-$\\frac{3}{2}$ currents for $M=3$ are the same as the one for $M=2$ and other terms.",
"This implies that for example, the OPE between $\\hat{G}_{11}(z)$ and $\\hat{G}_{12}$ , where the numerical factor 2 inside of the square root is replaced by 3 and the indices $1,2,3$ are replaced by $1,4,7$ , does not lead to the one of the known relation in the large ${\\cal N}=4$ nonlinear superconformal algebra due to the contribution from other terms we mentioned above.",
"Therefore, we do not have ${\\cal N}=4$ supersymmetry for $M=3$ and we expect that this holds for $M >2$ .",
"In summary, the currents of ${\\cal N}=4$ nonlinear superconformal algebra are given by (REF ) together with other currents appearing in the footnotes REF and REF .",
"Moreover, the higher spin-$\\frac{3}{2}, 2$ currents are described in (REF ) and (REF ).",
"The spin-$\\frac{5}{2}$ currents are in Appendix $H$ for fixed $N=5$ .",
"Finally, the spin-3 current is given by (REF ) implicitly."
],
[
" Conclusions and outlook",
"The highest component of the first ${\\cal N}=2$ multiplet is found in (REF ).",
"The second ${\\cal N}=2$ multiplet is found by (REF ), (REF ), (REF ) and (REF ).",
"The third ${\\cal N}=2$ multiplet is obtained from (REF ), (REF ) and (REF ).",
"Their OPEs between the currents of the ${\\cal N}=2$ superconformal algebra and these above currents are determined completely in the sections $3, 4$ and 5.",
"Moreover the OPEs between the first ${\\cal N}=2$ multiplet and itself are described with the observation of three kinds of new primary operators.",
"Similarly, we describe the OPE between the lowest singlet spin-2 current and itself in the second ${\\cal N}=2$ multiplet.",
"Finally, the extension of the large ${\\cal N}=4$ nonlinear superconformal algebra for $M=2$ is realized from the coset fields living in (REF ).",
"Therefore, we have obtained some currents (and their OPEs) in the supersymmetric coset model for generic $k, N$ and $M$ which correspond to the generators of the ${\\cal N}=2$ “rectangular” $W$ -algebra in the $AdS_3$ bulk theory according to a holography [30].",
"We have obtained the ${\\cal N}=2$ version of [12] and obtained the generalization of [73] (which is for $M=2$ case) to a generic $M$ .",
"In this paper, we have $f$ , $d$ symbols or Kronecker delta in the structure constants of the OPEs compared with the ones of [73] with various invariant tensors.",
"For generic $M > 2$ , we do not have to worry about the identities between these invariant tensors although the supersymmetry is given by ${\\cal N}=2$ .",
"We present the future directions related to the results of this paper as follows: $\\bullet $ More supersymmetric cases It is obvious that when we further restrict the level for $SU(N+M)_k$ to be the dual Coxeter number of $SU(N+M)$ , there arises an enhancement of the ${\\cal N}=3$ supersymmetry [83].",
"It would be interesting to see how we can construct the relevant coset fields explicitly by using the additional fermions.",
"Note that in the coset of (REF ), there is no $SU(M)_{k+N}$ factor in the denominator and this will lead to more general case because we should add this factor into the one of [83].",
"$\\bullet $ At the critical level with $M=2$ In this case (by fixing the $M$ value further in previous consideration), we should check whether the original ${\\cal N}=4$ supersymmetry is enhanced to more supersymmetric cases or not.",
"The point is how to obtain the additional supersymmetry generator by using the additional spin-$\\frac{1}{2}$ fermions without spoiling the present ${\\cal N}=4$ supersymmetry generators.",
"$\\bullet $ More OPEs between the ${\\cal N}=2$ multiplets So far we have considered the simplest OPEs in section 6.",
"However, there are other OPEs between the ${\\cal N}=2$ multiplets we should consider.",
"It is an open problem to study their OPEs systematically.",
"$\\bullet $ Are there any ${\\cal N}=2$ primary basis?",
"We have seen that the ${\\cal N}=2$ superconformal algebra we describe in this paper has the modified stress energy tensor.",
"It is an open problem to find whether there are any ${\\cal N}=2$ primary basis where the description of Appendix $B$ is satisfied or not.",
"What happens if we use the modified stress energy tensor rather than the stress energy tensor $T$ (REF )?",
"$\\bullet $ How do we construct the higher spin algebra in the $AdS_3$ gravity side via holography?",
"According to the result of [3] on the holography, it is natural to ask whether we can construct the corresponding ${\\cal N}=2$ higher spin algebra in the supersymmetric coset model in this paper.",
"For $M=2$ , there is a construction in [71].",
"For generic $M$ , it is an interesting problem to obtain them explicitly.",
"The additional $M$ dependence will play the role newly.",
"$\\bullet $ The orthogonal case Once we have found the higher spin currents in the bosonic orthogonal case, then its supersymmetric version can be obtained similarly.",
"If we use the embedding between the unitary and orthogonal groups, then maybe the $f$ and $d$ symbols in the unitary group can be used.",
"The previous works [84], [85], [86], [87] can be used.",
"See also [88].",
"$\\bullet $ Are there any systematical construction for the currents having more than two free indices?",
"We have seen the new primary currents in the section 6 having the two free indices.",
"Then the question is how we can determine the spin contents of this kind of primary currents.",
"As a first step, we should obtain the ${\\cal N}=2$ description for the above three spin-$\\frac{5}{2},3$ currents.",
"That is, can we obtain the possible lowest (and highest) components explicitly?",
"$\\bullet $ Extension of the large ${\\cal N}=4$ linear superconformal algebra We can think of the large ${\\cal N}=4$ linear superconformal algebra [89], [90], [91], [92], [93], [94], [95], [96], [97] by introducing the four spin-$\\frac{1}{2}$ operators and the spin-1 current with fixed $M=2$ .",
"For the spin-1 current we can think of some linear combination of the spin-1 operator $J^{u(1)}$ and $J^{u(1)}_f$ appearing in the spin-1 current of (REF ) as a candidate.",
"For the spin-$\\frac{1}{2}$ operators, it is not clear how to construct them by some contractions of the indices.",
"It would be interesting to study further.",
"Acknowledgments We thank Y. Hikida for the discussions.",
"This work was supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government(MSIT)(No.",
"2020R1F1A1066893)."
],
[
" The ${\\cal N}=2$ superconformal algebra with a modified\nstress energy tensor",
"In this Appendix, we describe the ${\\cal N}=2$ superconformal algebra discussed in section 2."
],
[
"The OPEs between the spin-$\\frac{1}{2}$ operators and\nthe spin-1 operators",
"By using (REF ) and (REF ) we obtain ${.",
"}1\\endcsname \\psi ^{(\\rho \\bar{i})}(z) J_f^{\\alpha }(w) &= & -\\frac{1}{(z-w)} \\,t^{\\alpha }_{\\sigma \\bar{\\tau }} \\, \\delta ^{\\rho \\bar{\\tau }}\\, \\psi ^{(\\sigma \\bar{i})}(w)+ \\cdots ,\\nonumber \\\\\\psi ^{(\\bar{\\sigma } j)}(z) J_f^{\\alpha }(w) &= & \\frac{1}{(z-w)} \\,t^{\\alpha }_{\\tau \\bar{\\nu }} \\, \\delta ^{\\tau \\bar{\\sigma }}\\, \\psi ^{(\\bar{\\nu } j)}(w)+ \\cdots ,\\nonumber \\\\\\psi ^{(\\rho \\bar{i})}(z) J_f^{a}(w) &= & \\frac{1}{(z-w)} \\,t^{a}_{ l \\bar{k}} \\, \\delta ^{l \\bar{i}}\\, \\psi ^{(\\rho \\bar{k})}(w)+ \\cdots ,\\nonumber \\\\\\psi ^{(\\bar{\\sigma } j)}(z) J_f^{a}(w) &= & -\\frac{1}{(z-w)} \\,t^{a}_{ l \\bar{k}} \\, \\delta ^{j \\bar{k}}\\, \\psi ^{(\\bar{\\sigma } l)}(w)+ \\cdots ,\\nonumber \\\\\\psi ^{(\\rho \\bar{i})}(z) J_f^{u(1)}(w) &= & - \\frac{1}{(z-w)} \\,\\psi ^{(\\rho \\bar{i})}(w) + \\cdots ,\\nonumber \\\\\\psi ^{(\\bar{\\sigma } j)}(z) J_f^{u(1)}(w) &= & \\frac{1}{(z-w)} \\,\\psi ^{(\\bar{\\sigma } j)}(w) + \\cdots .\\nonumber $"
],
[
" The OPEs between the spin-1 operators",
"Similarly, from the relations (REF ) and (REF ), the following OPEs satisfy ${.",
"}2\\endcsname J^{\\alpha }_f (z) \\, J^{\\beta }_f (w) & = & \\frac{1}{(z-w)^2} \\, M \\delta ^{\\alpha \\beta }+ \\frac{1}{(z-w)} \\, i \\, f^{\\alpha \\beta \\gamma } \\, J_f^{\\gamma }(w) +\\cdots ,\\nonumber \\\\J^{\\alpha }_f(z) \\, J^{a}_f(w) & = & 0 + \\cdots ,\\nonumber \\\\J_f^{\\alpha }(z) \\, J^{u(1)}_f(w) & = & 0 + \\cdots ,\\nonumber \\\\J^{a}_f (z) \\, J^{b}_f (w) & = & \\frac{1}{(z-w)^2} \\, N \\, \\delta ^{a b} +\\frac{1}{(z-w)}\\, i \\, f^{a b c} \\, J_f^c(w) +\\cdots ,\\nonumber \\\\J^{a}_f(z) \\, J^{u(1)}_f(w) & = & 0 + \\cdots ,\\nonumber \\\\J_f^{u(1)}(z) \\, J_f^{u(1)}(w) & = & \\frac{1}{(z-w)^2}\\, M N + \\cdots .\\nonumber $ We can read off various levels in these OPEs."
],
[
"The OPEs between the spin-$\\frac{1}{2}$ operators and\nthe currents of {{formula:46ecf4a7-24c7-41ba-961e-42f48502f4ce}} superconformal algebra ",
"By using (REF ), (REF ), (REF ) and (REF ) we determine the following OPEs ${.",
"}3\\endcsname \\psi ^{(\\rho \\bar{i})}(z) \\, K(w) &= &\\frac{1}{(z-w)} \\, \\frac{k}{(k+M+N)}\\, \\psi ^{(\\rho \\bar{i})}(w)+\\cdots ,\\nonumber \\\\\\psi ^{(\\bar{\\sigma } j)}(z) \\, K(w) &= &-\\frac{1}{(z-w)} \\, \\frac{k}{(k+M+N)}\\, \\psi ^{(\\bar{\\sigma } j)}(w)+\\cdots ,\\nonumber \\\\\\psi ^{(\\rho \\bar{i})}(z) \\, G^{+}(w) &= &\\frac{1}{(z-w)} \\, J^{(\\rho \\bar{i})}(w) + \\cdots ,\\nonumber \\\\\\psi ^{(\\bar{\\sigma } j)}(z) \\, G^{+}(w) &= & 0 + \\cdots ,\\nonumber \\\\\\psi ^{(\\rho \\bar{i})}(z) \\, G^{-}(w) &= &0 + \\cdots ,\\nonumber \\\\\\psi ^{(\\bar{\\sigma } j)}(z) \\, G^{-}(w) &= & \\frac{1}{(z-w)} \\,J^{(\\bar{\\sigma } j)}(w)+ \\cdots ,\\nonumber \\\\\\psi ^{(\\rho \\bar{i})}(z) \\, T(w) & = & \\frac{1}{(z-w)^2}\\, \\frac{(k M+M^2-1)}{2 M (k+M+N)} \\,\\psi ^{(\\rho \\bar{i})}(w) \\nonumber \\\\& + &\\frac{1}{(z-w)} \\, \\Bigg [\\frac{1}{(k+M+N)} \\,t^{\\alpha }_{\\sigma \\bar{\\tau }}\\,\\delta ^{\\rho \\bar{\\tau }}\\, J^{\\alpha } \\, \\psi ^{(\\sigma \\bar{i} )} \\nonumber \\\\& + & \\frac{1}{(k+M+N)}\\, t^{\\alpha }_{\\sigma \\bar{\\tau }}\\,\\delta ^{\\rho \\bar{\\tau }}\\, \\psi ^{(\\bar{\\sigma } i)} \\, J_f^{\\alpha }+\\frac{1}{(k+M+N)}\\, \\sqrt{\\frac{M+N}{M N}}\\,\\, J^{u(1)}\\, \\psi ^{(\\bar{\\sigma } j)} \\nonumber \\\\&+&\\frac{1}{(k+M+N)}\\, \\frac{(M+N)}{M N}\\, \\psi ^{(\\rho \\bar{i})} \\, J_f^{u(1)}- \\frac{(k M+M^2-1)}{2 M (k+M+N)} \\partial \\, \\psi ^{(\\rho \\bar{i})}\\Bigg ](w) + \\cdots ,\\nonumber \\\\\\psi ^{(\\bar{\\sigma } j)}(z) \\, T(w) & = & \\frac{1}{(z-w)^2}\\, \\frac{(k M+M^2-1)}{2 M (k+M+N)} \\,\\psi ^{(\\bar{\\sigma } j)}(w) \\nonumber \\\\& + &\\frac{1}{(z-w)} \\, \\Bigg [-\\frac{1}{(k+M+N)} \\,t^{\\alpha }_{\\tau \\bar{\\nu }}\\,\\delta ^{\\tau \\bar{\\sigma }}\\, J^{\\alpha } \\, \\psi ^{(\\bar{\\nu } j )} \\nonumber \\\\& - & \\frac{1}{(k+M+N)}\\, t^{\\alpha }_{\\tau \\bar{\\nu }}\\,\\delta ^{\\tau \\bar{\\sigma }}\\, \\psi ^{(\\bar{\\nu } j)} \\, J_f^{\\alpha }-\\frac{1}{(k+M+N)}\\, \\sqrt{\\frac{M+N}{M N}}\\,\\, J^{u(1)}\\, \\psi ^{(\\bar{\\sigma } j)} \\nonumber \\\\&-&\\frac{1}{(k+M+N)}\\, \\frac{(M+N)}{M N}\\, \\psi ^{(\\bar{\\sigma } j)} \\, J_f^{u(1)}- \\frac{(k M+M^2-1)}{2 M (k+M+N)} \\partial \\, \\psi ^{(\\bar{\\sigma } j)}\\Bigg ](w) + \\cdots .\\nonumber $"
],
[
"The OPEs between the spin-1 operators and\nthe currents of ${\\cal N}=2$ superconformal algebra",
"We have some OPEs between the spin-1 operators and the currents of ${\\cal N}=2$ superconformal algebra as follows: ${.",
"}4\\endcsname J_f^{u(1)}(z) \\, K(w) & = &- \\frac{1}{(z-w)^2}\\,\\frac{k M N}{(k+M+N)} +\\cdots ,\\nonumber \\\\J_f^{u(1)}(z) \\, G^{+}(w) & = &- \\frac{1}{(z-w)}\\,G^{+} + \\cdots ,\\nonumber \\\\J_f^{u(1)}(z) \\, G^{-}(w) & = &\\frac{1}{(z-w)}\\,G^{-} + \\cdots ,\\nonumber \\\\J_f^{u(1)}(z) \\, T(w) & = & -\\frac{1}{(z-w)^2} \\,K(w)+ \\cdots ,\\nonumber \\\\J^{u(1)}(z) \\, K(w) & = & \\frac{1}{(z-w)^2} \\,\\frac{k M N}{(k+M+N)} \\, \\sqrt{\\frac{M+N}{M N}} + \\cdots ,\\nonumber \\\\J^{u(1)}(z) \\, G^{+}(w) & = &\\frac{1}{(z-w)}\\, \\sqrt{\\frac{M+N}{M N} }\\,G^{+} + \\cdots ,\\nonumber \\\\J^{u(1)}(z) \\, G^{-}(w) & = &-\\frac{1}{(z-w)}\\, \\sqrt{\\frac{M+N}{M N} }\\,G^{-} + \\cdots ,\\nonumber \\\\J^{u(1)}(z) \\, T(w) & = & \\frac{1}{(z-w)^2} \\,\\sqrt{\\frac{M+N}{M N} }\\, K(w)+ \\cdots ,\\nonumber \\\\J^{(\\rho \\bar{i})}(z) \\, K(w) & = & -\\frac{1}{(z-w)} \\, \\frac{(M+N)}{(k+M+N)}\\,J^{(\\rho \\bar{i})}(w) +\\cdots ,\\nonumber \\\\J^{(\\rho \\bar{i})}(z) \\, G^{+}(w) & = & 0 + \\cdots ,\\nonumber \\\\J^{(\\rho \\bar{i})}(z) \\, G^{-}(w) & = & \\frac{1}{(z-w)^2} \\,k \\, \\psi ^{(\\rho \\bar{i})}(w) \\nonumber \\\\& +& \\frac{1}{(z-w)} \\,\\Bigg [ \\sqrt{\\frac{M+N}{M N}} \\, \\psi ^{(\\rho \\bar{i})}\\, J^{u(1)}+t^{\\alpha }_{\\sigma \\bar{\\rho }}\\,\\delta ^{\\rho \\bar{\\rho }}\\, \\psi ^{(\\sigma \\bar{i})} \\, J^{\\alpha }- t^{a}_{i \\bar{k}}\\,\\delta ^{i \\bar{i}}\\, \\psi ^{(\\rho \\bar{k})} \\, J^{a} \\Bigg ](w) \\nonumber \\\\& + & \\cdots ,\\nonumber \\\\J^{(\\rho \\bar{i})}(z) \\, T(w) & = & \\frac{1}{(z-w)^2}\\, \\Bigg [ \\frac{2 k M+2 M^2+M N-1}{2 M (k+M+N)} \\Bigg ] \\,J^{(\\rho \\bar{i})}(w) \\nonumber \\\\& + &\\frac{1}{(z-w)} \\, \\Bigg [\\frac{1}{(k+M+N)}\\, \\sqrt{\\frac{M+N}{M N}}\\,J^{(\\rho \\bar{i})} \\, J^{u(1)} \\nonumber \\\\& + &\\frac{(M+N)}{M N (k+M+N)} \\, J^{(\\rho \\bar{i})} \\, J_f^{u(1)}+ \\frac{1}{(k+M+N)} \\, t^{\\alpha }_{\\sigma \\bar{\\rho }}\\,\\delta ^{\\rho \\bar{\\rho }}\\, J^{\\alpha } \\, J^{(\\sigma \\bar{i})} \\nonumber \\\\&+&\\frac{1}{(k+M+N)}\\, t^{\\alpha }_{\\sigma \\bar{\\rho }}\\,\\delta ^{\\rho \\bar{\\rho }}\\, J^{(\\sigma \\bar{i})} \\, J_f^{\\alpha }-\\frac{(M N^2-2 M-N)}{2 M N (k+M+N)} \\, \\partial \\, J^{(\\rho \\bar{i})}\\Bigg ](w) \\nonumber \\\\& + & \\cdots ,\\nonumber \\\\J^{(\\bar{\\sigma } j)}(z) \\, K(w) & = & \\frac{1}{(z-w)} \\,\\frac{(M+N)}{(k+M+N)}\\,J^{(\\bar{\\sigma } j)}(w) +\\cdots ,\\nonumber \\\\J^{(\\bar{\\sigma } j)}(z) \\, G^{+}(w) & = &\\frac{1}{(z-w)^2} \\,k \\, \\psi ^{(\\bar{\\sigma } j)}(w) \\nonumber \\\\& +& \\frac{1}{(z-w)} \\,\\Bigg [ -\\sqrt{\\frac{M+N}{M N}} \\, \\psi ^{(\\bar{\\sigma } j)}\\, J^{u(1)}-t^{\\alpha }_{\\rho \\bar{\\tau }}\\,\\delta ^{\\rho \\bar{\\sigma }}\\, \\psi ^{(\\bar{\\tau } j)} \\, J^{\\alpha }+ t^{a}_{k \\bar{k}}\\,\\delta ^{j \\bar{k}}\\, \\psi ^{(\\bar{\\sigma } k)} \\, J^{a} \\Bigg ](w) \\nonumber \\\\& + & \\cdots ,\\nonumber \\\\J^{(\\bar{\\sigma } j)}(z) \\, G^{-}(w) & = & 0 +\\cdots ,\\nonumber \\\\J^{(\\bar{\\sigma } j)}(z) \\, T(w) & = & \\frac{1}{(z-w)^2}\\, \\Bigg [ \\frac{2 k M+2 M^2+M N-1}{2 M (k+M+N)} \\Bigg ] \\,J^{(\\bar{\\sigma } j)}(w) \\nonumber \\\\& + &\\frac{1}{(z-w)} \\, \\Bigg [-\\frac{1}{(k+M+N)}\\, \\sqrt{\\frac{M+N}{M N}}\\,J^{(\\bar{\\sigma } j)} \\, J^{u(1)} \\nonumber \\\\& - &\\frac{(M+N)}{M N (k+M+N)} \\, J^{(\\bar{\\sigma } j)} \\, J_f^{u(1)}- \\frac{1}{(k+M+N)} \\, t^{\\alpha }_{\\sigma \\bar{\\tau }}\\,\\delta ^{\\sigma \\bar{\\sigma }}\\, J^{\\alpha } \\, J^{(\\bar{\\tau } j)} \\nonumber \\\\&-&\\frac{1}{(k+M+N)}\\, t^{\\alpha }_{\\sigma \\bar{\\tau }}\\,\\delta ^{\\sigma \\bar{\\sigma }}\\, J^{(\\bar{\\tau } j)} \\, J_f^{\\alpha }-\\frac{(M N^2-2 M-N)}{2 M N (k+M+N)} \\, \\partial \\, J^{(\\bar{\\sigma } j)}\\Bigg ](w) + \\cdots ,\\nonumber \\\\J^a_f(z) \\, K(w) & = & 0 +\\cdots ,\\nonumber \\\\J^a_f(z) \\, G^{+}(w) & = & -\\frac{1}{(z-w)} \\, G^{+,a}(w) +\\cdots ,\\nonumber \\\\J^a_f(z) \\, G^{-}(w) & = & \\frac{1}{(z-w)} \\, G^{-,a}(w) +\\cdots ,\\nonumber \\\\J^{a}_f(z) \\, T(w) & = & \\frac{1}{(z-w)^2}\\, J^a_f(w) +\\cdots ,\\nonumber \\\\J^{\\alpha }_f(z) \\, K(w) & = & 0 +\\cdots ,\\nonumber \\\\J^{\\alpha }_f(z) \\, G^{+}(w) & = & -\\frac{1}{(z-w)} \\, t^{\\alpha }_{\\rho \\bar{\\nu }}\\, \\delta _{j \\bar{i}} \\, J^{(\\rho \\bar{i})}\\, \\psi ^{(\\bar{\\nu } j)}(w)+\\cdots ,\\nonumber \\\\J^{\\alpha }_f(z) \\, G^{-}(w) & = &\\frac{1}{(z-w)} \\, t^{\\alpha }_{\\sigma \\bar{\\sigma }}\\, \\delta _{j \\bar{i}} \\, \\psi ^{(\\sigma \\bar{i})}\\, J^{(\\bar{\\sigma } j)}\\,(w)+\\cdots ,\\nonumber \\\\J^{\\alpha }_f(z) \\, T(w) & = & \\frac{1}{(z-w)^2} \\,\\frac{1}{(k+M+N)} \\, \\Bigg [ k \\, J_f^{\\alpha } - M \\, J^{\\alpha }\\Bigg ](w)\\nonumber \\\\& - &\\frac{1}{(z-w)} \\,\\frac{1}{(k+M+N)} \\, i \\,f^{\\alpha \\beta \\gamma } \\, J^{\\beta } \\, J_f^{\\gamma }(w)+\\cdots .\\nonumber $ We can calculate the following OPEs ${.",
"}5\\endcsname J_f^{u(1)}(z) \\,\\delta _{\\rho \\bar{\\sigma }} \\, \\delta _{j \\bar{i}} \\,\\psi ^{(\\rho \\bar{i})} \\, \\partial \\, \\psi ^{(\\bar{\\sigma } j)}(w) &=&\\frac{1}{(z-w)^3} \\, M N - \\frac{1}{(z-w)^2} \\, J_f^{u(1)}(w) +\\cdots ,\\nonumber \\\\J_f^{u(1)}(z) \\,\\delta _{\\rho \\bar{\\sigma }} \\, \\delta _{j \\bar{i}} \\,\\partial \\, \\psi ^{(\\rho \\bar{i})} \\, \\psi ^{(\\bar{\\sigma } j)}(w) &=&\\frac{1}{(z-w)^3} \\, M N + \\frac{1}{(z-w)^2} \\, J_f^{u(1)}(w) +\\cdots ,\\nonumber \\\\J_f^{u(1)}(z) \\, J_f^{u(1)} \\, J_f^{u(1)}(w) & = & \\frac{1}{(z-w)^2}\\,2 M N \\, J_f^{u(1)}(w)+ \\cdots .\\nonumber $"
],
[
"The ${\\cal N}=2$ superconformal algebra",
"As before, we determine the following OPEs from (REF ), (REF ) and (REF ) ${.",
"}6\\endcsname K(z) \\, K(w) & = & \\frac{1}{(z-w)^2} \\, \\frac{k M N}{(k+M+N)} + \\cdots ,\\nonumber \\\\K(z) \\, G^{+}(w) & = &\\frac{1}{(z-w)} \\, G^{+}(w) + \\cdots ,\\nonumber \\\\K(z) \\, G^{-}(w) & = &-\\frac{1}{(z-w)} \\, G^{-}(w) + \\cdots ,\\nonumber \\\\K(z) \\, T(w) &= & \\frac{1}{(z-w)^2} \\, K(w) + \\cdots ,\\nonumber \\\\G^{+}(z) \\, G^{+}(w) &= & 0 + \\cdots ,\\nonumber \\\\G^{+}(z) \\, G^{-}(w) & = & \\frac{1}{(z-w)^3}\\, k M N+\\frac{1}{(z-w)^2}\\, (k+M+N) \\, K(w) \\nonumber \\\\& + &\\frac{1}{(z-w)}\\, \\Bigg [ \\frac{1}{2}\\, (k+M+N) \\, \\partial \\,K \\nonumber \\\\& + & (k +M+N) \\, \\Bigg ( T -\\frac{1}{2(k+M+N)} (J^a+J^a_f)(J^a+J^a_f)\\Bigg ) \\Bigg ](w) + \\cdots ,\\nonumber \\\\G^{+}(z) \\, T(w) & = & \\frac{1}{(z-w)^2} \\, \\frac{3}{2}\\, G^{+}(w)+\\frac{1}{(z-w)} \\, \\frac{1}{2}\\, \\partial \\, G^{+}(w)+ \\cdots ,\\nonumber \\\\G^{-}(z) \\, T(w) & = & \\frac{1}{(z-w)^2} \\, \\frac{3}{2}\\, G^{-}(w)+\\frac{1}{(z-w)} \\, \\frac{1}{2}\\, \\partial \\, G^{-}(w)+ \\cdots ,\\nonumber \\\\T(z) \\, T(w) & = &\\frac{1}{(z-w)^4} \\, \\frac{1}{2} \\,\\frac{(k M^2+3 k M N-k+M^2 N-N)}{(k+M+N)}\\nonumber \\\\& + & \\frac{1}{(z-w)^2}\\, 2 T(w) +\\frac{1}{(z-w)}\\, \\partial T(w) + \\cdots .\\nonumber $ The central term in the OPE of $K(z) \\, K(w)$ is the same as the standard expression $\\frac{c}{3}$ for only $M=1$ value.",
"We can multiply $\\frac{1}{(k+M+N)}$ in the OPE of $G^{+}(z)\\, G^{-}(w)$ in order to have the same central term as the one of the OPE of $K(z) \\, K(w)$ .",
"In obtaining the above result, we observe that there exist the relations ${.",
"}7\\endcsname J^{\\alpha }_f \\, J^{\\alpha }_f & = & t^{\\alpha }_{\\rho \\bar{\\sigma }} \\, t^{\\alpha }_{\\tau \\bar{\\nu }}\\,\\delta _{j \\bar{i}} \\, \\delta _{l \\bar{k}}\\psi ^{(\\rho \\bar{i})} \\, \\psi ^{(\\bar{\\sigma } j)} \\, \\psi ^{(\\tau \\bar{k})}\\, \\psi ^{(\\bar{\\nu } l)} + \\frac{(N^2-1)}{N} \\,\\delta _{\\rho \\bar{\\sigma }} \\, \\delta _{j \\bar{i}} \\,\\partial \\, \\psi ^{(\\rho \\bar{i})} \\, \\psi ^{(\\bar{\\sigma } j)}\\nonumber \\\\&-&\\frac{(N^2-1)}{N} \\,\\delta _{\\rho \\bar{\\sigma }} \\, \\delta _{j \\bar{i}} \\,\\, \\psi ^{(\\rho \\bar{i})} \\, \\partial \\, \\psi ^{(\\bar{\\sigma } j)},\\nonumber \\\\J_f^{u(1)} \\, J_f^{u(1)} & = &\\delta _{\\rho \\bar{\\sigma }} \\, \\delta _{\\tau \\bar{\\nu }}\\,\\delta _{j \\bar{i}} \\, \\delta _{l \\bar{k}}\\psi ^{(\\rho \\bar{i})} \\, \\psi ^{(\\bar{\\sigma } j)} \\, \\psi ^{(\\tau \\bar{k})}\\, \\psi ^{(\\bar{\\nu } l)} +\\delta _{\\rho \\bar{\\sigma }} \\, \\delta _{j \\bar{i}} \\,\\partial \\, \\psi ^{(\\rho \\bar{i})} \\, \\psi ^{(\\bar{\\sigma } j)}\\nonumber \\\\&-&\\delta _{\\rho \\bar{\\sigma }} \\, \\delta _{j \\bar{i}} \\,\\, \\psi ^{(\\rho \\bar{i})} \\, \\partial \\, \\psi ^{(\\bar{\\sigma } j)},\\nonumber \\\\J^{a}_f \\, J^{a}_f & = &t^a_{j \\bar{i}} \\, t^a_{l \\bar{k}} \\,\\delta _{\\rho \\bar{\\sigma }} \\, \\delta _{\\tau \\bar{\\nu }}\\,\\psi ^{(\\rho \\bar{i})} \\, \\psi ^{(\\bar{\\sigma } j)} \\, \\psi ^{(\\tau \\bar{k})}\\, \\psi ^{(\\bar{\\nu } l)} + \\frac{(M^2-1)}{M} \\,\\delta _{\\rho \\bar{\\sigma }} \\, \\delta _{j \\bar{i}} \\,\\partial \\, \\psi ^{(\\rho \\bar{i})} \\, \\psi ^{(\\bar{\\sigma } j)}\\nonumber \\\\&-&\\frac{(M^2-1)}{M} \\,\\delta _{\\rho \\bar{\\sigma }} \\, \\delta _{j \\bar{i}} \\,\\, \\psi ^{(\\rho \\bar{i})} \\, \\partial \\, \\psi ^{(\\bar{\\sigma } j)}.$ We can check the following identity between the spin-1 currents in (REF ) by using (REF ) ${.",
"}8\\endcsname && J^{\\alpha }_f \\, J^{\\alpha }_f + (\\frac{1}{M} + \\frac{1}{N})\\,J_f^{u(1)} \\, J_f^{u(1)} + J^{a}_f \\, J^{a}_f =\\nonumber \\\\&& (M+N) \\,(\\delta _{\\rho \\bar{\\sigma }} \\, \\delta _{j \\bar{i}} \\,\\partial \\, \\psi ^{(\\rho \\bar{i})} \\, \\psi ^{(\\bar{\\sigma } j)}-\\delta _{\\rho \\bar{\\sigma }} \\, \\delta _{j \\bar{i}} \\,\\, \\psi ^{(\\rho \\bar{i})} \\, \\partial \\, \\psi ^{(\\bar{\\sigma } j)}).\\nonumber $"
],
[
"The ${\\cal N}=2$ primary conditions",
"The ${\\cal N}=2$ primary conditions for the currents $(\\Phi ^{-(h)}, \\Phi ^{+(h+\\frac{1}{2})}, \\Phi ^{-(h+\\frac{1}{2})},\\Phi ^{+(h+1)})$ are summarized by ${.",
"}1\\endcsname K(z) \\, \\Phi ^{-(h)}(w) & = & 0 + \\cdots ,\\nonumber \\\\K(z) \\, \\Phi ^{+(h+\\frac{1}{2})}(w) & = & \\frac{1}{(z-w)}\\,\\Phi ^{+(h+\\frac{1}{2})}(w)+ \\cdots ,\\nonumber \\\\K(z) \\, \\Phi ^{-(h+\\frac{1}{2})}(w) & = & -\\frac{1}{(z-w)}\\,\\Phi ^{-(h+\\frac{1}{2})}(w)+ \\cdots ,\\nonumber \\\\K(z) \\, \\Phi ^{+(h+1)}(w) & = & \\frac{1}{(z-w)^2}\\, h\\,\\Phi ^{-(h)}(w)+ \\cdots ,\\nonumber \\\\G^+(z) \\, \\Phi ^{-(h)}(w) & = & -\\frac{1}{(z-w)}\\,\\Phi ^{+(h+\\frac{1}{2})}(w) + \\cdots ,\\nonumber \\\\G^+(z) \\, \\Phi ^{+(h+\\frac{1}{2})}(w) & = & 0 + \\cdots ,\\nonumber \\\\G^+(z) \\, \\Phi ^{-(h+\\frac{1}{2})}(w) & = & \\frac{1}{(z-w)^2}\\,h\\, \\Phi ^{-(h)}(w)+\\frac{1}{(z-w)}\\,\\Bigg [\\Phi ^{+(h+1)}+ \\frac{1}{2} \\, \\partial \\, \\Phi ^{-(h)} \\Bigg ](w)+ \\cdots ,\\nonumber \\\\G^+(z) \\, \\Phi ^{+(h+1)}(w) & = & \\frac{1}{(z-w)^2}\\, (h+\\frac{1}{2})\\,\\Phi ^{+(h+\\frac{1}{2})}(w) + \\frac{1}{(z-w)}\\, \\frac{1}{2}\\,\\partial \\, \\Phi ^{+(h+\\frac{1}{2})}(w) + \\cdots ,\\nonumber \\\\G^-(z) \\, \\Phi ^{-(h)}(w) & = & \\frac{1}{(z-w)}\\,\\Phi ^{-(h+\\frac{1}{2})}(w) + \\cdots ,\\nonumber \\\\G^-(z) \\, \\Phi ^{+(h+\\frac{1}{2})}(w) & = & -\\frac{1}{(z-w)^2}\\,h\\, \\Phi ^{-(h)}(w)+\\frac{1}{(z-w)}\\,\\Bigg [\\Phi ^{+(h+1)}- \\frac{1}{2} \\, \\partial \\, \\Phi ^{-(h)} \\Bigg ](w)+ \\cdots ,\\nonumber \\\\G^-(z) \\, \\Phi ^{-(h+\\frac{1}{2})}(w) & = & 0 + \\cdots ,\\nonumber \\\\G^-(z) \\, \\Phi ^{+(h+1)}(w) & = & \\frac{1}{(z-w)^2}\\, (h+\\frac{1}{2})\\,\\Phi ^{-(h+\\frac{1}{2})}(w) + \\frac{1}{(z-w)}\\, \\frac{1}{2}\\,\\partial \\, \\Phi ^{-(h+\\frac{1}{2})}(w) + \\cdots ,\\nonumber \\\\T(z) \\, \\Phi ^{-(h)}(w) & = & \\frac{1}{(z-w)^2}\\, h\\,\\Phi ^{-(h)}(w) +\\frac{1}{(z-w)}\\,\\partial \\, \\Phi ^{-(h)}(w) + \\cdots ,\\nonumber \\\\T(z) \\, \\Phi ^{+(h+\\frac{1}{2})}(w) & = & \\frac{1}{(z-w)^2}\\,(h+\\frac{1}{2}) \\, \\Phi ^{+(h+\\frac{1}{2})}(w)+\\frac{1}{(z-w)}\\,\\partial \\, \\Phi ^{+(h+\\frac{1}{2})}(w)+ \\cdots ,\\nonumber \\\\T(z) \\, \\Phi ^{-(h+\\frac{1}{2})}(w) & = &\\frac{1}{(z-w)^2}\\,(h+\\frac{1}{2}) \\, \\Phi ^{-(h+\\frac{1}{2})}(w)+\\frac{1}{(z-w)}\\,\\partial \\, \\Phi ^{-(h+\\frac{1}{2})}(w)+ \\cdots ,\\nonumber \\\\T(z) \\, \\Phi ^{+(h+1)}(w) & = & \\frac{1}{(z-w)^2}\\, (h+1)\\,\\Phi ^{+(h+1)}(w)+ \\frac{1}{(z-w)}\\,\\partial \\, \\Phi ^{+(h+1)}(w)+ \\cdots .\\nonumber $ In this paper, we observe that the currents we are considering do not satisfy these OPEs.",
"That is, some of the higher order terms arise and the structure constants appear differently."
],
[
"Some OPEs between the spin-1 currents and the\nsinglet and nonsinglet spin-3 currents in sections 4 and 5\n",
"In order to obtain the OPEs between the supersymmetry generators and the nonsinglet and singlet spin-3 currents obtained in the bosonic coset model, we need to calculate the following OPEs between the spin-1 currents and the singlet and nonsinglet spin-3 currents."
],
[
"The OPE between $J^{(\\rho \\bar{i})}(z) \\, W^{(3)}(w)$",
"The OPE between the spin-1 current and the spin-3 current is summarized by ${.",
"}1\\endcsname && J^{(\\rho \\bar{i})}(z) \\, W^{(3)}(w) =\\nonumber \\\\&& -\\frac{1}{(z-w)^3} \\,\\frac{ (k^2-1) (k^2-4) (k+M+N) (2 k+M+N) (3 k+2 M+2 N)}{k^2 M (k+M) (k+2 M)} \\, b_1 \\, J^{(\\rho \\bar{i})}(w)\\nonumber \\\\&& + \\frac{1}{(z-w)^2} \\frac{(k^2-4) (k+M+N) ( 3 k+2 M+2 N)}{k M (k+2 M)} \\, b_1 \\nonumber \\\\&& \\times \\, \\Bigg [-\\sqrt{\\frac{M+N}{M N}}\\, \\frac{\\, 3 (k+N) }{k} \\, J^{u(1)} \\, J^{(\\rho \\bar{i})}+ \\frac{3(k+N)}{(k+M)} \\, t^a_{i \\bar{k}} \\, \\delta ^{i \\bar{i}}\\,J^a \\, J^{(\\rho \\bar{k})}-3 \\, t^{\\alpha }_{\\sigma \\bar{\\rho }} \\, \\delta ^{\\rho \\bar{\\rho }}\\, J^{\\alpha }\\,J^{(\\sigma \\bar{i})}\\nonumber \\\\&& + \\frac{(k^3+2 k^2 M+2 k^2 N+3 k M N+2 k+M+N)}{k(k+M)}\\partial \\, J^{(\\rho \\bar{i})} \\Bigg ](w)\\nonumber \\\\&&+ \\frac{1}{(z-w)} \\, \\frac{ (k+M+N)}{M}\\, b_1\\, \\Bigg [- 3\\,d^{\\alpha \\beta \\gamma } \\, t^{\\gamma }_{\\sigma \\bar{\\rho }}\\, \\delta ^{\\rho \\bar{\\rho }}\\,J^{\\alpha } \\, J^{\\beta }\\, J^{(\\sigma \\bar{i})}\\nonumber \\\\&& -\\frac{3 (k+N) (k+2 N)}{ (k+M) (k+2 M)} \\,d^{a b c}\\, t^{c}_{i \\bar{k}}\\, \\delta ^{i \\bar{i}}\\,J^{a} \\, J^{b}\\, J^{(\\rho \\bar{k})}-\\frac{6 (k+N) (k+2 N) (M+N) }{k^2 M N}J^{u(1)}\\, J^{u(1)}\\, J^{(\\rho \\bar{i})}\\nonumber \\\\&& -\\frac{6 (k+N)}{k N}\\, J^{\\alpha }\\, J^{\\alpha }\\, J^{(\\rho \\bar{i})}-\\sqrt{\\frac{M+N}{M N}}\\, \\frac{12 (k+N) }{k } \\,t^{\\alpha }_{\\sigma \\bar{\\sigma }}\\, \\delta ^{\\rho \\bar{\\sigma }}\\,J^{\\alpha }\\, J^{u(1)} \\, J^{(\\sigma \\bar{i})} \\nonumber \\\\&&-\\frac{6 (k+N) (k+2 N) }{k M (k+2 M)}\\, J^a \\, J^a \\,J^{(\\rho \\bar{i})}+\\sqrt{\\frac{M+N}{M N}}\\,\\frac{12 (k+N) (k+2 N) }{k (k+2 M)}\\,t^a_{i \\bar{k}}\\, \\delta ^{i\\bar{i}}\\, J^a \\, J^{u(1)}\\, J^{(\\rho \\bar{k})}\\nonumber \\\\&& +\\frac{12 (k+N) }{ (k+2 M)}\\, \\delta _{\\nu \\bar{\\tau }} \\,\\delta _{k \\bar{j}} \\, J^{(\\nu \\bar{i})}\\, J^{(\\rho \\bar{j})}\\,J^{(\\bar{\\tau }k )}+\\frac{24 (k+N) }{k (k+2 M)}\\,\\delta _{\\sigma \\bar{\\tau }} \\,\\delta _{k \\bar{j}} \\, J^{(\\rho \\bar{i})}\\, J^{(\\sigma \\bar{j})}\\,J^{(\\bar{\\tau }k )} \\nonumber \\\\&& -\\sqrt{\\frac{M+N}{M N}}\\, \\frac{12 (k+N) }{k }\\,t^{\\alpha }_{\\sigma \\bar{\\rho }} \\,\\delta ^{\\rho \\bar{\\rho }} \\, J^{\\alpha }\\, J^{u(1)}\\,J^{(\\sigma \\bar{j} )}+\\frac{12 (k+N) }{ (k+2 M)}\\, t^{a}_{i \\bar{j}} \\,\\delta ^{i \\bar{i}} \\,t^{\\alpha }_{\\sigma \\bar{\\rho }} \\,\\delta ^{\\rho \\bar{\\rho }} \\,J^{\\alpha }\\, J^a \\, J^{(\\sigma \\bar{j} )}\\nonumber \\\\&&+ \\frac{3 (k^2+2 k N+4) }{k }\\,t^{\\alpha }_{\\tau \\bar{\\rho }} \\,\\delta ^{\\rho \\bar{\\rho }} \\,J^{\\alpha }\\, \\partial \\, J^{(\\tau \\bar{i})}-\\frac{3 (k^2+2 k M+4) (k+N) (k+2 N) }{k (k+M) (k+2 M)}\\,t^{a}_{i \\bar{l}} \\,\\delta ^{i \\bar{i}} \\,J^{a}\\, \\partial \\, J^{(\\rho \\bar{l})}\\nonumber \\\\&& + \\sqrt{\\frac{M+N}{M N}} \\,\\frac{3 (k+N) (k^3+2 k^2 M+2 k^2 N+8 k M N+4 k+8 M+8 N)}{k^2 (k+2 M)}\\, J^{u(1)}\\, \\partial \\, J^{(\\rho \\bar{i})}\\nonumber \\\\&& -3 \\, (k+N) \\, t^{\\alpha }_{\\tau \\bar{\\rho }} \\,\\delta ^{\\rho \\bar{\\rho }} \\,\\partial \\, J^{\\alpha } \\, J^{(\\tau \\bar{i})}+ \\frac{3 (k+N) (k+2 N) }{ (k+2 M)}\\,t^{a}_{i \\bar{j}} \\,\\delta ^{i \\bar{i}} \\,\\partial \\, J^{a} \\, J^{(\\rho \\bar{j})} \\nonumber \\\\&&- \\sqrt{\\frac{M+N}{M N}}\\, \\frac{3 (k+N) (k+2 N) }{k }\\,\\partial \\, J^{u(1)}\\, J^{(\\rho \\bar{i})} \\nonumber \\\\&&-\\frac{(k^2-4) }{2 k^2 (k+M) (k+2 M)}\\,(k^3 M+k^3 N+2 k^2 M^2-2 k^2 M N+2 k^2 N^2-6 k^2-7 k M\\nonumber \\\\&& -7 k N-2 M^2-4 M N-2 N^2) \\,\\partial ^2 \\, J^{(\\rho \\bar{i})}\\Bigg ](w) + \\cdots .\\nonumber $"
],
[
"The OPE between $J^{(\\bar{\\sigma } j)}(z) \\, W^{(3)}(w)$",
"Similarly, the OPE between the other spin-1 current and the spin-3 current is summarized by ${.",
"}2\\endcsname && J^{(\\bar{\\sigma } j)}(z) \\, W^{(3)}(w) =\\nonumber \\\\&& \\frac{1}{(z-w)^3} \\,\\frac{ (k^2-1) (k^2-4) (k+M+N) (2 k+M+N) (3 k+2 M+2 N)}{k^2 M (k+M) (k+2 M)} \\, b_1 \\, J^{(\\bar{\\sigma } j)}(w)\\nonumber \\\\&& + \\frac{1}{(z-w)^2} \\frac{(k^2-4) (k+M+N) ( 3 k+2 M+2 N)}{k M (k+2 M)} \\, b_1 \\nonumber \\\\&& \\times \\, \\Bigg [-\\sqrt{\\frac{M+N}{M N}}\\, \\frac{\\, 3 (k+N) }{k} \\, J^{u(1)} \\, J^{(\\bar{\\sigma } j)}+ \\frac{3(k+N)}{(k+M)} \\, t^a_{l \\bar{j}} \\, \\delta ^{j \\bar{j}}\\,J^a \\, J^{(\\bar{\\sigma } l)}-3 \\, t^{\\alpha }_{\\rho \\bar{\\tau }} \\, \\delta ^{\\rho \\bar{\\rho }}\\, J^{\\alpha }\\,J^{(\\bar{\\tau } j)}\\nonumber \\\\&& - \\frac{(k^3+2 k^2 M+2 k^2 N+3 k M N+2 k+M+N)}{k(k+M)}\\partial \\, J^{(\\bar{\\sigma } j)} \\Bigg ](w)\\nonumber \\\\&&+ \\frac{1}{(z-w)} \\,\\frac{ (k+M+N)}{M}\\, b_1\\, \\Bigg [3 \\,d^{\\alpha \\beta \\gamma } \\, t^{\\gamma }_{\\rho \\bar{\\tau }}\\, \\delta ^{\\rho \\bar{\\sigma }}\\,J^{\\alpha } \\, J^{\\beta }\\, J^{(\\bar{\\tau } j)}\\nonumber \\\\&& \\, +\\frac{3 (k+N) (k+2 N) }{ (k+M) (k+2 M)}\\,d^{a b c}\\, t^{c}_{k \\bar{j}}\\, \\delta ^{j \\bar{j}}\\,J^{a} \\, J^{b}\\, J^{(\\bar{\\sigma } k)}+\\frac{6 (k+N) (k+2 N) (M+N) }{k^2 M N}J^{u(1)}\\, J^{u(1)}\\, J^{(\\bar{\\sigma } j)}\\nonumber \\\\&&\\, + \\frac{6 (k+N) }{k N}\\,J^{\\alpha }\\, J^{\\alpha }\\, J^{(\\bar{\\sigma } j)}+ \\sqrt{\\frac{M+N}{M N}}\\, \\frac{12 (k+N) }{k }\\,t^{\\alpha }_{\\sigma \\bar{\\tau }}\\, \\delta ^{\\sigma \\bar{\\sigma }}\\,J^{\\alpha }\\, J^{u(1)} \\, J^{(\\bar{\\tau } j)} \\nonumber \\\\&&+\\frac{6 (k+N) (k+2 N) }{k M (k+2 M)}\\, J^a \\, J^a \\,J^{(\\bar{\\sigma } j)}-\\sqrt{\\frac{M+N}{M N}}\\,\\frac{12 (k+N) (k+2 N) }{k (k+2 M)}\\,t^a_{k \\bar{j}}\\, \\delta ^{j\\bar{j}}\\, J^a \\, J^{u(1)}\\, J^{(\\bar{\\sigma } k)}\\nonumber \\\\&& -\\frac{12 (k+N) }{ (k+2 M)}\\, \\delta _{\\sigma \\bar{\\nu }} \\,\\delta _{k \\bar{j}} \\, J^{(\\sigma \\bar{j})}\\, J^{(\\bar{\\nu } j)}\\,J^{(\\bar{\\sigma }k)}-\\frac{24 (k+N) }{k (k+2 M)}\\,\\delta _{\\sigma \\bar{\\tau }} \\,\\delta _{k \\bar{j}} \\, J^{(\\sigma \\bar{j})}\\, J^{(\\bar{\\sigma } j)}\\,J^{(\\bar{\\tau }k )} \\nonumber \\\\&& +\\sqrt{\\frac{M+N}{M N}}\\,\\frac{12 (k+N) }{k }\\,t^{\\alpha }_{\\sigma \\bar{\\tau }} \\,\\delta ^{\\sigma \\bar{\\sigma }} \\, J^{\\alpha }\\, J^{u(1)}\\,J^{(\\bar{\\tau }j )}-\\frac{12 (k+N) }{ (k+2 M)}\\, t^{a}_{k \\bar{j}} \\,\\delta ^{j \\bar{j}} \\,t^{\\alpha }_{\\sigma \\bar{\\tau }} \\,\\delta ^{\\sigma \\bar{\\sigma }} \\,J^{\\alpha }\\, J^a \\, J^{(\\bar{\\tau }k )}\\nonumber \\\\&&+\\frac{3 (k^2+2 k N+4) }{k }\\,t^{\\alpha }_{\\rho \\bar{\\tau }} \\,\\delta ^{\\rho \\bar{\\sigma }} \\,J^{\\alpha }\\, \\partial \\, J^{(\\bar{\\tau } j)}-\\frac{3 (k^2+2 k M+4) (k+N) (k+2 N) }{k (k+M) (k+2 M)}\\,t^{a}_{k \\bar{j}} \\,\\delta ^{j \\bar{j}} \\,J^{a}\\, \\partial \\, J^{(\\bar{\\sigma } k)}\\nonumber \\\\&& +\\sqrt{\\frac{M+N}{M N}}\\,\\frac{3 (k+N) (k^3+2 k^2 M+2 k^2 N+8 k M N+4 k+8 M+8 N)}{k^2 (k+2 M)}\\, J^{u(1)}\\, \\partial \\, J^{(\\bar{\\sigma } j)}\\nonumber \\\\&& -\\frac{3 (k^2-2 k M-8) (k+N) }{k (k+2 M)}\\,t^{\\alpha }_{\\rho \\bar{\\tau }} \\,\\delta ^{\\rho \\bar{\\sigma }} \\,\\partial \\, J^{\\alpha } \\, J^{(\\bar{\\tau } j)}-\\frac{3 (k+N) (-k^2+2 k N+8) }{k (k+2 M)}\\,t^{a}_{k \\bar{j}} \\,\\delta ^{j \\bar{j}} \\,\\partial \\, J^{a} \\, J^{(\\bar{\\sigma } k)} \\nonumber \\\\&&-\\sqrt{\\frac{M+N}{M N}}\\, \\frac{3 (k+N) (k+M+N)(k^2-2 k M-2 k N-4 M N-8)}{k (k+2 M)}\\,\\partial \\, J^{u(1)}\\, J^{(\\bar{\\sigma } j)} \\nonumber \\\\&&+ \\frac{1}{2 k^2 (k+M) (k+2 M)} \\,(k^5 M+k^5 N+2 k^4 M^2+10 k^4 M N+2 k^4 N^2+6 k^4+12 k^3 M^2 N\\nonumber \\\\&& +12 k^3 M N^2+25 k^3 M+25 k^3 N+12 k^2 M^2 N^2+14 k^2 M^2+64 k^2 M N+14 k^2 N^2+24 k^2\\nonumber \\\\&& +24 k M^2 N+24 k M N^2+28 k M+28 k N+8 M^2+16 M N+8 N^2)\\,\\partial ^2 \\, J^{(\\bar{\\sigma } j)}\\Bigg ](w) + \\cdots .\\nonumber $"
],
[
"The OPE between $J^{(\\rho \\bar{i})}(z) \\, P^{a}(w)$",
"The OPE between the spin-1 current and the other spin-3 current is summarized by ${.",
"}3\\endcsname && J^{(\\rho \\bar{i})}(z) \\, P^{a}(w) =\\nonumber \\\\&& \\frac{1}{(z-w)^3} \\,\\frac{ (k^2-1) (k^2-4) (2 k+M+N) (3 k+2 M+2 N)}{2 k^2 (k+M) (3 k+2 M)} \\, a_1 \\,t^a_{i \\bar{k}} \\, \\delta ^{i \\bar{i}} \\, J^{(\\rho \\bar{k})}(w)\\nonumber \\\\&&+ \\frac{1}{(z-w)^2} \\, \\frac{ (k^2-4)(3 k+2 M+2 N)}{2 k (k+M)}\\,a_1\\,\\Bigg [ \\sqrt{\\frac{M+N}{M N}}\\,\\frac{ (k+N)}{ k} \\, t^a_{i \\bar{k}} \\, \\delta ^{i \\bar{i}} \\, J^{u(1)} \\, J^{(\\rho \\bar{k})}\\nonumber \\\\&&-\\frac{ (3 k^2+3 k M+3 k N+M^2+M N)}{k M (3 k+2 M)}\\, J^a \\, J^{(\\rho \\bar{i})}+ t^{\\alpha }_{\\sigma \\bar{\\rho }} \\, \\delta ^{\\rho \\bar{\\rho }}\\,t^a_{i \\bar{k}} \\, \\delta ^{i \\bar{i}}\\, J^{\\alpha }\\, J^{(\\sigma \\bar{k})}\\nonumber \\\\&&+\\frac{ (-k-M+N) }{2 (3 k+2 M)}\\, i\\, f^{b a c} \\, t^c_{i \\bar{k}}\\,\\delta ^{i \\bar{i}}\\, J^b \\, J^{(\\rho \\bar{k})}-\\frac{ (3 k+M+3 N)}{2 (3 k+2 M)}\\,d^{ a b c} \\, t^c_{i \\bar{k}}\\,\\delta ^{i \\bar{i}}\\, J^b \\, J^{(\\rho \\bar{k})}\\nonumber \\\\&&-\\frac{ (k^3+k^2 M+2 k^2 N+2 k M N+2 k+M+N)}{ k (3 k+2 M)}\\,t^a_{i \\bar{k}} \\, \\delta ^{i \\bar{i}}\\, \\partial \\, J^{(\\rho \\bar{k})}\\Bigg ](w)\\nonumber \\\\&& + \\frac{1}{(z-w)} \\,J^{(\\rho \\bar{i})}(z) \\,P^{a}(w)\\Bigg |_{\\frac{1}{(z-w)}} + \\cdots .\\nonumber $ The first order pole above is given by ${.",
"}4\\endcsname && J^{(\\rho \\bar{i})}(z) \\, P^{a}(w)\\Bigg |_{\\frac{1}{(z-w)}} =-a_1 \\, t^{a}_{k \\bar{j}}\\, \\delta _{\\nu \\bar{\\tau }}\\,J^{(\\rho \\bar{j})}\\, J^{(\\nu \\bar{i})}\\, J^{(\\bar{\\tau } k)}+\\frac{a_1}{N}\\,t^{a}_{k \\bar{j}}\\, \\delta _{\\sigma \\bar{\\tau }}\\,J^{(\\sigma \\bar{j})}\\, J^{(\\rho \\bar{i})}\\, J^{(\\bar{\\tau } k)}\\nonumber \\\\&& + (\\sqrt{\\frac{M+N}{M N}} \\,a_1+2 \\, a_{7})\\,t^{\\alpha }_{\\sigma \\bar{\\rho }}\\, \\delta ^{\\rho \\bar{\\rho }}\\,t^a_{i \\bar{k}}\\, \\delta ^{i \\bar{i}} \\,J^{\\alpha }\\, J^{u(1)}\\, J^{(\\sigma \\bar{k})}+(\\frac{a_1}{N}+a_2)\\, t^{a}_{i \\bar{k}}\\, \\delta ^{i \\bar{i}} \\,J^{\\alpha }\\, J^{\\alpha }\\, J^{(\\rho \\bar{k})}\\nonumber \\\\&& + \\frac{a_1}{2}\\, (i \\, f+ d)^{\\alpha \\beta \\gamma }\\,t^{\\gamma }_{\\sigma \\bar{\\rho }}\\, \\delta ^{\\rho \\bar{\\rho }}\\,t^a_{i \\bar{k}}\\, \\delta ^{i \\bar{i}} \\,J^{\\alpha }\\, J^{\\beta }\\, J^{(\\sigma \\bar{k})}+(-\\frac{a_1}{M}-2 a_2+ 2 a_8)\\,t^{\\alpha }_{\\sigma \\bar{\\rho }}\\, \\delta ^{\\rho \\bar{\\rho }} \\,J^{\\alpha }\\, J^{a}\\, J^{(\\sigma \\bar{i})}\\nonumber \\\\&&- \\frac{a_1}{2}\\,(i \\, f+ d)^{b a c}\\,t^{\\alpha }_{\\sigma \\bar{\\rho }}\\, \\delta ^{\\rho \\bar{\\rho }}\\,t^c_{i \\bar{k}}\\, \\delta ^{i \\bar{i}} \\,J^{\\alpha }\\, J^{b}\\, J^{(\\sigma \\bar{k})}+(a_3 + \\frac{6}{M}\\, a_{17})\\,t^{a}_{i \\bar{k}}\\, \\delta ^{i \\bar{i}} \\,J^{b}\\, J^{b}\\, J^{(\\rho \\bar{k})}\\nonumber \\\\&& +(2 a_3 - 2 a_8 + \\frac{6}{M}\\, a_{17})\\,t^{b}_{i \\bar{k}}\\, \\delta ^{i \\bar{i}} \\,J^{a}\\, J^{b}\\, J^{(\\rho \\bar{k})}+(a_4 + 2 \\,\\sqrt{\\frac{M+N}{M N}} \\, a_7 )\\,t^{a}_{i \\bar{k}}\\, \\delta ^{i \\bar{i}} \\,J^{u(1)}\\, J^{u(1)}\\, J^{(\\rho \\bar{k})}\\nonumber \\\\&& +(-2 \\,\\sqrt{\\frac{M+N}{M N}} \\,a_4-\\frac{2}{M}\\, a_7+2\\sqrt{\\frac{M+N}{M N}} \\, a_8)\\,J^{a}\\, J^{u(1)}\\, J^{(\\rho \\bar{i})}\\nonumber \\\\&&+ a_5 \\, d^{a b c}\\,t^{c}_{i \\bar{k}}\\, \\delta ^{i \\bar{i}}\\,t^b_{k \\bar{j}}\\, \\delta _{\\sigma \\bar{\\sigma }} \\,J^{(\\rho \\bar{k})}\\, J^{(\\sigma \\bar{j})}\\, J^{( \\bar{\\sigma } k)}+ a_5 \\, d^{a b c}\\,t^{c}_{i \\bar{k}}\\, \\delta ^{i \\bar{i}}\\,t^b_{k \\bar{j}}\\, \\delta _{\\sigma \\bar{\\sigma }} \\,J^{(\\rho \\bar{k})}\\, J^{( \\bar{\\sigma } k)} \\, J^{(\\sigma \\bar{j})}\\nonumber \\\\&& - \\sqrt{\\frac{M+N}{M N}} \\, a_7 \\,t^{a}_{k \\bar{j}}\\,\\delta _{\\sigma \\bar{\\tau }} \\,J^{(\\rho \\bar{i})}\\, J^{(\\sigma \\bar{j})}\\, J^{( \\bar{\\tau } k)}-\\sqrt{\\frac{M+N}{M N}} \\, a_7 \\,t^{a}_{k \\bar{j}}\\,\\delta _{\\sigma \\bar{\\tau }} \\,J^{(\\rho \\bar{i})}\\, J^{( \\bar{\\tau } k)} \\, J^{(\\sigma \\bar{j})}\\nonumber \\\\&&+( 2 \\,\\sqrt{\\frac{M+N}{M N}} \\, a_5 -a_7+2 \\,a_9)\\, d^{a b c}\\,t^{b}_{i \\bar{k}}\\, \\delta ^{i \\bar{i}}\\,J^{c}\\, J^{u(1)} \\, J^{(\\rho \\bar{k})}+ 2 \\, a_5 \\, d^{a b c}\\,t^{\\alpha }_{\\sigma \\bar{\\rho }} \\, \\delta ^{\\rho \\bar{\\rho }}\\,t^{b}_{i \\bar{k}}\\, \\delta ^{i \\bar{i}}\\,J^{\\alpha }\\, J^c \\, J^{(\\sigma \\bar{k})}\\nonumber \\\\&& +(-\\frac{2}{M}\\, a_5 -\\sqrt{\\frac{M+N}{M N}} \\, a_9 )\\, d^{a b c}\\,J^{c}\\, J^{b} \\, J^{(\\rho \\bar{i})}-a_5\\, (i \\, f+d)^{e b d}\\,d^{a b c}\\, t^d_{i \\bar{k}}\\,\\delta ^{i \\bar{i}}\\,J^{c}\\, J^{e} \\, J^{(\\rho \\bar{k})}\\nonumber \\\\&&+ a_7 \\, i \\, f^{a b c}\\, t^c_{i \\bar{k}}\\,\\delta ^{i \\bar{i}} \\, J^{b}\\, J^{u(1)} \\, J^{(\\rho \\bar{k})}+ a_8 \\, t^a_{i \\bar{l}}\\,\\delta ^{i \\bar{i}} \\, \\delta _{\\sigma \\bar{\\tau }} \\, \\delta _{k \\bar{j}}\\,J^{(\\rho \\bar{l})}\\, J^{(\\sigma \\bar{j})} \\, J^{(\\bar{\\tau } k)}\\nonumber \\\\&&+a_8 \\, t^a_{i \\bar{l}}\\,\\delta ^{i \\bar{i}} \\, \\delta _{\\sigma \\bar{\\tau }} \\, \\delta _{k \\bar{j}}\\,J^{(\\rho \\bar{l})}\\, J^{(\\bar{\\tau } k)} \\, J^{(\\sigma \\bar{j})}+ \\frac{3}{2}\\, a_{17}\\,(i \\, f+d)^{a b e}\\,d^{e c d}\\, t^b_{i \\bar{k}}\\,\\delta ^{i \\bar{i}} \\, J^{c}\\, J^{d} \\, J^{(\\rho \\bar{k})}\\nonumber \\\\&&+\\frac{3}{2}\\, a_{17}\\,(i \\, f+d)^{a d e}\\,d^{e b c}\\, t^b_{i \\bar{k}}\\,\\delta ^{i \\bar{i}} \\, J^{c}\\, J^{d} \\, J^{(\\rho \\bar{k})}+\\frac{3}{2}\\, a_{17}\\,(i \\, f+d)^{a c e}\\,d^{e b d}\\, t^b_{i \\bar{k}}\\,\\delta ^{i \\bar{i}} \\, J^{c}\\, J^{d} \\, J^{(\\rho \\bar{k})}\\nonumber \\\\&&+ \\frac{6}{M}\\, a_{17}\\,t^b_{i \\bar{k}}\\,\\delta ^{i \\bar{i}} \\,J^{b}\\, J^{a} \\, J^{(\\rho \\bar{k})}+\\Big (\\frac{1}{2}\\, (N -\\frac{1}{N})\\, a_2+\\frac{1}{2}\\, (M -\\frac{1}{M})\\, a_3+\\frac{1}{2}\\, \\frac{M+N}{M N}\\, a_4\\nonumber \\\\&& - \\sqrt{\\frac{M+N}{M N}} \\,\\frac{(M^2-4)}{2 M}\\, a_9-\\frac{M}{2}\\, a_{11}-\\frac{N}{2}\\, a_{12}+ a_{16}+ \\frac{18-6M^2+M^4}{2 M^2} \\,a_{17} \\Big )\\,\\nonumber \\\\& & \\times t^a_{i \\bar{k}}\\,\\delta ^{i \\bar{i}} \\, \\partial ^2 \\, J^{(\\rho \\bar{k})}+(-2 \\, a_2 + a_{12})\\,t^{\\alpha }_{\\sigma \\bar{\\rho }}\\,\\delta ^{\\rho \\bar{\\rho }} \\, t^a_{i \\bar{k}} \\, \\delta ^{i \\bar{i}}\\,J^{\\alpha }\\, \\partial \\, J^{(\\sigma \\bar{k})} +\\Big ( (N-\\frac{1}{N})\\, a_2 + \\frac{(M+N)}{M N}\\,a_4 \\nonumber \\\\&& -(M+N)\\, a_8 -\\frac{1}{M}\\, a_{12}+ \\frac{6(2M^2-3)}{M^2}\\, a_{17} + (M+\\frac{1}{M})\\, a_3\\Big )\\, J^a \\, \\partial \\, J^{(\\rho \\bar{i})}\\nonumber \\\\&&+(a_3 -2 \\, a_3 -a_{11}+\\frac{1}{2}\\, a_{12})\\,i\\, f^{a b c} \\, t^c_{i \\bar{k}}\\,\\delta ^{i \\bar{i}} \\, J^b \\, \\partial \\, J^{(\\rho \\bar{k})}\\nonumber \\\\&& + \\Big ( a_3 - N\\, a_5 -2 \\, \\sqrt{\\frac{M+N}{M N}} \\,a_9-\\frac{1}{2}\\, a_{12}+ \\frac{3(M^2-6)}{M} \\, a_{17} \\Big )\\,d^{a b c} \\, t^c_{i \\bar{k}}\\,\\delta ^{i \\bar{i}} \\, J^b \\, \\partial \\, J^{(\\rho \\bar{k})}\\nonumber \\\\&& +(-2 \\,a_3-a_{11}-\\frac{1}{2}\\, a_{13})\\,i\\, f^{a b c} \\, t^b_{i \\bar{k}}\\,\\delta ^{i \\bar{i}} \\, \\partial \\, J^c \\, J^{(\\rho \\bar{k})}\\nonumber \\\\&& +(-2\\sqrt{\\frac{M+N}{M N}} \\,a_4 -N \\, a_7 + \\frac{(M^2-4)}{M}\\,a_9 + \\sqrt{\\frac{M+N}{M N}} \\,a_{12})\\,t^a_{i \\bar{k}}\\,\\delta ^{i \\bar{i}} \\, J^{u(1)}\\, \\partial \\, J^{(\\rho \\bar{k})}\\nonumber \\\\&& +\\sqrt{\\frac{M+N}{M N}} \\,a_{13} \\,t^a_{i \\bar{k}}\\,\\delta ^{i \\bar{i}} \\, \\partial \\, J^{u(1)} \\, J^{(\\rho \\bar{k})}+a_{13}\\,t^{\\alpha }_{\\sigma \\bar{\\rho }}\\,\\delta ^{\\rho \\bar{\\rho }} \\, t^a_{i \\bar{k}} \\, \\delta ^{i \\bar{i}}\\,\\partial \\, J^{\\alpha } \\, J^{(\\sigma \\bar{k})}-\\frac{1}{M}\\, a_{13}\\, \\partial \\, J^{a} \\,J^{(\\rho \\bar{i})}\\nonumber \\\\&&-\\frac{1}{2}\\, a_{13}\\,d^{a b c} \\, t^b_{i \\bar{k}}\\,\\delta ^{i \\bar{i}} \\, \\partial \\, J^c \\, J^{(\\rho \\bar{k})},\\nonumber $ where we do not substitute the coefficients appearing in (REF ) and then we can observe each contribution by looking at each coefficient term."
],
[
"The OPE between $J^{(\\bar{\\sigma } j)}(z) \\, P^{a}(w)$",
"The OPE between the other spin-1 current and the other spin-3 current is summarized by ${.",
"}5\\endcsname && J^{(\\bar{\\sigma } j)}(z) \\, P^{a}(w) =\\nonumber \\\\&& -\\frac{1}{(z-w)^3} \\,\\frac{ (k^2-1) (k^2-4) (2 k+M+N) (3 k+2 M+2 N)}{2 k^2 (k+M) (3 k+2 M)} \\, a_1 \\,t^a_{k \\bar{j}} \\, \\delta ^{j \\bar{j}} \\, J^{(\\bar{\\sigma } k)}(w)\\nonumber \\\\&&+ \\frac{1}{(z-w)^2} \\, \\frac{ (k^2-4)(3 k+2 M+2 N)}{2 k (k+M)}\\,a_1\\,\\Bigg [ \\sqrt{\\frac{M+N}{M N}}\\,\\frac{ (k+N)}{ k} \\, t^a_{k \\bar{j}} \\, \\delta ^{j \\bar{j}} \\, J^{u(1)} \\, J^{(\\bar{\\sigma } k)}\\nonumber \\\\&&-\\frac{ (3 k^2+3 k M+3 k N+M^2+M N)}{k M (3 k+2 M)}\\, J^a \\, J^{(\\bar{\\sigma } j)}+ t^{\\alpha }_{\\sigma \\bar{\\tau }} \\, \\delta ^{\\sigma \\bar{\\sigma }}\\,t^a_{l \\bar{k}} \\, \\delta ^{j \\bar{k}}\\, J^{\\alpha }\\, J^{(\\bar{\\tau } l)}\\nonumber \\\\&&+\\frac{ (-k-M+N) }{2 (3 k+2 M)}\\, i\\, f^{ a b c} \\, t^c_{k \\bar{l}}\\,\\delta ^{j \\bar{l}}\\, J^b \\, J^{(\\bar{\\sigma } k)}-\\frac{ (3 k+M+3 N)}{2 (3 k+2 M)}\\,d^{ a b c} \\, t^c_{k \\bar{l}}\\,\\delta ^{j \\bar{l}}\\, J^b \\, J^{(\\bar{\\sigma } k)}\\nonumber \\\\&&+\\frac{ (k^3+k^2 M+2 k^2 N+2 k M N+2 k+M+N)}{ k (3 k+2 M)}\\,t^a_{k \\bar{l}} \\, \\delta ^{j \\bar{l}}\\, \\partial \\, J^{(\\bar{\\sigma } k)}\\Bigg ](w)\\nonumber \\\\&& + \\frac{1}{(z-w)}\\,J^{(\\bar{\\sigma } j)}(z) \\, P^{a}(w)\\Bigg |_{\\frac{1}{(z-w)}}+ \\cdots .\\nonumber $ The first order pole can be summarized by ${.",
"}6\\endcsname &&J^{(\\bar{\\sigma } j)}(z) \\, P^{a}(w)\\Bigg |_{\\frac{1}{(z-w)}}=(t^a_{l \\bar{k}} \\, \\delta _{\\sigma \\bar{\\nu }} \\,J^{(\\bar{\\nu } j)}\\, J^{(\\sigma \\bar{k})}\\, J^{(\\bar{\\rho } l)}-\\frac{1}{N}\\, t^a_{l \\bar{k}} \\, \\delta _{\\sigma \\bar{\\tau }} \\,J^{(\\bar{\\rho } j)}\\, J^{(\\sigma \\bar{k})}\\, J^{(\\bar{\\tau } l)}\\nonumber \\\\&& -\\sqrt{\\frac{M+N}{M N}}\\,t^a_{l \\bar{k}} \\, t^{\\alpha }_{\\sigma \\bar{\\tau }} \\,\\delta ^{\\sigma \\bar{\\rho }} \\, \\delta ^{j \\bar{k}} \\,J^{\\alpha }\\, J^{u(1)}\\, J^{(\\bar{\\tau } l)}-\\frac{1}{N}\\, t^a_{l \\bar{k}} \\,\\delta ^{j \\bar{k}}\\, J^{\\alpha }\\, J^{\\alpha }\\, J^{(\\bar{\\rho } l)}\\nonumber \\\\&& -\\frac{1}{2}\\, (i \\, f +d)^{\\beta \\alpha \\gamma } \\,t^a_{l \\bar{k}} \\, t^{\\gamma }_{\\nu \\bar{\\tau }} \\,\\delta ^{\\nu \\bar{\\rho }} \\, \\delta ^{j \\bar{k}} \\,J^{\\alpha }\\, J^{\\beta }\\, J^{(\\bar{\\tau } l)}+ \\frac{1}{M} \\, t^{\\alpha }_{\\sigma \\bar{\\tau }} \\,\\delta ^{\\sigma \\bar{\\rho }} \\,J^{\\alpha }\\, J^{a}\\, J^{(\\bar{\\tau } j)} \\nonumber \\\\&& +\\frac{1}{2}\\, (i \\, f +d)^{a b c} \\,t^c_{l \\bar{i}} \\, t^{\\alpha }_{\\sigma \\bar{\\tau }} \\,\\delta ^{\\sigma \\bar{\\rho }} \\, \\delta ^{j \\bar{i}} \\,J^{\\alpha }\\, J^{b}\\, J^{(\\bar{\\tau } l)})\\, a_1 + (-t^{a}_{k \\bar{j}} \\, \\delta ^{j \\bar{j}} \\,J^{(\\bar{\\rho } k)} \\, J^{\\alpha }\\, J^{\\alpha }+ t^{\\alpha }_{\\rho \\bar{\\sigma }} \\, \\delta ^{\\rho \\bar{\\rho }} \\,J^a \\, J^{(\\bar{\\sigma } j)} \\, J^{\\alpha }\\nonumber \\\\&& + t^{\\alpha }_{\\rho \\bar{\\sigma }} \\, \\delta ^{\\rho \\bar{\\rho }} \\,J^a \\, J^{\\alpha } \\, J^{(\\bar{\\sigma } j)})\\, a_2 + (- t^{a}_{k \\bar{j}} \\, \\delta ^{j \\bar{j}} \\,J^{(\\bar{\\rho } k)} \\, J^{b}\\, J^{b} -t^{b}_{k \\bar{j}} \\, \\delta ^{j \\bar{j}} \\,J^a \\, J^{(\\bar{\\rho } k)} \\, J^{b}-t^{b}_{k \\bar{j}} \\, \\delta ^{j \\bar{j}} \\,J^a \\, J^b\\, J^{(\\bar{\\rho } k)} \\nonumber \\\\&& + M \\,t^{a}_{k \\bar{j}} \\, \\delta ^{j \\bar{j}} \\,\\partial ^2 \\, J^{(\\bar{\\rho } k)}-2 \\, i\\, f^{b a c}\\,t^{b}_{k \\bar{j}} \\, \\delta ^{j \\bar{j}} \\,J^{(\\bar{\\rho } k)}\\, \\partial \\, J^c-2 \\, i\\, f^{b a c}\\,t^{c}_{k \\bar{j}} \\, \\delta ^{j \\bar{j}} \\,J^b\\, \\partial \\, J^{(\\bar{\\rho } k)})\\, a_3 \\nonumber \\\\&& (-t^{a}_{k \\bar{j}} \\, \\delta ^{j \\bar{j}} \\,J^{(\\bar{\\rho } k)}\\, J^{u(1)}\\, J^{u(1)}+\\sqrt{\\frac{M+N}{M N}}\\,J^a \\, J^{(\\bar{\\rho } j)}\\, J^{u(1)}+ \\sqrt{\\frac{M+N}{M N}}\\,J^a \\, J^{u(1)} \\,J^{(\\bar{\\rho } j)})\\, a_4 \\nonumber \\\\&& + (- d^{ a b c}\\,t^{c}_{l \\bar{l}} \\, \\delta ^{j \\bar{l}} \\,t^b_{k \\bar{j}}\\, \\delta _{\\sigma \\bar{\\sigma }}\\,J^{(\\bar{\\rho } l)} \\, J^{(\\sigma \\bar{j})} \\, J^{(\\bar{\\sigma } k)}-d^{ a b c}\\,t^{c}_{l \\bar{l}} \\, \\delta ^{j \\bar{l}} \\,t^b_{k \\bar{j}}\\, \\delta _{\\sigma \\bar{\\sigma }}\\,J^{(\\bar{\\rho } l)} \\, J^{(\\bar{\\sigma } k)} \\, J^{(\\sigma \\bar{j})}\\nonumber \\\\&& -2 \\, \\sqrt{\\frac{M+N}{M N}}\\,d^{ a b c}\\,t^{b}_{k \\bar{l}} \\, \\delta ^{j \\bar{l}} \\,J^{c} \\, J^{(\\bar{\\rho } k)} \\, J^{u(1)}-2 \\,d^{ a b c}\\,t^{\\alpha }_{\\rho \\bar{\\sigma }} \\, \\delta ^{\\rho \\bar{\\rho }} \\,t^b_{k \\bar{l}}\\, \\delta ^{j \\bar{l}}\\,J^{c} \\, J^{(\\bar{\\sigma } k)} \\, J^{\\alpha }\\nonumber \\\\&& +\\frac{2}{M}\\,d^{ a b c}\\,J^{c} \\, J^{(\\bar{\\rho } j)} \\, J^{b}+(i \\, f +d)^{b e d} \\, d^{a b c}\\,t^d_{k \\bar{j}} \\, \\delta ^{j \\bar{j}} \\,J^{c}\\, J^{(\\bar{\\rho } k)}\\, J^e+ N \\,d^{a b c}t^b_{k \\bar{j}} \\, \\delta ^{j \\bar{j}} \\,J^{c}\\, \\partial \\, J^{(\\bar{\\rho } k)})\\, a_5\\nonumber \\\\&& +(\\sqrt{\\frac{M+N}{M N}}\\,t^{a}_{l \\bar{k}} \\,\\delta _{\\sigma \\bar{\\tau }}\\,J^{(\\bar{\\rho } j)} \\, J^{(\\sigma \\bar{k})} \\, J^{(\\bar{\\tau } l)}+\\sqrt{\\frac{M+N}{M N}}\\,t^{a}_{l \\bar{k}} \\,\\delta _{\\sigma \\bar{\\tau }}\\,J^{(\\bar{\\rho } j)} \\, J^{(\\bar{\\tau } l)}\\, J^{(\\sigma \\bar{k})}\\nonumber \\\\&& -2 \\, \\sqrt{\\frac{M+N}{M N}}\\,t^{a}_{k \\bar{l}} \\,\\delta ^{j \\bar{l}}\\,J^{u(1)} \\, J^{(\\bar{\\rho } k)}\\, J^{u(1)}-2 \\,t^{\\alpha }_{\\rho \\bar{\\sigma }} \\, \\delta ^{\\rho \\bar{\\rho }} \\, \\delta ^{j \\bar{l}} \\,t^a_{k \\bar{l}}\\, J^{u(1)}\\, J^{(\\bar{\\sigma } k)}\\, J^{\\alpha }+ \\frac{2}{M}\\, J^{u(1)}\\, J^{(\\bar{\\rho } j)}\\, J^{a}\\nonumber \\\\&& + (i \\, f +d)^{a b c} \\,t^c_{k \\bar{j}} \\, \\delta ^{j \\bar{j}} \\,J^{u(1)}\\, J^{(\\bar{\\rho } k)}\\, J^b+N \\,t^{a}_{k \\bar{j}} \\,\\delta ^{j \\bar{j}}\\,J^{u(1)} \\, \\partial \\, J^{(\\bar{\\rho } k)}) \\, a_7\\nonumber \\\\&& + (t^{a}_{m \\bar{j}} \\,\\delta ^{j \\bar{j}}\\, \\delta _{\\sigma \\bar{\\tau }}\\, \\delta _{k \\bar{j}}\\,J^{(\\bar{\\rho } m)} \\, J^{(\\sigma \\bar{j})} \\, J^{(\\bar{\\tau } k)}+ t^{a}_{m \\bar{j}} \\,\\delta ^{j \\bar{j}}\\, \\delta _{\\sigma \\bar{\\tau }}\\, \\delta _{k \\bar{j}}\\,J^{(\\bar{\\rho } m)} \\, J^{(\\bar{\\tau } k)}\\, J^{(\\sigma \\bar{j})}\\nonumber \\\\&& - 2 \\,\\sqrt{\\frac{M+N}{M N}}\\,J^{a} \\, J^{(\\bar{\\rho } j)}\\, J^{u(1)}-2 \\, t^{\\alpha }_{\\rho \\bar{\\tau }} \\,\\delta ^{\\rho \\bar{\\rho }}\\,J^{a} \\, J^{(\\bar{\\tau } j)}\\, J^{\\alpha }+ 2 \\,t^{b}_{k \\bar{j}} \\,\\delta ^{j \\bar{j}}\\,J^{a} \\, J^{(\\bar{\\rho } k)}\\, J^b\\nonumber \\\\&& +(M+N)\\, J^a \\, \\partial \\, J^{(\\bar{\\rho } j)})\\, a_8+(\\sqrt{\\frac{M+N}{M N}}\\,d^{a b c}\\delta ^{\\tau \\bar{\\rho }} \\, \\delta ^{j \\bar{k}} \\,J^{(\\bar{\\tau } k)} \\, J^b \\, J^c\\nonumber \\\\&&- d^{a b c}t^b_{k \\bar{j}} \\, \\delta ^{j \\bar{j}} \\,J^{u(1)} \\, J^{(\\bar{\\rho } k)} \\, J^c-d^{a b c}t^c_{k \\bar{j}} \\, \\delta ^{j \\bar{j}} \\,J^{u(1)} \\, J^b\\, J^{(\\bar{\\rho } k)})\\, a_9 \\nonumber \\\\&& +(- i\\, f^{a b c}t^b_{k \\bar{j}} \\, \\delta ^{j \\bar{j}} \\,\\partial \\, J^{(\\bar{\\rho } k)} \\, J^c-i\\, f^{a b c}t^c_{k \\bar{j}} \\, \\delta ^{j \\bar{j}} \\,\\partial \\,J^b\\, J^{(\\bar{\\rho } k)}) \\, a_{11}\\nonumber \\\\&& + (-\\sqrt{\\frac{M+N}{M N}}\\,t^a_{l \\bar{k}} \\, \\delta ^{j \\bar{k}} \\,J^{(\\bar{\\rho } l)} \\, \\partial \\, J^{u(1)} -t^{\\alpha }_{\\rho \\bar{\\tau }} \\, \\delta ^{\\rho \\bar{\\rho }} \\, \\delta ^{j \\bar{k}} \\,t^a_{l \\bar{k}}\\, J^{(\\bar{\\tau } l)}\\, \\partial \\, J^{\\alpha }+ \\frac{1}{M}\\, J^{(\\bar{\\rho } j)}\\, \\partial \\, J^a\\nonumber \\\\&& + \\frac{1}{2}\\,(i \\, f +d)^{a b c} \\,t^c_{l \\bar{j}} \\, \\delta ^{j \\bar{j}} \\,J^{(\\bar{\\rho } l)}\\, \\partial \\, J^b+\\frac{N}{2}\\,t^a_{k \\bar{j}}\\, \\delta ^{j \\bar{j}} \\,\\partial ^2 \\, J^{(\\bar{\\rho } k)})\\, a_{12}\\nonumber \\\\&&+ (-\\sqrt{\\frac{M+N}{M N}}\\,t^a_{k \\bar{l}} \\, \\delta ^{j \\bar{l}} \\,\\partial \\, J^{(\\bar{\\rho } k)} \\, J^{u(1)}-t^{\\alpha }_{\\rho \\bar{\\sigma }} \\, \\delta ^{\\rho \\bar{\\rho }} \\, \\delta ^{j \\bar{l}} \\,t^a_{k \\bar{l}}\\, \\partial \\, J^{(\\bar{\\rho } k)} \\, J^{\\alpha }+ \\frac{1}{M}\\, \\partial \\, J^{(\\bar{\\rho } j)} \\, J^a\\nonumber \\\\&& +\\frac{1}{2}\\,(i \\, f +d)^{a b c} \\,t^c_{k \\bar{j}} \\, \\delta ^{j \\bar{j}} \\,\\partial \\, J^{(\\bar{\\rho } k)}\\, J^b)\\, a_{13} -t^a_{k \\bar{j}}\\, \\delta ^{j \\bar{j}} \\, \\partial ^2 \\,J^{(\\bar{\\rho } k)}\\, a_{16}\\nonumber \\\\&& + (-\\frac{2}{M}\\,t^a_{k \\bar{j}} \\, \\delta ^{j \\bar{j}} \\,J^{(\\bar{\\rho } k)}\\, \\, J^b\\, J^b-\\frac{1}{2}\\,(i \\, f +d)^{a b e} \\,d^{e c d}\\,t^b_{k \\bar{j}} \\, \\delta ^{j \\bar{j}} \\,J^{(\\bar{\\rho } k)}\\,J^c \\, J^d-\\frac{2}{M}\\,t^b_{k \\bar{j}} \\, \\delta ^{j \\bar{j}} \\,J^{(\\bar{\\rho } k)}\\, \\, J^b\\, J^a\\nonumber \\\\&& -\\frac{1}{2}\\,(i \\, f +d)^{a d e} \\,d^{e b c}\\,t^b_{k \\bar{j}} \\, \\delta ^{j \\bar{j}} \\,J^{(\\bar{\\rho } k)}\\,J^c \\, J^d-\\frac{2}{M}\\,t^b_{k \\bar{j}} \\, \\delta ^{j \\bar{j}} \\,J^{(\\bar{\\rho } k)}\\, \\, J^a\\, J^b\\nonumber \\\\&&-\\frac{1}{2}\\,(i \\, f +d)^{a c e} \\,d^{e b d}\\,t^b_{k \\bar{j}} \\, \\delta ^{j \\bar{j}} \\,J^{(\\bar{\\rho } k)}\\,J^c \\, J^d-\\frac{2}{M}\\,t^c_{k \\bar{j}} \\, \\delta ^{j \\bar{j}} \\,J^a\\, J^{(\\bar{\\rho } k)}\\, \\, J^c\\nonumber \\\\&&-\\frac{1}{2}\\,(i \\, f +d)^{a b e} \\,d^{e c d}\\,t^c_{k \\bar{j}} \\, \\delta ^{j \\bar{j}} \\,J^b\\, J^{(\\bar{\\rho } k)}\\, J^d-\\frac{2}{M}\\,t^b_{k \\bar{j}} \\, \\delta ^{j \\bar{j}} \\,J^b\\, J^{(\\bar{\\rho } k)}\\, \\, J^a\\nonumber \\\\&& -\\frac{1}{2}\\,(i \\, f +d)^{a d e} \\,d^{e b c}\\,t^c_{k \\bar{j}} \\, \\delta ^{j \\bar{j}} \\,J^b\\, J^{(\\bar{\\rho } k)}\\, J^d-\\frac{2}{M}\\,t^a_{k \\bar{j}} \\, \\delta ^{j \\bar{j}} \\,J^b\\, J^{(\\bar{\\rho } k)}\\, \\, J^b\\nonumber \\\\&&-\\frac{1}{2}\\,(i \\, f +d)^{a c e} \\,d^{e b d}\\,t^c_{k \\bar{j}} \\, \\delta ^{j \\bar{j}} \\,J^b\\, J^{(\\bar{\\rho } k)}\\, J^d-\\frac{2}{M}\\,t^c_{k \\bar{j}} \\, \\delta ^{j \\bar{j}} \\,J^a\\, J^c\\, J^{(\\bar{\\rho } k)}\\nonumber \\\\&& -\\frac{1}{2}\\,(i \\, f +d)^{a b e} \\,d^{e c d}\\,t^d_{k \\bar{j}} \\, \\delta ^{j \\bar{j}} \\,J^b\\, J^c\\, J^{(\\bar{\\rho } k)}-\\frac{2}{M}\\,t^a_{k \\bar{j}} \\, \\delta ^{j \\bar{j}} \\,J^b \\, J^b\\, J^{(\\bar{\\rho } k)}\\nonumber \\\\&& -\\frac{1}{2}\\,(i \\, f +d)^{a d e} \\,d^{e b c}\\,t^d_{k \\bar{j}} \\, \\delta ^{j \\bar{j}} \\,J^b\\, J^c\\, J^{(\\bar{\\rho } k)}-\\frac{2}{M}\\,t^b_{k \\bar{j}} \\, \\delta ^{j \\bar{j}} \\,J^b \\, J^a \\, J^{(\\bar{\\rho } k)}\\nonumber \\\\&& -\\frac{1}{2}\\,(i \\, f +d)^{a c e} \\,d^{e b d}\\,t^d_{k \\bar{j}} \\, \\delta ^{j \\bar{j}} \\,J^b\\, J^c\\, J^{(\\bar{\\rho } k)}) \\, a_{17},\\nonumber $ where the coefficients are in (REF )."
],
[
"Some OPEs in terms of coset fields in section 6",
"We present the coset realizations for the following OPEs ${.",
"}1\\endcsname G^{+,a}(z) \\, G^{-,b}(w) &= &\\frac{1}{(z-w)^3}\\, k N \\,\\delta ^{a b}\\nonumber \\\\& + &\\frac{1}{(z-w)^2} \\, \\Bigg [ \\frac{(k+M+N)}{M} \\, K +\\frac{N}{2} \\, (i f - d)^{a b c} \\, J^c +\\frac{k}{2} \\,( i f + d)^{a b c} \\, J_f^c \\Bigg ](w)\\nonumber \\\\&+& \\frac{1}{(z-w)}\\,\\Bigg [ (t^b \\, t^a)_{l\\bar{i}}\\,\\delta _{\\rho \\bar{\\nu }} \\, J^{(\\rho \\bar{i})}\\, J^{(\\bar{\\nu } l)}- k \\, (t^a \\, t^b)_{j\\bar{k}}\\,\\delta _{\\tau \\bar{\\sigma }} \\,\\psi ^{(\\tau \\bar{k})}\\, \\partial \\, \\psi ^{(\\bar{\\sigma } j)}\\nonumber \\\\&-& \\sqrt{\\frac{M+N}{M N}} \\,(t^a \\, t^b)_{j\\bar{k}}\\,\\delta _{\\tau \\bar{\\sigma }} \\, J^{u(1)}\\,\\psi ^{(\\tau \\bar{k})} \\, \\psi ^{(\\bar{\\sigma } j)}-t^{\\alpha }_{\\sigma \\bar{\\tau }}\\, (t^a \\, t^b)_{j\\bar{k}}\\,J^{\\alpha }\\,\\psi ^{(\\tau \\bar{k})} \\, \\psi ^{(\\bar{\\sigma } j)}\\nonumber \\\\&+ & (t^a \\, t^c\\, t^b)_{j\\bar{k}}\\,\\delta _{\\tau \\bar{\\sigma }} \\,\\psi ^{(\\tau \\bar{k})} \\, J^c\\, \\psi ^{(\\bar{\\sigma } j)}\\Bigg ](w) +\\cdots .\\nonumber $ The first order pole can be simplified and is given by (REF ).",
"The OPEs with the spin-2 currents are ${.",
"}2\\endcsname G^{+,a}(z) \\, K^b(w) &=&-\\frac{1}{(z-w)^2}\\, \\frac{2(k^2-1)(2k+M+N)}{k(2k+M)}\\,(t^a \\, t^b)_{j \\bar{i}}\\, \\delta _{\\rho \\bar{\\sigma }} \\, J^{(\\rho \\bar{i})}\\,\\psi ^{(\\bar{\\sigma } j)}\\nonumber \\\\&+& \\frac{1}{(z-w)}\\, \\Bigg [-\\frac{2(k^2-1)(2k+M+N)}{k(2k+M)}\\,(t^a \\, t^b)_{j \\bar{i}}\\, \\delta _{\\rho \\bar{\\sigma }} \\, \\partial \\, (J^{(\\rho \\bar{i})}\\,\\psi ^{(\\bar{\\sigma } j)})\\nonumber \\\\&+&2(k+N) \\, (t^a \\, t^b)_{j \\bar{i}}\\, \\delta _{\\rho \\bar{\\sigma }} \\,\\partial \\,J^{(\\rho \\bar{i})}\\,\\psi ^{(\\bar{\\sigma } j)} -\\frac{2}{k M}\\, (k+M+N)\\, J^b \\, G^{+,a}\\nonumber \\\\&-& \\frac{2}{k}\\, (k+N)\\, \\sqrt{\\frac{M+N}{M N}}\\,(t^a \\, t^b)_{j \\bar{l}}\\, \\delta _{\\rho \\bar{\\sigma }} \\,J^{u(1)}\\, J^{(\\rho \\bar{l})}\\,\\psi ^{(\\bar{\\sigma } j)}\\nonumber \\\\&-& (i \\, f -\\frac{2k+M+2N}{2k+M})^{b c d}\\,(t^a \\, t^d)_{j \\bar{l}} \\,\\delta _{\\rho \\bar{\\sigma }}\\,J^{c}\\, J^{(\\rho \\bar{l})}\\,\\psi ^{(\\bar{\\sigma } j)}\\nonumber \\\\&-& 2 \\, t^{\\alpha }_{\\sigma \\bar{\\sigma }}\\,(t^a \\, t^b)_{j \\bar{l}} \\,J^{\\alpha }\\, J^{(\\sigma \\bar{l})}\\,\\psi ^{(\\bar{\\sigma } j)}\\Bigg ](w) + \\cdots ,\\nonumber $ and ${.",
"}3\\endcsname G^{-,a}(z) \\, K^b(w) &=&-\\frac{1}{(z-w)^2}\\, \\frac{2(k^2-1)(2k+M+N)}{k(2k+M)}\\,(t^b \\, t^a)_{k \\bar{i}}\\, \\delta _{\\rho \\bar{\\sigma }} \\,J^{(\\bar{\\sigma } k)}\\,\\psi ^{(\\rho \\bar{i})}\\nonumber \\\\&+& \\frac{1}{(z-w)}\\, \\Bigg [-\\frac{2(k^2-1)(2k+M+N)}{k(2k+M)}\\,(t^b \\, t^a)_{k \\bar{i}}\\, \\delta _{\\rho \\bar{\\sigma }} \\,\\partial \\, (J^{(\\bar{\\sigma } k)}\\,\\psi ^{(\\rho \\bar{i})})\\nonumber \\\\&+&2(k+N) \\, (t^b \\, t^a)_{k \\bar{i}}\\, \\delta _{\\rho \\bar{\\sigma }} \\,\\partial \\,J^{(\\bar{\\sigma } k)}\\,\\psi ^{(\\rho \\bar{i})} +\\frac{2}{k M}\\, (k+M+N)\\, J^b \\, G^{-,a}\\nonumber \\\\&+& \\frac{2}{k}\\, (k+N)\\, \\sqrt{\\frac{M+N}{M N}}\\,(t^b \\, t^a)_{k \\bar{l}}\\, \\delta _{\\rho \\bar{\\sigma }} \\,J^{u(1)}\\, J^{(\\bar{\\sigma } k)}\\,\\psi ^{(\\rho \\bar{i})}\\nonumber \\\\&-& (i \\, f +\\frac{2k+M+2N}{2k+M})^{b c d}\\,(t^d \\, t^a)_{k \\bar{i}} \\,\\delta _{\\rho \\bar{\\sigma }}\\,J^{c}\\, J^{(\\bar{\\sigma } k)}\\,\\psi ^{(\\rho \\bar{i} )}\\nonumber \\\\&+& 2 \\, t^{\\alpha }_{\\rho \\bar{\\sigma }}\\,(t^b \\, t^a)_{k \\bar{i}} \\,J^{\\alpha }\\, J^{(\\bar{\\sigma } k)}\\,\\psi ^{(\\rho \\bar{i} )}\\Bigg ](w) + \\cdots .\\nonumber $ They are simplified and are described in (REF ) and (REF )."
],
[
" The OPEs between the nonsinglet spin-2 operators in section 6 ",
"In the computation of the OPE between the nonsinglet spin-2 current (REF ) and itself, we should use the following OPEs.",
"$\\bullet $ The OPEs between the first term and the remaining terms ${.",
"}1\\endcsname && K^a(z) \\, J^b\\, J^{u(1)}_f(w) = \\frac{1}{(z-w)}\\,i \\, f^{a b c}\\, J^{u(1)}_f \\, K^c(w)+ \\cdots ,\\nonumber \\\\&& K^a(z) \\, (i \\, f+ d)^{c b d} \\, J^c\\, J^{d}_f(w) =-\\frac{1}{(z-w)}\\,(i \\, + d)^{c b d} \\, i\\, f^{c a e} \\, J^{d}_f \\, K^e(w)+ \\cdots ,\\nonumber \\\\&& K^a(z) \\, d^{b c d} \\, J^c\\, J^{d}(w) =\\frac{1}{(z-w)^2}\\, M \\, d^{a b c}\\, K^c \\nonumber \\\\&& +\\frac{1}{(z-w)}\\, \\Big [ i \\, f^{a c e} \\, d^{b c d}\\,K^e \\, J^d + i \\, f^{a d e}\\, d^{b c d}\\, J^c \\, K^e\\Big ](w) + \\cdots ,\\nonumber \\\\&& K^a(z) \\, J^b\\, J^{u(1)}(w) =\\frac{1}{(z-w)}\\, i\\, f^{a b c} \\, J^{u(1)}\\, K^c(w)+\\cdots .\\nonumber $ $\\bullet $ The OPEs between the second term and the remaining terms ${.",
"}2\\endcsname && t^a_{j \\bar{i}} \\, \\delta _{\\rho \\bar{\\sigma }} \\,\\psi ^{(\\rho \\bar{i})}\\,\\partial \\psi ^{(\\bar{\\sigma } j)}(z) \\,t^b_{k \\bar{l}} \\, \\delta _{\\mu \\bar{\\nu }} \\,\\psi ^{(\\mu \\bar{l})}\\,\\partial \\psi ^{(\\bar{\\nu } k)} =-\\frac{1}{(z-w)^4}\\,N \\, \\delta ^{a b} \\nonumber \\\\&& +\\frac{1}{(z-w)^2}\\, \\Big [-\\frac{2}{M}\\, \\delta ^{a b}\\,\\delta _{k \\bar{i}} \\, \\delta _{\\rho \\bar{\\nu }}\\, \\psi ^{(\\rho \\bar{i})}\\,\\partial \\psi ^{(\\bar{\\nu } k)} - d^{a b c} \\, t^c_{k \\bar{i}}\\, \\delta _{\\rho \\bar{\\nu }}\\, \\psi ^{(\\rho \\bar{i})}\\,\\partial \\psi ^{(\\bar{\\nu } k)}\\Big ](w)\\nonumber \\\\&&+ \\frac{1}{(z-w)}\\, \\Big [-\\frac{1}{M}\\, \\delta ^{a b}\\,\\delta _{k \\bar{i}} \\, \\delta _{\\rho \\bar{\\nu }}\\,\\partial \\, \\psi ^{(\\rho \\bar{i})}\\,\\partial \\psi ^{(\\bar{\\nu } k)}-\\frac{1}{2}\\, (i \\, f+d)^{b a c}\\, t^c_{k \\bar{i}}\\,\\delta _{\\rho \\bar{\\nu }}\\, \\partial \\, \\psi ^{(\\rho \\bar{i})}\\,\\partial \\, \\psi ^{(\\bar{\\nu } k)}\\nonumber \\\\&& - \\frac{1}{M}\\, \\delta ^{a b}\\,\\delta _{k \\bar{i}} \\, \\delta _{\\rho \\bar{\\nu }}\\,\\psi ^{(\\rho \\bar{i})}\\,\\partial ^2 \\psi ^{(\\bar{\\nu } k)}-\\frac{1}{2}\\, (i \\, f+d)^{ a b c}\\, t^c_{k \\bar{i}}\\,\\delta _{\\rho \\bar{\\nu }}\\, \\psi ^{(\\rho \\bar{i})}\\,\\partial ^2\\, \\psi ^{(\\bar{\\nu } k)}\\Big ](w) + \\cdots ,\\nonumber \\\\&& t^a_{j \\bar{i}} \\, \\delta _{\\rho \\bar{\\sigma }} \\,\\psi ^{(\\rho \\bar{i})}\\,\\partial \\psi ^{(\\bar{\\sigma } j)}(z) \\,J^{u(1)}\\, J_f^b(w) =\\frac{1}{(z-w)^3}\\, N \\, \\delta ^{a b}\\, J^{u(1)}(w)\\nonumber \\\\&&+\\frac{1}{(z-w)^2}\\, \\Big [\\frac{1}{M}\\, \\delta ^{a b} \\, J^{u(1)}\\, J_f^{u(1)}-\\frac{1}{2}\\, (i \\, f+ d)^{b a c}\\, J^{u(1)}\\,J_f^{c}\\Big ](w)\\nonumber \\\\&&+ \\frac{1}{(z-w)}\\, \\Big [\\frac{1}{M}\\, \\delta ^{a b} \\, J^{u(1)}\\, \\partial \\, J_f^{u(1)}-\\frac{1}{2}\\, (i \\, f+ d)^{b a c}\\, J^{u(1)}\\,\\partial \\, J_f^{c}\\nonumber \\\\&& +\\frac{1}{2}\\,(i \\, f+ d)^{a b c}\\, t^c_{j \\bar{k}}\\, \\delta _{\\rho \\bar{\\sigma }}\\, J^{u(1)}\\,\\psi ^{(\\rho \\bar{k})}\\,\\partial \\psi ^{(\\bar{\\sigma } j)}\\nonumber \\\\&& -\\frac{1}{2}\\,(i \\, f+ d)^{ b a c}\\, t^c_{l \\bar{i}}\\, \\delta _{\\rho \\bar{\\sigma }}\\, J^{u(1)}\\,\\psi ^{(\\rho \\bar{i})}\\,\\partial \\psi ^{(\\bar{\\sigma } l)}\\Big ](w)+ \\cdots ,\\nonumber \\\\&& t^a_{j \\bar{i}} \\, \\delta _{\\rho \\bar{\\sigma }} \\,\\psi ^{(\\rho \\bar{i})}\\,\\partial \\psi ^{(\\bar{\\sigma } j)}(z) \\,t^{\\alpha }_{\\mu \\bar{\\nu }} \\, t^b_{k \\bar{l}} \\, J^{\\alpha } \\,\\psi ^{(\\mu \\bar{l})}\\,\\psi ^{(\\bar{\\nu } k)}(w)=\\nonumber \\\\&& -\\frac{1}{(z-w)^2}\\, \\Big [ \\frac{1}{M}\\, \\delta ^{b a}\\, \\delta _{k \\bar{i}}+\\frac{1}{2} \\, (i \\, f + d)^{b a c}\\, t^c_{k \\bar{i}}\\Big ]t^{\\alpha }_{\\rho \\bar{\\nu }} \\, J^{\\alpha } \\,\\psi ^{(\\rho \\bar{i})}\\,\\psi ^{(\\bar{\\nu } k)}(w)\\nonumber \\\\&&+\\frac{1}{(z-w)}\\, \\Big [-\\frac{1}{M}\\, \\delta ^{b a}\\, \\delta _{k \\bar{i}}t^{\\alpha }_{\\rho \\bar{\\nu }} \\, J^{\\alpha } \\,\\partial \\, \\psi ^{(\\rho \\bar{i})}\\,\\psi ^{(\\bar{\\nu } k)}- \\frac{1}{2} \\, (i \\, f + d)^{b a c}\\, t^c_{k \\bar{i}}\\,t^{\\alpha }_{\\rho \\bar{\\nu }} \\, J^{\\alpha } \\,\\partial \\, \\psi ^{(\\rho \\bar{i})}\\,\\psi ^{(\\bar{\\nu } k)}\\nonumber \\\\&& -\\frac{1}{M}\\, \\delta ^{b a}\\, \\delta _{k \\bar{i}}t^{\\alpha }_{\\rho \\bar{\\nu }} \\, J^{\\alpha } \\,\\psi ^{(\\rho \\bar{i})}\\,\\partial \\, \\psi ^{(\\bar{\\nu } k)}- \\frac{1}{2} \\, (i \\, f + d)^{a b c}\\, t^c_{k \\bar{i}}\\,t^{\\alpha }_{\\rho \\bar{\\nu }} \\, J^{\\alpha } \\,\\psi ^{(\\rho \\bar{i})}\\,\\partial \\, \\psi ^{(\\bar{\\nu } k)}\\Big ](w)+\\cdots ,\\nonumber \\\\&& t^a_{j \\bar{i}} \\, \\delta _{\\rho \\bar{\\sigma }} \\,\\psi ^{(\\rho \\bar{i})}\\,\\partial \\psi ^{(\\bar{\\sigma } j)}(z) \\, J^b \\, J_f^{u(1)}(w) =\\frac{1}{(z-w)^2}\\, J^b \\, J^a_f(w) +\\frac{1}{(z-w)}\\, J^b \\, \\partial \\, J^a_f(w) + \\cdots ,\\nonumber \\\\&& t^a_{j \\bar{i}} \\, \\delta _{\\rho \\bar{\\sigma }} \\,\\psi ^{(\\rho \\bar{i})}\\,\\partial \\psi ^{(\\bar{\\sigma } j)}(z) \\, (i \\, f+d)^{c b d}\\, J^c \\, J^d_f(w)= \\frac{1}{(z-w)^3}\\, N \\, (i\\, f +d)^{c b a}\\, J^c\\nonumber \\\\&&+\\frac{1}{(z-w)^2}\\, \\Big [\\frac{1}{M}\\, \\delta ^{d a}\\, \\delta _{l \\bar{i}}+\\frac{1}{2}\\,(i\\, f +d)^{d a e}\\, t^e_{l \\bar{i}}\\Big ]\\, \\delta _{\\rho \\bar{\\sigma }}\\, (i\\, f +d)^{c b d}\\,J^c\\, \\psi ^{(\\rho \\bar{i})}\\, \\psi ^{(\\bar{\\sigma } l)}(w)\\nonumber \\\\&& +\\frac{1}{(z-w)} \\Big [\\frac{1}{M}\\, \\delta ^{d a}\\, \\delta _{l \\bar{i}}\\,(i\\, f +d)^{c b d}\\, t^e_{l \\bar{i}}\\, \\delta _{\\rho \\bar{\\sigma }}\\,J^c\\, \\partial \\, (\\psi ^{(\\rho \\bar{i})}\\, \\psi ^{(\\bar{\\sigma } l)})\\nonumber \\\\&& +\\frac{1}{2} \\, (i\\, f +d)^{ d a e }\\, t^e_{l \\bar{i}}\\, \\delta _{\\rho \\bar{\\sigma }}\\,(i \\, f +d)^{c b d}\\, J^c \\,\\partial \\, (\\psi ^{(\\rho \\bar{i})}\\, \\psi ^{(\\bar{\\sigma } l)})\\nonumber \\\\&& +\\frac{1}{2} \\, (i\\, f +d)^{ a d e }\\,t^e_{j \\bar{k}}\\, \\delta _{\\rho \\bar{\\sigma }}\\,(i \\, f +d)^{c b d}\\, \\delta _{\\rho \\bar{\\sigma }} \\, J^c \\,\\psi ^{(\\rho \\bar{k})}\\, \\partial \\, \\psi ^{(\\bar{\\sigma } j)}\\nonumber \\\\&& -\\frac{1}{2} \\, (i\\, f +d)^{ d a e }\\,t^e_{l \\bar{i}}\\, \\delta _{\\rho \\bar{\\sigma }}\\,(i \\, f +d)^{c b d} \\, J^c \\,\\psi ^{(\\rho \\bar{i})}\\, \\partial \\, \\psi ^{(\\bar{\\sigma } l)}\\Big ](w) +\\cdots ,\\nonumber \\\\&& t^a_{j \\bar{i}} \\, \\delta _{\\rho \\bar{\\sigma }} \\,\\psi ^{(\\rho \\bar{i})}\\,\\partial \\psi ^{(\\bar{\\sigma } j)}(z) \\,\\partial \\, J_f^b(w)=\\frac{1}{(z-w)^4}\\, 3N \\, \\delta ^{a b}\\nonumber \\\\&& + \\frac{1}{(z-w)^3} \\Big [ \\frac{2}{M} \\, \\delta ^{b a}\\, J_f^{u(1)}- (i \\, f + d)^{b a c}\\, J_f^c \\Big ](w)\\nonumber \\\\&& +\\frac{1}{(z-w)^2}\\, \\Big [ \\frac{2}{M}\\, \\delta ^{b a}\\,\\partial \\, J_f^{u(1)} - (i \\, f + d )^{b a c}\\, \\partial \\, J_f^c + i \\, f^{a b c}\\, t^c_{j \\bar{k}}\\,\\delta _{\\rho \\bar{\\sigma }}\\, \\psi ^{(\\rho \\bar{k})}\\, \\partial \\,\\psi ^{(\\bar{\\sigma } j)}\\Big ](w)\\nonumber \\\\&& + \\frac{1}{(z-w)}\\, \\Big [\\frac{1}{M}\\, \\delta ^{b a}\\,\\partial ^2 \\, J_f^{u(1)} - \\frac{1}{2} \\,(i \\, f + d )^{b a c}\\, \\partial ^2 \\, J_f^c+ i \\, f^{a b c}\\, t^c_{j \\bar{k}}\\,\\delta _{\\rho \\bar{\\sigma }}\\, \\partial \\, (\\psi ^{(\\rho \\bar{k})}\\, \\partial \\,\\psi ^{(\\bar{\\sigma } j)})\\Big ](w) + \\cdots .\\nonumber $ $\\bullet $ The OPEs between the third term and the remaining terms ${.",
"}3\\endcsname && J^{u(1)}\\, J_f^a(z) \\, J^{u(1)}\\, J_f^b(w)=\\frac{1}{(z-w)^4}\\, k \\, N \\, \\delta ^{a b}+\\frac{1}{(z-w)^3}\\, k \\, i \\, f^{a b c}\\, J_f^c\\nonumber \\\\&& + \\frac{1}{(z-w)^2}\\, \\Big [ k \\,J_f^a \\, J_f^b + N \\, \\delta ^{a b} \\, J^{u(1)}\\,J^{u(1)} \\Big ](w)\\nonumber \\\\&& + \\frac{1}{(z-w)}\\, \\Big [k \\,\\partial \\,J_f^a \\, J_f^b + N \\, \\delta ^{a b} \\, J^{u(1)}\\,\\partial \\, J^{u(1)} + i \\, f^{a b c}\\,J^{u(1)}\\, J^{u(1)}\\, J_f^c\\Big ](w)+\\cdots ,\\nonumber \\\\&& J^{u(1)}\\, J_f^a(z)\\,t^{\\alpha }_{\\mu \\bar{\\nu }} \\, t^b_{k \\bar{l}} \\, J^{\\alpha } \\,\\psi ^{(\\mu \\bar{l})}\\,\\psi ^{(\\bar{\\nu } k)}(w)=\\frac{1}{(z-w)}\\,\\Big [-\\frac{1}{M}\\, \\delta ^{a b}\\,\\delta _{j \\bar{k}}\\, t^{\\alpha }_{\\rho \\bar{\\sigma }}\\,J^{\\alpha } (( J^{u(1)}\\, \\psi ^{(\\rho \\bar{k})})\\psi ^{(\\bar{\\sigma } j)})\\nonumber \\\\&& -\\frac{1}{2}\\, (i \\, f+ d)^{b a c}\\,t^c_{j \\bar{k}}\\, t^{\\alpha }_{\\rho \\bar{\\sigma }}\\,J^{\\alpha } (( J^{u(1)}\\, \\psi ^{(\\rho \\bar{k})})\\psi ^{(\\bar{\\sigma } j)})+\\frac{1}{M}\\, \\delta ^{a b}\\,\\delta _{l \\bar{i}}\\, t^{\\alpha }_{\\rho \\bar{\\sigma }}\\,J^{\\alpha } \\, \\psi ^{(\\rho \\bar{i})}\\, J^{u(1)} \\, \\psi ^{(\\bar{\\sigma } l)}\\nonumber \\\\&&+\\frac{1}{2}\\, (i \\, f+ d)^{a b c}\\,t^c_{l \\bar{i}}\\, t^{\\alpha }_{\\rho \\bar{\\sigma }}\\,J^{\\alpha } \\, \\psi ^{(\\rho \\bar{i})}\\, J^{u(1)}\\,\\psi ^{(\\bar{\\sigma } l)} \\Big ](w) +\\cdots ,\\nonumber \\\\&& J^{u(1)}\\, J_f^a(z) \\, (i \\, f +d)^{c b d}\\, J^c \\,J_f^d(w) =\\frac{1}{(z-w)^2}\\, N \\, \\delta ^{d a}\\, (i \\, f +d)^{c b d} \\, J^c \\, J^{u(1)}(w)\\nonumber \\\\&& + \\frac{1}{(z-w)}\\, \\Big [N \\, \\delta ^{d a}\\, (i \\, f +d)^{c b d} \\, J^c \\, \\partial \\, J^{u(1)}- i \\, f^{d a e}\\, (i \\, f +d)^{c b d}\\,J^c \\, J^{u(1)}\\, J_f^e \\Big ](w)+ \\cdots ,\\nonumber \\\\&& J^{u(1)}\\, J_f^a(z) \\, \\partial \\,J_f^b(w) =\\frac{1}{(z-w)^3}\\, 2 N \\, \\delta ^{a b}\\, J^{u(1)} +\\frac{1}{(z-w)^2}\\, \\Big [2 N \\, \\delta ^{a b}\\, \\partial \\, J^{u(1)} - i \\, f^{a b c}\\, J^{u(1)}\\,J_f^c \\Big ](w)\\nonumber \\\\&& + \\frac{1}{(z-w)}\\, \\Big [N \\, \\delta ^{a b}\\, \\partial ^2 \\, J^{u(1)} - i \\, f^{a b c}\\, \\partial (J^{u(1)}\\,J_f^c) \\Big ](w)+\\cdots ,\\nonumber \\\\&&J^{u(1)}\\, J_f^a(z) \\,J^b \\, J^{u(1)}(w) = \\frac{1}{(z-w)^2}\\, k \\, J^b \\, J^a_f(w)\\frac{1}{(z-w)}\\, k \\, J^b \\, \\partial \\, J^a_f(w) + \\cdots .\\nonumber $ $\\bullet $ The OPEs between the fourth term and the remaining terms ${.",
"}4\\endcsname && t^{\\alpha }_{\\rho \\bar{\\sigma }} \\, t^a_{j \\bar{i}} \\, J^{\\alpha } \\,\\psi ^{(\\rho \\bar{i})}\\,\\psi ^{(\\bar{\\sigma } j)}(z) \\,t^{\\beta }_{\\mu \\bar{\\nu }} \\, t^b_{k \\bar{l}} \\, J^{\\beta } \\,\\psi ^{(\\mu \\bar{l})}\\,\\psi ^{(\\bar{\\nu } k)}(w) =\\frac{1}{(z-w)^4}\\, k \\, (N^2-1)\\, \\delta ^{a b}\\nonumber \\\\&& -\\frac{1}{(z-w)^3} \\, k \\, (N -\\frac{1}{N})\\, i \\, f^{a b c}\\,J_f^c(w) \\nonumber \\\\&& +\\frac{1}{(z-w)^2}\\, \\Big [ k \\, (\\delta _{\\rho \\bar{\\nu }} \\, \\delta _{\\mu \\bar{\\sigma }} -\\frac{1}{N} \\,\\delta _{\\rho \\bar{\\sigma }} \\, \\delta _{\\mu \\bar{\\nu }} )\\, t^a_{j \\bar{i}}\\, t^b_{k \\bar{l}}\\, (\\psi ^{(\\rho \\bar{i})}\\,\\psi ^{(\\bar{\\sigma } j)})(\\psi ^{(\\mu \\bar{l})}\\, \\psi ^{(\\bar{\\nu } k)})\\nonumber \\\\&& -N \\, (\\frac{2}{M}\\, \\delta ^{a b}\\, \\delta _{k \\bar{i}}\\, t^{\\gamma }_{\\rho \\bar{\\nu }}+ d^{a b c}\\, t^{\\gamma }_{\\rho \\bar{\\nu }} \\, t^c_{k \\bar{i}})\\, J^{\\gamma }\\, \\psi ^{(\\rho \\bar{i})}\\, \\psi ^{(\\bar{\\nu } k)}+ \\delta ^{a b}\\, J^{\\alpha }\\, J^{\\alpha } \\Big ](w)\\nonumber \\\\&&+ \\frac{1}{(z-w)}\\, \\Big [k \\, (\\delta _{\\rho \\bar{\\nu }} \\, \\delta _{\\mu \\bar{\\sigma }} -\\frac{1}{N} \\,\\delta _{\\rho \\bar{\\sigma }} \\, \\delta _{\\mu \\bar{\\nu }} ) \\, t^a_{j \\bar{i}}\\, t^b_{k \\bar{l}}\\, (\\partial \\,(\\psi ^{(\\rho \\bar{i})}\\,\\psi ^{(\\bar{\\sigma } j)}))(\\psi ^{(\\mu \\bar{l})}\\, \\psi ^{(\\bar{\\nu } k)})\\nonumber \\\\&& - i\\, f^{\\beta \\alpha \\gamma }\\, t^{\\alpha }_{\\rho \\bar{\\sigma }}\\, t^{\\beta }_{\\mu \\bar{\\nu }} \\, t^a_{j \\bar{i}}\\, t^b_{k \\bar{l}}\\,(J^{\\gamma }\\,\\psi ^{(\\rho \\bar{i})}\\,\\psi ^{(\\bar{\\sigma } j)} )(\\psi ^{(\\mu \\bar{l})}\\, \\psi ^{(\\bar{\\nu } k)})+ \\delta ^{a b}\\, J^{\\alpha }\\, \\partial \\, J^{\\alpha } \\nonumber \\\\&& +(\\frac{1}{N}\\, \\delta ^{\\alpha \\beta }\\, \\delta _{\\rho \\bar{\\nu }} +\\frac{1}{2} \\, (i \\, f+ d)^{\\alpha \\beta \\gamma } \\, t^{\\gamma }_{\\rho \\bar{\\nu }})(\\frac{1}{M}\\, \\delta ^{b a}\\, \\delta _{k \\bar{i}} +\\frac{1}{2} \\, (i \\, f+ d)^{b a c} \\, t^{c}_{k \\bar{i}})\\, J^{\\beta }\\, J^{\\alpha }\\, \\psi ^{(\\rho \\bar{i})}\\, \\psi ^{(\\bar{\\nu } k)}\\nonumber \\\\&& -(\\frac{1}{N}\\, \\delta ^{\\alpha \\beta }\\, \\delta _{\\mu \\bar{\\sigma }} +\\frac{1}{2} \\, (i \\, f+ d)^{\\beta \\alpha \\gamma } \\, t^{\\gamma }_{\\mu \\bar{\\sigma }})(\\frac{1}{M}\\, \\delta ^{b a}\\, \\delta _{j \\bar{l}} +\\frac{1}{2} \\, (i \\, f+ d)^{ a b c} \\, t^{c}_{j \\bar{l}})\\, J^{\\beta }\\, J^{\\alpha }\\, \\psi ^{(\\mu \\bar{l})}\\, \\psi ^{(\\bar{\\sigma } j)}\\Big ](w) \\nonumber \\\\&& +\\cdots ,\\nonumber \\\\&& t^{\\alpha }_{\\rho \\bar{\\sigma }} \\, t^a_{j \\bar{i}} \\, J^{\\alpha } \\,\\psi ^{(\\rho \\bar{i})}\\,\\psi ^{(\\bar{\\sigma } j)}(z) \\, (i \\, f +d)^{c b d} \\, J^c \\, J_f^d(w) =\\nonumber \\\\&& \\frac{1}{(z-w)}\\,(i \\, f+ d)^{ c b d} \\, i \\, f^{a d e} \\, t^{\\alpha }_{\\rho \\bar{\\sigma }}\\,t^{e}_{j \\bar{k}}\\, J^{c}\\, J^{\\alpha }\\, \\psi ^{(\\rho \\bar{k})}\\, \\psi ^{(\\bar{\\sigma } j)}+ \\cdots ,\\nonumber \\\\&& t^{\\alpha }_{\\rho \\bar{\\sigma }} \\, t^a_{j \\bar{i}} \\, J^{\\alpha } \\,\\psi ^{(\\rho \\bar{i})}\\,\\psi ^{(\\bar{\\sigma } j)}(z) \\, \\partial \\, J_f^b(w) =\\frac{1}{(z-w)^2}\\, i \\, f^{a b c}\\, t^{\\alpha }_{\\rho \\bar{\\sigma }}\\,t^c_{j \\bar{k}}\\, J^{\\alpha }\\, \\psi ^{(\\rho \\bar{k})}\\,\\psi ^{(\\bar{\\sigma } j)}\\nonumber \\\\&&+\\frac{1}{(z-w)}\\, i \\, f^{a b c}\\, t^{\\alpha }_{\\rho \\bar{\\sigma }}\\,t^c_{j \\bar{k}}\\, \\partial \\, (J^{\\alpha }\\, \\psi ^{(\\rho \\bar{k})}\\,\\psi ^{(\\bar{\\sigma } j)}) + \\cdots .\\nonumber $ $\\bullet $ The OPEs between the fifth term and the remaining terms ${.",
"}5\\endcsname && J^a \\, J^{u(1)}_f(z) \\,J^b \\, J^{u(1)}_f(w) =\\frac{1}{(z-w)^4}\\, k \\, M \\, N\\, \\delta ^{a b} +\\frac{1}{(z-w)^3}\\, M \\, N \\, i \\,f^{a b c}\\, J^c(w)\\nonumber \\\\&& + \\frac{1}{(z-w)^2}\\, \\Big [ k \\, \\delta ^{a b}\\, J_f^{u(1)} \\,J_f^{u(1)} + i \\, f^{a b c}\\, M \\, N \\, \\partial \\, J^c+M \\, N\\, J^b \\, J^a \\Big ](w)\\nonumber \\\\&&+ \\frac{1}{(z-w)} \\, \\Big [k \\, \\delta ^{a b}\\, \\partial \\, J_f^{u(1)} \\,J_f^{u(1)} + i \\, f^{a b c}\\, ((J^c \\, J_f^{u(1)})\\, J_f^{u(1)})+M \\, N\\, J^b \\, \\partial \\, J^a\\Big ](w) + \\cdots ,\\nonumber \\\\&& J^a \\, J^{u(1)}_f(z) \\, (i \\, f+ d)^{c b d}\\, J^c \\, J^{d}_f(w)=\\frac{1}{(z-w)^2}\\, k \\, (i \\, f +d)^{a b c}\\, J_f^c \\, J_f^{u(1)}(w)\\nonumber \\\\&&+ \\frac{1}{(z-w)}\\, \\Big [k \\, (i \\, f +d)^{a b c}\\, J_f^c \\, \\partial \\, J_f^{u(1)}-(i \\, f +d)^{c b d}\\, i\\, f^{c a e}\\, J^d_f \\, J^e\\, J_f^{u(1)}\\Big ](w) +\\cdots ,\\nonumber \\\\&& J^a \\, J^{u(1)}_f(z) \\, d^{b c d}\\, J^c \\, J^d(w)=\\frac{1}{(z-w)^2}\\, (2k +M)\\, d^{a b c}\\, J_f^{u(1)}\\, J^c(w)\\nonumber \\\\&&+ \\frac{1}{(z-w)} \\Big [(2k +M)\\, d^{a b c}\\, \\partial \\, (J_f^{u(1)}\\, J^c)-(2k +M)\\, d^{a b c}\\, J_f^{u(1)}\\, \\partial \\, J^c\\nonumber \\\\&& + i \\, f^{a b c} \\, d^{c d e}\\, J_f^{u(1)}\\, J^d \\, J^e \\Big ](w) +\\cdots ,\\nonumber \\\\&& J^a \\, J^{u(1)}_f(z) \\, J^b \\, J^{u(1)}(w)=\\frac{1}{(z-w)^2}\\, k \\, \\delta ^{a b}\\, J^{u(1)}\\, J_f^{u(1)}(w)\\nonumber \\\\&& + \\frac{1}{(z-w)}\\, \\Big [k \\, \\delta ^{a b}\\, J^{u(1)}\\, \\partial \\, J_f^{u(1)}+i \\, f^{a b c}\\, J^{u(1)}\\, J^c \\,J_f^{u(1)}\\Big ](w) + \\cdots .\\nonumber $ $\\bullet $ The OPEs between the sixth term and the remaining terms ${.",
"}6\\endcsname && (i \\, f+ d)^{f a c}\\, J^f \\, J^{c}_f(z)\\,(i \\, f+ d)^{d b e}\\, J^d \\, J^{e}_f(w)\\,= \\frac{1}{(z-w)^4}\\, k \\, N\\, (i \\, f + d)^{f a c}(i \\, f +d)^{f b c}\\nonumber \\\\&& + \\frac{1}{(z-w)^3} \\, \\Big [k \\, i \\, f^{c e g} \\, (i \\, f + d)^{f a c}(i \\, f +d)^{f b e}\\, J_f^g- N \\,i \\, f^{d f g} \\, (i \\, f + d)^{f a c}(i \\, f +d)^{d b c}\\, J^g\\Big ](w)\\nonumber \\\\&& + \\frac{1}{(z-w)^2}\\, \\Big [k\\, (i \\, f + d)^{f a c}(i \\, f +d)^{f b e}\\, J_f^c\\, J_f^e-N\\, i \\, f^{d f g}\\, (i \\, f + d)^{f a c}(i \\, f +d)^{d b c}\\,\\partial \\,J^g\\nonumber \\\\&& -f^{d h f} \\, f^{e c g}\\,(i \\, f + d)^{h a c}(i \\, f +d)^{d b e}\\, J^f\\, J_f^g+ N \\,(i \\, f + d)^{f a c}(i \\, f +d)^{d b c}\\, J^d\\, J^f\\Big ](w)\\nonumber \\\\&&+ \\frac{1}{(z-w)} \\, \\Big [k\\, (i \\, f + d)^{f a c}(i \\, f +d)^{f b e}\\, \\partial \\, J_f^c\\, J_f^e+ N \\,(i \\, f + d)^{f a c}(i \\, f +d)^{d b c}\\, J^d\\, \\partial \\, J^f\\nonumber \\\\&& - i\\, f^{d g f} \\, (i \\, f + d)^{g a c}(i \\, f +d)^{d b e}\\,((J^f \\, J_f^c) J_f^e)- i \\, f^{e c g}\\, (i \\, f + d)^{h a c}(i \\, f +d)^{d b e}\\,J^d \\, J^h \\, J_f^g\\Big ](w)\\nonumber \\\\&& + \\cdots ,\\nonumber \\\\&& (i \\, f+ d)^{f a c}\\, J^f \\, J^{c}_f(z)\\,\\partial \\, J_f^b(w)=\\frac{1}{(z-w)^3}\\, 2 \\, N \\, \\delta ^{b c}\\, (i \\, f+ d)^{d a c }\\, J^d(w)\\nonumber \\\\&& +\\frac{1}{(z-w)^2}\\, \\Big [ 2 \\, N \\, \\delta ^{b c}\\, (i \\, f+d)^{d a c}\\,\\partial \\, J^d - i \\, f^{b c g}\\, (i \\, f+d)^{d a c}\\, J^d \\, J_f^g\\Big ](w)\\nonumber \\\\&& + \\frac{1}{(z-w)}\\,\\Big [ N \\, \\delta ^{b c}\\, (i \\, f+d)^{d a c}\\,\\partial ^2 \\, J^d - i \\, f^{b c g}\\, (i \\, f+d)^{d a c}\\,\\partial \\, (J^d \\, J_f^g)\\Big ](w) + \\cdots ,\\nonumber \\\\&& (i \\, f+ d)^{f a c}\\, J^f \\, J^{c}_f(z)\\,d^{b d e}\\, J^d \\, J^e(w)=\\frac{1}{(z-w)^2}\\, (2k+M)\\, d^{d b c} \\, (i \\, f+d)^{d a e}\\,J^c \\, J_f^e(w) \\nonumber \\\\&& +\\frac{1}{(z-w)}\\, \\Big [(2k+M)\\, d^{d b c} \\, (i \\, f+d)^{d a e}\\,\\partial \\, (J^c \\, J_f^e)-(2k+M)\\, d^{d b c} \\, (i \\, f+d)^{d a e}\\,\\partial \\, J^c \\, J_f^e\\nonumber \\\\&& + i \\, f^{f b c}\\, d^{c d e}\\, (i \\, f +d)^{f a g}\\, J^d \\, J^e \\, J_f^g\\Big ](w) +\\cdots ,\\nonumber \\\\&& (i \\, f+ d)^{f a c}\\, J^f \\, J^{c}_f(z)\\,J^b \\, J^{u(1)}(w) =\\frac{1}{(z-w)^2}\\, k \\, (i \\, f +d)^{b a c}\\, J^{u(1)}\\,J_f^c \\nonumber \\\\&& +\\frac{1}{(z-w)}\\,\\Big [k \\, (i \\, f +d)^{b a c}\\, \\partial \\, (J^{u(1)}\\,J_f^c) -k \\, (i \\, f +d)^{b a c}\\, \\partial \\, J^{u(1)}\\,J_f^c \\nonumber \\\\&& +i \\, f^{e b d}\\, (i\\, f +d)^{e a c}\\, J^d \\, J^{u(1)}\\,J_f^c\\Big ](w)+\\cdots .\\nonumber $ $\\bullet $ The OPEs between the seventh term and the remaining terms ${.",
"}7\\endcsname && \\partial \\, J_f^a(z)\\, \\partial \\, J_f^b(w) =-\\frac{1}{(z-w)^4}\\, 6 \\, N\\, \\delta ^{a b}- \\frac{1}{(z-w)^3}\\, 2 \\, i \\, f^{a b c}\\, J_f^c(w)\\nonumber \\\\&& - \\frac{1}{(z-w)^2}\\, i \\, f^{a b c}\\, \\partial \\, J_f^c(w)+\\cdots .\\nonumber $ $\\bullet $ The OPEs between the eighth term and the remaining terms ${.",
"}8\\endcsname && d^{a c d}\\, J^c\\, J^d(z) \\, J^b \\, J^{u(1)}(w) =\\frac{1}{(z-w)^2}\\, (2k+M)\\, d^{b a c}\\, J^c \\, J^{u(1)}(w) \\nonumber \\\\&& +\\frac{1}{(z-w)}\\, \\Big [ (2k+M)\\, d^{b a c}\\, \\partial \\, J^c \\, J^{u(1)}-i \\, f^{b a c}\\, d^{c d e}\\, J^d \\, J^e \\, J^{u(1)} \\Big ](w) + \\cdots .\\nonumber $ $\\bullet $ The OPE between the last term and itself ${.",
"}9\\endcsname &&J^a \\, J^{u(1)}(z) \\, J^b \\, J^{u(1)}(w) =\\frac{1}{(z-w)^4}\\, k^2 \\, \\delta ^{a b}+\\frac{1}{(z-w)^3}\\, k \\, i \\, f^{a b c}\\, J^c(w)\\nonumber \\\\&& +\\frac{1}{(z-w)^2}\\, \\Big [k \\, \\delta ^{a b}\\, J^{u(1)}\\, J^{u(1)}+k \\, i \\, f^{a b c}\\, \\partial \\, J^c+ k \\, J^b \\, J^a\\Big ](w)\\nonumber \\\\&& + \\frac{1}{(z-w)}\\, \\Big [k \\, \\delta ^{a b}\\, \\partial \\, J^{u(1)}\\, J^{u(1)}+ i \\, f^{a b c}\\, ((J^c \\, J^{u(1)}) J^{u(1)})+ k \\, J^b \\, \\partial \\, J^a \\Big ](w) +\\cdots .\\nonumber $ The other OPEs where the order of the operators in the left hand side is reversed can be calculated by using the above OPEs with the appropriate contributions from the higher order poles."
],
[
"\nThe OPEs between the singlet spin-2 operators in section 7\n",
"In order to calculate the OPE in (REF ), we need to compute the following OPEs ${.",
"}1\\endcsname && \\delta _{\\rho \\bar{\\sigma }} \\delta _{j \\bar{i}} \\,J^{(\\rho \\bar{i})} J^{(\\bar{\\sigma } j)}(z) \\,\\delta _{\\mu \\bar{\\nu }} \\delta _{l \\bar{k}} \\,J^{(\\mu \\bar{k})} J^{(\\bar{\\nu } l)}(w)= \\frac{1}{(z-w)^4}\\,k N M \\, (k-N-M)\\nonumber \\\\&& + \\frac{1}{(z-w)^2}\\, \\Bigg [2 (k+N+M)^2\\, T_{boson}- 2(M+k)\\, (k+M+N)\\, J^a \\, J^a\\nonumber \\\\&& + (\\frac{M}{2(k+N)})^2 \\, 4 \\, (N+k)\\,J^{\\alpha }\\, J^{\\alpha }+ (\\frac{M+N}{2k})^2\\, 4 \\, k\\, J^{u(1)}\\,J^{u(1)}\\nonumber \\\\&& + k(M+N)\\, M N\\, \\sqrt{\\frac{M+N}{M N}}\\partial \\, J^{u(1)} \\Bigg ](w)+ {\\cal O}(\\frac{1}{(z-w)})+ \\cdots ,\\nonumber \\\\&& J^{\\alpha }\\, J^{\\alpha }(z) \\,J^{\\beta }\\, J^{\\beta }(z) =\\frac{1}{(z-w)^4} \\, 2k\\, (N+k)\\, (N^2-1)\\nonumber \\\\&& +\\frac{1}{(z-w)^2}\\, 4 \\, (N+k)\\, J^{\\beta }\\, J^{\\beta }+ {\\cal O}(\\frac{1}{(z-w)})+ \\cdots ,\\nonumber \\\\&& J^{\\alpha } \\, J_f^{\\alpha }(z)\\,J^{\\beta } \\, J_f^{\\beta }(w) =\\frac{1}{(z-w)^4}\\, M k \\, (N^2-1)+\\frac{1}{(z-w)^2}\\, \\Bigg [k \\, J_f^{\\alpha }\\, J_f^{\\alpha }\\nonumber \\\\&& - 2 N \\, J^{\\alpha }\\, J_f^{\\alpha }+ M \\, J^{\\alpha }\\, J^{\\alpha }\\Bigg ](w) + {\\cal O}(\\frac{1}{(z-w)})+ \\cdots ,\\nonumber \\\\&& J^{u(1)} J^{u(1)}_f(z) \\,J^{u(1)} J^{u(1)}_f(w) =\\frac{1}{(z-w)^4}\\, k M N \\nonumber \\\\&& +\\frac{1}{(z-w)^2}\\, \\Bigg [k\\, J_f^{u(1)}\\, J_f^{u(1)}+ M N \\, J^{u(1)}\\, J^{u(1)}\\Bigg ](w)+ {\\cal O}(\\frac{1}{(z-w)})+ \\cdots ,\\nonumber \\\\&& J^a \\, J^a_f(z) \\,J^b \\, J^b_f(w) =\\frac{1}{(z-w)^4}\\, k N (M^2-1)\\nonumber \\\\&& + \\frac{1}{(z-w)^2}\\,\\Bigg [ k J_f^a \\,J_f^a- 2 M \\, J^a \\, J^a_f +N \\, J^a \\, J^a \\Bigg ](w)+ {\\cal O}(\\frac{1}{(z-w)})+ \\cdots ,\\nonumber \\\\&& \\delta _{\\rho \\bar{\\sigma }} \\, \\delta _{j \\bar{i}} \\,\\partial \\, \\psi ^{(\\rho \\bar{i})} \\, \\psi ^{(\\bar{\\sigma } j)}(z)\\,\\delta _{\\mu \\bar{\\nu }} \\, \\delta _{l \\bar{k}} \\,\\partial \\, \\psi ^{(\\mu \\bar{k})} \\, \\psi ^{(\\bar{\\nu } l)}(w)= -\\frac{1}{(z-w)^4}\\, N M\\nonumber \\\\&& + \\frac{1}{(z-w)^2}\\, 2 \\,\\delta _{\\mu \\bar{\\nu }} \\, \\delta _{l \\bar{k}} \\,\\partial \\, \\psi ^{(\\mu \\bar{k})} \\, \\psi ^{(\\bar{\\nu } l)}(w)+ {\\cal O}(\\frac{1}{(z-w)})+ \\cdots ,\\nonumber \\\\&& J^a \\, J^a(z) \\,J^b \\, J^b(w) =\\frac{1}{(z-w)^4}\\,2k \\, (M+k)\\, (M^2-1)\\nonumber \\\\&& + \\frac{1}{(z-w)^2}\\, 4 \\,(M+k)\\, J^a \\, J^a+ {\\cal O}(\\frac{1}{(z-w)})+ \\cdots ,\\nonumber \\\\&& \\delta _{\\rho \\bar{\\sigma }} \\, \\delta _{j \\bar{i}} \\,\\psi ^{(\\rho \\bar{i})} \\, \\partial \\, \\psi ^{(\\bar{\\sigma } j)}(z)\\,\\delta _{\\mu \\bar{\\nu }} \\, \\delta _{l \\bar{k}} \\,\\psi ^{(\\mu \\bar{k})} \\, \\partial \\, \\psi ^{(\\bar{\\nu } l)}(w) =-\\frac{1}{(z-w)^4}\\, N M \\nonumber \\\\&& - \\frac{1}{(z-w)^2}\\, 2 \\,\\delta _{\\mu \\bar{\\nu }} \\, \\delta _{l \\bar{k}} \\,\\psi ^{(\\mu \\bar{k})} \\, \\partial \\, \\psi ^{(\\bar{\\nu } l)}(w)+ {\\cal O}(\\frac{1}{(z-w)})+ \\cdots ,\\nonumber \\\\&& J^{u(1)} \\, J^{u(1)}(z) \\,J^{u(1)} \\, J^{u(1)}(w) =\\frac{1}{(z-w)^4}\\, 2k^2+\\frac{1}{(z-w)^2}\\, 4k \\, J^{u(1)} \\, J^{u(1)}\\nonumber \\\\&& + {\\cal O}(\\frac{1}{(z-w)})+ \\cdots ,\\nonumber \\\\&& \\partial \\, J^{u(1)}(z) \\,\\partial \\, J^{u(1)}(w) =-\\frac{1}{(z-w)^4}\\, 6 k+ {\\cal O}(\\frac{1}{(z-w)})+ \\cdots .$ We only present the OPEs up to the second order pole in (REF ) where the bosonic stress energy tensor $T_{boson}$ is given by the footnote REF .",
"Other OPEs between the operators of $W^{-(2),0}$ can be performed similarly."
],
[
" Some of the defining relations\nin section 8",
"For convenience, we present the previous relations in [73].",
"The OPEs between the supersymmetry generators of spin-$\\frac{3}{2}$ are given by ${.",
"}1\\endcsname \\hat{G}_{11}(z) \\, \\hat{G}_{11}(w) & = &\\frac{1}{(z-w)} \\, \\frac{4}{(N+k+2)} \\left[ -\\hat{A}_{+} \\hat{B}_{-}\\right](w) + \\cdots ,\\nonumber \\\\\\hat{G}_{11}(z) \\, \\hat{G}_{12}(w) & = &\\frac{1}{(z-w)^2} \\left[ 4 i \\, \\gamma _A \\hat{A}_{+} \\right](w)\\nonumber \\\\& + &\\frac{1}{(z-w)} \\left[ 2 i \\, \\gamma _A \\partial \\hat{A}_{+} +\\frac{4}{(N+k+2)}\\hat{A}_{+} \\hat{B}_3\\right](w) +\\cdots ,\\nonumber \\\\\\hat{G}_{11}(z) \\, \\hat{G}_{21}(w) & = &\\frac{1}{(z-w)^2} \\left[ -4 i \\, \\gamma _B \\hat{B}_{-} \\right](w)\\nonumber \\\\&+ &\\frac{1}{(z-w)} \\left[ -2 i \\, \\gamma _B \\partial \\hat{B}_{-}+\\frac{4}{(N+k+2)}\\hat{A}_3 \\hat{B}_{-}\\right](w) + \\cdots ,\\nonumber \\\\\\hat{G}_{11}(z) \\, \\hat{G}_{22}(w) & = &\\frac{1}{(z-w)^3} \\, \\frac{2}{3}c_{\\mbox{Wolf}} +\\frac{1}{(z-w)^2} \\left[ 4 i \\left( \\gamma _A \\hat{A}_3 - \\gamma _B \\hat{B}_3 \\right) \\right](w)+\\frac{1}{(z-w)} \\left[ 2 \\hat{T} \\right.\\nonumber \\\\& + & \\left.",
"2 i \\partial \\left( \\gamma _A \\hat{A}_3 - \\gamma _B \\hat{B}_3 \\right)+\\frac{2}{(k+N+2)}\\left(\\hat{A}_i \\, \\hat{A}_i +\\hat{B}_i \\, \\hat{B}_i+2 \\hat{A}_3 \\, \\hat{B}_3 \\right) \\right] (w)+ \\cdots ,\\nonumber \\\\\\hat{G}_{12}(z) \\, \\hat{G}_{12}(w) & = &\\frac{1}{(z-w)} \\, \\frac{4}{(N+k+2)} \\left[ \\hat{A}_{+}\\hat{B}_{+} \\right](w) + \\cdots ,\\nonumber \\\\\\hat{G}_{12}(z) \\, \\hat{G}_{21}(w) & = &\\frac{1}{(z-w)^3} \\, \\frac{2}{3}c_{\\mbox{Wolf}} +\\frac{1}{(z-w)^2} \\left[ 4 i \\left( \\gamma _A \\hat{A}_3 + \\gamma _B \\hat{B}_3 \\right) \\right](w)+\\frac{1}{(z-w)} \\left[ 2 \\hat{T} \\right.\\nonumber \\\\& + & \\left.",
"2 i \\partial \\left( \\gamma _A \\hat{A}_3 + \\gamma _B \\hat{B}_3 \\right) +\\frac{2}{(k+N+2)}\\left(\\hat{A}_i \\, \\hat{A}_i +\\hat{B}_i \\, \\hat{B}_i-2 \\hat{A}_3 \\, \\hat{B}_3 \\right) \\right](w)+ \\cdots ,\\nonumber \\\\\\hat{G}_{12}(z) \\, \\hat{G}_{22}(w) & = &\\frac{1}{(z-w)^2} \\left[ -4i\\, \\gamma _B \\hat{B}_{+} \\right](w)\\nonumber \\\\&+&\\frac{1}{(z-w)} \\left[ -2 i\\, \\gamma _B \\partial \\hat{B}_{+}+ \\frac{4}{(N+k+2)} \\hat{A}_3 \\hat{B}_{+}\\right](w) + \\cdots ,\\nonumber \\\\\\hat{G}_{21}(z) \\, \\hat{G}_{21}(w) & = &\\frac{1}{(z-w)} \\frac{4}{(N+k+2)} \\left[ \\hat{A}_{-}\\hat{B}_{-}\\right](w) +\\cdots ,\\nonumber \\\\\\hat{G}_{21}(z) \\, \\hat{G}_{22}(w) & = &\\frac{1}{(z-w)^2} \\left[ 4i\\, \\gamma _A \\hat{A}_{-} \\right](w) +\\frac{1}{(z-w)} \\left[ 2 i\\, \\gamma _A \\partial \\hat{A}_{-}+ \\frac{4}{(N+k+2)} \\hat{A}_{-}\\hat{B}_3 \\right](w) \\nonumber \\\\& + & \\cdots ,\\nonumber \\\\\\hat{G}_{22}(z) \\, \\hat{G}_{22}(w) & = &\\frac{1}{(z-w)} \\frac{4}{(N+k+2)} \\left[ -\\hat{A}_{-} \\hat{B}_{+}\\right](w) +\\cdots ,$ where the two parameters are given by $\\gamma _A \\equiv \\frac{N}{N+k+2}$ , $\\gamma _B \\equiv \\frac{k}{N+k+2}$ , and we introduce the spin-1 currents $\\hat{A}_{\\pm }(z) \\equiv \\hat{A}_1 \\pm i \\hat{A}_2(z)$ and $\\hat{B}_{\\pm }(z) \\equiv \\hat{B}_1 \\pm i \\hat{B}_2(z)$ and the central term above is given by $c_{\\mbox{Wolf}} =\\frac{6 k N}{(2+k+N)}$ .",
"The higher spin-$\\frac{3}{2}$ currents can be obtained from the following OPEs ${.",
"}2\\endcsname \\hat{G}_{21} (z) \\, T^{(1)} (w) &=&\\frac{1}{(z-w)} \\left[ \\hat{G}_{21} + 2 T_{+}^{(\\frac{3}{2})} \\equiv \\hat{G}^{\\prime }_{21} \\right](w)+\\cdots ,\\nonumber \\\\\\hat{G}_{12} (z) \\, T^{(1)} (w) &=&\\frac{1}{(z-w)} \\left[ -\\hat{G}_{12} + 2 T_{-}^{(\\frac{3}{2})}\\equiv \\hat{G}^{\\prime }_{12}\\right](w)+\\cdots ,\\nonumber \\\\\\hat{G}_{11} (z) \\, T^{(1)} (w) &=&\\frac{1}{(z-w)} \\left[ \\hat{G}_{11} + 2 U^{(\\frac{3}{2})}\\equiv \\hat{G}^{\\prime }_{11}\\right](w)+\\cdots ,\\nonumber \\\\\\hat{G}_{22} (z) \\, T^{(1)} (w) &=&\\frac{1}{(z-w)} \\left[- \\hat{G}_{22} + 2 V^{(\\frac{3}{2})}\\equiv \\hat{G}^{\\prime }_{22}\\right](w)+\\cdots .$ The higher spin-2 currents are determined by ${.",
"}3\\endcsname \\hat{G}_{11}(z) \\, \\hat{G}^{\\prime }_{12} (w)&=&\\frac{1}{(z-w)} \\, \\left[ - 2 U_{-}^{(2)} -\\frac{4}{(N+k+2)}\\hat{A}_{+} B_3 \\right](w) +\\cdots ,\\nonumber \\\\\\hat{G}_{21}(z) \\, \\hat{G}^{\\prime }_{22} (w)&=&\\frac{1}{(z-w)} \\, \\left[ 2 V_{+}^{(2)} -\\frac{4}{(N+k+2)}\\hat{A}_{-} B_3 \\right](w) +\\cdots ,\\nonumber \\\\\\hat{G}_{11}(z) \\, \\hat{G}^{\\prime }_{21} (w)&=&\\frac{1}{(z-w)} \\, \\left[ - 2 U_{+}^{(2)} -\\frac{4}{(N+k+2)}\\hat{A}_3 \\hat{B}_{-} \\right](w) +\\cdots ,\\nonumber \\\\\\hat{G}_{12}(z) \\, \\hat{G}^{\\prime }_{22} (w)&=&\\frac{1}{(z-w)} \\, \\left[ 2 V_{-}^{(2)} -\\frac{4}{(N+k+2)}\\hat{A}_3 \\hat{B}_{+} \\right](w) +\\cdots ,\\nonumber \\\\\\hat{G}_{12}(z) \\, \\hat{G}^{\\prime }_{21} (w)&=&\\frac{1}{(z-w)^2} \\, 2 T^{(1)} (w) +\\frac{1}{(z-w)} \\, \\left[ -2 T^{(2)} +\\partial T^{(1)} + \\frac{2(k+N)}{(k+N+2kN)}\\hat{T} \\right.\\nonumber \\\\&+& \\left.",
"\\frac{2}{(N+k+2)} \\left( \\hat{A}_i \\hat{A}_i +\\hat{B}_i \\hat{B}_i-2 \\hat{A}_3 \\hat{B}_3 \\right)\\right](w) +\\cdots ,\\nonumber \\\\\\hat{G}_{11}(z) \\, \\hat{G}^{\\prime }_{22} (w)&=&\\frac{1}{(z-w)^2} \\, 2 T^{(1)} (w) +\\frac{1}{(z-w)}\\, \\left[ 2 W^{(2)} +\\partial T^{(1)} -2 \\hat{T} \\right.\\nonumber \\\\&-& \\left.",
"\\frac{2}{(N+k+2)} \\left( \\hat{A}_i \\hat{A}_i +\\hat{B}_i \\hat{B}_i+2 \\hat{A}_3 \\hat{B}_3 \\right)\\right](w) +\\cdots ,\\nonumber \\\\\\hat{G}_{21}(z) \\, \\hat{G}^{\\prime }_{12} (w)&=&\\frac{1}{(z-w)^2} \\, 2 T^{(1)} (w) +\\frac{1}{(z-w)} \\,\\left[ 2 T^{(2)} +\\partial T^{(1)} - \\frac{2(k+N)}{(k+N+2kN)} \\hat{T} \\right.\\nonumber \\\\&-& \\left.",
"\\frac{2}{(N+k+2)} \\left( \\hat{A}_i \\hat{A}_i +\\hat{B}_i \\hat{B}_i-2 \\hat{A}_3 \\hat{B}_3 \\right)\\right](w) +\\cdots ,\\nonumber \\\\\\hat{G}_{22}(z) \\, \\hat{G}^{\\prime }_{11} (w)&=&\\frac{1}{(z-w)^2} \\, 2 T^{(1)} (w) +\\frac{1}{(z-w)} \\, \\left[ -2 W^{(2)} +\\partial T^{(1)} + 2 \\hat{T} \\right.\\nonumber \\\\&+& \\left.",
"\\frac{2}{(N+k+2)} \\left( \\hat{A}_i \\hat{A}_i +\\hat{B}_i \\hat{B}_i+2 \\hat{A}_3 \\hat{B}_3 \\right)\\right](w) +\\cdots .$ Here $\\hat{G}^{\\prime }_{mn}$ is defined in (REF ).",
"Similarly, the higher spin-$\\frac{5}{2}$ currents can be obtained from ${.",
"}4\\endcsname \\hat{G}_{21} (z) \\, U^{(2)}_{-} (w)&=&\\frac{1}{(z-w)^2}\\, \\left[ \\frac{(N+2k)}{(N+k+2)} \\hat{G}_{11} +\\frac{2(N+2k+1)}{(N+k+2)} U^{(\\frac{3}{2})} \\right](w)\\nonumber \\\\&+&\\frac{1}{(z-w)} \\, \\left[ U^{ (\\frac{5}{2}) }+\\frac{1}{3} \\partial \\, (\\mbox{pole-2})\\right](w) + \\cdots ,\\nonumber \\\\\\hat{G}_{21} (z) \\, V^{(2)}_{-} (w)&=&\\frac{1}{(z-w)^2} \\, \\left[ - \\frac{(2N+k)}{(N+k+2)} \\hat{G}_{22} +\\frac{2(2N+k+1)}{(N+k+2)}V^{(\\frac{3}{2})} \\right](w)\\nonumber \\\\&+&\\frac{1}{(z-w)} \\, \\left[ V^{ (\\frac{5}{2}) }+\\frac{1}{3} \\partial \\, (\\mbox{pole-2})\\right] (w)+ \\cdots ,\\nonumber \\\\\\hat{G}_{21} (z) \\, W^{(2)} (w)&=&\\frac{1}{(z-w)^2} \\, \\left[ \\frac{(N+2k+1)}{(N+k+2)}\\hat{G}_{21} + \\frac{(k-N)}{(N+k+2)} T_{+}^{(\\frac{3}{2})} \\right](w)\\nonumber \\\\&+&\\frac{1}{(z-w)} \\, \\left[ W_{+}^{(\\frac{5}{2})}+\\frac{1}{3} \\partial \\, (\\mbox{pole-2})\\right] (w)+ \\cdots ,\\nonumber \\\\\\hat{G}_{12} (z) \\, W^{(2)} (w)&=&\\frac{1}{(z-w)^2} \\, \\left[ \\frac{(N+2k+1)}{(N+k+2)}\\hat{G}_{12} + \\frac{(N-k)}{(N+k+2)} T_{-}^{(\\frac{3}{2})} \\right](w)\\nonumber \\\\&+&\\frac{1}{(z-w)} \\, \\left[ W_{-}^{(\\frac{5}{2})}+\\frac{1}{3} \\partial \\, (\\mbox{pole-2})\\right] (w)+ \\cdots .$ In next Appendix, we express the higher spin-$\\frac{5}{2}$ currents for fixed $N=5$ with $M=2$ ."
],
[
" The higher spin-$\\frac{5}{2}$ currents\nfor fixed {{formula:69712eae-c249-4d7c-9ed9-1b990cdbeb24}} and {{formula:50fee963-8cb7-446a-a79d-11b4b39568d3}} in section 8",
"We present the higher spin-$\\frac{5}{2}$ currents as follows: ${.",
"}1\\endcsname U^{(\\frac{5}{2})} & =&\\left(\\frac{1}{k+7}\\right)^{3/2}\\, \\Bigg [i \\, f^{3 1 d} \\Big (\\frac{1}{2}\\, i \\, f^{3 d c}\\, V^{+(\\frac{5}{2}),c}+\\frac{(2 k+7)}{6 k}\\, i \\, f^{3 d c}\\, \\partial \\,G^{-,c}\\Big )\\nonumber \\\\& + & i\\, f^{3 2 d} \\Big (\\frac{1}{2} \\, f^{3 d c}\\, V^{+(\\frac{5}{2}),c}+\\frac{ (2 k+7)}{6 k}\\, f^{3 d c}\\, \\partial \\,G^{-,c}\\Big )\\nonumber \\\\& + & i\\, f^{3 1 d} \\Big (-\\frac{1}{2}\\, i \\, f^{3 d c}\\, V^{-(\\frac{5}{2}),c}-\\frac{7 }{6 k}\\, i \\, f^{3 d c}\\, \\partial \\,G^{-,c}\\Big )\\nonumber \\\\&+& i\\, f^{3 2 d} \\Big (-\\frac{1}{2} \\, f^{3 d c}\\, V^{-(\\frac{5}{2}),c}-\\frac{7 }{6 k}\\,f^{3 d c}\\, \\partial \\,G^{-,c}\\Big )- \\frac{(k+7)}{2 k}\\, f^{3 1 d}\\, f^{3 d e}\\,G^{+,e} \\, K\\nonumber \\\\& + & \\frac{i (k+7)}{2 k}\\,f^{3 2 d}\\, f^{3 d e}\\,G^{+,e} \\, K-\\frac{(k+7)}{2 k}\\,f^{3 1 d}\\, f^{3 d e}\\,G^{-,e} \\, K-i \\, f^{3 1 d}\\, J^d\\,G^{-,3}\\nonumber \\\\&+& \\frac{i (k+7)}{2 k}\\,f^{3 2 d}\\, f^{3 d e}\\,G^{-,e} \\, K-f^{3 2 d}\\, J^d\\,G^{-,3}+i \\, f^{3 1 d}\\, G^{+,d}\\, J_f^3\\nonumber \\\\& + & f^{3 2 d}\\, G^{+,d}\\, J_f^3+\\frac{ (2 k+7)}{k}\\,\\delta ^{3 3}\\, J^1 \\, G^+-\\frac{i (2 k+7)}{k}\\,\\delta ^{3 3}\\, J^2 \\, G^+\\nonumber \\\\&+&i\\, f^{3 1 d}\\, G^{-,d}\\, J_f^3+ f^{3 2 d}\\, G^{-,d}\\, J_f^3+ \\frac{(k+7)}{k}\\,\\delta ^{3 3}\\, J^1 \\, G^- -\\frac{i (k+7)}{k}\\,\\delta ^{3 3}\\, J^2 \\, G^- \\, \\Bigg ],\\nonumber \\\\V^{(\\frac{5}{2})} &=&\\left(\\frac{1}{k+7}\\right)^{3/2}\\, \\Bigg [-i \\, f^{3 1 d} \\, \\frac{1}{2}\\, f^{d e f} \\, f^{3 f g } \\, J^e \\, G^{+,g}\\nonumber \\\\&+& i \\, f^{3 2 d}\\, \\Big ( -\\frac{1}{2}\\, f^{d e f}\\, i \\, f^{3 f g } \\, J^e \\, G^{+,g}-\\frac{ \\left(k^2-7\\right)}{2 k}\\,f^{3 d e}\\, \\partial \\, G^{+,e}\\Big )\\nonumber \\\\&+& i \\, f^{3 1 d}\\, \\Big (\\frac{1}{2} \\,i \\, f^{3 d e}\\, V^{-(\\frac{5}{2}),e}- f^{d e f} \\, f^{3 f g } \\, J^e \\, G^{-,g}- \\frac{(k+7)}{2 k}\\,i\\, f^{3 d e}\\, J^e \\, G^{-}\\Big ) \\nonumber \\\\& + & i \\, f^{3 2 d}\\, \\Big (-\\frac{1}{2} \\, f^{3 d e}\\, V^{-(\\frac{5}{2}),e}-i \\,f^{d e f} \\, f^{3 f g } \\, J^e \\, G^{-,g}\\nonumber \\\\&-&\\frac{ \\left(k^2+10 k-14\\right)}{6 k}\\,f^{3 d e}\\, \\partial \\, G^{-,e}+\\frac{ (k+7)}{2 k}\\,f^{3 d e}\\, J^e\\, G^{-}\\Big )\\nonumber \\\\& +& \\frac{i (k+7)}{2 k}\\,f^{3 2 d} \\, f^{3 d e}\\,G^{+,e}\\, K+i \\, f^{3 1 d}\\, J^d \\, G^{-,3}- f^{3 2 d}\\, J^d \\, G^{-,3}\\nonumber \\\\&- &G^+\\, J_f^1-2 \\,t^1_{j \\bar{i}}\\, \\delta _{k \\bar{l}}\\, \\delta _{\\rho \\bar{\\mu }}\\,\\delta _{\\tau \\bar{\\sigma }}\\, \\psi ^{(\\bar{\\sigma } j)}\\, \\psi ^{(\\rho \\bar{i})}\\,\\psi ^{(\\bar{\\mu } k)} \\, J^{(\\tau \\bar{l})}+ k\\partial \\, G^{+,1}-2\\,G^-\\, J_f^1\\nonumber \\\\& - & 2 \\,t^1_{i \\bar{j}}\\, \\delta _{l \\bar{k}}\\, \\delta _{\\mu \\bar{\\rho }}\\, \\delta _{\\sigma \\bar{\\tau }}\\,\\psi ^{(\\sigma \\bar{j})}\\, \\psi ^{(\\bar{\\rho } i)}\\,\\psi ^{(\\mu \\bar{k})} \\, J^{(\\bar{\\tau } l)}+\\frac{ \\left(k^2+10 k-14\\right)}{3 k}\\,\\partial \\, G^{-,1}\\nonumber \\\\& + & \\sqrt{\\frac{7}{5}} \\,f^{3 2 c}\\, J^{u(1)}\\, G^{+,c}+ \\frac{1}{5} \\sqrt{2} (5 k+1)\\,f^{3 2 d}\\, t^d_{j \\bar{i}}\\, \\delta _{\\rho \\bar{\\sigma }}\\,\\partial \\, J^{(\\rho \\bar{i})}\\, \\psi ^{(\\bar{\\sigma } j)}\\nonumber \\\\& -&\\sqrt{2} \\,f^{3 2 c}\\, t^{\\alpha }_{\\rho \\bar{\\sigma }}\\, t^c_{j \\bar{i}}\\,J^{(\\rho \\bar{i})}\\, \\psi ^{(\\bar{\\sigma } j)}\\, J^{\\alpha }+i \\sqrt{2}\\,t^3_{l \\bar{k}}\\, \\delta _{\\tau \\bar{\\mu }}\\, (( \\psi ^{(\\bar{\\mu } l)}\\,J^{(\\tau \\bar{k})})\\, J_f^2)\\nonumber \\\\&- & \\frac{\\sqrt{\\frac{7}{5}} (k+5)}{k}\\,i \\, f^{3 1 c}\\, J^{u(1)}\\, G^{+,c}- \\frac{1}{5} \\sqrt{2} (5 k+1)\\,i \\, f^{3 1 c} \\, t^c_{j \\bar{i}}\\, \\delta _{\\rho \\bar{\\sigma }}\\,\\partial \\, J^{(\\rho \\bar{i})}\\, \\psi ^{(\\bar{\\sigma } j)}\\nonumber \\\\& + & \\sqrt{2}\\,i \\, f^{3 1 c}\\, t^{\\alpha }_{\\rho \\bar{\\sigma }}\\, t^c_{j \\bar{i}}\\, J^{(\\rho \\bar{i})}\\,\\psi ^{(\\bar{\\sigma } j)}\\, J^{\\alpha }+ 2 \\sqrt{2} \\,t^{\\alpha }_{\\rho \\bar{\\sigma }}\\, t^1_{j \\bar{i}}\\,t^{\\alpha }_{\\nu \\bar{\\mu }}\\, t^3_{k \\bar{l}}\\,\\psi ^{(\\bar{\\sigma } j)}\\,\\psi ^{(\\rho \\bar{i})}\\,\\psi ^{(\\nu \\bar{l})}\\,J^{(\\bar{\\mu } k)}\\nonumber \\\\& - & \\frac{1}{\\sqrt{2}}\\,f^{3 c e} \\, f^{c 1 d}\\, G^{+,e}\\, J_f^d- \\frac{ \\sqrt{35}}{k}\\,f^{3 2 c}\\, J^{u(1)}\\, G^{-,c}+\\frac{\\sqrt{35}}{k}\\,i \\, f^{3 1 c}\\, J^{u(1)}\\, G^{-,c}\\nonumber \\\\&-&\\frac{1}{5} \\sqrt{2} \\,i \\, f^{3 1 c}\\, t^c_{ \\bar{j} i}\\, \\delta _{\\sigma \\bar{\\rho }}\\,\\partial \\, J^{(\\bar{\\rho } i)}\\, \\psi ^{(\\sigma \\bar{j})} +\\frac{12}{5} \\sqrt{2} \\,t^3_{ \\bar{l} k}\\, \\delta _{\\mu \\bar{\\tau }}\\, ((\\psi ^{(\\mu \\bar{l})}\\, J^{(\\bar{\\tau } k)})\\, J_f^1) \\, \\Bigg ],\\nonumber \\\\W^{(\\frac{5}{2})}_{+} & = & \\left(\\frac{1}{k+7}\\right)^{3/2}\\,\\Bigg [\\frac{ (k-2) (k+1) }{2 k}\\,f^{1 2 d}\\, \\delta ^{3 d}\\, \\partial \\, G^++ \\frac{1}{10} (10 k+37)\\,f^{1 2 d}\\, \\delta ^{3 d}\\, \\partial \\, G^-\\nonumber \\\\& - & \\frac{ (k+7)}{7 k}\\,f^{1 2 d}\\, \\delta ^{3 d}\\, G^+\\, K-\\frac{1}{10} (k+7)\\,f^{1 2 d}\\, \\delta ^{3 d}\\, G^-\\, K+i \\, f^{b 3 c}\\, G^{+,b}\\, J_f^c\\nonumber \\\\&-&2 \\,t^3_{j \\bar{i}}\\, \\delta _{k \\bar{l}}\\, \\delta _{\\rho \\bar{\\mu }}\\,\\delta _{\\tau \\bar{\\sigma }}\\, \\psi ^{(\\bar{\\sigma } j)}\\, \\psi ^{(\\rho \\bar{i})}\\,\\psi ^{(\\bar{\\mu } k)} \\, J^{(\\tau \\bar{l})}-G^{+,3}\\, J_f^{u(1)}-\\frac{ \\left(k^2+13 k+7\\right)}{3 k}\\,\\partial \\, G^{+,3}\\nonumber \\\\& + &2 \\,i \\, f^{b 3 c}\\, G^{-,b}\\, J_f^c- 2 \\,\\delta _{l \\bar{k}}\\, \\delta _{\\mu \\bar{\\rho }}\\,\\delta _{\\sigma \\bar{\\tau }}\\, t^3_{ \\bar{i} j}\\,\\psi ^{(\\sigma \\bar{j})}\\,\\psi ^{(\\bar{\\rho } i)}\\,\\psi ^{(\\mu \\bar{k})}\\,J^{(\\bar{\\tau } l)}+G^{-,3}\\, J_f^{u(1)}\\nonumber \\\\&-&\\frac{ \\left(2 k^2+8 k-7\\right)}{3 k}\\,\\partial \\, G^{-,3}+\\frac{1}{70} \\sqrt{2} \\, (35 k+2)\\,\\delta ^{3 3}\\, \\delta _{\\rho \\bar{\\sigma }}\\, \\delta _{j\\bar{i}}\\,\\partial \\, J^{(\\rho \\bar{i})}\\, \\psi ^{(\\bar{\\sigma }) j}\\nonumber \\\\&- &\\frac{10}{7} \\sqrt{2} \\,t^{\\alpha }_{\\rho \\bar{\\sigma }}\\, t^3_{j \\bar{i}}\\, t^{\\alpha }_{\\mu \\bar{\\nu }}\\, t^3_{l \\bar{k}}\\,\\, \\psi ^{(\\bar{\\sigma } j)}\\, \\psi ^{(\\rho \\bar{i})}\\,\\psi ^{(\\bar{\\nu } l)} \\, J^{(\\mu \\bar{k})}- \\frac{ (7 k+10)}{\\sqrt{35} k}\\,\\delta ^{3 3}\\, J^{u(1)}\\, G^+\\nonumber \\\\& - & \\sqrt{2}\\,\\delta ^{3 3}\\, \\delta _{j \\bar{i}}\\, t^{\\alpha }_{\\rho \\bar{\\sigma }}\\, J^{(\\rho \\bar{i})}\\, \\psi ^{(\\bar{\\sigma }j)} \\, J^{\\alpha }+ \\frac{1}{35} \\sqrt{2}\\, (35 k+12)\\,\\delta ^{3 3}\\, \\delta _{i \\bar{j}}\\, \\delta _{\\sigma \\bar{\\rho }}\\,\\partial \\, J^{(\\bar{\\rho } i)}\\, \\psi ^{(\\sigma \\bar{j})}\\nonumber \\\\&+ & \\sqrt{2}\\,\\delta ^{3 3}\\, \\delta _{i \\bar{j}}\\, t^{\\alpha }_{\\sigma \\bar{\\rho }}\\,J^{(\\bar{\\rho } i)}\\, \\psi ^{(\\sigma \\bar{j})} \\, J^{\\alpha }+ \\frac{10}{7} \\sqrt{2} \\,t^{\\alpha }_{\\sigma \\bar{\\rho }}\\, t^3_{i \\bar{j}}\\,t^{\\alpha }_{\\nu \\bar{\\mu }}\\, t^3_{k \\bar{l}}\\,\\psi ^{(\\sigma \\bar{j})}\\,\\psi ^{(\\bar{\\rho } i)}\\,\\psi ^{(\\nu \\bar{l})}\\,J^{(\\bar{\\mu } k)}\\nonumber \\\\& + &\\frac{ (7 k+60)}{35 \\sqrt{2}}\\,\\delta ^{3 3}\\, G^-\\, J_f^{u(1)}- \\frac{1}{\\sqrt{2}}\\,f^{3 c d} \\, f^{3 d e}\\, J^c \\, G^{+,e}+\\sqrt{2}\\,J^3 \\, G^{+,3}+\\sqrt{2} \\,J^3 \\, G^{-,3}\\nonumber \\\\& - &V^{+(\\frac{5}{2}),3}+V^{-(\\frac{5}{2}),3} -\\frac{(k+7)}{k}\\,G^{+,3}\\, K - \\frac{(2 k+7)}{k}\\,J^3 \\, G^+\\nonumber \\\\&-& \\frac{(k+7)}{k}\\, G^{-,3}\\, K-\\frac{7 }{k}\\,J^3 \\, G^--i \\sqrt{2} \\,J^1 \\, G^{-,2}+ \\frac{5}{7} \\, f^{3 1 d} \\, G^{+,d}\\, J_f^2\\nonumber \\\\& + & \\frac{12}{7}\\, f^{3 1 d} \\, G^{-,d}\\, J_f^2+\\frac{5}{14} \\,f^{e 1 c} \\, f^{2 c d} \\, f^{3 e f} \\,G^{+,f}\\, J_f^d+\\frac{6}{7} \\,f^{e 1 c} \\, f^{2 c d} \\, f^{3 e f} \\,G^{-,f}\\, J_f^d\\nonumber \\\\& - & \\frac{i }{\\sqrt{2}}\\,f^{e 1 c} \\, f^{2 c d} \\,J^e \\, G^{-,d} \\Bigg ],\\nonumber \\\\W^{(\\frac{5}{2})}_{-} & = &\\left(\\frac{1}{k+7}\\right)^{3/2}\\, \\Bigg [\\frac{ \\left(k^2+3 k-6\\right)}{k}\\,f^{1 2 d}\\, \\delta ^{3 d}\\, \\partial \\, G^++ \\frac{1}{10} (5 k+2)\\,f^{1 2 d}\\, \\delta ^{3 d}\\, \\partial \\, G^-\\nonumber \\\\&- &\\frac{6 (k+7)}{7 k}\\,f^{1 2 d}\\, \\delta ^{3 d}\\, G^+\\, K-\\frac{1}{10} (k+7)\\,f^{1 2 d}\\, \\delta ^{3 d}\\, G^-\\, K+\\,2 \\,i \\, f^{b 3 c}\\, G^{+,b}\\, J_f^c\\nonumber \\\\&-& 2\\,t^3_{j \\bar{i}}\\, \\delta _{k \\bar{l}}\\, \\delta _{\\rho \\bar{\\mu }}\\,\\delta _{\\tau \\bar{\\sigma }}\\, \\psi ^{(\\bar{\\sigma } j)}\\, \\psi ^{(\\rho \\bar{i})}\\,\\psi ^{(\\bar{\\mu } k)} \\, J^{(\\tau \\bar{l})}-G^{+,3}\\, J_f^{u(1)}-\\frac{\\left(2 k^2+8 k-7\\right)}{3 k}\\,\\partial \\, G^{+,3}\\nonumber \\\\& + &\\,i \\, f^{b 3 c}\\, G^{-,b}\\, J_f^c- 2 \\,\\delta _{l \\bar{k}}\\, \\delta _{\\mu \\bar{\\rho }}\\,\\delta _{\\sigma \\bar{\\tau }}\\, t^3_{ i \\bar{j} }\\,\\psi ^{(\\sigma \\bar{j})}\\,\\psi ^{(\\bar{\\rho } i)}\\,\\psi ^{(\\mu \\bar{k})}\\,J^{(\\bar{\\tau } l)}+G^{-,3}\\, J_f^{u(1)}\\nonumber \\\\&-&\\frac{ \\left(k^2+13 k+7\\right)}{3 k}\\,\\partial \\, G^{-,3}+ \\frac{1}{35} \\sqrt{2} (35 k+12)\\,\\delta ^{3 3}\\, \\delta _{\\rho \\bar{\\sigma }}\\, \\delta _{j\\bar{i}}\\,\\partial \\, J^{(\\rho \\bar{i})}\\, \\psi ^{(\\bar{\\sigma }) j}\\nonumber \\\\&+ &\\frac{10}{7} \\sqrt{2}\\,t^{\\alpha }_{\\rho \\bar{\\sigma }}\\, t^3_{j \\bar{i}}\\, t^{\\alpha }_{\\mu \\bar{\\nu }}\\, t^3_{l \\bar{k}}\\,\\, \\psi ^{(\\bar{\\sigma } j)}\\, \\psi ^{(\\rho \\bar{i})}\\,\\psi ^{(\\bar{\\nu } l)} \\, J^{(\\mu \\bar{k})}- \\frac{(7 k+60)}{\\sqrt{35} k}\\,\\delta ^{3 3}\\, J^{u(1)}\\, G^+\\nonumber \\\\& - & \\sqrt{2}\\,\\delta ^{3 3}\\, \\delta _{j \\bar{i}}\\, t^{\\alpha }_{\\rho \\bar{\\sigma }}\\,J^{(\\rho \\bar{i})}\\, \\psi ^{(\\bar{\\sigma }j)} \\, J^{\\alpha }+\\frac{1}{35} \\sqrt{2} (35 k+2)\\,\\delta ^{3 3}\\, \\delta _{i \\bar{j}}\\, \\delta _{\\sigma \\bar{\\rho }}\\,\\partial \\, J^{(\\bar{\\rho } i)}\\, \\psi ^{(\\sigma \\bar{j})}\\nonumber \\\\&+ & \\sqrt{2} \\,\\delta ^{3 3}\\, \\delta _{i \\bar{j}}\\, t^{\\alpha }_{\\sigma \\bar{\\rho }}\\,J^{(\\bar{\\rho } i)}\\, \\psi ^{(\\sigma \\bar{j})} \\, J^{\\alpha }- \\frac{10}{7} \\sqrt{2}\\,t^{\\alpha }_{\\sigma \\bar{\\rho }}\\, t^3_{i \\bar{j}}\\,t^{\\alpha }_{\\nu \\bar{\\mu }}\\, t^3_{k \\bar{l}}\\,\\psi ^{(\\sigma \\bar{j})}\\,\\psi ^{(\\bar{\\rho } i)}\\,\\psi ^{(\\nu \\bar{l})}\\,J^{(\\bar{\\mu } k)}\\nonumber \\\\& + &\\frac{ (7 k+10)}{35 \\sqrt{2}}\\,\\delta ^{3 3}\\, G^-\\, J_f^{u(1)}-\\sqrt{2}\\,J^3 \\, G^{+,3}+\\frac{1}{\\sqrt{2}}\\,f^{3 c d}\\, f^{3 d e}\\, J^c \\, G^{-,e}\\nonumber \\\\& - & \\sqrt{2} \\,J^3 \\, G^{-,3}+V^{+(\\frac{5}{2}),3}-V^{-(\\frac{5}{2}),3} +\\frac{(k+7)}{k}\\,G^{+,3}\\, K \\nonumber \\\\& + & \\frac{7 }{k}\\,J^3 \\, G^++ \\frac{(k+7)}{k}\\, G^{-,3}\\, K+\\frac{(2 k+7)}{k}\\,J^3 \\, G^-\\nonumber \\\\& - & i \\sqrt{2} \\,J^1\\, G^{+,2}- \\frac{12}{7} \\,\\, f^{3 1 d} \\, G^{+,d}\\, J_f^2-\\frac{5}{7} \\, f^{3 1 d} \\, G^{-,d}\\, J_f^2\\nonumber \\\\& - &\\frac{6}{7} \\, f^{e 1 c} \\, f^{2 c d} \\, f^{3 e f} \\,G^{+,f}\\, J_f^d-\\frac{5}{14} \\, f^{e 1 c} \\, f^{2 c d} \\, f^{3 e f} \\,G^{-,f}\\, J_f^d\\nonumber \\\\& - &\\frac{i }{\\sqrt{2}}\\,f^{e 1 c} \\, f^{2 c d} \\,J^e \\, G^{+,d} \\Bigg ].$ The general $N$ dependence on these currents can be determined by calculating Appendix (REF ) explicitly.",
"Then the spin-$\\frac{5}{2}$ currents can be obtained from $V^{\\pm (\\frac{5}{2}),a}$ and other composite operators."
]
] | 2107.01781 |
[
[
"Towards Better Adversarial Synthesis of Human Images from Text"
],
[
"Abstract This paper proposes an approach that generates multiple 3D human meshes from text.",
"The human shapes are represented by 3D meshes based on the SMPL model.",
"The model's performance is evaluated on the COCO dataset, which contains challenging human shapes and intricate interactions between individuals.",
"The model is able to capture the dynamics of the scene and the interactions between individuals based on text.",
"We further show how using such a shape as input to image synthesis frameworks helps to constrain the network to synthesize humans with realistic human shapes."
],
[
"Introduction",
"Imagining the overall shape of persons based on a text description describing a scene is an easy task for humans even when it involves multiple humans interacting with each other such as in a football match.",
"However, this task is not as simple for machines since the human shape is highly articulated and is usually in interaction with other individuals and objects in the environment, which creates a huge space of versatility.",
"In this work, we propose a model that can generate 3D human shapes matching a high-level text description.",
"Motivations for creating the 3D human mesh include animating multiple 3D avatars based on text and guiding image-synthesis frameworks towards synthesizing more realistic human shapes.",
"In a similar work, Zhang et al.",
"[41] proposes a convolutional neural network (CNN) for generating a single person pose from text, which generates only the 2D joints but not the full shape and 3D pose.",
"The model also has a limited capacity such that it can only generate one pose for any given text, but cannot handle scenarios that contain multiple poses.",
"A 2D pose may be useful for guiding a pose transfer task as in [42], [21], [36], [35] where an image of the person is given, but is far too sparse to constrain a network to generate a realistic-looking unseen person from text.",
"Additionally, synthesizing multi-person shapes adds much more complexity to the task, since in typical scenarios the persons in the scene occlude or overlap with each other.",
"In this work, we address the problem of generating a variable number of 3D human shapes based on the SMPL model [20].",
"Figure REF describes the overall task.",
"In our scenario, the model should have the capability to deal with a variable number of shapes that match the text description, as well as infer the interactions between the different individuals in the scene.",
"Since a non-recurrent GAN cannot handle content generation with a variable number of outputs, we propose a recurrent generative adversarial network (GAN).",
"Our proposed recurrent GAN consists of a recurrent generator and recurrent discriminator (critic) such that the generator generates a single 3D body shape at each of its iterations, and the critic iterates over a set of shapes and outputs a final score as a many-to-one recurrent network.",
"To help the generator focus on the relevant words in the current iteration, we additionally learn words weights that indicate how much information each word is contributing to generating current shape.",
"The hidden state of the recurrent component maintains information that helps it remember for which words semantically consistent shapes have already been generated and for words more shapes have yet to be generated.",
"The critic has to fulfill two roles, first it has to assess the quality of the individual shapes and their semantic consistency with the given text.",
"Additionally it has to assess the matching between the generated shape and the text in a collective way, so that for example it can tell if certain words are still not satisfied, i.e.",
"not all words have a corresponding human shape.",
"When a text describes multiple people in the scene or even a single person, image generation methods synthesize images with good-looking scenes but with a disfigured human shape [38], [15], [37], [43].",
"Using 2D keypoints helps slightly as shown in [30] (figure REF ), however, it is too sparse to help guide these frameworks in generating a realistic-looking human.",
"To demonstrate the benefit of the proposed task, we integrate our model with an image synthesis framework while using COCO [18] as the training dataset and show how it substantially improves the person's appearance in the synthesized images.",
"Figure: The input to this GAN from is the 2D keypoints in addition to the text.",
"As can be seen, the generated person is incomplete and distorted."
],
[
"Related work",
"Generating high-fidelity visual content is a crucial task in computer vision.",
"Over the past few years, deep generative models such as [10], [33], [13], [5], [32] have improved the fidelity of generated content to a great extent.",
"Generative Adversarial Networks (GANs) which were first introduced in [5] have be been a powerful paradigm for learning high dimensional data distributions.",
"In the field of computer vision, GANs have been employed for different tasks for content synthesis, including unconditional image synthesis [5], [27], image synthesis conditioned on text [15], [17], [25], [26], [29], [38], [39], [43], [34], [37], [28], generating text description conditioned on images [3], style transfer between images [2],pose transfer in person images [22], [35], [36], [42], and human pose synthesis conditioned on text [41].",
"In [31], the authors generate images based on text and show that using a sparse set of keypoints makes synthesizing a higher resolution image possible.",
"Zhou et al.",
"[42] alter the pose of a person in a given image based on a text description.",
"The approach assumes that all poses can be represented by a set of clusters and is applied to a pedestrian dataset in which the poses are of standing or walking persons, and therefore cannot be applied to datasets with complex poses such as COCO.",
"Li et al.",
"[15] propose an object-driven attention module that relies on an object classifier loss.",
"However, the method fails to generate plausible human shapes despite improved results.",
"In fashion applications, Zhu et al.",
"[44] manipulate the clothing of a person in a given image based on a text description without altering the pose.",
"Prokudin et al.",
"[24] propose generating photorealistic images of humans using a sparse set of 3D vertices and a map of their corresponding RGB colors and depth values.",
"Other related works such as [6], [9], [16], [40] deal with searching for or synthesizing plausible human poses that match object affordances in a given scene.",
"Recurrent GAN architectures for generating sequential data was introduced in [23] and applied to music data.",
"In a subsequent work, Hyland et al.",
"[4] included a conditional input and applied it for generating time-series data in the medical domain."
],
[
"Approach",
"The task of predicting a variable-sized output requires a recurrent model such as a long short-term memory (LSTM) network [11].",
"We are interested in a generative model that can generate a variable number of SMPL shapes while learning the joint probability of all individuals in a scene based on text.",
"To achieve this, we rely on a recurrent adversarial learning scheme in which we design a recurrent iterative GAN that consists of two modules, namely the generator and the critic.",
"In this learning paradigm, the generator and critic are two neural networks trained simultaneously with an inverse objective, in which the critic aims to differentiate between real and generated samples while the generator aims to fool the critic."
],
[
"SMPL Body Model",
"For the 3D body model, we use the prevalent Skinned Multi-Person Linear (SMPL) model, which is a statistical human body model that allows a compact representation using shape and pose parameters.",
"The parameters consist of pose parameters $\\theta \\in \\mathbb {R}^{72}$ , shape parameters $\\beta \\in \\mathbb {R}^{10}$ and weak-perspective camera parameters $K\\in \\mathbb {R}^3$ .",
"The model outputs a 3D mesh using a learned differentiable function $M(\\theta , \\beta )$ trained on 3D scans of people, and produces vertices $V\\in R^{N_v\\times 3}, N_v=6890$ .",
"In order to find the SMPL fittings, we use the SPIN model [14] which exploits the good sides of both a deep network and an iterative optimization approach to estimate the 3D human pose and shape.",
"SPIN uses a deep network to initialize an iterative optimization algorithm and in turn uses its output to supervise the network."
],
[
"LSTM GAN",
"The proposed generative model is a Long Short Term Memory (LSTM) GAN that consists of a recurrent generator and two recurrent critics.",
"Such a recurrent design is critical for solving the task, since the output is a set of SMPL shapes with a variable size.",
"Additonally, when generating a set of SMPL shapes that are conditioned on a given text, the generator needs to have a memory component in order to monitor for which words the SMPL shapes have already been generated, and which words still need to be tended to in order to avoid repetitions.",
"The critic is designed to be recurrent in the same way as the generator.",
"This is because when the critic scores a given set of SMPL parameters conditioned on text, first it should be able to handle a variable-length set.",
"Additionally, several aspects need to be assessed simultaneously, these include the realism of the shape, its semantic consistency with the text and whether all parts of the text have a matching shape.",
"That is, when a pair of a SMPL shape set and text embedding $(s, x)$ is assessed, the critic needs a memory component in order to output a final score that takes into account all these criteria.",
"Since for example, it is possible that most shapes match the input text but perhaps a shape is missing or a duplicate shape exists or a non-matching shape is generated.",
"Therefore, the entire shape set needs to be seen by the critic in order to be evaluated as in a many-to-one recurrent LSTM.",
"The input to every iteration of the critic is a concatenation of the word embeddings and the SMPL shape generated in the same iteration $t$ in the generator or a single SMPL real shape .",
"The critics are illustrated in figure REF .",
"We use the Wasserstein GAN (WGAN) variant [1] which uses the earth mover's distance and makes the training more stable than the Vanilla GAN."
],
[
"Critics",
"The proposed model consists of two recurrent critics.",
"The first critic adds a direct supervisory signal on the set of SMPL parameters $s\\in R^{k\\cdot 85}$ where $k$ is the set size (number of person shapes).",
"It has an encoder network $(Enc(.",
"))$ for encoding the input which is a concatenation of a pair of the SMPL parameters set $s$ and text embedding $x$ .",
"It additionally has a recurrent fully connected LSTM network whose input in every iteration is a the encoded pair $Enc(s_t)$ corresponding to the SMPL parameters of one person $s_t$ generated in iteration $t$ , and the text embedding $x$ .",
"The LSTM is iterated $k$ times where $k$ is the number of persons appearing in the image.",
"Given the SMPL parameters alone which are simply represented as a real-valued vector, it is hard to distinguish whether a generated shape of the SMPL parameters ultimately corresponds to a realistic human shape or not and the network cannot learn the SMPL function which is responsible for generating the 3D body representation implicitly.",
"Accordingly, in order to simplify the generator's task, we add an additional critic whose input is the rendered SMPL UV map images.",
"These maps are downsampled using several convolutional layers and then each map's features are separately concatenated with the text embeddings and fed to the recurrent part of the critic.",
"The rendered map is obtained from the set of vertices $V$ using a differentiable renderer that projects them onto the $2D$ space.",
"This makes it possible to backpropagate through this part of the network as well, thereby making the entire pipeline differentiable.",
"The advantage of these two critics is shown in the ablative studies such that when dropping either of them, the model fails to learn.",
"Based on WGAN's earth mover's distance, the critic's objective is to maximize the distance between the true and generated samples.",
"This is reflected in the first term in eq.",
"REF .",
"Additionally, the critic should be able to discern when a pair of a SMPL set and text embeddings is real but is mismatched.",
"This is reflected in the second term of eq.",
"REF .",
"WGAN's objective requires the critic to be Lipschitz continuous, which is guaranteed when the critic's gradient norm is constrained to 1 [8].",
"The Lipschitz penalty term corresponds to the third term in eq.REF and its term is expressed in eq.",
"REF .",
"$\\begin{aligned}L_{D_{1}} = &-\\mathbb {E}_{(s,x)\\sim \\mathbb {P}_r,z\\sim \\mathbb {P}_{z}}[D(s,x)-D(G(z,x),x)]\\\\& -\\mathbb {E}_{(s,x)\\sim \\mathbb {P}_r,\\bar{x}\\sim \\mathbb {P}_{x}}[D(s,x)-D(s,\\bar{x})]\\\\& +\\lambda L_{LP},\\end{aligned}$ where $(s,x)\\sim \\mathbb {P}_r$ is pair of a SMPL set and the corresponding text embeddings from the training set $\\mathbb {P}_r$ .",
"$G(z,x)$ is the generated SMPL set for the same text embedding $x$ , and $z\\sim N(0,I)$ is the latent vector.",
"The mismatched text encoding $\\bar{x}$ is randomly sampled from the training set.",
"$L_{LP}=\\mathbb {E}_{(\\hat{s},x)\\sim \\mathbb {P}_{\\hat{s},x}}[(\\Vert \\nabla _{\\hat{s},x}D(\\hat{s},x)\\Vert _2-1)^2],$ where $\\hat{s}$ is a random interpolation between real and generated samples, a sampling which is motivated by the fact that an optical critic has gradient norm 1 at these coupled points [7].",
"$\\lambda $ is a regularization hyperparameter of the Lipschitz constraint.",
"The loss of the second critic $L_{D_2}$ is identical to the first critic loss $L_{D_1}$ , except that we replace the set of the SMPL parameters $s$ with the set of the rendered UV images $R(M(s))$ , where $M(.",
")$ is the SMPL function and $R(.",
")$ is the rendering function.",
"The total critic loss comprises the loss terms of both critics: $L_D=L_{D_1} + L_{D_2}$ and both terms are weighted equally."
],
[
"Generator",
"As stated earlier, the textual descriptions are the condition that guides the network in generating matching SMPL parameters.",
"In order to obtain the text features, the text descriptions are fed to a pre-trained bi-direction LSTM encoder trained with Deep Attentional modal Similarity Model (DAMSM) [38].",
"This component is optimized by maximizing the similarity between an image and text features at the word level, therefore guarantees fine-grained image-text matching.",
"Since a network can only take a fixed number of channels in the input, we take a maximum of $n\\in \\mathbb {N}$ words and pad with zeros if the sentence consists of less than $n$ words, making the size of the embedding $x\\in \\mathbb {R}^{n\\cdot L}$ where $L=256$ is the length of the embedding vector.",
"Having word-level encoding provides a fine level of detail for each word separately and allows the LSTM to focus its attention on the individual word when generating the SMPL shapes.",
"The text embeddings are concatenated with a latent vector $z\\in \\mathbb {R}^d$ that is sampled from the normal distribution.",
"$x\\oplus z$ is first fed through an encoder network that consists of several fully connected layers.",
"These features are concatenated with the LSTM hidden state of the previous time step $h_{1,t-1}$ and the previous output of the LSTM $o_{t-1}$ and fed through the LSTM part which generates the SMPL parameters $s_t\\in \\mathbb {R}^{1\\cdot 85}$ for one person.",
"The LSTM maintains a hidden state that helps with associating the generated shapes with the text selectively, as well as in monitoring for which text part a matching shape has already been generated.",
"Both $O_{t-1}$ and $h_{1,t-1}$ are initialized to zero in the first time step.",
"Generally, the text description that describes a scene does not refer to a distinct number of people and several numbers could match the text (for example, several people are skiing on the mountain).",
"Additionally, sometimes there are several people in the background which are ignored by text annotators.",
"During training, we iterate the LSTM based on the number of annotated persons in the scene.",
"During inference, to predict a reasonable number that can be used to halt the LSTM, we pretrain a shallow fully connected network that predicts a probability vector $p$ whose length can be set to the maximum number of annotated people in a given dataset.",
"During inference, the number of human shapes to be generated is estimated by $\\operatornamewithlimits{\\mathrm {argmax}}_i(p)$ .",
"Alternatively, since it is not always a distinct number, it can be set to the weighted average of $\\sum i\\cdot p_i$ .",
"The recurrent part of the generator consists of a fully connected LSTM network (FC-LSTM) with several hidden layers, which regresses a set of SMPL parameters corresponding to the shape of a single person in every iteration.",
"To aid the generator in paying attention to the words relevant to a given shape, we additionally learn attention weights $w_i$ s.t.",
"$\\sum _{i=1}^{n}w_i=1$ .",
"The weights are calculated in every LSTM iteration based on the hidden state and the encoded features of the text embeddings.",
"The encoded embeddings are then multiplied by these weights before being fed to the recurrent part of the generator.",
"These weights are weakly supervised using the critic's loss term.",
"The generator loss term is given by: $\\begin{aligned}L_{G_{1}}=&-\\mathbb {E}_{z\\sim \\mathbb {P}_{z},x\\sim \\mathbb {P}_{x}}\\left[D(G(z,x),x)\\right]\\\\\\end{aligned}$ $\\begin{aligned}L_{G_{2}}=&-\\mathbb {E}_{z\\sim \\mathbb {P}_{z},x\\sim \\mathbb {P}_{x}}\\left[D(R(M(G(z,x))),x)\\right]\\\\\\end{aligned}$ and the total generator loss is the sum of both loss term: $L_G=L_{G_{1}}+L_{G_{2}}$ where the first loss term refers to the generated SMPL parameter set $G(z,x)$ and the second refers to the rendering of this set.",
"An overview of the generator is shown in figure REF ."
],
[
"Generator",
"Here we provide a detailed description of the generator architecture which is described in figure REF .",
"First, the text encoder takes the text description and produces word embeddings of length $L=256$ for each word in the sentence.",
"The output is therefore $x\\in \\mathbb {R}^{n\\cdot L}$ where $n=17$ is the maximum number of words taken.",
"The vector $z\\in \\mathbb {R}^{120}\\sim N(0,I)$ is repeated $L$ times and concatenated with the embedding of each word $x_i$ .",
"The motivation is to maintain a separate encoding of each word so that the LSTM can select words for which it needs to generate the matching shape, and this is made easier when the different words encodings are not compounded.",
"The encoding of $x\\oplus z$ is fed to a small encoder network with two fully connected layers.",
"At this stage, we find the attention weights $w_i$ for each of the encoded words $i\\in [n]$ .",
"The weights are learned using a fully connected network consisting of 3 layers and a softmax operation for normalizing their sum to 1 for each generated shape.",
"The input to the attention network is a concatenation of the extracted features and the previous hidden state of the LSTM.",
"The extracted features are then multiplied by the attention weights and concatenated with the LSTM hidden state $h_{t-1}$ and the previous output $o_{t-1}$ and inputted to the LSTM network which has 3 hidden cells with gates consisting of fully connected layers as the output is a real-valued vector representing the SMPL parameters $s_t\\in \\mathbb {R}^{85}$ of one person."
],
[
"Critics",
"The first critic of the SMPL parameters (figure 4 in the manuscript) takes a pair of a set of SMPL parameters $s\\in R ^{k\\cdot 85}$ ($k$ is the number of person shapes) and the text embeddings $x\\in \\mathbb {R}^{17\\cdot 256}$ .",
"Each SMPL element $s_t$ in the set (corresponding to a person shape) is concatenated with the text embedding and the previous hidden state $h_{t-1}$ and then fed to a two-layer encoder network.",
"The number of extracted features correspond to the set size which is the number of human shapes to be assessed by the critic.",
"At this stage, each of the features corresponding to a shape is fed to the LSTM part and the LSTM is iterated $|s|$ times.",
"The final score that is taken into account in the loss function is the one outputted by the LSTM in the last iteration.",
"The LSTM consists of 3 hidden cells with fully connected gates where the hidden size is .",
"The second critic is similar to the first critic, but since now we have 2D images of resoltion $224\\times 224$ representing the rendered human shapes rather than vectors of the SMPL parameters, the images need to be downscaled from resolution $224\\times 224$ to resolution $14\\times 14$ before being concatenated with the text embeddings, where the text embeddings are repeated to match the set size and the downscaled resolution of 14.",
"Additionally, the LSTM part consists of 3 fully convolutional gates which further downscale the features until a real-value representing the score is produced.",
"The weight of the Lipschitz term in eq.",
"1 is set to $\\lambda =10$ .",
"To obtain the rendered shapes, we use an implementation of a differentiable renderer https://github.com/daniilidis-group/neural_renderer, which is based on Neural 3D Mesh Renderer [12]."
],
[
"Dataset.",
"We use the COCO (Common Objects in Context) [19] dataset for training and evaluating the model.",
"This dataset contains annotated images of everyday scenes and every image has five human-written text descriptions and 2D keypoints.",
"The SMPL fittings are extracted using SPIN [14] from images that contain humans which yields about $35k$ training sample.",
"The pseudo-ground truth of the SMPL parameters are obtained using the SPIN framework https://github.com/nkolot/SPIN.",
"SPIN expects a bounding box crop of each person appearing in the image.",
"The crops are augmented using small scale and rotation transformations.",
"These samples comprise the real pairs of SMPL shapes and text embeddings which are used to train the critics.",
"RMSprop optimizer is used with learning rate $\\eta =1e^{-5}$ .",
"Based on WGAN update strategy, the critic is updated 5 times for every generator update step.",
"After filtering out images without a valid SPIN prediction, there are in total $35k$ (image, text) sample pairs, in which the number of persons in each image varies.",
"The model is trained for 30 epochs.",
"Figure REF includes synthesized SMPL images.",
"As can be seen, the synthesized SMPL shapes match the text description.",
"Figure: Example generated SMPL shapes from the COCO validation set.",
"In the second row, even though the text description refers to a person jumping with a racket, the model infers an additional player since in COCO most images with a tennis match have two annotated players.",
"The model also infers the interaction between the players, such that the player on the left is striking the ball and the player on the right is preparing to return it.To show the benefit of the proposed task, we integrate our model with an image synthesis framework while using COCO as the training dataset.",
"We injected the generated SMPL shape during the training of [37], which is a state of the art image synthesis framework.",
"We then evaluated on captions from the COCO validation set.",
"Figure REF shows the results.",
"We can see how first generating the human shape using our pretrained model substantially improves the person's appearance in the synthesized images.",
"Figure: Images generated from several captions.",
"The images generated with DF GAN synthesize the scene well but fail to synthesize realistic looking humans.",
"When the generated SMPL image is injected along with the text features, it guides the network towards generating a realistic-looking person that reflects the relevant text description.",
"It can be observed that the network further learns to create body parts such as the hand at a fine level of detail, and can distinguish the different clothing parts and places the person reasonably within the scene."
],
[
"Quantitative evaluation",
"To evaluate our model quantitatively, we use measures that are similar to those introduced in [41] for single pose synthesis.",
"However, since we generate multiple shapes, we need to adapt the calculation such that shapes from two different samples are matched together.",
"In order to find an optimal matching between two samples of a generated and real SMPL set, we use the Hungarian algorithm where the cost matrix is calculated between each pair of shapes.",
"We sample 400 samples randomly from the validation set to make the computation time reasonable.",
"$\\bar{d}^p_{nn}$ denotes the average distance between the generated sample and nearest neighbor real sample.",
"$\\bar{d}^t_{p_{nn}}$ denotes the distance in the embedding space of the nearest neighbor, indicating that even if the nearest neighbor sample is not the ground-truth itself, the text distance to the nearest neighbor is still small.",
"This values is compared with $\\bar{d}^t_{all}$ which denotes the average unconditional distance between all text embeddings (this distance is fixed).",
"$\\bar{d}^p_{gt}$ denotes the average distance between the generated sample and the ground-truth real sample (the one corresponding to the input text).",
"We compute the distances in both the SMPL parameter space and UV space which are denoted by the superscript $p/r$ and the results are presented in tables REF , REF respectively.",
"Table: Quantitative evaluation of the generated SMPL shapes in the SMPL parameter space.",
"`wo critic2` refers to dropping the critic of the rendered SMPL shape.Table: Quantitative evaluation of the generated SMPL shapes in the SMPL UV space.",
"\"wo critic2\" refers to dropping the critic of the rendered SMPL shape."
],
[
"User study",
"In order to evaluate the realism of the generated shapes, we created a user study with 10 questions in which we display a set of generated shapes and 3 different captions that include one ground truth caption and another two captions.",
"Users were asked to select the caption that best matches the SMPL render set.",
"The average correct score was $68\\%$ .",
"Sometimes it is hard to determine distinctly what text the shape corresponds to if the action is generic (such as standing), unlike shapes for distinct activities such as batting a ball or skating.",
"However, such details can be determined more confidently once the context is clear from the whole scene as shown in the synthesized images in figure REF ."
],
[
"Ablative studies",
"If the SMPL parameters critic is dropped, then we lose the direct supervisory signal on the GAN output and the model fails to learn completely.",
"When the SMPL render critic is dropped, then we lose the supervisory signal on the human shape and the rendered shapes quality degrades.",
"This is also reflected in the quantitative measures in table REF and REF .",
"When perturbing the word ordering such as in an inverse or random ordering, then the model diverges completely, indicating that the word ordering is crucial for learning the relationship between the generated SMPL shapes and the text."
],
[
"Conclusion",
"In this work we have introduced the task of generating multiple 3D shapes of the human body conditioned on text.",
"We have evaluated the models using different metrics and a user study.",
"We additionally have demonstrated how it helps synthesizing more realistic and refined human shapes in image synthesis frameworks."
],
[
"Acknowledgement",
"We would like to thank Dimitris Tzionas for providing insightful and valuable feedback on this work."
]
] | 2107.01869 |
[
[
"On the image of Hitchin morphism for algebraic surfaces: The case ${\\rm\n GL}_n$"
],
[
"Abstract The Hitchin morphism is a map from the moduli space of Higgs bundles $\\mathscr{M}_X$ to the Hitchin base $\\mathscr{B}_X$, where $X$ is a smooth projective variety.",
"When $X$ has dimension at least two, this morphism is not surjective in general.",
"Recently, Chen-Ng\\^o introduced a closed subscheme $\\mathscr{A}_X$ of $\\mathscr{B}_X$, which is called the space of spectral data.",
"They proved that the Hitchin morphism factors through $\\mathscr{A}_X$ and conjectured that $\\mathscr{A}_X$ is the image of the Hitchin morphism.",
"We prove that when $X$ is a smooth projective surface, this conjecture is true for vector bundles.",
"Moreover, we show that $\\mathscr{A}_X$, for any dimension, is invariant under proper birational morphisms, and apply the result to study $\\mathscr{A}_X$ for ruled surfaces."
],
[
"Introduction",
"Throughout the paper, we work over an algebraically closed field $k$ of characteristic zero.",
"Let $\\mathcal {M}_X$ be the moduli space of semistable Higgs bundles of rank $n$ over a smooth projective variety $X$ over $k$ .",
"Let ${B}_X= \\bigoplus \\limits _{i=1}^n H^0(X, S^n T^*X)$ be the Hitchin base, where $T^*X$ is the cotangent bundle of $X$ .",
"The Hitchin morphism $h_X: \\mathcal {M}_X \\rightarrow {B}_X$ was introduced in Hitchin's seminal work [6] for algebraic curves.",
"The morphism was proved to be dominant by Beauville-Narasimhan-Ramanan [1], and was later on proved to be proper by Nisture [7] and Simpson [11], and hence surjective.",
"In a more general setting and for higher dimensional varieties, Simpson showed that the Hitchin morphism is still proper [10], [11], but it is not surjective in general.",
"Recently, Chen and Ngô took an attempt to understand the image of the Hitchin morphism and obtained a higher dimensional analogue of the BNR correspondence [1].",
"In [2], [3], they considered the moduli stack ${M}_X$ of Higgs bundles on $X$ , and introduced a closed subscheme $\\mathfrak {i}_X: {A}_X \\hookrightarrow {B}_X$ , where ${A}_X={\\rm Sect}(X,{\\rm Chow}^n(T^*_X/X))$ is called the space of spectral data (see §2).",
"They showed the Hitchin morphism factors through the space of spectral data MX [d,\"hX\"] [ld, dotted, \"sdX\" description] AX [r, hook, \"iX\"] BX, and conjectured that the morphism $sd_X$ is surjective.",
"Conjecture 1.1 (Conjecture 5.2 in [3]) For every point $a \\in {A}_X(k)$ , the fiber $sd^{-1}_X(a)$ is nonempty.",
"In fact, the conjecture is stated for $G$ -Higgs bundles, where $G$ is a split reductive group.",
"In case the dimension $d=2$ and $G={\\rm GL}_n$ , the conjecture is proved by Chen and Ngô but only for spectral data in an open subset ${A}^{\\heartsuit }_X(k) \\subseteq {A}_X(k)$ , and it is verified for some minimal surfaces, such as ruled surfaces and nonisotrivial elliptic surfaces for all $a\\in {A}_X(k)$ , see [3] for more precise statements.",
"In this article, we shall confirm the conjecture in case $d=2$ and $G={\\rm GL}_n$ : Theorem 1.2 (Theorem REF ) Let $X$ be a smooth projective surface.",
"Let ${M}_X$ be the moduli stack of Higgs bundles of rank $n$ over $X$ .",
"Then the image of the Hitchin morphism $h_X: {M}_X \\rightarrow {B}_X$ is ${A}_X$ , i.e., $sd_X : {M}_X \\rightarrow {A}_X$ is surjective.",
"Here is an overview of the proof of the main result and the structure of the paper.",
"Let ${\\rm Chow}^n(T^*_X/X)$ be the relative Chow variety of $n$ -points on $T^*_X$ over $X$ .",
"Chen and Ngô defined the space of spectral data ${A}_X$ to be the space of sections $X \\rightarrow {\\rm Chow}^n(T^*_X/X)$ , and a section $a$ is called a spectral datum.",
"Note that ${A}_X$ has a natural scheme structure, cf.",
"§2 for an elaboration.",
"For the relative Chow variety, there is a natural stratification, as in the absolute case, ${\\rm Chow}^{n}(T^*_X/X) = \\coprod _{\\mu } {\\rm Chow}^n_{\\mu }(T^*_X/X),$ where the union is taken over all partitions of $n$ .",
"Chen and Ngô showed that given a spectral datum $a: X \\rightarrow {\\rm Chow}^n(T^*_X/X)$ , if the generic point of $X$ under $a$ lies in ${\\rm Chow}^n_{(1^n)}(T^*_X/X)$ , then there is a finite flat cover $\\widetilde{X}_a \\rightarrow X$ such that the Higgs bundles over $X$ with spectral datum $a$ correspond to the Cohen-Macaulay sheaves of rank one on $\\widetilde{X}_a$ (see [3]).",
"In particular, for such general $a$ , then $sd_X^{-1}(a)$ is nonempty.",
"Now given an arbitrary $a \\in {A}_X(k)$ , we assume that $a$ maps the generic point of $X$ to the stratum ${\\rm Chow}^n_{\\mu }(T^*_X/X)$ , where $\\mu =(1^{\\alpha _1}\\dots s^{\\alpha _s})$ is a partition of $n$ .",
"Therefore, $a$ induces a morphism $X \\rightarrow \\overline{{\\rm Chow}^n_{\\mu }(T^*_X /X)}$ .",
"Since $X$ is smooth, the spectral datum $a$ factors through the normalization $\\overline{{\\rm Chow}^n_{\\mu }(T^*_X /X)}^{\\rm nor}$ .",
"Abusing the notation, we use the same notation $a: X \\rightarrow \\overline{{\\rm Chow}^n_{\\mu }(T^*_X /X)}^{\\rm nor}$ .",
"We observe the isomorphism, which is analogous to [4], $\\overline{{\\rm Chow}^n_{\\mu }(T^*_X/X)}^{\\rm nor} \\cong {\\rm Chow}^{\\alpha _1}(T^*_X/X)\\times _X \\cdots \\times _X {\\rm Chow}^{\\alpha _s}(T^*_X/X)$ still holds (Proposition REF ).",
"It follows that $a$ yields a collection of spectral data $(a_i)_{1\\le i \\le s}$ , $a_i: X \\rightarrow \\overline{{\\rm Chow}^{\\alpha _i}_{(1^{\\alpha _i})}(T^*_X/X)}={\\rm Chow}^{\\alpha _i}(T^*_X/X).$ This gives a decomposition of the given spectral datum $a$ , which is the key point to prove the main result.",
"Theorem 1.3 (Theorem REF ) Let $X$ a smooth projective surface.",
"Given a spectral datum $a: X \\rightarrow {\\rm Chow}^{n}(T^*_X /X)$ , suppose the generic point of $X$ is mapped to the stratum ${\\rm Chow}^{n}_{\\mu }(T^*_X /X)$ with $\\mu =(1^{\\alpha _1}2^{\\alpha _2}\\dots s^{\\alpha _s})$ .",
"Then there exist spectral data $a_i: X \\rightarrow \\overline{{\\rm Chow}^{\\alpha _i}_{(1^{\\alpha _i})}(T^*_X /X)}$ , $i=1, \\cdots , s$ , such that $a=\\sum ^s_{i=1} ia_i.$ Since $a_i$ maps the generic point of $X$ to ${\\rm Chow}^{\\alpha _i}_{(1^{\\alpha _i})}(T^*_X/X)$ , there exists a Higgs bundle $(E_i,\\phi _i)$ corresponds to $a_i$ according to [3].",
"Then the direct sum $(E,\\phi ):=\\bigoplus _{i=1}^s (E_i,\\phi _i)^{\\oplus i},$ is a Higgs bundle of $n$ corresponding to the spectral datum $a$ .",
"This gives the surjectivity of $sd_X: {M}_X(k) \\rightarrow {A}_X(k)$ .",
"In a different direction, in §5, we prove the birational invariance of ${A}_X$ for any dimensional variety $X$ , and apply the result to study the spaces of spectral data of ruled surfaces.",
"Theorem 1.4 (Theorem REF ) Let $\\chi : X^{\\prime }\\rightarrow X$ be a birational proper morphism of smooth varieties.",
"Then the natural morphism ${A}_X\\rightarrow {A}_{X^{\\prime }}$ induced by $\\chi $ is an isomorphism as $k$ -schemes.",
"There are a number of interesting questions to be addressed in the future.",
"As we stated above, in the surface case, we can construct a Higgs bundle for any given spectral data; however, we do not find a “cover\" $\\widetilde{X}_a \\rightarrow X$ such that we can get a correspondence between Higgs bundles with spectral data $a$ over $X$ and some special rank 1 sheaves over $\\widetilde{X}_a$ .",
"In Remark REF , we discuss a finite, flat cover $\\widetilde{X}_a \\rightarrow X$ based on an arbitrary spectral datum $a$ , which may help to give the correspondence.",
"In this article, we only consider Higgs bundles (${\\rm GL}_n$ -case).",
"It would be very interesting to generalize the result to $G$ -Higgs bundles over higher dimensional varieties.",
"This is of course the full statement of Chen and Ngô's conjecture.",
"The classical Hitchin morphism is given for the moduli space of semistable Higgs bundles, and the conjectural image of the Hitchin morphism is for the moduli stack of Higgs bundles.",
"It is natural to ask if ${A}_X$ is the image of the Hitchin morphism is for the moduli space $\\mathcal {M}^{ss}_X(P)$ of Higgs bundles with respect to some Hilbert polynomial $P$ ?",
"Acknowledgments.",
"The authors would like to thank X. Hu, G. Kydonakis, M. Li, J. Xu and L. Zhao for helpful discussions and conversations.",
"The authors are very grateful to G. Kydonakis for invaluable suggestions on an early draft.",
"During the preparation of the paper, L. S. and H. S. were partially supported respectively by the Guangdong Basic and Applied Basic Research Foundation 2020A1515010876 and 2019A1515110961."
],
[
"Moduli Stacks of Higgs Bundles, Hitchin Base and Spectral data",
"In this section, fix the rank $n$ , we discuss three moduli stacks on smooth quasi-projective variety $X$ over $k$ : the moduli stack of Higgs bundles ${M}_X$ ; the moduli stack of the Hitchin base ${B}_X$ ; the moduli stack of spectral data ${A}_X$ , and they are the main objects we work on.",
"In this paper, we only focus on the Higgs bundles, i.e.",
"${\\rm GL}_n$ -Higgs bundles, but some of the constructions can be extended to $G$ -Higgs bundles (see [2], [3] for more details).",
"Let $\\mathfrak {C}^d_{{\\rm GL}_n} \\subseteq \\mathfrak {gl}^d_n$ be the closed subscheme consisting of $d$ -tuples $(x_1,\\dots ,x_d) \\in \\mathfrak {gl}^d_n$ such that $x_i$ and $x_j$ commutes, i.e.",
"$[x_i,x_j]=0$ for all indices $i,j$ .",
"The scheme $\\mathfrak {C}^d_{{\\rm GL}_n}$ is also known as the commuting scheme (see [8]).",
"There are two group actions on $\\mathfrak {C}^d_{{\\rm GL}_n}$ : Given $g \\in {\\rm GL}_n$ , the ${\\rm GL}_n$ -action on $\\mathfrak {C}^d_{{\\rm GL}_n}$ is defined as $(x_1,\\dots ,x_d) \\rightarrow ({\\rm ad}(g)(x_1),\\dots ,{\\rm ad}(g)(x_d)).$ Given $h \\in {\\rm GL}_d$ , the ${\\rm GL}_d$ -action on $\\mathfrak {C}^d_{{\\rm GL}_n}$ is defined as $(x_1,\\dots ,x_d) \\rightarrow (x_1,\\dots ,x_d)h.$ We first consider the ${\\rm GL}_n$ -action on $\\mathfrak {C}^d_{{\\rm GL}_n}$ .",
"The Chevalley restriction theorem gives us a ${\\rm GL}_d$ -equivariant morphism $\\mathbb {C}[\\mathfrak {gl}_n^d]^{{\\rm GL}_n} \\rightarrow \\mathbb {C}[\\mathfrak {t}^d]^{\\mathfrak {S}_n},$ where $\\mathfrak {t} \\subseteq \\mathfrak {gl}_n$ is a Cartan subalgebra and $\\mathfrak {S}_n$ is the permutation group.",
"Restricting the morphism to $\\mathfrak {C}^d_{{\\rm GL}_n}$ , we have a ${\\rm GL}_d$ -equivariant lifting $[\\mathfrak {C}^{d}_{{\\rm GL}_n}/{\\rm GL}_n] \\rightarrow {\\rm Spec}(\\mathbb {C}[\\mathfrak {t}^d]^{\\mathfrak {S}_n})$ of $[\\mathfrak {C}^{d}_{{\\rm GL}_n}/{\\rm GL}_n] \\rightarrow \\mathfrak {C}^{d}_{{\\rm GL}_n}/{\\rm GL}_n$ along with the Chevalley restriction morphism.",
"Let $V$ be a vector space of dimension $d$ .",
"Therefore, we have ${\\rm Spec}(\\mathbb {C}[\\mathfrak {t}^d]^{\\mathfrak {S}_n}) \\cong {\\rm Chow}^n(V),$ where ${\\rm Chow}^n(V)$ is the Chow variety of $n$ points on $V$ .",
"Then we have the morphism $[\\mathfrak {C}^{d}_{{\\rm GL}_n}/{\\rm GL}_n] \\rightarrow {\\rm Chow}^n(V).$ By [2], we know that there is a closed embedding ${\\rm Chow}^n(V) \\hookrightarrow V \\times S^2 V \\times \\dots \\times S^n V,$ and this induces the following morphism $[\\mathfrak {C}^{d}_{{\\rm GL}_n}/{\\rm GL}_n] \\rightarrow {\\rm Chow}^n(V)\\hookrightarrow V \\times S^2 V \\times \\dots \\times S^n V.$ As a $d$ -dimensional vector space, there is a canonical ${\\rm GL}_d$ -action on $V$ .",
"We define the following quotient stacks ${A}:=[{\\rm Chow}^n(V)/{\\rm GL}_d], \\quad {B}:= [V \\times S^2V \\times \\dots \\times S^n V/ {\\rm GL}_d].$ By the definition of quotient stacks, for each $k$ -scheme $S$ , ${B}(S)$ is the set of pairs $(E,\\phi )$ , where $E$ is a rank $d$ vector bundle (as a ${\\rm GL}_d$ -principal bundle) over $S$ and $\\phi : E \\rightarrow V \\times S^2V \\times \\dots \\times S^n V$ is a ${\\rm GL}_d$ -equivariant map.",
"Note that the ${\\rm GL}_d$ -equivariant map $\\phi $ corresponds to a unique element in $\\bigoplus \\limits _{i=1}^{n}H^0(S, S^i E)$ .",
"Therefore, we have the following isomorphism as sets (or groupoids) by restricting to a fixed vector bundle $E$ over $S$ , ${B}(S)|_E \\cong \\bigoplus _{i=1}^{n}H^0(S, S^i E).$ This isomorphism also implies that ${B}(S)|_E$ has a scheme structure.",
"For the quotient stack ${A}$ , given a $k$ -scheme $S$ , ${A}(S)$ is the set of pairs $(E,\\varphi )$ , where $E$ is a rank $d$ vector bundle over $S$ and $\\varphi : E \\rightarrow {\\rm Chow}^n(V)$ is a ${\\rm GL}_d$ -equivariant map.",
"This map is equivalent to an element in ${\\rm Chow}^n(E/S)$ , where ${\\rm Chow}^n(E/S)$ is the relative Chow variety.",
"In other words, we have ${A}(S)|_E \\cong {\\rm Sect}(S,{\\rm Chow}^n(E/S)),$ where ${\\rm Sect}(S,{\\rm Chow}^n(E/S))$ is the set of sections."
],
[
"Moduli Stack of Higgs Bundles",
"Recall that there is a natural ${\\rm GL}_d$ -morphism on $[\\mathfrak {C}^{d}_{{\\rm GL}_n}/{\\rm GL}_n]$ .",
"Denote by ${M}:=[\\mathfrak {C}^{d}_{{\\rm GL}_n}/{\\rm GL}_n \\times {\\rm GL}_d]$ the quotient stack.",
"Let $S$ be a scheme, and then the $S$ -points of ${M}$ are triples $(V,E,\\theta )$ , where $V$ is a vector bundle of rank $d$ (${\\rm GL}_d$ -principal bundle) over $S$ ; $E$ is a vector bundle of rank $n$ (${\\rm GL}_n$ -principal bundle) over $S$ ; $\\theta \\in H^0(S, End(E) \\otimes V)$ is a $\\mathcal {O}_S$ -linear morphism such that $\\theta \\wedge \\theta =0$ .",
"Now we fix a smooth projective (or quasi-projective) variety $X$ of dimension $d$ .",
"Let $T^*_X$ be the cotangent bundle over $X$ .",
"We define a contravariant functor ${M}_X: ({\\rm Sch}/k)^{\\rm op} \\rightarrow {\\rm Sets}$ as follows.",
"Let $S$ be a scheme, and define $X_S:=X \\times _{k} S$ .",
"Denote by $\\pi : X_S \\rightarrow X$ the natural projection.",
"There is a natural morphism $[\\mathfrak {C}^{d}_{{\\rm GL}_n}/{\\rm GL}_n \\times {\\rm GL}_d] \\rightarrow [\\ast / {\\rm GL}_d]=\\mathbb {B}{\\rm GL}_d,$ which induces the morphism $[\\mathfrak {C}^{d}_{{\\rm GL}_n}/{\\rm GL}_n \\times {\\rm GL}_d](X_S) \\rightarrow \\mathbb {B}{\\rm GL}_d(X_S).$ Note that $\\pi ^* T^*_X$ is a dimension $d$ vector bundle over $X_S$ , which corresponds to a point $[\\pi ^* T^*_X] \\in \\mathbb {B}{\\rm GL}_d(X_S)$ .",
"We define ${M}_X(S)$ to be the preimage of $[\\pi ^* T^*_X]$ in $[\\mathfrak {C}^{d}_{{\\rm GL}_n}/{\\rm GL}_n \\times {\\rm GL}_d](X_S)$ , which parametrizes pairs $(E,\\theta )$ such that $E$ is a rank $n$ vector bundle over $X_S$ ; $\\theta \\in H^0(X_S, \\mathcal {E}nd(E) \\otimes \\pi ^* T^*_X)$ is a section such that $\\theta \\wedge \\theta =0$ .",
"It is easy to check that ${M}_X$ is a stack, which is actually the stack of Higgs bundles on $X$."
],
[
"Hitchin Base ${B}_X$",
"We have a natural morphism ${B} \\rightarrow \\mathbb {B}{\\rm GL}_d$ of stacks.",
"Fixing a dimension $d$ smooth projective variety $X$ , we define the following contravariant functor as follows ${B}_X: ({\\rm Sch}/k)^{\\rm op} \\rightarrow {\\rm Sets}$ For each scheme $S$ , we define the space ${B}_X(S):={B}(X_S)|_{[\\pi ^* T^*_X]} \\subseteq {B}(X_S)$ to be the preimage of the point $[\\pi ^*T^*_X] \\in \\mathbb {B}{\\rm GL}_d(X_S)$ under the map ${B}(X_S) \\rightarrow \\mathbb {B}{\\rm GL}_d(X_S)$ .",
"Thus, ${B}_X(S) \\cong \\bigoplus _{i=1}^{n}H^0(X_S, S^i \\pi ^* T^*_X),$ where $S^i \\pi ^* T^*_X$ is the $i$ -th symmetric product of $\\pi ^* T^*_X$ .",
"Based on the above definition, it is easy to check that ${B}_X$ has a natural stack structure.",
"Furthermore, we can rewrite $H^0(X_S, S^i \\pi ^* T^*_X)$ in the following way $& H^0(X_S, S^i \\pi ^* T^*_X) \\cong {\\rm Hom}(\\pi ^* T^*_X, S^i V)^{{\\rm GL}_d} \\cong {\\rm Hom}(X_S \\times _X T^*_X, S^i V)^{{\\rm GL}_d}\\\\\\cong & {\\rm Hom}(S \\times T^*_X, S^i V)^{{\\rm GL}_d} \\cong {\\rm Hom}(S, {\\rm Hom}(T^*_X, S^i V)^{{\\rm GL}_d}) \\cong {\\rm Hom}(S, H^0(X, S^i T^*_X)).$ Therefore, ${B}_X(S) \\cong \\bigoplus _{i=1}^{n} {\\rm Hom}(S, H^0(X, S^i T^*_X)).$ Then, the stack ${B}_X$ is representable by the scheme $\\bigoplus _{i=1}^{n} H^0(X, S^i T^*_X)$ , i.e.",
"${B}_X(\\ast ) \\cong {\\rm Hom}(\\ast , \\bigoplus _{i=1}^{n} H^0(X, S^i T^*_X)).$ This gives the following lemma.",
"Lemma 2.1 The moduli stack ${B}_X$ is represented by $\\bigoplus \\limits _{i=1}^{n} H^0(X, S^i T^*_X)$ .",
"By abusing notation, we also use ${B}_X$ for the scheme $\\bigoplus \\limits _{i=1}^{n} H^0(X, S^i T^*_X)$ , which is known as the Hitchin base."
],
[
"Space of Spectral Data ${A}_X$",
"Similar to ${B}_X$ , we can define the the functor ${A}_X: ({\\rm Sch}/k)^{\\rm op} \\rightarrow {\\rm Sets}$ such that for each scheme $S$ , define ${A}_X(S) \\subseteq {A}(X_S)$ to be the preimage of the point $[\\pi ^*T^*_X]$ under the map ${A}(X_S) \\rightarrow \\mathbb {B}{\\rm GL}_d(X_S)$ .",
"We know that ${A}_X(S) \\cong {\\rm Sect}(X_S,{\\rm Chow}^n(\\pi ^* T^*_X/X_S)).$ With the same discussion as for ${B}_X$ , we have ${A}_X(S) \\cong {\\rm Sect}(X_S, {\\rm Chow}^n( \\pi ^*T^*_X/X_S ) ) \\cong {\\rm Hom}(S, {\\rm Sect}(X, {\\rm Chow}^n( T^*_X/X ) )).$ This gives us the following lemma.",
"Lemma 2.2 The stack ${A}_X$ is representable by the scheme ${\\rm Sect}(X, {\\rm Chow}^n( T^*_X/X ) )$ , i.e.",
"${A}_X(\\ast ) \\cong {\\rm Hom}(\\ast , {\\rm Sect}(X, {\\rm Chow}^n( T^*_X/X ) )).$ We use the same notation ${A}_X$ for the corresponding scheme ${\\rm Sect}(X, {\\rm Chow}^n( T^*_X/X ) )$ .",
"An element $a \\in {A}_X$ is called a spectral datum and ${A}_X$ is called the space of spectral data.",
"Sometimes, we use the notation ${A}_X^n$ to highlight the integer $n$ .",
"Remark 2.3 ${B}_X$ and ${A}_X$ are claimed to be schemes [2], [3], and here we give the constructions from the viewpoint of moduli stack.",
"Remark 2.4 Let $U \\rightarrow X$ be a morphism of schemes.",
"Since ${M}$ is a stack, we have a natural map ${M}(X) \\rightarrow {M}(U)$ .",
"Furthermore, this map induces a morphism ${M}_X \\rightarrow {M}_U$ as stacks.",
"The same argument also works for morphisms ${A}_X \\rightarrow {A}_U$ and ${B}_X \\rightarrow {B}_U$ .",
"As a special case, if $U$ is an open subscheme of $X$ , then we have a natural morphism ${A}_X \\rightarrow {A}_U$ .",
"Remark 2.5 Recall that the closed embedding ${\\rm Chow}^n(V) \\hookrightarrow V \\times S^2 V \\times \\dots \\times S^n V,$ induces the closed embedding $\\mathfrak {i}_X: {A}_X \\hookrightarrow {B}_X$ ([3]).",
"Chen and Ngô showed that the Hitchin map $h_X: {M}_X \\rightarrow {B}_X$ factors through ${A}_X$ .",
"More precisely, there exists a map $sd_X: {M}_X \\rightarrow {A}_X$ such that $h_X=\\mathfrak {i}_X \\circ sd_X$ .",
"MX [d,\"hX\"] [ld, dotted, \"sdX\" description] AX [r, hook, \"iX\"] BX When $X$ is a smooth curve, we have ${A}_X \\cong {B}_X$ , and hence $sd_X$ is surjective.",
"But for higher dimensions, ${A}_X$ is smaller than ${B}_X$ in general."
],
[
"Decomposition of spectral data",
"In this section, we prove a decomposition theorem (Theorem REF ) for spectral datum $a \\in {A}_X$ , where $X$ is an algebraic surface.",
"This decomposition theorem is an important tool to prove the main theorem (Theorem REF ).",
"In §3.1, we briefly review some relevant facts about Chow varieties, and we refer the reader to [4] for more details.",
"In §3.2, we generalize some results to the relative case, and prove the decomposition theorem."
],
[
"Classical Case",
"Let $X$ be a smooth projective or quasi-projective surface.",
"Let ${\\rm Chow}^n(X)$ be the Chow variety of $n$ points on $X$ .",
"Let $\\mu $ be a partition of $n$ .",
"We will write $\\mu =(n_1,\\dots ,n_k)$ or $\\mu =(1^{\\alpha _1}2^{\\alpha _2}\\dots s^{\\alpha _s})$ interchangeably.",
"Define a locally closed subset ${\\rm Chow}^n_{\\mu }(X)$ of ${\\rm Chow}^n(X)$ as ${\\rm Chow}^n_{\\mu }(X):=\\lbrace \\sum _i^k n_i z_i \\text{ } | \\text{ } z_i \\in X \\text{ distinct points } \\rbrace .$ Then, there is a stratification of ${\\rm Chow}^{n}(X)$ : ${\\rm Chow}^{n}(X) = \\coprod _{\\mu } {\\rm Chow}^n_{\\mu }(X),$ where $\\mu $ ranges over all partitions of $n$ .",
"Given a partition $\\mu =(1^{\\alpha _1}2^{\\alpha _2}\\dots s^{\\alpha _s})$ , we have a natural morphism ${\\rm Chow}^{\\alpha _1}(X) \\times {\\rm Chow}^{\\alpha _2}(X) \\times \\dots \\times {\\rm Chow}^{\\alpha _s}(X) \\rightarrow {\\rm Chow}^n(X)$ defined as $(z_1,z_2,\\dots , z_s) \\mapsto z_1+ 2z_2 + \\dots + s z_s.$ Indeed this morphism gives the normalization of $\\overline{{\\rm Chow}^n_{\\mu }(X)}$ .",
"Lemma 3.1 (Exercise 7.4.2 in [4]) Given a partition $\\mu =(1^{\\alpha _1}, 2^{\\alpha _2}, \\dots , s^{\\alpha _s})$ , we have $\\overline{{\\rm Chow}^n_{\\mu }(X)}^{\\rm nor} \\cong {\\rm Chow}^{\\alpha _1}(X) \\times \\dots \\times {\\rm Chow}^{\\alpha _s}(X),$ where $\\overline{{\\rm Chow}^n_{\\mu }(X)}^{\\rm nor}$ is the normalization of the closure of ${\\rm Chow}^n_{\\mu }(X)$ ."
],
[
"Relative Case",
"Let $X$ be a smooth projective surface.",
"Denote by $T^*_X$ the cotangent bundle over $X$ .",
"The Chow variety ${\\rm Chow}^n(T^*_X/X)$ is defined as follows ${\\rm Chow}^n(T^*_X/X):= \\underbrace{T^*_X \\times _X \\dots \\times _X T^*_X}_n/ \\mathfrak {S}_n,$ where $\\mathfrak {S}_n$ is the symmetric group.",
"Similar to the classical case, we define a locally closed subset ${\\rm Chow}^n_{\\mu }(T^*_X/X)$ of ${\\rm Chow}^n(T^*_X/X)$ with respect to the partition $\\mu $ , and then we have a stratification of ${\\rm Chow}^{n}(T^*_X/X)$ , ${\\rm Chow}^{n}(T^*_X/X) = \\coprod _{\\mu } {\\rm Chow}^n_{\\mu }(T^*_X/X),$ where $\\mu $ ranges over all partitions of $n$ .",
"For ease of notation, set $\\prod ^s_{j=1, X}{\\rm Chow}^{\\alpha _j}(T^*_X/X):={\\rm Chow}^{\\alpha _1}(T^*_X/X) \\times _X {\\rm Chow}^{\\alpha _2}(T^*_X/X) \\times _X \\dots \\times _X {\\rm Chow}^{\\alpha _s}(T^*_X/X),$ where the subscript $X$ under the symbol $\\prod ^s_{j=1, X}$ indicates that the product is taken over $X$ .",
"Similarly set $\\prod ^n_{j=1, X}T^*_X:=\\underbrace{T^*_X \\times _X \\dots \\times _X T^*_X}_n.$ We summarize some facts: Lemma 3.2 For any $n\\ge 1$ and a partition $\\mu =(1^{\\alpha _1} \\dots s^{\\alpha _s})$ , the following hold: (i) $\\prod \\limits ^n_{j=1, X}T^*_X$ is smooth over $X$ .",
"(ii) ${\\rm Chow}^n_{\\mu }(T^*_X /X)$ is smooth over $X$ .",
"(iii) ${\\rm Chow}^n(T^*_X /X)$ has rational singularities, in particular it is normal and Cohen-Macaulay.",
"(iv) $\\prod \\limits ^s_{j=1, X}{\\rm Chow}^{\\alpha _j}(T^*_X/X)\\cong \\prod \\limits ^{\\alpha _1+\\dots +\\alpha _s}_{j=1, X}T^*_X/{\\mathfrak {S}_{\\alpha _1}\\times \\cdots \\times \\mathfrak {S}_{\\alpha _s}}$ has rational singularities, in particular, it is normal and Cohen-Macaulay.$\\Box $ Given a partition $\\mu =(1^{\\alpha _1}2^{\\alpha _2}\\dots s^{\\alpha _s})$ of $n$ , we define a morphism $\\tau _{\\mu }: \\prod ^s_{j=1, X}{\\rm Chow}^{\\alpha _j}(T^*_X/X) \\rightarrow {\\rm Chow}^n(T^*_X/X)\\qquad \\mathrm {(\\ast )}$ as $(z_1,z_2,\\dots , z_s) \\mapsto z_1+ 2z_2 + \\dots + s z_s.$ The following is a key observation, which is a relative version of Lemma REF .",
"Proposition 3.3 Given a partition $\\mu =(1^{\\alpha _1} \\dots s^{\\alpha _s})$ , the morphism $\\tau _{\\mu }$ gives the normalization $\\overline{{\\rm Chow}^n_{\\mu }(T^*_X/X)}$ .",
"The generic point of the product is mapped to the generic point of ${\\rm Chow}^n_{\\mu }(T^*_X/X)$ by $\\tau _{\\mu }$ .",
"Thus $\\tau _{\\mu }$ induces a proper, surjective morphism from $\\prod ^s_{j=1, X}{\\rm Chow}^{\\alpha _j}(T^*_X/X)$ to $\\overline{{\\rm Chow}^n_{\\mu }(T^*_X/X)}$ .",
"Also, note that by Lemma REF (iv), the product $\\prod ^s_{j=1, X}{\\rm Chow}^{\\alpha _j}(T^*_X/X)$ is a normal variety.",
"To see it is indeed the normalization, we shall utilize Lemma REF , which says one can check normalization locally.",
"Take an open covering $\\lbrace U_i\\rbrace $ of $X$ such that $T^*_X |_{U_i} \\cong U_i \\times \\mathbb {A}^2$ .",
"We have the commutative diagram with Cartesian squares ${&\\prod ^s_{j=1, U_i}{\\rm Chow}^{\\alpha _j}(T^*_{U_i}/{U_i})[d]^{\\tau ^{\\prime }_{\\mu }}[ld][rrd]&&\\\\\\prod ^s_{j=1, X}{\\rm Chow}^{\\alpha _j}(T^*_X/X)[rrd][d]^{\\tau _{\\mu }}& \\overline{{\\rm Chow}^n_{\\mu }(T^*_{U_i}/{U_i})}@{^(->}[rr] [ld][rrd] & & {\\rm Chow}^n(T^*_{U_i}/{U_i})[d][ld]\\\\\\overline{{\\rm Chow}^n_{\\mu }(T^*_X/X)}@{^(->}[rr][rrd]& & {\\rm Chow}^n(T^*_X/X)[d] & U_i[ld], \\\\& & X &}$ where $\\overline{{\\rm Chow}_{\\mu }^n(T^*_{U_i} /U_i)}$ is the closure of ${\\rm Chow}_{\\mu }^n(T^*_{U_i} /U_i)$ in ${\\rm Chow}^n(T^*_{U_i}/{U_i})$ .",
"We have the isomorphisms $\\overline{{\\rm Chow}_{\\mu }^n(T^*_{U_i} /U_i)} & \\cong \\overline{{\\rm Chow}_{\\mu }^n(\\mathbb {A}^2)} \\times U_i,\\\\\\prod ^s_{j=1, U_i}{\\rm Chow}^{\\alpha _j}(T^*_{U_i}/U_i) & \\cong \\big (\\prod ^s_{j=1}{\\rm Chow}^{\\alpha _j}(\\mathbb {A}^2)\\big )\\times U_i.$ Thus by Lemma REF , $\\tau ^{\\prime }_{\\mu }$ is the normalization of $\\overline{{\\rm Chow}^n_{\\mu }(T^*_{U_i}/{U_i})}$ .",
"Applying Lemma REF , we get $\\tau _{\\mu }$ is the normalization of $\\overline{{\\rm Chow}^n_{\\mu }(T^*_X/X)}$ .",
"Lemma 3.4 Let $X$ be an integral scheme and $\\tilde{X}$ a normal integral scheme.",
"Given a morphism $\\nu : \\tilde{X}\\rightarrow X$ , suppose there exists an open covering ${U_i}$ of $X$ such that $\\nu _i: \\tilde{U_i}=\\nu ^{-1}(U_i)\\rightarrow U_i$ is the normalization for each $i$ .",
"Then $\\nu $ is the normalization.",
"This follows from the universal property of normalization [5].",
"With respect to Proposition REF , we obtain the decomposition of spectral data.",
"It will be used to prove the main result (Theorem REF ) in §4.",
"Theorem 3.5 Let $X$ be a smooth projective surface.",
"Given a spectral datum $a: X \\rightarrow {\\rm Chow}^{n}(T^*_X /X)$ , suppose the generic point of $X$ is mapped to the stratum ${\\rm Chow}^{n}_{\\mu }(T^*_X /X)$ with $\\mu =(1^{\\alpha _1}2^{\\alpha _2}\\dots s^{\\alpha _s})$ .",
"Then there exist spectral data $a_i: X \\rightarrow \\overline{{\\rm Chow}^{\\alpha _i}_{(1^{\\alpha _i})}(T^*_X /X)}$ , $i=1, \\cdots , s$ , such that $a=\\sum ^s_{i=1} ia_i.$ By the assumption, $a$ induces a morphism $X \\rightarrow \\overline{{\\rm Chow}^n_{\\mu }(T^*_X /X)}$ over $X$ .",
"By Proposition REF , it holds that $\\overline{{\\rm Chow}^n_{\\mu }(T^*_X /X)}^{\\rm nor} \\cong {\\rm Chow}^{\\alpha _1}(T^*_X /X) \\times _X \\dots \\times _X {\\rm Chow}^{\\alpha _s}(T^*_X /X).$ Since $X$ is smooth, there exists a unique $a^{\\prime }$ fitting into the diagram Chown(T*X /X)nor [d] [rd, \"\"] X [r][ru,\"a'\", dotted] Chown(T*X /X) [r, hook] Chown(T*X /X) Here $a^{\\prime }$ is equivalent to data $(a_i)_{1 \\le i \\le s}$ , where $a_i: X \\rightarrow \\overline{{\\rm Chow}^{\\alpha _i}_{(1^{\\alpha _i})}(T^*_X /X)}.$ Finally by the commutativity of the diagram, we conclude that $a=\\sum \\limits _{i=1}^s i a_i$ ."
],
[
"The image of the Hitchin morphism for surfaces",
"Given a spectral datum $a: X \\rightarrow {\\rm Chow}^{n}(T^*_X/X)$ , we first briefly review the construction of the spectral cover $\\widetilde{X}_a \\rightarrow X$ , and the correspondence between Cohen-Macaulay sheaves of generic rank one over $\\widetilde{X}_a$ and Higgs bundles over $X$ [3].",
"Next, we show that given two Higgs bundles, their direct sum corresponds to the sum of the corresponding spectral data (Lemma REF ).",
"Finally, we prove the main theorem that $sd_X: {M}_X \\rightarrow {A}_X$ is surjective (Theorem REF ) with the help of Theorem REF .",
"Define $X_a$ as the pullback in the following diagram Xa [d,\"a\"] [rr] Cayleyn(T*X/X) [d,\"\"] X [rr,\"a\"] Chown(T*X/X) where ${\\rm Cayley}^n(T^*_X/X)$ is a subscheme of ${\\rm Chow}^n(T^*_X/X) \\times _X T^*_X$ ([3]).",
"Theorem 4.1 (Theorems 7.1 and 7.3 in [3]) Given a spectral datum $a: X \\rightarrow {\\rm Chow}^{n}(T^*_X/X)$ such that $a$ maps the generic point of $X$ into ${\\rm Chow}^n_{(1^n)}(T^*_X/X)$ , there exists a unique finite flat covering $\\widetilde{\\pi }_a: \\widetilde{X}_a \\rightarrow X$ such that there exists an open subset $U \\subseteq X$ of codimension at least 2 such that for every point $x \\in U$ , the fiber $(\\widetilde{\\pi }_a)^{-1}$ is a point of ${\\rm Hilb}^n(T^*_X/X)$ lying over the point $a(x) \\in {\\rm Chow}^n(T^*_X/X)$ ; there is a natural morphism $\\widetilde{\\iota }_a: \\widetilde{X}_a \\rightarrow T^*_X$ factoring through $\\iota _a: X_a \\hookrightarrow T^*_X$ such that the following diagram is commutative; Xa [rd,\"a\"] [r, \"\"] Xa [r,\"a\", hook] [d, \"a\"] T*X [ld, \"\"] X $\\widetilde{X}_a$ is the Cohen-Macaulayfication of $X_a$ ; the fiber $sd_X^{-1}(a)$ is isomorphic to the stack of Cohen-Macaulay sheaves of generic rank one over $\\widetilde{X}_a$ .",
"The correspondence demonstrated in the theorem can be understood in the following way.",
"Given a Cohen-Macaulay sheaf $L$ of generic rank one over $\\widetilde{X}_a$ , its pushfoward $\\iota _* L$ is a sheaf over $X_a$ .",
"Via $\\iota _a$ , the sheaf $(\\widetilde{\\iota }_a)_* L$ has a natural structure as a $S(T_X)$ -module, where $S(T_X)$ is the symmetric product of the tangent bundle $T_X$ .",
"Furthermore, it is a finite $S(T_X)$ -module.",
"Therefore, it corresponds to a spectral data in ${A}_X$ , which is exactly $a$ , and also corresponds to a Higgs bundle $(E,\\phi )$ over $X$ .",
"The correspondence between a finite $S(T_X)$ -module $L$ and a spectral data $a$ can be understood in the following way.",
"For each $x \\in X$ , we can equip $L|_{\\pi ^{-1}(x)}$ with a cycle ${\\rm cyc}(L|_{\\pi ^{-1}(x)}):=\\sum _{y \\in \\pi ^{-1}(x)} {\\rm len}_{y}(L_{\\pi ^{-1}(x)}) \\cdot y.$ Then, the $S(T_X)$ -module $L$ corresponds to $a$ if for each $x \\in X$ , we have $a(x)={\\rm cyc}(L|_{\\pi ^{-1}(x)}).$ Theorem REF only works for spectral data which map the generic point of $X$ into ${\\rm Chow}^n_{(1^n)}(T^*_X/X)$ .",
"Given an arbitrary spectral datum, we can follow the approach in the theorem to construct $X_a$ , but the correspondence is not clear.",
"We will discuss this issue in Remark REF .",
"Lemma 4.2 Given a pair of spectral data $a_i: X\\rightarrow {\\rm Chow}^{n_i}(T^*_X/X)$ , $i=1, 2$ , let $(E_i, \\phi _i)$ be Higgs bundles on $X$ whose spectral data are $a_i$ .",
"Then $(E_1\\oplus E_2, \\phi _1\\oplus \\phi _2)$ is a Higgs bundle with the spectral datum $a_1+a_2: X\\rightarrow {\\rm Chow}^{n_1+n_2}(T^*_X/X)$ .",
"The pair $(E_1\\oplus E_2, \\phi _1\\oplus \\phi _2)$ is obviously a Higgs bundle on $X$ .",
"By the correspondence [11], there exist coherent sheaves $F_i$ on $T^*_X$ such that $\\pi _* F_i\\simeq E_i$ .",
"Thus, $\\pi _*(F_1\\oplus F_2)\\simeq E_1\\oplus E_2$ .",
"For any $x\\in X$ , $a_1(x)+a_2(x)={\\rm cyc}( L_1 |_{\\pi ^{-1}(x)})+{\\rm cyc}(L_2|_{\\pi ^{-1}(x)})={\\rm cyc}((L_1+L_2)|_{\\pi ^{-1}(x)})=(a_1+a_2)(x).$ This finishes the proof.",
"Theorem 4.3 Let $X$ be a smooth projective surface.",
"The image of the Hitchin map $h_X: {M}_X \\rightarrow {B}_X$ is ${A}_X$ , i.e.",
"$sd_X : {M}_X \\rightarrow {A}_X$ is surjective.",
"We will show that given any spectral datum $a \\in {A}_X$ , we can construct a Higgs bundle $(E,\\phi ) \\in sd_X^{-1}(a) \\subseteq {M}_X$ .",
"By Theorem REF , this is equivalent to finding a $S(T_X)$ -module $L$ corresponding to the spectral datum $a$ .",
"Now given a spectral datum $a: X \\rightarrow {\\rm Chow}^n(T^*_X/X)$ , suppose that the generic point of $X$ is mapped into some stratum ${\\rm Chow}^n_{\\mu }(T^*_X/X)$ , where $\\mu =(1^{\\alpha _1}, 2^{\\alpha _2}, \\dots , s^{\\alpha _s})$ .",
"By Theorem REF , we have $a=a_1+2a_2+\\dots +sa_s,$ where the spectral datum $a_i$ maps the generic point of $X$ into ${\\rm Chow}^{\\alpha _i}_{(1^{\\alpha _i})}(T^*_X /X)$ .",
"Therefore, by Theorem REF , for each $i$ , one can construct a spectral cover $\\widetilde{X}_{a_i}$ of $X$ Xai [rd,\"i\"] [r] Xai [r,\"i\", hook] [d, \"i\"] T*X [ld, \"\"] X such that $\\widetilde{\\pi }_{a_i}: \\widetilde{X}_{a_i} \\rightarrow X$ is a finite flat morphism and the morphism $\\widetilde{\\iota }_{a_i}: \\widetilde{X}_{a_i} \\rightarrow T^*_X$ factors through $\\iota _{i}: X_{a_i} \\hookrightarrow T^*_X$ .",
"Then, the $S(T_X)$ -module $(\\widetilde{\\iota }_i)_* \\mathcal {O}_{\\widetilde{X}_{a_i}}$ corresponds to the spectral datum $a_i$ .",
"By Lemma REF , the sheaf $L:=\\bigoplus \\limits _{i=1}^s (\\widetilde{\\iota }_i)_* \\mathcal {O}_{\\widetilde{X}_{a_i}}^{\\oplus i}$ corresponds to the spectral data $\\sum \\limits _{i=1}^s i a_i$ , which is exactly $a$ .",
"Therefore, $sd_X^{-1}(a)$ is nonempty.",
"Remark 4.4 Although Theorem REF gives the surjectivity of the morphism $sd_X: {M}_X \\rightarrow {A}_X$ , we do not find a cover $\\widetilde{X}_a \\rightarrow X$ such that Higgs bundles over $X$ with spectral datum $a$ will correspond to some special sheaves over $\\widetilde{X}_a$ .",
"On the other hand, for any $a \\in {A}_X(k)$ , we can construct a finite, flat cover $\\widetilde{X}_a \\rightarrow X$ following Chen and Ngô's idea in [3].",
"However, the generic rank one sheaves over $\\widetilde{X}_a$ do not correspond to Higgs bundles of rank $n$ over $X$ .",
"We give the construction and explain the reason in the following.",
"Let $a: X \\rightarrow \\overline{{\\rm Chow}^n_{\\mu }(T^*_X/X)}$ be a spectral datum, where $\\mu =(1^{\\alpha _1}, \\dots , s^{\\alpha _s})$ .",
"As discussed in §3, it can be regarded as a morphism $a=(a_1,\\dots ,a_s): X \\rightarrow {\\rm Chow}^{\\alpha _1}(T^*_X /X) \\times _X \\dots \\times _X {\\rm Chow}^{\\alpha _s}(T^*_X /X),$ where $a_i$ maps the generic point of $X$ to ${\\rm Chow}^{\\alpha _i}_{(1^{\\alpha _i})}(T^*_X /X)$ .",
"Let $U^{\\prime }$ be an open subset of $X$ such that $a_i(U^{\\prime })\\subseteq {\\rm Chow}^{\\alpha _i}_{(1^{\\alpha _i})}(T^*_X /X)$ for all $i$ .",
"By the properness of the Hilbert-Chow morphism, one can find an open subset $U$ of $X$ with ${\\rm codim}(X\\backslash U, X)\\ge 2$ and there exists a unique morphism $\\tilde{a}: U \\rightarrow {\\rm Hilb}^{\\alpha _1}(T^*_X/X) \\times _X \\dots \\times _X {\\rm Hilb}^{\\alpha _s}(T^*_X/X)$ fitting into the commutative diagram U' [rr] [d] Hilb1(T*X/X) X ...X Hilbs(T*X/X) [d,\"Hilbert-Chow\"] U [urr, dotted] [rr,\"a|U\"] Chow1(T*X /X) X ...X Chows(T*X /X) and this morphism also induces an inclusion $U \\hookrightarrow T^*_X$ .",
"Now pulling back via $\\tilde{a}$ the product of universal families ${Z}_{\\alpha _1}(T^*_X/X)\\times _X \\cdots \\times _X {Z}_{\\alpha _s}(T^*_X/X),$ we get a finite, flat cover of $U$ , which is a closed subscheme of $T^*_U$ .",
"Then applying Serre's extension theorem [9], we get a finite, flat cover $\\widetilde{X}_a \\rightarrow X$ factoring through $T^*_X$ .",
"This cover has degree $\\sum ^s_{i=1}\\alpha _i<n$ in general.",
"Therefore generic rank one sheaves over $\\widetilde{X}_a$ do not correspond to rank $n$ Higgs bundles over $X$ .",
"We do not know whether one can consider some special sheaves over $\\widetilde{X}_a$ to give the correspondence."
],
[
"Birational invariance of the Space of spectral data",
"The purpose of this section is to show that for a smooth variety $X$ of any dimension, the space of spectral data ${A}_X$ is a birational invariant.",
"We apply the result to show the space of spectral data are affine spaces for ruled surfaces, building on [3], which assumes all fibres in a fibration are reduced.",
"Lemma 5.1 Let $\\chi : X^{\\prime }\\rightarrow X$ be a birational morphism of smooth varieties.",
"Then the natural morphism ${A}_X\\rightarrow {A}_{X^{\\prime }}$ is a closed immersion of schemes.",
"We have the commutative diagram ${{A}_X [r]@{^(->}[d]^{i_X}& {A}_{X^{\\prime }}@{^(->}[d]^{i_{X^{\\prime }}} \\\\{B}_X[r]^{\\chi ^*}& {B}_{X^{\\prime }},}$ where $\\mathfrak {i}_X$ and $\\mathfrak {i}_{X^{\\prime }}$ are closed immersions by [2] and $\\chi ^*: \\bigoplus ^n_{i=1} H^0(X, S^i T^*_X)\\rightarrow \\bigoplus ^n_{i=1} H^0(X^{\\prime }, S^i T^*_{X^{\\prime }})$ is an injection.",
"From the diagram, we conclude that the natural morphism is a closed immersion.",
"Lemma 5.2 Let $U$ be an open subset of a smooth variety $X$ with $\\text{codim}(X-U, X)\\ge 2$ .",
"Then ${A}_X\\rightarrow {A}_U$ is an isomorphism.",
"Recall for any $k$ -scheme $T$ , the $T$ -points of ${A}_X$ are ${A}_X(T)=\\big \\lbrace {X_T[r] [rd]_{id_{X_T}} & \\text{Chow}^n(T^*_{X_T/T}/{X_T})[d] \\\\& X_T,}\\big \\rbrace $ By Lemma REF and the Yoneda Lemma, we shall show that the natural map ${A}_X(T)\\rightarrow {A}_U(T)$ induced by $j$ is surjective for all such $T$ .",
"First we consider the case $T^*_X=T^*_{X/k}$ is trivial, i.e.",
"$T^*_X\\simeq X\\times V$ , where $\\dim V=r$ .",
"Then it holds that $\\text{Chow}^n(T^*_{X_T/T}/{X_T})\\simeq \\text{Chow}^n(V)\\times X_T$ and an analogous isomorphism for $U_T$ .",
"Given a spectral datum $a\\in {A}_U(T)$ , it is equivalent to a morphism $a: U_T\\rightarrow \\text{Chow}^n(V)$ , which for ease of notation will also be denoted by $a$ .",
"By the key lemma [2], it gives rise to a morphism $U_T\\rightarrow V\\times S^2V\\times \\cdots \\times S^nV,$ which in turn amounts to $U_T\\rightarrow T^*_{X_T/T}\\times S^2T^*_{X_T/T}\\times \\cdots \\times S^nT^*_{X_T/T}$ by the triviality of the relative cotangent bundle.",
"This is exactly an element in $\\bigoplus ^n_{i=1} H^0(U_T, S^i T^*_{U_T/T})$ Claim: for each $i>0$ , we have $H^0(X_T, S^i T^*_{X_T/T})\\simeq H^0(T, \\mathcal {O}_{T})\\otimes _k H^0(X, S^i T^*_{X/k}).$ Given this Claim and from Remark REF , we obtain that the natural map $H^0(X_T, S^i T^*_{X_T/T})\\rightarrow H^0(U_T, S^i T^*_{U_T/T})$ is an isomorphism for each $i>0$ .",
"Thus we get $\\bar{a}\\in \\bigoplus ^n_{i=1} H^0(X_T, S^i T^*_{X_T/T}).$ We can view $\\bar{a}$ as a morphism $ \\bar{a}: X_T\\rightarrow V\\times S^2V\\times \\cdots \\times S^nV,$ which is equivalent to a section $\\bar{a}\\in {A}_X(T)$ .",
"To prove the Claim above, consider the fibred product ${X_T [r]^{g}[d]_{\\varphi _T} & X[d]^{\\varphi } \\\\T[r]^{h}& {\\rm Spec}(k).",
"}$ It holds that $ T^*_{X_T/T}\\simeq g^* T^*_{X/k}.$ Since $X$ is smooth, $T^*_{X/k}$ is locally free, hence $S^i T^*_{X_T/T}\\simeq g^*S^i( T^*_{X/k})$ for all $i>0$ .",
"By flat base change, it therefore follows that $&&H^0(X_T, S^iT^*_{X_T/T})\\\\&\\simeq & H^0(X_T, g^*S^i( T^*_{X/k}))\\\\&\\simeq & H^0({\\rm Spec}{k}, h_*(\\varphi _T)_*g^*S^i( T^*_{X/k}))\\\\&\\simeq &H^0({\\rm Spec}{k}, h_*h^*\\varphi _*S^i(T^*_{X/k}))\\\\&\\simeq &H^0({\\rm Spec}{k}, h_*\\mathcal {O}_{T}\\otimes _k \\varphi _*S^i(T^*_{X/k}))\\\\&\\simeq & H^0(T, \\mathcal {O}_{T})\\otimes _k H^0(X, S^i(T^*_{X/k})).",
"$ This proves the Claim.",
"In general, we take an affine open covering $\\lbrace X_i\\rbrace $ of $X$ such that $T^*_{X/k}|_{X_i}$ is trivial for every $i$ .",
"Put $U_i=U\\cap X_i$ .",
"For all $i$ , it holds that $\\text{codim}(X_i-U_i, X_i)\\ge 2$ .",
"By the preceding argument, ${A}_{X_i}(T)\\rightarrow {A}_{U_i}(T)$ is surjective for all $i$ .",
"Since $j^*$ is actually injective, we can lift section locally and then glue the resulting sections together.",
"This finishes the proof of this lemma.",
"Theorem 5.3 Let $\\chi : X^{\\prime }\\rightarrow X$ be a birational proper morphism of smooth varieties.",
"Then the natural morphism ${A}_X\\rightarrow {A}_{X^{\\prime }}$ induced by $\\chi $ is an isomorphism as $k$ -schemes.",
"By the valuative criterion for properness, one can find an open subset $j: U\\rightarrow X$ with $\\text{codim}(X-U, X)\\ge 2$ and a morphism $\\nu : U\\rightarrow X^{\\prime }$ such that $\\chi \\circ \\nu =j$ .",
"As a result, we have the commutative diagram of schemes ${{A}_{X^{\\prime }} @{^(->}[dr] & \\\\{A}_X[r]^{\\simeq } @{^(->}[u] & {A}_{U},}$ where the bottom map is an isomorphism by Lemma REF .",
"Therefore ${A}_X\\rightarrow {A}_{X^{\\prime }}$ is an isomorphism.",
"Example 5.4 Let $Y \\hookrightarrow X$ be a smooth subvariety.",
"Denote by $\\pi : \\widetilde{X}={\\rm BL}_Y(X)\\rightarrow X$ the blow up of $X$ along $Y$ .",
"Then one has the injection of vector bundles $0 \\rightarrow \\pi ^* T^*_X \\rightarrow T^*_{\\widetilde{X}}.$ Note that ${\\rm Chow}^n(T^*_X/X) \\times _X \\widetilde{X} \\cong {\\rm Chow}^n(\\pi ^* T^*_X/ \\widetilde{X})$ , and the injection above induces a morphism ${\\rm Chow}^n(\\pi ^* T^*_X/ \\widetilde{X}) \\rightarrow {\\rm Chow}^n(T^*_{\\widetilde{X}} / \\widetilde{X})$ .",
"Now given a section $a_X: X \\rightarrow {\\rm Chow}_n(T^*_X /X)$ , we define the section $a_{\\widetilde{X}}: \\widetilde{X} \\rightarrow {\\rm Chow}_n(T^*_{\\widetilde{X}}/ \\widetilde{X})$ as the compositions of the following morphisms $a_{\\widetilde{X}}: \\widetilde{X} \\cong X \\times _X \\widetilde{X} \\xrightarrow{} {\\rm Chow}^n(T^*_X/X) \\times _X \\widetilde{X} \\cong {\\rm Chow}^n(\\pi ^* T^*_X/ \\widetilde{X}) \\rightarrow {\\rm Chow}^n(T^*_{\\widetilde{X}}/ \\widetilde{X}),$ which gives a map ${A}_X(k) \\rightarrow {A}_{\\widetilde{X}}(k)$ between the $k$ -points.",
"By Theorem REF , the map is bijective.",
"Corollary 5.5 For any projective ruled surface $X$ , ${A}_X={B}_X$ .",
"In particular it is an affine space.",
"By Theorem REF , we can assume $X$ is minimal, i.e.",
"it does not contain a smooth rational curve $\\Gamma $ with $\\Gamma ^2=-1$ .",
"If $X=\\mathbb {P}^{2}$ , then ${A}_X={B}_X$ is a point; otherwise $X\\simeq \\mathbb {P}_C(E)$ for some smooth curve $C$ and a vector bundle $E$ of rank two, so ${A}_X\\simeq {A}_C$ , and hence ${A}_X={B}_X$ , see [3] and the following remark.",
"School of Mathematics, Sun Yat-Sen University W. 135 Xingang Rd., Guangzhou, Guangdong 510275, P.R.",
"China E-mail address: [email protected] Department of Mathematics, South China University of Technology 381 Wushan Rd., Guangzhou, Guangdong 510641, P.R.",
"China E-mail address: [email protected]"
]
] | 2107.01679 |
[
[
"The $\\sigma_-$ Cohomology Analysis for Symmetric Higher-Spin Fields"
],
[
"Abstract In this paper, we present a complete proof of the so-called First On-Shell Theorem that determines dynamical content of the unfolded equations for free symmetric massless fields of arbitrary integer spin in any dimension and arbitrary integer or half-integer spin in four dimensions.",
"This is achieved by calculation of the respective $\\sigma_-$ cohomology both in the tensor language in Minkowski space of any dimension and in terms of spinors in $AdS_4$.",
"In the $d$-dimensional case $H^p(\\sigma_-)$ is computed for any $p$ and interpretation of $H^p(\\sigma_-)$ is given both for the original Fronsdal system and for the associated systems of higher form fields."
],
[
"Introduction",
"Higher-spin (HS) gauge theory is based on works of Fronsdal [1] and Fang and Fronsdal [2], where the action and equations of motion for massless gauge fields of any spin were originally obtained in flat four-dimensional Minkowski space.",
"Even earlier, important restrictions on low-energy HS vertices were obtained by Weinberg in [3], [4] and so-called no-go theorems restricting $S$ -matrix possessing too high symmetries in flat space-time were proven in [5], [6].",
"(For a review see [7].)",
"The no-go theorems implied the existence of the $s=2$ barrier suggesting that the construction of an interacting local HS theory in Minkowski space-time is impossible.",
"The proof of these theorems essentially uses the specific form of the algebra of isometries of Minkowski space.",
"The $s=2$ barrier in flat space can be overcome in the space-time with non-zero sectional curvature, for example, in the anti-de Sitter space [8].",
"In these spaces it becomes possible to formulate a consistent nonlinear theory of fields of all spins [9], [10].",
"The construction of a nonlinear HS theory is essentially based on the so-called unfolded approach [11], [12], which is a far-going generalization of the Cartan formulation of gravity ($s=2$ ) in terms of differential forms to fields of any spin $s>2$ .",
"Via introducing appropriate auxiliary variables, the unfolding procedure allows one to replace the system of partial differential equations of any order on a smooth manifold by a larger system of first-order equations on vector-valued differential forms.",
"One of the essential features of this approach, which is very useful for analysing symmetries of a given system, is that the variables in the equations are valued in one or another representation of the underlying symmetry algebra.",
"The dynamical content of the HS theory can be reconstructed from its unfolded formulation using the $\\sigma _-$ cohomology technique [13].",
"As is recalled below, the dynamical data of the theory are in one-to-one correspondence with the cohomology of certain linear nilpotent operator $\\sigma _-$ that can be read of the unfolded equations in question.",
"The statement that unfolded equations of free HS fields are equivalent to the Fronsdal equations was made in the original papers in the spinor [14] and tensor [15] formalisms.",
"In the tensor formulation of HS theory the idea of the proof was illustrated in [16], where however the analysis of the trace part of the Fronsdal equations was not completed, while general arguments for mixed-symmetry HS fields were given in [17].",
"In [18] the unfolded equations for massless fields were derived from the Fronsdal theory by the BRST methods.",
"To the best of our knowledge, no detailed analysis of the problem in the spinor formalism was available in the literature.",
"In this paper we present a complete proof of the so called First On-Shell Theorem by computation of the cohomology rings of $\\sigma _-$ for the physically important cases of the integer-spin symmetric fields both in flat space-time of any dimension and $AdS_d$ as well as for the fields of any integer and half-integer spin in ${AdS}_4$ .",
"The computation technique analogous to the Hodge theory for differential forms is performed in terms of so-called $\\sigma _-$ cohomology and provides a complete analysis of the dynamical content of the free unfolded equations for symmetric massless fields of any spin.",
"Giving a direct proof of the equivalence between the Fronsdal formulation of the HS gauge theory and its unfolded formulation this paper fills in some gaps in the literature also illustrating a general approach applicable to a broad class of unfolded systems.",
"In addition, in the tensor case we compute higher $\\sigma _-$ cohomology groups and interpret them in terms of higher Bianchi identities and more general dynamical systems.",
"In particular, we discuss the matching between the Bianchi identities in terms of one-form gauge fields and zero-form field strengths.",
"The rest of the paper is organized as follows.",
"In Section 2 we briefly recall different approaches to the description of HS massless fields.",
"Main idea of the $\\sigma _-$ cohomology approach is explained in Section 3.",
"Cohomology calculation method used in this paper is discussed in Section 4.",
"Section 5 contains derivation of $H^p(\\sigma _-)$ in Minkowski space of any dimension.",
"In particular, the cases of $\\mathrm {GL}(d)$ and $\\mathrm {O}(d)$ representations are analysed here.",
"In Section 6 calculation of the low-order cohomology groups in $AdS_4$ is performed.",
"Obtained results are discussed in Section 7.",
"Index conventions and normalisations of the tensor Young diagrams are presented in the Appendix A."
],
[
"Metric formulation",
"According to Fronsdal [1], a spin-$s$ massless symmetric field can be described in terms of two symmetric traceless tensors (for index conventions see Appendix A) $\\phi ^{a(s)} \\equiv \\phi ^{a_1..a_s}, \\quad \\phi ^{a(s-2)} \\equiv \\phi ^{a_1..a_{s-2}}, \\quad \\eta _{b_1 b_2}\\phi ^{b_1 b_2 a_3 .. a_s} \\equiv \\phi ^{a(s-2)b}{}_b = 0, \\quad \\phi ^{a(s-4)b}{}_b = 0\\,.$ These two fields can be combined into a single rank-$s$ totally symmetric tensor $\\varphi ^{a(s)} = \\phi ^{a(s)} + \\eta ^{aa}\\phi ^{a(s-2)}$ obeying the double-tracelessness condition $\\varphi ^{a(s-4)bc}{}_{bc} = 0\\,.$ The field equations in Fronsdal theory have the form $R^{a(s)}(\\varphi ) = \\square \\varphi ^{a(s)} - s \\partial ^a \\partial _k \\varphi ^{k a(s-1)} + \\frac{s(s-1)}{2} \\partial ^a \\partial ^a \\varphi ^{a(s-2)k}{}_k = 0\\,,$ where $\\partial _a = \\frac{\\partial }{\\partial x^a}$ .",
"The tensor $R^{a(s)}(\\varphi )$ is invariant under the gauge transformations with a rank-$(s-1)$ traceless gauge parameter $\\varepsilon (x)$ $\\delta \\varphi ^{a(s)} = \\partial ^a \\varepsilon ^{a(s-1)}, \\qquad \\varepsilon ^{a(s-3)k}{}_k = 0\\,.$ Fronsdal equation (REF ) is a generalization of the well-known equations of fields with spins $s = 0,1,2$ .",
"For the case of $s = 1$ the last term in the Fronsdal tensor disappears and Eq.",
"(REF ) reproduces Maxwell equations for the field $A^a$ .",
"Without the last two terms at $s = 0$ it gives Klein-Gordon equation for a massless scalar field.",
"The case of $s = 2$ reproduces the equations of linearized gravity [19].",
"Gauge transformation (REF ) gives the known gauge transformations of low-spin fields and its absence for a scalar field."
],
[
"Tensor formalism",
"The unified description of massless fields of arbitrary spin can be given in the so-called frame-like formalism that generalizes Cartan formulation of gravity, operating in terms of differential forms [14], [15], [20].",
"Frame-like formulation of the HS gauge theory in any dimension is given in terms of the one-form fields $ \\omega ^{a(s-1),b(t)} =dx^\\nu \\omega _\\nu ^{a(s-1),b(t)}$ valued in two-row Young diagrams corresponding to irreducible $\\mathfrak {o}(d-1,1)$ (i.e., traceless) modules [15], obeying conditions $& \\omega ^{a(s-1),a b(t-1)} = 0, \\\\& \\omega ^{a(s-3)k}{}_{k}{}^{,b(t)} = 0\\,.$ (For index conventions see Appendix A.)",
"By introducing auxiliary fields it is possible to put a system of partial differential equations into the first-order unfolded form [11], [12].",
"Generally, unfolded equations read as $d W^A = \\sum _{n = 1}^{\\infty } G^A_{B_1,..,B_n}W^{B_1}\\wedge ...\\wedge W^{B_n}\\,,\\qquad d:=dx^\\nu \\partial _\\nu \\,.$ Here $W^A$ is a set of differential forms over some manifold.",
"(Indices are treated formally and can take an infinite number of values.)",
"The coefficients $G^A_{B_1,..,B_n}$ satisfy the (anti)symmetry condition $G^A_{B_1,..,B_i,..,B_j,..,B_n} = (-1)^{|B_i||B_j|} G^A_{B_1,..,B_j,..,B_i,..,B_n}\\,,$ where $|B_i|$ denotes the form-degree of $W^{B_i}$ .",
"Also $G^A_{B_1,..,B_n}$ are restricted by the integrability conditions expressing that $d^2=0$ .",
"In the tensor language the unfolded HS equations in Minkowski space proposed in [15] read as $D_L\\omega ^{a(s-1),b(t)} + h_m \\wedge \\omega ^{a(s-1),b(t)m} = 0, \\quad t \\in \\lbrace 0,..,s-2\\rbrace ,$ $D_L\\omega ^{a(s-1),b(s-1)} = h_n \\wedge h_m \\wedge C^{a(s-1)n,b(s-1)m},$ where $h_n$ is a soldering form (vielbein, frame field, tetrad) and $D_L = d + \\varpi $ is the background Lorentz covariant derivative that satisfies relations $D_L h^a = 0, \\qquad D_L^2 = 0\\,.$ In the Cartesian coordinate system with $\\varpi =0$ the equations simplify to $d\\omega ^{a(s-1),b(t)} + h_m \\wedge \\omega ^{a(s-1),b(t)m} = 0, \\quad t \\in \\lbrace 0,..,s-2\\rbrace ,$ $d\\omega ^{a(s-1),b(s-1)} = h_n \\wedge h_m \\wedge C^{a(s-1)n,b(s-1)m}\\,,$ where $C$ satisfies the Lorentz irreducibility conditions $C^{a(n),a b(m-1)} = 0\\,, \\qquad C^{a(n-2)k}{}_k{}^{,b(m)} = 0\\,.$ The traceless tensor $C$ on the r.h.s.",
"of (REF ) is a generalized Weyl tensor.",
"There are also unfolded equations on $C$ and on additional auxiliary fields [10] (for reviews see [16], [21]).",
"This system constitutes an infinite chain of zero-form equations.",
"Zero-form sector, that contains equations on spin-zero and spin-one fields, will not be considered in this paper.",
"Equations (REF ) are invariant under the gauge transformations $\\delta \\omega ^{a(s-1),b(t)} = d\\varepsilon ^{a(s-1),b(t)} + h_m \\varepsilon ^{a(s-1),b(t)m}, \\quad t \\in \\lbrace 0,..,s-2\\rbrace $ and eq.",
"(REF ) is invariant under $&\\delta \\omega ^{a(s-1),b(s-1)} = d \\varepsilon ^{a(s-1),b(s-1)},\\\\&\\delta C^{a(s),b(s)} = 0,$ where $\\varepsilon $ are zero-forms valued in the appropriate two-row irreducible $\\mathfrak {o}(d-1,1)$ -modules obeying conditions analogous to (REF ).",
"The Fronsdal field is embedded into the frame-like one-form $e^{a(s-1)}\\equiv \\omega ^{a(s-1)}$ in the following manner.",
"Converting the form index into the fiber one using vielbein $h$ , $e^{a(s-1)|b} = e^{a(s-1)}_\\mu h^{\\mu b}\\,,$ the resulting tensor (REF ) can be decomposed into irreducible $\\mathfrak {o}(d-1,1)$ -modules.",
"In terms of traceless Young diagrams this decomposition is $\\begin{picture}(15,10)(0,7){\\put (05,10){\\line (1,0){10}}\\put (05,20){\\line (1,0){10}}\\put (05,10){\\line (0,1){10}}\\put (15,10){\\line (0,1){10}}}\\end{picture} \\,\\,\\,\\otimes _{so}\\,\\,\\,\\begin{picture}(65,18)(10,7){\\put (25,13){\\scriptsize {s-1}}\\put (05,20){\\line (1,0){60}}\\put (05,10){\\line (1,0){60}}\\put (05,10){\\line (0,1){10}}\\put (65,10){\\line (0,1){10}}}\\end{picture}\\,\\ \\cong \\,\\,\\,\\ \\begin{picture}(65,18)(10,7){\\put (30,13){\\scriptsize {s}}\\put (05,20){\\line (1,0){60}}\\put (05,10){\\line (1,0){60}}\\put (05,10){\\line (0,1){10}}\\put (65,10){\\line (0,1){10}}}\\end{picture}\\,\\ \\oplus \\,\\,\\,\\ \\begin{picture}(65,18)(10,7){\\put (25,13){\\scriptsize {s-2}}\\put (05,20){\\line (1,0){60}}\\put (05,10){\\line (1,0){60}}\\put (05,10){\\line (0,1){10}}\\put (65,10){\\line (0,1){10}}}\\end{picture}\\,\\,\\,\\ \\oplus \\,\\,\\,\\ \\begin{picture}(10,18)(0,7){\\put (05,00){\\line (1,0){10}}\\put (05,10){\\line (1,0){10}}\\put (05,0){\\line (0,1){10}}\\put (15,0){\\line (0,1){10}}}\\end{picture}$ ${s-1}$ .",
"The first two components give the Fronsdal field, while the third one is an excess of the components of the frame-like field in comparison with the Fronsdal field.",
"At the tensor level, this decomposition is represented as: $e^{a(s-1)|b} = \\psi _1^{a(s-1)b} + \\beta _1\\eta ^{aa}\\psi _2^{a(s-3)b} + \\beta _2\\eta ^{ab}\\psi _2^{a(s-2)} + \\psi _3^{a(s-1),b},$ where $\\psi _i$ are traceless and correspond to the $i$ -th diagram.",
"The relative coefficient $\\frac{\\beta _2}{\\beta _1}$ is fixed by the tracelessness condition with respect to indices $a$ .",
"This decomposition shows that the Fronsdal field identifies with the symmetric part of the frame-like field, since the contribution of the third diagram disappears upon symmetrization.",
"The resulting field $\\varphi ^{a(s)} := e^{a(s-1)|a}\\,$ is symmetric and double-traceless.",
"The extra term $\\psi _3^{a(s-1),b}$ is pure gauge.",
"Its contribution can be canceled by the gauge transformation $\\delta e^{a(s-1)|b} = \\varepsilon ^{a(s-1)|b} $ with suitable gauge parameter.",
"For detailed discussion of Fronsdal field embedding see [16], [20], [21].",
"It is not difficult to check [15], [20] (for reviews see [16], [21]) that the Fronsdal equations and gauge transformations follow from the unfolded system (REF ), (REF ).",
"A more complicated question is whether the Fronsdal fields and equations are the only ones that result from (REF ), (REF ).",
"The answer can be obtained via the $\\sigma _-$ cohomology technique [13]."
],
[
"Spinor language in $AdS_4$",
"The physically important case of the unfolded system for HS connection (REF ), (REF ) is that of $AdS_4$ space-time in which case the language of two-component spinors is most appropriate.",
"In this language instead of using Lorentz indices $a,b,...= 0,1,2,3$ , one uses two pairs of dotted and undotted spinor indices $\\alpha ,\\beta ,...$ and $\\dot{\\alpha },\\dot{\\beta },...$ taking values $\\lbrace 1,2\\rbrace $ .",
"The two languages are related via Pauli matrices.",
"The $AdS_4$ background geometry is described in terms of the Lorentz connection $\\varpi $ and frame field $h$ , that satisfy equations $\\begin{array}{l}d h^{\\alpha \\dot{\\beta }}+\\varpi ^{\\alpha }{ }_{\\gamma } \\wedge h^{\\gamma \\dot{\\beta }}+\\overline{\\varpi }^{\\dot{\\beta }}{}_{\\dot{\\gamma }} \\wedge h^{\\alpha \\dot{\\gamma }}=0\\,, \\\\d \\varpi ^{\\alpha \\beta }+\\varpi ^{\\alpha }{ }_{\\gamma } \\wedge \\varpi ^{\\gamma \\beta }=-\\lambda ^{2} h^{\\alpha }{}_{\\dot{\\gamma }} \\wedge h^{\\beta \\dot{\\gamma }}\\,, \\\\d \\overline{\\varpi }^{\\dot{\\alpha } \\dot{\\beta }}+\\overline{\\varpi }^{\\dot{\\alpha }}{}_{\\dot{\\gamma }} \\wedge \\overline{\\varpi }^{\\dot{\\gamma } \\dot{\\beta }}=-\\lambda ^{2} h_{\\gamma }{ }^{\\dot{\\alpha }} \\wedge h^{\\gamma \\dot{\\beta }}\\,,\\end{array}$ where $\\lambda ^2$ is proportional to the curvature of $AdS_4$ and we adopt the following rules $A_{\\alpha }=A^{\\beta } \\epsilon _{\\beta \\alpha }\\,, \\quad A^{\\alpha }=\\epsilon ^{\\alpha \\beta } A_{\\beta }\\,, \\quad \\epsilon _{\\alpha \\beta } \\epsilon ^{\\gamma \\beta }=\\epsilon _{\\alpha }{}^{\\gamma }=\\delta _{\\alpha }^{\\gamma }=-\\epsilon ^{\\gamma }{}_{\\alpha }\\,,$ where $\\epsilon _{\\alpha \\beta } = -\\epsilon _{\\beta \\alpha }\\,, \\quad \\epsilon _{12} = 1\\,.$ The spinor version of the unfolded system for one-form $\\omega $ reads as follows.",
"First, the HS curvatures in the spinor language are [14] $R^{\\alpha (n),\\dot{\\alpha }(m)}=D_L\\omega ^{\\alpha (n),\\dot{\\alpha }(m)} + \\lambda ^2(n h^\\alpha _{\\ \\dot{\\gamma }}\\wedge \\omega ^{\\alpha (n-1),\\dot{\\gamma }\\dot{\\alpha }(m)} + m h_\\gamma ^{\\ \\dot{\\alpha }}\\wedge \\omega ^{\\gamma \\alpha (n),\\dot{\\alpha }(m-1)})$ and $D_L=d+\\varpi +\\overline{\\varpi }$ is a Lorentz-covariant derivative with Cartan's spin-connection $(\\varpi \\oplus \\overline{\\varpi })$ $D_L \\omega ^{\\alpha (n),\\dot{\\alpha }(m)}=d\\omega ^{\\alpha (n),\\dot{\\alpha }(m)}+n\\varpi _\\alpha ^{\\ \\beta }\\wedge \\omega _{\\beta \\alpha (n-1),\\dot{\\alpha }(m)} + m\\overline{\\varpi }_{\\dot{\\alpha }}^{\\ \\dot{\\beta }}\\wedge \\omega _{\\alpha (n),\\dot{\\beta }\\dot{\\alpha }(m-1)}\\,.$ The $AdS_4$ deformation of the unfolded equations (REF ), (REF ) then takes the form [12] $R^{\\alpha (n),\\dot{\\alpha }(m)}=\\delta _{0,n}\\,h_{\\beta \\dot{\\alpha }}\\wedge h^{\\beta }_{\\ \\dot{\\alpha }}\\overline{C}^{\\dot{\\alpha }(m+2)} + \\delta _{0, m}\\, h_{\\alpha \\dot{\\beta }}\\wedge h_{\\alpha }^{\\ \\dot{\\beta }} C^{\\alpha (n+2)}.$ The main advantage of the two-component spinor notation is that it makes the representation theory of the Lorentz group very simple.",
"Namely, every Lorentz irreducible multispinor representing a traceless tensor is totally symmetric in its spinor indices.",
"Since the only Lorentz invariant objects are antisymmetric bispinors $\\epsilon _{\\alpha \\beta }$ and $\\overline{\\epsilon }_{\\dot{\\alpha }\\dot{\\beta }}$ irreducible multispinors $X^{\\alpha (n),\\dot{\\alpha }(m)}$ are necessarily symmetric with respect to the indices in the groups $\\alpha (n)$ and $\\dot{\\alpha }(m)$ separately.",
"Thus, working with the two-component spinor notation one can happily forget about painful calculations with the traces of Lorentz-tensors."
],
[
"The idea of $\\sigma _-$ cohomology analysis: example of integer spin massless fields",
"The l.h.s.",
"'s of unfolded HS equations and gauge transformations in $d$ -dimensional Minkowski space are [15], [16] $&R^{a(s-1),b(k)} = D_L \\omega ^{a(s-1),b(k)} + \\sigma _-(\\omega )^{{a(s-1),b(k)}},\\\\&\\delta \\omega ^{a(s-1),b(k)} = D_L \\varepsilon ^{a(s-1),b(k)} + \\sigma _-(\\varepsilon )^{{a(s-1),b(k)}},$ where $(\\sigma _-\\omega )^{{a(s-1),b(k)}} := h_c \\wedge \\omega ^{a(s-1),b(k)c}\\,,\\qquad (\\sigma _-\\varepsilon )^{{a(s-1),b(k)}} := h_c \\wedge \\varepsilon ^{a(s-1),b(k)c} .$ $R^{a(s-1),b(k)}$ is referred to as (linearized) HS curvature.",
"For simplicity we study the Minkowski case.",
"Since $\\sigma _-$ in $AdS_d$ is defined analogously, our analysis applies to that case as well.",
"Due to their definition, HS curvatures obey the Bianchi identities $D_L R^{a(s-1),b(k)} + \\sigma _-(R)^{a(s-1),b(k)} = 0\\,.$ The appearance of $\\sigma _-$ allows one to clarify the role of the fields $\\omega ^{a(s-1),b(k)}$ and gauge parameters $\\varepsilon ^{a(s-1),b(k)}$ .",
"Working with the zero-forms $\\varepsilon ^{a(s-1),b(k)}$ and one-forms $\\omega ^{a(s-1),b(k)}$ valued in two-row Young diagrams, we consider the space $V^p$ of $p$ -forms valued in two-row Young diagrams with any $p$ .",
"Defining $\\sigma _-$ to annihilate the forms with an empty second row, we find that $\\sigma _- V^p\\subset V^{p+1}$ and $\\sigma _-\\,\\sigma _-=0$ .",
"As originally proposed in [13], the $\\sigma _-$ cohomology $H(\\sigma _-)=\\ker ({\\sigma _-})/{\\mathrm {im\\,}(\\sigma _-)}$ classifies fields, their equations and gauge symmetries.",
"Indeed, those components of the fields $\\omega ^{a(s-1),b(t)}$ , that are not annihilated by $\\sigma _-$ , can be expressed via derivatives of the fields with lower $t$ by setting suitable components of the HS curvatures to zero.",
"Such fields are called auxiliary.",
"Conversely, those components of the fields $\\omega ^{a(s-1),b(t)}$ , that cannot be expressed in terms of derivatives of lower fields via zero-curvature conditions, are in $\\ker (\\sigma _-)$ .",
"By Stueckelberg fields we mean $\\sigma _-$ -exact fields (i.e.",
"fields of the form $\\sigma _-\\chi $ ) as they can be eliminated by an appropriate $\\sigma _-$ -exact term in the gauge transformation (REF ).",
"Fields that are not expressed via derivatives of other fields and are not Stueckelberg are called dynamical.",
"These describe the physical degrees of freedom of the theory.",
"Thus, the dynamical HS fields are associated with $H^1(\\sigma _-)$ .",
"The classification for the gauge parameters is analogous.",
"The parameters, that are not annihilated by $\\sigma _-$ , describe algebraic Stueckelberg shifts.",
"The leftover symmetries are described by the parameters in $\\ker (\\sigma _-)$ .",
"$\\sigma _-$ -exact parameters correspond to the so called gauge for gauge transformations.",
"Parameters, which are $\\sigma _-$ -closed and not $\\sigma _-$ -exact, are referred to as genuine differential gauge parameters.",
"Note that since in the HS example in question the gauge parameters are zero-forms there is no room for gauge for gauge symmetries in that case.",
"Let $V$ be a graded vector space, $\\mathcal {C}$ be an element of $\\Lambda ^p(\\mathcal {M}^d)\\otimes V$ over some smooth $d$ -dimensional manifold $\\mathcal {M}^d$ .",
"We demand the grading of $V$ to be bounded from below, that is $V$ is $\\mathbb {N}$ -graded.",
"Let $\\sigma _ {\\pm }$ be operators that act ”vertically”, i.e.",
"do not affect the space-time coordinates, and shift grading by $\\pm 1$ , $D_L$ be the Grassmann-odd operator that does not affect the grading and is allowed to act non-trivially on the space-time coordinates.",
"Consider the covariant constancy condition of a general form along with the zero-curvature condition $\\mathcal {D} \\mathcal {C} = (D_L + \\sigma _- + \\sigma _+) \\mathcal {C} = 0, \\quad \\mathcal {D}^2 = 0.$ Notice that eq.",
"(REF ) remains invariant under the gauge transformations $\\delta \\mathcal {C} = \\mathcal {D} \\varepsilon ,$ where $\\varepsilon \\in \\Lambda ^{p-1}(\\mathcal {M}^d) \\otimes V$ .",
"One can prove the following proposition [13] (see also [16], [22], [23], [24]): Theorem 3.1 The following is true: 1) Differential gauge symmetry parameters $\\varepsilon $ span $H^{p-1}(\\sigma _-)$ 2) Nontrivial dynamical fields $\\mathcal {C}$ span $H^p(\\sigma _-)$ 3) Physically distinguishable differential field equations on the nontrivial dynamical fields, contained in $\\mathcal {D} \\mathcal {C} = 0$ , span $H^{p+1}(\\sigma _-)$ Thus, taking into account that HS gauge fields are described by the one-forms $\\omega $ , to prove that the Fronsdal metric formulation is equivalent to the unfolded one, we have to calculate $H^{0}(\\sigma _-), H^{1}(\\sigma _-)$ and $H^{2}(\\sigma _-)$ .",
"More generally, higher cohomology $H^{k}(\\sigma _-)$ with $k>p+1$ describes Bianchi identities for dynamical equations at $k=p+2$ and Bianchi for Bianchi identites at $k>p+2$ [24].",
"Similarly, the lower cohomology $H^{k}(\\sigma _-)$ with $k<p-1$ describes gauge for gauge differential symmetries."
],
[
"A method for calculating cohomology",
"Calculation of $\\sigma _-$ cohomology is of utter importance for the analysis of unfolded systems of the general form (REF ).",
"The straightforward calculation of the cohomology can be quite involved.",
"In this paper we find cohomology using a standard homotopy approach recalled below, that is a generalization of the Hodge theory for de Rham cohomology extendable to a more general class of (co)chain complexes.",
"Main details of the construction used in this paper follow those of [24], where the $\\sigma _-$ cohomology analysis was applied to the conformal HS theories of the bosonic fields of any symmetry type.",
"Unfortunately, some of the methods of [24], based on the fact that $\\sigma _-$ in conformal theories has the clear meaning in terms of the conformal algebra, are not directly applicable to the non-conformal HS theories discussed in this paper, which makes the analysis of the latter a bit more involved.",
"Let $V$ be a graded vector space and $d$ be a linear operator of degree +1 on $V$ (that is, it raises the grading of a homogeneous element by 1) such that $d^2 = 0$ .",
"Then $H(d) = \\ker (d)/ \\mathrm {im\\,}(d)$ .",
"Let $\\partial $ be another operator of degree $-1$ on $V$ (i.e., it lowers the grading by 1) such that $\\partial ^2 = 0$ .",
"The operators $d$ and $\\partial $ can be used to compose the degree 0 operator $\\Delta $ $\\Delta := \\lbrace d,\\partial \\rbrace = d\\,\\partial + \\partial \\,d.$ It is easy to see that $\\Delta $ satisfies $[d,\\Delta ] = [\\partial ,\\Delta ] = 0\\,.$ Lemma 4.1 If $\\Delta $ is diagonalizable on the (graded) vector space $V$ , then $H(d) \\hookrightarrow \\ker (\\Delta )$ .",
"First of all we should show that $\\ker (d)$ is an invariant subspace of $\\Delta $ .",
"Suppose $f \\in \\ker (d)$ .",
"Then $\\Delta f = (d \\partial + \\partial d) f = d \\partial f \\Rightarrow \\Delta f \\in \\ker (d)\\,, \\forall f \\in \\ker (d)\\,.$ Therefore, $\\ker (d)$ is an invariant subspace of $\\Delta $ , because linearity is obvious.",
"Since the operator $\\Delta $ is diagonalizable by assumption, we can consider eigenvectors of $\\Delta $ .",
"Let $g$ be $d$ -closed and $\\Delta g = \\lambda g, \\lambda \\ne 0$ .",
"Then $g = \\frac{1}{\\lambda }\\Delta g= \\frac{1}{\\lambda }\\,d\\, \\partial g\\,.$ Hence, $g$ is also $d$ -exact for $\\lambda \\ne 0$ , representing a trivial element of $H(d)$ .",
"Thus, every $d$ -closed form annihilated by $\\Delta $ is not $d$ -exact.",
"In other words, every $d$ -closed form $f$ can be written as $f = h + d\\alpha $ with some $h\\in \\ker (\\Delta )$ .",
"If $V$ is a Hilbert space with inner product $\\langle \\,,\\,\\rangle $ , there exists such $\\partial $ that the converse inclusion $H(d) \\hookleftarrow \\ker (\\Delta )$ takes place as well, which means that $H(d) = \\ker (\\Delta )$ .",
"Lemma 4.2 Let $(V,\\langle \\,,\\,\\rangle )$ be a Hilbert space, let $d^*$ be the operator conjugated to $d$ in the usual sense $\\langle \\alpha , d\\beta \\rangle = \\langle d^*\\alpha , \\beta \\rangle $ and $\\Delta = \\lbrace d\\,, d^*\\rbrace $ .",
"Then $\\ker (\\Delta ) \\hookrightarrow H(d)$ .",
"Take any $f \\in \\ker (\\Delta )$ .",
"Then $0 = \\langle f,\\Delta f\\rangle = \\langle df,df\\rangle + \\langle d^*f,d^*f\\rangle \\Leftrightarrow df = 0 \\quad \\text{and} \\quad d^*f = 0\\,.$ Hence, $f \\in \\ker (d)$ .",
"To show that $f \\notin \\mathrm {im\\,} (d)$ suppose the opposite.",
"Let $f = dg$ .",
"Then due to (REF ) $d^*dg = 0 \\Rightarrow 0 = \\langle g, d^* dg\\rangle = \\langle dg,dg\\rangle \\Rightarrow dg = 0\\,.$ Thus, $\\ker (\\Delta ) \\hookrightarrow H(d)$ From Lemmas REF and REF , it follows that, if all the requirements are met, $H(d) = \\ker (\\Delta )\\,.$ Thus, in a Hilbert space with a diagonalizable Laplace operator $\\Delta :=\\lbrace d\\,,d^*\\rbrace $ , finding the cohomology is equivalent to finding $\\ker (\\Delta )$ .",
"Further calculations of $\\sigma _-$ cohomology will rely on this fact.",
"The following important comment [24] is now in order.",
"In the case of interest, for every unfolded subsystem associated with a fixed spin $V= \\oplus _n V_n$ with finite-dimensional grade-$n$ subspaces $V_n$ .",
"In that case $\\Delta $ leaves invariant every $V_n$ and, being self-adjoint in the finite-dimensional Hilbert space, is diagonalizable.",
"It is worth noting the similarity of the above analysis with the Hodge theory mentioned at the beginning of this section.",
"Indeed, consider the (finite-dimensional) vector space $V$ endowed with some nilpotent operators $d$ and $\\partial $ , $d^2 = \\partial ^2 = 0$ .",
"The condition of disjointness is also imposed (see [25] for details), that is, $\\mathrm {im}(d)\\cap \\ker (\\partial ) = \\mathrm {im}(\\partial )\\cap \\ker (d) = \\lbrace 0\\rbrace $ .",
"In other words, it is demanded that $d\\partial x = 0 \\quad \\text{implies} \\quad \\partial x = 0\\,,\\\\\\partial d x = 0 \\quad \\text{implies} \\quad d x = 0.$ Define the Laplacian $\\Delta $ by (REF ).",
"Under these assumptions it can be shown that $\\ker (\\Delta ) = \\ker (d) \\cap \\ker (\\partial )$ .",
"The harmonic cocycles annihilated by $\\Delta $ are those and only those, that are $d$ -closed and $\\partial $ -closed simultaneously; $V = \\mathrm {im}(d) \\oplus \\mathrm {im}(\\partial ) \\oplus \\ker (\\Delta )$ .",
"In other words, for any vector $x\\in V$ there exists a unique Hodge decomposition $x = d\\alpha + \\partial \\beta + h$ , where $\\alpha $ and $\\beta $ are some vectors in $V$ , and $h$ is harmonic $\\Delta h = 0$ .",
"Since by (REF ) $\\partial \\beta \\ne 0$ implies $d\\partial \\beta \\ne 0$ , the kernel of $d$ consists of vectors of the type $d\\alpha + h$ , where $h$ is harmonic, $\\ker (d) = \\left\\lbrace x\\in V\\Big |\\ x = d\\alpha + h,\\quad \\Delta h =0\\right\\rbrace .$ This implies that the harmonic cocycles and cohomology classes of $d$ are isomorphic as vector spaces, that is (REF ) is true.",
"In the subsequent sections the operators $\\sigma _-$ and $\\sigma _+ := (\\sigma _-)^*$ will play the roles of $d$ and $\\partial $ .",
"Moreover, in the following calculations one can spot which Young diagram or multispinor belongs to $\\mathrm {im} (\\sigma _-), \\mathrm {im} (\\sigma _+)$ or $\\ker (\\Delta )$ due to the equivariance of the constructed Laplace operators $\\Delta $ with respect to the action of $\\mathrm {GL}(d)$ or $\\mathrm {O}(d)$ or $\\mathrm {SL}(2;\\mathbb {C})$ , depending on the problem in question."
],
[
"Generating functions",
"The problem of finding the $\\sigma _-$ cohomology in tensor spaces of one or another type can be conveniently reformulated in terms of differential operators.",
"To this end two-row Young diagrams in the symmetric basis can be described as a subset of polynomial ring $\\mathbb {R}[Y,Z]$ generated by the set of $2d$ commuting variables $Y^a, Z^b$ (see [16] for detail).",
"Consider the ring $\\Lambda ^p(\\mathcal {M}^d) \\otimes \\mathbb {R}[Y,Z]$ .",
"Its homogeneous elements are differential $p$ -forms valued in $\\mathbb {R}[Y,Z]$ $\\omega _{n,m}(x,dx,Y,Z) = \\omega _{a(n),b(m)}(x,dx)Y^{a(n)}Z^{b(m)}\\,.$ Consider the generating function $\\omega (x,dx\\,|Y,Z) = \\sum \\limits _{n,m\\ge 0}\\omega _{n,m}(x,dx\\,|Y,Z)=\\sum \\limits _{n,m\\ge 0} \\omega _{a(n),b(m)}(x,dx)Y^{a(n)}Z^{b(m)}\\,.$ Its expansion in powers of $Y$ and $Z$ yields the tensor-valued forms $\\omega _{a(n),b(m)}$ as the Taylor coefficients.",
"In this language the Young irreducibility condition reads as $Y^a {\\omega }{Z^a} = 0\\quad \\Longleftrightarrow \\quad \\omega _{a(n),a b(m-1)} = 0\\,.$ The tracelessness condition takes the form $\\eta ^{ab}\\partial _{Ya}\\partial _{Yb} \\omega = 0 \\Longleftrightarrow \\omega ^k{}_{k a(n-2),b(m)} = 0\\,.$ Note that all other traces are also zero as a consequence of (REF ) and (REF ), $\\eta ^{ab}\\partial _{Ya}\\partial _{Zb} \\omega = 0 \\Longleftrightarrow \\omega ^k{}_{a(n-1),b(m-1)k} = 0\\,,$ $\\eta ^{ab}\\partial _{Za}\\partial _{Zb} \\omega = 0 \\Longleftrightarrow \\omega _{a(n),b(m-2)k}{}^k = 0\\,.$ The generators of $\\mathfrak {u}(d)$ and $\\mathfrak {so}(d)$Following [24], in this section we do not distinguish between different real forms of the same complex algebra freely going to their compact real (Euclidean) form since, not affecting the final results, this choice simplifies the analysis by allowing a positive-definite invariant scalar product on the space of tensors.",
"Results of the Euclidean case coincide with those of the Lorentz one due to the equivalence of their representation theory on finite-dimensional modules.",
"Indeed, suppose that some Lorentz-irreducible tensor $T^L$ represented $\\sigma _-$ cohomology in the Lorentz case.",
"Then analogous $\\mathfrak {o}(d)$ -irreducible tensor $T^E$ represents $\\sigma _-$ cohomology in the compact case and vice versa.",
"The only potential difference could be related to (anti)self-dual tensors that may exist in one signature but not in the other.",
"However, these do not play a role in our analysis where (anti)self-dual tensors always appear in pairs or do not appear at all in sufficiently high dimensions $d>4$ .",
"are now realized by the first-order differential operators $\\left(t_{\\mathfrak {gl}(d)}{}\\right)^a_b = Y^a \\partial _{Yb} + Z^a \\partial _{Zb} + \\theta ^a \\partial _{\\theta ^b}, \\quad \\left(t^{\\mathfrak {so}(d)}{}\\right)_{ab} = \\frac{1}{2}\\bigg (\\eta _{ac}t_{\\mathfrak {gl}}{}^c_b - \\eta _{bc}t_{\\mathfrak {gl}}{}^c_a\\bigg )\\,,$ where $\\theta ^c$ is a Grassmann-odd element of the exterior algebra associated with the frame one-form $e^a$ .",
"In these terms $\\sigma _-$ acts as $\\sigma _- \\omega = \\theta ^a {\\omega }{Z^a} = m \\theta ^c\\omega _{a(n),c b(m-1)}(x,\\theta )Y^{a(n)}Z^{b(m-1)}\\,.$ It differs from the definition of Section 3 by an additional numerical factor introduced for future convenience.",
"In the sequel we sometimes do not write variables $Y,Z,\\theta $ explicitly, that are always assumed to be present implicitly.",
"We adopt the convention that index $a$ is contracted with $Y$ , $b$ with $Z$ and $c_i$ with $\\theta ^{c_i}$ with $\\theta $ s ordered as $c_1,...,c_p$ .",
"The space $\\Lambda (\\mathcal {M}^d) \\otimes \\mathbb {R}[Y,Z]$ can be equipped with the scalar product $\\langle f,g\\rangle = \\frac{1}{\\pi ^{2n}}\\int _{\\mathbb {C}^d \\times \\mathbb {C}^d} d^{2d}Z\\, d^{2d}Y\\, d^d \\theta \\, d^d \\overline{\\theta }\\, f(Z,Y,\\theta )\\,\\overline{g(Z,Y,\\theta )}\\, e^{-|Z|^2-|Y|^2-\\overline{\\theta }\\theta },$ where $f,g \\in \\Lambda (\\mathcal {M}^d) \\otimes \\mathbb {R}[Y,Z]$ with complex $Y,Z,\\theta $ and Berezin integral over anticommuting variables.",
"(We work with the polynomials of complex variables with real coefficients).",
"The space $\\Lambda (\\mathcal {M}^d) \\otimes \\mathbb {R}[Y,Z]$ with the scalar product (REF ) is a Hilbert space in the Euclidean metric signature case used in this section.",
"This scalar product yields the following conjugation rules: $(Z^a)^* = \\partial _{Z}{}_{a}\\,,\\qquad (Y^a)^* = \\partial _{Y}{}_a\\,,\\qquad (\\theta ^a)^* = \\partial _{\\theta }{}_a\\,.$"
],
[
"$\\mathrm {GL}(d)$ example",
"To illustrate the idea of our construction let us first consider a simpler case where fields and gauge parameters take values in the irreps of $\\mathfrak {gl}(d)$ described by two-row Young diagrams (no tracelessness conditions are imposed).",
"Define the following operators, that form $\\mathfrak {gl}(2)$ $t_{1} = Y^{a}\\frac{\\partial }{\\partial Z^{a}}, \\quad t_{2} = Z^{a}\\frac{\\partial }{\\partial Y^{a}}, \\quad t_{0} = Y^{a}\\frac{\\partial }{\\partial Y^{a}} - Z^{a}\\frac{\\partial }{\\partial Z^{a}},$ ${t_{1}}{t_{2}} = t_{0}, \\quad {t_{0}}{t_{1}} = 2 t_{1}, \\quad {t_{0}}{t_{2}} = -2 t_{2},$ $h_{1} = Y^{a}\\frac{\\partial }{\\partial Y^{a}}, \\quad h_2= Z^{a}\\frac{\\partial }{\\partial Z^{a}}\\,.$ Namely, $t_i$ form $\\mathfrak {sl}(2)$ while $h_1+h_2$ is central.",
"In terms of these operators the space of $p$ -forms valued in two-row Young diagrams is identified as $\\ker (t_1)$ $V^p = \\lbrace F \\in \\Lambda ^p(\\mathcal {M}^d) \\otimes \\mathbb {R}[Y,Z]| t_1 F = 0 \\rbrace \\,.$ Here $\\Lambda ^p(\\mathcal {M}^d)$ is generated by the Grassmann variables $\\theta ^a$ .",
"Let us introduce auxiliary operators $Z_{\\theta } = Z^{a}\\frac{\\partial }{\\partial \\theta ^{a} }, \\quad Y_{\\theta } = Y^{a}\\frac{\\partial }{\\partial \\theta ^{a} }, \\quad D = \\theta ^a \\frac{\\partial }{\\partial \\theta ^{a} }, \\quad \\theta _Y = \\theta ^a \\frac{\\partial }{\\partial Y^{a} }, \\quad \\theta _Z = \\theta ^a \\frac{\\partial }{\\partial Z^{a} }\\,.$ Among the auxiliary operators $D$ plays the most important role as it gives differential form degree.",
"Now we should construct $\\sigma _+: \\sigma _+^2 = 0$ on $V^p$ and $\\Im (\\sigma _+) \\subset V^p$ .",
"Consider the following operator: $\\sigma _+ = f(t_{0})Z_{\\theta } + g(t_{0})Y_{\\theta }t_2\\,,$ where $f(t_{0}) = \\sum _{n=0}^{\\infty }f_{n}t_{0}^{n}$ and $g(t_{0}) = \\sum _{n=0}^{\\infty }g_{n}t_{0}^{n}$ .",
"Functions $f$ and $g$ have to be found from the conditions $\\sigma _+^2F=0\\,,\\qquad t_{1}\\sigma _+ F =0\\,, \\qquad \\forall F \\in V^p\\,.$ After some re-ordering of operators this yields two equations $& 0 = Y_{\\theta } \\bigg (f(t_{0} - 1)F + g(t_{0}-1)t_{0}F \\bigg ),\\\\& 0 = Z_{\\theta }Y_{\\theta } \\bigg (f(t_{0})g(t_{0}+1)t_{2}F - g(t_{0})f(t_{0}+1)t_{2}F - g(t_{0})g(t_{0}+1)t_{2}F \\bigg )\\,$ verified by $f(t_0) = -(t_0+1)g(t_0)$ giving $\\sigma _+ = -(t_0+1)g(t_0)Z_{\\theta }+g(t_0)Y_{\\theta }t_2\\,.$ The free coefficient $g(t_0)$ is determined from the conjugacy requirement: $(f,\\sigma _+ g) = -(\\theta _Z \\bigg (g(t_0)(t_0+1)+g(t_0)\\bigg )f, g) = (\\sigma _- f,g)\\,$ giving $g(t_0) = - \\frac{1}{t_0+2}$ and hence $\\sigma _+:=(\\sigma _-)^*= \\frac{t_0+1}{t_0+2}Z_{\\theta }-\\frac{1}{t_0+2}Y_{\\theta }t_2\\,.$ One can notice, that $\\sigma _+$ in (REF ) differs from what one would expect from the conjugation rules (REF ).",
"The reason is that in (REF ) we work with $C := \\Lambda ^p(\\mathcal {M}^d) \\otimes \\mathbb {R}[Y,Z]$ complex.",
"In the $\\mathrm {GL}(d)$ -case we deal with $\\big (\\Lambda ^p(\\mathcal {M}^d) \\otimes \\mathbb {R}[Y,Z]\\big ) \\cap \\ker (t_1)$ complex, therefore one should project on the highest weight vectors of the underlying $\\mathfrak {sl}(2)$ in the complex C. The same procedure applies to the $\\mathrm {O}(d)$ -case.",
"Though general formulae for extremal projectors are known for any simple Lie algebraWe are grateful to the referee for bringing this fact to our attention.",
"[26], [27], [28], [29](for reviews see [30], [31]), to keep the paper self-contained we derive the relevant projectors straightforwardly.",
"Knowing $\\sigma _+$ , it remains to construct the Laplace operator $\\Delta = \\lbrace \\sigma _-,\\sigma _+\\rbrace $ and find its zeros.",
"Elementary computation gives $\\Delta = \\frac{t_0}{t_0+1}(D+h_2-1) + \\frac{1}{t_0+1}Y_\\theta \\theta _Y - \\frac{1}{(t_0+1)(t_0+2)}t_2 \\theta _Z Y_\\theta \\,.$ Being built from the manifestly $\\mathfrak {gl}(d)$ -invariant operators, $\\Delta $ commutes with $\\mathfrak {gl}(d)$ hence being diagonal on its irreducible submodules.",
"Thus, it suffices to analyze zeros of $\\Delta $ on $\\mathfrak {gl}(d)$ irreducible components of the forms."
],
[
"$H^0(\\sigma _-)$",
"Any element of $V^0$ has the form $F = F_{a(n),b(m)}Y^{a(n)}Z^{b(m)}$ .",
"It is easy to see that $\\Delta F = h_2 F\\,.$ Therefore $H^0(\\sigma _-) = \\lbrace F = F_{a(n)}Y^{a(n)}| \\forall F_{a(n)}\\in \\mathbb {R}\\rbrace \\,.$"
],
[
"$H^p(\\sigma _-)$ , {{formula:0ba4be03-0dbe-402f-9169-c5a2a3811fb6}}",
"For $p>0$ , a general element of $V^p$ is $F = F_{a(n),b(m)|c_1,..,c_p}Y^{a(n)}Z^{b(m)}\\theta ^{c_1}..\\theta ^{c_p}$ .",
"Generally it forms a reducible $\\mathfrak {gl}(d)$ -module associated with the tensor product of two diagrams.",
"In terms of Young diagrams it decomposes into the following irreducible components: $\\begin{picture}(15,10)(0,7){\\put (05,0){\\line (1,0){10}}\\put (05,20){\\line (1,0){10}}\\put (05,0){\\line (0,1){20}}\\put (15,0){\\line (0,1){20}}\\put (8,10){\\scriptsize {p}}}\\end{picture} \\,\\,\\,\\otimes _{gl}\\,\\,\\,\\begin{picture}(10,18)(0,7){\\put (05,00){\\line (1,0){35}}\\put (05,10){\\line (1,0){35}}\\put (05,0){\\line (0,1){10}}\\put (40,0){\\line (0,1){10}}\\put (17,3){\\scriptsize {m}}}\\end{picture}\\begin{picture}(65,18)(10,7){\\put (35,13){\\scriptsize {n}}\\put (05,20){\\line (1,0){60}}\\put (05,10){\\line (1,0){60}}\\put (05,10){\\line (0,1){10}}\\put (65,10){\\line (0,1){10}}}\\end{picture}\\cong \\begin{picture}(10,18)(0,7){\\put (05,00){\\line (1,0){35}}\\put (05,10){\\line (1,0){35}}\\put (05,0){\\line (0,1){10}}\\put (40,0){\\line (0,1){10}}\\put (17,3){\\scriptsize {m}}\\put (05,-23){\\line (1,0){10}}\\put (05,-23){\\line (0,1){23}}\\put (15,-23){\\line (0,1){23}}\\put (7,-21){\\begin{turn}{90}\\scriptsize p-1\\end{turn}}}\\end{picture}\\begin{picture}(65,18)(10,7){\\put (26,13){\\scriptsize {n+1}}\\put (05,20){\\line (1,0){60}}\\put (05,10){\\line (1,0){60}}\\put (05,10){\\line (0,1){10}}\\put (65,10){\\line (0,1){10}}}\\end{picture}\\oplus \\begin{picture}(10,18)(0,7){\\put (05,00){\\line (1,0){35}}\\put (05,10){\\line (1,0){35}}\\put (05,0){\\line (0,1){10}}\\put (40,0){\\line (0,1){10}}\\put (12,3){\\scriptsize {m+1}}\\put (05,-23){\\line (1,0){10}}\\put (05,-23){\\line (0,1){23}}\\put (15,-23){\\line (0,1){23}}\\put (7,-21){\\begin{turn}{90}\\scriptsize p-1\\end{turn}}}\\end{picture}\\begin{picture}(65,18)(10,7){\\put (35,13){\\scriptsize {n}}\\put (05,20){\\line (1,0){60}}\\put (05,10){\\line (1,0){60}}\\put (05,10){\\line (0,1){10}}\\put (65,10){\\line (0,1){10}}}\\end{picture}\\oplus \\begin{picture}(10,18)(0,7){\\put (05,00){\\line (1,0){35}}\\put (05,10){\\line (1,0){35}}\\put (05,0){\\line (0,1){10}}\\put (40,0){\\line (0,1){10}}\\put (17,3){\\scriptsize {m}}\\put (05,-23){\\line (1,0){10}}\\put (05,-23){\\line (0,1){23}}\\put (15,-23){\\line (0,1){23}}\\put (7,-14){\\begin{turn}{90}\\scriptsize p\\end{turn}}}\\end{picture}\\begin{picture}(65,18)(10,7){\\put (35,13){\\scriptsize {n}}\\put (05,20){\\line (1,0){60}}\\put (05,10){\\line (1,0){60}}\\put (05,10){\\line (0,1){10}}\\put (65,10){\\line (0,1){10}}}\\end{picture}\\oplus \\begin{picture}(10,18)(0,7){\\put (05,00){\\line (1,0){35}}\\put (05,10){\\line (1,0){35}}\\put (05,0){\\line (0,1){10}}\\put (40,0){\\line (0,1){10}}\\put (12,3){\\scriptsize {m+1}}\\put (05,-23){\\line (1,0){10}}\\put (05,-23){\\line (0,1){23}}\\put (15,-23){\\line (0,1){23}}\\put (7,-21){\\begin{turn}{90}\\scriptsize p-2\\end{turn}}}\\end{picture}\\begin{picture}(65,18)(10,7){\\put (26,13){\\scriptsize {n+1}}\\put (05,20){\\line (1,0){60}}\\put (05,10){\\line (1,0){60}}\\put (05,10){\\line (0,1){10}}\\put (65,10){\\line (0,1){10}}}\\end{picture}\\,.\\\\$ At $p = 1$ the last diagram is absent.",
"The manifest decomposition of $F_{a(n),b(m)|c_1,c_2,..,c_p}$ into irreducible components is $F_{a(n),b(m)|c_1,c_2,..,c_p} = F_1{}_{a(n)c_1,b(m),c_2,..,c_p} + \\frac{m}{n-m+2} F_1{}_{a(n)b,b(m-1)c_1,c_2,..,c_p} + F_2{}_{a(n),b(m)c_1,c_2,..,c_p} + \\\\ + F_3{}_{a(n),b(m),c_1,..,c_p} + F_4{}_{a(n)c_1,b(m)c_2,c_3,..,c_p},$ where $F_i$ corresponds to the $i$ -th diagram.",
"There are no restrictions on $n,m,p$ in (REF ) except for $n \\ge m$ .",
"If for some $n,m$ tensor expression has a wrong Young shape, it is zero.",
"To simplify calculations we derive restrictions on $n,m,p$ for each diagram from the condition of being $\\sigma _-$ -closed.",
"For the second and fourth diagrams we find no restrictions, but for others we have $\\sigma _- \\bigg (\\text{1st diagram}\\bigg ) = -m\\bigg (1-\\frac{1}{n-m+2}\\bigg )F_1{}_{a(n)c_0,b(m-1)c_1,c_2,..,c_p} \\Rightarrow m = 0,$ $\\sigma _- F_3{}_{a(n),b(m),c_1,..,c_p} = m F_3{}_{a(n),b(m-1)c_0,c_1,..,c_p} \\Rightarrow m = 0\\,.$ Using this we obtain the action of $\\Delta $ on the rest diagrams: $\\Delta F_1{}_{a(n)c_1,c_2,..,c_p} = \\frac{1}{n+1}\\Delta \\theta _Y F_1(Y,Z,\\theta ) = \\frac{n(p-1)}{(n+1)^2}\\theta _Y F_1(Y,Z,\\theta ),$ $\\Delta F_2{}_{a(n),b(m)c_1,c_2,..,c_p} = \\frac{1}{m+1}\\Delta \\theta _Z F_2(Y,Z,\\theta ) = \\frac{m+p}{m+1}\\theta _Z F_2(Y,Z,\\theta ),$ $\\Delta F_4{}_{a(n)c_1,b(m)c_2,c_3,..,c_p} = \\frac{1}{(m+1)(n+1)}\\Delta \\theta _Y\\theta _Z F_4(Y,Z,\\theta )\\\\ = \\frac{(n-m)(p+m-1)}{(n-m+1)(m+1)(n+1)} \\theta _Y\\theta _Z F_4(Y,Z,\\theta )\\,.$ As a result, $H^1(\\sigma _-) = \\lbrace \\phi = F_{a(n)c}\\theta ^c Y^{a(n)}| h_2 F = 0, F\\in V^1\\rbrace \\,,$ $H^p(\\sigma _-) = \\lbrace W = \\theta _Y\\theta _Z C(Y,Z,\\theta )| t_0 C = 0, C \\in V^{p-2} \\,,\\quad p>1\\rbrace ,$ $C =C{}_{a(n),b(n),c_1,..,c_{p-2}}Y^{a(n)}Z^{b(n)}\\theta ^{c_1}..\\theta ^{c_{p-2}} \\in V^{p-2}\\,.$ The dynamical interpretation of the obtained results is as follows.",
"The system has one symmetric gauge field with gauge transformation described by a symmetric parameter.",
"The second cohomology group $H^2(\\sigma _-)$ is spanned by a single tensor corresponding to the generalized (traceful) Weyl tensor.",
"If the latter is set to zero, the system becomes topological with the zero-curvature field equations.",
"Otherwise the unfolded equations encode a set of constraints expressing all fields and Weyl tensor via derivatives of the physical fields.",
"Proceeding further with the equations on the Weyl tensor and its descendants results in an infinite set of constraints with no differential equations on the physical field.",
"Such off-shell unfolded equations were considered in [32].",
"The off-shell systems are for interest in many contexts such as, e.g., construction of actions and quantization [33], [34].",
"The lower cohomology groups (REF ), (REF ) and (REF ) match with those obtained, e.g., in [16]."
],
[
"Irreducibility conditions",
"The $\\mathrm {O}(d)$ case is in many respects analogous to that of $\\mathrm {GL}(d)$ .",
"The difference is due to the tracelessness condition (REF ).",
"The algebra of the operators encoding irreducibility conditions is extended since the metric allows the new types of contractions between $\\theta , Y, Z$ and their derivatives.",
"From the representation theory perspective new terms associated with traces appear in the diagram decomposition of the form coefficients, affecting the cohomology analysis.",
"The following operators form the algebra $\\mathfrak {sp}(4)$ : $&t_{1} = Y^{a}\\frac{\\partial }{\\partial Z^{a}}, \\quad t_{2} = Z^{a}\\frac{\\partial }{\\partial Y^{a}}, \\quad h_{1} = Y^{a}\\frac{\\partial }{\\partial Y^{a}}, \\quad h_{2}= Z^{a}\\frac{\\partial }{\\partial Z^{a}},\\\\&t_{0} = Y^{a}\\frac{\\partial }{\\partial Y^{a}} - Z^{a}\\frac{\\partial }{\\partial Z^{a}},\\\\&f_{1} = \\partial _{Y}^a\\partial _{Y a}, \\quad f_{2} = \\partial _{Z}^a\\partial _{Z a}, \\quad f_{3} = \\partial _{Y}^a\\partial _{Z a}, \\\\&e_{1} = Y^a Y_a, \\quad e_{2} = Z^a Z_a, \\quad e_{3} = Y^a Z_a\\,.$ Evidently, these operators commute with the $\\mathfrak {so}(d)$ generators (REF ).",
"$\\mathfrak {sp}(4)$ and $\\mathfrak {so}(d)$ form a Howe-dual pair [35].",
"Young condition (REF ) and tracelessness condition (REF ) impose highest weight conditions on a $\\mathfrak {sp}(4)$ -module.",
"In addition, we introduce the following $\\mathrm {O}(d)$ invariant operators: $&Z_{\\theta } = Z^a \\partial _{\\theta a}, \\quad Y_{\\theta } = Y^a \\partial _{\\theta a},\\\\&\\partial _{\\theta Z} = \\partial _{\\theta a}\\partial _{Z}^a, \\quad \\partial _{\\theta Y} = \\partial _{\\theta a}\\partial _{Y}^a,\\\\&\\theta _Z = \\theta ^a \\partial _{Z a}, \\quad \\theta _Y = \\theta ^a \\partial _{Y a}\\,,$ which, along with $D$ (REF ) counting differential form degree, extend $\\mathfrak {sp}(4)$ to $\\mathfrak {osp}(2|4)$ .",
"The simplest way to see this is to let index $a$ take a single value, treating the operators $Z,Y,\\partial _Z, \\partial _Y, \\theta , \\partial _\\theta $ as creation and annihilation operators, and apply the oscillator realization of $\\mathfrak {osp}(2|4)$ .",
"In the problem in question, the form space is $V^p = \\lbrace F \\in \\Lambda ^p(\\mathcal {M}^d) \\otimes \\mathbb {R}[Y,Z]| t_1 F = 0, f_1 F = 0 \\rbrace \\,.$ Note that these restrictions imply the tracelessness over indices $(Y,Z)$ and $(Z,Z)$ as a consequence of the form of commutators of $t_1$ with $f_{1,2}$ ."
],
[
"$\\sigma _+$",
"Let us look for $\\sigma _+ = \\sigma _-^* $ in the form $\\sigma _+ = g_1(h_1,h_2)Z_{\\theta } + g_2(h_1,h_2)t_2Y_{\\theta } + g_3(h_1,h_2)e_3 \\partial _{\\theta Y} + g_4(h_1,h_2)e_1t_2 \\partial _{\\theta Y} + g_5(h_1,h_2)e_2 \\partial _{\\theta Z}+\\\\+g_6(h_1,h_2)e_3 t_2\\partial _{\\theta Z} + g_7(h_1,h_2)e_1 t_2^2 \\partial _{\\theta Z}\\,.$ The condition $\\Im (\\sigma _+) \\subset V^p$ gives $0 = f_1 \\sigma _+ F = \\bigg (2 g_2(h_1+2,h_2)+2g_3(h_1+2,h_2)+2g_4(h_1+2,h_2)(d + 2h_1)\\bigg )t_2 \\partial _{\\theta Y} F + \\\\+ \\bigg (2g_6(h_1+2,h_2)+2g_7(h_1+2,h_2)(d+2h_1)\\bigg )t_2^2\\partial _{\\theta Z}F,$ $0 = t_1 \\sigma _+ F = \\bigg (g_1(h_1-1,h_2+1)+g_2(h_1-1,h_2+1)t_0\\bigg )Y_\\theta F + \\bigg (-g_3(-1,1)+2g_5(-1,1)+ \\\\ + g_6(-1,1)t_0 \\bigg )e_3 \\partial _{\\theta Z} + \\bigg (g_3(-1,1)+g_4(-1,1)(t_0-2) \\bigg )e_1 \\partial _{\\theta Y} + \\bigg (-g_4(-1,1)+g_6(-1,1)+ \\\\ + 2g_7(-1,1)(t_0-1)\\bigg )e_1 t_2 \\partial _{\\theta Z}\\,.$ This imposes the following six equations on seven coefficients $g_1(h_1,h_2)+g_2(h_1,h_2)(t_0+2) = 0, \\\\-g_3(h_1,h_2)+2g_5(h_1,h_2)+g_6(h_1,h_2)(t_0+2) = 0, \\\\g_3(h_1,h_2)+g_4(h_1,h_2) t_0 = 0, \\\\-g_4(h_1,h_2)+g_6(h_1,h_2)+2g_7(h_1,h_2)(t_0+1) = 0, \\\\g_2(h_1,h_2)+g_3(h_1,h_2)+g_4(h_1,h_2)(d+2h_1-4) = 0, \\\\g_6(h_1,h_2)+g_7(h_1,h_2)(d+2h_1-4) = 0\\,.$ Choosing $g_7(h_1,h_2)$ as a free parameter, we obtain $g_1(h_1,h_2) = -(t_0+2)(d-4+h_1+h_2)(d-6+2h_2)g_7(h_1,h_2),\\\\g_2(h_1,h_2) = (d-4+h_1+h_2)(d-6+2h_2)g_7(h_1,h_2),\\\\g_3(h_1,h_2) = t_0(d-6+2h_2)g_7(h_1,h_2),\\\\g_4(h_1,h_2) = -(d-6+2h_2)g_7(h_1,h_2),\\\\g_5(h_1,h_2) = (t_0+1)(d-4+h_1+h_2)g_7(h_1,h_2),\\\\g_6(h_1,h_2) = -(d-4+2h_1)g_7(h_1,h_2)\\,.$ Now using the conjugation rules (REF ) and highest weight conditions (REF ) we get $(F_1,\\sigma _+ F_2) = \\Big ( -g_7(h_1,h_2)(t_0+2)(d-4+h_1+h_2)(d-6+2h_2) \\Big ) (\\sigma _{-}F_1,F_2)\\,.$ The condition $\\sigma _+ = \\sigma ^*_-$ demands $g_7(h_1,h_2) = - \\frac{1}{(t_0+2)(d-4+h_1+h_2)(d-6+2h_2)}$ giving $\\sigma _+ = Z_{\\theta } - \\frac{1}{t_0+2} t_2 Y_{\\theta } - \\frac{t_0}{(t_0+2)(d-4+h_1+h_2)} e_3 \\partial _{\\theta Y} + \\frac{1}{(t_0+2)(d-4+h_1+h_2)}e_1t_2 \\partial _{\\theta Y} - \\\\-\\frac{t_0+1}{(t_0+2)(d-6+2h_2)} e_2 \\partial _{\\theta Z} +\\frac{d-4+2h_1}{(t_0+2)(d-4+h_1+h_2)(d-6+2h_2)} e_3 t_2\\partial _{\\theta Z} - \\\\-\\frac{1}{(t_0+2)(d-4+h_1+h_2)(d-6+2h_2)} e_1 t_2^2 \\partial _{\\theta Z}\\,.$ This yields operator $\\sigma _+$ such that $\\sigma _+^2 = 0$ on $V$ , $\\Im (\\sigma _+) \\subset V$ and $\\sigma _-^* = \\sigma _+$ .",
"To calculate the Laplace operator $\\Delta = \\sigma _- \\sigma _+ + \\sigma _+ \\sigma _-$ on $V^p$ we obtain straightforwardly that $\\sigma _- \\sigma _+ = \\theta _Z Z_\\theta - \\frac{1}{t_0+1}t_2 \\theta _Z Y_\\theta - \\frac{1}{t_0+1}\\theta _Y Y_\\theta - \\frac{t_0-1}{(t_0+1)(d-3+h_1+h_2)}\\bigg (e_3 \\theta _Z \\partial _{\\theta Y} + \\theta ^a Y_a \\partial _{\\theta Y}\\bigg ) \\\\+ \\frac{1}{(t_0+1)(d-3+h_1+h_2)}\\bigg (e_1 t_2 \\theta _Z \\partial _{\\theta Y} + e_1 \\theta _Y \\partial _{\\theta Y} \\bigg ) - \\frac{t_0}{(t_0+1)(d-4+2h_2)}\\bigg (e_2 \\theta _Z \\partial _{\\theta Z} + 2 \\theta ^a Z_a \\partial _{\\theta Z} \\bigg )\\\\+ \\frac{d-4+2h_1}{(t_0+1)(d-3+h_1+h_2)(d-4+2h_2)}\\bigg (e_3 t_2 \\theta _Z \\partial _{\\theta Z} + e_3 \\theta _Y \\partial _{\\theta Z} + t_2 \\theta ^a Y_a \\partial _{\\theta Z} - \\theta ^a Z_a \\partial _{\\theta Z}\\bigg )\\\\- \\frac{1}{(t_0+1)(d-3+h_1+h_2)(d-4+2h_2)}\\bigg (e_1 t_2^2\\theta _Z \\partial _{\\theta Z} + 2e_1 t_2 \\theta _Y \\partial _{\\theta Z} \\bigg )\\,.$ $\\sigma _+ \\sigma _-= Z_\\theta \\theta _Z + \\frac{1}{t_0+2}t_2 \\theta _Z Y_\\theta + \\frac{t_0}{(t_0+2)(d-4+h_1+h_2)}e_3 \\theta _Z \\partial _{\\theta Y}- \\frac{1}{(t_0+2)(d-4+h_1+h_2)}\\times \\\\ \\times e_1 t_2 \\theta _Z \\partial _{\\theta Y}+ \\frac{t_0+1}{(t_0+2)(d-6+2h_2)}e_2 \\theta _Z \\partial _{\\theta Z}- \\frac{d-4+2h_1}{(t_0+2)(d-4+h_1+h_2)(d-6+2h_2)}e_3 t_2 \\theta _Z \\partial _{\\theta Z}\\\\+ \\frac{1}{(t_0+2)(d-4+h_1+h_2)(d-6+2h_2)}e_1 t_2^2\\theta _Z \\partial _{\\theta Z}\\,.$ Since, by construction, both $\\sigma _- \\sigma _+$ and $\\sigma _+ \\sigma _-$ and hence $\\Delta $ are $\\mathrm {O}(d)$ invariant, $\\Delta $ is diagonal on irreducible $\\mathrm {O}(d)$ -modules and, to compute $H(\\sigma _-)$ , it suffices to find its zeros on the irreducible components."
],
[
"$H^0(\\sigma _-)$",
"In the sector of zero-forms, all terms that contain $\\frac{\\partial }{\\partial \\theta ^c}$ trivialize.",
"Hence, $\\Delta F = h_2 F = m F \\,$ and $H^0(\\sigma _-) = \\lbrace F = F_{a(n)}Y^{a(n)}| \\forall F_{a(n)}\\in \\mathbb {R}\\rbrace \\,.$ Comparing the resulting differential gauge parameters with (REF ), we find that, as anticipated, differential gauge symmetries in the unfolded formulation coincide with those of the Fronsdal theory."
],
[
"$H^p(\\sigma _-)$ , {{formula:b937a896-0a47-49ab-98e2-b9277cda9f42}}",
"The main difference between $\\mathfrak {o}(d)$ - and $\\mathfrak {gl}(d)$ - cases is due to the traceful terms in the decomposition of the $p$ -forms into the irreducible parts depicted as $\\begin{picture}(15,10)(0,7){\\put (05,0){\\line (1,0){10}}\\put (05,20){\\line (1,0){10}}\\put (05,0){\\line (0,1){20}}\\put (15,0){\\line (0,1){20}}\\put (8,10){\\scriptsize {p}}}\\end{picture} \\,\\,\\,\\otimes _{so}\\,\\,\\,\\begin{picture}(10,18)(0,7){\\put (05,00){\\line (1,0){35}}\\put (05,10){\\line (1,0){35}}\\put (05,0){\\line (0,1){10}}\\put (40,0){\\line (0,1){10}}\\put (17,3){\\scriptsize {m}}}\\end{picture}\\begin{picture}(65,18)(10,7){\\put (35,13){\\scriptsize {n}}\\put (05,20){\\line (1,0){60}}\\put (05,10){\\line (1,0){60}}\\put (05,10){\\line (0,1){10}}\\put (65,10){\\line (0,1){10}}}\\end{picture}\\cong \\begin{picture}(10,18)(0,7){\\put (05,00){\\line (1,0){35}}\\put (05,10){\\line (1,0){35}}\\put (05,0){\\line (0,1){10}}\\put (40,0){\\line (0,1){10}}\\put (17,3){\\scriptsize {m}}\\put (05,-25){\\line (1,0){10}}\\put (05,-25){\\line (0,1){25}}\\put (15,-25){\\line (0,1){25}}\\put (7,-15){\\begin{turn}{90}\\scriptsize p\\end{turn}}}\\end{picture}\\begin{picture}(65,18)(10,7){\\put (35,13){\\scriptsize {n}}\\put (05,20){\\line (1,0){60}}\\put (05,10){\\line (1,0){60}}\\put (05,10){\\line (0,1){10}}\\put (65,10){\\line (0,1){10}}}\\end{picture}\\oplus \\begin{picture}(10,18)(0,7){\\put (05,00){\\line (1,0){35}}\\put (05,10){\\line (1,0){35}}\\put (05,0){\\line (0,1){10}}\\put (40,0){\\line (0,1){10}}\\put (12,3){\\scriptsize {m+1}}\\put (05,-25){\\line (1,0){10}}\\put (05,-25){\\line (0,1){25}}\\put (15,-25){\\line (0,1){25}}\\put (7,-21){\\begin{turn}{90}\\scriptsize p-1\\end{turn}}}\\end{picture}\\begin{picture}(65,18)(10,7){\\put (35,13){\\scriptsize {n}}\\put (05,20){\\line (1,0){60}}\\put (05,10){\\line (1,0){60}}\\put (05,10){\\line (0,1){10}}\\put (65,10){\\line (0,1){10}}}\\end{picture}\\oplus \\begin{picture}(10,18)(0,7){\\put (05,00){\\line (1,0){35}}\\put (05,10){\\line (1,0){35}}\\put (05,0){\\line (0,1){10}}\\put (40,0){\\line (0,1){10}}\\put (17,3){\\scriptsize {m}}\\put (05,-25){\\line (1,0){10}}\\put (05,-25){\\line (0,1){25}}\\put (15,-25){\\line (0,1){25}}\\put (7,-21){\\begin{turn}{90}\\scriptsize p-1\\end{turn}}}\\end{picture}$ ${n+1}$ $\\oplus \\begin{picture}(10,18)(0,7){\\put (05,00){\\line (1,0){35}}\\put (05,10){\\line (1,0){35}}\\put (05,0){\\line (0,1){10}}\\put (40,0){\\line (0,1){10}}\\put (12,3){\\scriptsize {m+1}}\\put (05,-25){\\line (1,0){10}}\\put (05,-25){\\line (0,1){25}}\\put (15,-25){\\line (0,1){25}}\\put (7,-21){\\begin{turn}{90}\\scriptsize p-2\\end{turn}}}\\end{picture}\\begin{picture}(65,18)(10,7){\\put (26,13){\\scriptsize {n+1}}\\put (05,20){\\line (1,0){60}}\\put (05,10){\\line (1,0){60}}\\put (05,10){\\line (0,1){10}}\\put (65,10){\\line (0,1){10}}}\\end{picture}\\oplus \\begin{picture}(10,18)(0,7){\\put (05,00){\\line (1,0){35}}\\put (05,10){\\line (1,0){35}}\\put (05,0){\\line (0,1){10}}\\put (40,0){\\line (0,1){10}}\\put (17,3){\\scriptsize {m}}\\put (05,-25){\\line (1,0){10}}\\put (05,-25){\\line (0,1){25}}\\put (15,-25){\\line (0,1){25}}\\put (7,-21){\\begin{turn}{90}\\scriptsize p-1\\end{turn}}}\\end{picture}\\begin{picture}(65,18)(10,7){\\put (26,13){\\scriptsize {n-1}}\\put (05,20){\\line (1,0){60}}\\put (05,10){\\line (1,0){60}}\\put (05,10){\\line (0,1){10}}\\put (65,10){\\line (0,1){10}}}\\end{picture}\\oplus \\begin{picture}(10,18)(0,7){\\put (05,00){\\line (1,0){35}}\\put (05,10){\\line (1,0){35}}\\put (05,0){\\line (0,1){10}}\\put (40,0){\\line (0,1){10}}\\put (12,3){\\scriptsize {m-1}}\\put (05,-25){\\line (1,0){10}}\\put (05,-25){\\line (0,1){25}}\\put (15,-25){\\line (0,1){25}}\\put (7,-21){\\begin{turn}{90}\\scriptsize p-1\\end{turn}}}\\end{picture}\\begin{picture}(65,18)(10,7){\\put (35,13){\\scriptsize {n}}\\put (05,20){\\line (1,0){60}}\\put (05,10){\\line (1,0){60}}\\put (05,10){\\line (0,1){10}}\\put (65,10){\\line (0,1){10}}}\\end{picture}\\oplus \\begin{picture}(10,18)(0,7){\\put (05,00){\\line (1,0){35}}\\put (05,10){\\line (1,0){35}}\\put (05,0){\\line (0,1){10}}\\put (40,0){\\line (0,1){10}}\\put (12,3){\\scriptsize {m-1}}\\put (05,-25){\\line (1,0){10}}}\\put (05,-25){\\line (0,1){25}}\\put (15,-25){\\line (0,1){25}}\\put (7,-21){\\begin{turn}{90}\\scriptsize p-2\\end{turn}}\\end{picture}$ ${n+1}$ ${m+1}$90$p-2$ ${n-1}$ $\\oplus \\begin{picture}(10,18)(0,7){\\put (05,00){\\line (1,0){35}}\\put (05,10){\\line (1,0){35}}\\put (05,0){\\line (0,1){10}}\\put (40,0){\\line (0,1){10}}\\put (12,3){\\scriptsize {m-1}}\\put (05,-25){\\line (1,0){10}}\\put (05,-25){\\line (0,1){25}}\\put (15,-25){\\line (0,1){25}}\\put (7,-21){\\begin{turn}{90}\\scriptsize p-2\\end{turn}}}\\end{picture}\\begin{picture}(65,18)(10,7){\\put (26,13){\\scriptsize {n-1}}\\put (05,20){\\line (1,0){60}}\\put (05,10){\\line (1,0){60}}\\put (05,10){\\line (0,1){10}}\\put (65,10){\\line (0,1){10}}}\\end{picture}\\oplus \\begin{picture}(10,18)(0,7){\\put (05,00){\\line (1,0){35}}\\put (05,10){\\line (1,0){35}}\\put (05,0){\\line (0,1){10}}\\put (40,0){\\line (0,1){10}}\\put (17,3){\\scriptsize {m}}\\put (05,-25){\\line (1,0){10}}\\put (05,-25){\\line (0,1){25}}\\put (15,-25){\\line (0,1){25}}\\put (7,-21){\\begin{turn}{90}\\scriptsize p-2\\end{turn}}}\\end{picture}\\begin{picture}(65,18)(10,7){\\put (35,13){\\scriptsize {n}}\\put (05,20){\\line (1,0){60}}\\put (05,10){\\line (1,0){60}}\\put (05,10){\\line (0,1){10}}\\put (65,10){\\line (0,1){10}}}\\end{picture}\\oplus \\begin{picture}(10,18)(0,7){\\put (05,00){\\line (1,0){35}}\\put (05,10){\\line (1,0){35}}\\put (05,0){\\line (0,1){10}}\\put (40,0){\\line (0,1){10}}\\put (17,3){\\scriptsize {m}}\\put (05,-25){\\line (1,0){10}}\\put (05,-25){\\line (0,1){25}}\\put (15,-25){\\line (0,1){25}}\\put (7,-21){\\begin{turn}{90}\\scriptsize p-2\\end{turn}}}\\end{picture}\\begin{picture}(65,18)(10,7){\\put (35,13){\\scriptsize {n}}\\put (05,20){\\line (1,0){60}}\\put (05,10){\\line (1,0){60}}\\put (05,10){\\line (0,1){10}}\\put (65,10){\\line (0,1){10}}}\\end{picture}\\oplus \\begin{picture}(10,18)(0,7){\\put (05,00){\\line (1,0){35}}\\put (05,10){\\line (1,0){35}}\\put (05,0){\\line (0,1){10}}\\put (40,0){\\line (0,1){10}}\\put (17,3){\\scriptsize {m}}\\put (05,-25){\\line (1,0){10}}\\put (05,-25){\\line (0,1){25}}\\put (15,-25){\\line (0,1){25}}\\put (7,-21){\\begin{turn}{90}\\scriptsize p-3\\end{turn}}}\\end{picture}\\begin{picture}(65,18)(10,7){\\put (26,13){\\scriptsize {n+1}}\\put (05,20){\\line (1,0){60}}\\put (05,10){\\line (1,0){60}}\\put (05,10){\\line (0,1){10}}\\put (65,10){\\line (0,1){10}}}\\end{picture}\\oplus $ $\\oplus \\begin{picture}(10,18)(0,7){\\put (05,00){\\line (1,0){35}}\\put (05,10){\\line (1,0){35}}\\put (05,0){\\line (0,1){10}}\\put (40,0){\\line (0,1){10}}\\put (12,3){\\scriptsize {m+1}}\\put (05,-25){\\line (1,0){10}}\\put (05,-25){\\line (0,1){25}}\\put (15,-25){\\line (0,1){25}}\\put (7,-21){\\begin{turn}{90}\\scriptsize p-3\\end{turn}}}\\end{picture}\\begin{picture}(65,18)(10,7){\\put (35,13){\\scriptsize {n}}\\put (05,20){\\line (1,0){60}}\\put (05,10){\\line (1,0){60}}\\put (05,10){\\line (0,1){10}}\\put (65,10){\\line (0,1){10}}}\\end{picture}\\oplus \\begin{picture}(10,18)(0,7){\\put (05,00){\\line (1,0){35}}\\put (05,10){\\line (1,0){35}}\\put (05,0){\\line (0,1){10}}\\put (40,0){\\line (0,1){10}}\\put (17,3){\\scriptsize {m}}\\put (05,-25){\\line (1,0){10}}\\put (05,-25){\\line (0,1){25}}\\put (15,-25){\\line (0,1){25}}\\put (7,-21){\\begin{turn}{90}\\scriptsize p-3\\end{turn}}}\\end{picture}\\begin{picture}(65,18)(10,7){\\put (26,13){\\scriptsize {n-1}}\\put (05,20){\\line (1,0){60}}\\put (05,10){\\line (1,0){60}}\\put (05,10){\\line (0,1){10}}\\put (65,10){\\line (0,1){10}}}\\end{picture}\\oplus \\begin{picture}(10,18)(0,7){\\put (05,00){\\line (1,0){35}}\\put (05,10){\\line (1,0){35}}\\put (05,0){\\line (0,1){10}}\\put (40,0){\\line (0,1){10}}\\put (12,3){\\scriptsize {m-1}}\\put (05,-25){\\line (1,0){10}}\\put (05,-25){\\line (0,1){25}}\\put (15,-25){\\line (0,1){25}}\\put (7,-21){\\begin{turn}{90}\\scriptsize p-3\\end{turn}}}\\end{picture}\\begin{picture}(65,18)(10,7){\\put (35,13){\\scriptsize {n}}\\put (05,20){\\line (1,0){60}}\\put (05,10){\\line (1,0){60}}\\put (05,10){\\line (0,1){10}}\\put (65,10){\\line (0,1){10}}}\\end{picture}\\oplus \\begin{picture}(10,18)(0,7){\\put (05,00){\\line (1,0){35}}\\put (05,10){\\line (1,0){35}}\\put (05,0){\\line (0,1){10}}\\put (40,0){\\line (0,1){10}}\\put (17,3){\\scriptsize {m}}\\put (05,-25){\\line (1,0){10}}\\put (05,-25){\\line (0,1){25}}\\put (15,-25){\\line (0,1){25}}\\put (7,-21){\\begin{turn}{90}\\scriptsize p-4\\end{turn}}}\\end{picture}\\begin{picture}(65,18)(10,7){\\put (35,13){\\scriptsize {n}}\\put (05,20){\\line (1,0){60}}\\put (05,10){\\line (1,0){60}}\\put (05,10){\\line (0,1){10}}\\put (65,10){\\line (0,1){10}}}\\end{picture}$ For $1 \\le p \\le 3$ , the diagrams carrying negative $p$ -dependent labels are absent.",
"It is important to note that the diagram $(n,m,p-2)$ is present twice: one copy results from the contraction of one of the form indices with the first row followed by the symmetrization of another form index with the same row.",
"Another one results from the application of the same procedure to the second row.",
"This fact leads to two different tensor implementations.",
"Also note that some of the diagrams vanish for special dimensions by virtue of the Two-Column Theorem: Theorem 5.1 $\\mathfrak {so}(d)$ traceless tensors with the symmetry properties of such Young diagrams that the sum of the heights of the first two columns exceeds $d$ , are identically zero.",
"[36] The cohomology $H^p(\\sigma _-)$ is empty for $p>d$ .",
"Analogously, some potential elements of $H^p(\\sigma _-)$ are zero by the Two-Column Theorem for large $p \\le d$ .",
"Now we are in a position to consider the action of the Laplace operator on each of the diagrams (REF ) separately.",
"In the following restrictions on $n,m,p$ will be imposed: if for some $n,m,p$ a tensor has a wrong Young shape, it is zero.",
"In most cases we will simplify calculation by demanding $p$ -forms be $\\sigma _-$ -closed.",
"Another simplification is due to the fact that $\\Delta F = 0$ is equivalent to the two equations $\\sigma _- F = 0$ and $\\sigma _+ F = 0$ .",
"Diagram (n,m;p), $n\\ge 0, m\\ge 0, p\\ge 0$ has the tensor form $T_{a(n),b(m),c_1,..,c_p}$ .",
"It is $\\sigma _-$ -closed, if $m = 0$ .",
"Then $\\Delta T_{a(n),c_1,..,c_p} = p T_{a(n),c_1,..,c_p}\\,.$ This diagram is in $\\ker (\\Delta )$ at $p = 0$ that reproduces the already obtained result for $H^0(\\sigma _-)$ .",
"Diagram (n+1,m+1;p-2), $n\\ge 0, m\\ge 0, p\\ge 2$ has the tensor form $T_{a(n)c_1,b(m)c_2,..,c_p}$ .",
"This diagram is $\\sigma _-$ -closed.",
"The action of $\\Delta $ is $\\Delta T_{a(n)c_1,b(m)c_2,..,c_p} = \\frac{(n-m)(p+m-1)}{(n-m+1)}T_{a(n)c_1,b(m)c_2,..,c_p}\\,.$ This diagram is in $\\ker (\\Delta )$ , if $n=m$ , thus belonging to $H^p(\\sigma _-)$ with $p\\ge 2$ .",
"Diagram (n,m+1;p-1), $n\\ge 1, m\\ge 0, p\\ge 1$ has the tensor form $T_{a(n),b(m)c_0,c_1,..,c_{p-1}}$ .",
"Then $\\Delta T_{a(n),b(m)c_0,c_1,..,c_{p-1}} = (m+p) T_{a(n),b(m)c_0,c_1,..,c_{p-1}}\\,.$ This equation admits no solutions since $p\\ge 1$ in the case in question.",
"Diagram (n+1,m;p-1), $n\\ge 0, m\\ge 0, p\\ge 1$ has the tensor form $T_{a(n)c_0,b(m),c_1,..,c_{p-1}} + \\frac{m}{n-m+2}T_{a(n)b,b(m-1)c_0,c_1,..,c_{p-1}}\\,.$ It is $\\sigma _-$ -closed iff $m = 0$ .",
"Then $\\Delta T_{a(n)c_0,c_1,..,c_{p-1}} = \\frac{n(p-1)}{n+1} T_{a(n)c_0,c_1,..,c_{p-1}}\\,.$ This expression vanishes at $p = 1$ .",
"Hence, $T_{a(n)c_0} \\in H^1(\\sigma _-)\\,.$ As one can see, $n=0$ in (REF ) also leads to zero Laplace action.",
"However, this is not a new result, since it has been already accounted in the diagrams $(n,m;p)$ for $n=m=p=0$ (REF ), $(n+1,m+1;p-2)$ for $n=m=0,p\\ge 2$ (REF ) and the $n=m=0,p=1$ case of $(n+1,m;p-1)$ (REF ).",
"This fact is a simple consequence of the tensor multiplication of a column by a scalar.",
"Diagram (n-1,m;p-1), $n\\ge 1, m\\ge 0, p\\ge 1$ has the tensor form $T = \\eta _{ac_1}\\rho _{a(n-1),b(m),c_2,..,c_p} - \\frac{n-1}{d-4+2n}\\eta _{aa}\\rho _{a(n-2)c_1,b(m),c_2,..,c_p} + \\frac{m(n-1)}{(d-4+m+n)(d-4+2n)}\\times \\\\\\times \\eta _{aa}\\rho _{a(n-2)b,b(m-1)c_1,c_2,..,c_p} - \\frac{m}{d-4+m+n}\\eta _{ab}\\rho _{a(n-1),b(m-1)c_1,c_2,..,c_p}\\,,$ where $\\rho $ is an arbitrary $(n-1,m;p-1)$ tensor.",
"One can check that $T \\in \\ker (\\sigma _-)$ demands $m = 0$ with $T = \\eta _{ac_1}\\rho _{a(n-1),c_2,..,c_p} - \\frac{n-1}{d-4+2n}\\eta _{aa}\\rho _{a(n-2)c_1,c_2,..,c_p}.$ After some calculation we obtain $\\Delta T = \\frac{(p-1)(d-2+n)}{d-3+n} T\\,.$ This implies that $\\Delta $ has zero at $p = 1$ contributing to $H^1(\\sigma _-)$ .",
"The case of $d=2,n=1$ must be considered separately because of the divergent denominator.",
"The seeming divergence emerges due to the second term in (REF ), which is absent at $d=2,n=1$ , $T = \\eta _{ac_1}\\rho _{c_2,..,c_p} \\Rightarrow \\sigma _+ T =\\frac{p-1}{2}\\Big (\\eta _{bc_1}\\rho _{a,c_2,..,c_{p-1}} - \\eta _{ac_1}\\rho _{b,c_2,..,c_{p-1}}\\Big )\\,,$ leading to the same answer with $p=1$ .",
"Diagram (n,m-1;p-1) $n\\ge 1, m\\ge 1, p\\ge 1$ has the tensor form $T = (n-1)\\eta _{aa} \\rho _{a(n-2)bc_1,b(m-1),c_2,..,c_p} - \\frac{(n-1)(m-1)}{d-6+2m} \\eta _{aa} \\rho _{a(n-2)bb,b(m-2)c_1,c_2,..,c_p} - (d-4+m+n) \\times \\\\\\times \\eta _{ac_1} \\rho _{a(n-1)b,b(m-1),c_2,..,c_p} - (n-m)\\eta _{ab} \\rho _{a(n-1)c_1,b(m-1),c_2,..,c_p} + \\frac{(m-1)(d-4+2n)}{d-6+2m}\\eta _{ab} \\times \\\\\\times \\rho _{a(n-1)b,b(m-2)c_1,c_2,..,c_p} + \\frac{(n-m+1)(d-4+m+n)}{n} \\eta _{bc_1} \\rho _{a(n),b(m-1),c_2,..,c_p} - \\\\-\\frac{(m-1)(n-m+1)(d-4+m+n)}{(d-6+2m)n}\\eta _{bb} \\rho _{a(n),b(m-2)c_1,c_2,..,c_p}\\,,$ where $\\rho $ is an arbitrary $(n,m-1;p-1)$ tensor.",
"Let us show that this diagram can never be annihilated by $\\sigma _-$ .",
"Indeed, $\\sigma _- T = \\bigg (d-4+m+n-(n-m)\\bigg )\\eta _{ac_0} \\rho _{a(n-1)c_1,b(m-1),c_2,..,c_p} + (m-1)(\\emph {lit}),$ where (lit) denotes some terms that are linearly independent from the first one.",
"If $m=1$ the above expression reduces to $\\sigma _- T = \\big (d-2\\big )\\eta _{ac_0} \\rho _{a(n-1)c_1,c_2,..,c_p}.$ The expression in brackets vanishes at $d = 2$ .",
"However, such diagram is zero by virtue of the Two-column theorem 5.1, since the heights of the first two columns sum up to $3>d$ .",
"Thus, the nontrivial $T$ is never in $\\ker (\\sigma _-)$ .",
"Diagram (n+1,m-1;p-2) $n\\ge 1, m\\ge 1, p\\ge 2$ has the tensor form $T = \\eta _{ac_1}\\rho _{a(n-1)bc_2,b(m-1),c_3,..c_p}- \\frac{n-m+1}{n}\\eta _{bc_1}\\rho _{a(n)c_2,b(m-1),c_3,..c_p} + \\\\ + \\frac{m-1}{n-m+3}\\eta _{ac_1}\\rho _{a(n-1)bb,b(m-2)c_2,c_3,..c_p}-\\frac{(m-1)(n-1)}{(d-6+2m)(n-m+3)}\\eta _{aa}\\rho _{a(n-2)bbc_1,b(m-2)c_2,c_3,..c_p} + \\\\ + \\frac{2(m-1)(n-m+1)}{(d-6+2m)(n-m+3)}\\eta _{ab}\\rho _{a(n-1)bc_1,b(m-2)c_2,c_3,..c_p}- \\frac{(m-1)(n-m+1)}{n(n-m+3)}\\times \\\\ \\times \\eta _{bc_1}\\rho _{a(n)b,b(m-2)c_2,c_3,..c_p} - \\frac{(m-1)(n-m+2)(n-m+1)}{(d-6+2m)(n-m+3)n} \\eta _{bb}\\rho _{a(n)c_1,b(m-2)c_2,c_3,..c_p}\\,,$ where $\\rho $ is an arbitrary $(n+1,m-1;p-2)$ tensor.",
"Though the tensor realization (REF ) may look complicated, the problem is simplified by the observation that all terms except for the first and second ones carry a factor of $(m-1)$ .",
"The action of $\\sigma _-$ on the first and second terms produces a factor of $(m-1)$ in front of each $\\eta \\rho $ combination.",
"It can be checked that $\\sigma _- T$ has an overall factor of $(m-1)$ so that the only possible solution for $T \\in \\ker (\\sigma _-)$ is at $m=1$ in which case the tensor decomposition acquires the form $T = \\eta _{ac_1}\\rho _{a(n-1)bc_2,..,c_p} - \\eta _{bc_1}\\rho _{a(n)c_2,..,c_p}.$ At $p = 2$ , after some calculations one can check that $\\sigma _+ T = 0\\,.$ For $p>2$ it is not difficult to see that $\\sigma _+ T \\ne 0$ .",
"Indeed, $\\sigma _+ T = (p-2)\\bigg (1 + \\frac{1}{n}\\bigg )\\eta _{ac_1}\\rho _{a(n-1)bc_2,b,c_3,..,c_{p-1}} + (\\emph {lit})\\,,$ where (lit) denotes other linearly independent terms.",
"The first term is never zero.",
"Thus, $T = \\eta _{ac_1}\\rho _{a(n-1)bc_2} - \\eta _{bc_1}\\rho _{a(n)c_2} \\in H^2(\\sigma _-)\\,.$ Diagram (n-1,m+1;p-2), $n\\ge 1, m\\ge 0, p\\ge 2$ has the tensor form $T = \\eta _{ac_1} \\rho _{a(n-1),b(m)c_2,..,c_p} - \\frac{n-1}{d-4+2n}\\eta _{aa} \\rho _{a(n-2)c_1,b(m)c_2,..,c_p}\\,,$ where $\\rho $ is an arbitrary $(n-1,m+1;p-2)$ tensor.",
"Though it is obviously in $\\ker ( \\sigma _-)$ for any $m$ , it is not hard to see that it is never in $\\ker (\\sigma _+)$ .",
"$\\sigma _+ T = -\\bigg (1 + \\frac{p-2}{m+1} \\bigg ) \\eta _{ac_1} \\rho _{a(n-1),b(m+1),..,c_{p-1}} + (\\emph {lit} )\\,.$ Hence this diagram does not contribute to $H^p(\\sigma _-)$ .",
"Diagram (n-1,m-1;p-2), $n\\ge 1, m\\ge 1, p\\ge 2$ has an involved tensor form.",
"Since the coefficients in the expression below are complicated, we extract the factor of $(m-1)$ once present denoting the leftover coefficients as $\\alpha _i$ , 0.75em $T = \\eta _{ac_1}\\eta _{bc_2}\\rho _{a(n-1),b(m-1),..,c_p} + (m-1)\\alpha _1 \\eta _{aa}\\eta _{aa}\\rho _{a(n-4)bbc_1,b(m-2)c_2,..,c_p} + \\\\ + \\alpha _2 \\eta _{aa}\\eta _{ac_1}\\rho _{a(n-3)bc_2,b(m-1),..,c_p}+ (m-1)\\alpha _3\\eta _{aa}\\eta _{ac_1}\\rho _{a(n-3)bb,b(m-2)c_2,..,c_p} + \\\\ + (m-1)\\alpha _4 \\eta _{aa}\\eta _{ab}\\rho _{a(n-3)bc_1,b(m-2)c_2,..,c_p}+\\alpha _5 \\eta _{aa}\\eta _{bc_1}\\rho _{a(n-2)c_2,b(m-1),..,c_p} + \\\\ + (m-1)\\alpha _6\\eta _{aa}\\eta _{bc_1}\\rho _{a(n-2)b,b(m-2)c_2,..,c_p}+ (m-1)\\alpha _7 \\eta _{aa}\\eta _{bb}\\rho _{a(n-2)c_1,b(m-2)c_2,..,c_p} + \\\\ + \\alpha _8 \\eta _{ac_1}\\eta _{ab}\\rho _{a(n-2)c_2,b(m-1),..,c_p}+ (m-1)\\alpha _9 \\eta _{ac_1}\\eta _{ab}\\rho _{a(n-2)b,b(m-2)c_2,..,c_p} + \\\\ + (m-1)\\alpha _{10} \\eta _{ac_1}\\eta _{bb}\\rho _{a(n-1),b(m-2)c_2,..,c_p}+ (m-1) \\alpha _{11} \\eta _{ab}\\eta _{ab}\\rho _{a(n-2)c_1,b(m-2)c_2,..,c_p} + \\\\ + (m-1) \\alpha _{12} \\eta _{ab}\\eta _{bc_1}\\rho _{a(n-1),b(m-2)c_2,..,c_p}\\,,$ where $\\rho $ is an arbitrary $(n-1,m-1;p-2)$ tensor.",
"The explicit form of $\\alpha _i$ is given in the Appendix A.",
"Now we observe that the action of $\\sigma _-$ on the terms free of the factor of $(m-1)$ produces such factor.",
"Hence, $\\sigma _-(T)$ has the form of the sum of linearly independent terms with the common factor of $(m-1)$ .",
"Consequently, $\\sigma _- T = 0, \\text{ iff\\quad } m = 1.$ At $m=1$ , the only terms that remain are $T = \\eta _{ac_1}\\eta _{bc_2}\\rho _{a(n-1),..,c_p} + \\frac{(n-2)(n-1)}{(d-3+n)(d-4+2n)} \\eta _{aa}\\eta _{ac_1}\\rho _{a(n-3)bc_2,..,c_p} + \\\\ + \\frac{(n-1)(d-2+n)}{(d-3+n)(d-4+2n)} \\eta _{aa}\\eta _{bc_1}\\rho _{a(n-2)c_2,..,c_p} - \\frac{n-1}{d-3+n} \\eta _{ac_1}\\eta _{ab}\\rho _{a(n-2)c_2,..,c_p}.$ It can be checked that for $p = 2$ $\\sigma _+ T = 0\\,.$ For $p > 2$ it is not difficult to see that $\\sigma _+ T = \\frac{(p-2)d}{(d-2)(d-3+n)(d-2+n)} e_3^2 \\theta _Y \\rho + (\\emph {lit} ),$ where $\\rho = \\rho _{a(n-1),c_1,..,c_{p-2}}Y^{a(n-1)}\\theta ^{c_1}..\\theta ^{c_{p-2}}$ .",
"Therefore, $H^2(\\sigma _-) \\ni T = \\eta _{ac_1}\\eta _{bc_2}\\rho {}_{a(n-1)} + \\frac{(n-2)(n-1)}{(d-3+n)(d-4+2n)} \\eta _{aa}\\eta _{ac_1}\\rho {}_{a(n-3)bc_2} + \\\\+ \\frac{(n-1)(d-2+n)}{(d-3+n)(d-4+2n)}\\eta _{aa}\\eta _{bc_1}\\rho {}_{a(n-2)c_2} -\\frac{n-1}{d-3+n} \\eta _{ac_1}\\eta _{ab}\\rho {}_{a(n-2)c_2} \\,.$ Diagram (n+1,m;p-3), $n\\ge 1, m\\ge 1, p\\ge 3$ has the tensor form $T = \\eta _{bc_1}\\rho _{a(n)c_2,b(m-1)c_3,..,c_p} - \\frac{n}{n-m+1}\\eta _{ac_1}\\rho _{a(n-1)bc_2,b(m-1)c_3,..,c_p}\\,,$ where $\\rho $ is an arbitrary $(n+1,m;p-3)$ tensor.",
"Explicit computation gives $\\Delta T = \\frac{t_0}{t_0+1}\\bigg (p + h_2 - 1\\bigg )T - \\frac{2 t_0}{(t_0+1)(d-4+2h_2)}\\theta ^a Z_a \\partial _{\\theta Z} T + \\\\ + \\frac{d-4+2h_1}{(t_0+1)(d-3+h_1+h_2)(d-4+2h_2)} \\bigg (t_2 \\theta ^a Y_a \\partial _{\\theta Z} - \\theta ^a Z_a \\partial _{\\theta Z}\\bigg )T - \\\\ - \\frac{t_0-1}{(t_0+1)(d-3+h_1+h_2)}\\theta ^a Y_a \\partial _{\\theta Y}T\\,.$ This expression vanishes at $n = m$ .",
"Indeed, in this case $t_0 T = 0, h_1 T= h_2 T = n T, t_2 T = 0$ so that $\\Delta T = \\frac{1}{d-3+2n} \\bigg (t_2 \\theta ^a Y_a \\partial _{\\theta Z} - \\theta ^a Z_a \\partial _{\\theta Z}\\bigg )T + \\frac{1}{d-3+2n}\\theta ^a Y_a \\partial _{\\theta Y}T\\,.$ Since $t_2 \\theta ^a Y_a \\partial _{\\theta Z}T = \\theta ^a Y_a \\partial _{\\theta Z}t_2 T - \\theta ^a Y_a \\partial _{\\theta Y}T + \\theta ^a Z_a \\partial _{\\theta Z}T = - \\theta ^a Y_a \\partial _{\\theta Y}T + \\theta ^a Z_a \\partial _{\\theta Z}T\\,,$ it follows that $\\Delta T=0$ at $n=m$ .",
"To check that $\\Delta T\\ne 0$ at $n\\ne m$ one should substitute the expression for $T$ noticing that different linearly independent terms have no common factor to vanish, that implies nontriviality of $\\Delta T$ .",
"Diagram (n,m+1;p-3), $n\\ge 1, m\\ge 0, p\\ge 3$ has the tensor form $T = \\eta _{ac_1}\\rho _{a(n-1)c_2,b(m)c_3,..,c_p}$ .",
"It belongs to $\\ker (\\sigma _-)$ , but not to $\\ker (\\sigma _+)$ .",
"$\\sigma _+ T = \\bigg (1 + \\frac{p-3}{m+1}\\bigg )\\eta _{ac_1}\\rho _{a(n-1)c_2,b(m+1),..,c_p} + (\\emph {lit})\\,.$ Diagram (n,m;p-2) $n\\ge 1, m\\ge 0, p\\ge 2$ admits two tensor realizations $T_i$ due to the double presence of this diagram in the result of tensor product.",
"The tensors $T_1 = \\eta _{ac_1} \\rho _1{}_{a(n-1)c_2,b(m),c_3,..,c_p} + \\frac{m}{n}\\eta _{bc_1} \\rho _1{}_{a(n),b(m-1)c_2,c_3,..,c_p},$ $T_2 = \\eta _{aa} \\rho _2{}_{a(n-2)bc_1,b(m-1)c_2,c_3,..,c_p} - \\frac{n-m}{n-1}\\eta _{ab} \\rho _2{}_{a(n-1)c_1,b(m-1)c_2,c_3,..,c_p} - \\frac{d-4+n+m}{n-1}\\eta _{ac_1} \\times \\\\\\times \\rho _2{}_{a(n-1)b,b(m-1)c_2,c_3,..,c_p} + \\frac{(n-m+1)(d-4+m+n)}{n(n-1)}\\eta _{bc_1} \\rho _2{}_{a(n),b(m-1)c_2,c_3,..,c_p}$ are linearly independent.",
"That Laplace operator acts diagonally on $T_i$ , $\\Delta T_i = \\lambda _i(n,m,p) T_i$ , allows us to separately consider each of these diagrams.",
"Firstly, we check if these are in $\\ker (\\sigma _-)$ computing $&\\sigma _- T_1 = m \\eta _{ac_1} \\rho _1{}_{a(n-1)c_2,b(m-1)c_3,..,c_{p+1}},\\\\&\\sigma _- T_2 = \\frac{d-4+2m}{n-1} \\eta _{ac_1} \\rho _2{}_{a(n-1)c_2,b(m-1)c_3,..,c_{p+1}}.$ $T_1 \\in \\ker (\\sigma _-)$ at $m = 0$ .",
"Formally, $T_2$ is annihilated by $\\sigma _-$ at $d=2, m = 1$ , but this is not allowed by the Two-column theorem.",
"So, the only candidate for cohomology is $T_1$ .",
"However, $\\sigma _+ T_1 = \\frac{1}{n+1}\\Big (1 + \\frac{p-2}{n} \\Big )\\eta _{bc_1}\\rho _1{}_{a(n),c_2,..,c_{p-1}} + (\\emph {lit})\\,,$ which is never zero.",
"Diagram (n-1,m;p-3), $n\\ge 2, m\\ge 1, p\\ge 3$ has the tensor form $T = \\eta _{ac_1}\\eta _{bc_2} \\rho _{a(n-1),b(m-1)c_3,..,c_p} - \\frac{(n-1)(d-3+m+n)}{(d-4+n+m)(d-4+2n)}\\eta _{aa}\\eta _{bc_1} \\rho _{a(n-2)c_2,b(m-1)c_3,..,c_p} -\\\\- \\frac{(n-1)(n-2)}{(d-4+m+n)(d-4+2n)} \\eta _{aa}\\eta _{ac_1} \\rho _{a(n-3)bc_2,b(m-1)c_3,..,c_p} + \\frac{n-1}{d-4+m+n} \\times \\\\\\times \\eta _{ab}\\eta _{ac_1} \\rho _{a(n-2)c_2,b(m-1)c_3,..,c_p}\\,,$ where $\\rho $ is an arbitrary $(n-1,m;p-3)$ tensor.",
"Obviously, $T \\in \\ker (\\sigma _-)$ .",
"However, $T \\notin \\ker (\\sigma _+)$ .",
"$\\sigma _+ T = \\bigg (1 + \\frac{p-3}{m}\\bigg ) \\eta _{ac_1}\\eta _{bc_2} \\rho _{a(n-1),b(m),c_3,..,c_{p-1}} + (\\emph {lit})\\,,$ hence not contributing to cohomology.",
"Diagram (n,m-1;p-3), $n\\ge 1, m\\ge 1, p\\ge 3$ has the tensor form $T = \\eta _{ac_1}\\eta _{bc_2}\\rho _{a(n-1)c_3,b(m-1),..,c_p} + (m-1)(\\emph {lit})\\,$ with all terms except for the first one carrying a factor of $(m-1)$ .",
"The action of $\\sigma _-$ on the first term brings a factor of $(m-1)$ in front of $\\eta \\rho $ .",
"Since all $\\eta \\rho $ terms in the decomposition are linearly independent we conclude that $\\sigma _- T = 0, \\text{ if } m = 1.$ At $p = 3$ one can check that $\\sigma _+ T = 0$ .",
"However, for $p>3$ $T$ does not belong to $\\ker (\\sigma _+)$ , $\\sigma _+ T = - (p-3) \\eta _{ac_1}\\eta _{bc_2} \\rho _{a(n-1)c_3,b,..,c_{p-1}} + (\\emph {lit})\\,.$ Consequently, the only contribution to $H^3(\\sigma _-)$ is $T = \\eta _{ac_1}\\eta _{bc_2}\\rho _{a(n-1)c_3} \\in H^3(\\sigma _-)\\,.$ Diagram (n,m;p-4), $n\\ge 1, m\\ge 1, p\\ge 4$ has the tensor form $T = \\eta _{ac_1}\\eta _{bc_2}\\rho _{a(n-1)c_3,b(m-1)c_4,..,c_p}\\,.$ This is obviously annihilated by $\\sigma _-$ , but not by $\\sigma _+$ .",
"$\\sigma _+ T = - \\bigg (1 + \\frac{p-4}{m} \\bigg ) \\eta _{ac_1}\\eta _{bc_2}\\rho _{a(n-1)c_3,b(m),..,c_{p-1}} + (\\emph {lit})\\,.$ Hence it does not contribute to $H^p(\\sigma _-)$ ."
],
[
"Summary",
"Summarizing the results of Sections REF and REF we found the following cohomology groups: $H^0(\\sigma _-) = \\lbrace F = F_{a(n)}Y^{a(n)}| F \\in V^0\\rbrace \\,,$ $H^1(\\sigma _-) = \\lbrace \\phi = F_1{}_{a(n)c}Y^{a(n)}\\theta ^c, F_1 \\in V^1;\\\\ \\phi ^{tr} = \\Big [(n-1)\\eta _{aa}F_2{}_{a(n-2)c} - (d-4+2n)\\eta _{ac}F_2{}_{a(n-1)}\\Big ]Y^{a(n)}\\theta ^c \\in V^1 \\rbrace \\,,$ $H^2(\\sigma _-) = \\lbrace W = \\theta _Y\\theta _Z C(Y,Z): t_0 C = 0, C \\in V^{0}; \\\\ \\mathcal {E}_A = \\Big [\\eta _{ac_1}\\rho _1{}_{a(n-1)bc_2} - \\eta _{bc_1}\\rho _1{}_{a(n)c_2}\\Big ]Y^{a(n)}Z^b\\theta ^{c_1}\\theta ^{c_2} \\in V^2; \\\\ \\mathcal {E}_B = \\Big [\\eta _{ac_1}\\eta _{bc_2}\\rho _2{}_{a(n-1)} + \\frac{(n-2)(n-1)}{(d-3+n)(d-4+2n)} \\eta _{aa}\\eta _{ac_1}\\rho _2{}_{a(n-3)bc_2} +\\\\+ \\frac{(n-1)(d-2+n)}{(d-3+n)(d-4+2n)}\\eta _{aa}\\eta _{bc_1}\\rho _2{}_{a(n-2)c_2} - \\frac{n-1}{d-3+n} \\eta _{ac_1}\\eta _{ab}\\rho _2{}_{a(n-2)c_2}\\Big ]Y^{a(n)}Z^b\\theta ^{c_1}\\theta ^{c_2} \\in V^2 \\rbrace \\,,$ $H^3(\\sigma _-) = \\lbrace B^{fr} = \\eta _{ac_1}\\eta _{bc_2}\\rho _{a(n-1)c_3} Y^{a(n)}Z^{b}\\theta ^{c_1}\\theta ^{c_2}\\theta ^{c_3} \\in V^3; \\\\B_1 = \\theta _Y\\theta _Z C(Y,Z,\\theta ): t_0 C = 0, C = C_{a(n),b(n),c}Y^{a(n)}Z^{b(n)}\\theta ^{c} \\in V^1; \\\\B_2 = \\Big [\\eta _{bc_1}\\rho _{a(n)c_2,b(n-1)c_3} - n\\eta _{ac_1}\\rho _{a(n-1)bc_2,b(n-1)c_3}\\Big ]Y^{a(n)}Z^{b(n)}\\theta ^{c_1}\\theta ^{c_2}\\theta ^{c_3} \\in V^3\\rbrace \\,.$ At $p>3$ $H^p(\\sigma _-) = \\Big \\lbrace B_1 = \\theta _Y\\theta _Z C(Y,Z,\\theta ): t_0 C = 0, C = C_{a(n),b(n),c_1,..,c_{p-2}}Y^{a(n)}Z^{b(n)}\\theta ^{c_1}..\\theta ^{c_{p-2}} \\in V^{p-2}; \\\\B_2 = \\Big [\\eta _{bc_1}\\rho _{a(n)c_2,b(n-1)c_3,..,c_p} - n\\eta _{ac_1}\\rho _{a(n-1)bc_2,b(n-1)c_3,..,c_p}\\Big ]Y^{a(n)}Z^{b(n)}\\theta ^{c_1}..\\theta ^{c_{p}} \\in V^{p} \\Big \\rbrace .$ According to Theorem REF , the differential gauge transformation parameters are described by $H^0(\\sigma _-)$ (REF ).",
"The gauge parameter in the Fronsdal theory is known to be a symmetric traceless tensor.",
"Since tensors $(n,0,0)$ constitute the cohomology group $H^0(\\sigma _-)$ , the unfolded differential gauge transformation is shown to coincide with the Fronsdal one.",
"As recalled in Section REF , the Fronsdal field consists of two symmetric traceless fields (REF ).",
"These fields are represented by the cohomology groups $H^1(\\sigma _-)$ (REF ).",
"Cohomology group $H^1(\\sigma _-)$ consists of two elements $(n+1,0,0)$ and $(n-1,0,0)$ matching the components of the Fronsdal field.",
"Thus, the physical fields in the unfolded approach indeed coincide with the Fronsdal field.",
"The cohomology group $H^2(\\sigma _-)$ (REF ) describes gauge invariant combinations of derivatives of the physical fields that can be used to impose differential equations on the latter.",
"The Fronsdal cohomology classes $\\mathcal {E}_A$ and $\\mathcal {E}_B$ match with the Fronsdal equations: $\\mathcal {E}_A$ is associated with the traceless part of the Fronsdal equations, while $\\mathcal {E}_B$ with the trace part, that is the equations $\\mathcal {E}_A = 0, \\mathcal {E}_B = 0$ just reproduce the Fronsdal equations.",
"Note that the number of resulting equations is the same as the number of fields, as it should be in a Lagrangian system.",
"$W$ in (REF ) represents \"Weyl\" cohomology.",
"Imposing $W = 0$ in the case of gravity one gets conformally flat metrics and in the case of higher spins \"conformally flat\" fields.",
"In Einstein gravity and HS theory, the equation $W=0$ is not imposed.",
"Instead, elements of $W$ are interpreted as new fields $C$ that describe generalized Weyl tensors by virtue of the unfolded equations (REF ).",
"Thus, calculation of the cohomology group $H^2(\\sigma _ -)$ shows that unfolded equations (REF ), (REF ) contain Fronsdal equations along with constraints on auxiliary fields.",
"In accordance with the general discussion of Section elements of $H^3(\\sigma _-)$ (REF ) correspond to Bianchi identities.",
"Class $B^{fr}$ describes the Bianchi identities for the Fronsdal equations.",
"Note that their number coincides with the number of differential gauge parameters.",
"The remaining classes $B_1$ and $B_2$ correspond to the Bianchi identities on the Weyl tensor.",
"It is noteworthy that the latter can be checked to coincide with the first $\\widetilde{\\sigma }_-$ cohomology in the Weyl sector of zero-forms of [10] for $s>1$ .",
"This fact exhibits the connection between the gauge and Weyl sectors.",
"For $p>3$ cohomology groups $H^p(\\sigma _-)$ describe the higher Bianchi identities for Bianchi identities on the Weyl tensor also interpreted as syzygies [24].",
"Obtained lower cohomology groups match with the results of [16], [17], [18].",
"In HS theory the fields are realized by one-forms.",
"Formally, one can consider field equations (REF ), (REF ) for $p$ -forms $\\omega _{a(n),b(m)}$ valued in a two-row irreducible $\\mathfrak {o}(d)$ -module.",
"From our results and physical interpretation of the $\\sigma _-$ cohomology groups it follows that for $p>1$ the unfolded system in the gauge sector is off-shell.",
"To answer the question whether the full unfolded system including both the gauge $p$ -form sector and the Weyl $(p-1)$ -form sector is off-shell, the analysis of $H(\\tilde{\\sigma }_{-})$ has to be performed in the Weyl sector.",
"The case of $p>1$ may be somewhat similar to the $s=1$ case, where the equation on $A_\\mu $ lies in the Weyl sector.",
"Finally, let us stress that the results of this section for HS fields in Minkowski space admit a straightforward deformation to $AdS_d$ with the same operator $\\sigma _-$ .",
"This is because in that case dynamical fields are described by rectangular diagrams of the $AdS_d$ algebra $\\mathfrak {o}(d-1,2)$ [37].",
"In general, in the flat limit, irreducible massless (gauge) fields in $AdS_d$ decompose into nontrivial sets of irreducible flat space massless fields [38], [39], [40] and there is no one-to-one correspondence between massless fields in Minkowski space and $AdS_d$ .",
"Namely, a generic irreducible field in Minkowski space may admit no deformation to $AdS_d$ (see also [41])."
],
[
"$\\sigma _-$ cohomology in {{formula:6304f7a9-7e1d-44fd-b47c-3df99bd81cdd}} in the spinor language",
"$4d$ HS theories admit a description in terms of two-component spinors instead of tensors.",
"That is, instead of using the generating functions in the tensor form $\\omega (x,dx\\,|Y,Z)$ , where $Y^a$ and $Z^a$ carried vector Lorentz indices $ a=\\lbrace 0,1,2,3\\rbrace $ , we will use $\\omega (y,\\overline{y}\\,|x,dx) = \\sum \\limits _{k,m}\\omega ^{\\alpha _1\\dots \\alpha _k,\\dot{\\alpha }_1\\dots \\dot{\\alpha }_m}(x,dx)\\, y_{\\alpha _1}\\dots y_{\\alpha _k}\\,\\overline{y}_{\\dot{\\alpha }_1}\\dots \\overline{y}_{\\dot{\\alpha }_m}\\,,$ where the indices $\\alpha $ and $\\dot{\\alpha }$ of the two-component commuting spinors $y^\\alpha $ and $\\overline{y}^{\\dot{\\alpha }}$ take two values $\\lbrace 1,2\\rbrace $ .",
"Analogously to Sections 3 and 4, we have to introduce the grading on the space of $\\Lambda ^\\bullet (M)\\otimes \\mathbb {C}[[y,\\overline{y}]]$ .",
"Consider the homogeneous element of $\\Lambda ^\\bullet (M)\\otimes \\mathbb {C}[[y,\\overline{y}]]$ of degree $N$ and $\\overline{N}$ in $y$ and $\\overline{y}$ , respectively $\\omega (\\mu y, \\overline{\\mu }\\overline{y}\\,|x,dx) = \\mu ^N\\,\\overline{\\mu }{}^{\\overline{N}}\\,\\omega (y,\\overline{y}\\,|x,dx).$ Define the grading operator $G$ on $\\Lambda ^\\bullet (M)\\otimes \\mathbb {C}[[y,\\overline{y}]]$ as follows: $G\\omega (y,\\overline{y}\\,|x,dx) = |N-\\overline{N}|\\,\\omega (y,\\overline{y}\\,|x,dx) \\equiv |\\deg _y(\\omega (y,\\overline{y}\\,|x,dx))-\\deg _{\\overline{y}}(\\omega (y,\\overline{y}\\,|x,dx))|.$ Note that in the bosonic sector the frame-like fields $e^{m_1\\dots m_{s-1}} \\ \\leftrightarrow \\ e^{\\alpha _1\\dots \\alpha _{s-1},\\dot{\\alpha }_1\\dots \\dot{\\alpha }_{s-1}}$ have the lowest possible grading $G=0$ .",
"For our later computations to match with the Fronsdal theory, we define the action of $\\sigma _-$ on $\\omega (y,\\overline{y})$ to decrease the $G$ -grading: $\\sigma _-\\omega (y,\\overline{y}) &:= i\\, \\overline{y}^{\\dot{\\alpha }}h^\\alpha _{\\ \\dot{\\alpha }}\\partial _\\alpha \\ \\omega (y,\\overline{y}), \\quad \\quad \\text{at } \\deg _y(\\omega )>\\deg _{\\overline{y}}(\\omega ),\\\\\\sigma _-\\omega (y,\\overline{y}) &:= i\\, y^{\\alpha }h_\\alpha ^{\\ \\dot{\\alpha }}\\overline{\\partial }_{\\dot{\\alpha }}\\ \\omega (y,\\overline{y}), \\quad \\quad \\text{at } \\deg _y(\\omega )<\\deg _{\\overline{y}}(\\omega ),\\\\\\sigma _-\\omega (y,\\overline{y}) &:= 0\\qquad \\qquad \\qquad \\qquad \\quad \\,\\,\\text{at } \\deg _y(\\omega )=\\deg _{\\overline{y}}(\\omega )\\,,$ where $_\\alpha :=\\frac{}{y^\\alpha }\\,,\\qquad \\overline{}_{\\dot{\\alpha }}:=\\frac{}{\\overline{y}^{\\dot{\\alpha }}}\\,,$ and the dependence on $x$ and $dx$ in $\\omega (y,\\overline{y}) = \\omega (y,\\overline{y}\\,|x,dx)$ is always implicit.",
"It is easy to check that so defined $\\sigma _-$ is nilpotent, $(\\sigma _-)^2=0$ .",
"Note that $\\sigma _\\pm $ change the grading $G$ by 2.",
"This agrees in particular with the fact that the bosonic and fermionic sectors, where the grading is even and odd respectively, are independent.",
"We consider in detail the more complicated bosonic case, observing in the end that the computation for fermionic fields is quite similar.",
"Note that the analysis of $\\sigma _-$ cohomology in the $4d$ conformal HS theory was also performed in terms of two-component spinors in [42], [43].",
"It is more complicated since the generating functions of conformal HS theory depend on twice as many independent spinors, but simpler since it is free of the module factors like $|N- N|$ in the grading definition (REF ).",
"Next, we define a scalar productBeing $\\mathrm {SL}(2,\\mathbb {C})$ -invariant this scalar product is not positive-definite.",
"Analogously to the tensorial case, without affecting the $\\sigma _-$ cohomology analysis it can be made positive-definite by going to the $\\mathfrak {su}(2)\\oplus \\mathfrak {su}(2)$ algebra, which is the compact real form of $\\mathfrak {sl}(2,\\mathbb {C})\\oplus \\mathfrak {sl}(2,\\mathbb {C})$ with altered conjugation rules $\\overline{y}^\\alpha =y_\\alpha $ , $\\overline{y}^{\\dot{\\alpha }} =y_{\\dot{\\alpha }}$ .",
"on generating elements of $\\Lambda ^\\bullet (M)\\otimes \\mathbb {C}[[y,\\overline{y}]]$ by $\\langle q^\\alpha |q^\\beta \\rangle =\\langle y^\\alpha |y^\\beta \\rangle =i\\epsilon ^{\\alpha \\beta }\\,, &&\\langle \\overline{p}_{\\dot{\\alpha }}|\\overline{p}_{\\dot{\\beta }}\\rangle =-\\langle \\overline{\\partial }_{\\dot{\\alpha }}|\\overline{\\partial }_{\\dot{\\beta }}\\rangle =i\\overline{\\epsilon }_{\\dot{\\alpha }\\dot{\\beta }} \\,, && \\langle h_{\\alpha \\dot{\\alpha }}|h_{\\beta \\dot{\\beta }}\\rangle =\\epsilon _{\\alpha \\beta }\\epsilon _{\\dot{\\alpha }\\dot{\\beta }}\\,,$ where $q^\\alpha =y^\\alpha $ , $p_\\alpha =i\\partial _\\alpha $ and $h^{\\alpha \\dot{\\beta }}$ is a vierbein one-form.",
"In some local coordinates $x^\\mu $ on the base manifold (which in our case is $AdS_4$ ) the vierbein one-forms $h^\\alpha _{\\ \\dot{\\alpha }}$ can be expressed as $h^\\alpha _{\\ \\dot{\\alpha }} = \\left(h_{\\mu }\\right)^\\alpha _{\\ \\dot{\\alpha }}\\,dx^\\mu .$ The $AdS_4$ vierbein $\\left(h_{\\mu }\\right)^\\alpha _{\\ \\dot{\\alpha }}$ is demanded to be non-degenerate at any point of $AdS_4$ .",
"The next step is to obtain $\\sigma _+:=\\sigma _-^\\dagger $ with respect to the scalar product $\\langle \\,,\\,\\rangle $ , i.e., $\\langle \\phi |\\sigma _-\\psi \\rangle = \\langle \\sigma _+\\phi |\\psi \\rangle $ .",
"It is not hard to check that $\\sigma _+\\omega (y,\\overline{y}) &:= -iy^\\alpha D_\\alpha ^{\\ \\dot{\\alpha }}\\overline{\\partial }_{\\dot{\\alpha }}\\ \\omega (y,\\overline{y}) \\quad \\quad \\text{at } \\deg _y(\\omega )>\\deg _{\\overline{y}}(\\omega )\\,,\\\\\\sigma _+\\omega (y,\\overline{y}) &:= -i \\overline{y}^{\\dot{\\alpha }}D^\\alpha _{\\ \\dot{\\alpha }}\\partial _\\alpha \\ \\omega (y,\\overline{y}) \\quad \\quad \\text{at } \\deg _y(\\omega )<\\deg _{\\overline{y}}(\\omega )\\,,\\\\\\sigma _+ \\omega (y,\\overline{y})&:= -i\\left(y^\\alpha D_\\alpha ^{\\ \\dot{\\alpha }}\\overline{\\partial }_{\\dot{\\alpha }}+\\overline{y}^{\\dot{\\alpha }}D^\\alpha _{\\ \\dot{\\alpha }}\\partial _\\alpha \\right)\\omega (y,\\overline{y})\\quad \\quad \\text{at } \\deg _y(\\omega )=\\deg _{\\overline{y}}(\\omega )\\,,$ where $D_\\alpha ^{\\ \\dot{\\alpha }} := \\frac{}{h^\\alpha _{\\ \\dot{\\alpha }}} \\,,\\qquad D^\\alpha _{\\ \\dot{\\alpha }} := \\frac{}{h_\\alpha ^{\\ \\dot{\\alpha }}}\\,.$ By $\\omega = \\omega (y,\\overline{y}\\,|x,h)$ we mean a general $p$ -form polynomial in $y$ and $\\overline{y}$ with the coordinate one-forms $dx$ replaced by $h$ via (REF ), that is $\\omega (y,\\overline{y}\\,|x,h) = \\sum _{n,m} \\omega _{\\alpha _1\\dots \\alpha _p|\\mu (n)|\\dot{\\alpha }_1\\dots \\dot{\\alpha }_p|\\dot{\\mu }(m)}(x)\\,h^{\\alpha _1\\dot{\\alpha }_1}\\wedge \\dots \\wedge h^{\\alpha _p\\dot{\\alpha }_p}\\,y^{\\mu (n)}\\,\\overline{y}^{\\dot{\\mu }(m)}\\,.$ So defined $\\sigma _+$ increases the grading.",
"The Laplace operator $\\Delta :=\\sigma _-\\sigma _++\\sigma _+\\sigma _-$ is by construction self-adjoint with respect to $\\langle \\,|\\,\\rangle $ and non-negative definite for the compact version of the space-time symmetry algebra."
],
[
"Bosonic case in $AdS_4$",
"To calculate cohomology of $\\sigma _-$ we have to compute the action of $\\Delta $ .",
"Since $\\sigma _-$ and $\\sigma _+$ are defined differently in the different regions of the $(N, \\overline{N})$ plane, we compute the Laplacian action in the these regions separately.",
"Direct computation yields: $\\Delta _{N>\\overline{N}+2}&= N(\\overline{N}+2) + y^\\beta _\\alpha h^\\alpha _{\\ \\dot{\\gamma }} D_\\beta ^{\\ \\dot{\\gamma }} + \\overline{y}^{\\dot{\\alpha }}\\overline{}_{\\dot{\\beta }}h_{\\gamma \\dot{\\alpha }}D^{\\gamma \\dot{\\beta }}\\,,\\\\\\Delta _{N<\\overline{N}-2}&=\\overline{N}(N+2)+y^\\alpha _\\beta h_{\\alpha \\dot{\\gamma }}D^{\\beta \\dot{\\gamma }}+\\overline{y}^{\\dot{\\alpha }}\\overline{}_{\\dot{\\beta }} h_\\gamma ^{\\ \\dot{\\beta }}D^{\\gamma }_{\\ \\dot{\\alpha }}\\,,\\\\\\Delta _{N=\\overline{N}+2} &= \\Delta _{N>\\overline{N}+2} + \\overline{y}^{\\dot{\\alpha }}\\overline{y}^{\\dot{\\beta }}_\\alpha _\\beta h^\\beta _{\\ \\dot{\\beta }}D^\\alpha _{\\ \\dot{\\alpha }}\\,,\\\\\\Delta _{N=\\overline{N}-2} &= \\Delta _{N<\\overline{N}-2} + y^\\alpha y^\\beta \\overline{}_{\\dot{\\alpha }}\\overline{}_{\\dot{\\beta }}h_\\beta ^{\\ \\dot{\\beta }}D_\\alpha ^{\\ \\dot{\\alpha }}\\,,\\\\\\Delta _{N=\\overline{N}}&=\\overline{y}^{\\dot{\\alpha }}\\overline{}_{\\dot{\\beta }} h_{\\gamma \\dot{\\alpha }} D^{\\gamma \\dot{\\beta }}+y^\\alpha _\\beta h_{\\alpha \\dot{\\gamma }}D^{\\beta \\dot{\\gamma }}-\\overline{y}^{\\dot{\\alpha }} y^\\beta _\\alpha \\overline{}_{\\dot{\\beta }}h^\\alpha _{\\ \\dot{\\alpha }} D_\\beta ^{\\ \\dot{\\beta }}- y^\\alpha \\overline{y}^{\\dot{\\beta }}_\\beta \\overline{}_{\\dot{\\alpha }}h_\\alpha ^{\\ \\dot{\\alpha }}D^\\beta _{\\ \\dot{\\beta }}\\,.$ The computation of the cohomology $H^p(\\sigma _-)$ will be performed as follows.",
"Taking a general $p$ -form $\\omega (y,\\overline{y}\\,|x)$ , we decompose it into Lorentz irreducible components.",
"As we will observe, the projectors onto irreducible parts will commute with the action of the Laplacian.",
"Thus, instead of involved calculation of the action of $\\Delta $ on all of the irreducible components of $\\omega (y,\\overline{y}\\,|x)$ we can first calculate its action on the general $\\omega (y,\\overline{y}\\,|x)$ and then project."
],
[
" $H^0(\\sigma _-)$",
"Evidently, $\\Delta _{N=\\overline{N}}\\Big |_\\text{0-forms}= 0$ since all the terms in (REF ) contain derivatives in $h$ 's.",
"At the same time, $\\Delta _{N\\ne \\overline{N}}\\Big |_\\text{0-forms} > 0$ .",
"Thus, we conclude $H^0(\\sigma _-)=\\ker \\left(\\Delta \\Big |_\\text{0-forms}\\right)= \\left\\lbrace F(y,\\overline{y})=F_{\\alpha (n),\\dot{\\alpha }(n)}y^{\\alpha (n)}\\overline{y}^{\\dot{\\alpha }(n)},\\ n\\in \\mathbb {N}_0\\right\\rbrace .$ By Theorem REF , elements of this cohomology space correspond to the parameters of differential (non-Stueckelberg) linearized HS gauge transformations.",
"This result fits the pattern of the spin-$s$ Fronsdal gauge symmetry parameters with $n=s-1$ ."
],
[
" $H^1(\\sigma _-)$",
"The decomposition of a one-form $\\Theta (y,\\overline{y}\\,|x)$ into Lorentz irreps reads as $\\Theta (y,\\overline{y}\\,|x)=\\underbrace{\\Theta _{\\mu (n+1)|\\dot{\\mu }(m+1)}\\ h^{\\mu \\dot{\\mu }}y^{\\mu (n)}y^{\\dot{\\mu }(m)}}_{\\Theta _\\mathrm {A}(y,\\overline{y})}-\\frac{1}{2}\\underbrace{\\Theta _{\\mu (n-1)|\\dot{\\mu }(m+1)}\\ h_\\nu ^{\\ \\dot{\\mu }}y^\\nu y^{\\mu (n-1)}\\overline{y}^{\\dot{\\mu }(m)}}_{\\Theta _\\mathrm {B}(y,\\overline{y})} -\\\\-\\frac{1}{2}\\underbrace{\\Theta _{\\mu (n+1)|\\dot{\\mu }(m-1)}\\ h^{\\mu }_{\\ \\dot{\\nu }}y^{\\mu (n)}\\overline{y}^{\\dot{\\nu }}\\overline{y}^{\\dot{\\mu }(m-1)}}_{\\Theta _\\mathrm {C}(y,\\overline{y})} +\\frac{1}{4}\\underbrace{\\Theta _{\\mu (n-1)|\\dot{\\mu }(m-1)}\\ h_{\\nu \\dot{\\nu }} y^\\nu y^{\\mu (n-1)}\\overline{y}^{\\dot{\\nu }}\\overline{y}^{\\dot{\\mu }(m-1)}}_{\\Theta _\\mathrm {D}(y,\\overline{y})}.$ Thus, for fixed $n$ and $m$ , there are four Lorentz-irreducible one-forms: $\\Theta _\\text{A}$ , $\\Theta _\\mathrm {B}$ , $\\Theta _\\mathrm {C}$ , $\\Theta _\\mathrm {D}$ .",
"For direct computations it will be convenient to separate two of the indices of the $y$ group: $\\Theta (y,\\overline{y}) = \\Theta _{\\lambda ,\\nu ,\\mu (n-1)|\\dot{\\lambda },\\dot{\\nu },\\dot{\\mu }(m-1)}\\ h^{\\lambda \\dot{\\lambda }}y^{\\nu }y^{\\mu (n-1)}\\overline{y}^{\\dot{\\nu }}\\overline{y}^{\\dot{\\mu }(m-1)}.$ In terms of the basis one-forms $h^{\\lambda \\dot{\\lambda }}y^{\\nu }y^{\\mu (n-1)}\\overline{y}^{\\dot{\\nu }}\\overline{y}^{\\dot{\\mu }(m-1)}$ the projectors onto irreducible components are $\\mathcal {P}_\\mathrm {A} &= {S}_{(\\lambda ,\\mu ,\\nu )}{S}_{(\\dot{\\lambda },\\dot{\\mu },\\dot{\\nu })}, && \\mathcal {P}_\\mathrm {C} = {S}_{(\\lambda ,\\mu ,\\nu )}\\epsilon _{\\dot{\\lambda }\\dot{\\nu }},\\\\\\mathcal {P}_\\mathrm {B} &= \\epsilon _{\\lambda \\nu }{S}_{(\\dot{\\lambda },\\dot{\\mu },\\dot{\\nu })}, &&\\mathcal {P}_\\mathrm {D} = \\epsilon _{\\lambda \\nu }\\epsilon _{\\dot{\\lambda }\\dot{\\nu }},$ where ${S}_{(\\lambda ,\\mu ,\\nu )}$ implies symmetrization over indices $\\lambda ,\\mu ,\\nu $ and similarly for the dotted indices."
],
[
"$H^1(\\sigma _-)$ in the diagonal sector {{formula:a426b6e9-6846-47b5-abc4-9ff268867159}}",
"In the diagonal sector with $n=m$ the Laplacian is a sum of the following four terms: $\\Delta _{N=\\overline{N}}\\Theta (y,\\overline{y}) =\\underbrace{\\overline{y}^{\\dot{\\alpha }}\\overline{}_{\\dot{\\beta }} h_{\\gamma \\dot{\\alpha }} D^{\\gamma \\dot{\\beta }} \\Theta (y,\\overline{y}) }_{T_1(y,\\overline{y})}+\\underbrace{y^\\alpha _\\beta h_{\\alpha \\dot{\\gamma }}D^{\\beta \\dot{\\gamma }}\\Theta (y,\\overline{y}) }_{T_2(y,\\overline{y})}-\\\\\\underbrace{-\\overline{y}^{\\dot{\\alpha }} y^\\beta _\\alpha \\overline{}_{\\dot{\\beta }}h^\\alpha _{\\ \\dot{\\alpha }} D_\\beta ^{\\ \\dot{\\beta }}\\Theta (y,\\overline{y}) }_{T_3(y,\\overline{y})}\\underbrace{-y^\\alpha \\overline{y}^{\\dot{\\beta }}_\\beta \\overline{}_{\\dot{\\alpha }}h_\\alpha ^{\\ \\dot{\\alpha }}D^\\beta _{\\ \\dot{\\beta }}\\Theta (y,\\overline{y}) }_{T_4(y,\\overline{y})}.$ Consider the first term in (REF ).",
"$\\overline{y}^{\\dot{\\alpha }}\\overline{}_{\\dot{\\beta }} h_{\\gamma \\dot{\\alpha }} D^{\\gamma \\dot{\\beta }} \\Theta (y,\\overline{y}) = \\Theta _{\\lambda ,\\nu ,\\mu (n-1)|\\dot{\\lambda },\\dot{\\nu },\\dot{\\mu }(m-1)}\\ \\underbrace{\\Big [\\overline{y}^{\\dot{\\alpha }}\\overline{}_{\\dot{\\beta }} h_{\\gamma \\dot{\\alpha }} D^{\\gamma \\dot{\\beta }} \\big ( h^{\\lambda \\dot{\\lambda }}y^{\\nu }y^{\\mu (n-1)}\\overline{y}^{\\dot{\\nu }}\\overline{y}^{\\dot{\\mu }(m-1)}\\big )\\Big ]}_{T_1^{\\lambda ,\\nu ,\\mu (n-1)|\\dot{\\lambda },\\dot{\\nu },\\dot{\\mu }(n-1)}}\\,.$ The expression in square brackets is denoted by $T_1^{\\lambda ,\\nu ,\\mu (n-1)|\\dot{\\lambda },\\dot{\\nu },\\dot{\\mu }(n-1)}$ .",
"The notation for other irreducible components $T_2$ , $T_3$ and $T_4$ is analogous.",
"Straightforward computation yields $T_1^{\\lambda ,\\nu ,\\mu (n-1)|\\dot{\\lambda },\\dot{\\nu },\\dot{\\mu }(n-1)} &=& -h^\\lambda _{\\ \\dot{\\alpha }}\\epsilon ^{\\dot{\\nu }\\dot{\\lambda }} y^\\nu y^{\\mu (n-1)}\\overline{y}^{\\dot{\\alpha }}\\overline{y}^{\\dot{\\mu }(n-1)} -{}\\nonumber \\\\&&- (n-1) h^\\lambda _{\\ \\dot{\\alpha }}\\epsilon ^{\\dot{\\mu }\\dot{\\lambda }}y^\\nu y^{\\mu (n-1)}\\overline{y}^{\\dot{\\alpha }}\\overline{y}^{\\dot{\\nu }}\\overline{y}^{\\dot{\\mu }(n-2)}\\,,\\\\T_2^{\\lambda ,\\nu ,\\mu (n-1)|\\dot{\\lambda },\\dot{\\nu },\\dot{\\mu }(n-1)} &=& -h_\\alpha ^{\\ \\dot{\\lambda }}\\epsilon ^{\\nu \\lambda }y^{\\alpha }y^{\\mu (n-1)}\\overline{y}^{\\dot{\\nu }}\\overline{y}^{\\dot{\\mu }(n-1)} -{}\\nonumber \\\\&&- (n-1)h_\\alpha ^{\\ \\dot{\\lambda }}\\epsilon ^{\\mu \\lambda }y^\\alpha y^\\nu y^{\\mu (n-2)}\\overline{y}^{\\dot{\\nu }}\\overline{y}^{\\dot{\\mu }(n-1)}\\,,\\\\T_3^{\\lambda ,\\nu ,\\mu (n-1)|\\dot{\\lambda },\\dot{\\nu },\\dot{\\mu }(n-1)} &=& -\\left(h^\\nu _{\\ \\dot{\\alpha }}y^\\lambda y^{\\mu (n-1)}+(n-1)h^\\mu _{\\ \\dot{\\alpha }}y^\\lambda y^\\nu y^{\\mu (n-2)}\\right){}\\nonumber \\times \\\\&&\\times \\left(\\epsilon ^{\\dot{\\nu }\\dot{\\lambda }}\\overline{y}^{\\dot{\\alpha }}\\overline{y}^{\\dot{\\mu }(n-1)} + (n-1)\\epsilon ^{\\dot{\\mu }\\dot{\\lambda }}\\overline{y}^{\\dot{\\alpha }}\\overline{y}^{\\dot{\\nu }}\\overline{y}^{\\dot{\\mu }(n-2)}\\right)\\,,\\\\T_4^{\\lambda ,\\nu ,\\mu (n-1)|\\dot{\\lambda },\\dot{\\nu },\\dot{\\mu }(n-1)} &=& -\\left(h_\\alpha ^{\\ \\dot{\\nu }}\\overline{y}^{\\dot{\\lambda }} \\overline{y}^{\\dot{\\mu }(n-1)}+(n-1)h_{\\alpha }^{\\ \\dot{\\nu }} \\overline{y}^{\\dot{\\lambda }} \\overline{y}^{\\dot{\\nu }} \\overline{y}^{\\dot{\\mu }(n-2)}\\right){}\\nonumber \\times \\\\&&\\times \\left(\\epsilon ^{\\nu \\lambda } y^{\\alpha } y^{\\mu (n-1)} + (n-1)\\epsilon ^{\\mu \\lambda } y^{\\alpha } y^{\\nu } y^{\\mu (n-2)}\\right).$ Projecting onto the irreducible parts of $\\Theta (y,\\overline{y})$ we find $\\Delta _{N=\\overline{N}}\\left(\\Theta _{\\mathrm {A}}\\right)&=0\\,,\\\\\\Delta _{N=\\overline{N}}\\left(\\Theta _{\\mathrm {B}}\\right)&=(n+1)^2\\Theta _{\\mathrm {B}}\\ne 0\\,,\\\\\\Delta _{N=\\overline{N}}\\left(\\Theta _{\\text{C}}\\right)&=(n+1)^2\\Theta _{\\text{C}}\\ne 0\\,,\\\\\\Delta _{N=\\overline{N}}\\left(\\Theta _{\\mathrm {D}}\\right)&=0.$ Thus, the only elements of the kernel of $\\Delta ^\\text{1-forms}\\Big |_{N=\\overline{N}}$ are $\\Theta _{\\mathrm {A}}(y,\\overline{y})$ and $\\Theta _{\\mathrm {D}}(y,\\overline{y})$ .",
"By the Hodge theorem of Section 4 this yields that $H^1(\\sigma _-) = \\ker \\left(\\Delta ^\\text{1-forms}\\Big |_{N=\\overline{N}}\\right)$ is $H^1(\\sigma _-) &= \\bigoplus _{n\\ge 0} H^1_{(n)}(\\sigma _-),\\\\H^1_{(n)}(\\sigma _-) &= \\Big \\lbrace \\phi _{(n)}(y,\\overline{y}\\,|x) + \\phi _{(n)}^\\text{tr}(y,\\overline{y}\\,|x),\\\\\\phi _{(n)}(y,\\overline{y}\\,|x) &:= \\phi _{\\mu (n+1),\\dot{\\mu }(n+1)}(x)\\, h^{\\mu \\dot{\\mu }} \\, y^{\\mu (n)}\\overline{y}^{\\dot{\\mu }(n)},\\\\\\phi _{(n)}^\\text{tr}(y,\\overline{y}\\,|x) &:= \\phi ^\\mathrm {tr}_{\\mu (n-1),\\dot{\\mu }(n-1)}(x)\\, h_{\\nu \\dot{\\nu }} \\, y^\\nu y^{\\mu (n-1)}\\overline{y}^{\\dot{\\nu }}\\overline{y}^{\\dot{\\mu }(n-1)}\\,, \\text{ if } n>0 \\Big \\rbrace ,$ where $n$ is the number of indices of the corresponding cocycles.",
"Equivalently, $H^1(\\sigma _-) = \\Big \\lbrace h^{\\mu \\dot{\\mu }}\\,\\partial _\\mu \\overline{\\partial }_{\\dot{\\mu }}\\,F_1(y,\\overline{y}\\,|x) + h_{\\mu \\dot{\\mu }}\\,y^\\mu \\overline{y}^{\\dot{\\mu }} F_2(y,\\overline{y}\\,|x)\\Big \\rbrace ,$ where $F_{1,2}(y,\\overline{y}\\,|x)$ belongs to the diagonal $N=\\overline{N}$ , that is, $\\left(y^\\alpha \\frac{\\partial }{\\partial y^\\alpha } - \\overline{y}^{\\dot{\\alpha }}\\frac{\\partial }{\\partial \\overline{y}^{\\dot{\\alpha }}}\\right)F_{1,2}(y,\\overline{y}\\,|x) = 0.$ The fields $\\phi (y,\\overline{y})$ and $\\phi ^\\text{tr}(y,\\overline{y})$ exactly correspond to the irreducible components of the double-traceless Fronsdal field.",
"It remains to prove that there are no other nontrivial cocycles in $H^1(\\sigma _- )$ at $N\\ne \\overline{N}$ ."
],
[
"$H^1(\\sigma _-)$ in the far-from-diagonal sector {{formula:f3550507-9a9d-493c-9968-364e086c8c2f}}",
"Consider the action of the Laplace operator $\\Delta _{N>\\overline{N}+2}$ on general one-forms at $N>\\overline{N}+2$ $\\Delta _{N>\\overline{N}+2}\\Theta (y, \\overline{y}) =\\left(n(m+2) + y^\\beta _\\alpha h^\\alpha _{\\ \\dot{\\gamma }} D_\\beta ^{\\ \\dot{\\gamma }} + \\overline{y}^{\\dot{\\alpha }}\\overline{}_{\\dot{\\beta }}h_{\\gamma \\dot{\\alpha }}D^{\\gamma \\dot{\\beta }}\\right)\\Theta (y, \\overline{y}) =\\\\=\\underbrace{n(m+2)\\Theta (y, \\overline{y})}_{T_1(y,\\overline{y})} +\\underbrace{y^\\beta _\\alpha h^\\alpha _{\\ \\dot{\\gamma }} D_\\beta ^{\\ \\dot{\\gamma }}\\Theta (y, \\overline{y})}_{T_2(y,\\overline{y})} +\\underbrace{\\overline{y}^{\\dot{\\alpha }}\\overline{}_{\\dot{\\beta }}h_{\\gamma \\dot{\\alpha }}D^{\\gamma \\dot{\\beta }}\\Theta (y,\\overline{y})}_{T_3(y,\\overline{y})}.$ Analogously to (REF ), we denote $T_2^{\\lambda ,\\nu ,\\mu (n-1)|\\dot{\\lambda },\\dot{\\nu },\\dot{\\mu }(m-1)} = y^\\beta _\\alpha h^\\alpha _{\\ \\dot{\\gamma }} D_\\beta ^{\\ \\dot{\\gamma }} \\big ( h^{\\lambda \\dot{\\lambda }}y^{\\nu }y^{\\mu (n-1)}\\overline{y}^{\\dot{\\nu }}\\overline{y}^{\\dot{\\mu }(m-1)}\\big )$ and similarly for $T_1$ and $T_3$ .",
"In this sector we obtain $T_1^{\\lambda ,\\nu ,\\mu (n-1)|\\dot{\\lambda },\\dot{\\nu },\\dot{\\mu }(m-1)} &=n(m+2)h^{\\lambda \\dot{\\lambda }}y^{\\nu }y^{\\mu (n-1)}\\overline{y}^{\\dot{\\nu }}\\overline{y}^{\\dot{\\mu }(m-1)}\\,,\\\\T_2^{\\lambda ,\\nu ,\\mu (n-1)|\\dot{\\lambda },\\dot{\\nu },\\dot{\\mu }(m-1)} &=-h^{\\nu \\dot{\\lambda }}y^{\\lambda }y^{\\mu (n-1)}\\overline{y}^{\\dot{\\nu }}\\overline{y}^{\\dot{\\mu }(m-1)} - (n-1)h^{\\mu \\dot{\\lambda }}y^{\\lambda }y^{\\nu }y^{\\mu (n-2)}\\overline{y}^{\\dot{\\nu }}\\overline{y}^{\\dot{\\mu }(m-1)}\\,,\\\\T_3^{\\lambda ,\\nu ,\\mu (n-1)|\\dot{\\lambda },\\dot{\\nu },\\dot{\\mu }(m-1)} &= - h^{\\lambda }_{\\ \\dot{\\alpha }}\\epsilon ^{\\dot{\\nu }\\dot{\\lambda }}y^{\\nu }y^{\\mu (n-1)}\\overline{y}^{\\dot{\\alpha }}\\overline{y}^{\\dot{\\mu }(m-1)} - (m-1) h^{\\lambda }_{\\ \\dot{\\alpha }}\\epsilon ^{\\dot{\\lambda }\\dot{\\mu }}y^{\\nu }y^{\\mu (n-1)}\\overline{y}^{\\dot{\\alpha }}\\overline{y}^{\\dot{\\nu }}\\overline{y}^{\\dot{\\mu }(m-2)}\\,.$ Projection onto the irreducible components according to (REF ) yields $\\Delta _{N>\\overline{N}+2}\\left(\\Theta _\\mathrm {A}\\right) &= n(m+1)\\Theta _\\mathrm {A}\\ne 0\\,,\\\\\\Delta _{N>\\overline{N}+2}\\left(\\Theta _\\mathrm {B}\\right) &= (nm+2n+1)\\Theta _\\mathrm {B}\\ne 0\\,,\\\\\\Delta _{N>\\overline{N}+2}\\left(\\Theta _\\mathrm {C}\\right) &= (nm-n+2m)\\Theta _\\mathrm {C}\\ne 0\\,,\\\\\\Delta _{N>\\overline{N}+2}\\left(\\Theta _\\mathrm {D}\\right) &= (nm+2n-m+4)\\Theta _\\mathrm {D}\\ne 0\\,.$ Thus, there are no nontrivial cocycles in this sector.",
"For $N<\\overline{N}-2$ the computation is analogous.",
"Thus, $H^1(\\sigma _-) =0$ in the far-from-diagonal sector."
],
[
"Subtlety in the near-diagonal sector $|N-\\overline{N}|=2$",
"In this case we face certain peculiarity.",
"Denote the space of $p$ -forms ($p=1$ for $H^{1}$ ) with $N$ chiral and $\\overline{N}$ anti-chiral indices by $\\mathcal {V}_{(N,\\overline{N})}$ .",
"Recall that the grading operator is $G=|N-\\overline{N}|$ .",
"Consider the case with $N-\\overline{N}=2$ .",
"Namely, let $N=n+1$ and $\\overline{N}=n-1$ .",
"At $G=2$ the operator $\\sigma _-$ maps a state $X\\in \\mathcal {V}_{(n+1,n-1)}$ onto the diagonal, $\\sigma _-(X)\\in \\mathcal {V}_{(n,n)}$ , where in accordance with (REF ), $\\sigma _+$ acts 'both up and down': $\\mathcal {V}_{(n+1,n-1)} \\xrightarrow{} \\mathcal {V}_{(n,n)} \\xrightarrow{} \\mathcal {V}_{(n-1,n+1)} \\oplus \\mathcal {V}_{(n+1,n-1)}\\,.$ Thus, $\\mathcal {V}_{(n+1,n-1)} \\xrightarrow{}\\mathcal {V}_{(n-1,n+1)} \\oplus \\mathcal {V}_{(n+1,n-1)}\\,.$ As a result, $\\Delta _{(n+1,n-1)} : \\mathcal {V}_{(n+1,n-1)} \\longrightarrow \\mathcal {V}_{(n-1,n+1)} \\oplus \\mathcal {V}_{(n+1,n-1)}\\,,\\\\\\Delta _{(n-1,n+1)} : \\mathcal {V}_{(n-1,n+1)} \\longrightarrow \\mathcal {V}_{(n-1,n+1)} \\oplus \\mathcal {V}_{(n+1,n-1)}\\,.$ Consequently, $\\ker (\\Delta )$ should be searched in the form of a linear combination of the vectors both from $\\mathcal {V}_{(n+1,n-1)}$ and from $\\mathcal {V}_{(n-1,n+1)}$ .",
"Indeed, let $X$ be a vector in $\\mathcal {V}_{(n+1,n-1)}$ .",
"Consider the complex conjugated vector $\\overline{X}\\in \\mathcal {V}_{(n-1,n+1)}$ and compute the action of the Laplacian on them.",
"Let $\\Delta X = \\Delta _{(n+1,n-1)}X=\\alpha (n) X + \\beta (n) \\overline{X}\\,,\\\\\\Delta \\overline{X} = \\Delta _{(n-1,n+1)}\\overline{X}= \\gamma (n) X + \\delta (n) \\overline{X}$ with some coefficients $\\alpha $ , $\\beta $ , $\\gamma $ , and $\\delta $ .",
"That $X$ and $\\overline{X}$ are conjugated and operator $\\Delta $ is self-adjoint implies the relations $\\alpha = \\overline{\\delta }$ and $\\beta =\\overline{\\gamma }$ .",
"Looking for $\\ker (\\Delta )$ in the form $Y=F(n)X+G(n)\\overline{X}\\in \\ker (\\Delta )$ and acting on $Y$ by the Laplace operator we find that the condition $\\Delta Y =0$ yields $\\Delta Y = F(n)\\Delta _{(n+1,n-1)}X + G(n)\\Delta _{(n-1,n+1)}\\overline{X} =\\\\=\\big (\\alpha (n)F(n)+\\overline{\\beta (n)}G(n)\\big )X + \\big (\\beta (n)F(n)+\\overline{\\alpha (n)}G(n)\\big )\\overline{X} = 0\\,.$ Since $X$ and $\\overline{X}$ are linearly independent, the problem of finding such $Y=\\alpha X+\\beta \\overline{X}$ that $\\Delta Y=0$ , amounts to the linear system $\\begin{bmatrix}\\alpha (n) & \\overline{\\beta (n)} \\\\\\beta (n) & \\overline{\\alpha (n)}\\end{bmatrix}\\begin{bmatrix}F(n) \\\\G(n)\\end{bmatrix}=\\begin{bmatrix}0 \\\\0\\end{bmatrix}\\,,$ which admits non-trivial solutions iff $\\det \\begin{bmatrix}\\alpha (n) & \\overline{\\beta (n)} \\\\\\beta (n) & \\overline{\\alpha (n)}\\end{bmatrix}=|\\alpha (n)|^2 - |\\beta (n)|^2 = 0.$ Hence, we conclude that $\\alpha (n)=\\beta (n)\\cdot e^{i\\chi }, \\quad \\quad \\chi \\in [0,2\\pi )\\,.$ In the next section coefficients $\\alpha (n)$ and $\\beta (n)$ will be shown to be real, i.e., $e^{i\\chi }=\\pm 1$ .",
"Summarizing, if we find that the coefficients $\\alpha (n)$ and $\\beta (n)$ coincide up to a sign, $\\alpha (n)=\\pm \\beta (n)$ , this would imply the existence of a non-trivial $\\sigma _-$ -cocycle $Y = X \\mp \\overline{X}\\,.$ Otherwise the cohomology is trivial."
],
[
"$H^1(\\sigma _-)$ in the near-diagonal sector {{formula:1655b858-2c9a-47bf-8194-18e799b3a71f}}",
"To compute $H^1(\\sigma _-)$ in the leftover sector of $N=\\overline{N}+2$ (analysis at $N=\\overline{N}-2$ is analogous) consider a general one-form $\\Theta (y,\\overline{y})$ (REF ) with $N=\\overline{N}+2$ .",
"In this sector, the Laplacian differs form that at $N>\\overline{N}+2$ by the $T_4(y,\\overline{y})$ term in $\\Delta _{N=\\overline{N}+2}\\Theta (y, \\overline{y}) = \\Big (\\underbrace{N(\\overline{N}+2) + y^\\beta _\\alpha h^\\alpha _{\\ \\dot{\\gamma }} D_\\beta ^{\\ \\dot{\\gamma }} + \\overline{y}^{\\dot{\\alpha }}\\overline{}_{\\dot{\\beta }}h_{\\gamma \\dot{\\alpha }}D^{\\gamma \\dot{\\beta }}}_{\\Delta _{N>\\overline{N}+2}}\\Big )\\Theta (y, \\overline{y}) +\\underbrace{\\overline{y}^{\\dot{\\alpha }}\\overline{y}^{\\dot{\\beta }}_\\alpha _\\beta h^\\beta _{\\ \\dot{\\beta }}D^\\alpha _{\\ \\dot{\\alpha }}\\Theta (y, \\overline{y})}_{T_4(y,\\overline{y})}.$ Consequently, it is essential to compute the action of this additional term.",
"As before (cf.",
"(REF )), denote $T_4^{\\lambda ,\\nu ,\\mu (n-1)|\\dot{\\lambda },\\dot{\\nu },\\dot{\\mu }(m-1)} = \\overline{y}^{\\dot{\\alpha }}\\overline{y}^{\\dot{\\beta }}_\\alpha _\\beta h^\\beta _{\\ \\dot{\\beta }}D^\\alpha _{\\ \\dot{\\alpha }} \\big (h^{\\lambda \\dot{\\lambda }}y^{\\nu }y^{\\mu (n-1)}\\overline{y}^{\\dot{\\nu }}\\overline{y}^{\\dot{\\mu }(m-1)}\\big )\\,.$ This yields $T_4^{\\lambda ,\\nu ,\\mu (n-1)|\\dot{\\lambda },\\dot{\\nu },\\dot{\\mu }(m-1)}=(n-1)\\epsilon ^{\\mu \\lambda }h^{\\nu }_{\\ \\dot{\\beta }}y^{\\mu (n-2)}\\overline{y}^{\\dot{\\lambda }}\\overline{y}^{\\dot{\\beta }}\\overline{y}^{\\dot{\\nu }}\\overline{y}^{\\dot{\\mu }(m-1)} + (n-1)\\epsilon ^{\\nu \\lambda }h^{\\mu }_{\\ \\dot{\\beta }}y^{\\mu (n-2)}\\overline{y}^{\\dot{\\lambda }}\\overline{y}^{\\dot{\\beta }}\\overline{y}^{\\dot{\\nu }}\\overline{y}^{\\dot{\\mu }(m-1)}+\\\\+(n-1)(n-2)\\epsilon ^{\\mu \\lambda }h^{\\mu }_{\\ \\dot{\\beta }}y^{\\nu }y^{\\mu (n-3)}\\overline{y}^{\\dot{\\lambda }}\\overline{y}^{\\dot{\\beta }}\\overline{y}^{\\dot{\\nu }}\\overline{y}^{\\dot{\\mu }(m-1)}.$ Projecting $T_4$ onto the irreducible parts, we find: $\\text{(A)}:&& {S}_{(\\lambda ,\\nu ,\\mu )}{S}_{(\\dot{\\lambda },\\dot{\\nu },\\dot{\\mu })}T_4^{\\lambda ,\\nu ,\\mu (n-1)|\\dot{\\lambda },\\dot{\\nu },\\dot{\\mu }(m-1)}&=0\\,,\\\\\\text{(B)}:&& \\ \\ {S}_{(\\dot{\\lambda },\\dot{\\nu },\\dot{\\mu })}\\epsilon _{\\lambda \\nu }T_4^{\\lambda ,\\nu ,\\mu (n-1)|\\dot{\\lambda },\\dot{\\nu },\\dot{\\mu }(m-1)} &= -(n-1)(2n-1)h^{\\mu }_{\\ \\dot{\\beta }} y^{\\mu (n-2)}\\overline{y}^{\\dot{\\beta }}\\overline{y}^{\\dot{\\mu }(m+1)}\\,,\\\\\\text{(C)}:&& \\ \\ \\epsilon _{\\dot{\\lambda }\\dot{\\nu }}{S}_{(\\lambda ,\\nu ,\\mu )}T_4^{\\lambda ,\\nu ,\\mu (n-1)|\\dot{\\lambda },\\dot{\\nu },\\dot{\\mu }(m-1)}&=0\\,,\\\\\\text{(D)}:&& \\ \\ \\epsilon _{\\dot{\\lambda }\\dot{\\nu }}\\epsilon _{\\lambda \\nu }T_4^{\\lambda ,\\nu ,\\mu (n-1)|\\dot{\\lambda },\\dot{\\nu },\\dot{\\mu }(m-1)}&=0\\,.$ We observe that the action of the Laplacian in this sector differs from the previously computed one only in the type-(B) sector, namely, $\\Delta _{N=\\overline{N}+2}\\left(\\Theta _\\mathrm {B}\\right) = \\left(n(m+3)\\Theta _\\mathrm {B}+(2n^2-3n+1)\\Theta _\\mathrm {C}\\right)\\Big |_{m=n-2}=\\underbrace{(n^2+n)}_{\\alpha (n)}\\Theta _\\mathrm {B}+\\underbrace{(2n^2-3n+1)}_{\\beta (n)}\\Theta _\\mathrm {C}\\,.$ That $|\\alpha (n)|\\ne |\\beta (n)|$ at integer $n$ implies triviality of $H^1(\\sigma _-)$ in the near-diagonal sector."
],
[
"$H^2(\\sigma _-)$",
"The calculation of $H^2(\\sigma _-)$ is in main features analogous to that of $H^1(\\sigma _-)$ .",
"To decompose a general two-form $\\Omega (y,\\overline{y}\\,|x)$ into irreducible parts we use the following useful identity $h^{\\nu \\dot{\\nu }}\\wedge h^{\\lambda \\dot{\\lambda }} = \\frac{1}{2}H^{\\nu \\lambda }\\epsilon ^{\\dot{\\nu }\\dot{\\lambda }}+\\frac{1}{2}\\overline{H}^{\\dot{\\nu }\\dot{\\lambda }}\\epsilon ^{\\nu \\lambda },$ where $H^{\\nu \\lambda }=H^{(\\nu \\lambda )}:=h^{\\nu }_{\\ \\dot{\\gamma }}\\wedge h^{\\lambda \\dot{\\gamma }}\\,,\\qquad \\overline{H}^{\\dot{\\nu }\\dot{\\lambda }}=H^{(\\dot{\\nu }\\dot{\\lambda })}:=h^{\\ \\dot{\\nu }}_{\\gamma }\\wedge h^{\\gamma \\dot{\\lambda }}\\,.$ The decomposition reads $\\Omega (y,\\overline{y}\\, |x) = \\underbrace{\\Omega ^\\mathrm {A}_{\\mu (n+2)|\\dot{\\mu }(m)}\\ H^{\\mu \\mu }y^{\\mu (n)}\\overline{y}^{\\dot{\\mu }(m)}}_{\\Phi _{\\mathrm {A}(n,m)}}+\\underbrace{\\overline{\\Omega }^\\mathrm {A}_{\\mu (n)|\\dot{\\mu }(m+2)}\\ \\overline{H}^{\\dot{\\mu }\\dot{\\mu }}y^{\\mu (n)}\\overline{y}^{\\dot{\\mu }(m)}}_{\\overline{\\Phi }_{\\mathrm {A}(n,m)}}+\\\\+\\underbrace{\\Omega ^\\mathrm {B}_{\\mu (n-2)|\\dot{\\mu }(m)}\\ H_{\\nu \\nu }y^\\nu y^\\nu y^{\\mu (n-2)}\\overline{y}^{\\dot{\\mu }(m)}}_{\\Phi _{\\mathrm {B}(n,m)}}+\\underbrace{\\overline{\\Omega }^\\mathrm {B}_{\\mu (n)|\\dot{\\mu }(m-2)}\\overline{H}_{\\dot{\\nu }\\dot{\\nu }} y^{\\mu (n)}\\overline{y}^{\\dot{\\nu }}\\overline{y}^{\\dot{\\nu }}\\overline{y}^{\\dot{\\mu }(m-2)}}_{\\overline{\\Phi }_{\\mathrm {B}(n,m)}}+\\\\+\\underbrace{\\Omega ^\\mathrm {C}_{\\mu (n)|\\dot{\\mu }(m)}\\ H_{\\nu }^{\\ \\mu }y^{\\nu }y^{\\mu (n-1)}\\overline{y}^{\\dot{\\mu }(m)}}_{\\Phi _{\\mathrm {C}(n,m)}}+\\underbrace{\\overline{\\Omega }^\\mathrm {C}_{\\mu (n)|\\dot{\\mu }(m)}\\ \\overline{H}_{\\dot{\\nu }}^{\\ \\dot{\\mu }}y^{\\mu (n)}\\overline{y}^{\\dot{\\nu }}\\overline{y}^{\\dot{\\mu }(m-1)}}_{{\\overline{\\Phi }_{\\mathrm {C}(n,m)}}}.$ Consider now the reducible two-forms $\\Phi ^{\\lambda (2),\\nu (2),\\mu (n-2)|\\dot{\\mu }(m)} = H^{\\lambda \\lambda }y^{\\nu }y^{\\nu }y^{\\mu (n-2)}\\overline{y}^{\\dot{\\mu }(m)},\\\\\\overline{\\Phi }^{\\mu (n)|\\dot{\\lambda }(2),\\dot{\\nu }(2),\\dot{\\mu }(m-2)} = \\overline{H}^{\\dot{\\lambda }\\dot{\\lambda }}y^{\\mu (n)}\\overline{y}^{\\dot{\\nu }}\\overline{y}^{\\dot{\\nu }}\\overline{y}^{\\dot{\\mu }(m-2)}.",
"$ In these terms, the projectors onto irreducible components are $\\mathcal {P}_\\mathrm {A}&={S}_{(\\lambda ,\\nu ,\\mu )}\\,, && \\mathcal {P}_\\mathrm {B} = \\epsilon _{\\lambda \\nu }\\epsilon _{\\lambda \\nu }\\,, && \\mathcal {P}_\\mathrm {C} = {S}_{(\\lambda ,\\nu ,\\mu )}\\circ \\epsilon _{\\lambda \\nu }\\,,\\\\\\overline{\\mathcal {P}}_\\mathrm {A}&={S}_{(\\dot{\\lambda },\\dot{\\nu },\\dot{\\mu })}\\,, && \\overline{\\mathcal {P}}_\\mathrm {B}=\\epsilon _{\\dot{\\lambda }\\dot{\\nu }}\\epsilon _{\\dot{\\lambda }\\dot{\\nu }}\\,, && \\overline{\\mathcal {P}}_\\mathrm {C}={S}_{(\\dot{\\lambda },\\dot{\\nu },\\dot{\\mu })}\\circ \\epsilon _{\\dot{\\lambda }\\dot{\\nu }}\\,$ and the decomposition (REF ) reads as $\\Phi _\\mathrm {A}&=\\mathcal {P}_\\mathrm {A}\\Phi \\,, && \\Phi _\\mathrm {B}=\\mathcal {P}_\\mathrm {B}\\Phi \\,, && \\Phi _\\mathrm {C}=\\mathcal {P}_\\mathrm {C}\\Phi \\,, \\\\ \\overline{\\Phi }_\\mathrm {A}&=\\overline{\\mathcal {P}}_\\mathrm {A}\\overline{\\Phi }\\,, && \\overline{\\Phi }_\\mathrm {B}=\\overline{\\mathcal {P}}_\\mathrm {B}\\overline{\\Phi }\\,, && \\overline{\\Phi }_\\mathrm {C}=\\overline{\\mathcal {P}}_\\mathrm {C}\\overline{\\Phi }\\,.$ For practical calculations we have to find the result of the action of the operator $D=\\frac{\\partial }{\\partial h}$ on the two-form $H$ .",
"The result is $D_{\\alpha \\dot{\\beta }}H^{\\nu \\lambda } = D_{\\alpha \\dot{\\beta }}\\left(h^\\nu _{\\ \\dot{\\gamma }}\\wedge h^{\\lambda \\dot{\\gamma }}\\right) = \\epsilon _\\alpha ^{\\ \\nu }\\epsilon _{\\dot{\\beta }\\dot{\\gamma }} h^{\\lambda \\dot{\\gamma }} - h^{\\nu }_{\\ \\dot{\\gamma }}\\,\\epsilon _\\alpha ^{\\ \\lambda }\\epsilon _{\\dot{\\beta }}^{\\ \\dot{\\gamma }} = - \\epsilon _\\alpha ^{\\ \\nu } h^\\lambda _{\\ \\dot{\\beta }} - \\epsilon _\\alpha ^{\\ \\lambda }h^\\nu _{\\ \\dot{\\beta }} = -2\\epsilon _\\alpha ^{\\ (\\nu } h^{\\lambda )}_{\\ \\dot{\\beta }}\\,,$ or, in the condensed notation for symmetrized indices, $D_{\\alpha \\dot{\\beta }}H^{\\nu \\nu }=-2\\epsilon _\\alpha ^{\\ \\nu }h^{\\nu }_{\\ \\dot{\\beta }}.$"
],
[
"$H^2(\\sigma _-)$ in the far-from-diagonal sector {{formula:76b17993-ba2a-4ad4-89f9-a21723194d83}}",
"Compute $\\Delta _{N>\\overline{N}+2}$ on the general two-forms $\\Phi $ and $\\overline{\\Phi }$ , $\\Delta _{N>\\overline{N}+2}(\\Phi )=\\underbrace{n(m+2)\\Phi (y,\\overline{y})}_{T_1(y,\\overline{y})} + \\underbrace{y^\\beta _\\alpha h^\\alpha _{\\ \\dot{\\gamma }} D_\\beta ^{\\ \\dot{\\gamma }}\\Phi (y,\\overline{y})}_{T_2(y,\\overline{y})} + \\underbrace{\\overline{y}^{\\dot{\\alpha }}\\overline{}_{\\dot{\\beta }}h_{\\gamma \\dot{\\alpha }}D^{\\gamma \\dot{\\beta }}\\Phi (y,\\overline{y})}_{T_3(y,\\overline{y})}\\,.$ As in (REF ), denote $T_2^{\\lambda (2),\\nu (2),\\mu (n-2)|\\dot{\\mu }(m)} = y^\\beta _\\alpha h^\\alpha _{\\ \\dot{\\gamma }} D_\\beta ^{\\ \\dot{\\gamma }} H^{\\lambda \\lambda }y^{\\nu }y^{\\nu }y^{\\mu (n-2)}\\overline{y}^{\\dot{\\mu }(m)}$ and similarly for $T_1$ and $T_3$ .",
"Straightforward computation yields $T_1^{\\lambda (2),\\nu (2),\\mu (n-2)|\\dot{\\mu }(m)} &= n(m+2) \\, H^{\\lambda \\lambda }y^{\\nu }y^{\\nu }y^{\\mu (n-2)}\\overline{y}^{\\dot{\\mu }(m)}\\,,\\\\T_2^{\\lambda (2),\\nu (2),\\mu (n-2)|\\dot{\\mu }(m)} &=-4 H^{\\nu \\lambda } y^{\\lambda }y^{\\nu }y^{\\mu (n-2)}\\overline{y}^{\\dot{\\mu }(m)} - 2(n-2)\\,H^{\\mu \\lambda }y^{\\lambda }y^{\\nu (2)}y^{\\mu (n-3)}\\overline{y}^{\\dot{\\mu }(m)}\\,,\\\\T_3^{\\lambda (2),\\nu (2),\\mu (n-2)|\\dot{\\mu }(m)} &=m\\, H^{\\lambda \\lambda }y^{\\nu (2)} y^{\\mu (n-2)}\\overline{y}^{\\dot{\\mu }(m)}.$ Projecting onto the irreducible part $\\Phi _\\mathrm {A}$ we obtain $\\Delta _{N>\\overline{N}+2}\\Phi _\\mathrm {A}=\\mathcal {P}_\\mathrm {A}\\left(T_1+T_2+T_3\\right)=\\left[n(m+2)-2n+m\\right]\\Phi _\\mathrm {A}=m(n+1)\\Phi _\\mathrm {A}.$ We see that $\\Phi _\\mathrm {A}\\in \\ker (\\Delta )$ whenever $m=0$ .",
"This gives a 2-cocycle of the form $H^{\\mu \\mu }y^{\\mu (n)}$ .",
"It can be represented in terms of the generating function as follows.",
"Contract all the indices in $H^{\\mu \\mu }y^{\\mu (n)}$ with some symmetric coefficients $\\Omega ^\\mathrm {A}_{\\mu \\mu \\mu (n)}$ to obtain $\\Omega ^\\mathrm {A}_{\\mu (n+2)}H^{\\mu \\mu }y^{\\mu (n)}\\equiv h^{(\\lambda }_{\\ \\dot{\\gamma }}\\wedge h^{\\nu \\dot{\\gamma })}\\Omega _{(\\lambda \\nu \\mu (n))}y^{\\mu (n)}\\Rightarrow h^{\\lambda }_{\\ \\dot{\\gamma }}\\wedge h^{\\nu \\dot{\\gamma }}\\,_\\lambda _\\nu C(y,0\\, |x) \\in \\ker \\left(\\Delta _{N>\\overline{N}+2}\\Big |_\\text{2-forms}\\right),$ where $C(y,0\\, |x)=\\Omega _{\\mu \\mu \\mu (n)} y^\\mu y^{\\mu } y^{\\mu (n)}$ .",
"Summarizing, we found a part of the kernel of $\\Delta $ represented by the two-forms $W(y,0\\,|x)=H^{\\mu \\nu }_\\mu _\\nu C(y,0\\,|x)$ with $C(y,0|x)$ being a general polynomial of $y$ 's of degree $\\ge 4$ .",
"Let us now project (REF ) onto the second irreducible part $\\Phi _\\mathrm {B}$ , $\\Delta _{N>\\overline{N}+2}\\Phi _\\mathrm {B}=\\mathcal {P}_\\mathrm {B}\\left(T_1+T_2+T_3\\right)=\\left[n(m+2)+0+m\\right]\\Phi _\\mathrm {B}\\ne 0\\quad \\forall n,m\\in \\mathbb {N}_0.$ Since $\\Phi _{B(n,m)}$ is proportional to $H_{\\nu \\nu }y^\\nu y^\\nu y^{\\mu (n-2)}\\overline{y}^{\\dot{\\mu }(m)}$ , the case $n=m=0$ is beyond the allowed region.",
"Thus, $\\Phi _\\mathrm {B}$ does not contribute to $H^2(\\sigma _-)$ .",
"Projecting (REF ) onto $\\Phi _\\mathrm {C}$ , we find $\\Delta _{N>\\overline{N}+2}\\Phi _\\mathrm {C}=\\mathcal {P}_\\mathrm {C}\\left(T_1+T_2+T_3\\right) = [n(m+2)-2-(n-2)+m]\\Phi _\\mathrm {C} = (nm+n+m)\\Phi _\\mathrm {C}\\,.$ Again, $\\Phi _\\mathrm {C}$ does not contribute to $H^2(\\sigma _-)$ since $m>0$ , $n\\ge 0$ .",
"Next, we consider the anti-holomorphic two-form $\\overline{\\Phi }$ .",
"The action of the Laplacian yields $\\Delta _{N>\\overline{N}+2}(\\overline{\\Phi })=\\underbrace{n(m+2)\\overline{\\Phi }(y,\\overline{y})}_{T_1(y,\\overline{y})} + \\underbrace{y^\\beta _\\alpha h^\\alpha _{\\ \\dot{\\gamma }} D_\\beta ^{\\ \\dot{\\gamma }}\\overline{\\Phi }(y,\\overline{y})}_{T_2(y,\\overline{y})} + \\underbrace{\\overline{y}^{\\dot{\\alpha }}\\overline{}_{\\dot{\\beta }}h_{\\gamma \\dot{\\alpha }}D^{\\gamma \\dot{\\beta }}\\overline{\\Phi }(y,\\overline{y})}_{T_3(y,\\overline{y})}\\,.$ As in (REF ) we set $T_2^{\\mu (n)|\\dot{\\lambda }(2),\\dot{\\nu }(2),\\dot{\\mu }(m-2)} = y^\\beta _\\alpha h^\\alpha _{\\ \\dot{\\gamma }} D_\\beta ^{\\ \\dot{\\gamma }}\\,\\overline{H}^{\\dot{\\lambda }\\dot{\\lambda }}y^{\\mu (n)}\\overline{y}^{\\dot{\\nu }}\\overline{y}^{\\dot{\\nu }}\\overline{y}^{\\dot{\\mu }(m-2)}$ and analogously for $T_1$ and $T_3$ .",
"The computation in components yields $T_1^{\\mu (n)|\\dot{\\lambda }(2),\\dot{\\nu }(2),\\dot{\\mu }(m-2)} &= n(m+2) \\overline{H}^{\\dot{\\lambda }\\dot{\\lambda }}y^{\\mu (n)}\\overline{y}^{\\dot{\\nu }}\\overline{y}^{\\dot{\\nu }}\\overline{y}^{\\dot{\\mu }(m-2)},\\\\T_2^{\\mu (n)|\\dot{\\lambda }(2),\\dot{\\nu }(2),\\dot{\\mu }(m-2)} &=-n\\,\\overline{H}^{\\dot{\\lambda }\\dot{\\lambda }} y^{\\mu (n)}\\overline{y}^{\\dot{\\nu }(2)}\\overline{y}^{\\dot{\\mu }(m-2)},\\\\T_3^{\\mu (n)|\\dot{\\lambda }(2),\\dot{\\nu }(2),\\dot{\\mu }(m-2)} &=-4\\,\\epsilon ^{\\dot{\\nu }\\dot{\\lambda }}\\overline{H}_{\\dot{\\alpha }}^{\\ \\dot{\\lambda }} y^{\\mu (n)}\\overline{y}^{\\dot{\\alpha }}\\overline{y}^{\\dot{\\nu }}\\overline{y}^{\\dot{\\mu }(m-2)} - 2(m-2)\\,\\epsilon ^{\\dot{\\mu }\\dot{\\lambda }}\\overline{H}_{\\dot{\\alpha }}^{\\ \\dot{\\lambda }}y^{\\mu (n)}\\overline{y}^{\\dot{\\alpha }}\\overline{y}^{\\dot{\\nu }(2)}\\overline{y}^{\\dot{\\mu }(m-3)}.$ Projecting onto the irreducible components we obtain $\\Delta _{N>\\overline{N}+2}\\overline{\\Phi }_\\mathrm {A} &= \\overline{\\mathcal {P}}_\\mathrm {A}\\left(T_1+T_2+T_3\\right)=[n(m+1)]\\overline{\\Phi }_\\mathrm {A}\\,,\\\\\\Delta _{N>\\overline{N}+2}\\overline{\\Phi }_\\mathrm {B} &= \\overline{\\mathcal {P}}_\\mathrm {B}\\left(T_1+T_2+T_3\\right)=(nm+n+4m)\\overline{\\Phi }_\\mathrm {B}\\,,\\\\\\Delta _{N>\\overline{N}+2}\\overline{\\Phi }_\\mathrm {C} &= \\overline{\\mathcal {P}}_\\mathrm {C}\\left(T_1+T_2+T_3\\right) =(nm+n-m)\\overline{\\Phi }_\\mathrm {C}\\,.$ The condition $n>m+2$ valid in the far-from-diagonal sector does not allow $\\overline{\\Phi }_\\mathrm {A,B,C}$ to be in the kernel of $ \\Delta $ .",
"The analysis of the opposite sector $N<\\overline{N}-2$ is analogous via swapping dotted and undotted indices.",
"As a result, the final answer for the under-diagonal sector is $W(0,\\overline{y}\\, |x)=\\overline{H}^{\\dot{\\mu }\\dot{\\nu }}\\overline{}_{\\dot{\\mu }}\\overline{}_{\\dot{\\nu }}C(0,\\overline{y}\\, |x).$ This completes the analysis of $H^2(\\sigma _-)$ in the sector $|N-\\overline{N}|>2$ .",
"The cohomology is represented by the two-forms $W(y,\\overline{y}\\, |x) = h^{\\mu }_{\\ \\dot{\\gamma }}\\wedge h^{\\nu \\dot{\\gamma }}\\,_\\mu _\\nu C(y,0\\, |x) + h_\\gamma ^{\\ \\dot{\\mu }}\\wedge h^{\\gamma \\dot{\\nu }}\\,\\overline{}_{\\dot{\\mu }}\\overline{}_{\\dot{\\nu }} C(0,\\overline{y}\\, |x).$ These two-forms are known to represent the so-called Weyl cocycle in the HS theory.",
"It is thus shown that there are no other non-trivial 2-cocycles in this sector."
],
[
"$H^2(\\sigma _-)$ on the diagonal {{formula:bf16bd86-cdaa-48f9-9718-beef40a13be5}}",
"Now we prove that there are no non-trivial cocycles at $N=\\overline{N}$ except for the Weyl cohomology (REF ).",
"As before, act by the operator $\\Delta _{N=\\overline{N}}$ on the two-form $\\Phi ^{\\lambda (2)|\\nu (2)|\\mu (n-2)|\\dot{\\mu }(n)}(y,\\overline{y})=H^{\\lambda \\lambda }y^{\\nu }y^{\\nu }y^{\\mu (n-2)}\\overline{y}^{\\dot{\\mu }(n)}$ $\\Delta _{N=\\overline{N}}\\Phi (y,\\overline{y})=\\underbrace{\\overline{y}^{\\dot{\\alpha }}\\overline{}_{\\dot{\\beta }} h_{\\gamma \\dot{\\alpha }} D^{\\gamma \\dot{\\beta }} \\Phi (y,\\overline{y})}_{T_1(y,\\overline{y})}+\\underbrace{y^\\alpha _\\beta h_{\\alpha \\dot{\\gamma }}D^{\\beta \\dot{\\gamma }}\\Phi (y,\\overline{y})}_{T_2(y,\\overline{y})}-\\\\\\underbrace{-\\overline{y}^{\\dot{\\alpha }} y^\\beta _\\alpha \\overline{}_{\\dot{\\beta }}h^\\alpha _{\\ \\dot{\\alpha }} D_\\beta ^{\\ \\dot{\\beta }}\\Phi (y,\\overline{y})}_{T_3(y,\\overline{y})}\\underbrace{-y^\\alpha \\overline{y}^{\\dot{\\beta }}_\\beta \\overline{}_{\\dot{\\alpha }}h_\\alpha ^{\\ \\dot{\\alpha }}D^\\beta _{\\ \\dot{\\beta }}\\Phi (y,\\overline{y})}_{T_4(y,\\overline{y})}\\,.$ Denoting $T_1^{\\lambda (2),\\nu (2),\\mu (n-2)|\\dot{\\mu }(n)} = \\overline{y}^{\\dot{\\alpha }}\\overline{}_{\\dot{\\beta }} h_{\\gamma \\dot{\\alpha }} D^{\\gamma \\dot{\\beta }}\\, H^{\\lambda \\lambda }y^{\\nu (2)}y^{\\mu (n-2)}\\overline{y}^{\\dot{\\mu }(n)}$ and analogously for $T_2$ , $T_3$ and $T_4$ , straightforward computation yields $T_1^{\\lambda (2),\\nu (2),\\mu (n-2)|\\dot{\\mu }(n)} & = & n\\, H^{\\lambda \\lambda }y^{\\nu (2)}y^{\\mu (n-2)}\\overline{y}^{\\dot{\\mu }(n)}\\,,\\\\T_2^{\\lambda (2),\\nu (2),\\mu (n-2)|\\dot{\\mu }(n)} & = & -4 \\epsilon ^{\\nu \\lambda } H_{\\alpha }^{\\ \\lambda } y^{\\alpha }y^{\\nu }y^{\\mu (n-2)}\\overline{y}^{\\dot{\\mu }(n)} -{}\\nonumber \\\\&& - 2(n-2)\\epsilon ^{\\mu \\lambda } H_{\\alpha }^{\\ \\lambda }y^{\\alpha }y^{\\nu (2)}y^{\\mu (n-3)}\\overline{y}^{\\dot{\\mu }(n)}\\,,\\\\T_3^{\\lambda (2),\\nu (2),\\mu (n-2)|\\dot{\\mu }(n)} & = & 4n\\, h^{\\nu }_{\\ \\dot{\\alpha }}\\wedge h^{\\lambda \\dot{\\mu }}\\, y^{\\lambda } y^{\\nu } y^{\\mu (n-2)}\\overline{y}^{\\dot{\\alpha }}\\overline{y}^{\\dot{\\mu }(n-2)} +{}\\nonumber \\\\& &+ 2n(n-2)\\, h^{\\mu }_{\\ \\dot{\\alpha }}\\wedge h^{\\lambda \\dot{\\mu }} y^{\\lambda }y^{\\nu (2)}y^{\\mu (n-3)}\\overline{y}^{\\dot{\\alpha }}\\overline{y}^{\\dot{\\mu }(n-1)}\\,,\\\\T_4^{\\lambda (2),\\nu (2),\\mu (n-2)|\\dot{\\mu }(n)} & = & 4n\\, \\epsilon ^{\\nu \\lambda }\\, h_{\\alpha }^{\\ \\dot{\\mu }}\\wedge h^{\\lambda }_{\\ \\dot{\\beta }}\\,y^{\\alpha }y^{\\nu }y^{\\mu (n-2)}\\overline{y}^{\\dot{\\beta }}\\overline{y}^{\\dot{\\mu }(n-1)} +{}\\nonumber \\\\& & + 2n(n-2)\\, \\epsilon ^{\\mu \\lambda }\\,h_{\\alpha }^{\\ \\dot{\\mu }}\\wedge h^{\\lambda }_{\\ \\dot{\\beta }}\\,y^{\\alpha }y^{\\nu (2)}y^{\\mu (n-3)}\\overline{y}^{\\dot{\\beta }}\\overline{y}^{\\dot{\\mu }(n-1)}.$ and $\\Delta _{N=\\overline{N}}\\Phi _\\mathrm {A}&= \\mathcal {P}_\\mathrm {A}(T_1+T_2+T_3+T_4) = n^2\\Phi _\\mathrm {A},\\\\\\Delta _{N=\\overline{N}}\\Phi _\\mathrm {B}&= \\mathcal {P}_\\mathrm {B}(T_1+T_2+T_3+T_4) =(2n^2+5n+4)\\Phi _\\mathrm {B}\\ne 0,\\\\\\Delta _{N=\\overline{N}}\\Phi _\\mathrm {C}&= \\mathcal {P}_\\mathrm {C}(T_1+T_2+T_3+T_4) = 2(n^2+n+1)\\Phi _\\mathrm {C} -2n(n+1)\\overline{\\Phi }_\\mathrm {C}.$ We observe that the only way for some of $\\Phi _\\text{A,B,C}$ to be in $H^2(\\sigma _-)$ is at $n=0$ .",
"But in the diagonal sector with $N=\\overline{N} = n$ this implies $N=\\overline{N}=0$ .",
"This case extends formula (REF ) to the spin-one $y,\\overline{y}$ -independent sector.",
"The analysis of the anti-holomorphic part $\\overline{\\Phi }$ is analogous.",
"The resulting cohomology parameterizes the spin-one field strength, i.e., Faraday field strength."
],
[
"$H^2(\\sigma _-)$ in the near-diagonal sector {{formula:f4e6b1a7-8a76-4307-9345-3c00844fb11a}}",
"In the near-diagonal sector a subtlety considered in Section 7.2.3 takes place.",
"We should search for a kernel of $\\Delta $ in the form of a linear combination of the two-forms lying under the diagonal and above the diagonal.",
"Our strategy is to act separately on the general holomorphic (REF ) and anti-holomorphic () two-forms placed below the diagonal $N=\\overline{N}-2$ and then determine which two-forms are in $\\ker (\\Delta )$ .",
"(The computation with $N>\\overline{N}$ only differs by the complex conjugation.)",
"We start with the general holomorphic two-form below the diagonal $\\Phi _{(n-1,n+1)}^{\\lambda (2)|\\nu (2)|\\mu (n-3)|\\dot{\\mu }(n+1)}(y,\\overline{y})=H^{\\lambda \\lambda }y^{\\nu (2)}y^{\\mu (n-3)}\\overline{y}^{\\dot{\\mu }(n+1)}.$ Firstly, we set $n\\ge 4$ considering the cases of $n\\le 3$ , that are special in our computation scheme, because $n-3$ is the number of indices $\\mu $ , later.",
"This yields $\\Delta _{N=\\overline{N}-2}\\Phi _{(n-1,n+1)}(y,\\overline{y}) = \\underbrace{\\Delta _{N<\\overline{N}- 2}\\Phi _{(n-1,n+1)}(y,\\overline{y})}_{T_1(y,\\overline{y})} +\\underbrace{y^\\alpha y^\\beta \\overline{}_{\\dot{\\alpha }}\\overline{}_{\\dot{\\beta }}h_\\beta ^{\\ \\dot{\\beta }}D_\\alpha ^{\\ \\dot{\\alpha }}\\Phi _{(n-1,n+1)}(y,\\overline{y})}_{T_2(y,\\overline{y})}\\,.$ The first term is computed the same way as in (REF ) giving $T_1^{\\lambda (2),\\nu (2),\\mu (n-3)|\\dot{\\mu }(n+1)}= (n+1)^2 \\, H^{\\lambda \\lambda }y^{\\nu }y^{\\nu }y^{\\mu (n-3)}\\overline{y}^{\\dot{\\mu }(n+1)} -4 H^{\\nu \\lambda } y^{\\lambda }y^{\\nu }y^{\\mu (n-3)}\\overline{y}^{\\dot{\\mu }(n+1)} -\\\\-2(n-3)\\,H^{\\mu \\lambda }y^{\\lambda }y^{\\nu (2)}y^{\\mu (n-4)}\\overline{y}^{\\dot{\\mu }(n+1)} +(n-1)\\, H^{\\lambda \\lambda }y^{\\nu (2)} y^{\\mu (n-3)}\\overline{y}^{\\dot{\\mu }(n+1)}.$ The computation of the additional term $T_2(y,\\overline{y})$ yields $T_2^{\\lambda (2),\\nu (2),\\mu (n-3)|\\dot{\\mu }(n+1)} = y^\\alpha y^\\beta \\overline{}_{\\dot{\\alpha }}\\overline{}_{\\dot{\\beta }}h_\\beta ^{\\ \\dot{\\beta }}D_\\alpha ^{\\ \\dot{\\alpha }}H^{\\lambda \\lambda }y^{\\nu (2)}y^{\\mu (n-3)}\\overline{y}^{\\dot{\\mu }(n+1)} =\\\\=-2n(n+1)\\, h_{\\beta }^{\\ \\dot{\\mu }}\\wedge h^{\\lambda \\dot{\\mu }} y^{\\lambda }y^{\\beta }y^{\\nu (2)}y^{\\mu (n-3)}\\overline{y}^{\\dot{\\mu }(n-1)}=\\\\=-n(n+1)\\, \\overline{H}^{\\dot{\\mu }\\dot{\\mu }}y^{\\lambda (2)}y^{\\nu (2)}y^{\\mu (n-3)}\\overline{y}^{\\dot{\\mu }(n-1)}.$ Projection onto the irreducible parts $A$ , $B$ and $C$ yields $\\Delta \\Phi _{\\mathrm {A}(n-1,n+1)} &= \\underbrace{n(n+1)}_{\\alpha (n)}\\Phi _{\\mathrm {A}(n-1,n+1)} \\underbrace{ - n(n+1)}_{\\beta (n)}\\overline{\\Phi }_{\\mathrm {A}(n+1,n-1)}\\,,\\\\\\Delta \\Phi _{\\mathrm {B}(n-1,n+1)} &= (n^2+5n-4)\\Phi _{\\mathrm {B}(n-1,n+1)}\\,,\\\\\\Delta \\Phi _{\\mathrm {C}(n-1,n+1)} &= (n^2+2n-1)\\Phi _{\\mathrm {C}(n-1,n+1)}\\,.$ Let us stress that the complex conjugation denoted by $\\dagger $ swaps dotted and undotted indices $(y^\\alpha )^\\dagger = \\overline{y}^{\\dot{\\alpha }}, \\quad \\quad (H^{\\alpha \\alpha })^\\dagger = \\overline{H}^{\\dot{\\alpha }\\dot{\\alpha }}$ and relates $\\Phi $ and $\\overline{\\Phi }$ in the following way: $(\\Phi _{\\mathrm {A,B,C}(n-1,n+1)})^\\dagger = \\overline{\\Phi }_{\\mathrm {A,B,C}(n+1,n-1)}.$ The computation for the complex-conjugated objects $\\overline{\\Phi }_{\\mathrm {A,B,C}(n+1,n-1)}$ is analogous giving $\\Delta \\overline{\\Phi }_{\\mathrm {A}(n+1,n-1)} &= n(n+1)\\,\\overline{\\Phi }_{\\mathrm {A}(n+1,n-1)} - n(n+1)\\,\\Phi _{\\mathrm {A}(n-1,n+1)}\\,,\\\\\\Delta \\overline{\\Phi }_{\\mathrm {B}(n+1,n-1)} &= (n^2+5n-4)\\overline{\\Phi }_{\\mathrm {B}(n+1,n-1)}\\,,\\\\\\Delta \\overline{\\Phi }_{\\mathrm {C}(n+1,n-1)} &= (n^2+2n-1)\\overline{\\Phi }_{\\mathrm {C}(n+1,n-1)}\\,.$ From (REF ) and (REF ) we observe that there is a non-trivial 2-cocycle $\\mathcal {E}_\\mathrm {A}=\\Phi _{\\mathrm {A}(n-1,n+1)}+\\overline{\\Phi }_{\\mathrm {A}(n+1,n-1)}= \\mathcal {E}^\\mathrm {A}_{\\mu (n+1),\\dot{\\mu }(n+1)}\\,\\Big (H^{\\mu \\mu }y^{\\mu (n-1)}\\overline{y}^{\\dot{\\mu }(n+1)} + \\overline{H}^{\\dot{\\mu }\\dot{\\mu }}y^{\\mu (n+1)}\\overline{y}^{\\dot{\\mu }(n-1)}\\Big )$ with arbitrary coefficients $\\mathcal {E}^\\mathrm {A}_{\\mu (n+1),\\dot{\\mu }(n+1)}(x)$ .",
"This answer agrees with the analysis of Section 7.2.3.",
"Indeed, the coefficients on the r.h.s.",
"of (REF ) coincide up to a sign $\\alpha (n) = -\\beta (n)$ , and by (REF ) of Section 7.2.3 this implies a non-trivial 2-cocycle (REF ).",
"This cocycle represents the traceless part of the free Fronsdal HS equations.",
"The irreducible representations of types $(B)$ and $(C)$ do not contribute to cohomology since they are not in $\\ker (\\Delta )$ (recall that we are assuming $n\\ge 4$ ).",
"Now consider the cases of $n=1,2,3$ .",
"Computing the action of the Laplace operator on the following objects: $\\Phi _{(0,2)}^{\\lambda \\lambda |\\dot{\\mu }\\dot{\\mu }}(y,\\overline{y}) &= H^{\\lambda \\lambda }\\overline{y}^{\\dot{\\mu }}\\overline{y}^{\\dot{\\mu }},\\\\\\Phi _{(1,3)}^{\\lambda \\lambda |\\mu |\\dot{\\mu }(3)}(y,\\overline{y}) &= H^{\\lambda \\lambda }y^\\mu \\overline{y}^{\\dot{\\mu }(3)},\\\\\\Phi _{(2,4)}^{\\lambda \\lambda |\\nu \\nu |\\dot{\\mu }(4)}(y,\\overline{y}) &= H^{\\lambda \\lambda }y^\\nu y^\\nu \\overline{y}^{\\dot{\\mu }(4)}\\,,$ it is not difficult to obtain $(\\Delta \\Phi _{(0,2)})^{\\lambda \\lambda |\\dot{\\mu }\\dot{\\mu }}(y,\\overline{y}) &= 2 H^{\\lambda \\lambda }\\overline{y}^{\\dot{\\mu }(2)} - 2\\overline{H}^{\\dot{\\mu }\\dot{\\mu }}y^{\\lambda (2)},\\\\(\\Delta \\Phi _{(1,3)})^{\\lambda \\lambda |\\mu |\\dot{\\mu }(3)}(y,\\overline{y}) &= 8H^{\\lambda \\lambda }y^\\mu \\overline{y}^{\\dot{\\mu }(3)} - 2H^{\\mu \\lambda }y^\\lambda \\overline{y}^{\\dot{\\mu }(3)} - 6\\overline{H}^{\\dot{\\mu }\\dot{\\mu }}y^{\\lambda (2)}y^\\mu \\overline{y}^{\\dot{\\mu }},\\\\(\\Delta \\Phi _{(2,4)})^{\\lambda \\lambda |\\nu \\nu |\\dot{\\mu }(4)}(y,\\overline{y}) &= 16H^{\\lambda \\lambda }y^{\\nu \\nu }\\overline{y}^{\\dot{\\mu }(4)} - 4H^{\\nu \\lambda }y^\\lambda y^\\nu \\overline{y}^{\\dot{\\mu }(4)} - 12\\overline{H}^{\\dot{\\mu }\\dot{\\mu }}y^{\\lambda (2)}y^{\\nu (2)}\\overline{y}^{\\dot{\\mu }(2)}.$ We see that these results for $n=1,2,3$ extend the traceless part of the Fronsdal cohomology (REF ) to spins $s=2,3,4$ .",
"It remains to analyze the case of anti-holomorphic two-form below the diagonal $N=\\overline{N}-2$ $\\overline{\\Phi }_{(n-1,n+1)}^{\\dot{\\nu }(2)|\\mu (n-1)|\\dot{\\lambda }(2)|\\dot{\\mu }(n-1)}(y,\\overline{y}) = \\overline{H}^{\\dot{\\lambda }\\dot{\\lambda }}y^{\\mu (n-1)}\\overline{y}^{\\dot{\\nu }(2)}\\overline{y}^{\\dot{\\mu }(n-1)}.$ Unlike Eq.",
"(REF ), the number of indices $\\mu $ and $\\dot{\\mu }$ in (REF ) is $n-1$ , not $n-3$ .",
"Hence, there is no need to consider separately the cases of $n\\ge 4$ and $n\\le 3$ .",
"Instead, we set $n\\ge 2$ and then analyze the $n=1$ case separately.",
"Let $n\\ge 2$ .",
"The action of the corresponding Laplace operator on (REF ) yields $\\Delta _{N=\\overline{N}-2}\\overline{\\Phi }_{(n-1,n+1)}(y,\\overline{y}) = \\underbrace{\\Delta _{N<\\overline{N}+2}\\overline{\\Phi }_{(n-1,n+1)}(y,\\overline{y})}_{T_3(y,\\overline{y})} +\\underbrace{y^\\alpha y^\\beta \\overline{}_{\\dot{\\alpha }}\\overline{}_{\\dot{\\beta }}h_\\beta ^{\\ \\dot{\\beta }}D_\\alpha ^{\\ \\dot{\\alpha }}\\overline{\\Phi }_{(n-1,n+1)}(y,\\overline{y})}_{T_4(y,\\overline{y})}.$ The computation is completely analogous to that for the holomorphic two-form.",
"After projecting onto the irreducible components it gives $\\Delta \\overline{\\Phi }_{\\mathrm {A}(n-1,n+1)} &= (n^2+n-4)\\overline{\\Phi }_{\\mathrm {A}(n-1,n+1)}\\,,\\\\\\Delta \\overline{\\Phi }_{\\mathrm {B}(n-1,n+1)} &= \\underbrace{(n^2+4n-1)}_{\\alpha (n)}\\overline{\\Phi }_{\\mathrm {B}(n-1,n+1)} \\underbrace{ - (n^2+4n-1)}_{\\beta (n)}\\Phi _{\\mathrm {B}(n+1,n-1)}\\,,\\\\\\Delta \\overline{\\Phi }_{\\mathrm {C}(n-1,n+1)} &= (n^2+2n+1)\\overline{\\Phi }_{\\mathrm {C}(n-1,n+1)}\\,.$ Applying once again the result (REF ) of Section 7.2.3 to (REF ), on the r.h.s.",
"of which the coefficients coincide up to a sign, $\\alpha (n) = -\\beta (n)$ , we obtain the 2-cocycle of the form $\\mathcal {E}_\\mathrm {B}=\\Phi _{\\mathrm {B}(n+1,n-1)}+\\overline{\\Phi }_{\\mathrm {B}(n-1,n+1)} =\\\\= \\mathcal {E}^\\mathrm {B}_{\\mu (n-1)\\dot{\\mu }(n-1)}\\,\\Big (H_{\\nu \\nu }y^{\\nu (2)} y^{\\mu (n-1)}\\overline{y}^{\\dot{\\mu }(n-1)} + \\overline{H}_{\\dot{\\nu }\\dot{\\nu }}y^{\\mu (n-1)}\\overline{y}^{\\dot{\\nu }(2)}\\overline{y}^{\\dot{\\mu }(n-1)}\\big )\\,,$ that represents the trace part of the Fronsdal equations.",
"Having considered $n\\ge 2$ , now consider the case of $n=1$ .",
"Computation of the action of the Laplace operator on the following two-form: $\\overline{\\Phi }_{(0,2)}^{\\dot{\\lambda }\\dot{\\lambda }|\\dot{\\mu }\\dot{\\mu }}(y,\\overline{y}) = \\overline{H}^{\\dot{\\lambda }\\dot{\\lambda }}\\overline{y}^{\\dot{\\mu }}\\overline{y}^{\\dot{\\mu }}$ yields $(\\Delta \\overline{\\Phi }_{(0,2)})^{\\dot{\\lambda }\\dot{\\lambda }|\\dot{\\mu }\\dot{\\mu }}(y,\\overline{y}) = 4 \\overline{H}^{\\dot{\\lambda }\\dot{\\lambda }}\\overline{y}^{\\dot{\\mu }(2)}-4\\overline{H}^{\\dot{\\mu }\\dot{\\lambda }}\\overline{y}^{\\dot{\\lambda }}\\overline{y}^{\\dot{\\mu }} - 2 H_{\\alpha \\alpha }y^{\\alpha }y^{\\alpha }\\epsilon ^{\\dot{\\mu }\\dot{\\lambda }}\\epsilon ^{\\dot{\\mu }\\dot{\\lambda }}.$ After projecting onto the irreducible components, we find that the case of $n=1$ extends the trace part of the Fronsdal cocycle $\\mathcal {E}_\\mathrm {B}$ (REF ) to spin $s=2$ .",
"In addition, (REF ) also contributes to the antiholomorphic part of the Weyl cocycle represented by the second term on the r.h.s.",
"of (REF ).",
"The holomorphic part of the latter lies in the opposite (complex-conjugated) region, in which the analysis is completely analogous.",
"This 2-cocycle represents the Weyl tensor for the linearized gravity ($s=2$ ) in $AdS_4$ .",
"This completes the analysis of $H^2(\\sigma _-)$ in the near-diagonal sector $N=\\overline{N} \\pm 2$ ."
],
[
"Summary for bosonic $H^{0,1,2}(\\sigma _-)$",
"Here we collect the final results for the cocycles associated with the bosonic HS gauge parameters, fields and field equations in $AdS_4$ .",
"Recall that $H^0(\\sigma _-)$ represents parameters of the differential HS gauge symmetries.",
"It is spanned by the zero-forms $F(y,\\overline{y}|\\,x) = F_{\\alpha (n)\\,\\dot{\\alpha }(n)}(x)\\,y^{\\alpha (n)}\\overline{y}^{\\dot{\\alpha }(n)}\\,,\\qquad n\\in \\mathbb {N}_0\\,.$ $H^1(\\sigma _-)$ represents the dynamical HS fields.",
"For the bosonic HS fields in $AdS_4$ it is spanned by the two 1-cocycles $\\phi (y,\\overline{y}\\,|x)$ and $\\phi ^\\text{tr}(y,\\overline{y}\\,|x)$ corresponding, respectively, to the traceless and trace components of the original Fronsdal field in the metric formalism: $\\phi (y,\\overline{y}\\,|x) = h^{\\mu \\dot{\\mu }}\\,\\partial _\\mu \\overline{\\partial }_{\\dot{\\mu }}\\, F_1(y,\\overline{y}\\,|x),\\\\\\phi ^\\text{tr}(y,\\overline{y}\\,|x) = h_{\\mu \\dot{\\mu }}\\,y^\\mu \\overline{y}^{\\dot{\\mu }}\\, F_2(y,\\overline{y}\\,|x),$ where $F_{1,2}(y,\\overline{y}\\,|x)$ are $(N,\\overline{N})$ -diagonal, that is $\\left(y^\\alpha \\frac{\\partial }{\\partial y^\\alpha } - \\overline{y}^{\\dot{\\alpha }}\\frac{\\partial }{\\partial \\overline{y}^{\\dot{\\alpha }}}\\right)F_{1,2}(y,\\overline{y}\\,|x) = 0.$ Finally, $H^2(\\sigma _-)$ , which represents gauge invariant differential operators on the bosonic HS fields, are spanned by three different 2-cocycles: the so-called Weyl cocycle $W(y,\\overline{y}\\,|x)$ and two irreducible components of the Fronsdal cocycle $\\mathcal {E}_\\mathrm {A}(y,\\overline{y}\\,|x)$ (REF ) and $\\mathcal {E}_\\mathrm {B}(y,\\overline{y}\\,|x)$ (REF ).",
"The latter correspond to the ${\\it l.h.s.}",
"$ 's of the dynamical equations for the fields of spin $s>1$ (spin $s\\le 1$ field equations are in the zero-form sector of unfolded equations [12]).",
"Note that these cocycles are real since they contain equal numbers of dotted and undotted indices.",
"$W(y,\\overline{y}\\, |x) &= H^{\\mu \\nu } _\\mu _\\nu C(y,0\\, |x) + \\overline{H}^{\\dot{\\mu }\\dot{\\nu }} \\overline{}_{\\dot{\\mu }}\\overline{}_{\\dot{\\nu }} C(0,\\overline{y}\\, |x)\\,,\\\\\\mathcal {E}_\\mathrm {A}(y,\\overline{y}\\,|x) &= \\Big (H^{\\mu \\nu }\\partial _\\mu \\partial _\\nu + \\overline{H}^{\\dot{\\mu }\\dot{\\nu }}\\overline{\\partial }_{\\dot{\\mu }}\\overline{\\partial }_{\\dot{\\nu }}\\Big ) C^\\text{diag}(y,\\overline{y}\\,|x)\\,,\\\\\\mathcal {E}_\\mathrm {B}(y,\\overline{y}\\,|x) &= \\Big (H^{\\mu \\nu } y_\\mu y_\\nu + \\overline{H}^{\\dot{\\mu }\\dot{\\nu }}\\overline{y}_{\\dot{\\mu }}\\overline{y}_{\\dot{\\nu }}\\Big ) C^\\text{diag}(y,\\overline{y}\\,|x),$ where $C^\\text{diag}(y,\\overline{y})$ obey (REF )."
],
[
"Fermionic HS fields in $AdS_4$",
"So far we considered the bosonic case with even grading $G=|N-\\overline{N}|$ .",
"By (REF ) odd $G$ corresponds to fields of half-integer spins, i.e., oddness of $G$ determines the field statistics.",
"To extend the results for $H^p(\\sigma _-)$ to fermionic fields we should first define the operator $\\sigma _-$ on multispinors of odd ranks.",
"In the fermionic case the lowest possible odd grading is $G=|N-\\overline{N}|=1$ .",
"In this sector we define the action of $\\sigma _-$ to vanish.",
"In all other gradings $\\sigma _\\pm $ is defined analogously to the bosonic case.",
"Namely, $\\sigma _-\\omega (y,\\overline{y}) &:= i\\, \\overline{y}^{\\dot{\\alpha }}h^\\alpha _{\\ \\dot{\\alpha }}\\partial _\\alpha \\ \\omega (y,\\overline{y}), \\quad \\quad \\text{at } N\\ge \\overline{N}+3\\,,\\\\\\sigma _-\\omega (y,\\overline{y}) &:= i\\, y^{\\alpha }h_\\alpha ^{\\ \\dot{\\alpha }}\\overline{\\partial }_{\\dot{\\alpha }}\\ \\omega (y,\\overline{y}), \\quad \\quad \\text{at } N\\le \\overline{N}-3\\,.$ Analogously, the operator $\\sigma _+$ is defined as $\\sigma _+\\omega (y,\\overline{y}) &:= -i\\, y^\\alpha D_\\alpha ^{\\ \\dot{\\alpha }}\\overline{\\partial }_{\\dot{\\alpha }}\\ \\omega (y,\\overline{y}), \\quad \\quad \\text{at } N\\ge \\overline{N}+3\\,,\\\\\\sigma _+\\omega (y,\\overline{y}) &:= -i\\, \\overline{y}^{\\dot{\\alpha }}D^\\alpha _{\\ \\dot{\\alpha }}\\partial _\\alpha \\ \\omega (y,\\overline{y}), \\quad \\quad \\text{at } N\\le \\overline{N}-3\\,.$ For the lowest grading $|N-\\overline{N}|=1$ , $\\sigma _+$ is defined by $\\sigma _+ = -i\\left(y^\\alpha D_\\alpha ^{\\ \\dot{\\alpha }}\\overline{\\partial }_{\\dot{\\alpha }}+\\overline{y}^{\\dot{\\alpha }}D^\\alpha _{\\ \\dot{\\alpha }}\\partial _\\alpha \\right), \\quad \\quad \\text{at } |N-\\overline{N}|=1.$ Notice that the action of the fermionic Laplace operator is analogous to that of the bosonic one () with the grading shifted by one, $\\Delta ^\\text{fermionic}_G=\\Delta ^\\text{bosonic}_{G-1}$ .",
"The final result is $\\bullet \\quad \\Delta ^\\text{fermionic}_{N>\\overline{N}+3}&= \\Delta ^\\text{bosonic}_{N>\\overline{N}+2} = N(\\overline{N}+2) + y^\\beta _\\alpha h^\\alpha _{\\ \\dot{\\gamma }} D_\\beta ^{\\ \\dot{\\gamma }} + \\overline{y}^{\\dot{\\alpha }}\\overline{}_{\\dot{\\beta }}h_{\\gamma \\dot{\\alpha }}D^{\\gamma \\dot{\\beta }}\\,,\\\\\\bullet \\quad \\Delta ^\\text{fermionic}_{N=\\overline{N}+3}&= \\Delta ^\\text{bosonic}_{N=\\overline{N}+2} = \\Delta _{N>\\overline{N}+2} + \\overline{y}^{\\dot{\\alpha }}\\overline{y}^{\\dot{\\beta }}_\\alpha _\\beta h^\\beta _{\\ \\dot{\\beta }}D^\\alpha _{\\ \\dot{\\alpha }}\\,,\\\\\\bullet \\quad \\Delta ^\\text{fermionic}_{N=\\overline{N}+1}&= \\Delta ^\\text{fermionic}_{N=\\overline{N}-1} = \\Delta ^\\text{bosonic}_{N=\\overline{N}} =\\\\ \\qquad \\qquad \\qquad \\qquad =y^\\alpha _\\beta h_{\\alpha \\dot{\\gamma }}D^{\\beta \\dot{\\gamma }}+ \\overline{y}^{\\dot{\\alpha }}\\overline{}_{\\dot{\\beta }} &h_{\\gamma \\dot{\\alpha }} D^{\\gamma \\dot{\\beta }}-\\overline{y}^{\\dot{\\alpha }} y^\\beta _\\alpha \\overline{}_{\\dot{\\beta }}h^\\alpha _{\\ \\dot{\\alpha }} D_\\beta ^{\\ \\dot{\\beta }}- y^\\alpha \\overline{y}^{\\dot{\\beta }}_\\beta \\overline{}_{\\dot{\\alpha }}h_\\alpha ^{\\ \\dot{\\alpha }}D^\\beta _{\\ \\dot{\\beta }}\\,.\\nonumber $ This allows us do deduce the fermionic cohomology from the bosonic one arriving at the following final results."
],
[
"Fermionic $H^0(\\sigma _-)$",
"The space $H^0(\\sigma _-)$ for fermionic HS fields is spanned by two independent zero-forms with $N-\\overline{N} = \\pm 1:$ $H^0(\\sigma _-) = \\Big \\lbrace F(y,\\overline{y}\\,|x) + \\overline{F}(y,\\overline{y}\\,|x) = F_{\\alpha (n+1),\\dot{\\alpha }(n)}(x)\\,y^{\\alpha (n+1)}\\overline{y}^{\\dot{\\alpha }(n)}+\\overline{F}_{\\alpha (n),\\dot{\\alpha }(n+1)}(x)\\,y^{\\alpha (n)}\\overline{y}^{\\dot{\\alpha }(n+1)}\\Big \\rbrace \\,.$ Recall that, by Theorem 3.1, $H^0(\\sigma _-)$ represents parameters of differential HS gauge transformations."
],
[
"Fermionic $H^1(\\sigma _-)$",
"In the bosonic case, we had two physically different cocycles in $H^1$ (REF ) corresponding to traceless $\\phi (y,\\overline{y}\\,|x)$ and trace $\\phi ^\\text{tr}(y,\\overline{y}\\,|x)$ parts of the Fronsdal field.",
"These belong to the diagonal $N=\\overline{N}$ .",
"For the fermionic case the situation is analogous.",
"The lowest grading is now $G = |N-\\overline{N}| = 1$ .",
"So, in this sector there are four (not two) different 1-cocycles: $\\psi $ , $\\psi ^\\text{tr}$ , $\\overline{\\psi }$ and $\\overline{\\psi }^\\text{tr}$ given by $\\psi (y,\\overline{y}\\,|x) &= \\psi _{\\mu (n+2),\\dot{\\mu }(n+1)}(x)\\,h^{\\mu \\dot{\\mu }}\\,y^{\\mu (n+1)}\\overline{y}^{\\dot{\\mu }(n)},\\\\\\overline{\\psi }(y,\\overline{y}\\,|x) &= \\overline{\\psi }_{\\mu (n+1),\\dot{\\mu }(n+2)}(x)\\,h^{\\mu \\dot{\\mu }}\\,y^{\\mu (n)}\\overline{y}^{\\dot{\\mu }(n+1)},\\\\\\psi ^\\text{tr}(y,\\overline{y}\\,|x) &= \\psi ^\\text{tr}_{\\mu (n),\\dot{\\mu }(n-1)}(x)\\,h_{\\nu \\dot{\\nu }}\\,y^\\nu y^{\\mu (n)}\\overline{y}^{\\dot{\\nu }}\\overline{y}^{\\dot{\\mu }(n-1)},\\\\\\overline{\\psi }^\\text{tr}(y,\\overline{y}\\,|x) &= \\overline{\\psi }^\\text{tr}_{\\mu (n-1),\\dot{\\mu }(n)}(x)\\,h_{\\nu \\dot{\\nu }}\\,y^\\nu y^{\\mu (n-1)}\\overline{y}^{\\dot{\\nu }}\\overline{y}^{\\dot{\\mu }(n)}$ with a non-negative integer $n$ (positive for $\\psi ^\\text{tr}$ and $\\overline{\\psi }^\\text{tr}$ ).",
"Cocycles $\\psi $ and $\\psi ^\\text{tr}$ belong to the upper near-diagonal line $N=\\overline{N}+1$ , whereas $\\overline{\\psi }$ and $\\overline{\\psi }^\\text{tr}$ belong to the lower near-diagonal line $N=\\overline{N} -1$ .",
"All four of them have grading $G=1$ .",
"$\\psi $ and $\\overline{\\psi }$ are mutually conjugated.",
"These results can be put into the following concise form $\\psi (y,\\overline{y}\\,|x) &= h^{\\mu \\dot{\\mu }}\\,\\partial _\\mu \\overline{\\partial }_{\\dot{\\mu }}\\,F_1(y,\\overline{y}\\,|x),\\\\\\psi ^\\text{tr}(y,\\overline{y}\\,|x) &= h_{\\mu \\dot{\\mu }}\\,y^\\mu \\overline{y}^{\\dot{\\mu }}\\,F_2(y,\\overline{y}\\,|x),$ where $F_{1,2}(y,\\overline{y}\\,|x)$ are of the homogeneity degree $N-\\overline{N}=1$ , i.e., $\\left(y^\\alpha \\frac{\\partial }{\\partial y^\\alpha } - \\overline{y}^{\\dot{\\alpha }}\\frac{\\partial }{\\partial \\overline{y}^{\\dot{\\alpha }}}\\right) F_{1,2}(y,\\overline{y}\\,|x) = F_{1,2}(y,\\overline{y}\\,|x)\\,.$ For $\\overline{\\psi }$ and $\\overline{\\psi }^\\text{tr}$ the results are analogous except that $\\overline{F}_{1,2}(y,\\overline{y}\\,|x)$ have degree $N -\\overline{N} = -1$ ."
],
[
"Fermionic $H^2(\\sigma _-)$",
"By the same arguments the fermionic $H^2$ is analogous to the bosonic one.",
"Recall that the bosonic 2-cocycles are represented by three different two-forms: Weyl tensor, traceless and traceful parts of the generalized Einstein tensors (near diagonal, $G=3$ ).",
"The fermionic Weyl cohomology is given by the same formula as the bosonic one: $W^\\text{ferm}(y,\\overline{y}\\,|x) = H^{\\mu \\nu }\\,\\partial _\\mu \\partial _\\nu C(y,0\\,|x) + \\overline{H}^{\\dot{\\mu }\\dot{\\nu }}\\,\\overline{\\partial }_{\\dot{\\mu }}\\overline{\\partial }_{\\dot{\\nu }}C(0,\\overline{y}\\,|x),$ where $C(y,0\\,|x)$ and $C(0,\\overline{y}\\,|x)$ are polynomials of $y$ and $\\overline{y}$ , respectively.",
"The two bosonic Fronsdal cocycles (REF ) were represented by the two zero-forms $C^\\text{diag}(y,\\overline{y})$ with the support on the diagonal $N=\\overline{N}$ .",
"In the fermionic case the two Fronsdal cocycles split into four.",
"The bosonic diagonal polynomial $C^\\text{diag}(y,\\overline{y})$ is replaced by a pair of near-diagonal $C^\\text{near-diag}(y,\\overline{y})$ and $\\overline{C}^\\text{near-diag}(y,\\overline{y})$ satisfying the relations $\\left(y^\\alpha \\frac{\\partial }{\\partial y^\\alpha } - \\overline{y}^{\\dot{\\alpha }}\\frac{\\partial }{\\partial \\overline{y}^{\\dot{\\alpha }}}\\right) C^\\text{near-diag}(y,\\overline{y}\\,|x) &= C^\\text{near-diag}(y,\\overline{y}\\,|x),\\\\\\left(y^\\alpha \\frac{\\partial }{\\partial y^\\alpha } - \\overline{y}^{\\dot{\\alpha }}\\frac{\\partial }{\\partial \\overline{y}^{\\dot{\\alpha }}}\\right) \\overline{C}^\\text{near-diag}(y,\\overline{y}\\,|x) &= -\\overline{C}^\\text{near-diag}(y,\\overline{y}\\,|x)\\,$ These support the fermionic 2-cocycles associated with the ${\\it l.h.s.}",
"$ 's of the fermionic field equations for spin $s\\ge 3/2$ massless fields as follows $\\mathcal {E}^\\text{ferm}_\\mathrm {A}(y,\\overline{y}\\,|x) &= \\Big (H^{\\mu \\nu }\\partial _\\mu \\partial _\\nu + \\overline{H}^{\\dot{\\mu }\\dot{\\nu }}\\overline{\\partial }_{\\dot{\\mu }}\\overline{\\partial }_{\\dot{\\nu }}\\Big ) C^\\text{near-diag}(y,\\overline{y}\\,|x),\\\\\\overline{\\mathcal {E}}^\\text{ferm}_\\mathrm {A}(y,\\overline{y}\\,|x) &= \\Big (H^{\\mu \\nu }\\partial _\\mu \\partial _\\nu + \\overline{H}^{\\dot{\\mu }\\dot{\\nu }}\\overline{\\partial }_{\\dot{\\mu }}\\overline{\\partial }_{\\dot{\\nu }}\\Big ) \\overline{C}^\\text{near-diag}(y,\\overline{y}\\,|x),\\\\\\mathcal {E}^\\text{ferm}_\\mathrm {B}(y,\\overline{y}\\,|x) &= \\Big (H^{\\mu \\nu } y_\\mu y_\\nu + \\overline{H}^{\\dot{\\mu }\\dot{\\nu }}\\overline{y}_{\\dot{\\mu }}\\overline{y}_{\\dot{\\nu }}\\Big ) C^\\text{near-diag}(y,\\overline{y}\\,|x),\\\\\\overline{\\mathcal {E}}^\\text{ferm}_\\mathrm {B}(y,\\overline{y}\\,|x) &= \\Big (H^{\\mu \\nu } y_\\mu y_\\nu + \\overline{H}^{\\dot{\\mu }\\dot{\\nu }}\\overline{y}_{\\dot{\\mu }}\\overline{y}_{\\dot{\\nu }}\\Big ) \\overline{C}^\\text{near-diag}(y,\\overline{y}\\,|x)\\,.$"
],
[
"Conclusion",
"In this paper, free unfolded equations for massless HS fields are analyzed in detail in terms of $\\sigma _-$ cohomology.",
"This is done both in flat space of arbitrary dimension in the tensor formalism for bosonic fields and in $AdS_4$ in the spinor formalism for both bosonic and fermionic fields.",
"Not surprisingly, the final results agree with those stated long ago in the original papers [12], [15].",
"Our aim is to present the detailed analysis of the $\\sigma _-$ cohomology providing an exhaustive proof of the so-called First On-Shell Theorem of the form of free unfolded HS equations, allowing the interested reader to check every step.",
"In the tensor case the full set of cohomology groups $H^p(\\sigma _-)$ was found both for the groups $\\mathrm {GL}(d)$ and $\\mathrm {O}(d)$ .",
"Our results for $\\mathrm {GL}(d)$ and $p<3$ coincide with those found in [16].",
"For the $\\mathrm {O}(d)$ case of traceless fields lower cohomology groups matched against those in [16], [17], [18].",
"In $AdS_4$ we used spinor formalism to analyze $H^{0,1,2}(\\sigma _-)$ for both bosonic and fermionic HS fields.",
"To the best of our knowledge such analysis was not available in the literature.",
"Practically, to compute $H^k(\\sigma _-)$ in both $\\mathrm {Mink}^d$ and $AdS_4$ cases we used the analogue of the Hodge theorem.",
"Namely, the problem of finding the cohomology $H^k(\\sigma _-) = \\mathrm {ker}(\\sigma ^{k}_-)/\\mathrm {im}(\\sigma ^{k-1}_-)$ was reduced to the calculation of the kernel of an appropriate positive-definite Laplace-Hodge operator $\\Delta $ invariant under the action of compact version of the space-time symmetry algebra.",
"This technique was shown to be lucid and efficient.",
"Having found the cohomology groups $H^k(\\sigma _-)$ for $k=0,1,2$ , in accordance with [13] we obtained the exhaustive information about the differential HS gauge parameters, dynamical HS gauge fields and their field equations.",
"Thus, we have explicitly proven the so-called First On-Shell Theorem for bosonic HS fields in $\\mathrm {Mink}^d$ (which case is straightforwardly extendable to $AdS_d$ ) and all massless fields in $AdS_4$ .",
"The technique used in this paper can be further applied to the calculation of $H^p(\\sigma _-)$ in the zero-form sector of HS fields studied in [13], [22] that describes dynamics of a scalar field and $s=1$ particle as well as to more general systems considered in [44], [45], [46].",
"One of the byproduct results of this paper is the interpretation of the matching between $\\sigma _-$ cohomology of the one-form sector against zero-form sector expressing the matching between Bianchi identities in the two sectors."
],
[
"Acknowledgement",
"We are grateful to Vyacheslav Didenko, Anatoly Korybut and Alexander Tarusov for helpful and stimulating discussions and Maxim Grigoriev for a useful comment.",
"We are particularly grateful to the referee for useful suggestions and the correspondence.",
"We acknowledge a partial support from the Russian Basic Research Foundation Grant No 20-02-00208.",
"Since most of the tensors encountered in the course of this paper are Young tensors in the symmetric basis, it is convenient to accept the following notation.",
"A tensor without a certain type of index symmetry will be denoted as $T^{a|b|c|..}$ , where the vertical line $|$ separates groups of indices not related by any symmetries to each other.",
"A tensor that has a symmetric set of $n$ indices, say, $(a_1,a_2,\\dots ,a_n)$ will be denoted $T^{a(n)|...}\\equiv T^{(a_1a_2\\dots a_n)|...}$ .",
"A tensor corresponding to a certain Young diagram in the symmetric basis then has the form: $T^{a(n), b(m), c(k),...}$ .",
"Symmetrization over $n$ indices is performed by the formula $\\mathrm {Sym} = \\frac{1}{n!",
"}\\sum _{\\text{all permutations}}$ .",
"We will denote all symmetrized tensor indices by the same letter.",
"For example, $T^{a(n)|a} \\equiv \\frac{1}{(n+1)!}",
"\\sum _{\\sigma \\in S_{n+1}} T^{\\sigma (a_1..a_n|a_{n+1})}\\,.$ In Section we omit $Y,Z,\\theta $ and assume that all indices are contracted with the corresponding variables.",
"The rules for raising and lowering $\\mathfrak {sl}(2)$ -indices are $A_\\alpha = A^\\beta \\epsilon _{\\beta \\alpha }, \\quad \\quad A^\\alpha = \\epsilon ^{\\alpha \\beta }A_\\beta , \\quad \\quad \\epsilon _{\\alpha \\beta }\\epsilon ^{\\gamma \\beta } = \\epsilon _\\alpha ^{\\ \\gamma } = \\delta _\\alpha ^{\\ \\gamma } = -\\epsilon ^\\gamma _{\\ \\alpha }$ with $\\epsilon _{\\alpha \\beta } = -\\epsilon _{\\beta \\alpha },\\quad \\quad \\quad \\epsilon _{12} = 1.$ 0.75em $\\alpha _1 = \\tfrac{(-3 + n) (-2 + n) (-1 + n) (-3 + d + 2 n)}{(-5 + d +m + n) (-4 + d + m + n) (-4 + d + 2 n) (d^2 + d (-8 + m + 3 n) -2 (-8 + m + m^2 + 7 n - 3 m n))},$ 0.75em $\\alpha _2 = \\tfrac{(-2 + n) (-1 + n)}{(-4 + d + m + n) (-4 + d + 2 n)},$ 0.75em $\\alpha _3 = -\\tfrac{2 (-2 + n) (-1 + n) (-3 + d + 2 n)}{(-4 + d + m +n) (-4 + d + 2 n) (d^2 + d (-8 + m + 3 n) -2 (-8 + m + m^2 + 7 n - 3 m n))},$ 0.75em $\\alpha _4 = \\tfrac{2(1 + m - n) (-2 + n) (-1 + n) (-3 + d + 2 n)}{(-5 +d + m + n) (-4 + d + m + n) (-4 + d + 2 n) (d^2 +d (-8 + m + 3 n) - 2 (-8 + m + m^2 + 7 n - 3 m n))},$ 0.75em $\\alpha _5 = \\tfrac{ (-1 + n) (-3 + d + m + n)}{(-4 + d + m + n) (-4 + d + 2 n)},$ 0.75em $\\alpha _6 = -\\tfrac{(-2 + d + 2 m) (-1 + n)}{(-4 + d + 2 n) (d^2 +d (-8 + m + 3 n) - 2 (-8 + m + m^2 + 7 n - 3 m n))},$ 0.75em $\\alpha _7 = \\tfrac{(-1 + n) (-3 + d + 2 n) (16 + d^2 - 7 m - 9 n +2 d (-4 + m + n) + (m + n)^2)}{(-5 + d + m + n) (-4 + d + m +n) (-4 + d + 2 n) (d^2 + d (-8 + m + 3 n) -2 (-8 + m + m^2 + 7 n - 3 m n))},$ 0.75em $\\alpha _8 = -\\tfrac{(-1 + n)}{(-4 + d + m + n)},$ 0.75em $\\alpha _9 = \\tfrac{2 (-1 + n) (-3 + d + 2 n)}{(-4 + d + m + n) (d^2 +d (-8 + m + 3 n) - 2 (-8 + m + m^2 + 7 n - 3 m n))},$ 0.75em $\\alpha _{10} = -\\tfrac{(-3 + d + 2 n)}{d^2 + d (-8 + m + 3 n) - 2 (-8 + m + m^2 + 7 n - 3 m n)},$ 0.75em $\\alpha _{11} = -\\tfrac{(-4 + d + 2 m) (-1 + n) (-3 + d + 2 n)}{(-5 + d + m + n) (-4 + d + m + n) (d^2 + d (-8 + m + 3 n) - 2 (-8 + m + m^2 + 7 n - 3 m n))},$ 0.75em $\\alpha _{12} = \\tfrac{(14 + d^2 - 6 n + 2 m (-5 + m + n) +d (-8 + 3 m + n))}{(-4 + d + m + n) (d^2 + d (-8 + m + 3 n) -2 (-8 + m + m^2 + 7 n - 3 m n))}\\,.$"
]
] | 2107.01736 |
[
[
"Linear-Time Model Checking Branching Processes"
],
[
"Abstract (Multi-type) branching processes are a natural and well-studied model for generating random infinite trees.",
"Branching processes feature both nondeterministic and probabilistic branching, generalizing both transition systems and Markov chains (but not generally Markov decision processes).",
"We study the complexity of model checking branching processes against linear-time omega-regular specifications: is it the case almost surely that every branch of a tree randomly generated by the branching process satisfies the omega-regular specification?",
"The main result is that for LTL specifications this problem is in PSPACE, subsuming classical results for transition systems and Markov chains, respectively.",
"The underlying general model-checking algorithm is based on the automata-theoretic approach, using unambiguous B\\\"uchi automata."
],
[
"Introduction",
"Checking whether a (labelled) transition system satisfies a linear-time specification is a staple in verification.",
"The specification is often given as a formula of linear temporal logic (LTL).",
"While early procedures for LTL model checking work directly with the formula [24], the automata-theoretic approach translates LTL formulas into finite automata on infinite words, such as Büchi automata, and analyzes a product of the system and the automaton [36].",
"This approach can lead to clean and modular model-checking algorithms.",
"Although LTL captures only a subset of $\\omega $ -regular languages, model-checking algorithms based on the automata-theoretic approach can be made optimal from the point of view of computational complexity.",
"In particular, model checking finite transition systems against LTL specifications is PSPACE-complete [31], and the algorithm [36] that, loosely speaking, translates (the negation of) the LTL formula into a Büchi automaton and checks the product with the transition system for emptiness can indeed be implemented in PSPACE.",
"The same approach does not directly work for probabilistic systems modelled as finite Markov chains: intuitively, the nondeterminism in a Büchi automaton causes issues in a stochastic setting where the specification should hold with probability 1, i.e., almost surely but not necessarily surely.",
"A possible remedy is to translate the nondeterministic Büchi automaton further into a deterministic automaton, e.g., a deterministic Rabin automaton (deterministic Büchi automata are less expressive), with which the Markov chain can be naturally instrumented and subsequently analyzed.",
"This determinization step causes a (second) exponential blowup and does not lead to algorithms that are optimal from a computational-complexity point of view.",
"However, for Markov decision processes (MDPs), which allow for nondeterminism in the probabilistic system, this approach is adequate and leads to an optimal, double-exponential time, model-checking algorithm.",
"Checking whether a Markov chain satisfies an LTL specification with probability 1 is PSPACE-complete, but membership in PSPACE was proved only in [10], [11], not using the automata-theoretic approach but by a recursive procedure on the formula.",
"This raised the question if there is also an optimal algorithm based on the automata-theoretic approach; see [35] for a survey of the state of the art at the end of the 90s.",
"The answer is yes and was first given in [12], using a single-exponential translation from LTL to separated Büchi automata.",
"Such automata are special unambiguous Büchi automata, which restrict nondeterministic Büchi automata by requiring that every word have at most one accepting run.",
"Another algorithm, using alternating Büchi automata, was proposed in [6], exploiting reverse determinism, a property also related to unambiguousness.",
"A polynomial-time (even NC) model-checking algorithm for Markov chains against general unambiguous Büchi automata was given in [2].",
"These works all imply optimal PSPACE algorithms for LTL model checking of Markov chains via the automata-theoretic approach.",
"In this paper we exhibit an LTL model checking algorithm that has the following features: (1) it applies to (multi-type) branching processes, a well established model for random trees, generalizing both nondeterministic transition systems and Markov chains; (2) it runs in PSPACE, which is the optimal complexity both for nondeterministic transition systems and Markov chains; and (3) it is based on the automata-theoretic approach (using unambiguous Büchi automata).",
"The fact that there exists an algorithm with the first two features might seem surprising, as one might think that any system model that encompasses both nondeterminism and probability will generalize MDPs, for which LTL model checking is 2EXPTIME-complete [11].",
"Branching processes (BPs) are a well-studied model in mathematics with applications in numerous fields including biology, physics and natural language processing; see, e.g., [23], [1], [22].",
"BPs randomly generate infinite trees, and, from a computer-science point of view, they might be the most natural model to do so: (multi-type) BPs can be thought of as a version of stochastic context-free grammars without terminal symbols, randomly generating infinite derivation trees.",
"For example, consider the following BP, taken from [8], with 3 types $I, B, D$ : $&I {0.9} I && B {0.2} D && D {1} D \\nonumber \\\\[-1mm]&I {0.1} I B && B {0.5} B \\\\[-1mm]& && B {0.3} B B \\nonumber $ This BP might generate a tree with the following prefix: [yscale=0.8] 0) at (0,0) $I$ ; 1) at (-2,-1) $I$ ; 2) at (2,-1) $B$ ; 11) at (-2,-2) $I$ ; 111) at (-3,-3) $I$ ; 112) at (-1,-3) $B$ ; 21) at (1,-2) $B$ ; 22) at (3,-2) $B$ ; 211) at (1,-3) $B$ ; 221) at (3,-3) $D$ ; (0)–(1); (0)–(2); (1)–(11); (11)–(111); (11)–(112); (2)–(21); (2)–(22); (21)–(211); (22)–(221); [fill] (-3,-3.4) circle (0.7pt); [fill] (-3,-3.6) circle (0.7pt); [fill] (-3,-3.8) circle (0.7pt); [fill] (-1,-3.4) circle (0.7pt); [fill] (-1,-3.6) circle (0.7pt); [fill] (-1,-3.8) circle (0.7pt); [fill] (1,-3.4) circle (0.7pt); [fill] (1,-3.6) circle (0.7pt); [fill] (1,-3.8) circle (0.7pt); [fill] (3,-3.4) circle (0.7pt); [fill] (3,-3.6) circle (0.7pt); [fill] (3,-3.8) circle (0.7pt); The probability that the BP generates a tree with the shown prefix is the product of the probabilities of the fired transition rules, i.e., (in breadth-first order) $0.1 \\cdot 0.9 \\cdot 0.3 \\cdot 0.1 \\cdot 0.5 \\cdot 0.2$ .",
"BPs generalize transition systems.",
"Consider the following transition system: MDPrand] (X) at (0,0) $X$ ; MDPrand] (Y) at (2,0) $Y$ ; [->] (-1,0) to (X); [->] (X) edge[bend left] (Y); [->] (Y) edge[bend left] (X); [->] (Y) edge[loop right,looseness=12] (Y); It is equivalent to the BP with $X {1} Y$ and $Y {1} X Y$ , which generates with probability 1 the following unique tree: [yscale=0.8] 0) at (0,0) $X$ ; 1) at (0,-1) $Y$ ; 2l) at (-2,-2) $X$ ; 2r) at ( 2,-2) $Y$ ; 3l) at (-2,-3) $Y$ ; 3rl) at (1,-3) $X$ ; 3rr) at (3,-3) $Y$ ; [fill] (-2,-3.4) circle (0.7pt); [fill] (-2,-3.6) circle (0.7pt); [fill] (-2,-3.8) circle (0.7pt); [fill] ( 1,-3.4) circle (0.7pt); [fill] ( 1,-3.6) circle (0.7pt); [fill] ( 1,-3.8) circle (0.7pt); [fill] ( 3,-3.4) circle (0.7pt); [fill] ( 3,-3.6) circle (0.7pt); [fill] ( 3,-3.8) circle (0.7pt); (0) – (1); (1) – (2l); (1) – (2r); (2l) – (3l); (2r) – (3rl); (2r) – (3rr); The branches of this unique tree are exactly the executions of the transition system.",
"As a consequence, any LTL formula holds on all executions of the transition system if and only if it holds (with probability 1) on all branches of the generated tree.",
"BPs also generalize Markov chains.",
"Consider the following Markov chain: MDPrand] (X) at (0,0) $X$ ; MDPrand] (Y) at (2,0) $Y$ ; [->] (-1,0) to (X); [->] (X) edge[bend left] node[above] 1 (Y); [->] (Y) edge[bend left] node[below] $0.3$ (X); [->] (Y) edge[loop right,looseness=12] node[right] $0.7$ (Y); It is equivalent to the BP with $X {1} Y$ and $Y {0.3} X$ and $Y {0.7} Y$ , which generates, with probabilities $0.3$ , $0.7 \\cdot 0.3$ , $0.7 \\cdot 0.7$ , respectively, the following prefixes of (degenerated) trees: [yscale=0.8] 00) at (0, 0) $X$ ; 10) at (2, 0) $X$ ; 20) at (4, 0) $X$ ; 01) at (0,-1) $Y$ ; 11) at (2,-1) $Y$ ; 21) at (4,-1) $Y$ ; 02) at (0,-2) $X$ ; 12) at (2,-2) $Y$ ; 22) at (4,-2) $Y$ ; 03) at (0,-3) $Y$ ; 13) at (2,-3) $X$ ; 23) at (4,-3) $Y$ ; [fill] (0,-3.4) circle (0.7pt); [fill] (0,-3.6) circle (0.7pt); [fill] (0,-3.8) circle (0.7pt); [fill] (2,-3.4) circle (0.7pt); [fill] (2,-3.6) circle (0.7pt); [fill] (2,-3.8) circle (0.7pt); [fill] (4,-3.4) circle (0.7pt); [fill] (4,-3.6) circle (0.7pt); [fill] (4,-3.8) circle(0.7pt); (00) – (01) – (02) – (03); (10) – (11) – (12) – (13); (20) – (21) – (22) – (23); Here, each possible “tree” has only a single branch, and the possible “trees” are distributed in the same way as the possible executions of the Markov chain.",
"As a consequence, any LTL formula holds with probability 1 on a random execution of the Markov chain if and only if it holds with probability 1 on the (single) branch of the generated tree.",
"Hence, both for the transition system and for the Markov chain, the respective model-checking question reduces to the BP model-checking problem which asks whether with probability 1 the property holds on all branches.",
"For LTL specifications, we refer to this BP model-checking problem as $\\mathbb {P}(\\textup {LTL})=1$ .",
"Our main result is that it is in PSPACE, generalizing the corresponding classical results on transition systems and Markov chains.",
"As mentioned, our model-checking algorithm is based on the automata-theoretic approach, in particular on unambiguous Büchi automata.",
"Another important technical ingredient is the algorithmic analysis of certain nonnegative matrices in terms of their spectral radius.",
"The latter points to the fact that the numbers in the system generally matter, even though we only consider the qualitative problem of comparing the satisfaction probability with 1.",
"For example, for the BP given in (REF ), one can show that the probability that all branches eventually hit a node of type $D$ is less than 1 (in fact, it is 0).",
"Intuitively, this is because the probability of “branching” via $B {0.3} B B$ is larger than the probability of “dying” via $B {0.2} D$ .",
"Were the probabilities $0.3$ and $0.2$ swapped, the probability that all branches eventually hit a node of type $D$ would be 1; cf. [8].",
"We also consider the problem $\\mathbb {P}(\\textup {LTL}= 0)$ , which asks whether the probability that all branches satisfy a given LTL formula is 0.",
"Even though it is trivial to negate an LTL formula, this problem is (unlike in Markov chains) not equivalent to the complement of $\\mathbb {P}(\\textup {LTL}= 1)$ , because even when the probability is less than 1 that the formula holds on all branches, the probability may still be 0 that the negated formula holds on all branches.",
"We will show that $\\mathbb {P}(\\textup {LTL}= 0)$ is much more computationally complex than $\\mathbb {P}(\\textup {LTL}=1)$ : it is 2EXPTIME-complete.",
"Besides LTL, we also consider automata-based specifications.",
"Büchi automata are relevant from a verification point of view, as a way of specifying desired or undesired executions of the system.",
"Unambiguous Büchi automata are useful from a technical point of view, in particular, to facilitate our main result on $\\mathbb {P}(\\textup {LTL}=1)$ .",
"See sec:prelims,tab:map for definitions of our problems and a map of our results.",
"Readers familiar with MDPs may wonder how the problem $\\mathbb {P}(\\textup {LTL})=1$ can have lower computational complexity than the problem whether all schedulers of an MDP satisfy an LTL specification almost surely.",
"Consider the BP $X &{1} Y_1 Y_2 & \\qquad Y_1 &{0.7} X & \\qquad Y_1 &{0.3} Z & \\qquad Y_2 &{0.5} X & \\qquad Y_2 &{0.5} Z& \\qquad Z &{1} Z\\,,$ which might be depicted graphically as follows: MDPrand] (X) at (0,0) $X$ ; MDPrand] (Y1) at (3,1) $Y_1$ ; MDPrand] (Y2) at (3,-1) $Y_2$ ; MDPrand] (Z) at (6,0) $Z$ ; [->] (-1,0) to (X); [->] (X) edge[bend left=20] (Y1); [->] (X) edge[bend right=20] (Y2); [->] (Y1) edge[bend left=0] node[pos=0.4,below] $0.7$ (X); [->] (Y2) edge[bend right=0] node[pos=0.4,above] $0.5$ (X); [->] (Y1) edge node[pos=0.4,below] $0.3$ (Z); [->] (Y2) edge node[pos=0.4,above] $0.5$ (Z); [->] (Z) edge[loop right,looseness=12] node[right] 1 (Z); One might view this BP as an MDP where in an $X$ -node the scheduler nondeterministically picks either the $Y_1$ - or the $Y_2$ -successor, and in an $Y_i$ -node, the $X$ - or the $Z$ -successor is chosen randomly.",
"In such an MDP, regardless of the scheduler, a random run reaches with probability 1 a $Z$ -node.",
"However, in the BP above, the probability is positive that some branch of a random tree never reaches a $Z$ -node.",
"Although each branch of a random tree could be thought of as being witnessed by at least one scheduler, this is not a contradiction, as there are uncountably many schedulers (over which one cannot take a sum).",
"Hence, if an MDP is interpreted as a BP in the way sketched above, then the requirement that the BP satisfy an LTL formula almost surely on all branches is stronger, and computationally less complex to check, than the requirement that the MDP satisfy, for each scheduler, the formula almost surely."
],
[
"Related work.",
"We have already discussed related work concerning model checking transition systems and Markov chains.",
"In addition to the mentioned applications of BPs in various fields, there has also been work on BPs in computer science, especially in the last 10 years.",
"This paper builds on [8], where specifications in terms of deterministic parity tree automata are considered.",
"The work [8] implies decidability of the problems considered in this paper and some basic upper complexity bounds.",
"For example, it is not hard to derive from [8] that $\\mathbb {P}(\\textup {LTL}=1)$ is in 2EXPTIME.",
"Lowering this to PSPACE is the main achievement of this paper.",
"A related strand of work considers regular tree languages; i.e., the specification is not in terms of a word automaton that is run on each branch but in terms of tree automata.",
"Even measurability is not easy to show in this case [20], and fundamental decidability questions around computing the measure have been answered positively only for subclasses of regular tree languages [25], [26].",
"Fundamental results on the complexity of algorithmically analyzing BPs have been obtained in [18].",
"Indeed, in sec:as-finite we build on and improve results from [18] on finiteness (more often called “extinction” in the literature) of BPs.",
"Another recent line of work considers extensions of BPs with nondeterminism, focusing on algorithmic questions about properties such as reachability.",
"Branching MDPs, which are BPs where a controller chooses actions to influence the evolution of the tree, have been investigated, e.g., in [16], [17].",
"Even branching games, featuring two adversarial controllers, have been studied recently [14].",
"The work [21] also considers BPs with “internal” nondeterminism (as opposed to the “external” nondeterminism manifested as branching in the generated tree), along with model-checking problems against the logic GPL.",
"This expressive, $\\mu $ -calculus based modal logic had been introduced in [9].",
"The system model therein, called reactive probabilistic labeled transition systems (RPLTSs), is essentially equivalent to BPs as considered in this paper.",
"BPs are related to models for probabilistic programs with recursion, such as Recursive Markov chains, for which model-checking problems have been studied in detail; see, in particular, [19].",
"Very loosely speaking, a run of a (“1-exit”) Recursive Markov chain can be viewed as a depth-first traversal of a tree generated by a BP.",
"Indeed, for a lower bound in the present paper (thm:NBA-0) we adapt a proof from [19].",
"However, most qualitative model-checking problems for Recursive Markov chains are EXPTIME-complete [19], and so many of the BP problems we study turn out to have different computational complexity.",
"As a key technical tool we use unambiguous Büchi automata, as recently proposed for Markov chains [2].",
"It is non-trivial to extend their use to random trees, as the branching behaviour of BPs interferes with the spectral-radius based analysis from [2].",
"One may view as the main technical insight of this paper that the limited nondeterminism in unambiguous automata can be combined with the tree branching of BPs, so that, in a sense, BP model checking reduces to comparing the spectral radius of a certain nonnegative matrix with 1 (prop:coUBA-1)."
],
[
"Preliminaries",
"Let $\\mathbb {N}$ and $\\mathbb {N}_0$ denote the set of positive and nonnegative integers, respectively.",
"For a finite set $\\Gamma $ , we write $\\Gamma ^*$ (resp., $\\Gamma ^+$ ) for the set of words (resp., nonempty words) over $\\Gamma $ ."
],
[
"Branching processes.",
"A (multi-type) branching process (BP) is a tuple $\\mathcal {B}= (\\Gamma , \\mathord {{}}, \\mathit {Prob}, X_0)$ , where $\\Gamma $ is a finite set of types, $\\mathord {{}} \\subseteq \\Gamma \\times \\Gamma ^+$ is a finite set of transition rules, $\\mathit {Prob}$ is a function assigning positive rational probabilities to transition rules so that for every $X\\in \\Gamma $ we have $\\sum _{X {} w} \\mathit {Prob}(X {} w)=1$ , and $X_0\\in \\Gamma $ is the start type.",
"We write $X {p} w$ to denote that $\\mathit {Prob}(X {} w) = p$ .",
"Given a BP $\\mathcal {B}$ and a type $X \\in \\Gamma $ we write $\\mathcal {B}[X]$ for the BP obtained from $\\mathcal {B}$ by making $X$ the start type.",
"For $X, Y \\in \\Gamma $ we call $Y$ a successor of $X$ if there is a rule $X {} u Y v$ for some $u, v \\in \\Gamma ^*$ .",
"A BP with $\\varepsilon $ -rules allowed relaxes the requirement $\\mathord {{}} \\subseteq \\Gamma \\times \\Gamma ^+$ to $\\mathord {{}} \\subseteq \\Gamma \\times \\Gamma ^*$ , i.e., there may be rules of the form $X {} \\varepsilon $ , where $\\varepsilon $ denotes the empty word.",
"In the following, we disallow $\\varepsilon $ -rules unless specified otherwise; but the definitions generalize in a natural way.",
"Fix a BP $\\mathcal {B}= (\\Gamma , \\mathord {{}}, \\mathit {Prob}, X_0)$ for the rest of the section.",
"Write $\\llbracket \\mathcal {B} \\rrbracket $ for the set of trees generated by $\\mathcal {B}$ ; i.e., $\\llbracket \\mathcal {B} \\rrbracket $ denotes the set of ordered $\\Gamma $ -labelled trees $t$ such that for each $X \\in \\Gamma $ and each $X$ -labelled node $v$ in $t$ , there is a rule $X {} X_1 \\cdots X_k$ , denoted by $\\mathit {rule}(v)$ , such that the $k$ ordered children of $v$ are labelled with $X_1, \\ldots , X_k$ , respectively.",
"We say a node has type $X \\in \\Gamma $ if the node is labelled with $X$ .",
"A finite prefix of a tree $t \\in \\llbracket \\mathcal {B} \\rrbracket $ is an ordered $\\Gamma $ -labelled finite tree obtained from $t$ by designating some nodes as leaves, and removing all their children, grandchildren, etc.",
"Write $\\llparenthesis \\mathcal {B} \\rrparenthesis $ for the set of finite prefixes of trees generated by $\\mathcal {B}$ .",
"For $t \\in \\llparenthesis \\mathcal {B} \\rrparenthesis $ write $t{\\downarrow } \\subseteq \\llbracket \\mathcal {B} \\rrbracket $ for the (“cylinder”) set of trees $t^{\\prime } \\in \\llbracket \\mathcal {B} \\rrbracket $ such that $t$ is a finite prefix of $t^{\\prime }$ .",
"For $X \\in \\Gamma $ write $\\llbracket \\mathcal {B} \\rrbracket _X \\subseteq \\llbracket \\mathcal {B} \\rrbracket $ and $\\llparenthesis \\mathcal {B} \\rrparenthesis _X \\subseteq \\llparenthesis \\mathcal {B} \\rrparenthesis $ for the subsets of trees whose root has type $X$ ; the trees in $\\llbracket \\mathcal {B} \\rrbracket _X$ are called $X$ -trees.",
"A branch of a tree $t$ is a sequence $v_0 v_1 \\cdots $ of nodes in $t$ , where $v_0$ is the root of $t$ and $v_{i+1}$ is a child of $v_i$ for all $i \\in \\mathbb {N}_0$ .",
"See [8] for equivalent, more formal tree-related definitions.",
"For each $X \\in \\Gamma $ we define the probability space $(\\llbracket \\mathcal {B} \\rrbracket _X,\\Sigma _X,\\mathbb {P}_X)$ , where $\\Sigma _X$ is the $\\sigma $ -algebra generated by $\\lbrace t{\\downarrow } \\mid t\\in \\llparenthesis \\mathcal {B} \\rrparenthesis _X\\rbrace $ , and $\\mathbb {P}_X$ is the probability measure generated by $\\mathbb {P}_X(t{\\downarrow }) := \\prod _{v} \\mathit {Prob}(\\mathit {rule}(v))$ for all $t \\in \\llparenthesis \\mathcal {B} \\rrparenthesis _X$ , where the product extends over all non-leaf nodes $v$ in $t$ .",
"This is analogous to the standard definition of the probability space of a Markov chain.",
"We may write $\\mathbb {P}_\\mathcal {B}$ for $\\mathbb {P}_{X_0}$ , omitting the subscript when $\\mathcal {B}$ is understood.",
"We often talk about events (i.e., measurable sets of trees) and their probability in text form.",
"For example, by saying “a $\\mathcal {B}$ -tree has with positive probability infinitely many nodes of type $X$ ” we mean that $\\mathbb {P}_\\mathcal {B}(E) > 0$ where $E \\subseteq \\llbracket \\mathcal {B} \\rrbracket _{X_0}$ is the set of $X_0$ -trees with infinitely many nodes of type $X$ .",
"We are particularly interested in sets of trees all whose branches (more precisely, their associated sequences of types) satisfy an $\\omega $ -regular linear-time property $L \\subseteq \\Gamma ^\\omega $ .",
"Given $L \\subseteq \\Gamma ^\\omega $ , we write $\\mathbb {P}_\\mathcal {B}(L)$ for the probability that all branches of a $\\mathcal {B}$ -tree satisfy $L$ .",
"Linear temporal logic (LTL) formulas specify linear-time properties; see, e.g., [33] for a definition of LTL.",
"An important example for us are formulas of the form $\\mathsf {F}T$ , where $T \\subseteq \\Gamma $ , which denotes the linear-time property $\\lbrace u X w \\mid u \\in \\Gamma ^*,\\ X \\in T,\\ w \\in \\Gamma ^\\omega \\rbrace $ .",
"Accordingly, $\\mathbb {P}_\\mathcal {B}(\\mathsf {F}T)$ denotes the probability that all branches of a $\\mathcal {B}$ -tree have a node whose type is in $T$ (equivalently, the probability that a $\\mathcal {B}$ -tree has a finite prefix all whose leaves have a type in $T$ ).",
"We use finite automata on infinite words over $\\Gamma $ , where $\\Gamma $ is the set of types of a BP.",
"We use deterministic parity automata (DPAs), deterministic Büchi automata (DBAs), nondeterministic Büchi automata (NBAs), and unambiguous Büchi automata (UBAs).",
"The definitions are standard; see, e.g., [33].",
"In the following we fix some terms and notation.",
"Let $\\mathcal {A}= (Q, \\Gamma , \\delta , Q_0, F)$ be an NBA, where $Q$ is a finite set of states, $\\Gamma $ is the alphabet, $\\delta \\subseteq Q \\times \\Gamma \\times Q$ is the transition relation, $Q_0 \\subseteq Q$ is the set of initial states, and $F \\subseteq Q$ is the set of accepting states.",
"We write $q \\xrightarrow{} r$ to denote that $(q,X,r) \\in \\delta $ .",
"A finite sequence $q_0 \\xrightarrow{} q_1 \\xrightarrow{} \\cdots \\xrightarrow{} q_n$ is called a path and can be summarized as $q_0 {\\;\\xrightarrow{}{}\\hspace{0.0pt}^{*}\\;} q_n$ .",
"An infinite sequence $q_0 \\xrightarrow{} q_1 \\xrightarrow{} \\cdots $ is called a run of $X_1 X_2 \\cdots $ .",
"We call the run accepting if $q_0 \\in Q_0$ and $q_i \\in F$ holds for infinitely many $q_i$ .",
"The NBA $\\mathcal {A}$ accepts (resp., rejects) an infinite word $w \\in \\Gamma ^\\omega $ if $w$ has (resp., does not have) an accepting run in $\\mathcal {A}$ .",
"The NBA $\\mathcal {A}$ is called an unambiguous Büchi automaton (UBA) if every $w \\in \\Gamma ^\\omega $ has at most one accepting run.",
"An automaton $\\mathcal {A}$ defines $\\omega $ -regular linear-time properties $\\lbrace w \\in \\Gamma ^\\omega \\mid \\mathcal {A}\\text{ accepts } w\\rbrace $ and $\\lbrace w \\in \\Gamma ^\\omega \\mid \\mathcal {A}\\text{ rejects } w\\rbrace $ .",
"In keeping with previous definitions, we write $\\mathbb {P}_\\mathcal {B}(\\mathcal {A}\\text{ accepts})$ (resp., $\\mathbb {P}_\\mathcal {B}(\\mathcal {A}\\text{ rejects})$ ) for the probability that all branches of a $\\mathcal {B}$ -tree (more precisely, their associated sequences of types) are accepted (resp., rejected) by $\\mathcal {A}$ .",
"We consider the following computational problems.",
"The problem $\\mathbb {P}(\\text{finite}) = 1$ asks, given a BP $\\mathcal {B}$ with $\\varepsilon $ -rules allowed, whether the probability that a $\\mathcal {B}$ -tree is finite is 1.",
"The problem $\\mathbb {P}(\\textup {LTL}) = 1$ asks, given a BP $\\mathcal {B}$ and an LTL formula $\\varphi $ , whether $\\mathbb {P}_\\mathcal {B}(\\varphi ) = 1$ .",
"The problems $\\mathbb {P}(\\textup {DPA}) = 1$ (resp., $\\mathbb {P}(\\textup {NBA}) = 1$ ) ask, given a BP $\\mathcal {B}$ and a DPA (resp., NBA) $\\mathcal {A}$ , whether $\\mathbb {P}_\\mathcal {B}(\\mathcal {A}\\text{ accepts}) = 1$ .",
"The problems $\\mathbb {P}(\\textup {coNBA}) = 1$ (resp., $\\mathbb {P}(\\textup {coUBA}) = 1$ )We do not explicitly define or use a notion of “co-Büchi automata” to avoid possible confusion about accepting/rejecting.",
"If one were to do so, one would define a “co-NBA” $\\mathcal {A}$ like an NBA $\\mathcal {A}$ , but the “co-NBA” $\\mathcal {A}$ would accept a word $w \\in \\Gamma ^\\omega $ if and only if $\\mathcal {A}$ viewed as an NBA rejects $w$ .",
"Similarly for “co-UBAs”.",
"ask, given a BP $\\mathcal {B}$ and an NBA (resp., UBA) $\\mathcal {A}$ , whether $\\mathbb {P}_\\mathcal {B}(\\mathcal {A}\\text{ rejects}) = 1$ .",
"The problems $\\mathbb {P}(\\textup {LTL}) = 0, \\mathbb {P}(\\textup {DPA}) = 0, \\ldots $ are defined similarly, where “${=}\\,1$ ” is replaced with “${=}\\,0$ ”.",
"Table: Results and organization of the paper.The complexity classes indicate completeness results, except “in NC”, which only means membership in NC.See tab:map for a map of our results in those terms, as well as for an overview of the rest of the paper.",
"As explained in the introduction, the problem $\\mathbb {P}(\\textup {LTL})=1$ is of particular interest from a model-checking point of view, and the technically most challenging one.",
"In addition to standard complexity classes between P and 2EXPTIME, we use the class NC, the subclass of P comprising those problems solvable in polylogarithmic time by a parallel random-access machine using polynomially many processors; see, e.g., [27].",
"To prove membership in PSPACE in a modular way, we will use the following pattern: Let $P_1, P_2$ be two problems, where $P_2$ is in NC.",
"Suppose there is a reduction from $P_1$ to $P_2$ implemented by a PSPACE transducer, i.e., a Turing machine whose work tape (but not necessarily its output tape) is PSPACE-bounded.",
"Then $P_1$ is in PSPACE.",
"Note that the output of the transducer is (at most) exponential.",
"Problems in NC can be decided in polylogarithmic space [4].",
"Using standard techniques for composing space-bounded transducers (see, e.g., [27]), it follows that $P_1$ is in PSPACE.",
"We use finite sets $S$ to index matrices $M \\in \\mathbb {R}^{S \\times S}$ and vectors $v \\in \\mathbb {R}^S$ .",
"The graph of a nonnegative matrix $M \\in [0, \\infty )^{S \\times S}$ is the directed graph $(S,E)$ with $E = \\lbrace (s,t) \\in S \\times S \\mid M_{s,t} > 0\\rbrace $ .",
"The spectral radius of a matrix is the largest absolute value of its eigenvalues.",
"The following lemma allows to efficiently compare the spectral radius of a nonnegative matrix with 1.",
"Given a nonnegative rational matrix $M$ , one can determine in NC whether $\\rho < 1$ or $\\rho = 1$ or $\\rho > 1$ , where $\\rho $ denotes the spectral radius of $M$ .",
"Use the algorithm from [13], but not with Gaussian elimination as suggested there, but by solving the systems of linear equations described in [13] in NC.",
"The latter is possible in NC [5]."
],
[
"Basic Results",
"In this section we develop the more basic results indicated in tab:map, on finiteness (sec:as-finite), deterministic parity automata (sec:DPA), and Büchi automata (sec:NBA), on the one hand rounding off the complexity map in tab:map, and on the other hand building the foundation for more challenging results in the following sections.",
"In particular, prop:as-finite-NC is indirectly used throughout the paper."
],
[
"Finiteness",
"In this section we consider BPs with $\\varepsilon $ -rules allowed, i.e., rules of the form $X {} \\varepsilon $ .",
"Such BPs may generate finite trees.",
"We are interested in the almost-sure finiteness problem, also denoted as $\\mathbb {P}(\\text{finite})=1$ , i.e., the problem whether the probability that a given BP with $\\varepsilon $ -rules allowed generates a finite tree is equal to 1.",
"In prop:as-finite-NC below we show that this problem is in NC.",
"All upper bounds on the complexity of $\\mathbb {P}(\\cdot ) = 1$ problems in this paper build directly or indirectly on this result.",
"While the almost-sure finiteness (or “extinction”) problem has often been studied and is known to be in (strongly) polynomial time [18], [13], its membership in NC is, to the best of the authors' knowledge, new.",
"For instance, since linear programming is P-complete, one cannot use linear programming (as in [18]) to show membership in NC.",
"Nor can one directly use the strongly polynomial-time algorithm of [13], as it computes, in a sub-procedure, the set of types $X$ for which there exists a finite $X$ -tree.",
"But the latter problem is P-complete.",
"For the rest of the section, fix a BP $\\mathcal {B}= (\\Gamma , \\mathord {{}}, \\mathit {Prob}, X_0)$ with $\\varepsilon $ -rules allowed.",
"Define a directed graph $G = (\\Gamma ,E)$ (i.e., the types of $\\mathcal {B}$ are the vertices of $G$ ) with an edge $(X,Y) \\in E$ if and only if $Y$ is a successor of $X$ (i.e., there is a rule $X {} u Y v$ for some $u,v \\in \\Gamma ^*$ ).",
"Given a strongly connected component (SCC) $S \\subseteq \\Gamma $ of $G$ and $X \\in S$ , define a BP $\\mathcal {B}[S,X] = (S,\\mathord {{}_S},\\mathit {Prob}_S,X)$ obtained from $\\mathcal {B}$ by restricting the types to $S$ and deleting on all right-hand sides of the rules those types not in $S$ .",
"The following lemma is straightforward: A $\\mathcal {B}$ -tree is infinite with positive probability if and only if there exist an SCC $S \\subseteq \\Gamma $ of $G$ and $X \\in S$ such that $X$ is reachable from $X_0$ in $G$ and a $\\mathcal {B}[S,X]$ -tree is infinite with positive probability.",
"Let $M \\in \\mathbb {Q}^{\\Gamma \\times \\Gamma }$ be the nonnegative $\\Gamma \\times \\Gamma $ -matrix with $M_{X,Y} = \\sum _{X {p} w} p |w|_Y$ , where $|w|_Y \\in \\mathbb {N}_0$ is the number of occurrences of $Y$ in $w$ .",
"That is, $M_{X,Y}$ is the expected number of direct $Y$ -successors of the root of a $\\mathcal {B}[X]$ -tree.",
"By induction, $M^i$ , the $i$ th power of $M$ , is such that $(M^i)_{X,Y}$ is the expected number of $Y$ -nodes that are exactly $i$ levels under the root of a $\\mathcal {B}[X]$ -tree.",
"The graph of $M$ is exactly the previously defined graph $G$ .",
"Let $S \\subseteq \\Gamma $ be an SCC of $G$ .",
"Denote by $M_S \\in \\mathbb {Q}^{S \\times S}$ the (square) principal submatrix obtained from $M$ by restricting it to the rows and columns indexed by elements of $S$ .",
"Let $\\rho _S$ denote the spectral radius of $M_S$ .",
"Call $S$ supercritical if $\\rho _S > 1$ .",
"Call $S$ linear if for all rules $X {} w$ with $X \\in S$ there is exactly one occurrence in $w$ of a type in $S$ .",
"Observe that if $S$ is linear then $M_S$ is stochastic, i.e., $M_S \\vec{1} = \\vec{1}$ where $\\vec{1}$ is the all-1 vector, i.e., the element of $\\lbrace 1\\rbrace ^S$ .",
"In that case, by the Perron-Frobenius theorem [3], we have $\\rho _S = 1$ and, thus, $S$ is not supercritical.",
"The following characterization can be proved using [13] (which builds on [18]): lemmalemasfinitenesschar A $\\mathcal {B}$ -tree is infinite with positive probability if and only if there exist an SCC $S \\subseteq \\Gamma $ of $G$ and $X \\in S$ such that $X$ is reachable from $X_0$ in $G$ and $S$ is supercritical or linear.",
"It follows: propositionpropasfiniteNC The problem $\\mathbb {P}(\\text{finite})=1$ is in NC."
],
[
"Deterministic Parity Automata",
"In this section we consider deterministic parity automata (DPAs) on words.",
"In [8] it was shown that the problem $\\mathbb {P}(\\textup {DPA}) = 1$ can be decided in polynomial time.",
"We improve this to membership in NC.",
"By the following lemma we can check in NC whether a $\\mathcal {B}$ -tree almost surely has a finite prefix all whose leaves have types in a given set $T$ .",
"The proof is by reduction to almost-sure finiteness.",
"lemmalemAFTone Given a BP $\\mathcal {B}= (\\Gamma , \\mathord {{}}, \\mathit {Prob}, X_0)$ and a set of types $T \\subseteq \\Gamma $ , the problem whether $\\mathbb {P}_{X_0}(\\mathsf {F}T) = 1$ is in NC.",
"By combining lem:AFT-1 with results from [8] we obtain: theoremthmDPAone The problem $\\mathbb {P}(\\textup {DPA}) = 1$ is in NC.",
"The hardness result in the following theorem highlights the different complexities of $\\mathbb {P}(\\cdot ) = 0$ and $\\mathbb {P}(\\cdot ) = 1$ problems in this paper.",
"theoremthmDPAzero The problem $\\mathbb {P}(\\textup {DPA}) = 0$ is P-complete.",
"It is P-hard even for deterministic Büchi automata with two states, the accepting state being a sink."
],
[
"Büchi Automata",
"The problem $\\mathbb {P}(\\textup {NBA}) = 1$ is PSPACE-complete.",
"PSPACE-hardness is immediate in two different ways.",
"It follows from the PSPACE-hardness of model checking Markov chains against NBAs [34].",
"It also follows from the PSPACE-hardness of model checking transition systems against NBAs.",
"(The latter follows easily from the PSPACE-hardness of NBA universality [31].)",
"Both model-checking problems are special cases of $\\mathbb {P}(\\textup {NBA}) = 1$ .",
"Towards membership in PSPACE, we use a translation from NBA to DPA [28].",
"This translation causes an exponential blow-up, but an inspection of the construction [28] reveals that it can be computed by a PSPACE transducer.",
"By thm:DPA-1 the problem $\\mathbb {P}(\\textup {DPA}) = 1$ is in NC.",
"By lem:PSPACE-transducer it follows that $\\mathbb {P}(\\textup {NBA}) = 1$ is in PSPACE.",
"theoremthmNBAzero The problem $\\mathbb {P}(\\textup {NBA}) = 0$ is EXPTIME-complete.",
"It is EXPTIME-hard even for NBAs whose only accepting state is a sink.",
"Towards membership in EXPTIME, an NBA can be translated, in exponential time, to a DPA of exponential size; see, e.g., [28].",
"Since $\\mathbb {P}(\\textup {DPA}) = 0$ is in P by thm:DPA-0, it follows that $\\mathbb {P}(\\textup {NBA}) = 0$ is in EXPTIME.",
"Concerning EXPTIME-hardness, we adapt the proof (in the online appendix) of [19] on model checking recursive Markov chains against NBAs.",
"The details are in app:NBA-0."
],
[
"Co-Büchi Automata",
"In this section we consider the problem $\\mathbb {P}(\\textup {coNBA}) = 1$ , which asks, given a BP $\\mathcal {B}$ and a Büchi automaton $\\mathcal {A}$ , whether $\\mathcal {B}$ almost surely generates a tree whose branches are all rejected by $\\mathcal {A}$ ; i.e., whether $\\mathbb {P}_{\\mathcal {B}}(\\mathcal {A}\\text{ rejects}) = 1$ .",
"Dually, one might ask whether the probability is positive that a $\\mathcal {B}$ -tree has a branch accepted by $\\mathcal {A}$ .",
"Intuitively, we view the Büchi automaton $\\mathcal {A}$ as specifying “bad” branches, and we would like the tree almost surely not to have any bad branches.",
"This problem is in PSPACE, which can be shown via a translation to DPAs, as in thm:NBA-1.",
"However, with a view on the following sections, in particular on LTL specifications, we pursue a different approach to the problem $\\mathbb {P}(\\textup {coNBA}) = 1$ .",
"In this section we lay the groundwork for arbitrary Büchi automata $\\mathcal {A}$ .",
"By building on these results, we will show in the next section that if $\\mathcal {A}$ is unambiguous then the problem is in NC, which will allow us to derive our headline result, namely that $\\mathbb {P}(\\textup {LTL})=1$ is in PSPACE.",
"Let $\\mathcal {B}= (\\Gamma , \\mathord {{}}, \\mathit {Prob}, X_0)$ be a BP and $\\mathcal {A}= (Q, \\Gamma , \\delta , Q_0, F)$ a (not necessarily unambiguous) Büchi automaton.",
"Define a Büchi automaton, $\\mathcal {A}\\times \\mathcal {B}$ , by $\\mathcal {A}\\times \\mathcal {B}:= (Q \\times \\Gamma , \\Gamma , \\delta _{\\mathcal {A}\\times \\mathcal {B}}, Q_0 \\times \\lbrace X_0\\rbrace , F \\times \\Gamma )$ , where $&\\delta _{\\mathcal {A}\\times \\mathcal {B}}((q_1, X_1), X_2)={\\left\\lbrace \\begin{array}{ll}\\delta (q_1, X_1) \\times \\lbrace X_2\\rbrace & \\text{if $X_2$ is a successor of~$X_1$}\\\\\\emptyset & \\text{otherwise.}\\end{array}\\right.",
"}$ The remainder of the section is organized as follows.",
"In sub:UBA-X1-f we show that the problem $\\mathbb {P}(\\textup {coNBA}) = 1$ reduces to the analysis of certain SCCs within $\\mathcal {A}\\times \\mathcal {B}$ .",
"In sub:Bdet we introduce a key lemma, lem:Bdet, which allows us to “forget” about the distinction between accepting and non-accepting states: the lemma reduces $\\mathbb {P}(\\textup {coNBA}) = 1$ to a pure reachability problem in an exponential-sized BP, $\\mathcal {B}_{\\mathit {det}}$ .",
"This leads us to prove PSPACE-completeness of $\\mathbb {P}(\\textup {coNBA}) = 1$ , but more importantly, lem:Bdet plays a key role in the rest of the paper.",
"We prove it in sub:Bdet-proof."
],
[
"The Automaton $\\mathcal {A}[f,X_f]$",
"For any $(f,X_f) \\in F \\times \\Gamma $ on a cycle of the transition graph of $\\mathcal {A}\\times \\mathcal {B}$ , define the Büchi automaton $\\mathcal {A}[f,X_f] \\ := \\ (\\lbrace \\bar{q}_0\\rbrace \\cup Q[f,X_f], \\Gamma , \\delta [f,X_f], \\lbrace \\bar{q}_0\\rbrace , \\lbrace (f,X_f)\\rbrace )$ as the Büchi automaton obtained from $\\mathcal {A}\\times \\mathcal {B}$ by making $(f,X_f)$ the only accepting state, restricting the set of states, $Q[f,X_f] \\subseteq Q \\times \\Gamma $ , to those $(q,X)$ that, in the transition graph of $\\mathcal {A}\\times \\mathcal {B}$ , are reachable from $(f,X_f)$ and can reach $(f,X_f)$ , i.e., those $(q,X)$ in the SCC containing $(f, X_f)$ , restricting the transition function $\\delta [f,X_f]$ accordingly, i.e., $\\delta [f,X_f]((q,X),Y) \\ := \\ \\delta _{\\mathcal {A}\\times \\mathcal {B}}((q,X),Y) \\cap Q[f,X_f]\\,,$ making $\\bar{q}_0$ the only initial state, and setting $\\delta [f,X_f](\\bar{q}_0,X_f) := \\lbrace (f,X_f)\\rbrace $ and $\\delta [f,X_f](\\bar{q}_0,X) := \\emptyset $ for all $X \\in \\Gamma \\setminus \\lbrace X_f\\rbrace $ .",
"The following lemma follows from the pigeonhole principle and basic probability arguments:lemmalemUBAXonef The probability that some branch of a $\\mathcal {B}$ -tree is accepted by $\\mathcal {A}$ is positive if and only if there are $q_0 \\in Q_0$ and $f \\in F$ and $X_f \\in \\Gamma $ such that $(f, X_f)$ is reachable from $(q_0, X_0)$ in the transition graph of $\\mathcal {A}\\times \\mathcal {B}$ and the probability that some branch of a $\\mathcal {B}[X_f]$ -tree is accepted by $\\mathcal {A}[f,X_f]$ is positive.",
"For the rest of the section let $(f,X_f) \\in F \\times \\Gamma $ be on a cycle of the transition graph of $\\mathcal {A}\\times \\mathcal {B}$ ."
],
[
"The Determinization $\\mathcal {A}_{\\mathit {det}}$ and the BP {{formula:82465720-6fff-453e-a5bf-44f8cad75b0e}}",
"Let $\\mathcal {A}_{\\mathit {det}}:= (2^{\\lbrace \\bar{q}_0\\rbrace \\cup Q[f,X_f]}, \\Gamma , \\delta _{\\mathit {det}}, \\lbrace \\bar{q}_0\\rbrace , 2^{\\lbrace \\bar{q}_0\\rbrace \\cup Q[f,X_f]} \\setminus \\lbrace \\emptyset \\rbrace )$ be the determinization of $\\mathcal {A}[f,X_f]$ obtained by the standard subset construction.",
"Which states are accepting will not actually be relevant.",
"Note that every state reachable via a nonempty path from $\\lbrace \\bar{q}_0\\rbrace $ is of the form $P \\times \\lbrace X\\rbrace $ with $P \\subseteq Q$ and $X \\in \\Gamma $ .",
"Define a BP $\\mathcal {B}_{\\mathit {det}}$ based on $\\mathcal {A}_{\\mathit {det}}$ as $\\mathcal {B}_{\\mathit {det}}\\ := \\ (\\Gamma ^{\\prime }, \\mathord {{}^{\\prime }}, \\mathit {Prob}^{\\prime }, \\lbrace (f,X_f)\\rbrace )\\,,$ where the set of types $\\Gamma ^{\\prime } \\subseteq 2^{Q[f,X_f]}$ is the set of those states in $\\mathcal {A}_{\\mathit {det}}$ that are reachable (in $\\mathcal {A}_{\\mathit {det}}$ ) from $\\lbrace \\bar{q_0}\\rbrace $ via a nonempty path (recall that they are of the form $P \\times \\lbrace X\\rbrace $ with $P \\subseteq Q$ and $X \\in \\Gamma $ ), and $X^{\\prime } &{\\;{p}{}\\hspace{0.0pt}{^{\\prime }}\\;} \\delta _{\\mathit {det}}(X^{\\prime },X_1) \\cdots \\delta _{\\mathit {det}}(X^{\\prime },X_k)$ for all $X^{\\prime } = P \\times \\lbrace X\\rbrace \\in \\Gamma ^{\\prime }$ with $P \\ne \\emptyset $ and all $X {p} X_1 \\cdots X_k$ , and $\\emptyset {\\;{1}{}\\hspace{0.0pt}{^{\\prime }}\\;} \\emptyset $ .",
"Here is the key lemma of this section: lemmalemBdet The following statements are equivalent: The probability that some branch of a $\\mathcal {B}[X_f]$ -tree is accepted by $\\mathcal {A}[f,X_f]$ is positive.",
"The probability that some branch of a $\\mathcal {B}_{\\mathit {det}}$ -tree does not have any nodes of type $\\emptyset $ is positive.",
"We prove lem:Bdet in sub:Bdet-proof.",
"It will be used in the proof of thm:coNBA-1 below; but more importantly, lem:Bdet is the foundation of sec:coUBA.",
"Given that lem:Bdet reflects the key insight of this section, let us comment further.",
"Considering that condition (ii) does not mention a notion of acceptance, one might have two concerns at this point: Condition (ii) does not obviously imply that with positive probability there is even a branch with infinitely many nodes of types containing $(f,X_f)$ .",
"Even if with positive probability there is such a branch, it is not obvious that such branches would necessarily correspond to branches of $\\mathcal {B}[X_f]$ that are accepted by $\\mathcal {A}[f,X_f]$ .",
"Even for the special case of Markov chains (i.e., every tree has only a single branch), lem:Bdet is not at all obvious, and both concerns (a) and (b) apply.",
"Indeed, for Markov chains, Courcoubetis and Yannakakis prove a statement related to lem:Bdet, namely [11], with a proof related to ours and dealing explicitly with concern (b) above.",
"For the special case of transition systems (i.e., the BP generates exactly one tree), lem:Bdet is simple though: consider the branch that follows a cycle around $(f,X_f)$ .",
"For the general case, we need a result on BPs from [8], dealing with concern (a) above.",
"The high-level principle behind the proof of lem:Bdet is often used in the analysis of Markov chains: if it is possible, infinitely often, to reach a state with a probability bounded away from 0, then this state is almost surely reached infinitely often.",
"See sub:Bdet-proof for a full proof of lem:Bdet.",
"We can now derive a PSPACE procedure for the problem $\\mathbb {P}(\\textup {coNBA}) = 1$ without resorting to DPAs: theoremthmcoNBAone The problem $\\mathbb {P}(\\textup {coNBA}) = 1$ is PSPACE-complete.",
"thm:NBA-0 (for NBAs) has a coNBA-analogue: theoremthmcoNBAzero The problem $\\mathbb {P}(\\textup {coNBA}) = 0$ is EXPTIME-complete.",
"It is EXPTIME-hard even for NBAs all whose states are accepting."
],
[
"Co-Unambiguous Büchi Automata",
"In this section we build on the previous section, in particular on lem:Bdet, to derive our main technical result: given a BP $\\mathcal {B}$ and an unambiguous Büchi automaton (UBA) $\\mathcal {A}$ , one can decide in NC whether $\\mathcal {B}$ almost surely generates a tree all whose branches are rejected by $\\mathcal {A}$ : The problem $\\mathbb {P}(\\textup {coUBA}) = 1$ is in NC.",
"The rest of the section is devoted to the proof of this theorem.",
"Fix a BP $\\mathcal {B}$ and a UBA $\\mathcal {A}$ .",
"Since NC is closed under complement, we can focus on the problem whether the probability is positive that a $\\mathcal {B}$ -tree has some branch accepted by $\\mathcal {A}$ .",
"We use lem:UBA-X1-f.",
"Since reachability in a graph is in NL and, hence, in NC, it suffices to decide in NC whether the probability that some branch of a $\\mathcal {B}[X_f]$ -tree is accepted by $\\mathcal {A}[f,X_f]$ is positive.",
"By lem:Bdet it suffices to decide in NC whether the probability that some branch of a $\\mathcal {B}_{\\mathit {det}}$ -tree does not have any nodes of type $\\emptyset $ is positive.",
"The challenge is that $\\mathcal {B}_{\\mathit {det}}$ may be exponentially larger than $\\mathcal {A}$ , so we need to exploit the unambiguousness of $\\mathcal {A}$ and the regular structure it gives to $\\mathcal {B}_{\\mathit {det}}$ .",
"Let $\\mathcal {B}_{\\mathit {det}}^{\\prime \\prime }$ be the BP (with $\\varepsilon $ -rules allowed) obtained from $\\mathcal {B}_{\\mathit {det}}$ by removing the type $\\emptyset $ and eliminating all occurrences of type $\\emptyset $ from all right-hand sides.",
"The probability that a $\\mathcal {B}_{\\mathit {det}}$ -tree has an infinite branch of non-$\\emptyset $ nodes is equal to the probability that a $\\mathcal {B}_{\\mathit {det}}^{\\prime \\prime }$ -tree is infinite.",
"Hence, it remains to show that one can decide in NC whether the probability that a $\\mathcal {B}_{\\mathit {det}}^{\\prime \\prime }$ -tree is infinite is positive.",
"Define a matrix $M \\in \\mathbb {Q}^{Q[f,X_f] \\times Q[f,X_f]}$ whose rows and columns are indexed with the non-$\\bar{q}_0$ states of $\\mathcal {A}[f,X_f]$ : $M_{(q,X),(r,Y)} \\ := \\ {\\left\\lbrace \\begin{array}{ll} \\displaystyle \\sum _{X {p} u} p |u|_Y & \\text{if } (q,X) \\xrightarrow{} (r,Y) \\ \\text{ in } \\mathcal {A}[f,X_f] \\\\[1mm] 0 & \\text{otherwise\\,,}\\end{array}\\right.",
"}$ where $|u|_Y \\in \\mathbb {N}_0$ is the number of occurrences of $Y$ in $u$ .",
"(Think of $M_{(q,X),(r,Y)}$ as the expected number of $(r,Y)$ -“successors” of $(q,X)$ .)",
"The graph of $M$ is equal to the transition graph of $\\mathcal {A}[f,X_f]$ (excluding $\\bar{q}_0$ ), which is strongly connected.",
"Say that $\\mathcal {A}[f,X_f]$ has proper branching if there exist $(q,Y) \\xrightarrow{} (r_1,Z_1)$ and $(q,Y) \\xrightarrow{} (r_2,Z_2)$ in $\\mathcal {A}[f,X_f]$ and a rule $Y {p} u_1 Z_1 u_2 Z_2 u_3$ in $\\mathcal {B}$ with $u_1, u_2, u_3 \\in \\Gamma ^*$ .",
"Now we can state the key lemma: lemmalemkey Let $\\rho $ be the spectral radius of $M$ .",
"The probability that a $\\mathcal {B}_{\\mathit {det}}^{\\prime \\prime }$ -tree is infinite is positive if and only if either $\\rho > 1$ or $\\rho = 1$ and $\\mathcal {A}[f,X_f]$ does not have proper branching.",
"Observe the similarity between lem:key,lem:as-finiteness-char.",
"In fact, the proof of lem:key, given below, is based on lem:as-finiteness-char.",
"lem:key shows that properties of $\\mathcal {A}[f,X_f]$ and $M$ (which are polynomial-sized objects) determine a property of the exponential-sized BP $\\mathcal {B}_{\\mathit {det}}^{\\prime \\prime }$ .",
"Unambiguousness of $\\mathcal {A}[f,X_f]$ is crucial for that connection.",
"Given that lem:key reflects the key insight of this section (if not of this paper), let us comment further.",
"Suppose $\\mathcal {A}[f,X_f]$ has two outgoing transitions in a state $(q,Y)$ , say $(q,Y) \\xrightarrow{} (r_1,Z_1)$ and $(q,Y) \\xrightarrow{} (r_2,Z_2)$ .",
"This branching could be “proper branching” as defined before lem:key, or the original UBA $\\mathcal {A}$ could be nondeterministic when reading $Y$ in $q$ and have transitions $q \\xrightarrow{} r_1$ and $q \\xrightarrow{} r_2$ .",
"Either type of branching causes non-0 entries in the matrix $M$ and, intuitively, increases its spectral radius $\\rho $ .",
"lem:key tells us that the probability that a $\\mathcal {B}_{\\mathit {det}}^{\\prime \\prime }$ -tree is infinite is governed by the combined effect on $\\rho $ of both types of branching: if $\\rho > 1$ then a $\\mathcal {B}_{\\mathit {det}}^{\\prime \\prime }$ -tree is infinite with positive probability; only in the borderline case, $\\rho =1$ , the type of branching matters.",
"Again, this characterization is only correct if the nondeterminism in $\\mathcal {A}$ does not cause ambiguousness.",
"Let us consider what lem:key states for the special case of Markov chains.",
"In that case, clearly there is no proper branching.",
"One can show, using unambiguousness, that for Markov chains the spectral radius $\\rho $ of the matrix $M$ is at most 1.",
"Hence, lem:key states for Markov chains that the probability that a $\\mathcal {B}_{\\mathit {det}}^{\\prime \\prime }$ -tree (consisting of a single branch) is infinite is positive if and only if $\\rho = 1$ .",
"Indeed, a related statement can be found in [2].",
"To finish the proof of prop:coUBA-1 it suffices to show that we can check the conditions of lem:key in NC.",
"Indeed, for comparing the spectral radius with 1, we employ lem:determine-spectral-radius.",
"One can check for proper branching in logarithmic space, hence in NC.",
"This completes the proof of prop:coUBA-1."
],
[
"LTL",
"With prop:coUBA-1 from the previous section, we can now show our headline result: The problem $\\mathbb {P}(\\textup {LTL})=1$ is PSPACE-complete.",
"PSPACE-hardness is immediate in two different ways.",
"It follows both from the PSPACE-hardness of model checking Markov chains against LTL and from the PSPACE-hardness of model checking transition systems against LTL [30].",
"Both model-checking problems are special cases of $\\mathbb {P}(\\textup {LTL}) = 1$ .",
"Towards membership in PSPACE, there is a classical PSPACE procedure that translates an LTL formula into an (exponential-sized) Büchi automaton [36].",
"As noted by several authors (e.g., [12], [7]), this procedure can easily be adapted to ensure that the Büchi automaton be a UBA.",
"By applying this translation to the negation $\\lnot \\varphi $ of the input formula $\\varphi $ , we obtain a UBA that rejects exactly those words that satisfy $\\varphi $ .",
"By prop:coUBA-1 the problem $\\mathbb {P}(\\textup {coUBA}) = 1$ is in NC.",
"By lem:PSPACE-transducer it follows that $\\mathbb {P}(\\textup {LTL}) = 1$ is in PSPACE.",
"Finally we show the following result, exhibiting a big complexity gap between the problems $\\mathbb {P}(\\textup {LTL})=1$ and $\\mathbb {P}(\\textup {LTL})=0$ .",
"theoremthmLTLzero The problem $\\mathbb {P}(\\textup {LTL})=0$ is 2EXPTIME-complete.",
"For membership in 2EXPTIME, we use again the classical procedure that translates an LTL formula into an exponential-sized Büchi automaton [36] and then invoke thm:NBA-0.",
"For 2EXPTIME-hardness we adapt the reduction from [11] for MDPs.",
"The details are in app:LTL-0."
],
[
"Conclusions",
"We have devised a PSPACE procedure for $\\mathbb {P}(\\textup {LTL}) = 1$ , i.e., qualitative LTL model checking of BPs.",
"The best previously known procedure ran in 2EXPTIME [8].",
"Since BPs naturally generalize both transition systems and Markov chains (for both of which LTL model checking is PSPACE-complete), one might view our model-checking algorithm as an optimal general procedure.",
"The same holds for NBA-specifications instead of LTL.",
"The main technical ingredients have been the automata-theoretic approach and the algorithmic analysis of UBAs, nonnegative matrices, and finiteness of BPs.",
"Our proofs were inspired by the observation that the spectral radii of certain nonnegative matrices are central to model checking Markov chains against UBAs, and also determine fundamental properties of BPs.",
"Very loosely speaking, when model checking Markov chains against UBAs, the spectral radius measures the amount of nondeterministic branching in the UBA, whereas when analyzing BPs, the spectral radius measures the amount of tree branching.",
"The “general case”, i.e., model checking BPs, features both kinds of branching.",
"Serendipitously, an analysis of spectral radii still leads, as we have seen, to optimal algorithms.",
"We have also established the complexities of related problems, partially as a tool for the mentioned LTL and NBA problems and partially to map out the landscape.",
"We have shown that the $\\mathbb {P}(\\cdot ) = 0$ variants are more complex than their $\\mathbb {P}(\\cdot ) = 1$ counterparts.",
"An intuitive explanation of this phenomenon is that for an instance of an $\\mathbb {P}(\\cdot )=1$ problems to be negative, tree branching and probabilistic branching “work together” to falsify the specification on some branch.",
"In contrast, for $\\mathbb {P}(\\cdot )=0$ problems, tree branching and probabilistic branching are “adversaries”, like in MDPs.",
"Indeed, for lower bounds on $\\mathbb {P}(\\cdot )=0$ problems we have encoded alternation in various forms.",
"One might ask about the complexity of $\\mathbb {P}(\\textup {UBA}) = 1$ .",
"Indeed, in trying to solve $\\mathbb {P}(\\textup {LTL}) = 1$ efficiently, the authors set out to solve $\\mathbb {P}(\\textup {UBA})=1$ efficiently (perhaps in P or even NC), with the PSPACE transduction from LTL to UBA in mind.",
"However, the complexity of UBA universality is an open problem [29]; only membership in PSPACE is known.",
"So even for the fixed transition system with $a {1} a b$ and $b {1} a b$ the problem $\\mathbb {P}(\\textup {UBA})=1$ cannot be placed in P without improving the complexity of UBA universality.",
"A PSPACE-hardness proof of $\\mathbb {P}(\\textup {UBA})=1$ might have to make use of both types of branching in BPs, as $\\mathbb {P}(\\textup {UBA})=1$ is in NC for Markov chains [2].",
"Model checking BPs quantitatively, i.e., computing the satisfaction probability, comparing it with a threshold, or approximating it, is left for future work.",
"Exact versions of these problems are computationally complex, as they are at least as hard as the corresponding $\\mathbb {P}(\\cdot )=0$ problem.",
"The paper [8] describes, for DPAs, nonlinear equation systems whose least nonnegative solution characterizes the satisfaction probabilities.",
"Newton's method is efficient for approximating the solution of such equation systems; see [32], [15].",
"sectionappendix"
],
[
"Proof of lem:as-finiteness-char",
"* By lem:as-finite-SCC it suffices to show that if $G$ is strongly connected then a $\\mathcal {B}$ -tree is infinite with positive probability if and only if $\\Gamma $ is supercritical or linear.",
"So, let $G$ be strongly connected.",
"Call a type $X \\in \\Gamma $ immortal if there is no finite $X$ -tree.",
"If a type $X$ is immortal, then the probability that a $\\mathcal {B}[X]$ -tree is finite is 0, and for all $Y \\in \\Gamma $ the probability that a $\\mathcal {B}[Y]$ -tree is finite is less than 1 (as $G$ is strongly connected).",
"If $\\Gamma $ is linear then all types are immortal.",
"So we can assume in the following that $\\Gamma $ is not linear.",
"Since $\\Gamma $ is not linear and $G$ is strongly connected, for all $X \\in \\Gamma $ there is a finite prefix of an $X$ -tree that has at least two leaves of type $X$ ; and an $X$ -tree has, with positive probability, that finite prefix.",
"Suppose some type, say $X$ , is immortal.",
"Then every $X$ -tree that has a finite prefix with $k$ ($k \\in \\mathbb {N}$ ) leaves of type $X$ has at least $k$ (infinite) branches that go through those $k$ nodes of type $X$ .",
"By strong connectedness, it follows that an $X$ -tree that has a finite prefix with $k$ leaves of type $X$ has, with probability 1 (i.e., $\\mathbb {P}_X$ -almost surely), a finite prefix with $k+1$ leaves of type $X$ .",
"Hence, with probability 1 we have $\\lim _{i \\rightarrow \\infty } Z_i = \\infty $ , where $Z_i$ is the number of nodes that are exactly $i$ levels under the root of an $X$ -tree.",
"Denoting by $\\mathbb {E}$ the expectation with respect to $\\mathbb {P}_X$ , it follows with Fatou's lemma that $\\lim _{i \\rightarrow \\infty } \\mathbb {E}Z_i = \\infty $ .",
"We can write $\\mathbb {E}Z_i = (M^i \\vec{1})_X$ , where $\\vec{1}$ denotes the vector in $\\lbrace 1\\rbrace ^\\Gamma $ .",
"Then $ \\lim _{i \\rightarrow \\infty } \\left\\Vert M^i \\vec{1}\\right\\Vert \\ = \\ \\infty \\,,$ where $\\left\\Vert \\cdot \\right\\Vert $ denotes the 1-norm.",
"Since $G$ is strongly connected, by the Perron-Frobenius theorem [3], $M$ has an eigenvector $v \\in \\mathbb {R}^\\Gamma $ with $v_Y > 0$ for all $Y \\in \\Gamma $ and $M v = \\rho v$ , where $\\rho $ denotes the spectral radius of $M$ .",
"Since $M^i v = \\rho ^i v$ , it follows that $\\lim _{i \\rightarrow \\infty } \\rho ^i \\left\\Vert v\\right\\Vert \\ = \\ \\lim _{i \\rightarrow \\infty } \\left\\Vert M^i v\\right\\Vert \\ = \\ \\infty \\,.$ Hence $\\rho >1$ .",
"We conclude that if some type is immortal, then we have $\\rho >1$ and the probability that a $\\mathcal {B}$ -tree is finite is less than 1.",
"On the other hand, if no type is immortal, then it follows from [13] (building on [18]) that a $\\mathcal {B}$ -tree is infinite with positive probability if and only if $\\rho > 1$ .",
"We conclude that, regardless of whether there exists an immortal type, a $\\mathcal {B}$ -tree is infinite with positive probability if and only if $\\Gamma $ is supercritical."
],
[
"Proof of prop:as-finite-NC",
"* We use the characterization from lem:as-finiteness-char.",
"One can compute in NL, and hence in NC, the directed acyclic graph of SCCs of $G$ and the set of types that are reachable from $X_0$ .",
"Therefore, we assume without loss of generality that $G$ is strongly connected.",
"By lem:determine-spectral-radius one can check in NC whether $\\Gamma $ is supercritical.",
"Whether $\\Gamma $ is linear can be checked in logarithmic space, hence in NC."
],
[
"Proof of lem:AFT-1",
"* The problem can be rephrased as almost-sure finiteness of BPs with $\\varepsilon $ -rules allowed.",
"Indeed, given $\\mathcal {B}$ and $T$ , one can eliminate all occurrences of types in $T$ from all right-hand sides; then, $\\mathsf {F}T$ in the original BP corresponds to finiteness in the new BP.",
"Hence, the result follows from prop:as-finite-NC."
],
[
"Proof of thm:DPA-1",
"* In [8] it was shown that the problem $\\mathbb {P}(\\textup {DPA}) = 1$ can be decided in polynomial time.",
"We show how to implement this approach in NC.",
"In [8] a product of the BP and the DPA is computed and analyzed; the product is a BP whose types are coloured with priorities (natural numbers).",
"The product, call it $\\mathcal {B}= (\\Gamma , \\mathord {{}}, \\mathit {Prob}, X_0)$ , can be computed in logarithmic space.",
"The question is then whether the probability is 1 that all branches of a $\\mathcal {B}$ -tree are such that the highest priority that appears infinitely often is even.",
"It is shown in [8] that this is the case if and only if all types $X$ that are reachable from $X_0$ and are associated with an odd priority satisfy $\\mathbb {P}_X(\\mathsf {F}N_X) = 1$ , where $N_X \\subseteq \\Gamma $ is a certain set of types that can be computed in NL by a simple reachability analysis in $\\mathcal {B}$ .",
"By lem:AFT-1 one can determine in NC whether $\\mathbb {P}_X(\\mathsf {F}N_X) = 1$ .",
"The theorem follows."
],
[
"Proof of thm:DPA-0",
"* Membership in P was shown in [8].",
"For P-hardness we reduce from the monotone circuit value problem.",
"Given a monotone circuit with output gate $g_{\\mathit {out}}$ , we construct a BP $\\mathcal {B}= (\\Gamma ,\\mathord {{}},\\mathit {Prob},X_{g_{\\mathit {out}}})$ and a DBA $\\mathcal {A}$ such that $\\mathbb {P}_{X_{g_{\\mathit {out}}}}(\\mathcal {A}\\text{ accepts}) > 0$ if and only if $g_{\\mathit {out}}$ evaluates to 1.",
"For each $\\wedge $ - and each $\\vee $ -gate $g$ include a type $X_g \\in \\Gamma $ .",
"Also include types $X_0, X_1 \\in \\Gamma $ for the inputs $0, 1$ , respectively.",
"For each $\\wedge $ -gate $g$ with children $g_1, \\ldots , g_k$ include a rule $X_g &{1} X_{g_1} \\cdots X_{g_k}\\,.\\multicolumn{2}{l}{\\text{For each $\\vee $-gate~$g$ with children $g_1, \\ldots , g_k$ include rules}}\\\\X_g &{1/k} X_{g_1},\\ \\ldots ,\\ X_g {1/k} X_{g_k}\\,.\\multicolumn{2}{l}{\\text{Include rules}}\\\\X_0 &{1} X_0 \\text{ and }X_1 {1} X_1\\,.$ Construct a two-state DBA $\\mathcal {A}$ that accepts exactly those $w \\in \\Gamma ^\\omega $ that contain $X_1$ .",
"It follows from a straightforward induction over the gate height (longest distance to an input) that, for any gate $g$ , it evaluates to 1 if and only if $\\mathbb {P}_{X_g}(\\mathcal {A}\\text{ accepts}) > 0$ ."
],
[
"Additional Definitions Concerning Alternating Turing Machines",
"An alternating Turing machine is a 6-tuple $(S_\\exists , S_\\forall , \\Sigma , T, s_0, s_{\\mathit {acc}})$ , where $S = S_\\exists \\cup S_\\forall \\cup \\lbrace s_{\\mathit {acc}}\\rbrace $ is a finite set of (control) states partitioned into existential states $S_\\exists $ and universal states $S_\\forall $ and the (only) accepting state $s_{\\mathit {acc}}$ , $\\Sigma $ is a finite alphabet, $T \\subseteq (S_\\exists \\cup S_\\forall ) \\times \\Sigma \\times \\Sigma \\times \\lbrace -1,+1\\rbrace \\times S$ is a transition relation, and $s_0$ is the initial state.",
"A transition $(s,a,a^{\\prime },D,s^{\\prime }) \\in T$ means that if $M$ is in state $s$ and its head reads letter $a$ , then it rewrites the content of the current cell with the letter $a^{\\prime }$ , it moves the head in direction $D$ (either left if $D=-1$ , or right if $D=+1$ ), and it changes its state to $s^{\\prime }$ .",
"We assume that for all $s \\in S_\\exists \\cup S_\\forall $ and $a \\in \\Sigma $ there is at least one outgoing transition.",
"A configuration of an alternating Turing machine is given by a 3-tuple $(i, s, w)$ where $i \\in \\mathbb {N}$ indicates the header position, $s \\in S$ is the current state of the Turing machine, and $w \\in \\Sigma ^*$ is the contents of the memory tape.",
"For a word $w \\in \\Sigma ^*$ we will write $w(i)$ to denote its $i$ 'th component, and $w[i=a]$ to denote the string $w(1)\\ldots w(i-1) a w(i+1)\\ldots w(n)$ .",
"The initial configuration of an alternating Turing machine on a word $w \\in \\Sigma ^*$ is $(1, s_0, w)$ , and given a transition $t = (s, a, a^{\\prime }, D, s^{\\prime })$ and a configuration $c = (i, s^{\\prime \\prime }, w)$ we will use $c \\triangleright t$ to denote the configuration $(i+D, s^{\\prime }, w[i=a^{\\prime }])$ if $s^{\\prime \\prime } = s$ and $w_i = a$ (and leave it undefined otherwise).",
"We extend this notation to sequences of transitions $t_1\\ldots t_n \\in T^*$ by writing $c \\triangleright t_1\\ldots t_n = (c \\triangleright t_1) \\triangleright t_2\\ldots t_n$ .",
"We will use $\\pi _\\mathbb {N}$ , $\\pi _S$ , and $\\pi _{\\Sigma ^*}$ to denote the projections of a configuration $(i, s, w)$ to $i$ , $s$ , and $w$ , respectively.",
"For any $(s, a) \\in (S_\\exists \\cup S_\\forall ) \\times \\Sigma $ , let $T_{s,a} := \\lbrace (s,a,a^{\\prime },D,s^{\\prime }) \\in T \\mid a^{\\prime } \\in \\Sigma ,\\ D \\in \\lbrace -1,+1\\rbrace ,\\ s^{\\prime } \\in S\\rbrace $ .",
"A string $r = r_1r_2\\ldots \\in T^*\\cup T^\\omega $ is called a run of $M$ on $w$ if for all $i$ , $r_{i+1} \\in T_{\\pi _S(c_{i}), \\pi _{\\Sigma ^*}(c_{i})(\\pi _\\mathbb {N}(c_{i}))}$ , where $c_i = c_0\\triangleright r_1\\ldots r_i$ whenever $r_{i+1}$ exists.",
"Any run $r$ is called an accepting run if $r \\in T^*$ and $\\pi _S(c_0\\triangleright r) = s_{\\mathit {acc}}$ .",
"A strategy is a function $\\sigma : \\lbrace (i, s, w, r) \\in \\mathbb {N}\\times S \\times \\Sigma ^* \\times T^* \\mid s \\in S_\\exists \\rbrace \\rightarrow T$ such that $\\sigma (i, s, w, r) \\in T_{s, w_i}$ for any $i, s, w, r$ .",
"For a configuration $c$ and a run $r$ , by abuse of notation, we will write $\\sigma (c, r)$ to mean $\\sigma (\\pi _\\mathbb {N}(c), \\pi _S(c), \\pi _{\\Sigma ^*}(c), r)$ .",
"A run is consistent with a strategy $\\sigma $ if for all $i$ , $r_{i+1} = \\sigma (c_0\\triangleright r_1\\ldots r_i, r_1\\ldots r_i)$ whenever the latter is defined.",
"The behaviour of $\\sigma $ is the set of runs consistent with $\\sigma $ .",
"A winning strategy is a strategy $\\sigma $ such that every run $r$ consistent with $\\sigma $ is an accepting run.",
"Note that whenever $\\sigma $ is winning, its behaviour is finite.",
"Let $\\sigma $ be a strategy, then we call $\\sigma $ $N$ -bounded if for any $r$ in the behavior of $\\sigma $ , and any prefix $r_1\\ldots r_k$ of $r$ , $|\\pi _{\\Sigma ^*}(c_0\\triangleright r_1\\ldots r_k)| < N$ .",
"A Turing machine is $N$ -bounded if it has an $N$ -bounded winning strategy."
],
[
"Proof of thm:NBA-0",
"* Given the main body, it remains to prove EXPTIME-hardness of $\\mathbb {P}(\\textup {NBA}) = 0$ .",
"Recall that alternating PSPACE equals EXPTIME.",
"We give a polynomial-time reduction from the problem of acceptance of a word by a PSPACE-bounded alternating Turing machine.",
"Without loss of generality, we can assume that the Turing machine is linear-bounded, i.e., uses only the space occupied by the input word.",
"Let $M = (S_\\exists , S_\\forall , \\Sigma , T, s_0, s_{\\mathit {acc}})$ be a linear-bounded alternating Turing machine.",
"Let $w = a_1 \\cdots a_n \\in \\Sigma ^*$ be the input word.",
"As mentioned before, we can assume that $M$ uses exactly $n$ tape cells.",
"We construct a BP $\\mathcal {B}$ and an NBA $\\mathcal {A}$ such that $\\mathbb {P}_\\mathcal {B}(\\mathcal {A}\\text{ accepts}) > 0$ if and only $M$ accepts $w$ .",
"The BP $\\mathcal {B}$ has the following set of types: $\\Gamma \\ = \\ (\\lbrace 1, \\ldots , n\\rbrace \\times S \\times \\Sigma ) \\ \\cup \\ (\\lbrace 1, \\ldots , n\\rbrace \\times T) \\ \\cup \\ (\\lbrace 1, \\ldots , n\\rbrace \\times \\lbrace \\mathit {chk}\\rbrace ) \\ \\cup \\ \\lbrace E\\rbrace $ Intuitively, a type $(i,s,a)$ means that the head is at position $i$ , the current state is $s$ , and the head is reading letter $a$ ; a type $(i,t)$ means that the head is at position $i$ , and transition $t$ is being executed; a type $(i,\\mathit {chk})$ means that the accepting state has been reached, and cell $i$ is being “checked” (in a sense to be explained later); type $E$ indicates an error.",
"The type $(1,s_0,a_1)$ is the start type of $\\mathcal {B}$ .",
"For all $(i,s,a) \\in \\Gamma $ with $s \\in S_\\exists $ , include rules $(i,s,a) {1/k} (i,t_j) \\quad (1 \\le j \\le k)\\,,$ where $\\lbrace t_1, \\ldots , t_k\\rbrace = T_{s,a}$ .",
"(Intuitively, a transition going out of an existential state is chosen as the only child, uniformly at random.)",
"For all $(i,s,a) \\in \\Gamma $ with $s \\in S_\\forall $ , include a rule $(i,s,a) {1} (i,t_1) \\cdots (i,t_k) \\,,$ where $\\lbrace t_1, \\ldots , t_k\\rbrace = T_{s,a}$ .",
"(Intuitively, all possible transitions going out of a universal state are children.)",
"For all $(i,(s,a,a^{\\prime },D,s^{\\prime })) \\in \\Gamma $ with $1 \\le i{+}D \\le n$ , include rules $(i,(s,a,a^{\\prime },D,s^{\\prime })) {1/|\\Sigma |} (i{+}D, s^{\\prime }, b_j) \\quad (1 \\le j \\le |\\Sigma |)\\,,$ where $\\lbrace b_1, \\ldots , b_{|\\Sigma |}\\rbrace = \\Sigma $ .",
"(Intuitively, the letter in the cell at the new head position $i{+}D$ is guessed uniformly at random.)",
"For all $(i,(s,a,a^{\\prime },D,s^{\\prime })) \\in \\Gamma $ with $i{+}D \\in \\lbrace 0,n{+}1\\rbrace $ , include a rule $(i,(s,a,a^{\\prime },D,s^{\\prime })) {1} E\\,.$ (Intuitively, when the space bound is exceeded, move to the error type $E$ .)",
"Include a rule $E {1} E$ (i.e., a self-loop).",
"For all $(i, s_{\\mathit {acc}}, a) \\in \\Gamma $ , include a rule $(i, s_{\\mathit {acc}}, a) {1} (1,\\mathit {chk}) \\cdots (n,\\mathit {chk})\\,.$ (Intuitively, after reaching $s_{\\mathit {acc}}$ all $n$ cells are “checked”.)",
"For all $(i,\\mathit {chk}) \\in \\Gamma $ , include a rule $(i,\\mathit {chk}) {1} (i,\\mathit {chk})$ (i.e., a self-loop).",
"The NBA $\\mathcal {A}= (Q, \\Gamma , \\delta , Q_0, \\lbrace f\\rbrace )$ has the following set of states: $Q \\ = \\ (\\lbrace 1, \\ldots , n\\rbrace \\times \\Sigma ) \\ \\cup \\ \\lbrace f\\rbrace $ The set of initial states is $Q_0 = \\lbrace (1,a_1), \\ldots , (n,a_n)\\rbrace $ (recall that $a_1 \\cdots a_n$ is the input word).",
"The idea is that if a prefix $X_1 \\cdots X_k \\in \\Gamma ^*$ of a tree branch corresponds to a prefix of a correct computation of $M$ , then the set of automaton states in $\\delta (Q_0,X_1 \\cdots X_k)$ corresponds to the tape after this computation prefix.",
"In fact, the transition relation $\\delta \\subseteq Q \\times \\Gamma \\times Q$ is deterministic, i.e., for all $q \\in Q$ and $X \\in \\Gamma $ there is at most one $q^{\\prime }$ with $(q,X,q^{\\prime }) \\in \\delta $ .",
"Moreover, for any transition $((i,a), X, (j,a^{\\prime })) \\in \\delta $ we will have $i=j$ .",
"For $(i,a) \\in Q$ and all $(j,s,b) \\in \\Gamma $ with $i \\ne j$ or $a=b$ , include a self-loop $((i,a),(j,s,b),(i,a)) \\in \\delta \\,.$ (Intuitively, letter $a$ in cell $i$ is compatible with the head being on cell $j$ and reading letter $b$ .)",
"For all $(i,a) \\in Q$ and all $(j,t) \\in \\Gamma $ with $i \\ne j$ , include a self-loop $((i,a),(j,t),(i,a)) \\in \\delta \\,.$ (Intuitively, the content of cell $i$ stays unchanged when the head is at position $j$ .)",
"For all $i,s,a,a^{\\prime },D,s^{\\prime }$ with $(i,a) \\in Q$ and $(i,(s,a,a^{\\prime },D,s^{\\prime })) \\in \\Gamma $ , include a transition $((i,a), (i,(s,a,a^{\\prime },D,s^{\\prime })), (i,a^{\\prime }))\\in \\delta \\,.$ (Intuitively, the transition $(s,a,a^{\\prime },D,s^{\\prime })$ changes the content of cell $i$ from $a$ to $a^{\\prime }$ .)",
"For all $(i,a) \\in Q$ , include a transition $((i,a),(i,\\mathit {chk}),f) \\in \\delta \\,.$ (Intuitively, the type $(i,\\mathit {chk})$ checks if the computation has been consistent in cell $i$ .)",
"For all $X \\in \\Gamma $ , include a self-loop $(f,X,f) \\in \\delta $ .",
"We will show that in this case, $\\mathbb {P}_\\mathcal {B}(\\mathcal {A}\\text{ accepts}) > 0$ if and only if $M$ accepts $w$ .",
"Firstly note that $f$ is a sink state in $\\mathcal {A}$ , and hence, if $\\mathcal {A}$ reaches $f$ after reading some prefix of a branch, it will accept any branch with this prefix.",
"This means that a tree $t$ is accepted by $\\mathcal {A}$ if and only if there exists a finite prefix of $t$ such that $\\mathcal {A}$ reaches $f$ on all of its branches.",
"Assume that $M$ accepts $w$ .",
"Then there exists a linear-bounded winning strategy $\\sigma $ for the existential player.",
"We will write $c_0$ for the initial configuration of $M$ on $w$ , $R$ for the behaviour of $\\sigma $ , and $N$ for the length of the longest run in $R$ .",
"We will show that there exists a finite prefix generated by $\\mathcal {B}$ with nonzero probability that precisely models this strategy, and that is such that $\\mathcal {A}$ reaches $f$ on all of its branches.",
"Since any tree with this prefix is accepted, this implies that $\\mathbb {P}_\\mathcal {B}(\\mathcal {A}\\text{ accepts}) > 0$ if $M$ accepts $w$ .",
"Let $t$ be the tree defined as follows: the root of $t$ is $(1, s_0, a_1)$ , for any node $(p_i, s_i, b_i)$ on level $2i$ with $s_i \\in S_\\exists $ , let $t_1\\ldots t_{i}$ be such that the branch from the root to $(p_i, s_i, b_i)$ has nodes $(p_j, t_{j+1})$ on level $2j+1$ for each $0 \\le j < i$ .",
"Then $(p_i, s_i, b_i)$ has a child $(p_i, \\sigma (c_0 \\triangleright t_1\\ldots t_{i}, t_1\\ldots t_i))$ , for any node $(p_i, s_i, b_i)$ on level $2i$ with $s_i \\in S_\\forall $ , let $t_1\\ldots t_{i}$ be such that the branch from the root to $(p_i, s_i, b_i)$ has nodes $(p_j, t_{j+1})$ on level $2j+1$ for each $0 \\le j < i$ .",
"Let $w_i = \\pi _{\\Sigma ^*}(c_0 \\triangleright t_1\\ldots t_{i})$ .",
"Then $(p_i, s_i, b_i)$ has children $(p_i, t_{i+1})$ for each $t_{i+1} \\in T_{s_i, w_i(p_i)}$ , for any node $(p_i, t_{i+1})$ with $t_{i+1} = (s, a, a^{\\prime }, D, s^{\\prime })$ on level $2i+1$ , let $t_1\\ldots t_{i}$ be such that the branch from the root to $(p_i, t_{i+1})$ has nodes $(p_j, t_{j+1})$ on level $2j+1$ for each $0 \\le j < i$ .",
"Let $c_{i+1} = c_0 \\triangleright t_1\\ldots t_{i+1}$ .",
"Then $(p_i, t_{i+1})$ has a child $(p_i+D, s^{\\prime }, \\pi _{\\Sigma ^*}(c_{i+1})(p_i+D))$ , any node $(p_i, s_{\\mathit {acc}}, b_i)$ has children $(j, \\mathit {chk})$ for each $1 \\le j \\le n$ , and any node $(j, \\mathit {chk})$ has a child $(j, \\mathit {chk})$ .",
"By construction, $t$ is now such that for any node $(p_i, t_{i+1})$ with $t_{i+1} = (s, a, a^{\\prime }, D, s^{\\prime })$ on level $2i+1$ , there exist runs in $R$ prefixed by $t_1 \\ldots t_{i+1}$ , where $t_1\\ldots t_{i}$ are such that the branch from the root to $(p_i, t_{i+1})$ has nodes $(p_j, t_{j+1})$ on level $2j+1$ for each $0 \\le j < i$ .",
"Hence, by the fact that $\\sigma $ is linear-bounded, $1 \\le p_i+D \\le n$ and thus all the transitions in $t$ are according to the rules of $\\mathcal {B}$ .",
"Moreover, since the length of runs in $R$ is bounded by $N$ , all the nodes at level $2N+1$ must be of the form $(i, \\mathit {chk})$ , and since all states $(i, \\mathit {chk})$ are sink states, $t$ is generated with a nonzero probability.",
"Finally pick any state $(i, \\mathit {chk})$ in $t$ .",
"The only types from which $(i, a) \\in Q$ does not have an outgoing edge are of the form $(i, s, b)$ .",
"However, for any state $(i, s, b)$ at level $2j$ on the branch to $(i, \\mathit {chk})$ , let $t_1\\ldots t_{j}$ be such that the branch from the root to $(i,s,b)$ has nodes $(p_k, t_{k+1})$ on level $2k+1$ for each $0 \\le k < j$ .",
"Then $(i, b) = (i, \\pi _{\\Sigma ^*}(c_0 \\triangleright t_1\\ldots t_{j})(i)) = \\delta ((i,w(i)), (p_0, t_1)\\ldots (p_{j-1}, t_{j}))$ and hence $(i, a)$ survives, and reaches $f$ upon reading $(i, \\mathit {chk})$ .",
"For the other direction, assume that $\\mathbb {P}_\\mathcal {B}(\\mathcal {A}\\text{ accepts}) > 0$ .",
"Then there exists a prefix $t$ generated by $\\mathcal {B}$ with branches of length $N$ for some $N$ such that $\\mathcal {A}$ reaches $f$ on all of its branches.",
"W.l.o.g.",
"we can assume that $\\pi _S(c_0) \\ne s_{\\mathit {acc}}$ , because otherwise any strategy is winning.",
"Note that for any accepted tree $t$ and any branch prefix $(1, s_0, a_1)(1, t_1)\\ldots (p_i, s_i, b_i)(p_i, t_{i+1})$ in $t$ , the set of reachable states in the automaton from $Q_0$ reflects the tape contents of $M$ , ie.",
"$\\delta (Q_0, (1, s_0, a_1)(1, t_1)\\ldots (p_i, s_i, b_i)(p_i, t_{i+1}))=\\lbrace (j, w(j))\\mid 1 \\le j \\le n, w = \\pi _{\\Sigma ^*}(c_0 \\triangleright t_1\\ldots t_{i+1})\\rbrace $ .",
"Also $s_i = \\pi _S(c_0 \\triangleright t_1\\ldots t_i)$ .",
"If this was not the case, then there would exist $j$ such that $\\delta ((j, a_j), (1, s_0, a_1)(1, t_1)\\ldots (p_i, s_i, b_i)(p_i, t_{i+1})) = \\emptyset $ and hence the branch reaching $(j, \\mathit {chk})$ prefixed by $(1, s_0, a_1)(1, t_1)\\ldots (p_i, s_i, b_i)(p_i, t_{i+1})$ does not reach $f$ .",
"Let $n_0n_1\\ldots $ be any branch in $t$ .",
"Pick $m$ such that $n_{2i+1} = (a_i, t_{i+1})$ for all $0 \\le i < m$ and $n_{2m+1} = (a_m, t_{m+1})$ .",
"Let $c = c_0 \\triangleright t_1\\ldots t_m$ .",
"Then we define $\\sigma $ to be any strategy such that for any such $c$ , if $\\pi _S(c) \\in S_\\exists $ , then $\\sigma (c, t_1\\ldots t_m) = t_{m+1}$ .",
"This is a valid strategy since for any branch in $t$ , the set of reachable states of the automaton after reading a branch prefix reflects the tape contents of $M$ and the alphabet characters contained in the nodes of the branch prefix have to reflect the automaton states.",
"Note that $\\sigma $ is $n$ -bounded.",
"We claim that $\\sigma $ is a winning strategy and hence $M$ accepts $w$ .",
"Let $r = t_1t_2\\ldots $ be a run consistent with $\\sigma $ .",
"We will show that there exists a branch $(1, s_0, a_1)(1, t_1)\\ldots (p_i, t_{i+1})(p_{i+1}, s_{\\mathit {acc}}, w_{i+1}(p_{i+1}))$ in $t$ with $p_i = \\pi _\\mathbb {N}(c_0\\triangleright t_1\\ldots t_i)$ , $s_i = \\pi _S(c_0\\triangleright t_1\\ldots t_i)$ , and $w_i = \\pi _{\\Sigma ^*}(c_0\\triangleright t_1\\ldots t_i)$ (and hence $|r| = i+1$ and $\\pi _S(c_0\\triangleright r) = s_{\\mathit {acc}}$ ).",
"For any branch prefix $(1, s_0, a_1)(1, t_1)\\ldots (p_j, t_{j+1})$ where $t_{j+1} = (s, a, a^{\\prime }, D, s^{\\prime })$ , according to the rules of $\\mathcal {B}$ , $(p_j, t_{j+1})$ has a single child $(p_i+D, s^{\\prime }, b)$ .",
"Let $c_{j+1} = c_0 \\triangleright t_1\\ldots t_{j+1}$ .",
"Since trees prefixed by $t$ are accepted, $p_i+D = \\pi _\\mathbb {N}(c_{j+1})$ , $s^{\\prime } = \\pi _S(c_{j+1})$ , and $b = \\pi _{\\Sigma ^*}(c_{j+1})(\\pi _\\mathbb {N}(c_{j+1}))$ .",
"For any branch prefix $(1, s_0, a_1)(1, t_1)\\ldots (p_{j-1}, t_{j})(p_j, s_j, b_j)$ where $s_j \\in S_\\forall $ , according to the rules of $\\mathcal {B}$ , $(p_j, s_j, b_j)$ has children $(p_j, t^{\\prime })$ for each $t^{\\prime } \\in T_{s_j, b_j}$ .",
"Let $c_j = c_0\\triangleright t_1\\ldots t_j$ .",
"Since trees prefixed by $t$ are accepted, $s_j = \\pi _S(c_j)$ and $b_j = \\pi _{\\Sigma ^*}(c_j)(\\pi _\\mathbb {N}(c_j))$ .",
"Hence, $t_{j+1} \\in T_{s_j, b_j}$ and $(p_j, s_j, b_j)$ has a child $(p_j, t_{j+1})$ .",
"For any branch prefix $(1, s_0, a_1)(1, t_1)\\ldots (p_{j-1}, t_{j})(p_j, s_j, b_j)$ where $s_j \\in S_\\exists $ , according to the rules of $\\mathcal {B}$ , $(p_j, s_j, b_j)$ has a single child $(p_j, t^{\\prime })$ .",
"By definition of $\\sigma $ , $\\sigma (c_0\\triangleright t_1\\ldots t_j, t_1\\ldots t_j) = t^{\\prime }$ .",
"Since $t$ is finite, every branch reaches states of the form $(p, s_{\\mathit {acc}}, b)$ in a finite number of steps.",
"Hence, $r$ is finite and reaches $s_{\\mathit {acc}}$ and since $r$ is any run consistent with $\\sigma $ , $\\sigma $ is an winning strategy.",
"Thus, $M$ accepts $w$ ."
],
[
"Proof of lem:UBA-X1-f",
"* Consider a branch $X_0 X_1 X_2 \\cdots $ of a $\\mathcal {B}$ -tree accepted by $\\mathcal {A}$ .",
"Then $\\mathcal {A}$ has an accepting run $q_0 \\xrightarrow{} q_1 \\xrightarrow{} q_2 \\xrightarrow{} \\cdots $ with $q_0 \\in Q_0$ .",
"By the pigeonhole principle, there are $f \\in F$ and $X_f \\in \\Gamma $ such that this run contains the segment $f \\xrightarrow{}$ infinitely often.",
"By its construction, the Büchi automaton $\\mathcal {A}\\times \\mathcal {B}$ has the accepting run $(q_0,X_0) \\xrightarrow{} (q_1,X_1) \\xrightarrow{} \\cdots \\,,$ which contains the state $(f,X_f)$ infinitely often.",
"Let $k \\ge 1$ be such that $(q_k,X_k) = (f,X_f)$ .",
"Then $\\mathcal {A}[f,X_f]$ accepts $X_k X_{k+1} \\cdots $ via the run $\\bar{q}_0 \\xrightarrow{} (q_k,X_k) \\xrightarrow{} (q_{k+1},X_{k+1}) \\cdots \\,.$ Therefore, denoting by $E(f,X_f,n)$ for $n \\in \\mathbb {N}$ the event that there exists a branch $X_k X_{k+1} \\cdots $ emanating from the $n$ th (in a breadth-first order) node in the tree (necessarily a node of type $X_k = X_f$ ) such that $\\mathcal {A}[f,X_f]$ has an accepting run $\\bar{q}_0 \\xrightarrow{} (q_k,X_k) \\xrightarrow{} (q_{k+1},X_{k+1}) \\cdots \\,,$ we have $\\mathbb {P}_{X_0}(\\mathcal {A}\\text{ accepts some branch})\\ \\le \\ &\\sum _{f \\in F} \\sum _{X_f \\in \\Gamma } \\sum _{n \\in \\mathbb {N}} \\mathbb {P}_{X_0}(E(f,X_f,n))\\,.$ Further, $&\\ \\mathbb {P}_{X_0}(E(f,X_f,n)) \\\\=&\\ \\mathbb {P}_{X_0}(\\text{the $n$th node has type~$X_f$}) \\cdot \\mathbb {P}_{X_f}(\\mathcal {A}[f,X_f] \\text{ accepts some branch}) \\\\\\le &\\ \\mathbb {P}_{X_f}(\\mathcal {A}[f,X_f] \\text{ accepts some branch})\\,.$ The “only if” direction follows.",
"Towards the “if” direction, suppose that $\\mathcal {A}\\times \\mathcal {B}$ has a path $(q_0,X_0) {\\;\\xrightarrow{}{}\\hspace{0.0pt}^{*}\\;} (f,X_f)$ with $q_0 \\in Q_0$ and $f \\in F$ such that the probability that some branch of a $\\mathcal {B}[X_f]$ -tree is accepted by $\\mathcal {A}[f,X_f]$ is positive.",
"Then there is a successor, say $X_f^{\\prime }$ , of $X_f$ such that the probability that some branch of a $\\mathcal {B}[X_f^{\\prime }]$ -tree is accepted by $\\mathcal {A}\\times \\mathcal {B}$ when started in $(f,X_f)$ is positive.",
"Thus, the probability that some branch of a $\\mathcal {B}$ -tree (starts with $X_0 w X_f^{\\prime }$ and) is accepted by $\\mathcal {A}$ is positive."
],
[
"Proof of lem:Bdet",
"In this subsection we prove lem:Bdet, which is instrumental for the main results of the paper.",
"* Fix a word $w \\in \\Gamma ^+$ such that $\\delta [f,X_f]((f,X_f),w)$ is maximal, i.e., there is no $w^{\\prime } \\in \\Gamma ^+$ such that $\\delta [f,X_f]((f,X_f),w^{\\prime }) \\supsetneq \\delta [f,X_f]((f,X_f),w)$ .",
"Let $v \\in \\Gamma ^*$ be a word such that $\\delta [f,X_f]((f,X_f),v) \\ni (f,X_f)$ .",
"Then, $\\delta [f,X_f]((f,X_f),v w) = \\delta [f,X_f]((f,X_f), w)$ ; i.e., for any path $(f,X_f) {\\;\\xrightarrow{}{}\\hspace{0.0pt}^{*}\\;} (q,X)$ there is a path $(f,X_f) {\\;\\xrightarrow{}{}\\hspace{0.0pt}^{*}\\;} (f,X_f) {\\;\\xrightarrow{}{}\\hspace{0.0pt}^{*}\\;} (q,X)$ .",
"Since $\\delta [f,X_f]((f,X_f),v) \\ni (f,X_f)$ , we have $\\delta [f,X_f]((f,X_f),v w) \\ \\supseteq \\ \\delta [f,X_f]((f,X_f), w)\\,.$ But $w$ is maximal.",
"We enrich $\\mathcal {A}_{\\mathit {det}}$ to obtain a DBA, $\\mathcal {A}_{\\mathit {det}}^{\\prime }$ , whose states have an additional component keeping track of whether the word $w \\in \\Gamma ^+$ from above is being seen.",
"The accepting runs of $\\mathcal {A}_{\\mathit {det}}^{\\prime }$ contain infinitely many $w$ -labelled segments that, loosely speaking, “start from $(f,X_f)$ ”.",
"Formally, let $W = \\lbrace 0\\rbrace \\cup \\lbrace v \\in \\Gamma ^* \\mid v \\text{ is a suffix of } w\\rbrace \\,,$ including $w$ and the empty word $\\varepsilon $ .",
"We assume $0 \\notin \\Gamma $ .",
"Define $\\mathcal {A}_{\\mathit {det}}^{\\prime } \\ :=\\ (2^{\\lbrace \\bar{q}_0\\rbrace \\cup Q[f,X_f]} \\times W, \\delta _{\\mathit {det}}^{\\prime }, (\\lbrace \\bar{q}_0\\rbrace ,0), (2^{Q[f,X_f]} \\setminus \\lbrace \\emptyset \\rbrace ) \\times \\lbrace \\varepsilon \\rbrace )\\,,$ where $\\delta _{\\mathit {det}}^{\\prime }((U,X v), X) \\ &= \\ (\\delta _{\\mathit {det}}(U,X), v) \\\\\\delta _{\\mathit {det}}^{\\prime }((U,X v), Y) \\ &= \\ (\\delta _{\\mathit {det}}(U,X), 0) \\quad &&\\text{for } X \\ne Y \\\\\\delta _{\\mathit {det}}^{\\prime }((U,v), X) \\ &= \\ (\\delta _{\\mathit {det}}(U,X), w) \\quad &&\\text{for } v \\in \\lbrace 0, \\varepsilon \\rbrace ,\\ (f,X_f) \\in \\delta _{\\mathit {det}}(U,X) \\\\\\delta _{\\mathit {det}}^{\\prime }((U,v), X) \\ &= \\ (\\delta _{\\mathit {det}}(U,X), 0) \\quad &&\\text{for } v \\in \\lbrace 0, \\varepsilon \\rbrace ,\\ (f,X_f) \\notin \\delta _{\\mathit {det}}(U,X)\\,.$ It follows from lem:UBA-w and the construction of $\\mathcal {A}_{\\mathit {det}}^{\\prime }$ that $\\mathcal {A}_{\\mathit {det}}^{\\prime }$ has a single accepting state reachable from $(\\lbrace \\bar{q}_0\\rbrace ,0)$ , namely $(\\delta _{\\mathit {det}}(\\lbrace (f,X_f)\\rbrace ,w), \\varepsilon )$ .",
"The following statements are equivalent: The probability that some branch of a $\\mathcal {B}[X_f]$ -tree is accepted by $\\mathcal {A}[f,X_f]$ is positive.",
"The probability that some branch of a $\\mathcal {B}[X_f]$ -tree has a run (accepting or not) in $\\mathcal {A}[f,X_f]$ is positive.",
"The probability that some branch of a $\\mathcal {B}[X_f]$ -tree is accepted by $\\mathcal {A}_{\\mathit {det}}^{\\prime }$ is positive.",
"lem:UBA-3-equivalences implies lem:Bdet, as it follows from the definition of $\\mathcal {B}_{\\mathit {det}}$ that the probability that some branch of a $\\mathcal {B}[X_f]$ -tree has a run in $\\mathcal {A}[f,X_f]$ (cf.",
"condition (ii) in lem:UBA-3-equivalences) equals the probability that some branch of a $\\mathcal {B}_{\\mathit {det}}$ -tree does not have any nodes of type $\\emptyset $ (cf.",
"condition (ii) of lem:Bdet).",
"So it remains to prove lem:UBA-3-equivalences.",
"[Proof of lem:UBA-3-equivalences] (i) $\\Longrightarrow $ (ii).",
"Trivial.",
"(iii) $\\Longrightarrow $ (i).",
"Let $X_f X_1 X_2 \\cdots $ be accepted by $\\mathcal {A}_{\\mathit {det}}^{\\prime }$ .",
"Then $X_1 X_2 \\cdots $ can be decomposed in $v_1 w v_2 w \\cdots $ with $v_1, v_2, \\ldots \\in \\Gamma ^*$ such that $\\mathcal {A}[f,X_f]$ has paths $(f,X_f) {\\;\\xrightarrow{}{}\\hspace{0.0pt}^{*}\\;} (f,X_f)$ for all $i \\ge 1$ .",
"Let $i \\ge 2$ .",
"There is $(q,X)$ with $(f,X_f) {\\;\\xrightarrow{}{}\\hspace{0.0pt}^{*}\\;} (q,X) {\\;\\xrightarrow{}{}\\hspace{0.0pt}^{*}\\;} (f,X_f)\\,.$ By lem:UBA-w there is a path $(f,X_f) {\\;\\xrightarrow{}{}\\hspace{0.0pt}^{*}\\;} (f,X_f) {\\;\\xrightarrow{}{}\\hspace{0.0pt}^{*}\\;} (q,X)\\,.$ Thus also $(f,X_f) {\\;\\xrightarrow{}{}\\hspace{0.0pt}^{*}\\;} (f,X_f)$ .",
"Since $i \\ge 2$ was arbitrary, it follows that $\\mathcal {A}[f,X_f]$ has an accepting run $\\bar{q}_0 \\xrightarrow{} (f,X_f) {\\;\\xrightarrow{}{}\\hspace{0.0pt}^{*}\\;} (f,X_f) {\\;\\xrightarrow{}{}\\hspace{0.0pt}^{*}\\;} (f,X_f) {\\;\\xrightarrow{}{}\\hspace{0.0pt}^{*}\\;} \\cdots \\,.$ (ii) $\\Longrightarrow $ (iii).",
"For this part we use results from [8], which considers the problem of model checking BPs against DPAs.",
"We need only a special case of such automata: (a) the same (word) automaton is run on every branch of the tree, and (b) our automaton $\\mathcal {A}_{\\mathit {det}}^{\\prime }$ is a DBA, which can be viewed as a DPA whose states are labelled with only 2 priorities: priority 0 for non-accepting states and priority 1 for accepting states.",
"The paper [8] constructs a product BP from the BP and the automaton, and subsequently considers BPs whose types are coloured with a priority.",
"In this way the model-checking problem reduces to computing the probability that there is a branch on which the highest priority that occurs infinitely often is odd.",
"Since our automaton $\\mathcal {A}_{\\mathit {det}}^{\\prime }$ already embeds a BP, instead of taking another product, we define a BP, $\\mathcal {B}_{\\mathit {det}}^{\\prime }$ , more directly based on $\\mathcal {A}_{\\mathit {det}}^{\\prime }$ , in the same way that $\\mathcal {B}_{\\mathit {det}}$ was defined based on $\\mathcal {A}_{\\mathit {det}}$ in sub:Bdet.",
"More explicitly, $\\mathcal {B}_{\\mathit {det}}^{\\prime } \\ := \\ (\\Gamma ^{\\prime }, \\mathord {{}^{\\prime }}, \\mathit {Prob}^{\\prime }, (\\lbrace (f,X_f)\\rbrace ,w))\\,,$ where the set of types $\\Gamma ^{\\prime } \\subseteq 2^{Q[f,X_f]} \\times W$ is the set of those states in $\\mathcal {A}_{\\mathit {det}}^{\\prime }$ that are reachable (in $\\mathcal {A}_{\\mathit {det}}^{\\prime }$ ) from $(\\lbrace \\bar{q}\\rbrace ,0)$ via a nonempty path (recall that they are of the form $(P \\times \\lbrace X\\rbrace ,v)$ with $P \\subseteq Q$ and $X \\in \\Gamma $ and $v \\in W$ ), and $X^{\\prime } &{\\;{p}{}\\hspace{0.0pt}{^{\\prime }}\\;} \\delta _{\\mathit {det}}^{\\prime }(X^{\\prime },X_1) \\cdots \\delta _{\\mathit {det}}^{\\prime }(X^{\\prime },X_k)\\multicolumn{2}{l}{\\text{for all $X^{\\prime } = (P \\times \\lbrace X\\rbrace ,v) \\in \\Gamma ^{\\prime }$ with $P \\ne \\emptyset $ and all $X {p} X_1 \\cdots X_k$, and}}\\\\(\\emptyset ,v) &{\\;{1}{}\\hspace{0.0pt}{^{\\prime }}\\;} (\\emptyset ,v)$ for all $v \\in W$ .",
"Recall that $(\\delta _{\\mathit {det}}(\\lbrace (f,X_f)\\rbrace ,w), \\varepsilon ) =: X_w^{\\prime }$ is the single accepting state in $\\mathcal {A}_{\\mathit {det}}^{\\prime }$ that is reachable from $(\\lbrace \\bar{q}_0\\rbrace ,0)$ .",
"We call a branch of $\\mathcal {B}_{\\mathit {det}}^{\\prime }$ accepting if it contains $X_w^{\\prime }$ infinitely often.",
"Suppose “(ii)”, i.e., the probability that some branch of a $\\mathcal {B}[X_f]$ -tree has a (non-accepting or accepting) run in $\\mathcal {A}[f,X_f]$ is positive.",
"Thus, the probability that some branch of a $\\mathcal {B}[X_f]$ -tree has a run in $\\mathcal {A}_{\\mathit {det}}$ that does not enter the state $\\emptyset $ is positive.",
"Let $X_1 w^{\\prime } \\in \\Gamma ^+$ be such that $(f,X_f) \\in \\delta _{\\mathit {det}}(\\lbrace (f,X_f)\\rbrace ,w X_1 w^{\\prime })\\,.$ Then, also the probability that some branch of a $\\mathcal {B}[X_f]$ -tree starts with $X_f w X_1 w^{\\prime }$ and has a run in $\\mathcal {A}_{\\mathit {det}}$ that does not enter $\\emptyset $ is positive.",
"Thus, the probability that some branch of a $\\mathcal {B}[X_1]$ -tree starts with $X_1 w^{\\prime }$ and has a run in $\\mathcal {A}_{\\mathit {det}}$ , started in $\\delta _{\\mathit {det}}(\\lbrace (f,X_f)\\rbrace ,w)$ , that does not enter $\\emptyset $ is positive.",
"Hence, the probability that some branch of a $\\mathcal {B}[X_1]$ -tree has a run in $\\mathcal {A}_{\\mathit {det}}^{\\prime }$ , started in $(\\delta _{\\mathit {det}}(\\lbrace (f,X_f)\\rbrace ,w), \\varepsilon )$ , that does not enter a state of the form $(\\emptyset ,v)$ is positive.",
"From the construction of $\\mathcal {B}_{\\mathit {det}}^{\\prime }$ it follows that the probability that some branch of a $\\mathcal {B}_{\\mathit {det}}^{\\prime }[X_w^{\\prime }]$ -tree (i.e., with $X_w^{\\prime } = (\\delta _{\\mathit {det}}(\\lbrace (f,X_f)\\rbrace ,w), \\varepsilon )$ as start type) does not have a node of a type of the form $(\\emptyset ,v)$ is positive.",
"Consider any type $(U_0,v_0)$ in $\\mathcal {B}_{\\mathit {det}}^{\\prime }$ with $U_0 \\ne \\emptyset $ and view it as a state in $\\mathcal {A}_{\\mathit {det}}^{\\prime }$ .",
"Then there is $u_1 \\in \\Gamma ^*$ with $(U_0,v_0) {\\;\\xrightarrow{}{}\\hspace{0.0pt}^{*}\\;} (U_1,v_1)$ where $U_1 \\ne \\emptyset $ and $v_1 \\in \\lbrace 0, \\varepsilon \\rbrace $ .",
"Let $u_2 \\in \\Gamma ^*$ be a shortest word such that there is $(q_1,X_1) \\in U_1$ with $(q_1,X_1) {\\;\\xrightarrow{}{}\\hspace{0.0pt}^{*}\\;} (f,X_f)$ in $\\mathcal {A}[f,X_f]$ .",
"It follows that in $\\mathcal {A}_{\\mathit {det}}^{\\prime }$ we have $(U_0,v_0) {\\;\\xrightarrow{}{}\\hspace{0.0pt}^{*}\\;} (U_2,w) {\\;\\xrightarrow{}{}\\hspace{0.0pt}^{*}\\;} (U_3,\\varepsilon )$ with $(f,X_f) \\in U_2$ .",
"By lem:UBA-w we have $U_3 = \\delta _{\\mathit {det}}(\\lbrace (f,X_f)\\rbrace ,w)\\,.$ Hence $(U_0,v_0) {\\;\\xrightarrow{}{}\\hspace{0.0pt}^{*}\\;} X_w^{\\prime }$ .",
"Combining this reachability fact with the previous argument, we infer that the probability that some branch of a $\\mathcal {B}_{\\mathit {det}}^{\\prime }[X_w^{\\prime }]$ -tree has only nodes of types from which $X_w^{\\prime }$ is reachable (in $\\mathcal {B}_{\\mathit {det}}^{\\prime }$ ) is positive.",
"It follows from [8] that the probability that some branch of a $\\mathcal {B}_{\\mathit {det}}^{\\prime }[X_w^{\\prime }]$ -tree is accepting is positive.In terms of the notation therein, we instantiate [8] with $X := X_w^{\\prime }$ .",
"By the reachability argument above, $N_X = N_{X_w^{\\prime }}$ does not include types of the form $(U,v)$ with $U \\ne \\emptyset $ .",
"Thus, we have argued that the probability of $\\mathsf {F}N_{X_w^{\\prime }}$ is $p < 1$ .",
"Hence, [8] asserts that the probability that a $\\mathcal {B}_{\\mathit {det}}^{\\prime }[X_w^{\\prime }]$ -tree has an $X_w^{\\prime }$ -branch equals $1-p > 0$ , where $X_w^{\\prime }$ -branch means accepting path in terms of our definition.",
"Hence, the probability that some branch of a $\\mathcal {B}_{\\mathit {det}}^{\\prime }$ -tree is accepting is positive.",
"From the construction of $\\mathcal {B}_{\\mathit {det}}^{\\prime }$ it follows that the probability that some branch of a $\\mathcal {B}[X_f]$ -tree is accepted by $\\mathcal {A}_{\\mathit {det}}^{\\prime }$ is positive, i.e., “(iii)”."
],
[
"Proof of thm:coNBA-1",
"* Towards membership in PSPACE, fix a BP $\\mathcal {B}$ and an NBA $\\mathcal {A}$ .",
"Since PSPACE is closed under complement, we can focus on the problem whether the probability is positive that a $\\mathcal {B}$ -tree has a branch accepted by $\\mathcal {A}$ .",
"We use lem:UBA-X1-f.",
"Since reachability in a graph is in NL and, hence, in PSPACE, it suffices to decide in PSPACE whether the probability that some branch of a $\\mathcal {B}[X_f]$ -tree is accepted by $\\mathcal {A}[f,X_f]$ is positive.",
"In order to check that, by lem:Bdet it suffices to construct the BP $\\mathcal {B}_{\\mathit {det}}$ and then invoke the NC procedure of lem:AFT-1 to check if $\\mathbb {P}_{\\mathcal {B}_{\\mathit {det}}}(\\mathsf {F}\\lbrace \\emptyset \\rbrace ) < 1$ .",
"The BP $\\mathcal {B}_{\\mathit {det}}$ has exponential size but can be computed with a PSPACE transducer.",
"(In particular, whether a state in $\\mathcal {A}_{\\mathit {det}}$ is reachable from $\\lbrace \\bar{q}\\rbrace $ via a nonempty path can be determined in NPSPACE $=$ PSPACE.)",
"By lem:PSPACE-transducer it follows that $\\mathbb {P}(\\textup {coNBA}) = 1$ is in PSPACE.",
"PSPACE-hardness follows from the PSPACE-hardness [34] of the probabilistic emptiness problem, which, given a Markov chain and an NBA, asks if the probability is 0 that the Markov chain generates a word accepted by the NBA.",
"(We remark that, in contrast, the problem whether a given transition system has a run accepted by a given NBA is in NL: search the product for an accepting cycle)."
],
[
"Proof of thm:coNBA-0",
"* Towards membership in EXPTIME, an NBA can be translated, in exponential time, to a DPA of exponential size; see, e.g., [28].",
"By shifting the priorities (colours) in the DPA by 1, we can make the DPA accept exactly those words that are rejected by the NBA.",
"Since $\\mathbb {P}(\\textup {DPA}) = 0$ is in P by thm:DPA-0, it follows that $\\mathbb {P}(\\textup {coNBA}) = 0$ is in EXPTIME.",
"Concerning EXPTIME-hardness, we adapt the construction of the proof of thm:NBA-0.",
"As in that proof, let $M = (S_\\exists , S_\\forall , \\Sigma , T, s_0, s_{\\mathit {acc}})$ be a linear-bounded alternating Turing machine.",
"Let $w = a_1 \\cdots a_n \\in \\Sigma ^*$ be the input word, and assume again that $M$ uses exactly $n$ tape cells.",
"We construct a BP $\\mathcal {B}$ and an NBA $\\mathcal {A}$ such that the probability that all branches of the random tree are rejected by $\\mathcal {A}$ is positive if and only $M$ accepts $w$ .",
"For $\\mathcal {B}$ we use almost the same construction as in thm:NBA-0, except that we do not need the types $(1,\\mathit {chk}), \\ldots , (n,\\mathit {chk})$ .",
"We replace them by a single type $\\mathit {end}$ .",
"Accordingly, for all $(i, s_{\\mathit {acc}}, a) \\in \\Gamma $ , we replace the rule $(i, s_{\\mathit {acc}}, a) &{1} (1,\\mathit {chk}) \\cdots (n,\\mathit {chk})\\multicolumn{2}{l}{\\text{by a rule}}\\\\(i, s_{\\mathit {acc}}, a) &{1} \\mathit {end}\\,,$ and include a rule $\\mathit {end}{1} \\mathit {end}$ (i.e., a self-loop).",
"We want to construct the NBA $\\mathcal {A}$ so that it accepts exactly those branches that correspond to infinite computations that do not arrive at $s_{\\mathit {acc}}$ , or to “non-computations”, i.e., where the “guessing” in a rule $(i,(s,a,a^{\\prime },D,s^{\\prime })) {1/|\\Sigma |} (i{+}D, s^{\\prime }, b_j) \\quad (1 \\le j \\le |\\Sigma |)\\,,$ has been wrong.",
"The NBA $\\mathcal {A}= (Q, \\Gamma , \\delta , Q_0, Q)$ has the same set of states as in thm:NBA-0: $Q \\ = \\ (\\lbrace 1, \\ldots , n\\rbrace \\times \\Sigma ) \\ \\cup \\ \\lbrace f\\rbrace $ As in thm:NBA-0, the set of initial states is $Q_0 = \\lbrace (1,a_1), \\ldots , (n,a_n)\\rbrace $ (recall that $a_1 \\cdots a_n$ is the input word).",
"As in thm:NBA-0, the idea is that if a prefix $X_1 \\cdots X_k \\in \\Gamma ^*$ of a tree branch corresponds to a prefix of a correct computation of $M$ , then the set of automaton states in $\\delta (Q_0,X_1 \\cdots X_k)$ corresponds to the tape after this computation prefix.",
"In fact, the transition relation $\\delta \\subseteq Q \\times \\Gamma \\times Q$ is deterministic, i.e., for all $q \\in Q$ and $X \\in \\Gamma $ there is at most one $q^{\\prime }$ with $(q,X,q^{\\prime }) \\in \\delta $ .",
"Moreover, for any transition $((i,a), X, (j,a^{\\prime })) \\in \\delta $ we will have $i=j$ .",
"Unlike in thm:NBA-0, all states are accepting.",
"For $(i,a) \\in Q$ and all $(j,s,b) \\in \\Gamma $ with $i \\ne j$ or $a=b$ , include a self-loop $((i,a),(j,s,b),(i,a)) \\in \\delta \\,.$ (Intuitively, letter $a$ in cell $i$ is compatible with the head being on cell $j$ and reading letter $b$ .)",
"For $(i,a) \\in Q$ and all $(i,s,b) \\in \\Gamma $ with $a \\ne b$ , include a transition $((i,a),(i,s,b),f) \\in \\delta \\,.$ (Intuitively, letter $a$ in cell $i$ is not compatible with the head being on cell $i$ and reading letter $b$ ; i.e., the prefix of the branch does not correspond to a correct computation.)",
"For all $(i,a) \\in Q$ and all $(j,t) \\in \\Gamma $ with $i \\ne j$ , include a self-loop $((i,a),(j,t),(i,a)) \\in \\delta \\,.$ (Intuitively, the content of cell $i$ stays unchanged when the head is at position $j$ .)",
"For all $i,s,a,a^{\\prime },D,s^{\\prime }$ with $(i,a) \\in Q$ and $(i,(s,a,a^{\\prime },D,s^{\\prime })) \\in \\Gamma $ , include a transition $((i,a), (i,(s,a,a^{\\prime },D,s^{\\prime })), (i,a^{\\prime }))\\in \\delta \\,.$ (Intuitively, the transition $(s,a,a^{\\prime },D,s^{\\prime })$ changes the content of cell $i$ from $a$ to $a^{\\prime }$ .)",
"For all $X \\in \\Gamma $ , include a self-loop $(f,X,f) \\in \\delta $ .",
"Note that $(f,\\mathit {end},f)$ is the only transition labeled with $\\mathit {end}$ .",
"In this way: If a tree branch does not correspond to a correct computation, the automaton $\\mathcal {A}$ enters the state $f$ , remains there forever, and, thus, accepts.",
"If a tree branch corresponds to an infinite computation not entering $s_{\\mathit {acc}}$ , the set of states that $\\mathcal {A}$ can be in always reflects the tape.",
"Thus, $\\mathcal {A}$ accepts.",
"If a tree branch corresponds to a computation entering $s_{\\mathit {acc}}$ , the branch also enters $\\mathit {end}$ , and $\\mathcal {A}$ does not enter $f$ .",
"Thus, $\\mathcal {A}$ rejects.",
"It follows that the probability that all branches of the random tree are rejected by $\\mathcal {A}$ is positive if and only $M$ accepts $w$ .",
"A more detailed argument would follow very similar lines as the proof of thm:NBA-0."
],
[
"Proof of lem:key",
"* The automaton $\\mathcal {A}[f,X_f]$ is unambiguous, as $\\mathcal {A}$ is unambiguous, and has $(f,X_f)$ as (the only) accepting state.",
"Recall also that $(f,X_f)$ is reachable from all states in $\\mathcal {A}[f,X_f]$ .",
"It follows that $\\mathcal {A}[f,X_f]$ does not have diamonds, i.e., $\\mathcal {A}[f,X_f]$ does not have states $(q,X), (q^{\\prime },X^{\\prime })$ and a word $u \\in \\Gamma ^*$ such that $\\mathcal {A}[f,X_f]$ has two different paths $(q,X) {\\;\\xrightarrow{}{}\\hspace{0.0pt}^{*}\\;} (q^{\\prime },X^{\\prime })$ .",
"By the Perron-Frobenius theorem [3], $M$ has an eigenvector $v \\in (0,\\infty )^{Q[f,X_f]}$ (all entries positive) with $M v = \\rho v$ .",
"(Think of the entries of $v$ as “weights” of the states in $\\mathcal {A}[f,X_f]$ .",
"Loosely speaking, the equality $(M v)_{(q,X)} = \\rho v_{(q,X)}$ expresses that the expected combined weight of the “successors” of $(q,X)$ is equal to the weight of $(q,X)$ multiplied by $\\rho $ .)",
"We “lift” $v$ to define a vector $\\bar{v} \\in [0,\\infty )^{\\Gamma ^{\\prime }}$ where $\\Gamma ^{\\prime }$ is the set of types in $\\mathcal {B}_{\\mathit {det}}$ (recall that they are of the form $P \\times \\lbrace X\\rbrace $ with $P \\subseteq Q$ and $X \\in \\Gamma $ ): $\\bar{v}_{P \\times \\lbrace X\\rbrace } \\ := \\ \\sum _{q \\in P} v_{(q,X)}$ Denote by $\\bar{M} \\in \\mathbb {Q}^{\\Gamma ^{\\prime } \\times \\Gamma ^{\\prime }}$ the matrix defined before lem:as-finiteness-char, but for $\\mathcal {B}_{\\mathit {det}}$ .",
"Then we have for all $P \\times \\lbrace X\\rbrace \\in \\Gamma ^{\\prime }$ : $(\\bar{M} \\bar{v})_{P \\times \\lbrace X\\rbrace }&\\ =\\ \\sum _{X {p} X_1 \\cdots X_k} p \\sum _{i=1}^k \\bar{v}_{\\delta _{\\mathit {det}}(P \\times \\lbrace X\\rbrace , X_i)} \\\\&\\ =\\ \\sum _{X {p} X_1 \\cdots X_k} p \\sum _{i=1}^k \\ \\sum _{(r,X_i) \\in \\delta _{\\mathit {det}}(P \\times \\lbrace X\\rbrace , X_i)} v_{(r,X_i)} \\\\&\\ \\mathop {=}^*\\ \\sum _{X {p} X_1 \\cdots X_k} p \\sum _{i=1}^k \\ \\sum _{q \\in P} \\ \\sum _{r : (q,X) \\xrightarrow{} (r,X_i)} v_{(r,X_i)} \\\\&\\ =\\ \\sum _{q \\in P} \\ \\sum _{X {p} X_1 \\cdots X_k} p \\sum _{i=1}^k \\ \\sum _{r : (q,X) \\xrightarrow{} (r,X_i)} v_{(r,X_i)} \\\\&\\ =\\ \\sum _{q \\in P} (M v)_{(q,X)}\\\\&\\ =\\ \\sum _{q \\in P} \\rho v_{(q,X)}\\\\&\\ =\\ \\rho \\bar{v}_{P \\times \\lbrace X\\rbrace }\\;,$ where the third equality (marked with $\\displaystyle \\mathop {=}^*$ ) holds as for any $(r,X_i) \\in \\delta _{\\mathit {det}}(P \\times \\lbrace X\\rbrace , X_i)$ there is exactly one $q \\in P$ with $(q,X) \\xrightarrow{} (r,X_i)$ in $\\mathcal {A}[f,X_f]$ .",
"Indeed, towards a contradiction, suppose there are $q_1, q_2 \\in Q$ with $q_1 \\ne q_2$ and $(q_j,X) \\xrightarrow{} (r,X_i)$ for both $j \\in \\lbrace 1,2\\rbrace $ .",
"Since $P \\times \\lbrace X\\rbrace $ is reachable in $\\mathcal {A}_{\\mathit {det}}$ from $\\lbrace (f,X_f)\\rbrace $ , there is $u \\in \\Gamma ^*$ with $(f,X_f) {\\;\\xrightarrow{}{}\\hspace{0.0pt}^{*}\\;} (q_j,X) \\xrightarrow{} (r,X_i)$ in $\\mathcal {A}[f,X_f]$ for both $j \\in \\lbrace 1,2\\rbrace $ , contradicting the absence of diamonds in $\\mathcal {A}[f,X_f]$ .",
"We conclude from the above computation that $\\bar{M} \\bar{v} = \\rho \\bar{v}$ ; i.e., $\\bar{v}$ is an eigenvector of $\\bar{M}$ with eigenvalue $\\rho $ .",
"In the following, for subsets $\\Delta \\subseteq \\Gamma ^{\\prime }$ , we write $\\bar{M}_{\\Delta } \\in \\mathbb {Q}^{\\Delta \\times \\Delta }$ for the (square) principal submatrix obtained from $\\bar{M}$ by restricting it to the rows and columns indexed by elements of $\\Delta $ .",
"Similarly, define $\\bar{v}_{\\Delta } \\in [0,\\infty )^{\\Delta }$ by restricting $\\bar{v}$ to the entries indexed by elements of $\\Delta $ .",
"Since $\\bar{M}$ and $\\bar{v}$ are nonnegative, we have $\\bar{M}_{\\Delta } \\bar{v}_{\\Delta } \\le \\rho \\bar{v}_{\\Delta }$ (the inequality is meant componentwise).",
"Note that $\\bar{v}_\\emptyset = 0$ .",
"Define $\\Gamma ^{\\prime \\prime } := \\Gamma ^{\\prime } \\setminus \\lbrace \\emptyset \\rbrace $ .",
"Thus we have $\\bar{M}_{\\Gamma ^{\\prime \\prime }} \\bar{v}_{\\Gamma ^{\\prime \\prime }} = \\rho \\bar{v}_{\\Gamma ^{\\prime \\prime }}$ .",
"All entries of $\\bar{v}_{\\Gamma ^{\\prime \\prime }}$ are positive, as all entries of $v$ are.",
"By Perron-Frobenius theory [3] it follows that $\\rho $ is the spectral radius of $\\bar{M}_{\\Gamma ^{\\prime \\prime }}$ .",
"The matrix $\\bar{M}_{\\Gamma ^{\\prime \\prime }}$ is equal to the matrix defined before lem:as-finiteness-char, but for $\\mathcal {B}_{\\mathit {det}}^{\\prime \\prime }$ .",
"We complete the proof with the following case distinction.",
"Suppose $\\rho > 1$ .",
"Let $\\Delta \\subseteq \\Gamma ^{\\prime \\prime }$ be a bottom SCC of the graph of $\\bar{M}_{\\Gamma ^{\\prime \\prime }}$ .",
"As $\\Delta $ is bottom, $\\bar{M}_\\Delta \\bar{v}_\\Delta = \\rho \\bar{v}_\\Delta $ .",
"So the spectral radius of $\\bar{M}_\\Delta $ is at least $\\rho > 1$ (in fact, it must be equal to $\\rho $ ).",
"It follows that $\\Delta $ is supercritical in $\\mathcal {B}_{\\mathit {det}}^{\\prime \\prime }$ .",
"Thus, by lem:as-finiteness-char, a $\\mathcal {B}_{\\mathit {det}}^{\\prime \\prime }$ -tree is infinite with positive probability.",
"Suppose that $\\rho = 1$ and that $\\mathcal {A}[f,X_f]$ does not have proper branching.",
"Let $\\Delta \\subseteq \\Gamma ^{\\prime \\prime }$ be a bottom SCC of the graph of $\\bar{M}_{\\Gamma ^{\\prime \\prime }}$ .",
"Then $\\bar{M}_\\Delta \\bar{v}_\\Delta = \\bar{v}_\\Delta $ , so by the Perron-Frobenius theorem [3] the spectral radius of $\\bar{M}_\\Delta $ is 1.",
"By the absence of proper branching we also have $\\bar{M}_\\Delta \\vec{1} \\le \\vec{1}$ , where $\\vec{1}$ denotes the all-1 vector, i.e., the element of $\\lbrace 1\\rbrace ^\\Delta $ .",
"By Perron-Frobenius theory [3] it follows that $\\bar{M}_\\Delta \\vec{1} = \\vec{1}$ .",
"Thus, $\\Delta $ is linear in $\\mathcal {B}_{\\mathit {det}}^{\\prime \\prime }$ .",
"Hence, by lem:as-finiteness-char, a $\\mathcal {B}_{\\mathit {det}}^{\\prime \\prime }$ -tree is infinite with positive probability.",
"Suppose that $\\rho =1$ and that $\\mathcal {A}[f,X_f]$ has proper branching, i.e., there exist $(q,Y) \\xrightarrow{} (r_1,Z_1) \\text{ and } (q,Y) \\xrightarrow{} (r_2,Z_2) \\text{ in } \\mathcal {A}[f,X_f]$ and a rule $Y {p} u_1 Z_1 u_2 Z_2 u_3$ with $u_1, u_2, u_3 \\in \\Gamma ^*$ .",
"Consider any SCC $\\Delta \\subseteq \\Gamma ^{\\prime \\prime }$ of the graph of $\\bar{M}_{\\Gamma ^{\\prime \\prime }}$ .",
"Denote by $\\rho _\\Delta $ the spectral radius of $\\bar{M}_\\Delta $ .",
"As $\\bar{M}_\\Delta $ is a principal submatrix of $\\bar{M}$ , we have $\\rho _\\Delta \\le \\rho = 1$ [3].",
"So $\\Delta $ is not supercritical.",
"$\\rho _\\Delta < 1$ .",
"We have argued before lem:as-finiteness-char that $\\Delta $ being linear would imply $\\rho _\\Delta =1$ .",
"Hence, $\\Delta $ is not linear.",
"$\\rho _\\Delta = 1$ .",
"Recall that $\\bar{M}_{\\Delta } \\bar{v}_{\\Delta } \\le \\bar{v}_{\\Delta }$ , so by Perron-Frobenius theory [3] we must have $\\bar{M}_{\\Delta } \\bar{v}_{\\Delta } = \\bar{v}_{\\Delta }$ .",
"Let $P \\times \\lbrace X\\rbrace \\in \\Delta $ and $p \\in P$ .",
"Let $u_0 \\in \\Gamma ^*$ with $(p,X) {\\;\\xrightarrow{}{}\\hspace{0.0pt}^{*}\\;} (q,Y)$ in $\\mathcal {A}[f,X_f]$ .",
"Hence $(p,X) {\\;\\xrightarrow{}{}\\hspace{0.0pt}^{*}\\;} (q,Y) \\xrightarrow{} (r_j,Z_j)$ in $\\mathcal {A}[f,X_f]$ for both $j \\in \\lbrace 1,2\\rbrace $ .",
"Towards a contradiction, suppose $\\Delta $ is linear.",
"If $\\delta _{\\mathit {det}}(P \\times \\lbrace X\\rbrace , u_0)$ is not in $\\Delta $ , then neither are $\\delta _{\\mathit {det}}(P \\times \\lbrace X\\rbrace , u_0 Z_1)$ or $\\delta _{\\mathit {det}}(P \\times \\lbrace X\\rbrace , u_0 Z_2)$ .",
"Otherwise (i.e., $\\delta _{\\mathit {det}}(P \\times \\lbrace X\\rbrace , u_0) \\in \\Delta $ ) there is $j \\in \\lbrace 1,2\\rbrace $ such that $\\delta _{\\mathit {det}}(P \\times \\lbrace X\\rbrace , u_0 Z_j) \\notin \\Delta $ , as $\\Delta $ is linear.",
"Either way there is $j \\in \\lbrace 1,2\\rbrace $ such that $\\delta _{\\mathit {det}}(P \\times \\lbrace X\\rbrace , u_0 Z_j) \\notin \\Delta \\,.$ Note that $(r_j,Z_j) \\in \\delta _{\\mathit {det}}(P \\times \\lbrace X\\rbrace , u_0 Z_j) \\ne \\emptyset $ .",
"Let $x Z$ (with $x \\in \\Gamma ^*$ and $Z \\in \\Gamma $ ) be the shortest prefix of $u_0 Z_j$ such that $U := \\delta _{\\mathit {det}}(P \\times \\lbrace X\\rbrace , x) \\in \\Delta $ but $\\delta _{\\mathit {det}}(U,Z) \\notin \\Delta $ .",
"Then we have: $\\bar{v}_U&\\ =\\ (\\bar{M}_\\Delta \\bar{v}_\\Delta )_U \\\\&\\ <\\ (\\bar{M}_\\Delta \\bar{v}_\\Delta )_U + \\bar{M}_{U,\\delta _{\\mathit {det}}(U,Z)} \\bar{v}_{\\delta _{\\mathit {det}}(U,Z)} \\\\&\\ \\le \\ (\\bar{M} \\bar{v})_U \\\\&\\ =\\ \\bar{v}_U\\,,$ a contradiction.",
"Thus, $\\Delta $ is not linear.",
"We conclude that in both cases $\\Delta $ is neither linear nor supercritical.",
"Since $\\Delta $ was an arbitrary SCC, we conclude from lem:as-finiteness-char that a $\\mathcal {B}_{\\mathit {det}}^{\\prime \\prime }$ -tree is almost surely finite.",
"Suppose $\\rho < 1$ .",
"Then, for any SCC $\\Delta $ the spectral radius of $\\bar{M}_{\\Delta }$ is also less than 1 [3], so $\\Delta $ is neither supercritical nor linear.",
"Thus, by lem:as-finiteness-char, a $\\mathcal {B}_{\\mathit {det}}^{\\prime \\prime }$ -tree is almost surely finite.",
"Hence, the probability that a $\\mathcal {B}_{\\mathit {det}}^{\\prime \\prime }$ -tree is infinite is positive if and only if either $\\rho > 1$ or $\\rho = 1$ and $\\mathcal {A}[f,X_f]$ does not have proper branching."
],
[
"Proof of thm:LTL-0",
"* Given the main body, it remains to show 2EXPTIME-hardness of $\\mathbb {P}(\\textup {LTL}) = 0$ .",
"Our construction is inspired by the proof of 2EXPTIME-hardness of model-checking a concurrent probabilistic program (another name for MDP) against an LTL formula, see Theorem 3.2.1 in [11].",
"We will use the fact that 2EXPTIME is equal to alternating EXPSPACE.",
"Let $M = (S_\\exists , S_\\forall , \\Sigma , T, s_0, s_{\\mathit {acc}})$ be an alternating Turing machine whose work tape usage is bounded by $2^n$ on any input of length $n$ .",
"Without loss of generality, we can assume that the machine has two possible next moves for each configuration and that it halts when it reaches the accepting state $s_{\\mathit {acc}}$ .",
"For a given alternating TM $M$ and an input $w$ of length $n$ , we will construct a BP $\\mathcal {B}$ and an LTL formula $\\varphi $ both of size $\\mathcal {O}(n)$ such that $\\mathbb {P}_\\mathcal {B}(\\varphi ) > 0$ if and only if $M$ accepts $w$ .",
"The branching process $\\mathcal {B}$ is defined by the diagram in Fig.",
"REF .",
"Every node in the diagram has a unique label that corresponds to a type of $\\mathcal {B}$ , although not all nodes are explicitly labelled.",
"The $$ -nodes represent randomising branching and $$ -nodes represent tree branching.",
"Namely, if a $$ -node $x$ has $k$ successors $y_1,\\ldots , y_k$ : $@1@R=10pt@C=10pt{ & *+[l]{x\\,} [dr] [dl] &\\\\y_1 & \\cdots &y_k}$$, then $ B$ has the rules $ x 1/k yi$ for $ i=1,...,k$.",
"On the other hand, if a $$-node $ x$ has $ k$ successors $ y1,..., yk$: $$, then $ B$ has the rule $ x 1 y1yk$.$ The start type of $\\mathcal {B}$ is $a$ .",
"The detailed diagram of the blocks $I$ , $O$ , $N$ , $D_1$ , $D_2$ and $D_3$ is shown in Fig.",
"REF on the left.",
"All nodes in these blocks are $$ -nodes.",
"The diagram of the blocks $C_1$ and $C_2$ is shown in Fig.",
"REF on the right.",
"These blocks have $$ -nodes in the first $n+1$ levels, and the rest are $$ -nodes.",
"The initial and final nodes are labelled by $u$ and $v$ , respectively.",
"The nodes below $u$ are labelled with $(\\ell _i,0)$ and $(\\ell _i,1)$ , $i=1,\\ldots ,n$ , as shown in the picture.",
"Every block has its own unique set of labels $u$ , $v$ , $(\\ell _i,0)$ and $(\\ell _i,1)$ , $i=1,\\ldots ,n$ ; however we do not distinguish them in the diagram for simplicity.",
"In every block, except for $I$ , the nodes above $v$ are labelled by $(\\ell _{n+1},\\delta )$ , where $\\delta \\in \\Sigma \\;\\cup \\; (S_\\exists \\cup S_\\forall )\\times \\Sigma $ ranges over the symbols of the extended work tape alphabet.",
"In block $I$ , there are $n+1$ nodes above $v$ which are labelled by $(\\ell _{n+1},\\gamma _i)$ for $i=1,\\ldots ,n+1$ such that for an input word $w=w_1\\ldots w_n$ we have $\\gamma _1=(s_0,w_1)$ , $\\gamma _i=w_i$ for $1<i\\le n$ , and $\\gamma _{n+1}=\\Box $ is the blank symbol.",
"These nodes will define the initial configuration of the Turing machine $M$ .",
"The intended behaviour of process $\\mathcal {B}$ is as follows.",
"It starts generating a tree $T$ with a root node $a$ .",
"Then it tries to constructs the initial configuration of the Turing machine $M$ on input $w=w_1\\ldots w_n$ by looping through block $I$ for $2^n$ many times.",
"Each iteration of the block $I$ produces a string of the form $u(\\ell _1,b_1)\\ldots (\\ell _n,b_n)(\\ell _{n+1},z)v$ , where $b_1\\ldots b_n$ is the address of a work tape cell in binary, and $z$ is the content of that cell.",
"$\\mathcal {B}$ is supposed to construct the initial configuration by specifying the content of the work tape starting with 0 and ending with cell $2^n-1$ .",
"Each iteration of the loop, closed by the arc $c\\rightarrow b$ , corresponds to a move from one configuration of the Turing machine to the next.",
"First, in block $O$ , process $\\mathcal {B}$ tries to reproduce the current configuration (in the first iteration of the loop, it is the initial configuration) by making $2^n$ iterations.",
"Then, using tree branching in block $C_1$ , the process produces a full binary tree of height $n$ , each branch of which looks like $u(\\ell _1,b_1)\\ldots (\\ell _n,b_n)$ .",
"Note that the last type $(\\ell _n,b_n)$ is randomising, and after it $\\mathcal {B}$ tries to correctly reproduce the content of the cell $b_1\\ldots b_n$ in the current configuration.",
"If the current state of $M$ is existential, then $\\mathcal {B}$ is expected to move to $m_1$ ; if it is universal, it is expected to move to $m_2$ .",
"Two successors of $m_1$ and $m_2$ correspond to two possible moves out of the current configuration.",
"In $m_1$ , $\\mathcal {B}$ randomly chooses the next move; in $m_2$ , $\\mathcal {B}$ makes a tree branching with two children corresponding to two possible next moves.",
"Then, in block $N$ , $\\mathcal {B}$ tries to reproduce the next configuration of $M$ in the same way as it produced the current configuration in block $O$ .",
"In block $C_2$ , the process produces a full binary tree of height $n$ (in the same way as it does it in block $C_1$ ), and after each branch of the form $u(\\ell _1,b_1)\\ldots (\\ell _n,b_n)$ it tries to correctly reproduce the content of the cell $b_1\\ldots b_n$ in the next configuration.",
"Then, on every branch $u(\\ell _1,b_1)\\ldots (\\ell _n,b_n)(\\ell _{n+1},z)v$ produced by $C_2$ , in blocks $D_1$ , $D_2$ , $D_3$ , the process tries to reproduce the content of the cell $b_1\\ldots b_n$ from the old configuration (block $O$ ) and its two adjacent cells.",
"Finally, in state $c$ , $\\mathcal {B}$ is expected to move from $c$ to the sink state $d$ if the new configuration if accepting.",
"Otherwise, $\\mathcal {B}$ is expected to move to $b$ .",
"We now define an LTL formula $\\varphi $ that describes the expected behaviour of process $\\mathcal {B}$ .",
"The formula $\\varphi $ is the conjunction of the following parts: In blocks $I$ , $O$ and $N$ , the process constructs a configuration of $M$ cell-by-cell in order starting from cell 0 and ending with cell $2^n-1$ .",
"In block $I$ , the process constructs the initial configuration.",
"On a branch produced by $C_1$ that corresponds to index $k$ , the cell content specified by $C_1$ is equal to that of cell $k$ defined in block $O$ , in block $I$ (if this is the first iteration of block $C_1$ ) and in the previous iteration of block $N$ (if there was any).",
"If the current configuration if existential, then $\\mathcal {B}$ moves to $m_1$ .",
"Otherwise, it moves to $m_2$ .",
"On a branch produced by $C_2$ that corresponds to index $k$ , the cell content specified by $C_2$ is equal to that of cell $k$ defined in block $N$ , and the indices in blocks $D_1$ , $D_2$ , $D_3$ are $k-1$ , $k$ , $k+1$ , respectively.",
"The cell contents in blocks $D_1$ , $D_2$ , $D_3$ are equal to those defined in block $O$ .",
"The cell content specified by $C_2$ follows directly from the contents of the cells specified by $D_1$ , $D_2$ , $D_3$ and the rule of $M$ that was chosen on the current branch.",
"If the new configuration is accepting, then $\\mathcal {B}$ moves from $c$ to $d$ .",
"Otherwise, it moves to $b$ .",
"$\\mathsf {F}\\mathsf {G}\\, d$ , that is, eventually $d$ always holds.",
"The above properties can be expressed using LTL formulas.",
"We will not explicitly write them down but the details of these formulas are very similar to those defined in the proof of Theorem 3.2.1 from [11].",
"Now suppose that Turing machine $M$ accepts an input $w$ .",
"Hence there is a $2^n$ -bounded winning strategy for the existential player.",
"In this case, with some positive probability, $\\mathcal {B}$ can generate a tree $T$ all whose branches satisfy $\\varphi $ as follows: First, it generates the initial configuration in blocks $I$ and $O$ .",
"On every branch produced by $C_1$ , process $\\mathcal {B}$ generates the correct cell content.",
"Then it moves to either $m_1$ , if the current configuration is existential, or to $m_2$ , if it is universal.",
"In the former case, $\\mathcal {B}$ chooses the next move that agrees with the winning strategy of the existential player.",
"Then in block $N$ , it generates a new configuration of $M$ that follows from $O$ according to the chosen move.",
"(If the tree branching node $m_2$ was chosen, then $\\mathcal {B}$ generates correct new configurations on every branch.)",
"Next, $\\mathcal {B}$ generates correct cell content on each branch produced by $C_2$ and chooses correct indices and cell contents in blocks $D_1$ , $D_2$ , $D_3$ .",
"Finally, $\\mathcal {B}$ moves from $c$ to $d$ , if an accepting configuration is reached, or it moves back to $b$ otherwise.",
"In the latter case, $\\mathcal {B}$ generates in block $O$ the same configuration that was generated in $N$ and continues the process.",
"Note that since the existential player has a winning strategy, state $d$ will eventually appear on every branch of $T$ .",
"Since $T$ is finitely branching, it follows by König's lemma that the above process will reach $d$ on every branch of $T$ in a finite number of steps.",
"After that, $\\mathcal {B}$ repeats the rule $d {1} d$ forever.",
"Clearly, all these events can happen with some positive probability.",
"Hence, $\\mathbb {P}_\\mathcal {B}(\\varphi ) > 0$ .",
"Conversely, suppose there is a positive probability that $\\mathcal {B}$ generates a tree $T$ all whose branches satisfy $\\varphi $ .",
"Hence every branch of $T$ eventually reaches state $d$ (and then always repeats it).",
"Since $T$ is finitely branching, it follows by König's lemma that there exists a finite prefix of $T$ whose every leaf is labelled by $d$ .",
"This prefix encodes the moves of the existential player (after each appearance of state $m_1$ in the prefix) that allow him to reach the accepting configuration, no matter what the universal player does.",
"In other words, the existential player has a winning strategy, and hence $M$ accepts $w$ .",
"Formally, this can be shown along similar lines as in the proof of thm:NBA-0.",
"Therefore, we proved that $M$ accepts $w$ if and only if $\\mathbb {P}_\\mathcal {B}(\\varphi ) > 0$ .",
"Figure: Diagrams of the randomising (left) and tree (right) branching blocks.Figure: Diagram of the branching process ℬ\\mathcal {B}."
]
] | 2107.01687 |
[
[
"The Discovery of a Remnant Radio Galaxy in A2065 Using GMRT"
],
[
"Abstract The upgraded Giant Metrewave Radio Telescope (GMRT) has been used to map the cluster A2065 at z = 0.0726.",
"We report the discovery of a remnant radio galaxy at the peripheral cluster region.",
"The spatially resolved radio emission from the remnant radio galaxy shows an elongated, bar-shaped structure, whose size is $\\approx$ 52$^{\\prime\\prime}$ $\\times$ 110$^{\\prime\\prime}$ ($\\simeq$ 72 $\\times$ 152 kpc$^2$).",
"Our study with the multiwavelength GMRT data and \\textit{Chandra} data shows that across the remnant radio galaxy there is a hint of a surface-brightness edge in the hot X-ray gas.",
"We detect tentative flattening of the radio spectral index as the old plasma at the near end of the surface-brightness edge is reinvigorated by the passage of possible shock front and shows the expected change in radio emission characteristics.",
"We suggest that the remnant radio galaxy has been seeded by the lobes of the active galactic nucleus (AGN), hosted by the WISEA J152228.01$+$274141.3 source, demonstrating the connection between AGNs and remnant radio sources.",
"Although the number of known remnant radio sources is beginning to increase, we emphasize the need for better data to understand the physics and nature of poorly understood remnant radio sources."
],
[
"Introduction",
"The hierarchical structure formation scenario suggests that cluster mergers are the mechanisms by which galaxy clusters form and these mergers are the most energetic phenomena in the universe since the Big Bang.",
"A large number of clusters of galaxies are known to contain large-scale, low-surface-brightness diffuse radio emission whose origin is not often related to the active galactic nuclei (AGNs) but to the intracluster medium (ICM).",
"According to the location with respect to the cluster center, these sources are of two key types, radio halos and relics.",
"Radio halos show a regular, amorphous emission around the cluster center of a size from a few hundred kiloparsecs (minihalo) to megaparsecs (halo), whereas radio relics show an irregular emission located in peripheral cluster regions of a size from several hundreds of kiloparsecs to megaparsecs.",
"Another third broad type of sources comprises of revived fossil plasma sources and phoenices.",
"These do not have any preferred location in the ICM are of size from several tens of kpc to a few hundred kpc and trace AGN radio plasma that has somehow been reenergised through the processes in the ICM [27], [16], [9].",
"The radio jets emanating from an AGN arise from the accretion onto a supermassive black hole.",
"These form synchrotron-emitting radio lobes in a radio-loud AGN in the environments of their host galaxies.",
"The active phase of an AGN can last several tens of megayears, following this the radio jets stop and their features, namely an unresolved radio core, radio jets, hotspots, and radio lobes, all start to fade away due to synchrotron cooling.",
"This phase once the jets have switched off is called as the remnant phase of a radio galaxy, or merely the remnant radio galaxy.",
"The remnant radio galaxy falls into the third broad type of sources in the ICM, which is poorly understood and presently very few sources are known [25], [2], [23].",
"The high spatial resolution of the Chandra and upgraded Giant Metrewave Radio Telescope (GMRT) allow us to study the interplay between X-ray-emitting hot gas and nonthermal radio emission.",
"We have obtained new upgraded GMRT data, and have used archive GMRT and Chandra data of the nearby $z$ = 0.0726 [26] richness class 2 cluster, A2065, which is one of the 10 galaxy clusters that make up the Corona Borealis supercluster.",
"[24] noted that two cD galaxies dominate the cluster center in the optical and their radial velocities differ by $\\sim $ 600 km s$^{-1}$ .",
"Table: Summary of the radio observations and image properties.Previous radio study using 100 m Green Bank Telescope at 1.4 GHz [7] detected a smooth diffuse structure $\\simeq $ 1 Mpc in extent, a giant radio halo, whereas a recent study showed a slightly smaller extent for the radio halo using Low-Frequency Array (LOFAR) at 153 MHz [6].",
"Similarly, previous X-ray studies led to detections of two surface-brightness peaks coincident with the two cD galaxies observed in the cluster's center.",
"The data indicated that the cluster is an unequal-mass merger and only one of the two cooling cores has survived the merger.",
"The northern cluster seems to have fallen into the more massive southern cluster from the southeast, and having lost its gas content, it has now moved to its present location to the northwest.",
"Evidence of shock with $M$ $\\approx $ 1.7 at $\\sim $ 140 kpc from the southern cD galaxy too is noted and the cooling flow is displaced to the southeast of the southern cD galaxy [3], [20].",
"In this study, we report the discovery of a remnant radio galaxy in cluster A2065 using the GMRT.",
"We also present an X-ray analysis of archive Chandra data, which enhances the interpretation of the current discovery.",
"We define spectral index, $\\alpha $ as, $S_\\nu $ $\\propto $ $\\nu ^{\\alpha }$ , where S$_\\nu $ is the flux density at frequency, $\\nu $ .",
"To estimate the intrinsic parameters, we use a $\\Lambda $ CDM cosmology with $\\Omega _{\\rm m}$ = 0.27, $\\Omega _{\\Lambda }$ = 0.73, and H$_0$ = 70 km s$^{-1}$ Mpc$^{-1}$ .",
"At the redshift of the cluster, 1 arcsec corresponds to 1.384 kpc.",
"All uncertainties are at 1$\\sigma $ error bars unless otherwise stated.",
"The position angles (P.A.s) are measured from north to east.",
"Throughout, positions are given in J2000 coordinates.",
"The organization of the paper is as follows.",
"We present the X-ray and radio observations and their data analyses in Sec. .",
"Our results, i.e., radio morphology, spectral structure, and X-ray gas properties are presented in Sec. .",
"In Sec.",
", we discuss our observational results, source diagnostics, and dynamical interpretation of the remnant radio galaxy.",
"Sec.",
"summarizes our conclusions.",
"A2065 was observed by Chandra with the acis-i detector on 2002 August 18 and November 24 for 28 and 22 ks (Obs_ID 3182) respectively, and on 2007 September 13 for 5 ks (Obs_ID 7689).",
"The data reduction of these three archival data sets has been performed independently.",
"We used the calibration databases, caldb version 4.7.6 and ciao version 4.9.",
"After the initial calibration, i.e., standard filtering and removal of bad pixels, the event files of the three observations were projected to a common reference point and then merged to create Gaussian-smoothed, background-subtracted, exposure-corrected image in the broadband 0.5–7.0 keV energy range.",
"We excised the central AGN and background point sources.",
"Using the ciao tool specextract, we extracted the source, background spectra, and the response files of selected regions in the Chandra images separately for the three data sets.",
"The spectra for each data set were grouped so that each bin contained at least 20 counts.",
"We restricted our spectral extraction in the energy range 0.5–7.5 keV, and fitted for the thermal gas emission with xspec [1] using the isothermal apec model.",
"The abundance was fixed to 0.3 $\\times $ solar [3] and all spectral fits include absorption by the Galactic column [5].",
"Figure: Exposure-corrected, background-subtracted, broadband (0.5–7.0 keV) Chandra image of A2065.",
"The image was binned (bin = 16) and smoothed using a Gaussian kernel of 10 '' ^{\\prime \\prime }.",
"The color bar shows the surface-brightness, in 10 -6 ^{-6} counts s -1 ^{-1} per 8 ×\\times 8 pixel binning.",
"The X-ray peak is located at R.A. 15:22:29.18, decl.",
"27:42:21.05.",
"The peak surface-brightness and the faintest regions transitioning between blue and green have a surface-brightness of 6.0 ×\\times 10 -5 ^{-5} counts s -1 ^{-1} arcsec -2 ^{-2} and 1.6 ×\\times 10 -6 ^{-6} counts s -1 ^{-1} arcsec -2 ^{-2} respectively.The surface-brightness is displayed in logarithmic scales to emphasize low-surface-brightness emission."
],
[
"GMRT data",
"A summary of the radio observations is presented in Table REF .",
"We inspected the data for bad antennas, bad time stamps and data impacted by radio frequency interference.",
"The data were analyzed using the standard reduction methodologies [17].",
"After the initial steps of flux density and bandpass calibration, all corrupted data were removed using standard routines.",
"The bandpass calibrated data for wide-bandwith GMRT data, i.e., the 250–500 MHz band and 1050–1450 MHz band data were split into five 40 MHz (from 300–500 MHz) and eight 50 MHz (from 1050–1450 MHz) subbands respectively.",
"Subsequently, each subband data was analyzed separately, similar to narrowband “classic\" GMRT data.",
"Several cycles of phase-only self-calibration and imaging were performed and the calibration solutions were applied to the target data to correct for residual phase errors.",
"To take care of the wide-field imaging at low GMRT frequencies, we performed faceting imaging.",
"Problematic bright sources in the field of view whose side lobes affected the area of the target were subtracted, a.k.a.",
"peeled; this (direction-dependent) calibration step improved the dynamic range of images.",
"The calibrated data for five 40 MHz 250–500 MHz band data and eight 50 MHz 1050–1450 MHz band data were joined to form full 200 MHz and 400 MHz, respectively, wide-bandwidth-calibrated visibility data sets.",
"The combined wide-bandwidth-calibrated data (and similarly the narrowband-calibrated “classic\" GMRT data) were imaged using the casa task tclean.",
"We used 3D imaging (gridder = `widefield'), two Taylor coefficients (nterms = 2, tt0 and tt1) and Briggs weighting (robust = 0.5) in task tclean.",
"A final amplitude-and-phase self-calibration with solution interval equal to the length of observation was then carried out using the gaincal and applycal tasks in casa.",
"The two Stokes polarizations, RR and LL, were combined to obtain the final total intensity Stokes I image, and corrected for the GMRT primary beam response.",
"This provides images at center frequencies (tt0) and the corresponding in-band spectral index maps (tt1/tt0).",
"The amplitude errors are estimated to be within 5% or less for archival narrow-bandwidth and wide-bandwidth GMRT data.",
"Table REF also provides image properties, including the restoring beams and root mean square (rms) noise levels at half-power points of the final images.",
"The uncertainty in the flux density measurements are estimated as $\\left[(S_\\nu \\times f)^2 + (\\textsc {rms})^2 \\times N_{\\rm beams}\\right]^{0.5},$ where $S_\\nu $ is the flux density, $f$ is an absolute flux density calibration uncertainty, rms is the noise, and $N_{\\rm beams}$ is the number of synthesized beams.",
"We use these error estimates for the flux density measurements when computing the error estimates for spectral index measurements.",
"Figure: The central 3 ' ^{\\prime } ×\\times 3 ' ^{\\prime } cluster region with 250–500 MHz band radio contours overlaid on the Digitized Sky Survey image of A2065.",
"Note the northern cD galaxy registers well with the northern radio peak, but the southern cD galaxy is displaced by ∼\\sim 24 north of the radio peak.",
"The X-ray peak at the cluster center is coincident with the southern radio peak (see also Figure ).The radio emission from the two cD galaxies (the brighter southern galaxy 2MASS J152229.17++274227.5 and the fainter northern galaxy 2MASX J152228.92++274244.1) of the cluster is shown on the east of the remnant radio galaxy emission.The radio contour levels are --3, 3, 6, 9, 15, 42 and 84 ×\\times 80 μ\\mu Jy beam -1 ^{-1}.The cluster-member galaxies are marked and a majority (≈\\approx 60%) show radio emission.",
"The two bright cluster galaxies, SDSS ++230.6++27.7++0.08 (R.A.: 15:22:24.0, decl.",
"++27:42:51; zz = 0.0723) and 2MASS J152229.17++274222.75 (zz = 0.074875) are marked in blue colour ."
],
[
"Results",
"The exposure-corrected background-subtracted X-ray binned (bin = 16) and Gaussian-smoothed (10$^{\\prime \\prime }$ kernel) image in the 0.5–7.0 keV band is shown in Figure REF .",
"Figure REF shows Chandra X-ray contours from Figure REF and 250–500 MHz band radio contours overlaid on the exposure-corrected, background-subtracted broadband (0.5–7.0 keV) Chandra (bin = 8) image (left panel) and Digitized Sky Survey image (right panel) of A2065.",
"The central 3$^{\\prime }$ $\\times $ 3$^{\\prime }$ cluster region with 250–500 MHz band radio contours overlaid on the Digitized Sky Survey image of A2065 is shown in Figure REF .",
"Figure REF also marks the cluster member galaxies, including the two bright cluster galaxies in marked in blue colour, SDSS $+$ 230.6$+$ 27.7$+$ 0.08 (NED; R.A.: 15:22:24.0, decl.",
"$+$ 27:42:51; $z$ = 0.0723, $m$ = 16.1) and 2MASS J152229.17$+$ 274222.75 [24], and a majority of them, $\\approx $ 60%, show radio emission.",
"Figure REF shows clear detections for the southern cD galaxy and remnant radio galaxy at the GMRT bands, namely the 240 MHz band (top-left panel), 250–500 MHz band (top-right panel), 610 MHz band (bottom-left panel), and 1050–1450 MHz band (bottom-right panel).",
"The northern cD galaxy is only detected in our 250–500 MHz band data.",
"The images have rms noise values in the range 0.56–0.06 mJy beam$^{-1}$ at the half-power points.",
"The 150 MHz band and 1050–1450 MHz band data sets had very short on-source integration times, $\\lesssim $ 0.9 hr, and therefore these images have relatively high rms noise values (see Table REF ).",
"Note that we also made images at several angular resolutions, including imaging data sets using the same range of baseline lengths, and do not find the presence or absence of additional morphological features.",
"The integrated flux densities computed thus are also consistent, within error bars with those reported in Table REF ."
],
[
"Southern and northern cD galaxies",
"Figure REF shows a diffuse X-ray tail extending to the north at the center and that the cluster emission is elongated along northwest-southeast direction.",
"The hot gas emission (i) shows a compact, bright region and a diffuse tail extending to the north at the center, and (ii) is extended along the northwest–southeast direction showing pieces of evidence for surface-brightness edges.",
"Though the X-ray peak emission and peak of the radio emission register well (see Figure REF , left), the position of the optical host of the southern cD galaxy (R.A.: 15:22:29.17, decl.",
": $+$ 27:42:27.5) is displaced by $\\sim $ 24 from the compact, bright, X-ray peak emission at the cluster center (see Figure REF , right, and also Figure REF ).",
"A low-surface-brightness diffuse tail, both in X-ray as well as in radio images, extending to the north at the center of the image, reaches out to the second cD galaxy (R.A.: 15:22:28.92, decl.",
": $+$ 27:42:44.1; Figures REF and REF , top-right panels).",
"The X-ray peak corresponding to the second northern cD galaxy and the peak of the second weaker component of radio emission at this location also register well.",
"The southern cD galaxy seems to have retained a cool core residing at the center of its subcluster and there is no evidence for a cool core corresponding to the northern cD galaxy [3], [20].",
"The closely associated radio and X-ray morphology at the center and its connection with the southern cD galaxy suggest that both X-ray and radio emissions are due to ram pressure stripping as the southern cD galaxy moves relative to the ICM.",
"The two blue plus signs in the 250–500 MHz band image marked in Figure REF (top-right panel) correspond to the hosts of the brighter southern cD galaxy, 2MASS J152229.17$+$ 274222.75 at $z$ = 0.074875 and fainter northern cD galaxy, 2MASX J152228.92$+$ 274244.1 at $z$ = 0.072854 (see also Figure REF ).",
"The northern cD galaxy is only detected in our 250–500 MHz band image.",
"However, we not only detected the southern cD galaxy in our long observations, we also detected the peak radio emissions at the location of the southern cD galaxy at R.A.: 15:22:29.17, decl.",
": $+$ 27:42:27.5 in our short, 150 MHz band, and 1050–1450 MHz band data sets.",
"The two cD galaxies are barely resolved using LOFAR at 153 MHz [6].",
"The southern cD galaxy also had a 2.8$\\sigma $ (upper limit) detection at 74 MHz [4].",
"It seems that around the southern cD radio galaxy there is some extended radio emission, which is more clearly seen in our 250–500 MHz band image (Figure REF , top-right panel).",
"Since several cluster-member galaxies also show radio emission, it is unclear if this extended radio emission is indeed associated with the southern cD radio galaxy or with the two adjacent cluster-member galaxies southeast of it (see Figure REF ).",
"Table: Radio flux densities for the remnant radio galaxy and the southern cD galaxy"
],
[
"Remnant radio galaxy",
"Figures REF and REF also suggest that the remnant radio galaxy is hosted by the 19.19 magnitude WISEA J152228.01$+$ 274141.3 source that is 78 away from the optical galaxy V1CG 136 (R.A.: 15:22:28.5, decl.",
": $+$ 27:41:46) at redshift $z$ = 0.072 [12].",
"The cluster contains gas not only with an extremely wide range of temperatures and complex structure but that also shows a cold front at 30$^{\\prime \\prime }$ ($\\simeq $ 41.5 kpc) and a shock-like bow-shaped feature beyond $\\sim $ 100$^{\\prime \\prime }$ ($\\simeq $ 140 kpc) on the southeast of the southern cD [3].",
"Figure REF shows that the northeastern boundary of the remnant radio galaxy coincides with the 1.3 $\\times $ 10$^{-7}$ counts s$^{-1}$ arcsec$^{-2}$ surface-brightness contour level; gas in false color transitioning from the green region to the blue region at a distance of $\\sim $ 33$^{\\prime \\prime }$ ($\\simeq $ 45.7 kpc) from the southern cD galaxy.",
"All four panels of Figure REF suggest that the radio emission from the remnant radio galaxy has an elongated, bar-shaped structure.",
"The remnant radio galaxy, buried in the radio halo emission that is barely detected using LOFAR at 153 MHz seems to possess an elongated, bar-shaped structure [6], consistent with our results.",
"This suggests that the remnant radio galaxy has a steep spectrum, and its consistency in radio morphology gives us additional confidence in our image fidelity, imaging, calibration and data reduction processes.",
"The best sensitivity image for the low surface-brightness was obtained from the 250–500 MHz band data and we now discuss below the morphology of the remnant radio galaxy.",
"The remnant radio galaxy broadly consists of a high-surface-brightness southeastern part and a fainter low-surface-brightness extended northwestern part.",
"The two oppositely directed radio jets emanating from the apex of the WISEA J152228.01$+$ 274141.3 source initially traverse toward the southeast and northwest directions.",
"As the galaxy plows through the dense intracluster gas, these jets traversing in opposite directions form a trail, after sharp bends in the jets, behind the WISEA J152228.01$+$ 274141.3 source due to interaction with the ICM.",
"The width of the remnant radio galaxy increases from $\\sim $ 17$^{\\prime \\prime }$ ($\\simeq $ 23.5 kpc) at the southeastern end to $\\sim $ 27$^{\\prime \\prime }$ ($\\simeq $ 37.4 kpc) and reduces again before flaring to reach a maximum width of $\\sim $ 35$^{\\prime \\prime }$ ($\\simeq $ 48.4 kpc) and finally fading completely.",
"The average width of the narrower southeastern part is a factor of $\\sim $ 2.3 smaller than the broader northwestern part.",
"It seems that a pinch is present at $\\sim $ 35$^{\\prime \\prime }$ ($\\simeq $ 48.4 kpc) of the $\\sim $ 162 ($\\simeq $ 22.4 kpc) width from the southeastern end, i.e., between the two narrower and the broader parts.",
"The northeastern boundary toward the two cD galaxies of the remnant radio galaxy is sharp, while the emission fades more slowly at the southwestern boundary.",
"The overall source size, assuming elongation, of the bar-shaped structure is $\\approx $ 52$^{\\prime \\prime }$ $\\times $ 110$^{\\prime \\prime }$ , corresponding to 72 $\\times $ 152 kpc$^2$ at the assumed distance.",
"Figure: Integrated radio spectrum of the remnant radio galaxy using the data in Table .Peak flux density is reported from our low-resolution, 15 '' ^{\\prime \\prime } image at the 1050–1450 MHz band for the remnant radio galaxy (see also Sec.",
"for a discussion).We also show the spectra of the southern cD galaxy for comparison; note that the (nearly) straight spectra of it provides a good check of our calibration.",
"A 2.8σ\\sigma limit is reported at the 74 MHz , peak flux densities at the 150 MHz band, and 1050–1450ṀHz band are our measurements; the 1400 MHz data is from the VLA FIRST survey for the southern cD galaxy."
],
[
"Radio spectra",
"The total flux densities were determined from integration in polygons with measured in areas encompassing the remnant radio galaxy, i.e., we determined the surface-brightness and divided it by the number of pixels in the polygon at each frequency.",
"These results are presented in Table REF along with the total intensity spectra of the remnant radio galaxy, and the southern cD galaxy is shown in Figure REF .",
"A 2.8$\\sigma $ upper limit at the 74 MHz [4] and the peak flux density from our 150 MHz band data for the southern cD galaxy are also reported.",
"We see that all these radio measurements, within the error bars, fall nearly along with a `straight' power law, $\\alpha $ = $-$ 1.2 $\\pm $ 0.2 with some hints of energy losses by synchrotron cooling for the dominant southern cD galaxy.",
"This provides confidence in our data calibration and its reduction methodologies.",
"The remnant radio galaxy is not detected in our 150 MHz band short observation, and is barely detected using LOFAR at 153 MHz (see also Sec.",
"REF ).",
"However in the 1050–1450 MHz band short observation, we see hints of radio emission from the remnant radio galaxy.",
"We made a low-resolution, 15$^{\\prime \\prime }$ image at the 1050–1450 MHz band and report peak flux density for the radio emission from the remnant radio galaxy in Figure REF .",
"Since this 1050–1450 MHz band measurement comes from a short $\\lesssim $ 0.9 hr duration observation, has relatively high rms noise (see also Sec.",
"), and the observation does not sample similar ($u,v$ )-spacings as the rest of the low-frequency observations, we henceforth no longer include it in our spectral index analysis.",
"It appears that the integrated spectrum for the remnant radio galaxy cannot be represented by a single power law.",
"The segment of the spectrum with the best data is $<$ 1250 MHz, and the low-frequency spectral index $\\alpha $ (240–610 MHz) = $-$ 1.4 $\\pm $ 0.2.",
"Our data cannot rule out the possibility of spectral shape to be due to two populations of the relativistic electrons: a low-frequency component characterized by $\\alpha $ (240–610 MHz) $\\approx $ $-$ 1.4 and a second component having $\\alpha $ ($>$ 610 MHz) $<$ $-$ 1.4 with a cutoff or a spectral break between 400 and 1250 MHz.",
"Figure: The plot shows the radio spectra between the 240 MHz and 250–500 MHz band data sets (red upper triangles) and between the 250–500 MHz band and 610 MHz band data sets (blue lower triangles) as a function of distance from the head.",
"The location of the pinch is shown as a vertical line.Figure: The averaged in-band spectral index image showing spectra between 300 MHz and 500 MHz using 250–500 MHz band data.",
"The total intensity surface-brightness contours are at 0.23 and 0.46 mJy beam -1 ^{-1} from the 300–500 MHz radio image (Figure , top-right panel).",
"We used three irregular polygon regions, marked in the image, and determined the mean (and rms) spectral indices that are denoted at the upper-right corner of the image.",
"The averaged integrated in-band spectral index for the remnant radio galaxy is denoted at the lower-left corner of the image.",
"The cross marks correspond to the positions of two cD galaxies and the position of the optical host galaxy, discussed in Sec.",
"and .",
"The dashed line indicates the best-fitting position of the X-ray surface-brightness edge discussed in Sec.",
".Figure: Background-subtracted and exposure-corrected Chandra image at ∼\\sim 4 '' ^{\\prime \\prime } angular resolution in the broad (0.5–7.0 keV) energy band.",
"The color bar shows the surface-brightness, in photon cm -2 ^{-2} s -1 ^{-1} pixels -1 ^{-1}.",
"The black surface-brightness contours are displayed using the 250–500 MHz band image.",
"The inset shows the enlargement of the inner region.",
"Also shown are two sets of sectors (in cyan and blue colors) that were used for the surface-brightness and temperature measurements.The spectral index measurements between the 240 MHz band and 250–500 MHz band data sets, and between the 250–500 MHz band and 610 band MHz band data sets show that the radio spectrum steepens with distance from the head, from $-$ 1.2 $\\pm $ 0.1 to $-$ 1.4 $\\pm $ 0.2, and from $-$ 1.3 $\\pm $ 0.1 to $-$ 1.6 $\\pm $ 0.2, respectively, due to synchrotron and inverse Compton losses (Figure REF ).",
"The location after the pinch, where the radio plasma from the AGN connects to the remnant radio galaxy, the spectrum, between the 240 MHz band and 250–500 MHz band data, remains flattened, $\\alpha $ = $-$ 1.3 $\\pm $ 0.1.",
"The wide-bandwith data sets were imaged to make multiscale multifrequency synthesis images, which provides images at center frequencies and the corresponding in-band spectral index maps (see also Sec.",
"REF ).",
"The in-band spectral index map averaged over a region of interest, to increase signal-to-noise ratio, for the 250–500 MHz band data is shown in Figure REF .",
"We divided the remnant radio galaxy into three distinct regions: (1) the high-surface-brightness narrower southeastern part ahead of the pinch (in cyan).",
"The latter low-surface-brightness broader northwestern part is divided into two; (2) the northeastern region (in yellow), and (3) the southwestern far end region (in magenta).",
"The size of the regions is chosen such that the signal in the total intensity image is (approximately) more than five times the rms noise.",
"The southern cD galaxy has an in-band spectral index of $-$ 0.9 $\\pm $ 0.2.",
"Similarly, the spectra of three regions of the remnant radio galaxy are $-$ 1.2 $\\pm $ 0.2, $-$ 1.3 $\\pm $ 0.2, and $-$ 1.4 $\\pm $ 0.2 for regions (1), (2), and (3) respectively.",
"This is consistent with the averaged in-band spectral index of $-$ 1.3 $\\pm $ 0.2 within the error bars for the remnant radio galaxy.",
"It is also consistent with the crude estimates of two 50 MHz subband images, 300–350 MHz and 450–500 MHz, at two extreme ends of the 250–500 MHz band data.",
"The region ahead of the pinch (in cyan) and the northeastern region close to two cD galaxies (in yellow) have comparable spectral indices.",
"It appears that the high-surface-brightness narrower region (in cyan) ahead of the pinch, is flatter than the two regions, together forming low-surface-brightness broader northwestern region (in yellow and magenta) after the pinch, consistent with our result discussed above.",
"Furthermore, the area between the two regions of the broader northwestern part of the source, i.e., the southwestern far-end region (in magenta) have a significantly steeper spectrum than the northeastern region close to two cD galaxies (in yellow)."
],
[
"ICM gas properties",
"[3] reported that the cluster contains gas with an extremely wide range of temperatures and a wealth of structure.",
"They also reported a discontinuity at $\\sim $ 30$^{\\prime \\prime }$ ($\\simeq $ 41.5 kpc) in the southeast region from the southern cD galaxy.",
"It is at nearly this same distance where we report an enhancement of radio surface-brightness, toward the two cD galaxies, showing a sharp feature.",
"We aim to search an X-ray surface-brightness edge, i.e., the shock front that is coincident with the northeastern sharp edge toward two cD galaxies of the remnant radio galaxy.",
"In order to understand the gas physics at this region of the cluster, we have extracted the surface-brightness profile from two box-shaped regions, one encompassing the remnant radio galaxy (called “outside\" here) and the other adjacent to it toward cD galaxies (called “inside\" here) as shown in Figure REF (in magenta).",
"We extracted the surface-brightnesses and the spectra from these regions, which are displayed in Figure REF .",
"A single temperature model with Galactic absorption and the abundance fixed provided a good fit to the spectrum.",
"We noticed the X-ray surface-brightness edge, where the surface-brightness drops (see Figures REF and REF ) and therefore implies a discontinuity, or jump, in the gas density.",
"The surface-brightnesses are $S_{\\rm x}$ (inside) and $S_{\\rm x}$ (outside) $\\approx $ 8.2 $\\pm $ 0.3 $\\times $ 10$^{-6}$ counts s$^{-1}$ arcsec$^{-2}$ and 3.7 $\\pm $ 0.3 $\\times $ 10$^{-6}$ counts s$^{-1}$ arcsec$^{-2}$ , respectively.",
"Consequently, our modeled densities in the two regions, $\\rho $ (inside) and $\\rho $ (outside) are approximately 2.9$^{+0.2}_{-0.1}$ $\\times $ 10$^{-3}$ cm$^{-3}$ and 1.9$^{+0.2}_{-0.2}$ $\\times $ 10$^{-3}$ cm$^{-3}$ , respectively.",
"Our corresponding derived temperatures for the two regions, $kT_{\\rm (inside)}$ and $kT_{\\rm (outside)}$ are approximately 6.1$^{+0.9}_{-0.8}$ keV ($\\chi ^2$ = 0.93 for 221 degrees of freedom) and 5.5$^{+0.5}_{-0.4}$ keV ($\\chi ^2$ = 1.08 for 462 degrees of freedom), respectively.",
"We also repeated this exercise for a pie-circular aperture (shown in blue in Figure REF ) centered on the southern cD galaxy and lying toward the radio emission from the remnant radio galaxy with two regions containing the radio emission from the remnant radio galaxy and the smaller adjacent region toward the cD galaxy.",
"The quantitative estimates from this are nearly identical and the fit statistics are at 90% confidence intervals.",
"Although with large error bars, these density and temperature measurements are hinting that the surface-brightness edge toward the two cD galaxies is possibly a shock front.",
"Consequently, the observed density jump and the temperature jump across the edge, $\\rho _{\\rm (inside)}$ /$\\rho _{\\rm (outside)}$ = 1.5$^{+0.2}_{-0.2}$ and $kT_{\\rm (inside)}$ /$kT_{\\rm (outside)}$ = 1.1$^{+0.7}_{-0.5}$ suggest the Mach numbers, $M$ = 1.3$^{+0.2}_{-0.2}$ and 1.1$^{+0.4}_{-0.3}$ , respectively under the Rankine-Hugoniot shock conditions for the intracluster gas, $\\gamma $ = 5/3, which are all obviously in agreement, are within the error bars of [3].",
"We also believe that a smaller measure for the temperature jump is due to the contribution of cool thermal emission from the core to the adjacent region toward two cD galaxies, $kT_{\\rm (inside)}$ .",
"Note, it is equally possible that the surface-brightness edge is a cold front.",
"Here, we assume it to be a shock front because of consistencies between our measurements and those of [3] and [20].",
"Figure: Plot of the radial surface-brightness profile and the temperature profile for the two red sectors (shown in Figure ) as a function of distance from the southern cD galaxy.",
"The error bars represent 90% confidence intervals."
],
[
"Discussion",
"In passing, we note that we do not detect the radio halo of A2065 in our current $\\approx 8^{\\prime \\prime }$ resolution 250–500 MHz band image.",
"This is likely to be due to the fairly short duration of our observing run (just $\\approx 1.3$ hr), which results in poor ($u,v$ )-coverage at the short spacings that are critical to detect $\\approx 10^\\prime $ -sized structures like the radio halo [7].",
"However, the large fractional bandwidth at 250–500 MHz band implies that it has excellent ($u,v$ )-coverage at the intermediate and long baselines that are needed to accurately map the arcminute-sized structures like our target, the remnant radio galaxy (angular size $\\approx 1^\\prime \\times 2^\\prime $ ).",
"In addition, we have performed deep clean, where we have used the multiscale multifrequency synthesis process of combining data from multiple spectral channels onto the same spatial-frequency grid during imaging to take advantage of the increased ($u,v$ )-coverage and imaging sensitivity, and hence we do not foresee any loss of flux density that could be linked to the remnant radio galaxy in the data presented above.",
"Below we discuss the nature of this newly discovered remnant radio galaxy in A2065.",
"As the remnant radio galaxy ages, its electrons will lose so much energy that their spectra will eventually steepen.",
"This steepening would reach to a point where they do not emit significant radiation at high frequencies, instead, the radio emission may only be observable at low frequencies.",
"However, [2] argues that in addition to the aged population of remnant radio galaxies, there should be younger population as well in which lobes have not had time to steepen over the observed frequency range [8], [25].",
"It is possible that the remnant phase of a radio galaxy is governed by a Sedov-like expansion, which will cause the dimming of radio emission due to a decrease in the magnetic field strength and a decrease in particle energies due to adiabatic expansion losses.",
"Unfortunately, the models of the spectra of remnant radio galaxy are highly degenerate, and the modeling of an individual remnant radio galaxy cannot constrain the remnant phase of radio galaxy [15].",
"Therefore, a possible way to constrain the remnant phase is via a statistical approach, and sadly, due to their small number, the physics of remnant radio galaxies remains poorly understood.",
"Of course, alongside their must exist a significant population of remnant radio galaxies, which show the absence of detectable radio emission in many clusters associated with merger activity [11].",
"Thus, searching for low-surface-brightness profiles, say $\\lesssim $ 50 mJy arcmin$^{-2}$ , with an absence of radio cores and hotspots, and with no well-defined radio morphologies, is a possible way to identify remnant radio sources in large survey images.",
"Radio morphology of the remnant radio galaxy in A2065 shows that it is an elongated, bar-shaped structure (52$^{\\prime \\prime }$ $\\times $ 110$^{\\prime \\prime }$ $\\simeq $ 72 $\\times $ 152 kpc$^2$ ), which is buried within a radio halo emission of size $\\lesssim $ 1 Mpc in size [6] and [7].",
"The remnant radio galaxy is located in the near vicinity of the southern cD galaxy; the northeastern boundary of the remnant radio galaxy is merely at a distance of $\\sim $ 33$^{\\prime \\prime }$ ($\\simeq $ 45.7 kpc) from the southern cD galaxy.",
"The possible radio core of the remnant radio galaxy is coincident with the WISEA J152228.01$+$ 274141.3 source that is 78 away from the optical galaxy V1CG 136.",
"Our spectral index measurements show that the radio spectrum steepens with distance from the possible radio core from $\\alpha $ = $-$ 1.2 $\\pm $ 0.1 to $\\alpha $ = $-$ 1.4 $\\pm $ 0.2 in our low-frequency (between the 240 MHz band and the 250–500 MHz band) spectra, and from $\\alpha $ = $-$ 1.3 $\\pm $ 0.1 to $\\alpha $ = $-$ 1.6 $\\pm $ 0.2 in our high-frequency (between the 250–500 MHz band and the 610 MHz band) spectra (Figure REF ), as is expected for synchrotron and inverse Compton losses.",
"We also find evidence for spectral steepening across the remnant radio galaxy in our in-band spectral index measurements, from $\\alpha $ = $-$ 1.3 $\\pm $ 0.2 to $\\alpha $ = $-$ 1.4 $\\pm $ 0.2 (Figure REF ), in the direction away from the two cD galaxies.",
"Our X-ray measurements show a surface-brightness edge that is coincident with the northeastern boundary of the remnant radio galaxy and its nature (shock or cold front) is not clear from the data.",
"If it is a shock front, it is rather weak, with $M$ being between 1.1 and 1.3; however it is unclear if an (inward?)",
"traveling shock is responsible for spectral steepening across the remnant radio galaxy.",
"Below we discuss the possible progenitor of the remnant radio galaxy in A2065."
],
[
"Remnant source diagnostic",
"Remnant radio sources are old, and are bound by model-dependent physical parameters because of the required assumptions of volume-filling factors, etc.",
"We use the parameter $N$ , which is the number of relativistic electrons found by integrating the energy spectrum of radiating electrons between Lorentz factors of 1000 and 10,000 for a typical equipartition magnetic field of 5 $\\mu $ G in the 2–200 MHz band for the remnant radio galaxy.",
"The number distribution of relativistic electrons drops rapidly with energy for a steep spectrum remnant radio galaxy.",
"Hence the number of low-energy electrons is a good measure of the total acceleration of relativistic electrons during the lifetime of $\\sim $ 10$^8$ yr for the small redshift of the source, and indicates the entire history of the source.",
"We thus determine $N$ $\\approx $ 2.1 $\\times $ 10$^{60}$ and the total energy stored in the magnetic field $\\approx $ 1.1 $\\times $ 10$^{57}$ erg assuming cylindrical geometry for the size of the remnant radio galaxy [21], [22].",
"Here, we have assumed the volume filling factor is unity, and hence the volume and the number of relativistic electrons $N$ are thus the upper limits.",
"The emitting plasma will most likely occupy a smaller volume, leading to larger energy densities and therefore larger values of the magnetic field strength and will require a smaller number of electrons to produce the observed luminosity.",
"These basic energetics are similar to the Coma radio relic source 1253$+$ 275 [10].",
"Moreover, the number of relativistic electrons, $N$ , is similar to that of the typical FR I radio galaxy 3C 31 but two orders of magnitude smaller than the FR II radio galaxy Cygnus A [13].",
"This suggests that the FR I radio galaxy is possibly the progenitor of the remnant radio galaxy in A2065.",
"Thus, our high-resolution, high-sensitivity radio image at the 250–500 MHz band along with the presence of the WISEA J152228.01$+$ 274141.3 source suggest that the source is a wide-angle-tailed radio source (see also Sec.",
"REF ).",
"Furthermore, the position of the WISEA J152228.01$+$ 274141.3 source exactly coincides with the likely radio core position of the wide-angle-tailed radio source.",
"The source diagnostic based on the number of low-energy relativistic electrons also suggests that the remnant radio galaxy is fed by the small radio tail of the wide-angle-tailed FR I radio galaxy.",
"If this proposition is correct, the two oppositely directed radio jets emanating from the apex of the WISEA J152228.01$+$ 274141.3 source initially traverse toward the northwest and southeast.",
"As the galaxy plows through the dense ICM, these jets traversing in opposite directions form a trail after a sharp bend in the southeastern jet due to interaction with the ICM.",
"The jets probably overlap and twist forming a pinch and then expand, and the jet becomes weaker [18].",
"The sharpness in surface-brightness toward the northeast, i.e., the side facing two cD galaxies and diffuse extensions toward the far southwest side, along with corresponding spectral gradients, suggests the passage of an (inward?)",
"traveling shock front."
],
[
"Summary and conclusions",
"The remnant radio source seems to trace AGN radio plasma that has somehow been reenergised through processes in the ICM.",
"It is possible that, similar to radio relic sources, remnant radio sources are also believed to be tracers of merger shocks, unfortunately, only a few clusters have shown to possess corresponding features in hot X-ray gas.",
"The remnant radio galaxy in A2065 may belong to this (rare) class possessing an X-ray surface-brightness edge across the remnant radio galaxy, indicating a possible connection with the shock front.",
"Using our deep high-resolution low-frequency observations, we discovered a bar-shaped remnant radio galaxy, whose size is $\\approx $ 52$^{\\prime \\prime }$ $\\times $ 110$^{\\prime \\prime }$ (= 72 $\\times $ 152 kpc$^2$ ), remnant radio galaxy in the massive unequal-mass-merger galaxy cluster A2065 at $z$ = 0.072.",
"It has a steep spectral index $\\alpha $ (240–610 MHz) = $-$ 1.4 $\\pm $ 0.2 and $\\alpha $ ($>$ 610 MHz) $<$ $-$ 1.4 $\\pm $ 0.2, corresponding to the line representing the best-fitting regression to the radio data.",
"The Chandra data, and the upgraded and “classic\" GMRT data, show (i) the presence of an X-ray surface-brightness edge attributed to a shock front, though a possibility of cold front cannot be ruled out, at the remnant radio galaxy's edge toward the two cD galaxies, (ii) the tentative flattening of the radio spectral index, in our 250–500 MHz in-band data, of the remnant radio galaxy at the near end of the X-ray surface-brightness edge, and (iii) possibly a wide-angle-tailed, FR I radio galaxy is the progenitor of the remnant radio galaxy.",
"Thus, the remnant radio galaxy has possibly been reinvigorated by the passage of the shock front and adiabatic compression, and shows the expected change in radio emission.",
"We also suggest that the remnant radio galaxy was seeded by the AGN during its past active phase, and has been hosted by the WISEA J152228.01$+$ 274141.3 source, demonstrating the connection between AGNs and remnant radio galaxy, similar to the connection between AGNs and radio relic sources [28].",
"Although low-frequency observations are starting to reveal more and more of these type of sources, presently there are a very few known remnant radio systems; clearly an outstanding difficulty is the small sample size.",
"Although the number of known remnant radio sources is increasing, better data are necessary for several of them to allow a detailed study.",
"Fortunately, forthcoming radio surveys using the LOFAR, MeerKAT, Square Kilometre Array, etc., will greatly increase the sample size, along with deep multifrequency radio data, e.g., using the upgraded GMRT, which will constrain the physics and nature of poorly understood remnant radio sources.",
"We are undertaking an upgraded GMRT study at the 125–250 MHz, 250–500 MHz, and 550–850 MHz bands of several clusters in order to detect such ultra-steep-spectrum sources, map radio halo emission, and provide exact flux density estimates, models, and statistics."
],
[
"Acknowledgments",
"I would like to thank the anonymous referee for helpful remarks.",
"D.V.L.",
"acknowledges the support of the Department of Atomic Energy, Government of India, under project No.",
"12-R&D-TFR-5.02-0700, and thanks the staff of the GMRT who made these observations possible.",
"The GMRT is run by the National Centre for Radio Astrophysics of the Tata Institute of Fundamental Research.",
"This research made use of the radio astronomical database GalaxyClusters.com, maintained by the Observatory of Hamburg.",
"This research has made use of the NED, which is operated by the Jet Propulsion Laboratory, Caltech, under contract with the NASA, and NASA's Astrophysics Data System.",
"Facilities: Chandra, GMRT Data Availability: The GMRT and Chandra data underlying this article are available via the GMRT Online Archive, naps.ncra.tifr.res.in/goa and the Chandra Data Archive, cxc.harvard.edu/cda.",
"All data analyses packages used in this work are publicly available."
]
] | 2107.01795 |
[
[
"Restless and Uncertain: Robust Policies for Restless Bandits via Deep\n Multi-Agent Reinforcement Learning"
],
[
"Abstract We introduce robustness in \\textit{restless multi-armed bandits} (RMABs), a popular model for constrained resource allocation among independent stochastic processes (arms).",
"Nearly all RMAB techniques assume stochastic dynamics are precisely known.",
"However, in many real-world settings, dynamics are estimated with significant \\emph{uncertainty}, e.g., via historical data, which can lead to bad outcomes if ignored.",
"To address this, we develop an algorithm to compute minimax regret -- robust policies for RMABs.",
"Our approach uses a double oracle framework (oracles for \\textit{agent} and \\textit{nature}), which is often used for single-process robust planning but requires significant new techniques to accommodate the combinatorial nature of RMABs.",
"Specifically, we design a deep reinforcement learning (RL) algorithm, DDLPO, which tackles the combinatorial challenge by learning an auxiliary \"$\\lambda$-network\" in tandem with policy networks per arm, greatly reducing sample complexity, with guarantees on convergence.",
"DDLPO, of general interest, implements our reward-maximizing agent oracle.",
"We then tackle the challenging regret-maximizing nature oracle, a non-stationary RL challenge, by formulating it as a multi-agent RL problem between a policy optimizer and adversarial nature.",
"This formulation is of general interest -- we solve it for RMABs by creating a multi-agent extension of DDLPO with a shared critic.",
"We show our approaches work well in three experimental domains."
],
[
"Introduction",
"The restless multi-armed bandit (RMAB) problem models sequential decision making tasks under a limited resource constraint.",
"An RMAB instance consists of a budget $B$ and a set of $N$ arms, where an arm corresponds to an action-dependent Markov state-transition process.",
"At every timestep, a planner selects a subset of $N$ arms and decides what actions to take on each selected arm such that the total cost of actions per timestep is less than a budget $B$ .",
"The arms transition to new states and generate rewards, depending on the action taken on them.",
"The goal is to plan a policy that maximizes the total expected reward.",
"This problem naturally captures a wide range of real-world problems.",
"For example, in a healthcare intervention problem [26], a health worker (planner) is in charge of the well-being of a set of patients (arms) who have enrolled in a health program, e.g., by ensuring adherence to daily tuberculosis medication.",
"The healthcare worker provides timely interventions to the patients via phone call or in-person visit to ensure that the patients remain in the adhering state.",
"However, phone calls and in-person visits require dedicated time from the health worker, thereby imposing a budget $B$ on the number of patients she can call or visit in-person each day.",
"The problem is challenging because of three key reasons: (1) the state of each patient representing the extent of adherence evolves over time; (2) the intervention decisions at each timestep influence the state transitions of all patients and, consequently, affect future rewards; and (3) different actions incur different costs, and the total cost of allocation must be at most $B$ at every timestep.",
"The combinatorial nature of the problem makes it computationally expensive to solve optimally [31].",
"Prior works have considered specific variants of the RMAB problem that are applicable to real-world scenarios, including healthcare intervention planning [23], [26], anti-poaching patrol planning [34], sensor monitoring tasks [11], [17], machine replacement [35], and many more.",
"A common assumption across existing literature on RMABs is that the state transition dynamics of all the arms are known beforehand.",
"Unfortunately, in real-world scenarios, even in the presence of prior data, there is still often significant uncertainty in transition dynamics.",
"A natural way to encode this uncertainty is interval uncertainty—rather than assuming that the probability of transitioning from one state to another is fixed to one value, we assume that it can take any value from a given closed continuous interval.",
"This additional uncertainty makes the RMAB problem even more challenging since any solution must be robust to variations in transition probabilities within the given interval.",
"In the presence of interval uncertainty, we formulate a generalization of RMAB that we call Robust Restless Bandits.",
"Our goal is to achieve minimax regret—minimize the worst-case regret against the optimal policy over all possible values of transition probabilities within the given uncertainty intervals.",
"Our contributions are as follows: We introduce the Robust RMAB problem for interval uncertainty settings and optimize for minimax regret.",
"We are the first to provide a minimax-regret solution for the restless bandits problem, while also generalizing to multi-action RMABs.",
"We develop a double oracle algorithm for solving Robust RMABs and demonstrate its effectiveness on experimental domains, including two inspired by real-world problems.",
"To enable our double oracle approach, we propose RMABPPO, a deep reinforcement learning (RL) algorithm for solving multi-action RMABs.",
"RMABPPO hinges on learning an auxiliary “$\\lambda $ -network” allowing us to decouple each arm's learning, greatly reducing sample complexity.",
"Under minimax regret, the adversary in the double oracle approach is notoriously difficult to implement due to non-stationarity.",
"To address this, we formulate the adversary oracle as a multi-agent reinforcement learning problem and solve it with a multi-agent extension of RMABPPO."
],
[
"Restless bandits",
"[40] introduced RMABs, following which, there is a vast literature on binary-action RMABs.",
"In the classic setting, a planner selects $k$ out of $N$ state-transitioning arms, resulting in a two-action decision problem with the goal of maximizing reward.",
"Although the problem is PSPACE-Hard [31], heuristics based on Lagrangian relaxation of the optimization have been proposed [40] which turns out to be optimal under the indexability criterion.",
"However, the Whittle index solution does not extend to budgeted multi-action settings, where there are more than two actions, each with an associated cost.",
"[12] and [16] have extended index-based solutions to multi-action RMABs for instances with special monotonic structure.",
"Along similar lines, [20] proposed a method that leverages the convexity of an approximate Lagrangian version of the multi-action RMAB problem.",
"Another relevant category of research is taking decisions on weakly coupled Markov decision processes (WCMDP).",
"[15] proposed a Lagrangian relaxation of the general WCMDP problem and proposed an LP for minimizing the Lagrange bound.",
"[3] and [13] provided less scalable solutions that provide better approximations to WCMDPs.",
"All these papers have assumed perfect knowledge of the state transition dynamics to compute policies.",
"A few recent works have studied RMABs in online settings with unknown transition probabilities but make the limiting assumptions that the planner takes only one non-combinatorial action [8] or that states periodically reset [18].",
"None of the above papers consider robust planning under environment uncertainty, which we address.",
"Despite the recent successes of reinforcement learning (RL) for solving large-scale games [28], [37], RL has so far seen little application to RMABs, except for a few recent works that learn Whittle indices for indexable binary-action RMABs using (i) deep RL [29] and (ii) Q-learning when states are observable [5], [7] or when arms are homogeneous [4].",
"In contrast, our deep RL approach provides a more general solution to binary and multi-action RMAB domains that performs well regardless of indexability.",
"One recent work does also address learning in the multi-action setting [19], but their approach is based on tabular Q-learning which can be sample inefficient and scale poorly, unlike function-approximation methods like ours."
],
[
"Robust planning",
"The RL literature has a large body of work on robust planning, mainly focused on maximizing the minimum (maximin) reward through robust adversarial RL [33] or multi-agent RL settings [22], [24].",
"The minimax regret criterion [6] has been offered as a compelling alternative to maximin reward, which often leads to overly conservative policies that buffer against worst-possible realizations.",
"A challenge to computing minimax regret–optimal strategies is that such games often involve very large and sometimes continuous strategy spaces, rendering it impossible to explore the entire strategy space.",
"The double oracle approach [27] has been applied as an effective means by which to optimally explore only a small subset of these large spaces while still guaranteeing optimal performance [30], [10].",
"Double oracle has been extended to optimize multi-agent RL problems with multiple selfish agents [22].",
"More recently, [41] offered an algorithm for minimax regret planning for a non-RMAB problem using RL to instantiate the two oracles.",
"However, their approach cannot handle budget-constrained discrete actions, which we have in the RMAB setting.",
"Additionally, we formulate the nature oracle as a multi-agent RL problem, allowing us to handle the joint discrete and continuous action spaces that arise in the minimax regret nature oracle.",
"Robustness objectives have been considered for bandit applications in the two-action stochastic setting, where each pull of an arm draws a reward sampled from an unknown Bernoulli distribution.",
"[39] address minimax regret of time-varying reward shifts using heuristics to trade off remembering vs. forgetting, and [9] consider a maximin objective to guide exploration in Monte Carlo Tree Search.",
"However, RMABs are significantly harder than the stochastic settings since, in RMABs, the rewards are dependent on the current state which in turn depends on the actions taken on the arms."
],
[
"Problem Statement",
"We consider the multi-action RMAB setting with $N$ arms.",
"Each arm $n\\in [N]$ follows a Markov decision process (MDP) $(\\mathcal {S}_n, \\mathcal {A}_n, \\mathcal {C}_n, T_n, R_n, \\beta )$ , where $\\mathcal {S}_n$ is a set of finite, discrete states, $\\mathcal {A}_n$ is a set of finite, discrete actions, $\\mathcal {C}_n : \\mathcal {A}_n \\xrightarrow{} \\mathbb {R}$ corresponds to action costs, $T_n: \\mathcal {S}_n\\times \\mathcal {A}_n \\times \\mathcal {S}_n \\xrightarrow{} [0,1]$ gives the probability of transitioning from one state to another given an action, $R_n:\\mathcal {S}_n\\times \\mathcal {A}_n \\times \\mathcal {S}_n \\xrightarrow{} \\mathbb {R}$ is a reward function, and $\\beta \\in [0, 1)$ is the discount parameter.",
"The agent must select actions for each arm, each round, such that the sum cost of actions does not exceed a per-round budget $B$ .",
"The aim of multi-action RMABs is to maximize total reward over a fixed number of $H$ rounds, subject to this budget constraint, generalizing the well-studied binary-action RMAB.",
"In this work, we extend multi-action RMABs to the robust setting in which the exact transition probabilities are unknown.",
"Instead, the transitions of each arm are determined by a set of parameters $\\mathcal {P}_n$ .",
"For a given arm $n \\in [N]$ and parameter $p_n \\in \\mathcal {P}_n$ , let $\\langle {\\omega _{n,p_n}}\\rangle :=[\\underline{\\omega }_{n,p_n}, \\overline{\\omega }_{n,p_n}]$ represent the range, i.e., uncertainty over transition probabilities.",
"We ask the following question: how to find a solution to the multi-action RMAB problem that is robust to such uncertainties?",
"We consider an objective that incorporates our uncertainty over $\\mathcal {P}_n$ .",
"Let $\\omega $ be a given realization of the transition probabilities such that $\\omega _{n,p_n} \\in \\langle {\\omega _{n,p_n}}\\rangle $ for all $n \\in [N]$ and $p_n \\in \\mathcal {P}_n$ .",
"Let $G(\\pi ,\\omega )$ be the planner's expected reward under policy $\\pi $ and a realization $\\omega $ of the uncertainty.",
"Regret is defined: $L(\\pi ,\\omega ) = G(\\pi ^*_{\\omega },\\omega ) - G(\\pi ,\\omega ) \\ ,$ where $\\pi ^*_{\\omega }$ is the optimal policy under $\\omega $ .",
"In our robust planning formulation, our objective is to compute a policy $\\pi $ that minimizes the maximum regret $L$ possible for any realization of $\\omega $ , leading to the following minimax objective: $\\min _{\\pi }\\max _{{\\omega }}{L(\\pi ,{\\omega })} \\ .$ This problem is computationally expensive to solve since simply computing a policy $\\pi $ that maximizes the expected reward $G(\\pi ,\\omega )$ is PSPACE-Hard [31] even for the special case when the transition rules are known i.e., $\\omega $ is given.",
"This challenge is likely a key reason why the robust formulation has not yet been addressed in the literature.",
"To overcome the complexity of the minimax optimization, we take a double oracle approach [27], which requires key innovations to work in the RMAB setting."
],
[
"Preliminaries",
"The double oracle approach achieves the minimax regret objective in Eq.",
"REF ) by casting the optimization problem as a zero-sum game between two players, the agent and nature [41].",
"The agent selects a policy that minimizes regret for some realization of the transition probabilities.",
"Nature then adversarially selects the values of $\\mathcal {P}_n$ that maximize regret for a given policy of the planner.",
"This framework is desirable since it converges to an $\\varepsilon $ –optimal solution [2], [41], assuming there are oracles that return the best response for both players.",
"The key technical contributions of this paper arise from the design of the agent and nature oracles for best response computations.",
"For the agent, minimizing regret with respect to a fixed nature strategy is equivalent to maximizing reward w.r.t.",
"that strategy, so the agent objective is the same as solving a multi-action RMAB to find the best policy.",
"A policy $\\pi $ maps states to decision matrices $\\mathbf {A} \\in \\lbrace 0,1\\rbrace ^{N\\times |\\mathcal {A}|}$ where the total number of active actions is constrained by a budget for each round.",
"Let $\\mathbf {s} = (s^1, \\ldots , s^{N})$ represent the initial state of each arm.",
"Then, for a given parameter $\\omega $ , the optimal policy $\\pi ^*_{\\omega }$ maximizes the expected discounted sum of rewards of all arms as given by the constrained Bellman equation: $\\begin{aligned}J(\\mathbf {s}) &= \\max _{\\mathbf {A}}\\left\\lbrace \\sum _{n=1}^{N} R_n(s_n) + \\beta \\mathbb {E}_{\\omega }[J(\\mathbf {s}^\\prime ) \\mid \\mathbf {s}, \\mathbf {A}]\\right\\rbrace \\\\\\text{s.t. }",
"& \\sum _{j=1}^{|\\mathcal {A}|} a_{nj} = 1 \\hspace{8.53581pt} \\forall n \\in [N]\\qquad \\sum _{n=1}^{N}\\sum _{j=1}^{|\\mathcal {A}|} a_{nj}c_{nj} \\le B\\end{aligned}$ where $a_{nj} \\in \\mathbf {A}_n$ and $c_{nj}$ are the corresponding action costs in $\\mathcal {C}_n$ .",
"We then relax the problem by taking the Lagrangian relaxation of the budget constraint [15], giving: $\\begin{aligned}&J(\\mathbf {s}, \\lambda ^*) = \\min _{\\lambda } \\left( \\frac{\\lambda B}{1-\\beta } + \\sum _{n=1}^{N}\\max _{a_{nj}\\in \\mathcal {A}_n}Q_n(s_n, a_{nj}, \\lambda ) \\right)\\end{aligned}$ $\\begin{aligned}\\text{where }\\hspace{2.84526pt} Q_n(s_n, a_{nj}, \\lambda ) =R_n(s_n) - \\lambda c_{nj} + \\beta \\mathbb {E}_{\\omega } \\left[ Q_n(s^{\\prime }, a_{nj}, \\lambda ) \\mid \\pi ^{La}_{\\omega }(\\lambda ) \\right] \\ .\\end{aligned}$ Here, $Q$ is the value function and $\\pi ^{La}_{\\omega }(\\lambda )$ is the optimal policy for a given $\\lambda $ .",
"See [3] for a detailed derivation.",
"Note that for a given value of $\\lambda $ , Eq.",
"REF could be solved using $N$ individual value iterations.",
"However, setting $\\lambda :=$ $\\lambda ^*$ is critical to finding good policies for a multi-action RMAB and is known to be asymptotically optimal in the binary-action case [38], i.e., $\\pi ^{La}_{\\omega }(\\lambda ^*) \\xrightarrow{} \\pi ^{*}_{\\omega }$ .",
"Given this relationship, in the remainder of the paper, we focus of computing $\\pi ^{La}_{\\omega }(\\lambda ^*)$ and denote it as $\\pi ^{*}_{\\omega }$ for convenience."
],
[
"Solving Robust Restless Bandits",
"We now build up our approach for finding robust RMAB policies.",
"The underlying pure strategy space for the agent is the set of all feasible policies $\\pi : \\mathcal {S}_1\\times \\cdots \\times \\mathcal {S}_N \\mapsto \\mathcal {A}_1\\times \\cdots \\times \\mathcal {A}_N$ .",
"The pure strategy space for nature is a closed set of parameters $\\omega $ within the given uncertainty intervals.",
"The agent oracle's goal is to find a policy $\\pi $ , or pure strategy, to minimize regret (Eq.",
"REF ) given a mixed strategy $\\tilde{\\omega }$ , where a mixed strategy is a probability distribution over a set of pure strategies.",
"That is, the agent minimizes $L(\\pi ,\\tilde{\\omega })$ w.r.t.",
"$\\pi $ , while $\\tilde{\\omega }$ is constant.",
"Recall that $L(\\pi ,\\tilde{\\omega }) = G(\\pi ^{*}_{\\tilde{\\omega }},\\tilde{\\omega }) - G(\\pi ,\\tilde{\\omega })$ .",
"Since the first term $G(\\pi ^{*}_{\\tilde{\\omega }},\\tilde{\\omega })$ is constant, minimizing $L(\\pi ,\\tilde{\\omega })$ is equivalent to maximizing the second term $G(\\pi ,\\tilde{\\omega })$ , which is maximal at $\\pi =\\pi ^{*}_{\\tilde{\\omega }}$ .",
"In other words, the agent oracle must compute an optimal reward-maximizing policy w.r.t.",
"$\\tilde{\\omega }$ .",
"The objective of maximizing reward is standard in reinforcement learning, enabling us to build off existing RL techniques as we show in Section REF .",
"On the other hand, the nature oracle's goal is to find a parameter setting $\\omega $ , or pure strategy, that maximizes the agent's regret given a mixed strategy $\\tilde{\\pi }$ , i.e., maximize $L(\\tilde{\\pi },\\omega )$ with respect to $\\omega $ , while $\\tilde{\\pi }$ is fixed.",
"This objective is far more challenging because both $G(\\tilde{\\pi }^*_{\\omega },\\omega )$ and $G(\\tilde{\\pi },\\omega )$ are functions of $\\omega $ .",
"Moreover, computing $G(\\tilde{\\pi }^*_{\\omega },\\omega )$ requires obtaining an optimal policy $\\tilde{\\pi }^*_{\\omega }$ as $\\omega $ changes in the optimization.",
"Unfortunately, $\\omega $ comes from an infinite continuous strategy space, making this problem difficult.",
"However, as one of our main contributions, we propose a novel method for implementing the regret-maximizing nature oracle by casting it as a multi-agent reinforcement learning problem, simultaneously solving for policies $\\pi ^*_{\\omega }$ while computing worst-case parameters $\\omega $ to maximize $L(\\tilde{\\pi },\\omega )$ .",
"Solving a multi-agent reinforcement learning problem first requires an RL algorithm to optimize the underlying policy; hence we first introduce a novel RL approach, RMABPPO, to solve RMABs (Sec.",
"REF ) as a part of our agent oracle and then use the algorithm as the backbone of our nature oracle (Sec.",
"REF )."
],
[
"Agent Oracle: Deep RL for RMABs via RMABPPO",
"Existing deep RL approaches can be applied to the objective in Eq.",
"REF , but they fail to scale past trivially sized RMAB problems since the action and state spaces grow exponentially in $N$ .",
"For example, for a binary-action RMAB with $N=50$ and $B=20$ , the action space would be of size $\\binom{50}{20}\\approx 10^{12}$ , which is not feasible to learn, even with a neural network.",
"To overcome this, we develop a novel deep RL algorithm that instead solves the decoupled problem (Eq.",
"REF ).",
"The key benefit of decoupling is to render policies and $Q$ values of each arm independent, allowing us to learn $N$ independent networks with linearly sized state and action spaces, relieving the combinatorial burden of the learning problem — the above example would now only have $N \\times 2 = 100$ actions.",
"However, this approach introduces a new technical challenge in solving the dual objective which maximizes over policies but minimizes over $\\lambda $ , as discussed in Sec. .",
"We derive a dual gradient update procedure that iteratively optimizes each objective as follows: (1) holding $\\lambda $ constant, learn $N$ independent policies via a policy gradient procedure, augmenting the state space to include $\\lambda $ as input, as in Eq.",
"REF ; (2) use sampled trajectories from those learned policies as an estimate to update $\\lambda $ towards its minimizing value via a novel gradient update rule.",
"Another challenge is that $\\lambda ^*$ of Eq.",
"REF depends on the current state of each arm — therefore, a key element of our approach is to learn this function concurrently with our iterative optimization, using a neural network we call the $\\lambda $ -network that is parameterized by $\\Lambda $ .",
"To train the $\\lambda $ -network, we use the following gradient update rule.",
"[]propositionlambdaUpdate A gradient rule for updating the $\\lambda $ -network, parameterized by $\\Lambda $ , such that for a state $\\mathbf {s}$ , the $\\lambda $ -network predicts the value $\\lambda $ that minimizes Eq.",
"REF is as follows: $\\begin{aligned}\\Lambda _t = \\Lambda _{t-1} - \\alpha \\left( \\frac{B}{1-\\beta } + \\sum _{n=1}^{N}D_n(s_n, \\lambda _{t-1}(\\mathbf {s})) \\right)\\end{aligned}$ where $\\alpha $ is the learning rate and $D_n(s_n, \\lambda )$ is the negative of the expected $\\beta $ -discounted sum of action costs for arm $n$ starting at state $s_n$ under the optimal policy for arm $n$ for a given value of $\\lambda $ .",
"Although $D_n$ cannot be computed exactly as we do not know the optimal policy, it can be estimated from samples of multiple rollouts of the policy during training.",
"As long as arm policies are trained for adequate time on the given value of $\\lambda $ , the gradient estimate will be accurate, i.e., $D_n(s_n, \\lambda _{t-1}(\\mathbf {s})) \\approx -\\sum _{k=0}^{K-1} \\beta ^k c_{nk}$ where $K$ is the number of samples collected in an epoch and $c_{nk}$ is the action cost of arm $n$ in round $k$ .",
"Moreover, this procedure will converge to the optimal parameters $\\Lambda $ if the arm policies are optimal.",
"[]propositionlambdaConvergence Given arm policies corresponding to optimal $Q$ -functions, the gradient update rule of Prop.",
"REF will lead $\\Lambda $ to converge to the optimal as the number of training epochs and $K\\xrightarrow{}\\infty $ .",
"Note that to collect samples that reflect the proper gradient, the RMAB budget must not be imposed at training time — rather the policy networks and $\\lambda $ -network must learn to play the Lagrange policy of Eq.",
"REF which spends the correct budget in expectation.",
"At training time, actions are sampled randomly according to the actor network distributions.",
"At test time, actions are taken deterministically by greedily selecting the highest probability actions until the budget is spent.",
"In theory, the policy networks could be trained via any deep RL procedure, as long as the above characteristics for training the $\\lambda $ -network are ensured.",
"In practice, we train with proximal policy optimization (PPO) [36], which has been demonstrated to have state-of-the-art performance while being relatively simple to implement, and will allow flexibility to handle both discrete and continuous actions—the latter will be important for the nature oracle.",
"We also navigate the important trade-off between exploring new policy actions after an update to $\\Lambda $ and allowing the policy networks to converge before updating $\\Lambda $ , by using an entropy regularization term on the loss function of the policy networks for each arm that is controlled by a cyclical temperature parameter.",
"We call our algorithm RMABPPO, give pseudocode in Algorithm REF , and give more implementation details in the appendix."
],
[
"Nature Oracle: Multi-Agent RL for RMABs via MA-RMABPPO",
"Armed with a deep RL procedure for learning RMAB policies, we are now ready to create the multi-agent extension needed to implement the nature oracle.",
"Recall that the challenge of the nature oracle is to simultaneously optimize a policy as well as the environment parameters $\\omega $ .",
"We propose to treat this optimization as a multi-agent RL problem designed to handle this form of non-stationarity [25] via a centralized critic.",
"To implement the nature oracle, we introduce two agents $A$ and $B$ , where $A$ 's goal is to optimize the RMAB policy $\\pi ^*_{\\tilde{\\omega }}$ and $B$ 's goal is to find parameters $\\omega $ that maximize regret of the current agent mixed strategy $\\tilde{\\pi }$ .",
"The shared environment transition function is $T: \\mathcal {S} \\times \\mathcal {A}_A \\times \\mathcal {A}_B \\xrightarrow{} \\mathcal {S}$ .",
"The action space $\\mathcal {A}_A$ will be the same as the action space of multi-action RMAB.",
"At a given state $\\mathbf {s}$ , the action space $\\mathcal {A}_B$ will allow agent $B$ to select $\\omega $ which, in general, depends on $\\mathbf {s}$ .",
"That is, at each step, agent $B$ will select environment parameters $\\omega $ , and thus state/action transition probabilities that will determine the outcome of agent $A$ 's actions.",
"We adopt the centralized critic idea from multi-agent PPO [43] to our RMAB setting to create MA-RMABPPO.",
"Since the policy space of agent $A$ is discrete while that of agent $B$ is continuous, a notable benefit to PPO is that it offers a convenient way to train both policies.",
"The final step is to define the rewards for each agent $A$ and $B$ to match their respective objectives.",
"Since agent $A$ 's objective is to maximize $\\pi ^*_{\\tilde{\\omega }}$ , it simply adopts the reward defined by the RMAB, i.e., ${R}^{(A)} = \\sum _{n=1}^N R_n$ .",
"However, agent $B$ 's objective is to learn the regret-maximizing parameters $\\omega $ .",
"This objective is challenging because it requires computing and optimizing over the returns of the fixed input policy $\\tilde{\\pi }$ with respect to all possible $\\omega $ values, which in general is non-convex.",
"In practice, to estimate the returns of $\\tilde{\\pi }$ , we execute a series of single-step roll-outs for computational efficiency.",
"That is, given $\\mathbf {s}$ and $\\mathbf {a}$ at a given round, we sample the next state profile $\\mathbf {s^\\prime }$ , and define the reward of agent $B$ (i.e., the regret of input policy $\\tilde{\\pi }$ ) as ${R}^{(B)} = \\sum _{n=1}^N R_n(s_n,a_n,s^\\prime _n) - \\frac{1}{Y}\\sum _{y=1}^{Y}r_y^{\\tilde{\\pi }}$ , where $r_y^{\\tilde{\\pi }}$ is the reward obtained from each of $Y$ 1-step Monte Carlo simulations of the mixed strategy $\\tilde{\\pi }$ .",
"Since agent $A$ has the same policy network architecture as RMABPPO, i.e., N discrete policy networks and one $\\lambda $ -network.",
"The agent $B$ actor network is a single continuous-action policy network.",
"Since agent $A$ and $B$ have separate objectives, they each have their own centralized critic networks which take the actions of the other as inputs to their state space.",
"For agent $A$ , this is implemented as one critic network per arm, where each network is augmented with agent $B$ 's actions.",
"Finally, to ensure good gradient estimates for the $\\lambda $ -network in MA-RMABPPO, we keep agent $B$ 's network — and thus the environment — constant between $\\lambda $ updates, updating $B$ 's network at the same frequency as the $\\lambda $ -network updates.",
"Pseudocode for MA-RMABPPO and further details of its implementation are given in the appendix."
],
[
"The Minimax Regret–Robust RMAB Double Oracle",
"We now have all the pieces we need to present our robust algorithm RMAB Double Oracle (RMABDO), with pseudocode presented in Algorithm REF , adapted from the MIRROR framework [41].",
"We use RMABPPO to instantiate the agent oracle and MA-RMABPPO for the nature oracle.",
"The double oracle approach proceeds as follows.",
"The agent maintains strategy set $\\Pi $ , initially empty, and nature maintains strategy set $\\Omega $ , initialized with an arbitrary parameter setting.",
"In each iteration, we solve for a mixed Nash equilibrium in the regret game between the agent and nature to learn a mixed strategy ($\\tilde{\\pi }, \\tilde{\\omega }$ ) for each player.",
"We then call the agent and nature oracles to each compute a best response $\\pi $ and $\\omega $ to their opponent's strategy, which get added to their respective strategy sets $\\Pi $ and $\\Omega $ .",
"We repeat this process until the improvement in value for each player is within the tolerance $\\varepsilon $ , which is guaranteed to converge within finite steps to within $\\epsilon $ value of the minimax regret–optimal policy (Theorem 2 of [41]), or until a set number of iterations.",
"Additionally, we show that a policy that maximizes reward assuming a fixed parameter set can incur arbitrarily large regret when the parameters are changed (proof in appendix).",
"Formally, [tb] RMABPPO Input: Initial state $\\mathbf {s}_0$ , nature mixed strategy $\\tilde{\\omega }$ Parameters: num epochs n_epochs, num subepochs n_subepochs [1] Randomly initialize a policy network $\\pi _{\\theta _n}$ for each arm $n \\in [N]$ Randomly initialize $\\lambda $ -network Lambda Initialize an empty buffer $\\textit {epoch} = 1, 2, \\ldots , \\texttt {n\\_epochs}$ Sample $\\lambda = \\textsc {Lambda}(\\mathbf {s})$ $\\textit {subepoch} = 1, \\ldots , \\texttt {n\\_subepochs}$ timestep $t = 1, \\ldots , \\texttt {n\\_simulation\\_steps}$ Sample action $a_n = \\pi _{\\theta _n}(s_n, \\lambda )$ for all $n \\in [N]$ Add trajectories $(\\mathbf {s}, \\mathbf {a}, r, \\mathbf {s}^\\prime , \\lambda )$ to buffer Update policy network $\\pi _{\\theta _n}$ of all arms via PPO, using trajectories in buffer Update Lambda using discounted sum of trajectories of final subepoch return $\\pi _{\\theta _1}, \\ldots , \\pi _{\\theta _N}$ and Lambda [tb] RMABDO Input: Environment simulator and parameter uncertainty interval $\\langle \\omega _{n, p_n} \\rangle $ for all $n \\in [N]$ Parameters: Convergence threshold $\\varepsilon $ Output: Best agent mixed strategy $\\tilde{\\pi }$ [1] $\\Omega _0 = \\lbrace \\omega _0\\rbrace $ , with $\\omega _0$ selected at random $\\Pi _0 = \\lbrace \\pi _{B_1}, \\pi _{B_2}, \\ldots \\rbrace $ , where $\\pi _{B_i}$ are baseline and heuristic strategies epoch $e = 1, 2, \\ldots $ Solve for $(\\tilde{\\pi }_e, \\tilde{\\omega }_e)$ , mixed Nash equilibrium of regret game with strategy sets $\\Omega _{e-1}$ and $\\Pi _{e-1}$ $\\pi _e = \\textsc {RMABPPO}(\\tilde{\\omega }_e)$ $\\omega _e = \\textsc {MA-RMABPPO}(\\tilde{\\pi }_e)$ $\\Omega _e = \\Omega _{e-1} \\cup \\lbrace \\omega _e\\rbrace , \\Pi _e = \\Pi _{e-1} \\cup \\lbrace \\pi _e\\rbrace $ $L(\\tilde{\\pi }_e, \\omega _e) - L(\\tilde{\\pi }_{e-1}, \\tilde{\\omega }_{e-1}) \\le \\varepsilon $ and $L(\\pi _e, \\tilde{\\omega }_e) - L(\\tilde{\\pi }_{e-1}, \\tilde{\\omega }_{e-1}) \\le \\varepsilon $ return $\\tilde{\\pi }_e$ []propositionregretProposition In the RMAB problem with interval uncertainty, the max regret of a reward-maximizing policy can be arbitrarily large compared to a minimax regret–optimal policy."
],
[
"Experimental Evaluation",
"We compare our algorithm against five baselines.",
"Namely, we compare against three variations of the approach from [15], which computes a reward-maximizing Lagrange policy for each step of a multi-action RMAB problem for fixed transition probabilities.",
"The three variations are, pessimistic (HP), mean (HM), and optimistic (HO), which assume the transition probabilities have been set to lower bound, mean, and upper bound of the intervals, respectively.",
"We also compare against RLvMid, which learns a policy via RMABPPO assuming mean parameter values, and Rand, which acts randomly each round.",
"All results are averaged over 50 random seeds and were executed on a cluster running CentOS with Intel(R) Xeon(R) CPU E5-2683 v4 @ 2.1 GHz with 8GB of RAM using Python 3.7.10.",
"Our RMABPPO implementation builds on OpenAI Spinning Up [1] and RMABDO builds on the MIRROR implementation [41], computing Nash equilibria using Nashpy 0.0.21 [21].",
"Code is available on github.Code: https://github.com/killian-34/RobustRMAB Hyperparameter settings are included in the appendix.",
"We evaluate all the approaches on three experimental domains, which are as follows.",
"Synthetic domain: We create this setup to show that the reward-maximizing Lagrange policies (HP, HM, and HO) may incur large regret.",
"There are three arm types $\\lbrace U,V,W\\rbrace $ , each with two actions, $\\mathcal {C} = [0, 1]$ , two states $\\mathcal {S}=\\lbrace 0,1\\rbrace $ , $R(s)=s$ , and the following transition probabilities, with rows and columns corresponding to actions and next states, respectively: $T^n_{s=0}=\\begin{bmatrix}0.5 & 0.5 \\\\0.5 & 0.5\\end{bmatrix}, \\hspace{5.69054pt}T^n_{s=1}=\\begin{bmatrix}1.0 & 0.0 \\\\1-p_n & p_n\\end{bmatrix}\\hspace{5.69054pt}\\text{where}\\hspace{5.69054pt}\\begin{matrix}p_U \\in [0.00, 1.00] \\\\p_V \\in [0.05, 0.90] \\\\p_W \\in [0.10, 0.95]\\end{matrix} \\ .$ Note that, when an arm is at $s=0$ , both the actions have equal impact on the state transition.",
"When the arms are at $s=1$ , selecting the arm with higher $p_n$ is optimal.",
"Accordingly, $\\pi _\\textit {HP} = [W,V,U]$ , $\\pi _\\textit {HM} = [W,U,V]$ , and $\\pi _\\textit {HO} = [U,W,V]$ .",
"Observe that there exist values of $p_n$ that can make each of the policies incur large regret, e.g., $p=[0.0, 0.9, 0.1]$ for $\\pi _\\textit {HM}$ , which would induce an optimal policy $[1,2,0]$ that is the reverse of $\\pi _\\textit {HM}$ .",
"ARMMAN: The maternal healthcare intervention problem [5] is modeled as a binary-action RMAB that selects a subset of beneficiaries each week to intervene on with tailored maternal health messaging to encourage engagement.",
"The behavior of each enrolled woman is modeled by an MDP with three states: Self-motivated, Persuadable, and Lost Cause.",
"We use the summary statistics mentioned in their paper and assume uncertainty intervals of $0.5$ centered around the transition parameters, resulting in 6 uncertain parameters per arm (details in appendix).",
"Similar to the setup in [5], we assume 1:1:3 split of arms with high, medium, and low probability of increasing their engagement upon intervention.",
"In our experiments, we scale the value of $N$ in multiples of 5 to keep the same split of arm categories of 1:1:3.",
"SIS Epidemic Model: A discrete-state model in which arms represent distinct geographic regions and each member of an arm's population of size $N_{\\textit {p}}$ is tracked to measure whether they are already infected by a disease or are susceptible.",
"For a particular region, the fraction of susceptible members represents the state of the region.",
"The model is defined by parameters $\\lambda _{\\textit {c}}$ , the average number of contacts per timestep, and $r_{\\textit {infect}}$ , the probability of infection given contact with an infectious person.",
"Details on computing discrete state transition probabilities are derived from [42] and given in the appendix.",
"We augment the model to include three actions $\\lbrace a_0, a_1, a_2\\rbrace $ with costs $c=\\lbrace 0, 1, 2\\rbrace $ .",
"Action $a_0$ represents no action, $a_1$ divides the contacts per day $\\lambda _{\\textit {c}}$ by $a^{\\textit {eff}}_1$ , e.g., by communicating messaging about physical distancing, and $a_2$ divides the probability of infection given contact $r_{\\textit {infect}}$ by $a^{\\textit {eff}}_2$ , e.g., by distributing face masks.",
"We impose the following uncertainty intervals: $\\lambda _{\\textit {c}} \\in [1, 10]$ , $r_{\\textit {infect}} \\in [0.5, 0.99]$ , $a^{\\textit {eff}}_1 \\in [1, 10]$ , and $a^{\\textit {eff}}_2 \\in [1, 10]$ ."
],
[
"Robust Double Oracle",
"To evaluate the algorithms for robustness, we compute the regret of an agent's strategy against a nature pure strategy $\\omega $ as the difference between the average reward obtained by the agent's strategy on $\\omega $ and the average reward of the optimal strategy against $\\omega $ .",
"The average reward is computed as the discounted sum of rewards over all arms for a horizon of length 10, averaged over 25 simulations.",
"In each setting, double oracle runs for 6 iterations, using 100 rollout steps and 100 training epochs for each oracle.",
"After completion, each baseline strategy is evaluated, querying the nature oracle for the best response against that strategy.",
"We report the max regret against all nature's pure strategies.",
"Figure REF shows that our double oracle method has the lowest regret-per-arm and beats all the baselines across all experimental settings.",
"The top row shows the results on the synthetic domain, demonstrating that our approach can reduce regret by about $50\\%$ against existing benchmarks, across various values of $N$ and $B$ .",
"This is expected because the domain is designed to ensure large regret for HP, HM, and HO baselines.",
"Here, a benefit of $50\\%$ in the regret corresponds to, in the worst case, keeping $1/3$ of the arms in the good state for an extra round compared to the baseline.",
"Similarly, for the ARMMAN domain, our algorithm performs consistently better than the baselines, achieving regret that is around $50\\%$ lower than the best baselines.",
"The third domain (SIS) is interesting since the state space is large (number of states was restricted to 2 and 3 in the previous domains).",
"Similar to the earlier results, in the SIS model with populations size of 50, our algorithm outperforms in terms of regret.",
"We discuss the runtime of Figure 1 in Section REF .",
"Additional results included in the appendix."
],
[
"Agent Oracle",
"We evaluate the performance of RMABPPO, our novel RL approach to find a reward-maximizing policy for multi-action RMABs, against an upper bound, namely, the solution by Hawkins, given exact transition probabilities.",
"This will create an upper bound because, given access to the exact transition dynamics, Hawkins will compute the exact Lagrange policy, whereas RMABPPO must learn to approximate the Lagrange policy from samples.",
"Each setting instantiates the environment with a random sample of valid parameter settings for each seed.",
"Figure REF shows that the reward accumulated by our agent oracle is comparable to the Hawkins algorithm.",
"In the synthetic domain (top row), the policy learns to act on the $33\\%$ of arms who belong to category $W$ .",
"The mean reward of RMABPPO almost matches that of Hawkins algorithm as $N$ scales with a commensurate budget (Fig.",
"REF left).",
"As we fix $N$ and vary the budget (Fig.",
"REF right), the optimal policy accumulates more reward, and RMABPPO is almost equal to the optimal.",
"We observe similar results on the ARMMAN domain (middle row), where it is optimal to act on $20\\%$ of arms (that forms category $A$ ; details in appendix).",
"Additionally, on the third large-scale domain (SIS, bottom left), we show the strong performance of RMABPPO holds in a multi-action setting even as we increase the number of states from 50 to 500 to test the scalability.",
"Moreover, computationally, RMABPPO beats Hawkins: in the bottom-right of Fig.",
"REF , a single rollout (10 time steps) of the Hawkins policy takes around 100 seconds when there are 500 states (population size), and steeply increases with more states.",
"This demonstrates that it would be prohibitive to run Hawkins within the loop of the double oracle, since the agent policy needs to be evaluated thousands of times to compute the regret matrices—for merely 25 simulations, computation would take around 42 minutes to evaluate a single cell in the regret matrix over all pure strategy combinations, where the matrix has size $|\\Pi | \\times |\\Omega |$ ."
],
[
"Limitations",
"We believe these advancements have the potential to improve resource allocation in low-resource settings, but acknowledge they are not without tradeoffs.",
"For example, the baseline methods we compare against, while less robust, can provide interpretable `index' policies that capture the value for acting on an arm, whereas our solution's output can be difficult for a domain expert to interpret.",
"Further, such optimization tools have the risk of amplifying underlying biases in the data and translating that to unfair resource allocation.",
"However, by addressing the robust version of the problem, we directly address this concern by providing a flexible tool for mitigating biases, by allowing users to tune their uncertainties and thus, providing natural ways to develop good policies even when data availability is skewed."
],
[
"Conclusion",
"In this paper, we address a blocking limitation that inhibits restless bandits to be used for many real-world planning problems: we often lack perfect knowledge of the environment dynamics.",
"To plan effective and risk-averse policies, it is therefore essential to take a robust approach to account for this uncertainty.",
"Our approach enables us to learn minimax regret–optimal policies by providing RMABPPO, a novel RL algorithm to decouple the budget-constrained problem by training an auxiliary $\\lambda $ -network, and MA-RMABPPO, a new approach to instantiating the nature oracle of an adversarial double oracle game setup by using multi-agent RL to handle nonstationary in nature's challenging optimization problem.",
"Notably, RMABPPO can also extend to solving RMAB problems with continuous states and/or actions, a setting which has not previously been addressed in the literature.",
"We hope these contributions bring us closer to deploying restless bandits in the real world.",
"This work was supported in part by the Army Research Office by Multidisciplinary University Research Initiative (MURI) grant number W911NF1810208.",
"J.A.K.",
"was supported by an NSF Graduate Research Fellowship under grant DGE1745303.",
"A.B.",
"was supported by the Harvard Center for Research on Computation and Society."
],
[
"Proof of Proposition ",
"* This follows from taking the gradient of Eq.",
"REF with respect to $\\lambda $ .",
"To compute the gradient of $Q_n$ , we simply roll out $Q_n$ over time then take the derivative, yielding $\\frac{dQ_n}{d\\lambda } = \\sum _{t=0}^{1} \\beta ^t c_{n,t}$ , i.e., the negative of the expected discounted sum of action costs under the optimal policy."
],
[
"Proof of Proposition ",
"* This follows from the convexity of Eq.",
"REF with respect to $\\lambda $ .",
"The convexity of Eq.",
"REF can be seen from the definition of $Q_n$ , i.e., the max over piece-wise linear functions of $\\lambda $ is convex."
],
[
"Proof of Proposition ",
"* Consider a binary-action RMAB problem with two arms A and B.",
"Let the reward from each arm be $R$ when the arm is in a good state and 0 in a bad state.",
"Our problem is to plan the best action with a budget of 1 and horizon of 1.",
"Supposing the initial state is bad for each arm, the transition probabilities for the transition matrix for each arm $n$ is $\\begin{bmatrix} 1 & 0 \\\\ 1 - p_n & p_n \\end{bmatrix}$ where the uncertain variable $p_n$ is constrained to be within $p_A, p_B \\in [0, 1]$ .",
"Each value in the matrix corresponds to the probability of an arm at state bad transitioning to bad (column 1) or good (column 2) if we take the passive (row 1) or active action (row 2).",
"To compute a reward-maximizing policy that does not consider robustness to uncertainty, we must optimize for one instantiation of the uncertainty set, which requires making one of three assumptions.",
"Case 1: If we assume $p_A = p_B$ , then an optimal policy is to act with probability $a_A$ on arm A and $a_B$ on arm B as long as $a_A + a_B = 1$ .",
"W.l.o.g., suppose $a_A \\ge a_B$ ; then nature would set $p_A = 0$ and $p_B = 1$ , imposing regret at least $R/2$ .",
"Case 2: If $p_A > p_B$ , then the optimal policy would be to always act on arm A with probability $a_A = 1$ and never act on B ($a_B = 0$ ).",
"Nature would then set $p_A = 0$ and $p_B = 1$ to impose regret $R$ .",
"Case 3: If $p_A < p_B$ , the case is symmetric to Case 2 and result in regret $R$ .",
"Clearly, max regret is minimized when our action is such that $a_A + a_B = 1$ ; in this setting, we learn this optimal policy only under Case 1.",
"Following Case 2 or 3, the difference between our regret and the minimax regret is $R/2$ , which grows arbitrarily higher as $R \\rightarrow \\infty $ .",
"A slight modification to this problem renders Case 1 non-optimal.",
"Let the reward be $R$ when arm A is in a good state and $R-1$ for arm B, so the optimal policy learned under the assumption from Case 1 leads to $a_A = 1$ and $a_B = 0$ .",
"Then nature could respond with $p_A = 0$ and $p_B = 1$ , yielding reward 0 and regret $R-1$ , while the minimax regret–optimal policy achieves a minimum reward of $(R-1)/2$ (by playing $a_A = 0.5$ and $a_B = 0.5$ where nature responds with $p_A = 0$ and $p_B = 1$ ).",
"Thus, the gap again can grow arbitrarily high as $R \\rightarrow \\infty $ provided that $R > 1$ .",
"We therefore have that in all cases, any reward-maximizing policy can achieve arbitrarily bad performance in terms of regret."
],
[
"MA-RMABPPO: Nature Oracle Algorithm",
"We provide the pseudocode for MA-RMABPPO in Alg.",
", which is used to implement the nature oracle.",
"[tb] MA-RMABPPO Input: Agent mixed strategy $\\tilde{\\pi }$ Parameters: n_epochs, n_subepochs, n_simsteps, n_sims [1] Initialize agent A: arm policies $\\pi _{n}^{(A)}~ \\forall n \\in [N]$ , $\\lambda $ -network $\\textsc {Lambda}$ , arm critic networks $\\phi _{n}^{(A)}~\\forall n \\in [N]$ Initialize agent B: nature parameter policy $\\pi ^{(B)}$ , critic network $\\phi ^{(B)}$ Initialize empty buffer $\\textit {epoch} = 1, 2, \\ldots , \\texttt {n\\_epochs}$ Sample $\\mathbf {s}$ at random Sample $\\lambda = \\textsc {Lambda}(\\mathbf {s})$ $\\textit {subepoch} = 1, \\ldots , \\texttt {n\\_subepochs}$ $t = 1, \\ldots , \\texttt {n\\_simsteps}$ Sample agent A action $a_n^{(A)} \\sim \\pi _{n}^{(A)}(s_n, \\lambda )$ for each $n \\in [N]$ Sample agent B action $\\omega ^{(B)} \\sim \\pi ^{(B)}$ ($\\pi ^{(B)}$ may take $\\mathbf {s}$ as input) $r^{(A)}, \\mathbf {s}^\\prime = \\textsc {simulate}(\\mathbf {s}, \\mathbf {a}^{(A)}, \\omega ^{(B)})$ $\\tilde{r} = \\textsc {simulate}(\\mathbf {s}, \\tilde{\\pi }(\\mathbf {s}), \\omega ^{(B)}, \\texttt {n\\_sims})$ (mean of 1-step rollouts of $\\tilde{\\pi }$ ) $r^{(B)} = r^{(B)} - \\tilde{r}$ (agent regret) Store $(\\mathbf {s}, \\mathbf {a}^{(A)}, \\omega ^{(B)}, r^{(A)}, r^{(B)}, \\mathbf {s}^\\prime )$ in buffer $\\mathbf {s} = \\mathbf {s}^\\prime $ Update $\\pi _n^{(A)}$ , $\\phi _n^{(A)}$ for all $n$ using trajectories in buffer.",
"$\\pi _n^{(A)}$ gets $\\omega ^{(B)}$ as part of state Update Lambda via discounted trajectories in buffer Update $\\pi ^{(B)}, \\phi ^{(B)}$ from trajectories in buffer.",
"$\\phi ^{(B)}$ gets $a^{(A)}$ as part of state return $\\pi ^{(B)}$"
],
[
"ARMMAN",
"The MDPs in the ARMMAN domain [5] have three ordered states representing the level of engagement of the beneficiaries in the previous week.",
"Rewards are better for lower states, i.e., $R(0)=1, R(1)=0.5, R(2)=0$ .",
"At each step, the beneficiary may only change by one level, e.g., low-to-medium or high-to-medium but not low-to-high.",
"They also assume that beneficiaries follow one of three typical patterns, A, B, and C, resulting in three MDPs with different transition probabilities.",
"There are two patterns of effects present that differentiate the beneficiary types.",
"(1) For each of the above types, the planner can only make a difference when the patient is in state 1.",
"Type A responds very positively to interventions, but regresses to low reward states in absence.",
"Type B has a similar but less amplified effect, and type C is likely to stay in state 1, but can be prevented from regressing to state 2 when an action is taken.",
"(2) Further, types A and C have only a 10% chance of staying in the high reward state, while type B has a 90% chance of staying there.",
"We converted these patient types to robust versions where the transition probabilities are uncertain as follows: $T^i_{s=0}=\\begin{bmatrix}p^i_{000} & 1 - p^i_{000} & 0.0 \\\\p^i_{010} & 1 - p^i_{010} & 0.0\\end{bmatrix}, \\hspace{5.69054pt}T^i_{s=1}=\\begin{bmatrix}0.0 & 1 - p^i_{102} & p^i_{102} \\\\p^i_{110} & 1 - p^i_{110} & 0.0\\end{bmatrix}, \\ $ $T^i_{s=2}=\\begin{bmatrix}0.0 & 1 - p^i_{202} & p^i_{202} \\\\0.0 & 1 - p^i_{212} & p^i_{212}\\end{bmatrix},$ where $i$ indexes the type (i.e., A, B or C).",
"We then set each $p^i_{sas^\\prime }$ to be in a range of width 0.5 centered on the entries from each of the A, B, C beneficiary types for $s\\in \\lbrace 1,2\\rbrace $ .",
"To add additional heterogeneity to the experiments, for $s=0$ , we set the range to 1.0 so that any beneficiary type can be made to have some non-negligible chance of staying in the good state, rather than only type B beneficiaries.",
"The full set of parameter ranges are given in the Table REF below.",
"Table: Upper and lower parameter ranges for the robust ARMMAN environment.In all experiments, 20% of arms were sampled from type A, 20% from type B and 60% for type C. To add additional heterogeneity, for each of the 50 random seeds we uniformly sample a sub-range contained within the ranges given in Table REF .",
"In the agent oracle experiments, for each of the 50 random seeds, since these require fully instantiated transition matrices, we uniformly sample each parameter value for each arm according to its type such that the values are contained in the ranges given in Table REF ."
],
[
"SIS Epidemic Model",
"In this domain, each arm follows its own compartmental SIS epidemic model.",
"Each arm's SIS model tracks whether each of $N_p$ members of a population is in a susceptible (S) or infectious (I) state.",
"This can be tracked with $N_p$ states, since it can be computed how many people are in state I if only the number of people in state S and the population size $N_p$ is known.",
"To define a discrete SIS model, we instantiate the model given in [42] section 4.1 with a $\\Delta t$ of 1.",
"We also augment the model to include action effects and rewards.",
"Specifically, $R(N_S) = N_S/N_p$ , where $N_S$ is the number of susceptible (non-infected) people.",
"Further, there are three actions $\\lbrace a_0, a_1, a_2\\rbrace $ with costs $c=\\lbrace 0, 1, 2\\rbrace $ .",
"Action $a_0$ represents no action, $a_1$ divides the contacts per day $\\lambda _{\\textit {c}}$ ($\\lambda $ in [42]) by $a^{\\textit {eff}}_1$ , and $a_2$ divides the infectiousness $r_{\\textit {infect}}$ ($r(t)$ in [42]) by $a^{\\textit {eff}}_2$ .",
"That is, taking action $a_1$ will reduce the average number of contacts per day in a given arm, and taking action $a_2$ will reduce the probability of infection given contact in a given arm, thus reducing the expected number of people that will become infected in the next round.",
"However, to make this a robust problem, the relative effect sizes of each action for each arm will not be known to the planner, nor will the $\\lambda _\\textit {c}$ or $r_{\\textit {infect}}$ .",
"We impose the following uncertainty intervals for all arms: $\\lambda _{\\textit {c}} \\in [1, 10]$ , $r_{\\textit {infect}} \\in [0.5, 0.99]$ , $a^{\\textit {eff}}_1 \\in [1, 10]$ , and $a^{\\textit {eff}}_2 \\in [1, 10]$ .",
"In the robust double oracle experiments, to add additional heterogeneity, for each of the 50 random seeds we uniformly sample a sub-range contained within the ranges given above for each arm.",
"In the agent oracle experiments, for each of the 50 random seeds, since these require fully instantiated transition matrices, we uniformly sample each parameter value for each arm such that the values are contained in the ranges given above."
],
[
"Hyperparameter Settings and Implementation Details",
"Neural networks: All neural networks in experiments are implemented using PyTorch 1.3.1 [32] with 2 fully connected layers each with 16 units and tanh activation functions, and a final layer of appropriate size for the relevant output dimension with an identity activation function.",
"The output of discrete actor networks (i.e., the policy network from the agent oracle, and the policy network of agent A in the nature oracle) pass through a categorical distribution from which actions are randomly sampled at training time, without a budget imposed.",
"It is critical not to impose the budget at training time, so that the budget spent by the optimal policy under a given $\\lambda $ will result in a meaningful gradient for updating the $\\lambda $ -network.",
"The output of continuous actor networks (i.e., agent B in the nature oracle which selects environment parameter settings) instead are passed as the means of Gaussian distributions – with the log standard deviations learned as individual parameters separate from the network – from which continuous actions are sampled at training time.",
"At test time, actions are sampled from both types of networks deterministically.",
"For categorical distributions, we greedily select the highest probability actions.",
"For Gaussian distributions, we act according to the means.",
"All discount factors were set to 0.9.",
"The remaining hyperparameters that were constant for all experiments for the agent and nature oracles are indicated in Table REF .",
"For Robust Double Oracle experiments, all agent and nature oracles were run for 100 training epochs.",
"For Agent Oracle experiments, RMABPPO was run for 100 training epochs for the synthetic and ARMMAN domains and 200 epochs for the SIS domain.",
"$\\lambda $ -network: Critical to training the $\\lambda $ -network is cyclical control of the temperature parameter that weights the entropy term in the actor loss functions.",
"Recall that the $\\lambda $ -network is only updated every n_subepochs.",
"In general, after each update to the $\\lambda $ -network, we want to encourage exploration so that actor networks explore the new part of the state space defined by updated predictions of $\\lambda $ .",
"However, after n_subepochs rounds, we will use the cost of the sampled actor policies as a gradient for updating the $\\lambda $ -network, and that gradient will only be accurate if the actor policy has converged to the optimal policies for the given $\\lambda $ predictions.",
"Therefore, we also want to have little or no exploration in the round before we update the $\\lambda $ -network.",
"In general, we would also like the entropy of the policy network to reduce over time so that the actor networks and $\\lambda $ -networks eventually both converge.",
"To accomplish both of these tasks, the weight (temperature) of the entropy regularization term in the loss function of the actor network will decay/reset according to two processes.",
"The first process will linearly decay the temperature from some positive, but time-decaying starting value (see next process) $\\tau _t$ immediately after each $\\lambda $ -network update, down to 0 after $\\texttt {n\\_subepochs}$ .",
"The second process will linearly decay the temperature from a maximum $\\tau _0$ (start entropy coeff in Table REF ) down to $\\tau _{\\min }$ (end entropy coeff in Table REF ) by the end of training.",
"We found that it also helps to train the actor network with no entropy and with the $\\lambda $ -network frozen for some number of rounds before training is stopped (lambda freeze epochs in Table REF ).",
"Double Oracle: In all experiments in the main text, we initialize the agent strategy list with HO, HM, and HP, and the nature strategy list with pessimistic, mean, and optimistic nature strategies, then run RMABDO for 5 iterations.",
"This produces a set of 8 agent strategies, 8 nature strategies, a table where each entry represents the regret of each agent pure strategy (row) against each nature pure strategy (column), and an optimal mixed strategy over each set that represents a Nash equilibrium of the minimax regret game given in the table.",
"The regret table is computed by first computing the returns of each agent/nature pure strategy combination, then subtracting the max value of each column from all entries in that column (i.e., the best agent strategy for a given nature strategy gets 0 regret).",
"The regret of RMABDO is reported as the expected utility corresponding to the Nash equilibrium of the regret game given by the table, once that regret table is normalized to account for the returns of baselines (see next paragraph).",
"After this main loop completes, we then compute the regret of the baselines by evaluating each baseline policy against each pure strategy in the nature strategy list.",
"Then, we also run the nature oracle against each baseline policy to find a nature strategy that should maximize the regret of that baseline.",
"The regret for each baseline is reported as the max regret against this new nature strategy, as well as all pure nature strategies from the main RMABDO loop.",
"Hawkins Baselines: The Hawkins policies are implemented with gurobipy 9.1.2, a Python wrapper for Gurobi (9.0.3) [14] following the LP given in [15] equation 2.5 to compute $\\lambda $ and $Q(s,a,\\lambda )$ for each arm and the integer program in equation 2.12 to select actions.",
"RLvMid Baseline: We found that RLvMid found effective policies for the nature strategy it was trained against (as evidenced in Figure REF ), but that that learned policy could be brittle against other nature strategies.",
"This is likely because different nature strategies produce different distributions of states, meaning RLvMid would fit policies well to states seen when planning against the mean nature strategy, but underfit its policies for states seen more often in different distributions.",
"However, the lone RLvMid baseline policy can somewhat correct for this effect by training an ensemble of policies against slight perturbations of the mean nature strategy that adjust the parameter values output by nature by a small $\\epsilon $ .",
"In all experiments we train 3 RLvMid policies against 3 random perturbations of the mean nature strategy, then report the regret of RLvMid as the minimum of the max regrets returned by any of the 3.",
"Table: Hyperparameter settings for agent and nature oracles for all experiments."
],
[
"Additional Experimental Results",
"Fig.",
"REF shows regret results from additional robust experiments which scale up the number of arms for Synthetic (top left), ARMMAN (top right) and SIS (bottom left; $N_p=50$ ), as well as and the size of the state space of SIS (bottom right; $N=5, B=4$ ).",
"RMABDO continues to outperform all baselines by a large margin.",
"When the size of the state space is scaled up for SIS, it becomes infeasible to run the Hawkins baselines due to its long query time which grows quadratically with the size of the state space.",
"Because of this, we exclude the Hawkins baselines from the main double oracle loop in these experiments.",
"Further, Hawkins cannot even be evaluated as a baseline as the state space increases since to compute its maximum regret, we must get one best response from the nature oracle against the baseline, which requires querying the Hawkins baseline policies tens of thousands of times, which is prohibitive when the query time takes even ~$1s$ to run.",
"Figure: Additional experiments showing maximum policy regret in robust setting for Synthetic (top left), ARMMAN (top right) and SIS (bottom) domains, respectively.",
"Synthetic is scaled by 3 and ARMMAN by 5 to maintain the distributions of arm types specified in Section .",
"RMABDO beats all baselines by a large margin across various parameter settings.",
"When the state space is scaled up (bottom right) Hawkins baselines become infeasible to run due to its long query time (see Section for a discussion), even for a small number of arms (N=5,B=4N = 5, B = 4)."
]
] | 2107.01689 |
[
[
"Percolation transition for random forests in $d\\geq 3$"
],
[
"Abstract The arboreal gas is the probability measure on (unrooted spanning) forests of a graph in which each forest is weighted by a factor $\\beta>0$ per edge.",
"It arises as the $q\\to 0$ limit with $p=\\beta q$ of the $q$-state random cluster model.",
"We prove that in dimensions $d\\geq 3$ the arboreal gas undergoes a percolation phase transition.",
"This contrasts with the case of $d=2$ where all trees are finite for all $\\beta>0$.",
"The starting point for our analysis is an exact relationship between the arboreal gas and a non-linear sigma model with target space the fermionic hyperbolic plane $\\mathbb{H}^{0|2}$.",
"This latter model can be thought of as the $0$-state Potts model, with the arboreal gas being its random cluster representation.",
"Unlike the $q>0$ Potts models, the $\\mathbb{H}^{0|2}$ model has continuous symmetries.",
"By combining a renormalisation group analysis with Ward identities we prove that this symmetry is spontaneously broken at low temperatures.",
"In terms of the arboreal gas, this symmetry breaking translates into the existence of infinite trees in the thermodynamic limit.",
"Our analysis also establishes massless free field correlations at low temperatures and the existence of a macroscopic tree on finite tori."
],
[
"Introduction",
"This paper has two distinct motivations.",
"The first is to study the percolative properties of the arboreal gas, and the second is to understand spontaneously broken continuous symmetries.",
"We first present our results from the percolation perspective, and then turn to continuous symmetries."
],
[
"Main results for the arboreal gas",
"The arboreal gas is the uniform measure on (unrooted spanning) forests of a weighted graph.",
"More precisely, given an undirected graph $G=(\\Lambda ,E)$ , a forest $F=(\\Lambda ,E(F))$ is an acyclic subgraph of $G$ having the same vertex set as $G$ .",
"Given an edge weight $\\beta >0$ (inverse temperature) and a vertex weight $h\\ge 0$ (external field), the probability of a forest $F$ under the arboreal gas measure is $ ¶_{\\beta ,h}^{G}[F] \\frac{1}{Z_{\\beta ,h}^{G}} \\beta ^{|E(F)|} \\prod _{T\\in F} (1+h|V(T)|)$ where $T\\in F$ denotes that $T$ is a tree in the forest, i.e., a connected component of $F$ , $|E(F)|$ is the number of edges in $F$ , and $|V(T)|$ is the number of vertices in $T$ .",
"The arboreal gas is also known as the (weighted) uniform forest model, as Bernoulli bond percolation conditioned to be acyclic, and as the $q\\rightarrow 0$ limit of the $q$ -state random cluster model with $p/q$ converging to $\\beta $ , see [52].",
"We study the arboreal gas on a sequence of tori $\\Lambda _N = ^d/L^N^d$ with $L$ fixed and $N\\rightarrow \\infty $ .",
"To simplify notation, we will use $\\Lambda _N$ to denote both the graph and its vertex set.",
"From the percolation point of view, the most fundamental question concerns whether a typical forest $F$ under the law (REF ) contains a giant tree.",
"In all dimensions, elementary arguments show that giant trees can exist only if $h=0$ and if $\\beta $ is large enough, in the sense that connection probabilities decay exponentially whenever $h>0$ or $\\beta $ is small; see Appendix REF .",
"The existence of a percolative phase for $h=0$ and $\\beta $ large does not, however, follow from standard techniques.",
"The subtlety of the existence of a percolative phase is perhaps best evidenced by considering the case $d=2$ : in this case giant trees do not exist for any $\\beta >0$ [18].",
"Our main result is that for $d\\ge 3$ giant trees do exist for $\\beta $ large and $h=0$ , and that truncated correlations have massless free field decay.",
"To state our result precisely, let $\\lbrace 0 \\leftrightarrow x\\rbrace $ denote the event that 0 and $x$ are connected, i.e., in the same tree.",
"Let $d\\ge 3$ and $L \\ge L_0(d)$ .",
"Then there is $\\beta _0 \\in (0,\\infty )$ such that for $\\beta \\ge \\beta _0$ there exist $\\zeta _d(\\beta ) = 1-O(1/\\beta )$ , ${\\sf c}(\\beta ) = {\\sf c}+ O(1/\\beta )$ with ${\\sf c}>0$ , and $\\kappa >0$ such that $¶_{\\beta ,0}^{\\Lambda _N}[0\\leftrightarrow x]= \\zeta _d(\\beta ) + \\frac{{\\sf c}(\\beta )}{\\beta |x|^{d-2}} + O(\\frac{1}{\\beta |x|^{d-2+\\kappa }}) + O(\\frac{1}{\\beta L^{\\kappa N}}),$ where $|x|$ denotes the Euclidean norm.",
"Numerical evidence for this phase transition of the arboreal gas was given in [37].",
"More broadly our work was inspired by [34], [35], [19], [37], [54], [55], and we discuss further motivation later.",
"Although both the arboreal gas and Bernoulli bond percolation have phase transitions for $d\\ge 3$ , the supercritical phases of these models behave very differently: (REF ) shows that the arboreal gas behaves like a critical model even in the supercritical phase, in the sense that it has massless free field truncated correlation decay.",
"While this behaviour looks unusual when viewed through the lens of supercritical percolation, it is natural from the viewpoint of broken continuous symmetries.",
"We will return to this point in Section REF .",
"Theorem REF concerns the arboreal gas on large finite tori.",
"Another limit to consider the arboreal gas in is the weak infinite volume limit.",
"To this end, we consider the limit obtained by first taking $N\\rightarrow \\infty $ with $h>0$ and then taking $h\\downarrow 0$ .",
"In manner similar to that for Bernoulli bond percolation in [43] and [2], the external field is equivalent to considering the arboreal gas on an extended graph $G^{\\mathfrak {g}} = (\\Lambda \\cup \\lbrace \\mathfrak {g}\\rbrace , E\\cup E^{\\mathfrak {g}})$ where $E^{\\mathfrak {g}} = \\Lambda \\times \\lbrace \\mathfrak {g}\\rbrace $ and each edge in $E^{\\mathfrak {g}}$ has weight $h$ .",
"The additional vertex $\\mathfrak {g}$ is called the ghost vertex.",
"The measure (REF ) is then obtained by forgetting the connections to the ghost.",
"This rephrases that the product in (REF ) is equivalent to connecting a uniformly chosen vertex in each tree $T$ to $\\mathfrak {g}$ with probability $h|V(T)|/(1+h|V(T)|)$ .",
"For vertices $x,y \\in \\Lambda $ , we continue to denote by $\\lbrace x\\leftrightarrow y\\rbrace $ the event that $x$ and $y$ are connected in the random forest subgraph of $G$ with law (REF ), i.e., without using the edges in $E^\\mathfrak {g}$ .",
"We write $\\lbrace x\\leftrightarrow \\mathfrak {g}\\rbrace $ to denote the event that $x$ is connected to $\\mathfrak {g}$ .",
"The event $\\lbrace 0\\leftrightarrow \\mathfrak {g}\\rbrace $ is a finite volume proxy for the event that the tree $T_0$ containing 0 becomes infinite in the infinite volume limit when $h\\downarrow 0$ .",
"Indeed, let us define $\\theta _d(\\beta )\\lim _{h\\downarrow 0}\\lim _{N\\rightarrow \\infty } ¶^{\\Lambda _N}_{\\beta ,h}[0\\leftrightarrow \\mathfrak {g}],$ and let $¶_\\beta ^{^d}$ be any (possibly subsequential) weak infinite volume limit $\\lim _{h\\downarrow 0}\\lim _{N\\rightarrow \\infty }¶^{\\Lambda _N}_{\\beta ,h}$ .",
"Then $\\theta _d(\\beta )=¶_\\beta ^{^d}[|T_0|=\\infty ],$ see Proposition REF .",
"By a stochastic domination argument it is straightforward to show that $\\theta _d(\\beta ) =0 \\qquad \\text{for $0\\le \\beta <p_c(d)/(1-p_c(d))<\\infty $,}$ where $p_c(d)$ is the critical probability for Bernoulli bond percolation on $^d$ , see Proposition REF .",
"When $d=2$ , $\\theta _2(\\beta ) = 0$ for all $\\beta >0$ by [18].",
"The next theorem shows that for $d\\ge 3$ the arboreal gas also has a phase transition in this infinite volume limit.",
"Let $d\\ge 3$ and $L \\ge L_0(d)$ .",
"Then there is $\\beta _0 \\in (0,\\infty )$ such that for $\\beta \\ge \\beta _0$ the limit (REF ) exists and $\\theta _{d}(\\beta )^2 = \\zeta _d(\\beta )= 1-O(1/\\beta ),$ where $\\zeta _d(\\beta )$ is the finite volume density of the tree containing 0 from Theorem REF .",
"In fact, our proof shows that $\\theta _d(\\beta )\\sim 1-c/\\beta $ with $c=(-\\Delta ^{^d})^{-1}(0,0)>0$ the expected time a simple random walk spends at the origin.",
"This behaviour is different from that of Bernoulli bond percolation and more generally that of the random cluster model with $q>0$ .",
"For these models the percolation probability is governed by Peierls' contours and is $1-O((1-p)^{2d})$ by [63].",
"That the arboreal gas behaves critically within its supercritical phase can be further quantified in terms of the following truncated two-point functions: $\\tau _{\\beta }(x) = \\lim _{h\\downarrow 0} \\tau _{\\beta ,h}(x),& \\qquad \\tau _{\\beta ,h}(x)=\\lim _{N\\rightarrow \\infty }¶_{\\beta ,h}^{\\Lambda _N}[0\\leftrightarrow x, 0 \\lnot \\leftrightarrow \\mathfrak {g}],\\\\\\sigma _{\\beta }(x) = \\lim _{h\\downarrow 0} \\sigma _{\\beta ,h}(x),& \\qquad \\sigma _{\\beta ,h}(x) = \\lim _{N\\rightarrow \\infty }{¶_{\\beta ,h}^{\\Lambda _N}[0\\lnot \\leftrightarrow \\mathfrak {g}]^2 - ¶_{\\beta ,h}^{\\Lambda _N}[0\\lnot \\leftrightarrow x, 0\\lnot \\leftrightarrow \\mathfrak {g},x\\lnot \\leftrightarrow \\mathfrak {g}]}.$ Under the assumptions of Theorem REF , for $\\beta \\ge \\beta _0$ , the limits (REF )–() exist and there exist constants ${\\sf c}_i(\\beta ) = {\\sf c}_i+O(1/\\beta )$ and $\\kappa >0$ such that $\\tau _{\\beta }(x) &= \\frac{{\\sf c}_1(\\beta )}{\\beta |x|^{d-2}} +O(\\frac{1}{\\beta |x|^{d-2+\\kappa }}),\\\\\\sigma _{\\beta }(x) &= \\frac{{\\sf c}_2(\\beta )}{\\beta ^2|x|^{2d-4}} + O(\\frac{1}{\\beta ^2 |x|^{2d-4+\\kappa }}).$ The constants satisfy $({\\sf c}_2(\\beta )/{\\sf c}_1(\\beta )^2)\\theta _d(\\beta )^2=1$ and ${\\sf c}(\\beta ) = 2{\\sf c}_1(\\beta )$ , ${\\sf c}(\\beta )$ from Theorem REF .",
"Further results could be deduced from our analysis, but to maintain focus we have not carried these out in detail.",
"We mention some of them below in Section REF when discussing our results and open problems."
],
[
"The $^{0|2}$ model and continuous symmetries",
"In [34], [35], the arboreal gas was related to a fermionic field theory and a supersymmetric non-linear sigma model with target space one half of the degenerate super-sphere $\\mathbb {S}^{0|2}$ .",
"In [18] this was reinterpreted as a non-linear sigma model with hyperbolic target space $^{0|2}$ (which we refer to as the $^{0|2}$ model for short).",
"The reinterpretation was essential in [18]; it is less essential for the present work, but nevertheless, we continue to use the $^{0|2}$ formulation of the model.",
"Briefly, the $^{0|2}$ model is defined as follows; see [18] for further details.",
"For every vertex $x\\in \\Lambda $ , there are two (anticommuting) Grassmann variables $\\xi _x$ and $\\eta _x$ and we then set $z_x \\sqrt{1-2\\xi _x\\eta _x} 1-\\xi _x\\eta _x.$ Thus the $z_x$ commute with each other and with the odd elements $\\xi _x$ and $\\eta _x$ .",
"The formal triples $u_x(\\xi _x,\\eta _x,z_x)$ are supervectors with two odd components $\\xi _x,\\eta _x$ and an even component $z_x$ .",
"These supervectors satisfy the sigma model constraint $u_x\\cdot u_x=-1$ for the super inner product $u_x\\cdot u_y -\\xi _x\\eta _y-\\xi _y\\eta _x - z_xz_y.$ In analogy with the tetrahedral representation of the Potts model, the sigma model constraint can be thought of as $u_x\\cdot u_x = q-1$ with $q=0$ .",
"The constraint is also reminiscent of the embedding of the hyperbolic space $^2$ in $^3$ equipped with the standard quadratic form with Lorentzian signature $(1, 1, -1)$ .",
"Indeed, $-\\xi _x\\eta _y-\\xi _y\\eta _x$ is the fermionic analogue of the Euclidean inner product on $^2$ .",
"The expectation of the $^{0|2}$ model is ${F}_{\\beta ,h} \\frac{1}{Z_{\\beta ,h}}\\int (\\prod _{x\\in \\Lambda } \\partial _{\\eta _x}\\partial _{\\xi _x} \\frac{1}{z_x}) e^{\\frac{\\beta }{2} (u,\\Delta u) - h(1,z-1)} F.$ In this expression, $\\int \\prod _{x\\in \\Lambda } \\partial _{\\eta _x}\\partial _{\\xi _x}$ denotes the Grassmann integral (i.e., the coefficient of the top degree monomial of the integrand), $Z_{\\beta ,h}$ is a normalising constant, and $(u,\\Delta u) = -\\frac{1}{2} \\sum _{xy\\in E(\\Lambda ) }(u_x-u_y)\\cdot (u_x-u_y)= \\sum _{xy\\in E(\\Lambda ) }(u_x\\cdot u_y+1),\\qquad (1,z) = \\sum _{x\\in \\Lambda } z_x,$ where $xy\\in E(\\Lambda )$ denotes that $x$ and $y$ are nearest neighbours (counting every pair once), and the inner products are given by (REF ).",
"The factors $1/z_x$ in (REF ) are the canonical fermionic volume form invariant under the symmetries associated with (REF ) as discussed further below.",
"As explained in [18] (see also [34] where such relations were first observed) connection and edge probabilities of the arboreal gas are equivalent to correlation functions of the $^{0|2}$ model.",
"The following proposition summarises the ones we need, see Appendix .",
"For any finite graph $G$ , any $\\beta \\ge 0$ and $h \\ge 0$ , $ ¶_{\\beta ,h}[0\\leftrightarrow \\mathfrak {g}] &= {z_0}_{\\beta ,h},\\\\¶_{\\beta ,h}[0\\leftrightarrow x, 0 \\lnot \\leftrightarrow \\mathfrak {g}] &= {\\xi _0\\eta _x}_{\\beta ,h},\\\\¶_{\\beta ,h}[0\\leftrightarrow x]+¶_{\\beta ,h}[0\\lnot \\leftrightarrow x, 0\\leftrightarrow \\mathfrak {g}, x\\leftrightarrow \\mathfrak {g}] &= -{u_0\\cdot u_x}_{\\beta ,h},$ and the normalising constants in (REF ) and (REF ) are equal.",
"In particular, $ ¶_{\\beta ,0}[0\\leftrightarrow x] = -{u_0\\cdot u_x}_{\\beta ,0} = -{z_0z_x}_{\\beta ,0} = {\\xi _0\\eta _x}_{\\beta ,0} = 1-{\\xi _0\\eta _0\\xi _x\\eta _x}_{\\beta ,0}.$ These relations resemble those between the Potts model and the random cluster model, giving further credence to our heuristic that the $^{0|2}$ model may be interpreted as the 0-state Potts model, with the arboreal gas playing the role of the 0-state random cluster model.",
"Nevertheless, there are important differences from the $q$ -state Potts model with $q \\ge 1$ .",
"Chief amongst them is that the $^{0|2}$ model has continuous symmetries.",
"To make this precise, let $ T=\\sum _{x\\in \\Lambda } z_x \\partial _{\\xi _x}, \\qquad \\bar{T}=\\sum _{x\\in \\Lambda } z_x \\partial _{\\eta _x}.$ One way to understand the significance of $T, \\bar{T}$ is via the identities ${TF}_{\\beta ,0}={\\bar{T} F}_{\\beta ,0}=0$ for any (noncommutative) polynomial $F$ in the variables $\\xi $ and $\\eta $ .",
"For example, ${T\\xi _{0}}_{\\beta ,0}={z_{0}}_{\\beta ,0}=0$ .",
"Identities derived in this way are conventionally called Ward identities.",
"The maps $T$ and $\\bar{T}$ are infinitesimal generators of two global internal supersymmetries of the $^{0|2}$ model.",
"These supersymmetries are explicitly broken if $h \\ne 0$ .",
"They are analogues of infinitesimal Lorentz boosts or infinitesimal rotations.",
"Together with a further internal symmetry corresponding to rotations in the $\\xi , \\eta $ plane, these operators generate the symmetry algebra $\\mathfrak {osp}(1|2)$ of the $^{0|2}$ model.",
"For details and further explanations, see [18].",
"As generators of continuous symmetries, $T$ and $\\bar{T}$ imply Ward identities that are not available for the Potts model with $q\\ge 1$ .",
"These identities are crucial for our analysis and will be discussed below.",
"The phase transition of the arboreal gas corresponds to a spontaneous breaking of the above supersymmetries in the infinite volume limit.",
"This is shown in our next theorem for the $^{0|2}$ model from which Theorems REF and REF follow immediately by (REF )–() (except for the same statements relating the constants, which we omitted here); a similar reformulation applies to Theorem REF .",
"Let $d\\ge 3$ and $L\\ge L_0(d)$ .",
"There exists $\\beta _0\\in (0,\\infty )$ and constants $\\theta _d(\\beta )=1+O(1/\\beta )$ and ${\\sf c}_i(\\beta ) = {\\sf c}_i+O(1/\\beta )$ and $\\kappa >0$ (all dependent on $d$ ) such that for $\\beta \\ge \\beta _0$ , $\\lim _{h\\downarrow 0}\\lim _{N\\rightarrow \\infty }{z_0}_{\\beta ,h}&= \\theta _d(\\beta )\\\\\\lim _{h\\downarrow 0}\\lim _{N\\rightarrow \\infty }{\\xi _0\\eta _x}_{\\beta ,h}&= \\frac{{\\sf c}_1(\\beta )}{\\beta |x|^{d-2}} + O(\\frac{1}{\\beta |x|^{d-2+\\kappa }})\\\\\\lim _{h\\downarrow 0}\\lim _{N\\rightarrow \\infty }{{z_0z_x}_{\\beta ,h}-{z_0}_{\\beta ,h} {z_x}_{\\beta ,h}}&= -\\frac{{\\sf c}_2(\\beta )}{\\beta ^2 |x|^{2d-4}} + O(\\frac{1}{\\beta ^2|x|^{2d-4+\\kappa }}).$ In particular, $\\lim _{h\\downarrow 0}\\lim _{N\\rightarrow \\infty }{u_0\\cdot u_x}_{\\beta ,h} = -\\theta _d(\\beta )^2 -\\frac{2{\\sf c}_1(\\beta )}{\\beta |x|^{d-2}} + O(\\frac{1}{\\beta |x|^{d-2+\\kappa }}).$ In fact, the constants ${\\sf c_{i}}(\\beta )$ both satisfy ${\\sf c_{i}}(\\beta ) = c_{d}^{i} + O(1/\\beta )$ , where $c_{d}$ is the leading constant in the asymptotics of the Green function of the Laplacian $-\\Delta ^{^{d}}$ on $^{d}$ : $(-\\Delta ^{^d})^{-1}(0,x)= \\frac{c_{d}}{|x|^{d-2}} + O(|x|^{-(d-2)-1}).$ Our proof of Theorem REF is by a rigorous renormalisation group analysis aided by Ward identities.",
"The starting point is setting $\\psi =\\sqrt{\\beta }\\eta $ and $\\bar{\\psi }= \\sqrt{\\beta }\\xi $ ; the fermionic density in (REF ) is then equivalent to $\\exp {- (\\psi ,-\\Delta \\bar{\\psi })-\\frac{1}{\\beta }(1+h) \\sum _{x\\in \\Lambda }\\psi _x\\bar{\\psi }_x - \\frac{1}{2\\beta } \\sum _{x\\in \\Lambda } \\psi _x\\bar{\\psi }_x \\sum _{e\\in \\mathcal {E}_d} \\psi _{x+e} \\bar{\\psi }_{x+e}},$ where the 1 in the quadratic term arises from putting the volume form $1/z = e^{+\\xi \\eta }=e^{-\\eta \\xi }$ into the exponential, and $\\mathcal {E}_{d}=\\lbrace e_{1},\\dots , e_{2d}\\rbrace $ are the standard unit vectors (where $e_{d+j}=-e_{j}$ ).",
"The reformulation (REF ) looks very much like a fermionic version of the $\\varphi ^4$ spin model.",
"However, the following differences are important: The coupling constants of the quadratic and quartic terms are related.",
"This relation is due to the geometric origin of the model as a non-linear sigma model and analogous relations are present in intrinsic coordinates for other sigma models like the vector $O(n)$ model.",
"We will use the following Ward identity for the $^{0|2}$ model to address this point: $ {z_0}_{\\beta ,h} = {T\\xi _0}_{\\beta ,h}= -\\sum _{x\\in \\Lambda } h {\\xi _0 Tz_x}_{\\beta ,h}= h \\sum _{x\\in \\Lambda } {\\xi _0 \\eta _x}_{\\beta ,h},$ where $T$ is the symmetry generator (REF ).",
"Due to the fermionic nature of the field, the quartic term actually has gradients in it: denoting the discrete gradient in direction $e$ by $(\\nabla _e\\psi )_x = \\psi _{x+e}-\\psi _x$ , it can be written as $\\frac{1}{2}\\psi _x\\bar{\\psi }_x \\sum _{e\\in \\mathcal {E}_d} \\psi _{x+e}\\bar{\\psi }_{x+e}=\\frac{1}{2}\\psi _x\\bar{\\psi }_x \\sum _{e\\in \\mathcal {E}_d} (\\nabla _e \\psi )_x(\\nabla _e \\bar{\\psi })_x\\psi _x\\bar{\\psi }_x (\\nabla \\psi )_x(\\nabla \\bar{\\psi })_x,$ where we introduced the shorthand notation $(\\nabla \\psi )_x(\\nabla \\bar{\\psi })_x= \\frac{1}{2} \\sum _{e\\in \\mathcal {E}_d}(\\nabla _e \\psi )_x(\\nabla _e \\bar{\\psi })_x$ .",
"After taking into account the points above, power counting heuristics (which we expect can be generalised to all non-linear sigma models with continuous symmetry) predict that the lower critical dimension for spontaneous symmetry breaking with free field low temperature fluctuations is two for the $^{0|2}$ model.",
"In conjunction with [18], our results rigorously establish that the lower critical dimension is two for the $^{0|2}$ model."
],
[
"Background on non-linear sigma models and renormalisation",
"The low temperature renormalisation group analysis of non-linear sigma models with non-abelian continuous symmetry is a notorious problem that was famously considered by Balaban for the case of $O(n)$ symmetry, see [9], [10] and references therein.",
"Our comparatively simple analysis of the $^{0|2}$ model, which is a non-linear sigma model with non-abelian continuous $OSp(1|2)$ symmetry, is made possible mainly by the fact that it does not suffer a large field problem because it has a fermionic representation.",
"In addition to this, our approach to the $^{0|2}$ model differs from Balaban's approach to the $O(n)$ model on a conceptual level, in that it is based on intrinsic coordinates as opposed to extrinsic ones.",
"It is unclear to us how to implement an extrinsic approach in our situation of $OSp(1|2)$ symmetry.",
"Somewhat remarkably, despite its simplicity, the $^{0|2}$ model has all of the main features present in the non-abelian $O(n)$ models, including: absence of spontaneous symmetry breaking in 2d (proven in [18]); mass generation in 2d (conjectured in [35]); and a spontaneous symmetry breaking phase transition with massless low temperature fluctuations in $d\\ge 3$ (the main result of this work).",
"The $^{0|2}$ model is a member of the family of hyperbolic sigma models with target spaces $^{n|2m}$ , see [36] for a discussion of some aspects of this.",
"By supersymmetric localisation the observables of the $^{0|2}$ model considered in Theorem REF are equivalent to the analogous ones of the non-linear sigma model with target $^{2|4}$ .",
"While this relation does not play a role in this paper, it leads to a more direct representation of the continuous symmetry breaking observed here.",
"In brief, in the $^{2|4}$ model each vertex comes equipped with two real and four Grassmann fields.",
"By expressing these fields in horospherical coordinates one of the real fields and the four Grassmann fields can be integrated out.",
"The marginal distribution of the remaining real field, which is called the $t$ -field, may be viewed as a `$\\nabla \\phi $ ' random surface model, albeit with a nonconvex and nonlocal Hamiltonian.",
"By this we mean that the potential is invariant under the global translation $t_x\\mapsto t_x +r$ for $r\\in $ .",
"See [18] for more details, where this perspective was used to prove the absence of symmetry breaking in $d=2$ .",
"The full $^{n|2m}$ family has been important for advancing our understanding of other aspects of these models [18], [36].",
"Of particular note, we mention that the $^{2|2}$ model has received substantial prior attention due to its exact connection to linearly reinforced random walks and its motivation from random matrix theory, see [64], [71], [72], [70], [40].",
"For hyperbolic sigma models with target $^n$ , $n \\ge 1$ , spontaneous symmetry breaking for all $\\beta >0$ was shown in [70], and with target $^{2|2}$ for $\\beta $ large in [40] (see also [39]).",
"For motivation from random matrix theory and the Anderson transition see [68], [69].",
"These proofs make essential use of the horospherical coordinates mentioned above.",
"Moreover, the proof of symmetry breaking for the $^{2|2}$ model in [40] relies on an infinite number of Ward identifies resulting from supersymmetric localisation.",
"These identities are absent in the $^{0|2}$ model, limiting the applicability of the methods of [40] to our setting.",
"At the same time, the $^{2|2}$ model has no purely fermionic representation, and so our methods do not apply there, at least without significant further developments.",
"Introductions to fermionic renormalisation include [20], [59], [65], see also [47].",
"Recent probabilistic applications of these approaches to fermionic renormalisation include the study of interacting dimers [48], [49] and two-dimensional finite range Ising models [46], [45], [7].",
"Our organisation of the renormalisation group is instead based on a finite range decomposition, and follows [27] and its further developments in [30], [31], [15], [32], [33], [16], [28], [11].",
"This approach has its origins in [26].",
"For an introduction to this approach in a hierarchical bosonic context see [17].",
"Previous applications of this approach include the study of 4d weakly self-avoiding walks [14], [13]; the nearest-neighbour critical 4d $|\\varphi |^4$ model [12], [67] and long-range versions thereof [66], [56]; the ultraviolet $\\varphi ^4_3$ problem [25], [29]; analysis of the Kosterlitz–Thousless transition of the 2d Coulomb gas [38], [42]; the Cauchy–Born problem [1]; and others.",
"While the construction of the bulk renormalisation group flow is simpler for the intrinsic representation of the $^{0|2}$ model than in many of the previous references, a crucial novelty of our present work is the combination of the finite range renormalisation group approach with Ward identities, together with a precise analysis of a nontrivial zero mode.",
"This has enabled us to apply these methods to a non-linear sigma model in the phase of broken symmetry.",
"It would be extremely interesting to understand this approach for bosonic non-linear sigma models where, while `large fields' cause serious complications, the formal perturbative analysis is very much in parallel to the fermionic version we study in this paper.",
"Ward identities of a different type have previously been used in the renormalisation group analyses in [8] and [21] and many follow-up works including [48], [49].",
"Finally, we mention that Theorem REF yields quantitative finite volume statements.",
"The proof implements a rigorous finite size analysis along the lines of that proposed in [24].",
"It would be very interesting to extend this to even higher precision as discussed in Section REF below."
],
[
"Future directions for the arboreal gas",
"In this section we discuss several interesting open directions, including the geometric structure of the weak infinite volume limits of the arboreal gas and its relation to the uniform spanning tree, and a conjectural finite size universality similar to Wigner–Dyson universality from random matrix theory."
],
[
"Finite volume behaviour",
"The detailed finite volume behaviour of the arboreal gas would be very interesting to understand beyond the precision of Theorem REF .",
"On the complete graph at supercritical temperatures it is known that there is a unique macroscopic cluster, and that there are an unbounded number of clusters whose sizes are of order $|\\Lambda |^{2/3}$ [57].",
"The fluctuations of the macroscopic cluster are non-Gaussian of scale $|\\Lambda |^{2/3}$ and the distribution of the ordered cluster sizes of the mesoscopic clusters has been determined [57].",
"The joint law of the mesoscopic clusters can be characterised [58].",
"Intriguingly, $|\\Lambda |^{2/3}$ is the size of the largest tree at criticality on the complete graph.",
"The order statistics of the supercritical mesoscopic clusters follow the critical point order statistics [58].",
"Going beyond the complete graph, is this distribution of ordered cluster sizes universal, at least in sufficiently high dimensions?",
"This would be similar to the conjectured universality of Wigner–Dyson statistics from random matrix theory [60] or the conjectured universality of the distribution of macroscopic loops in loop representations of $O(n)$ (and other) spin systems [61], [51].",
"More generally it would be an instance of the universality of low temperature fluctuations in finite volume in models with continuous symmetries.",
"Finally, we mention that on expander graphs the existence of a phase transition for the arboreal gas is not difficult to show by using a natural split–merge dynamics [50].",
"It would be interesting if this dynamical approach could also be used to obtain information about the cluster size distribution.",
"As mentioned previously, the arboreal gas is also known as the uniform forest model [52].",
"We emphasise that the arboreal gas is not what is typically known as the uniform spanning forest (USF), which is in fact the weak limit as $\\Lambda _N \\uparrow ^d$ of a uniform spanning tree (UST) [62].",
"On a finite graph, the UST is the $\\beta \\rightarrow \\infty $ limit of the arboreal gas.",
"The correct scaling of the external field for this limit is $h=\\beta \\kappa $ and we thus write $¶_{\\text{UST},\\kappa } = \\lim _{\\beta \\rightarrow \\infty }¶_{\\beta ,\\beta \\kappa }$ for the UST on a finite graph (plus ghost vertex if $\\kappa >0$ ).",
"For $\\kappa >0$ , this measure is also known as the rooted spanning forest, because disregarding the connections to the ghost vertex disconnects the tree of the UST, with vertices previously connected to the ghost becoming roots.",
"The distributions of rooted and unrooted forests are not the same.",
"To help prevent confusion we will refer to the rooted spanning forests as (a special case of) the UST.",
"It is trivial that $¶_{\\text{UST},0}^{\\Lambda _N}[0\\leftrightarrow x]=1$ .",
"Nevertheless, the behaviour of the UST in the weak infinite volume limit depends on the dimension $d$ .",
"This limit can be defined as $¶_{\\rm UST}^{^d} =\\lim _{\\kappa \\downarrow 0}\\lim _{N\\rightarrow \\infty }¶_{\\text{UST},\\kappa }^{\\Lambda _N}$ and is independent of the finite volume boundary conditions (e.g.",
"free, wired, or periodic as above) imposed on $\\Lambda _N$ , see [62].",
"Even though the function $1_{0\\leftrightarrow x}$ is not continuous with respect to the topology of weak convergence, it is still true that $¶_{{\\rm UST}}^{^d}[0\\leftrightarrow x] = \\lim _{\\kappa \\downarrow 0}\\lim _{N\\rightarrow \\infty }¶_{{\\rm UST},\\kappa }^{\\Lambda _N}[0 \\leftrightarrow x].$ The order of limits here is essential.",
"In this infinite volume limit the UST disconnects into infinitely many infinite trees if $d>4$ , but remains a single connected tree if $d\\le 4$ , see [62].",
"Moreover, $¶_{\\rm UST}^{^d}[0 \\leftrightarrow x] + ¶_{\\rm UST}^{^d}[0\\lnot \\leftrightarrow x,|T_0|=\\infty , |T_x|=\\infty ] = 1.$ On the left-hand side, the second term vanishes if $d\\le 4$ whereas the first term tends to 0 as $|x|\\rightarrow \\infty $ if $d>4$ .",
"Furthermore, the geometric structure of the trees under $¶_{\\rm UST}^{^d}$ is well understood.",
"In particular, all trees are one-ended, meaning that removing one edge from a tree results in two trees, of which one is finite [62], [23].",
"For the arboreal gas, the existence and uniqueness of infinite volume limits is an open question.",
"Nonetheless, subsequential limits exist, and in such an infinite volume limit all trees are finite almost surely when $\\beta $ is small, while Theorem REF implies the existence of an infinite tree for $\\beta $ large.",
"Moreover, by Theorem REF , $ \\lim _{h\\downarrow 0}\\lim _{N\\rightarrow \\infty }{ ¶_{\\beta ,h}^{\\Lambda _N}[0 \\leftrightarrow x] + ¶_{\\beta ,h}^{\\Lambda _N}[0\\lnot \\leftrightarrow x, 0\\leftrightarrow \\mathfrak {g}, x\\leftrightarrow \\mathfrak {g}]}= \\theta _d(\\beta )^2 + \\frac{2{\\sf c}_1(\\beta )}{\\beta |x|^{d-2}} + O(\\frac{1}{\\beta |x|^{d-2+\\kappa }}).$ By analogy with the UST, we expect that only the first term on the left-hand side contributes for $d\\le 4$ and that only the second term contributes asymptotically as $|x|\\rightarrow \\infty $ for $d>4$ .",
"The tempting conjecture that the UST stochastically dominates the arboreal gas on the torus is consistent with these expectations.",
"The analogue of the left-hand side of (REF ) plays an important role in the proof of uniqueness of the infinite cluster in Bernoulli percolation in [5]; this is related to the vanishing of the second term.",
"As already mentioned, for the arboreal gas we only expect this to be true in $d\\le 4$ .",
"Beyond the questions above, it would be interesting to analyse more detailed geometric aspects of the arboreal gas.",
"For example, can one construct scaling limits as has been done for some spanning tree models [3], [4], [44], [6]?",
"Finally, we mention that a detailed analysis of the infinite volume behaviour of the arboreal gas on regular trees with wired boundary conditions has been carried out [41].",
"This infinite volume behaviour is consistent with the finite volume behaviour of the complete graph, e.g., at all supercritical temperatures the sizes of finite clusters have the same distribution as those of critical percolation.",
"Our analysis could be extended to a detailed study of the approach $h\\downarrow 0$ .",
"To keep the length of this paper within bounds, we do not carry this out, but here briefly comment on what we expect can be shown by extensions of our analysis.",
"As discussed above, a natural object is the magnetisation $M(\\beta ,h) = \\lim _{N\\rightarrow \\infty } M_{N}(\\beta ,h), \\qquad M_{N}(\\beta ,h) = ¶^{\\Lambda _{N}}_{\\beta ,h}[0\\leftrightarrow \\mathfrak {g}],$ and the corresponding susceptibility (neglecting questions concerning the order of limits) $\\chi (\\beta ,h)= {}{h}M(\\beta ,h)= \\sum _{x} \\sigma _{\\beta ,h}(x).$ Thus for the arboreal gas, the susceptibility is not the sum over $\\tau _{\\beta ,h}(x)$ as is the case for Bernoulli bond percolation, but the sum over $\\sigma _{\\beta ,h}(x)$ .",
"In terms of the sigma model, $\\chi $ maybe viewed as the longitudinal susceptibility, often denoted $\\chi _{||}$ .",
"In this interpretation, the sum over $\\tau _{\\beta ,h}(x)$ is the transversal susceptibility $\\chi _{\\perp }$ and satisfies the Ward identity $\\chi _{\\perp }(\\beta ,h)=\\sum _{x} \\tau _{\\beta ,h}(x) = h^{-1}M(\\beta ,h)$ .",
"This identity is crucial in our analysis.",
"For the longitudinal susceptibility, we expect that it would be possible to extend our analysis to show $\\chi (\\beta ,h) \\sim {\\left\\lbrace \\begin{array}{ll}C(\\beta ) h^{-1/2} & (d=3)\\\\C(\\beta ) |\\log h|& (d=4)\\\\C(\\beta ) & (d>4).\\end{array}\\right.",
"}$ Defining the free energy $f(\\beta ,h)=\\lim _{N\\rightarrow \\infty } |\\Lambda _N|^{-1}\\log Z_{\\beta ,h}^{\\Lambda _{N}}$ , for $\\beta \\ge \\beta _0$ the previous asymptotics suggest that $h\\mapsto f(\\beta ,h)$ is $C^2$ in $d>4$ but only $C^1$ for $d=3,4$ .",
"In fact, extrapolating from our renormalisation group analysis we believe that for $\\beta \\ge \\beta _0$ the free energy is $C^n$ but not $C^{n+1}$ as a function of $h\\ge 0$ for $n={\\frac{d-1}{2}}$ .",
"It is unclear how this is connected to the geometry of the component graph of the UST, which also changes as the dimension is varied [22], [53]."
],
[
"Organisation and notation",
"This paper is organised as follows.",
"In Section , we show how Theorem REF is reduced to renormalisation group results with the help of the Ward identity (REF ).",
"In Sections – we then prove these renormalisation group results.",
"Section is concerned with the construction of the bulk renormalisation group flow and Section uses this analysis to compute the susceptibility.",
"Section then constructs the renormalisation group flow for observables, which is used in Section to compute pointwise correlation functions.",
"The short Section then collects the results.",
"Finally, in Appendix we collect relations between the arboreal gas and the $^{0|2}$ model as well as basic percolation and high temperature properties of the arboreal gas, and in Appendix we include some background material about the renormalisation group method.",
"Throughout we use $a_{n}\\sim b_{n}$ to denote $\\lim _{n\\rightarrow \\infty }a_{n}/b_{n}=1$ , $a_{n}\\asymp b_{n}$ to denote the existence of $c,C>0$ such that $c a_{n}\\le b_{n}\\le Ca_{n}$ , $a_n \\lesssim b_n$ if $a_n \\le Cb_n$ , and $a_{n}=O(b_{n})$ if $|a_{n}|\\lesssim |b_{n}|$ .",
"We consider the dimension $d \\ge 3$ to be fixed, and hence allow implicit constants to depend on $d$ .",
"In Sections 1 and 2 we allow implicit constants to depend on $L$ as well, as this dependence does not play a role.",
"In subsequent sections $L$ -dependence is made explicit, though uniformity in $L$ is only crucial in the contractive estimate of Theorem REF .",
"Our main theorems hypothesise $L=L(d)$ is large, and for geometric convenience we will assume throughout that $L$ is at least $2^{d+2}$ ."
],
[
"Consequences of combining renormalisation and Ward\nidentities",
"In our renormalisation group analysis, which provides the foundation for the proofs of the theorems stated in Section , we will first drop the constraint between the coupling constants of the quadratic and quartic terms in (REF ).",
"The constraint will be restored in the end with the help of the Ward identity (REF ), i.e., $ {z_0}_{\\beta ,h}= h \\sum _{x\\in \\Lambda } {\\xi _0 \\eta _x}_{\\beta ,h},\\quad \\text{ and in particular }{z_0}_{\\beta ,0}= 0.$ The application of this Ward identity is the subject of this section.",
"In our analysis we distinguish between two orders of limits.",
"We first analyse the `infinite volume' limit $\\lim _{h\\downarrow 0} \\lim _{N\\rightarrow \\infty }$ , and prove Theorem REF (and thus Theorems REF –REF ).",
"Using results of this analysis (and with several applications of the Ward identity), we then also analyse the much more delicate `finite volume' limit $\\lim _{N\\rightarrow \\infty }\\lim _{h\\downarrow 0}$ in order to prove Theorem REF ."
],
[
"Infinite volume correlation functions",
"For $m^2 >0$ arbitrary and coupling constants $s_0,a_0,b_0$ , which eventually will be taken small, we consider the model with fermionic Gaussian reference measure with covariance $ C= (-\\Delta +m^2)^{-1}$ on $\\Lambda _N$ and interaction $V_0= V_0(\\Lambda _N)=\\sum _{x\\in \\Lambda _N}{s_0 (\\nabla \\psi )_x(\\nabla \\bar{\\psi })_x + a_0 \\psi _x\\bar{\\psi }_x + b_0 \\psi _x\\bar{\\psi }_x(\\nabla \\psi )_x(\\nabla \\bar{\\psi })_x},$ where we recall the squared gradient notation from (REF ).",
"Thus the corresponding expectation is ${F}_{m^2,s_0,a_0,b_0} =\\frac{1}{Z_{m^2,s_0,a_0,b_0}}\\frac{1}{\\det (-\\Delta +m^2)} \\int \\partial _\\psi \\partial _{\\bar{\\psi }}\\, e^{-(\\psi ,(-\\Delta +m^2)\\bar{\\psi }) - V_0} F,$ where $\\int \\partial _\\psi \\partial _{\\bar{\\psi }}$ denotes the Grassmann integral, and $Z_{m^2,s_0,a_0,b_0}$ is defined such that ${1}_{m^2,s_0,a_0,b_0}=1$ .",
"The following result resembles those in [14], [13], [67] for weakly self-avoiding walks in dimension 4.",
"Compared to the latter results, our analysis is substantially simplified since the $^{0|2}$ model can be studied in terms of only fermionic variables with a quartic interaction that is irrelevant in dimensions $d > 2$ .",
"However, in Section REF , we state an improvement of the following result that sees the full zero mode of the low temperature phase.",
"This analysis, which relies crucially on the interplay with Ward identities, goes beyond the analysis of [14], [13], [67].",
"Let $d \\ge 3$ and $L \\ge L_0(d)$ .",
"For $b_0$ sufficiently small and $m^2 \\ge 0$ , there are $s_0 = s_0^c(b_0,m^2)$ and $a_0= a_0^c(b_0,m^2)$ independent of $N$ so that the following hold: The functions $s_0^c$ and $a_0^c$ are continuous in both variables, differentiable in $b_0$ with uniformly bounded $b_0$ -derivatives, and satisfy the estimates $ s_0^c(b_0,m^2) = O(b_0), \\qquad a_0^c(b_0,m^2) = O(b_0)$ uniformly in $m^2\\ge 0$ .",
"There exists $\\kappa >0$ such that if the torus sidelength satisfies $L^{-N} \\le m$ , $\\sum _{x\\in \\Lambda _N} {\\bar{\\psi }_0\\psi _x}_{m^2,s_0,a_0,b_0} = \\frac{1}{m^2}+ \\frac{O(b_0L^{-(2+\\kappa )N})}{m^4}.$ Moreover, there are functions $\\lambda = \\lambda (b_0,m^2) = 1+O(b_0), \\qquad \\gamma = \\gamma (b_0,m^2) = (-\\Delta ^{^d}+m^2)^{-1}(0,0) + O(b_0),$ having the same continuity properties as $s_0^c$ and $a_0^c$ such that ${\\bar{\\psi }_0\\psi _0}_{m^2,s_0,a_0,b_0}&= \\gamma + O(b_0L^{-\\kappa N}),\\\\{\\bar{\\psi }_0\\psi _x}_{m^2,s_0,a_0,b_0}&= (-\\Delta +m^2)^{-1}(0,x) + O(b_0|x|^{-(d-2)-\\kappa })+ O(b_0L^{-\\kappa N}),\\\\{\\bar{\\psi }_0\\psi _0;\\bar{\\psi }_x\\psi _x}_{m^2,s_0,a_0,b_0}&= - \\lambda ^{2}(-\\Delta +m^2)^{-1}(0,x)^2 + O(b_0|x|^{-2(d-2)-\\kappa })+ O(b_0L^{-\\kappa N}).$ Here ${A;B}={AB}-{A}{B}$ .",
"The proof of this theorem is given in Sections – and occupies most of this paper.",
"We now show how to derive Theorem REF for the $^{0|2}$ model from it together with the Ward identity (REF ).",
"To this end, assuming $s_0>-1$ we further rescale $\\psi $ by $1/\\sqrt{1+s_0}$ (and likewise for $\\bar{\\psi }$ ) in (REF ), setting $ \\xi = \\sqrt{\\frac{1+s_0}{\\beta }} \\bar{\\psi }, \\qquad \\eta =\\sqrt{\\frac{1+s_0}{\\beta }} \\psi ,$ Up to a normalisation constant, the fermionic density with respect to the fermionic Gaussian measure with covariance $C=(-\\Delta +m^2)^{-1}$ becomes, see (REF ), $\\exp {-\\sum _{x\\in \\Lambda _N}{(1+s_0) (\\nabla \\psi )_x(\\nabla \\bar{\\psi })_x+\\frac{1+s_0}{\\beta }(1+h) \\psi _x\\bar{\\psi }_x + \\frac{(1+s_0)^2}{\\beta } \\psi _x\\bar{\\psi }_x (\\nabla \\psi )_x(\\nabla \\bar{\\psi })_x}}.$ For any $m^2 \\ge 0$ and $s_0>-1$ , (REF ) is of the form (REF ) with $a_0 = \\frac{1+s_0}{\\beta }(1+h) -m^2, \\qquad b_0 = \\frac{(1+s_0)^2}{\\beta }.$ To use Theorem REF we need to invert this implicit relation between $(\\beta ,h)$ and $(m^2,s_0,a_0,b_0)$ .",
"This is achieved by the following corollary.",
"A key observation is that the Ward identity (REF ) allows us to identify the critical point with $h=0$ .",
"To make this precise, with $s_0^c$ and $a_0^c$ as in Theorem REF , define the functions $\\beta (b_0,m^2) &= \\frac{(1+s_0^c(b_0,m^2))^2}{b_0},\\\\h(b_0,m^2) &= -1+ \\frac{a_0^c(b_0,m^2)+m^2}{b_0} (1+s_0^c(b_0,m^2)).$ By Theorem REF , both functions are continuous in $b_0>0$ small enough and $m^2 \\ge 0$ .",
"(i) Assume $b_0>0$ is small enough.",
"Then $h(b_0,m^2) = m^2 \\beta (b_0,m^2)(1+O(b_0)).$ In particular, $h(b_0,0) = 0$ and $h(b_0,m^2)>0$ if $m^2>0$ .",
"(ii) For $\\beta $ large enough and $h\\ge 0$ , there are functions $\\tilde{b}_0(\\beta ,h)>0$ and $\\tilde{m}^2(\\beta ,h)\\ge 0$ such that $h(\\tilde{b}_0,\\tilde{m}^2)=h$ and $\\beta (\\tilde{b}_0,\\tilde{m}^2)=\\beta $ .",
"Both functions are right-continuous as $h\\downarrow 0$ when $\\beta $ is fixed.",
"To prove (i), we use the Ward identity (REF ) with $(\\beta ,h)$ given by (REF )–().",
"The left- and right-hand sides of (REF ) are, respectively, ${z_0}_{\\beta ,h} &= 1-\\frac{1+s_0^c(b_0,m^2)}{\\beta } {\\bar{\\psi }_0\\psi _0}_{m^2,s_0,a_0,b_0} ,\\\\h \\sum _{x\\in \\Lambda _N} {\\xi _0 \\eta _x}_{\\beta ,h}&= \\frac{(1+s_0^c(b_0,m^2))h(b_0,m^2)}{\\beta (b_0,m^2)} \\sum _{x\\in \\Lambda _N} {\\bar{\\psi }_0\\psi _x}_{m^2,s_0,a_0,b_0} .$ By Theorem REF , in the limit $N\\rightarrow \\infty $ , we obtain from (REF ) that if $m^{2}>0$ , the identity $1-\\frac{1+s_0^c(b_0,m^2)}{\\beta (b_0,m^2)}\\gamma (b_0,m^2) = \\frac{(1+s_0^c(b_0,m^2))h(b_0,m^2)}{\\beta (b_0,m^2) m^2}$ holds.",
"Solving for $h$ , we have $h(b_0,m^2)= m^2 { \\frac{\\beta (b_0,m^2)}{1+s_0^c(b_0,m^2)} - \\gamma (b_0,m^2) }.$ Since $s_0^c(b_0,m^2)=O(b_0)$ , $\\beta (b_{0},m^{2}) \\asymp 1/b_0$ , and $\\gamma (b_0,m^2)=O(1)$ , all uniformly in $m^2\\ge 0$ , we obtain $h(b_0,m^2) = m^2 \\beta (b_0,m^2)(1+O(b_0))$ .",
"In particular, $h(b_0,0)=0$ .",
"Claim (ii) follows from an implicit function theorem argument that uses that $s_0^c$ and $a_0^c$ are continuous in $m^2\\ge 0$ and differentiable in $b_0$ if $m^2>0$ with $b_0$ -derivatives uniformly bounded in $m^2>0$ .",
"This argument is the same as the proof of [14] (with our notation $s_0$ instead of $z_0$ , $a_0$ instead of $\\nu _0$ , $b_0$ instead of $g_0$ , and with $1/\\beta $ instead of $g$ and $h$ instead of $\\nu $ ) and is omitted here.",
"Assuming Theorem REF , the proof of Theorem REF is immediate from the last corollary.",
"The main statements of Theorems REF and REF then follow immediately, except for the identifications $\\theta _{d}(\\beta )^{2}=\\zeta _{d}(\\beta )$ , $({\\sf c}_2(\\beta )/{\\sf c}_1(\\beta )^{2})\\theta _{d}(\\beta )^{2}=1$ , and ${\\sf c}(\\beta )=2{\\sf c}_2(\\beta )$ which we will obtain in Section REF .",
"Given $\\beta \\ge \\beta _0$ and $h>0$ we choose $b_0>0$ and $m^2>0$ as in Corollary REF (ii).",
"Since $z_x = 1-\\xi _x\\eta _x$ and using (REF ) we then have ${z_0}_{\\beta ,h}&= 1-{\\xi _0\\eta _0}_{\\beta ,h}=1- \\frac{1+s_0}{\\beta } {\\bar{\\psi }_0\\psi _0}_{m^2,s_0,a_0,b_0},\\\\{\\xi _0\\eta _x}_{\\beta ,h}&= \\frac{1+s_0}{\\beta } {\\bar{\\psi }_0\\psi _x}_{m^2,s_0,a_0,b_0},\\\\{z_0z_x}_{\\beta ,h}-{z_0}_{\\beta ,h}^2&= {\\xi _0\\eta _0\\xi _x\\eta _x}_{\\beta ,h} - {\\xi _0\\eta _0}_{\\beta ,h}^2= \\frac{(1+s_0)^2}{\\beta ^2} {\\bar{\\psi }_0\\psi _0;\\bar{\\psi }_x\\psi _x}_{m^2,s_0,a_0,b_0}.$ Taking $N\\rightarrow \\infty $ and then $h\\downarrow 0$ , the results follow from Corollary REF (i) and Theorem REF with $ \\theta _d(\\beta ) = 1- \\frac{b_0\\gamma }{1+s_0^c},\\qquad {\\sf c}_1(\\beta ) = (1+s_0^c)c_d,\\qquad {\\sf c}_2(\\beta ) = \\lambda ^{2} (1+s_0^c)^2 c_d^2,$ where the functions $\\lambda $ and $\\gamma $ are evaluated at $m^2=0$ and $b_0$ given as above, $c_d$ is the constant in the asymptotics of the free Green function on $^{d}$ , see (REF ), and we have used the simplification of the error terms $O(|x|^{-(d-2)-1})+O(b_0 |x|^{-(d-2+\\kappa )}) = O(|x|^{-(d-2+\\kappa )})$ and $O(|x|^{-2(d-2)-1})+O(b_0 |x|^{-2(d-2)-\\kappa )}) = O(|x|^{-2(d-2)-\\kappa )})$ ."
],
[
"Finite volume limit",
"The next theorem extends Theorem REF by more precise estimates valid in the limit $m^2\\downarrow 0$ with $\\Lambda _N$ fixed.",
"In these estimates $t_N \\in (0,1/m^{2})$ is a continuous function of $m^2>0$ that satisfies $t_N = \\frac{1}{m^2} - O(L^{2N}),\\qquad \\text{and}\\\\\\lim _{m\\downarrow 0}{ (-\\Delta +m^2)^{-1}(0,x) - \\frac{t_N}{|\\Lambda _N|}}= (-\\Delta ^{^d})^{-1}(0,x) + O(L^{-(d-2)N}),$ where on the right-hand side $\\Delta ^{^d}$ is the Laplacian on $^d$ , on the left-hand side $\\Delta $ is the Laplacian on $\\Lambda _N$ , and $|\\Lambda _N| = L^{dN}$ denotes the volume of the torus $\\Lambda _N$ .",
"We define $W_N(x) = W_{N,m^2}(x) = (-\\Delta +m^2)^{-1}(0,x) - \\frac{t_N}{|\\Lambda _N|}.$ In the following, $\\Lambda _{N}$ is fixed and the parameters $(\\beta ,h)$ are related to $(m^2,s_0,a_0,b_0)$ as in Corollary REF .",
"We will write ${\\cdot }_{m^2,b_0} ={\\cdot }_{m^2,s_0^c(b_0,m^2),a_0^c(b_0,m^2),b_0}$ for the corresponding expectation and similarly for the partition function $Z_{m^2,b_0}$ .",
"Under the conditions of Theorem REF except that we no longer restrict $L^{-N}\\le m$ , in addition to the functions $a^{c}_{0}$ , $s^{c}_{0}$ , $\\lambda $ , and $\\gamma $ , there are functions $\\tilde{a}_{N,N}^c = \\tilde{a}_{N,N}^c(b_0,m^2)$ and $u_N^c = u_N^c(b_0,m^2)$ , both continuous in $b_0$ small and $m^2\\ge 0$ , as well as $\\tilde{u}_{N,N}^c = \\tilde{u}_{N,N}^c(b_0,m^2)= t_N\\tilde{a}_{N,N}^c(b_0,m^2) + O(b_0L^{-\\kappa N}),$ continuous in $b_0$ small and $m^2> 0$ , such that, for $x\\in \\Lambda _N$ , $ \\sum _{x\\in \\Lambda _N} {\\bar{\\psi }_0\\psi _x}_{m^2, b_0}&= \\frac{1}{m^2}- \\frac{1}{m^4}\\frac{\\tilde{a}_{N,N}^c}{1+\\tilde{u}_{N,N}^c}, \\\\{\\bar{\\psi }_0\\psi _0}_{m^2,b_0}&=\\gamma + \\frac{\\lambda t_N |\\Lambda _N|^{-1}}{1+\\tilde{u}_{N,N}^c} +E_{00},\\\\{\\bar{\\psi }_0\\psi _0\\bar{\\psi }_x\\psi _x}_{m^2,b_0}&=-\\lambda ^2W_N(x)^2+\\gamma ^2+\\frac{-2\\lambda ^2W_N(x)+ 2\\lambda \\gamma }{1+\\tilde{u}_{N,N}^c} t_N|\\Lambda _N|^{-1}+E_{00xx},$ and $Z_{m^{2},b_{0}} = e^{-u_{N}^{c}|\\Lambda _N|}(1+\\tilde{u}_{N,N}^{c}).$ The remainder terms satisfy $E_{00}&=\\frac{O(b_0L^{-(d-2+\\kappa )N}+b_0L^{-\\kappa N}(m^2|\\Lambda _N|)^{-1})}{1+\\tilde{u}_{N,N}^c},\\\\E_{00xx}&= O(b_0|x|^{-2(d-2)-\\kappa }) + O(b_0L^{-(d-2+\\kappa )N})&\\qquad +(O(b_0|x|^{-(d-2+\\kappa )})+ O(b_0L^{-\\kappa N})) \\frac{(m^2|\\Lambda _N|)^{-1}}{1+\\tilde{u}_{N,N}^c}.$ The proof of this theorem is again given in Sections –.",
"The proof of Theorem REF is based on Theorem REF and several upcoming lemmas.",
"These lemmas exploit Ward identities to relate the functions given by the theorem to one another.",
"To orient the reader, the first two lemmas below can be viewed as preparatory for the key computation in Lemma REF .",
"Under the conditions of Theorem REF , $^{\\Lambda _{N}}_{\\beta ,0}|T_0|=\\frac{b_0}{1+s_0^c(b_0,0)} \\frac{1+O(b_0L^{-\\kappa N})}{\\tilde{a}_{N,N}^{c}(b_{0},0)} + O(b_0L^{2N}).$ In particular, if $b_0>0$ this implies $1/\\tilde{a}_{N,N}^{c}(b_{0},0) = O(|\\Lambda _{N}|/b_{0})$ and $\\tilde{a}_{N,N}^{c}(b_{0},0)>0$ .",
"From (REF ), we have that $^{\\Lambda _{N}}_{\\beta ,0}|T_0|=\\sum _{x\\in \\Lambda _N} ¶_{\\beta ,0}[0\\leftrightarrow x] = \\sum _{x\\in \\Lambda _N} {\\xi _0\\eta _x}_{\\beta ,0}=\\lim _{h\\rightarrow 0} \\sum _{x\\in \\Lambda _N} {\\xi _0\\eta _x}_{\\beta ,h}.$ Changing variables, $\\sum _{x\\in \\Lambda _N} {\\xi _0\\eta _x}_{\\beta ,h}= \\frac{b_0}{1+s_0^c(b_0,m^2)} \\sum _{x\\in \\Lambda _N} {\\bar{\\psi }_0\\psi _x}_{m^2,b_0},$ where $(\\beta , h)$ and $(b_0, m^2)$ are related as in (REF ) and ().",
"To evaluate the right-hand side we use (REF ).",
"Note that $\\frac{1}{m^2}- \\frac{1}{m^4}\\frac{\\tilde{a}_{N,N}^c}{1+\\tilde{u}_{N,N}^c}&= \\frac{1}{m^2}\\frac{1+\\tilde{u}_{N,N}^c - \\tilde{a}_{N,N}^cm^{-2}}{1+\\tilde{u}_{N,N}^c}&= \\frac{1+\\tilde{a}_{N,N}^c (t_N-m^{-2})+O(b_0L^{-\\kappa N})}{m^2+\\tilde{a}_{N,N}^ct_Nm^2 + O(b_0m^2L^{-\\kappa N})}&= \\frac{1+\\tilde{a}_{N,N}^c O(L^{2N})+O(b_0L^{-\\kappa N})}{m^2+\\tilde{a}_{N,N}^c(1+O(m^2L^{2N})) + O(b_0m^2L^{-\\kappa N})},$ where the second equality is due to (REF ) and the third follows from (REF ).",
"As $m^2\\downarrow 0$ , the right-hand side of the third equality behaves asymptotically as $\\frac{1+\\tilde{a}_{N,N}^c O(L^{2N})+O(b_0L^{-\\kappa N})}{\\tilde{a}_{N,N}^c}=\\frac{1+O(b_0L^{-\\kappa N})}{\\tilde{a}_{N,N}^c}+O(L^{2N}).$ Since $s_{0}^{c}(b_{0},0)=O(b_{0})$ by Theorem REF we therefore obtain the first claim: $_{\\beta ,0}^{\\Lambda _N}|T_0|=\\frac{b_0}{1+s_0^c(b_0,0)}\\lim _{m^2\\downarrow 0} \\sum _{x\\in \\Lambda _N} {\\bar{\\psi }_0\\psi _x}_{m^2,b_0}=\\frac{b_0}{1+s_0^c(b_0,0)} \\frac{1+O(b_0L^{-\\kappa N})}{\\tilde{a}_{N,N}^{c}(b_{0},0)} + O(b_0L^{2N})$ For the second claim, let us observe that on the one hand $ Z_{\\beta ,h}= {\\frac{\\beta }{1+s_0^c}}^{|\\Lambda _N|}(\\det (-\\Delta +m^2)) Z_{m^2, b_0}= {\\frac{\\beta e^{-u_{N}^{c}}}{1+s_0^c}}^{|\\Lambda _N|}(\\det (-\\Delta +m^2))(1+\\tilde{u}_{N,N}^{c}),$ where the first equality is by Proposition REF and (REF ), (REF ), and (REF ), and the second equality is (REF ).",
"On the other hand, by (REF ), $\\lim _{h\\rightarrow 0} Z_{\\beta ,h}= Z_{\\beta ,0}>0.$ Since, by Theorem REF , $u_{N}^{c}$ and $s_0^c$ remain bounded as $m^2\\downarrow 0$ with $\\Lambda _N$ fixed (and thus also $\\beta $ which is given by (REF )), from $\\det (-\\Delta +m^2) \\downarrow 0$ , we conclude that $1+\\tilde{u}_{N,N}^c$ diverges as $m^2\\downarrow 0$ .",
"By (REF ), this implies $\\tilde{a}_{N,N}^{c}(b_{0},0)>0$ .",
"The upper bound on $1/\\tilde{a}_{N,N}^c(b_0,0)$ follows by re-arranging (REF ) and using the trivial bound $|T_0| \\le |\\Lambda _N|$ .",
"Under the conditions of Theorem REF and if $b_{0}>0$ , $1=\\frac{b_0}{1+s_0^c(b_0,0)}{\\gamma (b_0,0) + \\frac{\\lambda (b_0,0)}{|\\Lambda _N|\\tilde{a}_{N,N}^c(b_0,0)}(1+O(b_0L^{-\\kappa N}))}.$ The Ward identity ${z_0}_{\\beta ,0}=0$ implies $0={z_0}_{\\beta ,0}= 1-{\\xi _0\\eta _0}_{\\beta ,0}&=1-\\lim _{m^2\\downarrow 0} \\frac{1+s_0^c(b_0,m^2)}{\\beta (b_0,m^2)}{\\bar{\\psi }_0\\psi _0}_{m^2,b_0}& =1-\\lim _{m^2\\downarrow 0} \\frac{b_0}{1+s_0^c(b_0,m^2)}{\\bar{\\psi }_0\\psi _0}_{m^2,b_0},$ where we used (REF ) and that $\\beta =\\beta (b_0,m^2)$ is as in (REF ).",
"To compute ${\\bar{\\psi }_0\\psi _0}_{m^2,b_0}$ , we apply ().",
"Since $\\tilde{u}_{N,N}^c = \\tilde{a}_{N,N}^ct_N +O(b_0L^{-\\kappa N})$ and $t_N = m^{-2}+O(L^{2N})$ , $\\lim _{m^2\\downarrow 0}{\\bar{\\psi }_0\\psi _0}_{m^2,b_0}&= \\gamma (b_{0},0) + \\lim _{m^{2}\\downarrow 0} \\frac{\\lambda (b_{0},m^{2})t_{N}|\\Lambda _{N}|^{-1}}{1+\\tilde{a}_{N,N}^{c}(b_{0},m^{2})t_{N}+O(b_{0}L^{-\\kappa N})} +\\lim _{m^{2}\\downarrow 0}E_{00}&= \\gamma (b_{0},0) +\\frac{\\lambda (b_{0},0)|\\Lambda _{N}|^{-1}}{\\tilde{a}_{N,N}^c(b_{0},0)} + \\lim _{m^{2}\\downarrow 0}E_{00}.$ The limits in the second line exist by Theorem REF and Lemma REF , which in particular implies $\\tilde{a}_{N,N}^c(b_0,0) > 0$ since $b_0>0$ .",
"As $m^2\\downarrow 0$ , the error term $E_{00}$ is bounded by $O(b_0L^{-\\kappa N}/(|\\Lambda _N|\\tilde{a}_{N,N}^c))= (\\lambda (b_{0},0) |\\Lambda _N|^{-1}/\\tilde{a}_{N,N}^c)O(b_0L^{-\\kappa N})$ since $\\lambda (b_{0},0) =1-O(b_0)\\ge 1/2$ , finishing the proof.",
"Given Theorem REF , the following lemma is the main step in the proof of Theorem REF .",
"Under the conditions of Theorem REF and if $b_{0}>0$ , $¶_{\\beta ,0}^{\\Lambda _N}[0\\leftrightarrow x]= \\theta _d(\\beta )^2+2 \\frac{b_0}{1+s_0^c}\\lambda \\theta _d(\\beta )(-\\Delta ^{^d})^{-1}(0,x)\\\\+ O(b_0^2|x|^{-(d-2)-\\kappa })+ O(b_0L^{-(d-2)N})+ O(b_0^2L^{-\\kappa N}),$ where $\\theta _d(\\beta )$ is defined in (REF ).",
"By the last expression for $¶_{\\beta ,0}[0\\leftrightarrow x]$ in (REF ) and (REF ), (REF ): $ ¶_{\\beta ,0}^{\\Lambda _N}[0\\leftrightarrow x]= 1-\\lim _{h\\downarrow 0}{\\xi _0\\eta _0\\xi _x\\eta _x}_{\\beta ,h}= 1- \\lim _{m^2\\downarrow 0}{ \\frac{b_0^2}{(1+s_0^c)^2}{\\bar{\\psi }_0\\psi _0\\bar{\\psi }_x\\psi _x}_{m^{2},b_{0}}}.$ To compute $\\lim _{m^2\\downarrow 0}{\\bar{\\psi }_0\\psi _0\\bar{\\psi }_x\\psi _x}_{m^{2},b_{0}}$ we start from ().",
"By Lemma REF , as $m^2\\downarrow 0$ with $\\Lambda _N$ fixed, $\\frac{1}{1+\\tilde{u}_{N,N}^c} \\sim \\frac{1}{m^{-2}\\tilde{a}_{N,N}^c(b_0,0)}=O(\\frac{m^2|\\Lambda _N|}{b_0}).$ This implies the error term in () is, as $m^2\\downarrow 0$ with $\\Lambda _N$ fixed, $|E_{00xx}|\\le O(|x|^{-(d-2)-\\kappa }) + O(L^{-\\kappa N}).$ For the main term we have $\\lim _{m^{2}\\downarrow 0}{\\bar{\\psi }_0\\psi _0\\bar{\\psi }_x\\psi _x}_{m^2,b_0}&=-\\lambda ^2 W_{N,0}(x)^2 +\\gamma ^2+\\lim _{m^{2}\\downarrow 0}\\frac{-2\\lambda ^2 W_{N}(x)+ 2\\lambda \\gamma }{1+\\tilde{u}_{N,N}^c} {t_N|\\Lambda _N|^{-1}}&= -\\lambda ^2 W_{N,0}(x)^2 +\\gamma ^2+ 2(-\\lambda W_{N,0}(x)+\\gamma )\\frac{\\lambda }{\\tilde{a}_{N,N}^c|\\Lambda _{N}|},$ where on the right-hand side the functions $\\lambda $ , $\\gamma $ , and $\\tilde{a}_{N,N}^c$ are evaluated at $m^2=0$ .",
"By Lemma REF , $\\frac{b_{0}}{1+s_{0}^{c}}\\frac{\\lambda }{\\tilde{a}_{N,N}^{c}|\\Lambda _{N}| } ={1-\\frac{b_{0}\\gamma }{1+s_{0}^{c}}}(1+O(b_0L^{-\\kappa N}))$ so that $-{\\frac{b_{0}}{1+s_{0}^{c}}}^{2}\\frac{2\\lambda ^2 W_{N,0}(x)}{\\tilde{a}_{N,N}^c|\\Lambda _{N}|}{ (1+O(b_0L^{-\\kappa N}))}&= -\\frac{2b_{0}}{1+s_{0}^{c}}(1-\\frac{b_{0}\\gamma }{1+s_{0}^{c}})\\lambda W_{N,0}(x)\\\\{\\frac{b_{0}}{1+s_{0}^{c}}}^{2}\\frac{2\\lambda \\gamma }{\\tilde{a}_{N,N}^c|\\Lambda _{N}|}{ (1+O(b_0L^{-\\kappa N}))}&= 2\\gamma \\frac{b_{0}}{1+s_{0}} - 2\\gamma ^{2}{\\frac{b_{0}}{1+s_{0}^{c}}}^{2}.$ Substituting these bounds into (REF ) and then (REF ) we obtain $¶_{\\beta ,0}^{\\Lambda _N}[0\\leftrightarrow x]&=1-{\\frac{b_{0}}{1+s_{0}^{c}}}^{2}\\lim _{m^{2}\\downarrow 0}{\\bar{\\psi }_0\\psi _0\\bar{\\psi }_x\\psi _x}_{m^2,b_0}&= (1-\\frac{\\gamma b_{0}}{1+s_{0}})^{2}+\\frac{2b_{0}\\lambda }{1+s_{0}^{c}} (1-\\frac{b_{0}\\gamma }{1+s_{0}^{c}}) W_{N,0}(x)+ {\\frac{b_{0}\\lambda W_{N,0}(x)}{1+s_{0}^{c}}}^{2}&\\qquad +O(b_0^2 L^{-\\kappa N} W_{N,0}(x)) + O(b_0^2 L^{-\\kappa N})+ O(b_0^2|E_{00xx}|).$ Using the definition (REF ) of $\\theta _d(\\beta )$ , that $W_{N,0}(x) = (-\\Delta ^{^d})^{-1}(0,x)+O(L^{-(d-2)N})$ by (), and in particular $W_{N,0}(x) = O(|x|^{-(d-2)})$ , the claim follows.",
"The next (and final) lemma is inessential for the main conclusions, but will allow us to identify the constants from the infinite volume and the finite volume analyses.",
"Under the conditions of Theorem REF and if $b_{0}>0$ , then $\\lambda \\theta _d(\\beta ) = 1$ .",
"Let $ w_{N} \\frac{b_{0}}{1+s^{c}_{0}} \\frac{1}{\\tilde{a}_{N,N}^{c}|\\Lambda _{N}|}= ^{\\Lambda _N}_{\\beta ,0} \\frac{|T_0|}{|\\Lambda _N|} + O(b_0L^{-\\kappa N}+b_0L^{-(d-2)N}),$ where the second equality is due to Lemma REF .",
"The density $^{\\Lambda _N}_{\\beta ,0} |T_0|/|\\Lambda _N|$ can also be computed by summing the estimate in Lemma REF and dividing by $|\\Lambda _N|$ .",
"Subtracting this result from (REF ) gives $w_N-\\theta _d(\\beta )^2=O(b_0L^{-\\kappa N}).$ On the other hand, (REF ) shows that $\\lambda w_N-\\theta _d(\\beta )= O(b_0L^{-\\kappa N}).$ The limit $w= \\lim _{N\\rightarrow \\infty } w_N$ thus exists and satisfies $\\lambda w = \\theta _d(\\beta )$ and $w=\\theta _d(\\beta )^2$ .",
"Since $\\theta _d(\\beta )=1- O(1/\\beta )\\ne 0$ this implies $\\lambda \\theta _d(\\beta ) = 1$ .",
"The proof follows by rewriting Lemma REF .",
"Let $c_d$ be the constant in the Green function asymptotics of (REF ), and recall the constants $\\theta _d(\\beta )$ and ${\\sf c}_i(\\beta )$ from (REF ).",
"Theorem REF then follows from Lemma REF by setting $\\zeta _d(\\beta )= \\theta _d(\\beta )^2, \\qquad {\\sf c}(\\beta )=(1+s_{0}^{c})2\\lambda \\theta _{d}(\\beta ) c_d,$ and simplifying the error terms using $O(b_0|x|^{-(d-2)-1})+O(b_0^2 |x|^{-(d-2+\\kappa )}) = O(\\beta ^{-1}|x|^{-(d-2+\\kappa )})$ and $O(b_0L^{-(d-2)N})+O(b_0^2 L^{-\\kappa N}) = O(\\beta ^{-1}L^{-\\kappa N})$ .",
"For Theorem REF , $\\zeta _d(\\beta ) =\\theta _d(\\beta )^2$ was established in the previous proof.",
"For Theorem REF , the identity $({\\sf c}_2(\\beta )/{\\sf c}_1(\\beta )^{2})\\theta _{d}(\\beta )^{2}=1$ is equivalent to $\\theta _{d}(\\beta )\\lambda =1$ , i.e., Lemma REF .",
"Similarly, ${\\sf c}(\\beta )=2\\lambda \\theta _{d}(\\beta ) {\\sf c}_1(\\beta ) = 2{\\sf c}_1(\\beta )$ .",
"To compute $¶_{\\beta ,0}[0\\leftrightarrow x]$ we started from the expression $1-{\\xi _0\\eta _0\\xi _x\\eta _x}_{\\beta ,0}$ in (REF ).",
"An alternative route would have been to start from ${\\xi _0\\eta _x}_{\\beta ,0}$ .",
"For technical reasons arising in Section it is, however, easier to obtain sufficient precision when working with ${\\xi _0\\eta _0\\xi _x\\eta _x}_{\\beta ,0}$ ."
],
[
"The bulk renormalisation group flow",
"We will prove Theorems REF and REF by a renormalisation group analysis that is set up following [27], [33] and [13], [14]; see also [17] for a conceptual introduction.",
"Our proof is largely self-contained.",
"The exceptions to self-containment concern general properties about finite range decomposition, norms, and approximation by local polynomials that were developed systematically in [11], [30], [31].",
"The properties we need are all reviewed in this section.",
"The first six subsections set up the framework of the analysis, and the remaining three define and analyse the renormalisation group flow.",
"Throughout $\\Lambda \\Lambda _N$ is the discrete torus of side length $L^N$ .",
"We leave $L$ implicit; it will eventually be chosen large.",
"We sometimes omit the $N$ when it does not play a role."
],
[
"Finite range decomposition",
"Let $\\Delta $ denote the lattice Laplacian on $\\Lambda _N$ , and let $m^2>0$ .",
"Our starting point for the analysis is the decomposition $ C = (-\\Delta +m^2)^{-1} = C_1 + \\cdots +C_{N-1}+C_{N,N}$ where the $C_j$ and $C_{N,N}$ are positive semidefinite $m^2$ -dependent matrices indexed by $\\Lambda _N$ .",
"These covariances can be chosen with the following properties, see [17] and Appendix REF ."
],
[
"Finite range property",
"For $j<N$ , the covariances $C_j$ satisfy the finite range property $ C_j(x,y) = 0 \\qquad \\text{ if } |x-y|_\\infty \\ge \\frac{1}{2} L^{j}.$ Moreover, they are invariant under lattice symmetries and independent of $\\Lambda _N$ in the sense that $C_j(x,y)$ can be identified as function of $x-y$ that is independent of the torus $\\Lambda _N$ .",
"They are defined and continuous for $m^2\\ge 0$ including the endpoint $m^2=0$ (and in fact smooth).",
"The covariances satisfy estimates consistent with the decay of the Green function: $ |\\nabla ^\\alpha C_{j+1}(x,y)| \\le O_{\\alpha ,s}(\\vartheta _j(m^2)) L^{-(d-2+|\\alpha |)j},$ where for an arbitrary fixed constant $s$ , $\\vartheta _j(m^2)=\\frac{1}{2d+m^2}{1+\\frac{m^2L^{2j}}{2d+m^2}}^{-s}.$ The discrete gradient in (REF ) can act on either the $x$ or the $y$ variable, and is defined as follows.",
"Recalling that $e_1, \\dots , e_d$ denote the standard unit vectors generating $^d$ , that $e_{d+j}=-e_j$ , and that $=\\lbrace e_1,\\dots ,e_{2d}\\rbrace $ , for any multiindex $\\alpha \\in _0^{}$ , we define $\\nabla ^\\alpha =\\prod _{i=1}^{2d} \\nabla ^{\\alpha (e_i)}_{e_i}$ as the discrete derivative in directions $\\alpha $ with order $|\\alpha |=\\sum _{i=1}^{2d} \\alpha (e_i)$ .",
"By the above independence of the covariances $C_j$ with $j<N$ from $\\Lambda _N$ , all finite volume torus effects are concentrated in the last covariance $C_{N,N}$ .",
"We further separate this covariance into a bounded part and the zero mode: $ C_{N,N} = C_{N} + t_NQ_N,$ where $t_N$ is an $m^2$ -dependent constant and $Q_N$ is the projection onto the zero mode, i.e., the matrix with all entries equal to $1/|\\Lambda _N|$ .",
"The bounded contribution $C_{N}$ (which does depend on $\\Lambda _N$ ) satisfies the estimates (REF ) with $j=N$ and also extends continuously to $m^2=0$ .",
"The constant $t_N$ satisfies $0 < t_N = \\frac{1}{m^2} - O(L^{2N}) < \\frac{1}{m^2}.$ In this section, we only consider the effect of $C_N$ (which is parallel to that of the $C_j$ with $j<N$ ) while the nontrivial finite volume effect of $t_N$ will be analysed in Sections –.",
"The above properties imply ()."
],
[
"Grassmann Gaussian integration",
"For $X \\subset \\Lambda =\\Lambda _N$ , we denote by $(X)$ the Grassmann algebra generated by $\\psi _x,\\bar{\\psi }_x$ , $x\\in X$ with the natural inclusions $(X) \\subset (X^{\\prime })$ for $X \\subset X^{\\prime }$ .",
"Moreover, we denote by $(X \\sqcup X)$ the doubled algebra with generators $\\psi _x,\\bar{\\psi }_x,\\zeta _x,\\bar{\\zeta }_x$ and by $\\theta \\colon (X) \\rightarrow (X \\sqcup X)$ the doubling homomorphism acting on the generators of $(X)$ by $\\theta \\psi _x = \\psi _x+\\zeta _x, \\qquad \\theta \\bar{\\psi }_x = \\bar{\\psi }_x + \\bar{\\zeta }_x.$ For a covariance matrix $C$ the associated Gaussian expectation $_{C}$ acts on $(X\\sqcup X)$ on the $\\zeta ,\\bar{\\zeta }$ variables.",
"Explicitly, when $C$ is positive definite, $F\\in (X\\sqcup X)$ maps to $_{C}F \\in (X)$ given by $_CF = (\\det C)\\int \\partial _{\\zeta }\\partial _{\\bar{\\zeta }}\\, e^{-(\\zeta ,C^{-1}\\bar{\\zeta })} F.$ Thus $_{C}\\theta \\colon (\\Lambda ) \\rightarrow (\\Lambda )$ is the fermionic convolution of $F \\in (\\Lambda )$ with the fermionic Gaussian measure with covariance $C$ .",
"It is well-known that this convolution operator can be written as $_{C}\\theta F = e^{_{C}}F$ where $_C = \\sum _{x,y\\in \\Lambda } C_{xy} \\partial _{\\psi _y} \\partial _{\\bar{\\psi }_x}$ .",
"In particular, it follows that $_{C}\\theta $ has the semigroup property $ _{C_2}\\theta \\circ _{C_1}\\theta = _{C_1+C_2}\\theta .$ This formula also holds for $C$ positive semidefinite if we take (REF ) as the definition of $_{C}\\theta F$ , which we will in the sequel.",
"See, for example, [30] for the elementary proofs.",
"These identities are fermionic versions of the relation between ordinary Gaussian convolution and the heat equation.",
"In particular, (REF ) allows for the evaluation of moments, e.g., $_{C}\\theta \\bar{\\psi }_{x}\\psi _{y}=\\bar{\\psi }_x\\psi _y+C_{xy}$ .",
"An important consequence of the finite range property (REF ) of $C_j$ is that if $F_i \\in (X_i)$ with $(X_1,X_2)>L^j$ then, by (REF ), $_{C_j}\\theta (F_1F_2) = (_{C_j}\\theta F_1)(_{C_j}\\theta F_2).$"
],
[
"Symmetries",
"We briefly discuss symmetries, which are important in extracting the relevant and marginal contributions in each renormalisation group step (see Section REF below).",
"To use the terminology of [30], [31], we call an element $F \\in (\\Lambda )$ (globally) gauge invariant if every monomial in its representation has the same number of factors of $\\bar{\\psi }$ and $\\psi $ .",
"Some readers may be more familiar with the terminology symplectically invariant or $U(1)$ invariant.",
"Similarly, $F \\in (\\Lambda \\sqcup \\Lambda )$ is gauge invariant if the combined number of factors of $\\bar{\\psi }$ and $\\bar{\\zeta }$ is the same as the combined number of factors of $\\psi $ and $\\zeta $ .",
"We denote by $_{\\rm sym}(X)$ the subalgebra of $(X)$ of gauge invariant elements and likewise for $_{\\rm sym}(\\Lambda \\sqcup \\Lambda )$ .",
"The maps $\\theta $ and $_{C}$ preserve gauge symmetry.",
"A bijection $E\\colon \\Lambda \\rightarrow \\Lambda $ is an automorphism of the torus $\\Lambda $ if it maps nearest neighbours to nearest neighbours.",
"Bijections act as homomorphisms on the algebra $(\\Lambda )$ by $E\\psi _x = \\psi _{Ex}$ and $E\\bar{\\psi }_x = \\bar{\\psi }_{Ex}$ and similarly for $(\\Lambda \\sqcup \\Lambda )$ .",
"If $C$ is invariant under lattice symmetries, i.e., $C(Ex,Ey)=C(x,y)$ for all automorphisms $E$ , then the convolution $_C\\theta $ commutes with automorphisms of $\\Lambda $ , i.e., $E _{C}\\theta F = _{C}\\theta E F$ .",
"In particular $E_{C_{j}}\\theta F = _{C_{j}}\\theta E F$ for the covariances of the finite range decomposition (REF ).",
"An important consequence of this discussion is that if $F\\in _{\\rm sym}(\\Lambda )$ and $F$ is invariant under lattice symmetries, then $_{C_{j}}\\theta F\\in _{\\rm sym}(\\Lambda )$ is also invariant under lattice symmetries."
],
[
"Polymer coordinates",
"We will use (REF ) and the decomposition (REF ) to study the progressive integration $ Z_{j+1} = _{C_{j+1}} \\theta Z_j,$ for a given $Z_{0}\\in (\\Lambda )$ .",
"To be concrete here, the reader may keep $Z_0=e^{-V_0(\\Lambda )}$ with $V_{0}(\\Lambda )$ from (REF ) in mind, but to compute correlation functions we will consider generalisations of this choice of $Z_{0}$ in Section .",
"The analysis is performed by defining suitable coordinates and norms that enable the progressive integration to be treated as a dynamical system: this is the renormalisation group.",
"Towards this end, this section defines local polymer coordinates as in [27], [33].",
"Figure: Illustration of jj-blocks when L=2L=2."
],
[
"Blocks and Polymers",
"Recall $\\Lambda \\Lambda _{N}$ denotes a torus of side length $L^{N}$ .",
"Partition $\\Lambda _N$ into nested scale-$j$ blocks $_j$ of side lengths $L^j$ where $j=0,\\dots , N$ .",
"Thus scale-0 blocks are simply the points in $\\Lambda $ , while the only scale-$N$ block is $\\Lambda $ itself, see Figure REF .",
"The set of $j$ -polymers $_j=_j(\\Lambda )$ consists of finite unions of blocks in $_j$ .",
"To define a notion of connectedness, say $X,Y \\in _j$ do not touch if $\\inf _{x\\in X,y\\in Y} |x-y|_\\infty > 1$ .",
"A polymer is connected if it is not empty and there is a path of touching blocks between any two blocks of the polymer.",
"The subset of connected $j$ -polymers is denoted $_j$ .",
"We will drop $j$ - prefixes when the scale is clear.",
"For a fixed $j$ -polymer $X$ , let $_j(X)$ denote the $j$ -blocks contained in $X$ and let $|_j(X)|$ be the number of such blocks.",
"Connected polymers $X$ with $|_j(X)| \\le 2^d$ are called small sets and the collection of all small sets is denoted $_j$ .",
"Polymers which are not small will be called large.",
"Finally, for $X \\in _j$ we define its small set neighbourhood $X^\\square \\in _j$ as the union over all small sets containing a block in $_j(X)$ , and its closure $\\bar{X}$ as the smallest $Y\\in _{j+1}$ such that $X\\subset Y$ ."
],
[
"Coordinates",
"For coupling constants $V_j=(z_j,y_j,a_j,b_j) \\in 4$ and a set $X \\subset \\Lambda _{N}$ , let $ V_j(X)=\\\\\\sum _{x\\in X} { y_j (\\nabla \\psi )_x (\\nabla \\bar{\\psi })_x + \\frac{z_j}{2}((-\\Delta \\psi )_x\\bar{\\psi }_x + \\psi _x(-\\Delta \\bar{\\psi })_x) + a_j \\psi _x\\bar{\\psi }_x + b_j \\psi _x\\bar{\\psi }_x (\\nabla \\psi )_x(\\nabla \\bar{\\psi })_x}.$ For the scale $j=0$ , if we set $ Z_0 = e^{-V_0(\\Lambda _{N})}$ , then the polymer coordinates take the simple form $ Z_0 = e^{-V_0(\\Lambda _{N})} = e^{-u_{0}|\\Lambda _{N}|}\\sum _{X \\subset \\Lambda _{N}} e^{-V_0(\\Lambda _{N} \\setminus X)} K_0(X), \\quad K_0(X) =1_{X=\\varnothing }, \\quad u_{0}=0.$ To study the recursion $ Z_{j+1}=_{C_{j+1}}\\theta Z_{j}$ at a general scale $j=1,\\dots ,N$ , we will make a choice of coupling constants $V_{j}$ and of polymer coordinates $(K_j(X))_{X\\in _j(X)}$ such that $ Z_{j}= e^{-u_{j}|\\Lambda _{N}|} \\sum _{X\\in _{j}} e^{-V_{j}(\\Lambda _{N}\\setminus X)} K_{j}(X).$ The $K_j(X)$ 's will be defined in such a way that they satisfy the locality and symmetry property $K_j(X) \\in _{\\rm sym}(X^\\square )$ and the following important component factorisation property: for $X,Y\\in _{j}$ that do not touch, $ K_j(X \\cup Y) = K_j(X)K_j(Y).$ Note that since they are gauge symmetric, the $K_j(X)$ are even elements of $$ , so they commute and the product on the right-hand side is unambiguous.",
"Using the previous identity, $K_j(X) = \\prod _{Y \\in \\operatorname{Comp}(X)} K_j(Y),$ where $\\operatorname{Comp}(X)$ denotes the set of connected components of the polymer $X$ .",
"In particular, each $K_j= (K_{j}(X))_{X\\in _{j}(\\Lambda _{N})}$ satisfying (REF ) can be identified with $K_j= (K_{j}(X))_{X\\in _{j}(\\Lambda _{N})}$ .",
"We say that $K_{j}$ is automorphism invariant if $EK_{j}(X) = K_{j}(E(X))$ and all $X\\in _{j}(\\Lambda _{N})$ for all torus automorphisms $E \\in {\\rm Aut}(\\Lambda _N)$ that map blocks in $_j$ to blocks in $_j$ .",
"Let $^\\varnothing _j(\\Lambda _N)$ be the linear space of automorphism invariant $K_j = (K_j(X))_{X\\in _j(\\Lambda _N)}$ with $K_j(X) \\in _{\\rm sym}(X^\\square )$ for every $X \\in _j$ .",
"Polymer coordinates at scale $j$ are thus a choice of $V_j$ together with an element of the space $^\\varnothing _{j}$ .",
"The renormalisation group map is a particular choice of a map $(V_j,K_j)\\mapsto (V_{j+1},K_{j+1})$ .",
"There is freedom in the choice because of the flexibility in the definition of coordinates.",
"The goal is to choose $V_{j}$ such that the size of the $K_{j}$ decrease rapidly as $j$ increases.",
"To achieve this goal we need norms to quantify the sizes of these expressions."
],
[
"Norms",
"We now define the $T_j(\\ell )$ norms we will use on $(\\Lambda )$ .",
"General properties of these norms were systematically developed in [30], to which we will refer for some proofs.",
"To help the reader, in places where we specialise the definitions of [30] we indicate the more general notation that is used in [30].",
"We start with some notation.",
"For any set $S$ , we write $S^*$ for the set of finite sequences in $S$ .",
"We write $\\Lambda _f = \\Lambda \\times \\lbrace \\pm 1\\rbrace $ and for $(x,\\sigma ) \\in \\Lambda _f$ we write $\\psi _{x,\\sigma } = \\psi _x$ if $\\sigma =+1$ and $\\psi _{x,\\sigma }=\\bar{\\psi }_x$ if $\\sigma =-1$ .",
"Then every element $F \\in (\\Lambda )$ can be written in the form $ F = \\sum _{z\\in \\Lambda _f^*} \\frac{1}{z!",
"}F_z \\psi ^z$ where $\\psi ^z = \\psi _{z_1}\\cdots \\psi _{z_n}$ if $z=(z_1,\\dots , z_n)$ .",
"We are using the notation that $z!=n!$ if the sequence $z$ has length $n$ .",
"The representation in (REF ) is in general not unique.",
"To obtain a unique representation we require that the $F_{z}$ are antisymmetric with respect to permutations of the components of $z$ (this is possible due to the antisymmetry of the Grassmann variables).",
"Antisymmetry implies that $F_{z}=0$ if $z$ has length exceeding $2|\\Lambda |$ or if $z$ has any repeated entries.",
"Let $p_{\\Phi }=2d$ .",
"The space of test functions $\\Phi _j(\\ell )$ is defined as the set of functions $g\\colon \\Lambda _f^* \\rightarrow $ , $z\\mapsto g_{z}$ together with norm $ \\Vert g\\Vert _{\\Phi _j(\\ell )} =\\sup _{n\\ge 0}\\sup _{z\\in \\Lambda _f^n} \\sup _{|\\alpha _i| \\le p_\\Phi }\\ell ^{-n}L^{j(|\\alpha _1|+\\cdots |\\alpha _n|)}|\\nabla ^{\\alpha _1}_{z_1}\\cdots \\nabla ^{\\alpha _n}_{z_n} g_z|.$ In this definition, $\\nabla _{z_i}^{\\alpha _i}$ denotes the discrete derivative $\\nabla ^{\\alpha _i}$ with multiindex $\\alpha _i$ acting on the spatial part of the $i$ th component of the finite sequence $z$ .",
"The $\\Phi _j(\\ell )$ norm measures spatial smoothness of test functions, which act as substitutes for fields.",
"Restricted to sequences of fixed length, it is a lattice $C^{p_\\Phi }$ norm at spatial scale $L^j$ and field scale $\\ell $ .",
"We will mainly use the following choice of $\\ell $ for $\\Phi _j(\\ell )$ : $\\ell _j = \\ell _0 L^{-\\frac{1}{2}(d-2)j}$ for a large constant $\\ell _0$ .",
"This choice captures the size of the covariances in the decomposition (REF ).",
"Indeed, since the covariances $C_{j}$ are functions of sequences of length 2, the bounds (REF ) imply $ \\Vert C_{j}\\Vert _{\\Phi _j(\\ell _j)} \\le 1,$ when $\\ell _0$ is chosen as a large ($L$ -dependent, due to the index $j+1$ on the left-hand side of (REF )) constant relative to the constants in (REF ) with $|\\alpha |\\le p_\\Phi $ ; we will always assume that $\\ell _0$ is fixed in this way.",
"In (REF ) we have made a slight abuse of notation to identify $C_{j}$ with the coefficient in (REF ) of $F= \\sum _{x,y} \\bar{\\psi }_{x}\\psi _{y}C_{j}(x,y)$ .",
"We define $T_j(\\ell )$ to be the algebra $ (\\Lambda )$ together with the dual norm $ \\Vert F\\Vert _{T_j(\\ell )} = \\sup _{\\Vert g\\Vert _{\\Phi _j(\\ell )} \\le 1}| {F,g}|,\\qquad \\text{where }{F,g} = \\sum _{z \\in \\Lambda _f^*} \\frac{1}{z!}",
"F_z g_z$ when $F\\in (\\Lambda )$ is expressed as in (REF ).",
"An analogous definition applies to $(\\Lambda \\sqcup \\Lambda )$ and we then write $T_j(\\ell \\sqcup \\ell ) T_j(\\ell )$ for this norm (with the first notation to emphasise the doubled algebra).",
"The $T_j(\\ell )$ norm measures smoothness of field functionals $F\\in (\\Lambda )$ with respect to fields whose size is measured by $\\Phi _j(\\ell )$ .",
"They therefore implement the power counting on which renormalisation relies.",
"Important, but relatively straightforwardly verified, properties of these norms are systematically developed in [30]; we summarise the ones we need now.",
"First, $T_j(\\ell )$ is a Banach algebra, i.e., the following product property holds (see [30]): $ \\Vert F_1F_2\\Vert _{T_j(\\ell )} \\le \\Vert F_1\\Vert _{T_j(\\ell )}\\Vert F_2\\Vert _{T_j(\\ell )}.$ Using the product property, we may gain some intuition regarding these norms by considering the following simple examples: $\\Vert \\psi _x\\bar{\\psi }_x\\Vert _{T_j(\\ell )} \\le \\Vert \\psi _x\\Vert _{T_j(\\ell )} \\Vert \\bar{\\psi }_x\\Vert _{T_j(\\ell )}= \\ell ^2, \\\\\\Vert (\\nabla _e\\psi )_x\\bar{\\psi }_x\\Vert _{T_j(\\ell )}\\le \\Vert \\nabla _e\\psi _x\\Vert _{T_j(\\ell )} \\Vert \\bar{\\psi }_x\\Vert _{T_j(\\ell )} = \\ell ^2 L^{-j}.$ The following more subtle example relies crucially on the antisymmetry of the coefficients $F_z$ and plays an important role for our model: $\\Vert \\psi _x\\bar{\\psi }_x\\psi _{x+e}\\bar{\\psi }_{x+e}\\Vert _{T_j(\\ell )}= \\Vert \\psi _x\\bar{\\psi }_x(\\nabla _e\\psi )_{x}(\\nabla _e\\bar{\\psi })_{x}\\Vert _{T_j(\\ell )}\\asymp \\ell ^4L^{-2j}.$ In general, and bearing in mind the antisymmetry, each factor of the fields contributes a factor $\\ell $ and each derivative a factor $L^{-j}$ .",
"Second, from the definition, the following monotonicity properties hold: for $\\ell \\le \\ell ^{\\prime }$ , $ \\Vert F\\Vert _{T_j(\\ell )} \\le \\Vert F\\Vert _{T_j(\\ell ^{\\prime })}, \\qquad \\Vert F\\Vert _{T_{j+1}(\\ell )} \\le \\Vert F\\Vert _{T_j(\\ell ^{\\prime })}.$ Third, the doubling map satisfies (see [30]): for $F \\in (\\Lambda )$ , $ \\Vert \\theta F\\Vert _{T_j(\\ell )} \\le \\Vert F\\Vert _{T_j(2\\ell )}$ where the norm on the left-hand side is the $T_j(\\ell ) T_j(\\ell \\sqcup \\ell )$ norm on $(\\Lambda \\sqcup \\Lambda )$ .",
"Finally, the following contraction bound for the fermionic Gaussian expectation is an application of the Gram inequality whose importance is well-known in fermionic renormalisation.",
"It is proved in [30].",
"Assume $C$ is a covariance matrix with $\\Vert C\\Vert _{T_j(\\ell )} \\le 1$ .",
"For $F \\in (\\Lambda \\sqcup \\Lambda )$ , then $ \\Vert _{C}F\\Vert _{T_j(\\ell )} \\le \\Vert F\\Vert _{T_j(\\ell )}.$ In particular, by (REF ) the fermionic Gaussian convolution satisfies $ \\Vert _{C} \\theta F\\Vert _{T_{j}(\\ell )}\\le \\Vert F\\Vert _{T_{j}(2\\ell )}.$ For our choices of $\\ell _j$ and of the finite range covariance matrices $C_j$ , the inequalities (REF ) and (REF ) in particular imply $ \\Vert F\\Vert _{T_{j+1}(\\ell _{j+1})} \\le \\Vert F\\Vert _{T_{j+1}(2\\ell _{j+1})} \\le \\Vert F\\Vert _{T_j(\\ell _j)},\\qquad \\Vert _{C_{j+1}}\\theta F\\Vert _{T_{j+1}(\\ell _{j+1})} \\le \\Vert F\\Vert _{T_j(\\ell _j)}.$ We remark that the existence of this contraction estimate for the expectation combined with (REF ) below is what makes renormalisation of fermionic fields much simpler than that of bosonic ones."
],
[
"Localisation",
"To define the renormalisation group map we need one more important ingredient: the localisation operators $\\text{Loc}_{X,Y}$ that will be used to extract the relevant and marginal terms from the $K_j$ coordinate to incorporate them in the renormalisation from $V_j$ into $V_{j+1}$ .",
"These operators are generalised Taylor approximations which take as inputs $F\\in (X)$ and produce best approximations of $F$ in a finite dimensional space of local field polynomials."
],
[
"Local field polynomials",
"Formal local field polynomials are formal polynomials in the symbols $\\psi ,\\bar{\\psi },\\nabla \\psi ,\\nabla \\bar{\\psi },\\nabla ^2\\psi ,...$ (without spatial index).",
"The dimension of a formal local field monomial is given by $(d-2)/2$ times the number of factors of $\\psi $ or $\\bar{\\psi }$ plus the number of discrete derivatives $\\nabla $ in its representation.",
"Concretely, we consider the following space of formal local field polynomials.",
"Let $^\\varnothing \\cong 4$ be the linear space of formal local field monomials of the form $ V= y (\\nabla \\psi ) (\\nabla \\bar{\\psi }) + \\frac{z}{2}((-\\Delta \\psi )\\bar{\\psi }+ \\psi (-\\Delta \\bar{\\psi }))+ a \\psi \\bar{\\psi }+ b \\psi \\bar{\\psi }(\\nabla \\psi )(\\nabla \\bar{\\psi }).$ We will identify elements $V\\in ^\\varnothing $ with their coupling constants $(z,y,a,b) \\in 4$ .",
"Sometimes we include a constant term and write $u+V \\in ^\\varnothing $ with $u+V\\cong (u,z,y,a,b)\\in 5$ .",
"Given a set $X\\subset \\Lambda $ , a formal local field polynomial $P$ can be specialised to an element of $(\\Lambda )$ by replacing formal monomials by evaluations.",
"For example, if $P=\\bar{\\psi }\\psi $ , $P(X) = \\sum _{x\\in X}\\bar{\\psi }_{x}\\psi _{x}$ .",
"We call polynomials arising in this way local polynomials.",
"The most important case is $V \\mapsto V(X)$ , with $ V(X) =\\sum _{x\\in X} { y (\\nabla \\psi )_x (\\nabla \\bar{\\psi })_x + \\frac{z}{2}((-\\Delta \\psi )_x\\bar{\\psi }_x + \\psi _x(-\\Delta \\bar{\\psi })_x) + a \\psi _x\\bar{\\psi }_x + b \\psi _x\\bar{\\psi }_x (\\nabla \\psi )_x(\\nabla \\bar{\\psi })_x},$ where $\\Delta -\\frac{1}{2} \\sum _{e\\in _d} \\nabla _{-e}\\nabla _{e}$ and $(\\nabla \\psi )_{x}(\\nabla \\bar{\\psi })_{x} \\frac{1}{2}\\sum _{e\\in _d}\\nabla _{e}\\psi _{x}\\nabla _{e}\\bar{\\psi }_{x}$ are the lattice Laplacian and the square of the lattice gradient; recall that $_d = \\lbrace e_1, \\dots , e_{2d}\\rbrace $ .",
"For a constant $u\\in we write$ u(X) = u|X|$, where $ |X|$ is the number of points in $ X $.",
"Thus $ (u+V)(X)= u(X)+V(X)=u|X|+V(X)$.$ For $X\\subset \\Lambda $ , define $^\\varnothing (X) = \\lbrace V(X): V\\in ^\\varnothing \\rbrace \\subset (\\Lambda )$ and analogously $(^\\varnothing )(X) = \\lbrace u|X|+V(X): u\\in \\,V \\in ^\\varnothing \\rbrace \\subset (\\Lambda )$ .",
"The space $^\\varnothing $ contains all formal local field polynomials of dimension at most $d$ that are (i) gauge invariant, (ii) respect lattice symmetries (if $EX=X$ for an automorphism $E$ , then $EV(X)=V(X)$ ) and (iii) have no constant terms.",
"Respecting lattice symmetries means that $^\\varnothing $ does not contain terms such as $(\\nabla _{e_1}\\psi )(\\nabla _{e_2}\\bar{\\psi })$ , which cannot occur if $X$ and $V(X)$ are invariant under lattice rotations.",
"We also emphasise that there is no $(\\bar{\\psi }\\psi )^2$ term, which would be consistent with power counting and symmetries, because it vanishes upon specialisation by anticommutativity of the fermionic variables.",
"Two further remarks are in order.",
"First, the monomial $\\psi \\bar{\\psi }(\\nabla \\psi )(\\nabla \\bar{\\psi })$ has dimension $2d-2>d$ for $d\\ge 3$ ; we include it in $^\\varnothing $ since it occurs in the initial potential.",
"Second, the monomials multiplying $z$ and $y$ are equivalent upon specialisation when $X=\\Lambda $ by summation by parts, and differ only by boundary terms for general $X \\subset \\Lambda $ .",
"This would allow us to keep only one of them, but it will be simpler to keep both.",
"The localisation operators $\\operatorname{Loc}_{X,Y}$ associate local field monomials to elements of $(X)$ .",
"In renormalisation group terminology, the image of $\\operatorname{Loc}$ projects onto the space of all relevant and marginal local polynomials.",
"The precise definitions of the localisation operators do not play a direct role in this paper.",
"Rather, only their abstract properties, summarised in the following Proposition REF , will be required.",
"We use the general framework developed in [31] to define these operators.",
"In short, the definitions of $\\operatorname{Loc}_X$ and $\\operatorname{Loc}_{X,Y}$ are those of [31].",
"These definitions require a choice of field dimensions, which we choose as $[\\psi ]=[\\bar{\\psi }] = (d-2)/2$ , a choice of maximal field dimension $d_+$ , which we choose as $d_{+}=d$ , and a choice of a space $\\hat{P}$ of test polynomials, which we define exactly as in [31] with the substitution $\\nabla _e\\nabla _e\\rightarrow -\\nabla _e\\nabla _{-e}$ explained in [31].",
"The following properties are then almost immediate from [31].",
"For $L=L(d)$ sufficiently large there is a universal $\\bar{C}>0$ such that: for $j<N$ and any small sets $Y \\subset X \\in _j$ , the linear maps $\\operatorname{Loc}_{X,Y}\\colon (X^\\square ) \\rightarrow (Y^\\square )$ have the following properties: (i) They are bounded: $ \\Vert \\operatorname{Loc}_{X,Y}F\\Vert _{T_{j}(\\ell _{j})} \\le \\bar{C} \\Vert F\\Vert _{T_j(\\ell _{j})}.$ (ii) The maps $\\operatorname{Loc}_X \\operatorname{Loc}_{X,X}\\colon (X^\\square ) \\rightarrow (X^\\square )$ satisfy the contraction bound $ \\Vert (1-\\operatorname{Loc}_X)F\\Vert _{T_{j+1}(2\\ell _{j+1})} \\le \\bar{C} L^{-d}{L^{-(\\frac{d-2}{2} \\wedge 1)}}\\Vert F\\Vert _{T_j(\\ell _j)}.$ (iii) If $X$ is the disjoint union of $X_1, \\dots , X_n$ then $\\operatorname{Loc}_X = \\sum _{i=1}^n \\operatorname{Loc}_{X,X_i}$ .",
"(iv) The maps are Euclidean invariant: if $E\\in {\\rm Aut}(\\Lambda _{N})$ then $E \\operatorname{Loc}_{X,Y} F = \\operatorname{Loc}_{EX,EY} EF$ .",
"(v) For a block $B$ , small polymers $X_1,\\dots , X_n$ , and any $F_i \\in _{\\rm sym}(X_i^\\square )$ such that $\\sum _{i=1}^n\\operatorname{Loc}_{X_i,B} F_i$ is invariant under automorphisms of $\\Lambda _N$ that fix $B$ , $ \\sum _{i=1}^n \\operatorname{Loc}_{X_i,B}F_i \\in (^\\varnothing )(B).$ Indeed, the bound (i) is [31], the contraction bound (ii) is [31], the decomposition property (iii) holds by the definition of $\\operatorname{Loc}_{X,Y}$ in [31], and the Euclidean invariance (iv) is [31].",
"Note that the parameter $A^{\\prime }$ in [31] does not appear here as it applies to the boson field $\\phi $ ; our fermionic context corresponds to $\\phi =0$ .",
"For the application of [31] we have used that $p_\\Phi $ was fixed to be $2d$ in Definition REF , and that we have only considered the action of $\\operatorname{Loc}$ on small sets.",
"Finally, property (v) follows from the fact that the space $^\\varnothing $ defined in Definition REF contains all local monomials of dimension at most $d$ invariant under lattice rotations.",
"We remark that the image of $\\operatorname{Loc}_{X,Y}$ is a larger space of local field monomials than $^\\varnothing (Y)$ , often denoted $$ in [31].",
"Since we will not use this space directly we have not assigned a symbol for it."
],
[
"Definition of the renormalisation group map",
"The renormalisation group map $\\Phi _{j+1} = \\Phi _{j+1,N}(m^2)$ is an $m^2$ - and $\\Lambda _N$ -dependent map $\\Phi _{j+1}\\colon (V_j,K_j) \\mapsto (u_{j+1},V_{j+1},K_{j+1})$ acting on $V_j \\in ^\\varnothing ,\\qquad K_j \\in ^\\varnothing _j(\\Lambda _N),$ with the space $^\\varnothing $ as in Definition REF and $^\\varnothing _j(\\Lambda _N)$ as in Definition REF .",
"We will identify $V_j \\in ^\\varnothing $ with the tuple $(V_j(B))_{B \\in _j(\\Lambda _N)}$ , and the tuple $(K_j(X))_{X\\in _j(\\Lambda _N)}$ with its extension $(K_j(X))_{X\\in _j(\\Lambda _N)}$ via the component factorisation property (REF ).",
"As indicated above the $u$ -coordinate does not influence the dynamics of the remaining coordinates.",
"Thus we can always explicitly assume that the incoming $u$ -component of $\\Phi _{j+1}$ is 0 and separate it from $V_{j+1}$ in the output.",
"This means that we will often regard $\\Phi _{j+1}$ as the map $(V_j,K_j) \\mapsto (V_{j+1},K_{j+1})$ where $u_j=u_{j+1}=0$ .",
"The precise definition of the map $\\Phi _{j+1}$ is in (REF ) and (REF ) below.",
"The definition is somewhat involved, and the explicit formulas could be skipped for a first reading.",
"One of the essential consequences of these definitions is Proposition REF : this is what enables the iterative analysis of the renormalisation group maps.",
"To define the maps $(K_j,V_j) \\mapsto (u_{j+1},V_{j+1},K_{j+1})$ , we first introduce, assuming $j+1<N$ , $ Q(B) &= \\sum _{X\\in _j: X \\supset B} \\operatorname{Loc}_{X,B}K_j(X), &\\qquad & (B \\in _j), \\\\J(B,B) &= - \\sum _{X\\in _j \\setminus _j: X \\supset B} \\operatorname{Loc}_{X,B}K_j(X), &\\qquad & (B \\in _j),\\\\J(B,X) &= \\operatorname{Loc}_{X,B}K_j(X), &\\qquad & (X \\in _j \\setminus _j, B \\in _j(X)),$ and $J(B,X)=0$ otherwise.",
"If $j+1=N$ we simply set $Q=J=0$ .",
"The map $(K_j,V_j) \\mapsto (u_{j+1},V_{j+1})$ is defined by $ u_{j+1}|B| + V_{j+1}(B) =_{C_{j+1}}\\theta (V_j(B) - Q(B)), \\quad B\\in _{j}.$ Let us emphasise that we evaluate $V_{j+1}$ on $B\\in _j$ here; $V_{j+1}$ can then be extended to $_{j+1}$ by additivity.",
"When $K_{j}$ is automorphism invariant, which is the case if $K_{j}\\in ^\\varnothing _{j}(\\Lambda _{N})$ , the right-hand side of (REF ) is in $(^\\varnothing )(B)$ and can thus be identified with an element of $^\\varnothing \\cong 5$ .",
"This can be checked by using Proposition REF (iv) and (v) and the properties of progressive integration discussed in Section REF .",
"Recall that we sometimes write the left-hand side as $(u+V)_{j+1}(B)$ .",
"Since $V_{j+1}(B)$ has no constant term by definition, the constant $u_{j+1}$ is unambiguously defined.",
"For $U \\in _{j+1}$ , the map $(V_j,K_j) \\mapsto K_{j+1}(U)$ is defined by $ K_{j+1}(U)=e^{u_{j+1}|U|}\\sum _{(,\\check{X}) \\in (U)} e^{-(u+V)_{j+1}(U \\setminus \\check{X} \\cup X_{})} _{C_{j+1}} \\check{K}_j(\\check{X})\\prod _{(B,X) \\in }\\theta J(B,X)$ where $ \\check{K}(X) &= \\prod _{Z \\in \\operatorname{Comp}(X)} \\check{K}(Z),&\\qquad \\check{K}(Z)&=\\sum _{Y \\in _j(Z)} (\\theta K_j(Z\\setminus Y)) (\\delta I)^Y- \\sum _{B\\in _j(Z)} \\theta J(B,Z),\\\\(\\delta I)^X &= \\prod _{B \\in _j(X)} \\delta I(B), &\\qquad \\delta I(B) &= \\theta e^{-V_j(B)}-e^{-(u+V)_{j+1}(B)}.$ The set $(U)$ (and the corresponding notation $$ and $X_$ ) is the same as in [27], and is for convenience indicated in Appendix REF .",
"The following proposition is essentially [27].",
"The only differences are that (i) we have factored out the factor $e^{-u_{j+1}|\\Lambda |}$ and (ii) the doubling map $\\theta $ is explicit (it is implicit in [27]).",
"We have included the proof in Appendix REF .",
"Given $(V_j,K_j)$ define $Z_j$ by (REF ) with $u_{j}=0$ .",
"Suppose $K_{j}$ has the factorisation property (REF ) with respect to $_{j}$ .",
"Then with the above choice of $(V_{j+1},K_{j+1},u_{j+1})$ and $Z_{j+1}$ given by (REF ) with $j+1$ in place of $j$ , we have $Z_{j+1}=_{C_{j+1}}\\theta Z_j$ , and $K_{j+1}$ has the factorisation property (REF ) with respect to $_{j+1}$ .",
"Moreover, if $K_j$ is automorphism invariant then so is $K_{j+1}$ .",
"Proposition REF implies in particular that if $K_{j}$ has the factorisation property (REF ), then we can identify $(K_{j+1}(X))_{X\\in _{j+1}(\\Lambda _{N})}$ with $(K_{j+1}(X))_{X\\in _{j+1}(\\Lambda _{N})}$ .",
"If further $K_{j}$ is automorphism invariant, then $K_{j+1}\\in ^\\varnothing _{j+1}(\\Lambda _{N})$ .",
"By construction and the consistency of the covariances $C_j$ with $j<N$ for different values of $N$ , the maps defined for different $\\Lambda _N$ are also consistent in the following sense: For $j+1<N$ and $U \\in _{j+1}(\\Lambda _N)$ , $V_{j+1}(U)$ and $K_{j+1}(U)$ above depend on $(V_j,K_j)$ only through $V_j(X),K_j(X)$ with $X \\in _j(U^\\square )$ .",
"Moreover, for $U \\in _{j+1}(\\Lambda _N) \\cap _{j+1}(\\Lambda _{M})$ with the natural local identification of $\\Lambda _N$ and $\\Lambda _M$ , the map $(V_j,K_j) \\mapsto (V_{j+1}(U),K_{j+1}(U))$ is independent of $N$ and $M$ .",
"Temporarily indicating the $N$ -dependence of $\\Phi _{j+1}=\\Phi _{j+1,N}$ explicitly, consistency implies the existence of an infinite volume limit $\\Phi _{j+1,\\infty }=\\lim _{N\\rightarrow \\infty } \\Phi _{j+1,N}$ defined for arguments $V_j \\in ^\\varnothing $ and $K_j =(K_j(X))_{X\\in _j(^d)} \\in ^\\varnothing _{j}(^{d})$ .",
"Explicitly, if we write $\\Phi _{j+1,N}(V_{j},K_{j})= (V^{N}_{j+1},K^{N}_{j+1})$ and omit the $N$ for the infinite volume map, $K_{j+1}(U) =\\lim _{N\\rightarrow \\infty } K^{N}_{j+1}(U)$ , and similarly for $V_{j+1}$ .",
"The limits exist as the sequences are constant after finitely many terms.",
"This infinite volume limit does not carry the full information from the $\\Phi _{j+1,N}$ because terms indexed by polymers that wrap around the torus are lost, but it does carry complete information about small sets at all scales and thus about the flow of $V_j$ ."
],
[
"Estimates for the renormalisation group map",
"The renormalisation group map $\\Phi _{j+1}= \\Phi _{j+1,N}$ is a function of $(V,K)\\in ^\\varnothing \\oplus ^\\varnothing _j(\\Lambda _N)$ .",
"The size of $V$ and $K$ will be measured in the norms $\\Vert V\\Vert _j &= \\sup _{B\\in _j} \\Vert V(B)\\Vert _{T_j(\\ell _j)}\\\\\\Vert K\\Vert _j &= \\sup _{X\\in _j} A^{(|\\mathcal {B}_j(X)|-2^d)_+} \\Vert K(X)\\Vert _{T_j(\\ell _j)}$ where $A>1$ is a parameter that will be chosen sufficiently large.",
"Note that $^\\varnothing \\oplus ^\\varnothing _j(\\Lambda _N)$ is a finite-dimensional complex normed vector space with the above norms since $N<\\infty $ .",
"As a consequence it is a Banach space.",
"Let $d \\ge 3$ , $L \\ge L_0(d)$ , and $A \\ge A_0(L,d)$ .",
"Assume that $u_j=0$ .",
"There exists $\\epsilon = \\epsilon (L,A)>0$ such that for $j+1<N$ if $ \\Vert V_j\\Vert _j + \\Vert K_j\\Vert _j \\le \\epsilon $ then $\\Vert u_{j+1} + V_{j+1} - _{C_{j+1}}\\theta V_j\\Vert _{j+1} &\\le O(L^d\\Vert K_j\\Vert _j)\\\\\\Vert K_{j+1}\\Vert _{j+1} &\\le O(L^{-(\\frac{d-2}{2} \\wedge 1)}+A^{-\\eta })\\Vert K_j\\Vert _j+O(A^{\\nu })( \\Vert V_j\\Vert _j^2+\\Vert K_j\\Vert _j^2 ),$ where $\\eta =\\eta (d)$ and $\\nu =\\nu (d)$ are positive geometric constants.",
"The maps $\\Phi _{j+1}$ are entire in $(V_j,K_j)$ and hence all derivatives of any order are uniformly bounded on $\\Vert V_j\\Vert _j+\\Vert K_j\\Vert _j \\le \\epsilon $ .",
"Moreover, the maps $\\Phi _{j+1}$ are continuous in $m^2 \\ge 0$ .",
"The last renormalisation group map $\\Phi _{N}$ satisfies the same bound with $L^{-(\\frac{d-2}{2} \\wedge 1)}$ replaced by 1.",
"The remainder of this section proves Theorem REF , and throughout the rest of this section the hypotheses of Theorem REF will be assumed to hold.",
"Theorem REF is the analogue of [32], [33] for the four-dimensional weakly self-avoiding walk, but much simpler since (i) we are only working with fermionic variables, and (ii) we are above the lower critical dimension (two for our model).",
"The factors $L^d$ and $A^\\nu $ in the error bounds are harmless.",
"On the other hand, it is essential that $O(L^{-(\\frac{d-2}{2} \\wedge 1)}+A^{-\\eta }) < 1$ for $L$ and $A$ large, as this estimate implies that $K$ is irrelevant (contracting) in renormalisation group terminology.",
"The substantive claims of Theorem REF are the estimates (REF ) and (): these quickly yield the claims regarding derivatives by a standard Cauchy estimate, as we now explain.",
"Recall that given two Banach spaces $X$ and $Y$ and a domain $D\\subset we say that a function $ gDX$ isanalytic if it satisfies the Cauchy-Riemann equation$ z g=0$.",
"For an open set $ OX$, we then saythat a function $ FO Y$ is analytic if $ Fg$ isanalytic for every analytic function $ gDX$.",
"After(possibly) adding some additional coordinates to ensure all necessarymonomial are in the domain, the maps$ (Vj,Kj) (Vj+1,Kj+1)$ are multivariate polynomials,and the norm estimates~(\\ref {e:step-V}) and~(\\ref {e:step-K}) extend tothis larger space.",
"Being multivariate polynomials, the $ j+1$are analytic functions.$ We use analyticity and the Cauchy integral formula to extract derivatives.",
"If $(V, K)$ and $(\\dot{V}^{(i)}, \\dot{K}^{(i)})_{i=1}^n$ are collections of polymer coordinates at scale $j$ satisfying $\\Vert V\\Vert _j+\\Vert K_j\\Vert \\le \\epsilon /2$ and $\\Vert \\dot{V}^{(i)}\\Vert _j+\\Vert \\dot{K}^{(i)}\\Vert _j \\le 1$ , then $D^{n} \\Phi _{j+1}|_{(V, K)} (\\dot{V}^{(i)}, \\dot{K}^{(i)})_{i=1}^{n}=\\oint \\cdots \\oint \\prod _{i=1}^k \\frac{ \\textrm {d} w_i}{w_i^2} \\Phi _{j+1}(V+ \\sum _{i=1}^{n} w_i \\dot{V}^{(i)}, K+\\sum _{i=1}^{n} w_i \\dot{K}^{(i)} )$ where the $k$ -tuple of contours are circles around 0 with radius $\\epsilon /(2 n)$ .",
"The statement of Theorem REF regarding boundedness of derivatives follows.",
"Continuity in $m^{2} \\ge 0$ follows from the explicit formulas for $(V_{j+1},K_{j+1})$ , that $\\operatorname{Loc}$ is linear, and that the covariances $C_{j}$ are continuous in $m^{2} \\ge 0$ ."
],
[
"Coupling constants",
"We begin the proof of Theorem REF with the simple bound (REF ) for $V_{j+1}$ .",
"The first term on the right-hand side in the definition (REF ) of $u_{j+1}+V_{j+1}$ produces the expectation term in (REF ).",
"For $B\\in _j$ , the remainder in (REF ) is bounded as follows: $\\Vert Q(B)\\Vert _{T_{j}(\\ell _{j})}&\\le \\sum _{X\\in _j: X \\supset B} \\Vert \\operatorname{Loc}_{X,B} K_j(X)\\Vert _{T_{j}(\\ell _{j})}&\\le O(1)\\sup _{B,X}\\Vert \\operatorname{Loc}_{X,B} K_j(X)\\Vert _{T_{j}(\\ell _{j})}\\le O(1)\\Vert K_j\\Vert _j$ where we have used that the number of small sets containing a fixed block is $O(1)$ in the first step, and (REF ) in the second.",
"Since each block in $_{j+1}$ contains $L^d$ blocks in $_j$ , and using (REF ) to bound the expectation and change of scale in the norm, the first claim (REF ) follows.",
"For the the subsequent bound of $K_{j+1}$ we note that by (REF ) the main term contributing to $u_{j+1}|B|+V_{j+1}(B)$ is bounded by, for $B\\in _j$ , $\\Vert _{C_{j+1}}\\theta V_j(B)\\Vert _{T_{j+1}(\\ell _{j+1})} \\le \\Vert V_j(B)\\Vert _{T_j(\\ell _j)}.$ Combining this with (REF ) we have that, for $B\\in _{j}$ , $u_{j+1}|B| \\le \\Vert V_j\\Vert _j + O( \\Vert K_j\\Vert _j),\\qquad \\Vert V_{j+1}(B)\\Vert _{T_{j+1}(\\ell _{j+1})} \\le \\Vert V_j\\Vert _j + O( \\Vert K_j\\Vert _j).$"
],
[
"Preparation for bound of the non-perturbative coordinate",
"We first separate from $K_{j+1}(U)$ the contribution from summands which are connected.",
"This contribution is: $ _{j+1}(U)\\sum _{X\\in _j: \\bar{X}=U} e^{-V_{j+1}(U\\setminus X)} e^{u_{j+1}|X|} _{C_{j+1}}{\\theta K_j(X) + (\\delta I)^X- \\sum _{B\\in _j} \\theta J(B,X)}.$ Note the sum is over connected polymers.",
"It includes the contribution to $K_{j+1}$ that is linear in $K_{j}$ .",
"We may divide the sum in (REF ) into the contributions from small sets $X \\in _j$ and large sets $X \\in _j\\setminus _j$ .",
"Large sets are easier to handle because they lose combinatorial entropy under change of scale (reblocking), i.e., $|_{j}(X)|$ will be significantly larger than $|_{j+1}(\\bar{X})|$ .",
"In renormalisation group terminology, they are irrelevant.",
"Small sets, on the other hand, require careful treatment."
],
[
"Small sets",
"We begin with the contribution to the expectation in (REF ) from the terms $X \\in _j$ in the sum.",
"By the definition of $J$ in (), for any $X \\in _j \\setminus _j$ , $\\sum _{B \\in _j(X)}_{C_{j+1}}\\theta J(B,X)=\\sum _{B \\in _j(X)}_{C_{j+1}}\\theta \\operatorname{Loc}_{X,B}K_j(X)=_{C_{j+1}}\\theta \\operatorname{Loc}_{X} K_j(X),$ the final equality by Proposition REF (iii).",
"Thus the contribution to (REF ) from $X \\in _j\\setminus _j$ is $ _{C_{j+1}}\\theta (1-\\operatorname{Loc}_X) K(X)+ _{C_{j+1}}(\\delta I)^X .$ The contribution to (REF ) from $X= B \\in _j$ is $ _{C_{j+1}}{\\theta K_j(B)+\\delta I(B)-\\theta J(B,B)}=_{C_{j+1}}\\theta (1-\\operatorname{Loc}_B)K_j(B)+ _{C_{j+1}}(\\delta I(B)+ \\theta Q(B)).$ The next two lemmas bound the collected $K$ and $\\delta I$ terms.",
"For any $U \\in _{j+1}$ , $\\sum _{X \\in _j: \\bar{X}=U} \\Vert _{C_{j+1}}\\theta (1-\\operatorname{Loc}_X)K_j(X)\\Vert _{T_{j+1}(\\ell _{j+1})} =O(L^{-(\\frac{d-2}{2} \\wedge 1)}) \\Vert K_j\\Vert _j.$ Note that $\\bar{X} \\in _{j+1}$ if $X \\in _j$ , so it suffices to prove the lemma for $U\\in _{j+1}$ .",
"Now for any $U \\in _{j+1}$ , since there are $O(L^d)$ small sets $X\\in _j$ such that $\\bar{X}=U$ we get $ \\sum _{X \\in _j: \\bar{X}=U} \\Vert _{C_{j+1}}\\theta (1-\\operatorname{Loc}_X) K_j(X)\\Vert _{T_{j+1}(\\ell _{j+1})}&\\le O(L^d) \\sup _{X \\in _j} \\Vert _{C_{j+1}}\\theta (1-\\operatorname{Loc}_X) K_j(X)\\Vert _{T_{j+1}(\\ell _{j+1})}&\\le O(L^d) \\sup _{X \\in _j} \\Vert (1-\\operatorname{Loc}_X) K_j(X)\\Vert _{T_{j+1}(2\\ell _{j+1})}&\\le O(L^d) O(L^{-d}) (L^{-(\\frac{d-2}{2} \\wedge 1)}) \\sup _{X \\in _j} \\Vert K_j(X)\\Vert _{T_{j}(\\ell _{j})}&\\le O( L^{-(\\frac{d-2}{2} \\wedge 1)}) \\Vert K_j\\Vert _j$ where we have used the contraction estimate for the expectation (REF ) in the second step and the contraction estimate (REF ) for $\\operatorname{Loc}_X$ in the third step.",
"For $B \\in _j$ , $\\Vert _{C_{j+1}}(\\delta I(B) + \\theta Q(B))\\Vert _{T_{j+1}(\\ell _{j+1})} = O(\\Vert V_j\\Vert _j^2 + \\Vert K_j\\Vert _j^2),$ By the definition of $(u+V)_{j+1}$ we have $ _{C_{j+1}}(\\delta I(B)+\\theta Q(B))= _{C_{j+1}}\\theta [e^{-V_j(B)}-1 + V_j(B)] - [e^{-(u+V)_{j+1}(B)}-1+ (u+V)_{j+1}(B)].$ By the product property (REF ), if for some $V$ and some $l$ we have $\\Vert V(B)\\Vert _{T_l(\\ell _l)} \\le 1$ then it follows that $\\Vert e^{-V(B)}-1+V(B)\\Vert _{T(\\ell )} \\le O(\\Vert V(B)\\Vert _{T(\\ell )}^2).$ Recall that $_{C_{j+1}}\\theta $ is contractive as a map from $T_{j+1}(\\ell _{j+1})$ to $T_{j}(\\ell _{j})$ by (REF ) and (REF ).",
"Applying these estimates to the $T_{j+1}(\\ell _{j+1})$ norm of (REF ) and using (REF ) gives the bound (REF ).",
"For $X \\in _j$ , $\\Vert _{C_{j+1}}(\\delta I)^X\\Vert _{T_{j+1}(\\ell _{j+1})} = (O(\\Vert V_j\\Vert _j+\\Vert K_j\\Vert _j))^{|_j(X)|}.$ Using that $_{C_{j+1}}$ satisfies the contraction estimate (REF ), it suffices to show $ \\Vert (\\delta I)^X\\Vert _{T_{j+1}(\\ell _{j+1})}= (O(\\Vert V_j\\Vert _j+\\Vert K_j\\Vert _j))^{|_j(X)|}.$ By the product property (REF ) it suffices to prove this estimate for a single block.",
"In this case, $\\Vert (\\delta I)(B)\\Vert _{T_{j+1}(\\ell _{j+1})}&\\le \\Vert \\theta (e^{-V_j(B)}-1)\\Vert _{T_{j+1}(\\ell _{j+1})}+\\Vert e^{-(u+V)_{j+1}(B)}-1\\Vert _{T_{j+1}(\\ell _{j+1})}&\\le O(\\Vert V_j(B)\\Vert _{T_{j+1}(2\\ell _{j+1})})+O(\\Vert (u+V)_{j+1}(B)\\Vert _{T_{j+1}(\\ell _{j+1})})$ by the product property (REF ) of the norms and (REF ).",
"Using $2\\ell _{j+1} \\le \\ell _j$ and (REF ) for the first term and (REF ) for the second term bounds the right-hand side by $O(\\Vert V_j\\Vert _j + \\Vert K_j\\Vert _j)$ as needed.",
"These lemmas will allow us to estimate the contribution of small sets to (REF ).",
"We need one further general estimate.",
"If $\\Vert K_j\\Vert _j + \\Vert V_j\\Vert _j \\le \\epsilon =\\epsilon (d,L)$ is sufficiently small, then if $\\bar{X} = U$ $\\Vert e^{-V_{j+1}(U\\setminus X)+u_{j+1}|X|}\\Vert _{T_{j+1}(\\ell _{j+1})}\\le 2^{|_{j}(X)|}.$ By the product property (REF ) and (REF ) to bound $V_{j+1}$ and $u_{j+1}$ , $\\Vert e^{-V_{j+1}(U\\setminus X)+u_{j+1}|X|}\\Vert _{T_{j+1}(\\ell _{j+1})} \\le (1+{ O(}\\epsilon ))^{|_{j}(U)|},$ and $|_{j}(U)|$ is at most $L^{d}|_{j+1}(U)|\\le L^{d}|_{j}(X)|$ .",
"The claim follows provided $(1+{ O(}\\epsilon ))^{L^{d}}\\le 2$ .",
"By the product property and Lemma REF , combining Lemma REF with (REF ) for $X\\in _j$ and with (REF ) for $X\\in _{j}\\backslash _j$ we obtain that the contribution of the small sets $X\\in _j$ to the right-hand side of (REF ) is $O(L^{-(\\frac{d-2}{2} \\wedge 1)} \\Vert K_j\\Vert _j+\\Vert V_j\\Vert _j^2 + \\Vert K_j\\Vert _j^2)$ in the $T_{j+1}(\\ell _{j+1})$ norm."
],
[
"Large sets",
"We will need the next combinatorial fact, see [27] or [33].",
"Recall that if $X\\in _{j}$ , then $\\bar{X}\\in _{j+1}$ denotes the smallest $(j+1)$ -polymer containing $X$ .",
"Let $L \\ge 2^d+1$ and $X \\in _j \\setminus _j$ .",
"There is a geometric constant $\\eta =\\eta (d)>0$ depending only on $d$ such that $ |_{j}(X)| \\ge (1+2\\eta ) |_{j+1}(\\bar{X})|.$ By (REF ), if $A=A(L)$ is large enough, $A^{|_{j+1}(U)|} \\sum _{X \\in _j\\setminus _{j}: \\bar{X}=U} (A/2)^{-|_j(X)|}\\le (2^{L^d} 2^{1+2\\eta }A^{-2\\eta })^{|_{j+1}(U)|} \\le A^{-\\eta |_{j+1}(U)|},$ as the set of $X\\in _{j}\\setminus _{j}$ with $\\bar{X}=U$ has size at most $2^{L^{d}|_{j+1}(U)|}$ .",
"Now, the contribution to (REF ) from large sets $X \\in _j \\setminus _j$ is $\\sum _{X\\in _j \\setminus _j: \\bar{X}=U} e^{-V_{j+1}(U\\setminus X)+u_{j+1}|X|} { _{C_{j+1}}\\theta K_j(X)+ _{C_{j+1}} (\\delta I)^X }.$ If $U\\in \\mathcal {C}_{j+1}\\backslash \\mathcal {S}_{j+1}$ , we proceed as follows: For $\\Vert K_j\\Vert _j + \\Vert V_j\\Vert _j \\le \\epsilon $ with $\\epsilon $ sufficiently small, by Lemma REF the $T_{j+1}(\\ell _{j+1})$ norm of the $K$ contribution to (REF ) is bounded by $ A^{|_{j+1}(U)|-2^d}\\sum _{X\\in _j \\setminus _j: \\bar{X}=U} 2^{|_j(X)|} \\Vert _{C_{j+1}}\\theta K_j(X)\\Vert _{T_{j+1}(\\ell _{j+1})}.$ By the definition of $\\Vert K_{j}\\Vert _{j}$ and noting that $(|_{j}(X)|-2^{d})_{+}=|_{j}(X)|-2^{d}$ since $X\\notin _{j}$ , $\\Vert _{C_{j+1}}\\theta K_j(X)\\Vert _{T_{j+1}(\\ell _{j+1})}\\le A^{-(|_{j}(X)|-2^{d})}\\Vert K_j\\Vert _j,$ where we have also used the contraction estimates (REF ), (REF ).",
"Inserting this bound into (REF ) and using (REF ) gives that the $K$ contribution to (REF ) is bounded by $ A^{|_{j+1}(U)|}\\sum _{X\\in _j \\setminus _j: \\bar{X}=U} (A/2)^{-|_j(X)|}\\Vert K_j\\Vert _{j} \\le A^{-\\eta } \\Vert K_j\\Vert _{j}.$ This is the desired bound.",
"Examining the $\\delta I$ contribution to (REF ), Lemmas REF and REF and the product property yield $& A^{|_{j+1}(U)|-2^d} \\Vert \\sum _{X\\in _j \\setminus _j: \\bar{X}=U} e^{-V_{j+1}(U\\setminus X)+u_{j+1}|X|} { _{C_{j+1}} (\\delta I)^X }\\Vert _{T_{j+1}(\\ell _{j+1})}&\\le A^{|_{j+1}(U)|-2^d}\\sum _{X\\in _j \\setminus _j: \\bar{X}=U} [2O(\\Vert V_j\\Vert _j + \\Vert K_j\\Vert _j)]^{|_j(X)|} .$ If $\\Vert V_j\\Vert _j + \\Vert K_j\\Vert _j<\\epsilon $ and $\\epsilon $ is sufficiently small (depending on $A$ ), then the quantity in brackets is less than $1/(4 A^{2})$ .",
"By the elementary inequality $(c^{2})^{n-2}\\le c^{n}$ for $c\\in (0,1)$ , $n>4$ and using that $|_{j}(X)|\\ge 2^{d}+1>4$ for each summand, we obtain the upper bound $[O(\\Vert V_j\\Vert _j + \\Vert K_j\\Vert _j)]^{2} A^{|_{j+1}(U)|} \\sum _{X\\in _j\\setminus _j: \\bar{X}=U} (A/2)^{-|_{j}(X)|}.$ Using again (REF ), it follows that the $\\delta I$ contribution to (REF ) is bounded by $O(A^{-\\eta }[\\Vert V_j\\Vert _j + \\Vert K_j\\Vert _j]^{2})$ for $A$ sufficiently large.",
"We have now completed the bound on (REF ) provided $U\\in \\mathcal {C}_{j+1}\\backslash \\mathcal {S}_{j+1}$ .",
"The argument is similar if $U \\in _{j+1}$ .",
"In this case the prefactor $A^{|_{j+1}(U)|-2^d}$ gets replaced by 1 in (REF ) and (REF ).",
"In place of (REF ) we obtain, since $1+ 2^d\\le |_j(X)|\\le L^d |_{j+1}(U)|\\le [2L]^d$ and the number of summands in this case is at most $2^{[2L]^d}$ , $\\Vert \\sum _{X\\in _j \\setminus _j: \\bar{X}=U} e^{-V_{j+1}(U\\setminus X)+u_{j+1}|X|} _{C_{j+1}}\\theta K_j(X)\\Vert _{T_{j+1}(\\ell _{j+1})}\\le A^{-1}2^{2[2L]^d}\\Vert K_j\\Vert _j=O(A^{-\\eta } \\Vert K_j\\Vert _j)$ for $A$ large enough depending on $L$ and $d$ .",
"On the other hand in place of (REF ) we have $\\Vert \\sum _{X\\in _j \\setminus _j: \\bar{X}=U} e^{-V_{j+1}(U\\setminus X)+u_{j+1}|X|} _{C_{j+1}}\\delta I^X\\Vert _{T_{j+1}(\\ell _{j+1})}\\le O(A^{-\\eta }[\\Vert V_j\\Vert _j + \\Vert K_j\\Vert _j]^{2})$ provided $\\epsilon $ is chosen sufficiently small depending on $L$ , since each summand on the left-hand side has $|_j(X))|\\ge 3$ .",
"Thus for $A=A(L,d)$ sufficiently large and $\\epsilon =\\epsilon (A,L)$ sufficiently small, the expression (REF ) is bounded in the $T_{j+1}(\\ell _{j+1})$ norm by $O(A^{-\\eta }(\\Vert K_j\\Vert _j+[\\Vert V_j\\Vert _j + \\Vert K_j\\Vert _j]^{2}))$ in all cases."
],
[
"Non-linear part",
"To conclude the proof of the theorem, we show $\\Vert K_{j+1}(U)-\\mathcal {L}_{j+1}(U)\\Vert _{j+1}\\le O(\\Vert K_j\\Vert _j^2+\\Vert V_j\\Vert _j^2)$ .",
"Recall the definition of $K_{j+1}$ from (REF ), see also Appendix REF .",
"We have $K_{j+1}(U)-\\mathcal {L}_{j+1}(U)=e^{u_{j+1}|U|} \\sum _{_2(U)}e^{-(u+V)_{j+1}(X_{I})} _{C_{j+1}} \\check{K}_j(\\check{X})\\prod _{(B,X)\\in }\\theta J(B,X) ,$ where $\\sum _{_{2}(U)}$ indicates that the system of polymers $(X_, \\check{X})$ contains at least two components.",
"In particular, if $=\\varnothing $ , $\\check{X}$ has least two components and if $\\check{X}=\\varnothing $ then $||\\ge 2$ , where $||$ is the number of pairs in $$ , see Appendix REF .",
"If $\\Vert V_j\\Vert _j + \\Vert K_j\\Vert _j$ is small enough, arguing as in (REF ) implies $\\Vert e^{u_{j+1}(U)}e^{-(u+V)_{j+1}( X_{I})}\\Vert _{T_{j+1}(\\ell _{j+1})}\\le 2^{|_{j}(U)|}$ .",
"By (REF ), $\\Vert J(B, X)\\Vert _{T_{j}(\\ell _{j})} =O(\\Vert K_j\\Vert _{j})$ .",
"Thus using that $_{C_{j+1}}\\theta $ contracts from $T_{j+1}(\\ell _{j+1})$ into $T_{j}(\\ell _{j})$ we obtain $\\Vert K_{j+1}(U)-\\mathcal {L}_{j+1}(U)\\Vert _{T_{j+1}(\\ell _{j+1})}\\le \\sum _{_2(U)}[O(\\Vert K_j\\Vert _j)]^{||}2^{|_j(U)|} \\Vert \\check{K}_j(\\check{X})\\Vert _{T_{j+1}(\\ell _{j+1})}.$ We first estimate the norm of $\\check{K}_j(\\check{X})$ , and then the resulting sum.",
"If $\\Vert V_{j}\\Vert _{j}+\\Vert K_{j}\\Vert _{j}\\le \\epsilon $ and $\\epsilon =\\epsilon (A, L)$ is sufficiently small, then $\\Vert \\check{K}(\\check{X})\\Vert _{T_{j+1}(\\ell _{j+1})} \\le [O( \\Vert V_j\\Vert _j +\\Vert K_j\\Vert _j)]^{|\\operatorname{Comp}(\\check{X})|}(\\frac{A}{2})^{-\\sum _{Y\\in \\operatorname{Comp}(\\check{X})}(|_j(Y)|-2^{d})_{+}}.$ For notational convenience, for $Y\\in \\mathcal {C}_j$ let $\\tilde{K}(Y)= \\sum _{W \\in _j(Y)} \\theta K_j(Y\\setminus W) (\\delta I)^W.$ The claimed bound follows from the definition of $\\check{K}(X)$ in (REF ) if we show $\\Vert \\tilde{K}(Y)-\\sum _{B \\in _j(Y)} {\\theta } J(B,Y)\\Vert _{T_{j+1}(\\ell _{j+1})} \\le 2^{d+1}O (\\Vert V_j\\Vert _j + \\Vert K_j\\Vert _j) (A/2)^{-(|_j(Y)|-2^{d})_{+}}$ for $Y$ a connected component of $\\check{X}$ .",
"We apply the triangle inequality.",
"Since $J(B,Y)=0$ if $Y\\notin _{j}$ , $\\Vert \\sum _{B \\in _j(Y)}_{C_{j+1}}\\theta J(B,Y)\\Vert _{T_{j+1}(\\ell _{j+1})} \\le O(\\Vert K_{j}\\Vert _{j})$ where we have used $\\Vert J(B, X)\\Vert _{T_{j}(\\ell _{j})}=O(\\Vert K_j\\Vert _{j})$ .",
"By (REF ), component factorisation of $K_j$ , and the contraction property of the norms and $\\theta $ , for $B\\in _j$ and $Z\\in _j$ , $\\Vert \\delta I(B)\\Vert _{T_{j+1}(\\ell _{j+1})}&\\le O(\\Vert V_j\\Vert _j+\\Vert K_j\\Vert _j), \\\\\\Vert \\theta K_j(Z)\\Vert _{T_{j+1}(\\ell _{j+1})} &\\le A^{-\\sum _{W\\in \\operatorname{Comp}(Z)} (|_j(W)|-2^d)_+}\\Vert K_j\\Vert _j^{|\\operatorname{Comp}(Z)|}.$ We now impose the condition that $\\epsilon \\le A^{-2^d}$ .",
"Then plugging the previous bounds into the expression for $\\tilde{K}(Y)$ we have $\\Vert \\tilde{K}(Y)\\Vert _{T_{j+1}(\\ell _{j+1})}& \\le \\sum _{Z \\in _j(Y)}\\Vert (\\delta I)^Z \\theta K_j(Y \\setminus Z)\\Vert _{T_{j+1}(\\ell _{j+1})}&\\le \\sum _{Z \\in _j(Y)}(O(\\Vert V_j\\Vert _{j}+ \\Vert K_j\\Vert _{j}))^{|_j(Z)|+|\\operatorname{Comp}(Y\\backslash Z)|} A^{-\\sum _{W\\in \\operatorname{Comp}(Y\\backslash Z)}(|_j(W)|-2^d)_+}&\\le \\sum _{Z \\in _j(Y)}\\left(A^{2^{d}}O(\\Vert V_j\\Vert _{j}+\\Vert K_j\\Vert _{j})\\right)^{|\\operatorname{Comp}(Y\\backslash Z)|}&\\qquad \\qquad \\qquad \\qquad \\times (O(\\Vert V_j\\Vert _{j}+\\Vert K_j\\Vert _{j}))^{|_j(Z)|}A^{-\\sum _{W\\in \\operatorname{Comp}(Y\\backslash Z)}|_{j}(W)|}&\\le (A^{2^{d}}O(\\Vert V_j\\Vert _{j}+ \\Vert K_j\\Vert _{j}))(O(\\Vert V_j\\Vert _{j}+ \\Vert K_j\\Vert _{j}) + A^{-1})^{|_{j}(Y)|}&\\le A^{2^{d}}\\left(\\frac{A}{2}\\right)^{-|_{j}(Y)|} O(\\Vert V_j\\Vert _{j}+\\Vert K_j\\Vert _{j}).$ (Note that if $Y$ is a small set, the factors of $A$ could have been omitted.)",
"Together with (REF ) this proves the lemma.",
"We are now ready to complete the bound on the non-linear part.",
"For brevity let us write $b$ for the factors $O(\\Vert V_j\\Vert _{j}+\\Vert K_j\\Vert _{j})$ above.",
"We have reduced verification of () to showing $A^{|_{j+1}(U)|}2^{|_{j}(U)|}\\sum _{_{2}(U)} (b2^{d+1}A^{2^{d}})^{||+|\\operatorname{Comp}(\\check{X})|}(\\frac{A}{2})^{-|_{j}(\\check{X})|} \\le O(b^2)$ for all $U\\in _{j+1}$ .",
"Since $|_{j}(U)|\\le L^{d}|_{j+1}(U)|$ , for any $c>0$ the prefactor can be bounded by $A^{|_{j+1}(U)|}2^{|_{j}(U)|} \\le (\\frac{A}{2})^{(1-c)|_{j+1}(U)|}2^{(L^{d}+1)|_{j}(U)|} (\\frac{A}{2})^{c|_{j+1}(U)|}.$ Taking $c>1$ , the product of the first two terms on the last right-hand side is less than 1 for $A$ sufficiently large.",
"It suffices to prove that for some $c>1$ , one has $(\\frac{A}{2})^{c|_{j+1}(U)|}\\sum _{_{2}(U)} (b2^{d+1}A^{2^{d}})^{||+|\\operatorname{Comp}(\\check{X})|}(\\frac{A}{2})^{-|_{j}(\\check{X})|} \\le O(b^2).$ At this point we appeal to [27]; this result estimates the same sum but over $(U)$ instead of $_{2}(U)$ .",
"As we are estimating $\\sum _{_{2}(U)}$ the supremum over $n\\ge 1$ in [27] becomes a supremum over $n\\ge 2$ since $|| + \\operatorname{Comp}(\\check{X})|\\ge 2$ .",
"This yields that if $A=A(L,d)$ is large enough, then there is a $m$ such that for all $U\\in _{j+1}$ $(\\frac{A}{2})^{c|_{j+1}(U)|}\\sum _{_{2}(U)} (b2^{d+1}A^{2^{d}})^{||+|\\operatorname{Comp}(\\check{X})|}(\\frac{A}{2})^{-|_{j}(\\check{X})|} = O( (bA^{m})^{2}),$ which is $A^{\\nu }O(b^{2})$ as needed."
],
[
"Flow of the renormalisation group",
"Recall the infinite volume limit of the renormalisation group maps $\\Phi _{j+1,\\infty }$ discussed below Proposition REF .",
"We equip $^\\varnothing _{j}(^{d})$ with the norm $\\Vert K\\Vert _{j}$ defined by ().",
"Next we study the iteration of the renormalisation group maps as a dynamical system.",
"In what follows $K_0=0$ means $K_0(X)=1_{X=\\varnothing }$ for $X\\in _j$ .",
"Let $d\\ge 3$ , $L\\ge L_0$ , and $A \\ge A_0(L)$ .",
"For $m^2 \\ge 0$ arbitrary and $b_0$ small, there exist $V_0^c(b_0,m^2)$ and $\\kappa >0$ such that if $(V_0,K_0)=(V_0^c(m^2,b_0),0)$ and $(V_{j+1},K_{j+1})= \\Phi _{j+1,\\infty }(V_j,K_j,m^2)$ is the flow of the infinite volume renormalisation group map then $ \\Vert V_j\\Vert _j = O(b_0L^{-\\kappa j}), \\qquad \\Vert K_j\\Vert _j = O(b_0^2 L^{-\\kappa j}).$ The components of $V_0^c(m^2,b_0)$ are continuous and uniformly bounded in $m^2 \\ge 0$ and differentiable in $b_0$ with uniformly bounded derivative.",
"The proof is by applying the well-known stable manifold theorem for smooth dynamical systems in the version given in [27].",
"This theorem applies to dynamical systems in Banach spaces, and hence two preparatory observations are needed.",
"First, for each $j$ , the linear space $^\\varnothing _{j}(^{d})$ equipped with $\\Vert \\cdot \\Vert _{j}$ is complete (when restricted to elements of finite norm), i.e., a Banach space.",
"Second, by the consistency of the finite volume renormalisation group maps (Proposition REF ), the estimates given in Theorem REF also hold for the infinite volume limit.",
"To prove the theorem we first write down the dynamical system corresponding to the renormalisation group map.",
"The definition of $V_{j+1}$ is (REF ).",
"We start with the contribution to $V_{j+1}$ arising from $_{C_{j+1}}\\theta V_{j}(B) (\\tilde{u}_{j+1}|B|+\\tilde{V}_{j+1}(B))$ .",
"This can be computed by the Wick formula (REF ): $\\tilde{z}_{j+1} = z_{j},\\qquad \\tilde{y}_{j+1} = y_{j} + \\kappa _{j}^{yb}b_{j},\\qquad \\tilde{a}_{j+1} = a_{j} + \\kappa _{j}^{ab}b_{j},\\qquad \\tilde{b}_{j+1} = b_{j},$ with $\\kappa _{j}^{yb} = -C_{j+1}(0)$ and $\\kappa _{j}^{ab}=\\Delta C_{j+1}(0)$ .",
"Since $\\Vert V_j(B)\\Vert _{T_j(\\ell _j)}$ is comparable with $|z_j| + |y_j| + L^{2j}|a_j| + L^{-(d-2)j} |b_j|$ , it is natural to rescale $\\hat{z}_{j}=z_{j}$ , $\\hat{y}_{j}=y_{j}$ , $\\hat{a}_{j}=L^{2j}a_{j}$ , $\\hat{b}_{j}=L^{-(d-2)j}b_{j}$ , $\\hat{\\kappa }_{j}^{ab}=L^{dj}\\kappa _{j}^{ab}$ , and $\\hat{\\kappa }_{j}^{yb}=L^{(d-2)j}\\kappa _{j}^{yb}$ .",
"The definition (REF ) of $V_{j+1}$ then becomes $\\hat{z}_{j+1} &= \\hat{z}_{j}+ \\hat{r}^{z}_{j},&\\qquad \\hat{y}_{j+1}&= \\hat{y}_{j} + \\hat{\\kappa }_{j}^{yb}\\hat{b}_{j} + \\hat{r}^{y}_{j}, \\\\\\hat{a}_{j+1}&= L^{2}\\hat{a}_{j} + \\hat{\\kappa }_{j}^{ab}\\hat{b}_{j} + \\hat{r}^{a}_{j}, &\\qquad \\hat{b}_{j+1}&= L^{-(d-2)}\\hat{b}_{j} + \\hat{r}^{b}_{j},$ where the $\\hat{r}_{j}$ are linear maps in $K_j$ and have size $O({K_{j}}_{j})$ by (REF ) of Theorem REF , and the $\\hat{\\kappa }_{j}$ are uniformly bounded in $j$ by the covariance estimates (REF ).",
"We set $v_j=(\\hat{y}_j,\\hat{z}_j,\\hat{a}_j)$ and $w_j=(\\hat{b}_j,K_j)$ and use $\\Vert \\cdot \\Vert $ for the norm given by maximum of the respective components.",
"By the computation above and Theorem REF the infinite volume renormalisation group map can be written in block diagonal form $\\begin{pmatrix}v_{j+1} \\\\ w_{j+1}\\end{pmatrix}= \\begin{pmatrix} E & B_j \\\\ 0 & D_j \\end{pmatrix}\\begin{pmatrix}v_{j} \\\\ w_{j}\\end{pmatrix}+ \\begin{pmatrix} 0 \\\\ g_{j+1}(v_j,w_j) \\end{pmatrix}$ with $\\Vert E^{-1}\\Vert = 1$ , $\\Vert B_j\\Vert $ bounded, and $\\Vert D_j\\Vert \\le \\max \\lbrace L^{-(d-2)}, O(L^{-3}+A^{-\\eta })\\rbrace \\le L^{-\\kappa }$ provided $A$ is large enough, and with $g_j(0,0)=0$ and $Dg_j(0,0)=0$ .",
"For every $m^2\\ge 0$ , the existence of $V_{0}^{c}(m^{2},b_{0})$ and its differentiability in $b_{0}$ now follow by [27].",
"To see that $V_0^c(m^2,b_0)$ is also continuous in $m^2$ , regard $v_j$ and $w_j$ as bounded continuous functions of $m^2$ , i.e., consider $v_j \\in C_b([0,\\infty ),^3)$ and $w_j\\in C_b([0,\\infty ), \\times ^\\varnothing _j(^d))$ .",
"Since all the estimates above are uniform in $m^2 \\ge 0$ , the previous argument then gives a solution $V_0^c \\in C_b((-\\epsilon ,\\epsilon ),C_b([0,\\infty ),^3))$ .",
"The bounds (REF ) are not part of the statement of [27], but are immediate from the proof.",
"By consistency, the finite volume renormalisation group flow for $V_j$ agrees with the infinite volume renormalisation group flow up to scale $j<N$ provided both have the same initial condition.",
"As a result we obtain the following corollary by iterating the recursion () for the $K$ -coordinate using the a priori knowledge that $\\Vert V_j\\Vert _j=O(b_0L^{-\\kappa j})$ due to Theorem REF .",
"Note that while Theorem REF assumes that $u_j=0$ and produces $u_{j+1}$ , it is trivial to extend the statement to $u_j\\ne 0$ by simply adding $u_j$ to the $u_{j+1}$ produced for $u_j=0$ .",
"Under the same assumptions as in Theorem REF , the same estimates hold for the finite volume renormalisation group flow for all $j \\le N$ , and the $V_j$ and $u_j$ produced by the finite volume renormalisation group flow are the same as those for the infinite volume flow when $j<N$ .",
"From this it follows that if $e^{-u_{N}|\\Lambda _N|}$ denotes the total prefactor accumulated in the renormalisation group flow up to scale $N$ , $u_{N}$ is uniformly bounded in $N$ and $m^2$ as $m^2\\downarrow 0$ if we begin with $V_0$ as in Theorem REF .",
"Indeed, up to scale $N-1$ this follows from the bounds (REF ) and (REF ).",
"In passing from the scale $N-1$ to $N$ , the renormalisation group step is $\\Lambda _N$ -dependent, but is nevertheless uniformly bounded by the last statement of Theorem REF ."
],
[
"Computation of the susceptibility",
"In the remainder of the paper, we will use the notation $ {F} = {F}_{V_0} = \\frac{_C(e^{-V_0(\\Lambda )}F)}{_C(e^{-V_0(\\Lambda )})}$ and assume that $(V_j,K_j)_{j=0,\\dots ,N}$ is a renormalisation group flow, i.e., $(V_{j+1},K_{j+1})=\\Phi _{j+1}(V_j,K_j)$ .",
"We begin with the computation of the susceptibility which can be computed directly from the bulk renormalisation group flow.",
"First recall that $Z_0=e^{-V_0(\\Lambda )}$ and that $C = (-\\Delta +m^2)^{-1} = C_1 + \\cdots + C_{N-1} + C_{N,N}, \\qquad C_{N,N} = C_N + t_N Q_N,$ where $\\Delta $ is the Laplacian on $\\Lambda _N$ .",
"Using (REF )–(REF ), with $u_N$ as in (REF ), we then set $ Z_{N,N} = _{t_NQ_N}\\theta Z_N = _C\\theta Z_0,\\qquad \\tilde{Z}_{N,N} = e^{u_N|\\Lambda _N|} Z_{N,N}$ where $_{t_NQ_N}\\theta $ is the fermionic Gaussian convolution with covariance $t_NQ_N$ defined in Section REF .",
"Thus $\\tilde{Z}_{N,N}$ is still a function of $\\psi ,\\bar{\\psi }$ , i.e., an element of $(\\Lambda )$ .",
"Note that $_{C}Z_{0}$ is the constant term of $Z_{N,N}$ .",
"The following technical device of restricting to constant fields $\\psi ,\\bar{\\psi }$ will be useful for extracting information.",
"By restriction to constant $\\psi ,\\bar{\\psi }$ we mean applying the homomorphism from $(\\Lambda )$ onto itself that acts on the generators by $\\psi _x \\mapsto \\frac{1}{|\\Lambda |}\\sum _{x\\in \\Lambda } \\psi _x$ and likewise for the $\\bar{\\psi }_x$ .",
"The result is an element in the subalgebra of $(\\Lambda )$ generated by $\\frac{1}{|\\Lambda |}\\sum _{x\\in \\Lambda } \\psi _x$ and $\\frac{1}{|\\Lambda |}\\sum _{x\\in \\Lambda } \\bar{\\psi }_x$ ; we will simply denote these generators by $\\psi $ and $\\bar{\\psi }$ when no confusion can arise.",
"Restricted to constant $\\psi ,\\bar{\\psi }$ , $ \\tilde{Z}_{N,N}=1+\\tilde{u}_{N,N} - |\\Lambda _N| \\tilde{a}_{N,N}\\psi \\bar{\\psi },\\qquad \\tilde{u}_{N,N} =k_N^0+\\tilde{a}_{N,N}t_N,\\qquad \\tilde{a}_{N,N} = a_N- \\frac{k_N^2}{|\\Lambda _N|}$ where $ k_N^0 = O(\\Vert K_N\\Vert _N), \\qquad k_N^2=O(\\ell _N^{-2}\\Vert K_N\\Vert _N).$ If $V_{0}, K_{0}$ are continuous in $m^{2}\\ge 0$ and $b_{0}$ small enough, then so are $k_{N}^{0}$ and $k^{2}_{N}$ .",
"Since the set of $N$ -polymers $_N(\\Lambda _N)$ is $\\lbrace \\varnothing ,\\Lambda _{N}\\rbrace $ and $e^{u_{N}|\\Lambda _N|}$ is a constant, (REF ) and (REF ) simplify to $\\tilde{Z}_{N,N} = _{t_NQ_N}\\theta (e^{-V_N(\\Lambda _{N})}+K_N(\\Lambda _{N})).$ We now evaluate the integral over the zero mode with covariance $t_NQ_N$ .",
"To this end, we restrict $V_N(\\Lambda _{N})$ and $K_N(\\Lambda _{N})$ to spatially constant $\\psi ,\\bar{\\psi }$ and denote these restrictions by $V_N^0(\\Lambda _{N})$ and $K_N^0(\\Lambda _{N})$ .",
"By anticommutativity, elements of the algebra that depend only on constant $\\psi ,\\bar{\\psi }$ are (noncommutative) polynomials in these generators of degree two.",
"In particular, since $V_{N}^{0}$ and $K_{N}^{0}$ are even, they are of the form $V_N^0(\\Lambda _{N},\\psi ,\\bar{\\psi }) &= |\\Lambda _{N}|a_N \\psi \\bar{\\psi }\\\\K_N^0(\\Lambda _{N},\\psi ,\\bar{\\psi }) &= k_{N}^{0} + k_N^{2} \\psi \\bar{\\psi },$ where the form of $V_N^0$ follows from the representation (REF ).",
"Thus $e^{-V_N^0(\\Lambda _{N})} + K_N^0(\\Lambda _{N})= 1 + k_N^0 - (|\\Lambda _{N}|a_N - k_N^2)\\psi \\bar{\\psi }.$ Therefore applying the fermionic Wick formula $_{t_NQ_N}\\theta \\psi \\bar{\\psi }= -t_N|\\Lambda _{N}|^{-1}+\\psi \\bar{\\psi }$ , we obtain (REF ).",
"The continuity claims for $k_{N}^{0}$ and $k_{N}^{2}$ follow as $(V_j,K_j)$ is a renormalisation group flow (see below (REF )) and since the renormalisation group map has this continuity.",
"The bounds (REF ) follow from the definition of the $T_N(\\ell _N)$ norm.",
"Indeed, since $k_0$ is the constant coefficient of $K_N(\\Lambda _{N})$ , clearly $k_N^0=O(\\Vert K_N\\Vert _N)$ .",
"Similarly, setting $g_{(x,1),(y,-1)} = 1$ for all $x,y\\in \\Lambda _{N}$ and $g_z=0$ for all other sequences, we have $\\Vert g\\Vert _{\\Phi _N(\\ell _N)} = \\ell _N^{-2}$ and $ |k_N^2| = |{K_N(\\Lambda _{N}),g}|\\le \\Vert g\\Vert _{\\Phi _N(\\ell _N)}\\Vert K_N\\Vert _N=\\ell _N^{-2}\\Vert K_N\\Vert _N$ where ${\\cdot ,\\cdot }$ is the pairing from Definition REF .",
"Using the notation of Proposition , $ \\sum _{x\\in \\Lambda _N} {\\bar{\\psi }_0\\psi _x} = \\frac{1}{m^2}- \\frac{1}{m^4}\\frac{\\tilde{a}_{N,N}}{1+\\tilde{u}_{N,N}}.$ We amend the algebra $(\\Lambda _{N})$ by two Grassmann variables $\\sigma $ and $\\bar{\\sigma }$ which we view as constant fields $\\sigma _x=\\sigma $ and $\\bar{\\sigma }_x=\\bar{\\sigma }$ .",
"We then consider the fermionic cumulant generating function (an element of the Grassmann algebra generated by $\\sigma $ and $\\bar{\\sigma }$ ) $ \\Gamma (\\sigma ,\\bar{\\sigma })= \\log _{C}{Z_0(\\psi ,\\bar{\\psi })e^{(\\sigma ,\\bar{\\psi })+(\\psi ,\\bar{\\sigma })}},$ where $C=(-\\Delta +m^2)^{-1}$ and $\\Delta $ is the Laplacian on $\\Lambda _{N}$ .",
"By translation invariance, the susceptibility can then be written as $\\sum _{x\\in \\Lambda _{N}}{\\bar{\\psi }_0\\psi _x}= \\frac{1}{|\\Lambda _{N}|}\\sum _{x,y\\in \\Lambda _N}{\\bar{\\psi }_x\\psi _y}= \\frac{1}{|\\Lambda _{N}|} \\partial _{\\bar{\\sigma }}\\partial _{\\sigma }\\Gamma (\\sigma ,\\bar{\\sigma }).$ The linear change of generators $\\psi \\mapsto \\psi +C\\sigma $ , $\\bar{\\psi }\\mapsto \\bar{\\psi }+ C\\bar{\\sigma }$ yields $\\Gamma (\\sigma ,\\bar{\\sigma })= (\\sigma ,C\\bar{\\sigma }) +\\log _{C}\\theta Z_{0}(C\\sigma ,C\\bar{\\sigma }).$ Since $\\sigma $ is constant in $x\\in \\Lambda _N$ , we have $C\\sigma = m^{-2} \\sigma $ .",
"With (REF ) thus $ \\Gamma (\\sigma ,\\bar{\\sigma })= m^{-2}|\\Lambda _{N}|\\sigma \\bar{\\sigma }+ \\log \\tilde{Z}_{N,N}(m^{-2}\\sigma ,m^{-2}\\bar{\\sigma }) - u_N|\\Lambda _N|.$ As a result, by (REF )–(REF ), $\\frac{1}{|\\Lambda _{N}|}\\partial _{\\bar{\\sigma }} \\partial _{\\sigma }\\Gamma (\\sigma ,\\bar{\\sigma })=\\frac{1}{m^2}- \\frac{1}{m^4}\\frac{\\tilde{a}_{N,N}}{1+\\tilde{u}_{N,N}} .$"
],
[
"The observable renormalisation group flow",
"Recall that ${\\cdot }$ denotes the expectation (REF ), in which we will ultimately choose $V_0=V_0^c(b_0,m^2)$ as in Theorem REF .",
"This section sets up and analyses the renormalisation group flow associated to observable fields.",
"This will enable the computation of correlation functions like ${\\bar{\\psi }_{a}\\psi _{b}}$ in Section .",
"Our strategy is inspired by that used in [13], [67], but with important differences arising due to the presence of a non-trivial zero mode in our setting."
],
[
"Observable coupling constants",
"As in the proofs in Section , we amend the Grassmann algebra by two observable fields.",
"Now, however, the additional fields are not constant in space but rather are localised at two points $a,b\\in \\Lambda $ .",
"For the two point function ${\\bar{\\psi }_a\\psi _b}$ (which we call `Case (1)'), the additional fields $\\sigma _a$ and $\\bar{\\sigma }_b$ are two additional Grassmann variables that anticommute with each other and the $\\psi ,\\bar{\\psi }$ .",
"For the quartic correlation function ${\\bar{\\psi }_a\\psi _a\\bar{\\psi }_b\\psi _b}$ (called `Case (2)'), the additional fields $\\sigma _a$ and $\\sigma _b$ are nilpotent commuting variables, i.e., they commute with each other and the $\\psi ,\\bar{\\psi }$ .",
"For convenience when discussing symmetries, we will assume these variables are realised by introducing two additional Grassmann variables $\\bar{\\vartheta }_x, \\vartheta _x$ at each vertex $x=a,b$ and letting $\\sigma _x=\\bar{\\vartheta }_x\\vartheta _x$ .",
"In both cases we relabel the initial potential $V_0$ from Section 3 by $V^\\varnothing _0$ and set $V_0 = V^\\varnothing _0+V^\\star _0$ where $V^\\star _0$ is an observable part to be defined.",
"In Case (1), $ V^\\star _0 = - \\lambda _{a,0} \\sigma _a \\bar{\\psi }_a 1_{x=a} - \\lambda _{b,0}\\psi _b \\bar{\\sigma }_b 1_{x=b}.$ The indicator functions signal the local nature of the observable fields, i.e., $V^\\star _{0}(X) = - \\lambda _{a,0} \\sigma _a \\bar{\\psi }_a 1_{a\\in X} - \\lambda _{b,0}\\psi _b \\bar{\\sigma }_b 1_{b\\in X}$ .",
"It follows that ${\\bar{\\psi }_a\\psi _b} = \\frac{1}{\\lambda _{a,0}\\lambda _{b,0}}\\partial _{\\bar{\\sigma }_b}\\partial _{\\sigma _a}\\log _{C} {e^{-V_0(\\Lambda )}}$ where we recall from (REF ) that $C=(-\\Delta +m^2)^{-1}$ with $\\Delta $ the Laplacian on $\\Lambda _N$ .",
"Obtaining (REF ) is just a matter of expanding $e^{-V_{0}^{\\star }}$ , using ${\\bar{\\psi }_{a}}={\\psi _{b}}=0$ , and applying the rules of Grassmann calculus.",
"Note the order of $\\partial _{\\bar{\\sigma }_b}$ and $\\partial _{\\sigma _a}$ , which is important to obtain the correct sign.",
"Note that although (REF ) holds for any constants $\\lambda _{a,0}, \\lambda _{b,0}$ , it is convenient for us to leave these as variables to be tracked with respect to the renormalisation group flow.",
"Similarly, in Case (2) we choose $ V^\\star _0 = - \\lambda _{a,0} \\sigma _a \\bar{\\psi }_a\\psi _a 1_{x=a} -\\lambda _{b,0} \\sigma _b\\bar{\\psi }_b\\psi _b 1_{x=b},$ so that ${\\bar{\\psi }_a\\psi _a}&= \\frac{1}{\\lambda _{a,0}} \\partial _{\\sigma _a} \\log _{C} {e^{-V_0(\\Lambda )}} \\Big |_{\\lambda _{b,0}=0}\\\\{\\bar{\\psi }_a\\psi _a\\bar{\\psi }_b\\psi _b}-{\\bar{\\psi }_a\\psi _a}{\\bar{\\psi }_b\\psi _b}&=\\frac{1}{\\lambda _{a,0}\\lambda _{b,0}} \\partial _{\\sigma _b}\\partial _{\\sigma _a} \\log _{C} {e^{-V_0(\\Lambda )}}.$ To distinguish the coupling constants in the two cases, we will sometimes write $\\lambda _{a,0}^{(p)}$ with $p=1$ or $p=2$ instead of $\\lambda _{a,0}$ , and analogously for the other coupling constants."
],
[
"The free observable flow",
"To orient the reader and motivate the discussion which follows, let us first consider the noninteracting case $V^\\varnothing _0=0$ , in which the microscopic model is explicitly fermionic Gaussian.",
"In this case, one may compute all correlations explicitly by applying the fermionic Wick rule.",
"The same computation can be carried out using the finite range decomposition of the covariance $C$ , and we review this now as it will be our starting point for our analysis of the interacting case.",
"To begin the discussion, observe that $\\sigma _a^2 = \\bar{\\sigma }_{b}^{2} = \\sigma _b^2 = 0$ .",
"This implies that $V^\\star _0(\\Lambda )^3=0$ since $V^\\star _{0}(\\Lambda )$ has no constant term and has at least one least observable field in each summand.",
"Given $V^\\star _{0}$ , we inductively define renormalised interaction potentials that share this property: $ u_{j+1}^{\\star }{(\\Lambda )}+ V^\\star _{j+1}(\\Lambda ) _{C_{j+1}}\\theta V^\\star _j(\\Lambda ) - \\frac{1}{2} _{C_{j+1}}(\\theta V^\\star _j(\\Lambda ); \\theta V^\\star _j(\\Lambda ))$ where $_{C_{j+1}}(\\theta V^\\star _j(\\Lambda ); \\theta V^\\star _j(\\Lambda ))_{C_{j+1}}(\\theta V^\\star _j(\\Lambda )^{2})-(_{C_{j+1}}\\theta V^\\star _j(\\Lambda ))^{2}$ and $u_{j+1}^{\\star }{(\\Lambda )}$ collects the terms that do not contain $\\psi $ or $\\bar{\\psi }$ .",
"Consequently, one can check that $_{C_{j+1}}\\theta e^{-V^\\star _j(\\Lambda )}=_{C_{j+1}}\\theta (1-V^\\star _j(\\Lambda ) + \\frac{1}{2} V^\\star _j(\\Lambda )^2)= e^{-u_{j+1}^{\\star }{(\\Lambda )}-V^\\star _{j+1}(\\Lambda )}.$ For convenience, in the last step when $j=N$ , we set $C_{N+1}=t_NQ_N$ .",
"This separation of the zero mode is not essential here but will be useful for our analysis in the interacting case.",
"For $j>0$ , the $V^\\star _j$ have terms not present in $V^\\star _{0}$ , for example the terms involving $q$ in the next definition.",
"The nilpotency of the observable fields $\\sigma _{a}, \\sigma _{b}$ limits the possibilities.",
"Let $^\\star $ be the space of formal field polynomials $u^\\star +V^\\star $ of the form: $&\\left.\\begin{aligned}V^\\star &=- \\lambda _a \\sigma _a \\bar{\\psi }_a - \\lambda _b \\psi _b\\bar{\\sigma }_b+ \\sigma _a \\bar{\\sigma }_b\\frac{r}{2}(\\bar{\\psi }_a\\psi _a+\\bar{\\psi }_b\\psi _b),\\;\\,\\quad \\\\\\nonumber u^\\star &= - { \\sigma _a\\bar{\\sigma }_b } q,\\end{aligned}\\right\\rbrace \\qquad \\text{in Case (1),}\\\\& \\left.\\begin{aligned}V^\\star &={ -}\\sigma _a \\lambda _a\\bar{\\psi }_a\\psi _a - \\sigma _b\\lambda _b\\bar{\\psi }_b\\psi _b{ -}\\sigma _a\\sigma _b\\frac{\\eta }{2} (\\bar{\\psi }_a\\psi _b+\\bar{\\psi }_b\\psi _a)\\\\& \\nonumber \\quad \\quad \\quad \\quad \\quad + \\sigma _a \\sigma _b \\frac{r}{2}(\\bar{\\psi }_a\\psi _a+\\bar{\\psi }_b\\psi _b)),\\\\\\nonumber u^\\star &=- \\sigma _a \\gamma _a- \\sigma _b \\gamma _b- { \\sigma _a\\sigma _b } q ,\\end{aligned}\\right\\rbrace \\qquad \\text{in Case (2),}$ for observable coupling constants $(\\lambda _{a},\\lambda _{b},q, r)\\in {4}$ respectively $(\\lambda _{a},\\lambda _{b},\\gamma _{a},\\gamma _b,q,\\eta , r)\\in {7}$ .",
"For $X \\subset \\Lambda $ , we define $(u^{\\star }+V^\\star )(X) \\in ^\\star (X \\cap \\lbrace a,b\\rbrace )$ by $ (u^\\star +V^\\star )(X) =-\\lambda _a \\sigma _a \\bar{\\psi }_a 1_{a\\in X} -\\lambda _b\\psi _b \\bar{\\sigma }_b 1_{b \\in X} + \\sigma _a \\bar{\\sigma }_b q 1_{a\\in X,b \\in X}+\\sigma _a \\bar{\\sigma }_b \\frac{r}{2}(\\bar{\\psi }_a\\psi _a+\\bar{\\psi }_b\\psi _b)1_{a\\in X,b\\in X}$ in Case (1), and analogously in Case (2).",
"The terms corresponding to $r$ do not appear at any step of the noninteracting iteration (REF ) if we start with them equal to 0.",
"We include them here in preparation for the interacting model.",
"The evolutions of $u^\\star _j+V^\\star _j \\rightarrow u^{\\star }_{j+1}$ and $V^\\star _j \\rightarrow V^\\star _{j+1}$ are equivalent to the evolution of the coupling constants $(\\lambda _a,\\lambda _b,q, r)$ respectively $(\\lambda _a,\\lambda _b,\\gamma _a,\\gamma _b,q,\\eta ,r)$ .",
"By computation of the fermionic Gaussian moments in (REF ), the flow of the observable coupling constants according to (REF ) is then given as follows.",
"Note that the evolution of couplings constants in $V^\\star $ is independent of the coupling constants in $u^\\star $ .",
"Let $V^\\varnothing _0=0$ , and let $u^\\star _{j}$ and $V^\\star _j$ be of the form as in Definition REF .",
"The map (REF ) is then given as follows.",
"In Case (1), for $x\\in \\lbrace a,b\\rbrace $ , $\\lambda _{x,j+1} &=\\lambda _{x,j}\\\\q_{j+1} &= q_j + \\lambda _{a,j}\\lambda _{b,j} C_{j+1}(a,b) + r_jC_{j+1}(0,0)\\\\r_{j+1} &= r_j,\\multicolumn{2}{l}{\\text{whereas in Case (2), for $x\\in \\lbrace a,b\\rbrace $,}}\\\\\\lambda _{x,j+1} &= \\lambda _{x,j}\\\\\\gamma _{x,j+1} &= \\gamma _{x,j} + \\lambda _{x,j}C_{j+1}(0,0)\\\\q_{j+1} &= q_j + \\eta _j C_{j+1}(a,b)+ r_{j} C_{j+1}(0,0)- \\lambda _{a,j}\\lambda _{b,j} C_{j+1}(a,b)^2\\\\\\eta _{j+1} &= \\eta _j - 2\\lambda _{a,j}\\lambda _{b,j} C_{j+1}(a,b)\\\\r_{j+1} & =r_{j}.$ This follows from straightforward evaluation of (REF ) using (REF ).",
"To continue the warm-up for the interacting case, we illustrate how these equations reproduce the direct computations of correlation functions.",
"When $r_{0}=0$ by a computation using Definition REF , the formulas (REF ) and (REF )–() imply the correlation functions in Cases (1) and (2) are given by ${\\bar{\\psi }_a\\psi _b} = \\frac{ q_{N+1}}{\\lambda _{a,0}\\lambda _{b,0}},\\qquad {\\bar{\\psi }_a\\psi _a} = \\frac{\\gamma _{a,N+1}}{\\lambda _{a,0}},\\qquad {\\bar{\\psi }_a\\psi _a;\\bar{\\psi }_b\\psi _b} = \\frac{q_{N+1}}{\\lambda _{a,0}\\lambda _{b,0}},$ with $qq^{(1)}$ and $\\lambda \\lambda ^{(1)}$ for the first equation $qq^{(2)}$ , $\\gamma \\gamma ^{(2)}$ , and $\\lambda \\lambda ^{(2)}$ for the last two.",
"Recalling the convention $C_{N+1}=t_{N}Q_{N}$ , for Case (2) with $r_{0}=0$ we have $q_{N+1}= - \\lambda _{0,a}\\lambda _{0,b}{ \\sum _{k \\le N} C_k(a,b) + t_NQ_N(a,b)}^2 = -\\lambda _{0,a}\\lambda _{0,b}(-\\Delta +m^2)^{-1}(a,b)^2,$ by (REF ).",
"When combined with (REF ) this gives the expected result.",
"In the preceding computation we kept the potential in the exponential for the entire computation, whereas in Sections and the zero mode is integrated out directly without rewriting the integrand in this form (see, e.g., (REF )).",
"We distinguish these two approaches by using $N+1$ subscripts for the former and $(N,N)$ for the latter, and by putting tildes on quantities associated with the $(N,N)$ th step as was done in Section .",
"Before moving to the interacting model, we introduce the coalescence scale $j_{ab}$ as the largest integer $j$ such that $C_{k+1}(a,b)=0$ for all $k < j$ , i.e., $j_{ab} = {\\log _L (2|a-b|_{{\\infty }})}.$ In the degenerate cases $\\lambda _a=0$ or $\\lambda _b=0$ when only one of the observable fields is present we use the convention $j_{ab}=+\\infty $ .",
"Note that the finite range property (REF ) implies that $q_j=\\eta _j=r_{j}=0$ for $j<j_{ab}$ provided they are all 0 when $j=0$ .",
"This will also be true in the interacting case."
],
[
"Norms with observables",
"To extend the above computation for $V^\\varnothing =0$ to the interacting case, we will extend the renormalisation group map to the Grassmann algebra amended by the observable fields.",
"In both Cases (1) and (2), this algebra has the decomposition $ (X) = ^\\varnothing (X) \\oplus ^a(X) \\oplus ^{b}(X) \\oplus ^{ab}(X)= ^\\varnothing (X) \\oplus ^\\star (X)$ where $^\\varnothing (X)$ is spanned by monomials with no factors of $\\sigma $ , $^a(X)$ is spanned by monomials containing a factor $\\sigma _a$ but no factor $\\bar{\\sigma }_b$ (respectively $\\sigma _b$ ), analogously for $^b(X)$ , and $^{ab}(X)$ is spanned by monomials containing $\\sigma _a\\bar{\\sigma }_b$ respectively $\\sigma _a\\sigma _b$ .",
"Thus any $F \\in (X)$ can be written as $ F= F^\\varnothing + F^\\star ={\\left\\lbrace \\begin{array}{ll}F_\\varnothing +\\sigma _aF_a+\\bar{\\sigma }_bF_b+\\sigma _a\\bar{\\sigma }_bF_{ab}, & \\text{Case (1)}\\\\F_\\varnothing +\\sigma _aF_a+\\sigma _bF_b+\\sigma _a\\sigma _bF_{ab}, & \\text{Case (2)},\\end{array}\\right.",
"}$ with $F_\\varnothing ,F_a,F_b,F_{ab} \\in ^\\varnothing (X)$ .",
"We denote by $\\pi _{\\varnothing },\\pi _{a},\\pi _b$ and $\\pi _{ab}$ the projections on the respective components, e.g., $\\pi _aF =\\sigma _aF_a$ , and $\\pi _\\star = \\pi _a+\\pi _b+\\pi _{ab}$ .",
"We will use superscripts instead of subscripts in the decomposition when the factors of $\\sigma $ are included, e.g., $F^a= \\sigma _aF_a$ and $F^\\varnothing =F_\\varnothing $ .",
"We say that $F$ is gauge invariant if the number of generators with a bar is equal to the number without a bar.",
"Explicitly, in Case (1) this means $F_{\\varnothing } $ and $F_{ab}$ are gauge invariant, $F_{a}$ has one more factor with a bar than without, and similarly for $F_{b}$ .",
"In Case (2) this means all of $F_{\\varnothing },F_{a},F_{b}$ and $F_{ab}$ are gauge invariant.",
"Denote by $_{\\rm sym}(X)$ the subalgebra of gauge invariant elements.",
"For $F$ decomposed according to (REF ) we define $ \\Vert F\\Vert _{T_j(\\ell _j)}= \\Vert F_\\varnothing \\Vert _{T_j(\\ell _j)}+ \\ell _{a,j}\\Vert F_a\\Vert _{T_j(\\ell _j)}+ \\ell _{b,j}\\Vert F_b\\Vert _{T_j(\\ell _j)}+ \\ell _{ab,j}\\Vert F_{ab}\\Vert _{T_j(\\ell _j)}$ where $ \\ell _{a,j}= \\ell _{b,j}= {\\left\\lbrace \\begin{array}{ll}\\ell _{j}^{-1}, & \\text{Case (1)}\\\\\\ell _j^{-2},& \\text{Case (2)},\\end{array}\\right.",
"}\\qquad \\ell _{ab,j}= {\\left\\lbrace \\begin{array}{ll}\\ell _{j}^{-2}, & \\text{Case (1)}\\\\\\ell _j^{-2} \\ell ^{-2}_{j\\wedge j_{ab}}, & \\text{Case (2)}.\\end{array}\\right.",
"}$ In particular, $\\Vert \\sigma _a\\Vert _{T_j(\\ell _j)} = \\ell _{a,j}$ and $\\Vert \\sigma _a\\sigma _b\\Vert _{T_j(\\ell _j)} = \\ell _{ab,j}$ and, in Cases (1) and (2), respectively, $ \\Vert \\sigma _a\\bar{\\psi }_a\\Vert _{T_j(\\ell _j)}= \\ell _{a,j}\\ell _j = 1,\\qquad \\Vert \\sigma _a\\bar{\\psi }_a\\psi _a\\Vert _{T_j(\\ell _j)} = \\ell _{a,j}\\ell _j^2 =1$ and, again in the two cases respectively, $ \\Vert \\sigma _a{\\bar{\\sigma }_b}\\bar{\\psi }_x\\psi _x\\Vert _{T_j(\\ell _j)} =\\ell _{ab,j}^2\\ell _j^2 = 1, \\qquad \\Vert \\sigma _a\\sigma _b\\bar{\\psi }_x\\psi _x\\Vert _{T_j(\\ell _j)} =\\ell _{ab,j}^2\\ell _j^2 = \\ell ^{-2}_{j\\wedge j_{ab}}.$ In both cases these terms do not change size under change of scale, provided that $j\\ge j_{ab}$ for the last term.",
"Thus they are marginal.",
"As will be seen in Section , the choices of $\\ell _{a,j}$ and $\\ell _{ab,j}$ are appropriate to capture the leading behaviour of the corresponding correlation functions.",
"The extended definition (REF ) of the $T_{j}(\\ell _{j})$ norm satisfies the properties discussed in Section REF , with the exception of the monotonicity estimate $\\Vert F_\\varnothing \\Vert _{T_{j+1}(2\\ell _{j+1})} \\le \\Vert F_\\varnothing \\Vert _{T_{j}(\\ell _{j})}$ .",
"Checking these properties is straightforward by using the properties of the bulk norm, and, in the case of the product property, using that $\\ell _{ab,j}\\le \\ell _{a,j}\\ell _{b,j}$ .",
"Similar reasoning also yields a weaker monotonicity-type estimate: by (REF ) and monotonicity in the bulk algebra, $\\Vert F\\Vert _{T_{j+1}(\\ell _{j+1})}\\le \\Vert F\\Vert _{T_{j+1}(2\\ell _{j+1})}\\le 16 L^{2(d-2)} \\Vert F\\Vert _{T_{j}(\\ell _{j})} .$"
],
[
"Localisation with observables",
"We combine the space $^\\varnothing $ of bulk coupling constants from Definition REF with the space $^\\star $ of observable coupling constants from Definition REF into $= ^\\varnothing \\oplus ^\\star .$ We extend the localisation operators $\\operatorname{Loc}_{X,Y}$ from Section REF to the amended Grassmann algebra (REF ) as follows.",
"As in the bulk setting, we will focus on the key properties of the extended localisation operators.",
"The extension of $\\operatorname{Loc}_{X,Y}$ is linear and block diagonal with respect to the decomposition (REF ), and so can be defined separately on each summand.",
"On $^\\varnothing (X)$ , the restriction $\\operatorname{Loc}_{X,Y}$ is defined to coincide with the operators from Proposition REF .",
"From now on we denote this restriction by $\\operatorname{Loc}_X^\\varnothing $ or $\\operatorname{loc}^{\\varnothing }_{X}$ if we want to distinguish it from the extended version.",
"To define the restriction $\\operatorname{Loc}^{\\star }_{X,Y}$ of $\\operatorname{Loc}_{X,Y}$ to $^\\star (X)$ , we continue to employ the systematic framework from [31].",
"Namely, in Case (1), for $\\sigma _a F_a\\in ^a(X)$ we set $\\operatorname{Loc}_{X, Y} (\\sigma _a F_a) = \\sigma _a \\operatorname{loc}^a_{X\\cap \\lbrace a\\rbrace , Y\\cap \\lbrace a\\rbrace } F_a$ and likewise for point $b$ , and for $\\sigma _a\\bar{\\sigma }_b F_{ab} \\in ^{ab}(X)$ we set $\\operatorname{Loc}_{X, Y} (\\sigma _a\\bar{\\sigma }_b F_{ab}) = \\sigma _a \\bar{\\sigma }_b \\operatorname{loc}^{ab}_{X\\cap \\lbrace a,b\\rbrace , Y\\cap \\lbrace a,b\\rbrace } F_{ab}$ where $\\operatorname{loc}^a$ , $\\operatorname{loc}^b$ , and $\\operatorname{loc}^{ab}$ are the localisation operators from [31] with respective maximal dimensions $ d_+^{a} = d_+^b = \\frac{p}{2}(d-2),\\qquad d_+^{ab} = d-2$ and $p=1$ .",
"Case (2) is defined analogously but with $\\bar{\\sigma }_{b}$ replaced by $\\sigma _{b}$ and with $p=2$ in the choice of $d_+^a=d_+^b$ .",
"The superscripts $\\varnothing ,a,b,ab$ are present to indicate that we have assigned different maximal dimensions to the summands in (REF ).",
"We use the same choice of field dimensions $[\\psi ]=[\\bar{\\psi }]=(d-2)/2$ as in Section REF .",
"We note that $\\operatorname{loc}^a_{X, \\varnothing }=\\operatorname{loc}^b_{X,\\varnothing }=\\operatorname{loc}^{ab}_{X,\\varnothing }=0$ .",
"The main difference between these operators and $\\operatorname{Loc}^\\varnothing $ is that the expressions produced by $\\operatorname{loc}^a$ , $\\operatorname{loc}^b$ , $\\operatorname{loc}^{ab}$ are local, i.e., supported near $a$ and $b$ .",
"A second difference is that the maximal dimensions vary.",
"The next proposition summarises the key properties of the operators $\\operatorname{Loc}_{X,Y}$ .",
"As with Proposition REF , these properties follow from [31].",
"That the choice of maximal dimensions (REF ) produce contractive estimates can intuitively be understood by considering the marginal monomials.",
"By (REF ) and (REF ), these are exactly the monomials with dimensions $d_+^a=d_+^b$ respectively $d_+^{ab}$ .",
"For $L=L(d)$ sufficiently large there is a universal $\\bar{C}>0$ such that: for $j<N$ and any small sets $Y \\subset X \\in _j$ , the linear maps $\\operatorname{Loc}_{X,Y}^\\star \\colon ^\\star (X^\\square ) \\rightarrow ^\\star (Y^\\square )$ have the following properties: (i) They are bounded: $ \\Vert \\operatorname{Loc}_{X,Y}^\\star F\\Vert _{T_{j}(\\ell _{j})} \\le \\bar{C} \\Vert F\\Vert _{T_j(\\ell _{j})}.$ (ii) For $j\\ge j_{ab}$ , the maps $\\operatorname{Loc}_X^{\\star } \\operatorname{Loc}_{X,X}^{\\star }\\colon ^{\\star }(X^\\square ) \\rightarrow ^{\\star }(X^\\square )$ satisfy the contraction bound: $ \\Vert (1-\\operatorname{Loc}_X^\\star )F\\Vert _{T_{j+1}(2\\ell _{j+1})} \\le \\bar{C} L^{-(\\frac{d-2}{2} \\wedge 1)}\\Vert F\\Vert _{T_j(\\ell _j)}.$ Moreover, the bound (REF ) holds also for $j<j_{ab}$ if $F^{ab}=0$ .",
"(iii) If $X$ is the disjoint union of $X_1, \\dots , X_n$ then $\\operatorname{Loc}_X^{\\star } = \\sum _{i=1}^n \\operatorname{Loc}_{X,X_i}^{\\star }$ .",
"(iv) For a block $B$ and polymers $X \\supset B$ , $\\operatorname{Loc}_{X,B}^\\star F \\in ^\\star (B)$ if $F \\in ^{\\star }_{\\rm sym}(X)$ .",
"Properties (i)–(iii) follow from [31] in the same way as corresponding properties in Proposition REF by making use of the observation that $\\Vert \\sigma _a\\Vert _{T_{j+1}(2\\ell _{j+1})}&\\le 2L^{d_+^a}\\Vert \\sigma _a\\Vert _{T_{j}(\\ell _{j})}, \\qquad \\Vert \\sigma _b\\Vert _{T_{j+1}(2\\ell _{j+1})} \\le 2L^{d_+^b}\\Vert \\sigma _b\\Vert _{T_{j}(\\ell _{j})}, \\\\\\Vert \\sigma _{a}\\sigma _b\\Vert _{T_{j+1}(2\\ell _{j+1})}&\\le 4L^{d_+^{ab}}\\Vert \\sigma _{a}\\sigma _b\\Vert _{T_{j}(\\ell _{j})}, \\qquad \\text{if $j\\ge j_{ab}$}$ in Case 2 and analogously in Case 1.",
"These factors of $L^{d_+}$ correspond to the missing $L^{-d_+}$ factors in (REF ) as compared to Proposition REF .",
"It only remains to verify (iv), i.e., to identify the image of $\\operatorname{Loc}_{X,B}^\\star $ when acting on $F \\in ^{\\star }_{\\rm sym}(X)$ .",
"Case (1).",
"By the choice of dimensions in its specification, the image of $\\sigma _{a}\\operatorname{loc}^{a}$ is spanned by the local monomials $\\sigma _{a}, \\sigma _{a}\\bar{\\psi }_{a}$ , $\\sigma _a\\psi _{a}$ .",
"The condition of gauge invariance then implies that if $\\sigma _a F_a\\in ^{a}_{\\rm sym}(X)$ only the monomial $\\sigma _{a}\\bar{\\psi }_{a}$ is admissible.",
"The situation is analogous for $\\operatorname{loc}^b$ .",
"Similarly, $\\sigma _{a}\\bar{\\sigma }_{b}\\operatorname{loc}^{ab}$ has image spanned by $\\sigma _{a}\\bar{\\sigma }_{b}$ and $\\sigma _{a}\\bar{\\sigma }_{b}\\bar{\\psi }_x\\psi _x$ for $x\\in \\lbrace a,b\\rbrace $ as well as further first order monomials with at most $(d-2)/2$ gradients, e.g., $\\sigma _a\\bar{\\sigma }_b\\nabla _{e_{1}}\\psi _x$ .",
"Only the even monomials $\\sigma _{a}\\bar{\\sigma }_{b}$ , $\\sigma _{a}\\bar{\\sigma }_{b}\\bar{\\psi }_{a}\\psi _{a}$ , and $\\sigma _{a}\\bar{\\sigma }_{b}\\bar{\\psi }_{b}\\psi _{b}$ are compatible with symmetry.",
"In summary, $\\operatorname{Loc}^\\star _{X,Y} F$ is contained in $^\\star $ if $F\\in ^{\\star }_{\\rm sym}(X)$ .",
"Case (2).",
"By the choice of dimensions, in this case $\\sigma _{a}\\operatorname{loc}^a$ has image spanned by the local monomials $\\sigma _{a},\\sigma _{a}\\bar{\\psi }_a\\psi _a$ as well as further first order monomials with at most $d-2$ gradients, and symmetry implies that only the even terms $\\sigma _{a}$ and $\\sigma _{a}\\bar{\\psi }_a\\psi _a$ arise in the image if $F \\in _{\\rm sym}(X)$ .",
"The analysis for $\\sigma _{a}\\operatorname{loc}^a$ is analogous, and $\\sigma _{a}\\sigma _{b}\\operatorname{loc}^{ab}$ has image spanned by $\\sigma _{a}\\sigma _{b}$ and the monomials $\\sigma _{a}\\sigma _{b}\\bar{\\psi }_x\\psi _x$ for $x\\in \\lbrace a,b\\rbrace $ and first order monomials with at most $(d-2)/2$ gradients.",
"Again only the even monomials are compatible with symmetry."
],
[
"Definition of the renormalisation group map with\nobservables",
"In this section the renormalisation group map $\\Phi _{j+1} =\\Phi _{j+1,N}$ is extended to include the observable components.",
"To this end, we now call the renormalisation group map from Section REF the bulk component and denote it by $\\Phi ^\\varnothing _{j+1}$ , and $\\Phi _{j+1}=(\\Phi ^\\varnothing _{j+1},\\Phi ^\\star _{j+1})$ will now refer to the renormalisation group map extended to the algebra with observables.",
"The map $\\Phi ^\\star _{j+1}$ is the observable component of the renormalisation group map.",
"This extension will be defined so that the bulk components of $K_{j+1}$ and $V_{j+1}$ only depend on the bulk components of $K_j$ and $V_j$ .",
"In other words, $\\pi _\\varnothing \\Phi _{j+1}(V_j,K_j) = \\Phi _{j+1}^\\varnothing (\\pi _\\varnothing V_j,\\pi _\\varnothing K_j).$ On the other hand, the observable components $V^\\star _{j+1}$ and $K^\\star _{j+1}$ will depend on both the observable and the bulk components of $(V_j,K_j)$ .",
"The observable component $\\Phi ^{\\star }_{j+1}$ is upper-triangular in the sense that the $a$ component of $\\Phi ^{\\star }_{j+1}(V_j,K_j)$ only depends on $(V^\\varnothing _{j},K^\\varnothing _{j})$ and $(V^a_{j},K^a_{j})$ but not on $(V^b_{j},K^b_{j})$ or $(V^{ab}_{j},K^{ab}_{j})$ , and similarly for the $b$ component.",
"The $ab$ component depends on all components from the previous scale.",
"We will use an initial condition $V_{0}\\in $ and $K_{0}(X)=1_{X=\\varnothing }$ as in (REF ).",
"We now give the precise definition of the observable component of the renormalisation group map $\\Phi _{j+1}^\\star \\colon (V_j,K_j)\\mapsto (u^\\star _{j+1}, V^\\star _{j+1}, K^\\star _{j+1})$ .",
"For $j+1<N$ , given $(V_j, K_j)$ and $B \\in _j$ , define $Q(B)$ and $J(B,X)$ as in (REF )–() using the extended version of $\\operatorname{Loc}$ from Section REF .",
"If $j+1=N$ set $Q=J=0$ .",
"We let $Q^\\star (B)=\\pi _\\star Q(B) $ and $J^\\star (B,X)=\\pi _\\star J(B, X)$ denote the observable components.",
"The new detail for the observable renormalisation group map is that, to define $V^\\star _{j+1}$ , we include the second order contribution from $V^\\star _{j}$ in order to maintain better control on the renormalisation group flow.",
"To this end, for $j+1\\le N$ and $B,B^{\\prime }\\in _{j}$ , let $ \\begin{split}P^\\star (B,B^{\\prime })&=\\frac{1}{2}_{C_{j+1}}(\\theta (V^\\star _j(B)-Q^\\star (B)); \\theta (V^\\star _j(B^{\\prime })-Q^\\star (B^{\\prime }))),\\\\P^\\star (B)&= \\sum _{B^{\\prime } \\in _j} P(B,B^{\\prime }).\\end{split}$ If $(B,B^{\\prime }) > L^{j+1}$ then $P^\\star (B,B^{\\prime })=0$ , by the finite range of the covariance $C_{j+1}$ .",
"As a result, $P^{\\star }(B)\\in ^{\\star }(B^\\square )$ .",
"With these definitions in place, $ u^\\star _{j+1}+V^\\star _{j+1}$ is defined in the same way as $ u_{j+1}+ V_{j+1}$ with the addition of the second order term $P^\\star $ , and $K^\\star _{j+1}$ is then defined in the same way as $K_{j+1}$ : The map $(V_j,K_j) \\mapsto ({ u^\\star _{j+1}},V^\\star _{j+1})$ is defined, for $B\\in _j$ , by $u^\\star _{j+1}(B)+V^\\star _{j+1}(B) =_{C_{j+1}}\\theta (V^\\star _j(B) - Q^\\star (B)) - P^\\star (B)$ where $u^\\star _{j+1}$ consists of all monomials that do not contain factors of $\\psi $ or $\\bar{\\psi }$ .",
"Explicitly, $u^\\star _{j+1}={\\left\\lbrace \\begin{array}{ll}-\\sigma _a\\bar{\\sigma }_b q_{j+1}, &\\text{Case (1)}, \\\\-\\sigma _a\\sigma _b q_{j+1} - \\sigma _a \\gamma _{a,j+1}-\\sigma _b\\gamma _{b,j+1}, &\\text{Case (2)},\\end{array}\\right.",
"}$ The map $(V_j,K_j)\\mapsto K^\\star _{j+1}$ is defined as in Definition REF except that $V^\\varnothing $ and $u^\\varnothing $ are replaced by $V=V^\\varnothing +V^\\star $ and $u=u^\\varnothing +u^\\star $ .",
"Propositions REF and REF also hold for this extended definition of the renormalisation group map.",
"The proofs are the same; for Proposition REF we provide a proof in Appendix REF ."
],
[
"Estimates for the renormalisation group map with observables",
"In this section, the $O$ -notation refers to scale $j+1$ norms, i.e., for $F,G \\in (\\Lambda )$ , we write $F=G+O(t)$ to denote that $\\Vert F-G\\Vert _{T_{j+1}(\\ell _{j+1})} \\le O(t)$ .",
"Under the assumptions of Theorem REF , if also $\\Vert V^\\star _j\\Vert _j+\\Vert K^\\star _j\\Vert _j \\le \\epsilon $ and $u^\\star _j=0$ , then for $j+1<N$ the observable components of the renormalisation group map $\\Phi ^\\star _{j+1}$ satisfy $u^\\star _{j+1}(\\Lambda ) +V^\\star _{j+1}(\\Lambda )= _{C_{j+1}}\\theta V^\\star _j(\\Lambda ) - \\frac{1}{2} _{C_{j+1}}(\\theta V^\\star _j(\\Lambda );\\theta V^\\star _j(\\Lambda ))+ O(L^{2(d-2)}\\Vert K^\\star _j\\Vert _j)\\\\\\Vert K^\\star _{j+1}\\Vert _{j+1} \\le O(L^{-(\\frac{d-2}{2} \\wedge 1)}+A^{-\\eta })\\Vert K^\\star _j\\Vert _j + O(A^{\\nu }) (\\Vert V^\\varnothing _j\\Vert _j +\\Vert K^\\varnothing _{j}\\Vert _j+\\Vert K^\\star _{j}\\Vert _{j})(\\Vert V^\\star _j\\Vert _j+\\Vert K^\\star _j\\Vert _j),$ provided that $K^{ab}_j(X)=0$ for $X\\in _j$ if $j<j_{ab}$ .",
"Both $\\eta =\\eta (d)$ and $\\nu =\\nu (d)$ are positive geometric constants.",
"For $j+1=N$ , $\\Phi ^\\star _{N}$ is bounded.",
"The proof of the theorem follows that of Theorem REF closely, with improvements for the leading terms that allow for $V^\\star $ to be tracked to second order."
],
[
"Coupling constants",
"We first give a bound on $u^{\\star }_{j+1}(\\Lambda )+ V^\\star _{j+1}(\\Lambda )$ .",
"By Proposition REF (iii), $Q^{\\star }(\\Lambda ) =\\sum _{X\\in _{j}}\\operatorname{Loc}^{\\star }_{X}K_{j}(X).$ Since only small sets $X$ that contain $a$ or $b$ contribute, Proposition REF (i), (REF ), and (REF ) imply $\\Vert Q^{\\star }(\\Lambda )\\Vert _{T_{j}(\\ell _{j})}\\le O(1)\\Vert K^\\star _j\\Vert _{j}.$ By algebraic manipulation, the product property, that $_{C_{j+1}}\\theta $ is a contraction, (REF ), and (REF ), $P^{\\star }(\\Lambda ) &= \\frac{1}{2}_{C_{j+1}}(\\theta V^{\\star }_{j}(\\Lambda );\\theta V^{\\star }_{j}(\\Lambda ))+_{C_{j+1}}(\\theta Q^{\\star }_{j}(\\Lambda );\\theta (V^{\\star }_{j}(\\Lambda )+Q^\\star _j(\\Lambda )))&= \\frac{1}{2}_{C_{j+1}}(\\theta V^{\\star }_{j}(\\Lambda );\\theta V^{\\star }_{j}(\\Lambda )) + O(L^{4(d-2)}(\\Vert V^\\star _{j}\\Vert _{j}+\\Vert K^\\star _{j}\\Vert _{j})\\Vert K^\\star _{j}\\Vert _{j}).$ Putting these pieces together establishes (REF ) as $L^{2(d-2)}(\\Vert V_{j}\\Vert _{j}+\\Vert K_{j}\\Vert _{j})\\le 1$ if $\\epsilon =\\epsilon (L)$ is small enough.",
"An immediate consequence is $\\Vert u^{\\star }_{j+1}(\\Lambda )\\Vert _{T_{j+1}(\\ell _{j+1})}&\\le O(\\Vert V^\\star _{j}\\Vert _{j} + L^{2(d-2)}\\Vert K^\\star _{j}\\Vert _{j}), \\\\\\Vert V^\\star _{j+1}(\\Lambda )\\Vert _{T_{j+1}(\\ell _{j+1})}&\\le O(\\Vert V^\\star _{j}\\Vert _{j} + L^{2(d-2)}\\Vert K^\\star _{j}\\Vert _{j}),$ which we will use in the bounds of the analysis of the $K$ coordinate."
],
[
"Small sets",
"The most significant improvement in the analysis concerns small sets, which we now analyse to second order.",
"To simplify notation, we write $\\hat{V}^\\star _j = V^\\star _j-Q, \\qquad \\tilde{V}^{\\star }_{j+1} = u^\\star _{j+1}+V^\\star _{j+1}.$ For any $B,B^{\\prime } \\in _j$ , $ P^\\star (B,B^{\\prime }) = \\frac{1}{2}_{C_{j+1}}(\\theta \\hat{V}^\\star _j(B)\\theta \\hat{V}^\\star _j(B^{\\prime }))-\\frac{1}{2} \\tilde{V}^{\\star }_{j+1}(B)\\tilde{V}^{\\star }_{j+1}(B^{\\prime }).$ Note that $P^{\\star }(B,B^{\\prime })=\\frac{1}{2}_{C_{j+1}}(\\theta \\hat{V}^\\star _{j}(B);\\theta \\hat{V}^\\star _{j}(B^{\\prime }))$ .",
"The definition of $P^\\star (B,B^{\\prime })$ implies that it can only contain monomials with a factor of $\\sigma _a\\bar{\\sigma }_b$ (Case (1)) or $\\sigma _a{\\sigma _b}$ (Case (2)).",
"Since $\\sigma _a^2=\\sigma _b^2=\\bar{\\sigma }_{b}^{2}=0$ , for any $U \\in ^\\star $ and $B,B^{\\prime },B^{\\prime \\prime } \\in _j$ , it follows that $P^\\star (B,B^{\\prime })U(B^{\\prime \\prime })=0$ .",
"The claim follows as this implies $(_{C_{j+1}}\\theta \\hat{V}^\\star _j(B))(_{C_{j+1}}\\theta \\hat{V}^\\star _j(B^{\\prime }))$ is the same as $(_{C_{j+1}}\\theta \\hat{V}^\\star _j(B) -P^\\star (B)) (_{C_{j+1}}\\theta \\hat{V}^\\star _j(B^{\\prime }) -P^\\star (B^{\\prime })).",
"$ The next lemmas are analogues of Lemmas REF –REF that apply to the observable components.",
"We begin with the replacement for Lemma REF .",
"Suppose that $\\Vert V_j^\\varnothing \\Vert _j+\\Vert K_{j}^\\varnothing \\Vert _j\\le 1$ .",
"Then for any $X \\in _j$ with $|_{j}(X)|\\ge 2$ , $ \\Vert \\pi _\\star _{C_{j+1}}(\\delta I)^X\\Vert _{T_{j+1}(\\ell _{j+1})} =O((1+|_j(X)|)^2 (\\Vert V_j\\Vert _j+ L^{2(d-2)}\\Vert K_j\\Vert _j)^2(\\Vert V^\\varnothing _j\\Vert _j+\\Vert K^\\varnothing _j\\Vert _j)^{|_j(X)|-2}).$ For each $B$ , write $\\delta I(B) = \\pi _{\\varnothing }\\delta I(B) + \\pi _{\\star }\\delta I(B)$ , and then expand the product defining $(\\delta I)^{X}$ .",
"Since $\\sigma _a^2=\\sigma _b^2=\\bar{\\sigma }_b^2=0$ , the non-zero terms involve at most 2 factors of $\\pi _{\\star } \\delta I(B)$ .",
"The lemma then follows since $\\Vert \\pi _{\\varnothing } \\delta I(B)\\Vert _{T_{j+1}(\\ell _{j+1})} = O(\\Vert V^\\varnothing _j\\Vert _j +\\Vert K^\\varnothing _j\\Vert _j)$ by Lemma REF and $\\Vert \\pi _{\\star }\\delta I(B)\\Vert _{T_{j+1}(\\ell _{j+1})}=O(\\Vert V^\\star _{j}\\Vert _{j}+L^{2(d-2)}\\Vert K^\\star _{j}\\Vert _{j})$ by ().",
"Next we replace Lemma REF .",
"Unlike before we explicitly consider terms arising from two blocks.",
"Suppose that $\\Vert V_j^\\varnothing \\Vert _j+\\Vert K_{j}^\\varnothing \\Vert _j\\le \\epsilon $ and $\\Vert V^\\star _{j}\\Vert _{j}+\\Vert K^\\star _{j}\\Vert _{j}\\le \\epsilon $ .",
"Then for $B\\in _j$ , ${\\pi _\\star _{C_{j+1}}{\\delta I(B)+ \\frac{1}{2}\\mathop {\\sum _{B^{\\prime } \\ne B}}_{{B^{\\prime }\\in (B^{\\square })}}\\delta I(B) \\delta I(B^{\\prime }) + \\theta Q(B)}}_{T_{j+1}(\\ell _{j+1})}\\\\ = O(L^{4d}(\\Vert V^\\star _{j}\\Vert _{j}+\\Vert K^\\star _{j}\\Vert )(\\Vert V^\\varnothing _j\\Vert _j+\\Vert K^\\varnothing _j\\Vert _j+\\Vert K^\\star _{j}\\Vert _{j}))$ Using the relation (REF ) to re-express $_{C_{j+1}}\\theta Q^\\star (B)$ , the bracketed term in (REF ) equals $\\pi _\\star _{C_{j+1}}{ \\delta I(B)+ \\frac{1}{2} \\sum _{B^{\\prime }\\ne B} \\delta I(B) \\delta I(B^{\\prime })}+ _{C_{j+1}}\\theta V^\\star _j(B)- \\sum _{B^{\\prime }} P^\\star (B,B^{\\prime })- \\tilde{V}^{\\star }_{j+1}(B),$ where both sums are over $B^{\\prime } \\in (B^\\square )$ , as $P(B,B^{\\prime })=0$ otherwise.",
"By Lemma REF for $P(B,B)$ and since $\\delta I(B)=\\theta e^{-V_j(B)}-e^{-(V_{j+1}+u_{j+1})(B)}$ , the one block terms $B^{\\prime }=B$ are $\\pi _\\star _{C_{j+1}}\\theta { e^{-V_j(B)}-1+ V^\\star _j(B) - \\frac{1}{2} \\hat{V}^\\star _j(B)^2}\\\\-\\pi _\\star { e^{-(V_{j+1}+u_{j+1})(B)}-1+\\tilde{V}^{\\star }_{j+1}(B)- \\frac{1}{2} \\tilde{V}^{\\star }_{j+1}(B)^2}.$ To estimate these terms let us first note that with $V=V_{j+1}+u_{j+1}$ , (REF ) and its consequences (REF )–(), already proven, imply $\\Vert V^\\varnothing \\Vert _{T_{j+1}(\\ell _{j+1})}\\le 1,\\Vert V^\\star \\Vert _{T_{j+1}(\\ell _{j+1})}\\le 1$ .",
"As this bound also holds for $V=V_j$ provided $\\epsilon $ is sufficiently small, we then have for $V=V_j$ or $V=u_{j+1}+V_{j+1}$ , $\\nonumber \\pi _{\\star }e^{-V(B)}&= \\pi _{\\star }(e^{-V^\\star (B)} + (e^{-V^\\varnothing (B)}-1)e^{-V^\\star (B)}) \\\\&= \\pi _{\\star }(1- V^\\star (B) + \\frac{1}{2} V^\\star (B)^2) +O((\\Vert V^\\star _{j}\\Vert _{j}+L^{2(d-2)}\\Vert K^\\star _{j}\\Vert ) (\\Vert V^\\varnothing _{j}\\Vert _j+L^d \\Vert K^\\varnothing _{j}\\Vert _j )),$ where we have used $V^\\star (B)^{3}=0$ , and in the case $V=u_{j+1}+V_{j+1}$ , also () and (REF ) to control $\\Vert u^{\\varnothing }_{j+1}+V^\\varnothing _{j+1}\\Vert _{j+1}$ in terms of $\\Vert V^\\varnothing _j\\Vert _j+L^d \\Vert K^\\varnothing _j\\Vert _j$ and similarly for $\\Vert u^\\star _{j+1}+K^\\star _{j+1}\\Vert _{j+1}$ .",
"Using also $\\hat{V}^\\star _j(B)^2= (V^\\star _j(B)-Q(B))^2= V^\\star _j(B)^2 + O(L^{4(d-2)}\\Vert K^\\star _j\\Vert _j (\\Vert V^\\star _j\\Vert _j +\\Vert K^\\star _j\\Vert _j)),$ the product property, (REF ), and the assumed norm bounds, the estimate for the one block terms follow.",
"For $B^{\\prime } \\ne B$ the two block terms are $\\frac{1}{2} \\pi _\\star {_{C_{j+1}} \\delta I(B)\\delta I(B^{\\prime })- _{C_{j+1}}(\\theta \\hat{V}^\\star _j(B)\\theta \\hat{V}^\\star _j(B^{\\prime })) + \\tilde{V}^{\\star }_{j+1}(B) \\tilde{V}^{\\star }_{j+1}(B^{\\prime })}.$ We start by rewriting this in a more convenient form.",
"Let $\\delta V_j = \\theta \\hat{V}_j-\\tilde{V}_{j+1}$ and $\\pi _{\\star }\\delta V_j =\\delta V^\\star _j$ .",
"By (REF ), $_{C_{j+1}}\\theta \\hat{V}^\\star _j= \\tilde{V}^{\\star }_{j+1} + P^\\star = \\tilde{V}^{\\star }_{j+1} +O(\\sigma _a\\sigma _b)$ , where $O(\\sigma _a\\sigma _b)$ denotes a monomial containing a factor $\\sigma _a\\bar{\\sigma }_b$ in Case (1) or a factor $\\sigma _a\\sigma _b$ in Case (2).",
"Since all terms in $\\delta V^\\star _j$ contain an observable field (that is, a $\\sigma $ -factor), nilpotency implies $_{C_{j+1}}\\delta V^\\star _j(B)\\delta V^\\star _j(B^{\\prime })&=_{C_{j+1}}\\theta \\hat{V}^\\star _j(B)\\hat{V}^\\star _j(B^{\\prime })+ \\tilde{V}^{\\star }_{j+1}(B)\\tilde{V}^{\\star }_{j+1}(B^{\\prime })&\\qquad -\\tilde{V}^{\\star }_{j+1}(B) _{C_{j+1}}\\theta \\hat{V}^\\star _j(B^{\\prime }) -\\tilde{V}^{\\star }_{j+1}(B^{\\prime }) _{C_{j+1}}\\theta \\hat{V}^\\star _j(B) &=_{C_{j+1}}\\theta \\hat{V}^\\star _j(B)\\hat{V}^\\star _j(B^{\\prime }) - \\tilde{V}^{\\star }_{j+1}(B)\\tilde{V}^{\\star }_{j+1}(B^{\\prime }).$ Therefore we need to estimate $\\frac{1}{2} \\pi _\\star _{C_{j+1}} \\delta I(B)\\delta I(B^{\\prime })- \\frac{1}{2} _{C_{j+1}}\\delta V^\\star _j(B)\\delta V^\\star _j(B^{\\prime }).$ First write $\\pi _\\star [ \\delta I(B)\\delta I(B^{\\prime })]=\\pi _\\star \\delta I(B)\\pi _\\star \\delta I(B^{\\prime })+\\pi _\\star \\delta I(B)\\pi _\\varnothing \\delta I(B^{\\prime })+\\pi _\\varnothing \\delta I(B)\\pi _\\star \\delta I(B^{\\prime }) .$ The second and third terms on the right-hand side are $O((\\Vert V^\\star _{j}\\Vert _{j}+L^{2(d-2)}\\Vert K^\\star _{j}\\Vert _{j})(\\Vert V^\\varnothing _j\\Vert _j+\\Vert K^\\varnothing _j\\Vert _j))$ using Lemma REF for $\\pi _\\varnothing \\delta I$ and $\\Vert \\pi _{\\star }\\delta I(B)\\Vert _{T_{j+1}(\\ell _{j+1})}=O(\\Vert V^\\star _{j}\\Vert _{j}+L^{2(d-2)}\\Vert K^\\star _{j}\\Vert _{j})$ by ().",
"Using (REF ), the term $\\pi _\\star \\delta I(B)\\pi _\\star \\delta I(B^{\\prime })$ can be estimated as $\\pi _\\star (\\delta V_j(B) -\\frac{1}{2} (\\theta V_j(B)^2-\\tilde{V}_{j+1}(B)^2))\\pi _\\star (\\delta V_j(B^{\\prime }) -\\frac{1}{2} (\\theta V_j(B^{\\prime })^2-\\tilde{V}_{j+1}(B^{\\prime })^2))\\\\\\qquad \\qquad \\qquad \\qquad \\qquad + O((\\Vert V^\\star _{j}\\Vert _{j}+L^{2(d-2)}\\Vert K^\\star _{j}\\Vert )(\\Vert V^\\varnothing _j\\Vert _j+L^d\\Vert K^\\varnothing _j\\Vert _j))\\\\ =\\delta V^\\star _j(B)\\delta V^\\star _j(B^{\\prime })+ O((\\Vert V^\\star _{j}\\Vert _{j}+L^{2(d-2)}\\Vert K^\\star _{j}\\Vert )(\\Vert V^\\varnothing _j\\Vert _j+L^d\\Vert K^\\varnothing _j\\Vert _j)),$ since $\\sigma _a^2=\\sigma _b^2=\\bar{\\sigma }_b^2=0$ .",
"Putting these bounds together gives (REF ), as there are at most $O(L^d)$ blocks $B^{\\prime }$ for which the two-block contribution is non-zero.",
"The factor $L^{4d}$ is a convenient common upper bound.",
"The next lemma replaces Lemma REF on the observable components.",
"For any $U \\in _{j+1}$ , if $K^{ab}_{j}(X)=0$ for $X\\in _j(X)$ and $j<j_{ab}$ , $\\sum _{X \\in _j: \\bar{X}=U} \\Vert _{C_{j+1}}\\theta (1-\\operatorname{Loc}_X^\\star ) K^\\star _{j} (X)\\Vert _{T_{j+1}(\\ell _{j+1})} = O(L^{-(\\frac{d-2}{2} \\wedge 1)}) \\Vert K^\\star \\Vert _j.$ The proof is the same as that of Lemma REF except for the following observation.",
"The sum over $X \\in _j$ that contributes a factor $O(L^d)$ in the proof of Lemma REF only contributes $O(1)$ on the observable components because for these only the small sets containing $a$ or $b$ contribute.",
"Thus the bound for $\\operatorname{Loc}^\\star $ from Proposition REF , which lacks a factor $L^{-d}$ compared to the bound for $\\operatorname{Loc}^\\varnothing $ , produces the same final bound.",
"The proof is analogous to that of Theorem REF , and we proceed in a similar manner, by beginning with an estimate of $\\pi _{\\star }_{j+1}(U)$ , where $_{j+1}(U)$ is defined by the formula (REF ) but with the extended coordinates introduced in Section REF .",
"We first consider the small set contributions to $_{j+1}(U)$ .",
"The bound (REF ) gets replaced by (see (REF )) $Q^{\\star }(B)=O( L^{d-2}\\Vert K^\\star _j\\Vert )$ , and we have that $\\Vert u_{j+1}^{\\star }\\Vert _{j+1}$ and $\\Vert V^\\star _{j+1}\\Vert _{j+1}$ are $O(\\Vert V^\\star _j\\Vert +L^{d-2}\\Vert K^{\\star }_j\\Vert _j).$ As stated previously, Lemma REF is replaced with Lemma REF whereas Lemmas REF and REF replace Lemmas REF and REF .",
"Arguing as in the proof of Theorem REF then gives that the small set contribution to $\\pi _{\\star }_{j+1}(U)$ is $O(L^{-(\\frac{d-2}{2} \\wedge 1)}\\Vert K^\\star _j\\Vert _j)$ .",
"The large set and non-linear contributions do not require any improved estimates: the only change is to extract the distinct factors of the bulk and observable terms.",
"That is, to deal with the large set contribution we use the bounds above together with the fact that for $F\\in $ , $ \\pi _\\star \\prod _{i=1}^k F_i= \\sum _{i} F_{i}^{\\star }\\prod _{l\\ne i} F_{l}^{\\varnothing }+ \\sum _{i\\ne k} F_{i}^{\\star } F_{k}^{\\star }\\prod _{l\\ne i,k}F_{l}^{\\varnothing }$ as the product of any three elements of $^{\\star }$ is zero.",
"In particular, Lemma REF can be replaced by $\\Vert _{C_{j+1}}\\pi _{\\star }(\\delta I)^X\\Vert _{T_{j+1}(\\ell _{j+1})} =O(\\sum _{r=1}^{2}L^{2r(d-2)}(\\Vert V^\\star _{j}\\Vert +\\Vert K^\\star _{j}\\Vert _{j})^{r}(\\Vert V^\\varnothing _j\\Vert _j+\\Vert K^\\varnothing _j\\Vert _j)^{|_j(X)|-r})$ for all $X \\in _j$ .",
"We have used that $\\pi _{\\star }\\delta I(B)$ is non-zero only if $a\\in B^{\\square }$ or $b\\in B^{\\square }$ , so the sums in (REF ) have at most $O(1)$ non-zero terms.",
"Arguing as in the proof of Theorem REF then yields (after possibly increasing $A$ ) that the large set contribution to $_{j+1}(U)$ is $O(A^{-\\eta }\\Vert K^\\star _{j}\\Vert _{j}) + O(A^{-\\eta }L^{2(d-2)}(\\Vert V^\\star _j\\Vert _j+ \\Vert K^\\star _j\\Vert _j) (\\Vert V^\\varnothing _j\\Vert _j + \\Vert K^\\varnothing _j\\Vert _j)).$ Using the above replacement for Lemma REF , the argument for the non-linear estimate is then the same as in the proof of Theorem REF ."
],
[
"Flow of observable coupling constants",
"With Theorem REF in place, the evolution of the observable coupling constants in $u^{\\star }+V^\\star $ is the same as the free one from Section REF up to the addition of remainder terms from the $K$ coordinate.",
"To avoid carrying an unimportant factor of $L^{2(d-2)}$ through equations, we write $O_L(\\cdot )$ to indicate bounds with constants possibly depending on $L$ (but we reemphasise that implicit constants are always independent of the scale $j$ ).",
"Suppose $j<N$ , $x\\in \\lbrace a,b\\rbrace $ , and that (REF ) holds.",
"If $j<j_{ab}$ , further suppose that $K^{ab}_{j}(X)=0$ for $X\\in _{j}$ .",
"In Case (1), $\\lambda _{x,j+1} &=\\lambda _{x,j} + O_L(\\ell _{x,j}^{-1}\\ell _{j}^{-1}\\Vert K_j^x\\Vert _j),\\\\q_{j+1} &= q_j + \\lambda _{a,j}\\lambda _{b,j} C_{j+1}(a,b) {+ r_j C_{j+1}(0,0)}+ O_L(\\ell _{ab,j}^{-1}\\Vert K_j^{ab}\\Vert _j 1_{j\\ge j_{ab}}),\\\\r_{j+1}&= r_j + O_L(\\ell _{ab,j}^{-1}\\ell _j^{-2}\\Vert K_j^{ab}\\Vert _j 1_{j \\ge j_{ab}}),\\multicolumn{2}{l}{\\text{and in Case (2),}}\\\\\\lambda _{x,j+1} &= \\lambda _{x,j}+ O_L(\\ell _{x,j}^{-1}\\ell _j^{-2}\\Vert K_j^x\\Vert _j),\\\\\\gamma _{x,j+1} &= \\gamma _{x,j} + \\lambda _{x,j}C_{j+1}(x,x) + O_L(\\ell _{x,j}^{-1}\\Vert K_j^x\\Vert _j),\\\\q_{j+1} &= q_j + \\eta _j C_{j+1}(a,b) - \\lambda _{a,j}\\lambda _{b,j} C_{j+1}(a,b)^2+r_{j}C_{j+1}(0,0)+ O_L(\\ell _{ab,j}^{-1}\\Vert K_j^{ab}\\Vert _j1_{j\\ge j_{ab}}),\\\\\\eta _{j+1} &= \\eta _j - 2\\lambda _{a,j}\\lambda _{b,j} C_{j+1}(a,b),\\\\r_{j+1}&= r_{j} + O_L(\\ell _{ab,j}^{-1}\\ell _j^{-2} \\Vert K_j^{ab}\\Vert _j 1_{j\\ge j_{ab}}).$ Moreover, for $j+1<N$ , all coupling constants are independent of $N$ .",
"Note that there is no error term in the equation for $\\eta $ , as the corresponding nonlocal field monomial is not contained in the image of $\\operatorname{Loc}$ .",
"For $j<N$ , the main contribution in (REF ) is identical to that in Lemma REF .",
"The indicator functions $1_{j\\ge j_{ab}}$ in the error terms are due to the assumption $K_{j}^{ab}(X)=0$ for $j<j_{ab}$ and $X\\in _{j}$ .",
"The bounds for the error terms follow from the definition of the norms as in obtaining (REF ).",
"Finally, that the couplings are independent of $N$ is a consequence of the consistency of the renormalisation group map, i.e., Proposition REF (applied to the renormalisation group map extended by observables).",
"The next lemma shows that if we maintain control of $\\Vert K_k^{\\star }\\Vert _k$ up to scale $j$ then we control the coupling constants in $V^\\star $ on scale $j$ .",
"Assume that $\\Vert K^\\star _k\\Vert _k = O_L(\\lambda _0b_0L^{-\\kappa k})$ for $k<j$ and that (REF ) holds for $k<j$ .",
"Then, in Case (1) if $q_0=r_0=0$ and $\\lambda _0>0$ , $\\lambda _{j} &= \\lambda _0 + O_L(\\lambda _0b_0)\\\\r_j &= O_L(\\lambda _0b_0|a-b|^{-\\kappa })1_{j\\ge j_{ab}}\\multicolumn{2}{l}{\\text{and, in Case (2), if $q_0=r_0=\\gamma _{x,0}=\\eta _0=0$ and $\\lambda _0>0$,}}\\\\\\lambda _{j} &= \\lambda _0 + O_L(\\lambda _0b_0)\\\\\\eta _j &= O_L(\\lambda _0^2|a-b|^{-(d-2)})1_{j\\ge j_{ab}}\\\\r_{j}&= O_L(\\lambda _0b_0|a-b|^{-(d-2)-\\kappa })1_{j\\ge j_{ab}},$ where $\\lambda _{j}=\\lambda _{x,j}$ for either $x=a$ or $x=b$ .",
"In both Cases (1) and (2), $\\Vert V^\\star _{j}\\Vert _{j} \\le \\lambda _0 + O_L(\\lambda _0^2) + O_L(\\lambda _0b_0).$ The bounds on the coupling constants follow from Lemma REF ; the hypothesis regarding $K_{j}(X)=0$ for $j<j_{ab}$ and $X\\in _{j}$ holds as Definition REF implies that for an iteration $(V_j,K_j)$ of the renormalisation group map, the $^{ab}$ components of $V_j(B)$ and $K_j(X)$ with $X\\in _j$ can only be nonzero for $j>j_{ab}$ since we have started the flow with $r_{0}=0$ in Case (1), and $q_{0}=\\eta _{0}=r_{0}=0$ in Case (2).",
"What remains is to analyse the recurrences.",
"For $\\lambda _{x,j}$ , since $\\ell _{x,j}^{-1}\\ell _j^{-p} = 1$ in Case ($p$ ), using (REF ), respectively (), $\\lambda _{x,j} = \\lambda _0 + \\sum _{k=0}^{j-1} O_L(\\Vert K^\\star _k\\Vert _k)= \\lambda _0 + \\sum _{k=0}^{j-1} O_L(\\lambda _0b_0L^{-\\kappa k})=\\lambda _{0} + O_L(\\lambda _{0}b_{0}).$ The bounds on $r_{j}$ follow from the fact that all contributions are 0 for scales $j<j_{ab}$ if $r_{0}=0$ .",
"For example, in Case (2), $|r_j|=\\lambda _0b_0O_L(\\sum _{k=j_{ab}}^{j-1} \\ell _{ab,j}^{-1} \\ell _j^{-2}L^{-\\kappa j})=\\lambda _0b_0\\ell _{j_{ab}}^2 O_L(\\sum _{k=j_{ab}}^{j-1} L^{-\\kappa j})=O_L(\\lambda _0b_0|a-b|^{-(d-2)-\\kappa }).$ Case (1) is similar, except no factor $\\ell _{j_{ab}}$ arises (see (REF )).",
"The bound on $\\eta _j$ in Case (2) follows from the preceding analysis of $\\lambda _{x,j}$ , the fact that $\\eta _{j}=0$ for $j<j_{ab}$ if $\\eta _{0}=0$ since $C_{j}$ has finite range ($C_{j}(a,b)=0$ if $|a-b|_\\infty \\ge \\frac{1}{2} L^{j}$ ), and that $C_{j+1}(a,b) \\le O_L(L^{-(d-2)j})$ : $|r_j|&=\\lambda _0b_0O_L(\\sum _{k=j_{ab}}^{j-1} \\ell _{ab,j}^{-1} \\ell _j^{-2}L^{-\\kappa j})=\\lambda _0b_0\\ell _{j_{ab}}^2 O_L(\\sum _{k=j_{ab}}^{j-1} L^{-\\kappa j})=O_L(\\lambda _0b_0|a-b|^{-(d-2)-\\kappa }),\\\\|\\eta _j|&=O_L(\\lambda _0^2 \\sum _{k=j_{ab}}^{j-1} L^{-(d-2)k})=O_L(\\lambda _0^2 |a-b|^{-(d-2)}).$ For the bound on the norm of $\\Vert V^\\star _{j}\\Vert _j$ recall that the $q$ and $\\gamma $ terms have been taken out of $V^\\star $ .",
"Thus in Case (1), $\\Vert V^\\star _j(B)\\Vert \\lesssim |\\lambda _j| + |r_j|\\ell _{ab,j}\\ell _j^2\\lesssim |\\lambda _j| + |r_j|= |\\lambda _j| + O_L(\\lambda _0b_0).$ Similarly, in Case (2), using that $\\ell _j^2\\ell _{ab,j}= \\ell _{j\\wedge j_{ab}}^{-2} = O_L(|a-b|^{d-2})$ for $j \\ge j_{ab}$ , $\\Vert V^\\star _j(B)\\Vert &\\lesssim |\\lambda _j| + |\\eta _j| \\ell _j^2\\ell _{ab,j}1_{j\\ge j_{ab}} +|r_{j}| \\ell _{ab,j}\\ell _j^2 1_{j\\ge j_{ab}}&\\lesssim |\\lambda _j| + |\\eta _j||a-b|^{d-2}1_{j\\ge j_{ab}} + |r_{j}| |a-b|^{d-2}1_{j\\ge j_{ab}}&\\lesssim |\\lambda _j| + O_L(\\lambda _0^2)1_{j\\ge j_{ab}} +O_L(\\lambda _0b_0 |a-b|^{-\\kappa })1_{j \\ge j_{ab}} = |\\lambda _j|+O_L(\\lambda _0^2)+O_L(b_0\\lambda _0).$ From now on, we assume that the bulk renormalisation group flow $(V_j^\\varnothing ,K_j^\\varnothing )_{j \\le N}$ is given by Theorem REF .",
"In particular, there is $\\alpha >0$ such that $\\Vert V^\\varnothing _j\\Vert _j = O(b_0L^{-\\alpha j}), \\qquad \\Vert K^\\varnothing _j\\Vert _j = O(b_0L^{-\\alpha j}).$ Using this as input, we iterate the observable flow (REF )–(), with initial condition $\\lambda _{a,0}=\\lambda _{b,0}=\\lambda _0$ small enough and all other observable coupling constants equal to 0.",
"Assume that the bulk renormalisation group flow $(V^\\varnothing _j,K^\\varnothing _j)$ obeys $\\Vert V^\\varnothing _j\\Vert _j+\\Vert K^\\varnothing _j\\Vert _j = O(b_0L^{-\\alpha j})$ for some $\\alpha >0$ .",
"Then there is $\\kappa >0$ such that for $\\lambda _{0,a}=\\lambda _{0,b}=\\lambda _0 >0$ sufficiently small and all other observable coupling constants initially 0, $ \\Vert V^\\star _j\\Vert _j \\le O_L(\\lambda _0), \\qquad \\Vert K^\\star _j\\Vert _j \\le O_L(\\lambda _0b_0L^{-\\kappa j}).$ The proof is by induction from a scale $j_0$ which will be determined below.",
"When $j=0$ we have $\\Vert V^\\star _{0}\\Vert _{0}\\asymp \\lambda _{0,a}+\\lambda _{0,b}$ and $\\Vert K^\\star _{0}\\Vert _{0}=0$ .",
"Now let us choose $C_j>0$ so that $\\Vert K^\\star _k\\Vert _{k} \\le C_j b_0\\lambda _0L^{-\\kappa k}$ for all $k \\le j$ .",
"We start the proof by studying the behaviour of $C_j$ as $j$ increases.",
"We may assume $\\kappa <\\alpha /2$ and also that $\\kappa $ is less than the exponents of $L$ and $A$ in ().",
"Then Lemma REF shows that $\\Vert V^\\star _{k}\\Vert _k \\le C_j\\lambda _0$ for $k\\le j+1$ provided $\\lambda _0<1$ and $b_0$ is sufficiently small (depending on $\\kappa $ ).",
"We now use Theorem REF to control $K^\\star _{j+1}$ .",
"Since $A \\gg L$ and taking $\\lambda _0$ sufficiently small we obtain $\\Vert K^\\star _{j+1}\\Vert _{j+1}&\\le \\frac{1}{2} L^{-\\kappa }\\Vert K^\\star _j\\Vert _j +O_L(1) C_j\\lambda _0 b_0 L^{-\\alpha j}& \\le C_jb_{0}\\lambda _{0} (\\frac{1}{2} L^{-\\kappa (j+1)} +\\frac{1}{2} O_L(1)L^{-(\\alpha /2)(j-1)}L^{-(\\alpha /2)(j+1)})&\\le C_jb_{0}\\lambda _{0} (\\frac{1}{2} +\\frac{1}{2} O_L(1)L^{-(\\alpha /2)(j-1)}) L^{-\\kappa (j+1)}$ since $\\alpha >2\\kappa $ .",
"Now there is $j_0\\in \\mathbb {N}$ so that for all $j\\ge j_0$ , the term $O_L(1)L^{-(\\alpha /2)(j-1)}$ on the right-hand side is bounded by 1.",
"Thus choosing $C=C_{j_0}$ , by induction for all $j\\ge j_0$ , $\\Vert K^\\star _j\\Vert _{j} \\le C b_0\\lambda _0L^{-\\kappa j}$ and $\\Vert V^\\star _{j}\\Vert _j \\le C\\lambda _0$ and the claim follows."
],
[
"Computation of pointwise correlation functions",
"In this section we use the results of Section to prove the following estimates for the pointwise correlation functions ${\\bar{\\psi }_a\\psi _b}$ , ${\\bar{\\psi }_a\\psi _a}$ , and ${\\bar{\\psi }_a\\psi _a\\bar{\\psi }_b\\psi _b}$ .",
"Recall the definition (REF ) of $W_{N}(x)=W_{N,m^{2}}(x)$ .",
"For $b_0$ sufficiently small and $m^2 \\ge 0$ , there exists continuous functions $\\lambda =\\lambda (b_0,m^2)= 1+O_L(b_0), \\qquad \\gamma = \\gamma (b_0,m^2) = (-\\Delta ^{^d}+m^2)^{-1}(0,0) + O_L(b_0),$ such that if $V^\\varnothing _{0}=V_{0}^{c}(m^{2},b_{0})$ is as in Theorem REF , $V^\\star _{0}$ is as in Proposition REF , and $\\tilde{u}_{N,N}^c=\\tilde{u}_{N,N}^c(b_0,m^2)$ is as in Proposition with initial condition $V^\\varnothing _0=V_0^c$ , $ {\\bar{\\psi }_a\\psi _a} =\\gamma + \\frac{\\gamma t_N |\\Lambda _N|^{-1}+ O_L(b_0L^{-(d-2+\\kappa )N}) + O_L(b_0L^{-\\kappa N}(m^{2}|\\Lambda _N|)^{-1})}{1+\\tilde{u}_{N,N}}.$ Under the same assumptions as in Proposition , ${\\bar{\\psi }_a\\psi _b}&=W_N(a-b)+ \\frac{t_N|\\Lambda _N|^{-1}}{1+\\tilde{u}_{N,N}}&\\qquad + O_L(b_0|a-b|^{-(d-2+\\kappa )})+\\frac{O_L(b_0|a-b|^{-\\kappa } (m^2|\\Lambda _N|)^{-1})}{1+\\tilde{u}_{N,N}}\\\\{\\bar{\\psi }_a\\psi _a\\bar{\\psi }_b\\psi _b}&= -\\lambda ^2 W_N(a-b)^2 + \\gamma ^2+\\frac{-2\\lambda ^2 W_N(a-b)+ 2\\lambda \\gamma }{1+\\tilde{u}_{N,N}}t_N|\\Lambda _N|^{-1}&\\qquad + O_L(b_0 |a-b|^{-2(d-2)-\\kappa })+ O_L(b_0L^{-(d-2+\\kappa )N})&\\qquad + (O_L(b_0L^{-\\kappa N}) + O_L(b_0 |a-b|^{-(d-2+\\kappa )}))\\frac{(m^{2}|\\Lambda _N|)^{-1}}{1+\\tilde{u}_{N,N}}.$ Throughout this section, we assume that the renormalisation group flow $(V_j,K_j)_{j \\le N}$ is given as in Corollary REF (bulk) and Proposition REF (observables)."
],
[
"Integration of the zero mode",
"As in the analysis of the susceptibility in Section , we treat the final integration over the zero mode explicitly.",
"Again we will only require the restriction to constant $\\psi ,\\bar{\\psi }$ (as discussed below (REF )) of $_{C} \\theta Z_0= _{t_N Q_N} \\theta Z_N = e^{-u^\\varnothing _N|\\Lambda _N|} \\tilde{Z}_{N,N},$ where the last equation defines $\\tilde{Z}_{N,N}$ .",
"We write $\\tilde{Z}_{N,N}=\\tilde{Z}_{N,N}^{\\varnothing }+\\tilde{Z}_{N,N}^{\\star }$ for its decomposition into bulk and observable parts.",
"The bulk term was already computed in Proposition .",
"The observable term $\\tilde{Z}_{N,N}^\\star $ is computed by the next lemma; in the lemma we only give explicit formulas for the terms that will be used in the proofs of Propositions and .",
"Restricted to constant $\\psi ,\\bar{\\psi }$ , in Case (1), $\\tilde{Z}_{N,N}^\\star &=\\sigma _a\\bar{\\psi }\\tilde{Z}_{N,N}^{\\sigma _a\\bar{\\psi }} + \\bar{\\sigma }_b\\psi \\tilde{Z}_{N,N}^{\\sigma _b\\psi }+\\sigma _a\\bar{\\sigma }_b \\tilde{Z}_{N,N}^{\\bar{\\sigma }_b\\sigma _a}+\\sigma _a\\bar{\\sigma }_b \\psi \\bar{\\psi }\\tilde{Z}_{N,N}^{\\bar{\\sigma }_b\\sigma _a\\psi \\bar{\\psi }}\\multicolumn{2}{l}{\\text{where}}\\\\\\tilde{Z}_{N,N}^{\\sigma _a\\bar{\\psi }}&= \\lambda _{a,N} +O_L(\\ell _{x,N}^{-1}\\ell _N^{-1}\\Vert K^\\star _N\\Vert _N)\\\\\\tilde{Z}_{N,N}^{\\bar{\\sigma }_b\\psi }&= \\lambda _{b,N} +O_L(\\ell _{x,N}^{-1}\\ell _N^{-1}\\Vert K^\\star _N\\Vert _N)\\\\\\tilde{Z}_{N,N}^{\\bar{\\sigma }_b\\sigma _a}&= q_N (1+\\tilde{u}_{N,N})+ \\lambda _{a,N}\\lambda _{b,N} t_N|\\Lambda _N|^{-1} { + r_N t_N |\\Lambda _N|^{-1}}&\\qquad \\qquad + O_L(m^{-2}|\\Lambda _N|^{-1}\\ell _N^{-2}\\ell _{ab,N}^{-1})\\Vert K^\\star _N\\Vert _N.$ In Case (2), $\\tilde{Z}_{N,N}^\\star &=\\sigma _a\\tilde{Z}_{N,N}^{\\sigma _a}+ \\sigma _a\\bar{\\psi }\\psi \\tilde{Z}_{N,N}^{\\sigma _a\\bar{\\psi }\\psi }+\\sigma _b \\tilde{Z}_{N,N}^{\\sigma _b} + \\sigma _b\\bar{\\psi }\\psi \\tilde{Z}_{N,N}^{\\sigma _b\\bar{\\psi }\\psi }+ \\sigma _a\\sigma _b \\tilde{Z}_{N,N}^{\\sigma _a\\sigma _b}+ \\sigma _a\\sigma _b \\psi \\bar{\\psi }\\tilde{Z}_{N,N}^{\\sigma _a\\sigma _b\\psi \\bar{\\psi }}\\multicolumn{2}{l}{\\text{where, setting $\\tilde{\\lambda }_{x,N,N} = \\lambda _{x,N}+O_L(\\ell _{x,N}^{-1} \\ell _N^{-2} \\Vert K^\\star _N\\Vert _N)$,}}\\\\\\tilde{Z}_{N,N}^{\\sigma _x}&=\\gamma _{x,N} (1+\\tilde{u}_{N,N}) +\\tilde{\\lambda }_{x,N,N}t_N|\\Lambda _N|^{-1}+ O_L(\\ell _{x,N}^{-1})\\Vert K^\\star _N\\Vert _N.\\\\\\tilde{Z}_{N,N}^{\\sigma _a\\sigma _b}&= (q_N+\\gamma _{a,N}\\gamma _{b,N}) (1+\\tilde{u}_{N,N})+ (\\eta _N+r_N +\\tilde{\\lambda }_{a,N,N}\\gamma _{b,N}+\\tilde{\\lambda }_{b,N,N}\\gamma _{a,N})t_N|\\Lambda _N|^{-1}&\\qquad \\qquad + O_L((|\\gamma _{a,N}|+|\\gamma _{b,N}|)\\ell _{x,N}^{-1}+ \\ell _{ab,N}^{-1}+ m^{-2}|\\Lambda _N|^{-1}\\ell _N^{-2}\\ell _{ab,N}^{-1})\\Vert K^\\star _N\\Vert _N.$ The error bounds above reveal the tension in the explicit choices of $\\ell _{x,j}^{-1}$ and $\\ell _{ab,j}^{-1}$ .",
"To obtain effective error estimates, we want $\\ell _{x,N}^{-1}$ and $\\ell _{ab,N}^{-1}$ to be as small as possible.",
"On the other hand, to control the iterative estimates of Theorem REF over the entire trajectory, i.e., to prove Proposition REF , we cannot have $\\ell _{x,j}$ and $\\ell _{ab,j}$ too large.",
"In particular, either of the more naive choices $\\ell _{ab,j}=\\ell _{x, j}^{2}$ and $\\ell _{ab,j} = \\ell _{x,j_{ab}}^{2}$ in Case (2) would lead to difficulties, both in terms of forcing us to track additional terms in the flow and in terms of controlling norms inductively, or to error bounds that are not strong enough to capture the zero mode sufficiently accurately.",
"Throughout the proof, we restrict to constant $\\psi ,\\bar{\\psi }$ .",
"Since ${e^{+u_N^\\varnothing (\\Lambda )}Z_N}^\\star &= {e^{-u_N^\\star (\\Lambda )}(e^{-V_N(\\Lambda )}+K_N(\\Lambda ))}^\\star &=(e^{-u_N^\\star (\\Lambda )}-1)(e^{-V^\\varnothing _N(\\Lambda )}+K^\\varnothing _N(\\Lambda ))+ e^{-u_N^\\star (\\Lambda )}(e^{-V_N(\\Lambda )}+K_N(\\Lambda ))^\\star ,$ by applying $_{t_NQ_N}\\theta $ we obtain $\\tilde{Z}_{N,N}^\\star = \\underbrace{(e^{-u^\\star _N(\\Lambda )}-1)(1+\\tilde{u}_{N,N}-|\\Lambda _N|\\tilde{a}_{N,N}\\psi \\bar{\\psi })}_{A}+ \\underbrace{e^{-u_N^\\star (\\Lambda )}_{t_NQ_N}\\theta (e^{-V_N(\\Lambda )}+K_N(\\Lambda ))^\\star }_{B}.$ In obtaining $A$ we used (REF ) which gives $_{t_NQ_N}\\theta (e^{-V_N^\\varnothing (\\Lambda )}+K_N^\\varnothing (\\Lambda ))= 1+\\tilde{u}_{N,N}-|\\Lambda _N|\\tilde{a}_{N,N}\\psi \\bar{\\psi }$ .",
"Since each term in $V_N^\\varnothing (\\Lambda )$ contains a factor $\\psi \\bar{\\psi }$ and each term in $V_N^\\star (\\Lambda )$ either $\\psi $ or $\\bar{\\psi }$ , we have $V_N^\\varnothing (\\Lambda )V_N^\\star (\\Lambda )=0$ .",
"Thus $B = e^{-u_N^\\star (\\Lambda )}_{t_NQ_N}\\theta (-V_N^\\star (\\Lambda )+\\frac{1}{2} V_N^\\star (\\Lambda )^2+K_N^\\star (\\Lambda )).$ Case (1).",
"Since $\\sigma _a^2=\\bar{\\sigma }_b^2=0$ , $e^{-u_N^\\star (\\Lambda )}-1=-u^\\star _N(\\Lambda ) = \\sigma _a\\bar{\\sigma }_b q_N,$ we get $A&=\\sigma _a\\bar{\\sigma }_b q_N (1+\\tilde{u}_{N,N}-|\\Lambda _N|\\tilde{a}_{N,N}\\psi \\bar{\\psi })\\\\B &=\\sigma _a\\bar{\\psi }(\\lambda _{a,N}+k^{\\sigma _a\\bar{\\psi }}_{N})+{\\psi \\bar{\\sigma }_{b}}(\\lambda _{b,N}+k^{\\bar{\\sigma }_b\\psi }_{N})+\\sigma _a\\bar{\\sigma }_b _{t_NQ_N}\\theta \\bar{\\psi }\\psi (\\lambda _{a,N}\\lambda _{b,N} - r_N + k^{\\sigma _a\\bar{\\sigma }_b \\bar{\\psi }\\psi }_{N}).$ The constants $k_N^\\#$ are given in terms of derivatives of $K_N(\\Lambda )$ and bounded analogously as in (REF ).",
"For example, $k_N^{\\sigma _a\\bar{\\psi }} = O_L(\\ell _{a,N}^{-1}\\ell _N^{-1}\\Vert K^\\star _N\\Vert _N)$ , and similarly for the other $k_N^\\#$ terms, the rule being that we have a factor $\\ell _{x,N}^{-1}$ if there is a superscript $\\sigma _a$ or $\\bar{\\sigma }_b$ but not both, a factor $\\ell _{ab,N}^{-1}$ for $\\sigma _a\\bar{\\sigma }_b$ and a factor $\\ell _N^{-1}$ for each superscript $\\psi $ or $\\bar{\\psi }$ .",
"These bounds follow from the definition of the $T_j(\\ell _j)$ norm.",
"Since $_{t_NQ_N}\\theta \\psi \\bar{\\psi }= -t_N|\\Lambda _N|^{-1}+\\psi \\bar{\\psi }$ the claim follows by collecting terms and using (REF ).",
"Case (2).",
"Using again that $\\sigma _a^2=\\sigma _b^2=0$ , but now taking in account that $u^\\star (\\Lambda )$ has additional terms compared to Case (1), $e^{-u_N^\\star (\\Lambda )}-1=-u^\\star _N(\\Lambda )+\\frac{1}{2} u^\\star _N(\\Lambda )^2 = \\sigma _a\\sigma _b (q_N+\\gamma _{a,N}\\gamma _{b,N}) + \\sigma _a \\gamma _{a,N} +\\sigma _b\\gamma _{b,N},$ and therefore $A =(\\sigma _a\\sigma _b(q_N+\\gamma _{a,N}\\gamma _{b,N})+\\sigma _a \\gamma _{a,N}+\\sigma _b\\gamma _{b,N})(1 + \\tilde{u}_{N,N} - |\\Lambda _N|\\tilde{a}_{N,N} \\psi \\bar{\\psi }).$ Since in Case (2) each term in $V^\\star _N(\\Lambda )$ contains a factor of $\\psi \\bar{\\psi }$ , we have $V^\\star _N(\\Lambda )^2=0$ and thus $B = (1-u_N^\\star (\\Lambda ))_{t_NQ_N}\\theta (-V^\\star _N(\\Lambda )+K^\\star _N(\\Lambda ))$ Therefore $B&= \\sigma _a k_N^{\\sigma _a}+\\sigma _b k_N^{\\sigma _b}+\\sigma _a\\sigma _b (\\gamma _{a,N} k_N^{\\sigma _b} + \\gamma _{b,N} k_N^{\\sigma _a} + k_N^{\\sigma _a\\sigma _b})&\\qquad +_{t_NQ_N}[\\sigma _a\\bar{\\psi }\\psi \\tilde{\\lambda }_{a,N}+ \\sigma _a \\sigma _b\\bar{\\psi }\\psi (\\eta _N+r_N +\\gamma _{a,N}\\tilde{\\lambda }_{b,N}+\\gamma _{b,N}\\tilde{\\lambda }_{a,N} + k^{\\sigma _a\\sigma _b\\bar{\\psi }\\psi })]$ where we have set $\\tilde{\\lambda }_{x,N} = \\lambda _{x,N} + k_N^{\\sigma _x\\bar{\\psi }\\psi }$ .",
"Taking the expectation and collecting all terms gives $\\tilde{Z}_{N,N}^{\\sigma _a\\sigma _b}&= (q_N+\\gamma _{a,N}\\gamma _{b,N}) (1+\\tilde{u}_{N,N})&\\qquad \\qquad + (\\eta _N+r_N +\\tilde{\\lambda }_{a,N}\\gamma _{b,N}+\\tilde{\\lambda }_{b,N}\\gamma _{a,N} + k^{\\sigma _a\\sigma _b\\bar{\\psi }\\psi })t_N|\\Lambda _N|^{-1}&\\qquad \\qquad +\\gamma _{a,N} k_N^{\\sigma _b} + \\gamma _{b,N} k_N^{\\sigma _a} + k_N^{\\sigma _a\\sigma _b}\\\\\\tilde{Z}_{N,N}^{\\sigma _a}&=\\gamma _{a,N} (1+\\tilde{u}_{N,N}) + \\tilde{\\lambda }_{a,N}t_N|\\Lambda _N|^{-1}+ k_N^{\\sigma _a}.$ The bounds on the constants $k_N^\\#$ are analogous to those in Case (1)."
],
[
"Analysis of one-point functions",
"We now analyse the observable flow given by Lemma REF to derive the asymptotics of the correlation functions.",
"Note that the coupling constants $\\lambda _{x,j}$ and $\\gamma _{x,j}$ can possibly depend on $x=a,b$ as the contributions from $K$ can depend on the relative position of the points in the division of $\\Lambda _N$ into blocks.",
"The following lemma shows that in the limit $j\\rightarrow \\infty $ they become independent of $x$ ; an analogous argument was used in [13].",
"Under the hypotheses of Proposition there are $\\lambda _\\infty ^{(p)} = \\lambda _0^{(p)} +O_L(\\lambda _0^{(p)}b_0)$ and $\\gamma _\\infty = O_L(\\lambda _0^{(2)})$ , all continuous in $m^{2}\\ge 0$ and $b_{0}$ small, such that for $x\\in \\lbrace a,b\\rbrace $ , $ \\lambda _{x,j}^{(p)} = \\lambda _\\infty ^{(p)} + O_L(\\lambda _0^{(p)} b_0 L^{-\\kappa j}),\\qquad \\gamma _{x,j}=\\gamma _\\infty + O_L(\\lambda _0^{(2)} b_0 L^{-(d-2+\\kappa )j}).$ In Case (1), $\\lambda _\\infty ^{(1)} = \\lambda _0^{(1)}$ .",
"In Case (2), $\\lambda _\\infty ^{(2)} = \\lambda _0^{(2)} + O_L(\\lambda _0^{(2)}b_0)$ and $\\gamma _\\infty ^{(2)} =\\lambda _\\infty ^{(2)}(-\\Delta ^{^d}+m^2)^{-1}(0,0)+O_L(\\lambda _0^{(2)}b_0)$ , and with the abbreviations $\\lambda \\lambda ^{(2)}$ and $\\gamma \\gamma ^{(2)}$ , $ {\\bar{\\psi }_a\\psi _a} =\\frac{\\gamma _\\infty }{\\lambda _0}+ \\frac{\\frac{\\lambda _{\\infty }}{\\lambda _0} t_N |\\Lambda _N|^{-1}+ O_L(b_0L^{-(d-2+\\kappa )N}) + O_L(b_0L^{-\\kappa N}(m^{2}|\\Lambda _N|)^{-1})}{1+\\tilde{u}_{N,N}}.$ We will typically drop the superscript $(p)$ .",
"In both cases, we have already seen that $\\lambda _{x,j} = \\lambda _0 + \\sum _{k=0}^{j-1} O_L(\\Vert K^\\star _k\\Vert _k) = \\lambda _0 + \\sum _{k=0}^{j-1} O_L(\\lambda _0b_0L^{-\\kappa k}) .$ Since the $K^\\star _k$ are independent of $N$ for $k<N$ (by Proposition REF for the extended renormalisation group map, see Section REF ), the limit $\\lambda _{x,\\infty }$ makes sense, exists, and $|\\lambda _{x,j}-\\lambda _{x,\\infty }| = O_L(\\lambda _0b_0L^{-\\kappa j})$ .",
"Similarly, in Case (2), by Lemma REF and $\\ell _{x,j}^{-1} = \\ell _{j}^2 =O_L(L^{-(d-2)j})$ , $\\gamma _{x,j}&= \\sum _{k=0}^{j-1} { \\lambda _{x,k} C_{k+1}(x,x) + O_L(L^{-(d-2)k}\\Vert K^\\star _k\\Vert _k)}&= \\sum _{k=0}^{j-1} { \\lambda _{x,\\infty } C_{k+1}(x,x) + O_L(\\lambda _0 b_0 L^{-(d-2+\\kappa )k})}= \\gamma _{x,\\infty } + O_L(\\lambda _0 b_0 L^{-(d-2+\\kappa )j}).$ In particular, we have $\\gamma _{x,\\infty }= \\lambda _{x,\\infty }\\sum _{k=0}^\\infty C_{k+1}(0,0) + O_L(\\lambda _0b_0)= \\lambda _{x,\\infty }(-\\Delta ^{^d}+m^2)^{-1}(0,0) + O_L(\\lambda _0b_0).$ The continuity claims follow from the continuity of the covariances $C_{j}$ in $m^{2}\\ge 0$ , of the renormalisation group coordinates $K_j$ , and that both $\\lambda _\\infty $ and $\\gamma _\\infty $ are uniformly convergent sums of terms continuous in $b_{0}$ and $m^{2}\\ge 0$ .",
"To show that in Case (1) $\\lambda _{x,\\infty }^{(1)}=\\lambda _0^{(1)}$ , which is in particular independent of $x$ , we argue as in the proof of [13].",
"On the one hand, Lemma REF implies as $N\\rightarrow \\infty $ with $m^2>0$ fixed, $\\partial _{\\bar{\\psi }} \\partial _{\\sigma _a} \\tilde{Z}_{N,N}|_{0}= \\lambda _{a,N} + O_L(\\ell _N^{-1}\\ell _{x,N}^{-1} \\Vert K^\\star _N\\Vert _N)= \\lambda _{a,N} + O_L(\\Vert K^\\star _N\\Vert _N)\\xrightarrow[N\\rightarrow \\infty ]{}\\lambda _{a,\\infty },$ where $|_0$ denotes projection onto the degree 0 part, i.e., $\\psi =\\bar{\\psi }=\\sigma =\\bar{\\sigma }=0$ .",
"On the other hand, we claim $\\partial _{\\bar{\\psi }} \\partial _{\\sigma _a} \\tilde{Z}_{N,N}|_{0}= \\lambda _{0,a}m^2(1+\\tilde{u}_{N,N})\\sum _{x\\in \\Lambda _N}{\\bar{\\psi }_0\\psi _x}= \\lambda _{0,a}{1+\\tilde{u}_{N,N}-\\frac{\\tilde{a}_{N,N}}{m^2}}.$ Indeed, the first equality in (REF ) follows analogously to [13]: let $\\Gamma (\\rho ,\\bar{\\rho })$ be as in (REF ), except that $Z_0$ now includes the observable terms $\\sigma _a$ and $\\bar{\\sigma }_b$ and we write $\\rho $ and $\\bar{\\rho }$ for the constant external field to distinguish them from $\\sigma _a$ and $\\bar{\\sigma }_b$ .",
"Then as in (REF ) $-\\sum _{x\\in \\Lambda _N} {\\psi _x}_{\\sigma _a,\\bar{\\sigma }_b} =\\partial _{\\bar{\\rho }} \\Gamma (\\rho ,\\bar{\\rho })|_{\\rho =\\bar{\\rho }=0} = m^{-2} \\frac{\\partial _{\\bar{\\psi }}\\tilde{Z}_{N,N}|_{\\psi =\\bar{\\psi }=0}}{\\tilde{Z}_{N,N}|_{\\psi =\\bar{\\psi }=0}}$ and ${\\cdot }_{\\sigma _a,\\bar{\\sigma }_b}$ denotes the expectation that still depends on the observable fields $\\sigma _a$ and $\\bar{\\sigma }_b$ .",
"Differentiating with respect to $\\sigma _a$ and setting $\\bar{\\sigma }_b=0$ gives $\\lambda _{0,a}\\sum _{x\\in \\Lambda _N} {\\bar{\\psi }_a\\psi _x}= -m^{-2} \\frac{\\partial _{\\sigma _a}\\partial _{\\bar{\\psi }}\\tilde{Z}_{N,N}|_{0}}{\\tilde{Z}_{N,N}|_0}= m^{-2} \\frac{\\partial _{\\bar{\\psi }}\\partial _{\\sigma _a}\\tilde{Z}_{N,N}|_{0}}{1+\\tilde{u}_{N,N}}$ which is the first equality of (REF ) upon rearranging.",
"The second equality in (REF ) follows from Proposition .",
"The right-hand side of (REF ) converges to $\\lambda _0$ in the limit $N\\rightarrow \\infty $ with $m^2>0$ fixed since $\\tilde{a}_{N,N}= a_N - k_N^2/|\\Lambda _N| =O_L(L^{-2N}\\Vert V_N\\Vert _N)+O_L(L^{-2N}\\Vert K_N\\Vert _N) \\rightarrow 0$ and $\\tilde{u}_{N,N}= k_N^0 + \\tilde{a}_{N,N}t_N = O(\\Vert K_N\\Vert _N) + \\tilde{a}_{N,N}t_N \\rightarrow 0$ when $m^2>0$ is fixed.",
"Since the left-hand sides of (REF )–(REF ) are equal, we conclude that $\\lambda _{a,\\infty }=\\lambda _0$ when $m^2>0$ .",
"By continuity this identity then extends to $m^2=0$ .",
"In Case (2), to show (REF ), we use (REF ), that Proposition REF implies $\\Vert K^\\star _N\\Vert _N = O_L(\\lambda _0b_0L^{-\\kappa N})$ , and $\\ell _{x,N}^{-1}\\ell _N^{-2} = 1$ and $\\ell _{x,N}^{-1}= \\ell _N^{2} = O_L(L^{-(d-2)N})$ to obtain $\\frac{\\tilde{Z}_{N,N}^{\\sigma _a}}{1+\\tilde{u}_{N,N}}= \\gamma _{a,N} +\\frac{\\lambda _{a,\\infty } t_N |\\Lambda _N|^{-1} + O_L(\\lambda _0 b_0L^{-(d-2+\\kappa )N}) + O_L( \\lambda _0b_0 L^{-\\kappa N} (m^{2}|\\Lambda _N|)^{-1})}{1+\\tilde{u}_{N,N}}.$ Since ${\\bar{\\psi }_a\\psi _a} =\\tilde{Z}_{N,N}^{\\sigma _a}/(\\lambda _0 (1+\\tilde{u}_{N,N}))$ this gives (REF ).",
"In particular, by the translation invariance of ${\\bar{\\psi }_a\\psi _a}$ , taking $N\\rightarrow \\infty $ implies both $\\gamma _{a,\\infty }$ and $\\lambda _{a,\\infty }$ are in fact independent of $a$ .",
"Taking $\\lambda _0>0$ small enough, the proposition follows immediately from Lemma REF with $\\lambda = \\lambda _\\infty ^{(2)}/\\lambda _0^{(2)}$ and $\\gamma = \\gamma _\\infty ^{(2)}/\\lambda _0^{(2)}$ ,"
],
[
"Analysis of two-point functions",
"Next we derive estimates for the two-point functions.",
"Under the hypotheses of Proposition , and in terms of the same $\\lambda _\\infty $ and $\\gamma _\\infty $ as in Lemma REF , ${\\bar{\\psi }_a\\psi _b}&=W_N(a-b)+ \\frac{t_N|\\Lambda _N|^{-1}}{1+\\tilde{u}_{N,N}}&\\qquad + O_L(\\frac{b_0}{\\lambda _0}|a-b|^{-(d-2+\\kappa )})+O_L(\\frac{b_0}{\\lambda _0} |a-b|^{-\\kappa })\\frac{(m^2|\\Lambda _N|)^{-1}}{1+\\tilde{u}_{N,N}}\\\\{\\bar{\\psi }_a\\psi _a\\bar{\\psi }_b\\psi _b}&= -\\frac{\\lambda _\\infty ^2}{\\lambda _0^2} W_N(a-b)^2 +\\frac{\\gamma _\\infty ^2}{\\lambda _0^2}+\\frac{-2\\lambda _\\infty ^2 W_N(a-b)+ 2\\lambda _{\\infty }\\gamma _{\\infty }}{\\lambda _0^2(1+\\tilde{u}_{N,N})}t_N|\\Lambda _N|^{-1}&\\qquad + O_L(\\frac{b_0}{\\lambda _0} |a-b|^{-2(d-2)-\\kappa })+ O_L(\\frac{b_0}{\\lambda _0}L^{-(d-2+\\kappa )N})&\\qquad +(O_L(\\frac{b_0}{\\lambda _0}|a-b|^{-(d-2+\\kappa )})+O_L(\\frac{b_0}{\\lambda _0}L^{-\\kappa N}))\\frac{(m^{2}|\\Lambda _N|)^{-1}}{1+\\tilde{u}_{N,N}}.$ The proofs of (REF ) and () corresponding to Cases (1) and (2) are again analogous.",
"Case (1).",
"By Lemma REF and Lemmas REF and REF , $ \\lambda _{x,j}= \\lambda _\\infty + O_L( \\lambda _0b_0 L^{-\\kappa })= \\lambda _0 + O_L( \\lambda _0b_0 L^{-\\kappa }),\\qquad r_j=O_L(\\lambda _0b_0 |a-b|^{-\\kappa })1_{j\\ge j_{ab}}.$ Using that $\\ell _{ab,j}^{-1}\\Vert K^{ab}_j\\Vert _j \\le O_L(\\lambda _0b_0L^{-(d-2+\\kappa )j})1_{j\\ge j_{ab}}$ and $C_{j+1}(a,b) \\le C_{j+1}(0,0) \\le O_L(L^{-(d-2)j})$ it then follows from Lemma REF that $q_{N}&= \\sum _{j=j_{ab}-1}^N { \\lambda _{a,j}\\lambda _{b,j} C_{j+1}(a,b) + r_j C_{j+1}(0,0) + O_L(\\lambda _0 b_0 L^{-(d-2+\\kappa )j}) }&= \\lambda _0^2 \\sum _{j=1}^N C_j(a,b)+ O_L(\\lambda _0 b_0 |a-b|^{-(d-2)-\\kappa })&= \\lambda _0^2 W_N(a-b) + O_L(\\lambda _0 b_0 |a-b|^{-(d-2)-\\kappa }),$ where we have used (REF ), $|a-b|\\le L$ , that $C_j(a,b)=0$ for $j<j_{ab}$ , and that $W_N(x-y)=C_1(x,y) + \\cdots + C_N(x,y)$ .",
"By (), using that $\\ell _{ab,N}^{-1}\\ell _N^{-2}=1$ and again (REF ), therefore $\\frac{\\tilde{Z}_{N,N}^{\\bar{\\sigma }_b\\sigma _a}}{1+\\tilde{u}_{N,N}}&= \\lambda _0^2 W_N(a-b)+ O_L(\\lambda _0b_0 |a-b|^{-(d-2)-\\kappa })&\\qquad + \\frac{\\lambda _0^2 t_N |\\Lambda _N|^{-1}+ O_L(\\lambda _0b_0 |a-b|^{-\\kappa } m^{-2}|\\Lambda _N|^{-1})}{1+\\tilde{u}_{N,N}}.$ Since ${\\bar{\\psi }_a\\psi _b} =\\tilde{Z}_{N,N}^{\\bar{\\sigma }_b\\sigma _a}/(\\lambda _0^2 (1+\\tilde{u}_{N,N}))$ and $|\\lambda _0| \\le 1$ , the claim for the two-point function follows.",
"Case (2).",
"Again, the analogue of (REF ) holds: $\\lambda _{x,j} = \\lambda _\\infty + O_L(b_0\\lambda _0 L^{-\\kappa j}),\\qquad r_{j} = O_L(b_0\\lambda _0|a-b|^{-(d-2+\\kappa )})1_{j\\ge j_{ab}}.$ The first estimate is by Lemma REF , the second by Lemma REF .",
"Since $C_{k}(a,b)=0$ for $k<j_{ab}$ and $C_{k+1}(a,b) \\le O_L(L^{-(d-2)k})$ , then by Lemma REF and as $|a-b|\\le L$ , $\\eta _{j}=-2 \\sum _{k=j_{ab}-1}^{j-1} \\lambda _{a,k}\\lambda _{b,k} C_{k+1}(a,b)=-2 \\lambda _\\infty ^2 \\sum _{k=1}^{j} C_k(a,b) + O_L(b_0\\lambda _0|a-b|^{-(d-2)-\\kappa }).$ Note that $\\sum _{k\\ge j_{ab}-1}|r_{k}| C_{k+1}(0,0)\\le O_L(b_0\\lambda _0|a-b|^{-(d-2+\\kappa )})\\sum _{k\\ge j_{ab}}L^{-(d-2)j}\\le O_L(b_0\\lambda _0|a-b|^{-2(d-2)-\\kappa }).$ As a result, again by Lemma REF , these bounds together then give $q_{N}&= \\sum _{k\\le N} [\\eta _{k-1} C_{k}(a,b) - \\lambda _\\infty ^2 C_k(a,b)^2]+ O_L(b_0\\lambda _0|a-b|^{-2(d-2)-\\kappa })&= - \\lambda _\\infty ^2\\sum _{k\\le N} [2\\sum _{l < k} C_{l}(a,b)C_k(a,b) + C_k(a,b)^2] + O_L(b_0\\lambda _0|a-b|^{-2(d-2)-\\kappa })&= - \\lambda _\\infty ^2 { \\sum _{k \\le N} C_k(a,b)}^2 + O_L(b_0\\lambda _0|a-b|^{-2(d-2)-\\kappa }).&= - \\lambda _\\infty ^2 W_N(a-b)^2 + O_L(b_0\\lambda _0|a-b|^{-2(d-2)-\\kappa }).$ We finally substitute these estimates into ().",
"Using also that $\\ell _{ab,N}^{-1}\\ell _N^{-2} = \\ell _{j_{ab}}^{2} = O_L(|a-b|^{-(d-2)})$ , that $\\ell _{x,N}^{-1}\\ell _N^{-2} = O_L(1)$ , that $\\gamma _{x,N}=\\gamma _\\infty +O_L(b_0\\lambda _0L^{-(d-2+\\kappa )N})$ by (REF ), and $\\Vert K^\\star _N\\Vert _N \\le O_L(b_0\\lambda _0L^{-\\kappa })$ , we obtain $\\frac{\\tilde{Z}^{\\sigma _a\\sigma _b}_{N,N}}{1+\\tilde{u}_{N,N}}&= -\\lambda _\\infty ^2 W_N(a-b)^2 + \\gamma _\\infty ^2+O_L(b_0\\lambda _0|a-b|^{-2(d-2)-\\kappa }) + O(b_0\\lambda _0L^{-(d-2+\\kappa )N})&\\qquad +\\frac{-2\\lambda _\\infty ^2 W_N(a-b)+ 2\\lambda _{\\infty }\\gamma _{\\infty }}{1+\\tilde{u}_{N,N}}t_N|\\Lambda _N|^{-1}&\\qquad + \\frac{O(b_0\\lambda _0L^{-\\kappa N}m^{-2}|\\Lambda _N|^{-1}) + O_L(b_0\\lambda _0|a-b|^{-(d-2+\\kappa )}m^{-2}|\\Lambda _N|^{-1})}{1+\\tilde{u}_{N,N}}$ which gives () since ${\\bar{\\psi }_a\\psi _a\\bar{\\psi }_b\\psi _b} = \\tilde{Z}_{N,N}^{\\sigma _a\\sigma _b}/(\\lambda _0^2(1+\\tilde{u}_{N,N}))$ .",
"The proposition follows immediately from Lemma REF with the same $\\lambda $ and $\\gamma $ as in Proposition ."
],
[
"Proof of Theorems ",
"By summation by parts on the whole torus $\\Lambda _N$ , we have $y_0(\\nabla \\psi ,\\nabla \\bar{\\psi }) + \\frac{z_0}{2}{ (-\\Delta \\psi ,\\bar{\\psi })+(\\psi ,-\\Delta \\bar{\\psi }) }= (y_0+z_0) (\\nabla \\psi ,\\nabla \\bar{\\psi }).$ Given $m^2\\ge 0$ and $b_0$ small, we choose $V_0^c(b_0,m^2)$ as in Theorem REF .",
"This defines the functions $s_0^c = y_0^c+z_0^c$ and $a_0^c$ in (REF ) with the required regularity properties.",
"The claims for the correlation functions and the partition function then follow from Propositions and –.",
"The continuity of $u_{N}^c$ follows from the continuity of $V_{0}^{c}$ and the continuity of the renormalisation group maps.",
"For Theorem REF , note that the statements simplify by the assumption $m^2 \\ge L^{-2N}$ .",
"Indeed, using that $(m^{2}|\\Lambda _N|)^{-1} \\le L^{-(d-2)N}$ and $|a_N| \\le O_L(b_0L^{-(2+\\kappa )N})$ , by Proposition , we have that $|\\tilde{a}_{N,N}|\\le O_L(b_0L^{-(2+\\kappa )N})$ and $|\\tilde{u}_{N,N}| \\le O_L(b_0L^{-\\kappa N})$ ."
],
[
"Proof of Proposition ",
"For any graph $G=(\\Lambda ,E)$ with edge weights $(\\beta _{xy})$ and vertex weights $(h_x)$ , the partition function appearing in (REF ) can be generalised to $ Z_{\\beta ,h}= \\sum _{F\\in } \\prod _{xy\\in F}\\beta _{xy}\\prod _{T\\in F} (1+\\sum _{x\\in T}h_{x}),$ where $$ is the set of forest subgraphs of $G$ .",
"Recall from the discussion above (REF ) that expanding the product over $T$ in (REF ) can be interpreted as choosing, for each $T$ , either (i) a root vertex $x\\in T$ with weight $h_{x}$ or (ii) leaving $T$ unrooted.",
"This interpretation will be used in Lemma REF .",
"By [18] (which follows [34]), $ Z_{\\beta ,h}=\\int \\prod _{x\\in \\Lambda } \\partial _{\\eta _x}\\partial _{\\xi _x} \\frac{1}{z_x} e^{\\sum _{xy}\\beta _{xy}(u_x\\cdot u_y+1) - \\sum _{x} h_x(z_x-1)}.$ Moreover, by [18], if $h=0$ then $ ¶_{\\beta ,0}[x\\leftrightarrow y] = -{u_0\\cdot u_x}_{\\beta ,0}= -{z_0z_x}_{\\beta ,0} = {\\xi _x\\eta _y}_{\\beta ,0}= 1-{\\xi _x\\eta _x\\xi _y\\eta _y}_{\\beta ,0}.$ Proposition REF follows easily from this.",
"For convenience, we restate the proposition as follows.",
"In the statement and throughout this appendix, inequalities like $\\beta \\ge 0$ are to be interpreted pointwise, i.e., $\\beta _{xy}\\ge 0$ for all edges $xy$ .",
"For any finite graph $G$ , any $\\beta \\ge 0$ and $h \\ge 0$ , $ ¶_{\\beta ,h}[0\\leftrightarrow \\mathfrak {g}]&= {z_0}_{\\beta ,h},\\\\¶_{\\beta ,h}[0\\leftrightarrow x, 0 \\lnot \\leftrightarrow \\mathfrak {g}]&= {\\xi _0\\eta _x}_{\\beta ,h},\\\\¶_{\\beta ,h}[0\\leftrightarrow x]+¶_{\\beta ,h}[0\\lnot \\leftrightarrow x, 0\\leftrightarrow \\mathfrak {g}, x\\leftrightarrow \\mathfrak {g}]&= -{u_0\\cdot u_x}_{\\beta ,h},$ and the normalising constants in (REF ) and (REF ) are equal.",
"In particular, $ ¶_{\\beta ,0}[0\\leftrightarrow x] = -{u_0\\cdot u_x}_{\\beta ,0} = -{z_0z_x}_{\\beta ,0} = {\\xi _0\\eta _x}_{\\beta ,0} = 1-{\\xi _0\\eta _0\\xi _x\\eta _x}_{\\beta ,0}.$ For notational ease, we write the proof for constant $h$ .",
"The equality of the normalising constants is a special case of (REF ).",
"To see (REF ), we use that $(z_{0}-1)^2=0$ so that $z_{0}=1-(1-z_{0})=e^{-(1-z_{0})}$ .",
"As a result ${z_{0}}_{\\beta ,h} = Z_{\\beta ,h-1_{0}} / Z_{\\beta ,h}$ , and (REF ) gives $ {z_{0}}_{\\beta ,h} = _{\\beta ,h} \\frac{h|T_0|}{1+h|T_0|} = ¶_{\\beta ,h}[0\\leftrightarrow \\mathfrak {g}].$ Similarly, ${z_0z_x} = Z_{\\beta ,h-1_0-1_x}/Z_{\\beta ,h}$ and thus (REF ) shows that $ {z_0z_x}_{\\beta ,h}&=_{\\beta ,h} \\frac{-1+h|T_{0}|}{1+h|T_{0}|}1_{0\\leftrightarrow x} + _{\\beta ,h} \\frac{h|T_{0}|}{1+h|T_{0}|} \\frac{h|T_{x}|}{1+h|T_{x}|}1_{0\\lnot \\leftrightarrow x}&= ¶_{\\beta ,h}[0\\leftrightarrow x] -2¶_{\\beta ,h}[0\\leftrightarrow x,0\\lnot \\leftrightarrow \\mathfrak {g}] + ¶_{\\beta ,h}[0\\lnot \\leftrightarrow x,0\\leftrightarrow \\mathfrak {g}, x\\leftrightarrow \\mathfrak {g}].$ To see (), we note that the left-hand side is the connection probability in the amended graph $G^{\\mathfrak {g}}$ .",
"From (REF ) with $\\beta _{xy} = \\beta $ for $x,y\\in \\Lambda $ and $\\beta _{x\\mathfrak {g}} = h$ for $x\\in \\Lambda $ we thus obtain the claim: $ -{u_0\\cdot u_x}_{\\beta ,h} =¶_{\\beta ,h}[0\\leftrightarrow x]+¶_{\\beta ,h}[0\\lnot \\leftrightarrow x, 0\\leftrightarrow \\mathfrak {g}, x\\leftrightarrow \\mathfrak {g}].$ To see (), we combine (REF ) and (REF ) to get $2{\\xi _0\\eta _x}_{\\beta ,h} = -{u_{0}\\cdot u_{x}}_{\\beta ,h} - {z_{0}z_{x}}_{\\beta ,h} = 2¶_{\\beta ,h}[0\\leftrightarrow x, 0\\lnot \\leftrightarrow \\mathfrak {g}].$ Finally, (REF ) is (REF ).",
"This completes the proof.",
"The amended graph $G^{\\mathfrak {g}}$ allows $z$ -observables to be interpreted in terms of edges connecting vertices in the base graph $G$ to $\\mathfrak {g}$ .",
"To state this, we denote by $\\lbrace x\\mathfrak {g}\\rbrace $ the event the edge between $x$ and $\\mathfrak {g}$ is present.",
"The next lemma will be used in Appendix REF .",
"$ h_{0}{z_0-1}_{\\beta ,h} &= ¶_{\\beta ,h}[0\\mathfrak {g}]\\\\h_{0}h_{x}{z_0-1;z_x-1}_{\\beta ,h} &=¶_{\\beta ,h}[0\\mathfrak {g},x\\mathfrak {g}]-¶_{\\beta ,h}[0\\mathfrak {g}]¶_{\\beta ,h}[x\\mathfrak {g}]$ As discussed above, after expanding the product in (REF ) the external fields $h_{x}$ can be viewed as edge weights for edges from $x$ to $\\mathfrak {g}$ .",
"With this in mind the formulas follow by differentiating (REF )."
],
[
"High-temperature phase and positive external field",
"If $\\beta <p_{c}(d)/(1-p_{c}(d))$ , then $\\theta _{d}(\\beta )=0$ .",
"In finite volume, Holley's inequality implies the stochastic domination $¶^{\\Lambda _{N}}_{\\beta ,h}\\preceq ¶^{\\Lambda _{N}}_{p,r}$ , where the latter measure is Bernoulli bond percolation on the amended graph with $p=\\beta /(1+\\beta )$ and $r=h/(1+h)$ , see [18].",
"In particular, $¶^{\\Lambda _{N}}_{\\beta ,h}[0\\leftrightarrow \\mathfrak {g}] \\le ¶^{\\Lambda _{N}}_{p,r}[0\\leftrightarrow \\mathfrak {g}].$ Since each edge to the ghost is chosen independently with probability $r$ , this latter quantity is $¶^{\\Lambda _{N}}_{p,r}[0\\leftrightarrow \\mathfrak {g}] = \\sum _{n=1}^{|\\Lambda _{N}|}¶^{\\Lambda _{N}}_{p,r}[{C_0}=n](1-(1-r)^{n}) \\le r^{\\Lambda _{N}}_{p,r}{C_0}$ since $1-(1-r)^{n}\\le rn$ for $0\\le r\\le 1$ .",
"Here $C_0$ is the cluster of the origin on the torus without the ghost site, so $^{\\Lambda _{N}}_{p,r}{C_0}=^{\\Lambda _{N}}_{p,0}{C_0}$ .",
"Now suppose $\\beta $ is such that $p<p_{c}(d)$ .",
"Then the right-hand side is finite and uniformly bounded in $N$ .",
"Hence $\\theta _d(\\beta ) =\\lim _{h\\rightarrow 0}\\lim _{N\\rightarrow \\infty }¶^{\\Lambda _{N}}_{\\beta ,h}[0\\leftrightarrow \\mathfrak {g}]\\le \\lim _{r\\rightarrow 0}r \\sup _{N}^{\\Lambda _{N}}_{p,0}{C_0} = 0.$ Let $h>0$ and suppose that for all $x$ , $h_x= h$ .",
"Then there are $c,C>0$ depending on $d,\\beta ,h$ such that $¶^{\\Lambda _{N}}_{\\beta ,h}[0\\leftrightarrow x,0\\mathfrak {g}] \\le Ce^{-c|x|},\\qquad ¶^{\\Lambda _{N}}_{\\beta ,h}[0\\leftrightarrow x,0\\lnot \\leftrightarrow \\mathfrak {g}] \\le Ce^{-c|x|}$ We begin with the inequality on the left of (REF ).",
"Define $(0\\leftrightarrow x)$ to be the set of forests in which both 0 is connected to $x$ and $T_{0}$ is rooted at 0, and $$ the set of all forests.",
"In this argument we treat $$ as being a set of (possibly) rooted forests, i.e., we identify edges to $\\mathfrak {g}$ with roots.",
"Without loss of generality we may assume $x\\cdot e_{1} \\ge \\alpha |x|$ for a fixed $\\alpha >0$ .",
"Note that if $F\\in (0\\leftrightarrow x)$ there is a unique path $\\gamma _{F}$ from 0 to $x$ in $F$ , and there are at least $\\alpha |x|$ edges of the form $\\lbrace u,u+e_{1}\\rbrace $ in $\\gamma _{F}$ .",
"We define a map $S\\colon (0\\leftrightarrow x) \\rightarrow 2^{}$ by, for $F\\in (0\\leftrightarrow x)$ , choosing a subset $\\lbrace u_{i},v_{i}\\rbrace $ of the edges $\\lbrace \\lbrace u,v\\rbrace \\in \\gamma _{F} \\mid v=u+e_{1}\\rbrace $ , and removing each $\\lbrace u_{i},v_{i}\\rbrace $ and rooting the tree containing $v_{i}$ at $v_{i}$ .",
"Thus $S(F)$ is the set of forests that results from all possible choices in the first step.",
"The second step does yield an element of $2^{}$ since $T_{0}$ is rooted at 0, so it cannot be the case that the tree containing $v_{i}$ is already rooted (connected to $\\mathfrak {g}$ ).",
"The map $S$ is injective, meaning that given $\\bar{F}\\in \\bigcup _{F\\in (0\\leftrightarrow x)}S(F)$ there is a unique $F$ such that $\\bar{F} \\in S(F)$ .",
"Indeed, given $\\bar{F}\\in S(F)$ , $F$ can be reconstructed as follows.",
"In $\\bar{F}$ , either the tree containing $x$ contains 0, or else it is rooted at a unique vertex $v^{\\prime }$ and it is not connected to $u^{\\prime }=v^{\\prime }-e_{1}$ .",
"Set $\\bar{F}^{\\prime }=\\bar{F}\\cup \\lbrace u^{\\prime },v^{\\prime }\\rbrace $ .",
"The previous sentence applies to $\\bar{F}^{\\prime }$ as well, and continuing until a connection to 0 is formed we recover $F$ .",
"This reconstruction was independent of $F$ , and hence if $\\bar{F}_{1}=\\bar{F}_{2}$ , $\\bar{F}_{i}\\in S(F_{i})$ , we have $F_{1}=F_{2}$ .",
"Let $w(F)= h \\beta ^{F}\\prod _{T\\ne T_0}(1+h|V(T)|)$ .",
"Then for $\\bar{F} \\in S(F)$ , $w(\\bar{F}) = w(F) (\\frac{h}{\\beta })^{k}$ if $\\bar{F}$ had $k$ edges removed.",
"Hence if the connection from 0 to $x$ in $F$ has $k$ edges of the form $\\lbrace u,v\\rbrace $ , $v=u+e_{1}$ , $\\sum _{\\bar{F}\\in S(F)}w(\\bar{F}) = (1+\\frac{h}{\\beta })^{k}w(F).$ Let $_{k}(x) \\subset (0\\leftrightarrow x)$ be the set of forests where the connection from 0 to $x$ contains $k$ edges of the form $\\lbrace u,v\\rbrace $ , $v=u+e_{1}$ .",
"We have the lower bound $Z_{\\beta ,h}^{\\Lambda _{N}} = \\sum _{F\\in } \\beta ^{F}\\prod _{T\\in F}(1+h|V(T)|) \\ge \\sum _{k\\ge 0} \\sum _{F\\in _{k}(x)} \\sum _{\\bar{F}\\in S(F)} w(\\bar{F})$ since $S$ is injective and all of the summands are non-negative.",
"Hence we obtain, using (REF ), $¶^{\\Lambda _{N}}_{\\beta ,h}[0\\leftrightarrow x,0\\mathfrak {g}]&\\le \\frac{\\sum _{k\\ge \\alpha |x|} \\sum _{F\\in _{k}(x)}w(F)}{\\sum _{k\\ge 0} \\sum _{F\\in _{k}(x)} \\sum _{\\bar{F}\\in S(F)} w(\\bar{F})} &= \\frac{\\sum _{k\\ge \\alpha |x|} \\sum _{F\\in _{k}(x)}(1+\\frac{h}{\\beta })^{-k}\\sum _{\\bar{F}\\in S(F)}w(\\bar{F})}{\\sum _{k\\ge 0} \\sum _{F\\in _{k}(x)} \\sum _{\\bar{F}\\in S(F)} w(\\bar{F})}\\le (1+\\frac{h}{\\beta })^{-\\alpha |x|}.$ A similar argument applies when $0\\lnot \\leftrightarrow \\mathfrak {g}$ ; this condition is used in the second step defining $S$ to ensure the trees containing the vertices $v_{i}$ are not already connected to $\\mathfrak {g}$ .",
"In this case the weight $w(F)$ does not have the factor $h$ , but the remainder of the argument is identical.",
"If $h>0$ , then there are $c,C>0$ depending on $d,\\beta ,h$ such that Let $h>0$ and suppose that for all $x$ , $h_x= h$ .",
"Then there are $c,C>0$ depending on $d,\\beta ,h$ such that $¶^{\\Lambda _{N}}_{\\beta ,h}[0\\leftrightarrow x] \\le Ce^{-c |x |}.$ Since $¶^{\\Lambda _{N}}_{\\beta ,h}[0\\leftrightarrow x] =¶^{\\Lambda _{N}}_{\\beta ,h}[0\\leftrightarrow x, 0\\leftrightarrow \\mathfrak {g}] +¶^{\\Lambda _{N}}_{\\beta ,h}[0\\leftrightarrow x, 0\\lnot \\leftrightarrow \\mathfrak {g}],$ it is enough to estimate the first term, as the second is covered by Lemma REF .",
"Note $¶^{\\Lambda _{N}}_{\\beta ,h}[0\\leftrightarrow x, 0\\leftrightarrow \\mathfrak {g}]=\\sum _{y} ¶^{\\Lambda _{N}}_{\\beta ,h} [1_{0\\leftrightarrow x}1_{0\\leftrightarrow y}1_{y\\mathfrak {g}}] = \\sum _{y} ¶^{\\Lambda _{N}}_{\\beta ,h} [1_{0\\leftrightarrow x}1_{0\\leftrightarrow y}1_{0\\mathfrak {g}}].$ where the first equality follows from the fact that the only one vertex per component may connect to $\\mathfrak {g}$ , and the second follows from exchangeability of the choice of root.",
"Examining the rightmost expression, there are most $c_{d} |x |^{d}$ summands in which $|y |\\le |x|$ ; for these terms we drop the condition $0\\leftrightarrow y$ .",
"For the rest we drop $0\\leftrightarrow x$ .",
"This gives, by Lemma REF , $¶^{\\Lambda _{N}}_{\\beta ,h}[0\\leftrightarrow x, 0\\leftrightarrow \\mathfrak {g}]\\le C |x|^{d}e^{-c |x |} + \\sum _{ |y |> |x |} Ce^{-c |y |} \\le Ce^{-c |x |},$ where $c,C$ are changing from location to location but depend on $d,\\beta ,h$ only."
],
[
"Infinite volume limit",
"We now discuss weak limits $¶^{^d}_{\\beta }$ obtained by (i) first taking a (possibly subsequential) infinite-volume weak limit $¶^{^d}_{\\beta ,h} = \\lim _{N}¶^{\\Lambda _N}_{\\beta ,h}$ and (ii) subsequently taking a (possibly subsequential) limit $¶^{^d}_{\\beta }=\\lim _{h\\downarrow 0 }¶^{^d}_{\\beta ,h}$ .",
"We do not explicitly indicate the convergent subsequence chosen as what follows applies to any fixed choice.",
"Define $\\theta _{d,N}(\\beta ,h) ¶^{\\Lambda _N}_{\\beta ,h}[0\\leftrightarrow \\mathfrak {g}]=1-h^{-1}¶^{\\Lambda _N}_{\\beta ,h}[0\\mathfrak {g}]$ where the second equality is due to (REF ).",
"Since this last display only involves cylinder events, $\\lim _{N\\rightarrow \\infty }\\theta _{d,N}(\\beta ,h) = 1-h^{-1}¶^{^d}_{\\beta ,h}[0\\mathfrak {g}] \\theta _{d}(\\beta ,h),$ where the last equality defines $\\theta _{d}(\\beta ,h)$ .",
"Assume $\\lim _{h\\downarrow 0}\\theta _{d}(\\beta ,h)=\\theta _{d}(\\beta )$ exists.",
"Then $¶^{^d}_{\\beta }[|T_{0}|=\\infty ] = \\theta _{d}(\\beta ).$ Write $¶_{\\beta ,h}=¶^{^d}_{\\beta ,h}$ .",
"We claim that $¶_{\\beta ,h}[0\\mathfrak {g}] = \\sum _{n\\ge 1} ¶_{\\beta ,h}[|T_{0}|=n] \\frac{h}{1+nh},$ and hence, since $\\theta _{d}(\\beta ,h) = 1-h^{-1}¶_{\\beta ,h}[0\\mathfrak {g}]$ , $\\theta _{d}(\\beta ,h) = 1-\\sum _{n\\ge 1} ¶_{\\beta ,h}[|T_{0}|=n] \\frac{1}{1+nh}.$ Granting the claim, by dominated convergence we obtain $¶_{\\beta ,0}[|T_{0}|<\\infty ] = \\sum _{n\\ge 1}¶_{\\beta ,0}[|T_{0}|=n]=1-\\theta _{d}(\\beta ),$ as desired.",
"To prove the claim, rewrite it as $¶_{\\beta ,h}[|T_{0}|=\\infty , 0\\mathfrak {g}]=\\lim _{r\\rightarrow \\infty }¶_{\\beta ,h}[|T_{0}|\\ge r, 0\\mathfrak {g}]=0.$ The probabilities inside the limit are probabilities of cylinder events, and hence are limits of finite volume probabilities.",
"For a fixed $r$ the probability is at most $h/(1+rh)$ in finite volume, which vanishes as $r\\rightarrow \\infty $ ."
],
[
"Finite range decomposition",
"In this appendix, we give the precise references for the construction of the finite range decomposition (REF ).",
"The general method we use was introduced in [11], and presented in the special case we use in [17] and we will use this reference.",
"For $t>0$ , first recall the polynomials $P_t$ from [17] (these polynomials are called $W_t^*$ in [11]).",
"These are polynomials of degree bounded by $t$ satisfying $ \\frac{1}{\\lambda } = \\int _{0}^\\infty t^2 P_t(\\lambda ) \\, \\frac{dt}{t},\\qquad 0 \\le P_t(u) \\le O_s(1+t^2u)^{-s}$ for any $s>0$ and $u \\in [0,2]$ .",
"Our decomposition (REF ) is defined by $C_1(x,y) &= \\frac{1}{(2d+m^2)|\\Lambda _N|}\\sum _{k \\in \\Lambda _N^*} e^{ik\\cdot (x-y)} \\int _{0}^{\\frac{1}{2} L} t^2 P_t(\\frac{\\lambda (k)+m^2}{2d+m^2}) \\, \\frac{dt}{t}\\\\C_j(x,y) &= \\frac{1}{(2d+m^2)|\\Lambda _N|}\\sum _{k \\in \\Lambda _N^*} e^{ik\\cdot (x-y)} \\int _{\\frac{1}{2} L^{j-1}}^{\\frac{1}{2} L^j} t^2 P_t(\\frac{\\lambda (k)+m^2}{2d+m^2}) \\, \\frac{dt}{t}\\\\C_{N,N}(x,y) &= \\frac{1}{(2d+m^2)|\\Lambda _N|}\\sum _{k \\in \\Lambda _N^*} e^{ik\\cdot (x-y)} \\int _{\\frac{1}{2} L^N}^\\infty t^2 P_t(\\frac{\\lambda (k)+m^2}{2d+m^2}) \\, \\frac{dt}{t},$ where $\\lambda (k) = 4\\sum _{j=1}^{d}\\sin ^{2}(k_{j}/2)$ and $\\Lambda _N^* \\subset [-\\pi ,\\pi )^d$ is the dual torus.",
"The estimates for $C_1, \\dots , C_{N-1}$ are straightforward from these Fourier representations and can be found in [17].",
"We remark that in [17], the torus covariances are defined by periodisation of the finite range covariances on $^d$ ; by Poisson summation this is equivalent to the above definition.",
"The decomposition of $C_{N,N}$ in (REF ) is defined by removing the zero mode from $C_{N,N}$ : $C_{N}(x,y) &= \\frac{1}{(2d+m^2)|\\Lambda _N|}\\sum _{k \\in \\Lambda _N^*:k \\ne 0} e^{ik\\cdot (x-y)} \\int _{\\frac{1}{2} L^N}^\\infty t^2 P_t(\\frac{\\lambda (k)+m^2}{2d+m^2}) \\, \\frac{dt}{t}\\\\t_N &= \\frac{1}{2d+m^2} \\int _{\\frac{1}{2} L^N}^\\infty t^2 P_t(\\frac{m^2}{2d+m^2}) \\, \\frac{dt}{t},$ from which (REF ) is immediate.",
"For $C_{N}$ estimates follows as in [17]: $|C_N(x,y)| &\\le \\frac{1}{|\\Lambda _N|} \\sum _{k \\in \\Lambda _N^*: k \\ne 0} {\\int _{\\frac{1}{2} L^N}^\\infty t^2 P_t(\\frac{\\lambda (k)+m^2}{2d+m^2}) \\, \\frac{dt}{t}}&\\lesssim \\frac{1}{|\\Lambda _N|} \\sum _{k \\in \\Lambda _N^*: k \\ne 0} {\\int _{\\frac{1}{2} L^N}^\\infty t^{2}t^{-2s}|k|^{-2s} \\, \\frac{dt}{t}}&\\lesssim \\frac{L^{2N}}{|\\Lambda _N|} \\sum _{k \\in \\Lambda _N^*: k \\ne 0}L^{-2sN}|k|^{-2s}\\lesssim L^{-(d-2)N} \\int _1^\\infty r^{-2s+d-1} dr \\lesssim L^{-(d-2)N}$ and analogously for the discrete gradients.",
"Finally, by (REF ), $t_N&= \\frac{1}{(2d+m^2)} \\int _{\\frac{1}{2} L^N}^\\infty t^2P_t(\\frac{m^2}{2d+m^2}) \\, \\frac{dt}{t}&= \\frac{1}{m^2}-\\frac{1}{(2d+m^2)} \\int _0^{\\frac{1}{2} L^N} t^2P_t(\\frac{m^2}{2d+m^2}) \\, \\frac{dt}{t}=\\frac{1}{m^2} - O(L^{2N}).$"
],
[
"Proof of Proposition ",
"Proposition REF is essentially [27], with the minor changes of the separation of the coupling constant $u_j$ and the explicit inclusion of $\\theta $ .",
"We include a proof here for convenience.",
"We begin by algebraically manipulating $Z_{j}$ .",
"These manipulations only rely on factorisation properties and not on the precise definitions of $I$ and $K$ .",
"We will clearly state below when we restrict to the context of Proposition REF .",
"Consider $Z_j = \\sum _{X\\in _j} I^{\\Lambda \\setminus X}K(X)$ where $I(B) = e^{-V_j(B)}$ and $K(X)= K_j(X)$ .",
"We will use that $I^Y = \\prod _{B \\in _j(Y)} I(B)$ factors over blocks and $K_{j}(X)$ factors over connected components of $X$ .",
"Given any $\\tilde{I}(B) \\in (B)$ for $B \\in _j$ , $ \\theta Z_j = \\sum _{X\\in _j} \\tilde{I}^{\\Lambda \\setminus X} \\tilde{K}(X),\\qquad \\tilde{K}(X)= \\sum _{Y \\in _j(X)} (\\delta I)^Y \\theta K(X\\setminus Y),$ where $\\delta I(B) = \\theta I(B)-\\tilde{I}(B)$ and $\\tilde{I}^{Y}\\prod _{B\\in _{j}(Y)}\\tilde{I}(B)$ .",
"To see this, insert the identity $\\theta I^{\\Lambda \\setminus X} = \\sum _{Y\\subset \\Lambda \\setminus X} \\tilde{I}^{Y}(\\delta I)^{\\Lambda \\setminus (X\\cup Y)}$ into (REF ); (REF ) then follows by changing the summation index.",
"Using that $\\tilde{K}$ factors over components since $K$ factors over components, we can then define $\\check{K}(Y)$ to be $\\tilde{K}(Y) - \\sum _{B\\in _{j}(Y)}\\theta J(B,Y)$ for any connected polymer $Y$ (and zero otherwise), where $J(B,Y)\\in $ are given.",
"This yields the formula $\\tilde{K}(X) = \\prod _{Y \\in \\operatorname{Comp}(X)} {\\check{K}(Y) + \\sum _{B \\in _j(Y)} \\theta J(B,Y)}.$ Expanding the product, this can be written as $\\tilde{K}(X) = \\sum _{\\check{X} \\subset \\operatorname{Comp}(X)} \\check{K}(\\check{X}) \\prod _{Y \\in \\operatorname{Comp}(X \\setminus \\check{X})} \\sum _{B \\in _j(Y)} \\theta J(B,Y).$ Given the polymer $X \\setminus \\check{X}$ , now define $=(X,\\check{X}) \\subset \\lbrace (B,Y): B\\in _{j}, Y\\in _{j}\\rbrace $ so that $\\prod _{Y \\in \\operatorname{Comp}(X \\setminus \\check{X})} \\sum _{B \\in _j(Y)}J(B,Y) = \\sum _{} \\prod _{(B,Y)\\in } J(B,Y).$ Then $\\tilde{K}(X) = \\sum _{(\\check{X}, )} \\check{K}(\\check{X}) \\prod _{(B,Y) \\in } \\theta J(B,Y).$ By (REF ), since $\\tilde{I}\\in (\\Lambda )$ is a constant with respect to $_{C_{j+1}}$ , $_{C_{j+1}} \\theta Z_j&= \\sum _{X \\in _{j}} \\tilde{I}^{\\Lambda \\setminus X} _{C_{j+1}}\\tilde{K}(X)&= \\sum _{U \\in _{j+1}} \\tilde{I}^{\\Lambda \\setminus U} \\sum _{X \\in _j: \\bar{X} = U} \\tilde{I}^{U\\setminus X} _{C_{j+1}} \\tilde{K}(X)= \\sum _{U \\in _{j+1}} \\tilde{I}^{\\Lambda \\setminus U} K^{\\prime }(U),$ where by (REF ) $K^{\\prime }(U) = \\sum _{X \\in _j: \\bar{X} = U} \\tilde{I}^{U\\setminus X}\\sum _{(\\check{X}, )} _{C_{j+1}} \\check{K}(\\check{X})\\prod _{(B,Y) \\in } \\theta J(B,Y).$ We now specialise to the setting of Proposition REF .",
"Thus we set $\\tilde{I}(B)= e^{-(u+V)_{j+1}(B)}$ and $K_{j+1}$ as in (REF ).",
"Then $_{C_{j+1}} \\theta Z_j$ is $e^{-u_{j+1}|\\Lambda |}\\sum _{U \\in _{j+1}} e^{-V_{j+1}(\\Lambda \\setminus U)} K_{j+1}(U).$ To confirm this we explain the notation used in (REF ).",
"In the definition of $K_{j+1}$ , $X_{}$ is by definition $\\bigcup _{(B,Y)\\in }Y$ .",
"Given $X$ and $\\check{X}$ a subset of components of $X$ , $(X,\\check{X})$ is the set of sets of pairs $(B,Y)$ where (i) $B$ is a block in $Y$ , (ii) $Y$ is a component of $X\\setminus \\check{X}$ , (iii) each component $Y$ occurs in exactly one pair, and (iv) the closure of $\\check{X}\\cup \\cup _{(B,Y)}Y$ is $U$ .",
"To conclude the proof we make the definition that $\\sum _{(U)}$ is a shorthand for the double sum $\\sum _{X\\in _{j}:\\bar{X}=U}\\sum _{(\\check{X},)}$ , as an explicit description of the set $(U)$ will not be needed.",
"What remains is to prove the claims regarding factorisation and automorphism invariance.",
"For factorisation, note that $\\sum _{(U_{1}\\cup U_{2})} = \\sum _{(U_{1})}\\sum _{(U_{2})}$ for $U_{1},U_{2}\\in _{j+1}$ that do not touch.",
"Since $U_{1}$ and $U_{2}$ are distance $L^{j+1}>\\frac{1}{2}L^{j+1}+2^{d+1}L^{j}$ apart in this case the Grassmann integrals in the definition of $K_{j+1}(U_{1}\\cup U_{2})$ factor.",
"Here we have used our standing assumption that $L>2^{d+2}$ , that $J(B,Y)=0$ if $Y\\notin _{j}$ , and that the range of $C_{j+1}$ is $\\frac{1}{2}L^{j+1}$ .",
"Automorphism invariance follows directly from the formula for $K_{j+1}$ ."
],
[
"Acknowledgements",
"We thank David Brydges and Gordon Slade.",
"This article would not have been possible in this form without their previous contributions to the renormalisation group method.",
"We also thank them for the permission to include Figure REF from [17].",
"R.B.",
"was supported by the European Research Council under the European Union's Horizon 2020 research and innovation programme (grant agreement No.",
"851682 SPINRG).",
"N.C. was supported by Israel Science Foundation grant number 1692/17."
]
] | 2107.01878 |
[
[
"Sub-Feller Semigroups Generated by Pseudodifferential Operators on\n Symmetric Spaces of Noncompact Type"
],
[
"Abstract We consider global pseudodifferential operators on symmetric spaces of noncompact type, defined using spherical functions.",
"The associated symbols have a natural probabilistic form that extend the notion of the characteristic exponent appearing in Gangolli's L\\'evy-Khinchine formula to a function of two variables.",
"The Hille-Yosida-Ray theorem is used to obtain conditions on such a symbol so that the corresponding pseudodifferential operator has an extension that generates a sub-Feller semigroup, generalising existing results for Euclidean space."
],
[
"Introduction",
"Pseudodifferential operator theory is a powerful tool in the study of Feller–Markov processes on Euclidean space (see for example [31], [37], [38], [39], [40], or [11] §5 for a summary).",
"Primarily developed by Niels Jacob and collaborators (see e.g.",
"[26], [27], [24]), this framework characterises sub-Feller semigroups and their generators as pseudodifferential operators ($\\Psi $ DOs) acting on $C_0(\\mathbb {R}^d)$ , the Banach space of continuous, real-valued functions on $\\mathbb {R}^d$ that vanish at infinity.",
"The associated symbols capture many properties of the sub-Feller processes and semigroups they represent, generalising the well-established relationship between Lévy processes and their characteristic exponents given by the Lévy–Khinchine formula.",
"The key difference is that the Feller–Markov symbols typically depend on two variables instead of one — in the case of Feller processes, this is sometimes described as the Lévy characteristics having gained spatial dependence.",
"Manifold-valued Feller–Markov processes have also attracted interest in recent years (see [13], [25] and [32] §7 for excellent summaries), though the absence of a global harmonic analysis on general manifolds has so far prevented a $\\Psi $ DO approach.",
"Lie groups and symmetric spaces come with their own harmonic analysis, however, in the form of the spherical transform (see Harish-Chandra [18], [19] and Helgason [22], [21]).",
"A natural question to ask then is to what extent a $\\Psi $ DO-based approach can be applied to the study of sub-Feller processes on Lie groups and symmetric spaces.",
"Much work has already been done in this area, especially in the “constant coefficient” case of Lévy processes, in which the symbol depends only on its second argument.",
"Here, the symbols are given by Gangolli's Lévy–Khinchine formula ([16] Theorem 6.2), a direct analogue to the classical result.",
"The first paper to use probabilistic $\\Psi $ DO methods on a Lie group was [6], where $\\Psi $ DOs are used to study Lévy processes on the Heisenberg group.",
"Pseudodifferential operator representations of semigroups and generators have also been found for Lévy processes on a general Lie group — see [2], [3] for the compact case and [4] Section 5 for the noncompact case.",
"For Feller processes, $\\Psi $ DO representations have been found when the Lie group or symmetric space is compact — see for example the final sections of [7] and [8].",
"This paper seeks to develop a more general theory of $\\Psi $ DOs for symmetric spaces of noncompact type, and apply it to seek conditions on a symbol so that the corresponding $\\Psi $ DO extends to the generator of some sub-Feller process.",
"For the $\\mathbb {R}^d$ case, this question has been studied thoroughly by both Jacob and Hoh — see [26], [27], as well as [24] Chapter 4.",
"Our approach is similar to Jacob [27] and Hoh [24], but where they have used Fourier transform, we use the spherical transform.",
"The spherical transform enjoys many of the same properties as the Fourier transform on $\\mathbb {R}^d$ , and we find that several of the arguments in [27] and [24] generalise directly to this setting.",
"However, there are significant differences between the two settings, and a direct transcription of Jacob and Hoh's arguments is certainly not possible.",
"One notable difference, for example, is the condition (2.2) in Jacob [27], which features a derivative that on a manifold makes little sense.",
"Even on a symmetric space, elements of a tangent space cannot be easily translated from point to point, and a different approach is needed.",
"We take a more operator-theoretic approach, and replace each of the partial derivatives $\\frac{\\partial }{\\partial x_i}$ ($i=1,\\ldots ,d$ ) with the fractional Laplacian $\\sqrt{-\\Delta }$ .",
"This is an exciting object to work with, and has the added advantage that spherical functions form a system of eigenfunctions for this operator.",
"The structure of this paper will be the following.",
"Section presents a summary of necessary concepts and results from harmonic analysis on symmetric spaces, and introduces the system of symbols and $\\Psi $ DOs that will be used later on.",
"We also introduce here the spherical anisotropic Sobolev spaces, a generalisation of the anisotropic Sobolev spaces first considered by Niels Jacob [26], [27].",
"In Section , we consider (a version of) the Hille–Yosida–Ray theorem (see Theorem REF ), and note that, thanks to the work of Applebaum and Ngan [7], [8], one of the conditionsNamely, that an operator must satisfy the positive maximum principle.",
"of the Hille–Yosida–Ray theorem has already been fully addressed.",
"This work motivates our introduction of a class of operators we will call Gangolli operators, which will automatically each be densely defined linear operators on $C_0(K|G|K)$ that satisfy the positive maximum principle.",
"We then prove that Gangolli operators are $\\Psi $ DOs in the sense of Section REF , and derive a formula for their symbols (Theorem REF ).",
"Section is concerned with seeking sufficient conditions for a Gangolli operator $q(\\sigma ,D)$ to extend to the generator a sub-Feller semigroup.",
"Informed by the work of the previous section, this amounts to finding conditions that ensure $\\overline{\\operatorname{Ran}(\\alpha +q(\\sigma ,D))}=C_0(K|G|K)$ for some $\\alpha >0$ (see Theorem REF (REF )).",
"This section perhaps most closely follows the approach of Jacob [27] and Hoh [24] Chapter 4, and where proofs are similar to these sources, we omit detail, and instead aim to emphasise what is different about the symmetric space setting.",
"Full proofs may also be found in my PhD thesis [42].",
"Finally, in Section we present a large class of examples of symbols that satisfy the conditions found in Sections and .",
"Notation.",
"For a topological space $X$ , $\\mathcal {B}(X)$ will denote the Borel $\\sigma $ -algebra associated with $X$ , and $B_b(X)$ the space of bounded, Borel functions from $X\\rightarrow \\mathbb {R}$ , a Banach space with respect to the supremum norm.",
"If $X$ is a locally compact Hausdorff space, then we write $C_0(X)$ for the closed subspace of $B_b(X)$ consisting of continuous functions vanishing at infinity, and $C_c(X)$ for the dense subspace of compactly supported continuous functions.",
"If $X$ is a smooth manifold and $M\\in \\mathbb {N}\\cup \\lbrace \\infty \\rbrace $ , then we write $C^M_c(X)$ for the space of compactly supported $M$ -times continuously differentiable functions on $X$ ."
],
[
"Preliminaries",
"Let $(G,K)$ is a Riemannian symmetric pair, so that $G$ is a connected, semisimple Lie group with finite centre, $K$ is a maximal compact subgroup of $G$ , and, for some nontrivial involution $\\Theta $ on $G$ , $G^\\Theta _0\\subseteq K\\subseteq G^\\Theta ,$ where $G^\\Theta $ is the fixed point set of $\\Theta $ , and $G^\\Theta _0$ is the identity component of $G^\\Theta $ .",
"Let $\\mathfrak {g}$ and $\\mathfrak {k}$ denote the Lie algebras of $G$ and $K$ , respectively.",
"Note that $\\mathfrak {k}$ is the $+1$ eigenspace of the differential $\\theta :=d\\Theta $ ; let $\\mathfrak {p}$ denote the $-1$ eigenspace.",
"In fact, $\\theta $ is a Cartan involution on $\\mathfrak {g}$ , and the corresponding Cartan decomposition is $\\mathfrak {g} = \\mathfrak {p} \\oplus \\mathfrak {k}.$ Let $B$ denote the Killing form of $G$ , defined for each for all $X,Y\\in \\mathfrak {g}$ by $B(X,Y) = \\operatorname{tr}(\\operatorname{ad}X\\operatorname{ad}Y)$ .",
"Since $(G,K)$ is of noncompact type, $B$ is nondegenerate, negative definite on $\\mathfrak {k}$ , and positive definite on $\\mathfrak {p}$ .",
"Fix an $\\operatorname{Ad}(K)$ -invariant inner product $\\langle \\cdot ,\\cdot \\rangle $ on $\\mathfrak {g}$ .",
"Associated with $(G,K)$ is the symmetric space $G/K$ , a smooth Riemannian manifold, with Riemannian structure induced by the restriction of $\\langle \\cdot ,\\cdot \\rangle $ to $\\mathfrak {p}$ .",
"There is a one to one correspondence between functions on $G/K$ and $K$ -right-invariant functions on $G$ , we denote both by $\\mathcal {F}(G/K)$ .",
"Similarly, we identify $K$ -invariant functions on $G/K$ with $K$ -bi-invariant functions on $G$ , and denote both by $\\mathcal {F}(K|G|K)$ .",
"Similar conventions will be used to denote standard subspaces of $\\mathcal {F}(G/K)$ and $\\mathcal {F}(K|G|K)$ ; for example $C(K|G|K)$ will denote both the space of continuous, $K$ -invariant functions on $G/K$ , and the space of continuous, $K$ -bi-invariant functions on $G$ ."
],
[
"Harmonic Analysis on Symmetric Spaces of Noncompact Type",
"For a thorough treatment of this topic, see Helgason [22], [21].",
"Let ${\\bf D}(G)$ denote the set of all left invariant differential operators on $G$ , and let ${\\bf D}_K(G)$ denote the subspace of those operators that are also $K$ -right-invariant.",
"A mapping $\\phi :G\\rightarrow \\mathbb {C}$ is called a spherical if it is $K$ -bi-invariant, satisfies $\\phi (e) =1$ , and is a simultaneous eigenfunction of every element of ${\\bf D}_K(G)$ .",
"Fix an Iwasawa decomposition $G=NAK$ , where $N$ is a nilpotent Lie subgroup of $G$ , and $A$ is Abelian.",
"Let $\\mathfrak {n}$ and $\\mathfrak {a}$ denote respectively the Lie algebras of $N$ and $A$ .",
"For each $\\sigma \\in G$ , let $A(\\sigma )$ denote the unique element of $\\operatorname{\\mathfrak {a}}$ such that $\\sigma \\in Ne^{A(\\sigma )}K$ .",
"Harish-Chandra's integral formula states that every spherical function on $G$ takes the form $\\phi _\\lambda (\\sigma ) = \\int _Ke^{(\\rho +i\\lambda )(A(k\\sigma ))}dk, \\hspace{20.0pt} \\forall \\sigma \\in G,$ for some $\\lambda \\in *$ .",
"Moreover, $\\phi _\\lambda =\\phi _{\\lambda ^{\\prime }}$ if an only if $s(\\lambda )=\\lambda ^{\\prime }$ for some element $s$ of the Weyl group $W$ .",
"A spherical function $\\phi _\\lambda $ is positive definite if and only if $\\lambda \\in $ .",
"The spherical transform of a function $f\\in L^1(K|G|K)$ is the function $\\hat{f}:\\rightarrow \\mathbb {C}$ given by $\\hat{f}(\\lambda ) = \\int _G\\phi _{-\\lambda }(\\sigma )f(\\sigma )d\\sigma , \\hspace{20.0pt} \\forall \\lambda \\in .$ Similarly, given a finite Borel measure $\\mu $ on $G$ , the spherical transform of $\\mu $ is the mapping $\\hat{\\mu }:\\rightarrow \\mathbb {C}$ given by $\\hat{\\mu }(\\lambda ) = \\int _G\\phi _{-\\lambda }(\\sigma )\\mu (d\\sigma ).$ The spherical transform enjoys many useful properties, the most powerful being that it defines an isomorphism of the Banach convolution algebra $L^1(K|G|K)$ with the space $L^1(,\\omega )^W$ of Weyl group invariant elements of $L^1(,\\omega )$ .",
"The Borel measure $\\omega $ is called Plancherel measure, and is given by $\\omega (d\\lambda ) = |\\operatorname{\\bf c}(\\lambda )|^{-2}d\\lambda ,$ where $\\operatorname{\\bf c}$ denotes Harish-Chandra's $\\operatorname{\\bf c}$ -function.",
"According to the spherical inversion theorem, for all $f\\in C_c^\\infty (K|G|K)$ and all $\\sigma \\in G$ , $f(\\sigma ) = \\int _{}\\phi _\\lambda (\\sigma )\\hat{f}(\\lambda )\\omega (d\\lambda ).$ There is also a version of Plancherel's identity for the spherical transform, namely $\\Vert f\\Vert _{L^2(K|G|K)} = \\Vert \\hat{f}\\Vert _{L^2(,\\omega )}, \\hspace{20.0pt} \\forall f\\in C_c^\\infty (K|G|K).$ Let $L^2(,\\omega )^W$ denote the subspace of $L^2(,\\omega )$ consisting of $W$ -invariants.",
"Then the image of $C_c^\\infty (K|G|K)$ under spherical transformation is a dense subspace of $L^2(,\\omega )^W$ , and as such the spherical transform extends to an isometric isomorphism between the Hilbert spaces $L^2(K|G|K)$ and $L^2(,\\omega )^W$ .",
"For more details, see for example Helgason [21] Chapter IV § 7.3, pp. 454.",
"Similarly to classical Fourier theory, the most natural setting for the spherical transform is Schwarz space.",
"A function $f\\in C^\\infty (G)$ is called rapidly decreasing if $\\sup _{\\sigma \\in G}(1+|\\sigma |)^q\\phi _0(\\sigma )^{-1}(Df)(\\sigma ) < \\infty , \\hspace{20.0pt} \\forall D\\in {\\bf D}(G), \\;q\\in \\mathbb {N}\\cup \\lbrace 0\\rbrace ,$ where $|\\sigma |$ denotes the geodesic distance between the double coset spaces $K\\sigma K$ and $KeK$ .",
"Equivalently, if $\\sigma \\in G$ is decomposed according to the Cartan decomposition $G=K\\overline{\\exp \\operatorname{\\mathfrak {a}}^+}K$ , so that $\\sigma =ke^Hk^{\\prime }$ for some unique $H\\in \\overline{\\operatorname{\\mathfrak {a}}^+}$ , then $|\\sigma |=|H|$ — see Gangolli and Varadarajan [17] pp.167 for more details on this object.",
"The ($K$ -bi-invariant) Schwarz space $\\mathcal {S}(K|G|K)$ is the Fréchet space comprising of all rapidly decreasing, $K$ -bi-invariant functions $f\\in C^\\infty (G)$ , together with the family of seminorms given by the left-hand side of (REF ).",
"By viewing the spaces $\\operatorname{\\mathfrak {a}}$ and $$ as finite dimensional vector spaces, we also consider the classical Schwartz spaces $\\mathcal {S}()$ and $\\mathcal {S}(\\operatorname{\\mathfrak {a}})$ , as well as $W$ -invariant subspaces $\\mathcal {S}(\\operatorname{\\mathfrak {a}})^W$ and $\\mathcal {S}()^W$ .",
"The Euclidean Fourier transform ${F}(f)(\\lambda ) = \\int _{\\operatorname{\\mathfrak {a}}}e^{-i\\lambda (H)}f(H)dH, \\hspace{20.0pt} \\forall f\\in \\mathcal {S}(\\operatorname{\\mathfrak {a}}),\\lambda \\in $ defines a topological isomorphism between the spaces $\\mathcal {S}(\\operatorname{\\mathfrak {a}})^W$ and $\\mathcal {S}()^W$ in the usual way.",
"Given $f\\in \\mathcal {S}(K|G|K)$ and $H\\in \\operatorname{\\mathfrak {a}}$ , the Abel transform is defined by ${A}f (H) = e^{\\rho (H)}\\int _Nf((\\exp H)n)dn.$ The Abel transform is fascinating in its own right, and we refer to Sawyer [36] for more information.",
"However, for our purposes we are mainly interested in its role in the following: Theorem 2.1 Writing ${H}$ for the spherical transform, the diagram [scale=2] 1) at (0,1)$\\mathcal {S}()^W$ ; 2) at (1,0)$\\mathcal {S}(\\operatorname{\\mathfrak {a}})^W$ ; 3) at (-1,0)$\\mathcal {S}(K|G|K)$ ; [->] (2) – (1); [->] (3) – (1); [->] (3) – (2); t (.63,.63) ${F}$ ; t (-.63,.63) ${H}$ ; t (0,-.2) ${A}$ ; commutes, up to normalizing constants.",
"Each arrow describes an isomorphism of Fréchet algebras.",
"This result will be extremely useful in later sections, especially when proving Theorem REF .",
"For more details, see Proposition 3 in Anker [1], Gangolli and Varadarajan [17] page 265, and Helgason [21] pp.",
"450."
],
[
"Probability on Lie Groups and Symmetric spaces",
"In this subsection, we summarise a few key notions from probability theory on Lie groups and symmetric spaces.",
"Books by Liao [33], [34] are excellent sources for this material, as is the paper [35] by Liao and Wang.",
"Fix a probability space $(\\Omega ,\\mathcal {F},P)$ .",
"Just as with functions on $G$ and $G/K$ , we may view stochastic processes on $G/K$ as projections of processes on $G$ whose laws are $K$ -right invariant.",
"Equipped with its natural filtration $\\lbrace \\mathcal {F}^X_t,t\\ge 0\\rbrace $ , $X$ is said to have independent increments if for all $t>s\\ge 0$ , $X(s)^{-1}X(t)$ is independent of $\\mathcal {F}^X_s$ , and stationary increments if $X(s)^{-1}X(t) \\sim X(0)^{-1}X(t-s) \\hspace{20.0pt} \\forall t>s\\ge 0.$ A process $X=(X(t),t\\ge 0)$ on $G$ is stochastically continuous if, for all $s\\ge 0$ and all $B\\in \\mathcal {B}(G)$ with $e\\notin B$ , $\\lim _{t\\rightarrow s}P(X(s)^{-1}X(t)\\in B)=0.$ A stochastically continuous process $X$ on $G$ with stationary and independent increments is called a Lévy process on $G$ .",
"A process on $G/K$ is called a Lévy process if it is the projection of a Lévy process on $G$ , under the canonical surjection $\\pi :G\\mapsto G/K$ .",
"The convolution product of two Borel measures $\\mu _1,\\mu _2$ on $G$ is defined for each $B\\in \\mathcal {B}(G)$ by $(\\mu _1\\ast \\mu _2)(B) = \\int _G\\int _G\\operatorname{\\bf 1}_B(\\sigma \\tau )\\mu _1(d\\sigma )\\mu _2(d\\tau ).$ Note that since $G$ is semisimple, it is unimodular, and hence this operation is commutative.",
"It is also clear from the definition that $\\mu _1\\ast \\mu _2$ is $K$ -bi-invariant whenever $\\mu _1$ and $\\mu _2$ are.",
"Definition 2.2 A family $(\\mu _t,t\\ge 0)$ of finite Borel measures on $G$ will be called a convolution semigroup (of probability measures) if $\\mu _t(G)=1$ for all $t\\ge 0$ , $\\mu _{s+t} = \\mu _s\\ast \\mu _t$ for all $s,t\\ge 0$ , and $\\mu _t\\rightarrow \\mu _0$ weakly as $t\\rightarrow 0$ .",
"Note that $\\mu _0$ must be an idempotent measure, in the sense that $\\mu _0\\ast \\mu _0=\\mu _0$ .",
"By Theorem 1.2.10 on page 34 of Heyer [23], $\\mu _0$ must coincide with Haar measure on a compact subgroup of $G$ .",
"We we will frequently take $\\mu _0$ to be normalised Haar measure on $K$ , so that the image of $\\mu _0$ after projecting onto $G/K$ is $\\delta _o$ , the delta mass at $o:=eK$ .",
"One may also define convolution of measures on $G/K$ , and convolution semigroups on $G/K$ are defined analogously — see Liao [34] Section 1.3 for more details.",
"In fact, the projection map $\\pi :G\\rightarrow G/K$ induces a bijection between the set of all convolution semigroups on $G/K$ and the set of all $K$ -bi-invariant convolution semigroups on $G$ — see Liao [34] Propositions 1.9 and 1.12, pp. 11–13.",
"We henceforth identify these two sets, but generally opt to perform calculations using objects defined on $G$ , for simplicity.",
"If $X=(X(t),t\\ge 0)$ is a Lévy process on $G$ (resp.",
"$G/K$ ), and if $\\mu _t$ denotes the law of $X$ at time $t$ , then $(\\mu _t,t\\ge 0)$ is a convolution semigroup on $G$ (resp.",
"$G/K$ ).",
"A well-known construction shows that all convolution semigroups on $G$ and $G/K$ arise this way — see for example Theorem 1.7 on page 8 of [34].",
"Let $(\\mu _t, t\\ge 0)$ be a $K$ -bi-invariant convolution semigroup of probability measures on $G$ associated with a Lévy process $X$ on $G/K$ .",
"The Feller semigroup $(T_t,t\\ge 0)$ associated with $X$ (also called the Hunt semigroup of $(\\mu _t,t\\ge 0)$ ) is given by $T_tf(\\sigma ) = \\int _Gf(\\sigma \\tau )\\mu _t(d\\tau ) \\hspace{20.0pt} \\forall f\\in C_0(K|G|K), \\;\\sigma \\in G.$ Definition 2.3 Let $X_1,\\ldots ,X_l$ be a basis of $\\mathfrak {g}$ , ordered so that $X_1,\\ldots ,X_d$ is a basis of $\\mathfrak {p}$ .",
"A collection $\\lbrace x_1,\\ldots ,x_l\\rbrace $ of smooth functions of compact support is called a system of exponential coordinate functions if there is a neighbourhood $U$ of $e$ for which $\\sigma = \\exp \\left(\\sum _{i=1}^lx_i(\\sigma )X_i\\right) \\hspace{20.0pt} \\forall \\sigma \\in U.$ The $x_i$ may be chosen so as to be $K$ -right-invariant for $i=1,\\ldots , m$ , and such that $\\sum _{i=1}^dx_i(k\\sigma )X_i = \\sum _{i=1}^dx_i(\\sigma )\\operatorname{Ad}(k)X_i \\hspace{20.0pt} \\forall k\\in K.$ For more details, see Liao [34] pp.36–37, 83.",
"The choice of basis $X_1,\\ldots ,X_d$ of $\\mathfrak {p}$ enables us to view $\\operatorname{Ad}k$ as a $m\\times m$ matrix, for each $k\\in K$ .",
"A vector $b\\in \\mathbb {R}^m$ is said to be $\\operatorname{Ad}(K)$ -invariant if $b=\\operatorname{Ad}(k)^Tb, \\hspace{20.0pt} \\forall k\\in K.$ Similarly, an $m\\times m$ real-valued matrix $a=(a_{ij})$ is $\\operatorname{Ad}(K)$ -invariant if $a=\\operatorname{Ad}(k)^Ta\\operatorname{Ad}(k) \\hspace{20.0pt} \\forall k\\in K.$ A Borel measure $\\nu $ on $G$ is called a Lévy measure if $\\nu (\\lbrace e\\rbrace ) = 0$ , $\\int _G\\sum _{i=1}^lx_i(\\sigma )^2\\nu (d\\sigma )$ , and $\\nu (B^c)<\\infty $ for any neighbourhood $B$ of $e$ .",
"We state a useful corollary of the famous Hunt formula.",
"For more details, including a proof, see Section 3.2 Liao [34], pp. 78.",
"Theorem 2.4 Let $\\mathcal {L}$ be the infinitesimal generator associated with a $K$ -bi-invariant Lévy process on $G$ .",
"Then $C_c^\\infty (G)\\subseteq \\operatorname{Dom}\\mathcal {L}$ , and there is an $\\operatorname{Ad}(K)$ -invariant vector $b\\in \\mathbb {R}^m$ , an $\\operatorname{Ad}(K)$ -invariant covariance matrix $a:=(a_{ij})$ , and a $K$ -bi-invariant Lévy measure $\\nu $ such that $\\begin{aligned}\\mathcal {L}f(\\sigma ) = \\sum _{i=1}^db_iX_if(\\sigma ) &+ \\frac{1}{2}\\sum _{j,k=1}^da_{jk}X_jX_kf(\\sigma ) \\\\&+ \\int _G\\left(f(\\sigma \\tau )-f(\\sigma )-\\sum _{i=1}^dx_i(\\tau )X_if(\\sigma )\\right)\\nu (d\\sigma ),\\end{aligned}$ for all $f\\in C_c^\\infty (G)$ and $\\sigma \\in G$ .",
"Moreover, the triple $(b,a,\\nu )$ is completely determined by $\\mathcal {L}$ , and independent of the choice of exponential coordinate functions $x_i,\\;i=1,\\ldots ,d$ .",
"Conversely, given a triple $(b,a,\\nu )$ of this kind, there is a unique $K$ -bi-invariant convolution semigroup of probability measures on $G$ with infinitesimal generator given by $\\mathcal {L}$ .",
"Let $\\mathcal {L}$ be the Lévy generator described in Theorem REF above, and let $\\mathcal {L}_D$ be the diffusion part of $\\mathcal {L}$ , so that $\\mathcal {L}_D = \\sum _{i=1}^db_iX_i + \\frac{1}{2}\\sum _{i,j=1}^da_{jk}X_jX_k.$ Now by the discussion surrounding equation (3.3) of Liao [34], pp.",
"75, $\\mathcal {L}_D\\in {\\bf D}_K(G)$ , and so for each $\\lambda \\in $ there is a constant $\\beta (\\mathcal {L}_D,\\lambda )\\in \\mathbb {C}$ such that $\\mathcal {L}_D\\phi _\\lambda = \\beta (\\mathcal {L}_D,\\lambda )\\phi _\\lambda .$ The mapping $\\lambda \\mapsto \\beta (\\mathcal {L}_D,\\lambda )$ is a $W$ -invariant quadratic polynomial function on $$ .",
"Theorem 2.5 (Gangolli's Lévy–Khinchine formula) Let $(\\mu _t,t\\ge 0)$ be a $K$ -bi-invariant convolution semigroup of probability measures on $G$ with infinitesimal generator $\\mathcal {L}$ , and let $\\mathcal {L}_D$ denote the diffusion part of $\\mathcal {L}$ .",
"Then $\\hat{\\mu }_t=e^{-t\\psi }$ , where $\\psi (\\lambda ) = -\\beta (\\mathcal {L}_D,\\lambda ) + \\int _G(1-\\phi _\\lambda (\\sigma ))\\nu (d\\sigma ) \\hspace{20.0pt} \\forall \\lambda \\in ,$ and $\\beta (\\mathcal {L}_D,\\lambda )$ is given by (REF ).",
"This result was first proven by Ramesh Gangolli in [16], see also Liao and Wang [35] .",
"For a proof of the specific statement above, see page 139 of Liao [34].",
"The function $\\psi $ given by (REF ) will be called the Gangolli exponent of the process $X$ .",
"Remark 2.6 If Definition REF (REF ) is relaxed so that each $\\mu _t$ need only satisfy $\\mu _t(G)\\le 1$ , all of the results described in this subsection continue to hold, except “sub-” must be added to some to the terms: convolution semigroups of sub-probability measures, sub-Lévy generators, sub-diffusion operators, and so on."
],
[
"Positive and Negative Definite Functions",
"By viewing $$ as a finite-dimensional real vector space, we may consider positive and negative definite functions on $$ , defined in the usual way.",
"Proposition 2.7 For all $\\sigma \\in G$ , $\\lambda \\mapsto \\phi _\\lambda (\\sigma )$ is positive definite.",
"Let $\\mu $ be a finite $K$ -bi-invariant Borel measure.",
"Then $\\hat{\\mu }$ is positive definite.",
"Let $\\sigma \\in G$ , $n\\in \\mathbb {N}$ , $\\lambda _1,\\ldots ,\\lambda _n\\in $ , and $c_1,\\ldots ,c_n\\in \\mathbb {C}$ , and note that $\\sum _{\\alpha ,\\beta =1}^nc_\\alpha \\overline{c_\\beta }e^{(i(\\lambda _\\alpha -\\lambda _\\beta )+\\rho )A(k\\sigma )} = \\left|\\sum _{\\alpha =1}^nc_\\alpha e^{(i\\lambda _\\alpha +\\frac{\\rho }{2})A(k\\sigma )}\\right|^2 \\ge 0.$ Therefore, by the Harish-Chandra integral formula, $\\sum _{\\alpha ,\\beta =1}^nc_\\alpha \\overline{c_\\beta }\\phi _{\\lambda _\\alpha -\\lambda _\\beta }(\\sigma ) = \\int _K\\sum _{\\alpha ,\\beta =1}^nc_\\alpha \\overline{c_\\beta }e^{(i(\\lambda _\\alpha -\\lambda _\\beta )+\\rho )A(k\\sigma )}dk \\ge 0.$ Part 1 follows.",
"For part 2, observe that since (REF ) holds for all $c_1,\\ldots ,c_n$ , we can replace each $c_j$ by its complex conjugate.",
"Therefore, $\\sum _{\\alpha ,\\beta =1}^n\\overline{c_\\alpha }c_\\beta \\phi _{\\lambda _\\alpha -\\lambda _\\beta }(\\sigma )\\ge 0$ for all $\\sigma \\in G$ , $n\\in \\mathbb {N}$ , $\\lambda _1,\\ldots ,\\lambda _n\\in $ , and $c_1,\\ldots ,c_n\\in \\mathbb {C}$ .",
"Taking complex conjugates and using the fact that $\\phi _{-\\lambda }=\\overline{\\phi _\\lambda }$ , it follows that for all $\\sigma \\in G$ , $n\\in \\mathbb {N}$ , $\\lambda _1,\\ldots ,\\lambda _n\\in $ , and $c_1,\\ldots ,c_n\\in \\mathbb {C}$ , $\\sum _{\\alpha ,\\beta =1}^nc_\\alpha \\overline{c_\\beta }\\phi _{-(\\lambda _\\alpha -\\lambda _\\beta )}(\\sigma ) = \\overline{\\sum _{\\alpha ,\\beta =1}^n\\overline{c_\\alpha }c_\\beta \\phi _{\\lambda _\\alpha -\\lambda _\\beta }}\\ge 0,$ and hence $\\sum _{\\alpha ,\\beta =1}^nc_\\alpha \\overline{c_\\beta }\\hat{\\mu }(\\lambda _\\alpha -\\lambda _\\beta ) = \\int _{}\\sum _{\\alpha ,\\beta =1}^nc_\\alpha \\overline{c_\\beta }\\phi _{-(\\lambda _\\alpha -\\lambda _\\beta )}(\\sigma )\\mu (d\\sigma )\\ge 0.$ By choosing a basis of $$ , we may identify it with $\\mathbb {R}^m$ , and apply classical results about positive (resp.",
"negative) definite functions on Euclidean space to functions on $$ , to obtain results about positive (resp.",
"negative) definite functions in this new setting.",
"One useful application of this is the Schoenberg correspondence, which states that a map $\\psi :\\rightarrow \\mathbb {C}$ is negative definite if and only if $\\psi (0)\\ge 0$ and $e^{-t\\psi }$ is positive definite for all $t>0$ .",
"This is immediate by the Schoenberg correspondence on $\\mathbb {R}^m$ — see Berg and Forst [10] page 41 for a proof.",
"Proposition 2.8 Let $\\psi :\\rightarrow \\mathbb {C}$ be the Gangolli exponent of a Lévy process on $G/K$ .",
"Then $\\psi $ is negative definite.",
"Let $X=(X(t),t\\ge 0)$ is a Lévy process on $G/K$ , and let $\\nu _t$ be the law of $X(t)$ , for all $t\\ge 0$ .",
"Then $(\\nu _t,t\\ge 0)$ forms a convolution semigroup on $G/K$ .",
"By Proposition 1.12 of Liao [34] (pp.",
"13), $(\\nu _t,t\\ge 0)$ arises as the projection onto $G/K$ of a $K$ -bi-invariant convolution semigroup $(\\mu _t,t\\ge 0)$ on $G$ .",
"By Proposition REF , the spherical transform of each $\\mu _t$ is positive definite, and by the Schoenberg correspondence, for each $t\\ge 0$ , there is a negative definite function $\\psi _t$ on $$ such that $\\psi _t(0)\\ge 0$ and $\\hat{\\mu }_t=e^{-\\psi _t}$ .",
"In fact, since $(\\mu _t,t\\ge 0)$ is a convolution semigroup, it must be the case that $\\hat{\\mu }_t = e^{-t\\psi _1}, \\hspace{20.0pt} \\forall t\\ge 0.$ By uniqueness of Gangolli exponents, $\\psi =\\psi _1$ , a negative definite function.",
"We finish this subsection with a collection of results about negative definite functions, which will be particularly useful for calculation in later sections.",
"Proposition 2.9 Let $\\psi :\\rightarrow \\mathbb {C}$ be a continuous negative definite function.",
"Then For all $\\lambda ,\\eta \\in $ , $\\left|\\sqrt{|\\psi (\\lambda )|}-\\sqrt{|\\psi (\\eta )|}\\right|\\le \\sqrt{|\\psi (\\lambda -\\eta )|}$ (Generalised Peetre inequality) For all $s\\in \\mathbb {R}$ and $\\lambda ,\\eta \\in $ , $\\left(\\frac{1+|\\psi (\\lambda )|}{1+|\\psi (\\eta )|}\\right)^s \\le 2^{|s|}(1+|\\psi (\\lambda -\\eta )|^2)^{|s|}.$ There is a constant $c_\\psi >0$ such that $|\\psi (\\lambda )|\\le c_\\psi (1+|\\lambda |^2) \\hspace{20.0pt} \\forall \\lambda \\in .$ These results follow from their analogues on $\\mathbb {R}^m$ — see Hoh [24] page 16."
],
[
"Spherical Anisotropic Sobolev Spaces",
"Suppose $\\psi $ is a real-valued continuous negative definite function, and let $s\\in \\mathbb {R}$ .",
"We define the (spherical) anisotropic Sobolev space associated with $\\psi $ and $s$ to be $H^{\\psi ,s} := \\left\\lbrace u\\in \\mathcal {S}^{\\prime }(K|G|K):\\int _G(1+\\psi (\\lambda ))^s|\\hat{u}(\\lambda )|^2\\omega (d\\lambda )<\\infty \\right\\rbrace ,$ where $\\mathcal {S}^{\\prime }(K|G|K)$ denotes the space of $K$ -bi-invariant tempered distributions.",
"One can check that each $H^{\\psi ,s}$ is a Hilbert space with respect to the inner product $\\langle u,v\\rangle _{\\psi ,s} := \\int _{}(1+\\psi (\\lambda ))^s\\hat{u}(\\lambda )\\overline{\\hat{v}(\\lambda )}\\omega (d\\lambda ), \\hspace{20.0pt} \\forall u,v\\in H^{\\psi ,s}.$ These spaces are a generalisation of the anisotropic Sobolev spaces first introduced by Niels Jacob, see [26], and developed further by Hoh, see [24].",
"For the special case $\\psi (\\lambda )=|\\rho |^2+|\\lambda |^2$ , we will write $H^{\\psi ,s}=H^s$ .",
"Note also that $H^{\\psi ,0}=L^2(K|G|K)$ , by the Plancherel theorem.",
"In this case, we will omit subscripts and just write $\\langle \\cdot ,\\cdot \\rangle $ for the inner product on $L^2(K|G|K)$ .",
"Note that $\\psi $ is a non-negative function, since it is negative definite and real-valued.",
"We impose an additional assumption, namely that there exist constants $r,c>0$ such that $\\psi (\\lambda )\\ge c\\left(|\\rho |^2+|\\lambda |^2\\right)^r \\hspace{20.0pt} \\forall \\lambda \\in .$ Analogous assumptions can be found in Jacob [27] (1.5) and Hoh [24] (4.4), and the role of (REF ) will be very similar.",
"Theorem 2.10 Let $\\psi $ be a real-valued, continuous negative definite symbol, satisfying (REF ).",
"Then $C_c^\\infty (K|G|K)$ and $\\mathcal {S}(K|G|K)$ are dense in each $H^{\\psi ,s}$ , and we have continuous embeddings $\\mathcal {S}(K|G|K)\\hookrightarrow H^{\\psi ,s}\\hookrightarrow \\mathcal {S}^{\\prime }(K|G|K)$ We have continuous embeddings $H^{\\psi ,s_2}\\hookrightarrow H^{\\psi ,s_1} $ whenever $s_1,s_2\\in \\mathbb {R}$ with $s_2\\ge s_1$ .",
"In particular, $H^{\\psi ,s}\\hookrightarrow L^2(K|G|K)$ for all $s\\ge 0$ .",
"Under the standard identification of $L^2(K|G|K)$ with its dual, the dual space of each $H^{\\psi ,s}$ is isomorphic to $H^{\\psi ,-s}$ , with $\\Vert u\\Vert _{\\psi ,-s} = \\sup \\left\\lbrace \\frac{|\\langle u,v\\rangle |}{\\Vert v\\Vert _{\\psi ,s}}:v\\in C_c^\\infty (K|G|K),\\;v\\ne 0\\right\\rbrace ,$ for all $s\\in \\mathbb {R}$ .",
"For $r>0$ as in equation (REF ), we have continuous embeddings $H^s \\hookrightarrow H^{\\psi ,s} \\hookrightarrow H^{rs},$ for all $s\\ge 0$ .",
"Let $s_3>s_2>s_1$ .",
"Then for all $\\epsilon >0$ , there is $c(\\epsilon )\\ge 0$ such that $\\Vert u\\Vert _{\\psi ,s_2} \\le \\epsilon \\Vert u\\Vert _{\\psi ,s_3} + c(\\epsilon )\\Vert u\\Vert _{\\psi ,s_1}$ for all $u\\in H^{\\psi ,s_3}$ .",
"There exist continuous embeddings $H^{\\psi ,s}\\hookrightarrow C_0(K|G|K)$ for all $s>\\frac{d}{r}$ , where $d=\\dim (G/K)$ .",
"For brevity, let $\\langle \\lambda \\rangle := \\sqrt{1+|\\lambda |^2}, \\hspace{20.0pt} \\forall \\lambda \\in ,$ and $\\Psi (\\lambda ) := \\sqrt{1+\\psi (\\lambda )}, \\hspace{20.0pt} \\forall \\lambda \\in .$ The proof of Theorem REF will be given after the next lemma.",
"Lemma 2.11 Let $M>d=\\dim (G/K)$ .",
"Then $\\langle \\cdot \\rangle ^{-M} \\in L^1(,\\omega )$ .",
"By standard arguments, one may check that $\\int _{\\mathbb {R}^d}\\langle \\xi \\rangle ^{-M}d\\xi <\\infty $ , for all $M>d$ .",
"Writing $p=\\frac{\\dim N}{2}$ , we have $d= \\dim + 2p$ , and hence $\\int _{}\\langle \\lambda \\rangle ^{-M+2p}d\\lambda < \\infty $ whenever $M>d$ .",
"By Proposition 7.2 on page 450 of Helgason [21], there are $C_1,C_2>0$ such that $|\\operatorname{\\bf c}(\\lambda )|^{-1} \\le C_1 + C_2|\\lambda |^p \\hspace{20.0pt} \\forall \\lambda \\in .$ Let $C>0$ be such that $(C_1 + C_2|\\lambda |^p)^2 < C(1+|\\lambda |^2)^p$ for all $\\lambda \\in $ .",
"Then $\\int _{}\\langle \\lambda \\rangle ^{-M}\\omega (d\\lambda ) = \\int _{}\\langle \\lambda \\rangle ^{-M}|\\operatorname{\\bf c}(\\lambda )|^{-2}d\\lambda \\le C\\int _{}\\langle \\lambda \\rangle ^{-M+2p}d\\lambda < \\infty ,$ whenever $M>d$ .",
"[Proof of Theorem REF ] Much of this theorem may be proved by adapting proofs from the $\\mathbb {R}^d$ case.",
"For example, to prove Theorem REF (REF ), let $\\mathcal {L}^{\\psi ,s}$ denote the space of all measurable functions $v$ on $$ for which $\\Psi ^sv\\in L^2(,\\omega )^W$ , a Hilbert space with respect to the inner product $\\langle u,v\\rangle = \\int _{\\operatorname{\\mathfrak {a}}^\\ast }\\Psi (\\lambda )^{2s}u(\\lambda )\\overline{v(\\lambda )}\\omega (d\\lambda ), \\hspace{20.0pt} \\forall u,v \\in \\mathcal {L}^{\\psi ,s}.$ By viewing $$ as a real vector space and using inequality (REF ) to relate $\\omega $ to Lebesgue measure, the proof of Theorem 3.10.3 on page 208 of Jacob [28] may be easily adapted to show that $\\mathcal {S}()^W\\hookrightarrow \\mathcal {L}^{\\psi ,s}\\hookrightarrow \\mathcal {S}^{\\prime }()^W$ is continuous.",
"Noting Theorem REF , Theorem REF (REF ) follows.",
"Proofs of Theorem REF (REF )–(REF ) are almost identical to their $\\mathbb {R}^d$ -based counterparts, see Jacob [27] §1, or Hoh [24] pp. 46–48.",
"By Theorem REF (REF ), if we can prove the existence of a continuous embedding $H^s \\hookrightarrow C_0(K|G|K)$ for all $s>\\frac{d}{2}$ , then Theorem REF (REF ) will follow.",
"Let $s>\\frac{d}{2}$ and $u\\in \\mathcal {S}(K|G|K)$ .",
"By Lemma REF , $\\langle \\cdot \\rangle ^{-s}\\in L^2(\\operatorname{\\mathfrak {a}}^\\ast ,\\omega )$ , and by spherical inversion, $|u(\\sigma )| = \\left|\\int _{}\\phi _\\lambda (\\sigma )\\hat{u}(\\lambda )\\omega (d\\lambda )\\right| \\le \\int _{}|\\hat{u}(\\lambda )|\\omega (d\\lambda ) = \\int _{}\\langle \\lambda \\rangle ^{-s}\\langle \\lambda \\rangle ^s|\\hat{u}(\\lambda )|\\omega (d\\lambda ),$ for all $\\sigma \\in G$ .",
"By the Cauchy–Schwarz inequality, $|u(\\sigma )| \\le \\Vert \\langle \\cdot \\rangle ^{-s}\\Vert _{L^2(,\\omega )}\\Vert \\langle \\cdot \\rangle ^s\\hat{u}\\Vert _{L^2(,\\omega )} = C\\Vert u\\Vert _s$ for all $\\sigma \\in G$ , where $C= \\Vert \\langle \\cdot \\rangle ^{-s}\\Vert _{L^2(,\\omega )}$ .",
"It follows that $\\Vert u \\Vert _{C_0(K|G|K)} := \\sup _{\\sigma \\in G}|u(\\sigma )| \\le C\\Vert u\\Vert _s.$ The embedding (REF ) may then be obtained using a density argument."
],
[
"Pseudodifferential Operators and Their Symbols",
"A measurable mapping $q:G\\times \\rightarrow \\mathbb {C}$ will be called a negative definite symbol if it is locally bounded, and if for each $\\sigma \\in G$ , $q(\\sigma ,\\cdot )$ is negative definite and continuous.",
"If in addition $q$ is continuous in its first argument, we will call $q$ a continuous negative definite symbol.",
"Let $\\mathcal {M}(G)$ denote the set of all measurable functions on $G$ .",
"Theorem 2.12 Let $q$ be a negative definite symbol, and for each $f\\in C^\\infty _c(K|G|K)$ and $\\sigma \\in G$ , define $q(\\sigma , D)f(\\sigma ) = \\int _{}\\hat{f}(\\lambda )\\phi _\\lambda (\\sigma )q(\\sigma ,\\lambda )\\omega (d\\lambda ).$ Then Equation (REF ) defines a linear operator $q(\\sigma ,D):C_c^\\infty (K|G|K)\\rightarrow \\mathcal {M}(G)$ .",
"If $q$ is a continuous negative definite symbol, then $q(\\sigma ,D):C_c^\\infty (K|G|K)\\rightarrow C(G)$ .",
"If $q$ is $K$ -bi-invariant in its first argument, then $q(\\sigma ,D)f$ is $K$ -bi-invariant for all $f\\in C_c^\\infty (K|G|K)$ .",
"Theorem REF (REF ) and (REF ) are proved in a similar manner to Theorem 4.5.7 of Jacob [28], while (REF ) is immediate from the $K$ -bi-invariance of each spherical function $\\phi _\\lambda $ .",
"Definition 2.13 Operators of the form (REF ), where $q$ is a negative definite symbol, will be called (spherical) pseudodifferential operators on $G$ .",
"Example 2.14 The following examples of pseudodifferential operators arise as the generators of Lévy processes on symmetric spaces.",
"Their symbols are constant in $\\sigma $ , a property that is well-known in the case of $G=\\mathbb {R}^d$ , $K=\\lbrace 0\\rbrace $ .",
"Section will introduce a large class of examples pseudodifferential operators with spatial dependence.",
"Diffusion operators with constant coefficients.",
"Consider the pure diffusion operator $\\mathcal {L} := \\sum _{i=1}^db_iX_i + \\sum _{i,j=1}^da_{ij}X_iX_j$ where $b$ and $a$ are as in Theorem REF .",
"Recall that such operators belong to ${\\bf D}_K(G)$ ; for each $\\lambda \\in $ , let $\\beta (\\mathcal {L},\\lambda )$ denote the $\\phi _\\lambda $ -eigenvalue of $\\mathcal {L}$ .",
"Then $(\\sigma ,\\lambda )\\mapsto -\\beta (T,\\lambda )$ is a continuous negative definite symbol, and the associated pseudodifferential operator is $-T$ .",
"To see this, let $(\\mu _t,t\\ge 0)$ denote the convolution semigroup generated by $\\mathcal {L}$ , and let $(T_t,t\\ge 0)$ be the associated Hunt semigroup, as defined in (REF ).",
"Then $\\mathcal {L}f(\\sigma ) = \\left.\\frac{d}{dt}T_tf(\\sigma )\\right|_{t=0},$ for all $f\\in C_c^\\infty (K|G|K)$ and $\\sigma \\in G$ .",
"By the spherical inversion formula (REF ), $T_tf(\\sigma ) = \\int _G\\int _{}\\hat{f}(\\lambda )\\phi _\\lambda (\\sigma \\tau )\\omega (d\\lambda )p_t(d\\tau ), \\hspace{20.0pt} \\forall f\\in C_c^\\infty (K|G|K),\\;\\sigma \\in G.$ Recalling that $\\hat{f}\\in \\mathcal {S}()$ whenever $f\\in C_c^\\infty (K|G|K)$ , a Fubini argument may be applied to conclude that $T_tf(\\sigma ) = \\int _{}\\hat{f}(\\lambda )T_t\\phi _\\lambda (\\sigma )\\omega (d\\lambda ) = \\int _{}\\hat{f}(\\lambda )e^{-t\\beta (\\mathcal {L},\\lambda )}\\phi _\\lambda (\\sigma )\\omega (d\\lambda ),$ for all $f\\in C_c^\\infty (K|G|K)$ and $\\sigma \\in G$ .",
"By (REF ), given $f\\in C_c^\\infty (K|G|K)$ and $\\sigma \\in G$ , $\\begin{aligned}\\mathcal {L}f(\\sigma ) &= \\left.\\frac{d}{dt}\\int _{}\\hat{f}(\\lambda )e^{-t\\beta (\\mathcal {L},\\lambda )}\\phi _\\lambda (\\sigma )\\omega (d\\lambda )\\right|_{t=0} \\\\&= \\lim _{t\\rightarrow 0}\\int _{}\\hat{f}(\\lambda )\\left(\\frac{e^{-t\\beta (\\mathcal {L},\\lambda )}-1}{t}\\right)\\phi _\\lambda (\\sigma )\\omega (d\\lambda ).\\end{aligned}$ Now, for all $t>0$ and $\\lambda \\in $ , $\\left|\\hat{f}(\\lambda )\\left(\\frac{e^{-t\\beta (\\mathcal {L},\\lambda )}-1}{t}\\right) \\phi _\\lambda (\\sigma )\\right| \\le \\left|\\hat{f}(\\lambda )\\right|\\left|\\frac{e^{-t\\beta (\\mathcal {L},\\lambda )}-1}{t}\\right| \\le \\left|\\hat{f}(\\lambda )\\right||\\beta (\\mathcal {L},\\lambda )|.$ Moreover, $|\\hat{f}||\\beta (\\mathcal {L},\\cdot )|\\in L^1(,\\omega )^W$ , since $\\hat{f}\\in \\mathcal {S}()^W$ , and $\\beta (\\mathcal {L},\\cdot )$ is a $W$ -invariant polynomial function.",
"By the dominated convergence theorem, we may bring the limit through the integral sign in (REF ) to conclude that $\\begin{aligned}\\mathcal {L}f(\\sigma ) &= \\int _{}\\hat{f}(\\lambda )\\lim _{t\\rightarrow 0}\\left(\\frac{e^{-t\\beta (\\mathcal {L},\\lambda )}-1}{t}\\right)\\phi _\\lambda (\\sigma )\\omega (d\\lambda ) \\\\&= -\\int _{}\\hat{f}(\\lambda )\\phi _\\lambda (\\sigma )\\beta (\\mathcal {L} ,\\lambda )\\omega (d\\lambda )\\end{aligned}$ for all $f\\in C_c^\\infty (K|G|K)$ and $\\sigma \\in G$ .",
"Brownian motion.",
"As a special case of the above, $-\\Delta $ is a pseudodifferential operator with symbol $|\\rho |^2+|\\lambda |^2$ .",
"Lévy generators.",
"More generally, if $\\mathcal {A}$ is the infinitesimal generator of a $K$ -bi-invariant Lévy process on $G$ , and if $\\psi $ is the corresponding Gangolli exponent, then $(\\sigma ,\\lambda )\\mapsto \\psi (\\lambda )$ is a continuous negative definite symbol, and $-\\mathcal {A}$ is the corresponding pseudodifferential operator — see Applebaum [4] Theorem 5.1."
],
[
"Gangolli Operators and the Hille–Yosida–Ray Theorem",
"We will soon define the class of pseudodifferential operators that will be of primary interest.",
"In this section, we motivate this definition with a short discussion of the Hille–Yosida–Ray theorem, and prove that our class of operators are pseudodifferential operators in the sense of Definition REF .",
"We finish the section with some examples.",
"Let $E$ be a locally compact, Hausdorff space, let $\\mathcal {C}$ be a closed subspace of $C_0(E)$ , and let $\\mathcal {F}(E)$ denote the space of all real-valued functions on $E$ .",
"A $C_0$ -semigroup $(T_t,t\\ge 0)$ defined on $C_0(K|G|K)$ is called sub-Feller if for all $f\\in C_0(E)$ , and all $t\\ge 0$ , $0\\le f\\le 1 ~\\Rightarrow ~ 0\\le T_tf\\le 1.$ A linear operator $A:\\mathcal {D}(A)\\rightarrow \\mathcal {F}(E)$ is said to satisfy the positive maximum principle, if for all $f\\in \\mathcal {D}(A)$ and $x_0\\in E$ such that $f(x_0)=\\sup _{x\\in E}f(x)\\ge 0$ , we have $Af(x_0)\\le 0$ .",
"The following theorem is an extended version of the Hille–Yosida–Ray theorem, which fully characterises the operators that extend to generators of sub-Feller semigroups on $C_0(E)$ .",
"Similar versions in which $E=\\mathbb {R}^d$ may found in [24], pp.",
"53, and [28], pp. 333.",
"For a proof, see Ethier and Kurz [14], pp. 165.",
"Theorem 3.1 (Hille–Yosida–Ray) A linear operator $(\\mathcal {A},\\operatorname{Dom}\\mathcal {A})$ on $C_0(E)$ is closable and its closure generates a strongly continuous, sub-Feller semigroup on $C_0(E)$ if and only if the following is satisfied: $\\operatorname{Dom}\\mathcal {A}$ is dense in $C_0(E)$ , $\\mathcal {A}$ satisfies the positive maximum principle, and There exists $\\alpha >0$ such that $\\operatorname{Ran}(\\alpha I-\\mathcal {A})$ is dense in $C_0(E)$ .",
"In their papers [7] and [8], Applebaum and Ngan found necessary and sufficient conditions for an operator defined on $C_c^\\infty (K|G|K)$ to satisfy Theorem REF (REF ), for the cases $E=G$ , $G/K$ and $K|G|K$ .",
"We will focus primarily on the case $E=K|G|K$ , since this is the realm in which the spherical transform is available.",
"A mapping $\\nu :G\\times \\mathcal {B}\\rightarrow [0,\\infty ]$ will be called a $K$ -bi-invariant Lévy kernel if it is $K$ -bi-invariant in its first argument, and if for all $\\sigma \\in G$ , $\\nu (\\sigma ,\\cdot )$ is a $K$ -bi-invariant Lévy measure.",
"Fix a system of exponential coordinate functions, as defined in Definition REF , and adopt all of the notation conventions from this definition.",
"Definition 3.2 An operator $\\mathcal {A}:C_c^\\infty (K|G|K)\\rightarrow \\mathcal {F}(G)$ will be called a Gangolli operator if there exist mappings $c, b_i, a_{jk}\\in C(K|G|K)$ ($1\\le i,j,k\\le d$ ), as well as a $K$ -bi-invariant Lévy kernel $\\nu $ , such that for all $f\\in C_c^\\infty (K|G|K)$ , $\\begin{aligned}\\mathcal {A}f(\\sigma ) = -c(\\sigma )f(\\sigma ) + \\sum _{i=1}^db_i&(\\sigma )X_if(\\sigma ) + \\sum _{j,k=1}^da_{jk}(\\sigma )X_jX_kf(\\sigma ) \\\\&+ \\int _G\\left(f(\\sigma \\tau )-f(\\sigma )-\\sum _{i=1}^dx_i(\\tau )X_if(\\sigma )\\right)\\nu (\\sigma ,d\\tau ),\\end{aligned}$ and if, in addition, $c$ is a non-negative mapping.",
"For all $\\sigma \\in G$ , $b(\\sigma ):=(b_1(\\sigma ),\\ldots ,b_d(\\sigma ))$ is an $\\operatorname{Ad}(K)$ -invariant vector, For all $\\sigma \\in G$ , $a(\\sigma ):=(a_{jk}(\\sigma ))$ is an $\\operatorname{Ad}(K)$ -invariant, non-negative definite, symmetric matrix.",
"For each $f\\in C_b(K|G|K)$ , the mappings $\\sigma \\mapsto \\int _Uf(\\tau )\\sum _{i=1}^dx(\\tau )^2\\nu (\\sigma ,d\\tau )$ and $\\sigma \\mapsto \\int _{U^c}f(\\tau )\\nu (\\sigma ,d\\tau )$ are continuous from $G$ to $[0,\\infty )$ .",
"$\\lim _{\\sigma \\rightarrow \\infty }\\nu (\\sigma ,U^c)=0$ .",
"The form (REF ) is necessary for $\\mathcal {A}$ to satisfy the positive maximum principle.",
"The various continuity assumptions, together with condition (REF ), ensure that $\\mathcal {A}$ maps into $C_0(K|G|K)$ .",
"By Theorem 3.2 (3) of [8] as well as Theorems 3.7 and 3.8 of Applebaum and Ngan [7], Gangolli operators are densely defined linear operators on $C_0(K|G|K)$ that satisfy the positive maximum principle.",
"Remarks 3.3 Gangolli operators were first introduced in [8] in compact symmetric spaces and for a more restrictive form of (REF ).",
"Equation (REF ) may be viewed as a spatially dependent generalisation of (REF ), with an additional killing term $c$ .",
"Indeed, if the spatial dependence of the coefficients $c$ , $b$ , $a$ and $\\nu $ is removed from (REF ), then the resulting operator extends to the generator of a killed Lévy process, with state space $G\\cup \\lbrace \\infty \\rbrace $ , the one-point compactification of $G$ .",
"For a more detailed discussion, see [7], pp. 149.",
"For a Gangolli operator $\\mathcal {A}$ given by (REF ), the local part of $\\mathcal {A}$ is defined $\\mathcal {L}^\\sigma = -c(\\sigma ) + \\sum _{i=1}^db_i(\\sigma )X_i + \\sum _{j,k=1}^da_{jk}(\\sigma )X_jX_k \\hspace{20.0pt} \\forall \\sigma \\in G.$ Holding $\\sigma \\in G$ fixed, $\\mathcal {L}^\\sigma +c(\\sigma )$ can be viewed as a diffusion operator with constant coefficients.",
"In particular, $\\mathcal {L}^\\sigma \\in {\\bf D}_K(G)$ for all $\\sigma \\in G$ .",
"Let $\\beta (\\mathcal {L}^\\sigma ,\\lambda )$ denote the $\\phi _\\lambda $ -eigenvalue of $\\mathcal {L}^\\sigma $ , so that $\\mathcal {L}^\\sigma \\phi _\\lambda = \\beta (\\mathcal {L}^\\sigma ,\\lambda )\\phi _\\lambda \\hspace{20.0pt} \\forall \\sigma \\in G,\\;\\lambda \\in .$ Lemma 3.4 Let $\\mathcal {A}$ be a Gangolli operator given by (REF ), and, using the notation introduced above, define $q:G\\times \\rightarrow \\mathbb {C}$ by $q(\\sigma ,\\lambda ) = -\\beta (\\mathcal {L}^\\sigma ,\\lambda ) + \\int _G(1-\\phi _\\lambda (\\tau ))\\nu (\\sigma ,d\\tau ), \\hspace{20.0pt} \\forall \\sigma \\in G,\\lambda \\in .$ Then $q$ is a continuous negative definite symbol.",
"That $q$ is continuous in its first argument is immediate from Definition REF .",
"Fix $\\sigma \\in G$ and consider $q(\\sigma ,\\cdot )-c(\\sigma )$ .",
"By Theorem REF , there is a convolution semigroup $(\\mu ^\\sigma _t,t\\ge 0)$ generated by $\\mathcal {A}+c(\\sigma )$ , and by Theorem REF , the corresponding Gangolli exponent is a continuous negative definite mapping on $$ , given by $\\psi ^\\sigma (\\lambda ) = q(\\sigma ,\\lambda )-c(\\sigma ) \\hspace{20.0pt} \\forall \\lambda \\in .$ Therefore $q(\\sigma ,\\cdot )$ is continuous, and negative definite since for fixed $\\sigma $ , $c(\\sigma )$ is a non-negative constant.",
"Definition 3.5 The symbols described by Lemma REF will be referred to as Gangolli symbols, due to their connection with Gangolli's Lévy–Khinchine formula.",
"Remarks 3.6 Gangolli exponents are precisely those Gangolli symbols constant in their first argument.",
"The set of all Gangolli symbols forms a convex cone.",
"Theorem 3.7 Let $\\mathcal {A}$ be a Gangolli operator of the form (REF ), and let $q$ be a Gangolli symbol given by (REF ).",
"Then $\\mathcal {A}$ is a pseudodifferential operator with symbol $q$ .",
"By Theorem REF and Lemma REF , $f\\mapsto -\\int _{}\\hat{f}(\\lambda )\\phi _\\lambda (\\sigma )q(\\sigma ,\\lambda )\\omega (d\\lambda )$ is a well-defined mapping from $C_c^\\infty (K|G|K)\\rightarrow C(G)$ .",
"We show that it is equal to $\\mathcal {A}$ .",
"Continue to denote the local part of $\\mathcal {A}$ by $\\mathcal {L}^\\sigma $ , as in (REF ), and let $\\mathcal {P}$ denote the non-local part, so that $\\mathcal {P}f(\\sigma ) = \\int _G\\left(f(\\sigma \\tau )-f(\\sigma )-\\sum _{i=1}^dx_i(\\tau )X_if(\\sigma )\\right)\\nu (\\sigma ,d\\tau )$ for all $f\\in C_c^\\infty (K|G|K)$ and $\\sigma \\in G$ .",
"By design, $\\mathcal {A}f(\\sigma ) = \\mathcal {L}^\\sigma f(\\sigma ) + \\mathcal {P}f(\\sigma ), \\hspace{20.0pt} \\forall f\\in C_c^\\infty (K|G|K),\\;\\sigma \\in G.$ Fix $\\sigma \\in G$ , and view $\\mathcal {L}^\\sigma + c(\\sigma )$ as a $K$ -bi-invariant diffusion operator with constant coefficients.",
"By (REF ) in Example REF (REF ), $\\left(\\mathcal {L}^\\sigma + c(\\sigma )\\right)f(\\tau ) = -\\int _{}\\hat{f}(\\lambda )\\phi _\\lambda (\\tau )\\beta (\\mathcal {L}^\\sigma +c(\\sigma ),\\lambda )\\omega (d\\lambda )$ for all $f\\in C_c^\\infty (K|G|K)$ and $\\tau \\in G$ , where $\\beta (\\mathcal {L}^\\sigma +c(\\sigma ),\\lambda )$ denotes the $\\phi _\\lambda $ -eigenvalue of $\\mathcal {L}^\\sigma +c(\\sigma )\\in {\\bf D}_K(G)$ .",
"Clearly $\\beta (\\mathcal {L}^\\sigma +c(\\sigma ),\\lambda ) = \\beta (\\mathcal {L}^\\sigma ,\\lambda )+c(\\sigma ), \\hspace{20.0pt} \\forall \\sigma \\in G,\\;\\lambda \\in ,$ and hence, expanding the right-hand side of (REF ) and applying the spherical inversion formula (REF ), $\\mathcal {L}^\\sigma f(\\tau ) = -\\int _{}\\hat{f}(\\lambda )\\beta (\\mathcal {L}^\\sigma ,\\lambda )\\phi _\\lambda (\\tau )\\omega (d\\lambda )$ for all $f\\in C_c^\\infty (K|G|K)$ and $\\sigma ,\\tau \\in G$ .",
"In particular, $\\mathcal {L}^\\sigma f(\\sigma ) = -\\int _{}\\hat{f}(\\lambda )\\beta (\\mathcal {L}^\\sigma ,\\lambda )\\phi _\\lambda (\\sigma )\\omega (d\\lambda ) \\hspace{20.0pt}\\forall f\\in C_c^\\infty (K|G|K),\\;\\sigma \\in G.$ Consider next the action of the non-local part $\\mathcal {P}$ .",
"By Lemma 2.3 on page 39 of Liao [34], for each fixed $\\sigma \\in G$ , and for all $f\\in C_b^2(K|G|K)$ , the integrand on the right-hand side of (REF ) is absolutely integrable with respect to $\\nu (\\sigma ,\\cdot )$ .",
"Therefore, (REF ) may be used to extend the domain of $\\mathcal {P}$ so as to include $C_b^2(K|G|K)$ .",
"We do so now, and (without any loss of precision) denote the extension by $\\mathcal {P}$ .",
"Let us proceed similarly to [8] Section 5, and define for each $\\sigma \\in G$ a linear functional $\\mathcal {P}^\\sigma :C_b^2(K|G|K)\\rightarrow \\mathbb {C}$ by $\\mathcal {P}^\\sigma f:= \\mathcal {P}\\left(L_\\sigma ^{-1}f\\right)(\\sigma ),$ for each $\\sigma \\in G$ and $f\\in C_b^2(K|G|K)$ .",
"Then $\\mathcal {P}f(\\sigma ) = \\mathcal {P}^\\sigma (L_\\sigma f)$ , for all $\\sigma \\in G$ and $f\\in C_b^2(K|G|K)$ , and hence $\\mathcal {P}^\\sigma \\phi _\\lambda = \\int _G\\left(L_\\sigma ^{-1}\\phi _\\lambda (\\sigma \\tau ) - L_\\sigma ^{-1}\\phi _\\lambda (\\sigma ) - \\sum _{i=1}^dx_i(\\tau )X_iL_\\sigma ^{-1}\\phi _\\lambda (\\sigma )\\right)\\nu (\\sigma ,d\\tau ),$ for all $\\lambda \\in $ and $\\sigma \\in G$ .",
"Moreover, by the above discussion, the integrand on the right-hand side of (REF ) is absolutely $\\nu (\\sigma ,\\cdot )$ -integrable, for all $\\lambda \\in $ and $\\sigma \\in G$ .",
"Now, $L_\\sigma ^{-1}\\phi _\\lambda (\\sigma \\tau ) - L_\\sigma ^{-1}\\phi _\\lambda (\\sigma ) - \\sum _{i=1}^dx_i(\\tau )X_iL_\\sigma ^{-1}\\phi _\\lambda (\\sigma ) &= \\phi _\\lambda (\\tau ) - \\phi _\\lambda (e) - \\sum _{i=1}^dx_i(\\tau )X_i\\phi _\\lambda (e) \\\\&= \\phi _\\lambda (\\tau ) - 1,$ since $\\phi _\\lambda (e)=1$ , and, by Theorem 5.3 (b) on page 139 of Liao [34], $X\\phi _\\lambda (e)=0$ for all $X\\in \\mathfrak {p}$ .",
"Thus, for all $\\lambda \\in $ and $\\sigma \\in G$ , $\\phi _\\lambda -1$ is absolutely $\\nu (\\sigma ,\\cdot )$ -integrable, and $\\mathcal {P}^\\sigma \\phi _\\lambda = \\int _G\\left(\\phi _\\lambda (\\tau ) - 1\\right)\\nu (\\sigma ,d\\tau ).$ A standard argument involving the functional equation $\\phi _\\lambda (\\sigma )\\phi _\\lambda (\\tau ) = \\int _K\\phi _\\lambda (\\sigma k\\tau )dk, \\hspace{20.0pt} \\forall \\lambda \\in ,\\sigma ,\\tau \\in G,$ for spherical functions (see [21] Proposition 2.2, pp.",
"400) may now be applied in precisely the same way as in [8] (5.3)–(5.7), to infer that $\\mathcal {P}\\phi _\\lambda (\\sigma ) = \\int _G(\\phi _\\lambda (\\sigma \\tau ) - \\phi _\\lambda (\\sigma ))\\nu (\\sigma ,d\\tau ) = \\int _G(\\phi _\\lambda (\\tau ) - 1)\\phi _\\lambda (\\sigma )\\nu (\\sigma ,d\\tau )$ for all $\\sigma \\in G$ and $\\lambda \\in $ .",
"Finally, let $f\\in C_c^\\infty (K|G|K)$ , and observe that by the spherical inversion formula $\\begin{aligned}\\mathcal {P}f(\\sigma ) &= \\int _G\\Bigg (\\int _{}\\phi _\\lambda (\\sigma \\tau )\\hat{f}(\\lambda )\\omega (d\\lambda ) - \\int _{}\\phi _\\lambda (\\sigma )\\hat{f}(\\lambda )\\omega (d\\lambda ) \\\\&\\hspace{100.0pt}-\\sum _{i=1}^dx_i(\\tau )X_i\\left[\\int _{}\\phi _\\lambda \\hat{f}(\\lambda )\\omega (d\\lambda )\\right](\\sigma )\\Bigg )\\nu (\\sigma ,d\\tau )\\end{aligned}$ Claim For all $X\\in \\mathfrak {p}$ and $f\\in C_c^\\infty (K|G|K)$ , $X\\left[\\int _{}\\phi _\\lambda \\hat{f}(\\lambda )\\omega (d\\lambda )\\right](\\sigma )=\\int _{}X\\phi _\\lambda (\\sigma )\\hat{f}(\\lambda )\\omega (d\\lambda ).$ Proof of Claim This is a fairly standard differentiation-through-integration-sign argument.",
"First note that by translation invariance of $X$ , it suffices to prove the claim for $\\sigma =e$ .",
"Now, $X\\left[\\int _{}\\phi _\\lambda \\hat{f}(\\lambda )\\omega (d\\lambda )\\right](e) &= \\left.\\frac{d}{dt}\\int _{}\\phi _\\lambda (\\exp tX)\\hat{f}(\\lambda )\\omega (d\\lambda )\\right|_{t=0} \\\\&= \\lim _{t\\rightarrow 0}\\int _{}\\frac{\\phi _\\lambda (\\exp tX)-1}{t}\\hat{f}(\\lambda )\\omega (d\\lambda ).$ By the mean value theorem, for each $t>0$ and $\\lambda \\in $ , $\\frac{\\phi _\\lambda (\\exp tX)-1}{t}=X\\phi _\\lambda (\\exp t^{\\prime }X),$ for some $0<t^{\\prime }<t$ , and hence $\\left|\\frac{\\phi _\\lambda (\\exp tX)-1}{t}\\right| \\le \\Vert X\\phi _\\lambda \\Vert _\\infty $ for all $t>0$ .",
"By Helgason [20] Theorem 1.1 (iii), $\\Vert X\\phi _\\lambda \\Vert _\\infty \\le C(1+|\\lambda |)$ , for some some constant $C>0$ .",
"Thus, for $f\\in C_c^\\infty (K|G|K)$ , $\\lambda \\in $ and $t>0$ , $\\left|\\frac{\\phi _\\lambda (\\exp tX)-1}{t}\\hat{f}(\\lambda )\\right| \\le C(1+|\\lambda |)|\\hat{f}(\\lambda )|,$ and clearly $C(1+|\\cdot |)\\hat{f} \\in L^1()^W$ , since $\\hat{f}\\in \\mathcal {S}()$ .",
"Hence by dominated convergence, $X\\left[\\int _{}\\phi _\\lambda \\hat{f}(\\lambda )\\omega (d\\lambda )\\right](e) = \\int _{}\\lim _{t\\rightarrow 0}\\frac{\\phi _\\lambda (\\exp tX)-1}{t}\\hat{f}(\\lambda )\\omega (d\\lambda ) = \\int _{}X\\phi _\\lambda (e)\\hat{f}(\\lambda )\\omega (d\\lambda ),$ which completes the proof of the claim.",
"Applying the claim to (REF ), for $f\\in C_c^\\infty (K|G|K)$ and $\\sigma \\in G$ , $\\mathcal {P}f(\\sigma ) &= \\int _G\\Bigg (\\int _{}\\phi _\\lambda (\\sigma \\tau )\\hat{f}(\\lambda )\\omega (d\\lambda ) - \\int _{}\\phi _\\lambda (\\sigma )\\hat{f}(\\lambda )\\omega (d\\lambda ) \\\\&\\hspace{150.0pt}-\\sum _{i=1}^dx_i(\\tau )\\int _{}X_i\\phi _\\lambda (\\sigma )\\hat{f}(\\lambda )\\omega (d\\lambda )\\Bigg )\\nu (\\sigma ,d\\tau ) \\\\&=\\int _G\\int _{}\\hat{f}(\\lambda )\\left(\\phi _\\lambda (\\sigma \\tau )-\\phi _\\lambda (\\sigma )-\\sum _{i=1}^dx_i(\\tau )X_i\\phi _\\lambda (\\sigma )\\right)\\omega (d\\lambda )\\nu (\\sigma ,d\\tau ).$ By the Fubini theorem, $\\mathcal {P}f(\\sigma ) &= \\int _{}\\hat{f}(\\lambda )\\int _G\\left(\\phi _\\lambda (\\sigma \\tau )-\\phi _\\lambda (\\sigma )-\\sum _{i=1}^dx_i(\\tau )X_i\\phi _\\lambda (\\sigma )\\right)\\nu (\\sigma ,d\\tau )\\omega (d\\lambda ) \\\\&= \\int _{}\\hat{f}(\\lambda )\\mathcal {P}\\phi _\\lambda (\\sigma )\\omega (d\\lambda )$ for all $f\\in C_c^\\infty (K|G|K)$ and $\\sigma \\in G$ .",
"It follows by (REF ) that $\\mathcal {P}f(\\sigma ) = \\int _{}\\hat{f}(\\lambda )\\phi _\\lambda (\\sigma )\\int _G(\\phi _\\lambda (\\tau ) - 1)\\nu (\\sigma ,d\\tau )\\omega (d\\lambda )$ for all $f\\in C_c^\\infty (K|G|K)$ and $\\sigma \\in G$ .",
"The result now follows by substituting (REF ) and (REF ) into (REF ).",
"Example 3.8 As already noted, all Gangolli exponents are Gangolli symbols.",
"Let $u\\in C(K|G|K)$ be non-negative, and let $v:\\rightarrow \\mathbb {C}$ be a Gangolli exponent.",
"Then $q:G\\times \\rightarrow \\mathbb {C}$ given by $q(\\sigma ,\\lambda ) = u(\\sigma )v(\\lambda ) \\hspace{20.0pt} \\forall \\sigma \\in G,\\;\\lambda \\in $ is a Gangolli symbol.",
"Indeed, by Theorem REF , there exists a sub-diffusion operator $T\\in {\\bf D}_K(G)$ and a $K$ -bi-invariant Lévy measure $\\nu $ such that, $v(\\lambda ) = -\\beta (T,\\lambda ) + \\int _G(1-\\phi _\\lambda (\\sigma ))\\nu (d\\tau )$ for all $\\lambda \\in $ .",
"But then, for each $\\sigma \\in G$ and $\\lambda \\in $ $q(\\sigma ,\\lambda ) = -\\beta (u(\\sigma )T,\\lambda ) + \\int _G(1-\\phi _\\lambda (\\sigma ))\\nu (d\\tau ).$ Writing $\\mathcal {L}^\\sigma =u(\\sigma )T$ and $\\nu (\\sigma ,d\\tau ) = u(\\sigma )\\nu (d\\tau )$ for each $\\sigma \\in G$ , $q$ resembles (REF ).",
"Since $T$ is a sub-diffusion operator, there is a non-negative constant $c$ , a vector $b\\in \\mathbb {R}^d$ and a non-negative definite symmetric matrix $(a_{jk})$ such that $T = -c + \\sum _{i=1}^db_iX_i+\\sum _{j,k=1}^da_{jk}X_jX_k.$ For each $\\sigma \\in G$ , and for $i,j,k=1,\\ldots ,d$ , let $c(\\sigma ) = cu(\\sigma ), \\hspace{20.0pt} b_i(\\sigma ) = b_iu(\\sigma ), \\hspace{20.0pt} a_{jk}(\\sigma ) = a_{jk}u(\\sigma )\\hspace{10.0pt} \\text{and} \\hspace{10.0pt} \\nu (\\sigma ,\\cdot )=\\nu .$ Since $u$ is non-negative, continuous and $K$ -bi-invariant, the conditions of Definition REF are easily verified for these characteristics."
],
[
"Construction of Sub-Feller Semigroups",
"In the previous section, we used results from Applebaum and Ngan [7] and [8] to introduce a class of pseudodifferential operators that satisfy all but the last condition of the Hille–Yosida–Ray theorem, Theorem REF (REF ), in the case where $E=K|G|K$ .",
"In this section we tackle this the third condition, and seek conditions on a symbol $q$ so that, for some $\\alpha >0$ , $\\overline{\\operatorname{Ran}(\\alpha +q(\\sigma ,D))}=C_0(K|G|K).$ We use an approach based primarily on Jacob [27] and Hoh [24] Section 4.",
"Now that we are on the level of operators, there are more arguments that closely resemble these sources.",
"In these cases, proofs are not expanded in great detail, and may be omitted entirely to save space.",
"Instead, we aim to emphasise what does not carry over from the Euclidean space setting.",
"As in previous work, let $\\psi :\\rightarrow \\mathbb {R}$ be a fixed real-valued, continuous negative definite function satisfying (REF ) for some fixed $r>0$ .",
"The next lemma will be needed later on for proving regularity properties of pseudodifferential operators — see [24] Lemma 4.2 on page 48 for comparison.",
"Before stating the result, we introduce some notation.",
"For a mapping $q:G\\times \\rightarrow \\mathbb {R}$ and for each $\\lambda ,\\eta \\in $ , $\\sigma \\in G$ , define $F_{\\lambda ,\\eta }(\\sigma ) = \\phi _{-\\lambda }(\\sigma )q(\\sigma ,\\eta ).$ Observe that if $q(\\cdot ,\\eta )\\in L^2(K|G|K)$ for all $\\eta \\in $ , then $F_{\\lambda ,\\eta }\\in L^2(K|G|K)$ , and we may consider the spherical transform $\\hat{F}_{\\lambda ,\\eta }\\in L^2(,\\omega )$ , given by $\\hat{F}_{\\lambda ,\\eta }(\\mu ) = \\int _G\\phi _{-\\mu }(\\sigma )\\phi _{-\\lambda }(\\sigma )q(\\sigma ,\\eta )d\\sigma , \\hspace{20.0pt} \\forall \\mu \\in .$ To give some intuition to the introduction of $F_{\\lambda ,\\eta }$ , consider the case $G=\\mathbb {R}^d$ , $K=\\lbrace 0\\rbrace $ .",
"In this case, the so-called frequency shift property for the Fourier transform says that $\\begin{aligned}\\hat{F}_{\\lambda ,\\eta }(\\mu ) &= \\frac{1}{(2\\pi )^{d/2}}\\int _{\\mathbb {R}^d}e^{-i\\mu \\cdot x}e^{-\\lambda \\cdot x}q(x,\\eta )dx \\\\&= \\frac{1}{(2\\pi )^{d/2}}\\int _{\\mathbb {R}^d}e^{-i(\\mu +\\lambda )\\cdot x}q(x,\\eta )dx = \\hat{q}(\\lambda +\\mu ,\\eta ),\\end{aligned}$ where $^{\\wedge }$ denotes the Fourier transform taken in the first argument of $q$ .",
"Hoh [24] and Jacob [29] make use of bounds on $\\hat{q}(\\lambda -\\mu ,\\eta )$ , and $\\hat{F}_{\\lambda ,\\eta }(-\\mu )$ will assume an analogous role in work to come.",
"The next lemma is an analogue of Lemma 2.1 of Jacob [27].",
"See also Hoh [24] Lemma 4.2, pp. 48.",
"It's proof is similar to these other lemmas, but some additional steps are required to work around some new difficulties that arise in this setting.",
"Lemma 4.1 Let $M\\in \\mathbb {N}$ , $q:G\\times \\rightarrow \\mathbb {R}$ and suppose $q(\\cdot ,\\lambda )\\in C^M_c(K|G|K)$ for all $\\lambda \\in $ .",
"Suppose further that for each $\\beta \\in \\lbrace 0,1,\\ldots ,M\\rbrace $ , there is a non-negative function $\\Phi _\\beta \\in L^1(K|G|K)$ such that, in the notation above, $\\left|(-\\Delta )^{\\beta /2}F_{\\lambda ,\\eta }(\\sigma )\\right| \\le \\Phi _\\beta (\\sigma )\\langle \\lambda \\rangle ^M(1+\\psi (\\eta )),$ for all $\\lambda ,\\eta \\in $ , $\\sigma \\in G$ .",
"Then there is a constant $C_M>0$ such that $\\left|\\hat{F}_{\\lambda ,\\eta }(\\mu )\\right| \\le C_M\\sum _{\\beta =0}^M\\Vert \\Phi _\\beta \\Vert _1\\langle \\lambda +\\mu \\rangle ^{-M}(1+\\psi (\\eta )),$ for all $\\lambda ,\\mu ,\\eta \\in $ , where $\\Vert \\cdot \\Vert _1$ denotes the usual norm on the Banach space $L^1(K|G|K)$ (taken with respect to Haar measure).",
"Remarks 4.2 Here, we are again using the notation $\\langle \\lambda \\rangle =\\sqrt{1+|\\lambda |^2}$ , introduced in (REF ).",
"The condition (REF ) may seem quite obscure.",
"The role of $\\langle \\lambda +\\mu \\rangle $ will hopefully become apparent in the proof of Theorem REF .",
"In Section , we will develop a class of examples where it is satisfied.",
"Under the conditions of the lemma, and using the Fubini theorem, we have the following: for all $u\\in C_c^\\infty (K|G|K)$ and $\\lambda \\in $ , $\\begin{aligned}(q(\\sigma ,D)u)^{\\wedge }(\\lambda ) &= \\int _C\\int _{}\\phi _{-\\lambda }(\\sigma )\\phi _\\eta (\\sigma )q(\\sigma ,\\eta )\\hat{u}(\\eta )\\omega (d\\eta )d\\sigma \\\\&= \\int _{}\\left(\\int _C\\phi _\\eta (\\sigma )F_{\\lambda ,\\eta }(\\sigma )d\\sigma \\right)\\hat{u}(\\eta )\\omega (d\\eta ) \\\\&= \\int _{}\\hat{F}_{\\lambda ,\\eta }(-\\eta )\\hat{u}(\\eta )\\omega (d\\eta ).\\end{aligned}$ Fubini's theorem does indeed apply here — a suitable bound for the integrand on the first line of (REF ) may be found by noting that for all $\\lambda ,\\eta \\in $ and $\\sigma \\in G$ , $|\\phi _{-\\lambda }(\\sigma )\\phi _\\eta (\\sigma )q(\\sigma ,\\eta )\\hat{u}(\\eta )|\\le |q(\\sigma ,\\eta )||\\hat{u}(\\eta )|\\le \\Phi _0(\\sigma )\\big (1+\\psi (\\eta )\\big )|\\hat{u}(\\eta )|,$ for some $\\Phi _0\\in L^1(K|G|K)$ , by (REF ).",
"By Theorem REF , $\\hat{u}\\in \\mathcal {S}()$ , and the usual bound (REF ) on the density of Plancherel measure may be applied, similarly to (REF ), to conclude that the right-hand side of (REF ) is $\\omega (d\\eta )\\times d\\sigma $ -integrable.",
"[Proof of Lemma REF ] Let $\\beta \\in \\lbrace 0,1,\\ldots ,M\\rbrace $ and $\\lambda ,\\eta \\in $ be fixed.",
"The fractional Laplacian $(-\\Delta )^{\\beta /2}$ satisfies a well-known eigenrelation $(-\\Delta )^{\\beta /2}\\phi _\\mu = \\left(|\\rho |^2+|\\mu |^2\\right)^{\\beta /2}\\phi _\\mu , \\hspace{20.0pt} \\forall \\mu \\in ,$ which may be proven using subordination methods and properties of the Laplace-Beltrami operator on a symmetric space, using similar techniques to Section 5.7 of [5], pp. 154–7.",
"One can also show using standard methods that $\\left((-\\Delta )^{\\beta /2}f\\right)^{\\wedge }(\\mu ) = \\left(|\\rho |^2+|\\mu |^2\\right)^{\\beta /2}\\hat{f}(\\mu ),$ for all $f\\in C^M_c(K|G|K)$ and $\\mu \\in $ .",
"Then, using the definition of the spherical transform, $(|\\rho |^2+|\\mu |^2)^{\\beta /2}\\hat{f}(\\mu ) = \\int _G\\phi _{-\\mu }(\\sigma )(-\\Delta )^{\\beta /2}f(\\sigma )d\\sigma , $ for all $f\\in C_c^M(K|G|K)$ and all $\\mu \\in $ .",
"Applying this to $f=F_{\\lambda ,\\eta }$ , we have for all $\\mu \\in \\operatorname{\\mathfrak {a}}^\\ast $ , $\\left|\\left(|\\rho |^2+|\\mu |^2\\right)^{\\beta /2}\\hat{F}_{\\lambda ,\\eta }(\\mu )\\right| &\\le \\int _G|\\phi _{-\\mu }(\\sigma )|\\left|(-\\Delta )^{\\beta /2}F_{\\lambda ,\\eta }(\\sigma )\\right|d\\sigma \\\\&\\le \\int _G\\Phi _\\beta (\\sigma )\\langle \\lambda \\rangle ^M(1+\\psi (\\eta ))d\\sigma = \\Vert \\Phi _\\beta \\Vert _1\\langle \\lambda \\rangle ^M(1+\\psi (\\eta )),$ and summing over $\\beta $ , $\\sum _{\\beta =0}^M\\left(|\\rho |^2+|\\mu |^2\\right)^{\\beta /2}\\left|\\hat{F}_{\\lambda ,\\eta }(\\mu )\\right| \\le \\sum _{\\beta =0}^M\\Vert \\Phi _\\beta \\Vert _1\\langle \\lambda \\rangle ^M(1+\\psi (\\eta )),$ for all $\\lambda ,\\mu ,\\eta \\in $ .",
"Let $C^{\\prime }_M>0$ be the smallest positive number such that $\\langle \\mu \\rangle ^M \\le C^{\\prime }_M\\sum _{\\beta =0}^M\\left(|\\rho |^2+|\\mu |^2\\right)^{\\beta /2} \\hspace{20.0pt} \\forall \\mu \\in .$ Then, rearranging (REF ), $\\left|\\hat{F}_{\\lambda ,\\eta }(\\mu )\\right| \\le C^{\\prime }_M\\sum _{\\beta =0}^M\\Vert \\Phi _\\beta \\Vert _1\\langle \\mu \\rangle ^{-M}\\langle \\lambda \\rangle ^M(1+\\psi (\\eta )),$ for all $\\lambda ,\\mu ,\\eta \\in $ .",
"Finally, observe that by Peetre's inequality (see Proposition REF (REF )), $\\langle \\lambda \\rangle ^M\\langle \\lambda +\\mu \\rangle ^{-M} = \\left(\\frac{1+|\\lambda |^2}{1+|\\lambda +\\mu |^2}\\right)^{M/2} \\le 2^{M/2}(1+|\\mu |^2)^{M/2} = 2^{M/2}\\langle \\mu \\rangle ^M$ for all $\\lambda ,\\mu \\in $ .",
"Therefore, for all $\\lambda ,\\mu \\in $ , $\\langle \\mu \\rangle ^{-M}\\langle \\lambda \\rangle ^M\\le 2^{M/2}\\langle \\lambda +\\mu \\rangle ^{-M}$ and by (REF ), $\\left|\\hat{F}_{\\lambda ,\\eta }(\\mu )\\right| \\le 2^{M/2}C^{\\prime }_M\\sum _{\\beta =0}^M\\Vert \\Phi _\\beta \\Vert _1\\langle \\lambda +\\mu \\rangle ^{-M}(1+\\psi (\\eta ))$ The result now follows by taking $C_M=2^{M/2}C^{\\prime }_M$ .",
"Remark 4.3 The constant $C_M:=2^{M/2}\\sup _{\\lambda \\in }\\frac{\\langle \\lambda \\rangle ^M}{\\sum _{\\beta =0}^M\\big (|\\rho |^2+|\\lambda |^2\\big )^{\\beta /2}}$ appearing in the proof of Lemma REF will remain relevant throughout this chapter.",
"Let now $q:G\\times \\rightarrow \\mathbb {R}$ be a continuous negative definite symbol, $K$ -bi-invariant in its first argument, and $W$ -invariant in its second (for example, $q$ could be taken to be a Gangolli symbol, as in (REF )).",
"Similarly to Jacob [27] §4 and Hoh [24] (4.26), we write $q(\\sigma ,\\lambda ) = q_1(\\lambda )+q_2(\\sigma ,\\lambda ), \\hspace{20.0pt} \\forall \\sigma \\in G,\\lambda \\in ,$ where $q_1(\\lambda )=q(\\sigma _0,\\lambda )$ and $q_2(\\sigma ,\\lambda )=q(\\sigma ,\\lambda )-q(\\sigma _0,\\lambda )$ , for some fixed $\\sigma _0\\in G$ .",
"Observe that $q_1$ is necessarily a negative definite symbol.",
"Though $q_2$ may not be, we may still define the operator $q_2(\\sigma ,D)$ in a meaningful way, by $q_2(\\sigma ,D) := q(\\sigma ,D) - q_1(D) = \\int _{}\\phi _\\lambda (\\sigma )q_2(\\sigma ,\\lambda )\\hat{f}(\\lambda )\\omega (d\\lambda ), \\hspace{20.0pt} \\forall \\sigma \\in G.$ By decomposing $q$ in this way, we view it as a perturbation of a negative definite function $q_1$ by $q_2$ .",
"The assumptions we place on $q$ will control the size of this perturbation, as well ensuring certain regularity properties of $q(\\sigma ,D)$ acting on the anisotropic Sobolev spaces introduced in Section REF .",
"Assumptions 4.4 In the notation above, we impose the following: There exist constants $c_0,c_1>0$ such that for all $\\lambda \\in $ with $|\\lambda |\\ge 1$ , $c_0(1+\\psi (\\lambda ))\\le q_1(\\lambda ) \\le c_1(1+\\psi (\\lambda )).$ Let $M\\in \\mathbb {N}$ , $M>\\dim (G/K)$ , and suppose that $q_2(\\cdot ,\\lambda )\\in C^M_c(K|G|K)$ for all $\\lambda \\in $ .",
"Suppose further that for $\\beta =0,1,\\ldots ,M$ , there exists $\\Phi _\\beta \\in L^1(K|G|K)$ such that $\\left|(-\\Delta )^{\\beta /2}F_{\\lambda ,\\eta }(\\sigma )\\right| \\le \\Phi _\\beta (\\sigma )\\langle \\lambda \\rangle ^M\\big (1+\\psi (\\eta )\\big ),$ for all $\\lambda ,\\eta \\in $ , $\\sigma \\in G$ , where $F_{\\lambda ,\\eta }(\\sigma )=\\phi _{-\\lambda }(\\sigma )q_2(\\sigma ,\\eta )$ (c.f.",
"(REF )).",
"Remarks 4.5 These assumptions are analogues to P.1, P.2.q of Jacob [27], pp.",
"156, or (A.1), (A.2.M) of Hoh [24], pp.54.",
"As noted in Remark REF (REF ), the conditions in Assumption REF (REF ) imply that $(q_2(\\sigma ,D)u)^{\\wedge }(\\lambda ) = \\int _{}\\hat{F}_{\\lambda ,\\eta }(-\\eta )\\hat{u}(\\eta )\\omega (d\\eta ),$ for all $\\lambda \\in $ and $u\\in C_c^\\infty (K|G|K)$ , a fact that will be useful several times more.",
"Theorem 4.6 Subject to Assumptions REF , for all $s\\in \\mathbb {R}$ , $q_1(D)$ extends to a continuous operator from $H^{\\psi ,s+2}$ to $H^{\\psi ,s}$ , and $q(\\sigma ,D)$ extends to a continuous operator from $H^{\\psi ,2}$ to $L^2(K|G|K)$ .",
"The proof of the first part is omitted, since it is an easy adaptation of the proof of Theorem 4.8 on page 55 of Hoh [24] — first proved as Corollary 3.1 in Jacob [27].",
"The second part is also proved similarly to Theorem 4.8 of [24], the main difference being that $\\hat{F}_{\\lambda ,\\eta }(-\\eta )$ takes the place of the transformed symbol, as discussed previously (see (REF )).",
"By (REF ) and the Plancherel theorem, $|\\langle q_2(\\sigma ,D)u,v\\rangle | &= \\left|\\int _{}(q_2(\\sigma ,D)u)^{\\wedge }(\\lambda )\\overline{\\hat{v}(\\lambda )}\\omega (d\\lambda )\\right| \\\\&= \\left|\\int _{}\\int _{}\\hat{F}_{\\lambda ,\\eta }(-\\eta )\\hat{u}(\\eta )\\overline{\\hat{v}(\\lambda )}\\omega (d\\eta )\\omega (d\\lambda )\\right| \\\\&\\le \\int _{}\\int _{}\\left|\\hat{F}_{\\lambda ,\\eta }(-\\eta )\\right||\\hat{u}(\\eta )||\\hat{v}(\\lambda )|\\omega (d\\eta )\\omega (d\\lambda ).$ Then, using (REF ), Lemma REF and Young's convolution inequalitySee Simon [43] Theorem 6.6.3, page 550.",
"Here, we are again identifying $$ with a Euclidean space., $|\\langle q_2(\\sigma ,D)u,v\\rangle | &\\le C_M\\sum _{\\beta =0}^M\\Vert \\Phi _\\beta \\Vert _1\\int _{}\\int _{}\\langle \\lambda -\\eta \\rangle ^{-M}\\Psi (\\eta )^2|\\hat{u}(\\eta )||\\hat{v}(\\lambda )|\\omega (d\\eta )\\omega (d\\lambda ) \\\\&= C_M\\sum _{\\beta =0}^M\\Vert \\Phi _\\beta \\Vert _1\\int _{}\\left[\\langle \\cdot \\rangle ^{-M}\\ast \\big (\\Psi ^2|\\hat{u}|\\big )\\right](\\lambda )|\\hat{v}(\\lambda )|\\omega (d\\lambda ) \\\\&\\le C_M\\sum _{\\beta =0}^M\\Vert \\Phi _\\beta \\Vert _1\\left\\Vert \\langle \\cdot \\rangle ^{-M}\\ast \\big (\\Psi ^2|\\hat{u}|\\big )\\right\\Vert _{L^2(,\\omega )}\\Vert \\hat{v}\\Vert _{L^2(,\\omega )} \\\\&\\le C_M\\sum _{\\beta =0}^M\\Vert \\Phi _\\beta \\Vert _1\\left\\Vert \\langle \\cdot \\rangle ^{-M}\\right\\Vert _{L^1(,\\omega )}\\Vert u\\Vert _{\\psi ,2}\\Vert v\\Vert ,$ for all $u,v\\in C_c^\\infty (K|G|K)$ .",
"Hence, for all $u\\in C_c^\\infty (K|G|K)$ , $\\Vert q_2(\\sigma ,D)u\\Vert &= \\sup _{\\begin{array}{c}v\\in C_c^\\infty (K|G|K) \\\\ \\Vert v\\Vert =1\\end{array}}|\\langle q_2(\\sigma ,D)u,v\\rangle | \\le C_M\\sum _{\\beta =0}^M\\Vert \\Phi _\\beta \\Vert _1\\left\\Vert \\langle \\cdot \\rangle ^{-M}\\right\\Vert _{L^1(,\\omega )}\\Vert u\\Vert _{\\psi ,2},$ and $q_2(\\sigma ,D)$ extends to a bounded linear operator $H^{\\psi ,2}\\rightarrow L^2(K|G|K)$ .",
"Under an additional assumption, we are able to obtain a more powerful result.",
"Theorem 4.7 Suppose Assumptions REF hold, and suppose further that $s\\in \\mathbb {R}$ satisfies $|s-1|+1+\\dim (G/K) < M$ .",
"Then $q(\\sigma ,D)$ extends to a continuous linear operator from $H^{\\psi ,s+2}\\rightarrow H^{\\psi ,s}$ .",
"We first need a technical lemma.",
"Lemma 4.8 Let $s\\in \\mathbb {R}$ and $M\\in \\mathbb {N}$ be such that $|s-1|+1+\\dim (G/K)<M$ .",
"Then for all $\\lambda ,\\eta \\in $ , $\\left|\\Psi (\\lambda )^s-\\Psi (\\eta )^s\\right| \\le C_{s,\\psi }\\langle \\lambda -\\eta \\rangle ^{|s-1|+1}\\Psi (\\eta )^{s-1},$ where $C_{s,\\psi } = 2^{(|s-1|+2)/2}(1+c_\\psi )^{(|s-1|+1)/2}|s|,$ and $c_\\psi $ is the constant from Proposition REF (REF ).",
"This is a special case of a bound obtained in Hoh [24] — see page 50, lines 5–11.",
"[Proof of Theorem REF ] By Theorem REF , it suffices to prove that $q_2(\\sigma ,D)$ extends to a continuous operator from $H^{\\psi ,s+2}\\rightarrow H^{\\psi ,s}$ .",
"Given $u\\in C^\\infty _c(K|G|K)$ , $\\begin{aligned}\\Vert q_2(\\sigma ,D)u\\Vert _{\\psi ,s} &= \\Vert \\Psi (D)^sq_2(\\sigma ,D)u\\Vert \\\\&\\le \\Vert q_2(\\sigma ,D)\\Psi (D)^su\\Vert + \\Vert [\\Psi (D)^s,q_2(\\sigma ,D)]u\\Vert .\\end{aligned}$ Also, by Theorem REF and Theorem REF (REF ), $\\Vert q_2(\\sigma ,D)\\Psi (D)^su\\Vert \\le C\\Vert \\Psi (D)^su\\Vert _{\\psi ,2} = C\\Vert u \\Vert _{\\psi ,s+2},$ where $C = C_M\\sum _{\\beta =0}^M\\Vert \\Phi _\\beta \\Vert _1\\left\\Vert \\langle \\cdot \\rangle ^{-M}\\right\\Vert _{L^1(,\\omega )}.$ We will estimate $\\left\\Vert [\\Psi (D)^s,q_2(\\sigma ,D)]u\\right\\Vert ,$ Our method is similar to that in Theorem 4.3 of Hoh [24], and so some details are omitted.",
"The map $F_{\\lambda ,\\eta }$ replaces the transformed symbol $\\hat{p}$ once again.",
"One can check using (REF ) that for all $\\lambda \\in $ , $([\\Psi (D)^s,q_2(\\sigma ,D)]u)^{\\wedge }(\\lambda ) = \\int _{}\\hat{F}_{\\lambda ,\\eta }(-\\eta )\\big \\lbrace \\Psi (\\lambda )^s-\\Psi (\\eta )^s\\big \\rbrace \\hat{u}(\\eta )\\omega (d\\eta ),$ and hence for all $u,v\\in C_c^\\infty (K|G|K)$ , $\\left|\\big \\langle [\\Psi (D)^s,q_2(\\sigma ,D)]u,v\\big \\rangle \\right|\\le \\int _{}\\int _{}\\left|\\hat{F}_{\\lambda ,\\eta }(-\\eta )\\right|\\left|\\Psi (\\lambda )^s-\\Psi (\\eta )^s\\right||\\hat{u}(\\eta )||\\hat{v}(\\lambda )|\\omega (d\\eta )\\omega (d\\lambda ).$ By Lemmas REF and REF , $ \\left|\\big \\langle [\\Psi (D)^s,q_2(\\sigma ,D)]u,v\\big \\rangle \\right| &\\le C_{s,\\psi ,M}\\int _{}\\int _{}\\langle \\lambda -\\eta \\rangle ^{-M+|s-1|+1}\\Psi (\\eta )^{s+1}|\\hat{u}(\\eta )||\\hat{v}(\\lambda )|\\omega (d\\eta )\\omega (d\\lambda ) \\\\&= C_{s,\\psi ,M}\\int _{}\\left(\\langle \\cdot \\rangle ^{-M+|s-1|+1}\\ast \\left[\\Psi ^{s+1}|\\hat{u}|\\right]\\right)(\\lambda )|\\hat{v}(\\lambda )|\\omega (d\\lambda ),$ where $C_{s,\\psi ,M} = C_{s,\\psi }C_M\\sum _{\\beta =0}^M\\Vert \\Phi _\\beta \\Vert _1$ .",
"By Lemma REF , $\\langle \\cdot \\rangle ^{-(M-|s-1|-2)}\\in L^1(,\\omega )$ , and one can check using the Cauchy–Schwarz and Young inequalities that $\\left|\\big \\langle [\\Psi (D)^s,q_2(\\sigma ,D)]u,v\\big \\rangle \\right| \\le C_{s,\\psi ,M}\\left\\Vert \\langle \\cdot \\rangle ^{-(M-|s-1|-1)}\\right\\Vert _{L^1(,\\omega )}\\Vert u\\Vert _{\\psi ,s+1}\\Vert v\\Vert .",
"$ Taking the supremum over $v\\in C_c^\\infty (K|G|K)$ , with $\\Vert v\\Vert =1$ , $\\Vert [\\Psi (D)^s,q_2(\\sigma ,D)]u\\Vert \\le C_{s,\\psi ,M}\\left\\Vert \\langle \\cdot \\rangle ^{-(M-|s-1|-1)}\\right\\Vert _{L^1(,\\omega )}\\Vert u\\Vert _{\\psi ,s+1}.$ Combining with (REF ) and (REF ), $\\Vert q_2(\\sigma ,D)u\\Vert _{\\psi ,s} \\le C_M\\sum _{\\beta =0}^M\\Vert \\Phi _\\beta \\Vert _{L^1(,\\omega )}\\Big (\\left\\Vert \\langle \\cdot \\rangle ^{-M}\\right\\Vert _{L^1(,\\omega )}\\Vert u\\Vert _{\\psi ,s+2} + C_{s,\\psi }\\Vert u\\Vert _{\\psi ,s+1}\\Big ).$ Theorem REF (REF ) may now be used to obtain the desired bound.",
"Recall that we wish to prove (REF ), for some $\\alpha >0$ .",
"To do this, we seek solutions $u$ to the equation $(q(\\sigma ,D)+\\alpha )u=f,$ for a given function $f$ and $\\alpha >0$ .",
"For $\\alpha >0$ , consider the bilinear form $B_\\alpha $ defined for all $u,v\\in C_c^\\infty (K|G|K)$ by $B_\\alpha (u,v) = \\langle (q(\\sigma ,D)+\\alpha )u,v\\rangle .$ The next result is an analogue of Jacob [27] Lemma 3.2, pp.",
"160, and Hoh [24] Theorem 4.9, pp. 56.",
"Theorem 4.9 Subject to Assumptions REF , $B_\\alpha $ extends continuously to $H^{\\psi ,1}\\times H^{\\psi ,1}$ .",
"This proof is very similar to those of Jacob [27] Lemma 3.2, pp.",
"160, and Hoh [24] Theorem 4.9, pp.",
"56, and so we give only a sketch.",
"Let $u,v\\in H^{\\psi ,1}$ .",
"Using Assumption REF (REF ) and the fact that $q_1$ is continuous, there is $\\kappa _1>0$ such that $|q_1|\\le \\kappa _1\\Psi ^2$ .",
"Plancherel's identity may then be used to show that $\\left|\\langle q_1(D)u,v\\rangle \\right| \\le \\int _{}|q_1(\\lambda )||\\hat{u}(\\lambda )||\\hat{v}(\\lambda )|\\omega (d\\lambda ) \\le \\kappa _1\\Vert u\\Vert _{\\psi ,1}\\Vert v\\Vert _{\\psi ,1}.$ Furthermore, methods similar to the proof of Theorem REF are used to show that $\\left|\\langle q_2(\\sigma ,D)u,v\\rangle \\right| \\le \\kappa _2\\left\\Vert \\langle \\cdot \\rangle ^{-M+1}\\right\\Vert _{L^1(,\\omega )}\\Vert u\\Vert _{\\psi ,1}\\Vert v\\Vert _{\\psi ,1},$ where $\\kappa _2=C_M\\sqrt{2(1+c_\\psi )}\\sum _{\\beta =0}^M\\Vert \\Phi _\\beta \\Vert _{L^1(,\\omega )}.$ By Theorem REF (REF ), there is $\\kappa _3>0$ such that $\\Vert u\\Vert \\le \\kappa _3\\Vert u\\Vert _{\\psi ,1}$ , and thus for all $u,v\\in H^{\\psi ,1}$ , $\\left|B_\\alpha (u,v)\\right| \\le \\left|\\langle q_1(D)u,v\\rangle \\right| + \\left|\\langle q_2(\\sigma , D)u,v\\rangle \\right| + \\alpha \\left|\\langle u,v\\rangle \\right| \\le \\left(\\kappa _1 + \\kappa _2 + \\alpha \\kappa _3^2\\right)\\Vert u\\Vert _{\\psi ,1}\\Vert v\\Vert _{\\psi ,2},$ which proves the theorem.",
"Recall that a bilinear form $B$ defined on a Hilbert space $(H,\\langle \\cdot ,\\cdot \\rangle )$ is coercive if there is $c>0$ such that $B(u,u)\\ge c\\langle u,u\\rangle \\hspace{20.0pt} \\forall u\\in H.$ Theorem 4.10 (Lax–Milgram) Let $B$ be a bounded bilinear form, defined on a Hilbert space $(H,\\langle \\cdot ,\\cdot ,\\rangle )$ , and suppose $B$ is coercive with constant $c$ .",
"Then given $f\\in H^{\\prime }$ , there is a unique $u\\in H$ such that $B(u,v) = f(v) \\hspace{20.0pt} \\forall v\\in H.$ See for example Theorem 1 of Evans [15], pp. 297–9.",
"We would like to apply the Lax–Milgram theorem to $B_\\alpha $ .",
"For this we impose an additional assumption, which will ensure $B_\\alpha $ is coercive on $H^{\\psi ,1}$ .",
"Assumption 4.11 Let $M\\in \\mathbb {N}$ , $M>\\dim (G/K)+1$ , and write $\\gamma _M = \\left(8C_M(2(1+c_\\psi ))^{1/2}\\Vert \\langle \\cdot \\rangle ^{-M+1}\\Vert _{L^1(,\\omega )}\\right)^{-1},$ where $c_{\\psi }$ and $C_M$ are constants given by (REF ) and (REF ), respectively.",
"For $c_0$ is as in Assumption REF (REF ), assume that $\\sum _{\\beta =0}^M\\Vert \\Phi _\\beta \\Vert _1 \\le \\gamma _Mc_0.$ Remark 4.12 See Jacob [27] P.3 and P.4, pp.",
"161, or Hoh [24] (A.3.M), pp.",
"54, for comparison.",
"Examples where Assumption REF is satisfied are considered in Section .",
"The next theorem is an analogue of Theorem 3.1 of Jacob [27].",
"Theorem 4.13 Suppose Assumptions REF and REF hold, with $M>\\dim (G/K)+1$ .",
"Then there is $\\alpha _0>0$ such that $B_\\alpha (u,u)\\ge \\frac{c_0}{2}\\Vert u\\Vert _{1,\\lambda }^2,$ for all $\\alpha \\ge \\alpha _0$ and $u\\in H^{\\psi ,1}$ .",
"Proceed exactly as in Hoh [24] page 57, lines 8–17.",
"By Assumption REF (REF ), there is $\\alpha _0>0$ such that $q_1(\\lambda ) \\ge c_0\\Psi (\\lambda )^2-\\alpha _0 \\hspace{20.0pt} \\forall \\lambda \\in .$ This may be used to prove that for all $u\\in H^{\\psi ,1}$ , $\\langle q_1(D)u,u\\rangle \\ge c_0\\Vert u\\Vert _{\\psi ,1}^2 - \\alpha _0\\Vert u\\Vert ^2,$ at which point we can apply (REF ) and (REF ), as well as Assumption REF , to conclude $\\left|\\langle q_2(\\sigma ,D)u,u\\rangle \\right| &\\le C_M\\sqrt{2(1+c_\\psi )}\\sum _{\\beta =0}^M\\Vert \\Phi _\\beta \\Vert _{L^1(,\\omega )}\\left\\Vert \\langle \\cdot \\rangle ^{-M+1}\\right\\Vert _{L^1(,\\omega )}\\Vert u\\Vert _{\\psi ,1}^2 \\\\&= \\frac{1}{8\\gamma _M}\\sum _{\\beta =0}^M\\Vert \\Phi _\\beta \\Vert _{L^1(,\\omega )}\\Vert u\\Vert _{\\psi ,1}^2 \\le \\frac{c_0}{8}\\Vert u\\Vert _{\\psi ,1},$ for all $u\\in H^{\\psi ,1}$ .",
"Thus, for all $u\\in H^{\\psi ,1}$ $\\langle q(\\sigma ,D)u,u\\rangle &\\ge \\langle q_1(D)u,u\\rangle - \\left|\\langle q_2(\\sigma ,D)u,u\\rangle \\right| \\\\&\\ge (c_0-\\frac{c_0}{8})\\Vert u\\Vert _{\\psi ,1}^2 - \\alpha _0\\Vert u\\Vert _{\\psi ,1}^2 \\ge \\frac{c_0}{2}\\Vert u\\Vert _{\\psi ,1}^2 - \\alpha _0\\Vert u\\Vert _{\\psi ,1}^2.$ Therefore, for all $\\alpha \\ge \\alpha _0$ and $u\\in H^{\\psi ,1}$ $B_\\alpha (u,u) &= \\langle q(\\sigma ,D)u,u\\rangle + \\alpha \\Vert u\\Vert \\ge \\langle q(\\sigma ,D)u,u\\rangle + \\alpha _0\\Vert u\\Vert \\ge \\frac{c_0}{2}\\Vert u\\Vert _{\\psi ,1}^2,$ We may now apply the Lax–Milgram theorem (Theorem REF ) to $B_\\alpha $ , together with the linear functional $v\\mapsto \\langle f,\\cdot \\rangle $ , and conclude the following: Theorem 4.14 Let $\\alpha \\ge \\alpha _0$ .",
"Then (REF ) has a weak solution in the following sense: for all $f\\in L^2(K|G|K)$ there is a unique $u\\in H^{\\psi ,1}$ such that for all $v\\in H^{\\psi ,1}$ , $B_\\alpha (u,v) = \\langle f,v\\rangle .$ Theorem REF provides a weak solution to (REF ).",
"The next task is to prove that this solution is in fact a strong solution that belongs to $C_0(K|G|K)$ .",
"This will be achieved using the Sobolev embedding of Theorem REF (REF ).",
"Just as in Jacob [27] Theorem 3.1 and Hoh [24] Theorem 4.11, we have a useful lower bound for the pseudodifferential operator $q(\\sigma ,D)$ acting on $H^{\\psi ,s}$ , when $s\\ge 0$ .",
"Theorem 4.15 Let $s\\ge 0$ , and suppose the symbol $q$ satisfies Assumptions REF and REF , for some $M>|s-1|+1+\\dim (G/K)$ .",
"Then there is $\\kappa >0$ such that for all $u\\in H^{\\psi ,s+2}$ , $\\Vert q(\\sigma ,D)u\\Vert _{\\psi ,s} \\ge \\frac{c_0}{4}\\Vert u\\Vert _{\\psi ,s+2}-\\kappa \\Vert u \\Vert .$ The proof is formally no different to the sources mentioned: let $u\\in H^{\\psi ,s+2}$ , and use (REF ) and Theorem REF (REF ) to prove that $\\Vert q_1(D)u\\Vert _{\\psi ,s} \\ge \\frac{c_0}{2}\\Vert u\\Vert _{\\psi ,s+2} - \\kappa _1\\Vert u\\Vert ,$ for some $\\kappa _1>0$ .",
"Recall the estimate (REF ) of $\\Vert q_2(\\sigma ,D)u\\Vert _{\\psi ,s}$ from the proof of Theorem REF .",
"In light of Assumption REF and the particular form chosen for $\\gamma _M$ , one can use (REF ) to show that $\\Vert q_2(\\sigma ,D)u\\Vert _{\\psi ,s} &\\le C_M\\sum _{\\beta =0}^M\\Vert \\Phi _\\beta \\Vert _{L^1(,\\omega )}\\Big (\\left\\Vert \\langle \\cdot \\rangle ^{-M}\\right\\Vert _{L^1(,\\omega )}\\Vert u\\Vert _{\\psi ,s+2} + C_{s,\\psi }\\Vert u\\Vert _{\\psi ,s+1}\\Big ) \\\\&\\le C_Mc_0\\gamma _M\\Big (\\left\\Vert \\langle \\cdot \\rangle ^{-M}\\right\\Vert _{L^1(,\\omega )}\\Vert u\\Vert _{\\psi ,s+2} + C_{s,\\psi }\\Vert u\\Vert _{\\psi ,s+1}\\Big ) \\\\&\\le \\frac{c_0}{8}\\Vert u\\Vert _{\\psi ,s+2} + c\\Vert u\\Vert _{\\psi ,s+1},$ where $c>0$ is a constant.",
"Using Theorem REF (REF ) once again, let $\\kappa _2>0$ such that $c\\Vert u\\Vert _{\\psi ,s+1} \\le \\frac{c_0}{8}\\Vert u\\Vert _{\\psi ,s+2} + \\kappa _2\\Vert u\\Vert .$ Then, by the above, $\\Vert q_2(\\sigma ,D)u\\Vert _{\\psi ,s} \\le \\frac{c_0}{4}\\Vert u\\Vert _{\\psi ,s+2} + \\kappa _2\\Vert u\\Vert .$ Combining (REF ) and (REF ), we get $\\Vert q(\\sigma ,D)u\\Vert _{\\psi ,s} \\ge \\Vert q_1(D)u\\Vert _{\\psi ,s} - \\Vert q_2(\\sigma ,D)u\\Vert _{\\psi ,s} \\ge \\frac{c_0}{4}\\Vert u\\Vert _{\\psi ,s+2} - (\\kappa _1+\\kappa _2)\\Vert u\\Vert .$ The proof of the next theorem makes use of a particular family $(J_\\epsilon ,0<\\epsilon \\le 1)$ of bounded linear operators $L^2(G)$ , which will play the role of a Friedrich mollifier, but in the noncompact symmetric space setting.",
"First note that by identifying $\\operatorname{\\mathfrak {a}}$ with $\\mathbb {R}^m$ via our chosen basis, it makes sense to consider Friedrich mollifiers on $\\operatorname{\\mathfrak {a}}$ .",
"For $0<\\epsilon \\le 1$ and $H\\in \\operatorname{\\mathfrak {a}}$ , let $l(H) := C_0e^{\\frac{1}{|H|^2-1}}\\operatorname{\\bf 1}_{B_1(0)}(H), ~~\\text{ and }~~ l_\\epsilon (H) := \\epsilon ^{-m}l(H/\\epsilon ),$ where $C_0>0$ is a constant chosen so that $\\int _{\\operatorname{\\mathfrak {a}}}l(H)dH=1$ .",
"This mollifier (acting on Euclidean space) is used frequently in Evans [15], where it is called the standard mollifier — see Appendix C.4, pp. 629.",
"Jacob [27] and Hoh [24] both use this mollifier to show that the weak solution $u$ in Theorem REF is also a strong solution, in the sense that it satisfies (REF ), and that $u\\in H^{\\psi ,s}$ for suitably large $s$ .",
"Observe that $l,l_\\epsilon \\in \\mathcal {S}(\\operatorname{\\mathfrak {a}})^W$ for all $0<\\epsilon \\le 1$ .",
"Let $j,j_\\epsilon \\in \\mathcal {S}(K|G|K)$ be the corresponding mappings obtained via the commutative diagram of Theorem REF , so that $\\hat{j_\\epsilon } = {F}(l_\\epsilon ), \\hspace{20.0pt} \\forall 0<\\epsilon \\le 1,$ where ${F}$ denotes the Euclidean Fourier transform (see equation (REF )).",
"For $0<\\epsilon \\le 1$ , let $J_\\epsilon $ be the convolution operator defined on $L^2(K|G|K)$ by $J_\\epsilon u = j_\\epsilon \\ast u.$ The most important properties of $(J_\\epsilon ,0<\\epsilon \\le 1)$ needed for the proof of Theorem REF are stated below, and proven in the appendix.",
"Proposition 4.16 $\\hat{j_\\epsilon }(\\lambda ) = \\hat{j}(\\epsilon \\lambda )$ for all $0<\\epsilon \\le 1$ and $\\lambda \\in $ .",
"For all $0<\\epsilon \\le 1$ , $J_\\epsilon $ is a self-adjoint contraction of $L^2(K|G|K)$ .",
"$J_\\epsilon u\\in H^{\\psi ,s}$ for all $s\\ge 0$ , $u\\in L^2(K|G|K)$ and $0<\\epsilon \\le 1$ , and if $u\\in H^{\\psi ,s}$ , then $\\Vert J_\\epsilon u\\Vert _{\\psi ,s} \\le \\Vert u\\Vert _{\\psi ,s}.$ $\\Vert J_\\epsilon u - u\\Vert _{\\psi ,s}\\rightarrow 0$ as $\\epsilon \\rightarrow 0$ .",
"The following commutator estimate will also be useful in the proof of Theorem REF .",
"Lemma 4.17 Let $s\\ge 0$ , and suppose $q$ is a continuous negative definite symbol satisfying Assumption REF (REF ) for $M>|s-1|+1+\\dim (G/K)$ .",
"Then there is $c>0$ such that for all $ 0<\\epsilon \\le 1$ and all $u\\in C_c^\\infty (K|G|K)$ , $\\Vert [J_\\epsilon ,q(\\sigma ,D)]u\\Vert _{\\psi ,s} \\le c\\Vert u\\Vert _{\\psi ,s+1}.$ Let $0<\\epsilon \\le 1$ and $u\\in C_c^\\infty (K|G|K)$ , and observe that by Proposition REF (REF ), $[J_\\epsilon ,q_1(D)]u)^\\wedge (\\lambda ) = \\hat{j}(\\epsilon \\lambda )q_1(\\lambda )\\hat{u}(\\lambda ) - q_1(\\lambda )\\hat{j}(\\epsilon \\lambda )\\hat{u}(\\lambda ) = 0,$ for all $\\lambda \\in $ , so $[J_\\epsilon ,q_1(D)]u = 0$ .",
"For $\\lambda ,\\eta \\in $ , let $F_{\\lambda ,\\eta }=\\phi _{-\\lambda }q_2(\\cdot ,\\eta )$ , as previously (c.f.",
"(REF )).",
"Then by (REF ), for all $\\lambda \\in $ , $\\begin{aligned}\\left([J_\\epsilon ,q(\\sigma ,D)]u\\right)^\\wedge (\\lambda ) &= (J_\\epsilon q_2(\\sigma ,D)u)^\\wedge (\\lambda ) - (q_2(\\sigma ,D)J_\\epsilon u)^\\wedge (\\lambda ) \\\\&=\\hat{j}(\\epsilon \\lambda )(q_2(\\sigma ,D)u)^\\wedge (\\lambda ) - \\int _{}\\hat{F}_{\\lambda ,\\eta }(-\\eta )\\hat{j}(\\epsilon \\eta )\\hat{u}(\\eta )\\omega (d\\eta ).\\end{aligned}$ Now, $(q_2(\\sigma ,D)u)^\\wedge (\\lambda ) &= \\int _G\\phi _{-\\lambda }(\\sigma )(q_2(\\sigma ,D)u)(\\sigma )d\\sigma \\\\&= \\int _G\\int _{}\\phi _{-\\lambda }(\\sigma )\\phi _{-\\eta }(\\sigma )q_2(\\sigma ,\\eta )\\hat{u}(\\eta )\\omega (d\\eta )d\\sigma .$ Also, since $q_2(\\sigma ,\\cdot )$ is continuous and $\\hat{u}\\in \\mathcal {S}()$ , we have $q_2(\\sigma ,\\cdot )\\hat{u}\\in L^1(,\\omega )$ .",
"Therefore, by the Fubini theorem, $(q_2(\\sigma ,D)u)^\\wedge (\\lambda ) &= \\int _{}\\int _G\\phi _{-\\eta }(\\sigma )\\phi _{-\\lambda }(\\sigma )q_2(\\sigma ,\\eta )\\hat{u}(\\eta )d\\sigma \\omega (d\\eta ) \\\\&=\\int _{}\\int _G\\phi _{-\\eta }(\\sigma )F_{\\lambda ,\\eta }(\\sigma )d\\sigma \\hat{u}(\\eta )\\omega (d\\eta ) = \\int _{}\\hat{F}_{\\lambda ,\\eta }(-\\eta )\\hat{u}(\\eta )\\omega (d\\eta ),$ and so, substituting back into (REF ), $\\left([J_\\epsilon ,q(\\sigma ,D)]u\\right)^\\wedge (\\lambda ) = \\int _{}\\hat{F}_{\\lambda ,\\eta }(-\\eta )\\left(\\hat{j}(\\epsilon \\lambda )-\\hat{j}(\\epsilon \\eta )\\right)\\hat{u}(\\eta )\\omega (d\\eta ),$ for all $\\lambda \\in $ .",
"From here, a straightforward adaptation to the proof of Hoh [24] Theorem 4.4, pp.",
"51–52, with (REF ) replacing Hoh [24] (4.23), completes the proof of the lemma.",
"We are now ready to state and prove that, subject to our conditions, a strong solution to (REF ) exists, and belongs to an anisotropic Sobolev space of suitably high order.",
"Theorem 4.18 Let $\\alpha _0$ be as in Theorem REF , let $\\alpha \\ge \\alpha _0$ , and let $s\\ge 0$ .",
"Suppose that the negative definite symbol $q$ satisfies Assumptions REF and REF , where $M>|s-1|+1+\\dim (G/K)$ .",
"Then for all $f\\in H^{\\psi ,s}$ , there is a unique $u\\in H^{\\psi ,s+2}$ such that $(q(\\sigma ,D)+\\alpha )u = f.$ Let $f\\in H^{\\psi ,s}$ .",
"By Theorem REF we also have $f\\in L^2(K|G|K)$ , and so by Theorem REF there is a unique $u\\in H^{\\psi ,1}$ such that $B_\\alpha (u,v) =\\langle f,v\\rangle \\hspace{20.0pt} \\forall v\\in C_c^\\infty (K|G|K).$ The proof follows that of [27] Theorem 4.3, pp.",
"163 and [24] Theorem 4.12, pp.",
"59, using induction to show that that $u\\in H^{\\psi ,t}$ for $1\\le t\\le s+2$ , and in particular, that $u\\in H^{\\psi ,s+2}$ .",
"The family of operators $(J_\\epsilon ,0<\\epsilon \\le 1)$ take over role of the Friedrich mollifiers of [27] and [24].",
"By Proposition REF these operators satisfy the properties needed for the proof to carry over with little alteration.",
"Lemma REF and Theorem REF replace [24] Theorem 4.4 and 4.11, respectively.",
"Theorem 4.19 Let $q$ be a continuous negative definite function on $$ , satisfying Assumptions REF and REF with $M>\\max \\left\\lbrace 1,\\frac{d}{r}\\right\\rbrace +d$ , where $d=\\dim (G/K)$ .",
"Then for all $\\alpha \\ge \\alpha _0$ , $\\overline{\\operatorname{Ran}(\\alpha +q(\\sigma ,D))}=C_0(K|G|K).$ Fix $s\\in \\mathbb {R}$ with $\\max \\left\\lbrace \\frac{d}{r},1\\right\\rbrace < s<M-d$ .",
"Let $\\mathcal {A}$ denote the linear operator on $C_0(K|G|K)$ with domain $H^{\\psi ,s+2}$ , defined by $\\mathcal {A}u = -q(\\sigma ,D)u$ for all $u\\in \\operatorname{Dom}\\mathcal {A}$ .",
"By a similar argument to that on page 60 of Hoh [24], one can show using that $C_c^\\infty (K|G|K)$ is a operator core for $\\mathcal {A}$ , with $\\overline{\\operatorname{Ran}(\\alpha +q(\\sigma ,D))} = \\overline{\\operatorname{Ran}(\\mathcal {A}-\\alpha )}$ for all $\\alpha \\in \\mathbb {R}$ .",
"Here, Theorem REF (REF ) replaces [24] Proposition 4.1, and Theorem REF replaces [24] Theorems 4.8 and 4.11.",
"Let $\\alpha _0$ be as in Theorem REF .",
"We show that $\\overline{\\operatorname{Ran}(\\mathcal {A}-\\alpha )}=C_0(K|G|K)$ for all $\\alpha \\ge \\alpha _0$ .",
"Given $f\\in C_0(K|G|K)$ , choose a sequence $(f_n)$ in $H^{\\psi ,s}$ such that $\\Vert f_n-f\\Vert _\\infty \\rightarrow 0$ as $n\\rightarrow \\infty $ .",
"Then $f_n\\in \\operatorname{Ran}(\\mathcal {A}-\\alpha )$ for all $\\alpha \\ge \\alpha _0$ , and thus $f\\in \\overline{\\operatorname{Ran}(\\mathcal {A}-\\alpha )}$ for all $\\alpha \\ge \\alpha _0$ .",
"Combining Theorem REF with the work of Section yields the following.",
"Corollary 4.20 Let $q$ be a Gangolli symbol that satisfies Assumptions REF and REF for some $M>\\min \\lbrace 1,d/r\\rbrace +d$ .",
"Then $-q(\\sigma ,D)$ extends to the infinitesimal generator of a strongly continuous sub-Feller semigroup on $C_0(K|G|K)$ .",
"By construction, $-q(\\sigma ,D)$ is a Gangolli operator, and therefore is a densely defined linear operator on $C_0(K|G|K)$ that satisfies the positive maximum principle.",
"By Theorems REF and REF , $-q(\\sigma ,D)$ is closable, and its closure generates a strongly continuous sub-Feller semigroup."
],
[
"A Class of Examples",
"We now present a class of Gangolli symbols for which the conditions of Corollary REF are satisfied.",
"Let $M\\in \\mathbb {N}$ such that $M>\\min \\lbrace 1,d/r\\rbrace +d+1$ .",
"We consider symbols $q:G\\times \\rightarrow \\mathbb {R}$ of the form $q(\\sigma ,\\lambda ) = \\kappa \\psi (\\lambda ) + u(\\sigma )v(\\lambda ), \\hspace{20.0pt} \\forall \\sigma \\in G,\\lambda \\in ,$ where $\\kappa $ is a positive constant, $u\\in C^M_c(K|G|K)$ is non-negative, and $v:\\rightarrow \\mathbb {R}$ is a negative definite function satisfying $|v(\\lambda )|\\le c_v(1+\\psi (\\lambda )) \\hspace{20.0pt} \\forall \\lambda \\in ,$ for some constant $c_v>0$ .",
"As in previous sections, $\\psi :\\rightarrow \\mathbb {R}$ is a fixed continuous negative definite function, satisfying (REF ).",
"In addition to this, we assume $\\psi $ is a Gangolli exponent.",
"By Example REF , the mappings $(\\sigma ,\\lambda )\\mapsto c_0\\psi (\\lambda )$ and $(\\sigma ,\\lambda )\\mapsto u(\\sigma )v(\\lambda )$ are both Gangolli symbols, and hence so is $q$ .",
"For each $\\lambda \\in $ and $\\sigma \\in G$ , $q_1(\\lambda ) = \\kappa \\psi (\\lambda ), \\hspace{20.0pt} \\forall \\lambda \\in ,$ and $q_2(\\sigma ,\\lambda ) = u(\\sigma )v(\\lambda ), \\hspace{20.0pt} \\forall \\sigma \\in G,\\;\\lambda \\in .$ Observe that since $v$ has compact support, $\\operatorname{Supp}(v)\\ne G$ , and for any $\\sigma _0\\in G\\setminus \\operatorname{Supp}(v)$ , $q_1(\\lambda ) = q(\\sigma _0,\\lambda ) \\hspace{20.0pt} \\forall \\lambda \\in ,$ and so $q$ is of the form (REF ).",
"Proposition 5.1 $q_1$ satisfies Assumption REF (REF ).",
"Let $c_0=\\frac{\\kappa }{2}\\min \\lbrace 1,c\\rbrace $ and $c_1=\\kappa $ .",
"Certainly we have $q_1(\\lambda ) \\le c_1(1+\\psi (\\lambda )) \\hspace{20.0pt} \\forall \\lambda \\in ;$ in particular this inequality holds when $|\\lambda |\\ge 1$ .",
"Also, by (REF ), if $|\\lambda |\\ge 1$ , then $\\psi (\\lambda )\\ge c|\\lambda |^r\\ge c$ and so, if $|\\lambda |\\ge 1$ , $q_1(\\lambda ) = \\frac{\\kappa }{2}(\\psi (\\lambda )+\\psi (\\lambda )) \\ge \\frac{\\kappa }{2}(c+\\psi (\\lambda )) \\ge c_0(1+\\psi (\\lambda )).$ That is, Assumption REF (REF ) holds.",
"Proving that Assumption REF (REF ) is satisfied will take more work.",
"However, once we have done so, Assumption REF will follow by taking $\\kappa $ sufficiently large.",
"Note that for the case we are considering, $F_{\\lambda ,\\eta }(\\sigma ) = \\phi _{-\\lambda }(\\sigma )u(\\sigma )v(\\eta ), \\hspace{20.0pt} \\forall \\sigma \\in G,\\;\\lambda ,\\eta \\in ,$ and so, for $\\beta =0,1,\\ldots ,M$ , $(-\\Delta )^{\\beta /2}F_{\\lambda ,\\eta }(\\sigma ) = v(\\eta )(-\\Delta )^{\\beta /2}(\\phi _{-\\lambda }u)(\\sigma ),$ for all $\\lambda ,\\eta \\in $ and $\\sigma \\in G$ .",
"By (REF ), $\\left|(-\\Delta )^{\\beta /2}F_{\\lambda ,\\eta }(\\sigma )\\right| = |v(\\eta )|\\left|(-\\Delta )^{\\beta /2}(\\phi _{-\\lambda }u)(\\sigma )\\right| \\le c_v\\left|(-\\Delta )^{\\beta /2}(\\phi _{-\\lambda }u)(\\sigma )\\right|\\big (1+\\psi (\\eta )\\big ).$ To verify Assumption REF (REF ), we seek for each $\\beta \\in \\lbrace 0,1,\\ldots ,M\\rbrace $ a mapping $\\Phi _\\beta \\in L^1(K|G|K)$ such that $\\left|(-\\Delta )^{\\beta /2}(\\phi _{-\\lambda }u)(\\sigma )\\right| \\le \\Phi _\\beta \\langle \\lambda \\rangle ^M, \\hspace{20.0pt} \\forall \\sigma \\in G,\\;\\lambda \\in .$ Let us first introduce some notation.",
"For each $n\\in \\mathbb {N}$ , a noncommutative version of the multinomial theorem tells us that $(-\\Delta )^n(\\phi _{-\\lambda }u) = (-1)^n\\left(\\sum _{j=1}^dX_j^2\\right)^n(\\phi _{-\\lambda }u) = \\sum _{\\begin{array}{c}\\alpha \\in \\mathbb {N}_0^d,\\\\|\\alpha |\\le r\\end{array}}c_\\alpha X^{\\alpha }(\\phi _{-\\lambda }u)$ for some coefficients $c_\\alpha $ , where $|\\alpha |=\\alpha _1+\\ldots +\\alpha _d$ and $X^{\\alpha }:=X_1^{\\alpha _1}\\ldots X_d^{\\alpha _d}$ .",
"This may also be seen by noting that $(-\\Delta )^n$ belongs to the universal enveloping algebra $\\mathcal {U}(\\mathfrak {p})$ , and applying the Poincaré-Birkoff–Witt theorem (see Knapp [30] Theorem 3.8, pp.",
"217) to write $(-\\Delta )^n$ in the basis $\\lbrace X^\\alpha :\\alpha \\in \\mathbb {N}_0^d\\rbrace $ of $\\mathcal {U}(\\mathfrak {p})$ .",
"Expanding the right-hand side of (REF ) using the fact that each $X_j$ is a derivation will give a large sum of terms of the form $\\kappa _{X,Y}X\\phi _{-\\lambda }Yu,$ where the $\\kappa _{X,Y}$ are constants, and $X,Y\\in {\\bf D}(G)$ are products of powers of $X_1,\\ldots , X_d$ , each with degree at most $2l$ .",
"Let ${U}_n$ be the set of all the $X$ 's and ${V}_n$ the set of all the $Y$ 's, so that $(-\\Delta )^n(\\phi _{-\\lambda }u) = \\sum _{\\begin{array}{c}X\\in {U}_n,\\\\Y\\in {V}_n\\end{array}}\\kappa _{X,Y}X\\phi _{-\\lambda }Yu.$ The following bound will be useful.",
"Lemma 5.2 For all $X\\in {\\bf D}(G)$ , there is a constant $C_X>0$ such that $|X\\phi _\\lambda (\\sigma )| \\le C_X\\langle \\lambda \\rangle ^{\\deg X}\\phi _0(\\sigma ),$ for all $\\lambda \\in $ and $\\sigma \\in G$ .",
"This is a straightforward corollary of Theorem 1.1 (iii) of [20] — see also Lemma 46 of Harish-Chandra [18], pp. 294.",
"Proposition 5.3 The mapping $q_2:G\\times \\rightarrow \\mathbb {R}$ given by (REF ) satisfies Assumption REF (REF ).",
"It is clear by construction that $q_2(\\cdot ,\\lambda )\\in C_c^M(K|G|K)$ for all $\\lambda \\in $ .",
"To verify the rest of Assumption REF (REF ), it will be useful to assume that $M$ is even.",
"Note that this is an acceptable assumption, since if $M$ is odd, we may replace it with $M-1$ .",
"We have chosen $M>\\min \\lbrace 1,d/r\\rbrace +d+1$ , and hence both $M$ and $M-1$ are strictly greater than $\\min \\lbrace 1,d/r\\rbrace +d$ , as required by Corollary REF .",
"Let $\\beta \\in \\lbrace 0,1,\\ldots ,M\\rbrace $ .",
"We seek $\\Phi _\\beta \\in L^1(K|G|K)$ for which (REF ) holds.",
"Let $n=\\lfloor \\beta \\rfloor $ .",
"Assume first that $\\beta $ is even, so that $n=\\beta /2$ .",
"By (REF ) and Lemma REF , $\\left|(-\\Delta )^{\\beta /2}(\\phi _{-\\lambda }u)\\right| \\le \\sum _{\\begin{array}{c}X\\in {U}_n,\\\\Y\\in {V}_n\\end{array}}|\\kappa _{X,Y}||X\\phi _{-\\lambda }||Yu| &\\le \\sum _{\\begin{array}{c}X\\in {U}_n,\\\\Y\\in {V}_n\\end{array}}C_X|\\kappa _{X,Y}||Yu|\\langle \\lambda \\rangle ^{\\deg Y}|\\phi _0| \\\\&\\le \\sum _{\\begin{array}{c}X\\in {U}_n,\\\\Y\\in {V}_n\\end{array}}C_X|\\kappa _{X,Y}||Yu|\\langle \\lambda \\rangle ^{\\deg X},$ since $|\\phi _0|\\le 1$ .",
"Now, $\\deg X\\le 2l=\\beta \\le M$ for all $X\\in {U}_n$ , and therefore, $\\left|(-\\Delta )^{\\beta /2}(\\phi _{-\\lambda }u)\\right| \\le \\kappa _\\beta \\sum _{Y\\in {V}_{\\beta /2}}|Yu|\\langle \\lambda \\rangle ^M$ where $\\kappa _\\beta = \\sup \\left\\lbrace C_X|\\kappa _{X,Y}|:X\\in {U}_{\\beta /2},Y\\in {V}_{\\beta /2}\\right\\rbrace .$ Let $\\Phi _\\beta :=\\kappa _\\beta \\sum _{Y\\in {V}_{\\beta /2}}|Yu|.$ Then $\\Phi _\\beta \\in L^1(K|G|K)$ , since each $Yu$ is a continuous function of compact support.",
"Moreover, $\\Vert \\Phi _\\beta \\Vert _1 \\le \\kappa _\\beta \\sum _{Y\\in {V}_{\\beta /2}}\\Vert Yu\\Vert _1$ In particular, we have verified (REF ) when $\\beta $ is even.",
"For the remainder of the proof of Proposition REF , assume $\\beta $ is an odd integer, so that $\\beta =2n+1$ .",
"Since we're assuming $M$ is even, note that $1\\le \\beta \\le M-1$ .",
"By (REF ), $\\left|(-\\Delta )^{\\beta /2}(\\phi _\\lambda u)\\right| = \\left|\\sqrt{-\\Delta }(-\\Delta )^n(\\phi _\\lambda u)\\right| \\le \\sum _{\\begin{array}{c}X\\in {U}_n,\\\\Y\\in {V}_n\\end{array}}|\\kappa _{X,Y}|\\left|\\sqrt{-\\Delta }\\big (X\\phi _{-\\lambda }Yu\\big )\\right|.$ The families ${U}_n$ and ${V}_n$ each consist of differential operators of degree at most $2n=\\beta -1$ .",
"Now, $-\\sqrt{-\\Delta }$ is the infinitesimal generator of the process obtained by subordinating Brownian motion on $G/K$ by the standard $\\frac{1}{2}$ -stable subordinator on $\\mathbb {R}$ .",
"By standard subordination theory (see [5] Section 5.7, pp.",
"154, or [41] §13 for more general theory) $\\sqrt{-\\Delta }$ may be expressed as a Bochner integral $\\sqrt{-\\Delta } = \\frac{1}{2\\sqrt{\\pi }}\\int _{0+}^\\infty t^{-3/2}(1-T_t)dt,$ where $(T_t,t\\ge 0)$ denotes the heat semigroup generated by $\\Delta $ .",
"Alternatively, equation (REF ) may also be obtained using the spectral theorem, see for example [9] §3.1.",
"Given $X\\in {U}_n$ and $Y\\in {V}_n$ and $\\sigma \\in G$ , $\\begin{aligned}\\left|\\sqrt{-\\Delta }(X\\phi _{-\\lambda }Yu)(\\sigma )\\right| &= \\frac{1}{2\\sqrt{\\pi }}\\left|\\int _{0+}^\\infty t^{-3/2}(1-T_t)\\big (X\\phi _{-\\lambda }Yu\\big )dt\\right| \\\\&\\le \\frac{1}{2\\sqrt{\\pi }}\\Bigg [\\left|\\int _{0+}^1 t^{-3/2}(1-T_t)\\big (X\\phi _{-\\lambda }Yu\\big )(\\sigma )dt\\right| \\\\&\\hspace{90.0pt}+ \\left|\\int _1^\\infty t^{-3/2}(1-T_t)\\big (X\\phi _{-\\lambda }Yu\\big )(\\sigma )dt\\right|\\Bigg ].\\end{aligned}$ Let $(h_t,t\\ge 0)$ denote the heat kernel associated with $(T_t,t\\ge 0)$ .",
"For the $\\int _1^\\infty $ term of (REF ), note that $\\int _1^\\infty t^{-3/2}dt=2$ , and so $&\\left|\\int _1^\\infty t^{-3/2}(1-T_t)\\big (X\\phi _{-\\lambda }Yu\\big )(\\sigma )dt\\right| \\\\&\\hspace{100.0pt}= \\left|\\int _1^\\infty t^{-3/2}X\\phi _{-\\lambda }(\\sigma )Yu(\\sigma )dt - \\int _1^\\infty t^{-3/2}T_t\\big (X\\phi _{-\\lambda }Yu\\big )(\\sigma )dt\\right| \\\\&\\hspace{100.0pt}\\le 2|X\\phi _{-\\lambda }(\\sigma )||Yu(\\sigma )| + \\left|\\int _1^\\infty t^{-3/2}\\int _GX\\phi _{-\\lambda }(\\sigma \\tau )Yu(\\sigma \\tau )h_t(\\tau )d\\tau dt\\right|.$ By Lemma REF and the fact that $\\deg X\\le \\beta -1$ , $|X\\phi _{-\\lambda }| \\le C_X\\langle \\lambda \\rangle ^{\\deg X} \\le C\\langle \\lambda \\rangle ^{\\beta -1},$ for all $\\lambda \\in $ , where $C_X$ is as in (REF ), and $C=\\max \\lbrace C_X:X\\in {U}_n\\rbrace $ .",
"Thus $&\\left|\\int _1^\\infty t^{-3/2}(1-T_t)\\big (X\\phi _{-\\lambda }Yu\\big )(\\sigma )dt\\right| \\\\&\\hspace{80.0pt}\\le 2|X\\phi _{-\\lambda }(\\sigma )||Yu(\\sigma )| + \\int _1^\\infty t^{-3/2}\\int _G|X\\phi _{-\\lambda }(\\sigma \\tau )||Yu(\\sigma \\tau )|h_t(\\tau )d\\tau dt \\\\&\\hspace{80.0pt}\\le C\\left(2|Yu(\\sigma )| + \\int _1^\\infty t^{-3/2}\\int _G|Yu(\\sigma \\tau )|h_t(\\tau )d\\tau dt\\right)\\langle \\lambda \\rangle ^{\\beta -1} \\\\&\\hspace{80.0pt}= \\Phi _{\\beta ,Y}^{(1)}(\\sigma )\\langle \\lambda \\rangle ^{\\beta -1},$ where $\\Phi _{\\beta ,Y}^{(1)} := C\\left(2|Yu| + \\int _1^\\infty t^{-3/2}T_t\\big (|Yu|\\big )dt\\right).$ Since $\\beta -1\\le M$ and $\\langle \\lambda \\rangle \\ge 1$ for all $\\lambda \\in $ , it follows that $\\left|\\int _1^\\infty t^{-3/2}(1-T_t)\\big (X\\phi _{-\\lambda }Yu\\big )dt\\right| \\le \\Phi _{\\beta ,Y}^{(1)}\\langle \\lambda \\rangle ^M,$ for all $\\lambda \\in $ .",
"We claim that $\\Phi _{\\beta ,Y}^{(1)}\\in L^1(K|G|K)$ .",
"Clearly $|Yu|\\in L^1(K|G|K)$ , since it is a continuous function of compact support.",
"Each of the operators $T_t$ is a positivity preserving contraction of $L^1(K|G|K)$ , and so $\\int _1^\\infty t^{-3/2}\\int _GT_t\\big (|Yu|\\big )(\\sigma )d\\sigma dt = \\int _1^\\infty t^{-3/2}\\left\\Vert T_t\\big (|Yu|\\big )\\right\\Vert _1 dt \\le \\int _1^\\infty t^{-3/2}\\Vert Yu\\Vert _1dt = 2\\Vert Yu\\Vert _1.$ By Fubini's theorem, $\\int _1^\\infty t^{-3/2}T_t\\big (|Yu|\\big )dt\\in L^1(K|G|K)$ , with $\\left\\Vert \\int _1^\\infty t^{-3/2}T_t\\big (|Yu|\\big )dt\\right\\Vert _{L_1(K|G|K)} \\le 2\\Vert Yu\\Vert _1.$ It follows by (REF ) that $\\Phi _{\\beta ,Y}^{(1)}\\in L^1(K|G|K)$ , and that $\\Vert \\Phi _{\\beta ,Y}^{(1)}\\Vert _1 \\le 4C\\Vert Yu\\Vert _1.$ For the $\\int _{0+}^1$ term of (REF ), observe that by Lemma 6.1.12 of Davies [12], pp.",
"169, as well as the Fubini theorem, $\\int _{0+}^1t^{-3/2}(1-T_t)\\big (X\\phi _{-\\lambda }Yu\\big )dt &= -\\int _{0+}^1t^{-3/2}\\int _0^t T_s\\Delta \\big (X\\phi _{-\\lambda }Yu\\big )dsdt \\\\&= -\\int _0^1\\int _s^1t^{-3/2}T_s\\Delta \\big (X\\phi _{-\\lambda }Yu\\big )dtds \\\\&= -\\int _0^12(s^{-1/2}-1)T_s\\Delta \\big (X\\phi _{-\\lambda }Yu\\big )ds.$ Hence, using the product formula for $\\Delta $ , $\\begin{aligned}\\int _{0+}^1t^{-3/2}(1-T_t)\\big (X\\phi _{-\\lambda }Yu\\big )dt &= -2\\int _0^1(s^{-1/2}-1)\\Big \\lbrace T_s\\big (X\\phi _{-\\lambda }\\Delta Yu\\big ) \\\\&\\hspace{6.0pt}+ 2\\sum _{j=1}^dT_s\\big (X_jX\\phi _{-\\lambda }X_jYu\\big ) + T_s\\big (\\Delta X\\phi _{-\\lambda }Yu\\big )\\Big \\rbrace ds.\\end{aligned}$ Let $C$ and $C_X$ be as in (REF ).",
"Then for all $\\sigma \\in G$ , $\\left|T_s\\big (X\\phi _{-\\lambda }\\Delta Yu\\big )(\\sigma )\\right| &= \\left|\\int _GX\\phi _{-\\lambda }(\\sigma \\tau )\\Delta Yu(\\sigma \\tau )h_s(\\tau )d\\tau \\right| \\\\&\\le \\int _G|X\\phi _{-\\lambda }(\\sigma \\tau )||\\Delta Yu(\\sigma \\tau )|h_s(\\tau )d\\tau \\\\&\\le C_X\\langle \\lambda \\rangle ^{\\deg X}\\int _G|\\Delta Yu(\\sigma \\tau )|h_s(\\tau )d\\tau \\le C\\langle \\lambda \\rangle ^{\\beta -1}T_s|\\Delta Yu|(\\sigma ).$ In exactly the same way, for $j=1,\\ldots ,d$ , $\\left|T_s\\big (X_jX\\phi _{-\\lambda }X_jYu\\big )\\right| \\le C_X^{(j)}\\langle \\lambda \\rangle ^{\\deg X+1}T_s|X_jYu| \\le C^{\\prime }\\langle \\lambda \\rangle ^{\\beta }T_s|X_jYu|,$ and also $\\left|T_s\\big (\\Delta X\\phi _{-\\lambda }Yu\\big )\\right| \\le C_X^{(0)}\\langle \\lambda \\rangle ^{\\deg X+2}T_s|Yu| \\le C^{\\prime }\\langle \\lambda \\rangle ^{\\beta +1}T_s|Yu|,$ where the constants $C_X^{(j)},C^{(j)}$ are chosen so that for all $\\lambda \\in $ and $j=1,\\ldots ,d$ , $|X\\phi _{-\\lambda }| \\le C_X^{(0)}\\langle \\lambda \\rangle ^{\\deg X +2}, \\hspace{40.0pt} |X_jX\\phi _{-\\lambda }| \\le C_X^{(j)}\\langle \\lambda \\rangle ^{\\deg X +1},$ and $C^{\\prime }:=\\max \\lbrace C_X^{(j)}:X\\in {U}_n,j=0,1,\\ldots ,d\\rbrace $ .",
"Such constants exist by Lemma REF .",
"Now, $\\langle \\lambda \\rangle ^{\\beta -1} \\le \\langle \\lambda \\rangle ^\\beta \\le \\langle \\lambda \\rangle ^{\\beta +1}$ for all $\\lambda \\in $ , and hence by (REF ), $&\\left|\\int _{0+}^1t^{-3/2}(1-T_t)\\big (X\\phi _{-\\lambda }Yu\\big )dt\\right| \\\\&\\hspace{20.0pt}\\le 2\\int _0^1(s^{-1/2}-1)\\Big \\lbrace \\left|T_s\\big (X\\phi _{-\\lambda }\\Delta Yu\\big )\\right| + 2\\sum _{j=1}^d\\left|T_s\\big (X_jX\\phi _{-\\lambda }X_jYu\\big )\\right| \\\\&\\hspace{240.0pt}+ \\left|T_s\\big (\\Delta X\\phi _{-\\lambda }Yu\\big )\\right|\\Big \\rbrace ds \\\\&\\hspace{20.0pt}\\le 2C^{\\prime }\\langle \\lambda \\rangle ^{\\beta +1}\\int _0^1(s^{-1/2}-1)T_s\\left(|\\Delta Yu|+2\\sum _{j=1}^d|X_jYu|+|Yu|\\right)ds.$ Since $\\beta \\le M-1$ , it follows that for all $X\\in {U}_n$ and $Y\\in {V}_n$ , $\\left|\\int _{0+}^1t^{-3/2}(1-T_t)\\big (X\\phi _{-\\lambda }Yu\\big )dt\\right| \\le \\Phi _{\\beta ,Y}^{(2)}\\langle \\lambda \\rangle ^M,$ where $\\Phi _{\\beta ,Y}^{(2)} = C^{\\prime }\\int _0^1(s^{-1/2}-1)T_s\\left(|\\Delta Yu|+2\\sum _{j=1}^d|X_jYu|+|Yu|\\right)ds.$ Observe that $\\Phi _{\\beta ,Y}^{(2)}\\in L^1(K|G|K)$ for all $Y\\in {V}_n$ .",
"Indeed, $u\\in C_c^M(K|G|K)$ , and $\\deg Y\\le \\beta -1\\le M-2$ , hence $|\\Delta Yu|$ , $\\sum _{j=1}^d|X_jYu|$ and $|Yu|$ are all continuous functions of compact support.",
"Thus $T_s\\big (|\\Delta Yu|+2\\sum _{j=1}^d|X_jYu|+|Yu|\\big )\\in L^1(K|G|K)$ , and, since $T_s$ is an $L^1(K|G|K)$ -contraction, $\\left\\Vert T_s\\left(|\\Delta Yu|+2\\sum _{j=1}^d|X_jYu|+|Yu|\\right)\\right\\Vert _1 \\le \\Vert \\Delta Yu\\Vert _1 +2\\sum _{j=1}^d\\Vert X_jYu\\Vert _1 +\\Vert Yu\\Vert _1.$ Noting that $\\int _0^1(s^{-1/2}-1)ds = 1$ , it follows by Fubini's theorem that $\\Phi _{\\beta ,Y}^{(2)}\\in L^1(K|G|K)$ , with $\\left\\Vert \\Phi _{\\beta ,Y}^{(2)}\\right\\Vert _1 \\le C_X^{\\prime }\\left(\\Vert \\Delta Yu\\Vert _1 + 2\\sum _{j=1}^d\\Vert X_jYu\\Vert _1 +\\Vert Yu\\Vert _1\\right).$ Substituting (REF ) and (REF ) into (REF ), we obtain the pointwise estimate $\\left|\\sqrt{-\\Delta }(X\\phi _{-\\lambda }Yu)\\right| \\le \\frac{1}{2\\sqrt{\\pi }}\\left(\\Phi _{\\beta ,Y}^{(1)}+\\Phi _{\\beta ,Y}^{(2)}\\right)\\langle \\lambda \\rangle ^M,$ for all $X\\in {U}_n$ , $Y\\in {V}_n$ and $\\lambda \\in $ , where the $\\Phi _{\\beta ,Y}^{(j)}$ ($j=1,2$ ) are given by (REF ) and (REF ).",
"Hence by (REF ), for all $\\lambda \\in $ , $\\left|(-\\Delta )^{\\beta /2}(\\phi _\\lambda u)\\right| \\le \\sum _{\\begin{array}{c}X\\in {U}_n,\\\\Y\\in {V}_n\\end{array}}|\\kappa _{X,Y}|\\left|\\sqrt{-\\Delta }\\big (X\\phi _{-\\lambda }Yu\\big )\\right| \\le \\Phi _\\beta \\langle \\lambda \\rangle ^M,$ where $\\Phi _\\beta :=\\frac{1}{2\\sqrt{\\pi }}\\sum _{\\begin{array}{c}X\\in {U}_n,\\\\Y\\in {V}_n\\end{array}}|\\kappa _{X,Y}|\\left(\\Phi _{\\beta ,Y}^{(1)}+\\Phi _{\\beta ,Y}^{(2)}\\right),$ and $\\beta $ is still assumed to be odd.",
"It was shown previously that $\\Phi _{\\beta ,Y}^{(1)},\\Phi _{\\beta ,Y}^{(2)}\\in L^1(K|G|K)$ , for all $Y\\in {V}_n$ , and hence $\\Phi _\\beta \\in L^1(K|G|K)$ .",
"Moreover, by (REF ) and (REF ), $\\begin{aligned}\\left\\Vert \\Phi _\\beta \\right\\Vert _1 &\\le \\kappa _\\beta ^{\\prime }\\sum _{Y\\in {V}_n}\\Bigg (\\Vert \\Delta Yu\\Vert _1+\\sum _{j=1}^d\\Vert X_jYu\\Vert _1 +\\Vert Yu\\Vert _1\\Bigg ),\\end{aligned}$ for some positive constant $\\kappa _\\beta ^{\\prime }$ .",
"In particular, we have verified (REF ) when $\\beta $ is odd.",
"Corollary 5.4 Let $q:G\\times \\rightarrow \\mathbb {R}$ be of the form (REF ).",
"Then for $\\kappa $ sufficiently large, the conditions of Corollary REF are satisfied.",
"In particular, $-q(\\sigma ,D)$ extends to the infinitesimal generator of a strongly continuous sub-Feller semigroup on $C_0(K|G|K)$ ."
],
[
"Appendix: The Friedrich Mollifier $J_\\epsilon $",
"Recall the Friedrich mollifier on $\\operatorname{\\mathfrak {a}}$ from Section , defined for each $0<\\epsilon \\le 1$ and $H\\in \\operatorname{\\mathfrak {a}}$ by $l_\\epsilon (H) := \\epsilon ^{-m}l(H/\\epsilon ), \\hspace{20.0pt} \\text{ where } \\hspace{20.0pt} l(H) := C_0e^{\\frac{1}{|H|^2-1}}\\operatorname{\\bf 1}_{B_1(0)}(H),$ where $C_0>0$ is a normalising constant, and the associated mappings $j,j_\\epsilon \\in \\mathcal {S}(K|G|K)$ are again given by $\\hat{j_\\epsilon } = {F}(l_\\epsilon ), \\hspace{20.0pt} \\forall 0<\\epsilon \\le 1.$ For $0<\\epsilon \\le 1$ , the operators $J_\\epsilon $ were defined $J_\\epsilon u = j_\\epsilon \\ast u \\hspace{20.0pt} \\forall f\\in L^2(K|G|K).$ This appendix will be devoted to proving Proposition REF , which we re-state below.",
"Proposition 6.1 $\\hat{j_\\epsilon }(\\lambda ) = \\hat{j}(\\epsilon \\lambda )$ for all $0<\\epsilon \\le 1$ and $\\lambda \\in $ .",
"For all $0<\\epsilon \\le 1$ , $J_\\epsilon $ is a self-adjoint contraction of $L^2(K|G|K)$ .",
"$J_\\epsilon u\\in H^{\\psi ,s}$ for all $s\\ge 0$ , $u\\in L^2(K|G|K)$ and $0<\\epsilon \\le 1$ , and if $u\\in H^{\\psi ,s}$ , then $\\Vert J_\\epsilon u\\Vert _{\\psi ,s} \\le \\Vert u\\Vert _{\\psi ,s}.$ $\\Vert J_\\epsilon u - u\\Vert _{\\psi ,s}\\rightarrow 0$ as $\\epsilon \\rightarrow 0$ .",
"Let $0<\\epsilon \\le 1$ and $\\lambda \\in $ .",
"Using a change of variable $H\\mapsto \\epsilon ^{-1}H$ , $\\hat{j_\\epsilon }(\\lambda ) = {F}(l_\\epsilon )(\\lambda ) &= \\int _{\\operatorname{\\mathfrak {a}}}e^{i\\lambda (H)}\\epsilon ^{-m}l(\\epsilon ^{-1}H)dH \\\\&= \\int _{\\operatorname{\\mathfrak {a}}}e^{i\\epsilon \\lambda (H)}l(H)dH = {F}(l)(\\epsilon \\lambda ) = \\hat{j}(\\epsilon \\lambda ).$ The map $l$ is symmetric under $H\\mapsto -H$ , and hence ${F}(l)=\\hat{j}$ is real-valued.",
"Therefore, given $u,v\\in L^2(K|G|K)$ and $0<\\epsilon \\le 1$ , $\\langle J_\\epsilon u,v\\rangle = \\int _{}\\hat{j}(\\epsilon \\lambda )\\hat{u}(\\lambda )\\overline{\\hat{v}(\\lambda )}\\omega (d\\lambda ) = \\int _{}\\hat{u}(\\lambda )\\overline{\\hat{j}(\\epsilon \\lambda )\\hat{v}(\\lambda )}\\omega (d\\lambda ) = \\langle u, J_\\epsilon v\\rangle .$ To see that $J_\\epsilon $ is a contraction, note that $|\\hat{j_\\epsilon }(\\lambda )|=|\\hat{j}(\\epsilon \\lambda )|\\le \\hat{j}(0)= 1$ for all $\\lambda \\in $ , and so by Plancherel's identity $\\Vert J_\\epsilon u\\Vert = \\Vert \\hat{j}_\\epsilon \\hat{u}\\Vert _{L^2(,\\omega )} \\le \\Vert \\hat{u}\\Vert _{L^2(K|G|K)} = \\Vert u \\Vert ,$ for all $u\\in L^2(K|G|K)$ and all $0<\\epsilon \\le 1$ .",
"Let $s\\ge 0$ , $0<\\epsilon \\le 1$ .",
"By Theorem REF , $\\hat{j}\\in \\mathcal {S}()^W$ , and hence there is $\\kappa >0$ such that $\\langle \\lambda \\rangle ^s\\left|\\hat{j}(\\epsilon \\lambda )\\right| \\le \\kappa , \\hspace{20.0pt} \\forall \\lambda \\in .$ Then, using Proposition REF (REF ), $\\Psi (\\lambda )^s\\left|\\hat{j}(\\epsilon \\lambda )\\right| \\le c_\\psi ^{s/2}\\langle \\lambda \\rangle ^s\\left|\\hat{j}(\\epsilon \\lambda )\\right|\\le c_\\psi ^{s/2}\\kappa ,$ for all $\\lambda \\in $ .",
"Let $u\\in L^2(K|G|K)$ .",
"By Plancherel's identity, $\\int _{}\\Psi (\\lambda )^{2s}\\left|\\hat{j}(\\epsilon \\lambda )\\right|^2|\\hat{u}(\\lambda )|^2\\omega (d\\lambda ) \\le c_\\psi ^s\\kappa ^2\\Vert u \\Vert ^2 < \\infty .$ By Proposition REF (REF ), $(J_\\epsilon u)^\\wedge (\\lambda ) = \\hat{j}(\\epsilon \\lambda )\\hat{u}(\\lambda )$ , for all $\\lambda \\in $ , and hence $\\int _{}\\Psi (\\lambda )^{2s}|(J_\\epsilon u)^\\wedge (\\lambda )|^2\\omega (d\\lambda )<\\infty .$ That is, $J_\\epsilon u\\in H^{\\psi ,s}$ .",
"Next, suppose $u\\in H^{\\psi ,s}$ .",
"Then, since $|\\hat{j_\\epsilon }|\\le 1$ , $\\Vert J_\\epsilon u\\Vert _{\\psi ,s} = \\Vert \\Psi ^s\\hat{j_\\epsilon }\\hat{u}\\Vert _{L^2(,\\omega )} \\le \\Vert \\Psi ^s\\hat{u}\\Vert _{L^2(,\\omega )} = \\Vert u\\Vert _{\\psi ,s},$ as desired.",
"By Theorem 1 on page 250 of Evans [15], for all $v\\in \\mathcal {S}(\\operatorname{\\mathfrak {a}})^W$ , $l_\\epsilon \\ast v\\rightarrow v$ as $\\epsilon \\rightarrow 0$ , in the classical Sobolev space $W^s()$ , for all $s\\ge 0$ .",
"Therefore, for all $s\\ge 0$ and $v\\in \\mathcal {S}(\\operatorname{\\mathfrak {a}})^W$ , $\\lim _{\\epsilon \\rightarrow 0}\\int _{}(1+|\\lambda |^2)^s|{F}(l_\\epsilon \\ast v-v)(\\lambda )|^2d\\lambda = 0,$ Let $u\\in C_c^\\infty (K|G|K)$ .",
"Then by Theorem REF , ${F}^{-1}(\\hat{u})\\in \\mathcal {S}(\\operatorname{\\mathfrak {a}})^W$ .",
"Observe that if $v={F}^{-1}(\\hat{u})$ , then ${F}(l_\\epsilon \\ast v - v) = (\\hat{j}_\\epsilon -1)\\hat{u} = (J_\\epsilon u - u)^\\wedge ,$ and so by (REF ), $\\lim _{\\epsilon \\rightarrow 0}\\int _{}(1+|\\lambda |^2)^s|(J_\\epsilon u - u)^\\wedge (\\lambda )|^2d\\lambda = 0,$ for all $s\\ge 0$ .",
"By (REF ), $\\int _{}(1+|\\lambda |^2)^s|(J_\\epsilon u - u)^\\wedge (\\lambda )|^2\\omega (d\\lambda )&\\le \\int _{}(1+|\\lambda |^2)^s|(J_\\epsilon u - u)^\\wedge (\\lambda )|^2(C_1+C_2|\\lambda |^p)^2d\\lambda \\\\&\\le \\kappa \\int _{}(1+|\\lambda |^2)^{s+p}|(J_\\epsilon u - u)^\\wedge (\\lambda )|^2d\\lambda ,$ for some constant $\\kappa >0$ , and where $p=\\frac{\\dim N}{2}$ .",
"Hence $\\lim _{\\epsilon \\rightarrow 0}\\int _{}(1+|\\lambda |^2)^s|(J_\\epsilon u - u)^\\wedge (\\lambda )|^2\\omega (d\\lambda ) = 0$ By Proposition REF (REF ), $\\Vert J_\\epsilon u-u\\Vert _{\\psi ,s}^2 &= \\int _{}(1+\\psi (\\lambda ))^s|(J_\\epsilon u - u)^\\wedge (\\lambda )|^2\\omega (d\\lambda ) \\\\&\\le c_\\psi \\int _{}(1+|\\lambda |^2)^s|(J_\\epsilon u - u)^\\wedge (\\lambda )|^2\\omega (d\\lambda ) \\rightarrow 0$ as $\\epsilon \\rightarrow 0$ .",
"Since $C_c^\\infty (K|G|K)$ is dense in $H^{\\psi ,s}$ , Proposition REF (REF ) follows.",
"Acknowledgement Many thanks to David Applebaum for his advice and support with writing this paper.",
"Thanks also to the University of Sheffield's School of Mathematics and Statistics, and to the EPSRC for providing PhD funding while this research was carried out."
]
] | 2107.01817 |
[
[
"End-to-end Neural Coreference Resolution Revisited: A Simple yet\n Effective Baseline"
],
[
"Abstract Since the first end-to-end neural coreference resolution model was introduced, many extensions to the model have been proposed, ranging from using higher-order inference to directly optimizing evaluation metrics using reinforcement learning.",
"Despite improving the coreference resolution performance by a large margin, these extensions add substantial extra complexity to the original model.",
"Motivated by this observation and the recent advances in pre-trained Transformer language models, we propose a simple yet effective baseline for coreference resolution.",
"Even though our model is a simplified version of the original neural coreference resolution model, it achieves impressive performance, outperforming all recent extended works on the public English OntoNotes benchmark.",
"Our work provides evidence for the necessity of carefully justifying the complexity of existing or newly proposed models, as introducing a conceptual or practical simplification to an existing model can still yield competitive results."
],
[
"Introduction",
"Coreference resolution is the task of clustering mentions in text that refer to the same real-world entities [1] (Figure REF ).",
"As a fundamental task of natural language processing, coreference resolution can be an essential component for many downstream applications [2], [3].",
"Many traditional coreference resolution systems are pipelined systems, each consists of two separate components: (1) a mention detector for identifying entity mentions from text (2) a coreference resolver for clustering the extracted mentions [4], [5], [6], [7], [8].",
"These models typically rely heavily on syntatic parsers and use highly engineered mention proposal algorithms.",
"Figure: An example of coreference resolution.",
"There are two coreference chains in this example.Figure: An overview of coreference resolution research in the last decade.",
"Pipelined systems were heavily used before the introduction of e2e-coref.",
"Since 2017, various extensions to the model have been proposed.In 2017, the first end-to-end coreference resolution model named e2e-coref was proposed [9].",
"It outperforms previous pipelined systems without using any syntactic parser or complicated hand-engineered features.",
"Since then, many extensions to the e2e-coref model have been introduced, ranging from using higher-order inference to directly optimizing evaluation metrics using reinforcement learning [10], [11], [12], [13], [14], [15], [16], [17] (Figure REF ).",
"Despite improving the coreference resolution performance by a large margin, these extensions add a lot of extra complexity to the original model.",
"Motivated by this observation and the recent advances in pre-trained Transformer language models, we propose a simple yet effective baseline for coreference resolution.",
"We introduce simplifications to the original e2e-coref model, creating a conceptually simpler model for coreference resolution.",
"Despite its simplicity, our model achieves promising performance, outperforming all aforementioned methods on the public English OntoNotes benchmark.",
"Our work provides evidence for the necessity of carefully justifying the complexity of existing or newly proposed models, as introducing a conceptual or practical simplification to an existing model can still yield competitive results.",
"The findings of our work agree with the results of several recent work [18], [19].",
"For example, in [19], the authors also introduced a minimalist approach that performs on par with more complicated models.",
"In Section , we first describe our baseline model for coreference resolution.",
"After that, we describe the conducted experiments and their results in Section .",
"Finally, we conclude this work and discuss future work in Section"
],
[
"Method",
"At a high level, our coreference resolution model is similar to the e2e-coref model (Figure REF ).",
"Given a sequence of tokens from an input document, the model first forms a contextualized representation for each token using a Transformer-based encoder.",
"After that, all the spans (up to a certain length) in the document are enumerated.",
"The model then assigns a score to each candidate span indicating whether the span is an entity mention.",
"A portion of top-scoring spans is extracted and fed to the next stage where the model predicts distributions over possible antecedents for each extracted span.",
"The final coreference clusters can be naturally constructed from the antecedent predictions.",
"In the following subsections, we go into more specific details.",
"Figure: An example illustrating the strategy of concatenating the speaker’s name with the corresponding utterance (assuming the model utilizes WordPiece for tokenization)."
],
[
"Notations and Preliminaries",
"Given an input document $D = (t_1, t_2, ..., t_n)$ consisting of $n$ tokens, the total number of possible text spans is $N = n(n+1)/2$ .",
"For each span $i$ , we denote the start and end indices of the span by $\\texttt {START}(i)$ and $\\texttt {END}(i)$ respectively.",
"We also assume an ordering of the spans based on $\\texttt {START}(i)$ ; spans with the same start index are ordered by $\\texttt {END}(i)$ .",
"Furthermore, we only consider spans that are entirely within a sentence and limit spans to a max length of $L$ .",
"Since the speaker information is known to contain useful information for coreference resolution, it has been extensively used in previous works [5], [9], [11], [15], [17].",
"For example, the original e2e-coref model converts speaker information into binary features indicating whether two candidate mentions are from the same speaker.",
"In this work, we employ a more intuitive strategy that directly concatenates the speaker’s name with the corresponding utterance [20].",
"This straightforward strategy is simple to implement and has been shown to be more effective than the feature-based method [20].",
"Figure REF illustrates the concatenation strategy."
],
[
"Encoder Layer",
"Given the input document $D = (t_1, t_2, ..., t_n)$ , the model simply forms a contextualized representation for each input token, using a Transformer-based encoder such as BERT [21] or SpanBERT [17].",
"These pretrained language models typically can only run on sequences with at most 512 tokens.",
"Therefore, to encode a long document (i.e, $n > 512$ ), we split the document into overlapping segments by creating a $n$ -sized segment after every $n/2$ tokens.",
"These segments are then passed on to the Transformer-based encoder independently.",
"The final token representations are derived by taking the token representations with maximum context.",
"Let $\\textbf {X} = (\\textbf {x}_1, \\textbf {x}_2, ..., \\textbf {x}_n)$ be the output of the Transformer encoder.",
"Note that the e2e-coref model uses the GloVe and Turian embeddings [22], [23] and character embeddings produced by 1-dimensional convolution neural networks.",
"From an implementation point of view, it is easier to use a Transformer encoder than combining these traditional embeddings.",
"For example, the Transformers library https://github.com/huggingface/transformers allows users to experiment with various state-of-the-art Transformer-based models by simply writing few lines of code.",
"Now, for each span $i$ , its span representation $\\textbf {g}_i$ is defined as: $\\textbf {g}_i = \\big [\\textbf {x}_{\\texttt {START}(i)}, \\textbf {x}_{\\texttt {END}(i)}, \\hat{\\textbf {x}}_{i} \\big ]$ where $\\textbf {x}_{\\texttt {START}(i)}$ and $\\textbf {x}_{\\texttt {END}(i)}$ are the boundary representations, consisting of the first and the last token representations of the span $i$ .",
"And $\\hat{\\textbf {x}}_{i}$ is computed using an attention mechanism [24] as follows: $\\begin{split}\\alpha _t & = \\text{FFNN}_{\\alpha }(\\textbf {x}_t) \\\\\\beta _{i,t} & = \\frac{\\exp {(\\alpha _t)}}{\\sum \\limits _{j=\\texttt {START}(i)}^{\\texttt {END}(i)} \\exp {(\\alpha _j)}} \\\\\\hat{\\textbf {x}}_{i} &= \\sum \\limits _{j=\\texttt {START}(i)}^{\\texttt {END}(i)} \\beta _{i,j} \\,\\textbf {x}_j\\end{split}$ where $\\text{FFNN}_{\\alpha }$ is a multi-layer feedforward neural network that maps each token-level representation $\\textbf {x}_t$ into an unnormalized attention score.",
"$\\hat{\\textbf {x}}_{i}$ is a weighted sum of token vectors in the span $i$ .",
"Our span representation generation process closely follows that of e2e-coref.",
"However, a simplification we make is that we do not include any additional features such as the size of span $i$ in its representation $\\textbf {g}_i$ ."
],
[
"Mentions Extractor Layer",
"In this layer, we first enumerate all the spans (up to a certain length $L$ ) in the document.",
"For each span $i$ , we simply use a feedforward neural network $\\text{FFNN}_\\text{m}$ to compute its mention score: $s_m(i) = \\text{FFNN}_\\text{m}(\\textbf {g}_i)$ After this step, we only keep up to $\\lambda n$ spans with the highest mention scores.",
"In previous works, to maintain a high recall of gold mentions, $\\lambda $ is typically set to be $0.4$ [9], [11].",
"These works do not directly train the mention extractor.",
"The mention extractor and the mention linker are jointly trained to only maximize the marginal likelihood of gold antecedent spans.",
"In coreference resolution datasets such as the OntoNotes benchmark [25], singleton mentions are not explicitly labeled, because the annotations contain only mentions that belong to a coreference chain.",
"However, these annotations of non-singleton mentions can still provide useful signals for training an efficient mention extractor [10].",
"Thus, we also propose to pre-train our mention extractor using these annotations.",
"In Section , we will empirically demonstrate that this pre-training step greatly improves the performance of our mention extractor layer.",
"As a result, we only need set the parameter $\\lambda $ to be 0.25 in order to maintain a high recall of gold mentions.",
"To this end, the pretraining loss is calculated as follows: $\\begin{split}\\mathcal {L}_{\\text{detect}}(i) &= y_i \\log {\\hat{y}_i} + (1-y_i) \\log {(1-\\hat{y}_i)} \\\\\\mathcal {L}_{\\text{detect}} &= - \\sum _{i\\,\\in \\,\\text{S}} \\mathcal {L}_{\\text{detect}}(i)\\end{split}$ where $\\hat{y}_i = \\text{sigmoid}(s_m(i))$ , and $y_i = 1$ if and only if the span $i$ is a mention in one of the coreference chains.",
"$S$ is the set of the top scoring spans (and so $|S| \\le \\lambda n$ )."
],
[
"Mentions Linker Layer",
"For each span $i$ extracted by the mention extractor, the mention linker needs to assign an antecedent $a_i$ from all preceding spans or a dummy antecedent $\\epsilon $ : $a_i \\in Y(i) = \\lbrace \\epsilon , 1, \\dots , i-1\\rbrace $ (the ordering of spans was discussed in Subsection REF ).",
"The dummy antecedent $\\epsilon $ represents two possible cases.",
"One case is the span itself is not an entity mention.",
"The other case is the span is an entity mention but it is not coreferent with any previous span extracted by the mention extractor.",
"The coreference score $s(i,j)$ of two spans $i$ and $j$ is computed as follows: $\\begin{split}s_a(i, j) &= \\text{FFNN}_\\text{a}\\big (\\big [\\textbf {g}_i, \\textbf {g}_j, \\textbf {g}_i \\circ \\textbf {g}_j \\big ]\\big ) \\\\s(i, j) &= s_m(i) + s_m(j) + s_a(i,j)\\end{split}$ where $\\text{FFNN}_\\text{a}$ is a feedforward network.",
"$s_m(i)$ and $s_m(j)$ are calculated using Equation REF .",
"The score $s(i,j)$ is affected by three factors: (1) $s_m(i)$ , whether span $i$ is a mention, (2) $s_m(j)$ , whether span $j$ is a mention, and (3) $s_a(i,j)$ whether $j$ is an antecedent of $i$ .",
"In the special case of the dummy antecedent, $s(i, \\epsilon )$ is fixed to 0.",
"In the e2e-coref model, when computing $s_a(i,j)$ , a vector encoding additional features such as genre information and the distance between the two spans is also used.",
"We do not use such feature vector when computing $s_a(i,j)$ to simplify the implementation.",
"We want to maximize the marginal log-likelihood of all antecedents in the correct coreference chain for each mention: $\\log {\\prod _{i \\in S}{\\sum _{\\hat{y}\\in Y(i) \\cap \\text{GOLD}(i)} P(\\hat{y})}}$ where $S$ is the set of the top scoring spans extracted by the mention extractor (i.e., the set of unpruned spans).",
"$\\text{GOLD}(i)$ is the set of spans in the gold cluster containing span $i$ .",
"If span $i$ does not belong to any coreference chain or all gold antecedents have been pruned, then $\\text{GOLD}(i) = \\lbrace \\epsilon \\rbrace $ .",
"Table: Performance on the OntoNotes coreference resolution benchmark.Table: Proportion of gold mentions covered in the development data by different mention extractors.To summarize, we first pre-train the mention extractor to minimize the loss function defined in Equation REF .",
"After that, we jointly train the mention extractor and the mention linker to optimize the objective function defined in Equation REF in an end-to-end manner."
],
[
"Experiments and Results",
"Dataset and Experiments Setup To evaluate the effectiveness of the proposed approach, we use the CoNLL-2012 Shared Task English data [25] which is based on the OntoNotes corpus.",
"This dataset has 2802/343/348 documents for the train/dev/test split.",
"Similar to previous works, we report precision, recall, and F1 of the MUC, $\\text{B}^3$ , and $\\text{CEAF}_{\\phi _4}$ metrics, and also average the F1 score of all three metrics.",
"We used SpanBERT (spanbert-large-cased) [17] as the encoder.",
"Two different learning rates are used, one for the lower pretrained SpanBERT encoder (5e-05) and one for the upper layers (1e-4).",
"We also use learning rate decay.",
"The number of training epochs is set to be 100.",
"The batch size is set to be 32.",
"We did hyper-parameter tuning using the provided dev set.",
"To train our model, we use two 16GB V100 GPUs and use techniques such as gradient checkpointing and gradient accumulation to avoid running out of GPUs' memory.",
"Comparison with Previous Methods Table REF compares our model with several state-of-the-art coreference resolution systems.",
"Overall, our model outperforms the original e2e-coref model and also all recent extended works.",
"For example, compared to the variant [c2f-coref + SpanBERT-large], our model achieves higher F1-scores for the MUC and $\\text{B}^3$ metrics.",
"Even though our model achieves a slightly lower F1-score for the $\\text{CEAF}_{\\phi _4}$ metric, the overall averaged F1 score of our model is still better.",
"It is worth mentioning that the variant [c2f-coref + SpanBERT-large] is more complex than our method, because it has some other additional components such as coarse-to-fine antecedent pruning and higher-order inference [11], [17].",
"Recently, a model named CorefQA has been proposed [20], and it achieves an averaged F1 score of $83.1$ on the English OntoNotes benchmark.",
"The work takes a complete departure from the paradigm used by the e2e-coref model, and instead, proposes to formulate the coreference resolution problem as a span prediction task, like in question answering.",
"Despite its impressive performance, the CorefQA model is very computationally expensive.",
"In order to predict coreference clusters for a single document, CorefQA needs to run a Transformer-based model on the same document many times (each time a different query is appended to the document).",
"Analysis on the Performance of the Mention Extractor As mentioned in Subsection REF , in our work, the value of the parameter $\\lambda $ for pruning is set to be 0.25.",
"On the other hand, it is set to be 0.4 in the e2e-coref model.",
"Table REF shows the comparison in more details.",
"Our mention extractor is able to extract 95.7% of all the gold mentions in the dev set, while the mention extractor of the e2e-coref model is only able to extract 92.7% of them.",
"Furthermore, by proposing less candidate spans, the workload of our mention linker is also reduced."
],
[
"Conclusions",
"In this work, we propose a simple yet effective baseline for the task of coreference resolution.",
"Despite its simplicity, our model still can achieve impressive performance, outperforming all recent extended works on the popular English OntoNotes benchmark.",
"In future work, we are interested in reducing the computational complexity of our baseline model using compression techniques [26], [27], [28].",
"We also plan to extend our work to address the task of event coreference resolution [29], [30]."
]
] | 2107.01700 |
[
[
"KAISA: An Adaptive Second-Order Optimizer Framework for Deep Neural\n Networks"
],
[
"Abstract Kronecker-factored Approximate Curvature (K-FAC) has recently been shown to converge faster in deep neural network (DNN) training than stochastic gradient descent (SGD); however, K-FAC's larger memory footprint hinders its applicability to large models.",
"We present KAISA, a K-FAC-enabled, Adaptable, Improved, and ScAlable second-order optimizer framework that adapts the memory footprint, communication, and computation given specific models and hardware to improve performance and increase scalability.",
"We quantify the tradeoffs between memory and communication cost and evaluate KAISA on large models, including ResNet-50, Mask R-CNN, U-Net, and BERT, on up to 128 NVIDIA A100 GPUs.",
"Compared to the original optimizers, KAISA converges 18.1-36.3% faster across applications with the same global batch size.",
"Under a fixed memory budget, KAISA converges 32.5% and 41.6% faster in ResNet-50 and BERT-Large, respectively.",
"KAISA can balance memory and communication to achieve scaling efficiency equal to or better than the baseline optimizers.",
"KAISA is open source and available at https://github.com/gpauloski/kfac_pytorch."
],
[
"Introduction",
"Deep neural networks (DNNs) have driven breakthroughs in many research domains, including image classification [22], [24], object detection and segmentation [21], machine translation [53], and language modeling [17].",
"DNNs are typically trained with stochastic gradient descent (SGD), or variants thereof.",
"As models and training datasets become larger, training must increasingly be performed in parallel on many CPUs, GPUs, or TPUs [56], [30].",
"For example, the BERT [17] model with 330 million parameters would take weeks to months to train on a single GPU, and GPT-3 [11] with 175 billion parameters cannot fit in the memory of any commercially available GPU.",
"While distributed training on many processors can reduce training time, the need to communicate weight updates and other information among processors can limit scalability [39].",
"Figure: SGD vs. K-FAC training for ResNet-32 with the CIFAR-10 dataset.",
"K-FAC reduces iterations needed for convergence.Recent theoretical [36], [20], [35] and empirical [8], [42], [50] studies have shown that Kronecker-factored Approximate Curvature (K-FAC) can accelerate training by enabling convergence with fewer iterations than SGD—for example, achieving baseline validation accuracy in 40% fewer epochs than SGD on ResNet-32 with the CIFAR10 dataset (see Figure REF ).",
"Technically, researchers use K-FAC to approximate the inverse of the Fisher Information Matrix (FIM)—an approximation of the Hessian—and precondition the gradients before parameter updates [36].",
"The K-FAC approximation is 1) compute intensive, due to the required inverse or eigen decomposition calculations with $O(N^3)$ complexity ($N$ is the number of parameters); 2) memory intensive, as it stores per-layer activations and gradients, which may take $O(N^2)$ space compared to $O(N)$ for SGD; and 3) communication intensive for distributed training because the Kronecker factors and accompanying inverses or eigen decompositions must be communicated to all workers.",
"To overcome these overheads and achieve faster training times than SGD at scale, K-FAC updates are often decoupled from gradient preconditioning so the expensive computations are performed less frequently.",
"Existing K-FAC systems either cache all Kronecker factor eigen decompositions needed to precondition gradients locally [44] or distribute gradient preconditioning across processors [42].",
"Neither approach enables K-FAC DNN deployments that are both memory and communication efficient, thus inhibiting scalability.",
"The first approach avoids communication when local memory is sufficient.",
"The second approach reduces memory footprint by not caching eigen decompositions locally but increases communication.",
"These various considerations make it difficult to use K-FAC efficiently in practice given the specific requirements of the DNN model or hardware and require users to choose between optimizing memory or communication.",
"To address these concerns, we propose KAISA , a K-FAC-enabled, Adaptable, Improved, and ScAlable second-order optimizer framework that can adapt execution to a given model size and memory limit.",
"KAISA allows tuning the ratio between communication and memory footprint to optimally apply K-FAC to distributed training by controlling the fraction of processes with local access to data.",
"This adaptation is made possible by grouping the processes, then distributing and communicating data within each group, as detailed in §REF and §REF .",
"K-FAC distribution strategies described in other work are effectively special cases of KAISA in which either all processes [44] or one process [42] cache data.",
"We evaluate KAISA on a range of classification, segmentation, and language modeling applications, including ResNet-50 [22], Mask R-CNN [21], U-Net [46], and BERT [17], on clusters of 448 NVIDIA Tesla V100s and 192 Ampere A100 GPUs.",
"Our results show that: 1) with fixed global batch size, KAISA trains deep neural networks 18.1–36.3% faster than the original optimizers for these applications; 2) with a fixed memory budget, KAISA trains ResNet-50 and BERT-Large phase 2 in 32.5% and 41.6% less time compared to momemtum SGD and Fused LAMB, respectively; 3) by varying the number of gradient workers, the K-FAC memory overhead can be reduced by 1.5–$2.9\\times $ ; 4) for high communication applications such as ResNet-50, extra processor memory can be used to train 24.4% faster; and 5) using state-of-the-art NVIDIA A100 GPUs, KAISA converges in fewer iterations at all scales, and KAISA 's scaling performance is on-par with SGD even with KAISA 's additional communication overhead.",
"Our code is open source with the MIT license and available at https://github.com/gpauloski/kfac_pytorch.",
"Our contributions in this paper are: An adaptive second-order optimizer framework that trains faster than SGD and its variants, while preserving convergence.",
"A quantitative study of the tradeoff between memory footprint and communication and its impact on training time in K-FAC design.",
"The first large scale evaluation of K-FAC convergence and speedup relative to SGD for Mask R-CNN, U-Net, and BERT on up to 128 A100 GPUs.",
"The rest of the paper is as follows.",
"We present the mathematical background and distributed implementation of K-FAC in §.",
"We describe KAISA 's design in § and implementation in §.",
"We present our experiments and results in §.",
"In §, we summarize existing DNN optimization frameworks and memory management techniques.",
"Finally, we conclude in §."
],
[
"Background",
"We first introduce K-FAC and its distributed implementation.",
"K-FAC is an efficient approximation of the Fisher Information Matrix (FIM), which has been shown to be equivalent to the Generalized Gauss-Newton (GGN) matrix in specific cases and can be viewed as an approximation of the Hessian [36].",
"In a standard SGD update step, the weights are updated using the gradients of the loss, as illustrated in Equation REF .",
"The K-FAC update step in Equation uses the FIM $F$ to precondition the gradients prior to update [44].",
"$w^{(k)}$ is the weight at iteration $k$ , ${\\alpha }^{(k)}$ is the learning rate at iteration $k$ , $n$ is the mini-batch size, $\\nabla {L_i}(w^{(k)})$ is the gradient of the loss function $L_i$ for the $i^\\text{th}$ example with regard to $w^{(k)}$ , and $F^{-1}(w^{(k)})$ is the inverse of the FIM.",
"$\\text{SGD: } w^{(k+1)} &= w^{(k)} - \\frac{{\\alpha }^{(k)}}{n}\\sum _{i=1}^{n}\\nabla {L_i}(w^{(k)}) \\\\\\text{K-FAC: } w^{(k+1)} &= w^{(k)} - \\frac{{\\alpha }^{(k)}F^{-1}(w^{(k)})}{n}\\sum _{i=1}^{n} \\nabla {L_i}(w^{(k)}) $ It has been shown empirically that training DNNs with the K-FAC second-order method enables convergence with fewer iterations than with SGD alone.",
"Theoretical understandings of the convergence rates of natural gradient methods, such as K-FAC, are an area of active research.",
"Previous work has shown that K-FAC [57] has linear convergence to the global minimum given a sufficiently over-parameterized model.",
"In strongly-convex problems, natural gradient methods have a quadratic convergence rate compared to the linear convergence of SGD [10].",
"The strongly-convex case provides some understanding of how K-FAC can improve convergence in non-convex cases.",
"Further, it has been shown that natural gradient methods enable larger learning rates improving the rate of convergence [57].",
"K-FAC makes greater per-iteration progress in minimizing the objective function at the cost of more computationally expensive iterations."
],
[
"K-FAC Approximation",
"K-FAC is based on the Kronecker product, a block matrix factorization that can reduce a large matrix inverse into two smaller matrix inverses.",
"K-FAC exploits the properties of the Kronecker product and the geometry of the FIM for DNNs to greatly reduce the complexity of computing the approximate FIM inverse.",
"The Kronecker product is written as $A \\otimes B$ where $A$ has size $m\\times n$ and $B$ has size $p \\times q$ .",
"The resulting matrix has shape $mp\\times nq$ .",
"$A \\otimes B=\\begin{bmatrix}a_{11}B & \\dots & a_{1n}B \\\\\\vdots & \\ddots & \\vdots \\\\a_{m1}B & \\dots & a_{mn}B\\end{bmatrix}$ The Kronecker product has two convenient properties: $(A \\otimes B)^{-1} &= A^{-1} \\otimes B^{-1} \\\\(A \\otimes B)\\vec{c} &= B^\\top \\vec{c} A .$"
],
[
"K-FAC Approximation",
"The FIM for a deep neural network is a block matrix where each block maps to layers in the model.",
"The block corresponding to the $i$ and $j^{\\text{th}}$ layers is approximated as $F_{i,j}\\approx a_{i-1}a_{j-1}^\\top \\otimes g_{i}g_{j}^\\top .$ Here, $a_{i-1}$ and $g_i$ are the activation of the $i-1^\\text{th}$ layer and the gradients of the $i^\\text{th}$ layer in the model, respectively [36], [44].Formally, $F$ represents an expected value, and the expectation of a Kronecker product is not equivalent to the Kronecker product of the expected factors.",
"However, this approximation still reasonably represents the structure of the FIM [36].",
"A fundamental assumption of K-FAC is the independence between layers [36], [44].This assumption is sufficient to produce an effective approximation for $F$ and necessary to produce a tractable algorithm.",
"Using this assumption, K-FAC approximates the FIM as a diagonal block matrix $\\hat{F}$ .",
"$\\hat{F}=\\texttt {diag}(\\hat{F}_1,...,\\hat{F}_i,...,\\hat{F}_L)$ As shown in Figure REF , the inverse of $F$ is a diagonal block matrix composed of the inverses of each diagonal block $\\hat{F}_i$ : $\\hat{F}^{-1}=\\texttt {diag}(\\hat{F}_1^{-1},...,\\hat{F}_i^{-1},...,\\hat{F}_L^{-1})$ where $\\hat{F}_{i}= a_{i-1}a_{i-1}^\\top \\otimes g_{i}g_{i}^\\top =A_{i-1}\\otimes G_i.",
"$ We refer to $A_{i-1}$ and $G_{i}$ as the Kronecker factors.",
"In practice, $A_{i-1}$ and $G_{i}$ are estimated with a running average of the factors computed over training batches [36], [44].",
"Due to the layer-wise independence of K-FAC, the gradient preconditioning and weight update for a single layer $i$ at iteration $k$ can be written as: $w_i^{(k+1)}=w_i^{(k)}-\\alpha ^{(k)}\\hat{F}_i^{-1}\\nabla L_i(w_i^{(k)}).$ We can apply properties REF and to reduce the gradient preconditioning, $\\hat{F}_i^{-1}\\nabla L_i(w_i^{(k)})$ , to an efficient form where the smaller Kronecker factors, rather than the large FIM, are inverted.",
"$\\hat{F}_i^{-1}\\nabla L_i(w_i^{(k)})=G_i^{-1}\\nabla L_i(w_i^{(k)})A_{i-1}^{-1}$ Tikhonov regularization is used to avoid ill-conditioned matrix inverses with K-FAC by adding a damping parameter $\\gamma $ to the diagonal of $\\hat{F}_i$ [20], [42].",
"In most implementations, instead of computing $\\hat{F}^{-1}$ , we compute $(\\hat{F}_i+\\gamma I)^{-1}$ as: $(\\hat{F}_i+\\gamma I)^{-1}=(A_{i-1}+\\gamma I)^{-1}\\otimes (G_i+\\gamma I)^{-1}.",
"$ Thus, the final update step for the parameters of layer $i$ at iteration $k$ is: $w_i^{(k+1)}=&w_i^{(k)}-\\alpha ^{(k)}(\\hat{F}_i+\\gamma I)^{-1}\\nabla L_i(w_i^{(k)}) \\\\=&w_i^{(k)}-\\alpha ^{(k)}(G_i+\\gamma I)^{-1}\\nabla L_i(w_i^{(k)})(A_{i-1}+\\gamma I)^{-1}.$"
],
[
"Alternative Approximation",
"It has been shown that an empirically more stableIn this context, stable means produces more consistent validation results across batch sizes and hyperparameter settings.",
"approximation for $(\\hat{F}+\\gamma I)^{-1}\\nabla L_i(w_i^{(k)})$ can be computed using an eigen decomposition of the Kronecker factors, $A$ and $G$ , from Equation REF [20], [44].",
"Given $Q_A$ and $Q_G$ , the eigenvectors of the factors, and $\\upsilon _A$ and $\\upsilon _G$ , the eigenvalues of the factors, the preconditioned gradient can be computed as follows.",
"$V_1&=Q_G^\\top \\nabla L_i(w_i^{(k)}) Q_A \\\\V_2&=V_1/(\\upsilon _G\\upsilon _A^\\top +\\gamma ) \\\\(\\hat{F}_i+\\gamma I)^{-1}\\nabla L_i(w_i^{(k)})&=Q_GV_2Q_A^\\top $ The composition of the factors $A$ and $G$ in Equation REF guarantees the factors are symmetric and therefore the factor eigen decompostions have real eigenvalues and orthogonal eigenvectors.",
"In this work, we use the eigen decomposition method for gradient preconditioning."
],
[
"Infrequent K-FAC Updates",
"A common strategy in second-order optimization methods is to update the second-order information every few iterations [36], [42], [44].",
"Intuitively, second-order information does not change as rapidly from one iteration to the next like first-order information.",
"The K-FAC update interval parameter controls the number of iterations between second-order updates, i.e., iterations between eigen decomposition recomputations.",
"Larger K-FAC update intervals result in more stale information, so tuning this parameter is key to achieving fast training with K-FAC.",
"Practically, K-FAC can maintain convergence with K-FAC update intervals of 100–2000 iterations [44].",
"Figure: Hybrid-parallel distributed K-FAC implementation overview.Blue boxes are standard computations in data-parallel training.Red boxes are computations required by K-FAC.Workers maintain identical copies of the model.The output of each layer zz is computed during the forward pass for the local batch xx.Then, the loss between the true output yy and predicted output z L z_L is calculated and used to compute gradients in the backward pass.The gradients are then allreduced across workers.During the forward/backward pass, K-FAC caches intermediate data for computing factors.In the K-FAC stage, workers compute factors AA and GG in data-parallel and allreduce the results.The eigen decompositions and preconditioned gradients 𝒢\\mathcal {G} are computed in the K-FAC approximation stage.The existing K-FAC methods, MEM-OPT and COMM-OPT, implement the model-parallel stage differently as shown at the bottom of the figure.After the K-FAC stage, all workers have 𝒢\\mathcal {G} and can update weights locally using 𝒢\\mathcal {G} and a standard optimizer (e.g., SGD or ADAM)."
],
[
"Distributed Implementation",
"Existing distributed K-FAC implementations are hybrid-parallel, with first-order information (e.g., gradients) computed in data-parallel and second-order information (e.g., K-FAC approximation) in model-parallel.",
"Figure REF outlines the model-parallel K-FAC computation performed between standard data-parallel training steps.",
"We refer to the two existing distributed K-FAC implementation strategies as MEM-OPT (memory optimized K-FAC) and COMM-OPT (communication optimized K-FAC).",
"Both work by replicating the DNN across all processes and assigning a random local batch of training data to each process at each iteration.",
"The data-parallel forward pass, backward pass, and SGD weight update stages outlined in Figure REF are the same in both methods.",
"The key differences between the two approaches are the model-parallel computation implementations in the gradient preconditioning stage."
],
[
"MEM-OPT",
"MEM-OPT [42] assigns each layer in the DNN to a different process during the preconditioning stage.",
"Each process computes the eigen decompositions of $A_{i-1}$ and $G_i$ needed for preconditioning the gradients using Equations REF –.",
"Each process then broadcasts the preconditioned gradients to all processes such that subsequent SGD weight updates can be done in data-parallel.",
"MEM-OPT has a low memory footprint because no eigen decompositions are duplicated: each layer's eigen decompositions are stored on only one process.",
"MEM-OPT has communication in three places: a) gradient allreduce, b) factor allreduce, and c) preconditioned gradient broadcast, all shown in Figure REF .",
"In non-K-FAC update steps (e.g., steps where the eigen decompositions are not updated), MEM-OPT avoids the factor allreduce because eigen decompositions from a previous step are reused; however, the preconditioned gradient broadcast is still required because the gradient being preconditioned changes every iteration."
],
[
"COMM-OPT",
"COMM-OPT [44] decouples the eigen decomposition from gradient preconditioning.",
"Instead of assigning each layer to a process, individual factors are assigned to a process to be eigen decomposed in parallel.",
"The eigen decompositions are broadcast back to all processes such that every process holds a copy of all eigen decompositions.",
"Each process computes all preconditioned gradients locally prior to weight updates.",
"COMM-OPT has a larger memory-footprint because every process must maintain a copy of the eigen decompositions.",
"COMM-OPT has communication in three places: a) gradient allreduce, b) factor allreduce, and c) eigen decomposition broadcast, also shown in Figure REF .",
"Decoupling the eigen decompositions from the gradient preconditioning achieves two goals: 1) $A_{i-1}$ and $G_i$ can be computed in different processes, doubling the maximum worker utilization, and 2) in non-K-FAC update steps, the communication in (b) and (c) can be avoided because every worker can locally precondition gradients with the cached eigen decompositions.",
"Thus, in non-K-FAC update intervals, COMM-OPT has no additional communication overhead compared to SGD.",
"COMM-OPT has been shown to be 4–16% faster than MEM-OPT for ResNet-50 training on 16–256 V100 GPUs [44].",
"These two implementations convey the fundamental tradeoff between caching the decompositions locally to avoid communication and communicating the preconditioned gradients every iteration to avoid additional memory overheads.",
"This tradeoff impacts the communication, computation, and memory overhead of K-FAC and motivates our research in this paper."
],
[
"Design",
"We introduce features of KAISA 's design that enable its tunable memory footprint and explain how KAISA performs load balancing and half precision training to improve scalability."
],
[
"Tunable Memory Footprint",
"KAISA introduces HYBRID-OPT, a distributed K-FAC strategy that achieves variable memory overhead in K-FAC.",
"In this strategy, each layer has a subset of the processes assigned to be gradient preconditioners, referred to as the gradient workers.",
"grad_worker_frac defines the size of this subset, i.e., the gradient worker count is $\\texttt {max}(1, \\textit {grad\\_worker\\_frac}\\times \\textit {world\\_size})$ .",
"One of the gradient workers is also responsible for computing the eigen decompositions for the layer and broadcasting the results to the remaining gradient workers.",
"At the end of the eigen decomposition stage, each gradient worker has a copy of the eigen decompositions and can precondition the gradient.",
"Gradient receivers are the processes not assigned as gradient workers for the layer.",
"Each gradient worker is responsible for broadcasting the preconditioned gradient to a subset of the gradient receivers.",
"With multiple gradient workers, the broadcast groups are smaller and simultaneous broadcasts are possible, reducing overall communication time.",
"For example, given two gradient workers and two gradient receivers, each gradient worker sends the preconditioned gradients to just one receiver; both gradient workers perform this communication at the same time.",
"After the gradient broadcast, all processes have a copy of the preconditioned gradient with which local weights can be updated.",
"Observe that KAISA unifies existing distributed K-FAC strategies because COMM-OPT and MEM-OPT are special cases of HYBRID-OPT.",
"COMM-OPT is the case where grad_worker_frac = 1 (all processes precondition the layer's gradient), and MEM-OPT is the case where grad_worker_frac = 1/world_size (a single process preconditions a layer's gradient and broadcasts the result).",
"Figure REF compares MEM-OPT, COMM-OPT, and HYBRID-OPT in an eight process environment.",
"As grad_worker_frac increases, more processes cache the eigen decompositions and the memory footprint increases.",
"Visually, the number of processes that cache the eigen decompositions is represented by the processes in the dashed red box in Figure REF .",
"Continuing with Figure REF , HYBRID-OPT has a lower preconditioned gradient broadcast cost than MEM-OPT because the broadcast groups are smaller and broadcasts are overlapped.",
"HYBRID-OPT requires four separate broadcasts to groups of size two in comparison to MEM-OPT which requires one large broadcast to a group of eight.",
"Since these broadcasts involve non-overlapping processes, we can execute all four broadcast calls simultaneously in the HYBRID-OPT case.",
"Given the complexity of broadcasting using the minimum spanning tree algorithm is $O(\\log p)$ , where $p$ is the number of processes in the broadcast group, the complexity is reduced from $O(\\log 8)$ in MEM-OPT to $O(\\log 2)$ in HYBRID-OPT.",
"Note in steps where the eigen decompositions are updated, HYBRID-OPT incurs an additional eigen decomposition broadcast with $O(\\log 4)$ complexity; however, as mentioned in §REF , eigen decompositions are updated infrequently so the average complexity for HYBRID-OPT is still less than that of MEM-OPT.",
"The problem addressed by KAISA's HYBRID-OPT strategy is akin to that of 2.5D matrix multiplication [48].",
"Both algorithms can utilize extra processor memory to reduce communication costs by controlling the replication factor of data.",
"In 2.5D matrix multiplication, a parameter $c$ determines the number of data copies, and in KAISA, the grad_worker_frac determines the number of workers that store the eigen decompositions."
],
[
"Greedy Factor Distribution",
"The eigen decompositions required for K-FAC are expensive to compute.",
"Distributed K-FAC optimizes this stage by computing eigen decompositions in a model-parallel fashion.",
"To most efficiently use available resources, the eigen decomposition computations should be distributed in a manner that minimizes the makespan $T$ , the time it takes for all processes to complete their assigned computations.",
"We use the longest processing time greedy algorithm which produces an assignment with makespan $T\\le \\frac{3}{2}T^*$ where $T^*$ is the optimal makespan [28].",
"The longest processing time algorithm sorts jobs by decreasing length and then iteratively assigns each job to the worker with the lowest current work load.",
"For each factor to be eigen decomposed, the processing time is approximated as $O(N^3)$ where each factor is an $N\\times N$ matrix [16].",
"Alternatively, memory usage can be optimized for by using $O(N^2)$ as the approximation since $N^2$ is the size of the factor.",
"KAISA uses the longest processing time strategy at the start of training to assign each factor to a process."
],
[
"Half Precision Storage and Computation",
"Mixed precision training is a common strategy to reduce training times and memory usage on supported hardware [37].",
"Popular frameworks for automatic mixed precision (AMP) training, such as NVIDIA AMP and PyTorch AMP, cast forward and backward pass operations to half precision where possible.",
"Gradient calculations in the backward pass can often produce very small values that would be clipped when cast to half precision, so these frameworks scale gradients to a larger value during the backward pass and unscale the gradients appropriately before the optimization step.",
"KAISA adapts to the precision being used for training.",
"When training with AMP, factors are stored in half precision, reducing memory costs.",
"K-FAC computations are performed in half precision where possible.",
"Eigen decompositions are generally unstable in half precision so factors are cast to single precision before decomposition.",
"KAISA can store eigen decompositions in half precision to further reduce memory consumption if needed.",
"Half precision training has become a staple in achieving state-of-the-art results in deep learning, and KAISA can particularly benefit from half precision training due to the K-FAC computation and communication overhead."
],
[
"K-FAC Usage",
"PythonStyle model = DistributedDataParallel(model) optimizer = optim.SGD(model.parameters(), ...) preconditioner = KFAC(model, grad_worker_frac=0.5) for data, target in train_loader: optimizer.zero_grad() output = model(data) loss = criterion(output, target) loss.backward() preconditioner.step() optimizer.step() We design KAISA to implement K-FAC as a preconditioner to standard optimizers with support for Conv2d and Linear layers.",
"KAISA has an easy-to-use interface and can be incorporated into existing training scripts in two lines: one to initialize and one to call KFAC.step() prior to optimizer.step() (see Listing ).",
"KAISA automatically registers the model and determines the distributed communication backend (e.g., Torch, Horovod, single-process).",
"A call to KFAC.step(): 1) computes the factors using the forward/backward pass data, 2) computes the eigen decompositions in parallel and broadcasts the results, 3) computes the preconditioned gradients and broadcasts the results if necessary, and 4) scales the preconditioned gradients.",
"We implement KAISA using PyTorch [43], with communication, interface, large-batch training, and gradient preconditioning implemented to collectively enable efficient second-order optimization."
],
[
"AMP and Distributed Training",
"We use PyTorch AMP for mixed precision training.",
"With PyTorch AMP, the GradScaler object responsible for scaling and unscaling the gradient in the backward pass can be passed to KAISA .",
"KAISA uses the GradScaler to correctly unscale the $G$ factors, since the scale factor can change from iteration to iteration, causing problems when computing the running average of $G$ over the course of training.",
"All communication operations are performed in the precision of the data to reduce bandwidth requirements.",
"KAISA supports torch.distributed and Horovod [47] for distributed training.",
"In this work, we use torch.distributed for all experiments because the DistributedDataParallel model wrapper overlaps gradient communication with backpropogation, works seamlessly with PyTorch AMP, and provides a broadcast group abstraction needed for HYBRID-OPT."
],
[
"Factor Accumulation",
"Processor memory (e.g., GPU VRAM) limits the maximum per-processor batch size during training.",
"A common strategy to achieve larger effective batch sizes is gradient accumulation, a method where gradient values for multiple forward and backward passes are accumulated between optimization steps.",
"For example, if processor memory limits the batch size to 8, an effective batch size of 32 can be achieved by accumulating gradients over four forward/backward passes prior to the optimization step.",
"This strategy is common in applications such as BERT, where effective batch sizes are often $>2^{15}$ , but modern GPUs are limited to local batch sizes $<2^7$ .",
"The forward/backward pass data needed by KAISA to compute the factors is accumulated over each mini-batch between calls to KFAC.step().",
"As the number of gradient accumulation steps increases, the memory needed to accumulate the forward/backward pass data grows linearly.",
"KAISA can efficiently support gradient accumulation by computing the factors for the current mini-batch during the forward/backward pass instead of during KFAC.step().",
"The factor communication is still performed during KFAC.step()."
],
[
"Triangular Factor Communication",
"Previous K-FAC work has exploited the symmetric nature of the Kronecker factors to reduce communication volume by sending only the upper triangle for each factor [49], [50].",
"Our implementation supports extracting the upper triangle for the factor allreduce and reconstructing the full factor before the eigen decomposition stage.",
"For the models studied in this work, this optimization did not yield performance improvements for two reasons.",
"First, network latency impacted overall communication time more than bandwidth; second, this optimization has an additional overhead for extracting the upper triangle and reconstructing the factor.",
"For models with larger individual layers—and therefore factors—this optimization could yield greater benefits."
],
[
"Gradient Preconditioning",
"The largest overhead of K-FAC in non-K-FAC update steps is computing the preconditioned gradients.",
"This process, described by Equations REF –, involves a series of matrix additions, divisions, and multiplications.",
"Observe that the gradient $\\nabla L_i(w_i^{(k)})$ is the only variable that changes between non-K-FAC update iterations.",
"In particular, the computation involving the outer product of the eigenvalues, $1/(\\upsilon _G\\upsilon _A^\\top +\\gamma )$ , in Equation does not need to be recomputed every iteration—only after the eigen decompositions are updated.",
"We also observe that when $\\emph {grad\\_worker\\_frac}>1/world\\_size$ , multiple processes perform this computation redundantly.",
"To reduce the total number of operations during the preconditioning stage, we move the computation of the outer product into the eigen decomposition stage.",
"The process assigned to eigen decompose $G$ computes $1/(\\upsilon _G\\upsilon _A^\\top +\\gamma )$ and broadcasts the result to all gradient workers instead of broadcasting $\\upsilon _A$ and $\\upsilon _G$ .",
"This ensures that $1/(\\upsilon _G\\upsilon _A^\\top +\\gamma )$ is computed once (on a single worker) and then reused many times by other workers.",
"In practice, this reduced the time to precondition the gradients for a single layer by up to 53%."
],
[
"Experiments",
"We report on experiments that address four issues.",
"1) Convergence to baseline evaluation metrics and 2) time to convergence with and without KAISA to validate the design and implementation.",
"3) An exploration of the memory and communication tradeoff using KAISA to develop a quantitative understanding of the impact of the grad_worker_frac configuration.",
"4) An evaluation of KAISA's scaling performance."
],
[
"Hardware and Software Stack",
"We performed experiments on two computers.",
"The first is the GPU subsystem of the Frontera supercomputer at the Texas Advanced Computing Center, which has 112 nodes, each powered by IBM Power9 processors, with four 16 GB NVIDIA V100 GPUs (448 GPUs in total).",
"Nodes are connected by an InfiniBand EDR network.",
"We use PyTorch 1.6, CUDA 10.2, CUDNN 7.6.5, and NCCL 2.5.6, and MVAPICH2-GDR 2.3.4 to launch processes on multiple nodes for distributed training.",
"The second is the GPU subsystem of the Theta supercomputer at Argonne National Laboratory.",
"This system has 24 NVIDIA DGXA100 nodes with eight 40GB A100 GPUs each (192 A100 GPUs in total).",
"On DGXA100 nodes, we use PyTorch 1.7, CUDA 11.0, CUDNN 8.0.4, and NCCL 2.7.8.",
"We distinguish between these two systems in the following text by specifying either V100 or A100 GPUs."
],
[
"Applications",
"We evaluate KAISA for classification, segmentation, and language modeling applications.",
"Classification: We use ResNet-50 [22] with the ImageNet-1k dataset [29], which has 1000 categories with approximately 1.3M training images and 50K validation images.",
"We use K-FAC to precondition all convolutional and linear layers in ResNet-50 and SGD for the weight updates.",
"Segmentation: We explore two segmentation tasks.",
"First, we use the NVIDIA reference PyTorch implementation of Mask R-CNN [21], [5] with the Common Objects in Context (COCO) 2014 dataset [32].",
"We use K-FAC to precondition the convolutional and linear layers in the region of interest (ROI) heads of Mask R-CNN and SGD for the weight updates.",
"Second, we use a U-Net [46] architecture for segmenting brain tumor sub-regions.",
"We extend a Kaggle competition implementation [2] to enable multi-GPU training.",
"The test case was run on the LGG Segmentation Dataset [3], which contains Magnetic Resonance (MR) images of the brain from 110 patients across five hospitals.",
"Images from a random subset of 100 patients are used as the training dataset and the remaining 10 patients are used for validation.",
"We apply K-FAC to all convolutional layers in the model.",
"Language Modeling: We train the BERT-Large Uncased model using a modified version of the NVIDIA reference PyTorch implementation for BERT [17], [5] with the English Wikipedia [52] and Toronto BookCorpus datasets [59].",
"Each transformer in BERT is implemented using a series of Linear layers, and we apply K-FAC to all linear layers.",
"We do not use K-FAC to precondition the embedding layer and prediction head because both of these layers have a Kronecker factor with shape $vocab\\_size\\times vocab\\_size$ , and since the vocab size for BERT-Large is 30K, these factors cannot be efficiently eigen decomposed.",
"Fused LAMB is used as the optimizer [56].",
"The strategy outlined in §REF is used to reduce memory consumption since gradient accumulation is used for training."
],
[
"Convergence with Fixed Batch Size",
"We compare KAISA performance (converged accuracy, epochs to convergence, and time to convergence) on ResNet-50, Mask R-CNN, U-Net, and BERT-Large against the baseline implementations listed in Table REF .",
"For ResNet-50 and Mask R-CNN, we use MLPerf benchmark target results [4].",
"For U-Net, we use the baseline validation Dice similarity coefficient (DSC) from the model's GitHub repository [2].",
"For the BERT-Large baseline, researchers have reported F1 scores of 91.08% [5], 91.0% [1], and 90.4% [56] for fine-tuning on the SQuAD v1.1 dataset.",
"We use the best reported F1 score (i.e., the NVIDIA implementation [5]) with the LAMB optimizer.",
"Due to the partial unavailability of the Toronto BookCorpus training dataset, our measurement only converges to 90.8%.",
"(The Toronto BookCorpus dataset is no longer available online as a holistic package; we could recover only 14,155 of the 16,846 books.)",
"Table: Baseline performance and hardware summary for ResNet-50, Mask R-CNN, U-Net, and BERT-Large.",
"“val acc” is validation accuracy.",
"“mAP” is mean average precision.",
"“DSC” is Dice similarity coefficient.In all comparisons, we use the same global batch size for KAISA and the original optimizers to isolate the improvement from second-order information.",
"Table REF summarizes the hyperparameters for each application.",
"For ResNet and K-FAC specific hyperparameters, we use values from [44] to provide direct performance comparisons to previous K-FAC works.",
"For Mask R-CNN and BERT-Large, we use the NVIDIA reference hyperparameters [5].",
"Further performance improvements could be gained through more extensive hyperparameter tuning; however, this was not necessary to achieve results better than the original optimizers.",
"Table: Summary of hyperparameters used for each application.",
"BS = global batch size, LR = learning rate, WU = warm up iterations, K_freq = number of iterations between eigen decomposition re-computations, F_freq = iterations between factor updates.",
"grad_worker_frac = 1 and damping = 0.003 for all cases.ResNet-50: We train ResNet-50 for 55 and 90 epochs for KAISA and momentum SGD, respectively, using FP16 precision on eight NVIDIA A100 GPUs.",
"Figure REF shows the validation accuracy curve using the two optimization methods.",
"KAISA converges to the baseline validation accuracy at epoch 46 and momentum SGD at epoch 65.",
"The time-to-convergence is 268.1 minutes for KAISA : 24.3% less than the 354.0 minutes for momentum SGD.",
"Figure: Validation Metric Curve Comparison between KAISA and SGD/ADAM on ResNet-50, Mask R-CNN, and U-Net.",
"The dotted lines represent the target metric.= 0pt Mask R-CNN: Our baseline measurement of Mask R-CNN on 32 NVIDIA V100 GPUs takes 25,640 iterations to converge to 0.377 bbox mAP and 0.342 segm mAP in FP32.",
"With K-FAC enabled for the ROI heads, training converges to 0.379 bbox mAP and 0.350 segm mAP in 21,000 iterations.",
"The bbox mAP comparision for SGD and KAISA is shown in Figure REF .",
"KAISA reduces the training time from 136.1 minutes to 115.8 minutes, a 14.9% improvement.",
"With a batch size of 128 on 64 V100 GPUs, KAISA converges in 12,000 iterations compared to 15,000 with SGD and reduces training time by 18.1%.",
"U-Net: The reference U-Net implementation [2] with ADAM converges to 91.0% validation DSC within 50 epochs with four NVIDIA A100 GPUs using FP32 training.",
"Using KAISA , the training converges above 91.0% in 30 epochs as seen in Figure REF .",
"KAISA reduces the training time by 25.4% (10.9 minutes vs. 14.6 minutes).",
"BERT: BERT pretraining has two phases.",
"The first trains with maximum sequence length of 128 for 7038 iterations and the second trains with maximum sequence length of 512 for 1,563 iterations.",
"We fine-tune the pretrained BERT model for three epochs using the SQuAD dataset.",
"All phases and fine-tuning are done in FP16.",
"Tuning the hyperparameters of BERT is expensive as each run can take over 130 hours with 8 A100 GPUs, so we showcase the effectiveness of KAISA with the second phase of BERT pretraining.",
"For phase two pretraining, we start with the same model pretrained with LAMB during phase one.",
"We train with KAISA for {800, 1,000, 1,200} iterations then fine-tune SQuAD.",
"Table REF summarizes the validation SQuAD F1 scores after fine-tuning and the pretraining performance improvements.",
"Table: BERT performance comparison: KAISA vs. LAMBKAISA converges to the 90.8 F1 baseline in 800 iterations, 47.9% less iterations than required for LAMB, and KAISA takes 36.3% less time than LAMB to converge."
],
[
"Convergence with Fixed Memory Budget",
"To understand how a grad_worker_frac can enable second-order optimization in memory-constrained environments, we compare KAISA's performance against baselines with fixed memory budgets.",
"In particular, we train ResNet-50 on 64 V100 GPUs and BERT-Large phase two on eight A100 GPUs.",
"For each experiment, we use the maximum possible local batch size and measure the convergence, epochs to convergence, and time to convergence.",
"Table REF summarizes the hyperparameters and time to convergence.",
"Table: Time to convergence and hyperparameters for each optimizer.",
"BS = global batch size, LR = learning rate, g_frac = gradient worker fraction, K_f = iterations between eigen decomposition re-computations, F_f = iterations between factor updates, T_conv is time to convergence in minutes.ResNet-50: With momentum SGD, the maximum local batch size is 128 per GPU and the global batch size is 8,192.",
"We train ResNet-50 for 90 epochs using momentum SGD which achieves 75.0% validation accuracy—0.9% lower than the MLPerf baseline.",
"Next, we use KAISA with grad_worker_frac = 1 and a local batch size of 80; however, the training runs out of memory.",
"Lowering grad_worker_frac to 1/2, training takes 83 minutes to converge to 75.9% in the 48th epoch.",
"The complete 55 epoch training takes 95 minutes and the validation accuracy reaches 76.0%.",
"So, even if momentum SGD converges to 75.9% by the 90th epoch, KAISA still reduces the time to convergence by 32.5%.",
"Finally, we run the same training process with MEM_OPT by setting grad_worker_frac to 1/64.",
"It converges at the 47th epoch and the time to convergence is 96 minutes.",
"The complete 55 epoch training takes 111 minutes.",
"This experiment highlights the benefit of KAISA in cases where compute resource can not be efficiently utilized with the original optimizers.",
"With a grad_worker_frac value of 1/2, KAISA offers benefits over COMM_OPT and MEM_OPT by enabling second-order optimization under a tight memory budget that is 13.9% faster than COMM_OPT and not feasible with MEM_OPT.",
"BERT: With LAMB, the maximum possible local batch size per GPU is 12 for the second phase of this BERT implementation [5], and the global batch size is 24,576.",
"For KAISA, we use a local batch size of 8 and global batch size of 32,768.",
"This experiment uses the same hyperparameters as in §REF , so all cases with LAMB and BERT should converge to the baseline, thus we only project the training time with the first 100 steps.",
"As the global batch size with LAMB is 24,576, 2,084 training steps are required to finish three epochs, and the training time is 2,917.6 minutes.",
"KAISA, with grad_worker_frac = 1/2, takes 3,268.8 minutes to finish the three epochs.",
"However, KAISA converges to baseline after 800 steps, as shown in Table REF , so the time to converge is 1,702.5 minutes—41.6% faster than LAMB.",
"Setting grad_worker_frac = 1 takes 1,703.5 minutes to converge for KAISA.",
"The performance is comparable to the case with grad_worker_frac = 1/2.",
"We conduct a detailed study on this convergences phenomena for different grad_worker_frac values in §REF ."
],
[
"Memory vs. Communication",
"To understand the impact of grad_worker_frac on training times, we train ResNet-{18, 50, 101, 152}, Mask R-CNN, and BERT-Large on 64 V100 GPUs for grad_worker_frac $\\in $ {1/64, 1/32, 1/16, 1/8, 1/4, 1/2, 1}.",
"For each experiment, we record the average iteration time, i.e., time between weight updates, and the GPU memory usage.",
"We refer to the K-FAC overhead as the memory required to store the factors and eigen decompositions, and the K-FAC overhead is computed as the difference between the K-FAC and SGD memory usage.",
"For all ResNet models, we use the same hyperparameters used for ResNet-50 (Section REF ) except for ResNet-152 where the local batch size was lowered to 24.",
"The results are presented in Figure REF , and a summary of the memory usage is provided in Table REF .",
"The K-FAC memory overhead increases linearly as a function of grad_worker_frac for all models.",
"KAISA requires 1.5–45.8% more memory than SGD depending on the application and value of grad_worker_frac (Table REF ).",
"The maximum K-FAC overhead (i.e., when grad_worker_frac = 1) is 1.5–2.9$\\times $ that of the minimum K-FAC overhead (i.e., when grad_worker_frac = $1/64$ ).",
"Table: Summary of per-GPU memory usage for training, in MB.Abs.",
"is the absolute memory required for training.Δ\\Delta is the %-increase in memory required over SGD.The K-FAC overhead is the K-FAC abs.",
"memory minus the SGD abs.",
"memory.With respect to iteration times, the ResNet models scale well with the number of gradient workers with ResNet-50 scaling the best.",
"For ResNet-50, the speedup from a gradient worker count of 1 to 64 is 24.4% for FP32 with a 22% increase in total memory usage.",
"The average iteration times for Mask R-CNN and BERT-Large remain constant as the number of gradient workers is increased.",
"For comparison, the same ResNet-50 experiment with 64 V100s in [44] only shows a 7.6% speedup when increasing the gradient worker count from 1 to 64.",
"This improvement in KAISA over previous work is due to the unique contributions presented in §REF , §REF , §REF , and §REF .",
"These promising results for ResNet-50, a de facto standard benchmark for deep learning systems, are important as the performance characteristics of ResNet-50 represent a large set of commonly used models (e.g.",
"VGG16, U-Nets, etc.).",
"We can understand why ResNet model performance varies across grad_worker_frac values while the Mask R-CNN and BERT-Large performance remains constant by considering the bandwidth requirements of these applications.",
"The bandwidth required by KAISA is a function of the size of the factors, eigen decompositions, and frequency of K-FAC updates.",
"With 64 V100s, ResNet-50 calls KFAC.step() frequently (4–6 calls/second) and incurs a K-FAC memory overhead between 634 MB and 1.8 GB.",
"In comparison, Mask R-CNN calls KFAC.step() with a lower frequency (3 calls/second) and has a much smaller K-FAC overhead (100–200 MB).",
"Thus, the changes in how KAISA communicates data with respect to grad_worker_frac are less apparent in Mask R-CNN.",
"BERT-Large has the lowest bandwidth requirements of all applications even though it has the largest K-FAC overhead.",
"BERT-Large uses gradient accumulation to achieve very large batch sizes (32K for phase 2) and as a result only calls KFAC.step() every $\\sim $ 120 seconds.",
"While the iteration times for low-communication models such as BERT and Mask R-CNN are invariant to the grad_worker_frac value in KAISA, KAISA still produces faster-than-SGD training times with small increases in memory-overhead.",
"This is due to KAISA's unique features outlined in § and §.",
"Further, practitioners training these models at larger scales, e.g., 100s or 1000s of GPUs, where communication becomes a greater bottleneck will benefit more from the flexibility KAISA provides to adapt training to environments with increasing communication costs.",
"Tuning the grad_worker_frac hyperparameter, to determine an optimal balance between iteration time and memory usage, is simple as it only requires profiling the average iteration time for each grad_worker_frac value over a few iterations.",
"We find that with respect to training times, grad_worker_frac has the most impact in applications that spend a larger proportion of time doing communication.",
"To further understand how grad_worker_frac impacts training times, we analyze the execution time for each section within KFAC.step() with ResNet-50 on 64 V100s.",
"Figure REF provides the time spent in each section for all layers in the model during a call to KFAC.step().",
"Times are averaged over 10,000 iterations and across all workers.",
"Eigen decompositions are updated every 500 iterations.",
"As shown in Figure REF , factor computation and communication, eigen decomposition, and scaling and updating the gradients, are invariant to the grad_worker_frac.",
"Figure: Average function execution time during calls to KFAC.step() for ResNet-50 on 64 GPUs.The time required to broadcast the eigen decompositions increases substantially as the number of gradient workers is increased.",
"Referring back to Figure REF , the more gradient workers there are, the more processes that need to receive the eigen decompositions.",
"However, the eigen decompositions, in this case, are only recomputed every 500 iterations so the eigen decomposition broadcast has a negligible effect on the average iteration time.",
"On the other hand, the gradient preconditioning and broadcast occur every iteration regardless of if the factors or eigen decompositons are updated and have the greatest influence on average iteration time.",
"We see that the time to precondition the gradients increases with the gradient worker count because each process is assigned as a gradient worker for more layers.",
"This discovery highlights the importance of the gradient preconditioning optimizations made in §REF .",
"The time to broadcast the preconditioned gradients decreases to 0 as the gradient worker count approaches world_size, and notably, the time decreases at a faster rate than the increase in time required in the preconditioning stage.",
"This trend is a result of each gradient worker needing to send the results to fewer other processes as the gradient worker count increases."
],
[
"Scaling",
"To examine the scaling characteristic of KAISA, we measure the average time per epoch for ResNet-50 and average time per iteration for BERT-Large phase 2.",
"We choose ResNet-50 and BERT-Large because they represent a high-communication and low-communication model, respectively (see §REF ).",
"We study three KAISA variants: COMM-OPT, MEM-OPT, and HYBRID-OPT with a grad_worker_frac of 1/2, and we report the projected end-to-end training time speedup for the KAISA variants over the base optimizers (SGD and LAMB) in Figure REF .",
"For ResNet-50, we project the training time to 90 epochs for SGD and 55 for KAISA.",
"For BERT-Large, we project the phase 2 training time to 1,563 steps for LAMB and 800 for KAISA based on the study in §REF .",
"The hyperparameters in Table REF are used, and for ResNet-50, the K-FAC update frequency is scaled inversely with the global batch size to keep the number of K-FAC updates per training samples constant.",
"We store and communicate the factors and eigen decompositions in FP16 for BERT-Large.",
"In Figure REF and REF , MEM-OPT maintains a constant speedup over SGD and LAMB, respectively, across all scales.",
"In contrast, the speedup for COMM-OPT improves in both cases as the scale increases indicating the tradeoff of more memory for reduced communication does have scaling benefits.",
"HYBRID-OPT sees performance improvements on par with COMM-OPT with BERT-Large while using less GPU memory.",
"Overall, HYBRID-OPT has the best balance of scaling and memory usage for large-scale BERT pretraining.",
"As a whole, KAISA achieves better speedups with BERT-Large which is likely due to ResNet-50 being more communication bound than BERT-Large as noted in §REF .",
"Further scaling experiments on a machine with more GPUs is needed to understand the characteristics and limits of KAISA's speedup over SGD.",
"The emergence of DNNs has motivated revisiting many aspects of system design.",
"TensorFlow [6], PyTorch [43], and MXNet [14] are examples of deep learning frameworks that support end-to-end training.",
"TensorFlow Serving [41], DLHub [13], and Clipper [15] focus on low-latency inference serving.",
"Ray [38] provides a platform to integrate simulation, training, and serving for reinforcement learning applications.",
"Researchers have also designed new scheduling approaches in GPU clusters with informed hardware heterogeneity [40], sharing capability [55], and early user feedback [54] to optimize cost and hardware utilization.",
"Our work here focuses on a lower-level question: given model architecture, communication bandwidth, processor count, and memory size, can the hybrid-parallel aspect of distributed K-FAC be adaptable and reduce training time?",
"In the rest of this section, we briefly review other works in optimization frameworks.",
"Optimizer Frameworks: Distributed optimization frameworks take many forms.",
"In a synchronous optimizer such as Horovod [47], all variables are updated in every iteration.",
"Asynchronous optimizers relax variable update consistency, for example by passing values via parameter servers [31], to achieve higher performance than synchronous methods [45], [33].",
"The pipeline parallel paradigm, such as GPipe [25] and PipeDream [39] which hold multiple versions of a model partition in a processor and exploit asynchronous optimizers, has shown comparable training convergence.",
"BytePS [27] proposes a unified interface for synchronous and asynchronous SGD.",
"Generally, asynchronous SGD has a non-linear slowdown compared to synchronous SGD [7].",
"KAISA implements K-FAC in a synchronous manner as a preconditioner such that it can work with any SGD variant.",
"Memory Shortage and Remedies: Training DNNs is memory intensive.",
"A common technique for training large models is to swap between processor memory and host memory, such as in SwapAdvisor [23] and SuperNeurons [51].",
"An alternative is to discard some activation tensors and rematerialize them when needed for back-propagation.",
"Checkmate [26] formulates this tradeoff as an optimization problem and provides an optimal rematerialization schedule.",
"In KAISA , we apply techniques such as precision relaxing to reduce the memory consumption of K-FAC and controlling the distribution of eigen decomposition results across processors to maintain a minimal cost in training time.",
"K-FAC Convergence: Prior work on distributed K-FAC has reported training convergence primarily on ResNet-like convolutional neural networks.",
"One study [8] used asynchronous distributed K-FAC to train ResNet-50 with ImageNet 2$\\times $ faster than standard SGD, but only achieved 70% validation accuracy, a 5.9% loss compared to the 75.9% MLPerf [4] baseline.",
"Another distributed K-FAC implementation [42] trained ResNet-50 with ImageNet to 74.9% validation accuracy in just 978 iterations; however, comparisons to SGD are not provided.",
"Later work iterated on this implementation to achieve 75.0% validation accuracy on ImageNet with ResNet-50 on 2048 V100 GPUs in 2 minutes by carefully optimizing the baseline SGD training and introducing a 21-bit floating point (FP21) specification for the factors along with other optimizations from previous works such as triangular matrix communication and infrequent K-FAC updates.",
"FP21 was introduced due to worse convergence when using FP16 for factor communication; however, with KAISA , we found similar validation accuracy with ResNet-50 for FP32 and FP16 factor communication.",
"Differences in the numerical stability of the eigen decomposition in KAISA rather than matrix inversion could be a possible reason for the discrepancies.",
"A fourth study [44] trained ResNet-50 to the 75.9% MLPerf baseline in 18-25% less time than with SGD by replacing the factor inverse with eigen decomposition and optimizing for reduced communication in non-K-FAC update steps, at the cost of a high memory footprint.",
"KAISA generalizes previous distributed K-FAC strategies via a tunable memory footprint to balance the memory and communication costs.",
"This design enables efficient, distributed K-FAC research across a wider range of hardware.",
"Further, we showcase KAISA 's effectiveness on a variety of domains and models.",
"Alternative K-FAC Methods: Eigenvalue-corrected Kronecker-factorization (EK-FAC) [18], a more accurate approximation of the FIM, can perform cheap, partial updates.",
"Noisy K-FAC [58] and noisy EK-FAC [9] are functionally similar to standard K-FAC but introduce adaptive weight noise.",
"K-BFGS and K-BFGS(L) [19] apply BFGS [12] and L-BFGS [34] in an analogous method to K-FAC (e.g., block-diagonal, Kronecker-factored).",
"These works have shown better-than-K-FAC performance in many small scale (e.g., single GPU) and small dataset/model (e.g., MNIST or CIFAR-10 with VGG16) cases.",
"Kronecker-factor based FIM approximation variants are a growing area of research; however, large-scale studies have largely been limited to standard K-FAC.",
"KAISA introduces a valuable, unified design paradigm that can be applied to these K-FAC variants to efficiently deploy and evaluate their effectiveness on large models at scale."
],
[
"Conclusion",
"We have presented KAISA , a K-FAC-enabled, Adaptable, Improved, and ScAlable second-order optimizer framework.",
"To enable scalable DNN training, KAISA adapts memory and communication usage, and appropriately distributes the complex K-FAC computations, to best suit the model and hardware characteristics.",
"We design KAISA to be adaptable to hardware with limited memory (e.g., gaming GPUs) or environments with high communication costs (e.g., Ethernet or massively parallel).",
"We study the fundamental tradeoff between data access on local and remote memory, and evaluate KAISA's correctness and impact on training time by using four real-world applications.",
"With the same global batch size, our experiments show a 18.1–36.3% training time reduction for ResNet-50, Mask R-CNN, U-Net, and BERT-Large, while preserving convergence to the baseline.",
"Under the same memory budget, ResNet-50 and BERT phase 2 converge to the baseline in 32.5% and 41.6% less time compared to momentum SGD and Fused LAMB.",
"In high-communication applications, such as ResNet models, we show that extra processor memory can be used to improve iteration times by reducing communication.",
"In low-communication applications, such as Mask R-CNN and BERT-Large, we show the optimal KAISA usage and efficient scaling on par with SGD up to 128 GPUs."
]
] | 2107.01739 |
[
[
"A precise bare simulation approach to the minimization of some\n distances. Foundations"
],
[
"Abstract In information theory -- as well as in the adjacent fields of statistics, machine learning, artificial intelligence, signal processing and pattern recognition -- many flexibilizations of the omnipresent Kullback-Leibler information distance (relative entropy) and of the closely related Shannon entropy have become frequently used tools.",
"To tackle corresponding constrained minimization (respectively maximization) problems by a newly developed dimension-free bare (pure) simulation method, is the main goal of this paper.",
"Almost no assumptions (like convexity) on the set of constraints are needed, within our discrete setup of arbitrary dimension, and our method is precise (i.e., converges in the limit).",
"As a side effect, we also derive an innovative way of constructing new useful distances/divergences.",
"To illustrate the core of our approach, we present numerous examples.",
"The potential for widespread applicability is indicated, too; in particular, we deliver many recent references for uses of the involved distances/divergences and entropies in various different research fields (which may also serve as an interdisciplinary interface)."
],
[
"Introduction",
"Directed (i.e., not necessarily symmetric) distances $D(\\mathbf {P},\\mathbf {Q})$ between two finite discretefor reasons of technicality, in this paper we only deal with such kind of distributions; for instance, these can be also achieved from more involved systems by quantizations of observations represented by finite partitions of the observation/data space, or by making use of the dual representation for CASM $\\varphi -$ divergences (cf.",
"Liese & Vajda [218], Broniatowski & Keziou [61]).",
"(probability) distributions $\\mathbf {P},\\mathbf {Q}$ or between two general Euclidean vectors $\\mathbf {P},\\mathbf {Q}$ are known as divergences; they serve as important (dis)similarity measures, proximity measures and discrepancy measures in various different research areas such as information theory, statistics, artificial intelligence, machine learning, signal processing, pattern recognition, physics, finance, etc.since there exists a vast literature on divergences and connected entropies in these fields, for the sake of brevity we will give in this introduction only some basic references; many corresponding concrete applications will be mentioned in the following sections, in the course of the method-illuminating examples..",
"Besides Bregman distances/divergences (with which we do not deal here), another major class are the $\\varphi -$ divergences $D_{\\varphi }( \\mathbf {P},\\mathbf {Q})$ of Csiszar-Ali-Silvey-Morimoto (in short CASM $\\varphi -$ divergences, cf.",
"[94], [11], [266]).",
"The latter covers — with corresponding choices of $\\varphi $ — e.g.",
"the omnipresent Kullback-Leibler information distance/divergence [200] (also known as relative entropy), the Jensen-Shannon distance/divergence, as well as the power divergences (also known as alpha-divergences, Cressie-Read measures, and Tsallis cross-entropies).",
"For some comprehensive overviews on CASM $\\varphi -$ divergences, the reader is referred to the insightful books of e.g.",
"Liese & Vajda [217], Read & Cressie [303], Vajda [371], Csiszar & Shields [99], Stummer [344], Pardo [282], Liese & Miescke [216], the survey articles of e.g.",
"Liese & Vajda [218], Vajda & van der Meulen [374], Reid & Williamson [304], Basseville [34], and the references therein; an imbedding of CASM $\\varphi -$ divergences to more general frameworks can be found e.g.",
"in Stummer & Vajda [350], Broniatowski & Vajda [65], Stummer & Kißlinger [346] and Broniatowski & Stummer [64].",
"0.5cm Frequently used special cases of the above-mentioned power divergences are e.g.",
"the (squared) Hellinger distance, the Pearson chi-square divergence, and the Neyman chi-square divergence.",
"Moreover, several deterministic transformations of power divergences are also prominently used in research, most notably the Bhattacharyya distance [48] and the more general Renyi divergences [309] (also known as Renyi cross-entropies); a comprehensive exposition of the latter is given e.g.",
"in van Erven & Harremoes [380].",
"Some other important deterministic transformations of power divergences include the Bhattacharyya coefficient (cf.",
"[48],[49],[50]) — which is also called affinity (cf.",
"Matusita [256]) and fidelity similarity (cf.",
"e.g.",
"Deza & Deza [113]) — as well as the Bhattacharyya arccos distance (cf.",
"[50]) and the Fisher distance (also known as Rao distance, geodesic distance, cf.",
"e.g.",
"Deza & Deza [113]).",
"As shown below, by further explicit transformations we can also recover Sundaresan’s divergence [352] [353].",
"The minimization $\\inf _{\\mathbf {Q}\\in \\mathbf {\\Omega }} D( \\mathbf {Q}, \\mathbf {P} )$ of divergences from one distribution (respectively, its equivalent vector of frequencies) $\\mathbf {P}$ to an appropriate set $\\mathbf {\\Omega }$ of distributions (frequency vectors) appears in a natural way in various different contexts, as indicated in the following.",
"For instance, let $\\mathbf {P} = \\mathbf {P}_{true}$ be the true distribution of a mechanism which generates non-deterministic data and $\\mathbf {\\Omega }$ is a pregiven model in the sense of a (parametric or non-parametric) family of distributions which serves as an “approximation” (in fact, a collection of approximations) of the “truth” $\\mathbf {P}_{true}$ .",
"If $\\mathbf {P}_{true} \\notin \\mathbf {\\Omega }$ — e.g.",
"since $\\mathbf {\\Omega }$ reflects some simplifications of $\\mathbf {P}_{true}$ which is in line with the general scientific procedure — then the positive quantity $\\Phi _{\\mathbf {P}_{true}}(\\mathbf {\\Omega }) := \\inf _{\\mathbf {Q}\\in \\mathbf {\\Omega }}D( \\mathbf {Q}, \\mathbf {P}_{true} )$ can be used as an index of model adequacy in the sense of a degree of departure between the model and the truth (cf.",
"Lindsay [222], see also e.g.",
"Lindsay et al.",
"[223], Markatou & Sofikitou [248], Markatou & Chen [249]); small index values should indicate high adequacy.",
"If $\\mathbf {P}_{true} \\in \\mathbf {\\Omega }$ , then $\\Phi _{\\mathbf {P}_{true}}(\\mathbf {\\Omega }) = 0$ which corresponds to full adequacy.",
"This index of model adequacy $\\Phi _{\\mathbf {P}_{true}}(\\mathbf {\\Omega })$ can also be seen as index of goodness/quality of approximation to the truth or as model misspecification error, and it can be used for model assessment as well as for model search (model selection, model hunting) by comparing the indices $\\Phi _{\\mathbf {P}_{true}}(\\mathbf {\\Omega _{1}}),\\Phi _{\\mathbf {P}_{true}}(\\mathbf {\\Omega _{2}}), \\ldots $ of competing models $\\mathbf {\\Omega _{1}}, \\mathbf {\\Omega _{2}}, \\ldots $ and choosing the one with the smallest index; this idea can be also used for classification (e.g.",
"analogously to Bilik & Khomchuk [54] who deal with continuous (rather than discrete) distributions) where the $\\mathbf {\\Omega _{i}}$ are interpreted as (possibly data-derived but fixed) classes which are disjoint and non-exhaustive.",
"Typically, in statistical analyses the true distribution $\\mathbf {P}_{true}$ is unknown and is either replaced by a hypothesis-distribution $\\mathbf {P}_{hyp}$ or by a distribution $\\mathbf {P}_{data}$ derived from data (generated by $\\mathbf {P}_{true}$ ) which converges to $\\mathbf {P}_{true}$ as the data/sample size tends to infinity (e.g.",
"$\\mathbf {P}_{data}$ may be the well-known empirical distribution or a conditional distribution).",
"Correspondingly, $\\Phi _{\\mathbf {P}_{data}}(\\mathbf {\\Omega })$ reflects a data-derived approximation (estimate) of the index of model adequacy (resp.",
"of the model misspecification error) from which one can cast corresponding model-adequacy tests and related goodness-of-fit tests.",
"Moreover, for the case of i.i.d.",
"data-generation and $\\mathbf {P}_{data}$ to be the corresponding empirical distribution, the (not necessarily existent or unique) best-model-member/element choice $\\arg \\min _{\\mathbf {Q}\\in \\mathbf {\\Omega }} D( \\mathbf {Q}, \\mathbf {P}_{data})$ amounts to the well-known minimum distance estimator which for the Kullback-Leibler information divergence $D$ is equal to the omnipresent maximum likelihood estimator; for comprehensive surveys on divergence-based statistical testing and estimation, the reader is referred to e.g.",
"the references in the second half of the first paragraph in the current introduction.",
"Most of the above-mentioned considerations also hold for deterministic (rather than non-deterministic) frameworks where $\\mathbf {P}$ is a general Euclidean vector (rather than a probability-distribution describing frequency vector in the probability simplex) and $\\mathbf {\\Omega }$ is a model in the sense of a family of general Euclidean vectors (which may be encodings of more complicated context descriptions).",
"Returning to the general context, let us mention that from CASM $\\varphi -$ divergences one can also derive the widely used $\\varphi -$ entropies $\\mathcal {E}_{\\varphi }(\\mathbf {Q})$ of a distribution $\\mathbf {Q}$ (and non-probability versions thereof) in the sense of Burbea & Rao [68] (see also Csiszar [95], Ben-Bassat [38], Ben-Tal & Teboulle [40], Kesavan & Kapur [187], Dacunha-Castelle & Gamboa [102], Teboulle & Vajda [357], Gamboa & Gassiat [132], Vajda & Zvarova [376]); these entropies can e.g.",
"be basically constructed from $D_{\\varphi }(\\mathbf {Q},\\mathbf {P}^{unif})$ where $\\mathbf {P}^{unif}$ denotes the uniform distribution.",
"Moreover, by use of certain deterministic transformations $h$ one can also deduce the more general $(h,\\varphi )-$ entropies (and non-probability versions thereof) in the sense of Salicru et al.",
"[314] (see also e.g.",
"Pardo [282]).",
"As will be worked out in detail in Subsection REF below, from this one can deduce as special cases a variety of prominently used quantities in research, such as for instance: the omnipresent Shannon entropy [328], the $\\gamma -$ order Renyi entropy [309], the $\\gamma -$ order entropy of Havrda-Charvat [157] (also called non-additive $\\gamma -$ order Tsallis entropy [363] in statistical physics), the $\\widetilde{\\gamma }-$ order entropy of Arimoto [16], Vajda's quadratic entropy [371], Sharma-Mittal entropies [329], the Euclidean $\\gamma -$ norms, as well as measures of diversity, heterogeneity and unevenness, like the Gini-Simpson diversity index, the diversity index of Hill [160], the Simpson-Herfindahl index (which is also known as index of coincidence, cf.",
"Harremoes & Topsoe [155] and its generalization in Harremoes & Vajda [156]), the diversity index of Patil & Taillie [286], the $\\gamma -$ mean heterogeneity index (see e.g.",
"van der Lubbe [379]); see also Nayak [271] and Jost [176] for some interrelations with the above-mentioned entropies.",
"Given that the constraint set $\\mathbf {\\Omega }$ reflects some incomplete/partial information about a system (e.g.",
"moment constraints), the maximization over $\\mathbf {Q}\\in \\mathbf {\\Omega }$ of the above-mentioned entropies, norms and diversity indices (and the more general $(h,\\varphi )-$ entropies) is important for many research topics, most notably manifested in Jaynes’s [165],[166] omnipresent, “universally applicable” maximum entropy principle (which employs the Shannon entropy), and its generalizations (see e.g.",
"the books of Kapur [184], Kapur & Kesavan [185], Arndt [17], and Gzyl et al.",
"[150] for comprehensive surveys).",
"Besides the above-mentioned principal overview, let us now briefly discuss some existing technical issues for the minimization of CASM $\\varphi -$ divergences $\\Phi _{\\mathbf {P}}(\\mathbf {\\Omega }) :=\\inf _{\\mathbf {Q}\\in \\mathbf {\\Omega }} D_{\\varphi }( \\mathbf {Q}, \\mathbf {P} )$ .",
"For (not necessarily discrete) probability distributions/measures $\\mathbf {P}$ and sets $\\mathbf {\\Omega }$ of probability distributions/measures satisfying a finite set of linear equality constraints, $\\Phi _{\\mathbf {P}}(\\mathbf {\\Omega })$ has been characterized in Csiszar [96] and more recently by Csiszar & Matus [98], Broniatowski & Keziou [60], Leonard [213], and Pelletier [288] among others, in various contexts; those results extend to inequality constraints.",
"Minimizations of $\\gamma -$ order Renyi divergences on $\\gamma -$ convex sets $\\mathbf {\\Omega }$ are studied e.g.",
"in Kumar & Sason [202], whereas Kumar & Sundaresan [203] [204] investigate minimizations of Sundaresan’s divergence on certain convex sets $\\mathbf {\\Omega }$ .",
"To our knowledge, no general representation for $\\Phi _{\\mathbf {P}}(\\mathbf {\\Omega })$ for a positive distribution/measure $\\mathbf {P}$ (respectively, for an Euclidean vector with positive components) and a general set $\\mathbf {\\Omega }$ of signed measures (respectively, of Euclidean vector with components of arbitrary sign) exists.",
"At the contrary, many algorithmic approaches for such minimization problems have been proposed; they mostly aim at finding minimizers more than at the evaluation of the minimum divergence itself, which is obtained as a by-product.",
"Moreover, it is well-known that such kind of CASM $\\varphi -$ divergence minimization problems may be hard to tackle or even intractable via usual methods such as the omnipresent gradient descent method and versions thereof, especially for non-parametric or semi-parametric $\\mathbf {\\Omega }$ in sufficiently high-dimensional situations.",
"For instance, $\\mathbf {\\Omega }$ may consist (only) of constraints on moments or on L-moments (see e.g.",
"Broniatowski & Decurninge [59]); alternatively, $\\mathbf {\\Omega }$ may be e.g.",
"a tubular neighborhood of a parametric model (see e.g.",
"Liu & Lindsay [225], Ghosh & Basu [135]).",
"The same intractability problem holds for the above-mentioned $(h,\\varphi )-$ entropy maximization problems.",
"In the light of this, the goals of this paper are: to solve constrained minimization problems of a large range of CASM $\\varphi -$ divergences and deterministic transformations thereof (respectively constrained maximization problems of $(h,\\varphi )-$ entropies including Euclidean norms and diversity indices), by means of a newly developed dimension-free bare (pure) simulation method which is precise (i.e., converges in the limit) and which needs almost no assumptions (like convexity) on the set $\\mathbf {\\Omega }$ of constraints; in doing so, for the sake of brevity we concentrate on finding/computing the minimum divergences themselves rather than the corresponding minimizers (to achieve the latter, e.g.",
"dichotomous search could be used in a subsequent step, however); to derive a method of constructing new useful distances/divergences; to present numerous examples in order to illuminate our method and its potential for wide-spread applicability; as we go along, we also deliver many recent references for uses of the outcoming distances/divergences and entropies (covering in particular all the above-mentioned ones).",
"This agenda is achieved in the following way.",
"In the next Section , we briefly introduce the principal idea of our new bare-simulation optimization paradigm.",
"After manifesting the fundamentally employed class of CASM $\\varphi -$ divergences in Section , we give in Section the main cornerstones, construction principles and theorems, for deterministic as well as for statistical divergence-minimization problems; the maximization of generalized entropies is addressed, too.",
"Section deals with the concrete determination of the involved simulation-weights, as well as with the interrelated issue of creating associated CASM $\\varphi -$ divergences.",
"Some sampling-concerning details for the principal implementation of our bare-simulation optimization approach are worked out in Section .",
"The main proofs are presented in the appendices.",
"A first simulation-based algorithm in vein with the present proposal has been developed by Broniatowski [58], in the restricted setup of risk estimation for power divergences.",
"The present paper extends this considerably by considering general CASM $\\varphi -$ divergences and related entropies, and by dealing with corresponding general optimization problems, of both deterministic respectively stochastic type."
],
[
"A new minimization paradigm ",
"We concern with minimization problems of the following type, where $\\mathcal {M}$ is a topological space and $\\mathcal {T}$ is the Borel $\\sigma -$ field over a given base on $\\mathcal {M}$ ; e.g.",
"take $\\mathcal {M} = \\mathbb {R}^{K}$ to be the $K-$ dimensional Euclidean space equipped with the Borel $\\sigma -$ field $\\mathcal {T}$ .",
"Definition 1 A measurable function $\\Phi : \\mathcal {M} \\mapsto [0,\\infty ]$ and measurable set $\\Omega \\subset \\mathcal {M}$ i.e.",
"$\\Omega \\in \\mathcal {T}$ are called “bare-simulation minimizable” (BS-minimizable) respectively “bare-simulation maximizable” (BS-maximizable) if for $\\Phi (\\Omega ):=\\inf _{Q\\in \\Omega }\\left\\lbrace \\Phi (Q) \\right\\rbrace < \\infty \\qquad \\textrm {respectively} \\qquad \\Phi (\\Omega ):=\\sup _{Q\\in \\Omega }\\left\\lbrace \\Phi (Q) \\right\\rbrace < \\infty $ there exists a measurable function $G: [0,\\infty [ \\mapsto [0,\\infty [$ as well as a sequence $\\left((\\mathfrak {X}_{n},\\mathcal {A}_{n},\\mathbb {\\Pi }_{n})\\right)_{n \\in \\mathbb {N}}$ of probability spaces and on them a sequence $(\\xi _{n})_{n\\in \\mathbb {N}}$ in order to emphasize the dependence on $\\Phi $ , one should use the notations $(\\xi _{\\Phi ,n})_{n\\in \\mathbb {N}}$ , $\\mathbb {\\Pi }_{\\Phi ,n}$ , etc.",
"; this is avoided for the sake of a better readability.",
"of $\\mathcal {M}-$ valued random variables such that $G\\left(-\\lim _{n\\rightarrow \\infty }\\frac{1}{n}\\log \\mathbb {\\Pi }_{n}\\left[\\xi _{n}\\in \\Omega \\right] \\right) =\\inf _{Q\\in \\Omega }\\Phi (Q)=\\Phi (\\Omega )$ respectively $G\\left(-\\lim _{n\\rightarrow \\infty }\\frac{1}{n}\\log \\mathbb {\\Pi }_{n}\\left[\\xi _{n}\\in \\Omega \\right] \\right) =\\sup _{Q\\in \\Omega }\\Phi (Q) =\\Phi (\\Omega );$ in situations where $\\Phi $ is fixed and different $\\Omega $ ’s are considered, we say that “$\\Phi $ is bare-simulation minimizable (BS-minimizable) on $\\Omega $ ” respectively “$\\Phi $ is bare-simulation maximizable (BS-maximizable) on $\\Omega $ ”.",
"Remark 2 (a) Even in situations where one can uniformly choose $(\\mathfrak {X}_{n},\\mathcal {A}_{n},\\mathbb {\\Pi }_{n}) \\equiv (\\widetilde{\\mathfrak {X}},\\widetilde{\\mathcal {A}},\\widetilde{\\mathbb {\\Pi }})$ , the sequence $(\\xi _{n})_{n\\in \\mathbb {N}}$ may be not “independent and identically distributed” .",
"(b) Throughout the paper, we shall mainly deal with $BS-$ minimizability.",
"The basic idea/incentive of this new approach is: if a minimization problem (REF ) has no explicit solution and is computationally intractable (or unfeasible) but can be shown to be BS-minimizable with concretely constructable $(\\xi _{n})_{n\\in \\mathbb {N}}$ and $(\\mathbb {\\Pi }_{n})_{n\\in \\mathbb {N}}$ , then one can basically simulate the log-probabilities $-\\frac{1}{n} \\mathbb {\\Pi }_{n}\\left[\\xi _{n}\\in \\Omega \\right]$ for large enough integer $n \\in \\mathbb {N}$ to obtain an approximation of (REF ) without having to evaluate the corresponding (not necessarily unique) minimizer, where the latter is typically time-costly.",
"Finding minimizers can be performed through dichotomic search, once an algorithm leading to the minimal value of the divergence on adequate families of sets $\\Omega $ is at hand; for the sake of brevity, this is omitted in the current paper.",
"For reasons of transparency, we start to demonstrate this approach for the following important/prominent class of minimization problems with the following components: $\\mathcal {M}$ is the $K-$ dimensional Euclidean space $\\mathbb {R}^{K}$ , i.e.",
"$\\Omega $ is a set of vectors $Q$ with a number of $K$ components (where $K$ may be huge, as it is e.g.",
"the case in big data contexts); $\\Phi (\\cdot ) := \\Phi _{P}(\\cdot )$ depends on some known vector $P$ in $\\mathbb {R}^{K}$ with $K$ nonnegative components; $\\Phi _{P}(\\cdot )$ is a “directed distance” (divergence) from $P$ into $\\Omega $ in the sense of $\\Omega \\ni Q \\mapsto \\Phi _{P}(Q):= D(Q, P)$ , where $D(\\cdot ,\\cdot )$ has the the two properties “$D(Q, P)\\ge 0$ ” and “$D(Q, P) = 0$ if and only if $Q=P$ ”.",
"In particular, $D(\\cdot ,\\cdot )$ needs neither satisfy the symmetry $D(Q,P)= D(P,Q)$ nor the triangular inequality.",
"In other words, (1) together with (i)-(iii) constitutes a distance/divergence-minimization problem; we design a “universal” method to solve such problems by constructing appropriate (cf.",
"(REF )) sequences $(\\xi _{n})_{n\\in \\mathbb {N}}$ of $\\mathbb {R}^{K}-$ valued random variables, for all directed distances $D(\\cdot ,\\cdot )$ from a large subclass of the important omnipresent Csiszar-Ali-Silvey-Morimoto CASM $\\varphi -$ divergences (also called $f-$ divergences).",
"As a second demonstration for the workability of our paradigm, we “extend” (i) to (iii) to the setup where $P$ is a random element of the simplex $\\mathbb {S}^{K}$ of $K-$ component probability (frequency) vectors (cf.",
"the exact definition below) and $\\Omega \\subset \\mathbb {S}^{K}$ ; for the sub-setup where $P$ corresponds to a data-observation-dependent probability distribution and $\\Omega $ corresponds to a pregiven model in the sense of a family of probability distributions, the formula (REF ) amounts to the corresponding (discrete) “minimization-distance estimation problem (MDEP)” of choosing the best model element/member under given data an alternative naming also used in literature is to call $\\Omega $ a model class (rather than model), and each $P \\in \\Omega $ a model (rather than model element).",
"This is important/prominent in statistics and in the adjacent research fields of artificial intelligence and machine learning; the concrete solving of the MDEP is especially “hard” for nonparametric respectively semiparametric problems, and our BS method is predestined for such kind of contexts."
],
[
"Directed distances ",
"In detail, concerning the above-mentioned point (i) we take the $K-$ dimensional Euclidean space $\\mathcal {M}=\\mathbb {R}^{K}$ , denote from now on — as usual — its elements (i.e.",
"vectors) in boldface letters, and also employ the subsets $&& \\mathbb {R}_{\\ne 0}^{K} := \\lbrace \\mathbf {Q}:= (q_{1},\\ldots ,q_{K}) \\in \\mathbb {R}^K: \\,q_{i} \\ne 0 \\ \\text{for all} \\ i=1,\\ldots ,K \\rbrace , \\\\&& \\mathbb {R}_{> 0}^{K} := \\lbrace \\mathbf {Q}:= (q_{1},\\ldots ,q_{K}) \\in \\mathbb {R}^K: \\,q_{i} > 0 \\ \\text{for all} \\ i=1,\\ldots ,K \\rbrace , \\\\&& \\mathbb {R}_{\\ge 0}^{K} := \\lbrace \\mathbf {Q}:= (q_{1},\\ldots ,q_{K}) \\in \\mathbb {R}^K: \\, q_{i} \\ge 0 \\ \\text{for all} \\ i=1,\\ldots ,K \\rbrace , \\\\&& \\mathbb {R}_{\\le 0}^{K} := \\lbrace \\mathbf {Q}:= (q_{1},\\ldots ,q_{K}) \\in \\mathbb {R}^K: \\, q_{i} \\le 0 \\ \\text{for all} \\ i=1,\\ldots ,K \\rbrace , \\\\&& \\mathbb {S}^{K} := \\lbrace \\mathbf {Q} := (q_{1},\\ldots ,q_{K}) \\in \\mathbb {R}_{\\ge 0}^{K}:\\, \\sum _{i=1}^{K} q_{i} =1 \\rbrace \\quad \\text{(simplex of probability vectors)},\\\\&& \\mathbb {S}_{> 0}^{K} := \\lbrace \\mathbf {Q}:= (q_{1},\\ldots ,q_{K}) \\in \\mathbb {R}_{>0}^{K}: \\, \\sum _{i=1}^{K} q_{i} =1 \\rbrace .",
"$ Concerning the directed distances $D(\\cdot ,\\cdot )$ in (ii) and (iii), we deal with the important omnipresent Csiszar-Ali-Silvey-Morimoto $\\varphi -$ divergences (CASM $\\varphi -$ divergences) — adapted to our context: Definition 3 (a) Let the “divergence-generator” be a lower semicontinuous convex function $\\varphi : \\, ]-\\infty ,\\infty [ \\rightarrow [0,\\infty ]$ satisfying $\\varphi (1)=0$ .",
"Furthermore, for the effective domain $dom(\\varphi ) := \\lbrace t \\in \\mathbb {R} : \\varphi (t) < \\infty \\rbrace $ we assume that its interior $int(dom(\\varphi ))$ is non-empty which implies that $int(dom(\\varphi )) = ]a,b[$ for some $-\\infty \\le a < 1 < b \\le \\infty $ .",
"Moreover, we suppose that $\\varphi $ is strictly convex in a non-empty neighborhood $]t_{-}^{sc},t_{+}^{sc}[ \\subseteq ]a,b[$ of one ($t_{-}^{sc} < 1 < t_{+}^{sc}$ ).",
"Also, we set $\\varphi (a) := \\lim _{t \\downarrow a} \\varphi (t)$ and $\\varphi (b) := \\lim _{t \\uparrow b} \\varphi (t)$ (these limits always exist).",
"The class of all such functions $\\varphi $ will be denoted by $\\widetilde{\\Upsilon }(]a,b[)$ .",
"A frequent choice is e.g.",
"$]a,b[=]0,\\infty [$ or $]a,b[=]-\\infty ,\\infty [$ .",
"(b) For $\\varphi \\in \\widetilde{\\Upsilon }(]a,b[)$ , $\\mathbf {P} := (p_{1},\\ldots ,p_{K}) \\in \\mathbb {R}_{\\ge 0}^{K}$ and $\\mathbf {Q} := (q_{1},\\ldots ,q_{K}) \\in \\mathbf {\\Omega } \\subset \\mathbb {R}^{K}$ , we define the Csiszar-Ali-Silvey-Morimoto $\\varphi -$ divergence $\\Phi _{\\mathbf {P}} \\left(\\mathbf {Q}\\right) := D_{\\varphi }( \\mathbf {Q}, \\mathbf {P} ) :=\\sum _{k=1}^{K} p_{k} \\cdot \\varphi \\left( \\frac{q_{k}}{p_{k}}\\right) \\, \\ge 0.$ As usual, in (REF ) we employ the three conventions that $p\\cdot \\varphi \\left( \\frac{0}{p}\\right) = p \\cdot \\varphi (0) >0$ for all $p > 0$ , and $0 \\cdot \\varphi \\left( \\frac{q}{0}\\right) = q \\cdot \\lim _{x\\rightarrow \\infty } \\frac{\\varphi (x \\cdot \\textrm {sgn}(q))}{x\\cdot \\textrm {sgn}(q)} >0$ for $q \\ne 0$ (employing the sign of $q$ ), and $0 \\cdot \\varphi \\left( \\frac{0}{0}\\right) :=0$ .",
"Throughout the paper, we only consider constellations $(\\varphi ,\\mathbf {P},\\mathbf {\\Omega })$ for which the very mild condition $\\Phi _{\\mathbf {P}}(\\Omega ) := \\inf _{\\mathbf {Q}\\in \\mathbf {\\Omega }}D_{\\varphi }( \\mathbf {Q}, \\mathbf {P} ) \\ne \\infty \\ \\ \\footnote {i.e.",
"dom(\\varphi ) covers (at least) a non-empty part of \\lbrace 1\\rbrace \\cup \\mathcal {R}\\big (\\frac{\\mathbf {\\Omega }}{\\mathbf {P}}\\big ), where \\mathcal {R}\\big (\\frac{\\mathbf {\\Omega }}{\\mathbf {P}}\\big ) := \\big \\lbrace \\frac{q_{k}}{p_{k}}: \\, k \\in \\lbrace 1,\\ldots ,K\\rbrace , \\mathbf {Q}:=(q_{1},\\ldots ,q_{K}) \\in \\Omega \\big \\rbrace is the range of all possibleentry-ratios.",
"}\\nonumber $ holds.",
"For probability vectors ${P}$ and ${Q}$ in $\\mathbb {S}^{K}$ , the $\\varphi -$ divergences $D_{\\varphi }( {Q}, {P} )$ were introduced by Csiszar [94], Ali & Silvey [11] and Morimoto [266] (where the first two references even deal with more general probability distributions); for some comprehensive overviews — including statistical applications to goodness-of-fit testing and minimum distance estimation — the reader is referred to the insightful books of e.g.",
"Liese & Vajda [217], Read & Cressie [303], Vajda [371], Csiszar & Shields [99], Stummer [344], Pardo [282], Liese & Miescke [216], the survey articles of e.g.",
"Liese & Vajda [218], Vajda & van der Meulen [374], Reid & Williamson [304], Basseville [34], and the references therein.",
"Some exemplary recent studies and applications of CASM $\\varphi -$ divergences appear e.g.",
"in Qiao & Minematsu [298] for invariances in speech recognition, Nguyen et al.",
"[273] in connection with empirical risk optimization, Feixas et al.",
"[124] for various different image processing tasks, Luo et al.",
"[232] for video clip segmentation and key frame generation, Kißlinger & Stummer [189] for model preselection (structure detection) in the context of nonlinear recursive models with additional exogenous inputs, Mahboubi & Kochenderfer [241] within a context of traffic-pattern learning from flight tracks, Guo et al.",
"[145] for local contrastive descriptors in image classification through e.g.",
"regional color distributions, Csiszar & Breuer [100] for modelling generalized-ball type constraints in expectation minimization problems, Kißlinger & Stummer [191] for the detection of distributional changes in random data (streams and clouds), Noh et al.",
"[275] within a context of generative local metric learning for nearest neighbor classification, Yu et al.",
"[416] for adversarial learning within oil spill segmentation, Arslan [19] for automated active reconfiguration in mobile sensor networks, Sason [320] in connection with with data-processing and majorization inequalities, Ciftci et al.",
"[89] for the optimization of multienergy microgrids in energy infrastructure systems, and Stummer [345] for solving some new optimal transport (OT) problems which flexibilize some Wasserstein-distance based OTs.",
"For the setup of $D_{\\varphi }( \\mathbf {Q}, \\mathbf {P} )$ for vectors $\\mathbf {P}$ , $\\mathbf {Q}$ with non-negative components the reader is referred to e.g.",
"Stummer & Vajda [349] (who deal with even more general nonnegative measures and giving some statistical as well as information-theoretic applications) and Gietl & Reffel [137] (including applications to iterative proportional fitting).",
"The case of $\\varphi -$ divergences for vectors with arbitrary components can be extracted from e.g.",
"Broniatowski & Keziou [60] who actually deal with finite signed measures.",
"For a comprehensive technical treatment, see also Broniatowski & Stummer [64].",
"Clearly, from (REF ) it is obvious that in general $D_{\\varphi }( \\mathbf {Q}, \\mathbf {P} ) \\ne D_{\\varphi }(\\mathbf {P}, \\mathbf {Q})$ (non-symmetry).",
"Moreover, it is straightforward to deduce that $D_{\\varphi }(\\mathbf {Q}, \\mathbf {P}) = 0$ if and only if $\\mathbf {Q}=\\mathbf {P}$ (reflexivity).",
"Very prominent and important examples of CASM $\\varphi -$ divergences are the power divergences in the scaling of e.g.",
"Liese & Vajda [217] (in other scalings also called Rathie & Kannapan’s non-additive directed divergences of order $\\gamma $ [302], Cressie-Read divergences [93] [303], relative Tsallis entropies or Tsallis cross-entropies [364] (see also Shiino [331]), Amari’s alpha-divergences [12]) where basically (up to technicalities) $\\varphi (t): = \\varphi _{\\gamma }(t) :=\\frac{t^\\gamma -\\gamma \\cdot t+ \\gamma - 1}{\\gamma \\cdot (\\gamma -1)}$ ($\\gamma \\in \\mathbb {R}\\backslash \\lbrace 0,1\\rbrace $ ), $\\varphi (t): = \\varphi _{0}(t) :=\\lim _{\\gamma \\rightarrow 0} \\varphi _{\\gamma }(t) = - \\log t + t - 1$ , $\\varphi (t): = \\varphi _{1}(t) :=\\lim _{\\gamma \\rightarrow 1} \\varphi _{\\gamma }(t) = t \\cdot \\log t + 1 - t$ .",
"Usually, in the literature one takes $t \\in \\, ]0,\\infty [$ (and the limit as $t \\rightarrow 0$ ), except for the case $\\gamma =2$ where one handles $t \\in \\, ]-\\infty ,\\infty [$ ; for our purposes, we have to essentially extend these divergence generators $\\varphi _{\\gamma }$ for $t<0$ , which will be carried out and discussed in detail below, namely in (REF ), (REF ) (see also Table 1), as well as at several other places in this paper.",
"Notice that $D_{\\varphi _{1}}(\\mathbf {Q},\\mathbf {P})$ basically corresponds to the (extended form of) the omnipresent Kullback-Leibler information resp.",
"relative entropy.",
"Below, we shall also consider the minimization/maximization of important transforms of power divergences such as Renyi divergences/entropies, Sundaresan’s divergence, etc., which are frequently used in information theory and its applications to e.g.",
"artificial intelligence, machine learning, and physics.",
"Remark 4 Since, in general, our methods work also for non-probability vectors $\\mathbf {Q},\\mathbf {P}$ , we can also deal with — plain versions and transformations of — weighted $\\varphi -$ divergences of the form $D_{\\varphi }^{wei}( \\mathbf {Q}, \\mathbf {P} ) :=\\sum _{k=1}^{K} c_{k} \\cdot p_{k} \\cdot \\varphi \\left( \\frac{q_{k}}{p_{k}}\\right) \\, \\ge 0$ where $c_{k} >0$ ($k=1,\\ldots ,K$ ) are weights which not necessarily add up to one.",
"Indeed, by means of (REF ) we formally end up with $\\inf _{\\mathbf {Q}\\in \\mathbf {\\Omega }}D_{\\varphi }^{wei}( \\mathbf {Q}, \\mathbf {P} ) =\\inf _{\\mathbf {Q}^{wei}\\in \\mathbf {\\Omega }^{wei}}D_{\\varphi }( \\mathbf {Q}^{wei}, \\mathbf {P}^{wei} )\\nonumber $ where $\\mathbf {P}^{wei} := (c_{1} \\cdot p_{1},\\ldots ,c_{K} \\cdot p_{K})$ , $\\mathbf {Q}^{wei} := (c_{1} \\cdot q_{1},\\ldots ,c_{K} \\cdot q_{K})$ and $\\mathbf {\\Omega }^{wei}$ is the corresponding rescaling of $\\mathbf {\\Omega }$ .",
"Of course, all the necessary technicalities for the $\\varphi -$ divergences (see below) have to be adapted to the weighted $\\varphi -$ divergences; for the sake of brevity, this will not be discussed in detail.",
"Notice that $\\mathbf {P}^{wei}$ , $\\mathbf {Q}^{wei}$ are generally not probability vectors anymore, even if $\\mathbf {Q},\\mathbf {P}$ are probability vectors.",
"In the latter case, and under the assumption $\\sum _{k=1}^{K} c_{k} = 1$ , the divergences (REF ) coincide with the discrete versions of the ($\\mathbf {c}-$ )local divergences of Avlogiaris et al.",
"[22], [23] who also give absolutely-continuous versions and beyond (see also Broniatowski & Stummer [64] for an imbedding in a general divergence framework)."
],
[
"The cornerstone ",
"In this Section , we show that a number of deterministic optimization problems and and problems in statistical minimum risk based approaches pertaining to non- or semi-parametric contexts are BS-minimizable/amenable in the sense of Definition REF .",
"The below-mentioned Sections and will draw conclusions, proposing effective solutions.",
"For the construction of the desired sequence $(\\xi _{n})_{n\\in \\mathbb {N}}$ of $\\mathbb {R}^{K}-$ valued random variables (viz.",
"random vectors) and a corresponding probability distribution $\\mathbb {\\Pi }$ (which will not depend on $n$ ), we will assume that the divergence generator $\\varphi \\in \\widetilde{\\Upsilon } (]a,b[)$ has the additional property that it can be represented as $\\varphi (t)=\\sup _{z\\in \\mathbb {R}}\\Big ( z\\cdot t-\\log \\int _{\\mathbb {R}}e^{z \\cdot y}d\\mathbb {} (y)\\Big ), \\qquad t\\in \\mathbb {R},\\ \\ $ for some probability distribution/measure $\\mathbb {} $ on the real line such that the function $z\\mapsto MGF_{\\mathbb {} }(z):=\\int _{\\mathbb {R}}e^{z \\cdot y}d\\mathbb {} (y)$ is finite on some open interval containing zero in particular, this implies that $\\mathbb {} $ has light tails.. From this, we shall construct — basically in Section below — a sequence $(W_{n})_{n\\in \\mathbb {N}}$ of i.i.d.",
"copies of a random variable $W$ whose distribution (under $\\mathbb {\\Pi }$ ) is $\\mathbb {}$ (i.e.",
"$\\mathbb {\\Pi }[W \\in \\cdot \\, ] = \\mathbb {}[ \\, \\cdot \\,]$ ), from which the desired $(\\xi _{n})_{n\\in \\mathbb {N}}$ will be constructed.",
"Since $\\varphi $ attains its minimal value at the point 1, fit follows that $\\varphi ^{\\prime }(1)=0.$ By (REF ), for all $t$ in $int(dom(\\varphi ))$ , $\\varphi ^{\\prime }(t)$ is the reciprocal of $\\psi (z):=\\left(d/dz\\right) \\log MGF(z)$ at point $t$ , whence $\\psi (0)=1$ , which is to say that the expectation $E_\\mathbb {\\Pi }[W]=1$ .",
"The class of functions $\\varphi \\in \\widetilde{\\Upsilon } (]a,b[)$ satisfying the representability (REF ) will be denoted by $\\Upsilon (]a,b[)$ .",
"Remark 5 The condition $\\varphi \\in \\Upsilon (]a,b[)$ implies that $\\mathbb {}$ can not be a one-point distribution (Dirac mass) $\\delta _{y}$ at some point $y$ , since for such a situation one can straightforwardly deduce from (REF ) that $\\varphi (y)=0$ and $\\varphi (t)=\\infty $ for all $t \\ne y$ , which leads to $int(dom(\\varphi )) = \\emptyset $ and thus $\\varphi \\notin \\widetilde{\\Upsilon } (]a,b[)$ (in fact, our requirement $\\varphi (y)=0$ would narrow down to $y=1$ anyway).",
"Let us remark that the class $\\Upsilon (]a,b[)$ contains many divergence generators; this together with $\\varphi -$ construction principles will be developed at length in Section below.",
"Also, for the minimization problems considered in Section REF hereunder, we mostly modify the generator $\\varphi $ into $\\widetilde{c}\\cdot \\varphi $ for strictly positive scales $\\widetilde{c}$ .",
"At this point, for the sake of transparency, we only present a summarizing Table 1 of a selection of concrete examples which will be treated in detail below: Table: NO_CAPTION Table 1.",
"Selection of concrete examples treated in this paper, included with some important features (to be explained in the course of method build-up).",
"As already explained above, the representability (REF ) is the cornerstone for our approach, and opens the gate to make use of simulation methods in appropriate contexts.",
"We first develop this approach for deterministic minimization problems (cf.",
"Subsection REF ); thereafter, in Subsection REF , we “extend” this to the setup where $\\mathbf {P}$ is identified with an unknown probability vector in the simplex $\\mathbb {S}^{K}$ which is supposed to be the limit (as $n$ tends to infinity) of the empirical distribution pertaining to a collection of observations $\\mathbf {X}_{n}$ $:=\\left(X_{1},..,X_{n}\\right) $ ; in the classical statistical setting, this amounts to the estimation of $\\Phi _{\\mathbf {P}}\\left( \\Omega \\right) $ based on $\\mathbf {X}_{n}$ , leading to the important “minimization-distance estimation problem” in statistics, artificial intelligence and machine learning.",
"Finally, we end up this Section by shortly dealing with divergences between fuzzy sets (cf.",
"Subsection REF ) and basic belief assignments (cf.",
"Subsection REF )."
],
[
"Deterministic minimization problems",
"Problem 6 For pregiven $\\varphi \\in \\Upsilon (]a,b[)$ , positive-entries vector $\\mathbf {P}:=\\left( p_{1},..,p_{K}\\right) \\in \\mathbb {R}_{>0}^{K}$ (or from some subset thereof), and subset $\\mathbf {\\Omega } \\subset \\mathbb {R}^{K}$ (also denoted in boldface letters, with a slight abuse of notation) with regularity properties $cl(\\mathbf {\\Omega } )=cl\\left( int\\left( \\mathbf {\\Omega } \\right) \\right) , \\qquad int\\left( \\mathbf {\\Omega } \\right) \\ne \\emptyset ,$ find $\\Phi _{\\mathbf {P}}(\\mathbf {\\Omega }) := \\inf _{\\mathbf {Q}\\in \\mathbf {\\Omega } } D_{\\varphi }(\\mathbf {Q},\\mathbf {P}),$ provided that $\\inf _{\\mathbf {Q}\\in \\mathbf {\\Omega } } D_{\\varphi }(\\mathbf {Q},\\mathbf {P}) < \\infty .$ An immediate consequence thereof is — for pregiven $\\varphi \\in \\Upsilon (]a,b[)$ — the treatment of the more flexible problem $\\Phi _{\\mathbf {P},h}(\\mathbf {\\Omega }) := \\inf _{\\mathbf {Q}\\in \\mathbf {\\Omega } }h\\Big (D_{\\varphi }(\\mathbf {Q},\\mathbf {P}) \\Big )= h\\Big (\\inf _{\\mathbf {Q}\\in \\mathbf {\\Omega } } D_{\\varphi }(\\mathbf {Q},\\mathbf {P}) \\Big )$ for any continuous strictly increasing function $h: \\mathcal {H} \\, \\mapsto \\mathbb {R}$ with $\\mathcal {H} := [0,\\infty [$ and extension $h(\\infty ) := \\sup _{y \\in \\mathcal {H}}(y)$ (depending on the problem, a sufficiently large $\\mathcal {H} \\subset [0,\\infty [$ may be enough), respectively of $\\sup _{\\mathbf {Q}\\in \\mathbf {\\Omega } }h\\Big (D_{\\varphi }(\\mathbf {Q},\\mathbf {P}) \\Big )= h\\Big (\\inf _{\\mathbf {Q}\\in \\mathbf {\\Omega } } D_{\\varphi }(\\mathbf {Q},\\mathbf {P}) \\Big )$ for any continuous strictly decreasing function $h: \\mathcal {H} \\, \\mapsto \\mathbb {R}$ and extension $h(\\infty ) := \\inf _{y \\in \\mathcal {H}}(y)$ .",
"Remark 7 (a) By the basic properties of $\\varphi $ , it follows that for all $c>0$ the level sets $\\mathbf {\\varphi }_{c}:=\\left\\lbrace x\\in \\mathbb {R}:\\varphi (x)\\le c\\right\\rbrace $ are compact and so are the level sets of $\\mathbf {Q\\rightarrow }D_{\\varphi }(\\mathbf {Q},\\mathbf {P})$ $\\Gamma _{c}:=\\left\\lbrace \\mathbf {Q}\\in \\mathbb {R}^{K}:D_{\\varphi }(\\mathbf {Q},\\mathbf {P})\\le c\\right\\rbrace \\nonumber $ for all $c>0$ .",
"(b) When $\\mathbf {\\Omega }$ is not closed but merely satisfies (REF ), then the infimum in (REF ) may not be reached in $\\mathbf {\\Omega }$ although being finite; however we aim for finding the infimum/minimum in (REF ).",
"Finding the minimizers in (REF ) is another question.",
"For instance, this can be solved whenever, additionally, $\\mathbf {\\Omega }$ is a closed set which implies the existence of minimizers in $\\mathbf {\\Omega }$ .",
"In this case, and when the number of such minimizers is finite, those can be approximated by dichotomic search.",
"For the sake of brevity, this will not be addressed in this paper.",
"(c) The purpose of the condition (REF ) is to get rid of the $\\lim \\sup $ type and $\\lim \\inf $ type results in our below-mentioned “bare-simulation” approach and to obtain simple limit-statements which motivate our construction.",
"In practice, it is enough to verify $\\mathbf {\\Omega }\\subseteq cl\\left(int\\left( \\mathbf {\\Omega }\\right) \\right) $ , which is equivalent to the left-hand part of (REF ).",
"Clearly, any open set $\\mathbf {\\Omega }\\subset \\mathbb {R}^{K}$ satisfies the left-hand part of (REF ).",
"In the subsetup where $\\mathbf {\\Omega }$ is a closed convex set and $int(\\mathbf {\\Omega })\\ne \\emptyset $ , (REF ) is satisfied and the minimizer $\\mathbf {Q}_{min}\\in \\mathbf {\\Omega }$ in (REF ) is attained and even unique.",
"When $\\mathbf {\\Omega }$ is open and satisfies (REF ), then the infimum in (REF ) exists but is reached at some generalized projection of $\\mathbf {P}$ on $\\mathbf {\\Omega }$ (see Csiszar [97] for the definition in the Kullback-Leibler case of probability measures, which extends to any $\\varphi -$ divergence in our framework).",
"(d) Without further mentioning, the regularity condition (REF ) is supposed to hold in the full topology.",
"Of course, $int\\left( \\mathbb {S}^{K} \\right) = \\emptyset $ and thus, for the important probability-vector setup $\\mathbf {\\Omega } \\subset \\mathbb {S}^{K}$ the condition (REF ) is violated which requires extra refinements (cf.",
"Subsection REF below).",
"The same is needed for $\\mathbf {\\Omega } \\subset A \\cdot \\mathbb {S}^{K}$ for some $A \\ne 1$ , since obviously $int\\left( A \\cdot \\mathbb {S}^{K} \\right) = \\emptyset $ ; such a context appears naturally e.g.",
"in connection with mass transportation problems (cf.",
"(REF ) below) and with distributed energy management (cf.",
"the paragraph after (REF )).",
"(e) Often, $\\mathbf {\\Omega }$ will present a (discrete) model recall that an alternative naming also used in literature is to call $\\Omega $ a model class (rather than model), and each $P \\in \\Omega $ a model (rather than model element).",
"Since $\\mathbf {\\Omega }$ is assumed to have a non-void interior (cf.",
"the right-hand part of (REF )), this will exclude (parametric) models $\\mathbf {\\Omega }:=\\lbrace \\mathbf {Q}_{\\theta }:\\theta \\in \\Theta \\rbrace $ for some $\\Theta \\subset \\mathbb {R}^{d}$ ($d<K-1$ ), for which $\\theta \\mapsto \\mathbf {Q}_{\\theta }$ constitutes a curve/surface in $\\mathbb {R}^{K}$ ; however, for such a situation, one can employ standard minimization principles.",
"Our approach is predestined for non- or semiparametric models, instead.",
"For instance, (REF ) is valid for appropriate tubular neighborhoods of parametric models or for more general non-parametric settings such as e.g.",
"shape constraints.",
"Let us now present our new bare-simulation approach (cf.",
"Definition REF ) for solving the distance-optimization Problem REF : Step 1: equivalently rewrite (REF ) such that the vector $\\mathbf {P}$ “turns into” a probability vector $\\widetilde{{P}}$ .",
"More exactly, define $M_{\\mathbf {P}}:=\\sum _{i=1}^{K}p_{i}>0$ and let $\\widetilde{{P}}:=\\mathbf {P}/M_{\\mathbf {P}},$ and for $\\mathbf {Q}$ in $\\mathbf {\\Omega }$ , let $\\widetilde{\\mathbf {Q}}:=\\mathbf {Q}/M_{\\mathbf {P}}$ (notice that $\\widetilde{\\mathbf {Q}}$ may be a non-probability vector).",
"With the function $\\widetilde{\\varphi } \\in \\Upsilon (]a,b[)$ defined through $\\widetilde{\\varphi }:=M_{\\mathbf {P}} \\cdot \\varphi $ , we obtain $D_{\\varphi }(\\mathbf {Q},\\mathbf {P})=\\sum _{k=1}^{K}p_{k}\\cdot \\varphi \\left( \\frac{q_{k}}{p_{k}}\\right) =\\sum _{k=1}^{K}M_{\\mathbf {P}}\\cdot \\widetilde{p_{k}}\\cdot \\frac{\\varphi \\left( \\frac{M_{\\mathbf {P}}\\cdot \\widetilde{q_{k}}}{M_{\\mathbf {P}}\\cdot \\widetilde{p_{k}}}\\right) }{M_{\\mathbf {P}}}=D_{\\widetilde{\\varphi }}(\\widetilde{\\mathbf {Q}},\\widetilde{{P}}).$ It follows that the solution of (REF ) coincides with the one of the problem of finding $\\widetilde{\\Phi }_{\\widetilde{{P}}}(\\widetilde{\\mathbf {\\Omega }}) := \\inf _{\\widetilde{\\mathbf {Q}}\\in \\widetilde{\\mathbf {\\Omega }} }D_{\\widetilde{\\varphi } }(\\widetilde{\\mathbf {Q}},\\widetilde{{P}}),\\qquad \\textrm {with } \\widetilde{\\mathbf {\\Omega }}:=\\mathbf {\\Omega } /M_{\\mathbf {P}};$ as a side remark, one can see that in such a situation the rescaling of the divergence generator $\\varphi $ is important, which is one incentive that we incorporate multiples of $\\varphi $ below.",
"As an important special case we get for the choice $\\mathbf {P} := (1, \\ldots , 1) := \\mathbf {1}$ that the “prominent/frequent” separable nonlinear optimization problem of finding the optimal value $\\inf _{\\mathbf {Q} \\in \\mathbf {\\Omega } } \\sum _{k=1}^{K}\\varphi (q_{k})$ — with objective (e.g.",
"cost, energy, purpose) function $\\varphi \\in \\Upsilon (]a,b[)$ and constraint set (choice set, search space) $\\mathbf {\\Omega }$ — can be imbedded into our BS-approach by $\\inf _{\\mathbf {Q} \\in \\mathbf {\\Omega } } \\sum _{k=1}^{K}\\varphi (q_{k}) =\\inf _{\\mathbf {Q} \\in \\mathbf {\\Omega } } D_{\\varphi }(\\mathbf {Q},\\mathbf {1}) =\\inf _{\\widetilde{\\mathbf {Q}} \\in \\mathbf {\\Omega }/K }D_{K \\cdot \\varphi }(\\widetilde{\\mathbf {Q}},{P}^{unif}),$ with ${P}^{unif} := (\\frac{1}{K}, \\ldots , \\frac{1}{K})$ being the probability vector of frequencies of the uniform distribution on $\\lbrace 1, \\ldots , K\\rbrace $ .",
"Notice that with our new BS approach one may even tackle more general optimization problems of the form $\\inf _{\\breve{Q} \\in \\breve{\\Omega }} \\sum _{k=1}^{K} \\breve{\\varphi } (\\breve{q}_{k})$ where $\\breve{\\varphi }$ is some function which is finite and convex in a non-empty neighborhood (say, $]t_{0} + a- 1, t_{0} + b- 1[$ with $a < 1 < b$ ) of some point $t_{0} \\in \\mathbb {R}$ as well as strictly convex in a non-empty sub-neighborhood of $t_{0}$ ; for this, the function $\\varphi (t) := \\breve{\\varphi }(t+t_{0}-1) - \\breve{\\varphi }^{\\prime }(t_{0})\\cdot \\Big ((t+t_{0}-1) - t_{0} \\Big ) - \\breve{\\varphi }(t_{0}), \\qquad t \\in ]a,b[,\\nonumber $ (which corresponds to shifting the argument and adding an affine-linear function) should be a member of $\\Upsilon (]a,b[)$ , and from the corresponding minimization problem $& &\\inf _{\\widetilde{\\mathbf {Q}} \\in \\mathbf {\\Omega }/K }D_{K \\cdot \\varphi }(\\widetilde{\\mathbf {Q}},{P}^{unif}) =\\inf _{\\mathbf {Q} \\in \\mathbf {\\Omega } } \\sum _{k=1}^{K} \\varphi (q_{k})= \\inf _{\\mathbf {Q} \\in \\mathbf {\\Omega } } \\sum _{k=1}^{K}\\Big (\\breve{\\varphi }(q_{k}+t_{0}-1) - \\breve{\\varphi }^{\\prime }(t_{0})\\cdot ((q_{k}+t_{0}-1) -t_{0}) - \\breve{\\varphi }(t_{0})\\Big )\\nonumber \\\\& & = \\inf _{\\breve{\\mathbf {Q}} \\in \\mathbf {\\Omega } + t_{0} -1} \\sum _{k=1}^{K} \\ \\Big (\\breve{\\varphi }(\\breve{q}_{k}) - \\breve{\\varphi }^{\\prime }(t_{0})\\cdot (\\breve{q}_{k} -t_{0}) - \\breve{\\varphi }(t_{0})\\Big )\\nonumber \\\\& & = K \\cdot \\Big (t_{0} \\cdot \\breve{\\varphi }^{\\prime }(t_{0}) - \\breve{\\varphi }(t_{0})\\Big )+ \\inf _{\\breve{\\mathbf {Q}} \\in \\breve{\\mathbf {\\Omega }} } \\left(\\sum _{k=1}^{K} \\breve{\\varphi }(\\breve{q}_{k})- \\breve{\\varphi }^{\\prime }(t_{0}) \\cdot \\sum _{k=1}^{K} \\breve{q}_{k}\\right),\\qquad \\textrm {with } \\breve{\\mathbf {\\Omega }} := \\mathbf {\\Omega } + t_{0} -1,$ the term $\\inf _{\\breve{Q} \\in \\breve{\\Omega }} \\sum _{k=1}^{K} \\breve{\\varphi } (\\breve{q}_{k})$ should be recoverable; for instance, later on we shall employ constraints sets $\\breve{\\mathbf {\\Omega }}$ which particularly include $\\sum _{k=1}^{K} \\breve{q}_{k} = A > 0$ , whereas another possibility would be to use a $\\breve{\\varphi }$ which satisfies $\\breve{\\varphi }^{\\prime }(t_{0}) =0$ .",
"As a different line of flexibilization of (REF ), we can also deal with the problem $\\inf _{\\mathbf {Q} \\in \\mathbf {\\Omega } } h\\Big (\\sum _{k=1}^{K}\\varphi (q_{k})\\Big )$ through $\\inf _{\\mathbf {Q} \\in \\mathbf {\\Omega } }h\\Big (\\sum _{k=1}^{K}\\varphi (q_{k})\\Big ) =h\\Big ( \\inf _{\\widetilde{\\mathbf {Q}} \\in \\mathbf {\\Omega }/K }D_{K \\cdot \\varphi }(\\widetilde{\\mathbf {Q}},{P}^{unif}) \\Big )$ for any $\\varphi \\in \\Upsilon (]a,b[)$ and any continuous strictly increasing function $h: \\mathcal {H} \\, \\mapsto \\mathbb {R}$ with $\\mathcal {H} := [0,\\infty [$ (or a sufficiently large subset thereof), and with the problem $\\sup _{\\mathbf {Q} \\in \\mathbf {\\Omega } } h\\Big (\\sum _{k=1}^{K}\\varphi (q_{k})\\Big )$ through $\\sup _{\\mathbf {Q} \\in \\mathbf {\\Omega } }h\\Big (\\sum _{k=1}^{K}\\varphi (q_{k})\\Big ) =h\\Big ( \\inf _{\\widetilde{\\mathbf {Q}} \\in \\mathbf {\\Omega }/K }D_{K \\cdot \\varphi }(\\widetilde{\\mathbf {Q}},{P}^{unif}) \\Big )$ for any $\\varphi \\in \\Upsilon (]a,b[)$ and any continuous strictly decreasing function $h: \\mathcal {H} \\, \\mapsto \\mathbb {R}$ .",
"Combining (REF ) with (REF ) (respectively, with (REF )) leads to a further flexibilization.",
"Of course, we can also apply our BS method to the maximization $\\sup _{\\mathbf {Q} \\in \\mathbf {\\Omega } } h\\Big (\\sum _{k=1}^{K}\\zeta (q_{k})\\Big )$ for any concave function $\\zeta $ with $-\\zeta \\in \\Upsilon (]a,b[)$ and any continuous strictly increasing function $h: \\mathcal {H} \\, \\mapsto \\mathbb {R}$ with $\\mathcal {H} := - [\\infty ,0]$ (or a sufficiently large subset thereof), via $\\sup _{\\mathbf {Q} \\in \\mathbf {\\Omega } }h\\Big (\\sum _{k=1}^{K}\\zeta (q_{k})\\Big ) =h\\Big ( - \\inf _{\\widetilde{\\mathbf {Q}} \\in \\mathbf {\\Omega }/K }D_{- K \\cdot \\zeta }(\\widetilde{\\mathbf {Q}},{P}^{unif}) \\Big ) ,$ and to $\\inf _{\\mathbf {Q} \\in \\mathbf {\\Omega } } h\\Big (\\sum _{k=1}^{K}\\zeta (q_{k})\\Big )$ for any concave function $\\zeta $ with $-\\zeta \\in \\Upsilon (]a,b[)$ and any continuous strictly decreasing function $h: \\mathcal {H} \\, \\mapsto \\mathbb {R}$ , via $\\inf _{\\mathbf {Q} \\in \\mathbf {\\Omega } }h\\Big (\\sum _{k=1}^{K}\\zeta (q_{k})\\Big ) =h\\Big ( - \\inf _{\\widetilde{\\mathbf {Q}} \\in \\mathbf {\\Omega }/K }D_{- K \\cdot \\zeta }(\\widetilde{\\mathbf {Q}},{P}^{unif}) \\Big ) .$ Moreover, we can tackle $\\sup _{\\breve{Q} \\in \\breve{\\Omega }} \\sum _{k=1}^{K} \\breve{\\zeta } (\\breve{q}_{k})$ where $\\breve{\\zeta }$ is some function which is finite and concave in a non-empty neighborhood $]t_{0} + a- 1, t_{0} + b- 1[$ (with $a < 1 < b$ ) of some point $t_{0} \\in \\mathbb {R}$ as well as strictly concave in a non-empty sub-neighborhood of $t_{0}$ ; for this, the function $- \\zeta (t) := - \\breve{\\zeta }(t+t_{0}-1) + \\breve{\\zeta }^{\\prime }(t_{0})\\cdot \\Big ((t+t_{0}-1) - t_{0} \\Big ) + \\breve{\\zeta }(t_{0}), \\qquad t \\in ]a,b[,\\nonumber $ should be a member of $\\Upsilon (]a,b[)$ , and from the corresponding minimization problem $& &- \\inf _{\\widetilde{\\mathbf {Q}} \\in \\mathbf {\\Omega }/K }D_{- K \\cdot \\zeta }(\\widetilde{\\mathbf {Q}},{P}^{unif})= \\sup _{\\mathbf {Q} \\in \\mathbf {\\Omega } } \\sum _{k=1}^{K} \\zeta (q_{k})= \\sup _{\\mathbf {Q} \\in \\mathbf {\\Omega } } \\sum _{k=1}^{K}\\Big (\\breve{\\zeta }(q_{k}+t_{0}-1) - \\breve{\\zeta }^{\\prime }(t_{0})\\cdot ((q_{k}+t_{0}-1) -t_{0}) - \\breve{\\zeta }(t_{0})\\Big )\\nonumber \\\\& & = \\sup _{\\breve{\\mathbf {Q}} \\in \\mathbf {\\Omega } + t_{0} -1} \\sum _{k=1}^{K} \\ \\Big (\\breve{\\zeta }(\\breve{q}_{k}) - \\breve{\\zeta }^{\\prime }(t_{0})\\cdot (\\breve{q}_{k} -t_{0}) - \\breve{\\zeta }(t_{0})\\Big )\\nonumber \\\\& & = K \\cdot \\Big (t_{0} \\cdot \\breve{\\zeta }^{\\prime }(t_{0}) - \\breve{\\zeta }(t_{0})\\Big )+ \\sup _{\\breve{\\mathbf {Q}} \\in \\breve{\\mathbf {\\Omega }} } \\left(\\sum _{k=1}^{K} \\breve{\\zeta }(\\breve{q}_{k})- \\breve{\\zeta }^{\\prime }(t_{0}) \\cdot \\sum _{k=1}^{K} \\breve{q}_{k}\\right),\\qquad \\textrm {with } \\breve{\\mathbf {\\Omega }} := \\mathbf {\\Omega } + t_{0} -1,$ the term $\\sup _{\\breve{Q} \\in \\breve{\\Omega }} \\sum _{k=1}^{K} \\breve{\\zeta } (\\breve{q}_{k})$ should be recoverable; the left-hand side of (REF ) corresponds to the special case $h(x) := x$ of the BS-minimizable (REF ).",
"A combination of (REF ) with (REF ) (respectively, with (REF )) leads to a further flexibilization.",
"Remark 8 (a) Since $\\mathbf {1}$ can be seen e.g.",
"as a reference vector with (normalized) equal components, the quantity $\\inf _{\\mathbf {Q} \\in \\mathbf {\\Omega } } D_{\\varphi }(\\mathbf {Q},\\mathbf {1})$ in (REF ) can be interpreted as an “index/degree of (in)equality of the set $\\mathbf {\\Omega }$ ”, respectively as an “index/degree of diversity of the set $\\mathbf {\\Omega }$ ”.",
"(b) The quantity $\\sum _{k=1}^{K}\\varphi (q_{k})$ in (REF ) can be interpreted as (non-probability extension of an) $\\varphi -$ entropy in the sense of Burbea & Rao [68] (see also Csiszar [95], Ben-Bassat [38], Ben-Tal & Teboulle [40], Kesavan & Kapur [187], Dacunha-Castelle & Gamboa [102], Teboulle & Vajda [357], Gamboa & Gassiat [132], Vajda & Zvarova [376]); for applications to scalar quantization for lossy coding of information sources see e.g.",
"György & Linder [149].",
"More generally, the quantity $h\\Big (\\sum _{k=1}^{K}\\varphi (q_{k})\\Big )$ in (REF ) can be seen as (non-probability extension of an) $(h,\\varphi )-$ entropy in the sense of Salicru et al.",
"[314] (see also e.g.",
"Pardo [282], Vajda & Vasek [375], as well as e.g.",
"Chen et al.",
"[78] for uses as supervised adaption criterion within stochastic information gradient algorithms and Ren et al.",
"[308] for applications to tracking in networked control systems).",
"Important special cases will be discussed in more detail, below.",
"Returning to the original distance-minimizing Problem REF , after the first step (REF ) and (REF ), we proceed as follows: Step 2: construct an appropriate sequence $(\\xi _{n})_{n\\in \\mathbb {N}}$ of $\\mathbb {R}^{K}-$ valued random variables/random vectors (cf.",
"(REF ) in Definition REF ): The following condition transposes the minimization problem (REF ) into a BS minimizable/amenable problem in the sense of Definition REF and it is required in order that Problem (REF ) is equivalent to Problem (REF ).",
"The connection of this condition with (REF ) will be discussed in Proposition REF and its surroundings, see Section .",
"Condition 9 With $M_{\\mathbf {P}} =\\sum _{i=1}^{K}p_{i}>0$ , the divergence generator $\\varphi $ in (REF ) (cf.",
"also (REF )) satisfies $\\widetilde{\\varphi } := M_{\\mathbf {P}} \\cdot \\varphi \\in \\Upsilon (]a,b[)$ , i.e.",
"$\\widetilde{\\varphi } \\in \\widetilde{\\Upsilon }(]a,b[)$ (which is equivalent to $\\varphi \\in \\widetilde{\\Upsilon }(]a,b[)$ ) and there holds the representation $\\widetilde{\\varphi }(t) =\\sup _{z \\in \\mathbb {R}} \\left( z\\cdot t - \\log \\int _{\\mathbb {R}} e^{zy} d\\widetilde{\\mathbb {}}(y) \\right),\\qquad t \\in \\mathbb {R},$ for some probability measure $\\widetilde{\\mathbb {}}$ on the real line such that the function $z \\mapsto MGF_{\\widetilde{\\mathbb {}}}(z) := \\int _{\\mathbb {R}} e^{zy} d\\widetilde{\\mathbb {}}(y)$ is finite on some open interval containing zero in particular, this implies that $\\int _{\\mathbb {R}} y d\\widetilde{\\mathbb {}}(y) =1$ (cf.",
"(G11i) below) and that $\\widetilde{\\mathbb {}}$ has light tails..",
"In the following, let us explain the above-mentioned Step 2 in detail: for any $n \\in \\mathbb {N}$ and any $k \\in \\left\\lbrace 1, \\ldots ,K\\right\\rbrace $ , let $n_{k}:=\\lfloor n \\cdot \\widetilde{p}_{k}\\rfloor $ where $\\lfloor x \\rfloor $ denotes the integer part of $x$ .",
"We assume $\\mathbf {P} \\in \\mathbb {R}_{> 0}^{K}$ , and since thus none of the $\\widetilde{p}_{k}$ 's is zero, one has $\\lim _{n\\rightarrow \\infty } \\frac{n_{k}}{n} = \\widetilde{p}_{k}.$ Moreover, we assume that $n \\in \\mathbb {N}$ is large enough, namely $n \\ge \\max _{k \\in \\lbrace 1, \\ldots , K\\rbrace } \\frac{1}{\\widetilde{p}_{k}}$ , and decompose the set $\\lbrace 1, \\ldots , n\\rbrace $ of all integers from 1 to $n$ into the following disjoint blocks: $I_{1}^{(n)}:=\\left\\lbrace 1,\\ldots ,n_{1}\\right\\rbrace $ , $I_{2}^{(n)}:=\\left\\lbrace n_{1}+1,\\ldots ,n_{1}+n_{2}\\right\\rbrace $ , and so on until the last block $I_{K}^{(n)} := \\lbrace \\sum _{k=1}^{K-1} n_{k} + 1, \\ldots , n \\rbrace $ which therefore contains all integers from $n_{1}+ \\ldots +n_{K-1}+1$ to $n$ .",
"Clearly, $I_{k}^{(n)}$ has $n_{k} \\ge 1$ elements (i.e.",
"$card(I_{k}^{(n)}) = n_{k}$ where $card(A)$ denotes the number of elements in a set $A$ ) for all $k \\in \\lbrace 1, \\ldots , K-1\\rbrace $ , and the last block $I_{K}^{(n)}$ has $n- \\sum _{k=1}^{K-1} n_{k} \\ge 1$ elements which anyhow satisfies $\\lim _{n\\rightarrow \\infty } card(I_{K}^{(n)})/n=\\widetilde{p}_{K}$ if all $\\widetilde{p}_{k}$ ($k=1,\\ldots ,K$ ) are rational numbers in $]0,1[$ with $\\sum _{k=1}^{K} \\widetilde{p}_{k} =1$ and $N$ is the (always existing) smallest integer such that all $N \\cdot \\widetilde{p}_{k}$ ($k=1,\\ldots ,K$ ) are integers (i.e.",
"$\\in \\mathbb {N}$ ), then for any multiple $n= \\ell \\cdot N$ ($\\ell \\in \\mathbb {N}$ ) one gets that all $n_{k} = n \\cdot \\widetilde{p}_{k}$ are integers and that $card(I_{K}^{(n)}) = n_{K}$ ..",
"Furthermore, consider a vector $\\mathbf {\\widetilde{W}}:=\\left( \\widetilde{W}_{1},\\ldots ,\\widetilde{W}_{n}\\right) $ where the $\\widetilde{W}_{i}$ 's are i.i.d.",
"copies of the random variable $\\widetilde{W}$ whose distribution is associated with the divergence-generator $\\widetilde{\\varphi }:=M_{\\textbf {P}} \\cdot \\varphi $ through (REF ), in the sense that $\\mathbb {\\Pi }[\\widetilde{W}\\in \\cdot \\,]=\\widetilde{\\mathbb {}}[\\,\\cdot \\,]$ .",
"We group the $\\widetilde{W}_{i}$ 's according the above-mentioned blocks and sum them up blockwise, in order to build the following $K-$ component random vector $\\xi _{n}^{\\mathbf {\\widetilde{W}}}:=\\Big (\\frac{1}{n}\\sum _{i\\in I_{1}^{(n)}}\\widetilde{W}_{i},\\ldots ,\\frac{1}{n}\\sum _{i\\in I_{K}^{(n)}}\\widetilde{W}_{i}\\Big );$ notice that the signs of its components may be negative, depending on the nature of the $\\widetilde{W}_{i}$ 's; moreover, the expectation of its $k-$ th component converges to $\\widetilde{p}_{k}$ as $n$ tends to infinity (since the expectation of $\\widetilde{W}_{1} $ is 1), whereas the $n-$ fold of the corresponding variance converges to $\\widetilde{p}_{k}$ times the variance of $\\widetilde{W}_{1}$ .",
"For such a context, we obtain the following assertion on BS-minimizability: Theorem 10 Let $\\mathbf {P} \\in \\mathbb {R}_{> 0}^{K}$ , $M_{\\mathbf {P}}:=\\sum _{i=1}^{K}p_{i}>0$ , and suppose that the divergence generator $\\varphi $ satisfies the Condition REF above, with $\\widetilde{\\mathbb {}}$ (cf.",
"(REF )).",
"Additionally, let $\\widetilde{W}:=(\\widetilde{W}_{i})_{i\\in \\mathbb {N}}$ be a sequence of random variables where the $\\widetilde{W}_{i}$ 's are i.i.d.",
"copies of the random variable $\\widetilde{W}$ whose distribution is $\\mathbb {\\Pi }[\\widetilde{W}\\in \\cdot \\,]=\\widetilde{\\mathbb {}}[\\,\\cdot \\,]$ and thus, $E_{\\mathbb {\\Pi }}[\\widetilde{W}_{i}]=1$.",
"Then, in terms of the random vectors $\\xi _{n}^{\\mathbf {\\widetilde{W}}}=\\Big (\\frac{1}{n}\\sum _{i\\in I_{1}^{(n)}}\\widetilde{W}_{i},\\ldots ,\\frac{1}{n}\\sum _{i\\in I_{K}^{(n)}}\\widetilde{W}_{i}\\Big )\\hspace{56.9055pt} \\text{(cf.",
"(\\ref {Xi_n^W vector}))}\\nonumber $ there holds $-\\lim _{n\\rightarrow \\infty }\\frac{1}{n}\\log \\,\\mathbb {\\Pi }\\left[\\xi _{n}^{\\mathbf {\\widetilde{W}}}\\in \\mathbf {\\Omega } /M_{\\mathbf {P}}\\right]=\\inf _{Q\\in \\mathbf {\\Omega }}D_{\\varphi }(\\mathbf {Q},\\mathbf {P}\\ )$ for any $\\mathbf {\\Omega }\\subset \\mathbb {R}^{K}$ with regularity properties (REF ) and finiteness property (REF ).",
"In particular, for each $\\mathbf {P} \\in \\mathbb {R}_{> 0}^{K}$ the function $\\Phi _{\\mathbf {P}} \\left( \\cdot \\right) := D_{\\varphi }( \\cdot , \\mathbf {P} )$ (cf.",
"(REF )) is bare-simulation minimizable (BS-minimizable, cf.",
"(REF ))) on any such $\\mathbf {\\Omega }\\subset \\mathbb {R}^{K}$ .",
"The proof of Theorem REF will be given in Appendix A.",
"Remark 11 (i) Whenever $int(\\mathbf {\\Omega }) \\ne \\emptyset $ , it clearly holds that $\\liminf _{n\\rightarrow \\infty }\\frac{1}{n}\\log \\,\\mathbb {\\Pi }\\left[\\xi _{n}^{\\mathbf {\\widetilde{W}}}\\in \\mathbf {\\Omega } /M_{\\mathbf {P}}\\right] > 0$ ; see the proof of Theorem REF .",
"Hence, the limit in (REF ) exists and is finite when $\\mathbf {\\Omega }$ satisfies (REF ).",
"(ii) For some contexts, one can explicitly give the distribution of each of the independent (non-deterministic parts of the) components $\\Big (\\sum _{i\\in I_{k}^{(n)}}\\widetilde{W}_{i}\\Big )_{k=1,\\ldots ,K}$ of the vector $\\xi _{n}^{\\mathbf {\\widetilde{W}}}$ ; this will ease the corresponding concrete simulations.",
"For instance, we shall give those in the Examples REF , REF , REF , REF and REF in Section below.",
"(iii) Let us emphasize that we have assumed $\\mathbf {P} \\in \\mathbb {R}_{> 0}^{K}$ in Theorem REF which excludes $\\mathbf {P}$ from having zero components.",
"However, in cases where $\\lim _{x\\rightarrow \\infty }\\left|\\frac{\\varphi \\left(x \\cdot sgn(q)\\right) }{x \\cdot sgn(q)}\\right|=+\\infty $ for $q\\ne 0$ then if $p_{k_{0}}=0$ for some $k_{0\\text{ }}$ it follows that $q_{k_{0}}=0$ , which proves that $\\mathbf {P}\\in \\mathbb {R}_{>0}^{K}$ imposes no restriction in Theorem 9, since the projection of $\\mathbf {P}$ in $\\mathbf {\\Omega }$ then belongs to the subspace of $\\mathbb {R}^{K}$ generated by the non-null components of $\\mathbf {P}$ ; such a situation appears e.g.",
"for power divergence generators $\\varphi _{\\gamma }$ with $\\gamma > 2$ .",
"So there is no loss of generality assuming $\\mathbf {P}\\in \\mathbb {R}_{>0}^{K}$ in this case.",
"As examples for the applicability of Theorem REF , one can e.g.",
"combine each of the divergence generators $\\varphi $ of Table 1 (except for the 9th row) with any of the optimization problems (REF ), (REF ), (REF ), (REF ), (REF ), (REF ); the needed distributions $\\mathbb {\\Pi }[\\widetilde{W}\\in \\cdot \\,]=\\widetilde{\\mathbb {}}[\\,\\cdot \\,]$ correspond to the entry in the second last column with the choice $\\widetilde{c} \\cdot M_{\\mathbf {P}}$ instead of $\\widetilde{c}$ .",
"By taking $\\zeta := - \\varphi $ instead, one can solve the corresponding problems (REF ) and (REF ).",
"Returning to the general context, the limit statement (REF ) provides the principle for the approximation of the solution of Problem REF .",
"Indeed, by replacing the left-hand side in (REF ) by its finite counterpart, we deduce for given large $n$ $- \\frac{1}{n}\\log \\mathbb {\\Pi } \\left[ \\xi _{n}^{\\mathbf {\\widetilde{W}}}\\in \\mathbf {\\Omega }/M_{\\mathbf {P}} \\right]\\approx \\inf _{Q\\in \\mathbf {\\Omega } }D_{\\varphi }(\\mathbf {Q},\\mathbf {P});$ it remains to estimate the left-hand side of (REF ).",
"The latter can be performed either by a naive estimator of the frequency of those replications of $\\xi _{n,\\mathbf {\\widetilde{x}}}^{\\mathbf {\\widetilde{W}}}$ which hit $\\Omega /M_{\\mathbf {P}}$ , or more efficiently by some improved estimator; this will be discussed in detail in Section below.",
"Remark 12 According to (REF ) of Theorem 9 as well as (REF ), we can principally tackle the (approximative) computation of the minimum value $\\inf _{Q\\in \\mathbf {\\Omega }}D_{\\varphi }(\\mathbf {Q},\\mathbf {P}\\ ) =\\inf _{Q\\in \\mathbf {\\Omega }}\\sum _{k=1}^{K}p_{k}\\cdot \\varphi \\left( \\frac{q_{k}}{p_{k}}\\right)\\nonumber $ and in particular of $\\inf _{\\mathbf {Q} \\in \\mathbf {\\Omega } } \\sum _{k=1}^{K}\\varphi (q_{k}) =\\inf _{\\mathbf {Q} \\in \\mathbf {\\Omega } } D_{\\varphi }(\\mathbf {Q},\\mathbf {1})\\qquad \\textrm {(cf.",
"(\\ref {min Pb one}))}\\nonumber $ by basically only employing a fast and accurate — pseudo, true, natural, quantum — random number generatorsee e.g.",
"Tucci [366], Teh et al.",
"[358], Aghamohammadi & Crutchfield [6], Herrero-Collantes & Garcia-Escartin [159], Balygin et al.",
"[31], Dang et al.",
"[103], Gong et al.",
"[138], Chandrasekaran et al.",
"[77], Drahi et al.",
"[117], Kollmitzer et al.",
"[194], Liu et al.",
"[228], Fischer & Gauthier [126], Kim et al.",
"[188], Stoller & Campbell [342], provided that the constraint set $\\mathbf {\\Omega }$ satisfies the mild assumptions (REF ) and (REF ).",
"Notice that (REF ) also covers (e.g.",
"high-dimensional) constraint sets $\\mathbf {\\Omega }$ which are non-convex and even highly disconnected, and for which other minimization methods (e.g.",
"pure enumeration, gradient or steepest descent methods, etc.",
"a detailed discussion and comparisons are beyond the scope of this paper, given its current length ) may be problematic or intractable.",
"For instance, (REF ) covers kind of “$K-$ dimensional (not necessarily regular) polka dot (leopard skin) pattern type” relaxations $\\mathbf {\\Omega } := \\dot{\\bigcup }_{i=1}^{N} \\mathcal {U}_{i}(Q_{i}^{dis})$ of finite discrete constraint sets $\\mathbf {\\Omega }^{dis} := \\lbrace Q_{1}^{dis}, \\ldots , Q_{N}^{dis}\\rbrace $ of high cardinality $N$ (e.g.",
"being exponential or factorial in a large $K$ ), where each $K-$ dimensional vector $Q_{i}^{dis}$ (e.g.",
"having pure integer components only) is surrounded by some small (in particular, non-overlapping/disjoint) neighborhood $\\mathcal {U}_{i}(Q_{i}^{dis})$ ; in such a context, e.g.",
"$\\inf _{\\mathbf {Q} \\in \\mathbf {\\Omega } } \\sum _{k=1}^{K}\\varphi (q_{k})$ can be regarded as a “BS-tractable” relaxation of the nonlinear discrete (e.g.",
"integer, combinatorial see e.g.",
"Schrijver [324], Bertsimas & Weismantel [45], Chen et al.",
"[83], Onn [279], Korte & Vygen [195], Wolsey [393] for comprehensive books on discrete, integer and combinatorial programming and their vast applications ) optimization program $\\inf _{\\mathbf {Q} \\in \\mathbf {\\Omega }^{dis} } \\sum _{k=1}^{K}\\varphi (q_{k})$ ."
],
[
"Minimum distance/risk estimation\n",
"In statistics of discrete data — and in the adjacent research fields of information theory, artificial intelligence and machine learning — one often encounters the following minimum distance estimation (MDE) problem which is often also named as estimation of the empirical risk: for index $i\\in \\mathbb {N}$ , let the generation of the $i-$ th (uncertainty-prone) data point be represented by the random variable $X_{i}$ which takes values in the discrete set $\\mathcal {Y}:=\\left\\lbrace d_{1},\\cdots ,d_{K}\\right\\rbrace $ of $K$ distinct values “of any kind”.",
"It is assumed that there exists a probability measure $\\mathbb {P}[\\cdot \\,]$ on $\\mathcal {Y}$ which is the a.s. limit of the empirical measures $\\mathbb {P}_{n}^{emp}$ defined by the collection of collected $\\left( X_{1},..,X_{n}\\right) $ as $n$ tends to infinity, in formula $\\lim _{n\\rightarrow \\infty }\\mathbb {P}_{n}^{emp}:=\\lim _{n\\rightarrow \\infty }\\frac{1}{n}\\sum _{i=1}^{n}\\delta _{X_{i}}=\\mathbb {P} \\qquad \\text{a.s.}$ where $\\delta _{y}$ denotes the one-point distribution (Dirac mass) at point $y$ notice that $\\mathbb {P}_{n}^{emp}$ a probability measure on the data space $\\mathcal {Y}$ , which is random due to its dependence on the $X_{i}$ ’s.",
"We will assume that none of the entries of $\\mathbb {P}$ bears zero mass so that $\\mathbb {P}$ is identified with a point in the interior of $\\mathbb {S}^{K}$ (see below).",
"The underlying probability space (say, $(\\mathfrak {X},\\mathcal {A},\\mathbb {\\Pi })$ ) where the above a.s. convergence holds, pertains to the random generation of the sequence $\\left( X_{n}\\right)_{n\\ge 1}$ , of which we do not need to know but for (REF ).",
"Examples include the i.i.d.",
"case (where the $X_{i}$ ’s are independent and have common distribution $\\mathbb {P}$ ), ergodic Markov chains on $\\mathcal {Y}$ with stationary distribution $\\mathbb {P}$ , more globally autoregressive chains with stationary measure $\\mathbb {P}$ , etc.",
"Let us briefly discuss our assumption (REF ) (resp.",
"its vector form (REF ) below) on the limit behavior of the empirical distribution of the observed sample $\\mathbf {X}_{n}:=\\left(X_{1},..,X_{n}\\right)$ as $n$ tends to infinity.",
"In the “basic” statistical context, the sample $\\mathbf {X}_{n}$ consists of i.i.d.",
"replications of a generic random variable $X$ with probability distribution ${P}$ .",
"However, our approach captures many other sampling schemes, where the distribution ${P}$ is defined implicitly through (REF ) for which we aim at some estimate of $\\Phi _{{P}}\\left(\\mathbb {\\Omega }\\right)$ of a family $\\mathbb {\\Omega }$ of probability distributions on $\\mathcal {Y}$ .",
"Sometimes the sequence of samples $\\left( \\mathbf {X}_{n}\\right) _{n\\ge 1}$ may stem from a triangular array so that $\\mathbf {X}_{n}=\\left(X_{1,n},..,X_{k_{n},n}\\right) $ with $k_{n}\\rightarrow \\infty $ and statement (REF ) is substituted by $\\lim _{n\\rightarrow \\infty }\\frac{1}{k_{n}}\\sum _{i=1}^{k_{n}}\\delta _{X_{i,n}} ={P} \\text{ \\ a.s.}$ which does not alter the results of this paper by any means.",
"given a model $\\mathbb {\\Omega }$ , i.e.",
"a family $\\mathbb {\\Omega }$ of probability distributions on $\\mathcal {Y}$ each of which serves as a potential description of the underlying (unknown) data-generating mechanism $\\mathbb {P}$ , one would like to find $\\Phi _{\\mathbb {P}}(\\mathbb {\\Omega }) := \\inf _{\\mathbb {Q}\\in \\mathbb {\\Omega }} D_{\\varphi }( \\mathbb {Q}, \\mathbb {P} )$ which quantifies the adequacy of the model $\\mathbb {\\Omega }$ for modeling $\\mathbb {P}$ , via the minimal distance/dissimilarity of $\\mathbb {\\Omega }$ to $\\mathbb {P}$ ; a lower $\\Phi _{\\mathbb {P}}-$ value means a better adequacy (in the sense of a lower departure between the model and the truth, cf.",
"Lindsay [222], Lindsay et al.",
"[223], Markatou & Sofikitou [248], Markatou & Chen [249]).",
"Hence, especially in the context of model selection within complex big-data contexts, for the search of appropriate models $\\mathbb {\\Omega }$ and model elements/members therein, the (fast and efficient) computation of $\\Phi _{\\mathbb {P}}(\\mathbb {\\Omega })$ constitutes a decisive first step, since if the latter is “too large” (respectively “much larger than” $\\Phi _{\\mathbb {P}}(\\overline{\\mathbb {\\Omega }})$ for some competing model $\\overline{\\mathbb {\\Omega }}$ ), then the model $\\mathbb {\\Omega }$ is “not adequate enough” (respectively “much less adequate than” $\\overline{\\mathbb {\\Omega }}$ ); in such a situation, the effort of computing the (not necessarily unique) best model element/member $\\arg \\inf _{\\mathbb {Q}\\in \\mathbb {\\Omega }} D_{\\varphi }( \\mathbb {Q}, \\mathbb {P} )$ within the model $\\mathbb {\\Omega }$ is “not very useful” and is thus a “waste of computational time”.",
"Because of such considerations, we concentrate ourselves to finding the infimum (REF ) rather than finding the corresponding minimizer(s).",
"Variants of (REF ) are of interest, too.",
"Since $int(\\mathbb {\\Omega })$ is supposed to be a non-empty set in the space of probability distribution on $\\mathcal {Y}$ , the present procedure is fitted for semi-parametric models $\\mathbb {\\Omega }$ , e.g.",
"such as defined through moment conditions (as extensions of the Empirical Likelihood paradigm, see e.g.",
"Broniatowski & Keziou [62]), or through L-moment conditions (i.e.",
"moment conditions pertaining to quantile measures, see Broniatowski & Decurninge [59]), or even more involved non-parametric models where the geometry of $\\mathbb {\\Omega }$ does not allow for ad-hoc procedures.",
"The measurement or the estimation of $\\Phi _{\\mathbb {P}}(\\mathbb {\\Omega })$ is a tool for the choice of pertinent putative models $\\mathbb {\\Omega }$ among a class of specifications.",
"The case when $\\Phi _{\\mathbb {P}}(\\mathbb {\\Omega })>0$ is interesting in its own, since it is quite common in engineering modelling to argue in favor of misspecified models (or (non-void) neighborhoods of such models for sake of robustness issues), due to quest for conservatism; the choice between them is a widely open field e.g.",
"in the practice of reliability.",
"This also opens the question of the choice of the divergence generator $\\varphi $ ; although this will not be discussed in this paper, as a motivating running example the reader may keep in mind the generator $\\varphi _{2}(x):=(x-1)^{2}/2$ which induces the divergence $D_{\\varphi _{2}}(\\mathbb {Q},\\mathbb {P})$ (see (REF ) below for details) which quantifies the expected square relative error when substituting the true distribution $\\mathbb {P}$ by the model $\\mathbb {Q}$ .",
"As examples of sets $\\mathbb {\\Omega }$ of probability distributions on $\\mathcal {Y}$ which obey (through their $K-$ vector of corresponding probability masses/frequencies) the global assumptions (REF ), one can consider semi-parametric models defined by moment conditions or defined through L-moment constraints (hence on the quantile functions), as well as more involved ones, for which no closed form of the divergence with respect to any probability distribution is available.",
"In the context of model selection, the choice of $\\mathbb {\\Omega }$ may be dictated by various considerations, and misspecification may be assumed as a requisite, for example for conservatism in reliability design.",
"An estimate of $\\Phi _{\\mathbb {P}}(\\mathbb {\\Omega })$ can be used as a statistics for some test of fit, and indeed the likelihood ratio test adapted to some semi-parametric models has been generalized to the divergence setting (see Broniatowski & Keziou [62]).",
"The statement of the limit distributions of our estimate, under the model and under misspecification, is postponed to future work.",
"In the following, we compute/approximate (REF ) — and some variants thereof — by our bare simulation (BS) method, by “mimicking” the deterministic minimization problem (REF ) respectively (REF ).",
"Let us first remark that, as usual, each probability distribution (probability measure) $\\mathbb {P}$ on $\\mathcal {Y}=\\left\\lbrace d_{1},\\ldots ,d_{K}\\right\\rbrace $ can be uniquely identified with the (row) vector ${P} := (p_{1}, \\ldots , p_{K}) \\in \\mathbb {S}^{K}$ of the corresponding probability masses (frequencies) $p_{k} = \\mathbb {P}[\\lbrace d_{k} \\rbrace ]$ via $\\mathbb {P}[A] = \\sum _{k=1}^{K} p_{k} \\cdot {1}_A(d_{k}) $ for each $A \\subset \\mathcal {Y}$ , where ${1}_A(\\cdot )$ denotes the indicator function on the set $A$ .",
"In particular, the probability distribution $\\mathbb {P}$ in (MDE1) can be identified with $(p_{1}, \\ldots , p_{K})$ in terms of $p_{k} = \\mathbb {P}[\\lbrace d_{k} \\rbrace ]$ (which in the i.i.d.",
"case turns into $p_{k} = \\mathbb {\\Pi }[X_{1} = d_{k}]$ ).",
"Along this line, the family $\\mathbb {\\Omega }$ of probability distributions in (MDE2) can be identified with a subset $\\Omega $$\\Omega \\subset \\mathbb {S}^{K}$ of probability vectors (viz.",
"of vectors of probability masses).",
"Analogously, each finite nonnegative measure $Q$ on $\\mathcal {Y}$ can be uniquely identified with a vector $\\mathbf {Q} := (q_{1}, \\ldots , q_{K}) \\in \\mathbb {R}_{\\ge 0}^{K}$ , and each finite signed measure $Q$ with a vector $\\mathbf {Q} := (q_{1}, \\ldots , q_{K}) \\in \\mathbb {R}^{K}$ .",
"The corresponding divergences between distributions/measures are then, as usual, defined through the divergences between their respective masses/frequencies: $D_{\\varphi }(Q,\\mathbb {P}) : = D_{\\varphi }(\\mathbf {Q},{P}).$ In particular, $\\mathbb {P}_{n}^{emp}$ can be identified with the vector ${P}_{n}^{emp} := (p_{n,1}^{emp}, \\ldots , p_{n,K}^{emp})$ where $p_{n,k}^{emp} \\ := \\ \\frac{1}{n} \\cdot n_{k} \\ := \\ \\frac{1}{n} \\cdot card(\\bigl \\lbrace i \\in \\lbrace 1, \\ldots , n\\rbrace : \\ X_{i} = d_{k} \\bigr \\rbrace )\\ =: \\ \\frac{1}{n} \\cdot card(I_{k}^{(n)}) , \\quad k \\in \\lbrace 1, \\ldots , K\\rbrace ,$ and accordingly the required limit behaviour (REF ) is equivalent to the vector-convergence $\\lim _{n\\rightarrow \\infty } \\Big ( \\frac{n_{1}}{n}, \\ldots , \\frac{n_{K}}{n} \\Big ) = (p_{1}, \\ldots , p_{K})\\qquad \\textrm {a.s.}$ Notice that, in contrast to the case handled in the above Subsection REF , the sets $I_{k}^{(n)}$ of indexes introduced in (REF ) and their numbers $n_{k} = card(I_{k}^{(n)})$ of elements are now random (due to their dependence on the $X_{i}$ ’s) and $M_{{P}_{n}^{emp}}=1$ .",
"In a batch procedure, when $D_{\\varphi }(\\textrm {$$\\hspace{-6.544pt}$$},{P}_{n}^{emp})$ is estimated once the sample $\\left(X_{1},..,X_{n}\\right)$ is observed, we may reorder this sample by putting the $n_{1}$ sample points $X_{i}$ which are equal to $d_{1}$ in the first places, and so on; accordingly one ends up with index sets $I_{k}^{(n)}$ as defined in Section REF .",
"When the online acquisition of the data $X_{i}$ ’s is required, then we usually do not reorder the sample, and the $I_{k}^{(n)}$ 's do not consist in consecutive indexes, which does not make any change with respect to the resulting construction nor to the estimator.",
"The above considerations open the gate to our desired “mimicking” of (REF ) and (REF ) to achieve (REF ) (and some variants thereof) by our bare simulation (BS) method.",
"To proceed, we employ a family of random variables $(W_{i})_{i \\in \\mathbb {N}}$ of independent and identically distributed $\\mathbb {R}-$ valued random variables with probability distribution $\\mathbb {}[ \\cdot \\, ] := \\mathbb {\\Pi }[W_{1} \\in \\cdot \\, ]$ (being connected with the divergence generator $\\varphi \\in \\Upsilon (]a,b[)$ via the representability (REF )), such that $(W_{i})_{i \\in \\mathbb {N}}$ is independent of $(X_{i})_{i \\in \\mathbb {N}}$ on the common underlying probability space $(\\mathfrak {X},\\mathcal {A},\\mathbb {\\Pi })$.",
"As a next step, notice that the “natural candidate” $\\xi _{n,\\mathbf {X}}^{\\mathbf {W}}:= \\frac{1}{n} \\cdot \\sum _{k=1}^{K}\\left(\\sum _{i \\in I_{k}^{(n)}}W_{i}\\right) \\cdot \\delta _{d_{k}}\\ = \\ \\frac{1}{n}\\sum _{i=1}^{n} W_{i} \\cdot \\delta _{X_{i}}$ is not a probability measure since its total mass is not 1 in general, since in terms of its equivalent vector version $\\xi _{n,\\mathbf {X}}^{\\mathbf {W}} := \\Big (\\frac{1}{n}\\sum _{i\\in I_{1}^{(n)}} W_{i}, \\ldots , \\frac{1}{n} \\sum _{i\\in I_{K}^{(n)}} W_{i} \\Big )$ the sum $\\sum _{k=1}^{K} \\frac{1}{n} \\sum _{i\\in I_{k}^{(n)}} W_{i} =\\frac{1}{n} \\sum _{j=1}^{n} W_{i}$ of the $K$ vector components of (REF ) is typically not equal to 1; this implies that no limit result of the form (REF ) with finite limit can hold, since $\\xi _{n,\\mathbf {X}}^{\\mathbf {W}}$ takes values in $\\mathbb {R}^{K}$ and $\\textrm {$$\\hspace{-6.544pt}$$}$ is a subset in the probability simplex $\\mathbb {S}^{K}$ which has void interior in $\\mathbb {R}^{K}$ causing a violation of condition (REF ) (cf.",
"Remark REF (c)); moreover, depending on the concrete form of the generator $\\varphi $ , the corresponding weights may take negative values.",
"Therefore, we need some “rescaling”.",
"Indeed, let us introduce the normalized weighted empirical measure $\\xi _{n,\\mathbf {X}}^{w\\mathbf {W}} &:=&{\\left\\lbrace \\begin{array}{ll}\\frac{1}{\\sum _{k=1}^{K}\\sum _{i \\in I_{k}^{(n)}}W_{i}} \\cdot \\sum _{k=1}^{K}\\left(\\sum _{i \\in I_{k}^{(n)}}W_{i}\\right) \\cdot \\delta _{d_{k}}= \\sum _{i=1}^{n} \\frac{W_{i}}{\\sum _{j=1}^{n} W_{j}} \\cdot \\delta _{X_{i}},\\qquad \\textrm {if } \\sum _{j=1}^{n} W_{j} \\ne 0, \\\\\\infty \\cdot \\sum _{k=1}^{K} \\delta _{d_{k}} =: \\underline{\\infty }, \\hspace{216.2411pt}\\textrm {if } \\sum _{j=1}^{n} W_{j} = 0,\\end{array}\\right.",
"}$ which will substitute $\\xi _{n,\\mathbf {X}}^{\\mathbf {W}}$ and which may belong to $\\textrm {$$\\hspace{-6.544pt}$$}$ with positive probability.",
"The equivalent vector version of $\\xi _{n,\\mathbf {X}}^{w\\mathbf {W}}$ is given by $\\xi _{n,\\mathbf {X}}^{w\\mathbf {W}} &:=&{\\left\\lbrace \\begin{array}{ll}\\left(\\frac{\\sum _{i \\in I_{1}^{(n)}}W_{i}}{\\sum _{k=1}^{K}\\sum _{i \\in I_{k}^{(n)}}W_{i}},\\ldots , \\frac{\\sum _{i \\in I_{K}^{(n)}}W_{i}}{\\sum _{k=1}^{K}\\sum _{i \\in I_{k}^{(n)}}W_{i}} \\right) ,\\qquad \\textrm {if } \\sum _{j=1}^{n} W_{j} \\ne 0, \\\\\\ (\\infty , \\ldots , \\infty ) =: \\infty , \\hspace{113.81102pt} \\textrm {if } \\sum _{j=1}^{n} W_{j} = 0,\\end{array}\\right.",
"}$ a point in the linear subset of $\\mathbb {R}^{K}$ spanned by $\\mathbb {S}^{K}$ at infinity.",
"Remark 13 (i) (Concerning e.g.",
"computer-program command availability) In case of $\\sum _{j=1}^{n}W_{j}=0$ , in (REF ) we may equivalently assign to $\\xi _{n,\\mathbf {X}}^{w\\mathbf {W}}$ instead of $\\underline{\\infty }$ any measure (e.g.",
"probability distribution) which does not belong to $\\mathbb {\\Omega }$ , respectively, in (REF ) we may equivalently choose for $\\xi _{n,\\mathbf {X}}^{w\\mathbf {W}}$ any vector outside of $\\Omega $$\\Omega $ instead of $\\infty $ .",
"(ii) By construction, in case of $\\sum _{j=1}^{n} W_{j} \\ne 0$ , the sum of the random $K$ vector components of (REF ) is now automatically equal to 1, but — as (depending on $\\varphi $ ) the $W_{i}$ ’s may take both positive and negative values — these random components may be negative (resp.",
"nonnegative) with probability strictly greater (resp.",
"smaller) than zero (resp.",
"one); in the framework of (REF ) this means that $\\xi _{n,\\mathbf {X}}^{w\\mathbf {W}}$ is in general a random signed measure with total mass 1, in case of $\\sum _{j=1}^{n} W_{j} \\ne 0$ .",
"However, $\\mathbb {\\Pi } [\\xi _{n,\\mathbf {X}}^{w\\mathbf {W}}\\in \\mathbb {S}_{>0}^{K}]>0$ since all the (identically distributed) random variables $W_{i}$ have expectation 1 (as a consequence of the assumed representability (REF )); in case of $\\mathbb {\\Pi }[W_{1}>0]=1$ one has even $\\mathbb {\\Pi }[\\xi _{n,\\mathbf {X}}^{w\\mathbf {W}}\\in \\mathbb {S}_{>0}^{K}]=1$ .",
"In the particular context of Example REF (c), one gets $\\mathbb {\\Pi }[\\xi _{n,\\mathbf {X}}^{w\\mathbf {W}}\\in \\mathbb {S}_{>0}^{K}]=\\left( \\mathbb {\\Pi }[ W_{1}>0]\\right)^{n}=\\left( \\int _{0}^{\\infty }\\sqrt{\\frac{\\widetilde{c}}{2\\pi }}\\cdot \\exp (-\\frac{\\widetilde{c}\\cdot (u-1)^{2}}{2})du\\right) ^{n}\\in ]0,1[$ .",
"Summing up things, the probability $\\mathbb {\\Pi }[\\xi _{n,\\mathbf {X}}^{w\\mathbf {W}}\\in \\Omega $$\\Omega ]$ is strictly positive and finite at least for large n, whenever $\\Phi _{{P}}(\\Omega $$\\Omega )= \\inf _{{Q}\\in \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega } }\\ D_{\\varphi }({Q},{P})$ is finite.",
"(iii) By generalizing the terminology of e.g.",
"Vajda [372], through the right-hand side of (REF ) one can interpret (for $\\sum _{j=1}^{n} W_{j} \\ne 0$ ) the normalized weighted empirical measure $\\xi _{n,\\mathbf {X}}^{w\\mathbf {W}}$ as response of an output neuron in a random perceptron consisting of random inputs $\\mathbf {X}$ , a layer with $n$ units having one-point-distribution-valued responses $\\delta _{X_{1}}, \\ldots , \\delta _{X_{n}}$ , and independent random synaptic weights $\\left(\\frac{W_{1}}{\\sum _{j=1}^{n} W_{j}}, \\ldots , \\frac{W_{n}}{\\sum _{j=1}^{n} W_{j}}\\right)$ .",
"With the above-mentioned ingredients, we are now in the position to tackle a variant of the distance minimization problem (REF ), by our bare simulation method through “mimicking” the deterministic minimization problem (REF ) respectively (REF ).",
"For this, we also employ the conditional distributions $\\mathbb {\\Pi }_{n}[\\, \\cdot \\, ] := \\mathbb {\\Pi }_{X_{1}^{n}}[\\, \\cdot \\, ] := \\mathbb {\\Pi }[ \\, \\cdot \\, | \\,X_{1}, \\ldots , X_{n} ]$ and obtain the following Theorem 14 Suppose that $(X_{i})_{i\\in \\mathbb {N}}$ is a sequence of random variables with values in $\\mathcal {Y}:=\\left\\lbrace d_{1},\\cdots ,d_{K}\\right\\rbrace $ such that (REF ) holds for some probability measure $\\mathbb {P}[\\cdot \\,]$ on $\\mathcal {Y}$ having no zero-mass frequencies (or equivalently, (REF ) holds for some probability vector ${P} \\in \\mathbb {S}_{> 0}^{K}$ ).",
"Moreover, let $(W_{i})_{i \\in \\mathbb {N}}$ be a family of independent and identically distributed $\\mathbb {R}-$ valued random variables with probability distribution $\\mathbb {}[ \\cdot \\, ] := \\mathbb {\\Pi }[W_{1} \\in \\cdot \\, ]$ being connected with the divergence generator $\\varphi \\in \\Upsilon (]a,b[)$ via the representability (REF ), such that $(W_{i})_{i \\in \\mathbb {N}}$ is independent of $(X_{i})_{i \\in \\mathbb {N}}$ .",
"Then there holds $-\\lim _{n\\rightarrow \\infty }\\frac{1}{n}\\log \\,\\mathbb {\\Pi }_{X_{1}^{n}}\\left[\\xi _{n,\\mathbf {X}}^{w\\mathbf {W}}\\in \\mathbb {\\Omega }\\right]& =\\inf _{\\mathbb {Q}\\in \\mathbb {\\Omega } }\\ \\inf _{m\\ne 0}D_{\\varphi }(m\\cdot \\mathbb {Q},\\mathbb {P})\\, \\\\& =\\inf _{m\\ne 0}\\ \\inf _{\\mathbb {Q}\\in \\mathbb {\\Omega } }D_{\\varphi }(m\\cdot \\mathbb {Q},\\mathbb {P})\\\\& =\\inf _{m\\ne 0}\\ \\inf _{{Q}\\in \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega } }D_{\\varphi }(m\\cdot {Q},{P})\\\\& = \\inf _{{Q}\\in \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega } }\\ \\inf _{m\\ne 0}D_{\\varphi }(m\\cdot {Q},{P})= -\\lim _{n\\rightarrow \\infty }\\frac{1}{n}\\log \\,\\mathbb {\\Pi }_{X_{1}^{n}}\\left[\\xi _{n,\\mathbf {X}}^{w\\mathbf {W}}\\in \\textrm {\\right.\\Omega \\hspace{-6.544pt}\\Omega }$ for all sets $\\mathbb {\\Omega }$ of probability distributions such that their equivalent probability-vector form $\\Omega $$\\Omega $ satisfies the regularity properties (REF ) in the relative topology and the finiteness property (REF ); notice that for the equality () we have used the “divergence link” (REF ).",
"In particular, for each ${P} \\in \\mathbb {S}_{>0}^{K}$ (respectively, its equivalent probability-distribution $\\mathbb {P}$ ) the function ${Q} \\mapsto \\inf _{m\\ne 0}D_{\\varphi }(m\\cdot {Q},{P}) $ (respectively, the function $\\mathbb {Q} \\mapsto \\inf _{m\\ne 0}D_{\\varphi }(m\\cdot \\mathbb {Q},\\mathbb {P})$ ) is BS-minimizable (cf.",
"(REF )) on all sets $\\textrm {$$\\hspace{-6.544pt}$$} \\subset \\mathbb {S}^{K}$ satisfying (7) in the relative topology and (9) (respectively, on their probability-distribution-equivalent $\\mathbb {\\Omega }$ ).",
"The proof of Theorem REF will be given in Appendix B. Analogously to Remark REF (iii), let us emphasize that we have assumed ${P} \\in \\mathbb {S}_{> 0}^{K}$ in Theorem REF .",
"Henceforth, for sets $\\textrm {$$\\hspace{-6.544pt}$$} \\subset \\mathbb {S}^{K}$ of probability vectors we deal with (REF ) only in the relative topology; thus, the latter will be unmentioned for the sake of brevity.",
"Remark REF (a),(b),(c),(e) applies accordingly.",
"Remark 15 (i) In strong contrast to Theorem REF , the above result does not provide a direct tool for the solution of Problem (REF ) since the limit in (REF ) bears no direct information on the minimum divergence $D_{\\varphi }\\left( \\mathbb {\\Omega },\\mathbb {P} \\right) :=\\inf _{\\mathbb {Q}\\in \\mathbb {\\Omega }} D_{\\varphi }( \\mathbb {Q}, \\mathbb {P})$ ; the link between the corresponding quantities can be emphasized and exploited e.g.",
"in the case of power type divergences, which leads to explicit minimization procedures as shown in the Subsection REF below.",
"For general divergences, Theorem REF allows for the estimation of upper and lower bounds of $D_{\\varphi }\\left( \\mathbb {\\Omega },\\mathbb {P} \\right)$ , as developed in the Subsection REF below.",
"(ii) Notice that $\\breve{D}_{\\varphi }(\\mathbb {Q},\\mathbb {P}):=\\inf _{m\\ne 0} D_{\\varphi }(m\\cdot \\mathbb {Q},\\mathbb {P})$ satisfies the axioms of a divergence, that is, $\\breve{D}_{\\varphi }(\\mathbb {Q},\\mathbb {P}) \\ge 0$ , as well as $\\breve{D}_{\\varphi }(\\mathbb {Q},\\mathbb {P}) = 0$ if and only if $\\mathbb {Q} = \\mathbb {P}$ (reflexivity).",
"Hence, in Theorem REF we are still within our framework of bare simulation of a divergence minimum w.r.t.",
"its first component (however, notice the difference to (i)).",
"(iii) Viewed from a “reverse” angle, Theorem REF gives a crude approximation for the probability for $\\xi _{n,\\mathbf {X}}^{w\\mathbf {W}}$ to belong to $\\mathbb {\\Omega }$ , conditionally upon $\\mathbf {X} = (X_{1}, \\ldots , X_{n})$ .",
"(iv) In the same spirit as Remark REF (ii), for some contexts one can explicitly give the distribution of each of the independent components $\\Big (\\sum _{i\\in I_{k}^{(n)}}W_{i}\\Big )_{k=1,\\ldots ,K}$ of the vector $\\xi _{n}^{w\\mathbf {W}}$ ; this will ease the corresponding concrete simulations in a batch procedure.",
"For instance, we shall give those in the Examples REF , REF , REF , REF and REF in the Section below.",
"(v) Consider the special “degenerate” case where all the data observations are certain and thus $(X_{i})_{i \\in \\mathbb {N}}$ is nothing but a purely deterministic sequence, say $(\\widetilde{x}_{i})_{i\\in \\mathbb {N}}$ , of elements $\\widetilde{x}_{i}$ from the arbitrary set $\\mathcal {Y}:=\\left\\lbrace d_{1},\\ldots ,d_{K}\\right\\rbrace $ of $K$ distinct values “of any kind” (e.g., $\\mathcal {Y}$ may consist of $K$ distinct numbers); then the corresponding empirical distribution $\\mathbb {P}_{n}^{emp}$ can be identified with the vector ${P}_{n}^{emp} := (p_{n,1}^{emp}, \\ldots , p_{n,K}^{emp})$ where $p_{n,k}^{emp} \\ := \\ \\frac{1}{n} \\cdot n_{k} \\ := \\ \\frac{1}{n} \\cdot card(\\bigl \\lbrace i \\in \\lbrace 1, \\ldots , n\\rbrace : \\ \\widetilde{x}_{i} = d_{k} \\bigr \\rbrace )\\ =: \\ \\frac{1}{n} \\cdot card(I_{k}^{(n)}) , \\quad k \\in \\lbrace 1, \\ldots , K\\rbrace ,$ and accordingly the required limit behaviour (REF ) is equivalent to the vector-convergence $\\lim _{n\\rightarrow \\infty } \\Big ( \\frac{n_{1}}{n}, \\ldots , \\frac{n_{K}}{n} \\Big ) = (p_{1}, \\ldots , p_{K})\\quad \\textrm {for some p_{1} >0, \\ldots , p_{K} >0 such that \\sum _{k=1}^{K} p_{k} = 1.",
"}$ Correspondingly, with the notations ${P} := (p_{1}, \\ldots , p_{K})$ and $\\mathbf {\\widetilde{x}} := (\\widetilde{x}_{1}, \\ldots , \\widetilde{x}_{n})$ , the vector-form part of the assertion (REF ) of Theorem REF becomes $-\\lim _{n\\rightarrow \\infty }\\frac{1}{n}\\log \\,\\mathbb {\\Pi }\\left[\\xi _{n,\\mathbf {\\widetilde{x}}}^{w\\mathbf {W}}\\in \\textrm {\\right.\\Omega \\hspace{-6.544pt}\\Omega }\\ = \\ \\inf _{{Q}\\in \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega } }\\ \\inf _{m\\ne 0}D_{\\varphi }(m\\cdot {Q},{P})\\ = \\ \\inf _{m\\ne 0}\\ \\inf _{{Q}\\in \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega } }D_{\\varphi }(m\\cdot {Q},{P})$ for all subsets $\\Omega $$\\Omega \\subset \\mathbb {S}^{K}$ satisfying the regularity properties (REF ) and the finiteness property (REF ); notice that the conditional probability $\\mathbb {\\Pi }_{X_{1}^{n}}[\\, \\cdot \\, ]$ has degenerated to the ordinary probability $\\mathbb {\\Pi }[\\, \\cdot \\, ]$ .",
"(vi) In a similar fashion to the proof of (the special degenerate case (v) of) Theorem REF , one can show $-\\lim _{n\\rightarrow \\infty }\\frac{1}{n}\\log \\,\\mathbb {\\Pi }\\left[\\xi _{n}^{w\\mathbf {W}}\\in \\textrm {\\right.\\Omega \\hspace{-6.544pt}\\Omega }\\ = \\ \\inf _{{Q}\\in \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega } }\\ \\inf _{m\\ne 0}D_{\\varphi }(m\\cdot {Q},{P})\\ = \\ \\inf _{m\\ne 0}\\ \\inf _{{Q}\\in \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega } }D_{\\varphi }(m\\cdot {Q},{P})$ for all subsets $\\Omega $$\\Omega \\subset \\mathbb {S}^{K}$ with regularity properties (REF ) and the finiteness property (REF ), where $\\xi _{n}^{w\\mathbf {W}} &:=&{\\left\\lbrace \\begin{array}{ll}\\left(\\frac{\\sum _{i \\in I_{1}^{(n)}}W_{i}}{\\sum _{k=1}^{K}\\sum _{i \\in I_{k}^{(n)}}W_{i}},\\ldots , \\frac{\\sum _{i \\in I_{K}^{(n)}}W_{i}}{\\sum _{k=1}^{K}\\sum _{i \\in I_{k}^{(n)}}W_{i}} \\right)= \\frac{n \\cdot \\xi _{n}^{\\mathbf {W}}}{\\sum _{i=1}^{n}W_{i}},\\qquad \\textrm {if } \\sum _{j=1}^{n} W_{j} \\ne 0, \\\\\\ (\\infty , \\ldots , \\infty ) =: \\infty , \\hspace{159.3356pt} \\textrm {if } \\sum _{j=1}^{n} W_{j} = 0,\\end{array}\\right.",
"}$ with $I_{1}^{(n)}:=\\left\\lbrace 1,\\ldots ,n_{1}\\right\\rbrace $ , $I_{2}^{(n)}:=\\left\\lbrace n_{1}+1,\\ldots ,n_{1}+n_{2}\\right\\rbrace $ , ..., $I_{K}^{(n)} := \\lbrace \\sum _{k=1}^{K-1} n_{k} + 1, \\ldots , n \\rbrace $ and $n_{k}:=\\lfloor n \\cdot p_{k}\\rfloor $ ($k \\in \\lbrace 1,\\ldots ,K\\rbrace $ ) for some pregiven known probability vector ${P} := (p_{1}, \\ldots , p_{K})$ .",
"Recall the definition of $\\xi _{n}^{\\mathbf {W}}$ in (REF ) (with $\\mathbf {W}$ instead of $\\mathbf {\\widetilde{W}}$ ).",
"The limit behaviour (REF ) contrasts to the one of Theorem REF , where $-\\lim _{n\\rightarrow \\infty }\\frac{1}{n}\\log \\,\\mathbb {\\Pi }\\left[\\xi _{n}^{\\mathbf {\\widetilde{W}}}\\in \\Omega /M_{\\mathbf {P}}\\right]=\\inf _{Q\\in \\mathbf {\\Omega }}D_{\\varphi }(\\mathbf {Q},\\mathbf {P})\\hspace{113.81102pt} \\text{(cf.",
"(\\ref {LDP Minimization}))}\\nonumber $ for any $\\mathbf {\\Omega }\\subset \\mathbb {R}^{K}$ with regularity properties (REF ) and the finiteness property (REF ); recall that $(\\widetilde{W}_{i})_{i \\in \\mathbb {N}}$ are i.i.d.",
"random variables with probability distribution $\\widetilde{\\mathbb {}}$ (being connected with the divergence generator $\\widetilde{\\varphi } := M_{\\mathbf {P}} \\cdot \\varphi $ via the representability (REF )), whereas $(W_{i})_{i \\in \\mathbb {N}}$ are i.i.d.",
"random variables with probability distribution $\\mathbb {}$ (being connected with the divergence generator $\\varphi $ via the representability (REF )).",
"Indeed, the construction leading to Theorem REF does not hold any longer when $\\mathbf {\\Omega } \\subset \\mathbb {S}^{K}$ is a set of vectors within the probability simplex $\\mathbb {S}^{K}$ and $\\mathbf {P}\\in \\mathbb {S}_{>0}^{K}$ is a known vector in this simplex with no zero entries.",
"In such a case, one has to use (REF ) and (REF ) instead.",
"Notice that for each constant $A >0$ , (REF ) can be rewritten as $& &\\hspace{-42.67912pt}-\\lim _{n\\rightarrow \\infty }\\frac{1}{n}\\log \\,\\mathbb {\\Pi }\\left[ \\xi _{n}^{w\\mathbf {W}}\\in \\textrm {\\right.\\Omega \\hspace{-6.544pt}\\Omega } = \\inf _{\\mathbf {Q}\\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega } }\\ \\inf _{m\\ne 0}D_{\\varphi }\\Big (\\frac{m}{A} \\cdot \\mathbf {Q},{P}\\Big )= \\inf _{\\mathbf {Q}\\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega } }\\ \\inf _{\\widetilde{m}\\ne 0}D_{\\varphi }(\\widetilde{m} \\cdot \\mathbf {Q},{P})= \\inf _{\\widetilde{m}\\ne 0}\\ \\inf _{\\mathbf {Q}\\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega } }D_{\\varphi }(\\widetilde{m} \\cdot \\mathbf {Q},{P}) ;$ therein, the constraint $\\mathbf {Q}\\in A \\cdot \\textrm {$$\\hspace{-6.544pt}$$}$ means geometrically that the vector $\\mathbf {Q}$ lives in a subset of a simplex which is parallel to the simplex $\\mathbb {S}^{K}$ of probability vectors and which is cut off at the edges of the first/positive orthant; in view of Remark REF (d) and (REF ), we can also handle such a situation.",
"Namely, in the light of the third expression in (REF ) in combination with (REF ) to (REF ) for the special case of $\\mathbf {\\Omega } := \\textrm {$$\\hspace{-6.544pt}$$}$ lying in the probability simplex, it makes sense to study e.g.",
"functional relationships between $\\inf _{\\widetilde{m}\\ne 0}D_{\\widetilde{c}\\cdot \\varphi }(\\widetilde{m} \\cdot \\mathbf {Q},{P})$ and $D_{\\widetilde{c}\\cdot \\varphi }(\\mathbf {Q},{P})$ ($\\widetilde{c} >0$ ) for $\\mathbf {Q} \\in A \\cdot \\mathbb {S}^{K}$ with arbitrary $A >0$ not necessarily being equal to 1 (i.e.",
"$\\mathbf {Q} = A \\cdot {Q}$ for some probability distribution ${Q}$ ).",
"Indeed, such a context appears naturally e.g.",
"in connection with mass transportation problems (cf.",
"(REF ) below) and with distributed energy management (cf.",
"the paragraph after (REF )); the special case $A=1/K$ of (REF ) will also be used below for the application of our BS method to solving (generalized) minimum/maximum entropy problems for probability vectors (and even for sub-/super-probability vectors) ${Q}$ with constraints.",
"Let us proceed with the main context.",
"As indicated in Remark REF (i), in a number of important cases the limit in the above Theorem REF can be stated in terms of an invertible function $G^{-1}$ (cf.",
"(REF )) of $\\inf _{{Q}\\in \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega }}D_{\\varphi }({Q},{P})$ by elimination of $m$ .",
"As explained above, for the degenerate case (cf.",
"Remark REF (v), (vi)) the search for $G^{-1}$ is even interesting for the more general infimum over non-probability vectors.",
"This is the scope of the following development."
],
[
"Construction principle for the estimation of the minimum\ndivergence, the power-type case ",
"Within the context of Theorem REF respectively Remark REF (v) and (vi), we obtain an explicit solution for the inner (i.e.",
"$m-$ concerning) minimization in (REF ) for the important case of power-divergence generators $\\varphi _{\\gamma } : \\mathbb {R} \\mapsto [0,\\infty ]$ defined by $\\varphi _{\\gamma }(t) \\hspace{-5.69046pt} &:=& \\hspace{-5.69046pt}{\\left\\lbrace \\begin{array}{ll}\\frac{t^\\gamma -\\gamma \\cdot t+ \\gamma - 1}{\\gamma \\cdot (\\gamma -1)}, \\hspace{170.71652pt} \\textrm {if } \\gamma \\in \\, ]-\\infty ,0[ \\ \\textrm {and } t \\in ]0,\\infty [,\\\\- \\log t + t - 1, \\hspace{156.49014pt} \\textrm {if } \\gamma = 0 \\ \\textrm {and } t \\in ]0,\\infty [,\\\\\\frac{t^\\gamma -\\gamma \\cdot t+ \\gamma - 1}{\\gamma \\cdot (\\gamma -1)}, \\hspace{170.71652pt} \\textrm {if } \\gamma \\in \\, ]0,1[ \\ \\textrm {and } t \\in [0,\\infty [,\\\\t \\cdot \\log t + 1 - t, \\hspace{156.49014pt} \\textrm {if } \\gamma = 1 \\ \\textrm {and } t \\in [0,\\infty [,\\\\\\frac{t^\\gamma -\\gamma \\cdot t+ \\gamma - 1}{\\gamma \\cdot (\\gamma -1)} \\cdot {1}_{]0,\\infty [}(t)+(\\frac{1}{\\gamma } - \\frac{t}{\\gamma -1}) \\cdot {1}_{]-\\infty ,0]}(t),\\hspace{28.45274pt} \\textrm {if } \\gamma \\in \\, ]1,2[ \\ \\textrm {and } t \\in \\, ]-\\infty ,\\infty [,\\\\\\frac{(t - 1)^2}{2}, \\hspace{193.47882pt} \\textrm {if } \\gamma = 2 \\ \\textrm {and } t \\in \\, ]-\\infty ,\\infty [,\\\\\\frac{t^\\gamma -\\gamma \\cdot t+ \\gamma - 1}{\\gamma \\cdot (\\gamma -1)} \\cdot {1}_{]0,\\infty [}(t)+(\\frac{1}{\\gamma } - \\frac{t}{\\gamma -1}) \\cdot {1}_{]-\\infty ,0]}(t),\\hspace{28.45274pt} \\textrm {if } \\gamma \\in \\, ]2,\\infty [ \\ \\textrm {and } t \\in \\, ]-\\infty ,\\infty [,\\\\\\infty , \\hspace{209.12791pt} \\textrm {else},\\end{array}\\right.",
"}$ which for arbitrary multiplier $\\widetilde{c} >0$ generate (the vector-valued form of) the generalized power divergences displayed in the first six rows of Table 1 (and beyond), i.e.",
"$D_{\\widetilde{c} \\cdot \\varphi _{\\gamma }}(\\mathbf {Q},\\mathbf {P}) \\hspace{-5.69046pt} &:=& \\hspace{-5.69046pt}{\\left\\lbrace \\begin{array}{ll}\\widetilde{c} \\cdot \\Big \\lbrace \\frac{ \\sum \\limits _{k=1}^{K} (q_{k})^{\\gamma } \\cdot (p_{k})^{1-\\gamma }}{\\gamma \\cdot (\\gamma -1)}- \\frac{1}{\\gamma -1} \\cdot \\sum \\limits _{k=1}^{K} q_{k} + \\frac{1}{\\gamma } \\cdot \\sum \\limits _{k=1}^{K} p_{k} \\Big \\rbrace ,\\hspace{56.9055pt} \\textrm {if } \\gamma \\in \\, ]-\\infty ,0[, \\ \\mathbf {P} \\in \\mathbb {R}_{\\ge 0}^{K} \\ \\textrm {and } \\mathbf {Q} \\in \\mathbb {R}_{> 0}^{K},\\\\\\widetilde{c} \\cdot \\Big \\lbrace \\sum \\limits _{k=1}^{K} p_{k} \\cdot \\log \\Big (\\frac{p_{k}}{q_{k}} \\Big )+ \\sum \\limits _{k=1}^{K} q_{k} - \\sum \\limits _{k=1}^{K} p_{k} \\Big \\rbrace ,\\hspace{88.2037pt} \\textrm {if } \\gamma = 0, \\ \\mathbf {P} \\in \\mathbb {R}_{\\ge 0}^{K} \\ \\textrm {and } \\mathbf {Q} \\in \\mathbb {R}_{> 0}^{K},\\\\\\widetilde{c} \\cdot \\Big \\lbrace \\frac{ \\sum \\limits _{k=1}^{K} (q_{k})^{\\gamma } \\cdot (p_{k})^{1-\\gamma }}{\\gamma \\cdot (\\gamma -1)}- \\frac{1}{\\gamma -1} \\cdot \\sum \\limits _{k=1}^{K} q_{k} + \\frac{1}{\\gamma } \\cdot \\sum \\limits _{k=1}^{K} p_{k} \\Big \\rbrace ,\\hspace{59.75095pt} \\textrm {if } \\gamma \\in \\, ]0,1[, \\ \\mathbf {P} \\in \\mathbb {R}_{\\ge 0}^{K} \\ \\textrm {and } \\mathbf {Q} \\in \\mathbb {R}_{\\ge 0}^{K},\\\\\\widetilde{c} \\cdot \\Big \\lbrace \\sum \\limits _{k=1}^{K} q_{k} \\cdot \\log \\Big (\\frac{q_{k}}{p_{k}} \\Big )- \\sum \\limits _{k=1}^{K} q_{k} + \\sum \\limits _{k=1}^{K} p_{k} \\Big \\rbrace ,\\hspace{91.04872pt} \\textrm {if } \\gamma = 1, \\ \\mathbf {P} \\in \\mathbb {R}_{> 0}^{K} \\ \\textrm {and } \\mathbf {Q} \\in \\mathbb {R}_{\\ge 0}^{K},\\\\\\widetilde{c} \\cdot \\Big \\lbrace \\sum \\limits _{k=1}^{K} \\frac{(q_{k})^{\\gamma } \\cdot (p_{k})^{1-\\gamma }}{\\gamma \\cdot (\\gamma -1)}\\cdot {1}_{[0,\\infty [}(q_{k})- \\frac{1}{\\gamma -1} \\cdot \\sum \\limits _{k=1}^{K} q_{k} + \\frac{1}{\\gamma } \\cdot \\sum \\limits _{k=1}^{K} p_{k} \\Big \\rbrace ,\\hspace{8.5359pt} \\textrm {if } \\gamma \\in \\, ]1,2[, \\ \\mathbf {P} \\in \\mathbb {R}_{>0}^{K} \\ \\textrm {and } \\mathbf {Q} \\in \\mathbb {R}^{K},\\\\\\widetilde{c}\\cdot \\sum \\limits _{k=1}^{K}\\frac{ (q_{k}-p_{k})^{2}}{2 \\cdot p_{k}} ,\\hspace{202.01474pt} \\textrm {if } \\gamma = 2, \\ \\mathbf {P} \\in \\mathbb {R}_{>0}^{K} \\ \\textrm {and } \\mathbf {Q} \\in \\mathbb {R}^{K},\\\\\\widetilde{c} \\cdot \\Big \\lbrace \\sum \\limits _{k=1}^{K} \\frac{(q_{k})^{\\gamma } \\cdot (p_{k})^{1-\\gamma }}{\\gamma \\cdot (\\gamma -1)}\\cdot {1}_{[0,\\infty [}(q_{k})- \\frac{1}{\\gamma -1} \\cdot \\sum \\limits _{k=1}^{K} q_{k} + \\frac{1}{\\gamma } \\cdot \\sum \\limits _{k=1}^{K} p_{k} \\Big \\rbrace ,\\hspace{11.38092pt} \\textrm {if } \\gamma \\in \\, ]2,\\infty [, \\ \\mathbf {P} \\in \\mathbb {R}_{>0}^{K} \\ \\textrm {and } \\mathbf {Q} \\in \\mathbb {R}^{K},\\\\\\infty , \\hspace{257.49751pt} \\textrm {else};\\end{array}\\right.",
"}$ notice that one has the straightforward relationship $D_{\\widetilde{c}\\cdot \\varphi _{\\gamma }}(\\cdot ,\\cdot )=\\widetilde{c}\\cdot D_{\\varphi _{\\gamma }}(\\cdot ,\\cdot )$ ; however, as a motivation for the introduction of $\\widetilde{c}>0$ , we shall show in the Examples REF , REF , REF , REF in Section below that the corresponding probability distribution $\\mathbb {\\mathbb {}}$ of the $W_{i}$ ’s depends on $\\widetilde{c}$ in a non-straightforward way (see also Remark REF (vi) for another motivation for $\\widetilde{c}$ ).",
"In the course of this, it turns out that $\\widetilde{c} \\cdot \\varphi _{\\gamma } \\in \\Upsilon (]a_{\\gamma },\\infty [)$ with $a_{\\gamma } =0$ for $\\gamma \\in ]-\\infty ,1]$ and $a_{\\gamma } = -\\infty $ for $\\gamma \\in [2,\\infty [$ .",
"For $\\widetilde{c}=1$ and probability vectors ${Q}$ , ${P}$ in $\\mathbb {S}^{K}$ respectively $\\mathbb {S}_{>0}^{K}$ , the divergences (REF ) simplify considerably, namely to the well-known power divergences $D_{\\varphi _{\\gamma }}({Q},{P})$ in the scaling of e.g.",
"Liese & Vajda [217] (in other scalings they are also called Rathie & Kannapan’s non-additive directed divergences of order $\\gamma $ [302], Cressie-Read divergences [93] [303], relative Tsallis entropies or Tsallis cross-entropies [364] (see also Shiino [331]), Amari’s alpha-divergences [12]); for some comprehensive overviews on power divergences $D_{\\varphi _{\\gamma }}({Q},{P})$ — including statistical applications to goodness-of-fit testing and minimum distance estimation — the reader is referred to the insightful books of e.g.",
"Liese & Vajda [217], Read & Cressie [303], Vajda [371], Stummer [344], Pardo [282], Liese & Miescke [216], the survey articles of e.g.",
"Liese & Vajda [218], Vajda & van der Meulen [374], and the references therein.",
"Prominent and widely used special cases of $D_{\\varphi _{\\gamma }}({Q},{P})$ are the omnipresent Kullback-Leibler information divergence (relative entropy) where $\\gamma =1$ , the equally important reverse Kullback-Leibler information divergence (reverse relative entropy) where $\\gamma =0$ , the Pearson chi-square divergence ($\\gamma =2$ ), the Neyman chi-square divergence ($\\gamma =-1$ ), the Hellinger divergence ($\\gamma =\\frac{1}{2}$ , also called squared Hellinger distance, squared Matusita distance [256] or squared Hellinger-Kakutani metric, see e.g.",
"Deza & Deza [113] in some literature, the (square root of the) Hellinger divergence (HD) is misleadingly called Bhattacharyya distance; however, the latter is basically some rescaled logarithm of HD, namely $R_{1/2}({Q},{P})$ (cf.",
"(REF ) with $\\gamma =1/2$ )).",
"Some exemplary (relatively) recent studies and applications of power divergences $D_{\\varphi _{\\gamma }}({Q},{P})$ — aside from the vast statistical literature (including in particular maximum likelihood estimation and Pearson’s chi-square test) — appear e.g.",
"in Matsuyama [253] for flexibilizations of the well-known expectation-maximization (EM) algorithm and their uses for big-data completion (cf.",
"[254]) and data credit computation in blockchain networks (cf.",
"[255]), Ku & Fine [199] in connection with blind source separation, Stummer & Vajda [348] as well as Stummer & Lao [347] for optimal decisions about some alternative financial models, Berend et al.",
"[41] for the derivation of a kind of reverse Pinsker’s inequality (with $\\gamma =1$ ), Verrelst et al.",
"[383] in geoscientific remote sensing via semiautomatic mapping of biophysical parameters from optical earth observations, Salem at al.",
"[315] for automatic alarm-triggering detection of events (e.g.",
"patient health degradations) from collected data by biomedical sensors, Fu et al.",
"[128] for the study of income distributions in China, Ha et al.",
"[151] for $x-$ ray spectrum reconstruction in computer tomography (CT) systems (with $\\gamma =1$ ), Iqbal & Seghouane [161] for robust sequential dictionary learning, Luppino et al.",
"[233] for unsupervised change detection in heterogeneous multi-temporal satellite images (with $\\gamma =\\frac{1}{2}$ ), Sason [320] in connection with data-processing and majorization inequalities, Krömer & Stummer [198] for the smoothing and error-correcting of crude mortality rates (where they even employ non-probability-type vectors), Bekhet & Ahmed [37] for effectiveness evaluations in video retrieval (with $\\gamma =-1$ , $\\gamma =\\frac{1}{2}$ ), Cai et al.",
"[71] for the stabilization of trainings of generative adversarial networks (GANs), Fu et al.",
"[129] for automatic molecule optimization, Görtler et al.",
"[139] for dimensionality reduction on uncertain data in visualization and computer graphics (with $\\gamma =\\frac{1}{2}$ ), Kammerer & Stummer [179] for optimal decision making in the presence of pandemics (e.g.",
"COVID-19), Kanapram et al.",
"[180] for the development of collective self-awareness in a network of connected and autonomous vehicles through agent-centered detection of abnormal situations (with $\\gamma =\\frac{1}{2}$ ), Kumbhakar [206] for modelling the streamwise velocity profile in open-channel flows, Sigmon et al.",
"[335] for the improvement of genetic quality control in mouse research for biomedical applications (with $\\gamma =2$ ), Zhang et al.",
"[420] for the design of a noise-adaptation adapted generative adversarial network for medical image analysis (with $\\gamma =\\frac{1}{2}$ ), Chen et al.",
"[79] for clustering high-dimensional microbial data from RNA sequencing (with $\\gamma =\\frac{1}{2}$ ), Dharmawan et al.",
"[114] for the development of improvements in long-term cell observations via semiconductor-chips-based lensless holographic microscopy, Liu & Sun [229] for analyzing approximate inferences in Bayesian neural networks, Rekavandi et al.",
"[307] for detections in functional magnetic resonance imaging (fMRI) as well as hyperspectral and synthetic aperture radar (SAR) data, Seghouane & Shokouhi [325] for adaptive learning within robust radial basis function networks (RBFN), and Wang et al.",
"[388] for recommender-system relevant collaborative filtering in sparse data.",
"For $\\widetilde{c}=1$ and nonnegative-component vectors $\\mathbf {Q}$ , $\\mathbf {P}$ in $\\mathbb {R}_{\\ge 0}^{K}$ respectively $\\mathbb {R}_{>0}^{K}$ , the generalized power divergences $D_{\\varphi _{\\gamma }}(\\mathbf {Q},\\mathbf {P})$ of (REF ) also (partially) simplify, and were treated by Stummer & Vajda [349] (for even more general probability measures, deriving e.g.",
"also generalized Pinsker’s inequalities); for a more general comprehensive technical treatment see also e.g.",
"Broniatowski & Stummer [64].",
"Returning to the general context, in Theorem REF we stated that for each ${P} \\in \\mathbb {S}_{>0}^{K}$ the function ${Q} \\mapsto \\inf _{m\\ne 0}D_{\\varphi }(m\\cdot {Q},{P}) $ is BS-minimizable (cf.",
"(REF )) on all sets $\\textrm {$$\\hspace{-6.544pt}$$} \\subset \\mathbb {S}^{K}$ satisfying (7) and (9).",
"The (corresponding subsetup of the) following Lemma REF is the cornerstone leading from this statement to BS-minimizability of the function ${Q} \\mapsto D_{\\varphi }({Q},{P}))$ on those same sets, for the special divergences in (REF ).",
"After giving the fundamental preparatory Lemma REF , we shall derive from it some BS-minimizability/BS-maximizability results for (extensions of) a variety of important, widely used, closely related divergences respectively entropy/diversity indices.",
"To achieve this in a transparent way, we employ the following three fundamental quantities $H_{\\gamma }(\\mathbf {Q},{P})$ , $I(\\mathbf {Q},{P})$ and $\\widetilde{I}(\\mathbf {Q},{P})$ .",
"To begin with, let $A >0$ be an arbitrary constant (notice that for the choice $A=1$ , all the following vectors $\\mathbf {Q}$ will turn into probability vectors ${Q}$ ).",
"Moreover — for any constellation $(\\gamma , {P},\\mathbf {Q}) \\in \\widetilde{\\Gamma }\\times \\widetilde{\\mathcal {M}}_{1}\\times \\widetilde{\\mathcal {M}}_{2}$ , where $\\widetilde{\\Gamma }\\times \\widetilde{\\mathcal {M}}_{1}\\times \\widetilde{\\mathcal {M}}_{2}:=]0,1[\\times \\mathbb {S}^{K}\\times A \\cdot \\mathbb {S}^{K}$ or $\\widetilde{\\Gamma }\\times \\widetilde{\\mathcal {M}}_{1}\\times \\widetilde{\\mathcal {M}}_{2}:=]-\\infty ,0[\\times \\mathbb {S}^{K}\\times A \\cdot \\mathbb {S}_{>0}^{K} $ or $\\widetilde{\\Gamma }\\times \\widetilde{\\mathcal {M}}_{1}\\times \\widetilde{\\mathcal {M}}_{2}:=]1,\\infty [\\times \\mathbb {S}_{>0}^{K}\\times A \\cdot \\mathbb {S}^{K}$ — let $0 < H_{\\gamma }(\\mathbf {Q},{P}) := \\sum \\displaylimits _{k=1}^{K} (q_{k})^{\\gamma } \\cdot (p_{k})^{1-\\gamma }\\ = \\ 1 + \\gamma \\cdot (A-1) +\\gamma \\cdot (\\gamma -1) \\cdot D_{\\varphi _{\\gamma }}(\\mathbf {Q}, {P}),\\qquad \\gamma \\in \\mathbb {R}\\backslash \\lbrace 0,1\\rbrace ,$ be the modified $\\gamma -$ order Hellinger integral of $\\mathbf {Q}$ and ${P}$ .",
"Furthermore, for any ${P} \\in \\mathbb {S}_{>0}^{K}$ , $\\mathbf {Q} \\in A \\cdot \\mathbb {S}^{K}$ , let $-1 \\ < \\ I(\\mathbf {Q},{P}) := \\sum \\displaylimits _{k=1}^{K} q_{k} \\cdot \\log \\left( \\frac{q_{k}}{p_{k}} \\right)\\ = \\ D_{\\varphi _{1}}(\\mathbf {Q}, {P}) + A - 1,$ be the modified Kullback-Leibler information (modified relative entropy).",
"Finally, for any ${P}\\in \\mathbb {S}^{K}$ , $\\mathbf {Q} \\in A \\cdot \\mathbb {S}_{>0}^{K}$ , let $1 - A \\ \\le \\ \\widetilde{I}(\\mathbf {Q},{P}) := \\sum \\displaylimits _{k=1}^{K} p_{k} \\cdot \\log \\left( \\frac{p_{k}}{q_{k}} \\right)\\ = \\ D_{\\varphi _{0}}(\\mathbf {Q}, {P}) + 1 - A,$ be the modified reverse Kullback-Leibler information (modified reverse relative entropy).",
"In terms of (REF ), (REF ) and (REF ) we obtain the following Lemma 16 Let $A >0$ be an arbitrary constant.",
"(a) Let $\\widetilde{c}>0$ be arbitrary and $(\\gamma , {P},\\mathbf {Q}) \\in \\widetilde{\\Gamma }\\times \\widetilde{\\mathcal {M}}_{1}\\times \\widetilde{\\mathcal {M}}_{2}$ as above.",
"Then one has $\\inf _{m\\ne 0}D_{\\widetilde{c}\\cdot \\varphi _{\\gamma }}(m \\cdot \\mathbf {Q},{P})=\\inf _{m>0}D_{\\widetilde{c}\\cdot \\varphi _{\\gamma }}(m\\cdot \\mathbf {Q},{P})&=&\\frac{\\widetilde{c}}{\\gamma }\\cdot \\left[ 1- A^{\\gamma /(\\gamma -1)} \\cdot \\left[ 1+ \\gamma \\cdot (A-1) + \\frac{\\gamma \\cdot \\left( \\gamma -1\\right) }{\\widetilde{c}}\\cdot D_{\\widetilde{c}\\cdot \\varphi _{\\gamma }}(\\mathbf {Q},{P})\\right]^{-1/\\left( \\gamma -1\\right) }\\right]\\nonumber \\\\& &\\\\&=&\\frac{\\widetilde{c}}{\\gamma }\\cdot \\left[ 1- A^{\\gamma /(\\gamma -1)} \\cdot H_{\\gamma }(\\mathbf {Q},{P})^{-1/\\left( \\gamma -1\\right) }\\right]\\nonumber $ and consequently for any subset $A \\cdot \\Omega $$\\Omega \\subset \\widetilde{\\mathcal {M}}_{2}$ $&&\\hspace{-19.91684pt}\\inf _{\\mathbf {Q}\\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega }}\\ \\inf _{m\\ne 0} D_{\\widetilde{c}\\cdot \\varphi _{\\gamma }}(m \\cdot \\mathbf {Q},{P})=\\frac{\\widetilde{c}}{\\gamma }\\cdot \\left[ 1-A^{\\gamma /(\\gamma -1)} \\cdot \\left[ 1+ \\gamma \\cdot (A-1) +\\frac{\\gamma \\cdot \\left( \\gamma -1\\right) }{\\widetilde{c}}\\cdot \\inf _{\\mathbf {Q}\\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega } }D_{\\widetilde{c}\\cdot \\varphi _{\\gamma }}(\\mathbf {Q},{P})\\right]^{-1/\\left( \\gamma -1\\right) }\\right] ,\\qquad \\ \\\\&&\\hspace{-19.91684pt}\\arg \\inf {}_{\\mathbf {Q}\\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega } }\\inf _{m\\ne 0}\\ D_{\\widetilde{c}\\cdot \\varphi _{\\gamma }}(m \\cdot \\mathbf {Q},{P})=\\arg \\inf {}_{\\mathbf {Q}\\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega } } \\ D_{\\widetilde{c}\\cdot \\varphi _{\\gamma }}(\\mathbf {Q},{P}),\\qquad \\ \\\\&&\\hspace{-19.91684pt}\\inf _{\\mathbf {Q}\\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega }}\\ \\inf _{m\\ne 0} D_{\\varphi _{\\gamma }}(m \\cdot \\mathbf {Q},{P})=\\frac{1}{\\gamma }\\cdot \\left[ 1-A^{\\gamma /(\\gamma -1)} \\cdot \\left[ \\inf _{\\mathbf {Q}\\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega } }H_{\\gamma }(\\mathbf {Q},{P})\\right]^{-1/\\left( \\gamma -1\\right) }\\right] ,\\qquad \\textrm {for \\gamma <0 and \\gamma >1}, \\\\&&\\hspace{-19.91684pt}\\arg \\inf {}_{\\mathbf {Q}\\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega } }\\inf _{m\\ne 0}\\ D_{\\varphi _{\\gamma }}(m \\cdot \\mathbf {Q},{P})=\\arg \\inf {}_{\\mathbf {Q}\\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega } } \\ H_{\\gamma }(\\mathbf {Q},{P}),\\hspace{96.73918pt} \\textrm {for \\gamma <0 and \\gamma >1}, \\\\&&\\hspace{-19.91684pt}\\inf _{\\mathbf {Q}\\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega }}\\ \\inf _{m\\ne 0} D_{\\varphi _{\\gamma }}(m \\cdot \\mathbf {Q},{P})=\\frac{1}{\\gamma }\\cdot \\left[ 1-A^{\\gamma /(\\gamma -1)} \\cdot \\left[ \\sup _{\\mathbf {Q}\\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega } }H_{\\gamma }(\\mathbf {Q},{P})\\right]^{-1/\\left( \\gamma -1\\right) }\\right] , \\hspace{22.76228pt}\\textrm {for \\gamma \\in ]0,1[}, \\\\&&\\hspace{-19.91684pt}\\arg \\inf {}_{\\mathbf {Q}\\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega } }\\inf _{m\\ne 0}\\ D_{\\varphi _{\\gamma }}(m \\cdot \\mathbf {Q},{P})=\\arg \\sup {}_{\\mathbf {Q}\\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega } } \\ H_{\\gamma }(\\mathbf {Q},{P}), \\hspace{96.73918pt} \\textrm {for \\gamma \\in ]0,1[},$ provided that the infimum on the right-hand side of (REF ) exists.",
"(b) For any ${P} \\in \\mathbb {S}_{>0}^{K}$ , $\\mathbf {Q} \\in A \\cdot \\mathbb {S}^{K}$ , $\\widetilde{c}>0$ one gets $\\inf _{m\\ne 0}D_{\\widetilde{c}\\cdot \\varphi _{1}}(m \\cdot \\mathbf {Q},{P})=\\inf _{m>0}D_{\\widetilde{c}\\cdot \\varphi _{1}}(m \\cdot \\mathbf {Q},{P})&=&\\widetilde{c}\\cdot \\left[ 1- A \\cdot \\exp \\left( -\\frac{1}{A \\cdot \\widetilde{c}}\\cdot D_{\\widetilde{c}\\cdot \\varphi _{1}}(\\mathbf {Q},{P})+ \\frac{1}{A} -1 \\right) \\right] \\\\&=&\\widetilde{c}\\cdot \\left[ 1- A \\cdot \\exp \\left( -\\frac{1}{A}\\cdot I(\\mathbf {Q},{P})\\right) \\right]\\nonumber $ and consequently for any subset $A \\cdot \\Omega $$\\Omega \\subset A \\cdot \\mathbb {S}^{K}$ $& &\\inf _{\\mathbf {Q}\\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega } } \\ \\inf _{m\\ne 0}D_{\\widetilde{c}\\cdot \\varphi _{1}}(m \\cdot \\mathbf {Q},{P})=\\widetilde{c}\\cdot \\left[ 1-A \\cdot \\exp \\left( -\\frac{1}{A \\cdot \\widetilde{c}}\\cdot \\inf _{Q\\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega } }D_{\\widetilde{c}\\cdot \\varphi _{1}}(\\mathbf {Q},{P}) + \\frac{1}{A} -1 \\right) \\right] , \\\\& & \\arg \\inf {}_{\\mathbf {Q}\\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega } } \\ \\inf _{m\\ne 0}D_{\\widetilde{c}\\cdot \\varphi _{1}}(m \\cdot \\mathbf {Q},{P}) =\\arg \\inf {}_{\\mathbf {Q}\\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega } } \\ D_{\\widetilde{c}\\cdot \\varphi _{1}}(Q,P),\\qquad \\ \\\\& &\\inf _{\\mathbf {Q}\\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega } } \\ \\inf _{m\\ne 0}D_{\\varphi _{1}}(m \\cdot \\mathbf {Q},{P})=\\left[ 1- A \\cdot \\exp \\left( -\\frac{1}{A}\\cdot \\inf _{Q\\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega } }I(\\mathbf {Q},{P}) \\right) \\right] , \\\\& & \\arg \\inf {}_{\\mathbf {Q}\\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega } } \\ \\inf _{m\\ne 0}D_{\\varphi _{1}}(m \\cdot \\mathbf {Q},{P}) =\\arg \\inf {}_{\\mathbf {Q}\\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega } } \\ I(\\mathbf {Q},{P}),\\qquad \\ $ provided that the infimum on the right-hand side of (REF ) exists.",
"(c) For any ${P}\\in \\mathbb {S}^{K}$ , $\\mathbf {Q} \\in A \\cdot \\mathbb {S}_{>0}^{K}$ , $\\widetilde{c}>0$ we obtain $\\inf _{m\\ne 0}D_{\\widetilde{c}\\cdot \\varphi _{0}}(m \\cdot \\mathbf {Q},{P})=\\inf _{m>0}D_{\\widetilde{c}\\cdot \\varphi _{0}}(m \\cdot \\mathbf {Q},{P})&=& D_{\\widetilde{c}\\cdot \\varphi _{0}}(\\mathbf {Q},{P})+ \\widetilde{c} \\cdot (1 - A + \\log A) \\\\&=& \\widetilde{c} \\cdot \\Big ( \\widetilde{I}(\\mathbf {Q},{P}) + \\log A \\Big )\\nonumber $ and consequently for any set subset $A \\cdot \\Omega $$\\Omega \\subset A \\cdot \\mathbb {S}_{>0}^{K}$ $&&\\hspace{-19.91684pt}\\inf _{\\mathbf {Q} \\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega }} \\ \\inf _{m\\ne 0} D_{\\widetilde{c}\\cdot \\varphi _{0}}(m \\cdot \\mathbf {Q},{P})= \\widetilde{c} \\cdot (1 - A + \\log A) +\\inf _{\\mathbf {Q} \\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega }} \\ D_{\\widetilde{c}\\cdot \\varphi _{0}}(\\mathbf {Q},{P}), \\\\&&\\hspace{-19.91684pt}\\arg \\inf {}_{\\mathbf {Q}\\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega } } \\ \\inf _{m\\ne 0}D_{\\widetilde{c}\\cdot \\varphi _{0}}(m \\cdot \\mathbf {Q},{P})=\\arg \\inf {}_{\\mathbf {Q}\\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega } } \\ D_{\\widetilde{c}\\cdot \\varphi _{0}}(\\mathbf {Q},{P}), \\qquad \\ \\\\&&\\hspace{-19.91684pt}\\inf _{\\mathbf {Q} \\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega }} \\ \\inf _{m\\ne 0} D_{\\varphi _{1}}(m \\cdot \\mathbf {Q},{P})= \\log A +\\inf _{\\mathbf {Q} \\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega }} \\ \\widetilde{I}(\\mathbf {Q},{P}), \\\\&&\\hspace{-19.91684pt}\\arg \\inf {}_{\\mathbf {Q}\\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega } } \\ \\inf _{m\\ne 0} D_{\\varphi _{0}}(m \\cdot \\mathbf {Q},{P})=\\arg \\inf {}_{\\mathbf {Q}\\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega } } \\ \\widetilde{I}(\\mathbf {Q},{P}), \\qquad \\ $ provided that the infimum on the right-hand side of (REF ) exists.",
"The proof of Lemma REF is given in Appendix C. Remark 17 Notice that for ${P} \\in \\mathbb {S}_{>0}^{K}$ and $\\mathbf {Q} \\in A \\cdot \\mathbb {S}^{K}$ , the modified Kullback-Leibler information has the property $I(\\mathbf {Q},{P}) \\ge 0$ if $A \\ge 1$ (cf.",
"(REF )); otherwise, $I(\\mathbf {Q},{P})$ may become negative, as can be easily seen from the case where ${P} := {P}^{unif} :=(\\frac{1}{K}, \\ldots , \\frac{1}{K})$ is the probability vector of frequencies of the uniform distribution on $\\lbrace 1, \\ldots , K\\rbrace $ , and $\\mathbf {Q} := (\\frac{1}{K+1}, 0, \\ldots , 0)$ .",
"Analogously, for ${P}\\in \\mathbb {S}^{K}$ and $\\mathbf {Q} \\in A \\cdot \\mathbb {S}_{>0}^{K}$ one gets $\\widetilde{I}(\\mathbf {Q},{P}) \\ge 0$ if $A \\le 1$ (cf.",
"(REF )); otherwise, $\\widetilde{I}(\\mathbf {Q},{P})$ may become negative (take e.g.",
"$\\mathbf {Q} = (\\frac{K+1}{K}, \\ldots , \\frac{K+1}{K})$ and ${P} := (1,0, \\ldots , 0)$ ).",
"Remark 18 (a) In the context of Remark REF (vi), according to (REF ) applied to $\\varphi := \\widetilde{c}\\cdot \\varphi _{\\gamma }$ , for all cases $\\gamma \\in \\, ]-\\infty ,0[ \\ \\cup \\ ]0,1[ \\ \\cup \\ [ \\ 2,\\infty [$ the left-hand side of each of (REF ), (), () is independent of $A >0$ and equal to $-\\lim _{n\\rightarrow \\infty }\\frac{1}{n}\\log \\,\\mathbb {\\Pi }\\left[\\xi _{n}^{w\\mathbf {W}}\\in \\textrm {\\right.$$\\hspace{-6.544pt}$$}$ where — as will be shown below — the corresponding $\\mathbf {W}$ 's have probability distribution $\\mathbb {}[\\cdot \\,]=\\mathbb {\\Pi }[W_{1}\\in \\cdot \\,]$ (cf.",
"(REF )) which varies “quite drastically” with $\\gamma $ (and the case $\\gamma \\in ]1,2[$ has to be even excluded for analytical difficulties because in this case there are some indications that the representation (REF ) only holds for some signed probability distribution $$ (e.g.",
"having a density with positive and negative values).).",
"Analogously, each of the left-hand sides of (REF ), (), (REF ), () is also independent of $A >0$ and equal to $-\\lim _{n\\rightarrow \\infty }\\frac{1}{n}\\log \\,\\mathbb {\\Pi }\\left[\\xi _{n}^{w\\mathbf {W}}\\in \\textrm {\\right.$$\\hspace{-6.544pt}$$}$ for some $\\mathbf {W}$ of respective distribution.",
"Hence, by inversion, all the extremum-describing target quantities $\\inf _{\\mathbf {Q}\\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega } }D_{\\widetilde{c}\\cdot \\varphi _{\\gamma }}(\\mathbf {Q},{P})$ ($\\gamma \\in \\mathbb {R}\\backslash ]1,2[$ ), $\\inf _{\\mathbf {Q}\\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega } }H_{\\gamma }(\\mathbf {Q},{P})$ ($\\gamma \\in ]-\\infty ,0[ \\, \\cup \\, [2,\\infty [$ ), $\\sup _{\\mathbf {Q}\\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega } }H_{\\gamma }(\\mathbf {Q},{P})$ ($\\gamma \\in ]0,1[$ ), $\\inf _{Q\\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega } }I(\\mathbf {Q},{P})$ and $\\inf _{Q\\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega } }\\widetilde{I}(\\mathbf {Q},{P})$ can be expressed as $G\\big (-\\lim _{n\\rightarrow \\infty }\\frac{1}{n}\\log \\,\\mathbb {\\Pi }\\left[\\xi _{n}^{w\\mathbf {W}}\\in \\textrm {\\right.$$\\hspace{-6.544pt}$$})\\, $ for some explicitly known ($A-$ dependent) function $G$ .",
"This means that — in the sense of Definition REF — all the corresponding four “cornerstone quantities” $D_{\\widetilde{c}\\cdot \\varphi _{\\gamma }}(\\mathbf {Q},{P})$ , $H_{\\gamma }(\\mathbf {Q},{P})$ , $I(\\mathbf {Q},{P})$ , $\\widetilde{I}(\\mathbf {Q},{P})$ are BS-minimizable, respectively BS-maximizable, on $\\mathbf {\\Omega } = A \\cdot \\textrm {$$\\hspace{-6.544pt}$$}$ .",
"The above-mentioned inversions (i.e.",
"constructions of $G(\\cdot )$ ) will be concretely carried out below — namely in the Propositions REF , REF , REF , REF , REF and REF .",
"In those, we also involve the BS-minimizability/maximizability of several other important closely related divergences and measures of entropy (measures of diversity, measures of heterogeneity/homogeneity, measures of concentration) which (i) are widely used in information theory and its applications to artificial intelligence, machine learning and physics, and which (ii) can be built from the above-mentioned four cornerstone quantities (power divergences, Hellinger integrals, Kullback-Leibler information divergences).",
"(b) The special case $\\varphi := \\widetilde{c}\\cdot \\varphi _{\\gamma }$ ($\\gamma \\in \\, ]-\\infty ,0[ \\ \\cup \\ ]0,1[ \\ \\cup \\ [ \\ 2,\\infty [$ ) of Theorem REF works analogously to (a), with the differences that we employ $A=1$ (instead of arbitrary $A>0$ ), () (instead of (REF )), $\\mathbb {\\Pi }_{X_{1}^{n}}[\\cdot ]$ (instead of $\\mathbb {\\Pi }[\\cdot ]$ ), and $\\xi _{n,\\mathbf {X}}^{w\\mathbf {W}}$ (instead of $\\xi _{n}^{w\\mathbf {W}}$ ).",
"(c) From the proof of Lemma REF in Appendix C below, one can see that for the important case $\\gamma =2$ the formulas (REF ) to () also hold for $A<0$ .",
"In the following, we further elaborate the three points (a),(b) and (c) of Remark REF “comprehensively and unifyingly”, where the expression “BS minimizable/maximizable” always has to be interpreted accordingly in terms of $-\\lim _{n\\rightarrow \\infty }\\frac{1}{n}\\log \\mathbb {\\Pi }[\\xi _{n}^{w\\mathbf {W}} \\in \\cdot \\ ]$ respectively $-\\lim _{n\\rightarrow \\infty }\\frac{1}{n}\\log \\mathbb {\\Pi }_{X_{1}^{n}}[\\xi _{n,\\mathbf {X}}^{w\\mathbf {W}}\\in \\cdot \\ ]$ (without explicit mentioning, for the sake of brevity).",
"Let us fix $\\widetilde{c}=1$ and an arbitrary triple $(\\gamma , {P},\\mathbf {Q})$ which satisfies the assumptions of Lemma REF (a) with $A := \\sum _{k=1}^{K} q_{k} >0$ .",
"For such a setup, we have obtained in (REF ) the $\\gamma -$ order Hellinger integral (of $\\mathbf {Q}$ and ${P}$ ) $H_{\\gamma }(\\mathbf {Q},{P}) > 0$ , which is not a divergence; as a terminology-concerning side remark, let us mention that $H_{\\gamma }(\\mathbf {Q},{P})$ ($\\gamma \\ge 1$ ) is called relative information generating function in Guiasu & Reischer [144], see e.g.",
"also Clark [90]; moreover, $H_{\\gamma }({Q},{P})$ is sometimes termed ($\\gamma -$ order) Chernoff coefficient being a component of the Chernoff distances/informations [85].",
"Torgersen [361] uses the name ($\\gamma -$ order) Hellinger transform.",
"Notice that the special case $\\gamma =\\frac{1}{2}$ is nothing but (a multiple of) the well-known important Bhattacharyya coefficient (cf.",
"[48],[49],[50]) $BC(\\mathbf {Q},{P}) :=H_{1/2}(\\mathbf {Q},{P}) =\\sum _{k=1}^{K} \\sqrt{q_{k} \\cdot p_{k}} =1 + \\frac{1}{2} \\cdot (A-1) + \\frac{1}{2} \\cdot (\\frac{1}{2}-1) \\cdot D_{\\varphi _{\\frac{1}{2}}}(\\mathbf {Q},{P})\\nonumber $ 0.3cm which is also known as affinity (cf.",
"Matusita [256], see e.g.",
"also Toussaint [362]) and (classic, non-quantum) fidelity similarity (cf.",
"e.g.",
"Deza & Deza [113]); for non-probability vectors $\\mathbf {P} \\in \\mathbb {R}_{\\ge 0}^{K}$ one can simply retransform ${P} := \\frac{\\mathbf {P}}{M_{\\mathbf {P}}}$ and thus imbed $BC(\\mathbf {Q},\\mathbf {P}) = \\sqrt{M_{\\mathbf {P}}} \\cdot BC(\\mathbf {Q},{P})$ into our BS context.",
"There is a vast literature on very recent applications of the Bhattacharyya coefficient, for instance it appears exemplarily in Peng & Li [289] for object tracking from successive video frames, Ayed et al.",
"[26] for efficient graph cut algorithms, Patra et al.",
"[287] for collaborative filtering in sparse data, El Merabet et al.",
"[119] for region classification in intelligent transport systems in order to compensate the lack of performance of Global Navigation Satellites Systems, Chiu et al.",
"[86] for the design of interactive mobile augmented reality systems, Noh et al.",
"[274] for dimension reduction in interacting fluid flow models, Bai et al.",
"[29] for material defect detection through ultrasonic array imaging, Dixit & Jain [115] for the design of recommender systems on highly sparse context aware datasets, Guan et al.",
"[143] for visible light positioning methods based on image sensors, Lin et al.",
"[220] for probabilistic representation of color image pixels, Chen et al.",
"[80] for distributed compressive video sensing, Jain et al.",
"[162] for the enhancement of multistage user-based collaborative filtering in recommendation systems, Pascuzzo et al.",
"[285] for brain-diffusion-MRI based early diagnosis of the sporadic Creutzfeldt–-Jakob disease, Sun et al.",
"[351] for the design of automatic detection methods multitemporal (e.g.",
"landslide) point clouds, Valpione et al.",
"[377] for the investigation of T cell dynamics in immunotherapy, Wang et al.",
"[387] for the tracking and prediction of downbursts from meteorological data, Xu et al.",
"[403] for adaptive distributed compressed video sensing for coal mine monitoring, Zhao et al.",
"[424] for the shared sparse machine learning of the affective content of images, Chen et al.",
"[82] for image segmentation and domain partitioning, De Oliveira et al.",
"[105] for the prediction of cell-penetrating peptides, Eshaghi et al.",
"[122] for the identification of multiple sclerosis subtypes through machine learning of brain MRI scans, Feng et al.",
"[125] for improvements of MRI-based detection of epilepsy-causing cortical malformations, Hanli et al.",
"[153] for designing pilot protection schemes for transmission lines, Jiang et al.",
"[170] for flow-assisted visual tracking through event cameras, Lysiak & Szmajda [235] for comparisons of selected feature quality evaluations, Joel & Sivakumar [172] for the despeckling enhancement of medical ultrasound image quality, Reising et al.",
"[305] for the design of security protection of Internet-of-Things (IoT) devices, Skrbic et al.",
"[338] for the uncovering of interplays between amino acid sequences and local structures in proteins, Tsiapoki et al.",
"[365] for the improvement of the detection performance of structural health monitoring frameworks, van Molle et al.",
"[381] for uncertainty quantification in deep neural networks, Yang et al.",
"[413] for the determination of the onset of transient signals, and Zhou & Yu [427] for the modelling of spatiotemporal human eye movements.",
"To proceed with the general context, for any $\\gamma \\in \\, ]-\\infty ,0[ \\ \\cup \\ ]0,1[ \\ \\cup \\ [ \\ 2,\\infty [$ let the function $h_{\\gamma } : \\, ]0,\\infty [ \\, \\mapsto \\, ]-\\infty ,\\infty [$ be such that $x \\mapsto h_{\\gamma }(1 + \\gamma \\cdot (A-1) + \\gamma \\cdot (\\gamma -1) \\cdot x)$ is continuous and strictly increasing (respectively, strictly decreasing) for all $x \\ge 0$ with $1 + \\gamma \\cdot (A-1) + \\gamma \\cdot (\\gamma -1) \\cdot x > 0$ ; since $D_{\\varphi _{\\gamma }}(\\mathbf {Q},{P})$ is BS-minimizable on $\\mathbf {\\Omega } = A \\cdot \\textrm {$$\\hspace{-6.544pt}$$}$ , then also the — not necessarily nonnegative — quantity $h_{\\gamma }\\Big (1 + \\gamma \\cdot (A-1) + \\gamma \\cdot (\\gamma -1) \\cdot D_{\\varphi _{\\gamma }}(\\mathbf {Q},{P})\\Big )= h_{\\gamma }\\Big (H_{\\gamma }(\\mathbf {Q},{P})\\Big )$ is BS-minimizable (respectively, BS-maximizable) on $\\mathbf {\\Omega } = A \\cdot \\textrm {$$\\hspace{-6.544pt}$$}$ .",
"If $h_{\\gamma }$ satisfies additionally $h_{\\gamma }(1)=0$ as well as $h_{\\gamma }(1 + \\gamma \\cdot (A-1) + \\gamma \\cdot (\\gamma -1) \\cdot x) \\ge 0$ for all $x \\ge 0$ with $1 + \\gamma \\cdot (A-1) + \\gamma \\cdot (\\gamma -1) \\cdot x > 0$ , then $D_{h_{\\gamma }}(\\mathbf {Q},{P}) :=h_{\\gamma }\\Big (1 + \\gamma \\cdot (A-1) + \\gamma \\cdot (\\gamma -1) \\cdot D_{\\varphi _{\\gamma }}(\\mathbf {Q},{P})\\Big )= h_{\\gamma }\\Big (H_{\\gamma }(\\mathbf {Q},{P})\\Big )$ constitutes a divergence in the usual sense that $D_{h_{\\gamma }}(\\mathbf {Q},{P}) \\ge 0 $ with equality iff ${Q}={P}$ .",
"which is BS-minimizable on $\\mathbf {\\Omega } = A \\cdot \\textrm {$$\\hspace{-6.544pt}$$}$ (respectively, BS-maximizable on $\\mathbf {\\Omega } = A \\cdot \\textrm {$$\\hspace{-6.544pt}$$}$ ).",
"Let us consider some important examples.",
"For the identity mapping $h_{\\gamma }^{Id}(y) := y$ ($y>0$ ) the function $x \\mapsto 1 + \\gamma \\cdot (A-1) + \\gamma \\cdot (\\gamma -1) \\cdot x$ is strictly increasing for $\\gamma <0$ and $\\gamma >1$ (on the required domain of $x$ ), and strictly decreasing for $\\gamma \\in ]0,1[$ .",
"Accordingly, $H_{\\gamma }(\\mathbf {Q},{P})$ is BS-minimizable on $\\mathbf {\\Omega } = A \\cdot \\textrm {$$\\hspace{-6.544pt}$$}$ for $\\gamma <0$ and $\\gamma \\ge 2$ and BS-maximizable on $\\mathbf {\\Omega } = A \\cdot \\textrm {$$\\hspace{-6.544pt}$$}$ for $\\gamma \\in \\, ]0,1[$ (this is consistent with (), ()); in particular, the Bhattacharyya coefficient $BC(\\mathbf {Q},{P})$ is BS-maximizable on $\\mathbf {\\Omega } = A \\cdot \\textrm {$$\\hspace{-6.544pt}$$}$ .",
"Some other important choices are $& & h_{\\gamma }(y) := h_{c_{1},c_{2},c_{3}}(y) := c_{1} \\cdot \\Big (y^{c_{2}} - c_{3} \\Big ),\\qquad y>0, \\, c_{1}, c_{2} \\in \\mathbb {R}\\backslash \\lbrace 0\\rbrace , \\,c_{3} \\in \\mathbb {R},\\\\& & h_{\\gamma }(y) := h_{c_{4},f}^{R}(y) := \\lim _{c_{2} \\rightarrow 0} h_{c_{4}/f(c_{2}),c_{2},1}(y) =\\frac{c_{4}}{f^{\\prime }(0)} \\cdot \\log (y),\\qquad y>0, \\, c_{4} \\in \\mathbb {R}\\backslash \\lbrace 0\\rbrace , \\,\\\\& & h_{\\gamma }(y) := h_{c_{5},c_{6}}^{GB2}(y) := c_{5} \\cdot (\\arccos (y))^{c_{6}},\\qquad \\gamma \\in \\,]0,1[, \\, y \\in \\, ]0,1], \\, c_{5}>0, \\, c_{6}>0,\\\\& & h_{\\gamma }(y) := h_{\\nu ,c_{7}}^{BB}(y) := c_{7} \\cdot \\frac{\\log (1-\\frac{1-y}{\\nu })}{\\log (1-\\frac{1}{\\nu })},\\qquad \\gamma \\in \\,]0,1[, \\, y \\in \\, ]0,1], \\, c_{7}>0, \\, \\nu \\in ]-\\infty ,0[ \\, \\cup \\, ]1,\\infty [,$ where the constants $c_{1}$ to $c_{7}$ may depend on $\\gamma $ , and $f$ is some (maybe $\\gamma -$ dependent) function which is differentiable in a neighborhood of 0 and satisfies $f(0)=0$ , $f^{\\prime }(0) \\ne 0$ (e.g.",
"$f(z)=c_{8} \\cdot z$ for some non-zero constant $c_{8}$ ).",
"Clearly, $h_{c_{1},c_{2},c_{3}}(\\cdot )$ is strictly increasing (respectively, strictly decreasing) if and only if $c_{1} \\cdot c_{2} >0$ (respectively, $c_{1} \\cdot c_{2} <0$ ).",
"Moreover, $h_{c_{4},f}^{R}(\\cdot )$ is strictly increasing (respectively, strictly decreasing) if and only if $\\frac{c_{4}}{f^{\\prime }(0)} >0$ (respectively, $\\frac{c_{4}}{f^{\\prime }(0)} < 0$ ).",
"Furthermore, both $h_{c_{5},c_{6}}^{GB2}(\\cdot )$ and $h_{\\nu ,c_{7}}^{GoBa}(\\cdot )$ are strictly decreasing.",
"For instance, the special case $h_{\\gamma }(y) = h_{c_{4},Id}^{R}(y)$ with $c_{4} := \\frac{1}{\\gamma \\cdot (\\gamma -1)}$ (recall that $\\gamma \\in \\, ]-\\infty ,0[ \\ \\cup \\ ]0,1[ \\ \\cup \\ [ \\ 2,\\infty [$ ) and identity function $f := Id$ leads to the quantities $\\hspace{-42.67912pt}R_{\\gamma }(\\mathbf {Q},{P}) &: =&D_{h_{c_{4},Id}^{R}}(\\mathbf {Q},{P}) =\\frac{\\log \\Big ( 1 + \\gamma \\cdot (A-1) +\\gamma \\cdot (\\gamma -1) \\cdot D_{\\varphi _{\\gamma }}(\\mathbf {Q},{P}) \\Big )}{\\gamma \\cdot (\\gamma -1)}= \\frac{\\log \\Big ( H_{\\gamma }(\\mathbf {Q},{P}) \\Big )}{\\gamma \\cdot (\\gamma -1)}\\nonumber \\\\&=& \\frac{\\log \\Big ( \\sum \\displaylimits _{k=1}^{K} (q_{k})^{\\gamma } \\cdot (p_{k})^{1-\\gamma } \\Big )}{\\gamma \\cdot (\\gamma -1)} \\, ,\\qquad \\gamma \\in \\, ]-\\infty ,0[ \\ \\cup \\ ]0,1[ \\ \\cup \\ [ \\ 2,\\infty [,$ (provided that all involved power divergences are finite), which are thus BS-minimizable on $\\mathbf {\\Omega } = A \\cdot \\textrm {$$\\hspace{-6.544pt}$$}$ ; notice that $R_{\\gamma }(\\mathbf {Q},{P}) \\ge 0$ if $\\gamma \\in \\ ]0,1[ \\ \\cup \\ [ \\ 2,\\infty [$ together with $A \\in [1,\\infty [$ , and if $\\gamma \\in \\, ]-\\infty ,0[$ together with $A \\in \\, ]0,1]$ .",
"The special subcase $A=1$ in (REF ) (and thus, $\\mathbf {Q}$ is a probability vector ${Q}$ ) corresponds to the prominent Renyi divergences/distances [309] (in the scaling of e.g.",
"Liese & Vajda [217] and in probability-vector form), see e.g.",
"van Erven & Harremoes [380] for a comprehensive study of their properties; as a side remark, $\\gamma \\cdot (\\gamma -1) \\cdot R_{\\gamma }({Q},{P})$ is also employed in the Chernoff distances/informations [85].",
"The special subcase $R_{1/2}({Q},{P})$ (i.e.",
"$\\gamma =1/2$ and $A=1$ in (REF )) corresponds to (a multiple of) the widely used Bhattacharyya distance (of type 1) between ${Q}$ and ${P}$ , cf.",
"[48] (see e.g.",
"also Kailath [178]).",
"Sometimes, $\\exp (R_{\\gamma }(\\mathbf {Q},{P}))$ is also called Renyi divergence/distance.",
"Some exemplary (relatively) recent studies and applications of Renyi divergences $R_{\\gamma }({Q},{P})$ (respectively, their multiple or exponential) — aside from the substantial statistical literature — appear e.g.",
"in Zhao et al.",
"[423] for the study of isomeric stability of fullerenes (which are e.g.",
"employed for state-of-the-art organic solar cells), in the papers of Sundaresan [353], Bunte & Lapidoth [67], Sason [318], Kumar et al.",
"[205] for (mismatch-cases of) coding and guessing as well as task partitioning, in the papers of Prest [296], Bai et al.",
"[30] for lattice-based cryptography, in He et al.",
"[158] for robot active olfaction search (by infotaxis) in turbulent flows, in Momeni et al.",
"[263] for the design of reprogrammable encrypted graphene-based coding metasurfaces, in Staszowska et al.",
"[340] for accurate and precise cluster analysis for super-resolution localization microscopy, in Yu & Tan [414] for distributed source simulation problems, in Zhang et al.",
"[419] for sensor control, in Yu & Tan [415] for the so-called random variable simulation problem, in Blanchet et al.",
"[55] for the robust treatment of extreme values in rainfall accumulation data, in Cai et al.",
"[70] for sensor tasking for search and catalog maintenance of geosynchronous space objects, in Gholami & Hodtani [134] for refinements of safety-and-security-targeted location verification systems in wireless communication networks (e.g in Intelligent Transportation Systems (ITSs) and vehicular technology), in Seweryn et al.",
"[326] for the assessment of similarity and diversity of expression profiles in single cell systems, in Zhou [426] for the study of secrecy constraints in key generation problems where side information might be present at untrusted users, in Makkawi et al.",
"[243] for the design of an automated decision-support framework for adaptive diagnosis of fault–tolerant multi–sensor data fusion for vehicle localization, in Mao et al.",
"[245] for privacy-preserving computation offloading for parallel deep neural networks training.",
"There is vast literature on recent applications of the above-mentioned special case $R_{1/2}({Q},{P})$ — that is, the Bhattacharyya distance (of type 1); for instance, it appears in Tarighati & Jalden [356] for rate balancing in wireless sensor networks, Bi et al.",
"[51], [52] for certain uncertainty quantifications respectively stochastic sensitivity analyses in mechanical systems and signal processing, Fu & He [130] for the design of multibit quantizers for cooperative spectrum sensing in cognitive radio networks, Cohen et al.",
"[91] for adaptive and causal random linear network coding with forward error correction for a point-to-point communication channel with delayed feedback, Xu et al.",
"[401] for cost minimization problems of big data analytics on geo-distributed data centers connected to renewable energy sources with unpredictable capacity, Xu et al.",
"[402] for community identification in networks, Arrigoni & Madsen [18] for automated discovering of low-energy defect configurations in materials, Fan et al.",
"[123] for region-merging-based methods for synthetic aperture radar (SAR) image segmentation, Mahfouz et al.",
"[242] for some refined ensemble classifications in microarray-based automated cancer diagnosis, Matchev & Shyamsundar [252] for some machine-learning based signal discovery in high energy physics (HEP) experiments, Wang et al.",
"[386] for the investigation of intratumoral heterogeneity (ITH) of some gastric cancer, Webster et al.",
"[389] for the characterization, identification, clustering and classification of disease, and Xiahou et al.",
"[396] for the prediction of remaining useful life (RUL) through fusion of expert knowledge and condition monitoring information.",
"As a further example, consider $\\mathcal {B}_{\\gamma ,c_{5},c_{6}}({Q},{P}) &: =&D_{h_{c_{5},c_{6}}^{GB2}}({Q},{P}) =c_{5} \\cdot \\Big (\\arccos \\Big (1 + \\gamma \\cdot (\\gamma -1) \\cdot D_{\\varphi _{\\gamma }}({Q},{P})\\Big ) \\, \\Big )^{c_{6}}= c_{5} \\cdot \\Big (\\arccos \\Big (H_{\\varphi _{\\gamma }}({Q},{P})\\Big ) \\, \\Big )^{c_{6}}\\nonumber \\\\&=& c_{5} \\cdot \\Big (\\arccos \\Big (\\sum \\displaylimits _{k=1}^{K} (q_{k})^{\\gamma } \\cdot (p_{k})^{1-\\gamma }\\Big ) \\, \\Big )^{c_{6}}\\ge 0 \\, ,\\qquad \\gamma \\in \\, ]0,1[, \\, c_5 >0, \\, c_{6} >0,\\nonumber $ which is BS-maximizable on $\\textrm {$$\\hspace{-6.544pt}$$}$ .",
"The case $\\mathcal {B}_{1/2,1,1}({Q},{P})$ corresponds to the well-known Bhattacharyya arccos distance (Bhattacharyya distance of type 2) in [50] (which is also called Wootters distance [395]), and $\\mathcal {B}_{1/2,1,2}({Q},{P})$ to its variant in [49]; the case $\\mathcal {B}_{1/2,2,1}({Q},{P})$ is known as Fisher distance or Rao distance or geodesic distance (see e.g.",
"Deza & Deza [113]); a nice graphical illustration of the geometric connection between the Fisher distance $\\mathcal {B}_{1/2,2,1}({Q},{P})$ and the Hellinger distance/metric $\\sqrt{\\frac{1}{2} \\cdot D_{\\varphi _{1/2}}({Q},{P})}$ can be found e.g.",
"on p.35 in Ay et al.",
"[25].",
"Some exemplary applications of the Bhattacharyya arccos distance $\\mathcal {B}_{1/2,1,1}({Q},{P})$ can be found e.g.",
"in Rao [301] and Juhasz [177] for cluster analysis of human populations, in Martin-Fernandez et al.",
"[251] for general hierarchical clustering, Greenacre [141] for metric scaling, and in Chen et al.",
"[79] for clustering high-dimensional microbial data from RNA sequencing.",
"Let us give another example, namely $\\widetilde{\\mathcal {B}}_{\\gamma ,\\nu ,c_{7}}({Q},{P}) &: =&D_{h_{\\nu ,c_{7}}^{BB}}({Q},{P}) =\\frac{c_{7}}{\\log (1-\\frac{1}{\\nu })} \\cdot \\log \\bigg (1-\\frac{1-\\Big ( 1 + \\gamma \\cdot (\\gamma -1) \\cdot D_{\\varphi _{\\gamma }}({Q},{P})\\Big )}{\\nu }\\bigg )\\nonumber \\\\&=& \\frac{c_{7}}{\\log (1-\\frac{1}{\\nu })} \\cdot \\log \\bigg (1-\\frac{1-H_{\\varphi _{\\gamma }}({Q},{P})}{\\nu }\\bigg )= \\frac{c_{7}}{\\log (1-\\frac{1}{\\nu })} \\cdot \\log \\bigg (1-\\frac{1-\\sum \\displaylimits _{k=1}^{K} (q_{k})^{\\gamma } \\cdot (p_{k})^{1-\\gamma }}{\\nu }\\bigg )\\in [0, c_{7}[ \\,, \\nonumber \\\\& & \\hspace{227.62204pt} \\gamma \\in \\,]0,1[, \\, \\, c_{7}>0, \\, \\nu \\in \\, ]-\\infty ,0[ \\, \\cup \\, ]1,\\infty [,\\nonumber $ which is BS-maximizable on $\\textrm {$$\\hspace{-6.544pt}$$}$ .",
"The case $\\widetilde{\\mathcal {B}}_{1/2,\\nu ,1}({Q},{P})$ corresponds to the Bounded Bhattacharyya Distance Measures of Jolad et al.",
"[174].",
"We can also employ divergences of the form $\\breve{R}_{\\gamma }({Q},{P}) :=R_{\\gamma }(T_{1}({Q}),T_{2}({P}))$ and analogously power divergences $\\breve{D}_{\\widetilde{c} \\cdot \\varphi _{\\gamma }}({Q},{P}) :=D_{\\widetilde{c} \\cdot \\varphi _{\\gamma }}(T_{1}({Q}),T_{2}({P}))$ etc.",
"where $T_{1}: \\mathcal {D}_{1} \\mapsto \\mathcal {R}_{1}$ , $T_{2}: \\mathcal {D}_{1} \\mapsto \\mathcal {R}_{2}$ are (say) invertible functions on appropriately chosen subsets $\\mathcal {D}_{1}, \\mathcal {D}_{2}, \\mathcal {R}_{1}, \\mathcal {R}_{2}$ of the probability-vector simplex $\\mathbb {S}^{K}$ .",
"For instance, consider the following special case (with a slight abuse of notation): $\\breve{R}_{\\gamma }({Q},{P}): = R_{\\gamma }(\\widetilde{{Q}},\\widetilde{{P}})=\\frac{1}{\\gamma \\cdot (\\gamma -1)} \\cdot \\log \\left( \\sum \\displaylimits _{k=1}^{K}\\left(\\frac{(q_{k})^{\\nu _{1}}}{\\sum _{j=1}^{K} (q_{j})^{\\nu _{1}}}\\right)^{\\gamma } \\cdot \\left(\\frac{(p_{k})^{\\nu _{2}}}{\\sum _{j=1}^{K} (p_{j})^{\\nu _{2}}}\\right)^{1-\\gamma }\\right)$ where (i) $\\widetilde{{Q}} := (\\widetilde{q}_{k})_{k=1}^{K}$ with $\\widetilde{q}_{k} := \\frac{(q_{k})^{\\nu _{1}}}{\\sum _{j=1}^{K} (q_{j})^{\\nu _{1}}}$ is the escort probability distribution (in vector form) associated with the probability distribution (in vector form) ${Q} := (q_{k})_{k=1}^{K} \\in \\mathbb {S}_{> 0}^{K}$ , and (ii) $\\widetilde{{P}} := (\\widetilde{p}_{k})_{k=1}^{K}$ with $\\widetilde{p}_{k} := \\frac{(p_{k})^{\\nu _{2}}}{\\sum _{j=1}^{K} (p_{j})^{\\nu _{2}}}$ is the escort probability distribution associated with the probability distribution ${P} := (p_{k})_{k=1}^{K} \\in \\mathbb {S}_{> 0}^{K}$ , in terms of some fixed escort parameters $\\nu _{1} >0$ , $\\nu _{2} >0$ .",
"In particular, for the special choice $\\nu _{1}= \\nu _{2} >0$ and $\\gamma := \\frac{\\nu }{\\nu _{1}}$ with $\\nu \\in ]0,\\nu _{1}[ \\, \\cup \\, [2\\nu _{1}, \\infty [$ we obtain from (REF ) $\\hspace{-28.45274pt}&& 0 \\le \\frac{\\nu }{\\nu _{1}} \\cdot R_{\\nu /\\nu _{1}}(\\widetilde{{Q}},\\widetilde{{P}})=\\frac{\\log \\Big ( \\sum \\displaylimits _{k=1}^{K} (\\widetilde{q}_{k})^{\\nu /\\nu _{1}} \\cdot (\\widetilde{p}_{k})^{1-(\\nu /\\nu _{1})} \\Big )}{\\frac{\\nu }{\\nu _{1}} -1}\\nonumber \\\\\\hspace{-28.45274pt}&& =\\frac{\\nu _{1}}{\\nu - \\nu _{1}} \\cdot \\log \\Big ( \\sum \\displaylimits _{k=1}^{K} (q_{k})^{\\nu } \\cdot (p_{k})^{\\nu _{1} - \\nu } \\Big )- \\frac{\\nu }{\\nu - \\nu _{1}} \\cdot \\log \\Big ( \\sum \\displaylimits _{k=1}^{K} (q_{k})^{\\nu _{1}} \\Big )+ \\log \\Big ( \\sum \\displaylimits _{k=1}^{K} (p_{k})^{\\nu _{1}} \\Big )=: \\breve{R}_{\\nu /\\nu _{1}}({Q},{P})$ which is BS-minimizable (in $\\widetilde{{Q}}$ ) on $\\textrm {$$\\hspace{-6.544pt}$$}$ .",
"Our divergence $\\breve{R}_{\\nu /\\nu _{1}}({Q},{P})$ in (REF ) is basically a multiple of a divergence which has been very recently used in Ghosh & Basu [136].",
"Moreover, $\\breve{R}_{1/\\nu _{1}}({Q},{P})$ (i.e.",
"the special case $\\nu =1$ in (REF )) is equal to Sundaresan’s divergence [352] [353] (see also Lutwak et al.",
"[234], Kumar & Sundaresan [203], [204], Yagli et al.",
"[408]); for our BS-approach, we need the restriction $\\nu _{1} \\in \\, ]0,\\frac{1}{2}] \\, \\cup \\, ]1,\\infty [$ .",
"Notice that Sundaresan’s divergence can be employed in mismatch-cases of (i) Campbell’s coding problem, (ii) Arikan’s guessing problem, (iii) memoryless guessing, and (iv) task partitioning problems; see e.g.",
"Sundaresan [353], Bunte & Lapidoth [67], Kumar et al.",
"[205].",
"Returning to the general context, functions of the modified Kullback-Leibler information $I(\\mathbf {Q},{P})$ and the modified reverse Kullback-Leibler information $\\widetilde{I}(\\mathbf {Q},{P})$ can be treated analogously.",
"For the sake of brevity, we only deal with the former and fix arbitrary ${P} \\in \\mathbb {S}_{>0}^{K}$ and $\\mathbf {Q} \\in A \\cdot \\mathbb {S}^{K}$ with $A := \\sum _{k=1}^{K} q_{k} >0$ .",
"For this, in (REF ) we have obtained $I(\\mathbf {Q},{P})$ which is generally not a divergence (cf.",
"Remark REF ).",
"In the following, let the function $h_{1} : \\, ]-1,\\infty [ \\, \\mapsto \\, ]-\\infty ,\\infty [$ be continuous and strictly increasing (respectively, strictly decreasing); since $D_{\\varphi _{1}}(\\mathbf {Q},{P})$ is BS-minimizable on $\\mathbf {\\Omega } = A \\cdot \\textrm {$$\\hspace{-6.544pt}$$}$ , also the quantity $h_{1}\\Big ( A - 1 + D_{\\varphi _{1}}(\\mathbf {Q},{P})\\Big )= h_{1}\\Big (I(\\mathbf {Q},{P})\\Big )$ is BS-minimizable on $\\mathbf {\\Omega } = A \\cdot \\textrm {$$\\hspace{-6.544pt}$$}$ (respectively, BS-maximizable on $\\mathbf {\\Omega } = A \\cdot \\textrm {$$\\hspace{-6.544pt}$$}$ ).",
"In particular, by using the negative identity mapping $h_{\\gamma }^{-Id}(y) := -y$ ($y>-1$ ) we get that $-I(\\mathbf {Q},{P})$ is BS-maximizable.",
"Another exemplary choice for $h_{1}$ is (cf.",
"Sharma & Mittal [330] in the scaling of e.g.",
"Morales et al.",
"[264]) $& & h_{1}(y) := h_{s}^{SM}(y) := \\frac{e^{(s-1) \\cdot y} -1}{s-1},\\qquad y \\in \\mathbb {R}, \\, s \\in \\, ]0,1[ \\, \\cup \\, ]1,\\infty [,$ which is strictly increasing; hence, $h_{s}^{SM}(I(\\mathbf {Q},{P}))$ (and also $h_{s}^{SM}(D_{\\varphi _{1}}(\\mathbf {Q},{P}))$ ) is BS-minimizable on $\\mathbf {\\Omega } = A \\cdot \\textrm {$$\\hspace{-6.544pt}$$}$ .",
"As another important application line, let us fix any $(\\gamma , \\mathbf {Q}) \\in (\\widetilde{\\Gamma }\\backslash ]1,2[) \\times \\widetilde{\\mathcal {M}}_{2}$ (cf.",
"Lemma REF (a)) with $A := \\sum _{k=1}^{K} q_{k} >0$ .",
"Moreover, we take ${P} := {P}^{unif} :=(\\frac{1}{K}, \\ldots , \\frac{1}{K})$ to be the probability vector of frequencies of the uniform distribution on $\\lbrace 1, \\ldots , K\\rbrace $ .",
"Then, for $\\gamma \\in \\, ]-\\infty ,0[ \\ \\cup \\ ]0,1[ \\ \\cup \\ [ \\ 2,\\infty [$ one gets $H_{\\gamma }(\\mathbf {Q},{P}^{unif}) =K^{\\gamma -1} \\cdot \\sum _{k=1}^{K} q_{k}^{\\gamma }$ .",
"One can rewrite $K^{1- \\gamma } \\cdot H_{\\gamma }(\\mathbf {Q},{P}^{unif})= \\sum _{k=1}^{K} q_{k}^{\\gamma }$ ; the latter is sometimes called heterogeneity index of type $\\gamma $, see e.g.",
"van der Lubbe [379], with $\\gamma =2$ being the Simpson-Herfindahl index which is also known as index of coincidence (cf.",
"Harremoes & Topsoe [155] and its generalization in Harremoes & Vajda [156]).",
"Alternatively, $\\sum _{k=1}^{K} q_{k}^{\\gamma }$ is also called Onicescu’s information energy in case of $\\gamma =2$ (cf.",
"Onicescu [278], see also Pardo & Taneja [283] for comprehensive investigations) and in general information energy of order $\\gamma $ (cf.",
"Theodorescu [359], see also e.g.",
"Pardo [281]); for exemplary applications to electron density functional theory (DFT) for quantum chemical reactivity, the reader may take (discretized versions of) e.g.",
"Liu et al.",
"[226], Lopez-Rosa et al.",
"[231] and Rong et al.",
"[311].",
"In some other literature (see e.g.",
"Clark [90]), $\\sum _{k=1}^{K} q_{k}^{\\gamma }$ is alternatively called Golomb’s [140] information generating function (of a probability distribution ${Q}$ ); yet another name is generalized information potential and for $\\gamma =2$ information potential (cf.",
"e.g.",
"Principe [297], Acu et al.",
"[4]).",
"From the above-mentioned investigations, we obtain that $\\sum _{k=1}^{K} q_{k}^{\\gamma }$ is BS-minimizable on $\\mathbf {\\Omega } = A \\cdot \\textrm {$$\\hspace{-6.544pt}$$}$ for $\\gamma <0$ and $\\gamma \\ge 2$ , and BS-maximizable on $\\mathbf {\\Omega } = A \\cdot \\textrm {$$\\hspace{-6.544pt}$$}$ for $\\gamma \\in ]0,1[$ .",
"More generally, by employing (REF ) and (), for the class of entropies (diversity indices) $\\mathcal {E}_{\\gamma ,c_{1},c_{2},c_{3}}(\\mathbf {Q}) &:=&h_{c_{1},c_{2},c_{3}}\\left(\\sum _{k=1}^{K} q_{k}^{\\gamma }\\right) =c_{1} \\cdot \\left(\\left(\\sum _{k=1}^{K} q_{k}^{\\gamma }\\right)^{c_{2}} - c_{3} \\right)\\nonumber \\\\&=&c_{1} \\cdot \\left(K^{c_{2} \\cdot (1- \\gamma )} \\cdot H_{\\gamma }(\\mathbf {Q},{P}^{unif})^{c_{2}} - c_{3} \\right),\\qquad c_{1}, c_{2} \\in \\mathbb {R}\\backslash \\lbrace 0\\rbrace , \\,c_{3} \\in \\mathbb {R},\\\\\\mathcal {E}_{c_{4},f}^{R}(\\mathbf {Q}) &:=&h_{c_{4},f}^{R}\\left(\\sum _{k=1}^{K} q_{k}^{\\gamma }\\right)= \\frac{c_{4}}{f^{\\prime }(0)} \\cdot \\log \\left(\\sum _{k=1}^{K} q_{k}^{\\gamma }\\right),\\nonumber \\\\&=& \\frac{c_{4}}{f^{\\prime }(0)} \\cdot \\Big ( \\log \\left(H_{\\gamma }(\\mathbf {Q},{P}^{unif})\\right) +(1- \\gamma ) \\cdot \\log (K)\\Big ),\\qquad c_{4} \\in \\mathbb {R}\\backslash \\lbrace 0\\rbrace , \\,$ (which is similar to the entropy-class of Morales et al.",
"[265] who use a different, more restrictive parametrization and probability distributions ${Q}$ ), one gets the following extremum-behaviour: $\\mathcal {E}_{\\gamma ,c_{1},c_{2},c_{3}}(\\mathbf {Q})$ is BS-minimziable if $\\gamma <0$ and $c_{1} \\cdot c_{2} >0$ ; $\\mathcal {E}_{\\gamma ,c_{1},c_{2},c_{3}}(\\mathbf {Q})$ is BS-minimizable if $\\gamma \\ge 2$ and $c_{1} \\cdot c_{2} >0$ ; $\\mathcal {E}_{\\gamma ,c_{1},c_{2},c_{3}}(\\mathbf {Q})$ is BS-minimizable if $\\gamma \\in \\, ]0,1[$ and $c_{1} \\cdot c_{2} < 0$ ; $\\mathcal {E}_{\\gamma ,c_{1},c_{2},c_{3}}(\\mathbf {Q})$ is BS-maximizable if $\\gamma <0$ and $c_{1} \\cdot c_{2} < 0$ ; $\\mathcal {E}_{\\gamma ,c_{1},c_{2},c_{3}}(\\mathbf {Q})$ is BS-maximizable if $\\gamma \\ge 2$ and $c_{1} \\cdot c_{2} < 0$ ; $\\mathcal {E}_{\\gamma ,c_{1},c_{2},c_{3}}(\\mathbf {Q})$ is BS-maximizable if $\\gamma \\in \\, ]0,1[$ and $c_{1} \\cdot c_{2} > 0$ ; $\\mathcal {E}_{c_{4},f}^{R}(\\mathbf {Q})$ is BS-minimizable if $\\gamma <0$ and $\\frac{c_{4}}{f^{\\prime }(0)} > 0$ ; $\\mathcal {E}_{c_{4},f}^{R}(\\mathbf {Q})$ is BS-minimizable if $\\gamma \\ge 2$ and $\\frac{c_{4}}{f^{\\prime }(0)} > 0$ ; $\\mathcal {E}_{c_{4},f}^{R}(\\mathbf {Q})$ is BS-minimizable if $\\gamma \\in \\, ]0,1[$ and $\\frac{c_{4}}{f^{\\prime }(0)} < 0$ ; $\\mathcal {E}_{c_{4},f}^{R}(\\mathbf {Q})$ is BS-maximizable if $\\gamma <0$ and $\\frac{c_{4}}{f^{\\prime }(0)} < 0$ ; $\\mathcal {E}_{c_{4},f}^{R}(\\mathbf {Q})$ is BS-maximizable if $\\gamma \\ge 2$ and $\\frac{c_{4}}{f^{\\prime }(0)} < 0$ ; $\\mathcal {E}_{c_{4},f}^{R}(\\mathbf {Q})$ is BS-maximizable if $\\gamma \\in \\, ]0,1[$ and $\\frac{c_{4}}{f^{\\prime }(0)} > 0$ .",
"From this, one can deduce that our new BS method works for the constrained minimization/maximization of the following well-known, prominently used measures of entropy respectively measures of diversity, and beyond: $c_{1}=1$ , $c_{2} = \\frac{1}{\\gamma }$ , $c_{3}=0$ : the Euclidean $\\gamma -$ norm (also known as $\\gamma -$ norm heterogeneity index, see e.g.",
"van der Lubbe [379]) $|| \\mathbf {Q} ||_{\\gamma } := \\Big (\\sum _{k=1}^{K} q_{k}^{\\gamma }\\Big )^{1/\\gamma }= K^{(1- \\gamma )/\\gamma } \\cdot \\Big (H_{\\gamma }(\\mathbf {Q},{P}^{unif})\\Big )^{1/\\gamma }$ is BS-minimizable on $\\mathbf {\\Omega } = A \\cdot \\textrm {$$\\hspace{-6.544pt}$$}$ for $\\gamma \\in ]0,1[$ and $\\gamma \\ge 2$ , and BS-maximizable on $\\mathbf {\\Omega } = A \\cdot \\textrm {$$\\hspace{-6.544pt}$$}$ for $\\gamma <0$ (note that $|| \\mathbf {Q} ||_{1} =A$ ) ; similarly, the $\\gamma -$ mean heterogeneity index (see e.g.",
"[379], as well as Jost [176] for its interpretation as “effective number of species” respectively as “numbers equivalent”) given by $\\mathcal {E}^{HI}(\\mathbf {Q}) := \\Big (\\sum _{k=1}^{K} q_{k}^{\\gamma }\\Big )^{1/(\\gamma -1)}= \\frac{1}{K} \\cdot \\Big (H_{\\gamma }(\\mathbf {Q},{P}^{unif})\\Big )^{1/(\\gamma -1)}$ is BS-minimizable on $\\mathbf {\\Omega } = A \\cdot \\textrm {$$\\hspace{-6.544pt}$$}$ for $\\gamma \\ge 2$ , and BS-maximizable on $\\mathbf {\\Omega } = A \\cdot \\textrm {$$\\hspace{-6.544pt}$$}$ for $\\gamma <0$ and $\\gamma \\in ]0,1[$ .",
"Alternatively, $\\mathcal {E}^{HI}(\\mathbf {Q})$ is also called ($\\gamma -$ order) Hill diversity index or Hill number [160], respectively ($\\gamma -$ order) Hannah-Kay index [154], respectively ($\\gamma -$ order) Renyi heterogeneity (cf.",
"Nunes et al.",
"[276]), respectively ($\\gamma -$ order) exponential Renyi entropy or exponential entropy (cf.",
"Campbell [72]) since it is equal to $\\exp (\\mathcal {E}^{gR}(\\mathbf {Q}))$ (cf.",
"(E6) below).",
"The $\\gamma -$ mean heterogeneity index (under one of the above-mentioned namings) was recently employed e.g.",
"by Greiff et al.",
"[142] for immunodiagnostic design of fingerprints of an individual’s ongoing immunological status (e.g., healthy, infected, vaccinated) — culminating in accurate and early detection of disease and infection, by Ma & Li [237] for the quantification of metagenome diversity and similarity, by Jasinska et al.",
"[164] for studying bacterial evolution — in particular evolution under sub-inhibitory antibiotic levels, by Ma et al.",
"[238] for the definition of individual-level genetic diversity and similarity profiles as well as their applications to datasets from the 1000-Genomes Project, and by Lassance & Vrins [209] for some optimal selection procedure of financial-asset portfolios.",
"$c_{1}=\\frac{1}{2^{1-\\gamma }-1}$ , $c_{2} = 1$ , $c_{3}=1$ : the entropy $\\mathcal {E}^{gHC}(\\mathbf {Q}) :=\\frac{1}{2^{1-\\gamma }-1} \\cdot \\left(\\sum _{k=1}^{K} q_{k}^{\\gamma } - 1 \\right)=\\frac{1}{2^{1-\\gamma }-1} \\cdot \\left(K^{1- \\gamma } \\cdot H_{\\gamma }(\\mathbf {Q},{P}^{unif}) - 1 \\right)$ is BS-minimizable on $\\mathbf {\\Omega } = A \\cdot \\textrm {$$\\hspace{-6.544pt}$$}$ for $\\gamma <0$ , and BS-maximizable on $\\mathbf {\\Omega } = A \\cdot \\textrm {$$\\hspace{-6.544pt}$$}$ for $\\gamma \\in ]0,1[$ and $\\gamma \\ge 2$ ; the special subcase $A=1$ in (REF ) (and thus, $\\mathbf {Q} ={Q}$ is a probability vector) corresponds to the $\\gamma -$ order entropy of Havrda-Charvat [157] (also called non-additive $\\gamma -$ order Tsallis entropy [363] in statistical physics) where the special case $\\gamma =2$ is (a multiple of) Vajda’s quadratic entropy [371] and Ahlswede’s identification entropy [7] (see also Ahlswede & Cai [8]).",
"Some exemplary (relatively) recent studies and applications of $\\mathcal {E}^{gHC}({Q})$ appear e.g.",
"in Peter & Rangarajan [291] for shape matching, in Liu et al.",
"[226] as well as in Rong et al.",
"[311] to electron density functional theory (DFT) for quantum chemical reactivity, in Yalcin & Beck [409] for the investigation of energy spectra of cosmic rays, in Wen & Jiang [390] for the quantification of complexity degrees in complex networks, in Bhandari [46] for fast multilevel thresholding for color image segmentation, in Erguzel et al.",
"[121] for the investigation of Electroencephalography (EEG) signals of subjects suffering from some psychiatric disorders, in Kang & Kim [182] for automatic synthetic aperture radar (SAR) image registration, in Namdari & Li [269] for the modelling of Lithium-Ion battery capacity fade, in Seweryn et al.",
"[326] for the assessment of similarity and diversity of expression profiles in single cell systems, in Zhang et al.",
"[418] for the search of functional relationships between groundwater depth and vegetation distribution, in Kumbhakar et al.",
"[207] for the modelling of streamwise velocity profiles in wide–open channel turbulent flows (e.g.",
"in rivers, streams, canals, ditches), and in Ramezani & Pourdarvish [300] for transfer learning for image classification of gravitational waves.",
"For the special case $\\gamma =2$ , a directly connected quantity is the measure of concentration (cf.",
"e.g.",
"De Wet et al.",
"[107]) $\\mathcal {E}^{gMC}({Q}) : = 1- \\frac{1}{K} - \\mathcal {E}^{gHC}({Q})= \\sum _{k=1}^{K} \\left(q_{k} - \\frac{1}{K} \\right)^{2}$ which (up to a multiple) was introduced by Brukner & Zeilinger [66] as an appropriate measure of information for quantum experiments.",
"$\\gamma := \\frac{1}{\\widetilde{\\gamma }}$ , $c_{1}=\\frac{1}{\\widetilde{\\gamma }-1}$ , $c_{2} = \\widetilde{\\gamma }$ , $c_{3}=1$ : the entropy $\\mathcal {E}^{gA}(\\mathbf {Q}) :=\\frac{1}{\\widetilde{\\gamma }-1} \\cdot \\left(\\left(\\sum _{k=1}^{K}q_{k}^{1/\\widetilde{\\gamma }}\\right)^{\\widetilde{\\gamma }} - 1 \\right)=\\frac{1}{\\widetilde{\\gamma }-1} \\cdot \\left(K^{\\widetilde{\\gamma } \\cdot (1- \\gamma )} \\cdot H_{1/\\widetilde{\\gamma }}(\\mathbf {Q},{P}^{unif})^{\\widetilde{\\gamma }} - 1 \\right)$ is BS-minimizable on $\\mathbf {\\Omega } = A \\cdot \\textrm {$$\\hspace{-6.544pt}$$}$ for $\\widetilde{\\gamma } <0$ and $\\widetilde{\\gamma } \\in \\, ]0,1[$ , and BS-maximizable on $\\mathbf {\\Omega } = A \\cdot \\textrm {$$\\hspace{-6.544pt}$$}$ for $\\gamma \\ge 2$ ; the special subcase $A=1$ in (REF ) (and thus, $\\mathbf {Q} ={Q}$ is a probability vector) corresponds to the $\\widetilde{\\gamma }-$ order entropy of Arimoto [16].",
"$s \\in \\mathbb {R}\\backslash \\lbrace 1\\rbrace $ , $c_{1}=\\frac{1}{1-s}$ , $c_{2} = \\frac{1-s}{1-\\gamma }$ , $c_{3}=1$ : the entropy $\\mathcal {E}^{gSM1}(\\mathbf {Q}):= \\frac{1}{1-s} \\cdot \\left(\\left(\\sum _{k=1}^{K} q_{k}^{\\gamma }\\right)^{(1-s)/(1-\\gamma )} - 1 \\right)= \\frac{1}{1-s} \\cdot \\left(K^{1- s} \\cdot H_{\\gamma }(\\mathbf {Q},{P}^{unif})^{(1-s)/(1-\\gamma )} - 1 \\right)$ is BS-minimizable on $\\mathbf {\\Omega } = A \\cdot \\textrm {$$\\hspace{-6.544pt}$$}$ for $\\gamma < 0$ and BS-maximizable on $\\mathbf {\\Omega } = A \\cdot \\textrm {$$\\hspace{-6.544pt}$$}$ for $\\gamma \\in \\, ]0,1[$ and $\\gamma \\ge 2$ ; the special subcase $A=1$ in (REF ) (and thus, $\\mathbf {Q} ={Q}$ is a probability vector) corresponds to the entropy of order $\\gamma $ and degree $s$ of Sharma & Mittal [329] in the scaling of e.g.",
"Salicru et al.",
"[314].",
"$s \\in \\mathbb {R}\\backslash \\lbrace 0\\rbrace $ , $\\gamma = s+1$ , $c_{1}= - \\frac{1}{s}$ , $c_{2} = 1$ , $c_{3}=1$ : the diversity index $\\mathcal {E}^{gPT}(\\mathbf {Q}):= - \\frac{1}{s} \\cdot \\left(\\sum _{k=1}^{K} q_{k}^{s+1} - 1 \\right)= - \\frac{1}{s} \\cdot \\left(K^{- s} \\cdot H_{s+1}(\\mathbf {Q},{P}^{unif}) - 1 \\right)$ is BS-minimizable on $\\mathbf {\\Omega } = A \\cdot \\textrm {$$\\hspace{-6.544pt}$$}$ for $s < -1$ and BS-maximizable on $\\mathbf {\\Omega } = A \\cdot \\textrm {$$\\hspace{-6.544pt}$$}$ for $s \\in \\, ]-1,0[$ and $s >0$ ; the special subcase $A=1$ in (REF ) (and thus, $\\mathbf {Q} ={Q}$ is a probability vector) corresponds to the diversity index of degree $s$ of Patil & Taillie [286]; the case $s=1$ for probability measures $\\mathbf {Q}={Q}$ gives the well-known Gini-Simpson diversity index.",
"$c_{4}=\\frac{1}{1-\\gamma }$ , $f(z) =z$ : the entropy $\\mathcal {E}^{gR}(\\mathbf {Q}):= \\frac{1}{1-\\gamma } \\cdot \\log \\left(\\sum _{k=1}^{K} q_{k}^{\\gamma }\\right)= \\frac{1}{1-\\gamma } \\cdot \\Big ( \\log \\left(H_{\\gamma }(\\mathbf {Q},{P}^{unif})\\right) + (1- \\gamma ) \\cdot \\log (K) \\Big )= \\frac{\\log 2}{1-\\gamma } \\cdot \\log _{2}\\left(\\sum _{k=1}^{K} q_{k}^{\\gamma }\\right)$ is BS-minimizable on $\\mathbf {\\Omega } = A \\cdot \\textrm {$$\\hspace{-6.544pt}$$}$ for $\\gamma <0$ , and BS-maximizable on $\\mathbf {\\Omega } = A \\cdot \\textrm {$$\\hspace{-6.544pt}$$}$ for $\\gamma \\in ]0,1[$ and $\\gamma \\ge 2$ ; the special subcase $A=1$ in (REF ) (and thus, $\\mathbf {Q} = {Q}$ is a probability vector) corresponds to the prominent (additive) $\\gamma -$ order Renyi entropy [309].",
"As well known, there is a vast literature on Renyi entropies $\\mathcal {E}^{gR}({Q})$ .",
"Some exemplary (mostly recent) studies and applications appear e.g.",
"in Nath [270] — as well as in Arikan [15], Sundaresan [353], Bunte & Lapidoth [67], Sason & Verdu [321], Kumar et al.",
"[205] — for coding and guessing, in Bennett et al.",
"[39] in connection with unconditionally secure secret-key agreement protocols and quantum cryptography, in Mayoral [257] for cluster sampling, in Aviyente et al.",
"[21] for information extraction in certain neurophysiological signals (so-called event-related potentials), in Tao et al.",
"[355] as well as in Jiao et al.",
"[171] for early defect/fault detection of rolling element bearings, in Pham et al.",
": [292] for blind source separation, in Liu et al.",
"[226] as well as in Rong et al.",
"[311] to electron density functional theory (DFT) for quantum chemical reactivity, in Sason [319] for data compression, in Carravilla et al.",
"[73] for the recognition of HIV-1 antibodies through STED microscopy and the corresponding design of therapeutic interventions, in Joshi et al.",
"[175] for the identification and tracking of relevant T cell receptors for adoptive immunotherapy, in Erguzel et al.",
"[121] for the investigation of Electroencephalography (EEG) signals of subjects suffering from some psychiatric disorders, in German–Sallo [133] for fault–characteristics extraction from discrete signals in manufacturing systems, in Schober et al.",
"[323] for investigations of some evolutions of the T cell antigen receptor (TCR) repertoire, in Seweryn et al.",
"[326] for the assessment of similarity and diversity of expression profiles in single cell systems, in Amezquita-Sanchez [13] for the detection of incipient damage in high-rise buildings subjected to dynamic vibrations, in Barennes et al.",
"[32] for comparing the accuracy of current T cell receptor sequencing methods employed for the understanding of adaptive immune responses, in Kumar et al.",
"[201] for the segmentation of digital images through multilevel iterative variational mode decomposition (VMD), and in Pandey [280] for the quantification of cosmic homogeneity.",
"Remark 19 (i) For Renyi entropies there are also matrix versions $\\mathcal {E}^{gR}(X) :=\\frac{1}{1-\\gamma } \\cdot \\log \\left(\\sum _{i=1}^{K_{1}}\\sum _{j=1}^{K_{2}} x_{ij}^{\\gamma }\\right)$ where $X:=(x_{ij})_{i=1,\\ldots ,K_{1}}^{j=1,\\ldots ,K_{2}}$ is a $K_{1} \\times K_{2}-$ matrix whose elements $x_{ij}$ are (say) strictly positive and sum up to $A$ .",
"Such a setup with $A=1$ is e.g.",
"used in time-frequency analyses of signals where the $i$ ’s correspond to discrete time points, the $j$ ’s to discrete frequencies, and $x_{ij}$ to the probability that $(i,j)$ occurs; see e.g.",
"Popescu & Aiordachioaie [295] for change detection in seismic signals.",
"Another line of application is to use as $X$ the normalized communicability matrix of a directed network (respectively the upper triangular part of $X$ in case of an unweighted and undirected network).",
"Of course, the matrix version $\\mathcal {E}^{gR}(X)$ can be easily and equivalently rewritten in our vector version $\\mathcal {E}^{gR}(\\mathbf {Q})$ by setting $\\mathbf {Q} := (q_{1}, \\ldots , q_{K_{1} \\cdot K_{2}})$ such that $x_{ij} = q_{(i-1)\\cdot K_{2} +j}$ ($i=1,\\ldots ,K_{1}$ , $j=1,\\ldots ,K_{2}$ and hence $K:= K_{1} \\cdot K_{2}$ ; accordingly, we can apply our BS method.",
"(ii) The latter conversion works analogously also for matrix versions of all the other entropies, divergences, etc.",
"of this paper; more flexible versions where $i \\in \\lbrace 1,\\ldots ,K_{1}\\rbrace $ , $j \\in J_{i}$ for some $J_{i} \\subseteq \\lbrace 1,\\ldots ,K_{2}\\rbrace $ as well as multidimensional-array/tensor versions can be transformed in a similar book-keeping manner, too.",
"For instance, within the above-mentioned framework of unweighted and undirected networks, Chen et al.",
"[81] and Shi et al.",
"[332] employ communicability matrix versions of the Shannon entropy and the Jensen-Shannon divergence (JSD), e.g.",
"in order to derive a new complexity measure of such kind of networks; see also Bagrow and Bollt [28] for similar network applications of the JSD.",
"Moreover, Jena et al.",
"[167] use “3D versions” of Tsallis entropies for brain magnetic resonance (MR) image segmentation.",
"Remark 20 All the above cases which are BS-maximizable can be interpreted as bare-simulation approach to the solution of generalized maximum entropy problems on $\\mathbf {\\Omega } = A \\cdot \\textrm {$$\\hspace{-6.544pt}$$}$.",
"Remark 21 (i) If (all) the above- and below-mentioned entropies are used for probability vectors ${Q} \\in \\mathbf {S}^{K}$ — i.e.",
"one employs $\\mathcal {E}({Q})$ — then typically the components $q_{k}$ of ${Q}$ represent a genuine probability mass (frequency) $q_{k} = \\mathbb {\\Pi }[\\lbrace d_{k} \\rbrace ]$ of some data point (state) $d_{k}$ .",
"However, ${Q} \\in \\mathbf {S}^{K}$ may alternatively be artificially generated.",
"For instance, for the purpose of fault detections of mechanical drives, Boskoski & Juricic [57] use Renyi entropies where the $q_{k}$ ’s are normalized squared energy-describing coefficients of the wavelet packet transform of measured vibration records.",
"Another exemplary “artificial” operation is concatenation, see e.g.",
"Subsection REF below.",
"(ii) An analogous statement holds for the employment of (all) the above- and below-mentioned divergences $D({Q},{P})$ — and their transformations — between genuine respectively artificially generated probability vectors ${Q}, {P} \\in \\mathbf {S}^{K}$ .",
"The remaining parameter cases $\\gamma =0$ and $\\gamma =1$ can be treated analogously.",
"For the sake of brevity, we only deal with the latter.",
"For this, let $\\mathbf {Q} \\in A \\cdot \\mathbb {S}^{K}$ with $A := \\sum _{k=1}^{K} q_{k} >0$ and ${P} := {P}^{unif}$ .",
"Clearly, $I(\\mathbf {Q},{P}^{unif}) - \\frac{\\log K}{K}=\\sum _{k=1}^{K} q_{k} \\cdot \\log (q_{k})$ ; thus the latter is BS-minimizable on $\\mathbf {\\Omega } = A \\cdot \\textrm {$$\\hspace{-6.544pt}$$}$ .",
"More generally, for any continuous strictly increasing (respectively strictly decreasing) function $h_{1} : [- \\frac{K}{e}, 0[ \\, \\mapsto \\mathbb {R}$ , the quantity $h_{1}\\Big (\\sum _{k=1}^{K} q_{k} \\cdot \\log (q_{k})\\Big )$ is BS-minimizable on $\\mathbf {\\Omega } = A \\cdot \\textrm {$$\\hspace{-6.544pt}$$}$ (respectively BS-maximizable on $\\mathbf {\\Omega } = A \\cdot \\textrm {$$\\hspace{-6.544pt}$$}$ ).",
"Important special cases are: $h_{1}(y) := h_{1}^{-Id}(y) = -y$ : the entropy $\\mathcal {E}^{Sh}(\\mathbf {Q}) :=h_{1}^{-Id}\\Big (\\sum _{k=1}^{K} q_{k} \\cdot \\log (q_{k})\\Big )= - \\sum _{k=1}^{K} q_{k} \\cdot \\log (q_{k})$ is BS-maximizable on $\\mathbf {\\Omega } = A \\cdot \\textrm {$$\\hspace{-6.544pt}$$}$ ; the special subcase $A=1$ in (REF ) (and thus, $\\mathbf {Q} = {Q}$ is a probability vector) corresponds to the omnipresent Shannon entropy; hence, by our bare-simulation approach we can particularly tackle maximum entropy problems on almost arbitrary sets $\\textrm {$$\\hspace{-6.544pt}$$}$ of probability vectors.",
"Analogously, we can treat $\\frac{1}{\\log (K)} \\cdot \\mathcal {E}^{Sh}({Q})$ which is called Pielou’s evenness index [293], and $1-\\frac{1}{\\log (K)} \\cdot \\mathcal {E}^{Sh}({Q}) \\in [0,1]$ which is sometimes used as clonality (clonotype diversity) index (see e.g.",
"Gabriel et al.",
"[131] for applications to HIV-connected T cell receptor repertoires, and Bashford-Rogers et al.",
"[33] (with supplementary private communication) for its use for comparative analyses of the BCR repertoire in immune-mediated diseases, for the sake of understanding pathological mechanisms and designing treatment strategies).",
"As a further example for Remark REF , Lyubushin [236] uses $q_{k}$ ’s which are normalized squared coefficients of an orthogonal wavelet decomposition of some seismic noise, and accordingly, $\\frac{1}{\\log (K)} \\cdot \\mathcal {E}^{Sh}({Q})$ can be interpreted as the entropy of the distribution of energy of oscillations at various frequency and time scales.",
"Some further exemplary studies and applications of the maximization of $\\mathcal {E}^{Sh}({Q})$ — aside from the vast physics literature — appear e.g.",
"in De Santis et al.",
"[106] for cryptanalytic guessing problems for breaking ciphertexts with probabilistic brute-force attacks, Johansson & Sternad [173] for tackling certain resource allocation problems under uncertainty, Marano & Franceschetti [246] for ray propagation in percolating lattices, Miao et al.",
"[260] for unsupervised mixed-pixel decomposition in image processing, Rodrigues et al.",
"[310] for modelling biological species geographic distribution, Xiong et al.",
"[400] for capturing desirable phrasal and hierarchical segmentations within a statistical machine translation context, Chan et al.",
"[76] for alignment-free DNA sequence comparison, Mann & Garnett [244] for capturing some collective behaviours of intelligent agents in social interactions, Singh et al.",
"[336] for the study of finite buffer queueing systems, Baddeley [27] for geoscientifical prediction of the occurrence of mineral deposits on regional scales, Einicke et al.",
"[118] for feature selection within change classification during running, and Han et al.",
"[152] for substructure imaging of blood cells by means of maximum entropy tomography (MET).",
"$s \\in \\, ]0,1[ \\, \\cup \\, ]1,\\infty [$ , $h_{1}(y) := h_{s}^{SM2}(y) := \\frac{e^{(s-1) \\cdot y} -1}{1-s}$ (cf.",
"(REF )) with $y \\in \\mathbb {R}$ : the entropy $\\mathcal {E}^{SM2}(\\mathbf {Q}) :=h_{s}^{SM2}\\Big (\\sum _{k=1}^{K} q_{k} \\cdot \\log (q_{k})\\Big )= \\frac{1}{1-s} \\cdot \\left( \\exp \\Big \\lbrace (s-1) \\cdot \\sum _{k=1}^{K} q_{k} \\cdot \\log (q_{k})\\Big \\rbrace -1 \\right)$ is BS-maximizable on $\\mathbf {\\Omega } = A \\cdot \\textrm {$$\\hspace{-6.544pt}$$}$ ; the special subcase $A=1$ in (REF ) (and thus, $\\mathbf {Q} = {Q}$ is a probability vector) corresponds to the (second type) entropy of Sharma & Mittal [329] in the scaling of e.g.",
"Pardo [282] (p.20).",
"Returning to the general context, we now (as already indicated above) state explicitly the corresponding bare-simulation-minimizations (respectively maximizations) of the power divergences $\\inf _{\\mathbf {Q}\\in \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega } }D_{\\widetilde{c}\\cdot \\varphi _{\\gamma }}(\\mathbf {Q},{P})$ ($\\gamma \\in \\mathbb {R}$ ), the Renyi divergences $\\inf _{\\mathbf {Q}\\in \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega } }R_{\\gamma }(\\mathbf {Q},{P})$ ($\\gamma \\in \\mathbb {R}$ ), the Hellinger integrals $\\inf _{\\mathbf {Q}\\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega } }H_{\\gamma }(\\mathbf {Q},{P})$ ($\\gamma \\in ]-\\infty ,0[ \\, \\cup \\, ]1,\\infty [$ ), $\\sup _{\\mathbf {Q}\\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega } }H_{\\gamma }(\\mathbf {Q},{P})$ ($\\gamma \\in ]0,1[$ ), the modified Kullback-Leibler information $\\inf _{Q\\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega } }I(\\mathbf {Q},{P})$ , the modified reverse Kullback-Leibler information $\\inf _{Q\\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega } }\\widetilde{I}(\\mathbf {Q},{P})$ , as well as the above-mentioned measures of entropy (diversity).",
"Since the corresponding probability distribution $\\mathbb {}[\\cdot \\,]=\\mathbb {\\Pi }[W_{1}\\in \\cdot \\,]$ of the $W_{i}$ 's (cf.",
"the representability (REF )) varies “quite drastically” with $\\gamma $ , we split this issue into several pieces.",
"Proposition 22 (a) Consider the context of Remark REF (vi) for $\\varphi := \\widetilde{c}\\cdot \\varphi _{\\gamma }$ with $\\gamma <0$ , and let ${P} \\in \\mathbb {S}_{>0}^{K}$ as well as $\\widetilde{c}>0$ be arbitrary but fixed.",
"Furthermore, let $W:=(W_{i})_{i\\in \\mathbb {N}}$ be an i.i.d.",
"sequence of non-negative real-valued random variables having densityin the classical sense, with respect to Lebesgue measure $f_{W_{1}}(y)\\ :=\\ \\frac{\\exp \\lbrace -\\frac{y \\cdot \\widetilde{c}}{1-\\gamma }\\rbrace }{\\exp \\lbrace \\widetilde{c}/\\gamma \\rbrace }\\cdot f_{Z}(y) \\cdot {1}_{]-\\infty ,0[}(y), \\qquad y \\in \\mathbb {R},$ where $f_{Z}$ is the density of a random variable $Z$ which has stable law with parameter-quadruple $(\\frac{-\\gamma }{1-\\gamma },1,0,-\\frac{\\widetilde{c}^{1/(1-\\gamma )} \\cdot (1-\\gamma )^{-\\gamma /(1-\\gamma )}}{\\gamma })$ in terms of “form-B notation” in Zolotarev [428], p.12.",
"Then for all $A>0$ and all $\\Omega $$\\Omega \\subset \\mathbb {S}_{>0}^{K}$ with (REF ) there holds $\\hspace{-19.91684pt}-\\lim _{n\\rightarrow \\infty }\\frac{1}{n}\\log \\,\\mathbb {\\Pi }\\left[\\xi _{n}^{w\\mathbf {W}}\\in \\textrm {\\right.\\Omega \\hspace{-6.544pt}\\Omega } =\\inf _{\\mathbf {Q}\\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega }}\\frac{\\widetilde{c}}{\\gamma }\\cdot \\left[ 1-A^{\\gamma /(\\gamma -1)} \\cdot \\left[ 1+ \\gamma \\cdot (A-1) +\\frac{\\gamma \\cdot \\left( \\gamma -1\\right) }{\\widetilde{c}}\\cdot D_{\\widetilde{c}\\cdot \\varphi _{\\gamma }}(\\mathbf {Q},{P})\\right]^{-1/\\left( \\gamma -1\\right) }\\right]$ as well as the BS minimizabilities/maximizabilites (cf.",
"Definition REF ) $& & \\hspace{-25.6073pt}\\inf _{\\mathbf {Q}\\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega }}D_{\\widetilde{c}\\cdot \\varphi _{\\gamma }}(\\mathbf {Q},{P})=\\lim _{n\\rightarrow \\infty }\\frac{\\widetilde{c}}{\\gamma \\cdot \\left(\\gamma -1\\right) }\\cdot \\left\\lbrace A^{\\gamma } \\cdot \\left( 1+\\frac{\\gamma }{\\widetilde{c}}\\cdot \\frac{1}{n}\\cdot \\log \\,\\mathbb {\\Pi }\\left[\\xi _{n}^{w\\mathbf {W}}\\in \\textrm {\\right.\\right.\\right.\\Omega \\hspace{-6.544pt}\\Omega } ^{1-\\gamma } + \\gamma \\cdot (1-A) -1 ,\\qquad \\ \\\\& & \\hspace{-25.6073pt}\\inf _{\\mathbf {Q}\\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega }}H_{\\gamma }(\\mathbf {Q},{P})=\\lim _{n\\rightarrow \\infty } A^{\\gamma } \\cdot \\left( 1+ \\gamma \\cdot \\frac{1}{n}\\cdot \\log \\,\\mathbb {\\Pi }\\left[\\breve{\\xi }_{n}^{w\\mathbf {W}}\\in \\textrm {\\right.\\right.\\Omega \\hspace{-6.544pt}\\Omega } ^{1-\\gamma } ,\\qquad \\ \\\\& & \\hspace{-25.6073pt}\\inf _{\\mathbf {Q}\\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega }}c_{1} \\cdot \\Big (H_{\\gamma }(\\mathbf {Q},{P})^{c_{2}} - c_{3} \\Big )=\\lim _{n\\rightarrow \\infty } c_{1} \\cdot \\left\\lbrace A^{c_{2} \\cdot \\gamma } \\cdot \\left( 1+\\frac{\\gamma }{n}\\cdot \\log \\,\\mathbb {\\Pi }\\left[\\breve{\\xi }_{n}^{w\\mathbf {W}}\\in \\textrm {\\right.\\right.\\right.\\Omega \\hspace{-6.544pt}\\Omega } ^{c_{2} \\cdot (1-\\gamma )} - c_{3} ,\\ \\ \\textrm {if c_{1} \\cdot c_{2} >0, c_{3} \\in \\mathbb {R}}, \\quad \\ \\\\& & \\hspace{-25.6073pt}\\sup _{\\mathbf {Q}\\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega }}c_{1} \\cdot \\Big (H_{\\gamma }(\\mathbf {Q},{P})^{c_{2}} - c_{3} \\Big )=\\lim _{n\\rightarrow \\infty } c_{1} \\cdot \\left\\lbrace A^{c_{2} \\cdot \\gamma } \\cdot \\left( 1+\\frac{\\gamma }{n}\\cdot \\log \\,\\mathbb {\\Pi }\\left[\\breve{\\xi }_{n}^{w\\mathbf {W}}\\in \\textrm {\\right.\\right.\\right.\\Omega \\hspace{-6.544pt}\\Omega } ^{c_{2} \\cdot (1-\\gamma )} - c_{3} ,\\ \\ \\textrm {if c_{1} \\cdot c_{2} < 0, c_{3} \\in \\mathbb {R}}, \\quad \\ \\\\& & \\hspace{-25.6073pt}\\inf _{\\mathbf {Q}\\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega }}R_{\\gamma }(\\mathbf {Q},{P})= \\lim _{n\\rightarrow \\infty } \\frac{1}{\\gamma \\cdot (\\gamma -1)} \\cdot \\log \\Big ( A^{\\gamma } \\cdot \\Big ( 1+ \\gamma \\cdot \\frac{1}{n}\\cdot \\log \\,\\mathbb {\\Pi }\\left[\\breve{\\xi }_{n}^{w\\mathbf {W}}\\in \\textrm {\\right.\\Omega \\hspace{-6.544pt}\\Omega } \\Big ) ^{1-\\gamma } \\Big ),\\\\& & \\hspace{-25.6073pt}\\inf _{\\mathbf {Q}\\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega }}c_{1} \\cdot \\Big (\\Big (\\sum _{k=1}^{K} q_{k}^{\\gamma }\\Big )^{c_{2}} - c_{3} \\Big )=\\lim _{n\\rightarrow \\infty } c_{1} \\cdot \\left\\lbrace K^{c_{2}\\cdot (1-\\gamma )} \\cdot A^{c_{2} \\cdot \\gamma } \\cdot \\left( 1+\\frac{\\gamma }{n}\\cdot \\log \\,\\mathbb {\\Pi }\\left[\\breve{\\breve{\\xi }}_{n}^{w\\mathbf {W}}\\in \\textrm {\\right.\\right.\\right.\\Omega \\hspace{-6.544pt}\\Omega } ^{c_{2} \\cdot (1-\\gamma )} - c_{3} ,\\ \\ \\nonumber \\\\& & \\hspace{369.88582pt}\\textrm {if c_{1} \\cdot c_{2} >0, c_{3} \\in \\mathbb {R}}, \\quad \\ $ $& & \\hspace{-25.6073pt}\\sup _{\\mathbf {Q}\\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega }}c_{1} \\cdot \\Big (\\Big (\\sum _{k=1}^{K} q_{k}^{\\gamma }\\Big )^{c_{2}} - c_{3} \\Big )=\\lim _{n\\rightarrow \\infty } c_{1} \\cdot \\left\\lbrace K^{c_{2}\\cdot (1-\\gamma )} \\cdot A^{c_{2} \\cdot \\gamma } \\cdot \\left( 1+\\frac{\\gamma }{n}\\cdot \\log \\,\\mathbb {\\Pi }\\left[\\breve{\\breve{\\xi }}_{n}^{w\\mathbf {W}}\\in \\textrm {\\right.\\right.\\right.\\Omega \\hspace{-6.544pt}\\Omega } ^{c_{2} \\cdot (1-\\gamma )} - c_{3} ,\\ \\ \\nonumber \\\\& & \\hspace{369.88582pt}\\textrm {if c_{1} \\cdot c_{2} < 0, c_{3} \\in \\mathbb {R}}, \\quad \\ \\\\& & \\hspace{-25.6073pt}\\inf _{\\mathbf {Q}\\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega }}\\frac{1}{1-\\gamma } \\cdot \\log \\Big (\\sum _{k=1}^{K} q_{k}^{\\gamma }\\Big )= \\lim _{n\\rightarrow \\infty } \\frac{1}{\\gamma \\cdot (\\gamma -1)} \\cdot \\left[\\log \\Big ( A^{\\gamma } \\cdot \\Big ( 1+ \\gamma \\cdot \\frac{1}{n}\\cdot \\log \\,\\mathbb {\\Pi }\\left[\\breve{\\breve{\\xi }}_{n}^{w\\mathbf {W}}\\in \\textrm {\\right.\\right.\\Omega \\hspace{-6.544pt}\\Omega } \\Big ) ^{1-\\gamma } \\Big ) + (1-\\gamma ) \\cdot \\log (K) ,$ where $\\xi _{n}^{w\\mathbf {W}}$ is the normalized randomly weighted empirical measure given in (REF ), $\\breve{\\xi }_{n}^{w\\mathbf {W}}$ is its special case for $\\widetilde{c}=1$ , and $\\breve{\\breve{\\xi }}_{n}^{w\\mathbf {W}}$ is its special case for $\\widetilde{c}=1$ together with ${P} = {P}^{unif}$ the latter two notations will be also used in the following Propositions REF to REF.",
"From this, the BS-minimizability/maximizability of the important norms/entropies/diversity indices (E1) to (E6) follow immediately as special cases.",
"(b) The special case $\\varphi := \\widetilde{c}\\cdot \\varphi _{\\gamma }$ ($\\gamma <0$ ) of Theorem REF works analogously to (a), with the differences that we employ (i) additionally a sequence $(X_{i})_{i\\in \\mathbb {N}}$ of random variables being independent of $(W_{i})_{i\\in \\mathbb {N}}$ and satisfying condition (REF ) (resp.",
"(REF )), (ii) $A=1$ (instead of arbitrary $A>0$ ), (iii) $\\mathbb {\\Pi }_{X_{1}^{n}}[\\cdot ]$ (instead of $\\mathbb {\\Pi }[\\cdot ]$ ), (iv) $\\xi _{n,\\mathbf {X}}^{w\\mathbf {W}}$ (instead of $\\xi _{n}^{w\\mathbf {W}}$ ), (v) $\\breve{\\xi }_{n,\\mathbf {X}}^{w\\mathbf {W}}$ (instead of $\\breve{\\xi }_{n}^{w\\mathbf {W}}$ ), and (vi) $\\breve{\\breve{\\xi }}_{n,\\mathbf {X}}^{w\\mathbf {W}}$ (instead of $\\breve{\\breve{\\xi }}_{n}^{w\\mathbf {W}}$ ).",
"The assertions of Proposition REF can be deduced from Theorem REF , Remark REF (vi), Lemma REF (a), (REF ), (REF ), (REF ), () and the below-mentioned $\\mathbb {}-$ concerning Example REF (d).",
"Employing Lemma REF (c) and Example REF (a) instead, one ends up with the following proposition on the reverse Kullback-Leibler divergence: Proposition 23 (a) Consider the context of Remark REF (vi) for $\\varphi := \\widetilde{c}\\cdot \\varphi _{\\gamma }$ with $\\gamma =0$ , and let ${P} \\in \\mathbb {S}_{>0}^{K}$ as well as $\\widetilde{c}>0$ be arbitrary but fixed.",
"Furthermore, let $W:=(W_{i})_{i\\in \\mathbb {N}}$ be an i.i.d.",
"sequence of non-negative real-valued random variables with Gamma distribution $\\mathbb {} =GAM(\\widetilde{c},\\widetilde{c})$ here and henceforth, we use the notation that a Gamma distribution $GAM(\\alpha ,\\beta )$ with rate parameter (inverse scale parameter) $\\alpha >0$ and shape parameter $\\beta >0$ has (Lebesgue-)density $f(y) := \\frac{\\alpha ^{\\beta } \\cdot y^{\\beta -1} \\cdot e^{-\\alpha \\cdot y} }{\\Gamma (\\beta )}\\cdot {1}_{]0,\\infty [}(y)$ , $y \\in \\mathbb {R}$ ; its cumulant generating function is $\\Lambda (z) = \\beta \\cdot \\log (\\frac{\\alpha }{\\alpha -z})$ for $z \\in ]-\\infty ,\\alpha [$ (and $\\Lambda (z) = \\infty $ for $z \\ge \\alpha $ ).",
"(where the subcase $\\widetilde{c}=1$ is the exponential distribution $\\mathbb {} =EXP(1)$ with mean 1).",
"Then for all $A >0$ and all $\\Omega $$\\Omega \\subset \\mathbb {S}_{>0}^{K}$ with (REF ) there holds the BS minimizabilites (cf.",
"(REF )) $& & \\inf _{{Q}\\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega }}D_{\\widetilde{c}\\cdot \\varphi _{0}}({Q},{P}) =-\\lim _{n\\rightarrow \\infty }\\frac{1}{n} \\log \\,\\mathbb {\\Pi }\\left[\\xi _{n}^{w\\mathbf {W}}\\in \\textrm {\\right.\\Omega \\hspace{-6.544pt}\\Omega } + \\widetilde{c} \\cdot (A-1- \\log A),\\\\& & \\inf _{{Q}\\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega }}\\widetilde{I}({Q},{P}) =\\inf _{{Q}\\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega }}\\sum \\displaylimits _{k=1}^{K} p_{k} \\cdot \\log \\left( \\frac{p_{k}}{q_{k}} \\right)= -\\lim _{n\\rightarrow \\infty }\\frac{1}{n} \\log \\,\\mathbb {\\Pi }\\left[\\xi _{n}^{w\\mathbf {W}}\\in \\textrm {\\right.\\Omega \\hspace{-6.544pt}\\Omega } - \\log A .\\nonumber $ (b) The special case $\\varphi := \\widetilde{c}\\cdot \\varphi _{\\gamma }$ ($\\gamma =0$ ) of Theorem REF works analogously to (a), with the differences that we employ (i) additionally a sequence $(X_{i})_{i\\in \\mathbb {N}}$ of random variables being independent of $(W_{i})_{i\\in \\mathbb {N}}$ and satisfying condition (REF ) (resp.",
"(REF )), (ii) $A=1$ (instead of arbitrary $A>0$ ), (iii) $\\mathbb {\\Pi }_{X_{1}^{n}}[\\cdot ]$ (instead of $\\mathbb {\\Pi }[\\cdot ]$ ), and (iv) $\\xi _{n,\\mathbf {X}}^{w\\mathbf {W}}$ (instead of $\\xi _{n}^{w\\mathbf {W}}$ ).",
"Proposition 24 (a) Consider the context of Remark REF (vi) for $\\varphi := \\widetilde{c}\\cdot \\varphi _{\\gamma }$ with $\\gamma \\in ]0,1[$ , and let ${P} \\in \\mathbb {S}_{>0}^{K}$ as well as $\\widetilde{c}>0$ be arbitrary but fixed.",
"Furthermore, let $W:=(W_{i})_{i\\in \\mathbb {N}}$ be an i.i.d.",
"sequence of non-negative real-valued random variables with Compound-Poisson-Gamma distribution $\\mathbb {} =C(POI(\\theta ),GAM(\\alpha ,\\beta ))$ having parameters $\\theta =\\frac{\\widetilde{c}}{\\gamma }>0$ , $\\alpha =\\frac{\\widetilde{c}}{1-\\gamma }>0$ , $\\beta =\\frac{\\gamma }{1-\\gamma }>0$ ; in other words, the $W_{i}$ are independent copies of a random variable $W_{1}:=\\sum _{j=1}^{N}\\overline{W}_{j}$ with the usual convention $\\sum _{i=1}^{0}\\overline{W}_{i}:=0$ constituted of some i.i.d.",
"sequence $(\\overline{W}_{j})_{j\\in \\mathbb {N}}$ of $Gamma(\\alpha ,\\beta )-$ distributed random variables and some independent $POI(\\theta )-$ distributed random variable $N$ .",
"Then for all $A >0$ and all $\\Omega $$\\Omega \\subset \\mathbb {S}^{K}$ with (REF ) there hold (REF ), (REF ), (), () as well as $& & \\hspace{-25.6073pt}\\sup _{\\mathbf {Q}\\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega }}H_{\\gamma }(\\mathbf {Q},{P})=\\lim _{n\\rightarrow \\infty } A^{\\gamma } \\cdot \\left( 1+ \\gamma \\cdot \\frac{1}{n}\\cdot \\log \\,\\mathbb {\\Pi }\\left[\\breve{\\xi }_{n}^{w\\mathbf {W}}\\in \\textrm {\\right.\\right.\\Omega \\hspace{-6.544pt}\\Omega } ^{1-\\gamma } ,\\qquad \\ \\\\& & \\hspace{-25.6073pt}\\sup _{\\mathbf {Q}\\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega }}c_{1} \\cdot \\Big (H_{\\gamma }(\\mathbf {Q},{P})^{c_{2}} - c_{3} \\Big )=\\lim _{n\\rightarrow \\infty } c_{1} \\cdot \\left\\lbrace A^{c_{2} \\cdot \\gamma } \\cdot \\left( 1+\\frac{\\gamma }{n}\\cdot \\log \\,\\mathbb {\\Pi }\\left[\\breve{\\xi }_{n}^{w\\mathbf {W}}\\in \\textrm {\\right.\\right.\\right.\\Omega \\hspace{-6.544pt}\\Omega } ^{c_{2} \\cdot (1-\\gamma )} - c_{3} ,\\ \\ \\textrm {if c_{1} \\cdot c_{2} >0, c_{3} \\in \\mathbb {R}}, \\quad \\ \\\\& & \\hspace{-25.6073pt}\\inf _{\\mathbf {Q}\\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega }}c_{1} \\cdot \\Big (H_{\\gamma }(\\mathbf {Q},{P})^{c_{2}} - c_{3} \\Big )=\\lim _{n\\rightarrow \\infty } c_{1} \\cdot \\left\\lbrace A^{c_{2} \\cdot \\gamma } \\cdot \\left( 1+\\frac{\\gamma }{n}\\cdot \\log \\,\\mathbb {\\Pi }\\left[\\breve{\\xi }_{n}^{w\\mathbf {W}}\\in \\textrm {\\right.\\right.\\right.\\Omega \\hspace{-6.544pt}\\Omega } ^{c_{2} \\cdot (1-\\gamma )} - c_{3} ,\\ \\ \\textrm {if c_{1} \\cdot c_{2} < 0, c_{3} \\in \\mathbb {R}}, \\quad \\ \\\\& & \\hspace{-25.6073pt}\\sup _{\\mathbf {Q}\\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega }}c_{1} \\cdot \\Big (\\Big (\\sum _{k=1}^{K} q_{k}^{\\gamma }\\Big )^{c_{2}} - c_{3} \\Big )=\\lim _{n\\rightarrow \\infty } c_{1} \\cdot \\left\\lbrace K^{c_{2}\\cdot (1-\\gamma )} \\cdot A^{c_{2} \\cdot \\gamma } \\cdot \\left( 1+\\frac{\\gamma }{n}\\cdot \\log \\,\\mathbb {\\Pi }\\left[\\breve{\\breve{\\xi }}_{n}^{w\\mathbf {W}}\\in \\textrm {\\right.\\right.\\right.\\Omega \\hspace{-6.544pt}\\Omega } ^{c_{2} \\cdot (1-\\gamma )} - c_{3} ,\\ \\ \\nonumber \\\\& & \\hspace{375.57628pt}\\textrm {if c_{1} \\cdot c_{2} >0, c_{3} \\in \\mathbb {R}}, \\quad \\ $ $& & \\hspace{-25.6073pt}\\inf _{\\mathbf {Q}\\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega }}c_{1} \\cdot \\Big (\\Big (\\sum _{k=1}^{K} q_{k}^{\\gamma }\\Big )^{c_{2}} - c_{3} \\Big )=\\lim _{n\\rightarrow \\infty } c_{1} \\cdot \\left\\lbrace K^{c_{2}\\cdot (1-\\gamma )} \\cdot A^{c_{2} \\cdot \\gamma } \\cdot \\left( 1+\\frac{\\gamma }{n}\\cdot \\log \\,\\mathbb {\\Pi }\\left[\\breve{\\breve{\\xi }}_{n}^{w\\mathbf {W}}\\in \\textrm {\\right.\\right.\\right.\\Omega \\hspace{-6.544pt}\\Omega } ^{c_{2} \\cdot (1-\\gamma )} - c_{3} ,\\ \\ \\nonumber \\\\& & \\hspace{375.57628pt}\\textrm {if c_{1} \\cdot c_{2} < 0, c_{3} \\in \\mathbb {R}} .",
"\\quad \\ $ From this, the BS-minimizability/maximizability of the important norms/entropies/diversity indices (E1) to (E6) follows immediately as special cases.",
"(b) The special case $\\varphi := \\widetilde{c}\\cdot \\varphi _{\\gamma }$ ($\\gamma \\in ]0,1[$ ) of Theorem REF works analogously to (a), with the differences that we employ (i) additionally a sequence $(X_{i})_{i\\in \\mathbb {N}}$ of random variables being independent of $(W_{i})_{i\\in \\mathbb {N}}$ and satisfying condition (REF ) (resp.",
"(REF )), (ii) $A=1$ (instead of arbitrary $A>0$ ), (iii) $\\mathbb {\\Pi }_{X_{1}^{n}}[\\cdot ]$ (instead of $\\mathbb {\\Pi }[\\cdot ]$ ), (iv) $\\xi _{n,\\mathbf {X}}^{w\\mathbf {W}}$ (instead of $\\xi _{n}^{w\\mathbf {W}}$ ), (v) $\\breve{\\xi }_{n,\\mathbf {X}}^{w\\mathbf {W}}$ (instead of $\\breve{\\xi }_{n}^{w\\mathbf {W}}$ ), and (vi) $\\breve{\\breve{\\xi }}_{n,\\mathbf {X}}^{w\\mathbf {W}}$ (instead of $\\breve{\\breve{\\xi }}_{n}^{w\\mathbf {W}}$ ).",
"This follows from Theorem REF , Remark REF (vi), Lemma REF (a), (REF ), (REF ), (REF ), () and the below-mentioned $\\mathbb {}-$ concerning Example REF (b).",
"Employing Lemma REF (b) and Example REF (a) instead, one ends up with the following proposition on the Kullback-Leibler divergence: Proposition 25 (a) Consider the context of Remark REF (vi) for $\\varphi := \\widetilde{c}\\cdot \\varphi _{\\gamma }$ with $\\gamma =1$ , and let ${P} \\in \\mathbb {S}_{>0}^{K}$ as well as $\\widetilde{c}>0$ be arbitrary but fixed.",
"Furthermore, let $W:=(W_{i})_{i\\in \\mathbb {N}}$ be an i.i.d.",
"sequence of non-negative real-valued random variables with distribution $\\mathbb {} =\\frac{1}{\\widetilde{c}}\\cdot POI(\\widetilde{c})$ being the “$\\frac{1}{\\widetilde{c}}-$ fold Poisson distribution with mean $\\widetilde{c}$ ” , which means that $W_{1}=\\frac{1}{\\widetilde{c}}\\cdot Z$ for a Poissonian $POI(\\widetilde{c})-$ distributed random variable $Z$ with mean $\\widetilde{c}$ (where the subcase $\\widetilde{c}=1$ amounts to $\\mathbb {} =POI(1)$ ).",
"Then for all $A >0$ and all $\\Omega $$\\Omega \\subset \\mathbb {S}^{K}$ with (REF ) there holds $\\hspace{-19.91684pt}-\\lim _{n\\rightarrow \\infty }\\frac{1}{n} \\log \\,\\mathbb {\\Pi }\\left[\\xi _{n}^{w\\mathbf {W}}\\in \\textrm {\\right.\\Omega \\hspace{-6.544pt}\\Omega } &=& \\inf _{{Q}\\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega } }\\ \\widetilde{c}\\cdot \\left[ 1-A \\cdot \\exp \\left( -\\frac{1}{A \\cdot \\widetilde{c}}\\cdot D_{\\widetilde{c}\\cdot \\varphi _{1}}(\\mathbf {Q},{P}) + \\frac{1}{A} -1 \\right) \\right]$ and the BS minimizabilities/maximizabilites (cf.",
"Definition REF ) $& & \\hspace{-34.14322pt}\\inf _{\\mathbf {Q}\\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega } }D_{\\widetilde{c}\\cdot \\varphi _{1}}(\\mathbf {Q},{P})=\\lim _{n\\rightarrow \\infty } \\widetilde{c}\\cdot \\left\\lbrace 1 - A \\cdot \\left[ 1 + \\log \\left( \\frac{1}{A} \\cdot \\Big (1+\\frac{1}{\\widetilde{c}}\\cdot \\frac{1}{n}\\cdot \\log \\,\\mathbb {\\Pi }\\left[\\xi _{n}^{w\\mathbf {W}}\\in \\textrm {\\right.\\right.\\right.\\right.\\Omega \\hspace{-6.544pt}\\Omega } \\Big ) ,\\\\& & \\hspace{-34.14322pt}\\inf _{\\mathbf {Q}\\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega } } \\,I(\\mathbf {Q},{P})=\\inf _{\\mathbf {Q}\\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega } } \\,\\sum \\displaylimits _{k=1}^{K} q_{k} \\cdot \\log \\left( \\frac{q_{k}}{p_{k}} \\right)= - \\lim _{n\\rightarrow \\infty } A \\cdot \\log \\left( \\frac{1}{A} \\cdot \\Big (1+\\frac{1}{n}\\cdot \\log \\,\\mathbb {\\Pi }\\left[\\breve{\\xi }_{n}^{w\\mathbf {W}}\\in \\textrm {\\right.\\right.\\Omega \\hspace{-6.544pt}\\Omega } \\Big ) ,\\nonumber \\\\& & \\hspace{-34.14322pt}\\max _{\\mathbf {Q}\\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega } } \\,\\mathcal {E}^{Sh}(\\mathbf {Q})= \\max _{\\mathbf {Q}\\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega } } \\,(-1) \\cdot \\sum _{k=1}^{K} q_{k} \\cdot \\log (q_{k})= \\lim _{n\\rightarrow \\infty }\\frac{\\log K}{K} + A \\cdot \\log \\left( \\frac{1}{A} \\cdot \\Big (1+\\frac{1}{n}\\cdot \\log \\,\\mathbb {\\Pi }\\left[\\breve{\\breve{\\xi }}_{n}^{w\\mathbf {W}}\\in \\textrm {\\right.\\right.\\Omega \\hspace{-6.544pt}\\Omega } \\Big ), \\\\& & \\hspace{-34.14322pt}\\max _{\\mathbf {Q}\\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega } } \\,\\mathcal {E}^{gSM2}(\\mathbf {Q})= \\max _{\\mathbf {Q}\\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega } } \\,\\frac{1}{1-s} \\cdot \\exp \\Big \\lbrace (s-1) \\cdot \\sum _{k=1}^{K} q_{k} \\cdot \\log (q_{k}) -1\\Big \\rbrace \\nonumber \\\\& & \\hspace{-34.14322pt}= \\lim _{n\\rightarrow \\infty }\\frac{1}{1-s} \\cdot \\exp \\left\\lbrace (1-s) \\cdot \\left[\\frac{\\log K}{K} + A \\cdot \\log \\left( \\frac{1}{A} \\cdot \\Big (1+\\frac{1}{n}\\cdot \\log \\,\\mathbb {\\Pi }\\left[\\breve{\\breve{\\xi }}_{n}^{w\\mathbf {W}}\\in \\textrm {\\right.\\right.\\right.\\right.\\Omega \\hspace{-6.544pt}\\Omega } \\Big ) -1 ,\\ \\ s \\in \\, ]0,1[ \\, \\cup \\, ]1,\\infty [ .$ The special subcase $A=1$ in () (and thus, $\\mathbf {Q}$ is a probability vector) corresponds to the maximum entropy problem for the Shannon entropy $\\mathcal {E}^{Sh}(\\cdot )$ .",
"This can hence be tackled by our bare-simulation approach for almost arbitrary sets $\\textrm {$$\\hspace{-6.544pt}$$}$ of probability vectors.",
"(b) The special case $\\varphi := \\widetilde{c}\\cdot \\varphi _{\\gamma }$ ($\\gamma =1$ ) of Theorem REF works analogously to (a), with the differences that we employ (i) additionally a sequence $(X_{i})_{i\\in \\mathbb {N}}$ of random variables being independent of $(W_{i})_{i\\in \\mathbb {N}}$ and satisfying condition (REF ) (resp.",
"(REF )), (ii) $A=1$ (instead of arbitrary $A>0$ ), (iii) $\\mathbb {\\Pi }_{X_{1}^{n}}[\\cdot ]$ (instead of $\\mathbb {\\Pi }[\\cdot ]$ ), (iv) $\\xi _{n,\\mathbf {X}}^{w\\mathbf {W}}$ (instead of $\\xi _{n}^{w\\mathbf {W}}$ ), (v) $\\breve{\\xi }_{n,\\mathbf {X}}^{w\\mathbf {W}}$ (instead of $\\breve{\\xi }_{n}^{w\\mathbf {W}}$ ), and (vi) $\\breve{\\breve{\\xi }}_{n,\\mathbf {X}}^{w\\mathbf {W}}$ (instead of $\\breve{\\breve{\\xi }}_{n}^{w\\mathbf {W}}$ ).",
"For the sake of completeness, let us mention here that we do not deal with the case $\\gamma \\in ]1,2[$ , for which we conjecture that $\\mathbb {}$ becomes a signed finite measure with total mass 1, i.e.",
"it has a density (with respect to some dominating measure) with positive and negative values which “integrates to 1” ; accordingly, our BS method can not be applied to this situation.",
"To proceed with further $\\gamma -$ cases, a combination of Theorem REF respectively Remark REF (vi), Lemma REF (a), (REF ), (REF ), (REF ), () and the below-mentioned $\\mathbb {}-$ concerning Example REF (c) leads to the following Proposition 26 (a) Consider the context of Remark REF (vi) for $\\varphi := \\widetilde{c}\\cdot \\varphi _{\\gamma }$ with $\\gamma =2$ , and let ${P} \\in \\mathbb {S}_{>0}^{K}$ as well as $\\widetilde{c}>0$ be arbitrary but fixed.",
"Furthermore, let $W:=(W_{i})_{i\\in \\mathbb {N}}$ be an i.i.d.",
"sequence of real-valued random variables with probability distribution $\\mathbb {} =NOR(1,\\frac{1}{\\widetilde{c}})$ being the Normal (Gaussian) law with mean 1 and variance $\\frac{1}{\\widetilde{c}}$ .",
"Then for all $A >0$ and $\\Omega $$\\Omega \\subset \\mathbb {S}^{K}$ with (REF ) there hold all the BS-extremizabilites (REF ) to () as well as (REF ) (below) with plugging-in $\\gamma =2$ .",
"From this, the BS-minimizability/maximizability of the important norms/entropies/diversity indices (E1) to (E6) follow immediately as special cases.",
"By Remark REF (c), one can even take $A<0$ in (REF ) to () and (REF ) as well as in (E1), (E2), (E4) and (E6).",
"(b) The special case $\\varphi := \\widetilde{c}\\cdot \\varphi _{\\gamma }$ ($\\gamma =2$ ) of Theorem REF works analogously to (a), with the differences that we employ (i) additionally a sequence $(X_{i})_{i\\in \\mathbb {N}}$ of random variables being independent of $(W_{i})_{i\\in \\mathbb {N}}$ and satisfying condition (REF ) (resp.",
"(REF )), (ii) $A=1$ (instead of arbitrary $A>0$ ), (iii) $\\mathbb {\\Pi }_{X_{1}^{n}}[\\cdot ]$ (instead of $\\mathbb {\\Pi }[\\cdot ]$ ), (iv) $\\xi _{n,\\mathbf {X}}^{w\\mathbf {W}}$ (instead of $\\xi _{n}^{w\\mathbf {W}}$ ), (v) $\\breve{\\xi }_{n,\\mathbf {X}}^{w\\mathbf {W}}$ (instead of $\\breve{\\xi }_{n}^{w\\mathbf {W}}$ ), and (vi) $\\breve{\\breve{\\xi }}_{n,\\mathbf {X}}^{w\\mathbf {W}}$ (instead of $\\breve{\\breve{\\xi }}_{n}^{w\\mathbf {W}}$ ).",
"For instance, the BS minimizability () of Proposition REF (a) can be employed to solve the following discrete Monge-Kantorovich-type optimal mass transportation problem (optimal coupling problem) with side (i.e.",
"additional) constraints: given two nonnegative-entries vectors $\\mu :=(\\mu _{1}, \\ldots \\mu _{K_{1}} ) \\in [0,\\infty [^{K_{1}}$ and $\\nu := (\\nu _{1}, \\ldots \\nu _{K_{2}} ) \\in [0,\\infty [^{K_{2}}$ with equal total “mass” $\\sum _{k=1}^{K_{1}} \\mu _{k} =\\sum _{k=1}^{K_{2}} \\nu _{k} = A >0$ , compute $&& \\inf _{K_{1} \\times K_{2}-\\textrm {matrices} \\ \\pi }K_{1} \\cdot K_{2} \\cdot \\sum _{u=1}^{K_{1}} \\sum _{v=1}^{K_{2}}\\left(\\pi _{u,v} - \\frac{1}{K_{1} \\cdot K_{2}} \\right)^{2}\\\\&&\\textrm {subject to}\\nonumber \\\\&& \\sum _{v=1}^{K_{2}} \\pi _{u,v} = \\mu _{u} \\quad \\textrm {for all } u \\in \\lbrace 1,\\ldots ,K_{1}\\rbrace ,\\\\&& \\sum _{u=1}^{K_{1}} \\pi _{u,v} = \\nu _{v} \\quad \\textrm {for all } v \\in \\lbrace 1,\\ldots ,K_{2}\\rbrace ,\\\\&& \\pi _{u,v} \\in [0,A] \\quad \\textrm {for all } u \\in \\lbrace 1,\\ldots ,K_{1}\\rbrace , \\, v \\in \\lbrace 1,\\ldots ,K_{2}\\rbrace ,\\\\&& \\textrm {side constraints on \\pi , \\mu , \\nu }.$ Indeed, this problem can be equivalently rewritten in terms $K_{1} \\cdot K_{2}-$ dimensional vectors as follows: given two nonnegative-entries vectors $\\mu $ ,$\\nu $ as above, compute $&& \\inf _{\\mathbf {Q}\\in \\mathbf {\\Omega }}K_{1} \\cdot K_{2} \\cdot \\sum _{k=1}^{K_{1} \\cdot K_{2}}\\left(q_{k} - \\frac{1}{K_{1} \\cdot K_{2}} \\right)^{2}\\ = \\ \\inf _{\\mathbf {Q}\\in \\mathbf {\\Omega }}K_{1} \\cdot K_{2} \\cdot \\sum _{k=1}^{K_{1} \\cdot K_{2}}q_{k}^{2} + 1 - 2A \\\\&& \\textrm {where\\mathbf {\\Omega } \\subset \\mathbb {R}^{K_{1}\\cdot K_{2}} is the set of allvectors \\mathbf {Q} = (q_{1}, \\ldots , q_{K_{1} \\cdot K_{2}})which satisfy the constraints}\\nonumber \\\\&& \\sum _{j=1}^{K_{2}} q_{(i-1)\\cdot K_{2} +j} \\ = \\ \\mu _{i} \\quad \\textrm {for all } i \\in \\lbrace 1,\\ldots ,K_{1}\\rbrace , \\\\&& \\sum _{i=1}^{K_{1}} q_{(i-1)\\cdot K_{2} +j} \\ = \\ \\nu _{j} \\quad \\textrm {for all } j \\in \\lbrace 1,\\ldots ,K_{2}\\rbrace , \\\\&& q_{k} \\in [0,A] \\quad \\textrm {for all } k \\in \\lbrace 1,\\ldots ,K_{1} \\cdot K_{2} \\rbrace , \\\\&& \\textrm {side constraints on \\mathbf {Q}, \\mu , \\nu }.$ Clearly, via divisions by $A$ , one can equivalently rewrite $\\mathbf {\\Omega }= A \\cdot \\textrm {$$\\hspace{-6.544pt}$$}$ for some $\\textrm {$$\\hspace{-6.544pt}$$} \\subset \\mathbb {S}^{K_{1} \\cdot K_{2}}$ in the $K_{1}\\cdot K_{2}-$ dimensional probability simplex.",
"Hence, we can employ () with $c_{1}=K_{1} \\cdot K_{2}$ , $c_{2}=1$ and $c_{3} = 1-2A$ , provided that the side constraints () are such that $\\mathbf {\\Omega }$ satisfies the regularity property (REF ) and the finiteness property (REF ).",
"Notice that (REF ) is equal to $\\inf _{\\mathbf {Q}\\in \\mathbf {\\Omega }}D_{2 \\cdot \\varphi _{2}}(\\mathbf {Q},{P}^{unif})\\nonumber $ where ${P}^{unif} := (\\frac{1}{K_{1} \\cdot K_{2}},\\ldots , \\frac{1}{K_{1} \\cdot K_{2}})$ is the probability vector of frequencies of the uniform distribution on $\\lbrace 1, \\ldots , K_{1} \\cdot K_{2}\\rbrace $ , and $\\widetilde{c}=2$ .",
"The special case $A=1$ with side constraint () of the form $K_{1} \\cdot \\min _{i \\in \\lbrace 1,\\ldots , K_{1}\\rbrace } \\mu _{i} +K_{2} \\cdot \\min _{j \\in \\lbrace 1,\\ldots , K_{2}\\rbrace } \\nu _{j} \\ge 1$ was explicitly solved by e.g.",
"Bertrand et al.",
"[43], [44], who also give applications to cryptographic guessing problems (spy problems), task partitioning and graph clustering.",
"The importance of the case $\\gamma =2$ stems also from the fact that one can equivalently rewrite separable quadratic minimization problems as minimization problems of Pearson chi-square divergences.",
"Indeed, by straightforward calculations one can derive that $\\inf _{\\mathbf {\\breve{Q}}\\in \\mathbf {\\breve{\\Omega }}} \\,\\sum _{k=1}^{K} ( \\, c_{1,k} + c_{2,k} \\cdot \\breve{q}_{k} + c_{3,k} \\cdot \\breve{q}_{k}^2 \\, ) \\, ,\\qquad c_{1,k} \\in \\mathbb {R}, \\ c_{2,k} \\in \\mathbb {R}\\backslash \\lbrace 0\\rbrace , \\ c_{3,k} \\in \\, ]0,\\infty [,$ is equal to (recall that $\\varphi _{2}(t) := \\frac{(t - 1)^2}{2}$ , cf.",
"(REF )) $c_{4} + \\inf _{\\mathbf {Q}\\in \\mathbf {\\Omega }}D_{2 \\cdot \\varphi _{2}}(\\mathbf {Q},\\mathbf {P}) \\, ,$ where $\\mathbf {Q} := (q_{1}, \\ldots , q_{K})$ with $q_{k} := - c_{2,k} \\cdot \\breve{q}_{k}$ , $\\mathbf {P} := (p_{1}, \\ldots , p_{K})$ with $p_{k}:= \\frac{c_{2,k}^{2}}{2 \\cdot c_{3,k}} >0$ , $c_{4} := \\sum _{k=1}^{K} \\big ( \\, c_{1,k} - \\frac{c_{2,k}^{2}}{4 \\cdot c_{3,k}} \\, \\big )$ , and $\\mathbf {\\Omega }$ is the corresponding reformulation of the constraint set $\\mathbf {\\breve{\\Omega }}$ .",
"To achieve the applicability of our BS method, we further transform (REF ) into its equal form (cf.",
"(REF )) $c_{4} + \\inf _{\\widetilde{\\mathbf {Q}}\\in \\mathbf {\\Omega }/M_{P}}D_{2 M_{\\mathbf {P}} \\cdot \\varphi _{2}}(\\mathbf {Q},\\widetilde{{P}}) \\,$ with $M_{\\mathbf {P}}:=\\sum _{k=1}^{K} p_{k}>0$ and $\\widetilde{{P}}:=\\mathbf {P}/M_{\\mathbf {P}}$ .",
"If $\\mathbf {\\Omega }/M_{\\mathbf {P}}$ satisfies (REF ) and (REF ) (e.g.",
"it may be highly disconnected), then we can apply Theorem REF .",
"In contrast, if $\\mathbf {\\Omega }/M_{\\mathbf {P}} = A \\cdot \\textrm {$$\\hspace{-6.544pt}$$}$ for some $A \\in \\mathbb {R}\\backslash \\lbrace 0\\rbrace $ and some $\\Omega $$\\Omega \\subset \\mathbb {S}_{>0}^{K}$ satisfying (REF ), then we can apply Proposition REF (a) together with Remark REF (c); for instance, this may appear if $\\mathbf {\\breve{\\Omega }}$ contains (amongst others) the original constraint $\\sum _{k=1}^{K} \\breve{q}_{k} = C$ for some constant $C>0$ , and $c_{2,k} =c_{2}$ does not depend on $k$ , which leads to the choice $A = - \\frac{c_{2} \\cdot C}{M_{\\mathbf {P}}}$ .",
"Notice that $A<0$ if $c_{2}>0$ .",
"For example, optimization problems (REF ) with $c_{1,k} >0$ , $c_{2,k} >0$ , $c_{3,k} >0$ and constraints $\\sum _{k=1}^{K} \\breve{q}_{k} = C$ , $\\breve{q}_{k} \\in [\\underline{\\breve{q}}_{k},\\overline{\\breve{q}}_{k}]$ appear in distributed energy management as economic dispatch problems in smart grids of power generators, where $\\breve{q}_{k}$ is the active power generation of the $k-$ th generator, $C$ is the total power demand, $\\underline{\\breve{q}}_{k}$ resp.",
"$\\overline{\\breve{q}}_{k}$ represent the lower resp.",
"upper bound of the $k-$ th generator's output, and the cost of power generation is $c_{1,k} + c_{2,k} \\cdot \\breve{q}_{k} + c_{3,k} \\cdot \\breve{q}_{k}^2$ (cf.",
"e.g.",
"Yang et al.",
"[412], Loia & Vaccaro [230], Wood et al.",
"[394], Xu et al.",
"[404]).",
"Another important special case of (REF ) to (REF ) is the omnipresent $L_{2}-$ minimization; indeed, with the choices $c_{3,k}=1$ , $c_{2,k}= - 2 v_{k}$ , and $c_{1,k}= v_{k}^2$ for some $\\mathbf {V} =(v_{1},\\ldots ,v_{K})$ , the minimization problem (REF ) is nothing but $\\inf _{\\mathbf {\\breve{Q}}\\in \\mathbf {\\breve{\\Omega }}} \\, || \\mathbf {\\breve{Q}} - \\mathbf {V} ||_{2}^{2}$ ; if $\\mathbf {\\breve{\\Omega }}$ depends on a pregiven $L-$ dimensional vector $\\mathbf {x}$ (with $L < K$ ), this can be regarded as a non-parametric regression problem in a wide sense.",
"To continue with our general investigations, by combining Theorem REF respectively Remark REF (vi), Lemma REF (a), (REF ), (REF ), (REF ), () and the below-mentioned $\\mathbb {}-$ concerning Example REF (e), we arrive at the following Proposition 27 (a) Consider the context of Remark REF (vi) for $\\varphi := \\widetilde{c}\\cdot \\varphi _{\\gamma }$ with $\\gamma >2$ , and let ${P} \\in \\mathbb {S}_{>0}^{K}$ as well as $\\widetilde{c}>0$ be arbitrary but fixed.",
"Furthermore, let $W:=(W_{i})_{i\\in \\mathbb {N}}$ be an i.i.d.",
"sequence of real-valued random variables having densityin the classical sense, with respect to Lebesgue measure $\\frac{\\exp \\lbrace \\frac{y \\cdot \\widetilde{c}}{\\gamma -1}\\rbrace }{\\exp \\lbrace \\widetilde{c}/\\gamma \\rbrace }\\cdot f_{Z}(- y), \\qquad y\\in ]-\\infty ,\\infty [,$ where $f_{Z}$ is the density of a random variable $Z$ which has stable law with parameter-quadruple $(\\frac{\\gamma }{\\gamma -1},1,0,\\frac{\\widetilde{c}^{1/(1-\\gamma )} \\cdot (\\gamma -1)^{\\gamma /(\\gamma -1)}}{\\gamma })$ in terms of the above-mentioned “form-B notation” in Zolotarev [428].",
"Then for all $A >0$ and $\\Omega $$\\Omega \\subset \\mathbb {S}^{K}$ with (REF ) there hold all the BS-extremizabilites (REF ) to (REF ) as well as $& & \\hspace{-42.67912pt}\\sup _{\\mathbf {Q}\\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega }}\\frac{1}{1-\\gamma } \\cdot \\log \\Big (\\sum _{k=1}^{K} q_{k}^{\\gamma }\\Big )= \\lim _{n\\rightarrow \\infty } \\frac{1}{\\gamma \\cdot (\\gamma -1)} \\cdot \\left[\\log \\Big ( A^{\\gamma } \\cdot \\Big ( 1+ \\gamma \\cdot \\frac{1}{n}\\cdot \\log \\,\\mathbb {\\Pi }\\left[\\breve{\\breve{\\xi }}_{n}^{w\\mathbf {W}}\\in \\textrm {\\right.\\right.\\Omega \\hspace{-6.544pt}\\Omega } \\Big ) ^{1-\\gamma } \\Big ) + (1-\\gamma ) \\cdot \\log (K) .$ From this, the BS-minimizability/maximizability of the important norms/entropies/diversity indices (E1) to (E6) follow immediately as special cases.",
"(b) The special case $\\varphi := \\widetilde{c}\\cdot \\varphi _{\\gamma }$ ($\\gamma >2$ ) of Theorem REF works analogously to (a), with the differences that we employ (i) additionally a sequence $(X_{i})_{i\\in \\mathbb {N}}$ of random variables being independent of $(W_{i})_{i\\in \\mathbb {N}}$ and satisfying condition (REF ) (resp.",
"(REF )), (ii) $A=1$ (instead of arbitrary $A>0$ ), (iii) $\\mathbb {\\Pi }_{X_{1}^{n}}[\\cdot ]$ (instead of $\\mathbb {\\Pi }[\\cdot ]$ ), (iv) $\\xi _{n,\\mathbf {X}}^{w\\mathbf {W}}$ (instead of $\\xi _{n}^{w\\mathbf {W}}$ ), (v) $\\breve{\\xi }_{n,\\mathbf {X}}^{w\\mathbf {W}}$ (instead of $\\breve{\\xi }_{n}^{w\\mathbf {W}}$ ), and (vi) $\\breve{\\breve{\\xi }}_{n,\\mathbf {X}}^{w\\mathbf {W}}$ (instead of $\\breve{\\breve{\\xi }}_{n}^{w\\mathbf {W}}$ ).",
"As mentioned above, in the Propositions REF to REF we have combined Theorem REF respectively Remark REF (vi), Lemma REF and explicitly solved representations (REF ).",
"The latter, important step will be discussed in a structured, comprehensive manner in the Section below.",
"By retransformation, we can even deal with optimizations of nonnegative linear objective functions with constraint sets on Euclidean $\\gamma $ -norm spheres.",
"Indeed, for nonnegative $\\breve{\\mathbf {Q}} := (\\breve{q}_{1}, \\ldots , \\breve{q}_{K})$ and $\\breve{\\mathbf {P}} := (\\breve{p}_{1}, \\ldots , \\breve{p}_{K})$ one can rewrite their scalar product as $\\gamma -$ order Hellinger integrals $& & \\hspace{-36.98866pt}\\sum _{k=1}^{K} \\breve{q}_{k} \\cdot \\breve{p}_{k}\\ = \\ c_{1} \\cdot \\sum _{k=1}^{K} q_{k}^{\\gamma } \\cdot p_{k}^{1-\\gamma }\\ = \\ c_{1} \\cdot H_{\\gamma }(\\mathbf {Q},{P}) \\, \\hspace{14.22636pt} \\textrm {where}\\\\& & \\hspace{-39.83368pt}\\textrm {\\gamma \\in \\, ]0,1[ \\, \\cup \\, [2,\\infty [ \\, if \\, \\breve{\\mathbf {Q}} \\in [0,\\infty [^{K},\\breve{\\mathbf {P}} \\in \\, ]0,\\infty [^{K}\\hspace{11.38092pt} respectively \\hspace{11.38092pt}\\gamma \\in \\, ]-\\infty ,0[ \\, if \\, \\breve{\\mathbf {Q}} \\in \\, ]0,\\infty [^{K},\\breve{\\mathbf {P}} \\in \\, ]0,\\infty [^{K},}\\\\& & \\hspace{-36.98866pt}q_{k} \\ := \\ \\breve{q}_{k}^{1/\\gamma } ,\\quad p_{k} \\ := \\ \\frac{\\breve{p}_{k}^{1/(1-\\gamma )}}{\\sum _{i=1}^{K} \\breve{p}_{i}^{1/(1-\\gamma )}} ,\\quad c_{1}:= \\Big ( \\sum _{i=1}^{K} \\breve{p}_{i}^{1/(1-\\gamma )} \\Big )^{1-\\gamma } = :|| \\breve{\\mathbf {P}} ||_{1-\\gamma } \\ .$ The required constraint $\\sum _{k=1}^{K} q_{k} = A >0$ retransforms to $|| \\breve{\\mathbf {Q}} ||_{\\gamma }=A^{1/\\gamma }$ and thus, $\\breve{\\mathbf {Q}}$ must lie on (the positive/nonnegative part of) the $\\gamma -$ norm-sphere $\\partial B_{\\gamma }(0,A^{1/\\gamma })$ around the origin with radius $A^{1/\\gamma }$ .",
"Accordingly, for $\\gamma \\in \\, [2,\\infty [$ we have $\\inf _{\\breve{\\mathbf {Q}} \\in \\breve{\\mathbf {\\Omega }}} \\,\\sum _{k=1}^{K} \\breve{q}_{k} \\cdot \\breve{p}_{k}\\ = \\ c_{1} \\cdot \\inf _{\\mathbf {Q}\\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega } }H_{\\gamma }(\\mathbf {Q},{P})$ and we can apply () of Proposition REF (a) respectively Proposition REF (a) here and analogously henceforth, by this we mean the condition () as it appears in the Proposition REF (a) respectively Proposition REF (a), as long as the original constraint set $\\breve{\\mathbf {\\Omega }} \\in \\partial B_{\\gamma }(0,A^{1/\\gamma }) \\, \\cap \\, [0,\\infty [^{K}$ transforms (via $q_{k} = \\breve{q}_{k}^{1/\\gamma }$ ) into a constraint set $A \\cdot \\textrm {$$\\hspace{-6.544pt}$$}$ which satisfies the regularity assumption (REF ) in the relative topology (as a side remark, notice that $int(\\partial B_{\\gamma }(0,A^{1/\\gamma })) = \\emptyset $ in the full topology).",
"For the case $\\gamma \\in \\, ]-\\infty ,0[$ we also have (REF ) and apply () of Proposition REF (a) for any original constraint set $\\breve{\\mathbf {\\Omega }} \\in \\partial B_{\\gamma }(0,A^{1/\\gamma }) \\, \\cap \\, ]0,\\infty [^{K}$ which transforms into $A \\cdot \\textrm {$$\\hspace{-6.544pt}$$}$ satisfying (REF ) in the relative topology.",
"In contrast, for the case $\\gamma \\in \\, ]0,1[$ we get $\\sup _{\\breve{\\mathbf {Q}} \\in \\breve{\\mathbf {\\Omega }}} \\,\\sum _{k=1}^{K} \\breve{q}_{k} \\cdot \\breve{p}_{k}\\ = \\ c_{1} \\cdot \\sup _{\\mathbf {Q}\\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega } }H_{\\gamma }(\\mathbf {Q},{P})\\nonumber $ and apply (REF ) of Proposition REF (a) for any original constraint set $\\breve{\\mathbf {\\Omega }} \\in \\partial B_{\\gamma }(0,A^{1/\\gamma }) \\, \\cap \\, [0,\\infty [^{K}$ which transforms into $A \\cdot \\textrm {$$\\hspace{-6.544pt}$$}$ satisfying (REF ) in the relative topology.",
"As a continuation of Remark REF , we can principally tackle all the optimization problems of this Subsection REF by basically only employing a fast and accurate — pseudo, true, natural, quantum — random number generator, provided that the constraint set $\\textrm {$ A $\\hspace{-6.544pt}$$}$ satisfies the mild assumptions (REF ) (in the relative topology) and (REF ).",
"Recall that $A>0$ (and for $\\varphi _{2}$ even $A \\in \\mathbb {R}\\backslash \\lbrace 0\\rbrace $ ) and that $\\mathbf {Q} \\in \\textrm {$ A $\\hspace{-6.544pt}$$}$ implies in particular the constraint $\\sum _{k=1}^{K} q_{k} = A$ .",
"The regularity assumption (REF ) allows for e.g.",
"high-dimensional constraint sets $\\textrm {$ A $\\hspace{-6.544pt}$$}$ which are non-convex and even highly disconnected, and for which other minimization methods (e.g.",
"pure enumeration, gradient or steepest descent methods, etc.)",
"may be problematic or intractable.",
"For example, (REF ) covers kind of “$K-$ dimensional (not necessarily regular) polka dot pattern type” relaxations $\\textrm {$ A $\\hspace{-6.544pt}$$} :=\\dot{\\bigcup }_{i=1}^{N} \\mathcal {U}_{i}(Q_{i}^{dis})$ of finite discrete constraint sets $\\textrm {$ A $\\hspace{-6.544pt}$ dis$}:= \\lbrace Q_{1}^{dis}, \\ldots , Q_{N}^{dis}\\rbrace $ of high cardinality $N$ (e.g.",
"being exponential or factorial in a large $K$ ), where each $K-$ dimensional vector $Q_{i}^{dis}$ has total-sum-of-components equal to $A$ and is surrounded by some small (“flat”, i.e.",
"in the relative topology) neighborhood $\\mathcal {U}_{i}(Q_{i}^{dis})$ .",
"For the sake of brevity, in the following discussion we confine ourselves to the deterministic setup (e.g.",
"Proposition REF (a) rather than (b)) which particularly involves $\\mathbb {\\Pi }[\\cdot ]$ (rather than $\\mathbb {\\Pi }_{X_{1}^{n}}[\\cdot ]$ ) and $\\xi _{n}^{w\\mathbf {W}}$ (rather than $\\xi _{n,\\mathbf {X}}^{w\\mathbf {W}}$ ).",
"In such a context, all the optimization problems of this Subsection REF , subsumed as (cf.",
"(REF ) to (REF )) $\\inf _{\\mathbf {Q}\\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega } } \\,\\Phi (\\mathbf {Q})\\qquad \\textrm {respectively} \\qquad \\sup _{\\mathbf {Q}\\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega } } \\,\\Phi (\\mathbf {Q})\\nonumber $ can be regarded as a “BS-tractable” relaxations of the corresponding nonlinear discrete (e.g.",
"integer, combinatorial) programming problems $\\inf _{\\mathbf {Q}\\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega ^{dis}} } \\,\\Phi (\\mathbf {Q})\\qquad \\textrm {respectively} \\qquad \\sup _{\\mathbf {Q}\\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega ^{dis}} } \\,\\Phi (\\mathbf {Q}) \\, ;\\nonumber $ as examples take e.g.",
"$\\Phi (\\mathbf {Q}) =c_{1} \\cdot \\Big ( \\Big (\\sum _{k=1}^{K} q_{k}^{\\gamma }\\Big )^{c_{2}}- c_{3} \\Big )$ (with $\\gamma \\ne 0,1$ ) or $\\Phi (\\mathbf {Q}) = \\Phi _{{P}}(\\mathbf {Q}) =D_{\\widetilde{c}\\cdot \\varphi _{\\gamma }}(\\mathbf {Q},{P})$ .",
"For instance, $A \\cdot \\textrm {$$\\hspace{-6.544pt}$ dis$}$ may contain only $K-$ dimensional vectors $Q_{i}^{dis}$ ($i=1,\\ldots ,N$ ) whose components stem from a finite set $\\mathcal {B}$ of nonnegative integers and add up to $A$ .",
"If $\\mathcal {B} = \\lbrace 0,1\\rbrace $ , then we can even deal with nonnegative linear objective functions $\\Phi (\\mathbf {Q}) = \\sum _{k=1}^{K} \\breve{p}_{k} \\cdot q_{k}$ where $\\mathbf {Q} := (q_{1}, \\ldots , q_{K})$ with $q_{k} \\in \\lbrace 0,1\\rbrace $ and $\\breve{\\mathbf {P}} := (\\breve{p}_{1}, \\ldots , \\breve{p}_{K})$ has components $\\breve{p}_{k} >0$ which reflect e.g.",
"the cost associated with the $k-$ th state.",
"Indeed, by noticing that $q_{k}^{1/\\gamma }=q_{k}$ for $\\gamma \\in \\, ]0,1[ \\, \\cup \\, [2,\\infty [$ , we can employ (REF ) and () to end up with $& & \\hspace{-31.2982pt}\\inf _{\\mathbf {Q}\\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega ^{dis}} } \\,\\sum _{k=1}^{K} q_{k} \\cdot \\breve{p}_{k}= || \\breve{\\mathbf {P}} ||_{1-\\gamma } \\cdot \\inf _{\\mathbf {Q}\\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega ^{dis}} } \\,\\sum _{k=1}^{K} q_{k}^{\\gamma } \\cdot \\left(\\frac{\\breve{p}_{k}^{1/(1-\\gamma )}}{\\sum _{i=1}^{K} \\breve{p}_{i}^{1/(1-\\gamma )}}\\right)^{1-\\gamma }= || \\breve{\\mathbf {P}} ||_{1-\\gamma } \\cdot \\inf _{\\mathbf {Q}\\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega ^{dis}} }H_{\\gamma }(\\mathbf {Q},{P}), \\ \\ \\gamma \\in \\, [2,\\infty [, \\ \\ \\ \\\\& & \\hspace{-31.2982pt}\\sup _{\\mathbf {Q}\\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega ^{dis}} } \\,\\sum _{k=1}^{K} q_{k} \\cdot \\breve{p}_{k}= || \\breve{\\mathbf {P}} ||_{1-\\gamma } \\cdot \\sup _{\\mathbf {Q}\\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega ^{dis}} }H_{\\gamma }(\\mathbf {Q},{P}), \\ \\ \\gamma \\in \\, ]0,1[ \\, .$ The corresponding relaxations are $& & \\hspace{-48.36958pt}\\inf _{\\mathbf {Q}\\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega } } \\,\\sum _{k=1}^{K} q_{k} \\cdot \\breve{p}_{k}\\ = \\ || \\breve{\\mathbf {P}} ||_{1-\\gamma } \\cdot \\inf _{\\mathbf {Q}\\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega } }H_{\\gamma }(\\mathbf {Q},{P}), \\ \\ \\gamma \\in \\, [2,\\infty [ \\, ,\\\\& & \\hspace{-48.36958pt}\\sup _{\\mathbf {Q}\\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega } } \\,\\sum _{k=1}^{K} q_{k} \\cdot \\breve{p}_{k}\\ = \\ || \\breve{\\mathbf {P}} ||_{1-\\gamma } \\cdot \\sup _{\\mathbf {Q}\\in A \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega } }H_{\\gamma }(\\mathbf {Q},{P}), \\ \\ \\gamma \\in \\, ]0,1[ \\, ;$ for (REF ) we can apply () of Proposition REF (a) respectively Proposition REF (a), whereas for () we apply (REF ) of Proposition REF (a) — as long as the relaxation constraint set $A \\cdot \\textrm {$$\\hspace{-6.544pt}$$}$ satisfies (REF ) in the relative topology.",
"For the sake of illustration, let us consider a sum-minimization-type linear assignment problem with side constraints (for a comprehensive book on assignment problems see e.g.",
"Burkard et al.",
"[69]).",
"Suppose that there are $K$ individuals (people, machines, etc.)",
"to carry out $K$ tasks (jobs, etc.).",
"Each individual is assigned to carry out exactly one task.",
"There is cost $c_{ij} >0$ if individual $i$ is assigned to (i.e., carries out) task $j$ .",
"We want to find the minimum total cost amongst all assignments.",
"There may be side constraints, e.g.",
"each assignment has a value $v_{ij} >0$ and the total value of the assignment should be above a pregiven threshold.",
"As usual, the problem can be formulated with the help of binary variables $x_{ij}$ where $x_{ij} =1$ if individual $i$ is assigned to task $j$ , and $x_{ij} =0$ otherwise.",
"Accordingly, we want to compute $&& \\inf _{K \\times K-\\textrm {matrices} \\ x=(x_{ij})} \\ \\sum _{i=1}^{K} \\sum _{j=1}^{K}c_{ij} \\cdot x_{ij}\\\\&&\\textrm {subject to}\\nonumber \\\\&& \\sum _{j=1}^{K} x_{ij} = 1 \\quad \\textrm {for all } i \\in \\lbrace 1,\\ldots ,K\\rbrace ,\\qquad \\textrm {(i.e.",
"each individual i does one task),}\\\\&& \\sum _{i=1}^{K} x_{ij} = 1 \\quad \\textrm {for all } j\\in \\lbrace 1,\\ldots ,K\\rbrace ,\\qquad \\textrm {(i.e.",
"each task j is done by one individual),}\\\\&& x_{ij} \\in \\lbrace 0,1\\rbrace \\quad \\textrm {for all } i \\in \\lbrace 1,\\ldots ,K\\rbrace , \\, j \\in \\lbrace 1,\\ldots ,K\\rbrace ,\\\\&& \\textrm {side (i.e.",
"additional) constraints on x=(x_{ij})_{i,j=1,\\ldots ,K}}.$ Analogously to (REF ), this problem can be equivalently rewritten in terms of $K^{2}-$ dimensional vectors as follows: let $\\mathbf {Q} := (q_{1}, \\ldots , q_{K^{2}})$ and $\\breve{\\mathbf {P}} := (\\breve{p}_{1}, \\ldots , \\breve{p}_{K^{2}})$ be such that $c_{ij}= \\breve{p}_{(i-1)\\cdot K +j}$ and $x_{ij}= q_{(i-1)\\cdot K +j}$ for $i,j \\in \\lbrace 1,\\ldots ,K\\rbrace $ and compute $&& \\inf _{\\mathbf {Q}\\in K \\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega ^{dis}} } \\,\\sum _{k=1}^{K} q_{k} \\cdot \\breve{p}_{k} \\\\&& \\textrm {whereK \\cdot \\textrm {\\Omega \\hspace{-6.544pt}\\Omega ^{dis}}\\subset \\mathbb {R}^{K^{2}} is the set of allvectors \\mathbf {Q} = (q_{1}, \\ldots , q_{K^{2}}) which satisfy the constraints}\\nonumber \\\\&& \\sum _{j=1}^{K} q_{(i-1)\\cdot K +j} \\ = \\ 1 \\quad \\textrm {for all } i \\in \\lbrace 1,\\ldots ,K\\rbrace ,\\\\&& \\sum _{i=1}^{K} q_{(i-1)\\cdot K +j} \\ = \\ 1 \\quad \\textrm {for all } j \\in \\lbrace 1,\\ldots ,K\\rbrace ,\\\\&& q_{k} \\in \\lbrace 0,1\\rbrace \\quad \\textrm {for all } k \\in \\lbrace 1,\\ldots ,K^{2} \\rbrace ,\\\\&& \\textrm {side constraints on \\mathbf {Q}}.$ As seen above, this can be rewritten as $\\gamma -$ order Hellinger-integral minimization problem (REF ), with $\\gamma \\ge 2$ .",
"We can obtain a highly disconnected “non-void-interior-type” relaxation of the binary integer programming problem (REF ) to () by replacing () with $q_{k} \\in [0,\\varepsilon _{1}] \\, \\cup \\, [1-\\varepsilon _{2},1]\\quad \\textrm {for all } k \\in \\lbrace 1,\\ldots ,K^{2} \\rbrace ,$ for some (possibly arbitrarily) small $\\varepsilon _{1}, \\varepsilon _{2} >0$ with $\\varepsilon _{1} + \\varepsilon _{2} < 1$ .",
"We denote by $K \\cdot \\textrm {$$\\hspace{-6.544pt}$$}$ the outcoming set manifested by the constraints (), (), () and (REF ), and accordingly we end up with a minimization problem of type (REF ), which we can tackle by () of Proposition REF (a) respectively Proposition REF (a), as long as (REF ) (in the relative topology) is satisfied.",
"For instance, we can take $\\gamma =2$ and basically solve the corresponding optimization problem by basically simulating $K^{2}-$ dimensional Gaussian random variables (even though the cardinality of $K \\cdot \\textrm {$$\\hspace{-6.544pt}$ dis$}$ may be high).",
"As a side remark, let us mention that our relaxation (REF ) contrasts considerably to the frequently used continuous linear programming (LP) relaxation $q_{k} \\in [0,1]\\quad \\textrm {for all } k \\in \\lbrace 1,\\ldots ,K^{2} \\rbrace .\\nonumber $ Let us finally mention that an important special case of a minimization problem (REF ) to () is — the integer programming formulation of — the omnipresent (asymmetric) traveling salesman problem (TSP) with possible side constraints see e.g.",
"Applegate et al.",
"[14], Gutin & Punnen [147], Cook [92] for comprehensive books on TSP, its variations and its applications to logistics, machine scheduling, printed circuit board drilling, communication-network frequencing, genome sequencing, data clustering, and many others..",
"There, one has $K$ cities and the cost of traveling from city $i$ to city $j\\ne i$ is given by $c_{ij}>0$ .",
"Moreover, one sets $x_{ij} =1$ if the traveler goes directly from city $i$ to city $j$ (in that order), and $x_{ij} =0$ otherwise.",
"For technical reasons, for $i=j$ we attribute a cost $c_{ii}>0$ (e.g.",
"hotel costs), but we require that always $x_{ii}=0$ which we subsume as the first part of the constraints ().",
"Then, the constraint () means that the traveler leaves from city $i$ exactly once, whereas () reflects that the traveler arrives at city $j$ exactly once.",
"The goal is to find a directed tour — i.e.",
"a directed cycle/circuit that visits all $K$ cities once — of minimum cost.",
"Within this context, the second part of the constraints () should basically exclude solutions which consist of disconnected subtours (subtour elimination constraints (of e.g.",
"the seminal Dantzig et al.",
"[104]), connectivity constraints, cut-set constraints).",
"Here, we also allow for additional/side constraints which we subsume as the third part () of the constraints.",
"Hence, by the above-mentioned considerations we can principally tackle such kind of TSP problems with our BS method.",
"For sum-maximization-type linear assignment problems with side constraints, where e.g.",
"$c_{ij}$ is a profit (rather than a cost) and the ultimate goal is total profit maximization, we can proceed analogously, by employing () and () (instead of (REF ) and (REF )).",
"Let us end this subsection with a comparison: suppose that we have a (sufficiently large) number $n$ of concrete data observations $X_{i} = x_{i}$ ($i=1,\\ldots ,n$ ) from the unknown probability distribution ${P}$ (in vector form), and from these we want to approximate/estimate the unknown distance $\\inf _{{Q}\\in \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega }}D_{\\varphi _{\\gamma }}( {Q}, {P} )$ from a family of probability models (in vector form) $\\Omega $$\\Omega $ (e.g.",
"for model-adequacy evaluations, for goodness-of-fit testing purposes): by the above-mentioned Propositions REF to REF (and especially, by (REF ), (REF ), (REF )) one can use $G\\Big (-\\frac{1}{n} \\cdot \\log \\,\\mathbb {\\Pi }_{x_{1}^{n}}\\left[\\xi _{n,\\mathbf {x}}^{w\\mathbf {W}}\\in \\textrm {\\right.\\Omega \\hspace{-6.544pt}\\Omega } \\Big )$ where $\\mathbb {\\Pi }_{x_{1}^{n}}[\\, \\cdot \\, ] := \\mathbb {\\Pi }[ \\, \\cdot \\, | \\,X_{1}=x_{1}, \\ldots , X_{n} = x_{n}]$ , $\\mathbf {x} := (x_{1},\\ldots ,x_{n})$ , and $G$ (cf.",
"(REF )) is e.g.",
"chosen as follows: $G(z) := -\\frac{\\widetilde{c}}{\\gamma \\cdot \\left( \\gamma -1\\right) }\\cdot \\left\\lbrace 1-\\left( 1-\\frac{\\gamma }{\\widetilde{c}} \\cdot z \\right) ^{1-\\gamma } \\right\\rbrace $ for the three cases $\\gamma <0$ , $\\gamma \\in ]0,1[$ and $\\gamma \\ge 2$ , $G(z) := z$ for $\\gamma =0$ (reversed Kullback-Leibler divergence), and $G(z):= - \\widetilde{c} \\cdot \\log (1- \\frac{1}{\\widetilde{c}} \\cdot z)$ for $\\gamma =1$ (Kullback-Leibler divergence).",
"Notice that (REF ) contrasts to the alternative approximation (of $\\inf _{{Q}\\in \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega }}D_{\\varphi _{\\gamma }}( {Q}, {P} )$ ) given by $\\inf _{{Q}\\in \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega }}D_{\\varphi _{\\gamma }}({Q}, \\mathbb {P}_{n}^{emp, co} )$ which is used in the context of “classical” statistical minimum distance estimation (MDE) with power divergences; in (REF ), we have employed $\\mathbb {P}_{n}^{emp, co} = \\frac{1}{n} \\cdot \\sum _{i=1}^{n} \\delta _{x_{i}}$ to be the realization of the empirical distribution $\\mathbb {P}_{n}^{emp} = \\frac{1}{n} \\cdot \\sum _{i=1}^{n} \\delta _{X_{i}}$ .",
"Indeed, especially in complicated high-dimensional non-parametric or semi-parametric big-data contexts, we have substituted a quite difficult optimization problem (REF ) by a much easier solvable counting problem (REF ).",
"The same holds analogously for Renyi distances/divergences, etc."
],
[
"Construction principle for bounds of the minimum divergence\nin the general case ",
"Turning back to Theorem REF , we now consider the general case when the divergence $\\varphi \\in \\Upsilon (]a,b[)$ is not of the power type (REF ).",
"Recall from (REF ) the crucial terms (with ${P} \\in \\mathbb {S}_{>0}$ ) $\\inf _{m\\ne 0} D_{\\varphi }(m\\cdot \\textrm {\\Omega \\hspace{-6.544pt}\\Omega },{P}):= \\inf _{m\\ne 0}\\ \\inf _{{Q}\\in \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega } }D_{\\varphi }(m\\cdot {Q},{P})= \\inf _{{Q}\\in \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega } }\\ \\inf _{m\\ne 0}D_{\\varphi }(m\\cdot {Q},{P}) \\, < \\, \\infty $ for all sets $\\Omega $$\\Omega $ satisfying the regularity properties (REF ) and the convenient, more restrictive finiteness property $\\inf _{{Q}\\in \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega }}\\ \\inf _{k=1,\\ldots ,K}\\frac{q_{k}}{p_{k}}\\in dom(\\varphi ),\\quad \\sup _{{Q}\\in \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega }}\\ \\sup _{k=1,\\ldots ,K}\\frac{q_{k}}{p_{k}}\\in dom(\\varphi )$ which implies (REF ); notice that $\\inf _{k=1,\\ldots ,K}\\frac{q_{k}}{p_{k}} \\le 1$ , $\\sup _{k=1,\\ldots ,K}\\frac{q_{k}}{p_{k}} \\ge 1$ with equalities if and only if ${Q} = {P}$ .",
"Since $\\textrm {$$\\hspace{-6.544pt}$$} \\ne \\lbrace {P} \\rbrace $ (cf.",
"the right-hand side of (REF )), the double infimum (supremum) in (REF ) is strictly smaller (larger) than 1.",
"In general, the inner minimization $\\inf _{m\\ne 0} D_{\\varphi }(m\\cdot {Q},{P})$ in (REF ) can not be performed in explicit closed form, but e.g.",
"in the specific case of power divergences (cf.",
"(REF ), (REF )) the optimization $\\inf _{m\\ne 0} D_{\\widetilde{c} \\cdot \\varphi _{\\gamma }}(m\\cdot {Q},{P})$ produces an explicit form, which in turn leads to a simple one-to-one correspondence between $D_{\\widetilde{c} \\cdot \\varphi _{\\gamma }}(\\textrm {$$\\hspace{-6.544pt}$$},{P})$ and $\\inf _{m\\ne 0}\\ D_{\\widetilde{c} \\cdot \\varphi _{\\gamma }}(m\\cdot \\textrm {$$\\hspace{-6.544pt}$$},{P})$ (cf.",
"Lemma REF ).",
"Under (REF ) and (REF ) it clearly holds $\\inf _{m\\ne 0} D_{\\varphi }(m\\cdot \\textrm {\\Omega \\hspace{-6.544pt}\\Omega },{P})\\ \\le \\ D_{\\varphi }(\\textrm {\\Omega \\hspace{-6.544pt}\\Omega },{P})\\ \\le \\ D_{\\varphi }({Q},{P}) .$ For transparency, we first investigate the (widely useable) subsetup where $dom(\\varphi )= ]0,\\infty [$ (and thus, $int(dom(\\varphi )) = \\, ]a,b[ \\, = \\, ]0,\\infty [$ ) and $\\textrm {$$\\hspace{-6.544pt}$$} \\subset \\mathbb {S}_{> 0}^{K}$ .",
"Let us start with the lower bound $\\inf _{m\\ne 0} D_{\\varphi }(m\\cdot \\textrm {$$\\hspace{-6.544pt}$$},{P})$ .",
"It can be proved that the minimizer in $m$ is a well defined constant, which belongs to a compact set in $\\mathbb {R}_{> 0}$ .",
"To see this, let us first observe that, obviously, from (REF ) one can obtain $\\Big [\\ \\inf _{\\mathbf {Q}\\in \\mathbf {\\Omega }}\\ \\inf _{k=1,\\ldots ,K}\\frac{m \\cdot q_{k}}{p_{k}}\\in dom(\\varphi ),\\quad \\sup _{\\mathbf {Q}\\in \\mathbf {\\Omega }}\\ \\sup _{k=1,\\ldots ,K}\\frac{m \\cdot q_{k}}{p_{k}}\\in dom(\\varphi ) \\Big ]\\ \\Longleftrightarrow \\ m \\in ]0,\\infty [ .$ Moreover, for any fixed ${Q}$ in $\\Omega $$\\Omega $ there is a unique number $m=m({Q}) > 0$ which satisfies the first-order optimality condition for $m \\in \\ ]0,\\infty [$ $\\psi _{{Q}}(m):=\\frac{d}{dm} D_{\\varphi }\\left( m \\cdot {Q},{P}\\right)=\\sum _{k=1}^{K}q_{k} \\cdot \\varphi ^{\\prime }\\left( \\frac{m \\cdot q_{k}}{p_{k}}\\right) =0$ and thus $D_{\\varphi }\\left( m({Q}) \\cdot {Q},{P}\\right) =\\inf _{m \\ne 0} D_{\\varphi }\\left( m \\cdot {Q},{P}\\right) ;$ indeed, the mapping $ ]0,\\infty [ \\ \\ni m \\rightarrow D_{\\varphi }\\left( m \\cdot {Q},{P}\\right) $ is strictly convex and infinitely differentiable (which follows straightforwardly from (G5),(G6) in the below-mentioned Section together with (C7ii), (C7iii) in Appendix D), and the strictly increasing function $\\psi _{{Q}}$ is such that $\\psi _{{Q}}(m)$ is strictly negative for all $m \\in ]0,1[$ for which $\\sup _{k=1,\\ldots ,K}\\frac{m \\cdot q_{k}}{p_{k}} <1$ whereas $\\psi _{{Q}}(m)$ is strictly positive for all $m > 1$ for which $\\inf _{k=1,\\ldots ,K}\\frac{m \\cdot q_{k}}{p_{k}} >1$ (recall the note right after (REF ) and $\\varphi ^{\\prime }(1)=0$ ).",
"Hence, for any ${Q} \\in \\textrm {$$\\hspace{-6.544pt}$$}$ the unique zero $m({Q})$ of (REF ) (and hence, unique minimizer in (REF )) is in the compact interval $\\Big [\\frac{1}{\\sup _{k=1,\\ldots ,K}\\frac{q_{k}}{p_{k}}}, \\, \\frac{1}{\\inf _{k=1,\\ldots ,K}\\frac{q_{k}}{p_{k}}}\\Big ]\\ \\subseteq \\ \\Big [\\frac{1}{\\sup _{{Q}\\in \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega }}\\ \\sup _{k=1,\\ldots ,K}\\frac{q_{k}}{p_{k}}}, \\,\\frac{1}{\\inf _{{Q}\\in \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega }}\\ \\inf _{k=1,\\ldots ,K}\\frac{q_{k}}{p_{k}}}\\Big ]\\ \\subset \\ \\Big ]\\frac{1}{b}, \\frac{1}{a}\\Big [ \\ = \\ ]0,\\infty [ .$ When $\\textrm {$$\\hspace{-6.544pt}$$}$ is closed in $\\mathbb {S}^{K}$ , then by continuity of the function ${Q} \\mapsto D_{\\varphi }\\left( m\\left( {Q}\\right)\\cdot {Q} ,{P}\\right) $ there exists a ${Q}^{\\ast }$ in $\\textrm {$$\\hspace{-6.544pt}$$}$ which achieves the infimum on $\\textrm {$$\\hspace{-6.544pt}$$}$ .",
"When $\\textrm {$$\\hspace{-6.544pt}$$}$ is not closed but satisfies (7), then the infimum exists anyway, possibly on the boundary $\\partial \\textrm {$$\\hspace{-6.544pt}$$}$ .",
"Anyhow, for such ${Q}^{\\ast }$ there holds $D_{\\varphi }\\left( m({Q}^{\\ast }) \\cdot {Q}^{\\ast },{P}\\right)\\le D_{\\varphi }\\left(\\textrm {\\right.\\Omega \\hspace{-6.544pt}\\Omega } ,{P}\\le D_{\\varphi }\\left( {Q}^{\\ast },{P}\\right) ,$ where we use the continuity of ${Q}\\mapsto D_{\\varphi }\\left( {Q},{P}\\right) $ and (REF ) to obtain the last inequality above, even when ${Q}^{\\ast }\\in \\partial \\textrm {$$\\hspace{-6.544pt}$$}$ and ${Q}^{\\ast }\\notin \\textrm {$$\\hspace{-6.544pt}$$}$ .",
"That (REF ) provides sharp bounds can be seen through the case of power divergences.",
"Indeed, for the latter one basically gets (cf.",
"Appendix C) $m({Q}) = (1 + \\frac{\\gamma \\cdot (\\gamma -1)}{\\widetilde{c}} \\cdot D_{\\widetilde{c} \\cdot \\varphi _{\\gamma }}({Q}, {P}) \\, )^{1/(1-\\gamma )}$ and $D_{\\varphi _{\\gamma }}\\left( m({Q}) \\cdot {Q},{P}\\right) =\\frac{\\widetilde{c}}{\\gamma } (1- m({Q}))$ for the case $\\gamma \\in \\mathbb {R}\\backslash \\lbrace 0,1\\rbrace $ , respectively, $m({Q}) = \\exp (-\\frac{1}{\\widetilde{c}} \\cdot D_{\\widetilde{c}\\cdot \\varphi _{1}}({Q}, {P}))$ and $D_{\\widetilde{c} \\cdot \\varphi _{1}}\\left( m({Q}) \\cdot {Q},{P}\\right) =\\frac{\\widetilde{c}}{\\gamma } (1- m({Q}))$ for the case $\\gamma =1$ , respectively, $m({Q}) = 1$ and $D_{\\widetilde{c} \\cdot \\varphi _{0}}\\left( m({Q}) \\cdot {Q},{P}\\right) =D_{\\widetilde{c} \\cdot \\varphi _{0}}\\left({Q},{P}\\right)$ for the remaining case $\\gamma =0$ .",
"In all cases, $D_{\\varphi _{\\gamma }}\\left( m({Q}) \\cdot {Q},{P}\\right)$ is an increasing function of $D_{\\varphi _{\\gamma }}\\left({Q},{P}\\right)$ and therefore, ${Q}^{\\ast }\\in \\arg \\inf _{{Q}\\in \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega } } D_{\\varphi }\\left(m({Q}) \\cdot {Q},{P}\\right)$ also satisfies ${Q}^{\\ast }\\in \\arg \\inf _{{Q}\\in \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega }}D_{\\varphi }\\left( {Q},{P}\\right)$ .",
"Hence, the right-hand side and the left-hand side of (REF ) coincide.",
"Now due to (REF ), the LHS of (REF ) can be estimated since by Theorem REF for each ${P}\\in \\mathbb {S}_{>0}^{K}$ the divergence $\\inf _{m\\ne 0}D_{\\varphi }\\left( m \\cdot {Q},{P}\\right)$ is BS-minimizable on sets $\\textrm {$$\\hspace{-6.544pt}$$} \\subset \\mathbb {S}^{K}$ .",
"We shall propose in Section an algorithm to handle the estimation of the RHS of (REF ), whenever ${P}$ is known (as in Remark 13(v)) or when ${P}$ is approximated by the empirical distribution of the data set $\\left(X_{1},..,X_{n}\\right)$ .",
"Also note that (REF ) holds also for $\\textrm {$$\\hspace{-6.544pt}$$}$ substituted by $A \\cdot \\textrm {$$\\hspace{-6.544pt}$$}$ for any $A\\ne 0$ .",
"Other cases of interest include when $dom\\left( \\varphi \\right) $ is not $\\left] 0,\\infty \\right[$ .",
"We list two cases which extend the above discussion.",
"Firstly, consider $\\varphi $ with $dom\\left( \\varphi \\right) =\\left[ 0,\\infty \\right[$ .",
"Then — since $\\varphi ^{\\prime }(0)=-\\infty $ in order that (REF ) should hold (see (G10ii) in Section below) — we may extend (REF ) to cases when $\\textrm {$$\\hspace{-6.544pt}$$} \\subset \\mathbb {S}^{K}$ instead of $\\textrm {$$\\hspace{-6.544pt}$$} \\subset \\mathbb {S}_{>0}^{K}$ , hence allowing for possible null entries in $\\textrm {$$\\hspace{-6.544pt}$$}$ .",
"When $dom\\left( \\varphi \\right) =\\left] a,b\\right[ $ for some $a<0$ , then clearly the same argument leading to (REF ) holds; this case is of interest, for instance, when extending a statistical model to signed measures (see e.g.",
"Broniatowski et al.",
"[63] for the important task of testing the number of components in a parametric probability mixture model).",
"Example 28 Consider the (non-probability version of the) Jensen-Shannon divergence defined by $J(\\mathbf {Q},\\mathbf {P}):=I(\\mathbf {Q},(\\mathbf {Q}+\\mathbf {P})/2)+I(\\mathbf {P},(\\mathbf {Q}+\\mathbf {P})/2),\\qquad \\mathbf {P}, \\mathbf {Q} \\in \\mathbb {R}_{\\ge 0}^{K},$ where $I(\\mathbf {Q},\\mathbf {P})$ denotes the modified Kullback-Leibler information between $\\mathbf {Q}$ and $\\mathbf {P}$ (cf.",
"(REF ) with $\\mathbf {P}$ instead of ${P}$ ).",
"In (REF ) and (REF ) of Example REF below, we shall show that $J(\\mathbf {Q},\\mathbf {P}) = D_{\\varphi _{snKL}}(\\mathbf {Q},\\mathbf {P})$ with (basically) divergence generator $\\varphi _{snKL}(t) := \\, t \\cdot \\log t + (t+1) \\cdot \\log \\Big ( \\frac{2}{t+1} \\Big )$ for $t > 0$ .",
"It is known that $J^{2}$ is a metric.",
"We explore the sharpness of the bounds for $J(\\textrm {$$\\hspace{-6.544pt}$$} ,{P})$ as defined in (REF ).",
"For this, we consider a given probability distribution ${P}$ on $\\mathcal {Y}$ with strictly positive entries; the set $\\textrm {$$\\hspace{-6.544pt}$$}$ consists of all probability distributions ${Q}$ on $\\mathcal {Y}$ whose total variation distance $V({Q},{P}):=\\sum _{k=1}^{K}\\left|q_{k}-p_{k}\\right|$ notice that $V({Q},{P})$ always takes values in the interval $[0,2[$ to ${P}$ lies between $v$ and $v+h$ for $v>0$ and small $h$ and which also satisfies $\\sup \\left( \\sup _{k=1,\\ldots ,K}\\frac{p_{k}}{q_{k}},\\sup _{k=1,\\ldots ,K}\\frac{q_{k}}{p_{k}}\\right)\\le L$ for some strictly positive finite $L$ .",
"This set $\\textrm {$$\\hspace{-6.544pt}$$}$ defines a class of distributions ${Q}$ away from ${P}$ still keeping some regularity w.r.t.",
"${P}$ .",
"Also, $\\textrm {$$\\hspace{-6.544pt}$$}$ satisfies (7).",
"We will prove that the bounds in (120) are sharp in this case.",
"Notice that $J(m \\cdot {Q},{P}) = \\infty $ for $m<0$ and hence $\\inf _{{Q}\\in \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega }}\\inf _{m \\ne 0}J(m \\cdot {Q},{P}) =\\inf _{{Q}\\in \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega }}\\inf _{m>0}J(m \\cdot {Q},{P})$ .",
"We first provide a lower bound for the latter.",
"It holds for all $m>0$ and ${Q}$ in $\\textrm {$$\\hspace{-6.544pt}$$}$ $& & J(m{Q},{P}) =(m+1) \\cdot \\log \\left( 2 \\right)- (m+1) \\cdot \\log \\left( m+1\\right)+m\\log m+I^{\\alpha }({P},{Q})+m \\cdot I^{1-\\alpha }({Q},{P})\\nonumber $ where $\\alpha :=1/\\left( m+1\\right) $ and $I^{\\alpha }({P},{Q})$ is the $\\alpha -$ skewed Kullback-Leibler divergence between ${P}$ and ${Q}$ defined through $I^{\\alpha }({P},{Q}):=I({P},\\alpha {P}+(1-\\alpha ){Q}).$ By Inequality (27) in Yamano [410] $I^{\\alpha }({P},{Q})\\ge -\\log \\left( 1-\\frac{\\alpha ^{2}}{4} V({Q},{P})^{2}\\right).$ Since $(m+1) \\cdot \\log \\left(2\\right) -(m+1) \\cdot \\log (m+1)+ m \\cdot \\log m$ is non-negative for all $m>0$ and takes its minimal value 0 for $m=1$ , we obtain $\\inf _{m>0} J(m {Q},{P})\\ge \\inf _{m>0}K(m)$ where $K(m):=-\\log \\left( 1-\\frac{1}{4\\left( m+1\\right) ^{2}} \\cdot V({Q},{P})^{2}\\right) -m \\cdot \\log \\left( 1-\\frac{m^{2}}{4\\left( m+1\\right)^{2}}\\cdot V({Q},{P})^{2}\\right) .$ Since $-\\log (1-x)\\ge x$ for all $x<1$ and both $\\frac{1}{4\\left(m+1\\right) ^{2}} \\cdot V({Q},{P})^{2}$ and $\\frac{m^{2}}{4\\left( m+1\\right)^{2}}\\cdot V({Q},{P})^{2}$ are less than 1, it follows that $K(m)\\ge \\frac{V({Q},{P})^{2}}{4} \\cdot \\frac{m^{3} +1}{\\left(m+1\\right)^{2}}$ where the right-hand side attains its minimal value on $]0,\\infty [$ at $m^{+}= \\sqrt{3}-1 \\approx 0.73$ .",
"Hence, we obtain $\\inf _{m>0}J(m\\cdot Q,P)\\ge \\frac{V({Q},{P})^{2}}{4}\\cdot (2\\sqrt{3}-3) > 0.116 \\, v^2$ Now by (19) in Yamano [410], for any ${Q}$ $J({Q},{P})\\le \\frac{1}{4}\\underline{J}({Q},{P})$ where $\\underline{J}({Q},{P}):=I({Q},{P})+I({P},{Q})$ is the Jensen divergence (also called symmetrized Kullback-Leibler divergence) between ${Q}$ and ${P}$ .",
"Since (see Dragomir [116]) $I({P},{Q})\\le \\sum _{k=1}^{K}\\sqrt{\\frac{p_{k}}{q_{k}}}\\cdot \\left|q_{k}-q_{k}\\right|,$ it follows that $J({Q}^{\\ast },{P})\\le \\frac{1}{2}\\sqrt{L} \\cdot V({Q}^{\\ast },{P})$ which provides $0.116 \\, v^2\\ \\le \\ \\inf _{m >0}J(m \\cdot {Q}^{\\ast },{P})=J(m({Q}^{\\ast })\\cdot {Q}^{\\ast },{P}) \\ \\le \\ J(\\textrm {$ $\\hspace{-6.544pt}$$},{P})\\ \\le \\ J({Q}^{\\ast },{P}) \\ \\le \\ \\frac{1}{2}\\sqrt{L} \\cdot (v+h).$$For small $ v$ thedifference between the RHS and the LHS in the above display is $ cstv + o(v) + 12 L h$which proves that the bounds are sharp locally, with non-trivial lower bound.Other upper bounds can be adapted to sets $$\\Omega $$\\Omega $$ defined throughtighter conditions on $ Q$\\Omega $$\\Omega $ k=1,...,Kpkqk$and $ Q$\\Omega $$\\Omega $ k=1,...,Kqkpk$ (of e.g.",
"Dragomir \\cite {Dra:00}).$ 0.3cm"
],
[
"On the difference between minimization problems of\ndeterministic nature and risk minimization",
"In the context of minimization of the functional $\\Phi _{\\mathbf {P}}(\\mathbf {Q})$ over $\\mathbf {\\Omega } \\subset \\mathbb {R}^{K}$ for known vector $\\mathbf {P}$ , due to Theorem REF our bare simulation approach allows for the approximate solution for any divergence $D_{\\varphi }$ satisfying the basic representation (REF ).",
"Indeed, any proxy of $\\mathbb {\\Pi }\\big [\\xi _{n}^{\\mathbf {\\widetilde{W}}}\\in \\Omega /M_{\\mathbf {P}}\\big ]$ yields a corresponding proxy for $\\Phi _{\\mathbf {P}}\\left[\\mathbf {\\mathbf {\\Omega }}\\right]$ .",
"This paves the way to the solution of numerous optimization problems, where the divergence $D_{\\varphi }$ is specifically suited to the problem at hand.",
"In the statistical context, when the probability distribution (in its vector-form) ${P}$ is unknown up to some indirect information provided by sampling or by any mean providing a sequence $(X_{i})_{i\\in \\mathbb {N}}$ satisfying condition (REF ) (resp.",
"(REF )), Theorem REF adds a complementary step of complexity; indeed, the estimation of $\\Phi _{\\mathbf {P}}(\\textrm {$$\\hspace{-6.544pt}$$})$ over $\\textrm {$$\\hspace{-6.544pt}$$}\\subset \\mathbb {S}^{K}$ results as its subproduct through the optimization upon $m$ which can be performed explicitly only in a number of specific divergences $D_{\\varphi }$ , e.g.",
"the power divergences $D_{\\varphi _{\\gamma }}$ , and which carries over also to their monotone transformations such as e.g.",
"the Renyi divergences.",
"It is of relevance to mention that — as already indicated above — these divergences cover a very broad range of statistical criteria, indeed most of them, from the (maximum-likelihood estimation connected) likelihood divergence ($\\gamma =0$ ) to the Kullback-Leibler one ($\\gamma =1$ ), the two standard Chi-square distances ($\\gamma =2$ , $\\gamma =-1$ ), the Hellinger distance ($\\gamma =1/2$ ), etc.",
"; in contrast with deterministic minimization problems, the choice of a statistical criterion (or risk function) is not imposed by the modelling of the problem at hand, but is dictated by the need for sharp measures of fit.",
"Other divergences are more difficult to handle and our general results in Section REF should still prove some usefulness, since estimation of upper and lower bounds for risk is of common use.",
"As a “preparatory” remark, recall first that each probability distribution (probability measure) $\\mathbb {P}$ on $\\mathcal {Y}=\\left\\lbrace d_{1},\\ldots ,d_{K}\\right\\rbrace $ has been uniquely identified with the vector ${P} := (p_{1}, \\ldots , p_{K}) \\in \\mathbb {S}^{K}$ of the corresponding probability masses (frequencies) $p_{k} = \\mathbb {P}[\\lbrace d_{k} \\rbrace ]$ via $\\mathbb {P}[A] = \\sum _{k=1}^{K} p_{k} \\cdot {1}_A(d_{k}) $ for each $A \\subset \\mathcal {Y}$ ; from this, we have measured the distance/divergence between two probability distributions $\\mathbb {P}, \\mathbb {Q}$ through the distance/divergence between their frequency vectors ${P}, {Q}$ : $D_{\\varphi }(\\mathbb {Q},\\mathbb {P}) : = D_{\\varphi }({Q},{P})\\qquad \\textrm {(cf.",
"(\\ref {measure divergence}))} .\\nonumber $ However, it has been noted in Kißlinger & Stummer [190] in a context of even more general divergences $D(\\mathbf {Q}, \\mathbf {P})$ between vectors $\\mathbf {P}, \\mathbf {Q}$ that — alternatively — the latter two may consist of components $p_{k} = \\mathbb {P}[\\lbrace E_{k} \\rbrace ]$ , $q_{k} = \\mathbb {Q}[\\lbrace E_{k} \\rbrace ]$ which are probabilities of only some selected (e.g.",
"increasing) events $(E_{k})_{k \\in \\lbrace 1,\\ldots ,M\\rbrace }$ of application-based concrete interest (within not necessarily discrete probability models).",
"Of course, we can apply our BS method to such a vector context.",
"As other alternatives, in the following we deal with divergences between non-probabilistic uncertainty quantifications."
],
[
"Minimization problems with fuzzy sets",
"Our BS framework also covers the — imprecise/inexact/vague information describing — fuzzy sets (cf.",
"Zadeh [417]) and optimization problems on divergences between those.",
"Indeed, let $\\mathcal {Y}=\\left\\lbrace d_{1},\\ldots ,d_{K}\\right\\rbrace $ be a finite set (called the universe (of discourse)), $A \\subset \\mathcal {Y}$ and $M^{A}: \\mathcal {Y} \\mapsto [0,1]$ be a corresponding membership function, where $M^{A}(d_{k})$ represents the degree/grade of membership of the element $d_{k}$ to the set $A$ ; accordingly, the object $A^{\\ast } := \\lbrace \\langle x,M^{A}(x) \\rangle | x \\in \\mathcal {Y}\\rbrace $ is called a fuzzy set in $\\mathcal {Y}$ (or fuzzy subset of $ \\mathcal {Y}$ ).",
"Moreover, if $A \\subset \\mathcal {Y}$ and $B \\subset \\mathcal {Y}$ are unequal, then the corresponding membership functions $M^{A}$ and $M^{B}$ should be unequal.",
"Furthermore, we model the vector of membership degrees to $A$ by $\\mathbf {P}^{A} := \\left(p_{k}^{A} \\right)_{k=1,\\ldots ,K}:= \\left(M^{A}(d_{k}) \\right)_{k=1,\\ldots ,K}$ which satisfies the key constraint $0 \\le p_{k}^{A} \\le 1$ for all $k \\in \\lbrace 1,\\ldots ,K\\rbrace $ and, consequently, the aggregated key constraint $0 \\le \\sum _{k=1}^{K} p_{k}^{A} \\le K$ (as a side remark, $\\sum _{k=1}^{K} M^{A}(d_{k})$ is called power of the fuzzy set $A^{\\ast }$).",
"For divergence generators $\\varphi $ in $\\Upsilon (]a,b[)$ (resp.",
"$\\widetilde{\\Upsilon }(]a,b[)$ ) with — say — $0 \\le a<1<b$ and for two sets $A, B \\subset \\mathcal {Y}$ we can apply (REF ) to the corresponding membership functions and define the $\\varphi -$ divergence $D_{\\varphi }(B^{\\ast }, A^{\\ast } )$ between the fuzzy sets $B^{\\ast }$ and $A^{\\ast }$ (on the same universe $\\mathcal {Y}$ ) as $D_{\\varphi }(B^{\\ast }, A^{\\ast } ) :=D_{\\varphi }( \\mathbf {P}^{B}, \\mathbf {P}^{A} ) =\\sum _{k=1}^{K} p_{k}^{A} \\cdot \\varphi \\left( \\frac{p_{k}^{B}}{p_{k}^{A}}\\right)= \\sum _{k=1}^{K} M^{A}(d_{k}) \\cdot \\varphi \\left( \\frac{M^{B}(d_{k})}{M^{A}(d_{k})}\\right)\\, \\ge 0$ (depending on $\\varphi $ , zero degree values may have to be excluded for finiteness).",
"For instance, we can take $\\varphi (t): = \\varphi _{1}(t) := t \\cdot \\log t + 1 - t \\in [0, \\infty [$ for $t \\in [0,\\infty [$ (cf.",
"(REF )) to end up with a generalized Kullback-Leibler divergence (generalized relative entropy) between $B^{\\ast }$ and $A^{\\ast }$ ; this contrasts the choice $\\varphi (t) := \\breve{\\varphi }(t) :=t \\cdot \\log t \\in [-\\frac{1}{e},\\infty [$ of Bhandari & Pal [47] for which $D_{\\breve{\\varphi }}(B^{\\ast }, A^{\\ast })$ (which they call fuzzy expected information for discrimination in favor of $B$ against $A$) may become negative (cf.",
"Stummer & Vajda [349] in a more general context).",
"In terms of (REF ), as a special case of the above-mentioned BS concepts, we can tackle optimization problems of the type $\\inf _{B^{\\ast } \\in \\Omega ^{\\ast }}D_{\\varphi }( B^{\\ast }, A^{\\ast } ) :=\\inf _{\\mathbf {P}^{B} \\in \\mathbf {\\Omega }}D_{\\varphi }( \\mathbf {P}^{B}, \\mathbf {P}^{A} )$ where $\\Omega ^{\\ast }$ is a collection of fuzzy sets (on the same universe $\\mathcal {Y}$ ) whose membership-degree vectors form the set $\\mathbf {\\Omega }$ satisfying (REF ) and (REF ).",
"Because of the inequality-type key constraint $0 \\le M^{B}(d_{k}) \\le 1\\quad \\textrm {for all k \\in \\lbrace 1, \\ldots , K\\rbrace }\\nonumber $ which is contained in $\\mathbf {\\Omega }$ and which implies $0 \\le \\sum _{k=1}^{K} p_{k}^{B} \\le K$ , Theorem REF and its consequences and derived examples will apply correspondingly — unless there is a more restrictive constraint which violates (REF ) such as e.g.",
"$\\sum _{k=1}^{K} p_{k}^{B} = C$ with $C\\le K$ for which Theorem REF (and its consequences and derived examples) can be employed.",
"The above-mentioned considerations can be extended to the recent concept of $\\nu -$ rung orthopair fuzzy sets (cf.",
"Yager [406]) and divergences between those.",
"Indeed, for $A \\subset \\mathcal {Y}$ , besides a membership function $M^{A}: \\mathcal {Y} \\mapsto [0,1]$ one additionally models a non-membership function $N^{A}: \\mathcal {Y} \\mapsto [0,1]$ , where $N^{A}(d_{k})$ represents the degree/grade of non-membership of the element $d_{k}$ to the set $A$ .",
"Moreover, if $A \\subset \\mathcal {Y}$ and $B \\subset \\mathcal {Y}$ are unequal, then the corresponding non-membership functions $N^{A}$ and $N^{B}$ should be unequal.",
"For fixed $\\nu \\in [1,\\infty [$ , the key constraint $0 \\le \\left(M^{A}(d_{k})\\right)^{\\nu } + \\left(N^{A}(d_{k})\\right)^{\\nu } \\le 1\\quad \\textrm {for all k \\in \\lbrace 1, \\ldots , K\\rbrace }$ is required to be satisfied, too.",
"Accordingly, the object $A^{\\ast \\ast } := \\lbrace \\langle x,M^{A}(x), N^{A}(x) \\rangle | x \\in \\mathcal {Y}\\rbrace $ is called a $\\nu -$ rung orthopair fuzzy set in $\\mathcal {Y}$ (or $\\ldots $ subset of $\\mathcal {Y}$ ).",
"The object $A^{\\ast \\ast }$ is called intuitionistic fuzzy set in $\\mathcal {Y}$ (cf.",
"Atanassov [20]) in case of $\\nu =1$ , and Pythagorean fuzzy set in $ \\mathcal {Y}$ (cf.",
"Yager [405], [407]) in case of $\\nu =2$ .",
"For the choice $\\nu =1$ together with $N^{A}(x) := 1-M^{A}(x)$ , the object $A^{\\ast \\ast }$ can be regarded as an extended representation of the fuzzy set $A^{\\ast }$ in $\\mathcal {Y}$ .",
"As is well known, there is a vast amount of recent literature on applications of fuzzy sets; for the sake of brevity, let us exemplarily mention the survey of Yanase & Triantaphyllou [411] on some recent uses in computer-aided medical diagnosis.",
"For any $\\nu -$ rung orthopair fuzzy set $A^{\\ast \\ast }$ in $\\mathcal {Y}$ , we model the corresponding vector of concatenated membership and non-membership degrees to $A$ by $\\mathbf {P}^{A} := \\left(p_{k}^{A} \\right)_{k=1,\\ldots ,2K}:= \\left(M^{A}(d_{1}), \\ldots M^{A}(d_{K}), N^{A}(d_{1}), \\ldots N^{A}(d_{K}) \\right)$ which due to (REF ) satisfies the aggregated key constraint $0 \\le \\sum _{k=1}^{2K} \\left(p_{k}^{A}\\right)^{\\nu } \\le K ;\\nonumber $ in other words, $\\mathbf {P}^{A}$ lies (within the $2K-$ dimensional Euclidean space) in the intersection of the first/positive orthant with the $\\nu -$ norm ball centered at the origin and with radius $K^{1/\\nu }$ .",
"Analogously to (REF ), we can define the $\\varphi -$ divergence $D_{\\varphi }(B^{\\ast \\ast }, A^{\\ast \\ast } )$ between the $\\nu -$ rung orthopair fuzzy sets $B^{\\ast \\ast }$ and $A^{\\ast \\ast }$ (on the same universe $\\mathcal {Y}$ ) as $D_{\\varphi }(B^{\\ast \\ast }, A^{\\ast \\ast } ) :=D_{\\varphi }( \\mathbf {P}^{B}, \\mathbf {P}^{A} ) =\\sum _{k=1}^{2K} p_{k}^{A} \\cdot \\varphi \\left( \\frac{p_{k}^{B}}{p_{k}^{A}}\\right)= \\sum _{k=1}^{K} \\left\\lbrace M^{A}(d_{k}) \\cdot \\varphi \\left( \\frac{M^{B}(d_{k})}{M^{A}(d_{k})}\\right) +N^{A}(d_{k}) \\cdot \\varphi \\left( \\frac{N^{B}(d_{k})}{N^{A}(d_{k})}\\right)\\right\\rbrace \\, \\ge 0$ respectively as its variant $& & \\hspace{-25.6073pt}D_{\\varphi }^{var}(B^{\\ast \\ast }, A^{\\ast \\ast } ) :=D_{\\varphi }( \\left(\\mathbf {P}^{B}\\right)^{\\nu }, \\left(\\mathbf {P}^{A}\\right)^{\\nu } )\\nonumber \\\\& & \\hspace{-25.6073pt}= \\sum _{k=1}^{2K} \\left(p_{k}^{A}\\right)^{\\nu } \\cdot \\varphi \\left( \\frac{\\left(p_{k}^{B}\\right)^{\\nu }}{\\left(p_{k}^{A}\\right)^{\\nu }}\\right)= \\sum _{k=1}^{K} \\left\\lbrace \\left(M^{A}(d_{k})\\right)^{\\nu } \\cdot \\varphi \\left( \\frac{\\left(M^{B}(d_{k})\\right)^{\\nu }}{\\left(M^{A}(d_{k})\\right)^{\\nu }}\\right)+\\left(N^{A}(d_{k})\\right)^{\\nu } \\cdot \\varphi \\left( \\frac{\\left(N^{B}(d_{k})\\right)^{\\nu }}{\\left(N^{A}(d_{k})\\right)^{\\nu }}\\right)\\right\\rbrace \\, \\ge 0 .$ For the special choice $\\nu =1$ , $N^{A}(x) := 1-M^{A}(x)$ and $\\varphi (t): = \\varphi _{1}(t) := t \\cdot \\log t + 1 - t \\in [0, \\infty [$ for $t \\in [0,\\infty [$ (cf.",
"(REF )), one can straightforwardly show that the outcoming divergence $D_{\\varphi _{1}}(B^{\\ast \\ast }, A^{\\ast \\ast })$ coincides with $D_{\\breve{\\varphi }}(B^{\\ast \\ast }, A^{\\ast \\ast })$ where $\\breve{\\varphi }(t) := t \\cdot \\log t$ ; the latter divergence was used e.g.",
"in Bhandari & Pal [47] under the name average fuzzy information for discrimination in favor of $B$ against $A$).",
"Moreover, the special choice $\\nu =1$ and $\\varphi (t):=\\varphi _{snKL,1}(t)$ (cf.",
"(REF )) leads to the Jensen-Shannon divergence between $B^{\\ast \\ast }$ and $A^{\\ast \\ast }$ given by $D_{\\varphi _{snKL,1}}(B^{\\ast \\ast }, A^{\\ast \\ast } ) :=D_{\\varphi _{snKL,1}}( \\mathbf {P}^{B}, \\mathbf {P}^{A} )$ ; from (REF ) and (REF ) one can see that this coincides with the symmetric information measure between $B^{\\ast \\ast }$ and $A^{\\ast \\ast }$ of Vlachos & Sergiadis [385].",
"In terms of the divergences (REF ) and (REF ), we can tackle — as a special case of the above-mentioned BS concepts — optimization problems of the type $& & \\inf _{B^{\\ast \\ast } \\in \\Omega ^{\\ast \\ast }}D_{\\varphi }( B^{\\ast \\ast }, A^{\\ast \\ast } ) :=\\inf _{\\mathbf {P}^{B} \\in \\mathbf {\\Omega }}D_{\\varphi }( \\mathbf {P}^{B}, \\mathbf {P}^{A} ) \\qquad \\textrm {respectively}\\nonumber \\\\& & \\inf _{B^{\\ast \\ast } \\in \\Omega ^{\\ast \\ast }}D_{\\varphi }^{var}( B^{\\ast \\ast }, A^{\\ast \\ast } ) :=\\inf _{\\mathbf {P}^{B} \\in \\mathbf {\\Omega }}D_{\\varphi }( \\left(\\mathbf {P}^{B}\\right)^{\\nu }, \\left(\\mathbf {P}^{A}\\right)^{\\nu } ) ,\\nonumber $ where $\\Omega ^{\\ast \\ast }$ is a collection of $\\nu -$ rung orthopair fuzzy sets whose concatenated-membership-nonmembership-degree vectors form the set $\\mathbf {\\Omega }$ satisfying (REF ) and (REF ) as well as (REF ) for $B$ in place of $A$ .",
"Because of the latter, Theorem REF and its consequences and derived examples will apply correspondingly — unless there is a more restrictive constraint which violates (REF ) such as e.g.",
"$\\sum _{k=1}^{2K} p_{k}^{B} = C$ with $C\\le K$ for which Theorem REF (and its consequences and derived examples) can be employed; such a situation appears e.g.",
"in the above-mentioned case $\\nu =1$ together with $N^{A}(x) := 1-M^{A}(x)$ which leads to $C=K$ .",
"For $\\nu -$ rung orthopair fuzzy sets $A^{\\ast \\ast }$ in $\\mathcal {Y}$ , we can also further “flexibilize” our divergences by additionally incorporating the hesitancy degree of the element $d_{k}$ to $A$ which is defined as $H^{A}(d_{k}):= \\left(1- \\left(M^{A}(d_{k})\\right)^{\\nu } -\\left(N^{A}(d_{k})\\right)^{\\nu } \\right)^{1/\\nu } \\in [0,1]$ (cf.",
"Yager [406]) and which implies the key constraint $\\left(H^{A}(d_{k})\\right)^{\\nu } + \\left(M^{A}(d_{k})\\right)^{\\nu } + \\left(N^{A}(d_{k})\\right)^{\\nu } =1\\quad \\textrm {for all k \\in \\lbrace 1, \\ldots , K\\rbrace .",
"}$ Accordingly, the object $A^{\\ast \\ast \\ast } :=\\lbrace \\langle x,M^{A}(x), N^{A}(x), H^{A}(x) \\rangle | x \\in \\mathcal {Y}\\rbrace $ can be regarded as an extended representation of the $\\nu -$ rung orthopair fuzzy set $A^{\\ast \\ast }$ in $\\mathcal {Y}$ .",
"For $A^{\\ast \\ast \\ast }$ , we model the corresponding vector of concatenated membership, non-membership and hesitancy degrees to $A$ by $\\mathbf {P}^{A} := \\left(p_{k}^{A} \\right)_{k=1,\\ldots ,3K}:= \\left(M^{A}(d_{1}), \\ldots M^{A}(d_{K}), N^{A}(d_{1}), \\ldots N^{A}(d_{K}),H^{A}(d_{1}), \\ldots H^{A}(d_{K}) \\right)$ which due to (REF ) satisfies the aggregated key constraint $\\sum _{k=1}^{3K} \\left(p_{k}^{A}\\right)^{\\nu } = K ;$ in other words, $\\mathbf {P}^{A}$ lies (within the $3K-$ dimensional Euclidean space) in the intersection of the first/positive orthant with the $\\nu -$ norm sphere centered at the origin and with radius $K^{1/\\nu }$ .",
"Analogously to (REF ) and (REF ), we can define the $\\varphi -$ divergence $D_{\\varphi }(B^{\\ast \\ast \\ast }, A^{\\ast \\ast \\ast } )$ between the extended-representation-type $\\nu -$ rung orthopair fuzzy sets $B^{\\ast \\ast \\ast }$ and $A^{\\ast \\ast \\ast }$ (on the same universe $\\mathcal {Y}$ ) as $& & \\hspace{-42.67912pt}D_{\\varphi }(B^{\\ast \\ast \\ast }, A^{\\ast \\ast \\ast } ) :=D_{\\varphi }( \\mathbf {P}^{B}, \\mathbf {P}^{A} )\\nonumber \\\\& & \\hspace{-42.67912pt}= \\sum _{k=1}^{3K} p_{k}^{A} \\cdot \\varphi \\left( \\frac{p_{k}^{B}}{p_{k}^{A}}\\right)= \\sum _{k=1}^{K} \\left\\lbrace M^{A}(d_{k}) \\cdot \\varphi \\left( \\frac{M^{B}(d_{k})}{M^{A}(d_{k})}\\right) +N^{A}(d_{k}) \\cdot \\varphi \\left( \\frac{N^{B}(d_{k})}{N^{A}(d_{k})}\\right) +H^{A}(d_{k}) \\cdot \\varphi \\left( \\frac{H^{B}(d_{k})}{H^{A}(d_{k})}\\right)\\right\\rbrace \\, \\ge 0\\nonumber $ respectively as its variant $& & \\hspace{-42.67912pt}D_{\\varphi }^{var}(B^{\\ast \\ast \\ast }, A^{\\ast \\ast \\ast } ) :=D_{\\varphi }( \\left(\\mathbf {P}^{B}\\right)^{\\nu }, \\left(\\mathbf {P}^{A}\\right)^{\\nu } )= \\sum _{k=1}^{3K} \\left(p_{k}^{A}\\right)^{\\nu } \\cdot \\varphi \\left( \\frac{\\left(p_{k}^{B}\\right)^{\\nu }}{\\left(p_{k}^{A}\\right)^{\\nu }}\\right)\\nonumber \\\\& & \\hspace{-42.67912pt}= \\sum _{k=1}^{K} \\left\\lbrace \\left(M^{A}(d_{k})\\right)^{\\nu } \\cdot \\varphi \\left( \\frac{\\left(M^{B}(d_{k})\\right)^{\\nu }}{\\left(M^{A}(d_{k})\\right)^{\\nu }}\\right) +\\left(N^{A}(d_{k})\\right)^{\\nu } \\cdot \\varphi \\left( \\frac{\\left(N^{B}(d_{k})\\right)^{\\nu }}{\\left(N^{A}(d_{k})\\right)^{\\nu }}\\right) +\\left(H^{A}(d_{k})\\right)^{\\nu } \\cdot \\varphi \\left( \\frac{\\left(H^{B}(d_{k})\\right)^{\\nu }}{\\left(H^{A}(d_{k})\\right)^{\\nu }}\\right)\\right\\rbrace \\, \\ge 0 .$ For instance, by taking the special choice $\\nu =2$ and $\\varphi (t):=\\varphi _{snKL,1}(t)$ (cf.",
"(REF )) in (REF ), we arrive at the Jensen-Shannon divergence between $B^{\\ast \\ast \\ast }$ and $A^{\\ast \\ast \\ast }$ of the form $D_{\\varphi _{snKL,1}}^{var}(B^{\\ast \\ast \\ast }, A^{\\ast \\ast \\ast } ) :=D_{\\varphi _{snKL,1}}^{var}( \\mathbf {P}^{B}, \\mathbf {P}^{A} )$ which — by the virtue of (REF ) and (REF ) — coincides with the (squared) Pythagorean-fuzzy-set Jensen-Shannon divergence measure between $B^{\\ast \\ast \\ast }$ and $A^{\\ast \\ast \\ast }$ of Xiao & Ding [399].",
"To continue with the general context, as a particular application of the above-mentioned BS concepts, we can tackle general optimization problems of the type $& & \\inf _{B^{\\ast \\ast \\ast } \\in \\Omega ^{\\ast \\ast \\ast }}D_{\\varphi }( B^{\\ast \\ast \\ast }, A^{\\ast \\ast \\ast } ) :=\\inf _{\\mathbf {P}^{B} \\in \\mathbf {\\Omega }}D_{\\varphi }( \\mathbf {P}^{B}, \\mathbf {P}^{A} ) \\qquad \\textrm {respectively}\\nonumber \\\\& & \\inf _{B^{\\ast \\ast \\ast } \\in \\Omega ^{\\ast \\ast \\ast }}D_{\\varphi }^{var}( B^{\\ast \\ast \\ast }, A^{\\ast \\ast \\ast } ) :=\\inf _{\\mathbf {P}^{B} \\in \\mathbf {\\Omega }}D_{\\varphi }( \\left(\\mathbf {P}^{B}\\right)^{\\nu }, \\left(\\mathbf {P}^{A}\\right)^{\\nu } )\\nonumber $ where $\\Omega ^{\\ast \\ast \\ast }$ is a collection of extended-representation-type $\\nu -$ rung orthopair fuzzy sets whose concatenated-membership-nonmembership-hesitancy-degree vectors form the set $\\mathbf {\\Omega }$ satisfying (REF ) and (REF ) as well as (REF ) for $B$ in place of $A$ .",
"Because of the latter and the implied aggregated key constraint (REF ) for $B$ in place of $A$ , Theorem REF (and its consequences and derived examples) can be employed.",
"Of course, we can also correspondingly adapt the transformations of $\\varphi -$ divergences and entropy-type special cases given in the sections above and below, to (classical respectively $\\nu -$ rung othopair) fuzzy sets, and apply our BS method for this.",
"For the sake of brevity, we only give a short example, namely the $\\gamma -$ order Renyi divergence between $\\nu -$ rung othopair fuzzy sets which we define by (cf.",
"(REF )) $& & \\hspace{-42.67912pt}R_{\\gamma }^{var}(B^{\\ast \\ast \\ast }, A^{\\ast \\ast \\ast } ) :=\\frac{1}{\\gamma \\cdot (\\gamma -1)} \\cdot \\left[\\log \\left( \\sum \\displaylimits _{k=1}^{3K}\\left(\\left(p_{k}^{B}\\right)^{\\nu }\\right)^{\\gamma } \\cdot \\left(\\left(p_{k}^{A}\\right)^{\\nu }\\right)^{1-\\gamma } \\right) - \\log (K) \\right]\\nonumber \\\\& & \\hspace{-42.67912pt}= \\frac{1}{\\gamma \\cdot (\\gamma -1)} \\cdot \\Bigg [\\log \\Bigg ( \\sum \\displaylimits _{k=1}^{K}\\bigg \\lbrace \\left(\\left(M^{B}(d_{k})\\right)^{\\nu }\\right)^{\\gamma } \\cdot \\left(\\left(M^{A}(d_{k})\\right)^{\\nu }\\right)^{1-\\gamma }+\\left(\\left(N^{B}(d_{k})\\right)^{\\nu }\\right)^{\\gamma } \\cdot \\left(\\left(N^{A}(d_{k})\\right)^{\\nu }\\right)^{1-\\gamma }\\nonumber \\\\& & \\hspace{-42.67912pt}+\\left(\\left(H^{B}(d_{k})\\right)^{\\nu }\\right)^{\\gamma } \\cdot \\left(\\left(H^{A}(d_{k})\\right)^{\\nu }\\right)^{1-\\gamma }\\bigg \\rbrace \\Bigg ) - \\log (K) \\Bigg ]\\, \\ge 0 ;\\qquad \\gamma \\in \\, ]-\\infty ,0[ \\ \\cup \\ ]0,1[ \\ \\cup \\ [ \\, 1,\\infty [,$ depending on $\\gamma $ , zero degree values may have to be excluded for finiteness.",
"As a side remark, let us mention that our divergence (REF ) contrasts to the recent (first) divergence of Verma [382] who basically uses a different scaling, the product $\\prod _{k=1}^{K}$ instead of the sum $\\sum _{k=1}^{K}$ , as well as $\\frac{\\left(p_{k}^{A}\\right)^{\\nu } + \\left(p_{k}^{B}\\right)^{\\nu }}{2}$ instead of $\\left(p_{k}^{A}\\right)^{\\nu }$ .",
"By equivalently rewriting (REF ), we can use () with $A=1$ together with the Propositions REF , REF , REF and REF to tackle for $\\gamma \\in \\, ]-\\infty ,0[ \\ \\cup \\ ]0,1[ \\ \\cup \\ [ \\, 2,\\infty [$ the minimization problem $& & \\inf _{B^{\\ast \\ast \\ast } \\in \\Omega ^{\\ast \\ast \\ast }}R_{\\gamma }^{var,nor}(B^{\\ast \\ast \\ast }, A^{\\ast \\ast \\ast } ) =\\inf _{{P}^{B} \\in \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega }}R_{\\gamma }({P}^{B},{P}^{A})\\nonumber $ with artificially generated probability vectors (cf.",
"(REF )) ${P}^{C} := \\left(\\frac{\\left(M^{C}(d_{1})\\right)^{\\nu }}{K}, \\ldots ,\\frac{\\left(M^{C}(d_{K})\\right)^{\\nu }}{K},\\frac{\\left(N^{C}(d_{1})\\right)^{\\nu }}{K}, \\ldots ,\\frac{\\left(N^{C}(d_{K})\\right)^{\\nu }}{K},\\frac{\\left(H^{C}(d_{1})\\right)^{\\nu }}{K}, \\ldots ,\\frac{\\left(H^{C}(d_{K})\\right)^{\\nu }}{K}\\right),\\quad C=A,B;$ here, $\\Omega ^{\\ast \\ast \\ast }$ is a collection of extended-representation-type $\\nu -$ rung orthopair fuzzy sets $B^{\\ast \\ast \\ast }$ whose corresponding normalized concatenated-membership-nonmembership-hesitancy-degree vectors ${P}^{B}$ form the set $\\Omega $$\\Omega $ satisfying (REF ) (in the relative topology).",
"The above-mentioned considerations can be carried over to (classical, intuitionistic, Pythagorean, $\\nu -$ rung orthopair) $L-$ fuzzy sets, where the range of the membership functions, non-membership functions and hesitancy functions is an appropriately chosen lattice $L$ (rather than $L=[0,1]$ ); for the sake of brevity, this is omitted here."
],
[
"Minimization problems with basic belief assignments",
"Our BS framework also covers — imprecise/inexact/vague information describing — basic belief assignments from Dempster-Shafer evidence theory (cf.",
"[109], [327]) and optimization problems on divergences between those.",
"Indeed, let $\\mathcal {Y}=\\left\\lbrace d_{1},\\ldots ,d_{K}\\right\\rbrace $ be a finite set (called the frame of discernment) of mutally exclusive and collectively exhaustive events $d_{k}$ .",
"The corresponding power set of $\\mathcal {Y}$ is denoted by $2^{\\mathcal {Y}}$ and has $2^{K}$ elements; we enumerate this by $2^{\\mathcal {Y}} := \\lbrace A_{1}, \\ldots , A_{2^{K}}\\rbrace $ where for convenience we set $A_{1} := \\emptyset $ .",
"A mapping $M: 2^{\\mathcal {Y}} \\mapsto [0,1]$ is called a basic belief assignment (BBA) sometimes alternatively called basic probability assignment (BPA) if it satisfies the two conditions $M(\\emptyset ) = 0 \\qquad \\textrm {and}\\qquad \\sum _{A \\in 2^{\\mathcal {Y}}}M(A) \\, = \\, 1.$ Here, the belief mass $M(A)$ reflects e.g.",
"the trust degree of evidence to proposition $A \\in 2^{\\mathcal {Y}}$ .",
"From this, one can build the belief function $Bel: 2^{\\mathcal {Y}} \\mapsto [0,1]$ by $Bel(A) := \\sum _{B: B \\subseteq A} M(B)$ and the plausability function $Pl: 2^{\\mathcal {Y}} \\mapsto [0,1]$ by $Pl(A) := \\sum _{B: B \\cap A \\ne \\emptyset } M(B)$ .",
"Moreover, we model the $2^{K}-$ dimensional vector of ($M-$ based) BBA values (vector of ($M-$ based) belief masses) by $\\mathbf {P}^{M} := \\left(p_{k}^{M} \\right)_{k=1,\\ldots ,2^{K}}:= \\left(M(A_{k}) \\right)_{k=1,\\ldots ,2^{K}}$ which satisfies the key constraint $0 \\le p_{k}^{M} \\le 1$ for all $k \\in \\lbrace 1,\\ldots ,2^{K}\\rbrace $ and, by virtue of (REF ), the aggregated key constraint $\\sum _{k=1}^{2^{K}} p_{k}^{M} =1$ .",
"Hence, $\\mathbf {P}^{M}$ lies formally in the $2^{K}-$ dimensional simplex $\\mathbb {S}^{2^{K}}$ (but generally not in the corresponding probability-vector-describing $\\mathbb {S}^{K}$ ).",
"For divergence generators $\\varphi $ in $\\Upsilon (]a,b[)$ with — say — $0 \\le a<1<b$ and for two BBAs $M_{1}, M_{2}$ on the same frame of discernment $\\mathcal {Y}$ , we can apply (REF ) to the corresponding vectors of BBA-values and define the $\\varphi -$ divergence $D_{\\varphi }(M_{2}, M_{1})$ between the BBAs $M_{2}$ and $M_{1}$ (in short, Belief$-\\varphi -$ divergence) as $D_{\\varphi }(M_{2}, M_{1}) :=D_{\\varphi }( \\mathbf {P}^{M_{2}}, \\mathbf {P}^{M_{1}} ) =\\sum _{k=1}^{2^{K}} p_{k}^{M_{1}} \\cdot \\varphi \\left( \\frac{p_{k}^{M_{2}}}{p_{k}^{M_{1}}}\\right)= \\sum _{k=1}^{2^{K}} M_{1}(A_{k}) \\cdot \\varphi \\left( \\frac{ M_{2}(A_{k})}{ M_{1}(A_{k})}\\right)\\, \\ge 0$ (depending on $\\varphi $ , zero belief masses may have to be excluded for finiteness).",
"For instance, we can take in (REF ) the special case $\\varphi (t):=\\varphi _{snKL,1}(t)$ (cf.",
"(REF )) to end up with the recent Belief-Jensen-Shannon divergence of Xiao [397], [398] who applies this to multi-sensor data fusion.",
"As another special case we can take $\\varphi (t):=\\varphi _{1/2}(t)$ (cf.",
"(REF )) to end up with the $4-$ times square of the recent Hellinger distance of BBAs of Li et al.",
"[224], who use this for characterizing the degree of conflict between BBAs.",
"To continue with the general context, as a particular application of the above-mentioned BS concepts, we can tackle general optimization problems of the type $& & \\inf _{M_{2} \\in \\Omega ^{BBA}}D_{\\varphi }( M_{2}, M_{1} ) :=\\inf _{\\mathbf {P}^{M_{2}} \\in \\mathbf {\\Omega }}D_{\\varphi }( \\mathbf {P}^{M_{2}}, \\mathbf {P}^{M_{2}} ) \\qquad \\textrm {respectively}\\nonumber $ where $\\Omega ^{BBA}$ is a collection of BBAs whose vectors of BBA-values form the set $\\mathbf {\\Omega } \\in \\mathbb {S}^{2^{K}}$ satisfying (REF ) and (REF ) as well as (REF ).",
"for $B$ in place of $A$ .",
"Hence, Theorem REF (and its consequences and derived examples) can be employed.",
"We can also apply our BS method to “crossover cases ” $D_{\\varphi }( \\mathbf {P}^{M}, \\mathbf {P})$ (respectively with interchanged components) where $\\mathbf {P}^{M}$ is a vector of $M-$ based BBA values and $\\mathbf {P}$ is a vector whose sum of components may not necessarily be 1.",
"For instance, for the special choice $\\varphi (t): = \\varphi _{1}(t) := t \\cdot \\log t + 1 - t \\in [0, \\infty [$ (cf.",
"REF ), $\\mathbf {P}^{M} := \\left(p_{k}^{M} \\right)_{k=1,\\ldots ,2^{K}}:= \\left(M(A_{k}) \\right)_{k=1,\\ldots ,2^{K}}$ , $\\mathbf {P} := \\left(p_{k} \\right)_{k=1,\\ldots ,2^{K}}$ with $p_{k} := 2^{|A_{k}|}-1$ employing the cardinality $|A_{k}|$ of $A_{k}$ , and the usual convention $0\\cdot \\log (\\frac{0}{0}) := 0$ , we end up with (cf.",
"(REF )) $D_{\\varphi _{1}}( \\mathbf {P}^{M}, \\mathbf {P}) =\\sum \\limits _{k=2}^{2^{K}} M(A_{k}) \\cdot \\log \\Big (\\frac{M(A_{k})}{2^{|A_{k}|}-1} \\Big )- 1 + \\sum \\limits _{k=2}^{2^{K}} (2^{|A_{k}|}-1)=: - \\mathcal {E}^{DE}(M) - 1 + \\sum \\limits _{k=2}^{2^{K}} (2^{|A_{k}|}-1)$ where $\\mathcal {E}^{DE}(M) := - \\sum _{k=2}^{2^{K}} M(A_{k}) \\cdot \\log \\Big (\\frac{M(A_{k})}{2^{|A_{k}|}-1} \\Big )\\ge 0$ is nothing but (a multiple of) Deng’s entropy of the BBA (BPA) $M$ (cf.",
"[110], see also e.g.",
"Kang & Deng [181]).",
"Our BS method can also be applied to divergences between rescalings of BBAs.",
"For instance, let $\\breve{M}(A) := \\frac{M(A)}{2^{|A|}-1}$ ($A \\in 2^{\\mathcal {Y}}$ ) with the convention that $\\frac{0}{0}:=0$ , and denote the corresponding vector $\\mathbf {P}^{\\breve{M}} := \\left(p_{k}^{\\breve{M}} \\right)_{k=1,\\ldots ,2^{K}}:= \\left(\\breve{M}(A_{k})\\right)_{k=1,\\ldots ,2^{K}}$ .",
"Accordingly, we define the $\\varphi -$ divergence $D_{\\varphi }(\\breve{M}_{2}, \\breve{M}_{1})$ between the rescaled BBAs $\\breve{M}_{2}$ and $\\breve{M}_{1}$ (in short, rescaled Belief$-\\varphi -$ divergence) as $D_{\\varphi }(\\breve{M}_{2}, \\breve{M}_{1}) :=D_{\\varphi }( \\mathbf {P}^{\\breve{M}_{2}}, \\mathbf {P}^{\\breve{M}_{1}} ) =\\sum _{k=1}^{2^{K}} p_{k}^{\\breve{M}_{1}} \\cdot \\varphi \\left( \\frac{p_{k}^{\\breve{M}_{2}}}{p_{k}^{\\breve{M}_{1}}}\\right)= \\sum _{k=2}^{2^{K}} \\breve{M}_{1}(A_{k}) \\cdot \\varphi \\left( \\frac{ \\breve{M}_{2}(A_{k})}{ \\breve{M}_{1}(A_{k})}\\right)=\\sum _{k=2}^{2^{K}} \\frac{M_{1}(A_{k})}{2^{|A_{k}|}-1} \\cdot \\varphi \\left( \\frac{ M_{2}(A_{k})}{ M_{1}(A_{k})}\\right)\\, \\ge 0$ where for $A_{1} := \\emptyset $ we have used the convention that $0 \\cdot \\varphi (\\frac{0}{0}) := 0$ (depending on $\\varphi $ , other zero rescaled belief masses may have to be excluded for finiteness); notice that Remark REF applies with $c_{k}:= \\frac{1}{2^{|A_{k}|}-1} >0$ .",
"As an example, for the special choice $\\varphi (t): = \\varphi _{1}(t) := t \\cdot \\log t + 1 - t \\in [0, \\infty [$ (cf.",
"REF ), we derive from (REF ) and (REF ) the divergence $& & 0 \\le D_{\\varphi _{1}}(\\breve{M}_{2}, \\breve{M}_{1}) =\\sum \\limits _{k=2}^{2^{K}} \\frac{M_{2}(A_{k})}{2^{|A_{k}|}-1} \\cdot \\log \\Big (\\frac{M_{2}(A_{k})}{M_{1}(A_{k})} \\Big )- \\sum \\limits _{k=2}^{2^{K}} \\frac{M_{2}(A_{k})}{2^{|A_{k}|}-1} + \\sum \\limits _{k=2}^{2^{K}} \\frac{M_{1}(A_{k})}{2^{|A_{k}|}-1}\\nonumber \\\\& & =: D^{SD}(M_{2}, M_{1}) -\\sum \\limits _{k=2}^{2^{K}} \\frac{M_{2}(A_{k})}{2^{|A_{k}|}-1} + \\sum \\limits _{k=2}^{2^{K}} \\frac{M_{1}(A_{k})}{2^{|A_{k}|}-1}\\nonumber $ where $D^{SD}(M_{2}, M_{1})$ has been recently developed by Song & Deng [339]; notice that $D^{SD}(M_{2}, M_{1})$ may be negative (cf.",
"Stummer & Vajda [349]) and then it is not a divergence anymore.",
"However, for applications to data fusion Song & Deng apply the symmetrization $\\frac{1}{2} \\cdot \\left(D^{SD}(M_{2}, M_{1}) + D^{SD}(M_{1}, M_{2}) \\right)$ which is equal to $\\frac{1}{2} \\cdot \\left( D_{\\varphi _{1}}(\\breve{M}_{2}, \\breve{M}_{1})+ D_{\\varphi _{1}}(\\breve{M}_{1}, \\breve{M}_{2}) \\right)$ and thus nonnegative.",
"Of course, we can also correspondingly adapt the transformations of $\\varphi -$ divergences (e.g.",
"Renyi divergences) and entropy-type special cases given in the sections above and below, to BBAs as well as crossover cases and rescalings, and apply our BS method for this.",
"Recall first that in Theorem REF , one crucial component is the sequence $(W_{n})_{n\\in \\mathbb {N}}$ of weights being i.i.d.",
"copies of a random variable $W$ whose probability distribution is $\\mathbb {}$ (i.e.",
"$\\mathbb {\\Pi }[W \\in \\cdot \\, ] = \\mathbb {}[ \\, \\cdot \\,]$ ), where the latter has to be connected with the divergence generator $\\varphi \\in \\Upsilon (]a,b[)$ through the representation $\\varphi (t)=\\sup _{z\\in \\mathbb {R}}\\left( z\\cdot t-\\log \\int _{\\mathbb {R}}e^{zy}d\\mathbb {} (y)\\right), \\qquad t\\in \\mathbb {R},\\ \\ \\hspace{28.45274pt} (cf.",
"\\ (\\ref {Phi Legendre of mgf(W)}))$ under the additional requirement that the function $z\\mapsto MGF_{\\mathbb {} }(z):=\\int _{\\mathbb {R}}e^{zy}d\\mathbb {} (y)$ is finite on some open interval containing zero (“light-tailedness”); for Theorem REF , we need the corresponding variant (REF ) for $M_{\\mathbf {P}} \\cdot \\varphi \\in \\Upsilon (]a,b[)$ (rather than $\\varphi $ ).",
"Hence, finding such “BS-associated pairs $(\\varphi ,\\mathbb {})$ ” is an important issue.",
"Subsequently, let us discuss the following direction: starting from a concrete optimization problem (REF ) — respectively (REF ) — with pregiven $\\varphi \\in \\widetilde{\\Upsilon } (]a,b[)$ (cf.",
"Definition REF ), as a first step one would like to verify whether indeed $\\varphi \\in \\Upsilon (]a,b[)$ (i.e.",
"it additionally satisfies (REF )) — respectively $M_{\\mathbf {P}} \\cdot \\varphi \\in \\Upsilon (]a,b[)$ ; as a second step, one would like to find the corresponding $\\mathbb {}$ explicitly.",
"As far as the above-mentioned first step is concerned, let us first present some fundamental properties of all $\\varphi \\in \\Upsilon (]a,b[)$ : Proposition 29 Let $\\varphi \\in \\Upsilon (]a,b[)$ .",
"Then the following assertions hold: $\\varphi : \\, ]-\\infty ,\\infty [ \\rightarrow [0,\\infty ]$ is lower semicontinuous and convex; $\\varphi (1)=0$ ; $int(dom(\\varphi )) = ]a,b[$ for some $-\\infty \\le a < 1 < b \\le \\infty $ ; $\\varphi $ is continuously differentiable on $]a,b[$ (i.e.",
"$\\varphi \\in C^{1}(]a,b[)$ ; $\\varphi $ is strictly convex only in a non-empty neighborhood $]t_{-}^{sc},t_{+}^{sc}[ \\subseteq ]a,b[$ of one ($t_{-}^{sc} < 1 < t_{+}^{sc}$ ); $\\varphi $ is infinitly differentiable on $]t_{-}^{sc},t_{+}^{sc}[$ (i.e.",
"$\\varphi \\in C^{\\infty }(]t_{-}^{sc},t_{+}^{sc}[$ ), and hence, $\\varphi ^{\\prime }(1) = 0$ , $\\varphi ^{\\prime \\prime }(t) > 0$ for all $t \\in ]t_{-}^{sc},t_{+}^{sc}[$ ; notice that the left-hand second derivative and the right-hand second derivative of $\\varphi $ may not coincide at $t_{-}^{sc}$ respectively at $t_{+}^{sc}$ (i.e.",
"possible non-second-differentiability at these two points); if $a > -\\infty $ , then $a=t_{-}^{sc}$ ; if $a = -\\infty $ , then either $t_{-}^{sc} = - \\infty $ or $\\varphi (t) = \\varphi (t_{-}^{sc}) + \\varphi ^{\\prime }(t_{-}^{sc}) \\cdot (t- t_{-}^{sc})$ for all $t \\in ]-\\infty , t_{-}^{sc}[$ (affine-linearity); notice that $\\varphi ^{\\prime }(t_{-}^{sc}) < 0$ ; if $b < \\infty $ , then $b=t_{+}^{sc}$ ; if $b = \\infty $ , then either $t_{+}^{sc} = \\infty $ or $\\varphi (t) = \\varphi (t_{+}^{sc}) + \\varphi ^{\\prime }(t_{+}^{sc}) \\cdot (t- t_{+}^{sc})$ for all $t \\in ]t_{+}^{sc},\\infty [$ (affine-linearity); notice that $\\varphi ^{\\prime }(t_{+}^{sc}) > 0$ ; the Fenchel-Legendre transform (also called convex conjugate) of $\\varphi $ $\\varphi ^{*} (z)=\\sup _{t\\in \\mathbb {R}}\\left( z\\cdot t- \\varphi (t) \\right)= \\sup _{t\\in ]a,b[}\\left( z\\cdot t- \\varphi (t) \\right), \\qquad z\\in \\mathbb {R},\\ \\ $ has the following properties: $int(dom(\\varphi ^{*})) = ]\\lambda _{-},\\lambda _{+}[$ , where $dom(\\varphi ^{*}) := \\lbrace z \\in \\mathbb {R} : -\\infty < \\varphi ^{*}(z) < \\infty \\rbrace $ , $\\lambda _{-} := \\inf _{t \\in ]a,b[} \\varphi ^{\\prime }(t) =\\lim _{t \\downarrow a} \\varphi ^{\\prime }(t) =: \\varphi ^{\\prime }(a) < 0$ and $\\lambda _{+} := \\sup _{t \\in ]a,b[} \\varphi ^{\\prime }(t) =\\lim _{t \\uparrow b} \\varphi ^{\\prime }(t) =: \\varphi ^{\\prime }(b) >0$ ; if $a >-\\infty $ , then $\\lambda _{-} = - \\infty $ ; the function $z \\mapsto e^{-a\\cdot z + \\varphi ^{*}(z)} =: M(z)$ is absolutely monotone on $]-\\infty ,0[$ , i.e.",
"all derivatives exist and satisfy $\\frac{\\partial ^{k}}{\\partial z^k} M(z) \\ge 0$ ($k\\in \\mathbb {N}_{0}$ , $z \\in ]-\\infty ,0[$ ); $\\lim _{z \\rightarrow 0-} M(z) =1$ ; if $b < \\infty $ , then $\\lambda _{+} = \\infty $ ; the function $z \\mapsto e^{b\\cdot z + \\varphi ^{*}(- z)} =: M(z)$ is absolutely monotone on $]-\\infty ,0[$ ; $\\lim _{z \\rightarrow 0-} M(z) =1$ ; if $a = -\\infty $ and $b = -\\infty $ , then the function $z \\mapsto e^{\\varphi ^{*}(z)} =: M(z)$ is exponentially convex on $]\\lambda _{-}, \\lambda _{+}[$ , i.e.",
"$M(\\cdot )$ is continuous and satisfies $\\sum _{i,j=1}^{n} c_{i} \\cdot c_{j} \\cdot M\\Big (\\frac{z_{i} + z_{j}}{2}\\Big ) \\ge 0 \\qquad \\textrm {for all n \\in \\mathbb {N},c_{i}, c_{j} \\in \\mathbb {R} and z_{i}, z_{j} \\in ]\\lambda _{-}, \\lambda _{+}[;}$ $\\lim _{z \\rightarrow 0-} M(z) =1$ ; as a side remark, notice the well-known fact that exponential-convexity is stronger than the usual log-convexity.",
"the endpoints of $int(dom(\\varphi )) =]a,b[$ have the following important “functioning” for the underlying probability distribution $\\mathbb {}$ (cf.",
"(REF )) respectively of an associated random variable $W$ with $\\mathbb {}[ \\cdot \\, ] := \\mathbb {\\Pi }[W \\in \\cdot \\, ]$ : $a = \\inf supp(\\mathbb {}) = \\inf supp(W)$ , $b = \\sup supp(\\mathbb {}) = \\sup supp(W)$ , where $supp(\\mathbb {})$ respectively $supp(W)$ denotes the support of $\\mathbb {}$ respectively $W$ ; consequently, $]a,b[ = int(conv(supp(\\mathbb {}))) = int(conv(supp(W)))$ where $conv(A)$ denotes the convex hull of a set $A$ ; if $a > -\\infty $ , then $\\varphi (a) = - \\log \\mathbb {}[\\lbrace a\\rbrace ]= - \\log \\mathbb {\\Pi }[W = a \\, ]$ ; consequently, there holds: $a = \\min supp(\\mathbb {}) = \\min supp(W)$ if and only if $\\mathbb {}[\\lbrace a\\rbrace ] = \\mathbb {\\Pi }[W = a \\, ] > 0$ if and only if $\\varphi (a) < \\infty $ if and only if $a \\in dom(\\varphi )$ ; if $b < \\infty $ , then $\\varphi (b) = - \\log \\mathbb {}[\\lbrace b\\rbrace ]= - \\log \\mathbb {\\Pi }[W = b \\, ]$ ; consequently, there holds: $b = \\max supp(\\mathbb {}) = \\max supp(W)$ if and only if $\\mathbb {}[\\lbrace b\\rbrace ] = \\mathbb {\\Pi }[W = b \\, ] > 0$ if and only if $\\varphi (b) < \\infty $ if and only if $b \\in dom(\\varphi )$ .",
"the first two derivatives of $\\varphi $ at the point 1 have the following important “functioning” for the underlying probability distribution $\\mathbb {}$ (cf.",
"(REF )) respectively of an associated random variable $W$ : $1 = \\varphi ^{\\prime -1}(0) = \\int _{\\mathbb {R}} y \\, d\\mathbb {} (y)= E_{\\mathbb {\\Pi }}[W]$ where $\\varphi ^{\\prime -1}(\\cdot )$ denotes the inverse of the first derivative $\\varphi ^{\\prime }(\\cdot )$ of $\\varphi (\\cdot )$ , $\\frac{1}{\\varphi ^{\\prime \\prime }(1)} =\\int _{\\mathbb {R}} \\Big (y - \\int _{\\mathbb {R}} \\widetilde{y} \\, d\\mathbb {} (\\widetilde{y})\\Big )^{2}\\, d\\mathbb {} (y)= E_{\\mathbb {\\Pi }}[W^{2}] - (E_{\\mathbb {\\Pi }}[W])^{2} = Var_{\\mathbb {\\Pi }}[W]$ ; in particular, scaling $\\widetilde{c} \\cdot \\varphi $ ($\\widetilde{c} >0$ ) does not change the mean 1 but the variance of $W$ .",
"The corresponding proof of Proposition REF will be given in Appendix D, except for the second items of (G9ii) and (G9iii) as well as the first item of (G9iv).",
"Those will be treated in the second next paragraph below, because the corresponding line of argumentation builds an insightful start for subsequently performed procedures to further track down the weight distribution $$ .",
"The properties (G1) to (G9iv) constitute necessary conditions for a pregiven function $\\varphi $ to belong to $\\Upsilon (]a,b[)$ ); accordingly, these should be verified first, in concrete situations where one aims to apply the BS approach.",
"An important role is played by the boundary points $a$ and $b$ of $int(dom(\\varphi ))$ through (G10i) to (G10iii), because their finiteness opens the gate to apply — via some straightforward transformations — a rich class of real-valued characterization theorems for probability distributions whose support lies in the positive real line $[0,\\infty [$ .",
"In contrast, there exist much less real-valued characterization theorems for probability distributions whose support is the whole real line $]-\\infty ,\\infty [$ ; typically, the involved conditions are also more difficult to verify.",
"Indeed, if $\\varphi \\in \\Upsilon (]a,b[)$ then one can deduce straightforwardly from the representation (REF ) that $e^{\\varphi ^{*}(z)} =\\int _{-\\infty }^{\\infty } e^{z\\cdot y} \\, \\mathrm {d}\\mathbb {} (y) =E_{\\mathbb {\\Pi }}[e^{z \\cdot W}] , \\quad z \\in ]\\lambda _{-}, \\lambda _{+}[,$ where $W$ is a random variable whose distribution is $\\mathbb {\\Pi }[W \\in \\cdot \\, ] = \\mathbb {}[ \\, \\cdot \\,]$ ; under the additional knowledge $a > -\\infty $ (and consequently $\\lambda _{-} = -\\infty $ ) employed together with (G10i) and thus $\\mathbb {\\Pi }[W \\ge a \\, ] =\\mathbb {}[ \\, [a,\\infty [ \\,] = 1$ , one arrives at $e^{\\varphi ^{*}(z) - a \\cdot z} =\\int _{a}^{\\infty } e^{z \\cdot (y-a)} \\, \\mathrm {d}\\mathbb {} (y) =\\int _{0}^{\\infty } e^{z\\cdot \\widetilde{y}} \\, \\mathrm {d}\\widetilde{\\widetilde{\\mathbb {}}} (\\widetilde{y})= E_{\\mathbb {\\Pi }}[e^{z \\cdot (W-a)}] , \\quad z \\in ]-\\infty , \\lambda _{+}[,$ where the probability distribution $\\widetilde{\\widetilde{\\mathbb {}}}[ \\, \\cdot \\, ] := \\mathbb {}[\\, \\cdot + a \\, ]$ is the $a-$ shifted companion of $\\mathbb {}$ ; recall that $\\lambda _{+} >0$ .",
"Put in other words, $\\mathbb {\\Pi }[\\widetilde{W} \\in \\cdot \\, ] = \\widetilde{\\widetilde{\\mathbb {}}}[ \\, \\cdot \\,]$ is the probability distribution of the (a.s.) nonnegative random variable $\\widetilde{W} := W-a$ .",
"Naturally, in this context, the interesting case is $-\\infty < a \\le 0$ .",
"Similarly, if $\\varphi \\in \\Upsilon (]a,b[)$ and $b < \\infty $ (and hence $\\lambda _{+} = \\infty $ ), one can derive from (G10i) and its consequence $\\mathbb {\\Pi }[W \\le b \\, ] =\\mathbb {}[ \\, ]-\\infty ,b] \\,]= 1$ that $e^{\\varphi ^{*}(-z) + b \\cdot z} =\\int _{-\\infty }^{b} e^{z \\cdot (b-y)} \\, \\mathrm {d}\\mathbb {} (y) =\\int _{0}^{\\infty } e^{z\\cdot \\widetilde{y}} \\, \\mathrm {d}\\widetilde{\\widetilde{\\mathbb {}}} (\\widetilde{y})= E_{\\mathbb {\\Pi }}[e^{z \\cdot (b-W)}] , \\quad z \\in ]- \\infty , - \\lambda _{-}[,$ where $- \\lambda _{-} >0$ and the probability distribution $\\widetilde{\\widetilde{\\mathbb {}}}[ \\, \\cdot \\, ] := \\mathbb {}[\\, b - \\, \\cdot \\, ]$ is the mirrored$-b-$ shifted companion of $\\mathbb {}$ .",
"This means that $\\mathbb {\\Pi }[\\widetilde{W} \\in \\cdot \\, ] = \\widetilde{\\widetilde{\\mathbb {}}}[ \\, \\cdot \\,]$ is the probability distribution of the (a.s.) nonnegative random variable $\\widetilde{W} := b-W$ .",
"Naturally, the interesting case is $0 < b \\le \\infty $ .",
"As already indicated above, the considerations (REF ) to (REF ) open the gate to the adaption of well-known real-valued (rather than complex-valued) characterizations from probability theory.",
"To begin with, the following assertions are very prominent: Theorem 30 (a) Let $M: ]-\\infty ,0] \\mapsto ]0,\\infty [$ be continuous on $]-\\infty ,0]$ with $M(0) =1$ .",
"Then one has $\\textrm {M is absolutely monotone on ]-\\infty ,0[}\\ \\Longleftrightarrow \\ \\exists \\ \\textrm {unique prob.",
"distr.",
"\\widetilde{\\widetilde{\\mathbb {}}} on [0,\\infty [s.t.}",
"\\ \\ M(z) = \\int _{0}^{\\infty } e^{z \\cdot y} \\mathrm {d}\\widetilde{\\widetilde{\\mathbb {}}}(y)\\ \\textrm {for all z \\in ]-\\infty ,0[.",
"}$ (b) Let $I$ be an open interval which contains 0, and $M: I \\mapsto [0,\\infty [$ be continuous with $M(0) =1$ .",
"Then one gets $\\textrm {M is exponentially convex}\\ \\Longleftrightarrow \\ \\exists \\ \\textrm {unique prob.",
"distr.",
"\\widetilde{\\widetilde{\\mathbb {}}} on ]-\\infty ,\\infty [such that} \\ \\ M(z) = \\int _{-\\infty }^{\\infty } e^{z \\cdot y} \\mathrm {d}\\widetilde{\\widetilde{\\mathbb {}}}(y)\\ \\ \\textrm {for all z \\in I.",
"}$ Assertion (a) of Theorem REF is known as (probability-version of) Bernstein’s theorem [42] (see e.g.",
"also Schilling et al.",
"[322]), whereas assertion (b) is known as (probability-version of) Widder’s theorem [391] for the relevant conversion between the involved Riemann-Stieltjes integral with nondecreasing (but not necessarily right-continuous) integrator into a measure integral, one can apply the general theory in e.g.",
"Chapter 6 of Chow & Teicher [88].",
"(see e.g.",
"also Widder [392], Akhiezer [9], Shucker [333], Jaksetic & Pecaric [163], Kotelina & Pevny [196]).",
"From Theorem REF (b) and (REF ), the first item in (G9iv) follows immediately by using the choice $I= ]\\lambda _{-}, \\lambda _{+}[$ .",
"Moreover, Theorem REF (a) together with (REF ) (respectively (REF )) implies the second item of (G9ii) (respectively of (G9iii)).",
"In fact, with the help of Theorem REF and some further considerations e.g.",
"on light-tailedness, one even gets assertions on the sufficiency of (G9ii), (G9iii) and (G9iv) for a “candidate generator” $\\varphi $ to belong to the BS-suitable class $\\Upsilon (]a,b[)$ .",
"More precisely, we obtain Proposition 31 Suppose that $\\varphi : ]-\\infty ,\\infty [ \\mapsto [0,\\infty ]$ satisfies (G1) to (G8), and recall the notations in (G9i).",
"Then, $\\varphi \\in \\Upsilon (]a,b[)$ if one of the following three conditions holds: (a) $a >-\\infty $ , $\\lambda _{-} = - \\infty $ , and the function $z \\mapsto e^{-a\\cdot z + \\varphi ^{*}(z)}$ is absolutely monotone on $]-\\infty ,0[$ , (b) $b < \\infty $ , $\\lambda _{+} = \\infty $ , and the function $z \\mapsto e^{b\\cdot z + \\varphi ^{*}(- z)}$ is absolutely monotone on $]-\\infty ,0[$ , (c) $a = -\\infty $ , $b = -\\infty $ , and the function $z \\mapsto e^{\\varphi ^{*}(z)}$ is exponentially convex on $]\\lambda _{-}, \\lambda _{+}[$ .",
"If one of the three conditions (a) to (c) holds, thenbasically by Theorem REF with $M(\\cdot )$ defined in G9(ii),(iii) or (iv); see Appendix D. the associated probability distribution $\\mathbb {}$ (cf.",
"(REF )) has expectation $\\int _{\\mathbb {R}} y d\\mathbb {}(y) =1$ and finite moments of all orders, i.e.",
"$\\int _{\\mathbb {R}} y^{j} d\\mathbb {}(y) < \\infty $ for all $j \\in \\mathbb {N}_{0}$ ; in terms of $\\mathbb {}[ \\cdot \\, ] := \\mathbb {\\Pi }[W \\in \\cdot \\, ]$ this means that $E_{\\mathbb {\\Pi }}[W] =1$ and $E_{\\mathbb {\\Pi }}[W^{j}] < \\infty $ .",
"The proof of Proposition REF will be given in Appendix D. As far as applicability is concerned, it is well known that, in general, verifying absolute monotonicity is typically more comfortable than verifying exponential convexity.",
"Fortunately, one can often use the former, since for many known divergence generators there holds $a > -\\infty $ (often $a=0$ ) or $b < \\infty $ or both, which by virtue of (G10i) is directly linked with the (endpoints of the) support of the potentially existing probability distribution $\\mathbb {}$ .",
"For a pregiven divergence generator $\\varphi $ , once its membership in $\\Upsilon (]a,b[)$ (and in particular, the representability (REF )) is verified, one would like to concretely find the underlying probability distribution $\\mathbb {}$ .",
"This may be quite challenging, but can be made more comfortable by systematically narrowing down the family of distributions where $\\mathbb {}$ belongs to.",
"In fact, we have already performed a first down-narrowing, in terms of identifying the endpoints of the support of $\\mathbb {}$ to be the endpoints of the effective domain of $\\varphi $ (cf.",
"(G10i)).",
"A further down-narrowing can be achieved from (REF ) to (REF ) in combination with real-valued characterization theorems which are more specific than Theorem REF .",
"This will be shown exemplarily for a few important sub-setups, in the following.",
"For the identification of light-tailed semi/half-lattice distributions, we obtain the following two sets of sufficient conditions, which even allow for the desired explicit determination of $\\mathbb {}$ : Proposition 32 Suppose that $\\varphi : ]-\\infty ,\\infty [ \\mapsto [0,\\infty ]$ satisfies (G1) to (G8), with some $a > -\\infty $ .",
"Furthermore, assume that there exists some constant $\\breve{c} >0$ as well as some function $H: [0,\\infty [ \\mapsto [0,\\infty [$ which is continuous on $[0,1]$ with $H(1)=1$ and absolutely monotone on $]0,1[$ , such that $e^{\\varphi ^{*}(\\frac{z}{\\breve{c}}) - a\\cdot \\frac{z}{\\breve{c}} } = H(e^{z}), \\qquad z \\in ]-\\infty , \\breve{c} \\cdot \\lambda _{+}[.$ Then one has $\\varphi \\in \\Upsilon (]a,b[)$ and $\\mathbb {} = \\sum _{n=0}^{\\infty } p_{n} \\cdot \\delta _{a + \\breve{c} \\cdot n}\\qquad \\textrm {with \\ } p_{n} := \\frac{1}{n !}",
"\\cdot \\frac{\\mathrm {d}^{n}H}{\\mathrm {d}t}(0),$ i.e.",
"$\\mathbb {\\Pi }[W = a + \\breve{c} \\cdot n \\, ] = p_{n}$ ($n \\in \\mathbb {N}_{0}$ ).",
"Proposition 33 Suppose that $\\varphi : ]-\\infty ,\\infty [ \\mapsto [0,\\infty ]$ satisfies (G1) to (G8), with some $b < \\infty $ .",
"Furthermore, assume that there exists some constant $\\breve{c} >0$ as well as some function $H: [0,\\infty [ \\mapsto [0,\\infty [$ which is continuous on $[0,1]$ with $H(1)=1$ and absolutely monotone on $]0,1[$ , such that $e^{\\varphi ^{*}(- \\frac{z}{\\breve{c}}) + b \\cdot \\frac{z}{\\breve{c}} } = H(e^{z}), \\qquad z \\in ]-\\infty , - \\breve{c} \\cdot \\lambda _{-}[.$ Then one has $\\varphi \\in \\Upsilon (]a,b[)$ and $\\mathbb {} = \\sum _{n=0}^{\\infty } p_{n} \\cdot \\delta _{b - \\breve{c} \\cdot n}\\qquad \\textrm {with \\ } p_{n} := \\frac{1}{n !}",
"\\cdot \\frac{\\mathrm {d}^{n}H}{\\mathrm {d}t}(0),$ i.e.",
"$\\mathbb {\\Pi }[W = b - \\breve{c} \\cdot n \\, ] = p_{n}$ ($n \\in \\mathbb {N}_{0}$ ).",
"The Propositions REF respectively REF follow from (REF ) respectively (REF ), some straightforward transformations, and a well-known characterization of probability generating functions $H$ (see e.g.",
"in Theorem 1.2.10 of Stroock [343]).",
"As an incentive for the following investigations, let us recall the discussion in the surroundings of Condition REF pertaining to the minimization problem (REF ), where we have addressed possible connections between the two representabilities (REF ) (needed e.g.",
"for Theorem REF ) and (REF ) (needed e.g.",
"for Theorem REF ); this strongly relates to the question, for which constants $\\widetilde{c} >0$ the validity $\\varphi \\in \\Upsilon (]a,b[)$ triggers the validity of $\\widetilde{c} \\cdot \\varphi \\in \\Upsilon (]a,b[)$ .",
"To begin with, it is straightforward to see that $\\varphi \\in \\Upsilon (]a,b[)$ always implies $\\widetilde{c} \\cdot \\varphi \\in \\Upsilon (]a,b[)$ for all integers $\\widetilde{c} \\in \\mathbb {N}$ ; indeed, if $\\varphi $ satisfies (REF ) for some $\\mathbb {} = \\mathbb {\\Pi }[ W \\in \\cdot \\, ]$ , then for each integer $\\widetilde{c} \\in \\mathbb {N}$ one gets that $\\widetilde{c} \\cdot \\varphi $ satisfies (REF ) for $\\widetilde{\\mathbb {}} = \\mathbb {\\Pi }[ \\sum _{j=1}^{\\widetilde{c}} \\frac{W_{j}}{\\widetilde{c}} \\in \\cdot \\, ]$ ; in the latter, the $W_{j}$ ’s are i.i.d.",
"copies from $W$ .",
"Clearly, $MGF_{\\widetilde{\\mathbb {}}}$ is then finite on some open interval containing zero (differing from the one for $MGF_{\\mathbb {}}$ only by a scaling with $1/\\widetilde{c}$ ).",
"For the following family of distributions, one can even trigger $\\widetilde{c} \\cdot \\varphi \\in \\Upsilon (]a,b[)$ for all $\\widetilde{c} >0$ : for the sake of a corresponding precise formulation, recall first the common knowledge that, generally speaking, a probability distribution $\\mathbb {}$ on $\\mathbb {R}$ with light tails — in the sense that its moment generating function $z\\mapsto MGF_{\\mathbb {} }(z):=\\int _{\\mathbb {R}}e^{z \\cdot y}d\\mathbb {} (y)$ is finite on some open interval $]\\lambda _{-},\\lambda _{+}[$ containing zero — is (said to be) infinitely divisible if there holds $\\textrm {for each n \\in \\mathbb {N} there existsa probability distribution \\mathbb {}_{n} on \\mathbb {R} such that}\\int _{-\\infty }^{\\infty } e^{z \\cdot y} d\\mathbb {} (y) =\\Big (\\int _{-\\infty }^{\\infty } e^{z \\cdot y} d\\mathbb {}_{n} (y) \\Big )^{n}, \\quad z \\in ]\\lambda _{-},\\lambda _{+}[ ;$ in fact, (REF ) means that the (light-tailed) moment generating function $MGF_{\\mathbb {} }$ is infinitely divisible in the sense that each $n-$ th root $(MGF_{\\mathbb {} })^{1/n}$ must be the moment generating function of some (light-tailed) probability distribution (denoted here by $\\mathbb {}_{n}$ ).",
"In particular, (REF ) implies that $\\mathbb {}_{n}$ is unique, and that $\\mathbb {}$ must necessarily have (one-sided or two-sided) unbounded support $supp(\\mathbb {})$ .",
"The latter may differ from $supp(\\mathbb {}_{n})$ .",
"In our BS context (REF ), (REF ) equivalently means that the associated random variable $W$ is infinitely divisible (with light-tailed distribution), in the sense that $\\textrm {for each n \\in \\mathbb {N} there exists a sequence of i.i.d.",
"random variables Y_{n,1}, \\cdots , Y_{n,n} such that }W \\stackrel{\\text{\\tiny d}}{=} Y_{n,1} + \\cdots + Y_{n,n} ,$ where $\\stackrel{\\text{\\tiny d}}{=}$ means “have equal probability distributions” and $\\mathbb {\\Pi }[W \\in \\cdot \\, ] = \\mathbb {}[ \\, \\cdot \\,]$ , $\\mathbb {\\Pi }[Y_{n,1} \\in \\cdot \\, ] = \\mathbb {}_{n}[ \\, \\cdot \\,]$ .",
"For the above-mentioned context, we obtain the useful Proposition 34 Suppose that $\\varphi \\in \\Upsilon (]a,b[)$ , with connected probability distribution $\\mathbb {}$ from (REF ) (recall that this implies that $\\mathbb {}$ is not a one-point distribution, cf.",
"Remark REF ).",
"Then there holds: $\\textrm {\\widetilde{c} \\cdot \\varphi \\in \\Upsilon (]a,b[)for all \\widetilde{c} > 0} \\ \\ \\Longleftrightarrow \\ \\ \\textrm {\\mathbb {} is infinitely divisible.",
"}$ The proof of Proposition REF is given in Appendix E. Notice that Proposition REF covers especially the important prominent power divergences (cf.",
"Examples REF and REF below) for which we provide the corresponding infinitely divisible distributions explicitly in the Examples REF and REF below, and for which the subsequent form of estimators (cf.",
"Chapter below) can be simplified.",
"For the identification of light-tailed infinitely divisible distributions, we obtain the following three sets of sufficient conditions: Proposition 35 Suppose that $\\varphi : ]-\\infty ,\\infty [ \\mapsto [0,\\infty ]$ satisfies (G1) to (G8), and recall the notations in (G9i) as well as $a = \\inf supp(\\mathbb {})$ , $b = \\sup supp(\\mathbb {})$ (cf.",
"(G10i)).",
"Then, $\\varphi \\in \\Upsilon (]a,b[)$ and the associated probability distribution $\\mathbb {}$ is infinitely divisible, if one of the following three conditions holds: (a) $a >-\\infty $ , $\\lambda _{-} = - \\infty $ , and the function $z \\mapsto \\varphi ^{* \\prime }(z) - a = (\\varphi ^{\\prime })^{-1}(z) - a $ is absolutely monotone on $]-\\infty ,0[$ , (b) $b < \\infty $ , $\\lambda _{+} = \\infty $ , and the function $z \\mapsto - \\varphi ^{* \\prime }(- z) + b= - (\\varphi ^{\\prime })^{-1}(-z) +b $ is absolutely monotone on $]-\\infty ,0[$ , (c) $a = -\\infty $ , $b = -\\infty $ , and the function $z \\mapsto \\frac{\\varphi ^{* \\prime \\prime }(z)}{\\varphi ^{* \\prime \\prime }(0)}= \\frac{\\varphi ^{\\prime \\prime }(1)}{\\varphi ^{\\prime \\prime }((\\varphi ^{\\prime })^{-1}(z))} $ is exponentially convex on $]\\lambda _{-}, \\lambda _{+}[$ .",
"In the first case (a) there automatically follows $b=\\infty $ , whereas in the second case (b) one automatically gets $a =- \\infty $ .",
"The proof of Proposition REF is given in Appendix E. So far, in the current section we have started from a given divergence generator $\\varphi \\in \\widetilde{\\Upsilon }(]a,b[)$ having some additional properties, switched to its Fenchel-Legendre transform $\\varphi ^{*}$ and some exponentially-linear transforms thereof, and presented some sufficient conditions for verifying that the outcome is a moment-generating function $MGF_{\\mathbb {}}$ of a unique probability distribution $\\mathbb {}$ which has light tails.",
"For finding the concrete $\\mathbb {}$ , one typically should know the explicit form of $\\varphi ^{*}$ .",
"However, it is well known that it can sometimes be hard to determine the explicit form of the Fenchel-Legendre transform of a convex function.",
"This issue also applies for the reverse direction of starting from a concrete probability distribution $\\mathbb {}$ with light tails, computing its log-moment-generating function (called cumulant-generating function) $z \\mapsto \\Lambda _{\\mathbb {}}(z) := \\log MGF_{\\mathbb {}}(z)$ and the corresponding Fenchel-Legendre transform $\\Lambda _{\\mathbb {}}^{*}$ which is nothing but the associated divergence generator $\\varphi $ (cf.",
"(REF )).",
"As will be illuminated in several examples below, the — “kind of intermediate” — construction method given in the below-mentioned Theorem REF can help to ease these two tasks.",
"To formulate this, we employ the class ${F}$ of functions $F: ]-\\infty ,\\infty [ \\mapsto [-\\infty ,\\infty ]$ with the following properties: $int(dom(F)) = ]a_{F},b_{F}[$ for some $-\\infty \\le a_{F} < 1 < b_{F} \\le \\infty $ ; $F$ is smooth (infinitely continuously differentiable) on $]a_{F},b_{F}[$ ; $F$ is strictly increasing on $]a_{F},b_{F}[$ .",
"Clearly, for any $F \\in {F}$ one gets the existence of $F(a_{F}) := \\lim _{t \\downarrow a_{F}} F(t) \\in [-\\infty , \\infty [$ and $F(b_{F}) := \\lim _{t \\uparrow b_{F}} F(t) \\in ]-\\infty , \\infty ]$ ; moreover, its inverse $F^{-1}: \\mathcal {R}(F) \\mapsto [a_{F},b_{F}]$ exists, where $\\mathcal {R}(F) := \\lbrace F(t): t \\in dom(F) \\rbrace $ .",
"Furthermore, $F^{-1}$ is strictly increasing and smooth (infinitely continuously differentiable) on the open interval $int(\\mathcal {R}(F)) = \\lbrace F(t): t \\in ]a_{F},b_{F}[ \\rbrace =]F(a_{F}),F(b_{F})[$ , and $F^{-1}(int(\\mathcal {R}(F))) = ]a_{F},b_{F}[$ .",
"Within such a context, we obtain Theorem 36 Let $F \\in {F}$ and fix an arbitrary point $c \\in int(\\mathcal {R}(F))$ .",
"Moreover, introduce the notationsfor the sake of brevity, we avoid here the more complete notation $\\lambda _{-}^{F,c}$ , $\\lambda _{+}^{F,c}$ , $t_{-}^{sc,F,c}$ , $t_{+}^{sc,F,c}$ indicating the dependence on $F$ and $c$ .",
"$]\\lambda _{-},\\lambda _{+}[ := int(\\mathcal {R}(F)) -c$ and $]t_{-}^{sc},t_{+}^{sc}[ := ]1+a_{F}-F^{-1}(c),1+b_{F}-F^{-1}(c)[$ (which implies $\\lambda _{-} < 0 < \\lambda _{+}$ and $t_{-}^{sc} < 1 < t_{+}^{sc}$ ).",
"Furthermore, define the functions $\\Lambda : \\ ]-\\infty ,\\infty [ \\ \\mapsto \\, [-\\infty , \\infty ]$ and $\\varphi : \\ ]-\\infty ,\\infty [ \\ \\mapsto \\, [0, \\infty ]$ by $\\hspace{-19.91684pt} \\Lambda (z) := \\Lambda ^{(c)}(z) \\hspace{-5.69046pt} &:=& \\hspace{-5.69046pt}{\\left\\lbrace \\begin{array}{ll}\\int \\displaylimits _{0}^{z} F^{-1}(u+c) \\, du+ z \\cdot (1-F^{-1}(c)) \\ \\in ]-\\infty , \\infty [,\\hspace{85.35826pt} \\textrm {if } z \\in ]\\lambda _{-},\\lambda _{+}[,\\\\\\int \\displaylimits _{0}^{\\lambda _{-}} F^{-1}(u+c) \\, du+ \\lambda _{-} \\cdot (1-F^{-1}(c))\\ \\in [-\\infty ,\\infty ],\\hspace{71.13188pt} \\textrm {if } z = \\lambda _{-} > -\\infty ,\\\\\\int \\displaylimits _{0}^{\\lambda _{+}} F^{-1}(u+c) \\, du+ \\lambda _{+} \\cdot (1-F^{-1}(c))\\ \\in \\ [-\\infty ,\\infty ],\\hspace{66.86414pt} \\textrm {if } z = \\lambda _{+} < \\infty ,\\\\\\infty , \\hspace{283.10483pt} \\textrm {else},\\end{array}\\right.",
"}$ where the second respectively third line are meant as $\\lim _{z \\downarrow \\lambda _{-}} \\big ( \\int \\displaylimits _{0}^{z} F^{-1}(u+c) \\, du+ z \\cdot (1-F^{-1}(c)) \\big )$ respectively $\\lim _{z \\uparrow \\lambda _{+}} \\big ( \\int \\displaylimits _{0}^{z} F^{-1}(u+c) \\, du+ z \\cdot (1-F^{-1}(c)) \\big )$ , and $\\varphi (t) := \\varphi ^{(c)}(t) \\hspace{-5.69046pt} &:=& \\hspace{-5.69046pt}{\\left\\lbrace \\begin{array}{ll}(t+F^{-1}(c)-1) \\cdot [ F\\left(t+F^{-1}(c)-1 \\right) - c ]-\\int \\displaylimits _{0}^{F\\left(t+F^{-1}(c)-1 \\right) - c} F^{-1}(u+c) du\\ \\in \\ [0,\\infty [,\\\\\\hspace{327.20668pt}\\quad \\textrm {if }t \\in ]t_{-}^{sc},t_{+}^{sc}[,\\\\(t_{-}^{sc}+F^{-1}(c)-1) \\cdot [ F\\left(t_{-}^{sc}+F^{-1}(c)-1 \\right) - c ]-\\int \\displaylimits _{0}^{F\\left(t_{-}^{sc}+F^{-1}(c)-1 \\right) - c} F^{-1}(u+c) du\\ \\in \\ ]0,\\infty ],\\\\\\hspace{327.20668pt}\\quad \\textrm {if }t = t_{-}^{sc} > -\\infty ,\\\\(t_{+}^{sc}+F^{-1}(c)-1) \\cdot [ F\\left(t_{+}^{sc}+F^{-1}(c)-1 \\right) - c ]-\\int \\displaylimits _{0}^{F\\left(t_{+}^{sc}+F^{-1}(c)-1 \\right) - c} F^{-1}(u+c) du\\ \\in \\ ]0,\\infty ],\\\\\\hspace{327.20668pt}\\quad \\textrm {if }t = t_{+}^{sc} < \\infty ,\\\\\\varphi (t_{-}^{sc}) +\\lambda _{-}\\cdot (t- t_{-}^{sc}) \\ \\in \\ ]0,\\infty ],\\hspace{99.58464pt} \\textrm {if }t_{-}^{sc} > - \\infty , \\ \\textrm {and} \\ t \\in \\ ]-\\infty , t_{-}^{sc}[,\\\\\\varphi (t_{+}^{sc}) +\\lambda _{+}\\cdot (t - t_{+}^{sc}) \\ \\in \\ ]0,\\infty ],\\hspace{99.58464pt} \\textrm {if }t_{+}^{sc} < \\infty , \\ \\textrm {and} \\ t \\in \\ ]t_{+}^{sc}, \\infty [,\\\\\\infty , \\hspace{321.51622pt} \\textrm {else},\\end{array}\\right.",
"}$ where the second respectively third line are again meant as lower respectively upper limit.",
"Then, $\\Lambda $ and $\\varphi $ have the following properties: (i) On $]\\lambda _{-},\\lambda _{+}[$ , the function $\\Lambda $ is smooth and strictly convex and consequently, $\\exp (\\Lambda )$ ) is smooth and strictly log-convex; moreover, there holds $\\Lambda (0) =0$ , $\\Lambda ^{\\prime }(0) =1$ ; (ii) $\\varphi \\in \\widetilde{\\Upsilon }(]a,b[)$ , where $a := t_{-}^{sc} \\cdot {1}_{ \\lbrace -\\infty \\rbrace }(\\lambda _{-}) - \\infty \\cdot {1}_{]-\\infty ,0[}(\\lambda _{-})$ , $b := t_{+}^{sc} \\cdot {1}_{ \\lbrace \\infty \\rbrace }(\\lambda _{+}) + \\infty \\cdot {1}_{]0,\\infty [}(\\lambda _{+})$ , and $\\varphi $ has the properties (G1) to (G8).",
"(iii) $\\varphi (t) =\\sup _{z \\in ]-\\infty ,\\infty [} \\left( z\\cdot t -\\Lambda (z)\\right) =\\sup _{z \\in ]\\lambda _{-},\\lambda _{+}[} \\left( z\\cdot t -\\Lambda (z)\\right)$ for all $t \\in \\mathbb {R}$ .",
"(iv) $\\Lambda (z) = \\varphi ^{*}(z) =\\sup _{t \\in ]-\\infty ,\\infty [} \\left( t\\cdot z -\\varphi (t)\\right) =\\sup _{t \\in ]a,b[} \\left( t\\cdot z -\\varphi (t) \\right)$ for all $z \\in \\mathbb {R}$ .",
"The proof of Theorem REF will be given Appendix F. Remark 37 Theorem REF indicates that the $F-$ constructed function $z \\mapsto \\exp (\\Lambda (z)) = \\exp (\\varphi ^{*}(z))$ is a good candidate for a moment generating function of a probability distribution $\\mathbb {}$ , and hence for the representability (REF ).",
"However, one still needs to verify one of the conditions (a) to (c) of Proposition REF .",
"This may go wrong, as the case of power divergences $\\varphi _{\\gamma }$ with $\\gamma \\in ]1,2[$ indicates (cf.",
"the conjecture of Example REF (f) below).",
"Notice that the newly constructed $\\Lambda $ and $\\varphi $ (cf.",
"(REF ), (REF )) depend on the choice of the anchor point $c$ ; this is e.g.",
"illustrated in Example REF (b) below.",
"Hence, as a side effect, by using whole families $(F_{\\vartheta })_{\\vartheta }$ together with different anchor points $c$ , via Theorem REF one can generate new classes (and new classifications) of $\\varphi -$ divergence generators — and thus of corresponding $\\varphi -$ divergences — which can be of great use, even in other contexts beyond our BS optimization framework.",
"If $F$ satisfies $F(1)=0$ and thus $F^{-1}(0)=1$ , then the natural choice $c:=0$ induces $]\\lambda _{-},\\lambda _{+}[ = int(\\mathcal {R}(F))$ and $]t_{-}^{sc},t_{+}^{sc}[ = ]a_{F},b_{F}[$ , and consequently (due to $F^{-1}(c)-1 =0$ ) leads to the simplification of “the first lines of” (REF ) and (REF ) to $& & \\hspace{-19.91684pt} \\Lambda (z) := \\Lambda ^{(0)}(z) :=\\int \\displaylimits _{0}^{z} F^{-1}(u) du,\\qquad z \\in int(\\mathcal {R}(F)),\\\\& & \\hspace{-19.91684pt} \\varphi (t) := \\varphi ^{(0)}(t) :=t \\cdot F\\left(t\\right)-\\int \\displaylimits _{0}^{F\\left(t\\right)} F^{-1}(u) du,\\qquad t \\in ]a_{F},b_{F}[ ;$ the simplifications of the respective other lines of (REF ) and (REF ) are straightforward.",
"Remark 38 Let $F \\in {F}$ with $a_{F} =0$ , $b_{F} = \\infty $ , $F(1)=0$ and hence, $int(\\mathcal {R}(F)) = \\, ]F(0),F(\\infty )[$ .",
"Then the transformation $\\widetilde{F}(t) &:=&{\\left\\lbrace \\begin{array}{ll}- \\int _{0}^{F(\\frac{1}{t})} F^{-1}(u) \\, du, \\qquad \\textrm {if } \\ t \\in ]0,\\infty [, \\\\- \\int _{0}^{F(\\infty )} F^{-1}(u) \\, du, \\qquad \\textrm {if } \\ t=0, \\\\-\\infty , \\qquad \\hspace{64.01869pt} \\textrm {if } \\ t \\in ]-\\infty , 0[,\\end{array}\\right.",
"}$ satisfies $\\widetilde{F} \\in {F}$ with $a_{\\widetilde{F}} =0$ , $b_{\\widetilde{F}} = \\infty $ , $\\widetilde{F}(1)=0$ and $int(\\mathcal {R}(\\widetilde{F})) =\\big ] - \\int _{0}^{F(\\infty )} F^{-1}(u) \\, du, - \\int _{0}^{F(0)} F^{-1}(u) \\, du \\, \\big [$ .",
"By choosing the natural anchor point $c=0$ (for both $F$ and $\\widetilde{F}$ ) and by using the relations $\\widetilde{F}(t) = - \\Lambda (F(\\frac{1}{t}))$ , $\\widetilde{F}^{-1}(z) = \\frac{1}{F^{-1}(\\Lambda ^{-1}(-z))}$ , as well as (REF ) in combination with (REF ) (for both contexts), it is straightforward to see that the corresponding quantities $\\widetilde{\\Lambda }$ and $\\widetilde{\\varphi }$ satisfy $\\widetilde{\\Lambda }(z) =- (-\\Lambda )^{-1}(z)$ ($z \\in int(\\mathcal {R}(\\widetilde{F}))$ ) and $\\widetilde{\\varphi }(t) = t \\cdot \\varphi (\\frac{1}{t})$ ($t \\in ]0,\\infty [$ ).",
"Hence, the corresponding divergences (cf.",
"(REF )) are “reciprocal to each other” in the sense that $D_{\\widetilde{\\varphi }}( \\mathbf {Q}, \\mathbf {P} ) =D_{\\varphi }( \\mathbf {P}, \\mathbf {Q} )$ for all $\\mathbf {P},\\mathbf {Q} \\in \\mathbb {S}_{>0}^{K}$ , and in case that $\\Lambda $ and $\\widetilde{\\Lambda }$ are indeed cumulant generating functions of some light-tailed distributions $\\mathbb {}$ and $\\widetilde{\\mathbb {}}$ (cf.",
"Remark REF ), then the latter two are said to be inverse to each other in the sense of Tweedie [368] (see also e.g.",
"Folks [127]).",
"As already indicated above, from Theorem REF one can comfortably generate various interesting examples, which we demonstrate in the following.",
"Example 39 (a) For $\\gamma \\in \\mathbb {R}\\backslash \\lbrace 1,2\\rbrace $ , $\\widetilde{c} \\in ]0,\\infty [$ and $]a_{F_{\\gamma ,\\widetilde{c}}},b_{F_{\\gamma ,\\widetilde{c}}}[ \\, = \\, ]0,\\infty [$ we define $F_{\\gamma ,\\widetilde{c}}(t) &:=&{\\left\\lbrace \\begin{array}{ll}\\frac{\\widetilde{c}}{\\gamma -1} \\cdot (t^{\\gamma -1}-1), \\qquad \\textrm {if } \\ t \\in \\, ]0,\\infty [, \\\\-\\frac{\\widetilde{c}}{\\gamma -1}, \\hspace{62.59596pt}\\textrm {if } \\ t=0 \\ \\textrm {and } \\ \\gamma \\in \\, ]1,2[ \\, \\cup \\, ]2,\\infty [, \\\\-\\infty , \\hspace{71.13188pt} \\textrm {if } \\ t=0 \\ \\textrm {and } \\ \\gamma < 1, \\\\- \\infty , \\hspace{71.13188pt} \\textrm {if } \\ t \\in \\, ]-\\infty ,0[,\\end{array}\\right.",
"}\\nonumber $ Clearly, $\\mathcal {R}(F_{\\gamma ,\\widetilde{c}})= \\big [-\\frac{\\widetilde{c}}{\\gamma -1},\\infty \\big [$ for $\\gamma \\in \\, ]1,2[ \\, \\cup \\, ]2,\\infty [$ , respectively $\\mathcal {R}(F_{\\gamma ,\\widetilde{c}})=\\big ]-\\infty , \\frac{\\widetilde{c}}{1-\\gamma }\\big [$ for $\\gamma <1$ ; notice that $0 \\in int(\\mathcal {R}(F_{\\gamma ,\\widetilde{c}}))$ for all $\\gamma \\in \\mathbb {R}\\backslash \\lbrace 1,2\\rbrace $ .",
"Furthermore, $F_{\\gamma ,\\widetilde{c}}(\\cdot )$ is strictly increasing and smooth on $]0,\\infty [$ , and thus, $F_{\\gamma ,\\widetilde{c}} \\in {F}$ .",
"Since $F_{\\gamma ,\\widetilde{c}}(1)=0$ , let us choose the natural anchor point $c:=0$ , which leads to $]\\lambda _{-},\\lambda _{+}[= int(\\mathcal {R}(F_{\\gamma ,\\widetilde{c}}))$ and $]t_{-}^{sc},t_{+}^{sc}[ = ]0,\\infty [$ .",
"By using $F_{\\gamma ,\\widetilde{c}}^{-1}(x) = (1+\\frac{(\\gamma -1) \\cdot x}{\\widetilde{c}})^{\\frac{1}{\\gamma -1}}$ for $x \\in int(\\mathcal {R}(F_{\\gamma ,\\widetilde{c}}))$ , we can derive from formula (REF ) (see also (REF )) for all $\\gamma \\in \\mathbb {R}\\backslash \\lbrace 0,1,2\\rbrace $ and $z \\in \\mathbb {R}$ $\\Lambda _{\\gamma ,\\widetilde{c}}(z) := \\Lambda _{\\gamma ,\\widetilde{c}}^{(0)}(z) &=&{\\left\\lbrace \\begin{array}{ll}\\frac{\\widetilde{c}}{\\gamma } \\cdot \\left\\lbrace \\left( \\frac{\\gamma -1}{\\widetilde{c}} \\cdot z +1 \\right)^{\\frac{\\gamma }{\\gamma -1}} -1 \\right\\rbrace , \\qquad \\textrm {if } \\ \\gamma \\in \\, ]1,2[ \\, \\cup \\, ]2,\\infty [\\ \\textrm {and } \\ z \\in \\big ]-\\frac{\\widetilde{c}}{\\gamma -1},\\infty \\big [ \\\\\\hspace{130.88284pt} \\textrm {or if } \\ \\gamma \\in ]-\\infty ,0[ \\cup ]0,1[ \\ \\textrm {and } \\ z \\in \\big ]-\\infty , \\frac{\\widetilde{c}}{1-\\gamma }\\big [, \\\\- \\frac{\\widetilde{c}}{\\gamma } < 0, \\hspace{105.2751pt} \\textrm {if } \\ \\gamma \\in \\, ]1,2[ \\, \\cup \\, ]2,\\infty [\\ \\textrm {and } \\ z = -\\frac{\\widetilde{c}}{\\gamma -1}, \\\\- \\frac{\\widetilde{c}}{\\gamma } >0, \\hspace{105.2751pt} \\textrm {if } \\ \\gamma < 0 \\ \\textrm {and } \\ z = \\frac{\\widetilde{c}}{1-\\gamma }, \\\\\\infty , \\hspace{128.0374pt} \\textrm {if } \\ \\gamma \\in \\, ]0,1[ \\ \\textrm {and } \\ z = \\frac{\\widetilde{c}}{1-\\gamma }, \\\\\\infty , \\hspace{128.0374pt} \\textrm {else} .\\end{array}\\right.",
"}$ Notice that $\\Lambda _{\\gamma ,\\widetilde{c}}(0) = 0$ for all $\\gamma \\in \\mathbb {R}\\backslash \\lbrace 0,1,2\\rbrace $ .",
"Moreover, for $\\gamma \\in \\, ]1,2[ \\, \\cup \\, ]2,\\infty [$ one has $\\Lambda _{\\gamma ,\\widetilde{c}}(\\infty ) = \\infty $ , $\\Lambda _{\\gamma ,\\widetilde{c}}^{\\prime }(-\\frac{\\widetilde{c}}{\\gamma -1}) = 0$ and $\\Lambda _{\\gamma ,\\widetilde{c}}^{\\prime }(\\infty ) = \\infty $ .",
"For $\\gamma < 0$ one gets $\\Lambda _{\\gamma ,\\widetilde{c}}(-\\infty ) = -\\infty $ , $\\Lambda _{\\gamma ,\\widetilde{c}}^{\\prime }(\\frac{\\widetilde{c}}{1-\\gamma }) = \\infty $ and $\\Lambda _{\\gamma ,\\widetilde{c}}^{\\prime }(-\\infty ) = 0$ .",
"In contrast, if $\\gamma \\in \\, ]0,1[$ then $\\Lambda _{\\gamma ,\\widetilde{c}}(-\\infty ) = - \\frac{\\widetilde{c}}{\\gamma } <0$ , $\\Lambda _{\\gamma ,\\widetilde{c}}^{\\prime }(\\frac{\\widetilde{c}}{1-\\gamma }) = \\infty $ and $\\Lambda _{\\gamma ,\\widetilde{c}}^{\\prime }(-\\infty ) = 0$ .",
"To proceed, from formula (REF ) (see also ()) we can deduce for all $\\gamma \\in \\mathbb {R}\\backslash \\lbrace 0,1,2\\rbrace $ and $t \\in \\mathbb {R}$ $\\varphi _{\\gamma ,\\widetilde{c}}(t) := \\varphi _{\\gamma ,\\widetilde{c}}^{(0)}(t)\\hspace{-5.69046pt} &=& \\hspace{-5.69046pt}{\\left\\lbrace \\begin{array}{ll}\\widetilde{c} \\cdot \\frac{t^\\gamma -\\gamma \\cdot t+ \\gamma - 1}{\\gamma \\cdot (\\gamma -1)}\\ \\in \\ [0,\\infty [,\\qquad \\quad \\textrm {if }t \\in ]0,\\infty [,\\\\\\frac{\\widetilde{c}}{\\gamma } > 0, \\hspace{108.12054pt} \\textrm {if } \\ \\gamma \\in \\, ]1,2[ \\, \\cup \\, ]2,\\infty [\\ \\textrm {and } \\ t = 0, \\\\\\infty , \\hspace{123.76965pt} \\textrm {if } \\ \\gamma < 0 \\ \\textrm {and } \\ t = 0, \\\\\\frac{\\widetilde{c}}{\\gamma } > 0, \\hspace{108.12054pt} \\textrm {if } \\ \\gamma \\in \\, ]0,1[ \\ \\textrm {and } \\ t = 0, \\\\\\frac{\\widetilde{c}}{\\gamma } -\\frac{\\widetilde{c}}{\\gamma -1} \\cdot t \\ \\in \\ ]0,\\infty [,\\hspace{45.52458pt} \\textrm {if } \\ \\gamma \\in \\, ]1,2[ \\, \\cup \\, ]2,\\infty [\\ \\textrm {and } \\ t < 0, \\\\\\infty , \\hspace{123.76965pt} \\textrm {else} ,\\end{array}\\right.",
"}$ which coincides with $\\widetilde{c} \\cdot \\varphi _{\\gamma }(t)$ for $\\varphi _{\\gamma }(t)$ from (REF ) and which generates the $\\gamma -$ corresponding power divergences given in (REF ); the first line in (REF ) can be proved by $& & \\hspace{-19.91684pt}\\varphi _{\\gamma ,\\widetilde{c}}(t) := \\varphi _{\\gamma ,\\widetilde{c}}^{(0)}(t) :=t \\cdot F_{\\gamma ,\\widetilde{c}}\\left(t\\right)-\\int \\displaylimits _{0}^{F_{\\gamma ,\\widetilde{c}}\\left(t\\right)} F_{\\gamma ,\\widetilde{c}}^{-1}(u) du\\nonumber \\\\& & \\hspace{-19.91684pt}= \\frac{t \\cdot \\widetilde{c}}{\\gamma -1} \\cdot (t^{\\gamma -1}-1)- \\frac{\\widetilde{c}}{\\gamma } \\cdot \\left\\lbrace \\left( \\frac{\\gamma -1}{\\widetilde{c}} \\cdot \\left[ \\frac{\\widetilde{c}}{\\gamma -1} \\cdot (t^{\\gamma -1}-1) \\right]+ 1 \\right)^{\\frac{\\gamma }{\\gamma -1}} -1 \\right\\rbrace \\nonumber \\\\& & \\hspace{-19.91684pt}= \\widetilde{c} \\cdot \\frac{t^\\gamma -\\gamma \\cdot t+ \\gamma - 1}{\\gamma \\cdot (\\gamma -1)},\\qquad t \\in ]0,\\infty [ \\, .$ Notice that for all $\\gamma \\in \\mathbb {R}\\backslash \\lbrace 0,1,2\\rbrace $ one has $\\varphi _{\\gamma ,\\widetilde{c}}(1) = 0$ , $\\varphi _{\\gamma ,\\widetilde{c}}^{\\prime }(1) = 0$ and $\\varphi _{\\gamma ,\\widetilde{c}}(\\infty ) = \\infty $ .",
"Moreover, for $\\gamma \\in \\, ]1,2[ \\, \\cup \\, ]2,\\infty [$ one has $\\varphi _{\\gamma ,\\widetilde{c}}^{\\prime }(0) = -\\frac{\\widetilde{c}}{\\gamma -1} < 0$ and $\\varphi _{\\gamma ,\\widetilde{c}}^{\\prime }(\\infty ) = \\infty $ .",
"In contrast, for $\\gamma < 0$ and $\\gamma \\in \\, ]0,1[$ one gets $\\varphi _{\\gamma ,\\widetilde{c}}^{\\prime }(0) = - \\infty $ and $\\varphi _{\\gamma ,\\widetilde{c}}^{\\prime }(\\infty ) = \\frac{\\widetilde{c}}{1-\\gamma } > 0$ .",
"(b) For $\\gamma =2$ , $\\widetilde{c} \\in ]0,\\infty [$ and $]a_{F_{\\gamma ,\\widetilde{c}}},b_{F_{\\gamma ,\\widetilde{c}}}[ \\, = \\, ]-\\infty ,\\infty [$ we define $F_{2,\\widetilde{c}}(t) &:=&\\widetilde{c} \\cdot (t-1), \\qquad t \\in \\, ]-\\infty ,\\infty [,\\nonumber $ Clearly, $\\mathcal {R}(F_{2,\\widetilde{c}})= \\, ]-\\infty , \\infty [$ , $0 \\in int(\\mathcal {R}(F_{2,\\widetilde{c}}))$ , and $F_{2,\\widetilde{c}}(\\cdot )$ is strictly increasing as well as smooth on $]-\\infty ,\\infty [$ .",
"Hence, $F_{2,\\widetilde{c}} \\in {F}$ .",
"Since $F_{2,\\widetilde{c}}(1)=0$ , let us choose the natural anchor point $c:=0$ , which leads to $]\\lambda _{-},\\lambda _{+}[= int(\\mathcal {R}(F_{2,\\widetilde{c}}))= \\, ]-\\infty , \\infty [$ and $]t_{-}^{sc},t_{+}^{sc}[ = \\, ]-\\infty , \\infty [$ .",
"By using $F_{2,\\widetilde{c}}^{-1}(x) = 1+\\frac{x}{\\widetilde{c}}$ for $x \\in int(\\mathcal {R}(F_{2,\\widetilde{c}}))$ , we can derive from formula (REF ) (see also (REF )) $\\Lambda _{2,\\widetilde{c}}(z) := \\Lambda _{2,\\widetilde{c}}^{(0)}(z) &=&\\frac{\\widetilde{c}}{2} \\cdot \\left\\lbrace \\left( \\frac{1}{\\widetilde{c}} \\cdot z +1 \\right)^{2} -1 \\right\\rbrace =\\frac{z^2}{2 \\widetilde{c}} + z,\\qquad z \\in \\, ]-\\infty ,\\infty [.$ Notice that $\\Lambda _{2,\\widetilde{c}}(0) = 0$ , $\\Lambda _{2,\\widetilde{c}}(-\\infty ) = \\Lambda _{2,\\widetilde{c}}(\\infty ) = \\infty $ , $\\Lambda _{2,\\widetilde{c}}^{\\prime }(-\\infty ) = -\\infty $ and $\\Lambda _{2,\\widetilde{c}}^{\\prime }(\\infty ) = \\infty $ .",
"From formula (REF ) (see also ()) we can deduce analogously to (REF ) $\\varphi _{2,\\widetilde{c}}(t) := \\varphi _{2,\\widetilde{c}}^{(0)}(t)&=&\\widetilde{c} \\cdot \\frac{(t-1)^{2}}{2}\\ \\in \\ [0,\\infty [,\\qquad t \\in \\, ]-\\infty ,\\infty [,$ which coincides with $\\widetilde{c} \\cdot \\varphi _{2}(t)$ for $\\varphi _{2}(t)$ from (REF ) which generates the corresponding power divergence given in the sixth line of (REF ).",
"Notice that $\\varphi _{2,\\widetilde{c}}(1) = 0$ , $\\varphi _{2,\\widetilde{c}}^{\\prime }(1) = 0$ and $\\varphi _{2,\\widetilde{c}}(-\\infty ) = \\varphi _{2,\\widetilde{c}}(\\infty ) = \\infty $ .",
"As an application of the reciprocity considerations of Remark REF , it is straightforward to see from the above-mentioned considerations (a) and (b) that for all $\\gamma \\in \\mathbb {R}\\backslash \\lbrace 0,1\\rbrace $ one has $\\widetilde{F}_{\\gamma ,\\widetilde{c}}(t) =- \\Lambda _{\\gamma ,\\widetilde{c}}(F_{\\gamma ,\\widetilde{c}}(\\frac{1}{t}))= F_{1-\\gamma ,\\widetilde{c}}(t)$ for all $t \\in ]0,\\infty [$ .",
"(c) Let us now continue with the remaining case $\\gamma =0$ (recall the natural anchor point $c:=0$ ).",
"By using $F_{0,\\widetilde{c}}^{-1}(x) =\\frac{1}{1- \\frac{x}{\\widetilde{c}}} $ for $x \\in int(\\mathcal {R}(F_{0,\\widetilde{c}})) = ]-\\infty , \\widetilde{c}[$ , we can derive from formula (REF ) (see also (REF )) $\\Lambda _{0,\\widetilde{c}}(z) := \\Lambda _{0,\\widetilde{c}}^{(0)}(z) &=&{\\left\\lbrace \\begin{array}{ll}- \\widetilde{c} \\cdot \\log \\left( 1 - \\frac{z}{\\widetilde{c}} \\right), \\qquad \\textrm {if } \\ z \\in \\big ]-\\infty ,\\widetilde{c} \\big [, \\\\\\infty , \\hspace{78.24507pt} \\textrm {if } \\ z \\in \\big [\\widetilde{c}, \\infty \\big [ .\\end{array}\\right.",
"}$ Notice that $\\Lambda _{0,\\widetilde{c}}(0) = 0$ , $\\Lambda _{0,\\widetilde{c}}(-\\infty ) = - \\infty $ , $\\Lambda _{0,\\widetilde{c}}^{\\prime }(\\widetilde{c}) = \\infty $ and $\\Lambda _{0,\\widetilde{c}}^{\\prime }(-\\infty ) = 0$ .",
"Moreover, from formula (REF ) (see also ()) we can deduce $\\varphi _{0,\\widetilde{c}}(t) := \\varphi _{0,\\widetilde{c}}^{(0)}(t)\\hspace{-5.69046pt} &=& \\hspace{-5.69046pt}{\\left\\lbrace \\begin{array}{ll}\\widetilde{c} \\cdot \\left(- \\log t + t -1 \\right)\\ \\in \\ [0,\\infty [,\\qquad \\quad \\textrm {if }t \\in \\, ]0,\\infty [,\\\\\\infty , \\hspace{145.10922pt} \\textrm {if } \\ t \\in \\, ]-\\infty , 0] ,\\end{array}\\right.",
"}$ which coincides with $\\widetilde{c} \\cdot \\varphi _{0}(t)$ for the generator $\\varphi _{0}(t)$ from (REF ) which generates the reverse Kullback-Leibler divergence (reverse relative entropy) given in (REF ) with $\\widetilde{c}=1$ ; the first line in (REF ) can be proved by $& & \\hspace{-19.91684pt}\\varphi _{0,\\widetilde{c}}(t) := \\varphi _{0,\\widetilde{c}}^{(0)}(t) :=t \\cdot F_{0,\\widetilde{c}}\\left(t\\right)-\\int \\displaylimits _{0}^{F_{0,\\widetilde{c}}\\left(t\\right)} F_{0,\\widetilde{c}}^{-1}(u) du\\nonumber \\\\& & \\hspace{-19.91684pt}= t \\cdot \\widetilde{c} \\cdot \\Big (1-\\frac{1}{t}\\Big )- (- \\widetilde{c}) \\cdot \\log \\left(1-\\frac{1}{\\widetilde{c}} \\cdot \\left[ \\widetilde{c} \\cdot \\Big ( 1-\\frac{1}{t}\\Big ) \\right]\\right)= \\widetilde{c} \\cdot \\left(- \\log t + t -1 \\right),\\qquad t \\in ]0,\\infty [ \\, .$ Notice that one has $\\varphi _{0,\\widetilde{c}}(1) = 0$ , $\\varphi _{0,\\widetilde{c}}(\\infty ) = \\infty $ , $\\varphi _{0,\\widetilde{c}}^{\\prime }(1) = 0$ , $\\varphi _{0,\\widetilde{c}}^{\\prime }(0) = -\\infty $ and $\\varphi _{0,\\widetilde{c}}^{\\prime }(\\infty ) = \\widetilde{c}$ .",
"Example 40 (a) For the remaining case $\\gamma =1$ , $\\widetilde{c} \\in ]0,\\infty [$ and $]a_{F_{1,\\widetilde{c}}},b_{F_{1,\\widetilde{c}}}[ = ]0,\\infty [$ we define $F_{1,\\widetilde{c}}(t) &:=&{\\left\\lbrace \\begin{array}{ll}\\widetilde{c} \\cdot \\log t =\\lim _{\\gamma \\rightarrow 1} F_{\\gamma ,\\widetilde{c}}(t), \\qquad \\textrm {if } \\ t \\in \\, ]0,\\infty [, \\\\-\\infty , \\hspace{108.12054pt} \\textrm {if } \\ t \\in ]-\\infty ,0].",
"\\\\\\end{array}\\right.",
"}\\nonumber $ Clearly, $\\mathcal {R}(F_{1,\\widetilde{c}})= ]-\\infty ,\\infty [$ .",
"Moreover, $F_{1,\\widetilde{c}}(\\cdot )$ is strictly increasing and smooth on $]0,\\infty [$ , and hence, $F_{\\gamma ,\\widetilde{c}} \\in {F}$ .",
"Since $F_{1,\\widetilde{c}}(1)=0$ , let us first choose the natural anchor point $c:=0$ , which leads to $]\\lambda _{-},\\lambda _{+}[= int(\\mathcal {R}(F_{1,\\widetilde{c}})) = ]-\\infty ,\\infty [$ and $]t_{-}^{sc},t_{+}^{sc}[ = ]0,\\infty [$ .",
"By using $F_{1,\\widetilde{c}}^{-1}(x) = \\exp (\\frac{x}{\\widetilde{c}})$ for $x \\in \\mathcal {R}(F_{1,\\widetilde{c}})$ , we can derive from formula (REF ) (see also (REF )) $& & \\hspace{-19.91684pt} \\Lambda _{1,\\widetilde{c}}(z) := \\Lambda _{1,\\widetilde{c}}^{(0)}(z) :=\\int \\displaylimits _{0}^{z} F_{1,\\widetilde{c}}^{-1}(u) \\, du= \\widetilde{c} \\cdot \\left( \\exp \\Big (\\frac{z}{\\widetilde{c}}\\Big ) - 1 \\right),\\ \\ z \\in ]-\\infty ,\\infty [ .$ Notice that $\\Lambda _{1,\\widetilde{c}}(0) = 0$ , $\\Lambda _{1,\\widetilde{c}}(-\\infty ) = - \\widetilde{c}$ , $\\Lambda _{1,\\widetilde{c}}(\\infty ) = \\infty $ , $\\Lambda _{1,\\widetilde{c}}^{\\prime }(-\\infty ) = 0$ and $\\Lambda _{0,\\widetilde{c}}^{\\prime }(\\infty ) = \\infty $ .",
"Moreover, from formula (REF ) (see also ()) we can deduce $\\varphi _{1,\\widetilde{c}}(t) := \\varphi _{1,\\widetilde{c}}^{(0)}(t)\\hspace{-5.69046pt} &:=& \\hspace{-5.69046pt}{\\left\\lbrace \\begin{array}{ll}\\widetilde{c} \\cdot \\left( t \\cdot \\log t + 1 - t \\right)\\ \\in \\ [0,\\infty [,\\qquad \\quad \\textrm {if }t \\in \\, ]0,\\infty [,\\\\1, \\hspace{150.79968pt} \\textrm {if } \\ t = 0, \\\\\\infty , \\hspace{145.10922pt} \\textrm {if } \\ t \\in \\, ]-\\infty ,0[ ,\\end{array}\\right.",
"}$ which coincides with $\\widetilde{c} \\cdot \\varphi _{1}(t)$ for the generator $\\varphi _{1}(t)$ from (REF ) which generates the Kullback-Leibler divergence (relative entropy) given in (REF ) with $\\widetilde{c}=1$ ; the first line in (REF ) can be proved by $& & \\hspace{-19.91684pt}\\varphi _{1,\\widetilde{c}}(t) := \\varphi _{1,\\widetilde{c}}^{(0)}(t) :=t \\cdot F_{1,\\widetilde{c}}\\left(t\\right)-\\int \\displaylimits _{0}^{F_{1,\\widetilde{c}}\\left(t\\right)} F_{1,\\widetilde{c}}^{-1}(u) du\\nonumber \\\\& & \\hspace{-19.91684pt}= t \\cdot \\widetilde{c} \\cdot \\log t- \\widetilde{c} \\cdot \\left( \\exp \\left(\\frac{1}{\\widetilde{c}} \\cdot \\left[ \\widetilde{c} \\cdot \\log t \\right]\\right) - 1 \\right)= \\widetilde{c} \\cdot \\left( t \\cdot \\log t + 1 - t \\right),\\qquad t \\in ]0,\\infty [ \\, ,$ Notice that one has $\\varphi _{1,\\widetilde{c}}(1) = 0$ , $\\varphi _{1,\\widetilde{c}}(\\infty ) = \\infty $ , $\\varphi _{1,\\widetilde{c}}^{\\prime }(1) = 0$ , $\\varphi _{1,\\widetilde{c}}^{\\prime }(0) = -\\infty $ and $\\varphi _{1,\\widetilde{c}}^{\\prime }(\\infty ) = \\infty $ .",
"As an application of the reciprocity considerations of Remark REF , it is straightforward to see that $\\widetilde{F}_{1,\\widetilde{c}}(t) =- \\Lambda _{1,\\widetilde{c}}(F_{1,\\widetilde{c}}(\\frac{1}{t}))= F_{0,\\widetilde{c}}(t)$ for all $t \\in ]0,\\infty [$ .",
"(b) For the choice $\\widetilde{c}=1$ , let us now fix a general anchor point $c \\in \\mathcal {R}(F_{1,\\widetilde{c}})= ]-\\infty ,\\infty [$ (rather than $c=0$ ), which leads to $]\\lambda _{-},\\lambda _{+}[= int(\\mathcal {R}(F_{1,1}))- c = ]-\\infty ,\\infty [$ and $]t_{-}^{sc},t_{+}^{sc}[ = ]1+a_{F_{1,1}}-F_{1,1}^{-1}(c),1+b_{F_{1,1}}-F_{1,1}^{-1}(c)[ \\,= \\, ]1-e^{c},\\infty [$ .",
"Accordingly, the formula (REF ) (see also (REF )) leads to $& & \\hspace{-19.91684pt} \\Lambda _{1,1}(z) := \\Lambda _{1,1}^{(c)}(z) :=\\int \\displaylimits _{0}^{z} F_{1,1}^{-1}(u+c) du+ z \\cdot (1-F_{1,1}^{-1}(c))\\nonumber \\\\& & \\hspace{59.75095pt}= e^{c} \\cdot \\left( e^{z} - 1 \\right) + z \\cdot (1- e^{c}),\\qquad z \\in ]-\\infty ,\\infty [, \\qquad \\ $ for which there holds $\\Lambda _{1,1}^{(c)}(0) = 0$ , $\\Lambda _{1,1}^{(c)}(-\\infty ) =\\infty \\cdot {1}_{]0,\\infty [}(c)- \\infty \\cdot {1}_{]-\\infty ,0[}(c)- 1 \\cdot {1}_{\\lbrace 0\\rbrace }(c)$ , $\\Lambda _{1,1}^{(c)}(\\infty ) = \\infty $ , $\\Lambda _{1,1}^{(c) \\prime }(-\\infty ) = 1- e^{c}$ and $\\Lambda _{1,1}^{(c) \\prime }(\\infty ) = \\infty $ .",
"Moreover, from formula (REF ) (see also ()) we can deduce $\\varphi _{1,1}(t) := \\varphi _{1,1}^{(c)}(t)\\hspace{-5.69046pt} &:=& \\hspace{-5.69046pt}{\\left\\lbrace \\begin{array}{ll}(t + e^{c} -1) \\cdot [\\log (t + e^{c} -1) -c] + 1 - t\\ \\in \\ [0,\\infty [,\\qquad \\quad \\textrm {if }t \\in \\, ]1-e^{c},\\infty [,\\\\e^{c}, \\hspace{240.42569pt} \\textrm {if } \\ t = 1-e^{c}, \\\\\\infty , \\hspace{239.00298pt} \\textrm {if } \\ t \\in \\, ]-\\infty ,1-e^{c}[ ;\\end{array}\\right.",
"}$ the first line in (REF ) can be proved by $& & \\hspace{-19.91684pt}\\varphi _{1,1}(t) := \\varphi _{1,1}^{(c)}(t) :=(t+F_{1,1}^{-1}(c)-1) \\cdot [ F_{1,1}\\left(t+F_{1,1}^{-1}(c)-1 \\right) - c ]-\\int \\displaylimits _{0}^{F_{1,1}\\left(t+F_{1,1}^{-1}(c)-1 \\right) - c}F_{1,1}^{-1}(u+c) du\\nonumber \\\\& & \\hspace{-19.91684pt}= (t + e^{c} -1) \\cdot [\\log (t + e^{c} -1) -c]- e^{c} \\cdot \\Big \\lbrace \\exp [\\log (t + e^{c} -1) -c] - 1 \\Big \\rbrace \\nonumber \\\\& & \\hspace{-19.91684pt}= (t + e^{c} -1) \\cdot [\\log (t + e^{c} -1) -c] + 1 - t ,\\qquad t \\in ]1-e^{c},\\infty [\\, .$ Clearly, one has $\\varphi _{1,1}^{(c)}(1) = 0$ , $\\varphi _{1,1}^{(c)}(\\infty ) = \\infty $ , $\\varphi _{1,1}^{(c) \\prime }(1) = 0$ , $\\varphi _{1,1}^{(c) \\prime }(1-e^{c}) = -\\infty $ and $\\varphi _{1,1}^{(c) \\prime }(\\infty ) = \\infty $ .",
"The corresponding divergence $D_{\\varphi _{1,1}^{(c)}}(\\mathbb {Q},\\mathbb {P})$ has been recently used in Broniatowski et al.",
"[63] for the important task of testing mixtures of probability distributions; in fact, in order to get considerable comfort in testing mixture-type hypotheses against corresponding marginal-type alternatives, they employ choices $c>0$ since then $\\varphi _{1,1}^{(c)}(t)$ is finite especially for some range of negative values $t<0$ .",
"The latter feature is also valid for the divergence generator $\\varphi _{bw,\\beta ,\\widetilde{c}}$ in the next example (cf.",
"(REF ) below).",
"Example 41 For $\\beta \\in \\, ]0,1]$ , $\\widetilde{c} \\in \\, ]0,\\infty [$ and $]a_{F_{bw,\\beta ,\\widetilde{c}}},b_{F_{bw,\\beta ,\\widetilde{c}}}[ \\,= \\, ]1-\\frac{1}{\\beta },\\infty [$ we define $F_{bw,\\beta ,\\widetilde{c}}(t) &:=&{\\left\\lbrace \\begin{array}{ll}\\frac{\\widetilde{c}}{2\\beta } \\cdot \\Big (1-\\frac{1}{(\\beta \\cdot t + 1 - \\beta )^2}\\Big ), \\qquad \\textrm {if } \\ t \\in \\, ]1-\\frac{1}{\\beta },\\infty [, \\\\- \\infty , \\hspace{93.89418pt} \\textrm {if } \\ t \\in \\, ]-\\infty ,1-\\frac{1}{\\beta }].\\end{array}\\right.",
"}\\nonumber $ Clearly, $\\mathcal {R}(F_{bw,\\beta ,\\widetilde{c}})= \\big ]-\\infty ,\\frac{\\widetilde{c}}{2\\beta }\\big [$ and $0 \\in int(\\mathcal {R}(F_{bw,\\beta ,\\widetilde{c}}))$ .",
"Moreover, $F_{bw,\\beta ,\\widetilde{c}}(\\cdot )$ is strictly increasing and smooth on $]1-\\frac{1}{\\beta },\\infty [$ , and thus, $F_{bw,\\beta ,\\widetilde{c}} \\in {F}$ .",
"Since $F_{bw,\\beta ,\\widetilde{c}}(1)=0$ , let us choose the natural anchor point $c:=0$ , which leads to $]\\lambda _{-},\\lambda _{+}[ \\, = int(\\mathcal {R}(F_{bw,\\beta ,\\widetilde{c}}))= \\big ]-\\infty ,\\frac{\\widetilde{c}}{2\\beta }\\big [$ and $]t_{-}^{sc},t_{+}^{sc}[ \\, = \\,]a_{F_{bw,\\beta ,\\widetilde{c}}},b_{F_{bw,\\beta ,\\widetilde{c}}}[\\, = \\, ]1-\\frac{1}{\\beta },\\infty [$ .",
"By using $F_{bw,\\beta ,\\widetilde{c}}^{-1}(x) =\\frac{1}{\\beta } \\cdot \\Big \\lbrace \\frac{1}{\\sqrt{1-2\\beta \\cdot x/\\widetilde{c}}} + \\beta -1 \\Big \\rbrace $ for $x \\in int(\\mathcal {R}(F_{bw,\\beta ,\\widetilde{c}}))$ , we can derive from formula (REF ) (see also (REF )) for all $\\beta \\in \\, ]0,1]$ and $z \\in \\mathbb {R}$ $\\Lambda _{bw,\\beta ,\\widetilde{c}}(z) :=\\Lambda _{bw,\\beta ,\\widetilde{c}}^{(0)}(z) &=&{\\left\\lbrace \\begin{array}{ll}-(\\frac{1}{\\beta }-1) \\cdot z + \\frac{\\widetilde{c}}{\\beta ^{2}} \\cdot \\Big \\lbrace 1 - \\sqrt{1-\\frac{2\\beta }{\\widetilde{c}} \\cdot z} \\ \\Big \\rbrace ,\\qquad \\textrm {if } z \\in \\big ]-\\infty ,\\frac{\\widetilde{c}}{2\\beta } \\big ], \\\\\\infty , \\hspace{176.407pt} \\textrm {else} .\\end{array}\\right.",
"}$ Notice that $\\Lambda _{bw,\\beta ,\\widetilde{c}}(0) = 0$ .",
"Moreover, $\\Lambda _{bw,\\beta ,\\widetilde{c}}(-\\infty ) = \\infty $ , $\\Lambda _{bw,\\beta ,\\widetilde{c}}(\\frac{\\widetilde{c}}{2\\beta })= \\frac{\\widetilde{c} \\cdot (\\beta +1)}{2 \\beta ^{2}}$ , $\\Lambda _{bw,\\beta ,\\widetilde{c}}^{\\prime }(-\\infty ) = - \\frac{1-\\beta }{\\beta } <0$ and $\\Lambda _{bw,\\beta ,\\widetilde{c}}^{\\prime }(\\frac{\\widetilde{c}}{2\\beta }) = \\infty $ .",
"Furthermore, from formula (REF ) (see also ()) we can straightforwardly deduce for all $t \\in \\mathbb {R}$ $\\varphi _{bw,\\beta ,\\widetilde{c}}(t) := \\varphi _{bw,\\beta ,\\widetilde{c}}^{(0)}(t)\\hspace{-5.69046pt} &:=& \\hspace{-5.69046pt}{\\left\\lbrace \\begin{array}{ll}\\widetilde{c} \\cdot \\frac{(t-1)^{2}}{2(\\beta \\cdot t +1 -\\beta )}\\ \\in \\ [0,\\infty [,\\qquad \\quad \\textrm {if }t \\in \\, ]1-\\frac{1}{\\beta },\\infty [,\\\\\\infty , \\hspace{120.92421pt} \\textrm {if } \\ t \\in \\, ]-\\infty ,1-\\frac{1}{\\beta }] .\\end{array}\\right.",
"}$ Note that $1-\\frac{1}{\\beta } <0$ so that negative $t$ are allowed here.",
"For $t\\ge 0$ , $\\varphi _{bw,\\beta ,\\widetilde{c}}(t)$ is known as Rukhin's generator (cf.",
"[312], see e.g.",
"also Marhuenda et al.",
"[247], Pardo [282]).",
"Obviously, one has $\\varphi _{bw,\\beta ,\\widetilde{c}}(1) = 0$ , $\\varphi _{bw,\\beta ,\\widetilde{c}}^{\\prime }(1) = 0$ , $\\varphi _{bw,\\beta ,\\widetilde{c}}^{\\prime }(1-\\frac{1}{\\beta }) = -\\infty $ and $\\varphi _{bw,\\beta ,\\widetilde{c}}^{\\prime }(\\infty ) = \\frac{\\widetilde{c}}{2\\beta }$ .",
"From the generator $\\varphi _{bw,\\beta ,\\widetilde{c}}$ given in (REF ), we build the corresponding divergence (cf.",
"(REF )) $& &\\hspace{-45.52458pt}D_{\\varphi _{bw,\\beta ,\\widetilde{c}}}(\\mathbf {Q},\\mathbf {P})= \\widetilde{c} \\cdot \\sum _{k=1}^{K} p_{k} \\cdot \\frac{(\\frac{q_{k}}{p_{k}}-1)^{2}}{2(\\beta \\cdot \\frac{q_{k}}{p_{k}} +1 -\\beta )}\\nonumber \\\\& & \\hspace{-45.52458pt}= \\frac{\\widetilde{c}}{2} \\cdot \\sum \\limits _{k=1}^{K}\\frac{(q_{k}-p_{k})^{2}}{\\beta \\cdot q_{k} + (1 -\\beta )\\cdot p_{k}},\\hspace{42.67912pt}\\textrm {if \\mathbf {P} \\in \\mathbb {R}^{K} and \\mathbf {Q} \\in \\mathbb {R}^{K}with \\mathbf {Q} \\in \\, ] \\, \\mathbf {P} \\cdot (1-\\frac{1}{\\beta }),\\infty [ component-wise;}$ for the special subcase $\\widetilde{c}=1$ and $\\mathbf {Q} \\in \\mathbb {R}_{> 0}^{K}$ , $D_{\\varphi _{bw,\\beta ,1}}(\\mathbf {Q},\\mathbf {P})$ can be interpreted as — “non-probability version” of — the well-known blended weight chi-square divergence of Lindsay [221] (see e.g.",
"also Basu & Lindsay [35], Györfy & Vajda [148], Basu et al.",
"[36]).",
"The special case $\\widetilde{c}=1$ and $\\beta =\\frac{1}{2}$ for probability vectors, i.e.",
"$D_{\\varphi _{bw,1/2,1}}({Q},{P})$ , is equal to (a multiple of the matrix-vector-converted (cf.",
"Remark REF )) Sanghvi’s genetic difference measure [316] and equal to the double of the so-called (squared) Vincze-Le Cam distance (cf.",
"Vincze [384], Le Cam [212], see also e.g.",
"Topsoe [360] — who used the alternative naming triangular discrimination — and Vajda [373]); this divergence $D_{\\varphi _{bw,1/2,1}}({Q},{P})$ has been used e.g.",
"in Liu et al.",
"[227] for a machine learning context of detecting salient objects, where ${Q}$ and ${P}$ are appropriate histograms of RGB color.",
"Remark 42 (a) By straightforward calculations, one can show that $\\varphi _{bw,1,\\widetilde{c}}$ (i.e.",
"with the choice $\\beta =1$ ) is equal to the $\\widetilde{c}-fold$ power-divergence generator $\\varphi _{\\gamma ,\\widetilde{c}} = \\widetilde{c} \\cdot \\varphi _{\\gamma }$ (cf.",
"(REF )) with $\\gamma =-1$ ; the corresponding divergence $D_{\\varphi _{bw,1,\\widetilde{c}}}(\\mathbf {Q},\\mathbf {P})$ is thus equal to the power divergence $D_{\\widetilde{c} \\cdot \\varphi _{-1}}(\\mathbf {Q},\\mathbf {P})$ (cf.",
"(REF )) which is nothing but the — “non-probability version” — of Neyman's chi-square divergence.",
"(b) For the case $\\beta =0$ — which has been excluded in Example REF for technical brevity — the divergence generator $\\varphi _{bw,0,\\widetilde{c}}$ corresponds to $\\widetilde{c}-fold$ power-divergence generator $\\varphi _{\\gamma ,\\widetilde{c}}$ with $\\gamma =2$ ; the corresponding divergence $D_{\\varphi _{bw,0,\\widetilde{c}}}(\\mathbf {Q},\\mathbf {P})$ is thus equal to the power divergence $D_{\\widetilde{c} \\cdot \\varphi _{2}}(\\mathbf {Q},\\mathbf {P})$ (cf.",
"(REF )) which is nothing but the — “non-probability version” — of Pearson's (i.e.",
"the classical) chi-square divergence.",
"Example 43 Let us give an interesting generalization of the Kullback-Leibler case of Example REF (a).",
"For $\\widetilde{c} >0$ and $\\alpha \\in \\, ]-1,0[ \\ \\cup \\ ]0,\\infty [$ let us define $F_{gKL,\\alpha ,\\widetilde{c}}(t) &:=&{\\left\\lbrace \\begin{array}{ll}\\widetilde{c} \\cdot \\log \\left( \\frac{(1+\\alpha ) \\cdot t}{1+ \\alpha \\cdot t} \\right), \\qquad \\textrm {if \\lbrace \\alpha \\in \\, ]0,\\infty [ and t \\in \\, ]0,\\infty [ \\rbrace or \\lbrace \\alpha \\in \\, ]-1,0[ and t \\in \\, ]0,-\\frac{1}{\\alpha }[ \\rbrace ,} \\\\- \\infty , \\hspace{71.13188pt} \\textrm {if \\alpha \\in \\, ]-1,0[ \\ \\cup \\ ]0,\\infty [ andt \\in \\, ]-\\infty ,0],} \\\\\\infty , \\hspace{78.52945pt} \\textrm {if \\alpha \\in \\, ]-1,0[ and t \\in \\, [-\\frac{1}{\\alpha },\\infty [,}\\end{array}\\right.",
"}\\nonumber $ (notice that $\\lim _{\\alpha \\rightarrow 0_{+}} F_{gKL,\\alpha ,\\widetilde{c}}(t)= F_{1,\\widetilde{c}}(t)$ , cf.",
"Example REF (a)).",
"Clearly, $]a_{F_{gKL,\\alpha ,\\widetilde{c}}},b_{F_{gKL,\\alpha ,\\widetilde{c}}}[ \\, := \\, ]0,\\infty [$ for $\\alpha \\in \\, ]0,\\infty [$ and $]a_{F_{gKL,\\alpha ,\\widetilde{c}}},b_{F_{gKL,\\alpha ,\\widetilde{c}}}[ \\, := \\, ]0,-\\frac{1}{\\alpha }[$ for $\\alpha \\in \\, ]-1,0[$ .",
"Moreover, $\\mathcal {R}(F_{gKL,\\alpha ,\\widetilde{c}})=\\, ]-\\infty , \\widetilde{c} \\cdot \\log (1+ \\frac{1}{\\alpha })[$ for $\\alpha \\in \\, ]0,\\infty [$ and $\\mathcal {R}(F_{gKL,\\alpha ,\\widetilde{c}})= \\, ]-\\infty ,\\infty [$ for $\\alpha \\in \\, ]-1,0[$ .",
"Furthermore, $F_{gKL,\\alpha ,\\widetilde{c}}(\\cdot )$ is strictly increasing and smooth on the respective $]a_{F_{gKL,\\alpha ,\\widetilde{c}}},b_{F_{gKL,\\alpha ,\\widetilde{c}}}[$ , and thus, $F_{gKL,\\alpha ,\\widetilde{c}} \\in {F}$ .",
"Since $F_{gKL,\\alpha ,\\widetilde{c}}(1)=0$ , let us choose the natural anchor point $c:=0$ , which leads to $]\\lambda _{-},\\lambda _{+}[ \\, = int(\\mathcal {R}(F_{gKL,\\alpha ,\\widetilde{c}})) = \\,]-\\infty , \\widetilde{c} \\cdot \\log (1+ \\frac{1}{\\alpha })[$ and $]t_{-}^{sc},t_{+}^{sc}[ \\, = \\, ]0,\\infty [$ for the case $\\alpha \\in \\, ]0,\\infty [$ , respectively, to $]\\lambda _{-},\\lambda _{+}[ \\, = int(\\mathcal {R}(F_{gKL,\\alpha ,\\widetilde{c}})) = \\, ]-\\infty ,\\infty [$ and $]t_{-}^{sc},t_{+}^{sc}[ \\, = \\, ]0,-\\frac{1}{\\alpha }[$ for the case $\\alpha \\in \\, ]-1,0[$ .",
"By employing $F_{gKL,\\alpha ,\\widetilde{c}}^{-1}(x) =\\frac{1}{(1+\\alpha ) \\cdot e^{-x/\\widetilde{c}} - \\alpha }$ for $x \\in ]\\lambda _{-},\\lambda _{+}[$ , one can deduce from formula (REF ) (see also (REF )) $& & \\Lambda _{gKL,\\alpha ,\\widetilde{c}}(z) := \\Lambda _{gKL,\\alpha ,\\widetilde{c}}^{(0)}(z)\\nonumber \\\\& & :={\\left\\lbrace \\begin{array}{ll}\\int \\displaylimits _{0}^{z} F_{gKL,\\alpha ,\\widetilde{c}}^{-1}(u) \\, du= - \\frac{\\widetilde{c}}{\\alpha } \\cdot \\log ((1+\\alpha ) - \\alpha \\cdot e^{z/\\widetilde{c}}),\\qquad \\textrm {if \\alpha \\in \\, ]0,\\infty [ and z \\in \\, ]-\\infty , \\widetilde{c} \\cdot \\log (1+ \\frac{1}{\\alpha })[}, \\\\\\int \\displaylimits _{0}^{z} F_{gKL,\\alpha ,\\widetilde{c}}^{-1}(u) \\, du= - \\frac{\\widetilde{c}}{\\alpha } \\cdot \\log ((1+\\alpha ) - \\alpha \\cdot e^{z/\\widetilde{c}}),\\qquad \\textrm {if \\alpha \\in \\, ]-1,0[ and z \\in \\, ]-\\infty , \\infty [}, \\\\\\infty , \\hspace{217.6634pt}\\textrm {if \\alpha \\in \\, ]0,\\infty [ and z \\in \\, [\\widetilde{c} \\cdot \\log (1+ \\frac{1}{\\alpha }), \\infty [}, \\\\\\end{array}\\right.",
"}$ for which there holds $\\Lambda _{gKL,\\alpha ,\\widetilde{c}}(0) = 0$ and $\\Lambda _{gKL,\\alpha ,\\widetilde{c}}(-\\infty ) = - \\frac{\\widetilde{c}}{\\alpha } \\cdot \\log (1+\\alpha )$ for $\\alpha \\in \\, ]-1,0[ \\ \\cup \\ ]0,\\infty [$ , as well as $\\Lambda _{gKL,\\alpha ,\\widetilde{c}}(\\widetilde{c} \\cdot \\log (1+ \\frac{1}{\\alpha })) = \\infty $ for $\\alpha \\in \\, ]0,\\infty [$ and $\\Lambda _{gKL,\\alpha ,\\widetilde{c}}(\\infty ) = \\infty $ for $\\alpha \\in \\, ]-1,0[$ .",
"The corresponding derivative satisfies $\\Lambda _{gKL,\\alpha ,\\widetilde{c}}^{\\prime }(- \\infty ) = 0$ for $\\alpha \\in \\, ]-1,0[ \\ \\cup \\ ]0,\\infty [$ , as well as $\\Lambda _{gKL,\\alpha ,\\widetilde{c}}^{\\prime }(\\widetilde{c}\\cdot \\log (1+ \\frac{1}{\\alpha })) = \\infty $ for $\\alpha \\in \\, ]0,\\infty [$ and $\\Lambda _{gKL,\\alpha ,\\widetilde{c}}^{\\prime }(\\infty ) = - \\frac{1}{\\alpha }$ for $\\alpha \\in \\, ]-1,0[$ .",
"Furthermore, from formula (REF ) (see also ()) one can derive $& & \\hspace{-19.91684pt} \\varphi _{gKL,\\alpha ,\\widetilde{c}}(t) := \\varphi _{gKL,\\alpha ,\\widetilde{c}}^{(0)}(t)\\nonumber \\\\& & \\hspace{-19.91684pt} :={\\left\\lbrace \\begin{array}{ll}\\widetilde{c} \\cdot \\left[ \\, t \\cdot \\log t + (t+\\frac{1}{\\alpha }) \\cdot \\log \\Big ( \\frac{1+\\alpha }{1+\\alpha \\cdot t} \\Big ) \\, \\right]\\ \\in \\ [0,\\infty [,\\quad \\textrm {if \\lbrace \\alpha \\in \\, ]0,\\infty [ and t \\in \\, ]0,\\infty [ \\rbrace or \\lbrace \\alpha \\in \\, ]-1,0[ and t \\in \\, ]0,-\\frac{1}{\\alpha }[ \\rbrace ,}\\\\\\frac{\\widetilde{c}}{\\alpha } \\cdot \\log (1+\\alpha )\\ \\in \\ ]0,\\infty [, \\hspace{106.69783pt}\\textrm {if \\alpha \\in \\, ]-1,0[ \\ \\cup \\ ]0,\\infty [ andt =0,} \\\\\\infty , \\hspace{197.74655pt}\\textrm {if \\alpha \\in \\, ]-1,0[ \\ \\cup \\ ]0,\\infty [ andt \\in \\, ]-\\infty ,0[,} \\\\\\infty , \\hspace{197.74655pt}\\textrm {if \\alpha \\in \\, ]-1,0[ andt \\in \\, [-\\frac{1}{\\alpha },\\infty [;}\\end{array}\\right.",
"}$ the first line in (REF ) can be proved by $& & \\hspace{-19.91684pt}\\varphi _{gKL,\\alpha ,\\widetilde{c}}(t) := \\varphi _{gKL,\\alpha ,\\widetilde{c}}^{(0)}(t) :=t \\cdot F_{gKL,\\alpha ,\\widetilde{c}}\\left(t\\right)-\\int \\displaylimits _{0}^{F_{gKL,\\alpha ,\\widetilde{c}}\\left(t\\right)}F_{gKL,\\alpha ,\\widetilde{c}}^{-1}(u) \\, du\\nonumber \\\\& & \\hspace{-19.91684pt}= \\widetilde{c} \\cdot t \\cdot \\log \\left( \\frac{(1+\\alpha ) \\cdot t}{1+ \\alpha \\cdot t} \\right)+ \\frac{\\widetilde{c}}{\\alpha } \\cdot \\log \\left( (1 +\\alpha ) -\\alpha \\cdot \\exp \\left[ \\log \\left( \\frac{(1+ \\alpha ) \\cdot t}{1+ \\alpha \\cdot t} \\right) \\right] \\right)\\nonumber \\\\& & \\hspace{-19.91684pt}= \\widetilde{c} \\cdot \\left[ \\, t \\cdot \\log t+ t \\cdot \\log \\Big ( \\frac{1+\\alpha }{1+ \\alpha \\cdot t} \\Big )+ \\frac{1}{\\alpha } \\cdot \\log \\Big ( \\frac{1+\\alpha }{1+ \\alpha \\cdot t} \\Big ) \\, \\right].$ Obviously, one has $\\varphi _{gKL,\\alpha ,\\widetilde{c}}(1) = 0$ , $\\varphi _{gKL,\\alpha ,\\widetilde{c}}^{\\prime }(1) = 0$ , $\\varphi _{gKL,\\alpha ,\\widetilde{c}}^{\\prime }(0) = -\\infty $ for $\\alpha \\in \\, ]-1,0[ \\ \\cup \\ ]0,\\infty [$ .",
"Moreover, for $\\alpha \\in \\, ]0,\\infty [$ there holds $\\varphi _{gKL,\\alpha ,\\widetilde{c}}(\\infty ) = \\infty $ , and $\\varphi _{gKL,\\alpha ,\\widetilde{c}}^{\\prime }(\\infty ) = \\widetilde{c} \\cdot \\log (1+\\frac{1}{\\alpha })$ , whereas for $\\alpha \\in \\, ]-1,0[$ we obtain $\\varphi _{gKL,\\alpha ,\\widetilde{c}}(-\\frac{1}{\\alpha }) = \\infty $ , and $\\varphi _{gKL,\\alpha ,\\widetilde{c}}^{\\prime }(-\\frac{1}{\\alpha }) = \\infty $ .",
"From the generator $\\varphi _{gKL,\\alpha ,\\widetilde{c}}$ given in (REF ), we build the corresponding divergence (cf.",
"(REF )) $& &\\hspace{-42.67912pt}D_{\\varphi _{gKL,\\alpha ,\\widetilde{c}}}(\\mathbf {Q},\\mathbf {P})= \\widetilde{c} \\cdot \\Big \\lbrace \\sum \\limits _{k=1}^{K}q_{k} \\cdot \\log \\Big (\\frac{q_{k}}{(1-\\frac{1}{1+\\alpha }) \\cdot q_{k} + \\frac{1}{1+\\alpha } \\cdot p_{k}} \\Big )+ \\frac{1}{\\alpha } \\cdot \\sum \\limits _{k=1}^{K} p_{k} \\cdot \\log \\Big (\\frac{p_{k}}{(1-\\frac{1}{1+\\alpha }) \\cdot q_{k} + \\frac{1}{1+\\alpha } \\cdot p_{k}} \\Big ) \\Big \\rbrace ,\\\\& & \\textrm {if \\lbrace \\alpha \\in \\, ]0,\\infty [,\\mathbf {P} \\in \\mathbb {R}_{> 0}^{K} and\\mathbf {Q} \\in \\mathbb {R}_{\\ge 0}^{K} \\rbrace or \\lbrace \\alpha \\in \\, ]-1,0[,\\mathbf {P} \\in \\mathbb {R}_{> 0}^{K} and\\mathbf {Q} \\in \\mathbb {R}_{\\ge 0}^{K}with \\mathbf {Q} \\le - \\frac{1}{\\alpha } \\cdot \\mathbf {P} \\rbrace .}",
"\\nonumber $ Notice that the symmetry $D_{\\varphi _{gKL,\\alpha ,\\widetilde{c}}}(\\mathbf {Q},\\mathbf {P})= D_{\\varphi _{gKL,\\alpha ,\\widetilde{c}}}(\\mathbf {P},\\mathbf {Q})$ generally holds only if $\\mathbf {P}, \\mathbf {Q} \\in \\mathbb {R}_{> 0}^{K}$ and $\\alpha =1$ ; indeed, this special case leads to $\\varphi _{snKL,\\widetilde{c}}(t):= \\varphi _{gKL,1,\\widetilde{c}}(t)\\hspace{-5.69046pt} &:=& \\hspace{-5.69046pt}{\\left\\lbrace \\begin{array}{ll}\\widetilde{c} \\cdot \\left[ \\, t \\cdot \\log t + (t+1) \\cdot \\log \\Big ( \\frac{2}{t+1} \\Big ) \\, \\right]\\ \\in \\ [0,\\infty [,\\qquad \\quad \\textrm {if }t \\in \\, ]0,\\infty [,\\\\\\widetilde{c} \\cdot \\log 2, \\hspace{187.78836pt} \\textrm {if } \\ t = 0, \\\\\\infty , \\hspace{209.12791pt} \\textrm {if } \\ t \\in \\, ]-\\infty ,0[ ,\\end{array}\\right.",
"}$ and $D_{\\varphi _{snKL,\\widetilde{c}}}(\\mathbf {Q},\\mathbf {P}):= D_{\\varphi _{gKL,1,\\widetilde{c}}}(\\mathbf {Q},\\mathbf {P})= \\widetilde{c} \\cdot \\Big \\lbrace \\sum \\limits _{k=1}^{K} q_{k} \\cdot \\log \\Big (\\frac{2 q_{k}}{q_{k} + p_{k}} \\Big )+ \\sum \\limits _{k=1}^{K} p_{k} \\cdot \\log \\Big (\\frac{2 p_{k}}{q_{k} + p_{k}} \\Big ) \\Big \\rbrace ,\\qquad \\mathbf {P} \\in \\mathbb {R}_{> 0}^{K}, \\mathbf {Q} \\in \\mathbb {R}_{\\ge 0}^{K}.$ For the special subcase that $\\widetilde{c} =1$ and that $\\mathbf {P} = {P}$ , $\\mathbf {Q} = {Q}$ are probability vectors, the divergence (REF ) can be rewritten as sum of two Kullback-Leibler divergences (cf.",
"(REF )) $D_{\\varphi _{snKL,1}}({Q},{P})= D_{\\varphi _{1}}({Q},({Q}+{P})/2)+ D_{\\varphi _{1}}({P},({Q}+{P})/2),\\qquad {P} \\in \\mathbb {S}_{> 0}^{K}, {Q} \\in \\mathbb {S}_{\\ge 0}^{K},$ which is the well-known (cf.",
"Burbea & Rao [68], Lin [219], Pardo & Vajda [284], Topsoe [360], Endres & Schindelin [120], Vajda [373], Sason [317]) Jensen-Shannon divergence (being also called symmetrized and normalized Kullback-Leibler divergence, symmetrized and normalized relative entropy, capacitory discrimination); this is equal to the $(2\\log 2)-$ fold of a special (namely, equally-weighted two-population) case of the Sibson information radius of order 1 (cf.",
"[334]) which has also been addressed e.g.",
"by Rao [301] for genetic cluster analysis.",
"By the way, for $\\alpha >0$ the divergence $D_{\\varphi _{gKL,\\alpha ,\\widetilde{c}}}({Q},{P})$ can also be interpreted as a multiple of a special non-equally-weighted Sibson information radius of order 1.",
"In a context of comparison of — not necessarily connected — networks where ${Q}$ , ${P}$ are probability vectors derived from matrices (cf.",
"Remark REF ) which are transforms of corresponding graph invariants (e.g.",
"network portraits), the (matrix-equivalent of the) Jensen-Shannon divergence $D_{\\varphi _{snKL,1}}({Q},{P})$ is also called the network portrait divergence, cf.",
"Bagrow and Bollt [28].",
"There is a vast literature on recent applications of the Jensen-Shannon divergence, for instance it appears exemplarily in Kvitsiani et al.",
"[208] for finding connections between the circuit-level function of different interneuron types in regulating the flow of information and the behavioural functions served by the cortical circuits, in Xu et al.",
"(2014) for browsing and exploration of video sequences, in Jenkinson et al.",
"[168] for the fundamental understanding of the epigenome that leads to a powerful approach for studying its role in disease and aging, in Martin et al.",
"[250] for the implementation of an evolutionary-based global localization filter for mobile robots, in Suo et al.",
"[354] for the revelation of critical regulators of cell identity in mice, in Abante et al.",
"[2] for the detection of biologically significant differences in DNA methylation between alleles associated with local changes in genetic sequences — for a better understanding of the mechanism of complex human diseases, in Afek et al.",
"[5] for revealing mechanisms by which mismatches can recruit transcription factors for modulating replication and repair activities in cells, in Alaiz-Rodriguez & Parnell [10] for the quantification of stability in feature selection and ranking algorithms, in Biau et al.",
"[53] for generative adversarial networks (GANs) in artificial intelligence and machine learning, in Carre et al.",
"[74] for the standardization of brain magnetic resonance (MR) images, in Chakraborty et al.",
"[75] for hierarchical clustering in foreign exchange FOREX markets (e.g.",
"in periods of major international crises), in Chong et al.",
"[87] as part of a web-based platform for comprehensive analysis of microbiome data outputs, in Cui et al.",
"[101] for modelling latent friend recommendation in online social media, in Gholami & Hodtani [134] for refinements of safety-and-security-targeted location verification systems in wireless communication networks (e.g in Intelligent Transportation Systems (ITSs) and vehicular technology), in Guo & Yuan [146] for accurate abnormality classification in semi-supervised Wireless Capsule Endoscopy (WCE) for digestive system cancer diagnosis, in Jiang et al.",
"[169] for the training of deep neural discriminative and generative networks used for designing and evaluating photonic devices, in Kartal et al.",
"[186] for uncovering the relationship between some genomic features and cell type-specific methylome diversity, in Laszlovszky et al.",
"[210] for investigating mechanisms of basal forebrain neurons which modulate synaptic plasticity,cortical processing, brain states and oscillations, in Lawson et al.",
"[211] for the improved understanding of some genetic circuits that allow cancer cells to evade destruction by the host immune system, in Li et al.",
"[215] for the search of causes of the progressive neurodevelopmental disorder Rett syndrome, in Machado et al.",
"[239] for discovering relations between distinct RNA viruses (including SARS-CoV-2), in Mohammadi et al.",
"[261] for the identification of cell states and their underlying topology, in Mohanty et al.",
"[262] for the design of implantable nanophotonic (i.e.",
"chip-scale optical circuit type) silicon probes for sub-millisecond deep-brain optical stimulation — e.g.",
"for the purpose of gaining a deeper understanding of the neural code, in Perera et al.",
"[290] for the quantification of the level of rationality in supply chain networks, in Pierri et al.",
"[294] for the study of growth of malicious/misleading information in some social media diffusion networks, in Rabadan et al.",
"[299] for the identification of gene mutations that lead to the genesis and progression of tumors, in Reiter et al.",
"[306] for quantifying metastatic phylogenetic diversity, in Van de Sande et al.",
"[378] as part of a computational toolbox for single-cell gene regulatory network analysis, in Skinnider et al.",
"[337] for the prediction of the chemical structures of genomically encoded antibiotics — in order to find means against the looming global crisis of antibiotic resistance, in Tuo et al.",
"[367] for the detection of high-order single nucleotide polymorphism (SNP) interactions, in Uttam et al.",
"[370] for predicting the risk of colorectal cancer recurrence and inferring associated tumor microenvironment networks, in Zhang et al.",
"[421] for incipient fault (namely, crack) detection, in Zhi et al.",
"[425] for the strengthening of information-centric networks against interest flooding attack (IFAs), in Acera Mateos et al.",
"[3] for deep-learning classification of SARS-CoV-2 and co-infecting RNA viruses, in Avsec et al.",
"[24] for uncovering the motifs and syntax of cis-regulatory sequences in genomics data, in Barennes et al.",
"[32] for comparing the accuracy of current T cell receptor sequencing methods employed for the understanding of adaptive immune responses, in Chen et al.",
"[79] for clustering high-dimensional microbial data from RNA sequencing, in Chen et al.",
"[84] for investigating key aspects of effective vocal social communication, in Koldobskiy et al.",
"[193] for investigations of genetic and epigenetic drivers of paediatric acute lymphoblastic leukaemia, in McGinnis et al.",
"[258] for evaluating RNA sequencing of pooled blood cell samples, in Mühlroth & Grottke [268] for the detection of emerging trends and technologies through artificial intelligence techniques, in Necci et al.",
"[272] for the assessment of protein intrinsic disorder predictions, in Okada et al.",
"[277] for the identification of genetic factors that cause individual differences in whole lymphocyte profiles and their changes after vaccination, and in Zhang et al.",
"[422] for the learning of functional magnetic resonance imaging (fMRI) time-series in a brain disease diagnosis context.",
"Remark 44 Let us transform $\\varphi _{gSH,\\alpha }(t) := \\frac{1-t}{\\alpha } \\cdot \\log (1+\\alpha )- \\varphi _{gKL,\\alpha ,1}(t)= - t \\cdot \\log t+ \\frac{1}{\\alpha } \\cdot (1 + \\alpha \\cdot t) \\cdot \\log (1 + \\alpha \\cdot t)- \\frac{1}{\\alpha } \\cdot (1+\\alpha ) \\cdot t \\cdot \\log (1+\\alpha )$ (for $t \\in [0,1]$ ).",
"The function $\\varphi _{gSH,\\alpha }(\\cdot )$ is strictly concave on $[0,1]$ with $\\varphi _{gSH,\\alpha }(0)= \\varphi _{gSH,\\alpha }(1)=0$ .",
"Hence, for probability vectors ${Q} =(q_{k})_{k=1,\\ldots ,K}$ , the $\\varphi -$ entropy $\\sum _{k=1}^{K}\\varphi _{gSH,\\alpha }(q_{k})$ is Kapur’s [183] generalization of the Shannon entropy (which corresponds to $\\alpha =0$ in the limit) whose maximization has been connected with generalizations of the Bose-Einstein statistics and the Fermi-Dirac statistics e.g.",
"in Kapur & Kesavan [185].",
"Example 45 Let us fix any $z_{1},z_{2} \\in \\mathbb {R}$ , $p \\in ]0,1[$ which satisfy $z_{1} < 1 < z_{2}$ and $z_{1} \\cdot p + z_{2} \\cdot (1-p) =1$ (and thus $p= \\frac{z_{2} -1}{z_{2} - z_{1}}$ ).",
"On $]a_{F_{twop}},b_{F_{twop}}[ := ]z_{1},z_{2}[$ we define $F_{twop}(t) &:=& \\frac{1}{z_{2}-z_{1}} \\cdot \\log \\left( \\frac{(t-z_{1})\\cdot p}{(z_{2}-t)\\cdot (1- p)} \\right)\\nonumber \\\\&=&\\frac{1}{z_{2}-z_{1}} \\cdot \\log \\left( \\frac{(t-z_{1})\\cdot (z_{2} -1)}{(z_{2}-t)\\cdot (1-z_{1})} \\right) \\, ,\\qquad t \\in \\, ]z_{1},z_{2}[ ,\\nonumber $ where for the last equality we have used the above constraint (in order to obtain a two-parameter representation).",
"Straightforwardly, we have $\\mathcal {R}(F_{twop})= ]-\\infty ,\\infty [$ .",
"Moreover, $F_{twop}(\\cdot )$ is strictly increasing and smooth on $]0,\\infty [$ , and thus, $F_{twop} \\in {F}$ .",
"Since $F_{twop}(1)=0$ , let us choose the natural anchor point $c:=0$ , which leads to $]\\lambda _{-},\\lambda _{+}[= int(\\mathcal {R}(F_{snKL,\\widetilde{c}})) = ]-\\infty , \\infty [$ and $]t_{-}^{sc},t_{+}^{sc}[ = ]z_{1},z_{2}[$ .",
"By using $F_{twop}^{-1}(x) = \\frac{p \\cdot z_{1} + (1- p) \\cdot z_{2} \\cdot e^{(z_{2}-z_{1}) \\cdot x}}{p + (1- p) \\cdot e^{(z_{2}-z_{1}) \\cdot x}} \\, , \\qquad x \\in ]-\\infty , \\infty [,$ we derive from formula (REF ) (see also (REF )) $& & \\hspace{-19.91684pt} \\Lambda _{twop}(z) := \\Lambda _{twop}^{(0)}(z) :=\\int \\displaylimits _{0}^{z} F_{twop}^{-1}(u) du= \\log \\Big ( p \\cdot e^{z_{1} \\cdot z} + (1- p) \\cdot e^{z_{2} \\cdot z} \\Big ),\\qquad z \\in ]-\\infty , \\infty [ ,$ which has the properties $\\Lambda _{twop}(0) = 0$ , $\\Lambda _{twop}(-\\infty ) =\\infty \\cdot {1}_{]-\\infty ,0[}(z_{1})- \\infty \\cdot {1}_{]0,\\infty [}(z_{1})+ \\log p \\cdot {1}_{\\lbrace 0\\rbrace }(z_{1})$ , $\\Lambda _{twop}(\\infty ) = \\infty $ , $\\Lambda _{twop}^{\\prime }(- \\infty ) = z_{1}$ and $\\Lambda _{twop}^{\\prime }(\\infty ) = z_{2}$ .",
"Furthermore, from formula (REF ) (see also ()) we deduce $\\varphi _{twop}(t) := \\varphi _{twop}^{(0)}(t)\\hspace{-5.69046pt} &:=& \\hspace{-5.69046pt}{\\left\\lbrace \\begin{array}{ll}\\frac{t-z_{1}}{z_{2}-z_{1}} \\cdot \\log \\left( \\frac{(t-z_{1})\\cdot (z_{2}-1)}{(z_{2}-t)\\cdot (1-z_{1})} \\right)- \\log \\left( \\frac{z_{2}-1}{z_{2}-t} \\right)\\ \\in \\ [0,\\infty [,\\qquad \\quad \\textrm {if }t \\in \\, ]0,\\infty [,\\\\\\log \\left( \\frac{z_{2} - z_{1}}{z_{2} -1} \\right), \\hspace{219.08612pt} \\textrm {if } \\ t = z_{1}, \\\\\\log \\left( \\frac{z_{2} - z_{1}}{1 - z_{1}} \\right), \\hspace{219.08612pt} \\textrm {if } \\ t = z_{2}, \\\\\\infty , \\hspace{233.3125pt} \\textrm {if } \\ t \\in \\, ]-\\infty ,z_{1}[ \\, \\cup \\, ]z_{2}, \\infty [ ;\\end{array}\\right.",
"}$ the first line in (REF ) can be proved by $& & \\hspace{-19.91684pt}\\varphi _{twop}(t) := \\varphi _{twop}^{(0)}(t) :=t \\cdot F_{twop}\\left(t\\right)-\\int \\displaylimits _{0}^{F_{twop}\\left(t\\right)} F_{twop}^{-1}(u) du\\nonumber \\\\& & \\hspace{-19.91684pt}= \\frac{t}{z_{2}-z_{1}} \\cdot \\log \\left( \\frac{(t-z_{1})\\cdot p}{(z_{2}-t)\\cdot (1- p)} \\right)\\nonumber \\\\& & \\hspace{-19.91684pt}- \\log \\left( p \\cdot \\left( \\frac{(t-z_{1})\\cdot p}{(z_{2}-t)\\cdot (1- p)} \\right)^{\\frac{z_{1}}{z_{2}-z_{1}}}+ (1- p) \\cdot \\left( \\frac{(t-z_{1})\\cdot p}{(z_{2}-t)\\cdot (1- p)} \\right)^{\\frac{z_{2}}{z_{2}-z_{1}}} \\right)\\nonumber \\\\& & \\hspace{-19.91684pt}= \\frac{t-z_{1}}{z_{2}-z_{1}} \\cdot \\log \\left( \\frac{(t-z_{1})\\cdot p}{(z_{2}-t)\\cdot (1- p)} \\right)- \\log \\left( \\frac{(z_{2}-z_{1})\\cdot p}{z_{2}-t} \\right)\\\\& & \\hspace{-19.91684pt}= \\frac{t-z_{1}}{z_{2}-z_{1}} \\cdot \\log \\left( \\frac{(t-z_{1})\\cdot (z_{2}-1)}{(z_{2}-t)\\cdot (1-z_{1})} \\right)- \\log \\left( \\frac{z_{2}-1}{z_{2}-t} \\right) , \\qquad t \\in \\, ]z_{1}, z_{2} [ \\, ,\\nonumber $ where for the last equality we have used the above constraint (to obtain a two-parameter representation).",
"Straightforwardly, one has $\\varphi _{twop}(1) = 0$ , $\\varphi _{twop}^{\\prime }(1) = 0$ , $\\varphi _{twop}^{\\prime }(z_{1}) = -\\infty $ and $\\varphi _{twop}^{\\prime }(z_{2}) = \\infty $ .",
"From the generator $\\varphi _{twop}$ given in (REF ), we build the corresponding divergence (cf.",
"(REF )) $D_{\\varphi _{twop}}(\\mathbf {Q},\\mathbf {P})=\\sum \\limits _{k=1}^{K} \\frac{q_{k} - z_{1} \\cdot p_{k}}{z_{2} - z_{1}} \\cdot \\log \\Big (\\frac{(z_{2}-1) \\cdot (q_{k} - z_{1} \\cdot p_{k})}{(1-z_{1}) \\cdot (z_{2} \\cdot p_{k} - q_{k})} \\Big )- \\sum \\limits _{k=1}^{K} p_{k} \\cdot \\log \\Big (\\frac{(z_{2}-1) \\cdot p_{k}}{z_{2} \\cdot p_{k} - q_{k}} \\Big ) .$ It is known that some types of robustness properties of minimum-divergence estimators are connected with the boundedness of the derivative $\\varphi ^{\\prime }$ of the divergence generator $\\varphi $ ; this property is satisfied for the next Example REF (and its $W-$ concerning continuation in Example REF ), which leads to the new classes of divergences (REF ), (REF ) and (REF ): Example 46 (a) For any parameter-quadrupel $\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c} \\in \\, ]0,\\infty [$ with $\\beta _{1} < \\beta _{2}$ , we choose $]a_{F},b_{F}[\\ \\, := \\ ]a_{F_{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}},b_{F_{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}}[ \\ \\, := \\ \\Big ]1 - \\alpha \\cdot \\frac{(\\beta _{1} - \\beta _{2})^{2} + \\beta _{1}^{2} + \\beta _{1} \\cdot \\beta _{2}}{2\\beta _{1}\\cdot \\beta _{2}\\cdot (\\beta _{2} - \\beta _{1} )}, \\,\\infty \\Big [ \\ \\ni 1\\nonumber $ and define with $\\breve{\\theta } := 1 + \\alpha \\cdot \\Big (\\frac{1}{\\beta _{2}}- \\frac{1}{\\beta _{1}} \\Big ) < 1$ $\\hspace{-17.07182pt}F_{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}(t)&:=& {\\left\\lbrace \\begin{array}{ll}\\widetilde{c} \\cdot \\frac{\\beta _{1}-\\beta _{2}}{2}+ \\frac{\\widetilde{c}}{\\frac{1-t}{\\alpha } + \\frac{1}{\\beta _{2}} - \\frac{1}{\\beta _{1}}}\\cdot \\Big (1 - \\frac{1}{2} \\cdot \\sqrt{4 + \\big (\\frac{1-t}{\\alpha } + \\frac{1}{\\beta _{2}} - \\frac{1}{\\beta _{1}}\\big )^{2} \\cdot (\\beta _{1}+\\beta _{2})^{2}}\\, \\Big ),\\quad \\textrm {if } \\ t \\in \\, ]a_{F},b_{F}[ \\backslash \\lbrace \\breve{\\theta }\\rbrace , \\\\\\widetilde{c} \\cdot \\frac{\\beta _{1}-\\beta _{2}}{2}, \\hspace{277.41437pt}\\textrm {if } \\ t= \\breve{\\theta } \\in \\, ]a_{F},b_{F}[, \\\\- \\widetilde{c} \\cdot \\beta _{1}, \\hspace{284.52756pt} \\textrm {if } \\ t=a_{F}, \\\\- \\infty , \\hspace{295.90848pt} \\textrm {if } \\ t \\in \\, ]-\\infty ,a_{F}[.\\end{array}\\right.",
"}$ Notice that $\\breve{\\theta } \\in \\, ]a_{F},b_{F}[$ if and only if $\\beta _{1} \\in \\,]\\frac{\\beta _{2}}{3},\\beta _{2}[$ ; if (say) the latter holds, then one has the continuity $\\lim _{t \\rightarrow \\breve{\\theta }}F_{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}(t) = \\widetilde{c} \\cdot \\frac{\\beta _{1}-\\beta _{2}}{2}$ .",
"For $\\beta _{1} \\le \\frac{\\beta _{2}}{3}$ one gets $]a_{F},b_{F}[ \\backslash \\lbrace \\breve{\\theta } \\rbrace = \\, ]a_{F},b_{F}[$ .",
"Returning to the general case, one can see in a straightforward way that $F_{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}(\\cdot )$ is strictly increasing and that $\\mathcal {R}(F_{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}})=[-\\widetilde{c}\\cdot \\beta _{1},\\widetilde{c}\\cdot \\beta _{1} [$ .",
"Furthermore, $F_{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}(\\cdot )$ is smooth on $]a_{F},b_{F}[$ , and thus $F_{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}} \\in {F}$ .",
"Since $F_{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}(1)=0$ , let us choose the natural anchor point $c:=0$ , which leads to $]\\lambda _{-},\\lambda _{+}[ \\,= int(\\mathcal {R}(F_{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}})= \\, ]-\\widetilde{c}\\cdot \\beta _{1},\\widetilde{c}\\cdot \\beta _{1} [$ and $]t_{-}^{sc},t_{+}^{sc}[ = \\, ]a_{F},b_{F}[$ .",
"Moreover, it is straightforward to see that the corresponding inverse is $F_{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}^{-1}(x) =1 + \\alpha \\cdot \\Big (\\frac{1}{\\beta _{2}}- \\frac{1}{\\beta _{1}} \\Big ) - \\alpha \\cdot \\frac{\\frac{1}{\\beta _{2}} - \\frac{1}{\\beta _{1}} -\\frac{2 x}{\\widetilde{c} \\cdot \\beta _{1} \\cdot \\beta _{2} }}{1+ \\frac{x}{\\widetilde{c}} \\cdot \\Big (\\frac{1}{\\beta _{2}} - \\frac{1}{\\beta _{1}} \\Big )- \\frac{x^{2}}{\\widetilde{c}^{2} \\cdot \\beta _{1} \\cdot \\beta _{2} }},\\qquad x \\in int(\\mathcal {R}(F_{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}})) ;$ from this, we can derive from formula (REF ) (see also (REF )) for all $z \\in \\mathbb {R}$ $\\hspace{-19.91684pt}\\Lambda _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}(z) :=\\Lambda _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}^{(0)}(z) &=&{\\left\\lbrace \\begin{array}{ll}\\breve{\\theta } \\cdot z - \\widetilde{c} \\cdot \\alpha \\cdot \\log \\Big (1+ \\frac{z}{\\widetilde{c}} \\cdot \\Big (\\frac{1}{\\beta _{2}} - \\frac{1}{\\beta _{1}} \\Big )- \\frac{z^{2}}{\\widetilde{c}^{2} \\cdot \\beta _{1} \\cdot \\beta _{2} }\\Big ),\\quad \\textrm {if } \\ z \\in \\, ]-\\widetilde{c}\\cdot \\beta _{1},\\widetilde{c}\\cdot \\beta _{1} [,\\\\- \\widetilde{c} \\cdot \\breve{\\theta } \\cdot \\beta _{1} -\\widetilde{c} \\cdot \\alpha \\cdot \\log \\Big (2 - 2 \\frac{\\beta _{1}}{\\beta _{2}}\\Big ),\\hspace{73.97733pt} \\textrm {if } \\ z = -\\widetilde{c}\\cdot \\beta _{1},\\\\\\infty , \\hspace{202.01474pt} \\textrm {else} .\\end{array}\\right.",
"}$ Notice that $\\Lambda _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}(0) = 0$ and $\\lim _{z \\rightarrow \\widetilde{c}\\cdot \\beta _{1}}\\Lambda _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}(z) = \\infty $ .",
"Moreover, $\\Lambda _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}^{\\prime }( -\\widetilde{c}\\cdot \\beta _{1})= a_{F}$ and $\\Lambda _{ -\\widetilde{c}\\cdot \\beta _{1}}^{\\prime }(\\widetilde{c}\\cdot \\beta _{1}) = \\infty = b_{F}$ (which have to be interpreted as limits, as usual).",
"To proceed, from formula (REF ) (see also ()) we can deduce for all $t \\in \\mathbb {R}$ $\\hspace{-5.69046pt}\\varphi _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}(t):= \\varphi _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}^{(0)}(t)\\hspace{-8.5359pt} &=& \\hspace{-8.5359pt}{\\left\\lbrace \\begin{array}{ll}\\widetilde{c} \\cdot \\alpha \\cdot \\Big \\lbrace \\frac{\\sqrt{4 + (\\beta _{1} + \\beta _{2})^{2}\\cdot (\\frac{1-t}{\\alpha } + \\frac{1}{\\beta _{2}} - \\frac{1}{\\beta _{1}})^2}\\, - \\, (\\frac{1-t}{\\alpha } + \\frac{1}{\\beta _{2}} - \\frac{1}{\\beta _{1}}) \\cdot (\\beta _{1} - \\beta _{2}) \\, - \\, 2}{2} \\\\+ \\log \\frac{\\sqrt{4 + (\\beta _{1} + \\beta _{2})^{2}\\cdot (\\frac{1-t}{\\alpha } + \\frac{1}{\\beta _{2}} \\, - \\, \\frac{1}{\\beta _{1}})^2} \\, - \\, 2}{\\beta _{1} \\beta _{2} \\cdot (\\frac{1-t}{\\alpha } + \\frac{1}{\\beta _{2}} \\, - \\, \\frac{1}{\\beta _{1}})^{2}} \\Big \\rbrace \\ \\in [0,\\infty [,\\hspace{71.13188pt} \\textrm {if }t \\in \\, ]a_{F},\\infty [,\\\\\\widetilde{c} \\cdot \\alpha \\cdot \\Big \\lbrace \\frac{3 \\beta _{1} - \\beta _{2}}{2 (\\beta _{2} - \\beta _{1})}+ \\log \\frac{2(\\beta _{2} - \\beta _{1})}{\\beta _{2}} \\Big \\rbrace - \\widetilde{c} \\cdot \\beta _{1} \\cdot (t - a_{F}) \\ \\in \\, ]0,\\infty [,\\hspace{19.91684pt} \\textrm {if } \\ t \\in ]-\\infty , a_{F}].\\end{array}\\right.",
"}$ The first subcase in (REF ) can be proved by $& & \\hspace{-19.91684pt}\\varphi _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}(t) :=\\varphi _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}^{(0)}(t) :=t \\cdot F_{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}\\left(t\\right)\\ - \\int \\displaylimits _{0}^{F_{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}\\left(t\\right)}F_{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}^{-1}(u) \\, du\\nonumber \\\\& & \\hspace{-19.91684pt}= (t - \\breve{\\theta }) \\cdot F_{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}\\left(t\\right)+ \\widetilde{c} \\cdot \\alpha \\cdot \\log \\Big (1+ \\frac{F_{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}\\left(t\\right)}{\\widetilde{c}} \\cdot \\Big (\\frac{1}{\\beta _{2}} - \\frac{1}{\\beta _{1}} \\Big )- \\frac{(F_{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}\\left(t\\right))^{2}}{\\widetilde{c}^{2} \\cdot \\beta _{1} \\cdot \\beta _{2} }\\Big )\\nonumber \\\\& & \\hspace{-19.91684pt}= \\widetilde{c} \\cdot \\alpha \\cdot \\frac{\\sqrt{4 + (\\beta _{1} + \\beta _{2})^{2}\\cdot (\\frac{1-t}{\\alpha } + \\frac{1}{\\beta _{2}} - \\frac{1}{\\beta _{1}})^2}\\, - \\, (\\frac{1-t}{\\alpha } + \\frac{1}{\\beta _{2}} - \\frac{1}{\\beta _{1}}) \\cdot (\\beta _{1} - \\beta _{2}) \\, - \\, 2}{2}\\nonumber \\\\& & \\hspace{-19.91684pt}\\ \\ \\ + \\ \\widetilde{c} \\cdot \\alpha \\cdot \\log \\bigg (1+ \\Big [\\frac{\\beta _{1}-\\beta _{2}}{2}+ \\frac{1}{\\frac{1-t}{\\alpha } + \\frac{1}{\\beta _{2}} - \\frac{1}{\\beta _{1}}}\\cdot \\Big (1 - \\frac{1}{2} \\cdot \\sqrt{4 + \\big (\\frac{1-t}{\\alpha } + \\frac{1}{\\beta _{2}} - \\frac{1}{\\beta _{1}}\\big )^{2} \\cdot (\\beta _{1}+\\beta _{2})^{2}}\\, \\Big ) \\Big ]\\cdot \\frac{\\beta _{1} - \\beta _{2}}{\\beta _{1} \\cdot \\beta _{2}}\\nonumber \\\\& & \\hspace{-19.91684pt}\\ \\ \\ - \\ \\Big [\\frac{\\beta _{1}-\\beta _{2}}{2}+ \\frac{1}{\\frac{1-t}{\\alpha } + \\frac{1}{\\beta _{2}} - \\frac{1}{\\beta _{1}}}\\cdot \\Big (1 - \\frac{1}{2} \\cdot \\sqrt{4 + \\big (\\frac{1-t}{\\alpha } + \\frac{1}{\\beta _{2}} - \\frac{1}{\\beta _{1}}\\big )^{2} \\cdot (\\beta _{1}+\\beta _{2})^{2}}\\, \\Big ) \\Big ]^{2} \\cdot \\frac{1}{\\beta _{1} \\cdot \\beta _{2} }\\bigg )\\nonumber $ and some straightforward calculations.",
"The second line in (REF ) follows by computing $\\varphi _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}(a_{F})=\\widetilde{c} \\cdot \\alpha \\cdot \\Big \\lbrace \\frac{3 \\beta _{1} - \\beta _{2}}{2 (\\beta _{2} - \\beta _{1})}+ \\log \\frac{2(\\beta _{2} - \\beta _{1})}{\\beta _{2}} \\Big \\rbrace $ .",
"Notice that $\\varphi _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}(1) = 0$ , $\\varphi _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}^{\\prime }(1) = 0$ , $\\varphi _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}(-\\infty ) = \\infty $ and $\\varphi _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}(\\infty ) = \\infty $ .",
"Moreover, $\\varphi _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}^{\\prime }(-\\infty ) =\\varphi _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}^{\\prime }(a_{F}) =-\\widetilde{c} \\cdot \\beta _{1}$ and $\\varphi _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}^{\\prime }(\\infty ) =\\widetilde{c} \\cdot \\beta _{1}$ .",
"From the generator $\\varphi _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}$ given in (REF ), we construct the corresponding divergence (cf.",
"(REF )) $& &D_{\\varphi _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}}(\\mathbf {Q},\\mathbf {P})= \\sum \\limits _{k=1}^{K} p_{k} \\cdot \\varphi _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}\\Big (\\frac{q_{k}}{p_{k}}\\Big )\\nonumber \\\\& &= \\sum \\limits _{k=1}^{K}p_{k} \\cdot \\bigg [{1}_{]a_{F},\\infty [}(\\frac{q_{k}}{p_{k}}) \\cdot \\widetilde{c} \\cdot \\alpha \\cdot \\Big \\lbrace \\frac{\\sqrt{4 + (\\beta _{1} + \\beta _{2})^{2}\\cdot (\\frac{1-\\frac{q_{k}}{p_{k}}}{\\alpha } + \\frac{1}{\\beta _{2}} - \\frac{1}{\\beta _{1}})^2}\\, - \\, (\\frac{1-\\frac{q_{k}}{p_{k}}}{\\alpha } + \\frac{1}{\\beta _{2}} - \\frac{1}{\\beta _{1}}) \\cdot (\\beta _{1} - \\beta _{2}) \\, - \\, 2}{2}\\nonumber \\\\& &+ \\log \\frac{\\sqrt{4 + (\\beta _{1} + \\beta _{2})^{2}\\cdot (\\frac{1-\\frac{q_{k}}{p_{k}}}{\\alpha } + \\frac{1}{\\beta _{2}} \\, - \\, \\frac{1}{\\beta _{1}})^2} \\, - \\, 2}{\\beta _{1} \\beta _{2} \\cdot (\\frac{1-\\frac{q_{k}}{p_{k}}}{\\alpha } + \\frac{1}{\\beta _{2}} \\, - \\, \\frac{1}{\\beta _{1}})^{2}} \\Big \\rbrace \\nonumber \\\\& &+ \\ {1}_{]-\\infty , a_{F}]}(\\frac{q_{k}}{p_{k}}) \\cdot \\widetilde{c} \\cdot \\Big \\lbrace \\alpha \\cdot \\Big \\lbrace \\frac{3 \\beta _{1} - \\beta _{2}}{2 (\\beta _{2} - \\beta _{1})}+ \\log \\frac{2(\\beta _{2} - \\beta _{1})}{\\beta _{2}} \\Big \\rbrace - \\beta _{1} \\cdot (\\frac{q_{k}}{p_{k}} - a_{F}) \\Big \\rbrace \\bigg ],\\qquad \\mathbf {P} \\in \\mathbb {R}_{\\ge 0}^{K}, \\mathbf {Q} \\in \\mathbb {R}^{K}.$ Notice that we can particularly include the case where $p_{k}=0$ in combination with $q_{k} \\ne 0$ , since $\\lim _{t \\rightarrow 0_{+}} t \\cdot \\varphi _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}(\\frac{1}{t})= \\widetilde{c} \\cdot \\beta _{1}$ and $\\lim _{t \\rightarrow 0_{-}} t \\cdot \\varphi _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}(\\frac{1}{t})= - \\widetilde{c} \\cdot \\beta _{1}$ are both finite, and hence $p_{k} \\cdot \\varphi _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}(\\frac{q_{k}}{p_{k}})= q_{k} \\cdot \\frac{p_{k}}{q_{k}} \\cdot \\varphi _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}(\\frac{q_{k}}{p_{k}}) $ stays finite as $p_{k}$ tends to zero.",
"(b) For any parameter-quadrupel $\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c} \\in \\, ]0,\\infty [$ with $\\beta _{1} > \\beta _{2}$ , one can proceed analogously to (a).",
"Let us start by choosing $]a_{F},b_{F}[\\ \\, := \\ ]a_{F_{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}},b_{F_{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}}[ \\ \\, := \\ \\Big ]-\\infty , \\, 1 + \\alpha \\cdot \\frac{(\\beta _{1} - \\beta _{2})^{2} + \\beta _{1} \\cdot \\beta _{2} + \\beta _{2}^{2}}{2\\beta _{1}\\cdot \\beta _{2}\\cdot (\\beta _{1} - \\beta _{2} )}, \\,\\Big [ \\ \\ni 1\\nonumber $ and defining with the same $\\breve{\\theta } := 1 + \\alpha \\cdot \\Big (\\frac{1}{\\beta _{2}}- \\frac{1}{\\beta _{1}} \\Big ) > 1$ $\\hspace{-17.07182pt}F_{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}(t)&:=& {\\left\\lbrace \\begin{array}{ll}\\widetilde{c} \\cdot \\frac{\\beta _{1}-\\beta _{2}}{2}+ \\frac{\\widetilde{c}}{\\frac{1-t}{\\alpha } + \\frac{1}{\\beta _{2}} - \\frac{1}{\\beta _{1}}}\\cdot \\Big (1 - \\frac{1}{2} \\cdot \\sqrt{4 + \\big (\\frac{1-t}{\\alpha } + \\frac{1}{\\beta _{2}} - \\frac{1}{\\beta _{1}}\\big )^{2} \\cdot (\\beta _{1}+\\beta _{2})^{2}}\\, \\Big ),\\quad \\textrm {if } \\ t \\in \\, ]a_{F},b_{F}[ \\backslash \\lbrace \\breve{\\theta }\\rbrace , \\\\\\widetilde{c} \\cdot \\frac{\\beta _{1}-\\beta _{2}}{2}, \\hspace{277.41437pt}\\textrm {if } \\ t= \\breve{\\theta } \\in \\, ]a_{F},b_{F}[, \\\\\\widetilde{c} \\cdot \\beta _{2}, \\hspace{293.06346pt} \\textrm {if } \\ t=b_{F}, \\\\\\infty , \\hspace{304.4444pt} \\textrm {if } \\ t \\in \\, ]b_{F}, \\infty [.\\end{array}\\right.",
"}$ Clearly, $\\breve{\\theta } \\in \\, ]a_{F},b_{F}[$ if and only if $\\beta _{1} \\in \\,]\\beta _{2}, 3\\beta _{2}[$ ; if (say) the latter holds, then one gets the continuity $\\lim _{t \\rightarrow \\breve{\\theta }}F_{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}(t) = \\widetilde{c} \\cdot \\frac{\\beta _{1}-\\beta _{2}}{2}$ .",
"For $\\beta _{1} \\le 3 \\beta _{2}$ there holds $]a_{F},b_{F}[ \\backslash \\lbrace \\breve{\\theta } \\rbrace = \\, ]a_{F},b_{F}[$ .",
"Returning to the general case, one can show comfortably that $F_{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}(\\cdot )$ is strictly increasing and that $\\mathcal {R}(F_{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}})=]-\\widetilde{c}\\cdot \\beta _{2},\\widetilde{c}\\cdot \\beta _{2} ]$ .",
"Moreover, $F_{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}(\\cdot )$ is smooth on $]a_{F},b_{F}[$ , and hence $F_{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}} \\in {F}$ .",
"In face of the validity of $F_{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}(1)=0$ , let us choose the natural anchor point $c:=0$ , which amounts to $]\\lambda _{-},\\lambda _{+}[ \\,= int(\\mathcal {R}(F_{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}})= \\, ]-\\widetilde{c}\\cdot \\beta _{2},\\widetilde{c}\\cdot \\beta _{2} [$ and $]t_{-}^{sc},t_{+}^{sc}[ = \\, ]a_{F},b_{F}[$ .",
"Since the first line in (REF ) coincides formally with that of (REF ) (with different $]a_{F},b_{F}[$ ), the corresponding inverse is formally the same as (REF ) (with different $]a_{F},b_{F}[$ ), and hence $\\hspace{-19.91684pt}\\Lambda _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}(z) :=\\Lambda _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}^{(0)}(z) &=&{\\left\\lbrace \\begin{array}{ll}\\breve{\\theta } \\cdot z - \\widetilde{c} \\cdot \\alpha \\cdot \\log \\Big (1+ \\frac{z}{\\widetilde{c}} \\cdot \\Big (\\frac{1}{\\beta _{2}} - \\frac{1}{\\beta _{1}} \\Big )- \\frac{z^{2}}{\\widetilde{c}^{2} \\cdot \\beta _{1} \\cdot \\beta _{2} }\\Big ),\\quad \\textrm {if } \\ z \\in \\, ]-\\widetilde{c}\\cdot \\beta _{2},\\widetilde{c}\\cdot \\beta _{2} [,\\\\\\widetilde{c} \\cdot \\breve{\\theta } \\cdot \\beta _{2} -\\widetilde{c} \\cdot \\alpha \\cdot \\log \\Big (2 - 2 \\frac{\\beta _{2}}{\\beta _{1}}\\Big ),\\hspace{73.97733pt} \\textrm {if } \\ z = \\widetilde{c}\\cdot \\beta _{2},\\\\\\infty , \\hspace{202.01474pt} \\textrm {else} .\\end{array}\\right.",
"}$ Notice that $\\Lambda _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}(0) = 0$ and $\\lim _{z \\rightarrow - \\widetilde{c}\\cdot \\beta _{2}}\\Lambda _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}(z) = -\\infty $ .",
"Furthermore, $\\Lambda _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}^{\\prime }( -\\widetilde{c}\\cdot \\beta _{2})= - \\infty = a_{F}$ and $\\Lambda _{ -\\widetilde{c}\\cdot \\beta _{1}}^{\\prime }(\\widetilde{c}\\cdot \\beta _{2})= b_{F}$ .",
"To proceed, from formula (REF ) (see also ()) we can derive — analogously to (REF ) — for all $t \\in \\mathbb {R}$ $\\hspace{-5.69046pt}\\varphi _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}(t):= \\varphi _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}^{(0)}(t)\\hspace{-8.5359pt} &=& \\hspace{-8.5359pt}{\\left\\lbrace \\begin{array}{ll}\\widetilde{c} \\cdot \\alpha \\cdot \\Big \\lbrace \\frac{\\sqrt{4 + (\\beta _{1} + \\beta _{2})^{2}\\cdot (\\frac{1-t}{\\alpha } + \\frac{1}{\\beta _{2}} - \\frac{1}{\\beta _{1}})^2}\\, - \\, (\\frac{1-t}{\\alpha } + \\frac{1}{\\beta _{2}} - \\frac{1}{\\beta _{1}}) \\cdot (\\beta _{1} - \\beta _{2}) \\, - \\, 2}{2} \\\\+ \\log \\frac{\\sqrt{4 + (\\beta _{1} + \\beta _{2})^{2}\\cdot (\\frac{1-t}{\\alpha } + \\frac{1}{\\beta _{2}} \\, - \\, \\frac{1}{\\beta _{1}})^2} \\, - \\, 2}{\\beta _{1} \\beta _{2} \\cdot (\\frac{1-t}{\\alpha } + \\frac{1}{\\beta _{2}} \\, - \\, \\frac{1}{\\beta _{1}})^{2}} \\Big \\rbrace \\ \\in [0,\\infty [,\\hspace{71.13188pt} \\textrm {if }t \\in \\, ]-\\infty , b_{F}[,\\\\\\widetilde{c} \\cdot \\alpha \\cdot \\Big \\lbrace \\frac{3 \\beta _{2} - \\beta _{1}}{2 (\\beta _{1} - \\beta _{2})}+ \\log \\frac{2(\\beta _{1} - \\beta _{2})}{\\beta _{1}} \\Big \\rbrace + \\widetilde{c} \\cdot \\beta _{2} \\cdot (t - b_{F}) \\ \\in \\, ]0,\\infty [,\\hspace{19.91684pt} \\textrm {if } \\ t \\in [b_{F}, \\infty [,\\end{array}\\right.",
"}$ where the last line in (REF ) follows by calculating $\\varphi _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}(b_{F})=\\widetilde{c} \\cdot \\alpha \\cdot \\Big \\lbrace \\frac{3 \\beta _{2} - \\beta _{1}}{2 (\\beta _{1} - \\beta _{2})}+ \\log \\frac{2(\\beta _{1} - \\beta _{2})}{\\beta _{1}} \\Big \\rbrace $ .",
"Notice that $\\varphi _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}(1) = 0$ , $\\varphi _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}^{\\prime }(1) = 0$ , $\\varphi _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}(-\\infty ) = \\infty $ and $\\varphi _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}(\\infty ) = \\infty $ .",
"Furthermore, $\\varphi _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}^{\\prime }(-\\infty ) =-\\widetilde{c} \\cdot \\beta _{2}$ and $\\varphi _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}^{\\prime }(\\infty ) =\\varphi _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}^{\\prime }(b_{F}) =\\widetilde{c} \\cdot \\beta _{2}$ .",
"From the generator $\\varphi _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}$ given in (REF ), we construct the corresponding divergence (cf.",
"(REF )) $& &D_{\\varphi _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}}(\\mathbf {Q},\\mathbf {P})= \\sum \\limits _{k=1}^{K} p_{k} \\cdot \\varphi _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}\\Big (\\frac{q_{k}}{p_{k}}\\Big )\\nonumber \\\\& &= \\sum \\limits _{k=1}^{K}p_{k} \\cdot \\bigg [{1}_{]-\\infty , b_{F}[}(\\frac{q_{k}}{p_{k}}) \\cdot \\widetilde{c} \\cdot \\alpha \\cdot \\Big \\lbrace \\frac{\\sqrt{4 + (\\beta _{1} + \\beta _{2})^{2}\\cdot (\\frac{1-\\frac{q_{k}}{p_{k}}}{\\alpha } + \\frac{1}{\\beta _{2}} - \\frac{1}{\\beta _{1}})^2}\\, - \\, (\\frac{1-\\frac{q_{k}}{p_{k}}}{\\alpha } + \\frac{1}{\\beta _{2}} - \\frac{1}{\\beta _{1}}) \\cdot (\\beta _{1} - \\beta _{2}) \\, - \\, 2}{2}\\nonumber \\\\& &+ \\log \\frac{\\sqrt{4 + (\\beta _{1} + \\beta _{2})^{2}\\cdot (\\frac{1-\\frac{q_{k}}{p_{k}}}{\\alpha } + \\frac{1}{\\beta _{2}} \\, - \\, \\frac{1}{\\beta _{1}})^2} \\, - \\, 2}{\\beta _{1} \\beta _{2} \\cdot (\\frac{1-\\frac{q_{k}}{p_{k}}}{\\alpha } + \\frac{1}{\\beta _{2}} \\, - \\, \\frac{1}{\\beta _{1}})^{2}} \\Big \\rbrace \\nonumber \\\\& &+ \\ {1}_{[b_{F}, \\infty [}(\\frac{q_{k}}{p_{k}}) \\cdot \\widetilde{c} \\cdot \\Big \\lbrace \\alpha \\cdot \\Big \\lbrace \\frac{3 \\beta _{2} - \\beta _{1}}{2 (\\beta _{1} - \\beta _{2})}+ \\log \\frac{2(\\beta _{1} - \\beta _{2})}{\\beta _{1}} \\Big \\rbrace + \\beta _{2} \\cdot (\\frac{q_{k}}{p_{k}} - b_{F}) \\Big \\rbrace \\bigg ],\\qquad \\mathbf {P} \\in \\mathbb {R}_{\\ge 0}^{K}, \\mathbf {Q} \\in \\mathbb {R}^{K}.$ As above, we can particularly include the case where $p_{k}=0$ in combination with $q_{k} \\ne 0$ , since $\\lim _{t \\rightarrow 0_{+}} t \\cdot \\varphi _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}(\\frac{1}{t})= \\widetilde{c} \\cdot \\beta _{2}$ and $\\lim _{t \\rightarrow 0_{-}} t \\cdot \\varphi _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}(\\frac{1}{t})= - \\widetilde{c} \\cdot \\beta _{2}$ are both finite.",
"(c) The analysis for the case $\\beta _{1} = \\beta _{2} =: \\beta $ can be obtained by taking $\\lim _{\\beta _1 \\rightarrow \\beta _{2}}$ in (a) respectively (b).",
"Alternatively, one can start afresh.",
"Due to its importance and its particularities, we nevertheless state the corresponding results explicitly.",
"To begin with, for any parameter-triple $\\alpha ,\\beta ,\\widetilde{c} \\in \\, ]0,\\infty [$ we choose $]a_{F},b_{F}[\\ \\, := \\ ]a_{F_{\\alpha ,\\beta ,\\widetilde{c}}},b_{F_{\\alpha ,\\beta ,\\widetilde{c}}}[ \\ \\, := \\ ]-\\infty , \\infty \\, [\\nonumber $ and define with $\\breve{\\theta } := 1$ $\\hspace{-17.07182pt}F_{\\alpha ,\\beta ,\\widetilde{c}}(t)&:=& {\\left\\lbrace \\begin{array}{ll}\\frac{\\widetilde{c} \\cdot \\alpha }{1-t}\\cdot \\Big (1 - \\sqrt{1 + \\big (\\frac{1-t}{\\alpha } \\big )^{2} \\cdot \\beta ^{2}}\\, \\Big ),\\qquad \\textrm {if } \\ t \\in \\, ]a_{F},b_{F}[ \\backslash \\lbrace \\breve{\\theta }\\rbrace , \\\\0, \\hspace{145.10922pt}\\textrm {if } \\ t= \\breve{\\theta }.\\end{array}\\right.",
"}$ Clearly, one has the continuity $\\lim _{t \\rightarrow \\breve{\\theta }}F_{\\alpha ,\\beta ,\\widetilde{c}}(t) = 0$ .",
"Moreover, one can see in a straightforward way that $F_{\\alpha ,\\beta ,\\widetilde{c}}(\\cdot )$ is strictly increasing and that $\\mathcal {R}(F_{\\alpha ,\\beta ,\\widetilde{c}})=]-\\widetilde{c}\\cdot \\beta ,\\widetilde{c}\\cdot \\beta [$ .",
"Furthermore, $F_{\\alpha ,\\beta ,\\widetilde{c}}(\\cdot )$ is smooth on $]a_{F},b_{F}[$ , and thus $F_{\\alpha ,\\beta ,\\widetilde{c}} \\in {F}$ .",
"Since $F_{\\alpha ,\\beta ,\\widetilde{c}}(1)=0$ , let us choose the natural anchor point $c:=0$ , which leads to the choice $]\\lambda _{-},\\lambda _{+}[ \\,= int(\\mathcal {R}(F_{\\alpha ,\\beta ,\\widetilde{c}})= \\, ]-\\widetilde{c}\\cdot \\beta ,\\widetilde{c}\\cdot \\beta [$ and $]t_{-}^{sc},t_{+}^{sc}[ = \\, ]a_{F},b_{F}[ = \\, ]-\\infty ,\\infty [$ .",
"The inverse in (REF ) collapses to $F_{\\alpha ,\\beta ,\\widetilde{c}}^{-1}(x) =1 + \\alpha \\cdot \\frac{\\frac{2 x}{\\widetilde{c} \\cdot \\beta ^{2} }}{1 - \\frac{x^{2}}{\\widetilde{c}^{2} \\cdot \\beta ^{2} }},\\qquad x \\in int(\\mathcal {R}(F_{\\alpha ,\\beta ,\\widetilde{c}})) ;$ from this, we can derive from formula (REF ) (see also (REF )) for all $z \\in \\mathbb {R}$ $\\hspace{-19.91684pt}\\Lambda _{\\alpha ,\\beta ,\\widetilde{c}}(z) :=\\Lambda _{\\alpha ,\\beta ,\\widetilde{c}}^{(0)}(z) &=&{\\left\\lbrace \\begin{array}{ll}\\breve{\\theta } \\cdot z - \\widetilde{c} \\cdot \\alpha \\cdot \\log \\Big (1 - \\frac{z^{2}}{\\widetilde{c}^{2} \\cdot \\beta ^{2} }\\Big ),\\qquad \\textrm {if } \\ z \\in \\, ]-\\widetilde{c}\\cdot \\beta ,\\widetilde{c}\\cdot \\beta [,\\\\\\infty , \\hspace{130.88284pt} \\textrm {else} .\\end{array}\\right.",
"}$ Notice that $\\Lambda _{\\alpha ,\\beta ,\\widetilde{c}}(0) = 0$ , $\\lim _{z \\rightarrow - \\widetilde{c}\\cdot \\beta }\\Lambda _{\\alpha ,\\beta ,\\widetilde{c}}(z) = -\\infty = $ , and $\\lim _{z \\rightarrow \\widetilde{c}\\cdot \\beta }\\Lambda _{\\alpha ,\\beta ,\\widetilde{c}}(z) = \\infty $ .",
"Furthermore, $\\lim _{z \\rightarrow - \\widetilde{c}\\cdot \\beta }\\Lambda _{\\alpha ,\\beta ,\\widetilde{c}}^{\\prime }(z) = -\\infty = a_{F}$ , and $\\lim _{z \\rightarrow \\widetilde{c}\\cdot \\beta }\\Lambda _{\\alpha ,\\beta ,\\widetilde{c}}^{\\prime }(z) = \\infty = b_{F}$ .",
"To proceed, the formula (REF ) (respectively, (REF )) collapses to $\\varphi _{\\alpha ,\\beta ,\\widetilde{c}}(t):=\\varphi _{\\alpha ,\\beta ,\\widetilde{c}}^{(0)}(t)= \\widetilde{c} \\cdot \\alpha \\cdot \\Big \\lbrace \\sqrt{1 + \\beta ^{2}\\cdot \\Big (\\frac{1-t}{\\alpha }\\Big )^2} \\, - \\, 1+ \\log \\frac{2 \\cdot \\Big (\\sqrt{1 + \\beta ^{2}\\cdot \\Big (\\frac{1-t}{\\alpha } \\Big )^2} \\, - \\, 1\\Big )}{\\beta ^{2} \\cdot \\Big (\\frac{1-t}{\\alpha }\\Big )^{2}} \\Big \\rbrace \\ \\in [0,\\infty [, \\ \\ t \\in \\, ]-\\infty , \\infty [ \\, = \\, ]a_{F}, b_{F}[.$ Notice that $\\varphi _{\\alpha ,\\beta ,\\widetilde{c}}(1) = 0$ , $\\varphi _{\\alpha ,\\beta }^{\\prime }(1) = 0$ , $\\varphi _{\\alpha ,\\beta ,\\widetilde{c}}(-\\infty ) = \\infty $ and $\\varphi _{\\alpha ,\\beta ,\\widetilde{c}}(\\infty ) = \\infty $ .",
"Moreover, $\\varphi _{\\alpha ,\\beta ,\\widetilde{c}}^{\\prime }(-\\infty ) =\\varphi _{\\alpha ,\\beta ,\\widetilde{c}}^{\\prime }(a_{F}) =-\\widetilde{c} \\cdot \\beta $ and $\\varphi _{\\alpha ,\\beta ,\\widetilde{c}}^{\\prime }(\\infty ) =\\varphi _{\\alpha ,\\beta ,\\widetilde{c}}^{\\prime }(b_{F}) =\\widetilde{c} \\cdot \\beta $ .",
"From the generator $\\varphi _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}$ given in (REF ), we construct the corresponding divergence (cf.",
"(REF )) $& & \\hspace{-19.91684pt}D_{\\varphi _{\\alpha ,\\beta ,\\widetilde{c}}}(\\mathbf {Q},\\mathbf {P})= \\sum \\limits _{k=1}^{K} p_{k} \\cdot \\varphi _{\\alpha ,\\beta ,\\widetilde{c}}\\Big (\\frac{q_{k}}{p_{k}}\\Big )\\nonumber \\\\& & \\hspace{-19.91684pt}= \\widetilde{c} \\cdot \\alpha \\cdot \\sum \\limits _{k=1}^{K}p_{k} \\cdot \\Big \\lbrace \\sqrt{1 + \\beta ^{2}\\cdot \\Big (\\frac{1-\\frac{q_{k}}{p_{k}}}{\\alpha }\\Big )^2} \\, - \\, 1+ \\log \\frac{2 \\cdot \\Big (\\sqrt{1 + \\beta ^{2}\\cdot \\Big (\\frac{1-\\frac{q_{k}}{p_{k}}}{\\alpha } \\Big )^2} \\, - \\, 1\\Big )}{\\beta ^{2} \\cdot \\Big (\\frac{1-\\frac{q_{k}}{p_{k}}}{\\alpha }\\Big )^{2}} \\Big \\rbrace ,\\qquad \\mathbf {P} \\in \\mathbb {R}_{\\ge 0}^{K}, \\mathbf {Q} \\in \\mathbb {R}^{K}.$ As above, we can particularly include the case where $p_{k}=0$ in combination with $q_{k} \\ne 0$ , since $\\lim _{t \\rightarrow 0_{+}} t \\cdot \\varphi _{\\alpha ,\\beta ,\\widetilde{c}}(\\frac{1}{t})= \\widetilde{c} \\cdot \\beta $ and $\\lim _{t \\rightarrow 0_{-}} t \\cdot \\varphi _{\\alpha ,\\beta ,\\widetilde{c}}(\\frac{1}{t})= - \\widetilde{c} \\cdot \\beta $ are both finite.",
"This ends the current Example REF .",
"As a side effect in the above-mentioned Example REF , for fixed $\\beta _{2}, \\alpha ,\\widetilde{c}$ notice the interesting behaviour (e.g.",
"with respect to $int(dom(F)) = ]a_{F},b_{F}[$ and the range of $\\varphi ^{\\prime }$ ) as $\\beta _{1}$ moves from $]0,\\beta _{2}[$ to $\\beta _{2}$ and further to $]\\beta _{2}, \\infty [$ .",
"Remark 47 The characterization of the probability distribution $$ in (REF ) which may result from Theorem REF — as seen through the above examples — considerably improves other approaches which make use of their identification through the concept of power variance functions of Natural Exponential Families, as developed by Tweedie [369], Morris [267], Letac & Mora [214], and others.",
"This approach has been used in Broniatowski [58] in a similar perspective as developed here, but can not be extended outside the range of power divergences, in contrast with the Examples REF , REF , REF and REF which can only be handled as a consequence of Theorem REF .",
"To continue with our general procedure, suppose now that for a divergence generator $\\varphi $ of interest we have concretely/explicitly found (e.g.",
"by direct calculations or via our $F-$ connection in Theorem REF , see also Remark REF ) its Fenchel-Legendre transform $\\Lambda = \\varphi ^{*}$ ; for this “candidate”, in order to achieve the desired representability (REF ) it remains to verify that $\\exp (\\Lambda (z)) = \\int _{\\mathbb {R}}e^{z \\cdot y} \\, d\\mathbb {} (y),\\qquad z \\in \\mathbb {R},$ for some probability distribution/measure $\\mathbb {} $ on the real line (the light-tailedness in the sense of finiteness on some open interval containing zero, will be typically guaranteed automatically by the assumptions on $\\varphi $ ); of course, this is equivalent to “the existence ” of a random variable $W$ whose moment generating function is equal to $\\exp (\\Lambda )$ (and thus, its cumulant generating function (log moment generating function) is $\\Lambda $ ), i.e.",
"$\\exp (\\Lambda (z)) = E_{\\mathbb {\\Pi }}[\\exp (z \\cdot W)]\\qquad z \\in \\mathbb {R},$ with $\\mathbb {\\Pi }[W \\in \\cdot \\, ] = \\mathbb {}[ \\, \\cdot \\,]$ ); recall that from this, we need to simulate a sequence $(W_{i})_{i\\in \\mathbb {N}}$ of i.i.d.",
"copies of $W$ which are the crucial building ingredients of $\\xi _{n}^{\\mathbf {W}}$ in Theorem REF , respectively, of $\\xi _{n,\\mathbf {X}}^{w\\mathbf {W}}$ in Theorem REF .",
"For the above-mentioned Examples REF to REF , we can give explicit solutions to the representabilities (REF ) respectively (REF ); this is achieved in the following Examples REF to REF (notice that the corresponding supports of $\\mathbb {}$ are explicitly mentioned in the summarizing Table 1 above): Example 48 for the power-divergence context of Example REF we obtain: (a) Case $\\gamma =0$ , $\\widetilde{c} >0$ : $\\Lambda _{0,\\widetilde{c}}(z)= - \\widetilde{c} \\cdot \\log \\left( 1 - \\frac{z}{\\widetilde{c}} \\right)$ (cf.",
"(REF )) is the cumulant generating function of the Gamma distribution $\\mathbb {} = GAM(\\widetilde{c},\\widetilde{c})$ with rate parameter (inverse scale parameter) $\\widetilde{c}$ and shape parameter $\\widetilde{c}$ ; hence, $\\varphi _{0,\\widetilde{c}} \\in \\Upsilon (]0,\\infty [)$ .",
"Prominent special case $\\widetilde{c} =1$ : $\\mathbb {} = GAM(1,1) = EXP(1)$ is the exponential distribution with mean 1.",
"Type: $\\mathbb {}$ is an infinitely divisible (cf.",
"Proposition REF ) continuous distribution with density $f(y) := \\frac{\\widetilde{c}^{\\widetilde{c}} \\cdot y^{\\widetilde{c}-1} \\cdot e^{-\\widetilde{c} \\cdot y} }{\\Gamma (\\widetilde{c})} \\cdot {1}_{]0,\\infty [}(y)$ ($y \\in \\mathbb {R}$ ).",
"Behaviour at zero: $\\mathbb {}[ \\, ]0,\\infty [ \\, ] = \\mathbb {\\Pi }[W>0]=1$ .",
"Corresponding generator: $\\varphi _{0,\\widetilde{c}} = \\widetilde{c} \\cdot \\varphi _{0}$ (cf.",
"(REF ), (REF )) of the $\\widetilde{c}-$ fold of the reversed Kullback-Leibler divergence (reversed relative entropy) given in the second line of (REF ).",
"Sums: for i.i.d.",
"copies $(W_{i})_{i \\in \\mathbb {N}}$ of $W$ , the probability distribution of $\\breve{W} := \\sum _{i\\in I_{k}^{(n)}} W_{i}$ (cf.",
"Remark REF (ii)) is $GAM(\\widetilde{c},\\widetilde{c} \\cdot card(I_{k}^{(n)}))$ .",
"(b) Case $\\gamma \\in \\, ]0,1[$ , $\\widetilde{c} >0$ : $\\Lambda _{\\gamma ,\\widetilde{c}}^{(0)}(z)= \\frac{\\widetilde{c}}{\\gamma } \\cdot \\Big \\lbrace \\left( \\frac{\\gamma -1}{\\widetilde{c}} \\cdot z + 1 \\right)^{\\frac{\\gamma }{\\gamma -1}} -1 \\Big \\rbrace $ (cf.",
"(REF )) is the cumulant generating function of the Compound-Poisson-Gamma distribution $\\mathbb {} = C(POI(\\theta ),GAM(\\alpha ,\\beta ))$ with $\\theta = \\frac{\\widetilde{c}}{\\gamma } > 0$ , rate parameter (inverse scale parameter) $\\alpha = \\frac{\\widetilde{c}}{1-\\gamma } >0$ , and shape parameter $\\beta = \\frac{\\gamma }{1-\\gamma } >0$ .",
"In other words, $W$ has the comfortably simulable form $W = \\sum _{i=1}^{N} \\widetilde{W}_{i}$ with the usual convention $\\sum _{i=1}^{0} \\widetilde{W}_{i} := 0$ for some i.i.d.",
"sequence $( \\widetilde{W}_{i})_{i\\in \\mathbb {N}}$ of Gamma $GAM(\\alpha ,\\beta )$ distributed random variables (with parameter-pair $(\\alpha ,\\beta )$ ) and some independent $POI(\\theta )-$ distributed random variable $N$ .",
"Hence, $\\varphi _{\\gamma ,\\widetilde{c}} \\in \\Upsilon (]0,\\infty [)$ .",
"Type: $\\mathbb {}$ is an infinitely divisible distribution (cf.",
"Proposition REF ), mixture of a one-point distribution at zero and a continuous distribution on $[0,\\infty [$ , with $\\mathbb {}[\\lbrace 0\\rbrace ] = \\mathbb {\\Pi }[W = 0]= e^{-\\theta }$ and $\\mathbb {}[B] = \\mathbb {\\Pi }[W \\in B] = \\int _{B} f_{\\widetilde{c},\\gamma }(u) \\, du$ for every (measurable) subset of $]0,\\infty [$ having density $& & \\hspace{-22.76228pt}f_{C(POI(\\theta ),GAM(\\alpha ,\\beta ))}(y): = \\frac{\\exp \\left(- \\alpha \\cdot y - \\theta \\right)}{y}\\cdot \\sum _{k=1}^{\\infty } \\frac{\\theta ^{k} \\cdot (\\alpha y)^{k\\beta }}{k!",
"\\cdot \\Gamma (k\\beta )}\\cdot {1}_{]0,\\infty [}(y)\\\\& & \\hspace{-22.76228pt}= \\frac{1}{y} \\cdot \\exp \\left(-\\widetilde{c} \\cdot \\left(\\frac{y}{1-\\gamma } + \\frac{1}{\\gamma } \\right) \\right)\\cdot \\sum _{k=1}^{\\infty } \\frac{a_{k}}{k!}",
"\\cdot \\widetilde{c}^{k/(1-\\gamma )} \\cdot \\gamma ^{-k}\\cdot (1-\\gamma )^{-k\\gamma /(1-\\gamma )} \\cdot y^{k\\gamma /(1-\\gamma )}\\cdot {1}_{[0,\\infty [}(y)=: f_{\\widetilde{c},\\gamma }(y),\\quad y \\in \\mathbb {R},\\nonumber $ where $a_{k} := 1/\\Gamma (\\frac{k \\cdot \\gamma }{1-\\gamma } )$ (see e.g.",
"Aalen [1] with a different parametrization).",
"Behaviour at zero: $\\mathbb {}[ \\, [0,\\infty [ \\, ] = \\mathbb {\\Pi }[W \\ge 0]=1$ , $\\mathbb {}[ \\, \\lbrace 0\\rbrace \\, ] = \\mathbb {\\Pi }[W = 0]= e^{-\\theta }$ .",
"Corresponding generator: $\\varphi _{\\gamma ,\\widetilde{c}}^{(0)} = \\widetilde{c} \\cdot \\varphi _{\\gamma }$ (cf.",
"(REF ), (REF )) of the power divergence given in the third line of (REF ); recall that the special case $\\gamma =\\frac{1}{2}$ corresponds to the prominent (multiple of the squared) Hellinger distance.",
"Sums: for i.i.d.",
"copies $(W_{i})_{i \\in \\mathbb {N}}$ of $W$ , the probability distribution of $\\breve{W} := \\sum _{i\\in I_{k}^{(n)}} W_{i}$ (cf.",
"Remark REF (ii)) is $C(POI(\\breve{\\theta }),GAM(\\alpha ,\\beta ))$ with $\\breve{\\theta } = \\frac{\\widetilde{c} \\cdot card(I_{k}^{(n)})}{\\gamma } > 0$ , $\\alpha = \\frac{\\widetilde{c}}{1-\\gamma } >0$ , $\\beta = \\frac{\\gamma }{1-\\gamma } >0$ .",
"(c) Case $\\gamma =2$ , $\\widetilde{c} >0$ : $\\Lambda _{2,\\widetilde{c}}^{(0)}(z)= \\frac{z^{2}}{2 \\widetilde{c}} + z$ (cf.",
"(REF ) ) is the well-known cumulant generating function of the Normal distribution (Gaussian distribution) $\\mathbb {} = N(1,\\frac{1}{\\widetilde{c}})$ with mean 1 and variance $\\frac{1}{\\widetilde{c}}$ .",
"Thus, $\\varphi _{2,\\widetilde{c}} \\in \\Upsilon (]-\\infty ,\\infty [)$ .",
"Type: $\\mathbb {}$ is an infinitely divisible (cf.",
"Proposition REF ) continuous distribution with density $f_{N(1,\\frac{1}{\\widetilde{c}})}(y) := \\sqrt{\\frac{\\widetilde{c}}{2 \\pi }} \\cdot \\exp (- \\frac{\\widetilde{c} \\cdot (y-1)^2}{2} )$ , ($y \\in \\mathbb {R}$ ).",
"Behaviour at zero: $\\mathbb {}[ \\, ]0,\\infty [ \\, ] = \\mathbb {\\Pi }[W > 0]=\\int _{0}^{\\infty } f_{N(1,\\frac{1}{\\widetilde{c}})}(u) \\, du \\in \\, ]0,1[$ , $\\mathbb {}[ \\, \\lbrace 0\\rbrace \\, ] = \\mathbb {\\Pi }[W = 0]= 0$ .",
"Corresponding generator: $\\varphi _{2,\\widetilde{c}}^{(0)} = \\widetilde{c} \\cdot \\varphi _{2}$ (cf.",
"(REF ), (REF )) is the generator of the $\\widetilde{c}-$ fold of the half Pearson-chisquare divergence given in the sixth line of (REF ).",
"Sums: for i.i.d.",
"copies $(W_{i})_{i \\in \\mathbb {N}}$ of $W$ , the probability distribution of $\\breve{W} := \\sum _{i\\in I_{k}^{(n)}} W_{i}$ (cf.",
"Remark REF (ii)) is $N(card(I_{k}^{(n)}),\\frac{card(I_{k}^{(n)})}{\\widetilde{c}})$ .",
"(d) Case $\\gamma <0$ , $\\widetilde{c} >0$ : $\\Lambda _{\\gamma ,\\widetilde{c}}^{(0)}(z) = \\frac{\\widetilde{c}}{\\gamma } \\cdot \\left\\lbrace \\left( \\frac{\\gamma -1}{\\widetilde{c}} \\cdot z +1 \\right)^{\\frac{\\gamma }{\\gamma -1}} -1 \\right\\rbrace $ (cf.",
"(REF )) is the cumulant generating function of a “tilted (i.e.",
"negatively distorted) stable distribution” $\\mathbb {}[ \\, \\cdot \\,] = \\mathbb {\\Pi }[W \\in \\cdot \\, ]$ of a random variable $W$ , which can be constructed as follows: let $Z$ be an auxiliary random variable (having density $f_{Z}$ and support $supp(Z) = [0,\\infty [$ ) of a stable law with parameter-quadruple $(\\frac{-\\gamma }{1-\\gamma },1,0,-\\frac{\\widetilde{c}^{1/(1-\\gamma )} \\cdot (1-\\gamma )^{-\\gamma /(1-\\gamma )}}{\\gamma })$ in terms of the “form-B notation” on p.12 in Zolotarev [428]; by applying a general Laplace-transform result on p.112 of the same text we can deduce $M_{Z}(z) := E_{\\mathbb {\\Pi }}[\\exp (z \\cdot Z)]= \\int _{0}^{\\infty } \\exp (z \\cdot y) \\cdot f_{Z}(y) \\, dy \\hspace{-5.69046pt} &=& \\hspace{-5.69046pt}{\\left\\lbrace \\begin{array}{ll}\\exp \\Big (\\frac{\\widetilde{c}^{1/(1-\\gamma )} \\cdot (1-\\gamma )^{-\\gamma /(1-\\gamma )}}{\\gamma }\\cdot (-z)^{\\alpha } \\Big ),\\quad \\textrm {if } \\ z \\in ]-\\infty ,0] , \\\\\\infty , \\hspace{150.79968pt} \\textrm {if } \\ z \\in \\, ]0,\\infty [, \\\\\\end{array}\\right.",
"}$ where $\\alpha := - \\frac{\\gamma }{1-\\gamma } \\in \\, ]0,1[$ .",
"Since $0 \\notin int(dom(M_{Z}))$ (and thus, $Z$ does not have light-tails) we have to tilt (dampen) the density in order to extend the effective domain.",
"Accordingly, let $W$ be a random variable having density $f_{W}(y)\\ :=\\ \\frac{\\exp \\lbrace -\\frac{y \\cdot \\widetilde{c}}{1-\\gamma }\\rbrace }{\\exp \\lbrace \\widetilde{c}/\\gamma \\rbrace }\\cdot f_{Z}(y)\\cdot {1}_{]0,\\infty [}(y),\\qquad y \\in \\mathbb {R},\\qquad \\text{(cf.",
"(\\ref {brostu3:fo.norweiemp7a}))}.\\nonumber $ Then one can straightforwardly deduce from (REF ) that $\\int _{0}^{\\infty } f_{W}(y) \\, dy =1$ and that $M_{W}(z) := E_{\\mathbb {\\Pi }}[\\exp (z \\cdot W)]= \\int _{0}^{\\infty } \\exp (z \\cdot y) \\cdot f_{W}(y) \\, dy \\hspace{-5.69046pt} &=& \\hspace{-5.69046pt}{\\left\\lbrace \\begin{array}{ll}\\exp \\left(\\frac{\\widetilde{c}}{\\gamma } \\cdot \\left\\lbrace \\left( \\frac{\\gamma -1}{\\widetilde{c}} \\cdot z +1 \\right)^{\\frac{\\gamma }{\\gamma -1}} -1 \\right\\rbrace \\right),\\qquad \\textrm {if } \\ z \\in ]-\\infty ,\\frac{\\widetilde{c}}{1-\\gamma }] , \\\\\\infty , \\hspace{156.49014pt} \\textrm {if } \\ z \\in \\, ]\\frac{\\widetilde{c}}{1-\\gamma },\\infty [ .",
"\\\\\\end{array}\\right.",
"}\\nonumber $ Hence, $\\varphi _{\\gamma ,\\widetilde{c}} \\in \\Upsilon (]0,\\infty [)$ .",
"Type: $\\mathbb {}$ is an infinitely divisible (cf.",
"Proposition REF ) continuous distribution with density $f_{W}$ .",
"Behaviour at zero: $\\mathbb {}[ \\, ] 0,\\infty [ \\, ] = \\mathbb {\\Pi }[W > 0]=1$ .",
"Corresponding generator: $\\varphi _{\\gamma ,\\widetilde{c}}^{(0)} = \\widetilde{c} \\cdot \\varphi _{\\gamma }$ (cf.",
"(REF ), (REF )) of the power divergence given in the first line of (REF ).",
"Sums: for i.i.d.",
"copies $(W_{i})_{i \\in \\mathbb {N}}$ of $W$ , the probability distribution of $\\breve{W} := \\sum _{i\\in I_{k}^{(n)}} W_{i}$ (cf.",
"Remark REF (ii)) has density $f_{\\breve{W}}(y)\\ :=\\ \\frac{\\exp \\lbrace -\\frac{y \\cdot \\widetilde{c}}{1-\\gamma }\\rbrace }{\\exp \\lbrace \\widetilde{c} \\cdot card(I_{k}^{(n)})/\\gamma \\rbrace }\\cdot f_{\\breve{Z}}(y)\\cdot {1}_{]0,\\infty [}(y),\\qquad y \\in \\mathbb {R},$ where $\\breve{Z}$ is a random variable with density $f_{\\breve{Z}}$ of a stable law with parameter-quadruple $(\\frac{-\\gamma }{1-\\gamma },1,0,-card(I_{k}^{(n)}) \\cdot \\frac{\\widetilde{c}^{1/(1-\\gamma )} \\cdot (1-\\gamma )^{-\\gamma /(1-\\gamma )}}{\\gamma })$ .",
"(e) Case $\\gamma >2$ , $\\widetilde{c} >0$ : $\\Lambda _{\\gamma ,\\widetilde{c}}^{(0)}(z) = \\frac{\\widetilde{c}}{\\gamma } \\cdot \\left\\lbrace \\left( \\frac{\\gamma -1}{\\widetilde{c}} \\cdot z +1 \\right)^{\\frac{\\gamma }{\\gamma -1}} -1 \\right\\rbrace $ (cf.",
"(REF )) is the cumulant generating function of a “distorted stable distribution” $\\mathbb {}[ \\, \\cdot \\,] = \\mathbb {\\Pi }[W \\in \\cdot \\, ]$ of a random variable $W$ , which can be constructed as follows: let $Z$ be an auxiliary random variable (having density $f_{Z}$ and support $supp(Z) = ]-\\infty ,\\infty ]$ ) of a stable law with parameter-quadruple $(\\frac{\\gamma }{\\gamma -1},1,0,\\frac{\\widetilde{c}^{1/(1-\\gamma )} \\cdot (\\gamma -1)^{\\gamma /(\\gamma -1)}}{\\gamma })$ in terms of the above-mentioned “form-B notation” ; by applying a general Laplace-transform result on p. 112 of Zolotarev [428], we can derive $M_{Z}(z) := E_{\\mathbb {\\Pi }}[\\exp (z \\cdot Z)]= \\int _{0}^{\\infty } \\exp (z \\cdot y) \\cdot f_{Z}(y) \\, dy \\hspace{-5.69046pt} &=& \\hspace{-5.69046pt}{\\left\\lbrace \\begin{array}{ll}\\exp \\Big (\\frac{\\widetilde{c}^{1/(1-\\gamma )} \\cdot (\\gamma -1 )^{\\gamma /(\\gamma -1)}}{\\gamma }\\cdot (-z)^{\\alpha } \\Big ),\\quad \\textrm {if } \\ z \\in ]-\\infty ,0] , \\\\\\infty , \\hspace{145.10922pt} \\textrm {if } \\ z \\in \\, ]0,\\infty [, \\\\\\end{array}\\right.",
"}$ where $\\alpha := \\frac{\\gamma }{\\gamma -1} \\in \\, ]1,2[$ .",
"Since $0 \\notin int(dom(M_{Z}))$ (and thus, $Z$ does not have light-tails) we have to distort the density in order to extend the effective domain.",
"Accordingly, let $W$ be a random variable having density $f_{W}(y)\\ :=\\ \\frac{\\exp \\lbrace \\frac{y \\cdot \\widetilde{c}}{\\gamma -1}\\rbrace }{\\exp \\lbrace \\widetilde{c}/\\gamma \\rbrace }\\cdot f_{Z}(- y), \\qquad y \\in \\mathbb {R},\\qquad \\text{(cf.",
"(\\ref {brostu3:fo.norweiemp7atwo}))}.\\nonumber $ Then one can straightforwardly deduce from (REF ) that $\\int _{-\\infty }^{\\infty } f_{W}(y) \\, dy =1$ and that $M_{W}(z) := E_{\\mathbb {\\Pi }}[\\exp (z \\cdot W)]= \\int _{-\\infty }^{\\infty } \\exp (z \\cdot y) \\cdot f_{W}(y) \\, dy \\hspace{-5.69046pt} &=& \\hspace{-5.69046pt}{\\left\\lbrace \\begin{array}{ll}\\exp \\left(\\frac{\\widetilde{c}}{\\gamma } \\cdot \\left\\lbrace \\left( \\frac{\\gamma -1}{\\widetilde{c}} \\cdot z +1 \\right)^{\\frac{\\gamma }{\\gamma -1}} -1 \\right\\rbrace \\right),\\qquad \\textrm {if } \\ z \\in [- \\frac{\\widetilde{c}}{\\gamma -1},\\infty [ , \\\\\\infty , \\hspace{156.49014pt} \\textrm {if } \\ z \\in \\, ]-\\infty , -\\frac{\\widetilde{c}}{\\gamma -1}[ .",
"\\\\\\end{array}\\right.",
"}\\nonumber $ Thus, $\\varphi _{\\gamma ,\\widetilde{c}} \\in \\Upsilon (]-\\infty ,\\infty [)$ .",
"Type: $\\mathbb {}$ is an infinitely divisible (cf.",
"Proposition REF ) continuous distribution with density $f_{W}$ .",
"Behaviour at zero: $\\mathbb {}[ \\, ]0,\\infty [ \\, ] = \\mathbb {\\Pi }[W > 0]=\\int _{0}^{\\infty } f_{W}(u) \\, du \\in \\, ]0,1[$ , $\\mathbb {}[ \\, \\lbrace 0\\rbrace \\, ] = \\mathbb {\\Pi }[W = 0]= 0$ .",
"Corresponding generator: $\\varphi _{\\gamma ,\\widetilde{c}}^{(0)} = \\widetilde{c} \\cdot \\varphi _{\\gamma }$ (cf.",
"(REF ), (REF )) of the power divergence given in the seventh line of (REF ).",
"Sums: for i.i.d.",
"copies $(W_{i})_{i \\in \\mathbb {N}}$ of $W$ , the probability distribution of $\\breve{W} := \\sum _{i\\in I_{k}^{(n)}} W_{i}$ (cf.",
"Remark REF (ii)) has density $f_{\\breve{W}}(y)\\ :=\\ \\frac{\\exp \\lbrace \\frac{y \\cdot \\widetilde{c}}{\\gamma -1}\\rbrace }{\\exp \\lbrace \\widetilde{c} \\cdot card(I_{k}^{(n)})/\\gamma \\rbrace }\\cdot f_{\\breve{Z}}(- y), \\qquad y \\in \\mathbb {R},\\nonumber $ where $\\breve{Z}$ is a random variable with density $f_{\\breve{Z}}$ of a stable law with parameter-quadruple $(\\frac{\\gamma }{\\gamma -1},1,0,card(I_{k}^{(n)})\\cdot \\frac{\\widetilde{c}^{1/(1-\\gamma )} \\cdot (\\gamma -1)^{\\gamma /(\\gamma -1)}}{\\gamma })$ .",
"(f) Case $\\gamma \\in ]1,2[$ , $\\widetilde{c} >0$ : one still has the (cumulant-generating-function) candidate $\\Lambda _{\\gamma ,\\widetilde{c}}^{(0)}(z) = \\frac{\\widetilde{c}}{\\gamma } \\cdot \\left\\lbrace \\left( \\frac{\\gamma -1}{\\widetilde{c}} \\cdot z +1 \\right)^{\\frac{\\gamma }{\\gamma -1}} -1 \\right\\rbrace $ (cf.",
"(REF )), but for the crucial exponent there holds $\\frac{\\gamma }{\\gamma -1} > 2$ .",
"From this, we conjecture that $\\mathbb {}$ becomes a signed finite measure with total mass 1, i.e.",
"it has a density (with respect to some dominating measure) with positive and negative values which “integrates to 1” ; accordingly, our BS method can not be applied to this situation.",
"Remark 49 As a continuation of Remark REF and the note in the third line after (REF ), we have shown as a side effect that for $\\gamma \\in \\, ]-\\infty ,-1] \\, \\cup \\, ]0,1[ \\, \\cup \\, [2,\\infty [$ the distributions $\\mathbb {}_{\\gamma }$ and $\\mathbb {}_{1-\\gamma }$ of Example REF (b)-(e) are inverse to each other.",
"Example 50 for the power-divergence context of Example REF we obtain: (a) Case $\\gamma =1$ , $\\widetilde{c} >0$ , anchor point $c=0$ : $\\Lambda _{1,\\widetilde{c}}(z) =\\widetilde{c} \\cdot \\left( \\exp (\\frac{z}{\\widetilde{c}}) - 1 \\right)$ (cf.",
"(REF )) is the cumulant generating function of $\\mathbb {} = \\frac{1}{\\widetilde{c}} \\cdot POI(\\widetilde{c})$ being the “$\\frac{1}{\\widetilde{c}}-$ fold Poisson distribution with mean $\\widetilde{c}$ ” which means that $W = \\frac{1}{\\widetilde{c}} \\cdot Z$ for a $POI(\\widetilde{c})-$ distributed random variable $Z$ .",
"Thus, $\\varphi _{1,\\widetilde{c}} \\in \\Upsilon (]0,\\infty [)$ .",
"Prominent special case $\\widetilde{c} =1$ : $\\mathbb {} = POI(1)$ is the Poisson distribution with mean 1.",
"Type: $\\mathbb {}$ is an infinitely divisible (cf.",
"Proposition REF ) discrete distribution with frequencies: $\\mathbb {\\Pi }[W=\\ell \\cdot \\frac{1}{\\widetilde{c}}]= \\exp (-\\widetilde{c}) \\cdot \\frac{ \\widetilde{c}^{\\ell }}{\\ell !",
"}$ for all nonnegative integers $\\ell \\in \\mathbb {N}_{0}$ (and zero elsewhere).",
"Behaviour at zero: $\\mathbb {\\Pi }[W\\ge 0]=1$ , $\\mathbb {\\Pi }[W=0]= \\exp (-\\widetilde{c})$ .",
"Corresponding generator: $\\varphi _{1,\\widetilde{c}} = \\widetilde{c} \\cdot \\varphi _{1}$ (cf.",
"(REF ), (REF )) of the $\\widetilde{c}-$ fold of the Kullback-Leibler divergence (relative entropy) given in the fourth line of (REF ).",
"Sums: for i.i.d.",
"copies $(W_{i})_{i \\in \\mathbb {N}}$ of $W$ , the probability distribution of $\\breve{W} := \\sum _{i\\in I_{k}^{(n)}} W_{i}$ (cf.",
"Remark REF (ii)) is $\\frac{1}{\\widetilde{c}} \\cdot POI(\\widetilde{c} \\cdot card(I_{k}^{(n)}))$ .",
"(b) Case $\\gamma =1$ , $\\widetilde{c} =1$ , anchor point $c \\in \\mathbb {R}$ : $\\Lambda _{1,1}^{(c)}(z)= e^{c} \\cdot \\left( e^{z} - 1 \\right) + z \\cdot (1- e^{c})$ (cf.",
"(REF )) is the well-known cumulant generating function of the “shifted Poisson distribution” $\\mathbb {} = POI(e^{c}) + 1-e^{c}$ , i.e.",
"$W := Z + 1-e^{c}$ with a $POI(e^{c})-$ distributed random variable $Z$ .",
"Hence, $\\varphi _{1,1}^{(c)} \\in \\Upsilon (]1- e^{c},\\infty [)$ .",
"Type: $\\mathbb {}$ is a discrete distribution with frequencies: $\\mathbb {\\Pi }[W=\\ell + 1- e^{c}]= \\exp (-e^{c}) \\cdot \\frac{ e^{c \\cdot \\ell }}{\\ell !",
"}$ for all $\\ell \\in \\mathbb {N}_{0}$ (and zero elsewhere).",
"Behaviour at zero: $\\mathbb {\\Pi }[W > 0]=1$ iff $c <0$ , $\\mathbb {\\Pi }[W < 0] >0$ iff $c >0$ , $\\mathbb {\\Pi }[W=0] \\ne 0$ iff “$c =\\log (1+k)$ for some $k \\in \\mathbb {N}_{0}$ ” .",
"Corresponding generator: $\\varphi _{1,1}^{(c)}$ (cf.",
"(REF )) of the divergence $& & D_{\\varphi _{1,1}^{(c)}}(\\mathbf {Q},\\mathbf {P}) :=\\sum \\limits _{k=1}^{K} \\Big (q_{k} + p_{k} \\cdot (e^{c}-1) \\Big )\\cdot \\Big \\lbrace \\log \\Big (\\frac{q_{k}}{p_{k}} + e^{c}-1 \\Big ) - c \\Big \\rbrace - \\sum \\limits _{k=1}^{K} q_{k} + \\sum \\limits _{k=1}^{K} p_{k} ,\\nonumber \\\\& & \\hspace{113.81102pt} \\textrm {if } \\ \\mathbf {P} \\in \\mathbb {R}_{\\ne 0}^{K} \\ \\textrm {and } \\mathbf {Q} \\in \\mathbb {R}^{K}\\textrm {with } \\mathbf {Q} \\in \\, [ (1-e^{c}) \\cdot \\mathbf {P}, \\mathbf {\\infty } [\\ \\textrm {component-wise},$ which for $c=0$ coincides with the Kullback-Leibler divergence (relative entropy) given in the fourth line of (REF ).",
"Sums: for i.i.d.",
"copies $(W_{i})_{i \\in \\mathbb {N}}$ of $W$ , the probability distribution of $\\breve{W} := \\sum _{i\\in I_{k}^{(n)}} W_{i}$ (cf.",
"Remark REF (ii)) is $POI(card(I_{k}^{(n)}) \\cdot e^{c}) + (1-e^{c}) \\cdot card(I_{k}^{(n)})$ .",
"Remark 51 (a) One can see from the Examples REF and REF the interesting effect that the “homogeneous” class of power-divergence generators $(\\varphi _{\\gamma })_{\\gamma \\in \\mathbb {R}}$ are connected to a “very inhomogeneous” family $(\\mathbb {}_{\\gamma })_{\\gamma \\in \\mathbb {R}}$ of $W-$ distributions: discrete, continuous, mixture of discrete and continuous, as the parameter $\\gamma $ varies.",
"Moreover, some cases satisfy $\\mathbb {\\Pi }[W=0] =0$ and some $\\mathbb {\\Pi }[W=0] >0$ , some $\\mathbb {\\Pi }[W>0]=1$ and some $\\mathbb {\\Pi }[W>0] \\in ]0,1[$ .",
"(b) As a continuation of Remark REF and the note in the last line of Example REF (a), we have shown as a side effect that for the the natural-anchor-point choice $c=0$ , the distributions $\\mathbb {}_{1}$ of of Example REF (a) and $\\mathbb {}_{0}$ of Example REF (a) are inverse to each other.",
"Example 52 for the context of Example REF we obtain: Case $\\widetilde{c} >0$ , anchor point $c=0$ : $\\Lambda _{bw,\\beta ,\\widetilde{c}}(z) =-(\\frac{1}{\\beta }-1) \\cdot z + \\frac{\\widetilde{c}}{\\beta ^{2}} \\cdot \\Big \\lbrace 1 - \\sqrt{1-\\frac{2\\beta }{\\widetilde{c}} \\cdot z} \\ \\Big \\rbrace $ (cf.",
"(REF )) is the cumulant generating function of a probability distribution $\\mathbb {}[ \\, \\cdot \\,] = \\mathbb {\\Pi }[\\check{W} \\in \\cdot \\, ]$ of a random variable $\\check{W}$ , which can be constructed as follows: $\\check{W} := \\frac{W}{\\beta } - (\\frac{1}{\\beta } - 1)$ , where $W$ is the random variable constructed in Example REF (d) with $\\gamma =-1$ and with $\\widetilde{c}$ replaced by $\\frac{\\widetilde{c}}{\\beta ^{2}}$ (recall that $W$ has a tilted stable distribution).",
"In other words, $\\mathbb {}$ is a special kind of modified tilted stable distribution.",
"Type: $\\mathbb {}$ is an infinitely divisible (cf.",
"Proposition REF ) continuous distribution with density $f_{\\check{W}}(u) := \\beta \\cdot f_{W}(\\beta \\cdot u + 1 -\\beta )\\cdot \\mathbf {1}_{]- (\\frac{1}{\\beta } -1),\\infty [}(u)$ ($u \\in \\mathbb {R}$ ), where $f_{W}(\\cdot )$ is given in (REF ) with $\\gamma =-1$ and with $\\widetilde{c}$ replaced by $\\frac{\\widetilde{c}}{\\beta ^{2}}$ .",
"Behaviour at zero: $\\mathbb {}[ \\, ] 0,\\infty [ \\, ] =\\mathbb {\\Pi }[\\check{W} > 0] >0$ .",
"Corresponding generator: $\\varphi _{bw,\\beta ,\\widetilde{c}}^{(0)}$ (cf.",
"(REF )) of the — “non-probability version” of — the well-known blended weight chi-square divergence given in (REF ).",
"Sums: for i.i.d.",
"copies $(\\check{W}_{i})_{i \\in \\mathbb {N}}$ of $\\check{W}$ , the probability distribution of $\\breve{\\check{W}} := \\sum _{i\\in I_{k}^{(n)}} \\check{W}_{i}= \\frac{1}{\\beta } \\cdot \\sum _{i\\in I_{k}^{(n)}} W_{i} - n_{k}\\cdot (\\frac{1}{\\beta } -1)$ (cf.",
"Remark REF (ii)) has density $f_{\\breve{\\check{W}}}(u) := \\beta \\cdot f_{\\breve{W}}(\\beta \\cdot u + (1 -\\beta )\\cdot n_{k})\\cdot \\mathbf {1}_{]- n_{k} \\cdot (\\frac{1}{\\beta } -1),\\infty [}(u)$ ($u \\in \\mathbb {R}$ ), where $f_{\\breve{W}}(\\cdot )$ is given in (REF ) (cf.",
"Example REF (d)) with $\\gamma =-1$ and with $\\widetilde{c}$ replaced by $\\frac{\\widetilde{c}}{\\beta ^{2}}$ .",
"Example 53 for the context of Example REF we obtain: (a) Case $\\alpha \\in \\, ]0,\\infty [$ , $\\widetilde{c} >0$ , anchor point $c=0$ : $\\Lambda _{gKL,\\alpha ,\\widetilde{c}}(z) =- \\frac{\\widetilde{c}}{\\alpha } \\cdot \\log ((1+\\alpha ) - \\alpha \\cdot e^{z/\\widetilde{c}})$ (cf.",
"(REF )) is the cumulant generating function of $\\mathbb {} = \\frac{1}{\\widetilde{c}} \\cdot NB(\\frac{\\widetilde{c}}{\\alpha },\\frac{1}{1+\\alpha })$ being the “$\\frac{1}{\\widetilde{c}}-$ fold Negative-Binomial distribution with parameters $\\frac{\\widetilde{c}}{\\alpha }$ and $\\frac{1}{1+\\alpha }$ ” which means that $W = \\frac{1}{\\widetilde{c}} \\cdot Z$ for a $NB(\\frac{\\widetilde{c}}{\\alpha },\\frac{1}{1+\\alpha })-$ distributed random variable $Z$ .",
"Thus, $\\varphi _{gKL,\\alpha ,\\widetilde{c}} \\in \\Upsilon (]0,\\infty [)$ .",
"Prominent special case $\\widetilde{c}=1$ , $\\alpha =1$ (see below): $\\mathbb {} = NB(1,\\frac{1}{2})$ is the Negative-Binomial distribution with parameters 1 and $\\frac{1}{2}$ .",
"Type: $\\mathbb {}$ is an infinitely divisible (cf.",
"Proposition REF ) discrete distribution with frequencies: $\\mathbb {\\Pi }[W=\\ell \\cdot \\frac{1}{\\widetilde{c}}]=(-1)^{\\ell } \\cdot \\binom{- \\frac{\\widetilde{c}}{\\alpha }}{\\ell }\\cdot {\\alpha }^{\\ell } \\cdot (1+\\alpha )^{-\\ell - \\widetilde{c}/\\alpha }$ for all nonnegative integers $\\ell \\in \\mathbb {N}_{0}$ (and zero elsewhere).",
"Behaviour at zero: $\\mathbb {\\Pi }[W\\ge 0]=1$ , $\\mathbb {\\Pi }[W=0]= \\frac{1}{(1+a)^{\\widetilde{c}/\\alpha }}$ .",
"Corresponding generator: $\\varphi _{gKL,\\alpha ,\\widetilde{c}}$ (cf.",
"(REF )) of the divergence (REF ); the special case $\\widetilde{c}=1$ , $\\alpha =1$ — i.e.",
"$\\varphi _{gKL,1,1} =: \\varphi _{snKL,1}$ (cf.",
"(REF )) — corresponds to the generator of the — “non-probability version” of the — Jensen-Shannon divergence (symmetrized and normalized Kullback-Leibler divergence, symmetrized and normalized relative entropy) given in (REF ).",
"Sums: for i.i.d.",
"copies $(W_{i})_{i \\in \\mathbb {N}}$ of $W$ , the probability distribution of $\\breve{W} := \\sum _{i\\in I_{k}^{(n)}} W_{i}$ (cf.",
"Remark REF (ii)) is $\\frac{1}{\\widetilde{c}} \\cdot NB(\\frac{\\widetilde{c}}{\\alpha }\\cdot card(I_{k}^{(n)}),\\frac{1}{1+\\alpha })$ .",
"(b) Case $\\alpha \\in \\, ]-1,0[$ , $\\widetilde{c} >0$ , anchor point $c=0$ : for any integer $m \\in \\mathbb {N}$ being strictly larger than $\\widetilde{c}$ and the choice $\\alpha = - \\frac{\\widetilde{c}}{m}$ , we obtain $\\Lambda _{gKL,-\\widetilde{c}/m,\\widetilde{c}}(z) =m \\cdot \\log ((1 - \\frac{\\widetilde{c}}{m}) + \\frac{\\widetilde{c}}{m} \\cdot e^{z/\\widetilde{c}})$ (cf.",
"(REF )) which is the cumulant generating function of $\\mathbb {} = \\frac{1}{\\widetilde{c}} \\cdot BIN(m,\\frac{\\widetilde{c}}{m})$ being the “$\\frac{1}{\\widetilde{c}}-$ fold Binomial distribution with parameters $m$ and $\\frac{\\widetilde{c}}{m}$ ” which means that $W = \\frac{1}{\\widetilde{c}} \\cdot Z$ for a $BIN(m,\\frac{\\widetilde{c}}{m})-$ distributed random variable $Z$ .",
"Thus, $\\varphi _{gKL,-\\widetilde{c}/m,\\widetilde{c}} \\in \\Upsilon (]0,\\infty [)$ .",
"Type: $\\mathbb {}$ is a non-infinitely divisible discrete distribution with frequencies: $\\mathbb {\\Pi }[W=\\ell \\cdot \\frac{1}{\\widetilde{c}}]=\\binom{m}{\\ell }\\cdot (\\frac{\\widetilde{c}}{m})^{\\ell } \\cdot (1-\\frac{\\widetilde{c}}{m})^{m-\\ell }$ for $\\ell \\in \\lbrace 0,1, \\ldots , m\\rbrace $ (and zero elsewhere).",
"Behaviour at zero: $\\mathbb {\\Pi }[W\\ge 0]=1$ , $\\mathbb {\\Pi }[W=0]= (1-\\frac{\\widetilde{c}}{m})^{m}$ .",
"Corresponding generator: $\\varphi _{gKL,\\alpha ,\\widetilde{c}}$ (cf.",
"(REF )) of the divergence (REF ).",
"Sums: for i.i.d.",
"copies $(W_{i})_{i \\in \\mathbb {N}}$ of $W$ , the probability distribution of $\\breve{W} := \\sum _{i\\in I_{k}^{(n)}} W_{i}$ (cf.",
"Remark REF (ii)) is $\\frac{1}{\\widetilde{c}} \\cdot BIN(m \\cdot card(I_{k}^{(n)}),\\frac{\\widetilde{c}}{m})$ .",
"Example 54 for the context of Example REF we obtain: Case of anchor point $c=0$ : $\\Lambda _{twop}(z)= \\log \\Big ( p \\cdot e^{z_{1} \\cdot z} + (1- p) \\cdot e^{z_{2} \\cdot z} \\Big )$ (cf.",
"(REF )) is the well-known cumulant generating function of the two-point probability distribution $\\mathbb {} = p \\cdot \\delta _{z_{1}} + (1-p) \\cdot \\delta _{z_{2}}$ , where $z_{1} < 1 < z_{2}$ and $p= \\frac{z_{2} -1}{z_{2} - z_{1}}$ .",
"Hence, $\\varphi _{twop} \\in \\Upsilon (]z_{1},z_{2}[)$ .",
"Type: $\\mathbb {}$ is a discrete distribution with frequencies: $\\mathbb {\\Pi }[W=z_{1}] = p$ , $\\mathbb {\\Pi }[W=z_{2}] = 1-p$ (and zero elsewhere).",
"Behaviour at zero: $\\mathbb {\\Pi }[W > 0]=1$ iff $z_{1} >0$ , $\\mathbb {\\Pi }[W=0] \\ne 0$ iff $z_{1}=0$ .",
"Corresponding generator: $\\varphi _{twop}$ (cf.",
"(REF )) of the divergence given in (REF ).",
"Sums: for i.i.d.",
"copies $(W_{i})_{i \\in \\mathbb {N}}$ of $W$ , the probability distribution of $\\breve{W} := \\sum _{i\\in I_{k}^{(n)}} W_{i}$ (cf.",
"Remark REF (ii)) is the distribution of the $card(I_{k}^{(n)})-$ th step of a generalized random walk starting at zero; this has a nice explicit (“binomial-type” ) expression in the special case $z_{1} = - z_{2}$ , namely $\\sum _{\\ell =0}^{card(I_{k}^{(n)})} \\binom{card(I_{k}^{(n)})}{\\ell }\\cdot p^{card(I_{k}^{(n)}) - \\ell } \\cdot (1-p)^{\\ell } \\cdot \\delta _{z_{2}\\cdot (2 \\ell - card(I_{k}^{(n)}))}$ .",
"Example 55 for the context of Example REF we obtain: Case $\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c} \\in \\, ]0,\\infty [$ , anchor point $c=0$ : by using $\\breve{\\theta } := 1 + \\alpha \\cdot \\Big (\\frac{1}{\\beta _{2}}- \\frac{1}{\\beta _{1}} \\Big )$ one can see that $\\Lambda _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}(z) =\\breve{\\theta } \\cdot z - \\widetilde{c} \\cdot \\alpha \\cdot \\log \\Big (1+ \\frac{z}{\\widetilde{c}} \\cdot \\Big (\\frac{1}{\\beta _{2}} - \\frac{1}{\\beta _{1}} \\Big )- \\frac{z^{2}}{\\widetilde{c}^{2} \\cdot \\beta _{1} \\cdot \\beta _{2} }\\Big )$ for $z \\in \\, ]- \\widetilde{c} \\cdot \\min \\lbrace \\beta _{1}, \\beta _{2}\\rbrace ,\\widetilde{c} \\cdot \\min \\lbrace \\beta _{1}, \\beta _{2}\\rbrace [$ — with different boundary behaviour for the three subcases $\\beta _{1} < \\beta _{2}$ resp.",
"$\\beta _{1} > \\beta _{2}$ resp.",
"$\\beta _{1} = \\beta _{2}$ (cf.",
"(REF ),(REF ),(REF )) — is the cumulant generating function of a generalized asymmetric Laplace distribution $\\mathbb {}[ \\, \\cdot \\,] = \\mathbb {\\Pi }[W \\in \\cdot \\, ]$ of a random variable $W:= \\breve{\\theta } + Z_{1} - Z_{2}$ , where $Z_{1}$ respectively $Z_{2}$ are auxiliary random variables which are independent and $GAM(\\widetilde{c} \\cdot \\beta _{1},\\widetilde{c} \\cdot \\alpha )-$ distributed respectively $GAM(\\widetilde{c} \\cdot \\beta _{2},\\widetilde{c} \\cdot \\alpha )-$ distributed.",
"In particular, $E_{\\mathbb {\\Pi }}[W] = \\breve{\\theta } + \\frac{\\widetilde{c} \\cdot \\alpha }{\\widetilde{c} \\cdot \\beta _{1}}- \\frac{\\widetilde{c} \\cdot \\alpha }{\\widetilde{c} \\cdot \\beta _{2}} =1$ (as required).",
"Thus, $\\varphi _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}\\in \\Upsilon (]-\\infty ,\\infty [)$ .",
"Prominent special case $\\widetilde{c} =1$ , $\\alpha =1$ , $\\beta _{1}=\\beta _{2} =: \\beta $ (and hence, $\\breve{\\theta }=1$ ): $\\mathbb {}$ is a classical Laplace distribution (two-tailed exponential distribution, bilateral exponential law) with location parameter 1 and scale parameter $\\frac{1}{\\beta }$ .",
"Type: $\\mathbb {}$ is an infinitely divisible (cf.",
"Proposition REF ) continuous distribution with density $f(u) := \\frac{\\sqrt{2} \\cdot \\exp \\lbrace \\frac{1}{\\sigma \\cdot \\sqrt{2}} \\cdot (\\frac{1}{\\kappa } - \\kappa ) \\cdot (u-\\theta ) \\rbrace }{\\sqrt{\\pi } \\cdot \\sigma ^{\\tau + 1/2} \\cdot \\Gamma (\\tau )} \\cdot \\left( \\frac{\\sqrt{2} \\cdot |u - \\theta |}{\\kappa + \\frac{1}{\\kappa }} \\right)^{\\tau - 1/2}\\cdot K_{\\tau - 1/2}\\left( \\frac{1}{\\sigma \\cdot \\sqrt{2}} \\cdot \\Big (\\kappa + \\frac{1}{\\kappa }\\Big ) \\cdot |u - \\theta | \\right),\\quad u \\ne \\theta ,$ where $(\\theta ,\\kappa ,\\sigma ,\\tau )$ is given in Remark REF below and $K_{\\lambda }$ is the modified Bessel function of the third kind with index $\\lambda $ .",
"For the above-mentioned special case of the classical Laplace distribution, this considerably simplifies to $f(u):= \\frac{\\beta }{2} \\exp \\lbrace - \\beta \\cdot |u -1 | \\rbrace $ .",
"Behaviour at zero: $\\mathbb {}[ \\, ]0,\\infty [ \\, ] = \\mathbb {\\Pi }[W > 0]=\\int _{0}^{\\infty } f(u) \\, du \\in \\, ]0,1[$ , $\\mathbb {}[ \\, \\lbrace 0\\rbrace \\, ] = \\mathbb {\\Pi }[W = 0]= 0$ .",
"Corresponding generator: $\\varphi _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}$ (cf.",
"(REF ) respectively (REF ) respectively (REF )) of the divergence given in (REF ) respectively (REF ) respectively (REF ).",
"Sums: for i.i.d.",
"copies $(W_{i})_{i \\in \\mathbb {N}}$ of $W$ , the probability distribution of $\\breve{W} := \\sum _{i\\in I_{k}^{(n)}} W_{i}$ (cf.",
"Remark REF (ii)) is the same as that of a random variable $\\breve{\\widetilde{W}} := \\breve{\\theta } \\cdot card(I_{k}^{(n)}) +\\breve{Z}_{1} - \\breve{Z}_{2}$ , where $\\breve{Z}_{1}$ respectively $\\breve{Z}_{2}$ are auxiliary random variables which are independent and $GAM(\\widetilde{c} \\cdot \\beta _{1},\\widetilde{c} \\cdot \\alpha \\cdot card(I_{k}^{(n)}))-$ distributed respectively $GAM(\\widetilde{c} \\cdot \\beta _{2},\\widetilde{c}\\cdot \\alpha \\cdot card(I_{k}^{(n)}))-$ distributed.",
"Remark 56 In the book of Kotz et al.",
"[197] one can find a very comprehensive study on generalized asymmetric Laplace distributions (also known as Bessel function distributions, McKay distributions), their close relatives (such as e.g.",
"the financial-econometric variance gamma model of Madan & Seneta [240]) as well as their applications; see also e.g.",
"Klar [192] for connections with some other Gamma difference distributions.",
"[197] use a different parametrization $(\\theta ,\\kappa ,\\sigma ,\\tau )$ which is one-to-one with our parametrization $(\\breve{\\theta },\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}=1)$ , as follows: $\\theta = \\breve{\\theta }$ , $\\tau = \\widetilde{c} \\cdot \\alpha $ , $\\sigma = \\frac{1}{\\widetilde{c}} \\cdot \\sqrt{\\frac{2}{\\beta _{1} \\cdot \\beta _{2}}}$ , $\\kappa = \\frac{\\sqrt{\\frac{4}{\\widetilde{c}^{2}} + (\\beta _{1} - \\beta _{2})^2} \\, + \\beta _{2}- \\beta _{1}}{2 \\cdot \\sqrt{\\beta _{1} \\cdot \\beta _{2}}} >0$ .",
"In particular, this implies that we cover all generalized asymmetric Laplace distributions with mean 1.",
"For better comparability, we have used the parametrization $(\\theta ,\\kappa ,\\sigma ,\\tau )$ in the above-mentioned representation (REF ) of the density (due to [197]).",
"Let us end this section by giving some further comments on the task of finding concretely the probability distribution (if existent) $\\mathbb {}[ \\, \\cdot \\,] = \\mathbb {\\Pi }[W \\in \\cdot \\, ]$ from the Fenchel-Legendre transform $\\Lambda = \\varphi ^{*}$ of a pregiven divergence generator $\\varphi $ , which should satisfy $\\exp (\\Lambda (z)) = \\int _{\\mathbb {R}}e^{z \\cdot y} \\, d\\mathbb {} (y)= E_{\\mathbb {\\Pi }}[\\exp (z \\cdot W)] ,\\qquad z \\in \\mathbb {R}, \\qquad \\text{(cf.",
"(\\ref {brostu3:repres lambda1}), (\\ref {brostu3:repres lambda2}))}.\\nonumber $ Recall that this is used for the simulation of the weights $(W_{i})_{i\\in \\mathbb {N}}$ which are i.i.d.",
"copies of $W$ and which are the crucial building ingredients of $\\xi _{n}^{\\mathbf {W}}$ in Theorem REF , respectively, of $\\xi _{n,\\mathbf {X}}^{w\\mathbf {W}}$ in Theorem REF .",
"The search for $\\mathbb {}$ can be done e.g.",
"by inversion of the moment generating function MGF, or by search in tables or computer software which list distributions and their MGF.",
"As already indicated above, we have eased/narrowed down this task by giving (additional) sufficient conditions for some deriving principal properties of $\\mathbb {}$ .",
"Also notice that $\\mathbb {}$ needs not necessarily to be explicitly known in full detail (e.g.",
"in terms of a computationally tractable density or frequency); for instance, as well known from insurance applications, for — comfortably straightforwardly simulable — doubly-random sums $W := \\sum _{i=1}^{N} A_{i}$ of nonnegative i.i.d.",
"random variables $(A_{i})_{i\\in \\mathbb {N}}$ with known law $\\Pi _A[\\, \\cdot \\, ]:= \\Pi [A \\in \\cdot \\, ]$ being independent of a counting-type random variable $N$ with known law $\\Pi _N$ , one can mostly compute explicitly $MGF_{\\mathbb {}}(z) = PGF_{\\Pi _N}(MGF_{\\Pi _A}(z))$ in terms of $\\mathbb {} := \\Pi _W$ and the probability generating function $PGF_{\\Pi _N}$ of $\\Pi _N$ , but the corresponding density/frequency of $\\mathbb {}$ may not be known explicitly in a tractable form.",
"The above-mentioned Example REF (b) of power divergences with generator $\\varphi _{\\gamma }$ ($\\gamma \\in ]0,1[$ ) manifests such a situation.",
"In the end, if no explicit distribution $\\mathbb {}$ and no comfortably simulable $W-$ construction are available, one can still try to simulate an i.i.d.",
"sequence $(W_{i})_{i\\in \\mathbb {N}}$ from the pregiven moment generating function (which is $\\exp (\\Lambda (z))$ here); see e.g.",
"McLeish [259] and references therein which also contains saddle point methods approximation techniques."
],
[
"Estimators",
"In the following, we demonstrate how one can principally implement our BS approach; a further, deeper analysis will be given in a follow-up paper."
],
[
"Estimators for the deterministic minimization problem",
"We address the minimization problem $D_{\\varphi }(\\mathbf {\\Omega },\\mathbf {P}):= \\inf _{\\mathbf {Q}\\in \\mathbf {\\Omega } } D_{\\varphi }(\\mathbf {Q},\\mathbf {P}) =\\inf _{\\widetilde{\\mathbf {Q}}\\in \\widetilde{\\mathbf {\\Omega }} }D_{\\widetilde{\\varphi } }(\\widetilde{\\mathbf {Q}},\\widetilde{{P}})=: D_{\\widetilde{\\varphi } }(\\widetilde{\\mathbf {\\Omega }},\\widetilde{{P}})\\qquad \\textrm {with } \\widetilde{\\mathbf {\\Omega }} :=\\mathbf {\\Omega } /M_{\\mathbf {P}}\\qquad \\textrm {(cf.",
"(\\ref {min Pb}) and (\\ref {min Pb prob2}))},$ whose numerical solution is based on Theorem REF which basically states that for large integer $n \\in \\mathbb {N}$ one has $\\inf _{\\mathbf {Q}\\in \\mathbf {\\Omega } } D_{\\varphi }(\\mathbf {Q},\\mathbf {P})\\approx -\\frac{1}{n}\\log \\,\\mathbb {\\Pi }\\left[\\xi _{n}^{\\mathbf {\\widetilde{W}}}\\in \\widetilde{\\mathbf {\\Omega }}\\right]$ in terms of $\\widetilde{\\varphi }:=M_{\\mathbf {P}} \\cdot \\varphi $ and the random vectors $\\xi _{n}^{\\mathbf {\\widetilde{W}}}=\\Big (\\frac{1}{n}\\sum _{i\\in I_{1}^{(n)}}\\widetilde{W}_{i},\\ldots ,\\frac{1}{n}\\sum _{i\\in I_{K}^{(n)}}\\widetilde{W}_{i}\\Big )\\hspace{56.9055pt} \\text{(cf.",
"(\\ref {Xi_n^W vector}))}\\nonumber $ with $n_{k}:=\\lfloor n \\cdot \\widetilde{p}_{k}\\rfloor $ leading to the disjoint index blocks $I_{1}^{(n)}:=\\left\\lbrace 1,\\ldots ,n_{1}\\right\\rbrace $ , $I_{2}^{(n)}:=\\left\\lbrace n_{1}+1,\\ldots ,n_{1}+n_{2}\\right\\rbrace $ , $\\ldots $ , $I_{K}^{(n)} := \\lbrace \\sum _{k=1}^{K-1} n_{k} + 1, \\ldots , n \\rbrace $ .",
"Recall that $\\mathbf {\\widetilde{W}} := (\\widetilde{W}_{1}, \\ldots , \\widetilde{W}_{n})$ is a random vector consisting of components $\\widetilde{W}_{i}$ which are i.i.d.",
"copies of the random variable $\\widetilde{W}$ whose distribution is $\\mathbb {\\Pi }[\\widetilde{W}\\in \\cdot \\,]=\\widetilde{\\mathbb {}}[\\,\\cdot \\,]$ obeying the representation $\\widetilde{\\varphi }(t) =\\sup _{z \\in \\mathbb {R}} \\left( z\\cdot t - \\log \\int _{\\mathbb {R}} e^{zy} d\\widetilde{\\mathbb {}}(y) \\right),\\qquad t \\in \\mathbb {R},\\qquad \\textrm {(cf.",
"(\\ref {brostu3:fo.link.var}))} .\\nonumber $ Hence, the estimation of $D_{\\varphi }(\\mathbf {\\Omega },\\mathbf {P})$ amounts to the estimation of $\\mathbb {\\Pi }\\left[\\xi _{n}^{\\mathbf {\\widetilde{W}}}\\in \\widetilde{\\mathbf {\\Omega }} \\right]$ .",
"For the rest of this subsection, we assume that $\\widetilde{{P}} \\in \\mathbb {S}_{> 0}^{K}$ , that $n$ is chosen such that all $n \\cdot \\widetilde{p}_{k}$ are integers (and hence, $n = \\sum _{k=1}^{K} n_{k}$ ), and that $\\widetilde{\\mathbf {\\Omega }} \\subset \\mathbb {R}^{K}$ satisfies the regularity property $cl(\\widetilde{\\mathbf {\\Omega }})=cl\\left( int\\left( \\widetilde{\\mathbf {\\Omega }} \\right) \\right) , \\qquad int\\left( \\widetilde{\\mathbf {\\Omega }} \\right) \\ne \\emptyset \\nonumber $ which implies that the same condition holds for $\\mathbf {\\Omega }$ ; moreover, we suppose that $D_{\\widetilde{\\varphi } }(\\widetilde{\\mathbf {\\Omega }},\\widetilde{{P}})$ is finite.",
"For the ease of the following discussions, we introduce the notations $T\\left( \\mathbf {x}\\right) :=\\left( \\frac{1}{n_{1}}\\sum _{i\\in I_{1}^{(n)}}x_{i},\\ldots ,\\frac{1}{n_{K}}\\sum _{i\\in I_{K}^{(n)}} x_{i}\\right)\\quad \\textrm {for any $ x:=( x1,..,xn) Rn$,}$$as well as $ D$ for the diagonal matrix with diagonal entries $ 1/p1,...,1/pK$and null entries off the diagonal.Accordingly, the probability in (\\ref {LDP Minimization approx}) becomes$$\\mathbb {\\Pi }\\left[\\xi _{n}^{\\mathbf {\\widetilde{W}}}\\in \\widetilde{\\mathbf {\\Omega }} \\right] \\ = \\ \\mathbb {\\Pi }\\left[ T(\\mathbf {\\widetilde{W}}) \\in \\mathbf {\\Lambda }\\right]$$where$$\\mathbf {\\Lambda } :=\\mathfrak {D} \\cdot \\widetilde{\\mathbf {\\Omega }}$$is a set of vectors in $ RK$ which is known/derived from the concrete context.The \\textit {naive estimator} $ Lnaive$ of$ [ nW ]$is constructedthrough the following procedure:simulate independently $ L$ copies$ W1,...,WL$ of the vector$ W:=( W1,...,Wn) $,with independent entries under $$, and define(with a slight abuse of notation)$$\\widehat{\\widetilde{\\Pi }}_{L}^{naive} :=\\frac{1}{L}\\sum _{\\ell =1}^{L} \\mathbf {1}_{\\mathbf {\\Lambda }}\\left( T\\left( \\mathbf {\\widetilde{W}}^{\\ell }\\right) \\right) ;$$however this procedure is time costly, since this estimate has a very bad hit rate.Thus, in the following, a so-called “efficient Importance Sampling (IS)”\\ scheme --- in the sense of Sadowsky\\& Bucklew \\cite {Sad:90} (denoted [SB] hereunder) ---is adapted for the sophisticated (i.e.",
"non-naive) estimation of$ [ nW ]$.The main property of IS schemes lays in the fact that the runtime for anestimate with a controlled relative error does \\textit {not} increase atexponential rate as $ n$ increases, in contrast to$ Lnaive$ which has exponential increase.In detail, let $ >0$ be agiven relative precision for an estimator $ PSLn$ of$ [ nW ]$, based on a number $ Ln$ of simulated samplesgenerated under some distribution $ S$, so that$$\\delta :=\\frac{var_{\\widetilde{S}}P_{\\widetilde{S}}^{L_{n}}}{\\left(\\mathbb {\\Pi }\\left[\\xi _{n}^{\\mathbf {\\widetilde{W}}}\\in \\widetilde{\\mathbf {\\Omega }}\\right]\\right)^{2}} \\, .$$Then $ Ln$ will grow exponentially as $ n$ tends to infinity if and only if$ S$ is not “asymptotically optimal”\\ ,the derivation of which is the scope of the current section.$ To start with the details, for the sake of brevity (to avoid certain substantial discussions on potential technical relaxations) we shall employ the following additional Assumption (OM) on the set $\\widetilde{\\mathbf {\\Omega }}$ : (OM) For any $\\widetilde{\\omega } \\in cl(\\widetilde{\\mathbf {\\Omega }})$ there exists a vector $\\mathbf {x} = \\left(x_{1},\\ldots ,x_{n}\\right) \\in \\left] t_{-}^{sc},t_{+}^{sc}\\right[^{n}$ such that $\\widetilde{\\omega } = \\left( \\frac{1}{n}\\sum _{i\\in I_{1}^{(n)}}x_{i},\\ldots ,\\frac{1}{n} \\sum _{i\\in I_{K}^{(n)}}x_{i}\\right)$ , or equivalently, for any $\\lambda \\in cl(\\mathbf {\\Lambda })$ there exists a vector $\\mathbf {x} = \\left(x_{1},\\ldots ,x_{n}\\right) \\in \\left] t_{-}^{sc},t_{+}^{sc}\\right[^{n}$ such that $\\lambda =T\\left(\\mathbf {x}\\right)$ .",
"For instance, in the common case $dom(\\widetilde{\\varphi })= dom(\\varphi ) = \\, ]a,b[ \\, =\\, \\left] t_{-}^{sc},t_{+}^{sc}\\right[ \\, = \\, ]0,\\infty [$ (e.g.",
"for the power-divergence generators $\\widetilde{\\varphi }= \\widetilde{c} \\cdot \\varphi _{\\gamma }$ , $\\gamma \\le 0$ , cf.",
"Example REF ) the Assumption (OM) is always feasible.",
"To proceed, for any distribution $\\widetilde{S}$ on $\\mathbb {R}^{n}$ with support included in the support of the product measure $\\widetilde{\\mathbb {}}^{\\otimes n}$ it holds $\\mathbb {\\Pi }\\left[\\xi _{n}^{\\mathbf {\\widetilde{W}}}\\in \\widetilde{\\mathbf {\\Omega }}\\right]=E_{\\widetilde{\\mathbb {}}^{\\otimes n}} \\left[\\mathbf {1}_{\\mathbf {\\Lambda } }( T( \\mathbf {\\widetilde{W}}))\\right] =E_{\\widetilde{S}}\\left[\\mathbf {1}_{\\mathbf {\\Lambda } }\\left( T\\left( \\mathbf {\\widetilde{V}}\\right) \\right) \\cdot \\frac{d\\widetilde{\\mathbb {}}^{\\otimes n}}{d\\widetilde{S}}\\left( \\mathbf {\\widetilde{V}}\\right) \\right]$ from where the improved IS estimator of $\\mathbb {\\Pi }\\left[\\xi _{n}^{\\mathbf {\\widetilde{W}}}\\in \\widetilde{\\mathbf {\\Omega }} \\right]$ is obtained by sampling $L$ i.i.d.",
"replications $\\mathbf {\\widetilde{V}}^{1},\\ldots ,\\mathbf {\\widetilde{V}}^{L}$ of the random vector $\\mathbf {\\widetilde{V}}$ with distribution $\\widetilde{S}$ and by defining $\\widehat{\\widetilde{\\Pi }}_{L}^{improved}:=\\frac{1}{L}\\sum _{\\ell =1}^{L}\\mathbf {1}_{\\mathbf {\\Lambda }}( T( \\mathbf {\\widetilde{V}}^{(\\ell )})) \\cdot \\frac{d\\widetilde{\\mathbb {}}^{\\otimes n}}{d\\widetilde{S}}\\left( \\mathbf {\\widetilde{V}}^{(\\ell )}\\right) \\ .$ The precise form of the efficient IS distribution $\\widetilde{S}^{opt}$ relies on the definition of a “dominating point” of $\\mathbf {\\Lambda }$ , which we recall now.",
"For $\\mathbf {x} := \\left( x_{1},..,x_{n}\\right)$ in $\\mathbb {R}^{n}$ we define $I_{\\mathbf {\\widetilde{W}}}(\\mathbf {x}):=\\sup _{\\mathbf {z} \\in \\mathbb {R}^{n}}\\left(\\left\\langle \\mathbf {z}, \\mathbf {x} \\right\\rangle -\\log E_{\\widetilde{\\mathbb {}}}[ \\, \\exp (\\langle \\mathbf {z},\\mathbf {\\widetilde{W}}\\rangle ) ]\\right) \\, ,$ and for $\\lambda $ in $\\mathbf {\\Lambda }$ we let $I(\\lambda ):=\\inf \\left\\lbrace I_{\\mathbf {\\widetilde{W}}}(\\mathbf {x}):T(\\mathbf {x)=\\lambda }\\right\\rbrace .$ Let $\\underline{\\lambda }:=\\left( \\underline{\\lambda }_{1},\\ldots ,\\underline{\\lambda }_{K}\\right) \\in \\partial \\mathbf {\\Lambda }$ .",
"We call $\\underline{\\lambda }$ a minimal rate point (mrp) of $\\mathbf {\\Lambda }$ if $I(\\underline{\\lambda })\\le I(\\lambda ) \\quad \\text{ for all }\\lambda \\in \\mathbf {\\Lambda } .$ A minimal rate point $\\underline{\\lambda }$ is called a dominating point of $\\mathbf {\\Lambda }$ if a) $\\underline{\\lambda }\\in \\partial \\mathbf {\\Lambda }$ , and b) $I\\left(\\underline{\\lambda }\\right)\\le I(\\lambda )$ for all $\\lambda \\in \\mathbf {\\Lambda }$ with attainment, namely there exists a vector $\\underline{\\mathbf {x}} \\in \\left] t_{-}^{sc},t_{+}^{sc}\\right[^{n}$ such that $I_{\\widetilde{W}}\\left( \\underline{\\mathbf {x}}\\right) =I\\left(\\underline{\\lambda }\\right)$ with $\\underline{\\lambda }=T\\left( \\underline{\\mathbf {x}}\\right)$ .",
"The characterization of the dominating point $\\underline{\\lambda }$ is settled in the following Lemma 57 Let $\\underline{\\lambda }$ be a mrp of $\\mathbf {\\Lambda }$ .",
"Then, under Assumption (OM), $\\underline{\\lambda }$ is a dominating point, and $\\inf \\left\\lbrace I_{\\mathbf {\\widetilde{W}}}\\left( \\mathbf {x}\\right) ,T(\\mathbf {x})=\\underline{\\lambda }\\right\\rbrace $ is reached at some vector $\\underline{\\mathbf {x}}$ in $\\left] t_{-}^{sc},t_{+}^{sc}\\right[^{n}$ such that for all $k \\in \\left\\lbrace 1,\\ldots ,K\\right\\rbrace $ and all $i \\in I_{k}^{(n)}$ there holds $\\underline{x}_{i}= \\underline{\\lambda }_{k}$ and $I_{\\mathbf {\\widetilde{W}}}(\\underline{\\mathbf {x}})=n \\cdot \\sum _{k=1}^{K}\\widetilde{p}_{k} \\cdot \\widetilde{\\varphi }\\left(\\underline{\\lambda }_{k}\\right)$ .",
"The proof Lemma REF is given in Appendix G. Notice that (OM) implies the existence of a dominating point $\\underline{\\lambda }$ , but uniqueness may not hold.",
"In the latter case, one can try to proceed as in Theorem 2 of [SB] and the discussion thereafter.",
"However, we assume now uniqueness of $\\underline{\\lambda }$ ; this allows for the identification of $\\widetilde{S}^{opt}$ .",
"By Theorem 1 of [SB] and Theorem 3.1 of Csiszar [96], the asymptotically optimal IS distribution $\\widetilde{S}^{opt}$ is obtained as the Kullback-Leibler projection of $\\widetilde{\\mathbb {}}^{n\\otimes }$ on the set of all probability distributions on $\\mathbb {R}^{n}$ centered at point $\\underline{\\mathbf {x}}$ , whose coordinates are — according to Lemma REF — functions of the coordinates of $\\underline{\\mathbf {\\widetilde{Q}}} := \\mathfrak {D}^{-1} \\underline{\\lambda }$ such that $T\\left( \\underline{\\mathbf {x}}\\right)= \\mathfrak {D} \\underline{\\mathbf {\\widetilde{Q}}}$ .",
"The above definition of $\\widetilde{S}^{opt}$ presumes the knowledge of $\\underline{\\lambda }$ , which cannot be assumed (otherwise the minimization problem is solved in advance).",
"The aim of the following construction is to provide a proxy $\\widetilde{S}$ to $\\widetilde{S}^{opt}$ , where $\\widetilde{S}$ is the Kullback-Leibler projection of $\\widetilde{\\mathbb {}}^{\\otimes n}$ on the set of all probability distributions on $\\mathbb {R}^{n}$ centered at some point $\\mathbf {x}^{\\ast }$ which is close to $\\underline{\\mathbf {x}}$ .",
"For this sake, we need to have at hand a proxy of $\\underline{\\lambda }$ or, equivalently, a preliminary guess $\\mathbf {\\widetilde{Q}}^{\\ast }$ of $\\underline{\\mathbf {\\widetilde{Q}}}:=\\arg \\inf _{\\mathbf {\\widetilde{Q}} \\in \\widetilde{\\mathbf {\\Omega }}}\\sum _{k=1}^{K} \\widetilde{p}_{k} \\cdot \\widetilde{\\varphi }(\\widetilde{q}_{k}/\\widetilde{p}_{k})$ .",
"This guess is by no means produced in order to provide a direct estimate of $D_{\\widetilde{\\varphi } }(\\widetilde{\\mathbf {\\Omega }},\\widetilde{{P}})$ but merely to provide the IS distribution $\\widetilde{S}$ which in turn leads to a sharp estimate of $D_{\\widetilde{\\varphi } }(\\widetilde{\\mathbf {\\Omega }},\\widetilde{{P}})$ .",
"Proxy method 1: in some cases we might have at hand some particular point $\\mathbf {\\widetilde{Q}}^{\\ast } :=\\left(\\widetilde{q}_{1}^{\\ast },..,\\widetilde{q}_{K}^{\\ast }\\right)$ in $\\widetilde{\\mathbf {\\Omega }}$ ; the resulting IS distribution $\\widetilde{S}$ with $\\mathbf {\\widetilde{Q}}$ substituted by $\\mathbf {\\widetilde{Q}^{\\ast }}$ is not optimal in the sense of [SB], but anyhow produces an estimator with good hitting rate, possibly with a loss in the variance.",
"A simple way to obtain such a point $\\mathbf {\\widetilde{Q}^{\\ast }}$ in $\\widetilde{\\mathbf {\\Omega }}$ is to simulate runs of (say) $M-$ variate i.i.d.",
"vectors $\\mathbf {\\widetilde{W}}$ under $\\widetilde{\\mathbb {}}^{\\otimes M}$ until the first time where $\\xi _{M}^{\\mathbf {\\widetilde{W}}}$ belongs to $\\widetilde{\\mathbf {\\Omega }}$ ; then we set $\\mathbf {\\widetilde{Q}^{\\ast }} := \\xi _{M}^{\\mathbf {\\widetilde{W}}}$ for the succeeding realization $\\mathbf {\\widetilde{W}}$ .",
"Before we proceed, it is useful to mention that the need for a drastic fall in the number of simulation runs pertains for cases when $D_{\\widetilde{\\varphi } }(\\widetilde{\\mathbf {\\Omega }},\\widetilde{{P}})$ is large.",
"The following construction is suited to this case, which is of relevance in applications both in optimization and in statistics when choosing between competing models none of which is assumed to represent the true one, but merely less inadequate ones.",
"Proxy method 2: when $D_{\\widetilde{\\varphi } }(\\widetilde{\\mathbf {\\Omega }},\\widetilde{{P}})$ is presumably large, we make use of asymptotic approximation to get a proxy of $\\underline{\\mathbf {\\widetilde{Q}}}$ .",
"For this, we define a sampling distribution on $\\mathbb {R}^{K}$ fitted to the divergence through $f(\\mathbf {\\widetilde{Q}}):=C \\cdot \\exp \\left( -\\sum _{k=1}^{K} \\widetilde{p}_{k} \\cdot \\widetilde{\\varphi }(\\widetilde{q}_{k}/\\widetilde{p}_{k})\\right) \\ = \\ C \\cdot \\exp \\left( -D_{\\widetilde{\\varphi }}\\left( \\mathbf {\\widetilde{Q}},\\widetilde{{P}}\\right)\\right)$ where $C$ is a normalizing constant.",
"Let $\\mathbf {T}$ be a $K-$ variate random variable with density $f$ .",
"The distribution of $\\mathbf {T}$ given $\\left( \\mathbf {T} \\in \\widetilde{\\mathbf {\\Omega }}\\right)$ concentrates around $\\arg \\inf _{\\widetilde{\\mathbf {Q}}\\in \\widetilde{\\mathbf {\\Omega }} }D_{\\widetilde{\\varphi } }(\\widetilde{\\mathbf {Q}},\\widetilde{{P}})$ when $D_{\\widetilde{\\varphi } }(\\widetilde{\\mathbf {\\Omega }},\\widetilde{{P}})$ is large.",
"Indeed, for any $\\widetilde{\\mathbf {Q}}\\in \\widetilde{\\mathbf {\\Omega }}$ denote by $\\mathbf {V}_{\\varepsilon }(\\widetilde{\\mathbf {Q}})$ a small neighborhood of $\\widetilde{\\mathbf {Q}}$ in $\\mathbb {R}^{K}$ with radius $\\varepsilon $ ; clearly, the probability of the event $\\left( \\mathbf {T} \\in \\mathbf {V}_{\\varepsilon }(\\widetilde{\\mathbf {Q}})\\right)$ when restricted to $\\widetilde{\\mathbf {Q}}\\in \\widetilde{\\mathbf {\\Omega }}$ is maximum when $\\widetilde{\\mathbf {Q}} = \\underline{\\widetilde{\\mathbf {Q}}}$ , where $\\underline{\\widetilde{\\mathbf {Q}}}$ is the “dominating point of $\\widetilde{\\mathbf {\\Omega }}$ ” in the sense that $\\underline{\\mathbf {\\widetilde{Q}}} := \\mathfrak {D}^{-1} \\underline{\\lambda }$ is the above-defined transform of the dominating point $\\underline{\\lambda }$ (assuming uniqueness); a precise argumentation under adequate conditions is postponed to Appendix G. Accordingly, we obtain a proxy $\\mathbf {\\widetilde{Q}}^{\\ast }$ of $\\underline{\\mathbf {\\widetilde{Q}}}$ by simulating a sequence of independent $K-$ variate random variables $\\mathbf {T}_{1},\\ldots $ with distribution (REF ) until (say) $\\mathbf {T}_{m}$ belongs to $\\widetilde{\\mathbf {\\Omega }}$ and set $\\mathbf {\\widetilde{Q}}^{\\ast }:=\\mathbf {T}_{m}$ .",
"To proceed with the derivation of the IS sampling distribution $\\widetilde{S}$ on $\\mathbb {R}^{n}$ , we fix $\\mathbf {\\widetilde{Q}}^{\\ast } :=\\left(\\widetilde{q}_{1}^{\\ast },..,\\widetilde{q}_{K}^{\\ast }\\right)$ to be a proxy of $\\underline{\\mathbf {\\widetilde{Q}}}$ or an initial guess in $\\widetilde{\\mathbf {\\Omega }}$ .",
"As an intermediate step, we construct the probability distribution $\\widetilde{U}_{k}$ on $\\mathbb {R}$ given by $d\\widetilde{U}_{k}(v) \\ := \\ \\exp \\left( \\tau _{k} \\cdot v -\\Lambda _{\\widetilde{\\mathbb {}}}(\\tau _{k})\\right) d\\widetilde{\\mathbb {}}(v)=\\frac{\\exp \\left(\\tau _{k} \\cdot v\\right)}{MGF_{\\widetilde{\\mathbb {}}}(\\tau _{k})} \\, d\\widetilde{\\mathbb {}}(v)$ where $\\tau _{k} \\in int(dom(MGF_{\\widetilde{\\mathbb {}}}))$ is the unique solution of the equation $\\Lambda _{\\widetilde{\\mathbb {}}}^{\\prime }\\left( \\tau _{k}\\right) =\\frac{\\widetilde{q}_{k}^{\\ast }}{\\widetilde{p}_{k}}$ and thus — by relation (REF ) of Appendix F — we can compute explicitly $\\tau _{k} = \\ \\widetilde{\\varphi }^{\\, \\prime }\\left(\\frac{\\widetilde{q}_{k}^{\\ast }}{\\widetilde{p}_{k}}\\right) \\ .$ Therefore, $\\widetilde{U}_{k}$ is the Kullback-Leibler projection of $\\widetilde{\\mathbb {}}$ on the class of all probability distributions on $\\mathbb {R}$ whose expectation is $\\widetilde{q}_{k}^{\\ast }$ .",
"As a side remark, notice that one possible way of obtaining an explicit form of the probability distribution $\\widetilde{U}_{k}$ may be by identification through its moment generating function $dom(MGF_{\\widetilde{\\mathbb {}}})-\\tau _{k} \\ \\ni \\ z \\ \\mapsto MGF_{\\widetilde{U}_{k}}(z) =\\frac{MGF_{\\widetilde{\\mathbb {}}}(z+\\tau _{k})}{MGF_{\\widetilde{\\mathbb {}}}(\\tau _{k})}$ of which all ingredients are principally available.",
"For instance, this will be used in Example REF below.",
"From (REF ), we define $\\widetilde{S}_{k}:=\\underbrace{\\widetilde{U}_{k} \\otimes \\cdots \\otimes \\widetilde{U}_{k}}_{n_{k}\\text{times}}\\qquad \\textrm {for all $ {1,...,K}$},$$whence\\begin{eqnarray}d\\widetilde{S}_{k}\\left( v_{k,1},\\ldots ,v_{k,n_{k}}\\right) =\\exp \\Big ( \\sum _{i\\in I_{k}^{(n)}}\\tau _{k} \\cdot v_{k,i}-n_{k} \\cdot \\Lambda _{\\widetilde{\\mathbb {}}}(\\tau _{k})\\Big ) \\ d\\widetilde{\\mathbb {}}\\left(v_{k,1}\\right)\\cdots d\\widetilde{\\mathbb {}}\\left(v_{k,n_{k}}\\right),\\\\\\end{eqnarray}which manifests $ Sk$ as the Kullback-Leibler projection of$ nktimes$on the class of all probability distributions on $ Rk$whose expectation vector is$ Q = (q1,...,qK) Rk$.",
"Let now\\begin{equation}\\widetilde{S} := \\widetilde{S}_{1} \\otimes \\cdots \\otimes \\widetilde{S}_{K} \\, ,\\end{equation}which therefore satisfies (recall that $ k=1K nk =n$)\\begin{equation}d\\widetilde{S}\\left( v_{1,1},\\ldots v_{1,n_{1}},\\ldots , v_{K,1},\\ldots v_{K,n_{K}}\\right)=\\exp \\Big (\\sum _{k=1}^{K}\\sum _{i\\in I_{k}^{(n)}} \\big ( \\tau _{k} \\cdot v_{k,i} - n_{k} \\cdot \\Lambda _{\\widetilde{\\mathbb {}}}(\\tau _{k}) \\big )\\Big ) \\,d\\widetilde{\\mathbb {}}^{\\otimes n}\\left( v_{1,1},\\ldots v_{1,n_{1}},\\ldots , v_{K,1},\\ldots v_{K,n_{K}}\\right).\\ \\end{equation}The same procedure with all $ qk$substituted by the coordinates $qk$of $Q$produces $ Sopt$.",
"Therefore, $ S$ is a substitutefor $ Sopt$ with the change in the centering fromthe unknown vector $Q$to its proxy $ Q$.$ As a straightforward consequence of (REF ) and (), we obtain the improved IS estimator of $\\mathbb {\\Pi }\\left[\\xi _{n}^{\\mathbf {\\widetilde{W}}}\\in \\widetilde{\\mathbf {\\Omega }}\\right]$ as $\\widehat{\\widetilde{\\Pi }}_{L}^{improved}\\ =\\ \\frac{1}{L}\\sum _{\\ell =1}^{L}\\mathbf {1}_{\\mathbf {\\Lambda }}( T( \\mathbf {\\widetilde{V}}^{(\\ell )})) \\cdot \\prod _{k=1}^{K} IS_{k}(\\mathbf {\\widetilde{V}}_{k}^{(\\ell )})$ where $\\mathbf {\\widetilde{V}}_{k}^{(\\ell )} := \\left( \\widetilde{V}_{i}^{(\\ell )} \\right)_{i \\in I_{k}^{(n)}}$ is the $k-$ th block of the $\\ell -$ th replication $\\mathbf {\\widetilde{V}}^{(\\ell )}$ of $\\mathbf {\\widetilde{V}}$ under $S$ , and the $k-$ th importance-sampling factor is $\\widetilde{IS}_{k}(v_{k,1},\\ldots ,v_{k,n_{k}}) \\ := \\ \\frac{d\\widetilde{\\mathbb {}}^{\\otimes n_{k}}}{d\\widetilde{S}_{k}}\\left( v_{k,1},\\ldots ,v_{k,n_{k}}\\right)= \\exp \\Big (n_{k} \\cdot \\Lambda _{\\widetilde{\\mathbb {}}}(\\tau _{k}) \\, - \\, \\tau _{k}\\cdot \\sum _{i=1}^{n_{k}} v_{k,i} \\Big )\\nonumber $ with $n_{k} = card(I_{k}^{(n)})$ .",
"Summing up things, we arrive at the following algorithm in case that $\\widetilde{\\mathbf {\\Omega }}$ has a unique dominating point (in the above-defined sense): Step D1 Exemplarily, we start with proxy method 2 (the other proxy method 1 works analogously): get a proxy $\\mathbf {\\widetilde{Q}}^{\\ast }$ of $\\underline{\\mathbf {\\widetilde{Q}}}$ by simulating a sequence of independent $K-$ variate random variables $\\mathbf {T}_{1},\\ldots $ with distribution (REF ) until (say) $\\mathbf {T}_{m}$ belongs to $\\widetilde{\\mathbf {\\Omega }}$ and set $\\mathbf {\\widetilde{Q}}^{\\ast }:=\\mathbf {T}_{m}$ .",
"Step D2 For all $k$ in $\\lbrace 1,\\ldots ,K\\rbrace $ compute $\\tau _{k} = \\ \\widetilde{\\varphi }^{\\, \\prime }\\left(\\frac{\\widetilde{q}_{k}^{\\ast }}{\\widetilde{p}_{k}}\\right)$ .",
"Step D3 For all $\\ell $ in $\\lbrace 1,\\ldots ,L\\rbrace $ perform a run of $\\mathbf {\\widetilde{V}}^{(\\ell )}$ under $\\widetilde{S}$ as follows: For all $k$ in $\\lbrace 1,\\ldots ,K\\rbrace $ simulate $n_{k}$ i.i.d.",
"random variables $\\widetilde{V}_{k_{1}}^{(\\ell )},\\ldots ,\\widetilde{V}_{k_{n_{k}}}^{(\\ell )}$ with common distribution $\\widetilde{U}_{k}$ defined in (REF ).",
"Set $\\mathbf {\\widetilde{V}}_{k}^{(\\ell )} :=(\\widetilde{V}_{k_{1}}^{(\\ell )},\\ldots ,\\widetilde{V}_{k_{n_{k}}}^{(\\ell )})$ to be the corresponding row vector.",
"Construct $\\mathbf {\\widetilde{V}}^{(\\ell )}$ as the row vector obtained by concatenating the $\\mathbf {\\widetilde{V}}_{k}^{(\\ell )}$ , i.e.",
"$\\mathbf {\\widetilde{V}}^{(\\ell )}:=\\left( \\mathbf {\\widetilde{V}}_{1}^{(\\ell )},\\ldots ,\\mathbf {\\widetilde{V}}_{K}^{(\\ell )}\\right) \\, ,$ and make use of $\\widehat{\\widetilde{\\Pi }}_{L}^{improved}$ given in (REF ) with the $\\tau _{k}$ 's obtained in Step D2 above to define (in the light of (REF ), (REF )) the BS minimum-distance estimator $\\widehat{D_{\\varphi }}(\\mathbf {\\Omega },\\mathbf {P})\\ := \\ \\widehat{D_{\\widetilde{\\varphi }}}(\\widetilde{\\mathbf {\\Omega }},\\widetilde{{P}})\\ := \\ - \\frac{1}{n}\\log \\widehat{\\widetilde{\\Pi }}_{L}^{improved} \\ .$ For many cases, the simulation burden needed for the computation of $\\widehat{\\widetilde{\\Pi }}_{L}^{improved}$ — and thus of $\\widehat{D_{\\varphi }}(\\mathbf {\\Omega },\\mathbf {P})$ — can be drastically reduced, especially for high dimensions $K$ and large sample size $n \\cdot L$ .",
"In fact, in terms of the notations $n_{k}:=card(I_{k}^{(n)})$ , $\\widehat{\\mathit {W}}_{k}^{(\\ell )}:=\\sum _{i\\in I_{k}^{(n)}}\\widetilde{V}_{i}^{(\\ell )}$ and $\\widetilde{ISF}_{k}(x) \\ := \\ \\frac{d\\widetilde{\\mathbb {}}^{\\ast n_{k}}}{d\\widetilde{U}_{k}^{\\ast n_{k}}}(x) \\ = \\ \\exp (n_{k} \\cdot \\Lambda _{\\widetilde{\\mathbb {}}}(\\tau _{k}) \\, - \\, x \\cdot \\tau _{k} )$ (where $\\widetilde{\\mathbb {}}^{\\ast n_{k}}$ is the $n_{k}-$ convolution of the measure $\\widetilde{\\mathbb {}}$ ), one can rewrite (REF ) as $\\widehat{\\widetilde{\\Pi }}_{L}^{improved}=\\frac{1}{L}\\sum _{\\ell =1}^{L}\\mathbf {1}_{\\mathbf {\\Lambda }}\\big ( \\,\\big (\\frac{1}{n_{1}} \\widehat{\\mathit {W}}_{1}^{(\\ell )},\\ldots , \\frac{1}{n_{K}} \\widehat{\\mathit {W}}_{K}^{(\\ell )}\\big ) \\, \\big ) \\cdot \\prod \\limits _{k=1}^{K}\\widetilde{ISF}_{k}(\\widehat{\\mathit {W}}_{k}^{(\\ell )}) .$ with $K-$ vector $\\big (\\frac{1}{n_{1}} \\widehat{\\mathit {W}}_{1}^{(\\ell )},\\ldots , \\frac{1}{n_{K}} \\widehat{\\mathit {W}}_{K}^{(\\ell )} \\big )$ .",
"Clearly, the random variable $\\widehat{\\mathit {W}}_{k}^{(\\ell )}$ ($k=1, \\ldots , K$ ) has distribution $\\widetilde{U}_{k}^{\\ast n_{k}}$ .",
"Hence, if $\\widetilde{U}_{k}^{\\ast n_{k}}$ can be explicitly constructed, then for the computation of $\\widehat{\\widetilde{\\Pi }}_{L}^{improved}$ it suffices to simulate the $K \\cdot L$ random variables $\\widehat{\\mathit {W}}_{k}^{(\\ell )}$ rather than the $n \\cdot L$ random variables $\\widetilde{V}_{i}^{(\\ell )}$ ; notice that according to the right-hand side of (REF ), one can explicitly compute $ISF_{k}\\left( \\cdot \\right)$ which can be interpreted as Importance Sampling Factor pertaining to the block $k$.",
"In the case that $\\widetilde{\\mathbb {}}$ is infinitely divisible, simulation issues may become especially comfortable.",
"In the following, we exemplarily demonstrate the tractability of this reduction effect, for the BS minimization of the important power divergences (for which the infinite divisibility holds): Example 58 Let $\\varphi _{\\gamma }$ ($\\gamma \\in \\mathbb {R}\\backslash ]1,2[$ ) be the power divergence generator from the Examples REF and REF , $\\mathbf {P} \\in \\mathbb {R}_{> 0}^{K}$ , $M_{\\mathbf {P}}:=\\sum _{i=1}^{K}p_{i}>0$ and $n_{k} = n \\cdot p_{k} \\in \\mathbb {N}$ where we have employed our notation $n_{k}= n \\cdot p_{k} \\in \\mathbb {N}$ for all $k \\in \\lbrace 1,\\ldots ,K\\rbrace $ .",
"Moreover, let $\\mathbf {\\widetilde{Q}}^{\\ast } :=\\left(\\widetilde{q}_{1}^{\\ast },\\ldots ,\\widetilde{q}_{K}^{\\ast }\\right)$ be a proxy obtained by either proxy method 1 or 2.",
"Case 1: Example REF (a): $\\gamma =0,\\widetilde{c}>0$ .",
"There holds $\\widetilde{U}_{k}^{\\ast n_{k}}=GAM\\left( \\widetilde{c} \\cdot M_{\\mathbf {P}} - \\tau _{k},n_{k}\\cdot \\widetilde{c} \\cdot M_{\\mathbf {P}}\\right)$ , with $\\tau _{k}=\\widetilde{c} \\cdot M_{\\mathbf {P}} \\cdot ( 1-\\frac{p_{k}}{M_{\\mathbf {P}} \\cdot \\widetilde{q}_{k}^{\\ast }})$ for $\\widetilde{q}_{k}^{\\ast } >0$ (the latter is equivalent to $\\tau _{k} < \\widetilde{c} \\cdot M_{\\mathbf {P}}$ ).",
"Moreover, for all $x >0$ one gets $\\widetilde{ISF}_{k}(x)=\\left( \\frac{\\widetilde{c} \\cdot M_{\\mathbf {P}}}{\\widetilde{c}\\cdot M_{\\mathbf {P}} - \\tau _{k}}\\right)^{n_{k}\\cdot \\widetilde{c}\\cdot M_{\\mathbf {P}}}\\cdot e^{-\\tau _{k} \\cdot x}$ .",
"Case 2: REF (b): $\\gamma \\in \\left( 0,1\\right) ,\\widetilde{c}>0$ .",
"We derive $\\widetilde{U}_{k}^{\\ast n_{k}}=C\\big ( POI( n_{k}\\cdot \\breve{\\theta }),GAM\\big (\\frac{\\widetilde{c} \\cdot M_{\\mathbf {P}}}{1-\\gamma } - \\tau _{k},\\frac{\\gamma }{1-\\gamma } \\big ) \\big )$ with $\\breve{\\theta }:= \\frac{\\widetilde{c} \\cdot M_{\\mathbf {P}}}{\\gamma }\\cdot \\big (\\frac{(\\gamma -1) \\cdot \\tau _{k}}{\\widetilde{c} \\cdot M_{\\mathbf {P}}}+1\\big )^{\\gamma /(\\gamma -1)}$ and $\\tau _{k} = \\widetilde{c} \\cdot M_{\\mathbf {P}} \\cdot \\frac{1-\\big (\\frac{\\widetilde{q}_{k}^{\\ast } \\cdot M_{\\mathbf {P}}}{p_{k}}\\big )^{\\gamma -1}}{1-\\gamma }$ for $\\widetilde{q}_{k}^{\\ast } >0$ .",
"Furthermore, $\\widetilde{ISF}_{k}(x)=e^{-\\tau _{k}x} \\cdot \\exp \\left( \\frac{n_{k}\\cdot \\widetilde{c} \\cdot M_{\\mathbf {P}}}{\\gamma }\\cdot \\left(\\left(1+ \\frac{\\gamma -1}{\\widetilde{c} \\cdot M_{\\mathbf {P}}} \\cdot \\tau _{k}\\right) ^{\\frac{\\gamma }{\\gamma -1}}-1 \\right)\\right), \\qquad x \\ge 0 ,$ (where $x=0$ covers the atom at zero).",
"Case 3: Example REF (c): $\\gamma =2,\\widetilde{c}>0$ .",
"One gets $\\widetilde{U}_{k}^{\\ast n_{k}}=N(n_{k}\\cdot (1+\\frac{\\tau _{k}}{\\widetilde{c} \\cdot M_{\\mathbf {P}}}),\\frac{n_{k}}{\\widetilde{c} \\cdot M_{\\mathbf {P}}})$ with $\\tau _{k}=\\widetilde{c} \\cdot M_{\\mathbf {P}} \\cdot ( \\frac{\\widetilde{q}_{k}^{\\ast } \\cdot M_{\\mathbf {P}}}{p_{k}} - 1 ) $ for $\\widetilde{q}_{k}^{\\ast } \\in \\mathbb {R}$ .",
"Moreover, for all $x \\in \\mathbb {R}$ one obtains $\\widetilde{ISF}_{k}(x)= \\exp \\big (\\frac{n_{k} \\cdot \\tau _{k}^2}{2\\widetilde{c} \\cdot M_{\\mathbf {P}}}- (x- n_{k}) \\cdot \\tau _{k} \\big )$ .",
"Case 4: Example REF (d): $\\gamma <0,\\widetilde{c}>0$ .",
"It holds that $\\widetilde{U}_{k}^{\\ast n_{k}}$ has the (Lebesgue-)density $f_{\\widetilde{U}_{k}^{\\ast n_{k}}}(x)\\ := \\ \\frac{\\exp ((\\tau _{k} - \\frac{\\widetilde{c}\\cdot M_{\\mathbf {P}}}{1-\\gamma })\\cdot x)}{\\exp \\left(n_{k} \\cdot \\frac{\\widetilde{c}\\cdot M_{\\mathbf {P}}}{\\gamma }\\cdot (1+\\frac{\\gamma -1}{\\widetilde{c} \\cdot M_{\\mathbf {P}}}\\cdot \\tau _{k})^{\\gamma /(\\gamma -1)}\\right)}\\cdot f_{\\breve{\\breve{Z}}}(x) \\cdot {1}_{]0,\\infty [}(x),\\qquad x \\in \\mathbb {R},$ where $\\tau _{k} = \\widetilde{c} \\cdot M_{\\mathbf {P}} \\cdot \\frac{1-\\big (\\frac{\\widetilde{q}_{k}^{\\ast } \\cdot M_{\\mathbf {P}}}{p_{k}}\\big )^{\\gamma -1}}{1-\\gamma }$ for $\\widetilde{q}_{k}^{\\ast } >0$ , and $\\breve{\\breve{Z}}$ is a random variable with density $f_{\\breve{\\breve{Z}}}$ of a stable law with parameter-quadruple $(\\frac{-\\gamma }{1-\\gamma },1,0,- n_{k} \\cdot \\frac{(\\widetilde{c}\\cdot M_{\\mathbf {P}})^{1/(1-\\gamma )} \\cdot (1-\\gamma )^{-\\gamma /(1-\\gamma )}}{\\gamma })$ (analogously to $\\breve{Z}$ of Example 40 (d) but with $\\widetilde{c}$ replaced by $\\widetilde{c}\\cdot M_{\\mathbf {P}}$ ).",
"Also, $\\widetilde{ISF}_{k}(x)=e^{-\\tau _{k}x} \\cdot \\exp \\left( \\frac{n_{k}\\cdot \\widetilde{c} \\cdot M_{\\mathbf {P}}}{\\gamma }\\cdot \\left(\\left(1+ \\frac{\\gamma -1}{\\widetilde{c} \\cdot M_{\\mathbf {P}}} \\cdot \\tau _{k}\\right) ^{\\frac{\\gamma }{\\gamma -1}}-1 \\right)\\right), \\qquad x >0.$ Case 5 : Example REF (e): $\\gamma >2,\\widetilde{c}>0$ .",
"We derive that $\\widetilde{U}_{k}^{\\ast n_{k}}$ has the (Lebesgue-)density $f_{\\widetilde{U}_{k}^{\\ast n_{k}}}(x)\\ := \\ \\frac{\\exp ((\\tau _{k} + \\frac{\\widetilde{c}\\cdot M_{\\mathbf {P}}}{\\gamma -1})\\cdot x)}{\\exp \\left(n_{k} \\cdot \\frac{\\widetilde{c}\\cdot M_{\\mathbf {P}}}{\\gamma }\\cdot (1+\\frac{\\gamma -1}{\\widetilde{c} \\cdot M_{\\mathbf {P}}}\\cdot \\tau _{k})^{\\gamma /(\\gamma -1)}\\right)}\\cdot f_{\\breve{\\breve{Z}}}(-x) ,\\qquad x \\in \\mathbb {R},$ where $\\tau _{k} = - \\frac{\\widetilde{c} \\cdot M_{\\mathbf {P}}}{\\gamma -1} \\cdot \\big (1- \\big (\\frac{\\widetilde{q}_{k}^{\\ast } \\cdot M_{\\mathbf {P}}}{p_{k}}\\big )^{\\gamma -1}\\cdot {1}_{]0,\\infty [}(\\widetilde{q}_{k}^{\\ast }) \\big )$ for $\\widetilde{q}_{k}^{\\ast } \\in \\mathbb {R}$ , and $\\breve{\\breve{Z}}$ is a random variable with density $f_{\\breve{\\breve{Z}}}$ of a stable law with parameter-quadruple $(\\frac{\\gamma }{\\gamma -1},1,0,n_{k} \\cdot \\frac{(\\widetilde{c}\\cdot M_{\\mathbf {P}})^{1/(1-\\gamma )} \\cdot (\\gamma -1)^{\\gamma /(\\gamma -1)}}{\\gamma })$ (analogously to $\\breve{Z}$ of Example 40 (e) but with $\\widetilde{c}$ replaced by $\\widetilde{c}\\cdot M_{\\mathbf {P}}$ ).",
"Furthermore, $\\widetilde{ISF}_{k}(x)=e^{-\\tau _{k}x} \\cdot \\exp \\left( \\frac{n_{k}\\cdot \\widetilde{c} \\cdot M_{\\mathbf {P}}}{\\gamma }\\cdot \\left(\\left(1+ \\frac{\\gamma -1}{\\widetilde{c} \\cdot M_{\\mathbf {P}}} \\cdot \\tau _{k}\\right) ^{\\frac{\\gamma }{\\gamma -1}}-1 \\right)\\right), \\qquad x \\in \\mathbb {R}.$ Case 6: Example REF (a): $\\gamma =1,\\widetilde{c}>0$ , anchor point $c=0$ .",
"It holds that $\\widetilde{U}_{k}^{\\ast n_{k}}$ is the probability distribution $\\frac{1}{\\widetilde{c} \\cdot M_{\\mathbf {P}}} \\cdot POI\\left(n_{k} \\cdot \\widetilde{c} \\cdot M_{\\mathbf {P}} \\cdot \\exp (\\frac{\\tau _{k}}{\\widetilde{c} \\cdot M_{\\mathbf {P}}})\\right) $ with support on the lattice $\\left\\lbrace \\frac{j}{\\widetilde{c} \\cdot M_{\\mathbf {P}}}, \\, j\\in \\mathbb {N}_{0}\\right\\rbrace $ , where $\\tau _{k}=\\widetilde{c} \\cdot \\log \\left( \\frac{M_{\\mathbf {P}} \\cdot \\widetilde{q}_{k}^{\\ast }}{p_{k}}\\right) $ for $\\widetilde{\\omega } _{k} >0$ .",
"Moreover, for all $j \\in \\mathbb {N}_{0}$ we obtain (by setting $x:= \\frac{j}{\\widetilde{c} \\cdot M_{\\mathbf {P}}}$ ) $\\widetilde{ISF}_{k}\\left(\\frac{j}{\\widetilde{c} \\cdot M_{\\mathbf {P}}}\\right)= \\exp \\left(n_{k} \\cdot \\widetilde{c} \\cdot M_{\\mathbf {P}} \\cdot \\left( \\exp \\left( \\frac{\\tau _{k}}{\\widetilde{c} \\cdot M_{\\mathbf {P}}}\\right) -1 \\right)- m \\cdot \\frac{\\tau _{k}}{\\widetilde{c} \\cdot M_{\\mathbf {P}}}\\right) .$ Case 7: Example REF (b): $\\gamma =1,\\widetilde{c}=1,$ anchor point $c\\in \\mathbb {R}$ .",
"For $M_{P}=1$ , $\\widetilde{U}_{k}^{\\ast n_{k}}$ is the shifted Poisson distribution $ POI\\left(n_{k} \\cdot e^{c +\\tau _{k}}\\right) + n_{k} \\cdot (1-e^{c})$ with support on the lattice $\\left\\lbrace j + n_{k} \\cdot (1-e^{c}), \\, j\\in \\mathbb {N}_{0}\\right\\rbrace $ , where $\\tau _{k}= \\log \\big (\\frac{\\widetilde{q}_{k}^{\\ast }}{p_{k}} + e^{c}-1\\big )-c$ for $\\widetilde{q}_{k}^{\\ast } > p_{k} \\cdot (1-e^{c})$ .",
"Furthermore, for all $j \\in \\mathbb {N}_{0}$ we obtain (by setting $x:= j + n_{k} \\cdot (1-e^{c})$ ) $\\widetilde{ISF}_{k}\\left(j + n_{k} \\cdot (1-e^{c})\\right)= \\exp \\left(n_{k} \\cdot e^{c} \\cdot (e^{\\tau _{k}} -1) - j \\cdot \\tau _{k} \\right).$ Notice that the mass of $\\widetilde{U}_{k}^{\\ast n_{k}}$ at zero depends on the value of the anchor point $c$ , since $\\widetilde{U}_{k}^{\\ast n_{k}}[\\lbrace 0\\rbrace ] >0$ if and only if $c=\\log (1+\\frac{\\ell }{n_{k}})$ for some $\\ell \\in \\mathbb {N}_{0}$ ; moreover, $\\widetilde{U}_{k}^{\\ast n_{k}}\\big [ \\ ]0,\\infty [ \\ \\big ] =1$ if $c <0$ and $\\widetilde{U}_{k}^{\\ast n_{k}}\\big [ \\ ]-\\infty ,0[ \\ \\big ] >0$ if $c >0$ .",
"Remark 59 (a) One can explicitly see in all cases of the above Example REF that all ingredients for computation are at hand.",
"(b) For both Cases 4 and 5 in the above Example REF , algorithms for simulation can be obtained by adapting e.g.",
"the works of Devroye [111] and Devroye & James [112].",
"In the previous Subsection REF , as a first step we have estimated $\\mathbb {\\Pi }\\left[\\xi _{n}^{\\mathbf {\\widetilde{W}}}\\in \\widetilde{\\mathbf {\\Omega }}\\right]$ in terms of the improved IS estimator $\\widehat{\\widetilde{\\Pi }}_{L}^{improved}$ .",
"From this, as a second step, we have derived — on the basis of Theorem REF — the estimator $\\widehat{D_{\\varphi }}(\\mathbf {\\Omega },\\mathbf {P})\\ := \\ - \\frac{1}{n}\\log \\widehat{\\widetilde{\\Pi }}_{L}^{improved} \\ \\nonumber \\qquad \\textrm {(cf.",
"(\\ref {estimator minimization}))}$ of the minimum distance $D_{\\varphi }(\\mathbf {\\Omega },\\mathbf {P}):= \\inf _{\\mathbf {Q}\\in \\mathbf {\\Omega } } D_{\\varphi }(\\mathbf {Q},\\mathbf {P})$ , where $\\mathbf {P} \\in \\mathbb {R}_{> 0}^{K}$ and $\\mathbf {\\Omega }\\subset \\mathbb {R}^{K}$ .",
"Recall that $\\widetilde{\\mathbf {\\Omega }} :=\\mathbf {\\Omega } /M_{\\mathbf {P}}$ with $M_{\\mathbf {P}}:=\\sum _{i=1}^{K}p_{i}>0$ , and that $\\xi _{n}^{\\mathbf {\\widetilde{W}}}=\\Big (\\frac{1}{n}\\sum _{i\\in I_{1}^{(n)}}\\widetilde{W}_{i},\\ldots ,\\frac{1}{n}\\sum _{i\\in I_{K}^{(n)}}\\widetilde{W}_{i}\\Big )\\hspace{56.9055pt} \\text{(cf.",
"(\\ref {Xi_n^W vector}))}\\nonumber $ where $\\mathbf {\\widetilde{W}} := (\\widetilde{W}_{1}, \\ldots , \\widetilde{W}_{n})$ is a random vector consisting of components $\\widetilde{W}_{i}$ which are i.i.d.",
"copies of the random variable $\\widetilde{W}$ whose distribution is $\\mathbb {\\Pi }[\\widetilde{W}\\in \\cdot \\,]=\\widetilde{\\mathbb {}}[\\,\\cdot \\,]$ obeying the representation (REF ).",
"In contrast, we now proceed as follows: as a first step, we derive an improved estimator $\\widehat{\\Pi }_{L}^{improved}$ of $\\mathbb {\\Pi }_{X_{1}^{n}}\\left[\\xi _{n,\\mathbf {X}}^{w\\mathbf {W}}\\in \\textrm {\\right.$ $\\hspace{-6.544pt}$$}$$where $$\\hspace{-6.544pt}$ SK$is a set of probability vectors which satisfies theregularity properties (\\ref {regularity}) and the finiteness property (\\ref {def fi wrt Omega}).Recall that\\begin{eqnarray}\\xi _{n,\\mathbf {X}}^{w\\mathbf {W}} &:=&{\\left\\lbrace \\begin{array}{ll}\\left(\\frac{\\sum _{i \\in I_{1}^{(n)}}W_{i}}{\\sum _{k=1}^{K}\\sum _{i \\in I_{k}^{(n)}}W_{i}},\\ldots , \\frac{\\sum _{i \\in I_{K}^{(n)}}W_{i}}{\\sum _{k=1}^{K}\\sum _{i \\in I_{k}^{(n)}}W_{i}} \\right) ,\\qquad \\textrm {if } \\sum _{j=1}^{n} W_{j} \\ne 0, \\\\\\ (\\infty , \\ldots , \\infty ) =: \\infty , \\hspace{113.81102pt} \\textrm {if } \\sum _{j=1}^{n} W_{j} = 0,\\end{array}\\right.",
"}\\hspace{42.67912pt}\\textrm {(cf.",
"(\\ref {brostu3:fo.norweiemp.vec}))}\\nonumber \\end{eqnarray}where $ (Xi)iN$ is a sequence of random variableswith values in $ Y:={ d1,,dK}$such that\\begin{equation}\\lim _{n\\rightarrow \\infty } \\Big ( \\frac{n_{1}}{n}, \\ldots , \\frac{n_{K}}{n} \\Big ) = (p_{1}, \\ldots , p_{K})\\qquad \\textrm {a.s.} \\qquad \\textrm {cf.",
"((\\ref {cv emp measure X to P vector}))}\\nonumber \\end{equation}holds for some probability vector $ P := (p1, ..., pK) S> 0K$,by employing the notation$$n_{k} \\ := \\ card(\\bigl \\lbrace i \\in \\lbrace 1, \\ldots , n\\rbrace : \\ X_{i} = d_{k} \\bigr \\rbrace )\\ =: \\ card(I_{k}^{(n)})\\qquad \\textrm {(cf.",
"(\\ref {I^(n)_k for stat case}));}$$hence, on the $ k$-th block of indexes $ Ik(n)$ all the $ Xi$^{\\prime }s share the same value $ dk$.Moreover, recall that $ (Wi)i N$ is a familyof independent and identically distributed $ R-$valued random variableswith probability distribution $ [ ] := [W1 ]$being connected with the divergence generator $ (]a,b[)$ via the representability(\\ref {Phi Legendre of mgf(W)}),such that $ (Wi)i N$ is independent of $ (Xi)i N$.$ As a second step (see Subsubsection REF below), for the important special case of the power-divergence generators $\\varphi _{\\gamma }$ (cf.",
"(REF )) we employ the Propositions REF to REF in order to deduce via the corresponding $\\widehat{\\Pi }_{L}^{improved}$ the estimators (e.g.",
"for $\\gamma <0$ ) $\\widehat{D_{\\widetilde{c}\\cdot \\varphi _{\\gamma }}(\\textrm {\\Omega \\hspace{-6.544pt}\\Omega },{P})} \\ := \\ \\frac{\\widetilde{c}}{\\gamma \\cdot \\left(\\gamma -1\\right) }\\cdot \\left\\lbrace \\left( 1+\\frac{\\gamma }{\\widetilde{c}}\\cdot \\frac{1}{n}\\cdot \\log \\,\\widehat{\\Pi }_{L}^{improved}\\right) ^{1-\\gamma } -1 \\right\\rbrace ,\\qquad \\ $ of the minimum power divergences $D_{\\widetilde{c}\\cdot \\varphi _{\\gamma }}(\\textrm {$ $\\hspace{-6.544pt}$$},{P}) :=\\inf _{{Q}\\in \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega }}D_{\\widetilde{c}\\cdot \\varphi _{\\gamma }}({Q},{P})$$as well as connected estimators of important deterministic transformations thereof.$ As a third step (see Subsubsection REF below), on the basis of Subsubsection REF we derive estimators of bounds of $D_{\\varphi }(\\textrm {$$\\hspace{-6.544pt}$$},{P})$ for more general divergence generators $\\varphi $ .",
"Let us start with the above-mentioned first step, by remarking that the development of the estimator $\\widehat{\\Pi }_{L}^{improved}$ works quite analogously to that of $\\widehat{\\widetilde{\\Pi }}_{L}^{improved}$ in the previous Subsection REF .",
"To make this even more transparent, we employ the notation $p_{n,k}^{emp}:=n_{k}/n$ (cf.",
"(REF )) and label all random vectors of length $n$ in the same way as above: we sort the already given and thus fixed data $X_{i}$ 's in such a way that the first $n_{1}$ of them share the same value $d_{1}$ , and so on, until the last block with length $n_{K}$ in which the data have common value $d_{K}$ .",
"In the light of the above considerations, we could achieve a naive estimate $\\widehat{\\Pi }_{L}^{naive}$ of $\\mathbb {\\Pi }_{X_{1}^{n}}[\\xi _{n,\\mathbf {X}}^{w\\mathbf {W}}\\in \\textrm {$$\\hspace{-6.544pt}$$}]$ through the following procedure.",
"We simulate independently $L$ replicates $\\mathbf {W}^{(1)},\\ldots ,\\mathbf {W}^{(L)}$ of the vector $\\mathbf {W}:=\\left( W_{1},\\ldots ,W_{n}\\right) $ , with independent entries under $\\mathbb {}$ (cf.",
"(REF )); those realizations do not depend on the $X_{i}$ 's.",
"Then we construct $\\widehat{\\Pi }_{L}^{naive} :=\\frac{1}{L}\\sum _{\\ell =1}^{L}\\mathbf {1}_{\\textrm {\\Omega \\hspace{-5.40608pt}\\Omega }}\\left(\\xi _{n,\\mathbf {X}}^{w\\mathbf {W}^{(\\ell )}}\\right) .$ However, this procedure is time costly, since the estimate given in (REF ) has a very bad hit rate.",
"Hence, analogously to Subsection REF we apply again an “efficient Importance Sampling (IS)” scheme in the sense of Sadowsky & Bucklew [313].",
"This will involve the simulation of $L$ independent $n-$ tuples $\\mathbf {V}^{(\\ell )}\\mathbf {:=}\\left(V_{n}^{(\\ell )},\\ldots ,V_{n}^{(\\ell )}\\right) $ with common distribution $S$ on $\\mathbb {R}^{n}$ , such that $\\mathbb {}^{\\otimes n}$ is (measure-)equivalent with respect to $S$ .",
"In fact, we rewrite $\\mathbb {\\Pi }_{X_{1}^{n}}[\\xi _{n,\\mathbf {X}}^{w\\mathbf {W}}\\in \\textrm {$$\\hspace{-6.544pt}$$}]$ as $\\mathbb {\\Pi }_{X_{1}^{n}}[\\xi _{n,\\mathbf {X}}^{w\\mathbf {W}}\\in \\textrm {\\Omega \\hspace{-6.544pt}\\Omega }]= E_{S}\\Big [\\frac{\\mathrm {d}\\mathbb {} ^{\\otimes n}}{\\mathrm {d}S}(V_{1},\\ldots ,V_{n})\\cdot \\mathbf {1}_{\\textrm {\\Omega \\hspace{-5.40608pt}\\Omega }}(\\xi _{n,\\mathbf {X}}^{w\\mathbf {V}})\\Big ]$ where $S$ designates any IS distribution of the vector $\\mathbf {V:=}(V_{1},\\ldots ,V_{n})$ , and $E_{S}[ \\, \\cdot \\, ]$ denotes the corresponding expectation operation.",
"Notice that $S$ is a random probability distribution on $\\mathbb {R}^{n}$ ; in fact, $S$ is a conditional probability distribution given $X_{1}^{n}$ , and thus it would be more precise to write $S | X_{1}^{n}$ instead of $S$ ; for the sake of brevity, we omit $| X_{1}^{n}$ .",
"As a consequence of (REF ), for adequately chosen $S$ , an improved estimator of $\\mathbb {\\Pi }_{X_{1}^{n}}[\\xi _{n,\\mathbf {X}}^{w\\mathbf {W}}\\in \\textrm {$$\\hspace{-6.544pt}$$}]$ is given by $\\widehat{\\Pi }_{L}^{improved}:= \\frac{1}{L}\\sum _{\\ell =1}^{L}\\frac{\\mathrm {d}\\mathbb {}^{\\otimes n}}{\\mathrm {d}S}(V_{1}^{(\\ell )},\\ldots ,V_{n}^{(\\ell )})\\cdot \\mathbf {1}_{\\textrm {\\Omega \\hspace{-5.40608pt}\\Omega }}(\\xi _{n,\\mathbf {X}}^{w\\mathbf {V}^{(\\ell )}}) \\, ,$ which also estimates $\\inf _{{Q}\\in \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega } }\\ \\inf _{m\\ne 0}D_{\\varphi }(m\\cdot {Q},{P})$ by the virtue of ().",
"Let us now deal with the concrete construction of a reasonable $S$ .",
"Given some (typically) large integer $M$ , we start with the realization $\\mathbf {W}^{\\ast }:=\\left( W_{1}^{\\ast },\\ldots ,W_{M}^{\\ast }\\right)$ such that $\\mathbf {Q}^{\\ast }:=\\xi _{M,\\mathbf {X}}^{w\\mathbf {W}^{\\ast }} \\in int(\\textrm {$$\\hspace{-6.544pt}$$})$ .",
"This may be given in advance or it may be achieved by drawing replicates $\\mathbf {W} = (W_{1}, \\ldots , W_{M})$ under $\\mathbb {}^{\\otimes M}$ until the first time where $\\xi _{M,\\mathbf {X}}^{w\\mathbf {W}}$ belongs to $int(\\textrm {$$\\hspace{-6.544pt}$$})$ .",
"Notice that by the nature of $\\textrm {$$\\hspace{-6.544pt}$$}$ , $\\mathbf {Q}^{\\ast }$ is a probability vector which has the $K$ components $q_{k}^{\\ast }:=\\sum _{i=1}^{M}\\frac{W_{i}^{\\ast }}{\\sum _{j=1}^{M}W_{j}^{\\ast }}\\mathbf {1}_{\\lbrace d_{k}\\rbrace }(X_{i}),\\hspace{42.67912pt} k=1,\\ldots ,K.$ Before we proceed, let us give the substantial remark that changing $\\left( V_{1},\\ldots ,V_{n}\\right) $ drawn under $S$ to $\\left(c \\cdot V_{1}, \\ldots ,c \\cdot V_{n}\\right) $ for any $c\\ne 0$ yields $\\xi _{n,\\mathbf {X}}^{w\\mathbf {V}}= \\xi _{n,\\mathbf {X}}^{w\\, c\\cdot \\mathbf {V}}$ so that the distribution $S$ is not uniquely determined.",
"Amongst all candidates, we choose the — uniquely determined — $S$ which is the Kullback-Leibler projection of $\\mathbb {}^{\\otimes n}$ on the set of all probability distributions on $\\mathbb {R}^{n}$ such that the $K$ “non-normalized” moment constraints $E_{S}[\\xi _{n,\\mathbf {X}}^{\\mathbf {V}}] =\\xi _{M,\\mathbf {X}}^{\\mathbf {W}^{\\ast }}$ (rather than the normalized $E_{S}[ \\xi _{n,\\mathbf {X}}^{w\\mathbf {V}}] =\\xi _{M,\\mathbf {X}}^{w\\mathbf {W}^{\\ast }}$ ) are satisfied, with the non-normalized vectors $\\xi _{M,\\mathbf {X}}^{\\mathbf {W}^{\\ast }} :=\\left( \\frac{1}{M}\\sum _{j=1}^{M}W_{j}^{\\ast }\\right) \\cdot Q^{\\ast } =: \\overline{W^{\\ast }} \\cdot Q^{\\ast },\\qquad \\xi _{n,\\mathbf {X}}^{\\mathbf {V}} :=\\left( \\frac{1}{n}\\sum _{j=1}^{n}V_{j}\\right) \\cdot \\xi _{n,\\mathbf {X}}^{w\\mathbf {V}} \\ .$ As already indicated above, this projection $S$ is a well-determined unique distribution on $\\mathbb {R}^{n}$ and — as we shall see in Proposition REF below — it is such that $\\xi _{n,\\mathbf {X}}^{w\\mathbf {V}}$ belongs to $\\textrm {$$\\hspace{-6.544pt}$$}$ with probability bounded away from 0 as $n$ increases, when $\\left( V_{1},\\ldots ,V_{n}\\right) $ are drawn under $S$ .",
"Therefore, this IS distribution produces an estimate of $\\mathbb {\\Pi }_{X_{1}^{n}}[\\xi _{n,\\mathbf {X}}^{w\\mathbf {W}}\\in \\textrm {$$\\hspace{-6.544pt}$$}]$ .",
"In order to justify the above construction of $S$ , we give the following result, which states that the IS sampling distribution $S$ yields a good hitting rate.",
"Its proof will be given in Appendix H. Proposition 60 With the above definition of $S$ , $\\lim \\inf _{n\\rightarrow \\infty }$ $S\\left[ \\xi _{n,\\mathbf {X}}^{w\\mathbf {V}} \\in \\textrm {\\right.$$\\hspace{-6.544pt}$$} $ is bounded away from 0.",
"We now come to the detailed construction of $S$ .",
"The constraints (REF ) can be written in explicit form as $E_{S}\\Big [ \\, \\frac{1}{n_{k}}\\sum _{i\\in I_{k}^{(n)}} V_{i} \\, \\Big ]\\ = \\ \\overline{W^{\\ast }} \\cdot \\frac{q_{k}^{\\ast }}{p_{n,k}^{emp}} \\, , \\qquad k=1,\\ldots ,K.$ The distribution $S$ can be obtained by blocks.",
"Indeed, let us define $S^{k}$ as the Kullback-Leibler (KL) projection of $\\mathbb {}^{\\otimes n_{k}}$ on the set of all distributions on $\\mathbb {R}^{n_{k}}$ such that (REF ) holds.",
"We define the resulting $S$ as the product distribution of those $S^{k}$ 's.",
"To obtain the latter, we start by defining $U_{k}$ as the KL projection of $\\mathbb {} $ on the set of all measures $Q$ on $\\mathbb {R}$ under (REF ).",
"Then, $dU_{k}(v)= \\exp (\\tau _{k}v-\\Lambda _{\\mathbb {} }\\left( \\tau _{k}\\right) )\\, d\\mathbb {} (v) \\, ,$ where $\\tau _{k} \\in int(dom(MGF_{\\mathbb {}}))$ is the unique solution of the equation $\\Lambda _{\\mathbb {} }^{\\prime }\\left( \\tau _{k}\\right) =\\overline{W^{\\ast }} \\cdot \\frac{q_{k}^{\\ast }}{p_{n,k}^{emp}}$ and thus — by relation (REF ) of Appendix F — we can compute explicitly $\\tau _{k} = \\varphi ^{\\prime }\\left( \\frac{\\overline{W^{\\ast }} \\cdot q_{k}^{\\ast }}{p_{n,k}^{emp}} \\right) .$ The distribution $S^{k}$ is then defined by $S^{k}:=\\underbrace{U_{k}\\otimes \\cdots \\otimes U_{k}}_{n_{k}\\text{times}}$ from which we obtain $S := S^{1}\\otimes \\cdots \\otimes S^{K}.$ With this construction, it holds $\\frac{dS}{d\\mathbb {} ^{\\otimes n}}(v_{1},\\ldots ,v_{n})=\\exp \\left(\\sum \\limits _{k=1}^{K}\\left( \\sum _{i\\in I_{k}^{(n)}}\\tau _{k} \\cdot v_{i}-\\Lambda _{\\mathbb {}}(\\tau _{k})\\right) \\right)$ which proves that $S$ is indeed the KL projection of $\\mathbb {} ^{\\otimes n}$ we aimed at.",
"Therefore, $\\mathbf {V}$ is composed of $K$ independent blocks of length $n_{k}$ each, and the $k-$ th subvector $\\mathbf {V}_{k}$ consists of all the random variables $V_{i}$ whose index $i$ satisfies $X_{i}=d_{k}.$ Within $\\mathbf {V}_{k}$ , all components are i.i.d.",
"with same distribution $U_{k}$ on $\\mathbb {R}$ defined through $\\frac{dU_{k}}{d\\mathbb {} }(u)=\\exp \\left\\lbrace \\tau _{k}\\cdot u-\\Lambda _{\\mathbb {}}(\\tau _{k})\\right\\rbrace =\\frac{\\exp \\left\\lbrace \\tau _{k}\\cdot u\\right\\rbrace }{MGF_{\\mathbb {} }(\\tau _{k})},$ which leads to the moment generating function $dom(MGF_{\\mathbb {}})-\\tau _{k} \\ \\ni \\ z \\ \\mapsto MGF_{U_{k}}(z):=\\int _{\\mathbb {R}}e^{zy}dU_{k}(y)=\\frac{MGF_{\\mathbb {}}(z+\\tau _{k})}{MGF_{\\mathbb {} }(\\tau _{k})}.$ Let us remark that $U_{k}$ can be interpreted as the distorted distribution of $\\mathbb {}$ with the distortion parameter $\\tau _{k}$ (in some cases, this distortion even becomes a tilting/dampening).",
"The estimator $\\widehat{\\Pi }_{L}^{improved}$ defined in (REF ) can be implemented through the following algorithm: Step S1 Choose some (typically large) $M$ and simulate repeatedly i.i.d.",
"vectors $\\left(W_{1},..,W_{M}\\right) $ — whose independent components have common distribution $\\mathbb {} $ — until $\\xi _{M,\\mathbf {X}}^{w\\mathbf {W}}$ belongs to $\\textrm {$$\\hspace{-6.544pt}$$}$ .",
"Call $\\left( W_{1}^{\\ast },..,W_{M}^{\\ast }\\right) $ the corresponding vector and $\\overline{W^{\\ast }}$ the arithmetic mean of its components.",
"Moreover, denote by $\\xi _{M,\\mathbf {X}}^{w\\mathbf {W}^{\\ast }}$ the corresponding normalized weighted empirical measure, identified with the $K-$ component vector $Q^{\\ast } := (q_{1}^{\\ast },..,q_{K}^{\\ast })$ with $q_{k}^{\\ast }$ defined in (REF ).",
"Step S2 For all $k \\in \\lbrace 1,\\ldots ,K\\rbrace $ compute $\\tau _{k} = \\varphi ^{\\prime }\\left( \\frac{\\overline{W^{\\ast }} \\cdot q_{k}^{\\ast }}{p_{n,k}^{emp}} \\right)$ .",
"Step S3 For all $\\ell \\in \\lbrace 1,\\ldots ,L\\rbrace $ simulate independently for all $k \\in \\lbrace 1,\\ldots ,K\\rbrace $ a row vector $\\mathbf {V}_{k}^{(\\ell )}$ $:=\\left(V_{k_{1}}^{(\\ell )},...,V_{k_{n_{k}}}^{(\\ell )}\\right) $ with independent components with common distribution $U_{k}$ defined in (REF ).",
"Concatenate these vectors to define the row vector $\\mathbf {V}^{(\\ell )}.$ Step S4 Compute the estimator $\\widehat{\\Pi }_{L}^{improved}$ by making use of the formula (REF ) which turns into the explicit form $& & \\widehat{\\Pi }_{L}^{improved} =\\frac{1}{L}\\sum _{\\ell =1}^{L}\\exp \\left(\\sum \\limits _{k=1}^{K}\\left(n_{k} \\cdot \\Lambda _{\\mathbb {} }(\\tau _{k}) -\\tau _{k} \\cdot \\sum _{i\\in I_{k}^{(n)}}V_{i}^{(\\ell )}\\right) \\right)\\cdot \\mathbf {1}_{\\textrm {\\Omega \\hspace{-5.40608pt}\\Omega }}\\left( \\xi _{n,\\mathbf {X}}^{w\\mathbf {V}^{(\\ell )}}\\right)$ Analogously to the paragraph right after (REF ) of the previous Subsection REF , in many cases we may improve the simulation burden needed for the computation of the estimator $\\widehat{\\Pi }_{L}^{improved}$ .",
"In fact, in terms of the notations $\\widehat{\\mathit {W}}_{k}^{(\\ell )}:=\\sum _{i\\in I_{k}^{(n)}}V_{i}^{(\\ell )}$ we can rewrite (REF ) as $\\widehat{\\Pi }_{L}^{improved} =\\frac{1}{L}\\sum _{\\ell =1}^{L}\\mathbf {1}_{\\textrm {\\Omega \\hspace{-5.40608pt}\\Omega }}\\left( \\xi _{n,\\mathbf {X}}^{w\\mathbf {V}^{(\\ell )}} \\right)\\cdot \\prod _{k=1}^{K}ISF_{k}\\left(\\widehat{\\mathit {W}}_{k}^{(\\ell )}\\right)$ with $ISF_{k}(x) \\ := \\ \\exp (n_{k} \\cdot \\Lambda _{\\mathbb {}}(\\tau _{k}) \\, - \\, x \\cdot \\tau _{k} )$ and $\\xi _{n,\\mathbf {X}}^{w\\mathbf {V}^{(\\ell )}} &=&{\\left\\lbrace \\begin{array}{ll}\\left(\\frac{\\widehat{\\mathit {W}}_{1}^{(\\ell )}}{\\sum _{k=1}^{K}\\widehat{\\mathit {W}}_{k}^{(\\ell )}},\\ldots , \\frac{\\widehat{\\mathit {W}}_{K}^{(\\ell )}}{\\sum _{k=1}^{K}\\widehat{\\mathit {W}}_{k}^{(\\ell )}} \\right) ,\\qquad \\textrm {if } \\sum _{k=1}^{K}\\widehat{\\mathit {W}}_{k}^{(\\ell )} \\ne 0, \\\\\\ (\\infty , \\ldots , \\infty ) =: \\infty , \\hspace{65.44142pt} \\textrm {if }\\sum _{k=1}^{K}\\widehat{\\mathit {W}}_{k}^{(\\ell )} = 0 \\, .\\end{array}\\right.",
"}$ Clearly, the random variable $\\widehat{\\mathit {W}}_{k}^{(\\ell )}$ ($k=1, \\ldots , K$ ) has distribution $U_{k}^{\\ast n_{k}}$ .",
"Hence, if $U_{k}^{\\ast n_{k}}$ can be explicitly constructed, then for the computation of $\\widehat{\\Pi }_{L}^{improved}$ it suffices to independently simulate the $K \\cdot L$ random variables $\\widehat{\\mathit {W}}_{k}^{(\\ell )}$ (rather than the $n \\cdot L$ random variables $V_{i}^{(\\ell )}$ ).",
"In the following subsubsection, we exemplarily demonstrate the tractability of this reduction effect."
],
[
"BS minimization of power divergences and related quantities",
"´ Consider the special case of power divergence generators $\\varphi := \\widetilde{c} \\cdot \\varphi _{\\gamma }$ ($\\gamma \\in \\mathbb {R}\\backslash ]1,2[$ ) of the Examples REF and REF .",
"The corresponding estimators $\\widehat{\\Pi }_{L}^{improved}$ can be obtained as follows: within the results of Example REF , set $M_{\\mathbf {P}}=1$ , and replace $\\widetilde{q}_{k}^{\\ast }$ by $\\overline{W^{\\ast }} \\cdot q_{k}^{\\ast }$ as well as $p_{k}$ by $p_{n,k}^{emp}$ ; accordingly, $\\widetilde{U}_{k}^{\\ast n_{k}}$ turns into $U_{k}^{\\ast n_{k}}$ and $\\widetilde{ISF}_{k}$ into $ISF_{k}$ ; simulate independently the random variables $\\widehat{\\mathit {W}}_{k}^{(\\ell )}$ from $U_{k}^{\\ast n_{k}}$ ($k \\in \\lbrace 1, \\ldots , K\\rbrace $ , $\\ell \\in \\lbrace 1,\\ldots ,L\\rbrace $ ); plug in the results of (i),(ii) into (REF ), (REF ), and (REF ) in order to concretely compute $\\widehat{\\Pi }_{L}^{improved}$ .",
"From this, we can easily generate improved estimators of the power divergences $\\inf _{\\mathbf {Q}\\in \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega }}D_{\\widetilde{c}\\cdot \\varphi _{\\gamma }}(\\mathbf {Q},{P})$ — and more generally, improved estimators of all the infimum-quantities (e.g.",
"Renyi divergences) respectively supremum-quantities in the parts (b) of the Propositions REF , REF , REF , REF , REF and REF with $A=1$ — by simply replacing $\\mathbb {\\Pi }_{X_{1}^{n}}[\\xi _{n}^{w\\mathbf {W}}\\in \\textrm {$$\\hspace{-6.544pt}$$} ]$ (respectively, its variants) by the corresponding estimator $\\widehat{\\Pi }_{L}^{improved}$ .",
"If — in the light of Remark REF (vi) — the ${P} = (p_{1}, \\ldots , p_{K})$ is a pregiven known probability vector e.g.",
"the uniform distribution ${P}^{unif}$ on $\\lbrace 1,\\ldots ,K\\rbrace $ (rather than the limit of the vector of empirical frequencies/masses of a sequence of random variables $X_{i}$ , cf.",
"(REF )), then we proceed analogously as above by replacing $p_{n,k}^{emp}$ with $p_{k}$ ; correspondingly, we obtain improved estimators of all the infimum-quantities respectively supremum-quantities (e.g.",
"Renyi entropies, diversity indices) in the parts (a) of the Propositions REF , REF , REF , REF , REF and REF with $A=1$ .",
"For the sake of brevity, in the following we only present explicitly the outcoming improved estimators for the power divergences (in the “$X_{i}-$ context” ).",
"Indeed, we simply replace the $\\mathbb {\\Pi }_{X_{1}^{n}}[\\xi _{n}^{w\\mathbf {W}}\\in \\textrm {$$\\hspace{-6.544pt}$$} ]$ in the formulas (REF ), (REF ), (REF ) (with $A=1$ ) by the improved estimator $\\widehat{\\Pi }_{L}^{improved}$ obtained through (i) to (iii); for arbitrarily fixed $\\widetilde{c} >0$ , this leads to the improved power-divergence estimators (BS estimators of power divergences) $&&\\hspace{-54.06006pt}\\widehat{D_{\\widetilde{c}\\cdot \\varphi _{\\gamma }}(\\textrm {\\Omega \\hspace{-6.544pt}\\Omega },{P})} \\ := \\ -\\frac{\\widetilde{c}}{\\gamma (\\gamma -1)}\\left\\lbrace 1-\\left( 1+\\frac{\\gamma }{\\widetilde{c}} \\cdot \\frac{1}{n} \\cdot \\log \\widehat{\\Pi }_{L}^{improved}\\right)^{1-\\gamma }\\right\\rbrace ,\\qquad \\gamma \\in \\, ]-\\infty ,0[ \\, \\cup \\, ]0,1[ \\, \\cup \\, [2,\\infty [,\\\\& &\\hspace{-54.06006pt}\\widehat{D_{\\widetilde{c}\\cdot \\varphi _{0}}(\\textrm {\\Omega \\hspace{-6.544pt}\\Omega },{P})} \\ := \\ -\\frac{1}{n}\\log \\widehat{\\Pi }_{L}^{improved} ,\\hspace{165.02606pt} \\gamma =0,\\\\& & \\hspace{-54.06006pt}\\widehat{D_{\\widetilde{c}\\cdot \\varphi _{1}}(\\textrm {\\Omega \\hspace{-6.544pt}\\Omega },{P})}\\ : = \\ - \\widetilde{c} \\cdot \\log \\left( 1+\\frac{1}{\\widetilde{c}} \\cdot \\frac{1}{n} \\cdot \\log \\widehat{\\Pi }_{L}^{improved}\\right) ,\\hspace{85.35826pt} \\gamma =1.$ Let us finally remark that from the above-mentioned Steps S1 to S4 (and analogously D1 to D4) one can see that for our BS method we basically need only a fast and accurate — pseudo, true, natural, quantum — random number generator.",
"The corresponding computations can be principally run in parallel, and require relatively moderate computer memory/storage; a detailed discussion is beyond the scope of this paper, given its current length."
],
[
"General case, part 2",
"The algorithm which is presented in this section aims at the evaluation of the bounds $\\inf _{m\\ne 0}D_{\\varphi }\\left( m \\cdot \\textrm {\\right.\\Omega \\hspace{-6.544pt}\\Omega },{P} =\\inf _{{Q}\\in \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega }}D_{\\varphi }\\left( m\\left( {Q}\\right) \\cdot {Q},{P}\\right)\\stackrel{(1)}{=} D_{\\varphi }\\left( m({Q}^{\\ast })\\cdot {Q}^{\\ast },{P}\\right) \\le D_{\\varphi }\\left( \\textrm {\\right.\\Omega \\hspace{-6.544pt}\\Omega },{P} \\le D_{\\varphi }\\left( {Q}^{\\ast },{P}\\right)$ obtained in Section REF , where ${Q}^{\\ast }$ satisfies the above equality $(1)$ .",
"The estimator of the lower bound in (REF ) is $\\widehat{D}:=-\\frac{1}{n}\\log \\widehat{\\text{ }\\Pi }_{L}^{improved}$ defined in (REF ).",
"We now turn to an estimate of the upper bound.",
"Consider for any fixed ${Q}:=\\left( q_{1},..,q_{K}\\right) $ in $\\mathbb {S}_{>0}^{K}$ the real number $m_{n}({Q})$ which satisfies $D_{\\varphi }\\left( m_{n}({Q}) \\cdot {Q},{P}_{n}^{emp} \\right) =\\inf _{m\\ne 0}D_{\\varphi }\\left( m\\cdot \\textrm {\\right.$ $\\hspace{-6.544pt}$$},{P}_{n}^{emp}$$where $ Pnemp$ was defined in the course of (\\ref {I^(n)_k for stat case}).Such $ mn(Q)$ is well defined for all $ Q$since it satisfies the equation (in $ m$)\\begin{equation}\\frac{d}{dm}D_{\\varphi }\\left( m \\cdot {Q},{P}_{n}^{emp}\\right) =\\sum _{k=1}^{K} q_{k} \\cdot \\varphi ^{\\prime }\\left( \\frac{m \\cdot q_{k}}{p_{n,k}^{emp}}\\right) =0 .\\end{equation}Since the mapping $ mD( m Q,P) $ is convexand differentiable, existence and uniqueness of $ mn(Q)$ hold; furthermore,$ mn(Q)] k pn,kemp/qk,k pn,kemp/qk[ $since $ ddmD( m Q,Pnemp) $is negative when $ m=k pn,kemp/qk$ and positive when $ m=k pn,kemp/qk$.$ An estimate of the distribution ${Q}^{\\ast }$ is required.",
"This can be achieved as follows: Estimate $\\inf _{m\\ne 0}D_{\\varphi }\\left( m \\cdot \\textrm {\\right.$$\\hspace{-6.544pt}$$}, {P}$ through $\\widehat{D} := -\\frac{1}{n}\\log \\widehat{\\Pi }_{L}^{improved}$ defined in (REF ).",
"Set $i=0.$ Get some ${Q}_{i} := (q_{i,1}, \\ldots , q_{i,K})$ in $\\textrm {$$\\hspace{-6.544pt}$$}$ ; this can be obtained by simulating runs of vectors $\\left( W_{1},..,W_{n}\\right) $ through i.i.d.",
"sampling under $\\mathbb {}$ .",
"Evaluate $m_{n}({Q}_{i})$ by solving () (with $q_{i,k}$ instead of $q_{k}$ ) for $m$ , which is a fast calculation by the bisection method.",
"If $D_{\\varphi }\\left( m_{n}({Q}_{i}) \\cdot {Q}_{i}, {P}_{n}^{emp} \\right) <\\widehat{D}+\\eta $ for some small $\\eta >0$ , then the proxy of ${Q}^{\\ast }$ is ${Q}_{i},$ denoted by $\\widehat{{Q}^{\\ast }}$ .",
"Else set $i\\leftarrow i+1$ and get ${Q}_{i}$ in $\\textrm {$$\\hspace{-6.544pt}$$}\\cap \\left\\lbrace {Q} \\, : \\, D_{\\varphi }\\left( {Q},{P}_{n}^{emp} \\right) <D_{\\varphi }\\left( {Q}_{i-1}, {P}_{n}^{emp}\\right) \\right\\rbrace $ and iterate.",
"That this algorithm converges in the sense that it produces some $\\widehat{{Q}^{\\ast }}$ is clear.",
"Since by (REF ) $D_{\\varphi }\\left( m({Q}^{\\ast }) \\cdot {Q}^{\\ast },{P}\\right)\\le D_{\\varphi }\\left(\\textrm {\\right.$ $\\hspace{-6.544pt}$$},{P}\\le D_{\\varphi }\\left( {Q}^{\\ast },{P}\\right) ,$$we have obtained both estimated lower and upper bounds for$ D( $\\Omega $$\\Omega $ , P)$.$ That the upper bound is somehow optimal can be seen from the power case developed in Section REF .",
"Indeed, in this case the solution of equation () is explicit and produces $m({Q})$ as a function of $D_{\\varphi }\\left( {Q},{P}\\right)$ through a Hellinger integral, and the mapping ${Q}\\rightarrow D_{\\varphi }\\left( m({Q})\\cdot {Q},{P}\\right)$ is increasing with respect to $D({Q},{P})$ .",
"Hence, ${Q} \\rightarrow \\inf _{m\\ne 0}D_{\\varphi }\\left( m \\cdot {Q},{P}\\right)$ is minimal when $D_{\\varphi }\\left( {Q},{P}\\right) $ is minimal as ${Q} \\in \\textrm {$$\\hspace{-6.544pt}$$}$ .",
"Therefore, ${Q}^{\\ast }\\in \\arg \\inf _{{Q}\\in \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega }}D_{\\varphi }\\left( m({Q}) \\cdot {Q},{P}\\right) $ also satisfies ${Q}^{\\ast }\\in \\arg \\inf _{{Q}\\in \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega }}D_{\\varphi }\\left( {Q},{P}\\right)$ ."
],
[
"Proofs — Part 1",
"Proof of Theorem REF .",
"This is a straightforward application of the classical Cramer-type Large Deviation Theorem in the vector case (see Theorem 2.2.30 and Corollary 6.1.6 in Dembo & Zeitouni [108]).",
"Recall that above we have transformed the original problem into a context where the second argument in $D_{\\varphi }(\\cdot ,\\cdot )$ is a probability vector, as follows: in terms of $M_{\\mathbf {P}}:=\\sum _{i=1}^{K}p_{i}>0$ we normalized $\\widetilde{{P}}:=\\mathbf {P}/M_{\\mathbf {P}},$ and $\\widetilde{\\mathbf {Q}}:=\\mathbf {Q}/M_{\\mathbf {P}}$ for $\\mathbf {Q}$ in $\\mathbf {\\Omega }$ .",
"With $\\widetilde{\\varphi } \\in \\Upsilon (]a,b[)$ defined through $\\widetilde{\\varphi }:=M_{\\mathbf {P}} \\cdot \\varphi $ , we have obtained $D_{\\varphi }(\\mathbf {Q},\\mathbf {P})=\\sum _{k=1}^{K}p_{k}\\cdot \\varphi \\left( \\frac{q_{k}}{p_{k}}\\right) =\\sum _{k=1}^{K}M_{\\mathbf {P}}\\cdot \\widetilde{p_{k}}\\cdot \\frac{\\varphi \\left( \\frac{M_{\\mathbf {P}}\\cdot \\widetilde{q_{k}}}{M_{\\mathbf {P}}\\cdot \\widetilde{p_{k}}}\\right) }{M_{\\mathbf {P}}}=D_{\\widetilde{\\varphi }}(\\widetilde{\\mathbf {Q}},\\widetilde{{P}})\\qquad \\textrm {(cf.",
"(\\ref {min Pb prob1}))}.$ It has followed that the solution of (REF ) coincides with the one of the problem of finding $\\widetilde{\\Phi }_{\\widetilde{{P}}}(\\widetilde{\\mathbf {\\Omega }}) := \\inf _{\\widetilde{\\mathbf {Q}}\\in \\widetilde{\\mathbf {\\Omega }} }D_{\\widetilde{\\varphi } }(\\widetilde{\\mathbf {Q}},\\widetilde{{P}}),\\qquad \\textrm {with } \\widetilde{\\mathbf {\\Omega }}:=\\mathbf {\\Omega } /M_{\\mathbf {P}}\\qquad \\textrm {(cf.",
"(\\ref {min Pb prob2}))}.$ So let us continue by tackling (REF ).",
"From the assumptions on $\\widetilde{\\varphi }$ and the requirement (REF ) one can see that $\\text{$\\widetilde{W}_{1}$ has moment generating function$t\\rightarrow E_{\\mathbb {\\Pi }}[e^{z \\cdot \\widetilde{W}_{1}}]= MGF_{\\widetilde{\\mathbb {}}}(z)$which is finite on a non-void neighborhood of $0$},$ $E_{\\mathbb {\\Pi }}[\\widetilde{W}_1] = 1 ,$ since $\\widetilde{\\varphi }(1)=0=\\widetilde{\\varphi }^{\\prime }(1)$ .",
"With the help of these, we obtain the following Proposition 61 Under the assumptions of Theorem REF , for any set $\\widetilde{\\mathbf {\\Omega }}\\subset \\mathcal {M} := \\mathbb {R}^{K}$ with (REF ) one has $-\\inf _{\\widetilde{\\mathbf {Q}} \\in int(\\widetilde{\\mathbf {\\Omega }}) }D_{\\varphi }\\left( \\widetilde{\\mathbf {Q}},\\widetilde{{P}}\\right) &\\le &\\lim \\inf _{n\\rightarrow \\infty }\\frac{1}{n}\\log \\mathbb {\\Pi }\\left[\\xi _{n}^{\\widetilde{\\textbf {W}}}\\in \\widetilde{\\mathbf {\\Omega }} \\right] \\nonumber \\\\&\\le &\\lim \\sup _{n\\rightarrow \\infty }\\frac{1}{n}\\log \\mathbb {\\Pi }\\left[\\xi _{n}^{\\widetilde{\\textbf {W}}}\\in \\widetilde{\\mathbf {\\Omega }} \\right]\\le -\\inf _{\\widetilde{\\mathbf {Q}}\\in cl(\\widetilde{\\mathbf {\\Omega }}) }D_{\\varphi }\\left( \\widetilde{\\mathbf {Q}},\\widetilde{{P}}\\right) .$ Proof of Proposition REF .",
"Recall from Remark REF (v) that $I_{k}^{(n)} := \\lbrace i \\in \\lbrace 1,\\ldots ,n\\rbrace : \\widetilde{x}_{i}=d_{k}\\rbrace $ and $n_{k} := card(I_{k}^{(n)})$ denotes the number of elements therein ($k \\in \\lbrace 1,\\ldots ,K\\rbrace $ ), i.e.",
"$n_{k}$ is the number of the $\\widetilde{x}_{i}$ ’s which equal $d_{k}$ .",
"We follow the line of proof of Theorem 2.2.30 in Dembo & Zeitouni [108], which states the large deviation principle (LDP) for the vector of partial sums of random vectors in $\\mathbb {R}^{K}$ , where we also use Corollary 6.1.6 in [108] in relation with condition (REF ).",
"Indeed, since the $k-$ th component of the vector $\\xi _{n}^{\\mathbf {\\widetilde{W}}}$ is the $1/n-$ fold of the sum of the $\\widetilde{W}_{i}$ ’s for which the corresponding $\\widetilde{x}_{i}$ ’s equal $d_{k}$ (i.e., $\\frac{1}{n} \\sum _{i\\in I_{k}^{(n)}} \\widetilde{W}_{i}$ ) the proof will follow from a similar treatment as for the standard Cramer LDP in $\\mathbb {R}^{K}.$ The only difference lies in two facts: the number of the summands for the coordinate $k$ is $n_{k}$ , the number of $\\widetilde{x}_{i}$ ’s which equal $d_{k}$ , instead of $n$ in the standard case.",
"Furthermore we will need to substitute $n_{k}$ by its equivalent $n \\cdot \\widetilde{p}_{k}$ , which adds an approximation step.",
"For the upper bound, the proof is based on the corresponding result for $B=B_{1}\\times \\cdots \\times B_{K}$ where the $B_{k}$ ’s are open bounded intervals on $\\mathbb {R}^{+}.$ Since the sequence $\\left( \\widetilde{x}_{1}, \\ldots \\right)$ satisfies $\\lim _{n\\rightarrow \\infty }\\frac{n_{k}}{n}= \\widetilde{p}_{k} ,\\qquad \\textrm {(cf.",
"(\\ref {fo.freqlim}))}$ there holds $&&\\frac{1}{n}\\log \\mathbb {\\Pi } \\left[\\xi _{n}^{\\mathbf {\\widetilde{W}}} \\in B\\right]=\\frac{1}{n}\\log \\mathbb {\\Pi } \\bigg [ \\bigcap \\limits _{k=1}^{K}\\bigg (\\frac{1}{n} \\sum _{i\\in I_{k}^{(n)}} \\widetilde{W}_{i}\\in B_{k}\\bigg ) \\bigg ]\\nonumber \\\\&=&\\frac{1}{n}\\sum \\limits _{k=1}^{K}\\log \\mathbb {\\Pi } \\bigg [ \\frac{1+o(1)}{n_{k}}\\sum _{i\\in I_{k}^{(n)}} \\widetilde{W}_{i}\\in \\frac{1}{\\widetilde{p}_{k}} B_{k}\\bigg ],$ and hence $\\lim \\sup _{n\\rightarrow \\infty }\\frac{1}{n}\\log \\mathbb {\\Pi } \\left[\\xi _{n}^{\\mathbf {\\widetilde{W}}}\\in B\\right]&\\le &\\sum \\limits _{k=1}^{K} \\widetilde{p}_{k} \\cdot \\lim \\sup _{n_{k}\\rightarrow \\infty }\\frac{1}{n_{k}}\\log \\mathbb {\\Pi } \\bigg [ \\frac{1}{n_{k}}\\sum _{i\\in I_{k}^{(n)}} \\widetilde{W}_{i}\\in \\frac{1}{p_{k}}B_{k}\\bigg ] \\nonumber \\\\&\\le &-\\sum \\limits _{k=1}^{K}\\inf _{x_{k}\\in cl(B_{k})} \\widetilde{p}_{k} \\cdot \\varphi \\left(\\frac{x_{k}}{\\widetilde{p}_{k}}\\right) .$ To deduce (REF ) from (REF ), we have used (i) the fact that for all $k$ the random variables $\\frac{1}{n_{k}}\\left( 1+o(1))\\right) \\cdot \\sum _{i\\in I_{k}^{(n)}} \\widetilde{W}_{i}$ and $\\frac{1}{n_{k}}\\sum _{i\\in I_{k}^{(n)}} \\widetilde{W}_{i}$ are exponentially equivalent in the sense that their difference $\\Delta _{n_{k}}$ satisfies $\\lim \\sup _{n_{k}\\rightarrow \\infty }\\frac{1}{n_{k}}\\log \\mathbb {\\Pi }\\left[ \\,\\left|\\Delta _{n_{k}}\\right|>\\eta \\, \\right] =-\\infty ,$ making use of the Chernoff inequality for all positive $\\eta $ , as well as (ii) Theorem 4.2.13 in [108].",
"Now the summation and the inf-operations can be permuted in (REF ) which proves the claim for the rectangle $B$ .",
"As in [108], for a compact set $\\Omega $ we consider its finite covering by such open sets $B$ and conclude; for $\\Omega $ being a closed set, a tightness argument holds, following [108] Theorem 2.2.30 verbatim.",
"For the lower bound consider the same rectangle $B$ .",
"The argument which locates the tilted distribution at the center of $B$ , together with the use of the LLN for the corresponding r.v’s as in [108], in combination with the same approximations as above to handle the approximation of $n_{k}$ by $n \\cdot \\widetilde{p}_{k}$ , complete the proof of Proposition REF .",
"We omit the details.",
"$\\blacksquare $ Let us continue with the proof of Theorem REF , by giving the following two helpful lemmas for $\\Phi _{{P}}(\\mathbf {A}) := \\inf _{\\mathbf {Q} \\in \\mathbf {A}} D_{\\varphi }\\left(\\mathbf {Q},{P}\\right),\\qquad \\mathbf {A} \\subset \\mathcal {M} := \\mathbb {R}^{K},$ Lemma 62 For any open set $\\mathbf {A} \\subset \\mathcal {M} := \\mathbb {R}^{K}$ one has $\\Phi _{{P}}(\\mathbf {A}) = \\Phi _{{P}}(cl(\\mathbf {A}))$ .",
"This is clear from the continuity of $\\Phi _{{P}}$ .",
"Lemma 63 For any $\\mathbf {A} \\subset \\mathcal {M} := \\mathbb {R}^{K}$ satisfying (REF ) one has $\\Phi _{{P}}(cl(\\mathbf {A})) = \\Phi _{{P}}(\\mathbf {A}) = \\Phi _{{P}}(int(\\mathbf {A}))$ .",
"Proof of Lemma REF .",
"Assume first that $\\Phi _{{P}}(\\mathbf {A})$ is finite.",
"Then suppose that $\\mathbf {A}$ satisfies (REF ) and $\\Phi _{{P}}(cl(\\mathbf {A})) < \\Phi _{{P}}(int(\\mathbf {A}))$ .",
"The latter implies the existence of a point $a \\in cl(\\mathbf {A})$ such that $a \\notin int(\\mathbf {A})$ and $\\Phi _{{P}}(a)= \\Phi _{{P}}(cl(\\mathbf {A}))$ .",
"But then, by Lemma REF and (REF ) one gets $\\Phi _{{P}}(int(\\mathbf {A})) = \\Phi _{{P}}(cl(int(\\mathbf {A}))) = \\Phi _{{P}}(cl(\\mathbf {A})) = \\Phi _{{P}}(a)$ which leads to a contradiction.",
"When $\\Phi _{{P}}(\\mathbf {A}) = \\infty $ then $\\Phi _{{P}} (cl(\\mathbf {A}))=\\Phi _{{P}} (int(\\mathbf {A}))=\\Phi _{{P}} (\\mathbf {A})=\\infty $ .",
"$\\blacksquare $ Putting things together, the required asymptotic assertion (REF ) follows from (REF ), (REF ) and Lemma REF .",
"This completes the proof of Theorem REF .",
"$\\blacksquare $"
],
[
"Proofs — Part 2",
"Before we tackle the proof of Theorem REF , let us introduce the following Lemma 64 If $\\Omega $$\\Omega \\subset \\mathbb {S}^{K}$ satisfies condition (REF ), then $\\widetilde{\\widetilde{\\textrm {\\Omega \\hspace{-6.544pt}\\Omega }}}:= \\bigcup \\limits _{m\\ne 0}cl(m \\cdot \\textrm {$$\\hspace{-6.544pt}$$})$ has the property (REF ).",
"This can be deduced in a straightforward way: the assumption implies that $cl(\\textrm {$$\\hspace{-6.544pt}$$})$ satisfies (REF ), and thus also $m \\cdot cl(\\textrm {$$\\hspace{-6.544pt}$$})$ satisfies (REF ).",
"But this implies the validity of (REF ) for the “cone” $\\bigcup \\limits _{m\\ne 0} m \\cdot cl(\\textrm {$$\\hspace{-6.544pt}$$})$ which is nothing but $\\bigcup \\limits _{m\\ne 0} cl(m \\cdot \\textrm {$$\\hspace{-6.544pt}$$})$ .",
"Proof of Theorem REF .",
"Recall the interpretations of the two vectors $\\xi _{n,\\mathbf {X}}^{\\mathbf {W}}$ respectively $\\xi _{n,\\mathbf {X}}^{w\\mathbf {W}}$ given in (REF ) respectively (REF ), and that the sum of their $k$ components are $\\sum _{k=1}^{K} \\frac{1}{n} \\sum _{i\\in I_{k}^{(n)}} W_{i} =\\frac{1}{n}\\sum _{i=1}^{n} W_{i}$ respectively $\\sum _{k=1}^{K}\\frac{\\sum _{i \\in I_{k}^{(n)}}W_{i}}{\\sum _{k=1}^{K}\\sum _{i \\in I_{k}^{(n)}}W_{i}}=1$ (in case of $\\sum _{i=1}^{n}W_{i} \\ne 0$ ).",
"In the light of these, for $\\textrm {$$\\hspace{-6.544pt}$$} \\subset \\mathbb {S}^{K}$ one gets the set identification $\\left\\lbrace \\xi _{n,\\mathbf {X}}^{w\\mathbf {W}} \\in \\textrm {\\right.$ $\\hspace{-6.544pt}$$} =\\bigcup \\limits _{m\\ne 0}\\left\\lbrace \\xi _{n,\\mathbf {X}}^{\\mathbf {W}}\\in m\\cdot \\textrm {\\right.$$\\hspace{-6.544pt}$$} ,\\frac{1}{n}\\sum _{i=1}^{n} W_{i} = m$$since $ { i=1nWi=0} $ amounts to $ m=0$, which cannothold when $ { n,XwW $\\Omega $$\\Omega $ }$.",
"Now\\begin{eqnarray*}\\mathbb {\\Pi }_{X_{1}^{n}}\\left[\\xi _{n,\\mathbf {X}}^{w\\mathbf {W}}\\in \\textrm {\\right.\\Omega \\hspace{-6.544pt}\\Omega } &=&\\mathbb {\\Pi }_{X_{1}^{n}} \\bigg [ \\bigcup \\limits _{m\\ne 0}\\bigg \\lbrace \\xi _{n,\\mathbf {X}}^{\\mathbf {W}} \\in m \\cdot \\textrm {\\Omega \\hspace{-6.544pt}\\Omega },\\frac{1}{n}\\sum _{i=1}^{n} W_{i} =m \\bigg \\rbrace \\bigg ] \\\\&=&\\mathbb {\\Pi }_{X_{1}^{n}}\\bigg [ \\bigcup \\limits _{m\\ne 0}\\bigg \\lbrace \\xi _{n,\\mathbf {X}}^{\\mathbf {W}} \\in m\\cdot \\textrm {\\Omega \\hspace{-6.544pt}\\Omega }\\big \\rbrace \\bigg ]= \\mathbb {\\Pi }_{X_{1}^{n}}\\bigg [ \\xi _{n,\\mathbf {X}}^{\\mathbf {W}}\\in \\bigcup \\limits _{m\\ne 0} m\\cdot \\textrm {\\Omega \\hspace{-6.544pt}\\Omega } \\bigg ]\\end{eqnarray*}since $ { n,XW m$\\Omega $$\\Omega $ } { 1ni=1n Wi =m }$.",
"Therefore\\begin{equation}\\frac{1}{n}\\log \\mathbb {\\Pi }_{X_{1}^{n}}\\left[\\xi _{n,\\mathbf {X}}^{w\\mathbf {W}}\\in \\textrm {\\right.\\Omega \\hspace{-6.544pt}\\Omega } =\\frac{1}{n}\\log \\mathbb {\\Pi }_{X_{1}^{n}}\\bigg [ \\xi _{n,\\mathbf {X}}^{\\mathbf {W}} \\in \\bigcup \\limits _{m\\ne 0}m\\cdot \\textrm {\\Omega \\hspace{-6.544pt}\\Omega } \\bigg ] .\\end{equation}Because of Proposition \\ref {PropLDPWEM-finitease} --- applied to$$\\Omega $$\\Omega $ := m0m $\\Omega $$\\Omega $$ --- one getsin terms of (\\ref {phishort})$ $-\\Phi _{{P}}\\Big (int\\Big (\\bigcup \\limits _{m\\ne 0}m\\cdot \\textrm {\\Omega \\hspace{-6.544pt}\\Omega }\\Big )\\Big )&\\le &\\lim \\inf _{n\\rightarrow \\infty }\\frac{1}{n}\\log \\mathbb {\\Pi }_{X_{1}^{n}}\\bigg [ \\xi _{n,\\mathbf {X}}^{\\mathbf {W}} \\in \\bigcup \\limits _{m\\ne 0}m\\cdot \\textrm {\\Omega \\hspace{-6.544pt}\\Omega } \\bigg ] \\nonumber \\\\&\\le &\\lim \\sup _{n\\rightarrow \\infty }\\frac{1}{n}\\log \\mathbb {\\Pi }_{X_{1}^{n}}\\bigg [\\xi _{n,\\mathbf {X}}^{\\mathbf {W}}\\in \\bigcup \\limits _{m\\ne 0}m\\cdot \\textrm {\\Omega \\hspace{-6.544pt}\\Omega } \\bigg ] \\le -\\Phi _{{P}}\\Big (cl\\Big (\\bigcup \\limits _{m\\ne 0}m\\cdot \\textrm {\\Omega \\hspace{-6.544pt}\\Omega }\\Big )\\Big ) .$ $\\hspace{-176.407pt}\\text{But}& & \\Phi _{{P}}\\Big (int\\Big (\\bigcup \\limits _{m\\ne 0}m\\cdot \\textrm {\\Omega \\hspace{-6.544pt}\\Omega }\\Big )\\Big ) \\le \\Phi _{{P}}\\Big (\\bigcup \\limits _{m\\ne 0}int\\Big (m\\cdot \\textrm {\\Omega \\hspace{-6.544pt}\\Omega }\\Big )\\Big ) =\\inf _{m\\ne 0} \\Phi _{{P}}(int(m\\cdot \\textrm {\\Omega \\hspace{-6.544pt}\\Omega }))\\\\\\hspace{-176.407pt}\\text{and}& & \\Phi _{{P}}\\Big (cl\\Big (\\bigcup \\limits _{m\\ne 0}m\\cdot \\textrm {\\Omega \\hspace{-6.544pt}\\Omega }\\Big )\\Big ) \\ge \\Phi _{{P}}\\Big (\\bigcup \\limits _{m\\ne 0}cl\\Big (m\\cdot \\textrm {\\Omega \\hspace{-6.544pt}\\Omega }\\Big )\\Big ) =\\inf _{m\\ne 0} \\Phi _{{P}}(cl(m\\cdot \\textrm {\\Omega \\hspace{-6.544pt}\\Omega })) .$ In fact, the inequality in (REF ) is straightforward because of $\\bigcup \\limits _{m\\ne 0}int(m\\cdot \\textrm {$$\\hspace{-6.544pt}$$})\\subset int(\\bigcup \\limits _{m\\ne 0}m\\cdot \\textrm {$$\\hspace{-6.544pt}$$})$ (since the latter is the largest open set contained in $\\bigcup \\limits _{m\\ne 0}m\\cdot \\textrm {$$\\hspace{-6.544pt}$$}$ ); the inequality in () follows from $& & \\Phi _{{P}}\\Big (cl\\Big (\\bigcup \\limits _{m\\ne 0}m\\cdot \\textrm {\\Omega \\hspace{-6.544pt}\\Omega }\\Big )\\Big ) \\ge \\Phi _{{P}}\\Big (cl\\Big (\\bigcup \\limits _{m\\ne 0}cl\\Big (m\\cdot \\textrm {\\Omega \\hspace{-6.544pt}\\Omega }\\Big )\\Big )\\Big ) =\\Phi _{{P}}\\Big (\\bigcup \\limits _{m\\ne 0}cl\\Big (m\\cdot \\textrm {\\Omega \\hspace{-6.544pt}\\Omega }\\Big )\\Big )\\nonumber $ An application of Lemma REF yields $\\Phi _{{P}}(int(m\\cdot \\textrm {$$\\hspace{-6.544pt}$$})) =\\Phi _{{P}}(m\\cdot \\textrm {$$\\hspace{-6.544pt}$$}) =\\Phi _{{P}}(cl(m\\cdot \\textrm {$$\\hspace{-6.544pt}$$}))$ for all $m \\ne 0$ , and hence $\\inf _{m\\ne 0} \\Phi _{{P}}(int(m\\cdot \\textrm {\\Omega \\hspace{-6.544pt}\\Omega })) =\\inf _{m\\ne 0} \\Phi _{{P}}(m\\cdot \\textrm {\\Omega \\hspace{-6.544pt}\\Omega })= \\inf _{m\\ne 0} \\Phi _{{P}}(cl(m\\cdot \\textrm {\\Omega \\hspace{-6.544pt}\\Omega })) .$ By combining (), (REF ), (REF ), () and (REF ), one arrives at $&&\\lim _{n \\rightarrow \\infty } \\frac{1}{n}\\log \\mathbb {\\Pi }_{X_{1}^{n}} \\left[\\xi _{n,\\mathbf {X}}^{w\\mathbf {W}}\\in \\textrm {\\right.\\Omega \\hspace{-6.544pt}\\Omega } = \\lim _{n \\rightarrow \\infty } \\frac{1}{n}\\log \\mathbb {\\Pi }_{X_{1}^{n}} \\bigg [ \\xi _{n,\\mathbf {X}}^{\\mathbf {W}}\\in \\bigcup \\limits _{m\\ne 0}m\\cdot \\textrm {\\Omega \\hspace{-6.544pt}\\Omega } \\bigg ]\\nonumber \\\\&& = - \\inf _{m\\ne 0} \\Phi _{{P}}(m\\cdot \\textrm {\\Omega \\hspace{-6.544pt}\\Omega })= - \\inf _{m\\ne 0} \\inf _{\\mathbf {Q} \\in m\\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega }}D_{\\varphi }\\left(\\mathbf {Q},{P}\\right)= - \\inf _{m\\ne 0} \\inf _{{Q} \\in \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega }}D_{\\varphi }\\left(m\\cdot {Q}, {P}\\right) ,\\nonumber $ where in the second last equality we have “reverted” the notation (REF ).",
"Note that we did not assume (REF ) for $\\bigcup \\limits _{m\\ne 0} m \\cdot \\textrm {$$\\hspace{-6.544pt}$$}$ .",
"$\\blacksquare $"
],
[
"Proofs — Part 3",
"Proof of Lemma REF .",
"From (REF ) one gets straightforwardly for arbitrary $\\widetilde{c} >0$ $D_{\\widetilde{c} \\cdot \\varphi _{\\gamma }}(m \\cdot \\mathbf {Q},{P}) \\hspace{-5.69046pt} &:=& \\hspace{-5.69046pt}{\\left\\lbrace \\begin{array}{ll}\\frac{\\widetilde{c} \\cdot \\left(m^{\\gamma } \\cdot H_{\\gamma } - m \\cdot A \\cdot \\gamma + \\gamma -1 \\right)}{\\gamma \\cdot (\\gamma -1)},\\hspace{75.39963pt} \\textrm {if } \\gamma \\in \\, ]-\\infty ,0[, \\ {P} \\in \\mathbb {S}_{\\ge 0}^{K}, \\ \\mathbf {Q} \\in A \\cdot \\mathbb {S}_{> 0}^{K} \\ \\ \\textrm {and } m >0, \\\\\\widetilde{c} \\cdot (- \\log m + \\widetilde{I} -1 + m \\cdot A),\\hspace{42.67912pt} \\textrm {if } \\gamma = 0, \\ {P} \\in \\mathbb {S}_{\\ge 0}^{K}, \\ A \\cdot \\mathbf {Q} \\in \\mathbb {S}_{> 0}^{K} \\ \\ \\textrm {and } m >0, \\\\\\frac{\\widetilde{c} \\cdot \\left(m^{\\gamma } \\cdot H_{\\gamma } - m \\cdot A \\cdot \\gamma + \\gamma -1 \\right)}{\\gamma \\cdot (\\gamma -1)},\\hspace{75.39963pt} \\textrm {if } \\gamma \\in \\, ]0,1[, \\ {P} \\in \\mathbb {S}_{\\ge 0}^{K}, \\ \\mathbf {Q} \\in A \\cdot \\mathbb {S}_{\\ge 0}^{K} \\ \\ \\textrm {and } m \\ge 0, \\\\\\widetilde{c} \\cdot (A \\cdot m \\cdot \\log m + m \\cdot (I-A) +1),\\hspace{14.22636pt} \\textrm {if } \\gamma = 1, \\ {P} \\in \\mathbb {S}_{> 0}^{K}, \\ \\mathbf {Q} \\in A \\cdot \\mathbb {S}_{\\ge 0}^{K} \\ \\ \\textrm {and } m \\ge 0, \\\\\\frac{\\widetilde{c} \\cdot \\left(m^{\\gamma } \\cdot H_{\\gamma } \\cdot {1}_{[0,\\infty [}(m)- m \\cdot A \\cdot \\gamma + \\gamma -1 \\right)}{\\gamma \\cdot (\\gamma -1)},\\hspace{39.83368pt} \\textrm {if } \\gamma \\in \\, ]1,2[, \\ {P} \\in \\mathbb {S}_{>0}^{K}, \\ \\mathbf {Q} \\in A \\cdot \\mathbb {S}^{K} \\ \\ \\textrm {and } m \\in ]-\\infty ,\\infty [,\\\\\\frac{\\widetilde{c} \\cdot \\left(m^{2} \\cdot H_{2} - m \\cdot A \\cdot 2 + 2 -1 \\right)}{2 \\cdot (2-1)},\\hspace{76.82234pt} \\textrm {if } \\gamma = 2, \\ {P} \\in \\mathbb {S}_{>0}^{K}, \\ \\mathbf {Q} \\in A \\cdot \\mathbb {S}^{K} \\ \\ \\textrm {and } m \\in ]-\\infty ,\\infty [,\\\\\\frac{\\widetilde{c} \\cdot \\left(m^{\\gamma } \\cdot H_{\\gamma } \\cdot {1}_{[0,\\infty [}(m)- m \\cdot A \\cdot \\gamma + \\gamma -1 \\right)}{\\gamma \\cdot (\\gamma -1)},\\hspace{39.83368pt} \\textrm {if } \\gamma \\in \\, ]2,\\infty [, \\ {P} \\in \\mathbb {S}_{>0}^{K}, \\ \\mathbf {Q} \\in A \\cdot \\mathbb {S}^{K} \\ \\ \\textrm {and } m \\in ]-\\infty ,\\infty [,\\\\\\infty , \\hspace{156.49014pt} \\textrm {else},\\end{array}\\right.",
"}$ where we have used the three $m-$ independent abbreviations $H_{\\gamma } := \\sum \\displaylimits _{k=1}^{K} (q_{k})^{\\gamma } \\cdot (p_{k})^{1-\\gamma }\\ = \\ 1 + \\gamma \\cdot (A-1) +\\frac{\\gamma \\cdot (\\gamma -1)}{\\widetilde{c}} \\cdot D_{\\widetilde{c} \\cdot \\varphi _{\\gamma }}(\\mathbf {Q}, {P}),\\qquad \\text{(cf.",
"(\\ref {brostu3:fo.divpow.hellinger1}))}\\nonumber $ $I:= \\sum \\displaylimits _{k=1}^{K} q_{k} \\cdot \\log \\left( \\frac{q_{k}}{p_{k}} \\right)\\ = \\ \\frac{1}{\\widetilde{c}} \\cdot D_{\\widetilde{c} \\cdot \\varphi _{1}}(\\mathbf {Q}, {P}) + A - 1,\\qquad \\text{(cf.",
"(\\ref {brostu3:fo.divpow.Kull1}))}\\nonumber $ $\\widetilde{I} := \\sum \\displaylimits _{k=1}^{K} p_{k} \\cdot \\log \\left( \\frac{p_{k}}{q_{k}} \\right)\\ = \\ \\frac{1}{\\widetilde{c}} \\cdot D_{\\widetilde{c} \\cdot \\varphi _{0}}(\\mathbf {Q}, {P}) + 1 - A.\\qquad \\text{(cf.",
"(\\ref {brostu3:fo.divpow.RevKull1}))}\\nonumber $ To proceed, let us fix an arbitrary constant $\\widetilde{c} >0$ .",
"(i) Case $\\gamma \\cdot (1-\\gamma ) \\ne 0$ .",
"(ia) Let us start with the subcase $\\gamma \\in ]-\\infty ,0[$ .",
"From the first and the last line of (REF ), it is clear that the corresponding $m-$ infimum can not be achieved for $m \\le 0$ ; since $H_{\\gamma } > 0$ one gets the unique minimizer $m_{min} = \\Big (\\frac{H_{\\gamma }}{A}\\Big )^{1/(1-\\gamma )} >0$ and the minimum $D_{\\widetilde{c} \\cdot \\varphi _{\\gamma }}(m_{min} \\cdot \\mathbf {Q},{P})=\\frac{\\widetilde{c}}{\\gamma } \\cdot (1- \\frac{H^{1/(1-\\gamma )}}{A^{\\gamma /(1-\\gamma )}} )$ .",
"Hence, (REF ) is established.",
"The assertions (REF ) and () follow immediately by monotonicity inspection of $x \\rightarrow \\frac{\\widetilde{c}}{\\gamma } \\cdot \\left[ 1-\\frac{1}{A^{\\gamma /(1-\\gamma )}} \\cdot \\left[ 1 + \\gamma \\cdot (A-1) + \\frac{\\gamma \\cdot \\left( \\gamma -1\\right)}{\\widetilde{c}}\\cdot x\\right] ^{-1/\\left( \\gamma -1\\right) }\\right]$ for $x \\ge 0$ such that $1 + \\gamma \\cdot (A-1) + \\frac{\\gamma \\cdot \\left( \\gamma -1\\right)}{\\widetilde{c}}\\cdot x \\ge 0$ .",
"(ib) The subcase $\\gamma \\in ]0,1[$ (cf.",
"the third line of (REF )) works analogously if $H_{\\gamma } >0$ ; furthermore, if $H_{\\gamma } =0$ — which can only appear when ${P}$ , $\\mathbf {Q}$ have disjoint supports (singularity)— then $\\inf _{m>0} D_{\\widetilde{c} \\cdot \\varphi _{\\gamma }}(m \\cdot \\mathbf {Q},{P}) =\\frac{\\widetilde{c}}{\\gamma }$ which is (the corresponding special case of) (REF ).",
"(ic) In the subcase $\\gamma \\in ]1,\\infty [$ (cf.",
"the fifth, sixth and seventh line of (REF )) it is straightforward to see that the desired infimum can not be achieved for $m < 0$ .",
"Hence, one can proceed analogously to subcase (ia).",
"(id) The assertions () to () are straightforward.",
"(ii) Case $\\gamma =1$ .",
"From the fourth line of (REF ), one obtains the unique minimizer $m_{min} = \\exp \\lbrace -I/A\\rbrace $ and the minimum $D_{\\widetilde{c} \\cdot \\varphi _{1}}(m_{min} \\cdot \\mathbf {Q},{P})= \\widetilde{c} \\cdot (1- A \\cdot m_{min})$ , which leads to (REF ).",
"The monotonicity of $x \\rightarrow \\widetilde{c} \\cdot (1- \\exp \\lbrace -x/\\widetilde{c}\\rbrace )$ for $x\\ge 0$ implies immediately (REF ) and (); moreover, () and () are immediate.",
"(iii) Case $\\gamma =0$ .",
"The second line of (REF ) implies the unique minimizer $m_{min} = 1/A$ , the minimum $D_{\\widetilde{c} \\cdot \\varphi _{0}}(m_{min} \\cdot \\mathbf {Q},{P})= \\widetilde{c} \\cdot (\\widetilde{I} + \\log A)$ , and hence (REF ).",
"The assertions (REF ) to () are obvious.",
"$\\blacksquare $"
],
[
"Proofs — Part 4",
"Proof of Proposition REF .",
"Clearly, (G1) and (G2) are part of the definition of $\\widetilde{\\Upsilon }(]a,b[)$ .",
"Recall our required representability (REF ).",
"The therein involved Laplace-Stieltjes transform (Laplace-Lebesgue transform) $z\\mapsto MGF_{\\mathbb {}}(z):=\\int _{\\mathbb {R}}e^{z \\cdot y} \\, d\\mathbb {} (y)= E_{\\mathbb {\\Pi }}[e^{z\\cdot W}]$ of a probability measure $\\mathbb {}$ on the real line respectively of an associated random variable $W$ (with $\\mathbb {}[ \\cdot \\, ] := \\mathbb {\\Pi }[W \\in \\cdot \\, ]$ ) has the following fundamental properties, according to well-known general theory: $MGF_{\\mathbb {}}$ takes values in $]0,\\infty ]$ ; the effective domain $dom(MGF_{\\mathbb {}})$ is an interval which contains 0 and which may be degenerated or even the whole real line; correspondingly, we denote its interior by $]\\lambda _{-},\\lambda _{+}[ := int(dom(MGF_{\\mathbb {}}))$ which may be the empty set (in case that $dom(MGF_{\\mathbb {}})=\\lbrace 0\\rbrace $ , i.e.",
"$\\lambda _{-}=\\lambda _{+}=0$ ); clearly, there holds $\\lambda _{-} \\in [-\\infty ,0]$ and $\\lambda _{+} \\in [0,\\infty ]$ ; $MGF_{\\mathbb {}}$ is continuous on $dom(MGF_{\\mathbb {}})$ and lower semicontinuous on $\\mathbb {R}$ ; if $\\lambda _{-} \\ne \\lambda _{+}$ , then $MGF_{\\mathbb {}}$ is real analytic and thus infinitely differentiable on $]\\lambda _{-},\\lambda _{+}[$ ; if $MGF_{\\mathbb {}}$ is finite in a neighborhood of zero, i.e.",
"$0 \\in ]\\lambda _{-},\\lambda _{+}[$ , then for all $k \\in \\mathbb {N}_{0}$ the $k-$ th moment of $\\mathbb {}$ respectively $W$ exists and is finite and can be computed in terms of the $k-$ th derivative $MGF_{\\mathbb {}}^{(k)}$ as $MGF_{\\mathbb {}}^{(k)}(0) =\\int _{\\mathbb {R}} y^{k} \\, d\\mathbb {} (y)= E_{\\mathbb {\\Pi }}[W^{k}],$ which, by the way, then allows the interpretation of $MGF_{\\mathbb {}}$ as “moment generating function of $\\mathbb {}$ resp.",
"$W$ ”since we assume $0 \\in ]\\lambda _{-},\\lambda _{+}[$ , we have already used the meaningful abbreviation $MGF$ (rather than LST) in (REF ); if $\\lambda _{-} \\ne \\lambda _{+}$ , then $MGF_{\\mathbb {}}$ is strictly convex on $]\\lambda _{-},\\lambda _{+}[$ .",
"Hence, the logarithm of the Laplace-Stieltjes transform $z\\mapsto \\Lambda _{\\mathbb {}}(z) := \\log MGF_{\\mathbb {} }(z):=\\log \\int _{\\mathbb {R}}e^{z \\cdot y} \\, d\\mathbb {} (y)= \\log E_{\\mathbb {\\Pi }}[e^{z\\cdot W}]$ (which in case of $0 \\in ]\\lambda _{-},\\lambda _{+}[$ can be interpreted as cumulant generating function) “carries over” (M1) to (M6), which partially can be even refined: $\\Lambda _{\\mathbb {}}$ takes values in $]-\\infty ,\\infty ]$ ; $dom(\\Lambda _{\\mathbb {}}) = dom(MGF_{\\mathbb {}})$ and thus $int(dom(\\Lambda _{\\mathbb {} })) = \\, ]\\lambda _{-},\\lambda _{+}[$ ; $\\Lambda _{\\mathbb {} }$ is continuous on $dom(\\Lambda _{\\mathbb {}})$ and lower semicontinuous on $\\mathbb {R}$ ; if $\\lambda _{-} \\ne \\lambda _{+}$ , then $\\lambda _{\\mathbb {}}$ is infinitely differentiable on $]\\lambda _{-},\\lambda _{+}[$ ; if $0 \\in ]\\lambda _{-},\\lambda _{+}[$ , then $& & \\Lambda _{\\mathbb {} }(0) = 0, \\quad \\Lambda _{\\mathbb {} }^{\\prime }(0) = \\int _{\\mathbb {R}} y \\, d\\mathbb {} (y)= E_{\\mathbb {\\Pi }}[W],\\\\& & \\Lambda _{\\mathbb {} }^{\\prime \\prime }(0) =\\int _{\\mathbb {R}} \\Big (y - \\int _{\\mathbb {R}} \\widetilde{y} \\, d\\mathbb {} (\\widetilde{y})\\Big )^{2}\\, d\\mathbb {} (y)= E_{\\mathbb {\\Pi }}[W^{2}] - (E_{\\mathbb {\\Pi }}[W])^{2} = Var_{\\mathbb {\\Pi }}[W];$ under the assumption $\\lambda _{-} \\ne \\lambda _{+}$ there holds: $\\Lambda _{\\mathbb {}}$ is strictly convex on $]\\lambda _{-},\\lambda _{+}[$ if and only if $\\mathbb {}$ is not a one-point distribution (Dirac mass) if and only if $W$ is not a.s. constant; otherwise, $\\Lambda _{\\mathbb {}}$ is linear; under the assumption that $\\mathbb {}$ is not a one-point distribution (Dirac mass) — with the notations $a := \\inf supp(\\mathbb {}) = \\inf supp(W)$ , $b := \\sup supp(\\mathbb {}) = \\sup supp(W)$ , $t_{-}^{sc} := \\inf \\lbrace \\Lambda _{\\mathbb {}}^{\\prime }(z) \\, : \\, z \\in ]\\lambda _{-},\\lambda _{+}[ \\rbrace = \\lim _{z \\downarrow \\lambda _{-}} \\Lambda _{\\mathbb {}}^{\\prime }(z)$ and $t_{+}^{sc} := \\sup \\lbrace \\Lambda _{\\mathbb {}}^{\\prime }(z) \\, : \\, z \\in ]\\lambda _{-},\\lambda _{+}[ \\rbrace = \\lim _{z \\uparrow \\lambda _{+}} \\Lambda _{\\mathbb {}}^{\\prime }(z)$ — one gets the following assertions: $]t_{-}^{sc},t_{+}^{sc}[ \\ \\subseteq \\ ]a,b[$ ; if $a > - \\infty $ , then $\\lambda _{-} = - \\infty $ , $t_{-}^{sc} = \\lim _{z \\rightarrow -\\infty } \\Lambda _{\\mathbb {}}^{\\prime }(z)= \\lim _{z \\rightarrow -\\infty } \\frac{\\Lambda _{\\mathbb {}}(z)}{z} = a$ ; if $b < \\infty $ , then $\\lambda _{+} = \\infty $ , $t_{+}^{sc} = \\lim _{z \\rightarrow \\infty } \\Lambda _{\\mathbb {}}^{\\prime }(z)= \\lim _{z \\rightarrow \\infty } \\frac{\\Lambda _{\\mathbb {}}(z)}{z} = b$ ; if $a = - \\infty $ and $\\lambda _{-} = - \\infty $ , then $t_{-}^{sc} = \\lim _{z \\rightarrow -\\infty } \\Lambda _{\\mathbb {}}^{\\prime }(z)= -\\infty = a$ ; if $b = \\infty $ and $\\lambda _{+} = \\infty $ , then $t_{+}^{sc} = \\lim _{z \\rightarrow \\infty } \\Lambda _{\\mathbb {}}^{\\prime }(z)= \\infty = b$ ; if $\\lambda _{-} \\in \\, ]-\\infty ,0[$ and $t_{-}^{sc} > -\\infty $ , then $a = - \\infty $ , $\\Lambda _{\\mathbb {}}(\\lambda _{-}) \\in \\, ]-\\infty ,\\infty [$ , $\\Lambda _{\\mathbb {}}(z) = \\infty $ for all $z < \\lambda _{-}$ , $\\Lambda _{\\mathbb {}}^{\\prime }(\\lambda _{-}) \\in \\, ]-\\infty ,\\infty [$ ; if $\\lambda _{+} \\in \\, ]0,\\infty [$ and $t_{+}^{sc} < \\infty $ , then $b = \\infty $ , $\\Lambda _{\\mathbb {}}(\\lambda _{+}) \\in \\, ]-\\infty ,\\infty [$ , $\\Lambda _{\\mathbb {}}(z) = \\infty $ for all $z > \\lambda _{+}$ , $\\Lambda _{\\mathbb {}}^{\\prime }(\\lambda _{+})\\in \\, ]-\\infty ,\\infty [$ ; if $\\lambda _{-} \\in \\, ]-\\infty ,0[$ and $t_{-}^{sc} = -\\infty $ , then $a= - \\infty $ ; if $\\lambda _{+} \\in \\, ]0,\\infty [$ and $t_{+}^{sc} = \\infty $ , then $b= \\infty $ .",
"Notice that (C7ii) to (C7ix) cover all possible constellations.",
"For a proof of (C7ii) to (C7vii) as well as further details, see e.g.",
"Section 9.1 in Borovkov [56].",
"By contradiction, (C7viii) follows from (C7ii) and (C7ix) follows from (C7iii).",
"Moreover, (C7i) is a consequence (C7ii) to (C7ix).",
"As a side remark, notice that (C6) refines (M6).",
"According to the representability requirement (REF ), one has $\\varphi (t)=\\sup _{z\\in \\mathbb {R}}\\left( z\\cdot t- \\Lambda _{\\mathbb {}}(z)\\right)=: \\Lambda _{\\mathbb {}}^{*}(t), \\qquad t\\in \\mathbb {R},\\ \\ $ (i.e.",
"the divergence generator $\\varphi $ must be equal to the Fenchel-Legendre transform $\\Lambda _{\\mathbb {}}^{*}$ of a cumulant generating function $\\Lambda _{\\mathbb {}}$ ) of some probability distribution $\\mathbb {}$ , such that $\\lambda _{-} < 0 < \\lambda _{+}$ holds.",
"Moreover, $\\varphi $ should satisfy $\\varphi (1)=0$ , and should be finite as well as strictly convex in a non-empty neighborhood $]t_{-}^{sc},t_{+}^{sc}[$ of 1 (cf.",
"the definition of $\\widetilde{\\Upsilon }(]a,b[)$ ).",
"The latter rules out that $\\mathbb {}$ is any one-point distribution (Dirac distribution), say $\\mathbb {} = \\delta _{y_{0}}$ for some $y_{0} \\in \\mathbb {R}$ , since in such a situation one gets $\\Lambda _{\\mathbb {}}(z) = z \\cdot y_{0}$ , and thus $\\varphi (t) = \\Lambda _{\\mathbb {}}^{*}(t) = 0$ for $t=y_{0}$ and $\\varphi (t) = \\Lambda _{\\mathbb {}}^{*}(t) = \\infty $ for all $t \\in \\mathbb {R}\\backslash \\lbrace y_{0}\\rbrace $ (even in the case $y_{0} =1$ for which $\\varphi (1)=0$ is satisfied).",
"Consequently, $\\Lambda _{\\mathbb {}}$ is strictly convex on $]\\lambda _{-},\\lambda _{+}[ \\ = int(dom(\\Lambda _{\\mathbb {} }))$ (cf.",
"(C6)) and (C7) applies.",
"Clearly, by continuity one gets $\\Lambda _{\\mathbb {}}^{*}(t) =\\sup _{z\\in ]\\lambda _{-},\\lambda _{+}[}\\left( t \\cdot z - \\Lambda _{\\mathbb {}}(z)\\right),\\qquad t\\in \\mathbb {R}.",
"\\ \\ $ For $t \\in ]t_{-}^{sc},t_{+}^{sc}[$ , the optimization problem (REF ) can be solved explicitly by the well-known “pure/original” Legendre transform, namely $\\Lambda _{\\mathbb {}}^{*}(t) = t \\cdot \\Lambda _{\\mathbb {}}^{\\prime -1}(t) -\\Lambda _{\\mathbb {}}\\Big (\\Lambda _{\\mathbb {}}^{\\prime -1}(t)\\Big ),\\qquad t \\in ]t_{-}^{sc},t_{+}^{sc}[.$ Let us inspect the further cases $t \\le t_{-}^{sc}$ .",
"In the contexts of (C7iv) and (C7viii), this is obsolete since $t_{-}^{sc} = a = -\\infty $ .",
"For (C7ii), where $t_{-}^{sc} = a > -\\infty $ , one can show $\\Lambda _{\\mathbb {}}^{*}(a) = - \\log \\mathbb {}[\\lbrace a\\rbrace ]= - \\log \\mathbb {\\Pi }[W = a \\, ]$ which together with (REF ) proves (G10ii); moreover, $\\Lambda _{\\mathbb {}}^{*}(t) = \\infty $ for all $t < a$ (see e.g.",
"Section 9.1 of Borovkov [56]).",
"In the setup (C7vi), where $t_{-}^{sc} > a = - \\infty $ it is clear that $\\Lambda _{\\mathbb {}}^{*}(t_{-}^{sc}) =t_{-}^{sc} \\cdot \\Lambda _{\\mathbb {}}^{\\prime -1}(t_{-}^{sc}) -\\Lambda _{\\mathbb {}}\\Big (\\Lambda _{\\mathbb {}}^{\\prime -1}(t_{-}^{sc})\\Big )= t_{-}^{sc} \\cdot \\lambda _{-} - \\Lambda _{\\mathbb {}}(\\lambda _{-})$ and $\\Lambda _{\\mathbb {}}^{*}(t)= t \\cdot \\lambda _{-} - \\Lambda _{\\mathbb {}}(\\lambda _{-})= \\Lambda _{\\mathbb {}}^{*}(t_{-}^{sc}) + \\lambda _{-} \\cdot (t- t_{-}^{sc})\\quad \\text{for all $t \\in ]-\\infty ,t_{-}^{sc}[$}.$ As far as the cases $t \\ge t_{+}^{sc}$ is concerned, in the situations of (C7v) and (C7ix), this is obsolete since $t_{+}^{sc} = b = \\infty $ .",
"For (C7iii), where $t_{+}^{sc} = b < \\infty $ , one can show $\\Lambda _{\\mathbb {}}^{*}(b) = - \\log \\mathbb {}[\\lbrace b\\rbrace ]= - \\log \\mathbb {\\Pi }[W = b \\, ]$ which together with (REF ) proves (G10iii); moreover, $\\Lambda _{\\mathbb {}}^{*}(t) = \\infty $ for all $t > b$ (see e.g.",
"Section 9.1 of Borovkov [56]).",
"In the setup (C7vii), where $t_{+}^{sc} < b = \\infty $ it is clear that $\\Lambda _{\\mathbb {}}^{*}(t_{+}^{sc}) =t_{+}^{sc} \\cdot \\Lambda _{\\mathbb {}}^{\\prime -1}(t_{+}^{sc}) -\\Lambda _{\\mathbb {}}\\Big (\\Lambda _{\\mathbb {}}^{\\prime -1}(t_{+}^{sc})\\Big )= t_{+}^{sc} \\cdot \\lambda _{+} - \\Lambda _{\\mathbb {}}(\\lambda _{+})$ and $\\Lambda _{\\mathbb {}}^{*}(t)= t \\cdot \\lambda _{+} - \\Lambda _{\\mathbb {}}(\\lambda _{+})= \\Lambda _{\\mathbb {}}^{*}(t_{+}^{sc}) + \\lambda _{+} \\cdot (t- t_{+}^{sc})\\quad \\text{for all $t \\in ]t_{+}^{sc}, \\infty [$}.$ As a side effect, we have thus also proved (G10i) and (G3) (notice that in (G3) we have started with $a, b$ to be the endpoints of the support of $\\mathbb {}$ respectively $W$ , in contrast to Definition REF where $a$ , $b$ are defined as the endpoints of the effective domain of $\\varphi $ ).",
"To proceed, from (REF ) and (REF ) we obtain $\\varphi ^{\\prime }(t) = (\\Lambda _{\\mathbb {}}^{*})^{\\prime }(t) = \\Lambda _{\\mathbb {}}^{\\prime -1}(t),\\qquad \\varphi ^{\\prime \\prime }(t) = (\\Lambda _{\\mathbb {}}^{*})^{\\prime \\prime }(t) =\\frac{1}{\\Lambda _{\\mathbb {}}^{\\prime \\prime }\\big (\\Lambda _{\\mathbb {}}^{\\prime -1}(t)\\big )} > 0,\\qquad t \\in ]t_{-}^{sc},t_{+}^{sc}[,$ which — together with the investigations below (REF ) — provides (G4) and (G5); moreover, (G6) is immediate since the infinite differentiability is straightforward and $\\varphi ^{\\prime }(1) =0$ because we have required both the nonnegativity of $\\varphi $ and (G2) (cf.",
"the definition of $\\widetilde{\\Upsilon }(]a,b[)$ ).",
"The property (G7) follows from (C7ii), (C7iv), (C7viii), (REF ), (REF ) and $\\varphi ^{\\prime }(t_{-}^{sc}) = \\Lambda _{\\mathbb {}}^{\\prime -1}(t_{-}^{sc}) = \\lambda _{-}$ .",
"Analogously, we get (G8) from (C7iii), (C7v), (C7ix), (REF ), (REF ) and $\\varphi ^{\\prime }(t_{+}^{sc}) = \\Lambda _{\\mathbb {}}^{\\prime -1}(t_{+}^{sc}) = \\lambda _{+}$ .",
"Let us continue with (G9).",
"By applying the general theory of double Fenchel-Legendre transforms (bi-conjugates), (REF ) turns into $\\varphi ^{*} (z) = \\Lambda _{\\mathbb {}}(z), \\qquad z\\in \\mathbb {R},\\ \\ $ which deduces (G9i).",
"The properties (G9ii), (G9iii) and (G9iv) follow from Theorem REF (cf.",
"the discussion thereafter).",
"Finally, we obtain (G11i) and (G11ii) from (REF ), (REF ) and ().",
"$\\blacksquare $ Proof of Proposition REF .",
"The assertions follow immediately from (REF ), (REF ), (REF ), Theorem REF , (REF ) (and the discussion thereafter) as well as (M5).",
"$\\blacksquare $"
],
[
"Proofs — Part 5",
"Proof of Proposition REF .",
"The assertion follows straightforwardly from the following two facts: (i) a moment generating function $MGF$ is infinitely divisible if and only if $MGF^{c}$ is a moment generating function for all $c >0$ (cf.",
"e.g.",
"(the MGF-version of) Prop.",
"IV.2.5 of Steutel & van Harn [341]).",
"(ii) $z \\mapsto MGF(z)$ is a moment generating function if and only if $z \\mapsto MGF(\\breve{c} \\cdot z) =: MGF_{\\breve{c}}(z)$ is a moment generating function for all $\\breve{c} >0$ .",
"Notice that for each $c >0$ , $\\breve{c} >0$ one has $int(dom(MGF)) = int(dom(MGF^{c}))$ and $int(dom(MGF_{\\breve{c}})) = \\frac{1}{\\breve{c}} \\cdot int(dom(MGF))$ , and hence the light-tailedness remains unchanged: $0 \\in int(dom(MGF))$ if and only if $0 \\in int(dom(MGF^{c}))$ if and only if $0 \\in int(dom(MGF_{\\breve{c}}))$ .",
"Since $\\varphi \\in \\Upsilon (]a,b[)$ , we have $\\varphi (t)=\\sup _{z\\in ]\\lambda _{-}, \\lambda _{+}[}\\left( z \\cdot t-\\log \\Big (\\int _{\\mathbb {R}} e^{z\\cdot y }\\, d\\mathbb {} (y) \\Big ) \\right), \\quad t\\in ]a,b[ \\, , \\ \\ $ and thus for the exponential of its Fenchel-Legendre transform $\\int _{\\mathbb {R}} e^{z \\cdot y} d\\mathbb {} (y) , \\qquad z \\in ]\\lambda _{-}, \\lambda _{+}[ .$ Now, let $\\widetilde{\\varphi } := \\widetilde{c} \\cdot \\varphi \\in \\Upsilon (]a,b[)$ for arbitrarily fixed $\\widetilde{c} >0$ .",
"From the application of (REF ) to $\\widetilde{\\varphi }$ we obtain $\\widetilde{\\varphi } (t)=\\sup _{\\widetilde{z} \\in ]\\widetilde{\\lambda }_{-}, \\widetilde{\\lambda }_{+}[}\\left( \\widetilde{z} \\cdot t-\\log \\int _{\\mathbb {R}}e^{\\widetilde{z} \\cdot \\widetilde{y}} d\\widetilde{\\mathbb {}}_{\\widetilde{c}} (\\widetilde{y})\\right), \\qquad t\\in ]a,b[ \\, , \\ \\ $ for some unique probability distribution $\\widetilde{\\mathbb {}}_{\\widetilde{c}}$ on $\\mathbb {R}$ .",
"Here, according to (G9i) for $\\widetilde{\\varphi }$ we have used $\\widetilde{\\lambda }_{-} := \\inf _{t \\in ]a,b[} \\widetilde{\\varphi }^{\\prime }(t) = \\widetilde{c} \\cdot \\lambda _{-}$ and $\\widetilde{\\lambda }_{+} := \\sup _{t \\in ]a,b[} \\widetilde{\\varphi }^{\\prime }(t) = \\widetilde{c} \\cdot \\lambda _{+}$ .",
"Dividing (REF ) by $\\widetilde{c}$ , we arrive at $\\varphi (t) = \\frac{\\widetilde{\\varphi } (t)}{\\widetilde{c}}&=&\\sup _{\\widetilde{z} \\in ]\\widetilde{c} \\cdot \\lambda _{-}, \\widetilde{c} \\cdot \\lambda _{+}[}\\left( \\frac{\\widetilde{z}}{\\widetilde{c}} \\cdot t-\\log \\Big (\\int _{\\mathbb {R}}e^{\\frac{\\widetilde{z}}{\\widetilde{c}} \\cdot \\widetilde{y} \\cdot \\widetilde{c}} d\\widetilde{\\mathbb {}}_{\\widetilde{c}} (\\widetilde{y})\\Big )^{1/\\widetilde{c}}\\right),\\nonumber \\\\&=&\\sup _{z \\in ]\\lambda _{-}, \\lambda _{+}[}\\left( z \\cdot t-\\log \\Big (\\int _{\\mathbb {R}}e^{z \\cdot \\widetilde{y} \\cdot \\widetilde{c}}d\\widetilde{\\mathbb {}}_{\\widetilde{c}} (\\widetilde{y})\\Big )^{1/\\widetilde{c}}\\right),\\qquad t\\in ]a,b[ \\, , \\ \\ $ and hence for the exponential of its Fenchel-Legendre transform $e^{\\varphi _{*}(z)} =\\Big ( \\int _{\\mathbb {R}} e^{z \\cdot \\widetilde{y} \\cdot \\widetilde{c}}d\\widetilde{\\mathbb {}}_{\\widetilde{c}} (\\widetilde{y}) \\Big )^{1/\\widetilde{c}}, \\qquad z \\in ]\\lambda _{-}, \\lambda _{+}[ .$ Here, according to (G9i) for $\\widetilde{\\varphi }$ we have used $\\widetilde{\\lambda }_{-} := \\inf _{t \\in ]a,b[} \\widetilde{\\varphi }^{\\prime }(t) = \\widetilde{c} \\cdot \\lambda _{-}$ and $\\widetilde{\\lambda }_{+} := \\sup _{t \\in ]a,b[} \\widetilde{\\varphi }^{\\prime }(t) = \\widetilde{c} \\cdot \\lambda _{+}$ .",
"From (REF ) and (REF ) we deduce for $\\widetilde{c} := \\frac{1}{n}$ the relation $MGF_{\\mathbb {}}(z) = (MGF_{\\widetilde{\\mathbb {}}_{1/n}}(\\frac{z}{n}))^{n}$ for all $n \\in \\mathbb {N}$ which (with the help of (ii)) implies the infinitely divisibility of $\\mathbb {}$ .",
"For the reverse direction, let us assume that $\\varphi \\in \\Upsilon (]a,b[)$ and that the corresponding $\\mathbb {}$ is infinitely divisible.",
"Recall that $]a,b[ = int(dom(\\varphi ))$ .",
"Moreover, we fix an arbitrary constant $\\widetilde{c} >0$ .",
"Of course, there holds $\\widetilde{c} \\cdot \\varphi \\in \\widetilde{\\Upsilon }(]a,b[)$ and $dom(\\widetilde{c} \\cdot \\varphi ) = dom(\\varphi )$ .",
"Furthermore, by multiplying (REF ) with $\\widetilde{c} >0$ and by employing (i), (ii) we get $\\widetilde{c} \\cdot \\varphi (t) &=&\\sup _{z\\in ]\\lambda _{-}, \\lambda _{+}[}\\left( \\widetilde{c} \\cdot z \\cdot t-\\log \\Big (\\int _{\\mathbb {R}} e^{\\widetilde{c} \\cdot z\\cdot \\frac{y}{\\widetilde{c}}} d\\mathbb {} (y) \\Big )^{\\widetilde{c}} \\right)= \\sup _{\\widetilde{z} \\in ]\\widetilde{c} \\cdot \\lambda _{-}, \\widetilde{c} \\cdot \\lambda _{+}[}\\left( \\widetilde{z} \\cdot t-\\log \\Big (\\int _{\\mathbb {R}}e^{\\frac{\\widetilde{z}}{\\widetilde{c}} \\cdot y} \\, d\\mathbb {} (y) \\Big )^{\\widetilde{c}} \\right)\\nonumber \\\\&=&\\sup _{\\widetilde{z} \\in ]\\widetilde{c} \\cdot \\lambda _{-}, \\widetilde{c} \\cdot \\lambda _{+}[}\\left( \\widetilde{z} \\cdot t-\\log \\Big ( \\int _{\\mathbb {R}}e^{\\widetilde{z} \\cdot y} d\\mathbb {}_{\\widetilde{c}} (y) \\Big ) \\right), \\quad t\\in ]a,b[; \\ \\ $ for some probability distribution $\\mathbb {}_{\\widetilde{c}}$ on $\\mathbb {R}$ .",
"$\\blacksquare $ Proof of Proposition REF .",
"It is well known that a candidate function $M: ]-\\infty ,0[ \\mapsto ]0,\\infty [$ is the moment-generating function of an infinitely divisible probability distribution if and only if $(\\log M)^{\\prime }$ is absolutely monotone (see e.g.",
"Theorem 5.11 of Schilling et al.",
"[322]).",
"By applying this to $M(z) := e^{-a\\cdot z + \\varphi ^{*}(z)}$ respectively $M(z) := e^{b\\cdot z + \\varphi ^{*}(- z)}$ , one gets straightforwardly the assertion (a) respectively (b); notice that the light-tailedness follows then from (G1) to (G8), and $b=\\infty $ respectively $a =- \\infty $ can be deduced from the fact that the support of an infinitely distribution is always (one-sided or two-sided) unbounded.",
"For the third case $a= - \\infty $ , $b= \\infty $ one can use the assertion (cf.",
"e.g.",
"Morris [267], p.73) that a candidate function $M: \\, ]\\lambda _{-}, \\lambda _{+}[ \\, \\mapsto \\, ]0,\\infty [$ is the moment-generating function of an infinitely divisible probability distribution if the connected function $z \\mapsto (\\log M)^{\\prime \\prime }(z)/(\\log M)^{\\prime \\prime }(0)$ is the moment-generating function of some auxiliary probability distribution; but the latter is equivalent to exponentially convexity (cf.",
"Theorem REF (b)).",
"By applying this to $M(z) := e^{\\varphi ^{*}(z)}$ , one ends up with (c).",
"$\\blacksquare $"
],
[
"Proofs — Part 6",
"Proof of Theorem REF .",
"(i) Clearly, on $]\\lambda _{-},\\lambda _{+}[$ the function $\\Lambda $ is differentiable with strictly increasing derivative $\\Lambda ^{\\prime }(z) = F^{-1}(z+c) + 1- F^{-1}(c) ,\\qquad z \\in ]\\lambda _{-},\\lambda _{+}[.$ Hence, $\\Lambda $ is strictly convex and smooth (because of the smoothness of $F^{-1}$ ), and satisfies $\\Lambda (0) =0$ as well as $\\Lambda ^{\\prime }(0) =1$ .",
"Also, the corresponding extensions of $\\Lambda $ to $z=\\lambda _{-}$ and $z=\\lambda _{+}$ are continuous.",
"(ii) It is straightforward to see that on $]t_{-}^{sc},t_{+}^{sc}[$ the function $\\varphi $ is differentiable with strictly increasing derivative $\\varphi ^{\\prime }(t) = F(t+ F^{-1}(c) - 1) - c ,\\qquad t \\in ]t_{-}^{sc},t_{+}^{sc}[.$ Hence, $\\varphi $ is strictly convex and smooth (because of the smoothness of $F$ ), and satisfies $\\varphi (1) =0$ as well as $\\varphi ^{\\prime }(1) =0$ .",
"Also, the corresponding extensions of $\\varphi $ to $t=t_{-}^{sc}$ and $t=t_{-}^{sc}$ are continuous.",
"Hence (G1), (G2), (G5) and (G6) hold.",
"To prove (G3) (and hence (G1)), let us first notice that obviously there holds $a \\le t_{-}^{sc}$ and $t_{+}^{sc} \\le b$ .",
"Moreover, the validity of $\\varphi (t) < \\infty $ for all $t \\in ]t_{-}^{sc},t_{+}^{sc}[$ is clear from (REF ) since $t+F^{-1}(c)-1 \\in ]a_{F},b_{F}[ = int(dom(F))$ and the involved integral over the continuous function $F^{-1}$ is taken over a compact interval.",
"For the subcase $t_{-}^{sc} =- \\infty = a$ we have thus shown $dom(\\varphi ) \\cap ]-\\infty ,1] = ]-\\infty ,1]= ]a,1]$ , whereas for the subcase $t_{+}^{sc} = \\infty = b$ we have verified $dom(\\varphi ) \\cap [1,\\infty [ = [1,\\infty [ = [1,b[$ .",
"Let us next examine the subcase “$t_{-}^{sc} > - \\infty $ and $\\varphi (t_{-}^{sc}) < \\infty $ ”: if $\\lambda _{-} > - \\infty $ then $a = -\\infty $ and (REF ) implies $\\varphi (t) = \\varphi (t_{-}^{sc}) +\\lambda _{-} \\cdot (t- t_{-}^{sc}) < \\infty $ for all $t \\in \\ ]-\\infty , t_{-}^{sc}] = ]a,t_{-}^{sc}]$ , which leads to $dom(\\varphi ) \\cap ]-\\infty ,1] = ]-\\infty ,1] = ]a,1]$ ; in contrast, if $\\lambda _{-} = - \\infty $ then $a = t_{-}^{sc}$ and (REF ) implies $\\varphi (t) = \\varphi (t_{-}^{sc}) +\\lambda _{-} \\cdot (t- t_{-}^{sc}) = \\infty $ for all $t \\in \\ ]-\\infty , t_{-}^{sc}[ = ]-\\infty ,a[$ , which leads to $dom(\\varphi ) \\cap ]-\\infty ,1] = [a,1]$ .",
"In the subcase “$t_{-}^{sc} > - \\infty $ and $\\varphi (t_{-}^{sc}) = \\infty $ ”, due to the strict convexity of $\\varphi $ one always has $\\lim _{t \\downarrow t_{-}^{sc}} \\varphi ^{\\prime }(t) = - \\infty $ ; this implies, by the below-mentioned (REF ), that $\\lambda _{-} = - \\infty $ and thus $a = t_{-}^{sc}$ ; from (REF ) we derive $\\varphi (t) = \\varphi (t_{-}^{sc}) +\\lambda _{-} \\cdot (t- t_{-}^{sc}) = \\infty $ for all $t \\in \\ ]-\\infty , t_{-}^{sc}[ = ]-\\infty ,a[$ , which leads to $dom(\\varphi ) \\cap ]-\\infty ,1] = ]a,1]$ .",
"As a further step, we deal with the subcase “$t_{+}^{sc} < \\infty $ and $\\varphi (t_{+}^{sc}) < \\infty $ ”: if $\\lambda _{+} < \\infty $ then $b = \\infty $ and (REF ) implies $\\varphi (t) = \\varphi (t_{+}^{sc}) +\\lambda _{+} \\cdot (t- t_{+}^{sc}) < \\infty $ for all $t \\in \\ [t_{+}^{sc},\\infty [ = [t_{+}^{sc},b[$ , which leads to $dom(\\varphi ) \\cap [1, \\infty [ = [1, \\infty [ = [1, b[$ ; in contrast, if $\\lambda _{+} = \\infty $ then $b = t_{+}^{sc}$ and (REF ) implies $\\varphi (t) = \\varphi (t_{+}^{sc}) +\\lambda _{+} \\cdot (t- t_{+}^{sc}) = \\infty $ for all $t \\in \\ ]t_{-}^{sc}, \\infty [ = ]b,\\infty [$ , which leads to $dom(\\varphi ) \\cap [1,\\infty [ = [1,b]$ .",
"In the subcase “$t_{+}^{sc} < + \\infty $ and $\\varphi (t_{+}^{sc}) = \\infty $ ”, due to the strict convexity of $\\varphi $ one always gets $\\lim _{t \\uparrow t_{+}^{sc}} \\varphi ^{\\prime }(t) = \\infty $ ; this implies, by the below-mentioned (), that $\\lambda _{+} = \\infty $ and thus $b = t_{+}^{sc}$ ; from (REF ) we deduce $\\varphi (t) = \\varphi (t_{+}^{sc}) +\\lambda _{+} \\cdot (t- t_{+}^{sc}) = \\infty $ for all $t \\in \\ ]t_{+}^{sc}, \\infty [ = ]b, \\infty [$ , which leads to $dom(\\varphi ) \\cap [1,\\infty [ = [1,b[$ .",
"Putting things together, we have proved (G3).",
"The property (G4) follows straightforwardly from (REF ), the continuity of $F$ and from $\\lim _{t \\downarrow t_{-}^{sc}} \\varphi ^{\\prime }(t) = \\lambda _{-}$ , $\\lim _{t \\uparrow t_{+}^{sc}} \\varphi ^{\\prime }(t) = \\lambda _{+}$ .",
"To see the latter two, from (REF ) we obtain $& & \\lim _{t \\downarrow t_{-}^{sc}} \\varphi ^{\\prime }(t) = \\lim _{t \\downarrow t_{-}^{sc}} F(t+ F^{-1}(c) - 1) - c= \\lim _{t \\downarrow t_{-}^{sc}} F(t+ a_{F} - t_{-}^{sc}) - c= \\inf \\lbrace F(\\widetilde{t}) - c: \\widetilde{t} \\in ]a_{F},b_{F}[ \\rbrace = \\lambda _{-},\\\\& & \\lim _{t \\uparrow t_{+}^{sc}} \\varphi ^{\\prime }(t) = \\lim _{t \\uparrow t_{+}^{sc}} F(t+ F^{-1}(c) - 1) - c= \\lim _{t \\uparrow t_{+}^{sc}} F(t+ b_{F} - t_{+}^{sc}) - c= \\sup \\lbrace F(\\widetilde{t}) - c: \\widetilde{t} \\in ]a_{F},b_{F}[ \\rbrace = \\lambda _{+}.$ The two properties (G7) and (G8) are clear form the above considerations.",
"(iii) From (REF ) and (REF ) one gets easily $\\Lambda ^{\\prime -1}(t) = F\\left(t+F^{-1}(c)-1 \\right) - c = \\varphi ^{\\prime }(t), \\qquad t \\in ]t_{-}^{sc},t_{+}^{sc}[,$ as well as $\\Lambda ^{\\prime -1}(1)=0$ .",
"From this, we derive $&& t \\cdot \\Lambda ^{\\prime -1}(t) - \\Lambda \\left(\\Lambda ^{\\prime -1}(t)\\right)\\nonumber \\\\&& = t \\cdot [F\\left(t+F^{-1}(c)-1 \\right) - c]+ [F^{-1}(c)-1] \\cdot [F\\left(t+F^{-1}(c)-1 \\right) - c]\\nonumber \\\\& & -\\int \\displaylimits _{0}^{F\\left(t+F^{-1}(c)-1 \\right) - c} F^{-1}(u+c) du\\nonumber \\\\& & = \\varphi (t), \\qquad t \\in ]t_{-}^{sc},t_{+}^{sc}[,$ and hence, with the help of (REF ) in combination with (REF ), () $\\varphi (t) = \\max _{z \\in ]\\lambda _{-},\\lambda _{+}[} \\left( z\\cdot t -\\Lambda (z)\\right), \\qquad t \\in ]t_{-}^{sc},t_{+}^{sc}[ ,$ i.e.",
"on $]t_{-}^{sc},t_{+}^{sc}[$ the divergence generator $\\varphi $ is the classical Legendre transform of the restriction of $\\Lambda $ to $]\\lambda _{-},\\lambda _{+}[$ .",
"If “$\\lambda _{-} > -\\infty $ , $\\Lambda (\\lambda _{-}) \\in ]-\\infty ,\\infty [$ and $\\Lambda ^{\\prime }(\\lambda _{-}) \\in ]-\\infty ,\\infty [$ ” respectively “$\\lambda _{+} < -\\infty $ , $\\Lambda (\\lambda _{+}) \\in ]-\\infty ,\\infty [$ and $\\Lambda ^{\\prime }(\\lambda _{+}) \\in ]-\\infty ,\\infty [$ ”, then one can apply classical facts of Fenchel-Legendre transformation to get the corresponding left-hand respectively right-hand linear extensions of $\\varphi $ on the complement of $]t_{-}^{sc},t_{+}^{sc}[$ , in order to obtain the desired $\\varphi (t) =\\sup _{z \\in ]-\\infty ,\\infty [} \\left( z\\cdot t - \\Lambda (z)\\right) ,\\qquad t \\in \\mathbb {R};$ notice that $t_{-}^{sc} = \\lim _{z \\downarrow \\lambda _{-}} \\Lambda ^{\\prime }(z)$ and $t_{+}^{sc} = \\lim _{z \\uparrow \\lambda _{+}} \\Lambda ^{\\prime }(z)$ .",
"(iv) This is just the inverse of (iii), by applying standard Fenchel-Legendre-transformation theory.",
"$\\blacksquare $"
],
[
"Further details and proofs for\nSubsection ",
"Proof of Lemma REF.",
"By Assumption (OM), one gets for all $\\lambda \\in cl(\\mathbf {\\Lambda })$ that $\\lbrace \\mathbf {x} \\in (dom(\\widetilde{\\varphi })^{n} : T(\\mathbf {x}) = \\lambda \\rbrace \\, \\cap \\, ]t_{-}^{sc},t_{+}^{sc}[^{n} \\ne \\emptyset $ .",
"Moreover, for any $\\mathbf {x}=\\left( x_{1},..,x_{n}\\right) $ in $\\mathbb {R}^{n}$ , by the independence of the components of $\\mathbf {\\widetilde{W}}$ as well as (REF ) and (REF ), we have $& & I_{\\mathbf {\\widetilde{W}}}(\\mathbf {x})=\\sup _{\\mathbf {z}=\\left( z_{1},\\ldots ,z_{n}\\right) \\in \\mathbb {R}^{n}}\\left( \\left\\langle \\mathbf {x},\\mathbf {z}\\right\\rangle -\\sum _{i=1}^{n}\\Lambda _{\\widetilde{\\mathbb {}}}(z_{i}) \\right)= \\sup _{\\mathbf {z}\\in ]\\lambda _{-},\\lambda _{+}[^{n}}\\left( \\sum _{i=1}^{n} \\left( x_{i} \\cdot z_{i}-\\Lambda _{\\widetilde{\\mathbb {}}}(z_{i}) \\right) \\right)\\nonumber \\\\& & = \\sum _{i=1}^{n} \\left( \\sup _{z_{i}\\in ]\\lambda _{-},\\lambda _{+}[}\\left( x_{i}\\cdot z_{i} - \\Lambda _{\\widetilde{\\mathbb {}}}(z_{i})\\right) \\right)= \\sum _{i=1}^{n} \\widetilde{\\varphi }(x_{i})=\\sum _{k=1}^{K}\\sum _{i\\in I_{k}^{(n)}} \\widetilde{\\varphi }(x_{i})$ which is finite if and only if $\\mathbf {x} \\in (dom(\\widetilde{\\varphi }))^{n}$ (recall that $\\widetilde{\\varphi }$ is a nonnegative function).",
"Hence, for each $\\lambda \\in \\mathbf {\\Lambda }$ we obtain $I(\\lambda ) &:=&\\inf _{\\mathbf {x} \\in \\mathbb {R}^{n}: \\, T(\\mathbf {x})=\\lambda }I_{\\mathbf {\\widetilde{W}}}(\\mathbf {x})= \\inf _{\\mathbf {x} \\in (dom(\\widetilde{\\varphi }))^{n}: \\, T(\\mathbf {x})=\\lambda }I_{\\mathbf {\\widetilde{W}}}(\\mathbf {x}) =\\inf _{\\mathbf {x} \\in (dom(\\widetilde{\\varphi }))^{n}: \\, T(\\mathbf {x})=\\lambda } \\ \\sum _{k=1}^{K}\\sum _{i\\in I_{k}^{(n)}} \\widetilde{\\varphi }(x_{i})\\\\&=& \\sum _{k=1}^{K} n_{k} \\cdot \\widetilde{\\varphi }(\\lambda _{k})\\ = \\ n \\cdot \\sum _{k=1}^{K} \\widetilde{p}_{k} \\cdot \\widetilde{\\varphi }(\\lambda _{k})= \\inf _{\\mathbf {x} \\in ]t_{-}^{sc},t_{+}^{sc}[^{n} \\, : \\, T(\\mathbf {x})=\\lambda }I_{\\mathbf {\\widetilde{W}}}(\\mathbf {x}) \\, ;$ here, we have employed the following facts: (i) the right-most infimum in (REF ) is achieved by minimizing each of the $K$ terms $\\sum _{i\\in I_{k}^{(n)}}\\widetilde{\\varphi }(x_{i})$ under the linear constraint $\\frac{1}{n_{k}} \\cdot \\sum _{i\\in I_{k}^{(n)}} x_{i}= \\lambda _{k}$ , and by the strict convexity of $\\widetilde{\\varphi }$ on $]t_{-}^{sc},t_{+}^{sc}[$ (cf.",
"(G5)) the minimum of this generic term is attained when all components $x_{i}$ are equal to $\\lambda _{k}$ , and (ii) the outcoming minimum does not depend on the particular (generally non-unique) choice of the $x_{i}$ ’s.",
"Notice that we have used the relation $n_{k} = n \\cdot \\widetilde{p}_{k}$ as well.",
"To proceed, let $\\underline{\\lambda }$ be a minimal rate point of $\\mathbf {\\Lambda }$ , which means that $\\underline{\\lambda } \\in \\partial \\mathbf {\\Lambda }$ and $I(\\underline{\\lambda }) \\le I(\\lambda )$ for all $\\lambda \\in \\mathbf {\\Lambda }$ .",
"By Assumption (OM) one can run all the steps in (REF ) and () with $\\underline{\\lambda }$ instead of $\\lambda $ , and hence $I(\\underline{\\lambda }) \\ = \\ \\inf _{\\mathbf {x} \\in \\mathbb {R}^{n}: \\, T(\\mathbf {x})=\\underline{\\lambda }}I_{\\mathbf {\\widetilde{W}}}(\\mathbf {x})\\ = \\inf _{\\mathbf {x} \\in ]t_{-}^{sc},t_{+}^{sc}[^{n} \\, : \\, T(\\mathbf {x})=\\underline{\\lambda }}I_{\\mathbf {\\widetilde{W}}}(\\mathbf {x})\\ = \\ n \\cdot \\sum _{k=1}^{K} \\widetilde{p}_{k} \\cdot \\widetilde{\\varphi }(\\underline{\\lambda }_{k})\\ = \\ n \\cdot \\sum _{k=1}^{K} \\widetilde{p}_{k} \\cdot \\widetilde{\\varphi }(\\underline{\\widetilde{q}}_{k}/\\widetilde{p}_{k})$ where for the last equality we have employed the vector $\\underline{\\mathbf {\\widetilde{Q}}} := \\mathfrak {D}^{-1} \\underline{\\lambda }$ which we have called the “dominating point of $\\widetilde{\\mathbf {\\Omega }}$ ”.",
"Also we have proved $I(\\underline{\\lambda })\\ = \\ n \\cdot \\inf _{\\mathbf {\\widetilde{Q}} \\in \\widetilde{\\mathbf {\\Omega }}}\\sum _{k=1}^{K} \\widetilde{p}_{k} \\cdot \\widetilde{\\varphi }(\\widetilde{q}_{k}/\\widetilde{p}_{k}).",
"\\qquad \\qquad \\blacksquare $ On the obtainment of proxies of minimal rate points by proxy method 2: For the rest of this section, besides (OM) we assume that $dom(\\widetilde{\\varphi }) = \\, ]a,b[ \\, = \\, ]t_{-}^{sc},t_{+}^{sc}[$ , and that in case of $a=-\\infty $ or $b=+\\infty $ the divergence generator $\\widetilde{\\varphi }$ is regularly varying at $a$ or $b$ accordingly, with positive index $\\beta $ , i.e.",
"(with a slight abuse of notation) if $a=-\\infty $ , then for all $\\lambda >0$ there holds $\\lim _{u\\rightarrow -\\infty }\\frac{\\widetilde{\\varphi } \\left( \\lambda u\\right) }{\\widetilde{\\varphi }\\left( u\\right) }=\\lambda ^{\\beta },$ if $b=+\\infty $ , then for all $\\lambda >0$ there holds $\\lim _{u\\rightarrow +\\infty }\\frac{\\widetilde{\\varphi } \\left( \\lambda u\\right) }{\\widetilde{\\varphi }\\left( u\\right) }=\\lambda ^{\\beta };$ this assumption is denoted by (H$\\widetilde{\\varphi }$ ).",
"A proxy of $\\underline{\\mathbf {\\widetilde{Q}}}$ can be obtained by sampling from a distribution on $\\mathbb {R}^{K}$ defined through $f(\\mathbf {\\widetilde{Q}}):=C \\cdot \\exp \\left( -\\sum _{k=1}^{K} \\widetilde{p}_{k} \\cdot \\widetilde{\\varphi }(\\widetilde{q}_{k}/\\widetilde{p}_{k})\\right) \\ = \\ C \\cdot \\exp \\left( -D_{\\widetilde{\\varphi }}\\left( \\mathbf {\\widetilde{Q}},\\widetilde{{P}}\\right)\\right)\\hspace{51.21504pt} \\textrm {(cf.",
"(\\ref {simul distr}))}\\nonumber $ where $C$ is a normalizing constant; strict convexity (cf.",
"(G5)) of $\\widetilde{\\varphi }$ together with (H$\\widetilde{\\varphi }$ ) prove that $f$ is a well-defined (Lebesgue-) density for a random variable $\\mathbf {T}$ on $\\mathbb {R}^{K}$ .",
"We denote by $\\mathbb {F}(\\cdot ) := \\mathbb {\\Pi }[\\mathbf {T} \\in \\cdot \\, ]$ the corresponding distribution on $\\mathbb {R}^{K}$ having density $f$ .",
"The distribution of $\\mathbf {T}$ given $\\left( \\mathbf {T} \\in \\widetilde{\\mathbf {\\Omega }} \\right)$ concentrates on the points in $\\widetilde{\\mathbf {\\Omega }}$ which minimize $D_{\\widetilde{\\varphi }}\\left(\\mathbf {\\widetilde{Q}},\\widetilde{{P}}\\right) $ as $\\mathbf {\\widetilde{Q}}$ runs in $\\widetilde{\\mathbf {\\Omega }}$ , when $D_{\\widetilde{\\varphi }}(\\widetilde{\\mathbf {\\Omega }},\\widetilde{{P}})$ is large.",
"This can be argued as follows.",
"We will consider the case when $\\widetilde{\\mathbf {\\Omega }}$ is a compact subset in $\\mathbb {R}_{> 0}^{K}$ and $\\widetilde{\\varphi }$ satisfies (H$\\widetilde{\\varphi }$ ) with $b=+\\infty $ .",
"For the case when $\\widetilde{\\mathbf {\\Omega }}$ is not compact, or belongs to $\\mathbb {R}^{K}/\\left\\lbrace \\mathbf {0}\\right\\rbrace $ , see the Remark REF hereunder.",
"Consider a compact set $\\mathbf {\\Gamma }$ in $\\widetilde{\\mathbf {\\Omega }}$ and let $\\mathbf {\\Gamma }_{t}$ be defined as deduced from $\\mathbf {\\Gamma } $ in a way that makes $D_{\\widetilde{\\varphi }}\\left( \\mathbf {\\Gamma }_{t},\\widetilde{{P}}\\right) $ increase with $t$ for sufficiently large $t$ .",
"For instance, set $\\mathbf {\\Gamma }_{t}:=t \\cdot \\mathbf {\\Gamma } .$ Hence, in case of $b=+\\infty $ the divergence $D_{\\widetilde{\\varphi }}\\left( \\mathbf {\\Gamma }_{t},\\widetilde{{P}}\\right) =\\inf _{g_{t}\\in \\mathbf {\\Gamma }_{t}}\\sum _{k=1}^{K} \\widetilde{p}_{k}\\cdot \\widetilde{\\varphi }\\left( \\frac{\\left(g_{t}\\right)_{k}}{\\widetilde{p}_{k}}\\right) =\\inf _{g \\in \\mathbf {\\Gamma } }\\sum _{k=1}^{K} \\widetilde{p}_{k}\\cdot \\widetilde{\\varphi }\\left( \\frac{t \\cdot g_{k}}{\\widetilde{p}_{k}}\\right)$ tends to infinity as $t \\rightarrow \\infty $ ; the case $a=-\\infty $ works analogously with $t\\rightarrow -\\infty $ .",
"In case of $b < \\infty $ we may consider $\\mathbf {\\Gamma } _{t}:=\\left\\lbrace b-g /t;g \\in \\mathbf {\\Gamma } \\right\\rbrace $ and indeed $D_{\\widetilde{\\varphi }}\\left( \\mathbf {\\Gamma } _{t},\\widetilde{{P}}\\right) \\rightarrow \\infty $ as $t\\rightarrow \\infty $ , with a similar statement when $a>-\\infty $ .",
"Assume that $\\mathbf {\\Gamma }$ has a dominating point $\\underline{g}$ .",
"Then $\\mathbf {\\Gamma }_{t}$ has dominating point $\\underline{g}_{t} := t \\cdot \\underline{g}$ .",
"We prove that $\\mathbf {T}$ with distribution (REF ) cannot be too far away (depending on $t$ ) from $\\underline{g}_{t}$ whenever $\\mathbf {T}$ belongs to $\\mathbf {\\Gamma } _{t}.$ This argument is valid in the present description of some asymptotics which makes $\\mathbf {\\Gamma } _{t}$ as a model for $\\widetilde{\\mathbf {\\Omega }}$ for large $t$ ; considering the case when $D_{\\widetilde{\\varphi } }\\left( \\widetilde{\\mathbf {\\Omega }} ,\\widetilde{{P}}\\right) $ is large is captured through the asymptotic statement $\\lim _{t\\rightarrow \\infty }D_{\\widetilde{\\varphi } }\\left( \\mathbf {\\Gamma } _{t},\\widetilde{{P}}\\right) =+\\infty .$ There holds the following Proposition 65 With the above notation and under condition (H$\\widetilde{\\varphi }$ ), denote by $\\mathbf {B}$ a neighborhood of $\\underline{g}$ and $\\mathbf {B}_{t}:=t \\cdot \\mathbf {B}$ .",
"Then $\\mathbb {F}[\\mathbf {\\Gamma }_{t}\\cap \\mathbf {B}_{t}^{c} \\, \\vert \\, \\mathbf {\\Gamma }_{t} \\, ]= \\mathbb {\\Pi }\\left[ \\left.",
"\\mathbf {T}\\in \\mathbf {\\Gamma }_{t}\\cap \\mathbf {B}_{t}^{c}\\, \\right|\\,\\mathbf {T}\\in \\mathbf {\\Gamma }_{t}\\right] \\rightarrow 0$ as $t\\rightarrow \\infty $ , which proves that simulations under (REF ) produce proxies of the dominating points $\\underline{g}_{t}$ in $\\mathbf {\\Gamma }_{t}$ .",
"Before we start with the proof of Proposition REF , we first quote the following Lemma 66 Let $\\widetilde{\\varphi } $ satisfy (H$\\widetilde{\\varphi }$ ) with $b=+\\infty $ .",
"Then for all $\\mathbf {A}$ in $\\mathbb {R}^{K}$ such that $\\breve{\\alpha } :=D_{\\widetilde{\\varphi }}(\\mathbf {A},\\widetilde{{P}}):=\\inf _{\\mathbf {v}\\in \\mathbf {A}}\\sum _{k=1}^{K} \\widetilde{p}_{k} \\cdot \\widetilde{\\varphi } \\left(\\frac{v_{k}}{\\widetilde{p}_{k}}\\right)$ is finite there holds $\\lim _{t\\rightarrow \\infty }\\frac{1}{t}\\log \\int _{\\mathbf {A}}\\exp \\left(-t\\sum _{k=1}^{K} \\widetilde{p}_{k} \\cdot \\widetilde{\\varphi }\\left( \\frac{v_{k}}{\\widetilde{p}_{k}}\\right) \\right) \\, dv_{1} \\ldots dv_{k}= - D_{\\widetilde{\\varphi }}\\left(\\mathbf {A},\\widetilde{{P}}\\right) .$ Proof of Lemma REF .",
"Let us first remark that according to the geometry of the set $\\mathbf {A}$ , various combinations for the asymptotics (REF ) or (REF ) may occur; for sake of brevity, we only handle the simplest ones, since all turn to be amenable through the same arguments.",
"Denote for positive $r$ $\\mathbf {B}(r):=\\left\\lbrace \\mathbf {v}\\in \\mathbb {R}^{K}:\\sum _{k=1}^{K} \\widetilde{p}_{k}\\widetilde{\\varphi } \\left( \\frac{v_{k}}{\\widetilde{p}_{k}}\\right) >r\\right\\rbrace .$ It holds, by making the change of variable $r=t\\cdot \\breve{\\alpha } +t \\cdot s$ , $& & \\int _{\\mathbf {A}}\\exp \\left( -t\\sum _{k=1}^{K} \\widetilde{p}_{k} \\cdot \\widetilde{\\varphi } \\left( \\frac{v_{k}}{\\widetilde{p}_{k}}\\right) \\right) \\,dv_{1} \\ldots dv_{k}=\\int \\cdots \\int 1_{\\mathbb {R}^{+}}(r)\\cdot 1_{\\mathbf {A}}(\\mathbf {v})\\cdot 1_{\\left]t\\sum _{k=1}^{K} \\widetilde{p}_{k}\\widetilde{\\varphi } \\left( \\frac{v_{k}}{\\widetilde{p}_{k}}\\right) ,\\infty \\right[}(r)\\cdot e^{-r} \\, dr \\, dv_{1}\\ldots dv_{K} \\\\& & = te^{-t\\breve{\\alpha } }\\int \\cdots \\int 1_{]- \\breve{\\alpha },\\infty [}(s) \\cdot 1_{\\mathbf {A}}(\\mathbf {v}) \\cdot 1_{\\mathbf {B}^{c}(\\breve{\\alpha }+s)}(\\mathbf {v})\\cdot e^{-ts} \\, ds \\, dv_{1}\\ldots dv_{K} \\ = \\ te^{-t\\breve{\\alpha } }\\int _{-\\breve{\\alpha }}^{\\infty }Vol\\left( \\mathbf {A}\\cap \\mathbf {B}^{c}(\\breve{\\alpha }+s)\\right) \\cdot e^{-ts} \\, ds.$ Let $I_{t}:=t \\cdot \\int _{0}^{\\infty }Vol\\left(\\mathbf {A}\\cap \\mathbf {B}^{c}(\\breve{\\alpha } +s)\\right) e^{-ts}ds $ .",
"We prove that $\\lim _{t\\rightarrow \\infty }\\frac{1}{t}\\log I_{t}=0.",
"$ When $a=-\\infty $ or $b=+\\infty $ , since $\\widetilde{\\varphi }$ satisfies (H$\\widetilde{\\varphi }$ ) there exists a polynomial $P$ such that $Vol\\left( \\mathbf {A}\\cap \\mathbf {B}^{c}(\\breve{\\alpha } +s)\\right) \\le P(s) \\, ;$ whence, assuming without loss of generality that $dom(\\widetilde{\\varphi }) =\\mathbb {R}^{+}$ , we obtain $\\frac{1}{t}\\log I_{t}\\le \\frac{1}{t}\\log \\int _{0}^{\\infty }P\\left(\\frac{u}{t}\\right) te^{-u}du$ which yields that for large $t$ $\\frac{1}{t}\\log I_{t}<0.$ When dealing with a context where $a$ or $b$ have finite value and the corresponding sets $\\mathbf {\\Gamma }_{t}$ are “far away” from $\\mathbf {\\Gamma } $ in terms of the distance measure $D_{\\widetilde{\\varphi }}\\left( \\cdot ,\\widetilde{{P}}\\right)$ , then $Vol\\left( \\mathbf {A}\\cap \\mathbf {B}^{c}(\\breve{\\alpha } +s)\\right) $ is bounded.",
"Hence, $\\lim \\sup _{t\\rightarrow \\infty }\\frac{1}{t}\\log I_{t}\\le 0$ .",
"Now fix $\\varepsilon >0.$ Then, since $Vol\\left( \\mathbf {A}\\cap \\mathbf {B}^{c}(a+s)\\right) $ is increasing in $s$ , we get $I_{t} &\\ge &t\\int _{\\varepsilon }^{\\infty }Vol\\left( \\mathbf {A}\\cap \\mathbf {B}^{c}(\\breve{\\alpha }+s)\\right) e^{-ts}ds \\\\&\\ge &Vol\\left( \\mathbf {A}\\cap \\mathbf {B}^{c}(\\breve{\\alpha } +\\varepsilon )\\right) e^{-t\\varepsilon }.$ Hence $\\frac{1}{t}\\log I_{t}\\ge \\frac{1}{t}\\log Vol\\left( \\mathbf {A}\\cap \\mathbf {B}^{c}(\\breve{\\alpha }+\\varepsilon )\\right) -\\varepsilon $ which yields $\\lim \\inf _{t\\rightarrow \\infty }\\frac{1}{t}\\log I_{t}\\ge 0.$ Therefore (REF ) holds, which concludes the proof.",
"$\\blacksquare $ We now turn to the Proof of Proposition REF .",
"Without loss of generality, let $b=+\\infty $ , $\\mathbf {\\Gamma }_{t}$ as in (REF ) and Condition (H$\\widetilde{\\varphi }$ ) hold.",
"Moreover, consider an arbitrary neighborhood $\\mathbf {B}$ of $\\underline{g}$ and the corresponding neighborhoods $\\mathbf {B}_{t} : =t \\cdot \\mathbf {B}$ of $\\underline{g}_{t} = t \\cdot \\underline{g}$ .",
"There holds $\\frac{1}{\\widetilde{\\varphi }(t)}\\log \\mathbb {\\Pi }\\left[ \\mathbf {T}\\in \\mathbf {\\Gamma }_{t}\\right] &=&\\frac{C}{\\widetilde{\\varphi }(t)}\\log \\int _{\\mathbf {\\Gamma } _{t}}\\exp \\left(-\\sum _{k=1}^{K}\\widetilde{p}_{k} \\cdot \\widetilde{\\varphi }\\left(\\frac{w_{k}}{\\widetilde{p}_{k}}\\right)\\right) \\, dw_{1}\\ldots dw_{K} \\\\&\\stackrel{(1)}{=}&\\frac{CK}{\\widetilde{\\varphi }(t)}\\log t+\\frac{C}{\\widetilde{\\varphi }(t)}\\log \\int _{\\mathbf {\\Gamma }}\\exp \\left(-t^{\\beta } \\cdot \\sum _{k=1}^{K}\\widetilde{p}_{k} \\cdot \\left( \\widetilde{\\varphi }\\left( \\frac{v_{k}}{\\widetilde{p}_{k}}\\right) \\cdot (1+o(1))\\right)\\right) \\, dv_{1} \\ldots dv_{K} \\\\&\\stackrel{(2)}{=}& \\frac{CK}{\\widetilde{\\varphi }(t)}\\log t+\\frac{C}{\\left( \\widetilde{\\varphi }(t)/t^{\\beta }\\right) } \\cdot \\frac{1}{t^{\\beta }}\\log \\left( (1+o(1)) \\cdot \\int _{\\mathbf {\\Gamma } }\\exp \\left( -t^{\\beta }\\sum _{k=1}^{K}\\widetilde{p}_{k} \\cdot \\widetilde{\\varphi }\\left( \\frac{v_{k}}{\\widetilde{p}_{k}}\\right) \\right)dv_{1} \\ldots dv_{K} \\right)\\\\&\\stackrel{(3)}{=}& -\\frac{Ct^{\\beta }}{\\widetilde{\\varphi }(t)}\\cdot D_{\\widetilde{\\varphi }}(\\mathbf {\\Gamma } ,\\widetilde{{P}}) \\cdot (1+o(1)) \\\\&\\stackrel{(4)}{=}& - \\breve{l}(t) \\cdot D_{\\widetilde{\\varphi }}(\\mathbf {\\Gamma } ,\\widetilde{{P}}) \\cdot (1+o(1))$ as $t$ tends to infinity.",
"In the above display, $(1)$ follows from $\\widetilde{\\varphi }(tx)=\\left( tx\\right)^{\\beta } \\cdot \\ell (tx)=t^{\\beta } \\cdot x^{\\beta } \\cdot \\ell (x) \\cdot \\frac{\\ell (tx)}{\\ell (x)} =t^{\\beta } \\cdot \\widetilde{\\varphi } (x) \\cdot \\left( 1+o(1)\\right) $ as $t$ tends to infinity and $x$ lies in a compact subset of $]0,\\infty [$ , where $\\ell $ is a slowly varying function.",
"The equality $(2)$ follows from compactness of $\\mathbf {\\Gamma }$ together with the fact that $\\widetilde{\\varphi }$ is a regularly varying function with index $\\beta $ , so that $\\lim _{t\\rightarrow \\infty }\\frac{\\widetilde{\\varphi }(tv)}{\\widetilde{\\varphi }(t)}=v^{\\beta }$ uniformly upon $v$ on compact sets in $]0,\\infty [$ .",
"The remaining equalities $(3)$ and $(4)$ follow from classical properties of regularly varying functions, where $\\breve{\\ell } := 1/\\ell $ is a slowly varying function at infinity, together with standard Laplace-Integral approximation.",
"In the same way we can show $\\frac{1}{\\widetilde{\\varphi }(t)}\\log \\mathbb {\\Pi } \\left[ \\, \\mathbf {T}\\in \\mathbf {\\Gamma } _{t}\\cap \\mathbf {B}_{t}^{c} \\, \\right]=- \\breve{l}(t) \\cdot D_{\\widetilde{\\varphi }}(\\mathbf {\\Gamma }\\cap \\mathbf {B}^{c},\\widetilde{{P}}) \\cdot (1+o(1))$ as $t$ tends to infinity.",
"Since $\\mathbf {B}$ is a neighborhood of the unique dominating point $\\underline{g}$ of $\\mathbf {\\Gamma }$ , one gets that $D_{\\widetilde{\\varphi }}(\\mathbf {\\Gamma }\\cap \\mathbf {B}^{c},\\widetilde{{P}}) > D_{\\widetilde{\\varphi }}(\\mathbf {\\Gamma } ,\\widetilde{{P}})$ .",
"This implies that $\\mathbb {\\Pi } \\left[ \\, \\mathbf {T}\\in \\mathbf {\\Gamma }_{t}\\cap \\mathbf {B}_{t}^{c} \\, \\big \\vert \\, \\mathbf {T}\\in \\mathbf {\\Gamma }_{t} \\, \\right] \\rightarrow 0\\qquad \\textrm {as $ $.",
"\\qquad $$}$$$ Remark 67 Firstly, let us quote that the case when $\\widetilde{\\mathbf {\\Omega }}$ is an unbounded subset in $\\mathbb {R}^{K}/\\left\\lbrace \\mathbf {0}\\right\\rbrace $ is somewhat immaterial for applications.",
"Anyhow, if compactness of $\\mathbf {\\Gamma } $ is lost, then in order to use the same line of arguments as above, it is necessary to strengthen the assumptions (H$\\widetilde{\\varphi }$ ) e.g.",
"as follows: when $b=+\\infty $ then $\\widetilde{\\varphi }$ has to be asymptotically homogeneous with degree $\\beta >0$ , in the sense that $\\widetilde{\\varphi }(tx)=t^{\\beta }\\widetilde{\\varphi }(x)\\cdot (1+o(1))$ as $t\\rightarrow \\infty $ ; for the subcase $a=-\\infty $ one employs an analogous assumption as $t\\rightarrow -\\infty $ .",
"The case when $\\widetilde{\\mathbf {\\Omega }}$ is a compact set in $\\mathbb {R}^{K}\\backslash \\lbrace \\mathbf {0}\\rbrace $ can be treated as above, by combining the asymptotics in $t$ in the neighborhood of $a$ and $b$ accordingly."
],
[
"Proof for Subsection ",
"Proof of Proposition REF.",
"Recall the weighted empirical measure $\\xi _{n,\\mathbf {X}}^{\\mathbf {V}}:=\\left( \\frac{1}{n}\\sum _{i\\in I_{1}^{(n)}} V_{i},\\ldots ,\\frac{1}{n}\\sum _{i\\in I_{K}^{(n)}} V_{i}\\right)$ which satisfies the $K$ linear constraints defined in (REF ) through $E_{S}[\\xi _{n,\\mathbf {X}}^{\\mathbf {V}}] =\\xi _{M,\\mathbf {X}}^{\\mathbf {W}^{\\ast }}= \\overline{W^{\\ast }} \\cdot \\xi _{M,\\mathbf {X}}^{w\\mathbf {W}^{\\ast }}$ where $\\mathbf {Q}^{\\ast }:= \\left(q_{1}^{\\ast }, \\ldots , q_{K}^{\\ast }\\right)= \\xi _{M,\\mathbf {X}}^{w\\mathbf {W}^{\\ast }}\\in int(\\textrm {$$\\hspace{-6.544pt}$$})$ and $\\overline{W^{\\ast }} = \\frac{1}{M}\\sum _{j=1}^{M}W_{j}^{\\ast }$ .",
"The probability distribution $S$ defined on $\\mathbb {R}^{n}$ is the Kullback-Leibler projection of $\\mathbb {}^{\\otimes n}$ on the class of all probability distributions on $\\mathbb {R}^{n}$ which satisfy (REF ).",
"We prove that $\\lim \\inf _{n\\rightarrow \\infty }S\\left[\\xi _{n,\\mathbf {X}}^{w\\mathbf {V}} \\in \\textrm {\\right.$$\\hspace{-6.544pt}$$}> 0$ .",
"To start with, we define for strictly positive $\\delta $ the set $A_{n,\\delta } := \\left\\lbrace \\left|\\frac{1}{n}\\sum _{i=1}^{n}V_{i}-\\overline{W^{\\ast }}\\right|\\le \\delta \\right\\rbrace $ and write $S\\left[\\xi _{n,\\mathbf {X}}^{w\\mathbf {V}} \\in \\textrm {\\right.$ $\\hspace{-6.544pt}$$}=S\\left[\\lbrace \\xi _{n,\\mathbf {X}}^{w\\mathbf {V}} \\in \\textrm {\\right.$$\\hspace{-6.544pt}$$} \\rbrace \\cap A_{n,\\delta }+ S\\left[\\lbrace \\xi _{n,\\mathbf {X}}^{w\\mathbf {V}} \\in \\textrm {\\right.$$\\hspace{-6.544pt}$$} \\rbrace \\cap A_{n,\\delta }^{c}=:I+II.$$By the law of large numbers, the second term $ II$ tends to $ 0$ as $ n$ tends to infinity.Moreover, one can rewrite$$I=S\\bigg [ \\bigcup \\limits _{m\\in \\left[ \\overline{W^{\\ast }}-\\delta ,\\overline{W^{\\ast }}+\\delta \\right] }\\left\\lbrace \\xi _{n,\\mathbf {X}}^{\\mathbf {V}}\\in m \\cdot \\textrm {\\right.$$\\hspace{-6.544pt}$$} \\bigg ]$$which entails$$I \\ \\ge \\ S\\bigg [ \\frac{1}{n_{k}}\\sum _{i\\in I_{k}^{(n)}}V_{i}\\in \\mathcal {V}_{\\eta }\\left( \\overline{W^{\\ast }}\\frac{q_{k}^{\\ast }}{p_{k}}\\right) \\text{ for all }k \\in \\lbrace 1,\\ldots ,K\\rbrace \\bigg ] ,$$where $ V( W qkpk) $ denotes a neighborhood of$Wqkpk$ with radius $$ being smallwhen $$ is small, for large enough $ n$, making use of the a.s. convergenceof $ nk/n$ to $ pk$.",
"Now, for any $ k {1,...,K}$ one has\\begin{equation}S\\bigg [ \\frac{1}{n_{k}}\\sum _{i\\in I_{k}^{(n)}}V_{i}\\notin \\mathcal {V}_{\\eta }\\left( \\overline{W^{\\ast }}\\frac{q_{k}^{\\ast }}{p_{k}}\\right)\\bigg ]\\ \\le \\ \\exp \\bigg (-n_{k} \\cdot \\inf _{x\\in \\mathcal {V}_{\\eta }\\left(\\overline{W^{\\ast }}\\frac{q_{k}^{\\ast }}{p_{k}}\\right)^{c}} \\,\\varphi \\left( x\\right) \\bigg )\\end{equation}since any margin of $ S$ with index in $ Ik(n)$ is a correspondingKullback-Leibler projection of $$ on the set of all distributionson $ R$ with expectation $W qkpM,kemp$ ---where $ pM,kemp$ denotes the fraction ofthe $ Xi$’s (within $ X1,...,XM$) which are equal to $ dk$(cf.",
"(\\ref {I^(n)_k for stat case})) ---and therefore has amoment generating function which is finite in a non-void neighborhood of $ 0$,which yields (\\ref {ineg}) by the Markov Inequality.",
"Note that the event$ { M,XwWint( $\\Omega $$\\Omega $ ) }$ is regenerative, so that $ M$ can be chosen large enough to make$ pM,kemp$ close to $ pk$ for all $ k {1,...,K}$.",
"This proves the claim.\\hspace{28.45274pt} $$$"
],
[
"Acknowledgment",
"W. Stummer is grateful to the Sorbonne Université Paris for its multiple partial financial support and especially the LPSM for its multiple great hospitality.",
"M. Broniatowski thanks very much the FAU Erlangen-Nürnberg for its partial financial support and hospitality.",
"Moreover, W. Stummer would like to thank Rene Schilling for an interesting discussion on complex-valued foundations of the Bernstein-Widder theorem."
],
[
"Finding/Constructing/On the distribution of the weights ",
"Recall first that in Theorem REF , one crucial component is the sequence $(W_{n})_{n\\in \\mathbb {N}}$ of weights being i.i.d.",
"copies of a random variable $W$ whose probability distribution is $\\mathbb {}$ (i.e.",
"$\\mathbb {\\Pi }[W \\in \\cdot \\, ] = \\mathbb {}[ \\, \\cdot \\,]$ ), where the latter has to be connected with the divergence generator $\\varphi \\in \\Upsilon (]a,b[)$ through the representation $\\varphi (t)=\\sup _{z\\in \\mathbb {R}}\\left( z\\cdot t-\\log \\int _{\\mathbb {R}}e^{zy}d\\mathbb {} (y)\\right), \\qquad t\\in \\mathbb {R},\\ \\ \\hspace{28.45274pt} (cf.",
"\\ (\\ref {Phi Legendre of mgf(W)}))$ under the additional requirement that the function $z\\mapsto MGF_{\\mathbb {} }(z):=\\int _{\\mathbb {R}}e^{zy}d\\mathbb {} (y)$ is finite on some open interval containing zero (“light-tailedness”); for Theorem REF , we need the corresponding variant (REF ) for $M_{\\mathbf {P}} \\cdot \\varphi \\in \\Upsilon (]a,b[)$ (rather than $\\varphi $ ).",
"Hence, finding such “BS-associated pairs $(\\varphi ,\\mathbb {})$ ” is an important issue.",
"Subsequently, let us discuss the following direction: starting from a concrete optimization problem (REF ) — respectively (REF ) — with pregiven $\\varphi \\in \\widetilde{\\Upsilon } (]a,b[)$ (cf.",
"Definition REF ), as a first step one would like to verify whether indeed $\\varphi \\in \\Upsilon (]a,b[)$ (i.e.",
"it additionally satisfies (REF )) — respectively $M_{\\mathbf {P}} \\cdot \\varphi \\in \\Upsilon (]a,b[)$ ; as a second step, one would like to find the corresponding $\\mathbb {}$ explicitly.",
"As far as the above-mentioned first step is concerned, let us first present some fundamental properties of all $\\varphi \\in \\Upsilon (]a,b[)$ : Proposition 29 Let $\\varphi \\in \\Upsilon (]a,b[)$ .",
"Then the following assertions hold: $\\varphi : \\, ]-\\infty ,\\infty [ \\rightarrow [0,\\infty ]$ is lower semicontinuous and convex; $\\varphi (1)=0$ ; $int(dom(\\varphi )) = ]a,b[$ for some $-\\infty \\le a < 1 < b \\le \\infty $ ; $\\varphi $ is continuously differentiable on $]a,b[$ (i.e.",
"$\\varphi \\in C^{1}(]a,b[)$ ; $\\varphi $ is strictly convex only in a non-empty neighborhood $]t_{-}^{sc},t_{+}^{sc}[ \\subseteq ]a,b[$ of one ($t_{-}^{sc} < 1 < t_{+}^{sc}$ ); $\\varphi $ is infinitly differentiable on $]t_{-}^{sc},t_{+}^{sc}[$ (i.e.",
"$\\varphi \\in C^{\\infty }(]t_{-}^{sc},t_{+}^{sc}[$ ), and hence, $\\varphi ^{\\prime }(1) = 0$ , $\\varphi ^{\\prime \\prime }(t) > 0$ for all $t \\in ]t_{-}^{sc},t_{+}^{sc}[$ ; notice that the left-hand second derivative and the right-hand second derivative of $\\varphi $ may not coincide at $t_{-}^{sc}$ respectively at $t_{+}^{sc}$ (i.e.",
"possible non-second-differentiability at these two points); if $a > -\\infty $ , then $a=t_{-}^{sc}$ ; if $a = -\\infty $ , then either $t_{-}^{sc} = - \\infty $ or $\\varphi (t) = \\varphi (t_{-}^{sc}) + \\varphi ^{\\prime }(t_{-}^{sc}) \\cdot (t- t_{-}^{sc})$ for all $t \\in ]-\\infty , t_{-}^{sc}[$ (affine-linearity); notice that $\\varphi ^{\\prime }(t_{-}^{sc}) < 0$ ; if $b < \\infty $ , then $b=t_{+}^{sc}$ ; if $b = \\infty $ , then either $t_{+}^{sc} = \\infty $ or $\\varphi (t) = \\varphi (t_{+}^{sc}) + \\varphi ^{\\prime }(t_{+}^{sc}) \\cdot (t- t_{+}^{sc})$ for all $t \\in ]t_{+}^{sc},\\infty [$ (affine-linearity); notice that $\\varphi ^{\\prime }(t_{+}^{sc}) > 0$ ; the Fenchel-Legendre transform (also called convex conjugate) of $\\varphi $ $\\varphi ^{*} (z)=\\sup _{t\\in \\mathbb {R}}\\left( z\\cdot t- \\varphi (t) \\right)= \\sup _{t\\in ]a,b[}\\left( z\\cdot t- \\varphi (t) \\right), \\qquad z\\in \\mathbb {R},\\ \\ $ has the following properties: $int(dom(\\varphi ^{*})) = ]\\lambda _{-},\\lambda _{+}[$ , where $dom(\\varphi ^{*}) := \\lbrace z \\in \\mathbb {R} : -\\infty < \\varphi ^{*}(z) < \\infty \\rbrace $ , $\\lambda _{-} := \\inf _{t \\in ]a,b[} \\varphi ^{\\prime }(t) =\\lim _{t \\downarrow a} \\varphi ^{\\prime }(t) =: \\varphi ^{\\prime }(a) < 0$ and $\\lambda _{+} := \\sup _{t \\in ]a,b[} \\varphi ^{\\prime }(t) =\\lim _{t \\uparrow b} \\varphi ^{\\prime }(t) =: \\varphi ^{\\prime }(b) >0$ ; if $a >-\\infty $ , then $\\lambda _{-} = - \\infty $ ; the function $z \\mapsto e^{-a\\cdot z + \\varphi ^{*}(z)} =: M(z)$ is absolutely monotone on $]-\\infty ,0[$ , i.e.",
"all derivatives exist and satisfy $\\frac{\\partial ^{k}}{\\partial z^k} M(z) \\ge 0$ ($k\\in \\mathbb {N}_{0}$ , $z \\in ]-\\infty ,0[$ ); $\\lim _{z \\rightarrow 0-} M(z) =1$ ; if $b < \\infty $ , then $\\lambda _{+} = \\infty $ ; the function $z \\mapsto e^{b\\cdot z + \\varphi ^{*}(- z)} =: M(z)$ is absolutely monotone on $]-\\infty ,0[$ ; $\\lim _{z \\rightarrow 0-} M(z) =1$ ; if $a = -\\infty $ and $b = -\\infty $ , then the function $z \\mapsto e^{\\varphi ^{*}(z)} =: M(z)$ is exponentially convex on $]\\lambda _{-}, \\lambda _{+}[$ , i.e.",
"$M(\\cdot )$ is continuous and satisfies $\\sum _{i,j=1}^{n} c_{i} \\cdot c_{j} \\cdot M\\Big (\\frac{z_{i} + z_{j}}{2}\\Big ) \\ge 0 \\qquad \\textrm {for all n \\in \\mathbb {N},c_{i}, c_{j} \\in \\mathbb {R} and z_{i}, z_{j} \\in ]\\lambda _{-}, \\lambda _{+}[;}$ $\\lim _{z \\rightarrow 0-} M(z) =1$ ; as a side remark, notice the well-known fact that exponential-convexity is stronger than the usual log-convexity.",
"the endpoints of $int(dom(\\varphi )) =]a,b[$ have the following important “functioning” for the underlying probability distribution $\\mathbb {}$ (cf.",
"(REF )) respectively of an associated random variable $W$ with $\\mathbb {}[ \\cdot \\, ] := \\mathbb {\\Pi }[W \\in \\cdot \\, ]$ : $a = \\inf supp(\\mathbb {}) = \\inf supp(W)$ , $b = \\sup supp(\\mathbb {}) = \\sup supp(W)$ , where $supp(\\mathbb {})$ respectively $supp(W)$ denotes the support of $\\mathbb {}$ respectively $W$ ; consequently, $]a,b[ = int(conv(supp(\\mathbb {}))) = int(conv(supp(W)))$ where $conv(A)$ denotes the convex hull of a set $A$ ; if $a > -\\infty $ , then $\\varphi (a) = - \\log \\mathbb {}[\\lbrace a\\rbrace ]= - \\log \\mathbb {\\Pi }[W = a \\, ]$ ; consequently, there holds: $a = \\min supp(\\mathbb {}) = \\min supp(W)$ if and only if $\\mathbb {}[\\lbrace a\\rbrace ] = \\mathbb {\\Pi }[W = a \\, ] > 0$ if and only if $\\varphi (a) < \\infty $ if and only if $a \\in dom(\\varphi )$ ; if $b < \\infty $ , then $\\varphi (b) = - \\log \\mathbb {}[\\lbrace b\\rbrace ]= - \\log \\mathbb {\\Pi }[W = b \\, ]$ ; consequently, there holds: $b = \\max supp(\\mathbb {}) = \\max supp(W)$ if and only if $\\mathbb {}[\\lbrace b\\rbrace ] = \\mathbb {\\Pi }[W = b \\, ] > 0$ if and only if $\\varphi (b) < \\infty $ if and only if $b \\in dom(\\varphi )$ .",
"the first two derivatives of $\\varphi $ at the point 1 have the following important “functioning” for the underlying probability distribution $\\mathbb {}$ (cf.",
"(REF )) respectively of an associated random variable $W$ : $1 = \\varphi ^{\\prime -1}(0) = \\int _{\\mathbb {R}} y \\, d\\mathbb {} (y)= E_{\\mathbb {\\Pi }}[W]$ where $\\varphi ^{\\prime -1}(\\cdot )$ denotes the inverse of the first derivative $\\varphi ^{\\prime }(\\cdot )$ of $\\varphi (\\cdot )$ , $\\frac{1}{\\varphi ^{\\prime \\prime }(1)} =\\int _{\\mathbb {R}} \\Big (y - \\int _{\\mathbb {R}} \\widetilde{y} \\, d\\mathbb {} (\\widetilde{y})\\Big )^{2}\\, d\\mathbb {} (y)= E_{\\mathbb {\\Pi }}[W^{2}] - (E_{\\mathbb {\\Pi }}[W])^{2} = Var_{\\mathbb {\\Pi }}[W]$ ; in particular, scaling $\\widetilde{c} \\cdot \\varphi $ ($\\widetilde{c} >0$ ) does not change the mean 1 but the variance of $W$ .",
"The corresponding proof of Proposition REF will be given in Appendix D, except for the second items of (G9ii) and (G9iii) as well as the first item of (G9iv).",
"Those will be treated in the second next paragraph below, because the corresponding line of argumentation builds an insightful start for subsequently performed procedures to further track down the weight distribution $$ .",
"The properties (G1) to (G9iv) constitute necessary conditions for a pregiven function $\\varphi $ to belong to $\\Upsilon (]a,b[)$ ); accordingly, these should be verified first, in concrete situations where one aims to apply the BS approach.",
"An important role is played by the boundary points $a$ and $b$ of $int(dom(\\varphi ))$ through (G10i) to (G10iii), because their finiteness opens the gate to apply — via some straightforward transformations — a rich class of real-valued characterization theorems for probability distributions whose support lies in the positive real line $[0,\\infty [$ .",
"In contrast, there exist much less real-valued characterization theorems for probability distributions whose support is the whole real line $]-\\infty ,\\infty [$ ; typically, the involved conditions are also more difficult to verify.",
"Indeed, if $\\varphi \\in \\Upsilon (]a,b[)$ then one can deduce straightforwardly from the representation (REF ) that $e^{\\varphi ^{*}(z)} =\\int _{-\\infty }^{\\infty } e^{z\\cdot y} \\, \\mathrm {d}\\mathbb {} (y) =E_{\\mathbb {\\Pi }}[e^{z \\cdot W}] , \\quad z \\in ]\\lambda _{-}, \\lambda _{+}[,$ where $W$ is a random variable whose distribution is $\\mathbb {\\Pi }[W \\in \\cdot \\, ] = \\mathbb {}[ \\, \\cdot \\,]$ ; under the additional knowledge $a > -\\infty $ (and consequently $\\lambda _{-} = -\\infty $ ) employed together with (G10i) and thus $\\mathbb {\\Pi }[W \\ge a \\, ] =\\mathbb {}[ \\, [a,\\infty [ \\,] = 1$ , one arrives at $e^{\\varphi ^{*}(z) - a \\cdot z} =\\int _{a}^{\\infty } e^{z \\cdot (y-a)} \\, \\mathrm {d}\\mathbb {} (y) =\\int _{0}^{\\infty } e^{z\\cdot \\widetilde{y}} \\, \\mathrm {d}\\widetilde{\\widetilde{\\mathbb {}}} (\\widetilde{y})= E_{\\mathbb {\\Pi }}[e^{z \\cdot (W-a)}] , \\quad z \\in ]-\\infty , \\lambda _{+}[,$ where the probability distribution $\\widetilde{\\widetilde{\\mathbb {}}}[ \\, \\cdot \\, ] := \\mathbb {}[\\, \\cdot + a \\, ]$ is the $a-$ shifted companion of $\\mathbb {}$ ; recall that $\\lambda _{+} >0$ .",
"Put in other words, $\\mathbb {\\Pi }[\\widetilde{W} \\in \\cdot \\, ] = \\widetilde{\\widetilde{\\mathbb {}}}[ \\, \\cdot \\,]$ is the probability distribution of the (a.s.) nonnegative random variable $\\widetilde{W} := W-a$ .",
"Naturally, in this context, the interesting case is $-\\infty < a \\le 0$ .",
"Similarly, if $\\varphi \\in \\Upsilon (]a,b[)$ and $b < \\infty $ (and hence $\\lambda _{+} = \\infty $ ), one can derive from (G10i) and its consequence $\\mathbb {\\Pi }[W \\le b \\, ] =\\mathbb {}[ \\, ]-\\infty ,b] \\,]= 1$ that $e^{\\varphi ^{*}(-z) + b \\cdot z} =\\int _{-\\infty }^{b} e^{z \\cdot (b-y)} \\, \\mathrm {d}\\mathbb {} (y) =\\int _{0}^{\\infty } e^{z\\cdot \\widetilde{y}} \\, \\mathrm {d}\\widetilde{\\widetilde{\\mathbb {}}} (\\widetilde{y})= E_{\\mathbb {\\Pi }}[e^{z \\cdot (b-W)}] , \\quad z \\in ]- \\infty , - \\lambda _{-}[,$ where $- \\lambda _{-} >0$ and the probability distribution $\\widetilde{\\widetilde{\\mathbb {}}}[ \\, \\cdot \\, ] := \\mathbb {}[\\, b - \\, \\cdot \\, ]$ is the mirrored$-b-$ shifted companion of $\\mathbb {}$ .",
"This means that $\\mathbb {\\Pi }[\\widetilde{W} \\in \\cdot \\, ] = \\widetilde{\\widetilde{\\mathbb {}}}[ \\, \\cdot \\,]$ is the probability distribution of the (a.s.) nonnegative random variable $\\widetilde{W} := b-W$ .",
"Naturally, the interesting case is $0 < b \\le \\infty $ .",
"As already indicated above, the considerations (REF ) to (REF ) open the gate to the adaption of well-known real-valued (rather than complex-valued) characterizations from probability theory.",
"To begin with, the following assertions are very prominent: Theorem 30 (a) Let $M: ]-\\infty ,0] \\mapsto ]0,\\infty [$ be continuous on $]-\\infty ,0]$ with $M(0) =1$ .",
"Then one has $\\textrm {M is absolutely monotone on ]-\\infty ,0[}\\ \\Longleftrightarrow \\ \\exists \\ \\textrm {unique prob.",
"distr.",
"\\widetilde{\\widetilde{\\mathbb {}}} on [0,\\infty [s.t.}",
"\\ \\ M(z) = \\int _{0}^{\\infty } e^{z \\cdot y} \\mathrm {d}\\widetilde{\\widetilde{\\mathbb {}}}(y)\\ \\textrm {for all z \\in ]-\\infty ,0[.",
"}$ (b) Let $I$ be an open interval which contains 0, and $M: I \\mapsto [0,\\infty [$ be continuous with $M(0) =1$ .",
"Then one gets $\\textrm {M is exponentially convex}\\ \\Longleftrightarrow \\ \\exists \\ \\textrm {unique prob.",
"distr.",
"\\widetilde{\\widetilde{\\mathbb {}}} on ]-\\infty ,\\infty [such that} \\ \\ M(z) = \\int _{-\\infty }^{\\infty } e^{z \\cdot y} \\mathrm {d}\\widetilde{\\widetilde{\\mathbb {}}}(y)\\ \\ \\textrm {for all z \\in I.",
"}$ Assertion (a) of Theorem REF is known as (probability-version of) Bernstein’s theorem [42] (see e.g.",
"also Schilling et al.",
"[322]), whereas assertion (b) is known as (probability-version of) Widder’s theorem [391] for the relevant conversion between the involved Riemann-Stieltjes integral with nondecreasing (but not necessarily right-continuous) integrator into a measure integral, one can apply the general theory in e.g.",
"Chapter 6 of Chow & Teicher [88].",
"(see e.g.",
"also Widder [392], Akhiezer [9], Shucker [333], Jaksetic & Pecaric [163], Kotelina & Pevny [196]).",
"From Theorem REF (b) and (REF ), the first item in (G9iv) follows immediately by using the choice $I= ]\\lambda _{-}, \\lambda _{+}[$ .",
"Moreover, Theorem REF (a) together with (REF ) (respectively (REF )) implies the second item of (G9ii) (respectively of (G9iii)).",
"In fact, with the help of Theorem REF and some further considerations e.g.",
"on light-tailedness, one even gets assertions on the sufficiency of (G9ii), (G9iii) and (G9iv) for a “candidate generator” $\\varphi $ to belong to the BS-suitable class $\\Upsilon (]a,b[)$ .",
"More precisely, we obtain Proposition 31 Suppose that $\\varphi : ]-\\infty ,\\infty [ \\mapsto [0,\\infty ]$ satisfies (G1) to (G8), and recall the notations in (G9i).",
"Then, $\\varphi \\in \\Upsilon (]a,b[)$ if one of the following three conditions holds: (a) $a >-\\infty $ , $\\lambda _{-} = - \\infty $ , and the function $z \\mapsto e^{-a\\cdot z + \\varphi ^{*}(z)}$ is absolutely monotone on $]-\\infty ,0[$ , (b) $b < \\infty $ , $\\lambda _{+} = \\infty $ , and the function $z \\mapsto e^{b\\cdot z + \\varphi ^{*}(- z)}$ is absolutely monotone on $]-\\infty ,0[$ , (c) $a = -\\infty $ , $b = -\\infty $ , and the function $z \\mapsto e^{\\varphi ^{*}(z)}$ is exponentially convex on $]\\lambda _{-}, \\lambda _{+}[$ .",
"If one of the three conditions (a) to (c) holds, thenbasically by Theorem REF with $M(\\cdot )$ defined in G9(ii),(iii) or (iv); see Appendix D. the associated probability distribution $\\mathbb {}$ (cf.",
"(REF )) has expectation $\\int _{\\mathbb {R}} y d\\mathbb {}(y) =1$ and finite moments of all orders, i.e.",
"$\\int _{\\mathbb {R}} y^{j} d\\mathbb {}(y) < \\infty $ for all $j \\in \\mathbb {N}_{0}$ ; in terms of $\\mathbb {}[ \\cdot \\, ] := \\mathbb {\\Pi }[W \\in \\cdot \\, ]$ this means that $E_{\\mathbb {\\Pi }}[W] =1$ and $E_{\\mathbb {\\Pi }}[W^{j}] < \\infty $ .",
"The proof of Proposition REF will be given in Appendix D. As far as applicability is concerned, it is well known that, in general, verifying absolute monotonicity is typically more comfortable than verifying exponential convexity.",
"Fortunately, one can often use the former, since for many known divergence generators there holds $a > -\\infty $ (often $a=0$ ) or $b < \\infty $ or both, which by virtue of (G10i) is directly linked with the (endpoints of the) support of the potentially existing probability distribution $\\mathbb {}$ .",
"For a pregiven divergence generator $\\varphi $ , once its membership in $\\Upsilon (]a,b[)$ (and in particular, the representability (REF )) is verified, one would like to concretely find the underlying probability distribution $\\mathbb {}$ .",
"This may be quite challenging, but can be made more comfortable by systematically narrowing down the family of distributions where $\\mathbb {}$ belongs to.",
"In fact, we have already performed a first down-narrowing, in terms of identifying the endpoints of the support of $\\mathbb {}$ to be the endpoints of the effective domain of $\\varphi $ (cf.",
"(G10i)).",
"A further down-narrowing can be achieved from (REF ) to (REF ) in combination with real-valued characterization theorems which are more specific than Theorem REF .",
"This will be shown exemplarily for a few important sub-setups, in the following.",
"For the identification of light-tailed semi/half-lattice distributions, we obtain the following two sets of sufficient conditions, which even allow for the desired explicit determination of $\\mathbb {}$ : Proposition 32 Suppose that $\\varphi : ]-\\infty ,\\infty [ \\mapsto [0,\\infty ]$ satisfies (G1) to (G8), with some $a > -\\infty $ .",
"Furthermore, assume that there exists some constant $\\breve{c} >0$ as well as some function $H: [0,\\infty [ \\mapsto [0,\\infty [$ which is continuous on $[0,1]$ with $H(1)=1$ and absolutely monotone on $]0,1[$ , such that $e^{\\varphi ^{*}(\\frac{z}{\\breve{c}}) - a\\cdot \\frac{z}{\\breve{c}} } = H(e^{z}), \\qquad z \\in ]-\\infty , \\breve{c} \\cdot \\lambda _{+}[.$ Then one has $\\varphi \\in \\Upsilon (]a,b[)$ and $\\mathbb {} = \\sum _{n=0}^{\\infty } p_{n} \\cdot \\delta _{a + \\breve{c} \\cdot n}\\qquad \\textrm {with \\ } p_{n} := \\frac{1}{n !}",
"\\cdot \\frac{\\mathrm {d}^{n}H}{\\mathrm {d}t}(0),$ i.e.",
"$\\mathbb {\\Pi }[W = a + \\breve{c} \\cdot n \\, ] = p_{n}$ ($n \\in \\mathbb {N}_{0}$ ).",
"Proposition 33 Suppose that $\\varphi : ]-\\infty ,\\infty [ \\mapsto [0,\\infty ]$ satisfies (G1) to (G8), with some $b < \\infty $ .",
"Furthermore, assume that there exists some constant $\\breve{c} >0$ as well as some function $H: [0,\\infty [ \\mapsto [0,\\infty [$ which is continuous on $[0,1]$ with $H(1)=1$ and absolutely monotone on $]0,1[$ , such that $e^{\\varphi ^{*}(- \\frac{z}{\\breve{c}}) + b \\cdot \\frac{z}{\\breve{c}} } = H(e^{z}), \\qquad z \\in ]-\\infty , - \\breve{c} \\cdot \\lambda _{-}[.$ Then one has $\\varphi \\in \\Upsilon (]a,b[)$ and $\\mathbb {} = \\sum _{n=0}^{\\infty } p_{n} \\cdot \\delta _{b - \\breve{c} \\cdot n}\\qquad \\textrm {with \\ } p_{n} := \\frac{1}{n !}",
"\\cdot \\frac{\\mathrm {d}^{n}H}{\\mathrm {d}t}(0),$ i.e.",
"$\\mathbb {\\Pi }[W = b - \\breve{c} \\cdot n \\, ] = p_{n}$ ($n \\in \\mathbb {N}_{0}$ ).",
"The Propositions REF respectively REF follow from (REF ) respectively (REF ), some straightforward transformations, and a well-known characterization of probability generating functions $H$ (see e.g.",
"in Theorem 1.2.10 of Stroock [343]).",
"As an incentive for the following investigations, let us recall the discussion in the surroundings of Condition REF pertaining to the minimization problem (REF ), where we have addressed possible connections between the two representabilities (REF ) (needed e.g.",
"for Theorem REF ) and (REF ) (needed e.g.",
"for Theorem REF ); this strongly relates to the question, for which constants $\\widetilde{c} >0$ the validity $\\varphi \\in \\Upsilon (]a,b[)$ triggers the validity of $\\widetilde{c} \\cdot \\varphi \\in \\Upsilon (]a,b[)$ .",
"To begin with, it is straightforward to see that $\\varphi \\in \\Upsilon (]a,b[)$ always implies $\\widetilde{c} \\cdot \\varphi \\in \\Upsilon (]a,b[)$ for all integers $\\widetilde{c} \\in \\mathbb {N}$ ; indeed, if $\\varphi $ satisfies (REF ) for some $\\mathbb {} = \\mathbb {\\Pi }[ W \\in \\cdot \\, ]$ , then for each integer $\\widetilde{c} \\in \\mathbb {N}$ one gets that $\\widetilde{c} \\cdot \\varphi $ satisfies (REF ) for $\\widetilde{\\mathbb {}} = \\mathbb {\\Pi }[ \\sum _{j=1}^{\\widetilde{c}} \\frac{W_{j}}{\\widetilde{c}} \\in \\cdot \\, ]$ ; in the latter, the $W_{j}$ ’s are i.i.d.",
"copies from $W$ .",
"Clearly, $MGF_{\\widetilde{\\mathbb {}}}$ is then finite on some open interval containing zero (differing from the one for $MGF_{\\mathbb {}}$ only by a scaling with $1/\\widetilde{c}$ ).",
"For the following family of distributions, one can even trigger $\\widetilde{c} \\cdot \\varphi \\in \\Upsilon (]a,b[)$ for all $\\widetilde{c} >0$ : for the sake of a corresponding precise formulation, recall first the common knowledge that, generally speaking, a probability distribution $\\mathbb {}$ on $\\mathbb {R}$ with light tails — in the sense that its moment generating function $z\\mapsto MGF_{\\mathbb {} }(z):=\\int _{\\mathbb {R}}e^{z \\cdot y}d\\mathbb {} (y)$ is finite on some open interval $]\\lambda _{-},\\lambda _{+}[$ containing zero — is (said to be) infinitely divisible if there holds $\\textrm {for each n \\in \\mathbb {N} there existsa probability distribution \\mathbb {}_{n} on \\mathbb {R} such that}\\int _{-\\infty }^{\\infty } e^{z \\cdot y} d\\mathbb {} (y) =\\Big (\\int _{-\\infty }^{\\infty } e^{z \\cdot y} d\\mathbb {}_{n} (y) \\Big )^{n}, \\quad z \\in ]\\lambda _{-},\\lambda _{+}[ ;$ in fact, (REF ) means that the (light-tailed) moment generating function $MGF_{\\mathbb {} }$ is infinitely divisible in the sense that each $n-$ th root $(MGF_{\\mathbb {} })^{1/n}$ must be the moment generating function of some (light-tailed) probability distribution (denoted here by $\\mathbb {}_{n}$ ).",
"In particular, (REF ) implies that $\\mathbb {}_{n}$ is unique, and that $\\mathbb {}$ must necessarily have (one-sided or two-sided) unbounded support $supp(\\mathbb {})$ .",
"The latter may differ from $supp(\\mathbb {}_{n})$ .",
"In our BS context (REF ), (REF ) equivalently means that the associated random variable $W$ is infinitely divisible (with light-tailed distribution), in the sense that $\\textrm {for each n \\in \\mathbb {N} there exists a sequence of i.i.d.",
"random variables Y_{n,1}, \\cdots , Y_{n,n} such that }W \\stackrel{\\text{\\tiny d}}{=} Y_{n,1} + \\cdots + Y_{n,n} ,$ where $\\stackrel{\\text{\\tiny d}}{=}$ means “have equal probability distributions” and $\\mathbb {\\Pi }[W \\in \\cdot \\, ] = \\mathbb {}[ \\, \\cdot \\,]$ , $\\mathbb {\\Pi }[Y_{n,1} \\in \\cdot \\, ] = \\mathbb {}_{n}[ \\, \\cdot \\,]$ .",
"For the above-mentioned context, we obtain the useful Proposition 34 Suppose that $\\varphi \\in \\Upsilon (]a,b[)$ , with connected probability distribution $\\mathbb {}$ from (REF ) (recall that this implies that $\\mathbb {}$ is not a one-point distribution, cf.",
"Remark REF ).",
"Then there holds: $\\textrm {\\widetilde{c} \\cdot \\varphi \\in \\Upsilon (]a,b[)for all \\widetilde{c} > 0} \\ \\ \\Longleftrightarrow \\ \\ \\textrm {\\mathbb {} is infinitely divisible.",
"}$ The proof of Proposition REF is given in Appendix E. Notice that Proposition REF covers especially the important prominent power divergences (cf.",
"Examples REF and REF below) for which we provide the corresponding infinitely divisible distributions explicitly in the Examples REF and REF below, and for which the subsequent form of estimators (cf.",
"Chapter below) can be simplified.",
"For the identification of light-tailed infinitely divisible distributions, we obtain the following three sets of sufficient conditions: Proposition 35 Suppose that $\\varphi : ]-\\infty ,\\infty [ \\mapsto [0,\\infty ]$ satisfies (G1) to (G8), and recall the notations in (G9i) as well as $a = \\inf supp(\\mathbb {})$ , $b = \\sup supp(\\mathbb {})$ (cf.",
"(G10i)).",
"Then, $\\varphi \\in \\Upsilon (]a,b[)$ and the associated probability distribution $\\mathbb {}$ is infinitely divisible, if one of the following three conditions holds: (a) $a >-\\infty $ , $\\lambda _{-} = - \\infty $ , and the function $z \\mapsto \\varphi ^{* \\prime }(z) - a = (\\varphi ^{\\prime })^{-1}(z) - a $ is absolutely monotone on $]-\\infty ,0[$ , (b) $b < \\infty $ , $\\lambda _{+} = \\infty $ , and the function $z \\mapsto - \\varphi ^{* \\prime }(- z) + b= - (\\varphi ^{\\prime })^{-1}(-z) +b $ is absolutely monotone on $]-\\infty ,0[$ , (c) $a = -\\infty $ , $b = -\\infty $ , and the function $z \\mapsto \\frac{\\varphi ^{* \\prime \\prime }(z)}{\\varphi ^{* \\prime \\prime }(0)}= \\frac{\\varphi ^{\\prime \\prime }(1)}{\\varphi ^{\\prime \\prime }((\\varphi ^{\\prime })^{-1}(z))} $ is exponentially convex on $]\\lambda _{-}, \\lambda _{+}[$ .",
"In the first case (a) there automatically follows $b=\\infty $ , whereas in the second case (b) one automatically gets $a =- \\infty $ .",
"The proof of Proposition REF is given in Appendix E. So far, in the current section we have started from a given divergence generator $\\varphi \\in \\widetilde{\\Upsilon }(]a,b[)$ having some additional properties, switched to its Fenchel-Legendre transform $\\varphi ^{*}$ and some exponentially-linear transforms thereof, and presented some sufficient conditions for verifying that the outcome is a moment-generating function $MGF_{\\mathbb {}}$ of a unique probability distribution $\\mathbb {}$ which has light tails.",
"For finding the concrete $\\mathbb {}$ , one typically should know the explicit form of $\\varphi ^{*}$ .",
"However, it is well known that it can sometimes be hard to determine the explicit form of the Fenchel-Legendre transform of a convex function.",
"This issue also applies for the reverse direction of starting from a concrete probability distribution $\\mathbb {}$ with light tails, computing its log-moment-generating function (called cumulant-generating function) $z \\mapsto \\Lambda _{\\mathbb {}}(z) := \\log MGF_{\\mathbb {}}(z)$ and the corresponding Fenchel-Legendre transform $\\Lambda _{\\mathbb {}}^{*}$ which is nothing but the associated divergence generator $\\varphi $ (cf.",
"(REF )).",
"As will be illuminated in several examples below, the — “kind of intermediate” — construction method given in the below-mentioned Theorem REF can help to ease these two tasks.",
"To formulate this, we employ the class ${F}$ of functions $F: ]-\\infty ,\\infty [ \\mapsto [-\\infty ,\\infty ]$ with the following properties: $int(dom(F)) = ]a_{F},b_{F}[$ for some $-\\infty \\le a_{F} < 1 < b_{F} \\le \\infty $ ; $F$ is smooth (infinitely continuously differentiable) on $]a_{F},b_{F}[$ ; $F$ is strictly increasing on $]a_{F},b_{F}[$ .",
"Clearly, for any $F \\in {F}$ one gets the existence of $F(a_{F}) := \\lim _{t \\downarrow a_{F}} F(t) \\in [-\\infty , \\infty [$ and $F(b_{F}) := \\lim _{t \\uparrow b_{F}} F(t) \\in ]-\\infty , \\infty ]$ ; moreover, its inverse $F^{-1}: \\mathcal {R}(F) \\mapsto [a_{F},b_{F}]$ exists, where $\\mathcal {R}(F) := \\lbrace F(t): t \\in dom(F) \\rbrace $ .",
"Furthermore, $F^{-1}$ is strictly increasing and smooth (infinitely continuously differentiable) on the open interval $int(\\mathcal {R}(F)) = \\lbrace F(t): t \\in ]a_{F},b_{F}[ \\rbrace =]F(a_{F}),F(b_{F})[$ , and $F^{-1}(int(\\mathcal {R}(F))) = ]a_{F},b_{F}[$ .",
"Within such a context, we obtain Theorem 36 Let $F \\in {F}$ and fix an arbitrary point $c \\in int(\\mathcal {R}(F))$ .",
"Moreover, introduce the notationsfor the sake of brevity, we avoid here the more complete notation $\\lambda _{-}^{F,c}$ , $\\lambda _{+}^{F,c}$ , $t_{-}^{sc,F,c}$ , $t_{+}^{sc,F,c}$ indicating the dependence on $F$ and $c$ .",
"$]\\lambda _{-},\\lambda _{+}[ := int(\\mathcal {R}(F)) -c$ and $]t_{-}^{sc},t_{+}^{sc}[ := ]1+a_{F}-F^{-1}(c),1+b_{F}-F^{-1}(c)[$ (which implies $\\lambda _{-} < 0 < \\lambda _{+}$ and $t_{-}^{sc} < 1 < t_{+}^{sc}$ ).",
"Furthermore, define the functions $\\Lambda : \\ ]-\\infty ,\\infty [ \\ \\mapsto \\, [-\\infty , \\infty ]$ and $\\varphi : \\ ]-\\infty ,\\infty [ \\ \\mapsto \\, [0, \\infty ]$ by $\\hspace{-19.91684pt} \\Lambda (z) := \\Lambda ^{(c)}(z) \\hspace{-5.69046pt} &:=& \\hspace{-5.69046pt}{\\left\\lbrace \\begin{array}{ll}\\int \\displaylimits _{0}^{z} F^{-1}(u+c) \\, du+ z \\cdot (1-F^{-1}(c)) \\ \\in ]-\\infty , \\infty [,\\hspace{85.35826pt} \\textrm {if } z \\in ]\\lambda _{-},\\lambda _{+}[,\\\\\\int \\displaylimits _{0}^{\\lambda _{-}} F^{-1}(u+c) \\, du+ \\lambda _{-} \\cdot (1-F^{-1}(c))\\ \\in [-\\infty ,\\infty ],\\hspace{71.13188pt} \\textrm {if } z = \\lambda _{-} > -\\infty ,\\\\\\int \\displaylimits _{0}^{\\lambda _{+}} F^{-1}(u+c) \\, du+ \\lambda _{+} \\cdot (1-F^{-1}(c))\\ \\in \\ [-\\infty ,\\infty ],\\hspace{66.86414pt} \\textrm {if } z = \\lambda _{+} < \\infty ,\\\\\\infty , \\hspace{283.10483pt} \\textrm {else},\\end{array}\\right.",
"}$ where the second respectively third line are meant as $\\lim _{z \\downarrow \\lambda _{-}} \\big ( \\int \\displaylimits _{0}^{z} F^{-1}(u+c) \\, du+ z \\cdot (1-F^{-1}(c)) \\big )$ respectively $\\lim _{z \\uparrow \\lambda _{+}} \\big ( \\int \\displaylimits _{0}^{z} F^{-1}(u+c) \\, du+ z \\cdot (1-F^{-1}(c)) \\big )$ , and $\\varphi (t) := \\varphi ^{(c)}(t) \\hspace{-5.69046pt} &:=& \\hspace{-5.69046pt}{\\left\\lbrace \\begin{array}{ll}(t+F^{-1}(c)-1) \\cdot [ F\\left(t+F^{-1}(c)-1 \\right) - c ]-\\int \\displaylimits _{0}^{F\\left(t+F^{-1}(c)-1 \\right) - c} F^{-1}(u+c) du\\ \\in \\ [0,\\infty [,\\\\\\hspace{327.20668pt}\\quad \\textrm {if }t \\in ]t_{-}^{sc},t_{+}^{sc}[,\\\\(t_{-}^{sc}+F^{-1}(c)-1) \\cdot [ F\\left(t_{-}^{sc}+F^{-1}(c)-1 \\right) - c ]-\\int \\displaylimits _{0}^{F\\left(t_{-}^{sc}+F^{-1}(c)-1 \\right) - c} F^{-1}(u+c) du\\ \\in \\ ]0,\\infty ],\\\\\\hspace{327.20668pt}\\quad \\textrm {if }t = t_{-}^{sc} > -\\infty ,\\\\(t_{+}^{sc}+F^{-1}(c)-1) \\cdot [ F\\left(t_{+}^{sc}+F^{-1}(c)-1 \\right) - c ]-\\int \\displaylimits _{0}^{F\\left(t_{+}^{sc}+F^{-1}(c)-1 \\right) - c} F^{-1}(u+c) du\\ \\in \\ ]0,\\infty ],\\\\\\hspace{327.20668pt}\\quad \\textrm {if }t = t_{+}^{sc} < \\infty ,\\\\\\varphi (t_{-}^{sc}) +\\lambda _{-}\\cdot (t- t_{-}^{sc}) \\ \\in \\ ]0,\\infty ],\\hspace{99.58464pt} \\textrm {if }t_{-}^{sc} > - \\infty , \\ \\textrm {and} \\ t \\in \\ ]-\\infty , t_{-}^{sc}[,\\\\\\varphi (t_{+}^{sc}) +\\lambda _{+}\\cdot (t - t_{+}^{sc}) \\ \\in \\ ]0,\\infty ],\\hspace{99.58464pt} \\textrm {if }t_{+}^{sc} < \\infty , \\ \\textrm {and} \\ t \\in \\ ]t_{+}^{sc}, \\infty [,\\\\\\infty , \\hspace{321.51622pt} \\textrm {else},\\end{array}\\right.",
"}$ where the second respectively third line are again meant as lower respectively upper limit.",
"Then, $\\Lambda $ and $\\varphi $ have the following properties: (i) On $]\\lambda _{-},\\lambda _{+}[$ , the function $\\Lambda $ is smooth and strictly convex and consequently, $\\exp (\\Lambda )$ ) is smooth and strictly log-convex; moreover, there holds $\\Lambda (0) =0$ , $\\Lambda ^{\\prime }(0) =1$ ; (ii) $\\varphi \\in \\widetilde{\\Upsilon }(]a,b[)$ , where $a := t_{-}^{sc} \\cdot {1}_{ \\lbrace -\\infty \\rbrace }(\\lambda _{-}) - \\infty \\cdot {1}_{]-\\infty ,0[}(\\lambda _{-})$ , $b := t_{+}^{sc} \\cdot {1}_{ \\lbrace \\infty \\rbrace }(\\lambda _{+}) + \\infty \\cdot {1}_{]0,\\infty [}(\\lambda _{+})$ , and $\\varphi $ has the properties (G1) to (G8).",
"(iii) $\\varphi (t) =\\sup _{z \\in ]-\\infty ,\\infty [} \\left( z\\cdot t -\\Lambda (z)\\right) =\\sup _{z \\in ]\\lambda _{-},\\lambda _{+}[} \\left( z\\cdot t -\\Lambda (z)\\right)$ for all $t \\in \\mathbb {R}$ .",
"(iv) $\\Lambda (z) = \\varphi ^{*}(z) =\\sup _{t \\in ]-\\infty ,\\infty [} \\left( t\\cdot z -\\varphi (t)\\right) =\\sup _{t \\in ]a,b[} \\left( t\\cdot z -\\varphi (t) \\right)$ for all $z \\in \\mathbb {R}$ .",
"The proof of Theorem REF will be given Appendix F. Remark 37 Theorem REF indicates that the $F-$ constructed function $z \\mapsto \\exp (\\Lambda (z)) = \\exp (\\varphi ^{*}(z))$ is a good candidate for a moment generating function of a probability distribution $\\mathbb {}$ , and hence for the representability (REF ).",
"However, one still needs to verify one of the conditions (a) to (c) of Proposition REF .",
"This may go wrong, as the case of power divergences $\\varphi _{\\gamma }$ with $\\gamma \\in ]1,2[$ indicates (cf.",
"the conjecture of Example REF (f) below).",
"Notice that the newly constructed $\\Lambda $ and $\\varphi $ (cf.",
"(REF ), (REF )) depend on the choice of the anchor point $c$ ; this is e.g.",
"illustrated in Example REF (b) below.",
"Hence, as a side effect, by using whole families $(F_{\\vartheta })_{\\vartheta }$ together with different anchor points $c$ , via Theorem REF one can generate new classes (and new classifications) of $\\varphi -$ divergence generators — and thus of corresponding $\\varphi -$ divergences — which can be of great use, even in other contexts beyond our BS optimization framework.",
"If $F$ satisfies $F(1)=0$ and thus $F^{-1}(0)=1$ , then the natural choice $c:=0$ induces $]\\lambda _{-},\\lambda _{+}[ = int(\\mathcal {R}(F))$ and $]t_{-}^{sc},t_{+}^{sc}[ = ]a_{F},b_{F}[$ , and consequently (due to $F^{-1}(c)-1 =0$ ) leads to the simplification of “the first lines of” (REF ) and (REF ) to $& & \\hspace{-19.91684pt} \\Lambda (z) := \\Lambda ^{(0)}(z) :=\\int \\displaylimits _{0}^{z} F^{-1}(u) du,\\qquad z \\in int(\\mathcal {R}(F)),\\\\& & \\hspace{-19.91684pt} \\varphi (t) := \\varphi ^{(0)}(t) :=t \\cdot F\\left(t\\right)-\\int \\displaylimits _{0}^{F\\left(t\\right)} F^{-1}(u) du,\\qquad t \\in ]a_{F},b_{F}[ ;$ the simplifications of the respective other lines of (REF ) and (REF ) are straightforward.",
"Remark 38 Let $F \\in {F}$ with $a_{F} =0$ , $b_{F} = \\infty $ , $F(1)=0$ and hence, $int(\\mathcal {R}(F)) = \\, ]F(0),F(\\infty )[$ .",
"Then the transformation $\\widetilde{F}(t) &:=&{\\left\\lbrace \\begin{array}{ll}- \\int _{0}^{F(\\frac{1}{t})} F^{-1}(u) \\, du, \\qquad \\textrm {if } \\ t \\in ]0,\\infty [, \\\\- \\int _{0}^{F(\\infty )} F^{-1}(u) \\, du, \\qquad \\textrm {if } \\ t=0, \\\\-\\infty , \\qquad \\hspace{64.01869pt} \\textrm {if } \\ t \\in ]-\\infty , 0[,\\end{array}\\right.",
"}$ satisfies $\\widetilde{F} \\in {F}$ with $a_{\\widetilde{F}} =0$ , $b_{\\widetilde{F}} = \\infty $ , $\\widetilde{F}(1)=0$ and $int(\\mathcal {R}(\\widetilde{F})) =\\big ] - \\int _{0}^{F(\\infty )} F^{-1}(u) \\, du, - \\int _{0}^{F(0)} F^{-1}(u) \\, du \\, \\big [$ .",
"By choosing the natural anchor point $c=0$ (for both $F$ and $\\widetilde{F}$ ) and by using the relations $\\widetilde{F}(t) = - \\Lambda (F(\\frac{1}{t}))$ , $\\widetilde{F}^{-1}(z) = \\frac{1}{F^{-1}(\\Lambda ^{-1}(-z))}$ , as well as (REF ) in combination with (REF ) (for both contexts), it is straightforward to see that the corresponding quantities $\\widetilde{\\Lambda }$ and $\\widetilde{\\varphi }$ satisfy $\\widetilde{\\Lambda }(z) =- (-\\Lambda )^{-1}(z)$ ($z \\in int(\\mathcal {R}(\\widetilde{F}))$ ) and $\\widetilde{\\varphi }(t) = t \\cdot \\varphi (\\frac{1}{t})$ ($t \\in ]0,\\infty [$ ).",
"Hence, the corresponding divergences (cf.",
"(REF )) are “reciprocal to each other” in the sense that $D_{\\widetilde{\\varphi }}( \\mathbf {Q}, \\mathbf {P} ) =D_{\\varphi }( \\mathbf {P}, \\mathbf {Q} )$ for all $\\mathbf {P},\\mathbf {Q} \\in \\mathbb {S}_{>0}^{K}$ , and in case that $\\Lambda $ and $\\widetilde{\\Lambda }$ are indeed cumulant generating functions of some light-tailed distributions $\\mathbb {}$ and $\\widetilde{\\mathbb {}}$ (cf.",
"Remark REF ), then the latter two are said to be inverse to each other in the sense of Tweedie [368] (see also e.g.",
"Folks [127]).",
"As already indicated above, from Theorem REF one can comfortably generate various interesting examples, which we demonstrate in the following.",
"Example 39 (a) For $\\gamma \\in \\mathbb {R}\\backslash \\lbrace 1,2\\rbrace $ , $\\widetilde{c} \\in ]0,\\infty [$ and $]a_{F_{\\gamma ,\\widetilde{c}}},b_{F_{\\gamma ,\\widetilde{c}}}[ \\, = \\, ]0,\\infty [$ we define $F_{\\gamma ,\\widetilde{c}}(t) &:=&{\\left\\lbrace \\begin{array}{ll}\\frac{\\widetilde{c}}{\\gamma -1} \\cdot (t^{\\gamma -1}-1), \\qquad \\textrm {if } \\ t \\in \\, ]0,\\infty [, \\\\-\\frac{\\widetilde{c}}{\\gamma -1}, \\hspace{62.59596pt}\\textrm {if } \\ t=0 \\ \\textrm {and } \\ \\gamma \\in \\, ]1,2[ \\, \\cup \\, ]2,\\infty [, \\\\-\\infty , \\hspace{71.13188pt} \\textrm {if } \\ t=0 \\ \\textrm {and } \\ \\gamma < 1, \\\\- \\infty , \\hspace{71.13188pt} \\textrm {if } \\ t \\in \\, ]-\\infty ,0[,\\end{array}\\right.",
"}\\nonumber $ Clearly, $\\mathcal {R}(F_{\\gamma ,\\widetilde{c}})= \\big [-\\frac{\\widetilde{c}}{\\gamma -1},\\infty \\big [$ for $\\gamma \\in \\, ]1,2[ \\, \\cup \\, ]2,\\infty [$ , respectively $\\mathcal {R}(F_{\\gamma ,\\widetilde{c}})=\\big ]-\\infty , \\frac{\\widetilde{c}}{1-\\gamma }\\big [$ for $\\gamma <1$ ; notice that $0 \\in int(\\mathcal {R}(F_{\\gamma ,\\widetilde{c}}))$ for all $\\gamma \\in \\mathbb {R}\\backslash \\lbrace 1,2\\rbrace $ .",
"Furthermore, $F_{\\gamma ,\\widetilde{c}}(\\cdot )$ is strictly increasing and smooth on $]0,\\infty [$ , and thus, $F_{\\gamma ,\\widetilde{c}} \\in {F}$ .",
"Since $F_{\\gamma ,\\widetilde{c}}(1)=0$ , let us choose the natural anchor point $c:=0$ , which leads to $]\\lambda _{-},\\lambda _{+}[= int(\\mathcal {R}(F_{\\gamma ,\\widetilde{c}}))$ and $]t_{-}^{sc},t_{+}^{sc}[ = ]0,\\infty [$ .",
"By using $F_{\\gamma ,\\widetilde{c}}^{-1}(x) = (1+\\frac{(\\gamma -1) \\cdot x}{\\widetilde{c}})^{\\frac{1}{\\gamma -1}}$ for $x \\in int(\\mathcal {R}(F_{\\gamma ,\\widetilde{c}}))$ , we can derive from formula (REF ) (see also (REF )) for all $\\gamma \\in \\mathbb {R}\\backslash \\lbrace 0,1,2\\rbrace $ and $z \\in \\mathbb {R}$ $\\Lambda _{\\gamma ,\\widetilde{c}}(z) := \\Lambda _{\\gamma ,\\widetilde{c}}^{(0)}(z) &=&{\\left\\lbrace \\begin{array}{ll}\\frac{\\widetilde{c}}{\\gamma } \\cdot \\left\\lbrace \\left( \\frac{\\gamma -1}{\\widetilde{c}} \\cdot z +1 \\right)^{\\frac{\\gamma }{\\gamma -1}} -1 \\right\\rbrace , \\qquad \\textrm {if } \\ \\gamma \\in \\, ]1,2[ \\, \\cup \\, ]2,\\infty [\\ \\textrm {and } \\ z \\in \\big ]-\\frac{\\widetilde{c}}{\\gamma -1},\\infty \\big [ \\\\\\hspace{130.88284pt} \\textrm {or if } \\ \\gamma \\in ]-\\infty ,0[ \\cup ]0,1[ \\ \\textrm {and } \\ z \\in \\big ]-\\infty , \\frac{\\widetilde{c}}{1-\\gamma }\\big [, \\\\- \\frac{\\widetilde{c}}{\\gamma } < 0, \\hspace{105.2751pt} \\textrm {if } \\ \\gamma \\in \\, ]1,2[ \\, \\cup \\, ]2,\\infty [\\ \\textrm {and } \\ z = -\\frac{\\widetilde{c}}{\\gamma -1}, \\\\- \\frac{\\widetilde{c}}{\\gamma } >0, \\hspace{105.2751pt} \\textrm {if } \\ \\gamma < 0 \\ \\textrm {and } \\ z = \\frac{\\widetilde{c}}{1-\\gamma }, \\\\\\infty , \\hspace{128.0374pt} \\textrm {if } \\ \\gamma \\in \\, ]0,1[ \\ \\textrm {and } \\ z = \\frac{\\widetilde{c}}{1-\\gamma }, \\\\\\infty , \\hspace{128.0374pt} \\textrm {else} .\\end{array}\\right.",
"}$ Notice that $\\Lambda _{\\gamma ,\\widetilde{c}}(0) = 0$ for all $\\gamma \\in \\mathbb {R}\\backslash \\lbrace 0,1,2\\rbrace $ .",
"Moreover, for $\\gamma \\in \\, ]1,2[ \\, \\cup \\, ]2,\\infty [$ one has $\\Lambda _{\\gamma ,\\widetilde{c}}(\\infty ) = \\infty $ , $\\Lambda _{\\gamma ,\\widetilde{c}}^{\\prime }(-\\frac{\\widetilde{c}}{\\gamma -1}) = 0$ and $\\Lambda _{\\gamma ,\\widetilde{c}}^{\\prime }(\\infty ) = \\infty $ .",
"For $\\gamma < 0$ one gets $\\Lambda _{\\gamma ,\\widetilde{c}}(-\\infty ) = -\\infty $ , $\\Lambda _{\\gamma ,\\widetilde{c}}^{\\prime }(\\frac{\\widetilde{c}}{1-\\gamma }) = \\infty $ and $\\Lambda _{\\gamma ,\\widetilde{c}}^{\\prime }(-\\infty ) = 0$ .",
"In contrast, if $\\gamma \\in \\, ]0,1[$ then $\\Lambda _{\\gamma ,\\widetilde{c}}(-\\infty ) = - \\frac{\\widetilde{c}}{\\gamma } <0$ , $\\Lambda _{\\gamma ,\\widetilde{c}}^{\\prime }(\\frac{\\widetilde{c}}{1-\\gamma }) = \\infty $ and $\\Lambda _{\\gamma ,\\widetilde{c}}^{\\prime }(-\\infty ) = 0$ .",
"To proceed, from formula (REF ) (see also ()) we can deduce for all $\\gamma \\in \\mathbb {R}\\backslash \\lbrace 0,1,2\\rbrace $ and $t \\in \\mathbb {R}$ $\\varphi _{\\gamma ,\\widetilde{c}}(t) := \\varphi _{\\gamma ,\\widetilde{c}}^{(0)}(t)\\hspace{-5.69046pt} &=& \\hspace{-5.69046pt}{\\left\\lbrace \\begin{array}{ll}\\widetilde{c} \\cdot \\frac{t^\\gamma -\\gamma \\cdot t+ \\gamma - 1}{\\gamma \\cdot (\\gamma -1)}\\ \\in \\ [0,\\infty [,\\qquad \\quad \\textrm {if }t \\in ]0,\\infty [,\\\\\\frac{\\widetilde{c}}{\\gamma } > 0, \\hspace{108.12054pt} \\textrm {if } \\ \\gamma \\in \\, ]1,2[ \\, \\cup \\, ]2,\\infty [\\ \\textrm {and } \\ t = 0, \\\\\\infty , \\hspace{123.76965pt} \\textrm {if } \\ \\gamma < 0 \\ \\textrm {and } \\ t = 0, \\\\\\frac{\\widetilde{c}}{\\gamma } > 0, \\hspace{108.12054pt} \\textrm {if } \\ \\gamma \\in \\, ]0,1[ \\ \\textrm {and } \\ t = 0, \\\\\\frac{\\widetilde{c}}{\\gamma } -\\frac{\\widetilde{c}}{\\gamma -1} \\cdot t \\ \\in \\ ]0,\\infty [,\\hspace{45.52458pt} \\textrm {if } \\ \\gamma \\in \\, ]1,2[ \\, \\cup \\, ]2,\\infty [\\ \\textrm {and } \\ t < 0, \\\\\\infty , \\hspace{123.76965pt} \\textrm {else} ,\\end{array}\\right.",
"}$ which coincides with $\\widetilde{c} \\cdot \\varphi _{\\gamma }(t)$ for $\\varphi _{\\gamma }(t)$ from (REF ) and which generates the $\\gamma -$ corresponding power divergences given in (REF ); the first line in (REF ) can be proved by $& & \\hspace{-19.91684pt}\\varphi _{\\gamma ,\\widetilde{c}}(t) := \\varphi _{\\gamma ,\\widetilde{c}}^{(0)}(t) :=t \\cdot F_{\\gamma ,\\widetilde{c}}\\left(t\\right)-\\int \\displaylimits _{0}^{F_{\\gamma ,\\widetilde{c}}\\left(t\\right)} F_{\\gamma ,\\widetilde{c}}^{-1}(u) du\\nonumber \\\\& & \\hspace{-19.91684pt}= \\frac{t \\cdot \\widetilde{c}}{\\gamma -1} \\cdot (t^{\\gamma -1}-1)- \\frac{\\widetilde{c}}{\\gamma } \\cdot \\left\\lbrace \\left( \\frac{\\gamma -1}{\\widetilde{c}} \\cdot \\left[ \\frac{\\widetilde{c}}{\\gamma -1} \\cdot (t^{\\gamma -1}-1) \\right]+ 1 \\right)^{\\frac{\\gamma }{\\gamma -1}} -1 \\right\\rbrace \\nonumber \\\\& & \\hspace{-19.91684pt}= \\widetilde{c} \\cdot \\frac{t^\\gamma -\\gamma \\cdot t+ \\gamma - 1}{\\gamma \\cdot (\\gamma -1)},\\qquad t \\in ]0,\\infty [ \\, .$ Notice that for all $\\gamma \\in \\mathbb {R}\\backslash \\lbrace 0,1,2\\rbrace $ one has $\\varphi _{\\gamma ,\\widetilde{c}}(1) = 0$ , $\\varphi _{\\gamma ,\\widetilde{c}}^{\\prime }(1) = 0$ and $\\varphi _{\\gamma ,\\widetilde{c}}(\\infty ) = \\infty $ .",
"Moreover, for $\\gamma \\in \\, ]1,2[ \\, \\cup \\, ]2,\\infty [$ one has $\\varphi _{\\gamma ,\\widetilde{c}}^{\\prime }(0) = -\\frac{\\widetilde{c}}{\\gamma -1} < 0$ and $\\varphi _{\\gamma ,\\widetilde{c}}^{\\prime }(\\infty ) = \\infty $ .",
"In contrast, for $\\gamma < 0$ and $\\gamma \\in \\, ]0,1[$ one gets $\\varphi _{\\gamma ,\\widetilde{c}}^{\\prime }(0) = - \\infty $ and $\\varphi _{\\gamma ,\\widetilde{c}}^{\\prime }(\\infty ) = \\frac{\\widetilde{c}}{1-\\gamma } > 0$ .",
"(b) For $\\gamma =2$ , $\\widetilde{c} \\in ]0,\\infty [$ and $]a_{F_{\\gamma ,\\widetilde{c}}},b_{F_{\\gamma ,\\widetilde{c}}}[ \\, = \\, ]-\\infty ,\\infty [$ we define $F_{2,\\widetilde{c}}(t) &:=&\\widetilde{c} \\cdot (t-1), \\qquad t \\in \\, ]-\\infty ,\\infty [,\\nonumber $ Clearly, $\\mathcal {R}(F_{2,\\widetilde{c}})= \\, ]-\\infty , \\infty [$ , $0 \\in int(\\mathcal {R}(F_{2,\\widetilde{c}}))$ , and $F_{2,\\widetilde{c}}(\\cdot )$ is strictly increasing as well as smooth on $]-\\infty ,\\infty [$ .",
"Hence, $F_{2,\\widetilde{c}} \\in {F}$ .",
"Since $F_{2,\\widetilde{c}}(1)=0$ , let us choose the natural anchor point $c:=0$ , which leads to $]\\lambda _{-},\\lambda _{+}[= int(\\mathcal {R}(F_{2,\\widetilde{c}}))= \\, ]-\\infty , \\infty [$ and $]t_{-}^{sc},t_{+}^{sc}[ = \\, ]-\\infty , \\infty [$ .",
"By using $F_{2,\\widetilde{c}}^{-1}(x) = 1+\\frac{x}{\\widetilde{c}}$ for $x \\in int(\\mathcal {R}(F_{2,\\widetilde{c}}))$ , we can derive from formula (REF ) (see also (REF )) $\\Lambda _{2,\\widetilde{c}}(z) := \\Lambda _{2,\\widetilde{c}}^{(0)}(z) &=&\\frac{\\widetilde{c}}{2} \\cdot \\left\\lbrace \\left( \\frac{1}{\\widetilde{c}} \\cdot z +1 \\right)^{2} -1 \\right\\rbrace =\\frac{z^2}{2 \\widetilde{c}} + z,\\qquad z \\in \\, ]-\\infty ,\\infty [.$ Notice that $\\Lambda _{2,\\widetilde{c}}(0) = 0$ , $\\Lambda _{2,\\widetilde{c}}(-\\infty ) = \\Lambda _{2,\\widetilde{c}}(\\infty ) = \\infty $ , $\\Lambda _{2,\\widetilde{c}}^{\\prime }(-\\infty ) = -\\infty $ and $\\Lambda _{2,\\widetilde{c}}^{\\prime }(\\infty ) = \\infty $ .",
"From formula (REF ) (see also ()) we can deduce analogously to (REF ) $\\varphi _{2,\\widetilde{c}}(t) := \\varphi _{2,\\widetilde{c}}^{(0)}(t)&=&\\widetilde{c} \\cdot \\frac{(t-1)^{2}}{2}\\ \\in \\ [0,\\infty [,\\qquad t \\in \\, ]-\\infty ,\\infty [,$ which coincides with $\\widetilde{c} \\cdot \\varphi _{2}(t)$ for $\\varphi _{2}(t)$ from (REF ) which generates the corresponding power divergence given in the sixth line of (REF ).",
"Notice that $\\varphi _{2,\\widetilde{c}}(1) = 0$ , $\\varphi _{2,\\widetilde{c}}^{\\prime }(1) = 0$ and $\\varphi _{2,\\widetilde{c}}(-\\infty ) = \\varphi _{2,\\widetilde{c}}(\\infty ) = \\infty $ .",
"As an application of the reciprocity considerations of Remark REF , it is straightforward to see from the above-mentioned considerations (a) and (b) that for all $\\gamma \\in \\mathbb {R}\\backslash \\lbrace 0,1\\rbrace $ one has $\\widetilde{F}_{\\gamma ,\\widetilde{c}}(t) =- \\Lambda _{\\gamma ,\\widetilde{c}}(F_{\\gamma ,\\widetilde{c}}(\\frac{1}{t}))= F_{1-\\gamma ,\\widetilde{c}}(t)$ for all $t \\in ]0,\\infty [$ .",
"(c) Let us now continue with the remaining case $\\gamma =0$ (recall the natural anchor point $c:=0$ ).",
"By using $F_{0,\\widetilde{c}}^{-1}(x) =\\frac{1}{1- \\frac{x}{\\widetilde{c}}} $ for $x \\in int(\\mathcal {R}(F_{0,\\widetilde{c}})) = ]-\\infty , \\widetilde{c}[$ , we can derive from formula (REF ) (see also (REF )) $\\Lambda _{0,\\widetilde{c}}(z) := \\Lambda _{0,\\widetilde{c}}^{(0)}(z) &=&{\\left\\lbrace \\begin{array}{ll}- \\widetilde{c} \\cdot \\log \\left( 1 - \\frac{z}{\\widetilde{c}} \\right), \\qquad \\textrm {if } \\ z \\in \\big ]-\\infty ,\\widetilde{c} \\big [, \\\\\\infty , \\hspace{78.24507pt} \\textrm {if } \\ z \\in \\big [\\widetilde{c}, \\infty \\big [ .\\end{array}\\right.",
"}$ Notice that $\\Lambda _{0,\\widetilde{c}}(0) = 0$ , $\\Lambda _{0,\\widetilde{c}}(-\\infty ) = - \\infty $ , $\\Lambda _{0,\\widetilde{c}}^{\\prime }(\\widetilde{c}) = \\infty $ and $\\Lambda _{0,\\widetilde{c}}^{\\prime }(-\\infty ) = 0$ .",
"Moreover, from formula (REF ) (see also ()) we can deduce $\\varphi _{0,\\widetilde{c}}(t) := \\varphi _{0,\\widetilde{c}}^{(0)}(t)\\hspace{-5.69046pt} &=& \\hspace{-5.69046pt}{\\left\\lbrace \\begin{array}{ll}\\widetilde{c} \\cdot \\left(- \\log t + t -1 \\right)\\ \\in \\ [0,\\infty [,\\qquad \\quad \\textrm {if }t \\in \\, ]0,\\infty [,\\\\\\infty , \\hspace{145.10922pt} \\textrm {if } \\ t \\in \\, ]-\\infty , 0] ,\\end{array}\\right.",
"}$ which coincides with $\\widetilde{c} \\cdot \\varphi _{0}(t)$ for the generator $\\varphi _{0}(t)$ from (REF ) which generates the reverse Kullback-Leibler divergence (reverse relative entropy) given in (REF ) with $\\widetilde{c}=1$ ; the first line in (REF ) can be proved by $& & \\hspace{-19.91684pt}\\varphi _{0,\\widetilde{c}}(t) := \\varphi _{0,\\widetilde{c}}^{(0)}(t) :=t \\cdot F_{0,\\widetilde{c}}\\left(t\\right)-\\int \\displaylimits _{0}^{F_{0,\\widetilde{c}}\\left(t\\right)} F_{0,\\widetilde{c}}^{-1}(u) du\\nonumber \\\\& & \\hspace{-19.91684pt}= t \\cdot \\widetilde{c} \\cdot \\Big (1-\\frac{1}{t}\\Big )- (- \\widetilde{c}) \\cdot \\log \\left(1-\\frac{1}{\\widetilde{c}} \\cdot \\left[ \\widetilde{c} \\cdot \\Big ( 1-\\frac{1}{t}\\Big ) \\right]\\right)= \\widetilde{c} \\cdot \\left(- \\log t + t -1 \\right),\\qquad t \\in ]0,\\infty [ \\, .$ Notice that one has $\\varphi _{0,\\widetilde{c}}(1) = 0$ , $\\varphi _{0,\\widetilde{c}}(\\infty ) = \\infty $ , $\\varphi _{0,\\widetilde{c}}^{\\prime }(1) = 0$ , $\\varphi _{0,\\widetilde{c}}^{\\prime }(0) = -\\infty $ and $\\varphi _{0,\\widetilde{c}}^{\\prime }(\\infty ) = \\widetilde{c}$ .",
"Example 40 (a) For the remaining case $\\gamma =1$ , $\\widetilde{c} \\in ]0,\\infty [$ and $]a_{F_{1,\\widetilde{c}}},b_{F_{1,\\widetilde{c}}}[ = ]0,\\infty [$ we define $F_{1,\\widetilde{c}}(t) &:=&{\\left\\lbrace \\begin{array}{ll}\\widetilde{c} \\cdot \\log t =\\lim _{\\gamma \\rightarrow 1} F_{\\gamma ,\\widetilde{c}}(t), \\qquad \\textrm {if } \\ t \\in \\, ]0,\\infty [, \\\\-\\infty , \\hspace{108.12054pt} \\textrm {if } \\ t \\in ]-\\infty ,0].",
"\\\\\\end{array}\\right.",
"}\\nonumber $ Clearly, $\\mathcal {R}(F_{1,\\widetilde{c}})= ]-\\infty ,\\infty [$ .",
"Moreover, $F_{1,\\widetilde{c}}(\\cdot )$ is strictly increasing and smooth on $]0,\\infty [$ , and hence, $F_{\\gamma ,\\widetilde{c}} \\in {F}$ .",
"Since $F_{1,\\widetilde{c}}(1)=0$ , let us first choose the natural anchor point $c:=0$ , which leads to $]\\lambda _{-},\\lambda _{+}[= int(\\mathcal {R}(F_{1,\\widetilde{c}})) = ]-\\infty ,\\infty [$ and $]t_{-}^{sc},t_{+}^{sc}[ = ]0,\\infty [$ .",
"By using $F_{1,\\widetilde{c}}^{-1}(x) = \\exp (\\frac{x}{\\widetilde{c}})$ for $x \\in \\mathcal {R}(F_{1,\\widetilde{c}})$ , we can derive from formula (REF ) (see also (REF )) $& & \\hspace{-19.91684pt} \\Lambda _{1,\\widetilde{c}}(z) := \\Lambda _{1,\\widetilde{c}}^{(0)}(z) :=\\int \\displaylimits _{0}^{z} F_{1,\\widetilde{c}}^{-1}(u) \\, du= \\widetilde{c} \\cdot \\left( \\exp \\Big (\\frac{z}{\\widetilde{c}}\\Big ) - 1 \\right),\\ \\ z \\in ]-\\infty ,\\infty [ .$ Notice that $\\Lambda _{1,\\widetilde{c}}(0) = 0$ , $\\Lambda _{1,\\widetilde{c}}(-\\infty ) = - \\widetilde{c}$ , $\\Lambda _{1,\\widetilde{c}}(\\infty ) = \\infty $ , $\\Lambda _{1,\\widetilde{c}}^{\\prime }(-\\infty ) = 0$ and $\\Lambda _{0,\\widetilde{c}}^{\\prime }(\\infty ) = \\infty $ .",
"Moreover, from formula (REF ) (see also ()) we can deduce $\\varphi _{1,\\widetilde{c}}(t) := \\varphi _{1,\\widetilde{c}}^{(0)}(t)\\hspace{-5.69046pt} &:=& \\hspace{-5.69046pt}{\\left\\lbrace \\begin{array}{ll}\\widetilde{c} \\cdot \\left( t \\cdot \\log t + 1 - t \\right)\\ \\in \\ [0,\\infty [,\\qquad \\quad \\textrm {if }t \\in \\, ]0,\\infty [,\\\\1, \\hspace{150.79968pt} \\textrm {if } \\ t = 0, \\\\\\infty , \\hspace{145.10922pt} \\textrm {if } \\ t \\in \\, ]-\\infty ,0[ ,\\end{array}\\right.",
"}$ which coincides with $\\widetilde{c} \\cdot \\varphi _{1}(t)$ for the generator $\\varphi _{1}(t)$ from (REF ) which generates the Kullback-Leibler divergence (relative entropy) given in (REF ) with $\\widetilde{c}=1$ ; the first line in (REF ) can be proved by $& & \\hspace{-19.91684pt}\\varphi _{1,\\widetilde{c}}(t) := \\varphi _{1,\\widetilde{c}}^{(0)}(t) :=t \\cdot F_{1,\\widetilde{c}}\\left(t\\right)-\\int \\displaylimits _{0}^{F_{1,\\widetilde{c}}\\left(t\\right)} F_{1,\\widetilde{c}}^{-1}(u) du\\nonumber \\\\& & \\hspace{-19.91684pt}= t \\cdot \\widetilde{c} \\cdot \\log t- \\widetilde{c} \\cdot \\left( \\exp \\left(\\frac{1}{\\widetilde{c}} \\cdot \\left[ \\widetilde{c} \\cdot \\log t \\right]\\right) - 1 \\right)= \\widetilde{c} \\cdot \\left( t \\cdot \\log t + 1 - t \\right),\\qquad t \\in ]0,\\infty [ \\, ,$ Notice that one has $\\varphi _{1,\\widetilde{c}}(1) = 0$ , $\\varphi _{1,\\widetilde{c}}(\\infty ) = \\infty $ , $\\varphi _{1,\\widetilde{c}}^{\\prime }(1) = 0$ , $\\varphi _{1,\\widetilde{c}}^{\\prime }(0) = -\\infty $ and $\\varphi _{1,\\widetilde{c}}^{\\prime }(\\infty ) = \\infty $ .",
"As an application of the reciprocity considerations of Remark REF , it is straightforward to see that $\\widetilde{F}_{1,\\widetilde{c}}(t) =- \\Lambda _{1,\\widetilde{c}}(F_{1,\\widetilde{c}}(\\frac{1}{t}))= F_{0,\\widetilde{c}}(t)$ for all $t \\in ]0,\\infty [$ .",
"(b) For the choice $\\widetilde{c}=1$ , let us now fix a general anchor point $c \\in \\mathcal {R}(F_{1,\\widetilde{c}})= ]-\\infty ,\\infty [$ (rather than $c=0$ ), which leads to $]\\lambda _{-},\\lambda _{+}[= int(\\mathcal {R}(F_{1,1}))- c = ]-\\infty ,\\infty [$ and $]t_{-}^{sc},t_{+}^{sc}[ = ]1+a_{F_{1,1}}-F_{1,1}^{-1}(c),1+b_{F_{1,1}}-F_{1,1}^{-1}(c)[ \\,= \\, ]1-e^{c},\\infty [$ .",
"Accordingly, the formula (REF ) (see also (REF )) leads to $& & \\hspace{-19.91684pt} \\Lambda _{1,1}(z) := \\Lambda _{1,1}^{(c)}(z) :=\\int \\displaylimits _{0}^{z} F_{1,1}^{-1}(u+c) du+ z \\cdot (1-F_{1,1}^{-1}(c))\\nonumber \\\\& & \\hspace{59.75095pt}= e^{c} \\cdot \\left( e^{z} - 1 \\right) + z \\cdot (1- e^{c}),\\qquad z \\in ]-\\infty ,\\infty [, \\qquad \\ $ for which there holds $\\Lambda _{1,1}^{(c)}(0) = 0$ , $\\Lambda _{1,1}^{(c)}(-\\infty ) =\\infty \\cdot {1}_{]0,\\infty [}(c)- \\infty \\cdot {1}_{]-\\infty ,0[}(c)- 1 \\cdot {1}_{\\lbrace 0\\rbrace }(c)$ , $\\Lambda _{1,1}^{(c)}(\\infty ) = \\infty $ , $\\Lambda _{1,1}^{(c) \\prime }(-\\infty ) = 1- e^{c}$ and $\\Lambda _{1,1}^{(c) \\prime }(\\infty ) = \\infty $ .",
"Moreover, from formula (REF ) (see also ()) we can deduce $\\varphi _{1,1}(t) := \\varphi _{1,1}^{(c)}(t)\\hspace{-5.69046pt} &:=& \\hspace{-5.69046pt}{\\left\\lbrace \\begin{array}{ll}(t + e^{c} -1) \\cdot [\\log (t + e^{c} -1) -c] + 1 - t\\ \\in \\ [0,\\infty [,\\qquad \\quad \\textrm {if }t \\in \\, ]1-e^{c},\\infty [,\\\\e^{c}, \\hspace{240.42569pt} \\textrm {if } \\ t = 1-e^{c}, \\\\\\infty , \\hspace{239.00298pt} \\textrm {if } \\ t \\in \\, ]-\\infty ,1-e^{c}[ ;\\end{array}\\right.",
"}$ the first line in (REF ) can be proved by $& & \\hspace{-19.91684pt}\\varphi _{1,1}(t) := \\varphi _{1,1}^{(c)}(t) :=(t+F_{1,1}^{-1}(c)-1) \\cdot [ F_{1,1}\\left(t+F_{1,1}^{-1}(c)-1 \\right) - c ]-\\int \\displaylimits _{0}^{F_{1,1}\\left(t+F_{1,1}^{-1}(c)-1 \\right) - c}F_{1,1}^{-1}(u+c) du\\nonumber \\\\& & \\hspace{-19.91684pt}= (t + e^{c} -1) \\cdot [\\log (t + e^{c} -1) -c]- e^{c} \\cdot \\Big \\lbrace \\exp [\\log (t + e^{c} -1) -c] - 1 \\Big \\rbrace \\nonumber \\\\& & \\hspace{-19.91684pt}= (t + e^{c} -1) \\cdot [\\log (t + e^{c} -1) -c] + 1 - t ,\\qquad t \\in ]1-e^{c},\\infty [\\, .$ Clearly, one has $\\varphi _{1,1}^{(c)}(1) = 0$ , $\\varphi _{1,1}^{(c)}(\\infty ) = \\infty $ , $\\varphi _{1,1}^{(c) \\prime }(1) = 0$ , $\\varphi _{1,1}^{(c) \\prime }(1-e^{c}) = -\\infty $ and $\\varphi _{1,1}^{(c) \\prime }(\\infty ) = \\infty $ .",
"The corresponding divergence $D_{\\varphi _{1,1}^{(c)}}(\\mathbb {Q},\\mathbb {P})$ has been recently used in Broniatowski et al.",
"[63] for the important task of testing mixtures of probability distributions; in fact, in order to get considerable comfort in testing mixture-type hypotheses against corresponding marginal-type alternatives, they employ choices $c>0$ since then $\\varphi _{1,1}^{(c)}(t)$ is finite especially for some range of negative values $t<0$ .",
"The latter feature is also valid for the divergence generator $\\varphi _{bw,\\beta ,\\widetilde{c}}$ in the next example (cf.",
"(REF ) below).",
"Example 41 For $\\beta \\in \\, ]0,1]$ , $\\widetilde{c} \\in \\, ]0,\\infty [$ and $]a_{F_{bw,\\beta ,\\widetilde{c}}},b_{F_{bw,\\beta ,\\widetilde{c}}}[ \\,= \\, ]1-\\frac{1}{\\beta },\\infty [$ we define $F_{bw,\\beta ,\\widetilde{c}}(t) &:=&{\\left\\lbrace \\begin{array}{ll}\\frac{\\widetilde{c}}{2\\beta } \\cdot \\Big (1-\\frac{1}{(\\beta \\cdot t + 1 - \\beta )^2}\\Big ), \\qquad \\textrm {if } \\ t \\in \\, ]1-\\frac{1}{\\beta },\\infty [, \\\\- \\infty , \\hspace{93.89418pt} \\textrm {if } \\ t \\in \\, ]-\\infty ,1-\\frac{1}{\\beta }].\\end{array}\\right.",
"}\\nonumber $ Clearly, $\\mathcal {R}(F_{bw,\\beta ,\\widetilde{c}})= \\big ]-\\infty ,\\frac{\\widetilde{c}}{2\\beta }\\big [$ and $0 \\in int(\\mathcal {R}(F_{bw,\\beta ,\\widetilde{c}}))$ .",
"Moreover, $F_{bw,\\beta ,\\widetilde{c}}(\\cdot )$ is strictly increasing and smooth on $]1-\\frac{1}{\\beta },\\infty [$ , and thus, $F_{bw,\\beta ,\\widetilde{c}} \\in {F}$ .",
"Since $F_{bw,\\beta ,\\widetilde{c}}(1)=0$ , let us choose the natural anchor point $c:=0$ , which leads to $]\\lambda _{-},\\lambda _{+}[ \\, = int(\\mathcal {R}(F_{bw,\\beta ,\\widetilde{c}}))= \\big ]-\\infty ,\\frac{\\widetilde{c}}{2\\beta }\\big [$ and $]t_{-}^{sc},t_{+}^{sc}[ \\, = \\,]a_{F_{bw,\\beta ,\\widetilde{c}}},b_{F_{bw,\\beta ,\\widetilde{c}}}[\\, = \\, ]1-\\frac{1}{\\beta },\\infty [$ .",
"By using $F_{bw,\\beta ,\\widetilde{c}}^{-1}(x) =\\frac{1}{\\beta } \\cdot \\Big \\lbrace \\frac{1}{\\sqrt{1-2\\beta \\cdot x/\\widetilde{c}}} + \\beta -1 \\Big \\rbrace $ for $x \\in int(\\mathcal {R}(F_{bw,\\beta ,\\widetilde{c}}))$ , we can derive from formula (REF ) (see also (REF )) for all $\\beta \\in \\, ]0,1]$ and $z \\in \\mathbb {R}$ $\\Lambda _{bw,\\beta ,\\widetilde{c}}(z) :=\\Lambda _{bw,\\beta ,\\widetilde{c}}^{(0)}(z) &=&{\\left\\lbrace \\begin{array}{ll}-(\\frac{1}{\\beta }-1) \\cdot z + \\frac{\\widetilde{c}}{\\beta ^{2}} \\cdot \\Big \\lbrace 1 - \\sqrt{1-\\frac{2\\beta }{\\widetilde{c}} \\cdot z} \\ \\Big \\rbrace ,\\qquad \\textrm {if } z \\in \\big ]-\\infty ,\\frac{\\widetilde{c}}{2\\beta } \\big ], \\\\\\infty , \\hspace{176.407pt} \\textrm {else} .\\end{array}\\right.",
"}$ Notice that $\\Lambda _{bw,\\beta ,\\widetilde{c}}(0) = 0$ .",
"Moreover, $\\Lambda _{bw,\\beta ,\\widetilde{c}}(-\\infty ) = \\infty $ , $\\Lambda _{bw,\\beta ,\\widetilde{c}}(\\frac{\\widetilde{c}}{2\\beta })= \\frac{\\widetilde{c} \\cdot (\\beta +1)}{2 \\beta ^{2}}$ , $\\Lambda _{bw,\\beta ,\\widetilde{c}}^{\\prime }(-\\infty ) = - \\frac{1-\\beta }{\\beta } <0$ and $\\Lambda _{bw,\\beta ,\\widetilde{c}}^{\\prime }(\\frac{\\widetilde{c}}{2\\beta }) = \\infty $ .",
"Furthermore, from formula (REF ) (see also ()) we can straightforwardly deduce for all $t \\in \\mathbb {R}$ $\\varphi _{bw,\\beta ,\\widetilde{c}}(t) := \\varphi _{bw,\\beta ,\\widetilde{c}}^{(0)}(t)\\hspace{-5.69046pt} &:=& \\hspace{-5.69046pt}{\\left\\lbrace \\begin{array}{ll}\\widetilde{c} \\cdot \\frac{(t-1)^{2}}{2(\\beta \\cdot t +1 -\\beta )}\\ \\in \\ [0,\\infty [,\\qquad \\quad \\textrm {if }t \\in \\, ]1-\\frac{1}{\\beta },\\infty [,\\\\\\infty , \\hspace{120.92421pt} \\textrm {if } \\ t \\in \\, ]-\\infty ,1-\\frac{1}{\\beta }] .\\end{array}\\right.",
"}$ Note that $1-\\frac{1}{\\beta } <0$ so that negative $t$ are allowed here.",
"For $t\\ge 0$ , $\\varphi _{bw,\\beta ,\\widetilde{c}}(t)$ is known as Rukhin's generator (cf.",
"[312], see e.g.",
"also Marhuenda et al.",
"[247], Pardo [282]).",
"Obviously, one has $\\varphi _{bw,\\beta ,\\widetilde{c}}(1) = 0$ , $\\varphi _{bw,\\beta ,\\widetilde{c}}^{\\prime }(1) = 0$ , $\\varphi _{bw,\\beta ,\\widetilde{c}}^{\\prime }(1-\\frac{1}{\\beta }) = -\\infty $ and $\\varphi _{bw,\\beta ,\\widetilde{c}}^{\\prime }(\\infty ) = \\frac{\\widetilde{c}}{2\\beta }$ .",
"From the generator $\\varphi _{bw,\\beta ,\\widetilde{c}}$ given in (REF ), we build the corresponding divergence (cf.",
"(REF )) $& &\\hspace{-45.52458pt}D_{\\varphi _{bw,\\beta ,\\widetilde{c}}}(\\mathbf {Q},\\mathbf {P})= \\widetilde{c} \\cdot \\sum _{k=1}^{K} p_{k} \\cdot \\frac{(\\frac{q_{k}}{p_{k}}-1)^{2}}{2(\\beta \\cdot \\frac{q_{k}}{p_{k}} +1 -\\beta )}\\nonumber \\\\& & \\hspace{-45.52458pt}= \\frac{\\widetilde{c}}{2} \\cdot \\sum \\limits _{k=1}^{K}\\frac{(q_{k}-p_{k})^{2}}{\\beta \\cdot q_{k} + (1 -\\beta )\\cdot p_{k}},\\hspace{42.67912pt}\\textrm {if \\mathbf {P} \\in \\mathbb {R}^{K} and \\mathbf {Q} \\in \\mathbb {R}^{K}with \\mathbf {Q} \\in \\, ] \\, \\mathbf {P} \\cdot (1-\\frac{1}{\\beta }),\\infty [ component-wise;}$ for the special subcase $\\widetilde{c}=1$ and $\\mathbf {Q} \\in \\mathbb {R}_{> 0}^{K}$ , $D_{\\varphi _{bw,\\beta ,1}}(\\mathbf {Q},\\mathbf {P})$ can be interpreted as — “non-probability version” of — the well-known blended weight chi-square divergence of Lindsay [221] (see e.g.",
"also Basu & Lindsay [35], Györfy & Vajda [148], Basu et al.",
"[36]).",
"The special case $\\widetilde{c}=1$ and $\\beta =\\frac{1}{2}$ for probability vectors, i.e.",
"$D_{\\varphi _{bw,1/2,1}}({Q},{P})$ , is equal to (a multiple of the matrix-vector-converted (cf.",
"Remark REF )) Sanghvi’s genetic difference measure [316] and equal to the double of the so-called (squared) Vincze-Le Cam distance (cf.",
"Vincze [384], Le Cam [212], see also e.g.",
"Topsoe [360] — who used the alternative naming triangular discrimination — and Vajda [373]); this divergence $D_{\\varphi _{bw,1/2,1}}({Q},{P})$ has been used e.g.",
"in Liu et al.",
"[227] for a machine learning context of detecting salient objects, where ${Q}$ and ${P}$ are appropriate histograms of RGB color.",
"Remark 42 (a) By straightforward calculations, one can show that $\\varphi _{bw,1,\\widetilde{c}}$ (i.e.",
"with the choice $\\beta =1$ ) is equal to the $\\widetilde{c}-fold$ power-divergence generator $\\varphi _{\\gamma ,\\widetilde{c}} = \\widetilde{c} \\cdot \\varphi _{\\gamma }$ (cf.",
"(REF )) with $\\gamma =-1$ ; the corresponding divergence $D_{\\varphi _{bw,1,\\widetilde{c}}}(\\mathbf {Q},\\mathbf {P})$ is thus equal to the power divergence $D_{\\widetilde{c} \\cdot \\varphi _{-1}}(\\mathbf {Q},\\mathbf {P})$ (cf.",
"(REF )) which is nothing but the — “non-probability version” — of Neyman's chi-square divergence.",
"(b) For the case $\\beta =0$ — which has been excluded in Example REF for technical brevity — the divergence generator $\\varphi _{bw,0,\\widetilde{c}}$ corresponds to $\\widetilde{c}-fold$ power-divergence generator $\\varphi _{\\gamma ,\\widetilde{c}}$ with $\\gamma =2$ ; the corresponding divergence $D_{\\varphi _{bw,0,\\widetilde{c}}}(\\mathbf {Q},\\mathbf {P})$ is thus equal to the power divergence $D_{\\widetilde{c} \\cdot \\varphi _{2}}(\\mathbf {Q},\\mathbf {P})$ (cf.",
"(REF )) which is nothing but the — “non-probability version” — of Pearson's (i.e.",
"the classical) chi-square divergence.",
"Example 43 Let us give an interesting generalization of the Kullback-Leibler case of Example REF (a).",
"For $\\widetilde{c} >0$ and $\\alpha \\in \\, ]-1,0[ \\ \\cup \\ ]0,\\infty [$ let us define $F_{gKL,\\alpha ,\\widetilde{c}}(t) &:=&{\\left\\lbrace \\begin{array}{ll}\\widetilde{c} \\cdot \\log \\left( \\frac{(1+\\alpha ) \\cdot t}{1+ \\alpha \\cdot t} \\right), \\qquad \\textrm {if \\lbrace \\alpha \\in \\, ]0,\\infty [ and t \\in \\, ]0,\\infty [ \\rbrace or \\lbrace \\alpha \\in \\, ]-1,0[ and t \\in \\, ]0,-\\frac{1}{\\alpha }[ \\rbrace ,} \\\\- \\infty , \\hspace{71.13188pt} \\textrm {if \\alpha \\in \\, ]-1,0[ \\ \\cup \\ ]0,\\infty [ andt \\in \\, ]-\\infty ,0],} \\\\\\infty , \\hspace{78.52945pt} \\textrm {if \\alpha \\in \\, ]-1,0[ and t \\in \\, [-\\frac{1}{\\alpha },\\infty [,}\\end{array}\\right.",
"}\\nonumber $ (notice that $\\lim _{\\alpha \\rightarrow 0_{+}} F_{gKL,\\alpha ,\\widetilde{c}}(t)= F_{1,\\widetilde{c}}(t)$ , cf.",
"Example REF (a)).",
"Clearly, $]a_{F_{gKL,\\alpha ,\\widetilde{c}}},b_{F_{gKL,\\alpha ,\\widetilde{c}}}[ \\, := \\, ]0,\\infty [$ for $\\alpha \\in \\, ]0,\\infty [$ and $]a_{F_{gKL,\\alpha ,\\widetilde{c}}},b_{F_{gKL,\\alpha ,\\widetilde{c}}}[ \\, := \\, ]0,-\\frac{1}{\\alpha }[$ for $\\alpha \\in \\, ]-1,0[$ .",
"Moreover, $\\mathcal {R}(F_{gKL,\\alpha ,\\widetilde{c}})=\\, ]-\\infty , \\widetilde{c} \\cdot \\log (1+ \\frac{1}{\\alpha })[$ for $\\alpha \\in \\, ]0,\\infty [$ and $\\mathcal {R}(F_{gKL,\\alpha ,\\widetilde{c}})= \\, ]-\\infty ,\\infty [$ for $\\alpha \\in \\, ]-1,0[$ .",
"Furthermore, $F_{gKL,\\alpha ,\\widetilde{c}}(\\cdot )$ is strictly increasing and smooth on the respective $]a_{F_{gKL,\\alpha ,\\widetilde{c}}},b_{F_{gKL,\\alpha ,\\widetilde{c}}}[$ , and thus, $F_{gKL,\\alpha ,\\widetilde{c}} \\in {F}$ .",
"Since $F_{gKL,\\alpha ,\\widetilde{c}}(1)=0$ , let us choose the natural anchor point $c:=0$ , which leads to $]\\lambda _{-},\\lambda _{+}[ \\, = int(\\mathcal {R}(F_{gKL,\\alpha ,\\widetilde{c}})) = \\,]-\\infty , \\widetilde{c} \\cdot \\log (1+ \\frac{1}{\\alpha })[$ and $]t_{-}^{sc},t_{+}^{sc}[ \\, = \\, ]0,\\infty [$ for the case $\\alpha \\in \\, ]0,\\infty [$ , respectively, to $]\\lambda _{-},\\lambda _{+}[ \\, = int(\\mathcal {R}(F_{gKL,\\alpha ,\\widetilde{c}})) = \\, ]-\\infty ,\\infty [$ and $]t_{-}^{sc},t_{+}^{sc}[ \\, = \\, ]0,-\\frac{1}{\\alpha }[$ for the case $\\alpha \\in \\, ]-1,0[$ .",
"By employing $F_{gKL,\\alpha ,\\widetilde{c}}^{-1}(x) =\\frac{1}{(1+\\alpha ) \\cdot e^{-x/\\widetilde{c}} - \\alpha }$ for $x \\in ]\\lambda _{-},\\lambda _{+}[$ , one can deduce from formula (REF ) (see also (REF )) $& & \\Lambda _{gKL,\\alpha ,\\widetilde{c}}(z) := \\Lambda _{gKL,\\alpha ,\\widetilde{c}}^{(0)}(z)\\nonumber \\\\& & :={\\left\\lbrace \\begin{array}{ll}\\int \\displaylimits _{0}^{z} F_{gKL,\\alpha ,\\widetilde{c}}^{-1}(u) \\, du= - \\frac{\\widetilde{c}}{\\alpha } \\cdot \\log ((1+\\alpha ) - \\alpha \\cdot e^{z/\\widetilde{c}}),\\qquad \\textrm {if \\alpha \\in \\, ]0,\\infty [ and z \\in \\, ]-\\infty , \\widetilde{c} \\cdot \\log (1+ \\frac{1}{\\alpha })[}, \\\\\\int \\displaylimits _{0}^{z} F_{gKL,\\alpha ,\\widetilde{c}}^{-1}(u) \\, du= - \\frac{\\widetilde{c}}{\\alpha } \\cdot \\log ((1+\\alpha ) - \\alpha \\cdot e^{z/\\widetilde{c}}),\\qquad \\textrm {if \\alpha \\in \\, ]-1,0[ and z \\in \\, ]-\\infty , \\infty [}, \\\\\\infty , \\hspace{217.6634pt}\\textrm {if \\alpha \\in \\, ]0,\\infty [ and z \\in \\, [\\widetilde{c} \\cdot \\log (1+ \\frac{1}{\\alpha }), \\infty [}, \\\\\\end{array}\\right.",
"}$ for which there holds $\\Lambda _{gKL,\\alpha ,\\widetilde{c}}(0) = 0$ and $\\Lambda _{gKL,\\alpha ,\\widetilde{c}}(-\\infty ) = - \\frac{\\widetilde{c}}{\\alpha } \\cdot \\log (1+\\alpha )$ for $\\alpha \\in \\, ]-1,0[ \\ \\cup \\ ]0,\\infty [$ , as well as $\\Lambda _{gKL,\\alpha ,\\widetilde{c}}(\\widetilde{c} \\cdot \\log (1+ \\frac{1}{\\alpha })) = \\infty $ for $\\alpha \\in \\, ]0,\\infty [$ and $\\Lambda _{gKL,\\alpha ,\\widetilde{c}}(\\infty ) = \\infty $ for $\\alpha \\in \\, ]-1,0[$ .",
"The corresponding derivative satisfies $\\Lambda _{gKL,\\alpha ,\\widetilde{c}}^{\\prime }(- \\infty ) = 0$ for $\\alpha \\in \\, ]-1,0[ \\ \\cup \\ ]0,\\infty [$ , as well as $\\Lambda _{gKL,\\alpha ,\\widetilde{c}}^{\\prime }(\\widetilde{c}\\cdot \\log (1+ \\frac{1}{\\alpha })) = \\infty $ for $\\alpha \\in \\, ]0,\\infty [$ and $\\Lambda _{gKL,\\alpha ,\\widetilde{c}}^{\\prime }(\\infty ) = - \\frac{1}{\\alpha }$ for $\\alpha \\in \\, ]-1,0[$ .",
"Furthermore, from formula (REF ) (see also ()) one can derive $& & \\hspace{-19.91684pt} \\varphi _{gKL,\\alpha ,\\widetilde{c}}(t) := \\varphi _{gKL,\\alpha ,\\widetilde{c}}^{(0)}(t)\\nonumber \\\\& & \\hspace{-19.91684pt} :={\\left\\lbrace \\begin{array}{ll}\\widetilde{c} \\cdot \\left[ \\, t \\cdot \\log t + (t+\\frac{1}{\\alpha }) \\cdot \\log \\Big ( \\frac{1+\\alpha }{1+\\alpha \\cdot t} \\Big ) \\, \\right]\\ \\in \\ [0,\\infty [,\\quad \\textrm {if \\lbrace \\alpha \\in \\, ]0,\\infty [ and t \\in \\, ]0,\\infty [ \\rbrace or \\lbrace \\alpha \\in \\, ]-1,0[ and t \\in \\, ]0,-\\frac{1}{\\alpha }[ \\rbrace ,}\\\\\\frac{\\widetilde{c}}{\\alpha } \\cdot \\log (1+\\alpha )\\ \\in \\ ]0,\\infty [, \\hspace{106.69783pt}\\textrm {if \\alpha \\in \\, ]-1,0[ \\ \\cup \\ ]0,\\infty [ andt =0,} \\\\\\infty , \\hspace{197.74655pt}\\textrm {if \\alpha \\in \\, ]-1,0[ \\ \\cup \\ ]0,\\infty [ andt \\in \\, ]-\\infty ,0[,} \\\\\\infty , \\hspace{197.74655pt}\\textrm {if \\alpha \\in \\, ]-1,0[ andt \\in \\, [-\\frac{1}{\\alpha },\\infty [;}\\end{array}\\right.",
"}$ the first line in (REF ) can be proved by $& & \\hspace{-19.91684pt}\\varphi _{gKL,\\alpha ,\\widetilde{c}}(t) := \\varphi _{gKL,\\alpha ,\\widetilde{c}}^{(0)}(t) :=t \\cdot F_{gKL,\\alpha ,\\widetilde{c}}\\left(t\\right)-\\int \\displaylimits _{0}^{F_{gKL,\\alpha ,\\widetilde{c}}\\left(t\\right)}F_{gKL,\\alpha ,\\widetilde{c}}^{-1}(u) \\, du\\nonumber \\\\& & \\hspace{-19.91684pt}= \\widetilde{c} \\cdot t \\cdot \\log \\left( \\frac{(1+\\alpha ) \\cdot t}{1+ \\alpha \\cdot t} \\right)+ \\frac{\\widetilde{c}}{\\alpha } \\cdot \\log \\left( (1 +\\alpha ) -\\alpha \\cdot \\exp \\left[ \\log \\left( \\frac{(1+ \\alpha ) \\cdot t}{1+ \\alpha \\cdot t} \\right) \\right] \\right)\\nonumber \\\\& & \\hspace{-19.91684pt}= \\widetilde{c} \\cdot \\left[ \\, t \\cdot \\log t+ t \\cdot \\log \\Big ( \\frac{1+\\alpha }{1+ \\alpha \\cdot t} \\Big )+ \\frac{1}{\\alpha } \\cdot \\log \\Big ( \\frac{1+\\alpha }{1+ \\alpha \\cdot t} \\Big ) \\, \\right].$ Obviously, one has $\\varphi _{gKL,\\alpha ,\\widetilde{c}}(1) = 0$ , $\\varphi _{gKL,\\alpha ,\\widetilde{c}}^{\\prime }(1) = 0$ , $\\varphi _{gKL,\\alpha ,\\widetilde{c}}^{\\prime }(0) = -\\infty $ for $\\alpha \\in \\, ]-1,0[ \\ \\cup \\ ]0,\\infty [$ .",
"Moreover, for $\\alpha \\in \\, ]0,\\infty [$ there holds $\\varphi _{gKL,\\alpha ,\\widetilde{c}}(\\infty ) = \\infty $ , and $\\varphi _{gKL,\\alpha ,\\widetilde{c}}^{\\prime }(\\infty ) = \\widetilde{c} \\cdot \\log (1+\\frac{1}{\\alpha })$ , whereas for $\\alpha \\in \\, ]-1,0[$ we obtain $\\varphi _{gKL,\\alpha ,\\widetilde{c}}(-\\frac{1}{\\alpha }) = \\infty $ , and $\\varphi _{gKL,\\alpha ,\\widetilde{c}}^{\\prime }(-\\frac{1}{\\alpha }) = \\infty $ .",
"From the generator $\\varphi _{gKL,\\alpha ,\\widetilde{c}}$ given in (REF ), we build the corresponding divergence (cf.",
"(REF )) $& &\\hspace{-42.67912pt}D_{\\varphi _{gKL,\\alpha ,\\widetilde{c}}}(\\mathbf {Q},\\mathbf {P})= \\widetilde{c} \\cdot \\Big \\lbrace \\sum \\limits _{k=1}^{K}q_{k} \\cdot \\log \\Big (\\frac{q_{k}}{(1-\\frac{1}{1+\\alpha }) \\cdot q_{k} + \\frac{1}{1+\\alpha } \\cdot p_{k}} \\Big )+ \\frac{1}{\\alpha } \\cdot \\sum \\limits _{k=1}^{K} p_{k} \\cdot \\log \\Big (\\frac{p_{k}}{(1-\\frac{1}{1+\\alpha }) \\cdot q_{k} + \\frac{1}{1+\\alpha } \\cdot p_{k}} \\Big ) \\Big \\rbrace ,\\\\& & \\textrm {if \\lbrace \\alpha \\in \\, ]0,\\infty [,\\mathbf {P} \\in \\mathbb {R}_{> 0}^{K} and\\mathbf {Q} \\in \\mathbb {R}_{\\ge 0}^{K} \\rbrace or \\lbrace \\alpha \\in \\, ]-1,0[,\\mathbf {P} \\in \\mathbb {R}_{> 0}^{K} and\\mathbf {Q} \\in \\mathbb {R}_{\\ge 0}^{K}with \\mathbf {Q} \\le - \\frac{1}{\\alpha } \\cdot \\mathbf {P} \\rbrace .}",
"\\nonumber $ Notice that the symmetry $D_{\\varphi _{gKL,\\alpha ,\\widetilde{c}}}(\\mathbf {Q},\\mathbf {P})= D_{\\varphi _{gKL,\\alpha ,\\widetilde{c}}}(\\mathbf {P},\\mathbf {Q})$ generally holds only if $\\mathbf {P}, \\mathbf {Q} \\in \\mathbb {R}_{> 0}^{K}$ and $\\alpha =1$ ; indeed, this special case leads to $\\varphi _{snKL,\\widetilde{c}}(t):= \\varphi _{gKL,1,\\widetilde{c}}(t)\\hspace{-5.69046pt} &:=& \\hspace{-5.69046pt}{\\left\\lbrace \\begin{array}{ll}\\widetilde{c} \\cdot \\left[ \\, t \\cdot \\log t + (t+1) \\cdot \\log \\Big ( \\frac{2}{t+1} \\Big ) \\, \\right]\\ \\in \\ [0,\\infty [,\\qquad \\quad \\textrm {if }t \\in \\, ]0,\\infty [,\\\\\\widetilde{c} \\cdot \\log 2, \\hspace{187.78836pt} \\textrm {if } \\ t = 0, \\\\\\infty , \\hspace{209.12791pt} \\textrm {if } \\ t \\in \\, ]-\\infty ,0[ ,\\end{array}\\right.",
"}$ and $D_{\\varphi _{snKL,\\widetilde{c}}}(\\mathbf {Q},\\mathbf {P}):= D_{\\varphi _{gKL,1,\\widetilde{c}}}(\\mathbf {Q},\\mathbf {P})= \\widetilde{c} \\cdot \\Big \\lbrace \\sum \\limits _{k=1}^{K} q_{k} \\cdot \\log \\Big (\\frac{2 q_{k}}{q_{k} + p_{k}} \\Big )+ \\sum \\limits _{k=1}^{K} p_{k} \\cdot \\log \\Big (\\frac{2 p_{k}}{q_{k} + p_{k}} \\Big ) \\Big \\rbrace ,\\qquad \\mathbf {P} \\in \\mathbb {R}_{> 0}^{K}, \\mathbf {Q} \\in \\mathbb {R}_{\\ge 0}^{K}.$ For the special subcase that $\\widetilde{c} =1$ and that $\\mathbf {P} = {P}$ , $\\mathbf {Q} = {Q}$ are probability vectors, the divergence (REF ) can be rewritten as sum of two Kullback-Leibler divergences (cf.",
"(REF )) $D_{\\varphi _{snKL,1}}({Q},{P})= D_{\\varphi _{1}}({Q},({Q}+{P})/2)+ D_{\\varphi _{1}}({P},({Q}+{P})/2),\\qquad {P} \\in \\mathbb {S}_{> 0}^{K}, {Q} \\in \\mathbb {S}_{\\ge 0}^{K},$ which is the well-known (cf.",
"Burbea & Rao [68], Lin [219], Pardo & Vajda [284], Topsoe [360], Endres & Schindelin [120], Vajda [373], Sason [317]) Jensen-Shannon divergence (being also called symmetrized and normalized Kullback-Leibler divergence, symmetrized and normalized relative entropy, capacitory discrimination); this is equal to the $(2\\log 2)-$ fold of a special (namely, equally-weighted two-population) case of the Sibson information radius of order 1 (cf.",
"[334]) which has also been addressed e.g.",
"by Rao [301] for genetic cluster analysis.",
"By the way, for $\\alpha >0$ the divergence $D_{\\varphi _{gKL,\\alpha ,\\widetilde{c}}}({Q},{P})$ can also be interpreted as a multiple of a special non-equally-weighted Sibson information radius of order 1.",
"In a context of comparison of — not necessarily connected — networks where ${Q}$ , ${P}$ are probability vectors derived from matrices (cf.",
"Remark REF ) which are transforms of corresponding graph invariants (e.g.",
"network portraits), the (matrix-equivalent of the) Jensen-Shannon divergence $D_{\\varphi _{snKL,1}}({Q},{P})$ is also called the network portrait divergence, cf.",
"Bagrow and Bollt [28].",
"There is a vast literature on recent applications of the Jensen-Shannon divergence, for instance it appears exemplarily in Kvitsiani et al.",
"[208] for finding connections between the circuit-level function of different interneuron types in regulating the flow of information and the behavioural functions served by the cortical circuits, in Xu et al.",
"(2014) for browsing and exploration of video sequences, in Jenkinson et al.",
"[168] for the fundamental understanding of the epigenome that leads to a powerful approach for studying its role in disease and aging, in Martin et al.",
"[250] for the implementation of an evolutionary-based global localization filter for mobile robots, in Suo et al.",
"[354] for the revelation of critical regulators of cell identity in mice, in Abante et al.",
"[2] for the detection of biologically significant differences in DNA methylation between alleles associated with local changes in genetic sequences — for a better understanding of the mechanism of complex human diseases, in Afek et al.",
"[5] for revealing mechanisms by which mismatches can recruit transcription factors for modulating replication and repair activities in cells, in Alaiz-Rodriguez & Parnell [10] for the quantification of stability in feature selection and ranking algorithms, in Biau et al.",
"[53] for generative adversarial networks (GANs) in artificial intelligence and machine learning, in Carre et al.",
"[74] for the standardization of brain magnetic resonance (MR) images, in Chakraborty et al.",
"[75] for hierarchical clustering in foreign exchange FOREX markets (e.g.",
"in periods of major international crises), in Chong et al.",
"[87] as part of a web-based platform for comprehensive analysis of microbiome data outputs, in Cui et al.",
"[101] for modelling latent friend recommendation in online social media, in Gholami & Hodtani [134] for refinements of safety-and-security-targeted location verification systems in wireless communication networks (e.g in Intelligent Transportation Systems (ITSs) and vehicular technology), in Guo & Yuan [146] for accurate abnormality classification in semi-supervised Wireless Capsule Endoscopy (WCE) for digestive system cancer diagnosis, in Jiang et al.",
"[169] for the training of deep neural discriminative and generative networks used for designing and evaluating photonic devices, in Kartal et al.",
"[186] for uncovering the relationship between some genomic features and cell type-specific methylome diversity, in Laszlovszky et al.",
"[210] for investigating mechanisms of basal forebrain neurons which modulate synaptic plasticity,cortical processing, brain states and oscillations, in Lawson et al.",
"[211] for the improved understanding of some genetic circuits that allow cancer cells to evade destruction by the host immune system, in Li et al.",
"[215] for the search of causes of the progressive neurodevelopmental disorder Rett syndrome, in Machado et al.",
"[239] for discovering relations between distinct RNA viruses (including SARS-CoV-2), in Mohammadi et al.",
"[261] for the identification of cell states and their underlying topology, in Mohanty et al.",
"[262] for the design of implantable nanophotonic (i.e.",
"chip-scale optical circuit type) silicon probes for sub-millisecond deep-brain optical stimulation — e.g.",
"for the purpose of gaining a deeper understanding of the neural code, in Perera et al.",
"[290] for the quantification of the level of rationality in supply chain networks, in Pierri et al.",
"[294] for the study of growth of malicious/misleading information in some social media diffusion networks, in Rabadan et al.",
"[299] for the identification of gene mutations that lead to the genesis and progression of tumors, in Reiter et al.",
"[306] for quantifying metastatic phylogenetic diversity, in Van de Sande et al.",
"[378] as part of a computational toolbox for single-cell gene regulatory network analysis, in Skinnider et al.",
"[337] for the prediction of the chemical structures of genomically encoded antibiotics — in order to find means against the looming global crisis of antibiotic resistance, in Tuo et al.",
"[367] for the detection of high-order single nucleotide polymorphism (SNP) interactions, in Uttam et al.",
"[370] for predicting the risk of colorectal cancer recurrence and inferring associated tumor microenvironment networks, in Zhang et al.",
"[421] for incipient fault (namely, crack) detection, in Zhi et al.",
"[425] for the strengthening of information-centric networks against interest flooding attack (IFAs), in Acera Mateos et al.",
"[3] for deep-learning classification of SARS-CoV-2 and co-infecting RNA viruses, in Avsec et al.",
"[24] for uncovering the motifs and syntax of cis-regulatory sequences in genomics data, in Barennes et al.",
"[32] for comparing the accuracy of current T cell receptor sequencing methods employed for the understanding of adaptive immune responses, in Chen et al.",
"[79] for clustering high-dimensional microbial data from RNA sequencing, in Chen et al.",
"[84] for investigating key aspects of effective vocal social communication, in Koldobskiy et al.",
"[193] for investigations of genetic and epigenetic drivers of paediatric acute lymphoblastic leukaemia, in McGinnis et al.",
"[258] for evaluating RNA sequencing of pooled blood cell samples, in Mühlroth & Grottke [268] for the detection of emerging trends and technologies through artificial intelligence techniques, in Necci et al.",
"[272] for the assessment of protein intrinsic disorder predictions, in Okada et al.",
"[277] for the identification of genetic factors that cause individual differences in whole lymphocyte profiles and their changes after vaccination, and in Zhang et al.",
"[422] for the learning of functional magnetic resonance imaging (fMRI) time-series in a brain disease diagnosis context.",
"Remark 44 Let us transform $\\varphi _{gSH,\\alpha }(t) := \\frac{1-t}{\\alpha } \\cdot \\log (1+\\alpha )- \\varphi _{gKL,\\alpha ,1}(t)= - t \\cdot \\log t+ \\frac{1}{\\alpha } \\cdot (1 + \\alpha \\cdot t) \\cdot \\log (1 + \\alpha \\cdot t)- \\frac{1}{\\alpha } \\cdot (1+\\alpha ) \\cdot t \\cdot \\log (1+\\alpha )$ (for $t \\in [0,1]$ ).",
"The function $\\varphi _{gSH,\\alpha }(\\cdot )$ is strictly concave on $[0,1]$ with $\\varphi _{gSH,\\alpha }(0)= \\varphi _{gSH,\\alpha }(1)=0$ .",
"Hence, for probability vectors ${Q} =(q_{k})_{k=1,\\ldots ,K}$ , the $\\varphi -$ entropy $\\sum _{k=1}^{K}\\varphi _{gSH,\\alpha }(q_{k})$ is Kapur’s [183] generalization of the Shannon entropy (which corresponds to $\\alpha =0$ in the limit) whose maximization has been connected with generalizations of the Bose-Einstein statistics and the Fermi-Dirac statistics e.g.",
"in Kapur & Kesavan [185].",
"Example 45 Let us fix any $z_{1},z_{2} \\in \\mathbb {R}$ , $p \\in ]0,1[$ which satisfy $z_{1} < 1 < z_{2}$ and $z_{1} \\cdot p + z_{2} \\cdot (1-p) =1$ (and thus $p= \\frac{z_{2} -1}{z_{2} - z_{1}}$ ).",
"On $]a_{F_{twop}},b_{F_{twop}}[ := ]z_{1},z_{2}[$ we define $F_{twop}(t) &:=& \\frac{1}{z_{2}-z_{1}} \\cdot \\log \\left( \\frac{(t-z_{1})\\cdot p}{(z_{2}-t)\\cdot (1- p)} \\right)\\nonumber \\\\&=&\\frac{1}{z_{2}-z_{1}} \\cdot \\log \\left( \\frac{(t-z_{1})\\cdot (z_{2} -1)}{(z_{2}-t)\\cdot (1-z_{1})} \\right) \\, ,\\qquad t \\in \\, ]z_{1},z_{2}[ ,\\nonumber $ where for the last equality we have used the above constraint (in order to obtain a two-parameter representation).",
"Straightforwardly, we have $\\mathcal {R}(F_{twop})= ]-\\infty ,\\infty [$ .",
"Moreover, $F_{twop}(\\cdot )$ is strictly increasing and smooth on $]0,\\infty [$ , and thus, $F_{twop} \\in {F}$ .",
"Since $F_{twop}(1)=0$ , let us choose the natural anchor point $c:=0$ , which leads to $]\\lambda _{-},\\lambda _{+}[= int(\\mathcal {R}(F_{snKL,\\widetilde{c}})) = ]-\\infty , \\infty [$ and $]t_{-}^{sc},t_{+}^{sc}[ = ]z_{1},z_{2}[$ .",
"By using $F_{twop}^{-1}(x) = \\frac{p \\cdot z_{1} + (1- p) \\cdot z_{2} \\cdot e^{(z_{2}-z_{1}) \\cdot x}}{p + (1- p) \\cdot e^{(z_{2}-z_{1}) \\cdot x}} \\, , \\qquad x \\in ]-\\infty , \\infty [,$ we derive from formula (REF ) (see also (REF )) $& & \\hspace{-19.91684pt} \\Lambda _{twop}(z) := \\Lambda _{twop}^{(0)}(z) :=\\int \\displaylimits _{0}^{z} F_{twop}^{-1}(u) du= \\log \\Big ( p \\cdot e^{z_{1} \\cdot z} + (1- p) \\cdot e^{z_{2} \\cdot z} \\Big ),\\qquad z \\in ]-\\infty , \\infty [ ,$ which has the properties $\\Lambda _{twop}(0) = 0$ , $\\Lambda _{twop}(-\\infty ) =\\infty \\cdot {1}_{]-\\infty ,0[}(z_{1})- \\infty \\cdot {1}_{]0,\\infty [}(z_{1})+ \\log p \\cdot {1}_{\\lbrace 0\\rbrace }(z_{1})$ , $\\Lambda _{twop}(\\infty ) = \\infty $ , $\\Lambda _{twop}^{\\prime }(- \\infty ) = z_{1}$ and $\\Lambda _{twop}^{\\prime }(\\infty ) = z_{2}$ .",
"Furthermore, from formula (REF ) (see also ()) we deduce $\\varphi _{twop}(t) := \\varphi _{twop}^{(0)}(t)\\hspace{-5.69046pt} &:=& \\hspace{-5.69046pt}{\\left\\lbrace \\begin{array}{ll}\\frac{t-z_{1}}{z_{2}-z_{1}} \\cdot \\log \\left( \\frac{(t-z_{1})\\cdot (z_{2}-1)}{(z_{2}-t)\\cdot (1-z_{1})} \\right)- \\log \\left( \\frac{z_{2}-1}{z_{2}-t} \\right)\\ \\in \\ [0,\\infty [,\\qquad \\quad \\textrm {if }t \\in \\, ]0,\\infty [,\\\\\\log \\left( \\frac{z_{2} - z_{1}}{z_{2} -1} \\right), \\hspace{219.08612pt} \\textrm {if } \\ t = z_{1}, \\\\\\log \\left( \\frac{z_{2} - z_{1}}{1 - z_{1}} \\right), \\hspace{219.08612pt} \\textrm {if } \\ t = z_{2}, \\\\\\infty , \\hspace{233.3125pt} \\textrm {if } \\ t \\in \\, ]-\\infty ,z_{1}[ \\, \\cup \\, ]z_{2}, \\infty [ ;\\end{array}\\right.",
"}$ the first line in (REF ) can be proved by $& & \\hspace{-19.91684pt}\\varphi _{twop}(t) := \\varphi _{twop}^{(0)}(t) :=t \\cdot F_{twop}\\left(t\\right)-\\int \\displaylimits _{0}^{F_{twop}\\left(t\\right)} F_{twop}^{-1}(u) du\\nonumber \\\\& & \\hspace{-19.91684pt}= \\frac{t}{z_{2}-z_{1}} \\cdot \\log \\left( \\frac{(t-z_{1})\\cdot p}{(z_{2}-t)\\cdot (1- p)} \\right)\\nonumber \\\\& & \\hspace{-19.91684pt}- \\log \\left( p \\cdot \\left( \\frac{(t-z_{1})\\cdot p}{(z_{2}-t)\\cdot (1- p)} \\right)^{\\frac{z_{1}}{z_{2}-z_{1}}}+ (1- p) \\cdot \\left( \\frac{(t-z_{1})\\cdot p}{(z_{2}-t)\\cdot (1- p)} \\right)^{\\frac{z_{2}}{z_{2}-z_{1}}} \\right)\\nonumber \\\\& & \\hspace{-19.91684pt}= \\frac{t-z_{1}}{z_{2}-z_{1}} \\cdot \\log \\left( \\frac{(t-z_{1})\\cdot p}{(z_{2}-t)\\cdot (1- p)} \\right)- \\log \\left( \\frac{(z_{2}-z_{1})\\cdot p}{z_{2}-t} \\right)\\\\& & \\hspace{-19.91684pt}= \\frac{t-z_{1}}{z_{2}-z_{1}} \\cdot \\log \\left( \\frac{(t-z_{1})\\cdot (z_{2}-1)}{(z_{2}-t)\\cdot (1-z_{1})} \\right)- \\log \\left( \\frac{z_{2}-1}{z_{2}-t} \\right) , \\qquad t \\in \\, ]z_{1}, z_{2} [ \\, ,\\nonumber $ where for the last equality we have used the above constraint (to obtain a two-parameter representation).",
"Straightforwardly, one has $\\varphi _{twop}(1) = 0$ , $\\varphi _{twop}^{\\prime }(1) = 0$ , $\\varphi _{twop}^{\\prime }(z_{1}) = -\\infty $ and $\\varphi _{twop}^{\\prime }(z_{2}) = \\infty $ .",
"From the generator $\\varphi _{twop}$ given in (REF ), we build the corresponding divergence (cf.",
"(REF )) $D_{\\varphi _{twop}}(\\mathbf {Q},\\mathbf {P})=\\sum \\limits _{k=1}^{K} \\frac{q_{k} - z_{1} \\cdot p_{k}}{z_{2} - z_{1}} \\cdot \\log \\Big (\\frac{(z_{2}-1) \\cdot (q_{k} - z_{1} \\cdot p_{k})}{(1-z_{1}) \\cdot (z_{2} \\cdot p_{k} - q_{k})} \\Big )- \\sum \\limits _{k=1}^{K} p_{k} \\cdot \\log \\Big (\\frac{(z_{2}-1) \\cdot p_{k}}{z_{2} \\cdot p_{k} - q_{k}} \\Big ) .$ It is known that some types of robustness properties of minimum-divergence estimators are connected with the boundedness of the derivative $\\varphi ^{\\prime }$ of the divergence generator $\\varphi $ ; this property is satisfied for the next Example REF (and its $W-$ concerning continuation in Example REF ), which leads to the new classes of divergences (REF ), (REF ) and (REF ): Example 46 (a) For any parameter-quadrupel $\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c} \\in \\, ]0,\\infty [$ with $\\beta _{1} < \\beta _{2}$ , we choose $]a_{F},b_{F}[\\ \\, := \\ ]a_{F_{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}},b_{F_{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}}[ \\ \\, := \\ \\Big ]1 - \\alpha \\cdot \\frac{(\\beta _{1} - \\beta _{2})^{2} + \\beta _{1}^{2} + \\beta _{1} \\cdot \\beta _{2}}{2\\beta _{1}\\cdot \\beta _{2}\\cdot (\\beta _{2} - \\beta _{1} )}, \\,\\infty \\Big [ \\ \\ni 1\\nonumber $ and define with $\\breve{\\theta } := 1 + \\alpha \\cdot \\Big (\\frac{1}{\\beta _{2}}- \\frac{1}{\\beta _{1}} \\Big ) < 1$ $\\hspace{-17.07182pt}F_{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}(t)&:=& {\\left\\lbrace \\begin{array}{ll}\\widetilde{c} \\cdot \\frac{\\beta _{1}-\\beta _{2}}{2}+ \\frac{\\widetilde{c}}{\\frac{1-t}{\\alpha } + \\frac{1}{\\beta _{2}} - \\frac{1}{\\beta _{1}}}\\cdot \\Big (1 - \\frac{1}{2} \\cdot \\sqrt{4 + \\big (\\frac{1-t}{\\alpha } + \\frac{1}{\\beta _{2}} - \\frac{1}{\\beta _{1}}\\big )^{2} \\cdot (\\beta _{1}+\\beta _{2})^{2}}\\, \\Big ),\\quad \\textrm {if } \\ t \\in \\, ]a_{F},b_{F}[ \\backslash \\lbrace \\breve{\\theta }\\rbrace , \\\\\\widetilde{c} \\cdot \\frac{\\beta _{1}-\\beta _{2}}{2}, \\hspace{277.41437pt}\\textrm {if } \\ t= \\breve{\\theta } \\in \\, ]a_{F},b_{F}[, \\\\- \\widetilde{c} \\cdot \\beta _{1}, \\hspace{284.52756pt} \\textrm {if } \\ t=a_{F}, \\\\- \\infty , \\hspace{295.90848pt} \\textrm {if } \\ t \\in \\, ]-\\infty ,a_{F}[.\\end{array}\\right.",
"}$ Notice that $\\breve{\\theta } \\in \\, ]a_{F},b_{F}[$ if and only if $\\beta _{1} \\in \\,]\\frac{\\beta _{2}}{3},\\beta _{2}[$ ; if (say) the latter holds, then one has the continuity $\\lim _{t \\rightarrow \\breve{\\theta }}F_{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}(t) = \\widetilde{c} \\cdot \\frac{\\beta _{1}-\\beta _{2}}{2}$ .",
"For $\\beta _{1} \\le \\frac{\\beta _{2}}{3}$ one gets $]a_{F},b_{F}[ \\backslash \\lbrace \\breve{\\theta } \\rbrace = \\, ]a_{F},b_{F}[$ .",
"Returning to the general case, one can see in a straightforward way that $F_{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}(\\cdot )$ is strictly increasing and that $\\mathcal {R}(F_{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}})=[-\\widetilde{c}\\cdot \\beta _{1},\\widetilde{c}\\cdot \\beta _{1} [$ .",
"Furthermore, $F_{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}(\\cdot )$ is smooth on $]a_{F},b_{F}[$ , and thus $F_{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}} \\in {F}$ .",
"Since $F_{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}(1)=0$ , let us choose the natural anchor point $c:=0$ , which leads to $]\\lambda _{-},\\lambda _{+}[ \\,= int(\\mathcal {R}(F_{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}})= \\, ]-\\widetilde{c}\\cdot \\beta _{1},\\widetilde{c}\\cdot \\beta _{1} [$ and $]t_{-}^{sc},t_{+}^{sc}[ = \\, ]a_{F},b_{F}[$ .",
"Moreover, it is straightforward to see that the corresponding inverse is $F_{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}^{-1}(x) =1 + \\alpha \\cdot \\Big (\\frac{1}{\\beta _{2}}- \\frac{1}{\\beta _{1}} \\Big ) - \\alpha \\cdot \\frac{\\frac{1}{\\beta _{2}} - \\frac{1}{\\beta _{1}} -\\frac{2 x}{\\widetilde{c} \\cdot \\beta _{1} \\cdot \\beta _{2} }}{1+ \\frac{x}{\\widetilde{c}} \\cdot \\Big (\\frac{1}{\\beta _{2}} - \\frac{1}{\\beta _{1}} \\Big )- \\frac{x^{2}}{\\widetilde{c}^{2} \\cdot \\beta _{1} \\cdot \\beta _{2} }},\\qquad x \\in int(\\mathcal {R}(F_{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}})) ;$ from this, we can derive from formula (REF ) (see also (REF )) for all $z \\in \\mathbb {R}$ $\\hspace{-19.91684pt}\\Lambda _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}(z) :=\\Lambda _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}^{(0)}(z) &=&{\\left\\lbrace \\begin{array}{ll}\\breve{\\theta } \\cdot z - \\widetilde{c} \\cdot \\alpha \\cdot \\log \\Big (1+ \\frac{z}{\\widetilde{c}} \\cdot \\Big (\\frac{1}{\\beta _{2}} - \\frac{1}{\\beta _{1}} \\Big )- \\frac{z^{2}}{\\widetilde{c}^{2} \\cdot \\beta _{1} \\cdot \\beta _{2} }\\Big ),\\quad \\textrm {if } \\ z \\in \\, ]-\\widetilde{c}\\cdot \\beta _{1},\\widetilde{c}\\cdot \\beta _{1} [,\\\\- \\widetilde{c} \\cdot \\breve{\\theta } \\cdot \\beta _{1} -\\widetilde{c} \\cdot \\alpha \\cdot \\log \\Big (2 - 2 \\frac{\\beta _{1}}{\\beta _{2}}\\Big ),\\hspace{73.97733pt} \\textrm {if } \\ z = -\\widetilde{c}\\cdot \\beta _{1},\\\\\\infty , \\hspace{202.01474pt} \\textrm {else} .\\end{array}\\right.",
"}$ Notice that $\\Lambda _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}(0) = 0$ and $\\lim _{z \\rightarrow \\widetilde{c}\\cdot \\beta _{1}}\\Lambda _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}(z) = \\infty $ .",
"Moreover, $\\Lambda _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}^{\\prime }( -\\widetilde{c}\\cdot \\beta _{1})= a_{F}$ and $\\Lambda _{ -\\widetilde{c}\\cdot \\beta _{1}}^{\\prime }(\\widetilde{c}\\cdot \\beta _{1}) = \\infty = b_{F}$ (which have to be interpreted as limits, as usual).",
"To proceed, from formula (REF ) (see also ()) we can deduce for all $t \\in \\mathbb {R}$ $\\hspace{-5.69046pt}\\varphi _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}(t):= \\varphi _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}^{(0)}(t)\\hspace{-8.5359pt} &=& \\hspace{-8.5359pt}{\\left\\lbrace \\begin{array}{ll}\\widetilde{c} \\cdot \\alpha \\cdot \\Big \\lbrace \\frac{\\sqrt{4 + (\\beta _{1} + \\beta _{2})^{2}\\cdot (\\frac{1-t}{\\alpha } + \\frac{1}{\\beta _{2}} - \\frac{1}{\\beta _{1}})^2}\\, - \\, (\\frac{1-t}{\\alpha } + \\frac{1}{\\beta _{2}} - \\frac{1}{\\beta _{1}}) \\cdot (\\beta _{1} - \\beta _{2}) \\, - \\, 2}{2} \\\\+ \\log \\frac{\\sqrt{4 + (\\beta _{1} + \\beta _{2})^{2}\\cdot (\\frac{1-t}{\\alpha } + \\frac{1}{\\beta _{2}} \\, - \\, \\frac{1}{\\beta _{1}})^2} \\, - \\, 2}{\\beta _{1} \\beta _{2} \\cdot (\\frac{1-t}{\\alpha } + \\frac{1}{\\beta _{2}} \\, - \\, \\frac{1}{\\beta _{1}})^{2}} \\Big \\rbrace \\ \\in [0,\\infty [,\\hspace{71.13188pt} \\textrm {if }t \\in \\, ]a_{F},\\infty [,\\\\\\widetilde{c} \\cdot \\alpha \\cdot \\Big \\lbrace \\frac{3 \\beta _{1} - \\beta _{2}}{2 (\\beta _{2} - \\beta _{1})}+ \\log \\frac{2(\\beta _{2} - \\beta _{1})}{\\beta _{2}} \\Big \\rbrace - \\widetilde{c} \\cdot \\beta _{1} \\cdot (t - a_{F}) \\ \\in \\, ]0,\\infty [,\\hspace{19.91684pt} \\textrm {if } \\ t \\in ]-\\infty , a_{F}].\\end{array}\\right.",
"}$ The first subcase in (REF ) can be proved by $& & \\hspace{-19.91684pt}\\varphi _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}(t) :=\\varphi _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}^{(0)}(t) :=t \\cdot F_{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}\\left(t\\right)\\ - \\int \\displaylimits _{0}^{F_{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}\\left(t\\right)}F_{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}^{-1}(u) \\, du\\nonumber \\\\& & \\hspace{-19.91684pt}= (t - \\breve{\\theta }) \\cdot F_{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}\\left(t\\right)+ \\widetilde{c} \\cdot \\alpha \\cdot \\log \\Big (1+ \\frac{F_{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}\\left(t\\right)}{\\widetilde{c}} \\cdot \\Big (\\frac{1}{\\beta _{2}} - \\frac{1}{\\beta _{1}} \\Big )- \\frac{(F_{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}\\left(t\\right))^{2}}{\\widetilde{c}^{2} \\cdot \\beta _{1} \\cdot \\beta _{2} }\\Big )\\nonumber \\\\& & \\hspace{-19.91684pt}= \\widetilde{c} \\cdot \\alpha \\cdot \\frac{\\sqrt{4 + (\\beta _{1} + \\beta _{2})^{2}\\cdot (\\frac{1-t}{\\alpha } + \\frac{1}{\\beta _{2}} - \\frac{1}{\\beta _{1}})^2}\\, - \\, (\\frac{1-t}{\\alpha } + \\frac{1}{\\beta _{2}} - \\frac{1}{\\beta _{1}}) \\cdot (\\beta _{1} - \\beta _{2}) \\, - \\, 2}{2}\\nonumber \\\\& & \\hspace{-19.91684pt}\\ \\ \\ + \\ \\widetilde{c} \\cdot \\alpha \\cdot \\log \\bigg (1+ \\Big [\\frac{\\beta _{1}-\\beta _{2}}{2}+ \\frac{1}{\\frac{1-t}{\\alpha } + \\frac{1}{\\beta _{2}} - \\frac{1}{\\beta _{1}}}\\cdot \\Big (1 - \\frac{1}{2} \\cdot \\sqrt{4 + \\big (\\frac{1-t}{\\alpha } + \\frac{1}{\\beta _{2}} - \\frac{1}{\\beta _{1}}\\big )^{2} \\cdot (\\beta _{1}+\\beta _{2})^{2}}\\, \\Big ) \\Big ]\\cdot \\frac{\\beta _{1} - \\beta _{2}}{\\beta _{1} \\cdot \\beta _{2}}\\nonumber \\\\& & \\hspace{-19.91684pt}\\ \\ \\ - \\ \\Big [\\frac{\\beta _{1}-\\beta _{2}}{2}+ \\frac{1}{\\frac{1-t}{\\alpha } + \\frac{1}{\\beta _{2}} - \\frac{1}{\\beta _{1}}}\\cdot \\Big (1 - \\frac{1}{2} \\cdot \\sqrt{4 + \\big (\\frac{1-t}{\\alpha } + \\frac{1}{\\beta _{2}} - \\frac{1}{\\beta _{1}}\\big )^{2} \\cdot (\\beta _{1}+\\beta _{2})^{2}}\\, \\Big ) \\Big ]^{2} \\cdot \\frac{1}{\\beta _{1} \\cdot \\beta _{2} }\\bigg )\\nonumber $ and some straightforward calculations.",
"The second line in (REF ) follows by computing $\\varphi _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}(a_{F})=\\widetilde{c} \\cdot \\alpha \\cdot \\Big \\lbrace \\frac{3 \\beta _{1} - \\beta _{2}}{2 (\\beta _{2} - \\beta _{1})}+ \\log \\frac{2(\\beta _{2} - \\beta _{1})}{\\beta _{2}} \\Big \\rbrace $ .",
"Notice that $\\varphi _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}(1) = 0$ , $\\varphi _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}^{\\prime }(1) = 0$ , $\\varphi _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}(-\\infty ) = \\infty $ and $\\varphi _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}(\\infty ) = \\infty $ .",
"Moreover, $\\varphi _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}^{\\prime }(-\\infty ) =\\varphi _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}^{\\prime }(a_{F}) =-\\widetilde{c} \\cdot \\beta _{1}$ and $\\varphi _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}^{\\prime }(\\infty ) =\\widetilde{c} \\cdot \\beta _{1}$ .",
"From the generator $\\varphi _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}$ given in (REF ), we construct the corresponding divergence (cf.",
"(REF )) $& &D_{\\varphi _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}}(\\mathbf {Q},\\mathbf {P})= \\sum \\limits _{k=1}^{K} p_{k} \\cdot \\varphi _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}\\Big (\\frac{q_{k}}{p_{k}}\\Big )\\nonumber \\\\& &= \\sum \\limits _{k=1}^{K}p_{k} \\cdot \\bigg [{1}_{]a_{F},\\infty [}(\\frac{q_{k}}{p_{k}}) \\cdot \\widetilde{c} \\cdot \\alpha \\cdot \\Big \\lbrace \\frac{\\sqrt{4 + (\\beta _{1} + \\beta _{2})^{2}\\cdot (\\frac{1-\\frac{q_{k}}{p_{k}}}{\\alpha } + \\frac{1}{\\beta _{2}} - \\frac{1}{\\beta _{1}})^2}\\, - \\, (\\frac{1-\\frac{q_{k}}{p_{k}}}{\\alpha } + \\frac{1}{\\beta _{2}} - \\frac{1}{\\beta _{1}}) \\cdot (\\beta _{1} - \\beta _{2}) \\, - \\, 2}{2}\\nonumber \\\\& &+ \\log \\frac{\\sqrt{4 + (\\beta _{1} + \\beta _{2})^{2}\\cdot (\\frac{1-\\frac{q_{k}}{p_{k}}}{\\alpha } + \\frac{1}{\\beta _{2}} \\, - \\, \\frac{1}{\\beta _{1}})^2} \\, - \\, 2}{\\beta _{1} \\beta _{2} \\cdot (\\frac{1-\\frac{q_{k}}{p_{k}}}{\\alpha } + \\frac{1}{\\beta _{2}} \\, - \\, \\frac{1}{\\beta _{1}})^{2}} \\Big \\rbrace \\nonumber \\\\& &+ \\ {1}_{]-\\infty , a_{F}]}(\\frac{q_{k}}{p_{k}}) \\cdot \\widetilde{c} \\cdot \\Big \\lbrace \\alpha \\cdot \\Big \\lbrace \\frac{3 \\beta _{1} - \\beta _{2}}{2 (\\beta _{2} - \\beta _{1})}+ \\log \\frac{2(\\beta _{2} - \\beta _{1})}{\\beta _{2}} \\Big \\rbrace - \\beta _{1} \\cdot (\\frac{q_{k}}{p_{k}} - a_{F}) \\Big \\rbrace \\bigg ],\\qquad \\mathbf {P} \\in \\mathbb {R}_{\\ge 0}^{K}, \\mathbf {Q} \\in \\mathbb {R}^{K}.$ Notice that we can particularly include the case where $p_{k}=0$ in combination with $q_{k} \\ne 0$ , since $\\lim _{t \\rightarrow 0_{+}} t \\cdot \\varphi _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}(\\frac{1}{t})= \\widetilde{c} \\cdot \\beta _{1}$ and $\\lim _{t \\rightarrow 0_{-}} t \\cdot \\varphi _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}(\\frac{1}{t})= - \\widetilde{c} \\cdot \\beta _{1}$ are both finite, and hence $p_{k} \\cdot \\varphi _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}(\\frac{q_{k}}{p_{k}})= q_{k} \\cdot \\frac{p_{k}}{q_{k}} \\cdot \\varphi _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}(\\frac{q_{k}}{p_{k}}) $ stays finite as $p_{k}$ tends to zero.",
"(b) For any parameter-quadrupel $\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c} \\in \\, ]0,\\infty [$ with $\\beta _{1} > \\beta _{2}$ , one can proceed analogously to (a).",
"Let us start by choosing $]a_{F},b_{F}[\\ \\, := \\ ]a_{F_{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}},b_{F_{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}}[ \\ \\, := \\ \\Big ]-\\infty , \\, 1 + \\alpha \\cdot \\frac{(\\beta _{1} - \\beta _{2})^{2} + \\beta _{1} \\cdot \\beta _{2} + \\beta _{2}^{2}}{2\\beta _{1}\\cdot \\beta _{2}\\cdot (\\beta _{1} - \\beta _{2} )}, \\,\\Big [ \\ \\ni 1\\nonumber $ and defining with the same $\\breve{\\theta } := 1 + \\alpha \\cdot \\Big (\\frac{1}{\\beta _{2}}- \\frac{1}{\\beta _{1}} \\Big ) > 1$ $\\hspace{-17.07182pt}F_{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}(t)&:=& {\\left\\lbrace \\begin{array}{ll}\\widetilde{c} \\cdot \\frac{\\beta _{1}-\\beta _{2}}{2}+ \\frac{\\widetilde{c}}{\\frac{1-t}{\\alpha } + \\frac{1}{\\beta _{2}} - \\frac{1}{\\beta _{1}}}\\cdot \\Big (1 - \\frac{1}{2} \\cdot \\sqrt{4 + \\big (\\frac{1-t}{\\alpha } + \\frac{1}{\\beta _{2}} - \\frac{1}{\\beta _{1}}\\big )^{2} \\cdot (\\beta _{1}+\\beta _{2})^{2}}\\, \\Big ),\\quad \\textrm {if } \\ t \\in \\, ]a_{F},b_{F}[ \\backslash \\lbrace \\breve{\\theta }\\rbrace , \\\\\\widetilde{c} \\cdot \\frac{\\beta _{1}-\\beta _{2}}{2}, \\hspace{277.41437pt}\\textrm {if } \\ t= \\breve{\\theta } \\in \\, ]a_{F},b_{F}[, \\\\\\widetilde{c} \\cdot \\beta _{2}, \\hspace{293.06346pt} \\textrm {if } \\ t=b_{F}, \\\\\\infty , \\hspace{304.4444pt} \\textrm {if } \\ t \\in \\, ]b_{F}, \\infty [.\\end{array}\\right.",
"}$ Clearly, $\\breve{\\theta } \\in \\, ]a_{F},b_{F}[$ if and only if $\\beta _{1} \\in \\,]\\beta _{2}, 3\\beta _{2}[$ ; if (say) the latter holds, then one gets the continuity $\\lim _{t \\rightarrow \\breve{\\theta }}F_{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}(t) = \\widetilde{c} \\cdot \\frac{\\beta _{1}-\\beta _{2}}{2}$ .",
"For $\\beta _{1} \\le 3 \\beta _{2}$ there holds $]a_{F},b_{F}[ \\backslash \\lbrace \\breve{\\theta } \\rbrace = \\, ]a_{F},b_{F}[$ .",
"Returning to the general case, one can show comfortably that $F_{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}(\\cdot )$ is strictly increasing and that $\\mathcal {R}(F_{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}})=]-\\widetilde{c}\\cdot \\beta _{2},\\widetilde{c}\\cdot \\beta _{2} ]$ .",
"Moreover, $F_{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}(\\cdot )$ is smooth on $]a_{F},b_{F}[$ , and hence $F_{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}} \\in {F}$ .",
"In face of the validity of $F_{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}(1)=0$ , let us choose the natural anchor point $c:=0$ , which amounts to $]\\lambda _{-},\\lambda _{+}[ \\,= int(\\mathcal {R}(F_{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}})= \\, ]-\\widetilde{c}\\cdot \\beta _{2},\\widetilde{c}\\cdot \\beta _{2} [$ and $]t_{-}^{sc},t_{+}^{sc}[ = \\, ]a_{F},b_{F}[$ .",
"Since the first line in (REF ) coincides formally with that of (REF ) (with different $]a_{F},b_{F}[$ ), the corresponding inverse is formally the same as (REF ) (with different $]a_{F},b_{F}[$ ), and hence $\\hspace{-19.91684pt}\\Lambda _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}(z) :=\\Lambda _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}^{(0)}(z) &=&{\\left\\lbrace \\begin{array}{ll}\\breve{\\theta } \\cdot z - \\widetilde{c} \\cdot \\alpha \\cdot \\log \\Big (1+ \\frac{z}{\\widetilde{c}} \\cdot \\Big (\\frac{1}{\\beta _{2}} - \\frac{1}{\\beta _{1}} \\Big )- \\frac{z^{2}}{\\widetilde{c}^{2} \\cdot \\beta _{1} \\cdot \\beta _{2} }\\Big ),\\quad \\textrm {if } \\ z \\in \\, ]-\\widetilde{c}\\cdot \\beta _{2},\\widetilde{c}\\cdot \\beta _{2} [,\\\\\\widetilde{c} \\cdot \\breve{\\theta } \\cdot \\beta _{2} -\\widetilde{c} \\cdot \\alpha \\cdot \\log \\Big (2 - 2 \\frac{\\beta _{2}}{\\beta _{1}}\\Big ),\\hspace{73.97733pt} \\textrm {if } \\ z = \\widetilde{c}\\cdot \\beta _{2},\\\\\\infty , \\hspace{202.01474pt} \\textrm {else} .\\end{array}\\right.",
"}$ Notice that $\\Lambda _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}(0) = 0$ and $\\lim _{z \\rightarrow - \\widetilde{c}\\cdot \\beta _{2}}\\Lambda _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}(z) = -\\infty $ .",
"Furthermore, $\\Lambda _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}^{\\prime }( -\\widetilde{c}\\cdot \\beta _{2})= - \\infty = a_{F}$ and $\\Lambda _{ -\\widetilde{c}\\cdot \\beta _{1}}^{\\prime }(\\widetilde{c}\\cdot \\beta _{2})= b_{F}$ .",
"To proceed, from formula (REF ) (see also ()) we can derive — analogously to (REF ) — for all $t \\in \\mathbb {R}$ $\\hspace{-5.69046pt}\\varphi _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}(t):= \\varphi _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}^{(0)}(t)\\hspace{-8.5359pt} &=& \\hspace{-8.5359pt}{\\left\\lbrace \\begin{array}{ll}\\widetilde{c} \\cdot \\alpha \\cdot \\Big \\lbrace \\frac{\\sqrt{4 + (\\beta _{1} + \\beta _{2})^{2}\\cdot (\\frac{1-t}{\\alpha } + \\frac{1}{\\beta _{2}} - \\frac{1}{\\beta _{1}})^2}\\, - \\, (\\frac{1-t}{\\alpha } + \\frac{1}{\\beta _{2}} - \\frac{1}{\\beta _{1}}) \\cdot (\\beta _{1} - \\beta _{2}) \\, - \\, 2}{2} \\\\+ \\log \\frac{\\sqrt{4 + (\\beta _{1} + \\beta _{2})^{2}\\cdot (\\frac{1-t}{\\alpha } + \\frac{1}{\\beta _{2}} \\, - \\, \\frac{1}{\\beta _{1}})^2} \\, - \\, 2}{\\beta _{1} \\beta _{2} \\cdot (\\frac{1-t}{\\alpha } + \\frac{1}{\\beta _{2}} \\, - \\, \\frac{1}{\\beta _{1}})^{2}} \\Big \\rbrace \\ \\in [0,\\infty [,\\hspace{71.13188pt} \\textrm {if }t \\in \\, ]-\\infty , b_{F}[,\\\\\\widetilde{c} \\cdot \\alpha \\cdot \\Big \\lbrace \\frac{3 \\beta _{2} - \\beta _{1}}{2 (\\beta _{1} - \\beta _{2})}+ \\log \\frac{2(\\beta _{1} - \\beta _{2})}{\\beta _{1}} \\Big \\rbrace + \\widetilde{c} \\cdot \\beta _{2} \\cdot (t - b_{F}) \\ \\in \\, ]0,\\infty [,\\hspace{19.91684pt} \\textrm {if } \\ t \\in [b_{F}, \\infty [,\\end{array}\\right.",
"}$ where the last line in (REF ) follows by calculating $\\varphi _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}(b_{F})=\\widetilde{c} \\cdot \\alpha \\cdot \\Big \\lbrace \\frac{3 \\beta _{2} - \\beta _{1}}{2 (\\beta _{1} - \\beta _{2})}+ \\log \\frac{2(\\beta _{1} - \\beta _{2})}{\\beta _{1}} \\Big \\rbrace $ .",
"Notice that $\\varphi _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}(1) = 0$ , $\\varphi _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}^{\\prime }(1) = 0$ , $\\varphi _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}(-\\infty ) = \\infty $ and $\\varphi _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}(\\infty ) = \\infty $ .",
"Furthermore, $\\varphi _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}^{\\prime }(-\\infty ) =-\\widetilde{c} \\cdot \\beta _{2}$ and $\\varphi _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}^{\\prime }(\\infty ) =\\varphi _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}^{\\prime }(b_{F}) =\\widetilde{c} \\cdot \\beta _{2}$ .",
"From the generator $\\varphi _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}$ given in (REF ), we construct the corresponding divergence (cf.",
"(REF )) $& &D_{\\varphi _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}}(\\mathbf {Q},\\mathbf {P})= \\sum \\limits _{k=1}^{K} p_{k} \\cdot \\varphi _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}\\Big (\\frac{q_{k}}{p_{k}}\\Big )\\nonumber \\\\& &= \\sum \\limits _{k=1}^{K}p_{k} \\cdot \\bigg [{1}_{]-\\infty , b_{F}[}(\\frac{q_{k}}{p_{k}}) \\cdot \\widetilde{c} \\cdot \\alpha \\cdot \\Big \\lbrace \\frac{\\sqrt{4 + (\\beta _{1} + \\beta _{2})^{2}\\cdot (\\frac{1-\\frac{q_{k}}{p_{k}}}{\\alpha } + \\frac{1}{\\beta _{2}} - \\frac{1}{\\beta _{1}})^2}\\, - \\, (\\frac{1-\\frac{q_{k}}{p_{k}}}{\\alpha } + \\frac{1}{\\beta _{2}} - \\frac{1}{\\beta _{1}}) \\cdot (\\beta _{1} - \\beta _{2}) \\, - \\, 2}{2}\\nonumber \\\\& &+ \\log \\frac{\\sqrt{4 + (\\beta _{1} + \\beta _{2})^{2}\\cdot (\\frac{1-\\frac{q_{k}}{p_{k}}}{\\alpha } + \\frac{1}{\\beta _{2}} \\, - \\, \\frac{1}{\\beta _{1}})^2} \\, - \\, 2}{\\beta _{1} \\beta _{2} \\cdot (\\frac{1-\\frac{q_{k}}{p_{k}}}{\\alpha } + \\frac{1}{\\beta _{2}} \\, - \\, \\frac{1}{\\beta _{1}})^{2}} \\Big \\rbrace \\nonumber \\\\& &+ \\ {1}_{[b_{F}, \\infty [}(\\frac{q_{k}}{p_{k}}) \\cdot \\widetilde{c} \\cdot \\Big \\lbrace \\alpha \\cdot \\Big \\lbrace \\frac{3 \\beta _{2} - \\beta _{1}}{2 (\\beta _{1} - \\beta _{2})}+ \\log \\frac{2(\\beta _{1} - \\beta _{2})}{\\beta _{1}} \\Big \\rbrace + \\beta _{2} \\cdot (\\frac{q_{k}}{p_{k}} - b_{F}) \\Big \\rbrace \\bigg ],\\qquad \\mathbf {P} \\in \\mathbb {R}_{\\ge 0}^{K}, \\mathbf {Q} \\in \\mathbb {R}^{K}.$ As above, we can particularly include the case where $p_{k}=0$ in combination with $q_{k} \\ne 0$ , since $\\lim _{t \\rightarrow 0_{+}} t \\cdot \\varphi _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}(\\frac{1}{t})= \\widetilde{c} \\cdot \\beta _{2}$ and $\\lim _{t \\rightarrow 0_{-}} t \\cdot \\varphi _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}(\\frac{1}{t})= - \\widetilde{c} \\cdot \\beta _{2}$ are both finite.",
"(c) The analysis for the case $\\beta _{1} = \\beta _{2} =: \\beta $ can be obtained by taking $\\lim _{\\beta _1 \\rightarrow \\beta _{2}}$ in (a) respectively (b).",
"Alternatively, one can start afresh.",
"Due to its importance and its particularities, we nevertheless state the corresponding results explicitly.",
"To begin with, for any parameter-triple $\\alpha ,\\beta ,\\widetilde{c} \\in \\, ]0,\\infty [$ we choose $]a_{F},b_{F}[\\ \\, := \\ ]a_{F_{\\alpha ,\\beta ,\\widetilde{c}}},b_{F_{\\alpha ,\\beta ,\\widetilde{c}}}[ \\ \\, := \\ ]-\\infty , \\infty \\, [\\nonumber $ and define with $\\breve{\\theta } := 1$ $\\hspace{-17.07182pt}F_{\\alpha ,\\beta ,\\widetilde{c}}(t)&:=& {\\left\\lbrace \\begin{array}{ll}\\frac{\\widetilde{c} \\cdot \\alpha }{1-t}\\cdot \\Big (1 - \\sqrt{1 + \\big (\\frac{1-t}{\\alpha } \\big )^{2} \\cdot \\beta ^{2}}\\, \\Big ),\\qquad \\textrm {if } \\ t \\in \\, ]a_{F},b_{F}[ \\backslash \\lbrace \\breve{\\theta }\\rbrace , \\\\0, \\hspace{145.10922pt}\\textrm {if } \\ t= \\breve{\\theta }.\\end{array}\\right.",
"}$ Clearly, one has the continuity $\\lim _{t \\rightarrow \\breve{\\theta }}F_{\\alpha ,\\beta ,\\widetilde{c}}(t) = 0$ .",
"Moreover, one can see in a straightforward way that $F_{\\alpha ,\\beta ,\\widetilde{c}}(\\cdot )$ is strictly increasing and that $\\mathcal {R}(F_{\\alpha ,\\beta ,\\widetilde{c}})=]-\\widetilde{c}\\cdot \\beta ,\\widetilde{c}\\cdot \\beta [$ .",
"Furthermore, $F_{\\alpha ,\\beta ,\\widetilde{c}}(\\cdot )$ is smooth on $]a_{F},b_{F}[$ , and thus $F_{\\alpha ,\\beta ,\\widetilde{c}} \\in {F}$ .",
"Since $F_{\\alpha ,\\beta ,\\widetilde{c}}(1)=0$ , let us choose the natural anchor point $c:=0$ , which leads to the choice $]\\lambda _{-},\\lambda _{+}[ \\,= int(\\mathcal {R}(F_{\\alpha ,\\beta ,\\widetilde{c}})= \\, ]-\\widetilde{c}\\cdot \\beta ,\\widetilde{c}\\cdot \\beta [$ and $]t_{-}^{sc},t_{+}^{sc}[ = \\, ]a_{F},b_{F}[ = \\, ]-\\infty ,\\infty [$ .",
"The inverse in (REF ) collapses to $F_{\\alpha ,\\beta ,\\widetilde{c}}^{-1}(x) =1 + \\alpha \\cdot \\frac{\\frac{2 x}{\\widetilde{c} \\cdot \\beta ^{2} }}{1 - \\frac{x^{2}}{\\widetilde{c}^{2} \\cdot \\beta ^{2} }},\\qquad x \\in int(\\mathcal {R}(F_{\\alpha ,\\beta ,\\widetilde{c}})) ;$ from this, we can derive from formula (REF ) (see also (REF )) for all $z \\in \\mathbb {R}$ $\\hspace{-19.91684pt}\\Lambda _{\\alpha ,\\beta ,\\widetilde{c}}(z) :=\\Lambda _{\\alpha ,\\beta ,\\widetilde{c}}^{(0)}(z) &=&{\\left\\lbrace \\begin{array}{ll}\\breve{\\theta } \\cdot z - \\widetilde{c} \\cdot \\alpha \\cdot \\log \\Big (1 - \\frac{z^{2}}{\\widetilde{c}^{2} \\cdot \\beta ^{2} }\\Big ),\\qquad \\textrm {if } \\ z \\in \\, ]-\\widetilde{c}\\cdot \\beta ,\\widetilde{c}\\cdot \\beta [,\\\\\\infty , \\hspace{130.88284pt} \\textrm {else} .\\end{array}\\right.",
"}$ Notice that $\\Lambda _{\\alpha ,\\beta ,\\widetilde{c}}(0) = 0$ , $\\lim _{z \\rightarrow - \\widetilde{c}\\cdot \\beta }\\Lambda _{\\alpha ,\\beta ,\\widetilde{c}}(z) = -\\infty = $ , and $\\lim _{z \\rightarrow \\widetilde{c}\\cdot \\beta }\\Lambda _{\\alpha ,\\beta ,\\widetilde{c}}(z) = \\infty $ .",
"Furthermore, $\\lim _{z \\rightarrow - \\widetilde{c}\\cdot \\beta }\\Lambda _{\\alpha ,\\beta ,\\widetilde{c}}^{\\prime }(z) = -\\infty = a_{F}$ , and $\\lim _{z \\rightarrow \\widetilde{c}\\cdot \\beta }\\Lambda _{\\alpha ,\\beta ,\\widetilde{c}}^{\\prime }(z) = \\infty = b_{F}$ .",
"To proceed, the formula (REF ) (respectively, (REF )) collapses to $\\varphi _{\\alpha ,\\beta ,\\widetilde{c}}(t):=\\varphi _{\\alpha ,\\beta ,\\widetilde{c}}^{(0)}(t)= \\widetilde{c} \\cdot \\alpha \\cdot \\Big \\lbrace \\sqrt{1 + \\beta ^{2}\\cdot \\Big (\\frac{1-t}{\\alpha }\\Big )^2} \\, - \\, 1+ \\log \\frac{2 \\cdot \\Big (\\sqrt{1 + \\beta ^{2}\\cdot \\Big (\\frac{1-t}{\\alpha } \\Big )^2} \\, - \\, 1\\Big )}{\\beta ^{2} \\cdot \\Big (\\frac{1-t}{\\alpha }\\Big )^{2}} \\Big \\rbrace \\ \\in [0,\\infty [, \\ \\ t \\in \\, ]-\\infty , \\infty [ \\, = \\, ]a_{F}, b_{F}[.$ Notice that $\\varphi _{\\alpha ,\\beta ,\\widetilde{c}}(1) = 0$ , $\\varphi _{\\alpha ,\\beta }^{\\prime }(1) = 0$ , $\\varphi _{\\alpha ,\\beta ,\\widetilde{c}}(-\\infty ) = \\infty $ and $\\varphi _{\\alpha ,\\beta ,\\widetilde{c}}(\\infty ) = \\infty $ .",
"Moreover, $\\varphi _{\\alpha ,\\beta ,\\widetilde{c}}^{\\prime }(-\\infty ) =\\varphi _{\\alpha ,\\beta ,\\widetilde{c}}^{\\prime }(a_{F}) =-\\widetilde{c} \\cdot \\beta $ and $\\varphi _{\\alpha ,\\beta ,\\widetilde{c}}^{\\prime }(\\infty ) =\\varphi _{\\alpha ,\\beta ,\\widetilde{c}}^{\\prime }(b_{F}) =\\widetilde{c} \\cdot \\beta $ .",
"From the generator $\\varphi _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}$ given in (REF ), we construct the corresponding divergence (cf.",
"(REF )) $& & \\hspace{-19.91684pt}D_{\\varphi _{\\alpha ,\\beta ,\\widetilde{c}}}(\\mathbf {Q},\\mathbf {P})= \\sum \\limits _{k=1}^{K} p_{k} \\cdot \\varphi _{\\alpha ,\\beta ,\\widetilde{c}}\\Big (\\frac{q_{k}}{p_{k}}\\Big )\\nonumber \\\\& & \\hspace{-19.91684pt}= \\widetilde{c} \\cdot \\alpha \\cdot \\sum \\limits _{k=1}^{K}p_{k} \\cdot \\Big \\lbrace \\sqrt{1 + \\beta ^{2}\\cdot \\Big (\\frac{1-\\frac{q_{k}}{p_{k}}}{\\alpha }\\Big )^2} \\, - \\, 1+ \\log \\frac{2 \\cdot \\Big (\\sqrt{1 + \\beta ^{2}\\cdot \\Big (\\frac{1-\\frac{q_{k}}{p_{k}}}{\\alpha } \\Big )^2} \\, - \\, 1\\Big )}{\\beta ^{2} \\cdot \\Big (\\frac{1-\\frac{q_{k}}{p_{k}}}{\\alpha }\\Big )^{2}} \\Big \\rbrace ,\\qquad \\mathbf {P} \\in \\mathbb {R}_{\\ge 0}^{K}, \\mathbf {Q} \\in \\mathbb {R}^{K}.$ As above, we can particularly include the case where $p_{k}=0$ in combination with $q_{k} \\ne 0$ , since $\\lim _{t \\rightarrow 0_{+}} t \\cdot \\varphi _{\\alpha ,\\beta ,\\widetilde{c}}(\\frac{1}{t})= \\widetilde{c} \\cdot \\beta $ and $\\lim _{t \\rightarrow 0_{-}} t \\cdot \\varphi _{\\alpha ,\\beta ,\\widetilde{c}}(\\frac{1}{t})= - \\widetilde{c} \\cdot \\beta $ are both finite.",
"This ends the current Example REF .",
"As a side effect in the above-mentioned Example REF , for fixed $\\beta _{2}, \\alpha ,\\widetilde{c}$ notice the interesting behaviour (e.g.",
"with respect to $int(dom(F)) = ]a_{F},b_{F}[$ and the range of $\\varphi ^{\\prime }$ ) as $\\beta _{1}$ moves from $]0,\\beta _{2}[$ to $\\beta _{2}$ and further to $]\\beta _{2}, \\infty [$ .",
"Remark 47 The characterization of the probability distribution $$ in (REF ) which may result from Theorem REF — as seen through the above examples — considerably improves other approaches which make use of their identification through the concept of power variance functions of Natural Exponential Families, as developed by Tweedie [369], Morris [267], Letac & Mora [214], and others.",
"This approach has been used in Broniatowski [58] in a similar perspective as developed here, but can not be extended outside the range of power divergences, in contrast with the Examples REF , REF , REF and REF which can only be handled as a consequence of Theorem REF .",
"To continue with our general procedure, suppose now that for a divergence generator $\\varphi $ of interest we have concretely/explicitly found (e.g.",
"by direct calculations or via our $F-$ connection in Theorem REF , see also Remark REF ) its Fenchel-Legendre transform $\\Lambda = \\varphi ^{*}$ ; for this “candidate”, in order to achieve the desired representability (REF ) it remains to verify that $\\exp (\\Lambda (z)) = \\int _{\\mathbb {R}}e^{z \\cdot y} \\, d\\mathbb {} (y),\\qquad z \\in \\mathbb {R},$ for some probability distribution/measure $\\mathbb {} $ on the real line (the light-tailedness in the sense of finiteness on some open interval containing zero, will be typically guaranteed automatically by the assumptions on $\\varphi $ ); of course, this is equivalent to “the existence ” of a random variable $W$ whose moment generating function is equal to $\\exp (\\Lambda )$ (and thus, its cumulant generating function (log moment generating function) is $\\Lambda $ ), i.e.",
"$\\exp (\\Lambda (z)) = E_{\\mathbb {\\Pi }}[\\exp (z \\cdot W)]\\qquad z \\in \\mathbb {R},$ with $\\mathbb {\\Pi }[W \\in \\cdot \\, ] = \\mathbb {}[ \\, \\cdot \\,]$ ); recall that from this, we need to simulate a sequence $(W_{i})_{i\\in \\mathbb {N}}$ of i.i.d.",
"copies of $W$ which are the crucial building ingredients of $\\xi _{n}^{\\mathbf {W}}$ in Theorem REF , respectively, of $\\xi _{n,\\mathbf {X}}^{w\\mathbf {W}}$ in Theorem REF .",
"For the above-mentioned Examples REF to REF , we can give explicit solutions to the representabilities (REF ) respectively (REF ); this is achieved in the following Examples REF to REF (notice that the corresponding supports of $\\mathbb {}$ are explicitly mentioned in the summarizing Table 1 above): Example 48 for the power-divergence context of Example REF we obtain: (a) Case $\\gamma =0$ , $\\widetilde{c} >0$ : $\\Lambda _{0,\\widetilde{c}}(z)= - \\widetilde{c} \\cdot \\log \\left( 1 - \\frac{z}{\\widetilde{c}} \\right)$ (cf.",
"(REF )) is the cumulant generating function of the Gamma distribution $\\mathbb {} = GAM(\\widetilde{c},\\widetilde{c})$ with rate parameter (inverse scale parameter) $\\widetilde{c}$ and shape parameter $\\widetilde{c}$ ; hence, $\\varphi _{0,\\widetilde{c}} \\in \\Upsilon (]0,\\infty [)$ .",
"Prominent special case $\\widetilde{c} =1$ : $\\mathbb {} = GAM(1,1) = EXP(1)$ is the exponential distribution with mean 1.",
"Type: $\\mathbb {}$ is an infinitely divisible (cf.",
"Proposition REF ) continuous distribution with density $f(y) := \\frac{\\widetilde{c}^{\\widetilde{c}} \\cdot y^{\\widetilde{c}-1} \\cdot e^{-\\widetilde{c} \\cdot y} }{\\Gamma (\\widetilde{c})} \\cdot {1}_{]0,\\infty [}(y)$ ($y \\in \\mathbb {R}$ ).",
"Behaviour at zero: $\\mathbb {}[ \\, ]0,\\infty [ \\, ] = \\mathbb {\\Pi }[W>0]=1$ .",
"Corresponding generator: $\\varphi _{0,\\widetilde{c}} = \\widetilde{c} \\cdot \\varphi _{0}$ (cf.",
"(REF ), (REF )) of the $\\widetilde{c}-$ fold of the reversed Kullback-Leibler divergence (reversed relative entropy) given in the second line of (REF ).",
"Sums: for i.i.d.",
"copies $(W_{i})_{i \\in \\mathbb {N}}$ of $W$ , the probability distribution of $\\breve{W} := \\sum _{i\\in I_{k}^{(n)}} W_{i}$ (cf.",
"Remark REF (ii)) is $GAM(\\widetilde{c},\\widetilde{c} \\cdot card(I_{k}^{(n)}))$ .",
"(b) Case $\\gamma \\in \\, ]0,1[$ , $\\widetilde{c} >0$ : $\\Lambda _{\\gamma ,\\widetilde{c}}^{(0)}(z)= \\frac{\\widetilde{c}}{\\gamma } \\cdot \\Big \\lbrace \\left( \\frac{\\gamma -1}{\\widetilde{c}} \\cdot z + 1 \\right)^{\\frac{\\gamma }{\\gamma -1}} -1 \\Big \\rbrace $ (cf.",
"(REF )) is the cumulant generating function of the Compound-Poisson-Gamma distribution $\\mathbb {} = C(POI(\\theta ),GAM(\\alpha ,\\beta ))$ with $\\theta = \\frac{\\widetilde{c}}{\\gamma } > 0$ , rate parameter (inverse scale parameter) $\\alpha = \\frac{\\widetilde{c}}{1-\\gamma } >0$ , and shape parameter $\\beta = \\frac{\\gamma }{1-\\gamma } >0$ .",
"In other words, $W$ has the comfortably simulable form $W = \\sum _{i=1}^{N} \\widetilde{W}_{i}$ with the usual convention $\\sum _{i=1}^{0} \\widetilde{W}_{i} := 0$ for some i.i.d.",
"sequence $( \\widetilde{W}_{i})_{i\\in \\mathbb {N}}$ of Gamma $GAM(\\alpha ,\\beta )$ distributed random variables (with parameter-pair $(\\alpha ,\\beta )$ ) and some independent $POI(\\theta )-$ distributed random variable $N$ .",
"Hence, $\\varphi _{\\gamma ,\\widetilde{c}} \\in \\Upsilon (]0,\\infty [)$ .",
"Type: $\\mathbb {}$ is an infinitely divisible distribution (cf.",
"Proposition REF ), mixture of a one-point distribution at zero and a continuous distribution on $[0,\\infty [$ , with $\\mathbb {}[\\lbrace 0\\rbrace ] = \\mathbb {\\Pi }[W = 0]= e^{-\\theta }$ and $\\mathbb {}[B] = \\mathbb {\\Pi }[W \\in B] = \\int _{B} f_{\\widetilde{c},\\gamma }(u) \\, du$ for every (measurable) subset of $]0,\\infty [$ having density $& & \\hspace{-22.76228pt}f_{C(POI(\\theta ),GAM(\\alpha ,\\beta ))}(y): = \\frac{\\exp \\left(- \\alpha \\cdot y - \\theta \\right)}{y}\\cdot \\sum _{k=1}^{\\infty } \\frac{\\theta ^{k} \\cdot (\\alpha y)^{k\\beta }}{k!",
"\\cdot \\Gamma (k\\beta )}\\cdot {1}_{]0,\\infty [}(y)\\\\& & \\hspace{-22.76228pt}= \\frac{1}{y} \\cdot \\exp \\left(-\\widetilde{c} \\cdot \\left(\\frac{y}{1-\\gamma } + \\frac{1}{\\gamma } \\right) \\right)\\cdot \\sum _{k=1}^{\\infty } \\frac{a_{k}}{k!}",
"\\cdot \\widetilde{c}^{k/(1-\\gamma )} \\cdot \\gamma ^{-k}\\cdot (1-\\gamma )^{-k\\gamma /(1-\\gamma )} \\cdot y^{k\\gamma /(1-\\gamma )}\\cdot {1}_{[0,\\infty [}(y)=: f_{\\widetilde{c},\\gamma }(y),\\quad y \\in \\mathbb {R},\\nonumber $ where $a_{k} := 1/\\Gamma (\\frac{k \\cdot \\gamma }{1-\\gamma } )$ (see e.g.",
"Aalen [1] with a different parametrization).",
"Behaviour at zero: $\\mathbb {}[ \\, [0,\\infty [ \\, ] = \\mathbb {\\Pi }[W \\ge 0]=1$ , $\\mathbb {}[ \\, \\lbrace 0\\rbrace \\, ] = \\mathbb {\\Pi }[W = 0]= e^{-\\theta }$ .",
"Corresponding generator: $\\varphi _{\\gamma ,\\widetilde{c}}^{(0)} = \\widetilde{c} \\cdot \\varphi _{\\gamma }$ (cf.",
"(REF ), (REF )) of the power divergence given in the third line of (REF ); recall that the special case $\\gamma =\\frac{1}{2}$ corresponds to the prominent (multiple of the squared) Hellinger distance.",
"Sums: for i.i.d.",
"copies $(W_{i})_{i \\in \\mathbb {N}}$ of $W$ , the probability distribution of $\\breve{W} := \\sum _{i\\in I_{k}^{(n)}} W_{i}$ (cf.",
"Remark REF (ii)) is $C(POI(\\breve{\\theta }),GAM(\\alpha ,\\beta ))$ with $\\breve{\\theta } = \\frac{\\widetilde{c} \\cdot card(I_{k}^{(n)})}{\\gamma } > 0$ , $\\alpha = \\frac{\\widetilde{c}}{1-\\gamma } >0$ , $\\beta = \\frac{\\gamma }{1-\\gamma } >0$ .",
"(c) Case $\\gamma =2$ , $\\widetilde{c} >0$ : $\\Lambda _{2,\\widetilde{c}}^{(0)}(z)= \\frac{z^{2}}{2 \\widetilde{c}} + z$ (cf.",
"(REF ) ) is the well-known cumulant generating function of the Normal distribution (Gaussian distribution) $\\mathbb {} = N(1,\\frac{1}{\\widetilde{c}})$ with mean 1 and variance $\\frac{1}{\\widetilde{c}}$ .",
"Thus, $\\varphi _{2,\\widetilde{c}} \\in \\Upsilon (]-\\infty ,\\infty [)$ .",
"Type: $\\mathbb {}$ is an infinitely divisible (cf.",
"Proposition REF ) continuous distribution with density $f_{N(1,\\frac{1}{\\widetilde{c}})}(y) := \\sqrt{\\frac{\\widetilde{c}}{2 \\pi }} \\cdot \\exp (- \\frac{\\widetilde{c} \\cdot (y-1)^2}{2} )$ , ($y \\in \\mathbb {R}$ ).",
"Behaviour at zero: $\\mathbb {}[ \\, ]0,\\infty [ \\, ] = \\mathbb {\\Pi }[W > 0]=\\int _{0}^{\\infty } f_{N(1,\\frac{1}{\\widetilde{c}})}(u) \\, du \\in \\, ]0,1[$ , $\\mathbb {}[ \\, \\lbrace 0\\rbrace \\, ] = \\mathbb {\\Pi }[W = 0]= 0$ .",
"Corresponding generator: $\\varphi _{2,\\widetilde{c}}^{(0)} = \\widetilde{c} \\cdot \\varphi _{2}$ (cf.",
"(REF ), (REF )) is the generator of the $\\widetilde{c}-$ fold of the half Pearson-chisquare divergence given in the sixth line of (REF ).",
"Sums: for i.i.d.",
"copies $(W_{i})_{i \\in \\mathbb {N}}$ of $W$ , the probability distribution of $\\breve{W} := \\sum _{i\\in I_{k}^{(n)}} W_{i}$ (cf.",
"Remark REF (ii)) is $N(card(I_{k}^{(n)}),\\frac{card(I_{k}^{(n)})}{\\widetilde{c}})$ .",
"(d) Case $\\gamma <0$ , $\\widetilde{c} >0$ : $\\Lambda _{\\gamma ,\\widetilde{c}}^{(0)}(z) = \\frac{\\widetilde{c}}{\\gamma } \\cdot \\left\\lbrace \\left( \\frac{\\gamma -1}{\\widetilde{c}} \\cdot z +1 \\right)^{\\frac{\\gamma }{\\gamma -1}} -1 \\right\\rbrace $ (cf.",
"(REF )) is the cumulant generating function of a “tilted (i.e.",
"negatively distorted) stable distribution” $\\mathbb {}[ \\, \\cdot \\,] = \\mathbb {\\Pi }[W \\in \\cdot \\, ]$ of a random variable $W$ , which can be constructed as follows: let $Z$ be an auxiliary random variable (having density $f_{Z}$ and support $supp(Z) = [0,\\infty [$ ) of a stable law with parameter-quadruple $(\\frac{-\\gamma }{1-\\gamma },1,0,-\\frac{\\widetilde{c}^{1/(1-\\gamma )} \\cdot (1-\\gamma )^{-\\gamma /(1-\\gamma )}}{\\gamma })$ in terms of the “form-B notation” on p.12 in Zolotarev [428]; by applying a general Laplace-transform result on p.112 of the same text we can deduce $M_{Z}(z) := E_{\\mathbb {\\Pi }}[\\exp (z \\cdot Z)]= \\int _{0}^{\\infty } \\exp (z \\cdot y) \\cdot f_{Z}(y) \\, dy \\hspace{-5.69046pt} &=& \\hspace{-5.69046pt}{\\left\\lbrace \\begin{array}{ll}\\exp \\Big (\\frac{\\widetilde{c}^{1/(1-\\gamma )} \\cdot (1-\\gamma )^{-\\gamma /(1-\\gamma )}}{\\gamma }\\cdot (-z)^{\\alpha } \\Big ),\\quad \\textrm {if } \\ z \\in ]-\\infty ,0] , \\\\\\infty , \\hspace{150.79968pt} \\textrm {if } \\ z \\in \\, ]0,\\infty [, \\\\\\end{array}\\right.",
"}$ where $\\alpha := - \\frac{\\gamma }{1-\\gamma } \\in \\, ]0,1[$ .",
"Since $0 \\notin int(dom(M_{Z}))$ (and thus, $Z$ does not have light-tails) we have to tilt (dampen) the density in order to extend the effective domain.",
"Accordingly, let $W$ be a random variable having density $f_{W}(y)\\ :=\\ \\frac{\\exp \\lbrace -\\frac{y \\cdot \\widetilde{c}}{1-\\gamma }\\rbrace }{\\exp \\lbrace \\widetilde{c}/\\gamma \\rbrace }\\cdot f_{Z}(y)\\cdot {1}_{]0,\\infty [}(y),\\qquad y \\in \\mathbb {R},\\qquad \\text{(cf.",
"(\\ref {brostu3:fo.norweiemp7a}))}.\\nonumber $ Then one can straightforwardly deduce from (REF ) that $\\int _{0}^{\\infty } f_{W}(y) \\, dy =1$ and that $M_{W}(z) := E_{\\mathbb {\\Pi }}[\\exp (z \\cdot W)]= \\int _{0}^{\\infty } \\exp (z \\cdot y) \\cdot f_{W}(y) \\, dy \\hspace{-5.69046pt} &=& \\hspace{-5.69046pt}{\\left\\lbrace \\begin{array}{ll}\\exp \\left(\\frac{\\widetilde{c}}{\\gamma } \\cdot \\left\\lbrace \\left( \\frac{\\gamma -1}{\\widetilde{c}} \\cdot z +1 \\right)^{\\frac{\\gamma }{\\gamma -1}} -1 \\right\\rbrace \\right),\\qquad \\textrm {if } \\ z \\in ]-\\infty ,\\frac{\\widetilde{c}}{1-\\gamma }] , \\\\\\infty , \\hspace{156.49014pt} \\textrm {if } \\ z \\in \\, ]\\frac{\\widetilde{c}}{1-\\gamma },\\infty [ .",
"\\\\\\end{array}\\right.",
"}\\nonumber $ Hence, $\\varphi _{\\gamma ,\\widetilde{c}} \\in \\Upsilon (]0,\\infty [)$ .",
"Type: $\\mathbb {}$ is an infinitely divisible (cf.",
"Proposition REF ) continuous distribution with density $f_{W}$ .",
"Behaviour at zero: $\\mathbb {}[ \\, ] 0,\\infty [ \\, ] = \\mathbb {\\Pi }[W > 0]=1$ .",
"Corresponding generator: $\\varphi _{\\gamma ,\\widetilde{c}}^{(0)} = \\widetilde{c} \\cdot \\varphi _{\\gamma }$ (cf.",
"(REF ), (REF )) of the power divergence given in the first line of (REF ).",
"Sums: for i.i.d.",
"copies $(W_{i})_{i \\in \\mathbb {N}}$ of $W$ , the probability distribution of $\\breve{W} := \\sum _{i\\in I_{k}^{(n)}} W_{i}$ (cf.",
"Remark REF (ii)) has density $f_{\\breve{W}}(y)\\ :=\\ \\frac{\\exp \\lbrace -\\frac{y \\cdot \\widetilde{c}}{1-\\gamma }\\rbrace }{\\exp \\lbrace \\widetilde{c} \\cdot card(I_{k}^{(n)})/\\gamma \\rbrace }\\cdot f_{\\breve{Z}}(y)\\cdot {1}_{]0,\\infty [}(y),\\qquad y \\in \\mathbb {R},$ where $\\breve{Z}$ is a random variable with density $f_{\\breve{Z}}$ of a stable law with parameter-quadruple $(\\frac{-\\gamma }{1-\\gamma },1,0,-card(I_{k}^{(n)}) \\cdot \\frac{\\widetilde{c}^{1/(1-\\gamma )} \\cdot (1-\\gamma )^{-\\gamma /(1-\\gamma )}}{\\gamma })$ .",
"(e) Case $\\gamma >2$ , $\\widetilde{c} >0$ : $\\Lambda _{\\gamma ,\\widetilde{c}}^{(0)}(z) = \\frac{\\widetilde{c}}{\\gamma } \\cdot \\left\\lbrace \\left( \\frac{\\gamma -1}{\\widetilde{c}} \\cdot z +1 \\right)^{\\frac{\\gamma }{\\gamma -1}} -1 \\right\\rbrace $ (cf.",
"(REF )) is the cumulant generating function of a “distorted stable distribution” $\\mathbb {}[ \\, \\cdot \\,] = \\mathbb {\\Pi }[W \\in \\cdot \\, ]$ of a random variable $W$ , which can be constructed as follows: let $Z$ be an auxiliary random variable (having density $f_{Z}$ and support $supp(Z) = ]-\\infty ,\\infty ]$ ) of a stable law with parameter-quadruple $(\\frac{\\gamma }{\\gamma -1},1,0,\\frac{\\widetilde{c}^{1/(1-\\gamma )} \\cdot (\\gamma -1)^{\\gamma /(\\gamma -1)}}{\\gamma })$ in terms of the above-mentioned “form-B notation” ; by applying a general Laplace-transform result on p. 112 of Zolotarev [428], we can derive $M_{Z}(z) := E_{\\mathbb {\\Pi }}[\\exp (z \\cdot Z)]= \\int _{0}^{\\infty } \\exp (z \\cdot y) \\cdot f_{Z}(y) \\, dy \\hspace{-5.69046pt} &=& \\hspace{-5.69046pt}{\\left\\lbrace \\begin{array}{ll}\\exp \\Big (\\frac{\\widetilde{c}^{1/(1-\\gamma )} \\cdot (\\gamma -1 )^{\\gamma /(\\gamma -1)}}{\\gamma }\\cdot (-z)^{\\alpha } \\Big ),\\quad \\textrm {if } \\ z \\in ]-\\infty ,0] , \\\\\\infty , \\hspace{145.10922pt} \\textrm {if } \\ z \\in \\, ]0,\\infty [, \\\\\\end{array}\\right.",
"}$ where $\\alpha := \\frac{\\gamma }{\\gamma -1} \\in \\, ]1,2[$ .",
"Since $0 \\notin int(dom(M_{Z}))$ (and thus, $Z$ does not have light-tails) we have to distort the density in order to extend the effective domain.",
"Accordingly, let $W$ be a random variable having density $f_{W}(y)\\ :=\\ \\frac{\\exp \\lbrace \\frac{y \\cdot \\widetilde{c}}{\\gamma -1}\\rbrace }{\\exp \\lbrace \\widetilde{c}/\\gamma \\rbrace }\\cdot f_{Z}(- y), \\qquad y \\in \\mathbb {R},\\qquad \\text{(cf.",
"(\\ref {brostu3:fo.norweiemp7atwo}))}.\\nonumber $ Then one can straightforwardly deduce from (REF ) that $\\int _{-\\infty }^{\\infty } f_{W}(y) \\, dy =1$ and that $M_{W}(z) := E_{\\mathbb {\\Pi }}[\\exp (z \\cdot W)]= \\int _{-\\infty }^{\\infty } \\exp (z \\cdot y) \\cdot f_{W}(y) \\, dy \\hspace{-5.69046pt} &=& \\hspace{-5.69046pt}{\\left\\lbrace \\begin{array}{ll}\\exp \\left(\\frac{\\widetilde{c}}{\\gamma } \\cdot \\left\\lbrace \\left( \\frac{\\gamma -1}{\\widetilde{c}} \\cdot z +1 \\right)^{\\frac{\\gamma }{\\gamma -1}} -1 \\right\\rbrace \\right),\\qquad \\textrm {if } \\ z \\in [- \\frac{\\widetilde{c}}{\\gamma -1},\\infty [ , \\\\\\infty , \\hspace{156.49014pt} \\textrm {if } \\ z \\in \\, ]-\\infty , -\\frac{\\widetilde{c}}{\\gamma -1}[ .",
"\\\\\\end{array}\\right.",
"}\\nonumber $ Thus, $\\varphi _{\\gamma ,\\widetilde{c}} \\in \\Upsilon (]-\\infty ,\\infty [)$ .",
"Type: $\\mathbb {}$ is an infinitely divisible (cf.",
"Proposition REF ) continuous distribution with density $f_{W}$ .",
"Behaviour at zero: $\\mathbb {}[ \\, ]0,\\infty [ \\, ] = \\mathbb {\\Pi }[W > 0]=\\int _{0}^{\\infty } f_{W}(u) \\, du \\in \\, ]0,1[$ , $\\mathbb {}[ \\, \\lbrace 0\\rbrace \\, ] = \\mathbb {\\Pi }[W = 0]= 0$ .",
"Corresponding generator: $\\varphi _{\\gamma ,\\widetilde{c}}^{(0)} = \\widetilde{c} \\cdot \\varphi _{\\gamma }$ (cf.",
"(REF ), (REF )) of the power divergence given in the seventh line of (REF ).",
"Sums: for i.i.d.",
"copies $(W_{i})_{i \\in \\mathbb {N}}$ of $W$ , the probability distribution of $\\breve{W} := \\sum _{i\\in I_{k}^{(n)}} W_{i}$ (cf.",
"Remark REF (ii)) has density $f_{\\breve{W}}(y)\\ :=\\ \\frac{\\exp \\lbrace \\frac{y \\cdot \\widetilde{c}}{\\gamma -1}\\rbrace }{\\exp \\lbrace \\widetilde{c} \\cdot card(I_{k}^{(n)})/\\gamma \\rbrace }\\cdot f_{\\breve{Z}}(- y), \\qquad y \\in \\mathbb {R},\\nonumber $ where $\\breve{Z}$ is a random variable with density $f_{\\breve{Z}}$ of a stable law with parameter-quadruple $(\\frac{\\gamma }{\\gamma -1},1,0,card(I_{k}^{(n)})\\cdot \\frac{\\widetilde{c}^{1/(1-\\gamma )} \\cdot (\\gamma -1)^{\\gamma /(\\gamma -1)}}{\\gamma })$ .",
"(f) Case $\\gamma \\in ]1,2[$ , $\\widetilde{c} >0$ : one still has the (cumulant-generating-function) candidate $\\Lambda _{\\gamma ,\\widetilde{c}}^{(0)}(z) = \\frac{\\widetilde{c}}{\\gamma } \\cdot \\left\\lbrace \\left( \\frac{\\gamma -1}{\\widetilde{c}} \\cdot z +1 \\right)^{\\frac{\\gamma }{\\gamma -1}} -1 \\right\\rbrace $ (cf.",
"(REF )), but for the crucial exponent there holds $\\frac{\\gamma }{\\gamma -1} > 2$ .",
"From this, we conjecture that $\\mathbb {}$ becomes a signed finite measure with total mass 1, i.e.",
"it has a density (with respect to some dominating measure) with positive and negative values which “integrates to 1” ; accordingly, our BS method can not be applied to this situation.",
"Remark 49 As a continuation of Remark REF and the note in the third line after (REF ), we have shown as a side effect that for $\\gamma \\in \\, ]-\\infty ,-1] \\, \\cup \\, ]0,1[ \\, \\cup \\, [2,\\infty [$ the distributions $\\mathbb {}_{\\gamma }$ and $\\mathbb {}_{1-\\gamma }$ of Example REF (b)-(e) are inverse to each other.",
"Example 50 for the power-divergence context of Example REF we obtain: (a) Case $\\gamma =1$ , $\\widetilde{c} >0$ , anchor point $c=0$ : $\\Lambda _{1,\\widetilde{c}}(z) =\\widetilde{c} \\cdot \\left( \\exp (\\frac{z}{\\widetilde{c}}) - 1 \\right)$ (cf.",
"(REF )) is the cumulant generating function of $\\mathbb {} = \\frac{1}{\\widetilde{c}} \\cdot POI(\\widetilde{c})$ being the “$\\frac{1}{\\widetilde{c}}-$ fold Poisson distribution with mean $\\widetilde{c}$ ” which means that $W = \\frac{1}{\\widetilde{c}} \\cdot Z$ for a $POI(\\widetilde{c})-$ distributed random variable $Z$ .",
"Thus, $\\varphi _{1,\\widetilde{c}} \\in \\Upsilon (]0,\\infty [)$ .",
"Prominent special case $\\widetilde{c} =1$ : $\\mathbb {} = POI(1)$ is the Poisson distribution with mean 1.",
"Type: $\\mathbb {}$ is an infinitely divisible (cf.",
"Proposition REF ) discrete distribution with frequencies: $\\mathbb {\\Pi }[W=\\ell \\cdot \\frac{1}{\\widetilde{c}}]= \\exp (-\\widetilde{c}) \\cdot \\frac{ \\widetilde{c}^{\\ell }}{\\ell !",
"}$ for all nonnegative integers $\\ell \\in \\mathbb {N}_{0}$ (and zero elsewhere).",
"Behaviour at zero: $\\mathbb {\\Pi }[W\\ge 0]=1$ , $\\mathbb {\\Pi }[W=0]= \\exp (-\\widetilde{c})$ .",
"Corresponding generator: $\\varphi _{1,\\widetilde{c}} = \\widetilde{c} \\cdot \\varphi _{1}$ (cf.",
"(REF ), (REF )) of the $\\widetilde{c}-$ fold of the Kullback-Leibler divergence (relative entropy) given in the fourth line of (REF ).",
"Sums: for i.i.d.",
"copies $(W_{i})_{i \\in \\mathbb {N}}$ of $W$ , the probability distribution of $\\breve{W} := \\sum _{i\\in I_{k}^{(n)}} W_{i}$ (cf.",
"Remark REF (ii)) is $\\frac{1}{\\widetilde{c}} \\cdot POI(\\widetilde{c} \\cdot card(I_{k}^{(n)}))$ .",
"(b) Case $\\gamma =1$ , $\\widetilde{c} =1$ , anchor point $c \\in \\mathbb {R}$ : $\\Lambda _{1,1}^{(c)}(z)= e^{c} \\cdot \\left( e^{z} - 1 \\right) + z \\cdot (1- e^{c})$ (cf.",
"(REF )) is the well-known cumulant generating function of the “shifted Poisson distribution” $\\mathbb {} = POI(e^{c}) + 1-e^{c}$ , i.e.",
"$W := Z + 1-e^{c}$ with a $POI(e^{c})-$ distributed random variable $Z$ .",
"Hence, $\\varphi _{1,1}^{(c)} \\in \\Upsilon (]1- e^{c},\\infty [)$ .",
"Type: $\\mathbb {}$ is a discrete distribution with frequencies: $\\mathbb {\\Pi }[W=\\ell + 1- e^{c}]= \\exp (-e^{c}) \\cdot \\frac{ e^{c \\cdot \\ell }}{\\ell !",
"}$ for all $\\ell \\in \\mathbb {N}_{0}$ (and zero elsewhere).",
"Behaviour at zero: $\\mathbb {\\Pi }[W > 0]=1$ iff $c <0$ , $\\mathbb {\\Pi }[W < 0] >0$ iff $c >0$ , $\\mathbb {\\Pi }[W=0] \\ne 0$ iff “$c =\\log (1+k)$ for some $k \\in \\mathbb {N}_{0}$ ” .",
"Corresponding generator: $\\varphi _{1,1}^{(c)}$ (cf.",
"(REF )) of the divergence $& & D_{\\varphi _{1,1}^{(c)}}(\\mathbf {Q},\\mathbf {P}) :=\\sum \\limits _{k=1}^{K} \\Big (q_{k} + p_{k} \\cdot (e^{c}-1) \\Big )\\cdot \\Big \\lbrace \\log \\Big (\\frac{q_{k}}{p_{k}} + e^{c}-1 \\Big ) - c \\Big \\rbrace - \\sum \\limits _{k=1}^{K} q_{k} + \\sum \\limits _{k=1}^{K} p_{k} ,\\nonumber \\\\& & \\hspace{113.81102pt} \\textrm {if } \\ \\mathbf {P} \\in \\mathbb {R}_{\\ne 0}^{K} \\ \\textrm {and } \\mathbf {Q} \\in \\mathbb {R}^{K}\\textrm {with } \\mathbf {Q} \\in \\, [ (1-e^{c}) \\cdot \\mathbf {P}, \\mathbf {\\infty } [\\ \\textrm {component-wise},$ which for $c=0$ coincides with the Kullback-Leibler divergence (relative entropy) given in the fourth line of (REF ).",
"Sums: for i.i.d.",
"copies $(W_{i})_{i \\in \\mathbb {N}}$ of $W$ , the probability distribution of $\\breve{W} := \\sum _{i\\in I_{k}^{(n)}} W_{i}$ (cf.",
"Remark REF (ii)) is $POI(card(I_{k}^{(n)}) \\cdot e^{c}) + (1-e^{c}) \\cdot card(I_{k}^{(n)})$ .",
"Remark 51 (a) One can see from the Examples REF and REF the interesting effect that the “homogeneous” class of power-divergence generators $(\\varphi _{\\gamma })_{\\gamma \\in \\mathbb {R}}$ are connected to a “very inhomogeneous” family $(\\mathbb {}_{\\gamma })_{\\gamma \\in \\mathbb {R}}$ of $W-$ distributions: discrete, continuous, mixture of discrete and continuous, as the parameter $\\gamma $ varies.",
"Moreover, some cases satisfy $\\mathbb {\\Pi }[W=0] =0$ and some $\\mathbb {\\Pi }[W=0] >0$ , some $\\mathbb {\\Pi }[W>0]=1$ and some $\\mathbb {\\Pi }[W>0] \\in ]0,1[$ .",
"(b) As a continuation of Remark REF and the note in the last line of Example REF (a), we have shown as a side effect that for the the natural-anchor-point choice $c=0$ , the distributions $\\mathbb {}_{1}$ of of Example REF (a) and $\\mathbb {}_{0}$ of Example REF (a) are inverse to each other.",
"Example 52 for the context of Example REF we obtain: Case $\\widetilde{c} >0$ , anchor point $c=0$ : $\\Lambda _{bw,\\beta ,\\widetilde{c}}(z) =-(\\frac{1}{\\beta }-1) \\cdot z + \\frac{\\widetilde{c}}{\\beta ^{2}} \\cdot \\Big \\lbrace 1 - \\sqrt{1-\\frac{2\\beta }{\\widetilde{c}} \\cdot z} \\ \\Big \\rbrace $ (cf.",
"(REF )) is the cumulant generating function of a probability distribution $\\mathbb {}[ \\, \\cdot \\,] = \\mathbb {\\Pi }[\\check{W} \\in \\cdot \\, ]$ of a random variable $\\check{W}$ , which can be constructed as follows: $\\check{W} := \\frac{W}{\\beta } - (\\frac{1}{\\beta } - 1)$ , where $W$ is the random variable constructed in Example REF (d) with $\\gamma =-1$ and with $\\widetilde{c}$ replaced by $\\frac{\\widetilde{c}}{\\beta ^{2}}$ (recall that $W$ has a tilted stable distribution).",
"In other words, $\\mathbb {}$ is a special kind of modified tilted stable distribution.",
"Type: $\\mathbb {}$ is an infinitely divisible (cf.",
"Proposition REF ) continuous distribution with density $f_{\\check{W}}(u) := \\beta \\cdot f_{W}(\\beta \\cdot u + 1 -\\beta )\\cdot \\mathbf {1}_{]- (\\frac{1}{\\beta } -1),\\infty [}(u)$ ($u \\in \\mathbb {R}$ ), where $f_{W}(\\cdot )$ is given in (REF ) with $\\gamma =-1$ and with $\\widetilde{c}$ replaced by $\\frac{\\widetilde{c}}{\\beta ^{2}}$ .",
"Behaviour at zero: $\\mathbb {}[ \\, ] 0,\\infty [ \\, ] =\\mathbb {\\Pi }[\\check{W} > 0] >0$ .",
"Corresponding generator: $\\varphi _{bw,\\beta ,\\widetilde{c}}^{(0)}$ (cf.",
"(REF )) of the — “non-probability version” of — the well-known blended weight chi-square divergence given in (REF ).",
"Sums: for i.i.d.",
"copies $(\\check{W}_{i})_{i \\in \\mathbb {N}}$ of $\\check{W}$ , the probability distribution of $\\breve{\\check{W}} := \\sum _{i\\in I_{k}^{(n)}} \\check{W}_{i}= \\frac{1}{\\beta } \\cdot \\sum _{i\\in I_{k}^{(n)}} W_{i} - n_{k}\\cdot (\\frac{1}{\\beta } -1)$ (cf.",
"Remark REF (ii)) has density $f_{\\breve{\\check{W}}}(u) := \\beta \\cdot f_{\\breve{W}}(\\beta \\cdot u + (1 -\\beta )\\cdot n_{k})\\cdot \\mathbf {1}_{]- n_{k} \\cdot (\\frac{1}{\\beta } -1),\\infty [}(u)$ ($u \\in \\mathbb {R}$ ), where $f_{\\breve{W}}(\\cdot )$ is given in (REF ) (cf.",
"Example REF (d)) with $\\gamma =-1$ and with $\\widetilde{c}$ replaced by $\\frac{\\widetilde{c}}{\\beta ^{2}}$ .",
"Example 53 for the context of Example REF we obtain: (a) Case $\\alpha \\in \\, ]0,\\infty [$ , $\\widetilde{c} >0$ , anchor point $c=0$ : $\\Lambda _{gKL,\\alpha ,\\widetilde{c}}(z) =- \\frac{\\widetilde{c}}{\\alpha } \\cdot \\log ((1+\\alpha ) - \\alpha \\cdot e^{z/\\widetilde{c}})$ (cf.",
"(REF )) is the cumulant generating function of $\\mathbb {} = \\frac{1}{\\widetilde{c}} \\cdot NB(\\frac{\\widetilde{c}}{\\alpha },\\frac{1}{1+\\alpha })$ being the “$\\frac{1}{\\widetilde{c}}-$ fold Negative-Binomial distribution with parameters $\\frac{\\widetilde{c}}{\\alpha }$ and $\\frac{1}{1+\\alpha }$ ” which means that $W = \\frac{1}{\\widetilde{c}} \\cdot Z$ for a $NB(\\frac{\\widetilde{c}}{\\alpha },\\frac{1}{1+\\alpha })-$ distributed random variable $Z$ .",
"Thus, $\\varphi _{gKL,\\alpha ,\\widetilde{c}} \\in \\Upsilon (]0,\\infty [)$ .",
"Prominent special case $\\widetilde{c}=1$ , $\\alpha =1$ (see below): $\\mathbb {} = NB(1,\\frac{1}{2})$ is the Negative-Binomial distribution with parameters 1 and $\\frac{1}{2}$ .",
"Type: $\\mathbb {}$ is an infinitely divisible (cf.",
"Proposition REF ) discrete distribution with frequencies: $\\mathbb {\\Pi }[W=\\ell \\cdot \\frac{1}{\\widetilde{c}}]=(-1)^{\\ell } \\cdot \\binom{- \\frac{\\widetilde{c}}{\\alpha }}{\\ell }\\cdot {\\alpha }^{\\ell } \\cdot (1+\\alpha )^{-\\ell - \\widetilde{c}/\\alpha }$ for all nonnegative integers $\\ell \\in \\mathbb {N}_{0}$ (and zero elsewhere).",
"Behaviour at zero: $\\mathbb {\\Pi }[W\\ge 0]=1$ , $\\mathbb {\\Pi }[W=0]= \\frac{1}{(1+a)^{\\widetilde{c}/\\alpha }}$ .",
"Corresponding generator: $\\varphi _{gKL,\\alpha ,\\widetilde{c}}$ (cf.",
"(REF )) of the divergence (REF ); the special case $\\widetilde{c}=1$ , $\\alpha =1$ — i.e.",
"$\\varphi _{gKL,1,1} =: \\varphi _{snKL,1}$ (cf.",
"(REF )) — corresponds to the generator of the — “non-probability version” of the — Jensen-Shannon divergence (symmetrized and normalized Kullback-Leibler divergence, symmetrized and normalized relative entropy) given in (REF ).",
"Sums: for i.i.d.",
"copies $(W_{i})_{i \\in \\mathbb {N}}$ of $W$ , the probability distribution of $\\breve{W} := \\sum _{i\\in I_{k}^{(n)}} W_{i}$ (cf.",
"Remark REF (ii)) is $\\frac{1}{\\widetilde{c}} \\cdot NB(\\frac{\\widetilde{c}}{\\alpha }\\cdot card(I_{k}^{(n)}),\\frac{1}{1+\\alpha })$ .",
"(b) Case $\\alpha \\in \\, ]-1,0[$ , $\\widetilde{c} >0$ , anchor point $c=0$ : for any integer $m \\in \\mathbb {N}$ being strictly larger than $\\widetilde{c}$ and the choice $\\alpha = - \\frac{\\widetilde{c}}{m}$ , we obtain $\\Lambda _{gKL,-\\widetilde{c}/m,\\widetilde{c}}(z) =m \\cdot \\log ((1 - \\frac{\\widetilde{c}}{m}) + \\frac{\\widetilde{c}}{m} \\cdot e^{z/\\widetilde{c}})$ (cf.",
"(REF )) which is the cumulant generating function of $\\mathbb {} = \\frac{1}{\\widetilde{c}} \\cdot BIN(m,\\frac{\\widetilde{c}}{m})$ being the “$\\frac{1}{\\widetilde{c}}-$ fold Binomial distribution with parameters $m$ and $\\frac{\\widetilde{c}}{m}$ ” which means that $W = \\frac{1}{\\widetilde{c}} \\cdot Z$ for a $BIN(m,\\frac{\\widetilde{c}}{m})-$ distributed random variable $Z$ .",
"Thus, $\\varphi _{gKL,-\\widetilde{c}/m,\\widetilde{c}} \\in \\Upsilon (]0,\\infty [)$ .",
"Type: $\\mathbb {}$ is a non-infinitely divisible discrete distribution with frequencies: $\\mathbb {\\Pi }[W=\\ell \\cdot \\frac{1}{\\widetilde{c}}]=\\binom{m}{\\ell }\\cdot (\\frac{\\widetilde{c}}{m})^{\\ell } \\cdot (1-\\frac{\\widetilde{c}}{m})^{m-\\ell }$ for $\\ell \\in \\lbrace 0,1, \\ldots , m\\rbrace $ (and zero elsewhere).",
"Behaviour at zero: $\\mathbb {\\Pi }[W\\ge 0]=1$ , $\\mathbb {\\Pi }[W=0]= (1-\\frac{\\widetilde{c}}{m})^{m}$ .",
"Corresponding generator: $\\varphi _{gKL,\\alpha ,\\widetilde{c}}$ (cf.",
"(REF )) of the divergence (REF ).",
"Sums: for i.i.d.",
"copies $(W_{i})_{i \\in \\mathbb {N}}$ of $W$ , the probability distribution of $\\breve{W} := \\sum _{i\\in I_{k}^{(n)}} W_{i}$ (cf.",
"Remark REF (ii)) is $\\frac{1}{\\widetilde{c}} \\cdot BIN(m \\cdot card(I_{k}^{(n)}),\\frac{\\widetilde{c}}{m})$ .",
"Example 54 for the context of Example REF we obtain: Case of anchor point $c=0$ : $\\Lambda _{twop}(z)= \\log \\Big ( p \\cdot e^{z_{1} \\cdot z} + (1- p) \\cdot e^{z_{2} \\cdot z} \\Big )$ (cf.",
"(REF )) is the well-known cumulant generating function of the two-point probability distribution $\\mathbb {} = p \\cdot \\delta _{z_{1}} + (1-p) \\cdot \\delta _{z_{2}}$ , where $z_{1} < 1 < z_{2}$ and $p= \\frac{z_{2} -1}{z_{2} - z_{1}}$ .",
"Hence, $\\varphi _{twop} \\in \\Upsilon (]z_{1},z_{2}[)$ .",
"Type: $\\mathbb {}$ is a discrete distribution with frequencies: $\\mathbb {\\Pi }[W=z_{1}] = p$ , $\\mathbb {\\Pi }[W=z_{2}] = 1-p$ (and zero elsewhere).",
"Behaviour at zero: $\\mathbb {\\Pi }[W > 0]=1$ iff $z_{1} >0$ , $\\mathbb {\\Pi }[W=0] \\ne 0$ iff $z_{1}=0$ .",
"Corresponding generator: $\\varphi _{twop}$ (cf.",
"(REF )) of the divergence given in (REF ).",
"Sums: for i.i.d.",
"copies $(W_{i})_{i \\in \\mathbb {N}}$ of $W$ , the probability distribution of $\\breve{W} := \\sum _{i\\in I_{k}^{(n)}} W_{i}$ (cf.",
"Remark REF (ii)) is the distribution of the $card(I_{k}^{(n)})-$ th step of a generalized random walk starting at zero; this has a nice explicit (“binomial-type” ) expression in the special case $z_{1} = - z_{2}$ , namely $\\sum _{\\ell =0}^{card(I_{k}^{(n)})} \\binom{card(I_{k}^{(n)})}{\\ell }\\cdot p^{card(I_{k}^{(n)}) - \\ell } \\cdot (1-p)^{\\ell } \\cdot \\delta _{z_{2}\\cdot (2 \\ell - card(I_{k}^{(n)}))}$ .",
"Example 55 for the context of Example REF we obtain: Case $\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c} \\in \\, ]0,\\infty [$ , anchor point $c=0$ : by using $\\breve{\\theta } := 1 + \\alpha \\cdot \\Big (\\frac{1}{\\beta _{2}}- \\frac{1}{\\beta _{1}} \\Big )$ one can see that $\\Lambda _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}(z) =\\breve{\\theta } \\cdot z - \\widetilde{c} \\cdot \\alpha \\cdot \\log \\Big (1+ \\frac{z}{\\widetilde{c}} \\cdot \\Big (\\frac{1}{\\beta _{2}} - \\frac{1}{\\beta _{1}} \\Big )- \\frac{z^{2}}{\\widetilde{c}^{2} \\cdot \\beta _{1} \\cdot \\beta _{2} }\\Big )$ for $z \\in \\, ]- \\widetilde{c} \\cdot \\min \\lbrace \\beta _{1}, \\beta _{2}\\rbrace ,\\widetilde{c} \\cdot \\min \\lbrace \\beta _{1}, \\beta _{2}\\rbrace [$ — with different boundary behaviour for the three subcases $\\beta _{1} < \\beta _{2}$ resp.",
"$\\beta _{1} > \\beta _{2}$ resp.",
"$\\beta _{1} = \\beta _{2}$ (cf.",
"(REF ),(REF ),(REF )) — is the cumulant generating function of a generalized asymmetric Laplace distribution $\\mathbb {}[ \\, \\cdot \\,] = \\mathbb {\\Pi }[W \\in \\cdot \\, ]$ of a random variable $W:= \\breve{\\theta } + Z_{1} - Z_{2}$ , where $Z_{1}$ respectively $Z_{2}$ are auxiliary random variables which are independent and $GAM(\\widetilde{c} \\cdot \\beta _{1},\\widetilde{c} \\cdot \\alpha )-$ distributed respectively $GAM(\\widetilde{c} \\cdot \\beta _{2},\\widetilde{c} \\cdot \\alpha )-$ distributed.",
"In particular, $E_{\\mathbb {\\Pi }}[W] = \\breve{\\theta } + \\frac{\\widetilde{c} \\cdot \\alpha }{\\widetilde{c} \\cdot \\beta _{1}}- \\frac{\\widetilde{c} \\cdot \\alpha }{\\widetilde{c} \\cdot \\beta _{2}} =1$ (as required).",
"Thus, $\\varphi _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}\\in \\Upsilon (]-\\infty ,\\infty [)$ .",
"Prominent special case $\\widetilde{c} =1$ , $\\alpha =1$ , $\\beta _{1}=\\beta _{2} =: \\beta $ (and hence, $\\breve{\\theta }=1$ ): $\\mathbb {}$ is a classical Laplace distribution (two-tailed exponential distribution, bilateral exponential law) with location parameter 1 and scale parameter $\\frac{1}{\\beta }$ .",
"Type: $\\mathbb {}$ is an infinitely divisible (cf.",
"Proposition REF ) continuous distribution with density $f(u) := \\frac{\\sqrt{2} \\cdot \\exp \\lbrace \\frac{1}{\\sigma \\cdot \\sqrt{2}} \\cdot (\\frac{1}{\\kappa } - \\kappa ) \\cdot (u-\\theta ) \\rbrace }{\\sqrt{\\pi } \\cdot \\sigma ^{\\tau + 1/2} \\cdot \\Gamma (\\tau )} \\cdot \\left( \\frac{\\sqrt{2} \\cdot |u - \\theta |}{\\kappa + \\frac{1}{\\kappa }} \\right)^{\\tau - 1/2}\\cdot K_{\\tau - 1/2}\\left( \\frac{1}{\\sigma \\cdot \\sqrt{2}} \\cdot \\Big (\\kappa + \\frac{1}{\\kappa }\\Big ) \\cdot |u - \\theta | \\right),\\quad u \\ne \\theta ,$ where $(\\theta ,\\kappa ,\\sigma ,\\tau )$ is given in Remark REF below and $K_{\\lambda }$ is the modified Bessel function of the third kind with index $\\lambda $ .",
"For the above-mentioned special case of the classical Laplace distribution, this considerably simplifies to $f(u):= \\frac{\\beta }{2} \\exp \\lbrace - \\beta \\cdot |u -1 | \\rbrace $ .",
"Behaviour at zero: $\\mathbb {}[ \\, ]0,\\infty [ \\, ] = \\mathbb {\\Pi }[W > 0]=\\int _{0}^{\\infty } f(u) \\, du \\in \\, ]0,1[$ , $\\mathbb {}[ \\, \\lbrace 0\\rbrace \\, ] = \\mathbb {\\Pi }[W = 0]= 0$ .",
"Corresponding generator: $\\varphi _{\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}}$ (cf.",
"(REF ) respectively (REF ) respectively (REF )) of the divergence given in (REF ) respectively (REF ) respectively (REF ).",
"Sums: for i.i.d.",
"copies $(W_{i})_{i \\in \\mathbb {N}}$ of $W$ , the probability distribution of $\\breve{W} := \\sum _{i\\in I_{k}^{(n)}} W_{i}$ (cf.",
"Remark REF (ii)) is the same as that of a random variable $\\breve{\\widetilde{W}} := \\breve{\\theta } \\cdot card(I_{k}^{(n)}) +\\breve{Z}_{1} - \\breve{Z}_{2}$ , where $\\breve{Z}_{1}$ respectively $\\breve{Z}_{2}$ are auxiliary random variables which are independent and $GAM(\\widetilde{c} \\cdot \\beta _{1},\\widetilde{c} \\cdot \\alpha \\cdot card(I_{k}^{(n)}))-$ distributed respectively $GAM(\\widetilde{c} \\cdot \\beta _{2},\\widetilde{c}\\cdot \\alpha \\cdot card(I_{k}^{(n)}))-$ distributed.",
"Remark 56 In the book of Kotz et al.",
"[197] one can find a very comprehensive study on generalized asymmetric Laplace distributions (also known as Bessel function distributions, McKay distributions), their close relatives (such as e.g.",
"the financial-econometric variance gamma model of Madan & Seneta [240]) as well as their applications; see also e.g.",
"Klar [192] for connections with some other Gamma difference distributions.",
"[197] use a different parametrization $(\\theta ,\\kappa ,\\sigma ,\\tau )$ which is one-to-one with our parametrization $(\\breve{\\theta },\\alpha ,\\beta _{1},\\beta _{2},\\widetilde{c}=1)$ , as follows: $\\theta = \\breve{\\theta }$ , $\\tau = \\widetilde{c} \\cdot \\alpha $ , $\\sigma = \\frac{1}{\\widetilde{c}} \\cdot \\sqrt{\\frac{2}{\\beta _{1} \\cdot \\beta _{2}}}$ , $\\kappa = \\frac{\\sqrt{\\frac{4}{\\widetilde{c}^{2}} + (\\beta _{1} - \\beta _{2})^2} \\, + \\beta _{2}- \\beta _{1}}{2 \\cdot \\sqrt{\\beta _{1} \\cdot \\beta _{2}}} >0$ .",
"In particular, this implies that we cover all generalized asymmetric Laplace distributions with mean 1.",
"For better comparability, we have used the parametrization $(\\theta ,\\kappa ,\\sigma ,\\tau )$ in the above-mentioned representation (REF ) of the density (due to [197]).",
"Let us end this section by giving some further comments on the task of finding concretely the probability distribution (if existent) $\\mathbb {}[ \\, \\cdot \\,] = \\mathbb {\\Pi }[W \\in \\cdot \\, ]$ from the Fenchel-Legendre transform $\\Lambda = \\varphi ^{*}$ of a pregiven divergence generator $\\varphi $ , which should satisfy $\\exp (\\Lambda (z)) = \\int _{\\mathbb {R}}e^{z \\cdot y} \\, d\\mathbb {} (y)= E_{\\mathbb {\\Pi }}[\\exp (z \\cdot W)] ,\\qquad z \\in \\mathbb {R}, \\qquad \\text{(cf.",
"(\\ref {brostu3:repres lambda1}), (\\ref {brostu3:repres lambda2}))}.\\nonumber $ Recall that this is used for the simulation of the weights $(W_{i})_{i\\in \\mathbb {N}}$ which are i.i.d.",
"copies of $W$ and which are the crucial building ingredients of $\\xi _{n}^{\\mathbf {W}}$ in Theorem REF , respectively, of $\\xi _{n,\\mathbf {X}}^{w\\mathbf {W}}$ in Theorem REF .",
"The search for $\\mathbb {}$ can be done e.g.",
"by inversion of the moment generating function MGF, or by search in tables or computer software which list distributions and their MGF.",
"As already indicated above, we have eased/narrowed down this task by giving (additional) sufficient conditions for some deriving principal properties of $\\mathbb {}$ .",
"Also notice that $\\mathbb {}$ needs not necessarily to be explicitly known in full detail (e.g.",
"in terms of a computationally tractable density or frequency); for instance, as well known from insurance applications, for — comfortably straightforwardly simulable — doubly-random sums $W := \\sum _{i=1}^{N} A_{i}$ of nonnegative i.i.d.",
"random variables $(A_{i})_{i\\in \\mathbb {N}}$ with known law $\\Pi _A[\\, \\cdot \\, ]:= \\Pi [A \\in \\cdot \\, ]$ being independent of a counting-type random variable $N$ with known law $\\Pi _N$ , one can mostly compute explicitly $MGF_{\\mathbb {}}(z) = PGF_{\\Pi _N}(MGF_{\\Pi _A}(z))$ in terms of $\\mathbb {} := \\Pi _W$ and the probability generating function $PGF_{\\Pi _N}$ of $\\Pi _N$ , but the corresponding density/frequency of $\\mathbb {}$ may not be known explicitly in a tractable form.",
"The above-mentioned Example REF (b) of power divergences with generator $\\varphi _{\\gamma }$ ($\\gamma \\in ]0,1[$ ) manifests such a situation.",
"In the end, if no explicit distribution $\\mathbb {}$ and no comfortably simulable $W-$ construction are available, one can still try to simulate an i.i.d.",
"sequence $(W_{i})_{i\\in \\mathbb {N}}$ from the pregiven moment generating function (which is $\\exp (\\Lambda (z))$ here); see e.g.",
"McLeish [259] and references therein which also contains saddle point methods approximation techniques."
],
[
"Estimators",
"In the following, we demonstrate how one can principally implement our BS approach; a further, deeper analysis will be given in a follow-up paper."
],
[
"Estimators for the deterministic minimization problem",
"We address the minimization problem $D_{\\varphi }(\\mathbf {\\Omega },\\mathbf {P}):= \\inf _{\\mathbf {Q}\\in \\mathbf {\\Omega } } D_{\\varphi }(\\mathbf {Q},\\mathbf {P}) =\\inf _{\\widetilde{\\mathbf {Q}}\\in \\widetilde{\\mathbf {\\Omega }} }D_{\\widetilde{\\varphi } }(\\widetilde{\\mathbf {Q}},\\widetilde{{P}})=: D_{\\widetilde{\\varphi } }(\\widetilde{\\mathbf {\\Omega }},\\widetilde{{P}})\\qquad \\textrm {with } \\widetilde{\\mathbf {\\Omega }} :=\\mathbf {\\Omega } /M_{\\mathbf {P}}\\qquad \\textrm {(cf.",
"(\\ref {min Pb}) and (\\ref {min Pb prob2}))},$ whose numerical solution is based on Theorem REF which basically states that for large integer $n \\in \\mathbb {N}$ one has $\\inf _{\\mathbf {Q}\\in \\mathbf {\\Omega } } D_{\\varphi }(\\mathbf {Q},\\mathbf {P})\\approx -\\frac{1}{n}\\log \\,\\mathbb {\\Pi }\\left[\\xi _{n}^{\\mathbf {\\widetilde{W}}}\\in \\widetilde{\\mathbf {\\Omega }}\\right]$ in terms of $\\widetilde{\\varphi }:=M_{\\mathbf {P}} \\cdot \\varphi $ and the random vectors $\\xi _{n}^{\\mathbf {\\widetilde{W}}}=\\Big (\\frac{1}{n}\\sum _{i\\in I_{1}^{(n)}}\\widetilde{W}_{i},\\ldots ,\\frac{1}{n}\\sum _{i\\in I_{K}^{(n)}}\\widetilde{W}_{i}\\Big )\\hspace{56.9055pt} \\text{(cf.",
"(\\ref {Xi_n^W vector}))}\\nonumber $ with $n_{k}:=\\lfloor n \\cdot \\widetilde{p}_{k}\\rfloor $ leading to the disjoint index blocks $I_{1}^{(n)}:=\\left\\lbrace 1,\\ldots ,n_{1}\\right\\rbrace $ , $I_{2}^{(n)}:=\\left\\lbrace n_{1}+1,\\ldots ,n_{1}+n_{2}\\right\\rbrace $ , $\\ldots $ , $I_{K}^{(n)} := \\lbrace \\sum _{k=1}^{K-1} n_{k} + 1, \\ldots , n \\rbrace $ .",
"Recall that $\\mathbf {\\widetilde{W}} := (\\widetilde{W}_{1}, \\ldots , \\widetilde{W}_{n})$ is a random vector consisting of components $\\widetilde{W}_{i}$ which are i.i.d.",
"copies of the random variable $\\widetilde{W}$ whose distribution is $\\mathbb {\\Pi }[\\widetilde{W}\\in \\cdot \\,]=\\widetilde{\\mathbb {}}[\\,\\cdot \\,]$ obeying the representation $\\widetilde{\\varphi }(t) =\\sup _{z \\in \\mathbb {R}} \\left( z\\cdot t - \\log \\int _{\\mathbb {R}} e^{zy} d\\widetilde{\\mathbb {}}(y) \\right),\\qquad t \\in \\mathbb {R},\\qquad \\textrm {(cf.",
"(\\ref {brostu3:fo.link.var}))} .\\nonumber $ Hence, the estimation of $D_{\\varphi }(\\mathbf {\\Omega },\\mathbf {P})$ amounts to the estimation of $\\mathbb {\\Pi }\\left[\\xi _{n}^{\\mathbf {\\widetilde{W}}}\\in \\widetilde{\\mathbf {\\Omega }} \\right]$ .",
"For the rest of this subsection, we assume that $\\widetilde{{P}} \\in \\mathbb {S}_{> 0}^{K}$ , that $n$ is chosen such that all $n \\cdot \\widetilde{p}_{k}$ are integers (and hence, $n = \\sum _{k=1}^{K} n_{k}$ ), and that $\\widetilde{\\mathbf {\\Omega }} \\subset \\mathbb {R}^{K}$ satisfies the regularity property $cl(\\widetilde{\\mathbf {\\Omega }})=cl\\left( int\\left( \\widetilde{\\mathbf {\\Omega }} \\right) \\right) , \\qquad int\\left( \\widetilde{\\mathbf {\\Omega }} \\right) \\ne \\emptyset \\nonumber $ which implies that the same condition holds for $\\mathbf {\\Omega }$ ; moreover, we suppose that $D_{\\widetilde{\\varphi } }(\\widetilde{\\mathbf {\\Omega }},\\widetilde{{P}})$ is finite.",
"For the ease of the following discussions, we introduce the notations $T\\left( \\mathbf {x}\\right) :=\\left( \\frac{1}{n_{1}}\\sum _{i\\in I_{1}^{(n)}}x_{i},\\ldots ,\\frac{1}{n_{K}}\\sum _{i\\in I_{K}^{(n)}} x_{i}\\right)\\quad \\textrm {for any $ x:=( x1,..,xn) Rn$,}$$as well as $ D$ for the diagonal matrix with diagonal entries $ 1/p1,...,1/pK$and null entries off the diagonal.Accordingly, the probability in (\\ref {LDP Minimization approx}) becomes$$\\mathbb {\\Pi }\\left[\\xi _{n}^{\\mathbf {\\widetilde{W}}}\\in \\widetilde{\\mathbf {\\Omega }} \\right] \\ = \\ \\mathbb {\\Pi }\\left[ T(\\mathbf {\\widetilde{W}}) \\in \\mathbf {\\Lambda }\\right]$$where$$\\mathbf {\\Lambda } :=\\mathfrak {D} \\cdot \\widetilde{\\mathbf {\\Omega }}$$is a set of vectors in $ RK$ which is known/derived from the concrete context.The \\textit {naive estimator} $ Lnaive$ of$ [ nW ]$is constructedthrough the following procedure:simulate independently $ L$ copies$ W1,...,WL$ of the vector$ W:=( W1,...,Wn) $,with independent entries under $$, and define(with a slight abuse of notation)$$\\widehat{\\widetilde{\\Pi }}_{L}^{naive} :=\\frac{1}{L}\\sum _{\\ell =1}^{L} \\mathbf {1}_{\\mathbf {\\Lambda }}\\left( T\\left( \\mathbf {\\widetilde{W}}^{\\ell }\\right) \\right) ;$$however this procedure is time costly, since this estimate has a very bad hit rate.Thus, in the following, a so-called “efficient Importance Sampling (IS)”\\ scheme --- in the sense of Sadowsky\\& Bucklew \\cite {Sad:90} (denoted [SB] hereunder) ---is adapted for the sophisticated (i.e.",
"non-naive) estimation of$ [ nW ]$.The main property of IS schemes lays in the fact that the runtime for anestimate with a controlled relative error does \\textit {not} increase atexponential rate as $ n$ increases, in contrast to$ Lnaive$ which has exponential increase.In detail, let $ >0$ be agiven relative precision for an estimator $ PSLn$ of$ [ nW ]$, based on a number $ Ln$ of simulated samplesgenerated under some distribution $ S$, so that$$\\delta :=\\frac{var_{\\widetilde{S}}P_{\\widetilde{S}}^{L_{n}}}{\\left(\\mathbb {\\Pi }\\left[\\xi _{n}^{\\mathbf {\\widetilde{W}}}\\in \\widetilde{\\mathbf {\\Omega }}\\right]\\right)^{2}} \\, .$$Then $ Ln$ will grow exponentially as $ n$ tends to infinity if and only if$ S$ is not “asymptotically optimal”\\ ,the derivation of which is the scope of the current section.$ To start with the details, for the sake of brevity (to avoid certain substantial discussions on potential technical relaxations) we shall employ the following additional Assumption (OM) on the set $\\widetilde{\\mathbf {\\Omega }}$ : (OM) For any $\\widetilde{\\omega } \\in cl(\\widetilde{\\mathbf {\\Omega }})$ there exists a vector $\\mathbf {x} = \\left(x_{1},\\ldots ,x_{n}\\right) \\in \\left] t_{-}^{sc},t_{+}^{sc}\\right[^{n}$ such that $\\widetilde{\\omega } = \\left( \\frac{1}{n}\\sum _{i\\in I_{1}^{(n)}}x_{i},\\ldots ,\\frac{1}{n} \\sum _{i\\in I_{K}^{(n)}}x_{i}\\right)$ , or equivalently, for any $\\lambda \\in cl(\\mathbf {\\Lambda })$ there exists a vector $\\mathbf {x} = \\left(x_{1},\\ldots ,x_{n}\\right) \\in \\left] t_{-}^{sc},t_{+}^{sc}\\right[^{n}$ such that $\\lambda =T\\left(\\mathbf {x}\\right)$ .",
"For instance, in the common case $dom(\\widetilde{\\varphi })= dom(\\varphi ) = \\, ]a,b[ \\, =\\, \\left] t_{-}^{sc},t_{+}^{sc}\\right[ \\, = \\, ]0,\\infty [$ (e.g.",
"for the power-divergence generators $\\widetilde{\\varphi }= \\widetilde{c} \\cdot \\varphi _{\\gamma }$ , $\\gamma \\le 0$ , cf.",
"Example REF ) the Assumption (OM) is always feasible.",
"To proceed, for any distribution $\\widetilde{S}$ on $\\mathbb {R}^{n}$ with support included in the support of the product measure $\\widetilde{\\mathbb {}}^{\\otimes n}$ it holds $\\mathbb {\\Pi }\\left[\\xi _{n}^{\\mathbf {\\widetilde{W}}}\\in \\widetilde{\\mathbf {\\Omega }}\\right]=E_{\\widetilde{\\mathbb {}}^{\\otimes n}} \\left[\\mathbf {1}_{\\mathbf {\\Lambda } }( T( \\mathbf {\\widetilde{W}}))\\right] =E_{\\widetilde{S}}\\left[\\mathbf {1}_{\\mathbf {\\Lambda } }\\left( T\\left( \\mathbf {\\widetilde{V}}\\right) \\right) \\cdot \\frac{d\\widetilde{\\mathbb {}}^{\\otimes n}}{d\\widetilde{S}}\\left( \\mathbf {\\widetilde{V}}\\right) \\right]$ from where the improved IS estimator of $\\mathbb {\\Pi }\\left[\\xi _{n}^{\\mathbf {\\widetilde{W}}}\\in \\widetilde{\\mathbf {\\Omega }} \\right]$ is obtained by sampling $L$ i.i.d.",
"replications $\\mathbf {\\widetilde{V}}^{1},\\ldots ,\\mathbf {\\widetilde{V}}^{L}$ of the random vector $\\mathbf {\\widetilde{V}}$ with distribution $\\widetilde{S}$ and by defining $\\widehat{\\widetilde{\\Pi }}_{L}^{improved}:=\\frac{1}{L}\\sum _{\\ell =1}^{L}\\mathbf {1}_{\\mathbf {\\Lambda }}( T( \\mathbf {\\widetilde{V}}^{(\\ell )})) \\cdot \\frac{d\\widetilde{\\mathbb {}}^{\\otimes n}}{d\\widetilde{S}}\\left( \\mathbf {\\widetilde{V}}^{(\\ell )}\\right) \\ .$ The precise form of the efficient IS distribution $\\widetilde{S}^{opt}$ relies on the definition of a “dominating point” of $\\mathbf {\\Lambda }$ , which we recall now.",
"For $\\mathbf {x} := \\left( x_{1},..,x_{n}\\right)$ in $\\mathbb {R}^{n}$ we define $I_{\\mathbf {\\widetilde{W}}}(\\mathbf {x}):=\\sup _{\\mathbf {z} \\in \\mathbb {R}^{n}}\\left(\\left\\langle \\mathbf {z}, \\mathbf {x} \\right\\rangle -\\log E_{\\widetilde{\\mathbb {}}}[ \\, \\exp (\\langle \\mathbf {z},\\mathbf {\\widetilde{W}}\\rangle ) ]\\right) \\, ,$ and for $\\lambda $ in $\\mathbf {\\Lambda }$ we let $I(\\lambda ):=\\inf \\left\\lbrace I_{\\mathbf {\\widetilde{W}}}(\\mathbf {x}):T(\\mathbf {x)=\\lambda }\\right\\rbrace .$ Let $\\underline{\\lambda }:=\\left( \\underline{\\lambda }_{1},\\ldots ,\\underline{\\lambda }_{K}\\right) \\in \\partial \\mathbf {\\Lambda }$ .",
"We call $\\underline{\\lambda }$ a minimal rate point (mrp) of $\\mathbf {\\Lambda }$ if $I(\\underline{\\lambda })\\le I(\\lambda ) \\quad \\text{ for all }\\lambda \\in \\mathbf {\\Lambda } .$ A minimal rate point $\\underline{\\lambda }$ is called a dominating point of $\\mathbf {\\Lambda }$ if a) $\\underline{\\lambda }\\in \\partial \\mathbf {\\Lambda }$ , and b) $I\\left(\\underline{\\lambda }\\right)\\le I(\\lambda )$ for all $\\lambda \\in \\mathbf {\\Lambda }$ with attainment, namely there exists a vector $\\underline{\\mathbf {x}} \\in \\left] t_{-}^{sc},t_{+}^{sc}\\right[^{n}$ such that $I_{\\widetilde{W}}\\left( \\underline{\\mathbf {x}}\\right) =I\\left(\\underline{\\lambda }\\right)$ with $\\underline{\\lambda }=T\\left( \\underline{\\mathbf {x}}\\right)$ .",
"The characterization of the dominating point $\\underline{\\lambda }$ is settled in the following Lemma 57 Let $\\underline{\\lambda }$ be a mrp of $\\mathbf {\\Lambda }$ .",
"Then, under Assumption (OM), $\\underline{\\lambda }$ is a dominating point, and $\\inf \\left\\lbrace I_{\\mathbf {\\widetilde{W}}}\\left( \\mathbf {x}\\right) ,T(\\mathbf {x})=\\underline{\\lambda }\\right\\rbrace $ is reached at some vector $\\underline{\\mathbf {x}}$ in $\\left] t_{-}^{sc},t_{+}^{sc}\\right[^{n}$ such that for all $k \\in \\left\\lbrace 1,\\ldots ,K\\right\\rbrace $ and all $i \\in I_{k}^{(n)}$ there holds $\\underline{x}_{i}= \\underline{\\lambda }_{k}$ and $I_{\\mathbf {\\widetilde{W}}}(\\underline{\\mathbf {x}})=n \\cdot \\sum _{k=1}^{K}\\widetilde{p}_{k} \\cdot \\widetilde{\\varphi }\\left(\\underline{\\lambda }_{k}\\right)$ .",
"The proof Lemma REF is given in Appendix G. Notice that (OM) implies the existence of a dominating point $\\underline{\\lambda }$ , but uniqueness may not hold.",
"In the latter case, one can try to proceed as in Theorem 2 of [SB] and the discussion thereafter.",
"However, we assume now uniqueness of $\\underline{\\lambda }$ ; this allows for the identification of $\\widetilde{S}^{opt}$ .",
"By Theorem 1 of [SB] and Theorem 3.1 of Csiszar [96], the asymptotically optimal IS distribution $\\widetilde{S}^{opt}$ is obtained as the Kullback-Leibler projection of $\\widetilde{\\mathbb {}}^{n\\otimes }$ on the set of all probability distributions on $\\mathbb {R}^{n}$ centered at point $\\underline{\\mathbf {x}}$ , whose coordinates are — according to Lemma REF — functions of the coordinates of $\\underline{\\mathbf {\\widetilde{Q}}} := \\mathfrak {D}^{-1} \\underline{\\lambda }$ such that $T\\left( \\underline{\\mathbf {x}}\\right)= \\mathfrak {D} \\underline{\\mathbf {\\widetilde{Q}}}$ .",
"The above definition of $\\widetilde{S}^{opt}$ presumes the knowledge of $\\underline{\\lambda }$ , which cannot be assumed (otherwise the minimization problem is solved in advance).",
"The aim of the following construction is to provide a proxy $\\widetilde{S}$ to $\\widetilde{S}^{opt}$ , where $\\widetilde{S}$ is the Kullback-Leibler projection of $\\widetilde{\\mathbb {}}^{\\otimes n}$ on the set of all probability distributions on $\\mathbb {R}^{n}$ centered at some point $\\mathbf {x}^{\\ast }$ which is close to $\\underline{\\mathbf {x}}$ .",
"For this sake, we need to have at hand a proxy of $\\underline{\\lambda }$ or, equivalently, a preliminary guess $\\mathbf {\\widetilde{Q}}^{\\ast }$ of $\\underline{\\mathbf {\\widetilde{Q}}}:=\\arg \\inf _{\\mathbf {\\widetilde{Q}} \\in \\widetilde{\\mathbf {\\Omega }}}\\sum _{k=1}^{K} \\widetilde{p}_{k} \\cdot \\widetilde{\\varphi }(\\widetilde{q}_{k}/\\widetilde{p}_{k})$ .",
"This guess is by no means produced in order to provide a direct estimate of $D_{\\widetilde{\\varphi } }(\\widetilde{\\mathbf {\\Omega }},\\widetilde{{P}})$ but merely to provide the IS distribution $\\widetilde{S}$ which in turn leads to a sharp estimate of $D_{\\widetilde{\\varphi } }(\\widetilde{\\mathbf {\\Omega }},\\widetilde{{P}})$ .",
"Proxy method 1: in some cases we might have at hand some particular point $\\mathbf {\\widetilde{Q}}^{\\ast } :=\\left(\\widetilde{q}_{1}^{\\ast },..,\\widetilde{q}_{K}^{\\ast }\\right)$ in $\\widetilde{\\mathbf {\\Omega }}$ ; the resulting IS distribution $\\widetilde{S}$ with $\\mathbf {\\widetilde{Q}}$ substituted by $\\mathbf {\\widetilde{Q}^{\\ast }}$ is not optimal in the sense of [SB], but anyhow produces an estimator with good hitting rate, possibly with a loss in the variance.",
"A simple way to obtain such a point $\\mathbf {\\widetilde{Q}^{\\ast }}$ in $\\widetilde{\\mathbf {\\Omega }}$ is to simulate runs of (say) $M-$ variate i.i.d.",
"vectors $\\mathbf {\\widetilde{W}}$ under $\\widetilde{\\mathbb {}}^{\\otimes M}$ until the first time where $\\xi _{M}^{\\mathbf {\\widetilde{W}}}$ belongs to $\\widetilde{\\mathbf {\\Omega }}$ ; then we set $\\mathbf {\\widetilde{Q}^{\\ast }} := \\xi _{M}^{\\mathbf {\\widetilde{W}}}$ for the succeeding realization $\\mathbf {\\widetilde{W}}$ .",
"Before we proceed, it is useful to mention that the need for a drastic fall in the number of simulation runs pertains for cases when $D_{\\widetilde{\\varphi } }(\\widetilde{\\mathbf {\\Omega }},\\widetilde{{P}})$ is large.",
"The following construction is suited to this case, which is of relevance in applications both in optimization and in statistics when choosing between competing models none of which is assumed to represent the true one, but merely less inadequate ones.",
"Proxy method 2: when $D_{\\widetilde{\\varphi } }(\\widetilde{\\mathbf {\\Omega }},\\widetilde{{P}})$ is presumably large, we make use of asymptotic approximation to get a proxy of $\\underline{\\mathbf {\\widetilde{Q}}}$ .",
"For this, we define a sampling distribution on $\\mathbb {R}^{K}$ fitted to the divergence through $f(\\mathbf {\\widetilde{Q}}):=C \\cdot \\exp \\left( -\\sum _{k=1}^{K} \\widetilde{p}_{k} \\cdot \\widetilde{\\varphi }(\\widetilde{q}_{k}/\\widetilde{p}_{k})\\right) \\ = \\ C \\cdot \\exp \\left( -D_{\\widetilde{\\varphi }}\\left( \\mathbf {\\widetilde{Q}},\\widetilde{{P}}\\right)\\right)$ where $C$ is a normalizing constant.",
"Let $\\mathbf {T}$ be a $K-$ variate random variable with density $f$ .",
"The distribution of $\\mathbf {T}$ given $\\left( \\mathbf {T} \\in \\widetilde{\\mathbf {\\Omega }}\\right)$ concentrates around $\\arg \\inf _{\\widetilde{\\mathbf {Q}}\\in \\widetilde{\\mathbf {\\Omega }} }D_{\\widetilde{\\varphi } }(\\widetilde{\\mathbf {Q}},\\widetilde{{P}})$ when $D_{\\widetilde{\\varphi } }(\\widetilde{\\mathbf {\\Omega }},\\widetilde{{P}})$ is large.",
"Indeed, for any $\\widetilde{\\mathbf {Q}}\\in \\widetilde{\\mathbf {\\Omega }}$ denote by $\\mathbf {V}_{\\varepsilon }(\\widetilde{\\mathbf {Q}})$ a small neighborhood of $\\widetilde{\\mathbf {Q}}$ in $\\mathbb {R}^{K}$ with radius $\\varepsilon $ ; clearly, the probability of the event $\\left( \\mathbf {T} \\in \\mathbf {V}_{\\varepsilon }(\\widetilde{\\mathbf {Q}})\\right)$ when restricted to $\\widetilde{\\mathbf {Q}}\\in \\widetilde{\\mathbf {\\Omega }}$ is maximum when $\\widetilde{\\mathbf {Q}} = \\underline{\\widetilde{\\mathbf {Q}}}$ , where $\\underline{\\widetilde{\\mathbf {Q}}}$ is the “dominating point of $\\widetilde{\\mathbf {\\Omega }}$ ” in the sense that $\\underline{\\mathbf {\\widetilde{Q}}} := \\mathfrak {D}^{-1} \\underline{\\lambda }$ is the above-defined transform of the dominating point $\\underline{\\lambda }$ (assuming uniqueness); a precise argumentation under adequate conditions is postponed to Appendix G. Accordingly, we obtain a proxy $\\mathbf {\\widetilde{Q}}^{\\ast }$ of $\\underline{\\mathbf {\\widetilde{Q}}}$ by simulating a sequence of independent $K-$ variate random variables $\\mathbf {T}_{1},\\ldots $ with distribution (REF ) until (say) $\\mathbf {T}_{m}$ belongs to $\\widetilde{\\mathbf {\\Omega }}$ and set $\\mathbf {\\widetilde{Q}}^{\\ast }:=\\mathbf {T}_{m}$ .",
"To proceed with the derivation of the IS sampling distribution $\\widetilde{S}$ on $\\mathbb {R}^{n}$ , we fix $\\mathbf {\\widetilde{Q}}^{\\ast } :=\\left(\\widetilde{q}_{1}^{\\ast },..,\\widetilde{q}_{K}^{\\ast }\\right)$ to be a proxy of $\\underline{\\mathbf {\\widetilde{Q}}}$ or an initial guess in $\\widetilde{\\mathbf {\\Omega }}$ .",
"As an intermediate step, we construct the probability distribution $\\widetilde{U}_{k}$ on $\\mathbb {R}$ given by $d\\widetilde{U}_{k}(v) \\ := \\ \\exp \\left( \\tau _{k} \\cdot v -\\Lambda _{\\widetilde{\\mathbb {}}}(\\tau _{k})\\right) d\\widetilde{\\mathbb {}}(v)=\\frac{\\exp \\left(\\tau _{k} \\cdot v\\right)}{MGF_{\\widetilde{\\mathbb {}}}(\\tau _{k})} \\, d\\widetilde{\\mathbb {}}(v)$ where $\\tau _{k} \\in int(dom(MGF_{\\widetilde{\\mathbb {}}}))$ is the unique solution of the equation $\\Lambda _{\\widetilde{\\mathbb {}}}^{\\prime }\\left( \\tau _{k}\\right) =\\frac{\\widetilde{q}_{k}^{\\ast }}{\\widetilde{p}_{k}}$ and thus — by relation (REF ) of Appendix F — we can compute explicitly $\\tau _{k} = \\ \\widetilde{\\varphi }^{\\, \\prime }\\left(\\frac{\\widetilde{q}_{k}^{\\ast }}{\\widetilde{p}_{k}}\\right) \\ .$ Therefore, $\\widetilde{U}_{k}$ is the Kullback-Leibler projection of $\\widetilde{\\mathbb {}}$ on the class of all probability distributions on $\\mathbb {R}$ whose expectation is $\\widetilde{q}_{k}^{\\ast }$ .",
"As a side remark, notice that one possible way of obtaining an explicit form of the probability distribution $\\widetilde{U}_{k}$ may be by identification through its moment generating function $dom(MGF_{\\widetilde{\\mathbb {}}})-\\tau _{k} \\ \\ni \\ z \\ \\mapsto MGF_{\\widetilde{U}_{k}}(z) =\\frac{MGF_{\\widetilde{\\mathbb {}}}(z+\\tau _{k})}{MGF_{\\widetilde{\\mathbb {}}}(\\tau _{k})}$ of which all ingredients are principally available.",
"For instance, this will be used in Example REF below.",
"From (REF ), we define $\\widetilde{S}_{k}:=\\underbrace{\\widetilde{U}_{k} \\otimes \\cdots \\otimes \\widetilde{U}_{k}}_{n_{k}\\text{times}}\\qquad \\textrm {for all $ {1,...,K}$},$$whence\\begin{eqnarray}d\\widetilde{S}_{k}\\left( v_{k,1},\\ldots ,v_{k,n_{k}}\\right) =\\exp \\Big ( \\sum _{i\\in I_{k}^{(n)}}\\tau _{k} \\cdot v_{k,i}-n_{k} \\cdot \\Lambda _{\\widetilde{\\mathbb {}}}(\\tau _{k})\\Big ) \\ d\\widetilde{\\mathbb {}}\\left(v_{k,1}\\right)\\cdots d\\widetilde{\\mathbb {}}\\left(v_{k,n_{k}}\\right),\\\\\\end{eqnarray}which manifests $ Sk$ as the Kullback-Leibler projection of$ nktimes$on the class of all probability distributions on $ Rk$whose expectation vector is$ Q = (q1,...,qK) Rk$.",
"Let now\\begin{equation}\\widetilde{S} := \\widetilde{S}_{1} \\otimes \\cdots \\otimes \\widetilde{S}_{K} \\, ,\\end{equation}which therefore satisfies (recall that $ k=1K nk =n$)\\begin{equation}d\\widetilde{S}\\left( v_{1,1},\\ldots v_{1,n_{1}},\\ldots , v_{K,1},\\ldots v_{K,n_{K}}\\right)=\\exp \\Big (\\sum _{k=1}^{K}\\sum _{i\\in I_{k}^{(n)}} \\big ( \\tau _{k} \\cdot v_{k,i} - n_{k} \\cdot \\Lambda _{\\widetilde{\\mathbb {}}}(\\tau _{k}) \\big )\\Big ) \\,d\\widetilde{\\mathbb {}}^{\\otimes n}\\left( v_{1,1},\\ldots v_{1,n_{1}},\\ldots , v_{K,1},\\ldots v_{K,n_{K}}\\right).\\ \\end{equation}The same procedure with all $ qk$substituted by the coordinates $qk$of $Q$produces $ Sopt$.",
"Therefore, $ S$ is a substitutefor $ Sopt$ with the change in the centering fromthe unknown vector $Q$to its proxy $ Q$.$ As a straightforward consequence of (REF ) and (), we obtain the improved IS estimator of $\\mathbb {\\Pi }\\left[\\xi _{n}^{\\mathbf {\\widetilde{W}}}\\in \\widetilde{\\mathbf {\\Omega }}\\right]$ as $\\widehat{\\widetilde{\\Pi }}_{L}^{improved}\\ =\\ \\frac{1}{L}\\sum _{\\ell =1}^{L}\\mathbf {1}_{\\mathbf {\\Lambda }}( T( \\mathbf {\\widetilde{V}}^{(\\ell )})) \\cdot \\prod _{k=1}^{K} IS_{k}(\\mathbf {\\widetilde{V}}_{k}^{(\\ell )})$ where $\\mathbf {\\widetilde{V}}_{k}^{(\\ell )} := \\left( \\widetilde{V}_{i}^{(\\ell )} \\right)_{i \\in I_{k}^{(n)}}$ is the $k-$ th block of the $\\ell -$ th replication $\\mathbf {\\widetilde{V}}^{(\\ell )}$ of $\\mathbf {\\widetilde{V}}$ under $S$ , and the $k-$ th importance-sampling factor is $\\widetilde{IS}_{k}(v_{k,1},\\ldots ,v_{k,n_{k}}) \\ := \\ \\frac{d\\widetilde{\\mathbb {}}^{\\otimes n_{k}}}{d\\widetilde{S}_{k}}\\left( v_{k,1},\\ldots ,v_{k,n_{k}}\\right)= \\exp \\Big (n_{k} \\cdot \\Lambda _{\\widetilde{\\mathbb {}}}(\\tau _{k}) \\, - \\, \\tau _{k}\\cdot \\sum _{i=1}^{n_{k}} v_{k,i} \\Big )\\nonumber $ with $n_{k} = card(I_{k}^{(n)})$ .",
"Summing up things, we arrive at the following algorithm in case that $\\widetilde{\\mathbf {\\Omega }}$ has a unique dominating point (in the above-defined sense): Step D1 Exemplarily, we start with proxy method 2 (the other proxy method 1 works analogously): get a proxy $\\mathbf {\\widetilde{Q}}^{\\ast }$ of $\\underline{\\mathbf {\\widetilde{Q}}}$ by simulating a sequence of independent $K-$ variate random variables $\\mathbf {T}_{1},\\ldots $ with distribution (REF ) until (say) $\\mathbf {T}_{m}$ belongs to $\\widetilde{\\mathbf {\\Omega }}$ and set $\\mathbf {\\widetilde{Q}}^{\\ast }:=\\mathbf {T}_{m}$ .",
"Step D2 For all $k$ in $\\lbrace 1,\\ldots ,K\\rbrace $ compute $\\tau _{k} = \\ \\widetilde{\\varphi }^{\\, \\prime }\\left(\\frac{\\widetilde{q}_{k}^{\\ast }}{\\widetilde{p}_{k}}\\right)$ .",
"Step D3 For all $\\ell $ in $\\lbrace 1,\\ldots ,L\\rbrace $ perform a run of $\\mathbf {\\widetilde{V}}^{(\\ell )}$ under $\\widetilde{S}$ as follows: For all $k$ in $\\lbrace 1,\\ldots ,K\\rbrace $ simulate $n_{k}$ i.i.d.",
"random variables $\\widetilde{V}_{k_{1}}^{(\\ell )},\\ldots ,\\widetilde{V}_{k_{n_{k}}}^{(\\ell )}$ with common distribution $\\widetilde{U}_{k}$ defined in (REF ).",
"Set $\\mathbf {\\widetilde{V}}_{k}^{(\\ell )} :=(\\widetilde{V}_{k_{1}}^{(\\ell )},\\ldots ,\\widetilde{V}_{k_{n_{k}}}^{(\\ell )})$ to be the corresponding row vector.",
"Construct $\\mathbf {\\widetilde{V}}^{(\\ell )}$ as the row vector obtained by concatenating the $\\mathbf {\\widetilde{V}}_{k}^{(\\ell )}$ , i.e.",
"$\\mathbf {\\widetilde{V}}^{(\\ell )}:=\\left( \\mathbf {\\widetilde{V}}_{1}^{(\\ell )},\\ldots ,\\mathbf {\\widetilde{V}}_{K}^{(\\ell )}\\right) \\, ,$ and make use of $\\widehat{\\widetilde{\\Pi }}_{L}^{improved}$ given in (REF ) with the $\\tau _{k}$ 's obtained in Step D2 above to define (in the light of (REF ), (REF )) the BS minimum-distance estimator $\\widehat{D_{\\varphi }}(\\mathbf {\\Omega },\\mathbf {P})\\ := \\ \\widehat{D_{\\widetilde{\\varphi }}}(\\widetilde{\\mathbf {\\Omega }},\\widetilde{{P}})\\ := \\ - \\frac{1}{n}\\log \\widehat{\\widetilde{\\Pi }}_{L}^{improved} \\ .$ For many cases, the simulation burden needed for the computation of $\\widehat{\\widetilde{\\Pi }}_{L}^{improved}$ — and thus of $\\widehat{D_{\\varphi }}(\\mathbf {\\Omega },\\mathbf {P})$ — can be drastically reduced, especially for high dimensions $K$ and large sample size $n \\cdot L$ .",
"In fact, in terms of the notations $n_{k}:=card(I_{k}^{(n)})$ , $\\widehat{\\mathit {W}}_{k}^{(\\ell )}:=\\sum _{i\\in I_{k}^{(n)}}\\widetilde{V}_{i}^{(\\ell )}$ and $\\widetilde{ISF}_{k}(x) \\ := \\ \\frac{d\\widetilde{\\mathbb {}}^{\\ast n_{k}}}{d\\widetilde{U}_{k}^{\\ast n_{k}}}(x) \\ = \\ \\exp (n_{k} \\cdot \\Lambda _{\\widetilde{\\mathbb {}}}(\\tau _{k}) \\, - \\, x \\cdot \\tau _{k} )$ (where $\\widetilde{\\mathbb {}}^{\\ast n_{k}}$ is the $n_{k}-$ convolution of the measure $\\widetilde{\\mathbb {}}$ ), one can rewrite (REF ) as $\\widehat{\\widetilde{\\Pi }}_{L}^{improved}=\\frac{1}{L}\\sum _{\\ell =1}^{L}\\mathbf {1}_{\\mathbf {\\Lambda }}\\big ( \\,\\big (\\frac{1}{n_{1}} \\widehat{\\mathit {W}}_{1}^{(\\ell )},\\ldots , \\frac{1}{n_{K}} \\widehat{\\mathit {W}}_{K}^{(\\ell )}\\big ) \\, \\big ) \\cdot \\prod \\limits _{k=1}^{K}\\widetilde{ISF}_{k}(\\widehat{\\mathit {W}}_{k}^{(\\ell )}) .$ with $K-$ vector $\\big (\\frac{1}{n_{1}} \\widehat{\\mathit {W}}_{1}^{(\\ell )},\\ldots , \\frac{1}{n_{K}} \\widehat{\\mathit {W}}_{K}^{(\\ell )} \\big )$ .",
"Clearly, the random variable $\\widehat{\\mathit {W}}_{k}^{(\\ell )}$ ($k=1, \\ldots , K$ ) has distribution $\\widetilde{U}_{k}^{\\ast n_{k}}$ .",
"Hence, if $\\widetilde{U}_{k}^{\\ast n_{k}}$ can be explicitly constructed, then for the computation of $\\widehat{\\widetilde{\\Pi }}_{L}^{improved}$ it suffices to simulate the $K \\cdot L$ random variables $\\widehat{\\mathit {W}}_{k}^{(\\ell )}$ rather than the $n \\cdot L$ random variables $\\widetilde{V}_{i}^{(\\ell )}$ ; notice that according to the right-hand side of (REF ), one can explicitly compute $ISF_{k}\\left( \\cdot \\right)$ which can be interpreted as Importance Sampling Factor pertaining to the block $k$.",
"In the case that $\\widetilde{\\mathbb {}}$ is infinitely divisible, simulation issues may become especially comfortable.",
"In the following, we exemplarily demonstrate the tractability of this reduction effect, for the BS minimization of the important power divergences (for which the infinite divisibility holds): Example 58 Let $\\varphi _{\\gamma }$ ($\\gamma \\in \\mathbb {R}\\backslash ]1,2[$ ) be the power divergence generator from the Examples REF and REF , $\\mathbf {P} \\in \\mathbb {R}_{> 0}^{K}$ , $M_{\\mathbf {P}}:=\\sum _{i=1}^{K}p_{i}>0$ and $n_{k} = n \\cdot p_{k} \\in \\mathbb {N}$ where we have employed our notation $n_{k}= n \\cdot p_{k} \\in \\mathbb {N}$ for all $k \\in \\lbrace 1,\\ldots ,K\\rbrace $ .",
"Moreover, let $\\mathbf {\\widetilde{Q}}^{\\ast } :=\\left(\\widetilde{q}_{1}^{\\ast },\\ldots ,\\widetilde{q}_{K}^{\\ast }\\right)$ be a proxy obtained by either proxy method 1 or 2.",
"Case 1: Example REF (a): $\\gamma =0,\\widetilde{c}>0$ .",
"There holds $\\widetilde{U}_{k}^{\\ast n_{k}}=GAM\\left( \\widetilde{c} \\cdot M_{\\mathbf {P}} - \\tau _{k},n_{k}\\cdot \\widetilde{c} \\cdot M_{\\mathbf {P}}\\right)$ , with $\\tau _{k}=\\widetilde{c} \\cdot M_{\\mathbf {P}} \\cdot ( 1-\\frac{p_{k}}{M_{\\mathbf {P}} \\cdot \\widetilde{q}_{k}^{\\ast }})$ for $\\widetilde{q}_{k}^{\\ast } >0$ (the latter is equivalent to $\\tau _{k} < \\widetilde{c} \\cdot M_{\\mathbf {P}}$ ).",
"Moreover, for all $x >0$ one gets $\\widetilde{ISF}_{k}(x)=\\left( \\frac{\\widetilde{c} \\cdot M_{\\mathbf {P}}}{\\widetilde{c}\\cdot M_{\\mathbf {P}} - \\tau _{k}}\\right)^{n_{k}\\cdot \\widetilde{c}\\cdot M_{\\mathbf {P}}}\\cdot e^{-\\tau _{k} \\cdot x}$ .",
"Case 2: REF (b): $\\gamma \\in \\left( 0,1\\right) ,\\widetilde{c}>0$ .",
"We derive $\\widetilde{U}_{k}^{\\ast n_{k}}=C\\big ( POI( n_{k}\\cdot \\breve{\\theta }),GAM\\big (\\frac{\\widetilde{c} \\cdot M_{\\mathbf {P}}}{1-\\gamma } - \\tau _{k},\\frac{\\gamma }{1-\\gamma } \\big ) \\big )$ with $\\breve{\\theta }:= \\frac{\\widetilde{c} \\cdot M_{\\mathbf {P}}}{\\gamma }\\cdot \\big (\\frac{(\\gamma -1) \\cdot \\tau _{k}}{\\widetilde{c} \\cdot M_{\\mathbf {P}}}+1\\big )^{\\gamma /(\\gamma -1)}$ and $\\tau _{k} = \\widetilde{c} \\cdot M_{\\mathbf {P}} \\cdot \\frac{1-\\big (\\frac{\\widetilde{q}_{k}^{\\ast } \\cdot M_{\\mathbf {P}}}{p_{k}}\\big )^{\\gamma -1}}{1-\\gamma }$ for $\\widetilde{q}_{k}^{\\ast } >0$ .",
"Furthermore, $\\widetilde{ISF}_{k}(x)=e^{-\\tau _{k}x} \\cdot \\exp \\left( \\frac{n_{k}\\cdot \\widetilde{c} \\cdot M_{\\mathbf {P}}}{\\gamma }\\cdot \\left(\\left(1+ \\frac{\\gamma -1}{\\widetilde{c} \\cdot M_{\\mathbf {P}}} \\cdot \\tau _{k}\\right) ^{\\frac{\\gamma }{\\gamma -1}}-1 \\right)\\right), \\qquad x \\ge 0 ,$ (where $x=0$ covers the atom at zero).",
"Case 3: Example REF (c): $\\gamma =2,\\widetilde{c}>0$ .",
"One gets $\\widetilde{U}_{k}^{\\ast n_{k}}=N(n_{k}\\cdot (1+\\frac{\\tau _{k}}{\\widetilde{c} \\cdot M_{\\mathbf {P}}}),\\frac{n_{k}}{\\widetilde{c} \\cdot M_{\\mathbf {P}}})$ with $\\tau _{k}=\\widetilde{c} \\cdot M_{\\mathbf {P}} \\cdot ( \\frac{\\widetilde{q}_{k}^{\\ast } \\cdot M_{\\mathbf {P}}}{p_{k}} - 1 ) $ for $\\widetilde{q}_{k}^{\\ast } \\in \\mathbb {R}$ .",
"Moreover, for all $x \\in \\mathbb {R}$ one obtains $\\widetilde{ISF}_{k}(x)= \\exp \\big (\\frac{n_{k} \\cdot \\tau _{k}^2}{2\\widetilde{c} \\cdot M_{\\mathbf {P}}}- (x- n_{k}) \\cdot \\tau _{k} \\big )$ .",
"Case 4: Example REF (d): $\\gamma <0,\\widetilde{c}>0$ .",
"It holds that $\\widetilde{U}_{k}^{\\ast n_{k}}$ has the (Lebesgue-)density $f_{\\widetilde{U}_{k}^{\\ast n_{k}}}(x)\\ := \\ \\frac{\\exp ((\\tau _{k} - \\frac{\\widetilde{c}\\cdot M_{\\mathbf {P}}}{1-\\gamma })\\cdot x)}{\\exp \\left(n_{k} \\cdot \\frac{\\widetilde{c}\\cdot M_{\\mathbf {P}}}{\\gamma }\\cdot (1+\\frac{\\gamma -1}{\\widetilde{c} \\cdot M_{\\mathbf {P}}}\\cdot \\tau _{k})^{\\gamma /(\\gamma -1)}\\right)}\\cdot f_{\\breve{\\breve{Z}}}(x) \\cdot {1}_{]0,\\infty [}(x),\\qquad x \\in \\mathbb {R},$ where $\\tau _{k} = \\widetilde{c} \\cdot M_{\\mathbf {P}} \\cdot \\frac{1-\\big (\\frac{\\widetilde{q}_{k}^{\\ast } \\cdot M_{\\mathbf {P}}}{p_{k}}\\big )^{\\gamma -1}}{1-\\gamma }$ for $\\widetilde{q}_{k}^{\\ast } >0$ , and $\\breve{\\breve{Z}}$ is a random variable with density $f_{\\breve{\\breve{Z}}}$ of a stable law with parameter-quadruple $(\\frac{-\\gamma }{1-\\gamma },1,0,- n_{k} \\cdot \\frac{(\\widetilde{c}\\cdot M_{\\mathbf {P}})^{1/(1-\\gamma )} \\cdot (1-\\gamma )^{-\\gamma /(1-\\gamma )}}{\\gamma })$ (analogously to $\\breve{Z}$ of Example 40 (d) but with $\\widetilde{c}$ replaced by $\\widetilde{c}\\cdot M_{\\mathbf {P}}$ ).",
"Also, $\\widetilde{ISF}_{k}(x)=e^{-\\tau _{k}x} \\cdot \\exp \\left( \\frac{n_{k}\\cdot \\widetilde{c} \\cdot M_{\\mathbf {P}}}{\\gamma }\\cdot \\left(\\left(1+ \\frac{\\gamma -1}{\\widetilde{c} \\cdot M_{\\mathbf {P}}} \\cdot \\tau _{k}\\right) ^{\\frac{\\gamma }{\\gamma -1}}-1 \\right)\\right), \\qquad x >0.$ Case 5 : Example REF (e): $\\gamma >2,\\widetilde{c}>0$ .",
"We derive that $\\widetilde{U}_{k}^{\\ast n_{k}}$ has the (Lebesgue-)density $f_{\\widetilde{U}_{k}^{\\ast n_{k}}}(x)\\ := \\ \\frac{\\exp ((\\tau _{k} + \\frac{\\widetilde{c}\\cdot M_{\\mathbf {P}}}{\\gamma -1})\\cdot x)}{\\exp \\left(n_{k} \\cdot \\frac{\\widetilde{c}\\cdot M_{\\mathbf {P}}}{\\gamma }\\cdot (1+\\frac{\\gamma -1}{\\widetilde{c} \\cdot M_{\\mathbf {P}}}\\cdot \\tau _{k})^{\\gamma /(\\gamma -1)}\\right)}\\cdot f_{\\breve{\\breve{Z}}}(-x) ,\\qquad x \\in \\mathbb {R},$ where $\\tau _{k} = - \\frac{\\widetilde{c} \\cdot M_{\\mathbf {P}}}{\\gamma -1} \\cdot \\big (1- \\big (\\frac{\\widetilde{q}_{k}^{\\ast } \\cdot M_{\\mathbf {P}}}{p_{k}}\\big )^{\\gamma -1}\\cdot {1}_{]0,\\infty [}(\\widetilde{q}_{k}^{\\ast }) \\big )$ for $\\widetilde{q}_{k}^{\\ast } \\in \\mathbb {R}$ , and $\\breve{\\breve{Z}}$ is a random variable with density $f_{\\breve{\\breve{Z}}}$ of a stable law with parameter-quadruple $(\\frac{\\gamma }{\\gamma -1},1,0,n_{k} \\cdot \\frac{(\\widetilde{c}\\cdot M_{\\mathbf {P}})^{1/(1-\\gamma )} \\cdot (\\gamma -1)^{\\gamma /(\\gamma -1)}}{\\gamma })$ (analogously to $\\breve{Z}$ of Example 40 (e) but with $\\widetilde{c}$ replaced by $\\widetilde{c}\\cdot M_{\\mathbf {P}}$ ).",
"Furthermore, $\\widetilde{ISF}_{k}(x)=e^{-\\tau _{k}x} \\cdot \\exp \\left( \\frac{n_{k}\\cdot \\widetilde{c} \\cdot M_{\\mathbf {P}}}{\\gamma }\\cdot \\left(\\left(1+ \\frac{\\gamma -1}{\\widetilde{c} \\cdot M_{\\mathbf {P}}} \\cdot \\tau _{k}\\right) ^{\\frac{\\gamma }{\\gamma -1}}-1 \\right)\\right), \\qquad x \\in \\mathbb {R}.$ Case 6: Example REF (a): $\\gamma =1,\\widetilde{c}>0$ , anchor point $c=0$ .",
"It holds that $\\widetilde{U}_{k}^{\\ast n_{k}}$ is the probability distribution $\\frac{1}{\\widetilde{c} \\cdot M_{\\mathbf {P}}} \\cdot POI\\left(n_{k} \\cdot \\widetilde{c} \\cdot M_{\\mathbf {P}} \\cdot \\exp (\\frac{\\tau _{k}}{\\widetilde{c} \\cdot M_{\\mathbf {P}}})\\right) $ with support on the lattice $\\left\\lbrace \\frac{j}{\\widetilde{c} \\cdot M_{\\mathbf {P}}}, \\, j\\in \\mathbb {N}_{0}\\right\\rbrace $ , where $\\tau _{k}=\\widetilde{c} \\cdot \\log \\left( \\frac{M_{\\mathbf {P}} \\cdot \\widetilde{q}_{k}^{\\ast }}{p_{k}}\\right) $ for $\\widetilde{\\omega } _{k} >0$ .",
"Moreover, for all $j \\in \\mathbb {N}_{0}$ we obtain (by setting $x:= \\frac{j}{\\widetilde{c} \\cdot M_{\\mathbf {P}}}$ ) $\\widetilde{ISF}_{k}\\left(\\frac{j}{\\widetilde{c} \\cdot M_{\\mathbf {P}}}\\right)= \\exp \\left(n_{k} \\cdot \\widetilde{c} \\cdot M_{\\mathbf {P}} \\cdot \\left( \\exp \\left( \\frac{\\tau _{k}}{\\widetilde{c} \\cdot M_{\\mathbf {P}}}\\right) -1 \\right)- m \\cdot \\frac{\\tau _{k}}{\\widetilde{c} \\cdot M_{\\mathbf {P}}}\\right) .$ Case 7: Example REF (b): $\\gamma =1,\\widetilde{c}=1,$ anchor point $c\\in \\mathbb {R}$ .",
"For $M_{P}=1$ , $\\widetilde{U}_{k}^{\\ast n_{k}}$ is the shifted Poisson distribution $ POI\\left(n_{k} \\cdot e^{c +\\tau _{k}}\\right) + n_{k} \\cdot (1-e^{c})$ with support on the lattice $\\left\\lbrace j + n_{k} \\cdot (1-e^{c}), \\, j\\in \\mathbb {N}_{0}\\right\\rbrace $ , where $\\tau _{k}= \\log \\big (\\frac{\\widetilde{q}_{k}^{\\ast }}{p_{k}} + e^{c}-1\\big )-c$ for $\\widetilde{q}_{k}^{\\ast } > p_{k} \\cdot (1-e^{c})$ .",
"Furthermore, for all $j \\in \\mathbb {N}_{0}$ we obtain (by setting $x:= j + n_{k} \\cdot (1-e^{c})$ ) $\\widetilde{ISF}_{k}\\left(j + n_{k} \\cdot (1-e^{c})\\right)= \\exp \\left(n_{k} \\cdot e^{c} \\cdot (e^{\\tau _{k}} -1) - j \\cdot \\tau _{k} \\right).$ Notice that the mass of $\\widetilde{U}_{k}^{\\ast n_{k}}$ at zero depends on the value of the anchor point $c$ , since $\\widetilde{U}_{k}^{\\ast n_{k}}[\\lbrace 0\\rbrace ] >0$ if and only if $c=\\log (1+\\frac{\\ell }{n_{k}})$ for some $\\ell \\in \\mathbb {N}_{0}$ ; moreover, $\\widetilde{U}_{k}^{\\ast n_{k}}\\big [ \\ ]0,\\infty [ \\ \\big ] =1$ if $c <0$ and $\\widetilde{U}_{k}^{\\ast n_{k}}\\big [ \\ ]-\\infty ,0[ \\ \\big ] >0$ if $c >0$ .",
"Remark 59 (a) One can explicitly see in all cases of the above Example REF that all ingredients for computation are at hand.",
"(b) For both Cases 4 and 5 in the above Example REF , algorithms for simulation can be obtained by adapting e.g.",
"the works of Devroye [111] and Devroye & James [112].",
"In the previous Subsection REF , as a first step we have estimated $\\mathbb {\\Pi }\\left[\\xi _{n}^{\\mathbf {\\widetilde{W}}}\\in \\widetilde{\\mathbf {\\Omega }}\\right]$ in terms of the improved IS estimator $\\widehat{\\widetilde{\\Pi }}_{L}^{improved}$ .",
"From this, as a second step, we have derived — on the basis of Theorem REF — the estimator $\\widehat{D_{\\varphi }}(\\mathbf {\\Omega },\\mathbf {P})\\ := \\ - \\frac{1}{n}\\log \\widehat{\\widetilde{\\Pi }}_{L}^{improved} \\ \\nonumber \\qquad \\textrm {(cf.",
"(\\ref {estimator minimization}))}$ of the minimum distance $D_{\\varphi }(\\mathbf {\\Omega },\\mathbf {P}):= \\inf _{\\mathbf {Q}\\in \\mathbf {\\Omega } } D_{\\varphi }(\\mathbf {Q},\\mathbf {P})$ , where $\\mathbf {P} \\in \\mathbb {R}_{> 0}^{K}$ and $\\mathbf {\\Omega }\\subset \\mathbb {R}^{K}$ .",
"Recall that $\\widetilde{\\mathbf {\\Omega }} :=\\mathbf {\\Omega } /M_{\\mathbf {P}}$ with $M_{\\mathbf {P}}:=\\sum _{i=1}^{K}p_{i}>0$ , and that $\\xi _{n}^{\\mathbf {\\widetilde{W}}}=\\Big (\\frac{1}{n}\\sum _{i\\in I_{1}^{(n)}}\\widetilde{W}_{i},\\ldots ,\\frac{1}{n}\\sum _{i\\in I_{K}^{(n)}}\\widetilde{W}_{i}\\Big )\\hspace{56.9055pt} \\text{(cf.",
"(\\ref {Xi_n^W vector}))}\\nonumber $ where $\\mathbf {\\widetilde{W}} := (\\widetilde{W}_{1}, \\ldots , \\widetilde{W}_{n})$ is a random vector consisting of components $\\widetilde{W}_{i}$ which are i.i.d.",
"copies of the random variable $\\widetilde{W}$ whose distribution is $\\mathbb {\\Pi }[\\widetilde{W}\\in \\cdot \\,]=\\widetilde{\\mathbb {}}[\\,\\cdot \\,]$ obeying the representation (REF ).",
"In contrast, we now proceed as follows: as a first step, we derive an improved estimator $\\widehat{\\Pi }_{L}^{improved}$ of $\\mathbb {\\Pi }_{X_{1}^{n}}\\left[\\xi _{n,\\mathbf {X}}^{w\\mathbf {W}}\\in \\textrm {\\right.$ $\\hspace{-6.544pt}$$}$$where $$\\hspace{-6.544pt}$ SK$is a set of probability vectors which satisfies theregularity properties (\\ref {regularity}) and the finiteness property (\\ref {def fi wrt Omega}).Recall that\\begin{eqnarray}\\xi _{n,\\mathbf {X}}^{w\\mathbf {W}} &:=&{\\left\\lbrace \\begin{array}{ll}\\left(\\frac{\\sum _{i \\in I_{1}^{(n)}}W_{i}}{\\sum _{k=1}^{K}\\sum _{i \\in I_{k}^{(n)}}W_{i}},\\ldots , \\frac{\\sum _{i \\in I_{K}^{(n)}}W_{i}}{\\sum _{k=1}^{K}\\sum _{i \\in I_{k}^{(n)}}W_{i}} \\right) ,\\qquad \\textrm {if } \\sum _{j=1}^{n} W_{j} \\ne 0, \\\\\\ (\\infty , \\ldots , \\infty ) =: \\infty , \\hspace{113.81102pt} \\textrm {if } \\sum _{j=1}^{n} W_{j} = 0,\\end{array}\\right.",
"}\\hspace{42.67912pt}\\textrm {(cf.",
"(\\ref {brostu3:fo.norweiemp.vec}))}\\nonumber \\end{eqnarray}where $ (Xi)iN$ is a sequence of random variableswith values in $ Y:={ d1,,dK}$such that\\begin{equation}\\lim _{n\\rightarrow \\infty } \\Big ( \\frac{n_{1}}{n}, \\ldots , \\frac{n_{K}}{n} \\Big ) = (p_{1}, \\ldots , p_{K})\\qquad \\textrm {a.s.} \\qquad \\textrm {cf.",
"((\\ref {cv emp measure X to P vector}))}\\nonumber \\end{equation}holds for some probability vector $ P := (p1, ..., pK) S> 0K$,by employing the notation$$n_{k} \\ := \\ card(\\bigl \\lbrace i \\in \\lbrace 1, \\ldots , n\\rbrace : \\ X_{i} = d_{k} \\bigr \\rbrace )\\ =: \\ card(I_{k}^{(n)})\\qquad \\textrm {(cf.",
"(\\ref {I^(n)_k for stat case}));}$$hence, on the $ k$-th block of indexes $ Ik(n)$ all the $ Xi$^{\\prime }s share the same value $ dk$.Moreover, recall that $ (Wi)i N$ is a familyof independent and identically distributed $ R-$valued random variableswith probability distribution $ [ ] := [W1 ]$being connected with the divergence generator $ (]a,b[)$ via the representability(\\ref {Phi Legendre of mgf(W)}),such that $ (Wi)i N$ is independent of $ (Xi)i N$.$ As a second step (see Subsubsection REF below), for the important special case of the power-divergence generators $\\varphi _{\\gamma }$ (cf.",
"(REF )) we employ the Propositions REF to REF in order to deduce via the corresponding $\\widehat{\\Pi }_{L}^{improved}$ the estimators (e.g.",
"for $\\gamma <0$ ) $\\widehat{D_{\\widetilde{c}\\cdot \\varphi _{\\gamma }}(\\textrm {\\Omega \\hspace{-6.544pt}\\Omega },{P})} \\ := \\ \\frac{\\widetilde{c}}{\\gamma \\cdot \\left(\\gamma -1\\right) }\\cdot \\left\\lbrace \\left( 1+\\frac{\\gamma }{\\widetilde{c}}\\cdot \\frac{1}{n}\\cdot \\log \\,\\widehat{\\Pi }_{L}^{improved}\\right) ^{1-\\gamma } -1 \\right\\rbrace ,\\qquad \\ $ of the minimum power divergences $D_{\\widetilde{c}\\cdot \\varphi _{\\gamma }}(\\textrm {$ $\\hspace{-6.544pt}$$},{P}) :=\\inf _{{Q}\\in \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega }}D_{\\widetilde{c}\\cdot \\varphi _{\\gamma }}({Q},{P})$$as well as connected estimators of important deterministic transformations thereof.$ As a third step (see Subsubsection REF below), on the basis of Subsubsection REF we derive estimators of bounds of $D_{\\varphi }(\\textrm {$$\\hspace{-6.544pt}$$},{P})$ for more general divergence generators $\\varphi $ .",
"Let us start with the above-mentioned first step, by remarking that the development of the estimator $\\widehat{\\Pi }_{L}^{improved}$ works quite analogously to that of $\\widehat{\\widetilde{\\Pi }}_{L}^{improved}$ in the previous Subsection REF .",
"To make this even more transparent, we employ the notation $p_{n,k}^{emp}:=n_{k}/n$ (cf.",
"(REF )) and label all random vectors of length $n$ in the same way as above: we sort the already given and thus fixed data $X_{i}$ 's in such a way that the first $n_{1}$ of them share the same value $d_{1}$ , and so on, until the last block with length $n_{K}$ in which the data have common value $d_{K}$ .",
"In the light of the above considerations, we could achieve a naive estimate $\\widehat{\\Pi }_{L}^{naive}$ of $\\mathbb {\\Pi }_{X_{1}^{n}}[\\xi _{n,\\mathbf {X}}^{w\\mathbf {W}}\\in \\textrm {$$\\hspace{-6.544pt}$$}]$ through the following procedure.",
"We simulate independently $L$ replicates $\\mathbf {W}^{(1)},\\ldots ,\\mathbf {W}^{(L)}$ of the vector $\\mathbf {W}:=\\left( W_{1},\\ldots ,W_{n}\\right) $ , with independent entries under $\\mathbb {}$ (cf.",
"(REF )); those realizations do not depend on the $X_{i}$ 's.",
"Then we construct $\\widehat{\\Pi }_{L}^{naive} :=\\frac{1}{L}\\sum _{\\ell =1}^{L}\\mathbf {1}_{\\textrm {\\Omega \\hspace{-5.40608pt}\\Omega }}\\left(\\xi _{n,\\mathbf {X}}^{w\\mathbf {W}^{(\\ell )}}\\right) .$ However, this procedure is time costly, since the estimate given in (REF ) has a very bad hit rate.",
"Hence, analogously to Subsection REF we apply again an “efficient Importance Sampling (IS)” scheme in the sense of Sadowsky & Bucklew [313].",
"This will involve the simulation of $L$ independent $n-$ tuples $\\mathbf {V}^{(\\ell )}\\mathbf {:=}\\left(V_{n}^{(\\ell )},\\ldots ,V_{n}^{(\\ell )}\\right) $ with common distribution $S$ on $\\mathbb {R}^{n}$ , such that $\\mathbb {}^{\\otimes n}$ is (measure-)equivalent with respect to $S$ .",
"In fact, we rewrite $\\mathbb {\\Pi }_{X_{1}^{n}}[\\xi _{n,\\mathbf {X}}^{w\\mathbf {W}}\\in \\textrm {$$\\hspace{-6.544pt}$$}]$ as $\\mathbb {\\Pi }_{X_{1}^{n}}[\\xi _{n,\\mathbf {X}}^{w\\mathbf {W}}\\in \\textrm {\\Omega \\hspace{-6.544pt}\\Omega }]= E_{S}\\Big [\\frac{\\mathrm {d}\\mathbb {} ^{\\otimes n}}{\\mathrm {d}S}(V_{1},\\ldots ,V_{n})\\cdot \\mathbf {1}_{\\textrm {\\Omega \\hspace{-5.40608pt}\\Omega }}(\\xi _{n,\\mathbf {X}}^{w\\mathbf {V}})\\Big ]$ where $S$ designates any IS distribution of the vector $\\mathbf {V:=}(V_{1},\\ldots ,V_{n})$ , and $E_{S}[ \\, \\cdot \\, ]$ denotes the corresponding expectation operation.",
"Notice that $S$ is a random probability distribution on $\\mathbb {R}^{n}$ ; in fact, $S$ is a conditional probability distribution given $X_{1}^{n}$ , and thus it would be more precise to write $S | X_{1}^{n}$ instead of $S$ ; for the sake of brevity, we omit $| X_{1}^{n}$ .",
"As a consequence of (REF ), for adequately chosen $S$ , an improved estimator of $\\mathbb {\\Pi }_{X_{1}^{n}}[\\xi _{n,\\mathbf {X}}^{w\\mathbf {W}}\\in \\textrm {$$\\hspace{-6.544pt}$$}]$ is given by $\\widehat{\\Pi }_{L}^{improved}:= \\frac{1}{L}\\sum _{\\ell =1}^{L}\\frac{\\mathrm {d}\\mathbb {}^{\\otimes n}}{\\mathrm {d}S}(V_{1}^{(\\ell )},\\ldots ,V_{n}^{(\\ell )})\\cdot \\mathbf {1}_{\\textrm {\\Omega \\hspace{-5.40608pt}\\Omega }}(\\xi _{n,\\mathbf {X}}^{w\\mathbf {V}^{(\\ell )}}) \\, ,$ which also estimates $\\inf _{{Q}\\in \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega } }\\ \\inf _{m\\ne 0}D_{\\varphi }(m\\cdot {Q},{P})$ by the virtue of ().",
"Let us now deal with the concrete construction of a reasonable $S$ .",
"Given some (typically) large integer $M$ , we start with the realization $\\mathbf {W}^{\\ast }:=\\left( W_{1}^{\\ast },\\ldots ,W_{M}^{\\ast }\\right)$ such that $\\mathbf {Q}^{\\ast }:=\\xi _{M,\\mathbf {X}}^{w\\mathbf {W}^{\\ast }} \\in int(\\textrm {$$\\hspace{-6.544pt}$$})$ .",
"This may be given in advance or it may be achieved by drawing replicates $\\mathbf {W} = (W_{1}, \\ldots , W_{M})$ under $\\mathbb {}^{\\otimes M}$ until the first time where $\\xi _{M,\\mathbf {X}}^{w\\mathbf {W}}$ belongs to $int(\\textrm {$$\\hspace{-6.544pt}$$})$ .",
"Notice that by the nature of $\\textrm {$$\\hspace{-6.544pt}$$}$ , $\\mathbf {Q}^{\\ast }$ is a probability vector which has the $K$ components $q_{k}^{\\ast }:=\\sum _{i=1}^{M}\\frac{W_{i}^{\\ast }}{\\sum _{j=1}^{M}W_{j}^{\\ast }}\\mathbf {1}_{\\lbrace d_{k}\\rbrace }(X_{i}),\\hspace{42.67912pt} k=1,\\ldots ,K.$ Before we proceed, let us give the substantial remark that changing $\\left( V_{1},\\ldots ,V_{n}\\right) $ drawn under $S$ to $\\left(c \\cdot V_{1}, \\ldots ,c \\cdot V_{n}\\right) $ for any $c\\ne 0$ yields $\\xi _{n,\\mathbf {X}}^{w\\mathbf {V}}= \\xi _{n,\\mathbf {X}}^{w\\, c\\cdot \\mathbf {V}}$ so that the distribution $S$ is not uniquely determined.",
"Amongst all candidates, we choose the — uniquely determined — $S$ which is the Kullback-Leibler projection of $\\mathbb {}^{\\otimes n}$ on the set of all probability distributions on $\\mathbb {R}^{n}$ such that the $K$ “non-normalized” moment constraints $E_{S}[\\xi _{n,\\mathbf {X}}^{\\mathbf {V}}] =\\xi _{M,\\mathbf {X}}^{\\mathbf {W}^{\\ast }}$ (rather than the normalized $E_{S}[ \\xi _{n,\\mathbf {X}}^{w\\mathbf {V}}] =\\xi _{M,\\mathbf {X}}^{w\\mathbf {W}^{\\ast }}$ ) are satisfied, with the non-normalized vectors $\\xi _{M,\\mathbf {X}}^{\\mathbf {W}^{\\ast }} :=\\left( \\frac{1}{M}\\sum _{j=1}^{M}W_{j}^{\\ast }\\right) \\cdot Q^{\\ast } =: \\overline{W^{\\ast }} \\cdot Q^{\\ast },\\qquad \\xi _{n,\\mathbf {X}}^{\\mathbf {V}} :=\\left( \\frac{1}{n}\\sum _{j=1}^{n}V_{j}\\right) \\cdot \\xi _{n,\\mathbf {X}}^{w\\mathbf {V}} \\ .$ As already indicated above, this projection $S$ is a well-determined unique distribution on $\\mathbb {R}^{n}$ and — as we shall see in Proposition REF below — it is such that $\\xi _{n,\\mathbf {X}}^{w\\mathbf {V}}$ belongs to $\\textrm {$$\\hspace{-6.544pt}$$}$ with probability bounded away from 0 as $n$ increases, when $\\left( V_{1},\\ldots ,V_{n}\\right) $ are drawn under $S$ .",
"Therefore, this IS distribution produces an estimate of $\\mathbb {\\Pi }_{X_{1}^{n}}[\\xi _{n,\\mathbf {X}}^{w\\mathbf {W}}\\in \\textrm {$$\\hspace{-6.544pt}$$}]$ .",
"In order to justify the above construction of $S$ , we give the following result, which states that the IS sampling distribution $S$ yields a good hitting rate.",
"Its proof will be given in Appendix H. Proposition 60 With the above definition of $S$ , $\\lim \\inf _{n\\rightarrow \\infty }$ $S\\left[ \\xi _{n,\\mathbf {X}}^{w\\mathbf {V}} \\in \\textrm {\\right.$$\\hspace{-6.544pt}$$} $ is bounded away from 0.",
"We now come to the detailed construction of $S$ .",
"The constraints (REF ) can be written in explicit form as $E_{S}\\Big [ \\, \\frac{1}{n_{k}}\\sum _{i\\in I_{k}^{(n)}} V_{i} \\, \\Big ]\\ = \\ \\overline{W^{\\ast }} \\cdot \\frac{q_{k}^{\\ast }}{p_{n,k}^{emp}} \\, , \\qquad k=1,\\ldots ,K.$ The distribution $S$ can be obtained by blocks.",
"Indeed, let us define $S^{k}$ as the Kullback-Leibler (KL) projection of $\\mathbb {}^{\\otimes n_{k}}$ on the set of all distributions on $\\mathbb {R}^{n_{k}}$ such that (REF ) holds.",
"We define the resulting $S$ as the product distribution of those $S^{k}$ 's.",
"To obtain the latter, we start by defining $U_{k}$ as the KL projection of $\\mathbb {} $ on the set of all measures $Q$ on $\\mathbb {R}$ under (REF ).",
"Then, $dU_{k}(v)= \\exp (\\tau _{k}v-\\Lambda _{\\mathbb {} }\\left( \\tau _{k}\\right) )\\, d\\mathbb {} (v) \\, ,$ where $\\tau _{k} \\in int(dom(MGF_{\\mathbb {}}))$ is the unique solution of the equation $\\Lambda _{\\mathbb {} }^{\\prime }\\left( \\tau _{k}\\right) =\\overline{W^{\\ast }} \\cdot \\frac{q_{k}^{\\ast }}{p_{n,k}^{emp}}$ and thus — by relation (REF ) of Appendix F — we can compute explicitly $\\tau _{k} = \\varphi ^{\\prime }\\left( \\frac{\\overline{W^{\\ast }} \\cdot q_{k}^{\\ast }}{p_{n,k}^{emp}} \\right) .$ The distribution $S^{k}$ is then defined by $S^{k}:=\\underbrace{U_{k}\\otimes \\cdots \\otimes U_{k}}_{n_{k}\\text{times}}$ from which we obtain $S := S^{1}\\otimes \\cdots \\otimes S^{K}.$ With this construction, it holds $\\frac{dS}{d\\mathbb {} ^{\\otimes n}}(v_{1},\\ldots ,v_{n})=\\exp \\left(\\sum \\limits _{k=1}^{K}\\left( \\sum _{i\\in I_{k}^{(n)}}\\tau _{k} \\cdot v_{i}-\\Lambda _{\\mathbb {}}(\\tau _{k})\\right) \\right)$ which proves that $S$ is indeed the KL projection of $\\mathbb {} ^{\\otimes n}$ we aimed at.",
"Therefore, $\\mathbf {V}$ is composed of $K$ independent blocks of length $n_{k}$ each, and the $k-$ th subvector $\\mathbf {V}_{k}$ consists of all the random variables $V_{i}$ whose index $i$ satisfies $X_{i}=d_{k}.$ Within $\\mathbf {V}_{k}$ , all components are i.i.d.",
"with same distribution $U_{k}$ on $\\mathbb {R}$ defined through $\\frac{dU_{k}}{d\\mathbb {} }(u)=\\exp \\left\\lbrace \\tau _{k}\\cdot u-\\Lambda _{\\mathbb {}}(\\tau _{k})\\right\\rbrace =\\frac{\\exp \\left\\lbrace \\tau _{k}\\cdot u\\right\\rbrace }{MGF_{\\mathbb {} }(\\tau _{k})},$ which leads to the moment generating function $dom(MGF_{\\mathbb {}})-\\tau _{k} \\ \\ni \\ z \\ \\mapsto MGF_{U_{k}}(z):=\\int _{\\mathbb {R}}e^{zy}dU_{k}(y)=\\frac{MGF_{\\mathbb {}}(z+\\tau _{k})}{MGF_{\\mathbb {} }(\\tau _{k})}.$ Let us remark that $U_{k}$ can be interpreted as the distorted distribution of $\\mathbb {}$ with the distortion parameter $\\tau _{k}$ (in some cases, this distortion even becomes a tilting/dampening).",
"The estimator $\\widehat{\\Pi }_{L}^{improved}$ defined in (REF ) can be implemented through the following algorithm: Step S1 Choose some (typically large) $M$ and simulate repeatedly i.i.d.",
"vectors $\\left(W_{1},..,W_{M}\\right) $ — whose independent components have common distribution $\\mathbb {} $ — until $\\xi _{M,\\mathbf {X}}^{w\\mathbf {W}}$ belongs to $\\textrm {$$\\hspace{-6.544pt}$$}$ .",
"Call $\\left( W_{1}^{\\ast },..,W_{M}^{\\ast }\\right) $ the corresponding vector and $\\overline{W^{\\ast }}$ the arithmetic mean of its components.",
"Moreover, denote by $\\xi _{M,\\mathbf {X}}^{w\\mathbf {W}^{\\ast }}$ the corresponding normalized weighted empirical measure, identified with the $K-$ component vector $Q^{\\ast } := (q_{1}^{\\ast },..,q_{K}^{\\ast })$ with $q_{k}^{\\ast }$ defined in (REF ).",
"Step S2 For all $k \\in \\lbrace 1,\\ldots ,K\\rbrace $ compute $\\tau _{k} = \\varphi ^{\\prime }\\left( \\frac{\\overline{W^{\\ast }} \\cdot q_{k}^{\\ast }}{p_{n,k}^{emp}} \\right)$ .",
"Step S3 For all $\\ell \\in \\lbrace 1,\\ldots ,L\\rbrace $ simulate independently for all $k \\in \\lbrace 1,\\ldots ,K\\rbrace $ a row vector $\\mathbf {V}_{k}^{(\\ell )}$ $:=\\left(V_{k_{1}}^{(\\ell )},...,V_{k_{n_{k}}}^{(\\ell )}\\right) $ with independent components with common distribution $U_{k}$ defined in (REF ).",
"Concatenate these vectors to define the row vector $\\mathbf {V}^{(\\ell )}.$ Step S4 Compute the estimator $\\widehat{\\Pi }_{L}^{improved}$ by making use of the formula (REF ) which turns into the explicit form $& & \\widehat{\\Pi }_{L}^{improved} =\\frac{1}{L}\\sum _{\\ell =1}^{L}\\exp \\left(\\sum \\limits _{k=1}^{K}\\left(n_{k} \\cdot \\Lambda _{\\mathbb {} }(\\tau _{k}) -\\tau _{k} \\cdot \\sum _{i\\in I_{k}^{(n)}}V_{i}^{(\\ell )}\\right) \\right)\\cdot \\mathbf {1}_{\\textrm {\\Omega \\hspace{-5.40608pt}\\Omega }}\\left( \\xi _{n,\\mathbf {X}}^{w\\mathbf {V}^{(\\ell )}}\\right)$ Analogously to the paragraph right after (REF ) of the previous Subsection REF , in many cases we may improve the simulation burden needed for the computation of the estimator $\\widehat{\\Pi }_{L}^{improved}$ .",
"In fact, in terms of the notations $\\widehat{\\mathit {W}}_{k}^{(\\ell )}:=\\sum _{i\\in I_{k}^{(n)}}V_{i}^{(\\ell )}$ we can rewrite (REF ) as $\\widehat{\\Pi }_{L}^{improved} =\\frac{1}{L}\\sum _{\\ell =1}^{L}\\mathbf {1}_{\\textrm {\\Omega \\hspace{-5.40608pt}\\Omega }}\\left( \\xi _{n,\\mathbf {X}}^{w\\mathbf {V}^{(\\ell )}} \\right)\\cdot \\prod _{k=1}^{K}ISF_{k}\\left(\\widehat{\\mathit {W}}_{k}^{(\\ell )}\\right)$ with $ISF_{k}(x) \\ := \\ \\exp (n_{k} \\cdot \\Lambda _{\\mathbb {}}(\\tau _{k}) \\, - \\, x \\cdot \\tau _{k} )$ and $\\xi _{n,\\mathbf {X}}^{w\\mathbf {V}^{(\\ell )}} &=&{\\left\\lbrace \\begin{array}{ll}\\left(\\frac{\\widehat{\\mathit {W}}_{1}^{(\\ell )}}{\\sum _{k=1}^{K}\\widehat{\\mathit {W}}_{k}^{(\\ell )}},\\ldots , \\frac{\\widehat{\\mathit {W}}_{K}^{(\\ell )}}{\\sum _{k=1}^{K}\\widehat{\\mathit {W}}_{k}^{(\\ell )}} \\right) ,\\qquad \\textrm {if } \\sum _{k=1}^{K}\\widehat{\\mathit {W}}_{k}^{(\\ell )} \\ne 0, \\\\\\ (\\infty , \\ldots , \\infty ) =: \\infty , \\hspace{65.44142pt} \\textrm {if }\\sum _{k=1}^{K}\\widehat{\\mathit {W}}_{k}^{(\\ell )} = 0 \\, .\\end{array}\\right.",
"}$ Clearly, the random variable $\\widehat{\\mathit {W}}_{k}^{(\\ell )}$ ($k=1, \\ldots , K$ ) has distribution $U_{k}^{\\ast n_{k}}$ .",
"Hence, if $U_{k}^{\\ast n_{k}}$ can be explicitly constructed, then for the computation of $\\widehat{\\Pi }_{L}^{improved}$ it suffices to independently simulate the $K \\cdot L$ random variables $\\widehat{\\mathit {W}}_{k}^{(\\ell )}$ (rather than the $n \\cdot L$ random variables $V_{i}^{(\\ell )}$ ).",
"In the following subsubsection, we exemplarily demonstrate the tractability of this reduction effect."
],
[
"BS minimization of power divergences and related quantities",
"´ Consider the special case of power divergence generators $\\varphi := \\widetilde{c} \\cdot \\varphi _{\\gamma }$ ($\\gamma \\in \\mathbb {R}\\backslash ]1,2[$ ) of the Examples REF and REF .",
"The corresponding estimators $\\widehat{\\Pi }_{L}^{improved}$ can be obtained as follows: within the results of Example REF , set $M_{\\mathbf {P}}=1$ , and replace $\\widetilde{q}_{k}^{\\ast }$ by $\\overline{W^{\\ast }} \\cdot q_{k}^{\\ast }$ as well as $p_{k}$ by $p_{n,k}^{emp}$ ; accordingly, $\\widetilde{U}_{k}^{\\ast n_{k}}$ turns into $U_{k}^{\\ast n_{k}}$ and $\\widetilde{ISF}_{k}$ into $ISF_{k}$ ; simulate independently the random variables $\\widehat{\\mathit {W}}_{k}^{(\\ell )}$ from $U_{k}^{\\ast n_{k}}$ ($k \\in \\lbrace 1, \\ldots , K\\rbrace $ , $\\ell \\in \\lbrace 1,\\ldots ,L\\rbrace $ ); plug in the results of (i),(ii) into (REF ), (REF ), and (REF ) in order to concretely compute $\\widehat{\\Pi }_{L}^{improved}$ .",
"From this, we can easily generate improved estimators of the power divergences $\\inf _{\\mathbf {Q}\\in \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega }}D_{\\widetilde{c}\\cdot \\varphi _{\\gamma }}(\\mathbf {Q},{P})$ — and more generally, improved estimators of all the infimum-quantities (e.g.",
"Renyi divergences) respectively supremum-quantities in the parts (b) of the Propositions REF , REF , REF , REF , REF and REF with $A=1$ — by simply replacing $\\mathbb {\\Pi }_{X_{1}^{n}}[\\xi _{n}^{w\\mathbf {W}}\\in \\textrm {$$\\hspace{-6.544pt}$$} ]$ (respectively, its variants) by the corresponding estimator $\\widehat{\\Pi }_{L}^{improved}$ .",
"If — in the light of Remark REF (vi) — the ${P} = (p_{1}, \\ldots , p_{K})$ is a pregiven known probability vector e.g.",
"the uniform distribution ${P}^{unif}$ on $\\lbrace 1,\\ldots ,K\\rbrace $ (rather than the limit of the vector of empirical frequencies/masses of a sequence of random variables $X_{i}$ , cf.",
"(REF )), then we proceed analogously as above by replacing $p_{n,k}^{emp}$ with $p_{k}$ ; correspondingly, we obtain improved estimators of all the infimum-quantities respectively supremum-quantities (e.g.",
"Renyi entropies, diversity indices) in the parts (a) of the Propositions REF , REF , REF , REF , REF and REF with $A=1$ .",
"For the sake of brevity, in the following we only present explicitly the outcoming improved estimators for the power divergences (in the “$X_{i}-$ context” ).",
"Indeed, we simply replace the $\\mathbb {\\Pi }_{X_{1}^{n}}[\\xi _{n}^{w\\mathbf {W}}\\in \\textrm {$$\\hspace{-6.544pt}$$} ]$ in the formulas (REF ), (REF ), (REF ) (with $A=1$ ) by the improved estimator $\\widehat{\\Pi }_{L}^{improved}$ obtained through (i) to (iii); for arbitrarily fixed $\\widetilde{c} >0$ , this leads to the improved power-divergence estimators (BS estimators of power divergences) $&&\\hspace{-54.06006pt}\\widehat{D_{\\widetilde{c}\\cdot \\varphi _{\\gamma }}(\\textrm {\\Omega \\hspace{-6.544pt}\\Omega },{P})} \\ := \\ -\\frac{\\widetilde{c}}{\\gamma (\\gamma -1)}\\left\\lbrace 1-\\left( 1+\\frac{\\gamma }{\\widetilde{c}} \\cdot \\frac{1}{n} \\cdot \\log \\widehat{\\Pi }_{L}^{improved}\\right)^{1-\\gamma }\\right\\rbrace ,\\qquad \\gamma \\in \\, ]-\\infty ,0[ \\, \\cup \\, ]0,1[ \\, \\cup \\, [2,\\infty [,\\\\& &\\hspace{-54.06006pt}\\widehat{D_{\\widetilde{c}\\cdot \\varphi _{0}}(\\textrm {\\Omega \\hspace{-6.544pt}\\Omega },{P})} \\ := \\ -\\frac{1}{n}\\log \\widehat{\\Pi }_{L}^{improved} ,\\hspace{165.02606pt} \\gamma =0,\\\\& & \\hspace{-54.06006pt}\\widehat{D_{\\widetilde{c}\\cdot \\varphi _{1}}(\\textrm {\\Omega \\hspace{-6.544pt}\\Omega },{P})}\\ : = \\ - \\widetilde{c} \\cdot \\log \\left( 1+\\frac{1}{\\widetilde{c}} \\cdot \\frac{1}{n} \\cdot \\log \\widehat{\\Pi }_{L}^{improved}\\right) ,\\hspace{85.35826pt} \\gamma =1.$ Let us finally remark that from the above-mentioned Steps S1 to S4 (and analogously D1 to D4) one can see that for our BS method we basically need only a fast and accurate — pseudo, true, natural, quantum — random number generator.",
"The corresponding computations can be principally run in parallel, and require relatively moderate computer memory/storage; a detailed discussion is beyond the scope of this paper, given its current length."
],
[
"General case, part 2",
"The algorithm which is presented in this section aims at the evaluation of the bounds $\\inf _{m\\ne 0}D_{\\varphi }\\left( m \\cdot \\textrm {\\right.\\Omega \\hspace{-6.544pt}\\Omega },{P} =\\inf _{{Q}\\in \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega }}D_{\\varphi }\\left( m\\left( {Q}\\right) \\cdot {Q},{P}\\right)\\stackrel{(1)}{=} D_{\\varphi }\\left( m({Q}^{\\ast })\\cdot {Q}^{\\ast },{P}\\right) \\le D_{\\varphi }\\left( \\textrm {\\right.\\Omega \\hspace{-6.544pt}\\Omega },{P} \\le D_{\\varphi }\\left( {Q}^{\\ast },{P}\\right)$ obtained in Section REF , where ${Q}^{\\ast }$ satisfies the above equality $(1)$ .",
"The estimator of the lower bound in (REF ) is $\\widehat{D}:=-\\frac{1}{n}\\log \\widehat{\\text{ }\\Pi }_{L}^{improved}$ defined in (REF ).",
"We now turn to an estimate of the upper bound.",
"Consider for any fixed ${Q}:=\\left( q_{1},..,q_{K}\\right) $ in $\\mathbb {S}_{>0}^{K}$ the real number $m_{n}({Q})$ which satisfies $D_{\\varphi }\\left( m_{n}({Q}) \\cdot {Q},{P}_{n}^{emp} \\right) =\\inf _{m\\ne 0}D_{\\varphi }\\left( m\\cdot \\textrm {\\right.$ $\\hspace{-6.544pt}$$},{P}_{n}^{emp}$$where $ Pnemp$ was defined in the course of (\\ref {I^(n)_k for stat case}).Such $ mn(Q)$ is well defined for all $ Q$since it satisfies the equation (in $ m$)\\begin{equation}\\frac{d}{dm}D_{\\varphi }\\left( m \\cdot {Q},{P}_{n}^{emp}\\right) =\\sum _{k=1}^{K} q_{k} \\cdot \\varphi ^{\\prime }\\left( \\frac{m \\cdot q_{k}}{p_{n,k}^{emp}}\\right) =0 .\\end{equation}Since the mapping $ mD( m Q,P) $ is convexand differentiable, existence and uniqueness of $ mn(Q)$ hold; furthermore,$ mn(Q)] k pn,kemp/qk,k pn,kemp/qk[ $since $ ddmD( m Q,Pnemp) $is negative when $ m=k pn,kemp/qk$ and positive when $ m=k pn,kemp/qk$.$ An estimate of the distribution ${Q}^{\\ast }$ is required.",
"This can be achieved as follows: Estimate $\\inf _{m\\ne 0}D_{\\varphi }\\left( m \\cdot \\textrm {\\right.$$\\hspace{-6.544pt}$$}, {P}$ through $\\widehat{D} := -\\frac{1}{n}\\log \\widehat{\\Pi }_{L}^{improved}$ defined in (REF ).",
"Set $i=0.$ Get some ${Q}_{i} := (q_{i,1}, \\ldots , q_{i,K})$ in $\\textrm {$$\\hspace{-6.544pt}$$}$ ; this can be obtained by simulating runs of vectors $\\left( W_{1},..,W_{n}\\right) $ through i.i.d.",
"sampling under $\\mathbb {}$ .",
"Evaluate $m_{n}({Q}_{i})$ by solving () (with $q_{i,k}$ instead of $q_{k}$ ) for $m$ , which is a fast calculation by the bisection method.",
"If $D_{\\varphi }\\left( m_{n}({Q}_{i}) \\cdot {Q}_{i}, {P}_{n}^{emp} \\right) <\\widehat{D}+\\eta $ for some small $\\eta >0$ , then the proxy of ${Q}^{\\ast }$ is ${Q}_{i},$ denoted by $\\widehat{{Q}^{\\ast }}$ .",
"Else set $i\\leftarrow i+1$ and get ${Q}_{i}$ in $\\textrm {$$\\hspace{-6.544pt}$$}\\cap \\left\\lbrace {Q} \\, : \\, D_{\\varphi }\\left( {Q},{P}_{n}^{emp} \\right) <D_{\\varphi }\\left( {Q}_{i-1}, {P}_{n}^{emp}\\right) \\right\\rbrace $ and iterate.",
"That this algorithm converges in the sense that it produces some $\\widehat{{Q}^{\\ast }}$ is clear.",
"Since by (REF ) $D_{\\varphi }\\left( m({Q}^{\\ast }) \\cdot {Q}^{\\ast },{P}\\right)\\le D_{\\varphi }\\left(\\textrm {\\right.$ $\\hspace{-6.544pt}$$},{P}\\le D_{\\varphi }\\left( {Q}^{\\ast },{P}\\right) ,$$we have obtained both estimated lower and upper bounds for$ D( $\\Omega $$\\Omega $ , P)$.$ That the upper bound is somehow optimal can be seen from the power case developed in Section REF .",
"Indeed, in this case the solution of equation () is explicit and produces $m({Q})$ as a function of $D_{\\varphi }\\left( {Q},{P}\\right)$ through a Hellinger integral, and the mapping ${Q}\\rightarrow D_{\\varphi }\\left( m({Q})\\cdot {Q},{P}\\right)$ is increasing with respect to $D({Q},{P})$ .",
"Hence, ${Q} \\rightarrow \\inf _{m\\ne 0}D_{\\varphi }\\left( m \\cdot {Q},{P}\\right)$ is minimal when $D_{\\varphi }\\left( {Q},{P}\\right) $ is minimal as ${Q} \\in \\textrm {$$\\hspace{-6.544pt}$$}$ .",
"Therefore, ${Q}^{\\ast }\\in \\arg \\inf _{{Q}\\in \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega }}D_{\\varphi }\\left( m({Q}) \\cdot {Q},{P}\\right) $ also satisfies ${Q}^{\\ast }\\in \\arg \\inf _{{Q}\\in \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega }}D_{\\varphi }\\left( {Q},{P}\\right)$ ."
],
[
"Proofs — Part 1",
"Proof of Theorem REF .",
"This is a straightforward application of the classical Cramer-type Large Deviation Theorem in the vector case (see Theorem 2.2.30 and Corollary 6.1.6 in Dembo & Zeitouni [108]).",
"Recall that above we have transformed the original problem into a context where the second argument in $D_{\\varphi }(\\cdot ,\\cdot )$ is a probability vector, as follows: in terms of $M_{\\mathbf {P}}:=\\sum _{i=1}^{K}p_{i}>0$ we normalized $\\widetilde{{P}}:=\\mathbf {P}/M_{\\mathbf {P}},$ and $\\widetilde{\\mathbf {Q}}:=\\mathbf {Q}/M_{\\mathbf {P}}$ for $\\mathbf {Q}$ in $\\mathbf {\\Omega }$ .",
"With $\\widetilde{\\varphi } \\in \\Upsilon (]a,b[)$ defined through $\\widetilde{\\varphi }:=M_{\\mathbf {P}} \\cdot \\varphi $ , we have obtained $D_{\\varphi }(\\mathbf {Q},\\mathbf {P})=\\sum _{k=1}^{K}p_{k}\\cdot \\varphi \\left( \\frac{q_{k}}{p_{k}}\\right) =\\sum _{k=1}^{K}M_{\\mathbf {P}}\\cdot \\widetilde{p_{k}}\\cdot \\frac{\\varphi \\left( \\frac{M_{\\mathbf {P}}\\cdot \\widetilde{q_{k}}}{M_{\\mathbf {P}}\\cdot \\widetilde{p_{k}}}\\right) }{M_{\\mathbf {P}}}=D_{\\widetilde{\\varphi }}(\\widetilde{\\mathbf {Q}},\\widetilde{{P}})\\qquad \\textrm {(cf.",
"(\\ref {min Pb prob1}))}.$ It has followed that the solution of (REF ) coincides with the one of the problem of finding $\\widetilde{\\Phi }_{\\widetilde{{P}}}(\\widetilde{\\mathbf {\\Omega }}) := \\inf _{\\widetilde{\\mathbf {Q}}\\in \\widetilde{\\mathbf {\\Omega }} }D_{\\widetilde{\\varphi } }(\\widetilde{\\mathbf {Q}},\\widetilde{{P}}),\\qquad \\textrm {with } \\widetilde{\\mathbf {\\Omega }}:=\\mathbf {\\Omega } /M_{\\mathbf {P}}\\qquad \\textrm {(cf.",
"(\\ref {min Pb prob2}))}.$ So let us continue by tackling (REF ).",
"From the assumptions on $\\widetilde{\\varphi }$ and the requirement (REF ) one can see that $\\text{$\\widetilde{W}_{1}$ has moment generating function$t\\rightarrow E_{\\mathbb {\\Pi }}[e^{z \\cdot \\widetilde{W}_{1}}]= MGF_{\\widetilde{\\mathbb {}}}(z)$which is finite on a non-void neighborhood of $0$},$ $E_{\\mathbb {\\Pi }}[\\widetilde{W}_1] = 1 ,$ since $\\widetilde{\\varphi }(1)=0=\\widetilde{\\varphi }^{\\prime }(1)$ .",
"With the help of these, we obtain the following Proposition 61 Under the assumptions of Theorem REF , for any set $\\widetilde{\\mathbf {\\Omega }}\\subset \\mathcal {M} := \\mathbb {R}^{K}$ with (REF ) one has $-\\inf _{\\widetilde{\\mathbf {Q}} \\in int(\\widetilde{\\mathbf {\\Omega }}) }D_{\\varphi }\\left( \\widetilde{\\mathbf {Q}},\\widetilde{{P}}\\right) &\\le &\\lim \\inf _{n\\rightarrow \\infty }\\frac{1}{n}\\log \\mathbb {\\Pi }\\left[\\xi _{n}^{\\widetilde{\\textbf {W}}}\\in \\widetilde{\\mathbf {\\Omega }} \\right] \\nonumber \\\\&\\le &\\lim \\sup _{n\\rightarrow \\infty }\\frac{1}{n}\\log \\mathbb {\\Pi }\\left[\\xi _{n}^{\\widetilde{\\textbf {W}}}\\in \\widetilde{\\mathbf {\\Omega }} \\right]\\le -\\inf _{\\widetilde{\\mathbf {Q}}\\in cl(\\widetilde{\\mathbf {\\Omega }}) }D_{\\varphi }\\left( \\widetilde{\\mathbf {Q}},\\widetilde{{P}}\\right) .$ Proof of Proposition REF .",
"Recall from Remark REF (v) that $I_{k}^{(n)} := \\lbrace i \\in \\lbrace 1,\\ldots ,n\\rbrace : \\widetilde{x}_{i}=d_{k}\\rbrace $ and $n_{k} := card(I_{k}^{(n)})$ denotes the number of elements therein ($k \\in \\lbrace 1,\\ldots ,K\\rbrace $ ), i.e.",
"$n_{k}$ is the number of the $\\widetilde{x}_{i}$ ’s which equal $d_{k}$ .",
"We follow the line of proof of Theorem 2.2.30 in Dembo & Zeitouni [108], which states the large deviation principle (LDP) for the vector of partial sums of random vectors in $\\mathbb {R}^{K}$ , where we also use Corollary 6.1.6 in [108] in relation with condition (REF ).",
"Indeed, since the $k-$ th component of the vector $\\xi _{n}^{\\mathbf {\\widetilde{W}}}$ is the $1/n-$ fold of the sum of the $\\widetilde{W}_{i}$ ’s for which the corresponding $\\widetilde{x}_{i}$ ’s equal $d_{k}$ (i.e., $\\frac{1}{n} \\sum _{i\\in I_{k}^{(n)}} \\widetilde{W}_{i}$ ) the proof will follow from a similar treatment as for the standard Cramer LDP in $\\mathbb {R}^{K}.$ The only difference lies in two facts: the number of the summands for the coordinate $k$ is $n_{k}$ , the number of $\\widetilde{x}_{i}$ ’s which equal $d_{k}$ , instead of $n$ in the standard case.",
"Furthermore we will need to substitute $n_{k}$ by its equivalent $n \\cdot \\widetilde{p}_{k}$ , which adds an approximation step.",
"For the upper bound, the proof is based on the corresponding result for $B=B_{1}\\times \\cdots \\times B_{K}$ where the $B_{k}$ ’s are open bounded intervals on $\\mathbb {R}^{+}.$ Since the sequence $\\left( \\widetilde{x}_{1}, \\ldots \\right)$ satisfies $\\lim _{n\\rightarrow \\infty }\\frac{n_{k}}{n}= \\widetilde{p}_{k} ,\\qquad \\textrm {(cf.",
"(\\ref {fo.freqlim}))}$ there holds $&&\\frac{1}{n}\\log \\mathbb {\\Pi } \\left[\\xi _{n}^{\\mathbf {\\widetilde{W}}} \\in B\\right]=\\frac{1}{n}\\log \\mathbb {\\Pi } \\bigg [ \\bigcap \\limits _{k=1}^{K}\\bigg (\\frac{1}{n} \\sum _{i\\in I_{k}^{(n)}} \\widetilde{W}_{i}\\in B_{k}\\bigg ) \\bigg ]\\nonumber \\\\&=&\\frac{1}{n}\\sum \\limits _{k=1}^{K}\\log \\mathbb {\\Pi } \\bigg [ \\frac{1+o(1)}{n_{k}}\\sum _{i\\in I_{k}^{(n)}} \\widetilde{W}_{i}\\in \\frac{1}{\\widetilde{p}_{k}} B_{k}\\bigg ],$ and hence $\\lim \\sup _{n\\rightarrow \\infty }\\frac{1}{n}\\log \\mathbb {\\Pi } \\left[\\xi _{n}^{\\mathbf {\\widetilde{W}}}\\in B\\right]&\\le &\\sum \\limits _{k=1}^{K} \\widetilde{p}_{k} \\cdot \\lim \\sup _{n_{k}\\rightarrow \\infty }\\frac{1}{n_{k}}\\log \\mathbb {\\Pi } \\bigg [ \\frac{1}{n_{k}}\\sum _{i\\in I_{k}^{(n)}} \\widetilde{W}_{i}\\in \\frac{1}{p_{k}}B_{k}\\bigg ] \\nonumber \\\\&\\le &-\\sum \\limits _{k=1}^{K}\\inf _{x_{k}\\in cl(B_{k})} \\widetilde{p}_{k} \\cdot \\varphi \\left(\\frac{x_{k}}{\\widetilde{p}_{k}}\\right) .$ To deduce (REF ) from (REF ), we have used (i) the fact that for all $k$ the random variables $\\frac{1}{n_{k}}\\left( 1+o(1))\\right) \\cdot \\sum _{i\\in I_{k}^{(n)}} \\widetilde{W}_{i}$ and $\\frac{1}{n_{k}}\\sum _{i\\in I_{k}^{(n)}} \\widetilde{W}_{i}$ are exponentially equivalent in the sense that their difference $\\Delta _{n_{k}}$ satisfies $\\lim \\sup _{n_{k}\\rightarrow \\infty }\\frac{1}{n_{k}}\\log \\mathbb {\\Pi }\\left[ \\,\\left|\\Delta _{n_{k}}\\right|>\\eta \\, \\right] =-\\infty ,$ making use of the Chernoff inequality for all positive $\\eta $ , as well as (ii) Theorem 4.2.13 in [108].",
"Now the summation and the inf-operations can be permuted in (REF ) which proves the claim for the rectangle $B$ .",
"As in [108], for a compact set $\\Omega $ we consider its finite covering by such open sets $B$ and conclude; for $\\Omega $ being a closed set, a tightness argument holds, following [108] Theorem 2.2.30 verbatim.",
"For the lower bound consider the same rectangle $B$ .",
"The argument which locates the tilted distribution at the center of $B$ , together with the use of the LLN for the corresponding r.v’s as in [108], in combination with the same approximations as above to handle the approximation of $n_{k}$ by $n \\cdot \\widetilde{p}_{k}$ , complete the proof of Proposition REF .",
"We omit the details.",
"$\\blacksquare $ Let us continue with the proof of Theorem REF , by giving the following two helpful lemmas for $\\Phi _{{P}}(\\mathbf {A}) := \\inf _{\\mathbf {Q} \\in \\mathbf {A}} D_{\\varphi }\\left(\\mathbf {Q},{P}\\right),\\qquad \\mathbf {A} \\subset \\mathcal {M} := \\mathbb {R}^{K},$ Lemma 62 For any open set $\\mathbf {A} \\subset \\mathcal {M} := \\mathbb {R}^{K}$ one has $\\Phi _{{P}}(\\mathbf {A}) = \\Phi _{{P}}(cl(\\mathbf {A}))$ .",
"This is clear from the continuity of $\\Phi _{{P}}$ .",
"Lemma 63 For any $\\mathbf {A} \\subset \\mathcal {M} := \\mathbb {R}^{K}$ satisfying (REF ) one has $\\Phi _{{P}}(cl(\\mathbf {A})) = \\Phi _{{P}}(\\mathbf {A}) = \\Phi _{{P}}(int(\\mathbf {A}))$ .",
"Proof of Lemma REF .",
"Assume first that $\\Phi _{{P}}(\\mathbf {A})$ is finite.",
"Then suppose that $\\mathbf {A}$ satisfies (REF ) and $\\Phi _{{P}}(cl(\\mathbf {A})) < \\Phi _{{P}}(int(\\mathbf {A}))$ .",
"The latter implies the existence of a point $a \\in cl(\\mathbf {A})$ such that $a \\notin int(\\mathbf {A})$ and $\\Phi _{{P}}(a)= \\Phi _{{P}}(cl(\\mathbf {A}))$ .",
"But then, by Lemma REF and (REF ) one gets $\\Phi _{{P}}(int(\\mathbf {A})) = \\Phi _{{P}}(cl(int(\\mathbf {A}))) = \\Phi _{{P}}(cl(\\mathbf {A})) = \\Phi _{{P}}(a)$ which leads to a contradiction.",
"When $\\Phi _{{P}}(\\mathbf {A}) = \\infty $ then $\\Phi _{{P}} (cl(\\mathbf {A}))=\\Phi _{{P}} (int(\\mathbf {A}))=\\Phi _{{P}} (\\mathbf {A})=\\infty $ .",
"$\\blacksquare $ Putting things together, the required asymptotic assertion (REF ) follows from (REF ), (REF ) and Lemma REF .",
"This completes the proof of Theorem REF .",
"$\\blacksquare $"
],
[
"Proofs — Part 2",
"Before we tackle the proof of Theorem REF , let us introduce the following Lemma 64 If $\\Omega $$\\Omega \\subset \\mathbb {S}^{K}$ satisfies condition (REF ), then $\\widetilde{\\widetilde{\\textrm {\\Omega \\hspace{-6.544pt}\\Omega }}}:= \\bigcup \\limits _{m\\ne 0}cl(m \\cdot \\textrm {$$\\hspace{-6.544pt}$$})$ has the property (REF ).",
"This can be deduced in a straightforward way: the assumption implies that $cl(\\textrm {$$\\hspace{-6.544pt}$$})$ satisfies (REF ), and thus also $m \\cdot cl(\\textrm {$$\\hspace{-6.544pt}$$})$ satisfies (REF ).",
"But this implies the validity of (REF ) for the “cone” $\\bigcup \\limits _{m\\ne 0} m \\cdot cl(\\textrm {$$\\hspace{-6.544pt}$$})$ which is nothing but $\\bigcup \\limits _{m\\ne 0} cl(m \\cdot \\textrm {$$\\hspace{-6.544pt}$$})$ .",
"Proof of Theorem REF .",
"Recall the interpretations of the two vectors $\\xi _{n,\\mathbf {X}}^{\\mathbf {W}}$ respectively $\\xi _{n,\\mathbf {X}}^{w\\mathbf {W}}$ given in (REF ) respectively (REF ), and that the sum of their $k$ components are $\\sum _{k=1}^{K} \\frac{1}{n} \\sum _{i\\in I_{k}^{(n)}} W_{i} =\\frac{1}{n}\\sum _{i=1}^{n} W_{i}$ respectively $\\sum _{k=1}^{K}\\frac{\\sum _{i \\in I_{k}^{(n)}}W_{i}}{\\sum _{k=1}^{K}\\sum _{i \\in I_{k}^{(n)}}W_{i}}=1$ (in case of $\\sum _{i=1}^{n}W_{i} \\ne 0$ ).",
"In the light of these, for $\\textrm {$$\\hspace{-6.544pt}$$} \\subset \\mathbb {S}^{K}$ one gets the set identification $\\left\\lbrace \\xi _{n,\\mathbf {X}}^{w\\mathbf {W}} \\in \\textrm {\\right.$ $\\hspace{-6.544pt}$$} =\\bigcup \\limits _{m\\ne 0}\\left\\lbrace \\xi _{n,\\mathbf {X}}^{\\mathbf {W}}\\in m\\cdot \\textrm {\\right.$$\\hspace{-6.544pt}$$} ,\\frac{1}{n}\\sum _{i=1}^{n} W_{i} = m$$since $ { i=1nWi=0} $ amounts to $ m=0$, which cannothold when $ { n,XwW $\\Omega $$\\Omega $ }$.",
"Now\\begin{eqnarray*}\\mathbb {\\Pi }_{X_{1}^{n}}\\left[\\xi _{n,\\mathbf {X}}^{w\\mathbf {W}}\\in \\textrm {\\right.\\Omega \\hspace{-6.544pt}\\Omega } &=&\\mathbb {\\Pi }_{X_{1}^{n}} \\bigg [ \\bigcup \\limits _{m\\ne 0}\\bigg \\lbrace \\xi _{n,\\mathbf {X}}^{\\mathbf {W}} \\in m \\cdot \\textrm {\\Omega \\hspace{-6.544pt}\\Omega },\\frac{1}{n}\\sum _{i=1}^{n} W_{i} =m \\bigg \\rbrace \\bigg ] \\\\&=&\\mathbb {\\Pi }_{X_{1}^{n}}\\bigg [ \\bigcup \\limits _{m\\ne 0}\\bigg \\lbrace \\xi _{n,\\mathbf {X}}^{\\mathbf {W}} \\in m\\cdot \\textrm {\\Omega \\hspace{-6.544pt}\\Omega }\\big \\rbrace \\bigg ]= \\mathbb {\\Pi }_{X_{1}^{n}}\\bigg [ \\xi _{n,\\mathbf {X}}^{\\mathbf {W}}\\in \\bigcup \\limits _{m\\ne 0} m\\cdot \\textrm {\\Omega \\hspace{-6.544pt}\\Omega } \\bigg ]\\end{eqnarray*}since $ { n,XW m$\\Omega $$\\Omega $ } { 1ni=1n Wi =m }$.",
"Therefore\\begin{equation}\\frac{1}{n}\\log \\mathbb {\\Pi }_{X_{1}^{n}}\\left[\\xi _{n,\\mathbf {X}}^{w\\mathbf {W}}\\in \\textrm {\\right.\\Omega \\hspace{-6.544pt}\\Omega } =\\frac{1}{n}\\log \\mathbb {\\Pi }_{X_{1}^{n}}\\bigg [ \\xi _{n,\\mathbf {X}}^{\\mathbf {W}} \\in \\bigcup \\limits _{m\\ne 0}m\\cdot \\textrm {\\Omega \\hspace{-6.544pt}\\Omega } \\bigg ] .\\end{equation}Because of Proposition \\ref {PropLDPWEM-finitease} --- applied to$$\\Omega $$\\Omega $ := m0m $\\Omega $$\\Omega $$ --- one getsin terms of (\\ref {phishort})$ $-\\Phi _{{P}}\\Big (int\\Big (\\bigcup \\limits _{m\\ne 0}m\\cdot \\textrm {\\Omega \\hspace{-6.544pt}\\Omega }\\Big )\\Big )&\\le &\\lim \\inf _{n\\rightarrow \\infty }\\frac{1}{n}\\log \\mathbb {\\Pi }_{X_{1}^{n}}\\bigg [ \\xi _{n,\\mathbf {X}}^{\\mathbf {W}} \\in \\bigcup \\limits _{m\\ne 0}m\\cdot \\textrm {\\Omega \\hspace{-6.544pt}\\Omega } \\bigg ] \\nonumber \\\\&\\le &\\lim \\sup _{n\\rightarrow \\infty }\\frac{1}{n}\\log \\mathbb {\\Pi }_{X_{1}^{n}}\\bigg [\\xi _{n,\\mathbf {X}}^{\\mathbf {W}}\\in \\bigcup \\limits _{m\\ne 0}m\\cdot \\textrm {\\Omega \\hspace{-6.544pt}\\Omega } \\bigg ] \\le -\\Phi _{{P}}\\Big (cl\\Big (\\bigcup \\limits _{m\\ne 0}m\\cdot \\textrm {\\Omega \\hspace{-6.544pt}\\Omega }\\Big )\\Big ) .$ $\\hspace{-176.407pt}\\text{But}& & \\Phi _{{P}}\\Big (int\\Big (\\bigcup \\limits _{m\\ne 0}m\\cdot \\textrm {\\Omega \\hspace{-6.544pt}\\Omega }\\Big )\\Big ) \\le \\Phi _{{P}}\\Big (\\bigcup \\limits _{m\\ne 0}int\\Big (m\\cdot \\textrm {\\Omega \\hspace{-6.544pt}\\Omega }\\Big )\\Big ) =\\inf _{m\\ne 0} \\Phi _{{P}}(int(m\\cdot \\textrm {\\Omega \\hspace{-6.544pt}\\Omega }))\\\\\\hspace{-176.407pt}\\text{and}& & \\Phi _{{P}}\\Big (cl\\Big (\\bigcup \\limits _{m\\ne 0}m\\cdot \\textrm {\\Omega \\hspace{-6.544pt}\\Omega }\\Big )\\Big ) \\ge \\Phi _{{P}}\\Big (\\bigcup \\limits _{m\\ne 0}cl\\Big (m\\cdot \\textrm {\\Omega \\hspace{-6.544pt}\\Omega }\\Big )\\Big ) =\\inf _{m\\ne 0} \\Phi _{{P}}(cl(m\\cdot \\textrm {\\Omega \\hspace{-6.544pt}\\Omega })) .$ In fact, the inequality in (REF ) is straightforward because of $\\bigcup \\limits _{m\\ne 0}int(m\\cdot \\textrm {$$\\hspace{-6.544pt}$$})\\subset int(\\bigcup \\limits _{m\\ne 0}m\\cdot \\textrm {$$\\hspace{-6.544pt}$$})$ (since the latter is the largest open set contained in $\\bigcup \\limits _{m\\ne 0}m\\cdot \\textrm {$$\\hspace{-6.544pt}$$}$ ); the inequality in () follows from $& & \\Phi _{{P}}\\Big (cl\\Big (\\bigcup \\limits _{m\\ne 0}m\\cdot \\textrm {\\Omega \\hspace{-6.544pt}\\Omega }\\Big )\\Big ) \\ge \\Phi _{{P}}\\Big (cl\\Big (\\bigcup \\limits _{m\\ne 0}cl\\Big (m\\cdot \\textrm {\\Omega \\hspace{-6.544pt}\\Omega }\\Big )\\Big )\\Big ) =\\Phi _{{P}}\\Big (\\bigcup \\limits _{m\\ne 0}cl\\Big (m\\cdot \\textrm {\\Omega \\hspace{-6.544pt}\\Omega }\\Big )\\Big )\\nonumber $ An application of Lemma REF yields $\\Phi _{{P}}(int(m\\cdot \\textrm {$$\\hspace{-6.544pt}$$})) =\\Phi _{{P}}(m\\cdot \\textrm {$$\\hspace{-6.544pt}$$}) =\\Phi _{{P}}(cl(m\\cdot \\textrm {$$\\hspace{-6.544pt}$$}))$ for all $m \\ne 0$ , and hence $\\inf _{m\\ne 0} \\Phi _{{P}}(int(m\\cdot \\textrm {\\Omega \\hspace{-6.544pt}\\Omega })) =\\inf _{m\\ne 0} \\Phi _{{P}}(m\\cdot \\textrm {\\Omega \\hspace{-6.544pt}\\Omega })= \\inf _{m\\ne 0} \\Phi _{{P}}(cl(m\\cdot \\textrm {\\Omega \\hspace{-6.544pt}\\Omega })) .$ By combining (), (REF ), (REF ), () and (REF ), one arrives at $&&\\lim _{n \\rightarrow \\infty } \\frac{1}{n}\\log \\mathbb {\\Pi }_{X_{1}^{n}} \\left[\\xi _{n,\\mathbf {X}}^{w\\mathbf {W}}\\in \\textrm {\\right.\\Omega \\hspace{-6.544pt}\\Omega } = \\lim _{n \\rightarrow \\infty } \\frac{1}{n}\\log \\mathbb {\\Pi }_{X_{1}^{n}} \\bigg [ \\xi _{n,\\mathbf {X}}^{\\mathbf {W}}\\in \\bigcup \\limits _{m\\ne 0}m\\cdot \\textrm {\\Omega \\hspace{-6.544pt}\\Omega } \\bigg ]\\nonumber \\\\&& = - \\inf _{m\\ne 0} \\Phi _{{P}}(m\\cdot \\textrm {\\Omega \\hspace{-6.544pt}\\Omega })= - \\inf _{m\\ne 0} \\inf _{\\mathbf {Q} \\in m\\cdot \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega }}D_{\\varphi }\\left(\\mathbf {Q},{P}\\right)= - \\inf _{m\\ne 0} \\inf _{{Q} \\in \\textrm {\\Omega \\hspace{-5.40608pt}\\Omega }}D_{\\varphi }\\left(m\\cdot {Q}, {P}\\right) ,\\nonumber $ where in the second last equality we have “reverted” the notation (REF ).",
"Note that we did not assume (REF ) for $\\bigcup \\limits _{m\\ne 0} m \\cdot \\textrm {$$\\hspace{-6.544pt}$$}$ .",
"$\\blacksquare $"
],
[
"Proofs — Part 3",
"Proof of Lemma REF .",
"From (REF ) one gets straightforwardly for arbitrary $\\widetilde{c} >0$ $D_{\\widetilde{c} \\cdot \\varphi _{\\gamma }}(m \\cdot \\mathbf {Q},{P}) \\hspace{-5.69046pt} &:=& \\hspace{-5.69046pt}{\\left\\lbrace \\begin{array}{ll}\\frac{\\widetilde{c} \\cdot \\left(m^{\\gamma } \\cdot H_{\\gamma } - m \\cdot A \\cdot \\gamma + \\gamma -1 \\right)}{\\gamma \\cdot (\\gamma -1)},\\hspace{75.39963pt} \\textrm {if } \\gamma \\in \\, ]-\\infty ,0[, \\ {P} \\in \\mathbb {S}_{\\ge 0}^{K}, \\ \\mathbf {Q} \\in A \\cdot \\mathbb {S}_{> 0}^{K} \\ \\ \\textrm {and } m >0, \\\\\\widetilde{c} \\cdot (- \\log m + \\widetilde{I} -1 + m \\cdot A),\\hspace{42.67912pt} \\textrm {if } \\gamma = 0, \\ {P} \\in \\mathbb {S}_{\\ge 0}^{K}, \\ A \\cdot \\mathbf {Q} \\in \\mathbb {S}_{> 0}^{K} \\ \\ \\textrm {and } m >0, \\\\\\frac{\\widetilde{c} \\cdot \\left(m^{\\gamma } \\cdot H_{\\gamma } - m \\cdot A \\cdot \\gamma + \\gamma -1 \\right)}{\\gamma \\cdot (\\gamma -1)},\\hspace{75.39963pt} \\textrm {if } \\gamma \\in \\, ]0,1[, \\ {P} \\in \\mathbb {S}_{\\ge 0}^{K}, \\ \\mathbf {Q} \\in A \\cdot \\mathbb {S}_{\\ge 0}^{K} \\ \\ \\textrm {and } m \\ge 0, \\\\\\widetilde{c} \\cdot (A \\cdot m \\cdot \\log m + m \\cdot (I-A) +1),\\hspace{14.22636pt} \\textrm {if } \\gamma = 1, \\ {P} \\in \\mathbb {S}_{> 0}^{K}, \\ \\mathbf {Q} \\in A \\cdot \\mathbb {S}_{\\ge 0}^{K} \\ \\ \\textrm {and } m \\ge 0, \\\\\\frac{\\widetilde{c} \\cdot \\left(m^{\\gamma } \\cdot H_{\\gamma } \\cdot {1}_{[0,\\infty [}(m)- m \\cdot A \\cdot \\gamma + \\gamma -1 \\right)}{\\gamma \\cdot (\\gamma -1)},\\hspace{39.83368pt} \\textrm {if } \\gamma \\in \\, ]1,2[, \\ {P} \\in \\mathbb {S}_{>0}^{K}, \\ \\mathbf {Q} \\in A \\cdot \\mathbb {S}^{K} \\ \\ \\textrm {and } m \\in ]-\\infty ,\\infty [,\\\\\\frac{\\widetilde{c} \\cdot \\left(m^{2} \\cdot H_{2} - m \\cdot A \\cdot 2 + 2 -1 \\right)}{2 \\cdot (2-1)},\\hspace{76.82234pt} \\textrm {if } \\gamma = 2, \\ {P} \\in \\mathbb {S}_{>0}^{K}, \\ \\mathbf {Q} \\in A \\cdot \\mathbb {S}^{K} \\ \\ \\textrm {and } m \\in ]-\\infty ,\\infty [,\\\\\\frac{\\widetilde{c} \\cdot \\left(m^{\\gamma } \\cdot H_{\\gamma } \\cdot {1}_{[0,\\infty [}(m)- m \\cdot A \\cdot \\gamma + \\gamma -1 \\right)}{\\gamma \\cdot (\\gamma -1)},\\hspace{39.83368pt} \\textrm {if } \\gamma \\in \\, ]2,\\infty [, \\ {P} \\in \\mathbb {S}_{>0}^{K}, \\ \\mathbf {Q} \\in A \\cdot \\mathbb {S}^{K} \\ \\ \\textrm {and } m \\in ]-\\infty ,\\infty [,\\\\\\infty , \\hspace{156.49014pt} \\textrm {else},\\end{array}\\right.",
"}$ where we have used the three $m-$ independent abbreviations $H_{\\gamma } := \\sum \\displaylimits _{k=1}^{K} (q_{k})^{\\gamma } \\cdot (p_{k})^{1-\\gamma }\\ = \\ 1 + \\gamma \\cdot (A-1) +\\frac{\\gamma \\cdot (\\gamma -1)}{\\widetilde{c}} \\cdot D_{\\widetilde{c} \\cdot \\varphi _{\\gamma }}(\\mathbf {Q}, {P}),\\qquad \\text{(cf.",
"(\\ref {brostu3:fo.divpow.hellinger1}))}\\nonumber $ $I:= \\sum \\displaylimits _{k=1}^{K} q_{k} \\cdot \\log \\left( \\frac{q_{k}}{p_{k}} \\right)\\ = \\ \\frac{1}{\\widetilde{c}} \\cdot D_{\\widetilde{c} \\cdot \\varphi _{1}}(\\mathbf {Q}, {P}) + A - 1,\\qquad \\text{(cf.",
"(\\ref {brostu3:fo.divpow.Kull1}))}\\nonumber $ $\\widetilde{I} := \\sum \\displaylimits _{k=1}^{K} p_{k} \\cdot \\log \\left( \\frac{p_{k}}{q_{k}} \\right)\\ = \\ \\frac{1}{\\widetilde{c}} \\cdot D_{\\widetilde{c} \\cdot \\varphi _{0}}(\\mathbf {Q}, {P}) + 1 - A.\\qquad \\text{(cf.",
"(\\ref {brostu3:fo.divpow.RevKull1}))}\\nonumber $ To proceed, let us fix an arbitrary constant $\\widetilde{c} >0$ .",
"(i) Case $\\gamma \\cdot (1-\\gamma ) \\ne 0$ .",
"(ia) Let us start with the subcase $\\gamma \\in ]-\\infty ,0[$ .",
"From the first and the last line of (REF ), it is clear that the corresponding $m-$ infimum can not be achieved for $m \\le 0$ ; since $H_{\\gamma } > 0$ one gets the unique minimizer $m_{min} = \\Big (\\frac{H_{\\gamma }}{A}\\Big )^{1/(1-\\gamma )} >0$ and the minimum $D_{\\widetilde{c} \\cdot \\varphi _{\\gamma }}(m_{min} \\cdot \\mathbf {Q},{P})=\\frac{\\widetilde{c}}{\\gamma } \\cdot (1- \\frac{H^{1/(1-\\gamma )}}{A^{\\gamma /(1-\\gamma )}} )$ .",
"Hence, (REF ) is established.",
"The assertions (REF ) and () follow immediately by monotonicity inspection of $x \\rightarrow \\frac{\\widetilde{c}}{\\gamma } \\cdot \\left[ 1-\\frac{1}{A^{\\gamma /(1-\\gamma )}} \\cdot \\left[ 1 + \\gamma \\cdot (A-1) + \\frac{\\gamma \\cdot \\left( \\gamma -1\\right)}{\\widetilde{c}}\\cdot x\\right] ^{-1/\\left( \\gamma -1\\right) }\\right]$ for $x \\ge 0$ such that $1 + \\gamma \\cdot (A-1) + \\frac{\\gamma \\cdot \\left( \\gamma -1\\right)}{\\widetilde{c}}\\cdot x \\ge 0$ .",
"(ib) The subcase $\\gamma \\in ]0,1[$ (cf.",
"the third line of (REF )) works analogously if $H_{\\gamma } >0$ ; furthermore, if $H_{\\gamma } =0$ — which can only appear when ${P}$ , $\\mathbf {Q}$ have disjoint supports (singularity)— then $\\inf _{m>0} D_{\\widetilde{c} \\cdot \\varphi _{\\gamma }}(m \\cdot \\mathbf {Q},{P}) =\\frac{\\widetilde{c}}{\\gamma }$ which is (the corresponding special case of) (REF ).",
"(ic) In the subcase $\\gamma \\in ]1,\\infty [$ (cf.",
"the fifth, sixth and seventh line of (REF )) it is straightforward to see that the desired infimum can not be achieved for $m < 0$ .",
"Hence, one can proceed analogously to subcase (ia).",
"(id) The assertions () to () are straightforward.",
"(ii) Case $\\gamma =1$ .",
"From the fourth line of (REF ), one obtains the unique minimizer $m_{min} = \\exp \\lbrace -I/A\\rbrace $ and the minimum $D_{\\widetilde{c} \\cdot \\varphi _{1}}(m_{min} \\cdot \\mathbf {Q},{P})= \\widetilde{c} \\cdot (1- A \\cdot m_{min})$ , which leads to (REF ).",
"The monotonicity of $x \\rightarrow \\widetilde{c} \\cdot (1- \\exp \\lbrace -x/\\widetilde{c}\\rbrace )$ for $x\\ge 0$ implies immediately (REF ) and (); moreover, () and () are immediate.",
"(iii) Case $\\gamma =0$ .",
"The second line of (REF ) implies the unique minimizer $m_{min} = 1/A$ , the minimum $D_{\\widetilde{c} \\cdot \\varphi _{0}}(m_{min} \\cdot \\mathbf {Q},{P})= \\widetilde{c} \\cdot (\\widetilde{I} + \\log A)$ , and hence (REF ).",
"The assertions (REF ) to () are obvious.",
"$\\blacksquare $"
],
[
"Proofs — Part 4",
"Proof of Proposition REF .",
"Clearly, (G1) and (G2) are part of the definition of $\\widetilde{\\Upsilon }(]a,b[)$ .",
"Recall our required representability (REF ).",
"The therein involved Laplace-Stieltjes transform (Laplace-Lebesgue transform) $z\\mapsto MGF_{\\mathbb {}}(z):=\\int _{\\mathbb {R}}e^{z \\cdot y} \\, d\\mathbb {} (y)= E_{\\mathbb {\\Pi }}[e^{z\\cdot W}]$ of a probability measure $\\mathbb {}$ on the real line respectively of an associated random variable $W$ (with $\\mathbb {}[ \\cdot \\, ] := \\mathbb {\\Pi }[W \\in \\cdot \\, ]$ ) has the following fundamental properties, according to well-known general theory: $MGF_{\\mathbb {}}$ takes values in $]0,\\infty ]$ ; the effective domain $dom(MGF_{\\mathbb {}})$ is an interval which contains 0 and which may be degenerated or even the whole real line; correspondingly, we denote its interior by $]\\lambda _{-},\\lambda _{+}[ := int(dom(MGF_{\\mathbb {}}))$ which may be the empty set (in case that $dom(MGF_{\\mathbb {}})=\\lbrace 0\\rbrace $ , i.e.",
"$\\lambda _{-}=\\lambda _{+}=0$ ); clearly, there holds $\\lambda _{-} \\in [-\\infty ,0]$ and $\\lambda _{+} \\in [0,\\infty ]$ ; $MGF_{\\mathbb {}}$ is continuous on $dom(MGF_{\\mathbb {}})$ and lower semicontinuous on $\\mathbb {R}$ ; if $\\lambda _{-} \\ne \\lambda _{+}$ , then $MGF_{\\mathbb {}}$ is real analytic and thus infinitely differentiable on $]\\lambda _{-},\\lambda _{+}[$ ; if $MGF_{\\mathbb {}}$ is finite in a neighborhood of zero, i.e.",
"$0 \\in ]\\lambda _{-},\\lambda _{+}[$ , then for all $k \\in \\mathbb {N}_{0}$ the $k-$ th moment of $\\mathbb {}$ respectively $W$ exists and is finite and can be computed in terms of the $k-$ th derivative $MGF_{\\mathbb {}}^{(k)}$ as $MGF_{\\mathbb {}}^{(k)}(0) =\\int _{\\mathbb {R}} y^{k} \\, d\\mathbb {} (y)= E_{\\mathbb {\\Pi }}[W^{k}],$ which, by the way, then allows the interpretation of $MGF_{\\mathbb {}}$ as “moment generating function of $\\mathbb {}$ resp.",
"$W$ ”since we assume $0 \\in ]\\lambda _{-},\\lambda _{+}[$ , we have already used the meaningful abbreviation $MGF$ (rather than LST) in (REF ); if $\\lambda _{-} \\ne \\lambda _{+}$ , then $MGF_{\\mathbb {}}$ is strictly convex on $]\\lambda _{-},\\lambda _{+}[$ .",
"Hence, the logarithm of the Laplace-Stieltjes transform $z\\mapsto \\Lambda _{\\mathbb {}}(z) := \\log MGF_{\\mathbb {} }(z):=\\log \\int _{\\mathbb {R}}e^{z \\cdot y} \\, d\\mathbb {} (y)= \\log E_{\\mathbb {\\Pi }}[e^{z\\cdot W}]$ (which in case of $0 \\in ]\\lambda _{-},\\lambda _{+}[$ can be interpreted as cumulant generating function) “carries over” (M1) to (M6), which partially can be even refined: $\\Lambda _{\\mathbb {}}$ takes values in $]-\\infty ,\\infty ]$ ; $dom(\\Lambda _{\\mathbb {}}) = dom(MGF_{\\mathbb {}})$ and thus $int(dom(\\Lambda _{\\mathbb {} })) = \\, ]\\lambda _{-},\\lambda _{+}[$ ; $\\Lambda _{\\mathbb {} }$ is continuous on $dom(\\Lambda _{\\mathbb {}})$ and lower semicontinuous on $\\mathbb {R}$ ; if $\\lambda _{-} \\ne \\lambda _{+}$ , then $\\lambda _{\\mathbb {}}$ is infinitely differentiable on $]\\lambda _{-},\\lambda _{+}[$ ; if $0 \\in ]\\lambda _{-},\\lambda _{+}[$ , then $& & \\Lambda _{\\mathbb {} }(0) = 0, \\quad \\Lambda _{\\mathbb {} }^{\\prime }(0) = \\int _{\\mathbb {R}} y \\, d\\mathbb {} (y)= E_{\\mathbb {\\Pi }}[W],\\\\& & \\Lambda _{\\mathbb {} }^{\\prime \\prime }(0) =\\int _{\\mathbb {R}} \\Big (y - \\int _{\\mathbb {R}} \\widetilde{y} \\, d\\mathbb {} (\\widetilde{y})\\Big )^{2}\\, d\\mathbb {} (y)= E_{\\mathbb {\\Pi }}[W^{2}] - (E_{\\mathbb {\\Pi }}[W])^{2} = Var_{\\mathbb {\\Pi }}[W];$ under the assumption $\\lambda _{-} \\ne \\lambda _{+}$ there holds: $\\Lambda _{\\mathbb {}}$ is strictly convex on $]\\lambda _{-},\\lambda _{+}[$ if and only if $\\mathbb {}$ is not a one-point distribution (Dirac mass) if and only if $W$ is not a.s. constant; otherwise, $\\Lambda _{\\mathbb {}}$ is linear; under the assumption that $\\mathbb {}$ is not a one-point distribution (Dirac mass) — with the notations $a := \\inf supp(\\mathbb {}) = \\inf supp(W)$ , $b := \\sup supp(\\mathbb {}) = \\sup supp(W)$ , $t_{-}^{sc} := \\inf \\lbrace \\Lambda _{\\mathbb {}}^{\\prime }(z) \\, : \\, z \\in ]\\lambda _{-},\\lambda _{+}[ \\rbrace = \\lim _{z \\downarrow \\lambda _{-}} \\Lambda _{\\mathbb {}}^{\\prime }(z)$ and $t_{+}^{sc} := \\sup \\lbrace \\Lambda _{\\mathbb {}}^{\\prime }(z) \\, : \\, z \\in ]\\lambda _{-},\\lambda _{+}[ \\rbrace = \\lim _{z \\uparrow \\lambda _{+}} \\Lambda _{\\mathbb {}}^{\\prime }(z)$ — one gets the following assertions: $]t_{-}^{sc},t_{+}^{sc}[ \\ \\subseteq \\ ]a,b[$ ; if $a > - \\infty $ , then $\\lambda _{-} = - \\infty $ , $t_{-}^{sc} = \\lim _{z \\rightarrow -\\infty } \\Lambda _{\\mathbb {}}^{\\prime }(z)= \\lim _{z \\rightarrow -\\infty } \\frac{\\Lambda _{\\mathbb {}}(z)}{z} = a$ ; if $b < \\infty $ , then $\\lambda _{+} = \\infty $ , $t_{+}^{sc} = \\lim _{z \\rightarrow \\infty } \\Lambda _{\\mathbb {}}^{\\prime }(z)= \\lim _{z \\rightarrow \\infty } \\frac{\\Lambda _{\\mathbb {}}(z)}{z} = b$ ; if $a = - \\infty $ and $\\lambda _{-} = - \\infty $ , then $t_{-}^{sc} = \\lim _{z \\rightarrow -\\infty } \\Lambda _{\\mathbb {}}^{\\prime }(z)= -\\infty = a$ ; if $b = \\infty $ and $\\lambda _{+} = \\infty $ , then $t_{+}^{sc} = \\lim _{z \\rightarrow \\infty } \\Lambda _{\\mathbb {}}^{\\prime }(z)= \\infty = b$ ; if $\\lambda _{-} \\in \\, ]-\\infty ,0[$ and $t_{-}^{sc} > -\\infty $ , then $a = - \\infty $ , $\\Lambda _{\\mathbb {}}(\\lambda _{-}) \\in \\, ]-\\infty ,\\infty [$ , $\\Lambda _{\\mathbb {}}(z) = \\infty $ for all $z < \\lambda _{-}$ , $\\Lambda _{\\mathbb {}}^{\\prime }(\\lambda _{-}) \\in \\, ]-\\infty ,\\infty [$ ; if $\\lambda _{+} \\in \\, ]0,\\infty [$ and $t_{+}^{sc} < \\infty $ , then $b = \\infty $ , $\\Lambda _{\\mathbb {}}(\\lambda _{+}) \\in \\, ]-\\infty ,\\infty [$ , $\\Lambda _{\\mathbb {}}(z) = \\infty $ for all $z > \\lambda _{+}$ , $\\Lambda _{\\mathbb {}}^{\\prime }(\\lambda _{+})\\in \\, ]-\\infty ,\\infty [$ ; if $\\lambda _{-} \\in \\, ]-\\infty ,0[$ and $t_{-}^{sc} = -\\infty $ , then $a= - \\infty $ ; if $\\lambda _{+} \\in \\, ]0,\\infty [$ and $t_{+}^{sc} = \\infty $ , then $b= \\infty $ .",
"Notice that (C7ii) to (C7ix) cover all possible constellations.",
"For a proof of (C7ii) to (C7vii) as well as further details, see e.g.",
"Section 9.1 in Borovkov [56].",
"By contradiction, (C7viii) follows from (C7ii) and (C7ix) follows from (C7iii).",
"Moreover, (C7i) is a consequence (C7ii) to (C7ix).",
"As a side remark, notice that (C6) refines (M6).",
"According to the representability requirement (REF ), one has $\\varphi (t)=\\sup _{z\\in \\mathbb {R}}\\left( z\\cdot t- \\Lambda _{\\mathbb {}}(z)\\right)=: \\Lambda _{\\mathbb {}}^{*}(t), \\qquad t\\in \\mathbb {R},\\ \\ $ (i.e.",
"the divergence generator $\\varphi $ must be equal to the Fenchel-Legendre transform $\\Lambda _{\\mathbb {}}^{*}$ of a cumulant generating function $\\Lambda _{\\mathbb {}}$ ) of some probability distribution $\\mathbb {}$ , such that $\\lambda _{-} < 0 < \\lambda _{+}$ holds.",
"Moreover, $\\varphi $ should satisfy $\\varphi (1)=0$ , and should be finite as well as strictly convex in a non-empty neighborhood $]t_{-}^{sc},t_{+}^{sc}[$ of 1 (cf.",
"the definition of $\\widetilde{\\Upsilon }(]a,b[)$ ).",
"The latter rules out that $\\mathbb {}$ is any one-point distribution (Dirac distribution), say $\\mathbb {} = \\delta _{y_{0}}$ for some $y_{0} \\in \\mathbb {R}$ , since in such a situation one gets $\\Lambda _{\\mathbb {}}(z) = z \\cdot y_{0}$ , and thus $\\varphi (t) = \\Lambda _{\\mathbb {}}^{*}(t) = 0$ for $t=y_{0}$ and $\\varphi (t) = \\Lambda _{\\mathbb {}}^{*}(t) = \\infty $ for all $t \\in \\mathbb {R}\\backslash \\lbrace y_{0}\\rbrace $ (even in the case $y_{0} =1$ for which $\\varphi (1)=0$ is satisfied).",
"Consequently, $\\Lambda _{\\mathbb {}}$ is strictly convex on $]\\lambda _{-},\\lambda _{+}[ \\ = int(dom(\\Lambda _{\\mathbb {} }))$ (cf.",
"(C6)) and (C7) applies.",
"Clearly, by continuity one gets $\\Lambda _{\\mathbb {}}^{*}(t) =\\sup _{z\\in ]\\lambda _{-},\\lambda _{+}[}\\left( t \\cdot z - \\Lambda _{\\mathbb {}}(z)\\right),\\qquad t\\in \\mathbb {R}.",
"\\ \\ $ For $t \\in ]t_{-}^{sc},t_{+}^{sc}[$ , the optimization problem (REF ) can be solved explicitly by the well-known “pure/original” Legendre transform, namely $\\Lambda _{\\mathbb {}}^{*}(t) = t \\cdot \\Lambda _{\\mathbb {}}^{\\prime -1}(t) -\\Lambda _{\\mathbb {}}\\Big (\\Lambda _{\\mathbb {}}^{\\prime -1}(t)\\Big ),\\qquad t \\in ]t_{-}^{sc},t_{+}^{sc}[.$ Let us inspect the further cases $t \\le t_{-}^{sc}$ .",
"In the contexts of (C7iv) and (C7viii), this is obsolete since $t_{-}^{sc} = a = -\\infty $ .",
"For (C7ii), where $t_{-}^{sc} = a > -\\infty $ , one can show $\\Lambda _{\\mathbb {}}^{*}(a) = - \\log \\mathbb {}[\\lbrace a\\rbrace ]= - \\log \\mathbb {\\Pi }[W = a \\, ]$ which together with (REF ) proves (G10ii); moreover, $\\Lambda _{\\mathbb {}}^{*}(t) = \\infty $ for all $t < a$ (see e.g.",
"Section 9.1 of Borovkov [56]).",
"In the setup (C7vi), where $t_{-}^{sc} > a = - \\infty $ it is clear that $\\Lambda _{\\mathbb {}}^{*}(t_{-}^{sc}) =t_{-}^{sc} \\cdot \\Lambda _{\\mathbb {}}^{\\prime -1}(t_{-}^{sc}) -\\Lambda _{\\mathbb {}}\\Big (\\Lambda _{\\mathbb {}}^{\\prime -1}(t_{-}^{sc})\\Big )= t_{-}^{sc} \\cdot \\lambda _{-} - \\Lambda _{\\mathbb {}}(\\lambda _{-})$ and $\\Lambda _{\\mathbb {}}^{*}(t)= t \\cdot \\lambda _{-} - \\Lambda _{\\mathbb {}}(\\lambda _{-})= \\Lambda _{\\mathbb {}}^{*}(t_{-}^{sc}) + \\lambda _{-} \\cdot (t- t_{-}^{sc})\\quad \\text{for all $t \\in ]-\\infty ,t_{-}^{sc}[$}.$ As far as the cases $t \\ge t_{+}^{sc}$ is concerned, in the situations of (C7v) and (C7ix), this is obsolete since $t_{+}^{sc} = b = \\infty $ .",
"For (C7iii), where $t_{+}^{sc} = b < \\infty $ , one can show $\\Lambda _{\\mathbb {}}^{*}(b) = - \\log \\mathbb {}[\\lbrace b\\rbrace ]= - \\log \\mathbb {\\Pi }[W = b \\, ]$ which together with (REF ) proves (G10iii); moreover, $\\Lambda _{\\mathbb {}}^{*}(t) = \\infty $ for all $t > b$ (see e.g.",
"Section 9.1 of Borovkov [56]).",
"In the setup (C7vii), where $t_{+}^{sc} < b = \\infty $ it is clear that $\\Lambda _{\\mathbb {}}^{*}(t_{+}^{sc}) =t_{+}^{sc} \\cdot \\Lambda _{\\mathbb {}}^{\\prime -1}(t_{+}^{sc}) -\\Lambda _{\\mathbb {}}\\Big (\\Lambda _{\\mathbb {}}^{\\prime -1}(t_{+}^{sc})\\Big )= t_{+}^{sc} \\cdot \\lambda _{+} - \\Lambda _{\\mathbb {}}(\\lambda _{+})$ and $\\Lambda _{\\mathbb {}}^{*}(t)= t \\cdot \\lambda _{+} - \\Lambda _{\\mathbb {}}(\\lambda _{+})= \\Lambda _{\\mathbb {}}^{*}(t_{+}^{sc}) + \\lambda _{+} \\cdot (t- t_{+}^{sc})\\quad \\text{for all $t \\in ]t_{+}^{sc}, \\infty [$}.$ As a side effect, we have thus also proved (G10i) and (G3) (notice that in (G3) we have started with $a, b$ to be the endpoints of the support of $\\mathbb {}$ respectively $W$ , in contrast to Definition REF where $a$ , $b$ are defined as the endpoints of the effective domain of $\\varphi $ ).",
"To proceed, from (REF ) and (REF ) we obtain $\\varphi ^{\\prime }(t) = (\\Lambda _{\\mathbb {}}^{*})^{\\prime }(t) = \\Lambda _{\\mathbb {}}^{\\prime -1}(t),\\qquad \\varphi ^{\\prime \\prime }(t) = (\\Lambda _{\\mathbb {}}^{*})^{\\prime \\prime }(t) =\\frac{1}{\\Lambda _{\\mathbb {}}^{\\prime \\prime }\\big (\\Lambda _{\\mathbb {}}^{\\prime -1}(t)\\big )} > 0,\\qquad t \\in ]t_{-}^{sc},t_{+}^{sc}[,$ which — together with the investigations below (REF ) — provides (G4) and (G5); moreover, (G6) is immediate since the infinite differentiability is straightforward and $\\varphi ^{\\prime }(1) =0$ because we have required both the nonnegativity of $\\varphi $ and (G2) (cf.",
"the definition of $\\widetilde{\\Upsilon }(]a,b[)$ ).",
"The property (G7) follows from (C7ii), (C7iv), (C7viii), (REF ), (REF ) and $\\varphi ^{\\prime }(t_{-}^{sc}) = \\Lambda _{\\mathbb {}}^{\\prime -1}(t_{-}^{sc}) = \\lambda _{-}$ .",
"Analogously, we get (G8) from (C7iii), (C7v), (C7ix), (REF ), (REF ) and $\\varphi ^{\\prime }(t_{+}^{sc}) = \\Lambda _{\\mathbb {}}^{\\prime -1}(t_{+}^{sc}) = \\lambda _{+}$ .",
"Let us continue with (G9).",
"By applying the general theory of double Fenchel-Legendre transforms (bi-conjugates), (REF ) turns into $\\varphi ^{*} (z) = \\Lambda _{\\mathbb {}}(z), \\qquad z\\in \\mathbb {R},\\ \\ $ which deduces (G9i).",
"The properties (G9ii), (G9iii) and (G9iv) follow from Theorem REF (cf.",
"the discussion thereafter).",
"Finally, we obtain (G11i) and (G11ii) from (REF ), (REF ) and ().",
"$\\blacksquare $ Proof of Proposition REF .",
"The assertions follow immediately from (REF ), (REF ), (REF ), Theorem REF , (REF ) (and the discussion thereafter) as well as (M5).",
"$\\blacksquare $"
],
[
"Proofs — Part 5",
"Proof of Proposition REF .",
"The assertion follows straightforwardly from the following two facts: (i) a moment generating function $MGF$ is infinitely divisible if and only if $MGF^{c}$ is a moment generating function for all $c >0$ (cf.",
"e.g.",
"(the MGF-version of) Prop.",
"IV.2.5 of Steutel & van Harn [341]).",
"(ii) $z \\mapsto MGF(z)$ is a moment generating function if and only if $z \\mapsto MGF(\\breve{c} \\cdot z) =: MGF_{\\breve{c}}(z)$ is a moment generating function for all $\\breve{c} >0$ .",
"Notice that for each $c >0$ , $\\breve{c} >0$ one has $int(dom(MGF)) = int(dom(MGF^{c}))$ and $int(dom(MGF_{\\breve{c}})) = \\frac{1}{\\breve{c}} \\cdot int(dom(MGF))$ , and hence the light-tailedness remains unchanged: $0 \\in int(dom(MGF))$ if and only if $0 \\in int(dom(MGF^{c}))$ if and only if $0 \\in int(dom(MGF_{\\breve{c}}))$ .",
"Since $\\varphi \\in \\Upsilon (]a,b[)$ , we have $\\varphi (t)=\\sup _{z\\in ]\\lambda _{-}, \\lambda _{+}[}\\left( z \\cdot t-\\log \\Big (\\int _{\\mathbb {R}} e^{z\\cdot y }\\, d\\mathbb {} (y) \\Big ) \\right), \\quad t\\in ]a,b[ \\, , \\ \\ $ and thus for the exponential of its Fenchel-Legendre transform $\\int _{\\mathbb {R}} e^{z \\cdot y} d\\mathbb {} (y) , \\qquad z \\in ]\\lambda _{-}, \\lambda _{+}[ .$ Now, let $\\widetilde{\\varphi } := \\widetilde{c} \\cdot \\varphi \\in \\Upsilon (]a,b[)$ for arbitrarily fixed $\\widetilde{c} >0$ .",
"From the application of (REF ) to $\\widetilde{\\varphi }$ we obtain $\\widetilde{\\varphi } (t)=\\sup _{\\widetilde{z} \\in ]\\widetilde{\\lambda }_{-}, \\widetilde{\\lambda }_{+}[}\\left( \\widetilde{z} \\cdot t-\\log \\int _{\\mathbb {R}}e^{\\widetilde{z} \\cdot \\widetilde{y}} d\\widetilde{\\mathbb {}}_{\\widetilde{c}} (\\widetilde{y})\\right), \\qquad t\\in ]a,b[ \\, , \\ \\ $ for some unique probability distribution $\\widetilde{\\mathbb {}}_{\\widetilde{c}}$ on $\\mathbb {R}$ .",
"Here, according to (G9i) for $\\widetilde{\\varphi }$ we have used $\\widetilde{\\lambda }_{-} := \\inf _{t \\in ]a,b[} \\widetilde{\\varphi }^{\\prime }(t) = \\widetilde{c} \\cdot \\lambda _{-}$ and $\\widetilde{\\lambda }_{+} := \\sup _{t \\in ]a,b[} \\widetilde{\\varphi }^{\\prime }(t) = \\widetilde{c} \\cdot \\lambda _{+}$ .",
"Dividing (REF ) by $\\widetilde{c}$ , we arrive at $\\varphi (t) = \\frac{\\widetilde{\\varphi } (t)}{\\widetilde{c}}&=&\\sup _{\\widetilde{z} \\in ]\\widetilde{c} \\cdot \\lambda _{-}, \\widetilde{c} \\cdot \\lambda _{+}[}\\left( \\frac{\\widetilde{z}}{\\widetilde{c}} \\cdot t-\\log \\Big (\\int _{\\mathbb {R}}e^{\\frac{\\widetilde{z}}{\\widetilde{c}} \\cdot \\widetilde{y} \\cdot \\widetilde{c}} d\\widetilde{\\mathbb {}}_{\\widetilde{c}} (\\widetilde{y})\\Big )^{1/\\widetilde{c}}\\right),\\nonumber \\\\&=&\\sup _{z \\in ]\\lambda _{-}, \\lambda _{+}[}\\left( z \\cdot t-\\log \\Big (\\int _{\\mathbb {R}}e^{z \\cdot \\widetilde{y} \\cdot \\widetilde{c}}d\\widetilde{\\mathbb {}}_{\\widetilde{c}} (\\widetilde{y})\\Big )^{1/\\widetilde{c}}\\right),\\qquad t\\in ]a,b[ \\, , \\ \\ $ and hence for the exponential of its Fenchel-Legendre transform $e^{\\varphi _{*}(z)} =\\Big ( \\int _{\\mathbb {R}} e^{z \\cdot \\widetilde{y} \\cdot \\widetilde{c}}d\\widetilde{\\mathbb {}}_{\\widetilde{c}} (\\widetilde{y}) \\Big )^{1/\\widetilde{c}}, \\qquad z \\in ]\\lambda _{-}, \\lambda _{+}[ .$ Here, according to (G9i) for $\\widetilde{\\varphi }$ we have used $\\widetilde{\\lambda }_{-} := \\inf _{t \\in ]a,b[} \\widetilde{\\varphi }^{\\prime }(t) = \\widetilde{c} \\cdot \\lambda _{-}$ and $\\widetilde{\\lambda }_{+} := \\sup _{t \\in ]a,b[} \\widetilde{\\varphi }^{\\prime }(t) = \\widetilde{c} \\cdot \\lambda _{+}$ .",
"From (REF ) and (REF ) we deduce for $\\widetilde{c} := \\frac{1}{n}$ the relation $MGF_{\\mathbb {}}(z) = (MGF_{\\widetilde{\\mathbb {}}_{1/n}}(\\frac{z}{n}))^{n}$ for all $n \\in \\mathbb {N}$ which (with the help of (ii)) implies the infinitely divisibility of $\\mathbb {}$ .",
"For the reverse direction, let us assume that $\\varphi \\in \\Upsilon (]a,b[)$ and that the corresponding $\\mathbb {}$ is infinitely divisible.",
"Recall that $]a,b[ = int(dom(\\varphi ))$ .",
"Moreover, we fix an arbitrary constant $\\widetilde{c} >0$ .",
"Of course, there holds $\\widetilde{c} \\cdot \\varphi \\in \\widetilde{\\Upsilon }(]a,b[)$ and $dom(\\widetilde{c} \\cdot \\varphi ) = dom(\\varphi )$ .",
"Furthermore, by multiplying (REF ) with $\\widetilde{c} >0$ and by employing (i), (ii) we get $\\widetilde{c} \\cdot \\varphi (t) &=&\\sup _{z\\in ]\\lambda _{-}, \\lambda _{+}[}\\left( \\widetilde{c} \\cdot z \\cdot t-\\log \\Big (\\int _{\\mathbb {R}} e^{\\widetilde{c} \\cdot z\\cdot \\frac{y}{\\widetilde{c}}} d\\mathbb {} (y) \\Big )^{\\widetilde{c}} \\right)= \\sup _{\\widetilde{z} \\in ]\\widetilde{c} \\cdot \\lambda _{-}, \\widetilde{c} \\cdot \\lambda _{+}[}\\left( \\widetilde{z} \\cdot t-\\log \\Big (\\int _{\\mathbb {R}}e^{\\frac{\\widetilde{z}}{\\widetilde{c}} \\cdot y} \\, d\\mathbb {} (y) \\Big )^{\\widetilde{c}} \\right)\\nonumber \\\\&=&\\sup _{\\widetilde{z} \\in ]\\widetilde{c} \\cdot \\lambda _{-}, \\widetilde{c} \\cdot \\lambda _{+}[}\\left( \\widetilde{z} \\cdot t-\\log \\Big ( \\int _{\\mathbb {R}}e^{\\widetilde{z} \\cdot y} d\\mathbb {}_{\\widetilde{c}} (y) \\Big ) \\right), \\quad t\\in ]a,b[; \\ \\ $ for some probability distribution $\\mathbb {}_{\\widetilde{c}}$ on $\\mathbb {R}$ .",
"$\\blacksquare $ Proof of Proposition REF .",
"It is well known that a candidate function $M: ]-\\infty ,0[ \\mapsto ]0,\\infty [$ is the moment-generating function of an infinitely divisible probability distribution if and only if $(\\log M)^{\\prime }$ is absolutely monotone (see e.g.",
"Theorem 5.11 of Schilling et al.",
"[322]).",
"By applying this to $M(z) := e^{-a\\cdot z + \\varphi ^{*}(z)}$ respectively $M(z) := e^{b\\cdot z + \\varphi ^{*}(- z)}$ , one gets straightforwardly the assertion (a) respectively (b); notice that the light-tailedness follows then from (G1) to (G8), and $b=\\infty $ respectively $a =- \\infty $ can be deduced from the fact that the support of an infinitely distribution is always (one-sided or two-sided) unbounded.",
"For the third case $a= - \\infty $ , $b= \\infty $ one can use the assertion (cf.",
"e.g.",
"Morris [267], p.73) that a candidate function $M: \\, ]\\lambda _{-}, \\lambda _{+}[ \\, \\mapsto \\, ]0,\\infty [$ is the moment-generating function of an infinitely divisible probability distribution if the connected function $z \\mapsto (\\log M)^{\\prime \\prime }(z)/(\\log M)^{\\prime \\prime }(0)$ is the moment-generating function of some auxiliary probability distribution; but the latter is equivalent to exponentially convexity (cf.",
"Theorem REF (b)).",
"By applying this to $M(z) := e^{\\varphi ^{*}(z)}$ , one ends up with (c).",
"$\\blacksquare $"
],
[
"Proofs — Part 6",
"Proof of Theorem REF .",
"(i) Clearly, on $]\\lambda _{-},\\lambda _{+}[$ the function $\\Lambda $ is differentiable with strictly increasing derivative $\\Lambda ^{\\prime }(z) = F^{-1}(z+c) + 1- F^{-1}(c) ,\\qquad z \\in ]\\lambda _{-},\\lambda _{+}[.$ Hence, $\\Lambda $ is strictly convex and smooth (because of the smoothness of $F^{-1}$ ), and satisfies $\\Lambda (0) =0$ as well as $\\Lambda ^{\\prime }(0) =1$ .",
"Also, the corresponding extensions of $\\Lambda $ to $z=\\lambda _{-}$ and $z=\\lambda _{+}$ are continuous.",
"(ii) It is straightforward to see that on $]t_{-}^{sc},t_{+}^{sc}[$ the function $\\varphi $ is differentiable with strictly increasing derivative $\\varphi ^{\\prime }(t) = F(t+ F^{-1}(c) - 1) - c ,\\qquad t \\in ]t_{-}^{sc},t_{+}^{sc}[.$ Hence, $\\varphi $ is strictly convex and smooth (because of the smoothness of $F$ ), and satisfies $\\varphi (1) =0$ as well as $\\varphi ^{\\prime }(1) =0$ .",
"Also, the corresponding extensions of $\\varphi $ to $t=t_{-}^{sc}$ and $t=t_{-}^{sc}$ are continuous.",
"Hence (G1), (G2), (G5) and (G6) hold.",
"To prove (G3) (and hence (G1)), let us first notice that obviously there holds $a \\le t_{-}^{sc}$ and $t_{+}^{sc} \\le b$ .",
"Moreover, the validity of $\\varphi (t) < \\infty $ for all $t \\in ]t_{-}^{sc},t_{+}^{sc}[$ is clear from (REF ) since $t+F^{-1}(c)-1 \\in ]a_{F},b_{F}[ = int(dom(F))$ and the involved integral over the continuous function $F^{-1}$ is taken over a compact interval.",
"For the subcase $t_{-}^{sc} =- \\infty = a$ we have thus shown $dom(\\varphi ) \\cap ]-\\infty ,1] = ]-\\infty ,1]= ]a,1]$ , whereas for the subcase $t_{+}^{sc} = \\infty = b$ we have verified $dom(\\varphi ) \\cap [1,\\infty [ = [1,\\infty [ = [1,b[$ .",
"Let us next examine the subcase “$t_{-}^{sc} > - \\infty $ and $\\varphi (t_{-}^{sc}) < \\infty $ ”: if $\\lambda _{-} > - \\infty $ then $a = -\\infty $ and (REF ) implies $\\varphi (t) = \\varphi (t_{-}^{sc}) +\\lambda _{-} \\cdot (t- t_{-}^{sc}) < \\infty $ for all $t \\in \\ ]-\\infty , t_{-}^{sc}] = ]a,t_{-}^{sc}]$ , which leads to $dom(\\varphi ) \\cap ]-\\infty ,1] = ]-\\infty ,1] = ]a,1]$ ; in contrast, if $\\lambda _{-} = - \\infty $ then $a = t_{-}^{sc}$ and (REF ) implies $\\varphi (t) = \\varphi (t_{-}^{sc}) +\\lambda _{-} \\cdot (t- t_{-}^{sc}) = \\infty $ for all $t \\in \\ ]-\\infty , t_{-}^{sc}[ = ]-\\infty ,a[$ , which leads to $dom(\\varphi ) \\cap ]-\\infty ,1] = [a,1]$ .",
"In the subcase “$t_{-}^{sc} > - \\infty $ and $\\varphi (t_{-}^{sc}) = \\infty $ ”, due to the strict convexity of $\\varphi $ one always has $\\lim _{t \\downarrow t_{-}^{sc}} \\varphi ^{\\prime }(t) = - \\infty $ ; this implies, by the below-mentioned (REF ), that $\\lambda _{-} = - \\infty $ and thus $a = t_{-}^{sc}$ ; from (REF ) we derive $\\varphi (t) = \\varphi (t_{-}^{sc}) +\\lambda _{-} \\cdot (t- t_{-}^{sc}) = \\infty $ for all $t \\in \\ ]-\\infty , t_{-}^{sc}[ = ]-\\infty ,a[$ , which leads to $dom(\\varphi ) \\cap ]-\\infty ,1] = ]a,1]$ .",
"As a further step, we deal with the subcase “$t_{+}^{sc} < \\infty $ and $\\varphi (t_{+}^{sc}) < \\infty $ ”: if $\\lambda _{+} < \\infty $ then $b = \\infty $ and (REF ) implies $\\varphi (t) = \\varphi (t_{+}^{sc}) +\\lambda _{+} \\cdot (t- t_{+}^{sc}) < \\infty $ for all $t \\in \\ [t_{+}^{sc},\\infty [ = [t_{+}^{sc},b[$ , which leads to $dom(\\varphi ) \\cap [1, \\infty [ = [1, \\infty [ = [1, b[$ ; in contrast, if $\\lambda _{+} = \\infty $ then $b = t_{+}^{sc}$ and (REF ) implies $\\varphi (t) = \\varphi (t_{+}^{sc}) +\\lambda _{+} \\cdot (t- t_{+}^{sc}) = \\infty $ for all $t \\in \\ ]t_{-}^{sc}, \\infty [ = ]b,\\infty [$ , which leads to $dom(\\varphi ) \\cap [1,\\infty [ = [1,b]$ .",
"In the subcase “$t_{+}^{sc} < + \\infty $ and $\\varphi (t_{+}^{sc}) = \\infty $ ”, due to the strict convexity of $\\varphi $ one always gets $\\lim _{t \\uparrow t_{+}^{sc}} \\varphi ^{\\prime }(t) = \\infty $ ; this implies, by the below-mentioned (), that $\\lambda _{+} = \\infty $ and thus $b = t_{+}^{sc}$ ; from (REF ) we deduce $\\varphi (t) = \\varphi (t_{+}^{sc}) +\\lambda _{+} \\cdot (t- t_{+}^{sc}) = \\infty $ for all $t \\in \\ ]t_{+}^{sc}, \\infty [ = ]b, \\infty [$ , which leads to $dom(\\varphi ) \\cap [1,\\infty [ = [1,b[$ .",
"Putting things together, we have proved (G3).",
"The property (G4) follows straightforwardly from (REF ), the continuity of $F$ and from $\\lim _{t \\downarrow t_{-}^{sc}} \\varphi ^{\\prime }(t) = \\lambda _{-}$ , $\\lim _{t \\uparrow t_{+}^{sc}} \\varphi ^{\\prime }(t) = \\lambda _{+}$ .",
"To see the latter two, from (REF ) we obtain $& & \\lim _{t \\downarrow t_{-}^{sc}} \\varphi ^{\\prime }(t) = \\lim _{t \\downarrow t_{-}^{sc}} F(t+ F^{-1}(c) - 1) - c= \\lim _{t \\downarrow t_{-}^{sc}} F(t+ a_{F} - t_{-}^{sc}) - c= \\inf \\lbrace F(\\widetilde{t}) - c: \\widetilde{t} \\in ]a_{F},b_{F}[ \\rbrace = \\lambda _{-},\\\\& & \\lim _{t \\uparrow t_{+}^{sc}} \\varphi ^{\\prime }(t) = \\lim _{t \\uparrow t_{+}^{sc}} F(t+ F^{-1}(c) - 1) - c= \\lim _{t \\uparrow t_{+}^{sc}} F(t+ b_{F} - t_{+}^{sc}) - c= \\sup \\lbrace F(\\widetilde{t}) - c: \\widetilde{t} \\in ]a_{F},b_{F}[ \\rbrace = \\lambda _{+}.$ The two properties (G7) and (G8) are clear form the above considerations.",
"(iii) From (REF ) and (REF ) one gets easily $\\Lambda ^{\\prime -1}(t) = F\\left(t+F^{-1}(c)-1 \\right) - c = \\varphi ^{\\prime }(t), \\qquad t \\in ]t_{-}^{sc},t_{+}^{sc}[,$ as well as $\\Lambda ^{\\prime -1}(1)=0$ .",
"From this, we derive $&& t \\cdot \\Lambda ^{\\prime -1}(t) - \\Lambda \\left(\\Lambda ^{\\prime -1}(t)\\right)\\nonumber \\\\&& = t \\cdot [F\\left(t+F^{-1}(c)-1 \\right) - c]+ [F^{-1}(c)-1] \\cdot [F\\left(t+F^{-1}(c)-1 \\right) - c]\\nonumber \\\\& & -\\int \\displaylimits _{0}^{F\\left(t+F^{-1}(c)-1 \\right) - c} F^{-1}(u+c) du\\nonumber \\\\& & = \\varphi (t), \\qquad t \\in ]t_{-}^{sc},t_{+}^{sc}[,$ and hence, with the help of (REF ) in combination with (REF ), () $\\varphi (t) = \\max _{z \\in ]\\lambda _{-},\\lambda _{+}[} \\left( z\\cdot t -\\Lambda (z)\\right), \\qquad t \\in ]t_{-}^{sc},t_{+}^{sc}[ ,$ i.e.",
"on $]t_{-}^{sc},t_{+}^{sc}[$ the divergence generator $\\varphi $ is the classical Legendre transform of the restriction of $\\Lambda $ to $]\\lambda _{-},\\lambda _{+}[$ .",
"If “$\\lambda _{-} > -\\infty $ , $\\Lambda (\\lambda _{-}) \\in ]-\\infty ,\\infty [$ and $\\Lambda ^{\\prime }(\\lambda _{-}) \\in ]-\\infty ,\\infty [$ ” respectively “$\\lambda _{+} < -\\infty $ , $\\Lambda (\\lambda _{+}) \\in ]-\\infty ,\\infty [$ and $\\Lambda ^{\\prime }(\\lambda _{+}) \\in ]-\\infty ,\\infty [$ ”, then one can apply classical facts of Fenchel-Legendre transformation to get the corresponding left-hand respectively right-hand linear extensions of $\\varphi $ on the complement of $]t_{-}^{sc},t_{+}^{sc}[$ , in order to obtain the desired $\\varphi (t) =\\sup _{z \\in ]-\\infty ,\\infty [} \\left( z\\cdot t - \\Lambda (z)\\right) ,\\qquad t \\in \\mathbb {R};$ notice that $t_{-}^{sc} = \\lim _{z \\downarrow \\lambda _{-}} \\Lambda ^{\\prime }(z)$ and $t_{+}^{sc} = \\lim _{z \\uparrow \\lambda _{+}} \\Lambda ^{\\prime }(z)$ .",
"(iv) This is just the inverse of (iii), by applying standard Fenchel-Legendre-transformation theory.",
"$\\blacksquare $"
],
[
"Further details and proofs for\nSubsection ",
"Proof of Lemma REF.",
"By Assumption (OM), one gets for all $\\lambda \\in cl(\\mathbf {\\Lambda })$ that $\\lbrace \\mathbf {x} \\in (dom(\\widetilde{\\varphi })^{n} : T(\\mathbf {x}) = \\lambda \\rbrace \\, \\cap \\, ]t_{-}^{sc},t_{+}^{sc}[^{n} \\ne \\emptyset $ .",
"Moreover, for any $\\mathbf {x}=\\left( x_{1},..,x_{n}\\right) $ in $\\mathbb {R}^{n}$ , by the independence of the components of $\\mathbf {\\widetilde{W}}$ as well as (REF ) and (REF ), we have $& & I_{\\mathbf {\\widetilde{W}}}(\\mathbf {x})=\\sup _{\\mathbf {z}=\\left( z_{1},\\ldots ,z_{n}\\right) \\in \\mathbb {R}^{n}}\\left( \\left\\langle \\mathbf {x},\\mathbf {z}\\right\\rangle -\\sum _{i=1}^{n}\\Lambda _{\\widetilde{\\mathbb {}}}(z_{i}) \\right)= \\sup _{\\mathbf {z}\\in ]\\lambda _{-},\\lambda _{+}[^{n}}\\left( \\sum _{i=1}^{n} \\left( x_{i} \\cdot z_{i}-\\Lambda _{\\widetilde{\\mathbb {}}}(z_{i}) \\right) \\right)\\nonumber \\\\& & = \\sum _{i=1}^{n} \\left( \\sup _{z_{i}\\in ]\\lambda _{-},\\lambda _{+}[}\\left( x_{i}\\cdot z_{i} - \\Lambda _{\\widetilde{\\mathbb {}}}(z_{i})\\right) \\right)= \\sum _{i=1}^{n} \\widetilde{\\varphi }(x_{i})=\\sum _{k=1}^{K}\\sum _{i\\in I_{k}^{(n)}} \\widetilde{\\varphi }(x_{i})$ which is finite if and only if $\\mathbf {x} \\in (dom(\\widetilde{\\varphi }))^{n}$ (recall that $\\widetilde{\\varphi }$ is a nonnegative function).",
"Hence, for each $\\lambda \\in \\mathbf {\\Lambda }$ we obtain $I(\\lambda ) &:=&\\inf _{\\mathbf {x} \\in \\mathbb {R}^{n}: \\, T(\\mathbf {x})=\\lambda }I_{\\mathbf {\\widetilde{W}}}(\\mathbf {x})= \\inf _{\\mathbf {x} \\in (dom(\\widetilde{\\varphi }))^{n}: \\, T(\\mathbf {x})=\\lambda }I_{\\mathbf {\\widetilde{W}}}(\\mathbf {x}) =\\inf _{\\mathbf {x} \\in (dom(\\widetilde{\\varphi }))^{n}: \\, T(\\mathbf {x})=\\lambda } \\ \\sum _{k=1}^{K}\\sum _{i\\in I_{k}^{(n)}} \\widetilde{\\varphi }(x_{i})\\\\&=& \\sum _{k=1}^{K} n_{k} \\cdot \\widetilde{\\varphi }(\\lambda _{k})\\ = \\ n \\cdot \\sum _{k=1}^{K} \\widetilde{p}_{k} \\cdot \\widetilde{\\varphi }(\\lambda _{k})= \\inf _{\\mathbf {x} \\in ]t_{-}^{sc},t_{+}^{sc}[^{n} \\, : \\, T(\\mathbf {x})=\\lambda }I_{\\mathbf {\\widetilde{W}}}(\\mathbf {x}) \\, ;$ here, we have employed the following facts: (i) the right-most infimum in (REF ) is achieved by minimizing each of the $K$ terms $\\sum _{i\\in I_{k}^{(n)}}\\widetilde{\\varphi }(x_{i})$ under the linear constraint $\\frac{1}{n_{k}} \\cdot \\sum _{i\\in I_{k}^{(n)}} x_{i}= \\lambda _{k}$ , and by the strict convexity of $\\widetilde{\\varphi }$ on $]t_{-}^{sc},t_{+}^{sc}[$ (cf.",
"(G5)) the minimum of this generic term is attained when all components $x_{i}$ are equal to $\\lambda _{k}$ , and (ii) the outcoming minimum does not depend on the particular (generally non-unique) choice of the $x_{i}$ ’s.",
"Notice that we have used the relation $n_{k} = n \\cdot \\widetilde{p}_{k}$ as well.",
"To proceed, let $\\underline{\\lambda }$ be a minimal rate point of $\\mathbf {\\Lambda }$ , which means that $\\underline{\\lambda } \\in \\partial \\mathbf {\\Lambda }$ and $I(\\underline{\\lambda }) \\le I(\\lambda )$ for all $\\lambda \\in \\mathbf {\\Lambda }$ .",
"By Assumption (OM) one can run all the steps in (REF ) and () with $\\underline{\\lambda }$ instead of $\\lambda $ , and hence $I(\\underline{\\lambda }) \\ = \\ \\inf _{\\mathbf {x} \\in \\mathbb {R}^{n}: \\, T(\\mathbf {x})=\\underline{\\lambda }}I_{\\mathbf {\\widetilde{W}}}(\\mathbf {x})\\ = \\inf _{\\mathbf {x} \\in ]t_{-}^{sc},t_{+}^{sc}[^{n} \\, : \\, T(\\mathbf {x})=\\underline{\\lambda }}I_{\\mathbf {\\widetilde{W}}}(\\mathbf {x})\\ = \\ n \\cdot \\sum _{k=1}^{K} \\widetilde{p}_{k} \\cdot \\widetilde{\\varphi }(\\underline{\\lambda }_{k})\\ = \\ n \\cdot \\sum _{k=1}^{K} \\widetilde{p}_{k} \\cdot \\widetilde{\\varphi }(\\underline{\\widetilde{q}}_{k}/\\widetilde{p}_{k})$ where for the last equality we have employed the vector $\\underline{\\mathbf {\\widetilde{Q}}} := \\mathfrak {D}^{-1} \\underline{\\lambda }$ which we have called the “dominating point of $\\widetilde{\\mathbf {\\Omega }}$ ”.",
"Also we have proved $I(\\underline{\\lambda })\\ = \\ n \\cdot \\inf _{\\mathbf {\\widetilde{Q}} \\in \\widetilde{\\mathbf {\\Omega }}}\\sum _{k=1}^{K} \\widetilde{p}_{k} \\cdot \\widetilde{\\varphi }(\\widetilde{q}_{k}/\\widetilde{p}_{k}).",
"\\qquad \\qquad \\blacksquare $ On the obtainment of proxies of minimal rate points by proxy method 2: For the rest of this section, besides (OM) we assume that $dom(\\widetilde{\\varphi }) = \\, ]a,b[ \\, = \\, ]t_{-}^{sc},t_{+}^{sc}[$ , and that in case of $a=-\\infty $ or $b=+\\infty $ the divergence generator $\\widetilde{\\varphi }$ is regularly varying at $a$ or $b$ accordingly, with positive index $\\beta $ , i.e.",
"(with a slight abuse of notation) if $a=-\\infty $ , then for all $\\lambda >0$ there holds $\\lim _{u\\rightarrow -\\infty }\\frac{\\widetilde{\\varphi } \\left( \\lambda u\\right) }{\\widetilde{\\varphi }\\left( u\\right) }=\\lambda ^{\\beta },$ if $b=+\\infty $ , then for all $\\lambda >0$ there holds $\\lim _{u\\rightarrow +\\infty }\\frac{\\widetilde{\\varphi } \\left( \\lambda u\\right) }{\\widetilde{\\varphi }\\left( u\\right) }=\\lambda ^{\\beta };$ this assumption is denoted by (H$\\widetilde{\\varphi }$ ).",
"A proxy of $\\underline{\\mathbf {\\widetilde{Q}}}$ can be obtained by sampling from a distribution on $\\mathbb {R}^{K}$ defined through $f(\\mathbf {\\widetilde{Q}}):=C \\cdot \\exp \\left( -\\sum _{k=1}^{K} \\widetilde{p}_{k} \\cdot \\widetilde{\\varphi }(\\widetilde{q}_{k}/\\widetilde{p}_{k})\\right) \\ = \\ C \\cdot \\exp \\left( -D_{\\widetilde{\\varphi }}\\left( \\mathbf {\\widetilde{Q}},\\widetilde{{P}}\\right)\\right)\\hspace{51.21504pt} \\textrm {(cf.",
"(\\ref {simul distr}))}\\nonumber $ where $C$ is a normalizing constant; strict convexity (cf.",
"(G5)) of $\\widetilde{\\varphi }$ together with (H$\\widetilde{\\varphi }$ ) prove that $f$ is a well-defined (Lebesgue-) density for a random variable $\\mathbf {T}$ on $\\mathbb {R}^{K}$ .",
"We denote by $\\mathbb {F}(\\cdot ) := \\mathbb {\\Pi }[\\mathbf {T} \\in \\cdot \\, ]$ the corresponding distribution on $\\mathbb {R}^{K}$ having density $f$ .",
"The distribution of $\\mathbf {T}$ given $\\left( \\mathbf {T} \\in \\widetilde{\\mathbf {\\Omega }} \\right)$ concentrates on the points in $\\widetilde{\\mathbf {\\Omega }}$ which minimize $D_{\\widetilde{\\varphi }}\\left(\\mathbf {\\widetilde{Q}},\\widetilde{{P}}\\right) $ as $\\mathbf {\\widetilde{Q}}$ runs in $\\widetilde{\\mathbf {\\Omega }}$ , when $D_{\\widetilde{\\varphi }}(\\widetilde{\\mathbf {\\Omega }},\\widetilde{{P}})$ is large.",
"This can be argued as follows.",
"We will consider the case when $\\widetilde{\\mathbf {\\Omega }}$ is a compact subset in $\\mathbb {R}_{> 0}^{K}$ and $\\widetilde{\\varphi }$ satisfies (H$\\widetilde{\\varphi }$ ) with $b=+\\infty $ .",
"For the case when $\\widetilde{\\mathbf {\\Omega }}$ is not compact, or belongs to $\\mathbb {R}^{K}/\\left\\lbrace \\mathbf {0}\\right\\rbrace $ , see the Remark REF hereunder.",
"Consider a compact set $\\mathbf {\\Gamma }$ in $\\widetilde{\\mathbf {\\Omega }}$ and let $\\mathbf {\\Gamma }_{t}$ be defined as deduced from $\\mathbf {\\Gamma } $ in a way that makes $D_{\\widetilde{\\varphi }}\\left( \\mathbf {\\Gamma }_{t},\\widetilde{{P}}\\right) $ increase with $t$ for sufficiently large $t$ .",
"For instance, set $\\mathbf {\\Gamma }_{t}:=t \\cdot \\mathbf {\\Gamma } .$ Hence, in case of $b=+\\infty $ the divergence $D_{\\widetilde{\\varphi }}\\left( \\mathbf {\\Gamma }_{t},\\widetilde{{P}}\\right) =\\inf _{g_{t}\\in \\mathbf {\\Gamma }_{t}}\\sum _{k=1}^{K} \\widetilde{p}_{k}\\cdot \\widetilde{\\varphi }\\left( \\frac{\\left(g_{t}\\right)_{k}}{\\widetilde{p}_{k}}\\right) =\\inf _{g \\in \\mathbf {\\Gamma } }\\sum _{k=1}^{K} \\widetilde{p}_{k}\\cdot \\widetilde{\\varphi }\\left( \\frac{t \\cdot g_{k}}{\\widetilde{p}_{k}}\\right)$ tends to infinity as $t \\rightarrow \\infty $ ; the case $a=-\\infty $ works analogously with $t\\rightarrow -\\infty $ .",
"In case of $b < \\infty $ we may consider $\\mathbf {\\Gamma } _{t}:=\\left\\lbrace b-g /t;g \\in \\mathbf {\\Gamma } \\right\\rbrace $ and indeed $D_{\\widetilde{\\varphi }}\\left( \\mathbf {\\Gamma } _{t},\\widetilde{{P}}\\right) \\rightarrow \\infty $ as $t\\rightarrow \\infty $ , with a similar statement when $a>-\\infty $ .",
"Assume that $\\mathbf {\\Gamma }$ has a dominating point $\\underline{g}$ .",
"Then $\\mathbf {\\Gamma }_{t}$ has dominating point $\\underline{g}_{t} := t \\cdot \\underline{g}$ .",
"We prove that $\\mathbf {T}$ with distribution (REF ) cannot be too far away (depending on $t$ ) from $\\underline{g}_{t}$ whenever $\\mathbf {T}$ belongs to $\\mathbf {\\Gamma } _{t}.$ This argument is valid in the present description of some asymptotics which makes $\\mathbf {\\Gamma } _{t}$ as a model for $\\widetilde{\\mathbf {\\Omega }}$ for large $t$ ; considering the case when $D_{\\widetilde{\\varphi } }\\left( \\widetilde{\\mathbf {\\Omega }} ,\\widetilde{{P}}\\right) $ is large is captured through the asymptotic statement $\\lim _{t\\rightarrow \\infty }D_{\\widetilde{\\varphi } }\\left( \\mathbf {\\Gamma } _{t},\\widetilde{{P}}\\right) =+\\infty .$ There holds the following Proposition 65 With the above notation and under condition (H$\\widetilde{\\varphi }$ ), denote by $\\mathbf {B}$ a neighborhood of $\\underline{g}$ and $\\mathbf {B}_{t}:=t \\cdot \\mathbf {B}$ .",
"Then $\\mathbb {F}[\\mathbf {\\Gamma }_{t}\\cap \\mathbf {B}_{t}^{c} \\, \\vert \\, \\mathbf {\\Gamma }_{t} \\, ]= \\mathbb {\\Pi }\\left[ \\left.",
"\\mathbf {T}\\in \\mathbf {\\Gamma }_{t}\\cap \\mathbf {B}_{t}^{c}\\, \\right|\\,\\mathbf {T}\\in \\mathbf {\\Gamma }_{t}\\right] \\rightarrow 0$ as $t\\rightarrow \\infty $ , which proves that simulations under (REF ) produce proxies of the dominating points $\\underline{g}_{t}$ in $\\mathbf {\\Gamma }_{t}$ .",
"Before we start with the proof of Proposition REF , we first quote the following Lemma 66 Let $\\widetilde{\\varphi } $ satisfy (H$\\widetilde{\\varphi }$ ) with $b=+\\infty $ .",
"Then for all $\\mathbf {A}$ in $\\mathbb {R}^{K}$ such that $\\breve{\\alpha } :=D_{\\widetilde{\\varphi }}(\\mathbf {A},\\widetilde{{P}}):=\\inf _{\\mathbf {v}\\in \\mathbf {A}}\\sum _{k=1}^{K} \\widetilde{p}_{k} \\cdot \\widetilde{\\varphi } \\left(\\frac{v_{k}}{\\widetilde{p}_{k}}\\right)$ is finite there holds $\\lim _{t\\rightarrow \\infty }\\frac{1}{t}\\log \\int _{\\mathbf {A}}\\exp \\left(-t\\sum _{k=1}^{K} \\widetilde{p}_{k} \\cdot \\widetilde{\\varphi }\\left( \\frac{v_{k}}{\\widetilde{p}_{k}}\\right) \\right) \\, dv_{1} \\ldots dv_{k}= - D_{\\widetilde{\\varphi }}\\left(\\mathbf {A},\\widetilde{{P}}\\right) .$ Proof of Lemma REF .",
"Let us first remark that according to the geometry of the set $\\mathbf {A}$ , various combinations for the asymptotics (REF ) or (REF ) may occur; for sake of brevity, we only handle the simplest ones, since all turn to be amenable through the same arguments.",
"Denote for positive $r$ $\\mathbf {B}(r):=\\left\\lbrace \\mathbf {v}\\in \\mathbb {R}^{K}:\\sum _{k=1}^{K} \\widetilde{p}_{k}\\widetilde{\\varphi } \\left( \\frac{v_{k}}{\\widetilde{p}_{k}}\\right) >r\\right\\rbrace .$ It holds, by making the change of variable $r=t\\cdot \\breve{\\alpha } +t \\cdot s$ , $& & \\int _{\\mathbf {A}}\\exp \\left( -t\\sum _{k=1}^{K} \\widetilde{p}_{k} \\cdot \\widetilde{\\varphi } \\left( \\frac{v_{k}}{\\widetilde{p}_{k}}\\right) \\right) \\,dv_{1} \\ldots dv_{k}=\\int \\cdots \\int 1_{\\mathbb {R}^{+}}(r)\\cdot 1_{\\mathbf {A}}(\\mathbf {v})\\cdot 1_{\\left]t\\sum _{k=1}^{K} \\widetilde{p}_{k}\\widetilde{\\varphi } \\left( \\frac{v_{k}}{\\widetilde{p}_{k}}\\right) ,\\infty \\right[}(r)\\cdot e^{-r} \\, dr \\, dv_{1}\\ldots dv_{K} \\\\& & = te^{-t\\breve{\\alpha } }\\int \\cdots \\int 1_{]- \\breve{\\alpha },\\infty [}(s) \\cdot 1_{\\mathbf {A}}(\\mathbf {v}) \\cdot 1_{\\mathbf {B}^{c}(\\breve{\\alpha }+s)}(\\mathbf {v})\\cdot e^{-ts} \\, ds \\, dv_{1}\\ldots dv_{K} \\ = \\ te^{-t\\breve{\\alpha } }\\int _{-\\breve{\\alpha }}^{\\infty }Vol\\left( \\mathbf {A}\\cap \\mathbf {B}^{c}(\\breve{\\alpha }+s)\\right) \\cdot e^{-ts} \\, ds.$ Let $I_{t}:=t \\cdot \\int _{0}^{\\infty }Vol\\left(\\mathbf {A}\\cap \\mathbf {B}^{c}(\\breve{\\alpha } +s)\\right) e^{-ts}ds $ .",
"We prove that $\\lim _{t\\rightarrow \\infty }\\frac{1}{t}\\log I_{t}=0.",
"$ When $a=-\\infty $ or $b=+\\infty $ , since $\\widetilde{\\varphi }$ satisfies (H$\\widetilde{\\varphi }$ ) there exists a polynomial $P$ such that $Vol\\left( \\mathbf {A}\\cap \\mathbf {B}^{c}(\\breve{\\alpha } +s)\\right) \\le P(s) \\, ;$ whence, assuming without loss of generality that $dom(\\widetilde{\\varphi }) =\\mathbb {R}^{+}$ , we obtain $\\frac{1}{t}\\log I_{t}\\le \\frac{1}{t}\\log \\int _{0}^{\\infty }P\\left(\\frac{u}{t}\\right) te^{-u}du$ which yields that for large $t$ $\\frac{1}{t}\\log I_{t}<0.$ When dealing with a context where $a$ or $b$ have finite value and the corresponding sets $\\mathbf {\\Gamma }_{t}$ are “far away” from $\\mathbf {\\Gamma } $ in terms of the distance measure $D_{\\widetilde{\\varphi }}\\left( \\cdot ,\\widetilde{{P}}\\right)$ , then $Vol\\left( \\mathbf {A}\\cap \\mathbf {B}^{c}(\\breve{\\alpha } +s)\\right) $ is bounded.",
"Hence, $\\lim \\sup _{t\\rightarrow \\infty }\\frac{1}{t}\\log I_{t}\\le 0$ .",
"Now fix $\\varepsilon >0.$ Then, since $Vol\\left( \\mathbf {A}\\cap \\mathbf {B}^{c}(a+s)\\right) $ is increasing in $s$ , we get $I_{t} &\\ge &t\\int _{\\varepsilon }^{\\infty }Vol\\left( \\mathbf {A}\\cap \\mathbf {B}^{c}(\\breve{\\alpha }+s)\\right) e^{-ts}ds \\\\&\\ge &Vol\\left( \\mathbf {A}\\cap \\mathbf {B}^{c}(\\breve{\\alpha } +\\varepsilon )\\right) e^{-t\\varepsilon }.$ Hence $\\frac{1}{t}\\log I_{t}\\ge \\frac{1}{t}\\log Vol\\left( \\mathbf {A}\\cap \\mathbf {B}^{c}(\\breve{\\alpha }+\\varepsilon )\\right) -\\varepsilon $ which yields $\\lim \\inf _{t\\rightarrow \\infty }\\frac{1}{t}\\log I_{t}\\ge 0.$ Therefore (REF ) holds, which concludes the proof.",
"$\\blacksquare $ We now turn to the Proof of Proposition REF .",
"Without loss of generality, let $b=+\\infty $ , $\\mathbf {\\Gamma }_{t}$ as in (REF ) and Condition (H$\\widetilde{\\varphi }$ ) hold.",
"Moreover, consider an arbitrary neighborhood $\\mathbf {B}$ of $\\underline{g}$ and the corresponding neighborhoods $\\mathbf {B}_{t} : =t \\cdot \\mathbf {B}$ of $\\underline{g}_{t} = t \\cdot \\underline{g}$ .",
"There holds $\\frac{1}{\\widetilde{\\varphi }(t)}\\log \\mathbb {\\Pi }\\left[ \\mathbf {T}\\in \\mathbf {\\Gamma }_{t}\\right] &=&\\frac{C}{\\widetilde{\\varphi }(t)}\\log \\int _{\\mathbf {\\Gamma } _{t}}\\exp \\left(-\\sum _{k=1}^{K}\\widetilde{p}_{k} \\cdot \\widetilde{\\varphi }\\left(\\frac{w_{k}}{\\widetilde{p}_{k}}\\right)\\right) \\, dw_{1}\\ldots dw_{K} \\\\&\\stackrel{(1)}{=}&\\frac{CK}{\\widetilde{\\varphi }(t)}\\log t+\\frac{C}{\\widetilde{\\varphi }(t)}\\log \\int _{\\mathbf {\\Gamma }}\\exp \\left(-t^{\\beta } \\cdot \\sum _{k=1}^{K}\\widetilde{p}_{k} \\cdot \\left( \\widetilde{\\varphi }\\left( \\frac{v_{k}}{\\widetilde{p}_{k}}\\right) \\cdot (1+o(1))\\right)\\right) \\, dv_{1} \\ldots dv_{K} \\\\&\\stackrel{(2)}{=}& \\frac{CK}{\\widetilde{\\varphi }(t)}\\log t+\\frac{C}{\\left( \\widetilde{\\varphi }(t)/t^{\\beta }\\right) } \\cdot \\frac{1}{t^{\\beta }}\\log \\left( (1+o(1)) \\cdot \\int _{\\mathbf {\\Gamma } }\\exp \\left( -t^{\\beta }\\sum _{k=1}^{K}\\widetilde{p}_{k} \\cdot \\widetilde{\\varphi }\\left( \\frac{v_{k}}{\\widetilde{p}_{k}}\\right) \\right)dv_{1} \\ldots dv_{K} \\right)\\\\&\\stackrel{(3)}{=}& -\\frac{Ct^{\\beta }}{\\widetilde{\\varphi }(t)}\\cdot D_{\\widetilde{\\varphi }}(\\mathbf {\\Gamma } ,\\widetilde{{P}}) \\cdot (1+o(1)) \\\\&\\stackrel{(4)}{=}& - \\breve{l}(t) \\cdot D_{\\widetilde{\\varphi }}(\\mathbf {\\Gamma } ,\\widetilde{{P}}) \\cdot (1+o(1))$ as $t$ tends to infinity.",
"In the above display, $(1)$ follows from $\\widetilde{\\varphi }(tx)=\\left( tx\\right)^{\\beta } \\cdot \\ell (tx)=t^{\\beta } \\cdot x^{\\beta } \\cdot \\ell (x) \\cdot \\frac{\\ell (tx)}{\\ell (x)} =t^{\\beta } \\cdot \\widetilde{\\varphi } (x) \\cdot \\left( 1+o(1)\\right) $ as $t$ tends to infinity and $x$ lies in a compact subset of $]0,\\infty [$ , where $\\ell $ is a slowly varying function.",
"The equality $(2)$ follows from compactness of $\\mathbf {\\Gamma }$ together with the fact that $\\widetilde{\\varphi }$ is a regularly varying function with index $\\beta $ , so that $\\lim _{t\\rightarrow \\infty }\\frac{\\widetilde{\\varphi }(tv)}{\\widetilde{\\varphi }(t)}=v^{\\beta }$ uniformly upon $v$ on compact sets in $]0,\\infty [$ .",
"The remaining equalities $(3)$ and $(4)$ follow from classical properties of regularly varying functions, where $\\breve{\\ell } := 1/\\ell $ is a slowly varying function at infinity, together with standard Laplace-Integral approximation.",
"In the same way we can show $\\frac{1}{\\widetilde{\\varphi }(t)}\\log \\mathbb {\\Pi } \\left[ \\, \\mathbf {T}\\in \\mathbf {\\Gamma } _{t}\\cap \\mathbf {B}_{t}^{c} \\, \\right]=- \\breve{l}(t) \\cdot D_{\\widetilde{\\varphi }}(\\mathbf {\\Gamma }\\cap \\mathbf {B}^{c},\\widetilde{{P}}) \\cdot (1+o(1))$ as $t$ tends to infinity.",
"Since $\\mathbf {B}$ is a neighborhood of the unique dominating point $\\underline{g}$ of $\\mathbf {\\Gamma }$ , one gets that $D_{\\widetilde{\\varphi }}(\\mathbf {\\Gamma }\\cap \\mathbf {B}^{c},\\widetilde{{P}}) > D_{\\widetilde{\\varphi }}(\\mathbf {\\Gamma } ,\\widetilde{{P}})$ .",
"This implies that $\\mathbb {\\Pi } \\left[ \\, \\mathbf {T}\\in \\mathbf {\\Gamma }_{t}\\cap \\mathbf {B}_{t}^{c} \\, \\big \\vert \\, \\mathbf {T}\\in \\mathbf {\\Gamma }_{t} \\, \\right] \\rightarrow 0\\qquad \\textrm {as $ $.",
"\\qquad $$}$$$ Remark 67 Firstly, let us quote that the case when $\\widetilde{\\mathbf {\\Omega }}$ is an unbounded subset in $\\mathbb {R}^{K}/\\left\\lbrace \\mathbf {0}\\right\\rbrace $ is somewhat immaterial for applications.",
"Anyhow, if compactness of $\\mathbf {\\Gamma } $ is lost, then in order to use the same line of arguments as above, it is necessary to strengthen the assumptions (H$\\widetilde{\\varphi }$ ) e.g.",
"as follows: when $b=+\\infty $ then $\\widetilde{\\varphi }$ has to be asymptotically homogeneous with degree $\\beta >0$ , in the sense that $\\widetilde{\\varphi }(tx)=t^{\\beta }\\widetilde{\\varphi }(x)\\cdot (1+o(1))$ as $t\\rightarrow \\infty $ ; for the subcase $a=-\\infty $ one employs an analogous assumption as $t\\rightarrow -\\infty $ .",
"The case when $\\widetilde{\\mathbf {\\Omega }}$ is a compact set in $\\mathbb {R}^{K}\\backslash \\lbrace \\mathbf {0}\\rbrace $ can be treated as above, by combining the asymptotics in $t$ in the neighborhood of $a$ and $b$ accordingly."
],
[
"Proof for Subsection ",
"Proof of Proposition REF.",
"Recall the weighted empirical measure $\\xi _{n,\\mathbf {X}}^{\\mathbf {V}}:=\\left( \\frac{1}{n}\\sum _{i\\in I_{1}^{(n)}} V_{i},\\ldots ,\\frac{1}{n}\\sum _{i\\in I_{K}^{(n)}} V_{i}\\right)$ which satisfies the $K$ linear constraints defined in (REF ) through $E_{S}[\\xi _{n,\\mathbf {X}}^{\\mathbf {V}}] =\\xi _{M,\\mathbf {X}}^{\\mathbf {W}^{\\ast }}= \\overline{W^{\\ast }} \\cdot \\xi _{M,\\mathbf {X}}^{w\\mathbf {W}^{\\ast }}$ where $\\mathbf {Q}^{\\ast }:= \\left(q_{1}^{\\ast }, \\ldots , q_{K}^{\\ast }\\right)= \\xi _{M,\\mathbf {X}}^{w\\mathbf {W}^{\\ast }}\\in int(\\textrm {$$\\hspace{-6.544pt}$$})$ and $\\overline{W^{\\ast }} = \\frac{1}{M}\\sum _{j=1}^{M}W_{j}^{\\ast }$ .",
"The probability distribution $S$ defined on $\\mathbb {R}^{n}$ is the Kullback-Leibler projection of $\\mathbb {}^{\\otimes n}$ on the class of all probability distributions on $\\mathbb {R}^{n}$ which satisfy (REF ).",
"We prove that $\\lim \\inf _{n\\rightarrow \\infty }S\\left[\\xi _{n,\\mathbf {X}}^{w\\mathbf {V}} \\in \\textrm {\\right.$$\\hspace{-6.544pt}$$}> 0$ .",
"To start with, we define for strictly positive $\\delta $ the set $A_{n,\\delta } := \\left\\lbrace \\left|\\frac{1}{n}\\sum _{i=1}^{n}V_{i}-\\overline{W^{\\ast }}\\right|\\le \\delta \\right\\rbrace $ and write $S\\left[\\xi _{n,\\mathbf {X}}^{w\\mathbf {V}} \\in \\textrm {\\right.$ $\\hspace{-6.544pt}$$}=S\\left[\\lbrace \\xi _{n,\\mathbf {X}}^{w\\mathbf {V}} \\in \\textrm {\\right.$$\\hspace{-6.544pt}$$} \\rbrace \\cap A_{n,\\delta }+ S\\left[\\lbrace \\xi _{n,\\mathbf {X}}^{w\\mathbf {V}} \\in \\textrm {\\right.$$\\hspace{-6.544pt}$$} \\rbrace \\cap A_{n,\\delta }^{c}=:I+II.$$By the law of large numbers, the second term $ II$ tends to $ 0$ as $ n$ tends to infinity.Moreover, one can rewrite$$I=S\\bigg [ \\bigcup \\limits _{m\\in \\left[ \\overline{W^{\\ast }}-\\delta ,\\overline{W^{\\ast }}+\\delta \\right] }\\left\\lbrace \\xi _{n,\\mathbf {X}}^{\\mathbf {V}}\\in m \\cdot \\textrm {\\right.$$\\hspace{-6.544pt}$$} \\bigg ]$$which entails$$I \\ \\ge \\ S\\bigg [ \\frac{1}{n_{k}}\\sum _{i\\in I_{k}^{(n)}}V_{i}\\in \\mathcal {V}_{\\eta }\\left( \\overline{W^{\\ast }}\\frac{q_{k}^{\\ast }}{p_{k}}\\right) \\text{ for all }k \\in \\lbrace 1,\\ldots ,K\\rbrace \\bigg ] ,$$where $ V( W qkpk) $ denotes a neighborhood of$Wqkpk$ with radius $$ being smallwhen $$ is small, for large enough $ n$, making use of the a.s. convergenceof $ nk/n$ to $ pk$.",
"Now, for any $ k {1,...,K}$ one has\\begin{equation}S\\bigg [ \\frac{1}{n_{k}}\\sum _{i\\in I_{k}^{(n)}}V_{i}\\notin \\mathcal {V}_{\\eta }\\left( \\overline{W^{\\ast }}\\frac{q_{k}^{\\ast }}{p_{k}}\\right)\\bigg ]\\ \\le \\ \\exp \\bigg (-n_{k} \\cdot \\inf _{x\\in \\mathcal {V}_{\\eta }\\left(\\overline{W^{\\ast }}\\frac{q_{k}^{\\ast }}{p_{k}}\\right)^{c}} \\,\\varphi \\left( x\\right) \\bigg )\\end{equation}since any margin of $ S$ with index in $ Ik(n)$ is a correspondingKullback-Leibler projection of $$ on the set of all distributionson $ R$ with expectation $W qkpM,kemp$ ---where $ pM,kemp$ denotes the fraction ofthe $ Xi$’s (within $ X1,...,XM$) which are equal to $ dk$(cf.",
"(\\ref {I^(n)_k for stat case})) ---and therefore has amoment generating function which is finite in a non-void neighborhood of $ 0$,which yields (\\ref {ineg}) by the Markov Inequality.",
"Note that the event$ { M,XwWint( $\\Omega $$\\Omega $ ) }$ is regenerative, so that $ M$ can be chosen large enough to make$ pM,kemp$ close to $ pk$ for all $ k {1,...,K}$.",
"This proves the claim.\\hspace{28.45274pt} $$$"
],
[
"Acknowledgment",
"W. Stummer is grateful to the Sorbonne Université Paris for its multiple partial financial support and especially the LPSM for its multiple great hospitality.",
"M. Broniatowski thanks very much the FAU Erlangen-Nürnberg for its partial financial support and hospitality.",
"Moreover, W. Stummer would like to thank Rene Schilling for an interesting discussion on complex-valued foundations of the Bernstein-Widder theorem."
]
] | 2107.01693 |
[
[
"Boosting Transferability of Targeted Adversarial Examples via\n Hierarchical Generative Networks"
],
[
"Abstract Transfer-based adversarial attacks can evaluate model robustness in the black-box setting.",
"Several methods have demonstrated impressive untargeted transferability, however, it is still challenging to efficiently produce targeted transferability.",
"To this end, we develop a simple yet effective framework to craft targeted transfer-based adversarial examples, applying a hierarchical generative network.",
"In particular, we contribute to amortized designs that well adapt to multi-class targeted attacks.",
"Extensive experiments on ImageNet show that our method improves the success rates of targeted black-box attacks by a significant margin over the existing methods -- it reaches an average success rate of 29.1\\% against six diverse models based only on one substitute white-box model, which significantly outperforms the state-of-the-art gradient-based attack methods.",
"Moreover, the proposed method is also more efficient beyond an order of magnitude than gradient-based methods."
],
[
"Introduction",
"Recent progress in adversarial machine learning demonstrates that deep neural networks (DNNs) are highly vulnerable to adversarial examples [42], [13], which are maliciously generated to mislead a model to produce incorrect predictions.",
"It has been demonstrated that adversarial examples possess an intriguing property of transferability [28], [45], [18], [5], [49] — the adversarial examples crafted for a white-box model can also mislead other unknown models, making black-box attacks feasible.",
"The threats of adversarial examples have raised concerns in numerous security-sensitive applications, such as autonomous driving [11] and face recognition [36], [48].",
"Figure: The targeted adversarial examples crafted by MIM and the conditional generative semantic pattern (C-GSP) crafted by our method for the Inception-v3 model given the target class Viaduct with perturbation budget 16 under the ℓ ∞ \\ell _{\\infty } norm constraint.",
"We also show the predicted labels and probabilities of these images by the black-box model DenseNet-201 .Tremendous efforts have been made to develop more effective black-box attack methods based on transferability since they can serve as an important surrogate to evaluate the model robustness in real-world scenarios [28], [9].",
"The current methods have achieved impressive performance of untargeted black-box attacks, intending to cause misclassification of the black-box models.",
"However, the targeted black-box attacks, aiming at misleading the black-box models by outputting the adversary-desired target class, perform unsatisfactorily [8] and have not been extensively explored [52].",
"The difficulty of fooling a black-box model by the existing targeted adversarial attacks could result in an over-estimation of model robustness under the challenging targeted black-box attack setting.",
"Existing efforts on targeted black-box attacks can be categorized as instance-specific and instance-agnostic attacks.",
"Specifically, the instance-specific attack methods [12], [30], [22], [9] craft adversarial examples by performing gradient updates iteratively, which achieve unsatisfactory performance for targeted black-box attacks due to easy overfitting to a white-box model [9], [46].",
"On the other hand, the instance-agnostic attack methods learn a universal adversarial perturbation [52] or a universal function [38], [31] on the data distribution independent of specific instances.",
"They can promote more general and transferable adversarial examples since the universal perturbation or function can alleviate the data-specific overfitting problem by training on an unlabeled dataset.",
"CD-AP [31], as one of the effective instance-agnostic methods, adopts a generative model as a universal function to obtain an acceptable performance when facing one specified target class.",
"However, CD-AP needs to learn a generative model for each target class while performing multi-target attack [14], , crafting adversarial examples targeted at different classes.",
"Thus it is not scalable to the increasing number of targets such as hundreds of classes, limiting practical efficiency.",
"To address the aforementioned issues and develop a targeted black-box attack in the practical scenario, in this paper we propose a conditional generative model as the universal adversarial function to craft adversarial perturbations.",
"Thus we can craft adversarial perturbations targeted at different classes, using a single model backbone with different class embeddings.",
"The proposed generative method is simple yet practical to obtain superior performance of targeted black-box attacks, meanwhile with two technical improvements including smooth projection mechanism that better helps the generator to probe targeted semantic knowledge from the classifier and adaptive Gaussian Smoothing with the focus of making generated results obtain adaptive ability against adversarially trained models.",
"The previous CD-AP requires costly training $N$ models while performing a multi-target attack with $N$ classes.",
"However, ours only trains one model and reaches an average success rate of 51.1% against six naturally trained models and 36.4% against three adversarially trained models based only on one substitute white-box model in NeurIPS ImageNet dataset, which outperforms CD-AP by a large margin of 6.0% and 31.3%, respectively.",
"While handling plenty of classes (, 1,000 classes in ImageNet), the effectiveness of generating targeted adversarial examples will be affected by a single generative model due to the difficulty of loss convergence in adversarial learning [47], [1].",
"Thus we train a feasible number of models (, 10$\\sim $ 20 models on ImageNet) to further promote the effectiveness beyond the single model backbone.",
"Specifically, each model is learned from a subset of classes specified by a designed hierarchical partition mechanism by considering the diversity property among subsets, for seeking a balance between effectiveness and scalability.",
"It reaches an average success rate of 29.6% against six different models, outperforming the state-of-the-art methods with an average success rate of $<$ 2% by a large margin, based only on one substitute white-box model in the NeurIPS 2017 competition.",
"Moreover, the proposed method achieves substantial speedup over gradient-based methods.",
"Furthermore, these adversarial perturbations generated by the proposed Conditional Generative models can arise as a result of strong Semantic Pattern (C-GSP) as shown in Fig.",
"REF .",
"We experimentally find that the generated adversarial semantic pattern itself achieves well-generalizing performance among the different models and is robust to the influence of data in Sec.",
"REF , which is very instructive for the understanding of adversarial examples.",
"Our main contributions can be summarized as follows: We present a systematical study on targeted black-box attacks involving instance-specific and instance-agnostic methods in ImageNet dataset with plenty of classes and face recognition.",
"We propose a simple yet practical conditional generative targeted attack method with a designed hierarchical partition mechanism, which can generate targeted adversarial examples without tuning the parameters.",
"Extensive experiments demonstrate that our method significantly improves the success rates of targeted black-box attacks over the existing methods."
],
[
"Related Work",
"In this section, we review related work on adversarial attacks belonging to different types.",
"Instance-specific attacks.",
"Some recent works [12], [30] adopt gradient-based optimization methods to generate the data-dependent perturbations.",
"MIM [9] introduces the momentum term into the iterative attack process to improve the black-box transferability.",
"DIM [46] and TIM [10] aim to achieve the better transferability by input or gradient diversity.",
"Instance-specific methods require iterative optimization for every instance, thus easily overfitting the current data point [9].",
"In contrast, we improve the transferability simultaneously with the inference-time efficiency.",
"Figure: An overview of our proposed generative method for crafting C-GSP, which includes modules of conditional generator and classifier.",
"The generator integrates the image and conditional class vector from Map network into a hidden incorporation.",
"Note that only the generator is trained in the whole pipeline to probe the target boundaries of the classifier.Instance-agnostic attacks.",
"Compared with instance-specific attacks, instance-agnostic attacks belong to image-independent (universal) methods.",
"The first pipeline is to learn a universal perturbation.",
"UAP [29] proposes to fool a model by adding a learned universal noise vector.",
"Another pipeline of attacks introduces learned generative models to craft adversarial examples.",
"GAP [33] and AAA [34] craft adversarial perturbations in a similar way based on target data directly and compress impressions, respectively.",
"Previous methods, including universal perturbation and function, require costly training the same number of models for multiple target classes.",
"Our method is capable of simultaneously generating adversarial samples for specifying multiple targets with better attack performance.",
"Multi-target attacks.",
"Instance-specific attacks have the ability for specifying any target in the optimization phase.",
"As elaborated in the introduction, these methods have degraded transferability and time-consuming iterative procedures.",
"Related MAN [14] trains a generative model in the ImageNet under the constraint of $\\ell _{2}$ norm to explore the targeted attacks, which specifies all 1,000 categories from ImageNet for seeking extreme speed and storage.",
"However, MAN does not demonstrate the effectiveness in terms of multi-target transferability against black-box models than previous instance-specific or instance-agnostic attacks, and the authors also claim that too many categories make it hard to transfer to another model.",
"Recent UAE [52] reveals better single-target transferability by learning universal perturbation, whereas they require to train multiple times while specifying multiple targets.",
"As a comparison, our method can generate adversarial samples for specifying multiple targets, meanwhile generated strong semantic patterns can outperform existing attacks by a significant margin."
],
[
"Method",
" In this section, we introduce a conditional generative model to learn a universal adversarial function, which can achieve effective multi-target black-box attacks.",
"While handing plenty of classes, we design a hierarchical partition mechanism to make the generative model capable of specifying any target class under a feasible number of models, regarding both the effectiveness and scalability."
],
[
"Problem Formulation",
"We use $\\mathbf {x}_{s}$ to denote an input image belonging to an unlabeled training set $\\mathcal {X}_{s} \\subset \\mathbb {R}^{d}$ , and use $c \\in \\mathcal {C}$ to denote a specific target class.",
"Let $\\mathcal {F}_{\\phi }: \\mathcal {X}_{s} \\rightarrow \\mathbb {R}^{K}$ denote a classification network that outputs a class probability vector with $K$ classes.",
"To craft a targeted adversarial example $\\mathbf {x}_{s}^*$ from a real example $\\mathbf {x}_{s}$ , the targeted attack aims to fool the classifier $\\mathcal {F}_{\\phi }$ by outputting a specific label $c$ as $\\operatornamewithlimits{arg\\,max}_{i\\in \\mathcal {C}}{\\mathcal {F}_{\\phi }(\\mathbf {x}_{s}^{*})}_{i} = c$ , meanwhile the $\\ell _{\\infty }$ norm of the adversarial perturbation is required to no more than value $\\epsilon $ as $\\Vert \\mathbf {x}_{s}^{*}-\\mathbf {x}_{s}\\Vert _{\\infty }\\le \\epsilon $ .",
"Although some generative methods [33], [31] can learn targeted adversarial perturbation, it does not take into account the effectiveness of multi-target generation, thus leading to inconvenience.",
"To make the generative model learn how to specify multiple targets, we propose a conditional generative network $\\mathcal {G}_{\\theta }$ that effectively crafts multi-target adversarial perturbations by modeling class-conditional distribution.",
"Different from previous single-target methods [31], [33], the target label $c$ is regarded as a discrete variable rather than a constant.",
"As illustrated in Fig.",
"REF , our model contains a conditional generator $\\mathcal {G}_{\\theta }$ and a classification network $\\mathcal {F}_{\\phi }$ parameterized by $\\theta $ and $\\phi $ , respectively.",
"The conditional generative model $\\mathcal {G}_{\\theta }: (\\mathcal {X}_{s}, \\mathcal {C}) \\rightarrow \\mathcal {P}$ learns a perturbation $\\mathbf {\\delta } = \\mathcal {G}_{\\theta }(\\mathbf {x}_{s}, c) \\in \\mathcal {P} \\subset \\mathbb {R}^{d}$ on the training data.",
"The output $\\mathbf {\\delta }$ of $\\mathcal {G}_{\\theta }$ is projected within the fixed $\\ell _{\\infty }$ norm, thus generating the perturbed image $\\mathbf {x}_{s}^{*} = \\mathbf {x}_{s} + \\mathbf {\\delta }$ .",
"Given a pretrained network $\\mathcal {F}_{\\phi }$ parameterized by $\\phi $ , we propose to generate the targeted adversarial perturbations by solving $\\begin{split}\\min \\limits _{\\theta }&\\mathbb {E}_{(\\mathbf {x}_{s}\\sim \\mathcal {X}_{s}, c\\sim \\mathcal {C})}[{\\mathbb {CE}\\big (\\mathcal {F}_{\\phi }(\\mathcal {G}_{\\theta }(\\mathbf {x}_{s}, c) + \\mathbf {x}_{s}\\big ), c)}], \\\\& \\text{s.t. }",
"\\Vert \\mathcal {G}_{\\theta }(\\mathbf {x}_{s}, c) \\Vert _{\\infty } \\le \\epsilon .\\end{split}$ By solving problem (REF ), we can obtain a targeted conditional generator by minimizing the loss of specific target class in the unlabeled training dataset.",
"Note that we only optimize the parameter $\\theta $ of the generator $\\mathcal {G}_{\\theta }$ using the training data $\\mathcal {X}_{s}$ , then the targeted adversarial example $\\mathbf {x}_{t}^{*}$ can be crafted by $\\mathbf {x}_{t}^{*} = \\mathbf {x}_{t} + \\mathcal {G}_{\\theta }(\\mathbf {x}_{t}, c) $ for any given image $\\mathbf {x}_{t}$ in the test data $\\mathcal {X}_{t}$ , which only requires an inference for this targeted image $\\mathbf {x}_{t}$ .",
"We experimentally find that the objective (REF ) can enforce the transferability for the generated perturbation $\\mathbf {\\delta }$ .",
"A reasonable explanation is that $\\mathbf {\\delta }$ can arise as a result of strong and well-generalizing semantic pattern inherent to the target class, which is robust to the influence of any training data.",
"In Sec.",
"REF , we illustrate and corroborate our claim by directly feeding scaled adversarial perturbationsThe perturbation is linearly scaled from [-$\\epsilon $ , $\\epsilon $ ] to [0, 255].",
"from different methods into the classifier.",
"Indeed, we find that our semantic pattern can be classified as the target class with a high degree of confidence while the perturbation from MIM [9] performs like the noise, meanwhile the scaled semantic pattern performs well transferability in different black-box models."
],
[
"Network Architecture",
"We now present the details of the conditional generative model for targeted attack, as illustrated in Fig.",
"REF .",
"Specifically, we design a mapping network to generate a target-specific vector in the implicit space of each target and train conditional generator $\\mathcal {G}_{\\theta }$ to reflect this vector by constantly misleading the classifier $\\mathcal {F}_{\\phi }$ .",
"Mapping network.",
"Given an one-hot class encoding $\\mathbb {1}_{c} \\in \\mathbb {R}^{K}$ from target class $c$ , the mapping network aims to generate the targeted latent vector $\\mathbf {w} = \\mathcal {W}(\\mathbb {1}_{c})$ , where $\\mathbf {w} \\in \\mathbb {R}^{M}$ and $\\mathcal {W}(\\cdot )$ consists of a multi-layer perceptron (MLP) and a normalization layer, which can construct diverse targeted vectors $\\mathbf {w}$ for a given target class $c$ .",
"Thus $\\mathcal {W}$ is capable of learning effective targeted latent vectors by randomly sampling different classes $c \\in \\mathcal {C}$ in training phase.",
"Generator.",
"Given an input image $\\mathbf {x}_{s}$ , the encoder first calculates the feature map $\\mathbf {F} \\in \\mathbb {R}^{N\\times H \\times W}$ , where $N$ , $H$ and $W$ refer to the number of channels, height and width of the feature map, respectively.",
"The target latent vector $\\mathbf {w}$ , derived from the mapping network $\\mathcal {W}$ by introducing a specific target class $c$ , is expanded along height and width directions to obtain the label feature map $\\mathbf {w}_s\\in \\mathbb {R}^{M\\times H \\times W}$ .",
"Then the above two feature maps are concatenated along the channels to obtain $\\mathbf {F}^{\\prime } \\in \\mathbb {R}^{(N+M)\\times H \\times W}$ .",
"The obtained mixed feature map is then fed to the subsequent network.",
"Therefore, our generator $\\mathcal {G}_{\\theta }$ translates an input image $\\mathbf {x}_{s}$ and latent target vector $\\mathbf {w}$ into an output image $\\mathcal {G}_{\\theta }(\\mathbf {x}_{s}, \\mathbf {w})$ , which enables $\\mathcal {G}_{\\theta }$ to synthesize adversarial images of a series of targets.",
"For the output of feature map $\\mathbf {f} \\in \\mathbb {R}^{d}$ in the decoder, we adopt a smooth projection $P(\\cdot )$ to perform a change of variables over $\\mathbf {f}$ rather than directly minimizing its $\\ell _2$ norm as [14] or clipping values outside the fixed norm [31], which can be denoted as $\\mathbf {\\delta } = P(\\mathbf {f}) = \\epsilon \\cdot \\mathrm {tanh}(\\mathbf {f}),$ where $\\epsilon $ is the strength of perturbation.",
"Since $-1 \\le \\mathrm {tanh}(\\mathbf {f}) \\le 1$ , $\\mathbf {\\delta }$ can automatically satisfy the $\\ell _{\\infty }$ -ball bound with perturbation budget $\\epsilon $ .",
"This transformation can be regarded as a better smoothing of gradient than directly clipping values outside the fixed norm, which is also instrumental for $\\mathcal {G}_{\\theta }$ to probe and learn the targeted semantic knowledge from $\\mathcal {F}_{\\phi }$ .",
"Training objectives.",
"The training objectives seek to minimize the classification error on the perturbed image of the generator as $\\vspace{-0.42502pt}\\theta ^{*} \\leftarrow \\operatornamewithlimits{arg\\,min}_{\\theta }{\\mathbb {CE} \\Big (F_{\\phi }\\big (\\mathbf {x}_{s} + \\mathcal {G}_{\\theta }(\\mathbf {x}_{s}, \\mathcal {W}(\\mathbb {1}_{c}))\\big ), c\\Big )},$ which adopts an end-to-end training paradigm with the goal of generating adversarial images to mislead the classifier the target label, and $\\mathbb {CE}$ is the cross entropy loss.",
"Previous studies attempt different classification losses in their works [52], [31], and we found that cross-entropy loss works well in our settings.",
"The detailed optimization procedure is summarized in Algorithm REF .",
"[t] Training Algorithm for the Conditional Generative Attack Training Data $\\mathcal {D}_{s}$ ; a generative network $\\mathcal {G}_{\\theta }$ ; a classification network $\\mathcal {F}_{\\phi }$ ; a mapping network $\\mathcal {W}$ .",
"Adversarial perturbations $\\mathbf {\\theta }$ .",
"iter in MaxIterations T Randomly sample $B$ images $\\lbrace \\mathbf {x}_{s_{i}}\\rbrace _{i=1}^{B}$ Randomly sample $B$ target classes $\\lbrace {c}_{i}\\rbrace _{i=1}^{B}$ Forward pass ${c}_{i}$ into $\\mathcal {W}$ to compute the targeted latent vectors $\\mathbf {w}_{i}$ Obtain the perturbed images by $\\mathbf {x}_{s_{i}}^{*} = \\epsilon \\cdot \\mathrm {tanh}(\\mathcal {G}(\\mathbf {x}_{s_{i}}, \\mathbf {w}_{i})) + \\mathbf {x}_{s_{i}}$ Forward pass $\\mathbf {x}_{s_{i}}^{*}$ to $\\mathcal {F}_{\\phi }$ and compute loss in Eq.",
"(REF ) Backward pass and update the $\\mathcal {G}_{\\theta }$ Table: Transferability comparison for multi-target attacks on ImageNet NeurIPS validation set (1k images) with the perturbation budget of ℓ ∞ ≤16\\ell _{\\infty } \\le 16.",
"The results are averaged on 8 different target classes.",
"Note that CD-AP † ^{\\dagger } indicates that training 8 models can obtain results, while our method only train one conditional generative model.",
"* indicates white-box attacks."
],
[
"Hierarchical Partition for Classes",
" While handling plenty of classes, the effectiveness of a conditional generative model will decrease as illustrated in Fig.",
"REF , because the representative capacity is limited with a single generator.",
"Therefore, we propose to divide all classes into a feasible number of subsets to train models when the class number $K$ is large, , 1,000 classes in ImageNet, with the aim of seeking the effectiveness of targeted black-box attack.",
"To obtain a good partition, we introduce a representative target class space, which is nearly equivalent to the original class space $\\mathcal {C}$ .",
"Specifically, we utilize the weights $\\phi _{cls} \\in \\mathbb {R}^{D\\times C}$ in the classifier layer for the classification network $\\mathcal {F}_{\\phi }$ .",
"Therefore, $\\phi _{cls}$ can be regarded as the alternative class space since the weight vector $\\mathbf {d}_{c} \\in \\mathbb {R}^{D}$ from $\\phi _{cls}$ can represent a class center of the feature embeddings of input images with same class $c$ .",
"Note that once those subsets with closer metric distance (, larger cosine similarity) in the target class space $\\phi _{cls}$ are regarded as conditional inputs of generative network, they obtain worse loss convergence and transferability than diverse them due to mutual influence among these input conditions, as illustrated in Fig.",
"REF .",
"Thus we focus on selecting target classes that do not tend to overlap or be close to each other as accessible subsets.",
"To capture more diverse examples in a given sampling space, we adopt K-determinantal point processes (DPP) [21], [20] to achieve a hierarchical partition, which can take advantage of the diversity property among subsets by assigning subset probabilities proportional to determinants of a kernel matrix.",
"First, we compute the RBF kernel matrix $L$ of $\\phi _{cls}$ and eigendecomposition of $L$ , and a random subset $V$ of the eigenvectors is chosen by regarding the eigenvalues as sampling probability.",
"Second, we select a new class $c_{i}$ to add to the set and update $V$ in a manner that de-emphaseizes items similar to the one selected.",
"Each successive point is selected and $V$ is updated by Gram-Schmidt orthogonalization, and the distribution shifts to avoid points near those already chosen.",
"The details are presented in Appendix A."
],
[
"Experiments",
" In this section, we present extensive experiments to demonstrate the effectiveness of proposed method for targeted black-box attacks."
],
[
"Experimental Settings",
" Datasets.",
"We consider the following datasets for training, including a widely used object detection dataset MS-COCO [26] and ImageNet training set [6].",
"We focus on standard and comprehensive testing settings, thus inference is performed on ImageNet validation set (50k samples), a subset (5k) of ImageNet proposed by [24] and ImageNet-NeurIPS (1k) proposed by [32].",
"Figure: Comparison of different projection functions and modes of Gaussian Smoothing.",
"Results are reported with Inc-v3 networkon ImageNet NeurIPS validation set.Networks.",
"We consider some naturally trained networks, , Inception-v3 (Inc-v3) [41], Inception-v4 (Inc-v4) [39], Resnet-v2-152 (Res-152) [15] and Inception-Resnet-v2 (IncRes-v2) [39], which are widely used for evaluating transferability.",
"Besides, we supplement DenseNet-201 (Dense-201) [16], GoogleNet [40] and VGG-16 [37] to fully evaluate the transferability.",
"Adversarially trained networks [43] are also selected to evaluate the performance, , ens3-adv-Inception-v3 ($\\textrm {Inc-v3}_\\textrm {ens3}$ ), ens4-adv-Inception-v3 ($\\textrm {Inc-v3}_\\textrm {ens4}$ ) and ens-adv-Inception-ResNet-v2 ($\\textrm {IncRes-v2}_\\textrm {ens}$ ).",
"All networks are publicly availablehttps://github.com/tensorflow/models/tree/master/research/slim https://github.com/tensorflow/models/tree/master/research/adv_imagenet_models https://github.com/pytorch/vision/tree/master/torchvision/models.",
"Implementation details.",
"We choose the same ResNet autoencoder architecture in [19], [31] as the basic generator networks, which consists of downsampling, residual and upsampling layers.",
"We initialize the learning rate as 2e-5 and set the mini-batch size as 32.",
"Smoothing mechanism is proposed to improve the transferability against adversarially trained models [10].",
"Instead of adopting smoothing for generated perturbation while the training is completed as CD-AP [31], we introduce adaptive Gaussian smoothing kernel to compute $\\mathbf {\\delta }$ from Eq.",
"(REF ) in the training phase, named adaptive Gaussian smoothing, with the focus of making generated results obtain adaptive ability.",
"More implementation details and discussion with other networks (, BigGAN [3]) are illustrated in Appendix B.",
"Table: Transferability results for targeted attacks on ImageNet validation set (50k images) with the perturbation budget of ℓ ∞ ≤10\\ell _{\\infty } \\le 10.",
"The attack is performed in same setting with the target class 'sea lion' and the training dataset MS-COCO."
],
[
"Transferability Evaluation",
"We consider 8 different target classes from [52] to form the multi-target black-box attack testing protocol with 8k times in 1k ImageNet NeurIPS set.",
"Efficiency of multi-target black-box attack.",
"Among comparable methods, instance-specific methods, , MIM, TI-DIM, DIM and TI-DIM, require iterative mechanism with $M$ steps by computing gradients to obtain adversarial examples.",
"Given the cost $t_{C}^{FP}$ and $t_{C}^{BP}$ of forward and backward passing the classifier, computing cost $T^{IS}$ of single data can be defined as $T^{IS} = t_{C}^{FP} * M + t_{C}^{BP} * M$ in Tab.",
"REF .",
"Instance-agnostic methods only require the inference cost from the trained generator as $T^{IA} = t_{G}^{FP}$ , thus possessing the priority for those attack scenarios within limited time.",
"However, instance-specific methods require to train 8 models to obtain all predictions from 8 different classes.",
"Due to time-consuming training and more storage, we only reproduce previous state-of-the-art generative method CD-AP [31] as a baseline, which already fully demonstrate the superior performance than other generative methods such as GAP [33] in their work.",
"As a comparison, our conditional generative method only trains one model to inference the results and outperforms in the aspect of efficiency.",
"Effectiveness of multi-target black-box attack.",
"Tab.",
"REF shows the transferability comparison of different methods on both naturally and adversarially trained models.",
"The success rate of instance-specific attacks are lower than 3%, possibly explained by the data-point overfitting that makes it hard to transfer another model.",
"The instance-agnostic attack CD-AP obtains acceptable performance, yet inferior to proposed method w.r.t black-box transferability.",
"The primary reason for such a trend lies in some distinctions as 1) direct clip projection in CD-AP and our smooth projection in Eq.",
"(REF ) and 2) their Gaussian Smoothing and our adaptive Gaussian Smoothing, as described in Sec.",
"REF and Appendix B.",
"Fig.",
"REF empirically shows the comparison results of single-target black-box attacks based on the CD-AP framework.",
"Thus proposed conditional generative method can be a reliable baseline w.r.t targeted black-box attacks, regarding both effectiveness and efficiency.",
"Results of single-target black-box attack.",
"Recent related work [52] has tried to solve the single-target transferable problem based on universal adversarial perturbation, and report an excellent single-target black-box performance.",
"We obtain single-target degraded version of our model by specifying an input target label during the training process.",
"We show the performance of black-box attack in terms of targeted attack success rate in Tab.",
"REF .",
"The promising results show generative semantic pattern from our method benefits black-box transferability than universal adversarial perturbations.",
"Some other instance-agnostic adversarial methods, , UAP [29], GAP [33] and RHP [24], have tendency towards the untargeted black-box problem.",
"Despite this, we follow the corresponding untargeted setting and compare different methods in Appendix C. Our method is steadily improved under untargeted black-box manner.",
"Table: Transferability comparison on NeurIPS 2017 competition with the perturbation budget of ℓ ∞ ≤16\\ell _{\\infty } \\le 16.",
"White-box substitute model is Inc-v3 for all attacks, following the standard protocol with 1,000 stochastic target classes.Table: The success rate of black-box impersonation attacks on face verification with the perturbation budget of ℓ ∞ ≤16\\ell _{\\infty } \\le 16.",
"ArcFace is chosen as white-box model.Figure: Generative examples of adversarial images with perturbation budget of ℓ ∞ ≤16\\ell _{\\infty }\\le 16.",
"We separately adopt the ImageNet and MS-COCO dataset as the training dataset to implement the generation of targeted perturbations.",
"Our method can generate semantic pattern independent of training dataset.Figure: Asr vs. numbers of conditional targets curve against Inc-v3 and VGG-16 models."
],
[
"Effectiveness on NeurIPS 2017 Competition",
" To illustrate the effectiveness of our proposed attack methods in practical 1,000 classification, we here follow the official setting from NeurIPS 2017 adversarial competition [23] for testing targeted black-box transferability.",
"Considering limited resource, previous instance-agnostic attacks are not required as comparable methods due to training 1,000 models, thus we focus on the instance-specific attacks, which are official top attack methods in NeurIPS 2017 adversarial competition.",
"Our hierarchical partition mechanism can make conditional generative networks be capable of specifying any target class via a feasible number of models for the scalability.",
"We consider 20 models, with each specifying 50 diverse classes from k-DPP hierarchical partition in this setting, to implement targeted attack by only once inference for each target image.",
"Our method outperforms all other baseline methods in Tab.",
"REF .",
"The results demonstrate that this method can be reliable in practical targeted attacks, regarding both effectiveness and efficiency."
],
[
"Effectiveness on Realistic Face Recognition",
" Adversarial perturbations added to original face have ability to evade being recognized or impersonate another individual [36], [50].",
"In this section, we consider the transferability of impersonation attack to further illustrate the generalization of our method, which is also corresponding to targeted attack in image classification.",
"Dataset and models.",
"We conduct the experiments on Labeled Faces in the Wild (LFW) [17] and introduce two test protocols.",
"For Protocol I defined as single-target impersonation attack, we choose 1 target identity and 1k source face images belonging with different identities from LFW as the attackers, thus forming 1k pairs.",
"For Protocol II named multi-target impersonation attack, 5 target identities and 1k source face images are selected to form 1k attack pairs, meaning that we need to implement 5k attacks.",
"We involve some excellent face recognition models for conducting black-box testing, including Sphereface [27], CosFace [44], FaceNet [35] and MobileFace [4].",
"These models lie in different model architectures and training objectives.",
"In all experiments, we only use one model ArcFace [7] as substitute model to craft adversarial samples, and test attack performance against other unknown models.",
"Figure: Comparison of loss convergence and transferability between diverse and close conditional subset with 10 target classes.Figure: Plots of logit vectors from the adversarial image L img L_{img} and scaled crafted perturbation L adv L_{adv} of MIM and proposed generative method, with their respective PCC values.Evaluation metrics.",
"We first compute the optimal threshold of every face recognition models from LFW dataset by following standard protocols.",
"If the similarity of a pair of images exceeds the threshold, we regard them as same identity, otherwise different identities.",
"Black-box attack results.",
"We adjust the optimization object function to adapt face recognition for chosen attack methods (detailed in Appendix C), and report the success rate of black-box impersonation attacks in Tab.",
"REF , which illustrates that our method can achieve nearly two times of the success rates than DIM in Protocol I and Protocol II.",
"The results indicate that our method is superior to other methods not only in image classification."
],
[
"Comparison Study about Target Classes",
" We conduct an extensive study to investigate two key points about target classes.",
"Different numbers of target classes.",
"We conduct effectiveness for different numbers of target classes in Fig.",
"REF .",
"It can be seen that the results perform well within a feasible number of targets, whereas to a certain extent effectiveness tend to decay.",
"Therefore, the effectiveness of conditional generative networks is influenced by the number of conditional classes, due to the representative capacity of single generator.",
"We aim to divide all classes into a feasible number of set while handling plenty of classes.",
"Comparison of different multi-target conditions.",
"We select closer conditional classes with larger cosine similarity in the target class space $\\phi _{cls}$ and diverse conditional classes from k-DPP method.",
"In Fig.",
"REF , closer conditional classes have worse loss convergence and transferability than diverse them due to mutual influence among conditions."
],
[
"More Analyses",
" Targeted adversarial samples from proposed generative method can produce semantic pattern inherent to the target class in Fig.",
"REF .",
"Why does generative semantic pattern work?",
"First, generative methods can produce strong targeted semantic pattern that is robust to the influence of data, which is obtained by minimizing the loss of specific target class in the training phase.",
"To corroborate our claim, we directly feed scaled crafted perturbations by instance-specific attack MIM and our generative method into the classifier.",
"Indeed, we find that our generative perturbation is considered as target class with a high degree of confidence whereas the perturbation from MIM performs like the noise, as shown in Fig.",
"REF .",
"We plot the logit relationship from scaled crafted perturbation and adversarial image in Fig.",
"REF , which is also consistent with previous claim.",
"Second, the generated adversarial semantic pattern achieves well-generalizing performance among the different models.",
"We feed 1k images from ImageNet test set into the generator trained by Inc-v3 model to obtain 1k semantic patterns, which are scaled to image pixel space and then fed into different classifiers.",
"We compute the mean confidence of $\\mathbf {0.46}$ for Dense-201, $\\mathbf {0.44}$ for Inc-v4, and $\\mathbf {0.35}$ for Res-152, whereas the perturbation from MIM is lower than $0.01$ .",
"The results show that our scaled semantic pattern can directly achieve well-generalizing performance among models, possibly explained by utilizing similar feature knowledge from the same class on different classifiers trained on same training data distribution.",
"Thus similar pattern can be instrumental for transferability among models."
],
[
"Discussion and Conclusion",
" Transferability of targeted black-box attack is simultaneously affected by data and model.",
"Therefore, instance-specific methods easily overfit the data point and white-box model, resulting in weak transferability.",
"As a comparison, proposed generative method with powerful learning capacity reduces the dependency for data point by adopting the unlabeled training data, thus enabling the model to learn semantic pattern and improve the transferability of targeted black-box attack.",
"Extensive experiments demonstrate that proposed generative method can significantly improve the success rates of targeted black-box attacks against naturally and adversarial trained models.",
"Th us we hope that crafting C-GSP can be regarded as a new reliable baseline method in terms of targeted black-box attacks."
],
[
"Sampling Algorithm",
"We summarize the overall sampling procedure based on k-DPP [20] in Algorithm .",
"Compute the RBF kernel matrix $L$ of $\\phi _{cls}$ and eigendecomposition of $L$ .",
"A random subset $V$ of the eigenvectors is chosen by regarding the eigenvalues as sampling probability.",
"Select a new class $c_{i}$ to add to the set and update $V$ in a manner that de-emphaseizes items similar to the one selected.",
"Update $V$ by Gram-Schmidt orthogonalization, and the distribution shifts to avoid points near those already chosen.",
"By performing the Algorithm , we can obtain a subset with $k$ size.",
"Thus while handling the conditional classes with K, we can hierarchically adopt this algorithm to get the final $K/k$ subsets, which are regarded as conditional variables of generative models to craft adversarial examples."
],
[
"Some Implementation Details",
"The study of smoothing mechanism.",
"Smoothing mechanism has been proved to improve the transferability against adversarially trained models.",
"CD-AP [31] uses direct clip projection to have a fixed norm $\\epsilon $ , and adopts smoothing for generated perturbation while the generator $\\mathcal {G}$ is trained, , ${\\begin{array}{c}\\textbf {Train: }\\mathbf {x}_{s_{i}}^{*} = \\mathrm {Clip}_{\\epsilon }(\\mathcal {G}(\\mathbf {x}_{s_{i}}),\\\\\\textbf {Test: }\\mathbf {x}_{s_{i}}^{*} = \\mathrm {W} * \\mathrm {Clip}_{\\epsilon }(\\mathcal {G}(\\mathbf {x}_{s_{i}}),\\end{array}}$ where $\\mathrm {W}$ indicates Gaussian smoothing of kernel size of 3, $*$ indicates the convolution operation, and Clip$_{\\epsilon }$ means clipping values outside the fixed norm $\\epsilon $ .",
"As a comparison, we introduce adaptive Gaussian smoothing kernel to compute adversarial images $\\mathbf {x}_{s_{i}}^{*}$ from in the training phase, named adaptive Gaussian smoothing as $\\textbf {Train \\& Test: } \\mathbf {x}_{s_{i}}^{*} = \\epsilon \\cdot \\mathrm {W} * \\mathrm {tanh}(\\mathcal {G}(\\mathbf {x}_{s_{i}}) + \\mathbf {x}_{s_{i}},$ which can make generated results obtain adaptive ability in the training phase.",
"We perform training in ImageNet dataset to report all results including comparable baselines.",
"[t] Sampling Algorithm by kDPP Weight Vector $\\mathbf {\\theta }_{cls}$ ; Subset size $k$ .",
"A subset $C$ .",
"Compute RBF kernel matrix $L$ of $\\mathbf {\\theta }_{cls}$ Compute eigenvector/value $\\lbrace v_{n}, \\lambda _{n}\\rbrace _{n=1}^N$ pairs of $L$ // Phase I: $J\\leftarrow \\phi $ , $e_{k}\\left(\\lambda _{1}, \\ldots , \\lambda _{N}\\right)=\\sum _{|J|=k} \\prod _{n \\in J} \\lambda _{n}$ n = N, ..., 1 $u\\sim U[0,1] < \\lambda _{n} \\frac{e^{n-1}_{k-1}}{e^{n}_{k}}$ and $k>0$ $J\\leftarrow J \\cup \\lbrace n\\rbrace $ ; $k\\leftarrow k-1$ // Phase II: $V \\leftarrow \\left\\lbrace v_{n}\\right\\rbrace _{n \\in J}, Y\\leftarrow \\phi $ $|V| > 0$ Select $c_i$ from $\\mathcal {C}$ with $\\operatorname{P}\\left(c_{i}\\right)=\\frac{1}{|V|} \\sum _{v \\in V}\\left(v^{\\top } e_{i}\\right)^{2}$ $C \\leftarrow C \\cup \\left\\lbrace c_{i}\\right\\rbrace $ $V \\leftarrow V_{\\perp },$ an orthonormal basis for the subspace of $V$ orthogonal to $e_{i}$ Figure: Some examples of adversarial images with perturbation budget of ℓ ∞ ≤16\\ell _{\\infty }\\le 16.",
"We separately adopt the ImageNet, MS-COCO and Comics dataset as the training dataset to implement the generation of targeted perturbations.Figure: Some examples of adversarial images with perturbation budget of ℓ ∞ ≤16\\ell _{\\infty }\\le 16.",
"We separately adopt the ImageNet, MS-COCO and Comics dataset as the training dataset to implement the generation of targeted perturbations.Network architecture of generator.",
"We adopt the same autoencoder architecture in [31] as the basic generator networks.",
"Besides, we also explore BigGAN [3] as conditional generator network.",
"An very weak testing performance is obtained even in the white-box attack scenario, possibly explained by the weak diversity of latent variable with the Gaussian distribution from BigGAN in the training phase, whereas autoencoder can take full advantage of large-scale training dataset, , ImageNet.",
"Furthermore, we also train the autoencoder with Gaussian noise as the training dataset and obtain similar inferior performance in the white-box attack scenario, indicating that a large-scale training dataset is very significant for generating transferable targeted adversarial examples.",
"Some details.",
"In our experiments of testing time, we apply NVIDIA 1080Ti GPUs.",
"Instance-specific methods, , MIM, TI-DIM, DIM and TI-DIM, adopt iterative steps $M = 20$ and follow their reported hyperparameters."
],
[
"Additional Experimental Results",
"Results on different datasets.",
"We craft adversarial examples on different datasets, including ImageNet training set, MS-COCO and Comics dataset [2], which consist of 1.2M, 82k and 50K images, respectively.",
"MS-COCO dataset can be applied to large-scale object detection and segmentation, and those images from Comics dataset are regarded as other domains different from normal ones in ImageNet.",
"Despite this diverse training types, we still find the common property of crafted adversarial examples by our method.",
"Specifically, we craft some examples of adversarial images with perturbation budget of $\\ell _{\\infty }\\le 16$ , and separately adopt the ImageNet, MS-COCO and Comics dataset as the training dataset to implement the generation of targeted perturbations.",
"As illustrated in Fig.",
"REF , we produce semantic pattern independent of any training dataset.",
"Table: Comparison results of targeted black-box attacks on different datasets.",
"Incv3 is the substitute model.We also report the success rate of targeted black-box attack, as shown in Tab.",
"REF .",
"We experimentally find that semantic pattern derived from ImageNet dataset achieves better performance of black-box performance, possibly explained by instructional effectiveness from more diverse data in ImageNet dataset.",
"Results of untargeted black-box attack.",
"We evaluate our method and other generative methods including UAP [29], GAP [33] and RHP [24].",
"Untargeted transferability from naturally trained models to adversarially trained models occurs due to differences in model sources, data types and other factors, thus enabling challenging comparison.",
"As illustrated in Tab.",
"REF , we report the untargeted attacks increase in error rate of adversarial and clean images to evaluate different methods.",
"Our method is steadily improved in different black-box models under untargeted black-box manner.",
"Table: Transferability results for untargeted attacks increase in error rate after attack on subset of ImageNet (5k images) with the perturbation budget of ℓ ∞ ≤16/32\\ell _{\\infty } \\le 16/32."
],
[
"Impersonation Attack of Face Recognition",
"We list attack methods of face recognition as follows.",
"Given an input $\\mathbf {x}$ and an image $\\mathbf {x}^r$ belonging with another identity, an attack method can generate an adversarial example $\\mathbf {x}^{adv}$ with perturbation budget $\\epsilon $ under the $\\ell _p$ norm ($\\Vert \\mathbf {x}^{adv} - \\mathbf {x}\\Vert _p \\le \\epsilon $ ).",
"Therefore, impersonation attack aims to perform this objective of $\\mathcal {C} (\\mathbf {x}^{adv}, \\mathbf {x}^{r}) = \\mathbb {I} (\\mathcal {D}_f (\\mathbf {x}^{adv}, \\mathbf {x}^{r}) < \\delta ),$ where $\\mathbb {I}$ is the indicator function, $\\delta $ is a threshold, and $\\mathcal {D}_f(\\mathbf {x}^{adv}, \\mathbf {x}^{r}) = \\Vert f(\\mathbf {x}^{adv}) - f(\\mathbf {x}^{r})\\Vert _2^2.$ Basic Iterative Method (BIM) [22] extends FGSM by iteratively taking multiple small gradient updates as $\\mathbf {x}_{t+1}^{adv} = \\mathrm {clip}_{\\mathbf {x},\\epsilon } \\big (\\mathbf {x}_t^{adv} - \\alpha \\cdot \\mathrm {sign}(\\nabla _{\\mathbf {x}}\\mathcal {D}_f(\\mathbf {x}_t^{adv},\\mathbf {x}^{r}))\\big ),$ where $\\mathrm {clip}_{\\mathbf {x},\\epsilon }$ projects the adversarial example to satisfy the $\\ell _{\\infty }$ constrain and $\\alpha $ is the step size.",
"Momentum Iterative Method (MIM) [9] introduces a momentum term into BIM for improving the transferability of adversarial examples as ${\\begin{array}{c}\\mathbf {g}_{t+1} = \\mu \\cdot \\mathbf {g}_t + \\frac{\\nabla _{\\mathbf {x}}\\mathcal {D}_f(\\mathbf {x}_t^{adv},\\mathbf {x}^{r})}{\\Vert \\nabla _{\\mathbf {x}}\\mathcal {D}_f(\\mathbf {x}_t^{adv},\\mathbf {x}^{r})\\Vert _1}; \\\\ \\mathbf {x}^{adv}_{t+1}=\\mathrm {clip}_{\\mathbf {x},\\epsilon }(\\mathbf {x}^{adv}_t - \\alpha \\cdot \\mathrm {sign}(\\mathbf {g}_{t+1})).\\end{array}}$ The training objectives of our generative method seek to minimize the classification error on the perturbed image of the generator as $\\vspace{-0.42502pt}\\min _{\\theta }\\mathbb {E}_{(\\mathbf {x}\\sim \\mathcal {X}, c\\sim \\mathcal {C})}[{\\mathcal {D}_f \\big (\\mathbf {x} + \\mathcal {G}_{\\theta }(\\mathbf {x}, c), \\mathbf {x}^{r}_{c}\\big )}],$ where $\\mathbf {x}^{r}_{c}$ refers to $\\mathbf {x}^{r}$ with the corresponding identity $c$ .",
"In the training phase, we randomly select $1,000$ identities from CASIA-WebFace [51] as training dataset to craft adversarial examples.",
"Therefore, our method can be applied not only in image classification."
],
[
"More Examples",
"We also show more semantic patterns from different target models, as illustrated in Fig.",
"REF ."
]
] | 2107.01809 |
[
[
"Autoencoder based Randomized Learning of Feedforward Neural Networks for\n Regression"
],
[
"Abstract Feedforward neural networks are widely used as universal predictive models to fit data distribution.",
"Common gradient-based learning, however, suffers from many drawbacks making the training process ineffective and time-consuming.",
"Alternative randomized learning does not use gradients but selects hidden node parameters randomly.",
"This makes the training process extremely fast.",
"However, the problem in randomized learning is how to determine the random parameters.",
"A recently proposed method uses autoencoders for unsupervised parameter learning.",
"This method showed superior performance on classification tasks.",
"In this work, we apply this method to regression problems, and, finding that it has some drawbacks, we show how to improve it.",
"We propose a learning method of autoencoders that controls the produced random weights.",
"We also propose how to determine the biases of hidden nodes.",
"We empirically compare autoencoder based learning with other randomized learning methods proposed recently for regression and find that despite the proposed improvement of the autoencoder based learning, it does not outperform its competitors in fitting accuracy.",
"Moreover, the method is much more complex than its competitors."
],
[
"Introduction",
"Feedforward neural networks (FNNs) have attracted a great deal of interest due to their excellence predictive performance and universal approximation capabilities.",
"The most popular learning methods involve some kind of gradient descent algorithm to learn FNN weights iteratively.",
"However, gradient-based methods suffer from many drawbacks making the learning process ineffective and time-consuming.",
"This is because they are sensitive to the initial values of the parameters.",
"The learning trajectory, starting with different initial parameters, leads to local minima of the loss function.",
"Thus, globally optimal parameters are not guaranteed.",
"Moreover, the learning process is time-consuming for complex target functions (TFs), big data sets and large FNN architectures.",
"Randomized learning, such as a random vector functional link (RVFL) network [1], has been proposed as an alternative to conventional FNNs iterative learning using gradients.",
"In randomized learning, the parameters of the hidden nodes are selected randomly and stay fixed.",
"They do not need to be tuned during the training stage.",
"The only parameters that need to be learned are the output weights.",
"This makes the optimization problem convex [2].",
"As such, it can be solved easily and quickly using a standard least-squares method.",
"Despite this simplification, randomized FNN learning still possesses universal approximation capabilities, provided there are a sufficient number of nonlinear hidden nodes [1], [3].",
"Many simulation studies reported in the literature show the high performance of the randomized FNN when compared to fully adaptable FNNs.",
"Randomization, which is cheaper than optimization, ensures simplicity of implementation and faster training.",
"However, a challenging and still open question in randomized learning is how to choose appropriate weights and biases for the hidden nodes to ensure best model performance [4], [5].",
"To deal with this problem, various methods of generating hidden node parameters have been proposed.",
"The simplest and most popular solution is to select both weights and biases from a uniform distribution over some symmetric interval $U=[-u, u]$ .",
"Usually, this interval is assigned as fixed, typically $[-1, 1]$ , regardless of the data, TF, and type of activation functions (AFs).",
"The independence of the hidden nodes from data is seen as an asset.",
"This overly-simplistic approach was criticized as illogical and misleading [6].",
"Therefore, to improve its performance, optimization of interval $U$ for a specified application is recommended.",
"For example, in [7], a supervisory mechanism which randomly assigns hidden node parameters from an adaptively selected interval, was proposed.",
"This paper clearly reveals that the selection of random parameters should be data dependent to ensure the universal approximation property of the resulting randomized FNN.",
"Data dependent random parameters were also recommended in [8].",
"The authors of this work noticed that if the hidden nodes are chosen at random and not subsequently trained, they are usually not placed in accordance with the density of the input data.",
"In such a case, training of linear parameters is less ineffective at reducing errors.",
"Therefore, in order to improve learning performance, the authors advise unsupervised placement of hidden nodes according to the input data density.",
"This recommendation was implemented in the methods proposed in [9] and [10].",
"In these works, it was noticed that as the weights and biases of hidden nodes have different functions, they should not be selected from the same interval.",
"The weights decide about AF slopes and should reflect TF complexity, while the biases decide about the placement of AF in the input space.",
"The biases should ensure the introduction of the most nonlinear fragments of AFs into the input hypercube.",
"These fragments are most useful for modeling TF fluctuations.",
"According to the methods described in [9] and [10], we first select the proper interval for the weights based on the AF features and TF properties.",
"Then, the biases are calculated based on the weights and data distribution.",
"In [11], a data-driven method was proposed to improve further the FNN randomized learning.",
"This method introduces the AFs into randomly selected regions of the input space and adjusts the slopes of individual AFs to the TF slopes in these regions.",
"As a result, the AFs mimic the TF locally and their linear combination approximates smoothly the entire TF.",
"An interesting method of generating parameters for RVFL was proposed recently in [12].",
"For this, the authors employ support-vector machines which in a supervised manner, by solving their corresponding optimization problems, generate the pre-trained weights.",
"These weights are used to initialize the hidden layer of the proposed RVFL architecture.",
"An alternative approach to generating hidden nodes in FNNs is unsupervised parameter learning using autoencoders (AEs), which was first introduced in [13].",
"AE, in the encoding phase, transforms input data into a meaningful feature representation obtained from the hidden layer.",
"Then, in the decoding phase, this feature representation is converted to the original inputs.",
"The information hidden in original data can be explored and encoded into the output weights of AE [14].",
"These output weights are then introduced to FNN as hidden node weights instead of randomly generated weights [13].",
"This approach was applied in [15] for classification tasks.",
"Here, RVFL uses a sparse AE with $\\ell _1$ -norm regularization to adaptively learn superior hidden node parameters for specific learning tasks.",
"The authors claim that the learned network parameters in their sparse pre-trained RVFL are embedded with the valuable information about input data, which alleviates the randomly generated parameter issue and improves algorithmic performance.",
"Another classifier based on RVFL with unsupervised parameter learning was proposed in [16].",
"In this solution, randomization based stacked AEs with a denoising criterion are used to extract better, higher-level representations.",
"Each randomization based AE acts as an independent feature extractor and a deep network is obtained by stacking several such AEs.",
"The network is built hierarchically with high level feature extraction followed by a final classification layer, which is RVFL with direct links.",
"The authors of both works on AE based randomized learning, [15] and [16], report experimental results on many real-world classification data sets from different domains.",
"The results confirm the excellent effectiveness of the proposed solutions.",
"Encouraged by these state-of-the-art results for FNN classifiers trained in an unsupervised manner using AEs, in this study, we analyze unsupervised parameter learning of FNN for regression.",
"We compare this approach with alternative methods, which were proposed recently in [10].",
"This paper makes the following contributions: We analyze AE based unsupervised parameter learning for a FNN regression model and find that this method has some drawbacks.",
"We show how to improve it.",
"We propose a learning method of AE that controls the produced random weights for FNN.",
"We also propose how to determine the biases for FNN.",
"We empirically compare AE based learning with other randomized learning methods proposed recently for regression and find that AE based learning does not outperform its competitors in fitting accuracy, and, in fact, it is much more complex.",
"The remainder of this paper is structured as follows.",
"In Section II, we describe randomized FNN learning and methods for generating random parameters.",
"AE based generating of random parameters is presented and critically analyzed from the perspective of AF distribution and shaping in Section III.",
"In Section IV, we analyze the complexity of randomized AE.",
"The performance of AE based randomized learning is evaluated in Section V. Finally, in Section VI, we conclude the work."
],
[
"Randomized Learning of FNN",
"Let us consider a shallow FNN architecture with $n$ inputs, a single-hidden layer including $m$ nonlinear nodes, and a single output.",
"AFs of hidden nodes, $h_i(\\mathbf {x})$ , map nonlinearly input vectors $\\mathbf {x}=[x_1, x_2,..., x_n]^T$ into $m$ -dimensional feature space.",
"An output node combines linearly $m$ nonlinear transformations of the inputs.",
"FNN expresses a function in the form: $\\varphi (\\mathbf {x}) = \\sum _{i=1}^{m}\\beta _ih_i(\\mathbf {x})$ where $\\beta _i$ is the output weight between the $i$ -th hidden node and the output node.",
"Such shallow architecture has a universal approximation property, even when the hidden layer parameters are not trained but generated randomly from the proper distribution [1], [3].",
"The output weights $ \\beta = [\\beta _1, \\beta _2, ..., \\beta _m]^T$ can be determined by solving the following linear problem: $\\mathbf {H}\\beta = \\mathbf {Y}$ , where $\\mathbf {H} = [\\mathbf {h}(\\mathbf {x}_1), \\mathbf {h}(\\mathbf {x}_2), ..., \\mathbf {h}(\\mathbf {x}_N)]^T \\in \\mathbb {R}^{N \\times m}$ is the hidden layer output matrix, and $ \\mathbf {Y} = [y_1, y_2, ..., y_N]^T $ is a vector of target outputs.",
"Using Moore-Penrose pseudoinverse, the optimal solution is given by: $\\beta = \\mathbf {H}^+\\mathbf {Y}$ where $ \\mathbf {H}^+ $ denotes the Moore–Penrose generalized inverse of matrix $ \\mathbf {H} $ .",
"The hidden node parameters, i.e.",
"weights $ \\mathbf {a} = [ a_{1}, a_{2}, ..., a_{n}]^T$ and bias $b$ , control AF slope, orientation and position in the input space.",
"For a sigmoid AF given by the formula: $h(\\mathbf {x}) = \\frac{1}{1 + \\exp \\left(-\\left(\\mathbf {a}^T\\mathbf {x} + b\\right)\\right)}$ weight $a_j$ expresses the sigmoid slope in the $j$ -th direction and bias $b$ decides about the sigmoid shift along a hyperplane containing all $x$ -axes.",
"An appropriate selection of the slopes and shifts of all sigmoids determine the approximation properties of the model.",
"As it was shown in [9] and [10], the standard way of selecting both the hidden node weights and biases randomly, from the same interval, $a_{i,j}, b_i \\sim U(-u, u)$ , is misguided.",
"This is because the optimal interval for weights, which determine the AF slope range, is not the optimal interval for biases, which represent an AF shift.",
"And vice versa.",
"With this in mind, in [9] and [10] separate methods for determining weights and biases were proposed.",
"The approach described in [9] first selects the weights $a_{i,j}$ from $U(-u, u)$ .",
"The bounds of the interval, $u$ , are adjusted to TF complexity.",
"For flat TFs we expect lower bounds, while for strongly fluctuating TFs we expect higher bounds.",
"Once the weights have been selected, the biases are determined in such a way as to ensure the steepest fragments of the sigmoids (which are around their inflection points) are introduced into input hypercube $ H = [x_{1,\\min }, x_{1,\\max }]\\times ... \\times [x_{n,\\min }, x_{n,\\max }] $ .",
"The resulting equation for the $i$ -th hidden node bias is as follows: $b_i = -\\mathbf {a}_i^T\\mathbf {x}_i^*$ where $\\mathbf {x}_i^*=[x_{i,1}^*, ..., x_{i,n}^*]$ is a point from $H$ where the $i$ -th sigmoid has one of its inflection points.",
"As you can see from (REF ), the biases are dependent on the weights.",
"Point $\\mathbf {x}_i^*$ can be selected as follows: this can be some point randomly selected from $H$ : $\\mathbf {x}_i^* \\sim U(H)$ .",
"This method is suitable when the input points are evenly distributed in $ H $ .",
"this can be some randomly selected training point: $\\mathbf {x}_i^* = \\mathbf {x}_\\xi \\in \\Phi $ , where $\\xi \\sim U\\lbrace 1, ..., N\\rbrace $ , and $\\Phi $ is a training set.",
"This method distributes the sigmoids according to the data density, avoiding empty regions.",
"this can be a prototype of the training point cluster: $\\mathbf {x}_i^* = \\mathbf {p}_i $ , where $ \\mathbf {p}_i $ is a prototype of the $ i $ -th cluster.",
"This method requires the clustering of training points into $ m =$ #nodes clusters.",
"It was noticed in [10] that the relationship between weights $a$ and the slope angles of sigmoids $\\alpha $ is highly nonlinear.",
"The standard interval for $a$ , $U=[-1, 1]$ corresponds to the interval $U_\\alpha =[-14^\\circ , 14^\\circ ]$ for $\\alpha $ , so only flat sigmoids are obtainable in such a case.",
"To get steep sigmoids, with $\\alpha $ near $90^\\circ $ , the bounds for $U$ should be $u>100$ .",
"For narrow $U$ , such as $[-1, 1]$ , the distribution of $\\alpha $ is similar to a uniform one.",
"When the interval for $a$ is extended, the $\\alpha $ distribution changes such that larger angles, near the bounds of $U_\\alpha $ , are more probable than smaller ones.",
"When $a \\in [-100, 100]$ , more than $77\\%$ of sigmoids are inclined at an angle greater than $80^\\circ $ , so they are very steep.",
"In such a case, there is a real threat of overfitting.",
"To generate sigmoids with uniformly distributed slope angles, first, we select randomly $|\\alpha _{i,j}| \\sim U(\\alpha _{\\min }, \\alpha _{\\max })$ , where the bound angles, $\\alpha _{\\min } \\in (0^\\circ , 90^\\circ )$ and $\\alpha _{\\max } \\in (\\alpha _{\\min }, 90^\\circ )$ , are adjusted to the TF complexity.",
"Then, we calculate the weights from: $a_{i,j}=4 \\tan \\alpha _{i,j}$ Finally, to introduce the sigmoids into the input hypercube $H$ , the biases are calculated from (REF ).",
"Both methods of generating random parameters of hidden nodes described above, we use in the experimental part of the work as comparative methods for AE based method.",
"We denote them as R$a$ M, i.e.",
"a random weights $a$ method, and R$\\alpha $ M, i.e.",
"a random slope angles $\\alpha $ method.",
"Fig.",
"REF illustrates an approximation of a highly nonlinear TF by FNN trained using R$\\alpha $ M (similar results were obtained for R$a$ M).",
"TF, shown as the dashed line in the left panel, is fitted accurately by the function built by FNN (red line).",
"This fitted function is composed of 25 hidden node sigmoids, which are shown in the right panel.",
"Note that the steepest fragments of the sigmoids are introduced by R$\\alpha $ M into the input interval, which is shown by the gray field in the panel on the right.",
"This fragments are the most useful for modeling the TF fluctuations.",
"The saturated AF fragments in the input interval are avoided.",
"The interval for sigmoid slope angles, $U_\\alpha =[0^\\circ , 83^\\circ ]$ , was adjusted to the TF complexity.",
"This gives a perfect fitting.",
"Figure: Fitted curve (left panel) and hidden node sigmoids (right panel) for Rα\\alpha M with |α|∼U(0 ∘ ,83 ∘ )|\\alpha | \\sim U(0^\\circ , 83^\\circ )."
],
[
"Autoencoder based Generation of Hidden Nodes Parameters",
"Unsupervised parameter learning using AEs was proposed in [13].",
"In this work, randomization based autoencoders (RAE) are used for unsupervised feature extraction for a multilayer FNN classifier.",
"RAE consists of two parts, an encoder and a decoder.",
"The encoder maps the input data randomly into some latent representation in such a way that the decoder is able to reconstruct the original input data form this representation.",
"RAE is a single hidden layer FNN with $n$ inputs, $n$ outputs, and $m$ nonlinear hidden nodes.",
"The sigmoid AFs, $g(\\mathbf {x})$ , are used for the hidden layer and linear AFs are used for the output layer.",
"The sigmoid AF is given by: $g(\\mathbf {x}) = \\frac{1}{1 + \\exp \\left(-\\left(\\mathbf {w}^T\\mathbf {x} + c\\right)\\right)}$ where $\\mathbf {w} = [ w_{1}, w_{2}, ..., w_{n}]^T $ are the hidden node weights and $ c$ is its bias.",
"In the first step, the learning method using RAE (RAEM) selects randomly the hidden node parameters for RAE, i.e.",
"weights $\\mathbf {w}_i$ and biases $ c_i, i = 1, 2, ..., m $ .",
"Typically, both are taken from the uniform distribution and interval $[-1,1]$ .",
"In the second step, the hidden layer output matrix is calculated: $\\mathbf {G} = [\\mathbf {g}(\\mathbf {x}_1), \\mathbf {g}(\\mathbf {x}_2), ..., \\mathbf {g}(\\mathbf {x}_N)]^T \\in \\mathbb {R}^{N \\times m}$ , where $\\mathbf {g}(\\mathbf {x}_l)= [g_1(\\mathbf {x}_l), g_2(\\mathbf {x}_l), ..., g_m(\\mathbf {x}_l)] $ is a vector of hidden node outputs for input pattern $\\mathbf {x}_l$ .",
"Finally, the output weight matrix, $\\mathbf {V} = [ \\mathbf {v}_1, \\mathbf {v}_2, ..., \\mathbf {v}_m]^T\\in \\mathbb {R}^{m \\times n}$ , where $ \\mathbf {v}_i=[v_{i,1}, v_{i,2}, ..., v_{i,n}]$ , is calculated from: $\\mathbf {V} = \\mathbf {G}^+\\mathbf {X}$ where $ \\mathbf {G}^+ $ denotes the Moore–Penrose generalized inverse of matrix $ \\mathbf {G} $ and $\\mathbf {X} \\in \\mathbb {R}^{N \\times n} $ is an input matrix.",
"Due to randomized learning of RAE, the output weights, $\\mathbf {V}$ , can be obtained easily using a standard linear least-squares method.",
"These weights are considered to be the latent features of input data [13], and are used as the hidden node weights for FNN instead of random weights.",
"Thus, $\\mathbf {A} = \\mathbf {V}^T$ , i.e.",
"$a_{j,i}=v_{i,j}$ .",
"As for the biases of hidden nodes $b_i$ , it is hard to find in the literature, how they are selected in RAE based FNN learning.",
"The exception is [15], where the authors outline the way of which the biases were determined.",
"They calculate the bias for the $i$ -th node as the mean value of this node weights: $b_i=1/n\\sum _{j=1}^{n}a_{j,i}$ .",
"Let us look at this case from the perspective of AF distribution in the input space.",
"When the bias is defined as the weight average, the inflection point of the sigmoid for the one-dimensional case, which is for $h(x)=0.5$ , can be obtained from: $h(x) = \\frac{1}{1 + \\exp \\left(-\\left(ax + b\\right)\\right)}=0.5 \\rightarrow x = -\\frac{b}{a} = -1$ where we substituted $b=a$ assuming that a bias is the mean value of the weights.",
"Thus, the inflection points of all sigmoids are in $x=-1$ .",
"This case is visualized in Fig.",
"REF .",
"Note that the steepest fragments of the sigmoids, which are around point $x=-1$ , are far from the input interval.",
"This interval includes many saturated fragments of the sigmoids, which results in poor fitting.",
"The accuracy does not improve with the number of hidden nodes.",
"Figure: Fitted curve (left panel) and hidden node sigmoids (right panel) for RAEM with b i =a i b_i=a_i.For the $n$ -dimensional case, the sigmoid value in the inflection points is: $h(\\mathbf {x}) = \\frac{1}{1 + \\exp \\left(-\\left(\\mathbf {a}^T\\mathbf {x} + b\\right)\\right)}=0.5 $ Substituting the mean value of weights for $b$ in this equation, after transformations we obtain: $\\mathbf {a}^T\\mathbf {x} + \\overline{a} = 0\\\\$ where $\\overline{a}=\\frac{1}{n}\\sum _{j=1}^{n}a_j$ .",
"This equation expresses the inflection hyperplane of the sigmoid.",
"As we can see, this hyperplane is totally dependent on weights $a_j$ .",
"So, the weights generated by the RAE determine all sigmoid features, slopes in all directions and shift.",
"The interdependence of the slopes and shift is an undoubted disadvantage of this approach.",
"It is unjustified and limits the model's flexibility.",
"The slopes should correspond to the TF complexity and the shift should be related to data distribution.",
"Unfortunately, RAEM proposed in [15] cannot control separately the slopes and shifts of the sigmoids when it generates them.",
"When ignoring biases and assuming $b_i=0$ for all hidden nodes, all sigmoids pass through $\\mathbf {x} = (0, ..., 0)$ .",
"This is also an unacceptable solution as it limits the approximation properties of the model.",
"In an attempt to improve the RAEM performance, we use the same solution for biases as in R$a$ M and R$\\alpha $ M. We calculate them from (REF ) on the basis of weights $\\mathbf {a}_i$ produced by RAE and randomly selected points $\\mathbf {x}^*$ .",
"Thus, we distribute the sigmoids in $H$ according to the data distribution.",
"The result for the one-dimensional case is shown in Fig.",
"REF .",
"As we can see, the accuracy did not improve significantly.",
"Moreover, when we add new hidden nodes (up to 2000), the RMSE remains unacceptable, i.e.",
"above 0.1.",
"This means that the problem is not only due to the mis-determination of the biases, but also due to the too flat sigmoids that do not correspond to TF complexity.",
"Figure: Fitted curve (left panel) and hidden node sigmoids (right panel) for RAEM with b i =-a i x i * b_i=-a_ix^*_i.The sigmoid slopes in FNN are determined by the RAE output weights, $v$ .",
"These weights are dependent on the RAE random projection $\\mathbf {G}$ , which in turn is dependent on the RAE hidden node random parameters: $w$ and $c$ .",
"In the above described simulations, these parameters were both selected from the standard interval $U_{AE}=[-1, 1]$ .",
"As we shown in Section , this is an incorrect approach.",
"Thus, to improve the regression model performance, we propose to use R$a$ M for generating $w$ and $c$ .",
"According to this method, we optimize the interval for $w$ , $U_{AE}=[-u_{AE}, u_{AE}]$ , and calculate the biases $c$ analogously to (REF ): $c = -\\mathbf {w}^T\\mathbf {x}^*$ As a result of this modified RAEM learning, we expect that RAE will produce the appropriate weights $\\mathbf {A} = \\mathbf {V}^T$ , which provide FNN sigmoids with the slopes adjusted to the TF complexity.",
"Fig.",
"REF shows the effect of the interval $U_{AE}$ bounds on the median of absolute values of weights $v$ (left panel) and on the fitting error (right panel) for TF shown in Figs.",
"REF -REF (the number of hidden nodes was $m=25$ , results are averaged over 100 runs).",
"Note that median of $|v|$ decreases quickly with $u_{AE}$ .",
"RMSE reaches its minimum for $u_{AE} \\in (0.05, 0.2)$ .",
"For such $U_{AE}$ bounds, RAE produces weights $v$ whose $median|v|$ is in the range from around 5 to 14 (for comparison, when weights $w$ were selected from the standard interval $U_{AE}=[-1, 1]$ , $median|v|$ was around 1.5).",
"For such a case, RMSE reaches an acceptable level of 0.005, which is similar to those obtained by R$a$ M and R$\\alpha $ M for the same number of hidden nodes.",
"Fig.",
"REF depicts fitting results for $U_{AE}=[-0.1, 0.1]$ .",
"Note steeper sigmoids than in the cases presented in Fig.",
"REF , resulting in good fitting.",
"It should be noted that the optimal interval $U_{AE}$ depends on the number of hidden nodes.",
"When, instead of 25, we used 200 hidden nodes, the optimal values of the interval bounds were $u_{AE} \\in (0.005, 0.022)$ .",
"In such a case $median|v|$ was from around 4 to 17.",
"Compare these values with $median(|v|) \\approx 0.19$ which we obtain for $U_{AE}=[-1, 1]$ .",
"Figure: The effect of the interval U AE U_{AE} bounds on the median of absolute values of weights vv (left panel) and on the fitting error (right panel).Figure: Fitted curve (left panel) and hidden node sigmoids (right panel) for RAEM with w i ∈[-0.1,0.1]w_i \\in [-0.1, 0.1] and c i =-w i x i * c_i=-w_ix^*_i.In the above analysis, RAE was trained without regularization.",
"However, $\\ell _1$ , $\\ell _2$ and elastic-net regularization are widely used in RAE to prevent overfitting [13], [15], [16].",
"Regularization in RAE decreases weights $v$ to improve the generalization property of the model, i.e.",
"generalization of mapping $\\mathbf {x}$ to themselves.",
"From the point of view of the regression FNN model trained using RAEM, regularisation in RAE is an unfavorable operation because it further flattens FNN sigmoids.",
"As we showed above, it is a disadvantage for strongly fluctuating TFs.",
"Note that RAEM in its standard version (without optimization of $U_{AE}$ ) produces weights $v$ dependent only on the distribution of points $\\mathbf {x}$ in the input space and independently of the TF.",
"So, for two TFs with different complexity (one of them flat and the second one with strong fluctuations), having the same distributed x-points, RAE can produce exactly the same weights $v$ .",
"This must be considered a serious drawback.",
"These findings on RAEM can be summarised in tree points: the output weights $v$ produced by RAE determine the sigmoid slopes in the FNN regression model and are crucial for accurate TF approximation.",
"These weights are dependent on the RAE hidden nodes parameters, i.e.",
"weights $w$ and biases $c$ .",
"The standard way of generating both these parameters from the same interval $U_{AE}$ is unjustified and misleading.",
"We recommend optimizing this interval for weights $w$ and determining biases $c$ from (REF ).",
"the optimal interval $U_{AE}$ for $w$ depends on the number of hidden nodes $m$ .",
"Thus, for each value of $m$ considered, interval $U_{AE}$ should be optimized.",
"the hidden nodes biases of the FNN regression model, $b$ , determine the sigmoid placement in the input space.",
"They should introduce the steepest fragments of the sigmoids into the input hypercube.",
"Incorrectly selected, such as $b_i=\\overline{a}_i$ or $b_i=0$ , they lead to the placement of the saturation parts of the sigmoids into $H$ .",
"To avoid this, we recommend determining biases $b$ from (REF )."
],
[
"Complexity of RAE",
"To generate weights $v$ , RAE can use linear least squares regression, lasso, ridge regression or elastic net algorithms [16].",
"In all these cases the total time complexity is $O(Nm^2+m^3)$ (this is true for lasso when least-angle regression (LARS) is used for fitting linear regression models [17]).",
"For $N>m$ the RAE runs in linear time with the size of the training set, $N$ , and in quadratic time with the number of nodes, $m$ .",
"For $N\\le m$ , it runs in cubic time with $m$ , although in [18] it was shown that for ridge regression this cost can be reduced to $O(N^2m)$ .",
"Taking into account the optimization process, i.e.",
"the selection of hyperparameters, the time complexity is as follows.",
"In RAE without regularization the number of hidden nodes $m$ and interval $U_{AE}$ need to be found using cross-validation.",
"Thus, the computational load increases linearly with the number of data splits used in cross-validation, $s$ , and also with the number of points of the grid which is $l_ml_u$ , where $l_m$ and $l_u$ are the number of searching values for $m$ and $u_{AE}$ , respectively.",
"So, the total complexity will be $O(sl_ml_uNm^2+sl_ml_um^3)$ .",
"In RAE with $\\ell _1$ and $\\ell _2$ regularization, the regularization parameter $\\lambda $ should also be selected in cross-validation.",
"So, the complexity of RAE with lasso or ridge regression is $O(sl_ml_ul_{\\lambda }Nm^2+sl_ml_ul_{\\lambda }m^3)$ , where $l_{\\lambda }$ is the number of searching points for $\\lambda $ .",
"Elastic net combines ridge and lasso, with the tuning parameter $\\alpha $ that balances the weights of ridge against lasso.",
"If the number of searching points for $\\alpha $ is $l_{\\alpha }$ , the total complexity of RAE with elastic net regularization is $O(sl_ml_ul_{\\lambda }l_{\\alpha }Nm^2+sl_ml_ul_{\\lambda }l_{\\alpha }m^3)$ ."
],
[
"Simulation Study",
"In this section, to demonstrate the fitting properties of RAEM, we report some simulation results over several regression problems.",
"They include an approximation of extremely nonlinear TFs: TF1 $g(\\mathbf {x}) = \\sum _{j=1}^{n}\\sin \\left(20\\cdot \\exp x_j\\right)\\cdot x_j^2, \\, x_i \\in [0, 1]$ TF2 $g(\\mathbf {x}) = -{\\sum _{i=1}^{n} \\sin (x_i) \\sin ^{20} \\left(\\frac{ix_i^2}{\\pi } \\right)}, \\, x_i \\in [0, \\pi ]$ TF3 $g(\\mathbf {x}) = 418.9829n -{\\sum _{i=1}^{n} x_i \\sin (\\sqrt{|x_i|})}, \\\\ x_i \\in [-500, 500]$ We considered these functions with $n=2, 5$ and 10 arguments.",
"The sizes of the training and test sets depended on the number of arguments.",
"They were 5000 for $n=2$ , 20,000 for $n=5$ , and 50,000 for $n=10$ .",
"All arguments for TF3-TF5 were normalized to $[0, 1]$ , and the function values were normalized to $[-1, 1]$ .",
"Two-argument functions TF1-TF3 are shown in Fig.",
"REF .",
"Note that TF1 combines flat regions with strongly fluctuating regions, TF2 expresses flat regions with perpendicular grooves, and TF3 fluctuates strongly, showing the greatest amplitude at the borders.",
"Figure: Target functions TF1-TF3 for n=2n=2.We use a modified version of RAEM.",
"That is, the interval for the hidden nodes in RAE, $U_{AE}$ , was optimized for each number of hidden nodes.",
"This is because, as we show in Section , the optimal interval is dependent on the hidden node number.",
"To introduce the sigmoids of RAE into input hypercube $H$ , biases $c$ were calculated from (REF ).",
"And analogously, to introduce the sigmoids of the FNN regression model into $H$ , we calculate biases $b$ from (REF ).",
"For comparison, we use R$a$ M and R$\\alpha $ M. For each experiment, we run 100 independent training sessions.",
"In Table REF , we present the fitting test errors (RMSE) of each method for each TF.",
"The optimal bounds of the intervals from which the model parameters ($w$ , $a$ and $\\alpha $ , respectively) were randomly selected are also shown.",
"In R$\\alpha $ M, we set fixed lower bounds, $\\alpha _{\\min }=0^\\circ $ , while the upper bounds, $\\alpha _{\\max }$ , were selected for each TF as $90^\\circ $ .",
"For RAEM and R$a$ M, we observe the difference in the optimal interval sizes between 2-argument and more than 2-argument TFs.",
"R$a$ M provides wide intervals, i.e.",
"steep sigmoids, for 2-argument TFs, and narrow intervals, i.e.",
"flat sigmoids, for 5- and 10-argument TFs.",
"Similarly RAEM provides steep sigmoids for 2-argument TFs (narrow $U_{AE}$ produces higher weights $a$ ), and flat sigmoids for 5- and 10-argument TFs.",
"This could be explained by the change in the TF landscape, which flattens with an increasing number of dimensions.",
"Interestingly, R$\\alpha $ M is insensitive to this phenomenon, giving the same broad interval for $\\alpha $ regardless of the number of arguments.",
"As can be seen from Table REF , R$\\alpha $ M demonstrates the highest fitting accuracy for all TFs.",
"This was confirmed by a Wilcoxon signed-rank test with $\\alpha =0.05$ .",
"Note that results for RAEM and R$a$ M are very similar.",
"Figs.",
"REF -REF depict fitting test errors depending on the number of hidden nodes $m$ .",
"Shaded regions are 10th and 90th percentiles, measured over 100 trials.",
"As can be seen from these figures, the confidence intervals of RAEM and R$a$ M overlap for higher $m$ .",
"This means that these two methods generate similar weights $a$ , which provide a similar set of basis functions for FNN.",
"To study this issue further, we show in Figs.",
"REF -REF the histograms of weights $a$ generated by RAEM for all TFs.",
"It is evident from these figures that weight $a$ distributions deviate from the uniform distribution, especially for $n=5$ and 10.",
"Thus, RAEM, unlike R$a$ M, does not generate uniformly distributed weights $a$ .",
"The weight distributions are unimodal, symmetrical, bell-shaped, and centered at 0.",
"Note that the intervals for $a$ observed in Figs.",
"REF -REF in most cases correspond to the intervals $U$ selected as optimal by R$a$ M (see hyperparameter $u$ for R$a$ M in Table REF ).",
"Table: Results for TF1-TF3.Figure: RMSE depending on the hidden node numbers for 2-argument TFs.Figure: RMSE depending on the hidden node numbers for 5-argument TFs.Figure: RMSE depending on the hidden node numbers for 10-argument TFs.Figure: Histograms of weights aa generated by RAEM for 2-argument TFs.Figure: Histograms of weights aa generated by RAEM for 5-argument TFs.Figure: Histograms of weights aa generated by RAEM for 10-argument TFs.It is clear from Table REF and Figs.",
"REF -REF that the best fitting was achieved for R$\\alpha $ M. Interestingly, for each TF this method generated weights $a$ from the same distribution, which is shown in Fig.",
"REF .",
"Figure: Histograms of weights aa generated by Rα\\alpha M for all TFs.To compare the performance of RAEM, R$a$ M and R$\\alpha $ M on real-world regression problems, we performed experiments on several data sets from different domains.",
"Data sets were collected from the KEEL repository, http://www.keel.es/ (stock, laser, treasury, dee, and machineCPU data sets) and from Delve repository, https://www.cs.toronto.edu/$\\sim $ delve/data/kin/desc.html (kin8nm data set).",
"The number of samples and arguments in each data set are shown in Table REF .",
"The input and output variables were normalized into $[0, 1]$ .",
"The data sets were divided into training sets containing 75% of the samples selected randomly, and test sets containing the remaining samples.",
"The optimal values of hyperparameters, i.e.",
"hidden node numbers and sizes of intervals for random parameters, were selected by 5-fold cross-validation.",
"Results are shown in Table REF .",
"The bold values indicate the lowest errors while the values in italics indicate the highest errors (Wilcoxon signed-rank test with $\\alpha =0.05$ was used to compare errors).",
"Note that RAEM demonstrate the highest errors for four data sets.",
"The best method is R$\\alpha $ M, which yielded the lowest errors for five out of six data sets.",
"Table: Results for real-word regression problems.Fig.",
"REF compares variants of RAEM: RAEM1 the improved RAEM proposed in this study (optimized interval for weights $w$ , $U_{AE}=[-u_{AE}, u_{AE}]$ ; biases $c$ and $b$ determined from (REF ) and (REF ), respectivelly), RAEM2 variant with the fixed interval for $w$ , $U_{AE}=[-1, 1]$ ; biases $c$ and $b$ determined from (REF ) and (REF ), respectivelly, RAEM3 variant with random selection of both $w$ and $c$ from the fixed interval $[-1, 1]$ ; biases $b$ determined from (REF ), RAEM4 variant with random selection of $w$ , $c$ and $b$ from the fixed interval $[-1, 1]$ , RAEM5 variant with random selection of both $w$ and $c$ from the fixed interval $[-1, 1]$ ; biases $b$ determined as mean values of weights, $b_i = \\overline{a}_i$ .",
"Note that the proposed modification of RAE in most cases gave the lowest errors compared to other RAE variants.",
"The laser and treasury data sets were insensitive to the method of generating weights and biases.",
"For these data sets, the errors for all RAE variants were similar.",
"Figure: Comparison of RAEM variants."
],
[
"Conclusion",
"In this work, we showed that RAE based randomized learning of FNN suffers from several drawbacks.",
"First, RAE produces the random weights for the FNN predictive model by taking into account just the input data.",
"This approach is questionable because the input data does not contain information about TF complexity.",
"TFs with strong fluctuations need higher weights than flat TFs to be modeled accurately.",
"Unfortunately, RAE for both these cases can generate similar sets of weights ignoring completely TF complexity.",
"Second, RAE does not generate the biases for the FNN predictive model.",
"These biases, which determine the distribution of the activation functions in the input space, are crucial for the approximation properties of the predictive model.",
"In this study, we propose improved, unsupervised parameter learning using RAEs.",
"First, we introduce the possibility of controlling the magnitude of the random weights produced by RAE.",
"This is realized by appropriately generating the RAE hidden node parameters.",
"Second, we determine the biases for the FNN predictive model so that the sigmoids have their steepest fragments introduced into an input hypercube.",
"These fragments are the most useful for modeling TF fluctuations.",
"The proposed modifications make the RAE method more flexible, more data dependent and more dependent on the complexity of the solved problem.",
"The experimental part of the work does not provide evidence that the improved RAEM outperforms in fitting accuracy other new methods of generating random parameters.",
"Moreover, its complexity is much greater as it requires additional learning of RAE.",
"Therefore, applying it to regression problems, rather than simpler and faster methods, may be questionable."
]
] | 2107.01711 |
[
[
"Convex optimization of bioprocesses"
],
[
"Abstract We optimize a general model of bioprocesses, which is nonconvex due to the microbial growth in the biochemical reactors.",
"We formulate a convex relaxation and give conditions guaranteeing its exactness in both the transient and steady state cases.",
"When the growth kinetics are modeled by the Monod function under constant biomass or the Contois function, the relaxation is a second-order cone program, which can be solved efficiently at large scales.",
"We implement the model on a numerical example based on a wastewater treatment system."
],
[
"Introduction",
"We optimize a model of dynamical bioprocesses consisting of a set of biochemical reactors interconnected by diffusion and mass flow.",
"The objectives include minimizing substrate outflow, maximizing biogas production, and tracking setpoints.",
"Within each reactor, several microbial reactions convert any number of biotic or abiotic reactants into biomass and/or products.",
"This setup describes a variety of physical systems such as wastewater treatment networks, the production of various chemicals, and compartmental approximations of bioprocesses in continuous media.",
"Here we focus on the case when the kinetics can be represented by a second-order cone (SOC) constraint, as recently shown in [1] for the Monod [2] and Contois [3] growth rates.",
"The most closely related topics to ours are chemical process optimization, control and optimization of wastewater systems, and control of bioprocesses.",
"Most existing approaches to process optimization [4] and wastewater [5] do not explicitly model the microbial growth, and often use either linear programming or general nonlinear solvers.",
"There have been many applications of nonlinear control [6] and optimization [7] to bioprocesses, but not convex relaxations or second-order cone programming (SOCP).",
"Our main results are generalizations of those in [1], which focused on the gradostat with a single reaction [8].",
"Here we allow for any number of substrates and biomasses, general convex objectives, and multiple biochemical reactions.",
"Our original theoretical contributions are as follows.",
"In Section , we formulate a convex relaxation for optimizing the trajectory and steady state solution of a general bioprocess.",
"In Section , we give conditions under which the relaxations are guaranteed to be exact in both the transient and steady state cases.",
"To streamline exposition, the only external inputs to the model are the influent concentrations, e.g., biochemical oxygen demand and ammonia.",
"Our main exactness results straightforwardly apply when the flow rates are also variable, but the resulting bilinearities make the problem nonconvex.",
"This can be handled using techniques like disjunctive programming, as in [1], or further convex relaxation, which we discuss at the end of Section REF .",
"We apply our results in two examples.",
"In Section REF , we show that our exactness conditions simplify to those in [1] when specialized to the gradostat.",
"In Section REF , we optimize the allocation of sewage to three wastewater treatment plants over two weeks.",
"The relaxation is exact, and takes roughly twenty minutes to solve using SOCP [9].",
"The system consists of $s$ well-mixed tanks interconnected by mass flow and diffusion.",
"We denote the set of tanks $\\mathcal {S}$ .",
"$V\\in \\mathbb {R}^{s\\times s}$ is a diagonal matrix in which $V_{ii}$ is the volume of tank $i$ .",
"Tank $i$ has water inlet flow rate $Q_i^{\\textrm {in}}$ and outlet flow rate $Q_i^{\\textrm {out}}$ .",
"We let $Q_{ij}$ denote the flow from tank $i$ to tank $j$ .",
"Let $d_{ij}$ denote the diffusion between tanks $i$ and $j$ , where $d_{ij}=d_{ji}$ .",
"Let $C=\\textrm {diag}\\left[Q_i^{\\textrm {in}}\\right]$ , $M_{ij}&=\\left\\lbrace \\begin{array}{ll}Q_{ji}, & i\\ne j\\\\-Q_i^{\\textrm {out}}-\\sum _{k\\in {\\mathcal {S}}}Q_{ik},& i= j\\end{array}\\right., \\quad L_{ij}&=\\left\\lbrace \\begin{array}{ll}d_{ij}, & i\\ne j\\\\-\\sum _{k\\in {\\mathcal {S}}}d_{ik}, & i= j\\end{array}\\right.,$ and $N=M+L$ .",
"$M$ and $L$ are respectively compartmental and Laplacian matrices.",
"$M$ is invertible if the network is outflow connected, which is to say that there is a directed path from every tank to some tank with outflow [10].",
"Because $L$ is negative semidefinite, $N$ is also invertible if $M$ is outflow connected, and potentially even if $M$ is not outflow connected."
],
[
"Microbial growth",
"We model the microbial growth in the tanks using the notation of Section 1.5 of [6].",
"There are $m$ substrates and biomasses in each perfectly mixed tank.",
"$\\xi _i\\in \\mathbb {R}^m_+$ is the process state vector of tank $i\\in \\mathcal {S}$ , which contains the concentrations of the substrates and biomasses, and $\\xi _i^{\\textrm {in}}\\in \\mathbb {R}^m_+$ is the corresponding influent concentration vector.",
"This model is minimal in that $\\xi $ includes intermediary products, e.g., substrates produced by one reaction and consumed by another, but not final products such as the $\\textrm {CH}_4$ ultimately produced by anaerobic digestion.",
"There are $r$ different types of reactions that convert substrates to other substrates and biomasses.",
"$\\phi _i(\\xi _i)\\in \\mathbb {R}^r_+$ is a vector of the reaction kinetics in tank $i$ .",
"We are interested in the case where the elements of $\\phi _i(\\xi _i)$ are concave functions, and in particular representable as SOC constraints.",
"We show how to do this for Monod and Contois kinetics later in Examples REF and REF .",
"Let $\\kappa _i\\in \\mathbb {R}^{m\\times r}$ be the stoichiometric matrix relating the reaction vector, $\\phi _i(\\xi _i)$ , to the evolution of the process state in tank $i$ .",
"The dynamics in tank $i\\in \\mathcal {S}$ are $V_{ii}\\dot{\\xi }_i=V_{ii}\\kappa _i\\phi _i(\\xi _i) -Q_{i}^{\\textrm {out}}\\xi _i- \\sum _{j\\in \\mathcal {S}}(Q_{ij}+d_{ij})\\xi _i+Q_{i}^{\\textrm {in}}\\xi _i^{\\textrm {in}}+\\sum _{j\\in \\mathcal {S}}(Q_{ji}+d_{ij})\\xi _j.$ The following example illustrates $\\xi _i$ and $\\kappa _i$ .",
"Example 1 (Two-step anaerobic digestion) There are two substrates, $S_i^a$ and $S_i^b$ , and two biomasses, $X_i^a$ and $X_i^b$ .",
"We let $\\xi _i=\\left[S_i^a,S_i^b,X_i^a,X_i^b\\right]^{\\top }$ .",
"$S_i^a$ is converted to both $X_i^a$ and $S_i^b$ at the rate $\\mu ^a\\left(S_i^a\\right)X_i^a$ .",
"$S_i^b$ is converted to $X_i^b$ at the rate $\\mu ^b\\left(S_i^b\\right)X_i^b$ .",
"Therefore, $\\phi _i(\\xi _i)=\\left[\\begin{array}{c}\\mu ^a\\left(S_i^a\\right)X_i^a\\\\\\mu ^b\\left(S_i^b\\right)X_i^b\\end{array}\\right]\\;\\textrm {and}\\;\\kappa _i=\\left[\\begin{array}{cc}-1&0\\\\1&-1\\\\1&0\\\\0&1\\end{array}\\right].$ We now write the dynamics in vector form.",
"We suppress subscripts to represent stacked vectors, i.e., $\\xi =[\\xi _1,...,\\xi _s]^{\\top }$ and $\\phi (\\xi )=[\\phi _1(\\xi _1),...,\\phi _s(\\xi _s)]^{\\top }$ .",
"Let $A\\otimes B$ denote the Kronecker product of $A$ and $B$ , $I_\\alpha \\in \\mathbb {R}^{\\alpha \\times \\alpha }$ the identity matrix, and $\\hat{A}=A\\otimes I_m$ .",
"Let $K$ be a block diagonal matrix with $\\kappa _1,...,\\kappa _s$ on its main diagonal.",
"If $\\kappa _i=\\kappa $ for all $i\\in \\mathcal {S}$ , then $K=I_s\\otimes \\kappa $ .",
"The dynamics of the full system are given in vector form by $\\hat{V}\\dot{\\xi }=\\hat{V}K \\phi (\\xi )+\\hat{N}\\xi +\\hat{C}\\xi ^{\\textrm {in}}.$ We allow the dynamics to be non-autonomous, in which case $\\hat{N}$ , $\\hat{C}$ , and $\\xi ^{in}$ can be time-varying."
],
[
"Discretization in time",
"To make (REF ) compatible with finite-dimensional optimization, we replace the derivatives with a numerical approximation, which we denote $\\mathcal {D}_n$ .",
"For example, in the case of the implicit Euler method with time step $\\Delta $ , $\\mathcal {D}_n[\\xi (\\cdot )]=(\\xi (n)-\\xi (n-1))/\\Delta $ .",
"$\\mathcal {D}_n$ could also be a more sophisticated approximation such as a Runge-Kutta scheme [11].",
"The time periods are indexed $n\\in \\mathcal {N}=\\lbrace 1,...,\\tau \\rbrace $ .",
"We have $\\hat{V}\\mathcal {D}_n[\\xi (\\cdot )]=\\hat{V}K \\phi (\\xi (n))+\\hat{N}(n)\\xi (n)+\\hat{C}(n)\\xi ^{\\textrm {in}}(n)$ for $n\\in \\mathcal {N}$ .",
"The initial condition is $\\xi (0)=\\xi _0$ ."
],
[
"Objectives",
"We consider objectives of the form $\\mathcal {F}(\\xi ,T)=\\sum _{n\\in \\mathcal {N}} \\mathcal {F}_{\\xi }(\\xi (n))+\\mathcal {F}_{\\phi }(T(n)),$ where $\\mathcal {F}_{\\xi }$ and $\\mathcal {F}_{\\phi }$ are convex and $T(n)=\\phi (\\xi (n))$ .",
"The following are examples.",
"Minimizing the outflow of substrates, $\\mathcal {F}_{\\xi }(\\xi (n))=\\sum _{i\\in \\mathcal {S}}Q_{i}^{\\textrm {out}}\\eta _i^{\\top }\\xi _i(n),$ where $\\eta _i$ is a vector that selects the entries of $\\xi _i(n)$ corresponding to pollutants.",
"The production of biogas in a tank is proportional to the kinetics that convert substrates to biomass.",
"Let $\\sigma _i\\in \\mathbb {R}^m_+$ be a vector that is only nonzero for entries of $T_i(n)$ corresponding to biogas production.",
"Let $\\mathcal {M}\\subseteq \\mathcal {S}$ be the subset of tanks that can capture biogas from anaerobic digestion.",
"We maximize biogas through the objective $\\mathcal {F}_{\\phi }(T(n))=-\\sum _{i\\in \\mathcal {M}}V_{ii}\\sigma _i^{\\top }T_i(n).$ Setpoint tracking, $\\mathcal {F}_{\\xi }(\\xi (n))=\\left(\\xi (n)-\\bar{\\xi }\\right)^{\\top }A\\left(\\xi (n)-\\bar{\\xi }\\right),$ where $A\\succeq 0$ and $\\bar{\\xi }$ is a desired operating point."
],
[
"Problem statement",
"We aim to solve the following optimization problem.",
"$\\mathcal {P}\\quad &\\min \\;\\mathcal {F}(\\xi ,T) \\\\\\textrm {such that}\\quad &T(n)=\\phi (\\xi (n)),\\quad n\\in \\mathcal {N}\\\\&\\hat{V}\\mathcal {D}_n[\\xi (\\cdot )]=\\hat{V}K T(n)+\\hat{N}(n)\\xi (n)+\\hat{C}(n)\\xi ^{\\textrm {in}}(n),\\quad n\\in \\mathcal {N}\\\\&\\left(\\xi ,\\xi ^{\\textrm {in}},T\\right)\\in \\Omega .$ $\\mathcal {P}$ models the optimization of a broad range of bioprocesses such as wastewater treatment.",
"We refer the reader to [6] for broad coverage of this topic.",
"A solution of $\\mathcal {P}$ is a trajectory $\\left(\\xi (n),\\xi ^{\\textrm {in}}(n),T(n)\\right)$ , $n\\in \\mathcal {N}$ .",
"The flows and diffusions between tanks, encoded by the matrices $\\hat{N}(n)$ and $\\hat{C}(n)$ , are not decision variables.",
"For this reason constraint () is linear.",
"The set $\\Omega $ in () consists of linear constraints such as the initial condition, total input matter, and maximum substrate concentrations; several other examples are given in [1].",
"Note that $\\Omega $ can constrain $T(\\cdot )$ so as to allow constraints on the growth without adding nonlinearities.",
"The only nonconvexity is therefore (), the growth constraint; this is the focus of the next two sections.",
"We note that in many applications, the flow rates are important decision variables.",
"They are parameters here because our focus is on incorporating the growth kinetics in a convex fashion.",
"In the case that the flow rates are variable, () becomes bilinear and hence nonconvex.",
"There are several ways to handle the bilinearity, including McCormick [12] and lift-and-project [13] relaxations; disjunctive programming reformulations if the flow variables are binary, as in [1]; and finding a local minimum via nonlinear programming, e.g., exploiting the biconvex structure with the Alternating Direction Method of Multipliers [14].",
"All three of the above techniques are viable because, as described in the next section, we have a tractable way to represent the growth constraint, ()."
],
[
"Convex relaxation",
"$\\mathcal {P}$ is nonconvex because constraint () is a nonlinear equality.",
"One way around this difficulty is to instead solve a convex relaxation of $\\mathcal {P}$ , as in [1].",
"If all elements of the vector $\\phi (\\cdot )$ are concave functions, we obtain a convex relaxation by replacing () with the inequality $T(n)\\le \\phi (\\xi (n)),\\quad n\\in \\mathcal {N}.$ We refer to the resulting optimization as $\\mathcal {P}_{\\textrm {R}}$ .",
"As mentioned earlier, we are interested in the case where (REF ) is concave and, ideally, representable as an SOC constraint.",
"Several such examples are given below.",
"Example 2 (Contois growth) Suppose that there is a substrate of concentration $S$ , a biomass of concentration $X$ , and the growth rate is Contois [3].",
"Then constraint (REF ) takes the form $T(n)\\le \\frac{\\mu S(n)X(n)}{k_{\\textrm {C}}X(n)+S(n)},$ where $\\mu $ and $k_{\\textrm {C}}$ are constant parameters.",
"As shown in [1], this is concave and can be written as the SOC constraint $\\left\\Vert \\left[\\begin{array}{c}\\mu S(n)\\\\k_{\\textrm {C}}T(n)\\\\\\mu kX(n)\\end{array}\\right]\\right\\Vert \\le \\mu k_{\\textrm {C}}X(n) + \\mu S(n) - kT(n).$ Example 3 (Monod growth with constant biomass) Consider Example REF , but now suppose that the growth rate is Monod [2].",
"Then constraint (REF ) takes the form $T(n)\\le \\frac{\\mu S(n)X(n)}{k_{\\textrm {M}}+S(n)}.$ This constraint is quasiconcave.",
"It becomes concave if we assume that the biomass in each time period is not an optimization variable, but an exogenous parameter, i.e., $X(n)=\\bar{X}(n)$ for $n\\in \\mathcal {N}$ .",
"This approximation is often valid because the biomass concentration is typically larger and varies more slowly than the substrate concentrations, and is therefore relatively insensitive to the substrates.",
"In this case, as shown in [1], it can be written as the SOC constraint $\\left\\Vert \\left[\\begin{array}{c}\\mu S(n) \\bar{X}(n)\\\\k_{\\textrm {M}}T(n)\\\\\\mu k\\bar{X}(n)\\end{array}\\right]\\right\\Vert \\le \\mu k_{\\textrm {M}}\\bar{X}(n) + \\mu S(n)\\bar{X}(n) - kT(n).$ Example 4 (Interactive and non-interactive growth) Suppose the growth of the biomass depends on two rates, $\\mu ^a(S(n),X(n))$ and $\\mu ^b(S(n),X(n))$ , and the individual kinetics constraints, $T^a(n)\\le \\mu ^a(S(n),X(n))X(n)$ and $T^a(n)\\le \\mu ^b(S(n),X(n))X(n)$ , both have SOC representations.",
"The dependency often takes one of two forms: non-interactive, $\\min \\lbrace \\mu ^a(S(n),X(n)),\\mu ^b(S(n),X(n))\\rbrace $ , which in ecological modeling is known as Liebig's Law, and interactive, $\\mu ^a(S(n),X(n))\\mu ^b(S(n),X(n))$ , which is common in models of bioprocesses.",
"The non-interactive case is enforced by the two individual kinetics constraints along with $T(n)\\le T^a(n)$ and $T(n)\\le T^b(n)$ .",
"The interactive case is in general nonconvex, and does not have an SOC representation.",
"However, the geometric mean of the growth rates, $\\sqrt{\\mu ^a(S(n),X(n))\\mu ^b(S(n),X(n))}$ , does lead to an SOC representation.",
"It is enforced by the two individual kinetics constraints along with $T(n)^2\\le T^a(n)T^b(n)$ .",
"The latter is hyperbolic, a type of SOC constraint [9]."
],
[
"Exactness",
"When (REF ) is satisfied with equality, $\\mathcal {P}_{\\textrm {R}}$ is exact, i.e., has an optimal solution that also solves $\\mathcal {P}$ ; this is the ideal outcome.",
"When (REF ) is not satisfied with equality, $\\mathcal {P}_{\\textrm {R}}$ might still provide a close approximation of $\\mathcal {P}$ , but this is hard to guarantee.",
"It is therefore useful to have conditions, even if narrow, under which the exactness of $\\mathcal {P}_{\\textrm {R}}$ is guaranteed.",
"For the rest of this section we let $\\mathcal {D}_n[\\xi (\\cdot )]=(\\xi (n)-\\xi (n-1))/\\Delta ,$ which corresponds to the implicit Euler step.",
"We assume that the only constraint specified by $\\Omega $ in () is an initial condition, $\\xi (0)=\\xi _0$ , and that $\\xi ^{\\textrm {in}}(\\cdot )$ is fixed; note that as long as exactness holds for all feasible values of $\\xi ^{\\textrm {in}}(\\cdot )$ , it holds when $\\xi ^{\\textrm {in}}(\\cdot )$ is a variable.",
"We also assume that strong duality holds for $\\mathcal {P}_{\\textrm {R}}$ ; this is a mild assumption that, e.g., holds as long as there is a feasible solution in which $\\xi (n)>0$ for all $n\\in \\mathcal {N}$ .",
"Let $\\mathcal {J}(\\xi (n))\\in \\mathbb {R}^{rs\\times ms}$ denote the Jacobian matrix of $\\phi (\\cdot )$ at $\\xi (n)$ .",
"For convenience, we define the following quantities: $\\Gamma (n)&=\\frac{1}{\\Delta }\\left(\\hat{V}/\\Delta -\\hat{N}(n)^{\\top }-\\mathcal {J}(\\xi (n))^{\\top }K^{\\top }\\hat{V}\\right)^{-1}\\hat{V}\\\\&=\\left(I_{ms}-\\Delta \\hat{V}^{-1}\\left(\\hat{N}(n)^{\\top }+\\mathcal {J}(\\xi (n))^{\\top }K^{\\top }\\hat{V}\\right)\\right)^{-1}\\\\\\Omega (n)&=-\\nabla \\mathcal {F}_{\\phi }(T(n))-\\Delta K^{\\top }\\hat{V}\\sum _{k=n}^{\\tau }\\left(\\prod _{l=n}^{k}\\Gamma (l)\\right)\\hat{V}^{-1}\\left(\\nabla \\mathcal {F}_{\\xi }(\\xi (k))+\\mathcal {J}(\\xi (k))^{\\top }\\nabla \\mathcal {F}_{\\phi }(T(k))\\right).$ Observe that if $\\Delta $ is small enough, $\\Gamma (n)$ is positive definite and close to the identify matrix.",
"Theorem 1 $\\mathcal {P}_{\\textrm {R}}$ is exact if at an optimal solution, $\\Omega (n)>0$ for all $n\\in \\mathcal {N}$ .",
"Let $\\lambda (n)\\in \\mathbb {R}^{ms}$ and $\\rho (n)\\in \\mathbb {R}^{rs}$ be the respective dual multipliers of constraints () and (REF ) for $n\\in \\mathcal {N}$ .",
"The Lagrangian of $\\mathcal {P}_{\\textrm {R}}$ is $\\mathcal {L}&=\\mathcal {F}(\\xi ,T) +\\sum _{n\\in \\mathcal {N}}\\rho (n)^{\\top }\\left(T(n)-\\phi (\\xi (n))\\right)\\\\&\\quad +\\lambda (n)^{\\top }\\left(\\hat{V}(\\xi (n-1)-\\xi (n))/\\Delta + \\hat{V}KT(n)+\\hat{N}(n)\\xi (n) + \\hat{C}(n)\\xi ^{\\textrm {in}}(n) \\right).$ Differentiating the Lagrangian by $T(n)$ and $\\xi (n)$ and setting it to zero gives $-\\rho (n) &=\\nabla \\mathcal {F}_{\\phi }(T(n))+ K^{\\top }\\hat{V}\\lambda (n),\\quad n\\in \\mathcal {N},\\\\\\mathcal {J}(\\xi (n))^{\\top }\\rho (n)&=\\nabla \\mathcal {F}_{\\xi }(\\xi (n)) -\\left(\\hat{V}/\\Delta -\\hat{N}(n)^{\\top }\\right)\\lambda (n)+\\hat{V}\\lambda (n+1)/\\Delta ,\\quad n\\in \\mathcal {N}\\setminus \\tau \\\\\\mathcal {J}(\\xi (\\tau ))^{\\top }\\rho (\\tau )&=\\nabla \\mathcal {F}_{\\xi }(\\xi (\\tau )) -\\left(\\hat{V}/\\Delta -\\hat{N}(n)^{\\top }\\right)\\lambda (\\tau ).$ We now solve for $\\rho (n)$ .",
"Premultiplying (REF ) by $\\mathcal {J}(\\xi (n))^{\\top }$ and summing with () and () gives $\\lambda (\\tau )=\\Delta \\Gamma (\\tau )\\hat{V}^{-1}\\left(\\nabla \\mathcal {F}_{\\xi }(\\xi (\\tau ))+\\mathcal {J}(\\xi (\\tau ))^{\\top }\\nabla \\mathcal {F}_{\\phi }(T(\\tau ))\\right),$ and, for $n\\in \\mathcal {N}\\setminus \\tau $ , $\\lambda (n)&=\\Delta \\Gamma (n)\\hat{V}^{-1}\\left(\\nabla \\mathcal {F}_{\\xi }(\\xi (n))+\\mathcal {J}(\\xi (n))^{\\top }\\nabla \\mathcal {F}_{\\phi }(T(n))\\right)+\\Gamma (n)\\lambda (n+1).$ Expanding the recursion yields $\\lambda (n)&=\\Delta \\sum _{k=n}^{\\tau }\\left(\\prod _{l=n}^{k}\\Gamma (l)\\right)\\hat{V}^{-1}\\left(\\nabla \\mathcal {F}_{\\xi }(\\xi (k))+\\mathcal {J}(\\xi (k))^{\\top }\\nabla \\mathcal {F}_{\\phi }(T(k))\\right).$ We now substitute this into (REF ) to obtain $\\rho (n)&=-\\nabla \\mathcal {F}_{\\phi }(T(n))-\\Delta K^{\\top }\\hat{V}\\sum _{k=n}^{\\tau }\\left(\\prod _{l=n}^{k}\\Gamma (l)\\right)\\hat{V}^{-1}\\left(\\nabla \\mathcal {F}_{\\xi }(\\xi (k))+\\mathcal {J}(\\xi (k))^{\\top }\\nabla \\mathcal {F}_{\\phi }(T(k))\\right)\\\\&=\\Omega (n).$ From here we can see that the conditions of the theorem guarantee that $\\rho (n)>0$ for all $n\\in \\mathcal {N}$ .",
"Theorem REF is of limited immediate use because we must know the optimal solution of $\\mathcal {P}_{\\textrm {R}}$ to test if $\\Omega (n)>0$ .",
"It can however be used to derive sufficient conditions for exactness that are easy to test.",
"We now derive two such conditions that do not require knowledge of the optimal solution.",
"For the rest of this section, assume that $\\mathcal {F}_{\\xi }$ and $\\mathcal {F}_{\\phi }$ are linear with gradients $f_{\\xi }\\in \\mathbb {R}^{ms}$ and $f_{\\phi }\\in \\mathbb {R}^{rs}$ .",
"In this case, we can write $&\\Omega (n)=\\\\&\\quad -\\Delta K^{\\top }\\hat{V}\\left(\\sum _{k=n}^{\\tau }\\prod _{l=n}^{k}\\Gamma (l)\\right)\\hat{V}^{-1}f_{\\xi }-\\left(I_{ms} +\\Delta K^{\\top }\\hat{V}\\sum _{k=n}^{\\tau }\\left(\\prod _{l=n}^{k}\\Gamma (l)\\right)\\hat{V}^{-1}\\mathcal {J}(\\xi (k))^{\\top } \\right)f_{\\phi }.$ We also assume that each element of the vector of reaction kinetics has bounded slope, so that all entries of $\\mathcal {J}(\\xi (n))$ , $n\\in \\mathcal {N}$ , are bounded.",
"Let $\\Psi (n)=\\hat{V}^{-1}\\hat{N}(n)^{\\top }+\\hat{V}^{-1}\\mathcal {J}(\\xi (n))^{\\top }K^{\\top }\\hat{V}.$ The first term of $\\Psi (n)$ is negative semidefinite because $N(n)$ is compartmental [10].",
"The latter term is bounded by assumption, and as we will see in the examples, usually negative semidefinite—we assume that this is the case.",
"We therefore assume that the eigenvalues of $\\Psi (n)$ are in the range $[-\\bar{\\psi },0]$ , where $\\bar{\\psi }>0$ is an upper bound on the magnitude.",
"We can use the push-through identity to write $\\Gamma (n)&=I_{ms}+\\Delta \\Psi (n)(I_{ms}-\\Delta \\Psi (n))^{-1}.$ The eigenvalues of the second term are in the range $[-\\Delta \\bar{\\psi },0]$ , and the eigenvalues of $\\Gamma (n)$ are in the range of $[1-\\Delta \\bar{\\psi },1]$ .",
"Corollary 1 If $f_{\\phi }=0$ and $K^{\\top }f_{\\xi }<0$ , then there exists a $\\Delta >0$ for which $\\mathcal {P}_{\\textrm {R}}$ is exact.",
"(Sketch) Observe that $\\prod _{l=n}^{k}\\Gamma (l)$ is equal to $I_{ms}$ plus terms that are norm-bounded by positive powers of $\\Delta \\bar{\\psi }$ .",
"Similarly, $\\sum _{k=n}^{\\tau }\\prod _{l=n}^{k}\\Gamma (l)$ is equal to $(\\tau -n+1)I_{ms}$ plus terms that are norm-bounded by positive powers of $\\Delta \\bar{\\psi }$ .",
"We can make these terms arbitrarily small by choosing $\\Delta $ small.",
"We therefore write $\\sum _{k=n}^{\\tau }\\prod _{l=n}^{k}\\Gamma (l)\\approx (\\tau -n+1)I_{ms}.$ Because $f_{\\phi }=0$ , we have that $\\Omega (n)\\approx &-(\\tau -n+1)\\Delta K^{\\top }f_{\\xi }.$ This is strictly positive for all $n\\in \\mathcal {N}$ if $K^{\\top }f_{\\xi }<0$ .",
"Corollary 2 If $f_{\\xi }=0$ and $f_{\\phi }>0$ , then there exists a $\\Delta >0$ for which $\\mathcal {P}_{\\textrm {R}}$ is exact.",
"(Sketch) Following the same logic as Corollary REF , we can choose $\\Delta >0$ such that $\\Omega (n)\\approx &\\left(I_{ms}-\\Delta K^{\\top }\\sum _{k=n}^{\\tau }\\mathcal {J}(\\xi (k))^{\\top }\\right)f_{\\phi }.$ The second term in the parentheses can be made arbitrarily small by choosing $\\Delta $ to be small.",
"In this case $\\Omega (n)\\approx f_{\\phi }$ , which is positive if $f_{\\phi }>0$ .",
"A shortcoming of Corollaries REF and REF is that they can be limited to short time intervals.",
"This is because if an interval is to remain constant, the number of time periods, $\\tau $ , must increase as the step, $\\Delta $ , decreases, and the approximations in the proofs of the corollaries do not hold for increasing $\\tau $ .",
"On the other hand, these are conservative sufficient conditions.",
"For example, the second term in the parentheses of (REF ) will often be positive semidefinite or nearly so.",
"This is because it is typical for $K$ to be lower triangular with a negative diagonal and for $\\mathcal {J}(\\xi (n))$ to be nonnegative and nearly diagonal.",
"We therefore expect that exactness will sometimes hold for larger $\\Delta $ over longer time intervals.",
"We view these theoretical results not as a complete characterization of when exactness is guaranteed, but rather as evidence that $\\mathcal {P}_\\textrm {R}$ is exact for a meaningful set of problems, and as guidance as to how to identify them.",
"While there are certainly problems of interest for which their conditions do not hold, $\\mathcal {P}_\\textrm {R}$ may nonetheless provide a useful and sometimes perfect approximation."
],
[
"Steady state",
"It may be of interest to optimize (REF ) in steady state, e.g., when the solution does not change significantly on the timescale of interest, to find the best operating point, or to reduce the number of variables.",
"We obtain a steady state optimization by dropping the time index and replacing the finite difference in () with zero.",
"The resulting (relaxed) optimization is: $\\mathcal {P}_{\\textrm {RS}}\\quad &\\min \\;\\mathcal {F}(\\xi ,T) \\\\\\textrm {such that}\\quad &T\\le \\phi (\\xi )\\\\&0=\\hat{V}K T+\\hat{N}\\xi +\\hat{C}\\xi ^{\\textrm {in}}\\\\&\\left(\\xi ,\\xi ^{\\textrm {in}},T\\right)\\in \\Omega .$ We remark that, in general, a solution to $\\mathcal {P}_{\\textrm {RS}}$ is not guaranteed to be an equilibrium of (REF ).",
"One special case in which guarantees do exist is the gradostat, which we discuss in Example REF .",
"We refer the reader to [1] for a brief summary.",
"Theorem 2 $\\mathcal {P}_{\\textrm {RS}}$ is exact if the network is outflow connected and at the optimal solution, $0<&\\left(I_{ms}+K^{\\top }\\hat{V}\\left(\\hat{N}^{\\top }\\right)^{-1}\\mathcal {J}(\\xi )^{\\top }\\right)^{-1}\\left(K^{\\top }\\hat{V}\\left(\\hat{N}^{\\top }\\right)^{-1}\\nabla \\mathcal {F}_{\\xi }(\\xi )-\\nabla \\mathcal {F}_{\\phi }(T)\\right).$ The Lagrangian of $\\mathcal {P}_{\\textrm {RS}}$ is $\\mathcal {L}&=\\mathcal {F}(\\xi ,T) +\\rho ^{\\top }\\left(T-\\phi (\\xi )\\right)+\\lambda ^{\\top }\\left( \\hat{V}KT+\\hat{N}\\xi + \\hat{C}\\xi ^{\\textrm {in}} \\right).$ Differentiating the Lagrangian and setting it to zero gives $0&=\\nabla \\mathcal {F}_{\\phi }(T) +\\rho + K^{\\top }\\hat{V}\\lambda \\\\\\mathcal {J}(\\xi )^{\\top }\\rho &=\\nabla \\mathcal {F}_{\\xi }(\\xi ) + \\hat{N}^{\\top }\\lambda .$ We now solve for $\\rho $ .",
"$\\hat{N}$ is invertible due to outflow-connectedness.",
"Then $\\lambda &=\\left(\\hat{N}^{\\top }\\right)^{-1}\\left(\\mathcal {J}(\\xi )^{\\top }\\rho - \\nabla \\mathcal {F}_{\\xi }(\\xi ) \\right),$ and $0&=\\nabla \\mathcal {F}_{\\phi }(T) +\\rho + K^{\\top }\\hat{V}\\left(\\hat{N}^{\\top }\\right)^{-1}\\left(\\mathcal {J}(\\xi )^{\\top }\\rho - \\nabla \\mathcal {F}_{\\xi }(\\xi ) \\right).$ Solving, we have $\\rho &=\\left(I_{ms}+K^{\\top }\\hat{V}\\left(\\hat{N}^{\\top }\\right)^{-1}\\mathcal {J}(\\xi )^{\\top }\\right)^{-1}\\left(K^{\\top }\\hat{V}\\left(\\hat{N}^{\\top }\\right)^{-1}\\nabla \\mathcal {F}_{\\xi }(\\xi )-\\nabla \\mathcal {F}_{\\phi }(T)\\right).$ By complementary slackness, $\\mathcal {P}_{\\textrm {RS}}$ is exact when $\\rho >0$ .",
"As in the latter part of the previous section, we now assume that $\\mathcal {F}_{\\xi }$ and $\\mathcal {F}_{\\phi }$ are linear with gradients $f_{\\xi }\\in \\mathbb {R}^{ms}$ and $f_{\\phi }\\in \\mathbb {R}^{rs}$ .",
"Corollary 3 Suppose that for all $\\xi \\ge 0$ , $\\left(I_{ms}+K^{\\top }\\hat{V}\\left(\\hat{N}^{\\top }\\right)^{-1}\\mathcal {J}(\\xi )^{\\top }\\right)^{-1}\\ge 0,$ and $K^{\\top }\\hat{V}\\left(\\hat{N}^{\\top }\\right)^{-1}f_{\\xi }-f_{\\phi }\\ge 0.$ If at least one of the inequalities is strict, then $\\mathcal {P}_{\\textrm {RS}}$ is exact.",
"Like Theorem REF , Corollary REF depends on the optimal solution, but to a lesser extent.",
"Whereas Theorem REF depends on $\\xi $ and $T$ through the objective and $\\mathcal {J}(\\xi )$ , Corollary REF only depends on $\\xi $ through $\\mathcal {J}(\\xi )$ .",
"This is more manageable because $\\mathcal {J}(\\xi )$ is often positive and nearly diagonal.",
"For example, if we assume assume that biomass is constant and all growth rates are Monod, as in Example REF , then $\\mathcal {J}(\\xi )$ is positive on the diagonal and zero elsewhere.",
"The gradostat is a special case of (REF ) where in each tank $i\\in \\mathcal {S}$ , a single substrate, $S_i$ , is converted to a single type of biomass, $X_i$ .",
"The conversion occurs at the rate $\\phi _i(S_i,X_i)/y$ , where $y$ is the yield.",
"Then $\\xi _i=[S_i,X_i]^{\\top }$ and $\\kappa _i=[-1/y,1]^{\\top }$ .",
"In [1], several examples are given in which the gradostat is optimized over time and in steady state.",
"Here we first apply Corollary REF to the gradostat.",
"Suppose $\\mathcal {F}_{\\xi }(\\xi (n))=f_S^{\\top }S(n)+f_X^{\\top }X(n)$ .",
"In this case, Corollary REF holds if $-f_S/y+f_X<0$ .",
"When dealing with the decontamination of undesirable substrates, we do not seek to minimize biomass, in which case $f_X=0$ and the condition is satisfied if $f_S>0$ .",
"We now apply Theorem REF to the gradostat in steady state.",
"Let $\\mathcal {J}_S(\\xi )\\in \\mathbb {R}^{r\\times s}$ be the Jacobian matrix of $\\phi (\\xi )$ with respect to $S$ , and let $\\mathcal {J}_X(\\xi )\\in \\mathbb {R}^{r\\times s}$ be the Jacobian matrix of $\\phi (\\xi )$ with respect to $X$ .",
"$\\mathcal {P}_{\\textrm {RS}}$ is exact if $&\\left(I_s+V\\left(N^{\\top }\\right)^{-1}\\left(-\\mathcal {J}_S(\\xi )^{\\top }/y+\\mathcal {J}_X(\\xi )^{\\top }\\right)\\right)^{-1}\\left(K^{\\top }V\\left(N^{\\top }\\right)^{-1}\\nabla \\mathcal {F}_{\\xi }(\\xi )-\\nabla \\mathcal {F}_{\\phi }(T)\\right)>0 .$ It is straightforward to verify that when $\\mathcal {F}_{\\xi }(\\xi )=0$ , this condition directly implies Theorem 1 in [1].",
"Similarly, we obtain Corollary 1 in [1] if we specialize Corollary REF to the gradostat."
],
[
"Wastewater treatment",
"In this example, we optimize an idealized wastewater treatment system consisting of three wastewater treatment plants of the city of Paris and its suburbs.",
"The model is based on that in [15], [16], and the influent data from the Inf_rain_2006 dataset of [17].",
"We implemented the model using the parser CVX [18] and the solver Gurobi [19] on a personal computer from 2014 with a 1.4 GHz dual-core processor.",
"In the present study, the flow rates are considered constant and given by $Q^{\\textrm {in}}_1=0.1\\;m^3/s$ , $Q^{\\textrm {in}}_2=0.4\\;m^3/s$ , and $Q^{\\textrm {in}}_3=0.2\\;m^3/s$ .",
"All tanks have volume $1000\\;m^3$ .",
"In each plant $i\\in \\mathcal {S}$ , the state vector $\\xi _i=\\left[\\xi ^{\\textrm {BOD}}_i,\\xi ^{\\textrm {NH}_4^+}_i,\\xi ^{\\textrm {NO}_2^-}_i,\\xi ^{\\textrm {NO}_3^-}_i\\right]^{\\top }\\in \\mathbb {R}^4$ consists of biochemical oxygen demand, ammonia nitrogen, nitrite, and nitrate.",
"The biomass in each time period is assumed to be an exogenous parameter, and therefore not a component of the process state.",
"This is an admissible assumption in the sense that the biomass concentration is typically much slower than that of the other process components, and has a larger amplitude.",
"In each tank $i\\in \\mathcal {S}$ , the elements of the process state have kinetics $\\phi _i^{\\textrm {BOD}},\\phi _i^{\\textrm {NH}_4^+},\\phi _i^{\\textrm {NO}_2^-}$ , and $\\phi _i^{\\textrm {NO}_3^-}$ .",
"All assume Monod growth rates with parameters given in Table REF , which comes from Table 1 in [20].",
"Because the biomass is constant, the growth kinetics can be represented as SOC constraints in $\\mathcal {P}_{\\textrm {R}}$ .",
"Table: Growth function parametersThe stochiometric matrix for each plant $i\\in \\mathcal {S}$ is $\\kappa _i=\\left[\\begin{array}{cccc}-1&0 & 0 & 0\\\\0 & -1 & 0 & 0\\\\0 & 1/y_i^{\\textrm {NH}_4^+,\\textrm {NO}_2^-} & -1 & 0\\\\0 & 0 & 1/y_i^{\\textrm {NO}_2^-,\\textrm {NO}_3^-} & -1\\end{array}\\right].$ We used the implicit Euler method, $\\mathcal {D}_n[\\xi (\\cdot )]=(\\xi (n)-\\xi (n-1))/\\Delta $ , with the stepsize $\\Delta =1$ , which corresponds to 15 minutes.",
"There are $\\tau =1345$ time periods, so that the total time is two weeks.",
"The boundary condition is $\\xi (0)=\\xi (\\tau )$ .",
"This could represent periodic operation, or exogenous conditions that are similar from week to week.",
"The process state must satisfy $\\xi ^{\\textrm {BOD}}_i(n)\\le 150$ mg/L and $\\xi ^{\\textrm {NH}_4^+}_i(n)\\le 60$ mg/L for each $i\\in \\mathcal {S}$ and $n\\in \\mathcal {N}$ .",
"Without these constraints, most of the substrate would be directed to Plant 1, which is more efficient than the others.",
"This constraint could represent, for example, regulatory limits on the pollution released by the plants.",
"In each time period, fixed quantities of $\\textrm {BOD}$ and $\\textrm {NH}_4^+$ , $\\Xi ^{\\textrm {BOD}}(n)$ and $\\Xi ^{\\textrm {NH}_4^+}(n)$ , must be allocated over the treatment plants.",
"These quantities are based on the Inf_rain_2006 dataset of [17], which covers 1345 15-minute intervals.",
"$\\Xi ^{\\textrm {BOD}}(\\cdot )$ is set to $S_{\\textrm {S}}$ (readily biodegradable substrate), and $\\Xi ^{\\textrm {NH}_4^+}(\\cdot )$ to $S_{\\textrm {NH}}$ ($\\textrm {NH}_4^+$ and $\\textrm {NH}_3$ nitrogen) in [17].",
"The allocation is represented by the linear constraints $\\Xi ^{\\textrm {BOD}}(n)&=\\sum _{i=1}^3Q^{\\textrm {in}}_i(n)\\xi ^{\\textrm {BOD},\\textrm {in}}_i(n)\\\\\\Xi ^{\\textrm {NH}_4^+}(n)&=\\sum _{i=1}^3Q^{\\textrm {in}}_i(n)\\xi ^{\\textrm {NH}_4^+,\\textrm {in}}_i(n),$ for each $n\\in \\mathcal {N}$ .",
"The other influent concentrations are $\\xi ^{\\textrm {NO}_2^-,\\textrm {in}}_i(n)=3$ and $\\xi ^{\\textrm {NO}_3^-,\\textrm {in}}_i(n)=10$ for $i\\in \\mathcal {S}$ and $n\\in \\mathcal {N}$ .",
"The plant biomass concentrations are set to $\\bar{X}_i(n)=100\\left(1+(-1)^i\\sin (10\\pi n/\\tau )\\right)$ for $i\\in \\mathcal {S}$ and $n\\in \\mathcal {N}$ .",
"Observe that this has larger magnitude and varies slower than the other influents.",
"For each $i\\in \\mathcal {S}$ and $n\\in \\mathcal {N}$ , the optimization variables are $\\xi _i(n)$ , $\\xi ^{\\textrm {BOD},\\textrm {in}}_i(n)$ , and $\\xi ^{\\textrm {NH}4,\\textrm {in}}_i(n)$ .",
"The objective is to minimize the untreated wastewater released by the plants, given by $\\sum _{n\\in \\mathcal {N}}\\sum _{i\\in \\mathcal {S}}Q^{\\textrm {out}}_i\\eta _i^{\\top }\\xi _i(n),$ where we note that $Q^{\\textrm {out}}_i=Q^{\\textrm {in}}_i$ , and $\\eta _i=[2,2,0.3,0.1,0]^{\\top }$ .",
"This objective was chosen to satisfy Corollary REF .",
"Note, however, that the result does not fully apply due to the boundary condition, $\\xi (0)=\\xi (\\tau )$ , and the upper limit on the process state.",
"The convex relaxation $\\mathcal {P}_{\\textrm {R}}$ contains 145272 variables (in standard form) and took 17 minutes to solve.",
"The 18 solver iterations accounted for only four seconds, and the rest of the time was used for preprocessing.",
"Despite not fully satisfying Corollary REF , the solution was exact in all time periods.",
"Figures REF and REF respectively show the optimal influent allocation, $\\xi ^{\\textrm {in}}(\\cdot )$ , and the resulting plant effluent concentrations, $\\xi (\\cdot )$ , between hours 175 and 275.",
"Figure: ξ in (·)\\xi ^{\\textrm {in}}(\\cdot ) from hour 175 to 275.",
"The units are mg/L.Figure: ξ(·)\\xi (\\cdot ) from hour 175 to 275.",
"The units are mg/L.The slower variation of the biomass dominates that of the diurnal variation in the Inf_rain_2006 dataset, leading to concentration increases whenever the biomass influent into each plant is high.",
"There is a spike in $\\Xi ^{\\textrm {BOD}}(\\cdot )$ and $\\Xi ^{\\textrm {NH}_4^+}(\\cdot )$ around hour 250.",
"This causes $\\xi ^{\\textrm {BOD}}_i(\\cdot )$ , and $\\xi ^{\\textrm {NH}4}_i(n)$ to hit their concentration limits in Plants 1 and 3.",
"When this happens, the remainder is allocated to Plant 2, which has little biomass at the time, and hence cannot transform the substrates as efficiently."
],
[
"Conclusion",
"We have formulated a convex relaxation for optimizing a broad class of bioprocesses.",
"We proved that the relaxation is exact under simple conditions, and implemented it on a wastewater treatment example with over one hundred thousand variables.",
"We believe that a wide range of problems can be approached in this manner due to the generality of the model and the tractability of SOCP.",
"One direction of future work is dealing with inexactness.",
"Two options are deriving general convex underestimators to limit the relaxation gap, as in [1], and finding local minima of the non-relaxed problem.",
"In particular, the concave-convex procedure [21] is well-suited to the nonconvexity encountered here and would entail solving a sequence of SOCPs.",
"We also intend to incorporate new elements into the model such as biomass death and recirculation, and to apply the relaxation in other contexts such as enzymes, where Michaelis-Menten kinetics [22] have the same form as Monod kinetics."
],
[
"Acknowledgments",
"Funding is acknowledged from the Natural Sciences and Engineering Research Council of Canada and the French LabEx NUMEV (Project ANR-10 LABX-20), incorporated into the I-Site MUSE, which partially funded the sabbatical of J.A.",
"Taylor at MISTEA lab."
]
] | 2107.01843 |
[
[
"Exergy of passive states: Waste energy after ergotropy extraction"
],
[
"Abstract Work extraction protocol is always a significant issue in the context of quantum batteries, in which the notion of ergotropy is used to quantify a particular amount of energy that can be extracted through unitary processes.",
"Given the total amount of energy stored in a quantum system, quantifying wasted energy after the ergotropy extraction is a question to be considered when undesired coupling with thermal reservoirs is taken into account.",
"In this paper, we show that some amount of energy can be lost when we extract ergotropy from a quantum system and quantified by the exergy of passive states.",
"Through a particular example, one shows that ergotropy extraction can be done by preserving the quantum correlations of a quantum system.",
"Our study opens the perspective for new advances in open system quantum batteries able to explore exergy stored as quantum correlations."
],
[
"Introduction",
"The dynamics of quantum systems that are in contact with an external environment, the so-called open system, arises from the interaction between the quantum degrees of freedom and the time evolution of coherence and system-environment coupling strength.",
"Recently, efforts have been made to design the environment-mediated charging process of quantum batteries by different scenarios [1], [2], [3], [4], [5], [6], [7], which is a path to the realization of new quantum batteries.",
"With the advent of quantum thermodynamics, studies on quantum devices able to use quantum advantages to store and extract useful energy from physical systems [8], [9], [10], [11], [12], [13], called quantum batteries (QBs), have allowed to define new physical quantities.",
"For example, the maximal work that can be extracted from a quantum system by unitary operations is called ergotropy [14].",
"However, when we take into account non-unitary effects on the QB due to the coupling of the system with external thermal baths [1], [2], the battery performance leads to ask how much energy is lost due to thermal effects on the system.",
"In particular, there are situations in which ergotropy can be stored [15], but some part of the total energy available in the system is not stored as ergotropy and, consequently, part of the total energy cannot be extracted from cyclic unitary transformations.",
"Under this point of view, it is worth considering the amount of residual energy that cannot be extracted as useful work from open quantum batteries by unitary processes.",
"In this direction, in this paper we explore the amount of energy lost due to the constraint of unitary processes for work extraction.",
"To this end, we show that the system exergy, the maximum amount of work extracted from the system bringing it into the equilibrium with a thermal bath, can be decomposed into two quantities: ergotropy and residual energy.",
"Such residual energy cannot be extracted through a unitary process, then it refers to the non-optimal performance of a cyclic thermodynamics process for work extraction from QBs.",
"In a general way, we show that the waste energy is quantified by the amount of exergy of the passive state (a state in which no energy can be extracted as ergotropy).",
"We then apply our discussion to Werner states, where we study how entanglement and discord are associated with ergotropy and exergy of Werner passive states.",
"We can quantify the extractable energy from a quantum system using the definition of ergotropy (for unitary process) and exergy (for non-unitary process).",
"This is motivated by the definition of the amount of energy that is accessible due to the presence of the thermal bath.",
"So, let us consider the following scenario: we have a quantum system of interest, the system is in an arbitrary non-equilibrium state $\\rho _{0}$ with a Hamiltonian $H_{S}$ .",
"One may regard only unitary evolution, where the system evolves through a cyclic process.",
"In this protocol, the Hamiltonian of the system is the same at beginning and at the end of the process, so that ergotropy is the maximum work that can be extracted from the battery under such a process as the following form [14] $\\mathcal {E}_{S}&=(H_{S}\\rho _{S})-\\min _{U\\in \\mathcal {U}}(H_{S}U\\rho _{S}U^{\\dagger })=(H\\rho _{0}) - (H\\varrho _{\\rho _{0}}),$ where $\\mathcal {U}$ covers the set of unitary operations derived from Hamiltonian $H_{S}$ , and $\\varrho _{\\rho _{0}}$ is the called passive state [14].",
"In addition, in situations where the system-reservoir interaction is taken into account, energy can be lost by thermalization.",
"To take into account such interaction, let us consider the thermalization process of the system, initially in state $\\rho _{0}$ , with a thermal bath at inverse temperature $\\beta $ .",
"It is known that during the thermalization process an amount of energy is exchanged between the system and the reservoir given by the variation of the free energy [16] $\\Sigma ^{\\rho _{0}\\rightarrow \\rho _{\\beta }} = \\mathcal {F}(\\rho _{0})- \\mathcal {F}(\\rho _{\\beta }) , $ where $\\mathcal {F}(\\rho _{0})\\!=\\!\\lbrace H\\rho _{0}\\rbrace -\\beta ^{-1}S(\\rho _{0})$ is the nonequilibrium free energy of the battery at the initial stage, $\\mathcal {F}(\\rho _{\\beta })$ being the free energy of the final equilibrium state $\\rho _{\\beta }=e^{-\\beta H}/Z$ , with the von Neumann entropy and the partition function given by $S(\\rho )\\!=\\!-{\\rho \\ln \\rho }$ and $Z=tr(e^{-\\beta H})$ , respectively.",
"Since the quantity $\\Sigma ^{\\rho _{0}\\rightarrow \\rho _{\\beta }}$ is the amount of extractable work from a system through a process that brings the system into the equilibrium with a thermal reservoir, it is worth mentioning here that we can call it exergy.",
"We use it in analogy with the thermodynamics classical definition of exergy, as defined by Zoran Rant [17], from Greek `ex' [$\\varepsilon \\xi $ ] and `ergon' [$\\varepsilon \\rho \\gamma o\\nu $ ].",
"Figure: Schematic representation of the three states addressed in our discussion.",
"The ergotropy extraction process, which does not change the system entropy, is followed by the thermalization process that brings the system into equilibrium with the thermal reservoir at temperature β\\beta ."
],
[
"Residual energy after ergotropy extraction",
"It is known that after the ergotropy extraction, the internal energy of the system is not zero [14], [18], so that a residual amount of energy is yet available in the system.",
"In order to quantify the amount of extractable energy after ergotropy extraction, we consider the dynamics as depicted in Fig.",
"REF .",
"The available work in an initial state $\\rho _{0}$ can be extracted in two ways: (i) a unitary process which brings the system into a passive state $\\varrho _{\\rho _{0}}$ , so that the extractable work is quantified by ergotropy [14], and (ii) a non-unitary process that leads the system to the thermal equilibrium state $\\rho _{\\beta }$ , quantified by the variation of the free energy [16].",
"As sketched in Fig.",
"REF , it is possible to identify some amount of energy $\\Sigma _{\\text{ex}}$ so that the exergy $\\Sigma (\\rho _{0}\\rightarrow \\rho _{\\beta })$ extracted in the process $\\rho _{0}\\rightarrow \\rho _{\\beta }$ , and the extractable ergotropy $\\mathcal {E}$ in the process $\\rho _{0}\\rightarrow \\varrho _{\\rho _{0}}$ , satisfy the balance equation $\\Sigma (\\rho _{0}\\rightarrow \\rho _{\\beta }) = \\mathcal {E} + \\Sigma _{\\text{ex}}$ , where $\\Sigma _{\\text{ex}}$ is an available work that cannot be extracted by unitary processes.",
"In fact, consider Eq.",
"(REF ) as $\\Sigma ^{\\rho _{0}\\rightarrow \\rho _{\\beta }} = \\lbrace H\\rho _{0}\\rbrace -\\beta ^{-1}S(\\rho _{0}) - [\\lbrace H\\rho _{\\beta }\\rbrace -\\beta ^{-1}S(\\rho _{\\beta })]$ where we can assume the sum-zero equation given by $\\lbrace H^{(0)}\\varrho _{\\rho _{0}}\\rbrace - \\lbrace H^{(0)}\\varrho _{\\rho _{0}}\\rbrace $ in order to write $\\Sigma ^{\\rho _{0}\\rightarrow \\rho _{\\beta }} &= \\lbrace H\\rho _{0}\\rbrace - \\lbrace H\\varrho _{\\rho _{0}}\\rbrace - [\\lbrace H\\rho _{\\beta }\\rbrace -\\beta ^{-1}S(\\rho _{\\beta })] \\nonumber \\\\ &+ \\lbrace H\\varrho _{\\rho _{0}}\\rbrace - \\beta ^{-1}S(\\rho _{0}) .$ Now, notice that the first two terms are associated with the ergotropy of the initial state $\\rho _{0}$ , so that $\\Sigma ^{\\rho _{0}\\rightarrow \\rho _{\\beta }} &= \\mathcal {E}(\\rho _{0}) - \\left[\\lbrace H\\rho _{\\beta }\\rbrace -\\beta ^{-1}S(\\rho _{\\beta })\\right] \\nonumber \\\\ &+ \\lbrace H\\varrho _{\\rho _{0}}\\rbrace - \\beta ^{-1}S(\\rho _{0}) .$ In addition, since the process that brings the system from state $\\rho _{0}$ to the passive state $\\varrho _{\\rho _{0}}$ is unitary, we also can write $S(\\rho _{0})\\!=\\!S(\\varrho _{\\rho _{0}})$ , and Eq.",
"(REF ) becomes $\\Sigma ^{\\rho _{0}\\rightarrow \\rho _{\\beta }} &= \\mathcal {E}(\\rho _{0}) + \\left[\\mathcal {F}(\\varrho _{\\rho _{0}})- \\mathcal {F}(\\rho _{\\beta })\\right] .$ In conclusion, as sketched in Fig.",
"REF , the additional amount of energy $\\Sigma _{\\text{ex}}$ is given by the free energy of the passive state associated to $\\rho _{0}$ , which is equivalent to exergy stored in the passive state $\\varrho _{\\rho _{0}}$ , mathematically $\\Sigma _{\\text{ex}} = \\Sigma ^{\\varrho _{\\rho _{0}}\\rightarrow \\rho _{\\beta }} = \\mathcal {F}(\\varrho _{\\rho _{0}})- \\mathcal {F}(\\rho _{\\beta }) .$ This result can be understood in two different ways.",
"The first interpretation refers to the uniqueness of an energetically efficient initial state for store ergotropy.",
"In fact, given a system that interacts with a reservoir, the initial ergotropy is stored in a non-pure state $\\rho _0$ .",
"Then, for a short time interval, we can drive the system in order to extract ergotropy through the optimal unitary operation $U_\\text{opt}$ .",
"By adequately choosing the initial state so that $U_\\text{opt}\\rho _{0}U_\\text{opt}^{\\dagger }\\rightarrow \\rho _{\\beta }$ , it is possible to see that $\\Sigma ^{\\varrho _{\\rho _{0}}\\rightarrow \\rho _{\\beta }}\\!=\\!0$ , leading to $\\Sigma ^{\\rho _{0}\\rightarrow \\rho _{\\beta }} \\!=\\!\\mathcal {E}(\\rho _{0})$ .",
"In conclusion, all available energy of the system can be extracted as ergotropy.",
"So, given the uniqueness of the thermal state $\\rho _{\\beta }$ , for the optimal unitary operation $U_\\text{opt}$ we have the uniqueness of $\\rho _{0}$ .",
"The second case refers to the way to efficiently extract energy.",
"In addition, given that the quantity $\\Sigma ^{\\varrho _{\\rho _{0}}\\rightarrow \\rho _{\\beta }}$ cannot be a negative number, this means that energy lost during the ergotropy extraction is expected as a natural process due to the entropy production.",
"This second case can be understood as an immediate application of the second law of thermodynamics to quantum batteries.",
"In fact, for $\\Sigma ^{\\varrho _{\\rho _{0}}\\rightarrow \\rho _{\\beta }}\\!>\\!0$ , the Eq.",
"(REF ) gives $\\Delta S_{\\text{ex}} > \\beta \\left( \\lbrace H\\rho _{\\beta }\\rbrace - \\lbrace H\\varrho _{\\rho _{0}}\\rbrace \\right) ,$ where $\\Delta S_{\\text{ex}}$ is the entropy production required to extract the exergy stored in the system.",
"Then, given that during the thermalization process the heat exchanged between the system and reservoir is given by the internal energy variation of the system [19], we conclude that $\\Delta S_{\\text{ex}}\\!>\\!\\beta Q$ ."
],
[
"Residual energy as quantum correlations",
"In this section, we show that quantum correlations after ergotropy extraction can preserve a non-zero amount of energy.",
"To this end, we consider a two-spin Werner state given by $\\rho _{\\text{w}} = \\frac{1-\\varepsilon }{4} \\mathbb {1} + \\varepsilon {\\beta }{\\beta } , $ with the Bell state ${\\beta }\\!=\\!",
"({\\uparrow \\uparrow }+{\\downarrow \\downarrow })/\\sqrt{2}$ .",
"The above state is adequate for our study because we can adequately choose $\\varepsilon $ to control the amount of quantum correlation of $\\rho _{\\text{w}}$ .",
"In particular, here we consider concurrence [20] as the measurement of entanglement, and quantum discord [21] as correlations beyond entanglement.",
"The concurrence is given by $\\mathcal {C}(\\rho )\\!=\\!\\mathrm {max} \\lbrace 0 , \\lambda _{1} - \\lambda _{2} - \\lambda _{3} - \\lambda _{4} \\rbrace $ , where $\\lambda _{n}$ are the eigenvalues of the operator $\\sqrt{\\rho ^{1/2}\\tilde{\\rho }\\rho ^{1/2}}$ in decreasing order, with $\\tilde{\\rho }=(\\sigma _{y}\\otimes \\sigma _{y})\\rho ^{\\ast }(\\sigma _{y}\\otimes \\sigma _{y})$ , being the complex conjugate $\\rho ^{\\ast }$ of $\\rho $ taken in the basis 0 and 1 of the system.",
"Quantum discord is computed from the one-sided trace distance discord (TDD) $\\mathcal {D}(\\rho )\\!=\\!D^{(\\rightarrow )}(\\rho )$ for $X$ -states as $\\mathcal {D}(\\rho ) = \\frac{1}{2}\\sqrt{\\frac{\\gamma ^{2}_{1} \\max \\lbrace \\gamma _{3}^{2}, \\gamma _{2}^{2} + x^2\\rbrace - \\gamma ^{2}_{2} \\max \\lbrace \\gamma _{3}^{2}, \\gamma _{1}^{2}\\rbrace }{\\max \\lbrace \\gamma _{3}^{2}, \\gamma _{2}^{2} + x^2\\rbrace - \\min \\lbrace \\gamma _{3}^{2}, \\gamma _{1}^{2}\\rbrace + \\gamma ^{2}_{1} - \\gamma ^{2}_{2}}} ,$ where $\\gamma _{1}\\!=\\!2(\\rho _{32}+\\rho _{41})$ , $\\gamma _{2}\\!=\\!2(\\rho _{32}-\\rho _{41}$ , $\\gamma _{3}\\!=\\!1-2(\\rho _{22}+\\rho _{33})$ , and $x\\!=\\!2 (\\rho _{11}+\\rho _{22})- 1$ .",
"Then, for the state in Eq.",
"(REF ), one gets $\\mathcal {D}(\\rho _{\\text{w}}) = \\frac{\\varepsilon }{2} , ~~ \\mathcal {C}(\\rho _{\\text{w}}) = \\max \\left[ 0 , \\frac{3\\varepsilon -1}{2} \\right] .",
"$ As we shall see, both quantum discord and entanglement of the passive state depend on the Hamiltonian of the system.",
"In general, the passive state of a system depends on the reference Hamiltonian used to set the system ergotropy.",
"In order to illustrate this situation, we consider here two different scenarios: the Ising and Heisenberg models.",
"As we shall see, while the energy eigenstates of the Ising Hamiltonian do not present entanglement, quantum correlations are observed for the energy eigenstates of the Heisenberg Hamiltonian."
],
[
"Ising Hamiltonian",
"First, we focus on the Ising model, whose reference Hamiltonian reads as $H_{Is}=H_{0}+H^{\\mathrm {Is}}_{int},$ with $H_{0}=\\sum _{i=1,2}\\hbar \\omega \\sigma ^{i}_{z}$ and $H^{\\mathrm {Is}}_{int}=J\\hbar \\sigma ^{1}_{z}\\sigma ^{2}_{z}$ being the free and interaction Hamiltonians of the coupled qubit system, respectively.",
"Where $\\sigma ^{i}_{z}~(i=1,2)$ belongs to the set of standard Pauli matrices $\\sigma _{j}$ with $j\\in \\lbrace x,~y,~z\\rbrace $ for each subsystem and, $J$ is the strength of two-body interaction.",
"The eigenenergies can be found by the direct diagonalization of the reference Hamiltonian $H_{Is}$ in following set $\\text{spectrum} = \\left\\lbrace -J,~-J,~J-2\\omega ,~J+2\\omega \\right\\rbrace ,$ in which the ordering depends on the values of $J$ and $\\omega $ , associated with eigenstates ${\\downarrow \\uparrow },~{\\uparrow \\downarrow },~{\\downarrow \\downarrow },~{\\uparrow \\uparrow }$ , respectively.",
"Here, ${\\downarrow }$ is the ground state and ${\\uparrow }$ is the excited state of a single qubit.",
"Consequently, we can be considered two feasible situations $2\\omega \\ge J$ and $2\\omega < J$ , to achieve an increasing order of energy levels.",
"In the former case, one can obtain $\\mathcal {E}(\\rho _{w})=2\\varepsilon \\omega $ , where the passive state is given by $\\varrho _{\\rho _{\\text{w}}}^{\\text{Is}}(\\omega \\!\\ge \\!J) = \\frac{1-\\varepsilon }{4} \\mathbb {1} + \\varepsilon {\\downarrow \\downarrow }{\\downarrow \\downarrow } ,$ and for the latter, we have $\\mathcal {E}(\\rho _{w})=2\\varepsilon J$ with the corresponding passive state $\\varrho _{\\rho _{\\text{w}}}^{\\text{Is}}(2\\omega \\!<\\!J) = \\frac{1-\\varepsilon }{4} \\mathbb {1} + \\varepsilon {\\downarrow \\uparrow }{\\downarrow \\uparrow } .$ Therefore, the maximum charge of the battery fulfill the following form $\\mathcal {E}^{\\text{Is}}(\\rho _{\\text{w}}) = \\mathcal {E}^{\\text{Is}}_{0} \\varepsilon , ~~ \\mathcal {E}^{\\text{Is}}_{0} = \\left\\lbrace \\begin{matrix}2 \\omega \\hbar & \\omega \\ge J\\\\2 J\\hbar & \\omega < J\\end{matrix} \\right.",
",$ for any desired value $\\varepsilon $ .",
"By comparing Eqs.",
"(REF ) and (REF ), we see that the ergotropy can be written in terms of the quantum discord as $\\mathcal {E}^{\\text{Is}}(\\rho _{\\text{w}}) = 2 \\mathcal {D}(\\rho _{\\text{w}}) \\mathcal {E}^{\\text{Is}}_{0}$ .",
"In addition, we can see that $\\mathcal {C}(\\rho _{\\text{w}})\\!=\\!0$ and $\\mathcal {E}(\\rho _{\\text{w}})\\!\\ne \\!0$ for $0\\!\\le \\!\\varepsilon \\!\\le \\!1/3$ .",
"This proves that, in general, entanglement is not an accessible resource for maximum work extraction compared to quantum discord in such a model.",
"On the other hand, it is straightforward to investigate that $\\mathcal {C}(\\varrho _{\\rho _{\\text{w}}}^{\\text{Is}})$ and $\\mathcal {D}(\\varrho _{\\rho _{\\text{w}}}^{\\text{Is}})$ are zero for both $2\\omega \\!<\\!J$ and $\\omega \\!\\ge \\!J$ .",
"So it can be argued that ergotropy is entirely stored in quantum discord.",
"According to Eq.",
"(REF ), after the ergotropy extraction procedure, we have an amount of energy that cannot be extracted through a coherent interaction with an external field, which leads to a unitary process.",
"Such residual amount of energy can be obtained for the Ising model as follow $\\Sigma ^{\\varrho _{\\rho _{0}}\\rightarrow \\rho _{\\beta }} = \\Sigma _{0} \\varepsilon + \\beta ^{-1}C , ~~ \\Sigma _{0} = \\left\\lbrace \\begin{matrix}(J-2 \\omega )\\hbar & \\omega \\ge J\\\\-J\\hbar & \\omega < J\\end{matrix} \\right.",
",$ where $C_{\\text{Is}}\\!=\\!\\text{ln}(Z_{\\text{Is}})-S(\\varrho _{\\rho _{\\text{w}}}^{\\text{Is}})$ , with $Z_{\\text{Is}}\\!=\\!2(e^{\\beta J}+e^{-\\beta J}\\cosh [2\\beta \\omega ])$ the partition function of thermal state $\\rho _{\\beta }$ and $S(\\varrho _{\\rho _{\\text{w}}}^{\\text{Is}})$ the entropy of the passive state given by $S(\\varrho _{\\rho _{\\text{w}}}^{\\text{Is}})=2 - \\left[\\frac{3(1-\\varepsilon )}{4}\\log _{2}(1-\\varepsilon )+\\frac{(1+3\\varepsilon )}{4}\\log _{2}(1+3\\varepsilon )\\right] .",
"$ It is important to mention that, besides the exergy depends on the parameter $\\varepsilon $ , it is not associated with any amount of correlation, since the passive state $\\varrho _{\\rho _{\\text{w}}}^{\\text{Is}}$ , does not present any quantum correlation.",
"Then, exergy of passive states for the Ising Hamiltonian is not stored as correlations."
],
[
"Heisenberg Hamiltonian",
"The Heisenberg Hamiltonian describes a system where the interaction part is given by $H_{\\text{int}}^{\\text{H}} = \\frac{J\\hbar }{\\sqrt{2}} \\left( \\sigma _{x}^{1} \\sigma _{x}^{2} + \\sigma _{y}^{1} \\sigma _{y}^{2} \\right) ,$ where factor $\\sqrt{2}$ is considered here to give a fair comparison with the Ising model, so that $||H_{\\text{int}}^{\\text{H}}||\\!=\\!||H_{\\text{int}}^{\\text{H}}||$ , being $||A||\\!=\\!",
"(\\text{tr} ( A A^{\\dagger } ) )^{1/2}$ the Hilbert-Schmidt norm of the operator $A$ .",
"This quantity allows for quantifying the thermodynamic cost of implementing the dynamics driven by an arbitrary Hamiltonian [22].",
"By computing the spectrum of the reference Hamiltonian $H_{\\text{H}}\\!=\\!",
"H_{0} + H_{\\text{int}}^{\\text{H}}$ , we find $\\text{spectrum} = \\left\\lbrace -\\sqrt{2} J, -2 \\omega , \\sqrt{2} J, 2 \\omega \\right\\rbrace ,$ with eigenstates $({\\downarrow \\uparrow }-{\\uparrow \\downarrow })/\\sqrt{2}$ , ${\\downarrow \\downarrow }$ , $({\\downarrow \\uparrow }+{\\uparrow \\downarrow })/\\sqrt{2}$ and ${\\uparrow \\uparrow }$ , respectively.",
"We stress that, for this case, whenever we have $|J|\\!>\\!\\!\\sqrt{2}|\\omega |$ , the ground state of the system is a maximally entangled state.",
"After a detailed analysis, it is possible to show that the ergotropy can be adequately written as $\\mathcal {E}^{\\text{H}}(\\rho _{\\text{w}}) = \\mathcal {E}^{\\text{H}}_{0} \\varepsilon , ~~ \\mathcal {E}^{\\text{H}}_{0} = \\left\\lbrace \\begin{matrix}2 \\omega \\hbar & \\omega \\ge J/\\sqrt{2}\\\\\\sqrt{2} J\\hbar & \\omega < J/\\sqrt{2}\\end{matrix} \\right.",
",$ Similar to the Ising case, given an arbitrary value for $\\varepsilon $ , it is not possible to write $\\mathcal {E}(\\rho _{\\text{w}})\\!\\propto \\!\\mathcal {C}(\\rho _{\\text{w}})$ , however, we can write $\\mathcal {E}^{\\text{H}}(\\rho _{\\text{w}})\\!=\\!2 \\mathcal {D}(\\rho _{\\text{w}}) \\mathcal {E}^{\\text{H}}_{0}$ .",
"Therefore, we conclude that the ergotropy stored in the Werner state is not stored as entanglement, but ergotropy is stored quantum discord.",
"Then, given recent results on discord-based ergotropy [23], we understand that entanglement is not the main quantum resource of quantum batteries in general and confirms the results presented in [11].",
"Then, after ergotropy extraction, the passive state for the Heisenberg Hamiltonian reads as $\\varrho _{\\rho _{\\text{w}}}^{\\text{H}}(\\omega \\!\\ge \\!J/\\sqrt{2}) = \\frac{1-\\varepsilon }{4} \\mathbb {1} + \\varepsilon {\\downarrow \\downarrow }{\\downarrow \\downarrow } ,$ for the regime where $\\omega \\!\\ge \\!J/\\sqrt{2}$ .",
"In this case, we can see the amount of energy $\\mathcal {E}(\\rho _{\\text{w}})\\!=\\!2\\varepsilon \\hbar \\omega $ is extracted from the system so that the passive state has no amount of correlation.",
"In fact, from concurrence and, discord we find $\\mathcal {C}(\\varrho _{\\rho _{\\text{w}}}^{\\text{H}})\\!=\\!\\mathcal {D}(\\varrho _{\\rho _{\\text{w}}}^{\\text{H}})\\!=\\!0$ .",
"Therefore, the maximum work extraction comes with fully “correlation extraction\".",
"On the other hand, by considering the case with $\\omega \\!<\\!J/\\sqrt{2}$ we get the passive state $\\varrho _{\\rho _{\\text{w}}}^{\\text{H}}(\\omega \\!<\\!J/\\sqrt{2}) = \\frac{1-\\varepsilon }{4} \\mathbb {1} + \\varepsilon {\\beta _{-}}{\\beta _{-}} ,$ where ${\\beta _{-}}\\!=\\!",
"({\\uparrow \\downarrow }-{\\downarrow \\uparrow })/\\sqrt{2}$ is the singlet-state of two-qubit, one of the Bell states.",
"It is worth highlighting this case for a particular motivation: the ergotropy extraction can be done without changing the amount of correlation in the system.",
"To the best of our knowledge, such a result was not observed so far and it has an implication different from our previous discussion.",
"Because the states $\\varrho _{\\rho _{\\text{w}}}^{\\text{H}}(\\omega \\!<\\!J/\\sqrt{2})$ and $\\rho _{\\text{w}}$ have the same amount of correlation, since we can obtain $\\rho _{\\text{w}}$ up to local rotations in $\\varrho _{\\rho _{\\text{w}}}^{\\text{H}}(\\omega \\!<\\!J/\\sqrt{2})$ , we understand that the ergotropy of the system with Heisenberg Hamiltonian, where $\\omega \\!<\\!J/\\sqrt{2}$ , is not stored as correlations.",
"It means that by increasing the interaction strength of the system, the ergotropy of Werner states can be extracted from the system without destroying correlations in the system, leading then to a Werner passive state.",
"According to the final interpretations of the previous section, it turns out that the passive state exergy can be a physical justification for such an event.",
"To clarify this idea underlying the waste energy, we find the exergy of passive state in the following form $\\Sigma ^{\\varrho _{\\rho _{0}}\\rightarrow \\rho _{\\beta }} = \\Sigma _{0} \\varepsilon + \\beta ^{-1}C_{\\text{H}} , ~~ \\Sigma _{0} = \\left\\lbrace \\begin{matrix}-2 \\omega \\hbar & \\omega \\ge \\frac{J}{\\sqrt{2}}\\\\-\\sqrt{2}J\\hbar & \\omega < \\frac{J}{\\sqrt{2}}\\end{matrix} \\right.",
",$ with $C_{\\text{H}}\\!=\\!\\text{ln}(Z_{\\text{H}})-S(\\varrho _{\\rho _{\\text{w}}}^{\\text{H}})$ , with $Z_{\\text{H}}\\!=\\!2(\\cosh [\\sqrt{2}\\beta J]+\\cosh [2\\beta \\omega ])$ and $S(\\varrho _{\\rho _{\\text{w}}}^{\\text{H}})$ given by Eq.",
"(REF ), since the passive state $\\varrho _{\\rho _{\\text{w}}}^{\\text{H}}$ can be obtained from $\\varrho _{\\rho _{\\text{w}}}^{\\text{Is}}$ through unitary operations for any $J$ and $\\omega $ .",
"As the main result, it is possible to see that part of the exergy is stored as discord, since we can rewrite the above equation as $\\Sigma ^{\\varrho _{\\rho _{0}}\\rightarrow \\rho _{\\beta }} = 2\\Sigma _{0}\\mathcal {D}(\\rho _{\\text{w}}) + \\beta ^{-1}C_{\\text{H}}$ ."
],
[
"conclusion",
"This paper dealt with the loss of energy from a (cyclic) unitary work extraction from a quantum system, where such an amount of energy is identified as the exergy of the quantum passive state.",
"We showed that, in general, in a real scenario the ergotropy leads to loss of energy due to the limitation of the unitary process.",
"In addition, given the system-bath interaction that leads to the thermalization process, we discussed the existence and uniqueness of an optimal passive state for ergotropy and exergy extraction.",
"From the point of view of the second thermodynamics law, we explain our main result as a natural consequence of the entropy production of the thermalization process for exergy extraction.",
"As an application of our results, it is possible to identify a family of ergotropy and exergy extraction where the total amount of quantum correlations (as quantified by the quantum discord) of the system is conserved.",
"Then, it implies that the exergy of a quantum passive state can be stored as quantum correlations.",
"Since exergy is the amount of energy extractable through a thermalization process, a new prospect is opening up for exploring protocols of operational open quantum batteries.",
"This work has been supported by the University of Kurdistan.",
"F. H. Kamin and S. Salimi thank Vice Chancellorship of Research and Technology, University of Kurdistan.",
"A.C.S.",
"acknowledges the financial support of the São Paulo Research Foundation (FAPESP) (Grant No.",
"2019/22685-1)."
]
] | 2107.01828 |
[
[
"Rates of Estimation of Optimal Transport Maps using Plug-in Estimators\n via Barycentric Projections"
],
[
"Abstract Optimal transport maps between two probability distributions $\\mu$ and $\\nu$ on $\\mathbb{R}^d$ have found extensive applications in both machine learning and statistics.",
"In practice, these maps need to be estimated from data sampled according to $\\mu$ and $\\nu$.",
"Plug-in estimators are perhaps most popular in estimating transport maps in the field of computational optimal transport.",
"In this paper, we provide a comprehensive analysis of the rates of convergences for general plug-in estimators defined via barycentric projections.",
"Our main contribution is a new stability estimate for barycentric projections which proceeds under minimal smoothness assumptions and can be used to analyze general plug-in estimators.",
"We illustrate the usefulness of this stability estimate by first providing rates of convergence for the natural discrete-discrete and semi-discrete estimators of optimal transport maps.",
"We then use the same stability estimate to show that, under additional smoothness assumptions of Besov type or Sobolev type, wavelet based or kernel smoothed plug-in estimators respectively speed up the rates of convergence and significantly mitigate the curse of dimensionality suffered by the natural discrete-discrete/semi-discrete estimators.",
"As a by-product of our analysis, we also obtain faster rates of convergence for plug-in estimators of $W_2(\\mu,\\nu)$, the Wasserstein distance between $\\mu$ and $\\nu$, under the aforementioned smoothness assumptions, thereby complementing recent results in Chizat et al.",
"(2020).",
"Finally, we illustrate the applicability of our results in obtaining rates of convergence for Wasserstein barycenters between two probability distributions and obtaining asymptotic detection thresholds for some recent optimal-transport based tests of independence."
],
[
"Introduction",
"Given two random variables $X\\sim \\mu $ and $Y\\sim \\nu $ , where $\\mu ,\\nu $ are probability measures on $\\mathbb {R}^d$ , $d\\ge 1$ , the problem of finding a “nice\" map $T_0(\\cdot )$ such that $T_0(X)\\sim \\nu $ has numerous applications in machine learning such as domain adaptation and data integration [67], [54], [38], [37], [41], [122], dimension reduction [72], [13], [98], generative models [66], [89], [96], [120], to name a few.",
"Of particular interest is the case when $T_0(\\cdot )$ is obtained by minimizing a cost function, a line of work initiated by Gaspard Monge [106] in 1781 (see (REF ) below), in which case $T_0(\\cdot )$ is termed an optimal transport (OT) map and has applications in shape matching/transfer problems [52], [131], [32], [117], Bayesian statistics [118], [51], [83], [88], econometrics [60], [16], [31], [56], [50], nonparametric statistical inference [44], [123], [124], [43], [42]; also see [139], [140], [121] for book-length treatments on the subject.",
"In this paper, we will focus on the OT map obtained using the standard Euclidean cost function, i.e., $T_0:=\\operatornamewithlimits{\\arg \\!\\min }\\limits _{T:T\\#\\mu =\\nu }\\mathbb {E}\\Vert X-T(X)\\Vert ^2,$ where $T\\#\\mu =\\nu $ means $T(X)\\sim \\nu $ for $X\\sim \\mu $ .",
"The estimation of $T_0$ has attracted a lot of interest in recent years due to its myriad applications (as stated above) and interesting geometrical properties (see [100], [62], [21] and def:otmpm below).",
"In practice, the main hurdle in constructing estimators for $T_0$ is that the explicit forms of the measures $\\mu ,\\nu $ are unknown; instead only random samples $X_1,\\ldots ,X_m\\sim \\mu \\qquad \\quad \\mbox{and} \\qquad \\quad Y_1,\\ldots ,Y_n\\sim \\nu $ are available.",
"A natural strategy in this scenario is to estimate $T_0$ using $\\widetilde{T}_{m,n}$ , where $\\widetilde{T}_{m,n}$ is computed as in (REF ) with $\\mu $ and $\\nu $ replaced by $\\widetilde{\\mu }_m$ and $\\widetilde{\\nu }_n$ which are empirical approximations of $\\mu $ and $\\nu $ based on $X_1,\\ldots ,X_m$ and $Y_1,\\ldots ,Y_n$ respectively (see def:bcenterproj).",
"Such estimators are often called plug-in estimators and have been used extensively, see [126], [102], [103], [8], [111], [73], [33].",
"The main goal of this paper is to study the rates of convergence of general plug-in estimators of $T_0$ under a unified framework.",
"We show that when $\\widetilde{\\mu }_m$ and $\\widetilde{\\nu }_n$ are chosen as $\\widehat{\\mu }_m$ and $\\widehat{\\nu }_n$ respectively, where $\\widehat{\\mu }_m$ and $\\widehat{\\nu }_n$ are the standard empirical distributions supported on $m$ and $n$ atoms, i.e., $\\widehat{\\mu }_m:=\\frac{1}{m}\\sum _{i=1}^m \\delta _{X_i}\\qquad \\quad \\mbox{and} \\qquad \\quad \\widehat{\\nu }_n:=\\frac{1}{n}\\sum _{j=1}^n \\delta _{Y_j},$ $\\widetilde{T}_{m,n}$ (appropriately defined using def:bcenterproj) converges at a rate of $m^{-2/d}+n^{-2/d}$ for $d\\ge 4$ .",
"This rate happens to be minimax optimal under minimal smoothness assumptions (see [80]) but suffers from the curse of dimensionality.",
"We next show that, if $\\mu $ and $\\nu $ are known to admit sufficiently smooth densities, it is possible to apply wavelet or kernel based smoothing techniques on $\\widehat{\\mu }_m$ and $\\widehat{\\nu }_n$ to obtain plug-in estimators that mitigate the aforementioned curse of dimensionality.",
"Our next contribution pertains to the estimation of $W_2^2(\\mu ,\\nu )$ (the squared Wasserstein distance), see (REF ) below, a quantity of independent interest in statistics and machine learning with applications in structured prediction [57], [97], image analysis [65], [19], nonparametric testing [17], [116], generative modeling [105], [11], etc.",
"In this paper, we also obtain rates of convergence for plug-in estimators $W_2^2(\\widetilde{\\mu }_m,\\widetilde{\\nu }_n)$ of $W_2^2(\\mu ,\\nu )$ .",
"We show that kernel smoothing $\\widehat{\\mu }_m$ and $\\widehat{\\nu }_n$ can be used to obtain plug-in estimators of $W_2^2(\\mu ,\\nu )$ that mitigate the curse of dimensionality as opposed to a direct plug-in approach using $\\widehat{\\mu }_m$ and $\\widehat{\\nu }_n$ (as used in [33]).",
"This provides an answer to the open question of estimating $W_2^2(\\mu ,\\nu )$ when $\\mu $ , $\\nu $ admit smooth densities laid out in [33].",
"In this section, we present some basic concepts and results associated with the OT problem that will play a crucial role in the sequel.",
"Let $\\mathcal {P}_{\\mathrm {ac}}(\\mathbb {R}^d)$ denote the set of all Lebesgue absolutely continuous probability measures on $\\mathbb {R}^d$ and $\\mathcal {P}_2(\\mathbb {R}^d)$ be the set of probability measures with finite second moments.",
"Then the 2-Wasserstein distance (squared) between $\\mu ,\\nu \\in \\mathcal {P}_2(\\mathbb {R}^d)$ is defined as: $W_2^2(\\mu ,\\nu ):=\\min \\limits _{\\pi \\in \\Pi (\\mu ,\\nu )}\\int \\Vert x-y\\Vert ^2\\,d\\pi (x,y),$ where $\\Pi (\\mu ,\\nu )$ is the set of probability measures on $\\mathbb {R}^d\\times \\mathbb {R}^d$ with marginals $\\mu $ and $\\nu $ .",
"The optimization problem in (REF ) is often called the Kantorovich relaxation (see [84], [85]) of the optimization problem in (REF ).",
"The existence of a minimizer in (REF ) follows from [140].",
"Proposition 1.1 (Brenier-McCann polar factorization theorem, see [139], [100]) Suppose $\\mu ,\\nu \\in \\mathcal {P}_{\\mathrm {ac}}(\\mathbb {R}^d)$ .",
"Then there exists a $\\mu $ -a.e.",
"(almost everywhere) unique function $T_0(\\cdot ):\\mathbb {R}^d\\rightarrow \\mathbb {R}^d$ , which is the gradient of a real-valued $d$ -variate convex function, say $\\varphi _0(\\cdot ):\\mathbb {R}^d\\rightarrow \\mathbb {R}$ , such that $T_0\\#\\mu =\\nu $ .",
"Further, the distribution defined as $\\pi (A\\times B)=\\mu (A\\cap (T_0)^{-1}(B))$ for all Borel sets $A,B\\subseteq \\mathbb {R}^d$ is the unique minimizer in (REF ) provided $\\mu ,\\nu \\in \\mathcal {P}_{\\mathrm {ac}}(\\mathbb {R}^d)\\cap \\mathcal {P}_2(\\mathbb {R}^d)$ .",
"[OT map and potential function] The function $T_0:\\mathbb {R}^d\\rightarrow \\mathbb {R}^d$ in prop:bmopt which satisfies $T_0\\#\\mu =\\nu $ will be called the OT map from $\\mu $ to $\\nu $ .",
"The convex function $\\varphi _0(\\cdot )$ in prop:bmopt satisfying $\\nabla \\varphi _0=T_0$ will be termed the OT potential.",
"The next and final important ingredient is the alternate dual representation of (REF ) which gives: $\\frac{1}{2}W_2^2(\\mu ,\\nu )&=\\frac{1}{2}\\int \\Vert x\\Vert ^2\\,d\\mu (x)+\\frac{1}{2}\\int \\Vert y\\Vert ^2\\,d\\nu (y)-\\min _{f\\in \\mathcal {F}} \\mathcal {S}_{\\mu ,\\nu }(f),\\,\\qquad \\mbox{where} \\\\ \\mathcal {S}_{\\mu ,\\nu }(f)&=\\int f\\,d\\mu +\\int f^*\\,d\\nu .$ Here $\\mathcal {F}$ denotes the space of convex functions on $\\mathbb {R}^d$ which are also elements of $L^1(\\mu )$ and $f^*(\\cdot )$ is the standard Legendre-Fenchel dual defined as: $f^*(x) := \\sup _{y \\in \\mathbb {R}^d} [y^\\top x - f(y)], \\qquad \\mbox{for } x \\in \\mbox{dom}(f).$"
],
[
"Estimating OT map via barycentric projection",
"Recall the setting from the Introduction.",
"Let $\\widetilde{\\mu }_m,\\widetilde{\\nu }_n\\in \\mathcal {P}_2(\\mathbb {R}^d)$ .",
"Here $\\widetilde{\\mu }_m,\\widetilde{\\nu }_n$ need not be absolutely continuous and can be very general.",
"Intuitively, $\\widetilde{\\mu }_m$ and $\\widetilde{\\nu }_n$ should be viewed as some empirical approximation of $\\mu $ and $\\nu $ respectively.",
"[Simple choices of $\\widetilde{\\mu }_m$ and $\\widetilde{\\nu }_n$ ] Let $X_1,\\ldots ,X_m\\overset{i.i.d.",
"}{\\sim }\\mu $ and $Y_1,\\ldots ,Y_n\\overset{i.i.d.",
"}{\\sim }\\nu $ ; in which case a natural choice would be to set $\\widetilde{\\mu }_m=\\widehat{\\mu }_m$ and $\\widetilde{\\nu }_n=\\widehat{\\nu }_n$ where $\\widehat{\\mu }_m$ and $\\widehat{\\nu }_n$ are the empirical distributions on $X_1,\\ldots ,X_m$ and $Y_1,\\ldots ,Y_n$ respectively, as defined in (REF ).",
"This is the standard choice adopted in the discrete-discrete Kantorovich relaxation; see [113].",
"Another popular choice is $\\widetilde{\\mu }_m=\\widehat{\\mu }_m$ , $\\widetilde{\\nu }_n=\\nu $ or $\\widetilde{\\mu }_m=\\mu $ , $\\widetilde{\\nu }_n=\\widehat{\\nu }_n$ .",
"This is the semi-discrete Kantorovich problem and is popular when one of the measures is fully specified; see [29], [61].",
"A natural way to estimate $T_0(\\cdot )$ , as defined in (REF ), would be to approximate it using the OT map from $\\widetilde{\\mu }_m$ to $\\widetilde{\\nu }_n$ .",
"However as $\\widetilde{\\mu }_m$ and $\\widetilde{\\nu }_n$ may not be elements of $\\mathcal {P}_{\\mathrm {ac}}(\\mathbb {R}^d)$ , prop:bmopt does not apply and an OT map may not exist from $\\widetilde{\\mu }_m$ to $\\widetilde{\\nu }_n$ .",
"Such is the case in ex:baseplug in the discrete-discrete case when $m\\ne n$ .",
"To circumvent this issue, we leverage the notion of barycentric projections (see [4]) defined below: [Barycentric projection] Define the set $\\widetilde{\\Gamma }_{\\mathrm {min}}:=\\operatornamewithlimits{\\arg \\!\\min }\\limits _{\\pi \\in \\Pi (\\widetilde{\\mu }_m,\\widetilde{\\nu }_n)} \\int \\Vert x-y\\Vert ^2\\,d\\pi (x,y).$ The optimization problem above is the plug-in analog of the optimization problem on the right hand side of (REF ).",
"Given any $\\gamma \\in \\widetilde{\\Gamma }_{\\mathrm {min}}$ , define the barycentric projection of $\\gamma $ as the conditional mean of $y$ given $x$ under $\\gamma $ , i.e., $\\widetilde{T}_{m,n}(x)\\equiv \\widetilde{T}_{m,n}^{\\gamma }(x):=\\frac{\\int _{y} y\\,d\\gamma (x,y)}{\\int _{y} d\\gamma (x,y)},\\qquad \\mbox{for}\\ x\\in \\mbox{supp}\\left({\\widetilde{\\mu }_m}\\right).$ In general, $\\widetilde{\\Gamma }_{\\mathrm {min}}$ need not be a singleton which is why we index the barycentric projection $\\widetilde{T}_{m,n}^{\\gamma }(\\cdot )$ by $\\gamma \\in \\widetilde{\\Gamma }_{\\mathrm {min}}$ .",
"Note that $\\widetilde{T}_{m,n}^{\\gamma }(\\cdot )$ need not be a transport map; however, if an OT map exists then it must be equal to $\\widetilde{T}_{m,n}^{\\gamma }(\\cdot )$ ($\\widetilde{\\mu }_m$ -a.e.).",
"Our goal is to obtain stochastic upper bounds for $\\sup \\limits _{\\gamma \\in \\widetilde{\\Gamma }_{\\mathrm {min}}} \\int \\big \\Vert \\widetilde{T}_{m,n}^{\\gamma }(x)-T_0(x)\\big \\Vert ^2\\,d\\widetilde{\\mu }_m(x).$ In addition, our proof techniques also yield rates of convergence for $\\big |W_2^2(\\widetilde{\\mu }_m,\\widetilde{\\nu }_n)-W_2^2(\\mu ,\\nu )\\big |.$ In this paper, we will focus on $d\\ge 2$ .",
"Due to the canonical ordering of $\\mathbb {R}$ , the case $d=1$ can be handled easily using the classical Hungarian embedding theorem [90]."
],
[
"Contributions",
" We provide a new and flexible stability estimate thm:newubd which yields a unified approach to obtaining rates of convergence for general plug-in estimators of the OT map $T_0(\\cdot )$ .",
"Unlike existing stability estimates, thm:newubd holds for the barycentric projection (which is the same as the OT map when it exists) and does not require any smoothness assumptions on $\\widetilde{\\mu }_m$ , $\\widetilde{\\nu }_n$ or $\\widetilde{T}_{m,n}^{\\gamma }(\\cdot )$ ; also see rem:comstab for a comparison with the existing literature.",
"in Sections REF and REF , we use thm:newubd to bound (REF ) and (REF ): In sec:nsmooth, we show that in both the discrete-discrete and semi-discrete Kantorovich relaxation problems (see ex:baseplug), the rate of convergence of (REF ) is $m^{-2/d}+n^{-2/d}$ for $d\\ge 4$ when $T_0$ is assumed to be Lipschitz (see thm:nsmooth), which is the minimax rate (see [80]).",
"To the best of our knowledge, rates of convergence for these natural estimators weren't previously established in the literature.",
"In sec:smooth, we show that the curse of dimensionality in the above rates can be mitigated provided $\\mu $ and $\\nu $ admit Besov smooth densities (see sec:wavelet) or (uniform) Sobolev smooth densities (see sec:kernel).",
"In sec:wavelet, our plug-in estimator is obtained using natural wavelet based density estimators.",
"The rate of convergence in (REF ) turns out to be $n^{-\\frac{1+s}{d+2s}}$ where $s$ denotes the degree of Besov smoothness (see thm:smoothwav).",
"Note that by choosing $s$ large enough, the exponent in the rate can be made arbitrarily close to $1/2$ , thereby reducing the curse of dimensionality.",
"In sec:kernel, our plug-in estimator is obtained by choosing $\\widetilde{\\mu }_m$ (and $\\widetilde{\\nu }_n$ ) as the convolution of $\\widehat{\\mu }_m$ (and $\\widehat{\\nu }_n$ ) and a smooth kernel with an appropriate bandwidth.",
"Under this choice, the rate of convergence in (REF ) is $m^{-\\left(\\frac{s+2}{d}\\wedge \\frac{1}{2}\\right)}+n^{-\\left(\\frac{s+2}{d}\\wedge \\frac{1}{2}\\right)}$ , where $s$ denotes the degree of Sobolev smoothness (see thm:smooth).",
"Clearly, if $2(s+2)\\ge d$ , the rate of convergence becomes dimension-free and mitigates the curse of dimensionality.",
"We also show the same rates of convergence mentioned above also hold for (REF ) (see e.g., prop:wassrate) which makes a strong case in favor of incorporating smoothness in the construction of plug-in estimators as was conjectured in [33].",
"In sec:dismooth, we use a discretization technique from [143] to construct discrete approximations to the smoothed $\\widetilde{\\mu }_m$ and $\\widetilde{\\nu }_n$ from the previous paragraph that in turn yield computable plug-in estimators for $T_0$ (provided one can sample from $\\widetilde{\\mu }_m$ and $\\widetilde{\\nu }_n$ ) that also achieve the same statistical guarantees as the smoothed plug-in estimator from sec:smooth (see thm:dismooth).",
"However the number of atoms required in the discretizations and correspondingly the computational complexity increases with the degree of smoothness; this highlights a statistical and computational trade off.",
"We provide implications of our results in popular applications of OT such as estimating the barycenter of two multivariate probability distributions (see thm:barate in sec:bar) and in nonparametric independence testing (see thm:dethresh in sec:nonptest)."
],
[
"Related work",
"Many recent works have focused on obtaining consistent estimators of $T_0$ using the plug-in principle, see [29], [61] (in the semi-discrete problem) and [75], [144], [44] (in the discrete-discrete problem).",
"In [61], the authors have studied the rate of convergence of the semi-discrete optimal transport map from $\\nu $ (absolutely continuous) to $\\widehat{\\mu }_m$ .",
"This paper complements the aforementioned papers by studying the rates of convergence for general plug-in estimators in a unified fashion.",
"In two other papers [10] and [95], the authors use a “Voronoi tessellation\" approach to estimate $T_0$ , however the rates obtained in this paper, even in the absence of smoothness, are strictly better than those in [10], [95].",
"Perhaps the most closely related paper to ours would be [73].",
"In [73], the author uses variational techniques to arrive at stability estimates while we exploit the Lipschitz nature of the OT map (see def:otmpm).",
"Further the rates in this paper have exponents $\\frac{s+2}{d}\\wedge \\frac{1}{2}$ which are strictly better than the exponents $\\frac{s+2}{2(s+2)+d}$ obtained in [73] under the same smoothness assumptions (Sobolev type of order $s$ , see def:holder).",
"In another line of work [80], the authors use theoretical wavelet based estimators (not of the plug-in type) of $T_0$ to obtain nearly minimax optimal rates of convergence.",
"However these estimators, by themselves, are not transport maps between two probability measures, which makes them harder to interpret.",
"In contrast, our focus is on obtaining rates of convergence for plug-in estimators, which are transport maps between natural aprroximations of $\\mu $ and $\\nu $ .",
"Such plug-in type strategies are a lot more popular in computational OT [126], [102], [103], [8], [111], [73], [33].",
"In terms of obtaining rates of convergence for (REF ), some attempts include [126], [119] where parametric rates are obtained when $\\mu ,\\nu $ are known to be finitely supported or are both Gaussian.",
"In a related problem, bounds for $W_2^2(\\widehat{\\mu }_m,\\mu )$ were obtained in [133], [7], [46], [55], [109], [143].",
"Using these bounds, it is easy to get a $n^{-1/d}$ rate of convergence for (REF ).",
"This rate was recently improved to $n^{-2/d}$ in [33] under no smoothness assumptions.",
"Our rates coincide with the $n^{-2/d}$ rate from [33] under no smoothness assumptions.",
"But further, we show in this paper that the curse of dimensionality in the above rate can be mitigated by incorporating smoothness into the plug-in procedure."
],
[
"Main results",
"Recall the definition of $\\varphi _0(\\cdot )$ from def:otmpm.",
"The following is our main result.",
"Theorem 2.1 (Stability estimate) Suppose that $\\mu ,\\nu \\in \\mathcal {P}_{\\mathrm {ac}}(\\mathbb {R}^d)\\cap \\mathcal {P}_2(\\mathbb {R}^d)$ and $\\widetilde{\\mu }_m,\\widetilde{\\nu }_n\\in \\mathcal {P}_2(\\mathbb {R}^d)$ .",
"Assume that $T_0(\\cdot )$ (as defined in (REF )) is $L$ -Lipschitz ($L >0$ ).",
"Then, $\\sup \\limits _{\\gamma \\in \\widetilde{\\Gamma }_{\\mathrm {min}}} \\int \\Vert \\widetilde{T}_{m,n}^{\\gamma }(x)-T_0(x)\\Vert ^2& \\,d\\widetilde{\\mu }_m(x)\\le L \\max \\left\\lbrace \\bigg |\\int \\Psi _{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n}^* \\,d(\\widetilde{\\nu }_n-\\overline{\\nu }_m)\\bigg |,\\bigg |\\int \\Psi _{\\widetilde{\\mu }_m,\\overline{\\nu }_m}^* \\,d(\\widetilde{\\nu }_n-\\overline{\\nu }_m)\\bigg |\\right\\rbrace \\nonumber \\\\ &+2L\\int \\varphi _0^*(y)\\,d(\\widetilde{\\nu }_n-\\overline{\\nu }_m)(y),$ where $\\overline{\\nu }_m=T_0\\#\\widetilde{\\mu }_m$ , $\\varphi _0^*(\\cdot )$ is defined as in (REF ), and with $\\mathcal {S}_{\\cdot ,\\cdot }(\\cdot )$ defined as in (), $\\Psi _{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n}(\\cdot ):=\\operatornamewithlimits{\\arg \\!\\min }_{f\\in \\mathcal {F}} \\mathcal {S}_{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n}(f),\\quad \\Psi _{\\widetilde{\\mu }_m,\\overline{\\nu }_m}(\\cdot ):=\\operatornamewithlimits{\\arg \\!\\min }_{f\\in \\mathcal {F}} \\mathcal {S}_{\\widetilde{\\mu }_m,\\overline{\\nu }_m}(f)$ .",
"The proof of thm:newubd (see sec:mainrespf) starts along the same lines as the proof of the curvature estimate in [62].",
"This is followed by some careful manipulations of $W_2^2(\\cdot ,\\cdot )$ (as in (REF )) and an application of the conditional version of Jensen's inequality, see (REF ).",
"The final step of the proof uses the dual representation in (REF ) with techniques similar to some intermediate steps in the proof of [101] and [33].",
"Remark 2.1 (Comparison with other stability estimates) thm:newubd provides some important advantages to existing stability estimates in the literature.",
"One of the earliest results in this direction can be found in [62] but their bound involves a push-forward constraint which makes it hard to use for rate of convergence analysis.",
"A bound similar to thm:newubd is presented in [61] but there the authors assume the existence of an OT map from $\\widetilde{\\mu }_m$ to $\\widetilde{\\nu }_n$ .",
"Therefore, it does not apply to the discrete-discrete problem where $\\widetilde{\\mu }_m=\\widehat{\\mu }_m$ and $\\widetilde{\\nu }_n=\\widehat{\\nu }_n$ with $m\\ne n$ .",
"Overcoming all these limitations is an important contribution of thm:newubd and allows us to deal with popular plug-in estimators all in one go.",
"The stability estimate in [80] on the other hand requires $\\widetilde{\\mu }_m$ , $\\widetilde{\\nu }_n$ to be sufficiently smooth and hence it does not hold for discrete-discrete or semi-discrete plug-in estimators (see ex:baseplug).",
"Further their result requires all the measures involved to be compactly supported unlike the much milder requirements of thm:newubd.",
"However, a shortcoming of thm:newubd is that it is hard to obtain rates faster than $n^{-1/2}$ using it directly, whereas [80] can obtain rates arbitrarily close to $n^{-1}$ .",
"This is a price we pay for analyzing natural and popular plug-in estimators as opposed to the (more intractable) wavelet based estimators in [80].",
"Remark 2.2 (How to use thm:newubd to obtain rates of convergence?)",
"Note that the second term on the right hand side of (REF ), under appropriate moment assumptions, is $O_p(m^{-1/2}+n^{-1/2})$ (free of dimension) by a direct application of Markov's inequality.",
"We therefore focus on the first term.",
"By (), $\\Psi _{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n}^*(\\cdot )$ , $\\Psi _{\\widetilde{\\mu }_m,\\overline{\\nu }_m}^*(\\cdot )\\in \\mathcal {F}$ .",
"Further, by Caffarelli's regularity theory [23], [24], [25], depending on the “smoothness\" of $\\widetilde{\\mu }_m$ , $\\widetilde{\\nu }_n$ , it can be shown that there exists a further class of functions $\\mathcal {F}_s$ (see Remarks REF and REF ) such that $\\Psi _{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n}^*(\\cdot )$ , $\\Psi _{\\widetilde{\\mu }_m,\\overline{\\nu }_m}^*(\\cdot )\\in \\mathcal {F}\\cap \\mathcal {F}_s$ .",
"Thus, we can bound the first term on the right hand side of (REF ) as: $\\max \\left\\lbrace \\bigg |\\int \\Psi _{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n}^* \\,d(\\widetilde{\\nu }_n-\\overline{\\nu }_m)\\bigg |,\\bigg |\\int \\Psi _{\\widetilde{\\mu }_m,\\overline{\\nu }_m}^* \\,d(\\widetilde{\\nu }_n-\\overline{\\nu }_m)\\bigg |\\right\\rbrace \\le \\sup _{f\\in \\mathcal {F}\\cap \\mathcal {F}_s}\\bigg |\\int f\\,d(\\widetilde{\\nu }_n-\\overline{\\nu }_m)\\bigg |.$ The right hand side of (REF ) can now be bounded using the corresponding Dudley's entropy integral bounds using empirical process techniques, see [137].",
"To conclude, the two main steps in our strategy are identifying the family of functions $\\mathcal {F}_s$ and computing Dudley's entropy integral.",
"Further, the more the smoothness of $\\widetilde{\\mu }_m$ , $\\widetilde{\\nu }_n$ , the smaller is the class of functions $\\mathcal {F}_s$ and smaller the supremum on the right hand side of (REF ).",
"This shows why better rates can be expected under smoothness assumptions."
],
[
"Natural non-smooth plug-in estimator",
"In this case, we discuss the rates of convergence for the discrete-discrete problem and the semi-discrete problem, where no smoothness is available on $\\widetilde{\\mu }_m$ and $\\widetilde{\\nu }_n$ .",
"Theorem 2.2 Suppose that $T_0(\\cdot )$ is $L$ -Lipschitz, $\\nu $ is compactly supported and $\\mathbb {E}\\exp (t\\Vert X_1\\Vert ^{\\alpha })<\\infty $ for some $t>0$ , $\\alpha >0$ .",
"(Discrete-discrete): Set $\\widetilde{\\mu }_m=\\widehat{\\mu }_m$ and $\\widetilde{\\nu }_n=\\widehat{\\nu }_n$ .",
"Then the following holds: $\\sup \\limits _{\\gamma \\in \\widetilde{\\Gamma }_{\\mathrm {min}}} \\int \\Vert \\widetilde{T}_{m,n}^{\\gamma }(x)-T_0(x)\\Vert ^2\\,d\\widetilde{\\mu }_m(x)=O_p\\left(r^{(m,n)}_d\\times (\\log {(1+\\max \\lbrace m,n\\rbrace )})^{t_{d,\\alpha }}\\right),$ $\\mbox{where}\\quad r^{(m,n)}_d:={\\left\\lbrace \\begin{array}{ll} m^{-1/2}+n^{-1/2} & \\mbox{for}\\ d=2,3,\\\\ m^{-1/2}\\log {(1+m)}+n^{-1/2}\\log {(1+n)} & \\mbox{for}\\ d=4,\\\\ m^{-2/d}+n^{-2/d} & \\mbox{for}\\ d\\ge 5, \\end{array}\\right.",
"}$ and $t_{d,\\alpha }:={\\left\\lbrace \\begin{array}{ll} (4\\alpha )^{-1}(4+((2\\alpha +2d\\alpha -d)\\vee 0)) & \\mbox{for}\\ d<4,\\\\ (\\alpha ^{-1}\\vee 7/2)-1 & \\mbox{for}\\ d=4,\\\\ 2(1+d^{-1}) & \\mbox{for}\\ d>4.\\end{array}\\right.",
"}$ The same bound holds for $|W_2^2(\\widetilde{\\mu }_m,\\widetilde{\\nu }_n)-W_2^2(\\mu ,\\nu )|$ without assuming $T_0(\\cdot )$ is Lipschitz.",
"(Semi-discrete): Set $\\widetilde{\\mu }_m=\\mu $ , $\\widetilde{\\nu }_n=\\widehat{\\nu }_n$ or $\\widetilde{\\mu }_m=\\widehat{\\mu }_m$ , $\\widetilde{\\nu }_n=\\nu $ .",
"Then the left hand side of (REF ) is $O_p(r_d^{(n,n)}\\times (\\log {(1+n)})^{t_{d,\\alpha }})$ or $O_p(r_d^{(m,m)}\\times (\\log {(1+m)})^{t_{d,\\alpha }})$ respectively.",
"A stronger result can be proved if both $\\mu $ and $\\nu $ are compactly supported.",
"Corollary 2.3 Consider the setting from thm:nsmooth and assume further that $\\mu $ is compactly supported.",
"Then, with $r_d^{(m,n)}$ defined as in (REF ), we have: $\\mathbb {E}\\left[\\sup \\limits _{\\gamma \\in \\widetilde{\\Gamma }_{\\mathrm {min}}} \\int \\Vert \\widetilde{T}_{m,n}^{\\gamma }(x)-T_0(x)\\Vert ^2\\,d\\widetilde{\\mu }_m(x)\\right]\\le C r^{(m,n)}_d,$ for some constant $C>0$ , in both the discrete-discrete and semi-discrete settings from thm:nsmooth.",
"A brief description of the proof technique of thm:nsmooth using thm:newubd is provided in rem:pfnstech below, and the actual proof is presented in sec:mainrespf.",
"Remark 2.3 (Proof technique) The proof of thm:nsmooth proceeds using the strategy outlined in rem:pftech.",
"We first show that $\\mathcal {F}_s$ (see rem:pftech) can be chosen as a certain class of convex functions which are in $L^2(\\nu )$ .",
"We then use Dudley's entropy integral type bounds which in turn requires the bracketing entropy [137] of $\\mathcal {F}_s$ , recently proved in [91].",
"This strategy is slightly different from that used in the proof of [33], where the authors assume that $\\mu $ is compactly supported whereas we only assume the finiteness of $\\mathbb {E}\\exp (t\\Vert X_1\\Vert ^{\\alpha })$ for some $t>0$ , $\\alpha >0$ .",
"The compactness assumption on $\\mu $ allows one to further restrict $\\mathcal {F}_s$ to the class of Lipschitz functions.",
"This additional restriction does not seem to be immediate without the compactness assumption.",
"As discussed in sec:contrib, the exponents obtained in thm:nsmooth are minimax optimal under bare minimal smoothness assumptions (see [80]).",
"To the best of our knowledge, rates for the discrete-discrete case for $m\\ne n$ and those for the semi-discrete case were not known previously in the literature.",
"Our rates are also strictly better than those (for different estimators, based on space tessellations) obtained in [10], [95] and require less stringent assumptions than those in [33].",
"In the next section, we show how smoothness assumptions can be leveraged to mitigate the curse of dimensionality in thm:nsmooth."
],
[
"Smooth plug-in estimator: mitigating the curse of dimensionality",
"In this section, we focus on two types of plug-in estimators for the densities associated with the probability measures $\\mu $ and $\\nu $ : (a) wavelet based estimators (see [143], [135], [47], [86], [142]) in sec:wavelet, and (b) kernel based estimators (see [63], [64], [112], [125], [108]) in sec:kernel.",
"In both these cases, we will show, using thm:newubd, that the corresponding estimators of $T_0(\\cdot )$ achieve (near) dimension-free rates under sufficient smoothness assumptions."
],
[
"Wavelet based estimators",
"We begin this subsection by defining the Besov class of functions which will play a pivotal role in the sequel.",
"[Besov class of functions] We describe Besov classes following the notation from [143].",
"Suppose $s>0$ and let $n>s$ be a positive integer.",
"Given $\\Omega \\subseteq \\mathbb {R}^d$ , $h\\in \\mathbb {R}^d$ and $f(\\cdot ):\\mathbb {R}^d\\rightarrow \\mathbb {R}^d$ , set $\\Delta _h^1 f(x):=f(x+h)-f(x),$ $\\Delta _h^k f(x):= \\Delta _h^1\\left(\\Delta _h^{k-1} f\\right)(x), \\quad \\forall \\ 2\\le k\\le n,$ where these functions are defined on $\\Omega _{h,n}:=\\lbrace x\\in \\Omega :x+nh\\in \\Omega \\rbrace $ .",
"For $t>0$ , we then define $\\omega _n(f,t):=\\sup _{\\Vert h\\Vert \\le t} \\Vert \\Delta _h^n f\\Vert _{L^2(\\Omega _{h,n})}.$ Finally, we define the space $\\mathcal {B}^s(\\Omega )$ to be the set of functions for which the quantity $\\Vert f\\Vert _{\\mathcal {B}^s(\\Omega )}:=\\Vert f\\Vert _{L^2(\\Omega )}+\\sum _{j\\ge 0} 2^{sj}\\omega _n(f,2^{-j})$ is finite.",
"The above expression can also be used to define Besov spaces (and norms) for $s<0$ ; see [36].",
"In this subsection, we assume that $\\mu $ and $\\nu $ admit Besov smooth densities $f_{\\mu }(\\cdot )$ and $f_{\\nu }(\\cdot )$ (see [36] and def:besov above for details).",
"Given $\\Omega \\subseteq \\mathbb {R}^d$ and $s>0$ , let $\\mathcal {B}^s(\\Omega )$ denote the set of Besov smooth functions on $\\Omega $ of order $s$ .",
"Assumption (A1) (Regularity of the densities) Suppose that: $f_{\\mu }$ and $f_{\\nu }$ are supported on compact and convex subsets of $\\mathbb {R}^d$ , say $\\mathcal {X}$ and $\\mathcal {Y}$ respectively.",
"There exists $s,M> 0$ such that $\\Vert f_{\\mu }\\Vert _{\\mathcal {B}^{s}(\\mathcal {X})}\\le M$ , $\\Vert f_{\\nu }\\Vert _{\\mathcal {B}^{s}(\\mathcal {Y})}\\le M$ and $f_{\\mu }(x),f_{\\nu }(y)\\ge M^{-1}$ for all $x\\in \\mathcal {X}$ , $y\\in \\mathcal {Y}$ .",
"We now present our wavelet based estimators for $f_{\\mu }(\\cdot )$ and $f_{\\nu }(\\cdot )$ .",
"Towards this direction, we begin with sets of functions in $L^2(\\mathcal {X})$ (set of square integrable functions on $\\mathcal {X}$ ), $\\Phi $ and $\\lbrace \\Psi _j\\rbrace _{j\\ge 0}$ , which form an orthonormal basis of $L^2(\\mathcal {X})$ and satisfy the standard regularity assumptions for a wavelet basis (see [104], [77], [143]).",
"We defer a formal discussion on these assumptions to def:wavelet in the Appendix so as not to impede the flow of the paper.",
"For the moment, it is worth noting that such sets of functions (e.g., Haar wavelets, Daubechies wavelets) are readily available in standard statistical softwares, see e.g., the R package wavelets.",
"Next, fix $J_m\\in \\mathbb {N}$ (a truncation parameter to be chosen later depending on the sample size $m$ ).",
"Consider the following: $\\widehat{f}_{\\mu }(x):=\\sum _{\\phi \\in \\Phi } a_{\\phi }\\phi (x)+\\sum _{j=0}^{J_m} \\sum _{\\psi \\in \\Psi _j} b_{\\psi }\\psi (x),$ where $a_{\\phi }:=\\frac{1}{m}\\sum _{i=1}^m \\phi (X_i), \\qquad b_{\\psi }:=\\frac{1}{m}\\sum _{i=1}^m \\psi (X_i).$ Unfortunately $\\widehat{f}_{\\mu }(\\cdot )$ as defined in (REF ) may not be a probability density and consequently cannot be used to obtain plug-in estimators for $\\widetilde{T}_{m,n}^{\\gamma }(\\cdot )$ .",
"We therefore take the same route as in [143] to define the following estimator for $f_{\\mu }(\\cdot )$ : $\\widetilde{f}_{\\mu }:=\\min _{g\\in \\mathcal {D}(\\mathcal {X})}\\Vert g-\\widehat{f}_{\\mu }\\Vert _{\\mathcal {B}^{-1}(\\mathcal {X})},$ where $\\mathcal {D}(\\mathcal {X})$ is the space of probability density functions on $\\mathcal {X}$ and $\\mathcal {B}^{-1}(\\mathcal {X})$ is the Besov norm on $\\mathcal {X}$ of order $-1$ as stated in def:besov.",
"We can define $\\widetilde{f}_{\\nu }(\\cdot )$ similarly.",
"Computing both $\\widetilde{f}_{\\mu }(\\cdot )$ and $\\widehat{f}_{\\mu }(\\cdot )$ (as it involves infinite sums) is challenging and we would refer the interested reader to [143] and the references therein, for details.",
"Further discussion of this aspect is beyond the scope of this paper.",
"We are now in a position to present the main theorem of this subsection.",
"Theorem 2.4 Suppose that $T_0(\\cdot )$ is $L$ -Lipschitz, and $\\widetilde{\\mu }_m$ and $\\widetilde{\\nu }_n$ are the probability measures corresponding to the probability densities $\\widetilde{f}_{\\mu }(\\cdot )$ and $\\widetilde{f}_{\\nu }(\\cdot )$ with $m^{\\frac{1}{d+2s}}\\le 2^{J_m}\\le m^{\\frac{1}{d}}$ and $n^{\\frac{1}{d+2s}}\\le 2^{J_n}\\le n^{\\frac{1}{d}}$ , then the following holds for some constant $C>0$ : $\\mathbb {E}\\left[\\sup \\limits _{\\gamma \\in \\widetilde{\\Gamma }_{\\mathrm {min}}} \\int \\Vert \\widetilde{T}_{m,n}^{\\gamma }(x)-T_0(x)\\Vert ^2\\,d\\widetilde{\\mu }_m(x)\\right]\\le C \\widetilde{r}^{(m,n)}_{d,s},$ $\\mbox{where}\\quad \\widetilde{r}^{(m,n)}_{d,s}:={\\left\\lbrace \\begin{array}{ll} m^{-1/2}\\log {(1+m)}+n^{-1/2}\\log {(1+n)} & \\mbox{for}\\ d=2,\\\\ m^{-\\frac{1+s}{d+2s}}+n^{-\\frac{1+s}{d+2s}} & \\mbox{for}\\ d\\ge 3, \\end{array}\\right.",
"}$ The same bound also holds for $\\mathbb {E}|W_2^2(\\widetilde{\\mu }_m,\\widetilde{\\nu }_n)-W_2^2(\\mu ,\\nu )|$ .",
"Note that $\\frac{1+s}{d+2s}\\rightarrow \\frac{1}{2}$ as $s\\rightarrow \\infty $ .",
"Therefore thm:smoothwav shows that, when $m=n$ , the rate of convergence for the wavelet based estimator is “close\" to $n^{-1/2}$ provided $s$ is large enough for each fixed $d$ .",
"This shows that $T_0(\\cdot )$ obtained using the wavelet estimators for $f_{\\mu }(\\cdot )$ and $f_{\\nu }(\\cdot )$ mitigates the curse of dimensionality, contrast this with the estimator in thm:barate.",
"To avoid repetition, we defer further discussions on the rates observed in thm:smoothwav to rem:curdim where a holistic comparison is drawn with two other “smooth” plug-in estimators."
],
[
"Kernel based estimators",
"We first introduce the Sobolev class of functions which we will exploit in this subsection to construct estimators that achieve rates of convergence which mitigate the curse of dimensionality under sufficient smoothness.",
"[Uniform Sobolev class of functions] Let $\\Omega \\subseteq \\mathbb {R}^d$ and $f(\\cdot )$ be uniformly continuous on $\\Omega $ and admits uniformly continuous derivatives up to order $s$ on $\\Omega $ for some $s\\in \\mathbb {N}$ .",
"For any $\\mathfrak {m}:=(m_1,\\ldots ,m_d)\\in \\mathbb {N}^d$ , let $\\partial ^{\\mathfrak {m}} f:=\\frac{\\partial }{\\partial _{x_1}^{m_1}}\\ldots \\frac{\\partial }{\\partial _{x_d}^{m_d}} f,\\quad |\\mathfrak {m}|:=\\sum _{i=1}^d m_i.$ For any $k\\le s$ , we further define, $\\Vert f\\Vert _{C^k(\\Omega )}:=\\sum _{|\\mathfrak {m}|\\le k} \\Vert \\partial ^{\\mathfrak {m}} f\\Vert _{L^{\\infty }(\\Omega )}.$ The space $C^{s}(\\Omega )$ is defined as the set of functions $f(\\cdot )$ for which $\\Vert f\\Vert _{C^k(\\Omega )}<\\infty $ for all $k\\le s$ .",
"For this subsection, assume that $\\mu $ and $\\nu $ admit Sobolev smooth densities $f_{\\mu }(\\cdot )$ and $f_{\\nu }$ in the uniform norm (see def:holder above).",
"Given $\\Omega \\subseteq \\mathbb {R}^d$ and $s\\in \\mathbb {N}$ , let $C^s(\\Omega )$ denote the set of Sobolev smooth functions on $\\Omega $ of order $s$ .",
"Assumption (A2) (Regularity of the densities) Suppose that $f_{\\mu }$ and $f_{\\nu }$ are supported on compact and convex subsets of $\\mathbb {R}^d$ , say $\\mathcal {X}$ and $\\mathcal {Y}$ respectively.",
"There exists $s,M> 0$ such that $f_{\\mu }(\\cdot )\\in C^{s}(\\mathcal {X};M)$ and $f_{\\nu }(\\cdot )\\in C^{s}(\\mathcal {Y};M)$ where $C^{s}(\\mathcal {X};M)$ is the space of real valued functions supported on $\\mathcal {X}$ such that for all $f(\\cdot )\\in C^s(\\mathcal {X};M)$ , we have $M^{-1}\\le f(x)\\le M$ for all $x\\in \\mathcal {X}$ and $\\Vert f\\Vert _{C^s(\\mathcal {X})}\\le M$ .",
"Here $\\Vert \\cdot \\Vert _{C^s(\\mathcal {X})}$ is the standard uniform Sobolev norm as defined in def:holder.",
"The space $C^s(\\mathcal {Y};M)$ is defined analogously.",
"We now define our estimators for $f_{\\mu }(\\cdot )$ and $f_{\\nu }(\\cdot )$ using the standard kernel density estimation technique (see [135]).",
"Set $\\widehat{f}_{\\mu }(x):=\\frac{1}{mh_m^d}\\sum _{i=1}^m K_d\\left(\\frac{X_i-x}{h_m}\\right),$ for some bandwidth parameter $h_m>0$ and $d$ -variate kernel $K_d(\\cdot )$ .",
"We assume that $K_d(\\cdot )$ is the $d$ -fold product of univariate kernels, i.e., there exists a kernel $K(\\cdot )$ such that for $u=(u_1,\\ldots ,u_d)\\in \\mathbb {R}^d$ , $K_d(u)=\\prod _{i=1}^d K(u_i)$ .",
"We define $\\widehat{f}_{\\nu }(\\cdot )$ similarly with the same univariate kernel and bandwidth.",
"Assumption (A3) (Regularity of the kernel) Assume that $K(\\cdot )$ is a symmetric, bounded, $s+1$ times differentiable kernel on $\\mathbb {R}^d$ with all $s+1$ derivatives bounded and integrable.",
"Further, suppose that $K(\\cdot )$ is of order $2s+2$ , i.e., $\\int u^j K(u)\\,du=\\mathbb {1}(j=0),\\qquad \\mbox{for }\\ j=\\lbrace 0,1,2,\\ldots ,2s+1\\rbrace ,\\qquad \\mbox{and} \\ \\int |u|^{2s+2}|K(u)|\\,du<\\infty .$ The above assumptions on $K(\\cdot )$ are standard for estimating smooth densities and their derivatives of different orders in the kernel density estimation literature; see e.g.",
"[76], [5], [63], [135], [64].",
"There are several natural ways to construct kernels satisfying as:kernel, see [135]; an example is also provided in ex:ordker below.",
"[Example of a kernel satisfying as:kernel] Let $\\psi _m(\\cdot )$ be the $m$ -th Hermite polynomial on $\\mathbb {R}$ (see [92]).",
"Then the kernel function defined as $K(u):=\\sum _{m=0}^{2s+2} \\psi _{m}(0)\\psi _m(u)\\exp (-u^2/2)$ satisfies as:kernel.",
"It is evident from as:kernel that $K(\\cdot )$ may take some negative values, in which case, $\\widehat{f}_{\\mu }(\\cdot )$ (respectively $\\widehat{f}_{\\nu }(\\cdot )$ ) may not be a probability density.",
"Consequently the barycentric projection (see def:bcenterproj) between $\\widehat{f}_{\\mu }(\\cdot )$ and $\\widehat{f}_{\\nu }(\\cdot )$ is not well-defined.",
"We get around this by resorting to the same (approximate) projection technique we used in (REF ) for the wavelet based estimators.",
"In this case however, instead of (approximately) projecting using an appropriate Besov norm as in (REF ), we use a integral probability metric (see def:ipm; also see [127], [107], [114] for examples, computational procedures and applications of such metrics).",
"The corresponding measure is defined below: [Integral probability metric] Given a class $\\mathcal {F}$ of bounded functions on $\\mathbb {R}^d$ and two probability densities $g_1(\\cdot )$ and $g_2(\\cdot )$ on $\\mathbb {R}^d$ , the integral probability metric/distance between $g_1(\\cdot )$ and $g_2(\\cdot )$ with respect to $\\mathcal {F}$ is defined as $d_{\\mathrm {IP}}(g_1,g_2;\\mathcal {F}):=\\sup _{\\psi (\\cdot )\\in \\mathcal {F}} \\bigg |\\int \\psi (x)(g_1(x)-g_2(x))\\,dx\\bigg |.$ Sufficient conditions on $\\mathcal {F}$ for $d_{\\mathrm {IP}}(\\cdot ,\\cdot ;\\mathcal {F})$ to be a metric on the space of probability measures (not on the space of probability densities as they can be altered on set of Lebesgue measure 0 without altering the underlying probability measures) on $\\mathbb {R}^d$ have been discussed in [107].",
"Observe that the measure $d_{\\mathrm {IP}}(g_1,g_2;\\mathcal {F})$ is well defined even when $g_1(\\cdot )$ and $g_2(\\cdot )$ are not probability densities.",
"In thm:smooth below, we use $\\mathcal {F}=C^{s+2}(\\mathcal {X},M^{\\prime })$ .",
"Note that any function in $C^{s+2}(\\mathcal {X},M^{\\prime })$ can be extended to a function in $C^{s+2}(\\mathbb {R}^d;M^{\\prime })$ (see [80] and [134]).",
"The fact that this choice of $\\mathcal {F}$ results in a metric follows from the argument in [107].",
"We are now in a position to describe the projection estimators for $f_{\\mu }(\\cdot )$ and $f_{\\nu }(\\cdot )$ , and the rates achieved by the corresponding plug-in estimator.",
"Theorem 2.5 Assume that $T_0(\\cdot )$ is $L$ -Lipschitz and $f_{\\mu }$ , $f_{\\nu }$ are Lebesgue densities satisfying as:smdensities.",
"Also suppose that $K(\\cdot )$ satisfies as:kernel.",
"Define $h_m:=m^{-\\frac{1}{d+2s}}\\log {m}$ , $h_n:=n^{-\\frac{1}{d+2s}}\\log {n}$ and $T:=\\int |K_d(u)|\\,du+1$ .",
"Fix any $M^{\\prime }>0$ .",
"Consider any probability density $\\widetilde{f}_{\\mu }^{M^{\\prime }}(\\cdot )\\in C^s(\\mathcal {X};TM)$ (where $M$ is defined as in as:smdensities) which satisfies $d_{\\mathrm {IP}}\\left(\\widetilde{f}_{\\mu }^{M^{\\prime }},\\widehat{f}_{\\mu };C^{s+2}(\\mathcal {X};M^{\\prime })\\right)\\le \\inf _{\\begin{array}{c}f(\\cdot )\\in C^s(\\mathcal {X};TM)\\\\ f\\ge 0,\\ \\int f=1\\end{array}} d_{\\mathrm {IP}}\\left(\\widehat{f}_{\\mu },f;C^{s+2}(\\mathcal {X};M^{\\prime })\\right)+r_{d,s}^{(m,n)}$ where $r_{d,s}^{(m,n)}$ is defined as in (REF ) and $d_{\\mathrm {IP}}(\\cdot ,\\cdot ;C^{s+2}(\\mathcal {X};M^{\\prime }))$ is the integral probability metric defined in def:ipm.",
"We define $\\widetilde{f}_{\\nu }^{M^{\\prime }}(\\cdot )$ analogously as in (REF ) with $\\mathcal {X}$ , $\\widehat{f}_{\\mu }(\\cdot )$ replaced by $\\mathcal {Y}$ , $\\widehat{f}_{\\nu }(\\cdot )$ .",
"Then the following conclusions hold.",
"There exists $M^{\\prime }>0$ (depending on $M$ ) such that, if, $\\widetilde{\\mu }_m$ and $\\widetilde{\\nu }_n$ are the probability measures corresponding to the probability densities $\\widetilde{f}_{\\mu }^{M^{\\prime }}(\\cdot )$ and $\\widetilde{f}_{\\nu }^{M^{\\prime }}(\\cdot )$ , then the following holds for some constant $C>0$ : $\\mathbb {E}\\left[\\sup \\limits _{\\gamma \\in \\widetilde{\\Gamma }_{\\mathrm {min}}} \\int \\Vert \\widetilde{T}_{m,n}^{\\gamma }(x)-T_0(x)\\Vert ^2\\,d\\widetilde{\\mu }_m(x)\\right]\\le C r_{d,s}^{(m,n)},$ $ \\mbox{where}\\quad r_{d,s}^{(m,n)} := {\\left\\lbrace \\begin{array}{ll} m^{-1/2}+n^{-1/2} & \\mbox{for}\\ d<2(s+2),\\\\ m^{-1/2}\\left(\\log {(1+m)}\\right)^d+n^{-1/2}\\left(\\log {(1+n)}\\right)^d & \\mbox{for}\\ d=2(s+2),\\\\ m^{-\\frac{s+2}{d}}+n^{-\\frac{s+2}{d}} & \\mbox{for}\\ d\\ge 2(s+2).",
"\\end{array}\\right.",
"}$ The same bound also holds for $\\mathbb {E}|W_2^2(\\widetilde{\\mu }_m,\\widetilde{\\nu }_n)-W_2^2(\\mu ,\\nu )|$ .",
"$\\widehat{f}_{\\mu }(\\cdot )$ satisfies $\\lim \\limits _{n\\rightarrow \\infty } \\max \\left\\lbrace \\mathbb {P}\\left(\\Vert \\widehat{f}_{\\mu }\\Vert _{C^s(\\widetilde{\\mathcal {X}})}\\ge TM\\right),\\mathbb {P}\\left(\\sup _{x\\in \\widetilde{\\mathcal {X}}} |\\widehat{f}_{\\mu }(x)-f_{\\mu }(x)|\\ge \\varepsilon \\right)\\right\\rbrace =0$ for any $\\varepsilon >0$ , where $\\widetilde{\\mathcal {X}}$ is any compact subset of $\\mathcal {X}^o$ .",
"The same conclusion holds for $\\widehat{f}_{\\nu }(\\cdot )$ with $\\mathcal {X}$ replaced by $\\mathcal {Y}$ .",
"In thm:smooth, $\\widetilde{f}_{\\mu }^{M^{\\prime }}(\\cdot )$ can be viewed as an approximate minimizer of $d_{\\mathrm {IP}}(\\widehat{f}_{\\mu },\\cdot ;C^{s+2}(\\mathcal {X},M^{\\prime }))$ over an appropriate class of Sobolev smooth probability densities.",
"This is carried out because $\\widehat{f}_{\\mu }(\\cdot )$ by itself may not be a probability density.",
"Further note that $\\widetilde{\\mu }_m,\\widetilde{\\nu }_n$ as specified in thm:smooth are both smooth, and consequently $\\widetilde{\\Gamma }_{\\mathrm {min}}$ is a singleton and the supremum in thm:smooth can be dropped.",
"A brief description of the proof technique for thm:smooth is presented in rem:pfstech below and the actual proof is given in sec:mainrespf.",
"Remark 2.4 (Proof technique) The proof of thm:smooth proceeds along the same lines as rem:pfnstech.",
"We first show that $\\mathcal {F}_s$ (see rem:pftech) can be chosen as a certain subset of $C^{s+2}(\\mathcal {Y}^{\\circ })$ .",
"We then use Dudley's entropy integral type bounds which in turn requires the bracketing entropy [137] of the class of compactly supported Sobolev smooth functions which can be found in [138].",
"We now explain the implications of both the parts of thm:smooth in the following two remarks.",
"Remark 2.5 (Mitigating the curse of dimensionality) thm:smooth shows that, under enough smoothness, i.e., when $2(s+2)>d$ , both the upper bounds for (REF ) and (REF ) are $O_p(n^{-1/2})$ .",
"This shows that, for large dimensions, provided $\\mu $ and $\\nu $ admit smooth enough densities, it is possible to construct plug-in estimators that mitigate the curse of dimensionality.",
"Note that a similar estimator was analyzed in [73] when $m=n$ .",
"However, the rates obtained in thm:smooth are strictly better than those in [73].",
"For $m=n$ , when $d<2(s+2)$ , [73] obtained a rate of $n^{-\\frac{s+2}{2(s+2)+d}}$ which is worse than $n^{-1/2}$ obtained in thm:smooth.",
"For the other regimes, [73] obtains rates (up to log factors) of $n^{-1/4}$ and $n^{-\\frac{1}{(s+2)(d+2(s+2))}}$ which are both worse than the respective rates of $n^{-1/2}$ and $n^{-\\frac{s+2}{d}}$ in thm:smooth.",
"In fact, the rates obtained in thm:smoothwav are also strictly better than the rates obtained in [73] (for every fixed $d$ ) described above, but they are strictly worse than the rates obtained in thm:smooth.",
"When the degree of smoothness $s$ is large, both Theorems REF and REF lead to rates of (approximately) $n^{-1/2}$ .",
"However when $s$ is small, the rate in thm:smooth is much faster than that in thm:smoothwav, e.g., if $s$ is close to 0, the rate in thm:smoothwav is approximately $n^{-\\frac{1}{d}}$ whereas that in thm:smooth is the faster rate of $n^{-\\frac{2}{d}}$ .",
"It must be noted however that the smoothness assumptions are different in Theorems REF and REF .",
"Remark 2.6 (Computational aspects of thm:smooth) In thm:smooth, we have shown that the plug-in estimator for $T_0(\\cdot )$ using $\\widetilde{f}_{\\mu }^{M^{\\prime }}(\\cdot )$ and $\\widetilde{f}_{\\nu }^{M^{\\prime }}(\\cdot )$ achieve rates that mitigate the curse of dimensionality under sufficient smoothness.",
"However, as is evident, $\\widetilde{f}_{\\mu }^{M^{\\prime }}(\\cdot )$ is hard to compute whereas $\\widehat{f}_{\\mu }(\\cdot )$ is computable easily in linear time.",
"Note that if $\\widehat{f}_{\\mu }(\\cdot )$ itself were a probability density in $C^s(\\mathcal {X};TM)$ , then we would have $\\widehat{f}_{\\mu }=\\widetilde{f}_{\\mu }^{M^{\\prime }}$ .",
"While thm:smooth does not establish that, it does come close in part 2, from which we can easily derive the following: $\\lim _{n\\rightarrow \\infty }\\mathbb {P}(\\widehat{f}_{\\mu }(\\cdot )\\notin C^s(\\widetilde{\\mathcal {X}};TM))=0.$ The above shows that $\\widehat{f}_{\\mu }(\\cdot )$ is indeed bounded below by $(TM)^{-1}$ on $\\widetilde{\\mathcal {X}}$ (any compact subset of the interior of $\\mathcal {X}$ ), and additionally belongs to $C^s(\\widetilde{\\mathcal {X}};TM)$ with probability converging to 1.",
"This leads us to conjecture that the natural density version of $\\widehat{f}_{\\mu }(\\cdot )$ , i.e., $\\frac{\\max \\lbrace \\widehat{f}_{\\mu }(\\cdot ),0\\rbrace }{\\int \\max \\lbrace \\widehat{f}_{\\mu }(x),0\\rbrace \\,dx}$ should serve as a good proxy for $\\widetilde{f}_{\\mu }^{M^{\\prime }}(\\cdot )$ and lead to rates of convergence that mitigate the curse of dimensionality.",
"From a computational perspective, the density specified above is easy to simulate from using an accept-reject algorithm without computing the integral in the denominator (see [110]).",
"However, our current proof technique does not provide rates of convergence for the above density estimator based on $\\widehat{f}_{\\mu }(\\cdot )$ .",
"Another important implication of thm:smooth is the bound obtained on $|W_2(\\widetilde{\\mu }_m,\\widetilde{\\nu }_n)-W_2(\\mu ,\\nu )|$ when $\\mu \\ne \\nu $ .",
"We first present the result and then describe the implication.",
"Proposition 2.6 Consider the setting in thm:smooth.",
"Then, provided $\\mu \\ne \\nu $ , the following holds: $|W_2(\\widetilde{\\mu }_m,\\widetilde{\\nu }_n)-W_2(\\mu ,\\nu )|=O_p(r_{d,s}^{(m,n)}).$ prop:wassrate (see sec:mainrespf for a proof) shows an interesting distinction between the $\\mu \\ne \\nu $ case and the $\\mu =\\nu $ case.",
"For $\\mu =\\nu $ , the best possible exponent is $n^{-\\frac{1+s}{2s+d}}$ for $d\\ge 3$ (see [143] where the result was established under more general Besov smoothness assumptions).",
"On the contrary, when $\\mu \\ne \\nu $ , prop:wassrate establishes a rate of $n^{-\\frac{s+2}{d}}$ for the Wasserstein distance which is strictly better than the minimax achievable rate mentioned above when $\\mu =\\nu $ .",
"This observation complements [33] where the authors make a similar observation for the special case of $s=0$ ."
],
[
"Discretized plug-in estimator under smoothness assumptions",
"In sec:nsmooth, we discussed how smoothness can be incorporated into the plug-in procedure to get faster rates of convergence.",
"Such plug-in estimators are popular in the computational OT literature (see [8], [9], [28], [39]).",
"However, even after $\\widetilde{f}_{\\mu }(\\cdot )$ , $\\widetilde{f}_{\\nu }(\\cdot )$ are calculated, $\\widetilde{T}_{m,n}^{\\gamma }$ as in thm:smooth cannot be computed explicitly from data if $\\widetilde{f}_{\\mu }(\\cdot )$ and $\\widetilde{f}_{\\nu }(\\cdot )$ are continuous densities.",
"This is in contrast to $\\widetilde{T}_{m,n}^{\\gamma }$ from thm:nsmooth in the discrete-discrete case which is explicitly computable using a standard linear program, but achieves worse rates of convergence.",
"This is not unexpected.",
"Thanks to the no free lunch principle, better statistical accuracy is naturally accompanied by heavier computational challenges.",
"Therefore, our goal here is to construct estimators, under smoothness assumptions as in sec:smooth, which are computable in polynomial time (with complexity increasing with smoothness) provided $\\widetilde{f}_{\\mu }(\\cdot )$ and $\\widetilde{f}_{\\nu }(\\cdot )$ can be sampled from, and also attain rates that mitigate the curse of dimensionality.",
"Construction: We will illustrate the discretized estimator using the kernel based estimator from sec:kernel.",
"Similar results also hold for the wavelet based estimator from sec:wavelet.",
"Recall the kernel density estimators $\\widetilde{f}_{\\mu }(\\cdot )$ and $\\widetilde{f}_{\\nu }(\\cdot )$ (see (REF )).",
"Sample $M\\ge 1$ random points from both $\\widetilde{f}_{\\mu }(\\cdot )$ and $\\widetilde{f}_{\\nu }(\\cdot )$ .",
"Let $\\widehat{\\mu }_{m,M}$ and $\\widehat{\\nu }_{n,M}$ denote the standard empirical measures on the $M$ points sampled from $\\widetilde{f}_{\\mu }(\\cdot )$ and $\\widetilde{f}_{\\nu }(\\cdot )$ respectively.",
"Finally construct $\\widetilde{T}_{m,n}\\equiv \\widetilde{T}_{m,n}^{\\gamma }$ as in def:bcenterproj with $\\widetilde{\\mu }_m=\\widehat{\\mu }_{m,M}$ and $\\widetilde{\\nu }_n=\\widehat{\\nu }_{n,M}$ .",
"It should be pointed out that a similar construction was also used in [143] for estimating probability densities under the Wasserstein loss.",
"Based on this construction, the main result of this section is as follows: Theorem 2.7 Consider the setting in thm:smooth and the same construction of $\\widetilde{T}_{m,n}^{\\gamma }$ as above.",
"For simplicity, let's also assume $m=n$ .",
"Accordingly set $M=n^{\\frac{s+2}{2}}$ .",
"Then $\\widetilde{\\Gamma }_{\\mathrm {min}}$ is a singleton and consequently the following conclusion holds for some constant $C>0$ : $\\mathbb {E}\\left[\\int \\Vert \\widetilde{T}_{m,n}(x)-T_0(x)\\Vert ^2\\,d\\widetilde{\\mu }_m(x)\\right]\\le C r_{d,s}^{(n,n)}.$ The same rates also hold for $\\mathbb {E}|W_2^2(\\widetilde{\\mu }_m,\\widetilde{\\nu }_n)-W_2^2(\\mu ,\\nu )|$ .",
"The proof of thm:dismooth is given in sec:mainrespf.",
"Once the empirical measures $\\widehat{\\mu }_{m,M}$ and $\\widehat{\\nu }_{n,M}$ have been obtained, an explicit computation of $\\widetilde{T}_{m,n}$ as described above requires $O(M^3)=O(n^{\\frac{3(s+2)}{2}})$ steps using the Hungarian algorithm, see [81].",
"This highlights the statistical versus computational trade off, i.e., in order to mitigate the curse of dimensionality in convergence rates by exploiting smoothness, the computational complexity gets progressively worse by polynomial factors in $n$ .",
"It should be mentioned that (approximate) algorithms faster than the Hungarian algorithm stated above, can be found in [59], [1], [39] to name a few.",
"Due to space constraints, we avoid a detailed discussion on this.",
"In the above construction, sampling from the smoothed kernel densities $\\widetilde{f}_{\\mu }(\\cdot )$ and $\\widetilde{f}_{\\nu }(\\cdot )$ is crucial.",
"If we would simply draw $M$ bootstrap samples from the empirical distributions $\\widehat{\\mu }_m$ and $\\widehat{\\nu }_n$ , the rates of convergence wouldn't improve from those observed in thm:nsmooth no matter how large $M$ is chosen.",
"In this section, we will apply our results to two popular problems, namely — estimating the Wasserstein barycenter between two probability distributions (see [2], [40], [15], [27]) in sec:bar, and obtaining detection thresholds in some recent optimal transport based independence testing procedures (see [44], [43], [61], [123], [124]) in sec:nonptest."
],
[
"Wasserstein barycenter estimation",
"Let $\\mu ,\\nu \\in \\mathcal {P}_2(\\mathbb {R}^d)$ .",
"The Wasserstein barycenter between $\\mu $ and $\\nu $ is then given by: $\\rho _0:=\\min \\limits _{\\rho \\in \\mathcal {P}_{\\mathrm {ac}}(\\mathbb {R}^d)} \\left(\\frac{1}{2}W_2^2(\\mu ,\\rho )+\\frac{1}{2}W_2^2(\\rho ,\\nu )\\right).$ In fact, by prop:bmopt, there exists an optimal transport map $T_0$ from $\\mu $ to $\\nu $ and by [2], [15], [18], an alternative characterization of $\\rho _0$ is as follows: $\\rho _0=\\left(\\frac{1}{2}\\mbox{Id}+\\frac{1}{2}T_0\\right)\\#\\mu ,\\qquad \\mbox{where}\\qquad \\mbox{Id}(x)=x.$ Estimating $\\rho _0$ as in (REF ) has attracted significant attention over the past few years in economics [31], [26], Bayesian learning [130], [129], dynamic formulations [35], [30], algorithmic fairness [68], [34], etc.",
"The most natural strategy employed in estimating $\\rho _0$ is to use the empirical plug-in estimator, i.e., replacing $\\mu ,\\nu $ in (REF ) with $\\widehat{\\mu }_m,\\widehat{\\nu }_n$ .",
"This strategy has been used, approximated and analyzed extensively in e.g., [40], [27], [93], [18].",
"Based on (REF ), the natural plug-in estimator of $\\rho _0$ would be: $\\widehat{\\rho }_0^{\\gamma }=\\left(\\frac{1}{2}\\mbox{Id}+\\frac{1}{2}\\widetilde{T}_{m,n}^{\\gamma }\\right)\\#\\widetilde{\\mu }_m$ where $\\widetilde{T}_{m,n}^{\\gamma }$ is the plug-in estimator of $T_0$ obtained by solving (REF ), with $\\mu $ and $\\nu $ replaced by $\\widetilde{\\mu }_m$ and $\\widehat{\\nu }_n$ respectively and $\\gamma \\in \\widetilde{\\Gamma }_{\\mathrm {min}}$ .",
"While the consistency of $\\widehat{\\rho }_0^{\\gamma }$ has been analyzed for $m=n$ in [93] and rates have been obtained for $d=1$ in [14], the more general question of obtaining rates of convergence for $\\widehat{\\rho }_0^{\\gamma }$ for general dimensions $d\\ge 1$ is yet unanswered.",
"We address this question in the following result (see sec:Appf for a proof).",
"Theorem 3.1 Suppose that the same assumptions from thm:nsmooth hold.",
"Then, with $\\widehat{\\rho }_0^{\\gamma }$ as defined in (REF ) and $r_d^{(m,n)}$ , $t_{d,\\alpha }$ defined in thm:nsmooth, the following holds: $\\sup \\limits _{\\gamma \\in \\widetilde{\\Gamma }_{\\mathrm {min}}}W_2^2(\\widehat{\\rho }_0^{\\gamma },\\rho _0)=O_p\\left(r^{(m,n)}_d\\times (\\log {(1+\\max \\lbrace m,n\\rbrace )})^{t_{d,\\alpha }}\\right).$"
],
[
"Nonparametric independence testing: Optimal transport based Hilbert-Schmidt independence criterion",
"Let $(X_1,Y_1),\\ldots ,(X_n,Y_n)\\overset{i.i.d.",
"}{\\sim }\\pi $ , a probability measure on $\\mathbb {R}^{d_1+d_2}$ , with marginals $\\mu \\in \\mathcal {P}_{\\mathrm {ac}}(\\mathbb {R}^{d_1})$ and $\\nu \\in \\mathcal {P}_{\\mathrm {ac}}(\\mathbb {R}^{d_2})$ .",
"Our problem of interest is the following hypothesis testing problem, given as: $\\mathrm {H}_0:\\pi =\\mu \\otimes \\nu \\qquad \\mbox{versus} \\qquad \\mathrm {H}_1:\\pi \\ne \\mu \\otimes \\nu .$ This is the classical nonparametric independence testing problem which has received a lot of attention in the statistics and machine learning literature (see [132], [71], [78], [12], and [48], [82] for a review).",
"In keeping with the overall theme of this paper, our focus here will be on a large class of OT based independence testing procedures, introduced first in [44] followed by recent developments in [123], [124], [43].",
"These tests bear resemblance to the Hilbert-Schmidt independence criterion (HSIC); see [71], [69], [70] and have attractive properties such as distribution-freeness (see prop:testprop), consistency without moment assumptions and robustness against heavy-tailed distributions and against contamination [44], [123].",
"Below, we describe this class of tests, see (REF ) and (REF ).",
"Our main theoretical contribution of this section will be to provide detection thresholds of these OT based tests.",
"Construction: Suppose $\\upsilon _1$ , $\\upsilon _2$ be two compactly supported probability distributions on $\\mathbb {R}^{d_1}$ and $\\mathbb {R}^{d_2}$ respectively (e.g., $\\upsilon _1\\equiv \\mathrm {Unif}[0,1]^{d_1}$ , $\\upsilon _2\\equiv \\mathrm {Unif}[0,1]^{d_2}$ ).",
"Let $U_1,\\ldots ,U_n\\overset{i.i.d.",
"}{\\sim }\\upsilon _1$ , $V_1,\\ldots ,V_n\\overset{i.i.d.",
"}{\\sim }\\upsilon _2$ , $\\widehat{u}_n:=n^{-1}\\sum _{i=1}^n \\delta _{U_i}$ and $\\widehat{v}_n:=n^{-1}\\sum _{j=1}^n \\delta _{V_j}$ .",
"Recall the definitions of $\\widehat{\\mu }_n$ (with $m=n$ ) and $\\widehat{\\nu }_n$ from (REF ).",
"Let $\\widehat{T}_{1,n}$ ($\\widehat{T}_{2,n}$ ) be obtained by solving (REF ), with $\\mu $ and $\\nu $ replaced by $\\widehat{\\mu }_n$ and $\\widehat{u}_n$ ($\\widehat{\\nu }_n$ and $\\widehat{v}_n$ ) respectively.",
"Consider two non negative definite, continuous, characteristic kernels (see [58], [128] for definitions) $K_1(\\cdot ,\\cdot )$ and $K_2(\\cdot ,\\cdot )$ on $(\\mbox{supp}(\\upsilon _1))^2$ and $(\\mbox{supp}(\\upsilon _2))^2$ .",
"Set $\\widehat{x}_{ij}:=K_1(\\widehat{T}_{1,n}(X_i),\\widehat{T}_{1,n}(X_j))$ and $\\widehat{y}_{ij}:=K_2(\\widehat{T}_{2,n}(Y_i),\\widehat{T}_{2,n}(Y_j))$ .",
"Our test statistic is as follows: $\\widehat{\\mathrm {rHSIC}}:=n^{-2}\\sum _{i,j}\\widehat{x}_{ij}\\widehat{y}_{ij}+n^{-4}\\sum _{i,j,r,s}\\widehat{x}_{ij}\\widehat{y}_{rs}-2n^{-3}\\sum _{i,j,r} \\widehat{x}_{ij}\\widehat{y}_{ir}.$ Proposition 3.2 (See [44], [124]) (Distribution-freeness) When $X_1$ and $Y_1$ are independent, the distribution of $n\\times \\widehat{\\mathrm {rHSIC}}$ is universal, i.e., it does not depend on $\\mu $ and $\\nu $ for every fixed $n$ .",
"(Consistency against fixed alternatives) Let $c_{n,\\alpha }$ be the upper $(1-\\alpha )$ -th quantile from the universal distribution in part 1 above.",
"Then $\\widehat{\\mathrm {rHSIC}}\\overset{P}{\\longrightarrow }\\mathrm {rHSIC}(\\pi |\\mu \\otimes \\nu )$ where $&\\mathrm {rHSIC}(\\pi | \\mu \\otimes \\nu ):=\\mathbb {E}[K_1(T_1(X_1),T_1(X_2))K_2(T_2(Y_1),T_2(Y_2))]+\\mathbb {E}[K_1(T_1(X_1),T_1(X_2))]\\nonumber \\\\&\\times \\mathbb {E}[K_2(T_2(Y_1),T_2(Y_2))]-2\\mathbb {E}[K_1(T_1(X_1),T_1(X_2))K_2(T_2(Y_1),T_2(Y_3))],$ where $T_{1}(\\cdot )$ (respectively $T_2(\\cdot )$ ) is the optimal transport map from $\\mu $ ($\\nu $ ) to $\\upsilon _1$ ($\\upsilon _2$ ); see def:otmpm.",
"Further $\\mathrm {rHSIC}(\\pi |\\mu \\otimes \\nu )=0$ if and only if $\\pi =\\mu \\otimes \\nu $ .",
"Define the following test function: $\\phi _{n,\\alpha }:=\\mathbb {1}(n\\times \\widehat{\\mathrm {rHSIC}}\\ge c_{n,\\alpha }).$ Then $\\mathbb {E}[\\phi _{n,\\alpha }]\\rightarrow 1$ as $n\\rightarrow \\infty $ under $\\mathrm {H}_1$ , i.e., when $\\pi \\ne \\mu \\otimes \\nu $ .",
"prop:testprop shows that the test based on $\\widehat{\\mathrm {rHSIC}}$ (see (REF )), i.e., $\\phi _{n,\\alpha }$ (see (REF )), can be carried out without resorting to the permutation principle as is necessary for the usual HSIC based test (see [69]).",
"Further, when the sampling distribution is fixed, prop:testprop shows that $\\widehat{\\mathrm {rHSIC}}$ consistently estimates $\\mathrm {rHSIC}(\\pi |\\mu \\otimes \\nu )$ , a quantity which equals 0 if and only if $\\pi =\\mu \\otimes \\nu $ (this yields the consistency of $\\phi _{n,\\alpha })$ against fixed alternatives.",
"While consistency against fixed alternatives is an attractive feature of $\\phi _{n,\\alpha }$ , a more intricate question of statistical interest is to understand the local power of $\\phi _{n,\\alpha }$ under “changing sequence of alternatives converging to the null\" as $n\\rightarrow \\infty $ .",
"To study the local power of $\\phi _{n,\\alpha }$ , we need to consider a triangular array setting, where the data distribution changes with $n$ , i.e., $(X_1,Y_1),\\ldots ,(X_n,Y_n)\\overset{i.i.d.",
"}{\\sim }\\pi ^{(n)}$ , a probability measure on $\\mathbb {R}^{d_1+d_2}$ , with marginals $\\mu ^{(n)}\\in \\mathcal {P}_{\\mathrm {ac}}(\\mathbb {R}^{d_1})$ and $\\nu ^{(n)}\\in \\mathcal {P}_{\\mathrm {ac}}(\\mathbb {R}^{d_2})$ .",
"As $\\mathrm {rHSIC}(\\cdot |\\cdot )$ characterizes independence, a mathematical formulation of “alternatives converging to null\" would be to say $\\mathrm {rHSIC}(\\pi ^{(n)}|\\mu ^{(n)}\\otimes \\nu ^{(n)})\\rightarrow 0$ as $n\\rightarrow \\infty $ .",
"Similar questions have attracted a lot of attention in modern statistics, featuring measures (other than $\\mathrm {rHSIC}(\\cdot |\\cdot )$ ) which characterize independence, see e.g., [12], [87], [94], [6].",
"In the following result (see sec:Appf for a proof), we show that if $\\mathrm {rHSIC}(\\pi ^{(n)}|\\mu ^{(n)}\\otimes \\nu ^{(n)})\\rightarrow 0$ slowly enough with $n$ , then $\\phi _{n,\\alpha }$ yields a consistent sequence of tests for problem (REF ).",
"Theorem 3.3 Consider problem (REF ) with $\\pi ^{(n)}$ , $\\mu ^{(n)}$ , $\\nu ^{(n)}$ (changing with $n$ ) and suppose $T_{1,n}(\\cdot )$ and $T_{2,n}(\\cdot )$ are both $L$ -Lipschitz ($L$ is free of $n$ ).",
"Also assume $K_1(\\cdot )$ , $K_2(\\cdot )$ are Lipschitz, $\\mu ^{(n)}$ , $\\nu ^{(n)}$ are supported on fixed compact sets (supports are free of $n$ ).",
"Set $r_{d_1,d_2}^{(n,n)}:=r_{d_1}^{(n,n)}+r_{d_2}^{(n,n)}$ where $r_{d_1}^{(n,n)},r_{d_2}^{(n,n)}$ is defined via (REF ).",
"Then, $\\mathbb {E}[\\phi _{n,\\alpha }]\\rightarrow 1\\qquad \\mbox{if} \\qquad (r_{d_1,d_2}^{(n,n)})^{-1/2}\\times \\mathrm {rHSIC}(\\pi ^{(n)}|\\mu ^{(n)}\\otimes \\nu ^{(n)})\\rightarrow \\infty ,$"
],
[
"Appendix",
"This section is devoted to proving our main results and is organized as follows: In sec:mainrespf, we present the proofs of results from sec:mainres and in sec:Appf, we present the proofs from sec:App.",
"Throughout this section, we will use the $\\lesssim $ sign to hide constants that are free of $m,n$ ."
],
[
"Proofs from sec:mainres",
"[Proof of thm:newubd] We begin the proof by observing that $\\varphi _0^*(\\cdot )$ is convex and finite on $\\mbox{supp}(\\nu )$ , and hence differentiable $\\nu $ almost everywhere (a.e.).",
"Further by lem:gradual, we also have: $\\nabla \\varphi _0^*(T_0(x))=x\\qquad \\mbox{$\\mu $-a.e.~$x$.",
"}$ Fix any arbitrary $\\gamma \\in \\widetilde{\\Gamma }_{\\mathrm {min}}$ and suppose that $\\gamma (y|x)$ denotes the conditional distribution of $y$ given $x$ under $\\gamma $ .",
"Define, $D_1:= \\int \\varphi _0^*(y)\\,d\\widetilde{\\nu }_n(y) - \\int \\varphi _0^*(y) d \\overline{\\nu }_m(y).$ As $\\gamma $ has marginals $\\widetilde{\\mu }_m$ and $\\widetilde{\\nu }_n$ , we have: $D_1=\\int _{x,y} \\varphi _0^*(y)\\,d\\gamma (y|x)\\,d\\widetilde{\\mu }_m(x)-\\int _{x} \\varphi _0^*(T_0(x))\\,d\\widetilde{\\mu }_m(x).$ Next, by applying the conditional version of Jensen's inequality, $\\int _{x} \\left(\\int _{y} \\varphi _0^*(y)\\,d\\gamma (y|x)\\right)\\,d\\widetilde{\\mu }_m(x)&\\ge \\int _{x}\\varphi _0^*\\left(\\int _{y} y\\,d\\gamma (y|x)\\right)\\,d\\widetilde{\\mu }_m(x)\\nonumber \\\\&=\\int _{x}\\varphi _0^*(\\widetilde{T}_{m,n}^{\\gamma }(x))\\,d\\widetilde{\\mu }_m(x).$ Using (REF ) with (REF ) yields, $D_1&\\ge \\int [\\varphi _0^*(\\widetilde{T}_{m,n}^{\\gamma }(x)) - \\varphi _0^*(T_0(x))] d \\widetilde{\\mu }_m(x) \\nonumber \\\\& \\overset{(a)}{\\ge } \\int \\left\\lbrace \\nabla \\varphi _0^*(T_0(x))^\\top (\\widetilde{T}_{m,n}^{\\gamma }(x) - T_0(x)) + \\frac{1}{2L} \\Vert \\widetilde{T}_{m,n}^{\\gamma }(x)-T_0(x)\\Vert ^2 \\right\\rbrace \\,d {\\widetilde{\\mu }_m}(x) \\nonumber \\\\& \\overset{(b)}{=} \\underbrace{\\int x^\\top (\\widetilde{T}_{m,n}^{\\gamma }(x) - T_0(x)) \\,d {\\widetilde{\\mu }_m}(x)}_{D_2} + \\frac{1}{2L} \\int \\Vert \\widetilde{T}_{m,n}^{\\gamma }(x)-T_0(x)\\Vert ^2 \\, d {\\widetilde{\\mu }_m}(x).$ Here (a) follows from the strong convexity of $\\varphi _0^*(\\cdot )$ with parameter $(1/L)$ (see lem:assm) and (b) follows from (REF ).",
"Next, we will simplify the term $D_2$ .",
"Towards this direction, observe that for every $\\gamma \\in \\widetilde{\\Gamma }_{\\mathrm {min}}$ , $W_2^2(\\widetilde{\\mu }_m,\\widetilde{\\nu }_n)&=\\int \\Vert x-y\\Vert ^2\\,d\\gamma (x,y)\\nonumber \\\\ &=\\int \\Vert x\\Vert ^2\\,d\\widetilde{\\mu }_m(x)+\\int \\Vert y\\Vert ^2\\,d\\widetilde{\\nu }_n(y)-2\\int _{x} \\left(x^{\\top }\\int _{y} y\\,d\\gamma (y|x)\\right)\\,d\\widetilde{\\mu }_m(x)\\nonumber \\\\&=\\int \\Vert x\\Vert ^2\\,d\\widetilde{\\mu }_m(x)+\\int \\Vert y\\Vert ^2\\,d\\widetilde{\\nu }_n(y)-2\\int _{x} x^{\\top }\\widetilde{T}_{m,n}^{\\gamma }(x)\\,d\\widetilde{\\mu }_m(x).$ Also, as $T_0$ is the gradient of a convex function, it is also an OT map from $\\widetilde{\\mu }_m$ to $\\overline{\\nu }_m$ (see [3]), we have: $W_2^2(\\widetilde{\\mu }_m,\\overline{\\nu }_m)&=\\int \\Vert x-T_0(x)\\Vert ^2\\,d\\widetilde{\\mu }_m(x)\\nonumber \\\\ &=\\int \\Vert x\\Vert ^2\\,d\\widetilde{\\mu }_m(x)+\\int \\Vert y\\Vert ^2\\,d\\overline{\\nu }_m(y)-2\\int _{x} x^{\\top }T_0(x)\\,d\\widetilde{\\mu }_m(x).$ Now (REF ) and (REF ) imply $D_2=\\frac{1}{2}\\big (W_2^2(\\widetilde{\\mu }_m,\\overline{\\nu }_m)-W_2^2(\\widetilde{\\mu }_m,\\widetilde{\\nu }_n)\\big )+\\frac{1}{2}\\int \\Vert y\\Vert ^2\\,d(\\widetilde{\\nu }_n-\\overline{\\nu }_m)(y).$ Finally by combining (REF ) and (REF ), we get: $&\\;\\;\\;\\;\\frac{1}{2L}\\int \\Vert \\widetilde{T}_{m,n}^{\\gamma }(x)-T_0(x)\\Vert ^2\\,d\\widetilde{\\mu }_m(x)\\nonumber \\\\ &\\le \\frac{1}{2}\\big (W_2^2(\\widetilde{\\mu }_m,\\widetilde{\\nu }_n)-W_2^2(\\widetilde{\\mu }_m,\\overline{\\nu }_m)\\big )+\\int (\\varphi _0^*(y)-(1/2)\\Vert y\\Vert ^2)\\,d(\\widetilde{\\nu }_n-\\overline{\\nu }_m)(y).$ Now note that the bound on the right hand side of the above display is free of the particular choice of $\\gamma \\in \\widetilde{\\Gamma }_{\\mathrm {min}}$ .",
"Therefore, the same bound holds if we take a supremum over $\\gamma \\in \\widetilde{\\Gamma }_{\\mathrm {min}}$ on the left hand side.",
"We will now provide an upper bound for the right hand side of (REF ).",
"The remainder of the proof proceeds as in the proof of [101].",
"By the dual representation presented in (REF ) and (), and the definitions of $\\Psi _{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n}(\\cdot )$ and $\\Psi _{\\widetilde{\\mu }_m,\\overline{\\nu }_m}(\\cdot )$ in the statement of thm:newubd, we have $\\frac{1}{2}W_2^2(\\widetilde{\\mu }_m,\\widetilde{\\nu }_n)=\\frac{1}{2}\\int \\Vert x\\Vert ^2\\,d\\widetilde{\\mu }_m(x)+\\frac{1}{2}\\int \\Vert y\\Vert ^2\\,d\\widetilde{\\nu }_n(y)-\\mathcal {S}_{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n}(\\Psi _{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n}),$ $\\mbox{and}\\qquad \\frac{1}{2}W_2^2(\\widetilde{\\mu }_m,\\overline{\\nu }_m)=\\frac{1}{2}\\int \\Vert x\\Vert ^2\\,d\\widetilde{\\mu }_m(x)+\\frac{1}{2}\\int \\Vert y\\Vert ^2\\,d\\overline{\\nu }_m(y)-\\mathcal {S}_{\\widetilde{\\mu }_m,\\overline{\\nu }_m}(\\Psi _{\\widetilde{\\mu }_m,\\overline{\\nu }_m}).$ By subtracting the two equations above, we get: $\\frac{1}{2}W_2^2(\\widetilde{\\mu }_m,\\widetilde{\\nu }_n)-\\frac{1}{2}W_2^2(\\widetilde{\\mu }_m,\\overline{\\nu }_m)=\\frac{1}{2}\\int \\Vert y\\Vert ^2\\,d(\\widetilde{\\nu }_n-\\overline{\\nu }_m)-\\mathcal {S}_{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n}(\\Psi _{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n})+\\mathcal {S}_{\\widetilde{\\mu }_m,\\overline{\\nu }_m}(\\Psi _{\\widetilde{\\mu }_m,\\overline{\\nu }_m}).$ Next, we use () to make the following observations: $\\mathcal {S}_{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n}(\\Psi _{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n})\\le \\mathcal {S}_{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n}(\\Psi _{\\widetilde{\\mu }_m,\\overline{\\nu }_m}),\\qquad \\mathcal {S}_{\\widetilde{\\mu }_m,\\overline{\\nu }_m}(\\Psi _{\\widetilde{\\mu }_m,\\overline{\\nu }_m})\\le \\mathcal {S}_{\\widetilde{\\mu }_m,\\overline{\\nu }_m}(\\Psi _{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n}).$ Note that (REF ) immediately yields the following conclusions: $\\mathcal {S}_{\\widetilde{\\mu }_m,\\overline{\\nu }_m}(\\Psi _{\\widetilde{\\mu }_m,\\overline{\\nu }_m})-\\mathcal {S}_{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n}(\\Psi _{\\widetilde{\\mu }_m,\\overline{\\nu }_m})\\le \\mathcal {S}_{\\widetilde{\\mu }_m,\\overline{\\nu }_m}(\\Psi _{\\widetilde{\\mu }_m,\\overline{\\nu }_m})-\\mathcal {S}_{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n}(\\Psi _{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n}),$ and $\\mathcal {S}_{\\widetilde{\\mu }_m,\\overline{\\nu }_m}(\\Psi _{\\widetilde{\\mu }_m,\\overline{\\nu }_m})-\\mathcal {S}_{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n}(\\Psi _{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n})\\le \\mathcal {S}_{\\widetilde{\\mu }_m,\\overline{\\nu }_m}(\\Psi _{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n})-\\mathcal {S}_{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n}(\\Psi _{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n}).$ By combining the above two displays, we have: $&\\;\\;\\;\\;\\left|\\mathcal {S}_{\\widetilde{\\mu }_m,\\overline{\\nu }_m}(\\Psi _{\\widetilde{\\mu }_m,\\overline{\\nu }_m})-\\mathcal {S}_{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n}(\\Psi _{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n})\\right|\\nonumber \\\\ &\\le \\max \\left\\lbrace \\left|\\mathcal {S}_{\\widetilde{\\mu }_m,\\overline{\\nu }_m}(\\Psi _{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n})-\\mathcal {S}_{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n}(\\Psi _{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n})\\right|,\\left|\\mathcal {S}_{\\widetilde{\\mu }_m,\\overline{\\nu }_m}(\\Psi _{\\widetilde{\\mu }_m,\\overline{\\nu }_m})-\\mathcal {S}_{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n}(\\Psi _{\\widetilde{\\mu }_m,\\overline{\\nu }_m})\\right|\\right\\rbrace .$ By () and some simple algebra, the following holds: $ \\left|\\mathcal {S}_{\\widetilde{\\mu }_m,\\overline{\\nu }_m}(\\Psi _{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n})-\\mathcal {S}_{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n}(\\Psi _{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n})\\right|=\\left|\\int \\Psi _{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n}^*\\,d(\\overline{\\nu }_m-\\widetilde{\\nu }_n)\\right|.$ A similar expression holds for $|\\mathcal {S}_{\\widetilde{\\mu }_m,\\overline{\\nu }_m}(\\Psi _{\\widetilde{\\mu }_m,\\overline{\\nu }_m})-\\mathcal {S}_{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n}(\\Psi _{\\widetilde{\\mu }_m,\\overline{\\nu }_m})|$ .",
"Using the above observation in (REF ), we get: $\\left|\\mathcal {S}_{\\widetilde{\\mu }_m,\\overline{\\nu }_m}(\\Psi _{\\widetilde{\\mu }_m,\\overline{\\nu }_m})-\\mathcal {S}_{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n}(\\Psi _{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n})\\right|\\le \\max \\left\\lbrace \\bigg |\\int \\Psi _{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n}^* \\,d(\\widetilde{\\nu }_n-\\overline{\\nu }_m)\\bigg |,\\bigg |\\int \\Psi _{\\widetilde{\\mu }_m,\\overline{\\nu }_m}^* \\,d(\\widetilde{\\nu }_n-\\overline{\\nu }_m)\\bigg |\\right\\rbrace .$ Combining the above display with (REF ), we further have: $&\\;\\;\\;\\;\\left|\\frac{1}{2}W_2^2(\\widetilde{\\mu }_m,\\widetilde{\\nu }_n)-\\frac{1}{2}W_2^2(\\widetilde{\\mu }_m,\\overline{\\nu }_m)-\\left(\\frac{1}{2}\\int \\Vert y\\Vert ^2\\,d(\\widetilde{\\nu }_n-\\overline{\\nu }_m)\\right)\\right|\\nonumber \\\\ &\\le \\max \\left\\lbrace \\bigg |\\int \\Psi _{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n}^* \\,d(\\widetilde{\\nu }_n-\\overline{\\nu }_m)\\bigg |,\\bigg |\\int \\Psi _{\\widetilde{\\mu }_m,\\overline{\\nu }_m}^* \\,d(\\widetilde{\\nu }_n-\\overline{\\nu }_m)\\bigg |\\right\\rbrace .$ Combining (REF ) with (REF ) then completes the proof.",
"[Proof of thm:nsmooth] First observe that $\\limsup _{M\\rightarrow \\infty }\\limsup _{m,n\\rightarrow \\infty } \\mathbb {P}\\left(\\Big |\\int \\varphi _0^*\\,d(\\widehat{\\nu }_n-\\overline{\\nu }_m)\\big |\\ge M \\left(r_d^{(m,m)}+r_d^{(n,n)}\\right)\\right)=0$ by the weak law of large numbers as $(r_{d}^{(n,n)})^{-1}n^{-1/2}=O(1)$ and $(r_{d}^{(m,m)})^{-1}m^{-1/2}=O(1)$ .",
"Combining the above observation with thm:newubd, we have: $&\\;\\;\\;\\;\\limsup _{M\\rightarrow \\infty }\\limsup _{m,n\\rightarrow \\infty }\\mathbb {P}\\left(\\sup \\limits _{\\gamma \\in \\widetilde{\\Gamma }_{\\mathrm {min}}} \\int \\Vert \\widetilde{T}_{m,n}^{\\gamma }(x)-T_0(x)\\Vert ^2 \\,d\\widetilde{\\mu }_m(x)\\ge Mr_d^{(m,n)}\\right)\\nonumber \\\\ &\\le \\limsup _{M\\rightarrow \\infty }\\limsup _{m,n\\rightarrow \\infty }\\mathbb {P}\\left(\\max \\left\\lbrace \\bigg |\\int \\Psi _{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n}^* \\,d(\\widetilde{\\nu }_n-\\overline{\\nu }_m)\\bigg |,\\bigg |\\int \\Psi _{\\widetilde{\\mu }_m,\\overline{\\nu }_m}^* \\,d(\\widetilde{\\nu }_n-\\overline{\\nu }_m)\\bigg |\\right\\rbrace \\ge \\frac{M}{2} r_d^{(m,n)}\\right)\\nonumber \\\\ &\\le \\limsup _{M\\rightarrow \\infty }\\limsup _{m,n\\rightarrow \\infty }\\Bigg [\\mathbb {P}\\left(\\left|\\int \\Psi _{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n}^* \\,d(\\widetilde{\\nu }_n-\\nu )\\right|\\ge \\frac{M}{2}r_d^{(n,n)}\\right)+\\mathbb {P}\\left(\\left|\\int \\Psi _{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n}^* \\,d(\\overline{\\nu }_m-\\nu )\\right|\\ge \\frac{M}{2}r_d^{(m,m)}\\right)\\nonumber \\\\&+\\mathbb {P}\\left(\\left|\\int \\Psi _{\\widetilde{\\mu }_m,\\overline{\\nu }_m}^* \\,d(\\widetilde{\\nu }_n-\\nu )\\right|\\ge \\frac{M}{2} r_d^{(n,n)}\\right)+\\mathbb {P}\\left(\\left|\\int \\Psi _{\\widetilde{\\mu }_m,\\overline{\\nu }_m}^* \\,d(\\overline{\\nu }_m-\\nu )\\right|\\ge \\frac{M}{2} r_d^{(m,m)}\\right)\\Bigg ].$ In the sequel, we will only discuss how to bound the first term on the right hand side of (REF ).",
"Once that is understood, the other terms can be bounded similarly.",
"Therefore, our focus is on bounding $\\limsup _{M\\rightarrow \\infty }\\limsup _{m,n\\rightarrow \\infty }\\mathbb {P}\\left(\\left|\\int \\Psi _{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n}^* \\,d(\\widetilde{\\nu }_n-\\nu )\\right|\\ge \\frac{M}{2}r_d^{(n,n)}(\\log {(1+\\max \\lbrace m,n\\rbrace )})^{t_{d,\\alpha }}\\right).$ For the next part, to simplify notation, let us begin with some notation.",
"Set $\\mathcal {Y}:=\\mbox{supp}(\\nu )$ and $\\mathcal {X}_{n,\\mu }$ denote the closure of the convex hull of $X_1,\\ldots ,X_n$ .",
"Note that if we replace $\\Psi _{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n}(\\cdot )$ by $\\Psi _{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n}(\\cdot )-C$ for some constant $C>0$ , then $\\Psi _{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n}^*(\\cdot )\\mapsto \\Psi _{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n}^*+C$ .",
"However replacing $\\Psi _{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n}^*(\\cdot )$ by $\\Psi _{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n}^*(\\cdot )+C$ in (REF ) doesn't change its value as $\\widetilde{\\nu }_n$ and $\\nu $ are both probability measures.",
"Therefore, without loss of generality, we can assume that $\\Psi _{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n}(X_1)=0$ for all $m,n$ .",
"We will stick to this convention for the rest of the proof.",
"Also note that $\\Psi _{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n}(\\cdot )$ is only determined at the data points $X_1,\\ldots ,X_n$ .",
"Without loss of generality, we extend $\\Psi _{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n}(\\cdot )$ to the whole of $\\mathbb {R}^d$ by linear interpolation for any $x\\in \\mathcal {X}_{n,\\mu }$ and setting $\\Psi _{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n}(x)=\\infty $ for $x\\in \\mathcal {X}_{n,\\mu }^c$ .",
"The proof now proceeds using the following steps: Step I: There exists a constant $C_1>0$ and $y_n\\in \\mbox{supp}(\\nu )=\\mathcal {Y}$ such that $|\\Psi _{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n}^*(y_n)|\\le \\max _{1\\le i\\le m}\\Vert X_i\\Vert .$ [Proof of step I] By Kantorovich duality, there exists $y_n$ such that $\\Psi _{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n}^*(y_n)+\\Psi _{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n}(X_1)=\\langle X_1,y_n\\rangle \\quad \\Rightarrow \\quad |\\Psi _{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n}^*(y_n)|\\le C_1\\Vert X_1\\Vert \\le C_1\\max _{1\\le i\\le m}\\Vert X_i\\Vert ,$ where $C_1:=\\sup \\lbrace \\Vert y\\Vert :\\ y\\in \\mathcal {Y}\\rbrace $ .",
"Step II: There exists a constant $C_2>0$ such that the following holds: $\\Vert \\Psi _{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n}^*\\Vert _{\\infty ,\\mathcal {Y}}\\le C_2\\max _{1\\le i\\le n} \\Vert X_i\\Vert ,$ where $\\Vert \\cdot \\Vert _{\\infty ,\\mathcal {Y}}$ is the uniform norm on the support of $\\nu $ .",
"[Proof of step II] As $\\Psi _{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n}(x)=\\infty $ for $x\\in \\mathcal {X}_{n,\\mu }^c$ , using (REF ), we can write $\\Psi _{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n}^*(y)=\\max _{x\\in \\mathcal {X}_{n,\\mu }} (\\langle x,y\\rangle -\\Psi _{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n}(x))$ for all $y\\in \\mathcal {Y}$ .",
"For any $y_0\\in \\mathcal {Y}$ , let $x_0\\in \\mathcal {X}_{n,\\mu }$ be such that $\\Psi _{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n}^*(y_0)=\\langle x_0,y_0\\rangle -\\Psi _{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n}(x_0)$ .",
"Then, for any $y\\in \\mathcal {X}$ , we have: $&{\\left\\lbrace \\begin{array}{ll} \\Psi _{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n}^*(y_0)=\\langle x_0,y_0\\rangle -\\Psi _{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n}(x_0)\\\\ \\Psi _{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n}^*(y)\\ge \\langle x_0,y\\rangle -\\Psi _{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n}(x_0)\\end{array}\\right.",
"}\\\\ \\Rightarrow &|\\Psi _{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n}^*(y_0)-\\Psi _{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n}^*(y)|\\le |\\langle x_0,y_0-y\\rangle |\\le \\left(\\max _{1\\le i\\le m}\\Vert X_i\\Vert \\right)\\Vert y_0-y\\Vert .$ where the last line uses the fact that $y_0,y$ are arbitrary.",
"In particular, by setting $y_0:=y_n$ from step I, we get: $\\Vert \\Psi _{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n}^*\\Vert _{\\infty ,\\mathcal {Y}}\\le |\\Psi _{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n}^*(y_n)|+ \\left(\\max _{1\\le i\\le m}\\Vert X_i\\Vert \\right)\\sup _{y\\in \\mathcal {Y}}\\Vert y_n-y\\Vert \\le C_2\\left(\\max _{1\\le i\\le m}\\Vert X_i\\Vert \\right),$ where $C_2:=3C_1$ with $C_1$ defined as specified in the proof of step I.",
"The above lemma allows us to bound (with high probability) the $L^{\\infty }$ -norm of $\\Psi _{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n}^*(\\cdot )$ on $\\mathcal {Y}$ , using the tail assumption $\\mathbb {E}\\exp (t\\Vert X_1\\Vert ^{\\alpha })<\\infty $ for some $t>0$ and $\\alpha >0$ .",
"This is the focus of the next step.",
"Step III: For $K>0$ , define the following two sets: $A_{m,n,K}:=\\left\\lbrace \\int (\\Psi _{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n}^*(u))^2\\,d\\nu (u)\\ge K\\right\\rbrace ,\\quad \\mathrm {and},$ $\\widetilde{A}_{m,n,K}:=\\left\\lbrace \\Vert \\Psi _{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n}^*\\Vert _{\\infty ,\\mathcal {Y}}\\ge K\\big (\\log {n}\\big )^{1/\\alpha }\\right\\rbrace .$ Then there exists $K_0>0$ such that for any $K\\ge K_0$ , we have: $\\lim _{m,n\\rightarrow \\infty }\\mathbb {P}(\\widetilde{A}_{m,n,K})=0.$ and $\\lim _{m,n\\rightarrow \\infty }\\mathbb {P}(A_{m,n,K})=0.$ [Proof of step III] By using the exponential Markov's inequality coupled with the standard union bound, we have: $\\mathbb {P}\\left(\\max _{1\\le i\\le m} \\Vert X_i\\Vert \\ge K(\\log {m})^{1/\\alpha }\\right)&\\le m\\mathbb {P}\\left(\\Vert X_1\\Vert \\ge K(\\log {m})^{1/\\alpha }\\right)\\\\ &\\le m\\exp (-tK^{\\alpha }(\\log {m}))\\mathbb {E}\\exp (t\\Vert X_1\\Vert ^{\\alpha })\\overset{m\\rightarrow \\infty }{\\longrightarrow }0$ provided $K>t^{-\\alpha }$ .",
"Using the above observation coupled with step II, (REF ) follows by choosing $K_0>C_2 t^{-\\alpha }$ .",
"For the next part, we define another set: $B_{m,n,\\varepsilon }:=\\left\\lbrace \\int \\big |\\Psi _{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n}^*(u)-\\Psi _{\\mu ,\\nu }^*(u)|^2\\,d\\nu (u)\\ge \\varepsilon \\right\\rbrace $ for $\\varepsilon >0$ , where, as in (), we have: $W_2^2(\\mu ,\\nu )=\\int \\Vert x\\Vert ^2\\,d\\mu (x)+\\int \\Vert y\\Vert ^2\\,d\\nu (y)-2\\left(\\int \\Psi _{\\mu ,\\nu }(x)\\,d\\mu (x)+\\int \\Psi _{\\mu ,\\nu }^*(y)\\,d\\nu (y)\\right).$ Now by using [45], we have $\\mathbb {P}(B_{m,n,\\varepsilon })\\rightarrow 0$ as $m,n\\rightarrow \\infty $ for all $\\varepsilon >0$ .",
"As $\\int (\\Psi _{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n}^*(u))^2\\,d\\nu (u)\\le 2\\int \\big |\\Psi _{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n}^*(u)-\\Psi _{\\mu ,\\nu }^*(u)|^2\\,d\\nu (u)+2\\int (\\Psi _{\\mu ,\\nu }^*(u))^2\\,d\\nu (u),$ (REF ) follows with $K_0>2\\int (\\Psi _{\\mu ,\\nu }^*(u))^2\\,d\\nu (u)+1$ if we choose $\\epsilon =1/2$ .",
"We are now in a position to complete the proof of thm:nsmooth using steps I-III.",
"Towards this direction, set $K^{\\prime }:=2K_0$ where $K_0$ is defined as in the proof of step III and observe that for any $M>0$ , $&\\;\\;\\;\\limsup _{M\\rightarrow \\infty }\\limsup _{m,n\\rightarrow \\infty }\\mathbb {P}\\left(\\Bigg |\\int \\Psi _{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n}^*(u)\\,d(\\widetilde{\\nu }_n-\\nu )\\Bigg |\\ge Mr_d^{(n,n)}(\\log {(1+m)})^{t_{d,\\alpha }}\\right)\\nonumber \\\\&\\le \\limsup _{M\\rightarrow \\infty }\\limsup _{m,n\\rightarrow \\infty }\\mathbb {P}\\left(\\Bigg |\\int \\Psi _{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n}^*(u)\\,d(\\widetilde{\\nu }_n-\\nu )\\Bigg |\\ge Mr_d^{(n,n)}(\\log {(1+m)})^{t_{d,\\alpha }}, A_{m,n,K^{\\prime }}^c\\cap \\widetilde{A}_{m,n,K^{\\prime }}^c\\right)\\nonumber \\\\ &+\\limsup \\limits _{n\\rightarrow \\infty }\\mathbb {P}(\\widetilde{A}_{m,n,K^{\\prime }})+\\limsup \\limits _{m,n\\rightarrow \\infty }\\mathbb {P}(A_{m,n,K^{\\prime }})\\nonumber \\\\&\\le \\limsup _{M\\rightarrow \\infty }\\limsup \\limits _{m,n\\rightarrow \\infty }\\mathbb {P}\\left(\\Bigg |\\int \\Psi _{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n}^*(u)\\,d(\\widetilde{\\nu }_n-\\nu )\\Bigg |\\ge Mr_d^{(n,n)}(\\log {(1+m)})^{t_{d,\\alpha }},A_{m,n,K^{\\prime }}^c\\cap \\widetilde{A}_{m,n,K^{\\prime }}^c\\right),$ where the last step follows from step III.",
"Observe that the left hand side of (REF ) is the same as (REF ).",
"Therefore, it is now enough to bound the right hand side of (REF ).",
"In order to achieve the above task, let us define the following class of functions: $\\mathcal {C}^{\\Gamma ,L}(\\mathcal {Y}):=\\lbrace f:\\mathcal {Y}\\rightarrow \\mathbb {R}, f\\ \\mathrm {is}\\ \\mathrm {convex,} \\ \\Vert f\\Vert _{\\infty ,\\mathcal {Y}}\\le \\Gamma , \\ \\Vert f\\Vert _{L^2(\\nu )}\\le L\\rbrace .$ By setting $\\Gamma :=K^{\\prime }(\\log {m})^{1/\\alpha }$ and $L:=K^{\\prime }$ , (REF ) yields the following conclusion: $&\\;\\;\\;\\limsup _{M\\rightarrow \\infty }\\limsup \\limits _{m,n\\rightarrow \\infty }\\mathbb {P}\\left(\\Bigg |\\int \\Psi _{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n}^*(u)\\,d(\\widetilde{\\nu }_n-\\nu )\\Bigg |\\ge Mr_d^{(n,n)}(\\log {(1+m)})^{t_{d,\\alpha }}\\right)\\\\ & \\le \\limsup _{M\\rightarrow \\infty }\\limsup _{m,n\\rightarrow \\infty }\\mathbb {P}\\left(\\sup _{f\\in \\mathcal {C}^{\\Gamma ,L}(\\mathcal {Y})}\\Bigg |\\int f\\,d(\\widetilde{\\nu }_n-\\nu )\\Bigg |\\ge Mr_d^{(n,n)}(\\log {(1+m)})^{t_{d,\\alpha }}\\right).$ By an application of Markov's inequality, it thus suffices to show that: $\\mathbb {E}\\left[\\sup _{f\\in \\mathcal {C}^{\\Gamma ,L}(\\mathcal {Y})}\\Bigg |\\int f\\,d(\\widetilde{\\nu }_n-\\nu )\\Bigg |\\right]=\\mathcal {O}\\left(r_d^{(n,n)}(\\log {(1+m)})^{t_{d,\\alpha }}\\right).$ In order to bound (REF ), we will use some standard empirical process techniques.",
"In particular, by using [136], the following bound holds: $&\\;\\;\\;\\mathbb {E}\\left[\\sup _{f\\in \\mathcal {C}^{\\Gamma ,L}(\\mathcal {Y})}\\Bigg |\\int f\\,d(\\widetilde{\\nu }_n-\\nu )\\Bigg |\\right]\\nonumber \\\\ &\\le D\\inf \\left\\lbrace a\\ge \\frac{\\Gamma }{\\sqrt{n}}:a\\ge \\frac{D}{\\sqrt{n}}\\int _a^{\\Gamma }\\sqrt{\\log N_{[]}(\\varepsilon ,\\mathcal {C}^{\\Gamma ,L}(\\mathcal {Y}),L^2(\\nu ))}\\,d\\varepsilon \\right\\rbrace ,$ for some positive constant $D>0$ , where $N_{[]}(\\varepsilon ,\\mathcal {C}^{\\Gamma ,L}(\\mathcal {Y}),L^2(\\nu ))$ is the $\\varepsilon $ -bracketing number of the class of functions $\\mathcal {C}^{\\Gamma ,L}(\\mathcal {Y})$ with respect to the $L^2(\\nu )$ norm.",
"Note that by [91], we have: $\\log N_{[]}(\\varepsilon ,\\mathcal {C}^{\\Gamma ,L}(\\mathcal {Y}),L^2(\\nu ))\\le \\gamma _d\\left(\\log {\\frac{\\Gamma }{\\varepsilon }}\\right)^{d+1}\\left(\\frac{L}{\\varepsilon }\\right)^{d/2}$ for some $\\gamma _d>0$ depending only on fand the diameter of $\\mathcal {Y}$ .",
"We will now bound the right hand side of (REF ).",
"Also we will use $D_d$ to denote changing constants which can depend on $d$ .",
"When $d=1,2,3$: Choose $a=D_d\\frac{(\\log {n})^{\\frac{1}{\\alpha }\\vee \\frac{2\\alpha +2d\\alpha -d+4}{4\\alpha }}}{\\sqrt{n}}$ .",
"Observe that: $\\frac{1}{\\sqrt{n}}\\int _a^{\\Gamma }\\sqrt{\\log N_{[]}(\\varepsilon ,\\mathcal {C}^{\\Gamma ,L}(\\mathcal {Y}),L^2(\\nu ))}\\,d\\varepsilon &\\le \\frac{(\\log {n})^{(d+1)/2}}{\\sqrt{n}}\\cdot \\left[\\frac{\\varepsilon ^{1-d/4}}{1-d/4}\\right]_{0}^{\\Gamma }\\\\ &\\lesssim \\frac{(\\log {n})^{(4-d)/(4\\alpha )}\\times (\\log {n})^{(d+1)/2}}{\\sqrt{n}}\\lesssim a.$ When $d=4$: Choose $a=D_d\\frac{(\\log {n})^{\\frac{1}{\\alpha }\\vee \\frac{7}{2}}}{\\sqrt{n}}$ .",
"Observe that: $\\frac{1}{\\sqrt{n}}\\int _a^{\\Gamma }\\sqrt{\\log N_{[]}(\\varepsilon ,\\mathcal {C}^{\\Gamma ,L}(\\mathcal {Y}),L^2(\\nu ))}\\,d\\varepsilon &\\le \\frac{(\\log {n})^{5/2}}{\\sqrt{n}}\\cdot \\left[\\log {\\varepsilon }\\right]_{D_d\\Gamma /\\sqrt{n}}^{\\Gamma }\\\\ &\\lesssim \\frac{(\\log {n})^{(7/2)}}{\\sqrt{n}}\\lesssim a.$ When $d>4$: Choose $a=D_d\\frac{(\\log {n})^{2(1+d^{-1})}}{n^{2/d}}$ .",
"Observe that: $\\frac{1}{\\sqrt{n}}\\int _a^{\\Gamma _0}\\sqrt{\\log N_{[]}(\\varepsilon ,\\mathcal {C}^{\\Gamma ,L}(\\mathcal {Y}),L^2(\\nu ))}\\,d\\varepsilon &\\le \\frac{(\\log {n})^{(d+1)/2}}{\\sqrt{n}}\\cdot \\left[\\frac{\\varepsilon ^{1-d/4}}{1-d/4}\\right]_{a}^{\\Gamma }\\\\ &\\lesssim \\frac{a^{1-d/4}(\\log {n})^{(d+1)/2}}{\\sqrt{n}}\\lesssim a.$ This completes the proof after applying the same technique on the other 3 terms on the right hand side of (REF ).",
"[Proof of cor:fsam] First observe that $\\mathbb {E}\\left[\\int \\varphi _0^*\\,d\\widehat{\\nu }_n\\right]=\\mathbb {E}\\left[\\int \\varphi _0^*\\,d\\overline{\\nu }_m\\right]=\\int \\varphi _0^*\\,d\\nu .$ Using the above observation and the same approach used as in the proof of thm:nsmooth, we will only focus on bounding $\\mathbb {E}\\bigg |\\int \\Psi _{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n}^* \\,d(\\widetilde{\\nu }_n-\\nu )\\bigg |.$ The general strategy to bound the term in (REF ) is derived from some intermediate steps in the proofs of [33].",
"We still present a sketch here for completeness.",
"By the same argument as in the proof of thm:nsmooth and using the fact that there exists fixed $R>0$ such that $\\max _{1\\le i\\le m}\\Vert X_i\\Vert \\le R$ , we have $\\Psi _{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n}^*(\\cdot )$ is a convex and $R$ -Lipschitz function on $\\mathcal {Y}$ .",
"This observation implies: $\\mathbb {E}\\bigg |\\int \\Psi _{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n}^* \\,d(\\widetilde{\\nu }_n-\\overline{\\nu }_m)\\bigg |\\le \\mathbb {E}\\left[\\sup _{\\psi \\in \\mathcal {F}_R(\\mathcal {Y})}\\left|\\int \\psi \\,d(\\widehat{\\nu }_n-\\nu )\\right|\\right]$ where $\\mathcal {F}_R(\\mathcal {Y})$ is the set of convex and $R$ -Lipschitz functions on $\\mathcal {Y}$ .",
"By [141], we then have: $\\mathbb {E}\\left[\\sup _{\\psi \\in \\mathcal {F}_R(\\mathcal {Y})}\\left|\\int \\psi \\,d(\\widehat{\\nu }_n-\\nu )\\right|\\right]\\lesssim \\inf _{\\delta >0}\\left(\\delta +n^{-1/2}\\int _{\\delta }^{R^2}\\sqrt{\\log {\\mathcal {N}_{\\infty }(\\mathcal {F}_R(\\mathcal {Y}),\\varepsilon )}}\\,d\\varepsilon \\right),$ where $\\mathcal {N}_{\\infty }(\\mathcal {F}_R(\\mathcal {Y}),\\varepsilon )$ is the $\\varepsilon $ -covering number of the set $\\mathcal {F}_R(\\mathcal {Y})$ with respect to the uniform metric.",
"By using [74] (also see [22]), there exists constants $C_1,C_2>0$ such that whenever $\\varepsilon /R^2\\le C_1$ , then $\\log {\\mathcal {N}_{\\infty }(\\mathcal {F}_R(\\mathcal {Y}),\\varepsilon )}\\le C_2(u/R^2)^{-d/2}$ .",
"By using this bound in (REF ), we get: $\\mathbb {E}\\left[\\sup _{\\psi \\in \\mathcal {F}_R(\\mathcal {Y})}\\left|\\int \\psi \\,d(\\widehat{\\nu }_n-\\nu )\\right|\\right]\\lesssim \\inf _{\\delta >0}\\left(\\delta +n^{-1/2}\\int _{\\delta }^1 \\varepsilon ^{-d/4}\\,d\\varepsilon \\right).$ Setting $\\delta =0$ for $d<4$ and $\\delta =n^{-2/d}$ for $d\\ge 4$ in (REF ), followed by a direct application of (REF ), we have: $\\mathbb {E}\\bigg |\\int \\Psi _{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n}^* \\,d(\\widetilde{\\nu }_n-\\overline{\\nu }_m)\\bigg |\\lesssim r_d^{(n,n)}.$ This completes the proof.",
"[Proof of thm:smoothwav] For this proof, we will use an intermediate step in the proof of thm:newubd, which is (REF ), that can alternatively be written as: $\\mathbb {E}\\left[\\int \\Vert \\widetilde{T}_{m,n}^{\\gamma }(x)-T_0(x)\\Vert ^2\\,d\\widetilde{\\mu }_m(x)\\right]\\lesssim \\mathbb {E}|W_2^2(\\widetilde{\\mu }_m,\\widetilde{\\nu }_n)-W_2^2(\\mu ,\\nu )|+\\mathbb {E}\\left|\\int h(y)\\,d(\\widetilde{\\nu }_n-\\overline{\\nu }_m)(y)\\right|$ where $h(y):=\\varphi _0^*(y)-(1/2)\\Vert y\\Vert ^2$ and $C>0$ is some constant.",
"As $\\mathcal {X}$ and $\\mathcal {Y}$ are compact sets, the function $h(\\cdot )$ is Lipschitz.",
"Therefore, $\\mathbb {E}\\left|\\int h(y)\\,d(\\widetilde{\\nu }_n-\\nu )(y)\\right|\\lesssim W_1(\\widetilde{\\nu }_n,\\nu )\\le W_2(\\widetilde{\\nu }_n,\\nu ).$ Further, as $T_0(\\cdot )$ is also Lipschitz, we further have: $\\mathbb {E}\\left|\\int h(y)\\,d(\\overline{\\nu }_m-\\nu )(y)\\right|\\lesssim W_1(T_0\\#\\widetilde{\\mu }_m,T_0\\#\\mu )\\lesssim W_1(\\widetilde{\\mu }_m,\\mu )\\le W_2(\\widetilde{\\mu }_m,\\mu ).$ Finally, by the triangle inequality, we also have: $\\mathbb {E}|W_2^2(\\widetilde{\\mu }_m,\\widetilde{\\nu }_n)-W_2^2(\\mu ,\\nu )|&\\lesssim \\mathbb {E}|W_2(\\widetilde{\\mu }_m,\\widetilde{\\nu }_n)-W_2(\\widetilde{\\mu }_m,\\nu )|+\\mathbb {E}|W_2(\\widetilde{\\mu }_m,\\nu )-W_2(\\mu ,\\nu )|\\\\ &\\le \\mathbb {E}W_2(\\widetilde{\\mu }_m,\\mu )+\\mathbb {E}W_2(\\widetilde{\\nu }_n,\\nu ).$ Combining the three displays above and plugging them back in (REF ), we get: $\\mathbb {E}\\left[\\int \\Vert \\widetilde{T}_{m,n}^{\\gamma }(x)-T_0(x)\\Vert ^2\\,d\\widetilde{\\mu }_m(x)\\right]\\lesssim \\mathbb {E}W_2(\\widetilde{\\mu }_m,\\mu )+\\mathbb {E}W_2(\\widetilde{\\nu }_n,\\nu ).$ The conclusion then follows from [143].",
"[Proof of thm:smooth] Part 1.",
"By the same arguments (see e.g., (REF )) as used in the proof of thm:nsmooth, it suffices to show that $\\mathbb {E}\\left|\\int \\Psi _{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n}^*(u)(\\widetilde{f}_{\\nu }^{M^{\\prime }}(u)-f_{\\nu }(u)) \\,du\\right|\\lesssim r_{d,s}^{(n,n)}$ for some $M^{\\prime }>0$ .",
"The general structure of the proof is similar to that of thm:nsmooth.",
"The crucial observation is that $\\widetilde{f}_{\\mu }^{M^{\\prime }}(\\cdot )$ and $\\widetilde{f}_{\\nu }^{M^{\\prime }}(\\cdot )$ are elements of $C^s(\\mathcal {X};TM)$ and $C^s(\\mathcal {Y};TM)$ respectively, for any $M^{\\prime }>0$ .",
"Note that, by Caffarelli regularity theory; see [80], there exists $M^{\\prime }>0$ such that $\\Vert \\Psi _{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n}^*(\\cdot )\\Vert _{C^{s+2}(\\mathcal {Y})}\\le M^{\\prime }.$ Next, let us define the following class of functions: $\\mathcal {G}_t^L(\\mathcal {Y}):=\\lbrace g:\\mathcal {Y}\\rightarrow \\mathbb {R},\\ g(\\cdot )\\ \\mbox{is}\\ \\mbox{convex},\\ \\Vert g\\Vert _{C^t(\\mathcal {Y})}\\le L\\rbrace .$ Observe that $\\mathbb {E}\\left|\\int \\Psi _{\\widetilde{\\mu }_m,\\widetilde{\\nu }_n}^*(u)(\\widetilde{f}_{\\nu }^{M^{\\prime }}(u)-f_{\\nu }(u)) \\,du\\right|&\\le \\mathbb {E}\\sup _{g\\in \\mathcal {G}_{s+2}^{M^{\\prime }}(\\mathcal {Y})}\\left|\\int g(u)(\\widetilde{f}_{\\nu }^{M^{\\prime }}(u)-f_{\\nu }(u)) \\,du\\right|\\nonumber \\\\ &\\le 2\\mathbb {E}\\sup _{g\\in \\mathcal {G}_{s+2}^{M^{\\prime }}(\\mathcal {Y})}\\left|\\int g(u)(\\widehat{f}_{\\nu }(u)-f_{\\nu }(u)) \\,du\\right|+r_{d,s}^{(n,n)}$ where the last line follows from (REF ).",
"Set $K_{d,h_n}(\\cdot ):=h_n^{-d}K_d(\\cdot /h_n)$ .",
"Following the same decomposition as in [115], we write: $&\\;\\;\\;\\;\\mathbb {E}\\sup _{g\\in \\mathcal {G}_{s+2}^{M^{\\prime }}(\\mathcal {Y})}\\left|\\int g(u)(\\widehat{f}_{\\nu }(u)-f_{\\nu }(u)) \\,du\\right|\\nonumber \\\\ &=\\mathbb {E}\\sup _{g\\in \\mathcal {G}_{s+2}^{M^{\\prime }}(\\mathcal {Y})}\\left|\\int g(u+u^{\\prime })K_{d,h_n}(u^{\\prime })\\,d\\widehat{\\nu }_n(u)\\,du^{\\prime }-\\int g(u)f_{\\nu }(u)\\,du\\right|\\nonumber \\\\ &\\le \\mathbb {E}\\sup _{g\\in \\mathcal {G}_{s+2}^{M^{\\prime }}(\\mathcal {Y})}\\left|\\int g(u+u^{\\prime })K_{d,h_n}(u^{\\prime })\\,d(\\widehat{\\nu }_n-\\nu )(u)\\,du^{\\prime }\\right|\\nonumber \\\\ &\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;+\\sup _{g\\in \\mathcal {G}_{s+2}^{M^{\\prime }}(\\mathcal {Y})}\\left|\\int g(u+u^{\\prime })K_{d,h_n}(u^{\\prime })f_{\\nu }(u)\\,du\\,du^{\\prime }-\\int g(u)f_{\\nu }(u)\\,du\\right|.$ We will now bound the two terms on the right hand side of (REF ).",
"For the first term, define $\\overline{g}_n(u):=\\int g(u+u^{\\prime })K_{d,h_n}(u)\\,du^{\\prime }.$ If $g\\in \\mathcal {G}_t^L(\\mathcal {Y}^o)$ , then by [53] and using as:kernel, we have $\\overline{g}_n\\in \\mathcal {G}_{s+2}^{cM^{\\prime }}(\\mathcal {Y}^o)$ , for some constant $c>0$ (depending on the constants involved in as:kernel and the diameter of $\\mathcal {Y}$ ).",
"Combining these observations with (REF ), we get: $&\\;\\;\\;\\;\\mathbb {E}\\sup _{g\\in \\mathcal {G}_{s+2}^{M^{\\prime }}(\\mathcal {Y})}\\left|\\int g(u+u^{\\prime })K_{d,h_n}(u^{\\prime })\\,d(\\widehat{\\nu }_n-\\nu )(u)\\,du^{\\prime }\\right|\\nonumber \\\\ &\\le \\mathbb {E}\\sup _{g\\in \\mathcal {G}_{s+2}^{cM^{\\prime }}(\\mathcal {Y})}\\left|\\int g(u)\\,d(\\widehat{\\nu }_n-\\nu )(u)\\right|\\nonumber \\\\ &\\le D\\inf \\left\\lbrace a\\ge \\frac{cM^{\\prime }}{\\sqrt{n}}:a\\ge \\frac{D}{\\sqrt{n}}\\int _a^{cM^{\\prime }}\\sqrt{\\log N_{[]}(\\varepsilon ,\\mathcal {G}_{s+2}^{cM^{\\prime }}(\\mathcal {Y}),L^2(\\nu ))}\\,d\\varepsilon \\right\\rbrace ,$ for some positive constant $D>0$ , where $N_{[]}(\\varepsilon ,\\mathcal {G}_{s+2}^{cM^{\\prime }}(\\mathcal {Y}),L^2(\\nu ))$ is the $\\varepsilon $ -bracketing entropy of the class of functions $\\mathcal {G}_{s+2}^{cM^{\\prime }}(\\mathcal {Y})$ with respect to the $L^2(\\nu )$ norm.",
"The last line follows from standard empirical process theory as used in the proof of thm:nsmooth; see (REF ).",
"Note that by [138], we have: $\\log N_{[]}(\\varepsilon ,\\mathcal {G}_{s+2}^{cL}(\\mathcal {Y}^o),L^2(\\nu ))\\le \\gamma _d\\left(\\frac{1}{\\varepsilon }\\right)^{d/(s+2)}$ for some $\\gamma _d>0$ depending only on dimension and the diameter of $\\mathcal {Y}$ .",
"We now plug-in the above bound into (REF ).",
"By using $D_d$ to denote constants that change with $d$ and choosing $a=D_d n^{-1/2}$ for $2(s+2)>d$ , $a=D_d n^{-1/2}\\log {(1+n)}$ for $2(s+2)=d$ and $a=D_d n^{-(s+2)/d}$ for $2(s+2)<d$ in (REF ), we have: $\\mathbb {E}\\sup _{g\\in \\mathcal {G}_{s+2}^{M^{\\prime }}(\\mathcal {Y})}\\left|\\int g(u+u^{\\prime })K_{d,h_n}(u^{\\prime })\\,d(\\widehat{\\nu }_n-\\nu )(u)\\,du^{\\prime }\\right|\\lesssim r_{d,s}^{(n,n)}.$ We now move on to the bounding the second term on the right hand side of (REF ).",
"For this part, our main technical tool will be the classical arguments for smoothed empirical processes developed in [63].",
"Towards this direction, set $\\overline{g}(u)=g(-u)$ (different from $\\overline{g}_n(\\cdot )$ defined earlier) for $g(\\cdot )\\in \\mathcal {G}_{s+2}^{M^{\\prime }}$ and note that by [63], we have: $\\left|\\int g(u+u^{\\prime })K_{d,h_n}(u^{\\prime })f_{\\nu }(u)\\,du\\,du^{\\prime }-\\int g(u)f_{\\nu }(u)\\,du\\right|=\\left|\\int K_d(u)\\left[(\\overline{g}*f_{\\nu })(h_n u)-(\\overline{g}*f_{\\nu })(0)\\right]\\,du\\right|,$ where $(\\overline{g}*f_{\\nu })(\\cdot )$ is the standard convolution between $\\overline{g}(\\cdot )$ and $f_{\\nu }(\\cdot )$ , and with a notational abuse 0 denotes the $d$ -dimensional zero vector.",
"The important observation now is to note that $(\\overline{g}*f_{\\nu })(\\cdot )$ belongs to a higher order Sobolev class compared to $\\overline{g}(\\cdot )$ and $f_{\\nu }(\\cdot )$ .",
"In particular, as $f_{\\nu }(\\cdot )\\in C^s(\\mathcal {Y};M)$ and $\\overline{g}(\\cdot )\\in \\mathcal {G}_{s+2}^{M^{\\prime }}(\\mathcal {Y})$ , we have $(\\overline{g}*f_{\\nu })(\\cdot )\\in \\mathcal {G}_{2s+2}^{M^{\\prime \\prime }}(\\mathcal {Y})$ where $M^{\\prime \\prime }$ depends on both $M^{\\prime }$ and $M$ .",
"Next, write $D^t(\\overline{g}*f_{\\nu })(\\cdot )$ to be the $t$ -th derivative of $(\\overline{g}*f_{\\nu })(\\cdot )$ and note that by a multivariate Taylor's approximation $&\\;\\;\\;\\;\\int K_d(u)\\left[(\\overline{g}*f_{\\nu })(h_n u)-(\\overline{g}*f_{\\nu })(0)\\right]\\,du\\\\ &=\\int K_d(u)\\sum _{r=1}^{2s+1} h_n^r\\sum _{(i_1,i_2,\\ldots ,i_r)\\in \\lbrace 1,2,\\ldots ,d\\rbrace ^r} [D^r(\\overline{g}*f_{\\nu })(0)]_{i_1,\\ldots ,i_r} u_{i_1}\\ldots u_{i_r}\\,du+O(h_n^{2s+2}).$ Recall that $K_d(u)=K(u_1)K(u_2)\\ldots K(u_d)$ .",
"As $K(\\cdot )$ is of order $2s+2$ (see as:kernel), all the integrals on the right hand side of the above display vanish.",
"We then appeal to (REF ) to get: $\\left|\\int g(u+u^{\\prime })K_{d,h_n}(u^{\\prime })f_{\\nu }(u)\\,du\\,du^{\\prime }-\\int g(u)f_{\\nu }(u)\\,du\\right|\\lesssim h_n^{2s+2}=\\lesssim n^{-\\frac{2s+2}{d+2s}}(\\log {n})^{2s+2}.$ We now compare the right hand side of the above display with $r_{d,s}^{(n,n)}$ .",
"When $d<2(s+2)$ : $d+2s<4(s+1)$ , and therefore $\\frac{2s+2}{d+2s}>\\frac{1}{2}$ .",
"This implies $n^{-\\frac{2s+2}{d+2s}}(\\log {n})^{2s+2}\\lesssim n^{-\\frac{1}{2}}=r_{d,s}^{(n,n)}$ .",
"When $d=2(s+2)$ : In this case $n^{-\\frac{2s+2}{d+2s}}(\\log {n})^{2s+2}\\lesssim n^{-\\frac{1}{2}}(\\log {n})^{2s+2}\\lesssim r_{d,s}^{(n,n)}$ .",
"When $d>2(s+2)$ : Note that $\\frac{2s+2}{d+2s}>\\frac{s+2}{d}\\Leftrightarrow 2ds+2d>sd+2s^2+2d+4s\\Leftrightarrow d>2(s+2).$ Therefore, once again $n^{-\\frac{2s+2}{d+2s}}(\\log {n})^{2s+2}\\lesssim n^{-\\frac{s+2}{d}}=r_{d,s}^{(n,n)}$ .",
"Therefore, combining the above observations, we have: $\\left|\\int g(u+u^{\\prime })K_{d,h_n}(u^{\\prime })f_{\\nu }(u)\\,du\\,du^{\\prime }-\\int g(u)f_{\\nu }(u)\\,du\\right|\\lesssim r_{d,s}^{(n,n)}.$ Combining the above display with (REF ) establishes (REF ).",
"Part 2.",
"This proof uses ideas from [76], [112] and [5].",
"First recall all the notation introduced in def:holder.",
"Next, we will prove the following sequence of displays: $\\limsup \\limits _{m,n\\rightarrow \\infty }\\max _{k\\le s}\\max _{|\\mathfrak {m}|=k}\\Vert \\partial ^{\\mathfrak {m}}\\mathbb {E}\\widehat{f}_{\\mu }\\Vert _{L^{\\infty }(\\widetilde{\\mathcal {X}})}\\le (T-1)M,$ $\\limsup _{m,n\\rightarrow \\infty }\\ \\Vert \\mathbb {E}\\widehat{f}_{\\mu }-f_{\\mu }\\Vert _{L^{\\infty }(\\widetilde{\\mathcal {X}})}=0$ $\\limsup _{m,n\\rightarrow \\infty }\\mathbb {P}\\left(\\Vert \\widehat{f}_{\\mu }-\\mathbb {E}\\widehat{f}_{\\mu }\\Vert _{C^{s}(\\widetilde{\\mathcal {X}})}\\ge \\varepsilon \\right)=0,$ for any arbitrary $\\varepsilon >0$ and $L^{\\infty }(\\mathcal {X})$ denotes the uniform norm on $\\mathcal {X}$ .",
"Clearly, (REF ), (REF ), and (REF ) together yield part 1 of the theorem.",
"Proof of (REF ).",
"Observe that $\\mathbb {E}\\widehat{f}_{\\mu }(x)=\\frac{1}{h_m^d}\\mathbb {E}K_d\\left(\\frac{x-X_1}{h_m}\\right)=\\frac{1}{h_m^d}\\int K_d\\left(\\frac{x-z}{h_m}\\right)f_{\\mu }(z)\\,dz.$ Since the maximums taken in (REF ) are over finite sets, it suffices to show that for any fixed $\\mathfrak {m}$ with $|\\mathfrak {m}|\\le s$ , we have: $\\sup _{x\\in \\widetilde{\\mathcal {X}}}|\\partial ^{\\mathfrak {m}}\\mathbb {E}\\widehat{f}_{\\mu }(x)|=\\sup _{x\\in \\mathcal {X}}\\bigg |\\frac{1}{h_n^d}\\int K_d\\left(\\frac{z}{h_n}\\right)\\partial ^{\\mathfrak {m}} f(x+z)\\,dz\\Bigg |\\le (T-1)M.$ Here the first equality in the above display follows from (REF ) and Fubini's Theorem.",
"Here $\\partial ^{\\mathfrak {m}}f_{\\mu }(\\cdot )$ is defined in the weak sense, i.e., it is defined naturally in the interior of the support of $f_{\\mu }(\\cdot )$ , denoted by $\\mathcal {X}$ ; it is set to be 0 outside $\\mathcal {X}$ and defined arbitrarily on the boundary of $\\mathcal {X}$ .",
"Note that the definition on the boundary doesn't matter as we are integrating with respect to the Lebesgue measure and the boundary of $\\mathcal {X}$ has Lebesgue measure 0.",
"Next note that, by (REF ), we have: $\\sup _{x\\in \\widetilde{\\mathcal {X}}}|\\partial ^{\\mathfrak {m}}\\mathbb {E}\\widehat{f}_{\\mu }(x)|\\le \\Vert f_{\\mu }\\Vert _{C^s(\\mathcal {X})}h_m^{-d}\\int |K_d(z/h_m)|\\,dz\\le (T-1)\\Vert f_{\\mu }\\Vert _{C^s(\\mathcal {X})}.$ This establishes (REF ).",
"Proof of (REF ).",
"First note that, as $\\widetilde{\\mathcal {X}}$ is a compact subset of $\\mathcal {X}^o$ , there exists $\\delta >0$ such that $\\widetilde{\\mathcal {X}}_{\\delta ^{\\prime }}:=\\lbrace x+z:\\ \\Vert z\\Vert \\le \\delta ;,\\ x\\in \\widetilde{\\mathcal {X}}\\rbrace \\subseteq \\mathcal {X}^o\\qquad \\forall \\ 0<\\delta ^{\\prime }\\le \\delta .$ Clearly, $\\widetilde{\\mathcal {X}}_{\\delta ^{\\prime }}$ is compact for all $\\delta ^{\\prime }>0$ .",
"Fix an arbitrary $\\delta ^{\\prime }\\le \\delta $ .",
"By using (REF ) and a change of variable formula, we have: $\\Vert \\mathbb {E}\\widehat{f}_{\\mu }-f_{\\mu }\\Vert _{L^{\\infty }(\\widetilde{\\mathcal {X}})}&=\\sup _{x\\in \\widetilde{\\mathcal {X}}}\\bigg |\\frac{1}{h_m^d}\\int K_d\\left(\\frac{z}{h_m}\\right)(f(x+z)-f(x))\\,dz\\bigg | \\\\& \\le (T-1)\\sup _{x\\in \\widetilde{\\mathcal {X}}}\\sup _{\\Vert z\\Vert \\le \\delta ^{\\prime }}|f(x+z)-f(x)|+2M\\int _{\\Vert z\\Vert >\\delta ^{\\prime } h_m^{-1}} |K_d(z)|\\,dz\\\\ & \\le (T-1)M\\delta ^{\\prime }+2M\\left(\\frac{h_m}{\\delta ^{\\prime }}\\right)^{2s+2}\\int \\Vert z\\Vert ^{2s+2}|K_d(z)|\\,dz.$ Observe that as $m,n\\rightarrow \\infty $ , the second term on the right hand side of the above display converges to 0.",
"This implies $\\limsup \\limits _{m,n\\rightarrow \\infty }\\Vert \\mathbb {E}\\widehat{f}_{\\mu }-f_{\\mu }\\Vert _{L^{\\infty }(\\widetilde{\\mathcal {X}})}\\le (T-1)M\\delta ^{\\prime }.$ As $\\delta ^{\\prime }$ can be chosen arbitrarily small, this completes the proof of (REF ).",
"Proof of (REF ).",
"The main technical tool for this part is lem:emproc which we borrow from [5] (also see [99]).",
"The proof is very similar to [5].",
"Consider the following class of functions: $\\mathcal {G}=\\left\\lbrace g_x(z,h):\\ g_x(z,h)=\\partial ^{\\mathfrak {m}}K_d\\left(\\frac{(x-z)}{h}\\right),\\ x\\in \\widetilde{\\mathcal {X}},\\ |\\mathfrak {m}|\\le s \\right\\rbrace .$ Observe that $\\sup _{|\\mathfrak {m}|\\le s}\\sup _{x\\in \\widetilde{\\mathcal {X}}}\\sup _{h\\in (0,1)} h^{-d} \\mathbb {E}\\left[\\partial ^{\\mathfrak {m}}K_d\\left(\\frac{(x-z)}{h}\\right)\\right]^2\\le \\Vert K\\Vert _{C^s(\\mathbb {R}^d)}\\Vert f\\Vert _{C^s(\\mathcal {X})}\\sup _{|\\mathfrak {m}|\\le s}\\int |\\partial ^{\\mathfrak {m}} K_d(v)|\\,dv<\\infty .$ Further, by as:kernel, $\\partial ^{\\mathfrak {m}} K_d(\\cdot )$ is differentiable for each $|\\mathfrak {m}|\\le s$ .",
"Consequently $\\mathcal {G}$ is point wise measurable and of VC-type (see [49]; also see [138] for definitions of point wise differentiability and VC classes).",
"This verifies the assumptions of lem:emproc.",
"Observe that $\\frac{1}{n}\\sum _{i=1}^m \\partial ^{\\mathfrak {m}} K\\left(\\frac{x-X_i}{h_m}\\right)=h_m^{d+|\\mathfrak {m}|}\\partial ^{\\mathfrak {m}} \\widehat{f}_{\\mu }(x),\\qquad \\mathbb {E}\\left[\\partial ^{\\mathfrak {m}}K\\left(\\frac{x-X}{h_m}\\right)\\right]=h_m^{d+|\\mathfrak {m}|}\\mathbb {E}\\left[\\partial ^{\\mathfrak {m}}\\widehat{f}_{\\mu }(x)\\right].$ A direct application of lem:emproc for all $|\\mathfrak {m}|\\le s$ , then implies $\\sup _{x\\in \\widetilde{\\mathcal {X}}}\\sqrt{\\frac{m}{h_m^d\\log {m}}}\\cdot h_m^{s+d} \\Vert \\widehat{f}_{\\mu }-\\mathbb {E}\\widehat{f}_{\\mu }\\Vert _{C^s(\\widetilde{\\mathcal {X}})}=O_p(1).$ Using the observation that $mh_m^{d+2s}/\\log {m}\\rightarrow 0$ as $m\\rightarrow \\infty $ then completes the proof.",
"[Proof of prop:wassrate] As $\\mu \\ne \\nu $ , we have $W_2(\\mu ,\\nu )>0$ .",
"Therefore, $|W_2(\\widetilde{\\mu }_m,\\widetilde{\\nu }_n)-W_2(\\mu ,\\nu )|=\\frac{W_2^2(\\widetilde{\\mu }_m,\\widetilde{\\nu }_n)-W_2^2(\\mu ,\\nu )|}{W_2(\\widetilde{\\mu }_m,\\widetilde{\\nu }_n)+W_2(\\mu ,\\nu )}\\le \\frac{W_2^2(\\widetilde{\\mu }_m,\\widetilde{\\nu }_n)-W_2^2(\\mu ,\\nu )|}{W_2(\\mu ,\\nu )}.$ The conclusion then follows from thm:smooth.",
"[Proof of thm:dismooth] Recall that $\\widetilde{\\mu }_m$ and $\\widetilde{\\nu }_n$ are defined as the empirical distributions induced by $M=n^{\\frac{s+2}{2}}$ random samples drawn from $\\widehat{f}_{\\mu }$ and $\\widehat{f}_{\\nu }$ respectively, where $\\widehat{f}_{\\mu }$ , $\\widehat{f}_{\\nu }$ are the kernel density estimates as presented in (REF ).",
"Let us write $\\mu _{h_n}$ and $\\nu _{h_n}$ for the probability measure induced by the kernel density estimates $\\widehat{f}_{\\mu }$ and $\\widehat{f}_{\\nu }$ respectively.",
"Once again, by using thm:nsmooth, (REF ), it suffices to prove the following: $\\mathbb {E}\\left|W_2^2(\\widetilde{\\mu }_m,\\widetilde{\\nu }_n)-W_2^2(\\mu ,\\nu )\\right|.$ Next note that by the triangle inequality, (REF ) can be bounded above by: $\\mathbb {E}\\big |W_2^2(\\widetilde{\\mu }_m,\\nu _{h_n})&-W_2^2(\\widetilde{\\mu }_m,\\widetilde{\\nu }_n)\\big |+\\mathbb {E}\\big |W_2^2(\\widetilde{\\mu }_m,\\nu _{h_n})-W_2^2(\\mu _{h_n},\\nu _{h_n})\\big |\\nonumber \\\\&+\\mathbb {E}\\big |W_2^2(\\mu _{h_n},\\nu _{h_n})-W_2^2(\\widetilde{\\mu }_m,\\widetilde{\\nu }_n)\\big |.$ Next note that, by thm:smooth, we have: $\\mathbb {E}\\left|W_2^2(\\mu _{h_n},\\nu _{h_n})-W_2^2(\\widetilde{\\mu }_m,\\widetilde{\\nu }_n)\\right|\\lesssim r_{d,s}^{(n,n)}.$ Next we show that $\\mathbb {E}\\left|W_2^2(\\widetilde{\\mu }_m,\\nu _{h_n})-W_2^2(\\mu _{h_n},\\nu _{h_n})\\right|\\lesssim r_{d,s}^{(n,n)}.$ The other term in (REF ) can be bounded similarly.",
"Note that, conditioned on $X_1,\\ldots ,X_n,Y_1,\\ldots ,Y_n$ , $\\mu _{h_n}$ and $\\nu _{h_n}$ are non-random measures and $\\widetilde{\\mu }_m$ and $\\widetilde{\\nu }_n$ are the empirical distributions on $M=n^{\\frac{s+2}{2}}$ random samples from the measures $\\mu _{h_n}$ and $\\nu _{h_n}$ , respectively.",
"Therefore, conditioned on $X_1,\\ldots ,X_n,Y_1,\\ldots ,Y_n$ (which have fixed compact supports), we can invoke cor:fsam to get: $\\mathbb {E}\\left|W_2^2(\\widetilde{\\mu }_m,\\nu _{h_n})-W_2^2(\\mu _{h_n},\\nu _{h_n})\\right| \\lesssim r_{d}^{(M,M)},$ with $M=n^{\\frac{s+2}{2}}$ .",
"Recall that: $r_d^{(M,M)}={\\left\\lbrace \\begin{array}{ll} n^{-\\frac{s+2}{4}} & \\mbox{if}\\ d\\le 3\\\\ n^{-\\frac{s+2}{4}}\\log {(1+n)} & \\mbox{if}\\ d=4\\\\ n^{-\\frac{s+2}{d}} & \\mbox{if}\\ d>4\\end{array}\\right.",
"}.$ It therefore only remains to compare $r_d^{(M,M)}$ and $r_{d,s}^{(n,n)}$ .",
"Case 1: $d\\le 2(s+2)$ .",
"In this case, if $d=1,2,3$ , then $r_d^{(M,M)}=n^{-\\frac{s+2}{4}}=n^{-\\frac{1}{2}}\\times n^{-\\frac{s}{4}}\\lesssim n^{-\\frac{1}{2}}$ .",
"If $d=4$ , then $r_d^{(M,M)}= n^{-\\frac{s+2}{4}}\\log {(1+n)}=n^{-\\frac{1}{2}}\\times \\left(n^{-\\frac{s}{4}}\\log {n}\\right)\\lesssim n^{-\\frac{1}{2}}$ .",
"If $d>4$ , then $r_d^{(M,M)}=n^{-\\frac{s+2}{d}}\\lesssim n^{-\\frac{1}{2}}$ as $\\frac{s+2}{d}\\ge \\frac{1}{2}$ .",
"Therefore, in all the cses, $r_d^{(M,M)}\\lesssim n^{-\\frac{1}{2}}=r_{d,s}^{(n,n)}$ for $d\\le 2(s+2)$ .",
"Case 2: $d>2(s+2)$ .",
"As $s>0$ , then $d>4$ .",
"In this case, once again $r_d^{(M,M)}=n^{-\\frac{s+2}{d}}=r_{d,s}^{(n,n)}$ .",
"This establishes (REF ) and completes the proof."
],
[
"Proofs from sec:App",
"[Proof of thm:barate] First define the following measure: $\\rho _0^{\\mathrm {OR}}:=\\left(\\frac{1}{2}\\mbox{Id}+\\frac{1}{2}T_0\\right)\\#\\widetilde{\\mu }_m.$ Fix any $\\gamma \\in \\widetilde{\\Gamma }_{\\mathrm {min}}$ .",
"By applying the triangle inequality followed by a power mean inequality, we have: $\\sup _{\\gamma \\in \\widetilde{\\Gamma }_{\\mathrm {min}}}W_2^2\\big (\\widehat{\\rho }_0^{\\gamma },\\rho _0\\big )\\lesssim W_2^2\\big (\\rho _0^{\\mathrm {OR}},\\rho _0\\big )+\\sup _{\\gamma \\in \\widetilde{\\Gamma }_{\\mathrm {min}}} W_2^2\\big (\\widehat{\\rho }_0^{\\gamma },\\rho _0^{\\mathrm {OR}}\\big ).$ Next observe that $\\rho _0^{\\mathrm {OR}}$ is the empirical distribution corresponding to $m$ random samples drawn according to $\\rho _0$ .",
"Therefore, by using [55], we get: $ W_2^2\\big (\\rho _0^{\\mathrm {OR}},\\rho _0\\big )\\lesssim r_d^{(m,m)}.$ Next we will bound the second term on the right hand side of (REF ).",
"Towards this direction, recall the definition of $\\Pi (\\cdot ,\\cdot )$ from sec:setting.",
"Consider the following coupling: $\\pi _0^{\\gamma }:=\\left(\\frac{1}{2}\\mbox{Id}+\\frac{1}{2}\\widetilde{T}_{m,n}^{\\gamma },\\frac{1}{2}\\mbox{Id}+\\frac{1}{2}T_0\\right)\\#\\widetilde{\\mu }_m.$ Observe that $\\pi _0^{\\gamma }\\in \\Pi \\big (\\widehat{\\rho }_0^{\\gamma },\\rho _0^{\\mathrm {OR}}\\big )$ .",
"By plugging the coupling $\\pi _0^{\\gamma }$ into the definition of 2-Wasserstein distance in (REF ), we further get: $\\sup _{\\gamma \\in \\widetilde{\\Gamma }_{\\mathrm {min}}} W_2^2\\big (\\widehat{\\rho }_0^{\\gamma },\\rho _0^{\\mathrm {OR}}\\big )&\\le \\sup _{\\gamma \\in \\widetilde{\\Gamma }_{\\mathrm {min}}} \\int \\Vert x-y\\Vert ^2\\,d\\pi _0^{\\gamma }(x,y)\\nonumber \\\\ &=\\sup _{\\gamma \\in \\widetilde{\\Gamma }_{\\mathrm {min}}} \\int \\Vert \\widetilde{T}_{m,n}^{\\gamma }(x)-T_0(x)\\Vert ^2\\,d\\widetilde{\\mu }_m(x)\\nonumber \\\\ &= O_p\\left(r^{(m,n)}_d\\times (\\log {(1+\\max \\lbrace m,n\\rbrace )})^{t_{d,\\alpha }}\\right)$ where the last inequality follows from thm:nsmooth.",
"Combining (REF ) and (REF ) with (REF ) completes the proof.",
"[Proof of thm:dethresh] Let $T_1^{(n)}(\\cdot )$ and $T_2^{(n)}(\\cdot )$ be the optimal transport maps from $\\mu ^{(n)}$ to $\\upsilon _1$ and $\\nu ^{(n)}$ to $\\upsilon _2$ .",
"Set $\\widehat{x}^{\\mathrm {OR}}_{ij}:=K_1(T_1^{(n)}(X_i),T_1^{(n)}(X_j)),\\qquad \\widehat{y}^{\\mathrm {OR}}_{ij}:=K_2(T_2^{(n)}(Y_i),T_2^{(n)}(Y_j))$ and define the oracle version of $\\widehat{\\mathrm {rHSIC}}$ as follows: $\\widehat{\\mathrm {rHSIC}}^{\\mathrm {OR}}:=\\underbrace{n^{-2}\\sum _{i,j}\\widehat{x}^{\\mathrm {OR}}_{ij}\\widehat{y}^{\\mathrm {OR}}_{ij}}_{\\widehat{A}_{n,1}^{\\mathrm {OR}}}+\\underbrace{n^{-4}\\sum _{i,j,r,s}\\widehat{x}^{\\mathrm {OR}}_{ij}\\widehat{y}^{\\mathrm {OR}}_{rs}}_{\\widehat{A}_{n,2}^{\\mathrm {OR}}}-2\\underbrace{n^{-3}\\sum _{i,j,r} \\widehat{x}^{\\mathrm {OR}}_{ij}\\widehat{y}^{\\mathrm {OR}}_{ir}}_{\\widehat{A}_{n,3}^{\\mathrm {OR}}}.$ The proof of thm:dethresh now proceeds using the following steps: Step I: We show that: $\\mathbb {E}\\big |\\widehat{\\mathrm {rHSIC}}^{\\mathrm {OR}}-\\mathrm {rHSIC}(\\pi ^{(n)}|\\mu ^{(n)}\\times \\nu ^{(n)})\\big |\\lesssim n^{-1/2},$ where $\\widehat{\\mathrm {rHSIC}}^{\\mathrm {OR}}(\\cdot |\\cdot )$ is defined in (REF ).",
"Step II: We prove that: $ \\mathbb {E}\\big |\\widehat{\\mathrm {rHSIC}}^{\\mathrm {OR}}-\\widehat{\\mathrm {rHSIC}}\\big |\\lesssim \\sqrt{r_d^{(n,n)}}.$ Step III: We combine steps I and II to prove thm:dethresh.",
"Let us begin with this step first.",
"Note that by using the triangle inequality, we have: $\\widehat{\\mathrm {rHSIC}}\\ge \\mathrm {rHSIC}(\\pi ^{(n)}|\\mu ^{(n)}\\times \\nu ^{(n)})-\\big |\\widehat{\\mathrm {rHSIC}}^{\\mathrm {OR}}-\\mathrm {rHSIC}\\big |-\\big |\\widehat{\\mathrm {rHSIC}}^{\\mathrm {OR}}-\\widehat{\\mathrm {rHSIC}}\\big |.$ Next observe that by steps I and II, $\\max \\bigg \\lbrace \\big |\\widehat{\\mathrm {rHSIC}}^{\\mathrm {OR}}-\\mathrm {rHSIC}\\big |,\\big |\\widehat{\\mathrm {rHSIC}}^{\\mathrm {OR}}-\\widehat{\\mathrm {rHSIC}}\\big |\\bigg \\rbrace =O_p\\big (\\sqrt{r_{d_1,d_2}^{(n,n)}}\\big ).$ Using the above display with (REF ) and the assumption $(r_{d_1,d_2}^{(n,n)})^{-1/2}\\mathrm {rHSIC}(\\pi ^{(n)}|\\mu ^{(n)}\\times \\nu ^{(n)})\\rightarrow \\infty $ , we have: $\\big (r_{d_1,d_2}^{(n,n)}\\big )^{-1/2}\\widehat{\\mathrm {rHSIC}}\\overset{P}{\\longrightarrow }\\infty .$ Therefore, as $n\\sqrt{r_{d_1,d_2}^{(n,n)}}\\rightarrow \\infty $ and $c_{n,\\alpha }=O(1)$ (see [44]), we have: $\\mathbb {E}\\phi _{n,\\alpha }=\\mathbb {P}(n\\times \\widehat{\\mathrm {rHSIC}}\\ge c_{n,\\alpha })\\rightarrow 1$ under $(r_{d_1,d_2}^{(n,n)})^{-1/2}\\mathrm {rHSIC}(\\pi ^{(n)}|\\mu ^{(n)}\\times \\nu ^{(n)})\\rightarrow \\infty $ .",
"This completes the proof.",
"It therefore remains to prove steps I and II.",
"For step I, let $(X_1^{\\prime },Y_1^{\\prime }),\\ldots ,(X_n^{\\prime },Y_n^{\\prime })\\overset{i.i.d.",
"}{\\sim }\\pi ^{(n)}$ .",
"Fix an arbitrary $1\\le j\\le n$ .",
"Let $\\widehat{A}_{n,1,j}^{\\mathrm {OR},^{\\prime }}$ be the same as $\\widehat{A}_{n,1}^{\\mathrm {OR}}$ except with $(X_j,Y_j)$ replaced by $(X_j^{\\prime },Y_j^{\\prime })$ .",
"It is easy to check by the compactness of supports of all distributions involved, that: $\\max \\limits _{1\\le j\\le n}\\big |\\widehat{A}_{n,1}^{\\mathrm {OR}}-\\widehat{A}_{n,1,j}^{\\mathrm {OR},^{\\prime }}\\big |\\lesssim n^{-1}.$ Therefore by using Mcdiarmid's inequality (see [20]), we have, for any $t>0$ , $\\mathbb {P}\\left(\\sqrt{n}(\\widehat{A}_{n,1}^{\\mathrm {OR}}-\\mathbb {E}\\widehat{A}_{n,1}^{\\mathrm {OR}})\\ge t\\right)\\le \\exp (-Ct^2)$ for some constant $C>0$ free of $n$ and $t$ .",
"Similar concentrations can be derived for $\\widehat{A}_{n,2}^{\\mathrm {OR}}$ and $\\widehat{A}_{n,3}^{\\mathrm {OR}}$ .",
"Combining these concentrations with the observation that $\\mathrm {rHSIC}(\\pi ^{(n)}|\\mu ^{(n)}\\times \\nu ^{(n)})=\\mathbb {E}\\widehat{A}_{n,1}^{\\mathrm {OR}}+\\mathbb {E}\\widehat{A}_{n,2}^{\\mathrm {OR}}-2\\mathbb {E}\\widehat{A}_{n,3}^{\\mathrm {OR}}$ completes the proof of step I.",
"We now move on to step II.",
"Recall the definition of $\\widehat{\\mathrm {rHSIC}}$ from (REF ) and write: $\\widehat{\\mathrm {rHSIC}}=\\underbrace{n^{-2}\\sum _{i,j}\\widehat{x}_{ij}\\widehat{y}_{ij}}_{\\widehat{A}_{n,1}}+\\underbrace{n^{-4}\\sum _{i,j,r,s}\\widehat{x}_{ij}\\widehat{y}_{rs}}_{\\widehat{A}_{n,2}}-2\\underbrace{n^{-3}\\sum _{i,j,r} \\widehat{x}_{ij}\\widehat{y}_{ir}}_{\\widehat{A}_{n,3}}.$ By the Lipschitzness of $K_1(\\cdot ,\\cdot )$ and $K_2(\\cdot ,\\cdot )$ , we have: $\\big |\\widehat{x}_{ij}-\\widehat{x}^{\\mathrm {OR}}_{ij}\\big |\\lesssim \\Vert \\widehat{T}_{1,n}(X_i)-T_1^{(n)}(X_i)\\Vert +\\Vert \\widehat{T}_{1,n}(X_j)-T_1^{(n)}(X_j)\\Vert ,$ $\\big |\\widehat{y}_{ij}-\\widehat{y}^{\\mathrm {OR}}_{ij}\\big |\\lesssim \\Vert \\widehat{T}_{2,n}(Y_i)-T_2^{(n)}(Y_i)\\Vert +\\Vert \\widehat{T}_{2,n}(Y_j)-T_2^{(n)}(Y_j)\\Vert .$ Therefore, by using the fact that the probability measures $\\upsilon _1$ and $\\upsilon _2$ are compactly supported, we get: $\\big |\\widehat{A}_{n,1}-\\widehat{A}_{n,1}^{\\mathrm {OR}}\\big |\\lesssim \\frac{1}{n}\\sum _{i=1}^n \\Vert \\widehat{T}_{1,n}(X_i)-T_1^{(n)}(X_i)\\Vert +\\frac{1}{n}\\sum _{j=1}^n \\Vert \\widehat{T}_{2,n}(Y_j)-T_2^{(n)}(Y_j)\\Vert .$ The same bound can similarly be verified for $|\\widehat{A}_{n,2}-\\widehat{A}_{n,2}^{\\mathrm {OR}}|$ and $|\\widehat{A}_{n,3}-\\widehat{A}_{n,3}^{\\mathrm {OR}}|$ .",
"Combining these observations, we have: $\\big |\\widehat{\\mathrm {rHSIC}}-\\widehat{\\mathrm {rHSIC}}^{\\mathrm {OR}}\\big |&\\lesssim \\frac{1}{n}\\sum _{i=1}^n \\Vert \\widehat{T}_{1,n}(X_i)-T_1^{(n)}(X_i)\\Vert +\\frac{1}{n}\\sum _{j=1}^n \\Vert \\widehat{T}_{2,n}(Y_j)-T_2^{(n)}(Y_j)\\Vert \\nonumber \\\\ &\\le \\sqrt{\\frac{1}{n}\\sum _{i=1}^n \\Vert \\widehat{T}_{1,n}(X_i)-T_1^{(n)}(X_i)\\Vert ^2}+\\sqrt{\\frac{1}{n}\\sum _{j=1}^n \\Vert \\widehat{T}_{2,n}(Y_j)-T_2^{(n)}(Y_j)\\Vert ^2}.$ Step II then follows by invoking cor:fsam."
],
[
"Auxiliary definitions and results",
"[Subdifferential set and subgradient] Given a convex function $f:\\mathbb {R}^d\\rightarrow \\mathbb {R}\\cup \\lbrace \\infty \\rbrace $ , we define the subdifferential set of $f(\\cdot )$ at $x\\in \\mbox{dom}(f):=\\lbrace z\\in \\mathbb {R}^d:f(z)<\\infty \\rbrace $ as follows: $\\partial f(x):=\\lbrace \\xi \\in \\mathbb {R}^d:\\ f(x)+\\langle \\xi , y-x\\rangle \\le f(y),\\quad \\mbox{for}\\ \\mbox{all}\\ y\\in \\mathbb {R}^d\\rbrace .$ Any element in the set $\\partial f(x)$ is called a subgradient of $f(\\cdot )$ at $x$ .",
"[Strong convexity] A function $f:\\mathbb {R}^d\\rightarrow \\mathbb {R}\\cup \\lbrace \\infty \\rbrace $ is strongly convex with parameter $\\lambda >0$ , if, for all $x,y\\in \\mbox{dom}(f)=\\lbrace z\\in \\mathbb {R}^d:f(z)<\\infty \\rbrace $ , the following holds: $f(y)\\ge f(x) + \\langle \\xi _x,y-x\\rangle +\\frac{\\lambda }{2}\\Vert y-x\\Vert ^2,$ where $\\xi _x\\in \\partial f(x)$ , the subgradient of $f(\\cdot )$ at $x$ as in def:subg.",
"[Wavelet basis] We present our main assumptions on the wavelet basis discussed in sec:wavelet only for the wavelets on the space $\\mathcal {X}$ .",
"The same assumptions are also required for the wavelets on $\\mathcal {Y}$ .",
"These are essentially a subset of the assumptions laid out in [143] as we heavily rely on [143] for proving thm:smoothwav.",
"(Regularity).",
"Fix $r>\\max \\lbrace s,1\\rbrace $ .",
"The functions in $\\Phi $ and $\\Psi _j$ , $j\\ge 0$ have $r$ continuous derivatives, and all polynomials of degree at most $r$ on $\\mathcal {X}$ lie in the span of the functions in $\\Phi $ .",
"(Tensor construction).",
"Each $\\psi (\\cdot )\\in \\Psi _j$ can be expressed as $\\psi (x)=\\prod _{i=1}^d \\psi _i(x_i)$ , where $x=(x_1,\\ldots ,x_d)$ , for some univariate functions $\\psi _i(\\cdot )$ 's.",
"(Locality).",
"For each $\\psi (\\cdot )\\in \\Psi _j$ there exists a rectangle $I_{\\psi }\\subseteq \\mathcal {X}$ such that $\\mbox{supp}(\\psi )\\subseteq I_{\\psi }$ , $\\mbox{diam}(I_{\\psi })\\le C_1\\cdot 2^{-j}$ , and $\\sup _{x\\in \\mathcal {X}}\\sum _{\\psi (\\cdot )\\in \\Psi _j}\\mathbb {1}(x\\in I_{\\psi })\\le C_2$ for some constants $C_1,C_2>0$ .",
"(Bernstein estimate).",
"$\\Vert \\nabla f\\Vert _{L^2(\\mathcal {X})}\\le C_3\\cdot 2^j\\Vert f\\Vert _{L^2(\\mathcal {X})}$ for any $f(\\cdot )$ in the span of the functions in $\\mbox{span}\\left(\\Phi \\cup \\left\\lbrace \\cup _{0\\le k<j}\\Psi _j\\right\\rbrace \\right)$ .",
"Here $C_3$ is some positive constant.",
"Lemma 5.1 (Strong convexity and Lipschitzness, see [79]) $\\varphi _0^*(\\cdot )$ is strongly convex with parameter $(1/L)$ if and only if $T_0(\\cdot )$ is $L$ -Lipschitz continuous.",
"Lemma 5.2 (Gradient of dual) Recall the definition of $f^*(\\cdot )$ from () and $\\partial f(\\cdot )$ from def:subg.",
"Then the following equivalence holds: $\\langle x,y\\rangle =f(x)+f^*(y) \\quad \\Longleftrightarrow \\quad y\\in \\partial f(x)\\quad \\Longleftrightarrow \\quad x\\in \\partial f^*(y).$ Lemma 5.3 (Bounding expected supremum of empirical process, see [5], [99]) Let $f(\\cdot )$ be a probability density supported on some subset of $\\mathbb {R}^d$ , and say $Z\\sim f(\\cdot )$ .",
"Let $\\mathcal {G}$ be a class of uniformly bounded measurable functions from $\\mathbb {R}^d\\times (0,1]$ to $\\mathbb {R}$ , such that: $\\sup _{g(\\cdot )\\in \\mathcal {G}}\\sup _{h\\in (0,1]} h^{-d}\\mathbb {E}[g^2(Z,h)]<\\infty ,$ and such that the class $\\mathcal {G}_0:=\\lbrace x\\mapsto g(x,h):\\ g(\\cdot )\\in \\mathcal {G},\\ h\\in (0,1)\\rbrace $ is point wise measurable and of VC-type (see [138] for relevant definitions of VC classes of sets/functions and point wise measurability).",
"Then there exists $b_0\\in (0,1)$ such that if $Z_1,Z_2,\\ldots $ is an i.i.d.",
"sequence of observations from the probability density $f(\\cdot )$ , we have: $\\sup _{g(\\cdot )\\in \\mathcal {G}}\\sup _{\\frac{\\log {n}}{n}\\le h^d\\le b_0} \\sqrt{\\frac{n}{h^d\\log {n}}}\\Bigg |\\frac{1}{n}\\sum _{i=1}^n g(Z_i,h)-\\mathbb {E}[g(Z,h)]\\Bigg |=O_p(1).$"
]
] | 2107.01718 |
[
[
"Symmetry, nodal structure, and Bogoliubov Fermi surfaces for nonlocal\n pairing"
],
[
"Abstract Multiband effects can lead to fundamentally different electronic behavior of solids, as exemplified by the possible emergence of Fermi surfaces of Bogoliubov quasiparticles in centrosymmetric superconductors which break time-reversal symmetry.",
"We extend the analysis of possible pairing symmetries, the corresponding nodal structure, and the Bogoliubov Fermi surfaces in two directions: We include nonlocal pairing and we consider internal degrees of freedom other than the effective angular momentum of length $j=3/2$ examined so far.",
"Since our main focus is on the Bogoliubov Fermi surfaces we concentrate on even-parity pairing.",
"The required symmetry analysis is illustrated for several examples, as a guide for the reader.",
"We find that the inclusion of nonlocal pairing leads to a much larger range of possible pairing symmetries.",
"For infinitesimal pairing strength, we find a simple yet powerful criterion for nodes in terms of a scalar product of form factors."
],
[
"Introduction",
"In condensed matter physics, we are used to study the electronic properties of materials by considering one band at a time.",
"Only when we are interested in excitations at higher energies, e.g., by electromagnetic waves, we include multiple bands.",
"However, it is not always true that the low-energy and equilibrium properties of a solid can be understood based on the single-band paradigm.",
"A case in point is the recent realization that in centrosymmetric multiband superconductors that break time-reversal symmetry (TRS) and have gap nodes, these nodes are generically two dimensional [1], [2].",
"We call these two-dimensional nodes Bogoliubov Fermi surfaces (BFSs).",
"This result has put meat on the bones of the proof [3], [4] that in such systems two-dimensional Fermi surfaces can be protected by a $\\mathbb {Z}_2$ topological invariant.",
"This invariant is related to the Pfaffian of the Bogoliubov–de Gennes (BdG) Hamiltonian, unitarily transformed into antisymmetric form [1], [2].",
"Another example is the belief that optical excitations across the gap are forbidden in clean superconductors, which is based on the single-band paradigm but does not hold for multiband superconductors, as recently shown by Ahn and Nagaosa [5].",
"One criterion for when optical excitations across the gap are allowed is the existence of BFSs—this holds both for centrosymmetric and for noncentrosymmetric superconductors.",
"Further experimental signatures of BFSs [6] and their instability against spontaneous breaking of inversion symmetry either electronically [7], [8], [9] or by lattice distortions [10] have been considered by several groups.",
"BFSs can be protected by multiple topological invariants [11], [2], which imposes constraints on how they can merge and gap out for strong coupling.",
"Furthermore, Herbut and Link [9] have recently revealed an analogy between the antisymmetric form of the BdG Hamiltonian—the existence of which is essential for the definition of the Pfaffian [1], [2]—and classical relativity.",
"In this analogy, one can understand the BFSs from a condition of orthogonal fictitious electric and magnetic fields in momentum space [9].",
"The analogy is useful for studying the interaction-induced instability of the BFSs.",
"However, it is restricted to four-dimensional internal Hilbert spaces.",
"Link and Herbut [12] have also presented a complementary study of BFSs in noncentrosymmetric multiband superconductors with broken TRS and gap nodes.",
"Although a $\\mathbb {Z}_2$ topological invariant does not exist, BFSs are typically present since the breaking of TRS causes bands shifts that are larger than the induced gaps [12].",
"In the language of tight-binding models, the presence of multiple bands in the vicinity of the Fermi energy results from the existence of multiple (Wannier) orbitals per unit cell that appreciably contribute to the Bloch states at the Fermi energy.",
"These orbitals can be located at the same site or, for structures with a basis, at different sites.",
"We will refer to these orbital and site degrees of freedom, together with the electron spin, as internal degrees of freedom.",
"Superconducting pairing states can be nontrivial with respect to these internal degrees of freedom (internally anisotropic [2]).",
"This allows nontrivial pairing states even for perfectly local pairing, which corresponds to a momentum-independent gap matrix, as we will discuss further below.",
"If the orbital degrees of freedom form a degenerate triplet, for example $p_x$ , $p_y$ , $p_z$ or $d_{yz}$ , $d_{zx}$ , $d_{xy}$ in a cubic crystal field, they can be combined with the spin to form states with effective angular momentum $j=1/2$ or $j=3/2$ [13], [1], [14], [15], [16], [2], [7], [8].",
"The latter leads to a natural description of the fourfold $\\Gamma _8$ band-touching points in cubic crystals.",
"While the results concerning the existence of BFSs and their $\\mathbb {Z}_2$ topological invariant were general, they were mostly illustrated by the example of the $j=3/2$ model [1], [2], [7], [8], [10].",
"It is important to realized that the possibilities for internal degrees of freedom are much richer.",
"Many superconductors that do not fit the $j=3/2$ description nevertheless have multiple bands close together and close to the Fermi energy.",
"Moreover, the examples studied so far were restricted to local pairing.",
"However, nonlocal pairing is generically present and is often necessary to obtain any superconductivity if local pairing is excluded by a repulsive Hubbard interaction.",
"We will see that nonlocal pairing typically allows for a large range of additional pairing states with symmetries that do not appear for local pairing.",
"In this paper, we extend the analysis of BFSs in centrosymmetric superconductors in two directions: We include nonlocal pairing and we consider internal degrees of freedom other than an effective angular momentum $j=3/2$ .",
"While we mainly discuss the physically most relevant case that the internal degrees of freedom include the spin and the time-reversal transformation squares to minus the identity operation, we also derive results for the opposite case.",
"We are mainly interested in the properties of BFSs in this more general setting and therefore concentrate on even-parity pairing.",
"We provide details on the symmetry analysis to help readers perform such an analysis for specific systems of interest.",
"Everything said here applies to three-dimensional crystals.",
"To be clear, we also specify what we do not consider: We do not address systems with a normal-state Fermi surface that is not topologically equivalent to a sphere enclosing the $\\Gamma $ point, for example quasi-two-dimensional systems such as $\\mathrm {Sr_2RuO_4}$ [17].",
"In such cases, directions in momentum space with symmetry-imposed nodes for infinitesimal pairing might not intersect with the normal-state Fermi surface so that these symmetry-imposed nodes would not be present.",
"Furthermore, we do not consider accidental gap nodes, pairing states that combine different irreducible representations (irreps) of the point group, and new BFSs that emerge for strong coupling away from the normal-state Fermi surface.",
"The description of such phenomena would only require straightforward extensions of the theory.",
"The remainder of this paper is organized as follows: In Sec.",
", we describe the symmetry analysis that produces all symmetry-allowed contributions to pairing of any symmetry for any crystallographic point group, together with the nodal structure for infinitesimal pairing strength and criteria for the inflation of nodes into BFSs beyond infinitesimal pairing.",
"In Sec.",
", we apply this general framework to a number of model systems of increasing complexity.",
"Finally, in Sec.",
", general insights gained by the preceding sections are discussed and an outlook on open questions is given.",
"Several formal points are presented in Appendices."
],
[
"General analysis",
"In this section, we describe the general symmetry analysis.",
"We assume the normal state to satisfy inversion symmetry and TRS.",
"The first condition implies that the only possible point groups are $C_i$ , $C_{2h}$ , $D_{2h}$ , $D_{3d}$ , $C_{4h}$ , $D_{4h}$ , $C_{6h}$ , $D_{6h}$ , $S_6 = C_{3i}$ , $T_h$ , and $O_h$ .",
"Of these groups, $C_i$ , $C_{2h}$ , and $D_{2h}$ have only one-dimensional irreps.",
"$C_{4h}$ , $C_{6h}$ , and $S_6$ in addition have two-dimensional real irreps which decompose into one-dimensional complex irreps.",
"The real irreps are relevant for the analysis of Hermitian irreducible tensor operators.",
"Finally, $D_{3d}$ , $D_{4h}$ , $D_{6h}$ , $T_h$ , and $O_h$ also have multidimensional complex irreps.",
"Pairing states described by multidimensional irreps are the most interesting for us since they lead to multicomponent order parameters, which naturally accommodate the breaking of TRS.",
"It is advantageous to consider the magnetic point group of the crystal since this allows us to treat point-group symmetries and TRS on equal footing [19], [20], [21], [22], [23], [18].",
"Since TRS is preserved in the normal state, the antiunitary time-reversal operator $\\mathcal {T}$ is an element of the magnetic point group.",
"Hence, we are dealing with a gray group: If $G$ is the structural point group, then the gray point group is $M = G + \\mathcal {T}G$ , where $\\mathcal {T}G$ contains all elements of $G$ multiplied by time reversal $\\mathcal {T}$ (the elements of $G$ commute with $\\mathcal {T}$ [23]).",
"The theory of complex corepresentations of magnetic groups [19], [20], [21], [22] has been reviewed for example by Bradley and Davies [23].",
"However, this theory is not the appropriate one for our purposes since it is based on the notions of unitary equivalence of matrices and reducibility of corepresentations by unitary transformations.",
"This leads to the result that two corepresentations that represent $\\mathcal {T}$ by $+$ and $-$ , respectively, can be equivalent.",
"Since we need to distinguish operators that are even or odd under time reversal this is not the appropriate equivalence relation.",
"We rather require real corepresentations based on orthogonal equivalence, which leaves the properties under time reversal invariant, and reducibility by orthogonal transformations.",
"By using Wigner's construction of corepresentations [20], [23] but restricting oneself to orthogonal transformations, it is fairly easy to see that for every irrep $\\Gamma $ of $G$ , the gray magnetic point group $M$ has two irreps $\\Gamma _+$ and $\\Gamma _-$ , which are distinguished by a (further) index $\\pm $ which indicates the sign under time reversal.",
"Character tables of real corepresentations of the magnetic point groups are given by Erb and Hlinka [24].",
"One more piece of group theory is needed: Since we are studying single-fermion Hamiltonians, we have to consider double groups, i.e., a rotation by $2\\pi $ does not give the group identity but only a rotation by $4\\pi $ does.",
"This leads to additional “double-valued” or spinor irreps [18].",
"However, the double-valued irreps do not play any role in our analysis for the following reason: We will expand Hamiltonians into sums of basis matrices with momentum-dependent coefficients which are basis functions of irreps.",
"The momentum components $k_x$ , $k_y$ , $k_z$ are basis functions of single-valued irreps, which implies that momentum-dependent functions can only be basis functions of single-valued irreps.",
"Hence, double-valued irreps do not occur in the expansion.",
"We now introduce relevant notations.",
"The superconductor is described by a BdG Hamiltonian of the form $\\mathcal {H}(\\mathbf {k}) = \\begin{pmatrix}H_N(\\mathbf {k}) & \\Delta (\\mathbf {k}) \\\\\\Delta ^\\dagger (\\mathbf {k}) & -H_N^T(-\\mathbf {k})\\end{pmatrix} ,$ where $H_N(\\mathbf {k})$ is the normal-state Hamiltonian and $\\Delta (\\mathbf {k})$ is the pairing potential.",
"Both are operators on the $N$ -dimensional Hilbert space of internal degrees of freedom and are given by $N\\times N$ matrices for a particular basis.",
"Antisymmetry of fermionic states implies that [25], [26], [27] $\\Delta ^T(-\\mathbf {k}) = -\\Delta (\\mathbf {k}) .$ By construction, the BdG Hamiltonian satisfies particle-hole (charge-conjugation) symmetry $\\mathcal {C}$ , which is expressed as $\\mathcal {U}_C\\, \\mathcal {H}^T(-\\mathbf {k})\\, \\mathcal {U}_C^\\dagger = - \\mathcal {H}(\\mathbf {k}) ,$ with $\\mathcal {U}_C = \\sigma _1 \\otimes $ .",
"We denote the Pauli matrices by $\\sigma _1$ , $\\sigma _2$ , $\\sigma _3$ and the $2\\times 2$ identity matrix by $\\sigma _0$ .",
"Identity matrices in any dimension are denoted by $$ .",
"Symmetries in the structural point group $G$ are expressed as $\\mathcal {U}\\, \\mathcal {H}(R^{-1} \\mathbf {k})\\, \\mathcal {U}^\\dagger = \\mathcal {H}(\\mathbf {k}) ,$ where $R$ is an appropriate three-dimensional generalized rotation matrix and $\\mathcal {U} = \\begin{pmatrix}U & 0 \\\\ 0 & U^*\\end{pmatrix}$ is unitary.",
"This form of $\\mathcal {U}$ follows from particle-hole symmetry.",
"The most important case of us is inversion symmetry or parity $P$ , which is implemented by a unitary matrix $U_P$ .",
"We assume inversion symmetry of the normal state, i.e., $U_P\\, H_N(-\\mathbf {k})\\, U_P^\\dagger = H_N(\\mathbf {k}) .$ Finally, TRS takes the form $\\mathcal {U}_T\\, \\mathcal {H}^T(-\\mathbf {k})\\, \\mathcal {U}_T^\\dagger = \\mathcal {H}(\\mathbf {k}) ,$ where $\\mathcal {U}_T = \\begin{pmatrix}U_T & 0 \\\\ 0 & U_T^*\\end{pmatrix}$ is (the matrix form of) the unitary part of the antiunitary time-reversal operator.",
"Time reversal $\\mathcal {T}$ can square to plus or minus the identity.",
"We denote the sign of $\\mathcal {T}^2$ by $s_T = \\pm 1$ .",
"Then $U_T U_T^* = s_T\\, $ and thus $U_T^T = s_T\\, U_T .$ If the internal degrees of freedom include the electron spin we have $s_T = -1$ and $U_T$ is antisymmetric.",
"The matrix $U_T$ is also unitary so that its dimension $N$ must be even since its spectrum consists of pairs $\\pm e^{i\\phi }$ .",
"The case $s_T = +1$ can only be realized if the spin does not occur explicitly, for example because electrons in one spin state are pushed to high energies by a strong magnetic field.",
"Then $\\mathcal {T}$ is not the physical TRS but an effective antiunitary symmetry.",
"For $s_T = +1$ , $U_T$ is symmetric and the dimension $N$ is not restricted.",
"Beyond these considerations, the specific form of $U_T$ as well as the specific form of structural point-group transformations depend on the physical nature of the internal degrees of freedom.",
"It is useful to write the pairing matrix as $\\Delta (\\mathbf {k}) = D(\\mathbf {k})\\, U_T .$ Under a general unitary symmetry transformation, Eq.",
"(REF ), the pairing matrix transforms as $\\Delta (\\mathbf {k}) \\mapsto U\\, \\Delta (R^ {-1}\\mathbf {k})\\, U^T ,$ i.e., not like a matrix.",
"One easily sees that $D(\\mathbf {k})$ transforms as $D(\\mathbf {k}) \\mapsto U\\, D(R^{-1}\\mathbf {k})\\, U^\\dagger ,$ i.e., like a matrix.",
"Analogously, one finds that under time reversal, $D(\\mathbf {k})$ transforms as $D(\\mathbf {k}) \\mapsto U_T\\, D^*(-\\mathbf {k})\\, U_T^\\dagger .$ Hence, $D(\\mathbf {k})$ transforms like $H_N(\\mathbf {k})$ under the magnetic point group, noting that $H_N^T(\\mathbf {k}) = H_N^*(\\mathbf {k})$ because of Hermiticity.",
"The condition (REF ) from fermionic antisymmetry together with Eqs.",
"(REF ) and (REF ) implies $U_T\\, D^T(-\\mathbf {k})\\, U_T^\\dagger = -s_T\\, D(\\mathbf {k}) .$ For the standard case of $s_T=-1$ , this relation is similar to TRS but differs from it since $D(\\mathbf {k})$ is generally not Hermitian so that $D^T(\\mathbf {k})$ is not the same as $D^*(\\mathbf {k})$ .",
"The general steps of the symmetry analysis are now as follows: (1) Construct a basis $\\lbrace h_\\nu \\rbrace $ of Hermitian matrices on the space of the internal degrees of freedom so that the $h_\\nu $ transform as irreducible tensor operators of the magnetic point group $M$ .",
"In the case of point groups with two-dimensional real irreps that decompose into two one-dimensional complex irreps, use the real irreps since this allows one to find Hermitian $h_\\nu $ ; the irreducible tensor operators of the corresponding one-dimensional complex irreps are generally not Hermitian.",
"We call the $h_\\nu $ basis matrices and normalize them in such a way that $\\mathop {\\textrm {Tr}}h_\\nu ^2 = N$ (the identity matrix is then normalized).",
"It is possible that not all irreps occur.",
"If the dimension of the internal Hilbert space is $N$ there are $N^2$ basis matrices.",
"To find the appropriate basis matrices and their irreps, it is necessary to determine the explicit forms of the symmetry operators, i.e., of the unitary matrices $U_T$ for time reversal and $U_g$ for at least a set of generators $g$ of the structural point group $G$ .",
"(2) Generate a list of all irreps of $M$ that possess basis functions of momentum.",
"These are all irreps that have the same parity under time reversal and inversion since the momentum $\\mathbf {k}$ is odd under both.",
"This allows $g+$ and $u-$ irreps but forbids $g-$ and $u+$ irreps.",
"It is also useful to obtain characteristic basis functions for those irreps that have them but it should be kept in mind that these are understood as placeholders for arbitrary sets of functions with the same symmetry under operations from $M$ .",
"In this paper, we will usually represent basis functions by the lowest-order polynomials.",
"(3) Construct the general form of the normal-state Hamiltonian $H_N(\\mathbf {k})$ by expanding it into the previously constructed basis, $H_N(\\mathbf {k}) = \\sum _n c_n(\\mathbf {k})\\, h_n .$ We enumerate all basis matrices by $\\nu $ but the subset that occurs in $H_N(\\mathbf {k})$ by $n$ .",
"The Hamiltonian and every term in the expansion must be invariant under $M$ , i.e., it must transform as an irreducible tensor operator belonging to the trivial irrep $A_\\mathrm {triv}$ of $M$ .",
"This irrep is even under inversion and under time reversal, i.e., it is $A_{g+}$ or $A_{1g+}$ depending on the group.",
"This requires the form factors $c_n(\\mathbf {k})$ to transform as basis functions of the same irrep to which $h_n$ belongs.",
"Moreover, for multidimensional irreps, $c_n(\\mathbf {k})$ and $h_n$ must transform as the same component of the irrep and all of them must have the same amplitude if the basis functions and tensor operators are properly normalized.",
"If there is no corresponding basis function for the irrep of some $h_\\nu $ , this matrix does not occur in Eq.",
"(REF ).",
"This excludes $h_\\nu $ belonging to $g-$ or $u+$ irreps.",
"Note that $h_0 \\equiv $ is always an allowed basis matrix since it is a reducible tensor operator of the trivial irrep $A_\\mathrm {triv}$ and $c_0(\\mathbf {k}) \\sim 1$ is always an appropriate basis function.",
"As noted above, for the standard case that the internal degrees of freedom include the electron spin, time reversal squares to $-1$ and the dimension $N$ of the internal Hilbert space must be even.",
"Then the number of allowed basis matrices $h_n$ appearing in $H_N(\\mathbf {k})$ is $N(N-1)/2$ , as shown in Appendix .",
"The case of $N=2$ corresponds to spin being the only internal degree of freedom.",
"There is only a single allowed basis matrix, namely $h_0=$ .",
"This means that the normal-state Hamiltonian is independent of spin, which is required by TRS.",
"For $N=4$ , there are 6 basis matrices $h_0=$ , $h_1$ , ..., $h_5$ .",
"These matrices have the special property that $h_1$ , ..., $h_5$ anticommute pairwise [28], [29], while all matrices commute with $h_0$ .",
"Results restricted to this case are discussed in Subsection REF .",
"(4) Construct the allowed pairing states.",
"Here, we have much greater freedom than in constructing $H_N(\\mathbf {k})$ since the superconducting state may break symmetries contained in $M$ .",
"We write the pairing matrix as $D(\\mathbf {k}) = \\sum _j \\delta _j D_j(\\mathbf {k}) ,$ where the $D_j(\\mathbf {k})$ are linearly independent matrix-valued functions transforming like (being irreducible tensor operators belonging to) components of a specific irrep $\\Gamma _s$ , $s=\\pm $ , of the magnetic point group $M$ and the $\\delta _j$ are complex pairing amplitudes.",
"Recall that $D(\\mathbf {k})$ transforms like a matrix.",
"$D_j(\\mathbf {k})$ can be chosen to be either Hermitian or anti-Hermitian.",
"This can be seen as follows: Note first that Eq.",
"(REF ) has to be satisfied by each component separately since the $D_j(\\mathbf {k})$ are independent functions, $U_T\\, D_j^T(-\\mathbf {k})\\, U_T^\\dagger = -s_T\\, D_j(\\mathbf {k}) .$ Depending on the irrep $\\Gamma _s$ , $D_j(\\mathbf {k})$ is either even ($s=+$ ) or odd ($s=-$ ) under time reversal.",
"Hence, Eq.",
"(REF ) implies that $U_T\\, D_j^*(-\\mathbf {k})\\, U_T^\\dagger = s\\, D_j(\\mathbf {k}) .$ It follows that $D_j^T(\\mathbf {k}) = -s_T\\, s\\, D_j^*(\\mathbf {k})$ and thus $D_j^\\dagger (\\mathbf {k}) = -s_T\\, s\\, D_j(\\mathbf {k}) .$ This equation states that the matrix functions $D_j(\\mathbf {k})$ are Hermitian or anti-Hermitian depending on the sign $s_T$ of $\\mathcal {T}^2$ and on the pairing state being even or odd under time reversal.",
"Since we consider pure-irrep pairing all appearing $D_j(\\mathbf {k})$ have the same sign $-s_T\\,s$ under Hermitian conjugation.",
"For the standard case of $s_T = -1$ , $D_j(\\mathbf {k})$ is Hermitian (anti-Hermitian) for time-reversal-even (time-reversal-odd) irreps.",
"For a time-reversal-odd irrep $\\Gamma _-$ , we can pull a common factor of $i$ out of all $D_j(\\mathbf {k})$ and absorb it into the order parameters $\\delta _j$ in Eq.",
"(REF ).",
"This makes $D_j(\\mathbf {k})$ Hermitian and changes the irrep from $\\Gamma _-$ to $\\Gamma _+$ since the behavior under spatial transformations is unaffected.",
"Hence, for $s_T = -1$ it is sufficient to consider only the time-reversal-even $g+$ and $u+$ irreps for pairing states.",
"This has the desirable consequence that the breaking of TRS is only encoded in the complex order parameters $\\delta _j$ .",
"If and only if all $\\delta _j$ can be made real by a global phase rotation the system respects TRS.",
"This is equivalent to the nonvanishing $\\delta _j$ having phase differences of 0 or $\\pi $ .",
"Conversely, for the nonstandard sign $s_T = +1$ , $D_j(\\mathbf {k})$ is anti-Hermitian (Hermitian) for time-reversal-even (time-reversal-odd) irreps.",
"Pulling out a factor of $i$ , we can make sure that $D_j(\\mathbf {k})$ is Hermitian and the irrep is odd under time reversal ($g-$ or $u-$ ).",
"Again, the breaking of TRS is only encoded in the order parameters $\\delta _j$ .",
"If and only if all $\\delta _j$ can be made purely imaginary by a global phase rotation the system respects TRS.",
"This is again equivalent to the nonvanishing $\\delta _j$ having phase differences of 0 or $\\pi $ .",
"A given model does not necessarily permit all time-reversal-even (or odd) irreps, though.",
"To see this, note that the Hermitian-matrix-valued functions $D_j(\\mathbf {k})$ can be expanded into the Hermitian basis matrices $h_\\nu $ as $D_j(\\mathbf {k}) = \\sum _\\nu d_{j\\nu }(\\mathbf {k})\\, h_\\nu ,$ with real functions $d_{j\\nu }(\\mathbf {k})$ .",
"Consider all products of momentum basis functions $g_l(\\mathbf {k})$ belonging to irreps $\\Gamma _l$ and of matrices $h_\\nu $ belonging to irreps $\\Gamma _\\nu $ .",
"Any such product transforms according to the product representation $\\Gamma _l \\otimes \\Gamma _\\nu $ , which is generally reducible.",
"A reduction into irreps by standard methods reveals which symmetries of pairing states can occur.",
"Recall that only $g+$ and $u-$ irreps possess momentum basis functions.",
"The possible irreps of basis matrices $h_\\nu $ have been obtained in step (1).",
"By reducing all possible products and keeping only those irreps that are even (odd) under time reversal for $s_T=-1$ ($s_T = +1$ ), we obtain the possible irreps.",
"The coefficients $d_{j\\nu }(\\mathbf {k})$ in Eq.",
"(REF ) can be constructed out of the functions $g_l(\\mathbf {k})$ by standard methods.",
"The full pairing matrix then has the form $D(\\mathbf {k}) = \\sum _\\nu \\sum _{j=1}^d \\delta _j d_{j\\nu }(\\mathbf {k})\\,h_\\nu \\equiv \\sum _\\nu f_\\nu (\\mathbf {k})\\, h_\\nu .$ The sum in Eq.",
"(REF ) generally contains many terms: While the number of possible basis matrices $h_\\nu $ is finite, there are infinitely many smooth functions of momentum that transform according to the same irrep.",
"This implies that TRS can be broken spontaneously for pairing belonging to any irrep, by having amplitudes $\\delta _j$ with nontrivial phase differences [30].",
"Consideration of energetics [31], [26], [13], [1], [2] is useful to find plausible modes of TRS breaking.",
"Spontaneous breaking of TRS occurs most naturally for multidimensional irreps, in the form of nontrivial phase factors of contributions belonging to different components of the irrep.",
"Table: Possible combinations of the signs under inversion (parity, gg for even, uu for odd) and under time reversal (++ for even, -- for odd) of irreps of pairing states for time reversal squaring to s T =±1s_T=\\pm 1.",
"Recall that s T =-1s_T=-1 is the standard case for electrons.Since the normal state is inversion symmetric the superconducting pairing is either even ($g$ irreps) or odd ($u$ irreps) under inversion (parity).",
"Table REF shows the possible combinations of signs under inversion and time reversal.",
"We note that for $s_T=-1$ and even-parity pairing, only basis matrices belonging to $g+$ and $u-$ irreps occur.",
"These are the same matrices $h_n$ that appear in the normal-state Hamiltonian $H_N(\\mathbf {k})$ in Eq.",
"(REF ).",
"On the other hand, for $s_T=-1$ and odd-parity pairing, only those basis matrices $h_\\nu $ that do not occur in $H_N(\\mathbf {k})$ can appear in $D(\\mathbf {k})$ .",
"The numbers of allowed basis matrices are given in Appendix .",
"(5) Analyze the nodal structure for each pairing symmetry (irrep) assuming infinitesimal pairing strength.",
"Since we are interested in the fate of BFSs we concentrate on the case $s_T=-1$ and even-parity pairing, while the other cases are briefly discussed in Appendix .",
"In the limit of infinitesimal pairing amplitudes $\\delta _j$ , superconductivity can be described for each band separately.",
"This is because the superconducting gap is of first order in $\\delta _j$ , whereas interband effects are of second order.",
"Moreover, there are at most point or line nodes but no BFSs since interband pairing is responsible for the latter [1], [2].",
"The normal-state bands are twofold degenerate because of inversion symmetry and TRS.",
"Hence, in an effective description of a single band, the dimension of the internal Hilbert space is $N=2$ and the internal degree of freedom can be described by a pseudospin of length $1/2$ [2], [10].",
"The superconducting pairing must be in the pseudospin-singlet channel since it is of even parity.",
"The pairing matrix is then $f_0(\\mathbf {k})\\, \\sigma _0$ .",
"Since $\\sigma _0$ belongs to a $g+$ irrep, namely the trivial one, $f_0(\\mathbf {k})$ must be a basis function of a $g+$ irrep; see Table REF .",
"The symmetry of the pairing state under the magnetic point group can then only be encoded in the function $f_0(\\mathbf {k})$ .",
"This function thus generically transforms like the pairing matrix $D(\\mathbf {k})$ of the full model.",
"Moreover, the symmetry-imposed zeros of $f_0(\\mathbf {k})$ in momentum space correspond to gap nodes for infinitesimal pairing (IP nodes).",
"One of the main messages of Refs.",
"[13], [1], [2] was that a momentum-independent but internally anisotropic pairing matrix can lead to a momentum-dependent function $f_0(\\mathbf {k})$ and to gap nodes.",
"A simple but powerful criterion for IP nodes can be obtained as follows: As noted above, for $s_T=-1$ and even-parity pairing, the same basis matrices $h_n$ appear in $H_N(\\mathbf {k})$ and $D(\\mathbf {k})$ .",
"From Eq.",
"(REF ), we know that the normal-state coefficients $c_n(\\mathbf {k})$ transform like the basis matrices $h_n$ under the magnetic point group.",
"Hence, the scalar function $F(\\mathbf {k}) \\equiv \\sum _n c_n(\\mathbf {k})\\, f_n(\\mathbf {k})$ transforms like the pairing matrix $D(\\mathbf {k})$ and thus also like the form factor $f_0(\\mathbf {k})$ in the single-band picture and, in particular, has the same symmetry-induced nodes.",
"Therefore, we can use $F(\\mathbf {k})$ as a proxy for the IP nodal structure.",
"In fact, instead of the normal-state coefficients $c_n(\\mathbf {k})$ , we could use any set of basis functions belonging to the same irreps.",
"We will see that an analogous measure emerges naturally for the case of $N=4$ .",
"(6) Check whether the nodes thereby obtained are inflated if the pairing amplitudes are not infinitesimal.",
"The main tool is the Pfaffian $\\mathop {\\textrm {Pf}}\\tilde{\\mathcal {H}}(\\mathbf {k})$ of an antisymmetric Hamiltonian $\\tilde{\\mathcal {H}}(\\mathbf {k})$ that is unitarily equivalent to the BdG Hamiltonian $\\mathcal {H}(\\mathbf {k})$ .",
"Such a unitary transformation is guaranteed to exist if the point group contains the inversion, as shown in [1].",
"A simpler version of the proof is presented in Appendix .",
"The square of the Pfaffian equals the determinant of the BdG Hamiltonian and thus the product of the quasiparticle energies.",
"Hence, nodes of any kind correspond to $\\mathop {\\textrm {Pf}}\\tilde{\\mathcal {H}}(\\mathbf {k}) = 0$ .",
"As shown in Appendix , the Pfaffian is real for even $N$ and imaginary for odd $N$ .",
"We define $P(\\mathbf {k}) \\equiv \\left\\lbrace \\begin{array}{ll}\\mathop {\\textrm {Pf}}\\tilde{\\mathcal {H}}(\\mathbf {k}) & \\mbox{for $N$ even,} \\\\[0.5ex]i\\mathop {\\textrm {Pf}}\\tilde{\\mathcal {H}}(\\mathbf {k}) & \\mbox{for $N$ odd}\\end{array} \\right.$ to obtain a real quantity.",
"We will simply call $P(\\mathbf {k})$ the Pfaffian in the following.",
"The sign of $P(\\mathbf {k})$ turns out to depend on the choice of unitary transformation which leads to the antisymmetric matrix $\\tilde{\\mathcal {H}}(\\mathbf {k})$ .",
"We choose this transformation in such a way that the Pfaffian is positive at some point far from the normal-state Fermi surface.",
"Since the Pfaffian is a smooth function of momentum this fixes the sign for all $\\mathbf {k}$ .",
"For $s_T = -1$ and preserved TRS, $P(\\mathbf {k})$ is nonnegative for all $\\mathbf {k}$ , the topological $\\mathbb {Z}_2$ invariant is thus trivial, and there are no topologically protected BFSs, as shown in [1], [2].",
"Conversely, such BFSs are expected for broken TRS.",
"The argument is reviewed in Appendix .",
"In addition, we there show that the Pfaffian can change sign and BFSs are expected also for $s_T = +1$ , regardless of symmetry."
],
[
"Four-dimensional internal Hilbert space",
"Even-parity superconductors with a four-dimensional internal Hilbert space (and time reversal squaring to $-1$ ) constitute the simplest case beyond the single-band paradigm.",
"According to Appendix , the normal-state Hamiltonian $H_N(\\mathbf {k}) = \\sum _n c_n(\\mathbf {k})\\, h_n$ is a superposition of six basis matrices $h_0$ , ..., $h_5$ .",
"As noted above, the same six basis matrices appear in the pairing matrix $D(\\mathbf {k})$ .",
"These matrices realize a nice algebraic structure of $4\\times 4$ gamma matrices: One can always choose the $h_n$ in such a way that $h_1$ , ..., $h_5$ anticommute pairwise, while $h_0=$ commutes with any matrix [28], [29]; see Appendix .",
"This implies that for any such model the eigenenergies in the normal state are $E_{N\\pm }(\\mathbf {k}) = c_0(\\mathbf {k}) \\pm \\sqrt{ c_1^2(\\mathbf {k}) + \\ldots + c_5^2(\\mathbf {k}) } ,$ both twofold degenerate.",
"The algebraic structure also allows to derive analytical results for the quasiparticle energies in the superconducting state and for the Pfaffian $P(\\mathbf {k})$ .",
"We obtain universal results when expressing these quantities in terms of coefficients of basis matrices.",
"Since we have found negative values of the Pfaffian and thus BFSs there, the same should be true for any model with $N=4$ in this class.",
"For this conclusion to hold, it is important that the prefactors of the matrices are not constrained by symmetries so that all values of the Pfaffian can actually occur.",
"Table: Commutation relations of the matrices defined in Eqs.",
"()–().",
"++ (--) denotes commutation (anticommutation).The BdG Hamiltonian in Eq.",
"(REF ) can be written as a linear combination of 18 Hermitian $8 \\times 8$ matrices, $\\mathcal {H}(\\mathbf {k}) = \\sum _{n=0}^5 c_n(\\mathbf {k})\\, H_n+ \\sum _{n=0}^5 f^1_n(\\mathbf {k})\\, \\Gamma _n+ \\sum _{n=0}^5 f^2_n(\\mathbf {k})\\, \\Phi _n .$ The coefficients are all real functions of momentum.",
"The 18 matrices are $H_n &= \\left(\\begin{array}{cc}h_n & 0 \\\\0 & - U_P^* h_n^T U_P^T\\end{array}\\right) , \\\\\\Gamma _n &= \\left(\\begin{array}{cc}0 & h_n U_T \\\\U_T^\\dagger h_n & 0\\end{array}\\right) , \\\\\\Phi _n &= \\left(\\begin{array}{cc}0 & ih_n U_T \\\\-i U_T^\\dagger h_n & 0\\end{array}\\right) ,$ where $n=0,\\ldots ,5$ .",
"The definition of $H_n$ requires some discussion.",
"The matrices $h_n$ are irreducible tensor operators of $g$ or $u$ irreps and $H_n = \\left\\lbrace \\begin{array}{ll}\\displaystyle \\left(\\begin{array}{cc}h_n & 0 \\\\0 & - h_n^T\\end{array}\\right) & \\mbox{for $g$ irreps,} \\\\[2.5ex]\\displaystyle \\left(\\begin{array}{cc}h_n & 0 \\\\0 & h_n^T\\end{array}\\right) & \\mbox{for $u$ irreps.",
"}\\end{array}\\right.$ For $g$ irreps, the minus sign of $-\\mathbf {k}$ in the lower right block of Eq.",
"(REF ) drops out and the first term in Eq.",
"(REF ) is obvious.",
"For $u$ irreps, the form factor $c_n(\\mathbf {k})$ is an odd function and the lower right block obtains an additional sign change.",
"Since in Eq.",
"(REF ) the coefficient of $H_n$ is $c_n(\\mathbf {k})$ this sign must be incorporated into $H_n$ .",
"The matrices $H_n$ , $\\Gamma _n$ , and $\\Phi _n$ are all Hermitian, traceless, square to $$ , and either commute or anticommute according to Table REF .",
"Moreover, if $A_\\alpha $ , $\\alpha =1,\\ldots ,18$ denote all 18 matrices, we have $\\mathop {\\textrm {Tr}}\\lbrace A_\\alpha , A_\\beta \\rbrace \\equiv \\mathop {\\textrm {Tr}}(A_\\alpha A_\\beta + A_\\beta A_\\alpha ) = 16\\, \\delta _{\\alpha \\beta } .$ The Pfaffian can be expressed in closed form, as discussed in more detail in Appendix .",
"We suppress momentum arguments for the rest of this section.",
"It is useful to define the real five-vectors $\\vec{c} &\\equiv (c_1,c_2,c_3,c_4,c_5) , \\\\\\vec{f}^{\\,1} &\\equiv (f^1_1,f^1_2,f^1_3,f^1_4,f^1_5) , \\\\\\vec{f}^{\\,2} &\\equiv (f^2_1,f^2_2,f^2_3,f^2_4,f^2_5)$ and the Minkowski-type scalar product $\\langle A,B\\rangle \\equiv A_0 B_0 - \\vec{A}\\cdot \\vec{B} .$ The Pfaffian can then be written as $P(\\mathbf {k}) &= \\langle c,c\\rangle ^2 + \\langle f^1,f^1\\rangle ^2+ \\langle f^2,f^2\\rangle ^2 \\nonumber \\\\&\\quad {}+ 4\\, \\big ( \\langle c,f^1\\rangle ^2 + \\langle f^1,f^2\\rangle ^2+ \\langle f^2,c\\rangle ^2 \\big ) \\nonumber \\\\&\\quad {}- 2\\, \\big ( \\langle c,c\\rangle \\, \\langle f^1,f^1\\rangle + \\langle f^1,f^1\\rangle \\, \\langle f^2,f^2\\rangle \\nonumber \\\\&\\quad {}+ \\langle f^2,f^2\\rangle \\, \\langle c,c\\rangle \\big ) .$ Another form that will prove useful is $P(\\mathbf {k}) &= \\big ( \\langle c,c\\rangle - \\langle f^1,f^1\\rangle - \\langle f^2,f^2\\rangle \\big )^2\\nonumber \\\\&\\quad {}+ 4\\, \\big ( \\langle c,f^1\\rangle ^2 + \\langle c,f^2\\rangle ^2 + \\langle f^1,f^2\\rangle ^2\\nonumber \\\\&\\quad {}- \\langle f^1,f^1\\rangle \\langle f^2,f^2\\rangle \\big ) .$ Nodes are signaled by $P(\\mathbf {k})=0$ .",
"If $P(\\mathbf {k})$ becomes negative for some momenta $\\mathbf {k}$ we obtain two-dimensional BFSs [1], [2].",
"The algebraic structure and the expressions for the Pfaffian are the same for all models with even-parity superconductors, time reversal squaring to $-1$ , and $N=4$ .",
"Hence, the conclusion of Refs.",
"[1], [2] that nodes are inflated into BFSs for TRS-breaking superconducting states applies to all such models.",
"For infinitesimal pairing, we can neglect terms of fourth order in the amplitudes $f^\\alpha _n$ compared to terms of second order in Eq.",
"(REF ).",
"The result $P(\\mathbf {k}) &\\cong \\big ( \\langle c,c\\rangle - \\langle f^1,f^1\\rangle - \\langle f^2,f^2\\rangle \\big )^2\\nonumber \\\\&\\quad {}+ 4\\, \\big ( \\langle c,f^1\\rangle ^2 + \\langle c,f^2\\rangle ^2 \\big )$ is nonnegative.",
"Hence, the momentum-space volume of the BFSs shrinks to zero for infinitesimal pairing, leaving only point and line nodes.",
"Since the expression in Eq.",
"(REF ) is a sum of squares IP nodes occur when three conditions hold simultaneously.",
"The first reads as $\\langle c,c\\rangle - \\langle f^1,f^1\\rangle - \\langle f^2,f^2\\rangle = 0 .$ Since $\\langle c,c\\rangle = E_{N+} E_{N-} = 0$ is a criterion for the normal-state Fermi surface we can say that Eq.",
"(REF ) describes a renormalized Fermi surface.",
"It will be close to the normal-state Fermi surface in the typical case that the pairing energy is small compared to the chemical potential.",
"The second and third condition read as $\\langle c,f^1\\rangle = \\langle c,f^2\\rangle = 0$ , which are equivalent to $\\langle c,f\\rangle = 0 ,$ where $f\\equiv f^1+if^2$ .",
"Explicitly, this condition reads as $c_0\\, (f^1_0 + i f^2_0) - \\sum _{n=1}^5 c_n\\, (f^1_n + i f^2_n)\\equiv c_0 f_0 - \\sum _{n=1}^5 c_n f_n = 0 .$ Except for the signs, which do not matter, this agrees with the function $F(\\mathbf {k})$ that we found above to encode the IP nodal structure; see Eq.",
"(REF )."
],
[
"Applications",
"In the following, we illustrate the general procedure for specific examples.",
"We will mainly consider a familiar setting: the dimension of the internal Hilbert space is $N=4$ , resulting from spin and either orbital or basis site, and the model is described by the cubic point group $O_h$ .",
"This point group has ten irreps, $A_{1g}$ , $A_{2g}$ , $E_g$ , $T_{1g}$ , $T_{2g}$ , $A_{1u}$ , $A_{2u}$ , $E_u$ , $T_{1u}$ , and $T_{2u}$ .",
"For the corresponding gray magnetic point group, the number of irreducible real corepresentations is doubled to $A_{1g+}$ , $A_{1g-}$ , $A_{2g+}$ , etc."
],
[
"Two ",
"We first consider a lattice without basis and with two orbitals per site that are invariant under all point-group transformations.",
"This means that they transform according to the trivial irrep $A_{1g}$ or, in other words, like s-orbitals.",
"The interesting point here is that even such a simple model supports nontrivial multiband superconductivity with BFSs.",
"For the internal Hilbert space, we use the basis $\\lbrace |1{\\uparrow }\\rangle , |1{\\downarrow }\\rangle , |2{\\uparrow }\\rangle , |2{\\downarrow }\\rangle \\rbrace $ , where 1, 2 refers to the orbital and $\\uparrow $ , $\\downarrow $ to the spin.",
"In this section, the first factor in Kronecker products refers to the orbital and the second to the spin.",
"The matrix representation of the inversion or parity operator $P$ has the trivial form $U_P = = \\sigma _0 \\otimes \\sigma _0 .$ The unitary part of the time-reversal operator is $U_T = \\sigma _0 \\otimes i\\sigma _2$ since the orbitals are invariant under time reversal, while in the spin sector we have the standard form $i\\sigma _2$ .",
"Table: Basis matrices on the internal Hilbert space for the case of two s-orbitals and point group O h O_h.",
"The basis matrices are irreducible tensor operators of the irreps listed in the second column.",
"For multidimensional irreps, the states transforming into each other under point-group operations are distinguished by the index in the third column.The 16 basis matrices $h_\\nu $ of the space of Hermitian $4\\times 4$ matrices obtained as Kronecker products are listed in Table REF , together with the corresponding irreps.",
"To understand the table, first consider the structural point group.",
"Since the orbitals transform trivially under all point-group elements the spin alone determines the irrep.",
"Then $\\sigma _0$ obviously transforms trivially, i.e., according to $A_{1g}$ , while $\\mbox{$\\sigma $}=(\\sigma _1,\\sigma _2,\\sigma _3)$ is a pseudovector, which transforms according to $T_{1g}$ .",
"Regarding time reversal, $\\sigma _0$ in the spin sector is of course even, whereas $\\mbox{$\\sigma $}$ is odd.",
"However, the time-reversal operator $\\mathcal {T}$ is antilinear so that the imaginary Pauli matrix $\\sigma _2$ in the orbital sector gives another sign change.",
"Note that although the orbital degree of freedom appears to be a trivial spectator, it does lead to the appearance of the additional irreps $A_{1g-}$ and $T_{1g+}$ .",
"Next, we consider momentum basis functions.",
"As noted above, they only exist for $g+$ and $u-$ irreps.",
"Low-order polynomial basis functions can be found in tables [18], [33].",
"It is important to note that for our purposes the constant function and the second-order function $k_x^2+k_y^2+k_z^2$ are allowed basis functions of $A_{1g+}$ but are not listed in some tables.",
"The tables usually do not show a basis function for $A_{1u-}$ since the simplest one is of order $l=9$ , specifically $k_xk_yk_z\\, [ k_x^4 (k_y^2-k_z^2) + k_y^4 (k_z^2-k_x^2) + k_z^4 (k_x^2-k_y^2) ]$ [34], [18].",
"The possible irreps of pairing states are now obtained by reducing all products of the allowed irreps of the momentum-dependent form factor and of pairing matrices and excluding the ones that are odd under time reversal and thus violate fermionic antisymmetry.",
"The reduction of the remaining combinations is shown in Table REF .",
"The normal-state Hamiltonian $H_N(\\mathbf {k})$ can, and generically does, contain all combinations that transform according to $A_{1g+}$ , set in bold face.",
"Only the first row of the table (form-factor irrep $A_{1g+}$ ) is compatible with purely local pairing, which can thus have $A_{1g+}$ or $T_{1g+}$ symmetry.",
"Note that the latter is impossible for a single-orbital system.",
"Table: Reduction of product representations of the allowed irreps of 𝐤\\mathbf {k}-dependent form factors (rows) and basis matrices h ν h_\\nu (columns) for two s-orbitals.",
"For the form factors, the minimum order of polynomial basis functions is given in the second column.",
"“∘\\circ ” indicates products that are forbidden since they violate fermionic antisymmetry.The normal-state Hamiltonian contains two types of terms, generated by $A_{1g+}\\otimes A_{1g+}$ and by $T_{1g+}\\otimes T_{1g+}$ , respectively.",
"For the first, there are three basis matrices belonging to $A_{1g+}$ according to Table REF , hence we get $H_{N1}(\\mathbf {k}) = c_{00}(\\mathbf {k})\\, \\sigma _0 \\otimes \\sigma _0+ c_{10}(\\mathbf {k})\\, \\sigma _1 \\otimes \\sigma _0+ c_{30}(\\mathbf {k})\\, \\sigma _3 \\otimes \\sigma _0 ,$ where $c_{00}$ , $c_{10}$ , and $c_{30}$ are generally distinct basis functions of $A_{1g+}$ .",
"In other words, they are invariant under all elements of the magnetic point group.",
"The leading polynomial terms read a [18], [33] $c_{m0}(\\mathbf {k}) &= c_{m0}^{(0)} + c_{m0}^{(2)}\\, (k_x^2+k_y^2+k_z^2)+ c_{m0}^{(4)}\\, (k_x^4+k_y^4+k_z^4) \\nonumber \\\\&\\quad {} + c_{m0}^{(6)}\\, k_x^2k_y^2k_z^2 + \\ldots $ for $m=0,1,3$ .",
"For the second type, we observe that there is a single triplet of matrices belonging to $T_{1g+}$ , namely $\\sigma _2 \\otimes \\mbox{$\\sigma $}$ .",
"To obtain an invariant Hamiltonian, they must each be multiplied by the corresponding momentum basis function, which gives $H_{N2}(\\mathbf {k}) &= c_{21}(\\mathbf {k})\\, \\sigma _2 \\otimes \\sigma _1+ c_{22}(\\mathbf {k})\\, \\sigma _2 \\otimes \\sigma _2 \\nonumber \\\\&\\quad {} + c_{23}(\\mathbf {k})\\, \\sigma _2 \\otimes \\sigma _3 .$ The functions $c_{2n}$ are not independent but must transform into each other under the magnetic point group.",
"The leading terms are [18], [33] $c_{21}(\\mathbf {k}) &= c_{2}^{(4)}\\, k_yk_z (k_y^2-k_z^2)+ c_{2}^{(6)}\\, k_yk_z (k_y^4-k_z^4) + \\ldots , \\\\c_{22}(\\mathbf {k}) &= c_{2}^{(4)}\\, k_zk_x (k_z^2-k_x^2)+ c_{2}^{(6)}\\, k_zk_x (k_z^4-k_x^4) + \\ldots , \\\\c_{23}(\\mathbf {k}) &= c_{2}^{(4)}\\, k_xk_y (k_x^2-k_y^2)+ c_{2}^{(6)}\\, k_xk_y (k_x^4-k_y^4) + \\ldots $ The full normal-state Hamiltonian $H_N(\\mathbf {k}) = H_{N1}(\\mathbf {k}) + H_{N2}(\\mathbf {k})= \\sum _{n=0}^5 c_n(\\mathbf {k})\\, h_n$ is a linear combination of the six basis matrices $h_0 &\\equiv \\sigma _0 \\otimes \\sigma _0 & A_{1g+}, \\\\h_1 &\\equiv \\sigma _1 \\otimes \\sigma _0 & A_{1g+}, \\\\h_2 &\\equiv \\sigma _3 \\otimes \\sigma _0 & A_{1g+}, \\\\h_3 &\\equiv \\sigma _2 \\otimes \\sigma _1 & T_{1g+}, \\\\h_4 &\\equiv \\sigma _2 \\otimes \\sigma _2 & T_{1g+}, \\\\h_5 &\\equiv \\sigma _2 \\otimes \\sigma _3 & T_{1g+}, $ where the irreps are also given.",
"$h_1$ , ..., $h_5$ anticommute pairwise; see Appendices and .",
"Table REF also provides useful information on superconductivity: (a) All ten irreps that are even under time reversal appear as symmetries of possible pairing states.",
"In fact, every $g+$ irrep $\\Gamma _{g+}$ is possible for any model since such a symmetry can be realized by combining an even momentum-space basis function belonging to $\\Gamma _{g+}$ with the $A_{1g+}$ basis matrix $h_0=$ .",
"On the other hand, $u+$ pairing state require $g-$ basis matrices, here belonging to $A_{1g-}$ and $T_{1g-}$ [35].",
"(b) For the even-parity pairing states, only the five $g+$ irreps are relevant.",
"From Table REF , we see that then only the basis matrices transforming according to $A_{1g+}$ or $T_{1g+}$ occur.",
"Further inspection shows that both types of basis matrices contribute to all $g+$ pairing states (all five occur in both columns).",
"We conclude that for pairing states belonging to any single $g+$ irrep, all $A_{1g+}$ and $T_{1g+}$ basis matrices can appear in the pairing matrix $D(\\mathbf {k}) = \\sum _{n=0}^5 f_n(\\mathbf {k})\\, h_n .$ What changes between different pairing states are the momentum-dependent form factors $f_n(\\mathbf {k})$ .",
"In the following, we analyze the nodal structure for several exemplary pairing symmetries.",
"We start with the simplest pairing symmetry, $A_{1g+}$ .",
"The construction of allowed terms in $D(\\mathbf {k})$ is analogous to the construction of $H_N(\\mathbf {k})$ .",
"From Table REF , we find two contributions to $A_{1g+}$ pairing: (a) form factors that transform according to $A_{1g+}$ combined with the three $A_{1g+}$ basis matrices and (b) a triplet of $T_{1g+}$ form factors combined with the triplet of $T_{1g+}$ basis matrices.",
"We discuss these two contributions in turn.",
"It will prove useful to separate momentum-independent pairing amplitudes denoted by $\\delta _{\\cdots }$ from suitably normalized momentum basis functions denoted by $d_{\\cdots }(\\mathbf {k})$ , as done in Eq.",
"(REF ).",
"(a) This contribution to the pairing matrix reads as $D_1(\\mathbf {k}) &= \\delta _{00} d_{00}(\\mathbf {k})\\, \\sigma _0 \\otimes \\sigma _0+ \\delta _{10} d_{10}(\\mathbf {k})\\, \\sigma _1 \\otimes \\sigma _0 \\nonumber \\\\&\\quad {}+ \\delta _{30} d_{30}(\\mathbf {k})\\, \\sigma _3 \\otimes \\sigma _0 ,$ where $d_{00}(\\mathbf {k})$ , $d_{10}(\\mathbf {k})$ , and $d_{30}(\\mathbf {k})$ are basis functions of $A_{1g+}$ , and $\\delta _{00}$ , $\\delta _{10}$ , and $\\delta _{30}$ denote the corresponding pairing amplitudes.",
"The leading polynomial forms of the basis functions are $d_{m0}(\\mathbf {k}) = d_{m0}^{(0)} + d_{m0}^{(2)}\\, (k_x^2+k_y^2+k_z^2) + \\ldots ,$ where we can set the three constants $d_{m0}^{(0)}$ to unity as a normalization.",
"The higher-order coefficients are then generally distinct for different $m$ .",
"These contributions can be interpreted as s-wave pairing since the minimum order of the basis functions is $l=0$ .",
"We use the terms s-wave, p-wave, etc.",
"to describe only the momentum dependence, not the symmetry of the full pairing state, for which we always use the irreps.",
"The contribution is evidently spin-singlet pairing because the matrix acting in spin space is $\\sigma _0$ .",
"(b) The reducible representation $T_{1g+}\\otimes T_{1g+}$ has nine matrix-valued basis functions $d_m(\\mathbf {k})\\, \\sigma _2 \\otimes \\sigma _n$ , $m,n=1,2,3$ , where $d_m(\\mathbf {k})$ are momentum basis functions of $T_{1g+}$ .",
"The reduction $T_{1g+}\\otimes T_{1g+} = A_{1g+} \\oplus E_{g+} \\oplus T_{1g+} \\oplus T_{2g+}$ tells us that a basis change to matrix basis functions of the four indicated irreps exists.",
"We here need to find the linear combination of the $d_m(\\mathbf {k})\\, \\sigma _2 \\otimes \\sigma _n$ that transforms according to $A_{1g+}$ .",
"This is simply the sum over products of corresponding components with identical coefficients, i.e., $D_2(\\mathbf {k}) &= \\delta _t\\, \\big [ d_{21}(\\mathbf {k})\\, \\sigma _2 \\otimes \\sigma _1+ d_{22}(\\mathbf {k})\\, \\sigma _2 \\otimes \\sigma _2 \\nonumber \\\\&\\quad {} + d_{23}(\\mathbf {k})\\, \\sigma _2 \\otimes \\sigma _3 \\big ] ,$ where $d_{21}(\\mathbf {k})$ , $d_{22}(\\mathbf {k})$ , and $d_{23}(\\mathbf {k})$ form a triplet of $T_{1g+}$ basis functions and $\\delta _t$ is their common amplitude.",
"The leading polynomials are $d_3(\\mathbf {k}) &\\equiv d_{21}(\\mathbf {k}) = d_t^{(4)}\\, k_y k_z (k_y^2-k_z^2) + \\ldots , \\\\d_4(\\mathbf {k}) &\\equiv d_{22}(\\mathbf {k}) = d_t^{(4)}\\, k_z k_x (k_z^2-k_x^2) + \\ldots , \\\\d_5(\\mathbf {k}) &\\equiv d_{23}(\\mathbf {k}) = d_t^{(4)}\\, k_x k_y (k_x^2-k_y^2) + \\ldots ,$ where we can choose $d_t^{(4)}=1$ as a normalization.",
"This is g-wave spin-triplet (hence the subscript “t”) pairing since the minimum order is $l = 4$ and the Pauli matrices $\\sigma _1$ , $\\sigma _2$ , $\\sigma _3$ act on the spin Hilbert space.",
"This combination is made possible by the nontrivial orbital content.",
"The full $A_{1g+}$ pairing matrix has the usual form $D(\\mathbf {k}) \\equiv D_1(\\mathbf {k}) + D_2(\\mathbf {k}) = \\sum _{n=0}^5f_n(\\mathbf {k})\\, h_n ,$ where the symmetry properties of the form factors $f_n(\\mathbf {k})$ have been obtained above.",
"We first consider pairing that respects TRS.",
"Then, all $f_n(\\mathbf {k})$ can be chosen real.",
"Equation (REF ) gives the condition for IP nodes.",
"Each pairing form factor $f_n(\\mathbf {k})$ is multiplied by the corresponding normal-state form factor $c_n(\\mathbf {k})$ .",
"The contribution (a) give, to leading order, $c_0&(\\mathbf {k}) f_0(\\mathbf {k}) - c_1(\\mathbf {k}) f_1(\\mathbf {k}) - c_2(\\mathbf {k}) f_2(\\mathbf {k})\\nonumber \\\\&= c_{00}^{(0)} \\delta _{00} - c_{10}^{(0)} \\delta _{10} - c_{30}^{(0)} \\delta _{30} + \\ldots ,$ which is generically nonzero and nodeless.",
"The expression can of course have accidental nodes from higher-order terms, which we disregard here.",
"For the contribution (b), $(c_3,c_4,c_5)$ and $(f_3,f_4,f_5)$ are corresponding basis functions of $T_{1g+}$ , and we find $- c_3&(\\mathbf {k}) f_3(\\mathbf {k}) - c_4(\\mathbf {k}) f_4(\\mathbf {k}) - c_5(\\mathbf {k}) f_5(\\mathbf {k})\\nonumber \\\\&= - c_2^{(4)} \\delta _t\\big [ k_y^2 k_z^2 (k_y^2-k_z^2)^2 + k_z^2 k_x^2 (k_z^2-k_x^2)^2 \\nonumber \\\\&\\quad {} + k_x^2 k_y^2 (k_x^2-k_y^2)^2 \\big ] + \\ldots $ This expression vanishes if the conditions $k_y^2k_z^2 (k_y-k_z)^2 (k_y+k_z)^2 &= 0 , \\\\k_z^2k_x^2 (k_z-k_x)^2 (k_z+k_x)^2 &= 0 , \\\\k_x^2k_y^2 (k_x-k_y)^2 (k_x+k_y)^2 &= 0$ hold simultaneously.",
"This is the case for the $6+8+12=26$ high-symmetry directions in the $O_h$ Brillouin zone.",
"Hence, there are 26 point nodes on a spheroidal normal-state Fermi surface around the $\\Gamma $ point.",
"The higher-order terms in the basis functions do not change this picture since the nodes are imposed by $T_{1g+}$ symmetry.",
"Since the conditions only contain squares, they are second-order (“double Weyl”) point nodes [26], [2], [36].",
"Together with the generically nodeless contribution (a), $\\langle c,f\\rangle $ can contain first-order line nodes provided that the amplitude $c_2^{(4)} \\delta _t$ is sufficiently large and not all terms have the same sign.",
"The location of these line nodes is not fixed by symmetries.",
"In this respect, the situation is similar to the case of mixed singlet-triplet pairing in noncentrosymmetric superconductors [37].",
"However, there is nothing that prevents a full gap, which is typically energetically favorable.",
"Breaking of TRS is possible for one-dimensional irreps, see Sec.",
".",
"However, since the $A_{1g+}$ time-reversal-symmetric state is generically nodeless, the breaking of TRS is not expected to lead to a reduction of the internal energy [26].",
"If TRS does break, then the condition (REF ) for IP nodes splits into two independent conditions for the real and imaginary parts of $\\langle c,f\\rangle $ .",
"Hence, TRS-breaking $A_{1g+}$ pairing states are even less likely to have nodes than time-reversal-symmetric ones."
],
[
"$A_{2g+}$ pairing",
"$A_{2g+}$ pairing is potentially interesting since it is governed by a nontrivial one-dimensional irrep.",
"It appears in two places in Table REF : (a) $A_{2g+}\\otimes A_{1g+}$ and (b) $T_{2g+}\\otimes T_{1g+}$ .",
"Hence, it is incompatible with purely local pairing, for which the first factor must be $A_{1g+}$ .",
"We discuss the two cases in turn.",
"(a) Each of the three $A_{1g+}$ basis matrices is combined with a $A_{2g+}$ form factor, giving $D_1(\\mathbf {k}) &= \\delta _{00} d_{00}(\\mathbf {k})\\, \\sigma _0 \\otimes \\sigma _0+ \\delta _{10} d_{10}(\\mathbf {k})\\, \\sigma _1 \\otimes \\sigma _0 \\nonumber \\\\&\\quad {}+ \\delta _{30} d_{30}(\\mathbf {k})\\, \\sigma _3 \\otimes \\sigma _0 .$ The leading-order polynomial form is $d_{m0}(\\mathbf {k}) &= d_{m0}^{(6)}\\, \\big [ k_x^4 (k_y^2-k_z^2) + k_y^4 (k_z^2-k_x^2) \\nonumber \\\\&\\quad {} + k_z^4 (k_x^2-k_y^2) \\big ] + \\ldots ,$ where we set $d_{m0}^{(6)}=1$ as a normalization.",
"These are i-wave ($l=6$ ) spin-singlet contributions.",
"Based on the rule of thumb that terms of lower order in $\\mathbf {k}$ are energetically favored since they have fewer nodes or nodes of lower order and thus lead to higher condensation energy, we expect this contribution to be weak compared to the following one.",
"(b) The reducible representation $T_{2g+}\\otimes T_{1g+}$ has nine matrix-valued basis functions $d_m(\\mathbf {k})\\, \\sigma _2 \\otimes \\sigma _n$ , $m,n=1,2,3$ , where $d_m(\\mathbf {k})$ are basis functions of $T_{2g+}$ .",
"The construction parallels the one for $A_{1g+}$ pairing.",
"The leading polynomial form factors read as $d_1(\\mathbf {k}) &\\equiv d_{21}(\\mathbf {k}) = d_2^{(2)}\\, k_y k_z + \\ldots , \\\\d_2(\\mathbf {k}) &\\equiv d_{22}(\\mathbf {k}) = d_2^{(2)}\\, k_z k_x + \\ldots , \\\\d_3(\\mathbf {k}) &\\equiv d_{23}(\\mathbf {k}) = d_2^{(2)}\\, k_x k_y + \\ldots $ We choose $d_2^{(2)}=1$ as normalization.",
"The $A_{2g+}$ part of $T_{2g+}\\otimes T_{1g+}$ has the matrix-valued basis function $D_{A_{2g+}}(\\mathbf {k}) &= d_1(\\mathbf {k})\\, h_3 + d_2(\\mathbf {k})\\, h_4 + d_3(\\mathbf {k})\\, h_5\\nonumber \\\\&\\cong k_y k_z\\, \\sigma _2 \\otimes \\sigma _1 + k_z k_x\\, \\sigma _2 \\otimes \\sigma _2+ k_x k_y\\, \\sigma _2 \\otimes \\sigma _3 ,$ which describes d-wave ($l=2$ ) spin-triplet pairing, allowed due to nontrivial orbital content.",
"Thus the second contribution to the pairing matrix is $D_2(\\mathbf {k}) \\cong \\delta _t ( k_y k_z\\, \\sigma _2 \\otimes \\sigma _1+ k_z k_x\\, \\sigma _2 \\otimes \\sigma _2 + k_x k_y\\, \\sigma _2 \\otimes \\sigma _3 ) .$ The leading order form factors $f_n(\\mathbf {k})$ can now be read off.",
"They are summarized in Table REF .",
"Table: Leading-order polynomial forms of the form factors f n (𝐤)f_n(\\mathbf {k}) describing A 2g+ A_{2g+} pairing for a model with two s-orbitals.Table: Leading-order polynomial forms of the products c n (𝐤)f n (𝐤)c_n(\\mathbf {k}) f_n(\\mathbf {k}) of form factors describing A 2g+ A_{2g+} pairing for a model with two s-orbitals.",
"The amplitudes of the leading terms in c n (𝐤)c_n(\\mathbf {k}) have been absorbed into new pairing amplitudes marked by a tilde.For time-reversal-symmetric pairing, we can choose all $f_n(\\mathbf {k})$ real.",
"For the condition for IP nodes, Eq.",
"(REF ), we require the products $c_n(\\mathbf {k}) f_n(\\mathbf {k})$ , which are listed in Table REF .",
"The amplitudes appearing the these products are distinguished by a tilde.",
"We obtain, to leading order, $c_0&(\\mathbf {k})\\, f_0(\\mathbf {k}) - \\vec{c}(\\mathbf {k})\\cdot \\vec{f}(\\mathbf {k}) \\nonumber \\\\&\\cong (\\tilde{\\delta }_{00} - \\tilde{\\delta }_{10} - \\tilde{\\delta }_{30} - \\tilde{\\delta }_t)\\,[ k_x^4 (k_y^2-k_z^2) + k_y^4 (k_z^2-k_x^2) \\nonumber \\\\&\\quad {}+ k_z^4 (k_x^2-k_y^2) ] \\nonumber \\\\&= -(\\tilde{\\delta }_{00} - \\tilde{\\delta }_{10} - \\tilde{\\delta }_{30} - \\tilde{\\delta }_t)\\,(k_x-k_y)(k_x+k_y) \\nonumber \\\\&\\quad {}\\times (k_y-k_z)(k_y+k_z)(k_z-k_x)(k_z+k_x) .$ This product clearly vanishes if two of the three components of $\\mathbf {k}$ are equal in magnitude.",
"The IP gap thus generically has six line nodes in the $\\lbrace 110\\rbrace $ planes.",
"They are of first order since $\\langle c,f\\rangle $ changes sign at the nodes.",
"The most obvious way to break TRS is to have a nontrivial phase difference between at least two of the amplitudes $\\tilde{\\delta }_{00}$ , $\\tilde{\\delta }_{10}$ , $\\tilde{\\delta }_{30}$ , and $\\tilde{\\delta }_t$ .",
"Then IP nodes exist where both the real part and the imaginary part of $\\langle c,f\\rangle $ vanish.",
"Equation (REF ) shows that the real and imaginary parts have the same symmetry-imposed line nodes so that the TRS-breaking state also has these line nodes for infinitesimal pairing.",
"To check whether these line nodes are inflated beyond infinitesimal pairing, we consider the Pfaffian.",
"It is useful to keep the full momentum dependence of the normal-state form factors, not just the leading terms.",
"The form factors $c_0(\\mathbf {k})$ , $c_1(\\mathbf {k})$ , and $c_2(\\mathbf {k})$ are independent functions with $A_{1g+}$ symmetry, while the remaining three form factors can be written as $c_3(\\mathbf {k}) &= a_T(\\mathbf {k})\\, k_y k_z (k_y^2-k_z^2) , \\\\c_4(\\mathbf {k}) &= a_T(\\mathbf {k})\\, k_z k_x (k_z^2-k_x^2) , \\\\c_5(\\mathbf {k}) &= a_T(\\mathbf {k})\\, k_x k_y (k_x^2-k_y^2) ,$ where $a_T(\\mathbf {k})$ is another function with $A_{1g+}$ symmetry.",
"Without loss of generality, we consider the plane $k_x=k_y$ , which is nodal for infinitesimal pairing.",
"In this plane, the generalized scalar products read as $\\langle c,c\\rangle &= c_0^2(\\mathbf {k}) - c_1^2(\\mathbf {k}) - c_2^2(\\mathbf {k}) \\nonumber \\\\&\\quad {}- 2 a_T^2(\\mathbf {k})\\, k_x^2 k_z^2 (k_x^2-k_z^2)^2 , \\\\\\langle c,f_1\\rangle &= \\langle c,f_2\\rangle = 0 , \\\\\\langle f^1,f^1\\rangle &= - (\\mathop {\\textrm {Re}}\\delta _t)^2\\, k_x^2 (2 k_z^2 + k_x^2) , \\\\\\langle f^2,f^2\\rangle &= - (\\mathop {\\textrm {Im}}\\delta _t)^2\\, k_x^2 (2 k_z^2 + k_x^2) , \\\\\\langle f^1,f^2\\rangle &= - \\mathop {\\textrm {Re}}\\delta _t \\mathop {\\textrm {Im}}\\delta _t\\, k_x^2 (2 k_z^2 + k_x^2) ,$ where $\\mathbf {k}=(k_x,k_x,k_z)$ .",
"Equation (REF ) then gives the Pfaffian $P(\\mathbf {k})&= \\big ( \\langle c,c\\rangle - \\langle f^1,f^1\\rangle - \\langle f^2,f^2\\rangle \\big )^2 \\nonumber \\\\&\\quad {}+ 4\\, \\big ( \\langle f^1,f^2\\rangle ^2 - \\langle f^1,f^1\\rangle \\langle f^2,f^2\\rangle \\big ) .$ The first term is a complete square and its zeros define the renormalized Fermi surface discussed above.",
"Using Eqs.",
"(REF )–(), the second term obviously vanishes, which can be attributed to the fact that in the plane $k_x=k_y$ only a single pairing channel ($T_{2g+}\\otimes T_{1g+}$ ) contributes to the pairing.",
"The phase of the corresponding amplitude $\\delta _t$ can always be chosen real so that TRS breaking does not affect the superconducting state.",
"The upshot is that for noninfinitesimal pairing the Pfaffian still has second-order zeros in the $\\lbrace 110\\rbrace $ planes and thus does not change sign.",
"At least within these planes the line nodes are shifted but neither gapped out nor inflated.",
"The question arises of what happens in the vicinity of these line nodes when we go off the high-symmetry planes.",
"The first term of the general Pfaffian given in Eq.",
"(REF ) has second-order zeros at the renormalized Fermi surface.",
"We expand the second term about a point on the plane $k_x=k_y$ by setting $\\mathbf {k} = (k_x+q/\\sqrt{2},k_x-q/\\sqrt{2},k_z)$ .",
"The leading form in $q$ reads as $4\\, &\\big ( \\langle c,f^1\\rangle ^2 + \\langle c,f^2\\rangle ^2+ \\langle f^1,f^2\\rangle ^2 - \\langle f^1,f^1\\rangle \\langle f^2,f^2\\rangle \\big ) \\nonumber \\\\&\\cong 32\\, k_x^2 (k_x^2-k_z^2)^4\\, \\Big (\\big | c_0 \\delta _{00} - c_1 \\delta _{10} - c_2 \\delta _{30} - a_T \\delta _t \\big |^2\\nonumber \\\\&\\quad {} + k_x^2 (k_x^2 + 2 k_z^2)\\, |\\delta _t|^2\\,\\big [ |\\delta _{00}|^2 \\sin ^2(\\phi _{00} - \\phi _t) \\nonumber \\\\&\\qquad {} - |\\delta _{10}|^2 \\sin ^2(\\phi _{10} - \\phi _t)- |\\delta _{30}|^2 \\sin ^2(\\phi _{30} - \\phi _t) \\big ] \\Big )\\, q^2 ,$ where $\\delta _{00} = |\\delta _{00}| e^{i\\phi _{00}}$ etc.",
"The expression contains contributions of second and fourth order in the pairing amplitudes.",
"At weak coupling, we can neglect the fourth-order contributions.",
"Then, the leading correction to the Pfaffian away from the $(110)$ plane is nonnegative and generically is strictly positive for $k_x \\ne k_z$ .",
"Hence, in this case, there is no BFS in the vicinity of the shifted line node, in any direction.",
"On the other hand, for strong coupling, the coefficient in Eq.",
"(REF ) can become negative.",
"In this case, BFSs can exist on both sides of the $\\lbrace 110\\rbrace $ planes and touching each other at these planes.",
"For the special case $k_x=k_z$ , the whole $q^2$ term in Eq.",
"(REF ) vanishes.",
"Since we already had assumed $k_x=k_y$ this corresponds to the threefold rotation axis $[111]$ .",
"Here, for infinitesimal pairing three nodal lines intersect.",
"We consider $\\mathbf {k} = (k_x+q/\\sqrt{2}, k_x-q/\\sqrt{2},k_x)$ .",
"The leading form in the expansion of the second term of the Pfaffian here reads as $12&8\\, k_x^6\\, \\Big (\\big | c_0 \\delta _{00} - c_1 \\delta _{10} - c_2 \\delta _{30} - a_T \\delta _d \\big |^2\\nonumber \\\\&{} + 3\\, k_x^4\\, |\\delta _t|^2\\, \\big [ |\\delta _{00}|^2 \\sin ^2(\\phi _{00} -\\phi _d) \\nonumber \\\\&\\quad {} - |\\delta _{10}|^2 \\sin ^2(\\phi _{10} - \\phi _d)- |\\delta _{30}|^2 \\sin ^2(\\phi _{30} - \\phi _d) \\big ] \\Big )\\, q^6 .$ This term is also nonnegative at weak coupling so that there are no BFSs close to the $\\langle 111\\rangle $ directions.",
"In conclusion, the line nodes for the TRS-breaking $A_{2g+}$ pairing state are not inflated into BFSs.",
"However, they are shifted away from the normal-state Fermi surface everywhere but remain within the high-symmetry (mirror) planes.",
"The lack of inflation within the high-symmetry planes can be understood on the basis that there is only a single relevant pairing amplitude, which can be chosen real."
],
[
"$E_{g+}$ pairing",
"Pairing conforming to the two-dimensional irrep $E_{g+}$ is of interest since the breaking of TRS occurs naturally for multidimensional irreps.",
"$E_{g+}$ pairing can emerge from the products (a) $E_{g+} \\otimes A_{1g+}$ , (b) $T_{1g+} \\otimes T_{1g+}$ , and (c) $T_{2g+} \\otimes T_{1g+}$ in Table REF .",
"Hence, it is incompatible with purely local pairing.",
"We discuss the contributions in turn.",
"(a) The three $A_{1g+}$ matrices are each combined with a doublet of $E_{g+}$ form factors, giving $D_1(\\mathbf {k})&= \\left[\\delta ^1_{00} d^1_{00}(\\mathbf {k}) + \\delta ^2_{00} d^2_{00}(\\mathbf {k})\\right]\\sigma _0 \\otimes \\sigma _0 \\nonumber \\\\&\\quad {} + \\left[\\delta ^1_{10} d^1_{10}(\\mathbf {k}) + \\delta ^2_{10} d^2_{10}(\\mathbf {k})\\right]\\sigma _1 \\otimes \\sigma _0 \\nonumber \\\\&\\quad {} + \\left[\\delta ^1_{30} d^1_{30}(\\mathbf {k}) + \\delta ^2_{30} d^2_{30}(\\mathbf {k})\\right]\\sigma _3 \\otimes \\sigma _0 .$ The leading polynomial terms are $d^1_{m0}(\\mathbf {k}) &= d^{(2)}_{m0}\\, (k_x^2-k_y^2) + d^{(4)}_{m0}\\, (k_x^4-k_y^4)+ \\ldots , \\\\d^2_{m0}(\\mathbf {k}) &= \\frac{d^{(2)}_{m0}}{\\sqrt{3}}\\,(2k_z^2-k_x^2-k_y^2) + \\frac{d^{(4)}_{m0}}{\\sqrt{3}}\\, (2k_z^4-k_x^4-k_y^4) \\nonumber \\\\&\\quad {} + \\ldots ,$ where we can set the coefficients $d^{(2)}_{m0}$ of the leading terms to unity as a normalization.",
"Recall that the higher-order coefficients are then generally distinct for different $m$ .",
"These contributions can be described as d-wave ($l=2$ ) spin-singlet pairing.",
"(b) The reducible representation $T_{1g+}\\otimes T_{1g+}$ has nine matrix-valued basis functions $d_m(\\mathbf {k})\\, \\sigma _2 \\otimes \\sigma _n$ , $m,n=1,2,3$ , where $d_m(\\mathbf {k})$ are momentum basis functions of $T_{1g+}$ .",
"The leading polynomial terms read as $d_1(\\mathbf {k}) &= d^{(4)}\\, k_y k_z (k_y^2-k_z^2) + \\ldots , \\\\d_2(\\mathbf {k}) &= d^{(4)}\\, k_z k_x (k_z^2-k_x^2) + \\ldots , \\\\d_3(\\mathbf {k}) &= d^{(4)}\\, k_x k_y (k_x^2-k_y^2) + \\ldots ,$ where $d^{(4)}$ can be chosen to be unity.",
"The reduction $T_{1g+}\\otimes T_{1g+} = A_{1g+} \\oplus E_{g+} \\oplus T_{1g+} \\oplus T_{2g+}$ implies that a basis change to matrix basis functions of the four indicated irreps exists.",
"The linear combinations of the functions $d_m(\\mathbf {k})\\, \\sigma _2 \\otimes \\sigma _n$ that transform according to $E_{g+}$ are $D_{x^2-y^2}(\\mathbf {k}) &= d_1(\\mathbf {k})\\, \\sigma _2 \\otimes \\sigma _1- d_2(\\mathbf {k})\\, \\sigma _2 \\otimes \\sigma _2 , \\\\D_{3z^2-r^2}(\\mathbf {k}) &= \\frac{2 d_3(\\mathbf {k})}{\\sqrt{3}}\\, \\sigma _2 \\otimes \\sigma _3- \\frac{d_1(\\mathbf {k})}{\\sqrt{3}}\\, \\sigma _2 \\otimes \\sigma _1 \\nonumber \\\\&\\quad {}- \\frac{d_2(\\mathbf {k})}{\\sqrt{3}}\\, \\sigma _2 \\otimes \\sigma _2 .$ These matrix basis functions are no longer simply the product of a scalar momentum-dependent form factor and a momentum-independent matrix.",
"Their contribution to the pairing matrix is $D_2(\\mathbf {k}) = \\delta ^1_{2t}\\, D_{x^2-y^2}(\\mathbf {k})+ \\delta ^2_{2t}\\, D_{3z^2-r^2}(\\mathbf {k}) .$ This describes g-wave ($l=4$ ) spin-triplet pairing, made possible by nontrivial orbital content.",
"(c) The analysis for $T_{2g+} \\otimes T_{1g+} = A_{2g+} \\oplus E_{g+} \\oplus T_{1g+} \\oplus T_{2g+}$ is analogous, except that now the momentum basis functions belong to $T_{2g+}$ , $d^{\\prime }_1(\\mathbf {k}) &= d^{\\prime (2)}\\, k_y k_z + \\ldots , \\\\d^{\\prime }_2(\\mathbf {k}) &= d^{\\prime (2)}\\, k_z k_x + \\ldots , \\\\d^{\\prime }_3(\\mathbf {k}) &= d^{\\prime (2)}\\, k_x k_y + \\ldots ,$ where we choose $d^{\\prime (2)}=1$ .",
"We have the matrix basis functions $D^{\\prime }_{x^2-y^2}(\\mathbf {k}) &= \\frac{2 d^{\\prime }_3(\\mathbf {k})}{\\sqrt{3}}\\, \\sigma _2 \\otimes \\sigma _3- \\frac{d^{\\prime }_1(\\mathbf {k})}{\\sqrt{3}}\\, \\sigma _2 \\otimes \\sigma _1 \\nonumber \\\\&\\quad {} - \\frac{d^{\\prime }_2(\\mathbf {k})}{\\sqrt{3}}\\, \\sigma _2 \\otimes \\sigma _2 , \\\\D^{\\prime }_{3z^2-r^2}(\\mathbf {k}) &= -d^{\\prime }_1(\\mathbf {k})\\, \\sigma _2 \\otimes \\sigma _1+ d^{\\prime }_2(\\mathbf {k})\\, \\sigma _2 \\otimes \\sigma _2 .$ Note that the forms of the expressions for the two components of $E_g$ , i.e., for the $x^2-y^2$ and the $3z^2-r^2$ matrix basis functions, are interchanged compared to case (b).",
"To determine the correct components, their behavior under twofold rotation about the $[110]$ direction has been examined.",
"Furthermore, to find the relative factor, which turns out to be $-1$ , the behavior under threefold rotation about $[111]$ has been considered.",
"The contribution to the pairing matrix is $D_3(\\mathbf {k}) = \\delta ^1_{2t^{\\prime }}\\, D^{\\prime }_{x^2-y^2}(\\mathbf {k})+ \\delta ^2_{2t^{\\prime }}\\, D^{\\prime }_{3z^2-r^2}(\\mathbf {k}) .$ This is d-wave spin-triplet pairing, again made possible by nontrivial orbital content.",
"The matrix $D(\\mathbf {k}) = D_1(\\mathbf {k}) + D_2(\\mathbf {k}) + D_3(\\mathbf {k})$ is evidently of the form of Eq.",
"(REF ).",
"The form factors $f_n(\\mathbf {k})$ are complicated functions of $\\mathbf {k}$ with nodes in different places.",
"The leading-order polynomial forms are listed in Table REF .",
"Table: Leading-order polynomial forms of the form factors f n (𝐤)f_n(\\mathbf {k}) describing E g+ E_{g+} pairing for a model with two s-orbitals.Table: Leading-order polynomial forms of the products c n (𝐤)f n (𝐤)c_n(\\mathbf {k}) f_n(\\mathbf {k}) of form factors describing E g+ E_{g+} pairing for a model with two s-orbitals.",
"The amplitudes of the leading terms in c n (𝐤)c_n(\\mathbf {k}) have been absorbed into new pairing amplitudes marked by a tilde.In the condition for IP nodes, Eq.",
"(REF ), each pairing form factor $f_n(\\mathbf {k})$ is multiplied by the normal-state form factor $c_n(\\mathbf {k})$ .",
"We list the leading-order polynomial form of these products in Table REF .",
"The contributions for $n=0,1,2$ have the same form and can be grouped together with new amplitudes $\\tilde{\\delta }^1_0$ and $\\tilde{\\delta }^2_0$ .",
"With this, we obtain, to leading order, $&c_0(\\mathbf {k})\\, f_0(\\mathbf {k}) - \\vec{c}(\\mathbf {k}) \\cdot \\vec{f}(\\mathbf {k})\\cong \\tilde{\\delta }^1_0\\, (k_x^2 - k_y^2) \\nonumber \\\\&{} + \\tilde{\\delta }^1_{2t}\\, k_z^2 (k_x^2 - k_y^2)(k_x^4 + k_y^4 + k_z^4 + k_x^2 k_y^2 - 2 k_x^2 k_z^2 - 2 k_y^2 k_z^2)\\nonumber \\\\&{} + \\frac{\\tilde{\\delta }^1_{2t^{\\prime }}}{\\sqrt{3}}\\, (k_x^2 - k_y^2)(k_z^4 - 2 k_x^2 k_y^2 - k_x^2 k_z^2 - k_y^2 k_z^2) \\nonumber \\\\&{} + \\frac{\\tilde{\\delta }^2_0}{\\sqrt{3}}\\, (2k_z^2 - k_x^2 - k_y^2) \\nonumber \\\\&{} + \\frac{\\tilde{\\delta }^2_{2t}}{\\sqrt{3}}\\, \\big ({-}2 k_x^6 k_y^2 + 4 k_x^4 k_y^4 - 2 k_x^2 k_y^6 + k_x^6 k_z^2+ k_y^6 k_z^2 \\nonumber \\\\&\\quad {} - 2 k_x^4 k_z^4 - 2 k_y^4 k_z^4 + k_x^2 k_z^6 + k_y^2 k_z^6\\big )\\nonumber \\\\&{} + \\tilde{\\delta }^2_{2t^{\\prime }}\\, k_z^2 (k_x^4 + k_y^4 - k_x^2 k_z^2 - k_y^2 k_z^2) .$ We first discuss time-reversal-symmetric pairing, for which all amplitudes can be chosen real.",
"Based on the order alone, we expect that typically $\\tilde{\\delta }^1_0$ and $\\tilde{\\delta }^2_0$ dominate.",
"Unless both amplitudes vanish these contributions give two first-order line nodes—the function changes sign at these nodes.",
"Note that all contributions belonging to the first component of $E_{g+}$ have nodal planes at $k_y=\\pm k_x$ .",
"This is because these nodes are symmetry induced and must lie in the diagonal mirror planes.",
"On the other hand, the contributions belonging to the second component also have two nodal surfaces (two line nodes on the normal-state Fermi surface) but these are not pinned to high-symmetry planes.",
"Hence, higher-order terms can shift them around.",
"Their only symmetry-enforced property is to pass through the $\\langle 111\\rangle $ directions, where they intersect with the nodes of the first component.",
"If both components have nonzero amplitudes there are still generically two line nodes, which have to pass through the $\\langle 111\\rangle $ directions [38].",
"If TRS is broken and only $\\tilde{\\delta }^1_0$ and $\\tilde{\\delta }^2_0$ are nonzero there must be a nontrivial phase difference between these two amplitudes and we find different line nodes in the real and imaginary parts, resulting in point nodes at their crossings.",
"Since the real and imaginary parts are zero in the $\\langle 111\\rangle $ directions, we obtain at least eight point nodes in these directions.",
"All higher-order terms are zero there so that they cannot shift or gap out these point nodes.",
"We next turn to the possibility of BFSs.",
"Without loss of generality, we consider the IP point node in the $[111]$ direction.",
"For $\\mathbf {k} = k\\,(1,1,1)/\\sqrt{3} \\equiv k\\, \\hat{\\mathbf {n}}_{111}$ , the normal-state form factors $c_0(\\mathbf {k}) \\equiv c_{00}(\\mathbf {k})$ , $c_1(\\mathbf {k}) \\equiv c_{10}(\\mathbf {k})$ , and $c_2(\\mathbf {k}) \\equiv c_{30}(\\mathbf {k})$ are independent even functions of $k$ ; see Eq.",
"(REF ).",
"On the other hand, $c_3(\\mathbf {k}) \\equiv c_{21}(\\mathbf {k})$ , $c_4(\\mathbf {k}) \\equiv c_{22}(\\mathbf {k})$ , and $c_5(\\mathbf {k}) \\equiv c_{23}(\\mathbf {k})$ vanish in this direction; see Eqs.",
"(REF )–().",
"Furthermore, Table REF shows that $f_0(\\mathbf {k}) &= f_1(\\mathbf {k}) = f_2(\\mathbf {k}) = 0 , \\\\f_3(\\mathbf {k}) &= - \\frac{\\delta ^1_{2t^{\\prime }}}{3\\sqrt{3}}\\, k^2 - \\frac{\\delta ^2_{2t^{\\prime }}}{3}\\, k^2 , \\\\f_4(\\mathbf {k}) &= - \\frac{\\delta ^1_{2t^{\\prime }}}{3\\sqrt{3}}\\, k^2 + \\frac{\\delta ^2_{2t^{\\prime }}}{3}\\, k^2 , \\\\f_5(\\mathbf {k}) &= \\frac{2\\delta ^1_{2t^{\\prime }}}{3\\sqrt{3}}\\, k^2 .$ This implies that $\\langle c,c\\rangle &= c_0^2(k) - c_1^2(k) - c_2^2(k) , \\\\\\langle c,f^1\\rangle &= \\langle c,f^2\\rangle = 0 , \\\\\\langle f^1,f^1\\rangle &= - \\frac{2k^2}{9} \\big [ (\\mathop {\\textrm {Re}}\\delta ^1_{2t^{\\prime }})^2 + (\\mathop {\\textrm {Re}}\\delta ^2_{2t^{\\prime }})^2 \\big ] , \\\\\\langle f^2,f^2\\rangle &= - \\frac{2k^2}{9} \\big [ (\\mathop {\\textrm {Im}}\\delta ^1_{2t^{\\prime }})^2 + (\\mathop {\\textrm {Im}}\\delta ^2_{2t^{\\prime }})^2 \\big ] , \\\\\\langle f^1,f^2\\rangle &= -\\frac{2k^2}{9} \\big [ \\mathop {\\textrm {Re}}\\delta ^1_{2t^{\\prime }} \\mathop {\\textrm {Im}}\\delta ^1_{2t^{\\prime }}+ \\mathop {\\textrm {Re}}\\delta ^2_{2t^{\\prime }} \\mathop {\\textrm {Im}}\\delta ^2_{2t^{\\prime }} \\big ] .$ Equation (REF ) then gives $P&(k\\, \\hat{\\mathbf {n}}_{111}) \\nonumber \\\\&= \\bigg [ c_0^2(k) - c_1^2(k) - c_2^2(k) + \\frac{2}{9}\\, k^2\\, |\\delta ^1_{2t^{\\prime }}|^2+ \\frac{2}{9}\\, k^2\\, |\\delta ^2_{2t^{\\prime }}|^2 \\bigg ]^{\\!2} \\nonumber \\\\&\\quad {}- \\frac{16}{81}\\, k^4\\, |\\delta ^1_{2t^{\\prime }}|^2 |\\delta ^2_{2t^{\\prime }}|^2 \\sin ^2(\\phi _1 - \\phi _2) ,$ where the two relevant pairing amplitudes are written as $\\delta ^{1,2}_{2t^{\\prime }} = |\\delta ^{1,2}_{2t^{\\prime }}|\\, e^{i\\phi _{1,2}}$ .",
"The first term has a second-order zero at the renormalized normal-state Fermi surface.",
"The second term is negative whenever the phase difference between $\\delta ^1_{2t^{\\prime }}$ and $\\delta ^2_{2t^{\\prime }}$ is not an integer multiple of $\\pi $ .",
"This is generically the case for broken TRS.",
"This means that in the vicinity of the renormalized normal-state Fermi surface we find a region with $P(k\\, \\hat{\\mathbf {n}}_{111})<0$ and thus the point node is inflated into a BFS pierced by the $[111]$ axis [39].",
"If the superconducting energy scale becomes comparable to normal-state energies the BFSs are no longer spheroidal pockets close to the IP point nodes.",
"The BFSs might then merge and could move either to the $\\Gamma $ point or to the edge of the Brillouin zone and annihilate there [2].",
"We now check whether this can happen.",
"On the unrenormalized normal-state Fermi surface in the $[111]$ direction, $c_0^2 - c_1^2 - c_2^2$ vanishes.",
"The Pfaffian can then be written as $P(k_F\\, \\hat{\\mathbf {n}}_{111}) &= \\frac{4}{81}\\, k_F^4 \\big ( |\\delta ^1_{2t^{\\prime }}|^2- |\\delta ^2_{2t^{\\prime }}|^2 \\big )^2 \\nonumber \\\\&\\quad {} + \\frac{16}{81}\\, k_F^4\\, |\\delta ^1_{2t^{\\prime }}|^2 |\\delta ^2_{2t^{\\prime }}|^2\\cos ^2(\\phi _1 - \\phi _2) .$ This means that for the special TRS-breaking state with $|\\delta ^1_{2t^{\\prime }}| = |\\delta ^2_{2t^{\\prime }}|$ and phase difference $\\pm \\pi $ or equivalent, the Pfaffian vanishes at $k=k_F$ so that the BFS must touch the normal-state Fermi surface.",
"In this case, the BFSs cannot annihilate for strong pairing.",
"The special conditions of equal amplitudes and phase difference of $\\pm \\pi $ are quite natural from the point of view of energetics [31], [26], [13], [1], [2].",
"In conclusion, at infinitesimal pairing, the gap generically has point nodes in the $\\langle 111\\rangle $ directions if TRS is broken.",
"These nodes are expected to be inflated into BFSs if the amplitudes $\\delta ^1_{2t^{\\prime }}$ and $\\delta ^2_{2t^{\\prime }}$ are both nonzero and have a nontrivial phase difference.",
"All other amplitudes do not contribute to the inflation of nodes along the $\\langle 111\\rangle $ directions since the corresponding form factors $f_n(\\mathbf {k})$ vanish there.",
"For the energetically favored $(1,i)$ state, the BFSs stick to the normal-state Fermi surface at the former point nodes."
],
[
"$T_{1g+}$ pairing",
"The analysis for the three-dimensional irrep $T_{1g+}$ is analogous and we will be brief.",
"All functions of momentum are represented by the lowest-order polynomials of correct symmetry.",
"$T_{1g+}$ pairing appears in the following products in Table REF : (a) $A_{1g+}\\otimes T_{1g+}$ , (b) $E_{g+}\\otimes T_{1g+}$ , (c) $T_{1g+}\\otimes A_{1g+}$ , (d) $T_{1g+}\\otimes T_{1g+}$ , (e) $T_{2g+}\\otimes T_{1g+}$ .",
"$T_{1g+}$ symmetry is possible even for purely local pairing due to the momentum-independent contribution (a).",
"(a) For the contribution $A_{1g+}\\otimes T_{1g+}$ , we find the matrix basis functions $D_{x,A_{1g+}}(\\mathbf {k}) &\\cong h_3 = \\sigma _2 \\otimes \\sigma _1 , \\\\D_{y,A_{1g+}}(\\mathbf {k}) &\\cong h_4 = \\sigma _2 \\otimes \\sigma _2 , \\\\D_{z,A_{1g+}}(\\mathbf {k}) &\\cong h_5 = \\sigma _2 \\otimes \\sigma _3 .$ These describe s-wave spin-triplet pairing, made possible by the nontrivial orbital structure.",
"(b) For $E_{g+}\\otimes T_{1g+}$ , the basis functions read as $D_{x,E_{g+}}(\\mathbf {k}) &\\cong \\frac{1}{3}\\, (2k_x^2-k_y^2-k_z^2)\\, h_3 \\nonumber \\\\&= \\frac{1}{3}\\, (2k_x^2-k_y^2-k_z^2)\\, \\sigma _2 \\otimes \\sigma _1 , \\\\D_{y,E_{g+}}(\\mathbf {k}) &\\cong \\frac{1}{3}\\, (2k_y^2-k_z^2-k_x^2)\\, h_4 \\nonumber \\\\&= \\frac{1}{3}\\, (2k_y^2-k_z^2-k_x^2)\\, \\sigma _2 \\otimes \\sigma _2 , \\\\D_{z,E_{g+}}(\\mathbf {k}) &\\cong \\frac{1}{3}\\, (2k_z^2-k_x^2-k_y^2)\\, h_5 \\nonumber \\\\&= \\frac{1}{3}\\, (2k_z^2-k_x^2-k_y^2)\\, \\sigma _2 \\otimes \\sigma _3 .$ This is d-wave spin-triplet pairing.",
"(c) $T_{1g+}\\otimes A_{1g+}$ involves three triplets of basis functions, $D_{x,m0}(\\mathbf {k}) &\\cong k_yk_z (k_y^2-k_z^2)\\, \\sigma _m \\otimes \\sigma _0 , \\\\D_{y,m0}(\\mathbf {k}) &\\cong k_zk_x (k_z^2-k_x^2)\\, \\sigma _m \\otimes \\sigma _0 , \\\\D_{z,m0}(\\mathbf {k}) &\\cong k_xk_y (k_x^2-k_y^2)\\, \\sigma _m \\otimes \\sigma _0 ,$ for $m=0,1,3$ .",
"This is g-wave spin-singlet pairing.",
"(d) For $T_{1g+}\\otimes T_{1g+}$ , we get the basis functions $&D_{x,T_{1g+}}(\\mathbf {k}) \\cong k_z k_x (k_z^2-k_x^2)\\, h_5 - k_x k_y (k_x^2-k_y^2)\\, h_4 \\nonumber \\\\&\\quad = k_z k_x (k_z^2-k_x^2)\\, \\sigma _2 \\otimes \\sigma _3- k_x k_y (k_x^2-k_y^2)\\, \\sigma _2 \\otimes \\sigma _2 , \\\\&D_{y,T_{1g+}}(\\mathbf {k}) \\cong k_x k_y (k_x^2-k_y^2)\\, h_3 - k_y k_z (k_y^2-k_z^2)\\, h_5 \\nonumber \\\\&\\quad = k_x k_y (k_x^2-k_y^2)\\, \\sigma _2 \\otimes \\sigma _1- k_y k_z (k_y^2-k_z^2)\\, \\sigma _2 \\otimes \\sigma _3 , \\\\&D_{z,T_{1g+}}(\\mathbf {k}) \\cong k_y k_z (k_y^2-k_z^2)\\, h_4 - k_z k_x (k_z^2-k_x^2)\\, h_3 \\nonumber \\\\&\\quad = k_y k_z (k_y^2-k_z^2)\\, \\sigma _2 \\otimes \\sigma _2- k_z k_x (k_z^2-k_x^2)\\, \\sigma _2 \\otimes \\sigma _1 .$ This is g-wave spin-triplet pairing.",
"(e) For $T_{2g+}\\otimes T_{1g+}$ , we get the basis functions $D_{x,T_{2g+}}(\\mathbf {k}) &\\cong k_z k_x\\, h_5 + k_x k_y\\, h_4 \\nonumber \\\\&= k_z k_x\\, \\sigma _2 \\otimes \\sigma _3 + k_x k_y\\, \\sigma _2 \\otimes \\sigma _2 , \\\\D_{y,T_{2g+}}(\\mathbf {k}) &\\cong k_x k_y\\, h_3 + k_y k_z\\, h_5 \\nonumber \\\\&= k_x k_y\\, \\sigma _2 \\otimes \\sigma _1 + k_y k_z\\, \\sigma _2 \\otimes \\sigma _3 , \\\\D_{z,T_{2g+}}(\\mathbf {k}) &\\cong k_y k_z\\, h_4 + k_z k_x\\, h_3 \\nonumber \\\\&= k_y k_z\\, \\sigma _2 \\otimes \\sigma _2 + k_z k_x\\, \\sigma _2 \\otimes \\sigma _1 .$ This is d-wave spin-triplet pairing.",
"Table: Leading-order polynomial forms of the form factors f n (𝐤)f_n(\\mathbf {k}) describing T 1g+ T_{1g+} pairing for a model with two s-orbitals.Table: Leading-order polynomial forms of the products c n (𝐤)f n (𝐤)c_n(\\mathbf {k}) f_n(\\mathbf {k}) of form factors describing T 1g+ T_{1g+} pairing for a model with two s-orbitals.",
"The amplitudes of theleading terms in c n (𝐤)c_n(\\mathbf {k}) have been absorbed into new pairing amplitudes marked by a tilde.The resulting form factors are summarized in Table REF .",
"To determine the IP nodes, we require the products $c_n(\\mathbf {k})\\, f_n(\\mathbf {k})$ , which are listed in Table REF .",
"Defining $\\tilde{\\delta }_{\\nu ,0} \\equiv \\tilde{\\delta }_{\\nu ,00} - \\tilde{\\delta }_{\\nu ,10} - \\tilde{\\delta }_{\\nu ,30} - \\tilde{\\delta }_{\\nu ,A_{1g+}}$ for $\\nu =x,y,z$ , we obtain $c_0&(\\mathbf {k})\\, f_0(\\mathbf {k}) - \\vec{c}(\\mathbf {k}) \\cdot \\vec{f}(\\mathbf {k}) \\nonumber \\\\&= \\bigg [ \\tilde{\\delta }_{x,0} - \\frac{\\tilde{\\delta }_{x,E_{g+}}}{3}\\, (2k_x^2-k_y^2-k_z^2)+ \\tilde{\\delta }_{x,T_{2g+}}\\, k_x^2 \\bigg ] \\nonumber \\\\&\\quad {}\\times k_y k_z (k_y^2-k_z^2) + \\ldots ,$ where two terms with cyclically permuted indices $x$ , $y$ , and $z$ have been suppressed.",
"Note that contribution (d) has dropped out.",
"This is an artifact of having used the same leading-order basis functions for $c_n(\\mathbf {k})$ and $f_n(\\mathbf {k})$ .",
"Using different ones, we see that the terms do not cancel.",
"They do not change the following discussion, though.",
"If only the $\\tilde{\\delta }_x$ amplitudes are different from zero, i.e., for pairing of $(1,0,0)$ type [26], [13], [2], we expect four first-order line nodes in the planes $k_y=0$ , $k_z=0$ , $k_y=k_z$ , and $k_y=-k_z$ .",
"This is a new example of a state that is necessarily nodal even for purely local pairing, in which case $\\tilde{\\delta }_{x,0}$ is the only nonvanishing amplitude.",
"Time-reversal-symmetric superpositions of $(1,0,0)$ , $(0,1,0)$ , and $(0,0,1)$ pairing generically also have four line nodes.",
"If TRS is broken, the states with $(1,i,0)$ and $(1,\\omega ,\\omega ^2)$ where $\\omega = e^{2\\pi i/3}$ are plausible [31], [26], [13].",
"The $(1,i,0)$ state has IP nodes where both the real and the imaginary part of $c_0 (\\mathbf {k})\\, f_0(\\mathbf {k}) - \\vec{c}(\\mathbf {k}) \\cdot \\vec{f}(\\mathbf {k})$ vanish.",
"This leads to 18 point nodes in the $\\langle 001\\rangle $ , $\\langle 101\\rangle $ , $\\langle 111\\rangle $ directions outside of the $k_z=0$ plane, and one line node in the $k_z=0$ plane.",
"Compare the $T_{2g+}$ , $(1,i,0)$ pairing state for the $j=3/2$ example [1], [2], where we found two point nodes and one line node in the $k_z=0$ plane.",
"Figure: Zeros of real (blue) and imaginary (red) parts of 〈c,f〉\\langle c,f\\rangle for the T 1g+ T_{1g+} pairing state with order parameter 1,ω,ω 2 1,\\omega ,\\omega ^2, with ω=e 2πi/3 \\omega =e^{2\\pi i/3}, as functions of the spherical polar angles θ\\theta , φ\\phi of 𝐤\\mathbf {k}.",
"Lowest-order polynomial basis functions and the parameter values |𝐤|=1|\\mathbf {k}|=1, δ ˜ i,E g+ /δ ˜ i,0 =0.5\\tilde{\\delta }_{i,E_{g+}}/\\tilde{\\delta }_{i,0}=0.5, and δ ˜ i,T 2g+ /δ ˜ i,0 =0.3\\tilde{\\delta }_{i,T_{2g+}}/\\tilde{\\delta }_{i,0}=0.3 for i=x,y,zi=x,y,z have been used.For the $(1,\\omega ,\\omega ^2)$ state, there are point nodes where both the real and the imaginary part vanish.",
"Figure REF illustrates the zeros of the real and imaginary parts for a typical parameter set.",
"For generic parameters, there are point nodes in the 26 cubic high-symmetry directions.",
"Six of these are special in that either the zero contours of the real and imaginary part are cotangent or in that the imaginary zero contour has a self crossing.",
"In these cases, the quasiparticle dispersion close to the point node is linear in all directions except along a single axis, where it is quadratic to leading order.",
"The other 20 point nodes show linear dispersion.",
"For noninfinitesimal pairing, we expect the nodes to be inflated.",
"We here only consider the $(1,i,0)$ state.",
"We write the pairing amplitudes as $\\delta _{x,00} = \\delta _{00}$ , $\\delta _{y,00} = i\\delta _{00}$ , and $\\delta _{z,00} = 0$ , etc.",
"The superconducting form factors then read as $f_0(\\mathbf {k}) &= \\delta _{00}\\, \\big [ k_y k_z (k_y^2-k_z^2)+ i\\, k_z k_x (k_z^2-k_x^2) \\big ] , \\\\f_1(\\mathbf {k}) &= \\delta _{10}\\, \\big [ k_y k_z (k_y^2-k_z^2)+ i\\, k_z k_x (k_z^2-k_x^2) \\big ] , \\\\f_2(\\mathbf {k}) &= \\delta _{30}\\, \\big [ k_y k_z (k_y^2-k_z^2)+ i\\, k_z k_x (k_z^2-k_x^2) \\big ] , \\\\f_3(\\mathbf {k}) &= \\delta _{A_{1g+}} + \\frac{\\delta _{E_{g+}}}{3}\\, (2k_x^2-k_y^2-k_z^2) \\nonumber \\\\&\\quad {}+ i\\, \\delta _{T_{1g+}}\\, k_x k_y (k_x^2-k_y^2)+ i\\, \\delta _{T_{2g+}}\\, k_x k_y , \\\\f_4(\\mathbf {k}) &= i\\delta _{A_{1g+}} + \\frac{i\\,\\delta _{E_{g+}}}{3}\\, (2k_y^2-k_z^2-k_x^2) \\nonumber \\\\&\\quad {}- \\delta _{T_{1g+}}\\, k_x k_y (k_x^2-k_y^2) + \\delta _{T_{2g+}}\\, k_x k_y , \\\\f_5(\\mathbf {k}) &= \\delta _{T_{1g+}}\\, \\big [ k_z k_x (k_z^2-k_x^2) - i\\,k_y k_z (k_y^2-k_z^2) \\big ]\\nonumber \\\\&\\quad {}+ \\delta _{T_{2g+}}\\, \\big [ k_z k_x + i\\, k_y k_z \\big ] .$ Equation (REF ) shows that $\\langle c,f^1\\rangle = \\langle c,f^2\\rangle = 0$ remains valid in the radial direction through all point nodes.",
"To go on, we have to distinguish between the inequivalent point nodes.",
"In the $[001]$ direction, we have $f_0(\\mathbf {k}) &= f_1(\\mathbf {k}) = f_2(\\mathbf {k}) = f_5(\\mathbf {k}) = 0 , \\\\f_3(\\mathbf {k}) &= \\delta _{A_{1g+}} - \\frac{\\delta _{E_{g+}}}{3}\\, k^2 , \\\\f_4(\\mathbf {k}) &= i\\delta _{A_{1g+}} - \\frac{i\\delta _{E_{g+}}}{3}\\, k^2 .$ We find $\\langle f^1,f^2\\rangle = 0$ and $\\langle f^1,f^1\\rangle = \\langle f^2,f^2\\rangle = -(\\delta _{A_{1g+}}-\\delta _{E_{g+}}k^2/3)^2$ and thus for the Pfaffian $P(\\mathbf {k}) = \\langle c,c\\rangle \\left[ \\langle c,c\\rangle + 4 \\left( \\delta _{A_{1g+}} - \\frac{\\delta _{E_{g+}}}{3}\\, k^2 \\right)^{\\!2} \\right] .$ The first factor changes sign at the normal-state Fermi surface.",
"The second is generically nonzero there and thus the Pfaffian also changes sign at the normal-state Fermi surface.",
"Hence, the point nodes in the $\\langle 001\\rangle $ directions are inflated and touch the normal-state Fermi surface at the IP point nodes.",
"In the $[101]$ direction, we have $\\mathbf {k}=k\\,(1,0,1)/\\sqrt{2}$ and $f_0(\\mathbf {k}) &= f_1(\\mathbf {k}) = f_2(\\mathbf {k}) = 0 , \\\\f_3(\\mathbf {k}) &= \\delta _{A_{1g+}} + \\frac{\\delta _{E_{g+}}}{6}\\, k^2 , \\\\f_4(\\mathbf {k}) &= i\\delta _{A_{1g+}} - \\frac{i\\delta _{E_{g+}}}{3}\\, k^2 , \\\\f_5(\\mathbf {k}) &= \\frac{\\delta _{T_{2g+}}}{2}\\, k^2 .$ This implies that $\\langle f^1,f^2\\rangle = 0$ but $\\langle f^1,f^1\\rangle $ and $\\langle f^2,f^2\\rangle $ are generally unequal.",
"The Pfaffian thus reads as $P(\\mathbf {k})= \\left[ \\langle c,c\\rangle - \\langle f^1,f^1\\rangle - \\langle f^2,f^2\\rangle \\right]^2- 4\\, \\langle f^1,f^1\\rangle \\langle f^2,f^2\\rangle .$ The first term goes to zero on a renormalized Fermi surface.",
"The second term $- 4\\, &\\langle f^1,f^1\\rangle \\langle f^2,f^2\\rangle = -4 \\left( \\delta _{A_{1g+}} - \\frac{\\delta _{E_{g+}}}{3}\\, k^2 \\right)^{\\!2} \\nonumber \\\\&{}\\times \\left[ \\left( \\delta _{A_{1g+}} + \\frac{\\delta _{E_{g+}}}{6}\\, k^2 \\right)^{\\!2}+ \\frac{\\delta _{T_{2g+}}^2}{4}\\, k^4 \\right]$ generically is strictly negative.",
"Then, the Pfaffian becomes negative and we find a BFS for a range of $k$ values in the vicinity of but not usually touching the normal-state Fermi surface.",
"The Pfaffian is identical for the directions $[\\bar{1}01]$ , $[011]$ , $[0\\bar{1}1]$ , and their negatives.",
"Finally, along $[111]$ , we have $\\mathbf {k}=k\\,(1,1,1)/\\sqrt{3}$ and $f_0(\\mathbf {k}) &= f_1(\\mathbf {k}) = f_2(\\mathbf {k}) = 0 , \\\\f_3(\\mathbf {k}) &= \\delta _{A_{1g+}} + \\frac{i\\delta _{T_{2g+}}}{3}\\, k^2 , \\\\f_4(\\mathbf {k}) &= i\\delta _{A_{1g+}} + \\frac{\\delta _{T_{2g+}}}{3}\\, k^2 , \\\\f_5(\\mathbf {k}) &= \\frac{1+i}{3}\\, \\delta _{T_{2g+}}\\, k^2 .$ We thus obtain $\\langle f^1,f^2\\rangle = -\\frac{1}{3}\\, \\delta _{T_{2g+}} k^2 \\left( 2\\, \\delta _{A_{1g+}}+ \\frac{1}{3}\\, \\delta _{T_{2g+}} k^2 \\right)$ and $\\langle f^1,f^1\\rangle = \\langle f^2,f^2\\rangle = - \\left(\\delta _{A_{1g+}}^2 + \\frac{2}{9}\\, \\delta _{T_{2g+}}^2 k^4 \\right) .$ The Pfaffian is $P(\\mathbf {k}) &= \\left[ \\langle c,c\\rangle - \\langle f^1,f^1\\rangle - \\langle f^2,f^2\\rangle \\right]^2 \\nonumber \\\\&\\quad {} + 4 \\left[ \\langle f^1,f^2\\rangle ^2 - \\langle f^1,f^1\\rangle \\langle f^2,f^2\\rangle \\right] ,$ wherein the second term evaluates to $4 &\\left[ \\langle f^1,f^2\\rangle ^2 - \\langle f^1,f^1\\rangle \\langle f^2,f^2\\rangle \\right] \\nonumber \\\\&= -\\frac{4}{27} \\left( 3\\, \\delta _{A_{1g+}} - \\delta _{T_{2g+}}\\, k^2 \\right)^2 \\nonumber \\\\&\\quad {}\\times \\left[ 2\\, \\delta _{A_{1g+}}^2 + ( \\delta _{A_{1g+}} + \\delta _{T_{2g+}}\\, k^2 )^2 \\right] .$ Since this is generally negative we also expect BFSs that do not touch the normal-state Fermi surface in the $\\langle 111\\rangle $ directions.",
"For the equatorial line node, we take $\\mathbf {k}=(k_x,k_y,0)$ .",
"The superconducting form factors are $f_0(\\mathbf {k}) &= f_1(\\mathbf {k}) = f_2(\\mathbf {k}) = f_5(\\mathbf {k}) = 0 , \\\\f_3(\\mathbf {k}) &= \\delta _{A_{1g+}} + \\frac{\\delta _{E_{g+}}}{3}\\, (2k_x^2-k_y^2) \\nonumber \\\\&\\quad {}+ i\\, \\delta _{T_{1g+}}\\, k_x k_y (k_x^2-k_y^2)+ i\\, \\delta _{T_{2g+}}\\, k_x k_y , \\\\f_4(\\mathbf {k}) &= i\\delta _{A_{1g+}} + \\frac{i\\,\\delta _{E_{g+}}}{3}\\, (2k_y^2-k_x^2) \\nonumber \\\\&\\quad {}- \\delta _{T_{1g+}}\\, k_x k_y (k_x^2-k_y^2) + \\delta _{T_{2g+}}\\, k_x k_y .$ This gives $\\langle f^1,&f^2\\rangle = -\\frac{1}{3}\\, k_x k_y \\big [ 6\\, \\delta _{A_{1g+}} \\delta _{T_{2g+}} \\nonumber \\\\&{}+ 3\\, \\delta _{E_{g+}} \\delta _{T_{1g+}} (k_x^2-k_y^2)^2+ \\delta _{E_{g+}} \\delta _{T_{2g+}} (k_x^2+k_y^2) \\big ] , \\\\\\langle f^1,&f^1\\rangle = -\\frac{1}{9} \\left[ 3 \\delta _{A_{1g+}}+ \\delta _{E_{g+}} (2k_x^2 - k_y^2) \\right]^2 \\nonumber \\\\&{}- \\left[ \\delta _{T_{1g+}} (k_x^2-k_y^2) - \\delta _{T_{2g+}} \\right]^2 k_x^2 k_y^2 , \\\\\\langle f^2,&f^2\\rangle = -\\frac{1}{9} \\left[ 3 \\delta _{A_{1g+}}+ \\delta _{E_{g+}} (2k_y^2 - k_x^2) \\right]^2 \\nonumber \\\\&{}- \\left[ \\delta _{T_{1g+}} (k_x^2-k_y^2) + \\delta _{T_{2g+}} \\right]^2 k_x^2 k_y^2 .$ The Pfaffian again has the form of Eq.",
"(REF ), where the second term $4 &\\left[ \\langle f^1,f^2\\rangle ^2 - \\langle f^1,f^1\\rangle \\langle f^2,f^2\\rangle \\right] \\nonumber \\\\&= -\\frac{4}{81}\\, \\Big [ 9\\, \\delta _{A_{1g+}}^2+ 2\\, \\delta _{A_{1g+}} \\delta _{E_{g+}} (k_x^2 + k_y^2) \\nonumber \\\\&\\quad {}+ \\delta _{E_{g+}}^2 (2k_x^2-k_y^2)(2k_y^2-k_x^2) \\nonumber \\\\&\\quad {}+ 9\\, \\delta _{T_{1g+}}^2\\, k_x^2 k_y^2 (k_x^2 - k_y^2)^2- 9\\, \\delta _{T_{2g+}}^2\\, k_x^2 k_y^2 \\Big ]^2$ is non-positive and generically negative so that the line node is inflated by noninfinitesimal pairing.",
"The resulting BFS is toroidal but may be pinched off, i.e., have self crossings, at some momenta, but these self crossings would be accidental.",
"Since the first term in Eq.",
"(REF ) becomes zero close to but not at the normal-state Fermi surface the BFS generically does not touch the normal-state Fermi surface.",
"In summary, the point and line nodes of the $T_{1g+}$ , $(1,i,0)$ pairing state are all inflated by noninfinitesimal pairing.",
"Only the inflated point nodes on the $k_z$ -axis touch the normal-state Fermi surface, the other pockets are shifted away from it and could thus be annihilated at strong coupling."
],
[
"Two orbitals of opposite parity",
"To have a single even and a single odd orbital per unit cell for the point group $O_h$ , the first must transform like a one-dimensional $g$ irrep and the second like a one-dimensional $u$ irrep.",
"The most natural possibilities are $A_{1g}$ (s-orbital) and $A_{2u}$ ($f_{xyz}$ -orbital).",
"Now the inversion or parity matrix is nontrivial: $U_P = \\sigma _3 \\otimes \\sigma _0 .$ Moreover, the $f_{xyz}$ -orbital is odd under fourfold rotations but even under threefold rotations.",
"Models of this symmetry have been analyzed in the context of Dirac and Weyl semimetals [32].",
"The unitary part of time reversal is $U_T = \\sigma _0 \\otimes i\\sigma _2$ since the orbitals are invariant.",
"The basis matrices can be written as Kronecker products, which transform according to the irreps as summarized in Table REF .",
"Compared to the example of two s-orbitals, Table REF , the Pauli matrices $\\sigma _1$ and $\\sigma _2$ for the orbital degree of freedom are now odd under inversion.",
"The new element is that $u$ irreps occur, due to the nontrivial parity operator.",
"This provides additional possibilities for the products of irreps of $\\mathbf {k}$ -dependent form factors and basis matrices.",
"Table REF shows all relevant reductions of product representations.",
"Table: Basis matrices on the internal Hilbert space for the case of one s-orbital and one f xyz f_{xyz}-orbital and point group O h O_h.",
"The basis matrices are irreducible tensor operators of the irreps listed in the second column.",
"For multidimensional irreps, the states transforming into each other under point-group operations are distinguished by the index in the third column.Table: Reduction of product representations of the allowed irreps of 𝐤\\mathbf {k}-dependent form factors (rows) and basis matrices h ν h_\\nu (columns) for one s-orbital and one f xyz f_{xyz}-orbital.",
"For the form factors, the minimum order of polynomial basis functions is given in the second column.",
"“∘\\circ ” indicates products that are forbidden since they violate fermionic antisymmetry.The normal-state Hamiltonian is a linear combination of all basis matrices that allow to form products with full $A_{1g+}$ symmetry, marked in bold face in Table REF .",
"These basis matrices, together with their irreps, are $h_0 &\\equiv \\sigma _0 \\otimes \\sigma _0 & A_{1g+}, \\\\h_1 &\\equiv \\sigma _3 \\otimes \\sigma _0 & A_{1g+}, \\\\h_2 &\\equiv \\sigma _2 \\otimes \\sigma _0 & A_{2u-}, \\\\h_3 &\\equiv \\sigma _1 \\otimes \\sigma _1 & T_{2u-}, \\\\h_4 &\\equiv \\sigma _1 \\otimes \\sigma _2 & T_{2u-}, \\\\h_5 &\\equiv \\sigma _1 \\otimes \\sigma _3 & T_{2u-} .$ There are thus six matrices that satisfy the same algebra as for the case of two s-orbitals; see Appendices and .",
"The normal-state Hamiltonian reads as $&H_N(\\mathbf {k}) \\nonumber \\\\&\\:= c_{00}(\\mathbf {k})\\, \\sigma _0 \\otimes \\sigma _0+ c_{30}(\\mathbf {k})\\, \\sigma _3 \\otimes \\sigma _0+ c_{20}(\\mathbf {k})\\, \\sigma _2 \\otimes \\sigma _0 \\nonumber \\\\&\\:\\quad {} + c_{11}(\\mathbf {k})\\, \\sigma _1 \\otimes \\sigma _1+ c_{12}(\\mathbf {k})\\, \\sigma _1 \\otimes \\sigma _2+ c_{13}(\\mathbf {k})\\, \\sigma _1 \\otimes \\sigma _3 ,$ where the leading polynomial forms are $c_{00}(\\mathbf {k}) &= c_{00}^{(0)} + c_{00}^{(2)}\\, (k_x^2 + k_y^2 + k_z^2) + \\ldots , \\\\c_{30}(\\mathbf {k}) &= c_{30}^{(0)} + c_{30}^{(2)}\\, (k_x^2 + k_y^2 + k_z^2) + \\ldots , \\\\c_{20}(\\mathbf {k}) &= c_{20}^{(3)}\\, k_x k_y k_z + \\ldots , \\\\c_{11}(\\mathbf {k}) &= c_1^{(3)}\\, k_x (k_y^2-k_z^2) + \\ldots , \\\\c_{12}(\\mathbf {k}) &= c_1^{(3)}\\, k_y (k_z^2-k_x^2) + \\ldots , \\\\c_{13}(\\mathbf {k}) &= c_1^{(3)}\\, k_z (k_x^2-k_y^2) + \\ldots $ Four of the form factors are odd in momentum.",
"They do not break inversion symmetry since they multiply orbital matrices that are also odd under inversion.",
"Turning to superconducting pairing, it is interesting that local pairing is now either trivial ($A_{1g+}$ ) or has odd parity ($A_{2u+}$ , $T_{2u+}$ ), as seen from the first row of Table REF .",
"Furthermore, for even-parity pairing ($g+$ irreps), only the basis matrices belonging to $A_{1g+}$ , $T_{2u-}$ , and $A_{2u-}$ can occur, i.e., the same matrices $h_0$ , ..., $h_5$ as in $H_N(\\mathbf {k})$ .",
"Since $\\mathcal {C}P$ squares to $+$ the Hamiltonian can be unitarily transformed into antisymmetric form, guaranteeing the existence of a Pfaffian [1].",
"In the present example, where $U_P = \\sigma _3 \\otimes \\sigma _0$ , the matrix $\\Omega $ mediating this transformation reads as $\\Omega = \\frac{1}{\\sqrt{2}} \\left(\\begin{array}{cc} 1 & 1 \\\\ i & -i \\end{array}\\right)\\otimes \\exp \\!\\left(-i\\,\\frac{\\pi }{2}\\, \\frac{\\sigma _3}{2}\\right) \\otimes \\sigma _0 .$ This specific form does not affect the eigenvalues, though.",
"Since the algebra of the basis matrices is unchanged, the expressions for the eigenvalues, the Pfaffian, and the condition for IP nodes remain unchanged.",
"In the following, we briefly discuss the $A_{2g+}$ and $E_{g+}$ pairing states and compare them to the case of two s-orbitals."
],
[
"$A_{2g+}$ pairing",
"$A_{2g+}$ appears in three places in Table REF : (a) $A_{2g+}\\otimes A_{1g+}$ , (b) $A_{1u-}\\otimes A_{2u-}$ , and (c) $T_{1u-}\\otimes T_{2u-}$ .",
"Note that the minimum orders of form factors are (a) 6, (b) 9, and (c) 1 so that one expects that the $T_{1u-}\\otimes T_{2u-}$ contribution typically dominates.",
"(a) For $A_{2g+}\\otimes A_{1g+}$ : $D_{A_{2g+}}(\\mathbf {k})= \\delta _{00} d_{00}(\\mathbf {k})\\, \\sigma _{0} \\otimes \\sigma _{0}+ \\delta _{30} d_{30}(\\mathbf {k})\\, \\sigma _{3} \\otimes \\sigma _{0} .$ To the leading order, $d_{m0}(\\mathbf {k})$ takes the form $d_{m0}(\\mathbf {k}) \\cong d_{m0}^{(6)}\\, \\big [ k_{x}^4 (k_{y}^2-k_{z}^2)+ k_{y}^4 (k_{z}^2-k_{x}^2) + k_{z}^4 (k_{x}^2-k_{y}^2) \\big ] .$ $d_{m0}^{(6)}$ is set to unity.",
"(b) For $A_{1u-}\\otimes A_{2u-}$ : $D_{A_{1u-}}(\\mathbf {k}) = \\delta _{A_{1u-}} d_{A_{1u-}}(\\mathbf {k})\\, \\sigma _{2} \\otimes \\sigma _{0} ,$ with $d_{A_{1u-}}(\\mathbf {k})$ to the leading order given by $d_{A_{1u-}}(\\mathbf {k}) &\\cong d_{A_{1u-}}^{(9)}\\, k_{x} k_{y} k_{z}\\, \\big [k_{x}^4 (k_{y}^2-k_{z}^2) + k_{y}^4 (k_{z}^2-k_{x}^2) \\nonumber \\\\&\\quad {}+ k_{z}^4 (k_{x}^2-k_{y}^2) \\big ] .$ We set $d_{A_{1u-}}^{(9)}$ to unity.",
"(c) For $T_{1u-}\\otimes T_{2u-}$ : $D_{T_{1u-}}(\\mathbf {k}) \\cong \\delta _{T_{1u-}} ( k_{x}\\, \\sigma _{1}\\otimes \\sigma _{1}+ k_{y}\\, \\sigma _{1}\\otimes \\sigma _{2} + k_{z}\\, \\sigma _{1}\\otimes \\sigma _{3} )$ to leading order.",
"The components are assigned such that the whole term $D_{T_{1u-}}(\\mathbf {k})$ changes sign under any four-fold rotation [40].",
"Table: Leading-order polynomial forms of the form factors f n (𝐤)f_n(\\mathbf {k}) describing A 2g+ A_{2g+} pairing for a model with one s-orbital and one f xyz f_{xyz}-orbital.Table: Leading-order polynomial forms of the products c n (𝐤)f n (𝐤)c_n(\\mathbf {k}) f_n(\\mathbf {k}) of form factors describing A 2g+ A_{2g+} pairing for a model with one s-orbital and one f xyz f_{xyz}-orbital.",
"The amplitudes of the leading terms in c n (𝐤)c_n(\\mathbf {k}) have been absorbed into new pairing amplitudes marked by a tilde.The resulting superconducting form factors $f_{n}$ and the products $c_{n} f_{n}$ , which are required to determine the IP nodes, are listed in Tables REF and REF , respectively, to the leading order.",
"The condition for IP nodes reads as $&c_0(\\mathbf {k})\\, f_0(\\mathbf {k}) - \\vec{c}(\\mathbf {k}) \\cdot \\vec{f}(\\mathbf {k})= \\big ( \\tilde{\\delta }_{00} - \\tilde{\\delta }_{30} - \\tilde{\\delta }_{A_{1u-}} k_{x}^2 k_{y}^2 k_{z}^2 \\big ) \\nonumber \\\\&\\qquad {}\\times \\big [ k_{x}^4 (k_{y}^2-k_{z}^2) + k_{y}^4 (k_{z}^2-k_{x}^2)+ k_{z}^4 (k_{x}^2-k_{y}^2) \\big ] = 0 .$ Contribution (c) has dropped out.",
"This is again an artifact of using the same basis functions for $c_n$ and $f_n$ .",
"Going beyond leading order, a contribution remains but does not affect the conclusions.",
"For example, the $T_{1u-}$ basis functions $k_x^3$ , $k_y^3$ , $k_z^3$ generate another term proportional to $k_{x}^4 (k_{y}^2-k_{z}^2) + k_{y}^4 (k_{z}^2-k_{x}^2) + k_{z}^4 (k_{x}^2-k_{y}^2)$ .",
"Equation (REF ) is satisfied whenever any two of the components of $\\mathbf {k}$ are equal.",
"Thus there are line nodes in the $\\lbrace 110\\rbrace $ planes for infinitesimal pairing.",
"Form factors in the $(110)$ plane read as $f_0(\\mathbf {k}) &= f_1(\\mathbf {k}) = f_2(\\mathbf {k}) = 0 , \\\\f_3(\\mathbf {k}) &= f_4(\\mathbf {k}) = \\delta _{T_{1u-}}k_{x} , \\\\f_5(\\mathbf {k}) &= \\delta _{T_{1u-}}k_{z} ,$ which gives $\\langle f^1 ,f^1 \\rangle &= -(\\mathop {\\textrm {Re}}\\delta _{T_{1u-}})^2 (2 k_{x}^2+k_{z}^2) , \\\\\\langle f^2 ,f^2 \\rangle &= -(\\mathop {\\textrm {Im}}\\delta _{T_{1u-}})^2 (2 k_{x}^2+k_{z}^2) , \\\\\\langle f^1 ,f^2 \\rangle &= -\\mathop {\\textrm {Re}}\\delta _{T_{1u-}}\\, \\mathop {\\textrm {Im}}\\delta _{T_{1u-}} (2 k_{x}^2+k_{z}^2)$ and also $\\langle c ,f^1 \\rangle =\\langle c ,f^2 \\rangle =0$ .",
"In this case, the Pfaffian simplifies to the form of Eq.",
"(REF ).",
"The second term $4\\,[\\langle f^1 ,f^2 \\rangle ^2-\\langle f^1 ,f^1 \\rangle \\langle f^2 ,f^2 \\rangle ]$ vanishes, which implies that there is no inflation of the line nodes in the mirror plane.",
"The vanishing can be attributed to the fact that in the $\\lbrace 110\\rbrace $ planes, only a single amplitude $\\delta _{T_{1u-}}$ leads to a superconducting gap, the phase of which can always be chosen real so that the TRS breaking is irrelevant.",
"On the other hand, $\\langle f^1 ,f^1 \\rangle +\\langle f^2 ,f^2 \\rangle $ is nonzero so that the nodes are shifted.",
"This is analogous to the case of two s-orbitals."
],
[
"$E_{g+}$ pairing",
"In Table REF , pairing with $E_{g+}$ symmetry occurs in (a) $E_{g+}\\otimes A_{1g+}$ , (b) $E_{u-}\\otimes A_{2u-}$ , (c) $T_{1u-}\\otimes T_{2u-}$ , and (d) $T_{2u-}\\otimes T_{2u-}$ .",
"The matrix-valued basis functions are given in the following to leading order only.",
"(a) For $E_{g+}\\otimes A_{1g+}$ : $D_{x^2-y^2,00}(\\mathbf {k}) &\\cong (k_x^2-k_y^2)\\, \\sigma _0 \\otimes \\sigma _0 , \\\\D_{3z^2-r^2,00}(\\mathbf {k}) &\\cong \\frac{1}{\\sqrt{3}}\\, (2k_z^2-k_x^2-k_y^2)\\, \\sigma _0 \\otimes \\sigma _0 , \\\\D_{x^2-y^2,30}(\\mathbf {k}) &\\cong (k_x^2-k_y^2)\\, \\sigma _3 \\otimes \\sigma _0 , \\\\D_{3z^2-r^2,30}(\\mathbf {k}) &\\cong \\frac{1}{\\sqrt{3}}\\, (2k_z^2-k_x^2-k_y^2)\\, \\sigma _3 \\otimes \\sigma _0 .$ (b) For $E_{u-}\\otimes A_{2u-}$ : $D_{x^2-y^2,20}(\\mathbf {k}) &\\cong k_xk_yk_z (k_x^2-k_y^2)\\, \\sigma _2 \\otimes \\sigma _0 , \\\\D_{3z^2-r^2,20}(\\mathbf {k}) &\\cong \\frac{1}{\\sqrt{3}}\\, k_xk_yk_z (2k_z^2-k_x^2-k_y^2)\\,\\sigma _2 \\otimes \\sigma _0 .$ Note that $k_xk_yk_z\\, \\sigma _2\\times \\sigma _0$ is invariant under $O_h$ .",
"(c) For $T_{1u-}\\otimes T_{2u-}$ : $D_{x^2-y^2,T_{1u-}}(\\mathbf {k}) &\\cong \\frac{1}{\\sqrt{3}}\\, (-k_x\\, \\sigma _1 \\otimes \\sigma _1- k_y\\, \\sigma _1 \\otimes \\sigma _2 \\nonumber \\\\&\\quad {}+ 2\\, k_z\\, \\sigma _1 \\otimes \\sigma _3 ) , \\\\D_{3z^2-r^2,T_{1u-}}(\\mathbf {k}) &\\cong - (k_x\\, \\sigma _1 \\otimes \\sigma _1- k_y\\, \\sigma _1 \\otimes \\sigma _2) .$ For this and the following contribution, the transformation properties under three- and four-fold rotations have been used to determine the two components.",
"(d) For $T_{2u-}\\otimes T_{2u-}$ : $&D_{x^2-y^2,T_{2u-}}(\\mathbf {k}) \\cong \\big [ k_x (k_y^2-k_z^2)\\, \\sigma _1\\otimes \\sigma _1 \\nonumber \\\\&\\quad {}- k_y (k_z^2-k_x^2)\\, \\sigma _1 \\otimes \\sigma _2 \\big ] , \\\\&D_{3z^2-r^2,T_{2u-}}(\\mathbf {k}) \\cong \\frac{1}{\\sqrt{3}}\\, \\big [ {-}k_x(k_y^2-k_z^2)\\,\\sigma _1 \\otimes \\sigma _1 \\nonumber \\\\&\\quad {}- k_y (k_z^2-k_x^2)\\, \\sigma _1 \\otimes \\sigma _2+ 2\\, k_z (k_x^2-k_y^2)\\, \\sigma _1 \\otimes \\sigma _3 \\big ] .$ The resulting superconducting form factors $f_n(\\mathbf {k})$ are given in Table REF and the products $c_n(\\mathbf {k}) f_n(\\mathbf {k})$ appearing in the condition (REF ) for IP nodes are shown in Table REF .",
"The total contribution to $\\langle c,f\\rangle $ from $x^2-y^2$ basis functions (with amplitudes $\\tilde{\\delta }^1_{\\cdots }$ ) has two symmetry-imposed first-order line nodes at $k_y=\\pm k_x$ .",
"In a time-reversal-symmetric state, the inclusion of $3z^2-r^2$ basis functions (with amplitudes $\\tilde{\\delta }^2_{\\cdots }$ ) generically leads to two line nodes elsewhere on the normal-state Fermi surface.",
"The nodes must intersect with the $\\langle 111\\rangle $ axes, though, since there the full expression vanishes.",
"TRS-breaking states generically lead to point nodes in the $\\langle 111\\rangle $ directions.",
"This is for example the case for the generalized $(1,i)$ -type state.",
"These point nodes are solely determined by symmetry and therefore agree with the case of two s-orbitals.",
"Table: Leading-order polynomial forms of the form factors f n (𝐤)f_n(\\mathbf {k}) describing E g+ E_{g+} pairing for a model with one s-orbital and one f xyz f_{xyz}-orbital.Table: Leading-order polynomial forms of the products c n (𝐤)f n (𝐤)c_n(\\mathbf {k}) f_n(\\mathbf {k}) of form factors describing E g+ E_{g+} pairing for a model with one s-orbital and one f xyz f_{xyz}-orbital.",
"The amplitudes of the leading terms in c n (𝐤)c_n(\\mathbf {k}) have been absorbed into new pairing amplitudes marked by a tilde.For noninfinitesimal pairing that breaks TRS, the point nodes are inflated.",
"This is seen by considering the Pfaffian on the high-symmetry axis $\\mathbf {k}=k\\,(1,1,1)/\\sqrt{3}$ through a IP point node.",
"On this axis, we have, to leading order, $f_0(\\mathbf {k}) &= f_1(\\mathbf {k}) = f_2(\\mathbf {k}) = 0 , \\\\f_3(\\mathbf {k}) &= -\\frac{\\delta ^1_{T_{1u-}}}{3}\\, k- \\frac{\\delta ^2_{T_{1u-}}}{\\sqrt{3}}\\, k , \\\\f_4(\\mathbf {k}) &= -\\frac{\\delta ^1_{T_{1u-}}}{3}\\, k+ \\frac{\\delta ^2_{T_{1u-}}}{\\sqrt{3}}\\, k , \\\\f_5(\\mathbf {k}) &= \\frac{2\\, \\delta ^1_{T_{1u-}}}{3}\\, k .$ For the $(1,i)$ pairing state with $\\delta ^2_{T_{1u-}} = i\\delta ^1_{T_{1u-}}$ , $\\delta ^1_{T_{1u-}}\\in \\mathbb {R}$ , we find $\\langle f^1,f^1\\rangle &= \\langle f^2,f^2\\rangle = -\\frac{2k^2}{3}\\, \\big (\\delta ^1_{T_{1u-}}\\big )^2 , \\\\\\langle f^1,f^2\\rangle &= 0 .$ On the other hand, the normal-state form factors are, to leading order, $c_0(\\mathbf {k}) &= c_{00}^{(0)} , \\\\c_1(\\mathbf {k}) &= c_{30}^{(0)} , \\\\c_2(\\mathbf {k}) &= \\frac{1}{3\\sqrt{3}}\\, c_{20}^{(3)}\\, k^3 , \\\\c_3(\\mathbf {k}) &= c_4(\\mathbf {k}) = c_5(\\mathbf {k}) = 0 .$ We thus find $\\langle c,f^1\\rangle = \\langle c,f^2\\rangle = 0$ and the analysis is analogous to the one for the $(1,i)$ state for two s-orbitals.",
"Hence, we expect BFSs that touch the normal-state Fermi surface at the IP nodes."
],
[
"Two-side basis: Diamond structure",
"Another origin of internal degrees of freedom is a nontrivial basis of the crystal.",
"This is a good place to consider an example: the diamond structure with one s-orbital per basis site.",
"The space group is 227, belonging to the point group $O_h$ .",
"We write matrices as Kronecker products of a matrix acting on site space and a matrix on spin space.",
"Table: Basis matrices on the internal Hilbert space for the case of s-orbitals forming a diamond structure.",
"The basis matrices are irreducible tensor operators of the irreps listed in the second column.",
"For multidimensional irreps, the states transforming into each other under point-group operations are distinguished by the index in the third column.The parity matrix $U_P = \\sigma _1 \\otimes \\sigma _0$ is nontrivial since inversion interchanges the basis sites.",
"This case has also been analyzed in the context of semimetals [32].",
"Moreover, the fourfold axes also interchange the basis sites, whereas the threefold axes do not.",
"Time reversal is unchanged, $U_T = \\sigma _0 \\otimes i\\sigma _2$ .",
"The basis matrices are listed in Table REF .",
"This is the same scheme as for two orbitals of opposite parity, see Table REF , except that the Pauli matrices $\\sigma _1$ and $\\sigma _3$ in the first (orbital/site) factor are interchanged.",
"Thus the results for the pairing can be mapped over from Sec.",
"REF without effort."
],
[
"Effective spin 3/2",
"Here, we consider electrons with effective angular momentum $j=3/2$ .",
"It is of interest to check whether the results obtained for local pairing in such a model [1], [2] are robust under nonlocal pairing and which additional pairing states are allowed for nonlocal pairing.",
"The Hilbert space for $j=3/2$ is four dimensional.",
"In this case, it is useful to express all matrices as polynomials of the standard angular-momentum-$3/2$ matrices $J_x &= \\left(\\begin{array}{cccc}0 & \\sqrt{3}/2 & 0 & 0 \\\\\\sqrt{3}/2 & 0 & 1 & 0 \\\\0 & 1 & 0 & \\sqrt{3}/2 \\\\0 & 0 & \\sqrt{3}/2 & 0\\end{array}\\right) , \\\\J_y &= \\left(\\begin{array}{cccc}0 & -i\\, \\sqrt{3}/2 & 0 & 0 \\\\i\\, \\sqrt{3}/2 & 0 & -i & 0 \\\\0 & i & 0 & -i\\,\\sqrt{3}/2 \\\\0 & 0 & i\\,\\sqrt{3}/2 & 0\\end{array}\\right) , \\\\J_z &= \\left(\\begin{array}{cccc}3/2 & 0 & 0 & 0 \\\\0 & 1/2 & 0 & 0 \\\\0 & 0 & -1/2 & 0 \\\\0 & 0 & 0 & -3/2\\end{array}\\right) ,$ and the $4\\times 4$ identity matrix $J_0 \\equiv $ .",
"Table: Basis matrices on the internal Hilbert space for the case of electrons with angular momentum j=3/2j=3/2.",
"The basis matrices are irreducible tensor operators of the irreps listed in the second column.",
"For multidimensional irreps, the states transforming into each other under point-group operations are distinguished by the index in the third column.The parity matrix is trivial, $U_P = = J_0$ , since the angular momentum is invariant under inversion.",
"The unitary part of the time-reversal operator now reads as $U_T = e^{i J_y \\pi }= \\begin{pmatrix}0 & 0 & 0 & 1 \\\\0 & 0 & -1 & 0 \\\\0 & 1 & 0 & 0 \\\\-1 & 0 & 0 & 0\\end{pmatrix} .$ The 16 basis matrices $h_\\nu $ of the space of Hermitian $4\\times 4$ matrices are listed in Table REF , together with the corresponding irreps.",
"We normalize the basis matrices in such a way that $\\mathop {\\textrm {Tr}}h_\\nu ^2 = 4$ .",
"Apart from this, the entries in the table follow from known basis functions [33], taking into account that $\\mathbf {J}=(J_x,J_y,J_z)$ is even under inversion and odd under time reversal, and symmetrizing products of angular-momentum matrices so as to generate Hermitian matrices [16].",
"A new feature is the presence of basis matrices belonging to the two-dimensional irrep $E_{g+}$ .",
"Table: Reduction of product representations of the allowed irreps of 𝐤\\mathbf {k}-dependent form factors (rows) and basis matrices h ν h_\\nu (columns) for electrons with angular momentum j=3/2j=3/2.",
"For the form factors, the minimum order of polynomial basis functions is given in the second column.",
"“∘\\circ ” indicates products that are forbidden since they violate fermionic antisymmetry.",
"For brevity, the symbols 𝒜≡A 1u+ ⊕E u+ ⊕T 1u+ ⊕T 2u+ \\mathcal {A} \\equiv A_{1u+} \\oplus E_{u+} \\oplus T_{1u+} \\oplus T_{2u+} and ℬ≡A 2u+ ⊕E u+ ⊕T 1u+ ⊕T 2u+ \\mathcal {B} \\equiv A_{2u+} \\oplus E_{u+} \\oplus T_{1u+} \\oplus T_{2u+} are used.Table REF shows all relevant reductions of product representations.",
"The normal-state Hamiltonian can only contain the highlighted $A_{1g+}$ combinations and local pairing is only compatible with the first row of the table—this reproduces the known three irreps $A_{1g+}$ , $T_{2g+}$ , and $E_{g+}$ [1], [2].",
"Again, all ten time-reversal-even pairing symmetries can occur and we restrict ourselves to $g+$ irreps (even parity).",
"All of these occur for any of the three irreps $A_{1g+}$ , $T_{2g+}$ , and $E_{g+}$ of basis matrices.",
"The normal-state Hamiltonian $H_N(\\mathbf {k})$ is a linear combination of the basis matrices $h_0 &\\equiv J_0 & A_{1g+}, \\\\h_1 &\\equiv \\frac{1}{\\sqrt{3}}\\, (J_yJ_z + J_zJ_y) & T_{2g+}, \\\\h_2 &\\equiv \\frac{1}{\\sqrt{3}}\\, (J_zJ_x + J_xJ_z) & T_{2g+}, \\\\h_3 &\\equiv \\frac{1}{\\sqrt{3}}\\, (J_xJ_y + J_yJ_x) & T_{2g+}, \\\\h_4 &\\equiv \\frac{1}{\\sqrt{3}}\\, (J_x^2 - J_y^2) & E_{g+}, \\\\h_5 &\\equiv \\frac{1}{3}\\, (2J_z^2 - J_x^2 - J_y^2) & E_{g+} ,$ which again satisfy the universal algebra; see Appendices and .",
"The normal-state Hamiltonian contains these matrices with form factors $c_0(\\mathbf {k}),\\ldots ,c_5(\\mathbf {k})$ , which must transform in the same way as $h_0,\\ldots ,h_5$ ."
],
[
"$A_{2g+}$ pairing",
"$A_{2g+}$ pairing appears in three places in Table REF : (a) $A_{2g+}\\otimes A_{1g+}$ , (b) $E_{g+}\\otimes E_{g+}$ , and (c) $T_{1g+}\\otimes T_{2g+}$ .",
"This is a potentially interesting pairing state since it is impossible for purely local pairing and it is an example of a nontrivial one-dimensional irrep.",
"In the following, we give the basis functions to the leading order only.",
"(a) For $A_{2g+}\\otimes A_{1g+}$ : $D_{A_{2g+}}(\\mathbf {k}) &\\cong \\big [ k_{x}^4 (k_{y}^2-k_{z}^2) + k_{y}^4 (k_{z}^2-k_{x}^2) \\nonumber \\\\&\\quad {} + k_{z}^4 (k_{x}^2-k_{y}^2) \\big ]\\, h_0 .$ (b) For $E_{g+}\\otimes E_{g+}$ : $D_{E_{g+}}(\\mathbf {k}) &\\cong (k_x^2 - k_y^2)\\, h_5 - \\frac{1}{\\sqrt{3}}\\, (2k_z^2 - k_x^2 - k_y^2)\\, h_4 .$ (c) For $T_{1g+}\\otimes T_{2g+}$ : $D_{T_{1g+}}(\\mathbf {k}) &\\cong k_y k_z (k_y^2 - k_z^2)\\, h_1 + k_z k_x (k_z^2 - k_x^2)\\, h_2 \\nonumber \\\\&\\quad {} + k_x k_y (k_x^2 - k_y^2)\\, h_3 .$ The assignment of the components is done in such a way that the form factor changes sign under any fourfold rotation.",
"Table: Leading-order polynomial forms of the form factors f n (𝐤)f_n(\\mathbf {k}) describing A 2g+ A_{2g+} pairing for electrons withangular momentum j=3/2j=3/2.Table: Leading-order polynomial forms of the products c n (𝐤)f n (𝐤)c_n(\\mathbf {k}) f_n(\\mathbf {k}) of form factors describing A 2g+ A_{2g+} pairing for electrons with angular momentum j=3/2j=3/2.",
"The amplitudes of the leading terms in c n (𝐤)c_n(\\mathbf {k}) have been absorbed into new pairing amplitudes marked by a tilde.The resulting superconducting form factors $f_{n}$ and the products $c_{n} f_{n}$ , which are required to determine the IP nodes, are listed in Tables REF and REF , respectively, to the leading order.",
"The condition for IP nodes reads as $c_0&(\\mathbf {k})\\, f_0(\\mathbf {k}) - \\vec{c}(\\mathbf {k}) \\cdot \\vec{f}(\\mathbf {k}) \\nonumber \\\\&= \\tilde{\\delta }_{A_{2g+}}\\, \\big [ k_{x}^4 (k_{y}^2-k_{z}^2) + k_{y}^4 (k_{z}^2-k_{x}^2)+ k_{z}^4 (k_{x}^2-k_{y}^2) \\big ] \\nonumber \\\\&\\quad {}- \\tilde{\\delta }_{T_{1g+}}\\, \\big [ k_y^2k_z^2 (k_y^2-k_z^2)+ k_z^2k_x^2 (k_z^2-k_x^2) \\nonumber \\\\&\\quad {}+ k_x^2k_y^2 (k_x^2-k_y^2)\\big ] = 0 ,$ where the contributions of type (b) cancel.",
"This is again an artifact of having used the same basis functions for $c_n$ and $f_n$ .",
"For general basis functions, the terms do not cancel, but they do not change the conclusions.",
"The above expression vanishes whenever any two of the three components of $\\mathbf {k}$ are equal.",
"There are line nodes in the $\\lbrace 110\\rbrace $ planes for infinitesimal pairing.",
"Next, we consider the TRS-breaking state where the amplitude from $T_{1g+}\\otimes T_{2g+}$ has a phase shift of $\\pi /2$ relative to the amplitude from $E_{g+}\\otimes E_{g+}$ .",
"The real and imaginary parts of the condition for IP nodes have the same momentum dependence.",
"Hence, the line nodes of the real and imaginary part coincide and the TRS-breaking state retains the six line nodes in the mirror planes.",
"The form factors in the $(110)$ plane read as $f_0(\\mathbf {k}) &= f_3(\\mathbf {k}) = f_5(\\mathbf {k}) = 0 , \\\\f_1(\\mathbf {k}) &=- i \\delta _{T_{1g+}} k_zk_x (k_z^2-k_x^2) , \\\\f_2(\\mathbf {k}) &= i \\delta _{T_{1g+}} k_zk_x (k_z^2-k_x^2) , \\\\f_4(\\mathbf {k}) &= -\\frac{2}{\\sqrt{3}}\\, \\delta _{E_{g+}} (k_z^2-k_x^2) ,$ with $\\delta _{T_{1g+}}$ and $\\delta _{E_{g+}}$ real, which gives $\\langle f^1 ,f^1 \\rangle &= -\\frac{4}{3}\\, \\delta _{E_{g+}}^2 (k_z^2-k_x^2)^2 , \\\\\\langle f^2 ,f^2 \\rangle &= -2\\, \\delta _{T_{1g+}}^2 k_z^2 k_x^2 (k_z^2-k_x^2)^2 , \\\\\\langle f^1 ,f^2 \\rangle &= 0 .$ and also $\\langle c ,f^1 \\rangle =\\langle c ,f^2 \\rangle =0$ .",
"In this case, the Pfaffian simplifies to $P(\\textbf {k}) &= (\\langle c ,c \\rangle -\\langle f^1 ,f^1 \\rangle -\\langle f^2 ,f^2 \\rangle )^2- 4\\langle f^1 ,f^1 \\rangle \\langle f^2 ,f^2 \\rangle .$ The second term $-4\\langle f^1 ,f^1 \\rangle \\langle f^2 ,f^2 \\rangle =-\\frac{32}{3}\\,\\delta _{E_{g+}}^2 \\delta _{T_{1g+}}^2 k_z^2 k_x^2 (k_z^2-k_x^2)^4$ is generically negative.",
"Thus we expect all the line nodes to inflate for strong coupling unlike for the two examples of $A_{2g+}$ pairing discussed above.",
"The inflated line nodes are not attached to the normal-state Fermi surface.",
"However, the inflation vanishes in special high-symmetry directions: On the $[111]$ axis, $\\langle f^1 ,f^1 \\rangle =\\langle f^2 ,f^2 \\rangle =0$ , thus the nodes are not inflated and stick to the normal-state Fermi surface.",
"For $[001]$ and $[110]$ , nodes are also not inflated because there is only a single amplitude from $E_{g+}\\otimes E_{g+}$ and its phase can be gauged away.",
"Moreover, along these directions $\\langle f^1 ,f^1 \\rangle $ and $\\langle f^2 ,f^2 \\rangle $ are not both zero.",
"Thus here the nodes are neither inflated nor attached to the normal-state Fermi surface.",
"The vanishing inflation in high-symmetry directions implies that the BFSs have self-touching points there.",
"Interestingly, $[111]$ is the direction in which three weak-coupling line nodes intersect whereas two intersect in the $[001]$ direction and there is no intersection in the $[110]$ direction."
],
[
"$E_{g+}$ pairing",
"We consider $E_{g+}$ pairing as an example for a symmetry that is also possible for purely local pairing.",
"The question is what changes for nonlocal pairing.",
"$E_{g+}$ appears in six places in Table REF : (a) $A_{1g+}\\otimes E_{g+}$ , (b) $A_{2g+}\\otimes E_{g+}$ , (c) $E_{g+}\\otimes A_{1g+}$ , (d) $E_{g+}\\otimes E_{g+}$ , (e) $T_{1g+}\\otimes T_{2g+}$ , and (f) $T_{2g+}\\otimes T_{2g+}$ .",
"The matrix-valued basis functions are given in the following to leading order only.",
"(a) For $A_{1g+}\\otimes E_{g+}$ , we find constants to leading order: $D_{x^2-y^2,A_{1g+}}(\\mathbf {k}) &\\cong h_4 = J_x^2 - J_y^2 , \\\\D_{3z^2-r^2,A_{1g+}}(\\mathbf {k}) &\\cong h_5 = \\frac{1}{\\sqrt{3}}\\, (2J_z^2 - J_x^2 - J_y^2) .$ These are of course the contributions from local pairing [1], [2].",
"(b) For $A_{2g+}\\otimes E_{g+}$ : $&D_{x^2-y^2,A_{2g+}}(\\mathbf {k}) \\nonumber \\\\&\\quad \\cong \\big [ k_x^4 (k_y^2-k_z^2) + k_y^4 (k_z^2-k_x^2) + k_z^4 (k_x^2-k_y^2) \\big ]\\, h_5 \\\\&D_{3z^2-r^2,A_{2g+}}(\\mathbf {k}) \\nonumber \\\\&\\quad \\cong -\\big [ k_x^4 (k_y^2-k_z^2) + k_y^4 (k_z^2-k_x^2) + k_z^4 (k_x^2-k_y^2) \\big ]\\, h_4 .$ (c) For $E_{g+}\\otimes A_{1g+}$ : $D_{x^2-y^2,0}(\\mathbf {k}) &\\cong (k_x^2 - k_y^2)\\, h_0 , \\\\D_{3z^2-r^2,0}(\\mathbf {k}) &\\cong \\frac{1}{\\sqrt{3}}\\, (2k_z^2 - k_x^2 - k_y^2)\\, h_0 .$ (d) For $E_{g+}\\otimes E_{g+}$ : $D_{x^2-y^2,E_{g+}}(\\mathbf {k}) &\\cong \\frac{1}{\\sqrt{3}}\\, (2k_z^2 - k_x^2 - k_y^2)\\, h_4+ (k_x^2 - k_y^2)\\, h_5 , \\\\D_{3z^2-r^2,E_{g+}}(\\mathbf {k}) &\\cong (k_x^2 - k_y^2)\\, h_4- \\frac{1}{\\sqrt{3}}\\, (2k_z^2 - k_x^2 - k_y^2)\\, h_5 .$ (e) For $T_{1g+}\\otimes T_{2g+}$ : $&D_{x^2-y^2,T_{1g+}}(\\mathbf {k}) \\cong \\frac{1}{\\sqrt{3}}\\, \\big [ k_y k_z(k_y^2-k_z^2)\\, h_1 \\nonumber \\\\&\\quad {}+ k_z k_x (k_z^2-k_x^2)\\, h_2 - 2\\, k_x k_y (k_x^2-k_y^2)\\, h_3 \\big ] , \\\\&D_{3z^2-r^2,T_{1g+}}(\\mathbf {k}) \\cong k_y k_z (k_y^2-k_z^2)\\, h_1 - k_zk_x (k_z^2-k_x^2)\\, h_2 .$ (f) For $T_{2g+}\\otimes T_{2g+}$ : $D_{x^2-y^2,T_{2g+}}(\\mathbf {k}) &\\cong - k_y k_z\\, h_1 + k_z k_x\\, h_2 , \\\\D_{3z^2-r^2,T_{2g+}}(\\mathbf {k}) &\\cong \\frac{1}{\\sqrt{3}}\\,( k_y k_z\\, h_1 + k_z k_x\\, h_2 - 2\\, k_x k_y\\, h_3 ) .$ The resulting superconducting form factors $f_n(\\mathbf {k})$ are given in Table REF and the products $c_n(\\mathbf {k}) f_n(\\mathbf {k})$ appearing in the condition (REF ) for IP nodes are shown in Table REF .",
"The analysis is analogous to the previous cases of $E_{g+}$ pairing: By inserting $k_y=\\pm k_x$ , one can see that the contribution to $\\langle c,f\\rangle $ from $x^2-y^2$ basis functions (with amplitudes $\\tilde{\\delta }^1_{\\cdots }$ ) has two symmetry-imposed first-order line nodes for $k_y=\\pm k_x$ .",
"In a time-reversal-symmetric state, the inclusion of $3z^2-r^2$ basis functions generically leads to two line nodes elsewhere on the normal-state Fermi surface.",
"The nodes must intersect with the $\\langle 111\\rangle $ axes.",
"TRS-breaking states generically lead to point nodes in the $\\langle 111\\rangle $ directions, for example for order parameters proportional to $(1,i)$ .",
"These point nodes are solely determined by symmetry.",
"The presence of these eight point nodes was also found for purely local pairing [2].",
"We thus find that the inclusion of nonlocal pairing does not change the nodal structure.",
"Table: Leading-order polynomial forms of the form factors f n (𝐤)f_n(\\mathbf {k}) describing E g+ E_{g+} pairing for electrons with angular momentum j=3/2j=3/2.Table: Leading-order polynomial forms of the products c n (𝐤)f n (𝐤)c_n(\\mathbf {k}) f_n(\\mathbf {k}) of form factors describing E g+ E_{g+} pairing for electrons with angular momentum j=3/2j=3/2.",
"The amplitudes of the leading terms in c n (𝐤)c_n(\\mathbf {k}) have been absorbed into new pairing amplitudes marked by a tilde.For purely local pairing, the point nodes are inflated into BFSs for noninfinitesimal pairing [2].",
"We briefly sketch the analysis when nonlocal pairing is included.",
"We consider the Pfaffian on the $[111]$ axis, $\\mathbf {k} = k\\,(1,1,1)/\\sqrt{3}$ .",
"Table REF then shows that $f_0(\\mathbf {k}) &= 0 , \\\\f_1(\\mathbf {k}) &= -\\frac{\\delta ^1_{T_{2g+}}}{3}\\, k^2 + \\frac{\\delta ^2_{T_{2g+}}}{3\\sqrt{3}}\\, k^2 , \\\\f_2(\\mathbf {k}) &= \\frac{\\delta ^1_{T_{2g+}}}{3}\\, k^2 + \\frac{\\delta ^2_{T_{2g+}}}{3\\sqrt{3}}\\, k^2 , \\\\f_3(\\mathbf {k}) &= -\\frac{2\\, \\delta ^2_{T_{2g+}}}{3\\sqrt{3}}\\, k^2 , \\\\f_4(\\mathbf {k}) &= \\delta ^1_{A_{1g+}} , \\\\f_5(\\mathbf {k}) &= \\delta ^2_{A_{1g+}} .$ For the generalized $(1,i)$ pairing state with $\\delta ^2_{A_{1g+}} = i\\delta ^1_{A_{1g+}}$ , $\\delta ^2_{T_{2g+}} = i\\delta ^1_{T_{2g+}}$ , and $\\delta ^1_{A_{1g+}}, \\delta ^1_{T_{2g+}} \\in \\mathbb {R}$ , we find $\\langle f^1,f^1\\rangle &= \\langle f^2,f^2\\rangle = -\\frac{2k^4}{9}\\, \\big ( \\delta ^1_{T_{2g+}} \\big )^2- \\big ( \\delta ^1_{A_{1g+}} \\big )^2 , \\\\\\langle f^1,f^2\\rangle &= 0 .$ On the other hand, the normal-state form factors are, to leading order, $c_0(\\mathbf {k}) &= c_0^{(0)} , \\\\c_1(\\mathbf {k}) &= c_2(\\mathbf {k}) = c_3(\\mathbf {k}) = \\frac{c_2^{(2)}}{3}\\, k^2 , \\\\c_4(\\mathbf {k}) &= c_5(\\mathbf {k}) = 0 .$ It follows that $\\langle c,c\\rangle &= \\big (c_0^{(0)}\\big )^2 - \\frac{\\big (c_2^{(2)}\\big )^2}{9}\\, k^4 , \\\\\\langle c,f^1\\rangle &= \\langle c,f^2\\rangle = 0 .$ The analysis is thus analogous to the previous two examples with $E_{g+}$ pairing.",
"The point nodes are inflated into BFSs, which touch the normal-state Fermi surface.",
"Hence, the inclusion of nonlocal pairing does not affect the phenomenology for this pairing state.",
"Note that only two of the six contributions (a)–(f) lead to inflation in the $[111]$ direction, namely the local $A_{1g+}\\otimes E_{g+}$ contribution and the $T_{2g+}\\otimes T_{2g+}$ contribution.",
"For this to happen, there must be a pair of nonzero amplitudes with nontrivial phase difference in at least one of these two channels."
],
[
"$T_{1g+}$ pairing",
"The irrep $T_{1g+}$ provides an example for a pairing state that cannot occur for local pairing in the $j=3/2$ model but unlike $A_{2g+}$ is multidimensional.",
"$T_{1g+}$ pairing emerges in seven places in Table REF : (a) $A_{2g+}\\otimes T_{2g+}$ , (b) $E_{g+}\\otimes T_{2g+}$ , (c) $T_{1g+}\\otimes A_{1g+}$ , (d) $T_{1g+}\\otimes T_{2g+}$ , (e) $T_{1g+}\\otimes E_{g+}$ , (f) $T_{2g+}\\otimes T_{2g+}$ , and (g) $T_{2g+}\\otimes E_{g+}$ .",
"We give the the matrix-valued basis functions to the leading order only.",
"(a) For $A_{2g+}\\otimes T_{2g+}$ : $& D_{x,A_{2g+}}(\\mathbf {k}) \\nonumber \\\\&\\quad \\cong \\big [ k_x^4 (k_y^2-k_z^2) + k_y^4 (k_z^2-k_x^2) + k_z^4 (k_x^2-k_y^2) \\big ]\\, h_1 , \\\\& D_{y,A_{2g+}}(\\mathbf {k}) \\nonumber \\\\&\\quad \\cong \\big [ k_x^4 (k_y^2-k_z^2) + k_y^4 (k_z^2-k_x^2) + k_z^4 (k_x^2-k_y^2) \\big ]\\, h_2 , \\\\& D_{z,A_{2g+}}(\\mathbf {k}) \\nonumber \\\\&\\quad \\cong \\big [ k_x^4 (k_y^2-k_z^2) + k_y^4 (k_z^2-k_x^2) + k_z^4 (k_x^2-k_y^2) \\big ]\\, h_3 .$ (b) For $E_{g+}\\otimes T_{2g+}$ : $D_{x,E_{g+}}(\\mathbf {k}) &\\cong (k_y^2-k_z^2)\\, h_1 , \\\\D_{y,E_{g+}}(\\mathbf {k}) &\\cong (k_z^2-k_x^2)\\, h_2 , \\\\D_{z,E_{g+}}(\\mathbf {k}) &\\cong (k_x^2-k_y^2)\\, h_3 .$ (c) For $T_{1g+}\\otimes A_{1g+}$ : $D_{x,T_{1g+}}(\\mathbf {k}) &\\cong k_y k_z (k_y^2-k_z^2)\\, h_0 , \\\\D_{y,T_{1g+}}(\\mathbf {k}) &\\cong k_z k_x (k_z^2-k_x^2)\\, h_0 , \\\\D_{z,T_{1g+}}(\\mathbf {k}) &\\cong k_x k_y (k_x^2-k_y^2)\\, h_0 .$ (d) For $T_{1g+}\\otimes T_{2g+}$ : $D^{\\prime }_{x,T_{1g+}}(\\mathbf {k}) &\\cong k_z k_x (k_z^2-k_x^2)\\, h_3 + k_x k_y (k_x^2-k_y^2)\\, h_2 ,\\\\D^{\\prime }_{y,T_{1g+}}(\\mathbf {k}) &\\cong k_x k_y (k_x^2-k_y^2)\\, h_1 + k_y k_z (k_y^2-k_z^2)\\, h_3 ,\\\\D^{\\prime }_{z,T_{1g+}}(\\mathbf {k}) &\\cong k_y k_z (k_y^2-k_z^2)\\, h_2 + k_z k_x (k_z^2-k_x^2)\\, h_1 .$ (e) For $T_{1g+}\\otimes E_{g+}$ : $D^{\\prime \\prime }_{x,T_{1g+}}(\\mathbf {k}) &\\cong k_y k_z (k_y^2-k_z^2)\\bigg (\\frac{\\sqrt{3}}{2}\\, h_4- \\frac{1}{2} \\, h_5\\bigg ) , \\\\D^{\\prime \\prime }_{y,T_{1g+}}(\\mathbf {k}) &\\cong -k_z k_x (k_z^2-k_x^2)\\bigg (\\frac{\\sqrt{3}}{2}\\, h_4+ \\frac{1}{2} \\, h_5\\bigg ) , \\\\D^{\\prime \\prime }_{z,T_{1g+}}(\\mathbf {k}) &\\cong k_x k_y (k_x^2-k_y^2)\\, h_5 .$ (f) For $T_{2g+}\\otimes T_{2g+}$ : $D_{x,T_{2g+}}(\\mathbf {k}) &\\cong k_z k_x \\, h_3 - k_x k_y \\, h_2 , \\\\D_{y,T_{2g+}}(\\mathbf {k}) &\\cong k_x k_y \\, h_1 - k_y k_z \\, h_3 , \\\\D_{z,T_{2g+}}(\\mathbf {k}) &\\cong k_y k_z \\, h_2 - k_z k_x \\, h_1 .$ (g) For $T_{2g+}\\otimes E_{g+}$ : $D^{\\prime }_{x,T_{2g+}}(\\mathbf {k}) &\\cong -k_y k_z\\bigg (\\frac{1}{2} \\, h_4 + \\frac{\\sqrt{3}}{2}\\, h_5\\bigg ) , \\\\D^{\\prime }_{y,T_{2g+}}(\\mathbf {k}) &\\cong k_z k_x\\bigg ({-}\\frac{1}{2} \\, h_4 + \\frac{\\sqrt{3}}{2}\\, h_5\\bigg ) , \\\\D^{\\prime }_{z,T_{2g+}}(\\mathbf {k}) &\\cong k_x k_y \\, h_4 .$ Table: Leading-order polynomial forms of the form factors f n (𝐤)f_n(\\mathbf {k}) describing T 1g+ T_{1g+} pairing for electrons with angular momentum j=3/2j=3/2.Table: Leading-order polynomial forms of the products c n (𝐤)f n (𝐤)c_n(\\mathbf {k}) f_n(\\mathbf {k}) of form factors describing T 1g+ T_{1g+} pairing for electrons with angular momentum j=3/2j=3/2.",
"The amplitudes of the leading terms in c n (𝐤)c_n(\\mathbf {k}) have been absorbed into new pairing amplitudes marked by a tilde.The analysis is analogous to the one for a system with two s-orbitals; see Sec.",
"REF .",
"The resulting superconducting form factors $f_n(\\mathbf {k})$ are given in Table REF and the products $c_n(\\mathbf {k}) f_n(\\mathbf {k})$ appearing in the condition (REF ) for IP nodes are shown in Table REF .",
"Hence, $c_0&(\\mathbf {k})\\, f_0(\\mathbf {k}) - \\vec{c}(\\mathbf {k}) \\cdot \\vec{f}(\\mathbf {k}) \\nonumber \\\\&= \\bigg [ \\tilde{\\delta }_{x,T_{1g+}} - \\tilde{\\delta }_{x,E_{g+}}+ \\tilde{\\delta }_{x,A_{2g+}} (k_z^2-k_x^2)(k_x^2-k_y^2) \\nonumber \\\\&{}+ \\tilde{\\delta }_{x,T_{1g+}^{\\prime }}\\, k_x^2- \\frac{\\tilde{\\delta }_{x,T_{1g+}^{\\prime \\prime }}}{\\sqrt{3}}\\, (2 k_x^2-k_y^2-k_z^2) \\nonumber \\\\&{}- \\tilde{\\delta }_{x,T_{2g+}^{\\prime }} \\bigg ]\\, k_y k_z (k_y^2-k_z^2) + \\ldots ,$ where two terms with cyclically permuted indices $x$ , $y$ , and $z$ have been suppressed.",
"For broken TRS, the pairing states $(1,i,0)$ and $(1,\\omega ,\\omega ^2)$ with $\\omega =e^{2\\pi i/3}$ , are plausible [31], [26], [13].",
"We here consider the simpler $(1,i,0)$ state, which has 18 point nodes in the $\\langle 001\\rangle $ , $\\langle 101\\rangle $ , and $\\langle 111\\rangle $ directions and one line node in the $k_z=0$ plane.",
"For noninfinitesimal pairing, we expect the nodes to be inflated.",
"For the nodes along the $[001]$ direction, the form factors read as $f_0(\\mathbf {k}) &= f_3(\\mathbf {k}) = f_4(\\mathbf {k}) = f_5(\\mathbf {k}) = 0 , \\\\f_1(\\mathbf {k}) &= -\\delta _{E_{g+}} k^2 , \\\\f_2(\\mathbf {k}) &= i\\delta _{E_{g+}} k^2 .$ We find $\\langle f^1, f^1 \\rangle =\\langle f^2, f^2 \\rangle =-\\delta ^2_{E_{g+}} k^4$ and $\\langle f^1, f^2 \\rangle =0$ as well as $\\langle c^1, f^1 \\rangle =\\langle c^2, f^2 \\rangle =0$ .",
"In this case, the Pfaffian simplifies to $P(\\mathbf {k})=\\langle c, c \\rangle \\, \\big (\\langle c, c \\rangle + 4 \\delta ^2_{E_{g+}} k^4 \\big ) .$ The first factor changes sign at the normal-state Fermi surface but the second factor does not and thus the Pfaffian changes sign at the normal-state Fermi surface.",
"Hence, the point nodes in the $\\langle 001 \\rangle $ directions are inflated and remain attached to the normal-state Fermi surfaces for arbitrarily strong coupling.",
"In the $[101]$ direction, we have $\\mathbf {k}=k(1,0,1)/\\sqrt{2}$ and the form factors read as $f_0(\\mathbf {k}) &= f_1(\\mathbf {k})=f_2(\\mathbf {k})= 0 , \\\\f_3(\\mathbf {k}) &= \\frac{\\delta _{T_{2g+}}}{2}\\, k^2 , \\\\f_4(\\mathbf {k}) &= - i\\, \\frac{\\delta _{T_{2g+}^{\\prime }}}{4}\\, k^2 , \\\\f_5(\\mathbf {k}) &= i\\, \\frac{\\sqrt{3}\\,\\delta _{T_{2g+}^{\\prime }}}{4}\\, k^2 .$ This implies $\\langle f^1, f^2 \\rangle =0$ but $\\langle f^1, f^1 \\rangle $ and $\\langle f^2, f^2 \\rangle $ are generally unequal.",
"The Pfaffian is thus $P(\\mathbf {k}) = \\left[ \\langle c,c\\rangle - \\langle f^1,f^1\\rangle - \\langle f^2,f^2\\rangle \\right]^2 - 4\\, \\langle f^1,f^1\\rangle \\langle f^2,f^2\\rangle $ .",
"The first term vanishes on a renormalized Fermi surface.",
"The second term $- 4\\, \\langle f^1,f^1\\rangle \\langle f^2,f^2\\rangle = -\\frac{1}{4}\\, \\delta _{T_{2g+}}^2 \\delta _{T_{2g+}^{\\prime }}^2 k^8$ is generically negative.",
"Hence, the Pfaffian generically becomes negative in the vicinity of the normal-state Fermi surface but the BFS does not usually touch it.",
"The Pfaffian is identical for all $\\langle 101\\rangle $ directions that are not in the $k_z=0$ plane.",
"Along $[111]$ , we have $\\mathbf {k}=k\\,(1,1,1)/\\sqrt{3}$ and $f_0(\\mathbf {k}) &= 0 , \\\\f_1(\\mathbf {k}) &= i\\, \\frac{\\delta _{T_{2g+}}}{3} \\,k^2 , \\\\f_2(\\mathbf {k}) &= - \\frac{\\delta _{T_{2g+}}}{3} \\,k^2 , \\\\f_3(\\mathbf {k}) &= \\frac{\\delta _{T_{2g+}}}{3} \\,k^2- i\\frac{\\delta _{T_{2g+}}}{3} \\,k^2 , \\\\f_4(\\mathbf {k}) &= - \\frac{\\delta _{T_{2g+}^{\\prime }}}{6} \\,k^2- i \\frac{\\delta _{T_{2g+}^{\\prime }}}{6} \\,k^2 , \\\\f_5(\\mathbf {k}) &= - \\frac{\\delta _{T_{2g+}^{\\prime }}}{2\\sqrt{3}} \\,k^2+ i \\frac{\\delta _{T_{2g+}^{\\prime }}}{2\\sqrt{3}} \\,k^2 .$ We thus obtain $\\langle f^1,f^2\\rangle = \\frac{k^4}{18}\\, \\big ( 2 \\delta _{T_{2g+}}^2+ \\delta _{T_{2g+}^{\\prime }}^2 \\big )$ and $\\langle f^1,f^1\\rangle = \\langle f^2,f^2\\rangle = -\\frac{k^4}{9}\\, \\big ( 2 \\delta _{T_{2g+}}^2 + \\delta _{T_{2g+}^{\\prime }}^2 \\big ) .$ The Pfaffian is $P(\\mathbf {k}) &= \\left[ \\langle c,c\\rangle - \\langle f^1,f^1\\rangle - \\langle f^2,f^2\\rangle \\right]^2\\nonumber \\\\&\\quad {} + 4 \\left[ \\langle f^1,f^2\\rangle ^2 - \\langle f^1,f^1\\rangle \\langle f^2,f^2\\rangle \\right] ,$ wherein the second term evaluates to $-\\frac{k^8}{27}\\, \\big ( 2 \\delta _{T_{2g+}}^2 + \\delta _{T_{2g+}^{\\prime }}^2 \\big )^2 .$ Since this is generally negative we also expect the nodes to inflate in the $\\langle 111\\rangle $ directions but they are not attached to the normal-state Fermi surface.",
"For the equatorial line node, we take $\\mathbf {k}=(k_x,k_y,0)$ .",
"The superconducting form factors are $f_0(\\mathbf {k}) &= f_3(\\mathbf {k}) = f_4(\\mathbf {k}) = f_5(\\mathbf {k}) = 0 , \\\\f_1(\\mathbf {k}) &= \\delta _{A_{2g+}} (k_x^4 k_y^2-k_y^4 k_x^2)+\\delta _{E_{g+}}\\, k_y^2+ i \\delta _{T_{2g+}} k_x k_y \\nonumber \\\\&\\quad {} + i \\delta _{T_{1g+}^{\\prime }} k_x k_y (k_x^2-k_y^2) \\\\f_2(\\mathbf {k}) &= i\\delta _{A_{2g+}} (k_x^4 k_y^2-k_y^4 k_x^2)-i \\delta _{E_{g+}}\\, k_x^2- \\delta _{T_{2g+}}\\, k_x k_y \\nonumber \\\\&\\quad {} + \\delta _{T_{1g+}^{\\prime }} k_x k_y (k_x^2-k_y^2) .$ We find $\\langle f^1,f^2\\rangle \\ne 0$ and $\\langle f^1,f^1\\rangle \\ne \\langle f^2,f^2\\rangle $ .",
"The Pfaffian thus again has the form of Eq.",
"(REF ).",
"The second term reads as $& - 4 k_x^2 k_y^2\\, \\big [ \\delta _{T_{2g+}}^2 - \\delta _{E_{g+}}^2- \\delta _{A_{2g+}} \\delta _{E_{g+}} (k_x^2-k_y^2)^2 \\nonumber \\\\&\\quad {}+ \\big ( \\delta _{A_{2g+}}^2 k_x^2 k_y^2- \\delta _{T_{1g+}^{\\prime }}^2 \\big ) (k_x^2 + k_y^2) (k_x^2 - k_y^2) \\big ]^2$ and is generically negative.",
"We conclude that the equatorial line node is inflated by noninfinitesimal pairing.",
"The resulting BFS is toroidal but pinched on the $k_x$ and $k_y$ axes since Eq.",
"(REF ) gives zero there.",
"Since the first term in Eq.",
"(REF ) becomes zero close to but not at the normal-state Fermi surface the BFS generically does not touch the normal-state Fermi surface.",
"This also holds on the $k_x$ and $k_y$ axes.",
"The behavior of the nodes is identical to the case of $T_{1g+}$ pairing for two s-orbitals."
],
[
"Orbital doublet",
"The discussion in Sec.",
"REF suggests how to construct further orbital models: the set of orbitals must be closed under the action of the point group.",
"This implies that the orbitals must transform like basis functions of one-dimensional irreps or as complete sets of basis functions of multidimensional irreps.",
"In the previous examples, we have considered two $A_{1g}$ orbitals and one $A_{1g}$ and one $A_{2u}$ orbital.",
"As the simplest example with a multidimensional irrep we here analyze the case of two orbitals transforming like basis functions of $E_g$ .",
"This is naturally realized by a doublet of $e_g$ orbitals ($d_{x^2-y^2}$ and $d_{3z^2-r^2}$ ) per site.",
"Since they are of even parity we have $U_P = \\sigma _0 \\otimes \\sigma _0$ .",
"Also, $U_T = \\sigma _0 \\otimes i\\sigma _2$ holds.",
"The new aspect here is that the orbital part alone can have higher-dimensional irreps.",
"The irreps of Pauli matrices in orbital space are the following: $\\eta _0 \\equiv \\sigma _0$ belongs to $A_{1g+}$ .",
"The two matrices $\\eta _1 \\equiv \\sigma _1$ and $\\eta _2 \\equiv -\\sigma _3$ form an $E_{g+}$ doublet.",
"It is easy to check that under rotations $\\eta _1$ and $\\eta _2$ transform like $k_x^2-k_y^2$ and $(2k_z^2-k_x^2-k_y^2)/\\sqrt{3}$ , respectively.",
"Finally, $\\eta _3 \\equiv \\sigma _2$ belongs to $A_{2g-}$ .",
"In spin space, $\\sigma _0$ of course transforms according to $A_{1g+}$ and $(\\sigma _1,\\sigma _2,\\sigma _3)$ form a $T_{1g-}$ triplet.",
"Combining higher-dimensional irreps from the orbital and spin parts, we obtain reducible product representations.",
"Thus Kronecker products of Pauli matrices in orbital and spin space have to be linearly combined to construct the proper basis matrices.",
"(Such a construction is also implicit in Table REF of basis matrices for the $j=3/2$ case above.)",
"Specifically, we require the nontrivial reduction $E_{g+} \\otimes T_{1g-} = T_{1g-} \\oplus T_{2g-}$ .",
"The resulting basis matrices are presented in Table REF .",
"Table: Basis matrices on the internal Hilbert spacefor the case of an E g E_g doublet of orbitals with spin and point group O h O_h.",
"The basis matrices are irreducible tensor operators of the irreps listed in the second column.",
"For multidimensional irreps, the states transforming into each other under point-group operations are distinguished by the index in the third column.The basis matrices relevant for the normal-state Hamiltonian and for even-parity pairing are $h_0 &\\equiv \\eta _0 \\otimes \\sigma _0 & A_{1g+}, \\\\h_1 &\\equiv \\eta _3 \\otimes \\sigma _1 & T_{2g+}, \\\\h_2 &\\equiv \\eta _3 \\otimes \\sigma _2 & T_{2g+}, \\\\h_3 &\\equiv \\eta _3 \\otimes \\sigma _3 & T_{2g+}, \\\\h_4 &\\equiv \\eta _1 \\otimes \\sigma _0 & E_{g+}, \\\\h_5 &\\equiv \\eta _2 \\otimes \\sigma _0 & E_{g+}.$ The symmetry properties are thus the same as for the $j=3/2$ case.",
"Hence, the analysis of pairing states is completely analogous to Sec.",
"REF , except for the different definition of the basis matrices $h_n$ ."
],
[
"Orbital triplet",
"We briefly consider the case of three $t_{2g}$ orbitals per site of a cubic lattice, i.e., $d_{yz}$ , $d_{zx}$ , and $d_{xy}$ orbitals.",
"In this example, the dimension of the internal Hilbert space is $N=6$ and according to Appendix there are 15 basis matrices for the normal-state Hamiltonian and even-parity superconductivity.",
"Their algebra is much more complicated than for $N=4$ .",
"In particular, they do not anticommute pairwise.",
"We require a basis of $3\\times 3$ matrices that act on the orbital space.",
"We take the Gell-Mann matrices [17] $\\lambda _0 = &\\begin{pmatrix} 1 & 0 & 0 \\\\ 0 & 1 & 0 \\\\ 0 & 0 & 1 \\end{pmatrix} \\!, \\\\\\lambda _1 = \\begin{pmatrix} 0 & 1 & 0 \\\\ 1 & 0 & 0 \\\\ 0 & 0 & 0 \\end{pmatrix} \\!,\\:\\lambda _2 = &\\begin{pmatrix} 0 & 0 & 1 \\\\ 0 & 0 & 0 \\\\ 1 & 0 & 0 \\end{pmatrix} \\!,\\:\\lambda _3 = \\begin{pmatrix} 0 & 0 & 0 \\\\ 0 & 0 & 1 \\\\ 0 & 1 & 0 \\end{pmatrix} \\!, \\\\\\lambda _4 = \\begin{pmatrix} 0 & -i & 0 \\\\ i & 0 & 0 \\\\ 0 & 0 & 0 \\end{pmatrix} \\!,\\:\\lambda _5 = &\\begin{pmatrix} 0 & 0 & -i \\\\ 0 & 0 & 0 \\\\ i & 0 & 0 \\end{pmatrix} \\!,\\:\\lambda _6 = \\begin{pmatrix} 0 & 0 & 0 \\\\ 0 & 0 & -i \\\\ 0 & i & 0 \\end{pmatrix} \\!, \\\\\\lambda _7 = \\begin{pmatrix} 1 & 0 & 0 \\\\ 0 & -1 & 0 \\\\ 0 & 0 & 0 \\end{pmatrix} \\!,\\hspace{3.50006pt}&\\lambda _8 = \\frac{1}{\\sqrt{3}} \\begin{pmatrix} 1 & 0 & 0 \\\\ 0 & 1 & 0 \\\\ 0 & 0 & -2 \\end{pmatrix} \\!.$ The associated irreps are given in Table REF .",
"The Gell-Mann matrices have to be combined with the Pauli matrices acting on spin space to form basis matrices for the combined internal degrees of freedom.",
"$\\sigma _0$ of course transforms according to $A_{1g+}$ and $(\\sigma _1,\\sigma _2,\\sigma _3)$ to $T_{1g-}$ .",
"Possible basis matrices for $H_N(\\mathbf {k})$ and even-parity pairing must belong to $g+$ irreps.",
"They can thus be constructed by combining $\\lambda _j$ for $j \\in \\lbrace 0,1,2,3,7,8\\rbrace $ with $\\sigma _0$ and by combining $\\lambda _j$ with $j \\in \\lbrace 4,5,6\\rbrace $ with $\\sigma _1$ , $\\sigma _2$ , or $\\sigma _3$ .",
"The relevant product representations are either trivial or involve the reduction $T_{1g-} \\otimes T_{1g-} = A_{1g+} \\oplus E_{g+} \\oplus T_{1g+} \\oplus T_{2g+}$ .",
"The 15 allowed basis matrices and the associated irreps are shown in Table REF .",
"Here, we have not chosen any special normalization, except that the entries belonging to the same multiplets have consistent numerical factors.",
"There are also 21 basis matrices belonging to $g-$ irreps, which are disallowed as pairing matrices and are not listed for simplicity.",
"Table: Basis matrices on the internal Hilbert space for the case of an T 2g T_{2g} triplet of orbitals with spin and point group O h O_h.",
"Unlike for the previous examples, only the basis matrices h n h_n allowed in the normal-state Hamiltonian and for even-parity pairing are listed.",
"The basis matrices are irreducible tensor operators of the irreps listed in the second column.",
"For multidimensional irreps, the states transforming into each other under point-group operations are distinguished by the index in the third column.The normal-state Hamiltonian reads as $H_N(\\mathbf {k}) = \\sum _{n=0}^{14} c_n(\\mathbf {k})\\, h_n ,$ where $c_n(\\mathbf {k})$ transforms like $h_n$ .",
"For even-parity superconductivity, we can combine the 15 matrices $h_n$ with form factors $f_n(\\mathbf {k})$ belonging to all the $g+$ irreps.",
"This obviously generates pairing states for all $g+$ irreps.",
"Any pairing state can be expressed in terms of a pairing matrix of the form $D(\\mathbf {k}) = \\sum _{n=0}^{14} f_n(\\mathbf {k})\\, h_n .$ Moreover, all even-parity pairing states other than $A_{2g+}$ can occur for purely local pairing, i.e., with constant form factors, because the basis matrices $h_n$ in Table REF cover all $g+$ irreps except $A_{2g+}$ .",
"Another interesting observation is that purely local pairing with trivial ($A_{1g+}$ ) symmetry now allows for an orbitally nontrivial contribution from $h_6$ .",
"All pairing states including nonlocal contributions can be constructed as described above.",
"For the present case of $N=6$ , we typically find a larger number of contributions than for $N=4$ .",
"For example, $E_{g+}$ pairing can result from the products (form factor times basis matrix) (a) $A_{1g+} \\otimes E_{g+}$ , (b) $A_{2g+} \\otimes E_{g+}$ , (c) $E_{g+} \\otimes A_{1g+}$ , (d) $E_{g+} \\otimes E_{g+}$ , (e) $T_{1g+} \\otimes T_{1g+}$ , (f) $T_{1g+} \\otimes T_{2g+}$ , (g) $T_{2g+} \\otimes T_{1g+}$ , and (h) $T_{2g+} \\otimes T_{2g+}$ .",
"The leading-order contribution of type (a) to the pairing matrix reads as $D_{x^2-y^2,A_{1g+}}(\\mathbf {k}) &\\cong \\delta _0\\, \\lambda _7 \\otimes \\sigma _0+ \\delta _1\\, ( \\lambda _6 \\otimes \\sigma _1 + \\lambda _5 \\otimes \\sigma _2 ) , \\\\D_{3z^2-r^2,A_{1g+}}(\\mathbf {k}) &\\cong -\\delta _0\\, \\lambda _8 \\otimes \\sigma _0+ \\frac{\\delta _1}{\\sqrt{3}}\\, ( 2\\lambda _4 \\otimes \\sigma _3 \\nonumber \\\\&\\quad {}- \\lambda _6 \\otimes \\sigma _1 + \\lambda _5 \\otimes \\sigma _2 ) ,$ where one of the constants $\\delta _0$ and $\\delta _1$ could be set to unity.",
"We omit the construction of the other contributions.",
"The expression $F(\\mathbf {k}) = \\sum _n c_n(\\mathbf {k})\\, f_n(\\mathbf {k})$ transforms like the pairing matrix $D(\\mathbf {k})$ and the condition $F(\\mathbf {k})=0$ determines the location of IP nodes, as shown in Sec.",
".",
"In principle, we can now obtain the form factors $c_n(\\mathbf {k})$ appearing in Eq.",
"(REF ) and $f_n(\\mathbf {k})$ in Eq.",
"(REF ) and thus $F(\\mathbf {k})$ and the IP nodal structure.",
"Pairing of not infinitesimal amplitude, in particular the existence of BFSs, could then be analyzed following Sec.",
".",
"This requires the calculation of the Pfaffian $\\mathop {\\textrm {Pf}}\\tilde{\\mathcal {H}}(\\mathbf {k})$ .",
"Due to the absence of simple algebraic relations between the basis matrices $h_n$ , there is no simple analytical expression in terms of the functions $c_n(\\mathbf {k})$ and $f_n(\\mathbf {k})$ so that such an analysis would likely require a numerical study of $\\mathop {\\textrm {Pf}}\\tilde{\\mathcal {H}}(\\mathbf {k})$ .",
"We do not execute this program here.",
"BFSs have been predicted for $E_{g+}$ pairing in an $N=6$ model for $\\mathrm {Sr}_2\\mathrm {RuO}_4$ [17], which has the point group $D_{4h}$ ."
],
[
"Discussion and conclusions",
"We have analyzed possible superconducting pairing symmetries for materials with local degrees of freedom beyond the electronic spin, such as orbital or basis site.",
"Local degrees of freedom can enable unconventional pairing, in particular with BFSs.",
"We find that this is the case even in the simple (and somewhat artificial) case of two s-orbitals at each lattice site, where the orbital index appears to be a spectator: Such a model permits orbitally nontrivial pairing of $T_{1g}$ symmetry.",
"The main step taken in this paper is to go beyond local pairing.",
"This implies that not only the normal state is characterized by momentum-dependent form factors $c_n(\\mathbf {k})$ but also the superconducting pairing is characterized by momentum-dependent form factors $f_n(\\mathbf {k})$ .",
"In case of even-parity pairing, the functions $f_n(\\mathbf {k})$ have the same symmetry properties as the $c_n(\\mathbf {k})$ .",
"There are three distinct types of contributions to pairing with nontrivial symmetry: (a) purely local pairing which is internally anisotropic, (b) internally isotropic pairing with nontrivial momentum-space form factors, and (c) contributions with nontrivial internal and momentum-space structure.",
"Type (a) has been studied in Refs.",
"[1], [2], whereas type (b) is exemplified by $d_{x^2-y^2}$ pairing in cuprates.",
"Types (a) and (c) are internally anisotropic, which means that the pairing matrix $\\Delta (\\mathbf {k}) = D(\\mathbf {k})\\, U_T$ acts nontrivially on the internal degrees of freedom.",
"$D(\\mathbf {k})$ then contains one or more basis matrices on the internal Hilbert space that are not proportional to the identity matrix.",
"Nonlocal pairing permits symmetries of pairing states that usually cannot all be realized for local pairing.",
"In fact, all even-parity ($g$ ) irreps of the magnetic point group occur for nonlocal pairing since they all have momentum-space basis functions and thus at least permit a contribution of type (b).",
"In particular, there is always at least one basis matrix belonging to the trivial irrep, which is even, namely the identity matrix.",
"Since the constant function of momentum also has full symmetry, local pairing with full, trivial symmetry is always allowed.",
"Of course, it can be energetically suppressed by a local repulsive interaction.",
"The example of two orbitals of opposite parity reveals that if parity acts nontrivially on the internal degrees of freedom, basis matrices become possible that are odd under parity; see Sec.",
"REF .",
"These basis matrices can combine with odd-in-momentum form factors to form even-parity superconducting states [41], [42].",
"Odd-parity superconductivity is characterized by $u+$ irreps (odd under inversion, even under time reversal), which requires $g-$ basis matrices and $u-$ form factors $f_\\nu (\\mathbf {k})$ .",
"All $u-$ irreps possess momentum-space basis functions and thus allow to write down such form factors.",
"Moreover, inspection of multiplication tables for irreps shows that for any magnetic point group with inversion, the existence of one $g-$ basis matrix is sufficient to generate pairing states belonging to all $u+$ irreps.",
"The IP nodes are determined by both the normal-state and the corresponding superconducting form factors, $c_n(\\mathbf {k})$ and $f_n(\\mathbf {k})$ , respectively, not by the superconducting factors alone.",
"Specifically, the simple criterion $\\sum _n c_n(\\mathbf {k})\\, f_n(\\mathbf {k})=0$ diagnoses the presence of IP nodes.",
"The position of the IP nodes is generically unaffected by nonlocal contributions to the extent that it is determined by symmetry.",
"It is then only determined by the irrep of the pairing state.",
"As a counterexample, for pairing belonging to the second ($3z^2-r^2$ ) component of $E_g$ for the point group $O_h$ , there are always two line nodes but these do not lie in high-symmetry planes and are only constrained to pass through the $\\langle 111\\rangle $ directions.",
"Pairing states belonging to one-dimensional irreps can break TRS if the pairing has more than one contribution, which allows nontrivial phase factors.",
"In the resulting TRS-breaking state, the symmetry-imposed line nodes of the time-reversal-symmetric state persist for infinitesimal pairing.",
"The same mechanism is also possible for multidimensional irreps, in addition to the more natural case of phase factors between different components.",
"If the pairing strength is not infinitesimal the point or line nodes can be inflated into BFSs.",
"Like for local pairing [1], [2], the BFSs are given by the zeros of the Pfaffian $P(\\mathbf {k})$ of the BdG Hamiltonian, unitarily transformed into antisymmetric form.",
"There are three possible cases for the inflated nodes: (1) they are forced to contain the original node and thus remain attached to the normal-state Fermi surface, (2) they do not remain attached to the original node and thus generically shift away from the normal-state Fermi surface with increasing pairing strength, or (3) inflated line nodes remain attached to the original line node only in high-symmetry directions (we have only observed the situation that the inflation also vanishes there).",
"For increasing pairing, the BFSs typically grow and eventually merge.",
"In case (2), the merged pockets can eventually shrink and finally disappear if this does not violate any remaining nonzero topological invariants they possess [11], [2].",
"In real materials, this only seems likely if the BFSs are located inside small normal-state Fermi pocket(s) of a poor metal.",
"It is an intriguing open question whether the resulting fully gapped superconducting state is topologically nontrivial.",
"In principle, new BFSs can emerge at strong coupling, when quasiparticle bands are shifted through the Fermi energy.",
"However, we expect that such BFSs are usually energetically disfavored and that the system can avoid them by developing a suitable momentum-dependent pairing amplitude.",
"The case of an internal Hilbert space of dimension $N=4$ is special [9].",
"We here only discuss the standard case where the internal degrees of freedom include the electron spin so that the transformation $P\\mathcal {T}$ squares to $-$ .",
"Then, there are exactly six basis matrices $h_n$ with simple algebraic properties: One of them, $h_0\\propto $ , commutes with all others, which anticommute pairwise.",
"This structures allows us to find relatively compact expressions for the Pfaffian $P(\\mathbf {k})$ in terms of the form factors $c_n(\\mathbf {k})$ and $f_n(\\mathbf {k})$ .",
"For $N>4$ , there is no comparable algebraic structure and the Pfaffian could only be given explicitly in terms of the components of the transformed Hamiltonian.",
"The number of terms in the resulting expression is exponentially large in $N$ .",
"It is useful to review and compare our results for one-dimensional $A_{2g+}$ pairing and multidimensional $T_{1g+}$ pairing, which illustrate some of our general remarks.",
"Note that it is hard to envision local degrees of freedom that realize an operator with $A_{2g+}$ symmetry, ultimately because of the high minimum order $l=6$ of basis functions.",
"Hence, purely local pairing is unlikely to exist and it indeed does not appear in the examples we have considered.",
"For infinitesimal pairing, the $A_{2g+}$ pairing state for any model has six symmetry-imposed line nodes in the $\\lbrace 110\\rbrace $ mirror planes.",
"Since they are present in the real and imaginary parts of the gap function, they persist for TRS-broken states.",
"We find that beyond infinitesimal pairing, the nodes of the TRS-broken $A_{2g+}$ states either persist as line nodes or are inflated into BFSs.",
"Specifically, the nodes are inflated in the mirror planes for the electrons with effective spin $j=3/2$ but not for the cases of two s-orbitals and of two orbitals of opposite parity.",
"The origin of this difference is that for the $j=3/2$ case, in the mirror planes, the two amplitudes $\\delta _{Eg+}$ and $\\delta _{T2g+}$ contribute to the Pfaffian whenever there is a phase difference between them.",
"For the other two cases, there is a single amplitude in the mirror planes and its phase can be gauged away.",
"Thus there is no inflation.",
"A general insight here is that if only a single amplitude contributes in some high-symmetry direction or plane the breaking of TRS cannot lead to the formation of a BFS or of a gap since the physics is (gauge) invariant under changes of the phase of this amplitude.",
"This argument also applies to multidimensional irreps.",
"TRS-breaking pairing states are more natural for multidimensional irreps.",
"In our context, $T_{1g+}$ is an interesting pairing symmetry.",
"Whereas it appears for purely local pairing in the case of two s-orbitals, it only appears for nonlocal pairing for effective-spin-$3/2$ fermions.",
"Our study suggests that for both cases all point and line nodes appearing for TRS-broken $T_{1g+}$ pairing state with order parameter $(1,i,0)$ are inflated for noninfinitesimal pairing.",
"The point nodes on the $k_z$ -axis are the only ones which remains attached to the normal-state Fermi surface, while all other point and line nodes are shifted away from the normal-state Fermi surface and thus could annihilate for strong coupling.",
"To conclude, nonlocal pairing typically permits a much larger number of possible pairing symmetries.",
"Their nodal structure, including the possibility of BFSs, can be analyzed based on symmetry.",
"For nodes at infinitesimal pairing, there is a simple yet powerful criterion in terms of a scalar product of form factors.",
"The known criterion for the appearance of BFSs in terms of a Pfaffian extends to nonlocal pairing and general internal degrees of freedom and shows that BFSs generically exist if the superconducting state breaks TRS.",
"The authors thank D. F. Agterberg, P. M. R. Brydon, I. F. Herbut, A. Knoll, S. Kobayashi, and J. M. Link for useful discussions.",
"Financial support by the Deutsche Forschungsgemeinschaft through the Collaborative Research Center SFB 1143, Project A04, the Research Training Group GRK 1621, and the Cluster of Excellence on Complexity and Topology in Quantum Matter ct.qmat (EXC 2147) is gratefully acknowledged."
],
[
"Enumeration of basis matrices",
"In this Appendix, we obtain the number of basis matrices that can occur and thus generically do occur in the normal-state Hamiltonian $H_N(\\mathbf {k})$ .",
"The same basis matrices can appear in the pairing matrix $D(\\mathbf {k})$ for $s_T=-1$ and even-parity superconductivity as well as for $s_T=+1$ and odd-parity superconductivity, while in the other two cases, only the remaining basis matrices can appear in $D(\\mathbf {k})$ .",
"The statements can be obtained in a more general framework but it might be useful to present them using representation theory of point groups.",
"The following theorem is proven: Let $N$ be the dimension of the Hilbert space describing the local degrees of freedom in the normal state.",
"Let $P$ be the unitary parity operator on this Hilbert space and let $\\mathcal {T}$ be the antiunitary time-reversal operator.",
"Then the number of Hermitian basis matrices $h_n$ that can appear in a normal-state Hamiltonian that respects $P\\mathcal {T}$ symmetry is $n_h = \\left\\lbrace \\begin{array}{ll}\\displaystyle \\frac{N(N+1)}{2} & \\mbox{for $(P\\mathcal {T})^2 = +1$} , \\\\[1.5ex]\\displaystyle \\frac{N(N-1)}{2} & \\mbox{for $(P\\mathcal {T})^2 = -1$} .\\end{array} \\right.$ As we shall see, the case $(P\\mathcal {T})^2 = -1$ can only occur for even $N$ .",
"Results for small $N$ are given in Table REF .",
"Table: Number of basis matrices h n h_n that appear in a normal-state Hamiltonian H N (𝐤)H_N(\\mathbf {k}) with P𝒯P\\mathcal {T} symmetry for the two cases that P𝒯P\\mathcal {T} squares to ±1\\pm 1.",
"NN is the dimension of the internal Hilbert space.",
"For (P𝒯) 2 =-1(P\\mathcal {T})^2=-1, which isthe standard case, NN has to be even.To show this, the normal-state Hamiltonian is expanded as $H_N(\\mathbf {k}) = \\sum _{n=1}^{n_h} c_n(\\mathbf {k})\\, h_n .$ Under the magnetic point group $M$ , the functions $c_n(\\mathbf {k})$ must transform like the corresponding $h_n$ to ensure that $H_N(\\mathbf {k})$ is invariant.",
"The operation $P\\mathcal {T}$ is an element of $M$ .",
"The irreps of $M$ can be uniquely and exhaustively divided into irreps that are even or odd under $P\\mathcal {T}$ .",
"Only the $P\\mathcal {T}$ -even irreps have momentum-space basis functions since the momentum $\\mathbf {k}$ is $P\\mathcal {T}$ even.",
"Hence, all $P\\mathcal {T}$ -even and no $P\\mathcal {T}$ -odd $h_\\nu $ can occur in Eq.",
"(REF ).",
"The proposition is thus a statement about the number $n_h$ of $N\\times N$ basis matrices $h_\\nu $ that are even under $P\\mathcal {T}$ .",
"Since $P\\mathcal {T}$ is antiunitary there exists a unitary $N\\times N$ matrix $U_{PT}$ with $P\\mathcal {T} = U_{PT} \\mathcal {K}$ , where $\\mathcal {K}$ is the complex conjugation.",
"Then we have $(P\\mathcal {T})^2 = U_{PT} \\mathcal {K} U_{PT} \\mathcal {K} = U_{PT} U_{PT}^* .$ Thus $(P\\mathcal {T})^2 = \\pm 1$ is equivalent to $U_{PT} U_{PT}^* = \\pm 1$ and, since $U_{PT}^*$ is unitary, to $U_{PT} = \\pm (U_{PT}^*)^{-1} = \\pm U_{PT}^T$ .",
"Case 1: $(P\\mathcal {T})^2 = +1$ and symmetric $U_{PT}$ .",
"For any unitary symmetric $N\\times N$ matrix $U_{PT}$ , there exists a unitary matrix $Q$ such that $U_{PT} = Q Q^T .$ $h_\\nu $ being even/odd under $P\\mathcal {T}$ means $U_{PT} h_\\nu ^* U_{PT}^\\dagger = \\pm h_\\nu $ , which due to the Hermiticity of $h_\\nu $ is equivalent to $U_{PT} h_\\nu ^T U_{PT}^\\dagger = \\pm h_\\nu .$ Equation (REF ) then gives $Q Q^T h_\\nu ^T Q^* Q^\\dagger = \\pm h_\\nu ,$ which is equivalent to $Q^T h_\\nu ^T Q^* = (Q^\\dagger h_\\nu Q)^T = \\pm Q^\\dagger h_\\nu Q .$ Note that $k_\\nu \\equiv Q^\\dagger h_\\nu Q$ is Hermitian.",
"The dimension of the vector space over $\\mathbb {R}$ of Hermitian $N\\times N$ matrices that are also symmetric is ${N(N+1)}/{2}$ .",
"Hence, the dimension of the vector space of Hermitian $P\\mathcal {T}$ -even $N\\times N$ and thus the number of $P\\mathcal {T}$ -even basis elements $h_\\nu $ also equals ${N(N+1)}/{2}$ .",
"Analogously, the dimension of the space of Hermitian $N\\times N$ matrices that are antisymmetric and thus the number of $P\\mathcal {T}$ -odd basis matrices equals ${N(N-1)}/{2}$ .",
"Note that the sum of the two numbers is $N^2$ , as expected.",
"Case 2: $(P\\mathcal {T})^2 = -1$ and antisymmetric $U_{PT}$ .",
"For any unitary antisymmetric $N\\times N$ matrix $U_{PT}$ , there exists a unitary matrix $Q$ such that $U_{PT} = Q \\Lambda Q^T ,$ with $\\Lambda = i\\sigma _2 \\otimes ,$ which clearly means that $N$ must be even.",
"We therefore write $N=2M$ .",
"$h_\\nu $ being even/odd under $P\\mathcal {T}$ means $U_{PT} h_\\nu ^T U_{PT}^\\dagger = \\pm h_\\nu .$ Equation (REF ) gives $Q \\Lambda Q^T h_\\nu ^T Q^* \\Lambda ^\\dagger Q^\\dagger = \\pm h_\\nu ,$ which is equivalent to $\\Lambda Q^T h_\\nu ^T Q^* \\Lambda ^\\dagger = \\Lambda (Q^\\dagger h_\\nu Q)^T\\Lambda ^\\dagger = \\pm Q^\\dagger h_\\nu Q .$ Let $k_\\nu \\equiv Q^\\dagger h_\\nu Q$ (Hermitian).",
"We write $\\Lambda $ and $k_\\nu $ in block form as $\\Lambda &= \\left(\\begin{array}{cc}0 & \\\\ -& 0\\end{array}\\right) , \\\\k_\\nu &= \\left(\\begin{array}{cc}\\kappa _{11} & \\kappa _{12} \\\\ \\kappa _{12}^\\dagger & \\kappa _{22}\\end{array}\\right) ,$ where $\\kappa _{11}$ and $\\kappa _{22}$ are Hermitian.",
"Equation (REF ) can then be written as $\\Lambda k_\\nu ^T \\Lambda ^\\dagger = \\begin{pmatrix}\\kappa _{22}^T & -\\kappa _{12}^T \\\\ -\\kappa _{12}^* & \\kappa _{11}^T\\end{pmatrix} = \\begin{pmatrix}\\pm \\kappa _{11} & \\pm \\kappa _{12} \\\\ \\pm \\kappa _{12}^\\dagger & \\pm \\kappa _{22}\\end{pmatrix} = \\pm k_\\nu .$ This yields the relations $\\kappa _{12}^T = \\mp \\kappa _{12}$ and $\\kappa _{22} = \\pm \\kappa _{11}^T$ , and $k_\\nu $ thus assumes the form $k_\\nu = \\begin{pmatrix}\\kappa _{11} & \\kappa _{12} \\\\ \\kappa _{12}^\\dagger & \\pm \\kappa _{11}^T\\end{pmatrix} ,$ with $\\kappa _{11}^\\dagger = \\kappa _{11}$ and $\\kappa _{12}^T = \\mp \\kappa _{12}$ .",
"The blocks are $M\\times M$ matrices.",
"The dimension of the vector space spanned by the $k_\\nu $ is $M^2 + M(M-1) = M(2M-1)$ for the upper sign and $M^2 + M(M+1) = M(2M+1)$ for the lower sign.",
"The sum is $4M^2 = N^2$ , as expected.",
"Hence, the dimension of the vector space of Hermitian $N\\times N$ matrices that are even under $P\\mathcal {T}$ is ${N(N-1)}/{2}$ , whereas the dimension for $P\\mathcal {T}$ -odd matrices is ${N(N+1)}/{2}$ .",
"Note that the two numbers are interchanged compared to the case of $(P\\mathcal {T})^2 = +1$ .",
"This completes the proof.",
"We are concerned with systems that satisfy TRS and inversion symmetry separately.",
"Then the $P\\mathcal {T}$ -even (odd) irreps are the $g+$ and $u-$ ($g-$ and $u+$ ) irreps.",
"Moreover, the two operations commute [19], [29] and $P$ squares to $+1$ .",
"Hence, $(P\\mathcal {T})^2 = \\mathcal {T}^2$ .",
"If the internal degrees of freedom include the electron spin we have $\\mathcal {T}^2 = -1$ [19] and even dimension $N$ .",
"The case $\\mathcal {T}^2 = +1$ can only be realized if the spin does not occur explicitly, for example because one spin state is pushed to high energies by a magnetic field.",
"Then $\\mathcal {T}$ is not the physical TRS but an effective antiunitary symmetry."
],
[
"Infinitesimal-pairing nodes",
"In this Appendix, we discuss the IP nodes for $s_T=-1$ and odd-parity pairing and for the unconventional sign $s_T=+1$ of time reversal squared.",
"For $s_T=-1$ and odd-parity pairing, it is still true that infinitesimal pairing can be described in a single-band, pseudospin picture.",
"However, it is now in the pseudospin-triplet channel.",
"The pairing matrix in the effective single-band picture thus has the form $D_\\mathrm {eff}(\\mathbf {k}) = \\mathbf {d}(\\mathbf {k}) \\cdot \\mbox{$\\sigma $}$ , where $\\mbox{$\\sigma $}$ is the vector of Pauli matrices representing the pseudospin.",
"Since the pseudospin is even under inversion and odd under time reversal its components belong to one or more $g-$ irreps.",
"One can use representation theory to work out which irreps the components of $\\mathbf {d}(\\mathbf {k})$ must belong to in order to obtain a pairing state of a certain symmetry.",
"The condition $\\mathbf {d}(\\mathbf {k})=0$ then gives the symmetry-imposed IP nodes.",
"Nodal gaps are thus less likely than for singlet pairing since they must satisfy three scalar conditions.",
"We now turn to the nonstandard case $s_T=+1$ .",
"According to Appendix , this allows an effective single-band model with Hilbert-space dimension $N=1$ .",
"Equation (REF ) then implies that $U_T=1$ and thus $\\Delta (\\mathbf {k})=D(\\mathbf {k})$ .",
"The only Hermitian basis matrix is $h_0=1$ .",
"Hence, for a single-band model, the full symmetry information is carried by the form factor $f_0(\\mathbf {k})$ .",
"$h_0$ belongs to the trivial irrep, which of course is a $g+$ irrep.",
"Table REF then shows that for $N=1$ only odd-parity pairing with $u-$ form factor $f_0(\\mathbf {k})$ is possible.",
"The analysis of possible pairing states is analogous to the case with $s_T=-1$ and even parity, except that $g+$ irreps are replaced by $u-$ irreps.",
"Even-parity pairing states for $s_T=+1$ cannot be described by an effective $N=1$ model but are possible for $N=2$ .",
"In fact, there are multiple possibilities to implement this case because Eq.",
"(REF ) now allows $U_T$ to be any symmetric unitary $2\\times 2$ matrix, while $U_P$ can be any unitary $2\\times 2$ matrix that squares to $$ .",
"The specific $U_T$ and $U_P$ and thus the symmetry properties of the $2\\times 2$ basis matrices $h_1$ , $h_2$ , $h_3$ (which are linear combinations of Pauli matrices) depend on the underlying system.",
"Universal properties are therefore unlikely and we do not pursue this here."
],
[
"Existence and properties of the Pfaffian",
"In this Appendix, we review the main results for the Pfaffian.",
"A simpler proof than in [1], [2] is presented.",
"The BdG Hamiltonian (REF ) satisfies the charge-conjugation symmetry $\\mathcal {U}_C \\mathcal {H}^T(-\\mathbf {k}) \\mathcal {U}_C^\\dagger = - \\mathcal {H}(\\mathbf {k})$ , where $\\mathcal {U}_C = \\sigma _1 \\otimes $ .",
"For it to also satisfy inversion symmetry, there must exist a unitary matrix $U_P$ such that $\\mathcal {U}_P\\, \\mathcal {H}(-\\mathbf {k})\\, \\mathcal {U}_P^\\dagger = \\mathcal {H}(\\mathbf {k}) ,$ where $\\mathcal {U}_P = \\begin{pmatrix}U_P & 0 \\\\ 0 & U_P^*\\end{pmatrix} .$ This is a special case of Eqs.",
"(REF ) and (REF ).",
"The two symmetries imply $\\mathcal {C}P$ symmetry, $\\mathcal {U}_{CP}\\, \\mathcal {H}^T(\\mathbf {k})\\, \\mathcal {U}_{CP}^\\dagger =- \\mathcal {H}(\\mathbf {k}) ,$ with $\\mathcal {U}_{CP} = \\mathcal {U}_C \\mathcal {U}_P^*$ .",
"We find that $\\mathcal {C}P$ squares to the identity since $(\\mathcal {U}_{CP} \\mathcal {K})^2 = \\mathcal {U}_{CP} \\mathcal {U}_{CP}^*= \\begin{pmatrix}U_P^2 & 0 \\\\ 0 & (U_P^*)^2\\end{pmatrix}= \\begin{pmatrix}& 0 \\\\ 0 & \\end{pmatrix} .$ This implies that $\\mathcal {U}_{CP} = (\\mathcal {U}_{CP}^*)^{-1} = (\\mathcal {U}_{CP}^*)^\\dagger = \\mathcal {U}_{CP}^T$ so that $\\mathcal {U}_{CP}$ is symmetric.",
"For any (complex) symmetric matrix $\\mathcal {U}_{CP}$ , there exists a unitary matrix $\\Omega $ such that $\\Lambda = \\Omega \\mathcal {U}_{CP} \\Omega ^T$ (note the transpose) is a diagonal matrix with real nonnegative components (Autonne-Takagi factorization [44], [45]).",
"$\\Lambda $ is evidently unitary.",
"A diagonal unitary matrix with nonnegative components must be $\\Lambda =$ .",
"We thus obtain $\\mathcal {U}_{CP} = \\Omega ^\\dagger \\Omega ^*$ and Eq.",
"(REF ) becomes $\\Omega ^\\dagger \\Omega ^*\\, \\mathcal {H}^T(\\mathbf {k})\\, \\Omega ^T \\Omega =- \\mathcal {H}(\\mathbf {k}) .$ This implies that $\\Omega ^*\\, \\mathcal {H}^T(\\mathbf {k})\\, \\Omega ^T= - \\Omega \\, \\mathcal {H}(\\mathbf {k})\\, \\Omega ^\\dagger .$ Hence, $\\tilde{\\mathcal {H}}(\\mathbf {k}) \\equiv \\Omega \\, \\mathcal {H}(\\mathbf {k})\\, \\Omega ^\\dagger $ is antisymmetric: $\\tilde{\\mathcal {H}}^T(\\mathbf {k}) = - \\tilde{\\mathcal {H}}(\\mathbf {k}) .$ This shows that for systems with inversion symmetry, the BdG Hamiltonian can always be unitarily transformed into antisymmetric form.",
"For the antisymmetric matrix $\\tilde{\\mathcal {H}}(\\mathbf {k})$ , the Pfaffian $\\mathop {\\textrm {Pf}}\\tilde{\\mathcal {H}}(\\mathbf {k})$ is well defined.",
"Note that $\\Omega $ is independent of $\\mathbf {k}$ (and is in fact solely determined by $\\mathcal {U}_{CP}$ ) so that the components of $\\tilde{\\mathcal {H}}(\\mathbf {k})$ are smooth functions of $\\mathbf {k}$ .",
"Hence, the Pfaffian, which is a polynomial of these components, is an smooth function of $\\mathbf {k}$ .",
"The sign of the Pfaffian is inverted under simultaneous interchange of two rows and the corresponding two columns.",
"This is a unitary transformation, which can be absorbed into $\\Omega $ .",
"Hence, the above derivation only determines the Pfaffian up to a sign.",
"If the dimension $N$ of the local Hilbert space is even the Pfaffian is real: The dimension $2N$ of the BdG Hamiltonian $\\tilde{\\mathcal {H}}(\\mathbf {k})$ is a multiple of four and thus the Pfaffian is a polynomial of even degree of its the components.",
"Since $\\tilde{\\mathcal {H}}(\\mathbf {k})$ is Hermitian and antisymmetric these components are purely imaginary.",
"Conversely, for odd $N$ , the Pfaffian is purely imaginary.",
"We define $P(\\mathbf {k}) \\equiv \\left\\lbrace \\begin{array}{ll}\\mathop {\\textrm {Pf}}\\tilde{\\mathcal {H}}(\\mathbf {k}) & \\mbox{for $N$ even,} \\\\i\\mathop {\\textrm {Pf}}\\tilde{\\mathcal {H}}(\\mathbf {k}) & \\mbox{for $N$ odd}\\end{array} \\right.$ so that $P(\\mathbf {k})$ is always real.",
"Due to $\\mathcal {C}P$ symmetry, the spectrum of the BdG Hamiltonian is symmetric.",
"We thus have $\\det \\mathcal {H}(\\mathbf {k}) = (-1)^N \\prod _{j=1}^N E_j^2(\\mathbf {k}) ,$ where $\\pm E_j(\\mathbf {k})$ are the quasiparticle energies.",
"We assume $E_j(\\mathbf {k})\\ge 0$ without loss of generality.",
"The determinant equals the square of the Pfaffian.",
"This implies that $P(\\mathbf {k}) = \\pm \\prod _{j=1}^N E_j(\\mathbf {k}) .$ At any momentum $\\mathbf {k}$ not on a node, $P(\\mathbf {k})$ is strictly positive or negative.",
"We now choose $\\Omega $ in such a way that in the normal state $P(\\mathbf {k}_\\infty )$ is positive at some momentum $\\mathbf {k}_\\infty $ far from the normal-state Fermi surface.",
"Then for not too large superconducting energy scale, the energies $E_j(\\mathbf {k}_\\infty )$ do not change sign when superconductivity is switched on and $P(\\mathbf {k}_\\infty )$ remains positive.",
"Hence, at $\\mathbf {k}_\\infty $ , the sign in Eq.",
"(REF ) is $+$ .",
"Since the Pfaffian and thus $P(\\mathbf {k})$ are smooth functions of momentum $P(\\mathbf {k})$ can only change sign at nodes.",
"Conversely, if $P(\\mathbf {k})$ is negative somewhere in the Brillouin zone there must be a surface of zeros, i.e., a BFS, separating the regions of positive and negative $P(\\mathbf {k})$ .",
"To determine under what conditions $P(\\mathbf {k})$ can actually become negative, we need to analyze Eq.",
"(REF ).",
"The eigenenergies $E_j(\\mathbf {k})$ are continuous functions and are smooth, except at zeros and potentially at crossing points.",
"If a single $E_j(\\mathbf {k})$ approaches zero linearly the smoothness of $P(\\mathbf {k})$ and the choice $E_j(\\mathbf {k})\\ge 0$ requires the explicit sign in Eq.",
"(REF ) to flip.",
"Hence, $P(\\mathbf {k})$ changes sign and there must be a closed BFS.",
"On the other hand, if two eigenenergies approach zero linearly and simultaneously, for example because they are degenerate, the explicit sign does not flip.",
"In this case, $P(\\mathbf {k})$ does not change sign at the zero but has a second-order zero there.",
"The same applies if a single eigenenergy approaches zero quadratically.",
"If the Pfaffian does not change sign the $\\mathbb {Z}_2$ topological invariant, which is the relative sign of $P(\\mathbf {k})$ [1], [2], exists but is trivial.",
"This makes BFSs unstable since for any low-symmetry momentum $\\mathbf {k}$ with $P(\\mathbf {k})=0$ , an infinitesimal change of parameters can make $P(\\mathbf {k})$ strictly positive.",
"For $s_T = -1$ , which implies even $N$ , and preserved TRS, Kramers' theorem [43], [19], [29] shows that the spectrum has twofold degeneracy for all $\\mathbf {k}$ .",
"Then, the latter case applies, $P(\\mathbf {k})$ does not change sign, and BFSs are not stable.",
"On the other hand, for $s_T = +1$ or broken TRS, there is no mechanism that leads to twofold degeneracy everywhere, the $P(\\mathbf {k})$ generically changes sign, and BFSs are stable."
],
[
"The algebra of basis matrices",
"As shown in Appendix , for $N=4$ and $(P\\mathcal {T})^2=-1$ , six basis matrices $h_0$ , ..., $h_5$ appear in the normal-state Hamiltonian.",
"One of them is (proportional to) the identity matrix, as discussed in Sec.",
".",
"We choose $h_0 = $ .",
"In this Appendix, we show that the basis matrices can always be chosen in such a way that they satisfy the generalized commutation relations $h_0 h_n = h_n h_0$ for $n = 1,2,3,4,5$ and $h_m h_n = -h_n h_m$ for $m, n = 1,2,3,4,5$ and $m\\ne n$ .",
"It is well known that (several sets of) six $4\\times 4$ matrices with these properties exist [28], [29].",
"The point here is that the basis matrices always realize this structure.",
"The matrices $h_n$ can be written as $k_n = Q^\\dagger h_n Q$ , with $Q$ unitary, where the $k_n$ satisfy Eq.",
"(REF ) with the upper sign and $\\kappa _{11}^\\dagger = \\kappa _{11}$ and $\\kappa _{12}^T = - \\kappa _{12}$ .",
"For $N=4$ , $\\kappa _{11}$ and $\\kappa _{12}$ are $2\\times 2$ matrices.",
"Then, $\\kappa _{11}$ has to be a linear combination of $\\sigma _0$ , $\\sigma _1$ , $\\sigma _2$ , $\\sigma _3$ with real coefficients and $\\kappa _{12}$ can be $\\sigma _2$ with an arbitrary complex prefactor.",
"A maximal set of linearly independent matrices is then $k_0 &= \\left(\\begin{array}{cc}\\sigma _0 & 0 \\\\ 0 & \\sigma _0\\end{array}\\right) = \\sigma _0 \\otimes \\sigma _0 , \\\\k_1 &= \\left(\\begin{array}{cc}\\sigma _1 & 0 \\\\ 0 & \\sigma _1\\end{array}\\right) = \\sigma _0 \\otimes \\sigma _1 , \\\\k_2 &= \\left(\\begin{array}{cc}\\sigma _2 & 0 \\\\ 0 & -\\sigma _2\\end{array}\\right) = \\sigma _3 \\otimes \\sigma _2 , \\\\k_3 &= \\left(\\begin{array}{cc}\\sigma _3 & 0 \\\\ 0 & \\sigma _3\\end{array}\\right) = \\sigma _0 \\otimes \\sigma _3 , \\\\k_4 &= \\left(\\begin{array}{cc}0 & \\sigma _2 \\\\ \\sigma _2 & 0\\end{array}\\right) = \\sigma _1 \\otimes \\sigma _2 , \\\\k_5 &= \\left(\\begin{array}{cc}0 & -i\\sigma _2 \\\\ i\\sigma _2 & 0\\end{array}\\right) = \\sigma _2 \\otimes \\sigma _2 .$ These matrices satisfy $k_0 k_n = k_n k_0$ for $n = 1,2,3,4,5$ and $k_m k_n = -k_n k_m$ for $m, n = 1,2,3,4,5$ and $m\\ne n$ .",
"Moreover, they are orthonormal with respect to the scalar product $\\mathop {\\textrm {Tr}}k_m k_n$ .",
"The basis matrices $h_n$ , $n=0,\\ldots ,5$ , are related to the $k_n$ by a unitary transformation.",
"Since the $k_n$ satisfy the generalized commutation relations so do the $h_n$ .",
"The upshot is that while the specific form of the basis matrices $h_n$ depends on the model, their algebra does not.",
"This result extends to general Hilbert-space dimension $N$ but the algebraic properties are more complicated for $N>4$ ."
],
[
"Pfaffian for the four-dimensional case",
"Here, we briefly discuss the analytical expression for the Pfaffian $P(\\mathbf {k})$ of the transformed BdG Hamiltonian $\\tilde{\\mathcal {H}}(\\mathbf {k})$ for the case of $s_T=-1$ , $N=4$ , and even-parity pairing.",
"The Pfaffian exists and can be chosen to be a smooth function of momentum $\\mathbf {k}$ , as shown in Appendix .",
"Then the property $P^2(\\mathbf {k}) = \\det \\mathcal {H}(\\mathbf {k})$ fixes $P(\\mathbf {k})$ up to an overall sign.",
"As discussed in Appendix , we choose this sign so that $P(\\mathbf {k})>0$ far from the normal-state Fermi surface.",
"If the superconducting energy scale is not too large, the Pfaffian is given in terms of the coefficients in Eq.",
"(REF ) as $P&(\\mathbf {k}) = \\langle c,c\\rangle ^2 + \\langle f^1,f^1\\rangle ^2+ \\langle f^2,f^2\\rangle ^2 \\nonumber \\\\&{}+ 4\\, \\big ( \\langle c,f^1\\rangle ^2 + \\langle f^1,f^2\\rangle ^2+ \\langle f^2,c\\rangle ^2 \\big ) \\nonumber \\\\&{}- 2\\, \\big ( \\langle c,c\\rangle \\, \\langle f^1,f^1\\rangle + \\langle f^1,f^1\\rangle \\, \\langle f^2,f^2\\rangle + \\langle f^2,f^2\\rangle \\, \\langle c,c\\rangle \\big ) ,$ with the Minkowski-type scalar product $\\langle A,B\\rangle \\equiv A_0 B_0 - \\sum _{n=1}^5 A_n B_n .$ This proposition is proved by evaluating $P^2(\\mathbf {k})$ and showing that it agrees with the determinant of the BdG Hamiltonian.",
"This cumbersome calculation can be simplified by realizing that the Pfaffian is invariant under simultaneous rotations of the five-vectors $\\vec{c} &= (c_1,c_2,c_3,c_4,c_5) , \\\\\\vec{f}^{\\,1} &= (f^1_1,f^1_2,f^1_3,f^1_4,f^1_5) , \\\\\\vec{f}^{\\,2} &= (f^2_1,f^2_2,f^2_3,f^2_4,f^2_5) .$ The sign of $P(\\mathbf {k})$ is also correct: The assumption of not too large superconducting energy scale means that the $f^1_n$ and $f^2_n$ for $n=0,\\ldots ,5$ are small compared to the $c_n$ far from the normal-state Fermi energy.",
"Then, at such momenta we get $P(\\mathbf {k}) \\cong \\langle c,c\\rangle ^2 > 0$ .",
"For large superconducting energy scale, the whole Brillouin zone is affected by superconductivity and we cannot choose the sign of $P(\\mathbf {k})$ by continuity from the normal state.",
"This simply means that there is no useful distinction between the inside and the outside of the BFS.",
"The conclusions of this paper remain valid, though."
]
] | 2107.01839 |
[
[
"Weinhold geometry and thermodynamics of Bardeen AdS black holes"
],
[
"Abstract Thermodynamics of Bardeen AdS black holes has attracted a great deal of attentions due to its intrinsic complications and rich phase structures.",
"However, the entropy and thermodynamic volume are incorrect in some literatures.",
"In this paper we revisit the thermodynamics of Bardeen AdS black holes and provide the correct entropy and thermodynamic volume.",
"Furthermore, thermodynamic geometries are a powerful tool to probe the microstructure of black holes.",
"Based on the Hessian matrix of black hole mass, we introduce a thermodynamic metric and give its scalar curvature in Weinhold's geometry.",
"The conformal relation between Weinhold's geometry and Ruppeiner's geometry will be changed due to the specific first law of thermodynamics for regular black holes like the Bardeen AdS black hole.",
"We also investigate the critical behaviour of phase transitions in an extended phase space, and find that the critical behaviour of the Bardeen AdS black hole coincides with that of liquid-gas systems.",
"In particular, based on the Weinhold geometry we unveil a repulsive interaction in the microstructure of the Bardeen AdS black hole under its small volume state, while it is known that only attractive interaction exists in the microstructure of the van der Waals fluid."
],
[
"Introduction",
"In the past decades, a complete statistical mechanics that describes the microstructure of black holes was still lacking although there was a wealth of work in the thermodynamics of black holes.",
"A geometric description of thermodynamic systems may provide an alternative attempt to probe the microstructure of black holes.",
"Weinhold developed [1] the mass (energy) representation by constructing a linear vector space with a full metric structure, where the vector space can be constructed directly from empirical laws of thermodynamics.",
"For each extensity $X^{\\mu }$ of a thermodynamic system, its conjugate field variable $v_{\\mu }$ is defined by $v_{\\mu }\\equiv \\partial M /\\partial X^{\\mu },$ which constitutes a thermodynamic phase space together with $X^{\\mu }$ that is required to have the properties of a vector space.",
"Thus, an abstract space having the full properties of inner product on the thermodynamic phase space is given, which implies that the thermodynamic laws can show the underlying structure of a geometric object.",
"Based on the fluctuation theory of thermodynamics, on the other hand, Ruppeiner introduced [2] the entropy representation, which shows that thermodynamic systems can be described by Riemannian manifolds.",
"Quite interesting is that there exists [3] a conformal equivalence between Weinhold'geometry and Ruppeiner's geometry.",
"In information geometry, a parameterized statistical model is considered as a Riemannian manifold in which the corresponding probability distributions can be defined.",
"In this view, Ruppeiner's geometry is one of information theories with the probability distribution [4], $\\mathcal {P}(X)\\propto \\sqrt{g^R}\\, e^{-\\frac{1}{2}g^R_{\\mu \\nu }X^{\\mu }X^{\\nu }},$ where $g^R_{\\mu \\nu }$ is the Ruppeiner metric and ${g^R}$ is the determinant of the metric.",
"Thermodynamic geometries have been extensively studied and applied [5], [6], [7], [8], [9], [10], [11], [12] to numerous systems.",
"In particular, the interpretations of thermodynamic curvatures are linked [13] to the interactions in the microstructure of ideal quantum gases.",
"In recent years, Ruppeiner's geometry has promoted substantially the connection between microstructure and thermodynamics of black holes, furthermore, such a connection has been shown [14], [15], [16], [17] to be analogous to that in the van der Waals fluid.",
"These interpretations are new attempts to extract information from the thermodynamic geometry of black holes.",
"The thermodynamic curvature does not depend [18] on the thermodynamic metric since it is an invariant for a given thermodynamic state, which has recently been verified [19], [20] again in two new phase spaces.",
"As is known, there is only an attractive interaction in the van der Waals fluid.",
"However, there exists [21] a weak repulsive interaction in the microstructure of charged AdS black holes.",
"We may argue whether the existence of a weak repulsive interaction has something to do with the singularity of charged AdS black hole spacetimes.",
"On the other hand, the current thermodynamic geometries are generally based on the first law of black hole mechanics associated with linear electrodynamics.",
"They cannot be applied directly to interpret the thermodynamic geometry of regular black holes that have no singularity.",
"We need to construct the thermodynamic geometry in the context of the nonlinear electrodynamics, which is able to give an interpretation of thermodynamic geometry for regular black holes.",
"It is challenging to generalize thermodynamic geometry to regular black holes due to the intrinsic complications in the configurations of the first law and the Lagrangian coupled with nonlinear electrodynamics.",
"In this paper, we study the Weinhold geometry of regular black holes in the context of nonlinear electrodynamics.",
"Our main task is to construct the Weinhold geometry by using the normalized scalar curvature [21] and reveal the behaviour of thermodynamic curvatures through macroscopic properties.",
"The outline of this paper is as follows.",
"In Sec.",
"we review briefly the properties of thermodynamic geometry and then give the general form of Weinhold scalar curvatures.",
"Next, in Sec.",
"we derive the entropy of Bardeen-AdS black holes in terms of the modified integration approach, and demonstrate the critical behaviour of thermodynamics in Bardeen-AdS black holes and the analogous behaviour in liquid-gas systems.",
"In Sec.",
"we compute the thermodynamic curvatures of Bardeen-AdS black holes in terms of Weinhold's geometry and Ruppeiner's geometry and make a comparison between our results and the thermodynamic curvature of the van der Waals system.",
"Finally, we give our conclusions in Sec. .",
"We start with reviewing some general properties of thermodynamic geometry theories.",
"The thermodynamic geometry links the statistical mechanics to thermodynamics, in which an appropriate line element is crucial in the equilibrium state space of a thermodynamic system.",
"In the Ruppeiner theory [2], [22], since the thermodynamic fluctuation theory originates from statistical mechanics, certain properties of thermodynamics and statistical mechanics for a thermodynamic system are encoded into a single thermodynamic manifold equipped with the line element, $ds^2_R=g^R_{\\mu \\nu }dX^\\mu dX^\\nu ,$ where the independent variables $X^\\mu $ with the Greek index, $\\mu =0,1,2,\\cdots $ , denote the extensive quantities of a thermodynamic system and $g^R_{\\mu \\nu }$ is the so-called Ruppeiner metric defined by the Hessian matrix of thermodynamic entropy, $g^R_{\\mu \\nu }=-\\frac{\\partial ^2S(X)}{\\partial X^\\mu \\partial X^ \\nu }.$ Here $S(X)$ is the entropy of the thermodynamic system with Boltzmann's constant, $k_B=1$ , and $g^R_{\\mu \\nu }$ must be a positive definite matrixThermodynamic metrics must be positive definite since the entropy has [18] a maximum value in equilibrium.",
"This is the condition to ensure thermodynamic stability.",
"However, the positive definiteness may fail [19] due to the non-independence between entropy and volume.",
"It is required [23], [24] to impose Sylvetser's criterion for a global thermodynamic stability.",
"in order to ensure thermodynamic stability.",
"Furthermore, we take note of a related metric which is defined by the Hessian matrix of black hole mass, known as the Weinhold metric [1], $g^W_{\\mu \\nu }=\\frac{\\partial ^2M(X)}{\\partial X^\\mu \\partial X^ \\nu },$ which can be obtained from the Ruppeiner metric via a transformation of fluctuation coordinates.",
"Technically, the two metrics that describe Ruppeiner's and Weinhold's geometries, respectively, are conformally related [3] to each other, $ds^2_R=\\frac{1}{T}ds^2_W,$ where $T$ is the temperature of the thermodynamic system under consideration."
],
[
"First law of black hole thermodynamics with nonlinear electrodynamics",
"For the stationary black holes in nonlinear electrodynamics, the first law of black hole thermodynamics has been established [25], [26] via a covariant approach, $dM=TdS+\\Phi dQ_e +\\Psi dQ_m, $ where $Q_e$ and $Q_m$ denote the electric charge and magnetic charge, respectively.",
"It was claimed [27] that the first law of black hole thermodynamics holds for the black holes with nonlinear electrodynamics, but the Smarr formula seems to be violated.",
"It is easy to check that Eq.",
"(REF ) is satisfied for Born-Infeld black holes but not for Bardeen black holes.",
"The reason why Eq.",
"(REF ) breaks down for Bardeen black holes has been clarified [28], where the Lagrangian should include the contribution of additional terms besides being a function of electromagnetic invariants when one tries to derive the first law of black hole thermodynamics.",
"Differing from Rasheed's formula, the first law of Bardeen black hole thermodynamics takes the form, $dM=TdS+\\Psi dq+\\mathcal {K} dq + \\Pi dM, $ which contains the extra terms,The extra terms originate from the partial derivative of action with respect to additional parameters that are introduced to give the correct Smarr's formula (see Ref.",
"[28] for the specific expression of extra terms).",
"For Bardeen black holes, $\\Pi $ takes the form of $1-r_h^3(q^2+r_h^2)^{-3/2}$ which is obviously a dimensionless constant with the value less than one when the charge is set to be a constant.",
"For more details on the first law and Smarr's formula of black hole thermodynamics in the context of nonlinear electrodynamics coupled to Einstein gravity, please refer to Refs.",
"[25], [26], [28], [29], [30], [31].",
"$\\mathcal {K} dq$ and $\\Pi dM$ , due to the variation with respect to mass and charge.",
"In an AdS spacetime, the cosmological constant is interpreted as thermodynamic pressure, $P=-\\Lambda /(8\\pi )$ .",
"When $PdV$ term is included, the first law can be rewritten to be $dM=\\frac{T}{1-\\Pi }dS+\\sum _{i}\\frac{Y_i}{1-\\Pi }dX^i, $ where the Latin index takes $1, 2, 3\\cdots $ , $X^i=(V,\\cdots )$ and $Y_i=(-P,\\cdots )$ .",
"When the notations, $X^\\mu =(S,V\\cdots )$ and $Y_\\mu =(T/(1-\\Pi ),Y_i/(1-\\Pi ))$ , are taken, Eq.",
"(REF ) becomes a compact form, $dM=Y_\\mu dX^\\mu .$ We note that the Einstein notation has been adopted in Eq.",
"(REF ) and will be used in the following contexts."
],
[
"Thermodynamic curvatures",
"Given the first law of black hole thermodynamics Eq.",
"(REF ), the thermodynamic line elements of Weinhold's geometry can be written as $ds^2_W=\\frac{1}{1-\\Pi }dTdS+\\frac{1}{1-\\Pi }dY_idX^i.$ Comparing with the line element of Ruppeiner's geometry [15], $ds^2_R=\\frac{1}{T}dTdS+\\frac{1}{T}dY_idX^i,$ we can see that there exists an conformal factor $(1-\\Pi )/T$ between the Weinhold line element Eq.",
"(REF ) and Ruppeiner line element Eq.",
"(REF ).",
"The conformal factor in Eq.",
"(REF ) is $1/T$ , now it changes to $(1-\\Pi )/T$ due to the introduction of extra terms in the first law of black hole thermodynamics.",
"When considering the phase space coordinates $(T,X^i)$ , we have $dS=\\left( \\frac{\\partial S}{\\partial T}\\right)dT+\\left( \\frac{\\partial S}{\\partial X^i}\\right)dX^i,$ $dY_j=\\left( \\frac{\\partial Y_j}{\\partial T}\\right)dT+\\left( \\frac{\\partial Y_j}{\\partial X^i}\\right)dX^i.$ Substituting Eqs.",
"(REF ) and (REF ) into Eq.",
"(REF ) and using the relation, $\\frac{\\partial S}{\\partial X^i}=-\\frac{\\partial Y_i}{\\partial T},$ we obtain the line element of Weinhold's geometry, $ds^2_W=\\frac{1}{1-\\Pi }\\frac{C_{X^i}}{T}dT^2+\\frac{1}{1-\\Pi }\\left( \\frac{\\partial Y_i}{\\partial X^j}\\right)_TdX^idX^j, $ where $C_{X^i}\\equiv T(\\partial S/\\partial T)_{X^i}$ is the heat capacity at constant $X^i$ .",
"Since $X^i$ can take anyone of extensive quantities, the metric Eq.",
"(REF ) is two dimensional.",
"In the coordinates $(T, X)$ , we obtain the general form of a scalar curvature by using its definition in the Riemannian geometry, $ \\mathbb {R}_W&=&\\frac{1-\\Pi }{2C_{X}^2(\\partial _{X}Y)^2}\\bigg \\lbrace C_{X}\\bigg [(\\partial _{X}C_{X})(\\partial _{X,X}Y)-T(\\partial _{T,X}Y)^2 \\bigg ]+(\\partial _{X}Y)\\bigg [(\\partial _{X}C_{X})^2 \\nonumber \\\\& &-T(\\partial _{T}C_{X})(\\partial _{T,X}Y)+C_{X} \\bigg (-2(\\partial _{X, X}C_{X})+\\partial _{T,X}Y+2T(\\partial _{T,T,X}Y) \\bigg ) \\bigg ] \\bigg \\rbrace ,$ where $\\partial _{T, X}Y\\equiv \\partial ^2 Y/(\\partial T \\partial X)$ as known in General Relativity.",
"We note that Eq.",
"(REF ) shares the same divergent points at $C_{X}=0$ or $\\partial _{X}Y=0$ with the scalar curvature of Ruppeiner's geometry given [15] by $ \\mathbb {R}_R&=&\\frac{1}{2C_{X}^2(\\partial _{X}Y)^2}\\bigg \\lbrace T(\\partial _{X}Y)\\bigg [(\\partial _{X}C_{X})^2+(\\partial _{T}C_{X})(\\partial _{X}Y-T(\\partial _{T,X}Y))\\bigg ] +C_{X}\\bigg [(\\partial _{X}Y)^2\\nonumber \\\\& &+T[(\\partial _{X}C_{X})(\\partial _{X,X}Y)-T(\\partial _{T,X}Y)^2]+2T(\\partial _{X}Y)(-(\\partial _{X,X}C_{X})+T(\\partial _{T,T,X}Y)) \\bigg ) \\bigg ] \\bigg \\rbrace .$ Following Refs.",
"[21], [15] for the treatment to the divergence of Ruppeiner scalar curvature, we introduce the normalized scalar curvatureSince the thermodynamic scalar curvature $\\mathbb {R}$ diverges at $C_{X}=0$ , $C_{X}$ is treated as a constant with the value of infinitely close to zero, namely, $C_{X}\\rightarrow 0^+$ .",
"Such a normalization just removes the divergence of $\\mathbb {R}$ at $C_{X}=0$ , see Ref.",
"[15] for the details, but not the divergence at $\\partial _{X}Y=0$ , which will be shown in Fig.",
"REF and Fig.",
"REF .",
"defined by $\\mathcal {R}=\\mathbb {R}\\,C_{X},$ and then reduce the Weinhold scalar curvature Eq.",
"(REF ) and Ruppeiner scalar curvature Eq.",
"(REF ) to be $ \\mathcal {R}_W=\\frac{(1-\\Pi )[\\partial _{T,X}Y-T(\\partial _{T,X}Y)^2+2T(\\partial _{T,T,X}Y)]}{2(\\partial _{X}Y)^2},$ $ \\mathcal {R}_R=\\frac{(\\partial _{X}Y)^2-T^2((\\partial _{T,X}Y)^2)+2T^2(\\partial _{X}Y)(\\partial _{T,T,X}Y)}{2(\\partial _{X}Y)^2},$ which are so-called normalized thermodynamic scalar curvatures because their divergent behaviourThere exists [22], [15] a relation between the divergence of correlation length and the divergence of thermodynamic curvature occurring at the critical point of phase transitions.",
"We shall discuss this issue in detail in section REF .",
"at the critical point of phase transitions has been removed.",
"In this section, we start with a brief review on the thermodynamic properties of Bardeen-AdS black holes and investigate the phase transition via $P-V$ criticality in an extended phase space.",
"The action of Bardeen-AdS black holes with the cosmological constant $\\Lambda $ in the four-dimensional spacetime reads [32], [33] $\\mathcal {S}=\\frac{1}{16\\pi }\\int d^4x\\sqrt{-g}\\left( R-2\\Lambda -4\\mathcal {L}(F) \\right),$ where $g$ is the determinant of the metric tensor, $R$ is the Ricci scalar, $\\Lambda $ is related to the AdS radius $l$ via the relation $\\Lambda =-3/l^2$ , and the function of electromagnetic invariant $F$ , $\\mathcal {L}(F)$ is given by $\\mathcal {L}=\\frac{M}{q^3}\\left( \\frac{\\sqrt{4q^2F}}{1+\\sqrt{4q^2F}}\\right) ^{5/2},$ where $M$ and $q$ denote mass and charge of Bardeen-AdS black holes, respectively.",
"In four dimensions, the line element of spherically symmetric Bardeen-AdS black holes takes the form, $ds^2=-f(r)dt^2+\\frac{dr^2}{f(r)}+r^2d\\Omega ^2,$ with the shape function, $f(r)=1-\\frac{2Mr^2}{(q^2+r^2)^3/2}+\\frac{r^2}{l^2},$ where $d\\Omega ^2$ is the line element of the unit two sphere.",
"The Hawking temperature of Bardeen-AdS black holes can be calculated, $T=\\frac{-2q^2+r_h^2+3r_h^4/l^2}{4\\pi r_h(q^2+r_h^2)},$ where $r_h$ stands for the horizon radius which is the solution of the equation $f(r_h)=0$ .",
"Note that we have to use the modified first law, Eq.",
"(REF ), to compute the entropy of the Bardeen black hole, which leads to $S=\\int \\frac{1-\\Pi }{T}\\,dM=\\pi r_h^2,$ where the form of $\\Pi $ has been given in footnote 2.",
"We see that the Bardeen black hole obeys the area law.",
"Moreover, one can derive the thermodynamic volume conjugated to the thermodynamic pressure, $V=\\left(\\frac{\\partial M}{\\partial P} \\right)_{S,q}= \\frac{4\\pi r_h^3}{3}\\left(1+\\frac{q^2}{r_h^2} \\right)^{3/2}.$ In the above considerations, the black hole mass has been identified with the enthalpy throughout the thermodynamic process in the extended phase space that includes $(P, V)$ .",
"With the clear representations of the pressure and volume, the heat capacity measuring the thermodynamic stability can be obtained, $C_{_V}&=&T\\left(\\frac{\\partial S}{\\partial T} \\right)_V=0,\\\\C_{_P}&=&T\\left(\\frac{\\partial S}{\\partial T} \\right)_P=\\frac{2\\pi (q^2+r_h^2)^{5/2}(-2q^2+r_h^2+8P\\pi r_h^4)}{2q^4r_h+7q^2r_h^3+(-1+24P\\pi q^2)r_h^5+8P\\pi r_h^7}.$ Unlike the van der Waals fluid, the heat capacity of Bardeen-AdS black holes at constant volume is zero, and the heat capacity at constant pressure is finite.",
"For Bardeen-AdS black holes, the equation of state $P=(T,V)$ can be derived from the Hawking temperature Eq.",
"(REF ), $P=\\frac{12q^2+\\left( \\frac{6V}{\\pi }\\right) ^{2/3} \\left( -1+2\\pi T\\sqrt{-4q^2+\\left( \\frac{6V}{\\pi }\\right) ^{2/3}} \\right) }{2\\pi \\left( -4q^2+\\left( \\frac{6V}{\\pi }\\right) ^{2/3}\\right)^{2} }.$ The behaviour of isotherms in $P-V$ diagram is shown in Fig.",
"REF .",
"Figure: P-VP-V diagram of Bardeen-AdS black holes.",
"Isotherms descend from top to bottom when the temperature is decreasing, where the black dashed curve corresponds to T>T c T>T_c, the red curve to the critical temperature T=T c T=T_c, and the two blue curves to T<T c T<T_c.Obviously, there exist oscillating parts (unstable regions) in the isotherms for the case of $T<T_c$ , which is similar to that of the van der Waals fluid [34], [35].",
"For the Bardeen-AdS black hole, the horizontal segments of the isothermsThese horizontal segments are determined by Maxwell's equal area law, see [36], [37] for the details.",
"They represent the co-existence curve of small and large black hole phases.",
"shorten when the temperature is increasing, similarly the specific volumes of gas and liquid phases approach to each other with the increasing of temperature for the van der Waals fluid.",
"Moreover, the left and right ends of the horizontal segments merge when the temperature reaches a certain limit of temperature, i.e., the critical temperature $T_c$ whose corresponding pressure is the critical pressure $P_c$ .",
"This critical point is located at the inflection point of the isotherm of $T=T_c$ , which is determined by $\\left( \\frac{\\partial P}{\\partial V}\\right)_T=0, \\qquad \\left( \\frac{\\partial ^2 P}{\\partial V^2}\\right)_T=0.$ We then have the critical thermodynamic quantities as follows, $T_{c}&=&-\\frac{\\left(\\sqrt{273}-17\\right) \\sqrt{\\frac{1}{2} \\left(\\sqrt{273}+15\\right)}}{24 \\pi q },\\\\V_{c}&=&\\frac{2}{3} \\left(19 \\sqrt{13}+15 \\sqrt{21}\\right) \\pi q^3,\\\\P_{c}&=&\\frac{\\sqrt{273}+27}{12 \\left(\\sqrt{273}+15\\right)^2 \\pi q^2 }.$ The small and large black hole phases merge [36], [37] at the critical point.",
"The equation of state can be rewritten to the reduced form, $\\tilde{P}=-\\frac{\\left(219+13 \\sqrt{273}\\right) \\left[-18+ \\left(51+3 \\sqrt{273}-4 \\sqrt{15+\\sqrt{273}} \\tilde{T} \\sqrt{-2+\\left(\\sqrt{273}+17\\right) \\tilde{V}^{2/3}}\\right)\\tilde{V}^{2/3}\\right]}{19 \\left(-2+\\left(\\sqrt{273}+17\\right) \\tilde{V}^{2/3}\\right)^2},$ where the dimensionless variables are defined by $\\tilde{P}\\equiv \\frac{P}{P_{c}},\\qquad \\tilde{V}\\equiv \\frac{V}{V_c},\\qquad \\tilde{T}\\equiv \\frac{T}{T_c}.$ We note that the charge does not appear manifestly in the reduced equation of state Eq.",
"(REF ).",
"It is crucial to give such an equation of state for us to compute the thermodynamic curvatures of Bardeen-AdS black holes in section ."
],
[
"Thermodynamic Geometry of Bardeen-AdS black holes",
"Now we turn to the thermodynamic geometry of Bardeen-AdS black holes which is a powerful tool to probe [38], [39], [40], [41], [42] the microstructure of black holes in an extended phase space.",
"We shall present the results in terms of the Ruppeiner's geometry and Weinhold's geometry which are constructed in the thermodynamic state space with the temperature and volume fluctuation coordinates, $(T,V)$ ."
],
[
"Ruppeiner's geometry",
"We consider the line element of Ruppeiner geometry in fluctuation coordinates $(T, V)$ [15], $ds^2_R=\\frac{C_{_V}}{T^2}dT^2-\\frac{1}{T}\\left( \\frac{\\partial P}{\\partial V}\\right)_TdV^2,$ where the heat capacity vanishes at constant volume, i.e., $C_{_V}=T(\\partial S/\\partial T)_V=0$ , which leads to a degenerate metric.",
"Nonetheless, we can deal with this issue by introducing [15] the normalized thermodynamic curvature.",
"Using Eq.",
"(REF ), we can obtain the normalized Ruppeiner thermodynamic scalar curvature, $\\mathcal {R}_R= \\bigg \\lbrace -6 \\bigg [2 \\sqrt{\\sqrt{273}+15} \\tilde{T} \\bigg (-7 (17 \\sqrt{273}+281) \\tilde{V}^{2/3}-4 (285 \\sqrt{273}+4709) \\tilde{V}^{4/3}+(4777 \\sqrt{273}+78929) \\tilde{V}^2\\nonumber \\\\+10 (\\sqrt{273}+17)\\bigg ) \\sqrt{-2+(\\sqrt{273}+17) \\tilde{V}^{2/3}}-105 (285 \\sqrt{273}+4709) \\tilde{V}^{2/3}+33 (4777 \\sqrt{273}+78929) \\tilde{V}^{4/3}\\nonumber \\\\-3 (80069 \\sqrt{273}+1322957) \\tilde{V}^2+75 (17 \\sqrt{273}+281)\\bigg ]\\bigg \\rbrace \\bigg / \\bigg \\lbrace 3 \\sqrt{(\\sqrt{273}+17) \\tilde{V}^{2/3}-2} \\bigg ((17 \\sqrt{273}+281) \\tilde{V}^{2/3}\\nonumber \\\\-5 (\\sqrt{273}+17)\\bigg )-2 \\sqrt{\\sqrt{273}+15} \\tilde{T} \\left((\\sqrt{273}+17) \\tilde{V}^{2/3}+(17 \\sqrt{273}+281) \\tilde{V}^{4/3}-4\\right)\\bigg \\rbrace ^2,\\nonumber \\\\ $ which is obviously independent of the charge of black holes when the reduced thermodynamic coordinates are adopted.",
"However, we note that it is a lengthy and cumbersome formula.",
"This is due in large part to the intrinsic complications of the Bardeen black hole in the contexts of its coupling with nonlinear electrodynamics and its rich thermodynamic phase structures [26].",
"We plot the behaviour of normalized Ruppeiner thermodynamic curvature in Fig.",
"REF .",
"We note that the normalized Ruppeiner curvature changes sign at $V=0.16V_c$ (round to the nearest hundredth), namely, it is positive for $V<0.16V_c$ but negative for $V>0.16V_c$ .",
"In general, we point out that the normalized Ruppeiner curvature of Bardeen-AdS black holes has a very similar microstructure to that of van der Waals fluids [15].",
"Figure: Ruppeiner thermodynamic curvature of Bardeen-AdS black holes.",
"This picture shows the characteristic behaviour of the Ruppeiner thermodynamic curvature as a function of the reduced temperature and volume."
],
[
"Weinhold's geometry",
"In $(T,V)$ fluctuation coordinates, the line element Eq.",
"(REF ) of Weinhold's geometry reads $ds^2_W=\\frac{1}{1-\\Pi }\\frac{C_{_V}}{T}dT^2+\\frac{1}{1-\\Pi }\\left( \\frac{\\partial P}{\\partial V}\\right)_TdV^2,$ where $C_{_V}=0$ still leads to a degenerate metric which can be cured by the normalization [15].",
"We note that $\\Pi $ satisfies $0<\\Pi <1$ and so it changes only the quantity rather than the sign of thermodynamic curvature.",
"This means that a repulsive interaction cannot be reversed to an attractive interaction by the presence of $\\Pi $ , and vice versa.",
"Without loss of generality, we set the dimensionless constant $\\Pi =0$ .",
"Using formula Eq.",
"(REF ), we obtain the normalized Weinhold thermodynamic scalar curvature straightforwardly, $\\mathcal {R}_W=\\bigg \\lbrace \\bigg ((\\sqrt{273}+17) \\tilde{V}^{2/3}-2\\bigg )^7 \\bigg ((\\sqrt{273}+17) \\tilde{V}^{2/3}+(17 \\sqrt{273}+281) \\tilde{V}^{4/3}-4\\bigg ) \\bigg [\\frac{-57 \\sqrt{\\sqrt{273}+15} \\@root 3 \\of {\\tilde{V}}}{[(\\sqrt{273}+17) \\tilde{V}^{2/3}-2]^{7/2}}\\nonumber \\\\-\\frac{8 (\\sqrt{273}+15) (13 \\sqrt{273}+219) \\tilde{T} [(\\sqrt{273}+17) \\tilde{V}^{2/3}+(17 \\sqrt{273}+281) \\tilde{V}^{4/3}-4]}{[(\\sqrt{273}+17) \\tilde{V}^{2/3}-2]^7} \\bigg ]\\bigg \\rbrace \\bigg /\\bigg \\lbrace 4 (13 \\sqrt{273}+219) \\nonumber \\\\ \\times \\bigg [3 \\sqrt{(\\sqrt{273}+17) \\tilde{V}^{2/3}-2} [(17 \\sqrt{273}+281) \\tilde{V}^{2/3}-5 (\\sqrt{273}+17)]-2 \\sqrt{\\sqrt{273}+15} T [(\\sqrt{273}+17) \\tilde{V}^{2/3}\\nonumber \\\\+(17 \\sqrt{273}+281) \\tilde{V}^{4/3}-4]\\bigg ]^2\\bigg \\rbrace .\\nonumber \\\\ $ We plot the behaviour of normalized Weinhold thermodynamic curvature in Fig.",
"REF .",
"Figure: Weinhold thermodynamic curvature of Bardeen-AdS black holes.",
"This picture shows the characteristic behaviour of the Weinhold thermodynamic curvature as a function of the reduced temperature and volume.From the two expressions in Eqs.",
"(REF ) and (REF ), it is obvious that the normalized thermodynamic scalar curvatures in the two geometry theories are different.",
"There is, however, a qualitative similarity between the structures of Ruppeiner's geometry and Weinhold's geometry as displayed in Fig.",
"REF and Fig.",
"REF .",
"The normalized thermodynamic curvatures diverge to negative infinity in very low temperatures.",
"In other words, for both the Weinhold's geometry and Ruppeiner's geometry there exists a strongly attractive interaction in the microstructure of Bardeen-AdS black holes in this region of temperature, which can be explicitly seen in the two figures.",
"Furthermore, it is worth noting that the normalized Weinhold curvature is negative and does not change sign, which is due to the existence of conformal relation between Weinhold's and Ruppeiner's geometries.",
"According to the microscopic interpretation of interactions [13], the positive and negative curvatures imply the repulsive and attractive interactions, respectively.",
"Hence, for Weinhold's geometry there are only attractive interactions but no repulsive interactions in the microstructure of Bardeen-AdS black holes, which coincides with the interacting behaviour in the van der Waals fluid."
],
[
"Critical phenomena",
"In order to get a deep understanding of the normalized thermodynamic curvatures of Bardeen-AdS black holes, we shall focus on the critical phenomena in which the Bardeen-AdS black hole has a markedly different behaviour from that of the charged AdS black holes with spacetime singularities, such as the Reissner-Nordström AdS and Gauss-Bonnet AdS black holes [16], [43], [14].",
"The normalized Ruppeiner thermodynamic curvature Eq.",
"(REF ) diverges and vanishes along the following curves on the $(V/V_c, T/T_c)$ plane, respectively, $\\tilde{T}|_{\\mathcal {R}_R\\rightarrow \\infty }&=& \\frac{3 \\sqrt{\\left(\\sqrt{273}+17\\right) \\tilde{V}^{2/3}-2} \\left(\\left(17 \\sqrt{273}+281\\right) \\tilde{V}^{2/3}-5 \\left(\\sqrt{273}+17\\right)\\right)}{2 \\sqrt{\\sqrt{273}+15} \\left(\\left(\\sqrt{273}+17\\right) \\tilde{V}^{2/3}+\\left(17 \\sqrt{273}+281\\right) \\tilde{V}^{4/3}-4\\right)},\\\\\\tilde{T}|_{\\mathcal {R}_R=0}&=&3 \\bigg [35 (285 \\sqrt{273}+4709) \\tilde{V}^{2/3}-11 (4777 \\sqrt{273}+78929) \\tilde{V}^{4/3}+(80069 \\sqrt{273}+1322957) \\tilde{V}^2\\nonumber \\\\& & -25 (17 \\sqrt{273}+281)\\bigg ]\\bigg /\\bigg [2 \\sqrt{\\sqrt{273}+15} \\sqrt{(\\sqrt{273}+17) \\tilde{V}^{2/3}-2}\\, \\bigg (-7 (17 \\sqrt{273}+281) \\tilde{V}^{2/3}\\nonumber \\\\& &-4 (285 \\sqrt{273}+4709) \\tilde{V}^{4/3}+(4777 \\sqrt{273}+78929) \\tilde{V}^2+10 (\\sqrt{273}+17)\\bigg )\\bigg ],$ which is plotted in Fig.",
"REF .",
"Figure: Critical behaviour of thermodynamic curvatures.",
"The upper (red) curve stands for the diverging case of ℛ R \\mathcal {R}_R and ℛ W \\mathcal {R}_W.",
"The lower (blue) curve corresponds to the vanishing case of ℛ R \\mathcal {R}_R only.From Eq.",
"(REF ) we find that the Weinhold curvature shares the same diverging curve as the Ruppeiner curvature, i.e., $\\tilde{T}|_{\\mathcal {R}_W\\rightarrow \\infty }=\\tilde{T}|_{\\mathcal {R}_R\\rightarrow \\infty }$ .",
"This happens due to the existence of conformal relation between the Weinhold's geometry and Ruppeiner's geometry.",
"However, the Weinhold curvature does not vanish at any finite positive temperature, which is quite different from the Ruppeiner curvature.",
"For Weinhold's geometry and Ruppeiner's geometry the normalized thermodynamic curvatures will diverge at the critical point of a phase transition as shown in Fig.",
"REF .",
"We can easily see that this divergent point is indeed located at the maximum of the red curve, where the temperature and volume just take their critical values, i.e., $T=T_c$ and $V=V_c$ .",
"This phenomenon was also noticed [44] for the charged AdS black holes with spacetime singularities, where the divergent points of specific heat match those of thermodynamic curvature.",
"Furthermore, a positive normalized thermodynamic curvature was obtained [15], [16] in the small volume phase of the charged AdS black holes with spacetime singularities, namely, a weak repulsive interaction appears in the microstructure, while it does not in the van der Waals fluid.",
"Interestingly, for the Bardeen-AdS black hole that has no spacetime singularities, the temperature converges to zero as the volume decreases, see Fig.",
"REF .",
"Such a behaviour has been observed [15] in the van der Waals fluid.",
"Hence, the Bardeen-AdS black hole has the critical phenomena similar to those of van der Waals fluid."
],
[
"Conclusions",
"In the present work, we revisit the thermodynamics and investigate the thermodynamic geometry for Bardeen black holes in the anti-de Sitter spacetime.",
"We find that the entropy of Bardeen-AdS black holes still obeys the area law if the modified first law of thermodynamics is considered.",
"We derive the normalized scalar curvature in the Weinhold's geometry and find that it has the same divergent behaviour as the Ruppeiner scalar curvature at the critical point of a phase transition, but it has no vanishing behaviour at any finite positive temperature while the Ruppeiner scalar curvature has.",
"Moreover, we notice that the additional parameter $\\Pi $ appeared in the conformal factor of Weinhold's geometry does not reverse the interaction in the microstructure of Bardeen-AdS black holes.",
"When compared to the charged AdS black holes with spacetime singularities, the Bardeen-AdS black hole as a regular black hole has a rather different behaviour, that is, it has no weak repulsive interaction in the microstructure.",
"When compared to the van der Waals fluid, the Bardeen-AdS black hole has a critical behaviour resembling the liquid-gas phase transition and both its Weinhold scalar curvature and Ruppeiner scalar curvature have a very similar microstructure to that of the van der Waals fluid.",
"Our results seem to justify the mathematical analogy of the Bardeen-AdS black hole with the van der Waals fluid.",
"Whether the vanishing of a weak repulsive interaction in the Bardeen-AdS black hole is universal or not in the other regular black holes, we shall deal with this issue by analyzing their thermodynamic geometries in detail."
],
[
"Acknowledgements",
"The authors would like to thank R.-G Cai and T. Vetsov for their helpful comments and disscussions.",
"This work was supported in part by the National Natural Science Foundation of China under Grant No.",
"11675081."
]
] | 2107.01866 |
[
[
"A theoretical analysis of one-dimensional discrete generation ensemble\n Kalman particle filters"
],
[
"Abstract Despite the widespread usage of discrete generation Ensemble Kalman particle filtering methodology to solve nonlinear and high dimensional filtering and inverse problems, little is known about their mathematical foundations.",
"As genetic-type particle filters (a.k.a.",
"sequential Monte Carlo), this ensemble-type methodology can also be interpreted as mean-field particle approximations of the Kalman-Bucy filtering equation.",
"In contrast with conventional mean-field type interacting particle methods equipped with a globally Lipschitz interacting drift-type function, Ensemble Kalman filters depend on a nonlinear and quadratic-type interaction function defined in terms of the sample covariance of the particles.",
"Most of the literature in applied mathematics and computer science on these sophisticated interacting particle methods amounts to designing different classes of useable observer-type particle methods.",
"These methods are based on a variety of inconsistent but judicious ensemble auxiliary transformations or include additional inflation/localisationtype algorithmic innovations, in order to avoid the inherent time-degeneracy of an insufficient particle ensemble size when solving a filtering problem with an unstable signal.",
"To the best of our knowledge, the first and the only rigorous mathematical analysis of these sophisticated discrete generation particle filters is developed in the pioneering articles by Le Gland-Monbet-Tran and by Mandel-Cobb-Beezley, which were published in the early 2010s.",
"Nevertheless, besides the fact that these studies prove the asymptotic consistency of the Ensemble Kalman filter, they provide exceedingly pessimistic meanerror estimates that grow exponentially fast with respect to the time horizon, even for linear Gaussian filtering problems with stable one dimensional signals.",
"In the present article we develop a novel self-contained and complete stochastic perturbation analysis of the fluctuations, the stability, and the long-time performance of these discrete generation ensemble Kalman particle filters, including time-uniform and non-asymptotic mean-error estimates that apply to possibly unstable signals.",
"To the best of our knowledge, these are the first results of this type in the literature on discrete generation particle filters, including the class of genetic-type particle filters and discrete generation ensemble Kalman filters.",
"The stochastic Riccati difference equations considered in this work are also of interest in their own right, as a prototype of a new class of stochastic rational difference equation."
],
[
"Introduction",
"The Ensemble Kalman filter (abbreviated EnKF) is a class of interacting particle system methodologies for solving nonlinear filtering and inverse problems.",
"They were introduced by Evensen in the seminal article [44] published in 1994, see also [45], [46] for a more recent overview.",
"In the last three decades the EnKF methodology has become one of the most used numerical techniques for solving high dimensional forecasting and data assimilation problems in a variety of applications including ocean and atmosphere sciences [1], [67], [68], [58], [73], fluid mechanics [14], [74], [78], image inverse problems [15], weather forecasting [3], [4], [30], [49], environmental and ecological statistics [43], [52] and oil reservoir simulations [47], [72], [82], [83], [94].",
"To connect these EnKF techniques with particle filtering methodology for solving high dimensional problems arising in fluid mechanics, we also refer to [59].",
"In contrast with genetic-type particle filters (a.k.a.",
"Sequential Monte Carlo, often abbreviated SMC), the EnKF is defined by a system of particles evolving as the signal in some state space with an interaction function that depends on the sample covariance matrices of the system.",
"For a detailed mathematical description of particle filters and ensemble Kalman filter methodologies in both continuous and discrete time settings we refer, for instance, to the book [39] and the references therein.",
"See also section REF in the present article dedicated to nonlinear Kalman-Bucy type Markov chains and their mean-field particle interpretations.",
"There exists a vast literature in data assimilation theory dedicated to the numerical analysis of the EnKF view as a useable observer-type algorithm with a very small sample size and equipped with a variety of judicious ensemble square-root type transformations, as well as inflation/localisation-type algorithmic innovations to control the unstable directions of the signal, see for instance [5], [6], [60], [61], [92], [93], as well as [21], [23] and the references therein.",
"From this point of view, the EnKF is no longer interpreted as a true approximation of the optimal filter but as an observer.",
"Here, the terminology “well-posedness\" is not understood as a traditional mathematical consistency-type property of some Monte Carlo statistical estimate but as a lack of divergence of the algorithm with respect to the time horizon.",
"Despite the widespread use of EnKF techniques to solve nonlinear and high dimensional filtering and inverse problems, little is known about their theoretical performance and their mathematical foundations.",
"The first rigorous mathematical analysis of discrete generation EnKF appeared in 2011 in the independent pioneering works of Le Gland, Monbet and Tran [48], and Mandel, Cobb and Beezley [70].",
"These two seminal articles provide mean error estimates for discrete generation EnKF and show that the EnKF converges towards the Kalman filter for linear-Gaussian problems as the number of samples, say $N$ , tends to infinity.",
"The article [48] also shows that the EnKF doesn't converge to the optimal filter for nonlinear or non-Gaussian filtering problems.",
"The consistent-type EnKF methodology for linear-Gaussian models can be extended to nonlinear filtering problems using the new class feedback particle filter methodology introduced by Mehta and Meyn and their co-authors in the series of seminal articles [84], [85], [86], [87], [88].",
"To obtain a consistent EnKF for nonlinear filtering problems one needs to solve a Poisson-type equation at every time step, which often requires an additional level of approximation.",
"The theoretical analysis of this type of sophisticated nonlinear EnKF is not discussed in the present article but we refer the reader to [88], [89], [90] for the analysis of its performance and the convergence.",
"Besides these fundamental theoretical advances in the understanding of EnKF filters in linear-Gaussian models, the non-asymptotic analysis developed in the aforementioned articles yields exceedingly pessimistic estimates that grow exponentially fast with respect to the time horizon.",
"Note that for a time horizon $t=39$ an innocent exponential bound of the form $55\\times e^{5 t}/N\\ge 10^{86}/N$ will require a sample size $N$ that is 10 times larger than the number $10^{86}$ of elementary particles of matter in the visible universe in order to obtain a poor $10\\%$ accuracy after 39 runs.",
"The mathematical analysis of the continuous-time version of the EnKF methodology has started more recently in [37], [38], followed by the series of articles [18], [20], [22], see also review article [23].",
"Some extensions to nonlinear filtering problems are also developed in the series of more recent articles [62], [63], [64], [65], [75].",
"In contrast with the feedback particle filter methodology discussed above, these articles are concerned with a class of EnKF particle filters that are only consistent for linear-Gaussian models.",
"As expected, for nonlinear or non-Gaussian filtering problems, none of these EnKF converge to the optimal filter as the number of particles tends to infinity.",
"From a probabilistic viewpoint, in the linear-Gaussian case, continuous time EnKF are represented by a Kalman-Bucy-type diffusion coupled with a stochastic Riccati matrix nonlinear diffusion equation (see for instance Theorem 3.1 in [37]).",
"In the series of articles [37], [38], [18], [20], [22], the authors present a refined stochastic stability analysis of these rather sophisticated diffusion-type perturbation models, including central limit theorems, as well as uniform bias and mean error estimates with respect to the time horizon for several classes of consistent EnKF stochastic models.",
"To describe briefly these results, as underlined in the review article [23], we emphasise that the continuous-time EnKF methodology (for linear-Gaussian models) may be broadly divided into three different classes of probabilistic models, according to the level of fluctuation added via sampling noise needed to ensure that the EnKF sample mean and covariance are consistent in the linear-Gaussian setting.",
"By chronological order of appearance in the literature, these three different classes of continuous-time EnKF are briefly discussed below.",
"The continuous time version of the EnKF discussed in section REF in the present article coincides with the so-called “Vanilla” discrete time original form of the EnKF [44], [30].",
"This first class of continuous-time EnKF exhibits the most fluctuations due to sampling both signal and observation noises (see for instance the evolution equation (REF )).",
"The second class of EnKF is a continuous-time version of the Sakov and Oke square root filter (a.k.a.",
"“deterministic EnKF”) introduced in [80], [81], see also [76], [13].",
"Unfortunately, this class of “deterministic EnKF” is only consistent for continuous time models.",
"In the discrete time situation, it fails to converge to the optimal filter as the number of particles tends to infinity, even in linear-Gaussian settings, see for instance [23] and references therein.",
"This class of discrete generation EnKF is not discussed in the present article.",
"Finally, the third class consists of purely deterministic transport-inspired EnKF with randomness coming only from the initial conditions.",
"By construction, this class of filters can be analysed directly using the evolution semigroup of the optimal filter and that of the Riccati matrix equations.",
"Further details on this subject can be found in the review article [23] on continuous time EnKF particle filters and in the articles [16], [17] dedicated to the stability of Kalman-Bucy diffusion processes and related Riccati matrix differential equations.",
"The stochastic analysis of the Riccati matrix diffusion associated with the sample covariance of these three classes of continuous-time EnKF is rather well understood for multivariate models.",
"The articles [37], [38], [18], [20], [22] present several multivariate fluctuation theorems for Riccati matrix diffusions, as well as a refined non-asymptotic analysis, including several time-uniform mean error estimates.",
"The extension of these uniform results to the Vanilla EnKF is developed in [37], [38] but only for stable and ergodic signals.",
"The stability analysis and the long-time performance of Vanilla EnKF with possibly unstable and multivariate transient signals remains partially understood.",
"To understand the difficulties that arise these problems it is worth mentioning that even for elementary one-dimensional problems, the stochastic Riccati diffusions describing the random evolution of the sample variances of the EnKF may exhibit surprisingly heavy tailed or Gaussian tailed invariant distributions, depending on the class of EnKF one uses as well as the number of samples [22], [23].",
"In this setting, some raw moments of the sample variances of the Vanilla continuous time EnKF for one-dimensional filtering problems are infinite, for any chosen finite sample size lower than some critical value.",
"To the best of our knowledge, the question of finding uniform mean error estimates for the Vanilla EnKF sample means remains an important open research question for both discrete generation and continuous-time multivariate models.",
"The only work in this direction appears to be the recent article [22].",
"This article is dedicated to the long time performance and the stability properties of the stochastic Riccati diffusion associated with the three different classes of continuous-time EnKF filters discussed above for one-dimensional and linear-Gaussian filtering problems with possibly unstable and transient signals.",
"The extension of these results to multivariate models remains open.",
"The main advantage in working with continuous-time models comes from the fact that the sample mean and the sample covariance of the EnKF filters satisfy a coupled nonlinear diffusion equation which can be handled using conventional stochastic analysis.",
"Note however that in the multivariate case, the sample covariance matrices satisfy a stochastic Riccati equation involving matrix valued martingales which require one to develop an appropriate and more sophisticated stochastic analysis in matrix spaces [18], [20].",
"Physical systems and most of the filtering problems arising in signal processing and tracking are typically defined by continuous time models.",
"Nevertheless, for obvious reasons, any numerical integration of these models including all known filtering sensors are defined by discrete-time models.",
"As a result, the stochastic analysis used for the continuous-time EnKF is no longer applicable without adding an additional level of approximation.",
"The analysis of discrete generation EnKF is more delicate.",
"To handle the structure and the statistical difficulty of discrete time models, new stochastic tools need to be developed.",
"For instance, in contrast with continuous-time models, discrete generation EnKF are not defined by a single coupled diffusion process but in terms of coupled two-step prediction-updating process (a.k.a.",
"forecasting-analysis steps in EnKF and data assimilation literature).",
"Furthermore, the Gaussian-nature of the diffusion models arising in the analysis of continuous-time EnKF theory is also lost.",
"Another inherent difficulty is that discrete generation EnKF involve more sophisticated non-central $\\chi $ -square nonlinear fluctuations (cf.",
"for instance Theorem REF and Corollary REF ).",
"We expect the multivariate version of the EnKF evolution to involve a similar two-step prediction-updating transition involving sophisticated non-central Wishart matrix local fluctuations.",
"Up to some additional technicalities and at the cost of multi-index and tensor theoretical notation, we believe that the analysis of multivariate and discrete generation stochastic Riccati equations can be conducted by extending the theory of discrete generation one-dimensional models developed in the present article in the spirit of the stochastic analysis of continuous time EnKF models developed in [18], [22].",
"As for continuous time models, we also believe that one cannot expect to directly deduce uniform state estimates from those at the level of the sample covariance matrices.",
"Due to these reasons, in the present article we have chosen to concentrate our study to one-dimensional models.",
"We intend to extend these results to more sophisticated multivariate models in a subsequent article.",
"The main objective of this article is to develop a novel and complete stochastic analysis on the fluctuations and the long time performance of discrete generation EnKF particle filters that applies to filtering problems involving possibly unstable signals.",
"Our main contributions are listed succinctly below.",
"For more precise statements of the results, we refer the reader to the series of theorems stated in section .",
"$\\bullet $ Up to some orthogonal transformations on the sample space, our first main result, Theorem REF , states that the sample variances of the EnKF satisfy an autonomous stochastic Riccati-type evolution equation, which is independent of the sequence of observations, and that the EnKF sample means evolve as a Kalman filter driven by sample variances.",
"In addition, the local perturbations of the EnKF and the sample variance Riccati equations are independent.",
"An alternative Markov chain realisation of the stochastic Riccati difference equation is also discussed in Corollary REF .",
"Theorem REF is an extended version of the stochastic perturbation theorem presented in [37].",
"The continuous time version of the sample variance equations stated in Theorem REF also coincides with the one-dimensional stochastic Riccati equation discussed in [22], see also [18] for the extended matrix version of these models in the context of multivariate filtering problems.",
"$\\bullet $ Section REF is concerned with the stability properties of the optimal filter and the stochastic Riccati equations describing the evolution of the sample variances of the EnKF for possibly unstable signals.",
"In this context, we show that the discrete generation stochastic Riccati equations converge exponentially fast to a single invariant measure.",
"The contraction theorem, Theorem REF , and the exponential semigroup estimates (REF ) stated in Theorem REF are partial extensions to discrete generation models of Theorem 2.2 in the article [22], dedicated to the stability of one dimensional stochastic Riccati equations.",
"In Theorem REF we also show that the sample variances and the difference between the sample means and the true (possibly transient) signal forms a Markov chain that converges exponentially fast to a unique invariant measure.",
"This seems to be the first result of this kind in the field of interacting particle-type filters, including EnKF and genetic-type particle filters.",
"$\\bullet $ Our third main result is concerned with uniform mean-error and bias estimates with respect to the time horizon for both the sample variances and the sample means delivered by the EnKF.",
"Theorem REF shows that the sample variances are uniformly close to the true optimal variances for all times, even when the signal and the ensemble are transient.",
"The EnKF sample mean uniform accuracy for possibly unstable signals is discussed in theorem REF .",
"This result ensures that the sample means are uniformly close to the true optimal filter at any time horizon.",
"The continuous time version of these results are discussed in Corollary 2.4 in [22].",
"$\\bullet $ Last, but not least, we present a novel fluctuation analysis of EnKF filters.",
"Theorems REF and REF present new multivariate central limit theorems for the sample mean and the sample variance processes.",
"These fluctuation results, at the level of the processes, are an extended multivariate version of central limit theorem presented in [22] in the context of continuous time models."
],
[
"The Kalman filter",
"Consider a one dimensional, time homogeneous linear-Gaussian filtering model of the following form $\\left\\lbrace \\begin{array}{rcl}Y_n&=&C\\,X_n+D\\,V_n\\\\X_{n+1}&=&A\\,X_{n}+B\\,W_{n+1},\\end{array}\\right.\\qquad n \\ge 0.$ In the above display, $(V_n,W_{n+1})$ is a sequence of 2-dimensional Gaussian centered random variables with unit variance, the initial condition of the signal $X_0$ is a Gaussian random variable with mean and variance denoted by $(\\widehat{X}^-_0,P_0)$ (independent of $(V_n,W_{n+1})$ ), and $(A,B,C,D)$ are some non-zero parameters.",
"The latter condition can be interpreted as a controllability and observability condition for one-dimensional filtering problems.",
"Let $ {\\cal Y }_n=\\sigma \\left(Y_k,~k\\le n\\right)$ be the filtration generated by the observation process.",
"The optimal filter is defined by the conditional distribution $\\widehat{\\eta }_n$ of the signal state $X_n$ given $ {\\cal Y }_n$ .",
"The distribution $\\widehat{\\eta }_n$ is a Gaussian distribution with conditional mean and variance $\\widehat{X}_n:=\\mathbb {E}(X_n~|~ {\\cal Y }_n)\\quad \\mbox{\\rm and}\\quad \\widehat{P}_n:=\\mathbb {E}\\left(\\left(X_n-\\widehat{X}_n\\right)^{2}\\right).$ The optimal one-step predictor is defined by the conditional distribution $\\eta _{n}$ of the signal state $X_n$ given $ {\\cal Y }_{n-1}$ .",
"The distribution $\\eta _n$ is also a Gaussian distribution with conditional mean and variance $\\widehat{X}^-_n:=\\mathbb {E}(X_n~|~ {\\cal Y }_{n-1})\\quad \\mbox{\\rm and}\\quad P_n:=\\mathbb {E}\\left(\\left(X_{n}-\\widehat{X}_{n}^{-}\\right)^{2}\\right).$ The updating-prediction steps are described by the following synthetic diagram $(\\widehat{X}_n^-,P_n)\\longrightarrow (\\widehat{X}_{n},\\widehat{P}_{n})\\longrightarrow (\\widehat{X}^-_{n+1},P_{n+1}),$ with the well-known updating-prediction Kalman filter equations: $\\left\\lbrace \\begin{array}{rcl}\\widehat{X}_n&=&\\widehat{X}_n^-+G_n~\\left(Y_n-C\\widehat{X}_n^-\\right) \\\\\\widehat{P}_n&=&(1-G_n C)P_n\\end{array}\\right.\\quad \\mbox{\\rm and} \\quad \\left\\lbrace \\begin{array}{rclcrcl}\\widehat{X}_{n+1}^-&=&A \\widehat{X}_{n}\\\\P_{n+1}&=& A^2\\widehat{P}_{n}+R,\\end{array}\\right.$ where $R = B^2$ .",
"In the above display, $G_n$ stands for the so-called gain parameter defined by the formula $G_n:=CP_n/(C^2P_n+D^2)\\Longrightarrow 1-G_nC=1/(1+SP_n)\\quad \\mbox{\\rm with}\\quad S:=(C/D)^2.$ The evolution equations of the variance parameters $(P_n,\\widehat{P}_n)$ satisfy the Riccati rational difference equations $P_{n+1}=\\phi \\left(P_n\\right):=\\frac{aP_n+b}{cP_n+d}\\quad \\mbox{\\rm and}\\quad \\widehat{P}_{n+1}=\\widehat{\\phi }\\left(\\widehat{P}_n\\right):=\\frac{\\widehat{a}\\,\\widehat{P}_n+\\widehat{b}}{\\widehat{c}\\,\\widehat{P}_n+\\widehat{d}}$ with the parameters $\\begin{array}{l}(a,b,c,d):=\\left(A^2+RS,R,S,1\\right)\\quad \\mbox{\\rm and}\\quad (\\widehat{a},\\widehat{b},\\widehat{c},\\widehat{d})=(A^2,R,A^2S,1+SR)\\\\\\\\\\Longrightarrow ad-bc=A^2=\\widehat{a}\\,\\widehat{d}-\\widehat{b}\\,\\widehat{c}>0.\\end{array}$ One-dimensional rational difference equations are rather well understood, see for instance [26].",
"Section REF also provides a refined stability analysis and several comparison properties of rational difference equations including non-asymptotic Taylor expansions any order."
],
[
"Ensemble Kalman filters",
"Let $({X}_0, {W}_n,{V}_n)$ be independent copies of $(X_0, W_n,V_n)$ .",
"Consider the nonlinear Markov chain starting at ${X}_0$ and defined sequentially for any $n\\ge 0$ by the updating-correction formulae $\\left\\lbrace \\begin{array}{rcl}\\widehat{{X}}_{n}&=&{X}_n+{G}_{\\eta _n}~(Y_n-(C\\,{X}_n+D\\,{V}_n))\\quad \\mbox{\\rm with}\\quad {G}_{\\eta _n}~:=C{Q}_{\\eta _n}/(C^2{Q}_{\\eta _n}+D^2)\\\\&&\\\\{X}_{n+1}&=& A\\,{X}_{n}+B\\,{W}_{n+1}.\\\\\\end{array}\\right.$ In the above display, ${Q}_{\\eta _n}$ stands for the variance parameter ${Q}_{\\eta _n}:=\\int \\left(x- \\int z~ \\eta _n(dz)\\right)^2\\eta _n(dx)\\quad \\mbox{\\rm indexed by}\\quad \\eta _n=\\mbox{\\rm Law}({X}_n~|~ {\\cal Y }_{n-1}).$ Using a simple induction argument, it is straightforward to show that $\\eta _n= {\\cal N }\\left(\\widehat{X}^-_n,P_n\\right)\\quad \\mbox{\\rm and}\\quad \\widehat{\\eta }_n= {\\cal N }\\left(\\widehat{X}_n,\\widehat{P}_n\\right)=\\quad \\mbox{\\rm Law}(\\widehat{{X}}_n~|~ {\\cal Y }_{n}).$ Also observe that the above stochastic model can be interpreted as a Markov chain with a two-step updating-prediction transition described by the following synthetic diagram ${X}_n\\longrightarrow \\widehat{{X}}_{n}\\longrightarrow {X}_{n+1}.$ Note that the updating transition ${X}_n\\longrightarrow \\widehat{{X}}_{n}$ depends on the observation $Y_n$ delivered by the sensor as well as on the conditional law of ${X}_n$ given the information $ {\\cal Y }_{n-1}$ .",
"The EnKF associated to the filtering problem (REF ) coincides with the mean-field particle interpretation of the nonlinear Markov chain (REF ).",
"To define these models, we let $\\xi _0=(\\xi _0^i)_{1\\le i\\le N+1}$ be a sequence of $(N+1)$ independent copies of $X_0$ , for some parameter $N\\ge 1$ .",
"We also denote by $ {\\cal W }_n=( {\\cal W }^i_n)_{i\\ge 1}$ and $ {\\cal V }_n=( {\\cal V }^i_n)_{i\\ge 1}$ a sequence of independent copies of $W_n$ and $V_n$ .",
"The mean field particle interpretation of the nonlinear Markov chain discussed above is given by the interacting particle system defined sequentially for any $1\\le i\\le N+1$ and $n\\ge 0$ by the formulae $\\left\\lbrace \\begin{array}{rcl}\\widehat{\\xi }^i_{n}&=&\\xi ^i_n+g_n~(Y_n-(C\\xi ^i_n+D {\\cal V }^i_n))\\quad \\mbox{\\rm with}\\quad g_n:=Cp_n/(C^2p_n+D^2)\\\\&&\\\\\\xi ^i_{n+1}&=& A\\,\\widehat{\\xi }^i_{n}+B\\, {\\cal W }^i_{n+1}.\\end{array}\\right.$ In the above display $p_n$ stands for the normalised sample variance $p_n:=\\frac{1}{N}\\sum _{1\\le i\\le N+1}(\\xi ^i_n-m_n)^2=\\left(1+\\frac{1}{N}\\right) {Q}_{\\eta ^N_n}\\quad \\mbox{\\rm with}\\quad \\eta ^N_n:=\\frac{1}{N+1}\\sum _{1\\le i\\le N+1}\\delta _{\\xi ^i_n}.$ We also consider the sample means $m_n:=\\frac{1}{N+1}\\sum _{1\\le i\\le N+1}\\xi ^i_n\\quad \\mbox{\\rm and}\\quad \\widehat{m}_n:=\\frac{1}{N+1}\\sum _{1\\le i\\le N+1}\\widehat{\\xi }^i_n$ and the $\\sigma $ -field filtrations $\\begin{array}{l} {\\cal G }_n:=\\sigma (\\xi _0, {\\cal W }_k, {\\cal V }_k, 0\\le k\\le n)\\supset \\sigma (\\xi _k,\\widehat{\\xi }_k, ~0\\le k\\le n),\\\\\\\\\\widehat{ {\\cal F }}_n:= {\\cal Y }_{n}\\vee {\\cal G }_n\\supset {\\cal F }_n:=\\widehat{ {\\cal F }}_{n-1}\\vee \\sigma ( {\\cal W }_n)\\supset \\widehat{ {\\cal F }}_{n-1}\\vee \\sigma (\\xi _n).\\end{array}$ Note that this implies that $\\widehat{ {\\cal F }}_n\\supset {\\cal Y }_{n}\\quad \\mbox{\\rm and}\\quad {\\cal F }_n\\supset {\\cal Y }_{n-1}.$ In the rest of the article we shall assume that $\\xi _0$ , $ {\\cal V }_n$ and $ {\\cal W }_n$ are column vectors.",
"We also write $X~\\underline{\\in }~ {\\cal F }$ when a random variable is $ {\\cal F }$ -measurable.",
"With this notation, at every time $n\\ge 0$ , the updating-prediction steps of the ensemble Kalman filter are described by the following synthetic diagram $(m_n,p_n)~\\underline{\\in }~ {\\cal F }_n\\longrightarrow (\\widehat{m}_{n},\\widehat{p}_{n})~\\underline{\\in }~ \\widehat{ {\\cal F }}_n\\longrightarrow (m_{n+1},p_{n+1})~\\underline{\\in }~ {\\cal F }_{n+1}.$"
],
[
"Some basic notation",
"This section presents some basic notation and preliminary results necessary for the statement of our main results.",
"Throughout the rest of the article, we write $\\tau $ , $\\tau _k$ as well as $\\tau (l)$ and $\\tau _k(l)$ for some collection of non-universal constants whose values may vary from line to line, but only depend on the parameters $k$ and $l$ in some given parameter spaces.",
"We use the letters $\\iota $ , $\\iota _k$ , $\\iota (l)$ , $\\iota _k(l)$ as well as $\\epsilon $ and $\\epsilon _k\\in ]0,1]$ when these constants may also depend on the parameters $(A,B,C,D)$ of the filtering model.",
"Importantly these constants do not depend on the time parameter nor on the number of particles.",
"We denote by $\\chi ^{2}_{N ,Nx}$ a collection a non-central $\\chi $ -square random variables indexed by the degree of freedom $N\\ge 1$ and non centrality parameter $N x\\ge 0$ .",
"We recall that for any $N \\ge 1$ , we have $\\begin{array}{l}\\displaystyle \\frac{1}{N}~\\chi ^2_{N,Nx}=\\frac{1}{N}~\\sum _{1\\le i\\le N}~\\left(Z_i+\\sqrt{x}\\right)^2=1+x+\\sqrt{x}~\\frac{2}{N}~\\sum _{1\\le i\\le N}~Z_i+\\frac{\\sqrt{2}}{N}~\\sum _{1\\le i\\le N}~\\frac{Z_i^2-1}{\\sqrt{2}},\\end{array}$ for some given sequence $(Z_i)_{i\\ge 1}$ of independent and centered Gaussian random variables.",
"Rewritten in a slightly different form, for any $u,v$ with $v\\ne 0$ we have $\\frac{v}{N}~\\chi ^2_{N,N(u/v)^2}&=&\\displaystyle v+\\frac{u^2}{v}+\\frac{2}{\\sqrt{N}}~u~ { {\\cal Z }}+\\sqrt{\\frac{2}{N}}~v~ \\widetilde{ {\\cal Z }},$ where ${ {\\cal Z }}:=\\frac{1}{\\sqrt{N}}~\\sum _{1\\le i\\le N}~Z_i\\quad \\mbox{\\rm and}\\quad \\widetilde{ {\\cal Z }}:=\\frac{1}{\\sqrt{N}}~\\sum _{1\\le i\\le N}~\\frac{Z_i^2-1}{\\sqrt{2}}.$ Note that these definitions imply that ${ {\\cal Z }}$ is a centred Gaussian random variable with unit variance and $\\widetilde{ {\\cal Z }}$ is a centered $\\chi $ -square variable.",
"We also consider the extended random vector $\\Delta :=\\left(\\begin{array}{c}\\widehat{Z}\\\\{ {\\cal Z }}\\\\\\widetilde{ {\\cal Z }}\\end{array}\\right)\\quad \\mbox{\\rm with}\\quad \\widehat{Z}:=\\frac{1}{\\sqrt{N+1}}~\\sum _{1\\le i\\le N+1}~Z_{N+i}.$ In a similar manner, we define $(\\widehat{\\Delta }_{n},\\Delta _{n+1})$ as the sequence of random vectors $\\widehat{\\Delta }_{n}:=\\left(\\begin{array}{c}\\widehat{\\Delta }^1_n\\\\\\widehat{\\Delta }^2_n\\\\\\widehat{\\Delta }^3_n\\end{array}\\right)=\\left(\\begin{array}{l}\\widehat{V}_{n}\\\\{ {\\cal V }}_{n}\\\\\\displaystyle \\widetilde{ {\\cal V }}_n\\end{array}\\right)\\quad \\mbox{\\rm and}\\quad \\Delta _{n+1}:=\\left(\\begin{array}{c}\\Delta _{n+1}^1\\\\\\Delta _{n+1}^2\\\\\\Delta _{n+1}^3\\end{array}\\right)=\\left(\\begin{array}{l}\\widehat{W}_{n+1}\\\\{ {\\cal W }}_{n+1}\\\\\\displaystyle \\widetilde{ {\\cal W }}_{n+1}\\end{array}\\right)$ indexed by the time parameter $n\\ge 0$ and defined as $\\Delta $ by replacing the sequence $(Z_i)_{i\\ge 1}$ by the sequences of Gaussian variables $ {\\cal V }_n=( {\\cal V }^i_n)_{i\\ge 1}$ and $ {\\cal W }_{n+1}=( {\\cal W }^i_{n+1})_{i\\ge 1}$ introduced in in the definition of the EnKF (REF ).",
"For $n=0$ we also set $\\Delta _0:=\\left(\\begin{array}{c}\\Delta ^1_0\\\\\\Delta ^2_0\\end{array}\\right)=\\left(\\begin{array}{c}\\widehat{W}_0\\\\\\widetilde{ {\\cal W }}_0\\end{array}\\right)$ with $\\widehat{W}_0$ and $\\widetilde{ {\\cal W }}_0$ defined as $\\widehat{Z}$ and $\\widetilde{ {\\cal Z }}$ by replacing the sequence $Z_i$ with the sequence of Gaussian variables $ {\\cal W }_0^i:=\\frac{\\xi ^i_0-\\widehat{X}^-_0}{\\sqrt{P_0}},$ where we recall that the $\\xi ^i_0$ are independent copies of the Gaussian initial condition $X_0$ of the signal used to initialise the EnKF in (REF ).",
"In this notation, up to a sequence of Helmert orthogonal transformations on the sequence of Gaussian vectors $ {\\cal W }_0^i$ , the initial condition of the ensemble Kalman filter is given by the stochastic perturbation formulae $m_0=\\widehat{X}^-_0+\\frac{1}{\\sqrt{N+1}}~\\upsilon _0\\quad \\mbox{and}\\quad p_0=P_0+\\frac{1}{\\sqrt{N}}~\\nu _0,$ with the initial local perturbations defined by $\\upsilon _0:=\\sqrt{P_0}~\\Delta ^1_0\\quad \\mbox{and}\\quad \\nu _0:=\\sqrt{2}~P_0~\\Delta ^2_0.$ Observe that the sample means are written in terms of random perturbations $\\upsilon _0$ which are independent of the perturbations $\\nu _0$ of the sample variances.",
"The independence between the complete sufficient statistic $m_0$ and the ancillary statistic $p_0$ can also be checked using Basu's theorem [11].",
"An alternative algebraic and direct approach based on Helmert orthogonal matrices can be conducted using the same arguments as those used in the proof of Theorem REF ."
],
[
"Local perturbation theorems",
"Consider the local stochastic perturbations defined for any $n\\ge 0$ by the formulae $\\left\\lbrace \\begin{array}{rcl}\\widehat{\\upsilon }_n&:=&-g_nD~\\widehat{\\Delta }^1_n\\\\&&\\\\\\widehat{\\nu }_n&:=&-2g_nD(1-g_nC)~\\sqrt{p_n}~\\widehat{\\Delta }^2_n+\\sqrt{2}~g_n^2D^2~\\widehat{\\Delta }^3_n\\end{array}\\right.~\\left\\lbrace \\begin{array}{rcl}\\upsilon _{n+1}&:=&B~\\Delta ^1_{n+1}\\\\&&\\\\\\nu _{n+1}&:=&2AB~\\sqrt{\\widehat{p}_n}~\\Delta ^{2}_{n+1}+\\sqrt{2}~R~\\Delta ^3_{n+1}.\\end{array}\\right.$ In the above display, $(\\Delta _n,\\widehat{\\Delta }_n)_{n\\ge 0}$ are the random vectors defined in (REF ), and $g_n$ is the particle gain defined in (REF ).",
"Theorem 3.1 Up to a sequence of orthogonal transformations on the sequence of Gaussian vectors $ {\\cal W }_n$ and $ {\\cal V }_n$ , the ensemble Kalman filter and the Riccati equation are given by the initial condition (REF ) and the updating-prediction transitions for $n \\ge 0$ $\\left\\lbrace \\begin{array}{rcl}\\displaystyle \\widehat{m}_{n}&=&\\displaystyle m_n+g_n~(Y_n-Cm_n)+\\frac{1}{\\sqrt{N+1}}~\\widehat{\\upsilon }_n\\\\&&\\\\\\widehat{p}_n&=&\\displaystyle (1-g_nC)~p_n+\\frac{1}{\\sqrt{N}}~ \\widehat{\\nu }_n\\end{array}\\right.~\\left\\lbrace \\begin{array}{rcl}\\displaystyle m_{n+1}&=&\\displaystyle A~\\widehat{m}_{n}+\\frac{1}{\\sqrt{N+1}}~\\upsilon _{n+1}\\\\&&\\\\p_{n+1}&=&\\displaystyle A^2~\\widehat{p}_{n}+R+\\frac{1}{\\sqrt{N}}~\\nu _{n+1}.\\end{array}\\right.$ The proof of the above theorem is provided in section REF .",
"Merging the updating and prediction steps we check the following corollary.",
"Corollary 3.2 The sample variance $p_n$ of the EnKF satisfies the stochastic Riccati rational difference equation given by the formula $p_{n+1}=\\phi (p_n)+\\frac{1}{\\sqrt{N}}~\\delta _{n+1}\\quad \\mbox{with}\\quad \\delta _{n+1}:=A^2~\\widehat{\\nu }_n+\\nu _{n+1}.$ In the above display, $\\phi (p)$ is the one step mapping of the Riccati equation defined in (REF ), and the random variables $(\\widehat{\\nu }_n,\\nu _{n+1})$ on the right hand side of (REF ) are the local fluctuations defined in (REF ).",
"Let $ {\\cal R }_n:=\\sigma (p_k,~0\\le k\\le n)\\vee {\\cal Y }_{n}$ be the $\\sigma $ -field generated by the stochastic Riccati equation $p_k$ and the observation sequence up to the time horizon $k\\le n$ and set $\\widehat{m}_{n}^r:=\\mathbb {E}(\\widehat{m}_{n}~|~ {\\cal R }_{n})\\quad \\mbox{\\rm and}\\quad m_{n+1}^r:= \\mathbb {E}(m_{n+1}~|~ {\\cal R }_{n}).$ Using this notation, the next corollary is a direct consequence of Theorem REF .",
"Corollary 3.3 Given the sample variances and the sequence of observations, for any $n \\ge 0$ , the updating-prediction steps of the sample mean of the EnKF satisfy the quenched Kalman-filter equations $\\widehat{m}_{n}^r =m_n^r+ g_n(Y_n-C\\,m_n^r)\\quad \\mbox{and}\\quad m_{n+1}^r=\\displaystyle A~\\widehat{m}_{n}^r,$ with the initial condition $m_{0}^r=\\widehat{X}^-_0$ .",
"The next corollary provides an alternative interpretation of the updating-prediction steps in terms of a two-steps Markov chain involving non-central $\\chi $ -square random variables.",
"Corollary 3.4 The updating-prediction steps of the stochastic Riccati difference equations stated in Theorem REF take the following form $\\widehat{p}_n=\\left(\\frac{p_n}{1+Sp_n}\\right)^2~\\frac{S}{N}~\\widehat{\\chi }^{(n,2)}_{N,N/(Sp_n)}\\quad \\mbox{and}\\quad p_{n+1}=\\frac{R}{N}~\\chi ^{(n+1,2)}_{N,N(A^2/R)\\widehat{p}_{n}},$ with the initial condition $p_0=\\frac{P_0}{N}~\\chi ^{(0,2)}_{N,0}.$ In the above display, $\\chi ^{(n,2)}_{N,x}$ and $\\widehat{\\chi }^{(n,2)}_{N,x}$ stand for a collection of independent non-central $\\chi $ -square random variables with $N$ degrees of freedom and non-centrality parameter $x$ , which are independent of the local perturbation sequence $(\\upsilon _{k+1},\\widehat{\\upsilon }_k)$ of the EnKF filter defined in Theorem REF .",
"The proof of the above corollary is a direct consequence of the decomposition (REF ), thus it is left as an exercise for the reader."
],
[
"Stability theorems",
"We denote by $\\phi ^n=\\phi ^{n-1}\\circ \\phi $ the composition evolution semigroup of the Riccati difference equation (REF ).",
"We also let $\\widehat{X}_n(x,p)$ be the solution of the Kalman filter associated with the initial values $(\\widehat{X}_0,P_0)=(x,p)$ .",
"The next theorem summarises the main stability properties of the Kalman filter and the Riccati difference equations.",
"Theorem 3.5 The Riccati equation (REF ) has an unique positive fixed point $P_{\\infty }=\\phi (P_{\\infty })>0$ and for any $n\\ge 0$ and any $p=(p_1,p_2)\\in \\mathbb {R}_+^2$ , we have the exponential contraction estimates $\\vert \\phi ^n(p_1)-\\phi ^n(p_2)\\vert \\le \\iota _1~\\left(1-\\epsilon _1\\right)^n~\\vert p_1-p_2\\vert .$ In addition, there exists some integer $k(p_1)\\ge 1$ such that for any $x=(x_1,x_2)\\in \\mathbb {R}^2$ we have $\\mathbb {E}\\left(\\left(\\widehat{X}_n(x_1,p_1)-\\widehat{X}_n(x_2,p_2)\\right)^2\\right)^{1/2}\\le ~\\iota (x,p)~(1-\\epsilon _2)^{n-k(p_1)}~\\left(\\vert p_1-p_2\\vert +\\vert x_1-x_2\\vert \\right).$ The proof of the above theorem is provided in section REF .",
"Let $ {\\cal P }(p, dq)$ denote the Markov transitions associated with the stochastic Riccati Markov chain $p_n$ discussed in (REF ).",
"Given some non-negative function $ {\\cal U }$ on $\\mathbb {R}_+ := [0, \\infty [$ we define the $ {\\cal U }$ -norm of some function $f: \\mathbb {R}_+ \\rightarrow \\mathbb {R}$ and some locally finite signed measure $\\mu (dp)$ on $\\mathbb {R}_+$ by $\\Vert f\\Vert _{\\tiny {\\cal U }}:=\\sup _{p\\ge 0}\\left|\\frac{f(p)}{ {\\cal U }(p)+1/2}\\right|\\quad \\mbox{\\rm and}\\quad \\Vert \\mu \\Vert _{\\tiny {\\cal U }}:=\\sup {\\lbrace \\vert \\mu (f)\\vert ~:~f~\\mbox{\\rm s.t.",
"}~\\Vert f\\Vert _{\\tiny {\\cal U }}\\le 1\\rbrace }.$ The $ {\\cal U }$ -Dobrushin ergodic coefficient of $ {\\cal P }$ is defined by $\\beta _{ {\\cal U }}( {\\cal P }) = \\sup _{(p,q) \\in \\mathbb {R}_+^2}\\frac{\\Vert {\\cal P }(p, \\cdot ) - {\\cal P }(q, \\cdot ) \\Vert _{ {\\cal U }}}{1 + {\\cal U }(p) + {\\cal U }(q)}.$ The next theorem concerns the stability properties of the stochastic Riccati rational difference equation presented in corollary REF .",
"Theorem 3.6 There exists an unique invariant measure $\\pi $ such that $\\pi = \\pi {\\cal P }$ and some Lyapunov function of the form $ {\\cal U }(p)=u p+v$ such that $\\beta _{ {\\cal U }}( {\\cal P }) < 1$ for some $u,v>0$ .",
"In addition, for any probability measures $\\mu _1$ and $\\mu _2$ on $\\mathbb {R}_+$ and for any $n\\ge 1$ we have $\\Vert \\mu _1 {\\cal P }^n - \\mu _2 {\\cal P }^n\\Vert _{ {\\cal U }} \\le \\beta _{ {\\cal U }}( {\\cal P })^n\\Vert \\mu _1 - \\mu _2\\Vert _{ {\\cal U }}.$ Moreover, for any function $f$ such that $|f(p)| \\le 1/2 + {\\cal U }(p)$ , for any $p \\in \\mathbb {R}_+$ , we have $| {\\cal P }^n(f)(p) - \\pi (f)| \\le \\beta _{ {\\cal U }}( {\\cal P })^n~(1 + {\\cal U }(p) + \\pi ( {\\cal U })).$ The above theorem comes from the fact that the Riccati map $\\phi (p)$ is a uniformly bounded function (cf.",
"for instance (REF ) and (REF )).",
"In addition, using (REF ) or (REF ) we readily check that $ {\\cal P }(p,dq)$ has a continuous density $k(p,q)>0$ with respect to the variables $(p,q)$ and the Lebesgue measure $dq$ on $\\mathbb {R}_+$ .",
"It is rather well-known that these two properties ensure the existence of a Lyapunov function of the form $ {\\cal U }(u,v)=u p+v$ such that $\\beta _{ {\\cal U }}( {\\cal P }) < 1$ .",
"For the convenience of the reader a detailed proof of the above theorem is provided in the appendix.",
"The above theorem is an extension of the stability theorem for stochastic Riccati diffusions presented in the article [22] (see also section 2.2 and Theorem 2.4 in [18] for the multivariate case) to the stochastic Riccati rational equations presented in Corollary REF .",
"We now turn to quantifying the stability properties of the EnKF sample means.",
"Since the one-step optimal predictor $\\widehat{X}_n^-$ is a minimal variance estimate of $X_n$ we already know that $M_n:=(m_n-X_n)\\Longrightarrow P_n= \\mathbb {E}((\\widehat{X}_n^--X_n)^2)\\le \\mathbb {E}(M_n^2).$ Thus for small sample sizes we expect $\\mathbb {E}((m_n-X_n)^2)$ to be much larger than $P_n$ .",
"To find some useful quantitative estimate, observe that $\\displaystyle M_{n+1}=\\displaystyle \\frac{A}{1+Sp_n}~M_n+\\Upsilon _{n+1},$ with the conditionally centered Gaussian random variables $\\Upsilon _{n+1}:=Ag_nDV_n-BW_{n+1}+\\frac{1}{\\sqrt{N+1}}~\\left(A~\\widehat{\\upsilon }_n+\\upsilon _{n+1}\\right).$ The decomposition (REF ) shows that the global stability of the stochastic process $M_n$ depends on the long-time behavior of the random products defined for any $0\\le l\\le n$ by the formula $ {\\cal E }_{l,n}:=\\prod _{l\\le k\\le n}\\frac{A}{1+Sp_k}.$ For stable signal drifts, i.e.",
"when $\\vert A\\vert <1$ , the exponential decays of these random products is immediate.",
"In the unstable case, that is when $\\vert A\\vert >1$ , none of the uniform estimates stated later in Theorem REF appear to be useful to quantify directly these random products.",
"Theorem 3.7 For any $k\\ge 1$ there exist some parameters $N_k\\ge 1$ and $\\epsilon _k\\in ]0,1]$ such that for any $N\\ge N_k$ and any time horizon $n\\ge l\\ge 1$ we have the uniform almost sure exponential decays and the time-uniform estimates $\\mathbb {E}\\left(\\left| {\\cal E }_{l,n}\\right|^k ~|~p_{l-1}\\right)\\le \\iota _k~\\left(1-\\epsilon _k\\right)^{n-l}\\quad \\mbox{and}\\quad \\mathbb {E}\\left(\\vert M_{n}\\vert ^k\\right)\\le \\iota _k.$ The proof of this theorem is provided at the end of section REF .",
"Let $ {\\cal M }((p,x), d(q,z))$ denote the Markov transitions associated with the coupled Markov chain $(p_n,M_n)$ defined by formulae (REF ) and (REF ).",
"The $ {\\cal V }$ -norms and the $ {\\cal V }$ -Dobrushin ergodic coefficient $\\beta _{ {\\cal V }}( {\\cal M })$ of $ {\\cal M }$ associated with some non-negative function $ {\\cal V }$ on $\\mathbb {R}_+\\times \\mathbb {R}$ are defined as in (REF ) and (REF ) by replacing the state space $\\mathbb {R}_+$ by $\\mathbb {R}_+\\times \\mathbb {R}$ and the function $ {\\cal U }$ by $ {\\cal V }$ .",
"For any $\\epsilon \\in ]0,1]$ there exists some time horizon $k\\ge 1$ such that $ {\\cal V }(p,x)= p+\\vert x\\vert \\Longrightarrow {\\cal M }^k( {\\cal V })(p,x)\\le \\epsilon ~ {\\cal V }(p,x)+\\iota .$ A proof of the above Lyapunov inequality is provided in section REF .",
"Arguing as in the proof of Theorem REF we readily prove the following theorem.",
"Theorem 3.8 There exists a unique invariant measure $\\mu $ such that $\\mu = \\mu {\\cal M }$ , a Lyapunov function of the form $ {\\cal V }(p,x)=u p+v \\vert x\\vert +w$ for some $u,v,w>0$ , and some integer $k\\ge 1$ such that $\\beta _{ {\\cal V }}( {\\cal M }^k) < 1$ .",
"In addition, for any probability measures $\\eta _1$ and $\\eta _2$ on $\\mathbb {R}_+\\times \\mathbb {R}$ and any $n\\ge 1$ we have $\\Vert \\eta _1 {\\cal M }^{nk} - \\eta _2 {\\cal M }^{nk}\\Vert _{ {\\cal V }} \\le \\beta _{ {\\cal V }}( {\\cal M }^k)^n\\Vert \\eta _1 - \\eta _2\\Vert _{ {\\cal V }}.$ Moreover, for any function $f$ on $\\mathbb {R}_+\\times \\mathbb {R}$ such that $|f(p,x)| \\le 1/2 + {\\cal V }(p,x)$ , for any $(p,x) \\in \\mathbb {R}_+\\times \\mathbb {R}$ , we have $| {\\cal M }^{nk}(f)(p,x) - \\mu (f)| \\le \\beta _{ {\\cal V }}( {\\cal M }^k)^n~(1 + {\\cal V }(p,x) + \\mu ( {\\cal V })).$"
],
[
"Uniform mean-error estimates",
"The next theorem provides uniform mean-error and bias estimates for the particle gain defined in (REF ) and the stochastic Riccati equations presented in Theorem REF .",
"Theorem 3.9 For any $k\\ge 1$ and any $N\\ge 1$ and $n\\ge 0$ we have the time uniform estimates $\\mathbb {E}\\left(\\vert p_n-P_n\\vert ^k\\right)^{1/k}\\vee \\mathbb {E}\\left(\\vert \\widehat{p}_n-\\widehat{P}_n\\vert ^k\\right)^{1/k}\\vee \\mathbb {E}\\left(\\vert g_n-G_n\\vert ^k\\right)^{1/k} \\le \\iota _k~(1\\vee P_0) /\\sqrt{N}.$ In addition, we have the uniform bias estimate $0\\le P_n-\\mathbb {E}(p_n)\\le \\frac{\\iota _1}{N}~\\left[1\\vee P_0\\right]^{2}\\quad \\mbox{and}\\quad 0\\le G_n-\\mathbb {E}(g_n)\\le \\frac{\\iota _2}{N}~\\left[1\\vee P_0\\right]^{2}.$ The same formula on the l.h.s.",
"of the above display holds by replacing $(p_n,P_n)$ by $(\\widehat{p}_n,\\widehat{P}_n)$ .",
"The proof of the uniform mean error estimates stated in the above theorem is provided in section REF .",
"The proof of the bias estimates is provided in the end of section REF .",
"Now we turn to quantifying the fluctuations of the sample means around the optimal filter.",
"We set $\\widehat{m}_{n}^o:=\\mathbb {E}(\\widehat{m}_{n}~|~ {\\cal Y }_{n})\\quad \\mbox{\\rm and}\\quad m_{n+1}^o:= \\mathbb {E}(m_{n+1}~|~ {\\cal Y }_{n}).$ The next theorem provides uniform mean-error and bias estimates for the sample means of the EnKF filter defined in (REF ).",
"Theorem 3.10 For any $k\\ge 1$ there exists some parameter $N_k\\ge 1$ such that for any $N\\ge N_k$ and $n\\ge 0$ we have the time-uniform estimates $\\mathbb {E}\\left(\\vert m_{n}-\\widehat{X}^-_{n}\\vert ^k\\right)^{1/k}\\vee \\mathbb {E}\\left(\\vert \\widehat{m}_{n}-\\widehat{X}_{n}\\vert ^k\\right)^{1/k}\\le {\\iota _k}/{\\sqrt{N}}.$ In addition, we have the time-uniform bias estimates $\\mathbb {E}\\left(\\vert m_{n}^o-\\widehat{X}^-_{n}\\vert ^k\\right)^{1/k}\\vee \\mathbb {E}\\left(\\vert \\widehat{m}^o_{n}-\\widehat{X}_{n}\\vert ^k\\right)^{1/k}\\le \\iota _k~(1\\vee P_0)^2/N.$ The proof of the mean error estimates (REF ) is provided in the beginning of section REF .",
"The uniform bias estimates (REF ) are proved in the end of section REF ."
],
[
"Central limit theorems",
"In the further development of this section, $Z^i_k$ and $\\widehat{Z}^i_k$ stand for a sequence of independent, centered Gaussian random variables with unit variance indexed by $i\\ge 1$ , $k\\ge 0$ .",
"We also consider the variables $(\\mathbb {U}_0,\\mathbb {V}_0):=(\\sqrt{P_0}~Z^1_0,\\sqrt{2}~P_0~Z^2_0)$ and the sequence of the Gaussian random variables $\\left\\lbrace \\begin{array}{rcl}\\widehat{\\mathbb {U}}_n&:=&\\displaystyle -G_nD~\\widehat{Z}^1_n\\\\&&\\\\\\widehat{\\mathbb {V}}_n&:=&\\displaystyle -2G_nD(1-G_nC)~\\sqrt{P_n}~\\widehat{Z}^2_n+\\sqrt{2}~G_n^2D^2~\\widehat{Z}^3_n\\end{array}\\right.\\left\\lbrace \\begin{array}{rcl}\\mathbb {U}_{n+1}&:=&\\displaystyle B~Z^1_{n+1}\\\\&&\\\\\\mathbb {V}_{n+1}&:=&\\displaystyle 2AB~\\sqrt{\\widehat{P}_n}~Z^{2}_{n+1}+\\sqrt{2}~R~Z^3_{n+1}.\\end{array}\\right.$ Observe that the above sequence of independent and centered Gaussian variables are defined in a similar manner to the local stochastic perturbations (REF ) by replacing the sample variances by their limiting values and the random vectors $(\\widehat{\\Delta }_n,\\Delta _{n+1})$ by the Gaussian random variables $(\\widehat{Z}_n,Z_{n+1})$ .",
"Theorem 3.11 For any time horizon $n\\ge 0$ we have the weak convergence $(\\upsilon _0,\\nu _0,(\\widehat{\\upsilon }_{k},\\widehat{\\nu }_{k},\\upsilon _{k+1},\\nu _{k+1})_{0\\le k\\le n})\\longrightarrow _{N\\rightarrow \\infty }~(\\mathbb {U}_0,\\mathbb {V}_0,(\\widehat{\\mathbb {U}}_{k},\\widehat{\\mathbb {V}}_{k},\\mathbb {U}_{k+1},\\mathbb {V}_{k+1})_{0\\le k\\le n}).$ The proof of the above theorem is provided in section REF .",
"Applying the continuous mapping theorem we obtain the following corollary.",
"Corollary 3.12 For any time horizon $n\\ge 0$ we have the weak convergence $(\\nu _0,(\\delta _k)_{1\\le k\\le n})\\longrightarrow _{N\\rightarrow \\infty }~(\\mathbb {Z}_0,(\\mathbb {Z}_k)_{1\\le k\\le n}),$ with the independent Gaussian random variables $\\mathbb {Z}_n$ defined by $\\mathbb {Z}_{n+1}:=A^2~\\widehat{\\mathbb {V}}_{n}+\\mathbb {V}_{n+1}\\quad \\mbox{and}\\quad \\mathbb {Z}_{0}=\\mathbb {V}_0.$ In the above display, $\\nu _0$ and $\\delta _k$ are the local fluctuations of the stochastic Riccati equation discussed in (REF ).",
"We now consider the collection of stochastic processes $(\\widehat{\\mathbb {Q}}_n^N,\\mathbb {Q}^N_{n+1})$ indexed by $N\\ge 1$ defined for any $n\\ge 0$ by the updating-prediction synthetic diagram $\\mathbb {Q}^N_{n}:=\\sqrt{N}(p_{n}-P_{n})\\longrightarrow \\widehat{\\mathbb {Q}}_n^N:=\\sqrt{N}(\\widehat{p}_n-\\widehat{P}_n)\\longrightarrow \\mathbb {Q}^N_{n+1}.$ Theorem 3.13 The stochastic processes $(\\widehat{\\mathbb {Q}}_n^N,\\mathbb {Q}^N_{n+1})$ converge in law in the sense of convergence of finite dimensional distributions, and as the number of particles $N\\rightarrow \\infty $ , to a sequence of centered stochastic processes $(\\widehat{\\mathbb {Q}}_n,\\mathbb {Q}_{n+1})$ with initial condition $\\mathbb {Q}_0 = \\mathbb {V}_0$ and updating-prediction transitions given by $\\left\\lbrace \\begin{array}{rcl}\\widehat{\\mathbb {Q}}_n&=&(1-G_nC)^{2}~\\mathbb {Q}_n+\\widehat{\\mathbb {V}}_n\\\\&&\\\\\\mathbb {Q}_{n+1}&=&A^2\\,\\widehat{\\mathbb {Q}}_n+\\mathbb {V}_{n+1}.\\end{array}\\right.$ The proof of the above theorem is provided in section REF .",
"We consider the collection of stochastic processes $(\\widehat{\\mathbb {X}}_n^N,\\mathbb {X}^N_{n+1})$ indexed by $N\\ge 1$ and defined for any $n\\ge 0$ by the updating-prediction synthetic diagram $\\mathbb {X}^N_{n}:=\\sqrt{N}(m_{n}-\\widehat{X}^-_{n})\\longrightarrow \\widehat{\\mathbb {X}}_n^N:=\\sqrt{N}(\\widehat{m}_n-\\widehat{X}_n)\\longrightarrow \\mathbb {X}^N_{n+1}.$ Theorem 3.14 The stochastic processes $(\\widehat{\\mathbb {X}}_n^N,\\mathbb {X}^N_{n+1})$ converge in law in the sense of convergence of finite dimensional distributions, as the number of particles $N\\rightarrow \\infty $ , to a sequence of centered stochastic processes $(\\widehat{\\mathbb {X}}_n,\\mathbb {X}_{n+1})$ with initial condition $\\mathbb {X}_0 = \\mathbb {U}_0$ and updating-prediction transitions given by $\\left\\lbrace \\begin{array}{rcl}\\widehat{\\mathbb {X}}_{n}&=&(1-G_nC)\\,\\mathbb {X}_n+\\mathbb {G}_n\\,(Y_n-C\\widehat{X}^-_n)+\\widehat{\\mathbb {U}}_n\\\\&&\\\\\\mathbb {X}_{n+1}&=&A\\,\\widehat{\\mathbb {X}}_{n}+\\mathbb {U}_{n+1}.\\end{array}\\right.$"
],
[
"Some comments and comparisons",
"As shown in section REF , the EnKF is a rather simple numerical filtering-type technique defined by an ensemble of particles mimicking the evolution of well-known Kalman filter equations, replacing `the true' covariances by the ensemble sample-covariances.",
"Besides the fact that the EnKF is only consistent for linear Gaussian filtering problems (cf.",
"[48]), these popular particle methodologies are applied to complex nonlinear and high dimensional filtering problems arising in fluid mechanics [14], [74], [78], weather forecasting [3], [4], [30], [49], geosciences and reservoir simulation [47], [72], [82], [83], [94], and many other data assimilation disciplines.",
"To reduce the computational cost, the ensemble sample size is generally chosen to be several orders of magnitude below the very large effective dimension of the underlying signal.",
"To prevent the ensemble covariance degeneracy in time, small sample EnKF requires one to combine several customisation techniques such as adaptive covariance/gain inflation and related localisation methodologies [12], [13], [21], [93].",
"The first rigorous analysis of the bias and the consistency of these additional levels of approximations for linear Gaussian models can be found in [21], [23].",
"As shown in [38], for nonlinear filtering problems, the EnKF can be interpreted as particle-type extended Kalman filter.",
"As any observer-type algorithm, the well-posedness is not connected to any kind of distance to the optimal filter for any small or large sample sizes, but in terms of non-degeneracy of the observer with respect to time.",
"As underlined by Majda and Tong in [69] “Why EnKF works well with a small ensemble has remained a complete mystery.",
"Practitioners often attribute the EnKF success to the existence of a low effective filtering dimension.",
"But the definition of the effective filtering dimension has remained elusive, as its associated subspace often evolves with the dynamics”.",
"In the article [69], the authors suggest several theoretical guidelines to choose judiciously the ensemble sample size using covariance inflation and spectral projection techniques on the unstable directions of the signal.",
"From a purely mathematical perspective, the first important question is clearly to find conditions that ensure that the EnKF is well-founded and consistent, in the sense that it converges to the optimal filter as the ensemble sample size, or equivalently the computational power, tends to infinity (cf.",
"for instance [48], [70] in discrete time settings and [37], [38] for continuous time EnKF models).",
"Of course, the consistency property and asymptotic analysis of any statistical Monte Carlo estimate such as the sample means or the sample covariances delivered by EnKF is reassuring but it only gives a partial answer to the performance of these ensemble filters when used with small sample sizes.",
"The non-asymptotic analysis for finite sample sizes is clearly a more delicate subject as the continuous or the discrete generation EnKF belong to unconventional classes of nonlinear Markov processes and mean-field particle processes interacting through the sample covariance of the system.",
"Here the interaction covariance function of the EnKF particle filter is not a bounded nor a Lipschitz function as is it is commonly assumed in traditional nonlinear and interacting diffusion theory.",
"As a result, none of the stochastic tools developed in this field, including the rather elementary variational methodology for nonlinear diffusions developed in [9], [10] nor the more recent powerful differential calculus developed in [31], can be used to analyse this class of continuous-time models equipped with a nonlinear quadratic-type interacting function.",
"Here, the sample covariance matrices satisfy a rather sophisticated quadratic-type stochastic Riccati matrix diffusion equation, which requires one to develop new stochastic analysis tools [18].",
"For a more thorough discussion on mean-field type particle systems and a probabilistic description of genetic type particle filters and Ensemble Kalman type particle filters within this framework we refer to the review article [23], the books [35], [37], [39] and references therein.",
"The analysis of the discrete generation EnKF discussed in the present article is more delicate and requires the development of new stochastic methodologies.",
"For instance, in discrete time settings, the EnKF particle system evolves as the Kalman filter (REF ) in terms of a two-step prediction-updating interacting Markov chain (REF ).",
"In contrast with the Gaussian nature of the continuous time EnKF and stochastic Riccati diffusions discussed above, Theorem REF presented in this article shows that these two transitions combine both Gaussian and $\\chi $ -square type perturbations.",
"To the best of our knowledge, the analysis of stochastic Riccati rational difference equations, such as those discussed in Corollary REF , has not been covered in the literature.",
"Corollary REF also provides an alternative description of the sample variances in terms of a Markov chain with non-central $\\chi $ -square nonlinear fluctuations.",
"Again, to the best of our knowledge, these stochastic perturbation theorems and the stability analysis of the analysis of stochastic Riccati rational difference equations are the first of this type in the applied probability literature.",
"In a study by Tong, Majda and Kelly [93], the authors analyse the long-time behaviour and ergodicity of discrete generation EnKF using Foster-Lyapunov techniques ensuring that the filter is asymptotically stable with respect to any erroneous initial condition as soon as the signal is stable.",
"These important properties ensure that the EnKF has a single invariant measure and initialisation errors of the EnKF will dissipate with respect to the time parameter.",
"Nevertheless, the only ergodicity of the signal and thus the ensemble process does not give any useful information on the convergence and the accuracy of the EnKF towards the optimal filter as the number of samples tends to infinity, nor does it allow one to quantify the fluctuations around the optimal filter for a given small or even very large ensemble sample size.",
"On the other hand, effective unstable directions are connected to unstable-transient signals that may grow exponentially fast.",
"In this context, for obvious reasons, any well tracking EnKF needs to be an unstable-transient particle process.",
"As a consequence, we cannot expect to obtain any type of filter ergodicity properties, nor any time-uniform estimates for the raw moments of the EnKF filter.",
"From our point of view, the terminology \"effective filtering dimensions\" discussed above should be understood as the unstable directions of the signal.",
"For time varying and multivariate linear-Gaussian filtering problems, these effective dimensions are rarely known as they are partially observed and they may change in time.",
"For a more detailed discussion on the stability properties of systems of time varying linear stochastic differential equations with random coefficients we refer to [19].",
"To avoid the time degeneracy of any type of filtering approximation, a crucial question in practice is to check wether or not the convergence is uniform with respect to the time parameter.",
"Equivalently, we need to obtain estimates of the error between the optimal filter and its approximation that are uniform with respect to any time horizon even if the signal and thus any well tracking approximating filter are both unstable and transient.",
"We emphasise that in this setting, any good approximating filter will diverge.",
"In this context, the filtering step is part of a control loop that allows one to track the transient signal at all times.",
"The main advantage of one-dimensional filtering problems over multivariate problems with unstable signals is that they have a single effective dimension.",
"In this context, the EnKF is also unstable and transient but the theorems REF and REF presented in this article show that the EnKF tracks and converges to the optimal filter uniformly in time.",
"These theorems also ensure that the precision error to the optimal filter can be quantified for any given finite sample size, and it will not degrade with respect to the time horizon.",
"Theorem REF also indicates that the sample variances satisfy an autonomous stochastic Riccati-type evolution equation, independent of the sequence of observations.",
"From a different angle, theorem REF also shows that the sample variances of the EnKF behaves as a Markov chain that converge exponentially fast as the time horizon tends to infinity to a single invariant measure, even if the signal of the filtering problem is transient.",
"To the best of our knowledge, this stability theorem and the non-asymptotic theorems discussed above are the first results of this type in the literature on particle filter theory, including the literature of discrete generation EnKF.",
"Next we provide a more detailed discussion on the stability of particle filters and their particle approximations, with precise reference pointers.",
"We also suggest an avenue of open research problems.",
"Most of the stability properties of genetic-type particle filters [34], [36], including continuous time interacting jump Feynman-Kac particle systems [8], are only valid for stable and ergodic signals.",
"It is clearly out the scope of this article to review in detail all contributions related to the stability of genetic type particle filters, thus we refer to the books [34], [35], [36] and references therein.",
"The analysis of these continuous-time or discrete generation genetic type processes for unstable signals remains an important and open research question.",
"Continuous time EnKF filters for stable signals are also discussed in [16], [38], including in the context of the extended Kalman filter.",
"The analysis of unstable and possibly transient signals is more recent, starting at the end of the 2010s in the context of continuous time filtering problems [20], [18], [22], see also [23] for a comprehensive review of this subject.",
"The articles [18], [20] analyse the stability and the fluctuation properties of the sample covariance matrices of continuous time EnKF filters for multivariate models under natural and minimal observability and controllability conditions.",
"To better understand the difficulty in handling unstable effective signal directions, we recall that under natural observability and controllability conditions the Kalman filtering theory developed by Kalman and Bucy in the end of the 1960s ensures that the optimal filter is able to track any possibly unstable signals uniformly with respect to the time horizon.",
"To answer any stability question related to EnKF particle filters or genetic type particle filters, we first need to extend the theory of Kalman and Bucy to this class of approximating particle filters.",
"This research project remains open for genetic type filters.",
"The present article provide a partial answer for discrete generation EnKF particle filters.",
"To better understand the difficulty in handling and extending the stability theory of Kalman and Bucy to particle filters, it is worth noting a brief of history.",
"The stability properties of continuous time or discrete generation Kalman filters and their associated Riccati equations for multidimensional filtering problems are usually conducted under some observability/controllability-type conditions using sophisticated and tedious matrix manipulations.",
"The first stability properties for continuous time models go back to the beginning of the 1960s with the celebrated work of Kalman and Bucy [57], with related prior work by Kalman [53], [54], [56] and later work by Bucy [27].",
"In [2], the stability of this filter was analysed under a relaxed controllability condition.",
"The first stability properties for discrete time models go back to to the end of the 1960s with the work of Deyst and Price [41], which was incorporated in the seminal lecture book of Jazwinski [51].",
"However, as noted in [50], there was in both cases a crucial and commonly made error in the proof which invalidated the results, see [71] and the more recent articles [16], [79] for a more detailed discussion and references on these issues.",
"This error was repeated in numerous subsequent works.",
"A first correction [28] was noted in a reply to [50]; see Bucy's reply [29] and a separate reply by Kalman [55].",
"However, a complete reworking of the result did not appear in entirety until 2001 by Delyon in [40].",
"The stability of Kalman filter and Riccati equations is nowadays rather well understood.",
"Nevertheless it is not always simple to find a self-contained, rigorous and easy-to-read study on these subjects.",
"For a complete and self-contained analysis on the stability and convergence of Kalman-Bucy filtering in continuous time settings, we also refer to [16].",
"A self-contained analysis of Riccati differential equations and discrete-time Kalman filters for one-dimensional models is provided in section REF and section REF in the present article, see also Theorem REF .",
"The stability analysis of continuous time stochastic Riccati matrix diffusions is also rather well understood, see for instance [16], [18], [20], [23].",
"The extension of these multivariate results in the discrete time case remains an important open question.",
"Theorem REF as well as (REF ) and Theorem REF provide a complete answer to the stability of discrete generation EnKF for one dimensional filtering problems, with possibly unstable signals, yielding what seems to be the first results of this type for this class of particle filters, including both EnKF and genetic type particle filters."
],
[
"Some preliminary results",
"In this section we give several technical results that will be useful in the further development of the article.",
"We start by considering the inverse moments of the non-central $\\chi $ -square random variables introduced in (REF ).",
"We then provide a self-contained analysis of the Riccati difference equations defined in (REF )."
],
[
"Non-central $\\chi $ -square moments",
"In this section, we will prove the following technical result that allows one to estimate the inverse raw moments of non-central $\\chi $ -square random variables.",
"Proposition 4.1 For any $k\\ge 0$ and $n>(k+1)$ we have the uniform estimate $0\\le \\mathbb {E}\\left(\\left(\\frac{2n}{\\chi ^2_{2n,2nx}}\\right)^{k}\\right)-\\frac{1}{\\left(1+x\\right)^{k}}\\le \\frac{k+1}{n}~k~\\left(1+\\frac{k+1}{n-(k+1)}\\right)^{k}.$ In addition, there exists a constant $\\omega _k$ such that, for any $n\\ge 2(k+2)$ we have $\\mathbb {E}\\left(\\left(\\frac{1+x}{\\frac{1}{2n}\\chi ^2_{2n,2nx}}-1\\right)^{k}\\right)\\le \\frac{\\omega _k}{n}~\\left(1+x\\right)^{k}.$ In order to prove this result, let us first consider the following Taylor series expansion, $e^{-x}=\\sum _{0\\le k\\le n}\\frac{(-x)^k}{k!}+\\frac{(-x)^{n+1}}{(n+1)!",
"}~\\tau _{n+1}(x), n \\ge 0,$ with the collection of integral remainders given by $\\tau _{n+1}(x)=(n+1)~\\int _{0}^1~(1-t)^ne^{-tx}~dt=1-\\frac{x}{n+2}~\\tau _{n+2}(x).$ For any $k\\ge 1$ we denote by $\\tau ^{(k)}_n$ s the $k$ -th differential of the function $\\tau _n$ .",
"In this notation, the inverse moment of a non-central $\\chi $ -square random variable (cf.",
"section 3 in [25]) are given, for any $k\\ge 0$ , by the formulae $\\begin{array}{rcl}\\displaystyle \\mathbb {E}\\left(\\left(\\frac{2}{\\chi ^2_{2(n+1),2x}}\\right)^{(k+1)}\\right)&=&\\displaystyle \\frac{(-1)^{k}}{k!",
"}~\\frac{1}{n-k}~\\tau ^{(k)}_{n-k}(x).\\end{array}$ Setting $k=0$ we find the formulae $\\frac{1}{n}~\\tau _{n}(x)=\\mathbb {E}\\left(\\frac{2}{\\chi ^2_{2(n+1),2x}}\\right)\\ge \\frac{2}{\\mathbb {E}(\\chi ^2_{2(n+1),2x})}=\\frac{1}{x+n+1},$ where we have used Jensen's inequality to obtain the lower bound.",
"This combined with another application of Jensen's inequality implies that $\\begin{array}{rcl}\\displaystyle \\frac{(-1)^{k}}{k!",
"}~\\frac{1}{n-k}~\\tau ^{(k)}_{n-k}(x) =\\mathbb {E}\\left(\\left(\\frac{2}{\\chi ^2_{2(n+1),2x}}\\right)^{(k+1)}\\right)&\\ge &\\displaystyle \\left(\\frac{1}{n}~\\tau _{n}(x)\\right)^{k+1}\\ge \\frac{1}{(x+n+1)^{k+1}}.\\end{array}$ On the other hand, note that for any $0\\le z<1$ we have $\\log {(1-z)}\\le -z\\Longrightarrow n\\log {\\left(1-\\frac{s}{m}\\right)}-sx\\le -\\left(x+\\frac{n}{m}\\right)~s.$ This yields the upper bound $m\\tau _{n}(mx) = \\int _0^{m}~\\left(1-\\frac{s}{m}\\right)^n~e^{-sx}~ds\\le \\int _0^{m}~e^{-\\left(x+\\frac{n}{m}\\right)s}~ds=\\frac{1}{x+\\frac{n}{m}}~\\left(1-e^{-\\left(x+\\frac{n}{m}\\right)m}\\right)\\le \\frac{1}{x+\\frac{n}{m}}.$ Thus, for any $m,n\\ge 1$ and any $x\\ge 0$ we have the estimate $\\frac{1}{x+(n+1)/m}\\le \\frac{m}{n}~\\tau _{n}(mx)\\le \\frac{1}{x+{(n-1)}/{m}}.$ More generally, for any $k,m,n\\ge 0$ we have $\\frac{(-1)^k}{n+1}~\\tau _{n+1}^{(k)}(mx)&=&\\int _0^1 t^k~(1-t)^n~e^{-mxt}~dt\\\\&=&\\frac{1}{m}\\int _0^{m}~\\left(\\frac{s}{m}\\right)^k~\\left(1-\\frac{s}{m}\\right)^n~e^{-sx}~ds\\\\&\\le & \\frac{1}{m^{k+1}}\\int _0^{m}~s^k~e^{-\\left(x+\\frac{n}{m}\\right)s}~ds\\le \\frac{1}{m^{k+1}}~\\frac{1}{\\left(x+\\frac{n}{m}\\right)}~\\frac{k!",
"}{\\left(x+\\frac{n}{m}\\right)^{k}}.$ This discussion can be summarised with the following lemma.",
"Lemma 4.2 For any $n> k\\ge 0$ , $m\\ge 1$ and any $x\\ge 0$ we have the estimate $\\frac{1}{\\left(x+\\frac{n+1}{m}\\right)^{k+1}}\\le \\frac{(-1)^{k}}{k!",
"}~\\frac{m^{k+1}}{n-k}~\\tau ^{(k)}_{n-k}(mx)& \\le & ~\\frac{1}{\\left(x+\\frac{n-k-1}{m}\\right)^{k+1}}.$ With this lemma in hand, we are now in a position to prove Proposition REF .",
"[Proof of Proposition REF ] By Lemma REF we have $\\frac{1}{\\left(x+\\frac{n+1}{m}\\right)^{k+1}}\\le \\mathbb {E}\\left(\\left(\\frac{2m}{\\chi ^2_{2(n+1),2mx}}\\right)^{(k+1)}\\right)& \\le & ~\\frac{1}{\\left(x+\\frac{n-k-1}{m}\\right)^{k+1}}.$ This yields the estimate $\\frac{1}{\\left(1+x\\right)^{k}}\\le \\mathbb {E}\\left(\\left(\\frac{2n}{\\chi ^2_{2n,2nx}}\\right)^{k}\\right)\\le ~\\frac{1}{\\left((1+x)-\\frac{k+1}{n}\\right)^{k}}.$ This implies that for any $n> k$ we have $0\\le \\mathbb {E}\\left(\\left(\\frac{2n}{\\chi ^2_{2n,2nx}}\\right)^{k}\\right)-\\frac{1}{\\left(1+x\\right)^{k}}\\le \\theta _k(x) \\le \\frac{k+1}{n}~\\frac{k}{\\left(x+\\frac{n-(k+1)}{n}\\right)^{k}\\left(1+x\\right)}$ with the function $\\theta _k(x):=~\\frac{1}{\\left((1+x)-\\frac{k+1}{n}\\right)^{k}}-\\frac{1}{\\left(1+x\\right)^{k}}.$ This ends the proof of the first assertion.",
"This first estimate also yields, for any $n\\ge k+2$ , the rather crude estimate $0\\le \\mathbb {E}\\left(\\left(\\frac{1+x}{\\frac{1}{2n}\\chi ^2_{2n,2nx}}\\right)^{k}\\right)-1\\le \\frac{k}{n}~(k+1)\\left(1+x\\right)^{k}\\left(k+2\\right)^{k}\\le \\frac{1}{n}~\\left(1+x\\right)^{k}\\left(k+2\\right)^{k+2}.$ This implies that $0\\le \\mathbb {E}\\left(\\left(\\frac{1+x}{\\frac{1}{2n}\\chi ^2_{2n,2nx}}\\right)^{k}\\right)-1\\le \\frac{\\omega _k}{n}~\\left(1+x\\right)^{k}\\quad \\mbox{\\rm with}\\quad \\omega _k:=\\left(k+2\\right)^{k+2}.$ Using the decomposition $\\left(y-1\\right)^k=\\sum _{0\\le l\\le k}\\left(\\begin{array}{c}k\\\\l\\end{array}\\right)~(-1)^{l}~y^{k-l}=\\sum _{0\\le l< k}\\left(\\begin{array}{c}k\\\\l\\end{array}\\right)~(-1)^{l}~\\left(y^{k-l}-1\\right),$ we check that $\\mathbb {E}\\left(\\left(\\frac{1+x}{\\frac{1}{2n}\\chi ^2_{2n,2nx}}-1\\right)^{k}\\right)=\\sum _{0\\le l< k}\\left(\\begin{array}{c}k\\\\l\\end{array}\\right)~(-1)^{l}~\\underbrace{\\left(\\mathbb {E}\\left(\\left(\\frac{1+x}{\\frac{1}{2n}\\chi ^2_{2n,2nx}}\\right)^{k-l}\\right)-1\\right)}_{\\ge 0}.$ Thus, for any $n\\ge 2(2k+2)$ we have the estimate $\\mathbb {E}\\left(\\left(\\frac{1+x}{\\frac{1}{2n}\\chi ^2_{2n,2nx}}-1\\right)^{2k}\\right)&\\le & \\sum _{1\\le l\\le k}\\left(\\begin{array}{c}2k\\\\2l\\end{array}\\right)~\\left(\\mathbb {E}\\left(\\left(\\frac{1+x}{\\frac{1}{2n}\\chi ^2_{2n,2nx}}\\right)^{2l}\\right)-1\\right).$ Using (REF ), this implies that $\\mathbb {E}\\left(\\left(\\frac{1+x}{\\frac{1}{2n}\\chi ^2_{2n,2nx}}-1\\right)^{2k}\\right)\\le \\frac{1}{n}~\\left(1+x\\right)^{2k}\\omega ^{\\prime }_{2k}.$ with $\\omega ^{\\prime }_{2k}:=\\sum _{1\\le l\\le k}\\left(\\begin{array}{c}2k\\\\2l\\end{array}\\right)~\\left(2(l+1)\\right)^{2(l+1)}.$ In the same vein, for any $n\\ge 2((2k+1)+2)$ we have $\\mathbb {E}\\left(\\left(\\frac{1+x}{\\frac{1}{2n}\\chi ^2_{2n,2nx}}-1\\right)^{2k+1}\\right)&\\le & \\sum _{0\\le 2l< 2k+1}\\left(\\begin{array}{c}2k+1\\\\2l\\end{array}\\right)~\\underbrace{\\left(\\mathbb {E}\\left(\\left(\\frac{1+x}{\\frac{1}{2n}\\chi ^2_{2n,2nx}}\\right)^{2(k-l)+1}\\right)-1\\right)}_{\\ge 0}\\\\&\\le & \\sum _{0\\le l\\le k}\\left(\\begin{array}{c}2k+1\\\\2(k-l)\\end{array}\\right)~\\underbrace{\\left(\\mathbb {E}\\left(\\left(\\frac{1+x}{\\frac{1}{2n}\\chi ^2_{2n,2nx}}\\right)^{2l+1}\\right)-1\\right)}_{\\le \\frac{\\omega _{2l+1}}{n}~ (1+x)^{2l+1}}.$ This implies that $\\mathbb {E}\\left(\\left(\\frac{1+x}{\\frac{1}{2n}\\chi ^2_{2n,2nx}}-1\\right)^{2k+1}\\right)\\le \\frac{\\omega ^{\\prime }_{2k+1}}{n}~ (1+x)^{2k+1}$ with $\\omega ^{\\prime }_{2k+1}=\\sum _{0\\le l\\le k}\\left(\\begin{array}{c}2k+1\\\\2(k-l)\\end{array}\\right)~(2l+3)^{2l+3}.$ This ends the proof of the proposition."
],
[
"Riccati difference equations ",
"We associate with some parameters $(a,b,c,d)\\in \\mathbb {R}_+^4$ satisfying $c>0$ and $ad>bc$ the Riccati difference equation on $\\mathbb {R}_+:=[0,\\infty [$ given by the recursion $P_n:=\\phi (P_{n-1})\\quad \\mbox{\\rm with}\\quad \\phi (x):=\\frac{a x+b}{cx+d}\\quad \\mbox{\\rm with}\\quad x\\ge 0.$ Observe that $\\partial \\phi (x)=\\frac{\\rho }{(cx+d)^2}> 0\\quad \\mbox{\\rm and}\\quad \\partial ^2 \\phi (x)=-\\frac{2\\rho c}{(cx+d)^3}< 0\\quad \\mbox{\\rm with}\\quad \\rho :=ad-bc>0.\\\\$ This shows that $\\phi $ is an increasing and concave function.",
"Rational difference equation of the form (REF ) can be solved explicitly.",
"This section provides a brief review on these equations.",
"We set $(u,v,w):=(a/c,d/c,b/c)\\in \\mathbb {R}_+^4\\quad \\mbox{\\rm and}\\quad \\lambda =\\lambda _1/\\lambda _2\\in ]0,1[$ with the parameters $\\lambda _2-v:=\\frac{u-v}{2}+\\sqrt{\\left(\\frac{v-u}{2}\\right)^2+w}>0 \\quad \\mbox{\\rm and}\\quad v-\\lambda _1:=\\frac{v-u}{2}+\\sqrt{\\left(\\frac{v-u}{2}\\right)^2+w}> 0.$ Observe that $(v-\\lambda _1) (\\lambda _2-v)=w \\qquad \\mbox{\\rm and}\\qquad r=\\phi (r)>0\\Longleftrightarrow r=\\lambda _2-v.$ Lemma 4.3 For any $x\\ge 0$ and $n\\ge 1$ the evolution semigroup $\\phi ^n=\\phi \\circ \\phi ^{n-1}$ of the Riccati difference equation (REF ) is positive and is given by the formula $\\phi ^n(x)=r+(x-r)~\\frac{\\left(\\lambda _2-\\lambda _1\\right)~\\lambda ^n}{\\left(x+(v-\\lambda _1)\\right)~\\left(1-\\lambda ^{n}\\right)+(\\lambda _2-\\lambda _1)~\\lambda ^{n}}$ with the parameters $\\lambda $ and $(\\lambda _1,\\lambda _2)$ defined in (REF ) and the unique positive fixed point $r$ given in (REF ).",
"In addition, for any $n\\ge 1$ and any $x\\in \\mathbb {R}_+$ we have the estimates ${b}/{d}\\le \\phi ^n(x)\\le {a}/{c}.$ For any $x,y\\in \\mathbb {R}_+$ and $n\\ge 0$ we have the exponential decay to equilibrium $\\vert \\phi ^n(x)-r\\vert \\le \\frac{\\lambda _2-\\lambda _1}{v-\\lambda _1}~\\lambda ^n~\\vert x-r\\vert \\quad \\mbox{and}\\quad \\vert \\phi ^n(x)-\\phi ^n(y)\\vert \\le \\vert x-y\\vert ~\\left(\\frac{\\lambda _2-\\lambda _1}{v-\\lambda _1}\\right)^2~\\lambda ^n.$ In addition, we have the first order derivative $\\partial \\phi ^n(y):=\\frac{(\\lambda _2-\\lambda _1)^2~\\lambda ^n}{\\big (\\left(y+(v-\\lambda _1)\\right)~\\left(1-\\lambda ^{n}\\right)+(\\lambda _2-\\lambda _1)~\\lambda ^{n}\\big )^2}=\\prod _{0\\le k<n}\\frac{\\rho }{(c\\phi ^{k}(y)+d)^2},$ where the parameter $\\rho $ was defined in (REF ), which satisfies the second order estimate $\\left|\\phi ^n(x)-\\phi ^n(y)-\\partial \\phi ^n(y) (x-y)\\right|\\le \\displaystyle \\frac{(\\lambda _2-\\lambda _1)^2}{(v-\\lambda _1)^3}~(x-y)^2~\\lambda ^n,$ and the Lipschitz estimate $\\begin{array}{l}\\displaystyle \\vert \\partial \\phi ^n(x)-\\partial \\phi ^n(y)\\vert \\le \\iota ~\\vert y-x\\vert ~\\lambda ^n.\\end{array}$ The proof of the above lemma is rather technical so it is housed in the appendix.",
"We end this short section with some comparison properties and Riccati differential inequalities.",
"For any $\\epsilon \\ge 0$ we have $\\phi _{\\epsilon }(x):=\\phi (x)+\\epsilon =\\frac{a_{\\epsilon }x+b_{\\epsilon }}{cx+d}$ with the parameters $(a_{\\epsilon },b_{\\epsilon })=((a+\\epsilon c),(b+\\epsilon d))\\in \\mathbb {R}_+^2\\Longrightarrow a_{\\epsilon }d-b_{\\epsilon }c=ad-bc>0.$ This shows that the boundedness properties of Riccati type differential inequalities of the form $r_n\\le \\phi _{\\epsilon }(r_{n-1})$ can be deduced from those of the solution to the evolution equation $s_n=\\phi _{\\epsilon }(s_{n-1})$ .",
"For instance, since $\\phi _{\\epsilon }$ is increasing we have $\\phi _{\\epsilon }(r_{n-1})\\le \\phi _{\\epsilon }^n(x)\\le x\\vee s^{\\epsilon }\\le x\\vee \\frac{a_{\\epsilon }}{c}= x\\vee (\\epsilon +\\frac{a}{c})\\quad \\mbox{\\rm with the fixed point}\\quad s^{\\epsilon }:= \\phi _{\\epsilon }(s^{\\epsilon })>0.$ Moreover, observe that for any $\\epsilon \\le \\epsilon _1$ we have $0<2c~s^{\\epsilon }&=&a-d+\\epsilon c+\\sqrt{\\left((a-d)+\\epsilon c\\right)^2+4c~(b+\\epsilon d)}\\\\&\\le &a-d+\\sqrt{2(a-d)^2+4bc}+2\\sqrt{\\epsilon _1c~\\left(d+\\frac{\\epsilon _1 c}{2}\\right)}+\\epsilon _1 c.$ In the same vein, for any $k\\ge 1$ we have $\\phi (x)^k=\\left(\\frac{a x+b}{cx+d}\\right)^k\\le \\phi _k(x^k)\\quad \\mbox{\\rm with}\\quad \\phi _k(x):=\\frac{a_{k}x+b_{k}}{c_kx+d_k}$ and where $(a_{k},b_{k},c_k,d_k):=(2^{k-1}a^k,2^{k-1}b,c^k,d^k)\\in \\mathbb {R}_+^4.$ Note that we have $a_{k}d_k-b_{k}c_k=2^{k-1}((ad)^k-(bc)^k)>0\\quad \\mbox{\\rm since}\\quad ad-bc>0.$ As above, this shows that the boundedness properties of Riccati type inequalities of the form $r_n\\le \\phi (r_{n-1})^k$ can be deduced from the solution of the recursion $s_n=\\phi _k(s_{n-1})$ .",
"Indeed, since $\\phi _k$ is increasing we have $r_n\\le \\phi (r_{n-1})^k\\le \\phi _{k}^n(r_0) \\le x\\vee s^{k}\\quad \\mbox{\\rm with the fixed point}\\quad s^{k}:= \\phi _{k}(s^{k})>0.$"
],
[
"Stability of Kalman filters",
"This section is mainly concerned with the proof of Theorem REF .",
"Note that Lemma REF yields the existence of the positive fixed point $P_{\\infty }=\\phi (P_{\\infty })>0$ , as well as the estimate (REF ).",
"The fixed point $P_\\infty $ can be written in closed form in terms of the parameters $(a,b,c)$ (defined in (REF )) by the formula $P_{\\infty }= \\frac{(a-1)+r}{2c}\\quad \\mbox{\\rm with}\\quad r:=\\sqrt{\\left( a-1\\right)^2+4bc} \\,<a+1.$ In order to prove (REF ) we need to consider the random products defined for any $0\\le k\\le n$ and $p\\ge 0$ by $\\displaystyle E_{k,n}(p):=\\prod _{k< l\\le n}\\frac{A}{1+S\\phi ^l(p)},$ since in this notation, for any given $p\\ge 0$ and for any $(x_1,x_2)\\in \\mathbb {R}^2$ and $n\\ge 0$ we have $\\widehat{X}_n(x_1,p)- \\widehat{X}_n(x_2,p)&=&E_{0,n}(p)~(x_1-x_2).$ When $k=0$ , we simplify notation and we write $E_{n}(p)$ instead of $E_{0,n}(p)$ .",
"When $k=n$ we use the convention $E_{n,n}(p)=1$ .",
"With this in mind, for any $k<l<n$ we readily check the semigroup properties $E_{k,n}(p)=E_{n-k}\\left(\\phi ^k(p)\\right)\\quad \\mbox{\\rm and}\\quad E_{n-k}(p)=E_{l-k}(p)~\\times ~E_{n-l}(\\phi ^{l-k}(p)).$ To obtain appropriate bounds for the products $E_{k, n}$ , recall that thanks to (REF ), for any $n\\ge 1$ and any starting point $P_0$ we have the uniform estimates $R\\le \\phi ^n(P_0)\\le A^2/S+R,$ which implies that $\\displaystyle E_{k, n}(p) \\le \\left(\\frac{\\vert A\\vert }{1+SB^2}\\right)^{n-k}.$ Also observe that $\\frac{A^2-1}{2}\\ge \\vert A\\vert -1\\Longrightarrow P_{\\infty }=\\frac{1}{S}~\\left(\\frac{A^2-1}{2}\\right)+\\frac{RS+r}{2S}>\\frac{\\vert A\\vert -1}{S}.$ This implies that $\\vert A\\vert (1-G_\\infty C)=:1- \\epsilon _{\\infty }<1,$ where $G_\\infty $ is he gain associated with the steady state $P_{\\infty }$ of the Riccati equation and is given by $G_{\\infty }=CP_{\\infty }/(C^2P_{\\infty }+D^2)\\Longleftrightarrow 1-G_{\\infty }C={1}/{(1+SP_{\\infty })}.$ On the other hand, using (REF ), for any $p\\ge 0$ there exists some integer $k(p)\\ge 1$ such that for any $k> k(p)$ , $\\frac{1+SP_{\\infty }}{1+S\\phi ^k(p)}=1+\\frac{S}{1+S\\phi ^k(p)}~(P_{\\infty }-\\phi ^k(p))\\le 1+S~\\kappa _1~\\left(1-\\epsilon _1\\right)^k~\\vert p-P_{\\infty }\\vert \\le 1+\\epsilon _{\\infty }.$ with the parameter $\\epsilon _{\\infty }$ defined in (REF ).",
"This yields the following technical lemma.",
"Lemma 4.4 For any $p\\ge 0$ there exists some integer $k(p)\\ge 1$ such that for any $n> k(p)$ we have $\\sqrt{\\vert (\\partial \\phi )(\\phi ^n(p))\\vert }=\\frac{\\vert A\\vert }{1+S\\phi ^{n}(p)} \\le 1-\\epsilon ^2_{\\infty },$ with the parameter $\\epsilon _{\\infty }$ defined in (REF ).",
"In addition, for any $n\\ge k\\ge k(p)\\ge l$ we have $\\vert E_{l,n}(p)\\vert \\le \\left(\\frac{\\vert A\\vert }{1+SB^2}\\right)^{k(p)-l}~(1-\\epsilon ^2_{\\infty })^{n-k(p)}\\quad \\mbox{\\rm and}\\quad \\vert E_{k,n}(p)\\vert \\le (1-\\epsilon ^2_{\\infty })^{n-k}.$ We are now in position to state and to prove the main result of this section, which also completes the proof of Theorem REF .",
"Theorem 4.5 For any $p_1\\ge 0$ there exists some integer $k(p_1)\\ge 1$ for any $n> k(p_1)$ and any $x_1,x_2\\in \\mathbb {R}$ we have the almost sure estimate $\\left|\\widehat{X}_n(x_1,p_1)- \\widehat{X}_n(x_2,p_1)\\right|\\le \\left(\\frac{\\vert A\\vert }{1+SB^2}\\right)^{k(p_1)}~(1-\\epsilon ^2_{\\infty })^{n-k(p_1)}~\\vert x_1-x_2\\vert .$ with the parameter $\\epsilon _{\\infty }$ defined in (REF ).",
"In addition, for any $p_2\\ge 0$ we have $\\mathbb {E}\\left(\\left(\\widehat{X}_n(x_1,p_1)- \\widehat{X}_n(x_2,p_2)\\right)^2\\right)^{1/2}\\le ~\\kappa (p_1,p_2,x_2)~(1-\\epsilon ^2_{\\infty })^{n-k(p_1)}~\\vert x_1-x_2\\vert .$ The estimate (REF ) is a direct consequence of the formula (REF ) and the exponential estimate (REF ).",
"On the other hand, we have $\\widehat{X}_n(x,p)-X_n&=&\\frac{1}{1+S\\phi ^n(p)} ~(A\\widehat{X}_{n-1}(x,p)-X_n)+\\frac{S\\phi ^n(p)}{1+S\\phi ^n(p)}~(C^{-1}~Y_n-X_n)\\\\&=&\\frac{A}{1+S\\phi ^n(p)} ~(\\widehat{X}_{n-1}(x,p)-X_{n-1})+\\widehat{U}_n(p),$ where $\\widehat{U}_n(p):=\\frac{S\\phi ^n(p)}{1+S\\phi ^n(p)}~\\frac{V_n}{\\sqrt{S}}-\\frac{1}{1+S\\phi ^n(p)}~BW_n\\Longrightarrow \\mathbb {E}(\\widehat{U}_n(p)^2)\\le R+1/S.$ Iterating, this implies that $\\widehat{X}_n(x,p)-X_n= E_{n}(p) (x-X_0)+\\sum _{1\\le k\\le n}~ E_{k,n}(p)~\\widehat{U}_k(p)$ and hence $\\begin{array}{l}\\displaystyle \\mathbb {E}\\left(\\left(\\widehat{X}_n(x,p)-X_n\\right)^2 \\right)=\\vert E_{0,n}(p)\\vert ~\\mathbb {E}((x-X_0)^2)+\\sum _{1\\le l\\le n}~\\vert E_{l,n}(p)\\vert ~ \\mathbb {E}(\\widehat{U}_l(p)^2).\\end{array}$ Using (REF ) we check the uniform estimate $\\mathbb {E}\\left(\\left(\\widehat{X}_n(x,p)-X_n\\right)^2 \\right)\\le \\kappa _1(p)~(1+\\mathbb {E}((x-X_0)^2)).$ In the same vein, for any given $q\\in \\mathbb {R}$ and for any $(p_1,p_2)\\in \\mathbb {R}_+^2$ we check that $\\widehat{X}_n(x,p_1)- \\widehat{X}_n(x,p_2)&=&\\sum _{1\\le l\\le n}~ E_{l,n}(p_1)~\\rho _l(p_1,p_2)~U_l(x,p_2)$ with the function $\\begin{array}{l}\\displaystyle \\rho _n(p_1,p_2):=\\frac{S(\\phi ^n(p_1)-\\phi ^n(p_2))}{(1+S\\phi ^n(p_1))(1+S\\phi ^n(p_2))} \\le S\\kappa _1^2~\\left(1-\\epsilon _1\\right)^n~\\vert p_1-p_2\\vert \\quad \\mbox{\\rm (by (\\ref {ricc-lower-cv-2})})\\end{array}$ and the collection of random variables $\\begin{array}{l}\\displaystyle U_n(x,p):= A(X_{n-1}-\\widehat{X}_{n-1}(x,p))+BW_n+V_n/\\sqrt{S}\\\\\\\\\\Longrightarrow \\mathbb {E}\\left(U_n(x,p)^2 \\right)\\le A^2\\kappa _1(p)~(1+\\mathbb {E}((x-X_0)^2))+R+1/S\\quad \\mbox{\\rm (by (\\ref {X-p-q-X}))}.\\end{array}$ This implies that $\\mathbb {E}\\left(\\left(\\widehat{X}_n(x,p_1)- \\widehat{X}_n(x,p_2)\\right)^2\\right)^{1/2}\\le \\left(\\sum _{1\\le l\\le n}~\\vert E_{l,n}(p_1)\\vert ~\\rho _l(p_1,p_2)~\\right)~\\kappa _2(x,p_2)$ with $\\kappa _2(x,p):=(A^2\\kappa _1(p)+B^2+1/S)~(1+\\mathbb {E}((x-X_0)^2)).$ Using (REF ), for any $n>k(p_1)$ we have $\\sum _{1\\le l\\le n}~\\vert E_{l,n}(p_1)\\vert ~\\rho _l(p_1,p_2)\\le \\kappa _3(p_1)~(1-\\epsilon ^2_{\\infty })^{n-k(p_1)}~\\vert p_1-p_2\\vert .$ This ends the proof of (REF ) and hence the theorem.",
"Summing the two estimates (REF ) and (REF ) we obtain the estimate (REF ) with $\\epsilon _2=\\epsilon ^2_{\\infty }$ ."
],
[
"Stochastic perturbation analysis",
"This section is mainly concerned with the proof of Theorem REF .",
"In addition, we also consider the bias and some time-uniform bounds for the sample covariances $p_n$ and $\\widehat{p}_n$ ."
],
[
"A local perturbation theorem",
"We fix some parameter $N\\ge 1$ and we let 1 be the $(N+1)$ column vector with unit entries, $I$ the $(N+1)\\times (N+1)$ identity matrix and $J={1}{1}^{\\prime }$ the $(N+1)\\times (N+1)$ matrix with unit entries, where $(\\cdot )^{\\prime }$ denotes the transpose operator.",
"We also let $\\epsilon $ be the $(N+1)\\times (N+1)$ matrix given by $\\epsilon = I-{J}/{(N+1)}\\quad \\Longrightarrow \\quad \\epsilon ^2=\\epsilon .$ For any given $(N+1)$ -column $p$ we have $\\overline{p}:=\\frac{1}{N+1}\\sum _{1\\le j\\le N+1} p_j\\quad \\Longrightarrow \\quad p-\\overline{p}\\,{1}:=\\left(\\begin{array}{c}p_1-\\overline{p}\\\\\\vdots \\\\p_{N+1}-\\overline{p}\\end{array}\\right)=\\epsilon p.$ From the mean-field evolution equations (REF ) and the definitions of $m_n$ and $\\widehat{m}_n$ in (REF ), it is straightforward to show that $\\left\\lbrace \\begin{array}{rcl}\\displaystyle (\\xi ^i_n-m_n)&=& A~(\\widehat{\\xi }^i_{n-1}-\\widehat{m}_{n-1})+BW^{i,\\epsilon }_n\\\\\\\\(\\widehat{\\xi }^i_n-\\widehat{m}_n)&=&(1-g_nC)(\\xi ^i_n-m_n)-g_nDV^{i,\\epsilon }_n,\\end{array}\\right.$ with the random variables $W^{i,\\epsilon }_n:=\\sum _{1\\le j\\le N+1}\\epsilon ^i_j~W^j_n\\quad \\mbox{\\rm and}\\quad V^{i,\\epsilon }_n:=\\sum _{1\\le j\\le N+1}\\epsilon ^i_j~V^j_n.$ Taking the square we obtain the formulae $\\left\\lbrace \\begin{array}{rcl}\\displaystyle (\\xi ^i_n-m_n)^2&=& A^2~(\\widehat{\\xi }^i_{n-1}-\\widehat{m}_{n-1})^2+B^2(W^{i,\\epsilon }_n)^2+2AB~(\\widehat{\\xi }^i_{n-1}-\\widehat{m}_{n-1})W^{i,\\epsilon }_n\\\\\\\\(\\widehat{\\xi }^i_n-\\widehat{m}_n)^2&=&(1-g_nC)^2(\\xi ^i_n-m_n)^2+g_n^2D^2(V^{i,\\epsilon }_n)^2-2g_n(1-g_nC)D(\\xi ^i_n-m_n)V^{i,\\epsilon }_n.\\end{array}\\right.$ This implies that $\\left\\lbrace \\begin{array}{rcl}\\displaystyle p_n&=&\\displaystyle A^2~\\widehat{p}_{n-1}+B^2+\\frac{1}{\\sqrt{N}}~\\nu _n\\\\\\\\\\widehat{p}_n&=&\\displaystyle (1-g_nC)^2p_n+g_n^2D^2+\\frac{1}{\\sqrt{N}}~ \\widehat{\\nu }_n=(1-g_nC)p_n+\\frac{1}{\\sqrt{N}}~ \\widehat{\\nu }_n\\end{array}\\right.$ with the orthogonal variables $\\nu _n:=2AB~\\chi _n+B^2~\\varsigma _n\\quad \\mbox{\\rm and}\\quad \\widehat{\\nu }_n:=-2g_n(1-g_nC)D~\\widehat{\\chi }_n+g_n^2D^2~\\widehat{\\varsigma }_n$ and the centered random variables $\\begin{array}{rclcrcl}\\chi _n&:=&\\displaystyle \\frac{1}{\\sqrt{N}}~\\sum _{1\\le i\\le N+1}~(\\widehat{\\xi }^i_{n-1}-\\widehat{m}_{n-1})W^{i,\\epsilon }_n&&\\varsigma _n&:=&\\displaystyle \\sqrt{N}~\\left(\\frac{1}{N}\\sum _{1\\le i\\le N+1}(W^{i,\\epsilon }_n)^2-1\\right)\\\\&&&&&&\\\\\\widehat{\\chi }_n&:=&\\displaystyle \\frac{1}{\\sqrt{N}}~\\sum _{1\\le i\\le N+1}(\\xi ^i_n-m_n)V^{i,\\epsilon }_n&&\\widehat{\\varsigma }_n&:=&\\displaystyle \\sqrt{N}\\left(\\frac{1}{N}\\sum _{1\\le i\\le N+1}(V^{i,\\epsilon }_n)^2-1\\right).\\end{array}$ Further, denote by $\\mathbb {H}$ the $((N+1)\\times (N+1))$ -Helmert matrix whose first row is defined by $\\mathbb {H}_{1,j}=1/\\sqrt{N+1}\\quad \\forall 1\\le j\\le (N+1)\\quad \\Longrightarrow \\quad \\mathbb {H}_{1,\\mbox{\\LARGE .",
"}}={1}^{\\prime }/\\sqrt{N+1}.$ and whose $i$ -th row for $2\\le i\\le (N+1)$ is given by $\\begin{array}{l}\\mathbb {H}_{i,j}:=\\overline{\\mathbb {H}}_{i,j}:=\\left\\lbrace \\begin{array}{ccl}1/\\sqrt{i(i-1)}&\\mbox{\\rm if}& 1\\le j<i\\\\&&\\\\-(i-1)/\\sqrt{i(i-1)}&\\mbox{\\rm if}& j=i\\\\&&\\\\0&\\mbox{\\rm if}& i<j\\le (N+1).\\end{array}\\right.\\end{array}$ This yields the matrix decomposition $\\mathbb {H}=\\left(\\begin{array}{c}{1}^{\\prime }/\\sqrt{N+1}\\\\\\overline{\\mathbb {H}}\\end{array}\\right)\\quad \\mbox{\\rm with}\\quad \\overline{\\mathbb {H}}=(\\overline{\\mathbb {H}}_{i,j})_{2\\le i\\le (N+1), 1\\le j\\le (N+1)}\\in \\mathbb {R}^{N\\times (N+1)}.$ We check that $\\mathbb {H}$ is an orthogonal matrix and we have the square root formula $\\mathbb {H}\\mathbb {H}^{\\prime }=I=\\mathbb {H}^{\\prime }\\mathbb {H}=\\frac{1}{N+1}~J+\\overline{\\mathbb {H}}^{\\prime }~\\overline{\\mathbb {H}} \\Longrightarrow \\overline{\\mathbb {H}}^{\\prime }~\\overline{\\mathbb {H}}=I-\\frac{1}{N+1}~J=\\epsilon .$ Also note that for any $(N+1)$ -column vectors $x,w$ we have $\\begin{array}{l}\\displaystyle (\\overline{\\mathbb {H}} x)^{\\prime }(\\overline{\\mathbb {H}}x)=(x-\\overline{x}\\,{1})^{\\prime }(w-\\overline{w}\\,{1})=x^{\\prime }\\epsilon \\,w.\\end{array}$ In this notation, we readily check that $\\displaystyle \\widehat{m}_{n}&=&\\displaystyle m_n+g_n~(Y_n-Cm_n)-\\frac{1}{\\sqrt{N+1}}~g_nD~\\mathbb {H}_{1,\\mbox{\\LARGE .}}",
"{\\cal V }_n^{(N)}\\\\m_{n+1}&=&\\displaystyle A~\\widehat{m}_{n}+\\frac{1}{\\sqrt{N+1}}~B~\\mathbb {H}_{1,\\mbox{\\LARGE .}}",
"{\\cal W }_n^{(N)}$ with the random vectors $ {\\cal W }_n^{(N)}:=\\left(\\begin{array}{c}W^1_n\\\\\\vdots \\\\W^{N+1}_n\\\\\\end{array}\\right)\\quad \\mbox{\\rm and}\\quad {\\cal V }_n^{(N)}:=\\left(\\begin{array}{c}V^1_n\\\\\\vdots \\\\V^{N+1}_n\\\\\\end{array}\\right).$ Using this notation, we may write $\\begin{array}{rclcrcl}\\chi _n&=&\\displaystyle \\sqrt{\\frac{\\widehat{p}_{n-1}}{N}}~\\widehat{\\zeta }_{n-1}^{\\prime }~ \\overline{\\mathbb {H}}~ {\\cal W }_n^{(N)}&&\\varsigma _n&=&\\displaystyle \\frac{1}{\\sqrt{N}}~\\left(\\left(\\overline{\\mathbb {H}}\\, {\\cal W }^{(N)}_n\\right)^{\\prime }\\left(\\overline{\\mathbb {H}}\\, {\\cal W }^{(N)}_n\\right)-N\\right)\\\\&&&&&&\\\\\\widehat{\\chi }_n&=&\\displaystyle \\sqrt{\\frac{p_n}{N}}~\\zeta _n^{\\prime }~\\overline{\\mathbb {H}}~ {\\cal V }_n^{(N)}&&\\widehat{\\varsigma }_n&=&\\displaystyle \\frac{1}{\\sqrt{N}}~\\left(\\left(\\overline{\\mathbb {H}}\\, {\\cal V }^{(N)}_n\\right)^{\\prime }\\left(\\overline{\\mathbb {H}}\\, {\\cal V }^{(N)}_n\\right)-N\\right),\\end{array}$ with the random vectors on the unit sphere defined by $\\widehat{\\zeta }_{n-1}:=\\frac{\\overline{\\mathbb {H}}\\,\\widehat{\\xi }_{n-1}}{\\sqrt{\\Vert \\overline{\\mathbb {H}}\\,\\widehat{\\xi }_{n-1}\\Vert }}\\quad \\mbox{\\rm and}\\quad \\zeta _n:=\\frac{\\overline{\\mathbb {H}}\\,\\xi _n}{\\sqrt{\\Vert \\overline{\\mathbb {H}}\\,\\xi _n\\Vert }}.$ Finally observe that $\\mathbb {H}\\, {\\cal V }_n^{(N)}=\\left(\\begin{array}{c}\\mathbb {H}_{1,\\mbox{\\LARGE .}}",
"{\\cal V }_n^{(N)}\\\\\\overline{\\mathbb {H}}\\, {\\cal V }_n^{(N)}\\end{array}\\right)\\quad \\mbox{\\rm and}\\quad \\mathbb {H}\\, {\\cal W }_n^{(N)}=\\left(\\begin{array}{c}\\mathbb {H}_{1,\\mbox{\\LARGE .}}",
"{\\cal W }_n^{(N)}\\\\\\overline{\\mathbb {H}}\\, {\\cal W }_n^{(N)}\\end{array}\\right)\\sim {\\cal N }\\left(0,I_{(N+1)\\times (N+1)}\\right)$ We end the proof of Theorem REF by replacing $ ( {\\cal V }_n^{(N)}, {\\cal W }_n^{(N)})$ by $\\left( \\overline{\\mathbb {H}}\\, {\\cal V }_n^{(N)}, \\overline{\\mathbb {H}}\\, {\\cal W }_n^{(N)}\\right)$ .",
"This ends the proof of the theorem.",
"Corollary 5.1 For any $n\\ge 0$ we have the biasedness properties $\\mathbb {E}(\\widehat{p}_n)< \\widehat{P}_n\\quad \\mbox{and}\\quad \\mathbb {E}(p_{n+1})< P_{n+1}.$ Since the function $p\\in \\mathbb {R}_+\\mapsto \\frac{p}{1+Sp}\\in \\mathbb {R}_+$ is strictly concave, we have $\\mathbb {E}((1-g_nC)~p_n)< \\frac{\\overline{p}_n}{1+S\\overline{p}_n}=(1-\\overline{g}_nC)~\\overline{p}_n$ with $\\overline{g}_n=C~\\overline{p}_n/(C^2~\\overline{p}_n+D^2)\\quad \\mbox{\\rm and}\\quad \\overline{p}_n:=\\mathbb {E}(p_n).$ This implies that $\\mathbb {E}(\\widehat{p}_n)< \\displaystyle (1-\\overline{g}_nC)~\\overline{p}_n$ from which we check the Riccati differential inequality $\\begin{array}[c]{rcl}\\overline{p}_n&=&\\displaystyle A^2~\\mathbb {E}(\\widehat{p}_n)+R<A^2~ \\frac{\\overline{p}_{n-1}}{1+S\\overline{p}_{n-1}}+R=\\phi (\\overline{p}_{n-1}).\\end{array}$ The proof of (REF ) then follows from comparison properties of Riccati inequalities."
],
[
"Uniform raw moments estimates",
"For any $k\\ge 1$ there exists some finite constant $\\tau _k$ such that for any $N\\ge 1$ and $n\\ge 0$ , the Burkholder-Davis-Gundy inequality yields the following uniform estimates $\\mathbb {E}\\left(\\Vert \\widehat{\\Delta }_{n}\\Vert ^{k}\\right)\\vee \\mathbb {E}\\left(\\Vert \\Delta _{n}\\Vert ^{k}\\right)\\le \\tau _k.$ black Proposition 5.2 For any $k\\ge 1$ , any $N\\ge 1$ and time horizon $n\\ge 0$ , we have the time-uniform estimates $\\begin{array}{l}\\mathbb {E}(\\widehat{p}_{n}^{k})\\vee \\mathbb {E}\\left(\\widehat{\\nu }_n^{k}\\right)\\le \\iota _k\\\\\\\\\\mathbb {E}(p_{n}^{k})\\le \\iota _k~(P_0\\vee 1)^k\\quad \\mbox{and}\\quad \\mathbb {E}\\left(\\vert \\nu _{n+1}\\vert ^{k}\\right)\\vee \\mathbb {E}(\\delta _{n+1}^{k})\\le \\iota _k~(P_0\\vee 1)^{k/2}.\\end{array}$ We first prove the last assertion.",
"To this end, for any $k\\in \\mathbb {R}$ and $n\\ge 1$ , set $q^{k}_n:=\\mathbb {E}(p_{n}^{k})$ and observe that for $k \\ge 1$ , $2^{k-1}\\phi (x)^k=2^{k-1}\\frac{((A^2+RS)x+R)^k}{(Sx+1)^k}\\le \\phi _{k}(x^k) \\quad \\mbox{\\rm where}\\quad \\phi _{k}(x):=\\frac{a_kx+b_k}{c_kx+1},$ with parameters $\\begin{array}{l}(a_k,b_k,c_k)=(2^{2(k-1)}(A^2+RS)^k,2^{2(k-1)}R^{k},S^k)\\in \\mathbb {R}_+^3\\\\\\\\\\Longrightarrow \\quad a_k-b_kc_k=2^{2(k-1)}\\left((A^2+RS)^k-R^{k}S^k\\right)>0.\\end{array}$ Combining the above with Corollary REF and the inequality $|a + b|^k \\le 2^{k-1}(|a|^k + |b|^k)$ , $a, b \\in \\mathbb {R}$ , $k \\ge 1$ , we obtain $p_{n+1}^{2k}\\le \\phi _{2k}(p_n^{2k})+\\frac{2^{2k-1}}{N^k}~\\left(A^{4k}~\\widehat{\\nu }_n^{2k}+\\nu ^{2k}_{n+1}\\right).$ By Jensen's inequality, we check that $q^{2k}_{n+1}\\le \\phi _{2k}(q^{2k}_n)+\\frac{2^{2k-1}}{N^k}~\\left(A^{4k}~\\mathbb {E}(\\widehat{\\nu }_n^{2k})+\\mathbb {E}(\\nu ^{2k}_{n+1})\\right).$ To control the expectations on the right hand side above, note that for any $k\\ge 1$ we have the crude estimates $(g_nD)^{2k}&=&\\left(\\frac{Sp_n}{1+Sp_n}~\\frac{p_n}{1+Sp_n}\\right)^{k}\\le S^{-k},\\\\(g_nD(1-g_nC))^{k}~p_{n}^{k/2}&=&\\left(\\frac{Sp_n}{1+Sp_n}~\\frac{1}{1+Sp_n}~\\left(\\frac{p_n}{1+Sp_n}\\right)^2\\right)^{k/2}\\le S^{-k}.$ Combining these estimates with (REF ) in (REF ), we check that $\\mathbb {E}\\left(\\widehat{\\nu }_n^{k}\\right)\\le \\iota _1(k)\\quad \\mbox{\\rm and}\\quad \\mathbb {E}(\\nu _{n+1}^{k})\\le \\iota _2(k)~\\left(1\\vee q_n^{k/2}\\right) = \\iota _2(k)~\\left(1\\vee q_n\\right)^{k/2}.$ Now, (REF ) and (REF ) imply that $q_n\\le \\iota _0~(P_0\\vee 1).$ Combining this with (REF ) yields the required bounds for $\\nu _n^k$ , $\\widehat{\\nu }_n^k$ and $\\delta _n^k$ .",
"Returning to the moments of $p_n$ , note that the previous estimates imply that $q^{2k}_{n+1}\\le \\phi _{2k}(q^{2k}_n)+\\frac{\\iota _5(k)}{N^k}~(q^k_n\\vee 1).$ Applying the above estimate to $k=1$ , along with (REF ), we have $q^{2}_{n+1}\\le \\phi _{2}(q^{2}_n)+\\frac{\\iota _1}{N}~( P_0\\vee 1).$ Using the comparison properties (REF ) and (REF ) we conclude that $\\sup _{N\\ge 1}\\sup _{n\\ge 0}q^{2}_{n}<\\iota _2~(P_0\\vee 1)^2.$ Iterating the argument for any $k\\ge 1$ we readily check that $\\sup _{N\\ge 1}\\sup _{n\\ge 0}q^{k}_{n}<\\iota _k~(P_0\\vee 1)^k$ and therefore $\\mathbb {E}\\left(\\vert \\nu _{n+1}\\vert ^{k}\\right)\\le \\iota _6(k)~\\left(1\\vee q_n\\right)^{k/2}\\le \\iota _7(k)~(P_0\\vee 1)^{k/2}\\quad \\mbox{\\rm and}\\quad \\mathbb {E}\\left(\\vert \\delta _{n+1}\\vert ^k\\right)\\le \\iota _k~(P_0\\vee 1)^{k/2}$ Finally, we have $\\widehat{p}_n=\\frac{p_n}{1+Sp_n}+\\frac{1}{\\sqrt{N}}~\\widehat{\\nu }_n\\Longrightarrow \\mathbb {E}(\\widehat{p}_{n}^{k})\\le S^{-k} +\\frac{1}{N^{k/2}}~ \\mathbb {E}(\\vert \\widehat{\\nu }_n\\vert ^{k})\\le \\iota _k,$ which ends the proof of the proposition."
],
[
"Stochastic Riccati equations",
"In this section we focus on the proof of Theorem REF , as well as the central limit theorems presented in section REF ."
],
[
"Perturbation analysis",
"We start by proving the time uniform estimates (REF ).",
"To this end, first note that we have the telescoping sum formula $p_n-P_n=\\left(\\phi ^n(p_0)-\\phi ^n(P_0)\\right)+\\sum _{1\\le k\\le n} \\left(\\phi ^{n-k}(p_k)-\\phi ^{n-(k-1)}(p_{k-1})\\right).$ To deal with the summands above, observe that $\\begin{array}{rcl}\\phi ^{n-k}(p_k)-\\phi ^{n-(k-1)}(p_{k-1})&=&\\phi ^{n-k}(p_k)-\\phi ^{n-k}(\\phi (p_{k-1}))\\\\&&\\\\&=&\\displaystyle \\phi ^{n-k}\\left(\\phi (p_{k-1})+\\frac{1}{\\sqrt{N}}~ \\delta _k\\right)-\\phi ^{n-k}\\left(\\phi (p_{k-1})\\right).\\end{array}$ Using the Lipschitz estimates (REF ) we check that $\\vert \\phi ^{n-k}(p_k)-\\phi ^{n-k}(\\phi (p_{k-1}))\\vert \\le \\frac{\\kappa _1}{\\sqrt{N}}~(1-\\epsilon _1)^{n-k}~\\left|\\delta _k\\right|,$ with the parameter $\\epsilon _1$ defined in (REF ).",
"Similarly, we have $\\vert \\phi ^n(p_0)-\\phi ^n(P_0)\\vert \\le \\frac{\\kappa _2}{\\sqrt{N}}~(1-\\epsilon _1)^{n}~\\left|\\nu _0\\right|.$ This yields the almost sure estimate $\\begin{array}{l}\\displaystyle \\sqrt{N}~\\vert p_n-P_n\\vert \\le \\kappa _3~\\left[(1-\\epsilon _1)^{n}\\vert \\nu _0\\vert +\\sum _{1\\le k\\le n} ~(1-\\epsilon _1)^{n-k}~\\left|\\delta _k\\right|\\right].\\end{array}$ Next we note that $\\vert \\widehat{p}_n-\\widehat{P}_n\\vert =\\left|\\frac{p_n-P_n}{(1+SP_n)(1+Sp_n)}\\right|+\\frac{1}{\\sqrt{N}}~\\vert \\widehat{\\nu }_n\\vert \\le \\left|p_n-P_n\\right|+\\frac{1}{\\sqrt{N}}~\\vert \\widehat{\\nu }_n\\vert .$ Finally, we note the following decomposition for the gain parameters $\\sqrt{N}(g_n - G_n)&= \\sqrt{N}C\\left(\\frac{p_n}{C^2p_n + D^2} - \\frac{P_n}{C^2P_n + D^2}\\right) \\\\&= CD^2 \\frac{\\sqrt{N}(p_n - P_n)}{(C^2p_n + D^2)(C^2P_n + D^2)}.$ The above estimates and decompositions allow one to derive several mean error bounds as soon as we control the moments of the local errors.",
"For instance, using the uniform moments estimates (REF ) we obtain the mean error estimates stated in Theorem REF ."
],
[
"Fluctuation analysis",
"This section is mainly concerned with the proofs of Theorem REF and Theorem REF .",
"We start with some technical results that will immediately lead to the proof of the former.",
"Lemma 6.1 The characteristic function of random vector $\\Delta $ defined in (REF ) satisfies for any $w\\in \\mathbb {R}^3$ we have the Gaussian type estimate $\\left|\\mathbb {E}\\left(e^{i w^{\\prime }\\Delta }\\right)-e^{-\\frac{\\Vert w\\Vert ^2}{2}}\\right|\\le \\epsilon (w)~e^{-\\frac{\\Vert w\\Vert ^2}{2}},$ with the function $\\epsilon (w):=\\vert w_3\\vert \\sqrt{\\frac{2}{N}}\\left( \\frac{w_2^2}{2}+w_3^2\\right)~\\exp {\\left(\\vert w_3\\vert \\sqrt{\\frac{2}{N}}\\left( \\frac{w_2^2}{2}+w_3^2\\right)\\right)}.$ The proof of the above lemma follows elementary calculations.",
"For the convenience of the reader a detailed proof is provided in the appendix.",
"Lemma 6.2 For any $n\\ge 0$ we have the weak convergence $\\left(\\Delta _0,(\\widehat{\\Delta }^i_k,\\Delta ^i_{k+1})_{0\\le k\\le n,~1\\le i\\le 3}\\right)\\hookrightarrow _{N\\rightarrow \\infty }~\\left((Z^i_0)_{1\\le i\\le 2},(\\widehat{Z}^i_k,Z^i_{k+1})_{0\\le k\\le n,~1\\le i\\le 3}\\right),$ where the $Z^i_k,\\widehat{Z}^i_k$ are independent sequences of independent and centered Gaussian random variables with unit variance.",
"Applying (REF ) for any $w\\in \\mathbb {R}^3$ we check the rather crude estimates $\\left|\\mathbb {E}\\left(e^{i~w^{\\prime }\\Delta _{n+1}}\\right)-e^{-\\frac{\\Vert w\\Vert ^2}{2}}\\right|\\le \\epsilon (w)~\\quad \\mbox{\\rm and}\\quad \\left|\\mathbb {E}\\left(e^{i~w^{\\prime }\\widehat{\\Delta }_{n}}\\right)-e^{-\\frac{\\Vert w\\Vert ^2}{2}}\\right|\\le \\epsilon (w)~$ with the function $\\epsilon (w)\\longrightarrow _{N\\rightarrow \\infty }0$ defined in (REF ).",
"For any $u,v\\in \\mathbb {R}^3$ and $n\\ge 0$ we have the decomposition $\\begin{array}{l}\\displaystyle \\mathbb {E}\\left(e^{i~u^{\\prime }\\Delta _{n+1}}~e^{i~v^{\\prime }\\widehat{\\Delta }_{n}}\\right)-e^{-\\frac{\\Vert u\\Vert ^2}{2}}e^{-\\frac{\\Vert v\\Vert ^2}{2}}\\\\\\\\\\displaystyle =\\mathbb {E}\\left(e^{i~v^{\\prime }\\widehat{\\Delta }_{n}}\\right)~\\left(\\mathbb {E}\\left(e^{i~u^{\\prime }\\Delta _{n+1}}\\right)-e^{-\\frac{\\Vert u\\Vert ^2}{2}}\\right)+\\left(\\mathbb {E}\\left(e^{i~v^{\\prime }\\widehat{\\Delta }_{n}}\\right)-e^{-\\frac{\\Vert v\\Vert ^2}{2}}\\right)e^{-\\frac{\\Vert u\\Vert ^2}{2}}.\\end{array}$ This yields the estimate $\\left|\\mathbb {E}\\left(e^{i~u^{\\prime }\\Delta _{n+1}}~e^{i~v^{\\prime }\\widehat{\\Delta }_{n}}\\right)-e^{-\\frac{\\Vert u\\Vert ^2+\\Vert v\\Vert ^2}{2}}\\right|\\le \\epsilon (u,v):=\\epsilon (u)+\\epsilon (v).$ Consider the sequence of random vectors $\\Lambda _n:=\\left(\\begin{array}{c}\\widehat{\\Delta }_{n}\\\\\\Delta _{n+1}\\end{array}\\right)\\in \\mathbb {R}^6.$ In this notation, for any $n\\ge 0$ and $w_n\\in \\mathbb {R}^6$ we have proven the following estimate $\\left|\\mathbb {E}\\left(e^{i~w_n^{\\prime }\\Lambda _n}\\right)-e^{-\\frac{\\Vert w_n\\Vert ^2}{2}}\\right|\\le \\epsilon (w_n).$ Using the decomposition $\\begin{array}{l}\\mathbb {E}\\left(e^{i~\\left(w_{n-1}^{\\prime }\\Lambda _{n-1}+w_n^{\\prime }\\Lambda _n\\right)}\\right)-e^{-\\frac{\\Vert w_{n-1}\\Vert ^2}{2}}e^{-\\frac{\\Vert w_n\\Vert ^2}{2}}\\\\\\\\=\\mathbb {E}\\left(e^{i~w_{n-1}^{\\prime }\\Lambda _{n-1}}\\right)~\\left(\\mathbb {E}\\left(e^{i~w_n^{\\prime }\\Lambda _n}\\right)-e^{-\\frac{\\Vert w_n\\Vert ^2}{2}}\\right)\\\\\\\\\\hspace{85.35826pt}+\\left(\\mathbb {E}\\left(e^{i~w_{n-1}^{\\prime }\\Lambda _{n-1}}\\right)-e^{-\\frac{\\Vert w_{n-1}\\Vert ^2}{2}}~\\right)e^{-\\frac{\\Vert w_n\\Vert ^2}{2}},\\end{array}$ we also check that $\\begin{array}{l}\\left|\\mathbb {E}\\left(e^{i~\\left(w_{n-1}^{\\prime }\\Lambda _{n-1}+w_n^{\\prime }\\Lambda _n\\right)}\\right)-e^{-\\frac{\\Vert w_{n-1}\\Vert ^2}{2}}e^{-\\frac{\\Vert w_n\\Vert ^2}{2}}\\right|\\le \\epsilon (w_{n-1})+\\epsilon (w_n).\\end{array}$ Iterating the argument, we conclude that $\\begin{array}{l}\\left|\\mathbb {E}\\left(e^{i~\\left(u_1\\Delta ^1_0+u_2\\Delta ^2_0+\\sum _{0\\le k\\le n}w_{k}^{\\prime }\\Lambda _k\\right)}\\right)-e^{-\\frac{ u_1^2/2+u_2^2/2+\\sum _{0\\le k\\le n}\\Vert w_{k}\\Vert ^2}{2}}\\right|.\\\\\\\\\\le \\epsilon (u_1,0,u_2)+ \\sum _{0\\le k\\le n}\\epsilon (w_{k})\\longrightarrow _{N\\rightarrow \\infty }0,\\end{array}$ which ends the proof of the lemma.",
"Combining Lemma REF with Theorem REF and Slutsky's lemma yields Theorem REF .",
"We are now in position to prove theorem REF and the bias estimates (REF ).",
"Proof of Theorem REF : By the second order estimate (REF ) we have $\\begin{array}{l}\\displaystyle \\vert \\phi ^{n-k}(p_k)-\\phi ^{n-(k-1)}(p_{k-1})-\\partial \\phi ^{n-k}\\left(\\phi (p_{k-1})\\right)\\frac{1}{\\sqrt{N}}~\\delta _k\\vert \\le \\frac{\\iota _1}{N}~(1-\\epsilon _1)^{n-k}~\\delta _k^2.\\end{array}$ Combining this with the telescoping sum formula (REF ) yields the decomposition $\\mathbb {Q}^N_n:=\\sqrt{N}(p_n-P_n)=\\partial \\phi ^n(P_0)~\\nu _0+\\sum _{1\\le k\\le n} \\partial \\phi ^{n-k}\\left(\\phi (p_{k-1})\\right)~\\delta _k+\\theta _1^N(1),$ with the remainder term $\\vert \\theta ^N_n(1)\\vert \\le \\epsilon _n^N(1):= \\frac{\\iota _2}{\\sqrt{N}}~(1-\\epsilon _1)^n~\\nu _0^2~+\\frac{\\iota _1}{\\sqrt{N}}~\\sum _{1\\le k\\le n} ~(1-\\epsilon _1)^{n-k}~\\delta _k^2\\longrightarrow _{N\\rightarrow \\infty }~0\\quad \\mbox{a.s.}.$ On the other hand, using the Lipschitz estimates (REF ) and (REF ), we have $\\begin{array}{l}\\displaystyle \\left|\\sum _{1\\le k\\le n} \\left[\\partial \\phi ^{n-k}(\\phi (p_{k-1}))-\\partial \\phi ^{n-k}(\\phi (P_{k-1}))\\right]~\\delta _k~\\right|\\\\\\\\\\displaystyle \\le \\epsilon _n^N(2):=\\iota _3 \\sum _{1\\le k\\le n}~(1-\\epsilon )^{n-k}~\\vert \\delta _k\\vert ~ \\vert p_{k-1}-P_{k-1}\\vert \\longrightarrow _{N\\rightarrow \\infty }~0\\quad \\mbox{a.s.}.\\end{array}$ This yields the formula $\\mathbb {Q}^N_n=F_n\\left(\\nu _0,\\delta _1,\\ldots ,\\delta _n\\right)+\\theta _n^N(2)$ with the function $\\displaystyle F_n\\left(\\nu _0,\\delta _1,\\ldots ,\\delta _n\\right):=\\partial \\phi ^n(P_0)~\\nu _0+\\sum _{1\\le k\\le n} \\partial \\phi ^{n-k}\\left(P_k\\right)~\\delta _k,$ and the remainder term $\\theta _n^N(2)$ satisfying $\\vert \\theta _n^N(2)\\vert \\le \\epsilon _n^N(1)+\\epsilon _n^N(2).$ Combining Slutsky's lemma with the continuous mapping theorem and Corollary REF , for any time horizon $n\\ge 0$ we conclude that $(F_k\\left(\\nu _0,\\delta _1,\\ldots ,\\delta _k\\right))_{0\\le k\\le n}\\longrightarrow _{N\\rightarrow \\infty }~(F_k\\left(\\mathbb {Z}_0,\\mathbb {Z}_1,\\ldots ,\\mathbb {Z}_k\\right))_{0\\le k\\le n}$ with the Gaussian random variables $\\mathbb {Z}_k$ defined in Corollary REF .",
"Thus, we have shown that $\\mathbb {Q}_n^N \\longrightarrow _{N \\rightarrow \\infty } F_n\\left(\\mathbb {Z}_0,\\mathbb {Z}_1,\\ldots ,\\mathbb {Z}_n\\right) =: \\mathbb {Q}_n.$ The recursive formulation (REF ) comes from the decomposition $\\widehat{\\mathbb {Q}}^N_n=\\frac{1}{(1+SP_n)(1+Sp_n)}~\\mathbb {Q}^N_n+\\widehat{\\nu }_n\\longrightarrow _{N\\rightarrow \\infty } \\widehat{\\mathbb {Q}}_n:=\\frac{1}{(1+SP_n)^2}~\\mathbb {Q}_n+\\widehat{\\mathbb {V}}_n.$ In the same vein, we have $\\mathbb {Q}^N_{n+1}=A^2~\\widehat{\\mathbb {Q}}^N_n+\\nu _{n+1}\\Longrightarrow \\mathbb {Q}_{n+1}=A^2\\widehat{\\mathbb {Q}}_n+\\mathbb {V}_{n+1}.$ The proof of the uniform bias estimates (REF ) follows the second order Taylor expansions discussed in the proof of Theorem REF as we will now demonstrate.",
"Proof of the bias estimate (REF ): Using the second order Taylor expansions discussed in the proof of Theorem REF we have the bias estimates $\\begin{array}{l}\\displaystyle 0\\le P_n-\\mathbb {E}(p_n)\\le \\frac{ \\iota _4}{N}~\\left[\\mathbb {E}(\\nu _0^2)~(1-\\epsilon )^n+~\\sum _{1\\le k\\le n}(1-\\epsilon )^{n-k}~ \\mathbb {E}(\\delta _k^2)\\right]\\\\\\\\\\displaystyle \\hspace{142.26378pt}+\\frac{\\iota _5}{\\sqrt{N}} \\sum _{1\\le k\\le n}~(1-\\epsilon )^{n-k}~\\mathbb {E}(\\vert \\delta _k\\vert ~ \\vert p_{k-1}-P_{k-1}\\vert ).\\end{array}$ The first bias estimte stated in (REF ) now follows from (REF ) and the estimates stated in (REF ).",
"blackIndeed, using (REF ) we check that $0\\le P_n-\\mathbb {E}(p_n)&\\le & \\frac{ \\iota _6}{N}~\\left[P_0~(1-\\epsilon )^n+~(1\\vee P_0)\\sum _{1\\le k\\le n}(1-\\epsilon )^{n-k}\\right] \\\\&&\\hspace{42.67912pt}+\\frac{\\iota _7}{\\sqrt{N}}~(1\\vee P_0)~ \\sum _{1\\le k\\le n}~(1-\\epsilon )^{n-k}~~\\mathbb {E}( \\vert p_{k-1}-P_{k-1}\\vert ^2)^{1/2}.$ The end of the proof of the l.h.s.",
"estimate in (REF ) is now a consequence of (REF ).",
"We also have the second order decomposition $g_n - G_n&=& \\frac{CD^2}{(C^2P_n + D^2)^2}~(p_n - P_n)-\\frac{CD^2}{(C^2P_n + D^2)^2(C^2p_n + D^2)}~(p_n - P_n)^2$ Using the bias estimate stated in the l.h.s.",
"of (REF ) we readily check that $0\\le G_n-\\mathbb {E}(g_n) \\le \\frac{CD^2}{(C^2P_n + D^2)^2}~\\frac{\\iota _1}{N}~\\left[1\\vee P_0\\right]^{2}+\\frac{C}{(C^2P_n + D^2)^2}~\\mathbb {E}\\left((p_n - P_n)^2\\right)$ Again, using the $\\mathbb {L}_k$ -mean error estimates stated in (REF ) we check the r.h.s.",
"estimate stated in (REF ).",
"This ends the proof of the bias estimate."
],
[
"Inverse raw moments",
"We end the section with some bounds on the moments of ratios of the form $\\phi (p_n)/p_{n+1}$ , $n \\ge 1$ .",
"Using (REF ) we have the formulae, which will be used throughout this section.",
"$\\frac{1}{\\widehat{p}_n}=\\left(S+\\frac{1}{p_n}\\right)~\\widehat{\\Upsilon }_n(p_n)\\quad \\mbox{\\rm and}\\quad \\frac{1}{p_{n+1}}=\\frac{1}{A^2\\widehat{p}_n+R}~\\Upsilon _{n+1}(\\widehat{p}_n),$ with the inverse non-central $\\chi $ -square random variables $\\widehat{\\Upsilon }_n(p_n):=\\frac{1+1/(Sp_n)}{\\frac{1}{N}~\\widehat{\\chi }^{(n,2)}_{N,N/(Sp_n)}}\\quad \\mbox{\\rm and}\\quad \\Upsilon _{n+1}(\\widehat{p}_n):=\\frac{1+\\left(A^2/R\\right)\\widehat{p}_n}{\\frac{1}{N}~\\chi ^{(n+1,2)}_{N,N\\left(A^2/R\\right)\\widehat{p}_n}}.$ Lemma 6.3 For any $n\\ge 0$ we have $\\begin{array}{l}\\displaystyle \\frac{\\phi (p_n)}{p_{n+1}}-1=\\frac{1}{N}\\frac{A^4}{\\phi (p_n)}~\\frac{\\widehat{\\nu }_n^2}{\\phi (p_n)+\\frac{A^2}{\\sqrt{N}}~\\widehat{\\nu }_n}-\\frac{A^2}{\\sqrt{N}}~ \\frac{\\widehat{\\nu }_n}{\\phi (p_n)}\\\\\\\\\\hspace{142.26378pt}\\displaystyle +\\frac{\\phi (p_n)}{A^2\\widehat{p}_n+R}~\\left(\\Upsilon _{n+1}(\\widehat{p}_n)-1\\right).\\end{array}$ First recall that $\\widehat{p}_n=\\displaystyle (1-g_nC)~p_n+\\frac{1}{\\sqrt{N}}~ \\widehat{\\nu }_n.$ Combining this with the second identity in (REF ), we have $\\begin{array}{l}\\displaystyle \\frac{\\phi (p_n)}{p_{n+1}}=\\frac{1}{1+\\frac{A^2}{\\sqrt{N}}~ \\frac{\\widehat{\\nu }_n}{\\phi (p_n)}}+\\frac{\\phi (p_n)}{A^2\\widehat{p}_n+R}~\\left(\\Upsilon _{n+1}(\\widehat{p}_n)-1\\right).\\end{array}$ The result now follows from the decomposition $\\frac{1}{1+\\frac{A^2}{\\sqrt{N}}~ \\frac{\\widehat{\\nu }_n}{\\phi (p_n)}}=1-\\frac{A^2}{\\sqrt{N}}~ \\frac{\\widehat{\\nu }_n}{\\phi (p_n)}+\\frac{1}{N}\\frac{A^4}{\\phi (p_n)}~\\frac{\\widehat{\\nu }_n^2}{\\phi (p_n)+\\frac{A^2}{\\sqrt{N}}~\\widehat{\\nu }_n}.$ Proposition 6.4 For any $n\\ge 0$ and any even number of particles $N>4$ we have the almost sure uniform drift estimates $1\\le \\mathbb {E}\\left(\\frac{\\phi (p_n)}{p_{n+1}}~|~p_{n}\\right)\\le 1+\\frac{\\iota }{N}.$ First note that from the previous lemma, we have $\\displaystyle \\mathbb {E}\\left(\\frac{\\phi (p_n)}{p_{n+1}}~\\bigg |~p_n,\\widehat{p}_n\\right)\\displaystyle =1-\\frac{A^2}{\\sqrt{N}}~ \\frac{\\widehat{\\nu }_n}{\\phi (p_n)}+\\frac{1}{N}\\frac{A^4}{\\phi (p_n)}~\\frac{\\widehat{\\nu }_n^2}{\\phi (p_n)+\\frac{A^2}{\\sqrt{N}}~\\widehat{\\nu }_n}\\\\+~\\frac{\\phi (p_n)}{A^2\\widehat{p}_n+R}~\\mathbb {E}\\left(\\left(\\Upsilon _{n+1}(\\widehat{p}_n)-1\\right)~|~\\widehat{p}_n\\right).$ Due to (REF ), for any even number of particles $N>4$ we have the estimate $0\\le \\mathbb {E}\\left(\\Upsilon _n(x)\\right)-1\\le \\frac{4}{N}~\\left(1+\\frac{A^2}{R}~x\\right)~\\left(1+\\frac{4}{N-4}\\right).$ Since both $\\widehat{p}_n$ and $\\phi (p_n)$ are bounded, it follows that $\\mathbb {E}\\left(\\frac{\\phi (p_n)}{p_{n+1}}~\\bigg |~p_n,\\widehat{p}_n\\right) \\le 1-\\frac{A^2}{\\sqrt{N}}~ \\frac{\\widehat{\\nu }_n}{\\phi (p_n)}+\\frac{1}{N}\\frac{A^4}{\\phi (p_n)}~\\frac{\\widehat{\\nu }_n^2}{\\phi (p_n)+\\frac{A^2}{\\sqrt{N}}~\\widehat{\\nu }_n}+\\frac{\\iota _1}{N}.$ for some finite constant $\\iota _1$ .",
"Taking the expectation we check that $\\displaystyle \\mathbb {E}\\left(\\frac{\\phi (p_n)}{p_{n+1}}~|~p_n\\right)\\displaystyle &\\le \\mathbb {E}\\left(1+\\frac{1}{N}\\frac{A^4}{\\phi (p_n)}~\\frac{\\widehat{\\nu }_n^2}{\\phi (p_n)+\\frac{A^2}{\\sqrt{N}}~\\widehat{\\nu }_n}~|~p_n\\right)+\\frac{\\iota _1}{N}\\\\\\ \\\\\\displaystyle &\\le 1+\\frac{1}{N}\\left(\\iota _1+\\frac{A^4}{R^2}~\\mathbb {E}\\left(~\\widehat{\\nu }_n^2~|~p_n\\right) \\right),$ where the last line follows from the estimates $\\phi (p_n)\\ge R\\quad \\mbox{\\rm and}\\quad R\\le A^2\\widehat{p}_n+R=\\phi (p_n)+\\frac{A^2}{\\sqrt{N}}~ \\widehat{\\nu }_n.$ The proposition now follows from the almost sure estimate $\\mathbb {E}\\left(\\widehat{\\nu }_n^2~|~p_n\\right)&=&(2g_n(1-g_nC)D~\\sqrt{p_n})^2+(\\sqrt{2}~g_n^2D^2)^2 \\\\&=&2S\\frac{p_n^3}{(1+Sp_n)^4}~\\left(2+Sp_n\\right)\\le 4S~\\frac{p_n^3}{(1+Sp_n)^3}\\le \\frac{4}{S^2}.$ Theorem 6.5 For any even $k\\ge 2$ there exists some finite constant $\\iota _k$ such that for any $n\\ge 0$ and any even parameter $N>2(k+2)$ we have the almost sure uniform estimates $1\\le \\mathbb {E}\\left(\\left(\\frac{\\phi (p_n)}{p_{n+1}}\\right)^k~|~p_n\\right)\\le 1+\\frac{\\iota _k}{N}.$ We will show that (REF ) holds for the cases $k = 2, 3$ and then outline how the general case may be obtained inductively.",
"By (REF ), for any $n\\ge 1$ any $k\\ge 0$ and any even $N>2(k+2)$ we have the uniform estimates $\\mathbb {E}\\left(\\left(\\frac{\\Upsilon _{n}(x)-1}{A^2x+R}\\right)^k\\right)\\le \\frac{\\iota _k}{N}.$ Now, recall that $\\frac{\\phi (p_n)}{p_{n+1}}-1=\\widehat{\\alpha }_n(p_n)+\\beta _{n+1}(\\widehat{p}_n),$ with the random variables $\\widehat{\\alpha }_n(p_n)=-\\frac{1}{\\sqrt{N}}~\\frac{A^2}{A^2\\widehat{p}_n+R}~ \\widehat{\\nu }_n\\quad \\mbox{\\rm and}\\quad \\beta _{n+1}(\\widehat{p}_n):=\\phi (p_n)~\\frac{\\Upsilon _{n+1}(\\widehat{p}_n)-1}{A^2\\widehat{p}_n+R}.$ Using the estimates (REF ), (REF ) and (REF ), we check the almost sure uniform estimate $\\begin{array}{l}\\displaystyle \\mathbb {E}\\left(\\left(\\widehat{\\alpha }_n(p_n)+\\beta _{n+1}(\\widehat{p}_n)\\right)~|~p_n\\right)\\\\\\\\\\displaystyle \\le \\frac{1}{N}\\frac{A^4}{R^2}~\\mathbb {E}(\\widehat{\\nu }_n^2~|~p_n)+\\phi (p_n)~\\mathbb {E}\\left(~\\frac{\\Upsilon _{n+1}(\\widehat{p}_n)-1}{A^2\\widehat{p}_n+R}~|~p_n\\right)\\le \\frac{\\iota _1}{N}.\\end{array}$ In the same vein, we have $\\begin{array}{l}\\displaystyle \\mathbb {E}\\left(\\left(\\widehat{\\alpha }_n(p_n)+\\beta _{n+1}(\\widehat{p}_n)\\right)^2~|~p_n,\\widehat{p}_n\\right)\\\\\\\\\\displaystyle =\\widehat{\\alpha }_n(p_n)^2+\\phi (p_n)^2~\\mathbb {E}\\left(\\left(\\frac{\\Upsilon _{n+1}(\\widehat{p}_n)-1}{A^2\\widehat{p}_n+R}\\right)^2~|~p_n,\\widehat{p}_n\\right)\\\\\\\\\\displaystyle \\hspace{85.35826pt}+2~\\phi (p_n)~\\widehat{\\alpha }_n(p_n)~\\mathbb {E}\\left(\\frac{\\Upsilon _{n+1}(\\widehat{p}_n)-1}{A^2\\widehat{p}_n+R}~|~p_n,\\widehat{p}_n\\right).\\end{array}$ On the other hand, we have $\\begin{array}{l}\\displaystyle \\widehat{\\alpha }_n(p_n)~\\mathbb {E}\\left(\\frac{\\Upsilon _{n+1}(\\widehat{p}_n)-1}{A^2\\widehat{p}_n+R}~|~p_n,\\widehat{p}_n\\right)\\\\\\\\\\displaystyle \\le \\widehat{\\alpha }_n(p_n)^2+\\mathbb {E}\\left(\\frac{\\Upsilon _{n+1}(\\widehat{p}_n)-1}{A^2\\widehat{p}_n+R}~|~p_n,\\widehat{p}_n\\right)^2\\le \\widehat{\\alpha }_n(p_n)^2+\\iota _2/N^2.\\end{array}$ Along with the almost sure uniform bound $\\mathbb {E}\\left(\\widehat{\\alpha }_n(p_n)^2~|~p_n\\right)\\le {\\iota _3}/{N},$ which follows from (REF ), we conclude that $\\mathbb {E}\\left(\\left(\\widehat{\\alpha }_n(p_n)+\\beta _{n+1}(\\widehat{p}_n)\\right)^2~|~p_n\\right)\\le {\\iota _4}/{N}.$ Combining this with (REF ), it follows that $1\\le \\mathbb {E}\\left(\\left(\\frac{\\phi (p_n)}{p_{n+1}}\\right)^2~|~p_n=x\\right)\\le 1+{\\iota _5}/{N}.$ For odd powers, for instance we have $\\begin{array}{l}\\displaystyle \\mathbb {E}\\left(\\left(\\widehat{\\alpha }_n(p_n)+\\beta _{n+1}(\\widehat{p}_n)\\right)^3~|~p_n,\\widehat{p}_n\\right)\\\\\\\\\\displaystyle =\\widehat{\\alpha }_n(p_n)^3+3~\\widehat{\\alpha }_n(p_n)^2\\phi (p_n)~\\mathbb {E}\\left(\\frac{\\Upsilon _{n+1}(\\widehat{p}_n)-1}{A^2\\widehat{p}_n+R}~|~\\widehat{p}_n\\right)\\\\\\\\\\displaystyle +3~\\widehat{\\alpha }_n(p_n)\\phi (p_n)^2~\\mathbb {E}\\left(\\left(\\frac{\\Upsilon _{n+1}(\\widehat{p}_n)-1}{A^2\\widehat{p}_n+R}\\right)^2~|~\\widehat{p}_n\\right)+\\phi (p_n)^3~\\mathbb {E}\\left(\\left(~\\frac{\\Upsilon _{n+1}(\\widehat{p}_n)-1}{A^2\\widehat{p}_n+B^2}\\right)^3~|~\\widehat{p}_n\\right).\\end{array}$ Now observe that $\\begin{array}{l}\\displaystyle 6~\\widehat{\\alpha }_n(p_n)\\phi (p_n)^2~\\mathbb {E}\\left(\\left(\\frac{\\Upsilon _{n+1}(\\widehat{p}_n)-1}{A^2\\widehat{p}_n+R}\\right)^2~|~\\widehat{p}_n\\right)\\\\\\\\\\displaystyle \\le \\left(3\\widehat{\\alpha }_n(p_n)\\phi (p_n)^2\\right)^2+\\mathbb {E}\\left(\\left(\\frac{\\Upsilon _{n+1}(\\widehat{p}_n)-1}{A^2\\widehat{p}_n+R}\\right)^4~|~\\widehat{p}_n\\right)\\le \\left(3\\widehat{\\alpha }_n(p_n)\\phi (p_n)^2\\right)^2+\\frac{\\iota _6}{N}.\\end{array}$ Using the estimates (REF ), this implies that $\\begin{array}{l}\\displaystyle \\mathbb {E}\\left(\\left(\\widehat{\\alpha }_n(p_n)+\\beta _{n+1}(\\widehat{p}_n)\\right)^3~|~p_n\\right)\\\\\\\\\\displaystyle \\le \\mathbb {E}\\left(\\widehat{\\alpha }_n(p_n)^3~|~p_n\\right)+\\frac{1}{2}\\left(3\\phi (p_n)^2\\right)^2~\\mathbb {E}\\left(\\widehat{\\alpha }_n(p_n)^2~|~p_n\\right)+\\frac{\\iota _6}{2N}\\\\\\\\\\displaystyle \\hspace{85.35826pt}+3\\,\\phi (p_n)~\\mathbb {E}\\left(\\widehat{\\alpha }_n(p_n)^2~|~p_n\\right)~\\frac{\\iota _1}{N}+\\phi (p_n)^3~\\frac{\\iota _3}{N}.\\end{array}$ Using the fact that $\\mathbb {E}\\left(\\vert \\widehat{\\alpha }_n(p_n)\\vert ^3~|~p_n\\right)\\le \\frac{\\iota _7}{N\\sqrt{N}},$ it follows that $\\begin{array}{l}\\displaystyle \\mathbb {E}\\left(\\left(\\widehat{\\alpha }_n(p_n)+\\beta _{n+1}(\\widehat{p}_n)\\right)^3~|~p_n\\right)\\le \\frac{\\iota _8}{N}.\\end{array}$ Using (REF ) and Jensen's inequality we check that $1\\le \\mathbb {E}\\left(\\left(\\frac{\\phi (p_n)}{p_{n+1}}\\right)^3~|~p_n\\right)\\le 1+\\frac{\\iota _9}{N}.$ More generally, for any $l\\ge 2$ and $k\\ge 1$ we have $\\mathbb {E}\\left(\\vert \\widehat{\\alpha }_n(p_n)\\vert ^l~|~p_n=x\\right)\\le \\frac{\\iota _l(1)}{N}\\quad \\mbox{\\rm and}\\quad \\mathbb {E}\\left(\\beta _{n+1}(\\widehat{p}_n)^{2k}~|~\\widehat{p}_n=x\\right)\\le \\frac{\\iota _2(k)}{N}$ for some finite constants $\\iota _1(k),\\iota _2(k)$ .",
"Also note that due to the binomial formula, we have $\\begin{array}{l}\\displaystyle \\mathbb {E}\\left(\\left(\\frac{\\phi (p_n)}{p_{n+1}}\\right)^k~|~p_n,\\widehat{p}_n\\right)\\\\\\\\\\displaystyle =1+\\mathbb {E}\\left(\\left(\\widehat{\\alpha }_n(p_n)+\\beta _{n+1}(\\widehat{p}_n)\\right)~|~p_n,\\widehat{p}_n\\right)+\\sum _{2\\le l\\le k}\\left(\\begin{array}{c}k\\\\l\\end{array}\\right)~\\mathbb {E}\\left(\\left(\\widehat{\\alpha }_n(p_n)+\\beta _{n+1}(\\widehat{p}_n)\\right)^{l}~|~p_n,\\widehat{p}_n\\right)\\end{array}.$ We may estimate the above summands, for any $l\\ge 2$ by $\\begin{array}{l}\\displaystyle \\mathbb {E}\\left(\\left(\\widehat{\\alpha }_n(p_n)+\\beta _{n+1}(\\widehat{p}_n)\\right)^{l}~|~p_n,\\widehat{p}_n\\right)\\\\\\\\\\displaystyle \\le \\vert \\widehat{\\alpha }_n(p_n)\\vert ^{l}+\\frac{1}{2}~\\sum _{0\\le i< l}\\left(\\begin{array}{c}l\\\\i\\end{array}\\right)\\left(\\widehat{\\alpha }_n(p_n)^{2l}+\\mathbb {E}\\left( \\left(\\beta _{n+1}(\\widehat{p}_n)\\right)^{2(l-i)}~|~p_n,\\widehat{p}_n\\right)\\right)\\\\\\\\\\displaystyle \\le \\vert \\widehat{\\alpha }_n(p_n)\\vert ^{l}+\\frac{1}{2}~\\sum _{0\\le i< l}\\left(\\begin{array}{c}.l\\\\i\\end{array}\\right)\\left(\\widehat{\\alpha }_n(p_n)^{2l}+\\frac{\\iota _{2}(l-i)}{N}\\right)\\end{array}$ The end of the proof of (REF ) is now a direct consequence of the moment estimates stated above."
],
[
"A Feynman-Kac formula",
"Theorem 7.1 Let $X_n$ be a Markov chain on some measurable state space $E$ with Markov transitions $M(x,dy)$ and starting at some state $X_0=x$ .",
"Also let $H(x)$ be some positive measurable function on $E$ .",
"Assume there exists some function $h$ and some parameters $\\epsilon _h\\in ]0,1[$ and $\\kappa _h \\in ]0, \\infty [$ such that $H^h(x):=H(x)~h(x)M(1/h)(x)\\le (1-\\epsilon _h)\\quad \\mbox{and}\\quad M(1/h)(x)\\le \\kappa _h ~M(1/h)(y).$ Then, for any $n\\ge 1$ and any bounded measurable function $F$ on the path space $E^n$ we have the Feynman-Kac change of measure formula $\\begin{array}{l}\\displaystyle \\mathbb {E}\\left(F(X_1,\\ldots ,X_n)~\\left(\\prod _{1\\le k\\le n}H(X_k)\\right)~|~X_0=x\\right)\\\\\\\\\\displaystyle =\\mathbb {E}\\left(F(X^{1/h}_1,\\ldots ,X^{1/h}_n)~~\\frac{M(h^{-1})(X^{1/h}_{0})}{M(h^{-1})(X^{1/h}_{n})}\\left(\\prod _{1\\le k\\le n}H^h(X^{1/h}_{k})\\right)~|~X^{1/h}_0=x\\right).\\end{array}$ In the above display, $X^{1/h}_n$ stands for the Markov chain with Markov transitions $M^{1/h}(x,dy)=M(x,dy)~\\frac{h^{-1}(y)}{M(h^{-1})(x)}.$ In addition, we have the uniform exponential decay estimate $\\sup _{x} \\mathbb {E}\\left(\\prod _{1\\le k\\le n}H(X_k)~|~X_0=x\\right)\\le \\kappa _h~(1-\\epsilon _h)^n.$ We set $h^{-1}(x)=1/h(x)$ .",
"Let $X^{1/h}_n$ be the Markov chain starting at $X^{1/h}_0=X_0=x_0$ whose Markov transitions $M^{1/h}$ are given by (REF ) so that $M^{1/h}(h)(x)=\\frac{1}{M(h^{-1})(x)}.$ Observe that $\\begin{array}{l}\\mathbb {P}^{1/h}_n(d(x_0,\\ldots ,x_n))\\\\\\\\:=\\mathbb {P}((X^{1/h}_0,\\ldots ,X^{1/h}_n)\\in d(x_0,\\ldots ,x_n))\\\\\\\\\\displaystyle =\\eta _0(dx_0)~M(x_0,dx_1)~\\frac{h^{-1}(x_1)}{M(h^{-1})(x_0)}\\ldots M(x_{n-1},dx_n)~\\frac{h^{-1}(x_n)}{M(h^{-1})(x_{n-1})}.\\end{array}$ Rewritten in terms of expectations we have $\\mathbb {E}\\left(F(X_0,\\ldots ,X_n)~\\prod _{1\\le k\\le n}\\frac{h^{-1}(X_k)}{M(h^{-1})(X_{k-1})}\\right)=\\mathbb {E}\\left(F(X_0^{1/h},\\ldots ,X_n^{1/h})\\right).$ Observe that $\\begin{array}{l}\\mathbb {P}^{1/h}_n(d(x_0,\\ldots ,x_n))\\\\\\\\\\displaystyle =\\mathbb {P}_n(d(x_0,\\ldots ,x_n))~\\left(\\frac{h^{-1}(x_1)}{M(h^{-1})(x_0)}\\ldots ~\\frac{h^{-1}(x_n)}{M(h^{-1})(x_{n-1})} \\right)\\end{array}$ with $\\mathbb {P}_n(d(x_0,\\ldots ,x_n)):=\\eta _0(dx_0)~M(x_0,dx_1)~\\ldots M(x_{n-1},dx_n).$ This clearly implies that $\\begin{array}{l}\\mathbb {P}_n(d(x_0,\\ldots ,x_n))\\\\\\\\\\displaystyle =\\mathbb {P}^{1/h}_n(d(x_0,\\ldots ,x_n))~\\left(\\left(h(x_1)~M(h^{-1})(x_0)\\right)\\ldots ~\\left(h(x_n)~M(h^{-1})(x_{n-1})\\right) \\right)\\\\\\\\\\displaystyle =\\mathbb {P}^{1/h}_n(d(x_0,\\ldots ,x_n))~\\left(\\frac{h(x_1)}{M^{1/h}(h)(x_0)}\\ldots ~\\frac{h(x_n)}{M^{1/h}(h)(x_{n-1})} \\right),\\end{array}$ where we have used (REF ) in the final step.",
"Rewritten in terms of expectations, the above formula takes the following form $\\mathbb {E}\\left(F(X_0^{1/h},\\ldots ,X_n^{1/h})\\prod _{1\\le k\\le n}\\frac{h(X_k^{1/h})}{M^{1/h}(h)(X^{1/h}_{k-1})}\\right)=\\mathbb {E}\\left(F(X_0,\\ldots ,X_n)\\right).$ Replacing $F(X_0,\\ldots ,X_n)$ by $F(X_0,\\ldots ,X_n)\\prod _{1\\le k\\le n}H(X_k),$ and again recalling (REF ), we obtain the formula $\\begin{array}{l}\\displaystyle \\mathbb {E}\\left(F(X_0,\\ldots ,X_n)\\prod _{1\\le k\\le n}H(X_k)\\right)\\\\\\\\\\displaystyle =\\mathbb {E}\\left(F(X_0^{1/h},\\ldots ,X^{1/h}_n)\\left(\\prod _{1\\le k\\le n}H(X^{1/h}_k)\\right)\\left(\\prod _{1\\le k\\le n}\\frac{h(X_k^{1/h})}{M^{1/h}(h)(X^{1/h}_{k-1})}\\right)\\right)\\\\\\\\\\displaystyle =\\mathbb {E}\\left(F(X^{1/h}_0,\\ldots ,X^{1/h}_n)\\left(\\prod _{1\\le k\\le n}(H h)(X^{1/h}_k)\\right)\\left(\\prod _{0\\le k< n}M(h^{-1})(X^{1/h}_{k})\\right)\\right)\\\\\\\\\\displaystyle =\\mathbb {E}\\left(F(X^{1/h}_0,\\ldots ,X^{1/h}_n)~M(h^{-1})(X^{1/h}_{0})(Hh)(X_{n}^{1/h})\\left(\\prod _{1\\le k< n}(Hh)(X^{1/h}_k)M(h^{-1})(X^{1/h}_{k})\\right)\\right)\\\\\\\\\\displaystyle =\\mathbb {E}\\left(F(X^{1/h}_0,\\ldots ,X^{1/h}_n)~\\frac{M(h^{-1})(X^{1/h}_{0})}{M(h^{-1})(X^{1/h}_n)}\\left(\\prod _{1\\le k\\le n}(Hh)(X^{1/h}_k)M(h^{-1})(X^{1/h}_{k})\\right)\\right).\\end{array}$ This ends the proof of the the Feynman-Kac change of measure formula.",
"The exponential decay estimate (REF ) is now a direct consequence of the regularity condition (REF ).",
"We are now in position to prove the uniform almost sure exponential decays stated in Theorem REF .",
"Proof of first inequality in (REF ) Up to a time change, it suffices to prove the result for $l=1$ and for any given initial condition $p_0=x$ .",
"Recall the Markov transition, $\\mathcal {P}$ , of the stochastic Riccati process $p_n$ .",
"We choose in Theorem REF the function $H(x)=\\left(\\frac{\\vert A\\vert }{1+Sx}\\right)^k\\quad \\mbox{\\rm and}\\quad h(x)=x^k.$ Suppose that $\\vert A\\vert <1$ .",
"In this situation, we have $\\mathbb {E}\\left(\\prod _{1\\le l\\le n}\\left(\\frac{\\vert A\\vert }{1+Sp_l}\\right)^k\\right)\\le (1-\\epsilon _k)^n\\quad \\mbox{\\rm with}\\quad \\epsilon _k:=1-\\vert A\\vert ^k.$ Now assume that $\\vert A\\vert \\ge 1$ .",
"In this situation, we have $\\vert A\\vert <(A^2+RS)$ .",
"Observe that $H^h(x)=H(x)~h(x)\\mathcal {P}(h^{-1})(x)=\\left(\\frac{\\vert A\\vert }{1+Sx}\\right)^k~\\mathbb {E}\\left(\\left(\\frac{p_0}{p_1}\\right)^k~|~p_0=x\\right).$ Using (REF ) we check that $H^h(x)=\\left(\\frac{\\vert A\\vert }{1+Sx}~\\frac{x}{\\phi (x)}\\right)^k~\\mathbb {E}\\left(\\left(\\frac{\\phi (p_0)}{p_1}\\right)^k~|~p_0=x\\right)\\le \\left(\\frac{\\vert A\\vert }{1+Sx}~\\frac{x}{\\phi (x)}\\right)^k~\\left(1+\\frac{\\iota _k}{N}\\right).$ This yields the estimate $H^h(x)\\le \\left(\\frac{\\vert A\\vert }{(A^2+RS)+R/x}\\right)^k~\\left(1+\\frac{\\iota _k}{N}\\right)\\le \\left(\\frac{\\vert A\\vert }{A^2+RS}\\right)^k~\\left(1+\\frac{\\iota _k}{N}\\right).$ Choosing $N$ such that $\\displaystyle N\\ge N_{k}:=\\iota _k \\left(1-\\left(\\frac{\\vert A\\vert }{A^2+RS}\\right)^k\\right)^{-1}$ yields $\\left(1+\\frac{\\iota _k}{N}\\right)\\le 1+\\sqrt{\\epsilon _k},\\quad \\mbox{\\rm with}\\quad \\sqrt{\\epsilon _k}:=\\left(1-\\left(\\frac{\\vert A\\vert }{A^2+RS}\\right)^k\\right).$ We readily check that $H^h(x)&\\le & \\left(1-\\sqrt{\\epsilon _k}\\right)\\left(1+\\sqrt{\\epsilon _k}\\right)=1-\\epsilon _k<1.$ This implies that $\\mathbb {E}\\left(\\left(\\prod _{1\\le l\\le n}\\frac{\\vert A\\vert }{1+Sp_l}\\right)^k~|~p_0=x\\right)\\le \\left(1-\\epsilon _k\\right)^{n-1}\\mathbb {E}\\left(\\frac{ {\\cal P }(h^{-1})(x)}{ {\\cal P }(h^{-1})(X^{1/h}_{n-1})}\\right).$ On the other hand, using (REF ) and (REF ), for any even parameter $N>2(k+2)$ we have $\\frac{1}{(A^2/S+R)^k}\\le {\\cal P }(h^{-1})(x)=\\frac{1}{\\phi (p_0)^k}~\\mathbb {E}\\left(\\left(\\frac{\\phi (p_0)}{p_1}\\right)^k~|~p_0=x\\right)\\le \\frac{1}{R^k}~\\left(1+\\frac{\\iota _k}{N}\\right)$ We conclude that $\\mathbb {E}\\left(\\left(\\prod _{1\\le l\\le n}\\frac{\\vert A\\vert }{1+Sp_l}\\right)^k\\right)\\le \\left(1-\\epsilon _k\\right)^{n-1}~\\left(A^2R/S+R^2\\right)^k \\left(1+\\frac{\\iota _k}{N}\\right).$ This ends the proof of the exponential decays estimates stated in the l.h.s.",
"of (REF )."
],
[
"Fluctuation analysis",
"Lemma 7.2 For any $k\\ge 1$ there exists some parameter $N_k\\ge 1$ such that for any $N\\ge N_k$ and $n\\ge 0$ we have the time uniform estimates $\\mathbb {E}\\left(\\vert m_{n}-X_{n}\\vert ^k\\right)\\vee \\mathbb {E}\\left(\\vert \\widehat{m}_{n}-X_{n}\\vert ^k\\right)\\vee \\mathbb {E}\\left(\\vert Y_n-Cm_{n}\\vert ^k\\right)\\le \\iota _k.$ Using (REF ) we check the recursion $(m_{n+1}-X_{n+1})=\\alpha _n~(X_n-m_n)+\\beta _{n+1},$ with the random variables $\\alpha _n&:=A(1-g_nC)=\\frac{A}{1+Sp_n} \\\\\\beta _{n+1}&:=Ag_nDV_n-BW_{n+1}+\\frac{1}{\\sqrt{N+1}}~\\left(A\\widehat{\\upsilon }_n+\\upsilon _{n+1}\\right).$ This yields the formula $(m_{n+1}-X_{n+1})= {\\cal E }_{0,n}~(X_0-m_0)+\\beta _{n+1}+\\sum _{0\\le k\\le n} {\\cal E }_{k,n}~\\beta _{k}.$ The raw moment estimates of the differences $(m_{n}-X_{n})$ and $(Y_n-Cm_{n})$ stated in (REF ) are now a direct consequence of the exponential decay estimate stated in (REF ).",
"The raw moment estimates of the difference $(\\widehat{m}_{n}-X_{n})$ is easily checked using the following decomposition $\\widehat{m}_{n}-X_{n}=\\frac{A}{1+Sp_n}~(m_n-X_n)+g_nCDV_n+\\frac{1}{\\sqrt{N+1}}~\\widehat{\\upsilon }_n$ This ends the proof of the lemma.",
"Note that this completes the proof of Theorem REF .",
"Now we come to the proof of the Lyapunov estimate (REF ).",
"Proof of (REF ): Combining (REF ) with the uniform almost sure exponential decays stated in (REF ) we check the uniform estimate $\\mathbb {E}\\left(\\vert M_{n+1}\\vert ~|~(M_0,p_0)\\right)\\le \\mathbb {E}( {\\cal E }_{0,n}~|~p_0)~\\vert M_0\\vert +\\iota _1\\le \\iota _1+\\iota _2 (1-\\epsilon _2)^{n}\\vert M_0\\vert .$ Thus, for any $\\epsilon \\in ]0,1]$ there exists some time horizon $n_{\\epsilon }\\ge 1$ such that for any $n\\ge n_{\\epsilon }$ we have $\\mathbb {E}\\left(\\vert M_{n}\\vert ~|~(M_0,p_0)\\right)\\le \\epsilon ~\\vert M_0\\vert +\\iota .$ The estimate (REF ) now comes from the fact that the one-step Riccati map $\\phi $ is uniformly bounded.",
"Proof of Theorem REF : First note that $m_{n+1}-\\widehat{X}_{n+1}^-=\\displaystyle A~(\\widehat{m}_{n}-\\widehat{X}_n)+\\frac{1}{\\sqrt{N+1}}~\\upsilon _{n+1}.$ We also have the decomposition $\\begin{array}{l}\\displaystyle (\\widehat{m}_{n}-\\widehat{X}_{n})\\\\\\\\=\\displaystyle (m_n-\\widehat{X}^-_{n})+(g_n-G_n)~(Y_n-Cm_n)-G_n~C(m_n-\\widehat{X}^-_{n})+\\frac{1}{\\sqrt{N+1}}~\\widehat{\\upsilon }_n\\\\\\\\=\\displaystyle (1-G_nC)(m_n-\\widehat{X}^-_{n})+\\frac{1}{\\sqrt{N}}~\\sqrt{N}(g_n-G_n)~(Y_n-Cm_n)+\\frac{1}{\\sqrt{N+1}}~\\widehat{\\upsilon }_n.\\end{array}$ This yields the recursion $(\\widehat{m}_{n}-\\widehat{X}_{n}) =\\alpha _n~(\\widehat{m}_{n-1}-\\widehat{X}_{n-1})+\\frac{1}{\\sqrt{N}}~ \\beta _n,$ with the parameters $\\alpha _n&:=&A(1-G_nC)= \\frac{A}{1+SP_n}\\\\\\beta _n&:=&\\frac{1}{\\sqrt{1+1/N}}~ \\left((1-G_nC) \\upsilon _{n}+\\widehat{\\upsilon }_n\\right)+\\sqrt{N}(g_n-G_n)~(Y_n-Cm_n).$ We conclude that $(\\widehat{m}_{n}-\\widehat{X}_{n}) =E_n(P_0)~(\\widehat{m}_{0}-\\widehat{X}_{0})+\\frac{1}{\\sqrt{N}}~\\sum _{1\\le l\\le n}~ E_{l,n}(P_0)~\\beta _l.$ Finally, observe that $(\\widehat{m}_{0}-\\widehat{X}_{0})=\\displaystyle (1-G_0C)(m_0-\\widehat{X}^-_{0})+\\frac{1}{\\sqrt{N}}~\\left(\\sqrt{N}(g_0-G_0)~(Y_0-Cm_0)+\\frac{1}{\\sqrt{1+1/N}}~\\widehat{\\upsilon }_0\\right).$ The estimate (REF ) is now a consequence of the exponential decays (REF ), the uniform bounds (REF ) and the mean error estimates stated in Theorem REF .",
"The proof of (REF ) follows similar arguments.",
"Combining (REF ) and (REF ) we check the decomposition $\\begin{array}{l}\\displaystyle m_{n+1}-\\widehat{X}_{n+1}^-=A~ (1-G_nC)(m_n-\\widehat{X}^-_{n})+A(g_n-G_n)~(Y_n-C\\widehat{X}_n^-)\\\\\\\\\\hspace{56.9055pt}\\displaystyle -\\frac{1}{N}~\\sqrt{N}~A(g_n-G_n)C~~\\sqrt{N}(m_n-\\widehat{X}^-_n)+\\frac{1}{\\sqrt{N+1}}~A\\widehat{\\upsilon }_n+\\frac{1}{\\sqrt{N+1}}~\\upsilon _{n+1}.\\end{array}$ Taking the expectations we obtain the formula $\\begin{array}{l}\\displaystyle (m_{n+1}^o-\\widehat{X}_{n+1}^-)=\\alpha _n~(m_n^o-\\widehat{X}^-_{n})+\\beta _n/N\\end{array}$ with the parameters $(\\alpha _n,\\beta _n)$ now given by $\\alpha _n&:=&A(1-G_nC)= {A}/{(1+SP_n)}\\\\\\beta _n&:=&A~N(\\mathbb {E}(g_n)-G_n)~(Y_n-C\\widehat{X}_n^-)-AC~\\mathbb {E}\\left(\\sqrt{N}~(g_n-G_n)\\sqrt{N}(m_n-\\widehat{X}^-_n)~|~ {\\cal Y }_{n-1}\\right).$ Using the uniform estimates stated in Theorem REF and the gain bias estimates (REF ), we obtain, for $k \\ge 1$ $\\sup _{n\\ge 0}\\mathbb {E}\\left(\\vert \\beta _n\\vert ^k\\right)^{1/k}\\le \\iota _k~(1\\vee P_0)^2.$ The end of the proof of (REF ) now follows the same lines of arguments as the proof of the estimates (REF ), thus we leave the details as an exercise for the reader.",
"Proof of Theorem REF : First note that from the proof of (REF ), we have the following decomposition $\\sqrt{N}(\\widehat{m}_n - \\widehat{X}_n) = \\mathcal {E}_n(P_0)\\sqrt{N}(\\widehat{m}_0 - \\widehat{X}_0) + \\sum _{1 \\le \\ell \\le n}\\mathcal {E}_{\\ell , n}(P_0)\\beta _\\ell ,$ where we recall that $\\mathcal {E}_{\\ell , n}(p) &= \\prod _{\\ell < k \\le n}\\frac{A}{1 + SP_\\ell (p)},\\\\\\beta _\\ell &= \\frac{1}{\\sqrt{1+1/N}}~ \\left((1-G_nC) \\upsilon _{n}+\\widehat{\\upsilon }_n\\right)+\\sqrt{N}(g_n-G_n)~(Y_n-Cm_n).$ We first deal with $\\sqrt{N}(\\widehat{m}_0 - \\widehat{X}_0)$ on the right hand side of (REF ).",
"Again, from the proof of (REF ), recall that $\\sqrt{N}(\\widehat{m}_{0}-\\widehat{X}_{0})=\\displaystyle (1-G_0C)\\sqrt{N}(m_0-\\widehat{X}^-_{0})+\\frac{1}{\\sqrt{1+1/N}}~\\widehat{\\upsilon }_0+\\sqrt{N}(g_0-G_0)~(Y_0-Cm_0).$ Further observe that $(Y_0-Cm_0)=(Y_0-C\\widehat{X}^-_0)+C(\\widehat{X}^-_0-m_0)\\quad \\mbox{\\rm and}\\quad (\\widehat{X}^-_0-m_0)\\longrightarrow _{N\\rightarrow \\infty } 0\\quad \\mbox{\\rm a.e.",
"}.$ Applying Slutsky's lemma, this yields the weak convergence $\\sqrt{N}(g_0-G_0)~C(\\widehat{X}^-_0-m_0)\\longrightarrow _{N\\rightarrow \\infty } 0.$ Using the fluctuation theorems REF and REF we also have $\\sqrt{N}(m_0-\\widehat{X}^-_{0}) &\\longrightarrow _{N\\rightarrow \\infty } \\mathbb {U}_0,\\\\\\sqrt{N}(g_0-G_0)(Y_0-C\\widehat{X}^-_0) &\\longrightarrow _{N\\rightarrow \\infty } \\mathbb {G}_0(Y_0-C\\widehat{X}^-_0),\\\\\\frac{1}{\\sqrt{1+1/N}}~\\widehat{\\upsilon }_0 &\\longrightarrow _{N\\rightarrow \\infty } \\widehat{\\mathbb {U}}_0.$ In the same vein, applying the continuous mapping theorem or the $\\delta $ -method (see for instance Theorem 9.3.3 and Lemma 9.3.1 in [36]) we check that $\\sqrt{N}(\\widehat{m}_{0}-\\widehat{X}_{0})\\longrightarrow _{N\\rightarrow \\infty } (1-G_0C) \\mathbb {U}_0+\\widehat{\\mathbb {U}}_0+\\mathbb {G}_0(Y_0-C\\widehat{X}^-_0).$ To deal with the sum on the right hand side of (REF ), again, thanks to Theorem REF , and arguing as above $\\beta _\\ell $ converges weakly to $(1-G_{\\ell }C)\\mathbb {U}_{\\ell } + \\widehat{\\mathbb {U}}_{\\ell }+\\mathbb {G}_{\\ell } (Y_{\\ell }-C\\widehat{X}_{\\ell }^-).$ In the same vein, we check that $\\widehat{\\mathbb {X}}_n^N &\\longrightarrow _{N\\rightarrow \\infty } \\widehat{\\mathbb {X}}_n:= \\sum _{0 \\le \\ell \\le n}E_{\\ell , n}(P_0)\\left((1-G_{\\ell }C)\\mathbb {U}_{\\ell } + \\widehat{\\mathbb {U}}_{\\ell }+\\mathbb {G}_{\\ell } (Y_{\\ell }-C\\widehat{X}_{\\ell }^-)\\right).$ The proof of the fluctuation theorem at the level of the process in the sense of convergence of finite dimensional distributions is conducted combining the continuous mapping theorem with Slutsky's lemma.",
"The proof of the recursive formulation (REF ) is also easily checked using the decompositions (REF ) and (REF ).",
"This ends the proof of the theorem."
],
[
"Proof of Theorem ",
"For the identity function $ {\\cal U }(p):=p$ , due to (REF ), we have $ {\\cal P }( {\\cal U })(p)=\\mathbb {E}( {\\cal U }(p_{n+1})~|~p_n=p)=\\phi (p)\\le \\alpha =A^2/S+R.$ In addition, for any $\\epsilon \\in ]0,1]$ we have $ {\\cal U }_{\\epsilon }:=1+\\frac{\\epsilon }{\\alpha }~ {\\cal U }\\ge 1\\Longrightarrow {\\cal P }( {\\cal U }_{\\epsilon })(p)\\le 1+\\epsilon \\Longrightarrow {\\cal P }( {\\cal U }_{\\epsilon })\\le 1+\\epsilon ~ {\\cal U }_{\\epsilon }.$ The above estimate implies that $ {\\cal U }_{\\epsilon }\\ge 1$ is a Lyapunov function.",
"On the other hand, $ {\\cal P }(p,dq)=k(p,q)~dq$ has a continuous density $k(p,q)>0$ with respect to the variables $(p,q)$ and the Lebesgue measure $dq$ on $\\mathbb {R}_+$ .",
"Thus, for any compact set $\\varpi \\subset \\mathbb {R}_+$ there exists some function $\\varphi $ from $[0,\\infty [$ into itself such that $k_{\\varpi }(q):=\\inf _{p\\in \\varpi }k(p,q)=k(\\varphi _{\\varpi }(q),q)>0 \\Longrightarrow 0<\\epsilon _{\\varpi }:=\\int ~k_{\\varpi }(q)~dq<1.$ This implies that $ {\\cal P }(p,dq) \\ge \\epsilon _{\\varpi }~\\nu _{\\varpi }(dq)\\quad \\mbox{\\rm with}\\quad \\nu _{\\varpi }(dq):=\\frac{k_{\\varpi }(q)~dq}{\\int ~k_{\\varpi }(p)~dp},$ from which we prove the Dobrushin local contraction property $\\varpi _r:=[0,r]\\Longrightarrow \\beta _{r}( {\\cal P }):=\\sup _{(p,q)\\in \\varpi _r^2}\\Vert {\\cal K }(p, \\mbox{\\LARGE .",
"})- {\\cal K }(q, \\mbox{\\LARGE .})",
"\\Vert _{\\tiny tv}<1-\\epsilon _{\\varpi }<1.$ For any $\\epsilon _1,\\epsilon _2\\in ]0,1]$ we set $ {\\cal U }_{\\epsilon _1,\\epsilon _2}:=\\epsilon _1 {\\cal U }_{\\epsilon _2}\\Longrightarrow {\\cal P }( {\\cal U }_{\\epsilon _1,\\epsilon _2})\\le \\epsilon _1\\left(1+\\epsilon _2 {\\cal U }_{\\epsilon _2}\\right)\\le \\epsilon _1+\\epsilon _2~ {\\cal U }_{\\epsilon _1,\\epsilon _2}.$ Choosing the parameters $(r,\\epsilon _2)$ such that $(1/r+\\epsilon _2)<1$ for any $\\epsilon _1\\in [0,1[$ we can check that $p,q\\ge r\\Longrightarrow \\frac{\\Vert {\\cal P }(p, \\cdot ) - {\\cal P }(q, \\cdot ) \\Vert _{ {\\cal U }_{\\epsilon _1,\\epsilon _2}}}{1 + {\\cal U }_{\\epsilon _1,\\epsilon _2}(p) + {\\cal U }_{\\epsilon _1,\\epsilon _2}(q)}<1.$ The detailed proof of the above assertion can be found in section 8.2.7 in [39].",
"In addition, for any $x,y\\le r$ we also have $\\epsilon _1<\\frac{1-\\beta _r( {\\cal K })}{4r}\\Longrightarrow \\frac{\\Vert {\\cal P }(x, \\cdot ) - {\\cal P }(y, \\cdot ) \\Vert _{ {\\cal U }_{\\epsilon _1,\\epsilon _2}}}{1 + {\\cal U }_{\\epsilon _1,\\epsilon _2}(x) + {\\cal U }_{\\epsilon _1,\\epsilon _2}(y)}\\le \\beta _r( {\\cal K })+4\\epsilon _1 r<1.$ We conclude that $(1/r+\\epsilon _2)<1\\quad \\mbox{\\rm and}\\quad \\epsilon _1<\\frac{1-\\beta _r( {\\cal P })}{4r}\\quad \\Longrightarrow \\quad \\beta _{ {\\cal U }_{\\epsilon _1,\\epsilon _2}}( {\\cal P })<1,$ which concludes the proof.",
"Following [26], we have $\\phi ^n(x)+v=\\lambda _2~\\frac{\\left(x+(v-\\lambda _1)\\right)~\\left(1-\\lambda ^{n+1}\\right)+(\\lambda _2-\\lambda _1)~\\lambda ^{n+1}}{\\left(x+(v-\\lambda _1)\\right)~\\left(1-\\lambda ^{n}\\right)+(\\lambda _2-\\lambda _1)~\\lambda ^{n}}.$ Next, we check that the r.h.s.",
"expression in the above display is non negative.",
"Notice that $\\phi ^n(x)=\\frac{\\left(x+(v-\\lambda _1)\\right)\\left(\\lambda _2~\\left(1-\\lambda ^{n+1}\\right)-v~\\left(1-\\lambda ^{n}\\right)\\right)-(v-\\lambda _1)(\\lambda _2-\\lambda _1)~\\lambda ^{n}}{\\left(x+(v-\\lambda _1)\\right)~\\left(1-\\lambda ^{n}\\right)+(\\lambda _2-\\lambda _1)~\\lambda ^{n}}.$ Rewritten in a slightly different form (REF ) takes the following form $\\phi ^n(x)=(\\lambda _2-v)+\\frac{\\left(x+(v-\\lambda _1)\\right)~-(\\lambda _2-\\lambda _1)}{\\left(x+(v-\\lambda _1)\\right)~\\left(1-\\lambda ^{n}\\right)+(\\lambda _2-\\lambda _1)~\\lambda ^{n}}~\\left(\\lambda _2-\\lambda _1\\right)~\\lambda ^n.$ This yields the formula $\\phi ^n(x)-r=\\frac{x-r}{\\left(x+(v-\\lambda _1)\\right)~\\left(1-\\lambda ^{n}\\right)+(\\lambda _2-\\lambda _1)~\\lambda ^{n}}~\\left(\\lambda _2-\\lambda _1\\right)~\\lambda ^n,$ which ends the proof of (REF ).",
"Recalling that $\\phi $ is an increasing function, we check that $\\phi (0)=\\frac{b}{d}\\le \\phi (x)\\le \\lim _{x\\rightarrow \\infty }\\phi (x)=\\frac{a}{c}\\quad \\mbox{\\rm and}\\quad r=\\phi (r)\\le \\frac{a}{c},$ which ends the proof of (REF ).",
"For any $x,y\\in \\mathbb {R}_+$ we also have $\\begin{array}{l}\\displaystyle \\frac{\\phi ^n(x)-\\phi ^n(y)}{x-y}\\\\\\\\\\displaystyle =\\frac{(\\lambda _2-\\lambda _1)^2~\\lambda ^n}{\\left[\\left(x+(v-\\lambda _1)\\right)~\\left(1-\\lambda ^{n}\\right)+(\\lambda _2-\\lambda _1)~\\lambda ^{n}\\right]\\left[\\left(y+(v-\\lambda _1)\\right)~\\left(1-\\lambda ^{n}\\right)+(\\lambda _2-\\lambda _1)~\\lambda ^{n}\\right]}.\\end{array}$ This ends the proof of the r.h.s.",
"estimate in (REF ).",
"The proof of the l.h.s.",
"estimate in (REF ) is a direct consequence of the easily checked estimate $\\vert \\phi ^n(x)-r\\vert \\le \\vert x-r\\vert ~\\frac{\\left(\\lambda _2-\\lambda _1\\right)}{\\left(x+(v-\\lambda _1)\\right)~\\wedge (\\lambda _2-\\lambda _1)}~~\\lambda ^n.$ We check (REF ) using the second order formula $\\begin{array}{l}\\displaystyle \\frac{\\phi ^n(x)-\\phi ^n(y)}{x-y}\\\\\\\\\\displaystyle =\\frac{(\\lambda _2-\\lambda _1)^2~\\lambda ^n}{\\left[\\left(y+(v-\\lambda _1)\\right)~\\left(1-\\lambda ^{n}\\right)+(\\lambda _2-\\lambda _1)~\\lambda ^{n}\\right]^2}\\\\\\\\\\displaystyle -\\frac{(\\lambda _2-\\lambda _1)^2~\\lambda ^n}{\\left[\\left(y+(v-\\lambda _1)\\right)~\\left(1-\\lambda ^{n}\\right)+(\\lambda _2-\\lambda _1)~\\lambda ^{n}\\right]^2}\\times \\left(\\frac{(x-y)\\left(1-\\lambda ^{n}\\right)}{\\left[\\left(x+(v-\\lambda _1)\\right)~\\left(1-\\lambda ^{n}\\right)+(\\lambda _2-\\lambda _1)~\\lambda ^{n}\\right]}\\right).\\end{array}$ Finally observe that $(\\ref {def-rho})\\Longrightarrow \\partial \\phi ^n(x)=\\partial (\\phi (\\phi ^{n-1}(x)))=(\\partial \\phi )(\\phi ^{n-1}(x))~\\partial \\phi ^{n-1}(x)=\\prod _{0\\le k<n}\\frac{\\rho }{(c\\phi ^{k}(x)+d)^2}$ with the convention $\\phi ^0(x)=x$ .",
"This ends the proof of (REF ) and the lemma.",
"It clearly suffices to prove (REF ) for $\\omega _1=0$ .",
"For any $u,v\\in \\mathbb {R}$ we have $\\mathbb {E}\\left(\\exp {\\left(i(uZ_1+vZ^2_1)\\right)}\\right)=(1-2iv)^{-1/2}~\\exp {\\left(-\\frac{u^2}{2(1-2iv)}\\right)}.$ Then, for any $u,v\\in \\mathbb {R}$ we have $\\mathbb {E}\\left(\\exp {\\left(\\frac{u}{N}\\sum _{1\\le i\\le N}Z_i+\\frac{v}{N}\\sum _{1\\le i\\le N}Z^2_i\\right)}\\right)=\\left(\\frac{1}{\\sqrt{1-2v/N}}\\right)^N~\\exp {\\left(-\\frac{u^2}{2N(1-2v/N)}\\right)}.$ This yields the characteristic function formula $\\mathbb {E}\\left(\\exp {\\left(i\\left(u~{ {\\cal Z }}+v~\\widetilde{ {\\cal Z }}\\right)\\right)}\\right)~\\exp {\\left(\\frac{u^2+v^2}{2}\\right)}=\\exp {\\left(\\epsilon _1(u,v)+\\epsilon _2(v)\\right)}.$ with the functions $\\epsilon _1(u,v)&:=&\\frac{u^2}{2}\\left(1-\\frac{1}{\\left(1-2i{v}/{\\sqrt{2N}}\\right)}\\right)=-i~\\frac{u^2v}{\\sqrt{2N}}~\\frac{1}{\\left(1-2i{v}/{\\sqrt{2N}}\\right)}\\\\&&\\\\\\epsilon _1(v)&:=&\\frac{v^2}{2}-i\\frac{Nv}{\\sqrt{2N}}-\\frac{N}{2}\\log {\\left(1-\\frac{2iv}{\\sqrt{2N}}\\right)}.$ Using the estimates $\\vert -\\log {(1-ix)}-\\left(ix-\\frac{x^2}{2}\\right)\\vert \\le 2~\\vert x \\vert ^3\\quad \\mbox{\\rm and}\\quad \\vert ix/(1-iy)\\vert =\\frac{\\vert x\\vert }{\\sqrt{1+y^2}}\\le \\vert x\\vert $ we also check that $\\vert \\epsilon _1(u,v)\\vert \\le \\frac{u^2\\vert v\\vert }{\\sqrt{2N}}\\quad \\mbox{\\rm and}\\quad \\vert \\epsilon _2(v)\\vert \\le \\frac{N}{2}~\\left(2\\frac{v}{\\sqrt{2N}}\\right)^3=\\vert v\\vert ^3\\sqrt{\\frac{2}{N}}.$ We conclude that $\\mathbb {E}\\left(e^{i\\left(u\\,{ {\\cal Z }}+v\\,\\widetilde{ {\\cal Z }}\\right)}\\right)-e^{-\\frac{u^2+v^2}{2}}=\\left(e^{\\epsilon _1(u,v)+\\epsilon _2(v)}-1\\right)~e^{-\\frac{u^2+v^2}{2}}.$ Finally, using the estimate $\\vert e^z-1\\vert \\le \\vert z\\vert ~\\int _0^1\\vert e^{tz}\\vert ~dt\\le \\vert z\\vert ~\\int _0^1\\vert e^{t~\\mbox{\\footnotesize Re}(z)}\\vert ~ dt \\le \\vert z\\vert ~e^{\\vert z\\vert },$ we check that $\\left|\\mathbb {E}\\left(e^{i\\left(u\\,{ {\\cal Z }}+v\\,\\widetilde{ {\\cal Z }}\\right)}\\right)-e^{-\\frac{u^2+v^2}{2}}\\right|\\le \\epsilon (u,v)~e^{-\\frac{u^2+v^2}{2}}.$ This ends the proof of the lemma."
]
] | 2107.01855 |
[
[
"Cosmological Constraints using the newest VLT-KMOS HII Galaxies and the\n full Planck CMB spectrum"
],
[
"Abstract We present novel cosmological constraints based on a joint analysis of our HII galaxies (HIIG) Hubble relation with the full Planck Cosmic Microwave Background anisotropy spectrum and the Baryon Acoustic Oscillations (BAO) probes.",
"The HII galaxies span a large redshift range $(0.088 \\le z \\le 2.5)$, reaching significantly higher redshifts than available SNIa and hence they probe the cosmic expansion at earlier times.",
"Our independent constraints compare well with those based on the Pantheon compilation of SNIa data, which we also analyse.",
"We find our results to be in agreement with the conformal $\\Lambda$CDM model within 1$\\sigma$.",
"We also use our HIIG data to examine the behaviour of the dark energy equation of state parameter under the CPL parameterisation, $w = w_0+w_a \\frac{z}{1+z}$, and find consistent results with those based on SNIa, although the degeneracy in the parameter space as well as the individual parameter uncertainties, when marginalizing one over the other, are quite large."
],
[
"Introduction",
"Since 1998 the scientific community of cosmology has been at a consensus over the expansion of the Universe [35], [29]; type Ia Supernovae (SNIa) observations have proven beyond doubt that we live in an expanding Universe, whose rate of expansion is accelerating.",
"Over the last two decades the combined analysis of Cosmic Microwave Background (CMB) anisotropies [24], [34], [43], [30], [1], [2] with Baryon Acoustic Oscillations (BAOs) [18], [9] and Hubble parameter measurements [13], [20], [37], [38], [19] have shown that the cosmic fluid appears to be dominated by an unknown component which is usually referred to as dark energy (hereafter DE).",
"From the phenomenological viewpoint, the unknown nature of dark energy is reflected in the equation of state parameter (hereafter EoS), namely $w=p_{\\rm D}/\\rho _{D}$ , where the quantities $\\rho _{D}$ and $p_{D}$ are the density and pressure of the DE fluid respectively.",
"It is well known that one of the main targets of observational cosmology is to constrain $w$ as well as to test its evolution.",
"In this context, the basic cosmological parameters (including the EoS) are constrained by a combination of (a) relatively low redshift ($z\\le 2$ ) cosmological probes (SNIa and BAOs: [35], [29], [23], [4], [36], [44], [7], [40]) and (b) high redshift probes ($z\\sim 1000$ Planck CMB fluctuations; eg., [30], [1], [2]).",
"Such a combination is essential in order to minimize degeneracies among the cosmological parameters; however, if we assume that $w$ is a function of redshift then strong degeneracies persist.",
"It is important to note that probing intermediate redshifts ($2\\le z \\le 10$ ), where the maximum difference in cosmological models take place [33], is a prerequisite for effectively constraining $w(z)$ .",
"HII galaxies are alternative and effective tracers of the Hubble relation for two reasons: (a) they can be observed up to high redshifts $z \\sim 3.5$ [45], [15], where the distance modulus is more sensitive to the cosmological parameters, and (b) there is a tight relation between the $H\\beta $ luminosity and the emission line velocity dispersion, which provides an effective HIIG distance indicator.",
"Indeed it has been proven [28], [42], [33], [13], [14], [15], [45], [21], [22] that this relation can be used as an alternative cosmological tracer.",
"Despite the fact that the scatter of the HIIG distance modulus is larger (by a factor of 2) than that of high-$z$ SNIa, this demerit is fully compensated by the fact that HIIG are observed to larger redshifts than SNIa, where, as discussed above, the degeneracies for different DE models are reduced [33].",
"It is important to remark that over the last ten years our team has published a series of papers on this subject.",
"In the current article, we use our new set of high spectral resolution observations of high-z HIIG obtained with VLT-KMOS [22] along with published HIIG data [21], [45], and combine them with the full Planck spectrum [2], Type 1a Supernovae data from the Pantheon dataset [40], Baryon Acoustic Oscillations data points [8], [25], [39], [3], and the value for the $H_0$ parameter reported in [38] in order to place constraints on the main cosmological parameters, as well as to check for dynamical DE.",
"In this work we consider the $\\Lambda $ CDM model as the fiducial model, and also explore the popular Chevallier-Polarski-Linder [16], [27] parameterisation (hereafter CPL).",
"This paper is organised as follows: in section 2 we provide the basic cosmological equations, in section 3 we briefly describe the data samples used in the current analysis.",
"In section 4 we present our analysis along with comments on our results, while in section 5 we draw our conclusions and discuss future research prospects."
],
[
"Basic cosmological equations",
"In the context of general relativity we consider that the universe is a self-gravitating fluid, endowed with a spatially flat homogeneous and isotropic geometry.",
"Furthermore, we also assume that the cosmic fluid is dominated by non-relativistic matter plus a dark energy component with an equation of state given by $p_{\\rm D}=w(a)\\rho _{\\rm D}$ which is responsible for the accelerated expansion of the universe.",
"Within this framework, the Hubble parameter takes the form: $E^{2}(a)=\\frac{H^{2}(a)}{H_{0}^{2}}= \\Omega _{m,0}a^{-3}+\\Omega _{\\rm D,0}X(a), $ where $X(a)={\\rm exp}\\left(3\\int ^{1}_{a} d{\\rm lny}[1+w(y)]\\right).$ with $E(a)$ denoting the normalized Hubble flow, $H_{0}$ the Hubble constant, $a(t)$ the scale factor and $w(a)$ the equation of state parameter.",
"Notice, that $\\Omega _{m,0}$ and $\\Omega _{D,0}(\\equiv 1-\\Omega _{m,0})$ are the fractional matter and dark energy density parameters at the present time, respectively.",
"For the concordance $\\Lambda $ CDM model we have $w(a)\\equiv - 1$ and thus $X(a)=1$ .",
"On the other hand we can parameterise the unknown form of the EoS parameter by using the CPL parameterisation.",
"This approach has been largely discussed in the literature, namely the equation of state parameter is expressed as a first order Taylor expansion around the present time, $a(t_{0})=1$ : $w(a)=w_{0}+w_{a}(1-a),$ hence $X(a)=a^{-3(1+w_{0}+w_{a})}{\\rm exp}\\left[3w_{a}(a-1)\\right],$ where $w_{0}$ and $w_{a}$ are constants.",
"Notice that for $a\\rightarrow 0$ we have $w\\simeq w_{0}+w_{a}$ , while prior to the present time ($a=1$ ) the EoS parameter tends to $w_{0}$ .",
"Finally, the luminosity distance corresponding to the spatially flat Friedmann-Lemaître-Robertson-Walker metric reads: $d_{L}(z)=c(1+z)\\int _{0}^{z} \\frac{du}{H(u)},$ and the distance modulus is given by $m-M=5{\\rm log}d_{L}+25,$ where the distance $d_L$ is given in units of Mpc, $z$ is the redshift and $1+z=a(z)^{-1}$ .",
"Figure: 1σ1\\sigma and 2σ2\\sigma contour plots for the Λ\\Lambda CDM model using CMB+HIIG data (red), CMB+HIIG+BAO data (blue), and CMB+HIIG+H0 data (green).Figure: Similar to , with SNIa data instead of HIIG."
],
[
"Cosmological Data",
"In the current work we use the publicly available Planck CMB spectrum in combination with HII galaxies data and other cosmological probes, described below, in order to constrain the standard set of cosmological parameters of the $\\Lambda $ CDM and CPL models, respectively.",
"Notice that for the former case the parameter space is $(\\Omega _{b}h^2, \\Omega _{c}h^2, 100\\theta _{MC}, \\ln (10^{10}A_s) , n_s , \\tau _{\\text{reio}})$ while for the latter we have $(\\Omega _{b}h^2, \\Omega _{c}h^2, 100\\theta _{MC}, \\ln (10^{10}A_s), n_s, \\tau _{\\text{reio}}, w_0, w_a),$ where $\\Omega _{b}$ and $\\Omega _{c}$ are the fractional baryonic and dark matter density parameters at the present time, respectively, $\\theta _{MC}$ is the angular size of the sound horizon at recombination, $A_{s}$ is the amplitude of the primordial power spectrum, $n_{s}$ is the spectral index and $\\tau _{\\text{reio}}$ is the optical depth at reionisation.",
"In the following we briefly present the type of cosmological data used in our current statistical analysis.",
"HII galaxies data: In a series of papers [22], [21], [19], [15], [45], [12], [32], [33], [28], [31] HII galaxies have been used as alternative tracers of the Hubble relation, hence extending Hubble relation on redshifts beyond the range of available SNIa data.",
"Here we utilize the HII galaxies dataset discussed in [22].",
"The current HIIG sample contains 181 objects, which can be separated in the local sample (107 HIIG with $z<0.16$ ) and a high-z sample based on 29 KMOS, 15 MOSFIRE, 6 XShooter and 24 literature objects (for details see [22]).",
"The luminosity of the HIIG in the sample is attributed mostly to a recent single starburst ($< 5Myr$ ).",
"Supernovae (SNIa): We also use the binned data from the Pantheon Supernova type Ia (SNIa) dataset [40], which combines the confirmed SNIa sources discovered by the Pan-STARRS1 (PS1) Medium Deep Survey with a large number of other surveys, to compare the two distinct and independent tracers of the Hubble function.",
"CMB: We use the full CMB spectrum from the Planck 2018 data release [2].",
"Specifically, we focus on the full (TT+TE+BB+lowE) spectrum, and compare results with those based on the CMB shift parameter likelihood, considered in [45] and [22].",
"To this end we use the publicly available CLASS software [10] within the Monte Python MCMC code [11], [5].",
"Baryon Acoustic Oscillations: We use the BAO probe with measurements from various sources: the WiggleZ Dark Energy Survey [25], the BOSS DR12 survey [3], one single 6DFGS point [8] and the Main Galaxy Sample of Data Release 7 of Sloan Digital SkySurvey (SDSS-MGS) [39].",
"Hubble constant: Lastly, we utilize the value $H_0 = 73.48 \\pm 1.66 \\text{\\rmfamily km s}^{-1}{}\\text{\\rmfamily Mpc}^{-1}$ as reported in [38].",
"Table: The results from the Λ\\Lambda CDM analysis using the HIIG data.",
"The 1-σ\\sigma error bars are quoted.",
"The parameter H 0 H_0 is displayed in kms -1 Mpc -1 \\text{\\rmfamily km s}^{-1}{}\\text{\\rmfamily Mpc}^{-1} units."
],
[
"Observational constraints",
"We perform a joint analysis of the HII galaxies Hubble relation with the full Planck CMB spectrum and the BAO probes in order to place constraints on the cosmological parameters of the concordance $\\Lambda $ CDM and the popular CPL models.",
"We shall consider that the above datasets can be treated as statistically independent probes of the models, which is a reasonable assumption since the individual probes are based on different cosmic objects which trace mostly different redshift ranges and are based on different physical mechanisms.",
"Only in one case, where we make combined use of both the HIIGs and the SNIa as tracers of the Hubble expansion, does the latter assumption come under question, given the fact that there is some spatial overlap between the two probes, which in turn could introduce correlations in the statistical analysis.",
"While this could be important, unfortunately at the moment there is no standard way to account for it, given the lack of the full correlation matrix between the samples.",
"Therefore, following the standard $\\chi ^{2}$ procedure, we have assumed in all cases that the different datasets are uncorrelated, which is equivalent to simply summing over their corresponding $\\chi ^{2}$ functions.",
"Table: The results from the Λ\\Lambda CDM analysis using the Pantheon data.",
"The 1-σ\\sigma error bars are quoted.",
"The parameter H 0 H_0 is displayed in kms -1 Mpc -1 \\text{\\rmfamily km s}^{-1}{}\\text{\\rmfamily Mpc}^{-1} units.Within this context, for the case of the concordance $\\Lambda $ CDM model, we constrain the standard parameter space, namely $(\\Omega _{b}h^2, \\Omega _{c}h^2, 100\\theta _{MC}, ln(10^{10}A_s), n_s , \\tau _{\\text{reio}})$ .",
"In Table REF we provide an overall presentation of the observational constraints imposed by the joint analysis of HIIG, CMB spectrum and BAO probes, while in Table REF we present the corresponding results based on using SNIa instead of HIIG.",
"Obviously, this is a comparison of the performance of the two independent tracers, HIIG and SNIa, in probing the cosmic expansion within this particular model.",
"Moreover in figures REF and REF we show the $1\\sigma $ and $2\\sigma $ contours in various parameter planes.",
"Inspecting the aforementioned constraints, we verify that the results based on HII galaxies are in excellent agreement with those based on SNIa.",
"It is noteworthy that this agreement extends also to the relative errors associated with the quoted values.",
"This, however, is because the majority of the sampled parameters are mostly constrained by the CMB spectrum.",
"Note that both joint analyses (based on either HIIG or SNIa) provide results which are consistent (within $1\\sigma $ ) with those provided by the Planck collaboration [1], [2].",
"Regarding the well known Hubble constant tension problem, ie., the fact that local measurements (ie.",
"$H_0 = 73.48 \\pm 1.66 \\text{\\rmfamily km s}^{-1}{}\\text{\\rmfamily Mpc}^{-1}$ as measured by SNIa [38]) and HIIG [13], [19] are in $\\sim 4\\sigma $ tension with the $z\\sim 1100$ value provided by Planck, ie., $H_{0}= 67.36 \\pm 0.54$ km/s/Mpc [2], our results are consistent with the latter measurements, as expected from the fact that the CMB spectrum probe dominates the $H_{0}-$ constraints imposed by the joint analysis.",
"We also note that such results are in agreement with those of [41], who found $H_{0}= 67.6 \\pm 1.52$ km/s/Mpc utilizing the GAIA parallax distances of Milky Way Cepheids.",
"We need to emphasize however all local Universe studies support larger values of $H_{0}$ [26], [6], [19], [17].",
"We now focus on the CPL parameterisation, which allows a dynamic evolution of the equation of state (EoS) parameter, $w = w_0 + w_a(1-a)$ , and hence it introduces two more parameters in the original parameter space.",
"We repeat our statistical analysis but now for the parameter space $(\\Omega _{b}h^2, \\Omega _{c}h^2, 100\\theta _{MC}, \\ln (10^{10}A_s), n_s, \\tau _{\\text{reio}}, w_0, w_a)$ .",
"Due to the large number of free parameters we restrict our analysis to the maximum data combinations of: CMB+HIIG+BAOs+$H_0$ and CMB+SNIa+BAOs+$H_0$ .",
"However, we also allow the combined use of both Hubble expansion tracers, with the caveats discussed previously, providing results based on the joint analysis CMB+HIIG+SNIa+BAOs+$H_0$ .",
"Table: The results from the CPL analysis using the HIIG data (second column), Pantheon data (third column), and both datasets (fourth column).The 1-σ\\sigma error bars are quoted.",
"The parameter H 0 H_0 is displayed in kms -1 Mpc -1 \\text{\\rmfamily km s}^{-1}{}\\text{\\rmfamily Mpc}^{-1}units.Our results are listed in table REF , while in figure REF we present the 1 and 2$\\sigma $ contours in the $w_0-w_a$ plane when using the combination of HIIG data and SNIa data.",
"It is worth noting that $\\Lambda $ CDM model lies within the 1-$\\sigma $ region.",
"Furthermore, we compare the aforementioned best fit values with those of [22].",
"This work is in alignment with the aforementioned, with the exception that the full Planck spectrum has been utilised in the analysis, in place of the CMB shift parameter, in an effort to include other important cosmological parameters overlooked by the shift parameter such as $\\sigma _8$ , $\\tau _{\\text{reio}}$ etc, as well as to have a more complete check of the tested models.",
"In all cases, [22] have found for the combination HIIG+CMB$_{\\rm shift}$ +BAOs $\\Omega _{m,0}=0.298^{+0.018}_{-0.021}$ , $H_{0}= 69.8\\pm 2.3$ km/s/Mpc, $w_0=-1.07^{+0.23}_{-0.32}$ and $w_{a}=0.16^{+0.96}_{-0.55}$ , while for SNIa+CMB$_{\\rm shift}$ +BAOs they found $\\Omega _{m,0}=0.3011 \\pm 0.0085$ , $H_{0}= 68.85\\pm 0.99$ km/s/Mpc, $w_{0}=-1.062\\pm 0.040$ and $w_{a}=0.25^{+0.22}_{-0.19}$ .",
"Evidently, our constraints are compatible within $1\\sigma $ with those of [22], [21] and [45].",
"Finally, our results are also consistent with those of [40], who found $\\Omega _{m,0}=0.300 \\pm 0.008$ , $H_{0}= 69.057\\pm 0.796$ km/s/Mpc, $w_{0}=-1.007\\pm 0.089$ and $w_{a}=-0.222\\pm 0.407$ for SNIa+CMB+BAOs.",
"Evidently, we can place constraints on both CPL parameters, marginalizing one over the other, but their corresponding uncertainties remain quite large.",
"However, what is important to highlight is that the parameter degeneracy in the $w_0-w_a$ plane is large, significantly larger than that of the corresponding SNIa analysis.",
"Future HII galaxies data are expected to improve the relevant constraints dramatically, since, based on Monte-Carlo simulations, our team has shown [15] that a significant improvement is expected when increasing the number of high-z HII galaxies to $\\sim 500$ , a goal that can be achieved in reasonable observing time with the existing large telescopes, as we have discussed in a number of our previous works.",
"Figure: 2D contours for the pair w 0 w_0-w a w_a from the dataset CMB+HIIG+BAOs+H0 (blue), CMB+SNIa+BAOs+H0 (green), CMB+HIIG+SNIa+BAOs+H0 (red)."
],
[
"Conclusions",
"We used HII galaxies as tracers of the Hubble expansion in a joint analysis, and combined them for the first time with the Planck CMB power spectrum (using CLASS) in order to constrain the parameters for the most popular cosmological models, namely $\\Lambda $ CDM and CPL.",
"We also compared the performance of HII galaxies with that of the widely used Type Ia Supernovae data and found that the best fit parameters of the explored cosmological models are mutually in good agreement.",
"Then we combined HIIG+CMB in a joint statistics with other cosmological probes (SNIa, BAOs, $H_{0}$ ) to place constraints on the CPL cosmological parameters.",
"We find that the degeneracy of the parameters $w_{0}$ and $w_{a}$ is quite large, which improves however (especially with the reduction of the $w_0$ uncertainty) when confronted with the full set of standard candles (SNIa+HIIG).",
"This attests to the necessity of further HIIG observations.",
"Indeed, according to [33] and [15], a deviation from $\\Lambda $ CDM could in principle be detected when using a few hundreds of high redshift HIIG galaxies ($1.5\\le z\\le 4$ ).",
"The current HIIG sample contains in total 181 objects, out of which 74 are high redshift galaxies ($0.6<z<2.6$ ), which are already yielding very promising results.",
"Our team, however, is designing the appropriate future KMOS-VLT observations of high-z objects to which we will add $\\sim 100$ additional HIIG, with intermediate redshifts, $z \\sim 0.7$ , to be observed with MEGARA-GTC as part of the INAOE guaranteed time, as well as to explore the Hubble diagram to redshift $z\\sim 3.5$ using MOSFIRE data (González Morán et al.",
"in preparation).",
"Hence, we argue that future HIIG data are expected to substantially improve the relevant constraints (especially on $w_{a}$ ) and thus the validity of dynamical dark energy on extragalactic scales will be effectively tested.",
"We are also planning to utilize the HII galaxies Hubble expansion probe to constrain modified gravity models in a future work."
],
[
"Data Availability",
"The main analysis of the present work is based on HII galaxy data, which have been made available in [22], at https://doi.org/10.1093/mnras/stab1385, [21] at https://doi.org/10.1093/mnras/stz1577, [19] at https://doi.org/10.1093/mnras/stx2710, [45] at https://doi.org/10.1093/mnras/stv1128[14] at https://doi.org/10.1093/mnras/stu987.",
"SB acknowledges support by the Research Center for Astronomy of the Academy of Athens in the context of the program “Tracing the Cosmic Acceleration”."
]
] | 2107.01749 |
[
[
"Data-driven enhancement of coherent structure-based models for\n predicting instantaneous wall turbulence"
],
[
"Abstract Predictions of the spatial representation of instantaneous wall-bounded flows, via coherent structure-based models, are highly sensitive to the geometry of the representative structures employed by them.",
"In this study, we propose a methodology to extract the three-dimensional (3-D) geometry of the statistically significant eddies from multi-point wall-turbulence datasets, for direct implementation into these models to improve their predictions.",
"The methodology is employed here for reconstructing a 3-D statistical picture of the inertial wall coherent turbulence for all canonical wall-bounded flows, across a decade of friction Reynolds number ($Re_{\\tau}$).",
"These structures are responsible for the $Re_{\\tau}$-dependence of the skin-friction drag and also facilitate the inner-outer interactions, making them key targets of structure-based models.",
"The empirical analysis brings out the geometric self-similarity of the large-scale wall-coherent motions and also suggests the hairpin packet as the representative flow structure for all wall-bounded flows, thereby aligning with the framework on which the attached eddy model (AEM) is based.",
"The same framework is extended here to also model the very-large-scaled motions, with a consideration of their differences in internal versus external flows.",
"Implementation of the empirically-obtained geometric scalings for these large structures into the AEM is shown to enhance the instantaneous flow predictions for all three velocity components.",
"Finally, an active flow control system driven by the same geometric scalings is conceptualized, towards favourably altering the influence of the wall coherent motions on the skin-friction drag."
],
[
"Introduction",
"Owing to its highly chaotic and random nature, an accurate prediction of the instantaneous velocity in a turbulent flow remains the most challenging demand from any turbulence model.",
"Despite its inherent complexities, turbulence flow modelling has remained an active area of research for over half a century [10], given the numerous incentives on offer.",
"In case of wall-bounded flows, for example, the ability to predict/replicate the instantaneous flow phenomena can greatly facilitate the design of active flow control techniques [12], and also aid in enhancing future high-fidelity numerical simulations, by providing realistic inflow boundary conditions and improving computational efficiency [55], [61].",
"One of the popular approaches of modelling wall-turbulence, amongst many, has been by focusing on the recurring and statistically significant `coherent' motions [48], [12] omnipresent in these flows.",
"These motions play a crucial role in both the kinematics as well as dynamics of wall-bounded flows [48], [30], [35] and have been directly associated with the behaviour of several flow statistics – both averaged [21], [27], [26] and instantaneous [2], [51], [52].",
"Hence, several studies proposing coherent structure-based models can be found in the literature [56], [24], [59], [57], [13], amongst which the attached eddy model (AEM; [46], [39]) based on Townsend's attached eddy hypothesis [57], is one of the most cited.",
"Here, the words `structures', `motions' and `eddies' are used interchangeably and essentially conform to the definition of a coherent motion given by [48].",
"According to Townsend's attached eddy hypothesis, an inviscid asymptotically high friction Reynolds number ($Re_{\\tau }$ $=$ $\\frac{{U_{\\tau }}{\\delta }}{\\nu }$ ) canonical wall-bounded flow can be modelled via a hierarchy of geometrically self-similar attached eddies, having population densities inversely proportional to their sizes, which vary over ${\\mathcal {O}}$ ($z$ ) – ${\\mathcal {O}}$ ($\\delta $ ).",
"The term `attached' here refers to any coherent motion whose geometric extent scales with its distance from the wall, given by $z$ , while the associated velocity fluctuations scale with the friction velocity $U_{\\tau }$ , but doesn't necessarily imply its velocity signatures physically extending down to the wall [39].",
"Further, $\\delta $ here corresponds to the outer scale of the respective canonical flow geometry, which is the boundary layer thickness for a zero-pressure gradient turbulent boundary layer (ZPG TBL), while it is the pipe radius and channel half-height for internal flows.",
"Increased access to high $Re_{\\tau }$ wall-turbulence datasets, over the past two decades, has confirmed that the kinetic energy production and transfer within these flows is mostly influenced by the inertial (inviscid) motions predominating in the inertial region [38], [35], [29], making AEM ideally suited for modelling these flows.",
"These characteristics also make the inertial motions a key target of active flow control schemes [1].",
"Studies in the past have already demonstrated that the AEM is able to predict ensemble/time-averaged statistics, such as the Reynolds stresses [40], [6], higher order moments [60], [53], two-point correlations [37] and the energy spectra [5], [16], [11] across the inertial layer (100 $\\lesssim $ $z^+$ $\\lesssim $ 0.15$Re_{\\tau }$ ) in high $Re_{\\tau }$ canonical flows.",
"While there have been a few studies focused on predicting the instantaneous flow phenomena in both the inertial and the wake region [51], [52], [20], success has mostly been limited to the wake region, predominated solely by the largest hierarchy of eddies ($\\sim $ $\\mathcal {O}$ ($\\delta $ )).",
"The primary challenge associated with accurately predicting the representative instantaneous flow in the inertial region, is its high sensitivity to the geometry of the coexisting motions, given that the region is populated by a host of statistically significant eddies, with sizes varying from ${\\mathcal {O}}$ ($z$ ) to ${\\mathcal {O}}$ ($\\delta $ ) [34], [16], [63].",
"These eddies are highly three-dimensional (3-D) in geometry, and hence vary considerably in terms of their spatial coherence, which also needs to be accounted for while attempting to replicate the instantaneous flow.",
"To this end, there are several studies in recent literature [63], [50], [11] which have envisioned and subsequently characterized the inertial region as a superposition of two types of inertial motions: (i) motions with their velocity signatures (i.e.",
"coherence) extending all the way down to the wall (i.e.",
"wall coherent or WC motions) and (ii) those with velocity signatures not extending to the wall (wall incoherent; WI).",
"Interestingly, both these eddy types (WC and WI) have been shown by the same set of studies to depict characteristics consistent with Townsend’s attached eddies, although with certain caveats; for example, apart from the hierarchy of self-similar eddies, the WC inertial motions also comprise $\\delta $ -scaled superstructures or very-large-scale motions (VLSMs), which do not conform to Townsend's attached eddies [16], [63].",
"These results, thus, showcase the prospect of predicting the instantaneous wall-bounded flow by using the AEM framework to individually model the WC and WI motions in the inertial region.",
"Figure: (a) Schematic depicting the representative Λ\\Lambda -eddy packet used in the AEM , , with the blue and red iso-contours respectively denoting the negative and positive streamwise velocity induced from the Λ\\Lambda -shaped vortex rods (indicated in green).",
"(b) Streamwise, (c) spanwise and (d) wall-normal velocity induced by the Λ\\Lambda -eddy packet in the reference wall-parallel plane at zz = 0.01z i z_{i} (highlighted in grey) shown in (a).",
"Labels in (a) are used to refer to the 3-D geometry of the packet, with Δy i {\\Delta }y_{i} representing the spanwise half-width, owing to the symmetry of the uu-distribution about the yy = 0 plane.",
"This figure has been adapted from .",
"Mean flow direction is along xx.In the present study, we take the first step towards achieving this by establishing an AEM-based methodology to model the inertial WC motions in all three canonical wall-bounded flows.",
"Specifically, to improve its predictive capability, the geometry of the attached eddies is based on estimates extracted directly from published experimental and numerical datasets.",
"It is worth noting here that although the inertial WC motions are only a subset of the full flow, these motions are responsible for the $Re_{\\tau }$ -dependence of the wall shear stress fluctuations (and hence, the skin-friction drag characteristics) in any wall-bounded flow [45], [14], [23], [1], [54].",
"Further to that, the WC motions are also known to superimpose onto and modulate the near-wall cycle [27], [41], which has a significant impact on the skin friction drag.",
"This makes the present modelling effort relevant for both fundamental as well as applied research.",
"We begin by first reviewing the current state of the AEM ($§$REF ) and its limitations ($§$), followed by proposal of the new methodology ($§$), which provides a data-driven basis to improve the AEM predictions.",
"Throughout this paper, we use the coordinate system $x$ , $y$ and $z$ to refer to the streamwise, spanwise and wall-normal directions respectively, with $u$ , $v$ and $w$ denoting the corresponding fluctuating velocity components.",
"${\\langle }{\\rangle }$ and capitalization indicates averaged quantities while the superscript $+$ refers to normalization in viscous units (eg., $u^+$ $=$ $u$ /$U_{\\tau }$ and $z^+$ $=$ $z{U_{\\tau }}/{\\nu }$ , where $\\nu $ is the kinematic viscosity).",
"In the original AEM conceptualized by [46], a single hairpin or a simple arch-shaped ($\\Lambda $ ) eddy, inclined forwards with respect to the flow direction ($x$ ) at $\\sim $ 45$^{\\circ }$ [25], [18], was considered as the representative coherent structure/eddy.",
"This shape was inspired from the seminal flow visualization studies of [25] in a low $Re_{\\tau }$ TBL.",
"The simplest version of this eddy is essentially made up of two vortex rods, arranged in a $\\Lambda $ -shape, with each rod containing a Gaussian distribution of vorticity about its core.",
"The corresponding velocity field for the eddy can be obtained by performing the Biot-Savart calculations (schematically depicted in figures 2, 6 and 7 of [46]).",
"Over time, the representative eddy shape has evolved as more detailed quantitative measurements, revealing the structure of wall-bounded flows were reported, such as the seminal PIV measurements of [2].",
"While this study highlighted the presence of $\\Lambda $ -type eddies in the TBL, it was found that the eddies are organized in the form of a packet in fully turbulent flows.",
"The use of a $\\Lambda $ -eddy packet (figure REF (a)), instead of a single eddy, as the representative attached eddy was subsequently incorporated into the AEM by [37], and this change was shown to further improve statistical predictions.",
"Figures REF (b-d) depict the near-wall ($z$ = 0.01$z_{i}$ ) planar flow field of the three velocity components associated with a single packet-like eddy, which is simply a superposition of the velocity fields obtained from multiple individual $\\Lambda $ -eddies.",
"In case of the AEM, it should be noted that $w$ = 0 $\\ne $ $u$ , $v$ at $z$ = 0 due to the impermeability condition being the only condition imposed at the wall, which is achieved by using packet structures with image packet pairs in the plane of the wall.",
"To model the inertially dominated (i.e.",
"outer) region, the flow is simply represented by the superposition of multiple such $\\Lambda $ -eddy packets (referred as hierarchies), of varying sizes and population density, randomly distributed in the flow domain.",
"Following Townsend's hypothesis, the geometry of these hierarchies is considered to vary self-similarly, a claim which has received both empirical [4], [16], [63], [28] and theoretical [47], [42] support in the literature.",
"The recent study by [20] is the latest published AEM flow-field configuration (henceforth referred simply as AEM) used to replicate the instantaneous flow-field for a ZPG TBL at $Re_{\\tau }$ $\\approx $ 3200.",
"In this study, the major axis of the $\\Lambda $ -eddy packet was oriented at various angles (with respect to $x$ ) along the $x$ -$y$ plane, to incorporate the `meandering' characteristics of the eddies noted in experiments [27], [31].",
"To summarize, while the meandering features and the eddy inclination angles have been chosen based on the empirical estimates [18], [31], the aspect ratios governing the representative eddy geometry (marked in figure REF (a)) have never had an empirical basis to date, which forms the motivation for the present study.",
"The eddy geometry plays a major role in the visual appearance of the instantaneous flow field generated by the AEM, as will be highlighted in the next section ($§$).",
"The present study, hence, follows in line with previous studies [53], [20] with the aim of enhancing the AEM framework, in order to improve spatial representation of a wall-bounded flow from the model.",
"We note that the typical shape of the representative eddy (of a $\\Lambda $ -eddy packet) has been kept the same with the intention to preserve its low-order complexity, and given its past success in replicating turbulent wall flow statistics [60], [6], [5], [16], [11].",
"Further, the present shape is also well suited to model the wall-coherent subset of the wall-bounded flows, given that all three velocity components generated from the eddy extend down to the near-wall region, (figures REF (b-d)).",
"Figure: Instantaneous (a,b) streamwise, (c,d) spanwise and (e,f) wall-normal velocity fluctuations on a wall-parallel plane at the lower bound of the inertial-region (zz ≈\\approx 0.05δ\\delta for the present Re τ Re_{\\tau }).",
"Data in (a,c,e) corresponds to the AEM of , while that in (b,d,e) corresponds to the ZPG TBL DNS of comprising solely of wall-coherent motions (represented by subscript wc).",
"Black dashed lines are used to highlight the largest spatial features of low momentum for a qualitative comparison.",
"Note the difference in axis limits between (a-d) and (e-f)."
],
[
"Motivation for a data-driven AEM",
"Here, we compare the instantaneous flow fields generated by the AEM [20] with the corresponding flow fields from the direct numerical simulation (DNS) of a ZPG TBL [49] to motivate a data-driven definition of the representative eddy geometry used in the former.",
"Given that the AEM simulates purely the WC portion of the TBL, a logical way to go about this would be by comparing it with the subset of the full DNS fields comprising solely the WC motions.",
"Figure REF presents this comparison between the instantaneous velocity fluctuations for all three components, considered along the wall-parallel plane at the lower bound of the inertial region.",
"It brings out significant differences in the geometry of the coherent motions, with the velocity features in the DNS fields substantially longer than those noted in the AEM fields.",
"A plausible reason behind this difference may be the geometry of a $\\Lambda $ -eddy packet not defined based on empirical estimates, which we aim to facilitate in the present study.",
"Here, we propose to obtain the geometric estimates by reconstructing the 3-D statistical picture of the wall-coherent turbulence from published datasets ($§$).",
"Another noteworthy limitation in the AEM is that all the coherent structures considered in the current model correspond to the hierarchy of self-similar eddies, following Townsend's hypothesis.",
"A true WC flow field, however, cannot be replicated without also considering the $\\delta $ -scaled superstructures or VLSMs [16], [63], which we also intend to include in the model.",
"Interestingly, these motions can also be modelled via the same representative $\\Lambda $ -eddy packet shown in figure REF (a), as demonstrated recently by [11] for modelling the 2-D spectra.",
"Readers may note here that the DNS flow fields in figures REF (b,d,f), which comprise solely the WC motions, have been computed using the publicly available full (WC+WI) fields via a spectral linear stochastic estimation (SLSE) based decomposition technique.",
"Interested readers may refer to appendix for more details on how the WC fields were estimated, and also see figure REF , where the full flow field instants corresponding to the WC fields in figure REF have been plotted.",
"Differences between the full (figure REF ) and the WC (figure REF ) flow fields are substantial, especially for the lateral velocity components, with large spatial coherence observed in case of the latter as compared to the former.",
"Further, the oblique features otherwise apparent in the full spanwise velocity field in the log-region (figure REF (b); [49],[52]), cannot be noted in its WC subset (figure REF (d)).",
"These differences underscore the importance of comparing the AEM fields with solely the WC motions, which the representative eddy models."
],
[
"Datasets and methodology",
"The present study utilizes five previously published multi-point datasets, spanning all three canonical flow geometries.",
"Key parameters of the datasets are summarised in table REF .",
"Dataset ${\\mathcal {T}_{1}}$ corresponds to DNS of ZPG TBL, while ${\\mathcal {T}_{2}}$ and ${\\mathcal {T}_{3}}$ correspond to higher $Re_{\\tau }$ experimental data for the same flow geometry.",
"${\\mathcal {C}_{1}}$ and ${\\mathcal {P}_{1}}$ are respectively the DNS and experimental datasets for a fully turbulent channel and pipe flow.",
"The selected datasets present a unique combination of synchronously acquired $u$ -fluctuations mapped across 3-D space in the wall-bounded shear flow (refer figures REF (a,b) and REF (a-c)).",
"${\\mathcal {T}_{1}}$ , ${\\mathcal {C}_{1}}$ and ${\\mathcal {P}_{1}}$ each comprise $u$ -fluctuations acquired using multiple near-wall fixed probes (placed at $x_w$ $=$ 0,$y_w$ ,$z_w$ ), distributed in log spacing along the spanwise/azimuthal directions, in conjunction with those acquired by the probe traversing along the wall-normal direction ($z$ ).",
"On the other hand, ${\\mathcal {T}_{2}}$ and ${\\mathcal {T}_{3}}$ comprise $u$ -fluctuations from a single near-wall fixed probe (placed at $x_w$ $=$ 0,$y_w$ $=$ 0,$z_w$ ), acquired synchronously with those measured farther from the wall by a probe traversing along the $z$ - and $y$ -directions, respectively.",
"Considering the fact that the canonical wall-bounded flows are statistically homogeneous along the span, the cumulative 3-D space mapped by combining the datasets ${\\mathcal {T}_{2}}$ and ${\\mathcal {T}_{3}}$ would be equivalent to that mapped in each of the individual datasets, ${\\mathcal {T}_{1}}$ , ${\\mathcal {C}_{1}}$ and ${\\mathcal {P}_{1}}$ .",
"In case of ${\\mathcal {P}}_{1}$ , it should be noted that the near-wall probe is essentially a hot-film sensor attached to the wall for measuring the instantaneous skin-friction velocity ${\\vec{U}}_{\\tau }$ (= ${U}_{\\tau }$ + ${u_{\\tau }}$ ), which is associated with the near-wall instantaneous streamwise velocity (${\\vec{U}}$ ) following ${\\vec{U}}_{\\tau }$ = $\\sqrt{{\\nu }({{\\partial }{\\vec{U}}}/{{\\partial }z})}$ .",
"Interested readers may refer to the individual references for more specific details regarding the measurements/simulations.",
"Table: Table summarizing the details of five previously published datasets comprising synchronized measurements of uu-fluctuations by various near-wall fixed probes (placed at x w x_{w} = 0,y w y_{w},z w z_{w}) and a traversing probe (xx = 0,yy,zz), in turn separated by relative spanwise (Δy{\\Delta }y == ∣y-y w ∣{\\mid {y}\\;{-}\\;{y_w}\\mid }) and wall-normal (z-z w {z}\\;{-}\\;{z_w} ≈\\approx zz) offsets.In case of dataset 𝒫 1 {\\mathcal {P}}_{1}, the hot-film attached to the wall measures the instantaneous skin-friction velocity, U → τ {\\vec{U}}_{\\tau } == ν(∂U →/∂z)\\sqrt{{\\nu }({{\\partial }{\\vec{U}}}/{{\\partial }z})}.The probe arrangements associated with each of these datasets have been schematically depicted in figures and , with probes in blue and red respectively denoting the traversing and near-wall fixed probes.We now move on to proposing the methodology used to reconstruct the 3-D statistical picture of the wall-coherent turbulence, from which the geometric estimates for the representative eddy (of the AEM) would be extracted.",
"There have been several studies in the past [21], [27], [32] which have used multi-point datasets to estimate the geometry of the coherent motions, by computing mostly space-time cross-correlations.",
"These cross-correlations, however, represent cumulative contributions from eddies of various length scales at a specific spatial offset (say streamwise offset, ${\\Delta }x$ ).",
"Here, since we are interested in estimating the 3-D geometry of individual hierarchies/length scales to be incorporated in the AEM, we compute the cross-correlations in the spectral domain to get a scale-specific estimate of the coherent structure geometry (i.e.",
"as a function of the streamwise wavelength, ${\\lambda }_{x}$ ).",
"Given our focus is on modelling the inertial wall-coherent motions coexisting in the outer region, we consider the cross-correlations specifically between $u$ -signals in the outer region (at $z$ ) and those acquired close to the wall (at $z^{+}_{w}$ $\\lesssim $ 15), via the cross-correlation spectra ($\\Gamma $ ) defined as [8], [19]: $\\begin{aligned}{{\\Gamma }}({z},{{\\Delta }{y}},{\\lambda _{x}}) &= \\frac{ {\\text{Re}}[\\lbrace \\widetilde{u}(z,y;\\lambda _{x}){{\\widetilde{u}}^{\\ast }}(z_{w},{y_{w}};\\lambda _{x}) \\rbrace ]}{ {\\sqrt{ \\lbrace {\\mid {\\widetilde{u}(z,y;\\lambda _{x})} \\mid }^{2} \\rbrace }}{\\sqrt{ \\lbrace {\\mid {\\widetilde{u}({z_{w}},{y_{w}};\\lambda _{x})} \\mid }^{2} \\rbrace }}}\\\\ &= \\frac{ \\text{Re}[ { {{\\phi }^{\\prime }_{{u}{u_{w}}}}(z,z_{w},{\\Delta }y;\\lambda _{x}) } ] }{ {\\sqrt{\\phi _{{u}{u}}({z},y;\\lambda _{x})}}{\\sqrt{\\phi _{{u_{w}}{u_{w}}}({z_{w}},{y_{w}};\\lambda _{x})}} },\\end{aligned}$ where ${\\tilde{u}}$ = ${\\mathcal {F}}$ ($u$ ) is the Fourier transform of $u$ in either time or $x$ depending on the dataset with ${\\lambda }_{x}$ $=$ 2${\\pi }/{k_x}$ , where $k_x$ is the streamwise wavenumber.",
"Further, the curly brackets ($\\lbrace \\rbrace $ ), asterisk ($\\ast $ ) and vertical bars ($\\mid \\mid $ ) indicate the ensemble averaging, complex conjugate and modulus, respectively while Re denotes the real component.",
"${\\phi }^{\\prime }_{{u}{u_{w}}}$ is, thus, the complex-valued 1-D cross-spectrum between $u$ -signals recorded at $z$ and $z_{w}$ , which quantifies the scale-specific cross-correlation, while $\\phi _{{u}{u}}$ and $\\phi _{{u_{w}}{u_{w}}}$ respectively are the conventional 1-D energy spectra at ${z}$ and ${z_{w}}$ used for the scale-specific normalization.",
"Such a definition forces $\\Gamma $ to vary between -1 to 1, with the former and latter respectively indicating perfect anti-correlation and correlation for each scale, ${\\lambda }_{x}$ .",
"In case of dataset ${\\mathcal {P}_{1}}$ , notably, $\\Gamma $ is computed by substituting the skin-friction velocity fluctuations ($u_{\\tau }$ ) measured by the wall-based hot-films, in place of $u$ ($z_w$ ).",
"As can be noted from its definition in (REF ), $\\Gamma $ is a function of the spatial offsets in all three directions, which is facilitated by computing the cross-correlations between $u$ -signals recorded at various relative spanwise offsets (${\\Delta }y$ $=$ ${\\mid }{y - {y_w}}{\\mid }$ ) and wall-normal offsets ($z$ $\\approx $ ${z - {z_w}}$ , given $z_{w}$ $\\ll $ $z$ ).",
"The offset along $x$ is obtained in the form of a streamwise wavelength, which is estimated by using the Taylor's hypothesis in case of the experimental datasets (where the local convection velocity, $U_{c}$ $=$ $U(z)$ ), or obtained directly from the spatial domain for the DNS datasets.",
"$\\Gamma $ ($z$ ,${\\Delta }y$ ,${\\lambda }_{x}$ ), thus, can be interpreted as a coherence metric which associates each WC scale/eddy (${\\lambda }_{x}$ ) with its corresponding spanwise (${\\Delta }y$ ) and wall-normal extent ($z$ ).",
"This information can be directly used to define the geometry of the representative eddy, for the respective coherent structure-based model, corresponding to the streamwise scale, ${\\lambda }_{x}$ .",
"The benefit in case of models such as the AEM, which incorporate vorticity-based structures, is that once the eddy geometry of the $\\Lambda $ -eddy packet is defined, velocity field for all three components can be simulated without needing to estimate $\\Gamma $ for the other (lateral) velocity components.",
"Figure: (a,b) Experimental setups corresponding to datasets (a) 𝒯 3 {\\mathcal {T}}_{3} and (b) 𝒯 2 {\\mathcal {T}}_{2}.",
"Hotwire sensors marked in blue correspond to traversing probes, which move along their respective traversing directions also marked in blue, while sensors marked in red correspond to the fixed near-wall probes.",
"(c) Cross-correlation spectra computed as a function of Δy{\\Delta }y and λ x {\\lambda }_{x} (i.e.",
"Γ\\Gamma (Δy{\\Delta }y,λ x {\\lambda }_{x})) for zz ≈\\approx 0.01δ\\delta from the dataset 𝒯 3 {\\mathcal {T}}_{3}.",
"(d) Γ\\Gamma (zz,λ x {\\lambda }_{x}) computed for Δy{\\Delta }y ≈\\approx 0 from the dataset 𝒯 2 {\\mathcal {T}}_{2}.",
"Black contour in (c,d) corresponds to Γ\\Gamma = 0.1.",
"(e-g) Reconstruction of the 3-D cross-correlation spectra, Γ\\Gamma (zz,Δy{\\Delta }y,λ x {\\lambda }_{x}) across the inertial region (0.01δ\\delta ≲\\lesssim zz ≲\\lesssim 0.15δ\\delta ) by fusing the empirical estimates from datasets 𝒯 2 {\\mathcal {T}}_{2} and 𝒯 3 {\\mathcal {T}}_{3}.",
"(e) is obtained by simply combining data in (c) and (d).",
"(f) and (g) are similarly obtained by fusing data in (d) and that estimated from dataset 𝒯 3 {\\mathcal {T}}_{3} over a range of zz: (f) 0.01δ\\delta ≲\\lesssim zz ≲\\lesssim 0.05δ\\delta and (g) 0.01δ\\delta ≲\\lesssim zz ≲\\lesssim 0.15δ\\delta .",
"Black iso-contour in (e-g) also corresponds to Γ\\Gamma = 0.1.",
"Dash-dotted lines in green, orange, indigo and grey colours represent the scalings: λ x {\\lambda }_{x} = 14zz, λ x {\\lambda }_{x} = 20Δy{\\Delta }y, Δy{\\Delta }y = 0.17δ\\delta and λ x {\\lambda }_{x} = 4δ\\delta , respectively.It is worth noting here that $\\Gamma $ being used here is different from the linear coherence spectrum (LCS; ${\\gamma }^{2}_{L}$ ) computed previously by [4] and [7], since $\\Gamma $ retains solely the real part of the cross-spectrum while ${\\gamma }^{2}_{L}$ uses the absolute value of the cross-spectrum.",
"Here, we prefer $\\Gamma $ over ${\\gamma }^2_{L}$ , to define the eddy geometry, given that the anti-correlated regions ($\\Gamma $ $<$ 0) are indicative of the relative placement of low-momentum ($-u$ ) regions with respect to the high-momentum ($+u$ ) regions, which otherwise can't be inferred from the LCS (0 $\\le $ ${\\gamma }^2_{L}$ $\\le $ 1).",
"For example, a recent study [19] focusing on the $\\Gamma $ -distribution over very large spatial extents has confirmed the periodic distribution of $\\delta $ -scaled $+u$ and $-u$ motions along the spanwise direction.",
"Investigating the $\\Gamma $ distribution in 3-D space, thus, could provide quantitative basis to the choice of the representative eddy for a coherent structure-based model, as will be discussed ahead in this section."
],
[
"Application to high $Re_{\\tau }$ ZPG TBL datasets",
"We begin the empirical analysis by first reconstructing $\\Gamma $ ($z,{{\\Delta }y},{{\\lambda }_{x}}$ ) for high $Re_{\\tau }$ ZPG TBL using the two experimental datasets, ${\\mathcal {T}}_{3}$ and ${\\mathcal {T}}_{2}$ .",
"In case of ${\\mathcal {T}}_{2}$ , the traversing probe is constrained to move vertically above the near-wall fixed probe (i.e.",
"maintaining ${\\Delta }y$ $=$ 0), owing to which this dataset yields the statistical picture solely in the streamwise wall-normal plane, given by $\\Gamma $ (${{\\Delta }y}$ $=$ 0$;z,{{\\lambda }_{x}}$ ) in figure REF (d).",
"This plot brings out the range of energetic WC scales/motions coexisting in the outer region, which correspond to the wavelength range, ${\\lambda }_{x}$ $\\gtrsim $ 0.1$\\delta $ .",
"For a specific energetic length scale, say ${\\lambda }_{x}$ $\\sim $ 4$\\delta $ , it indicates the presence of a positively correlated region up to $z$ $\\sim $ 0.3$\\delta $ , followed by a negatively correlated region between 0.3$\\delta $ $<$ $z$ $<$ 0.6$\\delta $ .",
"When interpreted physically in terms of the spatial extent of the $u$ -velocity distributions around a $\\Lambda $ -eddy packet of length $\\sim $ 4$\\delta $ (figure REF (a)), the aforementioned $z$ -ranges may be respectively associated with the wall-normal extent of the $-u$ region (represented by +$\\Gamma $ ) and of the $+u$ region (represented by -$\\Gamma $ ).",
"Consequently, the height of the $\\Lambda $ -eddy packet may be nominally defined based on the region where $\\Gamma $ $\\approx $ 0, which is approximately $z$ $\\sim $ 0.3$\\delta $ .",
"If we consider this for the entire range ${\\lambda }_{x}$ $\\gtrsim $ 0.1$\\delta $ , figure REF (d) indicates that the height ($z$ ) of the inertially dominated eddies varies self-similarly with respect to their streamwise length scale (${\\lambda }_{x}$ ), which is given by the linear relationship ${\\lambda }_{x}$ $=$ 14$z$ (indicated by a dashed-dotted green line fitted to $\\Gamma $ $\\approx $ 0).",
"Interestingly, the same linear relationship between ${\\lambda }_{x}$ and $z$ was noted by [4] for a ZPG TBL, although based on ${\\gamma }^2_{L}$ as the metric.",
"The same analysis is next conducted along the wall-parallel plane, at $z$ corresponding to the inertial-region, by using the dataset ${\\mathcal {T}}_{3}$ .",
"Figure REF (c) plots $\\Gamma $ ($z$ $=$ 0.01$\\delta $ ;${\\Delta }y$ ,${\\lambda }_{x}$ ) as an example, which corresponds to the 2-D statistical picture at the lower bound of the inertial region.",
"This picture, however, represents the spanwise coherence only along the $+{\\Delta }y$ axis and would be complete by considering a mirror image of the same plot about ${\\Delta }y$ = 0, owing to symmetry [27].",
"Given that the $\\Gamma $ -distribution in figure REF (c) is of the same nature as in figure REF (d), the physical interpretation discussed for the latter is now extended to the former.",
"By fitting a dashed-dotted orange line along $\\Gamma $ $\\approx $ 0, we can interpret from figure REF (c) that the spanwise half-width (${\\Delta }y$ ) of the $\\Lambda $ -eddy packet also varies self-similarly with respect to ${\\lambda }_x$ , which is represented by the linear relationship ${\\lambda }_{x}$ $=$ 20${\\Delta }y$ .",
"This self-similar trend, however, is only valid for streamwise wavelengths up to ${\\lambda }_{x}$ $\\lesssim $ 4$\\delta $ , as all the larger length scales are found to have a constant half-width, ${\\Delta }y$ $\\sim $ 0.17$\\delta $ .",
"These large eddies, hence, do not conform to Townsend's hierarchy of attached eddies but, in fact, correspond to the $\\delta $ -scaled very-large-scales or superstructures noted previously in the ZPG TBL [27], [32].",
"The spanwise widths of these structures, which is found from the present analysis to be 2${\\Delta }y$ $\\sim $ 0.35$\\delta $ , is consistent with previous findings based on PIV experiments [15], [22].",
"Following the analysis on the individual 2-D planes, the plots in figures REF (c,d) can now be stitched together to reconstruct the 3-D statistical picture of the WC motions in the ZPG TBL, as shown in figure REF (e).",
"By estimating the cross-correlation spectra at various $z$ : 0.01$\\delta $ $\\lesssim $ $z$ $\\lesssim $ 0.15$\\delta $ from dataset ${\\mathcal {T}}_{3}$ , $\\Gamma $ ($z$ ,${\\Delta }y$ ,${\\lambda }_{x}$ ) can be reconstructed across the inertial region, as depicted in figures REF (e-g).",
"These figures, thus, describe the scale-specific geometry of the range of energetic motions coexisting in the inertial region: the flow at the beginning of the inertial region ($z$ $\\sim $ 0.01$\\delta $ ) comprises contributions from the widest hierarchy of self-similar eddies, spanning 0.1$\\delta $ $\\lesssim $ ${\\lambda }_{x}$ $\\lesssim $ 4$\\delta $ , which reduces to 0.8$\\delta $ $\\lesssim $ ${\\lambda }_{x}$ $\\lesssim $ 4$\\delta $ at $z$ $\\approx $ 0.05$\\delta $ , and finally negligible contributions from the self-similar hierarchy at the upper bound ($z$ $\\approx $ 0.15$\\delta $ ).",
"The contributions from the $\\delta $ -scaled superstructures (${\\lambda }_{x}$ $\\gtrsim $ 4$\\delta $ ), on the other hand, is consistently present throughout the inertial region.",
"The corresponding spanwise and wall-normal extents for each eddy (${\\lambda }_{x}$ ) can also be inferred based on the scalings shown in figure REF (g), and have been summarized as a function of the flow $Re_{\\tau }$ below: ${{\\lambda }^{+}_x} {\\gtrsim } 4{Re_{\\tau }} \\; {\\left\\lbrace \\begin{array}{ll}{z^{+}} = {{\\lambda }^{+}_x}/14; {\\Delta }y^{+} = 0.17{Re_{\\tau }} & \\text{if true}\\\\\\end{array}{z^{+}} = {{\\lambda }^{+}_x}/14; {\\Delta }y^{+} = {{\\lambda }^{+}_x}/20 & \\text{otherwise}.\\right.",
"}$ This information can be directly used to facilitate definition of the representative eddy geometries in any coherent structure-based model.",
"It is worth noting here though, $\\Gamma $ ($z$ ,${\\Delta }y$ ,${\\lambda }_{x}$ ) in figure REF (g) looks consistent with the shape of a $\\Lambda $ -eddy packet when halved along the span (refer to figure REF or REF ), thus providing quantitative support to the choice of the representative eddy shape used in the AEM and making a strong case in favour of implementing (REF ) into AEM."
],
[
"Extension to various $Re_{\\tau }$ and flow geometries",
"Having described the methodology to reconstruct ${\\Gamma }$ ($z$ ,${\\Delta }y$ ,${{\\lambda }_{x}}$ ) and applied it to high $Re_{\\tau }$ ZPG TBL data, we now extend the same to datasets at different $Re_{\\tau }$ (${\\mathcal {T}}_{1}$ ) and other flow geometries (${\\mathcal {C}}_{1}$ and ${\\mathcal {P}}_{1}$ ) to check for the universality of the scalings noted in figure REF .",
"As discussed previously, each of these datasets are self-sufficient to reconstruct the 3-D statistical picture across the inertial region.",
"Figure REF (d) plots the same for the low $Re_{\\tau }$ ZPG TBL data using similar plotting style as in figure REF (g).",
"Interestingly, all the scalings noted previously from the high $Re_{\\tau }$ experimental data (expressed in (REF )) are also observed for the low $Re_{\\tau }$ case, confirming the $Re_{\\tau }$ -invariance of these empirical estimates and consequently, their direct utilization in data-driven coherent structure-based models.",
"Further, the spanwise half-width (${\\Delta }y$ ) is also found to tend towards a constant ($\\sim $ 0.17$\\delta $ ) at ${\\lambda }_{x}$ $\\gtrsim $ 4$\\delta $ for the DNS dataset, confirming that the wavelength range estimated for the $\\delta $ -scaled superstructures is independent of the Taylor's hypothesis assumption.",
"It is worth noting here that the low $Re_{\\tau }$ for the dataset ${\\mathcal {T}}_{1}$ reduces the thickness of the inertial region (0.05$\\delta $ $\\lesssim $ ${\\lambda }_{x}$ $\\lesssim $ 0.15$\\delta $ ) and consequently the range of energetic scales (${\\lambda }_{x}$ ) corresponding to the self-similar hierarchy, which is most evident at $z$ $\\approx $ 0.05$\\delta $ .",
"Figure: Reconstruction of the 3-D cross-correlation spectra, Γ\\Gamma (zz,Δy{\\Delta }y,λ x {\\lambda }_{x}) across the inertial region for datasets (d) 𝒯 1 {\\mathcal {T}}_{1}, (e) 𝒞 1 {\\mathcal {C}}_{1} and (f) 𝒫 1 {\\mathcal {P}}_{1} by utilizing the uu-fluctuations associated with the probe placements depicted for the respective flow geometries in (a-c).",
"Probes marked in blue correspond to traversing probe which move along their respective traversing directions also marked in blue, while those marked in red correspond to the fixed near-wall probes.",
"Here, only two near-wall probes are shown for representative purposes, with the actual number given in table .",
"Black iso-contour in (d-f) corresponds to Γ\\Gamma = 0.1.",
"Dash-dotted lines in green, orange, indigo and grey colours represent scalings indicated in each figure.Figures REF (e,f) respectively depict ${\\Gamma }$ ($z$ ,${\\Delta }y$ ,${{\\lambda }_{x}}$ ) reconstructed using the high $Re_{\\tau }$ channel (${\\mathcal {C}}_{1}$ ) and pipe flow (${\\mathcal {P}}_{1}$ ) datasets.",
"Remarkably, the same self-similar scalings for the WC motions, noted previously for ZPG TBL, are also noted for the case of internal flows.",
"Similarly, it is found that the contribution from the self-similar hierarchies becomes statistically insignificant beyond the upper bound of the inertial region, in both the internal flows.",
"Although not shown here, ${\\Gamma }$ ($z$ ,${\\Delta }y$ ,${{\\lambda }_{x}}$ ) (similar to figure REF (f)) was also reconstructed for the relatively low $Re_{\\tau }$ data ($\\approx $ 10000 and 20000) acquired at the same facility by [7], which exhibited the same scalings/behaviour.",
"This universality in the self-similar scaling, across all three canonical flows, is consistent with the observation of [28] based on low $Re_{\\tau }$ ($\\approx $ 1000) DNS datasets.",
"Given that the estimation of the exact linear scalings at low $Re_{\\tau }$ may be limited by the narrower range of the self-similar hierarchy, the present study confirms the existence as well as the $Re_{\\tau }$ -invariance of these self-similar scalings across all three canonical flows.",
"In case of the internal flows, however, the ${\\lambda }_{x}$ $=$ 20${\\Delta }y$ relationship is found to be valid up to a much larger length scale (${\\lambda }_{x}$ $\\approx $ 6$\\delta $ ), beyond which the constant spanwise half-width tends to a constant, ${\\Delta }y$ $\\sim $ 0.3$\\delta $ .",
"The scalings describing the 3-D spatial extent of the inertial motions, in case of the internal flows, can thus be expressed as a function of $Re_{\\tau }$ following: ${{\\lambda }^{+}_x} {\\gtrsim } 6{Re_{\\tau }} \\; {\\left\\lbrace \\begin{array}{ll}{z^{+}} = {{\\lambda }^{+}_x}/14; {\\Delta }y^{+} = 0.30{Re_{\\tau }} & \\text{if true}\\\\\\end{array}{z^{+}} = {{\\lambda }^{+}_x}/14; {\\Delta }y^{+} = {{\\lambda }^{+}_x}/20 & \\text{otherwise}.\\right.",
"}$ These trends suggest that the $\\delta $ -scaled VLSMs (${\\lambda }_{x}$ $\\gtrsim $ 6$\\delta $ ) conforming to the internal flow geometries are relatively wider than the superstructures in ZPG TBL [43].",
"These differences between the internal and external flows have been previously noted by [9], [44] and [33] and were attributed to the `persistent growth' of the energetic motions, beyond the inertial region, in case of internal flows, which is otherwise inhibited by the turbulent/non-turbulent interface (T/NTI) in the ZPG TBL [44], [33]."
],
[
"Incorporating the empirically-obtained scaling laws into the AEM",
"With the 3-D statistical picture of the wall-coherent turbulence now characterized for all three canonical flows, we move onto discussing how the empirically obtained scalings can be used in the data-driven AEM (dd-AEM).",
"In the remainder of this manuscript, we concentrate our efforts on the attached eddy modelling of solely the ZPG TBL flow.",
"The methodology, however, can also be applied to the internal flows by using the appropriate empirical estimates (given in (REF )).",
"Although a real ZPG TBL flow comprises of a continuous distribution of statistically energetic motions over the scale range: ${\\lambda }_{x}$ $\\gtrsim $ 0.1$\\delta $ (figure REF (g)), for the case of the AEM, we consider a discretized distribution of eddies across $n$ hierarchies for convenience in modelling.",
"Here, $n$ depends on the $Re_{\\tau }$ of the flow being modelled, with higher $Re_{\\tau }$ requiring the inclusion of more hierarchies/scales.",
"We follow the convention adopted by [46], of defining the wall-normal extent of the smallest hierarchy equal to the beginning of the inertial region ($z^+$ $\\sim $ 100), with subsequent larger hierarchies doubling in height.",
"Hence, the wall-normal extent of the `$i^{th}$ ' hierarchy (figure REF (b)) may be defined as $z_i$ = ${\\delta }$ (2$^{i-n}$ ), with $i$ varying from 1 (for the smallest hierarchy) to $n$ (for the largest hierarchy), with $z_{n}$ $=$ $\\delta $ by definition.",
"For the present study, we consider a ZPG TBL to be modelled via $n$ $=$ 6 hierarchies.",
"Hence, the height of the smallest ($i$ $=$ 1) hierarchy becomes $z_1$ $=$ ${\\delta }/{2^5}$ or $z^+_1$ $=$ ${Re_{\\tau }}/{2^5}$ .",
"Enforcing $z^+_1$ $\\approx $ 100 following [46] yields the $Re_{\\tau }$ $\\approx $ 3200 for the dd-AEM.",
"This $Re_{\\tau }$ is equal to that considered for the previously published AEM by [20], and is also close to the $Re_{\\tau }$ $\\approx $ 2000 of the ZPG TBL of [49].",
"To extract the geometric estimates for the eddy packet (figure REF (b)) corresponding to each of the six hierarchies, figure REF (a) shows the same 3-D statistical picture as in figure REF (g) discretized into six individual blocks.",
"Here, the $x$ -axis location of the $i^{th}$ block represents the streamwise length scale (${\\lambda }_{x_{i}}$ ) of the $i^{th}$ hierarchy, while the corresponding extents along the $y-$ and $z-$ axis respectively indicate its spanwise half-width (${\\Delta }y_{i}$ ) and the wall-normal height ($z_{i}$ ).",
"It is worth noting here that the eddy packet in figure REF (b), with seven $\\Lambda $ -eddies, is simply shown for representative purposes and the actual number of $\\Lambda $ -eddies in the packet may vary depending on ${\\lambda }_{x_{i}}$ /$z_{i}$ (table REF ), such that the inter-eddy spacing is maintained constant for both AEM and dd-AEM.",
"Table REF records the geometric estimates of all six hierarchies for the dd-AEM, and compares them with those considered in the previous AEM by [20].",
"The size of hierarchies 1 to 4 grows self-similarly in all three directions, in accordance to the empirically obtained scalings, which associates them with Townsend's attached eddies.",
"Hierarchies 5 and 6, on the other hand, have a constant spanwise half-width (${\\Delta }y$ $\\sim $ 0.17$\\delta $ ).",
"The two largest hierarchies, thus, represent the $\\delta $ -scaled superstructures, which were found to be the sole energetic WC motions coexisting beyond the upper bound of the inertial region (figure REF (g)).",
"The presence of these $\\delta $ -scaled eddies, which do not conform to the self-similar hierarchy, is another improvement of the dd-AEM over the AEM, given that all six hierarchies in the latter conformed to the self-similar hierarchy (table REF ).",
"Table: Comparing the geometry of the various eddy hierarchies in the AEM and dd-AEM.Here, SS and NSS respectively denote self-similar and non-self-similar.",
"Other terminologies have been defined in figure (b).Λ\\Lambda -eddy packets corresponding to the AEM and dd-AEM comprise of 7 and 30 Λ\\Lambda -eddies respectively, to account for the differing λ x i {\\lambda }_{x_{i}}/z i z_{i}.Figure: Cross-correlation spectra computed along (a-c) XZ-plane at Δy{\\Delta }y ≈\\approx 0, Γ\\Gamma (Δy{\\Delta }y ≈\\approx 0;zz,λ x {\\lambda }_{x}), (d-f) XY-plane at the lower bound of the inertial region, Γ\\Gamma (zz ≈\\approx 0.01δ\\delta ;Δy{\\Delta }y,λ x {\\lambda }_{x}) and (g-i) XY-plane at the upper bound of the inertial region, Γ\\Gamma (zz ≈\\approx 0.15δ\\delta ;Δy{\\Delta }y,λ x {\\lambda }_{x}) from the (a,d,g) AEM of , (b,e,h) data driven-AEM and (c,f,i) datasets 𝒯 2 {\\mathcal {T}}_{2} and 𝒯 3 {\\mathcal {T}}_{3}.",
"Dash-dotted lines in green, orange, indigo and grey colours represent the same scalings as described in figure .",
"The plots in (c,f) represent the same data as in figures (d,c) respectively, with the only difference being the change in axis limits to match with the plots from the AEM.The best way to quantify the impact of the changed eddy aspect ratios (table REF ) in the AEM would be to compute the cross-correlation spectra ($\\Gamma $ ) using both AEM and dd-AEM fields and comparing it with the empirical estimate.",
"This comparison is showcased in figure REF , where $\\Gamma $ is considered along the XZ plane (figures REF (a-c)), and that along two wall-parallel planes at the lower (figures REF (d-f)) and upper bound of the inertial region (figures REF (g-i)).",
"Here, for convenience in interpretation, $\\Gamma $ computed from both the AEMs is considered only in the ${\\lambda }_{x}$ -range encompassing all six hierarchies (table REF ).",
"We can note a significant difference between $\\Gamma $ estimated from the AEM and the dd-AEM fields, with the latter following the scaling trends consistent with those obtained empirically.",
"Based on the comparison along both the wall-parallel planes (figures REF (d-i)), we can expect the dd-AEM to perform much better than the AEM throughout the inertial region."
],
[
"Spatial representation in the inertial region",
"We now extend the comparison between the dd-AEM and the datasets to the instantaneous velocity fluctuations in the inertial region.",
"Figure REF presents this comparison between the instantaneous fields for all three velocity fluctuations estimated from the dd-AEM, AEM and the ZPG TBL DNS dataset (${\\mathcal {T}}_{1}$ ) at the lower bound of the inertial region.",
"Amongst these, the plots associated with the latter two are the same as shown previously in figure REF , with significant differences noted between them ($§$).",
"Now, with the geometry of the representative eddy defined based on data in the dd-AEM, along with the inclusion of the $\\delta $ -scaled superstructures, the corresponding instantaneous velocity fields (figures REF (b,e,h)) can be observed to closely match the DNS fields.",
"In particular, the extended coherence of the velocity features ($\\gtrsim $ 2$\\delta $ ) noted in all three velocity components in the DNS have been well replicated by the dd-AEM, highlighting the impact of basing the eddy geometry on the data.",
"This reaffirms the fact that all three components are essentially inter-dependent, and can be modelled via a representative eddy comprising multiple vortex structures (figure REF ).",
"The meandering very-large-scale motions or superstructures which extend beyond 6$\\delta $ in length [27], [32], as observed in the DNS (figure REF (c)), are also well represented by the dd-AEM (figure REF (b)).",
"The qualitative agreement between the dd-AEM and the DNS, showcased in figure REF , can be confirmed quantitatively by generating averaged flow fields conditioned on a statistically dominant feature in the flow.",
"One such statistical flow feature, which is predominant in the inertial region, is the negative value of the instantaneous momentum flux ($uw$ $<$ 0; [58], [17]), which is carried by coherent motions associated with ejection ($u$ < 0, $w$ > 0) and sweep ($u$ > 0, $w$ < 0) events.",
"Notably, the velocity field associated with the representative $\\Lambda $ -eddy packet (figure REF ) is also consistent with this notion and models features associated with both ejection and sweep events.",
"Hence, we compute the conditionally averaged wall-parallel flow fields for the coherent regions associated with strong ejections and sweeps by following the same methodology as adopted previously by [52] and [20].",
"This is performed by attaching a frame of reference to the centroid of each flow feature representing strong $u$ and $w$ fluctuations, which are used as the conditioning points.",
"Here, the strong fluctuations correspond to $u^+$ $\\lesssim $ -1 and $w^+$ $\\gtrsim $ 1 for ejections and vice versa for the sweeps.",
"The mean flow feature is obtained by averaging multiple such strong features extracted from several flow fields.",
"Figure: Instantaneous streamwise velocity fluctuations on a wall-normal plane.",
"Data in (a) corresponds to the dd-AEM developed in the present study, that in (b) is from the AEM of , while that in (c,d) corresponds to the same ZPG TBL DNS flow field , with difference being (c) comprises of solely the wall-coherent motions (represented by subscript wc) and (d) the full flow field (WC+WI).Figure REF presents the averaged velocity signatures conditioned for the strong ejection features, at the lower bound of the log-region, from flow fields obtained from the AEM, dd-AEM and the DNS.",
"Firstly, it can be noted that the conditionally averaged fields from the DNS (figures REF (c,f,i)) qualitatively resemble the velocity fields around a single $\\Lambda $ -eddy packet (figure REF ), giving further support to the choice of the representative eddy.",
"On comparing the geometric extents of the velocity features generated from the two AEMs and the DNS, a better match between the dd-AEM and the DNS can be noted, confirming the enhanced performance of the dd-AEM.",
"To list a few specific improvements, the long streamwise coherence observed particularly in ${\\langle }{w^+_{wc}}{\\rangle }$ (figure REF (i)) is well replicated by ${\\langle }w^+{\\rangle }$ generated from the dd-AEM (figure REF (h)).",
"Further, the extent of spanwise coherence for all three velocity components, noted in the dd-AEM, compares better with the DNS than that noted for the AEM.",
"The streamwise extent of the wall-parallel velocity features, however, are still falling short of the estimates from the DNS.",
"This may be an artefact of the assumption of scale-independent yaw angles, imposed onto the $\\Lambda $ -eddy packets, to replicate `meandering' phenomena in the TBL [20].",
"It seems plausible that the scales with the longest streamwise extent (i.e.",
"${\\lambda }_{x}$ ) `meander' with lesser intensity than those with relatively shorter extent, which may favourably alter the comparison being presented in figure REF .",
"This, however, remains a subject for future work.",
"It is also worth noting here that the close match of the conditioned fields from the DNS (figures REF (c,f,i)), with the corresponding flow fields from the $\\Lambda $ -eddy packet (figure REF ), is owing to the consideration of solely the WC motions.",
"Interested readers may refer to figures REF (d,e,f) where full flow fields (WC + WI) have been conditionally averaged based on the same criteria as in figure REF .",
"The significant mismatch between the velocity features from the WC and full fields, for the DNS, suggests the present representative eddy packet as a suitable choice to model solely the WC subset of the TBL (and not the full flow as a whole)."
],
[
"Discussion",
"The present effort promises enhancement in the spatial representation of an instantaneous wall flow by means of a data-driven coherent structure-based model.",
"One would hope that such models, which have the capability to predict very high $Re_{\\tau }$ flows in a computationally inexpensive way, will drive future investigations into flow phenomena otherwise difficult to measure experimentally, such as the instantaneous variation of the $w$ -component over the homogeneous wall-parallel plane (as in figure REF (h)) or that of $v$ -component over the wall-normal plane.",
"While the data driven-AEM developed here seems a promising prospect in this regard, more work needs to be done in terms of also incorporating the WI motions to be able to predict the full inertially-dominated turbulent flow field.",
"These WI motions not only are as statistically significant as the WC motions in the inertial region [11], [50], but are also the key driver of the flow phenomena in the wake region.",
"This is evident from the instantaneous $u$ -velocity fluctuations plotted in the wall-normal plane, which are compared for the AEM, dd-AEM and the DNS in figure REF .",
"As we would expect, the dd-AEM predictions (figure REF (a)) compare well with the instantaneous flow field from the DNS, comprising solely the WC motions (figure REF (c)).",
"These motions, however, lose their coherence in the wake region where the WI motions predominate, which can be deciphered from the full DNS field in figure REF (d).",
"Another aspect of coherent structure-based modelling requiring further work is the shape/form of the representative eddy.",
"The present choice of the $\\Lambda $ -eddy packet for the AEM, although consistent with the statistical picture ($\\Gamma $ ) obtained empirically, does not reflect the typical flow structure observed in an instantaneous flow.",
"Efficient machine learning-based algorithms [10], which have the capability to analyze big datasets, may be better placed to propose a solution for this.",
"Besides assisting with the modelling of the WC motions, the empirically determined geometric scalings in (REF ,REF ) can also be used to enhance the active flow control schemes operating based on real-time sensing.",
"Figure REF depicts a conceptual sketch showcasing the coherent motions in a ZPG TBL, which can be manipulated by cross-flow (wall-normal) jets controlled by a computer, based on data from a spanwise array of skin-friction sensors placed at a sufficiently upstream location [1].",
"These skin-friction sensors, which are used to detect the incoming high-momentum ($+u$ ) carrying coherent motions, are also capable of detecting the streamwise extent (${\\lambda }_{x}$ ) of these motions.",
"Utilizing the empirically estimated scalings, the computer controlling the jet actuation can decide how much momentum to inject through them, based on the estimated wall-normal extent ($z$ ) of the incoming motions (indicated as control system 1 in figure REF ).",
"Alternatively, in scenarios where sufficiently long separations between the sensors and the jets cannot be maintained, the spanwise sensor array can be used to estimate the spanwise coherence (${\\Delta }y$ ) of the incoming motions, through which the wall-normal extent could be predicted (indicated as control system 2 in figure REF ).",
"Such empirically-driven flow control systems, which also consider the wall-normal extent of the incoming motions, could potentially manipulate the turbulent inertial wall flow more efficiently, as demonstrated previously by [62].",
"Given the significant contribution of the attached eddy hierarchy to the mean skin friction at high $Re_{\\tau }$ [23], availability of their corresponding 3-D geometric scalings can also be used to decide the spanwise spacing between the cross-flow jets targeting these eddies.",
"Figure: Conceptualization of an active-flow control system aimed at efficiently manipulating the wall-coherent inertial motions in a ZPG TBL by utilizing the scalings determined from high Re τ Re_{\\tau } datasets.",
"Concept of the flow control system has been adapted from .",
"U ∞ U_{\\infty } denotes mean freestream speed."
],
[
"Concluding remarks",
"The present study analyses a unique set of multi-point datasets to reconstruct the 3-D statistical picture of the inertial wall-coherent (WC) turbulence in all three canonical wall-bounded flows.",
"Previous studies have found these motions to be responsible for both, the $Re_{\\tau }$ -dependence of the skin-friction drag [45], [14], [23], [54] as well as the bulk production and the inter-scale energy transfer in high $Re_{\\tau }$ flows [38], [35], [29].",
"The aforementioned characteristics make these motions a key target of coherent structure-based models, which the present study attempts to enhance.",
"Here, the statistical picture is reconstructed by computing the cross-correlation spectra ($\\Gamma $ ) using the streamwise velocity fluctuations mapped across the 3-D space in the wall-bounded shear flow.",
"The intermediate- and large-scaled inertial WC motions are found to exhibit geometric self-similarity with respect to the distance from the wall $z$ , expressible by simple linear relationships, which are universal across all canonical flows and independent of flow $Re_{\\tau }$ .",
"The geometry of the very-large-scaled motions, on the other hand, is found to exhibit $\\delta $ -scaling, associating them with the superstructures and VLSMs for external and internal flows, respectively.",
"The present study also confirms the 3-D geometry of the VLSMs to be much larger than the superstructures even in high $Re_{\\tau }$ flows, highlighting the role of the T/NTI in inhibiting the growth of these very-large-scaled motions in external flows [44], [33].",
"Alongside testing the universality of the 3-D statistical picture, the geometric scalings brought out from the data (given by REF and REF ) are also proposed to be used as a metric to estimate the spanwise and wall-normal extent corresponding to each WC scale/eddy (${\\lambda }_{x}$ ).",
"Application of these linear relationships to any coherent structure-based model can provide a data-driven basis to the geometry of the representative structure, which is demonstrated in this study using the attached eddy model (AEM; [46]) for a ZPG TBL.",
"Here, the choice of AEM (over other models) is also driven by the empirical analysis, given the latter provides evidence of geometric self-similarity of the inertial motions and also supports the $\\Lambda $ -eddy packet as the representative eddy (both of which are built into the AEM framework; [39]).",
"The present study further extends the scope of the AEM by using its framework to also model the $\\delta $ -scaled superstructures, which correspond to the very-large-scales beyond the self-similar hierarchy.",
"The data-driven AEM (or dd-AEM), which has the representative eddies defined based on the data, is shown to improve upon recent works on the AEM [20] in replicating instantaneous flow phenomena associated with all three velocity components.",
"This sets up the platform for future work, which would include both the WC and WI (wall-incoherent) motions modelled via the AEM framework, to replicate the full turbulent wall-bounded flow that can provide realistic inflow conditions for use in future numerical simulations [55], [61].",
"Figure: Instantaneous (a) streamwise, (b) spanwise and (c) wall-normal velocity fluctuations on a wall-parallel plane at the lower bound of the log-region (zz ≈\\approx 0.05δ\\delta for the present Re τ Re_{\\tau }) extracted from ZPG TBL DNS dataset (𝒯 1 {\\mathcal {T}}_{1}) of .",
"Unlike figures or , these flow fields comprise contributions from both WC and WI motions, i.e.",
"they are the raw flow fields directly from the dataset.",
"(d) Streamwise, (e) spanwise and (f) wall-normal velocity fluctuations conditioned for u + u^+ ≲\\lesssim -1 and w + w^+ ≳\\gtrsim 1 on the same wall-parallel plane as in (a-c)."
],
[
"Acknowledgement",
"The authors wish to acknowledge the Australian Research Council for financial support.",
"They are also grateful to the authors of [49], [4] and [7] for making their respective data available.",
"Sandia National Laboratories is a multimission laboratory managed and operated by National Technology and Engineering Solutions of Sandia, LLC., a wholly owned subsidiary of Honeywell International, Inc., for the U.S. Department of Energy's National Nuclear Security Administration under contract DE-NA0003525.",
"This paper describes objective technical results and analysis.",
"Any subjective views or opinions that might be expressed in the paper do not necessarily represent the views of the U.S. Department of Energy or the United States Government."
],
[
"SLSE-based decomposition of the instantaneous flow fields",
"As discussed previously in $§$, the flow field in the inertial region comprises contributions from both wall-coherent (WC) and wall-incoherent (WI) eddies.",
"Here, we demonstrate a methodology, based on the spectral linear stochastic estimate (SLSE) approach, to estimate the WC subset of a velocity fluctuation (say for $u$ -component) at any wall-normal location $z$ in the inertial region.",
"This procedure has been utilized in previous studies [3], [36], [17] for similar purposes (i.e.",
"to obtain a subset of the full flow field comprising selected coherent motions), which can be directly referred for an elaborate introduction to this technique.",
"According to SLSE, a scale-specific unconditional input at any near-wall location, $z_{w}$ (with $z^+_w$ $\\lesssim $ 15) can be used to obtain a scale-specific conditional input at $z$ following: ${\\widetilde{u}}^{E}(z^{+};{\\lambda }^{+}_{x},{\\lambda }^{+}_{y}) = {{H^{u}_{L}}(z^{+},z^{+}_{w};{\\lambda }^{+}_{x},{\\lambda }^{+}_{y})}\\widetilde{u}(z^{+}_{w};{\\lambda }^{+}_{x},{\\lambda }^{+}_{y}),$ where ${\\tilde{u}}({z_{w}};{\\lambda }_{x},{\\lambda }_{y})$ is the 2-D Fourier transform of the instantaneous wall-parallel flow field, $u$ (${z_w};x,y$ ) in space.",
"Here, the superscript $E$ and $H^{u}_{L}$ respectively represent the estimated quantity and the scale-specific linear transfer kernel (for $u$ -component).",
"$u^{E}$ in equation (REF ), essentially, corresponds to the energy distribution (at $z$ ) across wavelengths ${\\lambda }_{x}$ and ${\\lambda }_{y}$ , which are coherent across $z$ and $z_w$ .",
"This is facilitated by first computing $H^{u}_{L}$ from an ensemble of data as per: ${H^{u}_{L}}(z^{+},z^{+}_{w};{\\lambda }^{+}_{x},{\\lambda }^{+}_{y}) = \\frac{ \\lbrace \\widetilde{u}(z^{+};{\\lambda }^{+}_{x},{\\lambda }^{+}_{y}){{\\widetilde{u}}^{\\ast }}(z^{+}_{w};{\\lambda }^{+}_{x},{\\lambda }^{+}_{y}) \\rbrace }{ \\lbrace {\\widetilde{u}(z^{+}_{w};{\\lambda }^{+}_{x},{\\lambda }^{+}_{y})}{{\\widetilde{u}}^{\\ast }(z^{+}_{w};{\\lambda }^{+}_{x},{\\lambda }^{+}_{y})} \\rbrace },$ where the curly brackets ($\\lbrace \\rbrace $ ) and asterisk ($\\ast $ ) denote the ensemble averaging and complex conjugate, respectively.",
"Given that $z^+_{w}$ $\\lesssim $ 15 and $z^+_{w}$ $\\ll $ $z^+$ , ${\\widetilde{u}}^{E}(z^{+};{\\lambda }^{+}_{x},{\\lambda }^{+}_{y})$ thus represents energy contributions at $z$ from solely WC motions, that are taller than $z$ [36], [17], leading to: ${{{{\\widetilde{u}}^{E}}(z^{+};{\\lambda }^{+}_{x},{\\lambda }^{+}_{y})}{{\\big |}_{{z^{+}_{w}} {\\lesssim } 15}}}\\; {\\rightarrow }\\; {{{\\widetilde{u}}_{wc}}(z^{+};{\\lambda }^{+}_{x},{\\lambda }^{+}_{y})},$ where $u_{wc}$ ($z$ ) represents the subset of $u$ ($z$ ) comprising solely the WC motions.",
"On obtaining ${{\\widetilde{u}}_{wc}}$ , the corresponding flow field in physical space can be obtained by: ${{{u}_{wc}}(z;x,y)} = {\\mathcal {F}^{-1}}({{{\\widetilde{u}}_{wc}}(z;{\\lambda }_{x},{\\lambda }_{y})}),$ where ${\\mathcal {F}^{-1}}$ represents the inverse Fourier transform.",
"Here, although the entire procedure has been demonstrated solely for $u$ -fluctuations, the same can be applied to other (lateral) velocity fluctuations by estimating the respective linear transfer kernel ($H^{v}_{L}$ or $H^{w}_{L}$ ).",
"To give an example, figures REF (a-c) plot the instantaneous wall-parallel flow fields, comprising both WC + WI motions (i.e.",
"the full flow fields) at the lower bound of the inertial region.",
"While, figures REF (b,d,f) plot the subset of the corresponding full fields comprising solely the WC motions, estimated using the aforementioned procedure."
]
] | 2107.01750 |
[
[
"Supporting decisions by unleashing multiple mindsets using pairwise\n comparisons method"
],
[
"Abstract Inconsistency in pairwise comparison judgements is often perceived as an unwanted phenomenon and researchers have proposed a number of techniques to either reduce it or to correct it.",
"We take a viewpoint that this inconsistency unleashes different mindsets of the decision maker(s) that should be taken into account when generating recommendations as decision support.",
"With this aim we consider the spanning trees analysis which is a recently emerging idea for use with the pairwise comparison approach that represents the plurality of mindsets (in terms of a plurality of vectors corresponding to different spanning trees).",
"Until now, the multiplicity of the vectors supplied by the spanning trees approach have been amalgamated into a single preference vector, losing the information about the plurality of mindsets.",
"To preserve this information, we propose a novel methodology taking an approach similar to Stochastic Multi-criteria Acceptability Analysis.",
"Considering all the rankings of alternatives corresponding to the different mindsets, our methodology gives the probability that an alternative attains a given ranking position as well as the probability that an alternative is preferred to another one.",
"Since the exponential number of spanning trees makes their enumeration prohibitive, we propose computing approximate probabilities using statistical sampling of the spanning trees.",
"Our approach is also appealing because it can be applied also to incomplete sets of pairwise comparisons.",
"We demonstrate its usefulness with a didactic example as well as with an application to a real-life case of selecting a Telecom backbone infrastructure for rural areas."
],
[
"Introduction",
"In any decision of our life we have to compare a plurality of alternatives with respect to several points of views taking into account our preferences.",
"For example, if we have to choose a city car, we have to consider the models available on the market and we have to compare them with respect to their salient characteristics such as maximum speed, acceleration, space, price, fuel consumption.",
"To make a well thought-out decision it is necessary (1) to assign a priority to each alternative with respect to each criterion in order to evaluate its related performance, (2) to assign a priority to each criterion in order to define its importance (3) to aggregate the evaluations of performances on considered criteria obtained in point (1) taking into account the importance of criteria obtained in point (2).",
"In order to assign priorities to alternatives and criteria, the use of the pairwise comparison approach is quite common due to the fact that it allows decision makers to focus on one comparison at a time.",
"When prioritising alternatives in point (1), the decision maker focuses on two alternatives at a time, deciding which one is more preferred on the given criterion.",
"In most cases, the decision maker is also asked about the strength of his/her preference.",
"These two questions can be combined together as a single question, however, one may argue that establishing the strength of preference can only be established after deciding the direction of preference.",
"The same process can be repeated for prioritising criteria in point (2).",
"When there are only two options at hand, obviously there will be only one comparison required and therefore the prioritisation problem becomes trivial.",
"However, when there are three or more options to prioritise, the number of comparisons increases significantly.",
"When comparing three options, say A, B and C, we have three possibilities of comparing A to B, then B to C, and finally comparing A to C. One may argue that the last comparison is not required as the comparison of A to C can be mathematically derived from the other two comparisons.",
"However, the counter-argument is that we should still collect this apparently redundant information as the decision maker might have different opinion about options A and C when compared directly without involving option B.",
"If the derived judgement and the directly obtained judgement turn out to be same, then we can easily calculate the underlying preferences of the decision maker.",
"Instead, if these two judgements happen to be different, then we cannot straightforwardly calculate the underlying preferences.",
"Nevertheless, even if these judgements are different, in any case, from a psychological viewpoint, this gives us an additional piece of information about the problem.",
"In fact, we expose the inconsistency amongst the direct judgements, and there are a number of techniques for eliciting preferences from inconsistent judgements, for example, the row geometric mean [8], the logarithmic least squares method [9], [18], and the eigenvector method [19].",
"The basic idea of all these techniques for preference elicitation is to average and to amalgamate the preference information supplied by the decision maker (DM), trying to discover the \"true preferences\" behind the DM's inconsistent judgements.",
"We believe that with the existing methods a relevant wealth of information is lost.",
"In fact, we believe that more than merely identifying internal inconsistency, the judgements provided by the DM reveal the plurality of his/her mindsets.",
"Consequently, unleashing these multiple mindsets is an important area of investigation in an effective decision support process.",
"In this context, a recent work on the use of the spanning trees approach for analysing pairwise comparison judgements [23], [21] has gained our attention due to the fact that the generation of spanning trees does not involve any aggregation of underlying preferences.",
"The aim of this paper is to further investigate the use of the spanning trees approach to uncover all possible alternative preference vectors that represent the decision maker in one way or another.",
"Our aim in this paper is to use stochastic approaches to analyse all the possible preference vectors from inconsistent judgements.",
"In order to appreciate the use of stochastic approaches, one must realise that even a small number of alternatives and criteria can generate a large number of possible combination of priority vectors for the weights of the criteria (we call them weight vectors) and the evaluation of the alternatives on considered criteria (we call them evaluation vectors).",
"For example, a decision problem having only four criteria and four alternatives will still end up in generating over a million possible combinations of weight vectors and evaluation vectors.",
"We discuss this in more detail later in Section .",
"The stochastic approach proposed in this article can be used as a support in practical decision making situations.",
"A typical Multiple Criteria Decision Analysis (MCDA) support system aims at recommending a solution and to assess its robustness.",
"On the contrary, our proposed approach provides insights to the DM for better understanding of his/her preferences, which, in turn, will help the DM to make a well-informed decision.",
"The paper is organised as follows.",
"The next section provides background on the pairwise comparisons and the potential of the spanning trees approach; Section 3 then introduces the enumeration of all possible combinations of weight vectors and evaluation vectors, and the use of a stochastic approach for eliciting preferences; then the statistical sample selection of a well distributed family of spanning trees based on a random walk procedure is discussed in Section 4.",
"Section 5 discusses the Telecom backbone case study to demonstrate the use of random spanning trees, and then finally conclusions and future work are discussed in Section 6."
],
[
"Background",
"Pairwise comparisons are used to obtain DM's preferences using a \"one-at-a-time\" approach.",
"For example, in a car selection problem, we can either ask the DM to directly assign weights to the three criteria of Price, Speed, and Looks; or we can alternatively ask him/her to compare two criteria at a time and make these comparisons for all three possible pairs, that is, Price versus Speed, Speed versus Looks, and then Price versus Looks.",
"These comparative judgements can then be used to calculate the relative weights in the form of a weight vector (sometimes referred as preference vector).",
"Whilst comparing the three criteria, one would expect that if Price is considered more important than Speed and Speed more important than Looks, then Price must be more important than Looks.",
"This is often referred as the ordinal consistency requirement.",
"A much stronger requirement would be that if Price is preferred to Speed $p$ times, and Speed is preferred to Looks $q$ times, then Price should be preferred to Looks $pq$ times.",
"This is often termed as cardinal consistency.",
"Although these are rational and justified requirements from mathematical perspective, DMs often break these rules.",
"Obviously, if the DM is cardinally consistent, then the ordinal consistency will already be achieved.",
"However, the opposite is not true, that is, ordinal consistency does not ensure that the judgements are cardinally consistent.",
"This is demonstrated through an example shown in Fig.",
"REF where Price is considered twice as important as Speed, Speed is considered three times as important as Looks - which might suggest that that Price is 6 times as important as Looks.",
"If this is indeed the case then we can easily calculate the preference vector shown on the right side of this figure.",
"However, if instead of choosing 6, the DM suggests Price is 3 times as important as Looks, then this makes the three judgements cardinally inconsistent - though note that the order of preference is still preserved and so the judgements are not ordinally inconsistent.",
"In the case of cardinal inconsistency, eliciting the preference vector is not a straight-forward task.",
"There are numerous algorithms proposed to estimate the preference vector from an inconsistent set of judgements, however, it remains a widely debated issue as there is no single method that can be justified as the most appropriate method for eliciting preferences from inconsistent judgements.",
"Figure: The PC matrix acquired for the top-level criteriaSo far, we have assumed that the DMs provide a complete set of pairwise comparison judgements.",
"However, in practice, it is quite common to confront situations where DMs provide an incomplete set of judgements, and where it is not always possible to ask DMs to provide the missing data.",
"This leaves us in situations where preferences should be elicited from inconsistent, as well as, incomplete sets of judgements.",
"The most widely-used technique involving pairwise comparison judgements is the Analytic Hierarchy Process (AHP) [20] where DMs structure their criteria into a hierarchy and then evaluate alternatives with respect to each of these criteria.",
"Fig.",
"REF demonstrates this technique with the help of the car selection problem discussed earlier.",
"In this figure, the DM has evaluated four alternatives with respect to Price, Speed and Looks (therefore, providing three sets of pairwise comparison judgements).",
"Note that these judgements are shown in the form of a 4-by-4 matrix, which is a standard representation used to show pairwise comparisons.",
"On the top-right of this figure, the DM has also provided the relative importance of the three criteria, again by using pairwise comparison judgements.",
"These pairwise comparison judgements are then used to elicit preference vectors with the help of some elicitation technique.",
"Although there are many techniques for elicitation, the most widely used techniques are Right Eigenvector (REV) [20] and Row Geometric Mean (RGM) [7].",
"These elicited vectors are then used to construct a decision table as shown at the bottom right of this figure (Fig.",
"REF ).",
"At the bottom of this table, note that the criteria weights can also be elicited using the same REV or RGM method.",
"Finally, all these evaluation vectors and criteria weights can be compiled to generate some aggregated scores and/or to produce some form of recommendations to the decision maker(s).",
"Figure: The PC matrix acquired for the top-level criteriaHistorically, REV method has been widely used for eliciting preference vectors for both consistent and inconsistent judgements.",
"The inconsistency is usually measured in terms of the Consistency Ratio (CR) which is an Eigenvalue-based measure.",
"According to this measure, the PC matrix is usually considered acceptable when the CR value remains below a value of 0.1.",
"However, the REV method has been criticised due to its left-right eigenvector asymmetry, the use of arbitrary thresholds for inconsistency acceptability, as well as a few other further issues [2], [4].",
"Due to these shortcomings, several other methods have been proposed in the literature.",
"[6] analysed and numerically compared a variety of these prioritisation methods and concluded that there is no single best method that outperforms the others in every situation.",
"Although REV is the most commonly used method, the RGM approach has gained popularity due to its mathematical properties, and due to its ease of implementation [14].",
"While focusing on this \"single solution\" aspect, it can be argued that an in-depth analysis of the inconsistency gets neglected.",
"We contend that a prioritisation method must have the capabilities to focus on both aspects of the problem i.e.",
"production of a \"good quality\" preference vector and also facilitation of an in-depth analysis."
],
[
"Spanning tree approach",
"In this context, a graph-theoretic approach was proposed to generate a set of all possible preference vectors through enumeration ([23], see also [21]).",
"This approach is briefly summarised in Fig.",
"REF where the DM provides a set of PC judgements (shown on the top-left) which is then translated into a fully connected graph.",
"This graph is then analysed using spanning tree analysis to identify all possible combination of judgements, and eventually, generating all possible preference vectors.",
"Each of these alternative preference vectors essentially represents a mindset of the DM.",
"Figure: The PC matrix acquired for the top-level criteriaThe proposed method was shown to have a number of desirable properties, however, since the original method used the arithmetic mean to calculate the average of all these vectors, it was again focusing on the \"single solution\" aspect.",
"Also, one may argue that this computationally expensive enumeration will be overkill when the judgements are (fully) consistent, as all vectors will have identical values.",
"On the contrary, we argue that this rarely happens in real life; in the majority of cases, human judgements are found to be inconsistent, and therefore, unleashing these multiple mindsets can provide useful insights in practical decision making problems.",
"More recently, [17] proposed a geometric version of this spanning tree approach and established the mathematical equivalence of this approach with RGM.",
"They also proposed the possibility of using statistical techniques to gain insights into inconsistency.",
"In this paper, we develop this idea further and demonstrate its usefulness when we apply this technique at a meta-level, that is, applying it to the whole problem involving multiple criteria, instead of processing just one set of judgements."
],
[
"Stochastic approach to spanning trees",
"We have seen that until now the multiplicity of prioritisation vectors supplied by spanning trees approach have been amalgamated into a single priority vector either using the arithmetic mean or geometric mean.",
"However, with this approach the information about the plurality of mindsets is lost.",
"This is the price to pay to get a single overall ranking of alternatives.",
"A different way of thinking could be to consider the plurality of ranking supplied by the plurality of priority vectors corresponding to each combination of (evaluation and weighting) spanning trees, that in turn corresponds to a mindset."
],
[
"Problem formulation",
"Consider a situation where the DM has $n$ alternative options to choose from, and $m$ criteria to consider whilst making this decision.",
"These alternatives and criteria can be denoted as: Set of alternatives: $A=\\lbrace a_1,\\ldots ,a_i,\\ldots ,a_n\\rbrace ,$ Set of criteria: $G=\\lbrace g_1,\\ldots ,g_j,\\ldots ,g_m\\rbrace $ In order to assess these alternatives with respect to each criterion, $m$ sets of pairwise comparison judgements will be collected.",
"Let us denote each of these pairwise comparison by $M^j=[c^j_{i_1,i_2}], g_j \\in G$ with $c^j_{i_1,i_2}$ being the pairwise comparison judgement of alternative $a_{i_1}$ with alternative $a_{i_2}$ with respect to criterion $g_j$ .",
"The DM also needs to assess the relative importance of the criteria, and therefore another pairwise comparison matrix will be required for elicitation, say: $M^G=[c^G_{j_1,j_2}]$ with $c^G_{j_1,j_2}$ being the pairwise comparison judgement of criterion $g_{j_1}$ with criterion $g_{j_2}$ .",
"Considering a generic pairwise comparison matrix $M=[c_{rs}]$ , a spanning tree $\\tau _{k}$ consists of $(n-1)$ mutually independent judgements out of the total judgements, that is $\\tau _{k}=\\lbrace c^k_{r_1,s_1},\\ldots ,c^k_{r_{n-1},s_{n-1}}\\rbrace $ with $c^k_{r_1,s_1},\\ldots ,c^k_{r_{n-1},s_{n-1}}$ independent judgements.",
"Each of these spanning trees can be used to calculate a preference vector, and hence, we will interchangeably use the term \"spanning tree\" and \"spanning tree vector\".",
"For the matrices $M^j$ assessing alternatives with respect to considered criteria $g_j, j=1,\\ldots ,m$ , the set of spanning trees is denoted as: $\\mathcal {T}^j=\\lbrace \\tau ^j_{k}\\rbrace ,$ For the matrix $M^G$ assessing the importance of criteria, the set of spanning trees is denoted as: $\\mathcal {T}^G=\\lbrace \\tau _{k_r}^G\\rbrace ,$ This gives us a total of $m+1$ sets of spanning trees (spanning tree vectors), and therefore, $m+1$ sets of preference vectors.",
"As shown in Fig.",
"REF , we can pick one tree from each of these sets and construct a decision table that is traditionally used to calculate overall preferences.",
"These combinations of spanning trees can be formulated as $(\\tau _{k_1}^1,\\ldots ,\\tau _{k_m}^m,\\tau _{k_G}^G) \\in \\mathcal {T}^1\\times \\ldots \\mathcal {T}^m \\times \\mathcal {T}^G,$ Figure: Creating a number of decision tables by using different combinations of spanning tree vectors"
],
[
"Evaluating the combinations of spanning trees",
"The evaluation vector of alternatives $a_i \\in A$ corresponding to the spanning tree $\\tau ^j_{k_j} \\in \\mathcal {T}^j$ can thus be represented as: $(u^k_j(a_1),\\ldots ,u^k_j(a_n))$ while the weight vector corresponding to the spanning tree $\\tau ^G_{k} \\in \\mathcal {T}^G$ is denoted as: $(w^k_1,\\ldots ,w^k_m)$ Now the overall priority for alternatives $a_i \\in A$ given by the combination of spanning trees $\\mathbf {\\tau }=(\\tau _{k_1}^1,\\ldots ,\\tau _{k_m}^m,\\tau _{k_G}^G) \\in \\mathcal {T}^1\\times \\ldots \\mathcal {T}^m \\times \\mathcal {T}^G$ will be: $u_{\\mathbf {\\tau }}(a_i)=w^{{k_G}}_1 u^{k_1}_1(a_i)+ \\ldots +w^{k_G}_m u^{k_m}_m(a_i)$ From this formulation, we can deduce the set of combinations of spanning trees for which any $a_{i_1},a_{i_2} \\in A$ $a_{i1}$ is preferred to $a_{i2}$ : $B(a_{i_1}\\succ a_{i_2})=\\lbrace \\mathbf {\\tau } \\in \\mathcal {T}^1\\times \\ldots \\mathcal {T}^m \\times \\mathcal {T}^G:u_{\\mathbf {\\tau }}(a_{i_1})>u_{\\mathbf {\\tau }}(a_{i_2})\\rbrace $ $a_{i1}$ is indifferent with $a_{i2}$ : $B(a_{i_1}\\sim a_{i_2})=\\lbrace \\mathbf {\\tau }\\in \\mathcal {T}^1\\times \\ldots \\mathcal {T}^m \\times \\mathcal {T}^G:u_{\\mathbf {\\tau }}(a_{i_1})=u_{\\mathbf {\\tau }}(a_{i_2})\\rbrace $ The set of combinations of spanning trees $\\mathbf {\\tau } \\in \\mathcal {T}^1\\times \\ldots \\mathcal {T}^m \\times \\mathcal {T}^G$ for which alternative $a_i \\in A$ attains the $p-th$ position with respect to overall priority $u_{\\mathbf {\\tau }}(a_{i})\\rbrace $ can be formulated as: $R(a_i,p)=\\lbrace \\mathbf {\\tau } \\in \\mathcal {T}^1\\times \\ldots \\mathcal {T}^m \\times \\mathcal {T}^G:rank(a_i,\\mathbf {\\tau })=p\\rbrace $ with $rank(a_i,\\mathbf {\\tau })=1+\\sum _{i^\\prime \\ne i}\\rho (u_{\\mathbf {\\tau }}(a_{i^\\prime })>u_{\\mathbf {\\tau }}(a_{i}))$ where $\\rho (false)=0$ and $\\rho (true)=1$ .",
"Inspired by the Stochastic Multicriteria Acceptability Approach or SMAA [15], [16], the probability that alternative $a_{i_1}$ is preferred to alternative $a_{i_2}$ , with $a_{i1},a_{i2}\\in A$ can be represented as: $P(a_{i1} \\succ a_{i2})=\\frac{card(B(a_{i1} \\succ a_{i2}))}{card(\\mathcal {T}^1\\times \\ldots \\mathcal {T}^m \\times \\mathcal {T}^G)}$ that, remembering that the total number of spanning trees in a graph of $k$ nodes is $k^{k-2}$ [5], and, consequently $card(\\mathcal {T}^1\\times \\ldots \\mathcal {T}^m \\times \\mathcal {T}^G)=m^{m-2}n^{m(n-2)}$ , becomes $P(a_{i1} \\succ a_{i2})=\\frac{card(B(a_{i1} \\succ a_{i2}))}{m^{m-2}n^{m(n-2)}},$ the probability that alternative $a_{i} \\in A$ is attaining the $p-th$ ranking position will be: $P(rank(a_i,\\mathbf {\\tau })=p)=\\frac{R(a_i,p)}{card(\\mathcal {T}^1\\times \\ldots \\mathcal {T}^m \\times \\mathcal {T}^G)}=\\frac{R(a_i,p)}{m^{m-2}n^{m(n-2)}},$ [13] investigated incomplete sets of judgements where the DMs are allowed to respond with \"do not know\" or \"not sure\" to some judgements.",
"This is an important issue to investigate as the probability of acquiring an incomplete set of PC judgements increases with an increase in the total number of items for comparison [10].",
"Both the REV and the RGM methods are inappropriate in such cases due to the fact that the PC matrix cannot be constructed without estimating/imputing the missing judgements.",
"The spanning trees can be applied also to partial comparison matrix with the help of the Kirchoff formula (See details in [21]).",
"This implies that generating combinations of spanning trees is not restricted to only complete sets of judgements, and therefore, generating preference frequencies and rank-order frequencies are possible from incomplete pairwise comparisons, without any modification.",
"The implication of having an incomplete set of judgements is that the total number of combinations will be smaller than the number possible from a complete sets of judgements.",
"This permits the DM to only express values for those PC judgements for which he/she is sufficiently confident, avoiding forcing the DM to provide judgements when he/she is not sufficiently confident - in so doing maintaining the reliability of the preference information considered in the decision support process."
],
[
"Didactic example",
"Let us show how our approach works in practice by using the classic example of school selection proposed in [20].",
"The parents have to decide the high school for their son.",
"They consider six criteria being the following $g_1$ : Learning, $g_2$ : Friends, $g_3$ : School life, $g_4$ : Vocational training, $g_5$ : College preparation, $g_6$ : Music classes.",
"There are three alternatives corresponding to three schools denoted $A$ , $B$ and $C$ .",
"For each criterion, the parents compared in pairs the schools as shown in the following pairwise comparison matrices.",
"$M_{g_1}=\\begin{pmatrix}3 & \\frac{1}{3} & \\frac{1}{2} \\\\3 & 1 & 3 \\\\2 & \\frac{1}{3} & 1 \\\\\\end{pmatrix}$ $M_{g_2}=\\begin{pmatrix}1 & 1 & 1 \\\\1 & 1 & 1 \\\\1 & 1 & 1 \\\\\\end{pmatrix}$ $M_{g_3}=\\begin{pmatrix}1 & 5 & 1 \\\\\\frac{1}{5} & 1 & \\frac{1}{5} \\\\1 & 5 & 1 \\\\\\end{pmatrix}$ $M_{g_4}=\\begin{pmatrix}1 & 9 & 7 \\\\\\frac{1}{9} & 1 & \\frac{1}{5} \\\\\\frac{1}{7} & 5 & 1 \\\\\\end{pmatrix}$ $M_{g_5}=\\begin{pmatrix}1 & \\frac{1}{2} & 1 \\\\2 & 1 & 2 \\\\1 & \\frac{1}{2} & 1 \\\\\\end{pmatrix}$ $M_{g_6}=\\begin{pmatrix}1 & 6 & 4 \\\\\\frac{1}{6} & 1 & \\frac{1}{3} \\\\\\frac{1}{4} & 3 & 1 \\\\\\end{pmatrix}$ After comparing the alternatives with respect to these criteria, the parents also compared the criteria in terms of their importance as shown below.",
"$M_{criteria}=\\begin{pmatrix}1 & 4 & 3 & 1 & 3 & 4 \\\\\\frac{1}{4} & 1 & 7 & 3 & \\frac{1}{5} & 1 \\\\\\frac{1}{3} & \\frac{1}{7} & 1 & \\frac{1}{5} & \\frac{1}{5} & \\frac{1}{6} \\\\1 & \\frac{1}{3} & 5 & 1 & 1 & \\frac{1}{3} \\\\\\frac{1}{3} & 5 & 5 & 1 & 1 & 3 \\\\\\frac{1}{4} & 1 & 6 & 3 & \\frac{1}{3} & 1 \\\\\\end{pmatrix}$ The consistency ratios for the above pairwise comparison matrices are as follows: CR($M_{g_1}$ ) = 0.04, CR($M_{g_2}$ ) = 0, CR($M_{g_3}$ ) = 0, CR($M_{g_4}$ ) = 0.18, CR($M_{g_5}$ ) = 0, CR($M_{g_6}$ ) = 0.04, CR($M_{criteria}$ ) = 0.24 As we can see, there are four matrices which are not completely consistent and two of these, namely $M_{g_4}$ and $M_{criteria}$ have unacceptable levels of inconsistency as measured by the usual threshold of 0.1 for the CR value [20].",
"The priorities $u_j(X), X=A,B,C, j=1,\\ldots ,6$ , obtained from pairwise comparisons matrices $M_{g_1}-M_{g_6}$ representing the evaluations of schools with respect to the considered criteria are shown in Table REF , while the weights of the considered criteria, according to the priority obtained from pairwise comparison's matrix $M_{criteria}$ , are the following: $w_1=0.32, w_2=0.14,w_3=0.03,w_4=0.13,w_5=0.24,w_6=0.14.$ Table: Scores for alternatives with respect to each criterionUsing priorities $w_j$ and $u_j(X), X=A,B,C, j=1,\\ldots ,6$ , we can compute the overall priority $u(X)$ of each school, with $u(X)=\\sum _{j=1}^6 w_j u_j(X)$ obtaining the following results: $u(A)=0.37, u(B)=0.38, u(C)=0.25.$ Let us now handle the same problem with the spanning tree approach.",
"With this aim, we have to consider all the combinations of spanning trees from the pairwise comparison matrix of criteria $M_{criteria}$ and from the pairwise comparison matrices of alternatives with respect to the considered criteria $M_{g_j}, j=1,\\ldots ,m$ .",
"Remembering that with $m$ criteria and $n$ alternatives, we have ${n^{(n-2)}}{m^{(m-2)n}}$ combinations of spanning tress, for the decision problem at hand, we have $6^4\\cdot 3^6 = 944784$ combinations of spanning trees.",
"We compute weights of criteria $w^k_1,\\ldots ,w^k_6$ and evaluations of alternatives with respect to considered criteria $u^k_j(A),u^k_j(B),u^k_j(C)), j=1,\\ldots ,6,$ for each combination of spanning trees $\\mathbf {\\tau }=(\\tau _{k_1}^1,\\ldots ,\\tau _{k_6}^m,\\tau _{k_G}^G) \\in \\mathcal {T}^1\\times \\ldots \\mathcal {T}^6 \\times \\mathcal {T}^G$ .",
"Figure: Demonstrating the two different combinations of spanning trees, and the difference in their rankings and scores.Considering all the combinations of spanning trees from $\\mathcal {T}^1\\times \\ldots \\mathcal {T}^6 \\times \\mathcal {T}^G$ we can compute Table REF that shows the probability that each alternative attains a given rank (within parenthesis there is the total number of combinations of spanning trees for which there is the preference of the alternative on the row over the alternative on column), Table REF that for each pair of alternatives shows the probability that the alternative in the row is preferred to the alternative in the column (within parenthesis the total number of combinations of spanning tress for which the alternatives attains the considered ranking position).",
"Table: Preference frequency over 944784 combinations of spanning treesTable: Rank order frequency over 944784 combinations of spanning trees"
],
[
"Random spanning trees",
"We have seen that, for a problem with $n$ alternatives evaluated across $m$ criteria, the total number of combinations of spanning trees increases exponentially with the problem size - making it impractical to calculate the exact probabilities of $P(rank(a_i,\\mathbf {\\tau })=p)$ and of $P(a_{i1} \\succ a_{i2})$ for large problems.",
"We therefore propose using an approach which, combined with statistical sampling theory, allows us to provide estimates of the required probabilities of $P(rank(a_i,\\mathbf {\\tau })=p)$ and of $P(a_{i1} \\succ a_{i2})$ to within any user defined degree of accuracy according to any user defined level of confidence.",
"Of course we cannot physically select a statistical random sample from the population of spanning tree combinations since this would require the generation of the population itself and this is exactly what we aim to avoid.",
"We can however use the 'random walk' procedure to generate a tree and hence a sample of trees (see [1], [3].",
"Indeed, the 'random walk' procedure has been proven to generate a true statistical random sample in the sense that the generated sample is equivalent to selection of a statistical random sample from the population, that is, where each tree has the same uniform probability of being selected.",
"See [3], [1] for more details.",
"As we discussed the possibility of having incomplete sets of pairwise comparison judgements will arise in practice and so it is important to note that the concept of random walks is equally applicable to these incomplete sets without modification.",
"Figure: The PC matrix acquired for the top-level criteriaSince each iteration of the procedure generates a new member of the random sample, the total number of iterations used in the procedure is equivalent to sample size and we can use statistical large sample theory to determine the number of iterations required to generate sample parameters (including the probabilities of $P(rank(a_i,\\mathbf {\\tau })=p)$ and of $P(a_{i1} \\succ a_{i2})$ ) to any specified degree of accuracy and with any specified level of confidence.",
"That is, if we want the estimated probability to have an accuracy within $\\lambda $ and with a $C$ percent level of confidence, then the required number of iterations would be [22]: $It(\\lambda ,C) = \\frac{ {Z_C}^2 }{4\\lambda ^2}$ where $Z_C$ is the z-score calculated from the standardised normal distribution curve.",
"For example, if we want to achieve an accuracy within 0.01 and with a $99\\%$ level of confidence, then the required number of iterations would be $It(0.01,99\\%) = \\frac{ Z_{99}^2 }{4 \\times 0.01^2}$ As we know that the Z value for 99% confidence interval is equal to 2.58, therefore $It(0.01,99\\%) = \\frac{ {(2.58)}^2 }{4 \\times 0.01^2} = 16,641$ This suggests that we need to perform more than 16,641 iterations if we want to achieve an accuracy within $\\pm 0.01$ with $99\\%$ confidence."
],
[
"Applying random walk procedure to the school example",
"In the example below, we illustrate the use of random walks by repeating an experiment multiple times where each experiment itself uses the required number of iterations $It$ determined by the formula above.",
"We do this on the previously discussed School example as this is a tractable problem where the whole population of solutions is available.",
"Later, we will also extend this idea to a bigger problem where generating the whole population is impractical (see Section ).",
"Table REF shows the preference frequencies obtained through random walks, as well as the preference frequencies obtained by taking a sample from the whole population of solutions.",
"Recall that the whole population was generated by enumerating all possible combinations of spanning trees that was shown in Table REF earlier.",
"Here we can see that the estimated values from random walks and from the sample are both lying within 1% of the population values that were previously shown in Table REF .",
"Table: Preference frequencies of the school exampleTable: Rank order profiles generated for the school exampleWe performed the same analysis for rank-order frequencies as well, as summarised in Table REF .",
"The table on the left side shows that the rank-order frequencies obtained through random walks are also lying within 1% of the population values shown in Table REF .",
"The rank-order frequencies obtained by taking a sample are shown on the right side of this table.",
"This also illustrates that the accuracy and confidence level results for the generated random sample echo the same results had we been able to physically select a statistical random sample from the population of spanning tree combinations."
],
[
"Telecom backbone example",
"In order to demonstrate the use of spanning trees to explore multiple mindsets, we consider the practical data acquired in a recent study: the selection of a backbone infrastructure for telecommunication in rural areas [11].",
"This study focused on the use of a structured decision making approach towards selecting the telecommunication infrastructure for the rural areas of developing countries.",
"The lack of adequate telecommunications infrastructure in these parts of the world remains a major obstacle for providing affordable services.",
"The study considered four options of Fiber-optic cable (G1), Power-line communication (G2), Microwave link (G3) and Satellite communication (G4).",
"The authors of this study used AHP and Analytic Network Process (ANP) to structure the problem and acquired the necessary data on preferences and assessments from key stakeholders.",
"The criteria used to compare these alternatives were grouped into six major categories including technical, infrastructural, economic, social, regulatory and environmental factors.",
"These categories and their constituent criteria are presented in Fig.",
"REF .",
"Figure: Criteria to compare the available backbone infrastructuresFigure: The PC matrix acquired for the top-level criteriaThe PC matrix, $A_{top}$ , acquired for prioritising these six categories (top-level criteria) is shown in Fig.",
"REF .",
"Although $A_{top}$ is a transitive PC matrix, the estimated vectors produced by the widely used EV method does not preserve the original order of preferences in these judgements.",
"The final weights calculated using the EV and GM methods are found to be almost identical, as given in Table REF in normalised form.",
"Satellite communication (G4) is considered the most preferred alternative with a weight of 29.95% (using EV), followed by Microwave (G3) with a weight around 28.34% (using EV).",
"Table: Estimated weights for the available backbone infrastructure optionsMost criteria lie under the Technical and Infrastructure categories.",
"The Technical category includes nine criteria whilst the Infrastructure category has eight criteria used to compare the alternatives.",
"The two PC matrices for the Technical and Infrastructure categories, $A_{tech}$ and $A_{infra}$ , have been found to be intransitive and should be investigated along with $A_{top}$ for their impact on the final result."
],
[
"Spanning trees analysis using Random walks",
"In this case study, there are four alternatives and six top-level criteria, therefore, the number of “spanning-trees” solutions for criteria matrix is $m^{(m-2)}$ = $6^4$ = 1296, and the number of “spanning-trees” solutions for each alternative matrix is $n^{(n-2)}$ = $4^2$ = 16.",
"As we pick one “spanning-trees” solution from each matrix, there are $m^{(m-2)}n^{m(n-2)}$ = $1296\\times 16^6$ = $21,743,271,936$ combinations possible.",
"Generating these billions of combinations is computationally expensive, so sampling is the logical way to estimate probabilities.",
"Therefore we generated a sample of 20,000 solutions with the use of random walks.",
"We generated 20 such samples in order to check whether all these samples produce similar results.",
"In practise, only one sample should suffice but we took this approach for investigation only.",
"The average scores for the four alternatives are given in Table REF .",
"Each row in this table represents one iteration of generating a sample of 20,000 solutions.",
"Table: Scores calculated usign random walks; each iteration shows an average of 20k solutionsUnlike the School example where we calculated frequencies by generating all the combinations of spanning trees, here we will calculate the frequencies from the samples generated by random walks.",
"It can be argued that these frequencies may vary as they are based on randomly generated set of solutions.",
"However, the aim here is to gain statistical insights, and therefore, these findings should remain useful as long as they produce reliable statistical findings.",
"We investigated this reliability by repeating the same experiment twenty times and comparing the preference frequencies and rank-order frequencies.",
"The standard deviations of these values do not exceed 0.005, which supports the argument that these stochastic results are quite reliable.",
"Table REF shows the probability that one alternative is preferred to the other (for all possible pairs).",
"Looking at the first row, we can say that Fiber is preferred over Powerline in 63% (see $0.63\\pm 0.004$ under the Powerline column) of the random walk solutions, however, it is seldom preferred over Microwave and Satellite.",
"Microwave is clearly preferred over Fiber (with a probability of 87%) and Powerline (with a probability of 96%), however, it has been marginally preferred over Satellite (with a probability of 51%).",
"We can argue that the Fiber and Powerline are the two clearly dominated alternatives, while the Microwave and Satellite are the two dominating alternative.",
"However, when comparing Microwave and Satellite with each other, Microwave is slightly more preferred but there is no clear winner.",
"We are providing the standard deviations for each of these values to show that these scores didn't deviate much, as discussed earlier.",
"Table: Preference frequencies for Telecom backbone selectionTable: Rank-order frequencies for Telecom backbone selectionAnother interesting way to assess these options is to estimate the ranking of each alternative.",
"Table REF shows the probability that each alternative attains a given rank.",
"In this table, The probability of Fiber and Powerline taking the first rank is too low (only 4.9% and 2.5% respectively).",
"However, Microwave and Satellite have high probabilities of taking the first rank.",
"On the other end, although Fiber is more likely to attain the third rank, both Fiber and Powerline have high probabilities to attain the lowest rank (36.5% and 62.2% respectively).",
"The two alternatives of Microwave and Satellite are least likely to be placed on the lowest rank, as the percentage of combinations placing them at the 4th position were only 0.6% and 0.7%, respectively."
],
[
"Conclusions",
"The spanning trees analysis can help understand the plurality of mindsets in terms of a plurality of prioritisation vectors originating from a plurality of spanning trees of pairwise comparison matrices.",
"Considering all the rankings of alternatives corresponding to the different mindsets, we propose to estimate the probability that an alternative attains a given ranking position, and the probability that an alternative is preferred to another one.",
"Moreover, the proposed approach can be applied to incomplete sets of judgements without any modification.",
"Since the exponential number of spanning trees makes their enumeration prohibitive, we propose computing approximate probabilities using statistical sampling of spanning trees.",
"The usefulness of statistical sampling is further demonstrated with the case study of Telecom infrastructure selection for rural areas.",
"An interesting area of future research is to do performance analysis using the sigma-mu approach proposed by [12].",
"For each alternative, we compute the mean, $\\mu $ , and the standard deviation, $\\sigma $ , of all the solutions generated from combinations of spanning trees.",
"The objective would be to maximise the mean while minimising the standard deviation.",
"The use of interval judgements is also an important area to investigate as we know that decision makers often provide their judgements in intervals (for example, stating that X is about 3 to 5 times better than Y).",
"Therefore, the idea of interval judgements need to be combined with the use of spanning trees combinations.",
"Considering the lower and upper limit of intervals, the number of combinations will further increase exponentially, therefore the use of a stochastic approach might be the only practical approach for this purpose.",
"There is a further potential in the use of spanning trees analysis at group-level decision making where multiple judgements are collected for comparing the same pair of alternatives or weighing the same pair of criteria.",
"The concept of \"aggregated individual preferences\" and \"aggregated individual judgements\" has been widely debated in the context of group decision making, and we believe that the use of spanning tree analysis can offer better insights into these types of problems."
]
] | 2107.01731 |
[
[
"Age of Information in Relay-Assisted Status Updating Systems"
],
[
"Abstract In this paper we consider the age of information (AoI) of a status updating system with a relay, where the updates are delivered to destination either from the direct line or the two-hop link via the relay.",
"An updating packet generated at source is sent to receiver and the relay simultaneously.",
"When the direct packet transmission fails, the relay replaces the source and retransmits the packet until it is eventually obtained at the receiver side.",
"Assume that the propagation delay on each link is one time slot, we determine the stationary distribution of the AoI for three cases: (a) relay has no buffer and the packet delivery from relay cannot be preempted by fresher updates from source; (b) relay has no buffer but the packet substitution is allowable; (c) relay has size 1 buffer and the packet in buffer is refreshed when a newer packet is obtained.",
"The idea is invoking a multiple-dimensional state vector which contains the AoI as a part and constituting the multiple-dimensional AoI stochastic process.",
"We find the steady state of each multiple-dimensional AoI process by solving the system of stationary equations.",
"Once the steady-state distribution of larger-dimensional AoI process is known, the stationary AoI distribution is also obtained as it is one of the marginal distributions of that process's steady-state distribution.",
"For all the situations, we derive the explicit expression of AoI distribution, and calculate the mean and the variance of the stationary AoI.",
"All the results are compared numerically, including the AoI performance of the non-relay state updating system.",
"Numerical results show that adding the relay improves the system's timeliness dramatically, and no-buffer-and-preemption setting in relay achieves both minimal average AoI and AoI's variance.",
"Thus, for the system model discussed in this paper, to reduce the AoI at receiver there is no need to add the buffer in relay."
],
[
"Introduction",
"Apart from the traditional performance targets such as low propagation delay and large throughput, modern network designs put forward new requirements on information freshness and the timeliness of an information transmission system.",
"In many remote monitoring system, the source has to transmit the messages or signals via a channel fast to keep the receiver's knowledge about certain physical process in source side fresh enough.",
"These messages are time sensitive, whose freshness is of great importance in some applications.",
"For example, the controller of a driverless car must obtain sufficiently new messages to keep the car running safely.",
"The central node in a sensor network needs the fresh data from its neighbours to perform certain real time computations, such that an optimal resource scheduling policy is determined.",
"In general, this kind of information transmission system is abstracted into a status updating system, where the updates are generated in source and then sent to the receiver.",
"The state at the receiver is refreshed after an updating packet is obtained successfully.",
"To characterize the timeliness of the system or measure the freshness of an updating packets, a timeliness metric called age of information (AoI) was introduced in [1].",
"Since then, lots of articles has been published focusing on different aspects of AoI, such as the AoI performance analysis and designing various optimal status updating system who has minimal average AoI.",
"A detailed survey about AoI can be found in paper [2], in which the authors summarize the recent contributions of AoI in the aspect of theory and practice.",
"For simple queue models such as $M/M/1$ , $M/D/1$ , and $M/M/\\infty $ , the average AoI of state updating system were derived in papers [3], [4], [5].",
"In particular, the Stochastic Hybrid System (SHS) analysis which is a systematic method to calculate the mean of AoI was introduced in paper [5].",
"The method has also been used to determine the average AoI of some other status updating systems in [6], [7], [8], [9], [10], [11], [12] recently.",
"Packet managements problem was considered in [13], where the authors defined another metric called peak age of information (PAoI) and derived the average performance of both AoI and PAoI for three small-size updating systems.",
"Assume that the packet has a fixed or random deadline, the long term average value of steady-state AoI was calculated in work [14].",
"Apart from the analytical results of AoI described above, in practical applications the AoI has been considered widely in optimal system designs, either as the performance target or acting as one of constraints [15], [16], [17], [18], [19], [20], [21], [22].",
"Observing that in the majority of systems considered before, an updating packet takes only one hop from source to the destination.",
"Meanwhile, often there is only one path between the two nodes.",
"However, both the packet multihop transmissions and the packet multipath transmissions are existent everywhere in practical communication networks, especially in wireless environment.",
"The AoI of status updating system having two parallel servers was considered in [23], the authors computed the average AoI of this two-server system by sophisticate random events analysis.",
"In paper [24] the AoI in multihop networks was analyzed.",
"Recently, the AoI of a three node status updating system was considered in [25], where a relay was added to cooperate the packet transmission from source to receiver.",
"The authors showed that although two-hop transmission via the relay takes longer propagation delay, but the transmission success probability is raised, in particular when the direct line from source to destination is bad.",
"Summarize both aspects, they proved that adding a relay can reduce the average AoI at the receiver, thus improving the system's timeliness.",
"In this paper, we consider a more general relay-assisted status updating system than that was discussed in [25], and derive more fundamental results about system's AoI.",
"There, the situation that the relay has buffer was not mentioned and, for the non-relay cases, only the mean of steady-state AoI was calculated.",
"We investigate both non-relay cases and the situation where the relay has a size 1 buffer.",
"For each cae, the stationary distribution of AoI was derived explicitly, thus obtaining the complete description of the steady-state AoI.",
"In [25], the authors derived their results using the most initial formula for the limiting time average AoI, which is used to deal with continuous time AoI.",
"However, as the time is slotted, the AoI is discretized as well.",
"Thus, the AoI considered in that paper, and in many other papers where the time was slotted is actually discrete.",
"Here, we show that the analysis of AoI can be done by the idea and standard tools from queueing theory.",
"The idea is invoking a multiple-dimensional discrete state vector which contains the AoI as a part, and constituting the multiple-dimensional AoI stochastic process.",
"As long as the steady state of the constituted process is determined, i.e., all the stationary probabilities are solved, the stationary AoI distribution can be found by merging all the stationary probabilities with identical AoI state component, as it is in fact one of the marginal distributions of the larger-dimensional AoI process's steady-state distribution.",
"We point out that the idea that invoking a multiple-dimensional state vector to characterize the random transfers of AoI and analyzing the steady state of the constituted AoI process is also the idea of the SHS approach when the continuous AoI is discussed.",
"Differently, finding the steady-state of a AoI stochastic hybrid system, one has to solve a system of differential equations which is in general very hard.",
"While when the discrete AoI is analyzed, solving the system of stationary equations of constituted discrete AoI process is easier.",
"Just recently, in paper [26], [27], [28], [29], the authors proved that following above straightforward thinking, the stationary distribution of continuous AoI of status updating system with a few of simple queue models can also be determined, using AoI's Laplace Stieltjes Transform (LST) form.",
"Finding the stationary distribution of AoI gets more attention in recent two or three years, both for continuous and discrete form of AoI [30], [31], [7], [32].",
"The rest of the paper is organized as follows.",
"We describe the model of a relay-assisted state updating system in Section II and introduce the system parameters.",
"In Section III, assume that the relay has no buffer, we analyze the system's AoI of two cases where the packet retransmissions from relay can and cannot be preempted by later updates from the source node.",
"The explicit expressions of steady-state AoI are derived by constituting a two-dimensional stochastic process.",
"For the situation the relay is equipped with a size 1 buffer, the AoI of relay-assisted system is discussed in Section IV.",
"There, a three-dimensional state vector is defined and the steady state of a three-dimensional AoI stochastic process is analyzed.",
"The stationary AoI distribution is obtained as well.",
"Provided the AoI distributions obtained before, for each case, we calculate the mean and the variance of AoI in Section V. Numerical results are given in Section VI, in which we compare the average AoI and the AoI's variance of all three cases.",
"Besides, to demonstrate the improvement of adding the relay, we also calculate the AoI performance of an updating system without relay.",
"The paper is summarized in Section VII, where some possible generalizations of relay-assisted system are discussed as well."
],
[
"System Model and Problem Formulation",
"The system model is depicted in Figure REF .",
"The source $s$ observes certain physical process and samples the state of the process at random times.",
"The packet arrival is assumed to be a Bernoulli process, that is a packet arrives in each time slot independently with identical probability $p$ .",
"An updating packet is delivered to destination $d$ , so that the knowledge of receiver about the process is refreshed.",
"Since the existence of relay $R$ , the packet can be transmitted either from the direct line or the two-hop link via $R$ .",
"Specifically, when the source $s$ transmits a packet, the same packet is also sent to $R$ at the same time.",
"Figure: The model of a status updating system with a relay.Assume that $R$ gets a feedback signal when every time an updating packet is obtained at the receiver.",
"In this case, the packet stored in $R$ is deleted to avoid redundant transmissions.",
"In particular, if one packet is transmitted on direct link successfully, then all the packets in $R$ need to be deleted, including the one waiting in buffer when we consider the relay-has-buffer case, since all of these packets are out of date.",
"Otherwise, the relay will replace the source and retransmit the packet to $d$ .",
"Since $R$ is closer to $d$ , it is more likely that the transmission is successful.",
"Notice that the $s-R$ link is also unreliable, it is possible that both $s-R$ and $s-d$ transmissions fail at the same time.",
"The source $s$ triggers the next transmission when a new packet arrives.",
"We are interested in the stationary AoI of the relay-assisted system for three cases, i.e., when the relay $R$ has no buffer, we discuss the AoI performance assuming that the packet retransmission can and cannot be preempted by fresher packets from source, and the steady-state AoI of the system when the relay is equipped with a size 1 buffer.",
"The three parameters $P_1$ , $P_2$ , and $P_3$ denote the transmission success probabilities of $s-d$ , $s-R$ , and $R-d$ links, respectively.",
"Generally speaking, $P_1$ is less than $P_2$ and $P_3$ .",
"The propagation delays on all links are suppose to be one time slot.",
"In this paper, we shall prove that for all of three cases the stationary distribution of AoI can be determined, such that the complete description of the steady-state AoI for the considered relay-assisted status updating system is obtained."
],
[
"Analysis of Age of Information: Relay has no buffer",
"In this Section, assume that the relay has no buffer, we analyze the stationary AoI of the relay-assisted status updating system by constituting a two-dimensional discrete stochastic process.",
"We determine the explicit expressions of the AoI distribution for both the cases where the packet in relay can and cannot be preempted by the newer ones from thee source.",
"Observing that if no packet arrives to the destination, the AoI increases by 1.",
"The value of AoI is reduced to the age of the packet, which is actually equal to the system time of the packet until it is eventually received at receiver $d$ .",
"Define the two-dimensional discrete state vector $(n,m)$ , where $n$ denotes the AoI at $d$ , and $m$ represents the age of packet which is stored in relay $R$ .",
"When there is no packet in $R$ , we let $m=0$ .",
"In the following, for each case, we describe the random state transfers of each state vector, establish the stationary equations of the constituted AoI stochastic process and finally find its steady state, i.e., determining the stationary probabilities of every state vector $(n,m)$ .",
"Then, the stationary distribution of AoI is obtained by merging all the probabilities of those state vectors that have identical AoI-component $n$ .",
"Notice that in the following paragraphs we also call the state vector as the age-state of the system, or simply the state."
],
[
"Analysis of AoI: packet preemption is allowable in relay",
"Firstly, we analyze the AoI of the system assuming that the packet in $R$ is substituted when every time a fresher update is obtained from the source.",
"The random transfers of age-state $(n,m)$ are described in the following.",
"First of all, assume that the age-state is $(n,m)$ where $n> m\\ge 1$ .",
"In this case, a packet of age at least 1 is contained in $R$ .",
"If no packet arrives to source at next time slot, then the state transfers are dependent on the packet transmission on $R-d$ link.",
"With probability $P_3$ , the packet delivery succeeds and the state vector changes to $(m+1,0)$ .",
"The second state-component is zero because the packet in $R$ is deleted when it was sent to $d$ successfully.",
"Otherwise, both $n$ and $m$ will increase 1 at next time slot, i.e., the age-state turns to $(n+1,m+1)$ .",
"For the case a new packet is generated and sent out from source $s$ , we have to analyze all the packet transmissions on links $s-R$ , $R-d$ , and direct link $s-d$ .",
"Let $A$ be the random variable of packet arrivals, and we use $T_{s-d}$ , $T_{s-R}$ , $T_{R-d}$ to indicate whether the packet transmission is successful in three links, respectively.",
"$A=0$ denotes no packet arrives and $T_{s-d}=0$ means that the packet transmission on $s-d$ link fails.",
"Other variables are defined in similar manner.",
"We list all the possible state transitions in Table REF .",
"Table: Description of state vector transfers of AoI stochastic process AoI NB P AoI_{NB}^PTable: All the state transfers and corresponding transition probabilities of stochastic process AoI NB P AoI_{NB}^PIt is necessary to explain all the cases including $A=1$ and $T_{s-d}=1$ in Table REF .",
"When a new packet is delivered successfully over direct link $s-d$ , the value of AoI must reduce to 1 since a packet of age 1 is obtained in $d$ , although it is possible that an another packet is received on $R-d$ link.",
"At the same time, according to our assumption, a feedback signal is sent to $R$ and the relay deletes the packet received at current time slot or before.",
"Thus, $m$ is reduced to zero and the state vector jumps to $(1,0)$ .",
"Still we have to consider the state transfers when the initial age-state is $(n,0)$ , which means the relay $R$ is currently empty.",
"If no packet arrives, it is easy to see that the age-state transfers to $(n+1,0)$ at next time slot.",
"On the contrary, for the case $A=1$ , we need to consider the packet transmissions on $s-d$ and $s-R$ links.",
"All the possible state transitions from age-state $(n,0)$ are given in the last four lines in Table REF .",
"Constituting the two-dimensional AoI stochastic process $AoI_{NB}^P=\\lbrace (n_k,m_k):n_k> m_k\\ge 0,k\\in \\mathbb {N}\\rbrace ,$ where the subscript $NB$ denotes that at relay $R$ there is no buffer, and the superscript $P$ indicates that the packet retransmission in $R$ can be preempted.",
"Define $P_{(n,m),(i,j)}$ be the single step transition probability from $(n,m)$ to age-state $(i,j)$ .",
"Summarize above discussions, we list all the state transfers and corresponding transition probabilities of the AoI process $AoI_{NB}^P$ in Table REF .",
"Denote that $\\pi _{(n,m)}$ is the probability of age-state $(n,m)$ , $n> m\\ge 0$ when the AoI process runs in steady state.",
"In following Theorem 1, we establish the stationary equations of the process $AoI_{NB}^P$ .",
"Theorem 1 Assume that the stochastic process $AoI_{NB}^P$ reaches the steady state, then the stationary probabilities $\\pi _{(n,m)}$ of age-state $(n,m)$ , $n>m\\ge 0$ satisfy that ${\\left\\lbrace \\begin{array}{ll}\\pi _{(n,m)}=\\pi _{(n-1,m-1)}\\big [(1-p)(1-P_3) + p(1-P_1)(1-P_2)(1-P_3)\\big ] & \\quad (n>m\\ge 2) \\\\\\pi _{(n,1)}=\\pi _{(n-1,0)}p(1-P_1)P_2 + \\sum \\nolimits _{j=1}^{n-2}\\pi _{(n-1,j)}p(1-P_1)P_2(1-P_3) \\\\\\qquad \\qquad \\qquad + \\sum \\nolimits _{k=n}^{\\infty }\\pi _{(k,n-1)}p(1-P_1)P_2P_3 & \\quad (n\\ge 3) \\\\\\pi _{(2,1)}=\\pi _{(1,0)}p(1-P_1)P_2 + \\sum \\nolimits _{k=2}^{\\infty }\\pi _{(k,1)}p(1-P_1)P_2P_3 \\\\\\pi _{(n,0)}=\\pi _{(n-1,0)}\\left[(1-p)+p(1-P_1)(1-P_2)\\right] \\\\\\qquad \\qquad \\qquad + \\sum \\nolimits _{k=n}^{\\infty }\\pi _{(k,n-1)}\\left[(1-p)P_3+p(1-P_1)(1-P_2)P_3\\right] & \\quad (n\\ge 2) \\\\\\pi _{(1,0)}=\\left(\\sum \\nolimits _{n=1}^{\\infty }\\pi _{(n,0)}+ \\sum \\nolimits _{m=1}^{\\infty }\\sum \\nolimits _{n=m+1}^{\\infty }\\pi _{(n,m)} \\right) pP_1\\end{array}\\right.",
"}$ Equations in (1) are explained as follows.",
"First of all, begin with $(n-1,m-1)$ , as long as the packet transmission on $R-d$ link fails which occurs with probability $1-P_3$ , along with the condition that either no packet is delivered on $s-R$ and $s-d$ links or both transmissions are unsuccessful, then the age-state transfers to $(n,m)$ at next time slot.",
"This gives the first line of equations (1).",
"To obtain $(n,1)$ , more cases need to be considered.",
"From age-state $(n-1,0)$ which means $R$ is initially empty, at source $s$ a packet arrives and is transmitted to $R$ successfully makes the second parameter jump to 1.",
"On the other hand, let the packet delivery on $s-d$ link fail, such that the AoI in $d$ increases to $n$ at next time slot.",
"This state transitions occurs with probability $p(1-P_1)P_2$ .",
"Next, notice that the packet in $R$ is allowed to be refreshed, the second state-component 1 can also be created by packet replacement.",
"Assume that the initial state vector is $(n-1,j)$ , $1\\le j\\le n-2$ .",
"When the new packet is transmitted on $s-R$ link successfully but the packet transmissions on both $s-d$ and $R-d$ links fail, it was observed that the state also transfers to $(n,1)$ .",
"Thus, we obtain the second term of the RHS of second equation.",
"Now, suppose that a packet of age $n-1$ goes through $R-d$ link to the receiver.",
"As a result, the AoI in $d$ is reduced to $n$ at next time slot.",
"At the same time, the relay $R$ obtains a new packet from $s$ via $s-R$ link, which changes the second parameter to 1.",
"In this case, we can obtain the age-state $(n,1)$ again.",
"The initial state can be $(k,n-1)$ where $k$ is an arbitrary number greater than $n-1$ .",
"For the case $n=2$ , packet substitution in $R$ is nonexistent since $R$ is empty before one time slot.",
"With all above discussions, we explain how the state vector $(n,1)$ , $n\\ge 2$ is obtained and determine its stationary equation.",
"Then, the random transitions to $(n,0)$ is analyzed.",
"The relay $R$ is empty either it is empty initially, or is emptied as the contained packet is sent to $d$ successfully.",
"Therefore, two cases have to be analyzed.",
"Firstly, from age-state $(n-1,0)$ , as long as no packet arrives to $R$ and $d$ , obviously the state jumps to $(n,0)$ after one time slot.",
"On the other hand, let the initial state is $(k,n-1)$ , $k\\ge n$ , a packet of age $n-1$ can be transmitted to $d$ via $R-d$ link and assume that no other packet is obtained from $s$ through the direct link $s-d$ , which ensures that the AoI is set to $n$ at next time slot.",
"The second parameter maintains 0 if no packet comes to $s$ , or the arriving packet is not transmitted to $R$ successfully on $s-R$ link.",
"Combined with the transition probabilities in Table REF , we give the stationary equation for the state $(n,0)$ in the end.",
"The only way to get age-state $(1,0)$ is transmitting a packet via the direct link $s-d$ , no matter what the initial state is.",
"Actually, this last equation determines $\\pi _{(1,0)}=pP_1$ directly, since the probabilities in bracket add up to 1.",
"So far, all the state transfers are considered, and the proof of Theorem 1 completes.",
"We solve the system of equations (1) and determine all the stationary probabilities $\\pi _{(n,m)}$ in Theorem 2, whose proof is postponed to Appendix A. Theorem 2 The probabilities $\\pi _{(n,m)}$ of AoI process $AoI_{NB}^P$ are determined as follows.",
"Firstly, $\\pi _{(n,0)}=(1-\\delta )\\delta ^{n-1} - p(1-P_1)P_2[(1-P_3)\\delta ]^{n-1} \\qquad (n\\ge 1)$ and for $n>m\\ge 1$ $\\pi _{(n,m)}&=\\pi _{(2,1)}[(1-pP_1)(1-P_3)]^{n-2}\\left(\\frac{\\delta }{1-pP_1}\\right)^{m-1} + \\frac{p(1-P_1)P_2P_3(1-\\delta )\\delta }{\\delta -(1-pP_1)(1-P_3)} \\\\& \\qquad \\quad \\times \\left[ \\delta ^{n-2}(1-P_3)^{m-1} - [(1-pP_1)(1-P_3)]^{n-2}\\left(\\frac{\\delta }{1-pP_1} \\right)^{m-1} \\right]$ where $\\delta =1-p(P_1+P_2-P_1P_2), \\qquad \\pi _{(2,1)}=p^2(1-P_1)P_2[P_1+(1-P_1)P_2P_3] $ Provided all the stationary probabilities, the probability that the steady-state AoI equals $n$ can be obtained by $\\Pr \\lbrace \\Delta =n\\rbrace =\\pi _{(n,0)} + \\sum \\nolimits _{m=1}^{n-1}\\pi _{(n,m)} \\qquad (n\\ge 2)$ The probability that AoI takes value 1 is determined as $pP_1$ , which is known directly from the last line of equations (1).",
"We determine the explicit expression of AoI distribution in following Theorem 3 and give the calculations in Appendix B. Theorem 3 For the considered relay-assisted status updating system, the stationary distribution of AoI $\\Delta _{NB}^P$ is determined as $\\Pr \\lbrace \\Delta _{NB}^P=n\\rbrace =\\frac{(1-pP_1)P_3(1-\\delta )}{(1-pP_1)P_3-p(1-P_1)P_2}\\delta ^{n-1} +\\xi [(1-pP_1)(1-P_3)]^{n-1} \\quad (n\\ge 1)$ in which $\\xi =\\frac{p[P_1+(1-P_1)P_2P_3]}{1-P_3} - \\frac{P_3(1-\\delta )\\delta }{[(1-pP_1)P_3-p(1-P_1)P_2](1-P_3)} $"
],
[
"The stationary AoI: packet retransmission is not preempted in relay",
"In previous part of this Section, we have determined the stationary AoI distribution for the case where the packet in relay can be replaced.",
"Intuitively, substituting the large-age packet with fresher ones will result to lower AoI and enhance the timeliness of the system.",
"We will prove this conclusion by finding the steady-state AoI distribution for the $non-preemption$ case explicitly and do the comparisons.",
"The AoI stochastic process constituted for this case is denoted as $AoI_{NB}^{NP}$ , where the superscript $NP$ indicates that no packet preemption is allowed in relay $R$ .",
"When the packet in relay cannot be preempted before the receiver gets an updating packet eventually, which may be obtained from $R-d$ link or the $s-d$ link, the discussions of random state transitions are easier, since the relay will always reject the packet from the source as long as there has been one packet within it.",
"We first describe the possible state transitions when the initial age-state is $(n,m)$ , where $n>m\\ge 1$ .",
"If $A=1$ , that is a new packet is sent out from the source in this time slot.",
"Now, only the packet transmissions on $s-d$ and $R-d$ links need to be discussed, since the packet over $s-R$ link is always rejected.",
"For the case $A=0$ , we have to consider just the packet delivery over $R-d$ link.",
"All the possible state transfers are listed in Table REF , in which the state transfers from initial age-state $(n,0)$ are also given.",
"For each state transfer, we determine the corresponding transition probability in Table REF , from which the stationary equations of the AoI process $AoI_{NB}^{NP}$ can be established, when the process reaches the steady state.",
"Theorem 4 Assume that the AoI stochastic process $AoI_{NB}^{NP}$ reaches the steady-state, the stationary probabilities $\\pi _{(n,m)}$ satisfy following equations ${\\left\\lbrace \\begin{array}{ll}\\pi _{(n,m)}=\\pi _{(n-1,m-1)}[(1-p)(1-P_3)+p(1-P_1)(1-P_3)] & (n>m\\ge 2) \\\\\\pi _{(n,1)}=\\pi _{(n-1,0)}p(1-P_1)P_2 & (n\\ge 2) \\\\\\pi _{(n,0)}=\\pi _{(n-1,0)}[(1-p)+p(1-P_1)(1-P_2)] \\\\\\qquad \\quad \\quad + \\left(\\sum \\nolimits _{k=n}^{\\infty }\\pi _{(k,n-1)}\\right)[(1-p)P_3+p(1-P_1)P_3] & (n\\ge 2) \\\\\\pi _{(1,0)}= \\left(\\sum \\nolimits _{n=1}^{\\infty }\\pi _{(n,0)} + \\sum \\nolimits _{n=2}^{\\infty }\\sum \\nolimits _{m=1}^{n-1}\\pi _{(n,m)}\\right)pP_1\\end{array}\\right.",
"}$ Assume that the initial state is $(n-1,m-1)$ , to get age-state $(n,m)$ , it needs that no packet is obtained in both relay and destination, which is satisfied either no packet arrives at source and retransmission from relay fails, or a new packet comes to source but the packet transmissions on $s-d$ and $R-d$ links are unsuccessful.",
"Since the packet in relay cannot be substituted, the state $(n,1)$ can only be obtained from $(n-1,0)$ by letting a new packet arrive to relay $R$ but fail to transmit to receiver $d$ .",
"This gives the first two lines of equations (6).",
"Similarly to the explanations for the second to last equation in (1), both the state $(n-1,0)$ and $(k,n-1)$ can transfer to $(n,0)$ .",
"Beginning with $(n-1,0)$ , if no packet comes to source or the arriving packet is not sent to $R$ and $d$ successfully, no packet is obtained at $R$ or $d$ .",
"As a result, at the next time slot the AoI increases 1 while the second parameter remains 0.",
"On the other hand, a packet of age $n-1$ can be sent to $d$ from $R$ which reduces the AoI to $n$ .",
"Meanwhile, still we have to ensure that the receiver gets no packet from the direct link $s-d$ .",
"Combining these two cases, we determine the stationary equation of the state vector $(n,0)$ where $n\\ge 2$ .",
"Finally, it sees that still only one way can decrease the AoI to 1, i.e., the arriving packet is transmitted to $d$ through the direct link $s-d$ , no matter what the original age-state is.",
"As before, this equation yields that the stationary probability $\\pi _{(1,0)}$ is $pP_1$ directly.",
"So far, we have explained all the equations in (6).",
"This completes the proof of Theorem 4.",
"Table: Random state transfers and transtion probabilities: non-preemption caseSolving the system of equations (6) is easier than (1), since no-packet-preemption in relay $R$ dramatically simplifies how the age-state $(n,1)$ is obtained.",
"We give the solutions of all the stationary probabilities $\\pi _{(n,m)}$ in Theorem 5 and put the calculation details in Appendix C. Theorem 5 When the AoI process $AoI_{NB}^{NP}$ reaches the steady-state, the stationary probabilities $\\pi _{(n,m)}$ are solved as $\\pi _{(n,0)}=\\beta _1\\delta ^{n-1} + \\beta _2[(1-pP_1)(1-P_3)]^{n-1} \\quad (n\\ge 1)$ Other probabilities $\\pi _{(n,m)}$ , $n>m\\ge 1$ are determined by $\\pi _{(n,m)}&=p(1-P_1)P_2\\bigg \\lbrace \\beta _1\\delta ^{n-2}\\left(\\frac{(1-pP_1)(1-P_3)^{m-1}}{\\delta }\\right) + \\beta _2\\left[ (1-pP_1)(1-P_3) \\right]^{n-2}\\bigg \\rbrace ,$ in which the coefficients $\\delta $ , $\\beta _1$ are $\\beta _2$ are given by $\\delta &=1-p(P_1 + P_2 - P_1P_2), \\\\\\beta _1&= pP_1-\\beta _2, \\\\\\beta _2&= \\frac{[1-(1-pP_1)(1-P_3)]p(1-P_1)P_2P_3}{[1-(1-pP_1)(1-P_3)+p(1-P_1)P_2]} \\frac{1-pP_1}{(1-pP_1)(1-P_3)-\\delta }.$ Now, the stationary distribution of AoI, which we denote as $\\Delta _{NB}^{NP}$ can be determined.",
"Theorem 6 Assume that the packet in relay cannot be substituted by later packets from source, then for $n\\ge 1$ , the stationary AoI distribution of relay-assisted status updating system is given as $\\Pr \\lbrace \\Delta _{NB}^{NP}=n\\rbrace &=\\left(\\beta _1 + \\frac{p(1-P_1)P_2}{\\delta -(1-pP_1)(1-P_3)}\\beta _1\\right) \\delta ^{n-1} \\\\& \\quad + \\left(\\beta _2 - \\frac{p(1-P_1)P_2}{\\delta -(1-pP_1)(1-P_3)}\\beta _1\\right) [(1-pP_1)(1-P_3)]^{n-1} \\\\& \\quad + p(1-P_1)P_2\\beta _2(n-1)[(1-pP_1)(1-P_3)]^{n-2}$ As before, for $n\\ge 2$ , we first calculate the sum $&\\sum \\nolimits _{m=1}^{n-1}\\pi _{(n,m)} \\\\={}& \\sum \\nolimits _{m=1}^{n-1} p(1-P_1)P_2\\bigg \\lbrace \\beta _1\\delta ^{n-2}\\left(\\frac{(1-pP_1)(1-P_3)^{m-1}}{\\delta }\\right) + \\beta _2\\left[ (1-pP_1)(1-P_3) \\right]^{n-2}\\bigg \\rbrace \\\\={}& \\frac{p(1-P_1)P_2\\beta _1}{\\delta -(1-pP_1)(1-P_3)}\\left\\lbrace \\delta ^{n-1}-[(1-pP_1)(1-P_3)]^{n-1}\\right\\rbrace \\\\{}& \\qquad + p(1-P_1)P_2\\beta _2(n-1)[(1-pP_1)(1-P_3)]^{n-2}$ Thus, we have that $&\\Pr \\lbrace \\Delta _{NB}^{NP}=n\\rbrace \\\\={}& \\pi _{(n,0)} + \\sum \\nolimits _{m=1}^{n-1}\\pi _{(n,m)} \\\\={}& \\left(\\beta _1 + \\frac{p(1-P_1)P_2}{\\delta -(1-pP_1)(1-P_3)}\\beta _1\\right) \\delta ^{n-1} + \\left(\\beta _2 - \\frac{p(1-P_1)P_2}{\\delta -(1-pP_1)(1-P_3)}\\beta _1\\right) [(1-pP_1)(1-P_3)]^{n-1} \\\\{}& \\qquad + p(1-P_1)P_2\\beta _2(n-1)[(1-pP_1)(1-P_3)]^{n-2}$ where the probability expression (7) is substituted.",
"Let $n=1$ , equation (13) reduces to $\\beta _1 + \\beta _2=pP_1$ from (9), which equals $\\pi _{(1,0)}$ and is also the probability that AoI takes value 1.",
"Therefore, (11) is valid for all $n\\ge 1$ .",
"This completes the proof of Theorem 6."
],
[
"Characterizing the Age of Information When relay has size 1 buffer",
"In the previous Section, we have discussed the relay-assisted system where the relay has no buffer.",
"The explicit expressions of stationary AoI distribution was derived for two cases in which we assume that the new packet can and cannot preempt the packet retransmissions from relay.",
"In this Section, we will consider the steady-state AoI when the relay is equipped with a size 1 buffer, which is used to store the arriving packet if there is already one packet in relay.",
"In addition, the packet in buffer is allowed to be updated by later ones from source $s$ .",
"The purpose of this Section is two-fold.",
"On the one hand, we will prove that the AoI distribution can still be determined by constituting a multiple-dimensional stochastic process and solving its steady state.",
"Hence, not only the mean of AoI, but all the AoI performances can be computed.",
"On the other hand, finding out the influences of storing-another-packet on the system's AoI is practically meaningful, which directly guides the design of a superior statue updating system.",
"Table: The age-state transfers of relay-assisted status updating system: relay has size 1 bufferHere, a three-dimensional state vector $(n,m,l)$ is defined, and the three-dimensional stochastic process $AoI_B^{P}$ is constituted accordingly.",
"The added parameter $l$ denotes the age of packet in relay's buffer.",
"To describe the age-state transfers, again we list all the cases using a Table REF .",
"Table: Random state transfers and the corresponding transtion probabilities: relay has size 1 bufferThe state transfers are described assuming that the process $AoI_B^{P}$ has different initial state vector, i.e., $(n,m,l)$ , $(n,m,0)$ and $(n,0,0)$ .",
"Actually, we list all the state transitions in the order of number of packets in system.",
"For instance, age-state $(n,m,l)$ means that the current value of AoI is $n$ and there are two packets of age $m$ and $l$ in relay, one is transmitted on $R-d$ link and the other is waiting in buffer.",
"At the next time slot, if $A=0$ , that is no packet is obtained at source, only the packet transmission over $R-d$ link needs to be considered, since no packet is delivered on both $s-d$ and $s-R$ links.",
"When $A=1$ , we have to describe whether the new packet is successfully transmitted via $s-d$ and $s-R$ links and in relay $R$ , if the packet having age $m$ is obtained at destination $d$ .",
"This gives first eight rows of Table REF .",
"All the state tranfers from other initial age-states $(n,m,0)$ and $(n,0,0)$ are discussed similarly.",
"Notice that still only one way can reduce the AoI to 1, that is a new updating packet is sent to $d$ via the direct link $s-d$ .",
"Since we have assumed that when everytime a packet is obtained from $s-d$ link, all the packets in system are deleted, which makes the other two state components $m$ and $l$ jump to 0.",
"Therefore, we can determine that the first stationary probability $\\pi _{(1,0,0)}$ is equal to $pP_1$ immediately.",
"According to Table REF , in Table REF we summarize every state transfer and determine the transition probability, from which the stationary equations of AoI stochastic process $AoI_B^P$ can be established when the process reaches the steady state.",
"For the sake of simplicity, in this Section we denote that $\\delta =(1-p)+p(1-P_1)(1-P_2) =1-p(P_1+P_2-P_1P_2) $ and $\\eta =p(1-P_1)P_2 $ Theorem 7 Assume that the three-dimensional stochastic process $AoI_B^P$ runs in steady state, the stationary probabilities $\\pi _{(n,m,l)}$ of all the age-states satisfy following equations ${\\left\\lbrace \\begin{array}{ll}\\pi _{(n,m,l)}=\\pi _{(n-1,m-1,l-1)}\\delta (1-P_3) & \\qquad (n>m>l\\ge 2) \\\\\\pi _{(n,m,1)}=\\left(\\sum \\nolimits _{j=0}^{m-2}\\pi _{(n-1,m-1,j)}\\right)\\eta (1-P_3) & \\qquad (n>m\\ge 2) \\\\\\pi _{(n,m,0)}=\\pi _{(n-1,m-1,0)}\\delta (1-P_3) +\\left(\\sum \\nolimits _{k=n}^{\\infty }\\pi _{(k,n-1,m-1)}\\right)\\delta P_3 & \\qquad (n>m\\ge 2) \\\\\\pi _{(n,1,0)}=\\pi _{(n-1,0,0)}\\eta +\\left(\\sum \\nolimits _{k=n}^{\\infty }\\sum \\nolimits _{j=0}^{n-2}\\pi _{(k,n-1,j)}\\right)\\eta P_3 & \\qquad (n\\ge 2) \\\\\\pi _{(n,0,0)}=\\pi _{(n-1,0,0)}\\delta +\\left(\\sum \\nolimits _{k=n}^{\\infty }\\pi _{(k,n-1,0)}\\right)\\delta P_3 & \\qquad (n\\ge 2) \\\\\\pi _{(1,0,0)}=\\Big (\\sum \\nolimits _{n=1}^{\\infty }\\pi _{(n,0,0)}+ \\sum \\nolimits _{n=2}^{\\infty }\\sum \\nolimits _{m=1}^{n-1}\\pi _{(n,m,0)}\\\\\\qquad \\qquad \\qquad \\qquad \\qquad + \\sum \\nolimits _{n=3}^{\\infty }\\sum \\nolimits _{m=2}^{n-1}\\sum \\nolimits _{l=1}^{m-1}\\pi _{(n,m,l)}\\Big )pP_1\\end{array}\\right.",
"}$ We explain each line of equations (14) briefly.",
"For the cases $n>m>l\\ge 2$ , the state $(n,m,l)$ can only be obtained from $(n-1,m-1,l-1)$ when both $d$ and $R$ obtains no packet in the current time slot.",
"No packet comes to $d$ ensures that the AoI does not drop, while the age of packet stored in relay's buffer also increases 1 as long as no packet is sent to $R$ from $s$ successfully.",
"Since the packet contained in relay's buffer can be replaced by later updating packets, apart from $(n-1,m-1,0)$ , state vectors of form $(n-1,m-1,j)$ where $1\\le j\\le m-2$ can also transfer to $(n,m,1)$ .",
"These state transfers occur if the AoI at $d$ does not decrease, and the packet in relay's buffer is updated by a new packet at the same time.",
"This gives the second line of (14).",
"Age-state $(n,m,0)$ means that the buffer in $R$ is empty, which can be transferred to in two ways.",
"At first, if no packet arrives to $R$ and $d$ , then state vector $(n-1,m-1,0)$ transfers to $(n,m,0)$ after one time slot naturally.",
"Apart from this, notice that when the packet retransmission is successful via $R-d$ link, the AoI is reduced and this packet is then deleted from $R$ .",
"Next time the packet in buffer is transmitted.",
"As a result, any state having form $(k,n-1,m-1)$ , $k\\ge n$ will change to $(n,m,0)$ as well.",
"Combining both cases, we obtain the third equation in (14).",
"When considering the state $(n,1,0)$ , remember that the packet in buffer can be replaced, making the last parameter $l$ reduce to 1.",
"Similar to above discussions, we see that all the age-states of form $(k,n-1,j)$ where $k\\ge n$ and $1\\le j\\le n-2$ can transfer to $(n,1,0)$ , so that in this case we have a double sum in the stationary equation.",
"The empty state $(n,0,0)$ can be obtained from either $(n-1,0,0)$ when no packet arrives to $d$ and $R$ , or age-state $(k,n-1,0)$ , $k\\ge n$ assuming that the packet of age $n-1$ is sent to $d$ through $R-d$ link and no other packet comes to relay $R$ .",
"Substituting the transition probabilities in Table REF , the stationary equation corresponding to $(n,0,0)$ can be determined.",
"At last, the AoI is reduced to 1 from any initial age-state when a new packet is delivered to $d$ through the direct link $s-d$ , which yields the last line of equations (14) and in fact determines that $\\pi _{(1,0,0)}=pP_1$ .",
"This completes the proof of Theorem 7.",
"Compared with the no-buffer cases discussed before, here we have to find all the stationary probabilities $\\pi _{(n,m,l)}$ , $n>m>l\\ge 0$ .",
"Then, the probability that stationary AoI, $\\Delta _B^P$ , takes value $n$ is determined by the formula $\\Pr \\lbrace \\Delta _B^P=n\\rbrace =\\pi _{(n,0,0)} &+\\sum \\nolimits _{m=1}^{n-1}\\pi _{(n,m,0)} +\\sum \\nolimits _{m=2}^{n-1}\\sum \\nolimits _{l=1}^{m-1}\\pi _{(n,m,l)}$ Theorem 8 The solutions of equations (14), i.e., all the stationary probabilities $\\pi _{(n,m,l)}$ of three-dimensional stochastic process $AoI_B^P$ are given as follows.",
"First of all, $\\pi _{(n,0,0)}&=\\left(pP_1+\\eta \\right)\\delta ^{n-1} -\\left\\lbrace \\eta -(1-S)\\eta P_3\\right\\rbrace [\\delta (1-P_3)]^{n-1} \\\\& \\qquad \\qquad \\qquad \\qquad -(1-S)\\eta P_3 n[\\delta (1-P_3)]^{n-1} \\qquad (n\\ge 1)$ For $n>m\\ge 0$ , $&\\pi _{(n,m,0)} \\\\={}&\\eta \\left(pP_1+\\eta \\right)\\delta ^{n-2}\\left(1-P_3\\right)^{m-1}-\\eta ^2\\left[\\delta (1-P_3)\\right]^{n-2}- (1-S)\\eta ^2P_3(n-m-1)\\left[\\delta (1-P_3)\\right]^{n-2} \\\\{}& +\\widetilde{S}\\eta P_3 \\left[(\\delta +\\eta )(1-P_3)\\right]^{n-2} m \\left(\\frac{\\delta }{\\delta +\\eta }\\right)^{m-1} - (1-S)\\delta \\eta P_3^2 m[\\delta (1-P_3)]^{n-2}$ The probability $\\pi _{(n,m,l)}$ , $n>m>l\\ge 1$ is solved as $\\pi _{(n,m,l)}&= \\eta ^2(1-P_3)\\bigg \\lbrace \\left(pP_1+\\eta \\right)\\delta ^{n-3}\\left(\\frac{(\\delta +\\eta )(1-P_3)}{\\delta }\\right)^{m-2} \\left(\\frac{\\delta }{\\delta +\\eta }\\right)^{l-1} \\\\& \\qquad -\\eta [\\delta (1-P_3)]^{n-3}\\left(\\frac{\\delta +\\eta }{\\delta }\\right)^{m-2} \\left(\\frac{\\delta }{\\delta +\\eta }\\right)^{l-1} \\\\& \\qquad -(1-S)\\eta P_3(n-m-1) [\\delta (1-P_3)]^{n-3}\\left(\\frac{\\delta +\\eta }{\\delta }\\right)^{m-2}\\left(\\frac{\\delta }{\\delta +\\eta }\\right)^{l-1} \\bigg \\rbrace \\\\& \\quad + \\eta P_3\\widetilde{S}[(\\delta +\\eta )(1-P_3)]^{n-2}\\left(\\frac{\\delta }{\\delta +\\eta }\\right)^{l-1} - \\eta P_3\\widetilde{S}[(\\delta +\\eta )(1-P_3)]^{n-2}\\left(\\frac{\\delta }{\\delta +\\eta }\\right)^{m-1} \\\\& \\quad -\\delta \\eta P_3^2(1-S)[\\delta (1-P_3)]^{n-2}\\left(\\frac{\\delta +\\eta }{\\delta }\\right)^{m-1} \\left(\\frac{\\delta }{\\delta +\\eta }\\right)^{l-1} \\\\&\\quad + \\delta \\eta P_3^2(1-S)[\\delta (1-P_3)]^{n-2}$ where the numbers $S$ and $\\widetilde{S}$ are determined by $\\widetilde{S}&=S\\eta +(1-S)(\\delta +\\eta )P_3, \\\\S&=\\frac{pP_1[1-\\delta (1-P_3)]^2+\\delta \\eta P_3^2}{(1-\\delta )[1-\\delta (1-P_3)]^2-\\delta \\eta P_3(1-\\delta )(1-P_3)}$ We will solve the stationary equations (14) in Appendix D. Provided all the stationary probabilities, the distribution of steady-state AoI can be obtained by equation (15).",
"The calculation details are given in Appendix E. Theorem 9 Assume that the relay has a buffer of size 1, for $n\\ge 1$ , the steady-state AoI $\\Delta _B^P$ of the relay-assisted status updating system is distributed as $\\Pr \\lbrace \\Delta _B^P=n\\rbrace &=\\frac{(\\delta +\\eta )P_3(pP_1+\\eta )}{\\delta -(\\delta +\\eta )(1-P_3)}\\delta ^{n-1}-c_1[(\\delta +\\eta )(1-P_3)]^{n-1} +c_2[\\delta (1-P_3)]^{n-1} \\\\& \\quad +\\frac{(1-S)(\\delta +\\eta )P_3^2}{1-P_3}n[\\delta (1-P_3)]^{n-1} + \\frac{P_3\\widetilde{S}}{1-P_3}n[(\\delta +\\eta )(1-P_3)]^{n-1}$ where the coefficients $c_1$ and $c_2$ are given as $c_1&=\\frac{pP_1\\eta (1-P_3)+\\delta \\eta P_3}{[\\delta -(\\delta +\\eta )(1-P_3)](1-P_3)} + \\frac{\\delta P_3[\\eta +(\\delta +\\eta )P_3](1-S)+(\\delta +\\eta )P_3\\widetilde{S}}{\\eta (1-P_3)} \\\\c_2&=\\frac{P_3[\\eta (\\delta +\\eta )+P_3(1-S)(\\delta +\\eta )(2\\delta -\\eta )]}{\\eta (1-P_3)}$"
],
[
"Performance metrics of Age of Information",
"In Section III, for two cases where the packet retransmission from relay can and cannot be preempted by later packets, we derive the stationary AoI distribution of status updating system with a no-buffer relay.",
"To investigate the influence of storing another packet in relay on system's AoI, the distribution of steady state AoI is determined in Section IV.",
"Given the stationary AoI distributions, the following corollary shows the mean and the variance of AoI for all three cases, and the calculation details are provided in Appendix F. Corollary 1 For the relay-assisted status updating system, the average AoI for no-buffer-and-preemption case is equal to $\\mathbb {E}[\\Delta _{NB}^P]&=\\frac{(1-pP_1)P_3}{[(1-pP_1)P_3-p(1-P_1)P_2](1-\\delta )} + \\frac{\\xi }{[1-(1-pP_1)(1-P_3)]^2}$ while for the no-buffer-no-preemption case, it shows that $\\mathbb {E}[\\Delta _{NB}^{NP}]&= \\frac{\\beta _1}{(1-\\delta )^2}+\\frac{\\beta _2}{[1-(1-pP_1)(1-P_3)]^2} \\\\& \\quad + \\frac{p(1-P_1)P_2\\beta _1[2-\\delta -(1-pP_1)(1-P_3)]}{(1-\\delta )^2[1-(1-pP_1)(1-P_3)]^2} +\\frac{2p(1-P_1)P_2\\beta _2}{[1-(1-pP_1)(1-P_3)]^3}$ The average AoI of relay-assisted system when the relay is equipped with a buffer of size 1 is calculated as $\\mathbb {E}[\\Delta _B^P]&=\\frac{(\\delta +\\eta )P_3(pP_1+\\eta )}{[\\delta -(\\delta +\\eta )(1-P_3)](1-\\delta )^2}-\\frac{c_1}{[1-(\\delta +\\eta )(1-P_3)]^2} +\\frac{c_2}{[1-\\delta (1-P_3)]^2} \\\\& \\quad +\\frac{(1-S)(\\delta +\\eta )P_3^2[1+\\delta (1-P_3)]}{(1-P_3)[1-\\delta (1-P_3)]^3} +\\frac{P_3\\widetilde{S}[1+(\\delta +\\eta )(1-P_3)]}{(1-P_3)[1-(\\delta +\\eta )(1-P_3)]^3}$ where in all of three equations $\\delta =1-p(P_1+P_2-P_1P_2) $ Besides, $\\xi $ in (24) is given as $\\xi &=\\frac{p[P_1+(1-P_1)P_2P_3]}{1-P_3} - \\frac{P_3(1-\\delta )\\delta }{[(1-pP_1)P_3-p(1-P_1)P_2](1-P_3)} $ and $\\beta _1$ and $\\beta _2$ in expression (25) are defined in (9) and (10).",
"In (26), $\\eta =p(1-P_1)P_2$ , $S$ , $\\widetilde{S}$ are given in (19) and (20).",
"Other two coefficients $c_1$ , $c_2$ are introduced in equations (22) and (23).",
"For different situations, the second moments of AoI can also be obtained.",
"Then, we can calculate the variance of stationary AoI by $\\emph {Var}[\\Delta ]=\\mathbb {E}[\\Delta ^2]-\\left(\\mathbb {E}[\\Delta ]\\right)^2$ for all three cases.",
"The expression of $\\emph {Var}[\\Delta ]$ is lengthy but the calculations are direct.",
"Here, we provide only $\\emph {Var}[\\Delta _{NB}^{NP}]$ , AoI variances for other two cases can be obtained similarly.",
"From AoI distribution (11), we have $\\mathbb {E}\\left[\\left(\\Delta _{NB}^{NP}\\right)^2\\right] &= \\left(\\beta _1+\\frac{p(1-P_1)P_2\\beta _1}{\\delta -(1-pP_1)(1-P_3)}\\right)\\sum \\nolimits _{n=1}^{\\infty }n^2\\delta ^{n-1} \\\\& \\quad + \\left(\\beta _2 - \\frac{p(1-P_1)P_2\\beta _1}{\\delta -(1-pP_1)(1-P_3)}\\right) \\sum \\nolimits _{n=1}^{\\infty }n^2[(1-pP_1)(1-P_3)]^{n-1} \\\\& \\quad + p(1-P_1)P_2\\beta _2\\sum \\nolimits _{n=1}^{\\infty }n^2(n-1)[(1-pP_1)(1-P_3)]^{n-2} \\\\&= \\left(\\beta _1+\\frac{p(1-P_1)P_2\\beta _1}{\\delta -(1-pP_1)(1-P_3)}\\right)\\frac{1+\\delta }{(1-\\delta )^3} +\\left(\\beta _2 - \\frac{p(1-P_1)P_2\\beta _1}{\\delta -(1-pP_1)(1-P_3)}\\right) \\\\& \\quad \\times \\frac{1+(1-pP_1)(1-P_3)}{[1-(1-pP_1)(1-P_3)]^3} + p(1-P_1)P_2\\beta _2 \\frac{4+2(1-pP_1)(1-P_3)}{[1-(1-pP_1)(1-P_3)]^4}$ where the last sum $&\\sum \\nolimits _{n=1}^{\\infty }n^2(n-1)[(1-pP_1)(1-P_3)]^{n-2} \\\\={}& \\sum \\nolimits _{n=1}^{\\infty }(n+1)n(n-1)[(1-pP_1)(1-P_3)]^{n-2} -\\sum \\nolimits _{n=1}^{\\infty }n(n-1)[(1-pP_1)(1-P_3)]^{n-2} \\\\={}&\\frac{6}{[1-(1-pP_1)(1-P_3)]^4} - \\frac{2}{[1-(1-pP_1)(1-P_3)]^3} \\frac{4+2(1-pP_1)(1-P_3)}{[1-(1-pP_1)(1-P_3)]^4} $ can be derived using following formula $\\sum \\nolimits _{n=1}^{\\infty }n(n-1)\\cdots (n-k+1)x^{n-k}=\\frac{k!",
"}{(1-x)^{k+1}} \\text{ for }0<x<1, \\text{ and any }k\\ge 1.",
"$ Combining results (25) and (28), the AoI variance $\\emph {Var}\\left[\\Delta _{NB}^{NP}\\right]$ is obtained by formula (27)."
],
[
"Numerical Simulations",
"The numerical results of AoI are provided in this Section.",
"Specifically, we will depict the stationary distribution of three AoIs, i.e., $\\Delta _{NB}^P$ , $\\Delta _{NB}^{NP}$ and $\\Delta _B^P$ for relay-assisted status updating system.",
"For three situations, we also draw the curves of average AoI and the AoI's variance.",
"To demonstrate the improvement of system's timeliness by adding an intermediate relay, the AoI performance of a non-relay updating system is calculated and comparisons are carried out as well.",
"First of all, the age of information of a system without relay is analyzed briefly.",
"When there is no relay in system, the packet can only be transmitted to destination via the direct link $s-d$ .",
"Since the packet on channel $s-d$ is assumed to be erased independently at each time slot, and the propagation delay is 1, it is easy to describe the random dynamics of the AoI at destination $d$ .",
"Let the current AoI is $n$ , it is observed that either the value of AoI is reduced to 1 when a packet is obtained successfully on $s-d$ link, which occurs with probability $pP_1$ , or the AoI increases 1.",
"These random state transfers can be characterized by a one-dimensional stochastic process whose discrete state is simply defined as the AoI at $d$ .",
"When the process reaches the steady state, it is not hard to obtain that the AoI of non-relay system, $\\Delta $ , is distributed as $\\Pr \\lbrace \\Delta =n\\rbrace =pP_1(1-pP_1)^{n-1} \\qquad (n\\ge 1)$ which is a geometric distribution with parameter $pP_1$ .",
"The mean and variance of non-relay AoI are calculated as $\\mathbb {E}[\\Delta ]=\\frac{1}{pP_1}, \\qquad \\quad \\emph {Var}[\\Delta ]=\\frac{1-pP_1}{(pP_1)^2}$ Figure: Stationary AoI distributions of relay-assisted status updating system.We depict the AoI's distribution in Figure REF , including three relay-assisted cases and the AoI distribution of a non-relay system.",
"The packet generation probability is set as $p=2/5$ .",
"The transmission success probabilities of three links are $P_1=1/4$ , $P_2=1/3$ , and $P_3=1/3$ .",
"It was seen that when the relay is added, all of three distribution curves have a peak, and decrease rapidly as the value of AoI becomes larger.",
"While the AoI distribution of the non-relay system is decreasing in all the AoI range, whose tail drops much slower.",
"Therefore, compared with the non-relay system, it is less likely that the AoI of a relay-assisted status updating system takes large values.",
"The difference of three relay-assisted AoI distributions are tiny.",
"It is seen that three distribution curves take the peak value under approximately the same AoI, and almost coincide when the AoI get slightly large, for intance, when its value is greater than 5.",
"Figure: Average AoI of updating system with and without relay.To compare the timeliness performance of various updating systems, we depict the graphs of average AoI in Figure REF .",
"The relationship among three average AoI of relay-assisted situations are clear in Figure REF , in which we offer the numerical results of all the cases when the packet generation probability is high.",
"Firstly, it is apparent that introducing a relay can reduce the average AoI at the destination.",
"The average AoI of relay-assisted cases are smaller than that of a non-relay system.",
"As packet generation probability $p$ gets larger, the average performance of AoI are all decreasing, no matter there has a relay in system or not.",
"At the same time, the average AoI's gap between relay-assisted cases and non-relay case is getting small.",
"Numerical results in Figure REF shows that the average AoI $\\mathbb {E}\\left[\\Delta _{NB}^P\\right]$ is minimal among three relay-assisted situations.",
"It is understandable that average age of information $\\mathbb {E}\\left[\\Delta _{NB}^P\\right]$ is less than $\\mathbb {E}\\left[\\Delta _{NB}^{NP}\\right]$ , since replacing an “old” packet with a fresh one does not increase the AoI at the destination.",
"However, observing that $\\mathbb {E}\\left[\\Delta _{B}^P\\right]$ is above $\\mathbb {E}\\left[\\Delta _{NB}^P\\right]$ , which implies that under the system settings in this paper, storing another packet in relay does not help to reduce the system's AoI.",
"Equivalently, from the sense of average performance, adding buffer in relay cannot enhance the timeliness of a relay-assisted status updating system.",
"Mathematically, we have $\\mathbb {E}\\left[\\Delta _{NB}^P\\right]< \\mathbb {E}\\left[\\Delta _B^P\\right]< \\mathbb {E}\\left[\\Delta _{NB}^{NP}\\right]< \\mathbb {E}\\left[\\Delta \\right]$ Figure: Variance of AoI for the status updating system with and without relay.In Figure REF , we also provide the variance of stationary AoI for each case.",
"Simulation results show that $\\emph {Var}\\left[\\Delta _{NB}^P\\right]< \\emph {Var}\\left[\\Delta _B^P\\right]< \\emph {Var}\\left[\\Delta _{NB}^{NP}\\right]< \\emph {Var}\\left[\\Delta \\right]$ Notice that the relationships between various AoI variances are the same as that of average AoIs.",
"The age of information $\\Delta _{NB}^P$ also has the minimal variance.",
"The comparisons between three AoI variances for relay-assisted situations are illustrated in Figure REF .",
"When the packet generation probability is large, the relationships among them is clearer."
],
[
"Conclusion",
"In this paper, we considere a relay-assisted status updating system and analyze the performace of age of information at the destination.",
"We characterize the stationary AoI for three different settings at relay, i.e., no-buffer-and-preemption, no-buffer-no-preemption, and buffer-and-preemption.",
"The obtained results show that introducing an intermediate relay can reduce the average AoI and AoI's variance dramatically, thus proving the improvement of system's timeliness by adding the relay.",
"On the other hand, we show that the no-buffer-and-preemption setting of relay achieve the minimal average performance and variance of the AoI.",
"As a result, under the system model considered in this paper, there is no necessity to add the buffer in relay because this does not help to enhance the system's timeliness.",
"We attribute the idea that invoking a multi-dimensional state vector and constituting multi-dimensional AoI stochastic process to SHS method, which is a systematic approach to calculate the average AoI of status updating systems in continuous time model.",
"The methods used in the paper are the standard tools from queueing theory.",
"Notice that in a status updating system, it is not easy to describe the AoI itself, but indeed is easier to characterize the dynamics of a “bigger” state vector which contains the AoI as the part.",
"Therefore, as long as the steady state of the larger-dimensional process is solved, as one of marginal distributions, the stationary distribution of AoI is obtained as well.",
"Once all the stationary probabilities of larger-dimensional process are obtained, apart from AoI, other marginal distributions are all be determined.",
"In this paper, for example, we can calculate the age distribution of packet in relay by summing over AoI-component in state vector."
],
[
"Proof of Theorem 2",
"In this Appendix, we solve the following system of equations and find all the probabilities $\\pi _{(n,m)}$ , $n>m\\ge 0$ .",
"${\\left\\lbrace \\begin{array}{ll}\\pi _{(n,m)}=\\pi _{(n-1,m-1)}\\delta _1 & \\qquad (n>m\\ge 2) \\\\\\pi _{(n,1)}=\\pi _{(n-1,0)}p(1-P_1)P_2 + \\sum \\nolimits _{j=1}^{n-2}\\pi _{(n-1,j)}p(1-P_1)P_2(1-P_3) \\\\\\qquad \\qquad + \\sum \\nolimits _{k=n}^{\\infty }\\pi _{(k,n-1)}p(1-P_1)P_2P_3 & \\qquad (n\\ge 3) \\\\\\pi _{(2,1)}=\\pi _{(1,0)}p(1-P_1)P_2 + \\sum \\nolimits _{k=2}^{\\infty }\\pi _{(k,1)}p(1-P_1)P_2P_3 \\\\\\pi _{(n,0)}=\\pi _{(n-1,0)}\\delta _2 + \\sum \\nolimits _{k=n}^{\\infty }\\pi _{(k,n-1)}\\delta _3 & \\qquad (n\\ge 2) \\\\\\pi _{(1,0)}=\\left(\\sum \\nolimits _{n=1}^{\\infty }\\pi _{(n,0)} + \\sum \\nolimits _{m=1}^{\\infty }\\sum \\nolimits _{n=m+1}^{\\infty }\\pi _{(n,m)} \\right) pP_1\\end{array}\\right.",
"}$ where we denote $\\delta _2&=(1-p) + p(1-P_1)(1-P_2) = 1-p(P_1+P_2-P_1P_2) $ and $\\delta _1&=(1-p)(1-P_3)+p(1-P_1)(1-P_2)(1-P_3) =(1-P_3)\\delta _2, \\\\\\delta _3&=(1-p)P_3+p(1-P_1)(1-P_2)P_3=P_3\\delta _2 $ First of all, from the first line of (33) we have $\\pi _{(n,m)}&= \\pi _{(n-1,m-1)}\\delta _1=\\dots =\\pi _{(n-m+1,1)}\\delta _1^{m-1} \\\\&= {\\left\\lbrace \\begin{array}{ll}\\left[\\pi _{(1,0)}p(1-P_1)P_2 \\sum \\nolimits _{k=2}^{\\infty }\\pi _{(k,1)}p(1-P_1)P_2P_3 \\right]\\delta _1^{m-1} & (n-m=1) \\\\\\Big [\\pi _{(n-m,0)}p(1-P_1)P_2 \\sum \\nolimits _{j=1}^{n-m-1}\\pi _{(n-m,j)}p(1-P_1)P_2(1-P_3) \\\\\\qquad \\quad + \\sum \\nolimits _{k=n-m+1}^{\\infty }\\pi _{(k,n-m)}p(1-P_1)P_2P_3 \\Big ]\\delta _1^{m-1} & (n-m\\ge 2)\\end{array}\\right.",
"}$ In above expressions, we have substituted the second and third lines of (33).",
"For cases $n\\ge 2$ , the fourth equation of (33) says that $\\pi _{(n,0)}=\\pi _{(n-1,0)}\\delta _2 + \\left(\\sum \\nolimits _{k=n}^{\\infty }\\pi _{(k,n-1)}\\right) \\delta _3$ Using probability expressions obtained in (34), we deal with the sum in equation (35) as follows.",
"$&\\sum \\nolimits _{k=n}^{\\infty }\\pi _{(k,n-1)}= \\pi _{(n,n-1)}+\\sum \\nolimits _{k=n+1}^{\\infty }\\pi _{(k,n-1)} \\\\={}&\\left[\\pi _{(1,0)}p(1-P_1)P_2 + \\sum \\nolimits _{k=2}^{\\infty }\\pi _{(k,1)}p(1-P_1)P_2P_3 \\right]\\delta _1^{n-2} \\\\{}& \\quad + \\sum \\nolimits _{k=n+1}^{\\infty } \\bigg \\lbrace \\pi _{(k-n+1,0)}p(1-P_1)P_2 + \\sum \\nolimits _{j=1}^{k-n}\\pi _{(k-n+1,j)}p(1-P_1)P_2(1-P_3) \\\\{}& \\quad + \\sum \\nolimits _{y=k-n+2}^{\\infty }\\pi _{(y,k-n+1)}p(1-P_1)P_2P_3 \\bigg \\rbrace \\delta _1^{n-2} \\\\={}&\\left[\\pi _{(1,0)}p(1-P_1)P_2 + \\left(\\sum \\nolimits _{k=2}^{\\infty }\\pi _{(k,1)}\\right) p(1-P_1)P_2P_3 \\right]\\delta _1^{n-2} \\\\{}& \\quad + \\bigg [ \\left(\\sum \\nolimits _{n=2}^{\\infty }\\pi _{(n,0)}\\right) p(1-P_1)P_2 + \\left( \\sum \\nolimits _{k=n+1}^{\\infty } \\sum \\nolimits _{j=1}^{k-n}\\pi _{(k-n+1,j)}\\right) p(1-P_1)P_2(1-P_3) \\\\{}& \\quad + \\left(\\sum \\nolimits _{k=n+1}^{\\infty }\\sum \\nolimits _{y=k-n+2}^{\\infty }\\pi _{(y,k-n+1)}\\right) p(1-P_1)P_2P_3 \\bigg ]\\delta _1^{n-2} \\\\={}&\\left[\\pi _{(1,0)}p(1-P_1)P_2 + \\left(\\sum \\nolimits _{n=2}^{\\infty }\\pi _{(n,1)}\\right) p(1-P_1)P_2P_3 \\right]\\delta _1^{n-2} \\\\{}& \\quad + \\Big [ \\left(\\sum \\nolimits _{n=2}^{\\infty }\\pi _{(n,0)}\\right) p(1-P_1)P_2 + \\left( \\sum \\nolimits _{n=2}^{\\infty } \\sum \\nolimits _{m=1}^{n-1}\\pi _{(n,m)}\\right) p(1-P_1)P_2(1-P_3) \\\\{}& \\quad + \\left(\\sum \\nolimits _{m=2}^{\\infty }\\sum \\nolimits _{n=m+1}^{\\infty }\\pi _{(n,m)}\\right) p(1-P_1)P_2P_3 \\Big ]\\delta _1^{n-2} \\\\={}& \\left[\\left(\\sum \\nolimits _{n=1}^{\\infty }\\pi _{(n,0)}\\right)p(1-P_1)P_2 + \\left( \\sum \\nolimits _{m=1}^{\\infty } \\sum \\nolimits _{n=m+1}^{\\infty }\\pi _{(n,m)}\\right) p(1-P_1)P_2 \\right]\\delta _1^{n-2} \\\\={}& p(1-P_1)P_2\\delta _1^{n-2}$ Let $k-n+1=\\tilde{n}$ , $j=\\tilde{m}$ , then $\\sum \\nolimits _{k=n+1}^{\\infty } \\sum \\nolimits _{j=1}^{k-n}\\pi _{(k-n+1,j)}=\\sum \\nolimits _{\\tilde{n}=2}^{\\infty }\\sum \\nolimits _{\\tilde{m}=1}^{\\tilde{n}-1}\\pi _{(\\tilde{n},\\tilde{m})} $ Similarly, do the substitutions $y=\\tilde{n}$ and $k-n+1=\\tilde{m}$ , it shows that $\\sum \\nolimits _{k=n+1}^{\\infty }\\sum \\nolimits _{y=k-n+2}^{\\infty }\\pi _{(y,k-n+1)}=\\sum \\nolimits _{\\tilde{m}=2}^{\\infty }\\sum \\nolimits _{\\tilde{n}=\\tilde{m}+1}^{\\infty }\\pi _{(\\tilde{n},\\tilde{m})} $ Substituting these sums, we obtain the equation (36).",
"In addition, observing that $&\\sum \\nolimits _{n=2}^{\\infty }\\pi _{(n,1)} + \\sum \\nolimits _{m=2}^{\\infty } \\sum \\nolimits _{n=m+1}^{\\infty }\\pi _{(n,m)} = \\sum \\nolimits _{m=1}^{\\infty } \\sum \\nolimits _{n=m+1}^{\\infty }\\pi _{(n,m)} $ and $\\sum \\nolimits _{m=1}^{\\infty } \\sum \\nolimits _{n=m+1}^{\\infty }\\pi _{(n,m)}=\\sum \\nolimits _{n=2}^{\\infty } \\sum \\nolimits _{m=1}^{n-1}\\pi _{(n,m)} $ which yields the result (37).",
"Equation (38) holds because the probabilities contained in bracket of (37) add up to 1.",
"Finally, we derive the following recursive equation $\\pi _{(n,0)}=\\pi _{(n-1,0)}\\delta _2 + p(1-P_1)P_2\\delta _3\\delta _1^{n-2} \\qquad (n\\ge 2)$ Applying (39) iteratively, the general formula of $\\pi _{(n,0)}$ can be obtained.",
"We show that $\\pi _{(n,0)}&=\\pi _{(1,0)}\\delta _2^{n-1}+p(1-P_1)P_2\\delta _3 \\sum \\nolimits _{j=0}^{n-2}\\delta _2^j\\delta _1^{n-2-j} \\\\&=\\pi _{(1,0)}\\delta _2^{n-1}+p(1-P_1)P_2\\delta _3 \\frac{\\delta _1^{n-1}-\\delta _2^{n-1}}{\\delta _1-\\delta _2}$ The first probability $\\pi _{(1,0)}$ is equal to $pP_1$ , which can be determined directly from the last equation of (33).",
"Remember that $\\delta _2=1-p(P_1+P_2-P_1P_2)$ and $\\delta _1=(1-P_3)\\delta _2, \\quad \\delta _3=P_3\\delta _2 $ we have $\\pi _{(n,0)}&=pP_1\\delta _2^{n-1}+p(1-P_1)P_2\\delta _3\\frac{\\delta _1^{n-1}-\\delta _2^{n-1}}{(1-P_3)\\delta _2-\\delta _2} \\\\&=pP_1\\delta _2^{n-1}+p(1-P_1)P_2\\left(\\delta _2^{n-1}-\\delta _1^{n-1} \\right) \\\\&=\\left[pP_1+p(1-P_1)P_2\\right]\\delta _2^{n-1} - p(1-P_1)P_2\\delta _1^{n-1} \\\\&=(1-\\delta _2)\\delta _2^{n-1} - p(1-P_1)P_2[(1-P_3)\\delta _2]^{n-1}$ Let $n=1$ , equation (41) gives $\\pi _{(1,0)}=pP_1$ .",
"Therefore, we prove that expression (41) is valid for all $n\\ge 1$ .",
"Next, probabilities $\\pi _{(n,m)}$ , $n>m\\ge 1$ are solved.",
"We first calculate $\\pi _{(n,1)}$ , then by the relation $\\pi _{(n,m)}=\\pi _{(n-m+1,1)}\\delta _1^{m-1}$ all the probabilities $\\pi _{(n,m)}$ are obtained.",
"When $n\\ge 3$ , from the second line of equations (33), we show that $\\pi _{(n,1)}&= \\pi _{(n-1,0)}p(1-P_1)P_2 + \\sum \\nolimits _{j=1}^{n-2}\\pi _{(n-1,j)}p(1-P_1)P_2(1-P_3) + \\sum \\nolimits _{k=n}^{\\infty }\\pi _{(k,n-1)}p(1-P_1)P_2P_3 \\\\&=\\left\\lbrace (1-\\delta _2)\\delta _2^{n-2} - p(1-P_1)P_2\\delta _1^{n-2}\\right\\rbrace p(1-P_1)P_2 \\\\& \\qquad + \\sum \\nolimits _{j=1}^{n-2}\\pi _{(n-j,1)}\\delta _1^{j-1} p(1-P_1)P_2(1-P_3) + p(1-P_1)P_2\\delta _1^{n-2}p(1-P_1)P_2P_3 \\\\&= p(1-P_1)P_2(1-\\delta _2)\\delta _2^{n-2} - [p(1-P_1)P_2]^2(1-P_3)\\delta _1^{n-2} \\\\& \\qquad + p(1-P_1)P_2(1-P_3)\\sum \\nolimits _{j=1}^{n-2}\\pi _{(n-j,1)}\\delta _1^{j-1}$ In (43), we have substituted equations (41) and (38).",
"Using relation (42), we can rewrite $\\pi _{(n-1,j)}$ as $\\pi _{(n-j,1)}\\delta _1^{j-1}$ .",
"Computing the difference $\\pi _{(n,1)}-\\pi _{(n-1,1)}\\delta _1 = p(1-P_1)P_2(1-\\delta _2)\\delta _2^{n-3}(\\delta _2-\\delta _1) + p(1-P_1)P_2(1-P_3)\\pi _{(n-1,1)} $ which is equivalent to $\\pi _{(n,1)}&=\\pi _{(n-1,1)}[\\delta _1+p(1-P_1)P_2(1-P_3)] + p(1-P_1)P_2(1-\\delta _2)P_3\\delta _2^{n-2} \\\\&= \\pi _{(n-1,1)}(1-P_3)(1-pP_1) + p(1-P_1)P_2(1-\\delta _2)P_3\\delta _2^{n-2}$ where $\\delta _1 + p(1-P_1)P_2(1-P_3) &= (1-P_3)\\delta _2 + p(1-P_1)P_2(1-P_3) \\\\&=(1-P_3)\\left\\lbrace 1-p(P_1+P_2-P_1P_2) + p(1-P_1)P_2 \\right\\rbrace \\\\&=(1-P_3)(1-pP_1) $ Equation (45) gives a recursive relation of the probabilities $\\pi _{(n,1)}$ , $n\\ge 3$ , from which we can derive that $\\pi _{(n,1)}&= \\pi _{(2,1)}[(1-P_3)(1-pP_1)]^{n-2} + p(1-P_1)P_2(1-\\delta _2)P_3 \\sum \\nolimits _{j=0}^{n-3}[(1-P_3)(1-pP_1)]^j \\delta _2^{n-2-j} \\\\&=\\pi _{(2,1)}[(1-P_3)(1-pP_1)]^{n-2} + p(1-P_1)P_2P_3(1-\\delta _2)\\delta _2 \\frac{\\delta _2^{n-2}-[(1-P_3)(1-pP_1)]^{n-2}}{\\delta _2-(1-P_3)(1-pP_1)} \\\\&=\\frac{p(1-P_1)P_2P_3(1-\\delta _2)\\delta _2}{\\delta _2-(1-P_3)(1-pP_1)}\\delta _2^{n-2}+ \\left(\\pi _{(2,1)} - \\frac{p(1-P_1)P_2P_3(1-\\delta _2)\\delta _2}{\\delta _2-(1-P_3)(1-pP_1)} \\right) [(1-P_3)(1-pP_1)]^{n-2}$ Using the basic relation in (33), the probability $\\pi _{(2,1)}$ is calculated as $\\pi _{(2,1)}&=\\pi _{(1,0)}p(1-P_1)P_2 + \\sum \\nolimits _{k=2}^{\\infty }\\pi _{(k,1)}p(1-P_1)P_2P_3 \\\\&=pP_1p(1-P_1)P_2 + [p(1-P_1)P_2]^2P_3 \\\\&=p^2(1-P_1)P_2[P_1+(1-P_1)P_2P_3] $ In equation (46), it is easy to see that for the case $n=2$ , expression (47) reduces to $\\pi _{(2,1)}=\\pi _{(2,1)}$ .",
"Therefore, we show that (47) is actually valid for all $n\\ge 2$ .",
"Now, for $n>m\\ge 1$ , the probabilities $\\pi _{(n,m)}$ can be obtained eventually.",
"It shows that $\\pi _{(n,m)}&=\\pi _{(n-m+1,1)}\\delta _1^{m-1} \\\\&= \\bigg \\lbrace \\frac{p(1-P_1)P_2P_3(1-\\delta _2)\\delta _2}{\\delta _2-(1-P_3)(1-pP_1)}\\delta _2^{n-m-1} \\\\& \\quad + \\left(\\pi _{(2,1)} - \\frac{p(1-P_1)P_2P_3(1-\\delta _2)\\delta _2}{\\delta _2-(1-P_3)(1-pP_1)} \\right) [(1-P_3)(1-pP_1)]^{n-m-1} \\bigg \\rbrace \\delta _1^{m-1} \\\\&= \\frac{p(1-P_1)P_2P_3(1-\\delta _2)\\delta _2}{\\delta _2-(1-P_3)(1-pP_1)}\\delta _2^{n-2}(1-P_3)^{m-1} \\\\& \\quad + \\left(\\pi _{(2,1)} - \\frac{p(1-P_1)P_2P_3(1-\\delta _2)\\delta _2}{\\delta _2-(1-P_3)(1-pP_1)} \\right) [(1-P_3)(1-pP_1)]^{n-2}\\left(\\frac{\\delta _2}{1-pP_1}\\right)^{m-1}$ So far, we have obtained all the stationary probabilities of process $AoI_{NB}^P$ and complete the proof of Theorem 2."
],
[
"Proof of Theorem 2",
"In this part, according to equation (4), we calculation the stationary distribution of AoI.",
"For $n\\ge 2$ , we first compute $&\\sum \\nolimits _{m=1}^{n-1}\\pi _{(n,m)} \\\\={}&\\sum \\nolimits _{m=1}^{n-1}\\Bigg \\lbrace \\pi _{(2,1)}[(1-pP_1)(1-P_3)]^{n-2}\\left(\\frac{\\delta }{1-pP_1}\\right)^{m-1} + \\frac{p(1-P_1)P_2P_3(1-\\delta )\\delta }{\\delta -(1-pP_1)(1-P_3)} \\\\{}& \\quad \\times \\left[ \\delta ^{n-2}(1-P_3)^{m-1} - [(1-pP_1)(1-P_3)]^{n-2}\\left(\\frac{\\delta }{1-pP_1} \\right)^{m-1} \\right] \\Bigg \\rbrace \\\\={}&\\pi _{(2,1)}\\frac{[(1-pP_1)(1-P_3)]^{n-1}-[(1-P_3)\\delta ]^{n-1}}{(1-pP_1-\\delta )(1-P_3)} + \\frac{p(1-P_1)P_2P_3(1-\\delta )\\delta }{\\delta -(1-pP_1)(1-P_3)} \\\\{}& \\quad \\times \\left\\lbrace \\frac{\\delta ^{n-2}-(1-P_3)[(1-P_3)\\delta ]^{n-2}}{P_3} - \\frac{[(1-pP_1)(1-P_3)]^{n-1}-[(1-P_3)\\delta ]^{n-1}}{(1-pP_1-\\delta )(1-P_3)} \\right\\rbrace \\\\={}& \\frac{p(1-P_1)P_2(1-\\delta )}{\\delta -(1-pP_1)(1-P_3)}\\left\\lbrace \\delta ^{n-1}-[(1-P_3)\\delta ]^{n-1}\\right\\rbrace \\\\{}& \\quad + \\left(\\pi _{(2,1)} - \\frac{p(1-P_1)P_2P_3(1-\\delta )\\delta }{\\delta -(1-pP_1)(1-P_3)} \\right) \\frac{[(1-pP_1)(1-P_3)]^{n-1}-[(1-P_3)\\delta ]^{n-1}}{(1-pP_1-\\delta )(1-P_3)} \\\\={}&\\eta _1\\left\\lbrace \\delta ^{n-1}-[(1-P_3)\\delta ]^{n-1}\\right\\rbrace + \\eta _2\\left\\lbrace [(1-pP_1)(1-P_3)]^{n-1}-[(1-P_3)\\delta ]^{n-1}\\right\\rbrace $ where we denote $\\eta _1&=\\frac{p(1-P_1)P_2(1-\\delta )}{\\delta -(1-pP_1)(1-P_3)}, \\\\\\eta _2&=\\left(\\pi _{(2,1)}-\\frac{p(1-P_1)P_2P_3(1-\\delta )\\delta }{\\delta -(1-pP_1)(1-P_3)}\\right) \\frac{1}{(1-pP_1-\\delta )(1-P_3)} $ Therefore, when $n\\ge 2$ , the probability that AoI equals $n$ is calculated as $\\Pr \\lbrace \\Delta =n\\rbrace &=\\pi _{(n,0)}+\\sum \\nolimits _{m=1}^{n-1}\\pi _{(n,m)} \\\\&= [(1-\\delta )+\\eta _1]\\delta ^{n-1}+\\eta _2[(1-pP_1)(1-P_3)]^{n-1} \\\\&\\qquad \\quad -\\left(\\eta _1+\\eta _2+p(1-P_1)P_2\\right)[(1-P_3)\\delta ]^{n-1}$ To obtain equation (50), we have substituted the probability expression (2).",
"Since $\\delta -(1-pP_1)(1-P_3) &=1-p(P_1+P_2-P_1P_2)-(1-pP_1)(1-P_3) \\\\&=1-pP_1-p(1-P_1)P_2-(1-pP_1)(1-P_3) \\\\&=(1-pP_1)P_3-p(1-P_1)P_2 $ we have the first coefficient $(1-\\delta )+\\eta _1&=1-\\delta + \\frac{p(1-P_1)P_2(1-\\delta )}{\\delta -(1-pP_1)(1-P_3)} \\\\&=(1-\\delta ) \\left( 1 + \\frac{p(1-P_1)P_2}{(1-pP_1)P_3-p(1-P_1)P_2}\\right) \\\\&=\\frac{(1-pP_1)P_3(1-\\delta )}{(1-pP_1)P_3-p(1-P_1)P_2} $ Notice that $1-pP_1-\\delta =1-pP_1-[1-p(P_1+P_2-P_1P_2)] = p(1-P_1)P_2 $ Substituting $\\pi _{(2,1)}$ , it shows that $\\eta _2&=\\frac{p[P_1+(1-P_1)P_2P_3]}{1-P_3} - \\frac{P_3(1-\\delta )\\delta }{[(1-pP_1)P_3-p(1-P_1)P_2](1-P_3)} $ In following paragraph, we will prove that the coefficient before the last term, i.e., $\\eta _1+\\eta _2+p(1-P_1)P_2$ is zero.",
"Thus, for $n\\ge 2$ we obtain $\\Pr \\lbrace \\Delta _{NB}^P=n\\rbrace &=\\frac{(1-pP_1)P_3(1-\\delta )}{(1-pP_1)P_3-p(1-P_1)P_2}\\delta ^{n-1} + \\eta _2[(1-pP_1)(1-P_3)]^{n-1}$ From equation (49), it is easy to see that the sum in (4) is zero for the case $n=1$ .",
"So, the expression (51) is valid for all $n\\ge 1$ .",
"We now show that $\\eta _1+\\eta _2+p(1-P_1)P_2=0$ .",
"That is $&\\frac{p(1-P_1)P_2(1-\\delta )}{\\delta -(1-pP_1)(1-P_3)} + \\left(p[P_1+(1-P_1)P_2P_3]-\\frac{P_3(1-\\delta )\\delta }{\\delta -(1-pP_1)(1-P_3)}\\right) \\\\& \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\times \\frac{p(1-P_1)P_2}{(1-pP_1-\\delta )(1-P_3)} + p(1-P_1)P_2 =0 $ which is equivalent to $&\\frac{1-\\delta }{\\delta -(1-pP_1)(1-P_3)} \\\\& + \\left(p[P_1+(1-P_1)P_2P_3]-\\frac{P_3(1-\\delta )\\delta }{\\delta -(1-pP_1)(1-P_3)}\\right) \\frac{1}{(1-pP_1-\\delta )(1-P_3)} + 1 =0 \\\\\\Leftrightarrow {}& \\frac{1-\\delta }{\\delta -(1-pP_1)(1-P_3)} + \\frac{p[P_1+(1-P_1)P_2P_3]}{(1-pP_1-\\delta )(1-P_3)} \\\\& - \\frac{P_3(1-\\delta )\\delta }{[\\delta -(1-pP_1)(1-P_3)](1-pP_1-\\delta )(1-P_3)} +1 =0$ In (52), we have $\\delta -(1-pP_1)(1-P_3)=(1-pP_1)P_3 - p(1-P_1)P_2 $ and $(1-pP_1-\\delta )(1-P_3) =\\left[1-pP_1- \\left(1-p(P_1+P_2-P_1P_2) \\right)\\right](1-P_3) =p(1-P_1)P_2(1-P_3) $ Then, equation (52) can be rewritten as $&\\frac{1-\\delta }{(1-pP_1)P_3-p(1-P_1)P_2}\\left(\\frac{P_3\\delta }{p(1-P_1)P_2(1-P_3)}-1\\right) =\\frac{p[P_1+(1-P_1)P_2P_3]}{p(1-P_1)P_2(1-P_3)}+1$ where the RHS of (53) is equal to $&\\frac{p[P_1+(1-P_1)P_2P_3]+p(1-P_1)P_2(1-P_3)}{p(1-P_1)P_2(1-P_3)} = \\frac{p[P_1+(1-P_1)P_2]}{p(1-P_1)P_2(1-P_3)}= \\frac{1-\\delta }{p(1-P_1)P_2(1-P_3)} $ Therefore, to prove (52), it suffices to show that $&\\frac{1}{(1-pP_1)P_3-p(1-P_1)P_2}\\left(\\frac{P_3\\delta }{p(1-P_1)P_2(1-P_3)}-1\\right) = \\frac{1}{p(1-P_1)P_2(1-P_3)} \\\\\\Leftrightarrow {}&\\frac{P_3\\delta }{p(1-P_1)P_2(1-P_3)}-1 = \\frac{(1-pP_1)P_3-p(1-P_1)P_2}{p(1-P_1)P_2(1-P_3)} = \\frac{(1-pP_1)P_3-p(1-P_1)P_2P_3}{p(1-P_1)P_2(1-P_3)}-1 \\\\\\Leftrightarrow {}&\\delta =(1-pP_1)-p(1-P_1)P_2 =1-p(P_1+P_2-P_1P_2)$ Since equation (54) holds, we verify the original equation (52) and show that the coefficient $\\eta _1+\\eta _2+p(1-P_1)P_2$ is equal to zero.",
"This completes the proof of Theorem 3."
],
[
"Proof of Theorem 5",
"In this Appendix, we solve the system of stationary equations (6) and find all the stationary probabilities $\\pi _{(n,m)}$ of AoI stochastic process $AoI_{NB}^{NP}$ .",
"For convenience, we copy these equations in the following.",
"${\\left\\lbrace \\begin{array}{ll}\\pi _{(n,m)}=\\pi _{(n-1,m-1)}[(1-p)(1-P_3)+p(1-P_1)(1-P_3)] & \\qquad (n>m\\ge 2) \\\\\\pi _{(n,1)}=\\pi _{(n-1,0)}p(1-P_1)P_2 & \\qquad (n\\ge 2) \\\\\\pi _{(n,0)}=\\pi _{(n-1,0)}[(1-p)+p(1-P_1)(1-P_2)] \\\\\\qquad \\qquad + \\left(\\sum \\nolimits _{k=n}^{\\infty }\\pi _{(k,n-1)}\\right)[(1-p)P_3+p(1-P_1)P_3] & \\qquad (n\\ge 2) \\\\\\pi _{(1,0)}= \\left(\\sum \\nolimits _{n=1}^{\\infty }\\pi _{(n,0)} + \\sum \\nolimits _{n=2}^{\\infty }\\sum \\nolimits _{m=1}^{n-1}\\pi _{(n,m)}\\right)pP_1\\end{array}\\right.",
"}$ First of all, applying the first line of (55) repeatedly, we can write $\\pi _{(n,m)}$ as $\\pi _{(n,m)}&=\\pi _{(n-1,m-1)}[(1-pP_1)(1-P_3)] \\\\&=\\pi _{(n-2,m-2)}[(1-pP_1)(1-P_3)]^2 \\\\&\\qquad \\qquad \\quad \\quad \\vdots \\\\&=\\pi _{(n-m+1,1)}[(1-pP_1)(1-P_3)]^{m-1} \\\\&=\\pi _{(n-m,0)}p(1-P_1)P_2[(1-pP_1)(1-P_3)]^{m-1}$ where to obtain the last step (56), the second equation in (55) is applied.",
"Substituting (56), we first handle the sum in the third line of (55).",
"It shows that $\\sum \\nolimits _{k=n}^{\\infty }\\pi _{(k,n-1)}&= \\sum \\nolimits _{k=n}^{\\infty }\\pi _{(k-n+1,0)}p(1-P_1)P_2 [(1-pP_1)(1-P_3)]^{n-2}\\\\&= \\left(\\sum \\nolimits _{k=1}^{\\infty }\\pi _{(k,0)}\\right) p(1-P_1)P_2 [(1-pP_1)(1-P_3)]^{n-2} $ Since all the probabilities $\\pi _{(n,m)}$ have to add up to 1, we have the relation $1&=\\sum \\nolimits _{n=1}^{\\infty }\\pi _{(n,0)} + \\sum \\nolimits _{n=m+1}^{\\infty }\\sum \\nolimits _{m=1}^{\\infty }\\pi _{(n,m)} \\\\&=\\sum \\nolimits _{n=1}^{\\infty }\\pi _{(n,0)} + \\sum \\nolimits _{n=m+1}^{\\infty }\\sum \\nolimits _{m=1}^{\\infty }\\pi _{(n-m,0)} p(1-P_1)P_2[(1-pP_1)(1-P_3)]^{m-1} \\\\&=\\sum \\nolimits _{n=1}^{\\infty }\\pi _{(n,0)} + \\left(\\sum \\nolimits _{t=1}^{\\infty }\\pi _{(t,0)}\\right) p(1-P_1)P_2 \\sum \\nolimits _{m=1}^{\\infty } [(1-pP_1)(1-P_3)]^{m-1} \\\\&=\\left(\\sum \\nolimits _{k=1}^{\\infty }\\pi _{(k,0)}\\right)\\left[1 + \\frac{p(1-P_1)P_2}{1-(1-pP_1)(1-P_3)}\\right]$ To obtain (57), do the substitution $t=n-m$ .",
"From equation (58) we can derive that $\\sum \\nolimits _{k=1}^{\\infty }\\pi _{(k,0)}=\\frac{1-(1-pP_1)(1-P_3)}{1-(1-pP_1)(1-P_3)+p(1-P_1)P_2} $ Combining above results, for $n\\ge 2$ we can write that $\\pi _{(n,0)}=\\pi _{(n-1,0)}\\delta + \\xi \\left[(1-pP_1)(1-P_3)\\right]^{n-1}$ where $\\delta &=1-p(P_1+P_2-P_1P_2), \\\\\\xi &=\\frac{p(1-P_1)P_2P_3[1-(1-pP_1)(1-P_3)]}{[1-(1-pP_1)(1-P_3)+p(1-P_1)P_2](1-P_3)} $ Equation (59) gives a recursive relation of the probabilities $\\pi _{(n,0)}$ .",
"Using (59) iteratively yields that $\\pi _{(n,0)}&=\\pi _{(1,0)}\\delta ^{n-1} + \\xi \\sum \\nolimits _{j=0}^{n-2}\\delta ^j[(1-pP_1)(1-P_3)]^{n-1-j} \\\\&=\\pi _{(1,0)}\\delta ^{n-1} + \\frac{\\xi (1-pP_1)(1-P_3)}{(1-pP_1)(1-P_3)-\\delta } \\left\\lbrace [(1-pP_1)(1-P_3)]^{n-1} - \\delta ^{n-1}\\right\\rbrace \\\\&=\\beta _1 \\delta ^{n-1} + \\beta _2[(1-pP_1)(1-P_3)]^{n-1}$ Substituting $\\xi $ , it show that the coefficient $\\beta _2$ is equal to $\\beta _2&=\\frac{[1-(1-pP_1)(1-P_3)]p(1-P_1)P_2P_3}{1-(1-pP_1)(1-P_3)+p(1-P_1)P_2} \\frac{1-pP_1}{(1-pP_1)(1-P_3)-\\delta } $ Since $\\pi _{(1,0)}=pP_1$ , we have $\\beta _1=pP_1-\\beta _2$ .",
"It is easy to see that when $n=1$ , equation (61) reduces to $\\pi _{(1,0)}=\\pi _{(1,0)}$ , thus probability expression (61) is actually valid for all $n\\ge 1$ .",
"According to (56), for $n>m\\ge 1$ , the probability $\\pi _{(n,m)}$ is then determined as $\\pi _{(n,m)}&= \\pi _{(n-m,0)}p(1-P_1)P_2[(1-pP_1)(1-P_3)]^{m-1} \\\\={}& p(1-P_1)P_2\\left\\lbrace \\beta _1\\delta ^{n-m-1} + \\beta _2[(1-pP_1)(1-P_3)]^{n-m-1} \\right\\rbrace [(1-pP_1)(1-P_3)]^{m-1} \\\\={}& p(1-P_1)P_2 \\left\\lbrace \\beta _1\\delta ^{n-2}\\left(\\frac{(1-pP_1)(1-P_3)}{\\delta }\\right)^{m-1} + \\beta _2[(1-pP_1)(1-P_3)]^{n-2} \\right\\rbrace $ So far, we have completely solved the system of equations (55) and obtained all the stationary probabilities $\\pi _{(n,m)}$ ."
],
[
"Proof of Theorem 8",
"In this appendix, we determine all the stationary probabilities $\\pi _{(n,m,l)}$ of stochastic process $AoI_B^P$ by solving the following system of equations.",
"${\\left\\lbrace \\begin{array}{ll}\\pi _{(n,m,l)}=\\pi _{(n-1,m-1,l-1)}\\delta (1-P_3) & \\qquad (n>m>l\\ge 2) \\\\\\pi _{(n,m,1)}=\\left(\\sum \\nolimits _{j=0}^{m-2}\\pi _{(n-1,m-1,j)}\\right)\\eta (1-P_3) & \\qquad (n>m\\ge 2) \\\\\\pi _{(n,m,0)}=\\pi _{(n-1,m-1,0)}\\delta (1-P_3) +\\left(\\sum \\nolimits _{k=n}^{\\infty }\\pi _{(k,n-1,m-1)}\\right)\\delta P_3 & \\qquad (n>m\\ge 2) \\\\\\pi _{(n,1,0)}=\\pi _{(n-1,0,0)}\\eta +\\left(\\sum \\nolimits _{k=n}^{\\infty }\\sum \\nolimits _{j=0}^{n-2}\\pi _{(k,n-1,j)}\\right)\\eta P_3 & \\qquad (n\\ge 2) \\\\\\pi _{(n,0,0)}=\\pi _{(n-1,0,0)}\\delta +\\left(\\sum \\nolimits _{k=n}^{\\infty }\\pi _{(k,n-1,0)}\\right)\\delta P_3 & \\qquad (n\\ge 2) \\\\\\pi _{(1,0,0)}=\\Big (\\sum \\nolimits _{n=1}^{\\infty }\\pi _{(n,0,0)}+ \\sum \\nolimits _{n=2}^{\\infty }\\sum \\nolimits _{m=1}^{n-1}\\pi _{(n,m,0)}\\\\\\qquad \\qquad \\qquad \\qquad \\qquad + \\sum \\nolimits _{n=3}^{\\infty }\\sum \\nolimits _{m=2}^{n-1}\\sum \\nolimits _{l=1}^{m-1}\\pi _{(n,m,l)}\\Big )pP_1\\end{array}\\right.",
"}$ First of all, for $n>m>l\\ge 1$ , using first two lines of (63) we can obtain that $\\pi _{(n,m,l)}&= \\pi _{(n-1,m-1,l-1)}[\\delta (1-P_3)] \\\\{}& \\qquad \\qquad \\qquad \\vdots \\\\&= \\pi _{(n-l+1,m-l+1,1)}[\\delta (1-P_3)]^{l-1} \\\\&= \\left(\\sum \\nolimits _{j=0}^{m-l-1}\\pi _{(n-l,m-l,j)}\\right)\\eta (1-P_3)[\\delta (1-P_3)]^{l-1}$ Substituting expression (64), the sum in the third equation of (63) can be rewritten as $\\sum \\nolimits _{k=n}^{\\infty }\\pi _{(k,n-1,m-1)}&= \\sum \\nolimits _{k=n}^{\\infty }\\left(\\sum \\nolimits _{j=0}^{n-m-1}\\pi _{(k-m+1,n-m,j)} \\right) \\eta (1-P_3)[\\delta (1-P_3)]^{m-2} \\\\&= \\left( \\sum \\nolimits _{k=n-m+1}^{\\infty }\\sum \\nolimits _{j=0}^{n-m-1}\\pi _{(k,n-m,j)}\\right) \\eta (1-P_3)[\\delta (1-P_3)]^{m-2}\\\\&= t_{n-m}\\eta (1-P_3)[\\delta (1-P_3)]^{m-2}$ where we define $t_n= \\sum \\nolimits _{k=n+1}^{\\infty }\\sum \\nolimits _{j=0}^{n-1}\\pi _{(k,n,j)} \\qquad (n\\ge 1)$ which is actually the probability that the middle parameter equals $n$ .",
"Therefore, for $n>m\\ge 2$ we have $\\pi _{(n,m,0)}&=\\pi _{(n-1,m-1,0)}\\delta (1-P_3) + t_{n-m}\\eta (1-P_3)[\\delta (1-P_3)]^{m-2} \\delta P_3 \\\\&=\\pi _{(n-1,m-1,0)}\\delta (1-P_3) + t_{n-m}\\eta P_3[\\delta (1-P_3)]^{m-1}$ Equation (67) is a recursive formula of the probabilities $\\pi _{(n,m,0)}$ .",
"By repeatedly using (67), it shows that $\\pi _{(n,m,0)}&=\\pi _{(n-m+1,1,0)}[\\delta (1-P_3)]^{m-1} + (m-1)t_{n-m}\\eta P_3[\\delta (1-P_3)]^{m-1} \\\\&= \\left\\lbrace \\pi _{(n-m,0,0)}\\eta +t_{n-m}\\eta P_3\\right\\rbrace [\\delta (1-P_3)]^{m-1} + (m-1)t_{n-m}\\eta P_3[\\delta (1-P_3)]^{m-1} \\\\&=\\pi _{(n-m,0,0)}\\eta [\\delta (1-P_3)]^{m-1} + m t_{n-m}\\eta P_3[\\delta (1-P_3)]^{m-1}$ In (68), we have used the fourth line in (63) to represent $\\pi _{(n-m+1,1,0)}$ with $\\pi _{(n-m,0,0)}$ and $t_{n-m}$ .",
"Notice that equation (69) is valid for all $n>m\\ge 1$ .",
"Substituting equation (69), the summation within the fifth line of (63) can be transformed as follows.",
"$\\sum \\nolimits _{k=n}^{\\infty }\\pi _{(k,n-1,0)}&=\\sum \\nolimits _{k=n}^{\\infty }\\Big \\lbrace \\pi _{(k-n+1,0,0)}\\eta [\\delta (1-P_3)]^{n-2} +(n-1)t_{k-n+1}\\eta P_3[\\delta (1-P_3)]^{n-2} \\Big \\rbrace \\\\&= \\left(\\sum \\nolimits _{k=1}^{\\infty }\\pi _{(k,0,0)}\\right)\\eta [\\delta (1-P_3)]^{n-2} + \\left(\\sum \\nolimits _{k=1}^{\\infty }t_{k}\\right) \\eta P_3(n-1)[\\delta (1-P_3)]^{n-2} $ Since $t_n$ denotes the probability that the middle parameter equals $n$ , we have the relation $t_0 + \\sum \\nolimits _{n=1}^{\\infty }t_n =1 $ in which $t_0=\\sum \\nolimits _{k=1}^{\\infty }\\pi _{(k,0,0)}=S $ is the probability that the middle parameter takes value 0, which is represented by $S$ in following paragraphs.",
"Combining above results, for $n\\ge 2$ we have that $\\pi _{(n,0,0)}&=\\pi _{(n-1,0,0)}\\delta + \\Big \\lbrace S\\eta [\\delta (1-P_3)]^{n-2} + (1-S)\\eta P_3(n-1)[\\delta (1-P_3)]^{n-2}\\Big \\rbrace \\delta P_3$ which gives a recursive formula for $\\pi _{(n,0,0)}$ .",
"Before the general expression of $\\pi _{(n,0,0)}$ is given, we first determine $S$ .",
"From $n=2$ to $\\infty $ , adding up both sides of equation (70) yields that $S-\\pi _{(1,0,0)}=S\\delta +\\frac{S\\delta \\eta P_3}{1-\\delta (1-P_3)}+\\frac{(1-S)\\delta \\eta P_3^2}{[1-\\delta (1-P_3)]^2}$ Since we have known that $\\pi _{(1,0,0)}=pP_1$ , from (71) we can derive $S=\\frac{pP_1[1-\\delta (1-P_3)]^2+\\delta \\eta P_3^2}{(1-\\delta )[1-\\delta (1-P_3)]^2-\\delta \\eta P_3(1-\\delta )(1-P_3)}$ Applying equation (70) iteratively, we have $\\pi _{(n,0,0)}&=\\pi _{(1,0,0)}\\delta ^{n-1}+S\\delta \\eta P_3\\sum \\nolimits _{j=0}^{n-2}\\delta ^j[\\delta (1-P_3)]^{n-2-j} \\\\&\\qquad +(1-S)\\delta \\eta P_3^2\\sum \\nolimits _{j=0}^{n-2}\\delta ^j(n-1-j)[\\delta (1-P_3)]^{n-2-j} \\\\&= \\left(pP_1+\\eta \\right)\\delta ^{n-1}-\\eta [\\delta (1-P_3)]^{n-1} - (1-S)\\eta P_3(n-1)[\\delta (1-P_3)]^{n-1}$ in which we have substituted $\\pi _{(1,0,0)}=pP_1$ and omit some intermediate calculation details.",
"Equation (73) is valid for $n=1$ can be verified directly.",
"In order to determine probabilities $\\pi _{(n,m,0)}$ by (69), we next derive the general expression of $t_n$ .",
"For $n\\ge 2$ , it shows that $t_n&=\\sum \\nolimits _{k=n+1}^{\\infty }\\sum \\nolimits _{j=0}^{n-1}\\pi _{(k,n,j)} \\\\&=\\sum \\nolimits _{k=n+1}^{\\infty }\\pi _{(k,n,0)}+\\sum \\nolimits _{k=n+1}^{\\infty }\\sum \\nolimits _{j=1}^{n-1}\\pi _{(k,n,j)} \\\\&=\\sum \\nolimits _{k=n+1}^{\\infty }\\Big \\lbrace \\pi _{(k-n,0,0)}\\eta [\\delta (1-P_3)]^{n-1}+n t_{k-n}\\eta P_3[\\delta (1-P_3)]^{n-1}\\Big \\rbrace \\\\& \\qquad +\\sum \\nolimits _{k=n+1}^{\\infty }\\sum \\nolimits _{j=1}^{n-1}\\left(\\sum \\nolimits _{y=0}^{n-j-1}\\pi _{(k-j,n-j,y)}\\right)\\eta (1-P_3) [\\delta (1-P_3)]^{j-1} \\\\&=S\\eta [\\delta (1-P_3)]^{n-1}+(1-S)\\eta P_3n[\\delta (1-P_3)]^{n-1} \\\\&\\qquad + \\eta (1-P_3)\\sum \\nolimits _{j=1}^{n-1}t_{n-j}[\\delta (1-P_3)]^{j-1}$ where in (74) we have used equations (69) and (64).",
"To obtain (75), notice that $\\sum \\nolimits _{k=n+1}^{\\infty }\\sum \\nolimits _{y=0}^{n-j-1}\\pi _{(k-j,n-j,y)}=\\sum \\nolimits _{k=n-j+1}^{\\infty }\\sum \\nolimits _{y=0}^{n-j-1}\\pi _{(k,n-j,y)}= t_{n-j} $ Similarly, we also have $t_{n-1}&=S\\eta [\\delta (1-P_3)]^{n-2}+(1-S)\\eta P_3(n-1)[\\delta (1-P_3)]^{n-2} \\\\& \\quad + \\eta (1-P_3)\\sum \\nolimits _{j=1}^{n-2}t_{n-1-j}[\\delta (1-P_3)]^{j-1}$ Compute the following difference $t_n-t_{n-1}\\delta (1-P_3)=(1-S)\\eta P_3[\\delta (1-P_3)]^{n-1}+t_{n-1}\\eta (1-P_3) $ which is equivalent to $t_n=t_{n-1}(\\delta +\\eta )(1-P_3)+(1-S)\\eta P_3[\\delta (1-P_3)]^{n-1}$ Applying equation (77) repeatedly yields that $t_n&=t_1 [(\\delta +\\eta )(1-P_3)]^{n-1} +(1-S)\\eta P_3 \\sum \\nolimits _{j=0}^{n-2}[(\\delta +\\eta )(1-P_3)]^j[\\delta (1-P_3)]^{n-1-j} \\\\&=t_1 [(\\delta +\\eta )(1-P_3)]^{n-1} +(1-S)\\delta P_3 \\left\\lbrace [(\\delta +\\eta )(1-P_3)]^{n-1} - [\\delta (1-P_3)]^{n-1} \\right\\rbrace $ Using the general expression (69), we have $t_1=\\sum \\nolimits _{k=2}^{\\infty }\\pi _{(k,1,0)} =\\sum \\nolimits _{k=2}^{\\infty } \\left\\lbrace \\pi _{(k-1,0,0)}\\eta + t_{k-1}\\eta P_3 \\right\\rbrace =S\\eta +(1-S)\\eta P_3$ Combining (78) and (79), eventually we derive that $t_n&=\\left\\lbrace S\\eta +(1-S)(\\delta +\\eta )P_3\\right\\rbrace [(\\delta +\\eta )(1-P_3)]^{n-1} - (1-S)\\delta P_3[\\delta (1-P_3)]^{n-1} \\\\&=\\widetilde{S}[(\\delta +\\eta )(1-P_3)]^{n-1} - (1-S)\\delta P_3[\\delta (1-P_3)]^{n-1}$ where we denote $\\widetilde{S}=S\\eta +(1-S)(\\delta +\\eta )P_3 $ Provided equations (73) and (80), now the probabilities $\\pi _{(n,m,0)}$ can be determined by (69).",
"$\\pi _{(n,m,0)}&= \\pi _{(n-m,0,0)}\\eta [\\delta (1-P_3)]^{m-1}+ m t_{n-m}\\eta P_3[\\delta (1-P_3)]^{m-1} \\\\&=\\bigg \\lbrace \\left(pP_1+\\eta \\right)\\delta ^{n-m-1}-\\eta [\\delta (1-P_3)]^{n-m-1} \\\\& \\quad -(1-S)\\eta P_3 (n-m-1)[\\delta (1-P_3)]^{n-m-1}\\bigg \\rbrace \\eta [\\delta (1-P_3)]^{m-1} \\\\& \\quad +\\left\\lbrace \\widetilde{S}[(\\delta +\\eta )(1-P_3)]^{n-m-1} - (1-S)\\delta P_3 [\\delta (1-P_3)]^{n-m-1} \\right\\rbrace \\eta P_3 m[\\delta (1-P_3)]^{m-1} \\\\&= \\eta (pP_1+\\eta )\\delta ^{n-2}(1-P_3)^{m-1}-\\eta ^2[\\delta (1-P_3)]^{n-2} - (1-S)\\eta ^2 P_3(n-m-1)[\\delta (1-P_3)]^{n-2} \\\\& \\quad + \\widetilde{S}\\eta P_3[(\\delta +\\eta )(1-P_3)]^{n-2}m\\left(\\frac{\\delta }{\\delta +\\eta }\\right)^{m-1} - (1-S)\\delta \\eta P_3^2 m[\\delta (1-P_3)]^{n-2}$ At last, the probabilities $\\pi _{(n,m,l)}$ , $n>m>l\\ge 1$ remains to be determined.",
"Since $\\pi _{(n,m,l)}=\\pi _{(n-l+1,m-l+1,1)}[\\delta (1-P_3)]^{l-1} $ thus it suffices to find all the probabilities $\\pi _{(n,m,1)}$ for all the tuples $(n,m)$ .",
"From the general formula (64), we have $\\pi _{(n,m,1)}&=\\left(\\sum \\nolimits _{j=0}^{m-2}\\pi _{(n-1,m-1,j)}\\right)\\eta (1-P_3) \\\\&=\\left(\\pi _{(n-1,m-1,0)}+\\sum \\nolimits _{j=1}^{m-2}\\pi _{(n-1,m-1,j)}\\right)\\eta (1-P_3) \\\\&=\\pi _{(n-1,m-1,0)}\\eta (1-P_3) + \\left(\\sum \\nolimits _{j=1}^{m-2}\\pi _{(n-j,m-j,1)}[\\delta (1-P_3)]^{j-1}\\right)\\eta (1-P_3) $ Do once iteration, we can obtain the expansion expression of probability $\\pi _{(n-1,m-1,1)}$ .",
"The difference $&\\pi _{(n,m,1)}-\\pi _{(n-1,m-1,1)}\\delta (1-P_3) \\\\={}& \\left\\lbrace \\pi _{(n-1,m-1,0)}-\\pi _{(n-2,m-2,0)}\\delta (1-P_3)\\right\\rbrace \\eta (1-P_3) +\\pi _{(n-1,m-1,1)}\\eta (1-P_3) \\\\={}&t_{n-m}\\eta ^2 P_3(1-P_3)[\\delta (1-P_3)]^{m-2}+\\pi _{(n,m,1)}\\eta (1-P_3)$ gives the following recursive relation $\\pi _{(n,m,1)}=\\pi _{(n-1,m-1,1)}(\\delta +\\eta )(1-P_3) + t_{n-m}\\eta ^2 P_3(1-P_3)[\\delta (1-P_3)]^{m-2}$ In (82), we have substituted the probability expression (69).",
"Then, the general formula of $\\pi _{(n,m,1)}$ can be derived by applying (83) repeatedly.",
"It shows that $\\pi _{(n,m,1)}&=\\pi _{(n-m+2,2,1)}[(\\delta +\\eta )(1-P_3)]^{m-2} \\\\&\\quad + \\eta ^2P_3(1-P_3)t_{n-m}\\sum \\nolimits _{j=0}^{m-3}[(\\delta +\\eta )(1-P_3)]^j[\\delta (1-P_3)]^{m-2-j} \\\\&=\\pi _{(n-m+2,2,1)}[(\\delta +\\eta )(1-P_3)]^{m-2} \\\\&\\quad + \\delta \\eta P_3(1-P_3)t_{n-m} \\left\\lbrace [(\\delta +\\eta )(1-P_3)]^{m-2}-[\\delta (1-P_3)]^{m-2}\\right\\rbrace $ in which $\\pi _{(n-m+2,2,1)}&= \\pi _{(n-m+1,1,0)}\\eta (1-P_3) \\\\&=\\left\\lbrace \\pi _{(n-m,0,0)}\\eta +t_{n-m}\\eta P_3\\right\\rbrace \\eta (1-P_3) \\\\&=\\pi _{(n-m,0,0)}\\eta ^2(1-P_3) + t_{n-m}\\eta ^2 P_3(1-P_3)$ Combining equations (85) and (84) and merging the same terms gives $\\pi _{(n,m,1)}&=\\pi _{(n-m,0,0)}\\eta ^2(1-P_3)[(\\delta +\\eta )(1-P_3)]^{m-2} \\\\&\\qquad + \\eta P_3 t_{n-m} \\left\\lbrace [(\\delta +\\eta )(1-P_3)]^{m-1}-[\\delta (1-P_3)]^{m-1}\\right\\rbrace $ Since both $\\pi _{(n-m,0,0)}$ and $t_{n-m}$ have been obtained, by substituting (73) and (80), we show that the final result is $\\pi _{(n,m,1)}&= \\eta ^2(1-P_3)\\bigg \\lbrace (pP_1+\\eta )\\delta ^{n-3}\\left(\\frac{(\\delta +\\eta )(1-P_3)}{\\delta }\\right)^{m-2} - \\eta [\\delta (1-P_3)]^{n-3}\\left(\\frac{\\delta +\\eta }{\\delta }\\right)^{m-2} \\\\& \\quad -(1-S)\\eta P_3(n-m-1)[\\delta (1-P_3)]^{n-3}\\left(\\frac{\\delta +\\eta }{\\delta }\\right)^{m-2} \\bigg \\rbrace \\\\& \\quad +\\eta P_3\\widetilde{S}[(\\delta +\\eta )(1-P_3)]^{n-2} - \\eta P_3\\widetilde{S}[(\\delta +\\eta )(1-P_3)]^{n-2}\\left(\\frac{\\delta }{\\delta +\\eta }\\right)^{m-1} \\\\& \\quad - \\delta \\eta P_3^2(1-S)[\\delta (1-P_3)]^{n-2}\\left(\\frac{\\delta +\\eta }{\\delta }\\right)^{m-1} + \\delta \\eta P_3^2(1-S)[\\delta (1-P_3)]^{n-2}$ For $n>m>l\\ge 1$ , the last step is $\\pi _{(n,m,l)}&=\\pi _{(n-l+1,m-l+1,1)}[\\delta (1-P_3)]^{l-1} \\\\&=\\eta ^2(1-P_3)\\bigg \\lbrace \\left(pP_1+\\eta \\right)\\delta ^{n-3}\\left(\\frac{(\\delta +\\eta )(1-P_3)}{\\delta }\\right)^{m-2} \\left(\\frac{\\delta }{\\delta +\\eta }\\right)^{l-1} \\\\& \\qquad -\\eta [\\delta (1-P_3)]^{n-3}\\left(\\frac{\\delta +\\eta }{\\delta }\\right)^{m-2} \\left(\\frac{\\delta }{\\delta +\\eta }\\right)^{l-1} -(1-S)\\eta P_3(n-m-1) \\\\& \\qquad \\times [\\delta (1-P_3)]^{n-3}\\left(\\frac{\\delta +\\eta }{\\delta }\\right)^{m-2}\\left(\\frac{\\delta }{\\delta +\\eta }\\right)^{l-1} \\bigg \\rbrace \\\\& \\quad + \\eta P_3\\widetilde{S}[(\\delta +\\eta )(1-P_3)]^{n-2}\\left(\\frac{\\delta }{\\delta +\\eta }\\right)^{l-1} - \\eta P_3\\widetilde{S}[(\\delta +\\eta )(1-P_3)]^{n-2}\\left(\\frac{\\delta }{\\delta +\\eta }\\right)^{m-1} \\\\&\\quad -\\delta \\eta P_3^2(1-S)[\\delta (1-P_3)]^{n-2}\\left(\\frac{\\delta +\\eta }{\\delta }\\right)^{m-1} \\left(\\frac{\\delta }{\\delta +\\eta }\\right)^{l-1} \\\\&\\quad + \\delta \\eta P_3^2(1-S)[\\delta (1-P_3)]^{n-2}$ Collecting the results in equations (73), (81) and (87), we show that all the stationary probabilities of stochastic process $AoI_B^P$ are determined.",
"This completes the proof of Theorem 8."
],
[
"Proof of Theorem 9",
"The stationary distribution of AoI, $\\Delta _B^P$ , is calculated in this Appendix.",
"According to formula (15), for $n\\ge 3$ , we have $\\Pr \\lbrace \\Delta _B^P=n\\rbrace = \\pi _{(n,0,0)}+\\sum \\nolimits _{m=1}^{n-1}\\pi _{(n,m,0)}+\\sum \\nolimits _{m=2}^{n-1}\\sum \\nolimits _{l=1}^{m-1}\\pi _{(n,m,l)} $ where $\\pi _{(n,0,0)}$ is given in (16).",
"Substituting expression (17), the first sum is computed as $&\\sum \\nolimits _{m=1}^{n-1}\\pi _{(n,m,0)} \\\\={}&\\sum \\nolimits _{m=1}^{n-1}\\Big \\lbrace \\eta \\left(pP_1+\\eta \\right)\\delta ^{n-2}\\left(1-P_3\\right)^{m-1}-\\eta ^2\\left[\\delta (1-P_3)\\right]^{n-2} - (1-S)\\eta ^2P_3(n-m-1) \\\\{}& \\times \\left[\\delta (1-P_3)\\right]^{n-2} +\\widetilde{S}\\eta P_3 \\left[(\\delta +\\eta )(1-P_3)\\right]^{n-2} m \\left(\\frac{\\delta }{\\delta +\\eta }\\right)^{m-1} - (1-S)\\delta \\eta P_3^2 m[\\delta (1-P_3)]^{n-2} \\Big \\rbrace \\\\={}&\\eta \\left(pP_1+\\eta \\right)\\delta ^{n-2} \\frac{1-(1-P_3)^{n-1}}{P_3} - \\eta ^2(n-1)\\left[\\delta (1-P_3)\\right]^{n-2} - (1-S)\\eta ^2P_3 \\\\{}& \\times \\frac{(n-1)(n-2)}{2}\\left[\\delta (1-P_3)\\right]^{n-2} +\\widetilde{S}\\eta P_3 \\left[(\\delta +\\eta )(1-P_3)\\right]^{n-2}\\frac{(\\delta +\\eta )^2}{\\eta ^2} \\\\{}&\\times \\left[1- \\left(\\frac{\\delta }{\\delta +\\eta }\\right)^{n-1}- \\frac{\\eta }{\\delta +\\eta }(n-1) \\left(\\frac{\\delta }{\\delta +\\eta }\\right)^{n-1} \\right] -(1-S)\\delta \\eta P_3^2 \\frac{n(n-1)}{2}[\\delta (1-P_3)]^{n-2}$ We omit the further calculations and directly show that $\\sum \\nolimits _{m=1}^{n-1}\\pi _{(n,m,0)}&=\\frac{\\eta (pP_1+\\eta )}{\\delta P_3}\\delta ^{n-1} + \\frac{\\widetilde{S}(\\delta +\\eta )P_3}{\\eta (1-P_3)}[(\\delta +\\eta )(1-P_3)]^{n-1} \\\\&\\qquad - \\left\\lbrace \\frac{\\eta (pP_1+\\eta )}{\\delta P_3}+\\frac{\\widetilde{S}\\delta P_3}{\\eta (1-P_3)}-\\frac{\\eta ^2-(1-S)\\eta ^2 P_3}{\\delta (1-P_3)}\\right\\rbrace [\\delta (1-P_3)]^{n-1} \\\\&\\qquad - \\left\\lbrace \\frac{\\eta ^2-(1-S)\\eta ^2P_3}{\\delta (1-P_3)}+\\frac{\\widetilde{S}\\eta P_3}{\\eta (1-P_3)} \\right\\rbrace n[\\delta (1-P_3)]^{n-1} \\\\&\\qquad -\\frac{(1-S)\\eta P_3(\\delta P_3+\\eta )}{2\\delta (1-P_3)}n(n-1)[\\delta (1-P_3)]^{n-1}$ At last, $&\\sum \\nolimits _{m=2}^{n-1}\\sum \\nolimits _{l=1}^{m-1}\\pi _{(n,m,l)} \\\\={}&\\sum \\nolimits _{m=2}^{n-1}\\sum \\nolimits _{l=1}^{m-1} \\text{Equation (18)} \\\\={}&\\sum \\nolimits _{m=2}^{n-1}\\Bigg \\lbrace \\eta (pP_1+\\eta )\\delta ^{n-2} \\left[\\left(\\frac{(\\delta +\\eta )(1-P_3)}{\\delta }\\right)^{m-1}-(1-P_3)^{m-1}\\right] \\\\{}&\\qquad - \\eta ^2[\\delta (1-P_3)]^{n-2}\\left[\\left(\\frac{\\delta +\\eta }{\\delta }\\right)^{m-1} -1 \\right] \\\\{}&\\qquad \\quad -(1-S)\\eta ^2 P_3 [\\delta (1-P_3)]^{n-2} (n-m-1)\\left[\\left(\\frac{\\delta +\\eta }{\\delta }\\right)^{m-1} -1\\right] \\Bigg \\rbrace \\\\{}&\\quad +\\widetilde{S}(\\delta +\\eta )P_3[(\\delta +\\eta )(1-P_3)]^{n-2}\\left[1-\\left(\\frac{\\delta }{\\delta +\\eta }\\right)^{m-1} \\right] \\\\{}&\\quad -\\widetilde{S}\\eta P_3[(\\delta +\\eta )(1-P_3)]^{n-2}(m-1)\\left(\\frac{\\delta }{\\delta +\\eta }\\right)^{m-1} \\\\{}&\\quad -(1-S)\\delta (\\delta +\\eta )P_3^2[\\delta (1-P_3)]^{n-2}\\left[\\left(\\frac{\\delta +\\eta }{\\delta }\\right)^{m-1}-1 \\right] \\\\{}&\\quad +(1-S)\\delta \\eta P_3^2(m-1)[\\delta (1-P_3)]^{n-2} \\\\={}&\\eta (pP_1+\\eta )\\delta ^{n-2} \\bigg [ \\frac{\\eta (1-P_3)}{[\\delta -(\\delta +\\eta )(1-P_3)]P_3} \\\\{}& \\quad - \\frac{\\delta }{\\delta -(\\delta +\\eta )(1-P_3)} \\left(\\frac{(\\delta +\\eta )(1-P_3)}{\\delta }\\right)^{n-1} +\\frac{(1-P_3)^{n-1}}{P_3}\\bigg ] \\\\{}&-\\eta ^2[\\delta (1-P_3)]^{n-2}\\left[ \\frac{\\delta }{\\eta }\\left(\\frac{\\delta +\\eta }{\\delta }\\right)^{n-1}-\\frac{\\delta +\\eta }{\\eta }-(n-2) \\right] \\\\{}&-(1-S)\\eta ^2 P_3 [\\delta (1-P_3)]^{n-2}\\bigg [-(n-2)\\frac{\\delta +\\eta }{\\eta } \\\\{}& \\quad + \\frac{\\delta ^2}{\\eta ^2}\\left(\\frac{\\delta +\\eta }{\\delta }\\right)^{n-1}-\\frac{\\delta (\\delta +\\eta )}{\\eta ^2} -\\frac{(n-2)(n-3)}{2}\\bigg ] \\\\{}&+\\widetilde{S}(\\delta +\\eta )P_3[(\\delta +\\eta )(1-P_3)]^{n-2} \\left[(n-2)-\\frac{\\delta }{\\eta }+\\frac{\\delta +\\eta }{\\eta } \\left( \\frac{\\delta }{\\delta +\\eta }\\right)^{n-1} \\right] \\\\{}&-\\widetilde{S}\\eta P_3[(\\delta +\\eta )(1-P_3)]^{n-2}\\bigg [ \\frac{\\delta (\\delta +\\eta )}{\\eta ^2} -\\frac{(\\delta +\\eta )^2}{\\eta ^2} \\left(\\frac{\\delta }{\\delta +\\eta }\\right)^{n-1}-\\frac{\\delta +\\eta }{\\eta }(n-2)\\left(\\frac{\\delta }{\\delta +\\eta }\\right)^{n-1} \\bigg ] \\\\{}&-(1-S)\\delta (\\delta +\\eta )P_3^2[\\delta (1-P_3)]^{n-2} \\left[ \\frac{\\delta }{\\eta }\\left(\\frac{\\delta +\\eta }{\\delta }\\right)^{n-1}-\\frac{\\delta +\\eta }{\\eta }-(n-2) \\right] \\\\{}&+(1-S)\\delta \\eta P_3^2\\frac{(n-1)(n-2)}{2}[\\delta (1-P_3)]^{n-2}$ Collecting all the results obtained in (16), (89) and (90), after careful calculations and numerical verification, we derive that $&\\Pr \\lbrace \\Delta _B^P=n\\rbrace \\\\={}& \\frac{(\\delta +\\eta )P_3(pP_1+\\eta )}{\\delta -(\\delta +\\eta )(1-P_3)}\\delta ^{n-1} +\\frac{P_3[\\eta (\\delta +\\eta )+(1-S)(\\delta +\\eta )P_3(2\\delta -\\eta )]}{\\eta (1-P_3)}[\\delta (1-P_3)]^{n-1} \\\\{}& -\\bigg \\lbrace \\frac{pP_1\\eta (1-P_3)+\\delta \\eta P_3}{[\\delta -(\\delta +\\eta )(1-P_3)](1-P_3)} +\\frac{(1-S)\\delta P_3[\\eta +(\\delta +\\eta )P_3]+\\widetilde{S}(\\delta +\\eta )P_3}{\\eta (1-P_3)} \\bigg \\rbrace [(\\delta +\\eta )(1-P_3)]^{n-1} \\\\{}& +\\frac{(1-S)(\\delta +\\eta )P_3^2}{1-P_3}n[\\delta (1-P_3)]^{n-1} +\\frac{\\widetilde{S}P_3}{1-P_3}n[(\\delta +\\eta )(1-P_3)]^{n-1} \\qquad (n\\ge 3)$ It can be checked directly that equation (89) is zero when $n=1$ , and (90) equals zero for both cases $n=1$ and $n=2$ .",
"Therefore, the probability distribution (91) is in fact valid for all $n\\ge 1$ .",
"This completes the proof of Theorem 9."
],
[
"Proof of Corollary 1",
"Notice that for $0<x<1$ and an arbitrary integer $k\\ge 1$ , we have $\\sum \\nolimits _{n=1}^{\\infty }n(n-1)\\cdots (n-k+1)x^{n-k}=\\frac{k!",
"}{(1-x)^{k+1}}$ Then, according to equation (5), the mean of AoI $\\Delta _{NB}^P$ is given as $\\mathbb {E}[\\Delta _{NB}^P]&=\\sum \\nolimits _{n=1}^{\\infty }n\\Pr \\lbrace \\Delta =n\\rbrace \\\\&=\\sum \\nolimits _{n=1}^{\\infty }\\bigg \\lbrace \\frac{(1-pP_1)P_3(1-\\delta )}{(1-pP_1)P_3-p(1-P_1)P_2}n\\delta ^{n-1} + \\xi n[(1-pP_1)(1-P_3)]^{n-1} \\bigg \\rbrace \\\\&=\\frac{(1-pP_1)P_3}{[(1-pP_1)P_3-p(1-P_1)P_2](1-\\delta )} + \\frac{\\xi }{[1-(1-pP_1)(1-P_3)]^2}$ From AoI distribution (11), we can obtain that $\\mathbb {E}[\\Delta _{NB}^{NP}]&=\\left(\\beta _1 + \\frac{p(1-P_1)P_2}{\\delta -(1-pP_1)(1-P_3)}\\beta _1\\right) \\frac{1}{(1-\\delta )^2} \\\\& \\quad + \\left(\\beta _2 - \\frac{p(1-P_1)P_2}{\\delta -(1-pP_1)(1-P_3)}\\beta _1\\right) \\frac{1}{[1-(1-pP_1)(1-P_3)]^2}+\\frac{2p(1-P_1)P_2\\beta _2}{[1-(1-pP_1)(1-P_3)]^3} \\\\&=\\frac{\\beta _1}{(1-\\delta )^2}+\\frac{\\beta _2}{[1-(1-pP_1)(1-P_3)]^2} \\\\& \\quad + \\frac{p(1-P_1)P_2\\beta _1[2-\\delta -(1-pP_1)(1-P_3)]}{(1-\\delta )^2 [1-(1-pP_1)(1-P_3)]^2} + \\frac{2p(1-P_1)P_2\\beta _2}{[1-(1-pP_1)(1-P_3)]^3}$ Finally, the average value of AoI $\\Delta _B^P$ can be calculated using distribution expression (21).",
"We show that $\\mathbb {E}[\\Delta _B^P]&=\\frac{(\\delta +\\eta )P_3(pP_1+\\eta )}{\\delta -(\\delta +\\eta )(1-P_3)}\\frac{1}{(1-\\delta )^2} -\\frac{c_1}{[1-(\\delta +\\eta )(1-P_3)]^2} + \\frac{c_2}{[1-\\delta (1-P_3)]^2} \\\\& \\qquad + \\frac{(1-S)(\\delta +\\eta )P_3^2}{1-P_3}\\frac{1+\\delta (1-P_3)}{[1-\\delta (1-P_3)]^3} + \\frac{P_3\\widetilde{S}}{1-P_3}\\frac{1+(\\delta +\\eta )(1-P_3)}{[1-(\\delta +\\eta )(1-P_3)]^3} \\\\&=\\frac{(\\delta +\\eta )P_3(pP_1+\\eta )}{[\\delta -(\\delta +\\eta )(1-P_3)](1-\\delta )^2}-\\frac{c_1}{[1-(\\delta +\\eta )(1-P_3)]^2} +\\frac{c_2}{[1-\\delta (1-P_3)]^2} \\\\& \\qquad +\\frac{(1-S)(\\delta +\\eta )P_3^2[1+\\delta (1-P_3)]}{(1-P_3)[1-\\delta (1-P_3)]^3} +\\frac{P_3\\widetilde{S}[1+(\\delta +\\eta )(1-P_3)]}{(1-P_3)[1-(\\delta +\\eta )(1-P_3)]^3}$ where in (96) observing that $\\sum \\nolimits _{n=1}^{\\infty }n^2[\\delta (1-P_3)]^{n-1} &= \\sum \\nolimits _{n=1}^{\\infty }[n(n-1)+n][\\delta (1-P_3)]^{n-1} \\\\&=\\sum \\nolimits _{n=1}^{\\infty }n(n-1)[\\delta (1-P_3)]^{n-1} + \\sum \\nolimits _{n=1}^{\\infty }n[\\delta (1-P_3)]^{n-1} \\\\&=\\delta (1-P_3)\\frac{2}{[1-\\delta (1-P_3)]^3} + \\frac{1}{[1-\\delta (1-P_3)]^2} \\\\&= \\frac{1+\\delta (1-P_3)}{[1-\\delta (1-P_3)]^3} $ Similarly, we have $\\sum \\nolimits _{n=1}^{\\infty }n^2[(\\delta +\\eta )(1-P_3)]^{n-1} = \\frac{1+(\\delta +\\eta )(1-P_3)}{[1-(\\delta +\\eta )(1-P_3)]^3} $ All the parameters can be found in previous Theorem 3, Theorem 6 and Theorem 9.",
"This completes the proof of this Corollary."
]
] | 2107.01833 |
[
[
"Kelvin-Helmholtz instability in strongly coupled dusty plasma with\n rotational shear flows and tracer transport"
],
[
"Abstract Kelvin-Helmholtz (KH) instability plays a significant role in transport and mixing properties of any medium.",
"In this paper, we numerically explore this instability for a two-dimensional strongly coupled dusty plasma with rotational shear flows.",
"We study this medium using generalized hydrodynamic fluid model which treats it as viscoelastic fluid.",
"We consider the specific cases of rotating vorticity with abrupt radial profiles of rotation.",
"In particular: single-circulation, and multi-circulation vorticity shell profiles have been chosen.",
"We observe the KH vortices at each circular interface between two relative rotating flows along with a pair of ingoing and outgoing wavefronts of transverse shear waves.",
"Our studies show that due to the interplay between KH vortices and shear waves in the strongly coupled medium, the mixing and transport behaviour are much better than inviscid hydrodynamic fluids.",
"In interests of substantiating the mixing and transport behaviour, the generalized hydrodynamic fluid model is extended to include the Lagrangian tracer particles.",
"The numerical dispersion of these tracer particles in a flow provides an estimate of the diffusion in such a medium.",
"We present the preliminary observations of tracers distribution (cluster formation) and their diffusion (mean square displacement) across the medium."
],
[
"Introduction",
"The Kelvin-Helmholtz (KH) instability has been ubiquitously observed in hydrodynamic fluids [1], [2], plasmas [3], geophysical flows [4], and astrophysical situations [5].",
"This instability occurs in form of vortices at the interface between two flows in the presence of a velocity shear.",
"The interactions between these KH vortices govern the transport processes like mixing and diffusion [6].",
"The main objectives of this study are to explore the formation and evolution of KH vortices of a rotating dusty plasma and quantifying the mixing they generate using the tracer particles simulation.",
"The formation and evolution of these vortices depend on the shear and nature of the medium (here the viscoelasticity of the system).",
"The KH instability for dust flows has also been investigated theoretically [7], [8], [9], [10], [11] and numerically [12], [13], [14], [15] as well experimentally [16] for planar sheared flows.",
"The rotating vortex flows have been studied considerably [17], [18], [19], [20], but to our knowledge, no prior studies have explicitly examined the KH instability for such flows, except [21], [22].",
"Dharodi $et~al.$ [21] numerically studied shearless smooth rotating flows to avoid KH destabilization and also considered sharp rotating flows where KH arises in homogeneous medium.",
"Dharodi in [22] has explored these rotating sheared flows in heterogeneous medium.",
"A dusty plasma can exist in strong coupling state quite easily because of high charged dust particles, which is called strongly coupled dusty plasma (SCDP).",
"The SCDP has been modelled under the formalism of generalized hydrodynamic (GHD) fluid model.",
"This model treats the SCDP as a VE fluid and characterizes its VE effects through the coupling strength parameter which is often measured as the ratio ${\\eta /\\tau _m}$ , the coupling parameters $\\eta $ and $\\tau _m$ are the shear viscosity and the Maxwell relaxation time, respectively.",
"Thus, here, the effects of viscoelasticity on KH vortices are observed by varying the ratio ${\\eta /\\tau _m}$.",
"We consider the incompressible limit of GHD model which in addition to the evolution of hydrodynamic KH instability, also supports transverse shear (TS) waves that propagate at phase velocity $\\sqrt{\\eta /\\tau _m}$ [23], [21], [24].",
"These propagating shear waves have the same symmetry as that of their source structure, until there is no boundary effect or no interaction with other waves or obstacle like vortex.",
"Since our interest is in KH instability of rotating SCDPs, we consider the specific cases of sharp vorticity patches: (i) single-circulation, and (ii) multi-circulation vorticity shell profiles.",
"The single-circulation case has already been somewhat discussed in [21], in the present paper we explore it in more detail.",
"We observe the KH instability at each circular interface between two relative rotating flows in form of small vortices along with a pair of ingoing and outgoing wavefronts of TS waves.",
"The interactions between interacting KH vortices and TS waves help the VE fluid in better mixing than standard HD fluids where the only interactions between the KH vortices take place.",
"To substantiate this observation, the passive tracers are dispersed throughout the medium.",
"In the context of fluid mechanics the tracer transport has been studied extensively for flow visualization [25], [26] with the help of theoretical [27], [28], [29], [30], computational [31], [32], [33], [34], [35], [36], [37], [38], [39], [40], [41], [42] and experimental [43], [44], [45], [46], [47] approaches.",
"This technique is also used in complex fluids (polymers, colloids and biological materials) [48], [49].",
"An analysis of the separation of the particle trajectories with the 2D hydrodynamic fluid is also being carried out by Falkovich et al.",
"[50].",
"In dusty plasma, Schwabe et al.",
"[51] observed the vortex movements by adding some micro particles around the void.",
"Here, we consider two kinds of point-like tracer particles, (i) non-inertial, and (ii) inertial tracer particles.",
"The tracer dynamics is simulated using a one-way coupled Lagrangian point-particle approach [52], [53] which means the tracers are affected by the fluid flow, but not vice-versa.",
"In case of multi-circulation vortex profiles, at intermediate time range, a complete picture of a turbulent flow is observed which has a collection of small KH vortices and waves.",
"When the system is left for a very long time, it ultimately settles down to a single vortex faster than in HD fluid.",
"It is observed that the relaxing rate of such turbulent medium increases with the increasing coupling strength.",
"This paper has been organized as follows.",
"In section , the GHD model especially developed for the study of SCDPs is extended to include the transport of passive Lagrangian tracers.",
"This extended model is referred as incompressible Generalized Hydrodynamic Tracer Transport (i-GHTT) model.",
"Section presents the numerical procedure in order to solve the set of equations of GHTT model.",
"In Sec.",
", we numerically explore the evolution of different types of sharp rotating vorticities in VE fluids and quantifying the mixing they generate using the tracer particles simulations.",
"Finally, Section contains the discussion and the conclusions."
],
[
"Generalized Hydrodynamic Tracer Transport (GHTT) Model",
"A dusty plasma can be prepared or found as a strongly coupled dusty plasma (SCDP) rather easily because of high charge on the micron-sized dust particles.",
"Below the crystallization limit, a SCDP behaves like a viscoelastic fluid which favors both the incompressible transverse shear modes and the compressible longitudinal modes.",
"To study such SCDP the generalized hydrodynamic (GHD) fluid model is found to be quite suitable which takes into account both types of modes.",
"To scrutinize the effect of transverse modes and to abate the possible pairing with the longitudinal mode, we consider the incompressible limit of GHD (i-GHD) model.",
"Thus, i-GHD model represents the incompressible SCDPs which support transverse modes only.",
"The momentum and continuity equations for i-GHD of homogeneous strongly coupled dusty plasma can be written as: $\\left[1 + \\tau _m \\left(\\frac{\\partial }{\\partial t}+\\vec{v}_d \\cdot \\nabla \\right)\\right] \\left[{\\frac{\\partial \\vec{v}_d }{\\partial t}+\\vec{v}_d \\cdot \\nabla \\vec{v}_d } + \\frac{\\nabla P}{n_d} - {\\nabla \\phi _d} \\right] = {\\eta }{\\nabla ^2 }{\\vec{v}_d}{,}$ and $\\nabla \\cdot \\vec{v}_d= 0{,}$ respectively.",
"$n_d$ is the number density which is normalized by its respective equilibrium value $n_{d0}$ .",
"The scalar potential $\\phi _d$ in the dusty plasma system is normalized by ${{K_B}{T_i}}/{e}$ .",
"The parameters $e$ , $T_i$ and $K_B$ are the electronic charge, ion temperature and Boltzmann constant, respectively.",
"The charge fluctuation over each dust grain has been ignored.",
"The time and length are normalized by inverse of dust plasma frequency $\\omega ^{-1}_{pd}=\\left({4\\pi (Z_de)^{2}n_{d0}}/{m_{d0}}\\right)^{-1/2}$ , plasma Debye length $\\lambda _{d}=\\left({K_B T_i}/{4{\\pi } {Z_d}{n_{d0}}{e^2}}\\right)^{1/2}$ , respectively.",
"In incompressible limit the Poisson equation has been replaced by the quasineutrality condition.",
"The dust fluid velocity $\\vec{v}_d$ is normalized by ${\\lambda _d}{\\omega _{pd}}$ .",
"The term ${\\tau _{m}}({\\vec{v}_d}{\\cdot }{\\nabla })$ in the generalized momentum equation is responsible for introducing the collective behavior in the medium.",
"When this term becomes zero ($\\tau _m$ =0), the momentum equation becomes Navier-Stokes equation.",
"In other words, the GHD model turns into a standard hydrodynamic fluid model.",
"Moreover, the presence of this term conserves the Gallilean invariance [54].",
"In our earlier manuscript [21], we proposed an idea of extending the GHD model by including the passive tracer particles in interest to estimate of the diffusion in a medium.",
"In order to accomplish this, we consider two kinds of point-like tracer particles, (i) non-inertial tracers, and (ii) inertial tracers.",
"The non-inertial tracers follow the flow exactly while the velocity of inertial tracers differs from flow velocity due to viscous drag force [55].",
"The tracer particles dynamics is simulated using a one-way coupled Lagrangian point-particle approach.",
"One way coupled particle approach means the tracers do not affect the fluid motion, in other words the tracers are passive.",
"We also neglect any kind of interaction between particles and gravity effect on their dynamics.",
"We further assume that the inertial tracer particles with density $\\rho _p$ different than the density $\\rho _d$ of the fluid.",
"Under these assumptions, the particles are transported in the flow according to the equations: $\\frac{d{\\vec{v}_{pi}}}{dt} &= \\frac{1}{\\tau _s}({\\vec{v}_d}({\\vec{r}_{pi}})-{\\vec{v}_{pi}}){,}$ $\\frac{d{\\vec{r}_{pi}}}{dt} &= {\\vec{v}_{pi}}{,}$ where ${\\vec{r}_{pi}}$ and $\\vec{v}_{pi}$ are the position and velocity of the $i$ th particle, respectively.",
"${\\vec{v}_d}({\\vec{r}_{pi}})$ is the dust fluid velocity at the particle position, ${\\vec{r}_{pi}}$ which is obtained by solving the set of Eqs.",
"(REF ) and (REF ).",
"These equations are a simplified approximation of the Maxey-Riley equations [56].",
"Although, the neglected effects might have significant impacts in real flows, but these could be incorporated into the future study because even after neglecting them, the Eq.",
"(REF ) describe an enough complex system and it is worth studying to set the foundation for the future research.",
"The particle time-scale $\\tau _s={2{r^2_0}{\\rho _p}}/{(9{\\eta }{\\rho _d})}$ denotes the response time of the particles is known as Stokes time [57].",
"$r_0$ is the radius of a particle.",
"Although the particles are assumed as point particles, but they do have finite mass and therefore finite inertia.",
"The ratio of a particle time-scale to a fluid time-scale is known as Stokes number (St).",
"The effect of particle inertia is often given by using the $St$ or $\\tau _s$ .",
"On the basis of St or $\\tau _s$ or inertia: For low value, the particles are predicted to follow the fluid flow passively like fluid particles, while at very high value almost the particles remain unaffected by the medium fluctuations.",
"In-between these two limits, when the particle and fluid time-scales are comparable, the particles respond in fast and strong manner to the fluctuations.",
"The non-inertial tracers follow the fluid flow exactly and can be considered as attached to fluid surface.",
"They are characterised by their position ${\\vec{r}_{pi}}$ , and velocity ${\\vec{v}_{pi}} = {\\vec{v}_d}({\\vec{r}_{pi}})$ that is the dust fluid velocity at their position.",
"Their equation of motion corresponds to the limit $\\tau _s\\rightarrow $ 0 in the set of Eqs.",
"(REF ) and (REF ) becomes: $\\frac{d{\\vec{r}_{pi}}}{dt}={\\vec{v}_d}({\\vec{r}_{pi}}){,}$ The set of Eqs.",
"(REF )-(REF )-(REF )-(REF ) and Eqs.",
"(REF )-(REF )-(REF ) represent the viscoelastic model for inertial and non-inertial tracer particles, respectively, in which the tracers follow the evolution of fluid with time in Lagrangian way.",
"Both the set of equations would be referred as incompressible Generalized Hydrodynamic Tracer Transport (i-GHT2 or i-GHTT) model henceforth in the article.",
"It should be noted that in i-GHTT model, in interest to include both the compressible longitudinal and incompressible transverse modes just replace the i-GHD model ((set of Eqs.",
"(REF )-(REF )) with complete GHD model (set of Eqs.",
"(5)- (6)-(7) in [23]), say GHTT model ."
],
[
"Transport Properties",
"To quantify the average diffusion of tracer particles, the ensemble averaged mean square displacement of tracers is measured which is associated with the mixing of the fluid [58].",
"The MSD is defined as, $MSD(t)= {\\frac{1}{N}}{\\sum ^N_{j=1}}{(r_j(t)-r_j(0))^2}{,}\\nonumber $ Here, $r_j(0)$ is initial position of $j$ th particle at t=0 and $r_j(t)$ is position at time $t$ .",
"N is the total number of tracers in the ensemble.",
"The slope of the MSD versus time is proportional to the diffusion coefficient of tracer particles which in turn is supposed to measure the mixing performance of the carrier fluid.",
"Thus, the mixing performance of the carrier fluid can be quantified through the time-MSD slope, the larger slope means the carrier fluid is a better mixture."
],
[
"Simulation methodology",
"For the numerical simulations, first we need to express the model Eq.",
"(REF ) as per requirements of simulation software, LCPFCT (Boris $et al.$ [59]).",
"To fulfill these requirements split the Eq.",
"(REF ) in following two coupled equations, ${{\\frac{\\partial \\vec{v}_d }{\\partial t}+\\vec{v}_d\\cdot \\nabla \\vec{v}_d }+ \\frac{\\nabla P}{n_d} -\\nabla \\phi _d}={\\vec{\\psi }} {,}$ $\\frac{\\partial {\\vec{\\psi }}}{\\partial t}+\\vec{v}_d \\cdot \\nabla {\\vec{\\psi }}={\\frac{\\eta }{\\tau _m}}{\\nabla ^2}{\\vec{v}_d }-{\\frac{\\vec{\\psi }}{\\tau _m}}{.",
"}$ For two-dimensional (2D) studies all the variables are functions of $x$ and $y$ only.",
"The new introduced quantity ${\\vec{\\psi }}(x,y)$ in LHS of Eq.",
"(REF ) represents the strain induced in the elastic medium by the time-varying velocity fields.",
"Next, the gradient terms are eliminated by taking the curl of Eq.",
"(REF ) which yields an equation for the evolution of the vorticity field.",
"So the coupled set of Eqs.",
"(REF )-(REF ) has been recast in the following form: $\\frac{\\partial {\\xi }_z}{\\partial t}+\\left(\\vec{v}_d \\cdot \\vec{\\nabla }\\right){{\\xi }_z}={\\vec{\\nabla }}{\\times }{\\vec{\\psi }}{,}$ $\\frac{\\partial {\\vec{\\psi }}}{\\partial t}+\\vec{v}_d \\cdot \\nabla {\\vec{\\psi }}={\\frac{\\eta }{\\tau _m}}{\\nabla ^2}{\\vec{v}_d }-{\\frac{\\vec{\\psi }}{\\tau _m}}{.",
"}$ Here, ${\\xi _z}={\\nabla }{\\times }{\\vec{v}_d}$ is the vorticity.",
"${\\vec{\\xi }}$ is normalised with dust plasma frequency.",
"The LCPFCT software [59] is based on a finite difference method has been used to solve the coupled set of Eqs.",
"(REF ) and (REF ).",
"Taking the curl of relation ${\\xi _z}={\\nabla }{\\times }{\\vec{v}_d}$ and using incompressible condition given by Eq.",
"(REF ) i.e.",
"$\\nabla \\cdot \\vec{v}_d= 0$ , we get ${\\nabla }^2{\\vec{v}_d} =-{\\nabla }{\\times }{\\xi _z}$ This velocity-vorticity relation is used to update the fluid velocity at each time step using FISHPACK [60].",
"The validation of our numerical code has been done in our earlier papers [21], [22].",
"Further details on simulation procedure in advancing the tracer particles with flow of a VE fluid.",
"We have the dust fluid velocity for each particle at their respective position at each time step.",
"This dust velocity is going to be used in particle momentum Eq.",
"(REF ).",
"Equations (REF ) and (REF ) are numerically integrated together to find the new position and velocity at the end of each time step.",
"This integration is based on the first order Runge-Kutta method.",
"The particle velocity $\\vec{v}_p$ is calculated by interpolating the velocity defined on nearby grid points, based on first-order Lagrangian interpolation scheme.",
"The particles are advanced with fluid time step."
],
[
"Numerical results and discussion",
"The prime objectives of this section are to numerically explore the evolution of different types of sharp rotating vorticity flows in VE fluids and quantifying the mixing they generate using the tracer particles simulations.",
"For each type of flow, the simulations are performed for the varying coupling strength of VE fluid which is usually measured as the ratio ${\\eta /\\tau _m}$ .",
"All the simulations are performed with periodic boundary conditions in both the x and y directions on simulation box."
],
[
"KH instability of rotating SCDPs",
"Since our interest is in KH instability of rotating SCDPs, we consider the specific cases of sharp vorticity patches: (i) single-circulation, and (ii) multi-circulation vorticity shell profiles.",
"Before proceeding with the direct assessment of the evolution of KH instability through the numerical results, it is good to understand the process of formation of these vortices for sharp rotating flows through the schematic picture illustrated in Fig.",
"REF .",
"The multi-circulation vorticity profile can be depicted as core-shell fluid flows.",
"In Fig.",
"REF , one of rotating flows forms the inner core (yellow color regime) and the others make the outer shells (cyan and green color regimes).",
"Each rotating flow is divided by a sharp interface; the fluid on either side rotates in opposite directions.",
"The black circle with arrows in the flow regime indicate its direction of rotation.",
"Figure: Formation of KH vortices at the circular sharp interfaces of core-shell flow regions (without any corresponding scale).",
"One of flows forms the inner core and the others make the outer shells.",
"The black circles with arrows in the flow regions indicates the direction of rotation.",
"At core-shell interface, the KH votices rotate anti-clockwise (blue color vortices with blue curved arrows) while at shell-shell interface rotate clockwise (red color vortices with red curved arrows).At each of the interfaces, the counter-rotating flows create a region of high shear which immediately evolve into small co-rotating (like-sign) KH vortices.",
"The schematic diagram clearly illustrates how the direction of rotation of these vortices depend on the relative motion between the alternative flows.",
"At inner (core-shell) interface, the KH votices rotate anti-clockwise (blue color vortices with blue curved arrows) while at outer (shell-shell) interface rotate clockwise (red color vortices with red curved arrows).",
"Thus far, these appraisals are particularly true for an inviscid HD fluid where no source term exists.",
"Whereas an incompressible VE fluid, besides KH instability, would also support the TS waves that propagate through the medium at phase velocity $v_p= \\sqrt{\\eta /\\tau _m}$ .",
"A medium with higher coupling strength (ratio ${\\eta /\\tau _m}$ ) favors the faster and stronger TS waves.",
"A stronger wave shows a less fall in amplitude with time in comparison to the lower one (see Figs.",
"2 and 4 in [21], and Fig.",
"4 in [22]).",
"Since, the present simulations have been carried out in the x-y plane, which is the plane of rotation of vorticity profile, the shear waves emitted from each interface should be cylindrical in shape.",
"Thus, a stronger cylindrical wave tries to dominate over KH instability and attempts to keep the cylindrical symmetry during the evolution.",
"In evolution of these KH vortices the transport processes like merging and convection become important.",
"When two co-rotating (like-sign) vortices are brought sufficiently close to each other they start to rotate around one another and eventually merge to form a single vortex while in convection the counter-rotating (unlike-sign) propagate together as single structure (dipole) to convect the fluid.",
"The velocity profile for the single-circulation sharp vorticity vortex is given as follows ${\\vec{v}_{0}} = \\left\\lbrace \\begin{array}{ll}v_{x0}=-{\\phi _0}{\\frac{(y-y_c)}{b}};\\:v_{y0}={\\phi _0}{\\frac{(x-x_c)}{a}} & \\quad |r| \\le 6 \\\\0 & \\quad \\mathrm {otherwise}{.",
"}\\end{array}\\right.$ The vorticity corresponding to the above velocity profile is given below ${\\xi _{z0}} = \\left\\lbrace \\begin{array}{ll}{\\phi _0}{\\left(\\frac{1}{a}+\\frac{1}{b}\\right)} & \\quad |r| \\le 6.0\\\\0 & \\quad \\mathrm {otherwise}{.",
"}\\end{array}\\right.$ Here $|r|= {\\sqrt{{\\left({(x-x_c)/a}\\right)}^2+{\\left({(y-y_c)/b}\\right)}^2}}$ , $a$ and $b$ are the major and minor axes, respectively.",
"$x_c$ and $y_c$ are the $x$ and $y$ coordinates of the center of the vorticity profile.",
"The vorticity will have a clockwise rotation if amplitude ${\\phi _0}{>}0$ , or have a anti-clockwise rotation if ${\\phi _0}{<}0$ and have no rotation if ${\\phi _0}{=}0$ .",
"We consider the clockwise rotating vorticity profile has circular symmetry with parameters $a=b=1$ , amplitude ${\\phi _0}=1$ , sharp cutoff at radial distance $|r|$ =6 units away from the centre of the vortex $(0,0)$ .",
"This vorticity profile (yellow color) is shown in the following Fig.",
"REF (a) which has circular interface with surrounding stagnant fluid at $t=0$ .",
"The simulation region is a square box of length 12$\\pi $ units.",
"Figure: Initial rotating sharp vorticity profiles at t = 0 to study the KH instability.",
"The color scale corresponding to a particular vorticity profile has been given in the following respective figure which shows its evolution.",
"The simulation region is a square box of length 12π\\pi units for all these systems.The panels of Fig.",
"REF shows time evolution of vorticity profile given in Fig.",
"REF (a) in the inviscid HD fluid.",
"The sharpness of the clockwise rotating vorticity profile generates a strong rotational sheared flow.",
"This sheared flow results in creation of small co-rotating (anti–clockwise) KH vortices (dark blue color vortices) at the vorticity interface $|r|=6$ .",
"These like-sign vortices start merging as rotation progresses that leads the fluid to evolve into an anisotropic isolated structure.",
"Figure: An inviscid HD fluid.",
"The time evolution of a sharp clockwise rotating vorticity which has circular interface with surrounding stagnant fluid.",
"The sharpness of the vorticity profile generates small KH vortices at the interface that results in a anisotropic isolated structure.In Fig.",
"REF where ${\\eta }=5$ and ${\\tau _m}=20$ , once the vortex begins to rotate, we observe a pair of ingoing and outgoing cylindrical shear waves from the interface along with these small like-sign KH vortices.",
"During the evolution, it is observe that both the waves carry the like-sign vortices which interact with themselves in order to merge and have interplay with these waves as well.",
"The fluid within the inner region ($|r| \\le 6$ ) favors mixing due to the ingoing waves, while the stationary fluid in the outer region ($|r| \\ge 6$ ) gets mixing due to the outgoing waves.",
"Thus, the TS waves assist the process of fluid mixing by convecting it inside and outside the vortex structure.",
"Unlike the inviscid case, here, the KH vortices are confined to the radial emitted waves.",
"Since the wavefronts are cylindrical in shapes, the interplay between waves and vortices leads the evolution of the VE fluid toward an isotropic structure.",
"Figure: Viscoelastic fluid with coupling parameters η=5{\\eta }=5, and τ m =20{\\tau _m}=20.",
"The time evolution of a sharp rotating vorticity which has circular interface with surrounding stagnant fluid.",
"The sharpness of the vorticity profile generates small KH vortices at the interface along with a pair of ingoing and outgoing wavefronts of TS waves that assist in fluid mixing by convecting it inside and outside the vortex structure.In Fig.",
"REF , we have simulated another case of VE fluid which has less coupling strength ($\\eta /\\tau _m$ =0.125) with ${\\eta }=2.5$ and ${\\tau _m}$ =20.",
"Unlike Fig.",
"REF , since the TS waves are not strong enough that they can dominant over the KH instability, less confinement of KH vortices to the waves.",
"This is evident from Fig.",
"REF where the evolution of medium towards an isotropic structure and mixing process are much slower in comparison to Fig.",
"REF .",
"Figure: The time evolution of a sharp rotating vorticity in VE fluid with coupling parameters η=2.5{\\eta }=2.5, and τ m =20{\\tau _m}=20.",
"Due to the less coupling strength, the evolution of medium towards an isotropic structure and mixing process are much slower in comparison to Fig.",
".In conclusion, as a result of greater the coupling strength or stronger TS wave, the medium evolution attempts to possess radial symmetry and shows better mixing.",
"The mixing is found minimal in inviscid fluid.",
"The single-circulation sharp profile carries single types of anti-clockwise rotating KH vortices across its only interface.",
"While a multi-circulation profile with two or more than two interfaces produces clockwise as well as anti-clockwise KH vortices.",
"Thus, in multi-circulation case apart from merging between like-sign vortices, the propagation of unlike-sign vortices as a dipolar structure also becomes important which can assist to increasing the spatial domain of mixing fluids."
],
[
"Multi-circulation vorticity shell profile",
"Here, we first consider the simplest case of multiple shells of vorticity having two flows: Inner core flow and a outer shell flow, both flows have reversal circulation.",
"The velocity profile for this configuration is given below.",
"${\\vec{v}_{0}} = \\left\\lbrace \\begin{array}{ll}v_{x0}=-{\\phi _0}{(y-y_c)};\\: v_{y0}={\\phi _0}{(x-x_c)} & \\quad |r|\\le 5 \\\\v_{x0}={\\phi _0}{(y-y_c)};\\: v_{y0}=-{\\phi _0}{(x-x_c)} & \\quad 5<|r|\\le 10 \\\\0 & \\quad \\mathrm {otherwise}{.",
"}\\end{array}\\right.$ The vorticity corresponding to the above velocity profile is given below, ${\\xi _{z0}} = \\left\\lbrace \\begin{array}{ll}2{\\phi _0} & \\quad |r| \\le 5 \\\\-2{\\phi _0} & \\quad 5 < |r| \\le 10 \\\\0 & \\quad \\mathrm {otherwise}{.",
"}\\end{array}\\right.$ With the parameter $\\phi _0$ =1, we consider both flows rotate with equal rotation rates and in opposite directions (Fig.",
"REF (b)).",
"Figure REF shows the evolution of this vorticity profile for an inviscid fluid.",
"At inner interface, the KH vortices rotate anti-clockwise (blue color vortices) while at outer interface rotate clockwise (red color vortices).",
"As time goes on, the merging between like-sign vortices take place at both the interfaces, and simultaneously growing closeness between both the interfaces due to the radial gradient in vorticity results in interactions between the counter-rotating (red-blue) vortices as well.",
"These counter-rotating vortices results in the formation of propagating dipolar structures which help to convect the fluid across the wider domain than the single interface (Fig.",
"REF ) which only favors the merging process.",
"Figure: An inviscid HD fluid.",
"Evolution of of vorticity having two circular sharp flows, inner core and an outer shell flows have reversal circulation, in time generates small KH vortices at each interface that results in a anisotropic isolated structure.In Fig.",
"REF , we observe that a pair of ingoing and outgoing wavefronts emanates from each of the two sharp interfaces of the vortex structure along with KH vortices (see the first panel).",
"It is observe that both the wavefronts at inner interface carry the blue color (rotating anti-clockwise) vortices while at outer interface carry red color (rotating clockwise) vortices.",
"During the emission of these shear waves like-sign vortices interact with themselves in order to get merge.",
"Figure: Viscoelastic fluid with coupling parameters η=5{\\eta }=5, and τ m =20{\\tau _m}=20.",
"The time evolution of vorticity having two circular sharp interfaces, inner core and outer shell have reversal circulation, generates small KH vortices at each interface along with a pair of ingoing and outgoing TS waves fronts that results in well fluid mixing by convecting it inside and outside the vortex structure.The stagnant fluid in the outermost region ($|r| \\ge 10$ ) undergoes mixing due to the outgoing wave from the outermost interface at 10, while the innermost vortex region undergoes mixing due to the ingoing wave emanating from the sharp interface located at 5.",
"Interestingly, the vortex region confined within the two sharp interfaces ($5 <|r|\\le 10$ ) undergoes mixing due to the ingoing wave from the outermost interface and the outgoing wave from the innermost interface.",
"This region is probably convection dominating region due to the higher possibility of formation of propagating dipolar structures.",
"As the results of multiple interaction processes the mixing becomes fast and efficient than the cases discussed so far.",
"Next, we consider the another VE fluid in Fig.",
"REF having lower coupling strength i.e.",
"${\\eta }=2.5, {\\tau _m}=20, v_p$ =0.35, it is observed that at each time step in the spatial confinement of this medium is lower than Fig.",
"REF .",
"Moreover, the comparison manifests that the mixing and evolution symmetry of a medium are proportional to the coupling strength of that medium as observed for earlier cases.",
"Figure: Viscoelastic fluid with coupling parameters η=2.5{\\eta }=2.5, and τ m =20{\\tau _m}=20.",
"The time evolution of vorticity having two circular sharp interfaces, both have reversal circulation.",
"Due to the less coupling strength the spatial confinement of this medium is lower than Fig.",
".Next, in order to make a better comparative numerical analysis about the mixing rate, we have considered a more complex scenario of multiple circulations (each consecutive one having a reversal in its circulation) having the following velocity flow profile: ${\\vec{v}_{0}}= \\left\\lbrace \\begin{array}{ll}v_{x0}=-{\\phi _0}{(y-y_c)};\\: v_{y0}={\\phi _0}{(x-x_c)} & \\quad |r|\\le 2.5 \\\\v_{x0}={\\phi _0}{(y-y_c)};\\: v_{y0}=-{\\phi _0}{(x-x_c)} & \\quad 2.5<|r|\\le 5 \\\\v_{x0}=-{\\phi _0}{(y-y_c)};\\: v_{y0}={\\phi _0}{(x-x_c)} & \\quad 5<|r|\\le 7.5 \\\\v_{x0}={\\phi _0}{(y-y_c)};\\: v_{y0}=-{\\phi _0}{(x-x_c)} & \\quad 7.5<|r|\\le 10 \\\\0 & \\quad \\mathrm {otherwise}{.",
"}\\end{array}\\right.$ The vorticity corresponding to the above velocity profile is given below, ${\\xi _{z0}} = \\left\\lbrace \\begin{array}{ll}2{\\phi _0} & \\quad |r| \\le 2.5 \\\\-2{\\phi _0} & \\quad 2.5 < |r| \\le 5 \\\\2{\\phi _0} & \\quad 5 < |r| \\le 7.5 \\\\-2{\\phi _0} & \\quad 5 < |r| \\le 10 \\\\0 & \\quad \\mathrm {otherwise}{.",
"}\\end{array}\\right.$ For the parameter $\\phi _0$ =1, the initial vorticity profile is shown in Fig.",
"REF (c).",
"The complexity of this motion of multi-circulation structure is evident from the subplots of Fig.",
"REF for inviscid fluid.",
"In initial time period, the vortices of KH instability develop across the interface of each shell.",
"At intermediate time range, this evolution provides a complete picture of a turbulent flow throughout the entire system which is collection of several small symmetric and non-symmetric vortices.",
"The transport phenomena like merging between two co-rotating vortices, convection due to the evolution of dipolar (two counter-rotating ) vortices, collision between these dipolar vortices, and also the formation and evolution of triplor structures becomes more frequent than above discussed cases.",
"Figure: An inviscid HD fluid.",
"The time evolution of multi-circulation vorticity profile, each consecutive one having a reversal in its circulation, generates small KH vortices at each interface.",
"This evolution provides a complete picture of a turbulent flow which is collection of various kind of vortices, and exhibits transport properties like convection and merging.Figure REF , represents the evolution of same initial profile (Fig.",
"REF ) of vorticity for VE fluid with coupling parameters ${\\eta }=5$ , ${\\tau _m}$ =20.",
"From the comparative observations between Fig.",
"REF and Fig.",
"REF , it is interesting to notice that the presence of TS waves leads to the relaxing of the turbulent medium to a single vortex faster than in inviscid fluid.",
"Figure: The time evolution of multi-circulation vorticity profile in VE fluid with η=5{\\eta }=5, and τ m =20{\\tau _m}=20.",
"The strong interaction between KH vortices and TS waves results in the relaxing the medium into a single vortex faster than inviscid fluid.In Fig.",
"REF (${\\eta }=2.5, {\\tau _m}=20, v_p$ =0.35), since the TS waves are weaker than Fig.",
"REF , the less spatial confinement of the turbulent medium that results in slower merging process, due to which the relaxation time of the medium becomes longer.",
"Figure: The time evolution of multi-circulation vorticity profile in VE fluid with η=2.5{\\eta }=2.5, and τ m =20{\\tau _m}=20.",
"Since the TS waves are weaker than Fig.",
", the relaxation time of the medium becomes longer.The comparison of Fig.",
"REF and Fig.",
"REF clearly displays these observations.",
"In Section we have stated that tracer particles with very low inertia follow the flow passively, while particles with very high inertia will remain almost unaffected by the medium fluctuations.",
"In between these two limits particles show the strongest response to the medium fluctuations.",
"The simulations are performed for all these three inertial particles: very low ($\\tau _s$ =0.05), intermediate ($\\tau _s$ =1), and very high inertia ($\\tau _s$ =50).",
"To observe the exclusive effect of inertia, we transport these particles through a similar smooth rotating vorticity vortex in an inviscid fluid.",
"An inviscid fluid has no source term which favors the emission of TS waves or dissipative term like viscosity.",
"And also, we choose the smooth rotating vorticity vortex which does not satisfy the KH destabilization condition anywhere in the vorticity patch.",
"The equation of such smooth rotating vorticity is given as ${\\xi _{0}}={\\Omega _0}exp\\left(-\\left({x^2+y^2}\\right)/{a^2_c}\\right){.",
"}$ The evolution of same structure given by Eq.",
"REF for ${a_c}$ =1.0, ${\\Omega _0}=5$ in an inviscid fluid is shown in Figs.",
"(REF ), (REF ), and (REF ).",
"From these figures it is clear that the rotating vortex keeps rotating without any change.",
"Figure: An inviscid HD fluid carries tracer particles are shown as red dots.",
"The low inertial tracers (τ s \\tau _s=0.05) follow the dynamics along the smooth rotating vortex.Now, in order to see the response of tracer particles, initially $(t=0)$ , we have distributed 900 inertial particles (shown by red dots) homogeneously throughout the domain.",
"Figure: An inviscid HD fluid carries tracer particles are shown as red dots.",
"The high inertial tracers (τ s \\tau _s=50) show negligible response to the rotating vortex.From Fig.",
"REF , it is clear that low inertial particles ($\\tau _s$ =0.05) follow the dynamics along the rotating vortex, and the particles with higher inertia $\\tau _s$ = 50 (Fig.",
"REF ) show negligible response to the vorticity gradient.",
"Figure: An inviscid HD fluid carries tracer particles are shown by red dots.",
"Smooth rotating vorticity vortex with tracers having intermediate inertia i.e τ s \\tau _s=1 show high response to the vorticity gradient, the particles are pushed away where the flow is strong enough and get accumulate in strain-dominated region.In comparison to previous cases (Fig.",
"REF and Fig.",
"REF ), Fig.",
"REF shows that the particles with intermediate value of $\\tau _s$ =1 counter a significant outward push.",
"It is because the particle and fluid time- scales are comparable which results the particles experience a notable centrifugal force due to vorticity gradient.",
"Since, the inertial particles are pushed away from regions where the flow is strong enough, these particles get accumulate in strain-dominated regions.",
"Thus, particles tend to leave regions of high vorticity and cluster into regions of high strain [61].",
"Note, we have simulated a range of intermediate inertial particles, and observed the same effect with varying outward push.",
"With the identification of intermediate inertial particles and understanding of their evolution, next we compare the evolution of intermediate inertial (typical, $\\tau _s$ =1) with non-inertial particles for the inviscid HD and VE ($\\eta =5$ , $\\tau _m=20$ ) fluids in the following Fig.",
"REF and Fig.",
"REF , respectively.",
"For this, we choose a sharp rotating vorticity profile which is given by Eq.",
"(REF ).",
"The middle row in Fig.",
"REF and Fig.",
"REF represent the time evolution of this profile for inviscid and VE fluids, respectively.",
"As the sharp vortex starts to rotate, produces larger strain (deformation) in medium along the interface that results in formation of KH instability.",
"The evolution of this rotor has already been discussed in the earlier Section REF for inviscid fluid in Fig.",
"REF and for VE fluid in Fig.",
"REF .",
"Initially $(t=0)$ , here, we distribute 3600 tracer particles homogeneously throughout these fluids.",
"Figure: An inviscid HD fluid advect tracer particles.",
"First and third row show the temporal and spatial distribution of non-inertial particles (shown by magenta dots) and inertial particles (τ s \\tau _s=1, shown by red dots), respectively, corresponding to the sharp vorticity profile evolution shown in second row.In both Figs.",
"REF and REF , the first and the third row visualize the distribution of non-inertial particles (shown by magenta dots) and inertial particles (shown by red dots) respectively, advect by their respective fluid flow shown in the middle row.",
"The accumulation of non-inertial particles is observed in rotation-dominated regions over the vortex structures.",
"While the inertial particles accumulate in strain-dominated regions along the interfaces.",
"Figure: Viscoelastic fluid with coupling parameters η=5{\\eta }=5, and τ m =20{\\tau _m}=20 advect tracer particles.",
"First and third row show the spatiotemporal distribution of non-inertial particles (shown by magenta dots) and inertial particles (τ s \\tau _s=1, shown by red dots), respectively, corresponding to the sharp vorticity profile evolution given in second row.This accumulation process leads to the spatial inhomogeneous distribution of particles.",
"This inhomogeneous distribution of the particles is known as clustering or preferential concentration.",
"Clustering is well-studied in the case of inertial particles [62], [31], [26], [27], [63], [64], [65], [66], and several studies are available for non-inertial particles [67].",
"In real flows the tracer particles always have some inertial value, so we compute the ensemble averaged MSD of intermediate inertial particles to analyze the diffusion of particles in the carrier VE fluid.",
"This diffusion of particles is associated with the mixing of the fluid.",
"For this, we advect the inertial particles having $\\tau _s$ =1 using the sharp rotating flows discussed above, inviscid fluid in Fig.",
"REF /Fig.",
"REF , VE fluid with $\\eta =5$ , $\\tau _m=20$ in Fig.",
"REF /Fig.",
"REF , and another VE fluid with $\\eta =2.5$ , $\\tau _m=20$ in Fig.",
"REF .",
"Initially t=0, we disperse 3600 tracer particles homogeneously over these fluids.",
"Figure: The MSD as function of time for tracing particles.",
"The sharp rotating vorticity advect the particles with (a) τ s \\tau _s=1, and (b) τ s \\tau _s=0.5.",
"At later time, the MSD is proportional to the coupling strength.Figure REF (a) compares the ensemble averaged MSD values of inertial particles of $\\tau _s$ =1 for all three types of fluids.",
"The evolution of MSD occurs in three stages.",
"Initially (here, $0{\\ge }t{\\approx }1$ ), inertial particles do not respond until their time-scale (here, $\\tau _s=1$ ) is less than fluid time-scale or Stokes number (St) is less than unity.",
"During the second stage the particles start to respond the flow ($1{\\ge }t{\\le }3.8$ ), the slopes grow at the almost same rate for all fluids.",
"In the final stage ($t{\\ge }3.8$ ), the slope shows the larger value for the higher coupling strength and it is minimal for inviscid fluid.",
"The slope of the MSD versus time is proportional to the diffusion coefficient of the tracer particles.",
"Thus, the larger time-MSD slope means the carrier fluid shows a better mixing performance.",
"Thus, this result is found consistent with our earlier observations made from the comparative analysis of pictorial evolution of vorticities in Figs.",
"REF , REF , and REF .",
"In order to substantiate these observations we further compute the MSD of tracers with $\\tau _s$ =0.5 (Figure REF (b)) for the all three flows under the same flow conditions.",
"From the plot, again the diffusion of particles increasing with coupling strength.",
"It should be noted that the MSD has been calculated up to time before hitting the flow to the boundaries.",
"In the present work, although we have undertaken the simplest tracer model and preliminary investigation of tracers distribution (cluster formation) and their transport property (MSD), but it is worth attempting as a foundation for the future study."
],
[
"Summary and conclusions",
"In this paper, we numerically explore the KH instability for a two-dimensional rotating SCDP under the formalism of GHD fluid model.",
"This model treats the SCDP as a viscoelastic fluid.",
"Here, we consider the specific cases of vorticity with abrupt radial changes, in particular: single-circulation, and multi-circulation vorticity shell profiles.",
"We observe the KH instability in form of small vortices at each interface along with a pair of ingoing and outgoing wavefronts of TS waves.",
"The interactions between KH vortices, and with TS waves govern the mixing which has been quantified using the passive tracer particles simulation.",
"The advection of tracers with the flow has been noticed.",
"Some main observations are as follows.",
"The interplay between the TS waves and interacting KH vortices results in better mixing of VE fluids than standard hydrodynamic fluids where the only interaction between KH vortices happens.",
"By tweaking the coupling strength parameter (ratio $\\eta /\\tau _m$ ) which usually represents the strength of viscoelasticity of the medium, one can control the evolution of KH instability that results in control over the transport properties like mixing and diffusion.",
"The GHD model especially developed for the study of SCDPs is extended to include the transport of passive Lagrangian inertial and non-inertial tracer particles.",
"This extended model is referred as Generalized Hydrodynamic Tracer Transport (GHTT) model.",
"We observe that the diffusion of intermediate inertial tracer particles in VE fluids is proportional to the coupling strength.",
"It is least for an inviscid fluid.",
"The multi-circulation vortex profiles, we found that the relaxing rate of a turbulent medium increases with the increasing coupling strength.",
"The particle tracking model appears a suitable diagnostic to understand the associated mixing in a fluid through the diffusion process, and nonlinear dust fluid dynamics through the clustering of a tracers.",
"In present paper, we just discuss the preliminary simulations for tracers which are based on first order schemes like to advance the tracers we use first order RK scheme, and for the velocity the first order interpolation scheme.",
"Further improvisations of this model would be to include higher order schemes, finite size tracers, other forces e.g.",
"gravity, inter-particle interactions, feedback from tracers to the carrier fluid, etc.",
"These improvisations would be extremely fruitful which are left to future publications.",
"Although, the rotational dust flows are extensively studied, but to our knowledge, no prior studies have explicitly examined the KH instability for the rotating flows, except [21], [22].",
"An experimental research effort is required in order to valid this numerical work."
]
] | 2107.01831 |
[
[
"Controllable cardiac synthesis via disentangled anatomy arithmetic"
],
[
"Abstract Acquiring annotated data at scale with rare diseases or conditions remains a challenge.",
"It would be extremely useful to have a method that controllably synthesizes images that can correct such underrepresentation.",
"Assuming a proper latent representation, the idea of a \"latent vector arithmetic\" could offer the means of achieving such synthesis.",
"A proper representation must encode the fidelity of the input data, preserve invariance and equivariance, and permit arithmetic operations.",
"Motivated by the ability to disentangle images into spatial anatomy (tensor) factors and accompanying imaging (vector) representations, we propose a framework termed \"disentangled anatomy arithmetic\", in which a generative model learns to combine anatomical factors of different input images such that when they are re-entangled with the desired imaging modality (e.g.",
"MRI), plausible new cardiac images are created with the target characteristics.",
"To encourage a realistic combination of anatomy factors after the arithmetic step, we propose a localized noise injection network that precedes the generator.",
"Our model is used to generate realistic images, pathology labels, and segmentation masks that are used to augment the existing datasets and subsequently improve post-hoc classification and segmentation tasks.",
"Code is publicly available at https://github.com/vios-s/DAA-GAN."
],
[
"Introduction",
"Whilst large scale public datasets are available for traditional vision tasks, medical data are difficult to acquire.",
"Even in a large-scale medical training dataset, examples of rare diseases and anatomies are scarce.",
"As a result, generalisation to observations that are not seen during training will be reduced.",
"To increase the diversity of training data and for instance, to increase the incidence of rare characteristics, we would like the ability to mix and match factors that encode these variations [4] in a controllable way i.e.",
"perform controllable image synthesis.",
"Figure: Top: overview of the “disentangled anatomy arithmetic\" concept illustrated with 3 factors that represent 3 different anatomical parts of the heart (e.g.",
"left/right ventricle and myocardium).",
"Bottom: DAA-GAN generated example.",
"Given a healthy Subject A, we aim to generate an image A' which exhibits hypertrophic cardiomyopathy (HCM).",
"We select a Subject B with HCM and remove the anatomical factors from A (i.e.",
"the ones that encode the myocardium and left ventricular cavity) and add the corresponding factors of B (inner part of the red circle).",
"Arrows in A' point to local deformations showing the non-linear abilities of our arithmetic.",
"Arithmetic operations are denoted with ±\\pm .The idea of generating realistic images to augment existing limited data is not new in medical image analysis.",
"Generative Adversarial Networks (GANs) [16] have been used to generate variants for a given input image based on sampling from a random noise vector.",
"In fact, more recent GAN architectures pursue controllability by disentangling existing factors of variation, conditioning the generation process using semantic priors (e.g.",
"segmentation masks) [17], [10], [35], [22], [28], [18], [25] and class labels [20], [13], or learning cross-modality translations [29], [3], [24], [8], [11].",
"An alternative to relying on sampling from a noise vector for new medical images, is the idea of mixing existing information from different populations (e.g.",
"patients with different anatomical characteristics) to learn the intermediate latent space and generate more realistic data in a more controllable way.",
"A concept that approximates this idea is “vector arithmetic\" [33], where existing vector-based latent representations are combined using simple arithmetic operations to produce new images.",
"However, vector representations do not exploit the spatial equivariance of the image content and the respective task (e.g.",
"segmentation) and have shown poor reconstruction quality.",
"An alternative is to use to disentangled representations that use both spatial (tensor) and vector representations to capture factors of variation and permit decomposition of the input in spatially equivariant (and closely to be semantic) and imaging information [7], [9], [38].",
"Herein using disentangled representations we propose the concept of “disentangled anatomy arithmetic\" (DAA), visualized in Fig.",
"REF , that enables controllable image synthesis of plausible images with a target pathology, which we show to be useful for augmenting existing medical training data.",
"We design the DAA-GAN model that learns to combine and transform spatial anatomical factors –we provide a visual example of anatomy factors in Fig.",
"1 of the supplemental– from input images captured by different vendors or from different populations, and then re-entangle them with the chosen imaging factors (e.g.",
"MRI) to generate unseen intermediate representations.",
"Inspired by recent findings regarding the advantages of introducing spatial stochasticity in the generation process [23], [1], [14], [2], we propose a convolutional module for the combined anatomical factor representation transformation, in which structured noise is injected at the spatial locations where the arithmetic operations take place.",
"Our contributions are to: Introduce the concept of “disentangled anatomy arithmetic\" on spatial representations of the anatomy.",
"Propose DAA-GAN, a generative model that to the best of our knowledge, is the first to condition image generation using spatial anatomy factors as semantic priors.",
"Propose the noise injection module that encourages local deformations to realistically blend the new factors after the arithmetic step.",
"Evaluate the impact of using DAA-GAN for cardiac data augmentation in the context of a classification and a semantic segmentation post-hoc task."
],
[
"Generative Model Architecture",
"Our model assumes disentangled anatomy and imaging representations as inputs.",
"To obtain them, we use SDNet [7] as this model provides binary spatial factors that correspond to the whole anatomy and can be used as semantic priors.",
"Model overview.",
"As depicted in Fig.",
"REF , DAA-GAN has 4 distinct steps: 1) We combine anatomy factors, using disentangled anatomy arithmetic, to obtain a new mixed anatomical representation $\\mathbf {\\hat{C}}$ .",
"2) A noise injection network $\\mathcal {J}$ takes $\\mathbf {\\hat{C}}$ and aims to create a plausible and more refined anatomical representation $\\mathbf {\\tilde{C}}$ .",
"3) A generator $\\mathcal {G}$ reconstructs an image corresponding to $\\mathbf {\\tilde{C}}$ and a given imaging representation.",
"4) Two critics, namely a discriminator $\\mathcal {D}$ and a pathology classifier $\\mathcal {F}$ ensure good image fidelity, but also that the reconstructed image contains the right target characteristics.",
"We proceed detailing these steps.",
"Figure: DAA-GAN overview (from left to right): arithmetic operations are performed between anatomical factors 𝐂 a \\mathbf {C}^{a} and 𝐂 b \\mathbf {C}^{b} to produce a mixed representation 𝐂 ^\\mathbf {\\hat{C}} which is then refined by the noise injection network 𝒥\\mathcal {J}.",
"The generator 𝒢\\mathcal {G} receives this refined representation 𝐂 ˜\\mathbf {\\tilde{C}} and re-entangles it with the the input imaging factors to generate the new image I ˜\\tilde{I}.",
"Finally, a discriminator is responsible to judge if I ˜\\tilde{I} is real or fake, whilst a pathology classifier assesses if I ˜\\tilde{I} has the desired cardiac pathology.Disentangled anatomy arithmetic.",
"As shown in Fig.",
"REF (top), we consider an example with 3 cardiac anatomy populations $\\mathbf {C}^{a}, \\mathbf {C}^{b}, \\mathbf {C}^{c}$ (e.g.",
"patients from 3 imaging populations $a$ , $b$ , and $c$ ) with 3 anatomical factors per population, which –when combined– form a nested structure that corresponds to the heart region.",
"These factors are extracted from $I^{a}, I^{b}, I^{c}$ medical images of dataset $Z_{\\sim \\lbrace I^{a}, I^{b}, I^{c}\\rbrace }$ .",
"Based on this setup, we define factor arithmetic between populations any swapping operation between the corresponding factors.",
"Following this example, to create a new $\\mathbf {C}^{a}$ anatomy by mixing factors, we first swap $\\mathbf {C}_{1}^{a}$ with $\\mathbf {C}_{1}^{c}$ , and then swap $\\mathbf {C}_{2}^{a}$ with $\\mathbf {C}_{2}^{b}$ .",
"The result is an intermediate $\\mathbf {\\hat{C}}$ that is used as input to the next module of DAA-GAN.",
"Note that before swapping two factors, assuming upright anatomies, we perform a registration step (warping) to align the swapped-in factor with the center of mass location of the swapped-out one.",
"Noise injection.",
"Since cardiac anatomy is a nested structure, the output of the arithmetic step might be non-realistic, e.g.",
"have factors that overlap with each other.",
"This can lead to generated images with ambiguous pathology.",
"We tackle this problem with module $\\mathcal {J}$ , which receives $\\mathbf {\\hat{C}}$ and produces a refined representation $\\mathbf {\\tilde{C}}$ .",
"Inspired by recent work of Karras et al.",
"[23], we introduce stochastic variation (as Gaussian noise) at specific spatial locations of each convolutional (CONV) layer's activations.",
"This variation is exploited by the network to cause local deformations around the added (swapped-in) factor(s) in order to preserve the non-overlapping nested structure of the heart (see Fig.",
"REF (bottom)).",
"$\\mathcal {J}$ consists of 4 CONV layers, whilst the noise is injected in the form of noise patches (see Fig.",
"REF ) added to each CONV layer's activation features in an element-wise fashion.",
"The last CONV layer is followed by a Gumbel-Softmax operator that bounds $\\mathbf {\\tilde{C}}$ to $[0,1]$ .",
"Figure: Visualization of the localized Gaussian noise patch generation process.",
"⊗\\otimes and ⊙\\odot denote convolution and element-wise multiplication, respectively.Generator.",
"The generator is responsible for the re-entanglement of anatomy and imaging information, and by extension for the generation of a new image $\\tilde{I}$ .",
"$G$ consists of 4 CONV-ReLU layers followed by a hyperbolic tangent activation function.",
"The first CONV layer receives $\\mathbf {\\tilde{C}}$ as input, while after each CONV-ReLU block, there is an Adaptive Instance Normalization (AdaIN) layer [21] that scales and shifts the activations based on the input imaging factors.",
"Since $\\mathbf {\\tilde{C}}$ has the same dimensions as $I$ and $\\tilde{I}$ , there is no need for any upsampling.",
"Critics.",
"Since for $\\tilde{I}$ we have no ground truth we use a discriminator to guide reconstruction.",
"We adopt the architecture of the LSGAN discriminator [26] for faster convergence, better stability and ultimately better image quality (compared to a variant with spectral normalization layers adopted from [27]).",
"Additionally, since we want $\\tilde{I}$ to have desired properties, we use a VGG16 model [36] $\\mathcal {F}$ to classify the pathology represented in $\\tilde{I}$ .",
"$\\mathcal {F}$ has 7 CONV-BN-ReLU blocks and 3 fully-connected layers followed by a softmax function.",
"Note that $\\mathcal {F}$ is pre-trained on the original data, and is then used as a pathology predictor during training of the generative model.",
"Having presented each module in detail, we now proceed to describe the 4 losses used to train our model as a whole: Adversarial loss ($\\mathcal {L}_{adv})$ .",
"We use an adversarial loss to encourage our model to generate realistic images.",
"We choose to minimize the LSGAN loss [26] as it is more stable during training and leads to higher quality image generation compared to the traditional GAN loss.",
"Our adversarial loss is defined as: ${\\begin{array}{c}\\mathcal {L}_{adv} = \\mathcal {L}_{\\mathcal {D}} + \\mathcal {L}_{\\mathcal {G}},\\\\\\mathcal {L}_{\\mathcal {D}} = \\frac{1}{2}\\mathbb {E}_{I^{\\prime }\\sim p(Z)}[(\\mathcal {D}(M\\cdot I^{\\prime }) - 1)^2]\\ +\\frac{1}{2}\\mathbb {E}_{\\mathbf {C}\\sim p(\\mathbf {C}_{Z})}[(\\mathcal {D}(M\\cdot \\mathcal {G}(\\mathbf {\\hat{C}})))^2],\\\\\\mathcal {L}_{\\mathcal {G}} = \\frac{1}{2}\\mathbb {E}_{\\mathbf {C}\\sim p(\\mathbf {C}_{Z})}[(\\mathcal {D}(\\mathcal {G}(\\mathbf {\\hat{C}})) - 1)^2],\\end{array}}$ where $I^{\\prime }$ is the image that contributes only the anatomy factor(s) $\\mathbf {C^{\\prime }}$ which are used to form $\\mathbf {\\hat{C}}$ (e.g.",
"$I^{b}$ in Fig.",
"REF ).",
"$M$ is a binary mask produced by the union of the anatomical factors that contain information about the heart.",
"Note that $M(j)=0$ for the remaining (i.e.",
"non-heart) pixels.",
"Pathology classification ($\\mathcal {L}_{path}$ ).",
"Since we know that we add the anatomical factor $C_{k}$ to the mixed representation, we expect to be able to recognize the pathology corresponding to $C_{k}$ in the generated image $\\tilde{I}$ .",
"To achieve this we minimize the cross entropy loss, defined as $\\mathcal {L}_{path} = -\\sum _{i=1}^{\\Omega }y_{i}\\log (p(x_{i}))$ , where $y_{i}$ and $p(x_{i})$ are the ground truth and predicted pathology labels, and $\\Omega $ is the number of pathology classes.",
"Anatomical and background consistency ($\\mathcal {L}_{cons}$ and $\\mathcal {L}_{bg}$ ).",
"$\\mathcal {L}_{cons}$ encourages the anatomical factors which are not related with the heart to remain unaltered after the arithmetic and noise injection steps.",
"To find which pixels should not be changed, we use the blurred mask produced during the noise patch generation, denoted as $\\Phi (j)$ , with $\\Phi (j)=0$ for each pixel location $j$ that is not part of the anatomy mixing.",
"We define $\\mathcal {L}_{cons} = \\frac{1}{N} \\sum _{j}(1-\\Phi (j)) ||\\mathbf {\\hat{C}}(j) - \\mathbf {\\tilde{C}}(j)||_{1}$ .",
"$\\mathcal {L}_{bg}$ is the same as $\\mathcal {L}_{cons}$ , but on images.",
"Namely, $\\mathcal {L}_{bg} = \\frac{1}{N} \\sum _{j}(1-M(j)) ||(I^{a}(j) - \\tilde{I(j)}) ||_{1}$ where $M$ is defined previously, and $N$ is a the total number of pixels.",
"Total loss ($\\mathcal {L}_{total}$ ).",
"These losses discussed above, are combined as: $\\mathcal {L}_{total} = \\mathcal {L}_{adv} +\\mathcal {L}_{path} + \\lambda _{1}(\\mathcal {L}_{cons} + \\mathcal {L}_{bg}),$ where $\\lambda _{1}=10$ is a weighting hyperparameter for the consistency losses."
],
[
"Experiments",
"In this section we present key information on datasets and metrics, and discuss the results of our experiments (see training details in Sec.",
"1 of the supplemental).",
"Data.",
"We use the ACDC [5] dataset and data from the M$\\&$ Ms challenge [6].",
"ACDC consists of cardiac images acquired from MRI scanners across 100 subjects, and provides pathology annotations (5 classes) for all images and pixel-level segmentation masks for end-systole (ES) and end-diastole (ED) per subject.",
"M$\\&$ Ms has cardiac-MR images acquired from 4 different (known) sites (domains), across 345 subjects.",
"It provides ED/ES pixel-level segmentation masks annotations for 320 (of 345) subjects and pathology annotations (4 classes) for all images (for more dataset details see Sec.",
"2 of the supplemental).",
"Metrics.",
"We approach this in several levels.",
"To measure image quality we compute the Fréchet Inception Distance (FID) [19], which quantifies the distance between feature vectors from real and generated images.",
"To quantify the utility of the generated images we study if they help improving the performance of two post-hoc tasks: a) pathology classification, and b) semantic segmentation.",
"For the former, we measure the classification accuracy achieved by a VGG16 model, whilst for the latter, we measure the Dice score [12], [37] achieved by a U-Net [34] model.",
"The hardest is to assess controllability, i.e.",
"the ability to generate images that faithfully create combinations of anatomical factors.",
"We approximate this by examining the existence of the added or removed pathologies.",
"We train two VGG16 classifiers, one 5-class for ACDC data and one 4-class for M$\\&$ Ms data, and measure the pathology classification accuracy on the generated images.",
"Table: Comparing data augmentation methods in the context of the 4 questions defined in the Results section (see text for details).We report average (standard deviation as subscript) classification accuracy (Acc.)",
"and segmentation performance as average Dice score.",
"“n/a\" denotes not applicable experiment.",
"* and ** denote significant improvement over the 2nd best method with p<0.05p<0.05 and p<0.1p<0.1 (Wilcoxon non-parametric test), respectively.Setup.",
"We generate data with our DAA-GAN; SPADE [31] a model that conditions synthesis using segmentation masks; and AC-GAN [30] a model that conditions synthesis on class rather than semantic priors.",
"Then, we train the two posthoc task-specific models (i.e.",
"VGG16 and U-Net) with the following training sets (applies to both datasets): i) original data (OD), ii) OD augmented with data from DAA-GAN, iii) OD augmented with data from SPADE, and iv) OD augmented with data from AC-GAN.",
"For (i)-(iv) we further augment the training set using traditional intensity (blur, gamma correction) and geometric (crop, rotate, flip, elastic transformation) augmentations for fairness (validation and test sets remain unaltered).",
"Note that for each generated image, we extract and binarize only the heart-related $\\mathbf {\\tilde{C}}$ factors (output of $\\mathcal {J}$ ) and use them as near-ground truth masks for retraining U-Net in the context of post-hoc segmentation.",
"Results.",
"To demonstrate the effectiveness of our method, we answer several key questions referring to quantitative results presented in Table REF .",
"Does DAA-GAN augmentation improve learning on a balanced dataset?",
"For this experiment we use ACDC, which is balanced in terms of subjects per pathology class.",
"We split the OD into 70%, 15%, 15% subjects for training, validation, and testing, and use DAA-GAN to generate 50 new images for the 5 pathology classes (this corresponds to a 25% samples increase).",
"These images are picked based on $\\mathcal {F}$ confidence.",
"As Table REF shows (column “Balanced”) data generated by our model lead to a 3.1% absolute classification accuracy improvement compared to using the OD with traditional augmentations, whilst outperforming both AC-GAN and SPADE.",
"Regarding segmentation, compared to only using the OD, our model improves the Dice score by an absolute 1.6%.",
"Since AC-GAN does not condition on masks and SPADE does not have a mechanism to refine the fused segmentation masks, they cannot be used in this experiment.",
"Does DAA-GAN augmentation improve learning of underrepresented classes?",
"The M$\\&$ Ms dataset is imbalanced in terms of the Abnormal Right Ventricle (ARV) pathology class, which represents only 5% of the dataset.",
"We showcase here the impact of our generative model in augmenting underrepresented classes.",
"We split the data in a 70-15-15 percentage fashion (as with ACDC), and use DAA-GAN to generate 168 –high $\\mathcal {F}$ confidence– new images (matching the highest represented class) with the ARV pathology by mixing the factor that corresponds to the RV with healthy anatomies from other subjects.",
"We use this new balanced training set that is used to re-train VGG16 and U-Net.",
"As reported in Table REF , column “Un/ed Class”, augmentations improve both accuracy by 3.3% and Dice by 1.7% Dice outperforming AC-GAN and SPADE where applicable.",
"Does DAA-GAN augmentation improve learning of underrepresented or new domains?",
"As stated above, M$\\&$ Ms comprises data captured from 4 different sites (A-D), thus 4 different populations.",
"However, data from site D represent only 14% in the dataset.",
"Here we showcase the augmentation of site D. Thus, in this experiment we aim to balance the training set of M$\\&$ Ms by augmenting site D data.",
"We adopt the same split percentage with the previous experiments, and augment the original training set by mixing pathological factors from subjects of vendors A-C with anatomies of vendor D. Results in Table REF , column “Un/ed Vendor”, show that by augmenting site D with 101 generated images of high $\\mathcal {F}$ confidence value, we improve the two tasks' performance by 3.1% in accuracy and 1.2% Dice, respectively, while outperforming AC-GAN and SPADE.",
"Figure: Two examples of anatomy factor traversals.",
"For each step ii increases by 3: a) top row depicts the transformed factor CC, b) middle row depicts the generated images I ˜\\tilde{I}, and c) bottom row shows the factor difference between the current generated image I ˜\\tilde{I} and the input image.",
"LV, MYO denote the left ventricular cavity and the myocardium.Does DAA-GAN achieve good image quality?",
"Table REF (rightmost) reports generated image quality for each model.",
"Our model outperforms SPADE, which is unsurprising since SPADE has no mechanism to mix semantic priors, thus generating images with unusual heart representations due to overlapping anatomical factors (see examples generated using SPADE in Fig.",
"3 of the supplemental).",
"AC-GAN, has slightly better FID compared to DAA-GAN in both datasets quality but is the worst model in terms of controllability and utility.",
"Can we control DAA-GAN synthesis?",
"We explore this visually by generating images through manipulation of the anatomical factors of the same subject (i.e.",
"without mixing between subjects).",
"We erode and dilate only a single factor of each subject and generate images based on the altered anatomy.",
"From the examples in Fig.",
"REF , we observe that the generated cardiac parts do not correspond linearly to the generative factors when approaching extreme cases, i.e.",
"for large kernel size.",
"Thus, we argue that our model's controllability is constrained only by the dataset bias, i.e.",
"“unseen\" extreme (possibly unrealistic) combinations.",
"Ablations.",
"To evaluate the impact of $\\mathcal {J}$ and $\\mathcal {F}$ on the augmentation results, we replicate each experiment removing one or both modules.",
"From the results in Table REF , we conclude that the two modules have similar contribution to post-hoc classification improvement, $\\mathcal {F}$ slightly improves segmentation, whilst we also observe that $\\mathcal {J}$ plays an important role in the quality of the generated images."
],
[
"Conclusions",
"In this paper we introduced a novel framework for controllable cardiac image synthesis.",
"In particular, we presented a generative model that learns to produce unseen images from existing images using a user-selected combination of spatial anatomy factors.",
"We conducted experiments demonstrating the controllability of the generation process, whilst showcasing the potential of augmenting existing medical data with images generated using the concept of “disentangled anatomy arithmetic\".",
"Future work will focus on extending the capability of our model beyond simple mixing and morphology to richer anatomy arithmetic operations."
],
[
"Acknowledgement",
"This work was supported by the University of Edinburgh, the Royal Academy of Engineering and Canon Medical Research Europe.",
"This work was partially supported by the Alan Turing Institute under the EPSRC grant EP/N510129/1.",
"We thank Nvidia for donating a Titan-X GPU.",
"S. A. Tsaftaris acknowledges the support of Canon Medical and the Royal Academy of Engineering and the Research Chairs and Senior Research Fellowships scheme (grant RCSRF1819\\8\\25)."
],
[
"Experimental Setting",
"Both $\\mathcal {J}$ and $\\mathcal {D}$ were trained for 90 epochs, using the Adam optimizer with $\\beta _{1}=0$ , $\\beta _{2}=0.999$ , and a learning rate of 0.0001.",
"Our pathology classification model $\\mathcal {F}$ was pre-trained separately using the Adam optimizer with $\\beta _{1}=0.9$ , $\\beta _{2}=0.999$ , and a learning rate of 0.0001, with classification accuracy in the corresponding validation set as the early stopping criterion.",
"We use the the pre-trained decoder of SDNet as our generator $\\mathcal {G}$ .",
"During training of the generative model, the weights of $\\mathcal {F}$ and $\\mathcal {G}$ are frozenFor the ablation study, when we remove $\\mathcal {J}$ , we instead fine-tune $\\mathcal {G}$ during generative model training, with learning rate set to 0.0001..",
"Finally, the weights of $\\mathcal {D}$ were initialized using the Xavier method [15].",
"DAA-GAN is implemented using PyTorch [32], while all experiments were conducted on an Nvidia GTX 1080 Ti graphics processor.",
"For all experiments, when performing anatomy arithmetic we allow only one pathology per subject.",
"That is, we may add a factor exhibiting a certain pathology into an otherwise healthy subject, or swap one factor for another factor of the same pathology, but we do not combine two factors with different pathologies in the same subject.",
"During inference, our model uses $\\mathcal {J}$ and $\\mathcal {G}$ to generate an image with the targeted pathology and $\\mathcal {F}$ to generate the pathology label, while $\\mathcal {D}$ is discarded.",
"Figure: Example of MRI sample from ACDC cardiac dataset (top left), the predicted segmentation masks of the ROI (top right), and the 5 (out of 12) most semantic disentangled anatomical factors.",
"Blue, yellow, and red show the segmentation prediction for the myocardium (MYO), left ventricle (LV) and right ventricle (RV), respectively."
],
[
"Dataset Details",
"ACDC pathology class annotations: a) normal (NOR), b) myocardial infarction (MINF), c) dilated cardiomyopathy (DCM), d) hypertrophic cardiomyopathy (HCM), and e) abnormal right ventricle (ARV).",
"ACDC segmentation annotations (3 semantic classes): left ventricular cavity (LV), myocardium (MYO) of the LV, and right ventricle (RV).",
"All images are resampled to 1.37mm$^2$ /pixel resolution and cropped to $224\\times 224$ pixels.",
"M$\\&$ Ms (see also https://www.ub.edu/mnms/) comprises class annotations for 19 pathologies.",
"However, 15 of them represent the $12\\%$ of the subjects that corresponds to 1-3 subjects per class, thus we choose to experiment on the 4 most dominant classes that are: a) NOR, b) DCM, c) HCM, and d) ARV.",
"Note that ARV in our experiments is the underrepresented class, but with 26 subjects.",
"M$\\&$ M segmentation annotations are identical with ACDC ones.",
"All images are resampled to 1.2mm$^2$ /pixel resolution and cropped to $224\\times 224$ pixels.",
"For both datasets during both model training and evaluation we combine pairs of MR images from the same heart axis slice level.",
"Figure: Four SPADE-generated images using overlapped anatomical factors as input.",
"Arrows point to the areas where the performed arithmetic operations lead to such overlaps."
]
] | 2107.01748 |
[
[
"Toward Increased Airspace Safety: Quadrotor Guidance for Targeting\n Aerial Objects"
],
[
"Abstract As the market for commercially available unmanned aerial vehicles (UAVs) booms, there is an increasing number of small, teleoperated or autonomous aircraft found in protected or sensitive airspace.",
"Existing solutions for removal of these aircraft are either military-grade and too disruptive for domestic use, or compose of cumbersomely teleoperated counter-UAV vehicles that have proven ineffective in high-profile domestic cases.",
"In this work, we examine the use of a quadrotor for autonomously targeting semi-stationary and moving aerial objects with little or no prior knowledge of the target's flight characteristics.",
"Guidance and control commands are generated with information just from an onboard monocular camera.",
"We draw inspiration from literature in missile guidance, and demonstrate an optimal guidance method implemented on a quadrotor but not usable by missiles.",
"Results are presented for first-pass hit success and pursuit duration with various methods.",
"Finally, we cover the CMU Team Tartan entry in the MBZIRC 2020 Challenge 1 competition, demonstrating the effectiveness of simple line-of-sight guidance methods in a structured competition setting."
],
[
"Motivation",
"Micro Aerial Vehicles (MAVs), also referred to as drones, Unmanned Aerial Vehicles (UAVs) and small Unmanned Aerial Systems (sUAS), have seen a huge growth in various market sectors across the globe.",
"Business Insider projects the sale of drones to surpass $12 billion in 2021, of which consumer drone shipments will comprise 29 million units [5].",
"Enterprise and governmental sectors generally have strict regulations under which MAVs are operated; however, not only is the consumer sector's operation of these aircraft weakly regulated but there is strong community pushback against any such legislation.",
"Stronger oversight of private drone use is further motivated by numerous incidents involving small, typically teleoperated drones in public spaces.",
"In December 2018, dozens of drone sighting reports over Gatwick Airport, near London, affected 1,000 flights and required the help of both the police and military, neither of whom were able to capture the drone over a 24-hour period [6].",
"Worries over the potential of UAVs above crowds at the 2018 Winter Olympics prompted South Korean authorities to train to disable drones, including developing a teleoperated platform which disables others with a dropped net (Figure REF ) [7].",
"Beyond these documented examples, there are numerous videos online of recreational drone users losing control of their aircraft due to weather conditions, loss of GPS positioning, or low battery behavior.",
"Figure: Current possible domestic-use counter-UAV systems in development.While these issues could be mitigated by enforcing strict regulations and oversight on the consumer drone market, this may also drastically curb the independence of hobbyists and researchers.",
"A potential alternative may be to capture or disable rogue UAVs in a non-disruptive way.",
"Current anti-UAV technology exists primarily in the military sector, in the form of jammers (used to disrupt the teleoperation signal from a nearby radio controller) or Stinger missiles (meant to disable a UAV by impact).",
"Neither of these options are suitable for domestic use, where both noise and debris are of issue.",
"Therefore, we need a solution that minimizes destruction while being agile enough to capture a wide variety of MAVs in local airspaces.",
"Quadrotors benefit from high, 4-degree of freedom (DOF) maneuverability and can accelerate to high speeds quicker than some single-rotor or fixed-wing counterparts.",
"This implies a higher capability to stay on an impact trajectory towards a target with an unknown flight plan or characteristics.",
"Furthermore, recent research in aerial robotics has shown that a suite of obstacle avoidance, detection, planning, and control can run fully onboard on an autonomous quadrotor platform.",
"Common shortfalls of quadrotors include low battery life, but for this mission type, flights are short but with high accelerations (and therefore, higher energy throughput)."
],
[
"Challenges and Approach",
"The challenges of autonomously impacting an unknown small aerial vehicle with a UAV are numerous, involving fast flight through potentially cluttered environments, as well as the development of a mechanically sound method of capturing the target without damage to either agent.",
"However, the primary challenge addressed in this thesis surrounds guidance and control of a quadrotor UAV towards a target.",
"In this thesis, we describe two projects to address this challenge.",
"In the first study, multiple control and guidance methods derived and inspired from different fields of literature are reviewed and modified for use on a simulated UAV-target scenario.",
"Here, the perception task is simplified and environmental factors are eliminated to focus on the evaluation of several guidance methods.",
"The second study evaluates LOS guidance in a robotics competition setting, specifically comprising of the CMU Team Tartan entry in the Mohamed Bin Zayed International Robotics Challenge 2020 Challenge 1.",
"This effort includes work on (a) planning an adjustable and robust path around a fixed arena based on measured GPS coordinates, (b) control towards semi-stationary targets placed throughout the arena, and (c) detection of a small yellow ball moving at 8m/s against a cluttered background.",
"A further challenge in this work is the localization of the target in the world relative to the UAV.",
"Depending on the size and shape of target (e.g.",
"fixed wing, multirotor, helicopter, blimp) as well as its distance, it cannot be assumed that a 3D sensor, such as LIDAR or stereo vision, can be used to accurately localize the target in space.",
"For example, because of their sparse and sometimes mesh-structured frames, multirotors in particular can be notoriously difficult to localize with cheap and lightweight scanning LIDARs or stereo cameras at long range.",
"Therefore, the focus in this thesis is to use monocular vision and adapt guidance methods to use only approximate depth estimates when necessary."
],
[
"Contribution and Outline",
"The main contributions of this thesis are as follows.",
"An evaluation of various guidance and control methods and how they might be adapted for use on a quadrotor in a simulated environment.",
"A software system using LOS-guidance for finding and targeting semi-stationary and moving targets within a fixed arena.",
"Chapter presents a short summary of related work in various fields, including classical visual servoing, missile guidance, and trajectory generation and tracking.",
"The following two chapters, and , expand on the work done specifically towards the two contributions listed above, respectively.",
"Chapter describes conclusions drawn from this work, shortcomings of the approach, as well as suggested future directions."
],
[
"Classical Visual Servoing",
"Visual servoing spans a wide range of research focusing on controlling robot links relative to input from visual sensors.",
"The most common application of visual servoing is in pick-and-place operations done with robotic arms fitted with cameras.",
"These robots generally either have a eye-in-hand (closed-loop control) or eye-to-hand (open-loop control) setup [9].",
"Two of the most common approaches in this field are image-based visual servoing (IBVS) and pose-based visual servoing (PBVS), with the difference between the two involving the estimation of the target's relative pose to the robot [10].",
"IBVS, as described in Hutchinson, et al.",
"(1996), is only useful within a small region of the task space unless the image Jacobian is computed online with knowledge of the distance-to-target, which is further complicated by a monocular vision-based system.",
"Unless target velocity is constant, errors or lag in the image plane with a moving target introduces errors in servoing with either method, which would in turn have to be tuned out with a more complex control system.",
"Chaumette and Santos (1993) [11] tracked a moving target with IBVS but assumed a constant acceleration.",
"When the target maneuvered abruptly, the Kalman filter-based target motion predictor took some cycles of feedback to recalibrate to the new motion.",
"In [12], the major pitfall of PBVS is pointed out as the need for 3D information of the target, specifically the depth which may not be readily available."
],
[
"Missile Guidance",
"Homing air missiles are singularly focused on ensuring impact with an aerial target.",
"Since at least 1956, proportional navigation in some form has been a standard in missile guidance methods [1].",
"Adler (1956) describes a couple of such methods, including constant-bearing navigation and proportional navigation.",
"It is noted that constant-bearing navigation, which applies acceleration to maintain a constant bearing-to-target, requires instantaneous corrections to deviations in the line-of-sight (LOS) direction.",
"This renders it incompatible with the dynamics of missiles, which cannot directly satisfy lateral acceleration commands (similar to fixed-wing aircraft); therefore, Adler proposes using 3D proportional navigation (PN) which applies a turning rate proportional to the LOS direction change.",
"In later texts, the term proportional navigation is used interchangeably between these two schemes, and also extended to other similar methods.",
"In this thesis, PN will be used as a general term to refer to any control law using the LOS rotation rate to generate an acceleration command.",
"As noted in [4], PN, when assuming no autopilot lag, is an optimal control law that assumes very little about the acceleration characteristics of the target.",
"However, variations on classical PN have also been developed that adapt to different flight profiles, including constant-acceleration and constant-jerk [13].",
"PN is typically split into two categories, the “true\" variant and the “pure\" variant [2].",
"Though the naming is largely arbitrary, the primary difference lies in the reference frame in which the lateral acceleration is applied to the pursuing missile.",
"True PN applies this acceleration orthogonal to the current missile velocity; Pure PN applies the acceleration orthogonal to the current LOS towards the target.",
"Generalized True PN (as seen in Figure REF from [2]) is not covered in this thesis.",
"Figure: Comparison of pure and true proportional navigation.",
"A M A_M and V M V_M refer to the missile's desired acceleration and current velocity, respectively ."
],
[
"Trajectory Generation and Tracking",
"Trajectories provide robots with smooth or kinodynamically feasible paths to follow through its state space.",
"This is opposed to sending raw differential commands, which may exceed the robot's limitations and lead to controller instability or even to physical breakdown of the robot's actuators.",
"As such, there has been extensive work in the generation and following of trajectories for use with various types of robots and applications, primarily with robot arms for grasping and self-driving vehicles [14][15][16][17].",
"This has been extended to MAVs to ensure smooth and efficient flight.",
"Richter, et al.",
"(2013) [18] showed that polynomial trajectories eliminated the need for an extensive sample-based search over the state space of the vehicle.",
"This approach, while not providing asymptotic convergence to the global optimal path, ensured smooth and fast flight of the quadrotor.",
"In [3], it was shown that with continuous minimum-jerk trajectories and fast re-planning, they achieved higher trajectory smoothness compared to other, more reactive planners.",
"Figure REF shows the smooth trajectory generated by tracking motion primitive-generated paths.",
"Gao, et al.",
"(2018) [19] first finds a time-indexed minimal arrival path that may not be feasible for quadrotor flight, and then forms a surrounding free-space flight corridor in which they generate a feasible trajectory.",
"They use a Bernstein polynomial basis and represent the trajectory as a piecewise Bézier curve.",
"Figure: Example of concatenated trajectories in a cluttered environment.",
"Colored lines represent individual motion primitives; the quadrotor tracks the initial part of every generated trajectory, forming a complete, stitched trajectory represented by the black line.",
"To follow trajectories, controllers take in a desired trajectory typically composed of position waypoints each with an associated velocity, and issue actuator commands to the robot.",
"In the MAV case, an autopilot software may accept attitude or attitude-rate commands which come from such a controller.",
"Hoffman, et al.",
"(2008) [20] demonstrated a controller that took as input a non-feasible trajectory and outputted feasible attitude commands for a quadrotor that accurately followed the original path.",
"This was demonstrated outdoors with 50cm accuracy.",
"A similar approach was taken (but extended to 3D) for the path following controller implemented in [21].",
"Here, the desired position and desired velocity from the closest point on the trajectory are used to find the position and velocity errors of the robot.",
"These are used to calculate the desired robot acceleration with PD feedback and a feedforward term.",
"As described in Section , servoing towards moving targets is challenging with classical methods.",
"As such, LOS-based guidance principals (Section ) and trajectory following methods () may produce better results when target acceleration is nonzero.",
"In addition, the quadrotor platform's control limits might be avoided with smooth trajectory-based methods.",
"This chapter focuses on the development and evaluation of various such guidance methods to achieve impact with a generalized form of an aerial, mobile target.",
"No information is known about the target other than its color, which is used for segmentation in an RGB image to simplify the detection and tracking problem."
],
[
"Derivation of True Proportional Navigation Guidance Law",
"Line-of-sight (LOS) guidance methods are used to apply acceleration commands to the pursuer that minimize change in the LOS vector towards the target.",
"In this section, the basic LOS geometry is introduced and used to derive proportional navigation (PN) guidance.",
"Following subsections show how this is used, with target detections, to calculate quantities used for the applied PN guidance.",
"Figure: LOS coordinate system .As seen in Figure REF , the fixed world coordinate frame is specified by the unit vectors $\\bar{\\mathbf {1}}_{\\mathbf {x}}$ , $\\bar{\\mathbf {1}}_{\\mathbf {y}}$ , $\\bar{\\mathbf {1}}_{\\mathbf {z}}$ .",
"The LOS coordinate frame, attached to the moving missile, is specified by the unit vectors $\\bar{\\mathbf {1}}_{\\mathbf {r}}$ , $\\bar{\\mathbf {1}}_{\\mathbf {n}}$ , $\\bar{\\mathbf {1}}_{\\mathbf {\\omega }}$ ; $\\bar{\\mathbf {1}}_{\\mathbf {r}}$ points along the LOS $\\bar{\\mathbf {r}}$ ; $\\bar{\\mathbf {1}}_{\\mathbf {n}}$ is the change in direction (i.e.",
"a rotation) of the LOS vector; $\\bar{\\mathbf {1}}_{\\mathbf {\\omega }}$ is the cross product of the former two, in that order (forming a right-handed coordinate frame).",
"In general, the angular velocity of the LOS coordinate frame is given by: $\\bar{\\dot{\\phi }}= \\dot{\\Phi }_{\\mathbf {r}} \\bar{\\mathbf {1}}_{\\mathbf {r}}+ \\dot{\\Phi }_{\\mathbf {n}} \\bar{\\mathbf {1}}_{\\mathbf {n}}+ \\dot{\\Phi }_{\\mathbf {\\omega }} \\bar{\\mathbf {1}}_{\\mathbf {\\omega }}$ Where $\\dot{\\Phi }_{\\mathbf {r}}$ , $\\dot{\\Phi }_{\\mathbf {n}}$ , $\\dot{\\Phi }_{\\mathbf {\\omega }}$ are the magnitudes of the components of the angular velocity defined as: $\\dot{\\Phi }_{\\mathbf {r}} &= \\bar{\\dot{\\phi }} \\mathchoice{\\mathbin {\\hbox{\\scalebox {.5}{$\\m@th \\displaystyle \\bullet $}}}}{}{}{}$ 1r n = 1n = 1 As derived in [4] but not reproduced here, the components of the relative acceleration between the missile and target are: $(\\bar{\\mathbf {a}}_T - \\bar{\\mathbf {a}}_M) \\mathchoice{\\mathbin {\\hbox{\\scalebox {.5}{$\\m@th \\displaystyle \\bullet $}}}}{}{}{}$ 1r = R - R 2 (aT - aM) 1n = 2R + R (aT - aM) 1 = R r Where $\\bar{\\mathbf {a}}_T$ and $\\bar{\\mathbf {a}}_M$ are the target and missile accelerations, respectively.",
"From this result, specifically using the condition in Equation REF , we can list sufficient conditions to satisfy the equation and ensure intercept: (i) interceptor is able to achieve an acceleration along the LOS greater than that of the target ($(\\bar{\\mathbf {a}}_T - \\bar{\\mathbf {a}}_M) \\mathchoice{\\mathbin {\\hbox{\\scalebox {.5}{$\\m@th \\displaystyle \\bullet $}}}}{}{}{}$ 1r < 0$), (\\textit {ii}) the initial rate of change in the range $ R$ is negative ($ R < 0$), which then ensures $ R < 0$ given the first condition, and (\\textit {iii}) the rate of change in the LOS is $ 0$ ($ = 0$).",
"Condition (\\textit {i}) depends on the nature of the interceptor and target; condition (\\textit {ii}) implies that PN pursuit must be initialized with a positive closing velocity; condition (\\textit {iii}) implies that the interceptor must satisfy acceleration commands such that the LOS vector remains constant.",
"Palumbo, et al.",
"(2010) finds the following true PN (TPN) law that ensures system stability:$ $\\bar{\\mathbf {a}}_M \\mathchoice{\\mathbin {\\hbox{\\scalebox {.5}{$\\m@th \\displaystyle \\bullet $}}}}{}{}{}$ 1n = N Vc , N>2 Where $N$ is a proportional gain and $V_c$ is the closing velocity.",
"In other words, the interceptor acceleration $\\bar{\\mathbf {a}}_M$ must have a component, orthogonal to the LOS, proportional to the rotation rate of the LOS as specified in Equation REF ."
],
[
"True Proportional Navigation",
"To generate the desired acceleration vector with magnitude specified by Equation REF and direction orthogonal to the LOS, we first calculate both $\\dot{\\Phi }_{\\mathbf {\\omega }}$ (directly represented in Equation REF ) and $\\mathbf {1_{\\mathbf {n}}}$ (acceleration direction).",
"The target's centroid in image frame coordinates at times $t-1$ and $t$ is represented by $(u_{t-1},v_{t-1})$ and $(u_{t},v_{t})$ , respectively, as shown in Figure REF .",
"The camera principal point, specified in the calibrated camera's intrinsic matrix (typically denoted $\\mathbf {K}$ ), is represented by $(c_x,c_y)$ .",
"Figure: Top-down diagram showing intermediate quantities in calculation of desired acceleration command.The LOS vector $\\mathbf {r_t}$ in the camera's frame of reference at time $t$ is given by the following.",
"$\\mathbf {r_t} = \\begin{bmatrix}[1.75] \\dfrac{u_t-c_x}{f_x} \\\\ \\dfrac{v_t-c_y}{f_y} \\\\ 1 \\end{bmatrix}$ In the special case of the first iteration of the algorithm, at $t=0$ , the LOS vector is calculated according to Equation REF then stored for use as $\\mathbf {r_{t-1}}$ the upcoming iteration.",
"For the $t=0$ computation cycle the control output is set to $\\mathbf {0}$ .",
"The angle spanned by the two vectors $\\mathbf {r_{t}}$ and $\\mathbf {r_{t-1}}$ is as follows: $\\Phi _{\\mathbf {\\omega }} = \\arccos {\\dfrac{ <\\mathbf {r_{t}},\\mathbf {r_{t-1}}> }{ ||\\mathbf {r_{t}}||||\\mathbf {r_{t-1}}|| }}$ Therefore, if the difference in time for one cycle is represented by $\\Delta t$ , then the magnitude of the rotation rate of the LOS vector is: $\\dot{\\Phi }_{\\mathbf {\\omega }} = \\frac{\\Phi _{\\mathbf {\\omega }}}{\\Delta t}$ The direction of the acceleration (direction of the LOS rotation) $\\mathbf {1_{\\mathbf {n}}}$ is shown in Figure REF and calculated below.",
"$\\mathbf {n} &= \\mathbf {r_{t}} - \\texttt {proj}(\\mathbf {r_{t}}, \\mathbf {r_{t-1}})\\\\ &= \\mathbf {r_{t}} - <\\mathbf {r_{t}}, \\frac{\\mathbf {r_{t-1}}}{||\\mathbf {r_{t}}||}> \\dfrac{\\mathbf {r_{t-1}}}{||\\mathbf {r_{t}}||}\\\\\\mathbf {1_{\\mathbf {n}}} &= \\dfrac{\\mathbf {n}}{||\\mathbf {n}||}$ Where the vector projection of $\\mathbf {r_{t}}$ onto $\\mathbf {r_{t-1}}$ is represented by the red vector in Figure REF .",
"With the scalar $\\dot{\\Phi }_{\\mathbf {\\omega }}$ and the vector $\\mathbf {1_{\\mathbf {n}}}$ , we can compute the desired acceleration as follows.",
"$\\mathbf {a}_{LOS^{\\prime }} = N V_c \\dot{\\Phi }_{\\mathbf {\\omega }} \\mathbf {1_{\\mathbf {n}}}$ In application on a quadrotor, in this work, the acceleration vector is fed into a velocity controller by integration of the command, which submits a roll, pitch, yawrate, thrust command to the internal autopilot controllers.",
"Therefore, rather than adjusting heading to satisfy lateral accelerations, the application of TPN in this work relies on roll angle control.",
"This more direct method of achieving lateral accelerations (that does not require forward velocity) is not possible on a missile or fixed-wing aircraft."
],
[
"Proportional Navigation with Heading Control",
"The TPN algorithm presented above maintains the integrity of the algorithm commonly presented in missile guidance literature, but applies the control command more directly by controlling the roll angle of the UAV.",
"During a missile's flight, the vehicle fulfills desired acceleration commands by flying in an arc, gradually changing its heading by relying on forward motion and the use of thrust vectoring or control surfaces.",
"A quadrotor, however, has direct control over its heading by applying yaw-rate control.",
"In this section, we describe an algorithm that uses PN acceleration in all axes but the lateral axis, and instead controls the heading to achieve lateral acceleration.",
"Since it does not utilize PN in the lateral axis, we do not assign the label of “true\".",
"We define an inertial reference frame at the center of the UAV, with $x$ pointing forward, $y$ pointing to the left, and $z$ point upward.",
"The acceleration along the $x$ and $z$ axes are simply taken from Equation REF as the corresponding components: $a_{x} &= \\mathbf {a}_{LOS^{\\prime }} \\mathchoice{\\mathbin {\\hbox{\\scalebox {.5}{$\\m@th \\displaystyle \\bullet $}}}}{}{}{}$ 1x az = aLOS' 1z The heading control composes of a commanded yaw-rate, which includes the computation of the heading: $\\dot{yaw} &= K_{P,yaw} heading\\\\&= K_{P,yaw} \\arctan { \\dfrac{\\mathbf {r_t} \\mathchoice{\\mathbin {\\hbox{\\scalebox {.5}{$\\m@th \\displaystyle \\bullet $}}}}{}{}{}}{\\mathbin {\\hbox{\\scalebox {.5}{$\\m@th \\textstyle \\bullet $}}}}}{\\mathbin {\\hbox{\\scalebox {.5}{$\\m@th \\scriptstyle \\bullet $}}}}$ 1yrt 1x Where $K_{P,yaw}$ is a tuned proportional gain and $\\mathbf {r_t}$ is the current LOS vector."
],
[
"Hybrid TPN-Heading Control",
"There are potential benefits to both methods presented in Sections REF and REF .",
"TPN specifically applied to quadrotors via roll angle control might yield quicker reaction time for a moving object.",
"PN while keeping the target centered in the frame ensures that the target is not lost from frame; otherwise, in a full system, the pursuing UAV would have to return to a search pattern.",
"The goal of the hybrid algorithm is to capture the advantages of both methods.",
"This method switches between the two modes, PN and Heading.",
"The transition between them simply relies on a tuned threshold $k_{heading}$ on the heading towards the target.",
"If $|heading| < k_{heading}$ , enter state PN: $a_{x} &= \\mathbf {a}_{LOS^{\\prime }} \\mathchoice{\\mathbin {\\hbox{\\scalebox {.5}{$\\m@th \\displaystyle \\bullet $}}}}{}{}{}$ 1x ay = aLOS' 1y az = aLOS' 1z yaw = 0.2 KP,yaw heading If $|heading| \\ge k_{heading}$ , enter state Heading Control: $a_{x} &= \\mathbf {a}_{LOS^{\\prime }} \\mathchoice{\\mathbin {\\hbox{\\scalebox {.5}{$\\m@th \\displaystyle \\bullet $}}}}{}{}{}$ 1x ay = 0.2 aLOS' 1y az = 0 yaw = KP,yaw heading Where $k_{heading}$ may be tuned depending on certain factors of the UAV system, including the camera field-of-view (FOV) or the maximum yaw-rate.",
"Note that at all times, regardless of the heading, both $a_{y}$ and $\\dot{yaw}$ are nonzero, and are instead suppressed with a factor less than 1.",
"The factor of 0.2 was found empirically in this study to perform well and yield an appropriate influence of both acceleration and yaw-rate.",
"Using this hybrid method, the UAV may potentially be able to react to changes in target motion while also keeping it in view."
],
[
"Trajectory Following",
"All trajectory following methods were implemented with some replanning rate at which updated trajectories are published.",
"Replanning is constantly done from the look ahead point, which is maintained by the Trajectory Controller as described in Section REF ."
],
[
"$LOS^{\\prime }$ Acceleration Trajectory Following",
"These trajectories are formed by taking the desired acceleration command calculated with Equation REF and calculating position waypoints and velocities given the starting position set to the look ahead point.",
"The calculations for the positions and velocities are done with the following kinematic equations, set in the UAV's inertial reference frame.",
"$\\mathbf {p}_t &= \\mathbf {v}_0t + \\frac{1}{2} \\mathbf {a}_{LOS^{\\prime }} t^2\\\\\\mathbf {v}_t &= \\mathbf {v}_0 + \\mathbf {a}_{LOS^{\\prime }} t , \\, t=0,0.1,...,T$ Where $T$ is the time length of each trajectory.",
"As $T$ approaches 0, this method becomes equivalent to commanding the desired $LOS^{\\prime }$ acceleration directly.",
"The discretization of the timesteps is also a tunable parameter."
],
[
"Target Motion Forecasting",
"In its simplest form, target motion forecasting involves estimating the target velocity in 3D, calculating the future location of the target with a straight-line path, and generating a collision course trajectory towards that point in space at a velocity which completes the trajectory at the specified time.",
"This method makes three critical assumptions: (i) the target has zero acceleration, (ii) we can approximate time-to-collision by calculating the time along the current LOS vector, and (iii) the forecasted target position will change slowly, so generating straight-path trajectories is sufficient to result in a final, smooth stitched trajectory.",
"First, the LOS unit vectors at two times are calculated.",
"These are used along with the depth estimation to find the target's velocity: $\\mathbf {v}_{target} = \\dfrac{d_1(\\mathbf {1_{LOS}}_1)-d_0(\\mathbf {1_{LOS}}_0)}{t_1-t_0}$ $d_1$ and $d_0$ , $\\mathbf {1_{LOS}}_1$ and $\\mathbf {1_{LOS}}_0$ , and $t_1$ and $t_0$ are the estimated depth, calculated LOS unit vector, and time, at two timesteps.",
"Once we have the velocity, we find the approximate time-to-collision along the current LOS vector by using the UAV's velocity component along the LOS: $t_{collision} = \\dfrac{d_1}{<\\mathbf {v}_{UAV}, \\mathbf {1_{LOS}}_1>}$ Where $\\mathbf {v}_{UAV}$ is the current UAV velocity vector.",
"Therefore, the approximate point in space to plan the trajectory to is as follows, where $\\mathbf {p}_{collision}$ is in the UAV reference frame.",
"$\\mathbf {p}_{collision} = \\mathbf {v}_{target}t_{collision} + d_1(\\mathbf {1_{LOS}}_1)$"
],
[
"System",
"Figure REF shows a diagram of the most important parts of the system, including perception, control, and the simulation software.",
"Each block generally consists of one or two nodes in ROS (Robot Operation System), the framework used in this work."
],
[
"Object Segmentation",
"Since the focus for this chapter was on the comparison of guidance methods, the perception challenge was simplified.",
"The simulation does not include rich background visuals, there is no dropout in the camera feed, and the object was reduced from a potential UAV shape to a yellow sphere.",
"The segmentation node composes of color thresholding that creates a binary segmentation image from each camera frame in real-time (30Hz), as seen in Figure REF .",
"In addition, this node publishes the centroid $(C_x,C_y)$ of the detected object by using the image moments.",
"$\\begin{bmatrix}C_x \\\\ C_y\\end{bmatrix}=\\begin{bmatrix}[2.0]\\dfrac{M_{10}}{M_{00}} \\\\ \\dfrac{M_{01}}{M_{00}}\\end{bmatrix}$"
],
[
"Object Depth Estimation",
"Depth estimation with monocular camera images relies on prior knowledge of the target size as well as the assumption that the target is spherical (which is correct in this simulation, but extendable to real-life targets with approximate knowledge of target or aircraft type).",
"The estimate takes into account the camera intrinsic parameters and projects the 2D information (pixels) to 3D (rays starting at the camera center) to extract the depth.",
"Pixel coordinates at two points in the image need to be located which form a 2D line in the image that projects to a 3D line along the diameter of the object.",
"The easiest found such line is the line through the image principal point that passes through the centroid, since the target is a sphere.",
"This line is the dotted line passing through the target in Figure REF .",
"We must first find the pixel coordinates along this line that correspond to the object edges in the image.",
"First, we find the rotation angle formed by the centroid coordinates with respect to the image horizontal, labeled $\\theta $ in Figure REF .",
"$\\theta = \\arctan \\dfrac{C_y-c_y}{C_x-c_x}$ With this angle we can use a rotation matrix to rotate the segmented object onto the $y=c_y$ line.",
"Note that we use $-\\theta $ to achieve the correct rotation, and that we first compute new pixel coordinates $(u_i,v_i)$ relative to the image center, from the original coordinates $(x_i,y_i)$ .",
"$\\begin{bmatrix}u_{i} \\\\ v_{i}\\end{bmatrix}&=\\begin{bmatrix}x_i - c_x \\\\ y_i - c_y\\end{bmatrix}, \\; i=1,...,N \\\\\\begin{bmatrix}u_{i,R} \\\\ v_{i,R}\\end{bmatrix}&=\\begin{bmatrix}cos(-\\theta ) && -sin(-\\theta )\\\\sin(-\\theta ) && cos(-\\theta )\\end{bmatrix}\\begin{bmatrix}u_{i} \\\\ v_{i}\\end{bmatrix}, \\; i=1,...,N$ Where N is the total number of segmented pixels in the binary segmentation image.",
"Once the object is rotated onto the image horizontal, simple min and max operations yield the extreme pixel coordinates on the left and right of the target (or right and left, if $C_x<c_x$ ).",
"$\\begin{bmatrix}u_{left} \\\\ v_{left}\\end{bmatrix}&=\\begin{bmatrix}min_i(u_i) \\\\ c_y\\end{bmatrix}, \\; i=1,...,N\\\\\\begin{bmatrix}u_{right} \\\\ v_{right}\\end{bmatrix}&=\\begin{bmatrix}max_i(u_i) \\\\ c_y\\end{bmatrix}, \\; i=1,...,N$ These two points are shown in red in Figure REF .",
"We apply the inverse rotation and translation to get the original coordinates of these two identified pixel coordinates.",
"$\\begin{bmatrix}u_{left,R^{-1}} \\\\ v_{left,R^{-1}}\\end{bmatrix}&=\\begin{bmatrix}cos(\\theta ) && -sin(\\theta )\\\\sin(\\theta ) && cos(\\theta )\\end{bmatrix}\\begin{bmatrix}u_{left} \\\\ v_{left}\\end{bmatrix}+\\begin{bmatrix}c_x \\\\ c_y\\end{bmatrix}\\\\\\begin{bmatrix}u_{right,R^{-1}} \\\\ v_{right,R^{-1}}\\end{bmatrix}&=\\begin{bmatrix}cos(\\theta ) && -sin(\\theta )\\\\sin(\\theta ) && cos(\\theta )\\end{bmatrix}\\begin{bmatrix}u_{right} \\\\ v_{right}\\end{bmatrix}+\\begin{bmatrix}c_x \\\\ c_y\\end{bmatrix}$ These two pixels are projected to 3D rays similar to the LOS ray calculation in Equation REF .",
"They can also be seen in a similar top-down view in Figure REF .",
"Once the two rays are found, the depth (to the nearest point on the object) is as follows: $d = \\dfrac{w/2}{\\sin {(\\alpha /2)}} - \\dfrac{w}{2}$"
],
[
"$LOS$ and {{formula:d6696680-365c-4a40-8438-df53bb400fdf}} Computation",
"The calculation of $\\mathbf {r}_t$ , the LOS vector, can be found in Equation REF .",
"The calculations of $\\dot{\\Phi }_{\\mathbf {\\omega }}$ and $\\mathbf {1}_{\\mathbf {n}}$ , the scaling and directional components of LOS', can be found in Equations REF and REF .",
"Before being input to the Guidance node, the quantities undergo smoothing with a flat moving average filter.",
"The filtered values are used for trajectory-based guidance, while LOS-based guidance uses the raw values."
],
[
"Guidance",
"The Guidance node either executes LOS-based or trajectory-based guidance.",
"The LOS-based guidance, as described in Section , utilizes the LOS and LOS' computation from the Perception block.",
"In this case, the node outputs acceleration commands that are satisfied by the Velocity Controller.",
"Trajectory-based guidance utilizes both the LOS information as well as the depth estimate to generate 3D trajectories towards the target's current state or forecasted motion.",
"Here, the node outputs a trajectory composing of waypoints and corresponding speeds which is accepted by the Trajectory Controller.",
"Every method is initialized with two seconds of simple LOS guidance, where the UAV accepts velocity commands directly along the current LOS vector."
],
[
"Trajectory, Pose, and Velocity Controllers",
"The Trajectory Controller accepts a trajectory in the form of a list of waypoints, each with a corresponding $(x,y,z)$ position, yaw, and speed (scalar).",
"It outputs a tracking point, which the Pose Controller takes as input for tracking the trajectory, and a look ahead point, which is used as the replanning point.",
"The configurable look ahead time sets how far ahead the look ahead point is from the tracking point, and is approximated to be $1/f + \\Delta buffer$ , where $f$ is the replanning frequency and $\\Delta buffer$ is a buffer in case there is some lag in the system.",
"The Pose Controller composes of PID controllers with the tracking point from the Trajectory Controller as the reference and the odometry from the PX4 SITL interface as the actual.",
"This outputs a velocity reference (and a velocity feedforward term from the Trajectory Controller) to the Velocity Controller, which also accepts the odometry from the PX4 SITL interface as the actual.",
"The Velocity Controller also uses PID controllers, and outputs a roll, pitch, yawrate, thrust commands to the PX4 SITL interface."
],
[
"Gazebo and PX4 SITL",
"Simulation was used to develop, deploy, and evaluate many UAV guidance algorithms quickly.",
"In this environment, the behavior of the UAV and the target was easily modified and could be strained without the safety risk involved in real-world testing.",
"Gazebo [22], the simulator of choice, is commonly used for robotics testing and simulates robot dynamics with the ODE physics engine.",
"This was paired with PX4 autopilot [23] software-in-the-loop (SITL) for control of the quadrotor.",
"An example of the simulated world can be seen in Figure REF .",
"Obstacles, a visually rich backdrop, and other factors were eliminated in the simulation setup to reduce the impact of external factors on the evaluation of the guidance algorithms.",
"When deployed on a real-world UAS, the algorithms here can be merged within a larger autonomy architecture including robust detection and tracking, as well as obstacle avoidance and others.",
"The quadrotor is outfitted with a RGB camera that models a realistic vision sensor."
],
[
"Target Position Controller",
"A library of target paths was used to strain the UAV's capability of intercepting the target.",
"The first, simplest target motion is a straight path with constant velocity, which may mimic an autonomous aircraft on a search pattern or a fixed-wing plane.",
"The starting position places the target on either side of the UAV's FOV, and the path crosses in front of the UAV with some random variation in slope.",
"The third trajectory is a figure-8 with a randomized 3D tilt, similar to an evasive maneuver a small aircraft may take.",
"These trajectories can be seen in Figure REF .",
"The third trajectory composes of a knot shape filling a 2m$\\times $ 2m$\\times $ 2m space at a random location within a 10m$\\times $ 20m$\\times $ 10m area in front of the UAV.",
"This more rapid movement back and forth is similar to a multi-rotor hovering in a changing wind field."
],
[
"Results",
"In this section, we evaluate the five guidance algorithms described in the sections prior (Sections REF , REF , REF , REF , REF ).",
"An experiment configuration composed of a selection of one parameter from each of the following categories.",
"Each configuration underwent 50 trials.",
"UAV Guidance: True Proportional Navigation (TPN), Proportional Navigation with Heading Control (PN-Heading), Hybrid True Proportional Navigation with Heading Control (Hybrid TPN-Heading), LOS' Trajectory, Forecasting Trajectory UAV speed [m/s]: 2.0, 3.0, 4.0, 5.0 Target path: Straight, Figure-8, Knot Target speed [% of UAV speed]: 25%, 50%, 75%, 100% In the case of sinusoidal target paths (figure-8 and knot), the target speed was set by determining the length of the path and dividing by the desired speed to calculate the period of the sinusoids.",
"The primary metric for comparing different methods is the first-pass hit rate, presented in Section REF .",
"All of the following conditions must be met for a trial to be considered a successful first-pass hit on the target.",
"UAV is within 0.5m of the target (measured from the closest point on the target to the center of the UAV).",
"Duration of pursuit is less than 20s.",
"UAV stays within a 35m$\\times $ 100m$\\times $ 40m area surrounding the target.",
"Target is not outside the UAV camera's FOV for more than 3s.",
"These conditions were specified after consideration of the simulation scene, UAV model, and target model.",
"The target is a yellow sphere of diameter 1m.",
"The RGB camera has a horizontal FOV of 105$$ with a size 680$\\times $ 480.",
"In lieu of a gimbal camera in simulation, the UAV model had a rigidly fixed, angled camera that was adjusted for each speed, to compensate for the steady state pitch down when flying forward at high speeds.",
"The next metric presented is the mean pursuit durations, in Section REF .",
"These were computed by taking the mean of the time-to-hit measurements over each of (a) target speed, and (b) UAV velocity, respectively.",
"This was done using only the successful hits during each experiment."
],
[
"First-Pass Hit Rates",
"Datapoints in the following heatmaps that are inside a black square represent experiments that were not able to complete more than 95% of the trials without crashing due to UAV instability with the particular experiment configuration.",
"Figure: True Proportional Navigation hit rate across three target paths.Figure: Proportional Navigation with Heading Control hit rate across three target paths.Figure: Hybrid True Proportional Navigation-Heading Control hit rate across three target paths.Figure: LOS' Trajectory hit rate across three target paths.Figure: Forecasting Trajectory hit rate across three target paths.TPN achieves the highest hit rate across almost all configurations compared to the other methods in both classes (LOS Guidance and Trajectory Following).",
"Across all experiments there is a trend of lower UAV and target speeds resulting in higher hit rates, sometimes even of 1.0 (i.e., 50 of 50 trials result in a hit).",
"Moving down and to the right within each subfigure presents results at higher target and UAV speeds.",
"This can be seen as increasing the closing velocity, which reduces the time that the UAV has to react to changes in target motion.",
"As seen in the figures, this results in lower hit rates, which may be due to lag in the UAV controllers' ability to fulfill desired acceleration commands.",
"The quadorotor achieves a lateral acceleration more directly than a fixed-wing craft by inducing a roll angle and thereby shifting a component of the thrust to this axis.",
"However, the moment of the aircraft and the response time of the controller both contribute to lag in achieving the necessary roll angle.",
"Therefore, as the time allowed for this control decreases, with the increase in closing velocity, it is more unlikely that the necessary lateral acceleration will be achieved.",
"Similar logic applies to the thrust controller for achieving desired acceleration in the $z$ axis.",
"It is possible that a more accurate controller might increase the hit rates at high closing velocities.",
"The Hybrid TPN-Heading method generally had higher hit rates than PN-Heading, but lower than TPN.",
"However, Hybrid TPN-Heading consistently performed better than TPN at a low target speed (25%) in the straight and figure-8 target path, with the highest increase in hit rate as 0.28 among these configurations.",
"This suggests that similar to the behavior of PN-Heading at low target speeds (further described in a below section), Hybrid TPN-Heading is able to chase the target at these speeds even if the initial TPN-driven approach is unsuccessful.",
"The LOS Guidance class of methods generally has higher hit rates than the Trajectory Following methods implemented here.",
"When designing these algorithms and implementing them, it was found that significant parameter tuning and filtering was necessary to improve the results of the Trajectory Following methods.",
"For example, although the LOS' Trajectory method produces smooth trajectories and therefore smoother UAV flight, it has to use a filtered (smoothed with flat moving average filter) LOS' in order to create consistent trajectories.",
"This filtering introduces lag, which quickly becomes intractable when the target or UAV speed is increased past what was used for tuning the system.",
"This is amplified in the case of the Forecasting Trajectory.",
"While Equation REF is geometrically sound, in the simulated system imperfections in depth estimation, LOS computation, and corresponding frame transformations (especially during high roll- and pitch-rates) cause inaccuracies that require filtering and tuning to make the 3D target forecast feasible.",
"These limitations are largely a byproduct of using only monocular camera information to estimate 3D positioning.",
"The effect can be seen in the UAV instability in the majority of experiments shown in Figure REF .",
"While TPN generally achieves the highest hit rate compared to all other methods presented here, a notable exception is the slowest configuration of UAV velocity 2m/s and target speed 25% with the straight target path, which has a hit rate of 0.48.",
"Figure REF shows that the target progressively gets farther out of view as the UAV flies by.",
"The straight target path begins on one side of the pre-defined arena space and terminates on the other side, with randomized variation in path slope.",
"Since the target moves so slowly in this configuration, it remains on one side of the UAV's image.",
"The $y$ component of the body-frame acceleration command generated by TPN while the target is moving towards the edge of the image (due to the UAV's forward velocity) is not enough to keep the target in sight, and it slowly slips out of view.",
"This problem becomes less likely at higher UAV speeds, however, since the greater motion of the UAV creates more motion of the target in the image, thereby creating a larger lateral acceleration command that keeps the target in view.",
"This issue is only apparent in the straight path case since only with this path the starting target location had to be on the edges of the image to compensate for larger movement at higher target speeds.",
"This issue does not appear in the PN-Heading or Hybrid TPN-Heading methods, since they compensate by commanding yaw-rate to center the object in the image.",
"Figure: Demonstration of TPN, straight target path, UAV speed 2m/s and target speed 25% (0.5m/s); corresponding datapoint is represented in top left corner of Figure ."
],
[
"PN-Heading at High Speeds",
"PN-Heading has lower performance at high UAV or target speeds relative to both TPN or Hybrid TPN-Heading.",
"This was often observed to be due to the time required to effectively change the UAV's velocity direction through a yaw-rate, and this time delay increases further as the UAV speed increases.",
"A notable artifact in the results can be seen in Figure REF , where the results at 100% target speed is 0 at every UAV velocity.",
"This is due to the way in which the UAV “chases\" the target in the straight target path scenario.",
"Figure REF shows that the UAV might pass the target initially, but turns towards it via commanding a yawrate, and finds it again (within the 3s detection timeout).",
"Then it is able to pursue the target with PN acceleration commands in the $x$ and $z$ directions.",
"However, during this chase period, the UAV's velocity is in the same direction as the target's.",
"Therefore, the target's speed must be less than the UAV's, or it will be impossible to maintain a nonzero closing velocity.",
"This is the case specifically for the bottom row of Figure REF .",
"This does not occur with the other target motions since the target has non-zero acceleration.",
"If there were no hits across all experiments for a particular configuration, then those data points are missing in the figures, as seen in REF and REF .",
"Figure: Mean pursuit durations for straight target path.Figure: Mean pursuit durations for figure-8 target path.Figure: Mean pursuit durations for knot target path.The pursuit times generally decreased as UAV velocity increased, though this trend is not as apparent in some of the guidance methods in Figure REF .",
"In the straight target path scenario, because the target has constant velocity it will eventually be out of the UAV's FOV, restricting the time window in which an intercept is possible.",
"A strong downward trend is still apparent with TPN and PN-Heading.",
"The effect of target speed on pursuit duration is most observed in the straight target path case.",
"Here, the increase in target speed caused increased times for PN-Heading and Hybrid TPN-Heading.",
"Both of these methods utilize yaw-rate control to keep the target centered in the UAV's camera image, and as shown in Figure REF , this can result in the UAV turning and “chasing\" the target as it passes.",
"As the target speed increases, the probability of a successful hit on first approach goes down, which then increases the chance of this kind of chasing maneuver.",
"LOS' Trajectory, though using a smoothed version of the LOS' used for TPN, does not exhibit as strong of a decrease in pursuit time over increasing UAV velocity or target speed.",
"These results actually show that of all methods, LOS' Trajectory has the lowest pursuit durations with few exceptions.",
"However, when paired with the results in Figure REF , it seems more likely that these times are a byproduct of being most likely to hit a target that is initialized closer to the UAV starting point."
],
[
"Pursuit Behavior",
"In this section, we present each method's UAV path relative to target motion in hand-picked configurations for qualitative assessment.",
"Figure: TPN behavior.Figure: PN-Heading behavior.Figure: Hybrid TPN-Heading behavior.Figure: LOS' Trajectory behavior.Figure: Forecasting Trajectory behavior.Figure REF shows the UAV track the target's motion as it dips and then rises again as it moves through one half of the figure-8 path.",
"Figure REF shows the UAV approach the straight target path but make a sharp left turn via yaw-rate commands as it passes by.",
"It then goes on to implement PN to catch the target.",
"Figure REF starts with motion similar to TPN, then uses heading control to yaw towards the target when it is close and the heading becomes larger.",
"Figure REF shows a smoother path than TPN, as a result of smoothing the PN commands and utilizing trajectory following.",
"The trajectory generated at the timestamp shown in the image is shown in yellow.",
"Figure REF shows the path mimicking the motion of the target, but shifted in space due to forecasting of the target motion.",
"The current forecasted trajectory is shown in yellow."
],
[
"Introduction",
"The Mohamed Bin Zayed International Robotics Challenge 2020 is an outdoor robotics competition in which dozens of international teams, including many top robotics universities, demonstrate autonomous performance in different tasks.",
"“MBZIRC aims to provide an ambitious, science-based, and technologically demanding set of challenges in robotics, open to a large number of international teams.",
"It is intended to demonstrate the current state of the art in robotics in terms of scientific and technological accomplishments, and to inspire the future of robotics.\"",
"[24] Teams had the choice of competing in any or all of three challenges, differentiated by the types and number of robots allowed, the theme of the tasks involved, and physically separated arenas.",
"Challenge 1: Airspace Safety involved aerial robots targeting semi-stationary and moving targets; Challenge 2: Construction involved both ground and aerial robots building separate wall structures with provided blocks of varying size and weight; Challenge 3: Firefighting involved ground and aerial robots cooperating to extinguish real and fake fires surrounding and inside a three-story building.",
"The work towards competing in Challenge 1 and the associated competition results are the most relevant towards the thesis of airspace safety, and are therefore the focus in this report.",
"In Challenge 1, the two primary tasks are as follows: Task 1: Semi-stationary targets.",
"Pop five 60cm-diameter green balloons placed randomly throughout the arena.",
"Task 2: Moving target.",
"Capture and return a 15cm yellow foam ball hanging from a UAV flying throughout the arena in a figure-8 path.",
"The balloons were placed on rigid poles but filled with helium and attached to the pole with string; therefore, they can move in a limited range due to wind or downdraft from a UAV's propellers.",
"The moving target moved at speeds up to 8m/s along a figure-8 trajectory at some unknown altitude.",
"After 8 minutes, the speed of the target was reduced to 3m/s.",
"The ultimate goal is to remove the ball from the magnetic tether and return it to a dropoff location in the arena.",
"The total time allowed for this challenge was 15 minutes, without the possibility of pausing the clock during robot resets.",
"There was a choice of running the robots in autonomous or manual mode (scores in either category are compared separately), but all competition runs by Team Tartans were completed in autonomous mode.",
"Each team was allowed to have at most three aerial robots in action at any given time in the arena.",
"We used two UAVs, one for each of the two tasks listed above.",
"The discussion in this thesis focuses on the most developed work done in the efforts towards this competition; as such, the entire software stack is discussed for Task 1, but only the target capture (not return) portion of Task 2 is covered."
],
[
"Software System",
"Both UAVs used for Tasks 1 and 2, respectively, were very similar in their software stacks for perception, planning/guidance/control, and interface with the DJI SDK.",
"The system can be simplified to the diagram seen in Figure REF ."
],
[
"Object Detection",
"The target objects in both tasks (balloons and ball) have distinct color and shape.",
"For this reason, classical object detection and segmentation techniques such as color thresholding and ellipse fitting were tried due to their low computation and ease of implementation.",
"However, through multiple trials in different environments (sunny, cloudy, grassy or urban backgrounds) the performance of these detectors failed unless hand-tuned for each case.",
"We therefore turned to a deep learning approach, and evaluated different object detection architectures for accuracy and speed.",
"The result was use of the Tiny YOLOv3 architecture, which provided high accuracy and real-time inference when paired with the Intel OpenVino framework.",
"This framework optimizes the architecture to run on a CPU.",
"The accuracy of the network inference was increased by tuning hyperparameters during training, as well as augmenting the hand-collected dataset.",
"Data of sample balloons and balls were collected in different settings (indoor/outdoor, sunny/cloudy, with/without obstruction or multiple objects in scene).",
"Data augmentation also helped to increase the size and variation in the dataset.",
"Since there are multiple balloon targets placed throughout the arena, it is possible that in any given camera frame there may be multiple registered, true positive detections.",
"To eliminate ambiguity in the balloon detection and targeting pipeline, certain conditions must be met for a detection to be considered a valid target.",
"The primary condition is that the total area of the bounding box must be greater than some threshold.",
"This parameter was specified as a percentage of the total image area, and was tuned during competition according to the size of the balloons used.",
"Though two networks were trained and used for each of the two tasks, and each task's objects were colored differently (green balloons vs. yellow ball), false positives would often result from detections of the other task's targets.",
"The most common case was the ball-trained network detecting balloons, since the Task 2 UAV had a much wider view of the arena from the higher altitude than the Task 1 UAV, which stayed under a 5m artificial ceiling.",
"This problem was mitigated during the competition by filtering out detections from the ball-trained network based on location in the image; detections in the bottom 30% of the image were discarded since the ball was expected to be at or above the Task 2 UAV's altitude."
],
[
"$LOS$ Computation",
"This computation was performed with camera parameters obtained from the calibrated gimbal camera, published by the DJI SDK.",
"Please refer to Equation REF for the calculation.",
"The LOS vector was then passed to the LOS Guidance node."
],
[
"Global Planner",
"The Global Planner produced a global trajectory search pattern for both Tasks 1 and 2.",
"For Task 1, this consisted of a forward-backward lawnmower path running over a specified portion of the arena, at a preset altitude.",
"It stitches separate trajectory segments together to create a complete trajectory including takeoff, recovery after LOS Guidance, and landing.",
"The sweep width, altitude, and velocity were tuned during competition (e.g.",
"6m sweep width, 2.4m altitude, and 2m/s velocity).",
"After LOS Guidance is triggered due to a valid target detection, the Global Planner pauses the trajectory.",
"Once the state machine returns to the Global Planner, a segment is stitched to the original search pattern starting from the current UAV odometry, going up 1.5m, then going back to the original point at which the original detection was triggered.",
"This is a simple way to ensure robustness in the balloon-popping pipeline.",
"For Task 2, the Global Planner produced a trajectory consisting of a square path in the center of the arena at a higher altitude than the Task 1 UAV.",
"The orientation of the UAV was set to always align with the long side of the arena, facing towards the cross portion of the figure-8 path of the target.",
"This ensured that the UAV should have the target both in view and in close proximity for the longest time possible.",
"Once the target was identified, the state machine passes into LOS Guidance mode, and the UAV only returns to the Global Planner mode if the target has not been seen for the approximate time it takes for the target to complete two passes.",
"The altitude of the search pattern was set to 10-12m during competition, with a slower velocity of 1m/s compared to Task 1.",
"The wait time before transitioning from LOS Guidance mode to Global Planner was 60s, sufficient time for two passes of the target."
],
[
"$LOS$ Guidance",
"Both Tasks 1 and 2 use the LOS vector to guide the UAV towards targets.",
"For Task 1, once a valid detection is registered, as specified in Section REF , the UAV enters LOS Guidance mode which attempts to puncture the target balloon with the propellers.",
"This guidance mode runs through two states: Adjust and Attack.",
"In the first state, Adjust, the node outputs z-velocity (altitude) and yawrate commands to satisfy two conditions: (i) the upwards angle of the LOS vector (computed in the camera frame) is within 5 degrees of a set angle threshold, and (ii) the horizontal angle of the LOS vector is within 5 degrees of 0.",
"Once these conditions are met, the guidance transitions to the Attack state, where the UAV satisfies velocity commands along the LOS until a certain time period after the target has lost sight.",
"This relies on the assumption that no targets will be placed sufficiently close to any obstacles or boundaries to require obstacle avoidance immediately after popping the balloon.",
"The Adjust state satisfies these conditions so that when the UAV moves along the LOS, the upwards trajectory makes it more likely that the propellers will hit the balloon without causing a miss due to propeller downdraft.",
"After the Attack state is complete, control is passed back to the Global Planner.",
"In Task 2, once a valid detection is registered, LOS Guidance issues velocity proportional to the LOS vector in the y- (lateral) and z-direction (altitude).",
"The x-direction (forward) component of the velocity vector is 0.",
"This adjusts the UAV position to align with the target's path.",
"Once the UAV loses sight of the target, it transitions to the Wait state and waits in place for the next pass.",
"If the target is not seen within some period of time (tuned during competition to be approximately the duration of two cycles of the target's path) then it will pass control back to the Global Planner.",
"Otherwise, if the target is seen again, then velocity commands are again issued along the LOS to further refine the UAV position."
],
[
"Trajectory and Pose Controllers",
"These controllers are the same as used for the previous portion of this thesis, so please refer to Chapter for additional details.",
"Since the DJI SDK node contains an internal velocity controller and therefore accepts velocity commands, only Trajectory and Pose Controllers were used for this platform."
],
[
"DJI SDK",
"The DJI Onboard SDK was used for interfacing with the UAV sensors and autopilot.",
"Specifically, the $\\texttt {dji\\_sdk}$ ROS package was used with the specific SDK version corresponding to the DJI M210 V2.",
"It should be noted that while the SDK supported gimbal camera angle control, a custom gimbal angle controller was never successful in controlling the gimbal angle when the UAV was yawed 180 degrees from the initial UAV orientation.",
"Because of this, a custom gimbal angle controller was never able to be used, and the team relied on the internal SDK controller to maintain the preset angle set relative to the UAV.",
"This sometimes failed mid-flight, which on at least one occasion specifically caused missed targets in Task 1.",
"It was later found that there was a known bug in the SDK's handling of gimbal angle commands and angle wrapping past 180 degrees.",
"The competition trials were split over the course of three days, with the first two days focused on individual challenge trials and the third day hosting the grand challenge, when all three challenges are run at once under modified rules.",
"For the purpose of this thesis, we will consider every trial on any of these days equally.",
"Table: Success tally for Task 1 subtasks across all competition trials.Table REF shows the complete tally for three subtasks for Task 1 over all competition trials.",
"Note that this excludes only one trial run from competition, which is excluded because of bad gimbal angle initialization during system setup (refer to Section REF ).",
"Table REF specifies the failure cause for each of the Task 1 subtask failures.",
"The Target identification subtask was defined by the UAV reaching the closest point on the global search plan to a particular balloon and registering it as a valid target.",
"There were two failures in this subtask.",
"In both cases, though the incidents are from different trials, the reasoning was the same: the balloons happened to be outside the search path bounds and therefore appeared too small in the image to pass the required area threshold described in Section REF .",
"This was largely a failure in correctly setting global search path.",
"The Pop sequence subtask was defined as, after having registered a target detection, successfully adjusting and guiding the UAV such that the balloon is popped.",
"The overwhelming cause for a low success rate for this subtask was the detection of a balloon during a turn in the search path.",
"Turns were tuned to be executed quickly to ensure that the 40m$\\times $ 100m arena is covered efficiently.",
"However, registering a detection during these turns almost always resulted in a failed pop sequence, since the internal DJI SDK delay in camera feed of 0.1s resulted in a delayed notification of a target after the UAV had turned too far to recover.",
"Other failure cases for the pop sequence included the gimbal being off-center and thereby creating an inaccurate LOS vector, and instances of propeller downdraft moving the balloon away from the propellers.",
"Table: Causes of failure for the Task 1 subtasks.Figure: Snapshots of Task 1 subtasks.The Recover & pop subtask was defined as the pop sequence attempt after having failed the first pop attempt on a target and resuming the initial global search position from which the target was seen.",
"In the case of the pop failures resulting from high-rate turns, the LOS Guidance state machine never transitioned into Attack (refer to Section REF ); therefore, there was no recovery and second attempt in these cases.",
"However, the four failures caused by either an off-center gimbal camera and downdraft from the propellers did create an opportunity to recover, view the target again, and try again.",
"In all four recovery and pop attempts, the UAV was successful in popping the balloon."
],
[
"Task 2",
"Due to complications during the competition, there are only two trials from which to report results for Task 2.",
"As such, the analysis of Task 2 performance is confined to qualitative results.",
"During both of these trials, the UAV was able to find the yellow ball target via the square search pattern shown in Figure REF .",
"However, since the direction in which the target is flying the figure-8 path is not previously known, this particular search plan can spot the target either when it is approaching or receding from the UAV.",
"In both trials, the UAV successfully identified the target and implemented LOS guidance towards it, but did so when it was flying away from the UAV.",
"This prevented the UAV from ever achieving the position necessary to intercept the target, as described in geometric detail below.",
"Figure: Target ball moving away from UAV in Task 2.",
"Netting is part of a UAV-mounted ball catcher.",
"(a) LOS guidance issues a velocity command upwards, towards the target to intercept it on the next pass; (b) the LOS vector is pointing mostly forward; (c) the LOS vector is pointing downward as the target leaves view.As can be seen in Figure REF , as the frames progress, the LOS vector changes from pointing up towards the target, where the path of the target actually is, to having a smaller component upwards and instead pointing mostly forwards in the middle frame.",
"By the last frame, the LOS vector is actually pointing downwards, away from the closest point in the target path.",
"In general, an object moving away from a camera, in a direction perpendicular to the image plane, will project closer to the center of the image.",
"Since the LOS vectors had a diminishing upward and even downward component as the target moved away, the UAV was never able to achieve a successful intercept.",
"The global search pattern successfully found the target and guidance towards it was geometrically correct.",
"However, since there was no infrastructure developed for determining if the target was moving away or towards the UAV, the system was not able to adapt in the scenario presented here.",
"Further analysis of the camera stream suggests that the UAV was less than 50cm from the target."
],
[
"Discussion",
"In this thesis, a variety of guidance algorithms were designed and implemented in simulated and real environments.",
"Metrics used to assess the performance of each method include first-pass hit rate and pursuit duration.",
"LOS Guidance methods were derived from the True Proportional Navigation algorithm, which comes from missile guidance literature.",
"Implementation on a quadrotor allows greater control and maneuverability due to additional degrees of freedom.",
"While TPN was shown to have the highest success rate, the results presented here also suggest that there are benefits to utilizing the additional degrees of freedom when operating on a quadrotor.",
"For example, the Hybrid TPN-Heading method trades off between roll angle and yaw-rate control to implement lateral acceleration commands.",
"This adopts the behavior of the TPN method when the target is centered in frame, but also changes heading to re-center the target when necessary.",
"Hybrid TPN-Heading outperforms TPN at low target speeds.",
"This, paired with the qualitative results presented, suggest that this method may outperform TPN with additional tuning and under certain conditions (e.g., large target motions including straight line and figure-8).",
"TPN, the simplest method presented here, is preferred when a target is performing evasive maneuvers with high accelerations (e.g., tight turns).",
"We thus arrive at the first key conclusion of this work: when implemented on a quadrotor, a hybrid model of yaw control and lateral acceleration control (through roll angle commands) can outperform the original True Proportional Navigation algorithm under certain conditions.",
"All of these methods relied only on the information from a monocular RGB camera onboard a quadrotor UAV.",
"In Chapter this camera was rigidly fixed to the UAV frame, and in Chapter it was gimbal-mounted and set to point forward relative to the UAV heading.",
"This sensor information was sufficient for good first-pass hit rate performance for some algorithms but not for others.",
"As described in Section , LOS Guidance methods such as TPN, PN-Heading, and Hybrid TPN-Heading generally had much higher hit rates than the Trajectory Following methods such as LOS' Trajectory and Forecasting Trajectory.",
"In the simulated system, the vision was modeled realistically, so that a small target at long range suffered from a noisy LOS estimation due to pixel discretization.",
"The trajectory methods therefore relied on smoothing of the LOS and LOS' to avoid generating trajectories with high variation.",
"This smoothing, however, induced lag in the system and would sometimes lead to instability at high speeds or close ranges.",
"For these reasons, we can arrive at the second key conclusion of this work: trajectory following methods generally require less noisy measurements of the target; with noisy measurements, simpler methods perform better.",
"Chapter discussed the implementation of simple LOS following guidance on real systems, for both semi-stationary and moving targets, within a larger mission architecture.",
"The global plan was successful in both parts of the mission (Tasks 1 and 2) in positioning the UAV such that it can effectively use LOS guidance to move towards the targets.",
"In the semi-stationary targets case, even if the guidance did not result in a eliminated target on the first attempt, simply returning to the original point in space where the target detection was made and executing the guidance again always resulted in successful target elimination.",
"In the moving target case, while the UAV was positioned correctly for LOS guidance, the inability to determine the target's flight direction caused inaccurate guidance and resulted in missing the target by an estimated distance of less than 50cm.",
"From the competition results, we can arrive at a third key conclusion: in a scenario with repetitive target motions (semi-stationary or moving) simple LOS-direction guidance can effectively achieve target impact when within a larger system architecture including a global search plan and edge case infrastructure."
],
[
"Future Work",
"This work opens questions regarding what guidance algorithms can be developed for quadrotors through only the use of monocular RGB camera imagery.",
"As previously mentioned, it seems likely that further development of a hybrid method involving both proportional navigation and heading control might result in better performance.",
"This may lead to higher hit rate, but also lower pursuit duration as the need to “chase\" the target might disappear as the UAV maintains the target in view in the initial approach.",
"In addition, certain failure cases of the algorithms presented here suggest that more accurate controllers, particularly the thrust controller, might improve performance.",
"Due to the inability of Forecasting Trajectory to predict the 3D target position under moderate to high UAV speeds, suggesting that the UAV motion renders these estimates inaccurate, future work might limit the forecasting step to the 2D image plane.",
"This would avoid the multiplicative nature of projecting inaccurate pixel estimates into 3D for determining target motion.",
"Estimates of the target position in the image frame, paired with a LOS-based guidance method, might yield better results and greater stability at higher speeds."
]
] | 2107.01733 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.